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Register a new userTowards a theoretical description of dense QCD
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Authors
Owe Philipsen
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The properties of matter at finite baryon densities play an important role for the astrophysics of compact stars as well as for heavy ion collisions or the description of nuclear matter. Because of the sign problem of the quark determinant, lattice QCD cannot be simulated by standard Monte Carlo at finite baryon densities. I review alternative attempts to treat dense QCD with an effective lattice theory derived by analytic strong coupling and hopping expansions, which close to the continuum is valid for heavy quarks only, but shows all qualitative features of nuclear physics emerging from QCD. In particular, the nuclear liquid gas transition and an equation of state for baryons can be calculated directly from QCD. A second effective theory based on strong coupling methods permits studies of the phase diagram in the chiral limit on coarse lattices.
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We study the vacuum topology of 2+1 flavor QCD above the chiral crossover transition, at $T lesssim 1.2~T_c$, on lattices of size $32^3times 8$. Since overlap fermions have exact chiral symmetry and an index theorem even on a finite lattice, we use them to detect the topological content of gauge fields generated using domain wall fermion discretization for quarks. We further use different periodicity phases along the temporal direction for the valence overlap quarks, which allows us to probe different topological structures present in the gauge field ensembles, through its zero modes. This procedure provides strong evidences that fermion zero-modes can be quantitatively understood to arise due to different species of instanton-dyons. We estimate their relative abundances from the Dirac-eigenvalue density and resolve the so called topological clusters via multi-parameter fits to their density, providing therefore an understanding of the interactions between instanton-dyons. The typical separation between dyons we obtain, is $sim 0.3$ fm. Surprisingly, it emerges out from this study that a semi-classical description of the fermionic zero modes in the QCD vacuum is quite accurate just above $T_c$.
QCD is expected to have a rich phase structure. It is empirically known to be difficult to access low temperature and nonzero chemical potential $mu$ regions in lattice QCD simulations. We address this issue in a lattice QCD with the use of a dimensional reduction formula of the fermion determinant. We investigate spectral properties of a reduced matrix of the reduction formula. Lattice simulations with different lattice sizes show that the eigenvalues of the reduced matrix follow a scaling law for the temporal size $N_t$. The properties of the fermion determinant are examined using the reduction formula. We find that as a consequence of the $N_t$ scaling law, the fermion determinant becomes insensitive to $mu$ as $T$ decreases, and $mu$-independent at T=0 for $mu<m_pi/2$. The $N_t$ scaling law provides two types of the low temperature limit of the fermion determinant: (i) for low density and (ii) for high-density. The fermion determinant becomes real and the theory is free from the sign problem in both cases. In case of (ii), QCD approaches to a theory, where quarks interact only in spatial directions, and gluons interact via the ordinary Yang-Mills action. The partition function becomes exactly $Z_3$ invariant even in the presence of dynamical quarks because of the absence of the temporal interaction of quarks. The reduction formula is also applied to the canonical formalism and Lee-Yang zero theorem. We find characteristic temperature dependences of the canonical distribution and of Lee-Yang zero trajectory. Using an assumption on the canonical partition function, we discuss physical meaning of those temperature dependences and show that the change of the canonical distribution and Lee-Yang zero trajectory are related to the existence/absence of $mu$-induced phase transitions.
A previously derived three-dimensional effective lattice theory describing the thermodynamics of QCD with heavy quarks in the cold and dense region is extended through order $sim u^5kappa^8$ in the combined character and hopping expansion of the original four-dimensional Wilson action. The systematics of the effective theory is investigated to determine its range of validity in parameter space. We demonstrate the severe cut-off effects due to lattice saturation, which afflict any lattice results at finite baryon density independent of the sign problem or the quality of effective theories, and which have to be removed by continuum extrapolation. We then show how the effective theory can be solved analytically by means of a linked cluster expansion, which is completely unaffected by the sign problem, in quantitative agreement with numerical simulations. As an application, we compute the cold nuclear equation of state of heavy QCD. Our continuum extrapolated result is consistent with a polytropic equation of state for non-relativistic fermions.
QCD matter at finite temperature and density is a subject that has witnessed very impressive theoretical developments in the recent years. In this review I will discuss some new insights on the microscopic degrees of freedom of the QCD medium near the chiral crossover transition from lattice QCD. Latest high precision lattice data on the fluctuations and correlations between conserved charges like the baryon number, strangeness can help us to understand and distinguish between different models of interacting hadrons. Furthermore, the latest constraints on the location of the critical end-point and the curvature of the critical line will be discussed. In the later part of this review I will discuss about the insights on the thermal nature of the medium created in heavy ion collision experiments that have come from the theoretical analysis of the particle yields, and to what extent the lattice data on correlations and fluctuations of conserved charges can give us any information about the fireball at freezeout.
In this paper we carry out 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 $mu = m_{pi}/2$ 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 non-zero 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 SU($N_c$) theory at large $N_c$.
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