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During the last years it has become possible to address the cold and dense regime of QCD directly for sufficiently heavy quarks, where combined strong coupling and hopping expansions are convergent and a 3d effective theory can be derived, which allows to control the sign problem either in simulations or by fully analytic calculations. In this contribution we review the effective theory and study the $N_c$-dependence of the nuclear liquid gas transition, as well as the equation of state of baryonic matter in the strong coupling limit. We find the transition to become more strongly first order with growing $N_c$, suggesting that in the large $N_c$ limit its critical endpoint moves to high temperatures to connect with the deconfinement transition. Furthermore, to leading and next-to-leading order in the strong coupling and hopping expansions, respectively, the pressure is found to scale as $psim N_c$. This suggests that baryonic and quarkyonic matter might be the same at nuclear densities. Further work is needed to see whether this result is stable under gauge corrections.
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After combined character and hopping expansions and integration over the spatial gauge links, lattice QCD reduces to a three-dimensional $SU(3)$ Polyakov loop model with complicated interactions. A simple truncation of the effective theory is valid for heavy quarks on reasonably fine lattices and can be solved by linked cluster expansion in its effective couplings. This was used ealier to demonstrate the onset transition to baryon matter in the cold and dense regime. Repeating these studies for general $N_c$, one finds that for large $N_c$ the onset transition becomes first-order, and the pressure scales as $psim N_c$ through three consecutive orders in the hoppoing expansion. These features are consistent with the formal definition of quarkyonic matter given in the literature. We discuss the implications for $N_c=3$ and physical QCD.
Lattice QCD with heavy quarks reduces to a three-dimensional effective theory of Polyakov loops, which is amenable to series expansion methods. We analyse the effective theory in the cold and dense regime for a general number of colours, $N_c$. In particular, we investigate the transition from a hadron gas to baryon condensation. For any finite lattice spacing, we find the transition to become stronger, i.e. ultimately first-order, as $N_c$ is made large. Moreover, in the baryon condensed regime, we find the pressure to scale as $psim N_c$ through three orders in the hopping expansion. Such a phase differs from a hadron gas with $psim N_c^0$, or a quark gluon plasma, $psim N_c^2$, and was termed quarkyonic in the literature, since it shows both baryon-like and quark-like aspects. A lattice filling with baryon number shows a rapid and smooth transition from condensing baryons to a crystal of saturated quark matter, due to the Pauli principle, and is consistent with this picture. For continuum physics, the continuum limit needs to be taken before the large $N_c$ limit, which is not yet possible in practice. However, in the controlled range of lattice spacings and $N_c$-values, our results are stable when the limits are approached in this order. We discuss possible implications for physical QCD.
We study the cold and dense regime in the phase diagram of two-color QCD with heavy quarks within a three-dimensional effective theory for Polyakov loops. This theory is derived from two-color QCD in a combined strong-coupling and hopping expansion. In particular, we study the onset of diquark density as the finite-density transition of the bosonic baryons in the two-color world. In contrast to previous studies of heavy dense QCD, our zero-temperature extrapolations are consistent with a continuous transition without binding energy. They thus provide evidence that the effective theory for heavy quarks is capable of describing the characteristic differences between diquark condensation in two-color QCD and the liquid-gas transition of nuclear matter in QCD.
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.
We determine the magnetic susceptibility of thermal QCD matter by means of first principles lattice simulations using staggered quarks with physical masses. A novel method is employed that only requires simulations at zero background field, thereby circumventing problems related to magnetic flux quantization. After a careful continuum limit extrapolation, diamagnetic behavior (negative susceptibility) is found at low temperatures and strong paramagnetism (positive susceptibility) at high temperatures. We revisit the decomposition of the magnetic susceptibility into spin- and orbital angular momentum-related contributions. The spin term -- related to the normalization of the photon lightcone distribution amplitude at zero temperature -- is calculated non-perturbatively and extrapolated to the continuum limit. Having access to both the full magnetic susceptibility and the spin term, we calculate the orbital angular momentum contribution for the first time. The results reveal the opposite of what might be expected based on a free fermion picture. We provide a simple parametrization of the temperature- and magnetic field-dependence of the QCD equation of state that can be used in phenomenological studies.
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