Perturbative study of the QCD phase diagram for heavy quarksat nonzero chemical potential

# Perturbative study of the QCD phase diagram for heavy quarks at nonzero chemical potential

U. Reinosa Centre de Physique Théorique, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France    J. Serreau Astro-Particule et Cosmologie (APC), CNRS UMR 7164, Université Paris 7 - Denis Diderot
10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France
M. Tissier LPTMC, Laboratoire de Physique Théorique de la Matière Condensée, CNRS UMR 7600, Université Pierre et Marie Curie,
boite 121, 4 pl. Jussieu, 75252 Paris Cedex 05, France
July 12, 2019
###### Abstract

We investigate the phase diagram of QCD with heavy quarks at finite temperature and chemical potential in the context of background field methods. In particular, we use a massive extension of the Landau-DeWitt gauge which is motivated by previous studies of the deconfinement phase transition in pure Yang-Mills theories. We show that a simple one-loop calculation is able to capture the richness of the phase diagram in the heavy quark region, both at real and imaginary chemical potential. Moreover, dimensionless ratios of quantities directly measurable in numerical simulations are in good agreement with lattice results.

QCD phase diagram, quantum field theory at finite temperature, deconfinement transition
###### pacs:
12.38.Mh, 11.10.Wx, 12.38.Bx

## I Introduction

The study of the phase diagram of QCD is an important theoretical challenge with numerous phenomenological implications, e.g., for heavy-ion collision experiments, early Universe cosmology, and astrophysics. The properties of the theory along the temperature axis have been intensively studied by means of lattice calculations and it is by now well established that the first-order confinement-deconfinement phase transition of the pure gauge SU() theory becomes a crossover in the presence of light dynamical quarks with realistic masses Borsanyi:2013bia ().

The situation is much less under control at finite baryonic chemical potential, where lattice simulations suffer from a severe sign problem deForcrand:2010ys (); Philipsen:2010gj (). Calculations with realistic quark masses are limited to the region of small chemical potentials in units of the temperature. One hotly debated issue in this context is the possible existence of a line of first-order chiral transition with a critical endpoint, as predicted by various model calculations Stephanov:2004wx (); deForcrand:2007rq (); Schaefer:2007pw (). Efforts to tackle this question directly from the QCD action combine the investigation of various ways to circumvent the sign problem on the lattice deForcrand:2010ys (); Sexty:2014zya (); Aarts:2015kea (); Tanizaki:2015pua () and the use of nonperturbative continuum approaches Fischer:2011mz (), where the sign problem, although not completely absent, is much less severe.

Unlike lattice calculations, continuum approaches rely on identifying efficient approximations for the relevant dynamics. Standard perturbative tools are notoriously inadequate in this context because the typical transition temperatures are of the order of the intrinsic scale of the theory, where the coupling becomes large. This motivates the use of nonperturbative approaches, typically based on truncations of the functional renormalization group or Dyson-Schwinger equations Fischer:2011mz (); Fischer:2012vc (); Fischer:2013eca (); Herbst:2013ail (); Gutierrez:2013sta (); Fischer:2014ata (); Fischer:2014vxa (). Lattice simulations in situations where they are well under control provide then valuable benchmarks for testing the various methods and approximations.

Examples of such situations are the case of a purely imaginary chemical potential, where there is no sign problem, or the study of QCD with heavy quarks. A systematic expansion around the infinite mass (quenched) limit can be devised, for which the sign problem is mild enough so that numerical simulations can be performed up to large chemical potentials de Forcrand:2002ci (); de Forcrand:2003hx (); D'Elia:2002gd (); Fromm:2011qi (). Although these do not correspond to physically relevant situations, the phase diagram in these cases is interesting in its own and actually features a rich structure. This has been investigated both on the lattice and by means of effective models Pisarski:2000eq (); Dumitru:2003cf (); Dumitru:2012fw (); Kashiwa:2012wa (); Kashiwa:2013rm (); Nishimura:2014rxa (); Nishimura:2014kla (); Lo:2014vba ().

In the present work, we investigate the phase diagram of QCD with heavy quarks by means of a modified perturbative expansion in the context of background field methods. This is based on a simple massive extension of the standard Faddeev-Popov (FP) action in the Landau-DeWitt gauge which has been successfully applied to the confinement-deconfinement transition of the SU() and SU() Yang-Mills theories Reinosa:2014ooa (); Reinosa:2014zta (); Reinosa:su3 ().

The motivations for such a massive extension are twofold. First, gauge-fixed lattice calculations of the vacuum Yang-Mills correlators in the (minimal) Landau gauge have shown that the gluon propagator behaves as that of a massive field at small momentum, whereas the ghost remains massless. This suggests that the dominant aspect of the nontrivial gluon dynamics (in the Landau gauge) might be efficiently captured by a simple mass term, up to corrections that can be computed perturbatively. This scenario has been put to test in Refs. Tissier_10 (); Tissier_11 (), where a one-loop calculation of the gluon and ghost propagators in a simple massive extension of the Landau gauge—known as the Curci-Ferrari model Curci76 ()—has been indeed shown to describe the lattice results remarkably well. Similar successful results have been obtained for the three-point correlation functions Pelaez:2013cpa () as well as for the vacuum propagators of QCD Pelaez:2014mxa () and the Yang-Mills propagators at finite temperature Reinosa:2013twa (). An interesting aspect of this approach is that the gluon mass acts as an infrared regulator and perturbation theory is well defined down to deep infrared momentum scales Tissier_11 ().

The second motivation is more formal. It is related to the fact that the FP quantization procedure, which underlies standard perturbative tools, is plagued by the existence of Gribov ambiguities Gribov77 () and is, at best, a valid description at high energies. For instance, it is known that a nonperturbative implementation of the BRST symmetry of the FP action on the lattice leads to undefined zero over zero ratios for gauge-invariant observables Neuberger:1986vv (); Neuberger:1986xz (). In fact, existing gauge-fixing procedures on the lattice—e.g., the minimal Landau gauge—typically break the BRST symmetry. This suggests that a consistent quantization procedure, which correctly deals with the Gribov issue, is likely to induce effective BRST breaking terms. The simplest such term consistent with locality and renormalizability is a gluon mass term.111A more precise relation between the Gribov problem and the gluon mass term has been obtained in Ref. Serreau:2012cg (), where a new quantization procedure based on a particular average over the Gribov copies along each gauge orbit has been proposed. The bare gluon mass is related to the gauge-fixing parameter which lifts the degeneracy between the copies.

In Ref. Reinosa:2014ooa (), we have extended this approach to the Landau-DeWitt gauge—which generalizes the Landau gauge in the presence of a nontrivial background field—with the aim of studying the confinement-deconfinement phase transition of SU() Yang-Mills theories. Background field methods allow one to efficiently take into account the nontrivial order parameter of the transition Braun:2007bx (). We have shown that a simple one-loop calculation correctly describes a confined phase at low temperature and a phase transition of second order for the SU() theory and of first-order for the SU() theory, with transition temperatures in qualitative agreement with known values. Two-loop corrections, computed in Refs. Reinosa:2014zta (); Reinosa:su3 (), make this agreement more quantitative; see Ref. Serreau:2015saa () for a short review. We mention that two-loop contributions also resolve some spurious unphysical behaviors (e.g., negative entropy) observed in background field calculations at one-loop order; see, e.g., Sasaki:2012bi ().

The present work is a natural extension of these studies to QCD with heavy quarks flavors at finite chemical potential. We compute the background field potential at one-loop order, from which we can read the value of the order parameters—the averages of the traced Polyakov loop and of its Hermitic conjugate—as functions of the temperature and of the chemical potential. We show that this simple calculation accurately captures the rich structure of the phase diagram, both for real and imaginary chemical potential. Moreover, we obtain parameter-free values for the dimensionless ratios of the quark mass over the temperature at various critical points which compare well with lattice results.

The paper is organized as follows. We present the basic framework, i.e., the formulation of the QCD action in the massive extension of the Landau-DeWitt gauge in Sec. II and discuss in detail various symmetry properties of the relevant generating function in Sec. III. This generalizes known material to the case of a nontrivial background field. The one-loop calculation of the background field potential is straightforward and is detailed in Sec. IV. The rest of the paper is devoted to the discussion of our results at vanishing (Sec. V), imaginary (Sec. VI), and real (Sec. VII) chemical potential. In the latter case, we discuss how the sign problem also affects continuum approaches. Finally, Appendix A briefly presents some consequences of charge conjugation symmetry at vanishing chemical potential and Appendix B provides an approximate calculation of the background field potential which allows for a simple analytic understanding of some results presented in the main text.

## Ii The QCD action in the massive Landau-DeWitt gauge

The Euclidean action of QCD in dimensions with colors and quark flavors reads

 SQCD=∫x{14FaμνFaμν+Nf∑f=1¯ψf(Dto7.499886pt\hss\sl/\/\hss+Mf+μγ0)ψf}, (1)

where , with the inverse temperature, , with the coupling constant and the structure constants of the SU() group, and , with the SU() generators in the fundamental representation, normalized as . Finally, denotes the chemical potential. We leave the Dirac and color indices of the quark fields implicit and and are understood in the common sense as column and line bispinors, respectively. The Euclidean Dirac matrices222These are related to the standard Minkowski matrices as and . In the following, we work in the Weyl basis, where and . are Hermitian and satisfy the anticommutation relations .

The gauge field is decomposed in a background field and a fluctuating contribution as

 Aaμ=¯Aaμ+aaμ, (2)

and the Landau-DeWitt gauge is defined as

 (¯Dμaμ)a=0, (3)

 SFP=∫x{(¯Dμ¯c)a(Dμc)a+iha(¯Dμaμ)a}, (4)

where , and are anticommuting ghost fields, and is a Lagrange multiplier. Finally, as discussed in the Introduction, we also consider a massive extension of the Landau-DeWitt gauge, with mass term333 In principle, one could implement different bare masses for the gluon fields in the temporal and spatial directions, e.g., to account for Gribov ambiguities on asymmetric lattices at finite temperature. We chose here the minimal extension of the FP theory with a symmetric, vacuumlike mass term. Reinosa:2014ooa ()

 Sm=∫x12m2aaμaaμ. (5)

In the following, we grab together the fluctuating fields of the pure gauge sector, which all belong to the adjoint representation of the gauge group, as . The total gauge-fixed action is invariant under the combined transformation of the fluctuating fields

 ψ→Uψ,¯ψ→¯ψU†,φata→UφataU† (6)

and of the background field

 ¯Aaμta→U¯AaμtaU†+igU∂μU†, (7)

where is a local SU() matrix.

The background field is nothing but a gauge-fixing parameter and is used here as a device to capture nontrivial physics in an efficient way. We constrain it so as to break as few symmetries as possible. The symmetries of the Euclidean space with the boundary conditions implied by finite temperature field theory allow for a constant vector field in the temporal direction: . Moreover, it is always possible, through a global color transformation, to bring the color vector in the Cartan subalgebra of the gauge group, spanned by the diagonal generators. In the following, we denote the latter by . For SU(), these are and , where are the Gell-Mann matrices. Accordingly, we consider the generating function (we make explicit the dependence on the chemical potential for latter use)

 Z(J,¯A,μ)=∫D(φ,ψ,¯ψ)e−SQCD−SFP−Sm+Ssrc, (8)

with the source term

 Ssrc=∫xJaμAaμ (9)

restricted to a constant source in the temporal direction and in the Cartan subalgebra: , with . In the following, and denote vectors in the plane spanned by the SU() Cartan directions (to which we shall refer as the Cartan plane in the following):

 (10)

The main quantity of interest in the following is the Legendre transform , where is defined through

 Ak=1βΩ∂lnZ∂Jk∣∣∣J(A,¯A,μ). (11)

We shall evaluate this Legendre transform at , that is,

 V(r,μ)=−1βΩlnZ(~J(¯A,μ),¯A,μ)+~Jk(¯A,μ)¯Ak, (12)

where , is the spatial volume, and where we defined . A priori, one could leave independent of and extremize the effective potential with respect to at fixed . However, one obtains the same physics by identifying and extremizing444We shall see below that, for any imaginary chemical potential, one needs to minimize the potential (12). However, for real chemical potential, some amendments need to be made to this rule. with respect to . This is a particularly convenient choice since the fluctuating gauge field has vanishing expectation value, , even at nonvanishing source.

Finally, important physical observables to be considered below are the averages of the traced Polyakov loop in the fundamental representation and of its Hermitic conjugate:

 ℓ(μ) =13tr⟨Pexp(ig∫β0dτAa0ta)⟩, (13) ¯ℓ(μ) =13tr⟨¯Pexp(−ig∫β0dτAa0ta)⟩, (14)

where and denote path ordering and antiordering respectively. In general, is not equal to the complex conjugate , e.g., in the case of a complex action. For real chemical potential, the quantities (13) and (14) are real and are related to the free energies and of a static quark or antiquark as and Svetitsky:1985ye (); Philipsen:2010gj (). The physical loops and are to be evaluated at vanishing sources, that is, at an extremum of the background field potential (12). In order to discuss symmetries below, it is useful to introduce the following functions of the background field, defined at nonvanishing source ,

 ℓ(r,μ) =13tr⟨Pexp(irktk+ig∫β0dτaa0ta)⟩, (15) ¯ℓ(r,μ) =13tr⟨¯Pexp(−irktk−ig∫β0dτaa0ta)⟩, (16)

where it is understood that the averages on the right-hand side are evaluated at , that is, at . The physical loops and are obtained by evaluating the functions (15) and (16) at the relevant extremum of the potential (12).

## Iii Symmetries

In this section, we discuss various transformation properties of the generating function (8), background field potential (12) and Polyakov loops functions (15) and (16). For later purposes, it is useful to consider the general case where , , and are complex numbers. We first discuss the effect of charge conjugation and of complex conjugation which relate the cases and , respectively. Next, we analyze the consequences of the continuous global and local color transformations. Finally, we discuss the Roberge-Weiss symmetry which relates the theories with chemical potentials and .

### iii.1 Charge conjugation

We perform the change of variables

 ψ→C¯ψt,¯ψ→−ψtC†,φ→Cφ (17)

under the functional integral (8). Here the charge conjugation matrix only acts on the Dirac indices of fields in the fundamental representation whereas acts on the color indices of adjoint fields. The former satisfies

 C†γμC=−γtμ (18)

and is given by in the Weyl basis. It is such that . As for the latter, it acts on the generators of the color group as

 Cabtb=−(ta)t. (19)

Using the Gell-Mann basis, one easily checks that it is a diagonal matrix with eigenvalues on the lines 2, 5 and 7 and otherwise. It is thus an orthogonal matrix such that and which conserves the structure constants (). Notice, however, that it has determinant and it is, therefore, not a color transformation in the adjoint representation.

It is an easy exercise to check that the only effect of the change of variables (17) on the generating function (8) is to change

 J→CJ=−J,¯A→C¯A=−¯A,andμ→−μ, (20)

where we used the fact that in the Cartan subalgebra. We conclude that

 Z(J,¯A,μ)=Z(−J,−¯A,−μ). (21)

This implies the relation for the source defined in Eq. (11). It follows that and thus

 V(r,μ)=V(−r,−μ). (22)

Similarly, we deduce the relation

 ℓ(r,μ)=¯ℓ(−r,−μ). (23)

### iii.2 Complex conjugation

We now consider the change of variables555This is similar to the -transformation discussed in Refs. Nishimura:2014rxa (); Nishimura:2014kla ().

 ψ→Kψ,¯ψ→¯ψK†,φ→Kφ, (24)

where the matrix only acts on Dirac indices and is such that and

 K†γμK=γ⋆μ. (25)

As for the matrix acting on the color indices of the adjoint fields, it satisfies

 Kabtb=−(ta)⋆. (26)

Using the fact that the generators are Hermitian, we see that . Upon the additional change of integration variable and the transformation

 J→KJ⋆=−J⋆,¯A→K¯A⋆=−¯A⋆,andμ→μ⋆, (27)

one recovers the original form of the generating function with all complex coefficients replaced by their complex conjugate, that is,666Note that the integration variables are not affected by the complex conjugation.

 Z(J,¯A,μ)=Z⋆(−J⋆,−¯A⋆,μ⋆). (28)

Following similar steps as before, this implies and thus

 V(r,μ)=V⋆(−r⋆,μ⋆). (29)

One also shows that

 ℓ(r,μ)=ℓ⋆(−r⋆,μ⋆) (30)

and similarly for .

### iii.3 Global color symmetry

The gauge-fixed action is invariant under global SU() transformations

 ψ→Rψ,¯ψ→¯ψR†,φ→Rφ, (31)

where is a global SU() matrix and the corresponding SO(8) transformation

 Rab=2tr(R†taRtb). (32)

Of particular interest are those global color transformations which do not mix the Cartan subalgebra with the other elements of the Lie algebra (i.e., is block-diagonal), such that they leave the source and the background in the Cartan subalgebra. For instance, the SU() matrix

 R1=⎛⎜⎝$0$$1$$0$$−1$$0$$0$$0$$0$$1$⎞⎟⎠ (33)

induces such a block-diagonal color rotation whose restriction to the Cartan subalgebra reads

 RCartan1=diag(−1,1). (34)

In the Cartan plane, this corresponds to a reflexion about the -axis. We can therefore write

 Z(J,¯A,μ)=Z(R1J,R1¯A,μ). (35)

As before, this implies that the source transforms covariantly, i.e., , from which we conclude that

 V(r,μ)=V(R1r,μ). (36)

Similar considerations yield, for the functions (15) and (16),

 ℓ(r,μ)=ℓ(R1r,μ) (37)

and similarly for .

It is easy to check that the SU() matrices

 R2=⎛⎜⎝$0$$0$$−1$$0$$1$$0$$1$$0$$0$⎞⎟⎠andR3=⎛⎜⎝$1$$0$$0$$0$$0$$−1$$0$$1$$0$⎞⎟⎠ (38)

also generate color rotations in the adjoint representation such that the Cartan components do not mix with other color directions. In the Cartan plane, they correspond to mirror images about two axes passing through the origin and making an angle of with the -axis. Together with , they generate all the color rotations under which the Cartan subalgebra is stable. These form a group which also contains rotations by an angle around the origin. This is the Weyl group of the su() algebra.

### iii.4 Background gauge invariance

We now come to analyze the consequences of the invariance of the action under the local transformation (6)–(7). In order to maintain the background field constant, in the temporal direction, and in the Cartan subalgebra, we consider local color transformations of the form , which simply generate a translation of the background field in the Cartan plane: . True gauge transformations—which leave physical observables invariant—are periodic in time, , which implies that the eigenvalues of the matrix must be multiples of . This is solved by

 ϕ=j1u1+j2u2, (39)

with integers and

 u1=2π(1√3),u2=2π(1−√3). (40)

Both the integration measure and the action at vanishing source in (8) are left invariant by the transformation (6)-(7). The change of the source term is trivial and we get

 lnZ(J,¯A+ϕ/(βg),μ)βΩ=lnZ(J,¯A,μ)βΩ+Jkϕkβg. (41)

We deduce that and thus that

 V(r,μ)=V(r+ϕ,μ), (42)

with given in Eq. (39). One also shows that

 ℓ(r,μ)=ℓ(r+ϕ,μ) (43)

and similarly for .

### iii.5 Roberge-Weiss symmetry

Following Ref. Roberge:1986mm (), we now consider local SU() transformations of the type studied in the previous subsection with periodic in time up to a nontrivial element of the center of the gauge group: . This results in translations of the background field with integers and777The vectors and generate a lattice dual to that generated by the roots of the algebra; see below. This property generalizes to any compact Lie group with a simple Lie algebra, see Reinosa:su3 () for a detailed discussion.

 e1=2π(1−1/√3),e2=2π(−1−1/√3). (44)

As is well known, the pure gauge action—including the FP and mass terms—is invariant under such twisted gauge transformations, whereas the source term transforms trivially, as in Eq. (41). This symmetry is explicitly broken by the antiperiodic boundary conditions of the quark fields in the fundamental representation. To cope for this, we perform the additional change of variables

 ψ→e∓2iπ3τβψ,¯ψ→e±2iπ3τβ¯ψ. (45)

All terms in the functional integral (8) are invariant except for the quark kinetic term, whose variation, , can, however, be compensated by a shift of the chemical potential in the imaginary direction. A similar analysis as in the previous subsections leads to the following identities for the background field potential

 V(r,μ) =V(r+ej,μ+2iπ3β). (46)

where . As is well known, the traced Polyakov loop gets multiplied by an element of the center under the twisted gauge transformations considered here. This leads to

 ℓ(r,μ) =e−2iπ/3ℓ(r+ej,μ+2iπ3β), (47) ¯ℓ(r,μ) =e2iπ/3¯ℓ(r+ej,μ+2iπ3β). (48)

Observe that , which shows that if we use the property (46) twice so as to add and subtract to the chemical potential, we retrieve the original potential, up to a gauge transformation of the form (42). Similarly, using the property (46) thrice so as to add three times to the chemical potential and using the relation we see that we generate a gauge transformation of the form (42) up to a translation by of the chemical potential. We, thus, recover the well-known fact that the physics is unaffected by a change888This can be checked by performing the change of variables , , which preserves the antiperiodic boundary conditions. .

### iii.6 Vanishing sources

We end this section by discussing the case of vanishing sources , which corresponds to the physical point. For instance, the partition function is obtained as the generating function (8) at . In the present context, where we set , it is obtained from Eq. (12) at , namely,

 −lnZ(μ)βΩ=V(rext(μ),μ) (49)

where is any physical extremum of . In general, there are many such extrema, which form a set . The properties (22) and (46) imply999This is obviously true for the set of absolute minima which are the relevant extrema at vanishing and imaginary chemical potential. For a real chemical potential, the choice of the relevant extrema is more subtle, as discussed below. Our prescription, in Sec. VII, is also compatible with Eq. (51). that this set satisfies,

 {rext(μ)}={−rext(−μ)}={rext(μ±2iπ/(3β))∓ej}, (50)

with . Similarly, the relation (29) implies that the set of extrema of is . Using Eq. (49) and the relations (22), (29), and (46), we then retrieve the standard relations for the partition function Philipsen:2010gj ()

 Z(μ)=Z(−μ)=Z⋆(μ⋆)=Z(μ+2iπ3β). (51)

Similarly, using the definitions and and assuming that none of the symmetries mentioned here is spontaneously broken, we get the known relations for the averages of the traced Polyakov loop and of its Hermitic conjugate:

 ℓ(μ)=¯ℓ(−μ)=ℓ⋆(μ⋆)=e−2iπ/3ℓ(μ+2iπ3β). (52)

For imaginary chemical potential, the Roberge-Weiss symmetry can be spontaneously broken. In that case, the averaged Polyakov loops can be multivalued and the relations in (52) relate the sets of their possible values.

## Iv The background field potential at one-loop order

Here, we detail the calculation of the leading-order, one-loop effective potential (12). We consider a general compact Lie group with simple Lie algebra and we specify to SU() when needed. For completeness we also give the corresponding leading-order (tree-level) expressions of the Polyakov loop functions (15) and (16). Our approach is similar to that used in Dumitru:2012fw ().

The one-loop contribution to the potential only involves the action up to quadratic order in the fluctuating fields. The pure gauge contribution has been computed in Ref. Reinosa:2014ooa (). Denoting by the quark mass matrix, the relevant contribution from the quark sector is

 Vq(r,μ)=−TrLn(∂to7.499886pt\hss\sl/\/\hss+M+μγ0−igγ0¯Aktk), (53)

where the functional trace involves a sum over fermionic Matsubara frequencies, an integral over spatial momenta and a trace over Dirac, color, and flavor indices. The Cartan generators can be diagonalized simultaneously and their respective eigenvalues form a set of (possibly degenerate, i.e., identical) vectors in the space spanned by the Cartan directions, with the dimension of the fundamental representation.101010The present calculation easily generalizes to fields in any representation, provided one works with the corresponding weights. In group theory language, these are called the weights of the representation. The fundamental representation of the su() algebra has three nondegenerate weights, , which are related to the vectors introduced in Eq. (44) as and . Explicitly,

 ρ1 =12(−11/√3), (54) ρ2 =12(11/√3), (55) ρ3 =−(01/√3). (56)

Having diagonalized the color structure in Eq. (53), we see that the constant temporal background can be absorbed in a redefinition of the chemical potential , with , which lifts the degeneracy between the various color states. We thus get

 Vq(r,μ)=∑f,ρV0f(μ−irρ/β), (57)

where the sum runs over all flavors and all color states (weights ) in the fundamental representation and is the one-loop contribution from a single quark flavor in a definite color state at vanishing background field and nonzero chemical potential Kashiwa:2012wa (). This is given by the standard formula (written here for and letting aside a trivial - and -independent piece)

 V0f(μ)=−Tπ2∫∞0dqq2{ln[1+e−β(εfq+μ)]+(μ→−μ)}, (58)

where .

The pure gauge contribution has been computed in Reinosa:2014ooa () for SU() and SU(). It is instructive to derive the corresponding formula for a general compact Lie group with simple Lie algebra. After taking into account the contribution from the massive gluon modes and the partial cancelation between massless modes contributions from the ghost and the sectors, one has, in ,

 Vgauge(r)=32Fm(r)−12F0(r), (59)

where

 Fm(r)=TrLn(¯D2+m2) (60)

and , with the generators of the Lie algebra in the adjoint representation. Here, the functional trace involves a sum over bosonic Matsubara frequencies, a spatial integral and a trace over color indices.

As for the quark contribution, this trace can be easily evaluated using the weights of the adjoint representation. In the subspace spanned by the Cartan directions, these form a set of vectors whose components are the eigenvalues of the Cartan generators , where is the dimension of the adjoint representation. One sees that the temporal background enters as a shift of the Matsubara frequencies, which can be interpreted as an effective imaginary chemical potential , with . We thus get

 Fm(r)=∑κTπ2∫∞0dqq2ln[1−e−βεq+irκ], (61)

where the sum runs over all color states (weights ) in the adjoint representation. By definition, the generators have as many zero eigenvalues—corresponding to —as there are Cartan directions. Furthermore, one can show that the nonzero weights always come in pairs , which are called the roots of the Lie algebra. We obtain,

 Fm(r)=Tπ2 ∫∞0dqq2{dCln[1−e−βεq] + ∑αln[1+e−2βεq−2e−βεqcosrα]}, (62)

where is the dimension of the Cartan subalgebra and the sum runs over the possible pairs . For SU(), there are two Cartan directions and so there are six roots , where we can choose [see Eq. (40)] and . Explicitly,

 α1 =12(1√3), (63) α2 =12(1−√3), (64) α3 =−(10). (65)

The sum in (IV) runs only over the set . For completeness, we mention that, for , the function admits the closed form Weiss:1980rj (); Gross:1980br ()

 F0(r) =−π2T445dC +T46∑α[(rα−π)42π2−(rα−π)2+7π230]. (66)

Finally, the leading-order, tree-level expressions of the Polyakov loop functions (15) and (16) are trivially obtained as

 ℓ(r,μ)=¯ℓ(−r,μ)=tr(eirktk)dF=1dF∑ρeirρ (67)

and do not depend explicitly on . For SU(), we have

 ℓ(r,μ)=13[e−ir8√3+2eir82√3cos(r3/2)]. (68)

It is easy to check that the above expressions reproduce the SU() results of Ref. Reinosa:2014ooa (). Also, it is a simple exercise to verify that the leading-order expressions (57), (58), (59), (IV), and (68) satisfy all the symmetry properties derived in the previous section. To this aim, it is useful to note that the scalar products , , and are all multiples of , whereas and .

## V Vanishing chemical potential

We now apply the above calculations to various situations of interest. We begin our discussion with the case , where the lattice simulations are well under control. We first discuss how the symmetry considerations of Sec. III constrain the background field potential at . Then we study the phase structure of the theory as a function of the quark masses and we compare our findings with lattice results.

### v.1 Symmetries

A direct consequence of the relations (21) and (28) at is that the generating function is real if we choose the source and the background field components to be both real. Indeed, these relations imply

 Z(J,¯A,0)=Z(−J,−¯A,0)=Z⋆(J,¯A,0). (69)

In particular, this guarantees that the average value of the gauge field at nonvanishing source, defined in Eq. (11), is real. This is important in the present setup where we eventually choose such that . Moreover, the above relations, together with Eqs. (23) and (30) imply, for a real background field,

 V(r,0)=V(−r,0)=V⋆(r,0) (70)

and

 ℓ(r,0)=¯ℓ(−r,0)=ℓ⋆(−r,0). (71)

In this case, the physical point is obtained by minimizing the background field potential.111111The standard proof that the physical point corresponds to the absolute minimum of the potential is based on the convexity of which is easily shown in the case where the action in the functional integral (8) is real. We note that this is not guaranteed in the usual FP quantization procedure since the determinant of the FP operator—obtained by integrating out the ghost and antighost fields—is not positive definite. We expect the situation in the present massive extension to be more favourable to a convex since the mass term suppresses the large field configurations for which the FP operator develops negative eigenvalues. A more thorough investigation of these aspects goes beyond the scope of the present work and we postpone it to a future work (we stress that this is not particular to the present model but it is a general issue for background field methods). Here, we shall make the conservative assumption that in the cases where the background field potential is a real function of real variables, the physical point corresponds to an absolute minimum.

The inversion symmetry (70) implies that there are three more reflexion symmetry axes121212These are symmetries of the potential, not exactly of the function , which gets complex conjugated in the inversion through the origin; see Eq. (71). in the plane on top of those identified in Sec. III.3. One is the axis and the two other are the lines which pass through the origin and which make an angle with the axis. This is represented in Fig. 1. The origin and the points related to it by the translations described in Sec. III.4 now have the symmetry .

Using the symmetries of the potential, we can restrict the search for its minimum to the elementary cell depicted as a shaded triangle on Fig. 1. As discussed in Appendix A, the assumption that charge conjugation symmetry is not spontaneously broken at implies that the absolute minimum of the background potential lies on the axis in this cell. This is indeed, what we find from the detailed investigation of the one-loop potential. Furthermore, Eqs. (71) and (37) imply that on the axis so that the physical Polyakov loops, Eqs. (13) and (14), are equal and real,

 ℓ(μ=0)=¯ℓ(μ=0)∈\mathdsR, (72)

as expected. Indeed, the fact that they are real is consistent with their standard interpretation in terms of the free energy of a static quark or antiquark Svetitsky:1985ye (); Philipsen:2010gj (). The fact that they are equal simply tells that there is no distinction between the free energy of a quark and that of an antiquark at , as a result of the charge conjugation symmetry.

### v.2 One-loop results

To analyze the phase diagram at , we track the absolute minimum of the one-loop background field potential computed in Sec. IV as a function of temperature for different values of the quark masses. We consider the case of two degenerate flavors with mass and a third flavor with mass . In the following, we shall either present our results in units of for dimensionful quantities, or compute dimensionless quantities that can be compared with lattice results. To get a rough idea of scales, a typical value used to fit lattice propagators at zero temperature for the SU() Yang-Mills theory is  GeV Reinosa:2014ooa (); Tissier_11 ().

Depending on the values of the quark masses, we find different types of behaviors when we change the temperature, as illustrated on Fig. 2. For large masses, the absolute minimum presents a finite jump at some transition temperature, signaling a first-order transition. Instead, for small masses, there is always a unique minimum, whose location rapidly changes with temperature in some crossover regime. At the common boundary of these two mass regions, the system presents a critical behavior: there exists a unique minimum of the potential for all temperatures, which however behaves as a power-law around some critical temperature. The associated nonanalytic behavior of the minimum of the potential as a function of temperature is a consequence of the fact that, at the critical temperature, the first, second and third derivatives in the direction vanish, a criterion which we used in order to determine the critical line in the Columbia plot of Fig. 3. In the degenerate case we get, for the critical mass, and, for the critical temperature, . We thus have . This dimensionless ratio does not depend on the value of and can be directly compared to lattice results. For instance, the calculation of Ref. Fromm:2011qi () yields, for degenerate quarks, . We obtain similar good agreement for different numbers of degenerate quark flavors, as summarized in Table 1.

We observe that the critical temperature is essentially unaffected by the presence of quarks. It is actually close to the one obtained in the present approach for the pure gauge SU() theory Reinosa:2014ooa (). This is due to the fact that, for the typical values of near the critical line, the quark contribution to the potential is Boltzmann suppressed as compared to that of the gauge sector, as discussed in Appendix B.

Finally, we compare our results for the ratio to those of Ref. Kashiwa:2012wa () based on matrix models. We also mention that recent calculations in the Dyson-Schwinger approach Fischer:2014vxa () yield values of the ratio that are systematically smaller than the ones obtained on the lattice. The origin of this discrepancy remains to be clarified.

## Vi Imaginary chemical potential

The study of the QCD phase diagram for imaginary chemical potential is of great interest. There is no sign problem in this case and lattice simulations can be performed in a standard way de Forcrand:2003hx (); D'Elia:2002gd (). Thermodynamic potentials at real chemical potential can then be obtained, in principle, by analytic continuation. In practice, however, this is restricted to small values because the theory at imaginary chemical potential has a nontrivial phase structure with nonanalytical behaviors. The phase diagram at imaginary chemical potential is also interesting in itself as it presents bona fide properties of QCD Roberge:1986mm (); deForcrand:2010he (). In this section, we describe the predictions of our one-loop calculation for a purely imaginary chemical potential . As in the previous case, we begin the discussion by analyzing the consequences of the symmetry properties discussed in Sec. III.

### vi.1 Symmetries

As before, we seek conditions on the source and the background field such that the identification , with defined in Eq. (11) is consistent. The relations (21) and (28) imply that the generating function is real if we choose both the source and background field to be real. We have, in that case,

 Z(J,¯A,iμi)=Z(−J,−¯A,−iμi)=Z⋆(J,¯A,iμi). (73)

The average field of Eq. (11) is thus real and can be consistently identified with . The background field potential is also real, the Polyakov loop functions are related by complex conjugation, , and so are the physical Polyakov loops:

 ¯ℓ(iμi)=ℓ⋆(iμi). (74)

The Polyakov loop is complex for generic values of and its argument actually plays the role of an order parameter for the various transitions described below Roberge:1986mm (). Finally, the background field potential being a real function of real variables as in the case , we adopt the same prescription as before to choose the background field for the evaluation of physical observables, namely we minimize the potential .

The symmetry properties of the potential for a generic value of are summarized in Fig. 4. The white dots are all images of each others by the global color symmetries (37) and the gauge transformations (42) and are thus all physically equivalent. The same holds for the black dots but, contrarily to the case discussed in the previous section, the black and white dots are not equivalent. The elementary cell, to which one can restrict the analysis, is thus twice as large as that of the previous section. It can be chosen as the shaded equilateral triangle shown in Fig. 4. Observe that the origin [and the equivalent points obtained by the translations (42)] have the symmetry .

Remarkably, the elementary cell has an extra symmetry when the imaginary chemical potential is a multiple of . Using the general periodicity property , one can reduce the discussion to the interval . The case has been discussed before and possesses a symmetry around the origin [and all the equivalent points obtained by the translations (42)]. This is also true for . Indeed, using Eq. (21) and the -periodicity in , we have

 V(r,iπ/β)=V(−r,−iπ/β)=V(−r,iπ/β). (75)

This corresponds to an inversion about the origin and we thus retrieve the same set of symmetries as in the case , depicted in Fig. 1. Finally, the cases and , with integer, can be obtained from the cases and , respectively, by using the relation (46). The symmetries are similar to those depicted in Fig. 1 where, however, the point with symmetry is not the origin anymore but one of the two other vertices of the elementary triangle [and their images by the translations (42)]. For and , this is the upper vertex, located at , whereas for and , it is the lower vertex, located at . Using the axes of symmetry corresponding to global color transformations (36) (the plain lines in Fig. 1), one sees that the elementary triangle is rotated by each time the imaginary chemical potential is shifted by . The corresponding Polyakov loop acquires a phase .