density dependent hadron field theory - citeseerx
TRANSCRIPT
- 29 -
Density Dependent Hadron Field Theory
H. Lenske1 and C. Keil1
1Institut fur Theoretische Physik, Universitat Giessen
In symmetric nuclear matter and stable nuclei one usesvery often strictly phenomenological approaches, both innon-relativistic and relativistic formulation. In this con-tribution, a fully microscopic approach to nucleon-nucleon(NN) interactions in symmetric and asymmetric nuclearmatter and extensions to hypernuclear matter will bediscussed. Dirac-Brueckner theory is used to derive in-medium interactions [1] from well established free-spaceNN meson exchange potentials. By means of the DensityDependent Relativistic Hadron (DDRH) field theory [2, 3]the nuclear matter results for the in-medium interactionsare used in a theoretically well defined approach in calcula-tions for infinite nuclear matter, neutron stars and groundstates of finite nuclei [3, 4] and hypernuclei [5].
In DDRH field theory interactions of baryons in a nu-clear environment are described by density dependentmeson-baryon vertices. They are introduced as functionalsdepending on Lorentz-scalars of the baryon field operators.In this way correlations and many-body effects are takeninto account in an elegant and transparent formulation [3]by a Lagrangian L = LB +LM +Lint with baryonic (LB),mesonic (LM ) and interaction (Lint) parts which for non-strange systems is:
LB = Ψ [iγµ∂µ −M ]Ψ (1)
LM =12
∑i=σ,δ
(∂µΦi∂
µΦi −m2iΦ
2i
)−
12
∑κ=ω,ρ,γ
(12F (κ)µν F
(κ)µν −m2κA
(κ)µ A(κ)µ
)(2)
Lint = ΨΓσ(ρ)ΨΦσ −ΨΓω(ρ)γµΨA(ω)µ +
ΨΓδ(ρ)τΨΦδ −ΨΓρ(ρ)γµτΨA(ρ)µ −eΨQγµΨA(γ)µ . (3)
Basic principles as the covariance of the field equationsand thermodynamical consistency are retained. An im-portant contribution are rearrangement terms obtained bythe variational derivation of the field equations [2, 3]. Theresulting field equations are solved in mean-field approxi-mation.
The formulation is held general allowing for the flexibil-ity required for a unified description of interactions withinflavour multiplets. For baryon octet physics this means toinclude isospin and strangeness exchange channels as de-scribed by the pseudo-scalar and vector meson octets (or,more precisely, nonets). In addition, a more hypotheticalnonet of scalar mesons with S = 0,±1 is required. ForS = 0 the physically closest realization realized is possiblyfound in the (low energy) f0 spectrum. In practice, thevertex functionals are obtained from Dirac-Brueckner the-ory. The in-medium K-matrix K is represented in termsof renormalized meson exchange potentials [3]
K(q′, q|qs, kF ) =∑m
zm(qs|kF )Vm(q′, q) . (4)
As shown in [3] the renormalization factors zm are in factdetermined by the underlying interactions. In Fig.1 resultsfor the in-medium vertices Γ2(ρ) = z(kF )g2 are shown.Generally, the interaction strengths decline with increasingdensity, except in the scalar-isoscalar δ/a0(980) channelshowing a more complicated behavior. The DDRH the-
Figure 1: DDRH in-medium interactions. Dirac-Brueckner vertices for the mean-field producing mesonfield are shown [3].
ory provides a parameter-free and successful scheme forinteractions in nuclear environments once the free spaceNN interaction is chosen. Calculations for symmetric andasymmetric nuclear matter, binding energies of nuclei andhypernuclei reproduce data rather well and have showntheir potential for predictions in the unexplored mass re-gions up to neutron stars. [3, 4]. More recently, the ap-proach has been used by several groups on a phenomeno-logical level by assuming functional forms and adjustingthe parameters in data fits, e.g. [6].
References
[1] F. de Jong, H. Lenske, Phys. Rev. C 57 (1998) 3099 .[2] H. Lenske, F. Fuchs, Phys. Lett. B 345 (1995) 355 ;
Phys. Rev. C 52 (1995) 3043 .[3] H. Lenske, Springer Lecture Notes, (2004), (in print).[4] F. Hofmann, C. Keil, H. Lenske, Phys. Rev. C 64
(2001) 034314 ; Phys. Rev. C 64 (2001) 025804 .[5] C. Keil, F. Hofmann, H. Lenske, Phys. Rev. C 61
(2000) 06401 ; C. Keil, H. Lenske, Phys. Rev. C(2002) .
[6] T. Niksic, D. Vretenar, and P. Ring, Phys. Rev. C 66(2002) 064302 (2002); T. Niksic et al., Phys. Rev. C66 (2002) 024306 .
- 30 -
Relativistic nuclear model with point-couplings constrained byQCD and chiral symmetry
Paolo Finelli1,2, Norbert Kaiser3, Dario Vretenar4, and Wolfram Weise1,3
1ECT∗, I-38050 Villazzano (Trento), Italy; 2Physics Department, University of Bologna, and INFN - Bologna, I-40126Bologna, Italy; 3Physik-Department, Technische Universitat Munchen, D-85747 Garching, Germany; 4Physics
Department, Faculty of Science, University of Zagreb, 10 000 Zagreb, Croatia
We have derived a microscopic relativistic point-coupling model of nuclear many-body dynamics con-strained by in-medium QCD sum rules and chiral symme-try [1]. The effective Lagrangian is characterized by den-sity dependent coupling strengths, determined by chiralone- and two-pion exchange and by QCD sum rule con-straints for the large isoscalar nucleon self-energies thatarise through changes of the quark condensate and thequark density at finite baryon density. In comparison withpurely phenomenological mean-field approaches, the built-in QCD constraints and the explicit treatment of pion ex-change restrict the freedom in adjusting parameters andfunctional forms of density dependent couplings. It isshown that chiral (two-pion exchange) fluctuations play aprominent role for nuclear binding and saturation, whereasstrong scalar and vector fields of about equal magnitudeand opposite sign, induced by changes of the QCD vacuumin the presence of baryonic matter, generate the large ef-fective spin-orbit potential in finite nuclei.
The model is defined by the Lagrangian density
L = Lfree + L4f + Lder + Lem, (1)
with the four terms specified as follows:
Lfree = ψ(iγµ∂µ −M)ψ , (2)
L4f = −12GS(ρ)(ψψ)(ψψ)
−12GV (ρ)(ψγµψ)(ψγµψ) + . . . , (3)
Lder = −12DS(ρ)(∂ν ψψ)(∂νψψ)
−12DV (ρ)(∂ν ψγµψ)(∂νψγµψ) + . . . , (4)
Lem = eAµψ1 + τ3
2γµψ − 1
4FµνF
µν . (5)
This Lagrangian (1)-(5) is understood to be formally usedin the mean-field approximation, with fluctuations en-coded in density-dependent couplings Gi(ρ) and Di(ρ).Combining effects from in-medium QCD condensates [2],where the ratio of scalar and vector nucleon self-energiesat the leading order is:
Σ(0)S
Σ(0)V
=G
(0)S ρs
G(0)V ρ
= − σN4(mu +md)
ρsρ
−1 , (6)
and pionic fluctuations encoded in G(π)S,V , determined
within in-medium Chiral Perturbation Theory [3], thestrength parameters of the isoscalar four-fermion interac-tion terms in the Lagrangian (1) are:
GS,V (ρ) = G(0)S,V +G(π)
S,V (ρ) . (7)
-1 0 1 2 3 4 5 6 7 8-50
-40
-30
-20
-10
0
10
Sin
gle
part
icle
ene
rgy
leve
ls (
MeV
) ν π
16O
Theor. Exp. Theor. Exp.
1s1/2
1p3/2
1p1/2
6.10
6.306.02
5.91
0 2 4 6 8-40
-30
-20
-10
0
10
Sin
gle
part
icle
ene
rgy
leve
ls (
MeV
)
40Ca
Theor. Exp. Theor. Exp.
6.23
ν π
1p3/2
1p1/2
1d5/2
1d3/2
2s1/2 6.15
6.00 6.00
Figure 1: Neutron and proton single-particle energies for16O (left) and 40Ca (right)
The following table displays the calculated binding en-ergies per nucleon and charge radii of 16O and 40Ca incomparison with experimental data (in brackets).
E/A (exp) (MeV) rc (exp) (fm−3)16O 8.027 (7.976) 2.735 (2.730)40Ca 8.508 (8.551) 3.470 (3.485)
The neutron and proton single-particle energies are repro-duced, respectively, in comparison with the correspondingexperimental levels in Fig. 1.
We have demonstrated that an approach to nuclear dy-namics, constrained by the chiral symmetry breaking pat-tern and the condensate structure of low-energy QCD, candescribe properties of finite nuclei, at a quantitative levelcomparable with the best phenomenological relativisticand non-relativistic mean-field models. This is a promis-ing result which opens perspectives of bridging the gap be-tween basic features of low-energy, non perturbative QCDand the rich nuclear phenomenology.
Work supported in part by BMBF, GSI and MURST.
References
[1] P. Finelli, N. Kaiser, D. Vretenar and W. Weise, Eur.Phys. J. A17, 573 (2003); Nucl. Phys. A (2004) inprint.
[2] X. Jin, M. Nielsen, T.D. Cohen, R.J. Furnstahl andD.K. Griegel, Phys. Rev. C49, 464 (1994).
[3] N. Kaiser, S. Fritsch, and W. Weise, Nucl. Phys.A697, 255 (2002).
- 31 -
Nuclear structure calculations in FMD
T. Neff and H. FeldmeierGSI Darmstadt
In the Fermionic Molecular Dynamics (FMD) model [3] theA-body state is given as a Slater determinant
∣∣∣Q ⟩ of single-
particle states∣∣∣ qi
⟩∣∣∣Q ⟩ = A
∣∣∣ q1
⟩⊗ . . . ⊗ ∣∣∣ qA
⟩. (1)
The single-particle wave functions are described by Gaussianwave packets localized in phase-space
⟨x∣∣∣ q ⟩ =∑
i
ci exp
− (x − bi)2
2ai
∣∣∣ χi
⟩⊗ ∣∣∣ ξ ⟩ . (2)
The FMD many-body state is obtained by varying the energywith respect to all single-particle parameters (Variation/V). Themany-body state can be intrinsically deformed and has to beprojected to good angular momentum (Projection after Varia-tion/PAV). An improved description is obtained by minimiz-ing the energy of the projected many-body state (Variation afterProjection/VAP). We perform VAP calculations in the sense ofthe Generator Coordinate Method. The energy of the (unpro-jected) many-body state is first minimized under certain con-straints. The energy of the projected many-body states is thenminimized with respect to the constraint parameters. A furtherimprovement is achieved by diagonalizing the Hamiltonian in aset of projected FMD states (Multiconfig).
An effective interaction derived from the realistic Bonn inter-action is used. With the Unitary Correlation Operator Method[2] we explicitly include the short-range correlations inducedby the repulsive core and the tensor force. The correlated inter-action is then a low-momentum interaction that can be used di-rectly in the many-body spaces of the FMD. An additional 15%two-body correction term to the correlated interaction that sim-ulates the effects of missing three-body forces and three-bodycorrelations is fitted to reproduce the binding energies and radiiof 4He, 16O, 40Ca, 24O and 48Ca.
Within this framework we are able to calculate the propertiesof stable and exotic nuclei in the p- and sd-shell. This is illus-trated in the following for 12C. Further examples including theneutron rich He isotopes can be found in [1].
Varying the intrinsic energy of the FMD state (V) results ina spherical shell model many-body state as depicted in Fig. 1.Projection (PAV) does not change the result here. Comparedto the experimental observations the binding energy is too lowand the radius is too small. If we perform a VAP calcula-tion with constraints on the octupole moment and the radius
Eb [MeV] rcharge [fm] B(E2) [e2fm4]
V/PAV 84.7 2.30 -
VAP α-cluster 80.4 2.64 56.3
VAP 91.9 2.36 24.7
Multiconfig 93.4 2.48 40.0
Exp 92.2 2.47 39.7 ± 3.3
Table 1: Binding energies, charge radii and BE(2)-values.
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
V/PAV VAP α-cluster VAP
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
-4 -2 0 2 4x [fm]
-4
-2
0
2
4
y [fm
]
“ 3−1 ” “ 0+2 ” “ 0+3 ”
Figure 1: Intrinsic shapes used in the calculation.
where the FMD state is restricted to be of α-cluster type (VAPα-cluster) we obtain a triangle configuration that is higher inenergy than the shell model configuration. The full VAP cal-culation where we use two Gaussian wave packets per single-particle state gives a much improved result. The obtained in-trinsic state is an interpolation between the shell model and theα-cluster states. Binding energy and radii agree now much bet-ter with the experimental observations, see Tab. 1. In a multi-configuration calculation with four shapes whose energies areminimized with respect to the first three 0+ and the first 3− statethe results can be further improved. The admixture of moreextended configurations is especially important for the radiusand the B(E2) values. With the multiconfiguration calculationwe can also study the excited states, see Fig. 2. The structureespecially of the excited 0+ states is currently under debate.
-90.0
-85.0
-80.0
-75.0
-70.0
0+
0+
2+
4+
3–
0+
0+
0+
2+
2+2+
4+
4+
4+
1–2–
3–
0+
0+
0+
2+
2+
2+
4+4+
4+
1–2–
3–
0+
0+
0+
2+
2+
2+
4+
1–2–
3–
Variation VAP Multiconfig (4) Multiconfig (25) Exp
12C[MeV]
Figure 2: Calculated and experimental spectrum of 12C.
References
[1] T. Neff, H. Feldmeier, nucl-th/0312130,http://theory.gsi.de/˜fmd/
[2] T. Neff, H. Feldmeier, Nuc. Phys. A713 (2003) 311[3] H. Feldmeier, J. Schnack, Rev. Mod. Phys. 72 (2000) 655
- 32 -
Tensor correlations and P-wave pairing in neutron matter
Achim Schwenka and Bengt Frimanb
aDepartment of Physics, The Ohio State University, Columbus, OH 43210; bGSI, Darmstadt
Landau-Fermi liquid theory is a powerful effective the-ory for strongly interacting Fermi systems at low temper-atures. It has been successfully applied to liquid 3He, nu-clear matter and nuclei. However, in contrast to the inter-action between 3He atoms, nuclear forces are complicateddue to the large non-central spin-orbit and tensor compo-nents. Here, we report on the in-medium modification ofthe effective nucleon-nucleon interaction, paying particularattention to spin-dependent forces [1].
The form of the two-body interaction in vacuum is con-strained by symmetries. In particular, in non-relativistictheories, Galilean invariance implies that the two-body in-teraction is independent of the particle-pair (cm) momen-tum P ≡ p1 + p2 = p3 + p4. In this case, the possi-ble operators are scalar, spin-spin, spin-orbit, tensor andquadratic spin-orbit forces. In the medium, the Fermi seadefines a preferred frame, and the effective two-body in-teraction depends on the cm momentum. On the Fermisurface, this leads to novel non-central forces of the form
S12(P) cm tensor (1)D12(q,P) ≡ i(σ1 − σ2) · q×P diff vector (2)A12(q′,P) ≡ (σ1 × σ2) · (q′ ×P) cross vector, (3)
with q ≡ p1 − p3 and q′ ≡ p1 − p4. The antisymmetricoperatorsD12 and A12 do not conserve the spin of the par-ticle pair. Both cm tensor and A12 survive in the Landau(q → 0) limit and lead to novel Fermi liquid parameters.
In [1], we have studied the microscopic origin of these in-teractions. We have also computed particle-particle, hole-hole (pp/hh) and particle-hole (ph) contributions to thequasiparticle interaction and scattering amplitude to sec-ond order in the low-momentum interaction Vlow k [2]. It iswell-known that the second-order tensor force is importantfor nuclear binding, since the spin-recoupling of iteratedtensor operators gives a large contribution to the scalarinteraction. Similar recoupling arguments imply that theinterference of the large spin-spin part (G0 ≈ 0.6 − 0.8 inneutron matter) with the tensor force in the ph channelsleads to a substantial renormalization of the tensor inter-action [3]. (Similarly, we also find a significant renormal-ization of the spin-orbit force; this leads to the suppressionof P-wave pairing discussed below.) Moreover, the pres-ence of a third particle in intermediate states induces novelcontributions, which give rise to a particular coupling be-tween the spin and the angular motion and leads to anti-symmetric spin operators. The cm tensor emanates fromboth pp/hh and ph diagrams. There are also kinematical(boost) corrections to D12 and A12 [4], which however arerelatively small (of order k2
F/m2).
We find a substantial renormalization of the exchangetensor S12(q′) as well as significant contributions to thecm tensor. This implies that higher-order calculations,e.g., within the RG approach [3], are needed. We alsoexplore the effect of particle-hole screening on the 3P2
pairing gap in neutron star interiors. As for the S-wave
1.3 1.4 1.5 1.6 1.7 1.8
kF [fm
-1]
0.0001
0.001
0.01
0.1
1
∆ [M
eV]
A: direct, m*=mdirect, m*C: + ZS + ZS’ channels+ central induced+ spin-orbit inducedB: Baldo et al. (1998)
A
C
BA
C
Figure 1: The angle-averaged gap ∆3P2 versus Fermi mo-mentum in neutron matter. The direct (Vlowk) and thepairing gap including polarization effects on the pairinginteraction are shown (with only induced central, only in-duced spin-orbit or total to second-order). For reference,we give the results of Baldo et al. [5], obtained by solvingthe coupled channel BCS equation for different potentials.
gaps [3], large effects are expected. In weak couplingBCS theory, neglecting the coupling to the 3F2 channel,∆3P2 = k2
F/m exp[π/
(2 kFmVpairing;3P2
)]. As shown in
Fig. 1, the in-medium modification of the spin-orbit forceleads to a significant reduction of the 3P2 gap. Notethat when only central induced forces are included (dottedvs. dashed line), the gap is increased [6]. Neutron star cool-ing calculations require P-wave gaps below ∆ 30 keV [7].
Implications of the new interactions for nuclear spectra,for spin-isospin response and neutrino transport in super-novae, for spin-polarized systems, spin relaxation, mixingof spin and density waves, and for scattering with polarizedbeams remain to be investigated.
References
[1] A. Schwenk and B. Friman, Phys. Rev. Lett. 92(2004) 082501, nucl-th/0307089.
[2] S.K. Bogner, T.T.S. Kuo and A. Schwenk, Phys. Rep.386 (2003) 1, and references therein.
[3] A. Schwenk, B. Friman and G.E. Brown, Nucl. Phys.A713 (2003) 191, ibid. A703 (2002) 745.
[4] J.L. Forest, V.R. Pandharipande and J.L. Friar, Phys.Rev. C52 (1995) 568, and references therein.
[5] M. Baldo et al., Phys. Rev. C58 (1998) 1921.[6] C.J. Pethick and D.G. Ravenhall, Ann. N.Y. Acad.
Sci. 647 (1991) 503.[7] D.G Yakovlev and C.J. Pethick, Ann. Rev. Astron.
Astrophys. (2004) in press, astro-ph/0402143.
- 33 -
Shell Gaps in the Sn Isotopes
M.Bender1, T.Burvenich2, T.Cornelius3, P. Fleischer4, P.Klupfel4, J.A.Maruhn3, P.-G.Reinhard4
1 Department of Physics, University of Washington, Seattle / USA 2 Los Alamos National Laboratory, Los Alamos / USA3 Institut fur Theoretische Physik, Universitat Frankfurt / Germany4 Institut fur Theoretische Physik, Universitat Erlangen / Gemany
The exploration of nuclear shell structure is still asubject of great current interest, particularly in view ofnew information gathered by experiments with radioac-tive beams, see e.g. [1, 2]. A key feature are shell closuresrelative to the appearance of magic or semi-magic nuclei.A theoretical exploration of the proton shell closure in Pbisotopes was published in [3]. Here we want to present firstresults from a similar study for the chain of Sn isotopes.
The notion of a shell closure comes from a mean-field de-scription where one has full insight into the single-nucleonenergies εk. A shell closure is associated with a large gapin the spectrum of εk. However, these single-nucleon ener-gies are not directly observable in experiment. A quantitywhich is accessible from mass systematics is the so-calledtwo-proton shell gap
δ2p = E(Z+2, N)− 2E(Z,N) + E(Z−2, N) . (1)
In case that the single-particle energies do not changemuch within the three isotones considered and that thechange in the total binding energy comes from the varia-tion of occupations around the Fermi surface, Koopman’stheorem states that δ2p represents twice the gap in thesingle-nucleon spectrum. This requires, however, that nodramatic rearrangements happen amongst the three nucleiinvolved in δ2p. In the following, we will investigate theimpact of such rearrangements. To that end, we employ aself-consistent mean-field description in terms of Skyrme-Hartree-Fock eventually with some correlations added.
The theoretical description is handled at several levelsof refinement. We start from purely spherical calcula-tions mean-field for all involved nuclei. This minimizesrearrangement effects and comes close to the situation as-sumed in Koopman’s theorem. In a second step, we al-low for deformations. Most nuclei will sty spherical. Butthe softer ones may prefer a spontaneous transition to de-formed shape. Soft systems, however, are not well de-formed either. The fluctuate through a whole landscapeof deformations. This gives rise to collective correlationeffects which we include in the third and last step of ourtreatment (for details see [4]).
Figure 1 shows results for the two-proton shell gap alongthe chain of Sn isotopes, computed with the Skyrme forceSkI3 [5] at the three levels of approximation as discussedabove. The spherical calculations produce a more or lessconstant gap with a slight trend to increase towards de-creasing neutron number. This feature agrees in value andtrend nicely with the spectral gap at the Fermi energyfound in the single-proton spectra of the Sn isotopes. Butthe spherical results are far from the experimental values.The same happened in the Pb isotopes [3]. The pointis that the δ2p involves the Z ± 2 nuclei which have nomagic shell any more. Thus they are much softer and of-ten develop some deformation. This lowers the energies
4
6
8
10
12
14
16
50 54 58 62 66 70 74 78 82
δ2p/
MeV
neutrons
δ2p for the Tin-chain
GSC-ski3def. MF-ski3sph. MF-ski3
Exp.
Figure 1: The two-proton shell gap δ2p as defined in eq. (1)calculated for the Skyrme interaction SkI3 at various levelsof approximation: circles = purely spherical mean field,diamonds = deformed mean field, triangles = deformedmean field plus collective correlations, boxes = experimen-tal values.
E(Z±2, N) as compared to their spherical values whilethe magic isotopes stay spherical and their E(Z,N) re-mains unchanged. it is obvious that this rearrangement ofthe Z±2 neighbors will reduce the two-proton shell gap.And that is seen nicely in figure 1. All nuclei near themagic neutron number stay spherical and the freedom todeform is not exploited. Farther away from N = 82 , theneighbors Cd (Z = 48) and Te (Z = 52) develop deforma-tion and we see a substantial reduction of δ2p from that.Even though, the deformed results are still far from theexperimental values. Finally, we invoke the collective cor-relations which is, in fact, the correct treatment of the softnuclei next to Sn. This provides nicely a further push to-wards the experimental values. There remains, however,still some mismatch. It is not feature of this particularforce SkI3. The same results is found with several otherSkyrme forces and it hints at further correlation effects,yet missing.
Acknowledgment: This work was supported by theBMBF, projects 06 ER 808 and 06 ER 124.
References
[1] T. Aumann et al, Act.Phys.Hung.N.S. 12 (2001) 111[2] H. Emling, Prog.Part.Nucl.Phys. 46 (2001) 285-291[3] M. Bender etal, Eur.Phys.J. A, 14 (2002) 23[4] P. Fleischer, PhD thesis, Erlangen 2003[5] P.-G. Reinhard, H. Flocard, Nucl.Phys. A 584 (1995)
467
- 34 -
Spin-Isospin Resonances and the Neutron Skin of Nuclei
N. Paar1, P. Ring1, D. Vretenar2, and T. Niksic2
1Physik Department, Technische Universitat Munchen, 85748 Garching, Germany; 2Physics Department, Faculty ofScience, University of Zagreb, 10000 Zagreb, Croatia
Charge distributions in nuclei have been measured withextremely high precision since many years and they pro-vide an essential tools for our understanding of atomic nu-clei. It is much more difficult to measure the distribution ofneutrons. Therefore this quantity is at present in the cen-ter of many theoretical and experimental investigations.It places important additional constraints on effective in-teractions used in nuclear models. Various experimentalmethods have been used, or suggested, for the determi-nation of this quantity, but no existing measurement ofneutron densities or radii has an established accuracy ofone percent.
We suggest a new method for determining the differencebetween the radii of the neutron and proton density dis-tributions along an isotopic chain, based on measurementof the excitation energies of the Gamow-Teller resonancesrelative to the isobaric analog states.
Collective spin and isospin excitations in atomic nucleihave been the subject of many experimental and theoret-ical studies. Nucleons with spin up and spin down canoscillate either in phase (spin scalar S=0 mode) or out ofphase (spin vector S=1 mode). The spin vector, or spin-flip excitations can be of isoscalar (S=1, T=0) or isovector(S=1, T=1) nature. These collective modes provide di-rect information on the spin and spin-isospin dependenceof the effective nuclear interaction. Especially interest-ing is the collective spin-isospin oscillation with the excessneutrons coherently changing the direction of their spinsand isospins without changing their orbital motion – theGamow-Teller resonance (GTR) Jπ = 1+. The simplestcharge-exchange excitation mode, however, does not re-quire the spin-flip (i.e. S=0) and corresponds to the wellknown isobaric analog state (IAS) Jπ = 0+. The spin-isospin characteristics of the GTR and the IAS are relatedthrough the Wigner super multiplet scheme, which impliesthe degeneracy of the GTR and IAS. The Wigner SU(4)symmetry is, however, broken by the spin-orbit term ofthe effective nuclear potential.
It is implicit, therefore, that the energy difference be-tween the GTR and the IAS reflects the magnitude of theeffective spin-orbit potential. In Ref. [1] the framework ofrelativistic mean field theory was employed in an analy-sis of the isospin dependence of the spin-orbit term of theeffective single-nucleon potential for light neutron-rich nu-clei. It was shown that the magnitude of the spin-orbitpotential is considerably reduced in neutron-rich nuclei.
We found a direct connection between the increase ofthe neutron-skin thickness in neutron-rich nuclei, and thedecrease of the energy difference between the GTR and theIAS [2]. The calculation is performed in the framework ofthe fully self-consistent RHB plus proton-neutron relativis-tic QRPA model. The RHB model represents a relativisticextension of the Hartree-Fock-Bogoliubov framework, andit provides a unified description of particle-hole (ph) and
particle-particle (pp) correlations, that is essential for aquantitative analysis of ground-state properties and mul-tipole response of unstable, weakly-bound nuclei far fromthe line of β-stability [3].
In Fig. 1 the calculated and experimental energy spac-ings between the GTR and IAS are plotted as a functionof the calculated differences between the rms radii of theneutron and proton density distributions of even-even Snisotopes. The calculated radii correspond to the RHBNL3+D1S self-consistent ground-state solutions, on whichthe proton-neutron RQRPA calculations are performed.We notice a remarkable uniform dependence of the energyspacings between the GTR and IAS on the size of theneutron-skin. In principle, therefore, the value of rn − rpcan be directly determined from the theoretical curve fora given value of EGT − EIAS. This method is, of course,not completely model independent, but it does not requireadditional assumptions. Since the neutron-skin thicknessis determined in an indirect way from the measurement ofthe GTR and IAS excitation energies in a sequence of iso-topes, in practical applications at least one point on thetheoretical curve should be checked against independentdata on rn − rp.
0.10 0.15 0.20 0.25 0.30rn-r
p[fm]
0.5
1.0
1.5
2.0
2.5
EG
T-E
IAR[M
eV]
RQRPAEXP.
124Sn
112Sn
116Sn
Figure 1: The proton-neutron RQRPA(NL3) differencesbetween the excitation energies of the GTR and IAS asa function of neutron skin in Sn isotopes. Exp. data arefrom Refs. [4, 5]
References
[1] G. A. Lalazissis, D. Vretenar, and P. Ring, Phys. Rev.C57, 2294 (1998).
[2] D. Vretenar, N. Paar, T. Niksic, and P. Ring, Phys.Rev. Lett. 91, 262502 (2003).
[3] N. Paar, P. Ring, T. Niksic, and D. Vretenar, Phys.Rev. C67, 034312 (2003).
[4] K. Pham et al., Phys. Rev. C51, 526 (1995).[5] A. Krasznahorkay et al., Phys. Rev. Lett. 82, 3216
(1999).
- 35 -
Study of halo formation with (e,e’p) and (e,e’d) reactions
L.C. Liu, T. Kuhl, M. Tomaselli, S. Fritzsche, and D. UrsescuLANL GSI TUD Kassel
The separation of the longitudinal and transverse re-sponses in quasielastic electron scattering at intermediatevalues of momentum transfer has been firstly achieved in1980 [1]. Subsequent theoretical studies [2] have foundthat the experimentally observed reduction of the longitu-dinal response function with respect to Fermi-gas modelprediction may be a result of the existence of nuclear cor-relations. Essential result obained in these studies [2] isthat the short-range and the tensor parts of the nucleon-nucleon interaction deplete the shell-model orbitals andthis depletion affects the longitudinal and transverse re-sponse functions in a very different manner. The quasielas-tic scattering cross section for (e,e’p) is written in terms ofthe Coulomb form factor F 2
C and transversal form factorF 2T as [3]:
dσ
dΩdE2= 4π σMott(∆2/q2)
2[F 2
C(q, ω)
+(q2/∆2)(12
+ (q2/∆2) tan2 θ
2) F 2
T (q, ω)]
(1)
where ∆2 ≡ q2 − ω2. Using the nonrelativistic approx-imation to the Dirac spinor and keeping only terms tothe order of q2/m2, de Forest obtained for the quasielasticscattering from a nucleus [4]
F 2C(q, ω) = K
∑MiMf
|< JfMf |∑j
[ej + (q2/8m2)
(ej − 2µj)]ei q· rj | JiMi >|2 δ(Ef − Ei − ω)(2)
F 2T (q, ω) = K
∑MiMf
|< JfMf |∑j
[(ej/im)0(j)
t
−(iµj/2m)0q × 0σ(j)]ei q· rj | JiMi >|2 δ(Ef − Ei − ω)(3)
where K = f2(∆2)(2Ji + 1)2/4π with f(∆2) denotingthe electric and magnetic form factors of the proton andneutron (assumed to be all equal). In impulse approx-imation, the electron interacts with one single nucleonor nucleon cluster while leaving the rest of the nucleusas a spectator. Hence, the energy-conservation δ func-tion reduces, respectively, to δ(εp(0p+ 0q) − εp(0p) − ω) andδ(εd(0p+0q)− εd(0p−ω) for the (e,e’p) and (e,e’d) reactions.If the final nuclear states are not measured in the exper-iment, the F 2
C and F 2T will be combintations of functions
of the type∫ρ(p)H(p)d0pδ(ε(0p+ 0q)− ε(0p)−ω), with H(p)
being 1, p cosφ , p2cos2φ, or or p2. Here, φ is the anglebetween the momentum 0p of the active nucleon (or nu-cleon cluster) and the momentum transfer 0q. The ρ(p) isthe momentum-space density distribution. In our calcu-lation, we do not use the Fermi gas model. Instead, weuse the density distribution given by the BDCM [5] thattakes fully into account the nuclear core excitation and re-produces the measured energy levels of the nucleus. We
define the longitudinal and transverse nuclear reponses, re-spectively, as SL = F 2
C and ST = F 2T and we are interested
in the longitudinal-to-transverse ratio defined by
R = (∆2/q2)2SLST
=A
B− 1
2(∆2/q2). (4)
The removal of the kinematic singularity allows the nu-clear structure effect to be more clearly revealed. In Fig.1 we show calculations for 6Li. Since what is of inter-est is the density distribution of the valence pn pair, wepropose to consider the (e,e’d) process. The selection ofthis deuteron channel has also the added benefit of cuttingdown background events. As we can see from the figure,the configuration mixing causes the reduction of R. Thisreduction indicates that the core excitation (responsiblefor the halo formation) decreases the relative importanceof SL. Our result is in agreement with the findings ofRef. [2]. In conclusion, the quasielastic scattering can beused to probe the dynamics of nuclear correlations. Whilethis dynamics can only be probed at very large momen-tum transfers (of the order of several GeV/c ) in elasticscattering (e,e), it can already be studied in quasieleasticscattering at moderated momentum transfers (of the order100 MeV/c). In addition, the elastic scattering probes allthe nucleons in a nucleus while the quasielastic scatteringcan be channeled to probe selected nuclear clusters. Theratio of response functions is a prefered observable as itminimizes the final-state distortion effects. The use of Ras observable is further preferable as it does not containsingularities of kinematical origin.
q=150 MeV/c
ω [ M e V ]0 20 40 60 80 100 120 140 160 180 200 220240
(∆2 /q
2 )2 (SL/S
T)
0.00.20.40.60.81.01.21.41.61.82.02.22.42.62.83.03.23.43.63.84.04.24.44.64.85.0
no config. mixing with config. mixing
q=300 MeV/c
ω [ M e V ]0 20 40 60 80 100 120 140 160 180 200 220240
(∆2 /q
2 )2 (SL/S
T)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
no config. mixing with config. mixing
Figure 1: Calculated singularity-free longitudinal-to-transverse response ratio R. Left side: q=150 MeV/c.Right side: q=300 MeV/c.
References
[1] R. Altemus et al., Phys. Rev. Lett. 44 (1980) 965; P.Barreau et al., Nucl. Phys. A358 (1981) 287.
[2] L.S. Celenza et al., Phys. Rev. C 26, 320 (1982).[3] H. Uberall, Electron Scattering From Complex Nuclei,
Academic Press 1971[4] T. de Forest, Nucl. Phys. A132 (1969) 305.[5] M. Tomaselli, L.C. Liu et al., Journal of Optics B 5
(2003) 395.
- 36 -
Final-state effects in the electromagnetic breakup of exotic nuclei
S. Typel1 and G. Baur2
1GSI Darmstadt; 2IKP, Forschungszentrum Julich
Exotic nuclei have been studied extensively in recentyears by electromagnetic excitation with the help of theCoulomb breakup method. These unstable nuclei areweakly bound and the excitation process readily leads toa breakup of the projectile into fragments at intermedi-ate to high beam energies. Electromagnetic transitionsto the continuum with large strength at low energies areobserved. For a recent review see Ref. [1].
Experiments are usually analyzed in first order theorieswith simple structure models. However, the effects of thefinal-state interaction (FSI) between the fragments and be-tween target and fragments have to be assessed in order toobtain reliable information from experimental data. Thefinal state of the breakup process with at least three parti-cles is a complicated system. Various theoretical methodshave been developed to study final-state effects.
The FSI between target and fragments destroys thesimple factorization of the cross section into contribu-tions from nuclear structure and the excitation process infirst-order theories. Higher-order effects from the target-fragment interaction can be included in the semiclassicalapproach by extending time-dependent perturbation the-ory to higher orders or by applying the sudden approxi-mation The most general approach is studying the timeevolution of the projectile system by coupled-channel cal-culations or by solving the time-dependent Schrodingerequation (dynamical calculation).
A prime example is the well-studied breakup of the halonucleus 8B into a proton and 7Be with a very small pro-ton separation energy Sp = 0.137 MeV. E1-E2 interferenceleads to a marked asymmetry of longitudinal momentumdistributions at low beam energies. Higher-order effectsthat were considered in a dynamical calculation reducethis asymmetry and require an increased E2 strength ascompared to a simple potential model in order to describethe experimentally observed asymmetry [2]. The Coulombbreakup of 8B was also used as an indirect method to ex-tract the astrophysical S factor S17(E) of the radiative cap-ture reaction 7Be(p,γ)8B [3]. Higher-order effects of thetarget-fragment interaction lead to a reduction of the dis-sociation cross section. High projectile energies and largeimpact parameters reduce the effect considerably. A de-pendence of the S factor on the fragment-fragment FSIis also found in this case. It affects the extrapolation ofmeasured S17(E) values to zero energy [3]. The FSI is alsorelevant for the ANC method where the zero-energy S fac-tor is calculated theoretically using ground-state asymp-totic normalization coefficients that were experimentallydetermined from transfer or nucleon-removal reactions [4].The sensitivity of the zero-energy S factor on the potentialstrength increases with Sp.
For neutron+core nuclei the dependence of the final-state effects on characteristic quantities of the system isfound from simple but realistic models that can be solvedanalytically [5]. Matrix elements for electric multipole
01
20 20 40 60 80
0
0.5
1
1.5
2
2.5
01
2
11Be (2s1/2, 0.504 MeV)
E [MeV]
dB(E1)/dE
[e2fm2/MeV]
V0 [MeV]
Figure 1: Reduced transition probability dB(E1)/dE forthe breakup of 11Be into 10Be+neutron for various depthsV0 of the nuclear potential (of Woods-Saxon shape) in thefinal state.
transitions to the continuum are determined by the asymp-totics of the wave functions in the initial and final state.The reduced transition probability exhibits a characteristicuniversal shape when expressed in terms of the relevant pa-rameters. The effective-range expansion of the phase shiftsmotivates the introduction of reduced scattering lengths inorder to include effects of the fragment-fragment interac-tion in the model [6].
An increase of the strength of the FSI between the frag-ments from zero to realistic values leads to a significantvariation of the shape of the reduced transition proba-bility. These results are corroborated in more realisticcalculations with simple potential models for n+core sys-tems (see Fig. 1 for the case of 11Be with a 2s1/2 neu-tron and a separation energy of Sn = 0.504 MeV in theground state). The sensitivity of the transition probabilityon the fragment-fragment FSI increases with the neutronseparation energy. For n-halo nuclei with small neutronseparation energy plane-wave calculations neglecting thefragment-fragment interaction give similar results like cal-culations with realistic values of the potential.
References
[1] G. Baur, K. Hencken, and D. Trautmann,Prog. Part. Nucl. Phys. 51 (2003) 487;
[2] B. Davids and S. Typel, Phys. Rev. C68 (2003)045802;
[3] F. Schumann, F. Hammache, S. Typel et al.,Phys. Rev. Lett. 90 (2003) 232501;
[4] L. Trache et al., Phys. Rev. Lett. 87 (2001) 271102;Phys. Rev. C66 (2002) 035801; Phys. Rev. C67(2003) 062801;
[5] S. Typel and G. Baur, Phys. Rev.C64 (2001) 024601;
[6] S. Typel, G. Baur, Proceedings Hirschegg 2004.
- 37 -
Factorization of charge and nuclear formfactors for clusterized nuclei
Gridnev K.A.1, Ershov K.V.2, Kartavenko V.G.2, and Greiner W.3
1St Petersburg State University; 2JINR; 3Frankfurt am Main University
Considering high energy electrons as a probe it is worthnoting previous attempts to extract information on clusterstructure [1]. The role of cluster configurations in the nu-cleus was investigated in [2]. In [1] authors used nucleongaussian trial wave functions in the s-state to calculatecluster wave function. Only in this case, when momentumtransfer is high it is possible to factorize the formfactor.In general formfactors cannot be factorized. However thefirst attempt to factorize charge formfactor was done byR. Helm [3]. He used a folded charge distribution given by
ρ(r) =∫ρ0(0r ′)ρ1(0r − 0r ′)d30r ′ (1)
When Eq. (1) is substituted in the expression of formfac-tor:
F (0q) =∫ρ(0r)ei q rd3r (2)
it follows thatF (0q) = F0(q)F1(0q) (3)
where F0 and F1 result from the substitution of ρ0 and ρ1respectively, in Eq. (2):
F0(0q) =∫ρ0(0r)ei q rd3r, F1(0q) = e−g2q2/2 (4)
ρ1(r) had the gaussian form
ρ1(r) = (2πg2)3/2e−r2/2g2(5)
In his review [4] Ueberall considers the value in Eq. (5)as ”smearing”, because the transition charge density mustbe concentrated on a shell of about nuclear radius, in theform of delta-function smeared by convolution in Eq. (1).Helm’s model represents the transition moments of ρ0(r)as spherical shells at a radius R,
ρtrL (r) = J0βJ0JL R−2δ(r −R) (6)
with a strength parameter to be fitted by experiment. Thereduced matrix element reads
J −10 < J ||ML(q)||J0 >= βJ0J
L F1(q)jL(qR) (7)
If we consider a nucleus consisting of α-particles, theformfactor (spherical Bessel function) represents a spheri-cal shell made of α-particles. This result has been obtainedconsidering high energy electrons scattered on fullerenes [5]
F (q) = FA(q)∑J
ei q· rj = FA(q)n(q) (8)
where FA(q) is the formfactor of a single carbon atom andn(q) is the formfactor of the carbon atomic concentration.
We considered the model when alpha-particles liewhithin some surface shell. To certain extent this is a
30 40 50 60 70 80 90 100 110
1E-7
1E-6
1E-5
1E-4
1E-3
0,01
0,1
30 40 50 60 70 80 90 100 110
1E-6
1E-5
1E-4
1E-3
0,01
0,1
S
Mg
250MeV
250MeV
30 40 50 60 70 80 90 100 110
1E-6
1E-5
1E-4
1E-3
0,01
0,1
60 80 100 120
1E-8
1E-7
1E-6
1E-5
1E-4
Si
C
250MeV
300MeV
θ, grad θ, grad
θ, grad θ, grad
dσ/
dΩ, mBrn/sr
Figure 1: Electron scattering.
variant of the surface well model [6]. Assuming the quasi-cristaline structure of alpha-particles in nuclei we consid-ered two possible types of alpha-particle density distribu-tion, namely surface and volume distributions. In the firstcase the formfactor will be proportional to the Bessel func-tion of order zero j0 in the second case it will be the Besselfunction of order one j1. In FIG. 1 one can see scatteringof electrons on 12C, 28Si, 32S, 24Mg. At lower energies thesurface distribution of alpha-particles in light nuclei is fa-vorable. The Bessel function of order zero oscillates withthe period q = 2π/R, where R is the nuclear radius.
We have used the factorized formfactors for clusterizedlight nuclei assuming the alpha-particle density take thegaussian form. It is necessary to stress that in the micro-scopic approach charge densities of 12C were derived byfolding the finite proton size determined by gaussian [7].Factorized formfactors give us a possibility to scan nuclearsurface and some intrinsic layers of a nucleus.
References
[1] Inopin E.V., Kinchakov V.S., Lukyanov V.K., PolYu.S., Annals of Physics, 118 (1979) 307
[2] Gridnev K.A., Kartamishev M.P., Vaagen J.S.,Lukyanov V.K., Anagnastatos G.S., Inter. Journ ofModern Physics, 11E (2002) 359
[3] Helm R.H., Phys. Rev. 104 (1956)1466[4] Uberall H., Springer Tracts Modern Physics 49
(1969), 1[5] Gershikov L.G., Efimov P.V., Mikoushkin V.M.,
Solov’yov A.V., Phys. Rev. Lett., 81 (1998) 2707[6] Baz A.I., Goldberg V.Z., Gridnev K.A., Semjonov
V.M., Hefter E., Z. Physik 28A (1977) 171[7] Kamimura M., Nucl. Phys., 351A (1981) 456
- 38 -
Deformation influence on cold fusion reactions
R. A. Gherghescu1, G. Munzenberg2, D. N. Poenaru1,3, and W. Greiner3
1NIPNE, Bucharest; 2GSI - Darmstadt; 3Institut fur Theoretische Physik, Universitat Frankfurt am Main
Effects of target and projectile deformation on the shellcorrections are calculated within the deformed two-centershell model [1, 2]. The deformed two center oscillator po-tential which assures equipotentiality on the surface of thetwo intesected nuclei reads:
V (r)(ρ, z) =V1 = 1
2m0ω2ρ1ρ2 + 1
2m0ω2z1(z + z1)2 , v1
V2 = 12m0ω
2ρ2ρ2 + 1
2m0ω2z2(z − z2)2 , v2
(1)where v1 and v2 are the space regions where the two po-tentials are acting. The frequencies are shape dependent,thus one obtains from volume conservation and mass de-pendency of ω the relations:
m0ω2ρi
= (ai/bi)2/3 ·m0ω20i = (ai/bi)2/3 · 54.5/R2
i
m0ω2zi = (bi/ai)4/3 ·m0ω
20i = (bi/ai)4/3 · 54.5/R2
i
(2)In this way the two center oscillator potential for fusion likeshapes follows the changes of the two spheroidal partnerdeformations.
In order to find out each of the space regions v1 andv2, we rely on the assumption that the pass from V1 toV2 must be smooth; hence no abrupt cusp in the potentialvalue has to exist between v1 and v2. If the two regionscomply to this condition, they have to be bordered by thesame surface. Such a surface is the solution of the followingmatching condition between V1(ρ, z) and V2(ρ, z):
V1(ρ, z) = V2(ρ, z) (3)
Eq. (3) describes an ellipsoidal surface. On any of itspoints the two potentials, V1 and V2, match eachother.
The total Hamiltonian of the system is:
H = − h2
2m0∆ + V (r)(ρ, z) + VΩs + VΩ2 (4)
where VΩs and VΩ2 are the spin-orbit and the squaredangular momentum interaction potentials.
The spin-orbit operator is calculated as usual using cre-ation and anihilation components:
Ωs =12(Ω+s− +Ω−s+) +Ωzsz (5)
where:
Ω+ = −eiϕ[∂V r(ρ, z)
∂ρ
∂
∂z− ∂V r(ρ, z)
∂z
∂
∂ρ− i
ρ
∂V r(ρ, z)∂z
∂
∂ϕ
]Ω− = e−iϕ
[∂V r(ρ, z)
∂ρ
∂
∂z− ∂V r(ρ, z)
∂z
∂
∂ρ+i
ρ
∂V r(ρ, z)∂z
∂
∂ϕ
]Ωz = − i
ρ
∂V r
∂ρ
∂
∂ϕ(6)
In this way the angular momentum dependent operatorsare also shape-dependent. Details of calculation and ma-trix element formulae are given in [1, 3].
The superheavy synthesis reaction U+48Ca → 112 hasbeen analyzed for different isotopes of uranium spanningcontinuously the spheroidal target deformation. In Fig. 1the level schemes (uppper plot) and the shell effects forprotons E(
shellp) and total Eshell are drawn. Differencesare visible for the higher energy levels. Until the last partof the overlapping almost no difference exists within theproton shell correction. Divergence in the curve behaviourappears at the end of the fusion process. The two branchesin Eshell correspond to the two proton level scehme defor-mations of Z=112 (where the ratio of spheroid semiaxesis χ = 0.89 and χ = 1). The total shell correction Eshell
follows the same behaviour slightly lowering the 232U reac-tion shell correction in the first part of the process. Shellcorrection values less than those corresponding to spheresuggest a more deformed ground state for 112 isotopesthan those taken into account here.
0.0 0.4 0.80
20
40
E(M
eV)
48Ca+232U 280112(232U) = .754
s1/2
p3/2
p1/2
d5/2
s1/2
d3/2
f7/2
p3/2
f5/2p1/2
0.0 0.4 0.8
48Ca+238U 286112(238U) = 0.747
0.0 0.4 0.8
48Ca+240U 288112(240U) = 0.738
0.0 0.4 0.8
48Ca+244U 292112(244U) = 0.730
(R-Rf)/(Rt-Rf)
-505
10152025
Esh
ell(M
eV)
48Ca+244U 292112
48Ca+240U 288112
48Ca+238U 286112
48Ca+232U 280112
0.0 0.5 1.0 1.5(R-Rf)/(Rt-Rf)
-10
-5
0
5
E(p
) shel
l(M
eV)
Figure 1: Target deformation effects on the level scheme(upper plot) and on the proton and total shell correctionfor four isotopic reactions in the synthesis of Z=112.
References
[1] R. A. Gherghescu, Phys. Rev. C 67, 014309 (2003).[2] R. A. Gherghescu, D. N. Poenaru and W. Greiner, in
Fission and Properties of Neutron-Rich Nuclei Eds. J. H.Hamilton, A. V. Ramayya and H. K. Carter (World Sci.,Singapore, 2003) p. 177.
[3] R. A. Gherghescu, W. Greiner and G. Munzenberg, Phys.Rev. C 68, 054314 (2003).
- 39 -
An alternative explanation of nuclear fission mass asymmetry
D. N. Poenaru1,2, R. A. Gherghescu1,2, and W. Greiner2
1NIPNE, Bucharest; 2Institut fur Theoretische Physik, Universitat Frankfurt am Main
The asymmetric distribution of fragment masses from thespontaneous or low excitation energy induced fission can’tbe explained within a liquid drop model (LDM). P. Mollerand S. G. Nilsson in 1970 as well as V. V. Pashkevichin 1971 gave the first explanation of the mass asymme-try by showing that the outer fission barrier for reflection-asymmetric shapes is lower than for symmetric ones whenshell corrections are added to the LDM deformation en-ergy. J. A. Maruhn and W. Greiner successfully used in1972 the two center shell model for this purpose.
The shapes during a fission process from one parent nu-cleus to the two final fragments, have been studied eitherstatically (looking for the minimimum of potential energy),or dynamically (by choosing a path with the smallest valueof action integral).
In a static approach, the equilibrium nuclear shapes aredetermined by minimizing the energy functional on a classof trial functions representing the surface equation. Therequired number of independent shape parameters may beas high as nine values. We derived a method allowing toobtain a very general reflection asymmetric saddle pointshape as a solution of an integro-differential equation with-out a shape parametrization apriori introduced [1, 2]. Thismethod [3] allows to obtain straightforwardly the axiallysymmetric surface shape for which the liquid drop energy,ELDM = Es + EC , is minimum. By taking into accountthe shell corrections, δE, it was possible to obtain minimaat a finite value of the mass asymmetry parameter [2].
The deformation energy increases with the mass-asymmetry parameter η = (A1 − A2)/(A1 + A2), as isillustrated in Fig. 1, where η is replaced by an almost lin-ear dependent quantity (dL− dR) expressed in units of R0
— the radius of a spherical nucleus with the same volume.In this way the well known fission fragment mass asymme-try can not be explained. By adding the shell correctionsδE, Edef = ELDM+δE, one can obtain the minima shownin Fig. 2.
-0.05 0.0 0.05 0.1Asymmetry (dL - dR)
0.0
0.5
1.0
1.5
2.0
ES
P-
ES
P0
(MeV
) 228Th
Figure 1: Saddle-point deformation energy versus massasymmetry parameter for the binary fission of 228Thwithin Myers-Swiatecki’s liquid drop model.
-0.1 -0.05 0.0 0.05 0.1Asymmetry (dL - dR)
0.0
0.5
1.0
1.5
2.0
ES
P-
ES
Pm
in(M
eV)
238Th
236Th
234Th
232Th
230Th
228Th
226Th
Figure 2: Saddle-point deformation energy versus massasymmetry parameter for the binary fission of even-massThorium isotopes in the presence of shell corrections. Theminima at a finite value of mass asymmetry are clearlyseen.
The surface equation of an axially symmetric nucleusis a solution of an integro-differential equation with d asan input parameter (dL and dR for the left-hand side andright-hand side of the shape, respectively) which deter-mines the deformation and the mass asymmetry (whendL = dR). We included in the deformation energy E(R) =ELDM (R) + δE(R) − δE0 a phenomenological shell cor-rections δE inspired from [4]. In this way the minima ofdeformation energy illustrated in Fig. 2 for 226−238Th nu-clides are showing up at a finite mass asymmetry.
The saddle point shapes and the corresponding energybarrier heights (Eb) for 170Yb (X = 0.6), 204Pb (X = 0.7),210Po (X = 0.71), 226Th (X = 0.76), and 230U (X = 0.78)are in good agreement with those tabulated By Cohen andSwiatecki in 1963.
The variations of the saddle point energy with the massasymmetry parameter dL−dR (which is almost linear func-tion of the mass asymmetry η) for some even-mass isotopesof Th are plotted in figure 2. The minima of the saddle-point energy occur at nonzero mass asymmetry parame-ters dL−dR between about 0.04 and 0.085 for these nuclei.When the mass number of an isotope increases, the valueof the mass asymmetry corresponding to the minimum ofthe saddle point energy decreases.
References
[1] D.N. Poenaru, W. Greiner, Y. Nagame, R.A. Ghergh-escu, J. Nucl. Radiochem. Sci. Japan, 3 (2002) 43.
[2] D.N. Poenaru, W. Greiner, Europhysics Letters, 64(2003) 164.
[3] V.M. Strutinsky JETF 42 (1962) 1571.[4] W.D. Myers, W.J. Swiatecki, Nucl. Phys.
A81(1966)1.
- 40 -
Analytical expression for cranking inertia of a spheroidal harmonicoscillator
D. N. Poenaru1,2, R. A. Gherghescu1,2, and W. Greiner2
1NIPNE, Bucharest; 2Institut fur Theoretische Physik, Universitat Frankfurt am Main
The hybrid macroscopic-microscopic method can be suc-cessfully applied to study nuclear fission phenomena byadding, in a statical approach, to the liquid drop modeldeformation energy, ELDM , a small shell plus pairing cor-rection, δE = δU + δP , which is obtained from a micro-scopic model by using the Strutinski’s procedure [1]. In or-der to describe the dynamics one needs to know the tensorof inertial coefficients, Bij , which can be computed eitherphenomenologically (within irrotational hydrodynamics [2]or Werner-Wheeler approximation [3]) or microscopically— with the Inglis’s cranking model [1, 4].
In a realistic two center shell model the cranking iner-tia have to be computed numerically. As an intermediatestage, allowing to test complex computer codes one shoulddevelop, we present a simplified case of a spheroidal har-monic oscillator (the simplest Nilsson model) without spin-orbit interaction, for which an analytical result may be ob-tained. There is one deformation parameter expressed asε = 3(c− a)/(2c+ a) in terms of semiaxes a, c.
The eigenvalues [5], in units of hω00 = 41A−1/3 MeV, and
the eigenfunctions [4] of the spheroidal harmonic oscillatorare given by:
εi = [N+3/2+ε(n⊥−2N/3)][1−ε2(1/3+2ε/27)]−1/3 (1)
|nrmnz〉 =√
2α⊥ψmnr
(η)1√2πeimϕ 1
√αzψnz(ξ) (2)
where the quantum numbers nz, nr = 0, 1, 2, ...n⊥, m =n⊥ − 2nr, N = n⊥ + nz . The wave functions ψmnr
(η)and ψnz(ξ) are expressed in terms of associated Laguerrepolynomials and Hermite polynomials, respectively. Theundimensional variables are defined by: η = ρ2/α2
⊥ andξ = z/αz with α⊥ =
√h/(Mω⊥) and αz =
√h/(Mωz).
On the other hand the potential V (η, ξ; ε) = (hω⊥η +hωzξ
2)/2 derivative is given by
1hω0
0
dV
dε=
32
[f1(ε)η + f2(ε)ξ2
](3)
where
f1 =ε(ε+ 6) + 9
[27 − ε2(9 + 2ε)]4/3; f2 = 2
ε(2ε+ 3) − 9
[27 − ε2(9 + 2ε)]4/3(4)
For a single deformation parameter the inertia tensorbecomes a scalar
Bε = 2h2∑νµ
〈ν|∂V/∂ε|µ〉〈µ|∂V/∂ε|ν〉(Eν + Eµ)3
(uνvµ+uµvν)2 (5)
where Eν , vν , uν are the BCS quasiparticle energies andoccupation probabilities for quasiparticles and holes, andthe summation is performed over the whole number ofstates ν, µ around the Fermi level, which are considered in
pairing interactions. The contribution of the neutron levelscheme is added to that of proton levels.
Finally, in order to obtain Bε in units of h2/MeV, onehas to add bε = bε1 + bε2 + bε3 by multiplying each termwith δn′
rnrδm′m9/4 and the result should be divided byhω0
0 .
bε1 =
kf∑ν=ki
[f1(2nr + |m| + 1) + f2(nz + 1/2)]2(uνvν)2
E3ν
(6)
bε2 =f22
2
∑ν =µ
(nz + 1)(nz + 2)(uνvµ + uµvν)2
(Eν + Eµ)3δn′
znz+2 (7)
bε3 =f22
2
∑ν =µ
(nz − 1)nz(uνvµ + uµvν)2
(Eν + Eµ)3δn′
znz−2 (8)
Results for 240Pu are presented in figure 1.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-30
-20
-10
0
10
20
30
40
E(M
eV
)
Epair corr.shell corr. n+pshell corr. n
0
500
1000
1500
2000
2500
3000B
(h2 /M
eV
)
spheroid(irr)sphere(irr)cranking n+pcranking n
Figure 1: Top: comparison of the effective mass (in unitsof h2/MeV) calculated microscopically for the proton andneutron level scheme, only for neutrons, the irrotationalvalue and that of a spherical shape. Bottom: shell andpairing corrections. Nucleus 240Pu.
References
[1] M. Brack, J. Damgaard, A. Jensen, H.C. Pauli, V.M.Strutinsky, G.Y. Wong, Rev. Mod. Phys. 67 (1972) 320.
[2] A. Sobiczewski, E.Ch.A.Ya. 10 (1979) 1170.[3] R.A. Gherghescu, D.N. Poenaru, W. Greiner, Phys. Rev.
C 52 (1995) 2636; Z. Phys. A 354 (1996) 367.[4] J. Damgaard, H.C. Pauli, V.V. Pashkevich, V.M. Strutin-
ski, Nucl. Phys. A 135 (1969) 432.[5] D.N. Poenaru, W. Greiner (Eds) Nuclear Decay Modes,
(IOP Publishing, Bristol, 1996).
- 41 -
Does localization occur in collective nuclear motion?
S. Yan∗ and W. Norenberg
GSI
Large-amplitude collective nuclear motion is frequentlydescribed by a collective Hamiltonian [1], where the massparameter is determined within the cranking approxima-tion while the potential energy is defined by the ground-state expectation value of the many-body Hamiltonian asfunction of the collective variable q. However, due to quasi-crossings of levels the mass parameter exhibits large fluc-tuations as function of q. These fluctuations become dra-matic if the collective motion in excited adiabatic statesis considered. The reason is that the number of quasi-crossings of many-body levels becomes extremely large.We have studied this complexity of collective motion
within the diabatic description [1, 2], where the single-particle wavefunctions are smoothly varying as function ofq. Every diabatic many-body configuration, defined by aSlater determinant (Fock state) of diabatic single-particlestates, exhibits a minimum in the energy as function ofq, which determines the ground state of this diabatic con-figuration. These diabatic many-body states Φj(qj) arelocalized in q and serve as a basis for the description of col-lective motion. The non-orthogonality of the basis statesis not large and would lead to some tails in q-space afterorthogonalization.The local diabatic states are coupled by residual two-
body interactions. The eigenstates are determined by di-agonalization of the total Hamiltonian. These eigenstatesmay turn out to be localized around certain q-values orextended over the whole q-space according to the densityof states. The occurrence of localization would have sig-nificant qualitative and quantitative consequences for ob-servables in large-amplitude collective nuclear motion, e.g.in fusion and fission processes.We have investigated [3] the occurrence of localization
for a schematic diabatic single-particle model with ran-dom two-body interactions [4] between the Fock states.The diabatic single-particle levels are equidistant (distance∆) with constant slopes +a and −a. The magnitude ofthe non-vanishing two-body interactions are chosen froma Gaussian distribution multiplied with a strength factorλ. Within this model the Fock states are localized at adiscrete set of equidistant points qm (distance δ = ∆/a)and are coupled by two-body interactions within a class mand to the neighbouring classes m±1 and m±2. We treatabout 4000 Fock states which cover all states up to about5∆ excitation energy.The degree of localization is described by the correlation
function
R(∆q, E) =〈|〈Φj(qj = qm)| 1
H−E+iη |Φj′(qj′ = qm +∆q)〉|2〉j,j′,mwhich measures the mixing of two local Fock states at adistance ∆q in the same eigenstates with energies aroundE averaged over all Fock states. Since localized eigenstatesspread over only a few neighbouring Fock states, a correla-tion function indicates the occurrence of localization when
it is sharply peaked at ∆q = 0.
0.05
0.1
0.15
0.2
0.25
0.3
0.35
-4 -2 0 2 4R
(∆q,
E)
∆q/δ
λ=0.0036∆λ=0.0054∆λ=0.018∆λ=0.02∆λ=0.04∆λ=0.1∆λ=0.3∆
Figure 1: Correlation function R(∆q, E) as function of ∆qat the excitation energy E = 4∆ for different values of thecoupling strength λ.
Some numerical results are shown in Fig.1 for the cor-relation function R(∆q, E). We observe localization forsmall interaction strengths λ. However, with increasing λthe correlation function broadens continuously, indicatingthat there is a smooth transition to extended eigenstates.From the present study we are not able to perform thethermodynamic limit to macroscopic systems. From a sim-ple scaling estimate we expect that no localization survivesin that limit, and hence there is no indication for Andersonlocalization [5]. On the other hand, Rupp et al. [6] predictlocalization for our hierachies m of states which have cou-plings not depending on the hierarchy. This discrepancymay be due to the differences in the coupling schemes andstatistical assumptions (cf. [4, 6, 7]).
∗) Permanent address: Institute of Low Energy NuclearPhysics, Beijing Normal University, Beijing 100875
References
[1] D.L. Hill and J.A. Wheeler, Phys.Rev. 89 (1953) 1102[2] W. Norenberg and B. Milek, Nucl.Phys. A 545 (1992)
485c; W. Norenberg, in Heavy Ion Reaction Theory,eds. W.Q. Shen, J.Y. Liu and L.X. Ge (World Scien-tific, Singapore, 1989) p.1
[3] S. Yan and W. Norenberg, to be published[4] O. Bohigas and J. Flores, Phys.Lett. 34B (1971) 261;
35B (1971) 383[5] P.W. Anderson, Phys.Rev. 109 (1958) 1492[6] P. Rupp, H.A. Weidenmuller and J. Richert, Phys.Lett.
483B (2000) 331[7] B.L. Altshuler, Y. Gefen, A. Kamenev and L.S. Levi-
tov, Phys.Rev.Lett. 78 (1997) 2803
- 42 -
Isoscalar-isovector instabilities of a hot and dilute nuclear droplet
W. Norenberg1, G. Papp2, and P. Rozmej3
1GSI and TU Darmstadt; 2Eotvos University Budapest; 3University Zielona Gora
197Sm–like droplet
0.0 0.2 0.4 0.6 0.8 1.0ρ/ρ0
0
4
8
12
T (
MeV
)
0.0 0.2 0.4 0.6 0.8 1.00
4
8
12
-3 -2 -1 -0
.5 0
0.5
0.5
1
1
2
2
3
Figure 1: Contour plot (values in MeV) of largest growthrates in the unstable (shaded) region and smallest relax-ation rates in the stable region for a nuclear droplet withZ=62, A=197 near the neutron drip line.
In [1] we have introduced a collective model which allowsto study the bulk and surface modes of a nuclear dropletas function of its density and temperature T . The de-scription is based on the diabatic approach to dissipativecollective motion and – in the local density approximation– yields equations of motion for small amplitudes, wherethe mass and stiffness tensors are obtained analytically.The model is suited to explore systematically characteris-tic properties of hot nuclear droplets as function of theirdensities. Dissipation has been included and examined in[2]. This is essential, because realistic values for the re-laxation times are of the same order as the characteristictimes of collective motion in the region of the liquid-gasphase transition. While the spinodals remain unchangedthe growth rates of unstable modes are reduced from theiradiabatic (i.e. non-dissipative) values by factors 1/2 to 1/4in the region of interest.We have now extended our studies to the inclusion of
isovector modes by introducing two independent displace-ment fields for protons and neutrons, respectively [3]. Thisapproach still allows to obtain analytical expressions forthe mass and stiffness tensors and is applicable not only inthe region of instability, but also to vibrations of pure aswell as coupled isoscalar and isovector modes at arbitrarydensities and temperatures.We have calculated the eigenfrequencies of the coupled
isoscalar and isovector vibrations for a soft equation of
197Fr–like droplet
0.0 0.2 0.4 0.6 0.8 1.0ρ/ρ0
0
4
8
12
T (
MeV
)
0.0 0.2 0.4 0.6 0.8 1.00
4
8
12
-3.5
-3 -2 -1 -0
.5 0
0.5
0.5
1 1
2
3
Figure 2: Contour plot (values in MeV) of largest growthrates in the unstable (shaded) region and smallest relax-ation rates in the stable region for a nuclear droplet withZ=87, A=197 near the proton drip line.
state using the Skyrme energy functional SLy10 [4] whichdoes not give an unphysical isovector instability at largedensities like SkM∗. We have chosen the same values forthe relaxation time as in [2]. The results are summarizedas follows.
– Due to the coupling between isoscalar and isovectormodes the lowest eigenmode is pushed down in energy.
– However, the mixing between these modes is small forneutron-to-proton ratios within the drip lines. This isillustrated in the figures. Only for very asymmetricnuclear matter (e.g. relevant in some astrophysicalproblems) the coupling between isoscalar and isovec-tor modes becomes large [5].
References
[1] W. Norenberg, G. Papp and P. Rozmej, Eur. Phys. J. A9 (2000) 327
[2] W. Norenberg, G. Papp and P. Rozmej, Eur. Phys. J. A14 (2002) 43
[3] W. Norenberg, G. Papp and P. Rozmej, to be submittedto Eur. Phys. J. A
[4] E. Chabanat, P. Bonche, P. Haensel, J. Meyer and R. Scha-effer, Nucl. Phys. A 627, 710 (1997) and E. Chabanat,P. Bonche, P. Haensel, J. Meyer and R. Schaeffer, Nucl.Phys. A 635, 231 (1998).
[5] W. Norenberg, G. Papp and P. Rozmej, to be published
- 43 -
On evolution of the density fluctuation
V. G. Kartavenko, K. A. Gridnev, W. Greiner
Institut fur Theoretische Physik, J. W. Goethe Universitat, Frankfurt am Main
Nuclear cold fragmentation leads to the large density re-distribution and indicates the formation of a binary or pos-sibly multi-center quasistationary nuclear systems, whichlives ∼ 10−21 sec without reaching statistical equilibrium.The existence of a central region of constant density anda well shaped surface region makes it possible to describea clustering in the region of nuclear surface, including theneck region. Nuclear density falls considerably down in theregion of nuclear surface, the fluctuations of the nucleardensity may develop and lead to fragmentation processes.
In this short report, we present one possible way to de-scribe the density fluctuation in the framework of nonlinearnuclear hydrodynamics, The general quantum scheme toanalyze the large amplitude vibrational and vortical modesand their coupling is given [1]. In the semiclassical limitone derives the semi hydrodynamical equations of motion.The initial state with the density fluctuation in the surfaceregion could be simulated via the following simplest way:n(0r; t = 0) ≡ n + n, where n(r, z) is the density distribu-tion of a stationary state. The initial density fluctuationn(r, z, θ) ≡ Λn0 sech2 (r/b) , r ≡
√r2 − 2rz0 cos θ + z20 ,
where r, θ, z are the spherical polar coordinates of a point,z0 controls the distance between centers of the two densitywaves [2]. Our preliminary estimations show that, the ex-isting three dimensional hydrodynamical codes should berebuilt to deal correctly with the gradient terms. Thereforewe consider here to the simplest case: the small time in thebeginning of the evolution of the effective one dimensionalsystem. We selected the initial state with not moving den-sity waves (u(z, 0) = 0). Therefore, in the beginning ofthe evolution, at the first small time step δt, the densitydistribution will be the same, as the initial one, and thecorresponding velocity distribution is proportional to theeffective ”force”
n(z, δt) ≈ n(z, 0). u(z, δt) ≈ d
dz
( 1m
δWδn
)· δt
δWδn
=h2ξ2
2m
(−2
∆nn
+|∇n|2n2
)− 2γ∆n
+ 2A1n+ (2 + α)A7n1+α +
53κn2/3 +
83κβn5/3
where all parameters can be fixed using the integrals ofmotion A = A + A =
∫nd3r, A =
∫nd3r A =
∫nd3r
and the parameters of Skyrme type forces. The initialone dimensional density distribution simulating the threedimensional system with α-particle (A=4) in the surfaceregion of a large nucleus (A ∼ 100) are presented in Fig. 1for z0 = RA and in Fig. 2 for z0 = RA +RA respectively.The dashed curve describes the initial large nucleus n(z, 0),the dotted curve - the density fluctuation n(z, 0). Thesolid line corresponds to the total density. The respec-tive velocity distributions (u(z, δt) are presented (in thesame arbitrary units) in the right sides of these figures,.One can see the complicate flux leading to dispersion of
0 2 4 6 8 10 120,0
0,5
1,0
1,5
2,0
n(z)
/n0
z,fm0 2 4 6 8 10 12
-20
-15
-10
-5
0
5
10
15
20
u(z,
δt),
arb
.un.
z,fm
Figure 1: The initial one dimensional density distributionsimulating the three dimensional system with α-particle(A=4) in the surface region of a large nucleus (A=100) forz0 = RA. The corresponding velocity distribution in thebeginning of evolution (in arbitrary units).
0 2 4 6 8 10 12 140,0
0,5
1,0
n(z)
/n0
z,fm
0 2 4 6 8 10 12 14
-6
-4
-2
0
2
4
6
8
u(z,
δt),
10-1
4 arb.
un.
z,fm
Figure 2: The same as Fig. 1 for the z0 = RA +RA. Onecan see the practically negligible flux in the case of the”touching” distance.
an initial density ”bump”, in the case of large overlappingdensities Fig. 1, and the practically negligible flux at the”touching” distance Fig. 2. These preliminary results canbe considered only as the first estimations. The completethree dimensional calculations are in progress.
References
[1] V.G. Kartavenko, K.A. Gridnev, J. Maruhn andW. Greiner, Phys. Atomic Nucl. 66 (2003) 1439.
[2] K.A. Gridnev, V.G. Kartavenko, S.N. Fadeev andW. Greiner, Nucl. Phys. A 722C (2003) 409.
- 44 -
- 67 -
Chiral symmetry restoration in linear sigma models with differentnumbers of quark flavors
Dirk Roder, Jorg Ruppert and Dirk H. RischkeInstitut fur Theoretische Physik, Universitat Frankfurt a.M., Germany
In this work [1], we use several different chiral models,the O(4), the U(2)r × U(2)2, the U(3)r × U(3)2, and theU(4)r×U(4)2 linear sigma model, to compute the temper-ature dependence of meson masses and quark condensatesacross the chiral phase transition. The meson masses andcondensates are self-consistently calculated in the Hartreeapproximation, which we derived via the CJT formalism[2]. Moreover, we study several distinct patterns of sym-metry breaking within the different models.
We first consider the physically relevant case of explicitsymmetry breaking in the presence of the U(1)A anomaly(Fig.1) and compare the results of the different chiral mod-els in order to clarify how they change with the number ofquark flavors Nf .
Comparing the O(4) model with the U(2)r × U(2)2model, one first notices that the degrees of freedom havedoubled: in addition to the σ meson and the pions whichare already present in the O(4) model, one now has in ad-dition the η meson and the a0 mesons. This has the conse-quence that the meson masses grow more rapidly with tem-perature in the phase where chiral symmetry is restored.The reason for this are the tadpole contributions from theadditional degrees of freedom to the meson self-energies,which lead to an increase in the meson masses. This resultalso applies when adding the strange degree of freedomin the framework of the U(3)r × U(3)2 model. In fact,this picture holds in general, as long as the masses of theadditional degrees of freedom are of the same order of mag-nitude as the chiral phase transition temperature. On theother hand, adding the heavy charm quark degree of free-dom in the framework of the U(4)r×U(4)2 model does notsignificantly influence the results for the masses of the non-charmed mesons and the non-charmed condensates. Thereason is that the additional tadpole contributions fromthe heavy charmed mesons are exponentially suppressedwith the meson mass, ∼ exp(−M/T ). Vice versa, also themasses of the charmed mesons do not change appreciablyfrom their vacuum values over the range of temperaturesof interest for chiral symmetry restoration, simply becausethe tadpole contributions from the non-charmed meson aresmall compared to the large vacuum mass of the charmedmesons. This result is intuitively clear from the physicalpoint of view, but is still non-trivial: first, the equationsfor the in-medium masses are structurally different for theU(4)r × U(4)2 model as compared to the U(3)r × U(3)2model. Second, the set of coupled equations for the massesand condensates is a nonlinear system of equations, whichmeans that small perturbations could lead to large quan-titative changes in the solution.
We then studied the case of explicit chiral symmetrybreaking without U(1)A anomaly. The main difference tothe previous case is that the region of the chiral transitionis narrower and located at a somewhat smaller tempera-ture.
0 50 100 150 200 250 300 350 400T(MeV)
1800
2000
2200
2400
2600
2800
M(M
eV
)
Ds
DpseudoscalarDs0
D0
scalar (b)0
100
200
300
400
500
600
700
800
900
M(M
eV
)
U(4)r U(4)U(3)r U(3)U(2)r U(2)O4mass ofU(4)r U(4)U(3)r U(3)U(2)r U(2)O4mass of
(a)
Figure 1: The meson masses as a function of temperaturefor the different models studied here, for the case withU(1)A anomaly and explicit chiral symmetry breaking.
Finally, we considered the meson masses and quark con-densates in the chiral limit. The Hartree approximationcorrectly predicts the chiral transition to be of first or-der in the U(2)r × U(2)2 model without U(1)A anomalyand in the U(3)r × U(3)2 model. For the O(4) modeland the U(2)r × U(2)2 model with U(1)A anomaly theHartree approximation incorrectly produces a first orderinstead of a second order phase transition. The transitiontemperatures are surprisingly close to the ones obtainedin lattice QCD calculations. However, in the case withU(1)A anomaly the transition temperature increases withthe number of flavors, while in lattice QCD it decreases.This picture changes in the case without U(1)A anomaly,where the transition temperature shows the same behav-ior with the number of quark flavors as in lattice QCD.This may indicate that the U(1)A symmetry is, at leastpartially, restored at and above the chiral phase transitiontemperature.
References
[1] D. Roder, J. Ruppert, and D. H. Rischke, Phys. Rev.D68 (2003) 016003.
[2] J. M. Cornwall, R. Jackiw, and E. Tomboulis, Phys.Rev. D10 (1974) 2428-2445.
- 68 -
Chiral symmetry restorationwith nucleonic degrees of freedom
I.N. Mishustin1,2,3, L.M. Satarov1,2 and W. Greiner1
1Institut fur Theoretische Physik, J.W. Goethe Universitat, D–60054 Frankfurt am Main; 2The Kurchatov Institute,Russian Research Centre, 123182 Moscow, Russia; 3The Niels Bohr Institute,DK–2100 Copenhagen Ø, Denmark
Properties of strongly interacting matter away fromthe nuclear saturation point at ρ = ρ0 = 0.17 fm−3 remainrather uncertain. Relativistic mean–field models of theWalecka type turned out to be successful in describingproperties of medium and heavy nuclei. However, theyhave a serious drawback, namely, they do not respect thechiral symmetry of strong interactions.
Below we use the chirally symmetric Nambu–Jona-Lasinio (NJL) model to describe characteristics of coldisosymmetric nuclear matter in terms of nucleonic degreesof freedom. It was argued [1] that bound nucleonic matterwith spontaneously broken chiral symmetry is not possi-ble within the standard NJL. In fact, this conclusion wasbased on implicit assumption that the maximum (cut–off)momentum for constituent nucleons Λ is the same as nor-mally used in quark models (Λ ∼ 0.6GeV). It was shownin Ref. [2] that by assuming sufficiently low value of thecut–off momentum (Λ 0.3GeV) it is possible to achievea bound state at normal density even in the standard NJLmodel. Similarly to Ref. [1] we apply the generalized NJLmodel which includes an additional scalar–vector (SV) in-teraction. We also take into account explicit symmetrybreaking effects by introducing a bare nucleon mass m0 .
The model Lagrangian is written as
L = ψ (i ∂/−m0)ψ +12(GSA−GV B +GSVAB) , (1)
where ψ is the nucleon field and
A = (ψψ)2 − (ψγ5τψ)2 , B = (ψγµψ)2 + (ψγ5γµψ)2 . (2)
Within the mean–field approximation one obtains the ex-pression for the energy density of cold nuclear matter
ε = − 12π3
Λ∫pF
d3p√m2 + p2+
(m−m0)2
2 (GS +GSV ρ2)
+GV ρ
2
2+ε0 ,
(3)where ρ = 2p3F/3π
2 is the baryon density. The gap equa-tion for the effective nucleon mass m∗ follows from mini-mizing ε(m, ρ) with respect tom . The constant ε0 is foundfrom the requirement ε(mN , 0) = 0 .
By using Eqs. (3) we calculate the energy per baryonε/ρ as function of ρ (for details, see Ref. [3]). The modelparameters Λ, m0 and Gi (i = S, V, SV ) are chosen byfitting the nuclear saturation point (the binding energyB = 16MeV at ρ = ρ0) as well as nucleon mass and piondecay constant in vacuum (at ρ = 0). We also assume thevalue m∗(ρ0) = 0.6mN at which the properties of finitenuclei are well reproduced within relativistic mean–fieldmodels [4]. As a result of this fitting procedure we getthe model parameters Λ = 0.369GeV, m0 = 52.3MeV,GS = 1.105GV = 2.097GeVfm3, GSV = 2.58GeV fm9 .
01
23
45
6–1
0
1
0
1
2
3
ε/ρ
(GeV
)
ρ/ρ0
m(G
eV)
Figure 1: Energy per baryon of cold nuclear matter asa function of nucleon effective mass and baryon density.Thick line shows local minima given by the gap equation.The dot marks a global minimum corresponding to thenuclear saturating point.
With these parameters we obtain the compression mod-ulus of nuclear matter, K(ρ0) = 267MeV, which is in agood agreement with empirical values.
Figure 1 shows the resulting energy per baryon as func-tion of ρ and m . Physical values of m correspond to localminima of ε/ρ at fixed ρ . One can see that these min-ima become rather shallow at high densities. This impliesenhanced fluctuations of chiral condensate (the nucleonscalar density) ρs =< ψψ >. The calculation shows that|ρs| decreases almost linearly with ρ at ρ 2ρ0 . Ap-proximate restoration of chiral symmetry takes place atρ = ρc 2.3ρ0 where the linearly extrapolated ρs(ρ) van-ishes. Nucleon mass fluctuations are especially large atρ ∼ ρc , signalling a chiral phase transition of the crossovertype.
This work has been supported in part by GSI and theRFBR Grants 03-02-04007, NSH-1885.2003.2.
References
[1] V. Koch, T.S. Biro, J.Kunz, and U. Mosel, Phys. Lett.B 185, 1 (1987).
[2] I.N. Mishustin, in Proc. Int. Conf. Nuclear Physics atthe Turn of Millenium (Wilderness, 1996) 499.
[3] I.N. Mishustin, L.M. Satarov, and W. Greiner, Phys.Rep. 391, 363 (2004).
[4] T.J. Burvenich and D.G. Madland, Nucl. Phys.A729, 769 (2003).
- 69 -
Flavor-Mixing Effects on the QCD Phase Diagram at non-vanishingIsospin Chemical Potential: One or Two Phase Transitions?
M. Franka,b, M. Buballab,c, and M. Oerteld
aFachbereich Mathematik, TU Darmstadt; bInstitut fur Kernphysik, TU Darmstadt; cGSI; dIPN-Lyon, France
0
100
200
300
400
500
330 370 410
Mi [
MeV
]
µ [MeV]
α = 0
330 370 410
µ [MeV]
α = 0α = 0.05
330 370 410 450
µ [MeV]
α = 0.11
Figure 1: Constituent quark masses Mu (solid) and Md
(dashed) at T = 0 as functions of quark number chemicalpotential µ for µI = 60 MeV and α = 0 (left), α = 0.05(center), and α = 0.11 (right).
The structure of the QCD phase diagram at nonzerotemperature T and baryon chemical potential µB has beenintensively studied throughout the last decade, as well ontheoretical as on experimental side. A feature of particularinterest is the possible existence of a second-order endpointof the quark-hadron phase boundary, which in principlecould be detected in heavy-ion collisions. Whereas mosttheoretical investigations were restricted to zero isospinchemical potential µI , recently the µB-T phase diagramfor a fixed µI = 0 has been studied using a random ma-trix model [1] and a Nambu–Jona-Lasinio (NJL) model [2].The authors found the striking result that there are twofirst-order phase boundaries and thus two second-orderendpoints.
In the following we show that the existence of two sepa-rate phase transitions becomes unlikely, once flavor-mixingeffects due to instantons are taken into account [3]. Tothis end we consider a model Lagrangian for up and downquarks with two different chirally symmetric interactionparts,
L1 = G1
(qq)2 + (q 0τq)2 + (q iγ5q)2 + (q iγ50τq)2
(1)
and
L2 = G2
(qq)2 − (q 0τq)2 − (q iγ5q)2 + (q iγ50τq)2
, (2)
where G1 and G2 are coupling contants. L1 exhibits anadditional UA(1) symmetry, whereas this is not true forL2, which has the structure of a ’t-Hooft determinant andwhich leads to flavor mixing. To study its effect we writeG1 = (1 − α)G0 and G2 = αG0, and calculate the phasediagram in mean-field approximation for fixed G0 but dif-ferent values of α.
Our results are shown in Figs. 1 and 2. (For numericaldetails, see Ref. [3].) In Fig. 1 we display the values of thedynamically generated constituent quark masses Mu andMd at T = 0 as functions of the quark number chemicalpotential µ ≡ µB/3 for fixed isospin chemical potentialµI ≡ µu − µd = 60 MeV. The left panel corresponds to
0
20
40
60
80
100
300 350 400
T [
MeV
]
µ [MeV]
α = 0
300 350 400
µ [MeV]
α = 0.11
300 350 400
µ [MeV]
α = 0.15
Figure 2: Phase diagrams in the µ-T -plane for µI =60 MeV and α = 0 (left), α = 0.11 (center), and α = 0.15(right). The lines correspond to first-order phase bound-aries which end in second-order endpoints.
α = 0, i.e., to the case where up and down quarks com-pletely decouple. Consequently, we observe two indepen-dent phase transitions for up and down quarks (manifest-ing themselves by discontinuous drops of the constituentmasses) at critical quark number chemical potentials whichjust differ by the chosen value of µI . This is basically theresult reported in Refs. [1, 2]. However, with increasing α,i.e., with increasing flavor mixing, the difference betweenthe critical quark number chemical potentials diminishes(central panel) and eventually, there is only one single first-order phase transition (right panel).
Accordingly, when the analysis is extended to non-vanishing temperature, we find two phase boundaries (andthus two endpoints) for low values of α, but only one phaseboundary for higher values of α. This is illustrated inFig. 2, where the phase diagrams in the µ-T plane for fixedµI = 60 MeV and three different values of α are shown.Note that there is also an intermediate case (central panel),where we find a single phase boundary at low temperatureswhich splits into two lines above T = 25 MeV.
In this example, a rather small amount of flavor mixingis sufficient to remove the existence of the second phaseboundary. More generally, a crude estimate yields thattwo phase boundaries are only present if |µI | αMvac [3],where Mvac is the constituent quark mass in vacuum. In-serting typical values one finds |µI | 75 MeV, whereasin heavy-ion collisions |µI | is considerably smaller (e.g.,µI = −12 MeV for Si + Au collisions at AGS at freeze-out [4]). Thus, the phenomena described in Refs. [1, 2] areunlikely to be seen in heavy-ion collisions.
References
[1] B. Klein et al., Phys. Ref. D 68 (2003) 014009.[2] D. Toublan and J.B. Kogut, Phys. Lett. B 564 (2003)
212.[3] M. Frank, M. Buballa, and M. Oertel, Phys. Lett. B
562 (2003) 221.[4] I. Heppe, diploma thesis, Universitat Heidelberg 1998,
unpublished.
- 70 -
On meson resonances and chiral SU(3) symmetry
M.F.M. Lutz1,2 and E.E. Kolomeitsev3
1GSI, 2TU-Darmstadt, 3NBI
A profound understanding of the nature of meson andbaryon resonances is one of the outstanding issues of QCD.The discovery of meson and baryon resonance states withexotic quantum numbers turns this into a burning ques-tion deserving prime attention. The goal is to establisha systematic approach which is capable of describing andpredicting the resonance spectrum of QCD quantitatively.In this context the authors radical conjecture [1, 2, 3]that meson and baryon resonances that do not belong tothe large-Nc ground state of QCD should be generated interms of coupled-channel dynamics strongly questions thequantitative applicability of the constituent-quark modelto the resonance spectrum. In more explicit terms we ex-pect that the resonance spectrum of QCD in the (u, d, s)-sector should be described in terms of the baryon-octet12
+ and baryon-decuplet 32
+ fields together with the Gold-stone boson 0− and vector meson 1− fields. In the large-Nc and heavy-quark mass limit limit of QCD the latterbaryon fields and also the latter meson fields have degen-erate masses. If this change of paradigm is justified itrequires to provide an explanation of the many meson andbaryon resonances established experimentally.
It is long known that the spectrum of light scalar mesonscan be successfully reproduced in terms of coupled-channeldynamics as first pointed out by Tornqvist [4] and VanBeveren [5] in the 80s. In recent years this idea has beensystematized by applying the chiral Lagrangian (see e.g.[6, 7]). Guided by the heavy-quark mass limit in whichone expects a degeneracy of the 0+ and 1+ spectrum it isnaturally to expect a similar description of the axial-vectormeson spectrum [3].
The spectrum of axial-vector mesons as represented inthe two figures in terms of generalized speed plots [3] is ob-tained by studying the scattering of Goldstone bosons offvector mesons using the leading order chiral Lagrangian.
Since the intermediate vector mesons have in part a sub-stantial decay width we allow for spectral distributions ofthe broadest vector mesons, the ρµ- and Kµ-mesons. Ourresults are parameter-free once we insist on approximatecrossing symmetric scattering amplitudes. We find thatchiral symmetry predicts the existence of the axial-vectormeson resonances (h1(1170), h1(1380), f1(1285), a1(1260),b1(1235), K1(1270), K1(1400)) with a spectrum that issurprisingly close to the empirical one. A result analo-gous to the baryon sector [8, 9] is found: in the heavySU(3) limit with mπ,K,η 500 MeV and mρ,ω,K∗,φ 900MeV the resonance states turn into bound states formingtwo degenerate octets and one singlet of the SU(3) flavorgroup with masses 1360 MeV and 1280 MeV respectively.Taking the light SU(3) limit with mπ,K,η 140 MeV andmρ,ω,K∗,φ 700 MeV we do not observe any bound-statenor resonance signals anymore. Since the leading order in-teraction kernel scales with N−1
c the resonances disappearalso in the large-Nc limit. Using physical masses a patternarises that compares surprisingly well with the empiricalproperties of the JP =1− meson resonances.
References
[1] M.F.M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A700 (2002) 193.
[2] M.F.M. Lutz, GSI-Habil-2002-1.[3] M.F.M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A
730 (2004) 392.[4] N.A Tornqvist, Phys. Rev. Lett. 49 (1982) 624.[5] E. van Beveren et al., Z. Phys. C 30 (1986) 615.[6] J. Nieves, M.P. Valderrama and E. Ruiz Arriola,
Phys. Rev. D 65 (2002) 036002.[7] A.G. Nicola and J.R. Pelaez, Phys. Rev. D 65 (2002)
054009.[8] C. Garcıa-Recio, M.F.M. Lutz and J. Nieves, Phys.
Lett. B 582 (2004) 49.[9] E.E. Kolomeitsev and M.F.M. Lutz, Phys. Lett. B in
print, nucl-th/0305101.
- 71 -
On baryon resonances and chiral symmetry
E.E. Kolomeitsev1 and M.F.M. Lutz2,3
1NBI, 2GSI, 3TU-Darmstadt
We report on further tests of the conjecture [1, 2, 3] thatbaryon resonances not belonging to the large-Nc groundstates may be generated by coupled-channel dynamics. Be-fore the event of the quark-model it was already suggestedby Wyld [4] and also by Dalitz, Wong and Rajasekaran[5] that a t-channel vector meson exchange model for themeson-baryon scattering problem has the potential to dy-namically generate s-wave baryon resonances upon solvinga coupled-channel Schrodinger equation. In a more mod-ern language the t-channel exchange was rediscovered interms of the Weinberg-Tomozawa interaction, the leadingterm of the chiral Lagrangian that reproduces the firstterm of the vector meson exchange in an appropriate Tay-lor expansion. The main difference of the early attemptsfrom computations based on the chiral Lagrangian is theway the coupled-channel scattering equation is regularizedand renormalized. The crucial advance over the last yearsin this field is a significant improvement of the systemat-ics, i.e. how to implement corrections terms into coupled-channel dynamics consistently [1].
0
200
400
0
200
400
1200 1400 1600 18000
2
4
6
8
1200 1400 1600 1800
200
400
600
spee
d [ 1
/mπ ]
π Ξ K Λ K Σ η Ξ
(I,S)=(1/2,-2)
Ξ***(1690)
JP=(1/2)-
π Σ K N η Λ K Ξ
(I,S)=(0,-1)
Λ****(1670)Λ
****(1405)
s1/2 [MeV]
(I,S)=(1,-1)
π Λ π Σ K N η Σ K Ξ
N****(1535)
(I,S)=(1/2, 0) π N η N K Λ K Σ
In the above figure the spectrum of s-wave baryon reso-nances based on the leading order chiral Lagrangian is pre-sented in terms of generalized speed plots. Parameter-freeresult are obtained within the χ−BS(3) approach [1, 3],which insists on the perturbative nature of subthresholdscattering amplitudes and a smooth matching of s- andu-channel unitarized scattering amplitudes. The resultingglued amplitudes satisfy crossing symmetry exactly in thephysical region and too high precision at energies in be-tween the s- and u-channel unitarity branch points. Theexistence of a wealth of s-wave baryon resonances is ex-plained. The latter may be classified in the ’heavy’ SU(3)limit as forming two mass-degenerate octet and one singletstates [6, 7].
In the SU(6) quark-model approach the s-wave reso-nances belong to a 70-plet, that contains many more reso-nance states [8]. An interesting question arises: what is therole played by the d-wave resonances belonging to the verysame 70-plet as the s-wave resonances. The phenomeno-logical model [2] generated successfully also non-strange
d-wave resonances belonging to the 70-plet by coupled-channel dynamics describing a large body of pion and pho-ton scattering data.
Since a d-wave baryon resonance couples to s-wave me-son baryon-decuplet states chiral symmetry is quite predic-tive for such resonances under the assumption that the lat-ter channels are dominant [7]. This is in full analogy to theanalysis of the s-wave resonances [4, 5, 6] that neglects theeffect of the contribution of d-wave meson baryon-decupletstates. The empirical observation that the d-wave reso-nances N(1520), N(1700) and ∆(1700) have large branch-ing fractions (> 50%) into the inelastic Nππ channel,even though the elastic πN channel is favored by phasespace, supports our assumption. A parameter free schemearises since the Weinberg-Tomozawa theorem predicts theleading s-wave interaction strength of Goldstone bosonsnot only with baryon-octet but also with baryon-decupletstates. The resulting spectrum represented in terms of gen-eralized speed plots is shown below [7]. We find that thechiral dynamics predicts the existence of octet and decu-plet bound states in the ’heavy’ SU(3) limit. Those boundstates disappear once the current quark masses of QCDare sufficiently small.
References
[1] M.F.M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A700 (2002) 193.
[2] M.F.M. Lutz, Gy. Wolf and B. Friman, Nucl. Phys.A 706 (2002) 431.
[3] M.F.M. Lutz, GSI-Habil-2002-1.[4] H.W. Wyld, Phys. Rev. 155 (1967) 1649.[5] R.H. Dalitz, T.C. Wong and G. Rajasekaran, Phys.
Rev. 153 (1967) 1617.[6] C. Garcıa-Recio, M.F.M. Lutz and J. Nieves, Phys.
Lett. B 582 (2004) 49.[7] E. E. Kolomeitsev and M.F.M. Lutz, Phys. Lett. B in
print, nucl-th/0305101.[8] C.L. Schat, J.L. Goity and N.N. Scoccola, Phys. Rev.
Lett. 88 (2002) 102002.
- 72 -
On heavy-light meson resonances and chiral symmetry
E.E. Kolomeitsev1 and M.F.M. Lutz2,3
1NBI, 2GSI, 3TU-Darmstadt
Using the chiral SU(3) Lagrangian involving light-heavyJP = 0− and JP = 1− fields that transform non-linearlyunder the chiral SU(3) group a coupled-channel descrip-tion of the meson-meson scattering in the open charm andbeauty sector is developed [1]. We apply the χ-BS(3) ap-proach developed originally for meson-baryon scattering[2] but recently also applied successfully to meson-mesonscattering [3]. The possible importance of coupled-channeldynamics for the heavy-light meson states was empha-sized recently [4]. The major result of our work is theprediction that there exist states with JP =0+, 1+ quan-tum numbers forming anti-triplet and sextet representa-tions of the SU(3) group [1]. This differs from the resultsimplied by the chiral quark model leading to anti-tripletstates only [5, 6]. Our result suggests the existence ofJP =0+, 1+ states with unconventional quantum numbers(I, S) = (1, 1) and (I, S) = (0,−1). The possible existenceof a 0+ state with negative strangeness was also suggestedrecently by Terasaki [7] relying on a constituent diquarkpicture for four-quark states.
0
500
1000
D K D
s η
spee
d [fm
]
s1/2 [MeV]
[3*]
(0,1)D
s(2317)
0
2
4
Ds π
D K
(1,1)[6]
2000 2200 2400 26000
40
80
D π D η D
s K
(1/2,0)
[6]
[3*]× 5
2000 2200 2400 26000
10
D K
(0,-1)[6]
The spectrum of scalar and axial-vector mesons as repre-sented in the two figures in terms of generalized speed plots[3] is obtained by studying the scattering of Goldstonebosons off the open-charm 0− and 1− ground state mesonsusing the leading order chiral Lagrangian. A particular re-sult concerns the ’heavy’ SU(3) limit with mπ,η,K ∼ 500MeV and MD ∼ 1800 MeV in which the chiral coupled-channel dynamics predicts anti-triplet bound states ratherthan resonance states only. In the ’light’ SU(3) limit withmπ,η,K ∼ 140 MeV and MD ∼ 1800 MeV we do not findanymore resonances or bound states in the JP = 0+, 1+
sectors. Using physical mass parameters we predict nar-row JP =0+, 1+ states in the (I, S) = (0, 1), (1
2 , 0), (0,−1)channels with open charm as illustrated by the two figures.
We obtain a 0+ bound state of mass 2303 MeV in the(0, 1)-sector. This state should be identified with a nar-row resonance of mass 2317 MeV recently observed by theBABAR collaboration [8]. Since we do not consider isospin
violating processes like η → π0 the latter state is a truebound state in our present scheme. Similarly we generatea 1+ bound state in the (0, 1)-sector of mass 2440 MeV.Thus the mass splitting of the 1+ and 0+ states in thischannel agrees extremely well with the empirical value ofabout 140 MeV measured by the BABAR and CLEO col-laborations [8, 9].
Contrary to the open beauty sector in which already theleading order computation [1] predicts weakly bound K Bisospin-zero states the binding in the K D system is notquite sufficiently strong to form a bound state at leadingorder. However, chiral correction terms provide additionalattraction to form open-charm bound states with negativestrangeness [1, 10] if the D(2420) resonance is identified asa member of a SU(3)-sextet.
0
500
1000
Ds(2463)
D* K D
s* η
s1/2 [MeV]
spee
d [fm
]
(0,1)
[3*]
0
4
8
Ds* π
D* K(1,1)
[6]
2200 2400 2600 28000
75
150
D(2420)
D* π D* η D
s* K
[6]
[3*]
(1/2,0)
[6]
× 5
2200 2400 2600 28000
10
D* K(0,-1)
References
[1] E. E. Kolomeitsev and M.F.M. Lutz, Phys. Lett. B582 (2004) 39.
[2] M.F.M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A700 (2002) 193.
[3] M.F.M. Lutz and E. E. Kolomeitsev, Nucl. Phys. A730 (2004) 392.
[4] E. van Beveren and G. Rupp, hep-ph/0305035; help-ph/0306051.
[5] M.A. Nowak, M. Rho and I. Zahed, Phys. Rev. D 48(1993) 4370
[6] W.A. Bardeen and C.T. Hill, Phys. Rev. D 49 (1994)409.
[7] K. Terasaki, Phys. Rev. D 68 (2003) 011501.[8] BABAR Collaboration, B. Aubert et al., Phys. Rev.
Lett. 90 (2003) 242001.[9] CLEO Collaboration, D. Besson et al., hep-
ex/0305017, Phys. Rev. D 68 (2003) 032002.[10] J. Hofmann and M.F.M. Lutz, Nucl. Phys. A in print.
- 73 -
Exclusive φ production in proton-proton collisions in the resonancemodel
C. Fuchs, Amand Faessler, M.I. Krivoruchenko, B.V. MartemyanovInstitut fur Theoretische Physik, Universitat Tubingen
The production of φ mesons is strongly suppressed comparedto that of ω mesons. This fact is known under the name “OZIrule”. According to the OZI rule φ mesons can only be pro-duced due to a small admixture of non-strange light quarks intheir wave function. The corresponding mixing angle θmix isequal to θmix ≈ 3.7o. A naive estimate based on the OZI ex-pectation underestimates the measured ratio φ and ω mesonscross sections should at comparable energies by about one orderof magnitude.
In ref. [2] we determined the φ production in elementarynucleon-nucleon reactions witjin the framework of the reso-nance model. Their the φ production is described by a two stepmechanism, i.e. the excitaton of nucleon resonances and theirsubsequent decay pp → pR → ppφ. The cross sections for theresonance production were taken from ref.[3] where they werefitted to describe π, η, ρ, ππ production in NN collisions. TheφNR coupling is obtained from the known ωNR coupling ofthe ω meson [4] and the mixing angle between φ and ω mesons.The ωNR couplings, in turn, have been determined within theframework of the extended Vector Meson Dominance (eVMD)model by fitting the available data on electro- and photo- pro-duction of nucleon resonances as well as their mesonic decays[5]. The description of the φ and ω meson production in ele-mentary nucleon-nucleon reactions is then essentially parame-ter free since all model parameters have already been fixed byother sources. In [4] it was demonstrated that available data onthe exclusive ω production in pp reactions are very accuratelyreproduced by the present model over a wide energy range,i.e. from extremely close to threshold up to several GeV abovethreshold.
The pp→ pR → ppφ cross section is given as follows:
σ(s)pp→ppφ =∑
R
∫ (√
s−2mp)2
0
dM2
∫ (√
s−mp)2
(mp+M)2
dµ2
× dσ(s, µ)pp→pR
dµ2
dB(µ,M)R→pφ
dM2, (1)
with M being the running mass of φ meson. The cross sectionsfor the baryon resonances production are given by
dσ(s, µ)pp→pR =|MR|2
16pi√sπ2
Φ2(√s, µ,mp)dWR(µ) (2)
with Φ2(√s, µ,mp) = πp∗(
√s, µ,mp)/
√s being the two-body
phase space, p∗(√s, µ,mp) the final c.m. momentum, pi the
initial c.m. momentum. µ and mR are the running and polemasses of the resonances, respectively, mp the proton mass.The mass distribution dWR(µ) of the resonances is describedby the standard Breit-Wigner formula:
dWR(µ) =1
π
µΓR(µ)dµ2
(µ2 −m2R)2 + (µΓR(µ))2
. (3)
The sum in (1) runs over the same same set of nucleon res-onances which is responsible for the ω meson production [4].This includes all well established (4∗) N∗ resonances quotedby the PDG [?] with masses below 2 GeV. The branching tothe φ decay mode is given by
dB(µ,M)R→pφ =tan2 θmixΓR
Nω(µ,M)
ΓR(µ)dWφ(M) , (4)
with ΓRNω(µ,M) calculated the same way as in the case of ω
meson production[4].
0.01 0.1 1
Q [GeV]
1e-05
0.0001
0.001
0.01
φ/ω
Figure 1: The ratio of the σ(pp→ ppφ) over σ(pp→ ppω)cross sections as a function of the c.m. energy above thethreshold Q =
√s− 2mp −mφ (solid line) is compared to
experimental data. The dashed line corresponds to ideal-ized case of stable φ and ω mesons, i.e. the zero widthlimit.
The results for cross section σ(pp → ppφ) are presented inFig. 1 in form of the ratio of the φ over ω meson produc-tion cross sections. The ω mesons production cross section wasshown to agree well with experimental data if one uses a strongN∗(1535)Nω coupling [4] The dashed line in Fig. 1 correspondsto the idealized case of stable φ and ω mesons, i.e. to the limitof zero widths.
Important properties of the ΓRNω(µ,M) width are the follow-
ing ones: the M dependence of magnetic, electric and Coulombcouplings entering into the amplitude and the Blatt-Weisskopfsuppression factor which suppresses the width for large off-shellmasses µ of the resonances [4]. Their combined effects lead toan increase of the ΓR
Nω(µ,M) width with M increasing frommω to mφ for µ > 2GeV . This effect is finally responsible forthe violation of the naive OZI rule estimate.
References
[1] DISTO Collaboration, F. Balestra et al., Phys. ReV. C63, 024004 (2001).
[2] Amand Faessler, C. Fuchs, M.I. Krivoruchenko, B.V.Martemyanov, Phys. Rev. C 68, 068201 (2003).
[3] S. Teis, W. Cassing, M. Effenberger, A. Hombach, U.Mosel, and Gy. Wolf, Z. Phys. A356, 421 (1997).
[4] C. Fuchs, M.I. Krivoruchenko, H.L. Yadav, AmandFaessler, B.V. Martemyanov and K. Shekhter., Phys. Rev.C 67, 025202 (2003).
[5] M. I. Krivoruchenko, B. V. Martemyanov, A. Faessler andC. Fuchs, Annals Phys. (N.Y.) 296, 299 (2002).
- 74 -
Simulation of Multiple Interactions in High-Energy Hadron Collisions
S. Hochea, T. Gleisberga, F. Kraussa, A. Schalickea, S. Schumanna, J. Wintera, and G. Soffa
aInstitut fur theoretische Physik, Technische Universitat Dresden
High-energy particle collisions at hadron colliders suchas the Tevatron (FNAL) and the future Large Hadron Col-lider (CERN) provide an excellent opportunity to studythe dynamics of the standard model and allow for thesearch for new physics. In order to describe the QCD back-ground in high-p⊥ events at these colliders correctly it isnecessary to consider the possibility of multiple parton-parton scattering occuring when two protons collide. Amodel to describe such phenomena and some first applica-tions are subject of the current presentation.
One major problem in hadronic collisions is to take thecomposite nature of the beam particles into account. Toallow for a perturbative description within the scope of thestandard model, hadrons have to be decomposed into ele-mentary particles, namely quarks and gluons. This makesit possible to have several parton-parton interactions inone hadronic event. It has been shown [1] that especiallyfor large-p⊥ measurements the production of minijets as aresult of such multiple parton scattering can lead to ap-preciable effects on the overall charged particle multiplicityand the event shape. With the shift of the center of massenergy from values around
√s = 540 GeV for pp reactions
at the SppS collider at CERN to√s = 1.8 TeV at the
Tevatron these effects have become more pronounced. Infact, at future hadron colliders – the next one being LHC atCERN – a deeper understanding of the dynamics underly-ing the then copious production of minijets will be crucial.Due to the complicated structure of high-energy collisions,resulting in hundreds of particles hitting the detector, it isvirtually impossible to rely on analytical methods alone inorder to model such effects. Instead, one usually relies oncomputer programmes, also known as Monte-Carlo eventgenerators.
Over the last three months a formalism for the Monte-Carlo generation of multiple parton-parton interactions ac-cording to [2] has been incorporated into the event genera-tor Sherpa [3], which has been developed over the last fewyears at the Dresden University of Technology. Followingthe new programming paradigm in high-energy physics,this programme is written from scratch in the modern,object-oriented language C++. Its modular structure al-lows for easy additions, like the one reported here. Withinthe model named above multiple parton interactions aregenerated according to the differential cross section forleading order perturbative 2 → 2 QCD processes
f(p⊥) =1
σp⊥≥p⊥minhard
dσhard(p⊥)dp⊥
.
This results in the total probability
p(p⊥,i) = f(p⊥,i) exp
−
∫ p⊥,i−1
p⊥,i
dp′⊥ f(p′⊥)
to have the i’th parton-parton interaction at p⊥,i withthe i-1’st being at p⊥,i−1. The differential cross section
dσhard(p⊥) results as usual as a sum over convolutions ofhard scattering matrix elements with the correspondingparton distribution functions
dσhard(p⊥)dp⊥
=∑i,j,k,l
∫dx1
∫dx2
× fi(x1, Q2)fj(x2, Q
2)dσij→kl(p⊥)
dp⊥(p⊥) .
Figure 1 shows the multiplicity distribution of hard par-tonic subprocesses obtained with the current model fordifferent values of the cms energy and the lower p⊥ cutoff.
hardN2 4 6 8 10 12 14 16 18 20 22
har
dd
Nσ
d
to
t σ
1
-410
-310
-210
-110
1
SHERPA
SpSpCDFLHC
Figure 1: SppS:√s = 540 GeV, p⊥min = 1.6 GeV;
Tevatron:√s = 1.8 TeV, p⊥min = 1.9 GeV; LHC:
√s =
14 TeV, p⊥min = 2.6 GeV
This procedure still lacks a physically significant crite-rion for the regularization at low p⊥ where the leading or-der differential cross section dσ(p⊥) diverges like dp2⊥/p
4⊥
and nonperturbative effects set in. Thus in the futurehigher twist effects will be studied to find such criteria.Furthermore we plan to incorporate low-p⊥ scattering ofpartons by the exchange of reggeized gluons. An interfaceto the parton shower module APACIC++ which is includedin the event generation framework of Sherpa has also beenestablished, allowing for the subsequent showering of allpartons generated according to the formalism describedabove.
References
[1] N. Paver and D. Treleani, Nuovo Cim. A 73 (1983)392.
[2] T. Sjostrand and M. van Zijl, Phys. Rev. D 36 (1987)2019.
[3] T. Gleisberg, et al., arXiv:hep-ph/0311263.[4] T. Sjostrand, L. Lonnblad, S. Mrenna and P. Skands,
arXiv:hep-ph/0308153.
- 75 -
Moving rho-mesons at finite temperature
Robert Wurfel, Stefan Leupold, Ulrich Mosel
Institut fur Theoretische Physik, Universitat Giessen
An exciting aspect of hadron physics is the question howa hadron changes its properties once it is put in a stronglyinteracting environment. Here, the vector mesons deservespecial attention as they couple directly to (virtual) pho-tons. The latter can decay into dileptons which leave thestrongly interacting system untouched. Via that processinformation about possible in-medium modifications of thevector mesons can be carried to the detectors.
In the following we shall describe how ρ-mesons changetheir properties at finite temperature. For not too hightemperatures such a system consists mainly of pions as theby far lightest hadronic states. ρ-mesons are resonances,i.e. they can be observed in scattering events, e.g. in pion-pion scattering. Consequently we will explore pion-pionscattering in the presence of a thermal gas of pions.
At low energies pion-pion scattering is described by chi-ral perturbation theory (χPT). The scattering amplitudecalculated in χPT becomes invalid in the resonance re-gion. The resonances cause poles in the scattering ampli-tudes which terminates the region of convergence of χPT.In contrast, if one calculates the inverse scattering ampli-tude the poles turn to zeros. Here one can expect that theradius of convergence is larger. This approach — calledinverse amplitude method (IAM) — has been shown towork rather well in the vacuum [1]. The ρ-meson couldbe reconstructed among several other resonances from theχPT expressions for the inverse scattering amplitude.
Consequently this approach has been generalized to fi-nite temperature. In [2] the χPT expressions for pion-pionscattering were developed. In [3] the in-medium proper-ties of a ρ-meson which is at rest with respect to the pionicheat bath were determined within the IAM approach.
Here we take up the formalism developed in [2, 3] andgeneralize it to the situation of a ρ-meson which moveswith respect to the heat bath. Recall that the ρ-meson isreconstructed form pion-pion scattering. Hence we haveto study scattering events were the center-of-mass frameof the reaction moves with respect to the heat bath frame.In general, such a situation is characterized by five inde-pendent variables (in vacuum there are only two): Onecan choose e.g. the Mandelstam variables s and t and inaddition the relative velocity of the reaction plane withrespect to the heat bath and the two relative angles be-tween this velocity and the incoming and outgoing scat-tering lines, respectively. To extract the ρ-meson from ascattering amplitude one has to project on spin 1. Thisonly makes sense if the angular momentum is conserved inthe reaction. This is not fulfilled for the general kinemati-cal situation described above. It is fulfilled, however, if thereaction plane of the scattering pions (which form the ρ-meson) is perpendicular with respect to the boost vectorwhich connects the heat bath frame with the center-of-mass frame of the reaction. In this case rotations withinthe reaction plane do not change the physical situation:The lines of incoming and outgoing pions, respectively, re-
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]
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Figure 1: Mass (top) and width (bottom) of the ρ-mesonas functions of the temperature for various momenta of theρ-meson relative to the heat bath.
main perpendicular to the boost vector.The situation is now characterized by the temperature
T of the heat bath and the momentum Kρ of the ρ-meson(or the scattering system) with respect to the heat bath.In Fig. 1 we show how the properties of the ρ-meson (massand width) depend on T andKρ. We observe that the massfirst rises with increasing temperature before it drops downrapidly. The mass increases with growing momentum. Thewidth strongly rises with the temperature. This tendencyis to some degree diminished if the momentum grows.
The consequences of this intricate behavior for dileptonyields remains to be seen.
References
[1] A. Gomez Nicola and J. R. Pelaez, Phys. Rev. D 65,054009 (2002) [arXiv:hep-ph/0109056].
[2] A. Gomez Nicola, F. J. Llanes-Estrada andJ. R. Pelaez, Phys. Lett. B 550, 55 (2002) [arXiv:hep-ph/0203134].
[3] A. Dobado, A. Gomez Nicola, F. J. Llanes-Estradaand J. R. Pelaez, Phys. Rev. C 66, 055201 (2002)[arXiv:hep-ph/0206238].
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Selfconsistent description of vector-mesons in matter
Felix Riek1 and Jorn Knoll1
1GSI
The behaviour of hadrons in a dense hadronic medium isan interesting question. Ever since the early experimentson dilepton rates [1] which observed an enhanced dileptonyield below the masses of the light vector mesons comparedto simple extrapolations from e.g. proton proton scatter-ing, much work has been devoted to study the properties ofthe ρ-meson in matter, e.g. [2]. We would like to addressthe problem of the in medium behaviour of the ω-mesontaking medium modified pions into account [3]. The cal-culations were done using approximations according to theΦ-derivable approach in order to generate a set of coupledDyson equations which we then solve numerically for fi-nite energy and momentum by iteration. Our equilibriummodel consists of two parts. In the first part we calculatethe self-consistent spectral function of the pion in spin aswell as iso-spin symmetric nuclear matter for finite densityand temperature taking the coupling to the nucleon and∆(1232)-resonance into account. In addition we make useof Migdals short range correlations in order to avoid pioncondensation at normal nuclear density. In the second partwe use the so obtained pion spectral function to determinethe effects on the vector mesons. Here we include selfcon-sistently the processes ω → ρπ and ρ → ωπ which leadto a mixing of both mesons as well as the decay ρ → ππwhich is the dominating decay process for the ρ-meson.The four-transversality of the selfenergies is normally vio-lated by the use of Dyson equations. We circumvent thisproblem which would otherwise lead to the propagationof unphysical degrees of freedom by a projector technique.Concentrating on the medium changes of the width we didnot calculate the real parts of the meson selfenergies andset all in medium masses to their corresponding vacuumvalues. With the so obtained spectral functions for bothmesons we can make use of the vector dominance concept[4] to calculate the emission rate of e+e−-pairs. To isolatethe effect of the different in medium components of thepion we decomposed the total damping width of the ω-meson into partial widths Γω,tot = ΣiΓω,i which all resultfrom a different part of the pion spectra. We then broughtthe dilepton yield into a Breit-Wigner like form
dR
d4qd3xdt=
3(2π)4
nT (q0)∑i
4 q20Γω,i Γω,e+e−
(q2 −m2ω)2 + q20Γ
2ω,tot
(1)
(Hereby the tensor structure of the selfenergies is sup-pressed, nT (q0) is the Bose Occupation number andΓω,e+e− is the partial decay width ω → e+e−.). In Fig. 1we show the result for the decay of the ω-meson at T=120MeV and normal nuclear density. We observe that thescattering process ρN → ωN (S), which is mediatedby a space-like pion, contributes with equal strength com-pared with the fusion (F) of a ρ-meson and a quasi-realin medium pion - the inverse of the fusion determines thevacuum width of the ρ-meson - and the ρ-Dalitz decay (D)which dominates the low energy region. This shows thatthe width of the ω-meson is sensitive to the low energy
1e-13
1e-12
1e-11
1e-10
1e-09
1e-08
200 400 600 800
Rat
e
inv. mass [MeV]
SumFDS
Figure 1: e+e−-rate from the decay of the ω-meson atT=120 MeV, ρ = ρ0 divided into different contributions:F: πρ→ ω , D: ρ→ πω , S: ρN → ωN .
components of the pion spectral function caused by nu-cleon nucleon-hole excitations. In Fig. 2 we summariseour results for the width of the ω-meson at finite densityand temperature (only the mean values of ΓL and ΓT areshown). We observe a nonlinear dependence on the den-
020406080
100120140160180200
0 30 60 90 120
0 0.5 1 1.5 2 2.5
Wid
th [M
eV]
Temperature [MeV]
ρ/ρ0
Figure 2: Mean value Γω at T=30 MeV depending on thedensity (full line) and Γω at ρ = ρ0 depending on thetemperature (dashed line).
sity resulting from the self-consistent treatment. In addi-tion temperature leads to a strong increase of the widthlarger than predicted by Schneider and Weise using freepion states [5].
References
[1] J.P. Wessels et al., Nuc. Phys. A715 (2003) 262c[2] R.Rapp and J. Wambach, Adv. Nuc. Phys. 25
(2000) 1[3] F. Riek and J. Knoll, nucl-th 0402090[4] J.J. Sakurai, Currents and mesons, Chicago Uni-
versity Press 1969[5] R.A. Schneider and W. Weise, Phys. Lett. B515
(2001) 89
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Self-Consistent Approach to Off-Shell Transport
Yu.B. Ivanov1,2, J. Knoll1, and D.N. Voskresensky1,3
1GSI, Darmstadt; 2Kurchatov Institute, Moscow; 3Moscow Institute for Physics and Engineering, Moscow
The interest in transport beyond the quasiparticle ap-proximation recently has been revived in connection withstudies of transport properties of broad resonances [1] inheavy-ion collisions, as well as rapid thermalization atRHIC energies [2] which cannot be explained by binary on-shell parton scattering. Transport approaches for treatingsuch off-shell dynamics were proposed in refs. [1, 3, 4, 5, 6],based on the Kadanoff–Baym equations expanded up tothe first space–time gradients. Presently two slightlydifferent forms of the gradient-expanded Kadanoff–Baymequations are used: the original Kadanoff–Baym (KB)form as it follows right after the gradient expansion with-out any further approximations, and the Botermans–Malfliet (BM) one [7], which is derived from the KB formby omitting certain second-order space-time gradient cor-rections. In this work we compare these two forms of“quantum” kinetic equations and discuss their advantagesand disadvantages from the point of view of their conserv-ing properties, the possibility of numerical realization, etc.The details of this discussion can be found in [8]. Here wejust summarize the present status of the two consideredapproaches to off-shell transport.
From a consistency point of view, the BM-choice looksmore appealing, since it preserves the exact identity be-tween the kinetic and mass-shell equation, a property in-herent in the original KB equations [3]. For the KB-choicethis identity between the kinetic and mass-shell equation isonly approximately preserved, namely within the validityrange of the first-order gradient approximation. However,this disadvantage is not of great practical use, since onlythe kinetic equation is used in actual calculations.
Conservation laws related to global symmetries solelydepend on the properties of the first order gradient termsin the kinetic equation. In this respect the KB kineticequation has a conceptual advantage as it leads to exact[4] rather than approximate conservation laws, providedthe scheme is based on Φ-derivable approximations. Orig-inally derived for local interactions [4], we extended thisstatement also to systems with derivative couplings andwith interactions of finite range, like a non-relativistic po-tential. The reason for this property is that the KB kineticequation preserves certain contour symmetries among thevarious gradient terms, while they are violated for theBM-choice. Of course, within their range of applica-bility these two approaches are equivalent, because theBM kinetic equation conserves the charge and energy–momentum within the theoretical accuracy of the gradientapproximation. Still, the fact that the KB-choice possesexact conservation laws put this version to the level of ageneric equation, much like the Boltzmann or hydrody-namic equations, to be used as phenomenological dynam-ical equations for practical applications. Such conservingdynamical schemes may be useful even beyond their appli-cability range. E.g., such a situation happens at the initialstage of heavy-ion collisions. As the conservations are ex-
act, we can still use the gradient approximation, relying ona minor role of this rather short initial stage in the totalevolution of a system.
Although the KB kinetic equations posses exactly con-served Noether currents, a practical numerical approach(e.g., by a test-particle method) for its solution has notyet been established. The obstacle is the special Poisson-bracket term in the KB kinetic equation,
Γin,ReGR
,
where Γin and ReGR are the gain part of the collisionterm and the real part of the retarded Green function, re-spectively. This term lacks proper interpretation since thephase-space occupation function F (X, p) enters only indi-rectly through the gain-rate gradient terms. This problem,of course, does not exclude solution of the KB kinetic equa-tion, e.g within well adapted lattice methods [2], which are,however, much more complicated and time-consuming ascompared to the test-particle approach. For the BM ki-netic equation, on the other hand, an efficient test-particlemethod is already available [5, 6], for the price that it dealswith an alternative rather than Noether current.
An important feature of kinetic descriptions is the ap-proach to thermal equilibrium during evolution of a closedsystem. A sufficient (while not necessary!) condition isprovided by an H-theorem. As was demonstrated in ref.[3], at least, within simplest Φ-derivable approximationsfor the BM kinetic equation an H-theorem indeed can beproven. The so derived kinetic entropy merged the equilib-rium expression [3]. For the KB kinetic equation the resultis by far weaker. Here we were able to prove the H-theoremonly within simplest Φ-derivable approximations and for asystem very close to almost spatially homogeneous thermallocal equilibrium or stationary state. These results, in gen-eral, do not imply that the system does not approach equi-librium but suggests that equilibration should be tested inactual calculations.
References
[1] M. Effenberger and U. Mosel, Phys. Rev. C60 (1999),51901.
[2] S. Juchem, W. Cassing, and C. Greiner, nucl-th/0401046.
[3] Yu.B. Ivanov, J. Knoll, and D.N. Voskresensky, Nucl.Phys. A657 (1999) 413; A672 (2000), 313.
[4] J. Knoll, Yu.B. Ivanov, and D.N. Voskresensky, Ann.Phys. (NY) 293 (2001), 126.
[5] W. Cassing and S. Juchem, Nucl. Phys. A 665 (2000),377; 672 (2000), 417.
[6] S. Leupold, Nucl. Phys. A 672 (2000), 475.
[7] W. Botermans and R. Malfliet, Phys. Rep. 198(1990), 115.
[8] Yu.B. Ivanov, J. Knoll, and D.N. Voskresensky, Phys.Atom. Nucl. 66 (2003), 1902.
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Nonequilibrium Dynamics and Off-Shell Transport of RelativisticQuantum Fields
S. Juchem1, W. Cassing1, C. Greiner2
1University of Giessen, 2 University of FrankfurtThe dynamics of quantum many-body systems are a
challenging task for far-from-equilibrium conditions as ap-pearing in high energy heavy-ion collisions or cosmologicalproblems. It is of fundamental importance to describe ab-initio thermalization by taking into account all quantumaspects of the particles (or fields) rather than invokingsemi-classical approximation schemes – as inherent in theBoltzmann limit – from the beginning [1, 2] and to developtractable approximation schemes that retain the essentialquantum dynamics.
We here consider the equilibration of the scalar φ4-theory in 2+1 space-time dimensions. The evolution equa-tions for the Green function iG<(x, y) =< φ(y)φ(x) >are derived self-consistently by a loop expansion of thetwo-particle irreducible (2PI) action [3]. When consider-ing contributions up to the three-loop order one obtains aKadanoff-Baym equation incorporating two types of self-energies: a) the tadpole diagram that produces a meanfield while b) the sunset diagram contains the influenceof scattering processes and leads to a non-locality in time(memory integrals). We recall that equilibration requiresthe inclusion of collisions as inherent in the sunset diagram[2]. Due to the numerical expense we here restrict to ho-mogeneous systems in space and solve the Kadanoff-Baym(KB) equations in momentum space.
As an example for the quantum equilibration in 2+1space-time dimensions we show in the upper part of Fig.1 the time evolution of the occupation numbers for sev-eral momentum modes within the full KB-theory [3]. Forthe initial distribution D1 we have adopted two particleaccummulations separated on the px-axis. The thermal-ization within the KB-theory is characterized by three dif-ferent time scales given by i) an exponential damping ofthe initial oscillations, followed by ii) kinetic equilibrationand for large times by iii) chemical equilibration. In thelast phase the chemical potential µ decreases by 1 ↔ 3off-shell processes which become possible in case of broadspectral functions A. As demonstrated in Ref. [3] theasymptotic off-shell distribution function N determinedfrom iG<(0p, p0, t) = N(0p, p0, t)A(0p, p0, t) has the shapeof a Bose-function for all (off-shell) energies p0 with theproper chemical potential µ = 0. Furthermore, the quan-tum evolution of the occupation numbers shows an ’over-shooting’ for the low momentum modes at times tm ∼ 10,which does not occur in the particle number conservingBoltzmann limit (lower part of Fig. 1). In addition, thequasi-particle Boltzmann limit does not yield the properequilibrium distribution since the chemical potential µ re-mains finite in the respective equilibrium state [3] .
In order to overcome the difficulties of the quasi-particleBoltzmann limit we have formulated an off-shell transporttheory [4] that incorporates the evolution in the spectralfunction A of the field quanta. The numerical results ofthis off-shell theory (for the initial distribution D1) are
displayed in the middle part of Fig. 1 (KB-transport) andalso show the characteristic ’overshooting’ in the occupa-tion numbers of the low momentum modes. Moreover,the width and the pole position of the spectral functionsA(p0, t) are found to agree for all times tm > 5 with thosefrom the full KB-theory [5]. Also the approach to theequilibrium distribution is described properly due to theinclusion of off-shell 1 ↔ 3 processes in the KB-transportlimit. Thus the off-shell transport theory [4] provides asuitable approximation to the full quantum dynamics thatcan be solved conveniently within an extended testparticleansatz [4].
Figure 1: Time evolution of the occupation numbers of lowmomentum modes within the full Kadanoff-Baym theory[3] (upper part) in comparison to the results from the off-shell transport theory [5] (middle part) and the quasipar-ticle Boltzmann approximation [3] (lower part).
References
[1] P. Danielewicz, Annals of Physics 152 (1984) 305.[2] J. Berges, Nucl. Phys. A 699 (2002) 847.[3] S. Juchem, W. Cassing, C. Greiner, Phys. Rev. D69
(2004) 025006.[4] W. Cassing, S. Juchem, Nucl. Phys. A665 (2000) 377;
A672 (2000) 417.[5] S. Juchem, W. Cassing, C. Greiner, nucl-th/0401046.
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Causal Theories of Dissipative Relativistic Fluid Dynamics for NuclearCollisions
Azwinndini MurongaInstitut fur Theoretische Physik, J.W. Goethe–Universitat, 60325 Frankfurt am Main, Germany
In the early stages of heavy ion collisions, non–equilibrium effects play a dominant role. A com-plete description of the dynamics of heavy ion reactionsneeds to include the effects of dissipation through non–equilibrium/dissipative fluid dynamics. As is well–known[1], extended theories (which are hyperbolic and causal) ofdissipative fluids due to Grad [2], Muller [3], and Israel andStewart [4] were introduced to remedy some undesirablefeatures such as acausality. It seems appropriate thereforeto resort to hyperbolic theories instead of parabolic theo-ries (due to Eckart [5] and to Landau and Lifshitz [6]) indescribing the dynamics of heavy ion collisions.
The parabolic theories of dissipative fluid dynamics arebased on the assumption that the entropy four–currentcontains terms up to linear order in dissipative quantitiesand hence they are referred to as first order theories ofdissipative fluids. The resulting equations for the dissi-pative fluxes are linearly related to the thermodynamicforces, and the resulting equations of motion are parabolicin structure, from which we get the Fourier–Navier–Stokesequations. They have the undesirable feature that causal-ity may not be satisfied. That is, they may propagateviscous and thermal signals with speeds exceeding that oflight.
The causal theories are based on the assumption that theentropy four–current should include terms quadratic in thedissipative fluxes and hence they are referred to as secondorder theories of dissipative fluids. The resulting equationsfor the dissipative fluxes are hyperbolic and they lead tocausal propagation of signals. In second order theories thespace of thermodynamic quantities is expanded to includethe dissipative quantities for the particular system underconsideration. These dissipative quantities are treated asthermodynamic variables in their own right. That is, non–equilibrium effects are introduced by enlarging the spaceof basic independent variables through the introduction ofnon–equilibrium variables, such as dissipative fluxes ap-pearing in the conservation equations. The next step is tofind evolution equations for these extra variables. Whereasthe evolution equations for the equilibrium variables aregiven by the usual conservation laws, no general criteriaexist concerning the evolution equations of the dissipativefluxes, with the exception of the restriction imposed onthem by the second law of thermodynamics.
A natural way to obtain the evolution equations for thefluxes from a macroscopic basis is to generalize the equi-librium thermodynamic theories. That is, we assume theexistence of a generalized entropy which depends on thedissipative fluxes and on the equilibrium variables as well.Restrictions on the form of the evolution equations arethen imposed by the laws of thermodynamics. From theexpression for the generalized entropy one then can de-rive the generalized equations of state, which are of in-terest in the description of the system under considera-
tion. The phenomenological formulation of the transportequations for the first order and second order theories isaccomplished by combining the conservation of energy–momentum and particle number with the Gibbs equation.One then obtains an expression for the entropy 4–current,and its divergence leads to entropy production. Because ofthe enlargement of the space of variables the expressionsfor the energy-momentum tensor T µν , particle 4–currentNµ, entropy 4-current Sµ, and the Gibbs equation containextra terms. Transport equations for dissipative fluxes areobtained by imposing the second law of thermodynamics,that is, the principle of nondecreasing entropy.
The connection between the macroscopic theory and mi-croscopic theory enters through the transport coefficientsof the matter. The equation of state provides closure tothe system of conservation equations.
The consequences of non-ideal fluid dynamics, both firstorder (if applicable) and second order were demonstrated[1] in a simple situation, assuming the Bjorken scaling so-lution and a simple equation of state. A more careful studyof the effects of non-ideal fluid dynamics on the observablesis therefore important. Conversely, measurements of theobservables related to thermodynamic quantities would al-low us to determine the importance and strength of dissi-pative processes in heavy ion collisions.
For the complete description of the dynamics of viscous,heat conducting matter we need to consider more realis-tic situations: a system that expands in both the longi-tudinal and transverse directions [7] and we need a full(3+1)–dimensional solution to the conservation and evo-lution equations. Also a thorough study of the equationsof state and transport coefficients is needed [8]. This willrequire extensive numerical computations. This is a chal-lenging but interesting problem which is studied right now.
References
[1] A. Muronga, nucl-th/0309055, to appear in Phys.Rev. C.
[2] H. Grad, Commun. Pure Appl. Math. 2 (1949) 331.[3] I. Muller, Z. Phys. 198 (1967) 329.[4] W. Israel and J.M. Stewart, Ann. Phys. (N.Y.) 118
(1979) 341.[5] C. Eckart, Phys. Rev. 58 (1940) 919.[6] L.D. Landau and E.M. Lifshitz, Fluid Mechanics
(Pergamon, New York, 1959)[7] A. Muronga and D.H. Rischke, in preparation.[8] A. Muronga, nucl-th/0309056, to appear in Phys.
Rev. C.
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Three-Fluid Simulations of Relativistic Heavy-Ion Collisions
Yu.B. Ivanov1,2, E.G. Nikonov1,3, W. Norenberg1, V.N. Russkikh1,2, and V.D. Toneev1,3
1GSI, Darmstadt; 2Kurchatov Institute, Moscow; 3Joint Institute for Nuclear Research, Dubna
Probing Equation of State (EoS) of QCD matter in rel-ativistic heavy-ion collisions is still a fascinating probleminspiring new experimental projects [1]. Hydrodynamicmodels have an advantage that they directly address EoSof hot and dense nuclear matter, which is of prime inter-est for this domain of physics. We have developed 3D 3-fluid relativistic hydrodynamic model [2, 3] for simulatingheavy-ion collisions at incident energies from few to about200 A·GeV. The model involves possibility to use variousEoS including those with deconfinement phase transitionof different orders.
A specific feature of the dynamic 3-fluid description isa finite stopping power resulting in a counter-streamingregime of leading baryon-rich matter. This counter-streaming behavior is supported by experimental rapiditydistributions in nucleus–nucleus collisions and simulatedby introducing the multi–fluid concept. The basic idea ofa 3-fluid approximation to heavy-ion collisions is that ateach space-time point x = (t,x) the distribution functionof baryon-rich matter, can be represented by a sum of twodistinct contributions fb(x, p) = fp(x, p)+ft(x, p), initiallyassociated with constituent nucleons of the projectile (p)and target (t) nuclei. During their mutual friction thesefluids radiate mesons, gluons, quark-antiquark pairs, etc.,depending on the EoS phase, which form a third baryon-free (i.e. with zero net baryonic charge) fluid in the mid-rapidity region. Thus, in addition to the two baryon-richfluids (specifying 2-fluid models), a new model incorpo-rates this ”fireball” fluid which is treated on equal footingwith the baryon-rich ones. Its evolution is delayed due toa formation time τ , during which the baryon-free fluid nei-ther thermalizes nor interacts with the baryon-rich fluids.After formation it thermalizes and starts to interact withthe baryon-rich fluids. Mutual friction forces between dif-ferent fluids are estimated, based on available experimen-tal hadronic cross sections, and then tuned to reproducethe observed stopping power in nucleus-nucleus collisions.Details of this extension of the model can be found in [2].
In our hydrodynamic model we simulate the whole pro-cess of the reaction, i.e. from the formation of a hot anddense nuclear system to its subsequent decay. This is indistinction to numerous hydrodynamic simulations, whichtreat only the expansion stage of a fireball formed in thecourse of the reaction, while the initial state of this denseand hot nuclear system is constructed from either kineticsimulations or more general albeit model-dependent as-sumptions. The set of hydrodynamic equations is solvednumerically using the particle-in-cell method.
It is found that for τ = 0 the interaction with baryon-rich fluids strongly affects the baryon-free fluid. However,at reasonable finite formation time, τ 1 fm/c, the effectof this interaction turns out to be substantially reducedalthough still noticeable. Baryonic observables are onlyslightly affected by the interaction with the baryon-freefluid [2].
As an example, Fig.1 presents proton rapidity spectrafor Au + Au collisions at incident energies E0 =6, 8, and10.5 A · GeV . These preliminary calculations were per-formed for two EoS’s [3]. The impact parameters b relevantto each set of data were determined by a fraction of thetotal reaction cross section, corresponding to this set. Onecan see that evolution of the spectra shape with changingthe impact parameter is reproduced reasonably well for allconsidered energies. However, the rapidity spectra proveto be quite insensitive to the EoS used.
Figure 1: Proton rapidity spectra from Au + Au colli-sions at three bombarding energies E0 and different im-pact parameters b (given in fm). Solid and dashed linesare calculated for the hadron gas EoS and mixed phaseEoS with crossover phase transition, respectively. Experi-mental points are from [4].
Unfortunately, in spite of reasonable reproduction ofobservable proton rapidity spectra in the wide range ofbombarding energies and centrality parameters we are un-able to favor any of considered EoS’s. Careful analysisof more delicate characteristics, like excitation functionsfor directed and elliptic flows, transverse temperature andstrangeness abundance, as well as dilepton production, isneeded. This work is in progress now.
References
[1] Conceptual Design Report “ An International Accel-erator Facility for Beams of Ions and Antiprotons”,http://www.gsi.de/GSI-Future/cdr/ .
[2] V.N. Russkikh, Yu.B. Ivanov, E.G. Nikonov, W. No-renberg and V.D. Toneev, Phys. Atom. Nucl. 67(2004) 199.
[3] V.D. Toneev, Yu.B. Ivanov, E.G. Nikonov, W. Noren-berg and V.N. Russkikh, Proc. of XII Int. Conf. onSelected Problems of Modern Physics, Dubna, June8-11, 2003, [nucl-th/0309008].
[4] E917 Collaboration, Phys. Rev. Lett. 86 (2001) 1970.
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Density Perturbations in Heavy-Ion Collisions below the Critical Point
A. Dumitru, K. Paech and H. Stocker
Institut fur Theoretische Physik, Universitat Frankfurt a.M., Germany
Universality arguments suggest that the chiral phasetransition for two massless quark flavors is second-orderat baryon-chemical potential µB = 0 [1], which then be-comes a crossover for small quark masses. On the otherhand, a first-order phase transition is predicted by a vari-ety of low-energy effective theories for small temperatureT and large µB [2]. Hence, the first-order phase transi-tion line in the (µB , T ) plane must end in a second-ordercritical point [3]. For 2 + 1 quark flavors the critical pointhas been located at T = 160 MeV and µB = 725 MeV [4].However, a reliable extrapolation to the continuum limitand to physical pion mass has not been attempted so far.
There is an ongoing experimental effort to detect thatcritical point in heavy-ion collisions at high energies. It ishoped that by varying the beam energy, for example, onecan “switch” between the regimes of first-order transitionand cross over, respectively (higher energies correspond tolarger entropy per baryon or T/µB).
To investigate collective dynamics in the vicinity of thecritical endpoint we introduce a model for the real-timeevolution of a relativistic fluid of quarks coupled to non-equilibrium dynamics of the long wavelength (classical)modes of the chiral condensate [5]:
∂µ∂µφ+ ∂Veff/∂φ = 0 , ∂µ
(T µν
fl + T µνφ
)= 0. (1)
Here, T µνfl is the energy-momentum tensor of the fluid,
T µνφ that of the classical modes of the chiral condensate,
and Veff is the effective potential obtained by integratingout the thermalized degrees of freedom. We focus first onenergy-density inhomogeneities for vanishing baryon den-sity (the nature of the transition is then determined bythe effective quark-field coupling rather than the baryon-chemical potential [5, 6]). We allow for “primordial” Gaus-sian fluctuations of the condensate φ on length scales∼ 1 fm on top of a smoothly varying mean field. If prop-agated through a first-order chiral phase transition thesefluctuations give rise to a rather inhomogeneous (energy-)density distribution as seen in Fig. 1. Such effects werepreviously studied in the context of the QCD transitionin the early universe, where inhomogeneities of the en-tropy (or baryon to photon ratio) might affect BBN [7].However, in the cross-over regime we find much smalleramplitudes of density perturbations [5].
In heavy-ion collisions the scale of the density pertur-bations is too small for them to be resolved in rapidityspace. This would require a resolution ∆y < 1, whichis about the thermal width of the local particle momen-tum distributions. However, observable consequences oflarge density inhomogeneities created in a first-order tran-sition at beam energies below the critical endpoint maystill exist. (Inhomogeneities from fluctuations of particleproduction in the primary nucleon-nucleon collisions [8]should be largely washed out until decoupling.) For exam-ple, fluctuations of the energy-momentum tensor of mat-ter in coordinate space are uncorrelated to the reaction
y [fm]0 1 2 3 4 5 6 7
t [f
m/c
]0
2
4
6
8
10
12
14
16
0
1
2
3
4
5
g = 5.5
Figure 1: Fluid energy density distribution in space-timefor a first-order chiral phase transition [5].
plane and should therefore reduce out-of-plane collectiveflow (v2/〈pt〉) as compared to equilibrium hydrodynam-ics [5]. Moreover, by analogy to BBN, perturbations ofthe entropy per baryon s/ρB should affect abundances ofrare hadrons: B/B, Λ/p [9] and K+/π+ [10] are largerthan for a homogeneous system with the same total vol-ume, baryon number and entropy.
References
[1] R. D. Pisarski and F. Wilczek, Phys. Rev. D 29 (1984)338.
[2] M. A. Halasz et al., Phys. Rev. D 58 (1998) 096007;J. Berges and K. Rajagopal, Nucl. Phys. B 538 (1999)215; T. M. Schwarz, S. P. Klevansky and G. Papp,Phys. Rev. C 60 (1999) 055205; O. Scavenius et al.,Phys. Rev. C 64 (2001) 045202.
[3] M. Stephanov, K. Rajagopal and E. V. Shuryak,Phys. Rev. Lett. 81 (1998) 4816.
[4] Z. Fodor and S. D. Katz, JHEP 0203 (2002) 014.[5] K. Paech, H. Stocker and A. Dumitru, Phys. Rev. C
68 (2003) 044907.[6] O. Scavenius et al., Phys. Rev. D 63 (2001) 116003.[7] see e.g. D. J. Schwarz, Annalen Phys. 12 (2003) 220.[8] M. Bleicher et al., Nucl. Phys. A 638 (1998) 391.[9] B. B. Back et al. [E917 Collaboration], Phys. Rev.
Lett. 87 (2001) 242301.[10] C. Alt et al. [NA49 Collaboration], J. Phys. G 30
(2004) S119.
- 82 -
Kaon and pion production at CBM energies
Markus Wagner, Alexei Larionov∗ and Ulrich Mosel
Institut fur Theoretische Physik, Universitat Giessen
It has been argued that a strangeness enhancement inheavy ion collisions might be an indirect signal for thequark-gluon plasma [1]. This enhancement should be mosteasily seen in the most abundant strange particles, thekaons. Experiments at the AGS and SPS indeed found amaximum in the K+/π+ at about 30 GeV [2]. In [3] theauthors presented an excitation function of the K+/π+ ra-tio at these energies calculated by the two transport codesHSD and UrQMD. We will show the results of a thirdmodel, namely the BUU model and compare it to the othertwo models and the statistical model.The BUU model is based upon hadronic and string degreesof freedom and the dynamics is governed by the semiclassi-cal transport equation. The model is basically the versionof Ref. [4] with a few changes. In the case of a meson-meson collision with an invariant energy above twice themass of a kaon (∼ 1 GeV), we include direct channels forstrangeness-production
mm↔ KK. (1)
The reaction ππ ↔ KK has been implemented before. Iftwo isospin one mesons collide, we use the same cross sec-tion as in the ππ case. All other cross sections of the typein eqn.(1) are chosen to be constant 2 mb. The back reac-tions are implemented according to detailed balance.The baryon-meson reactions with
√s < 2 GeV are treated
via resonance production as described in [4]. Above thatthreshold we either excite the strings according to theFritiof model or with a probability of P (s) =max(0.85 −0.17
√s/GeV,0) the meson and the the baryon merge into
a single object, which then decays according to the stringfragmentation. The merging happens only if the antiquarkof the meson and one of the quarks of baryon share thesame flavour and, therefore, can annihilate. The remain-ing object is treated as a string with the quark of the mesonat the one end and a diquark of the baryon at the otherend. We assume that this string carries the total four-momentum of the initial meson and baryon. The proba-bility with which this process takes place, is determinedsuch that it describes the strangeness production in π + pcollisions.In fig.1 we see the K+/π+ ratio as a function of energy
in comparison to data, HSD and UrQMD [3] and the sta-tistical model [4]. All calculations by the transport codeshave been performed in the cascade mode. We see that ourmodel describes the data well in the energy range whichis shown. Only at the lower energies we overestimate theratio in an unsatisfying way, which is due to an overes-timation of kaons [6]. This is however expected becausewe overpredict already the K+ in the NN channel at thelower energies. Therefore we can say that we describe therise of the slopes and we reach the plateau, which startsat 40 GeV. At 10 GeV and 30 GeV we underestimate the
∗On leave from RRC ”I.V. Kurchatov Institute”, 123182 Moscow,Russia
0
0.05
0.1
0.15
0.2
0.25
0.3
5 10 15 20 25 30 35 40 45
K+/π
+ (
mid
)
Elab [GeV]
BUUThermo
dataHSD
UrQMD
Figure 1: TheK+/π+ ratio as a function of energy in com-parison to data [2], HSD, UrQMD [3] and the statisticalmodel [5]
ratio, but only by less than 30%. The reason for this dis-agreement is due to the pions, which we overpredict. Thisis in agreement with the calculations of HSD and UrQMD,which are off for the same reason, although their disagree-ment looks more pronounced.The BUU results are in line with the calculations per-formed with the statistical model, which might be an in-dication that the assumption of chemical equilibrium isjustified. However, none of the curves is able to reach the10 GeV and the 30 GeV point, although the discrepancyis less than in the comparable work [3]. It will be interest-ing to explore the region between 10 and 30 GeV in moredetail and to gather more data in that energy range. Sofar our studies of the K+/π+ ratio do not admit a safestatement about the existence or the exclusion of a quarkgluon plasma.
References
[1] J. Rafelski and B. Muller, Phys. Rev. Lett. 48, 1066(1982)
[2] V. Friese, nucl-ex/0305017 v1[3] H. Weber, E. L. Bratkovskaya, W. Cassing and H.
Stocker, Phys. Rev. C 67, 014904 (2003)[4] M. Effenberger, E.L. Bratkovskaya, and U. Mosel,
Phys. Rev. C 60, 044614 (1999)[5] P. Braun-Munzinger, J. Cleymans, H. Oeschler and
K.Redlich, Nucl. Phys. A 697, 902 (2002)[6] M. Wagner, Diploma thesis
- 83 -
Pions and kaons at 1-2 A GeV:Effect of in-medium NN → N∆ cross section
A.B. Larionov∗ and U. MoselInstitut fur Theoretische Physik, Universitat Giessen
Pion multiplicity in heavy-ion collisions at the beam en-ergies of about 1 A GeV [1, 2] is usually overpredictedby transport calculations with vacuum cross sections. Re-cently [3] we proposed to take into account the in-mediummodifications of the NN → N∆ cross section, namely, theexchange pion collectivity, vertex renormalization by thecontact nuclear interactions and Dirac effective masses ofthe baryons due to coupling with the scalar σ field. Itturned out that the NN → N∆ cross section is substan-tially reduced in nuclear medium, mainly due to the effec-tive mass corrections:
σin−mediumNN→N∆ ∝ (2m∗
N)32M∗∆ .
This leads to a reduced pion production and a better agree-ment with experimental data of the FOPI Collaboration[1, 2].
In this work we present BUU calculations of the pionand kaon multiplicity excitation functions for the inclusivereactions Au+Au at 0.56, 0.78, 0.96 1.1, 1.46 A GeV andC+C at 0.8, 1.0, 1.2, 1.5, 1.8, 2.0 A GeV measured bythe KaoS Collaboration [4]. It has been shown in [5] thatthe ratio of the kaon multiplicities from Au+Au and C+Cplotted vs the beam energy is sensitive to the compress-ibility of nuclear matter. The best description of the datahas been reached in [5] by applying the soft momentum-dependent mean field (SM). Thus, we have also performedour BUU calculations within the model [6] using the SMmean field (K = 220 MeV). At the collision energies of1-2 A GeV the processes πN → Y K+ and N∆ → NYK+
have the dominant contribution to the kaon production [7].We expect, therefore, the kaon yield to be quite sensitiveto the π and ∆(1232) abundancies in the colliding system.
Fig. 1 shows the pion and kaon multiplicities per nu-cleon vs the beam energy calculated without (dashed lines)and with (solid lines) the in-medium modifications of theNN → N∆ cross section. For the in-medium cross sectionwe have used the baryon effective masses given by the rela-tivistic Hartree approximation (RHA) [8], which producesa somewhat slower decreasing effective mass with densitythan the popular versions of the non-linear Walecka modelNL1 and NL2 [3]. We see that in the case of standardcalculation (without modifications) the pion multiplicityis overpredicted below 1.5 A GeV for the both systemsAu+Au and C+C.The kaon yield is somewhat overpre-dicted for Au+Au and well described for C+C by thestandard calculation. The in-medium reduction of theNN → N∆ cross section imroves the agreement with thedata on pion production. However, our RHA calculationis somewhat above the pion data on Au+Au [4]. We haveto point out that there is still an ambiguity in the experi-mental data on pion multiplicities [1, 2, 4]. In particular,
∗On leave of absence from RRC ”I.V. Kurchatov Institute”,123182 Moscow, Russia
10-7
10-6
10-5
10-4
10-3
0.6 0.8 1 1.2 1.4 1.6 1.8 2
Ebeam (A GeV)
π (x 10-2)
K+
M/A
Au+AuC+C
standardRHA
the latest analysis of the FOPI data [2] is in a good agree-ment with our calculations applying the in-medium crosssections [3]. For the RHA calculation, the kaon multiplic-ity agrees within errorbars with the data for both systemsAu+Au and C+C, excepting the points at 0.56 A GeV forAu+Au and at 1 A GeV for C+C.
In conclusion, the in-medium reduced NN → N∆ crosssection acts in the same way as a repulsive kaon potential,i.e. by reducing the kaon multiplicity. Therefore, in orderto get an information on the kaon potential we have tolook also on other observables, e.g. the in-plane and out-of-plane kaon flow.
References
[1] D. Pelte et al., Z. Phys. A 357, 215 (1997).[2] M.R. Stockmeier et al., GSI Scientific Report 2001, p.
35.[3] A.B. Larionov and U. Mosel, Nucl. Phys. A 728, 135
(2003).[4] C. Sturm et al., Phys. Rev. Lett. 86, 39 (2001).[5] C. Fuchs, Amand Faessler, E. Zabrodin, Yu-Ming
Zheng, Phys. Rev. Lett. 86, 1974 (2001).[6] M. Effenberger, E.L. Bratkovskaya, and U. Mosel,
Phys. Rev. C 60, 44614 (1999).[7] E.L. Bratkovskaya, W. Cassing and U. Mosel, Nucl.
Phys. A 622, 593 (1997).[8] B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16, 1
(1986).
- 84 -
Production of Ξ Hyperons near the threshold
M. Reiter, G. Zeeb, M. Bleicher, H. Stöcker, and W. GreinerInstitut für Theoretische Physik, Johann Wolfgang Goethe-Universität Frankfurt am Main
Strange particle yields and spectra are key probes tostudy excited nuclear matter and to detect the transi-tion of (confined) hadronic matter to quark-gluon-matter,i.e. QGP[1, 2]. The relative enhancement of strange andmulti-strange hadrons, as well as hadron ratios in centralheavy ion collisions with respect to peripheral or protoninduced interactions have been suggested as a signaturefor the transient existence of a QGP phase [2].For the present work, the Ultra-relativistic Quantum
Molecular Dynamics model (UrQMD 1.2) [3] was appliedto study the production of strange particles in heavy ionreactions at Elab = 6 AGeV. The results were comparedto data on the Ξ hyperon near threshold [4] which hasrecently become available.Figure 1 (top and middle panel) shows the centrality
dependence of the Ξ− and Λ yields in Au+Au interactionsat Elab = 6 AGeV.
0.0
0.1
0.2
0.3
-
Au+Au6 AGeV
0
5
10
15
20
0 +0
0 1 2 3 4 5 6 7 8b [fm]
0.00
0.01
0.02
- /(0 +
0 )
Figure 1: Centrality dependence of the ratio Ξ−/(Λ+Σ0)(bottom) and the Λ (middle) and Ξ− (top) yields inAu+Au interactions at Elab = 6 AGeV. Small symbols de-note the calculations, while large symbols show the datafrom [4].
One clearly observes a strong increase of the(multi-)strange baryon yields toward head-on collisions.However, the Cascade yield increases stronger than the sin-gle strange hyperon abundance toward central collisions.This leads to a moderate centrality dependence of theΞ−/Λ ratio as shown explicitly in Fig. 1 (bottom). The
centrality dependent yields as well as the ratios show goodagreement with the recently measured data [4] of the E895collaboration.In [5] we have performed a detailed analysis of the dy-
namics of Ξ production. It was demonstrated that Ξs areformed in the late stages of central heavy ion interactionsat AGS energies and that the production is strongly dom-inated by meson-baryon reactions over a time of nearly10 fm/c. Thus, multi-strange baryon production relies onlong living hadronic stages, which might allow for chemicalequilibration even in this exotic channel.In particular, it became evident that Ξ production is
mainly driven by meson-baryon interactions including a Λor Σ. In fact, 70% of all meson-baryon reactions resultingin the production of a Ξ include a hyperon. However,the present analysis indicates that only 11% of the meson-baryon interactions leading to Ξ production proceed viathe strangeness exchange reaction anti-Kaon + hyperon,in contrast to Ref. [6].It should be noted that the nice agreement between ex-
perimentally observed yield and the present calculation inthe AGS energy range is in stark contrast to the results ob-tained at SPS energies. There hadronic transport modelswithout non-standard modifications clearly fail to describethe measured Ξ yields.This work used computational resources provided by the
Centre Calcule at Lyon, France, and the Center for Scien-tific Computing (CSC) at Frankfurt, Germany. This workwas supported by GSI, BMBF, DFG.
References[1] S.A. Bass, M. Gyulassy, H. Stöcker and W. Greiner,
J. Phys. G25, R1 (1999);R. Stock, Phys. Lett. B456, (1999) 277
[2] J. Rafelski and B. Müller, Phys. Rev. Lett. 48, 1066(1982); J. Rafelski, Phys. Rep. 88, 331 (1982);P. Koch, B. Müller and J. Rafelski, Phys. Rep. 142,167 (1986).
[3] M. Bleicher, E. Zabrodin, C. Spieles, S.A. Bass, C.Ernst, S. Soff, L. Bravina, M. Belkacem, H. Weber,H. Stöcker, W. Greiner, J. Phys. G 25 (1999) 1859[arXiv:hep-ph/9909407];S.A. Bass, M. Belkacem, M. Bleicher, M. Brandstet-ter, L. Bravina, C. Ernst, L. Gerland, M. Hofmann,S. Hofmann, J. Konopka, G. Mao, L. Neise, S. Soff,C. Spieles, H. Weber, L.A. Winckelmann, H. Stöcker,W. Greiner, C. Hartnack, J. Aichelin, N. Amelin,Prog. Part. Nucl. Phys. 41 (1998) 225 [arXiv:nucl-th/9803035].
[4] P. Chung et al. (E895 collaboration), Phys. Rev. Lett.91 (2003) 202301 [arXiv:nucl-ex/0302021].
[5] G. Zeeb, M. Reiter and M. Bleicher, arXiv:nucl-th/0312015.
[6] S. Pal, C. M. Ko, J. M. Alexander, P. Chung andR. A. Lacey, arXiv:nucl-th/0211020.
- 85 -
Strangeness Excitation Functions and Nonhadronic Degrees of Freedom
S.Soff1, E.Bratkovskaya1, M.Bleicher1, M.Reiter1, H.Stocker1, M.vanLeeuwen2, S.Bass3, W.Cassing4
1Institut fur Theoretische Physik, Goethe-Universitat, 60054 Frankfurt, Germany; 2NIKHEF, Amsterdam,Netherlands; LBNL, Berkeley, CA 94720, USA; 3Duke University, Durham, NC 27708, USA; RIKEN BNL, Upton,
NY 11973, USA; 4Institut fur Theoretische Physik, Universitat Giessen, 35392 Giessen, Germany
We have studied hadron yields and transverse momen-tum spectra for p+p, p+A, and A+A collisions from2AGeV to 21.3ATeV with two microscopic transportmodels (HSD & UrQMD ) that are based on quark, di-quark, string and hadronic degrees of freedom. The com-parison to data for mt-spectra from pp, pA and C+C(or Si+Si) reactions has shown that the transport mod-els reasonably describe the elementary and light systems.However, for central Au+Au (Pb+Pb) collisions aboveElab ∼ 5AGeV, the kaon inverse slope parameters TK ,as extracted from the spectra,
(1/mt)dN/dmt ∼ exp(−mt/TK)
are considerably larger for the data than expected fromthe calculations. Various modifications of the model in-gredients were tested with respect to their effect on TK .
The figure shows the dependence of TK on√s in com-
parison to experimental data from the E866, E917, NA49,NA44, STAR, BRAHMS, and PHENIX collaborations. forcentral Au+Au (Pb+Pb) collisions and for pp collisions.The two solid lines (open circles) on the l.h.s. representthe HSD results, where the range between the lines is dueto fitting the slope T itself, an uncertainty in the repul-sive K±-pion potential or possible effects of string over-lap. Solid lines with stars include the Cronin effect. Thedashed lines show UrQMD results. The new UrQMD 2.0includes PYTHIA for the hard processes at high energies(√s > 50GeV) increasing the slope parameter. Dotted
lines with crosses show UrQMD 2.1 results that addition-ally incorporate high mass resonance states between 2 and3GeV. For pp, TK increases smoothly with energy (in thedata and calculations). At high energies the inclusion ofjets (via PYTHIA) becomes necessary.
The Cronin effect indeed improves the description of thedata in AA at RHIC energies. However, it definitely doesnot describe the enhancement at AGS energies. The highmass resonance states (as included in UrQMD 2.1) seemto improve the comparison of the inverse slope parametersystematics (in particular for K+).
Moreover, we have shown that the maximum in theK+/π+ ratio at 20 to 30 AGeV is missed and the approx-imately constant slope of the K± spectra at SPS energiesis not reproduced either. Hence, the excitation functionsmay suggest that the additional pressure - correspondingto lattice QCD calculations at finite quark chemical poten-tial µq and temperature T - is generated by strong interac-tions in the early pre-hadronic/partonic phase of centralAu+Au (Pb+Pb) collisions.
Acknowledgments: Work supported by DFG, BMBF,GSI, Humboldt-Foundation, CSC Frankfurt.
10 1000.10
0.15
0.20
0.25
0.30
0.35
E866 NA49 NA44 STAR BRAHMS PHENIX
Au+Au / Pb+Pb -> K++X
T [
GeV
]
HSD HSD with Cronin eff. UrQMD 1.3 UrQMD 2.0 UrQMD 2.1
10 1000.10
0.15
0.20
0.25
0.30
0.35 p+p -> K++X
T [
GeV
]
exp. data: K+; K0
S
FRITIOF-7.02 in HSD UrQMD 1.3 UrQMD 2.0
10 1000.10
0.15
0.20
0.25
0.30
0.35
E866 NA49 NA44 STAR BRAHMS PHENIX
HSD HSD with Cronin eff. UrQMD 1.3 UrQMD 2.0 UrQMD 2.1
Au+Au / Pb+Pb -> K−−−−+X
s1/2 [GeV]
10 1000.10
0.15
0.20
0.25
0.30
0.35 p+p -> K−−−−+X
exp. data: K−−−−; K0
S
FRITIOF-7.02 in HSD UrQMD 1.3 UrQMD 2.0
s1/2 [GeV]
Figure 1: Inverse slope parameters T for K+ and K−
mesons from central Au+Au (Pb+Pb) collisions (l.h.s.)and pp reactions (r.h.s.) at midrapidity as a function ofthe invariant energy
√s. Experimental data (big symbols)
are compared to UrQMD and HSD transport calculations.
[1] E. Bratkovskaya, M. Bleicher, M. Reiter, S. Soff,H. Stocker, M. van Leeuwen, S. Bass, W. Cassing.Submitted to Phys. Rev. C (2004); nucl-th/0402026.[2] E. Bratkovskaya, S. Soff, H. Stocker, M. vanLeeuwen,W. Cassing. Phys. Rev. Lett. 92, 032302 (2004).[3] S. Soff, J. Phys. G 30, 139 (2004).[4] M. Reiter, E. Bratkovskaya, M. Bleicher, W. Bauer,W. Cassing, H. Weber, H. Stocker, Nucl. Phys. A 722,142 (2003).[5] H. Weber, E. Bratkovskaya, W. Cassing, H. Stocker,Phys. Rev. C 67, 014904 (2003).[6] S.A. Bass et al., Prog.Part.Nucl.Phys. 41, 255 (1998),M. Bleicher et al., J. Phys. G25, 1859 (1999).[7] W. Cassing and E. L. Bratkovskaya, Phys. Rept. 308,65 (1999).
- 86 -
Evidence for ψ′ Regeneration in Heavy Ion Collisions
Andriy Kostyuk1,2 and Horst Stocker1
1Institut fur Theoretische Physik and Frankfurt Institute for Advanced Studies, J.W. Goethe-Universitat, Frankfurt amMain, Germany; 2Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine
The study of hidden charm production is an importantpart of the heavy ion program. The standard approachto this problem [1] assumes that cc bound states are cre-ated only at the initial stage of the reaction and then par-tially destroyed at later stages due to interactions with themedium [2, 3, 4].
The idea of the statistical J/ψ production [5] triggeredthe development of an alternative approach [6, 7] — thestatistical coalescence model (SCM): charmonia (as well asopen charm hadrons) are formed at hadronization due tocoalescence of charm quarks and antiquarks.
The standard approach (two its versions — the thresh-old suppression model [2] and comover model [3, 4])demonstrates reasonable agreement with the data on J/ψproduction in Pb+Pb collisions at SPS. Better agreement,but only for (semi)central collisions Np > 100 (Np is thenumber participant nucleons) can be obtained within theSCM [8].
It is interesting to check, which of the two scenarios isbetter suited for excited charmonium states. Recently theN50 collaboration presented new data on ψ′ production inPb+Pb collisions [9]. It appears that the both versions ofthe standard approach as well as SCM are able to fit thenew ψ′ data (see Fig. 1). But the observed ψ′ suppressionseems to be too weak to give reasonable values of the fitparameters of the standard scenario.
The free parameter of the threshold suppression modelis the threshold density of the participant nucleons in thetransverse plane at which the charmonium species underconsideration is suppressed. The J/ψ data can be rea-sonably fitted with nJ/ψ = 3.77 +0.09−0.10 fm−2 for primary
J/ψ’s and nχ = 1.95 +0.35−0.45 fm−2 for excited charmonia(mostly χ-states) contributing up to 40% to the total J/ψyield in p+p collisions. Because the ψ′ is much weakerbound than any of the χ states, one would expect thatits suppression begins at substantially lower density. Itappears, however, that the ψ′ data suggest approximatelythe same (or even larger) threshold for the ψ′ suppressionas for χ: nψ′ = 2.17 +0.33−0.43 fm−2.
The comover model does not alow to extract the sup-pression parameters for primary J/ψ and χ separately.The free parameter of the model is the effective suppres-sion cross section averaged over all charmonium states thatcontribute to the production of the species under consid-eration. There is no obvious contradiction between thefit results for J/ψ and ψ′: σcoJ/ψ = 1.01 ± 0.05 mb and
σcoψ′ = 2.84 +1.56−0.79 mb, i.e. the comover dissociation crosssection for ψ′ is by a factor of about 3 larger than that forJ/ψ. This ratio of the cross sections can be obtained, if oneassumes that the matrix element of the reaction is approx-imately the same for J/ψ and ψ′ [10] and the suppressionis dominated by exothermic (e.g. J/ψ + ρ → D +D) re-
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0 20 40 60 80 100 120
Bµµψ
’ σ(ψ
’) /
BµµJ/
ψ σ
(J/ψ
)
ET
Pb+Pb 1998Pb+Pb 2000co-mover modelthreshold suppressionSCM
Figure 1: The ψ′ to J/ψ ratio in Pb+Pb collisions at SPS.
actions. Still, model calculations of the matrix elementpredict much larger dissociation cross section for ψ′ thanfor J/ψ [12]. If that is indeed so, then the only possibleexplanation of the observed ψ′ yield is regeneration of ψ′
at late stages of the reaction. As can be seen from Fig. 1the new data (as well as the old ones [6, 11]) are consistentwith the assumption that the ψ′ to J/ψ ratio is nearly con-stant at Np > 100− 150. It is equal to the thermal value,corresponding to freeze-out temperatures T = 150 − 170MeV, which agrees with the SCM. This suggest that ex-cited charmonium states are likely to be produced at thefinal rather than at the initial stage of heavy ion reactions.
Measurement of the charmonium production in heavyion collisions at lower energies, e.g. at the new GSI accel-erator facility, would help to disentangle different charmo-nium production mechanisms.
We acknowlege the finantial support of GSI, DFG andBMBF.
References
[1] T. Matsui and H. Satz, Phys. Lett. B 178 (1986) 416.[2] J. Blaizot, M. Dinh and J. Ollitrault, Phys. Rev. Lett.
85 (2000) 4012.[3] A. Capella, A. B. Kaidalov and D. Sousa, Phys. Rev.
C 65 (2002) 054908.[4] C. Spieles et al., Phys. Rev. C 60 (1999) 054901.[5] M. Gazdzicki and M. I. Gorenstein, Phys. Rev. Lett.
83 (1999) 4009.[6] P. Braun-Munzinger and J. Stachel, Phys. Lett. B
490, (2000) 196.[7] M. I. Gorenstein, A. P. Kostyuk, H. Stocker and
W. Greiner, Phys. Lett. B 509 (2001) 277; J. Phys.G 27 (2001) L47.
[8] A. P. Kostyuk, M. I. Gorenstein, H. Stocker, andW. Greiner, J. Phys. G 28, 2297 (2002).
[9] H. Santos [NA50 Collaboration], Presented at QM2004, Oakland, 11-17 Jan 2004.
[10] E. L. Bratkovskaya et al., arXiv:nucl-th/0402042.[11] H. Sorge, E. Shuryak and I. Zahed, Phys. Rev. Lett.
79 (1997) 2775.[12] T. Barnes, arXiv:nucl-th/0306031.
- 87 -
Nuclear Attenuation in Hard Processes
B.Z. Kopeliovich1-3 and A. Schafer1
1University of Regensburg; 2MPI-Kernphysik Heidelberg; 3JINR Dubna
Nonexponential attenuation of hadrons in nucleiand renormalization of data for hard QCD probes.For classical particles the propagation through a mediumis exponentially attenuated with the path length L,i.e. the survival probability in a nucleus decreases asexp(−σhNin ρL), where σhNin is the cross section of inelas-tic hadron-nucleon collisions, and ρ is the nuclear den-sity. This kind of attenuation is actually what the Glaubermodel is based upon. However, it is known since Gribov’swork [1] that this behavior is subject to quantum correc-tions. The incoming hadron experiences quantum fluctu-ations which make the medium more transparent [2]. ForQCD, gauge invariance leads to the phenomenon of colortransparency [3, 4] which means that small size fluctu-ations have a cross section which vanishes quadraticallywith their transverse dimension. As a result, the depen-dence of the attenuation factor on L changes from an ex-ponential to a power in L [3]. Thus, the Glauber modelpredicts corrections known as Gribov’s inelastic shadow-ing. A deviation of the Glauber model calculations fromdata by about 10% has been well established experimen-tally up to the energy 280 GeV in the nuclear rest frame.It is a theoretical challenge to predict Gribov’s correctionsat the energies of RHIC and LHC which are two and fiveorders of magnitude higher, respectively.
In experiments studying heavy ion and proton-nucleuscollisions the Glauber model is widely used for calculationof inelastic cross sections, the so called number of colli-sions, the number of participants, etc. The current nor-malization of data for the cross sections of hard processes(high-pT , heavy flavors, etc.) relies on Glauber model cal-culations and therefore should be corrected accordingly.
The problem of calculating the inelastic shadowing cor-rections in all orders for multiple interactions was solved in[2] using the light-cone dipole approach. The impact pa-rameter dependence of Gribov’s corrections to the inelasticdeuteron-gold cross section is shown in Fig. 1. Togetherwith a correction for missed diffractive events in the taggedminimal bias event sample of PHENIX, this amounts toabout 20% reduction of all hard reaction cross sections,which is of the same order of magnitude as the detectedeffects (the Cronin enhancement in high-pT production,nuclear suppression of J/Ψ and open charm).
The lack of inelastic shadowing corrections in the anal-yses of experimental data can mimic real effects, as isdemonstrated for a few examples below.
Frequently nuclear effects are studied comparing periph-eral and central collisions. In such a case the cross sectionsare normalized to the number of collisions calculated inthe Glauber model. In this case inelastic shadowing cor-rections affect the ratio much more than in the minimalbias case. Indeed, according to Fig. 1 these correctionsare hardly visible in the nuclear center, while they reach amaximum of about 30-40% at the nuclear periphery. Thiscould explain at least part of the unusually large Cronin
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14
b(fm)
dσin (b
) / d
2 b
Figure 1: The impact parameter dependent inelastic crosssection for deuteron-gold collisions (three upper curves) in-cluding diffractive excitations (STAR trigger). The dashedcurve corresponds to the Glauber approximation. The up-per, thin solid curve includes inelastic shadowing relatedto excitation of the valence quark skeleton. The thick solidcurve just below shows the final result including gluonshadowing. The bottom solid thick curve shows the dif-ference between the Glauber approximation and the finalresult. The dotted curve is an example of strong gluonshadowing, RG = 0.3, as used in the HIJING code.
effect at negative rapidities in d − Au collisions reportedrecently by the PHENIX collaboration [5].
A weak Cronin effect of about 10% enhancement for pi-ons was found by the PHENIX collaboration. However,after correction for inelastic shadowing and missed diffrac-tion no enhancement is left [2].
The concentration of inelastic shadowing on the nuclearperiphery (see Fig. 1) could also explain why the Croninratio for pions in central relative to peripheral collisions atmid rapidities is about three times as large as in minimalbias events [7].
The observation of antishadowing for J/Ψ produced atnegative rapidities in deuteron-gold collisions was recentlyreported by the PHENIX collaboration [6]. However, aftera 20% correction for inelastic shadowing and diffraction isapplied to the normalization, no antishadowing effect isleft.
Attenuation of hadrons produced in DIS.Exciting results from RHIC and promising perspectivesfor forthcoming experiments at LHC on the suppressionof hadron production in heavy ion collisions have drawnmuch attention to the long standing problem of in-mediumhadronization. The best laboratory for that seems to beinclusive hadron production in deep-inelastic scattering
- 88 -
(DIS) off nuclei. In this case the spatial distribution ofthe density of the medium is well known. Experimentallythe jet energy and its fraction zh carried by the hadronare measured. The latest measurements of HERMES pro-vide plentiful high quality data. More measurements areongoing at Jefferson Lab.
A model of in-medium hadronization based on pertur-bative QCD was formulated in [8] where the correspondingmeasurements at HERMES were also proposed. The keyconcept of this approach is the existence of two time scales.One, called production time or length, is the time inter-val taken by color neutralization. This time contracts as∝ (1 − zh) at large zh. Another scale is the time intervalneeded for the formation of the hadronic wave function.This formation time rises linearly with zh.
Now first results from the HERMES experiment areavailable and a comparison with the theoretical predic-tions presented in [9, 10] demonstrates good agreement.The nuclear to nucleon ratio depicted in Fig. 2 rises withenergy. In contrast, the zh-dependence shown in Fig. 3 is
Figure 2: Comparison of the model predictions for theenergy dependence of the nuclear ratio for nitrogen andkrypton targets HERMES with data [9]. The solid anddashed curves correspond to calculations with and withoutthe effects of induced radiation respectively
falling. Both effects are related to the specific behavior
Figure 3: Same as in Fig. 2, but for zh- dependence.
of the production time, which increases with energy, butshortens with zh. Correspondingly, a colorless pre-hadronis produced outside or inside of the nucleus leading to aweaker or stronger absorption.
An alternative scenario which is able to fit the abovedata rather well, is based on an assumption that the pro-duction time, or rather the corresponding length, is al-ways longer than the nuclear size and hadrons attenuatedue to induced energy loss. To disentangle the two modelsone should rely on specific observables discussed in [10].One of them is the Q2 dependence of attenuation whichis predicted to be a falling function in the energy loss sce-nario. This is not supported by HERMES data depictedin Fig. 4 which demonstrate a rise with Q2, in good agree-ment with the model [10]. Other observables sensitive to
Figure 4: Same as in Fig. 2, but for Q2- dependence.
model assumptions are the flavor dependence and the pT -broadening [10].
Attenuation of hadrons produced in heavy ion col-lisions.In the center of mass system the nuclei are expected topass through each other leaving behind a dense medium ofquarks and qluons. The produced matter has a transversedimension similar to the nuclear size. A parton producedat mid rapidity with high transverse momentum has aboutthe same range of momenta as investigated in the HER-MES experiment.
In spite of this similarity, there is a principal differ-ence between hadronization processes in DIS and high-pThadron production at mid rapidities. In the case of 90o
parton scattering the jet energy and the parton virtualityare controlled by the same parameter, which is the trans-verse momentum of the parton, pT . Therefore, in this case,the production time tp ∝ 1/pT .
It is intuitively clear why the energy dependence of theproduction time inverts. The vacuum energy loss with arate proportional to p2T is so intensive, that in spite of theLorentz factor pT , the hadronization process must finishshortly after the hard partonic collision, otherwise the par-ton energy will degrade too much, making the productionof an energetic hadron impossible.
Since the production time is shrinking with pT in this
- 89 -
case, the energy dependence of the attenuation in themedium produced in heavy ion collision should be oppositeto what was observed in DIS. As long as the nuclear ratioin DIS rises with energy (Fig. 2), we should expect a fallingpT -dependence of the nuclear ratio for heavy ion collisions.Indeed, such an unusual effect has been observed at RHIC.
Cronin effect at large rapidities: Attenuation orColor Glass Condensate?The preliminary results for the Cronin effect at large for-ward pseudorapidities in d−Au collisions at RHIC releasedrecently by the BRAHMS collaboration [11] demonstratesurprisingly strong nuclear suppression. These data ac-cess the region of very small x < 10−3 where one mayexpect sizeable effects of gluon shadowing. This is why itis tempting to relate the observed strong suppression tothe effects of gluon saturation as described by the colorglass condensate concept. Such an explanation, however,does not go along with the fact the suppression of high-pThadrons in d − Au collisions at the mid rapidity [12, 13]was not correctly described, and with the fact of a quiteweak nuclear suppression of J/Ψ even at forward rapidities[6]. A quite weak gluon shadowing was predicted in [14]and indeed found recently in the next-to-leading analysisof parton distribution functions in nuclei [15].
The BRAHMS data are taken at pseudorapidity η = 3.2.The corresponding Bjorken x of the parton fragmentinginto the detected hadron ranges within x ≈ 0.16− 0.5 i.e.in the region completely dominated by valence quarks ofthe deuteron. Corresponding values of x for the nucleusare much smaller, less than 10−3.
The total hadron-nucleus inelastic cross section is knownto have an approximate A-dependence σhAin ∝ A2/3 forheavy nuclei, which is caused by shadowing related to thestrong attenuation of the incoming hadron in the nucleus.At the same time, the inclusive cross section of hadronproduction at mid rapidities is approximately proportionalto A. This is due to the fact that the gluon multiplicityrises proportionally to A1/3. Therefore, at mid-rapiditiesthe Cronin ration has values around 1 [16, 13]. However,the number of valence quarks is fixed, and the inclusivecross section at forward rapidities can rise only as A2/3.Therefore the Cronin ratio is quite suppressed by a factorA−1/3 ≈ 1/6. The pT -distribution of this cross sectiondescribes data rather well [17].
Nuclear attenuation of heavy flavors.This correction related to the nonzero separation of heavyQQ is treated as a higher twist effect and neglected withinthe parton model. However, it might be rather large onthe charm mass scale [18, 19, 20]. At high energies thisattenuation does not come as a simple absorption of acolorless cc pair in a medium. The fluctuation gluon→ cchas a long lifetime in the rest frame of the nuclear target,and this fluctuation is a color octet. Its attenuation lookslike shadowing, i.e. the result of a competition betweendifferent nucleons of the target in freeing this fluctuation.The dipole cross section which controls this process turnsout to correspond to a 3-body colorless system ccG. Thispart of nuclear shadowing for χ2 production is quite large,leading to nearly a factor of two suppression for minimalbias events [19]. The effect is less pronounced for J/Ψ
which has a smaller radius, and even less for open charmproduction [18, 19, 20]. In the latter case, however, it risessteeply with energy.
References
[1] V.N. Gribov, Sov. Phys. JETP 56 (1968) 892.[2] B.Z. Kopeliovich, Phys. Rev. C68 (2003) 044906.[3] B.Z. Kopeliovich and L.I. Lapidus, A.B. Zamolod-
chikov, Sov. Phys. JETP Lett. 33 (1981) 612.[4] J. Bertch, S.J. Brodsky, A.S. Goldhaber and J.G. Gu-
nion, Phys.Rev.Lett. 47 (1981) 297.[5] The PHENIX Collaboration: Meng Liu, a talk at the
international Conference Quark Matter 2004, Oak-land, January 2004.
[6] The PHENIX Collaboration: M. Brooks, a talk at theinternational Conference Quark Matter 2004, Oak-land, January 2004.
[7] The PHENIX Collaboration: D. d’Enterria, a talkat the international Conference Quark Matter 2004,Oakland, January 2004.
[8] B.Z. Kopeliovich, J. Nemchik and E. Predazzi, Pro-ceedings of the workshop on Future Physics at HERA,ed. by G. Ingelman, A. De Roeck and R. Klanner,DESY 1995/1996, v. 2, 1038 (nucl-th/9607036); Pro-ceedings of the ELFE Summer School on Confinementphysics, ed. by S.D. Bass and P.A.M. Guichon, Cam-bridge 1995, Editions Frontieres, p. 391
[9] HERMES Collaboration: A. Airapetian et al.,Eur. Phys. J. C20 (2001) 479 (hep-ex/0012049);A. Airapetian et al., hep-ex/0307023, to appear inPhys. Lett. B.
[10] B.Z. Kopeliovich, J. Nemchik, E. Predazzi,A. Hayashigaki, hep-ph/0311220, submitted toNucl. Phys. A. (hep-ph/9511214).
[11] The BRAHMS Collaboration: R. Debbe, a talk at theinternational Conference Quark Matter 2004, Oak-land, January 2004.
[12] D. Kharzeev, E. Levin, L. McLerran, Phys. Lett.B561 (2003) 93.
[13] The PHENIX Collaboration, S.S. Adler, et al., Phys.Rev. Lett. 91 (2003) 072303.
[14] B.Z. Kopeliovich, A. Schafer and A.V. Tarasov, Phys.Rev. D62 (2000) 054022
[15] D. de Florian, R. Sassot, hep-ph/0311227.[16] B.Z. Kopeliovich, J. Nemchik, A. Schafer,
A.V. Tarasov, Phys. Rev. Lett. 88 (2002) 232303.[17] M. Johnson and B.Z. Kopeliovich, paper in prepara-
tion.[18] B.Z. Kopeliovich, J. Raufeisen, Heavy flavor produc-
tion off protons and in a nuclear environment, hep-ph/0305094.
[19] B.Z. Kopeliovich, A.V. Tarasov, J. Hufner, Nucl.Phys. A696 (2001) 669.
[20] B.Z. Kopeliovich and A.V. Tarasov, Nucl. Phys.A710 (2002) 180.
- 90 -
Fermion pair production in time-dependent fields
Dennis D. Dietricha
a
Institut fur Theoretische Physik, Johann Wolfgang Goethe-Universitat, Frankfurt am Main, Deutschland
Particle production in classical bosonic fields is a topic ofcontinuing interest in QED and QCD. In strong fields pro-cesses with multiple couplings to the classical field are notparametrically suppressed by powers of the coupling con-stant g. In the absence of an additional scale, they have tobe taken into account to all orders. In the following let usconsider fermion production based on the exact retardedpropagator in a time-dependent external field. It is a so-lution of the equation of motion for the Green’s functionsof the Dirac operator. A threefold Fourier transformation(0x− 0y → 0k) leads to:[
iγ0∂t − 0γ · 0k + γ · A(t) −m]GR(t, t′, 0k) = δ(t− t′),
where the coupling constant g is included in the definitionfor the gauge field. With the retarded boundary conditionGR(t, t′, 0k) = 0 for all t < t′ the solution is given by [1]:
iGR(t, t′, 0k)γ0 =
= P expi
∫ t
t′dτγ0[γ · A(τ) − 0γ · 0k −m]
. (1)
The one-particle scattering operator can be written as:
T (q0, p0) = γ · A(q0 − p0) +
+∫dtdt′eiq0t−ip0t
′γ · A(t)GR(t, t′, 0p)γ ·A(t′). (2)
The expectation value for the number of produced pairscan be calculated according to:
〈n〉 =V
4(2π)3
∫d3p
ωp2|u(ωp,−0p)T (ωp,−ωp)v(ωp,+0p)|2 ,
where V stands for the volume in which the process occursand ωp =
√|0p|2 +m2.
Various approximation schemes for this expectationvalue can be obtained by approximating the retarded prop-agator. For the Born approximation one keeps merely thefirst term in (2). Further lowest order approximations fol-low from neglecting certain parts of the exponent of (1).Higher orders can be calculated with the help of a resum-mation formula [1]. In (1), dropping the field (for weakfields) leads to the free solution, keeping only the fieldterm (for strong fields) to a generalised Wilson line. With-out the path ordering we have the Abelian approximation.Here higher orders can be derived from the path-orderedexponentials’ group property [1]. Finally the different ap-proaches are to be compared for a special choice of thefield (see figure). Many other forms could have been taken.This choice was also inspired by a numerical study [2]. Thevalue chosen for the parameter Aint0 corresponds roughlyto one expected for an ultrarelativistic heavy-ion collisionat the LHC.
As a function of the transverse momentum kT the exactsolution peaks once and shows no further relative extremaor other distinct structures. For increasing values of theparameter Aint0 the peak in the transverse momentumspectrum increases in height and decreases in width. Actu-ally, the width of the transverse-momentum spectrum formassless particles at mid-rapidity scales exactly inverselyproportionally to t0. For fields of a functional form anal-ogous to that of the present special field the peak heightseems to be strictly monotonically decreasing with increas-ing longitudinal momentum. Further, the fermions andantifermions are never produced with momenta exactlyparallel to the direction of the field.
A comparison of the different approaches (see figure)shows that for large momenta all approximations tend to-ward the exact solution. This implies also that there theBorn result is satisfactory. This is due to the form ofthe one-particle scattering operator (2). All curves tendtoward zero for high particle energies. The weak-field ap-proximation is an improvement compared to the Born ap-proach. The strong-field is generally closer to the exactresult than the weak-field approximation. The Abelianapproximation scheme is closest to the exact values. Thelargest deviations are found for small energies and largevalues of the parameter Aint0. While for low values ofAint0 the Born approximation is reasonably good it is notappropriate for large values.
The generalisation to space-time dependent fields canbe found in [3].
References
[1] D. D. Dietrich, Phys. Rev. D 68 (2003) 105005.[2] R. S. Bhalerao and G. C. Nayak, Phys. Rev. C 61
(2000) 054907.[3] D. D. Dietrich, arXiv:hep-th/0402026.
- 91 -
Quasiparticle Description of Hot QCD at Finite Quark ChemicalPotential ∗)
M.A. Thalera, R.A. Schneidera, and W. Weisea,b
aTU Munchen, Garching; bECT∗, Trento
It is possible to describe the EOS of hot QCD at vanish-ing quark chemical potential µ to good approximation bythe EOS of a gas of quasiparticles with thermally gen-erated masses, incorporating confinement effectively bya temperature-dependent, reduced number of thermody-namically active degrees of freedom [1]. Lattice thermody-namical quantities such as the pressure, the energy densityand the entropy density are accurately reproduced in thetemperature range Tc < T ∼< 4Tc. This model is extendedto finite quark chemical potential in a thermodynamicallyself-consistent way [2].
The pressure of an ideal gas of quark and gluon quasi-particles with effective masses depending on temperatureand quark chemical potential, is given by
p(T, µ) = −B(T, µ) +νg6π2
∫ ∞
0
dkC(T, µ)fB(Egk)k4
Egk
+Nc
3π2
Nf∑q=1
∫ ∞
0
dkC(T, µ)[f+D (Eq
k) + f−D (Eqk)]
k
Eqk
.
Eik =
√k2 +m2
i (T, µ;G(T, µ)) is the quasiparticle en-ergy, G(T, µ) is the effective coupling strength and C(T, µ)parametrizes the reduction of thermally active degreesof freedom due to the onset of confinement close to Tc.B(T, µ) acts as a background field and is necessary tomaintain thermodynamic consistency. The expressions forthe effective coupling G(T, µ = 0) and the confinement fac-tor C(T, µ = 0) from Ref. [1] are generalized to finite quarkchemical potential using Maxwell relations. The charac-teristic curves of constant confinement C(T, µ) =const areshown in Fig.1.
In our quasiparticle model, the sudden decrease of thepressure, the energy density, the quark number density andthe entropy density when Tc is approached from above, isparametrized by the confinement factor C(T, µ). Conse-quently, it is natural to relate the critical line to the char-acteristic curve of the confinement factor through Tc(µ).The lattice phase boundary line and our result is shown inFig.1.
0 2 4 6 8 10 12µ/T
c
0
1
2
3
4
T/T
c
0 0.5 1 1.5 2 2.5 3µ
B/T
c
0.9
0.92
0.94
0.96
0.98
1
T/T
c
LatticeQPM
Figure 1: Left figure: C(T, µ) = const. Right figure: Thephase boundary line Tc(µ) calculated with the quasiparti-cle model for Nf = 3. The shaded band shows the 1 − σerror band obtained in lattice calculations in ref Ref. [3].
Thermodynamical quantities have been calculated in re-cent lattice computations using a p4-improved staggeredaction on a 163 × 4 lattice [4]. There, the Nτ dependenceis known to be small, in contrast to standard staggeredfermion actions which show substantially larger cut-off ef-fects. Our quasiparticle model accurately reproduces thepressure discrepancy ∆p(T, µ) = [p(T, µ)−p(T, 0)]/T 4 andthe quark number density nq = ∂p(T, µ)/∂µ. Results areshown in Figs.2 and 3.
0.8 1 1.2 1.4 1.6 1.8 2T/T
c
0
0.05
0.1
0.15
0.2 ∆p/T4
µ/Tc=0.6
µ/Tc=0.4
µ/Tc=0.2
0.8 1 1.2 1.4 1.6 1.8 2T/T
c
0
0.2
0.4
0.6∆p/T
4
µ/Tc=1.0
µ/Tc=0.8
Figure 2: The normalized pressure difference ∆p(T, µ)/T 4
as a function of temperature compared to lattice resultsfrom Ref. [4] (symbols).
0.8 1 1.2 1.4 1.6 1.8 2T/T
c
0
0.2
0.4
0.6
0.8 nq/T
3
µ/Tc=0.6
µ/Tc=0.4
µ/Tc=0.2
0.8 1 1.2 1.4 1.6 1.8 2T/T
c
0
0.5
1
1.5
2n
q/T
3
µ/Tc=0.8
µ/Tc=1.0
Figure 3: The normalized quark number densitynq(T, µ)/T 3 as a function of temperature compared to Ref.[4] (symbols).
∗) Supported in part by BMBF and GSI.
References
[1] R. A. Schneider and W. Weise, Phys. Rev. C 64,055201 (2001).
[2] M. A. Thaler, R. A. Schneider and W. Weise,arXiv:hep-ph/0310251; Phys. Rev. C (2004), in print.
[3] P. de Forcrand and O. Philipsen, Nucl. Phys. B 673,170 (2003).
[4] C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek,F. Karsch, E. Laermann and C. Schmidt, Phys. Rev.D 68, 014507 (2003).
- 92 -
Relativistic Fermi-liquid theory of dense matter
Kai Hebeler and Bengt Friman (GSI)
Low temperature many-fermion systems can be de-scribed by Landau Fermi-liquid theory [1]. The parametersof the theory can be determined either empirically or by di-rect calculation. However, for gauge theories (QED/QCD)there are problems. We explore the Fermi-liquid propertiesof relativistic electron and quark systems at small temper-atures and high densities. Polarization contributions tothe quasiparticle effective interaction are computed in therandom-phase approximation 1.
= +
= +
Figure 1: Effective interaction and quasiparticle propaga-tor in random-phase-approximation
At low temperatures longitudinal gauge bosons acquire amass in the static limit, the Thomas-Fermi massMTF . Onthe other hand, the transverse ones remain massless:
ΠL(k0,k) = −M2TF − iπM
2TF
2vfk0|k| coth
k02T
(1)
ΠT (k0,k) = +iπvfM
2TF
4k0|k| coth
k02T
. (2)
Here vf denotes the Fermi-velocity of the particles.Due to the static screening in the longitudinal part, the
exchange of a longitudinal boson is infrared finite. Con-versely, the transverse part diverges at zero temperaturedue to the lack of static screening. However, at finite tem-peratures the transverse channel is screened by the imag-inary part in (2). This dynamical screening renders thetransverse contribution finite. To leading order in T andω = (p0 − µ) one finds for the fermion self-energy
ImΣ+(ω) = −e2vf24π
ω (3)
The corresponding single-particle strength of the quasipar-ticle is given by:
Z−1 = 1 − ∂ReΣ+(ω)∂ω
∣∣∣∣ω=0
= 1 − e2vf12π2
ln cT , (4)
where c is a constant. Thus, at temperature strictly zero,the Z-factor and the discontinuity at the Fermi surface van-ish. Furthermore, the Landau parameters diverge. Con-sequently, such a system is not a normal Fermi liquid [2].However, at finite temperatures the singularity is washedout and the system may be described as a Fermi liquid.
At non-zero temperature also the Landau parametersare finite. For the one-photon-exchange interaction of Fig.2 one finds [3]:
1We present only the QED results here. The QCD results areobtained by a straightforward generalization.
p p+ q
p′ p′ + q
p
p′p′
p
Figure 2: Interaction processes on the Fermi-surface.The interaction is particle-hole reducible in the u-channel(right) and remains irreducible in the t-channel (left).
fgespσ,p′σ′ = fgesL
pσ,p′σ′ + fgesTpσ,p′σ′ (5)
fgesLpσ,p′σ′ =
e2
q2+ fLpp′ + gLpp′σ · σ′ +
jLpp′
p4fS12(p× p′)
+kLpp′
p2f(σ × σ′) · (p× p′)
fgesTpσ,p′σ′ =
e2
q2 + iε[p · p′]⊥q
µ2+ fTpp′ + gTpp′σ · σ′ +
+hTpp′
p2fS12 (q′) +
kTpp′
p2f(σ × σ′) · (p× p′)
where S12(p) = [σ · p] [σ′ · p] − p2
3 σ · σ′, q′ = p − p′
and
fL/Tpp′ , g
L/Tpp′ , h
Tpp′ , jLpp′ , k
L/Tpp′ ∝ 1
q′2 −ΠL/T (0,q′).
The Landau parameter F1 is related to the effectivemass [4]. To leading order in T one finds
m∗
µ≡ 1 +
F1
3= − e
2vf12π2
ln[e2
vf
T
µ
]. (6)
Thus, the leading terms of the effective mass and the Z-factor are identical as they should be. The dynamicalscreening at finite temperatures regularizes the singulari-ties in the self energy and the effective interaction in a con-sistent manner. In quark matter the temperature, abovewhich the singularity is effectively washed out, is 10−4µ,while for metals it is so small that the singularity can safelybe neglected [5].
References
[1] P. Noizieres and D. Pines, Addison-Wesley, Re-wood City (1988)
[2] D. Boyanovsky and H. J. de Vega, Phys. Rev. D63(2001) 034016; M. Bellac und C. Manuel, Phys.Rev. D55 (1997) 3215-3218
[3] K. Hebeler, Diplomathesis 2003; K. Hebeler and B.Friman, to be published
[4] G. Baym and S.A. Chin, Nucl. Phys. A262 (1976)527-53
[5] T. Holstein et al., Phys. Rev. D8 (1973) 2649
- 93 -
Recent progress in weak-coupling color superconductivity
A. Schmitt, D. Hou, Q. Wang, D.H. Rischke
Institut fur Theoretische Physik, J.W. Goethe-Universitat, D-60054 Frankfurt/Main, Germany
We have calculated the polarization tensor Πµνab (P ),
a, b = 1, . . . , 8, γ, for gluons and photons in different color-superconducting phases. We have explicitly computed itszero-energy, low-momentum limit for the two-flavor colorsuperconducting (2SC), color-flavor-locking (CFL), polar,and color-spin-locking (CSL) phases, which yield the De-bye and Meissner masses. These masses determine thescreening length of electric and magnetic fields. Parts ofthe results were already known in the literature, namelythe gluon Debye and Meissner masses for the spin-0 phases2SC and CFL [1, 2]. Our result for the photon Debyemass in the 2SC phase differs from that of Ref. [3] sinceour calculation shows that the photon-gluon mass matrixis already diagonal and thus electric gluons do not mixwith the photon (while in Ref. [3] the same diagonal ma-trix was rotated and a “mixed mass” was obtained). Themasses for the spin-1 phases have been computed here forthe first time. For the polar phase, we have shown thatthere is mixing between the magnetic gauge bosons but, asin the 2SC phase, no mixing of the electric gauge bosons.In a system of one quark flavor, this mixing leads to a van-ishing Meissner mass. However, for more than one quarkflavor, we have shown that, if the electric charges of thequark flavors are not identical, there is an electromagneticMeissner effect in the polar phase, contrary to both con-sidered spin-0 phases. For the CSL phase, we have foundthe remarkable result that, for any number of flavors, nei-ther electric nor magnetic gauge fields are mixed. Sincethere is no vanishing eigenvalue of Πµν
ab (0), all eight glu-ons and the photon (electric as well as magnetic modes)become massive and there is an electromagnetic Meiss-ner effect. For polar phase, although there is a massslessnew photon for each flavor, the electromagnetic Meissnereffect still exists for a color superconductor with three fla-vors of quarks because there is no unique mixing angle forthe new photon. We argued that, in spite of a suppres-sion of the gap of three orders of magnitude compared tothe spin-0 gaps [5, 6], spin-1 gaps might be preferred in acharge-neutral system. The reason is that a mismatch ofthe Fermi surfaces of different quark flavors has no effecton the spin-1 phases, where quarks of the same flavor formCooper pairs. Therefore we found that a compact stellarobject with a core consisting of quark matter in a spin-onecolor-superconducting state is, with respect to its electro-magnetic properties, different from an ordinary neutronstar: a spin-one color superconductor is an electromag-netic superconductor of type I, while an ordinary neutronstar is commonly believed to be of type II. We note that atype-I superconductor could provide one possible explana-tion for the observation of pulsars with precession periodsof order 1 year [7].
We also derived a generalized Ward identity from QCDfor dense, color-superconducting quark matter. The iden-tity implies that, on the quasi-particle mass shell, the gapfunction and the quasi-particle dispersion relation are in-
dependent of the gauge parameter in covariant gauge upto subleading order. We have shown that, to subleadingorder, the gauge dependence of the quark self-energy aris-ing from the gauge-dependent part of the gluon propagatorvanishes on the mass shell. In principle, however, othergauge-dependent terms arise from the gauge dependenceof the full vertex and of the full quark propagator , whencombined with the physical part of the gluon propagator.We will show that these two cases do not provide addi-tional subleading contribution to the gap function on themass-shell [10, 11]. Our result shows that in order to ob-tain a gauge-independent gap function up to subleadingorder, one has to use the full vertex as well as the fullfermion propagator in the Nambu-Gor’kov basis. A con-sequence is that the prefactor exp(3ξ/2) to the gap func-tion found in the mean-field approximation [8, 9] will beremoved by contributions from the full qqg vertex whentaking the gap function on the quasi-particle mass shell.An explicit diagrammatic proof of this statement will bepresented elsewhere [11].
References
[1] D.H. Rischke, Phys. Rev. D 62, 034007 (2000), Phys.Rev. D 62, 054017 (2000).
[2] D.T. Son and M.A. Stephanov, Phys. Rev. D 61,074012 (2000).
[3] D.F. Litim and C. Manuel, Phys. Rev. D 64, 094013(2001).
[4] A. Schmitt, Q. Wang, and D.H. Rischke, Phys. Rev.Lett. 91, 242301 (2003).
[5] T. Schafer, Phys. Rev. D 62, 094007 (2000).[6] A. Schmitt, Q. Wang, and D.H. Rischke, Phys. Rev.
D 66, 114010 (2002).[7] B. Link, astro-ph/0302441.[8] D.K. Hong, V.A. Miransky, I.A. Shovkovy, and
L.C.R. Wijewardhana, Phys. Rev. D 61, 056001(2000); Erratum, 62, 059903 (2000); I.A. Shovkovy,L.C.R. Wijewardhana, Phys. Lett. B 470, 189 (1999).
[9] D.K. Hong, T. Lee, D.-P. Min, D. Seo, and C. Song,Phys. Lett. B 565, 153 (2003).
[10] A. Mishra, Q. Wang, and D.H. Rischke, in prepara-tion.
[11] D. Hou, Q. Wang, and D.H. Rischke, in preparation.
- 94 -
Gapless color superconducting phases of dense quark matter
I.A. Shovkovy
Institut fur Theoretische Physik, J.W. Goethe-Universitat, 60054 Frankurt am Main, Germany
Sufficiently cold and dense baryonic matter is expectedto be a color superconductor. The corresponding groundstate is characterized by a condensate of Cooper pairsmade of quarks. Since the latter carry color charges, theSU(3)c color gauge group of strong interactions is partiallyor completely broken through the Anderson-Higgs mecha-nism. At asymptotic densities, this phenomenon was stud-ied from first principles, suggesting that dense quark mat-ter has a very rich phase structure.
It is natural to expect that in nature some color super-conducting phases may exist in the interior of compactstars. The estimated central densities of such stars mightbe as large as 10ρ0 (where ρ0 ≈ 0.15 fm−3 is the satura-tion density), while their temperatures are in the range oftens of keV. This could provide favorable conditions forthe diquark Cooper pairing of color superconductivity.
In theoretical studies, it is important to take into ac-count that matter in the bulk of compact stars shouldbe neutral (at least, on average) with respect to electri-cal as well as color charges. Also, such matter should re-main in β-equilibrium. Satisfying these requirements mayimpose nontrivial relations between the chemical poten-tials of different quarks. In turn, such relations could sub-stantially influence the pairing dynamics between quarks,for instance, by suppressing some color superconductingphases and by favoring others. This was the main motiva-tion for the study in Ref. [1]. The results, however, werebeyond all expectations.
In Refs. [1, 2] it was found that neutral 2-flavor quarkmatter in β-equilibrium can have a rather unusual, stableground state which was called a gapless color supercon-ductor (g2SC). While the symmetry of the g2SC groundstate is the same as that of a conventional 2-flavor colorsuperconductor, the low energy spectrum of the fermionicquasiparticles is different. In particular, two out of fourgapped quasiparticles of the conventional 2SC phase be-come gapless in the g2SC phase. In addition, the numberdensities of the pairing quarks in g2SC at zero tempera-ture are not equal. For example, the density of red (green)up quarks is different from the density of green (red) downquarks. This is in contrast to the conventional “enforced”pairing [3] where the corresponding densities are equal.
It appears that the g2SC phase has very unusual finitetemperature properties. These were studied in detail inRef. [2]. Among the most striking features of the gaplesssuperconductivity are (i) the phase transition between theg2SC phase and normal quark matter is a second orderphase transition; (ii) the gap parameter is not a mono-tonic function of temperature, see Fig. 1. In fact, at somevalues of the diquark coupling strength, a nonzero finitetemperature gap may appear even if its zero temperaturevalue is vanishing; (iii) the ratio of the critical temperatureand the zero temperature gap is not a universal numberin the gapless phase. Moreover, this ratio could becomearbitrarily large at some values of the coupling constant.
10 20 30 40 50T (MeV )
20
40
60
∆(MeV
)
η=0.64
η=0.66
η=0.684
η=0.70
η=0.75
Figure 1: The temperature dependence of the diquark gapin neutral quark matter calculated for several values of thediquark coupling strength η = GD/GS .
Recent studies suggest that gapless superconductingphases might be very common. For example, it was shownin Ref. [4] that 3-flavor quark matter can have a stablegapless color-flavor locked phase when the strange quarkmass is nonzero. Also, similar gapless phases may existin asymmetric nuclear matter [5] and in two-componentmixtures of cold fermionic atoms [6].
One may argue that the g2SC phase of quark mattershould appear inside central regions of compact stars. Thiswould be natural since this phase is neutral with respectto electrical and color charges, and it is β-equilibrated byconstruction. Also, this phase is more stable than normalquark matter under similar conditions [1, 2]. If this is in-deed the case, one may have a chance to detect the indirectsignatures of its presence by deciphering the observationaldata from stars. The study of the physical properties ofthe g2SC phase in Refs. [1, 2] is the first step in this di-rection. In future, one also needs to study the transportproperties of the g2SC phase. For example, the knowledgeof the neutrino emissivities and mean free paths would becrucial for the understanding of the cooling rates of starswith such a gapless phase in their interior.
References
[1] I.A. Shovkovy and M. Huang, Phys. Lett. B 564(2003) 205.
[2] M. Huang and I.A. Shovkovy, Nucl. Phys. A 729(2003) 835.
[3] K. Rajagopal and F. Wilczek, Phys. Rev. Lett. 86,3492 (2001).
[4] M. Alford, C. Kouvaris and K. Rajagopal, hep-ph/0311286.
[5] A. Sedrakian and U. Lombardo, Phys. Rev. Lett. 84,602 (2000).
[6] W. V. Liu and F. Wilczek, Phys. Rev. Lett. 90,047002 (2003).
- 95 -
Effect of color superconductivity on the mass and radius of a quark star
Stefan B. Ruster1 and Dirk H. Rischke1
1Institut fur Theoretische Physik, Johann Wolfgang Goethe-Universitat, 60054 Frankfurt am Main, Germany
At sufficiently high densities and sufficiently low tem-peratures quark matter is a color superconductor [1]. Innature, color-superconducting quark matter could exist incompact stars. Among the best known properties of thoseobjects are their masses and radii. We want to investigatehow much color-superconductivity changes these proper-ties of a compact star. Therefore we compare quark starsmade of color-superconducting quark matter to normal-conducting quark stars and focus on the most simple color-superconducting system, a two-flavor color superconduc-tor. As the transition to hadronic matter introduces an-other degree of freedom which may either mask or enhancethe effects of color superconductivity, we do not considerhybrid stars, but focus exclusively on pure quark stars.
We use a Nambu–Jona-Lasinio (NJL) model [2] to com-pute the gap parameter and the equation of state. Becausewe want to know how large the color-superconducting gapparameter has to be in order to see substantial changesin the mass and radius of a compact star, we vary thestrength of the four-fermion coupling of the NJL model.We also introduce a cutoff to render the momentum inte-gral finite.
Compact stars have to be neutral with respect to electriccharge. In order to achieve this, we have to add electrons.If the chemical potential for strange quarks exceeds thestrange quark mass, we also have to include strange quarksinto our consideration. We assume them to be noninter-acting. Compact stars not only have to be neutral withrespect to electric charge but also with respect to colorcharge due to confinement. In order to achieve this, wesolve the neutrality conditions in our equation of state.We also have to substract the pressure of the perturbativevacuum in the form of the MIT bag constant.
To compute the masses and radii of compact starsfrom the equation of state, we have to solve the Tolman-Oppenheimer-Volkoff (TOV) equation.
Our results [3] show that the color-superconducting gapparameter φ is of the order of 100 MeV. If the couplingconstant exceeds a critical value, the gap parameter doesnot vanish even at zero density, see Fig. 1. For couplingconstants below this critical value, mass and radius of acolor-superconducting quark star change at most by∼ 20%compared to a star consisting of normal-conducting quarkmatter. For coupling constants above the critical value,mass and radius may change by factors of two or more,see Fig. 2.
References
[1] D. Bailin, A. Love, Phys. Rept. 107, 325 (1984)[2] Y. Nambu, G. Jona-Lasinio, Phys. Rev. 122, 345
(1961); Phys. Rev. 124, 246 (1961)[3] S. B. Ruster and D. H. Rischke, nucl-th/0309022, to
be published in Phys. Rev. D
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
φ [G
eV]
µ [GeV]
50%100%150%200%
Figure 1: The gap φ as a function of the quark-chemicalpotential µ. Full line: the coupling constant is reduced bya factor of two as compared to the default value, for whichthe corresponding gap is shown as the dashed line. Short-dashed line: the coupling constant is 3/2 times the defaultvalue; dotted line: the coupling constant is twice the de-fault value. All gaps are for electric- and color-neutralmatter including strange quarks.
0
0.5
1
1.5
2
2.5
3
0 2 4 6 8 10 12 14 16
star
mas
s [M
° ⋅]
star radius [km]
φ=0φ>0, 50%φ>0, 100%φ>0, 150%φ>0, 200%
Figure 2: The mass-radius relation of electric- and color-neutral quark stars containing strange quarks. The full lineis for normal-conducting quark stars, all other lines are forcolor-superconducting quark stars. Dashed line (almost in-distinguishable from the full line): the coupling constantis 50% of its default value. Short-dashed line: the couplingconstant assumes its default value. Dotted line: the cou-pling constant is multiplied by a factor 3/2. Dash-dottedline: the coupling constant is twice as large as the defaultvalue.
- 96 -
How to detect the Quark-Gluon Plasma with Telescopes
M. Hanauske
Institut fur Theoretische Physik, J. W. Goethe–Universitat, D-60054 Frankfurt, Germany
Abstract
The appearance of the QCD - phase transition (QPT) atlow temperatures and high densities will change the prop-erties of neutron stars (NS). Whether this change will bevisible with telescopes and gravitational wave antennas de-pends strongly on the equation of state (eos) of hadronicand quark matter and on the construction of the phasetransition (PT).
1 Introduction
If the onset of the QPT at low temperatures is below≈ 5ρ0, ρ0 := 0.15 fm−3 a PT is going to happen in theinterior of NS. The accepted underlying theory of stronginteractions, QCD, is however not solvable in the nonper-turbative regime. So far numerical solutions of QCD on afinite space-time lattice are unable to describe infinite nu-clear matter at low temperatures and high densities. As analternative approach several effective models of hadronicand quark interactions have been proposed. By choosingdifferent models (and/or parameter sets inside the mod-els) for the hadronic and quark phase, the compositionof particles inside the stars, the eos and as a result, theproperties of the stars will change. The construction ofthe QPT between the models should be done by a Gibbs-construction, having two independent chemical potentials(µB, µQ). However the Gibbs-construction can be modi-fied due to surface tension and coulomb effects. Such ef-fects would shift the PT to be more ”maxwell-like”, havingan almost constant pressure during the mixed phase. Thestar properties depend strongly on the onset of the PT andespecially if it begins before or after the appearance of hy-peronic particles in the hadronic phase (ρB ≈ 2.5ρ0). De-pending on the eos the properties of the stars will change;but finally just one eos will have been realized in nature.
2 Properties of Compact Stars
Depending on the used model the following varieties ofcompact stars (CS) are possible:NS: no PT, solely hadrons; hybrid stars (HS): PT; nakedquark stars (QS):no PT, solely quarks and exotic stars(like stars with kaon condensation or strongly boundhyperon stars (HyS)). In Fig.1 the star properties areplotted for three hybrid models (NLZY-B180(Gibbs-PT),NLZY-B175(Maxwell-PT) [3] and CH-NJL-2SC(2 Gibbs-PT) [4]), a naked quark model (NJL (ξ = 0) [2]) and ahybrid and naked hyperon star model [1]. Some of thehybrid models show a twin star behavior, where the thirdsequence of HS is separated from the second one by a un-stable region (green part of the curves). Such a twin starbehavior is unusual in hybrid models using Gibbs-PT; inhybrid models using maxwell-PT twins appear quite often.Because of their selfboundness at ’low’ energy densities,the QS and HyS have lower radii than the NS and HS.
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
M
4 6 8 10 12 14 16 18R
RS = 2M 9
8RS
3
2RS
@@@R
[1]
[1]
[2] [3]
[4]
R1 = 8 10 12 14
Figure 1: Mass M [M] and radius R [km] for hybrid,quark and hyperon stars. The Schwarzschild radius RS =2M , the absolute threshold for stable stars (R = 9/8RS),the photon surface (R = 3/2RS) and R∞ =const lines.
3 Astrophysical Observables for the QGP
NS are usually bigger than QS and third sequence HS.Telescope are able to measure the radiation radius R∞from thermal radiating stars. The measure of two differ-ent radii of stars having the same mass would proof theexistence of a twin and therefore of a strong PT. QS canrotate faster than NS whereas in HS a spin up effect can oc-cur. Single quadrupodeformed CS will emit gravitationalwaves with twice their rotation frequency. During a twinstar collapse energy is emitted by neutrinos, high energeticphotons and gravitational waves. Mergers of two NS andtwo QS will emit different gravitational waves profiles.Acknowledgments
I thank the Gesellschaft fur Schwerionenforschung-forschung for their financial support. I also gratefully ac-knowledge the Saha Institute for Nuclear Physics (Kolkata,India) for their hospitality and the DAAD/DST fortheir financial support during the days of writing thisreport. I thank all authors for their work, Prof.Dr.D.Bandyopadhyay for fruitful discussions and HD Dr.J.Schaffner for his comments on this report.
References
[1] J. Schaffner-Bielich, M. Hanauske, H. Stocker, and W.Greiner, Phys. Rev. Lett.88(2002) 171101
[2] M.Hanauske, D.Zschiesche, S.Schramm et.al.,http://wave.xray.mpg.de/conferences/xeus-workshop
[3] I. N. Mishustin, M. Hanauske, A. Bhattacharyya, L.M. Satarov, et.al., Phys. Lett. B552(2003)
[4] Igor Shovkovy, Matthias Hanauske and Mei Huang,Phys. Rev. D66(2003)
- 127 -
Nuclear Sizes and Isotopic Shift
M. Tomaselli1,2, L.-C. Liu3, T. Kuhl1,4, S. Fritzsche5, D. Ursescu1,4, and P. Neumayer1,2
1Gesellschaft fur Schwerionenforschung mbH, Planckstr. 1, D-64291 Darmstadt, Germany; 2TU Darmstadt, Institutfur Kernphysik, Schlossgartenstr. 9, 64289 Darmstadt; 3Los Alamos National Laboratory, USA;
4Johannes-Guttenberg-Universitat Mainz,Germany; 5Universitat Kassel, Heinrich-Plett-Str.40 D-34132, Germany
We study the charge radii of exotic nuclei through nu-clear calculations and by non-perturbative isotopic-shift(IS) evaluations. Beside nuclear masses (or binding en-ergies), nuclear charge radii, spins, and nuclear momentsare informations on ground state properties of atomic nu-clei. The latter can be obtained by atomic spectroscopy.In fact, most information on the static properties of ex-otic systems have been determined in this way [1]. For agiven electronic transition, the IS is the sum of: A) themass shift (MS) originating from the finite mass of thenucleus and the electron-electron correlation, and B) thefield shift (FS) that reflects the differences in the nuclearcharge distributions. Although the information on nuclearground-state properties extracted from a study of hyper-fine structure and isotope shift is model-independent, it ishampered in complex neutral atoms by the accuracy withwhich the electron wave functions are known at the site ofthe nucleus. However, in the case of simple few-electronsystems the electron wave function can be precisely cal-culated. Recent advances in variational calculations forlithium and lithium-like ions using multiple basis set inHylleraas coordinates [2], made possible to calculate theMS in the 2S-3S and 2S-2P transitions of lithium witha very good accuracy. Therefore, if the overall isotopicshift can be measured with a comparable precision, therms charge radius can be extracted. Accordingly [2], thecharge radius of the isotopes is given by:
δ(IS)exp = δ(MS)the +2πZ3δ|ψ(0)|2δ < r2 > (1)
In this way, absolute charge radii can be determined. Fur-thermore, in combination with measurements of the mat-ter radius, neutron radii can be extracted. For the future,experiments on stable and long-lived lithium-like ions upto uranium are planned at the ESR and NESR. This re-quires the availability of a reliable soft-x-ray laser, whichwas recently demonstrated at PHELIX [3]. A recent ex-periment on radioactive lithium isotopes [4], aiming fora determination of the charge and the neutron radius of11Li, provides an excellent example for a test of nuclear andatomic theories. Our nuclear computations of the chargeradius are performed in the framework of the dynamic-correlation model DCM for nuclei with an odd numberof valence particles, and in the boson dynamic-correlationmodel (BCDM) for those with an even number of valenceparticles [5]. These nuclear models take fully into consid-eration the correlation between valence particles as well asbetween valence and core particles. Consequently, thesecomputations may reveal feature physics which is asso-ciated to the strong correlation between the valence andthe core polarized states. Moreover, we propose to testthe derived charge radii within the isotopic shift theory inwhich the electronic transitions for lithium and lithium-like ions are calculated by considering the three corre-
lated electrons described by a model similar to the nu-clear DCM. Within this nonlinear and non-perturbativemodel, the treatment of the halo of the proton distributioncan be performed self-consistently. The proposed theoret-ical method is applied to two specific problems: a) high-resolution isotope shift calculation on unstable lithiumisotopes, and b) measurement of the 2p-2s transition inlithium-like uranium. In both ranges of isotopes is per-formed non-perturbative IS calculation which is appropri-ate for calculating charge radii in halo- and exotic nuclei.Nuclear-model-independent rms charge radii can be thenobtained from IS calculations. The IS is evaluated as in [6]in terms of the rms radius using an electron distributioncalculated selfconsistently. At this stage of the calcula-tion, in order to test from one side the non perturbativeelectron dynamic-correlation model (eDCM) presented inRef. [7] and from the other side the charge radii as de-rived from the charge distributions [8], we propose to in-sert these calculated values in Eq. 1. The calculated MSand the theoretical charge radii should then reproduce theexperimental IS.
Preliminary results calculated using eDCM for the elec-tron space are given in table 1. One example is the bindingenergy in the atomic lithium system without nuclear cor-rections, the other the 2s-2p transition energy in lithium-like 235U91+.Energies in lithium (au) 1s22sDrake [9] -7.47806032310 (31)eDCM this work -7.478060733Transitions in 235U91+ (eV) 2s-2pYerokhin [10] 288.44(20)eDCM this work 288.33
References
[1] E.W. Otten, Treatise on Heavy-ion Science, D.A.Bromley ed. Vol. 8 (1988) 515, Plenum Press, NewYork 1988
[2] G.W.F. Drake, Z.C. Yan, Phys. Rev.A 46 (1992) 2378[3] S. Borneis, Th. Kuhl et al., Hyperfine Interactions
127 (2000) 315.[4] W. Nortershauser et al., Nucl. Instr. and Met. in Phys.
Res. B204 (2003) 644; W. Nortershauser et al., to bepublished.
[5] M. Tomaselli et al., J. Opt. B5 (2003) 395.[6] E.C. Seltzer, Phys. Rev. 188 (1969) 1916.[7] M. Tomaselli et al., Can.J. Phys. 80 (2002) 1347.[8] M. Tomaselli et al., Dynamical Aspect of Nuclear Fis-
sion, J. Kliman et al. eds. (2002) 445, World Scientific,New Jersey 2002.
[9] Z.-C. Yan and G.W.F Drake, Phys. Rev. A 61 (2000)022504.
[10] V.A. Yerokhin, A.N. Artemyev, V.M. Shabaev et al.,Phys. Rev. Lett. 85 (2000) 4699.
- 128 -
Ab-initio QED Treatment of Electron-Correlation Effects andDeduction of Nuclear Parameters in Highly-Charged Ions
I. Bednyakov1, T. Beyer1, F. Erler1, G. Schaller1, S. Schumann1, J. Winter1, G. Plunien1, G. Soff1,A.N. Artemyev2, K.V. Koshelev2, L.N. Labzowsky2, V.M. Shabaev2, and V.A. Yerokhin2
1Institut fur Theoretische Physik, TU Dresden; 2Department of Physics, St. Petersburg State University
The accuracy reached in experimental and theoreticalinvestigations of the low-lying states in lithiumlike ionsprovides a promising tool for probing QED corrections upto second order in the finestructure constant α and, inprinciple, for the determination of nuclear parameters viaatomic spectroscopy. Theoretical uncertainties in the de-scription of electron-correlation effects as they are inher-ently generated in evaluations based on relativistic many-body perturbation theory (MBPT) can be improved byperforming ab-initio QED calculations up to order α2
or higher. Within the framework of bound-state QEDinterelectron-interaction effects are mediated by the ex-change of virtual photons described by the fully relativis-tic photon propagator. In order to achieve proper renor-malization and to provide gauge-invariant and consistentresults for the energy shift one has to evaluate simultane-ously a suitable set of Feynman diagrams up to a givenorder in α.
In recent experiments performed at the ESR facilityat GSI the 2p1/2 − 2s splittings in very heavy lithium-like ions have been determined with high precision uti-lizing low-energy dielectric recombination [1]. A compari-son between experimental data and theoretical predictionsbased on rigorous QED calculations [2] is presented inTable 1. The theoretical values account for various cor-rections: finite-nuclear size and nuclear recoil, one-, two-and three-photon exchange corrections, one-electron self-energy and vacuum-polarization corrections of order α aswell as for screened self-energy and vacuum-polarizationeffects. So far an excellent agreement between theory andexperiment can be stated. Also the 2p3/2 − 2s transitionenergy in lithium-like bismuth has been determined exper-imentally with an accuracy of 0.04 eV [3]. Provided thatQED corrections have been calculated with sufficient accu-racy one can utilize the knowledge about the atomic struc-ture to probe nuclear physics (determination of nuclearparameters, test of specific nuclear models, etc.). E.g.,the magnetic dipole, octupole and the electric quadrupolemoments of the 209
83 Bi nucleus can be deduced from thehyperfine-structure splittings (HFS) of the 2p3/2 state.Preliminary studies have been performed recently [4] uti-lizing the dynamic proton model. It describes HFS as dueto the interaction between the electron and the valenceproton and takes into account simultaneously the electricand magnetic nuclear moment distributions.
Extensive ab-initio QED calculations for the 2p3/2 − 2stransition energy in lithium-like ions with nuclear chargenumbers 20 ≤ Z ≤ 100 (see [5] and references therein)have been performed recently. The results for the variouscontributions of the two-photon exchange corrections tothe energy shift of the (2s)22p3/2 state in Li-like ions arepresented in Table 2. The subscripts ”dir” and ”exch” indi-cate direct and exchange contribution, respectively, while
Table 1: Experimental [1] and theoretical results [2] forthe 2p1/2 − 2s splittings in Li-like ions (in eV).
Ion Experiment Total theory19779 Au76+ 216.134(29)(39)(28) 216.17(13)(11)20882 Pb79+ 230.650(30)(22)(29) 230.68(6)(13)23892 U89+ 280.516(34)(22)(43) 280.64(11)(21)
Table 2: Two-photon exchange correction for the (2s)2p3/2state in Li-like ions (in atomic units) [5].
Z −∆E2eldir −∆E2el
exch −∆E3el Total20 0.03876 0.03902 -0.45509 -0.3773130 -0.12453 0.03715 -0.29807 -0.3854540 -0.18304 0.03465 -0.24844 -0.3968350 -0.21185 0.03156 -0.23126 -0.4115660 -0.22960 0.02795 -0.22801 -0.4296770 -0.24292 0.02392 -0.23233 -0.4513383 -0.25831 0.01822 -0.24511 -0.4851992 -0.26936 0.01406 -0.25752 -0.51281100 -0.28022 0.01030 -0.27070 -0.54061
the superscripts ”2el” and ”3el” refer to the two- andthree-electron contributions, respectively. The differencebetween QED and MBPT results can be envisaged as the”nontrivial” QED contribution to the interelectron inter-action. The complete MBPT results can be obtained fromQED calculations performed within the Coulomb gaugebut restricting the summations over intermediate Diracstates to the positive-energy spectrum only and separatingout the contributions due to the exchange of Coulomb andBreit photons, respectively. For the case under considera-tion the nontrivial QED contribution is essentially largerthan that for the 2p1/2 − 2s transition [2]. Moreover, forthe 2p3/2 − 2s transition the total correction changes itssign in the region between Z = 92 and Z = 100. Thedetailed studies in [5] represent an important step towardsthe evaluation of all two-electron QED corrections of orderα2 to the 2p3/2 − 2s transition energy for the Li isoelec-tronic sequence
References
[1] C. Brandau et al., Phys. Rev. Lett. 91 (2003) 073202.[2] V.A. Yerokhin et al., Phys. Rev. A64 (2001) 032109.[3] P. Beiersdorfer et al., Phys. Rev. Lett. 80 (1998) 3022.[4] K.V. Kosholev et al., Phys. Rev. A68 (2003) 052504.[5] A.N. Artemyev et al., Phys. Rev. A67 (2003) 062506.
- 129 -
Electron interaction and isotope effects studied by dielectronicrecombination with heavy few-electron ions
Zoltan Harman1, Roxana Schiopu2, Norbert Grun1, and Werner Scheid1
1Institut fur Theoretische Physik, Justus-Liebig-Universitat Giessen; 2Institut fur Physik,Johannes-Gutenberg-Universitat Mainz
The dynamics of electrons and their interaction arestrongly influenced by relativistic effects in very heavyatomic systems. The investigation of dielectronic recombi-nation (DR), or, the analogous process of resonant transferand excitation, has proved to be a suitable tool to studythese phenomena in highly charged ions [1, 2, 3].
The electrons interact by exchanging virtual photons.Thus, in transversal gauge of the photon field and in loworder, the operator responsible for capture is the sum ofthe Coulomb and generalized Breit operators [4]. The lat-ter has been shown to give an important contribution todielectronic capture rates in highly charged very heavyions [1, 2, 3], especially for transitions where inner-shellelectrons are involved. Therefore, here we consider the ef-fects of higher-order perturbative terms to the interaction.
The expansion of the transition operator yields two-photon exchange corrections, namely, ladder and crossed-photons diagrams. We evaluate these terms in the limitwhen the frequencies of both transversal photons approachzero. Figure 1 shows the differential cross section for DRinto U91+ at an electron energy of 68556 eV within theKL1/2L3/2 resonance group as a function of the angle ofthe hypersatellite photon, which is emitted in the transi-tion from the intermediate state to a singly-excited state.The effects of the two-photon exchange are small relativeto the one-photon exchange.
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180
diff
. cro
ss s
ectio
n (b
arn/
sr)
theta (deg)
one-photon ex.with two-photon ex.
Figure 1: Differential cross section in the ionic frame as afunction of the emission angle of the hypersatellite photonfor recombination into U91+ at an energy of 68556 eV.The full curve is calculated with one-photon exchange, thedashed curve includes the two-photon exchange correction.
The investigation of the DR process can also provide anew approach to obtain information about the charge dis-tribution of nuclei. At the maxima of the resonances in thetotal DR cross section the energy of the continuum elec-tron is equal to the difference of the energies of the initialand final bound atomic states. By performing experimentswith different isotopes, the resonances are shifted due tothe change of the charge distribution.
We made calculations of the resonance energies using the
multiconfiguration Dirac-Fock package GRASP of Dyall etal. [5] for relativistic elements with Z ranging from 54 (Xe)to 94 (Pu). The spherical Fermi distribution was takenwith parameters obtained from a fit to experimental val-ues [6]. Figure 2 shows the dependence of resonance energyshifts on the charge number Z. We compare two differentscenarios, namely, recombination into H-like ions with theexcitation of the bound K-shell electron and recombinationinto Li-like ions evoking transitions within the L-shell.
Figure 2: Difference between the resonance energies of twodifferent isotopes with A and A− 5 in the case of DR intoH-and Li-like ions as a function of the charge number Z.
As one can see in figure 2 for the case of initially H-like ions, the resonance with both electrons in the 2p3/2state is shifted the most and the 2s21/2 resonance the least.For recombination into Li-like ions (lower curves), the iso-topic variation of the resonance energies is even smaller.This is due to the fact that the electrons in the n = 2and n = 6 shells, actively involved in the reaction, have asmaller overlap with the nucleus. Even though the shiftsare smaller for initially Li-like ions with intra-shell exci-tations than in H- and He-like ions, the better precisionin measuring the resonance positions makes these systemsthe most promising candidates for the experimental obser-vation of nuclear volume effects [7].
References
[1] M. Gail et al., J. Phys. B 31 4645 (1998)[2] X. Ma et al., Phys. Rev. A 68 042712 (2003)[3] S. Zakowicz et al., Phys. Rev. A 68 042711 (2003)[4] J.B. Mann and W.R. Johnson, Phys. Rev. A 4 41
(1971)[5] K.G. Dyall et al., Comput. Phys. Commun. 55 425
(1989)[6] W.R. Johnson and G. Soff, At. Data Nucl. Data Ta-
bles 33 405 (1985)[7] A. Muller, private communication
- 130 -
Negative-continuum dielectronic recombination into n=2 states
A. N. Artemyeva,b, A. E. Klasnikovb,c, T. Beiera, J. Eichlerc, C. Kozhuharova, V. M. Shabaeva,b,c,T. Stohlkera, and V. A. Yerokhina,b
a Gesellschaft fur Schwerionenforschung, Planckstr. 1, 64291 Darmstadt, Germanyb St. Petersburg State University, Oulianovskaya 1, St. Petersburg 198504, Russiac Abteilung Theoretische Physik, Hahn-Meitner Institut, 14109 Berlin, Germany
In a collision between an electron and a bare heavy nu-cleus, radiative recombination (RR) is the main reactionchannel in a wide range of collision energies. In a re-cent work [1], we have investigated the so-called negative-continuum dielectronic recombination (NCDR) into theground state of a He-like ion where the incident electronis captured into the 1s state with simultaneous creation ofa free-positron–1s-electron pair:
XZ+ + e− → X(Z−2)+ + e+ .
This process may occur if the energy of the incident elec-tron in the nuclear rest frame is larger than the ground-state energy of the corresponding He-like ion plus thepositron rest energy. The maximum total cross sectionfor this process was found to be about 27µb for U92+ andabout 11µb for Pb82+, corresponding to projectile energiesof about 2 GeV/u in the electron-rest frame. The signa-ture of positron emission together with a twofold change ofthe projectile ion’s charge forms a very distinct signaturefor the NCDR.
The differential cross section for the NCDR process isgiven by (h = me = c = 1)
dσ
dΩf=
4π3|pf |N2
εfp2i
∑JM
∑ma,mb
∑mi,mf
∣∣∣ ∑P,κi,κf ,Mf
(−1)P ili+l′f
× exp(i∆κi + i∆κf)√
2li + 1
×Cji mi
li 0, (1/2)miCjf Mf
l′fml′
f, (1/2)−mf
Y ∗l′f,ml′
f
(−pf/|pf |)
×〈PaPb|I(εi − εPa)|(εi, κi,mi)(−εf , κf ,Mf )〉∣∣∣2 ,
where (ε, κ,m) is the electron wave function with energy ε,angular momentum and parity determined by κ, and angu-lar momentum projectionm. P is a permutation operator,Y and C denote spherical harmonics and Clebsch-Gordancoefficients, εi, pi and εf , pf denote energy and momen-tum of the incoming electron and outgoing positron, ∆κ
is a phase shift, and I is an expression related to thephoton propagator [2]. J and M refer to total angu-lar momentum and its projection of the formed He-likeion, and a and b denote the one-electron wavefunctions.N indicates a normalization factor (cf. [1]). The mu-tual interactions of the two electrons is of order 1/Z andnot considered here. We have now enlarged our inves-tigations to captures of one of the electrons into exitedstates and present the first results for the capture of oneof the electrons in an n = 2 state in Figs. 1 and 2. De-tailed results will be published elsewhere. In particular,it is clear from Fig. 2 that there is a range of angle, intowhich the positron is emitted only in case of both electronsbound in the ground state, which experimentally even al-lows to distinguish this from a capture of one of the elec-trons into an n = 2 state. Also, it is possible to estimate
the background to the process by measuring the positronyield outside the angular range of the NCDR positrons.
1000 1500 2000 2500 30000
0,5
1
1,5
2
2,5
3
Pb
1000 1500 2000 2500 30000
2
4
6
8U
1000 1500 2000 2500 30000
3
6
9
12
15
1000 1500 2000 2500 30000
10
20
30
40
(1s2s)
(1s2p1/2
)
(1s2p3/2
) (1s2p3/2
)
(1s2p1/2
)
(1s2s)
(1s)2+(1s2v) (1s)
2+(1s2v)
(1s)2
(1s)2
(1s2v) (1s2v)
Kinetic energy of the electron [kev]
Cro
ss s
ectio
n [µ
barn
]
Fig. 1: Total cross section of NCDR into various boundstates and the sum of cross sections into the (1s)2 and(1s2v) states as function of the energy of the electron.The energies are equivalent to about 1.6 GeV/u up to 5.5GeV/u of the projectile in the electron-rest frame.
0 5 10 15 20 25
1e-06
0,0001
0,01
1Pb
0 5 10 15 20 25 30
1e-06
0,0001
0,01
1U
(1s)2
(1s2s)
(1s2p1/2
)
(1s2p3/2
)
(1s)2
(1s2s)
(1s2p1/2
)
(1s2p3/2
)
Dif
fere
ntia
l cro
ss s
ectio
n [µ
barn
/sr]
Angle of positron emission [deg]
Fig. 2: Differential cross section of NCDR in the nucleus-rest frame for kinetic energies of the electrons of 1200 keV.The cross section folded by any finite angular resolutionremains finite at the maximum scattering angle (cf. [1]).This work was supported by the DFG (Grant No. 436RUS 113/616), by RFBR (Grant No. 01-02-04011),and by the Russian Ministry of Education (Grant No.PD02-1.2-79). A. E. K. acknowledges support by theRussian Ministery of Education (grant No. A03-2.9-219).
References
[1] A. Artemyev et al., Phys. Rev. A 67, 052711 (2003).[2] V. A. Yerokhin et al., Phys. Rev. A 62, 042712 (2000).
- 131 -
Cooling of Ions with Magnetized Electrons in Traps
B. Mollers, C. Toepffer, G. ZwicknagelInstitut fur Theoretische Physik II, Universitat Erlangen
In precision experiments like the planned QED-testswith highly charged ions in HITRAP [1] it is necessaryto work with cool ions in a trap. One possibility is elec-tron cooling: The ions are mixed with cold electrons andlose their energy because of the Coulomb interaction. Thismethod is well established in storage rings, and it is alsoused in traps. Because of the presence of a strong mag-netic guiding field 0B in the trap the cooling force 0F onthe ion cannot be calculated analytically. We developedseveral methods to calculate the cooling force on ions in amagnetized electron plasma. For the calculation of coolingtimes of the ions we use the binary collision model, wherethe energy transfer from individual collisions of ions andmagnetized electrons are accumulated. The energy trans-fer is calculated by treating the Coulomb interaction as aperturbation to the helical motion of the electrons up tosecond order including a correction for hard collisions [2,3].To estimate cooling times we calculated the energy of theions and the electrons as a function of time. For that cal-culation three effects are taken into account [1]:
- Energy loss of the ions to the electrons
- Heating of the electrons due to the energy transferfrom the ions
- Cooling of the electrons by emission of synchrotronradiation
This leads to three coupled differential equations for theion velocity parallel (vi‖) and transverse (vi⊥) to the mag-netic field and the electron temperature Te which have tobe solved numerically:
dvi⊥dt
=1MF⊥ (vi⊥, vi‖, Te)
dvi‖dt
=1MF‖ (vi⊥, vi‖, Te)
dTedt
= − 23kB
ninevidEi
ds(vi⊥, vi‖, Te)
− 1τe
(Te − Te,0) .
M is the ion mass, ni the ion density, ne the electron den-sity, Te,0 the temperature to which the trap is cooled andτe the time constant for cooling of the electrons by theemission of synchrotron radiation.Fig. 1 shows the ion energy Ei = 1
2Mv2i for U92+ ions
as a function of time with ne = 107 cm−3, B = 6 T,α : = (0vi, 0B) = 30 and an initial electron temperatureTe = 4 K for different ratios ni/ne. If the heating of theelectrons can be neglected (ni/ne → 0) the cooling timeis about 0.35 seconds. The cooling time increases withgrowing ion density since the increasing electron temper-ature results in a reduction of the cooling force. Fig. 2shows the electron energy Ee = 3
2kBTe as a function of
time. Initially the electrons are heated very fast by thehighly charged ions until an equilibrium with the coolingby emission of synchrotron radiation is reached and theelectron energy remains almost constant. Because of thehigh ion charge a test of the perturbation treatment un-derlying these results is desirable. We currently calculatethe cooling force in the framework of the Vlasov-Poissonequation, which accounts for all nonlinearities as well ascollective response [4,5]. - This work has been supportedby a GSI collaboration contract.
0
2
4
6
8
10
12
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
ion
ener
gy (k
eV/Z
)
time (s)
Z=92
ni / ne → 0
ni / ne = 10-4
ni / ne = 10-3
Figure 1
0
5
10
15
20
25
30
35
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
elec
tron
ener
gy (e
V)
time (s)
Z=92
ni / ne = 10-4
ni / ne = 10-3
Figure 2
References
[1] W. Quint et al., Hyp. Int. 132 (2001) 457.[2] C. Toepffer, Phys. Rev. A 66, 022714 (2002).[3] B. Mollers et al., Nucl. Instr. and Meth. B 205, 285
(2003)[4] B. Mollers et al., Nucl. Instr. and Meth. B 207, 462
(2003)[5] M. Walter et al., Nucl. Instr. and Meth. B 168, 347
(2000)
- 132 -
Polarization transfer in heavy hydrogen–like ions followingthe radiative capture of electrons
Andrey Surzhykov1, Stephan Fritzsche1, Thomas Stohlker2, and Andreas Orsic Muthig2
1Universitat Kassel, D–34132 Kassel; 2Gesellschaft fur Schwerionenforschung (GSI), D–64291 Darmstadt
A large number of experiments have been performed re-cently at the GSI storage ring in order to explore the radia-tive electron capture (REC) by bare highly–charged ions.In these experiments, attention was placed, in particular,on the electron recombination into the excited ion statesand to their subsequent radiative decay. For instance, fromthe measurements of the angular distribution of the char-acteristic Lyman–α1 (2p3/2 → 1s1/2) radiation, the strongalignment of the 2p3/2 state was found for hydrogen–likeuranium ions U91+ following electron capture [1]. Onemay expect, of course, that such an alignment of the ex-cited ion may arise not only due to the direct capture ofelectrons into the 2p3/2 state but also due to the cascadefeeding from the higher–lying levels. Both of these pop-ulation mechanisms are now well understood within thetheoretical approach, based on Dirac’s relativistic theory[2, 3].
So far, however, theoretical studies on the alignmentof the excited ion states have dealt with ion beamsand atomic (or electronic) targets which are both spin–unpolarized. While such (theoretical) assumptions areappropriate for the present–days experimental set–up of”spin–independent” ion–atom collisions, there become newexperiments likely to be carried out in the near future inwhich use is made of either spin–polarized projectile ionsand/or target electrons. In this contribution, therefore,we like to address the question: How the alignment of theexcited ion states is affected by the ion or electron spin–polarization?
In this report, we present theoretical studies for the mag-netic sublevel population of the 2p3/2 state of hydrogen–like heavy ions following the radiative capture of free po-larized electrons. Similar to the previous studies, we con-sider the mechanisms for the population of the excited ionstates, following both the direct electron capture in thegiven levels as well as (cascade) feedings from the upperstates. For the (direct) electron capture into magneticsublevels |nb jb µb〉, the computation of the cross sectionsσRRnbjbµb
has been performed within the exact relativistic ap-proach and discussed in detail elsewhere [2, 4]. Then, byutilizing these partial cross sections as the initial popula-tion of the excited states Nnbjbµb
(0) = C ·σRRnbjbµba system
of so–called rate equations [5]
dNi
dt= −
<∑j
λij Ni +>∑k
λkiNk (1)
is solved which describes the decay dynamics of the ion.In the equations (1), λij is the decay rate for the |i〉 → |j〉transition and Λ is the total number of (excited) sublevelswhich are considered in the decay cascade; the index j runsover all those with Ek > Ei.
By performing an integration of the system (1), we mayfind the occupation of the magnetic sublevels of the 2p3/2
0 30 60 90 120 150 180-1.0
-0.5
0.0
0.5
1.0
P = 1.0 P = 0.7 P = 0.3D
egre
e of
circ
ular
pol
ariz
atio
n
Observation Angle (deg)
Figure 1: Circular polarization of the Lyman–α1 photonsfollowing radiative electron capture by bare uranium ionswith energy Tp = 1 MeV/u. Calculations are presented forthree different polarizations of the incident electrons.
state and, hence, evaluate the alignment parameters A1
and A2. In general, these parameters depend on the pro-jectile ion energy Tp and its nuclear charge Z [4]. Theybehave, however, in a rather different way as function ofthe polarization of the incident electrons. For example, thesecond–rank parameter A2, which completely determinesthe angular distribution as well as the linear polarizationof the Lyman–α1 characteristic decay [5], is not affectedby the spin–polarization of the incident electrons. There-fore, the (future) angular–distribution measurements onthe characteristic radiation will not bring any additionalinformation on the polarization properties of particles.
The first–rank (orientation) parameter
A1 =1√5
3σRR3/2 − 3σRR−3/2 + σRR1/2 − σRR−1/2
σRR3/2 + σRR1/2 + σRR−1/2 + σRR−3/2
, (2)
in contrast, is proportional to the (degree of) polariza-tion of the incident electrons: A1 ∝ P . The orientation(2) determines the circular polarization of the Lyman–α1
photons which, as seen from the Figure 1, may serve asa ”detector” for the spin–polarization of target electronsor projectile ions. In practice, however, such polarizationstudies seems to be hardly possible to be performed in thenear future since the detection of the circular polarizationof hard x–rays still remains an unsolved problem.
References
[1] Th. Stohlker et al., Phys. Rev. Lett. 79, 3270 (1997).[2] A. Surzhykov et al., Phys. Rev. Lett. 88, 15300
(2002).[3] A. Orsic Muthig et al., GSI Sci. Report, 90 (2002).[4] J. Eichler et al., Phys. Rev. A 58, 2128 (1998).[5] K. Blum, Density Matrix Theory and Appl., (1981).
- 133 -
On the measurement of the spin–polarization of highly–charged ions
Andrey Surzhykov1, Stephan Fritzsche1, Thomas Stohlker2, and Stanislav Tachenov2
1Universitat Kassel, D–34132 Kassel; 2Gesellschaft fur Schwerionenforschung (GSI), D–64291 Darmstadt
During the last decade, ion–atom and ion–electron col-lisions have been the subject of intense studies at the GSIstorage ring. A large number of measurements were per-formed in order to explore, for example, relativistic as wellas quantum electrodynamic (QED) phenomena in ener-getic collisions of high–Z projectile ions with low–Z targets.Until now, however, most of the collision experiments havedealt with ion beams and target atoms (or free electrons)which are both spin–unpolarized. While, of corse, such”spin–independent” measurements have brought a greatdeal of information on the structure and dynamics of heavyatomic systems, more details may be obtained from ex-periments with spin–polarized ions. Very recently, a num-ber of such polarization experiments have been proposedfor studying parity nonconservation phenomena in few–electron systems [1] or spin–dependent effects in electroncapture processes [2].
Obviously, however, any practical realization of ”spin–dependent” collision experiments will require the solutionof two key problems: (i) how to produce beams of polar-ized heavy ions and (ii) how their polarization can be mea-sured. The method for producing of polarized hydrogen–like heavy ions was recently discussed by Prozorov andco–workers [3]. In particular, it was proposed to applythe optical pumping of the hyperfine ground–state levelsof hydrogen–like europium ion Eu62+ with a nuclear spin I= 5/2 in order to obtain a predominant population of thestate |F = 2,MF = 2〉. Since the ion state |F MF 〉 resultsfrom the coupling of an electron in the (one–particle) state|jb µb〉 with the nuclear spin F = I+jb, a fully polarized F= 2 ground state may lead to a polarization of the nuclearspin of about 93 %. However, as mentioned in [3], themeasurement and, hence, the control of this polarizationremained up to the present a unresolved problem.
In this contribution, we suggest to utilize the radiativecapture of a target electron into a bound state of theprojectile ion as a ”probe” process for measuring thespin–polarization of ion beam. As recently shown, forexample, the linear polarization of the recombinationx–ray photons is strongly affected by the spin–polarizationof the target atoms [4]. Since, however, the electron andion occur rather symmetrical in the collision process,a similar effect on the polarization of recombinationlight can therefore be expected if the projectile ionsare themselves polarized. In order to investigate suchpolarization effects we calculated the linear polarizationof the photons as emitted in the radiative capture offree electrons into the ground state of spin–polarizedhydrogen–like heavy ions [5]. Most naturally, the polar-ization of the recombination photons is described in termsof the Stokes parameters, which are simply determinedby the intensities of the light Iχ, as measured under thedifferent angles with respect to the reaction plane [4, 5].While the parameter P1 = (I0− I90)/(I0 + I90) is obtainedfrom intensities within and perpendicular to the reac-
Figure 1: The Stokes parameter P2 of the photons whichare emitted in the electron capture into the K–shell ofcompletely polarized hydrogen–like europium ions.
tion plane, the parameter P2 follows a similar intensityratio which is taken at χ = 45 and χ = 135, respectively.
As shown by the theoretical analysis [5], the two Stokesparameters P1 and P2 behave in rather different ways withrespect to the spin–polarization of (hydrogen–like) projec-tile ions. While the parameter P1 does not depend onbeam polarization and, hence, can not be used for polar-ization studies, the second Stokes parameter P2 appearsto be proportional to the degree of the beam polarization
P2(θ) ∝ λF · f(θ) . (1)
In the Eq. (1), the beam polarization is defined by [3]:
λF =∑MF
nF,MFMF /F (2)
as the sum over the magnetic sublevels, where nF,MF refersto the corresponding population. The Stokes parameter P2
may serve, therefore, as a valuable tool for ”measuring” thepolarization properties of the heavy ion beams at storagerings. Figure 1 displays the parameter P2 as calculated, forexample, for radiative capture of electrons into the groundstate of completely polarized (λF = 1) hydrogen–like eu-ropium ions with energies in the range 200 MeV/u ≤ Tp ≤400 MeV/u. The effect of the ion polarization becomesparticularly remarkable around a photon emission angleof θ = 18, where the second Stokes parameter decreasesfrom the P2 = -0.05 for Tp = 200 MeV/u to almost -0.16for Tp = 400 MeV/u.
References
[1] L. Labzowsky et al., Phys. Rev. A 63, 054105 (2001).[2] A. Klasnikov et al., Phys. Rev. A 66, 042711 (2002).[3] A. Prozorov et al., Phys. Lett. B 574, 180 (2003).[4] A. Surzhykov et al., Phys. Rev. A 68, 022719 (2003).[5] A. Surzhykov et al., Phys. Rev. Lett. to be submitted.
- 134 -