Light front approach to correlations in hot quark matter

0.1 MB·English

by  S. Strauss

Preprint submitted to
HEAVY ION
Acta Phys.Hung.A 22/1 (2005) 000–000
PHYSICS
Light front approach to correlations in hot quark matter
6 S. Strauss1, M. Beyer1 and S. Mattiello1
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2 1 Institute of Physics, Universit¨at Rostock,
n 18051 Rostock, Germany
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Received

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Abstract. We investigatetwo-quarkcorrelationsinhotanddense quarkmat-

2 ter. To this end we use the light front field theory extended to finite tempera-
v ture T andchemicalpotentialµ. Thereforeitis necessaryto developquantum
5 statistics formulated on the light front plane. As a test case for light front
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quantizationatfiniteT andµwe considerthe NJL model. The solutionofthe
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in-medium gap equation leads to a constituent quark mass which depends on
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1 T andµ. Two-quarksystemsareconsideredinthepionicanddiquarkchannel.
5 We compute the masses of the two-body system using a T-matrix approach.
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/ Keywords: quark matter, light front quantization, finite temperature
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t PACS: 11.10.Wx, 12.38Mh 25.75
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1. Introduction
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Lightfront(LF)quantization[1]providesanovelpossibilitytoformulatequantum
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X fieldtheoryatfiniteT andµ. Especiallythe phasediagramofquantumchromody-
r namics(QCD),thatistackledexperimentallybyrelativisticheavyioncollisionsand
a astrophysical observations may become accessible by LFQCD [ 2]. Here we intro-
duce relativistic thermodynamics and quantum statistics in light front coordinates
and as an application of light front techniques, we consider the pionic and scalar
diquark channel of the Nambu Jona-Lasinio (NJL) model at finite temperature.
For the description of quark matter quantized on the light front plane one
introduces coordinates x± = x0 ±x3 and x1,2 = x1,2 (see e.g. [ 2]). Contrary to
the instant form in the front form all initial and boundary conditions of the fields
are expressed on the plane x+ = x0 +x3 = 0. The metric in the LF form has an
off-diagonalformwithg++ =g−− =0,g+− =2andgii =−1fori=1,2. Amajor
consequenceofthe frontformisthe on-shellrelation,i.e. LFenergyk− isgivenby
on
k2 +m2
− ⊥
k = (1)
on k+
2 S. Strauss, M. Beyer and S. Mattiello
with k+ 6=0. Note that there arisesno signambiguity for the energy in contrastto
the instant form.
The NJL Lagrangianfor the two flavor case reads
L =ψ¯(iγ ∂µ−m )ψ+G (ψ¯ψ)2+(ψ¯iγ τψ)2 , (2)
NJL µ 0 5
(cid:0) (cid:1)
thatis(approximately)invariantunderchiraltransformationdue the smallcurrent
quark mass m . Dynamical symmetry breaking leads to a gap equation for the
0
quark mass. An extensive discussion on the NJL model and its implications for
light mesons at finite temperature and density can be found in [ 3].
2. Quantum statistics on the light front
Our derivation of the statistical operator for a grand canonical ensemble is along
Refs. [ 4, 5]. One starts with the von-Neumann equation for the LF time x+, viz.
i∂−̺ˆ= Pˆ−,̺ˆ . (3)
h i
Inthermodynamicalequilibriumtheleft-handsideof(3)vanishesandthestatistical
operator ̺ˆ depends only on the Poincar´e generators that commute with the LF
Hamiltonian Pˆ−. The explicit form of the operator ̺ˆfollows from demanding the
extremum of the entropy S = −Tr{̺ˆln̺ˆ} under the constraints Tµν = Tr(̺ˆTˆµν)
for the energy-momentum tensor and jµ = Tr(̺ˆˆjµ) for all conserved currents of
ℓ ℓ
the system, i.e. the variation of entropy δS under the constraint with respect to ̺ˆ
vanishes. Furthermore one identifies the Lagrange multipliers by comparing to the
relativistic generalization of the Gibbs-Duhem relation [ 6]
σµ =θ T̺µ+Pθµ+ α jµ, (4)
̺ A A
XA
where σµ, P is the entropy flux and the invariant pressure. The four vector θ is
µ
givenbyθ =u /T. ConsideringonlytheparticlenumberN asconservedquantity
µ µ
one has
1 1
̺ˆ= exp − u Pˆν −µNˆ , (5)
ν
Z (cid:26) T (cid:16) (cid:17)(cid:27)
where u is the velocity of the medium.
ν
Analogous to the instant form we compute the Fermi-Dirac distribution for an
ideal gas of fermions. For the medium at rest, i.e. u=(1,1,0,0) one obtains
−1
1 1 1
f±(k+,k⊥)=(cid:20)exp(cid:26)T (cid:18)2ko−n+ 2k+∓µ(cid:19)(cid:27)+1(cid:21) , (6)
where f+(f−) is the distribution for quarks (antiquarks).
In the following we will need the in-medium quark propagator to take into
account medium effects. Therefore one has to evaluate
iG(x−y)=θ(x+−y+)hψ(x)ψ¯(y)i−θ(y+−x+)hψ¯(y)ψ(x)i. (7)
Light front approach to correlations in hot quark matter 3
700 700
µ = 0 MeV µ = 0 MeV
600 µ = 100 MeV 600 µ = 100 MeV
µ = 200 MeV µ = 200 MeV
µ = 300 MeV diquark mass µ = 300 MeV
500 500 µ = 330 MeV
MeV]400 2 x quark mass MeV]400 2 x quark mass
m [300 m [300

200 pion mass 200
100 A 100 B
0 0
0 50 100 150 200 250 0 50 100 150 200 250
T [MeV] T [MeV]
Fig. 1. (A)ThepionmassasafunctionofT fordifferentµ. (B)Thediquarkmass
as a function of T for different µ. The continuum is given by 2m. The lines of the
two-body masses end at the Mott dissociation points.
The expectation value h...i indicates averaging over the grand canonical ensemble
h...i=Tr(̺ˆ...). A longer calculation (cf. [ 5]) leads to
G(k)= γµkoµn+m θ(k+) 1−f+(k+,k⊥) + f+(k+,k⊥)
k+ n (cid:16) k−−ko−n+iε k−−ko−n−iε(cid:17)
+θ(−k+) f−(−k+,k⊥) + 1−f−(−k+,k⊥) . (8)
(cid:16) k−−ko−n+iε k−−ko−n−iε (cid:17)o
Thein-mediumgapequationfollowsfromm=m −2Ghψ¯ψibyasimilarevaluation
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m(T,µ)=m0+24GZ dkk++(d22πk)3⊥m(1−f+(k+,k⊥)−f−(k+,k⊥)) (9)
LB
which is regularized by an invariant Lepage-Brodsky (LB) cut-off [ 7].
3. Two-body systems
The two-body correlations are evaluated within a Bethe-Salpeter approach. We
use a T-matrix equation with an appropriate interaction kernel induced by the
Lagrangian(2). Boundstates of mass M show up as poles ofthe T-matrixT(k)at
k2 =M2. Firstweconsiderthepionicchannelandduetothezero-rangeinteraction
the T-matrix reduces to
−2iG
t (k)= (10)
π 1+2GΠ (k2)
π
with the pion loop integral Π (k2) evaluated using (8)
π
Π (k2)=−6 dxd2q⊥ M220(x,q⊥)(1−f+(M20)−f−(M20)). (11)
π Z x(1−x)(2π)3 M220(x,q⊥)−k2
LB
Here M220(x,q⊥) = (q2⊥+m2)/x(1−x) is the mass of the virtual two-body state.
For the determination of the pion mass for given T and µ one has to vary k2 until
4 S. Strauss, M. Beyer and S. Mattiello
the pole condition 1+2GΠ (k2 = m2) = 0 holds. The results are presented in
π π
Fig. 1 (A).
A similar treatment like the pion case leads to the loop integral for a scalar,
isospin singulet, color-antitripletdiquark [ 9]
Π (k2)=−6 dxd2q⊥ M220(x,q⊥)(1−2f+(M20)). (12)
s Z x(1−x)(2π)3 M220(x,q⊥)−k2
LB
Here the pole condition takes the form 1+ 2G Π (k2 = m2) = 0, where G is
s s d s
the coupling in the diquark channel which we treat as a parameter. The medium
dependence of the diquark mass is shown in Fig. 1 (B).
4. Conclusions
Wehaveshownthatwell-knownpropertiesoftheNJLmodelarerecoveredinthermo
field theory quantized on the light front. This includes the restoration of chiral
symmetry and the dissociation of pion and diquark states. Since LF quantization
is capable to treat the perturbative and nonpertubative part of QCD there is a
perspectivetoexplorethewholephasediagramofQCDatallvaluesoftemperature
and chemical potential.
Acknowledgments
One of the authors (S.S.) thanks the organizers of the International Quarkmatter
Conference 2005 in Budapest for financial support and the inspiring meeting. This
work is supported by the Deutsche Forschungsgemeinschaft.
References
1. P.A.M. Dirac, Rev. Mod. Phys. 21, 392 (1949).
2. S. J. Brodsky, H. C. Pauli and S. S. Pinsky, Phys. Rept. 301 (1998) 299
3. S. P. Klevansky, Rev. Mod. Phys. 64 (1992) 649.
4. J. Raufeisen and S. J. Brodsky, Phys. Rev. D 70 (2004) 085017
5. M. Beyer, S. Mattiello, T. Frederico and H. J. Weber, J. Phys. G 31, 21
(2005).
6. W. Israel, Annals Phys. 100 (1976) 310; H. A. Weldon, Phys. Rev. D 26
(1982) 1394.
7. G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22 (1980) 2157.
8. N. Ishii, W. Bentz and K. Yazaki, Nucl. Phys. A 587 (1995) 617; W. Bentz,
T. Hama, T. Matsuki and K. Yazaki, Nucl. Phys. A 651 (1999) 143.
9. K. Rajagopaland F. Wilczek, arXiv:hep-ph/0011333.