reviewarticle finite temperature qcd sum rules: a...

Review ArticleFinite Temperature QCD Sum Rules A Review

Alejandro Ayala12 C A Dominguez2 and M Loewe234

1 Instituto de Ciencias Nucleares Universidad Nacional Autonoma de Mexico Apartado Postal 70-54304510 Mexico City Mexico2Centre for Theoretical and Mathematical Physics and Department of Physics University of Cape TownRondebosch 7700 South Africa3Instituto de Fisica Pontificia Universidad Catolica de Chile Casilla 306 Santiago 22 Chile4Centro Cientfico-Tecnologico de Valparaiso Casilla 110-V Valparaiso Chile

Correspondence should be addressed to C A Dominguez cesareodominguezuctacza

Received 24 August 2016 Revised 16 November 2016 Accepted 14 December 2016 Published 5 February 2017

Academic Editor Lokesh Kumar

Copyright copy 2017 Alejandro Ayala et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

The method of QCD sum rules at finite temperature is reviewed with emphasis on recent results These include predictions forthe survival of charmonium and bottonium states at and beyond the critical temperature for deconfinement as later confirmed bylattice QCD simulations Also included are determinations in the light-quark vector and axial-vector channels allowing analysingthe Weinberg sum rules and predicting the dimuon spectrum in heavy-ion collisions in the region of the rho-meson Also in thissector the determination of the temperature behaviour of the up-down quark mass together with the pion decay constant will bedescribed Finally an extension of the QCD sum rule method to incorporate finite baryon chemical potential is reviewed

1 Introduction

The purpose of this article is to review progress over the pastfew years on the thermal behaviour of hadronic and QCDmatter obtained within the framework of QCD sum rules(QCDSR) [1 2] extended to finite temperature 119879 = 0 Thesethermal QCDSR were first proposed long ago by Bochkarevand Shaposhnikov [3] leading to countless applications withthe most recent ones being reviewed here The first stepin the thermal QCDSR approach is to identify the relevantquantities to provide information on the basic phase tran-sitions (or crossover) that is quark-gluon deconfinementand chiral-symmetry restoration This is done below to befollowed in Section 2 by a brief description of the QCDsum rule method at 119879 = 0 which relates QCD to hadronicphysics by invoking Cauchyrsquos theorem in the complex square-energy plane Next in Section 3 the extension to finite119879 will be outlined in the light-quark axial-vector channelleading to an intimate relation between deconfinement andchiral-symmetry restoration In Section 4 the thermal light-quark vector channel is described with an application tothe dimuon production rate in heavy-ion collisions at high

energies which can be predicted in the 120588-meson region inexcellent agreement with data Section 5 is devoted to thethermal behaviour of the Weinberg sum rules and the issueof chiral-mixing In Section 6 very recent results on thethermal behaviour of the up-down quarkmass will be shownSection 7 is devoted to the thermal behaviour of heavy-quarksystems that is charmonium and bottonium states which ledto the prediction of their survival at and above the criticaltemperature for deconfinement confirmed by lattice QCD(LQCD) results In Section 8 we review an extension of thethermal QCDSR method to finite baryon chemical potentialFinally Section 9 provides a short summary of this review

Figure 1 illustrates a typical hadronic spectral functionImΠ(119904) in terms of the square-energy 119904 equiv 1198642 in the time-like region 119904 gt 0 at 119879 = 0 (curve (a)) First there could bea delta-function corresponding to a stable particle present asa pole (zero-width state) in the spectral function This couldbe for example the pion pole entering the axial-vector or thepseudoscalar correlator with the spectral function

ImΠ (119904)|POLE = 21205871198912120587120575 (119904 minus 1198722

120587) (1)

HindawiAdvances in High Energy PhysicsVolume 2017 Article ID 9291623 24 pageshttpsdoiorg10115520179291623

2 Advances in High Energy Physics

(s) (a)

(b)

s equiv E2

Realistic spectral function (T)

Im Π

s0(T) s0(0)

Figure 1 Typical hadronic spectral function ImΠ(119904) at 119879 = 0curve (a) showing a pole and three resonancesThe squared-energy1199040(0) is the threshold for PQCD At finite 119879 curve (b) the stablehadron develops a width the resonances become broader (onlyone survives in this example) and the PQCD threshold 1199040(119879)approaches the origin Eventually at 119879 ≃ 119879119888 there will be no traceof resonances and 1199040(119879119888) rarr 0

where 119891120587 ≃ 93MeV is the pion (weak-interaction) decayconstant defined as ⟨0|119860120583(0)|120587(119901)⟩ = radic2119891120587119901120583 and 119872120587 isits mass This is followed by resonances of widths increasingin size with increasing 119904 and corresponding to poles in thesecond Riemann sheet in the complex 119904-plane For instancefor narrow resonances the Breit-Wigner parametrization isnormally adequate

ImΠ (119904)|RES = 1198912119877

1198723119877Γ119877(119904 minus 1198722

119877)2 + 1198722119877Γ2

119877

(2)

where 119891119877 is the coupling of the resonance to the currententering a correlation function 119872119877 being its mass and Γ119877being its (hadronic) width At high enough squared-energy1199040(0) ≃ 2-3GeV2 the spectral function becomes smooth andshould be well approximated by perturbative QCD (PQCD)In the sequel this parameter will be indistinctly referred to asthe perturbative QCD threshold the continuum thresholdor the deconfinement parameter At finite 119879 this spectrumgets distorted The pole in the real axis moves down into thesecond Riemann sheet thus generating a finite width Thewidths of the rest of the resonances increase with increasing119879 and some states begin to disappear from the spectrum asfirst proposed in [4] Eventually close to or at the criticaltemperature for deconfinement 119879 ≃ 119879119888 there will be notrace of the resonances as their widths would be very largeand their couplings to hadronic currents would approachzero At the same time 1199040(119879) would approach the originThus 1199040(119879) becomes a phenomenological order parameterfor quark-deconfinement as first proposed by Bochkarevand Shaposhnikov [3] This order parameter associated withQCD deconfinement is entirely phenomenological and quitedifferent from the Polyakov-loop used by LQCD Never-theless and quite importantly qualitative and quantitativeconclusions regarding this phase transition (or crossover)and the behaviour of QCD and hadronic parameters as119879 rarr 119879119888 obtained from QCDSR and LQCD should agreeIt is reassuring that this turns out to be the case as willbe reviewed here In this scenario whatever happens to

the mass is totally irrelevant it could either increase ordecrease with temperature providing no information aboutdeconfinementThe crucial parameters are the width and thecoupling but not the mass In fact if a particle mass wouldapproach the origin or even vanish with increasing temper-ature this in itself is not sufficient to signal deconfinementas a massless particle with a finite coupling and width wouldstill contribute to the spectrum What is required is that thewidths diverge and the couplings vanish In all applicationsof QCD sum rules at finite 119879 the hadron masses in somechannels decrease and in other cases they increase slightlywith increasing 119879 At the same time for all light- and heavy-light-quark bound states the widths are found to divergeand the couplings to vanish close to or at 119879119888 thus signallingdeconfinement However in the case of charmonium andbottonium hadronic states after an initial surge the widthsdecrease considerably with increasing temperature while thecouplings are initially independent of 119879 and eventually growsharply close to 119879119888 This survival of charmonium states wasfirst predicted from thermal QCD sum rules [5 6] and laterextended to bottonium [7] in qualitative agreement withLQCD [8 9]

In addition to 1199040(119879) there is another important thermalQCD quantity the quark-condensate this time a funda-mental order parameter of chiral-symmetry restoration Itis well known that QCD in the light-quark sector possessesa chiral 119878119880(2) times 119878119880(2) symmetry in the limit of zero-massup and down quarks This chiral-symmetry is realized inthe Nambu-Goldstone fashion as opposed to the Wigner-Weyl realization [10] In other words chiral-symmetry is adynamical as opposed to a classification symmetry Hencethe pion mass squared vanishes as the quark mass

1198722120587 = 119861119898119902 (3)

and the pion decay constant squared vanishes as the quark-condensate

1198912120587 = 1119861 ⟨119902119902⟩ (4)

where 119861 is a constant While from (3) 1198722120587 can vanish as

the quark mass at 119879 = 0 the vanishing of 119891120587 can onlytake place at finite temperature as ⟨119902119902⟩119879 rarr 0 at 119879 =119879119888 the critical temperature for chiral-symmetry restorationThe correct meaning of chiral-symmetry restoration is aphase transition from a Nambu-Goldstone realization ofchiral 119878119880(2) times 119878119880(2) to a Wigner-Weyl realization of thesymmetry In other words there is a clear distinction betweena symmetry of the Lagrangian and a symmetry of the vacuum[10]

In summary at finite temperature the hadronic param-eters to provide relevant information on deconfinement arethe hadron width and its coupling to the correspondinginterpolating current On the QCD side we have (a) thechiral condensate ⟨119902119902⟩119879 providing information on chiral-symmetry restoration and (b) the onset of PQCD as deter-mined by the squared-energy 1199040(119879) providing informationon quark-deconfinement The next step is to relate these twosectors This is currently done by considering the complex

Advances in High Energy Physics 3

squared-energy plane and invoking Cauchyrsquos theorem asdescribed first at119879 = 0 in Section 2 and at finite119879 in Section 3However to keep a historical perspective a summary ofthe original approach [3] not entirely based on Cauchyrsquostheorem will be provided first

2 QCD Sum Rules at 119879 = 0The primary object in the QCD sum rule approach is thecurrent-current correlation function

Π (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119869 (119909) 119869dagger (0)) |0⟩ (5)

where 119869(119909) is a local current built either from the QCDquarkgluon fields or from hadronic fields In the case ofQCD and invoking the Operator Product Expansion (OPE)of current correlators at short distances beyond perturbationtheory [1 2] one of the two pillars of the QCD sum rulemethod one has

Π (1199022)10038161003816100381610038161003816QCD = 1198620 + sum119873=1

1198622119873 (1199022 1205832)(minus1199022)119873 ⟨O2119873 (1205832)⟩ (6)

where ⟨O2119873(1205832)⟩ equiv ⟨0|O2119873(1205832)|0⟩ 1205832 is a renormalizationscale and the Wilson coefficients 119862119873 depend on the Lorentzindexes and quantum numbers of the currents and on thelocal gauge invariant operators O119873 built from the quarkand gluon fields in the QCD Lagrangian These operatorsare ordered by increasing dimensionality and the Wilsoncoefficients are calculable in PQCD The unit operator abovehas dimension 119889 equiv 2119873 = 0 and 1198620 stands for the purelyperturbative contribution At 119879 = 0 the dimension 119889 equiv2119873 = 2 term in the OPE cannot be constructed from gaugeinvariant operators built from the quark and gluon fields ofQCD (apart fromnegligible light-quarkmass corrections) Inaddition there is no evidence from such a term from analysesusing experimental data [11 12] so that the OPE starts atdimension 119889 equiv 2119873 = 4 The contributions at this dimensionarise from the vacuum expectation values of the gluon fieldsquared (gluon condensate) and of the quark-antiquark fields(the quark-condensate) times the quark mass

While the Wilson coefficients in the OPE (6) can becomputed in PQCD the values of the vacuum condensatescannot be obtained analytically from first principles asthis would be tantamount to solving QCD analytically andexactly These condensates can be determined from theQCDSR themselves in terms of some input experimentalinformation for example spectral function data from 119890+119890minusannihilation into hadrons or hadronic decays of the 120591-leptonAlternatively they may be obtained by LQCD simulationsAn exception is the value of the quark-condensate which isrelated to the pion decay constant through (4) As an examplelet us consider the conserved vector current correlator

Π120583] (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119881120583 (119909) 119881dagger] (0)) |0⟩

= (minus119892120583]1199022 + 119902120583119902]) Π (1199022) (7)

where 119881120583(119909) = (12)[ 119906(119909)120574120583119906(119909) minus 119889(119909)120574120583119889(119909) ] is the(electric charge neutral) conserved vector current in thechiral limit (119898119906119889 = 0) and 119902120583 = (120596 ) is the four-momentumcarried by the current The function Π(1199022) in PQCD isnormalized as

ImΠ (1199022) = 18120587 [1 + O (120572119904 (1199022))] (8)

where the first term in brackets corresponds to the one-loopcontribution andO(120572119904(1199022)) stands for the multiloop radiativecorrections The leading nonperturbative term of dimension119889 equiv 2119873 = 4 is given by

1198624 ⟨O4⟩ = 1205873 ⟨1205721199041198662⟩ + 41205872 (119898119906 + 119898119889) ⟨119902119902⟩ (9)

a renormalization group invariant quantity where119898119906119889 are theQCD current quark masses in the 119872119878 regularization schemeand ⟨119906119906⟩ = ⟨119889119889⟩ equiv ⟨119902119902⟩ No radiative corrections to vacuumcondensates will be considered hereThe scale dependence ofthe quark-condensate cancels with the corresponding depen-dence of the quark masses In general the numerical valuesof the vacuumcondensates cannot be determined analyticallyfrom first principles as mentioned earlier An importantexception is the quark-condensate term above whose valuefollows from the Gell-Mann-Oakes-Renner relation in chiral119878119880(2) times 119878119880(2) symmetry [13]

(119898119906 + 119898119889) ⟨119902119902⟩ = 11989121205871198722

120587 (10)

where119891120587 = 9221plusmn002MeV is the experimentally measuredpion decay constant [14] Corrections to this relation essen-tially hadronic are small and at the level of a few percent [13]

Turning to the hadronic representation of the currentcorrelation functionΠ(1199022) in the time-like region 1199022 equiv 119904 ge 0in (7) it is given by the rho-meson resonance at leading orderTo a good approximation this is well described by a Breit-Wigner form

1120587 ImΠ1003816100381610038161003816100381610038161003816HAD (119904) = 11205871198912120588

1198723120588Γ120588

(119904 minus 1198722120588)2 + 1198722

120588Γ2120588

(11)

where 119891120588 = 497 plusmn 007 is the coupling of the 120588-meson tothe vector current measured in its leptonic decay [14] and119872120588 = 77526 plusmn 025MeV and Γ120588 = 1478 plusmn 09MeV are theexperimental mass and width of the 120588-meson respectivelyThis parametrization has been normalized such that the areaunder it equals the area under a zero-width expression thatis

ImΠ|(0)HAD (119904) = 11987221205881198912

120588120575 (119904 minus 1198722120588) (12)

The next step is to find away to relate theQCD representationof Π(119904) to its hadronic counterpart Historically at 119879 = 0one of the first attempts was made in [1] using as a first step


a dispersion relation (Hilbert transform) which follows fromCauchyrsquos theorem in the complex squared-energy 119904-plane

120593119873 (11987620) equiv 1119873 (minus 1198891198891198762

)119873 Π (1198762)1003816100381610038161003816100381610038161003816100381610038161198762=11987620= 1120587 intinfin

0

ImΠ (119904)(119904 + 1198762

0)119873+1119889119904

(13)

where 119873 equals the number of derivatives required for theintegral to converge asymptotically 1198762

0 is a free parameterand1198762 equiv minus1199022 gt 0 As it stands the dispersion relation (13) is atautology In the early days of high energy physics the opticaltheorem was invoked in order to relate the spectral functionImΠ(119904) to a total hadronic cross section together with someassumptions about its asymptotic behaviour and thus relatethe integral to the real part of the correlator or its derivativesThe latter could in turn be related to for example scatteringlengthsThe procedure proposed in [1] was to parametrize thehadronic spectral function as

ImΠ (119904)|HAD = ImΠ (119904)|POLE+ ImΠ (119904)|RES 120579 (1199040 minus 119904)+ ImΠ (119904)|PQCD 120579 (119904 minus 1199040)

(14)

where the ground-state pole (if present) is followed by the res-onances whichmerge smoothly into the hadronic continuumabove some threshold 1199040 This continuum is expected to bewell represented by PQCD if 1199040 is large enough Subsequentlythe left-hand side of this dispersion relation is written interms of the QCD OPE (6) The result is a sum rule relatinghadronic to QCD information Subsequently in [1] a specificasymptotic limiting process in the parameters 119873 and 1198762 wasperformed that is lim1198762 rarr infin and lim119873 rarr infin with1198762119873 equiv 1198722 fixed leading to Laplace transform QCD sumrules expected to be more useful than the original Hilbertmoments

119872 [Π (1198762)]equiv lim

1198762 119873rarrinfin

1198762119873equiv1198722

(minus)119873(119873 minus 1) (1198762)119873 ( 1198891198891198762)119873 Π (1198762)

equiv Π (1198722) = 11198722intinfin

0

1120587 ImΠ (119904) 119890minus1199041198722119889119904(15)

Notice that this limiting procedure leads to the transmutationof 1198762 into the Laplace variable 1198722 This equation is stilla tautology In order to turn it into something with usefulcontent one still needs to invoke (14) In applications ofthese sum rules [2] Π(1198722) was computed in QCD byapplying the Laplace operator 119872 to the OPE expressionof Π(1198762) (6) and the spectral function on the right-handside was parametrized as in (14) The function Π(1198722) inPQCD involves the transcendental function 120583(119905 120573 120572) [15]as first discussed in [16] This feature largely ignored for

a long time has no consequences in PQCD at the two-loop level However at higher orders ignoring this relationleads to wrong results It was only after the mid 1990s thatthis situation was acknowledged and higher order radiativecorrections in Laplace transform QCDSR were properlyevaluated

This novelmethod had an enormous impact as witnessedby the several thousand publications to date on analyticsolutions to QCD in the nonperturbative domain [2] How-ever in the past decade and as the subject moved towardshigh precision determinations to compete with LQCD theseparticular sum rules have fallen out of favour for a variety ofreasons as detailed next Last but not least Laplace transformQCDSR are ill-suited to deal with finite temperature asexplained below

The first thing to notice in (15) is the introductionof an ad hoc new parameter 1198722 the Laplace variablewhich determines the squared-energy regions where theexponential kernel would have a minormajor impact It hadbeen regularly advertised in the literature that a judiciouschoice of 1198722 would lead to an exponential suppression ofthe often experimentally unknown resonance region beyondthe ground-state as well as to a factorial suppression ofhigher order condensates in the OPE In practice thoughthis was hardly factually achieved thus not supportingexpectations Indeed since the parameter1198722 has no physicalsignificance other than being amathematical artefact resultsfrom these QCDSR would have to be independent of 1198722 in ahopefully broad region In applications this so-called stabilitywindow is often unacceptably narrow and the expectedexponential suppression of the unknown resonance regiondoes not materialize Furthermore the factorial suppressionof higher order condensates only starts at dimension 119889 =6 with a mild suppression by a factor 1Γ(3) = 12 Butbeyond 119889 = 6 little if anything is numerically knownabout the vacuum condensates to profit from this featureAnother serious shortcoming of these QCDSR is that therole of the threshold for PQCD in the complex 119904-plane1199040 that is the radius of the circular contour in Figure 2is exponentially suppressed This is rather unfortunate as1199040 is a parameter which unlike 1198722 has a clear physicalinterpretation and which can be easily determined fromdata in some instances for example 119890+119890minus annihilation intohadrons and 120591-lepton hadronic decays When dealing withQCDSR at finite temperature this exponential suppressionof 1199040 is utterly unacceptable as 1199040(119879) is the phenomenologicalorder parameter of deconfinement A more detailed criticaldiscussion of Laplace transform QCDSR may be found in[17] In any case and due to the above considerations no usewill be made of these sum rules in the sequel

A different attempt at relating QCD to hadronic physicswas made by Shankar [18] (see also [19ndash21]) by consideringthe complex squared-energy 119904-plane shown in Figure 2 Thenext step is the observation that there are no singularitiesin this plane except on the positive real axis where theremight be a pole (stable particle) and a cut which introducesa discontinuity across this axis This cut arises from thehadronic resonances (on the second Riemann sheet) present


Im (s)

Re (s)

Figure 2 The complex squared-energy 119904-plane used in Cauchyrsquostheorem The discontinuity across the positive real axis is given bythe hadronic spectral function and QCD is valid on the circle ofradius 1199040 the threshold for PQCD

in any given correlation function Hence from Cauchyrsquostheorem in this plane (quark-hadron duality) one obtains

∮ Π (119904) 119889119904 = 0= int1199040

0Π (119904 + 119894120598) 119889119904 + int0

1199040

Π (119904 minus 119894120598) 119889119904+ ∮

119862(|1199040|)Π (119904) 119889119904

(16)

which becomes finite energy sum rules (FESR)

int1199040

0

1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD 119875 (119904) 119889119904= minus 12120587119894 ∮

119862(|1199040|)Π (119904)QCD 119875 (119904) 119889119904

(17)

where an analytic function 119875(119904) has been inserted withoutchanging the result and the radius of the circle 119904 = |1199040| isunderstood to be large enough for QCD to be valid thereThe function 119875(119904) need not be an analytic function in whichcase the contour integral instead of vanishing would beproportional to the residue(s) of the integrand at the pole(s)In some cases this is deliberately considered especially ifthe residue of the singularity is known independently orconversely if the purpose is to determine this residue Thefunction 119875(119904) above is introduced in order to for examplegenerate a set of FESR projecting each and every vacuum

condensate of different dimensionality in the OPE (6) Forinstance choosing 119875(119904) = 119904119873 with 119873 ge 1 leads to the FESR

(minus)(119873minus1) 1198622119873 ⟨O2119873⟩= 81205872 int1199040

0119889119904119904119873minus1 1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD minus 1199041198730119873 [1 + O (120572119904)]

(119873 = 1 2 ) (18)

where the leading order vacuum condensates in the chirallimit (119898119902 = 0) are the dimension 119889 equiv 2119873 = 4 condensate(9) and the dimension 119889 equiv 2119873 = 6 four-quark-condensate

1198626 ⟨O6⟩= minus81205873120572119904 [⟨(1199021205741205831205745120582119886119902)2⟩ + 29 ⟨(119902120574120583120582119886119902)2⟩] (19)

where 120582119886 are 119878119880(3) Gell-Mann matrices A word of cautionfirst brought up in [18] is important at this point havingto do with the validity of QCD on the circle of radius |1199040|in Figure 2 Depending on the value of this radius QCDmay not be valid on the positive real axis a circumstancecalled quark-hadron duality violation (DV) This is currentlya contentious issue which however has no real impact onfinite temperature QCD sum rules to wit At 119879 = 0 oneway to deal with potential DV is to introduce in the FESR(17) weight functions 119875(119904) which vanish on the positive realaxis (pinched kernels) [11 12 22 23] or alternatively designspecific models of duality violations [24] The size of thiseffect is relatively small becoming important only at higherorders (four- to five-loop order) in PQCD Thermal QCDsum rules are currently studied only at leading one-loop orderin PQCD so that DV can be safely ignored In additionresults at finite 119879 are traditionally normalized to their 119879 = 0values so that only ratios are actually relevant

In order to verify that the FESR (18) give the right orderof magnitude results one can choose for example the vectorchannel use the zero-width approximation for the hadronicspectral function ignore radiative corrections and consider119873 = 0 FESR to determine 1199040 The result is 1199040 ≃ 19GeV2or radic1199040 ≃ 14GeV which lies above the 120588-meson and slightlybelow its very broad first radial excitation 1198721205881015840 ≃ 15GeVAn accurate determination using theBreit-Wigner expression(11) together with radiative corrections up to five-loop orderin QCD gives instead 1199040 = 144GeV2 or radic1199040 = 12GeVa very reassuring result Among recent key applications ofthese QCD-FESR are high precision determinations of thelight- and heavy-quarkmasses [17 25ndash28] now competing inaccuracy with LQCD results and the hadronic contributionto the muon magnetic anomaly (119892 minus 2)120583 [29ndash31]

Turning to the case of heavy-quarks instead of FESR itis more convenient to use Hilbert moment sum rules [32] asdescribed next The starting point is the standard dispersionrelation or Hilbert transform which follows from Cauchyrsquostheorem in the complex 119904-plane (13) In order to obtainpractical information one invokes Cauchyrsquos theorem in the


complex 119904-plane (quark-hadron duality) so that the Hilbertmoments (13) become effectively FESR

120593119873 (11987620)10038161003816100381610038161003816HAD = 120593119873 (1198762

0)10038161003816100381610038161003816QCD (20)

where

120593119873 (11987620)10038161003816100381610038161003816HAD equiv 1120587 int1199040

0

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|HAD

120593119873 (11987620)10038161003816100381610038161003816QCD

equiv 1120587sdot int1199040

41198982119876

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|PQCD + 120593119873 (1198762

0)10038161003816100381610038161003816NP (21)

In principle these sum rules are not valid for all values ofthe free parameter 1198762

0 In practice though a reasonably wideand stable window is found allowing for predictions to bemade [32] Traditionally these sum rules have been usedin applications involving heavy-quarks (charm bottom)while FESR are usually restricted to the light-quark sectorHowever there is no a priori reason against departing fromthis approach In the light-quark sector the large parameteris 1198762 (and 1199040 the onset of PQCD) with the quark massesbeing small at this scale Hence the PQCD expansion involvesnaturally inverse powers of 1198762 In the heavy-quark sectorthere is knowledge of PQCD in terms of the expansionparameter 11987621198982

119902 leading to power series expansions interms of this ratio Due to this most applications of QCDSRhave been restricted to FESR in the light-quark sector andHilbert transforms for heavy-quarks

The nonperturbative moments above 120593119873(11987620)|NP involve

the vacuum condensates in the OPE (6) One importantdifference is that there is no quark-condensate as there is nounderlying chiral-symmetry for heavy-quarksThe would-bequark-condensate ⟨119876119876⟩ reduces to the gluon condensate forexample at leading order in the heavy-quark mass 119898119876 onehas [1]

⟨119876119876⟩ = minus 112119898119876

⟨120572119904120587 1198662⟩ (22)

where 119898119876 is the heavy-quark mass (charm bottom) Writingseveral FESR one obtains for example information onheavy-quark hadronmasses couplings and hadronic widthsAlternatively using some known hadronic information onecan find the values of QCD parameters such as heavy-quarkmasses [17 25ndash28] and the gluon condensate [33 34] Fora review see for example [32] Their extension to finitetemperature will be discussed in Section 7

The techniques required to obtain the QCD expressionsof current correlators both perturbative and nonperturbative(vacuum condensates) at 119879 = 0 are well described in detailin [35]

3 Light-Quark Axial-VectorCurrent Correlator at Finite 119879Relating Deconfinement toChiral-Symmetry Restoration

The first thermal QCDSR analysis was performed byBochkarev and Shaposhnikov in 1986 [3] using mostlythe light-quark vector current correlator (120588- and 120601-mesonchannels) at finite temperature in the framework of Laplacetransform QCD sum rules Additional field-theory supportfor such an extension was given later in [36] in responseto baseless criticisms of the method at the time LaplacetransformQCDSRwere in fashion in those days [2] but theirextension to finite 119879 turned out to be a major breakthroughopening up a new area of research (for early work see eg[37ndash44]) The key results of this pioneer paper [3] were thetemperature dependence of the masses of 120588 and 120601 vectormesons as well as the threshold for PQCD 1199040(119879) Withhindsight instead of the vector mesons masses it wouldhave been better to determine the vector meson couplingsto the vector current However at the time there were someproposals to consider the hadron masses as relevant thermalparameters We have known for a long time now thatthis was an ill-conceived idea In fact the 119879-dependenceof hadron masses is irrelevant to the description of thebehaviour of QCD and hadronic matter and the approachto deconfinement and chiral-symmetry restoration This wasdiscussed briefly already in Section 1 and in more detailbelow Returning to [3] its results for the 119879-dependence of1199040(119879) that is the deconfinement phenomenological orderparameter clearly showed a sharp decrease with increasing119879 Indeed 1199040(119879) dropped from 1199040(0) ≃ 2GeV2 to 1199040(119879119888) ≃02GeV2 at 119879119888 ≃ 150MeV A similar behaviour was alsofound in the 120601-meson channel The masses in both cases haddecreased only by some 10

The first improvement of this approach was proposedin [45] where QCD-FESR instead of Laplace transformQCDSR were used for the first timeThe choice was the light-quark axial-vector correlator

Π120583] (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119860120583 (119909) 119860dagger] (0)) |0⟩

= minus119892120583]Π1 (1199022) + 119902120583119902]Π0 (1199022) (23)

where 119860120583(119909) š 119906(119909)1205741205831205745119889(119909) is the (electrically charged)axial-vector current and 119902120583 = (120596 ) is the four-momentumcarried by the current The functions Π01(1199022) are free ofkinematical singularities a key property needed in writingdispersion relations and sum rules with Π0(1199022) normalizedas

ImΠ0 (1199022)10038161003816100381610038161003816QCD= 14120587 [1 + O (120572119904 (1199022))] (24)

Notice the difference in a factor-two with the normalizationin (8) This is due to the currents in (23) being electricallycharged and those in (7) being electrically neutral (thusinvolving an overall factor 12 as stated after (7))The reasonfor this choice of correlation function was that since the


axial-vector correlator involves the pion decay constant 119891120587on the hadronic sector the thermal FESR would providea relation between 119891120587(119879) and 1199040(119879) Since the former isrelated to the quark-condensate ⟨119902119902⟩(119879) (4) one would thenobtain a relation between chiral-symmetry restoration anddeconfinement the latter being encapsulated in 1199040(119879) Avery recent study [46] of the relation between 1199040(119879) and thetrace of the Polyakov-loop in the framework of a nonlocal119878119880(2) chiral quark model concludes that both parametersprovide the same information on the deconfinement phasetransition This conclusion holds for both zero and finitechemical potential This result validates the thirty-year-oldphenomenological assumption of [3] and its subsequent usein countless thermal QCD sum rule applicationsWewill firstassume pion-saturation of the hadronic spectral function inorder to follow closely [45] Subsequently we shall describerecent precision results in this channel [47] Starting at119879 = 0the pion-pole contribution to the hadronic spectral functionin the FESR (18) is given by

ImΠ0 (1199022)10038161003816100381610038161003816HAD = 21205871198912120587120575 (119904) (25)

where 120575(119904 minus 1198982120587) above was approximated in the chiral limit

With 1198622⟨O2⟩ = 0 (see (6)) the first FESR (18) for 119873 = 1simply reads

1199040 = 812058721198912120587 (26)

Numerically 1199040 ≃ 07GeV2 which is a rather small valuethe culprit being the pion-pole approximation to the spectralfunction In fact as it will be clear later when additionalinformation is incorporated into (25) in the form of thenext hadronic state 1198861(1260) the value of 1199040 increasessubstantially In any case thermal results will be normalizedto the 119879 = 0 values

The next step in [45] was to use the Dolan-Jackiw [48]thermal quark propagators equivalent to the Matsubara for-malism at the one-loop level to find the QCD and hadronicspectral functions For instance at the QCD one-loop levelthe thermal quark propagator becomes

119878119865 (119896 119879) = 119894119896 minus 119898minus 2120587

(119890|1198960|119879 + 1) (119896 + 119898) 120575 (1198962 minus 1198982) (27)

and an equivalent expression for bosons except for a positiverelative sign between the two terms above and the obviousreplacement of the Fermi by the Bose thermal factor Anadvantage of this expression is that it allows for a straightfor-ward calculation of the imaginary part of current correlatorswhich is the function entering QCDSR It turns out that thereare two distinct thermal contributions as first pointed outin [3] One in the time-like region 119904 = 1199022 ge 0 calledthe annihilation term and the other one in the space-likeregion 119904 = 1199022 le 0 referred to as the scattering term Here1199022 = 1205962 minus |q2| where 120596 is the energy and q is the three-momentum with respect to the thermal bath The scattering

term can be visualized as due to the scattering of quarksand hadrons entering spectral functions with quarks andhadrons in the hot thermal bath In the complex energy 120596-plane (see Figure 26) the correlation functions have cuts inboth the positive and the negative real axes folding into onesingle cut along the positive real axis in the complex 119904 = 1199022planeThese singularities survive at119879 = 0 On the other handthe space-like contributions nonexistent at 119879 = 0 if presentat 119879 = 0 are due to cuts in the 120596-plane centred at 120596 = 0 withextension minus|q| le 120596 le |q| In the limit |q| rarr 0 that is in therest-frame of the medium this contribution either vanishesentirely or becomes proportional to a delta-function 120575(1205962)in the spectral function depending on 1199022 behaviour of thecurrent correlator A detailed derivation of a typical scatteringterm is done in the Appendix

Proceeding to finite 119879 the thermal version of the QCDspectral function (24) in the time-like (annihilation) regionand in the chiral limit (119898119902 = 0) becomes

ImΠ1198860 (120596 119879)1003816100381610038161003816QCD = 14120587 [1 minus 2119899119865 ( 1205962119879)] 120579 (1205962)

= 14120587 tanh( 1205964119879) 120579 (1205962) (28)

and the counterpart in the space-like (scattering) region is

ImΠ1199040 (120596 119879)1003816100381610038161003816QCD = 4120587120575 (1205962) intinfin

0119910119899119865 ( 119910119879) 119889119910

= 1205873 1198792120575 (1205962) (29)

where 119899119865(119911) = 1(1 + 119890119911) is the Fermi thermal factorA detailed derivation for finite quark masses is given inthe Appendix On the hadronic side the scattering term atleading order is a two-loop effect as the axial-vector currentcouples to three pions This contribution is highly phase-space suppressed and can be safely ignoredThe leading orderthermal FESR is then

812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)] (30)

which relates chiral-symmetry restoration encapsulated in1198912120587(119879) prop minus⟨119902119902⟩(119879) to deconfinement as described by 1199040(119879)

At the time of this proposal [45] there was no LQCD infor-mation on the thermal behaviour of the quark-condensate (or119891120587) One source of information on 119891120587(119879) was available fromchiral perturbation theory CHPT [49] whose proponentsclaimed it was valid up to intermediate temperatures Usingthis information the deconfinement parameter 1199040(119879) wasthus obtained in [45] It showed a monotonically decreasingbehaviour with temperature similar to that of 119891120587(119879) butvanishing at a much lower temperature Quantitatively thiswas somewhat disappointing as it was expected that bothcritical temperatures will be similar The culprit turned outto be the CHPT temperature behaviour of 119891120587(119879) whichcontrary to those early claims is now known to be validonly extremely close to 119879 = 0 say only a few MeV Shortlyafter this proposal [45] the thermal behaviour of 119891120587(119879) valid


06 07 08 09 10 11 12 13 14 15 1605

s (GeV2)

000

001

002

003

004

005

006

007

008

1120587

Im0

(s)

Π

Figure 3 The experimental data points of the axial-vector spectralfunction from the ALEPH Collaboration [54] together with the fitusing (31) (solid curve)

in the full temperature range as obtained in [50ndash52] wasused in [53] to solve the FESR (30) The result showed aremarkable agreement between the ratios 119891120587(119879)119891120587(0) and[1199040(119879)1199040(0)] over thewhole range119879 = (0ndash119879119888)This result wasvery valuable as it supported the method Formal theoreticalvalidation has been obtained recently in [46]

Further improved results along these lines were obtainedmore recently [47] as summarized next

The first improvement on the above analysis is theincorporation into the hadronic spectral function of theaxial-vector three-pion resonance state 1198861(1260) At 119879 = 0there is ample experimental information in this kinematicalregion from hadronic decays of the 120591-lepton as measured bythe ALEPH Collaboration [54ndash56] Clearly there is no suchinformation at finite 119879 The procedure is to first fit the dataon the spectral function using some analytical expressioninvolving hadronic parameters for example mass and widthand coupling to the axial-vector current entering the currentcorrelator Subsequently the QCDSR will fix the temperaturedependence of these parameters together with that of 1199040(119879)An excellent fit to the data (see Figure 3) was obtained in [47]with the function

1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587120575 (119904)

+ 119862119891 exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(31)

where 1198721198861= 10891GeV and Γ1198861 = 56878MeV are the

experimental values [14] and 119862119891 = 0048326 is a fittedparameter Notice that there is a misprint of (31) in [47]where the argument of the exponential was not squaredCalculations there were donewith the correct expression (31)The dimension 119889 equiv 2119873 = 4 condensate entering the FESRis given in (9) after multiplying by a factor-two to accountfor the different correlator normalization The next term inthe OPE (6) is the dimension 119889 equiv 2119873 = 6 condensate(19) As it stands it is useless as it cannot be determinedtheoretically It has been traditional to invoke the so-called

vacuum saturation approximation [1] a procedure to saturatethe sum over intermediate states by the vacuum state leadingto

1198626 ⟨O6⟩10038161003816100381610038161003816119860 prop 1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (32)

which is channel dependent and has a very mild dependenceon the renormalization scaleThe numerical coefficient aboveis not important as it cancels out in the ratio with respectto 119879 = 0 This approximation has no solid theoreticaljustification other than its simplicity Hence there is noreliable way of estimating corrections which in fact appearto be rather large from comparisons between (32) and directdeterminations from data [57 58] This poses no problemfor the finite temperature analysis where (32) is only usedto normalize results at 119879 = 0 and one is usually interestedin the behaviour of ratios Next the pion decay constant 119891120587is related to the quark-condensate through the Gell-Mann-Oakes-Renner relation

211989121205871198982

120587 = minus (119898119906 + 119898119889) ⟨0| 119906119906 + 119889119889 |0⟩ (33)

Chiral corrections to this relation are at the 5 level [13] andat finite 119879 deviations are negligible except very close to thecritical temperature [59]

Starting at 119879 = 0 the first three FESR (18) after dividingby a factor-two the first term on the right-hand side can beused to determine 1199040(0) and 119889 equiv 2119873 = 4 6 condensatesThese values will be used later to normalize all results at finite119879 The value thus obtained for 1199040(0) is 1199040(0) = 115GeV2 afar more realistic result than that from using only the pion-pole contribution (26) Next 119879 = 0 values of 119889 equiv 2119873 = 4 6condensates obtained from the next two FESR are in goodagreement with determinations from data [57 58]

Moving to finite 119879 in principle there are six unknownquantities to be determined from three FESR to wit (1)1199040(119879) (2) 119891(119879) and (3) Γ1198861(119879) on the hadronic side and(4) 119891120587(119879) prop minus⟨119902119902⟩(119879) and (5) 1198624⟨O4⟩ = (1205873)⟨1205721199041198662⟩ (inthe chiral limit) and (6) 1198626⟨O6⟩ on the QCD side The lattercan be determined using vacuum saturation thus leaving fiveunknown quantities for which there are three FESR In [47]the strategy was to use LQCD results for the thermal quarkand gluon condensates thus allowing the determination of1199040(119879) 119891(119879) and Γ1198861(119879) from the three FESR The LQCDresults are shown in Figure 4 for the gluon condensate [60]and in Figure 5 for the quark-condensate [61ndash63]

The three FESR to be solved are then

812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)]

minus 41205872 int1199040(119879)

0119889119904 1120587 ImΠ0 (119904 119879)10038161003816100381610038161003816100381610038161198861

minus1198624 ⟨O4⟩ (119879) = 41205872 int1199040(119879)

0119889119904119904 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

0119889119904119904 [1 minus 2119899119865 (radic1199042119879)]


00 01 02 03 04 05 06 07 08 09 10

TTc

16

14

12

10

08

06

04

02

00

C4⟨O

4⟩(T)C4⟨O

4⟩(0)

Figure 4 The normalized thermal behaviour of the gluon conden-sate (solid curve) together with LQCD results (dots) [60] for 119879119888 =197MeV

1412100806040200

12

10

08

06

04

02

00

TTc

f2 120587(T)f2 120587(0)

Figure 5The quark-condensate ⟨119902119902⟩(119879)⟨119902119902⟩(0) = 1198912120587 (119879)1198912

120587 (0) asa function of 119879119879119888 in the chiral limit (119898119902 = 119872120587 = 0) with 119879119888 =197MeV [61] (solid curve) and for finite quark masses from a fit tolattice QCD results [62 63] (dotted curve)

1198626 ⟨O6⟩ (119879) = 41205872 int1199040(119879)

01198891199041199042 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

01198891199041199042 [1 minus 2119899119865 (radic1199042119879)]

(34)

The result for 1199040(119879) is shown in Figure 6 together withthat of 119891120587(119879) both normalized to their values at 119879 =0 The difference in the behaviour of the two quantitieslies well within the accuracy of the method In fact thecritical temperatures for chiral-symmetry restoration and fordeconfinement differ by some 10 In any case it is reassuringthat deconfinement precedes chiral-symmetry restoration asexpected from general arguments [3] Next the behaviourof 1198861(1260) resonance coupling to the axial-vector current119891(119879) is shown in Figure 7 As expected it vanishes sharplyas 119879 rarr 119879119888 1198861(1260) resonance width is shown in Figure 8One should recall that at 119879 = 0 this resonance is quite broadeffectively some 500MeV as seen from Figure 3 Hence a30 increase in width as indicated in Figure 8 together

00 01 02 03 04 05 06 07 08 09 1110

TTc

s 0(T)s 0(0)

14

12

10

08

06

04

02

00

Figure 6 Results from the FESR (34) for the continuum thresh-old 1199040(119879)1199040(0) in the light-quark axial-vector channel signallingdeconfinement (solid curve) as a function of 119879119879119888 together with1198912120587 (119879)1198912

120587 (0) = ⟨119902119902⟩(119879)⟨119902119902⟩(0) signalling chiral-symmetry restora-tion (dotted curve)

00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

08

06

04

02

00

f(T)f(0)

Figure 7 Results from the FESR (34) for the coupling of 1198861(1260)resonance 119891(119879)119891(0) as a function of 119879119879119888

with the vanishing of its coupling renders this resonanceunobservable

This completes the thermal analysis of the light-quarkaxial-vector channel and we proceed to study the thermalbehaviour of the corresponding vector channel

4 Light-Quark Vector Current Correlator atFinite Temperature and Dimuon Productionin Heavy-Ion Collisions at High Energy

Thefinite119879 analysis in the vector channel [64] follows closelythat in the axial-vector channel except for the absence of thepion pole However we will summarize the results here asthey have an important impact on the dimuon productionrate in heavy nuclei collisions at high energies to be discussedsubsequently This rate can be fully predicted using theQCDSR results for the 119879-dependence of the parametersentering the vector channel followed by an extension to finitechemical potential (density)


00 01 02 03 04 05 06 07 08 09 10

TTc

Γ(T)Γ(0)

15

14

13

12

11

10

09

Figure 8 Results from the FESR (34) for the hadronic width of1198861(1260) resonance Γ1198861 (119879)Γ1198861 (0) as a function of 119879119879119888

Beginning with the QCD sector the annihilation andscattering spectral functions in the chiral limit are identicalto those in the axial-vector channel (28)-(29) An importantdifference is due to the presence of a hadronic scattering terma leading two-pion one-loop order instead of a three-piontwo-loop order as in the axial-vector channel This is givenby [64]

1120587 ImΠ1199041003816100381610038161003816100381610038161003816HAD (120596 119879) = 231205872

120575 (1205962) intinfin

0119910119899119861 ( 119910119879) 119889119910 (35)

where 119899119861(119911) = 1(119890119911 minus 1) is the Bose thermal function Onceagain there are three FESR (18) to determine six quantities119891120588(119879)119872120588(119879) Γ120588(119879) 1199040(119879)1198624⟨O4⟩(119879) and1198626⟨O6⟩(119879) Start-ing with the latter it can be related to the quark-condensatein the vacuum saturation approximation [1]

1198626 ⟨O6⟩10038161003816100381610038161003816119881 prop minus1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (36)

where the sign is opposite to that in the axial-vector channel(32)

The 119879-dependence of the quark-condensate was takenfrom LQCD [62 63] Next for the gluon condensate onceagain the LQCD results of [60] were used (see Figure 4)Finally the remaining four-parameter space was mappedimposing as a constraint that the width Γ120588(119879) shouldincrease with increasing 119879 and that both of the couplings119891120588(119879) and 1199040(119879) should decrease with temperature In thisway the following thermal behaviour was obtained (for moredetails see [64])

Γ120588 (119879) = Γ120588 (0)1 minus (119879119879119888)119886 (37)

where 119886 = 3 and 119879119888 = 197MeV

1198626 ⟨O6⟩ (119879) = 1198626 ⟨O6⟩ (0) [[

1 minus ( 119879119879lowast119902

)119887]]

(38)

with 119887 = 8 and 119879lowast119902 = 187MeV and

119872120588 (119879) = 119872120588 (0) [1 minus ( 119879119879lowast119872

)119888] (39)

where 119888 = 10 and119879lowast119872 = 222MeV constrained to satisfy119879lowast

119872 gt119879119888 The slight 5 difference between 119879119888 and 119879lowast119902 is well within

the accuracy of the method The remaining quantities are

1199040 (119879) = 1199040 (0)sdot [1 minus 05667 ( 119879119879119888

)1138 minus 4347 ( 119879119879119888

)6841] (40)

1198624 ⟨O4⟩ (119879) = 1198624 ⟨O4⟩ (0)sdot [1 minus 165 ( 119879119879119888

)8735 + 004967 ( 119879119879119888

)07211] (41)

119891120588 (119879)119891120588 (0) = 1 minus 03901 ( 119879119879119888

)1075 + 004155 ( 119879119879119888

)1269 (42)

The behaviour of 1199040(119879) is very similar to that in the axial-vector channel Figure 6 The results for the coupling119891120588(119879) the width Γ120588(119879) the mass 119872120588(119879) and 1198626⟨O6⟩(119879)all normalized to 119879 = 0 are shown in Figures 9ndash12 Theirbehaviour is fully consistent with deconfinement taking placeat a critical temperature 119879119888 ≃ 190ndash200MeV Of particularimportance is the behaviour of the hadron mass As shownin Figure 11 the hadron mass hardly changes with increasing119879 particularly in relation to the behaviour of the hadronicwidth and coupling A similar situation was found in thecase of the heavy-light-quark pseudoscalar and vector-mesonchannels [65] In fact in the former channel the hadronmassincreases by some 20 at 119879119888 while the coupling vanishes andthe width increases by a factor 1000 In the latter channelthe mass decreases by some 30 while the coupling vanishesand the width increases by a factor 100 This should put torest the ill-conceived idea that the 119879-behaviour of hadronmasses is of any relevance to physics at finite temperatureThe hadronic vector spectral function is shown in Figure 13at119879 = 0 (solid curve) and close to the critical temperature fordeconfinement (dotted curve) The resonance broadeningtogether with the strong decrease of its peak value as wellas the decrease of the coupling points to the disappearanceof the 120588-meson from the spectrum It should be pointed outthat the correct parametrization of the 120588-spectral function isas written in (11) as there is a misprint in [64]

To complete this section we describe how to obtainthe dimuon production rate in heavy-ion collisions at highenergy in particular for the case of In + In (at 158AGeV)into 120583+120583minus as measured by CERN NA60 Collaboration [66ndash70] The issues in dimuon production were discussed longago in [71ndash73] In particular in [73] a detailed analysis ofthis process using Bjorkenrsquos scaling solution for longitudinalhydrodynamic expansion [74] was discussed Predictions forthe dimuon production rate were also made in [73] assumingthe pion form factor entering the production rate to bedominated by the 120588-meson with parameters strictly from119879 = 0 in (11) A proposal to use instead a 119879-dependenthadronic width in the 120588-meson spectral function was firstmade in [4] and rediscovered several years later in [75 76] Itmust be mentioned that at the time of this proposal [4] thisidea was truly innovative It was shown in [4] using some


14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)

Figure 9 Results from the FESR in the vector channel for thehadronic coupling of the 120588-meson 119891120588(119879)119891120588(0) (42) as a functionof 119879119879119888

00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)

Figure 10 Results from the FESR in the vector channel for thehadronic width of the 120588-meson Γ120588(119879)Γ120588(0) (37) as a function of119879119879119888

00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)

Figure 11 Results from the FESR in the vector channel for the massof the 120588-meson 119872120588(119879)119872120588(0) (39) as a function of 119879119879119888

00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)

Figure 12Thedimension119889 equiv 2119873 = 6 four-quark-condensate in thevector channel and in the vacuum saturation approximation (38) asa function of 119879119879119888

1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00

Figure 13 The vector spectral function at 119879 = 0 (11) (solid curve)and at 119879 = 175MeV (dotted curve) with thermal parameters givenin (37) (39) and (42)

primitive model for Γ120588(119879) that there would be importantdetectable changes in the production rate such as a flatteningof the rate around the 120588-peak together with a broadening ofthis peak This prediction was made in 1991 way before anyexperimental data were available and 119879-dependent hadronwidths hardly used By the time data became available theproposal had been forgotten but recent experimental resultsfully confirmed the idea of a119879-dependent120588-mesonwidth andthe prediction of a flattening rate with increasing119879 as shownnext

The dimuon production rate involves several kinematicaland dynamical factors (see [4 73]) including the 120588-mesonhadronic spectral function Recently in a reanalysis of thisprocess [77] the latter was parametrized as in (11) butwith 119879-dependent parameters given in (37) (39) and (42)Furthermore in addition to the temperature it turns out thatthe chemical potential (density) 120583 needs to be introducedThis topic will be discussed in Section 8 following [78]where a QCDSR analysis at finite 120583 was first proposedThe parameter-free prediction of the dimuon invariant mass


06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)

Figure 14 The dimuon invariant mass distribution in In + In colli-sions in the region of the 120588-meson using (11) with predeterminedvalues of thermal parameters from QCDSR (37) (39) and (42)(solid curve) Dash curve is for all 120588-parameters independent of119879 The predicted resonance broadening and the flattening of theinvariant mass distribution near the peak are clearly observed Datais from [66ndash70] Results are for 120583 = 0 Finite chemical potentialresults change slightly in off-peak regions (see [77])

distribution is shown in Figure 14 (solid curve) togetherwith the NA60 data [66ndash70] and the prediction using a 119879independent spectral function (dash curve) The predictedresonance broadening essentially from (37) and the flatten-ing of the spectrum around the peak are fully confirmed Itmust be pointed out that this determination is only valid inthe region around the 120588-peak At lower as well as at higherenergies this approximation breaks down and the dynamicswould involve a plethora of processes hardly describable inQCD Intermediate energymodels of various kinds have beeninvoked to account for the data in those regions with varyingdegrees of success (for a recent review see [79])

5 Weinberg Sum Rules and Chiral-Mixing atFinite Temperature

The twoWeinberg sum rules (WSR) (at 119879 = 0) [80] were firstderived in the framework of chiral 119878119880(2) times 119878119880(2) symmetryand current algebra and read

1198821 equiv intinfin

0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)

Given that both the vector and the axial-vector spectralfunctions enter in theWSR unfortunately we need to changethe previous normalization of the vector correlator (8) to turnit equal to that of the axial-vector one (24) that is we choose

ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)

In the framework of perturbative QCD (PQCD) where bothspectral functions have the same asymptotic behaviour (in

the chiral limit) these WSR become effectively QCD finiteenergy sum rules (FESR)

119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)

where 1199040 ≃ 1ndash3GeV2 is the squared-energy beyond whichQCD is valid and both sum rules have been combined Thisresult also follows from Cauchyrsquos theorem in the complex 119904-plane together with the assumption of quark-hadron duality(17)The latter is not expected to hold in the resonance regionwhere QCD is not valid on the positive real 119904-axis This leadsto duality violations (DV) first identified long ago in [18] andcurrently a controversial subject of active research [22ndash24]In relation to the WSR it was pointed out long ago [81] thatthese sum rules were hardly satisfied by saturating them withthe ALEPH data on hadronic 120591-lepton decays [54ndash56] Thiswas and still can be interpreted as a signal for DV A proposalwas made in [81] (see also [82]) to introduce the nontrivialkernel 119875(119904) in (17) leading to

119882119875 (1199040)equiv int1199040

0119889119904 (1 minus 1199041199040 ) 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 21198912120587

(47)

This expression is fully satisfied [22 23 81] thus validatingthe procedure It turns out that this is also the case in othersum rules [81] that is pinched kernels quench or eveneliminate DV

Turning to theWSR at finite temperature [83] all param-eters in the vector channel have been already determined in(37) (39) and (40) The axial-vector channel parameters at119879 = 0 require a slight update as they were obtained in thesquared-energy region below 119904 ≃ 15GeV2 Going above thisvalue and up to 119904 ≃ 20GeV2 the resonance hadronic spectralfunction at 119879 = 0 fitted to the ALEPH 120591-decay data is [83]

1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[

[minus (12GeV2 minus 1198722

1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)


Table 1 The values of the coefficients entering (49)

Parameter Coefficients in equation (49)1198861 1198862 1198871 11988721199040(119879) minus285 minus06689 3560 393119891120587(119879) minus02924 minus07557 7343 11081198911198861(119879) minus1934 1427 7716 6153Γ1198861 (119879) 2323 1207 2024 7869

02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π

Figure 15 Solid curve is the axial-vector (1198861-resonance) spectralfunction at 119879 = 0 fitted to the ALEPH data [54] shown with errorbars the size of the data points Dotted curve is the spectral functionat 119879 = 175MeV with thermal parameters given in (49)

where 1198721198861= 10891GeV Γ1198861 = 56878MeV 119862 = 0662 and1198911198861

= 0073 (the latter two parameters were split to facilitatecomparison between 1198911198861

and 119891120588 for readers used to zero-width resonance saturation of theWSR)The results from theFESR for the thermal parameters can be written as

119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)

where the various coefficients are given in Table 11198861 mass hardly changes with temperature so that it waskept constantThis behaviour can be ascribed to the very largewidth of 1198861 resonance

A comparison of the vector spectral functions at 119879 =0 and at 119879 = 175MeV is shown in Figure 13 Resonancebroadening with a slight decrease of the mass is clearly seenIn the axial-vector case the spectral function is shown inFigure 15 where the solid curve is the fit to the ALEPH dataat 119879 = 0 and the dotted curve corresponds to 119879 = 175MeVAt such temperature there is no trace of 1198861

Proceeding to the WSR at finite 119879 the first obvious thingto notice is the dramatic difference between the vector and theaxial-vector spectral functionsThese spectral functions havevery different evolution with increasing temperature for theobvious reason that they are already so different at 119879 = 0perhaps with the exception that 1199040(0) is the same in bothchannels With increasing 119879 the parameters of each channelevolve independently thus keeping both spectral functionsdistinct Eventually this asymmetry is expected to vanish at

00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)

Figure 16 The first WSR (43) at finite T Solid (dash) line is theleft (right)-hand side of (43) The divergence at high 119879 is caused bythe asymmetric hadronic scattering contribution in the space-likeregion (1199022 lt 0) which disappears at deconfinement (119879 = 119879119888)

deconfinement when 120588 and 1198861 mesons disappear from thespectrum This implies no chiral-mixing at any temperatureexcept obviously at 119879 ≃ 119879119888 In addition to these differencesthere is an additional asymmetry due to the hadronic (pionic)scattering term present in the vector channel at the leadingone-loop level and strongly two-loop level suppressed in theaxial-vector case This is manifest very close to the criticaltemperature where this term is important as it increasesquadratically with temperature This can be appreciated inFigure 16 which shows the 119879-dependence of the first WSR1198821(119879) (43) The behaviour of the pinchedWSR 119882119875(119879) (47)is essentially the same except close to 119879119888 where the scatteringterm causes 119882119875(119879) to decrease rather than increase slightlyIn both cases the scattering term disappears at 119879 = 119879119888as the pions would have melted To be more specific letus consider the vector and axial-vector correlators (7) and(23) respectively In a thermal bath and in the hadronicrepresentation one has (schematically)

Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)

To the extent that Isospin and 119866-parity remain valid symme-tries at finite temperature the chiral asymmetry is manifestto wit The leading term in the vector channel is the two-pion one-loop term and in the axial-vector channel it is thetree-level pion-to-vacuum term (119891120587) followed by a highlyphase-space suppressed three-pion two-loop term In otherwords the matrix element ⟨120587|119860120583(0)|120587⟩ invoked by chiral-mixing proposers [84] vanishes identically at leading order


The correct matrix element beyond the pion pole is thephase-space suppressed second term in (51) In principle thisterm could have a resonant subchannel contribution fromthe matrix element ⟨120588120587|119860120583(0)|0⟩ which again is phase-spacesuppressed (see results from [85 86] which can be easilyadapted to this channel) An alternative argument clearlyshowing the nonexistence of chiral-mixing at finite119879 is basedon the chiral Lagrangian to leading order [87] with all termsrespecting Isospin and 119866-parity6 Temperature Dependence of

the Up-Down Quark Mass

In this section we discuss a recent determination of thethermal dependence of the up-down quark mass [88] At119879 = 0 the values of the light-quark masses are determinedfrom QCD sum rules usually involving the correlator of theaxial-vector divergences [17 25ndash28]

1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)

where 119898119906119889 are the running quark masses in the 119872119878-schemeat a scale 120583 = 2GeV [17 25ndash28 89] and 119906(119909) 119889(119909) are thecorresponding quark fields As usual the relation between theQCD and the hadronic representation of current correlatorsis obtained by invoking Cauchyrsquos theorem in the complexsquare-energy plane Figure 2 which leads to the FESR

int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)

where 1205955(0) is the residue of the pole generated by thedenominator in (56) that is

1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)

The radius of the contour 1199040 in Figure 2 is large enoughfor QCD to be valid on the circle Information on thehadronic spectral function on the left-hand side of (55) allowsdetermining the quark masses entering the contour integralCurrent precision determinations of quark masses [17 25ndash28] require the introduction of integration kernels on bothsides of (55) These kernels are used to enhance or quench

hadronic contributions depending on the integration regionand on the quality of the hadronic information availableTheyalso deal with the issue of potential quark-hadron dualityviolations as QCD is not valid on the positive real axis in theresonance region This will be of no concern here as we aregoing to determine only ratios for example 119898119906119889(119879)119898119906119889(0)to leading order in the hadronic and the QCD sectors To thisorder theQCDexpression of the pseudoscalar correlator (52)is

1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)

where ⟨119902119902⟩ = (minus267 plusmn 5MeV)3 from [13] and ⟨(120572119904120587)1198662⟩ =0017 plusmn 0012GeV4 from [90] The gluon- and quark-condensate contributions in (58) are respectively one andtwo orders of magnitude smaller than the leading pertur-bative QCD term Furthermore at finite temperature bothcondensates decrease with increasing 119879 so that they can besafely ignored in the sequel

The QCD spectral function in the time-like region atfinite 119879 obtained from the Dolan-Jackiw formalism [48] inthe rest-frame of the medium (1199022 = 1205962 minus |q|2 rarr 1205962) is givenby

Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)

TheQCD scattering term present in the axial-vector correla-tor (29) is absent in 1205955(1199022 119879) due to the overall factor of 1199022in the first term in (58)This factor prevents the appearance ofthe delta-function 120575(1205962) in (29) In the hadronic sector thescattering term is due to a phase-space suppressed two-loopthree-pion contribution which is negligible in comparisonwith the pion-pole term

Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)

The two FESR (55)-(56) at finite 119879 become

21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)

Equation (62) is the thermal Gell-Mann-Oakes-Renner rela-tion incorporating a higher order QCD quark-mass correc-tion O(1198982

119906119889) While at 119879 = 0 this correction is normally


9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)

Figure 17 The ratio of the quark masses 119898119906119889(119879)119898119906119889(0) as afunction of 119879 from the FESR (61)-(62) Curve (a) is for a 119879-dependent pion mass from [94] and curve (b) is for a constant pionmass

f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)

Figure 18 The ratio of the pion decay constant 119891120587(119879)119891120587(0) asa function of 119879 from the FESR (61)-(62) Curve (a) is for a 119879-dependent pion mass from [94] and curve (b) is for a constant pionmass

neglected [13] at finite temperature this cannot be done asit is of the same order in the quark mass as the right-handside of (61)

As discussed previously in Section 3 the thermal quark-condensate (signalling chiral-symmetry restoration) and1199040(119879) (signalling deconfinement) are related through

1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)

Further support for this relation is provided by LQCD results[91 92] One does not expect this relation to be valid veryclose to the critical temperature 119879119888 as the thermal quark-condensate for finite quark masses is nonvanishing closeto 119879119888 With 1199040(119879)1199040(0) as input in the FESR (61)-(62)together with LQCD results for ⟨119902119902⟩(119879) for finite quarkmasses [93] and independent determinations of 119872120587(119879) [94]the ratios 119898119906119889(119879)119898119906119889(0) and 119891120587(119879)119891120587(0) were obtainedin [88] The results are shown in Figures 17 and 18 Thequark mass remains constant up to 119879 ≃ 150MeV andincreases sharply thereafter As expected from the discussion

on chiral-symmetry in Section 1 leading to (3) the quarkmass is intimately related to the pion mass The behaviourof the quark mass is also consistent with the expectationthat at deconfinement free light-quarks would acquire amuch higher constituent mass Figure 18 shows the thermalbehaviour of119891120587 which is fully consistent with the expectationfrom chiral-symmetry (4) that is that 119891120587(119879) is independentof 119872120587(119879) and 119891120587(119879) prop ⟨119902119902⟩(119879)7 Quarkonium at Finite Temperature and Its

Survival atbeyond 119879119888

In 1986 Matsui and Satz [95] invoking colour screeningin charmonium concluded that this effect would preventbinding in the deconfined interior of the interaction regionin heavy-ion collisions This scenario became an undisputedmantra for more than two decades until 2010 when it wasshown [5] that thermal QCD sum rules clearly predict thesurvival of charmonium (119869120595) at and beyond 119879119888 Subse-quently this was supported by an analysis of scalar andpseudoscalar charmonium states [6] and pseudoscalar andvector bottonium states [7] all behaving similarly to 119869120595The results for bottonium were in qualitative agreementwith LQCD simulations [8 9] An interesting aspect of thelatter is the result for the widths In fact the qualitativetemperature behaviour of hadronic widths from LQCDagrees with that from QCDSR This is reassuring given thatthese two approaches employ very different parameters todescribe deconfinement Recent work [46] shows that thesetwo parameters 1199040(119879) for the thermal QCDSR and thePolyakov thermal loop for LQCD are in fact related as theyprovide the same information on deconfinement

We proceed to discuss the thermal behaviour of char-monium in the vector channel [5] that is 119869120595 state Thecase of scalar and pseudoscalar charmonium [6] as well asbottonium [7] follows along similar lines so the reader isreferred to the original papers for details The vector currentcorrelator is given by (7) with the obvious replacement ofthe light- by the heavy-quark fields in the vector current119881120583(119909) š 119876(119909)120574120583119876(119909) where 119876(119909) is the charm-quark fieldA straightforward calculation in the time-like region toleading order in PQCD gives

1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)

sdot [1 minus 119899119865 (|q| 119909 + 1205962119879 ) minus 119899119865 (|q| 119909 minus 1205962119879 )] (64)

where V2 = 1 minus 411989821198761199022 119898119876 is the heavy-quark mass 1199022 =1205962 minus q2 ge 41198982

119876 and 119899119865(119911) is the Fermi thermal function Inthe rest-frame of the thermal bath |q| rarr 0 the above resultreduces to

1120587 ImΠ119886 (120596 119879)= 181205872

V (3 minus V2) [1 minus 2119899119865 ( 1205962119879)] 120579 (120596 minus 2119898119876) (65)


The quarkmass is assumed independent of119879 which is a goodapproximation for temperatures below 200MeV [96] In thespace-like region the QCD scattering term (35) needs to bereevaluated to take the quark mass into account This gives

1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)

In the hadronic sector the spectral function is given by theground-state pole 119869120595 followed by PQCD

1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)

where 119904 equiv 1199022 = 1205962 minus q2 and the leptonic decay constant isdefined as

⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)

Next considering a finite (total) width the following replace-ment will be understood

120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)

where the constant is fixed by requiring equality of areasfor example if the integration is in the interval (0ndashinfin) thenconst = 119872119881(119879)Γ119881(119879)120587

To complete the hadronic parametrization one needsthe hadronic scattering term due to the current scatteringoff heavy-light-quark pseudoscalar mesons (119863-mesons) Theexpression in (35) needs to be reobtained in principle asit is valid for massless pseudoscalar hadrons (pions) In themassive case it becomes

1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)

It is easy to verify that this term is exponentially suppressednumerically being two to three orders of magnitude smallerthan its QCD counterpart (65)

Turning to the sum rules the vector correlation functionΠ(1199022 119879) (7) satisfies a once-subtracted dispersion relation

12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)

Figure 19 The ratio 1199040(119879)1199040(0) as a function of 119879119879119888 for 119869120595channel from thermal Hilbert moment QCD sum rules

Hence one can use Hilbert moments (20)-(21) The nonper-turbative QCD term of dimension 119889 = 4 corresponding tothe gluon condensate is given by

120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2

sdot 119865 (119873 + 2 minus12 119873 + 72 120588)times 2119873119873 (119873 + 1)2 (119873 + 2) (119873 + 3) (119873 minus 1)(2119873 + 5) (2119873 + 3) Φ

(71)

where 119865(119886 119887 119888 119911) is the hypergeometric function 120585 equiv119876241198982119876 120588 equiv 120585(1 + 120585) and

Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)

The thermal behaviour of the gluon condensate needed as aninput was obtained from LQCD results available at the time[97 98] Those results are in good agreement with the mostrecent ones [60] shown in Figure 4 The first three Hilbertmoments and four ratios were considered in [5] to determinethe thermal behaviour of the four quantities 1199040(119879) 119872119881(119879)Γ119881(119879) and 119891119881(119879) Details of the procedure are thoroughlydiscussed in [5] sowe proceed to discuss the results Figure 19shows the behaviour of the normalized continuum threshold1199040(119879)1199040(0) Unlike the situation in the light-quark sectorwhere this ratio approaches zero quite rapidly close to 119879119888

(see Figure 6) in 119869120595 channel 1199040(119879) shows a dramaticallydifferent behaviour In fact 1199040(119879) decreases by only some10 at 119879 = 119879119888 as shown in Figure 19 At 119879 ≃ 12119879119888 thedecrease is only close to 40 Above this temperature thesum rules no longer have solutions as there is no supportfor the integrals in the Hilbert moments This is somethingwhich happens generally regardless of the type of currententering the correlation functions for light- or heavy-quarksThe unequivocal interpretation of this result is that 119869120595survives above the critical temperature for deconfinement


4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)

Figure 20 The ratio Γ119881(119879)Γ119881(0) as a function of 119879119879119888 for 119869120595channel from thermal Hilbert moment QCD sum rules

12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)


This puts to rest the historical expectation [95] of the meltingof charmonium at or close to 119879 = 119879119888 Further evidence isprovided by the behaviour of the width Figure 20 Whileinitially the width behaves as in light- and heavy-light-quarksystems by increasing with increasing 119879 just above 119879119888 thewidth has a sharp turnaround decreasing substantially thussuggesting survival of 119869120595 Finally the behaviour of thecoupling increasing (rather than decreasing) sharply withtemperature as shown in Figure 21 provides an unambiguousevidence for the survival of this state Contrary to the thermalbehaviour of these quantities the mass hardly changes withtemperature as shown in Figure 22

The thermal behaviours of these four parameters in thescalar and pseudoscalar charmonium [6] as well as in thevector and pseudoscalar bottonium [7] are very similar to1198691205958 QCD Phase Diagram at Finite 119879 and

Baryon Chemical Potential

In this section we outline the extension of the analysis of thethermal axial-vector current correlator Section 3 to finite

12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)

Figure 22 The ratio 119872119881(119879)119872119881(0) as a function of 119879119879119888 for 119869120595channel from thermal Hilbert moment QCD sum rules This ratiois basically the same in zero width as in finite width

baryon chemical potential [78]The starting point is the light-quark axial-vector current correlator (23) and the two-pointfunction Π0(1199022) In the static limit (q rarr 0) to leading orderin PQCD for finite 119879 and quark chemical potential 120583119902 with120583119902 = 1205831198613 the function Π0(1199022) now becomes Π0(1205962 119879 120583119902)and is given by

1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD

= 141205872[1 minus + (radic1199042 ) minus minus (radic1199042 )]

minus 212058721198792120575 (119904) [Li2 (minus119890120583119861119879) + Li2 (minus119890minus120583119861119879)]

(73)

where Li2(119909) is the dilogarithm function 119904 = 1205962 and theFermi-Dirac thermal distributions for particles (antiparti-cles) are given by

plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)

In the limit where 119879 andor 120583119861 are large compared to a massscale for example the quark mass (73) becomes

1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)

The hadronic spectral function (25) is

1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)


Turning to the FESR (18) with119873 = 1 and using (73) and (76)one finds

int1199040(119879120583119902)

0119889119904 [1 minus + (radic1199042 ) minus minus (radic1199042 )]

= 812058721198912120587 (119879 120583119902)

+ 81198792 [Li2 (minus119890120583119902119879) + Li2 (minus119890minus120583119902119879)] (77)

This transcendental equation determines 1199040(119879 120583119902) in terms of119891120587(119879 120583119902) The latter is related to the light-quark-condensatethrough the Gell-Mann-Oakes-Renner relation [59]

1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)

where the quark and pionmasses were assumed independentof 119879 and 120583119902 in [78] In view of the results obtained in [88]as discussed in Section 6 it would seem important to revisitthis issue It is easy to see that a119879-dependent quarkmass doesnot affect the validity of (78) In fact the thermal quark massfollows the thermal pionmass independently of119891120587(119879)whichin turn follows ⟨119902119902⟩(119879)

A good closed form approximation to the FESR (77) forlarge 119879 andor 120583119902 is obtained using (75) with +(radic1199042) ≃minus(radic1199042) ≃ 0 in which case

1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)

Using (78) this can be rewritten as

1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)

The quark-condensate can be computed from the in-mediumquark propagator whose nonperturbative properties canbe obtained for example from known solutions to theSchwinger-Dyson equations (SDE) as discussed in detail in[78] The result is

⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)

where 1198701(119909) is a Bessel function and for convenience onedefines 119887119894 = 1 for 119894 = 1 2 3 and 1198874 = 119887 The values ofthe parameters 119898119894 119903119894 and 1198874 equiv 119887 are given in Table I andTable II in [78] In the limit 120583119861 = 0 the result for the quark-condensate using (81) is shown in Figure 23 together withLQCD data [62 63]

The expressions for 1199040(119879 120583119902) and ⟨120595120595⟩(119879 120583119902) (80) (81)characterizing deconfinement and chiral-symmetry restora-tion transitions are the central results of this analysis Theyare used next to explore the phase diagram To this end oneneeds the corresponding susceptibilities proportional to theheat capacities minus120597⟨120595120595⟩(119879 120583119902)120597119879 and minus1205971199040120597119879 for a given

Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)

ParametrizationLattice data

Figure 23 LQCD data (dots) [62 63] and absolute value of thequark-condensate ⟨120595120595⟩(119879) (81) (solid curve) as a function of 119879 inthe phase transition (or crossover) region

120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0

Figure 24 Heat capacity for the quark-condensate as a function of119879 for 120583119861 = 0 (solid line) and 120583119861 = 300MeV (dash line) The criticaltemperature119879119888 corresponds to themaximumof the heat capacity fora given value of 120583119861

value of 120583119861 The transition temperature is then identified asthe value of119879 forwhich the heat capacity reaches amaximumFigure 24 shows the behaviour of the heat capacity for thequark-condensate (normalized to its value in the vacuum) asa function of 119879 for 120583119861 = 0 (solid line) and 120583119861 = 300MeV(dash line) The PQCD threshold 1199040(119879 120583119861) is somewhatbroader than the quark-condensate (see [78]) but it peaksat essentially the same temperature that is 119879 = 185MeVwithin 3MeVThe results for the phase diagram 119879119888 versus 120583119861

are shown in Figure 25 where the solid dots correspond to 119879119888

for chiral-symmetry restoration (quark-condensate) and thesolid triangles refer to deconfinement (1199040)


025

0225

02

Chirally symmetric phase Deconfined phase

0175

015Broken chiral-symmetry phase Confined phase

0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)

Tc (condensate)Tc (s0)

Figure 25 Transition temperatures for the quark-condensate⟨120595120595⟩(119879 120583119902) and the PQCD threshold 1199040(119879 120583119902) as functions of thebaryon chemical potential

Im 120596

Re 120596minus|q| |q|

Figure 26 The complex energy 120596-plane showing the central cutaround the origin (scattering term) extending between 120596 = minus|q|and 120596 = |q| The standard (time-like) annihilation right-hand andleft-hand cuts at 120596 = plusmn[|q|2 + 1205962

th]12 are not shown (120596th is somechannel dependent mass threshold)

9 Summary

The extension of the QCD sum rule programme at 119879 = 0[2] to finite temperature was first proposed in [3] in theframework of Laplace transform QCDSR [1] There are twomain assumptions behind this extension (i) the OPE ofcurrent correlators at short distances remains valid exceptthat the vacuum condensates will acquire a temperaturedependence and (ii) the notion of quark-hadron duality canbe invoked in order to relate QCD to hadronic physics Thelatter is known to be violated at 119879 = 0 in the low energyresonance region DV albeit by a relatively small amountThis is unimportant at finite 119879 not only because of the small

relative size of DV but also because all determinations arenormalized to their values at 119879 = 0 Next the starting pointis the identification of the basic object at finite 119879 This isthe retarded (advanced) two-point function after appropriateGibbs averaging

Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899

exp (minus119864119899119879) ⟨119899| 119860 sdot 119861 |119899⟩Tr (exp (minus119867119879)) (83)

and |119899⟩ is a complete set of eigenstates of the (QCD) Hamil-tonian The OPE of Π(119902 119879) is now written as

Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)

It is essential to stress that the states |119899⟩ entering (83)can be any states as long as they form a complete set Inother words they could be hadronic states or quark-gluonbasis and so forth The hadronic (mostly pionic) basis wasadvocated to obtain thermal information on some quantitiesfor example vacuum condensates [99]These determinationsare constrained to very low temperatures in the domainof thermal chiral perturbation theory way below 119879119888 Thisapproach does not invoke quark-hadron duality thus ithas little relationship to the QCD sum rule programmeIn addition being restricted to very low temperatures itprovides no useful thermal information on for exampleQCD condensates which is currently provided by LQCDAlternatively another complete set is the quark-gluon ofQCD as first advocated in [3]This choice allows for a smoothextension of theQCDSRmethod to finite119879The only thermalrestriction has to do with the support of the integrals enteringthe sum rules In most cases this extends up to the criticaltemperature an exception being charmonium which goeseven further Field-theory arguments fully supporting thisapproach were given in [36]

Another key element in this programme is the identifica-tion of the relevant QCD and hadronic parameters character-izing the transition to deconfinement and chiral-symmetryrestoration While the latter is universally understood to bethe thermal quark-condensate an order parameter in thecase of deconfinement the parameter is purely phenomeno-logical It also differs from that used by LQCD that is theso-called Polyakov-loop Thermal QCD sum rules invokeinstead the onset of perturbative QCD in the square-energy119904-plane so-called 1199040(119879) as first proposed in [3] This choiceis supported a posteriori by all applications in the light-quark and the heavy-light-quark sector resulting in 1199040(119879)decreasing monotonically with increasing temperature andeventually vanishing at a critical temperature 119879 = 119879119888An important exception to this behaviour is the heavy-heavy-quark system that is charmonium (vector scalar andpseudoscalar channels) [5 6] and bottonium [7] (vector andpseudoscalar) for which 1199040(119879) remains well above zero at orbeyond 119879119888 Crucial theoretical validation of the role playedby 1199040(119879) has been obtained recently in [46] where a direct


relation was found between 1199040(119879) and LQCDrsquos Polyakovthermal loop

On the hadronic sector the relevant parameters are thecurrent-hadron coupling and the hadronic width both ofwhich underpin the conclusions derived from the behaviourof 1199040(119879) to wit For light- and heavy-light-quark systems thecurrent-hadron coupling decreases and the hadronic widthincreases monotonically with increasing 119879 thus signallingdeconfinement Instead for the heavy-heavy-quark systemsthe coupling actually increases and the width while initiallygrowing reverses behaviour decreasing close to 119879119888 indicatingthe survival of these hadrons at and above 119879119888 This predictionwas later confirmed for bottonium by LQCD [8 9]

Another fundamental issue to which this method con-tributed was the relation between the two phase transitionsthat is deconfinement and chiral-symmetry restorationAfter preliminary indications of the approximate equality ofboth critical temperatures [45] a later analysis [53] supportedthis conclusion Recently a more refined updated analysis[47] fully confirmed earlier results

The extension of the well known Weinberg sum rules[80] to finite119879 without prejudice on some preexisting chiral-mixing scenario [84] clearly shows their full saturationexcept very close to 119879119888 albeit returning to full saturation at119879 = 119879119888 These deviations are caused by the thermal space-like cut in the energy plane arising at leading order in thevector channel but loop suppressed in the axial-vector caseThis asymmetric contribution growing with the square ofthe temperature vanishes at 119879 = 119879119888 Hence this featurehas no relation whatsoever with a potential chiral-mixingscenario In fact an inspection of the thermal behaviour ofthe hadronic parameters in the vector and the axial-vectorchannel fully disproves this idea These spectral functionsremain quite distinct at all temperatures except at 119879 = 119879119888

where they vanish for obvious reasons In any case and asshown in Section 5 as well as in [87] in a hadronic thermalbath there is a chiral asymmetry due to Isospin and 119866-paritypreventing any mixing

On a separate issue thermal QCD sum rules allowdetermining the behaviour of the light-quark masses 119898119906119889

together with the pion decay constant 119891120587(119879) [88] Thetwo sum rules for the light-quark pseudoscalar axial-vectorcurrent divergence require as input the 119879-dependence of thepion mass [94] and the quark-condensate [93] The resultfor 119891120587(119879) is fully consistent with chiral-symmetry in that itfollows the behaviour of |⟨119902119902⟩(119879)| independently of 119872120587(119879)(see (3) (4)) It is also consistent with the expectation thatclose to 119879119888 the quark mass should increase becoming theconstituent mass at deconfinement Finally QCDSR havebeen extended to finite119879 togetherwith finite baryon chemicalpotential 120583119861 [78] This has allowed obtaining the phasediagram (119879119888 120583119861) It should be possible in future to extendthe explored range of 120583119861 and study other applications at finite119879 and 120583119861

A topic not discussed here is that of nondiagonal(Lorentz noninvariant) condensates Clearly the existence ofa medium that is the thermal bath breaks trivially Lorentzinvariance However after choosing a reference system at restwith respect to the medium one can ignore this issue and

continue to use a covariant formulation Nevertheless theremight exist new terms in the OPE absent at 119879 = 0 In thecase of nongluonic operators it has been shown that they arehighly suppressed [65 100] so that they can be ignored Agluonic twist-two term in the OPE was considered in [101]and computed on the lattice in [102 103] Once again thecontribution of such a term is negligible in comparison withall regular (diagonal) terms as shown in [5]

In closing wewish to brieflymention a few applications ofthermal QCDSR which were not covered here An indepen-dent validation of this method was obtained by determiningthe thermal behaviour of certain three-point functions (formfactors) and in particular their associated root-mean-squared(rms) radii In the case of the electromagnetic form factor ofthe pion119865120587(1199022 119879) it was found in [104] that it decreases withincreasing 119879 almost independently of 1198762 The pion radius⟨119903120587⟩(119879) increases with temperature doubling at 119879119879119888 ≃ 08and diverging at 119879 ≃ 119879119888 thus signalling deconfinementOn a separate issue the axial-vector coupling of the nucleon119892119860(119879) was found to be essentially constant in most of thetemperature range except very close to 119879119888 where it startsto grow [105] The associated rms ⟨1199032119860⟩(119879) was also foundto be largely constant but diverging close to 119879119888 consistentwith deconfinementThis information was used to determinethe thermal behaviour of 119878119880(2) times 119878119880(2) Goldberger-Treimanrelation (GTR) and its deviation Δ120587 defined in [105] as

119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)

which is different from the standard definition Δ120587 = 1 minus119872119892119860119891120587119892120587119873119873 Given that the nucleon mass is basically inde-pendent of 119879 except very close to 119879119888 [106 107] and similarlyfor 119892119860(119879) the deviation Δ120587(119879) decreases with increasing 119879and the GTR ceases to be valid

Another thermal three-point function analysis dealt withthe coupling 119892120588120587120587 the associated rms radius and the issue ofthe Vector-Meson Dominance (VMD) at finite temperature[108ndash110] Results from [108] indicated the approximatevalidity of an extension of VMD where the strong coupling119892120588120587120587 becomes a function of the momentum transfer Thisextended coupling decreases with increasing temperaturevanishing just before 119879 = 119879119888 and the associated rms radiusdiverges close to the critical temperature thus signallingdeconfinement

Finally the Adler-Bell-Jackiw axial anomaly [111 112] atfinite 119879 was studied at low temperatures in [113] and in thewhole 119879 range in [114] Results from [113] showed that theamplitude of 1205870 rarr 120574120574 decreased with increasing119879The samebehaviour was found in [114] leading to the vanishing of thatamplitude provided VMD remains valid

Appendix

In this Appendix we derive the QCD expression of the QCDscattering term for a vector current correlation function ofnonzero (equal mass) quarks Extensions to other currentsandor unequal quark masses should be straightforward We


begin with the correlator (7) in the time-like region Substi-tuting in (7) the current119881120583(119909) š 119876119886(119909)120574120583119876119886(119909) where119876(119909)is a quark field of mass 119898119876 and 119886 is the colour index resultsin

Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)

= minus1198943119873119888 int 1198894119909119890119894119902119909Tr [120574120583119878119865 (119909) 120574]119878119865 (minus119909)] (A1)

where 119878119865(119909) is the quark propagator in space-time and119873119888 = 3 Transforming the propagators to momentum-spaceperforming the integrations and taking the imaginary part ofΠ(1199022) giveImΠ119886 (1199022) = 316120587 int+V

minusV119889119909 (1 minus 1199092) = 18120587V (3 minus V2) (A2)

where the variable V equiv V(1199022) is given by

V (1199022) = (1 minus 411989821198761199022 )

12

(A3)

Notice that because of the particular form of the currentin this case the normalization factor of Π(1199022) for masslessquarks is ImΠ(1199022) = 1(4120587) instead of 1(8120587) as in (8)

The extension to finite 119879 can be performed using theDolan-Jackiw thermal propagators (27) in (A1) to obtain

ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)

sdot [1 minus 119899119865 (|q| 119909 + 1205962119879 ) minus 119899119865 (|q| 119909 minus 1205962119879 )] (A4)

In the rest-frame of the medium |q| rarr 0 this reduces to

ImΠ119886 (120596 119879)= 316120587 int+V

minusV119889119909 (1 minus 1199092) [1 minus 2119899119865 ( 1205962119879)]

= 316120587 int+V

minusV119889119909 (1 minus 1199092) tanh( 1205964119879)

(A5)

Proceeding to the scattering term the equivalent to (A4) is

ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)

sdot [119899119865 (|q| 119909 + 1205962119879 ) minus 119899119865 (|q| 119909 minus 1205962119879 )] (A6)

where the integration limits arise from the bounds in theangular integration in momentum-space Notice that thisterm vanishes identically at 119879 = 0 and the overall multi-plicative factor is twice the one in (A4) Next the thermaldifference in the integrand can be converted into a derivative

ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)

where 119910 = |q|119909(2119879) This expression reduces to

ImΠ119904 (1199022 119879) = 34120587 120596|q| [minus119899119865 (|q| V2119879 ) (1 minus V2)+ 81198792

|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)

In the limit |q| rarr 0 this result becomes

ImΠ119904 (1199022 119879) = 3120587 lim|q|rarr0120596rarr0

120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)

After careful performance of the limit in the order indicatedthe singular term 120596|q|3 above becomes a delta-function

lim|q|rarr0120596rarr0

120596|q|3 = 23120575 (1205962) (A10)

and the final result for the scattering term is

ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)

Depending on the correlator the limiting function (A10)could instead be less singular in |q| in which case thescattering term vanishes identically

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work of Alejandro Ayala was supported in part byUNAM-DGAPA-PAPIIT Grant no IN101515 and by ConsejoNacional deCiencia y TecnologiaGrant no 256494Theworkof M Loewe was supported in part by Fondecyt 1130056Fondecyt 1150847 (Chile) and Proyecto Basal (Chile) FB0821 This work was also supported by NRF (South Africa)and the Research Administration University of Cape Town

References

[1] M A Shifman A I Vainshtein and V I Zakharov ldquoQCD andresonance physics Theoretical foundationsrdquo Nuclear Physics Bvol 147 no 5 pp 385ndash447 1979

[2] P Colangelo and A Khodjamirian ldquoQCD sum rules a modernperspectiverdquo in At the Frontier of Particle Physics M ShifmanEd vol 3 pp 1495ndash1576 World Scientific Singapore 2001

[3] A I Bochkarev and M E Shaposhnikov ldquoThe spectrum ofhot hadronic matter and finite-temperature QCD sum rulesrdquoNuclear Physics Section B vol 268 no 1 pp 220ndash252 1986


[4] C A Dominguez andM Loewe ldquoDimuon production in ultra-relativistic nuclear collisions and QCD phase transitionsrdquo Par-ticles amp Fields vol 49 no 3 pp 423ndash430 1991

[5] C A Dominguez M Loewe J C Rojas and Y Zhang ldquoChar-monium in the vector channel at finite temperature from QCDsum rulesrdquo Physical Review D vol 81 no 1 Article ID 0140072010

[6] C A Dominguez M Loewe J C Rojas and Y Zhangldquo(Pseudo)scalar charmonium in finite temperatureQCDrdquoPhys-ical Review D vol 83 no 3 2011

[7] C A Dominguez M Loewe and Y Zhang ldquoBottonium inQCD at finite temperaturerdquo Physical Review D vol 88 no 5Article ID 054015 2013

[8] G Aarts C Allton S Kim et al ldquoS wave bottomonium statesmoving in a quark-gluon plasma from lattice NRQCDrdquo Journalof High Energy Physics vol 2013 article 84 2013

[9] G Aarts C Allton S Kim M Lombardo S Ryan and JSkullerud ldquoMelting of Pwave bottomonium states in the quark-gluon plasma from lattice NRQCDrdquo Journal of High EnergyPhysics vol 12 article 064 2013

[10] H Pagels ldquoDepartures from chiral symmetryrdquo Physics Reportsvol 16 no 5 pp 219ndash311 1975

[11] C A Dominguez ldquoPhenomenological analysis of a dimension-two operator in QCD and its impact on 120572119904(119872119879)rdquo Physics LettersB vol 345 no 3 pp 291ndash295 1995

[12] C A Dominguez and K Schilcher ldquoIs there evidence fordimension-two corrections in QCD two-point functionsrdquoPhysical Review D vol 61 no 11 2000

[13] J Bordes C A Dominguez P Moodley J Penarrocha andK Schilcher ldquoChiral corrections to the SU(2) times SU(2) Gell-Mann-Oakes-Renner relationrdquo Journal of High Energy Physicsvol 2010 no 5 article 064 2010

[14] K A Olive K Agashe C Amsler et al ldquoReview of particlephysicsrdquo Chinese Physics vol 38 no 9 Article ID 090001 2014

[15] A Erdelyi EdHigher Transcendental Functions McGraw-HillNew York NY USA 1955

[16] E de Rafael ldquoCentre de physique theorique Marseillerdquo ReportCPT-81P1344 1981 Proceedings of the French-American Sem-inar Theoretical Aspects of Quantum Chromodynamics J WDash editor CPT-81P1345 1981

[17] C A Dominguez ldquoAnalytical determination of QCD quarkmassesrdquo in Fifty Years of Quarks H Fritzsch andMGell-MannEds pp 287ndash313 World Scientific Publishing Co Singapore2015

[18] R Shankar ldquoDetermination of the quark-gluon coupling con-stantrdquo Physical Review D vol 15 no 3 pp 755ndash758 1977

[19] A Bramon E Etim andM Greco ldquoA vectormeson dominanceapproach to scale invariancerdquo Physics Letters B vol 41 no 5 pp609ndash612 1972

[20] M Greco ldquoDeep-inelastic processesrdquoNuclear Physics B vol 63pp 398ndash412 1973

[21] E Etim and M Greco ldquoDuality sum rules in e+eminus annihilationfrom canonical trace anomaliesrdquo Lettere al Nuovo Cimento vol12 no 3 pp 91ndash95 1975

[22] C A Dominguez L A Hernandez K Schilcher and HSpiesberger ldquoQuarkndashhadron duality pinched kernel approachrdquoModern Physics Letters A vol 31 no 27 article 1630026 2016

[23] C A Dominguez L A Hernandez K Schilcher and H Spies-berger ldquoTests of quarkndashhadron duality in 120591-decaysrdquo ModernPhysics Letters A vol 31 no 31 Article ID 1630036 2016

[24] M Gonzalez Alonso A Pich and A Rodriguez-SanchezldquoDetermination of the QCD coupling from ALEPH 120591 decaydatardquo Physical Review D vol 94 no 3 Article ID 034027 2016

[25] CADominguez ldquoDetermination of light quarkmasses in qcdrdquoInternational Journal of Modern Physics A vol 25 no 29 pp5223ndash5234 2010

[26] C A Dominguez ldquoQuark masses in QCD a progress reportrdquoModern Physics Letters A vol 26 no 10 pp 691ndash710 2011

[27] C A Dominguez ldquoQuark mass determinations in QCDrdquoModern Physics Letters A vol 29 no 28 article 1430031 2014

[28] C A Dominguez ldquoAnalytical determination of the QCD quarkmassesrdquo International Journal of Modern Physics A vol 29 no29 24 pages 2014

[29] S Bodenstein C A Dominguez and K Schilcher ldquoHadroniccontribution to the muon 119892 minus 2 factor a theoretical determina-tionrdquo Physical Review D vol 85 no 1 Article ID 014029 2012

[30] S Bodenstein C A Dominguez K Schilcher and H Spies-berger ldquoHadronic contribution to the muon gminus2 factorrdquo Physi-cal Review D vol 88 no 1 Article ID 014005 2013

[31] C ADominguez K Schilcher andH Spiesberger ldquoTheoreticaldetermination of the hadronic g minus 2 of the muonrdquo ModernPhysics Letters A vol 31 no 32 Article ID 1630035 2016

[32] L J Reinders H Rubinstein and S Yazaki ldquoHadron propertiesfrom QCD sum rulesrdquo Physics Reports vol 127 no 1 pp 1ndash971985

[33] C A Dominguez L A Hernandez andK Schilcher ldquoDetermi-nation of the gluon condensate from data in the charm-quarkregionrdquo Journal of High Energy Physics vol 2015 article 1102015

[34] C A Dominguez L A Hernandez K Schilcher and HSpiesberger ldquoChiral sum rules and vacuum condensates fromtau-lepton decay datardquo Journal of High Energy Physics vol 20152015

[35] P Pascual and R Tarrach QCD Renormalization for thePractitioner vol 194 of Lecture Notes in Physics Springer 1984

[36] C A Dominguez and M Loewe ldquoComment on lsquoCurrentcorrelators in QCD at finite temperaturersquordquo Physical Review Dvol 52 no 5 p 3143 1995

[37] R J Furnstahl T Hatsuda and S H Lee ldquoApplications of QCDsum rules at finite temperaturerdquo Physical Review D vol 42 no5 article 1744 1990

[38] C Adami T Hatsuda and I Zahed ldquoQCD sum rules at lowtemperaturerdquo Physical Review D vol 43 no 3 article 921 1991

[39] C Adami and I Zahed ldquoFinite-temperature QCD sum rules forthe nucleonrdquo Physical Review D vol 45 no 11 pp 4312ndash43221992

[40] THatsuda Y Koike and S H Lee ldquoPattern of chiral restorationat low temperature from QCD sum rulesrdquo Physical Review Dvol 47 no 3 pp 1225ndash1230 1993

[41] T Hatsuda Y Koike and S-H Lee ldquoFinite-temperature QCDsum rules reexamined 120588 120596 and A1 mesonsrdquo Nuclear Physics Bvol 394 no 1 pp 221ndash264 1993

[42] Y Koike ldquoOctet baryons at finite temperature QCD sum rulesversus chiral symmetryrdquo Physical Review D vol 48 no 5 pp2313ndash2323 1993

[43] C Song ldquoMasses of vector and axial-vector mesons at finitetemperaturerdquo Physical Review D vol 48 no 3 pp 1375ndash13891993

[44] C Song ldquoPions at finite temperaturerdquo Physical Review D vol49 no 3 pp 1556ndash1565 1994


[45] C Dominguez and M Loewe ldquoDeconfinement and chiral-symmetry restoration at finite temperaturerdquo Physics Letters Bvol 233 no 1-2 pp 201ndash204 1989

[46] J P Carlomagno and M Loewe ldquoComparison between thecontinuum threshold and the Polyakov loop as deconfinementorder parametersrdquo httpsarxivorgabs161005429

[47] C A Dominguez M Loewe and Y Zhang ldquoChiral symmetryrestoration and deconfinement in QCD at finite temperaturerdquoPhysical Review D vol 86 no 3 Article ID 034030 2012

[48] L Dolan and R Jackiw ldquoSymmetry behavior at finite tempera-turerdquo Physical Review D vol 9 no 12 pp 3320ndash3341 1974

[49] J Gasser and H Leutwyler ldquoLight quarks at low temperaturesrdquoPhysics Letters B vol 184 no 1 pp 83ndash88 1987

[50] A Barducci R Casalbuoni S deCurtis RGatto andG PettinildquoPion decay constant at finite temperaturerdquo Physics Letters Bvol 240 no 3-4 pp 429ndash437 1990

[51] A Barducci R Casalbuoni S De Curtis R Gatto and G Pet-tini ldquoChiral-symmetry breaking in QCD at finite temperatureand densityrdquo Physics Letters B vol 231 no 4 pp 463ndash470 1989

[52] A Barducci R Casalbuoni S deCurtis RGatto andG PettinildquoChiral phase transitions in QCD for finite temperature anddensityrdquo Physical Review D vol 41 no 5 pp 1610ndash1619 1990

[53] A Barducci R Casalbuoni S De Curtis R Gatto and G Pet-tini ldquoHeuristic argument for coincidence or almost coincidenceof deconfinement and chirality restoration in finite temperatureQCDrdquo Physics Letters B vol 244 no 2 pp 311ndash315 1990

[54] M Davier A Hocker B Malaescu C Z Yuan and Z ZhangldquoUpdate of the ALEPH non-strange spectral functions fromhadronic 120591 decaysrdquo The European Physical Journal C vol 74article 2803 2014

[55] R Barate et al ldquoMeasurement of the axial-vector 120591 spec-tral functions and determination of 120572119904(1198722

120591) from hadronic 120591decaysrdquoThe European Physical Journal C vol 4 no 3 pp 409ndash431 1998

[56] S Schael R Barate R Bruneliere et al ldquoBranching ratios andspectral functions of 120591 decays final ALEPHmeasurements andphysics implicationsrdquo Physics Reports vol 421 no 5-6 pp 191ndash284 2005

[57] C A Dominguez and K Schilcher ldquoQCD vacuum condensatesfrom tau-lepton decay datardquo Journal of High Energy Physics vol2007 no 1 article no 93 2007

[58] S Bodenstein C A Dominguez S I Eidelman H Spiesbergerand K Schilcher ldquoConfronting electron-positron annihilationinto hadrons with QCD an operator product expansion analy-sisrdquo Journal of High Energy Physics vol 2012 article 39 2012

[59] C A Dominguez M S Fetea and M Loewe ldquoPions at finitetemperature from QCD sum rulesrdquo Physics Letters B vol 387no 1 pp 151ndash154 1996

[60] M Cheng N H Christ S Datta et al ldquoQCD equation of statewith almost physical quark massesrdquo Physical Review D vol 77no 1 Article ID 014511 2008

[61] S Qin L Chang H Chen Y Liu and C D Roberts ldquoPhasediagram and critical end point for strongly interacting quarksrdquoPhysical Review Letters vol 106 no 17 2011

[62] A Bazavov T Bhattacharya M Cheng et al ldquoEquation of stateand QCD transition at finite temperaturerdquo Physical Review Dvol 80 no 1 Article ID 014504 2009

[63] M Cheng S Ejiri P Hegde et al ldquoEquation of state for physicalquark massesrdquo Physical Review D vol 81 no 5 Article ID054504 2010

[64] A Ayala C A Dominguez M Loewe and Y Zhang ldquoRho-meson resonance broadening in QCD at finite temperaturerdquoPhysical Review D vol 86 no 11 Article ID 114036 2012

[65] CADominguezM Loewe and J C Rojas ldquoHeavy-light quarkpseudoscalar and vector mesons at finite temperaturerdquo Journalof High Energy Physics vol 2007 no 8 article 040 2007

[66] R Arnaldi et al ldquoFirst measurement of the rho spectralfunction in high-energy nuclear collisionsrdquo Physical ReviewLetters vol 96 Article ID 16302 2006

[67] R Arnaldi K Banicz J Castor et al ldquoEvidence for radial flowof thermal dileptons in high-energy nuclear collisionsrdquo PhysicalReview Letters vol 100 no 2 Article ID 022302 2008

[68] R Arnaldi K Banicz K Borer et al ldquoEvidence for theproduction of thermal muon pairs with masses above 1 GeVc2in 158AGeV indium-indium collisionsrdquoThe European PhysicalJournal C vol 59 no 3 pp 607ndash623 2009

[69] S Damjanovic ldquoThermal dileptons at SPS energiesrdquo Journal ofPhysics G Nuclear and Particle Physics vol 35 no 10 Article ID104036 2008

[70] S Damjanovic ldquoNA60 results on thermal dimuonsrdquoThe Euro-pean Physical Journal C vol 61 no 4 pp 711ndash720 2009

[71] G Domokos and J I Goldman ldquoDiagnosing quark matterby measuring the total entropy and the photon or dileptonemission ratesrdquo Physical Review D vol 32 no 5 p 1109 1985

[72] K Kajantie J Kapusta L McLerran and A Mekjian ldquoDileptonemission and the QCD phase transition in ultrarelativisticnuclear collisionsrdquo Physical Review D vol 34 no 9 pp 2746ndash2754 1986

[73] J Cleymans J Fingberg and K Redlich ldquoTransverse-momentum distribution of dileptons in different scenarios forthe QCD phase transitionrdquo Physical Review D vol 35 no 7 pp2153ndash2165 1987

[74] J D Bjorken ldquoHighly relativistic nucleus-nucleus collisions thecentral rapidity regionnrdquo Physical ReviewD vol 27 no 1 article140 1983

[75] V Eletsky B Ioffe and J Kapusta ldquoIn-medium modification of120587-mesons produced in heavy ion collisionsrdquo Nuclear Physics Avol 642 no 1-2 pp c155ndashc164 1998

[76] V L Eletsky B L Ioffe and J I Kapusta ldquoMass shift and widthbroadening of 120588-mesons produced in heavy ion collisionsrdquoTheEuropean Physical Journal A vol 3 no 4 pp 381ndash387 1998

[77] A Ayala C A Dominguez L A Hernandez M Loewe andA J Mizher ldquoDimuon production from in-medium rho decaysfromQCD sum rulesrdquo Physical Review D vol 88 no 11 ArticleID 114028 2013

[78] A Ayala A Bashir C A Dominguez E Gutierrez M Loeweand A Raya ldquoQCD phase diagram from finite energy sumrulesrdquo Physical Review D vol 84 no 5 2011

[79] P M Hohler and R Rapp ldquoDileptons and chiral symmetryrestorationrdquo Nuclear and Particle Physics Proceedings vol 276-278 pp 253ndash256 2016

[80] S Weinberg ldquoPrecise relations between the spectra of vectorand axial-vector mesonsrdquo Physical Review Letters vol 18 no 13article 507 1967

[81] C A Dominguez and K Schilcher ldquoChiral sum rules andduality in QCDrdquo Physics Letters B vol 448 no 1-2 pp 93ndash981999

[82] K Maltman ldquoConstraints on hadronic spectral functions fromcontinuous families of finite energy sum rulesrdquo Physics LettersB vol 440 no 3-4 pp 367ndash374 1998


[83] A Ayala C Dominguez M Loewe and Y Zhang ldquoWeinbergsum rules at finite temperaturerdquo Physical Review D vol 90 no3 Article ID 034012 2014

[84] M Dey V Eletsky and B Ioffe ldquoMixing of vector and axialmesons at finite temperature an indication towards chiralsymmetry restorationrdquo Physics Letters B vol 252 no 4 pp 620ndash624 1990

[85] A Pich and E de Rafael ldquoKminus 119870 mixing in the standard modelrdquoPhysics Letters B vol 158 no 6 pp 477ndash484 1985

[86] J Prades C A Dominguez J A Penarrocha A Pich andE Rafael ldquoThe K0minusK0B-factor in the QCD-hadronic dualityapproachrdquo Zeitschrift fur Physik C Particles and Fields vol 51no 2 pp 287ndash295 1991

[87] SMallik and S Sarkar ldquoVector and axial-vectormesons at finitetemperaturerdquoTheEuropean Physical Journal C vol 25 no 3 pp445ndash452 2002

[88] C A Dominguez and L A Hernandez ldquoDetermination of thetemperature dependence of the up- and down-quark massesin QCDrdquo Modern Physics Letters A vol 31 no 36 Article ID1630042 2016

[89] S Aoki Y Aoki C Bernard et al ldquoReview of lattice resultsconcerning low energy particle physicsrdquo httpsarxivorgabs13108555

[90] C A Dominguez L A Hernandez K Schilcher and HSpiesberger ldquoChiral sum rules and vacuum condensates fromtau-lepton decay datardquo Journal of High Energy Physics vol 2015no 53 2015

[91] S Borsanyi Z Fodor C Hoelbling et al ldquoIs there still any T119888

mystery in lattice QCD Results with physical masses in thecontinuum limit IIIrdquo The Journal of High Energy Physics vol2010 article 73 2010

[92] T Bhuttacharya M I Buchoff N H Christ et al ldquoQCDphase transition with chiral quarks and physical quark massesrdquoPhysical Review Letters vol 113 no 8 Article ID 082001 2014

[93] G S Bali F Bruckmann G Endrodi Z Fodor S D Katz andA Schafer ldquoQCDquark condensate in externalmagnetic fieldsrdquoPhysical Review D vol 86 no 7 2012

[94] M Heller and M Mitter ldquoPion and 120578-meson mass splitting atthe two-flavor chiral crossoverrdquo Physical Review D vol 94 no7 2016

[95] T Matsui and H Satz ldquoJ120595 suppression by quark-gluon plasmaformationrdquo Physics Letters B vol 178 no 4 pp 416ndash422 1986

[96] T Altherr and D Seibert ldquoThermal quark production inultrarelativistic nuclear collisionsrdquo Physical Review C vol 49no 3 pp 1684ndash1692 1994

[97] G Boyd and D E Miller ldquoThe temperature dependence ofthe SU(N) gluon condensate from lattice gauge theoryrdquo httpsarxivorgabshep-ph9608482

[98] D E Miller ldquoGluon condensates at finite temperaturerdquo httpsarxivorgabshep-ph0008031

[99] V L Eletsky ldquoFour-quark condensates at T =0rdquo Physics LettersB vol 299 no 1-2 pp 111ndash114 1993

[100] V L Eletsky ldquoBaryon masses from QCD current correlators at119879 = 0rdquo Physics Letters B vol 352 no 3-4 pp 440ndash444 1995[101] F Klingl S Kim S H Lee P Morath andWWeise ldquoMasses of119869120595 and 120578119888 in the nuclear medium QCD sum rule approachrdquo

Physical Review Letters vol 82 no 17 p 3396 1999[102] K Morita and S H Lee ldquoMass shift and width broadening

of 119869120595 in hot gluonic plasma from QCD sum rulesrdquo PhysicalReview Letters vol 100 no 2 Article ID 022301 2008

[103] K Morita and S H Lee ldquoCritical behavior of charmonia acrossthe phase transition a QCD sum rule approachrdquo PhysicalReview C vol 77 no 6 Article ID 064904 2008

[104] C A Dominguez M Loewe and J S Rozowsky ldquoElectromag-netic pion form factor at finite temperaturerdquo Physics Letters Bvol 335 no 3-4 pp 506ndash509 1994

[105] C A Dominguez M Loewe and C van Gend ldquoQCD sum ruledetermination of the axial-vector coupling of the nucleon atfinite temperaturerdquo Physics Letters B vol 460 no 3-4 pp 442ndash446 1999

[106] C A Dominguez and M Loewe ldquoNucleon propagator at finitetemperaturerdquo Zeitschrift fur Physik C Particles and Fields vol58 no 2 pp 273ndash277 1993

[107] H Leutwyler andA V Smilga ldquoNucleons at finite temperaturerdquoNuclear Physics Section B vol 342 no 2 pp 302ndash316 1990

[108] C A Dominguez M S Fetea and M Loewe ldquoVector mesondominance and g984858120587120587 at finite temperature fromQCDsumrulesrdquoPhysics Letters B vol 406 no 1-2 pp 149ndash153 1997

[109] R D Pisarski ldquoThermal rhorsquos in the quark-gluon plasmardquoNuclear Physics A vol 590 no 1-2 pp 553Cndash556C 1995

[110] R D Pisarski ldquoWhere does the 120588 go Chirally symmetric vectormesons in the quark-gluon plasmardquo Physical Review D vol 52no 7 pp R3773ndashR3776 1995

[111] S L Adler ldquoAxial-vector vertex in spinor electrodynamicsrdquoPhysical Review vol 177 no 5 article 2426 1969

[112] J S Bell and R Jackiw ldquoA PCAC puzzle 1205870 rarr 120574120574 in the 120590-modelrdquo Il Nuovo Cimento A vol 60 no 1 pp 47ndash61 1969

[113] R D Pisarski T L Trueman and M H G Tytgat ldquoHow 1205870 rarr120574120574 changes with temperaturerdquo Physical Review DmdashParticlesFields Gravitation and Cosmology vol 56 no 11 article 70771997

[114] C Dominguez and M Loewe ldquoAxial anomaly vector mesondominance and 1205870 rarr 120574120574 at finite temperaturerdquo Physics LettersB vol 481 no 2-4 pp 295ndash298 2000

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



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Journal of



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Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


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Journal of

Biophysics


ThermodynamicsJournal of


(s) (a)

(b)

s equiv E2

Realistic spectral function (T)

Im Π

s0(T) s0(0)

Figure 1 Typical hadronic spectral function ImΠ(119904) at 119879 = 0curve (a) showing a pole and three resonancesThe squared-energy1199040(0) is the threshold for PQCD At finite 119879 curve (b) the stablehadron develops a width the resonances become broader (onlyone survives in this example) and the PQCD threshold 1199040(119879)approaches the origin Eventually at 119879 ≃ 119879119888 there will be no traceof resonances and 1199040(119879119888) rarr 0

where 119891120587 ≃ 93MeV is the pion (weak-interaction) decayconstant defined as ⟨0|119860120583(0)|120587(119901)⟩ = radic2119891120587119901120583 and 119872120587 isits mass This is followed by resonances of widths increasingin size with increasing 119904 and corresponding to poles in thesecond Riemann sheet in the complex 119904-plane For instancefor narrow resonances the Breit-Wigner parametrization isnormally adequate

ImΠ (119904)|RES = 1198912119877

1198723119877Γ119877(119904 minus 1198722

119877)2 + 1198722119877Γ2

119877

(2)

where 119891119877 is the coupling of the resonance to the currententering a correlation function 119872119877 being its mass and Γ119877being its (hadronic) width At high enough squared-energy1199040(0) ≃ 2-3GeV2 the spectral function becomes smooth andshould be well approximated by perturbative QCD (PQCD)In the sequel this parameter will be indistinctly referred to asthe perturbative QCD threshold the continuum thresholdor the deconfinement parameter At finite 119879 this spectrumgets distorted The pole in the real axis moves down into thesecond Riemann sheet thus generating a finite width Thewidths of the rest of the resonances increase with increasing119879 and some states begin to disappear from the spectrum asfirst proposed in [4] Eventually close to or at the criticaltemperature for deconfinement 119879 ≃ 119879119888 there will be notrace of the resonances as their widths would be very largeand their couplings to hadronic currents would approachzero At the same time 1199040(119879) would approach the originThus 1199040(119879) becomes a phenomenological order parameterfor quark-deconfinement as first proposed by Bochkarevand Shaposhnikov [3] This order parameter associated withQCD deconfinement is entirely phenomenological and quitedifferent from the Polyakov-loop used by LQCD Never-theless and quite importantly qualitative and quantitativeconclusions regarding this phase transition (or crossover)and the behaviour of QCD and hadronic parameters as119879 rarr 119879119888 obtained from QCDSR and LQCD should agreeIt is reassuring that this turns out to be the case as willbe reviewed here In this scenario whatever happens to

the mass is totally irrelevant it could either increase ordecrease with temperature providing no information aboutdeconfinementThe crucial parameters are the width and thecoupling but not the mass In fact if a particle mass wouldapproach the origin or even vanish with increasing temper-ature this in itself is not sufficient to signal deconfinementas a massless particle with a finite coupling and width wouldstill contribute to the spectrum What is required is that thewidths diverge and the couplings vanish In all applicationsof QCD sum rules at finite 119879 the hadron masses in somechannels decrease and in other cases they increase slightlywith increasing 119879 At the same time for all light- and heavy-light-quark bound states the widths are found to divergeand the couplings to vanish close to or at 119879119888 thus signallingdeconfinement However in the case of charmonium andbottonium hadronic states after an initial surge the widthsdecrease considerably with increasing temperature while thecouplings are initially independent of 119879 and eventually growsharply close to 119879119888 This survival of charmonium states wasfirst predicted from thermal QCD sum rules [5 6] and laterextended to bottonium [7] in qualitative agreement withLQCD [8 9]

In addition to 1199040(119879) there is another important thermalQCD quantity the quark-condensate this time a funda-mental order parameter of chiral-symmetry restoration Itis well known that QCD in the light-quark sector possessesa chiral 119878119880(2) times 119878119880(2) symmetry in the limit of zero-massup and down quarks This chiral-symmetry is realized inthe Nambu-Goldstone fashion as opposed to the Wigner-Weyl realization [10] In other words chiral-symmetry is adynamical as opposed to a classification symmetry Hencethe pion mass squared vanishes as the quark mass

1198722120587 = 119861119898119902 (3)

and the pion decay constant squared vanishes as the quark-condensate

1198912120587 = 1119861 ⟨119902119902⟩ (4)

where 119861 is a constant While from (3) 1198722120587 can vanish as

the quark mass at 119879 = 0 the vanishing of 119891120587 can onlytake place at finite temperature as ⟨119902119902⟩119879 rarr 0 at 119879 =119879119888 the critical temperature for chiral-symmetry restorationThe correct meaning of chiral-symmetry restoration is aphase transition from a Nambu-Goldstone realization ofchiral 119878119880(2) times 119878119880(2) to a Wigner-Weyl realization of thesymmetry In other words there is a clear distinction betweena symmetry of the Lagrangian and a symmetry of the vacuum[10]

In summary at finite temperature the hadronic param-eters to provide relevant information on deconfinement arethe hadron width and its coupling to the correspondinginterpolating current On the QCD side we have (a) thechiral condensate ⟨119902119902⟩119879 providing information on chiral-symmetry restoration and (b) the onset of PQCD as deter-mined by the squared-energy 1199040(119879) providing informationon quark-deconfinement The next step is to relate these twosectors This is currently done by considering the complex




Π (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119869 (119909) 119869dagger (0)) |0⟩ (5)


Π (1199022)10038161003816100381610038161003816QCD = 1198620 + sum119873=1

1198622119873 (1199022 1205832)(minus1199022)119873 ⟨O2119873 (1205832)⟩ (6)



Π120583] (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119881120583 (119909) 119881dagger] (0)) |0⟩

= (minus119892120583]1199022 + 119902120583119902]) Π (1199022) (7)


ImΠ (1199022) = 18120587 [1 + O (120572119904 (1199022))] (8)


1198624 ⟨O4⟩ = 1205873 ⟨1205721199041198662⟩ + 41205872 (119898119906 + 119898119889) ⟨119902119902⟩ (9)


(119898119906 + 119898119889) ⟨119902119902⟩ = 11989121205871198722

120587 (10)



1120587 ImΠ1003816100381610038161003816100381610038161003816HAD (119904) = 11205871198912120588

1198723120588Γ120588

(119904 minus 1198722120588)2 + 1198722

120588Γ2120588

(11)


ImΠ|(0)HAD (119904) = 11987221205881198912

120588120575 (119904 minus 1198722120588) (12)




120593119873 (11987620) equiv 1119873 (minus 1198891198891198762

)119873 Π (1198762)1003816100381610038161003816100381610038161003816100381610038161198762=11987620= 1120587 intinfin

0

ImΠ (119904)(119904 + 1198762

0)119873+1119889119904

(13)




(14)


119872 [Π (1198762)]equiv lim

1198762 119873rarrinfin

1198762119873equiv1198722

(minus)119873(119873 minus 1) (1198762)119873 ( 1198891198891198762)119873 Π (1198762)

equiv Π (1198722) = 11198722intinfin

0

1120587 ImΠ (119904) 119890minus1199041198722119889119904(15)







Im (s)

Re (s)



∮ Π (119904) 119889119904 = 0= int1199040

0Π (119904 + 119894120598) 119889119904 + int0

1199040

Π (119904 minus 119894120598) 119889119904+ ∮

119862(|1199040|)Π (119904) 119889119904

(16)


int1199040

0

1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD 119875 (119904) 119889119904= minus 12120587119894 ∮

119862(|1199040|)Π (119904)QCD 119875 (119904) 119889119904

(17)



(minus)(119873minus1) 1198622119873 ⟨O2119873⟩= 81205872 int1199040


(119873 = 1 2 ) (18)


1198626 ⟨O6⟩= minus81205873120572119904 [⟨(1199021205741205831205745120582119886119902)2⟩ + 29 ⟨(119902120574120583120582119886119902)2⟩] (19)






120593119873 (11987620)10038161003816100381610038161003816HAD = 120593119873 (1198762

0)10038161003816100381610038161003816QCD (20)

where

120593119873 (11987620)10038161003816100381610038161003816HAD equiv 1120587 int1199040

0

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|HAD

120593119873 (11987620)10038161003816100381610038161003816QCD


41198982119876

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|PQCD + 120593119873 (1198762

0)10038161003816100381610038161003816NP (21)






⟨119876119876⟩ = minus 112119898119876

⟨120572119904120587 1198662⟩ (22)






Π120583] (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119860120583 (119909) 119860dagger] (0)) |0⟩

= minus119892120583]Π1 (1199022) + 119902120583119902]Π0 (1199022) (23)


ImΠ0 (1199022)10038161003816100381610038161003816QCD= 14120587 [1 + O (120572119904 (1199022))] (24)




ImΠ0 (1199022)10038161003816100381610038161003816HAD = 21205871198912120587120575 (119904) (25)



1199040 = 812058721198912120587 (26)



119878119865 (119896 119879) = 119894119896 minus 119898minus 2120587

(119890|1198960|119879 + 1) (119896 + 119898) 120575 (1198962 minus 1198982) (27)




ImΠ1198860 (120596 119879)1003816100381610038161003816QCD = 14120587 [1 minus 2119899119865 ( 1205962119879)] 120579 (1205962)

= 14120587 tanh( 1205964119879) 120579 (1205962) (28)


ImΠ1199040 (120596 119879)1003816100381610038161003816QCD = 4120587120575 (1205962) intinfin

0119910119899119865 ( 119910119879) 119889119910

= 1205873 1198792120575 (1205962) (29)


812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)] (30)




06 07 08 09 10 11 12 13 14 15 1605

s (GeV2)

000

001

002

003

004

005

006

007

008

1120587

Im0

(s)

Π





1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587120575 (119904)

+ 119862119891 exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(31)




1198626 ⟨O6⟩10038161003816100381610038161003816119860 prop 1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (32)


211989121205871198982

120587 = minus (119898119906 + 119898119889) ⟨0| 119906119906 + 119889119889 |0⟩ (33)





812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)]

minus 41205872 int1199040(119879)

0119889119904 1120587 ImΠ0 (119904 119879)10038161003816100381610038161003816100381610038161198861

minus1198624 ⟨O4⟩ (119879) = 41205872 int1199040(119879)

0119889119904119904 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

0119889119904119904 [1 minus 2119899119865 (radic1199042119879)]


00 01 02 03 04 05 06 07 08 09 10

TTc

16

14

12

10

08

06

04

02

00

C4⟨O

4⟩(T)C4⟨O

4⟩(0)


1412100806040200

12

10

08

06

04

02

00

TTc

f2 120587(T)f2 120587(0)



1198626 ⟨O6⟩ (119879) = 41205872 int1199040(119879)

01198891199041199042 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

01198891199041199042 [1 minus 2119899119865 (radic1199042119879)]

(34)


00 01 02 03 04 05 06 07 08 09 1110

TTc

s 0(T)s 0(0)

14

12

10

08

06

04

02

00



00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

08

06

04

02

00

f(T)f(0)







00 01 02 03 04 05 06 07 08 09 10

TTc

Γ(T)Γ(0)

15

14

13

12

11

10

09



1120587 ImΠ1199041003816100381610038161003816100381610038161003816HAD (120596 119879) = 231205872

120575 (1205962) intinfin

0119910119899119861 ( 119910119879) 119889119910 (35)


1198626 ⟨O6⟩10038161003816100381610038161003816119881 prop minus1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (36)



Γ120588 (119879) = Γ120588 (0)1 minus (119879119879119888)119886 (37)

where 119886 = 3 and 119879119888 = 197MeV

1198626 ⟨O6⟩ (119879) = 1198626 ⟨O6⟩ (0) [[

1 minus ( 119879119879lowast119902

)119887]]

(38)


119872120588 (119879) = 119872120588 (0) [1 minus ( 119879119879lowast119872

)119888] (39)




1199040 (119879) = 1199040 (0)sdot [1 minus 05667 ( 119879119879119888

)1138 minus 4347 ( 119879119879119888

)6841] (40)

1198624 ⟨O4⟩ (119879) = 1198624 ⟨O4⟩ (0)sdot [1 minus 165 ( 119879119879119888

)8735 + 004967 ( 119879119879119888

)07211] (41)

119891120588 (119879)119891120588 (0) = 1 minus 03901 ( 119879119879119888

)1075 + 004155 ( 119879119879119888

)1269 (42)




14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)


1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00





06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Π (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119869 (119909) 119869dagger (0)) |0⟩ (5)


Π (1199022)10038161003816100381610038161003816QCD = 1198620 + sum119873=1

1198622119873 (1199022 1205832)(minus1199022)119873 ⟨O2119873 (1205832)⟩ (6)



Π120583] (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119881120583 (119909) 119881dagger] (0)) |0⟩

= (minus119892120583]1199022 + 119902120583119902]) Π (1199022) (7)


ImΠ (1199022) = 18120587 [1 + O (120572119904 (1199022))] (8)


1198624 ⟨O4⟩ = 1205873 ⟨1205721199041198662⟩ + 41205872 (119898119906 + 119898119889) ⟨119902119902⟩ (9)


(119898119906 + 119898119889) ⟨119902119902⟩ = 11989121205871198722

120587 (10)



1120587 ImΠ1003816100381610038161003816100381610038161003816HAD (119904) = 11205871198912120588

1198723120588Γ120588

(119904 minus 1198722120588)2 + 1198722

120588Γ2120588

(11)


ImΠ|(0)HAD (119904) = 11987221205881198912

120588120575 (119904 minus 1198722120588) (12)




120593119873 (11987620) equiv 1119873 (minus 1198891198891198762

)119873 Π (1198762)1003816100381610038161003816100381610038161003816100381610038161198762=11987620= 1120587 intinfin

0

ImΠ (119904)(119904 + 1198762

0)119873+1119889119904

(13)




(14)


119872 [Π (1198762)]equiv lim

1198762 119873rarrinfin

1198762119873equiv1198722

(minus)119873(119873 minus 1) (1198762)119873 ( 1198891198891198762)119873 Π (1198762)

equiv Π (1198722) = 11198722intinfin

0

1120587 ImΠ (119904) 119890minus1199041198722119889119904(15)







Im (s)

Re (s)



∮ Π (119904) 119889119904 = 0= int1199040

0Π (119904 + 119894120598) 119889119904 + int0

1199040

Π (119904 minus 119894120598) 119889119904+ ∮

119862(|1199040|)Π (119904) 119889119904

(16)


int1199040

0

1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD 119875 (119904) 119889119904= minus 12120587119894 ∮

119862(|1199040|)Π (119904)QCD 119875 (119904) 119889119904

(17)



(minus)(119873minus1) 1198622119873 ⟨O2119873⟩= 81205872 int1199040


(119873 = 1 2 ) (18)


1198626 ⟨O6⟩= minus81205873120572119904 [⟨(1199021205741205831205745120582119886119902)2⟩ + 29 ⟨(119902120574120583120582119886119902)2⟩] (19)






120593119873 (11987620)10038161003816100381610038161003816HAD = 120593119873 (1198762

0)10038161003816100381610038161003816QCD (20)

where

120593119873 (11987620)10038161003816100381610038161003816HAD equiv 1120587 int1199040

0

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|HAD

120593119873 (11987620)10038161003816100381610038161003816QCD


41198982119876

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|PQCD + 120593119873 (1198762

0)10038161003816100381610038161003816NP (21)






⟨119876119876⟩ = minus 112119898119876

⟨120572119904120587 1198662⟩ (22)






Π120583] (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119860120583 (119909) 119860dagger] (0)) |0⟩

= minus119892120583]Π1 (1199022) + 119902120583119902]Π0 (1199022) (23)


ImΠ0 (1199022)10038161003816100381610038161003816QCD= 14120587 [1 + O (120572119904 (1199022))] (24)




ImΠ0 (1199022)10038161003816100381610038161003816HAD = 21205871198912120587120575 (119904) (25)



1199040 = 812058721198912120587 (26)



119878119865 (119896 119879) = 119894119896 minus 119898minus 2120587

(119890|1198960|119879 + 1) (119896 + 119898) 120575 (1198962 minus 1198982) (27)




ImΠ1198860 (120596 119879)1003816100381610038161003816QCD = 14120587 [1 minus 2119899119865 ( 1205962119879)] 120579 (1205962)

= 14120587 tanh( 1205964119879) 120579 (1205962) (28)


ImΠ1199040 (120596 119879)1003816100381610038161003816QCD = 4120587120575 (1205962) intinfin

0119910119899119865 ( 119910119879) 119889119910

= 1205873 1198792120575 (1205962) (29)


812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)] (30)




06 07 08 09 10 11 12 13 14 15 1605

s (GeV2)

000

001

002

003

004

005

006

007

008

1120587

Im0

(s)

Π





1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587120575 (119904)

+ 119862119891 exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(31)




1198626 ⟨O6⟩10038161003816100381610038161003816119860 prop 1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (32)


211989121205871198982

120587 = minus (119898119906 + 119898119889) ⟨0| 119906119906 + 119889119889 |0⟩ (33)





812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)]

minus 41205872 int1199040(119879)

0119889119904 1120587 ImΠ0 (119904 119879)10038161003816100381610038161003816100381610038161198861

minus1198624 ⟨O4⟩ (119879) = 41205872 int1199040(119879)

0119889119904119904 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

0119889119904119904 [1 minus 2119899119865 (radic1199042119879)]


00 01 02 03 04 05 06 07 08 09 10

TTc

16

14

12

10

08

06

04

02

00

C4⟨O

4⟩(T)C4⟨O

4⟩(0)


1412100806040200

12

10

08

06

04

02

00

TTc

f2 120587(T)f2 120587(0)



1198626 ⟨O6⟩ (119879) = 41205872 int1199040(119879)

01198891199041199042 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

01198891199041199042 [1 minus 2119899119865 (radic1199042119879)]

(34)


00 01 02 03 04 05 06 07 08 09 1110

TTc

s 0(T)s 0(0)

14

12

10

08

06

04

02

00



00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

08

06

04

02

00

f(T)f(0)







00 01 02 03 04 05 06 07 08 09 10

TTc

Γ(T)Γ(0)

15

14

13

12

11

10

09



1120587 ImΠ1199041003816100381610038161003816100381610038161003816HAD (120596 119879) = 231205872

120575 (1205962) intinfin

0119910119899119861 ( 119910119879) 119889119910 (35)


1198626 ⟨O6⟩10038161003816100381610038161003816119881 prop minus1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (36)



Γ120588 (119879) = Γ120588 (0)1 minus (119879119879119888)119886 (37)

where 119886 = 3 and 119879119888 = 197MeV

1198626 ⟨O6⟩ (119879) = 1198626 ⟨O6⟩ (0) [[

1 minus ( 119879119879lowast119902

)119887]]

(38)


119872120588 (119879) = 119872120588 (0) [1 minus ( 119879119879lowast119872

)119888] (39)




1199040 (119879) = 1199040 (0)sdot [1 minus 05667 ( 119879119879119888

)1138 minus 4347 ( 119879119879119888

)6841] (40)

1198624 ⟨O4⟩ (119879) = 1198624 ⟨O4⟩ (0)sdot [1 minus 165 ( 119879119879119888

)8735 + 004967 ( 119879119879119888

)07211] (41)

119891120588 (119879)119891120588 (0) = 1 minus 03901 ( 119879119879119888

)1075 + 004155 ( 119879119879119888

)1269 (42)




14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)


1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00





06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120593119873 (11987620) equiv 1119873 (minus 1198891198891198762

)119873 Π (1198762)1003816100381610038161003816100381610038161003816100381610038161198762=11987620= 1120587 intinfin

0

ImΠ (119904)(119904 + 1198762

0)119873+1119889119904

(13)




(14)


119872 [Π (1198762)]equiv lim

1198762 119873rarrinfin

1198762119873equiv1198722

(minus)119873(119873 minus 1) (1198762)119873 ( 1198891198891198762)119873 Π (1198762)

equiv Π (1198722) = 11198722intinfin

0

1120587 ImΠ (119904) 119890minus1199041198722119889119904(15)







Im (s)

Re (s)



∮ Π (119904) 119889119904 = 0= int1199040

0Π (119904 + 119894120598) 119889119904 + int0

1199040

Π (119904 minus 119894120598) 119889119904+ ∮

119862(|1199040|)Π (119904) 119889119904

(16)


int1199040

0

1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD 119875 (119904) 119889119904= minus 12120587119894 ∮

119862(|1199040|)Π (119904)QCD 119875 (119904) 119889119904

(17)



(minus)(119873minus1) 1198622119873 ⟨O2119873⟩= 81205872 int1199040


(119873 = 1 2 ) (18)


1198626 ⟨O6⟩= minus81205873120572119904 [⟨(1199021205741205831205745120582119886119902)2⟩ + 29 ⟨(119902120574120583120582119886119902)2⟩] (19)






120593119873 (11987620)10038161003816100381610038161003816HAD = 120593119873 (1198762

0)10038161003816100381610038161003816QCD (20)

where

120593119873 (11987620)10038161003816100381610038161003816HAD equiv 1120587 int1199040

0

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|HAD

120593119873 (11987620)10038161003816100381610038161003816QCD


41198982119876

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|PQCD + 120593119873 (1198762

0)10038161003816100381610038161003816NP (21)






⟨119876119876⟩ = minus 112119898119876

⟨120572119904120587 1198662⟩ (22)






Π120583] (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119860120583 (119909) 119860dagger] (0)) |0⟩

= minus119892120583]Π1 (1199022) + 119902120583119902]Π0 (1199022) (23)


ImΠ0 (1199022)10038161003816100381610038161003816QCD= 14120587 [1 + O (120572119904 (1199022))] (24)




ImΠ0 (1199022)10038161003816100381610038161003816HAD = 21205871198912120587120575 (119904) (25)



1199040 = 812058721198912120587 (26)



119878119865 (119896 119879) = 119894119896 minus 119898minus 2120587

(119890|1198960|119879 + 1) (119896 + 119898) 120575 (1198962 minus 1198982) (27)




ImΠ1198860 (120596 119879)1003816100381610038161003816QCD = 14120587 [1 minus 2119899119865 ( 1205962119879)] 120579 (1205962)

= 14120587 tanh( 1205964119879) 120579 (1205962) (28)


ImΠ1199040 (120596 119879)1003816100381610038161003816QCD = 4120587120575 (1205962) intinfin

0119910119899119865 ( 119910119879) 119889119910

= 1205873 1198792120575 (1205962) (29)


812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)] (30)




06 07 08 09 10 11 12 13 14 15 1605

s (GeV2)

000

001

002

003

004

005

006

007

008

1120587

Im0

(s)

Π





1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587120575 (119904)

+ 119862119891 exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(31)




1198626 ⟨O6⟩10038161003816100381610038161003816119860 prop 1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (32)


211989121205871198982

120587 = minus (119898119906 + 119898119889) ⟨0| 119906119906 + 119889119889 |0⟩ (33)





812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)]

minus 41205872 int1199040(119879)

0119889119904 1120587 ImΠ0 (119904 119879)10038161003816100381610038161003816100381610038161198861

minus1198624 ⟨O4⟩ (119879) = 41205872 int1199040(119879)

0119889119904119904 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

0119889119904119904 [1 minus 2119899119865 (radic1199042119879)]


00 01 02 03 04 05 06 07 08 09 10

TTc

16

14

12

10

08

06

04

02

00

C4⟨O

4⟩(T)C4⟨O

4⟩(0)


1412100806040200

12

10

08

06

04

02

00

TTc

f2 120587(T)f2 120587(0)



1198626 ⟨O6⟩ (119879) = 41205872 int1199040(119879)

01198891199041199042 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

01198891199041199042 [1 minus 2119899119865 (radic1199042119879)]

(34)


00 01 02 03 04 05 06 07 08 09 1110

TTc

s 0(T)s 0(0)

14

12

10

08

06

04

02

00



00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

08

06

04

02

00

f(T)f(0)







00 01 02 03 04 05 06 07 08 09 10

TTc

Γ(T)Γ(0)

15

14

13

12

11

10

09



1120587 ImΠ1199041003816100381610038161003816100381610038161003816HAD (120596 119879) = 231205872

120575 (1205962) intinfin

0119910119899119861 ( 119910119879) 119889119910 (35)


1198626 ⟨O6⟩10038161003816100381610038161003816119881 prop minus1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (36)



Γ120588 (119879) = Γ120588 (0)1 minus (119879119879119888)119886 (37)

where 119886 = 3 and 119879119888 = 197MeV

1198626 ⟨O6⟩ (119879) = 1198626 ⟨O6⟩ (0) [[

1 minus ( 119879119879lowast119902

)119887]]

(38)


119872120588 (119879) = 119872120588 (0) [1 minus ( 119879119879lowast119872

)119888] (39)




1199040 (119879) = 1199040 (0)sdot [1 minus 05667 ( 119879119879119888

)1138 minus 4347 ( 119879119879119888

)6841] (40)

1198624 ⟨O4⟩ (119879) = 1198624 ⟨O4⟩ (0)sdot [1 minus 165 ( 119879119879119888

)8735 + 004967 ( 119879119879119888

)07211] (41)

119891120588 (119879)119891120588 (0) = 1 minus 03901 ( 119879119879119888

)1075 + 004155 ( 119879119879119888

)1269 (42)




14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)


1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00





06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Im (s)

Re (s)



∮ Π (119904) 119889119904 = 0= int1199040

0Π (119904 + 119894120598) 119889119904 + int0

1199040

Π (119904 minus 119894120598) 119889119904+ ∮

119862(|1199040|)Π (119904) 119889119904

(16)


int1199040

0

1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD 119875 (119904) 119889119904= minus 12120587119894 ∮

119862(|1199040|)Π (119904)QCD 119875 (119904) 119889119904

(17)



(minus)(119873minus1) 1198622119873 ⟨O2119873⟩= 81205872 int1199040


(119873 = 1 2 ) (18)


1198626 ⟨O6⟩= minus81205873120572119904 [⟨(1199021205741205831205745120582119886119902)2⟩ + 29 ⟨(119902120574120583120582119886119902)2⟩] (19)






120593119873 (11987620)10038161003816100381610038161003816HAD = 120593119873 (1198762

0)10038161003816100381610038161003816QCD (20)

where

120593119873 (11987620)10038161003816100381610038161003816HAD equiv 1120587 int1199040

0

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|HAD

120593119873 (11987620)10038161003816100381610038161003816QCD


41198982119876

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|PQCD + 120593119873 (1198762

0)10038161003816100381610038161003816NP (21)






⟨119876119876⟩ = minus 112119898119876

⟨120572119904120587 1198662⟩ (22)






Π120583] (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119860120583 (119909) 119860dagger] (0)) |0⟩

= minus119892120583]Π1 (1199022) + 119902120583119902]Π0 (1199022) (23)


ImΠ0 (1199022)10038161003816100381610038161003816QCD= 14120587 [1 + O (120572119904 (1199022))] (24)




ImΠ0 (1199022)10038161003816100381610038161003816HAD = 21205871198912120587120575 (119904) (25)



1199040 = 812058721198912120587 (26)



119878119865 (119896 119879) = 119894119896 minus 119898minus 2120587

(119890|1198960|119879 + 1) (119896 + 119898) 120575 (1198962 minus 1198982) (27)




ImΠ1198860 (120596 119879)1003816100381610038161003816QCD = 14120587 [1 minus 2119899119865 ( 1205962119879)] 120579 (1205962)

= 14120587 tanh( 1205964119879) 120579 (1205962) (28)


ImΠ1199040 (120596 119879)1003816100381610038161003816QCD = 4120587120575 (1205962) intinfin

0119910119899119865 ( 119910119879) 119889119910

= 1205873 1198792120575 (1205962) (29)


812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)] (30)




06 07 08 09 10 11 12 13 14 15 1605

s (GeV2)

000

001

002

003

004

005

006

007

008

1120587

Im0

(s)

Π





1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587120575 (119904)

+ 119862119891 exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(31)




1198626 ⟨O6⟩10038161003816100381610038161003816119860 prop 1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (32)


211989121205871198982

120587 = minus (119898119906 + 119898119889) ⟨0| 119906119906 + 119889119889 |0⟩ (33)





812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)]

minus 41205872 int1199040(119879)

0119889119904 1120587 ImΠ0 (119904 119879)10038161003816100381610038161003816100381610038161198861

minus1198624 ⟨O4⟩ (119879) = 41205872 int1199040(119879)

0119889119904119904 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

0119889119904119904 [1 minus 2119899119865 (radic1199042119879)]


00 01 02 03 04 05 06 07 08 09 10

TTc

16

14

12

10

08

06

04

02

00

C4⟨O

4⟩(T)C4⟨O

4⟩(0)


1412100806040200

12

10

08

06

04

02

00

TTc

f2 120587(T)f2 120587(0)



1198626 ⟨O6⟩ (119879) = 41205872 int1199040(119879)

01198891199041199042 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

01198891199041199042 [1 minus 2119899119865 (radic1199042119879)]

(34)


00 01 02 03 04 05 06 07 08 09 1110

TTc

s 0(T)s 0(0)

14

12

10

08

06

04

02

00



00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

08

06

04

02

00

f(T)f(0)







00 01 02 03 04 05 06 07 08 09 10

TTc

Γ(T)Γ(0)

15

14

13

12

11

10

09



1120587 ImΠ1199041003816100381610038161003816100381610038161003816HAD (120596 119879) = 231205872

120575 (1205962) intinfin

0119910119899119861 ( 119910119879) 119889119910 (35)


1198626 ⟨O6⟩10038161003816100381610038161003816119881 prop minus1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (36)



Γ120588 (119879) = Γ120588 (0)1 minus (119879119879119888)119886 (37)

where 119886 = 3 and 119879119888 = 197MeV

1198626 ⟨O6⟩ (119879) = 1198626 ⟨O6⟩ (0) [[

1 minus ( 119879119879lowast119902

)119887]]

(38)


119872120588 (119879) = 119872120588 (0) [1 minus ( 119879119879lowast119872

)119888] (39)




1199040 (119879) = 1199040 (0)sdot [1 minus 05667 ( 119879119879119888

)1138 minus 4347 ( 119879119879119888

)6841] (40)

1198624 ⟨O4⟩ (119879) = 1198624 ⟨O4⟩ (0)sdot [1 minus 165 ( 119879119879119888

)8735 + 004967 ( 119879119879119888

)07211] (41)

119891120588 (119879)119891120588 (0) = 1 minus 03901 ( 119879119879119888

)1075 + 004155 ( 119879119879119888

)1269 (42)




14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)


1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00





06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120593119873 (11987620)10038161003816100381610038161003816HAD = 120593119873 (1198762

0)10038161003816100381610038161003816QCD (20)

where

120593119873 (11987620)10038161003816100381610038161003816HAD equiv 1120587 int1199040

0

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|HAD

120593119873 (11987620)10038161003816100381610038161003816QCD


41198982119876

119889119904(119904 + 1198762

0)(119873+1)ImΠ (119904)|PQCD + 120593119873 (1198762

0)10038161003816100381610038161003816NP (21)






⟨119876119876⟩ = minus 112119898119876

⟨120572119904120587 1198662⟩ (22)






Π120583] (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (119860120583 (119909) 119860dagger] (0)) |0⟩

= minus119892120583]Π1 (1199022) + 119902120583119902]Π0 (1199022) (23)


ImΠ0 (1199022)10038161003816100381610038161003816QCD= 14120587 [1 + O (120572119904 (1199022))] (24)




ImΠ0 (1199022)10038161003816100381610038161003816HAD = 21205871198912120587120575 (119904) (25)



1199040 = 812058721198912120587 (26)



119878119865 (119896 119879) = 119894119896 minus 119898minus 2120587

(119890|1198960|119879 + 1) (119896 + 119898) 120575 (1198962 minus 1198982) (27)




ImΠ1198860 (120596 119879)1003816100381610038161003816QCD = 14120587 [1 minus 2119899119865 ( 1205962119879)] 120579 (1205962)

= 14120587 tanh( 1205964119879) 120579 (1205962) (28)


ImΠ1199040 (120596 119879)1003816100381610038161003816QCD = 4120587120575 (1205962) intinfin

0119910119899119865 ( 119910119879) 119889119910

= 1205873 1198792120575 (1205962) (29)


812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)] (30)




06 07 08 09 10 11 12 13 14 15 1605

s (GeV2)

000

001

002

003

004

005

006

007

008

1120587

Im0

(s)

Π





1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587120575 (119904)

+ 119862119891 exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(31)




1198626 ⟨O6⟩10038161003816100381610038161003816119860 prop 1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (32)


211989121205871198982

120587 = minus (119898119906 + 119898119889) ⟨0| 119906119906 + 119889119889 |0⟩ (33)





812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)]

minus 41205872 int1199040(119879)

0119889119904 1120587 ImΠ0 (119904 119879)10038161003816100381610038161003816100381610038161198861

minus1198624 ⟨O4⟩ (119879) = 41205872 int1199040(119879)

0119889119904119904 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

0119889119904119904 [1 minus 2119899119865 (radic1199042119879)]


00 01 02 03 04 05 06 07 08 09 10

TTc

16

14

12

10

08

06

04

02

00

C4⟨O

4⟩(T)C4⟨O

4⟩(0)


1412100806040200

12

10

08

06

04

02

00

TTc

f2 120587(T)f2 120587(0)



1198626 ⟨O6⟩ (119879) = 41205872 int1199040(119879)

01198891199041199042 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

01198891199041199042 [1 minus 2119899119865 (radic1199042119879)]

(34)


00 01 02 03 04 05 06 07 08 09 1110

TTc

s 0(T)s 0(0)

14

12

10

08

06

04

02

00



00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

08

06

04

02

00

f(T)f(0)







00 01 02 03 04 05 06 07 08 09 10

TTc

Γ(T)Γ(0)

15

14

13

12

11

10

09



1120587 ImΠ1199041003816100381610038161003816100381610038161003816HAD (120596 119879) = 231205872

120575 (1205962) intinfin

0119910119899119861 ( 119910119879) 119889119910 (35)


1198626 ⟨O6⟩10038161003816100381610038161003816119881 prop minus1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (36)



Γ120588 (119879) = Γ120588 (0)1 minus (119879119879119888)119886 (37)

where 119886 = 3 and 119879119888 = 197MeV

1198626 ⟨O6⟩ (119879) = 1198626 ⟨O6⟩ (0) [[

1 minus ( 119879119879lowast119902

)119887]]

(38)


119872120588 (119879) = 119872120588 (0) [1 minus ( 119879119879lowast119872

)119888] (39)




1199040 (119879) = 1199040 (0)sdot [1 minus 05667 ( 119879119879119888

)1138 minus 4347 ( 119879119879119888

)6841] (40)

1198624 ⟨O4⟩ (119879) = 1198624 ⟨O4⟩ (0)sdot [1 minus 165 ( 119879119879119888

)8735 + 004967 ( 119879119879119888

)07211] (41)

119891120588 (119879)119891120588 (0) = 1 minus 03901 ( 119879119879119888

)1075 + 004155 ( 119879119879119888

)1269 (42)




14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)


1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00





06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





ImΠ0 (1199022)10038161003816100381610038161003816HAD = 21205871198912120587120575 (119904) (25)



1199040 = 812058721198912120587 (26)



119878119865 (119896 119879) = 119894119896 minus 119898minus 2120587

(119890|1198960|119879 + 1) (119896 + 119898) 120575 (1198962 minus 1198982) (27)




ImΠ1198860 (120596 119879)1003816100381610038161003816QCD = 14120587 [1 minus 2119899119865 ( 1205962119879)] 120579 (1205962)

= 14120587 tanh( 1205964119879) 120579 (1205962) (28)


ImΠ1199040 (120596 119879)1003816100381610038161003816QCD = 4120587120575 (1205962) intinfin

0119910119899119865 ( 119910119879) 119889119910

= 1205873 1198792120575 (1205962) (29)


812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)] (30)




06 07 08 09 10 11 12 13 14 15 1605

s (GeV2)

000

001

002

003

004

005

006

007

008

1120587

Im0

(s)

Π





1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587120575 (119904)

+ 119862119891 exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(31)




1198626 ⟨O6⟩10038161003816100381610038161003816119860 prop 1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (32)


211989121205871198982

120587 = minus (119898119906 + 119898119889) ⟨0| 119906119906 + 119889119889 |0⟩ (33)





812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)]

minus 41205872 int1199040(119879)

0119889119904 1120587 ImΠ0 (119904 119879)10038161003816100381610038161003816100381610038161198861

minus1198624 ⟨O4⟩ (119879) = 41205872 int1199040(119879)

0119889119904119904 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

0119889119904119904 [1 minus 2119899119865 (radic1199042119879)]


00 01 02 03 04 05 06 07 08 09 10

TTc

16

14

12

10

08

06

04

02

00

C4⟨O

4⟩(T)C4⟨O

4⟩(0)


1412100806040200

12

10

08

06

04

02

00

TTc

f2 120587(T)f2 120587(0)



1198626 ⟨O6⟩ (119879) = 41205872 int1199040(119879)

01198891199041199042 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

01198891199041199042 [1 minus 2119899119865 (radic1199042119879)]

(34)


00 01 02 03 04 05 06 07 08 09 1110

TTc

s 0(T)s 0(0)

14

12

10

08

06

04

02

00



00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

08

06

04

02

00

f(T)f(0)







00 01 02 03 04 05 06 07 08 09 10

TTc

Γ(T)Γ(0)

15

14

13

12

11

10

09



1120587 ImΠ1199041003816100381610038161003816100381610038161003816HAD (120596 119879) = 231205872

120575 (1205962) intinfin

0119910119899119861 ( 119910119879) 119889119910 (35)


1198626 ⟨O6⟩10038161003816100381610038161003816119881 prop minus1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (36)



Γ120588 (119879) = Γ120588 (0)1 minus (119879119879119888)119886 (37)

where 119886 = 3 and 119879119888 = 197MeV

1198626 ⟨O6⟩ (119879) = 1198626 ⟨O6⟩ (0) [[

1 minus ( 119879119879lowast119902

)119887]]

(38)


119872120588 (119879) = 119872120588 (0) [1 minus ( 119879119879lowast119872

)119888] (39)




1199040 (119879) = 1199040 (0)sdot [1 minus 05667 ( 119879119879119888

)1138 minus 4347 ( 119879119879119888

)6841] (40)

1198624 ⟨O4⟩ (119879) = 1198624 ⟨O4⟩ (0)sdot [1 minus 165 ( 119879119879119888

)8735 + 004967 ( 119879119879119888

)07211] (41)

119891120588 (119879)119891120588 (0) = 1 minus 03901 ( 119879119879119888

)1075 + 004155 ( 119879119879119888

)1269 (42)




14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)


1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00





06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




06 07 08 09 10 11 12 13 14 15 1605

s (GeV2)

000

001

002

003

004

005

006

007

008

1120587

Im0

(s)

Π





1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587120575 (119904)

+ 119862119891 exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(31)




1198626 ⟨O6⟩10038161003816100381610038161003816119860 prop 1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (32)


211989121205871198982

120587 = minus (119898119906 + 119898119889) ⟨0| 119906119906 + 119889119889 |0⟩ (33)





812058721198912120587 (119879) = 4312058721198792 + int1199040(119879)

0119889119904 [1 minus 2119899119865 (radic1199042119879)]

minus 41205872 int1199040(119879)

0119889119904 1120587 ImΠ0 (119904 119879)10038161003816100381610038161003816100381610038161198861

minus1198624 ⟨O4⟩ (119879) = 41205872 int1199040(119879)

0119889119904119904 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

0119889119904119904 [1 minus 2119899119865 (radic1199042119879)]


00 01 02 03 04 05 06 07 08 09 10

TTc

16

14

12

10

08

06

04

02

00

C4⟨O

4⟩(T)C4⟨O

4⟩(0)


1412100806040200

12

10

08

06

04

02

00

TTc

f2 120587(T)f2 120587(0)



1198626 ⟨O6⟩ (119879) = 41205872 int1199040(119879)

01198891199041199042 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

01198891199041199042 [1 minus 2119899119865 (radic1199042119879)]

(34)


00 01 02 03 04 05 06 07 08 09 1110

TTc

s 0(T)s 0(0)

14

12

10

08

06

04

02

00



00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

08

06

04

02

00

f(T)f(0)







00 01 02 03 04 05 06 07 08 09 10

TTc

Γ(T)Γ(0)

15

14

13

12

11

10

09



1120587 ImΠ1199041003816100381610038161003816100381610038161003816HAD (120596 119879) = 231205872

120575 (1205962) intinfin

0119910119899119861 ( 119910119879) 119889119910 (35)


1198626 ⟨O6⟩10038161003816100381610038161003816119881 prop minus1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (36)



Γ120588 (119879) = Γ120588 (0)1 minus (119879119879119888)119886 (37)

where 119886 = 3 and 119879119888 = 197MeV

1198626 ⟨O6⟩ (119879) = 1198626 ⟨O6⟩ (0) [[

1 minus ( 119879119879lowast119902

)119887]]

(38)


119872120588 (119879) = 119872120588 (0) [1 minus ( 119879119879lowast119872

)119888] (39)




1199040 (119879) = 1199040 (0)sdot [1 minus 05667 ( 119879119879119888

)1138 minus 4347 ( 119879119879119888

)6841] (40)

1198624 ⟨O4⟩ (119879) = 1198624 ⟨O4⟩ (0)sdot [1 minus 165 ( 119879119879119888

)8735 + 004967 ( 119879119879119888

)07211] (41)

119891120588 (119879)119891120588 (0) = 1 minus 03901 ( 119879119879119888

)1075 + 004155 ( 119879119879119888

)1269 (42)




14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)


1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00





06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00 01 02 03 04 05 06 07 08 09 10

TTc

16

14

12

10

08

06

04

02

00

C4⟨O

4⟩(T)C4⟨O

4⟩(0)


1412100806040200

12

10

08

06

04

02

00

TTc

f2 120587(T)f2 120587(0)



1198626 ⟨O6⟩ (119879) = 41205872 int1199040(119879)

01198891199041199042 1120587 ImΠ0 (119904)10038161003816100381610038161003816100381610038161198861

minus int1199040(119879)

01198891199041199042 [1 minus 2119899119865 (radic1199042119879)]

(34)


00 01 02 03 04 05 06 07 08 09 1110

TTc

s 0(T)s 0(0)

14

12

10

08

06

04

02

00



00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

08

06

04

02

00

f(T)f(0)







00 01 02 03 04 05 06 07 08 09 10

TTc

Γ(T)Γ(0)

15

14

13

12

11

10

09



1120587 ImΠ1199041003816100381610038161003816100381610038161003816HAD (120596 119879) = 231205872

120575 (1205962) intinfin

0119910119899119861 ( 119910119879) 119889119910 (35)


1198626 ⟨O6⟩10038161003816100381610038161003816119881 prop minus1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (36)



Γ120588 (119879) = Γ120588 (0)1 minus (119879119879119888)119886 (37)

where 119886 = 3 and 119879119888 = 197MeV

1198626 ⟨O6⟩ (119879) = 1198626 ⟨O6⟩ (0) [[

1 minus ( 119879119879lowast119902

)119887]]

(38)


119872120588 (119879) = 119872120588 (0) [1 minus ( 119879119879lowast119872

)119888] (39)




1199040 (119879) = 1199040 (0)sdot [1 minus 05667 ( 119879119879119888

)1138 minus 4347 ( 119879119879119888

)6841] (40)

1198624 ⟨O4⟩ (119879) = 1198624 ⟨O4⟩ (0)sdot [1 minus 165 ( 119879119879119888

)8735 + 004967 ( 119879119879119888

)07211] (41)

119891120588 (119879)119891120588 (0) = 1 minus 03901 ( 119879119879119888

)1075 + 004155 ( 119879119879119888

)1269 (42)




14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)


1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00





06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




00 01 02 03 04 05 06 07 08 09 10

TTc

Γ(T)Γ(0)

15

14

13

12

11

10

09



1120587 ImΠ1199041003816100381610038161003816100381610038161003816HAD (120596 119879) = 231205872

120575 (1205962) intinfin

0119910119899119861 ( 119910119879) 119889119910 (35)


1198626 ⟨O6⟩10038161003816100381610038161003816119881 prop minus1205721199041003816100381610038161003816⟨119902119902⟩10038161003816100381610038162 (36)



Γ120588 (119879) = Γ120588 (0)1 minus (119879119879119888)119886 (37)

where 119886 = 3 and 119879119888 = 197MeV

1198626 ⟨O6⟩ (119879) = 1198626 ⟨O6⟩ (0) [[

1 minus ( 119879119879lowast119902

)119887]]

(38)


119872120588 (119879) = 119872120588 (0) [1 minus ( 119879119879lowast119872

)119888] (39)




1199040 (119879) = 1199040 (0)sdot [1 minus 05667 ( 119879119879119888

)1138 minus 4347 ( 119879119879119888

)6841] (40)

1198624 ⟨O4⟩ (119879) = 1198624 ⟨O4⟩ (0)sdot [1 minus 165 ( 119879119879119888

)8735 + 004967 ( 119879119879119888

)07211] (41)

119891120588 (119879)119891120588 (0) = 1 minus 03901 ( 119879119879119888

)1075 + 004155 ( 119879119879119888

)1269 (42)




14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)


1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00





06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




14

12

10

08

06

04

02

0000 01 02 03 04 05 06 07 08 09 10

TTc

f120588(T)f120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

20

18

16

14

12

10

8

6

4

2

0

Γ 120588(T)Γ 120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

M120588(T)M

120588(0)


00 01 02 03 04 05 06 07 08 09 10

TTc

14

12

10

08

06

04

02

00

C6⟨119978

6⟩(T)C6⟨119978

6⟩(0)


1412100806040200

s (GeV2)

Im Π

V(s

)

05

04

03

02

01

00





06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




06 07 08 09 1005

M (GeV)

0

1

2

3

4

5

6

7

dNdM

(in10

minus7

GeV

minus1)






0119889119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 21198912

120587 (43)


0119889119904119904 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)] = 0 (44)


ImΠ119881 (1199022) = ImΠ119860 (1199022) = 14120587 [1 + O (120572119904 (1199022))] (45)



119882119899+1 (1199040) equiv int1199040

0119889119904119904119899 1120587 [ImΠ119881 (119904) minus ImΠ119860 (119904)]

= 211989121205871205751198990

(46)


119882119875 (1199040)equiv int1199040


= 21198912120587

(47)



1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861exp[

[minus (119904 minus 1198722

1198861Γ21198861

)2]]

(0 le 119904 le 12GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861

= 1198621198911198861exp[


1198861Γ21198861

)2]]

(12GeV2 le 119904 le 145GeV2) 1120587 ImΠ119860 (119904)10038161003816100381610038161003816100381610038161198861 = 1198621198911198861

exp[[

minus (119904 minus 11987221198861Γ2

1198861

)2]]

(145GeV2 le 119904 le 1198722120591)

(48)




02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






02 04 06 08 10 12 14 16 18 2000

s (GeV2)

000

002

004

006

008

010

012

014

016

018

020

A(s

)Im

Π





119884 (119879)119884 (0) = 1 + 1198861 ( 119879119879119888

)1198871 + 1198862 ( 119879119879119888

)1198871 (49)




00 01 02 03 04 05 06 07 08

TTc

005

004

003

002

001

000

W1(T)



Π120583]10038161003816100381610038161003816119881 = ⟨120587| 119881120583 (0) 119881] (119909) |120587⟩

= ⟨120587| 119881120583 (0) |120587⟩ ⟨120587| 119881] (119909) |120587⟩+ ⟨120587120587| 119881120583 (0) |120587120587⟩ ⟨120587120587| 119881] (119909) |120587120587⟩ + sdot sdot sdot

(50)

Π120583]10038161003816100381610038161003816119860 = ⟨120587| 119860120583 (0) 119860] (119909) |120587⟩

= ⟨120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587⟩+ ⟨120587120587120587| 119860120583 (0) |0⟩ ⟨0| 119860] (119909) |120587120587120587⟩ + sdot sdot sdot

(51)






1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







1205955 (1199022) = 119894 int 1198894119909119890119894119902119909 ⟨0| 119879 (120597120583119860120583 (119909) 120597]119860dagger] (0)) |0⟩ (52)

with

120597120583119860120583 (119909) = 119898119906119889 119889 (119909) 1198941205745119906 (119909) (53)

and the definition

119898119906119889 equiv (119898119906 + 119898119889) ≃ 10MeV (54)


int1199040

0119889119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD

= minus 12120587119894 ∮119862(|1199040|)

119889119904 1205955 (119904)1003816100381610038161003816QCD (55)

int1199040

0

119889119904119904 1120587 Im1205955 (119904)1003816100381610038161003816100381610038161003816HAD + 12120587119894 ∮119862(|1199040|)

119889119904119904 1205955 (119904)1003816100381610038161003816QCD

= 1205955 (0) (56)


1205955 (0) = Residue [1205955 (119904)119904 ]119904=0

(57)



1205955 (1199022)10038161003816100381610038161003816QCD= 1198982

119906119889 minus 3812058721199022ln(minus11990221205832

) + 119898119906119889 ⟨119902119902⟩1199022

minus 181199022 ⟨120572119904120587 1198662⟩ + O(O61199024 ) (58)



Im1205955 (1199022 119879)10038161003816100381610038161003816QCD

= 381205871198982119906119889 (119879) 1205962 [1 minus 2119899119865 ( 1205962119879)] (59)


Im1205955 (1199022 119879)HAD

= 21205871198912120587 (119879) 1198724

120587 (119879) 120575 (1199022 minus 1198722120587) (60)


21198912120587 (119879) 1198724

120587 (119879)= 31198982

119906119889 (119879)81205872int1199040(119879)

0119904 [1 minus 2119899119865 (radic1199042119879)] 119889119904 (61)

21198912120587 (119879) 1198722

120587 (119879)= minus2119898119906119889 (119879) ⟨119902119902⟩ (119879)

+ 3812058721198982

119906119889 (119879) int1199040(119879)

0[1 minus 2119899119865 (radic1199042119879)] 119889119904

(62)




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




9

7

5

3

1

0

0 50 100 150

T (MeV)

(a)

(b)mud(T)mud(0)


f120587(T)f120587(0)

12

10

08

06

04

02

00

0 50 100 150

T (MeV)

(a)

(b)




1199040 (119879)1199040 (0) ≃ [⟨119902119902⟩ (119879)⟨119902119902⟩ (0) ]23 (63)






1120587 ImΠ119886 (1199022 119879) = 3161205872intV

minusV119889119909 (1 minus 1199092)




1120587 ImΠ119886 (120596 119879)= 181205872




1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





1120587 ImΠ119904 (120596 119879) = 212058721198982

119876120575 (1205962)sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (66)


1120587 ImΠ (119904 119879)1003816100381610038161003816100381610038161003816HAD = 21198912119881 (119879) 120575 (119904 minus 1198722

119881 (119879))+ 1120587 ImΠ (119904 119879)119886 120579 (119904 minus 1199040)

(67)


⟨0| 119881120583 (0) |119881 (119896)⟩ = radic2119872119881119891119881120598120583 (68)


120575 (119904 minus 1198722119881 (119879))

997904rArr const 1(119904 minus 1198722

119881 (119879))2 + 1198722119881 (119879) Γ2

119881 (119879) (69)



1120587 ImΠ119904 (120596 119879)1003816100381610038161003816100381610038161003816HAD = 2312058721198722

119863120575 (1205962)sdot [119899119861 (119872119863119879 ) + 21198792

1198722119863

intinfin

119898119863119879119910119899119861 (119910) 119889119910]

(70)



12100806040200

TTc

14

13

12

11

10

06

05

07

08

09

s 0(T)s 0(0)



120593119873 (1198762 119879)10038161003816100381610038161003816NP = minus 341205872

1(41198982

119876)1198731

(1 + 120585)119873+2


(71)


Φ equiv 41205872

9 1(41198982

119876)2 ⟨120572119904120587 1198662⟩100381610038161003816100381610038161003816100381610038161003816100381610038161003816119879

(72)




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




4000

3000

2000

1000

012100806040200

TTc

ΓV(T)ΓV(0)


12100806040200

14

13

12

11

10

TTc

fV(T)fV(0)






12100806040200

TTc

14

13

12

11

10

09

08

07

06

MV(T)M

V(0)



1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD



(73)


plusmn (119909) = 1119890(119909∓120583119902)119879 + 1 (74)


1120587 ImΠ0 (119904)1003816100381610038161003816100381610038161003816PQCD


+ 11205872120575 (119904) (1205832

119902 + 12058721198792

3 ) (75)


1120587 ImΠ (119904)1003816100381610038161003816100381610038161003816HAD = 21198912120587 (119879 120583119902) 120575 (119904) (76)



int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





int1199040(119879120583119902)


= 812058721198912120587 (119879 120583119902)



1198912120587 (119879 120583119902)1198912120587 (0 0) = ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) (78)



1199040 (119879 120583119902) ≃ 812058721198912120587 (119879 120583119902) minus 4312058721198792 minus 41205832

119902 (79)


1199040 (119879 120583119902)1199040 (0 0) ≃ ⟨120595120595⟩ (119879 120583119902)⟨120595120595⟩ (0 0) minus (11987923 minus 12058321199021205872)

21198912120587 (0 0) (80)


⟨120595120595⟩ (119879 120583119902)10038161003816100381610038161003816matt

= minus81198791198731198881205872

infinsum119897=1

(minus1)119897119897 cosh(120583119897119902119879 ) 4sum

119894=1

1199031198941198982119894100381610038161003816100381611988711989410038161003816100381610038163 1198701 (119897 1003816100381610038161003816119898119894

1003816100381610038161003816119879 ) (81)



Con

dens

ate (

GeV

)3

0015

0012

0009

0006

0003

0016 018 02 022 024 026

T (GeV)



120583B = 00GeV120583B = 03GeV

015 016 017 018 019 02 021 022 023 024

T (GeV)

Hea

t cap

acity

(con

dens

ate)

(GeV

minus1)

30

25

20

15

10

5

0






025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




025

0225

02


0175


0 01 02 03 04 05 06 07 08 09 1

Tc

(GeV

)

120583B (GeV)



Im 120596




9 Summary



Π (119902 119879) = 119894 int 1198894119909119890119894119902119909120579 (1199090) ⟨⟨[119869 (119909) 119869dagger (0)]⟩⟩ (82)

where

⟨⟨119860 sdot 119861⟩⟩ = sum119899



Π (119902 119879) = 119862119868 ⟨⟨119868⟩⟩ + 119862119903 (119902) ⟨⟨O119903⟩⟩ (84)














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














119891120587 (119879) 119892120587119873119873 (119879)119872119873 (119879) 119892119860 (119879) equiv 1 + Δ120587 (119879) (85)




Appendix




Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





Π119886120583] (1199022) equiv (minus119892120583]1199022 + 119902120583119902]) Π (1199022)





V (1199022) = (1 minus 411989821198761199022 )

12

(A3)



ImΠ119886 (1199022 119879) = 316120587 int+V

minusV119889119909 (1 minus 1199092)



ImΠ119886 (120596 119879)= 316120587 int+V


= 316120587 int+V


(A5)


ImΠ119904 (1199022 119879) = 38120587 intinfin

V119889119909 (1 minus 1199092)



ImΠ119904 (1199022 119879) = 38120587 120596119879 intinfin

V119889119909 (1 minus 1199092) 119889119889119910119899119865 (119910) (A7)



|q|2 intinfin

|q|V2119879119910119899119865 (119910) 119889119910]

(A8)



120596|q|3 1198982

119876

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910]

(A9)



120596|q|3 = 23120575 (1205962) (A10)


ImΠ119904 (120596 119879) = 21205871198982119876120575 (1205962)

sdot [119899119865 (119898119876119879 ) + 21198792

1198982119876

intinfin

119898119876119879119910119899119865 (119910) 119889119910] (A11)


Competing Interests


Acknowledgments


References





























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




























































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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