Landau gauge gluon and ghost propagators from lattice QCD with twisted mass fermions at finite temperature
We investigate the temperature dependence of the Landau gauge gluon and ghost propagators in lattice QCD with two flavors of maximally twisted mass fermions. For these propagators we provide and analyze data which corresponds to pion mass values between 300 and 500 MeV. For the gluon propagator we find that both the longitudinal and transversal component change smoothly in the crossover region, while the ghost propagator exhibits only a very weak temperature dependence. For momenta between 0.4 and 3.0 GeV we give a parametrization for our lattice data. It may serve as input to studies which employ continuum functional methods.
pacs:11.15.Ha, 12.38.Gc, 12.38.Aw
Hadronic matter may experience a phase transition or crossover from a confined phase, with broken chiral symmetry, to a deconfined chirally symmetric phase. The latter phase is characterized by a state called quark gluon plasma. It is widely believed this state of matter has been passed in the early universe, and efforts are undertaken to reproduce it experimentally in heavy-ion collisions as in present collider experiments at RHIC (BNL) or with the ALICE and CMS detectors at LHC (CERN).
The possible existence of such a phase transition was first discussed Cabibbo and Parisi (1975) in the context of Hagedorn’s thermodynamic model Hagedorn (1968, 1970) as a way to evade the consequence of a maximal hadronic temperature. First numerical evidence has been found already in the early days of lattice QCD (LQCD) McLerran and Svetitsky (1981); Kuti et al. (1981). In fact, the lattice formulation offers an ab-initio approach to study such aspects of QCD nonperturbatively. This remains true as long as the chemical potential is small compared to the temperature. Over the last decade immense computational resources were dedicated to reach at a consistent picture of QCD at finite temperature and zero chemical potential (for recent reviews see Kanaya (2010); Levkova (2011); Philipsen (2012); Lombardo (2012)).
LQCD, however, is not the only framework to tackle nonperturbative problems of QCD at zero or non-zero temperature. Powerful continuum functional methods exist as well, like for instance in the context of Dyson-Schwinger (DS) equations von Smekal et al. (1998); Hauck et al. (1998); Roberts and Schmidt (2000); Maris and Roberts (2003) or functional renormalization group (FRG) equations Gies (2002); Pawlowski et al. (2004); Braun et al. (2009), which also allow to address such problems.
Within these frameworks (see, e.g., the recent reviews Fischer et al. (2009); Boucaud et al. (2011)) the Landau gluon and ghost propagators appear—together with the corresponding vertices—as the main building blocks in the formulation of the DS or FRG equations; they constitute part of the solutions of the latter. These functional methods though come with a potential source of error: To solve the (infinite) tower of equations it has to be truncated appropriately, and so the solutions beyond the far-infrared regime, depend on how the truncations are done, in particular in the momentum range around GeV. Therefore, independent information, at best from first principles, is welcome to improve these (unavoidable) truncations.
LQCD calculations allow to provide the Landau gluon and ghost propagators in an ab-initio way. The available momentum range, however, is restricted from above by the lattice spacing and from below by the available lattice volume (up to a further uncertainty related to so-called Gribov copies Bornyakov et al. (2009); Bornyakov and Mitrjushkin (2010)). Despite these restrictions, an impressive amount of data has been produced over the last years for these propagators at zero (see, e.g., Bornyakov et al. (2010); Bogolubsky et al. (2009) and references therein) and non-zero temperature Heller et al. (1995, 1998); Cucchieri et al. (2001a, b); Cucchieri et al. (2007); Sousa et al. (2007); Maas (2010); Bornyakov et al. (2010); Bornyakov and Mitrjushkin (2010); Fischer et al. (2010); Bornyakov and Mitrjushkin (2011); Cucchieri and Mendes (2011a); Aouane et al. (2012); Maas et al. (2012); Cucchieri and Mendes (2011b); Cucchieri et al. (2012). This data has allowed for a variety of cross-checks with corresponding results from the continuum functional methods.
Conventional lattice calculations of QCD at finite temperature often employ the Polyakov loop and the chiral condensate to probe for the (de)confinement and chiral phase transition, respectively. In recent years it turned out that these and other observables can be also calculated from DS and FRG equations, where the Landau gauge gluon and ghost propagators are used as input Braun et al. (2010); Fischer et al. (2008); Fischer (2009); Fischer et al. (2010); Fischer and Mueller (2011); Fischer and Luecker (2012); Fister and Pawlowski (2011a, b). Even an extrapolation to the notoriously difficult regime of nonzero chemical potential seems to be possible Fischer et al. (2011); Lucker and Fischer (2012). Appropriate lattice data for the propagators close to the continuum limit is therefore essential to assist those efforts.
In a recent study Aouane et al. (2012) we have provided such data for quenched QCD (see, e.g., Fukushima and Kashiwa (2012) for a first application). We could show that the longitudinal (electric) component of the gluon propagator may be used to probe the thermal phase transition of pure gauge theory. Similar was shown in Maas et al. (2012). The present article further complements the available lattice data with finite-temperature data for the Landau gauge gluon and ghost propagators from LQCD with dynamical fermion flavors. To the best of our knowledge, there are only two studies which have provided such data in the past Furui and Nakajima (2007); Bornyakov and Mitrjushkin (2011).
For our study we adopt a lattice formulation that employs the
Symanzik-improved gauge action for the gluonic field and the twisted
mass Wilson-fermion action for the fermionic part. The latter ensures
an automatic improvement provided the Wilson -parameter
is tuned to maximal twist (for further details we refer to
Shindler (2008); Urbach (2007)). Our data is based on the vast
set of gauge field configurations that has been generated by the tmfT
Collaboration. The tmfT Collaboration has explored the complicated
phase structure of the theory Ilgenfritz et al. (2009) and is still
investigating the smooth crossover region from the confining and
chirally broken regime at low temperature to the deconfinement and
chirally restored phase at high Burger et al. (2010, 2011).
These studies are restricted so far to pseudo-scalar meson (pion)
masses from MeV to MeV.
To set the scale we use results at of the European Twisted Mass
(ETM) Collaboration Baron et al. (2010). Specifically, our data for the
gluon and ghost propagators covers the whole crossover regime at three
pion mass values: , 398 and 469 MeV. At these values
the crossover regime is characterized by a very smooth behavior of the
chiral condensate and the Polyakov loop as well as their susceptibilities.
Moreover, one observes for these settings the breakdown of chiral symmetry
and the deconfinement phase transition occur at slightly
different temperatures and ,
respectively, in agreement with observations reported in
Borsanyi et al. (2010).
The paper is organized as follows. In Section II we give all lattice parameters and outline the setup of our Monte Carlo simulations. Section III recalls the definitions of the gluon and ghost propagators on the lattice in the Landau gauge. Data and fits for various temperature and pion mass values are presented in Section IV. Conclusions are drawn in Section V.
Ii Lattice action and simulation parameters
Our study is based on gauge field configurations provided by the tmfT Collaboration. These configurations were generated on a four-dimensional periodic lattice of spatial linear size and a temporal extent of for a mass-degenerate doublet of twisted mass fermions, cf. the review in Ref. Shindler (2008). The corresponding gauge action is the tree-level Symanzik improved action defined as
with , , and being the bare coupling constant. represents quadratic and rectangular Wilson loops built from the link variables . One important feature of this gauge action is its inherent improvement. For more details see Refs. Symanzik (1983a, b). The Wilson fermion action with an additional parity-flavor symmetry violating improvement term reads Frezzotti et al. (2001)
where the Pauli matrix acts in flavor space and the fermionic fields are expressed in terms of the twisted basis , which is related to the physical fields by
The Wilson covariant derivative acts on these as
and the quark mass is set by the twisted mass parameter and the hopping parameter , parameterizing the untwisted bare quark mass component. Here is the lattice spacing and . Note that for any finite the value for gets corrections through mass renormalization.
It must be noted that maximal twist is accomplished by tuning the hopping parameter to its critical value , where the untwisted theory would become massless. At maximal twist one achieves an automatic -improved fermion formulation Frezzotti and Rossi (2004). The hopping parameter entering the simulation is based on a set of -values for which was provided by the ETM Collaboration Baron et al. (2010). For intermediate the corresponding values are obtained through an interpolation as described in Ref. Burger et al. (2011). The bare twisted mass parameter has been adjusted such as to keep the physical pion mass constant along our scans in the bare inverse coupling .
As usual in finite temperature QCD, the imaginary time extent corresponds to the inverse temperature . To quote it in physical units we use interpolated—as well as slightly extrapolated—data for the lattice spacing reported for , 4.05 and 4.20 by the ETM Collaboration Baron et al. (2010) (see also Ref. Burger et al. (2011)). We restrict our analysis to lattice spacings fm.
For the reader’s convenience all parameters, like , the corresponding lattice spacings, temperatures, pion masses, the number of independent configurations and other relevant values are collected in Table 1. Note that there, for definiteness, “pion mass” corresponds to the charged pion. In Table 2 we provide also the respective pseudo-critical couplings and the corresponding temperatures and for the three pion mass values we use. These temperatures were obtained from fits around the maxima of the chiral susceptibility and from the behavior of the (renormalized) Polyakov loop , respectively (see the revised version of Ref. Burger et al. (2011)).
Iii The gluon and ghost propagators on the lattice
Gluon and ghost propagators are gauge dependent quantities. As in Ref. Aouane et al. (2012) we focus on Landau gauge and therefore have to transform the (unfixed) tmfT gauge ensemble until it satisfies the corresponding gauge condition. In differential form it reads
with the lattice gauge potentials
To render the link variables satisfying this condition one maximizes the gauge functional
by successive local gauge transformations acting on the link variables as follows
In order to achieve this, we subsequently apply two methods, first simulated annealing () and then over-relaxation (). is applied to finally satisfy the gauge condition (Eq. (6)) with a local accuracy of
while to reduce the Gribov ambiguity of lattice Landau gauge, by favoring gauge-fixed (Gribov) copies with large values for , see Parrinello and Jona-Lasinio (1990); Zwanziger (1990); Bakeev et al. (2004); Sternbeck et al. (2005); Bogolubsky et al. (2006, 2008); Bornyakov et al. (2010); Bogolubsky et al. (2009). For this the algorithm generates gauge transformations randomly by a Monte Carlo chain with a statistical weight . The “simulated annealing temperature” is a technical parameter which is monotonously lowered. Our annealing schedule is specified by a hot start at , after which is continuously lowered in equal steps until is reached. We apply 3500 steps between these two temperatures, and, for better performance, also added a few microcanonical steps to each (heatbath) step. Since we apply large number of steps to maximize , we restrict ourselves to one Gribov copy per configuraton.
Our first quantity of interest is the gluon propagator defined in momentum space as the ensemble average
where represents the average over configurations. denotes the Fourier transform of the gauge potential (7) and is the lattice momentum (), which relates to the physical momentum as
Henceforth we will use the notation where .
For non-zero temperature Euclidean invariance is broken, and it is useful to split into two components, the transversal (“chromomagnetic”) and the longitudinal (“chromoelectric”) propagator, respectively,
The fourth momentum component conjugate to the Euclidean time (Matsubara frequency) will be restricted to zero later on. For Landau gauge represent projectors transversal and longitudinal relative to the time-direction :
For the propagators [or their respective dimensionless dressing functions ] we find
where and . The zero-momentum propagator values are then defined as
Note that we have neglected a possible improvement related to the use of the improved gauge action Eq. (1).
The Landau gauge ghost propagator is given by
where the four-vector . denotes the ghost dressing function. The matrix is the lattice Faddeev-Popov operator
written in terms of , , i.e. the Hermitian generators of the su(3) Lie algebra normalized according to . For the inversion of we use the pre-conditioned conjugate gradient algorithm of Sternbeck et al. (2005) with plane-wave sources with color and position components .
To reduce lattice artifacts, we apply cylinder and cone cuts to our data Leinweber et al. (1999). Specifically we consider only diagonal and slightly off-diagonal momenta for the gluon propagator and diagonal momenta for the ghost propagator. Moreover, only modes with zero Matsubara frequency () are used.
Iv Gluon and ghost propagator results
iv.1 Momentum dependence
Data for the unrenormalized transverse () and longitudinal () gluon dressing functions and also for the ghost dressing function () is shown in Fig. 1. We show it versus the physical momentum for selected temperatures and for three pion masses (panels from top to bottom are for , 398 and 469 MeV, respectively). The corresponding renormalized functions, in momentum subtraction (MOM) schemes, can be obtained from
with the -factors being defined such that . For a renormalization scale of the -factors are quoted in Table 1.
Fig. 1 also shows curves connecting data points of same temperature. These were obtained from fits to the data for momenta . These curves may serve as input to studies of the corresponding DS or FRG equations.
which has been also used in Cucchieri et al. (2003); Cucchieri and Mendes (2011a) and appears in the context of the so-called “Refined Gribov-Zwanziger” framework Dudal et al. (2008, 2011). But we found it sufficient to set and . This fits well the data for and gives excellent values. The latter together with the results for the fit parameters are listed in Table 3. Note again that Fig. 1 shows data and the corresponding fits only for a selected range of temperatures, but Table 3 gives the fit parameters for all available temperatures. We cannot exclude that the -term is needed for smaller momenta. If true, it would indicate the occurrence of a pair of complex-conjugate poles.
For momenta above 3 GeV the fit fails to describe the data. In this range logarithmic corrections are expected to become important.
For the ghost dressing function we propose to use a fit formula like
In a first attempt we also tried for the last term with as a free parameter, but this was always found being consistent with . We therefore omit such infrared mass parameter and only keep a constant term in the ultraviolet limit.
Fit results for the fitting range are presented also in Table 3. One notes that our values are far from being optimal, in particular for the lower temperatures. Deviations typically occur at the lowest momenta. But this could not be cured, e.g., by a mass term alone. However note, the maximal deviations of fit and data points do not exceed .
iv.2 Temperature dependence
We now look at the temperature dependence of the dressing functions, where our temperature values cover the chiral restoration and the deconfinement phase transition, with the latter being signaled by a peak in the Polyakov loop susceptibility. These two crossover phenomena typically occur at different temperatures and will be denoted and , respectively, in what follows (see Table 2 for their values).
Looking once again at Fig. 1, we see that the momentum dependences of and change differently with temperature, irrespective of . In fact, while the (unrenormalized) transverse dressing function seems to be relatively insensitive to the temperature, the curves describing fan out for momenta below the renormalization scale . A stronger temperature dependence we also observed for for pure gauge theory Aouane et al. (2012), though there it was found to be much more pronounced due to the existence of a first order phase transition Aouane et al. (2012).
These observations are seen more clearly in Fig. 2, where we show ratios of the renormalized dressing functions or propagators
as functions of the temperature for 6 fixed (interpolated) momentum values , and for different pion masses (panels from top to bottom). For better visibility, ratios are normalized with respect to the respective left-most shown temperature in Fig. 2.
Looking at Fig. 2, we see decrease more or less monotonously with temperature in the crossover region, and this decrease is stronger the smaller the momentum. instead signals a slight increase within the same range, and the ghost propagator (at fixed low momenta) seems to rise a bit around .
Fig. 2 does not show ratios at zero momentum, but we show it for in Fig. 3 (upper row), again versus and from left to right for different pion masses. For , for example, clearly rises towards , whereas there are only weak indications for such a behavior for for the other two data sets. Much more statistics is necessary to resolve that.
The lower panels of Fig. 3, show data for the inverse renormalized longitudinal propagator at zero momentum, again versus temperature and from left to right for different pion masses. This quantity can be identified with an infrared gluon screening mass, and we clearly see it to rise with temperature in the crossover region. This again shows that this infrared gluon screening mass may serve as an useful indicator for the finite-temperature crossover of the quark-gluon system. However, we should keep in mind that the zero-momentum results for the gluon propagators are influenced also by strong finite-size and Gribov copy effects, which we could not analyze here.
We have presented data for the Landau gauge gluon and ghost propagators for lattice QCD at finite temperature with twisted mass fermion flavors. Our data is for a momentum range of 0.4 GeV to 4.0 GeV and was obtained on gauge field configurations produced by the tmfT Collaboration. This has allowed us to explore the propagator’s momentum dependence over the whole temperature range of the crossover region, and this separately for three (charged) pion mass values between 300 MeV and 500 MeV. We find that the propagators change smoothly passing through the crossover region and the most significant change is seen for the longitudinal (i.e., electric) component of the gluon propagator.
We also provide fitting functions for our data. These and the corresponding fit parameters, given in Table 3, may serve as interpolation functions of our data when used as input to studies which employ continuum functional methods to address problems of QCD at finite temperature. Actually, for both the transversal and longitudinal gluon dressing function these interpolation functions give quite a good description of our data for all temperatures. For the ghost dressing function, this is achieved only for selected temperatures (see Table 3 for details).
We hope our results will help these (continuum-based) studies to get further predictions of the behavior of hadronic matter close to the transition region that would be too difficult if addressed on the lattice directly.
We thank the members of the tmfT Collaboration for giving us access to their gauge field configurations produced for fermion degrees of freedom with the Wilson-twisted mass approach. We express our gratitude to the HLRN supercomputer centers in Berlin and Hannover for generous supply with computing time. R. A. expresses thanks to L. Zeidlewicz for his help in carrying out the data analysis and gratefully acknowledges financial support by the Yousef Jameel Foundation. A. S. acknowledges support from the European Reintegration Grant (FP7-PEOPLE-2009-RG No.256594), and F. B. from the DFG-funded graduate school GK 1504.
- preprint: HU-EP-12/45
- Detailed results are presented in a recent update to Burger et al. (2011). We thank the tmfT Collaboration for providing us their -data prior to publication.
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