Bosonic partition functions at nonzero (imaginary) chemical potential

International audience

We discuss the spectrum of the QCD Dirac operator both at zero and at nonzero baryon chemical potential. We show that, in the ergodic domain of QCD, the Dirac spectrum can be obtained from the replica limit of a Toda lattice equation. At zero chemical potential this method explains the factorization of known results into compact and noncompact integrals, and at nonzero chemical potential it allows us to derive the previously unknown microscopic spectral density.

We consider the phase diagram of twisted mass Wilson fermions of two-flavor\nQCD in the parameter space of the quark mass, the isospin chemical potential,\nthe twist angle and the lattice spacing. This work extends earlier studies in\nthe continuum and those at zero chemical potential. We evaluate the phase\ndiagram as well as the spectrum of the (pseudo-)Goldstone bosons using the\nchiral Lagrangian for twisted mass Wilson fermions at non-zero isospin chemical\npotential. The phases are obtained from a mean field analysis. At zero twist\nangle we find that already an infinitesimal isospin chemical potential destroys\nthe Aoki phase. The reason is that in this phase we have massless Goldstone\nbosons with a non-zero isospin charge. At finite twist angle only two different\nphases are present, one phase which is continuously connected to the Bose\ncondensed phase at non-zero chemical potential and another phase which is\ncontinuously connected to the normal phase. For either zero or maximal twist\nthe phase diagram is more complicated as the saddle point equations allow for\nmore solutions.\n

The thermodynamics of massless ideal gas of overlap quarks has been investigated both analytically and numerically for both zero and nonzero baryon chemical potential. Any ${\ensuremath{\mu}}^{2}$ divergence is shown analytically to be absent for a class of actions with nonzero chemical potential. All such actions are shown to violate chiral invariance. While the parameter $M$ can be shown to be irrelevant in the continuum limit, as expected, it is shown numerically that the continuum limit can be reached with relatively coarser lattices for $1.5\ensuremath{\le}M\ensuremath{\le}1.6$. Numerical limitations of the existing method of introduction of chemical potential are demonstrated. Finally we also show that the energy density for the massive overlap fermions has the correct continuum limit.

We present a subset method that solves the sign problem for QCD at nonzero quark chemical potential in 0+1 dimensions. The subsets of gauge configurations are constructed using the center symmetry of the SU(3) group. These subsets completely solve the sign problem for up to five flavors. For a larger number of flavors the sign problem slowly reappears, and we propose an extension of the subsets that also solves the sign problem for these cases. The subset method allows for numerical simulations of the model at nonzero chemical potential. We also present some preliminary results on subsets for QCD in two, three, and four dimensions.

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.

The general issue of large ${N}_{c}$ QCD at nonzero chemical potential is considered with a focus on understanding the difference between large ${N}_{c}$ QCD with an isospin chemical potential and large ${N}_{c}$ QCD with a baryon chemical potential. A simple diagrammatic analysis analogous to 't Hooft's analysis at $\ensuremath{\mu}=0$ implies that the free energy with a given baryon chemical potential is equal to the free energy with an isospin chemical potential of the same value plus $1/{N}_{c}$ corrections. Phenomenologically, these two systems behave quite differently. A scenario to explain this difference in light of the diagrammatic analysis is explored. This scenario is based on a phase transition associated with pion condensation when the isospin chemical potential exceeds ${m}_{\ensuremath{\pi}}/2$; associated with this transition there is breakdown of the $1/{N}_{c}$ expansion---in the pion condensed phase there is a distinct $1/{N}_{c}$ expansion including a larger set of diagrams. While this scenario is natural, there are a number of theoretical issues which at least superficially challenge it. Most of these can be accommodated. However, the behavior of quenched QCD which raises a number of apparently analogous issues cannot be easily understood completely in terms of an analogous scenario. Thus, the overall issue remains open.

At nonzero quark chemical potential lattice QCD simulations with dynamical quarks are hampered by the sign problem caused by the complex fermion determinant. In this talk we discuss the relation between QCD and random matrix theory and present a novel solution to the sign problem in dynamical random matrix simulations. The sign problem is solved by gathering matrices in subsets, whose sums of complex determinants are real and positive. Markov chains of relevant subsets were generated in Monte Carlo simulations and various observables, like the chiral condensate and quark number, were accurately computed. The subset formula, which is at the basis of the positivity proof for the subset weights, also provides a simple solution to the Silver Blaze puzzle.

Symmetries and spectrum of SU(2) lattice gauge theory at finite chemical potential

Higher order cumulants of net baryon-number distributions at non-zero μB

We derive a general expression for the absorptive part of the one-loop photon polarization tensor in a strongly magnetized quark-gluon plasma at nonzero baryon chemical potential. To demonstrate the application of the main result in the context of heavy-ion collisions, we study the effect of a nonzero baryon chemical potential on the photon emission rate. The rate and the ellipticity of photon emission are studied numerically as a function the transverse momentum (energy) for several values of temperature and chemical potential. When the chemical potential is small compared to the temperature, the rates of the quark and antiquark splitting processes (i.e., qrightarrow q +gamma and {bar{q}}rightarrow {bar{q}} +gamma , respectively) are approximately the same. However, the quark splitting gradually becomes the dominant process with increasing the chemical potential. We also find that increasing the chemical potential leads to a growing total photon production rate but has only a small effect on the ellipticity of photon emission. The quark-antiquark annihilation (q+{bar{q}}rightarrow gamma ) also contributes to the photon production, but its contribution remains relatively small for a wide range of temperatures and chemical potentials investigated.

We study the free fermion gas at finite temperature and chemical potential in the lattice regularized version proposed by Hasenfratz and Karsch and by Kogut et al. Special emphasis is placed on the identification of the particle and antiparticle contributions to the partition function. In the case of naive fermions we show that the partition function no longer separates into particle-antiparticle contributions in the way familiar from the continuum formulation. The use of Wilson fermions, on the other hand, eliminates this unpleasant feature, and leads, after subtracting the vacuum contributions, to the familiar expressions for the average energy and charge densities.

We discuss the distribution of the quark number over the gauge fields for QCD at nonzero quark chemical potential. As the quark number operator is non-hermitian, the distribution is over the complex plane. Moreover, because of the fermion determinant, the distribution is not real and positive. The computation is carried out within leading order chiral perturbation theory and gives direct insight into the delicate cancellations that take place in contributions to the total baryon number.

We present results for fluctuations of the baryon number for QCD at nonzero temperature and chemical potential. These are extracted from solutions to a coupled set of truncated Dyson-Schwinger equations for the quark and gluon propagators of Landau-gauge QCD with $N_f = 2 + 1$ quark flavors, that has been studied previously. We discuss the changes of fluctuations and ratios thereof up to fourth order for several temperatures and baryon chemical potential up to and beyond the critical endpoint. In the context of preliminary STAR data for the skewness and kurtosis ratios, the results are compatible with the scenario of a critical endpoint at large chemical potential and slightly offset from the freeze-out line. We also discuss the caveats involved in this comparison.

We consider QCD at strong coupling with scalar quarks coupled to a chemical potential. Performing the link integrals we present a diagrammatic representation of the path integral weight. It is based on mesonic and baryonic building blocks, in close analogy to fermionic QCD. Likewise, the baryon loops are subject to a manifest conservation of the baryon number. The sign problem is expected to disappear in this representation and we do confirm this for three flavors, where a scalar baryon can be built and thus a dependence on the chemical potential occurs. For higher flavor number we analyse examples for a potential sign problem in the baryon sector and conjecture that all weights are positive upon exploring the current conservation of each flavor.