Effective SU(2) Polyakov Loop Theories with Heavy Quarks on the Lattice
Philipp Scior, David Scheffler, Dominik Smith, Lorenz von Smekal
EEffective SU(2) Polyakov Loop Theories with HeavyQuarks on the Lattice
Philipp Scior ∗ , David Scheffler, Dominik Smith Theoriezentrum, Institut für Kernphysik, TU Darmstadt, 64289 Darmstadt, GermanyE-mail: [email protected],[email protected],[email protected]
Lorenz von Smekal
Theoriezentrum, Institut für Kernphysik, TU Darmstadt, 64289 Darmstadt, GermanyInstitut für Theoretische Physik, Justus-Liebig-Universität Gießen, 35392 Gießen, GermanyE-mail: [email protected]
We compare SU(2) Polyakov loop models with different effective actions with data from full two-color QCD simulations around and above the critical temperature. We then apply the effectivetheories at finite temperature and density to extract quantities like Polyakov loop correlators,effective Polyakov loop potentials and baryon density.
The 32nd International Symposium on Lattice Field Theory23-28 June, 2014Columbia University New York, NY ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] D ec ffective SU(2) Polyakov Loop Theories with Heavy Quarks on the Lattice Philipp Scior
1. Introduction
Phase structure and thermodynamics of QCD matter are important to understand for the phe-nomenology of heavy-ion collisions or the properties of neutron stars. We have significant insightinto the QCD phase structure at vanishing chemical potential. Unfortunately, in the case of finitebaryon chemical potential, lattice simulations are hampered by the fermion sign-problem in QCD.There has been a lot of effort to solve or circumvent this sign problem, such as the development ofthe complex Langevin algorithm for QCD [1, 2], for example. Other works have avoided the signproblem by switching to QCD-like theories, with gauge groups like SU(2) [3, 4] or G [5], in orderlearn about the more qualitative properties of strong interaction matter at finite density.In recent years we have also seen the development of effective Polyakov loop theories onthe lattice in order to investigate the deconfinement transition of QCD. For pure gauge theoriesthese Polyakov loop models lie in the same universality class as the underlying Yang-Mills theory.Explicit results have demonstrated that versions of these models are able to predict the location ofthe deconfinement transition of SU(3) Yang-Mills theory within less than 6% [6]. It is possible toincorporate dynamical fermions in the effective theory and it has been shown that the sign problemat finite chemical potential can be dealt with, e.g. by a complex Langevin algorithm [7].To check the range of applicability we can compare the simulations of our effective Polyakovloop theory to simulations of the underlying gauge theory. Therefore we investigate two-coloreffective Polyakov loop theories to be able to compare results of full two-color QCD simulationswith the effective SU(2) Polyakov loop theory at all chemical potentials.One basic quantity to calculate is the effective Polyakov loop potential. It is a crucial inputfor effective theories in the continuum like Polyakov–Quark-Meson or Polyakov–Nambu–Jona-Lasinio models. Usually one assumes the Polyakov loop potential to have no explicit dependenceon the baryon chemical potential, with only implicit dependencies originating from sea quarkswhich are incorporated in a chemical potential dependence of the model parameters [8]. Whilethis approximation is valid for small values of the chemical potential it is not quite clear if this isalso true at larger chemical potentials. Here we will show first steps towards such a calculation.We present unquenched results for the Polyakov loop potential obtained from full two-color latticesimulations and compare them to simulations of different effective Polyakov loop models. Fur-ther we will apply the effective theory to the cold and dense regime of two-color QCD where wemeasure the baryon density.
2. Effective Polyakov Loop Theory
The most general form of an effective action in terms of Polyakov loops is S eff = ∑ i j L i K ( ) ( i , j ) L j + ∑ i jkl L i L j K ( ) ( i , j , k , l ) L k L l + · · · + ∑ i h ( ) ( i ) L i + . . . , (2.1)with L x being the Polyakov loop at position x . The goal is now to find the effective kernels K ( n ) andcouplings h ( n ) of the effective theory in terms of the parameters of the underlying theory. This canbe done by non-perturbative methods like inverse Monte-Carlo [9] or the relative weights method[10], or one can calculate the kernels and couplings analytically in a combined strong-coupling2 ffective SU(2) Polyakov Loop Theories with Heavy Quarks on the Lattice Philipp Scior P ( L ) L λ =0.073 λ =0.180 λ =0.190YM: β c Figure 1:
Distributions of the SU(2) Polyakov loopfrom the simple effective model at subcritical cou-plings λ < λ c = . P ( L ) L λ =0.073 λ =0.180 λ =0.213YM: β c Figure 2:
Distributions of the SU(2) Polyakovloop from the resummed effective model at subcriti-cal couplings λ < λ c = . and hopping expansion. To compare these methods we will match results from SU(2) Yang-Millstheory to results both from the simplest ansatz for an effective action as well as the leading orderresult of the effective theory derived by a strong-coupling expansion. The action for the simplestAnsatz is given by S eff = − λ ∑ i j L i L j , (2.2)describing a nearest-neighbor interaction between Polyakov loops. The effective coupling λ interms of the original lattice coupling β and time-like extension of the lattice N t can be obtainedby inverse Monte-Carlo methods. The leading order effective action from the strong-couplingapproach is given by S eff = − ∑ i j log ( + λ L i L j ) . (2.3)Here one resums generalized Polyakov loops, winding several times around the lattice to producethe logarithm. The effective coupling in terms of the original lattice parameters is given by λ ( u , N t ≥ ) = u N t exp (cid:20) N t (cid:18) u − u + u − u + O ( u ) (cid:19)(cid:21) , (2.4) -06 -05 < L ( x ) L ( x + R ) > R YM: β =2.0 λ =0.073 Figure 3:
Polyakov-loop correlator of theeffective model (2.3) compared to the puregauge theory at β = . × where u ( β ) is the ratio of the first two modified Besselfunctions u ( β ) = I ( β ) / I ( β ) , as usual. Terms involv-ing more Polyakov loops or longer range interactionsare of higher order in β and are therefore suppressed inthe strong coupling limit. To compare the effective the-ories with pure gauge theory results we compare mea-sured Polyakov loop distributions which we found tobe much more sensitive to parameter changes than e.g.the expectation values of the Polyakov loop. In thepure gauge theory the distribution remains symmetricand unmodified throughout the symmetric phase. Atcouplings above the critical coupling β c center sym-metry is spontaneously broken and the Polyakov loopdistributions get skewed, resulting in a finite value of (cid:104) L (cid:105) . Figures 1 and 2 show the results of the3 ffective SU(2) Polyakov Loop Theories with Heavy Quarks on the Lattice Philipp Scior P ( L ) LN t =12 T/T c =0.83N t =10 T/T c =1.00N t =8 T/T c =1.25N t =6 T/T c =1.67 Figure 4:
Polyakov-loop distributions from simula-tions of two-color QCD with two flavors of staggeredquarks of various masses at β = . V e ff ( L ) LN t =12 T/T c =0.83N t =10 T/T c =1.00N t =8 T/T c =1.25N t =6 T/T c =1.67 Figure 5:
Effective Polyakov-loop potentials fortwo-color QCD with two flavors of staggered quarksfrom the Polyakov-loop distributions in Fig. 4. simulations at subcritical values of λ for both effective models compared to results from pure SU(2)gauge theory simulations at the critical coupling β c for N t =
10. In the strong coupling regime orrespectively at small effective coupling λ , the distributions match the ‘exact’ distribution from theSU(2) gauge theory simulations very well. At larger couplings but still in the symmetric phase, theeffective theory distributions get deformed, however. This effect is strongest the in simple effectivemodel of Eq. (2.2) whose distribution shows strong deformations even developing a double peakstructure close to λ c . The distributions from the resummed model, Eq. (2.3), also get deformed buttheir overall shape remains qualitatively almost unchanged, and the deviations from the full gaugetheory result are generally much smaller. We therefore conclude that the resummed model is bettersuited for couplings close to the critical coupling. The Polyakov-loop correlator of this model inthe strong coupling regime is compared with the gauge theory result in Fig. 3.
3. Effective Polyakov Loop Potential
We can use the Polyakov loop distribution P ( L ) to calculate the on-site effective Polyakov-loop potential. First, we calculate the constrained effective potential V and then obtain the effectivePolyakov loop potential V eff via Legendre transformation: V ( L ) = − log P ( L ) , W ( h ) = log (cid:90) dL (cid:48) exp ( − V ( L (cid:48) ) + hL (cid:48) ) , (3.1) V eff ( L ) = sup h ( Lh − W ( h )) . Note that the minimum of V eff is correctly located at the expectation value of the Polyakov loop (cid:104) L (cid:105) .Figures 4 and 5 show Polyakov loop distributions and the corresponding effective potentials fromfull two-color QCD simulations with two flavors of staggered quarks at β = . (cid:104) L (cid:105) . This effect gets stronger for higher temperaturesand smaller quark masses. The effect on the effective potential is similar but not quite as drastic:The minimum of the potential moves to larger L with higher temperature and lower quark mass.4 ffective SU(2) Polyakov Loop Theories with Heavy Quarks on the Lattice Philipp Scior P ( L ) Lam=0.5am=0.1h=8.5e-4h=2.3e-3
Figure 6:
Polyakov-loop distributions from the ef-fective theory compared to two-color QCD simula-tions at T = . T c with am = . am = . -1 0 1 2 3 4 5 6 7 8 9-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 V e ff LQC D: T/T c =1.67, am=0.01e ff theory: λ =0.215687, h=0.0032 Figure 7:
Effective Polyakov loop potentials fromthe unquenched effective model and two-color QCDsimulations at T = . T c and am = . Since we now want to analyze the unquenched effective Polyakov loop potential with oureffective theory we have to include dynamical fermions into the theory. The structure of the fermionaction can be derived from a combined strong coupling and hopping expansion analogous to [7].As for the pure gauge theory on can resum generalized Polyakov loops to obtain, S ferm = − N f ∑ i log ( + hL i + h ) , (3.2)and the leading order hopping expansion result for h reads h ( u , κ , N t ) = ( κ e a µ ) N t + O ( κ u ) . (3.3)In the range where the hopping expansion is valid one can check that the resummation is not as im-portant as in the gauge action because h is relatively small. This changes for smaller quark massesor finite chemical potential µ , however, where h increases and resummation becomes important.We choose (2.3) together with (3.2) as the action for our Polyakov-loop effective theory. How-ever, to compare our results to the full two-color results from above we have to apply the theoryoutside the regime where the strong coupling and hopping expansions are valid. Therefore, rela-tions (2.4) and (3.3) for the effective couplings are no longer valid. Our strategy for comparing ourfull two-color simulations with the ones from the effective theory is a two step procedure: First,we compare distributions from pure gauge simulations and the effective theory without quarks tomatch distributions with the same expectation value of the Polyakov loop, (cid:104) L (cid:105) = (cid:90) − dP ( L ) L , (3.4)in order to find the effective gauge coupling λ associated with a particular temperature T . In thesecond step we then match the effective theory with dynamical quarks at that λ or T for differentvalues of h to the distributions from full two-color QCD simulations with a particular quark mass am , again identifying distributions with the same (cid:104) L (cid:105) . Figures 6 and 7 show Polyakov-loop distri-butions and effective potentials from the effective model compared to two-color QCD results fora temperature of T = . T c . The overall shape of the distribution is reproduced quite well, butsome deviations from the full two-color QCD results remain. One can show, that these deviationsoriginate in the gauge part of the action. We believe that this is due to neglecting the interactions5 ffective SU(2) Polyakov Loop Theories with Heavy Quarks on the Lattice Philipp Scior a n < | L | > µ in GeVa n<|L|> Figure 8:
Quark density in lattice units a n togetherwith the Polyakov loop (cid:104)| L |(cid:105) as a funtion of µ inphysical units. a n µ in GeVa n ∼ exp (cid:0) µ − mT (cid:1) ∼ exp (cid:0) µ − mT (cid:1) Figure 9:
Logarithmic plot of the density in latticeunits a n . The lines indicate the two regions withdifferent exponential increases. between Polyakov loops at larger distances which become increasingly important with higher tem-peratures, above T c . The effective potential shows much better agreement than the distributions.It seems to be a generic effect of the Legendre transformation that distributions with the sameexpectation value (cid:104) L (cid:105) lead to quite similar effective potentials.
4. Effective Theory for the Cold and Dense Regime
We will now apply the effective Polyakov-loop theory in the cold and dense regime of twocolor QCD with heavy quarks. Here we are well inside the region where the strong coupling andhopping expansions are applicable. In fact, the effective gauge coupling λ from (2.4) is negligi-ble even at β = . T ∼ −
10 MeV). We have λ ( β = . , N t = ) ∼ · − . We therefore end up with a completely fermionic partitionfunction. Due to the large number of time slices, with N t between 200 and 600, we have to includemore terms in the hopping expansion of the fermion determinant. The effective action up to order κ in the hopping exansion reads − S eff = ∑ (cid:126) x log ( + hL i + h ) − h ∑ (cid:126) x , i Tr hW (cid:126) x + hW (cid:126) x Tr hW (cid:126) x + i + hW (cid:126) x + i + κ N t N c ∑ (cid:126) x , i Tr hW (cid:126) x ( + hW (cid:126) x ) Tr hW (cid:126) x + i ( + hW (cid:126) x + i ) + κ N t N c ∑ (cid:126) x , i , j Tr hW (cid:126) x ( + hW (cid:126) x ) Tr hW (cid:126) x − i + hW (cid:126) x − i Tr hW (cid:126) x − j + hW (cid:126) x − j + κ N t N c ∑ (cid:126) x , i , j Tr hW (cid:126) x ( + hW (cid:126) x ) Tr hW (cid:126) x − i + hW (cid:126) x − i Tr hW (cid:126) x + j + hW (cid:126) x + j + κ N t N c ∑ (cid:126) x , i , j Tr hW (cid:126) x ( + hW (cid:126) x ) Tr hW (cid:126) x + i + hW (cid:126) x + i Tr hW (cid:126) x + j + hW (cid:126) x + j + κ N t ∑ x , i h ( + hL x + h )( + hL x + i + h ) , (4.1) where W (cid:126) x stands for an untraced Polyakov loop. The only leftovers from the Yang-Mills partof the original theory in the effective action are gauge corrections to the effective fermion couplings h = exp (cid:20) N t (cid:18) a µ + ln 2 κ + κ − u N t − u (cid:19)(cid:21) , h = κ N t N c (cid:20) + u − u N t − u + . . . (cid:21) . (4.2)In order to set a physical scale we again use the scale from [11] together with √ σ =
440 MeV andwe calculate the diquark mass in the combined strong coupling and hopping expansion to be am d = − ( κ ) − κ − κ u − u + κ + . . . . (4.3)6 ffective SU(2) Polyakov Loop Theories with Heavy Quarks on the Lattice Philipp Scior
We are now able to determine the fermion number density of the system, n = TV ∂∂ µ log Z . (4.4)Figures 8 and 9 show the results for a simulation in the cold and dense regime. The parametersfor this simulation are β = . a = .
081 fm, κ = . m d =
20 GeV, N t = T = µ = m d / µ < .
96 GeV the slope is well described by a freequark gas. This comes from the small but finite Polyakov-loop expectation value at finite T due tothe presence of the dynamical quarks which break center symmetry explicitly. At around µ ≈ . T (cid:54) =
0. At very large chemicalpotentials the density saturates when every lattice site is occupied by the maximum number ofquarks, 2 N f · N c = -QCD [5] simulations. Acknowledgments
This work was supported by the Helmholtz International Center for FAIR within the LOEWEinitiative of the State of Hesse. All results were obtained on Nvidia GTX or Tesla graphics cards.
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