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 experiments2-4 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 results5-16 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 quarks22,23. 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
L=12(∂μσ)2+12(∂μπ⃗)2+a22(σ2+π⃗2)-λ4(σ2+π⃗2)2+iψ¯γμ∂μψ-gψ¯(σ+iγ5τ⃗⋅π⃗)ψ,
(1)
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
σ→σ+v,
(2)
which can later be taken as the order parameter of the theory. After this shift, the Lagrangian can be rewritten as
L=-12[σ∂μ2σ]-123λv2-a2σ2
-12[π⃗∂μ2π⃗]-12λv2-a2π⃗2+a22v2
-λ4v4+iψ¯γμ∂μψ-gvψ¯ψ+LIb+LIf,
(3)
where LIb and LIf are given by
LIb=-λ4[(σ2+(π0)2)2
+4π+π-(σ2+(π0)2+π+π-)],
LIf=-gψ¯(σ+iγ5τ⃗⋅π⃗)ψ.
(4)
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
mσ2=3λv2-a2,mπ2=λv2-a2,mf=gv,
(5)
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
Vtree(v)=-a22v2+λ4v4,
(6)
whose minimum is given by
v0=a2λ,
(7)
since v0≠0, we notice that the symmetry is spontaneously broken. We also notice that
d2Vtreedv2=3λv2-a2=mσ2,
(8)
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
Vtree=-a22v2+λ4v4→-(a2+δa2)2v2+(λ+δλ)4v4.
(9)
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
V(1)b(v,T)=T∑n∫d3k(2π)3lnD(ωn,k⃗)1/2,
(10)
where
D(ωn,k⃗)=1ωn2+k2+mb2,
(11)
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
V1fv,T,μq=-T∑n∫d3k(2π)3
×Tr[lnS(ω~n-iμq,k→)-1,
(12)
where
Sω~n-k→=1γ0ω~n+k+mf,
(13)
is the free fermion propagator and ω̃n=(2n+1)πT are the Matsubara frequencies for fermion fields.
The ring diagrams term is given by
VRing(v,T,μq)=T2∑n∫d3k(2π)3×ln(1+Π(mb,T,μq)D(ωn,k⃗)),
(14)
where Π(mb,T,μq) is the boson’s self-energy.
3.1. Self-energy.
We start by computing the self-energy for one boson field. For this purpose we need to include all the contribution from the Feynman rules in Eq.(4). The diagrams representing the bosons’ self-energy are depicted in Fig. 1. Each boson has a self-energy with two kinds of terms, one corresponds to a loop made by a boson field and other one corresponding to a loop made by a fermion anti-fermion pair. Therefore, the self-energy is written as
Π(T,μq)=∑i=σ,π0,π±Πi(T)+∑j=u,dΠj(T,μq),
(15)
where
Πσ(T)=λ4[12Imσ2+Πσ+4Imπ02+Ππ0+8Imπ±2+Ππ±],Ππ±(T)=λ4[4Imσ2+Πσ+4Imπ02+Ππ0+16Imπ±2+Ππ±],Ππ0(T)=λ4[4Imσ2+Πσ+12Imπ02+Ππ0+8Imπ±2+Ππ±],
(16)
with
I(x)=12π2∫dkk2k2+xnk2+x,
(17)
and n(x) being the Bose-Einstein distribution.
The leading temperature approximation to the boson self-energy is given by
Πσ(T)=Ππ±(T)=Ππ0(T)=λ4[24I(0)]=6λ2π2∫dkk1ek/T-1=λT26.
(18)
This approximation, where the boson’s mass is neglected with respect to the temperature, is a good approximation around the phase transition where the boson’s mass (including its thermal correction) vanishes, namely, mi2+Πi=0.
On the other hand, the fermion contribution is given by
Πj(T,μq)=-g2T∑n∫d3k(2π)3Tr[S(ω̃n-iμq,k⃗,mf)×S(ω̃n-iμq-ω̃m,k⃗-p⃗,mf)].
(19)
Equation (19) can be computed without resorting to assuming a hierarchy between T and μq. Also, since we work close to the phase transition, we take mf=0. The fermion self-energy contribution becomes
ΠjT,μq=-Ncg2T2π2[Li2(-eμq/T)
+Li2-e-μq/T].
(20)
With Eqs. (18) and (20), the total self-energy for one boson is
Π(T,μq)=-NfNcg2T2π2[Li2-eμq/T+Li2-e-μq/T]+λT22.
(21)
With the boson self-energy at hand we can study the properties of the effective potential. In order to work with analytical expressions we turn to study two cases: first the high temperature approximation, i.e.T≫mb, μq and then the low temperature approximation i.e.T≪mb, μq. In the following we compute explicitly both regimes.
3.2. High temperature approximation
For small μB and the transition temperature for chiral symmetry restoration found by LQCD computations21, we observe that T is the largest of the energy scales. Therefore, a high temperature approximation is suited to study the chiral symmetry restoration. Let’s start from Eq. (10), the one-loop correction for boson fields. The first step is to compute the sum over Matsubara frequencies. On doing so we obtain
V(1)b(v,T)=12π2∫dk k2{k2+mb22+Tln1-e-k2+mb2/T}.
(22)
Notice that Eq. (22) has two pieces, the first one is the vacuum contribution and the second one is the matter contribution, namely, the T-dependent correction. In order to compute the vacuum term, we need to regularize and renormalize the former. For this purpose, we employ dimensional regularization and the Minimal Subtraction scheme (MS), with the renormalization scale μ̃=e-1/2a. For the matter term, we take the approximation mb/T≪1 and we include only the most dominant terms (for more details see Appendix A). Taking all this into account, the one-loop contribution to the effective potential from boson fields is given by
VHT(1)b(v,T)=-mb464π2[ln(4πa2mb2)-γE+12]-mb464π2ln(mb2(4πT)2)-π2T490+mb2T224-mb3T12π.
(23)
For the case of the fermion one-loop contribution, we follow the procedure outlined for the boson case. Thus, we start by computing the sum over the Matsubara frequencies to obtain
V(1)f(v,T,μq)=-1π2∫dk k2{k2+mf2-Tln(1-e-(k2+mb2-μq)/T)-Tln(1-e-(k2+mb2+μq)/T)}.
(24)
As for the boson case, we find that Eq. (24) contains two pieces, one corresponding to the vacuum contribution and the other one to the matter contribution. The latter has the contribution of the quark chemical potential and for this reason we now have two terms corresponding to the particle and the anti-particle contributions. The vacuum contribution is computed exactly in the same manner for the boson case. For the matter term, we compute the integral in momentum taking into account the approximation where mf/T≪1 and μq/T<1, and we consider only the dominant terms (for more details see Appendix B). After we compute the momentum integral in Eq. (24) we get
VHT(1)f(v,T)=mf416π2[ln(4πa2mf2)-γE+12]+mf416π2[ln(mf2(4πT)2)-ψ0(12+iμ2πT)-ψ0(12-iμ2πT)]-8mf2T2[Li2(-eμq/T)+Li2(-e-μq/T)]+32T4[Li4(-eμq/T)+Li4(-e-μq/T)].
(25)
In order to go beyond the mean field (one-loop) approximation, we need to consider the plasma screening effects. These can be accounted for by means of the ring diagrams. Since we are working in the high temperature approximation, we notice that the lowest Matsubara mode is the most dominant term24. Therefore we do not need to compute the other modes and Eq. (14) becomes
VRing(v,T,μq)=T2∫d3k(2π)3ln(1+Π(T,μq)D(k⃗))=T4π2∫dk k2{ln(k2+mb2+Π(T,μq))-ln(k2+mb2)}.
(26)
From Eq. (26), we see that both integrands are almost the same except that one is modified by the self-energy and the other one is not. Thus, after integration, we obtain that the ring diagrams contribution is
VRing(v,T,μq)=T12π(mb3-(mb2+Π(T,μq))3/2).
(27)
With these pieces at hand, we can write the effective potential up to the ring diagrams contribution in the high temperature approximation. This is given by
VHTeff(v,T,μq)=-(a2+δa2)2v2+(λ+δλ)4v4+∑b=σ,π¯{-mb464π2[ln(a24πT2)-γE+12]-π2T490+mb2T224-(mb2+Π(T,μq))3/2T12π}+∑f=u,d{mf416π2[ln(a24πT2)-γE+12-ψ0(12+iμq2πT)-ψ0(12-iμq2πT)]-8mf2T2[Li2(-eμq/T)+Li2(-e-μq/T)]+32T4[Li4(-eμq/T)+Li4(-e-μq/T)]}.
(28)
Notice that the potentially dangerous pieces coming from linear or cubic powers of the boson mass, that could become imaginary for certain values of v, are removed or replaced by the contribution of the ring diagrams25.
3.3. Low temperature approximation
To have access to the region in the QCD phase diagram where μB is large and T is small, we compute the effective potential in the approximation where T is the soft scale in the system. We call this the low temperature approximation. The approximation is applied both to the contribution of boson and fermion fields.
In the case of boson fields, we include a boson chemical potential. We relate this to the energy required to add or remove one boson to the system. We associate this term to the description of high density in the analysis, in other words, the bosons’ chemical potential μb, is related to the conservation of an average number of particles and not to a conserved charge. The introduction of the boson’s chemical potential is used to account for the possible onset of meson condensates as the quark chemical potential increases. This phenomenon has been described since long ago in the context of processes taking place in the core of neutron stars, where an excess of negative pions appears when the electron chemical potential approaches the pion rest mass26. In the present context, since the relevant interactions are between mesons and quarks, an excess of pions is bound to appear when the quark chemical potential approaches the pion mass. Therefore, the one-loop contribution for boson fields after the sum over Matsubara frequencies is
VLT(1)b(v,T,μb)=12π2∫dk k2{k2+mb2+2Tln(1-e-(k2+mb2-μb)/T)}.
(29)
In this approximation, it is not necessary to compute the vacuum and matter contributions
separately, in fact the full expression can be computed at once. In this work,
we follow the procedure used in Ref. 27.
The general idea consists on developing a Taylor series around T=0 of the following expression
VLT(1)b(v,T,μb)=∫μb-mbT∞V0b(v,μb+xT)hB(x)dx,
(30)
where hB(x) is the first derivative of the Bose-Einstein distribution and V0b(v,μb+xT) is the one-loop boson contribution evaluated at T = 0, which is given explicitly by
V0(1)b(v,μb)=-mb464π2[ln4πa2(μb+μb2-mb2)2-γE+12]+μbμb2-mb296π2(2μb2-5mb2).
(31)
Notice that the one loop contribution from boson fields in the limit T = 0, that appears in Eq. (30), is evaluated at μb→μb+xT. Then the expression of one-loop matter contribution from one boson field in the low temperature approximation becomes
VLT(1)b(v,T,μb)=V0b(v,μb+xT)|T=0+π2T212∂2∂(xT)2V0b(v,μb+xT)|T=0+7π4T41260∂4∂(xT)4V0b(v,μb+xT)|T=0.
(32)
For more details see Appendix C.
For fermion fields, we start from Eq. (24), such that we implement the low temperature approximation in the same way as we did for boson fields. We now develop a Taylor series around T=0 of the following expression
VLT(1)f(v,T,μq)=∫μq-mfT∞V0f(v,μq+xT)hF(x)dx,
(33)
with hF(x) is the first derivative of the Fermi-Dirac distribution and V0f(v,μq+xT) is the one-loop potential for one fermion field evaluated at T = 0. This can be written as follows
V0(1)f(v,μq)=mb416π2[ln4πa2(μq+μq2-mf2)2-γE+12]-μqμq2-mf224π2(2μq2-5mf2).
(34)
Once again, we notice that the one-loop contribution from fermion fields in the limit T = 0 that appears in Eq. (33) is evaluated at μq→μq+xT. The one-loop contribution for one fermion field in the low temperature approximation then becomes
VLT(1)f(v,T,μq)=V0f(v,μq+xT)|T=0+π2T26∂2∂(xT)2V0f(v,μq+xT)|T=0+π4T4360∂4∂(xT)4V0f(v,μq+xT)|T=0.
(35)
For more details see Appendix D.
Equations (6), (9), (32) and (35) provide the full expression for the effective potential in the low temperature approximation, which is given by
VLTeff(v,T,μq,μb)=-(a2+δa2)2v2+(λ+δλ)4v4-∑i=σ,π¯{mi464π2[ln(4π2a2(μb+μb2-mi2)2)-γE+12]-μbμb2-mi224π2(2μb2-5mi2)-T2μb122μb2-5mi2-π2T4μb180(2μb2-3mi2)(μb2-mi2)3/2}+Nc∑f=u,d{mf416π2[ln(4π2a2(μq+μq2-mf2)2)-γE+12]-μqμq2-mf224π2(2μq2-5mf2)-T2μq6μq2-mf2-7π2T4μq360(2μq2-3mf2)(μq2-mf2)3/2} .
(36)
We are now in position to explore the QCD phase transition in the regions of the QCD phase diagram where the temperature is larger than the quark chemical potential and where the temperature is smaller than the quark chemical potential. However, before exploring the phase diagram, we need to determine the value of all the parameters involved in the linear sigma model, appropriate for the conditions of the analysis. In the following section we proceed in this direction to determine the values of those parameters and in particular of the couplings λ and g.
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
a=mσ2-3mπ22.
(37)
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
mπ2(0,T0c,μq=0)=-a2+Π(T0c,μq=0)=0.
(38)
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
mπ2(v1,0,μqc)=λv1-a2+Π(0,μqc)=0,
(39)
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
∂Veff∂v(v=0,T=0,μq=μqc)=0,∂Veff∂v(v=v1,T=0,μq=μqc)=0,Veff(v=0,T=0,μq=μqc)=Veff(v=v1,T=0,μq=μqc).
(40)
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.