We study $SU(N_c)$ lattice gauge theories with $N_f$ flavors of massless staggered fermions in the presence of quark chemical potential $μ$. A recent exact result that in the strong coupling limit (vanishing inverse gauge coupling $β$) and for sufficiently large $μ$ the theory is in a chiral symmetric phase is here extended into the finite gauge coupling region. A cluster expansion combining a fermion spacelike hopping expansion and a strong coupling plaquette expansion is shown to converge for sufficiently large $μ$ and small $β$ at any temperature $T$. All expectations of chirally non-invariant local fermion operators vanish identically, or, equivalently, their correlations cluster exponentially within the expansion implying absence of spontaneous chiral symmetry breaking. The resulting phase at low $T$ may be described as a "quarkyonic" matter phase. Some implications for the phase diagram of $SU(N_c)$ theories are discussed.
It is argued on general ground and demonstrated in the particular example of the Chiral Schwinger Model that there is nothing wrong with apparently anomalous chiral gauge theory. If quantised correctly, there should be no gauge anomaly and chiral gauge theory should be renormalisable and unitary, even in higher dimensions and with non-abelian gauge groups. Furthermore, mass terms for gauge bosons and chiral fermions can be generated without spoiling the gauge invariance.
A recently proposed formulation of chiral lattice gauge theories is reviewed, in which the locality and gauge invariance of the theory can be preserved if the fermion representation of the gauge group is anomaly-free.
After a brief introduction to the overlap two examples relating to topological properties of chiral fermion systems in interaction with gauge fields are presented: It is shown how the overlap preserves the continuum structure of exact fermionic zero modes in gauge backgrounds that are instanton-like and why chiral anomalies are inevitable.
We analyze the dynamics of an $SU_L(2)\otimes U_R(1)$ chiral gauge theory on a lattice with a large multifermion coupling $1\ll g_2 < \infty$. It is shown that no spontaneous symmetry breaking occurs; the ``spectator'' fermion $ψ_R(x)$ is a free mode; doublers are decoupled as massive Dirac fermions consistently with the chiral gauge symmetry. Whether right-handed three-fermion states disappear and chiral fermions emerge in the low-energy limit are discussed. Provided right-handed three-fermion states disappear, we discuss the chiral gauge coupling, Ward identities, the gauge anomaly and anoma…
We unveil the dynamics of four dimensional chiral gauge-Yukawa theories featuring several scalar degrees of freedom transforming according to distinct representations of the underlying gauge group. We consider generalized Georgi-Glashow and Bars-Yankielowicz theories. We determine, to the maximum known order in perturbation theory, the phase diagram of these theories and further disentangle their ultraviolet asymptotic nature according to whether they are asymptotically free or safe. We therefore extend the number of theories that are known to be fundamental in the Wilsonian sense to the case …
We describe how an SU(N) chiral gauge theory can be put on the lattice using non-perturbative gauge fixing. In particular, we explain how the Gribov problem is dealt with. Our construction is local, avoids doublers, and weak-coupling perturbation theory applies at the critical point which defines the continuum limit of our lattice chiral gauge theory.
We propose a nonperturbative formulation of chiral gauge theories. The method involves a `pre-regulation' of the gauge fields, which may be implemented on a lattice, followed by a computation of the chiral fermion determinant in the form of a functional integral which is regularized in the continuum. Our result for the chiral determinant is expressed in terms of the vector-like Dirac operator and hence can be realized in lattice simulations. We investigate the local and global anomalies within our regularization scheme. We also compare our result for the chiral determinant to previous exact $ζ…
The overlap formula proposed by Narayanan and Neuberger in chiral gauge theories is examined. The free chiral and Dirac Green's functions are constructed in this formalism. Four dimensional anomalies are calculated and the usual anomaly cancellation for one standard family of quarks and leptons is verified.
I review the substantial progress which has been made recently with the non-perturbative construction of chiral gauge theories on the lattice. In particular, I discuss three different approaches: a gauge invariant method using fermions which satisfy the Ginsparg-Wilson relation, and two gauge non-invariant methods, one using different cutoffs for the fermions and the gauge fields, and one using gauge fixing. Open problems within all three approaches are addressed.
We discuss the ultraviolet (UV) to infrared (IR) evolution of asymptotically free chiral gauge theories. Various types of IR behavior are considered, including confinement without spontaneous chiral symmetry breaking, and formation of bilinear fermion condensates that preserve or dynamically break the chiral gauge invariance. We compare different schemes for the study of this evolution and the resultant IR structure of the theories, including a conjectured inequality based on the counting of degrees of freedom. We note some new patterns of UV to IR evolution.
Path integration over Euclidean chiral fermions is replaced by the quantum mechanics of an auxiliary system of non--interacting fermions. Our construction avoids the no--go theorem and faithfully maintains all the known important features of chiral fermions, including the violation of some perturbative conservation laws by gauge field configurations of non--trivial topology.
We study the gauge-fixing approach to the construction of lattice chiral gauge theories in one-loop weak-coupling perturbation theory. We show how infrared properties of the gauge degrees of freedom determine the nature of the continuous phase transition at which we take the continuum limit. The fermion self-energy and the vacuum polarization are calculated, and confirm that, in the abelian case, this approach can be used to put chiral gauge theories on the lattice in four dimensions. We comment on the generalization to the nonabelian case.
After many years of investigations, our understanding of the dynamics of strongly-coupled chiral gauge theories is still quite unsatisfactory today. Conventional wisdom about strongly-coupled gauge theories, successfully applied to QCD, is not always as useful in chiral gauge theories. Recently some new ideas and techniques have been developed, which involve concepts of generalized symmetries, of gauging a discrete center symmetry, and of generalizing the 't Hooft anomaly matching constraints to include certain mixed symmetries. This new development has been applied to chiral gauge theories, l…
In the framework of perturbation theory, it is possible to put chiral gauge theories on the lattice without violating the gauge symmetry or other fundamental principles, provided the fermion representation of the gauge group is anomaly-free. The basic elements of this construction (which starts from the Ginsparg-Wilson relation) are briefly recalled and the exact cancellation of the gauge anomaly, at any fixed value of the lattice spacing and for any compact gauge group, is then proved rigorously through a recursive procedure.
Analyzing an $SU_L(2)$-chiral gauge theory with external multifermion couplings, we find a possible scaling region where doublers decouple by acquiring chiral-invariant masses and $ψ_R$ is free mode owing to the $ψ_R$-shift-symmetry, the chiral continuum theory of $ψ^i_L$ can be defined. This is not in agreement with the general belief of the failure of theories so constructed .
Dynamics and symmetry realization in various chiral gauge theories in four dimensions are investigated, generalizing a recent work by M. Shifman and the present authors, by relying on the standard 't Hooft anomaly matching conditions and on some other general ideas. These requirements are so strong that the dynamics of the systems are severely constrained. Color-flavor or color-flavor-flavor locking, dynamical Abelianization, and combinations of these, are powerful ideas which often leads to solutions of the anomaly matching conditions. Moreover, a conjecture is made on generation of a mass hi…
We study two well-known $SU(N)$ chiral gauge theories with fermions in the symmetric, anti-symmetric and fundamental representations. We give a detailed description of the global symmetry, including various discrete quotients. Recent work argues that these theories exhibit a subtle mod 2 anomaly, ruling out certain phases in which the theories confine without breaking their global symmetry, leaving a gapless composite fermion in the infra-red. We point out that no such anomaly exists. We further exhibit an explicit path to the gapless fermion phase, showing that there is no kinematic obstructi…
We compute the low lying spectrum of the overlap Dirac operator in the deconfined phase of finite-temperature quenched gauge theory. It suggests the existence of a chiral condensate which we confirm with a direct stochastic estimate. We show that the part of the spectrum responsible for the chiral condensate can be understood as arising from a dilute gas of instantons and anti-instantons.
We study a concrete lattice regularization of a U(1) chiral gauge theory. We use Wilson fermions, and include a Lorentz gauge-fixing term and a gauge-boson mass counterterm. For a reduced version of the model, in which the gauge fields are constrained to the trivial orbit, we show that there are no species doublers, and that the fermion spectrum contains only the desired states in the continuum limit, namely charged left-handed (LH) fermions and neutral right-handed (RH) fermions.
The lattice formulation of gauge theories in a renormalizable gauge is discussed. The formulation invokes a new phase diagram, and it may allow for a lattice definition of Chiral Gauge Theories.
Noncompact chiral abelian gauge theories are defined on the lattice using the overlap formalism. The main definitions are presented, the role of anomaly cancelation is discussed, and the triviality issue in four dimensions is explained.
We show that, the lattice regularization of chiral gauge theories proposed by Kaplan, when applied to a (2+1)-dimensional domain wall, produces a (1+1)-dimensional theory at low energy even if gauge anomaly produced by chiral fermions does not cancel. But the corresponding statement is not true in higher dimensions.
We study $N=1$ SUSY gauge theories in four dimensions with gauge group $Spin(7)$ and $N_f$ flavors of quarks in the spinorial representation. We find that in the range $6< N_f < 15$, this theory has a long distance description in terms of an $SU(N_f-4)$ gauge theory with a symmetric tensor and $N_f$ antifundamentals. As a spin-off, we obtain by deforming along a flat direction a dual description of the theories based on the exceptional gauge group $G_2$ with $N_f$ fundamental flavors of quarks.
We analyze patterns of dynamical symmetry breaking in strongly coupled chiral gauge theories with direct-product gauge groups $G$. If the gauge coupling for a factor group $G_i \subset G$ becomes sufficiently strong, it can produce bilinear fermion condensates that break the $G_i$ symmetry itself and/or break other gauge symmetries $G_j \subset G$. Our comparative study of a number of strongly coupled direct-product chiral gauge theories elucidates how the patterns of symmetry breaking depend on the structure of $G$ and on the relative sizes of the gauge couplings corresponding to factor group…
We investigate a U(1) lattice chiral gauge theory with domain wall fermions and compact gauge fixing. In the reduced model limit, our perturbative and numerical investigations show that there exist no extra mirror chiral modes. The longitudinal gauge degrees of freedom have no effect on the free domain wall fermion spectrum consisting of opposite chiral modes at the domain wall and at the anti-domain wall which have an exponentially damped overlap.
The axion particle may or may not exist, but the axion field can be used, as shown here, in an explicitly local formulation of a chiral U(1) gauge theory with both classical and quantum gauge invariance. Nonabelian analogues of axion fields, which have recently been introduced, can be used, together with their special symmetries, in a similar construction of nonabelian chiral gauge theories. As in known cases, the gauge symmetry is broken and the gauge boson acquires a mass by swallowing the axions which are therefore not physical in this construction.
We show that the 3450 U(1) chiral fermion theory can appear as the low energy effective field theory of a 1+1D local lattice model, with an on-site U(1) symmetry and finite-range interactions. The on-site U(1) symmetry means that the U(1) symmetry can be gauged (gaugeable for both background probe and dynamical fields), which leads to a non-perturbative definition of chiral gauge theory --- a chiral fermion theory coupled to U(1) gauge theory. Our construction can be generalized to regularize any U(1)-anomaly-free 1+1D gauged chiral fermion theory with a zero chiral central charge (thus no gra…
We propose a novel gauge-invariant regularization for the perturbative chiral gauge theory.Our method consists of the two ingredients: use of the domain-wall fermion to describe a chiral fermion with Pauli-Villars regulators and application of the di- mensional regularization only to the gauge field. This regularization is implemented in the Lagrangian level, unlike other gauge-invariant regularizations (eg. the covariant regularizations). We show that the Abelian (fermion number) anomaly is reproduced correctly in this formulation. We also show that once we add the counter terms to the full t…
In this more technical part we give additional details on the gauge-fixing approach presented in hep-lat/9709113. We also explain how the gauge-fixing approach evades the Nielsen-Ninomiya no-go theorem.
In this letter we uncover a new facet of chiral symmetry and the implications of its breaking in some theories. By generalizing the concept of chiral symmetry, tensor theories naturally arise. This novel approach adds to the known uses of tensor theories (quantum gravity, holography, entanglement,etc.) a possible link to QCD phenomena.
Motivated by the search for possible CP violating terms in the trace of the energy-momentum tensor in theories coupled to gravity we revisit the problem of trace anomalies in chiral theories. We recalculate the latter and ascertain that in the trace of the energy-momentum tensor of theories with chiral fermions at one-loop the Pontryagin density appears with an imaginary coefficient. We argue that this may break unitarity, in which case the trace anomaly has to be used as a selective criterion for theories, analogous to the chiral anomalies in gauge theories. We analyze some remarkable consequ…
We consider the chiral ring of the pure N=1 supersymmetric gauge theory with SU(N) gauge group and show that the classical relation S^{N^2}=0 is modified to the exact quantum relation (S^N-Λ^{3N})^N=0.
For two decades it was believed that chiral symmetries cannot be realized in lattice field theory but this has changed now. Highlights of these new developments will be presented with emphasis on the mathematical structure of the so called ``overlap''.
Gauge field flow for chiral gauge theories on a disk boundary
Peking University（北京大学）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；Iowa State University（爱荷华州立大学）
A formulation of abelian and non-abelian chiral gauge theories is presented together with arguments for the unitarity and renormalisability in four dimensions. IASSNS-HEP-94/70, UM-P-94/96, and RCHEP-94/26.
We report on recent progress with the definition of lattice chiral gauge theories, using a lattice action that includes a discretized Lorentz gauge-fixing term. This gauge-fixing term has a unique global minimum, and allows us to use perturbation theory in order to study the influence of the gauge degrees of freedom on the fermions. For the abelian case, we find, both in perturbation theory and numerically, that the fermions remain chiral, and that there are no doublers. More details on our approach, including a discussion of how we evade the Nielsen-Ninomiya theorem, can be found in hep-lat/9…
We show that noncommutative U(r) gauge theories with a chiral fermion in the adjoint representation can be constructed on the lattice with manifest star-gauge invariance in arbitrary even dimensions. Chiral fermions are implemented using a Dirac operator which satisfies the Ginsparg-Wilson relation. A gauge-invariant integration measure for the fermion fields can be given explicitly, which simplifies the construction as compared with lattice chiral gauge theories in ordinary (commutative) space-time. Our construction includes the cases where continuum calculations yield a gauge anomaly. This r…
We consider the dynamics of a strongly coupled SU(N) chiral gauge theory. By using its large-N equivalence with N=1 super-Yang-Mills theory we find the vacuum structure of the former. We also consider its finite-N dynamics.
The pattern of spontaneous chiral symmetry breaking (CSB) in confining background fields is analyzed. It is explicitly demonstrated how to get the inverse square root large proper time asymptotic of the operator product expansion which is needed for CSB.
We investigate a proposal for the construction of models with chiral fermions on the lattice using staggered fermions. In this approach the gauge invariance is broken by the coupling of the staggered fermions to the gauge fields. Motivated by previous results in the non-gauge invariant massive Yang-Mills theory and certain gauge-fermion models we aim at a dynamical restoration of the gauge invariance in the full quantum model. If the gauge symmetry breaking is not too severe, this procedure could lead in the continuum limit to the desired gauge invariant chiral gauge theory. This scenario is v…
We discuss how to construct anomaly-free chiral gauge theories on the lattice with exact gauge invariance in the framework of domain wall fermion. Chiral gauge coupling is realized by introducing a five-dimensional gauge field which interpolates between two different four-dimensional gauge fields at boundaries. The five-dimensional dependence is compensated by a local and gauge-invariant counter term. The cohomology problem to obtain the counter term is formulated in 5+1 dimensional space, using the Chern-Simons current induced from the five-dimensional Wilson fermion. We clarify the connectio…
We formulate a manifestly supersymmetric gauge covariant regularization of supersymmetric chiral gauge theories. In our scheme, the effective action in the superfield background field method above one-loop is always supersymmetric and gauge invariant. The gauge anomaly has a covariant form and can emerge only in one-loop diagrams with all the external lines being the background gauge superfield. We also present several illustrative applications in the one-loop approximation: the self-energy part of the chiral multiplet and of the gauge multiplet; the super-chiral anomaly and the superconformal…
Strong coupling dynamics of Yang--Mills theories with chiral fermion content remained largely elusive despite much effort over the years. In this work, we propose a dynamical framework in which we can address non-perturbative properties of chiral, non-supersymmetric gauge theories, in particular, chiral quiver theories on $S_1 \times R_3 $. Double-trace deformations are used to stabilize the center-symmetric vacuum. This allows one to smoothly connect small-$r(S_1)$ to large-$r(S_1)$ physics ($R_4$ is the limiting case) where the double-trace deformations are switched off. In particular, occur…
In this letter we show how the covariant anomaly emerges in the overlap scheme. We also prove that the overlap scheme correctly reproduces the anomaly in the flavour currents such as $j^5_μ$ in vector like theories like QCD.
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 demonstrate that $SO(N_{c})$ gauge theories with matter fields in the vector representation confine due to monopole condensation and break the $SU(N_{F})$ chiral symmetry to $SO(N_{F})$ via the quark bilinear. Our results are obtained by perturbing the ${\cal N}=1$ supersymmetric theory with anomaly-mediated supersymmetry breaking.
We study dynanics of $SU(N-4)$ gauge theories with fermions in rank-2 symmetric tensor and $N$ anti-fundamental representations, by perturbing supersymmetric theories with anomaly-mediated supersymmetry breaking. We find the $SU(N)\times U(1)$ global symmetry is dynamically broken to $SO(N)$ for $N\geq 17$, a different result from conjectures in the literature. For $N<17$, theories flow to infrared fixed points.
We derive the finite one-loop counterterm required to restore the Ward Identities broken by the regularization scheme in chiral gauge theories. Our result is an analytic expression applicable to a wide class of regularizations satisfying a few general properties. We adopt the background field method, which ensures background gauge invariance in the quantized theory, and focus on renormalizable chiral theories with arbitrary gauge group and fermions in general representations. Our approach can be extended to theories involving scalars, such as the Standard Model, or to non-renormalizable theori…
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.
We present scheme-independent calculations of the derivative of the beta function, denoted $β'_{IR}$, at a conformally invariant infrared (IR) fixed point, in several asymptotically free chiral gauge theories, namely SO($4k+2$) with $2 \le k \le 4$ with respective numbers $N_f$ of fermions in the spinor representation, and E$_6$ with fermions in the fundamental representation.
Chirality transfer between fermions and gauge fields plays a crucial role for understanding the dynamics of anomalous transport phenomena such as the Chiral Magnetic Effect. In this proceeding we present a first principles study of these processes based on classical-statistical real-time lattice simulations of strongly coupled QED $(e^2N_f=64)$. Our simulations demonstrate that a chirality imbalance in the fermion sector triggers chiral plasma instabilities in the gauge field sector, which ultimately lead to the generation of long range helical magnetic fields via a self-similar turbulent casc…
Monopole operators play a central role in 3 dimensional supersymmetric dualities: a careful understanding of their spectrum is necessary to match chiral operators on either sides of a conjectured duality. In Chern-Simons theories ($k\neq0$), monopole operators acquire an electric charge, thus they need to be "dressed" by chiral matter superfields to be made gauge-invariant. Here we present strong evidence that "dressed" monopoles appear in $SU(N)$ chiral theories even for $k=0$ because of mixed CS terms generated along certain Coulomb branch directions. Our analysis is based on the dimensional…
The overlap formula for the chiral determinant is presented and the realization of gauge anomalies and gauge field toplogy in this context is discussed. The ability of the overlap formalism to deal with supersymmetric theories and Majorana-Weyl fermions is outlined. Two applications of the overlap formalism are discussed in some detail. One application is the computation of a fermion number violating process in a two dimensional U(1) chiral gauge theory. The second application is a measurement of the probability ditribution of the index of the chiral Dirac operator in four dimensional pure SU(…
Chiral U(1) anomaly is derived with mathematical rigor for a Euclidean fermion coupled to a smooth external U(1) gauge field on an even dimensional torus as a continuum limit of lattice regularized fermion field theory with the Wilson term in the action. The present work rigorously proves for the first time that the Wilson term correctly reproduces the chiral anomaly.
We investigate the zero temperature chiral phase transition in an SU(N) gauge theory as the number of fermions $N_f$ is varied. We argue that there exists a critical number of fermions $N_f^c$, above which there is no chiral symmetry breaking or confinement, and below which both chiral symmetry breaking and confinement set in. We estimate $N_f^c$ and discuss the nature of the phase transition.
Considering as an example a simple lattice ansatz for the chiral fermion determinant, we demonstrate that even very mild violation of gauge invariance by the determinant at finite lattice spacing leads to the need for another scale in the full gauge theory. This new scale is much grater than the lattice spacing and is associated with the gauge variables.
