Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userChiral condensate and Dirac spectrum of one- and two-flavor QCD at nonzero $theta$-angle
71
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In the $epsilon$-domain of QCD we have obtained exact analytical expressions for the eigenvalue density of the Dirac operator at fixed $theta e 0$ for both one and two flavors. These results made it possible to explain how the different contributions to the spectral density conspire to give a chiral condensate at fixed $theta$ that does not change sign when the quark mass (or one of the quark masses for two flavors) crosses the imaginary axis, while the chiral condensate at fixed topological charge does change sign. From QCD at nonzero density we have learnt that the discontinuity of the chiral condensate may move to a different location when the spectral density increases exponentially with the volume with oscillations on the order of the inverse volume. This is indeed what happens when the product of the quark masses becomes negative, but the situation is more subtle in this case: the contribution of the quenched part of the spectral density diverges in the thermodynamic limit at nonzero $theta$, but this divergence is canceled exactly by the contribution from the zero modes. We conclude that the zero modes are essential for the continuity of the chiral condensate and that their contribution has to be perfectly balanced against the contribution from the nonzero modes. Lattice simulations at nonzero $theta$-angle can only be trusted if this is indeed the case.
rate research
Read More
In a sector of fixed topological charge, the chiral condensate has a discontinuity given by the Banks-Casher formula also in the case of one-flavor QCD. However, at fixed theta-angle, the chiral condensate remains constant when the quark mass crosses zero. To reconcile these contradictory observations, we have evaluated the spectral density of one-flavor QCD at theta=0. For negative quark mass, it becomes a strongly oscillating function with a period that scales as the inverse space-time volume and an amplitude that increases exponentially with the space-time volume. As we have learned from QCD at nonzero chemical potential, if this is the case, an alternative to the Banks-Casher formula applies, and as we will demonstrate in this talk, for one-flavor QCD this results in a continuous chiral condensate. A special role is played by the topological zero modes which have to be taken into account exactly in order to get a finite chiral condensate in the thermodynamic limit.
We analyze the mass dependence of the chiral condensate for QCD at nonzero $theta$-angle and find that in general the discontinuity of the chiral condensate is not on the support of the Dirac spectrum. To understand this behavior we decompose the spectral density and the chiral condensate into contributions from the zero modes, the quenched part, and a remainder which is sensitive to the fermion determinant and is referred to as the dynamical part. We obtain general formulas for the contributions of the zero modes. Expressions for the quenched part, valid for an arbitrary number of flavors, and for the dynamical part, valid for one and two flavors, are derived in the microscopic domain of QCD. We find that at nonzero $theta$-angle the quenched and dynamical part of the Dirac spectral density are strongly oscillating with an amplitude that increases exponentially with the volume $V$ and a period of order of $1/V$. The quenched part of the chiral condensate becomes exponentially large at $theta e0$, but this divergence is canceled by the contribution from the zero modes. The oscillatory behavior of the dynamical part of the density is essential for moving the discontinuity of the chiral condensate away from the support of the Dirac spectrum. As important by-products of this work we obtain analytical expressions for the microscopic spectral density of the Dirac operator at nonzero $theta$-angle for both one- and two-flavor QCD with nonzero quark masses.
We derive exact analytical expressions for the spectral density of the Dirac operator at fixed theta-angle in the microscopic domain of one-flavor QCD. These results are obtained by performing the sum over topological sectors using novel identities involving sums of products of Bessel functions. Because the fermion determinant is not positive definite for negative quark mass, the usual Banks-Casher relation is not valid and has to be replaced by a different mechanism first observed for QCD at nonzero chemical potential. Using the exact results for the spectral density we explain how this mechanism results in a chiral condensate that remains constant when the quark mass changes sign.
We investigate the nature of the chiral phase transition in the massless two-flavor QCD using the renormalization group improved gauge action and the Wilson quark action on $32^3times 16$, $24^3times 12$, and $16^3times 8$ lattices. We calculate the spacial and temporal propagators of the iso-triplet mesons in the pseudo-scalar ($PS$), scalar ($S$), vector ($V$) and axial-vector ($AV$) channels on the lattice of three sizes. We first verify that the RG scaling is excellently satisfied for all cases. This is consistent with the claim that the chiral phase transition is second order. Then we compare the spacial and temporal effective masses between the axial partners, i.e. $PS$ vs $S$ and $V$ vs $AV$, on each of the three size lattices. We find the effective masses of all of six cases for the axial partners agree remarkably. This is consistent with the claim that at least $Z_4$ subgroup of the $U_A(1)$ symmetry in addition to the $SU_A(2)$ symmetry is recovered at the chiral phase transition point.
We study the phase diagram and the thermodynamic properties of QCD at nonzero isospin asymmetry at physical quark masses with staggered quarks. In particular, continuum results for the phase boundary between the normal and the pion condensation phases and the chiral/deconfinement transition are presented. Our findings indicate that the pion condensation phase is restricted to $Tlesssim170$~MeV for isospin chemical potentials up to 325~MeV. We also use the data to test the range of validity of the Taylor expansion method and show first results for the equation of state.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
