We investigate the effect of a uniform background magnetic field on the chiral symmetry breaking in SU(2) Yang-Mills theory on the lattice. We observe that the chiral condensate grows linearly with the field strength B up to \sqrt{e B} = 3 GeV as predicted by chiral perturbation theory for full QCD. As the temperature increases the coefficient in front of the linear term gets smaller. In the magnetic field near-zero eigenmodes of the Dirac operator tend to have more regular structure with larger (compared to zero-field case) Hausdorff dimensionality. We suggest that the delocalization of near-zero eigenmodes plays a crucial role in the enhancement of the chiral symmetry breaking.
In the Abelian projection the Yang--Mills theory contains Abelian gauge fields (diagonal degrees of freedom) and the Abelian matter fields (off-diagonal degrees) described by a complicated action. The matter fields are essential for the breaking of the adjoint string. We obtain numerically the effective action of the Abelian gauge and the Abelian matter fields in quenched SU(2) QCD and show that the Abelian matter fields provide an essential contribution to the total action even in the infrared region. We also observe the breaking of an Abelian analog of the adjoint string using Abelian operat…
Using the lattice gauge field theory, we study the relation among the local chiral condensate, monopoles, and color magnetic fields in quantum chromodynamics (QCD). First, we investigate idealized Abelian gauge systems of 1) a static monopole-antimonopole pair and 2) a magnetic flux without monopoles, on a four-dimensional Euclidean lattice. In these systems, we calculate the local chiral condensate on quasi-massless fermions coupled to the Abelian gauge field, and find that the chiral condensate is localized in the vicinity of the magnetic field. Second, using SU(3) lattice QCD Monte Carlo ca…
We present clear numerical evidence for string breaking in three-dimensional SU(2) gauge theory with fundamental bosonic matter through a mixing analysis between Wilson loops and meson operators representing bound states of a static source and a dynamical scalar. The breaking scale is calculated in the continuum limit. In units of the lightest glueball we find $r_{\rm b} m_G\approx13.6$. The implications of our results for QCD are discussed.
We analyze the vacuum structure of SU(2) QCD with multiple massless adjoint representation fermions formulated on a small spatial $S^1 \times \R^3$. The absence of thermal fluctuations, and the fact that quantum fluctuations favoring the vacuum with unbroken center symmetry in a weakly coupled regime renders the interesting dynamics of these theories analytically calculable. Confinement, the area law behavior for large Wilson loops, and the generation of the mass gap in the gluonic sector are shown analytically. By abelian duality transformation, the long distance effective theory of QCD is ma…
I review recent progress achieved on the lattice in the quantitative comprehension of chiral symmetry breaking in QCD. Emphasis is given to the recent precise computations of the spectral density of the Dirac operator in the continuum limit, and of the topological susceptibility.
We analytically study the relation between quark confinement and spontaneous chiral-symmetry breaking in QCD. In terms of the Dirac eigenmodes, we derive some formulae for the Polyakov loop, its fluctuations, and the string tension from the Wilson loop. We also investigate the Polyakov loop in terms of the eigenmodes of the Wilson, the clover and the domain wall fermion kernels, respectively. For the confinement quantities, the low-lying Dirac/fermion eigenmodes are found to give negligible contribution, while they are essential for chiral symmetry breaking. These relations indicate no direct …
The breaking of chiral symmetry in holographic light-front QCD is encoded in its longitudinal dynamics with its chiral limit protected by the superconformal algebraic structure which governs its transverse dynamics. The scale in the longitudinal light-front Hamiltonian determines the confinement strength in this direction: It is also responsible for most of the light meson ground state mass consistent with the Gell-Mann-Oakes-Renner constraint. Longitudinal confinement and the breaking of chiral symmetry are found to be different manifestations of the same underlying dynamics as found in 't Ho…
We discuss spontaneous chiral symmetry breaking and quark bilinear condensate in large $N_c$ QCD. It has been known that the existence of $η'$ meson is implied in large $N_c$ QCD as pointed out by Witten and Veneziano. First, we show that the existence of $η'$ and Ward-Takahashi identities imply the existence of the Nambu-Goldstone bosons from chiral symmetry breaking $SU(N_f)_L \times SU(N_f)_R \to SU(N_f)_V$. Second, we show that a QCD inequality implies a nonzero lower bound on the quark bilinear condensate.
We present numerical evidence for the hypothesis that, in the planar limit, four dimensional Euclidean Yang-Mills theory on a finite symmetrical four-torus breaks chiral symmetry spontaneously when the length of the sides l is larger than a critical value l_c with a bilinear condensate whose value is independent of l. Therefore spontaneous symmetry breaking occurs at finite volume and infinite N_c reduction holds for the chiral condensate.
Calculations of the chiral condensate on the lattice using staggered fermions and the Lanczos algorithm are presented. Four gauge fields are considered: the quenched non-Abelian field, an Abelian projected field, and monopole and photon fields further decomposed from the Abelian field. Abelian projection is performed in maximal Abelian gauge and in Polyakov gauge. The results show that monopoles in maximal Abelian gauge largely reproduce the chiral condensate values of the full non-Abelian theory, in both SU(2) and SU(3) color.
Chiral symmetry breaking parameters are calculated in quenched SU(2) gauge theory and with Abelian gauge fields projected in maximal Abelian gauge and in field strength gauge. Maximal Abelian gauge projected fields lead to chiral condensate values which are quite similar to those of the full nonabelian theory. Pseudoscalar and vector meson correlators are calculated and found to be reproduced by the use of maximal Abelian gauge fields for small quark masses. In field-strength gauge, Abelian-projected fields give a chiral condensate which closely resembles the results of strongly coupled ($β<1$…
Lüscher's recent formulation of Abelian chiral gauge theories on the lattice, in the vacuum (or perturbative) sector in infinite volume, is reinterpreted in terms of the lattice covariant regularization. The gauge invariance of the effective action and the integrability of the gauge current in anomaly-free cases become transparent. The real part of the effective action is simply one-half that of the Dirac fermion and, when the Dirac operator behaves properly in the continuum limit, the imaginary part in this limit reproduces the $η$-invariant.
Calculations of the chiral condensate $\langle \barψ ψ\rangle$ on the lattice using staggered fermions and the Lanczos algorithm are presented. Three gauge fields are considered: the quenched non-Abelian field, the Abelian field projected in the maximal Abelian gauge, and the monopole field further decomposed from the Abelian field. The results show that the Abelian monopoles largely reproduce the chiral condensate values of the full non-Abelian theory, both in SU(2) and in SU(3).
We study the breaking of the string spanned between test charges in the three dimensional Abelian Higgs model with compact gauge field and fundamentally charged Higgs field at zero temperature. In agreement with current expectations we demonstrate that string breaking is associated with pairing of monopoles. However, the string breaking is not accompanied by an ordinary phase transition.
We report on an exploratory lattice study on the phenomenon of chiral instabilities in non-Abelian gauge theories at high temperature. It is based on a recently constructed anomalous Langevin-type effective theory of classical soft gauge fields in the presence of a chiral number density $n_5=n_{\rm R}-n_{\rm L}$. Evaluated in thermal equilibrium using classical lattice techniques it reveals that the fluctuating soft fields indeed exhibit a rapid energy increase at early times and we observe a clear dependence of the diffusion rate of topological charge (sphaleron rate) on the the initial $n_5$…
Defining chiral lattice gauge theories in the Ginsparg-Wilson formalism is complicated by the so-called fermion measure problem. It has been proven for the abelian theories that smooth well-behaved fermion measure exists if and only if the anomaly-free condition is granted, and the same was shown to hold in perturbative theories for non-abelian gauge groups, but the non-perturbative proof is absent. In this paper, we consider a simpler problem in 2-d and present a proof for the existence of smooth and gauge invariant fermion measure on the gauge field configuration space with zero field streng…
After an introduction in which we review the fundamental difficulty in constructing lattice chiral gauge theories, we summarize the analytic and numerical evidence that abelian lattice chiral gauge theories can be nonperturbatively constructed through the gauge-fixing approach. In addition, we indicate how we believe that the method may be extended to nonabelian chiral gauge theories.
Considering a self-interaction only of mirror fermions in the context of a lattice-regularized fermion field theory, we show that the system undergoes spontaneous breaking of chiral symmetry and mirror-fermion masses are generated. However, it is explicitly shown that there are no Goldstone bosons appearing together with this spontaneous symmetry breaking phenomenon, since Lorentz invariance, one of very general prerequisites of the Goldstone theorem, is violated. The result and its possible application are briefly discussed.
It is shown that U(1) chiral gauge theories with anomaly-free multiplets of Weyl fermions can be put on the lattice without breaking the gauge invariance or violating any other fundamental principle. The Ginsparg-Wilson relation plays a key role in this construction, which is non-perturbative and includes all topological sectors of the theory in finite volume. In particular, the cancellation of the gauge anomaly and the absence of global topological obstructions can be established on the basis of this relation and the lattice symmetries alone.
We propose a method for implementing the large-Nc, large-Nf limit of QCD at the effective Lagrangian level. Depending on the value of the ratio Nf/Nc, different patterns of chiral symmetry breaking can arise, leading in particular to different behaviors of the eta-prime mass in the combined large-N limit.
Chiral symmetry breaking is calculated as a function of cooling in quenched lattice QCD. A non-zero signal is found for the chiral condensate beyond one hundred cooling steps, suggesting that there is chiral symmetry breaking associated with instantons. Quantitatively, the chiral condensate in cooled gauge field configurations is small compared to the value without cooling.
We investigate the effects of several Abelian external fluxes $ϕ$, on the Euclidean Dirac spectrum of light quarks in QCD with two colors. Our results provide for a novel way of assessing the pion weak decay constant directly from spectral fluctuations, and discriminating a flux-rich from a flux-poor vacuum, using QCD lattice simulations.
We report on a mean-field study of spontaneous breaking of chiral symmetry for Dirac fermions with contact interactions in the presence of chiral imbalance, which is modelled by nonzero chiral chemical potential. We point out that chiral imbalance lowers the vacuum energy of Dirac fermions, which leads to the increase of the renormalized chiral chemical potential upon chiral symmetry breaking. The critical coupling strength for the transition to the broken phase is slightly lowered as the chiral chemical potential is increased, and the transition itself becomes milder. Furthermore, we study th…
The CP symmetry is not manifestly implemented for the local and doubler-free Ginsparg-Wilson operator in lattice chiral gauge theory. We precisely identify where the effects of this CP breaking appear.
We propose a novel lattice calculation of spontaneous chiral symmetry breaking in QED model with 2+1 dimensional fermion brane. Considering the relativistic action with gauge symmetry we rigorously carry out path integral in Monte-Carlo simulation with Fermi-velocity relevant to effective coupling constant. We numerically show the evidence of spontaneous chiral symmetry breaking in strong coupling region with chiral condensate, low-lying mode distribution and Nambu-Goldstone boson spectrum in bare Fermi-velocty $v=0.1$. This is a feasible study to investigate the phase structure of Graphene.
A key question to QCD is what mechanism generates the hadron mass in the light quark sector, where both confinement and chiral symmetry breaking are in the game. Are confinement and chiral symmetry breaking in the vacuum uniquely interconnected? Can hadrons survive chiral symmetry restoration? If yes, what happens with their mass and what symmetries beyond the chiral symmetry are there? We review our recent insights. In particular, in a dynamical lattice simulation we artificially restore chiral symmetry by removing the low-lying Dirac modes of the valence quark propagators, which is a well de…
We review important aspects of QCD in the continuum and on the lattice and take a look at the fate of its symmetries with an emphasis on chiral symmetry breaking on the lattice.
In this paper we study how dynamical chiral symmetry breaking is affected by nonzero chiral chemical potential in Dirac semimetals. To perform this study we applied lattice quantum Monte Carlo simulations of Dirac semimetals. Within lattice simulation we calculated the chiral condensate for various fermion masses, the chiral chemical potentials and effective coupling constants. For all parameters under consideration we have found that the chiral condensate is enhanced by chiral chemical potential. Thus our results confirms that in Dirac semimetals the chiral chemical potential plays a role of …
One of the most challenging issues in QCD is the investigation of spontaneous chiral-symmetry breaking, which is characterized by the non-vanishing chiral condensate when the bare fermion mass is zero. In standard methods, one has to perform expensive lattice simulations at multiple bare quark masses, and employ some modeled function to extrapolate the data to the chiral limit. This paper applies the probability distribution function method to computing the chiral condensate in lattice QCD with massless dynamical quarks, without any ambiguous mass extrapolation. The results for staggered quark…
We investigate which momentum components of gluons induce color confinement and spontaneous chiral symmetry breaking in lattice QCD. For this purpose, we formulate a lattice framework to introduce the momentum cutoff of the gluon field. Using this framework, we calculate the quark-antiquark potential, the color flux tube, the chiral condensate, and the Dirac spectrum. Our results suggest that confinement and chiral symmetry breaking are induced by somehow different momentum components of gluons.
We study the dynamical breaking of chiral symmetry in the 3-d Thirring model for a small number of fermion species. The critical point is identified by fitting lattice data to an equation of state. The spectrum of the theory is studied to confirm the phase structure of the model.
We examine the feasibility of the proposition that there is a temperature range T$_c$ < T < T$_{ch}$ in N$_f$=0 QCD, where real Polyakov line (deconfined) vacuum exhibits valence spontaneous chiral symetry breaking and dynamical chiral polarization of Dirac eigenmodes. Detailed finite-volume analysis convincingly demonstrates the existence of such phase at fixed cutoff (a=0.085 fm). Moreover, it is found that this behavior also takes place closer to the continuum limit (a=0.060 fm) without qualitative change in its properties.
A chiral fermion action allows one to do very clean studies of chiral symmetry breaking in QCD. I will briefly describe how to compute with the overlap action (relatively) cheaply, and then turn to physics: Low modes of the Dirac operator show a ``lumped'' chiral density which peaks at the locations of instantons and anti-instantons. These modes dominate correlation functions at small quark mass in many channels. The picture qualitatively (and in some cases quantitatively) resembles an instanton liquid model.
We calculate the spectral density of the Dirac operator over an ensemble of configurations composed of overlapping instantons and anti-instantons. We find evidence that the spectral density diverges in the limit of small eigenvalues. This indicates the breaking of chiral symmetry and also provides evidence that quenched QCD may be pathological in nature.
We investigate spontaneous chiral symmetry breaking in SU(N) gauge theories at large N using overlap fermions. The exact zero modes and the low-lying modes of the Dirac operator provide the tools to gain insight into the interplay between chiral symmetry breaking and topology. We find that topology indeed drives chiral symmetry breaking at N=3 as well as at large N. By comparing the results on various volumes and at different lattice spacings we are able to show that our conclusions are not affected by finite volume effects and also hold in the continuum limit. We then address the question whe…
In the quenched approximation we use the abelian and monopole fields from abelian projection in SU(2) lattice gauge theory to numerically compute the value of the chiral condensate. The condensate calculated using abelian projection is observed to vanish at the same critical temperature as the full SU(2) theory predicts.
We analyse - within the hamiltonian formalism with staggered fermions - the patterns of chiral symmetry breaking for the strongly coupled Schwinger and $U({\cal N}_c)$-color `t Hooft models with one and two flavor of fermions. Using the correspondence between these strongly coupled gauge models and antiferromagnetic spin chains, we provide a rather intuitive picture of their ground states, elucidate their patterns of chiral symmetry breaking, and compute the pertinent chiral condensates. Our analysis evidences an intriguing relationship between the values of the lattice chiral condensates of t…
According to the Nielsen-Ninomiya No-Go theorem, the doubling of fermions on the lattice cannot be suppressed in a chiral theory. Whereas Wilson and staggered fermions suppress doublers with explicit breaking of chiral symmetry, the random lattice does so by spontaneous chiral symmetry breaking even in the free theory. I present results for meson masses, the chiral condensate and fermionic eigenvalues from simulations of quenched QCD on random lattices in four dimensions, focusing on chiral symmetry breaking.
Partially quenched chiral perturbation theory assumes that valence quarks propagating on gauge configurations prepared with sea quarks of different masses will form a chiral condensate as the valence quark mass goes to zero. I present a counterexample involving non-degenerate sea quarks where the valence condensate does not form.
Using matrix product states, we explore numerically the phenomenology of string breaking in a non-Abelian lattice gauge theory, namely 1+1 dimensional SU(2). The technique allows us to study the static potential between external heavy charges, as traditionally explored by Monte Carlo simulations, but also to simulate the real-time dynamics of both static and dynamical fermions, as the latter are fully included in the formalism. We propose a number of observables that are sensitive to the presence or breaking of the flux string, and use them to detect and characterize the phenomenon in each of …
