We derive some exact results concerning the anomalous U(1)$_A$ symmetry in the chirally symmetric phase of QCD at high temperature. We discuss the importance of topology and finite-volume effects on the U(1)$_A$ symmetry violation characterized by the difference of chiral susceptibilities. In particular, we present a reliable method to measure the anomaly strength in lattice simulations with fixed topology. We also derive new spectral sum rules and a novel Banks-Casher-type relation. Through our spectral analysis we arrive at a simple alternative proof of the Aoki-Fukaya-Taniguchi "theorem" on the effective restoration of the U(1)$_A$ symmetry at high temperature.
We evaluate for arbitrary even dimensions the classical continuum limit of the lattice axial anomaly defined by the overlap-Dirac operator. Our calculational scheme is simple and systematic. In particular, a powerful topological argument is utilized to determine the value of a lattice integral involved in the calculation. When the Dirac operator is free of species doubling, the classical continuum limit of the axial anomaly in various dimensions is combined into a form of the Chern character, as expected.
A recent proposal suggests that even if a Ginsparg-Wilson lattice Dirac operator does not possess any topological zero modes in topologically-nontrivial gauge backgrounds, it can reproduce correct axial anomaly for sufficiently smooth gauge configurations, provided that it is exponentially-local, doublers-free, and has correct continuum behavior. In this paper, we calculate the axial anomaly of this lattice Dirac operator in weak coupling perturbation theory, and show that it recovers the topological charge density in the continuum limit.
The chiral susceptibility, or the first derivative of the chiral condensate with respect to the quark mass, is often used as a probe for the QCD phase transition since the chiral condensate is an order parameter of $SU(2)_L \times SU(2)_R$ symmetry breaking. However, the chiral condensate also breaks the axial $U(1)$ symmetry, which is usually not paid attention to as it is already broken by anomaly and apparently gives little impact on the transition. We investigate the susceptibilities in the scalar and pseudoscalar channels in order to quantify how much the axial $U(1)$ breaking contributes…
The axial-vector form factor of the nucleons is considered in the framework of hard-wall model of holographic QCD. A new interaction term between the bulk gauge and matter fields was included into the interaction Lagrangian. We obtain the axial-vector form factor of nucleons in the boundary QCD from the bulk action using AdS/CFT correspondence. The momentum square dependence of the axial-vector form factor is analysed numerically.
We provide the renormalization constants of the axial current operators, both singlet and non-singlet, in dimensional regularization up to four-loop order in QCD, determined using the off-shell Ward-Takahashi identity for an axial current with a non-anticommuting $γ_5$. A possible application of the result for the singlet axial current operator is the extraction of the non-decoupling mass logarithms in the axial quark form factors.
The aim of this work is to study some possible local and global $U(1)$ axial condensates in the high-temperature chirally restored phase of QCD, by means of two nonperturbative analytical techniques: (i) by expressing the functional averages in terms of the spectral density of the Euclidean Dirac operator and (ii) by evaluating the functional integrals in the instanton-background approximation. In this way, besides proving that these condensates are indeed different from zero in the high-temperature regime, we shall also derive their asymptotic temperature dependence and compare it with that o…
An investigation of singular fields emerging in the process of transforming QCD to the axial gauge is presented. The structure of the singularities is analyzed. It is shown that apart from well known neutral magnetic monopole singularities, the field configurations also exhibit singularities in their charged and transverse components. This complex singularity structure guarantees finite non-Abelian field strength and thus finite action if expressed in terms of gauge fixed fields. A relation between the monopole charges of singular field configurations and their topological charge is derived. Q…
We reconsider constraints on the eigenvalue density of the Dirac operator in the chiral symmetric phase of 2 flavor QCD at finite temperature. To avoid possible ultra-violet(UV) divergences, we work on a lattice, employing the overlap Dirac operator, which ensures the exact "chiral" symmetry at finite lattice spacings. Studying multi-point correlation functions in various channels and taking their thermodynamical limit (and then taking the chiral limit), we obtain stronger constraints than those found in the previous studies: both the eigenvalue density at the origin and its first and second d…
We study the chiral effective theory in the presence of QCD vortices. Gauge invariance requires novel terms from vortex singularities in the gauged Wess-Zumino-Witten action, which incorporate anomaly induced currents along the vortices. We examine these terms for systems with QCD axial domain-walls bounded by vortices (vortons) under magnetic fields. We discuss how the baryon and the electric charge conservations are satisfied in these systems through interplay between domain-walls and vortices, which manifests Callan-Harvey's mechanism of the anomaly inflow.
Magnetically charged vortex defects are shown to arise in canonically quantised modified axial gauge QCD_{3+1} on a torus. This gauge -- being an Abelian projection -- keeps only the eigenphases as dynamical variables of the Wilson line in x_3-direction. A mismatch between the identification of large gauge transformations before and after gauge fixing is indicated.
We provide the $\bar{\mathrm{MS}}$ factor of the renormalization constant of the singlet axial-current operator in dimensional regularization at $\mathcal{O}(α_s^5)$ in perturbative Quantum Chromodynamics (QCD). The result is obtained from a formula derived using the Adler-Bell-Jackiw equation in terms of renormalized operators. The required input consists of the complete four-loop results for operator renormalization constants obtained by explicit diagrammatic computation in a previous publication.
The dynamics of topological domains which break parity (${\cal P}$) and charge-parity (${\cal CP}$) symmetry of QCD are studied. We derive in a general setting that those local domains will generate an axial current and quantify the strength of the induced axial current. Our findings are verified in a top-down holographic model. The relation between the real time dynamics of those local domains and chiral magnetic effect is also elucidated. We finally argue that such an induced axial current would be phenomenologically important if topological domains are created in heavy-ion collisions experi…
We show that holographic models of QCD predict the presence of a Chern-Simons coupling between vector and axial-vector mesons at finite baryon density. In the AdS/CFT dictionary, the coefficient of this coupling is proportional to the baryon number density, and is fixed uniquely in the five-dimensional holographic dual by anomalies in the flavor currents. For the lightest mesons, the coupling mixes transverse $ρ$ and $a_1$ polarization states. At sufficiently large baryon number densities, it produces an instability, which causes the $ρ$ and $a_1$ mesons to condense in a state breaking both ro…
A previous formal derivation of the effective chiral Lagrangian for low-lying pseudoscalar mesons from first-principles QCD without approximations [Wang et al., Phys. Rev. D61, (2000) 54011] is generalized to further include scalar, vector, and axial-vector mesons. In the large Nc limit and with an Abelian approximation, we show that the properties of the newly added mesons in our formalism are determined by the corresponding underlying fundamental homogeneous Bethe--Salpeter equation in the ladder approximation, which yields the equations of motion for the scalar, vector, and axial-vector mes…
The Axial Magnetic Effect is the generation of an equilibrium dissipationless energy flow of chiral fermions in the direction of the axial (chiral) magnetic field. At finite temperature the dissipationless energy transfer may be realized in the absence of any chemical potentials. We numerically study the temperature behavior of the Axial Magnetic Effect in quenched SU(2) lattice gauge theory. We show that in the confinement (hadron) phase the effect is absent. In the deconfinement transition region the conductivity quickly increases, reaching the asymptotic $T^2$ behavior in a deep deconfineme…
A comparative dynamical study of axial gauge QED and QCD is presented. Elementary excitations associated with particular field configurations are investigated. Gluonic excitations analogous to linearly polarized photons are shown to acquire infinite energy. Suppression of this class of excitations in QCD results from quantization of the chromelectric flux and is interpreted as a dual Meissner effect, i.e. as expulsion from the QCD vacuum of chromo-electric fields which are constant over significant distances. This interpretation is supported by a comparative evaluation of the interaction energ…
We investigate the Dirac eigenvalue spectrum ($ρ(λ,m_l)$) to study the microscopic origin of axial anomaly in high temperature phase of QCD. We propose novel relations between the derivatives ($\partial^n ρ(λ,m_l)/\partial m_l^n$) of the Dirac eigenvalue spectrum with respect to the quark mass ($m_l$) and the $(n+1)$-point correlations among the eigenvalues ($λ$) of the massless Dirac operator. Based on these relations, we present lattice QCD results for $\partial^n ρ(λ,m_l)/\partial m_l^n$ ($n=1, 2, 3$) with $m_l$ corresponding to pion masses $m_π=160-55$ MeV, and at a temperature of about 1.…
We introduce novel relations between the derivatives [$\partial^{n}ρ(λ, m_{l})/\partial m_{l}^n$] of the Dirac eigenvalue spectrum [$ρ(λ, m_{l})$] with respect to the light sea quark mass ($m_{l}$) and the $(n+1)$-point correlations among the eigenvalues ($λ$) of the massless Dirac operator. Using these relations we present lattice QCD results for $\partial^{n}ρ(λ, m_{l})/\partial m_{l}^n$ ($n=1, 2, 3$) for $m_{l}$ corresponding to pion masses $m_π=160-55$ MeV, and at a temperature of about 1.6 times the chiral phase transition temperature. Calculations were carried out using (2+1) flavors of …
An alternative pattern of the chiral symmetry breaking, suggested recently by Stern, is investigated. It could be self-consistent provided that the chiral $SU(N_f) \times SU(N_f)$ symmetry is broken spontaneously down to $SU(N_f) \times Z_{N_f}$ rather than to $SU(N_f)_V$. The discrete axial $Z_{N_f}$ then would play a custodial role preventing the quark bilinears from condensation. It is shown that this pattern of the chiral symmetry breaking is ruled out in QCD by exact inequalities. It is not ruled out, however, in other gauge theories with scalar quarks and/or Yukawa couplings.
We study the two-color QCD matter within two fundamental quark flavors via both chiral perturbation theory and Nambu--Jona-Lasinio model methods. The effective Lagrangian described by low lying meson and baryon, i.e., diquark, is derived, where the excitations locate in the extended $\mathrm{SU}(4)$ flavor symmetry space. We determine the leading order terms on the dependence of the baryon and axial isospin densities. Then, the two-color NJL model is employed to run the numerical simulation and the phase diagram in the plane of $μ-ν_{5}$ is plotted.
In this article, we carry out the Hamiltonization in the axial gauge, of the t'Hooft-Polyakov monopole field outside the localized region, which represents the monopole's core. One feature of the treatment here, is using the Higgs vacuum condition as both strong and weak equation instead of using it in the degree of freedom reduction.
An explicit, detailed evaluation of the classical continuum limit of the axial anomaly/index density of the overlap Dirac operator is carried out in the infinite volume setting, and in a certain finite volume setting where the continuum limit involves an infinite volume limit. Our approach is based on a novel power series expansion of the overlap Dirac operator. The correct continuum expression is reproduced when the parameter $m_0$ is in the physical region $0<m_0<2$. This is established for a broad range of continuum gauge fields. An analogous result for the fermionic topological charge, giv…
Canonical quantization of a gauge theory in the spatial axial gauge produces an anisotropic Hamiltonian and matter particles surrounded by physically unrealistic asymmetric electric or chromoelectric fields. We show how to restore rotational symmetry for a nonabelian theory with a gauge fixing condition $A^a_3=0$. We also discuss similarities between recovering isotropy in the spatial axial gauge and finding gauge invariant quantities in the Weyl gauge in both abelian and nonabelian field theories.
Whereas the theoretical results for the dominant contributions to hadronic light-by-light scattering coming from pseudoscalar meson exchange have converged over the past years, the various published estimates of the contribution due to axial vector meson exchange differ wildly. Since holographic AdS/QCD models have proved to provide rather good models of singly and doubly virtual pion transition form factors, which reproduce remarkably well the known low-energy data and also the asymptotic leading-order pQCD behavior, it is of interest to consider their predictions for axial vector transition …
General Axial Gauges within a perturbative approach to QCD are plagued by 'spurious' propagator singularities. Their regularisation has to face major conceptual and technical problems. We show that this obstacle is naturally absent within a Wilsonian or 'Exact' Renormalisation Group approach and explain why this is so. The axial gauge turns out to be a fixed point under the flow, and the universal 1-loop running of the gauge coupling is computed.
We investigate an impact of the axial-vector interaction on spatial modulation of quark matter. A magnetic field coupled with baryon density leads to a topological axial current, so that the effect of the axial-vector interaction is crucially enhanced then. Using the Sakai-Sugimoto model we have found that, contrary to a naive expectation, the spatially modulated phase is less favored for a stronger magnetic field, which is realized by the presence of topological current.
We calculate the constitutive equations of the stress-energy tensor and the currents of the free massless Dirac field at thermodynamic equilibrium with acceleration and rotation and a conserved axial charge by using the density operator approach. We carry out an expansion in thermal vorticity to the second order with finite axial chemical potential $μ_A$. The obtained coefficients of the expansion are expressed as correlators of angular momenta and boost operators with the currents. We confirm previous observations that the axial chemical potential induces non-vanishing components of the stres…
The Berry phase is analyzed for Weyl and Dirac fermions in a phase space representation of the worldline formalism. Kinetic theories are constructed for both at a classical level. Whereas the Weyl fermion case reduces in dimension, resembling a theory in quantum mechanics, the Dirac fermion case takes on a manifestly Lorentz covariant form. To achieve a classical kinetic theory for the non-Abelian Dirac fermion Berry phase a spinor construction of Barut and Zanghi is utilized. The axial anomaly is also studied at a quantum level. It is found that under an adiabatic approximation, which is nece…
We consider a Dirac fermion in a metric-axial-tensor (MAT) background. By regulating it with Pauli-Villars fields we analyze and compute its full anomaly structure. Appropriate limits of the MAT background allows to recover the anomalies of Dirac and Weyl fermions in the usual curved spacetime, obtaining in particular the trace anomaly of a chiral fermion, which has been the object of recent analyses.
Two real vector fields are revealed as spin connections of the spinor field, which is introduced as a representation of the local Lorentz group by Dirac spinors. One of these fields is identified as the Maxwell field. Another one is the axial field, which is sufficient to describe parity nonconservation effects in atoms and it is sensitive to the second internal degree of freedom of the Dirac spinor field. The polarization associated with this degree of freedom is a distinctive characteristic of the Dirac spinor field in spatially closed localized states. The short-distance behavior of the axi…
The leading, planar diagrams of the $1/N_c$ expansion and the usual string description suggest that quarks propagate on the boundary of a two-dimensional world surface. We restrict the quarks to the boundary of the world surface by giving them infinitely large mass on the interior of the surface and zero mass on its boundary and show that in this picture the QCD $θ$--vacua can be represented by the self-intersection number (or equivalently by the first Chern number of the normal bundle) of the surface.
We investigate on the plane the axial anomaly for euclidean Dirac fermions in the presence of a background Aharonov--Bohm gauge potential. The non perturbative analysis depends on the self--adjoint extensions of the Dirac operator and the result is shown to be influenced by the actual way of understanding the local axial current. The role of the quantum mechanical parameters involved in the expression for the axial anomaly is discussed. A derivation of the effective action by means of the stereographic projection is also considered.
We show that as a result of the axial anomaly, massless fermions at zero temperature define a relativistic quantum superfluid. The anomaly pole implies the existence of a gapless Chiral Density Wave (CDW), i.e. an axion-like acoustic mode of an irrotational and dissipationless Hamiltonian perfect fluid, that is a correlated fermion/anti-fermion pair excitation of the Fermi-Dirac sea. In $D\!=\!2$ dimensions the chiral superfluid effective action coincides with that of the Schwinger model as $e\rightarrow 0$, and the CDW acoustic mode is precisely the Schwinger boson. Since this identity holds …
We compute the axial anomaly for the Taub-NUT metric on $R^4$. We show that the axial anomaly for the generalized Taub-NUT metrics introduced by Iwai and Katayama is finite, although the Dirac operator is not Fredholm. We show that the essential spectrum of the Dirac operator is the whole real line.
Fermion-number fractionalization without breaking of time-reversal symmetry was recently demonstrated for a field theory in $(2+1)$-dimensional space and time that describes the couplings between massive Dirac fermions, a complex-valued Higgs field carrying an axial gauge charge of 2, and a U(1) axial gauge field. Charge fractionalization occurs whenever the Higgs field either supports vortices by itself, or when these vortices are accompanied by half-vortices in the axial gauge field. The fractional charge is computed by three different techniques. A formula for the fractional charge is given…
