The Quantum and Local Polyakov loop in Chiral Quark Models at

The Quantum and Local Polyakov loop in Chiral Quark Models at Finite Temperature 1

arXiv:hep-ph/0610095v1 9 Oct 2006

E. Megías 2, E. Ruiz Arriola and L.L. Salcedo Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada. 18071-Granada (Spain) Abstract. We describe results for the confinement-deconfinement phase transition as predicted by the Nambu–Jona-Lasinio model where the local and quantum Polyakov loop is coupled to the constituent quarks in a minimal way (PNJL). We observe that the leading correlation of two Polyakov loops describes the chiral transition accurately. The effects of the current quark mass on the transition are also analysed.

The simultaneous ocurrence of two phase transitions where chiral symmetry is restored and hadrons become deconfined seems a unique and misterious feature of QCD matter at finite temperature (for an early review see e.g. [1]). Besides direct QCD lattice simulations, there exist theoretical constraints below and above the deconfinement phase transition. At low temperatures the leading thermal excitations correspond to a gas of weakly interacting pions [2]. Moreover, in the large Nc limit with the temperature T kept fixed, if a chiral phase transition takes place it should be first order [3]. The use of resonance hadron Lagrangians implies that thermal corrections are 1/Nc suppressed [4]. At high temperatures one has a weakly interacting quark-gluon plasma (for a review see e.g. [5]). However, the previous powerful constraints assume from the start a given phase and do not provide a clue on how chiral symmetry restoration and deconfinement are intertwined. The coupling of relevant order parameters such as the quark condensate for chiral symmetry breaking and the Polyakov loop for deconfinement at finite temperature can be made explicit in Polyakov chiral quark models, an amalgamate of colour and flavour degrees of freedom where the simultaneous chiral-deconfinement crossover can be quantitatively studied with an acceptable phenomenological success [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Actually, in our recent work [10] we have shown why and how Polyakov loops must be coupled to Chiral Quark Models (CQM) to comply with large gauge invariance at finite temperature and how the quantum and local nature of the Polyakov loop generates a rather sharp crossover at about the observed critical temperature, although uncertainties are expected. More specifically, ChPT and large Nc constraints are naturally accomodated [13] within those models, hence solving a long standing puzzle 1

Supported by funds provided by the Spanish DGI and FEDER founds with Grant No. FIS2005-00810, Junta de Andalucía Grant No. FM-225, and EURIDICE Grant No. HPRN-CT-2003-00311. 2 Speaker at Quark Confinement and the Hadron Spectrum VII, Ponta Delgada, Portugal, 2-7 IX, 2006.

which was ignored for a long time; traditional CQM did produce a chiral phase transition while violating those restrictions. An immediate consequence of this new coupling is an upward shift of the critical temperature referred to as Polyakov cooling in Ref. [10]. In this work we will deal with the PNJL model for definiteness. After bosonization the NJL Lagrangian reads 1 ∂ + Mˆ 0 + S + iγ P + ΓQ [S, P] = −Tr log i/ 4GS 

5



Z


d 4 x tr f S2 + P2 .

(1)

We use Tr for the full functional trace, tr f for the trace in flavour space, and trc for the trace in colour space. The UV divergencies in Eq. (1) from the Dirac determinant only affect in practice the zero temperature contributions [16]. In the Polyakov gauge Ω = eiA4/T , where A4 is time independent and diagonal, and the minimal coupling is made ∂4 → ∂4 − iA4 . Integrating further over the A4 gluon field in a gauge invariant manner [17] yields a generic partition function of the form Z=

Z

DS DP DΩ e−ΓG [Ω] e−ΓQ [S,P;Ω] ,

(2)

where DΩ is the Haar measure of the SU(Nc ) colour group, ΓG is the effective gluon action and ΓQ stands for the quark effective action. In general Ω(x) is a local and quantum variable since the gluon field itself depends on the point. As argued in [10] mean field approximations [7, 8, 11, 12] generate a spurious gauge orbit dependence and a possibly complex Ω (violating colour charge conjugation) and they necessarily imply a non-vanishing value of the Polyakov loop in the adjoint representation, in contradiction with lattice results [18]. For a constant value of the Polyakov loop, Ω, and the scalar field S = M (which we identify with the constituent quark mass) the quark effective action is given by " Z 3k 1 d T ΓQ (M, Ω, T ) = Tr f (M − Mˆ 0 )2 − 2N f Nc εk V 4GS (2π )3 # n    o +T Trc log 1 + e−εk /T Ω + log 1 + e−εk /T Ω† , (3) where we have only retained the vacuum contribution, so there is no contribution of 4 meson fields (S, P) (see √ [13] for a chiral expansion up to O(p )), V is three dimensional volume and εk = + k2 + M 2 is the energy of a constituent quark with mass M. We define the Polyakov-loop averaged action −ΓQ (M,T )

e

=

Z

dΩ e−ΓG [Ω] e−ΓQ (M,Ω,T ) .

(4)

The value of M is determined by minimization of ΓQ (M, T ) with respect to M, ∂ ΓQ (M, T )/∂ M = 0, which corresponds to computing the integration in DSDP at the mean field level and determines M at a given temperature T , denoted as M ∗ = M(T ). In

addition, the relation between the (single flavour) chiral quark condensate, hqqi, ¯ and the constituent quark mass, reads 2GS N f hqqi ¯ ∗ = −(M ∗ − mq ) .

(5)

Any observable is obtained by using M ∗ and averaging over Ω. The integral in dΩ in the case Nc = 3 and in the Polyakov gauge was computed numerically in Ref. [10]. Here we show a much simpler method which is based on evaluating the integral analytically for any Nc inR the low temperature limit and
corresponds to take Ω small. (To see this use M /T 3 − ε 2 k = 4π M T K2 T , where K2 (x) is the modified Bessel function the formula d k e p −x and K2 (x) ∼ π /2 x e for x → ∞). From Eq. (3) we get ∞

(−1)n ΓQ (M, Ω, T ) = ΓQ (M, 0) + 2N f ∑ n n=1

Z

d 3 x d 3 k −nεk /T Trc [Ωn + Ω†n ] . (6) e 3 (2π )

Expanding the exponent in Eq. (2) one obtains a power series in terms of Ω and Ω† . The simplest correlation of two Polyakov loops is taken to be [10] Z

dΩ Trc Ω(~x)Trc Ω† (~y) = e−σ |~x−~y|/T ,

(7)

with σ the string tension, and yields the leading thermal contribution to the effective action  2 Z T d 3 k −εk /T T ΓQ (M, T ) = ΓQ (M, 0) − TVσ 2N f e +··· , (8) V V (2π )3

where Vσ = 8π T 3 /σ 3 is the equivalent confinement correlation volume. As we see, the effect of quantum corrections on the Polyakov loop lowers the vacuum energy as it should. Moreover, they are 1/Nc suppressed, as one would expect in Chiral Perturbation Theory or a resonance gas model but unlike traditional chiral quark models without Polyakov loop. Minimizing with respect to the mass we get the effective temperature dependent mass M ∗ and from Eq. (5) the corresponding hqqi ¯ ∗ condensate can be evaluated. The approximate result is presented and compared to the full result [10] in Fig. 1 and, as we can see the approximation is quite efficient and very easy to implement in standard chiral quark models. For the Polyakov loop expectation value similar manipulations hold, yielding the leading order contribution r   N f M2T N f Vσ M 3 T 5 −M/T 1 trc Ω = Vσ 2 K2 (M/T ) + · · · ∼ e . (9) L= Nc Nc π Nc T 2π 3

The full result and the approximated formula are compared in Fig. 1. In this case the agreement is only up to temperatures about 0.75TD . The analysis above is done with physical current quark masses, mq = 5.5 MeV where one obtains Tχ = TD = 256(1) MeV. This remarkable coincidence between transitions is not accidental nor depends on the particular choice of mq as can be seen in Fig. 1, where we show the temperature dependence of hqqi ¯ ∗ and L for mq = 0, 5.5, 40, 80, 120 and 300 MeV. The corresponding susceptibilities are displayed in Fig. 2.

1

1

L

L

*

*

/

/ 0.8

Order Parameters

Order Parameters

0.8

0.6

0.4

Full Integration Leading order

0.2

0.6

mq = 0 MeV mq = 5.5 MeV mq = 40 MeV mq = 80 MeV mq = 120 MeV mq = 300 MeV

0.4

0.2

0

0 0

50

100

150

200

250

300

350

400

450

0

50

100

150

T (MeV)

200

250

300

350

400

450

T (MeV)

FIGURE 1. Chiral condensate hqqi ¯ ∗ and Polyakov loop L = htrc Ωi /Nc . Left: Leading √ Polyakov loop correlation approximation (see Eqs. (8) and (9)). We take the 2-flavor PNJL model, and σ = 425 MeV, fπ = 93 MeV, M = 300 MeV, mu = md ≡ mq = 5.5 MeV. Right: Current quark mass dependence. 18 1

16

*

0.6

0.4

14 12 −1

2

∂ /∂ T [GeV ]

0.8

∂ L/∂ T [GeV ]

mq = 0 MeV mq = 5.5 MeV mq = 40 MeV mq = 80 MeV mq = 120 MeV mq = 300 MeV

10

mq = 0 MeV mq = 5.5 MeV mq = 40 MeV mq = 80 MeV mq = 120 MeV mq = 300 MeV

8 6 4

0.2

2 0

0

220

240

260

280

300

320

150

T (MeV)

200

250

300

350

T (MeV)

FIGURE 2. Temperature dependence of ∂ L/∂ T (left) and ∂ hqqi ¯ ∗ /∂ T (right) for several values of the current quark mass. The transition temperatures are Tχ = TD = 255(1), 256(1), 266(1) MeV for mq = 0, 5.5, 40 MeV respectively, and Tχ = 297(1) MeV, TD = 290(1) MeV for mq = 300 MeV.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

F. Karsch, PoS Corfu98, 008 (1999). P. Gerber, and H. Leutwyler, Nucl. Phys. B321, 387 (1989). A. Gocksch, and F. Neri, Phys. Rev. Lett. 50, 1099 (1983). D. Toublan, and J. B. Kogut, Phys. Lett. B605, 129–136 (2005), hep-ph/0409310. U. Kraemmer, and A. Rebhan, Rept. Prog. Phys. 67, 351 (2004), hep-ph/0310337. P. N. Meisinger, and M. C. Ogilvie, Phys. Lett. B379, 163–168 (1996). P. N. Meisinger, T. R. Miller, and M. C. Ogilvie, Nucl. Phys. Proc. Suppl. 129, 563–565 (2004). K. Fukushima, Phys. Lett. B591, 277–284 (2004), hep-ph/0310121. E. Megías, E. Ruiz Arriola, and L. L. Salcedo (2004), hep-ph/0410053. E. Megías, E. Ruiz Arriola, and L. L. Salcedo, Phys. Rev. D74, 065005 (2006), hep-ph/0412308. C. Ratti, M. A. Thaler, and W. Weise, Phys. Rev. D73, 014019 (2006), hep-ph/0506234. S. K. Ghosh, T. K. Mukherjee, M. G. Mustafa, and R. Ray, Phys. Rev. D73, 114007 (2006). E. Megías, E. Ruiz Arriola, and L. L. Salcedo (2006), hep-ph/0607338. H. Hansen, et al. (2006), hep-ph/0609116. C. Ratti, S. Roessner, M. A. Thaler, and W. Weise (2006), hep-ph/0609218. C. V. Christov, E. Ruiz Arriola, and K. Goeke, Acta Phys. Polon. B22, 187–202 (1991). H. Reinhardt, Mod. Phys. Lett. A11, 2451–2462 (1996), hep-th/9602047. A. Dumitru, Y. Hatta, J. Lenaghan, K. Orginos, and R. D. Pisarski, Phys. Rev. D70, 034511 (2004).