phase transitions in qcd

Phase Transitions in QCDPhase Transitions in QCD

Eduardo S. FragaEduardo S. Fraga

Instituto de Física Instituto de Física Universidade Federal do Rio de JaneiroUniversidade Federal do Rio de Janeiro

2nd Rio-Saclay Meeting - Rio de Janeiro, 20072nd Rio-Saclay Meeting - Rio de Janeiro, 2007 22

OutlineOutline

3rd Lecture3rd Lecture

• Finite T x finite Finite T x finite : pQCD, lattice, sign problem, : pQCD, lattice, sign problem, etcetc

• Nuclear EoS: relativistic and non-relativistic Nuclear EoS: relativistic and non-relativistic (brief)(brief)

• pQCD at nonzero T and pQCD at nonzero T and (brief)(brief)

• Cold pQCD at high density for massless quarksCold pQCD at high density for massless quarks

• Nonzero mass effectsNonzero mass effects

• Compact stars and QCD at high densityCompact stars and QCD at high density

• SummarySummary


• So far we have focused on the finite-temperature zero-So far we have focused on the finite-temperature zero-density sector of the phase diagram for QCD. density sector of the phase diagram for QCD. How about How about finite-density?finite-density?

• We know that naïve perturbative finite-temperature QFT is We know that naïve perturbative finite-temperature QFT is plagued by infrared divergences, so that one has to improve plagued by infrared divergences, so that one has to improve the series by some sort of resummationthe series by some sort of resummation [François’ lectures][François’ lectures]

[Andersen, Braaten & Strickland, 2000][Andersen, Braaten & Strickland, 2000]

[From Fodor, Lattice 2007][From Fodor, Lattice 2007]

Link between lattice QCD and Link between lattice QCD and pQCD at finite T !!pQCD at finite T !!

Finite T x finite Finite T x finite : pQCD, lattice, sign problem, : pQCD, lattice, sign problem, etcetc


• This is due to sum-integrals that yield a series involving This is due to sum-integrals that yield a series involving ss1/21/2

instead of instead of ss (+ logs). At the end, it generates huge (+ logs). At the end, it generates huge oscillations in sign for the pressure as we include (large) oscillations in sign for the pressure as we include (large) corrections, the radius of convergence being essentially zero corrections, the radius of convergence being essentially zero [Braaten, 2000][Braaten, 2000]

• For T = 0 and For T = 0 and > 0, though, the perturbative series seems > 0, though, the perturbative series seems to be much more well-behaved, without oscillations and with to be much more well-behaved, without oscillations and with decreasing correctionsdecreasing corrections [there is still a bit of wishful thinking here, [there is still a bit of wishful thinking here, and the computation of and the computation of ss

33 corrections would help to corroborate it corrections would help to corroborate it or not]or not]

• Anyway, there seems to be room for sensible pQCD Anyway, there seems to be room for sensible pQCD calculations for T = 0 and calculations for T = 0 and large (and not ridiculously large!) large (and not ridiculously large!)


• So, this is good news. So, this is good news. Bad news is:Bad news is: it is really tough to it is really tough to compute QCD at finite compute QCD at finite on the lattice due to the on the lattice due to the Sign ProblemSign Problem::

- On the lattice, one does Monte Carlo calculations that - On the lattice, one does Monte Carlo calculations that rely on Srely on Seucleucl being positive-definite to sample “relevant” being positive-definite to sample “relevant” contributions weighted by contributions weighted by exp(-Sexp(-Seucleucl):):

- This method (“importance sampling”) reduces - This method (“importance sampling”) reduces dramatically the number of field configurations one has to dramatically the number of field configurations one has to consider [ the total would go as exp (consider [ the total would go as exp (##V) ]V) ]


- For QCD with - For QCD with ≠ 0, the effective S ≠ 0, the effective Seucleucl is complex, so is complex, so that one does not have a well-defined weight factor any more…that one does not have a well-defined weight factor any more…

The Sign Problem comes about in the following way:The Sign Problem comes about in the following way:

Including fermions at finite Including fermions at finite , one has the action, one has the action

Integrating over the fermions, one obtainsIntegrating over the fermions, one obtains


- Using D- Using Dtt = -D, one can write: = -D, one can write:

so that positivity and reality are lost, unless one so that positivity and reality are lost, unless one assumes assumes ff to be imaginary (analytic continuation) to be imaginary (analytic continuation) or that quarks are such that mor that quarks are such that mu u = m= mdd & & u u = -= -dd (isospin symmetry)(isospin symmetry)

No well-defined criteria to choose configurations!No well-defined criteria to choose configurations!

ExerciseExercise:: fill in the gaps, i.e., do the calculations in detail. fill in the gaps, i.e., do the calculations in detail.


• There are many techniques, developed over the last few There are many techniques, developed over the last few years, to try to bypass the Sign Problem. But there is no final years, to try to bypass the Sign Problem. But there is no final established method for the general case yet, and the road is established method for the general case yet, and the road is long and winding…long and winding…

• Some techniques at finiteSome techniques at finite :: Fodor et al.:Fodor et al.: Multi-parameter reweightingMulti-parameter reweighting

Bielefeld-Swansea:Bielefeld-Swansea: Taylor exp. around Taylor exp. around =0=0

de Focrand, Philipsen; D’Elia, Lombardo; de Focrand, Philipsen; D’Elia, Lombardo; …… Analytic continuation from imaginaryAnalytic continuation from imaginary


[From: de Forcrand, Lattice 2007][From: de Forcrand, Lattice 2007]

[Allton et al., 2002][Allton et al., 2002]

• Some results and possibilitiesSome results and possibilities::


• The basic message is:The basic message is: at finite at finite , we , we do notdo not have yet the kind have yet the kind of guidance from the lattice as we have at finite T (of guidance from the lattice as we have at finite T (=0). And =0). And most likely it will take a while…most likely it will take a while…

• So, one has to compare different schemes of computation of So, one has to compare different schemes of computation of thermodynamic quantities: pQCD, resummed pQCD (different thermodynamic quantities: pQCD, resummed pQCD (different possibilities), quasiparticle models, low-energy effective possibilities), quasiparticle models, low-energy effective models, quantum hadrodynamics (QHD), non-relativistic models, quantum hadrodynamics (QHD), non-relativistic many-body nuclear theory, etc.many-body nuclear theory, etc.

• In general, one has to study the limits of low and high In general, one has to study the limits of low and high densities separately, matching results in the critical region densities separately, matching results in the critical region where the phase transition happens (Lecture I).where the phase transition happens (Lecture I).

• Observables have to be searched in astrophysics, more Observables have to be searched in astrophysics, more precisely in compact stars, where densities can be high precisely in compact stars, where densities can be high enough to allow for a deconfined phase.enough to allow for a deconfined phase.


Nuclear EoS: relativistic and non-relativisticNuclear EoS: relativistic and non-relativistic

• Very different from the Very different from the bag model EoSbag model EoS

• Hard to trust for Hard to trust for > > 2200

• Has to be matched Has to be matched onto high density EoSonto high density EoS

• Kodama’s lecturesKodama’s lectures

[From Reddy, SEWM 2005][From Reddy, SEWM 2005]


pQCD at nonzero T and pQCD at nonzero T and (in-medium QCD)(in-medium QCD)

Freedman & McLerran, 1977-1978Freedman & McLerran, 1977-1978Baluni, 1978 ; Toimela, 1980’sBaluni, 1978 ; Toimela, 1980’sBraaten & Pisarski, Frenkel & Taylor, 1990-1992Braaten & Pisarski, Frenkel & Taylor, 1990-1992Arnold & Zhai, 1994-1995Arnold & Zhai, 1994-1995Andersen, Braaten & Strickland, 1999-2001Andersen, Braaten & Strickland, 1999-2001Kajantie et al, 2001 ; Peshier et al, 1999-2003Kajantie et al, 2001 ; Peshier et al, 1999-2003Blaizot, Iancu & Rebhan, 1999-2003Blaizot, Iancu & Rebhan, 1999-2003ESF, Pisarski & Schaffner-Bielich, 2001-2004ESF, Pisarski & Schaffner-Bielich, 2001-2004Rebhan & Romatschke, 2003Rebhan & Romatschke, 2003Vuorinen, 2004-2007Vuorinen, 2004-2007ESF & Romatschke, 2005ESF & Romatschke, 2005… … ((manymany omissions) omissions)

Problems from theory side:Problems from theory side: really tough calculations, really tough calculations, resummation methods, valid for high energy scales, misses resummation methods, valid for high energy scales, misses vacuum, pushing perturbative methods to the limit,…vacuum, pushing perturbative methods to the limit,…


Pressure (diagrammatically speaking…)Pressure (diagrammatically speaking…)

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

(Vuorinen, 2004)(Vuorinen, 2004)



+ HDL resummation & mass effects (really tough!)+ HDL resummation & mass effects (really tough!)


Cold pQCD at high density for massless quarks Cold pQCD at high density for massless quarks

• Gas of massless u, d, s quarksGas of massless u, d, s quarks [charge and [charge and equilibrium achieved (no need for electrons)] equilibrium achieved (no need for electrons)]

• TTstarstar << typical << typical in the core region -> T in the core region -> T ~~ 0 0

• mmss ~~ 100 MeV << typical 100 MeV << typical -> massless quarks (well, -> massless quarks (well,…)…)

• Interaction taken into account perturbatively up toInteraction taken into account perturbatively up to ss22

• ss22 runs according to the renormalization group eqn.runs according to the renormalization group eqn.

• No bag constantNo bag constant

• Charge neutrality and Charge neutrality and -equilibrium:-equilibrium: ssdduu


: renormalization scale: renormalization scale

From PDG: From PDG: ss(2 GeV)= 0.3089 -> (2 GeV)= 0.3089 -> MS MS = 365 MeV for N= 365 MeV for Nff = 3 = 3

GG00 = 10.374 ± 0.13 = 10.374 ± 0.13

Thermodynamic potential in MSbar scheme for NThermodynamic potential in MSbar scheme for Nff flavors and N flavors and Ncc=3 =3

In principle can be freely chosen (only freedom left). can be freely chosen (only freedom left). However, the choice is tightly constrained by physics to 2 However, the choice is tightly constrained by physics to 2 3 3

Results are sensitive to the choiceResults are sensitive to the choice


Results for the pressure:Results for the pressure:

(ESF, Pisarski & Schaffner-Bielich, 2001)(ESF, Pisarski & Schaffner-Bielich, 2001)

(Peshier, K(Peshier, Käämpfer & Soff, 2002)mpfer & Soff, 2002)

(Andersen & Strickland, 2002)(Andersen & Strickland, 2002)

+ similar results from + similar results from

Blaizot, Iancu, Rebhan (2001)Blaizot, Iancu, Rebhan (2001)Rebhan, Romatschke (2003)Rebhan, Romatschke (2003)……

pQCD, HDL, quasiparticle models, ...pQCD, HDL, quasiparticle models, ...


Two possibilitiesTwo possibilities

Matching to “low” density - Two scenarios:Matching to “low” density - Two scenarios:

Physical picturePhysical picture

• Possible scenarios for the intensity of the Possible scenarios for the intensity of the chiral transition can be related to the choice of chiral transition can be related to the choice of • Moreover:Moreover:

- Pure neutron matter up to - Pure neutron matter up to ~~2n2n00:: [Akmal, Pandharipande, Ravelhall, 1998][Akmal, Pandharipande, Ravelhall, 1998]

p/pp/pfreefree ≈ 0.04 (n/n ≈ 0.04 (n/n00))22

- Dilute nuclear matter in - Dilute nuclear matter in PT at PT at >0>0 [ESF, Hatta, Pisarski, Schaffner-Bielich, 2003/2004][ESF, Hatta, Pisarski, Schaffner-Bielich, 2003/2004]

So: So: hadronic phase with small pressure hadronic phase with small pressure viable!viable!


Mixed phase structures (“pasta”):Mixed phase structures (“pasta”):

ExoticNuclear

• Typical size of defects ~ 5-10 fm

[From Reddy, SEWM 2005][From Reddy, SEWM 2005]


Crust Crust structure:structure:


Nonzero quark mass effectsNonzero quark mass effects

• Original approach: bag model + corrections Original approach: bag model + corrections ~ ~ ss from pQCD. In from pQCD. In this case, corrections cancel out in the EoS for massles quarks.this case, corrections cancel out in the EoS for massles quarks. • Quark mass effects were then estimated to modify the EoS by Quark mass effects were then estimated to modify the EoS by less than 5% and were essentially ignored for less than 5% and were essentially ignored for ~~20 years. 20 years.

• Quark masses & color SUC gaps neglected compared to typical Quark masses & color SUC gaps neglected compared to typical values of values of in compact stars, in compact stars, ~~400 MeV and higher. 400 MeV and higher.

• However, it was recently argued that both effects should However, it was recently argued that both effects should matter in the lower-density sector of the EoS matter in the lower-density sector of the EoS [Alford et al. (2004)][Alford et al. (2004)].. • Although quarks are essentially massless in the core of quark Although quarks are essentially massless in the core of quark stars, mstars, mss runs up as one approaches the surface of the star ! runs up as one approaches the surface of the star !

This suggests the analysis of finite mass effects on the EoS for This suggests the analysis of finite mass effects on the EoS for pQCD at high density including RG running of pQCD at high density including RG running of ss and m and mss ! !


Thermodynamic potential (1 massive flavor)Thermodynamic potential (1 massive flavor)

Leading-order piece:Leading-order piece:

The exchange term:The exchange term:


Results:Results:Using standard QFT methods, one obtains the complete Using standard QFT methods, one obtains the complete renormalized renormalized exchange energy for a massive quark in the MSbar scheme:exchange energy for a massive quark in the MSbar scheme:

• depends on the quark chemical potential depends on the quark chemical potential and on the and on the renormalizationrenormalization subtraction point subtraction point both explicitly and implicitly through the scale both explicitly and implicitly through the scale dependence of the strong coupling constant dependence of the strong coupling constant ss(() and the mass m() and the mass m().).

• The scale dependencies of both The scale dependencies of both ss and m and mss are known up to 4-loop are known up to 4-loop order inorder in the MSbar scheme the MSbar scheme [Vermaseren, 1997][Vermaseren, 1997]. Since we have only determined . Since we have only determined the freethe free energy to first order in energy to first order in ss, we choose, we choose

+ rest fixed by PDG data+ rest fixed by PDG data


Thermodynamic potential for one massive flavorThermodynamic potential for one massive flavor

• Finite quark mass effects can dramatically modify the EoS for Finite quark mass effects can dramatically modify the EoS for cold and dense QCD -> astrophysical effects!cold and dense QCD -> astrophysical effects!

• Numbers here are just illustrative of the strength of the effect. Numbers here are just illustrative of the strength of the effect. O(O(ss

22) corrections will modify them significantly) corrections will modify them significantly

(ESF & Romatschke, 2005)(ESF & Romatschke, 2005)

2nd Rio-Saclay Meeting - Rio de Janeiro, 20072nd Rio-Saclay Meeting - Rio de Janeiro, 2007 2424(F. Weber, 2000)(F. Weber, 2000)

(NASA)(NASA)

Compact stars and QCD at high densityCompact stars and QCD at high density

• Created by supernova Created by supernova explosionexplosion

• One possible final stage of One possible final stage of evolution of massive starsevolution of massive stars

• Extreme central densitiesExtreme central densities

• More than a thousand pulsars More than a thousand pulsars known todayknown today

• Several different scenarios for Several different scenarios for the core, including stable the core, including stable strange quark matter (Bodmer-strange quark matter (Bodmer-Witten)Witten)


Cooling:Cooling:

Mass-radius:Mass-radius:

Glitches in Glitches in

Some observables in compact starsSome observables in compact stars



Mass:Mass:

(Thorsett & Chakrabarty, 1999)(Thorsett & Chakrabarty, 1999)


Quark star structureQuark star structure

Tolman-Oppenheimer-Volkov equations:Tolman-Oppenheimer-Volkov equations:

• Einstein’s GR field equationsEinstein’s GR field equations• Spherical symmetrySpherical symmetry• Hydrostatic equilibriumHydrostatic equilibrium

• Given the EoS p = p(Given the EoS p = p(), one can integrate the TOV ), one can integrate the TOV equations from the origin until the pressure vanishes p(R) = equations from the origin until the pressure vanishes p(R) = 00

• Different EoS’s define different types of stars: white Different EoS’s define different types of stars: white dwarfs, neutron stars, quark stars, strange stars, …dwarfs, neutron stars, quark stars, strange stars, …


Results for massless quarks (pQCD) + hadronic mantleResults for massless quarks (pQCD) + hadronic mantle

• Interaction plays a role. One can still fit p=p(Interaction plays a role. One can still fit p=p() with an effective ) with an effective bag model, but needs to modify not only B but also correct the bag model, but needs to modify not only B but also correct the free termfree term

• Matching in the critical region is difficult because both Matching in the critical region is difficult because both calculations have problems for calculations have problems for ~~cc -> still: for a strongly 1st -> still: for a strongly 1st order transition, there can be a new class of compact stars (3rd order transition, there can be a new class of compact stars (3rd family)family)


New class of stars and quark star twinsNew class of stars and quark star twins

[Glendenning & Kettner, 2000; Schertler et al, 2000] [Glendenning & Kettner, 2000; Schertler et al, 2000]

[Papasotiriou, 2006][Papasotiriou, 2006]

• Quark core with a Quark core with a hadronic mantlehadronic mantle

• Smaller, denser Smaller, denser companion (twin) to a companion (twin) to a hybrid (more hadronic) hybrid (more hadronic) star star

• Not so large differences in Not so large differences in mass and radiusmass and radius

• Different density profileDifferent density profile

• Not self-bound down to Not self-bound down to R=0 as strangeletsR=0 as strangelets


Results for pure massive quark matter + electrons (mResults for pure massive quark matter + electrons (mss > 0)> 0)


Color superconductivityColor superconductivity

[From Shovkovy, SEWM 2005][From Shovkovy, SEWM 2005]

• Only quarks with different Only quarks with different colors and flavors participate in colors and flavors participate in Cooper pairing (attractive Cooper pairing (attractive channel) -> diquark condensatechannel) -> diquark condensate


SummarySummary

• Lattice QCD is just starting to explore the finite density region, Lattice QCD is just starting to explore the finite density region, still far away from the high-density low-temperature sector.still far away from the high-density low-temperature sector.

• pQCD at finite density seems to provide sensible results, even pQCD at finite density seems to provide sensible results, even for not so large values of for not so large values of Mass and gap effects provide Mass and gap effects provide important contributions to the EoS near the critical region.important contributions to the EoS near the critical region.

• The phase diagram can be very rich in the high-The phase diagram can be very rich in the high- sector, with sector, with different possibilities for pairing and color superconductivity.different possibilities for pairing and color superconductivity.

• Astrophysical measurements are becoming increasingly Astrophysical measurements are becoming increasingly precise, and will start killing models soon. Some signatures (for precise, and will start killing models soon. Some signatures (for strange, quark or hybrid neutron stars) are still very similar, strange, quark or hybrid neutron stars) are still very similar, though.though.

• The interior of compact stars is a very rich and intricate The interior of compact stars is a very rich and intricate medium, which may contain all sorts of condensates as well as medium, which may contain all sorts of condensates as well as deconfined quark matter.deconfined quark matter.

phase transitions in qcd

Documents

lecture finite t x finite

lattice qcd

sign problem

nonzero t

sensible pqcd calculations

brief cold pqcd

large corrections

nonrelativistic brief