Phase diagram of dense two-color QCD within lattice simulations

We present the results of a low-temperature scan of the phase diagram of dense two-color QCD with N f = 2 quarks. The study is conducted using lattice simulation with rooted staggered quarks. At small chemical potential we observe the hadronic phase, where the theory is in a confining state, chiral symmetry is broken, the baryon density is zero and there is no diquark condensate. At the critical point μ = mπ/2 we observe the expected second order transition to Bose-Einstein condensation of scalar diquarks. In this phase the system is still in confinement in conjunction with nonzero baryon density, but the chiral symmetry is restored in the chiral limit. We have also found that in the first two phases the system is well described by chiral perturbation theory. For larger values of the chemical potential the system turns into another phase, where the relevant degrees of freedom are fermions residing inside the Fermi sphere, and the diquark condensation takes place on the Fermi surface. In this phase the system is still in confinement, chiral symmetry is restored and the system is very similar to the quarkyonic state predicted by S U(Nc) theory at large Nc.


Introduction.
The phase diagram of QCD is of high importance for several fields of observational physics like cosmology and astrophysics.One field of experimental physics, located between nuclear physics and high energy physics, is the study of hadronic matter created by relativistic heavy ion collisions.Such experiments are addressing the structure of the phase diagram, although the understanding and modeling of an actual collision requires much more than the knowledge of the equilibrium phase diagram.However, equilibrium observables like the equation of state and transport coefficients are highly needed to be used in hydrodynamical approaches which serve to probe various scenarios.
Unfortunately, lattice simulation of QCD can not be applied today to arbitrary chemical potential because of the sign problem [1].The origin of the sign problem is that the fermion determinant becomes complex-valued which makes impossible the direct simulation by importance sampling of gauge field configurations.An alternative to lattice simulation of SU(3) QCD with µ 0 is the simulation of SU(2) QCD (also called QC 2 D).Introduction of a chemical potential to the latter theory does not lead to a sign problem, so one can apply the standard lattice approach to study this theory.Although a two-color world differs from the tree-color world, lattice study of QC 2 D with chemical potential can provide us with important information about the properties of QCD with non-zero baryon density (EoS, generation of the fermion mass gap, etc.).We would like also to note that it is interesting to study the QC 2 D by its own due to a rich structure of the phase diagram.
The first lattice study of QC 2 D with chemical potential and Wilson fermions was performed by A. Nakamura in [14].Futher lattice investigation of dense two-color QCD was continued by J. Kogut and collaborators using staggered quarks.In [15] eight-flavor theory was investigated and in [16][17][18] the authors studied N f = 4 theory.What concerns a low temperature scan of the phase diagram, these authors observed the succession of a hadronic phase and the BEC phase, with their properties well described by ChPT, but they didn't find a BCS phase.
The main activity in two-color QCD was later continued by the Swansea group (S.Hands and collaborators) for the two-flavor theory with Wilson fermions [19][20][21][22].In a low temperature scan of the phase diagram the authors observed a hadronic phase, followed by the BCS phase with deconfinement, but did not encounter the BEC phase.
In this paper we are going to study the QC 2 D phase diagram with N f = 2 flavors going back to the lattice simulation of staggered fermions using the rooting procedure.In the present paper we are going to carry out a µ 1 scan at low temperature of the QC 2 D phase diagram.

The lattice set-up
In our simulations we used the Wilson action for the SU(2) gauge fields For the fermionic degrees of freedom we used staggered fermions with an action of the form where ψ, ψ are staggered fermion fields, a is the lattice spacing, m is the bare quark mass, and η µ (x) are the standard staggered phase factors: η 1 (x) = 1, η µ (x) = (−1) x 1 +...+x µ−1 , µ = 2, 3, 4. The chemical potential µ is introduced into equation (3) through the multiplication of the links along and opposite to the temporal direction by factors e ±µa respectively.This way of introducing the chemical potential makes it possible to avoid additional divergences and to reproduce well known continuum results [23].
In addition to the standard staggered fermion action we add a diquark source term [15] to equation (2).The diquark source term explicitly violates U V (1) and allows to observe diquark condensation even on finite lattices.The results presented in this paper are obtained as follows: we carry out simulations at small but nonzero parameter λ ma, and then extrapolate obtained data to λ → 0. Notice that similar to the diquark source term an additional pion term was introduced to the fermion action during the studies of QCD phase diagram with isospin chemical potential [24][25][26].
Integrating out the fermion fields the partition function for the theory can be written as where M is the staggered Dirac operator and λ is the diquark source.Note that the pfaffian P f is strictly positive, such that one can use Monte-Carlo methods to study this system.Partition function (4) corresponds to N f = 4 in the continuum limit.In the present paper we are going to study which corresponds to the theory with N f = 2 dynamical fermions in the continuum limit.

The diquark condensate
In Fig. 1 we plot the diquark condensate qq , obtained by linear extrapolation to λ = 0, as a function of µ in the region µ ∈ [0.0; 440] MeV (µa ∈ [0.0; 0.25]).It may be seen, that for µ ≤ 141 MeV (µa ≤ 0.08) the diquark condensate qq is compatible with zero, i.e. the system is in the hadronic phase.However, for µ ≥ 176 MeV (µa ≥ 0.1) the diquark condensate starts to deviate from zero.If we are sufficiently far from the position of the phase transition, one can try to use ChPT [2,3] to describe the data.In particular, ChPT predicts that at µ c = m π /2 there is the transition from the hadronic phase to the phase of Bose-Einstein condensation of scalar diquarks with qq 0, and the behaviour of the diquark condensate above the transition would be given by the formula qq = qq 0 1 − (µ c /µ) 4 , where qq 0 is the chiral condensate at zero chemical potential.Fit with this formula in the region µ ∈ [263; 352] MeV (µa ∈ [0.15; 0.20]) provides µ c = 215(10) MeV (aµ c = 0.122( 6)) with χ 2 /do f = 2.5.We plot this fit in the Fig. 1.Let us consider the region of larger chemical potential µ > 352 MeV (µa > 0.2).To understand what happens in this region, we plot in Fig. 2 the linearly extrapolated diquark condensate, divided by T µ 2 , as a function of µ.As it is visible from this plot, in the region µ ∈ [528; 1055] MeV (µa ∈ [0.3; 0.6]) there is a plateau, i.e. the value of the diquark condensate is proportional to the surface of a sphere with the radius µ: qq ∼ µ 2 .This is a characteristic property of the BCS theory, where the condensate appears on the Fermi surface and is proportional to the density of states on this surface.Thus we conclude, that for µ > 528 MeV (µa > 0.3) the system reveals properties of the BCS phase, and that the transition from the BEC to the BCS phase is smooth.

The chiral condensate
In the Fig. 3 we plot the chiral condensate calculated for the smallest diquark source value λ = 0.0005 as a function of µ (during the simulations we discovered that dependence of the chiral condensate on the source λ was very weak).The first observation is that up to µ < 176 MeV (µa < 0.1), which is approximately m π /2, the chiral condensate does not depend on the chemical potential.In the region µ > 176 MeV, where the system is in the vicinity of the transition to the BEC phase, the chiral condensate starts to decrease.These properties are in agreement with ChPT predictions (see the Figures 4 and 5 in paper [2]).
According to ChPT, at µ > µ c the chiral condensate drops as qq = qq 0 (µ c /µ) 2 .To check this prediction in the region µ ∈ [263; 352] MeV (µa ∈ [0.15; 0.20]) we fitted our data by a power law qq = A/µ α , which provided α = 0.78(2) with χ 2 /do f = 0.3.It is interesting to note, that this fit gives a satisfactory description of the data up to µ ∼ 1055 MeV (µa ∼ 0.6).Thus, one sees that the chiral condensate drops slower with increasing chemical potential than ChPT predicts (similar slower decrease of the form qq ∼ 1/µ was observed in [21] with N f = 2 Wilson quarks).Finally, it is interesting to study the chiral symmetry breaking in the chiral limit for different regions of the chemical potential.In Fig. 4 we plot the chiral condensate for different values of the chemical potential as a function of the quark mass and the results of the linear extrapolation to ma = 0.It may be seen from Fig. 4, that chiral symmetry breaking exists in the chiral limit within the hadronic phase (values µ = 0, 70 and 141 MeV), whereas there is no chiral symmetry breaking in the chiral limit in the BCS phase (µ = 615 MeV) and in the BEC phase (µ = 246, 281, 352 MeV).However, it is difficult to claim, that there is no chiral symmetry breaking in the whole BEC phase: when we take the chiral limit, we change the pion mass and thus shift the critical point µ c closer to zero.This effect might be important near the phase transition.Note, that the absence of chiral symmetry breaking in the chiral limit within the BEC phase agrees with ChPT predictions.

The baryon density
In Fig. 5 we plot the baryon density 2 in the region µ ∈ [0.0; 528] MeV (µa ∈ [0.0; 0.3]).It is clear, that for all µ < 176 MeV (µa < 0.1) the baryon density is vanishing within the uncertainty of the calculations.In the vicinity of the phase transition (µ ≥ 176 MeV) the baryon density starts to deviate from zero, and for larger values of the chemical potential it rises with the increasing of µ.ChPT predicts, that the dependence of the baryon density on the chemical potential above µ c is given by a formula n B ∼ µ − µ 4 c /µ 3 .Fit of the data by this formula in the region µ ∈ [263; 352] MeV (µa ∈ [0.15; 0.20]) provides µ c = 207(7) MeV (aµ c = 0.118(4)) with χ 2 /do f = 1.2.This value is in agreement with our previous results for µ c , obtained from the qq fits.From Fig. 5 it is visible, that for bigger chemical potential, µa > 0.2, our data deviate from the ChPT prediction.
Next, let us consider the baryon density at larger values of the chemical potential.In Fig. 6 we plot the ratio n B /n 0 as a function of µ, where for the square points the reference density n 0 is the baryon density for free continuum fermions at T = 0, n 0 = (2µ 3 )/(3π 2 ), and for the circle points n 0 is the baryon number density for the free lattice fermions.It can be seen, that in the region µ ∈ [528; 1055] MeV (µa ∈ [0.3; 0.6]) these ratios are slowly varying functions of the chemical potential, whereas the measured baryon density changes by an order of magnitude.We believe, that the scaling of the 2 The ansatz n B (λ) = A + Bλ 2 was employed for the λ → 0 extrapolation.baryon density n B ∼ n 0 confirms the conclusion that in the region µ ∈ [528; 1055] MeV the system is in a BCS-like phase.The relevant degrees of freedom in this phase are quarks, which mostly live inside the Fermi sphere with a condensate of Cooper pairs on the Fermi surface.The fact that n B /n 0 ∼ 2.0 . . .2.5, but not ∼ 1.0, can be attributed to UV and IR effects (similar artifact effects for n B scaling were observed in [21]).

The gluon observables
In this section we study the gluon observables: Polyakov loop and Wilson loops.Similarly to the chiral condensate the gluon observables are not sensitive to the value of the λ, thus we take these observables calculated at the smallest value λ = 0.0005 as their values at the λ = 0. We measured the average of the Polyakov loop as a function of the chemical potential.The result of this measurement is that for all values of the chemical potential studied in this paper the average Polyakov loop is vanishing within the uncertainty of the calculation.
Furthermore, in order to investigate the confinement properties of the system, we have calculated time-like Wilson loops for the quadratic contours of the size 8 × 8 and 10 × 10 (for larger Wilson loops we obtained results compatible with zero) as functions of the chemical potential.The results are shown in Fig. 7.One learns from this plot, that for µ > 352 MeV (µ > 0.2) the Wilson loops decrease with the growth of the chemical potential.At small µ, for µ ∈ [0; 263] MeV (µa ∈ [0.0; 0.15]), a plateau for both Wilson loops may be noticed.From these results one can conclude, that the system is in a confined phase for all values of the chemical potential under consideration.The possible explanation for this behaviour may be the absence of the Debye screening in two-color QCD at zero temperature [4,28].

Discussion and conclusion
Our results can be summarized as follows.At small chemical potential µ < µ c = m π /2 ∼ 200 MeV we observe a hadronic phase.In this phase QC 2 D matter is in confinement, chiral symmetry is broken, the diquark condensate vanishes and the baryon number density is also zero.Relevant degrees of freedom in this phase are Goldstone bosons.
In the region µ c < µ < µ d ∼ 352 MeV we observe the BEC phase.Characteristic feature of this phase is Bose-Einstein condensation of scalar diquarks.The order parameter for the transition to the BEC phase is the diquark condensate, which develops a non-zero value in the region µ > µ c .Within the uncertainty of the calculation µ c = m π /2, where m π is the pion mass at zero chemical potential.In this phase, QC 2 D matter has also confining properties, whereas the baryon density is non-zero.Relevant degrees of freedom in the BEC phase are Goldstone bosons as well.
It is important to notice, that for all values of the chemical potential µ < µ d our results are in good agreement with the predictions of ChPT.An exception is the chiral condensate, which drops with increasing chemical potential slower than ChPT predicts in the leading order.This behaviour of the chiral condensate might be explained by higher order radiative corrections.
If we further increase the chemical potential, starting from µ ∼ 500 − 600 MeV one can observe that the diquark condensate scales as qq ∝ µ 2 and the baryon density scales as n B ∝ µ 3 .Physically, this implies that the relevant degrees of freedom are quarks, which are mostly living inside the Fermi sphere with a condensate of Cooper pairs on the Fermi surface.These properties are clear hints in favor of the BCS phase.In this phase the chiral symmetry is restored in the chiral limit.The BCS phase extends up to µ ∼ 1000 − 1100 MeV.We believe, that the BCS phase of the QC 2 D theory may be similar to the "quarkyonic phase" of the SU(N c ) QCD at large N c [29].

Figure 5 .Figure 6 .
Figure 5.The baryon density n B in the physical units as a function of µ.

Figure 7 .
Figure 7.The time-like Wilson loops for the contours 8 × 8 and 10 × 10 as a functions of the chemical potential.