1. Introduction

Among the important subjects of study in the realm of high-energy/nuclear physics, both from the theoretical and experimental points of view, are the properties of strongly interacting matter under extreme conditions of temperature and baryon density. Of particular interest is the location of the Critical End Point (CEP) in the QCD phase diagram. To this aim, the STAR BES-I program has recently analyzed collisions of heavy-nuclei in the energy range 200 GeV >sNN> 7.7 GeV¹. Future experiments^2-4 will keep on conducting a thorough exploration of the transition from confined/chiral-symmetry broken hadron matter to the deconfined/chiral-symmetry restored state, varying the temperature and baryon density by changing the collision energy down to about sNN≃5 GeV and the system size in hadron and heavy-ion reactions. From the theoretical side, efforts to locate the CEP employing a variety of techniques such as Schwinger-Dyson equations, finite energy sum rules, functional renormalization methods, holography, and effective models, have produced a wealth of results^5-16 ranging from low to large values of the baryon chemical potential (μB) and temperature (T). Recent lattice QCD (LQCD) analyses¹⁷ have resorted to using the imaginary baryon chemical potential technique, to later extrapolate to real values, to study the chiral transition near the T-axis. Albeit with still large uncertainties, this technique has shown that the transition keeps being a smooth crossover¹⁸. The Taylor expansion LQCD technique has also been employed to restrict the CEP’s location to values μB/T>2 for the temperature range 135 MeV <T< 155 MeV. Its location for temperatures larger than 0.9 Tc(μB=0) seems to also be highly disfavored¹⁹ (see also²⁰).

Effective models have proven to be useful tools to gain insight into the phase structure of QCD. Given the dual nature of the QCD phase transition, at least for low values of μB, one can ask whether models that incorporate both chiral symmetry breaking and deconfinement are better suited to describe the transition features. However, since LQCD results show that for 2+1 light flavors, the crossover chiral and deconfinenent transitions are indistinguishable²¹, one may resort to a simplified analysis whereby one or the other feature is emphasized. Recently, we have made use of the linear sigma model coupled to quarks^22,23. We have shown that this tool can be successfully employed provided one accounts for the screening properties of the plasma, which makes the analysis effectively go beyond the mean field approximation, and one finds the values of the couplings from the physical values of the model parameters.

In this work we use the linear sigma model coupled to quarks, including the plasma screening effects, to explore the effective QCD phase diagram from the point of view of chiral symmetry restoration. Our strategy is to fix the coupling constants using the physical values of the model parameters, such as the vacuum pion and sigma masses, the critical temperature Tc at μB=0 and the conjectured end point value of μB (≃1 GeV) of the transition line at T=0. For the present purposes we compute an analytical, leading order in T approximation for the effective potential, both at high and low temperatures, for finite values of the baryon chemical potential. We show that this strategy can be used to locate the CEP. The work is organized as follows: In Sec. 2. we introduce the linear sigma model coupled to quarks. In Sec. 3. we compute the effective potential up to the contribution of the ring diagrams. We work out the high and low temperature analytical approximation for the effective potential and show explicitly how in the high temperature domain, the ring diagrams contribution cures the non-analyticities that appear at one-loop order. In Sec. 4. we spell out the conditions that give rise to the equations to find the values of the model coupling constants. In Sec. 5. we use these couplings to compute the critical T and μB values that define the transition curves and locate the CEP. In Sec 6. we finally summarize and conclude. We reserve for the appendices the calculation details for the boson and fermion contributions to the one-loop effective potential. In a sequel, to be reported elsewhere, we will study the case where the analytical approximation is extended to cover a larger set of possible μB and T values as well as to include the case where the couplings are allowed to bear the dependence on μB and T.

2. Linear Sigma Model coupled to quarks

In order to explore the QCD phase diagram, we study the restoration of chiral symmetry using an effective model that accounts for the physics of spontaneous symmetry breaking at finite temperature and density, the Linear Sigma Model. In order to account for the fermion degrees of freedom around the phase transition, we also include quarks in this model. The Lagrangian for the linear sigma model when the two lightest quarks are included is given by

where ψ is an SU(2) isospin doublet, π⃗=(π1,π2,π3) is an isospin triplet and σ is an isospin singlet. λ is the boson’s self-coupling and g is the fermion-boson coupling. a2>0 is the mass parameter.

To allow for an spontaneous breaking of symmetry, we let the σ field to develop a vacuum expectation value v

which can later be taken as the order parameter of the theory. After this shift, the Lagrangian can be rewritten as

where LIb and LIf are given by

Equation (4) describes the interactions among the σ, π⃗ and ψ fields after symmetry breaking. From Eq. (3) one can see that the sigma, the three pions and the quarks have masses given by

respectively.

In order to determine the chiral symmetry restoration conditions as function of temperature and quark chemical potential, we study the behavior of the effective potential, which we now proceed to deduce in detail.

3. Effective potential

Chiral symmetry restoration can be identified by means of the finite temperature and density effective potential, which in turn is computed order by order. In this work we include the classical potential or tree-level contribution, the one-loop correction both for bosons and fermions and the ring diagrams contribution, which accounts for the plasma screening effects.

The tree level potential is given by

whose minimum is given by

since v0≠0, we notice that the symmetry is spontaneously broken. We also notice that

which means that the curvature of the classical potential is equal to the sigma mass squared. This property is maintained even when corrections due to finite temperature and density are included in the effective potential.

However, in order to make sure that the quantum corrections at finite temperature and density maintain the general properties of the effective potential, we need to add counter-terms δa2 and δλ to the bare constants a2 and λ, respectively, and write

These counter-terms are needed to make sure that the phase transition at the critical temperature Tc for μB=0 is second order and that this transition is first order at the critical baryon density μBc for T=0. We will come back to these conditions when we introduce the analysis to determine the parameters of the model.

To include quantum corrections at finite temperature and density, we work within the imaginary-time formalism of thermal field theory. The general expression for the one-loop boson contribution can be written as

where

is the free boson propagator with mb2 being the square of the boson’s mass and ωn=2nπT the Matsubara frequencies for boson fields.

For a fermion field with mass mf, the general expression for the one-loop correction at finite temperature and quark chemical potential μq is

where

is the free fermion propagator and ω̃n=(2n+1)πT are the Matsubara frequencies for fermion fields.

The ring diagrams term is given by

where Π(mb,T,μq) is the boson’s self-energy.

4. Coupling Constants

Regardless of the approximation to the effective potential that is being considered, Eq. (28) or Eq. (36), we observe that we have five free parameters which should be fixed. These are the two coupling constants λ and g, the square mass parameter a ² and the counter-terms δa2 and δλ. In order to determine a ², we use that the vacuum boson masses, Eq. (5), satisfy

We can fix a using the physical vacuum sigma and pion masses. This analysis is shown in Figs. 4-6. Alternatively, we can work in the strict chiral limit, taking mπ=0. This analysis is shown in Figs. 7-9. We notice that the two kinds of phase diagrams obtained are very similar, in particular the CEP’s location changes very little.

We now need to use two conditions to fix the values of the coupling constants, the main idea is to use physical inputs such that the relations which satisfy λ and g are consistent with the realistic behavior of QCD matter around the phase transition in the high and low temperature domains.

From LQCD computations²¹, we know that at μq≡μB/3=0, the QCD phase transition is a crossover, hereby described as a second order transition, and happens for 2+1 light flavors at T0c≃155 MeV and for only 2 light flavors at T0c≃170 MeV. In a second order phase transition, the vacuum expectation value (vev) continuously transits from the broken phase to the restored phase and thus there is only one minimum. On the other hand, from the analysis using effective models²⁸ it is found that at very low values of T and high values of μq the transition is first order. From the analysis based on Hagedorn’s limiting temperature²⁹ at finite μB, we know that the critical value for the transition curve to intersect the horizontal axis in the QCD diagram is μB≃mB, where mB≃ 1 GeV is the typical value of the baryon mass. The vev transits from the broken phase to the restored phase in a discontinuous way. This means that at the phase transition, the effective potential develops two degenerate minima. In one or the other case, the thermal pion mass evaluated at the minima of the potential always vanishes, since this field is a Goldstone mode.

In order to fix the coupling constants we use as inputs the values of temperature and quark chemical potential in two extreme points along the transition curve, namely, when the restoration of chiral symmetry is at μq=0 and when it is at T = 0. Hereafter we refer to these extreme points of the diagram as points (A) and (B), respectively.

At point (A), the phase transition is second order, hence the square of the pion thermal mass, evaluated at v = 0 and T=Tc0, is given by

In other words, Eq. (38) tells us that the curvature at v = 0 and T=T0c is zero. Therefore the shape of the potential near v=0 is flat both in the σ and the pion directions. This is depicted in Fig. 2.

At point (B), the phase transition is first order, therefore we expect that at μq≃mB/3 the effective potential develops two degenerate minima. This is depicted in Fig. 3. Notice that the fermion contribution to the effective potential is responsible for the order of the phase transition. At low densities, this contribution is not strong enough to produce a hump in the effective potential whereas at high densities this contribution produces the barrier between minima at the critical temperature.

Since the analysis we carry out describes the transit from the broken to the restored phase, the minimum we are following is the one with a vev different from zero, which we call v1. This last condition can be written as

In Eq. (39), we notice that a new unknown appears: v1, that is, the value of the non-vanishing minimum. The set of conditions necessary to determine all the unknowns is

The three expressions in Eq. (40) indicate that the effective potential has two degenerated minima at the phase transition and thus that the transition is first order when T = 0 and the quark chemical potential is finite and equal to its critical value.

5. Results

The above set of conditions, Eqs. (38), (39) and (40), represent the five algebraic equations that determine the values of λ and g. These equations provide four pairs of solutions, out of which we pick the pair that corresponds to real positive solutions for λ and g. We are therefore in the position to explore the QCD phase diagram.

Figures 4-6 show the phase diagram obtained for the case when the mass parameter a is computed using the physical pion mass in vacuum. These are computed using μq=μb, 2μb, 0.5μb, respectively. In each figure, the band’s upper line is computed with T0c(μq=0)=175 MeV and μqc(T=0)=350 MeV and the lower line with T0c(μq=0)=165 MeV and μqc(T=0)=330 MeV. These ranges produce corresponding ranges to the solutions given by (0.77<λ<0.86, 1.53<g<1.63), (0.45<λ<0.49, 1.59<g<1.68), and (0.99<λ<1.10, 1.50<g<1.59), respectively.

Figures 7-9 show the phase diagram obtained for the case when the mass parameter a is computed setting mπ=0, that is in the chiral limit. These are computed using μq=μb, 2μb, 0.5μb, respectively. In each figure, the band’s upper line is computed with T0c(μq=0)=175 MeV and μqc(T=0)=350 MeV and the lower line with T0c(μq=0)=165MeV and μqc(T=0)=330 MeV. These ranges produce corresponding ranges to the solutions given by (1.02<λ<1.13, 1.78<g<1.89), (0.58<λ<0.64, 1.84<g<1.96), and (1.15<λ<1.30, 1.74<g<1.85), respectively. Notice that the CEP location does not change significantly regardless of weather we set the pion mass either to its physical value or to zero.

We find that at high (low) temperature and low (high) quark chemical potential the phase transitions are second (first) order. The second order transitions are indicated by the shaded red areas and the first order transitions by the blue shaded areas. These areas represent the results directly obtained from our analysis. The intermediate green shaded area is a Padé approximation that interpolates between the high and low temperature regimes. In all cases, we locate the CEP’s region at low temperatures and high quark chemical potential.

6. Summary and conclusions

In this work we have used the linear sigma model with quarks to explore the QCD phase diagram from the point of view of chiral symmetry restoration. We have computed the finite temperature effective potential up to the contribution of the ring diagrams to account for the plasma screening effects and have introduced a quark and a boson chemical potentials. The latter is related to the high density of the system that is in turn linked to the high baryon abundance at large values of the quark chemical potential.

Our approach was to determine the model’s couplings using physical inputs such as the vacuum pion and sigma masses, the LQCD value for the critical temperature at μq=0 and the conjectured end point value of μB of the transition line at T = 0. We have also performed the analysis using mπ=0 instead of its physical vacuum value. The set of conditions that determine the couplings enforce the requirement that at high temperature the transition is second order whereas at low temperature is first order. Of particular importance is the observation that at the minima of the effective potential, the pion thermal mass vanishes, since this particle is a Goldstone boson.

We find that when varying the values of T0c(μq=0) and μqc(T=0) by 10% from the chosen central values, the procedure allows to locate the phase transition lines in narrow band. The CEP is however stable when varying the relation between μq and μb and even when the pion mass is set to either its physical value or to zero. The CEP location corresponds to low temperatures and high values of the quark chemical potential. Table I summarizes the CEP location found in some recent works together with our findings.

In order to provide a more robust CEP’s location, we need to extend the analytical expansion of the effective potential to a larger temperature range. Perhaps even more important will be to include the temperature and density modifications to the couplings which has been shown useful to describe the inverse magnetic catalysis phenomenon³³. This is work for the future and will be reported elsewhere.