Paper Info

Title: Nuclear Liquid-Gas Transition in the Strong Coupling Regime of Lattice QCD
Publish Date: 28 Mar 2023
Author List: J Kim (from Institute for Advanced Simulation (IAS-4), Forschungszentrum Jülich), P Pattanaik (from Fakultät für Physik, Bielefeld University), W Unger (from Fakultät für Physik, Bielefeld University)

Figure

FIG. 1.Typical 2-dimension configuration at β = 1.0, at non-zero quark mass, temperature, chemical potential.The black dots are monomers, the blue lines are dimers, the red arrows are baryon loop segments (or triplets g b + f b = ±3 if adjacent to a non-trivial plaquette), and the green squares are plaquette occupations ±1.The actual configurations are 3+1-dimensional.
FIG.2.Chiral susceptibility on a 2 4 volume for various quark masses, as a function of the bare anisotropy γ (with aT = γ 2 /2), analytic results from enumeration compared to numerical data from simulations via the worm algorithm.
FIG.3.Various observables in the µB-T plane on a 2 4 volume at amq = 0.1.The back-bending of the first order transition at temperatures below aT = 0.5 in all observables is an artifact of the small volume, and vanishes in the thermodynamic limit.The temperature aT = 1/2 corresponds to the isotropic lattice here.
FIG. 4. The chiral condensate (left) and the baryon density (right) for quark mass m = 1.5 as a function of the chemical potential and for various temperatures.
FIG. 7. ∆f at amq = 0.2 as a function of chemical potential and β the on a 6 3 × 4 lattice
FIG. 8. Baryon mass from ∆E as a function of the quark mass amq, and contributions from different dual variables: monomers, dimers and baryon segments.
FIG. 9. Baryon density for volume 4 3 × 8 in the full µB − mq plane, illustrating the strong quark mass dependence of the onset to nuclear matter.
FIG. 10.Baryonic observables on various volumes in the first order region amq = 1.5.Vertical bands indicate the mean and error of the nuclear transition.
FIG. 12. Left: Extrapolation of the pseudo-critical values of µB for the various volumes into the thermodynamic limit.Right: Critical baryon chemical potential for different quark masses.The first order transition region is shown in blue, the crossover region is shown in red and the range for critical end point is marked in black.
FIG. 17. Nuclear interaction scaled with baryon mass.As the quark mass increases, it tends to zero.
FIG. 18. Critical baryon chemical potential and baryon mass from different approaches.
Parameters for the Monte Carlo runs to determine the nuclear transition at strong coupling, with statistics after thermalization.

abstract

The nuclear liquid-gas transition from a gas of hadrons to a nuclear phase cannot be determined numerically from conventional lattice QCD due to the severe sign problem at large values of the baryon chemical potential. In the strong coupling regime of lattice QCD with staggered quarks, the dual formulation is suitable to address the nuclear liquid gas transition.
We determine this first order transition at low temperatures and as a function of the quark mass and the inverse gauge coupling β. We also determine the baryon mass and discuss the nuclear interactions as a function of the quark mass, and compare to mean field results. It is known from experiments that at low temperatures, there is a phase transition between dilute hadron gas and dense nuclear matter as the baryon chemical potential increases.
This transition is of first order and terminates at about T c = 16 MeV in a critical end point. The value of the chemical potential µ 1st B at zero temperature is given roughly by the baryon mass m B , where the difference of µ 1st B −m B is due to nuclear interactions. For a review on nuclear interactions see .
As the nuclear force between baryons to form nuclear matter is due to the residual strong interactions between quarks and gluons, it should be accurately described by QCD. We choose to study the nuclear transition and nuclear interaction via lattice QCD , with its Lagrangian being a function of the quark mass and the inverse gauge coupling.
In order to understand the nature of the transition, it is helpful to study its dependence on these parameters. However, at finite baryon density, lattice QCD has the infamous sign problem which does not allow us to perform direct Monte Carlo simulations on the lattice. Various methods have been proposed to overcome the numerical sign problem, but they are either limited to µ B /T 3 or can not yet address full QCD in 3+1 dimensions in the whole µ B − T plane , in particular the nuclear transition is out of reach.
An alternative method is to study lattice QCD via the strong coupling expansion. There are two established effective theories for lattice QCD based on this: (1) the 3-dim. effective theory for Wilson fermions in terms of Polyakov loops, arising from a joint strong coupling and hopping parameter expansion , the dual representation for staggered fermions in 3+1 dimensions, with dual degrees of freedom describing mesons and baryons.
Both effective theories have their limitations: is limited to rather heavy quarks (but is valid for large values of β) whereas ( ) is limited to the strong coupling regime β 1 (but is valid for any quark mass). We study lattice QCD in the dual formulation, both at infinite bare gauge coupling, β = 0, and at leading order of the strong coupling expansion in the regime β < 1, which is far from the continuum limit.
But since strong coupling lattice QCD shares important features with QCD, such as confinement, and chiral symmetry breaking and its restoration at the chiral transition temperature, and a nuclear liquid gas transition, we may get insights into the mechanisms, in particular as the dual variables give more information in terms of its world lines, as compared to the usual fermion determinant that depends on the gauge variables.
To establish a region of overlap of both effective theories, we have chosen to perform the Monte Carlo simulations in the dual formulation extending to rather large quark masses. This paper is organized as follows: in the first part we explain the dual formulation in the strong coupling regime, in the second part we provide analytic results based on exact enumeration and mean field theory, in the third part we explain the setup of our Monte Carlo simulations and present result on the m q -and β-dependence of the nuclear transition.
Since the strong coupling regime does not have a well defined lattice spacing, we also determine the baryon mass am B to set the parameters of the grand-canonical partition function, aT and aµ B , in units of am B . We conclude by discussing the resulting nuclear interactions, and compare our findings with other results.

Staggered action of strong coupling QCD and its dual representation

In the strong coupling regime, the gauge integration is performed first, followed by the Grassmann integration to obtain a dual formulation. This was pioneered for the strong coupling limit in and has been extended by one of us to include gauge corrections . The sign problem is mild in the strong coupling limit and still under control for β < 1, where we can apply sign reweighting.
The dual degrees of freedom are color-singlet mesons and baryons, which are point-like in the strong coupling limit, and become extended about a lattice spacing by incorporating leading order gauge corrections. The partition function of lattice QCD is given by where DU is the Haar measure, U ∈ SU(3) are the gauge fields on the lattice links (x, μ) and { χx , χ x } are the unrooted staggered fermions at the lattice sites x.
The gauge action S G [U] is given by the Wilson plaquette action and the staggered fermion action S F [ χ, χ, U] is: where the gauge action depends on the inverse gauge coupling β = 2Nc g 2 and the fermion action depends on the quark chemical potential aµ q which favors quarks in the positive temporal direction, and the bare quark mass am q .
First we consider the strong coupling limit where the inverse gauge coupling β=0 and hence the gauge action S G [U] drops out from the partition function in this limit. The gauge integration is over terms depending only on the individual links (x, μ) so the partition function factorizes into a product of one-link integrals and we can write it as:
with z(x, μ) the one-link gauge integral that can be eval-uated from invariant integration, as discussed in , where we write the one-link integral in terms of new hadronic variables: Only terms of the form (M (x)M (y)) k x, μ (with k x,μ called dimers which count the number of meson hoppings) and B(y)B(x) and B(x)B(y) (called baryon links) are present in the solution of the one-link integral.
The sites x and y = x + μ are adjacent lattice sites. It remains to perform the Grassmann integral of the fermion fields χ, χ. This requires to expand the exponential containing the quark mass in Eq. (4) (left), which results in the terms (2am q M (x)) nx (with n x called monomers). To obtain non-vanishing results, at every site, the 2N c Grassman variables χ x,i and χx,i have to appear exactly once, resulting in the Grassmann constraint (GC):
where n x is the number of monomers, k x,μ is the number of dimers and the baryons form self-avoiding loops x,μ , which due to the constraint cannot coexist with monomers or dimers. With this, we obtain an exact rewriting of the partition function Eq. ( ) for N c = 3, in terms of integer-valued dual degrees of freedom {n, k, }:
where the sum over valid configurations has to respect the constraint (GC). The first term in the partition function is the contribution from dimers and the second term is the contribution from monomers. The weight factor w( ) for each baryon loop depends on the baryon chemical potential µ B = 3µ q and induces a sign factor σ( ) which depends on the geometry of :
Here, ω is the winding number of the loop . The total sign factor σ( ) ∈ {±1} is explicitly calculated for every configuration. We apply sign reweighting as the dual formulation has a mild sign problem: baryons are non-relativistic and usually have loop geometries that have a positive signs. The dual partition function of the strong coupling limit is simulated with the worm algorithm (see Section III A) and the sign problem is essentially solved in this limit.

Extension to finite β

The leading order gauge corrections O(β) to the strong coupling limit are obtained by expanding the Wilson gauge action Eq. ( ) before integrating out the gauge links. A formal expression is obtained by changing the order of integration (first gauge links, then Grassmann-valued fermions) within the QCD partition function:
With this the O (β) partition function is The challenge in computing Z (1) is to address the SU(N c ) integrals that receive contributions from the elementary plaquette U P . Link integration no longer factorizes, however the tr[U P ] can be decomposed before integration: Integrals of the type J ij with two open color indices -as compared to link integration at strong coupling -have been derived from generating functions
for either J = 0 or for G = U(N c ) . The SU(3) result was discussed in , in terms of the dual variables, neglecting rotation and reflection symmetries, there are 19 distinct diagrams to be considered. The resulting partition function, valid to O(β), is with q P ∈ {0, ±1}, and the site weights w x → ŵx , bond weights w b → ŵb and baryon loop weights w → ŵ receive modifications compared to the strong coupling limit Eq. ( ) for sites and bonds adjacent to an excited plaquette q P = 1.
The weights are given in , and are rederived for any gauge group in . The configurations {n, k, , q p } must satisfy at each site x the constraint inherited from Grassmann integration: which is the modified version of Eq. ( ) with q x = 1 if located at the corner of an excited plaquette q p = 0, otherwise q x = 0.
A more general expression that we obtained via group theory and is valid to higher orders of the strong coupling expansion is discussed in terms of tensor networks . A typical 2-dimensional configuration that arises at β = 1 in the Monte Carlo simulations is given in Fig. . Note that if a baryon loop enters a non-trivial plaquette, one quark is separated from the two other quarks, resulting in the baryon being extended object, rather being point-like in the strong coupling limit.
The O(β) partition function has been used in the chiral limit to study the full µ B − T plane via reweighting from the strong coupling ensemble. Whereas the second order chiral transition for small values of the aµ B decreased up to the tri-critical point, the first order nuclear transition was invariant: aµ 1st B 1.78(1) at zero temperature has no β-dependence.
For the ratio T (µ B = 0)/µ 1st B (T 0) we found the values 0.787 for β = 0 and 0.529 β = 1, which should be compared to T c / 0.165 for full QCD . However, since reweighting cannot be fully trusted across a first order boundary, direct simulations at nonzero β are necessary. The Monte Carlo technique to update plaquette variables is discussed in Section III A.
In this section, we provide analytic results from exact enumeration for small volumes, and mean field results based on the 1/d expansion, valid in the thermodynamic limit. The main purpose is to compare our Monte Carlo results to these analytic predictions.

Exact enumeration

To establish that our Monte Carlo simulations indeed sample the partition functions Eq. ( ) and Eq. ( ), we have obtained analytic results on a 2 4 volume at strong coupling, and at finite beta in two dimensions on a 4 × 4 volume, comparing O (β) and O β 2 truncations. Our strategy to obtain an exact enumeration of the partition function Z is to enumerate plaquette configurations first, then fixing the fermion fluxes which together with the gauge fluxes that are induced by the plaquettes form a singlet, a triplet or anti-triplet, i.e. on a given bond b, g b + f b ∈ {−3, 0, 3}, and last we perform the monomerdimer enumeration on the available sites not saturated by fermions yet by a depth-first algorithm .
At strong coupling, with no plaquettes, g b = 0 and f b are baryonic fluxes. All observables that can be written in terms of derivatives of log(z), such as the baryon density, the chiral condensate, the energy density, and also the average sign, are shown in Fig.

Expectations from mean field theory

Another analytical method to study strong coupling lattice QCD is the mean field approach, where the partition function is expanded in 1 d (d is the spatial dimension) and then a Hubbard-Stratonovich transformation performed . After this procedure, the free energy is a function of temperature T , the chiral condensate σ and chemical potential µ B :
here E[m] is one-dimensional quark excitation energy which is a function of the quark mass m = am q . For N c = 3 and d = 3 we determined the minimum of the free energy with respect to the chiral condensate. This gives us the equilibrium chiral condensate as a function of (T, m, µ B ). The chiral condensate and the baryon density as a function of the baryon chemical potential in lattice units aµ B and for various temperatures at quark mass m = 1.5 is shown in Fig. . We have determined the critical temperature to be aT c = 0.23 , which is characterized by an infinite slope of the chiral condensate.
For lower temperatures, there is a clear discontinuity of the chiral con-densate, separating the low density phase from the high density phase. For temperatures above and in the vicinity of aT c the chiral condensate and baryon density has no discontinuity but rapidly changes, corresponding to a crossover transition.
With this method, the phase diagram is plotted for different quark masses in Fig. . The second order phase transition in the chiral limit is plotted in solid blue line, the dotted lines show the first order phase transition for different quark masses and the solid red line indicates the critical end point for the different quark masses.
Mean field theory also gives an expression for the pion mass am π and the baryon mass am B : The mean field baryon mass for N c = 3, d = 3 is also plotted in red in Fig. . Whereas the baryon mass is around N c in the chiral limit (am B 3.12 for N c = 3), it approximately doubles at m = 3.5 (am B 6.28) which corresponds to the pion mass am π = 4.45, i.e. m π /m B = 0.708.
Hence, at around bare mass m = 3.5, the valence quark mass of the baryon corresponds roughly to 1/3 of the chiral limit value of the baryon mass. The first Monte Carlo simulations that could extend in the µ B − T plane was the MDP algorithm , but it required the introduction of the worm algorithm to make substantial progress.
First studies of the worm algorithm applied to the strong coupling limit QCD (with gauge group U(3)) are , and for gauge group SU . Monte Carlo simulations to extend the worm to incorporate leading order corrections were first proposed in . We will shortly review the setup of or Monte Carlo strategy for the nuclear transition, with an emphasis on the challenges to address large quark masses.

Strong Coupling

Without any further resummation, there is a mild sign problem in the dual formulation of lattice QCD in the strong coupling limit. When the average sign σ is not too small (close to zero), it implies that most of the configurations have a positive weight thus allowing us to perform sign reweighting strategies.
In Fig. , ∆f is plotted as a function of the baryon chemical potential and the quark masses. It is seen that ∆f is close to zero for most cases except near the critical chemical potential and for small quark masses, but never exceeds 5 × 10 −4 . Hence sign reweighting can be performed in the full parameter space.
The result that the sign problem becomes even milder when increasing the mass is related to the fact that larger critical chemical potentials result in a larger fraction of static baryons (spatial baryon hoppings become rare). FIG. . ∆F at strong coupling as a function of chemical potential and quark mass on a 6 3 × 8.
The sign problem becomes milder as the quark mass increases.

Finite β

All runs at finite β have been obtained for N τ = 4, which corresponds to a moderately low temperature aT = 0.25 compared to the value of the chiral transition aT 1.54. Those simulations were too expensive to attempt N τ = 8 runs, in particular as a higher statistics was required. The spatial volumes are 4 3 , 6 3 and 8 3 .
For β values are from 0.0 to 1.0 with step size 0.1, and for am q values from 0.00 to 1.00 with step size 0.01. The values of aµ were chosen close to the nuclear transition, the scanning range is shifted to large values as am q increases. At small quark masses the scanning range is from aµ = 0.4 to 1.0 and for the large quark masses, it is from 0.6 to 1.2 with step size 0.01.
The statistics used for are 15 × 10 4 measurements and between measurement, 40 × N 3 s worm updates.

Residual sign problem

Although it is possible to resum the sign problem at strong coupling with a resummation of baryon and pion world lines, this is not possible when including gauge corrections. In order to compare both sign problems, we kept the original dual formulation to monitor the severity of the sign problem. This is done via the relation
between the average sign σ and the difference of the free energy density ∆f between the full ensemble f and of the sign-quenched ensemble f || .

Nuclear interactions

We have found that aµ 1st B is very different from the baryon mass. This must be due to strong attractive interactions of nucleons. In contrast to continuum physics, in the strong coupling limit there is no pion exchange due to the Grassmann constraint. Instead, nucleons are point like and hard core repulsive.
However, the pion bath, which is modified by the presence of static baryons, results in an attractive interaction. In , this has been analyzed in the chiral limit using the snake algorithm, and it has been found that the attractive force is of entropic origin. Here, we do not quantify the nuclear interaction via the nuclear potential, but via the difference between critical baryon chemical potential and baryon mass, in units baryon mass, as shown in Fig. , given the am B as measured in Section III C.
This compares better to the 3dim. effective theory. The nuclear interaction is maximal and more than 40% in the chiral limit, which is related to pions being massless: the modification of the pion bath is maximal. We clearly find that the nuclear interaction decreases drastically and almost linearly until it almost approaches zero at about am q = 2.0, corresponding to a pion mass am π = 3.36, see Section II B. The large error bars for larger quark masses, that are due to the subtraction of almost same magnitudes, makes it difficult to extract a non-zero nuclear interaction at the largest quark masses.
In this work, we have determined the baryon mass and the nuclear transition via Monte Carlo: the worm algorithm based on the dual formulation, at finite β equipped with additional updates. All those numerical results and various analytic expressions are summarized in Fig. . We find that as the quark mass becomes large, spatial mesons hoppings (i.e.
spatial dimers) become rare, which makes this 3+1-dimensional system closer to 1dim. QCD . Also, both the baryon mass and the baryon chemical potential obtained in our dual representation, i.e. for staggered fermions, approaches the baryon mass of the 3-dim. effective theory which is based on Wilson fermions.
Another comparison that summarizes the validity of the mean field approach discussed in Section II B is shown in Fig. . It is evident that mean field theory has strong deviations for small quark masses, but this discrepancy becomes smaller for larger quark masses. The extension of the study of the nuclear transition to finite inverse gauge coupling β is summarized in Fig. , which shows the β-dependence of aµ c B for various quark masses.
For all quark masses ranging from am q = 0 to am q = 1.0, there is only a very weak β-dependence, confirming the expectation from mean field theory . This works was restricted to isotropic lattices ξ = a/a t = 1, i.e. we performed simulations at fixed temperature. Non-isotropic lattices are necessary to vary the temperature at fixed values of β.
This requires to include two bare anisotropies, γ for the fermionic action and γ G for the gauge action. Finite β has only been studied by us in the chiral limit . Clearly, it is interesting to study the location of the nuclear critical point also including higher order gauge corrections and at finite quark mass.
Simulations including O(β 2 ) are under preparation.

Question: What is the main focus of the research paper?
Answer: Nuclear liquid-gas transition in lattice QCD.