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The $SU(3)\otimes SU(2) \otimes U(1)$ standard model maps smoothly onto a conventional lattice gauge formulation, including the parity violation of the weak interactions. The formulation makes use of the pseudo-reality of the weak group and requires the inclusion a full generation of both leptons and quarks. As in continuum discussions, chiral eigenstates of the Dirac operator generate known anomalies, although with rough gauge configurations these are no longer exact zero modes of the Dirac operator., Comment: 12 pages, 2 figures. Revisions include numerous added references and brief comments on comparisons with other methods

QCD, the theory of the strong interactions, involves quarks interacting with non-Abelian gluon fields. This theory has many features that are difficult to impossible to see in conventional diagrammatic perturbation theory. This includes quark confinement, mass generation, and chiral symmetry breaking. This paper is a colloquium level overview of the framework for understanding how these effects come about., Comment: 21 pages, 12 figures. Revision adds new section and various edits

A central idea in general relativity is that physics should not depend on the space-time coordinates in use \cite{einstein}. But the qualitative description of various phenomena can appear superficially quite different. Here we consider falling into a classical black hole using four distinct but equivalent metrics. First is the Schwarzchild case, with extreme time dilation at the horizon. Second, rescaling the dilation allows falling into the hole in finite proper time. Third, time and space are rescaled into a Penrose motivated picture where light trajectories all have unit slope. Fourth, a white hole variation of the second metric allows passage out through the horizon, with reentry forbidden., Comment: 9 pages, 4 figures. Honorable mention in the Gravity Research Foundation 2020 Awards for Essays on Gravitation

Among the parameters of QCD is one that results in CP violation when non-vanishing. This is closely related to possible quark mass terms. It is conventionally interpreted in terms of gauge field topology or alternatively in terms of phases in the quark masses. There is no experimental evidence for this parameter having a non-zero value, a puzzle for theories involving unification., Comment: 11 pages, 9 figures, contribution to conference XIII Quark Confinement and the Hadron Spectrum - Confinement2018

Using the Sigma model to explore the lowest order pseudo-scalar spectrum with SU(3) breaking, this talk considers an additional exact "taste" symmetry to mimic species doubling. Rooting replicas of a valid approach such as Wilson fermions reproduces the desired physical spectrum. In contrast, extra symmetries of the rooted staggered approach leave spurious states and a flavor dependent taste multiplicity., Comment: 7 pages, 2 figures. Contribution to Lattice2017, 18-24 June, Granada, Spain

Quantized Yang-Mills fields lie at the heart of our understanding of the strong nuclear force. To understand the theory at low energies, we must work in the strong coupling regime. The primary technique for this is the lattice. While basically an ultraviolet regulator, the lattice avoids the use of a perturbative expansion. I discuss the historical circumstances that drove us to this approach, which has had immense success, convincingly demonstrating quark confinement and obtaining crucial properties of the strong interactions from first principles., Comment: 13 pages, 3 figures. Contribution to the Conference on 60 Years of Yang-Mills Gauge Field Theories, 25-28 May 2015. arXiv admin note: substantial text overlap with arXiv:hep-lat/0406007, arXiv:hep-lat/0306024, arXiv:hep-lat/0010047

We investigate the 2-flavor QCD with non-degenerate quark masses at low-energy, using the chiral perturbation theory including the $\eta$ meson and anomaly effects. For the fixed $m_d\not=0$, the neutral pion becomes massless at two values of $m_u$, between which a spontaneously CP broken phase appears with the neutral pion condensation. We then show that the topological susceptibility diverges at these two critical points. We also consider the case of $m_u=m_d$ but $\theta=\pi$, equivalently $m_u= - m_d$ with $\theta = 0$ by the chiral rotation and show that the CP symmetry is spontaneously broken by the $\eta$ condensation at small $m$. Around $m=0$, three pions become Nambu-Goldstone modes, showing non-standard behavior that $m_\pi^2 = O(m^2)$, which, however, is consistent with the chiral Ward-Takahashi identities.}, Comment: 7 pages, 2 figures, talk presented at the 32nd International Symposium on Lattice Field Theory, 23-28 June, 2014, Columbia University New York, NY

Topologically non-trivial gauge field configurations are an interesting aspect of non-abelian gauge theories. These become particularly important upon quantizing the theory, especially through their effect on the pseudo-scalar spectrum. These effects are closely tied to chiral anomalies and the possibility of CP violation in the strong interactions., Comment: 11 pages, 7 figures. Contribution to Quark Confinement and the Hadron Spectrum XI, 8-12 September 2014, Saint-Petersburg State University, Russia

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., Comment: 4 pages, 1 figure. Contribution to the 32nd International Symposium on Lattice Field Theory 23-28 June, 2014, Columbia University

We investigate some aspects of 2-flavor QCD with $m_u\not= m_d$ at low-energy, using the leading order chiral perturbation theory including anomaly effects. While nothing special happens at $m_u=0$ for the fixed $m_d\not=0$, the neutral pion mass becomes zero at two critical values of $m_u$, between which the neutral pion field condenses, leading to a spontaneously CP broken phase, the so-called Dashen phase. We also show that the "topological susceptibility" in the ChPT diverges at these two critical points. We briefly discuss a possibility that $m_u=0$ can be defined by the vanishing the "topological susceptibility. We finally analyze the case of $m_u=m_d=m$ with $\theta=\pi$, which is equivalent to $m_u= - m_d=-m$ with $\theta = 0$ by the chiral rotation. In this case, the $\eta$ condensation occurs at small $m$, violating the CP symmetry spontaneously. Deep in the $\eta$ condensation phase, three pions become Nambu-Goldstone bosons, but they show unorthodox behavior at small $m$ that $m_\pi^2 = O(m^2)$, which, however, is shown to be consistent with the chiral Ward-Takahashi identities., Comment: 10 pages, 2 figures

Lattice gauge theory and chiral perturbation theory are among the primary tools for understanding non-perturbative aspects of QCD. I review several subtle and sometimes controversial issues that arise when combining these techniques. Among these are one failure of partially quenched chiral perturbation theory when the valence quarks become lighter than the average sea quark mass and a potential ambiguity in comparisons of perturbative and lattice properties of non-degenerate quarks., Comment: 11 pages, 5 figures. Contribution to the conference "From quarks and gluons to hadronic matter: A bridge too far?" 2-6 Sept. 2013. European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), Villazzano, Trento (Italy)}

Quantum mechanics and relativity in the continuum imply the well known spin-statistics connection. However for particles hopping on a lattice, there is no such constraint. If a lattice model yields a relativistic field theory in a continuum limit, this constraint must "emerge" for physical excitations. We discuss a few models where a spin-less fermion hopping on a lattice gives excitations which satisfy the continuum Dirac equation. This includes such well known systems such as graphene and staggered fermions., Comment: 14 pages, 3 figures. Revision has the four dimensional discussion substantially rewritten

The CP violating Dashen phase in QCD is predicted by chiral perturbation theory to occur when the up-down quark mass difference becomes sufficiently large at fixed down-quark mass. Before reaching this phase, all physical hadronic masses and scattering amplitudes are expected to behave smoothly with the up-quark mass, even as this mass passes through zero. In Euclidean space, the topological susceptibility of the gauge fields is positive at positive quark masses but diverges to negative infinity as the Dashen phase is approached. A zero in this susceptibility provides a tentative signal for the point where the mass of the up quark vanishes. I discuss potential ambiguities with this determination., Comment: 14 pages, 2 figures. Revised to match version to appear in Annals of Physics

Considered as a function of the quark mases, two-flavor QCD depends on three parameters, including one that is CP violating. As the masses vary to unphysical values, regions of both first- and second-order phase transitions are expected. For non-degenerate quarks, non-perturbative effects leave individual quark mass ratios with a renormalization scheme dependence. This complicates matching lattice results with perturbative schemes and clarifies the tautology with attacking the strong CP problem via a vanishing up quark mass., Comment: 10 pages, 6 figures. Contribution to the International Workshop on QCD Green's Functions, Confinement and Phenomenology, 5-9 September 2011, Trento, Italy

We present a theoretical foundation for the index theorem in naive and minimally doubled lattice fermions by studying the spectral flow of a Hermitean version of Dirac operators. We utilize the point splitting method to implement flavored mass terms, which play an important role in constructing proper Hermitean operators. We show the spectral flow correctly detects the index of the would-be zero modes which is determined by gauge field topology and the number of species doublers. Using the flavored mass terms, we present new types of overlap fermions from the naive fermion kernels, with a number of flavors that depends on the choice of the mass terms., Comment: 7 pages, 3 figures, presented at 29th International Symposium on Lattice Field Theory, Lattice2011, July 10-16, 2011, Squaw Valley, California, USA

We investigate the parity-broken phase (Aoki phase) for staggered-Wilson fermions by using the Gross-Neveu model and the strong-coupling lattice QCD. In the both cases the gap equations indicate the parity-broken phase exists and the pion becomes massless on the phase boundaries. We also show we can take the chiral and continuum limit in the Gross-Neveu model by tuning mass and gauge-coupling parameters. This supports the idea that the staggered-Wilson fermions can be applied to the lattice QCD simulation by taking a chiral limit, as with Wilson fermions., Comment: 7 pages, 4 figures, presented at 29th International Symposium on Lattice Field Theory, Lattice2011, July 10-16, 2011, Squaw Valley, California, USA

Two crucial properties of QCD, confinement and chiral symmetry breaking, cannot be understood within the context of conventional Feynman perturbation theory. Non-perturbative phenomena enter the theory in a fundamental way at both the classical and quantum level. Over the years a coherent qualitative picture of the interplay between chiral symmetry, quantum mechanical anomalies, and the lattice has emerged and is reviewed here., Comment: 126 pages, 36 figures. Revision corrects additional typos and renumbers equations to be more consistent with the published version

We investigate the parity-broken phase structure for staggered and naive fermions in the Gross-Neveu model as a toy model of QCD. We consider a generalized staggered Gross-Neveu model including two types of four-point interactions. We use generalized mass terms to split the doublers for both staggered and naive fermions. The phase boundaries derived from the gap equations show that the mass splitting of tastes results in an Aoki phase both in the staggered and naive cases. We also discuss the continuum limit of these models and explore taking the chirally-symmetric limit by fine-tuning a mass parameter and two coupling constants. This supports the idea that in lattice QCD we can derive one- or two-flavor staggered fermions by tuning the mass parameter, which are likely to be less expensive than Wilson fermions in QCD simulation., Comment: 31 pages, 14 figures; typos corrected

Two flavor QCD involves three independent mass parameters for which non-perturbative effects are not universal. This precludes matching lattice and perturbative results for non-degenerate quarks and eliminates a vanishing up quark mass as a viable solution to the strong CP problem., Comment: 1 page, 1 figure; poster contribution to the conference "Quark confinement and the hadron spectrum IX," August 30 - September 10, 2010

We present a theoretical foundation for the Index theorem in naive and minimally doubled lattice fermions by studying the spectral flow of a Hermitean version of Dirac operators. We utilize the point splitting method to implement flavored mass terms, which play an important role in constructing proper Hermitean operators. We show the spectral flow correctly detects the index of the would-be zero modes which is determined by gauge field topology. Using the flavored mass terms, we present new types of overlap fermions from the naive fermion kernels, with a number of flavors that depends on the choice of the mass terms. We succeed to obtain a single-flavor naive overlap fermion which maintains hypercubic symmetry., Comment: 27 pages, 17 figures; references added, version accepted in JHEP

I explore the rich phase diagram of two-flavor QCD as a function of the quark masses. The theory involves three parameters, including one that is CP violating. As the masses vary, regions of both first and second order transitions are expected. For non-degenerate quarks, non-perturbative effects cease to be universal, leaving individual quark mass ratios with a renormalization scheme dependence. This raises complications in matching lattice results with perturbative schemes and demonstrates the tautology of attacking the strong CP problem via a vanishing up quark mass., Comment: 16 pages, 6 figures. Revision adds discussion of robustness for the effective potential model

We propose a method to control the number of species of lattice fermions, which yields new classes of minimally doubled lattice fermions with one exact chiral symmetry and exact locality. We classify all the known minimally doubled fermions into two types based on the locations of the propagator poles in the Brillouin zone. We also study higher-dimensional extension of them and show it tends to be more difficult to realize minimal-doubling in higher dimensions., Comment: 7 pages, presented at 28th International Symposium on Lattice Field Theory, Lattice2010, June 14-19, 2010, Villasimius, Italy

Minimally doubled fermions have been proposed as a strictly local discretization of the QCD quark action, which also preserves chiral symmetry at finite cut-off. We study the renormalization and mixing properties of two particular realizations of minimally doubled fermions in lattice perturbation theory at one loop, and we construct conserved axial currents which have a simple form involving only nearest-neighbours sites. We also introduce a notation which allows a unified description of the renormalization properties of both actions., Comment: 14 pages; combined talk and poster contributions to the 28th International Symposium on Lattice Field Theory (Lattice 2010)

Minimally-doubled chiral fermions have the unusual property of a single local field creating two fermionic species. Spreading the field over hypercubes allows construction of combinations that isolate specific modes. Combining these fields into bilinears produces meson fields of specific quantum numbers., Comment: 6 pages; poster contribution to The XXVIII International Symposium on Lattice Field Theory Revised version corrects typos

Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I use the fourth power of the naive Dirac operator to define a local lattice measure of topological charge. For smooth gauge fields this reduces to the usual topological density. For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. On cooling with respect to the Wilson gauge action, instanton like structures do emerge. As cooling proceeds, these objects tend shrink and finally "fall through the lattice." Modifying the action can block the shrinking at the expense of a loss of reflection positivity. The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility., Comment: 22 pages, 12 figures, final version to appear in Annals of Physics

We propose a method to control the number of species of lattice fermions which yields new classes of minimally doubled lattice fermions. We show it is possible to control the number of species by handling $O(a)$ Wilson-term-like corrections in fermion actions, which we will term ``Twisted-ordering Method". Using this method we obtain new minimally doubled actions with one exact chiral symmetry and exact locality. We classify the known minimally doubled fermions into two types based on the locations of the propagator poles in the Brillouin zone., Comment: 23 pages, 6 figures; version accepted in Phys.Rev.D

We investigate the renormalization properties of minimally doubled fermions, at one loop in perturbation theory. Our study is based on the two particular realizations of Borici-Creutz and Karsten-Wilczek. A common feature of both formulations is the breaking of hyper-cubic symmetry, which requires that the lattice actions are supplemented by suitable counterterms. We show that three counterterms are required in each case and determine their coefficients to one loop in perturbation theory. For both actions we compute the vacuum polarization of the gluon. It is shown that no power divergences appear and that all contributions which arise from the breaking of Lorentz symmetry are cancelled by the counterterms. We also derive the conserved vector and axial-vector currents for Karsten-Wilczek fermions. Like in the case of the previously studied Borici-Creutz action, one obtains simple expressions, involving only nearest-neighbour sites. We suggest methods how to fix the coefficients of the counterterms non-perturbatively and discuss the implications of our findings for practical simulations., Comment: 23 pages, 1 figure

The quantum anomaly that breaks the U(1) axial symmetry of massless multi-flavored QCD leaves behind a discrete flavor-singlet chiral invariance. With massive quarks, this residual symmetry has a close connection with the strong CP-violating parameter theta. One result is that if the lightest quarks are degenerate, then a first order transition will occur when theta passes through pi. The resulting framework helps clarify when the rooting prescription for extrapolating in the number of flavors is valid., Comment: 12 pages, 7 figures. Contribution to the International Workshop on QCD Green's Functions, Confinement and Phenomenology September 7-11, 2009, ECT Trento, Italy. Revised version adds references

I review some aspects of the interplay between anomalies and chiral symmetry. The quantum anomaly that breaks the U(1) axial symmetry of massless QCD leaves behind a flavor-singlet discrete chiral invariance. When the mass is turned on this residual symmetry has a close connection with the strong CP violating parameter theta. One result is that a first order transition is usually expected when the strong CP violating angle passes through pi. This symmetry can be understood either in terms of effective chiral Lagrangians or in terms of the underlying quark fields., Comment: 19 pages, 12 figures. Revision adds references and corrects minor typos

The rooting procedure commonly used with staggered fermions does not correctly treat non-perturbative effects associated with gauge field topology. In practice these effects are small for the physics of flavor non-singlet particles. However large uncontrolled systematic errors are expected for flavor singlet issues, such as the mass of the eta prime meson. While the relative speed of the algorithm in large scale simulations may justify its use, the method is an approximation and should not be promoted as a first principles approach to the strong interactions., Comment: 5 pages, 2 figures, contribution to the 8th Conference on Quark Confinement and the Hadron Spectrum, September 1-6, 2008, Mainz, Germany

Borici's construction of minimally doubled chiral fermions builds on a linear combination of two unitarily related naive fermion actions. Being strictly local, extremely efficient numerical implementation should be possible. The resulting system is symmetric under the subgroup of the hypercubic group that preserves a major hypercube diagonal. The symmetry includes both parity even and odd transformations, but allows for an anisotropy to appear at finite lattice spacing., Comment: 7 pages, for proceedings of The XXVI International Symposium on Lattice Field Theory, July 14 - 19, 2008, Williamsburg, Virginia, USA; revision corrects typos and adds references

A recent criticism of the proof of the failure of the rooting procedure with staggered fermions is shown to be incorrect., Comment: 4 pages, no figures; revised version as to be published in Phys. Rev. D

I look at the rooting controversy from a historical point of view and review how I have come to the conclusion that these simulations involving staggered quarks must be discarded., Comment: HTML, about 5 pages, 2 figures. Revision adds some references and mollifies the discussion slightly

Motivated by the description of the graphene electronic structure in terms of the relativistic Dirac equation, a generalization to four dimensions yields a strictly local fermion action describing two species and possessing an exact chiral symmetry. This is the minimum number of species required by the well known ``no-go'' theorems., Comment: 10 pages, 1 figure. Revision adds more references and adds minor clarifications. Version to appear in JHEP

In 1976 't Hooft introduced an elegant approach towards understanding the physical consequences of the topological structures that appear in non-Abelian gauge theories. These effects are concisely summarized in terms of an effective multi-fermion interaction. These old arguments provide a link between a variety of recent and sometimes controversial ideas including discrete chiral symmetries appearing in some models for unification, ambiguities in the definition of quark masses, and flaws with some simulation algorithms in lattice gauge theory., Comment: 22 pages, 5 figures

I explore the origins of the unphysical predictions from rooted staggered fermion algorithms. Before rooting, the exact chiral symmetry of staggered fermions is a flavored symmetry among the four "tastes." The rooting procedure averages over tastes of different chiralities. This averaging forbids the appearance of the correct 't Hooft vertex for the target theory., Comment: 10 pages, 5 figures. Invited talk at the XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg, Germany

Using effective Lagrangian arguments, I explore the qualitative behavior expected at finite temperature for two flavor lattice QCD formulated with Wilson fermions and a twisted mass term. A rather rich phase structure is predicted, exhibiting Aoki's parity violating phase along with a deconfinement region forming a conical structure in the space of coupling, hopping parameter, and twisted mass variables., Comment: 19 pages, 15 figures; version to appear in Phys. Rev. D

I respond to the Bernard et al. comment on my letter ``Chiral anomalies and rooted staggered fermions.'', Comment: 3 pages. Rebuttal to arXiv:hep-lat/0603027. To appear in Physics Letters B

A popular approximation in lattice gauge theory is an extrapolation in the number of fermion species away from the four fold degeneracy natural with the staggered fermion formulation. I show that the extrapolation procedure mutilates the expected continuum holomorphic behavior in the quark masses. The conventional resolution proposes canceling the unphysical singularities with a plethora of extra states appearing at finite lattice spacing. This unproven conjecture requires an explicit loss of unitarity and locality. Even if correct, the approach implies large cutoff effects in the low-energy flavor-neutral sector., Comment: 10 pages, no figures; revision includes various clarifications, a changed title, and an additional reference; version to appear in Physics Letters B

For this after dinner talk I intersperse images of real lattices with a discussion of the motivations for lattice gauge theory and some current unresolved issues., Comment: 6 pages, 7 figures. After dinner talk at Quark confinement and the hadron spectrum VII, Ponta Delgada, Azores, Portugal, 2-7 September 2006

One flavor QCD is a rather intriguing variation on the underlying theory of hadrons. In this case quantum anomalies remove all chiral symmetries. This paper discusses the qualitative behavior of this theory as a function of its basic parameters, exploring the non-trivial phase structure expected as these parameters are varied. Comments are made on the expected changes to this structure if the gauge group is made larger and the fermions are put into higher representations., Comment: 29 pages, 8 figures; revision adds references and minor clarifications

Calculations using staggered quarks augmented with a root of the fermion determinant to reduce doubling give a qualitatively incorrect behavior in the small quark mass region. Attempts to circumvent this problem for the continuum limit involve an unproven combination of unphysical states, a loss of unitarity, and a rather peculiar non-commutation of limits., Comment: 5 pages, 1 figure; poster at LATTICE '06, Tucson, Arizona, July 23-28, 2006

A popular approximation in lattice gauge theory is an extrapolation in the number of fermion species away from the four fold degeneracy natural with the staggered fermions formulation. I show that at finite lattice spacing and for an odd number of flavors this extrapolation misses terms which, on general principles, must be present in the continuum theory. For a correct continuum limit, this forces unphysical singularities in parameter regions where continuum physics is smooth and all physical particles are massive. These singularities are not expected with other lattice regulators. Finally, I argue that unnatural constraints on certain correlation functions appear even when all quarks are massive., Comment: 9 pages, no figures. Latest revision comments on peculiarities persisting at non-zero quark masses

It is popular to discuss low energy physics in lattice gauge theory in terms of the small eigenvalues of the lattice Dirac operator. I play with some ensuing pitfalls in the interpretation of these eigenvalue spectra., Comment: 10 pages, 6 figures. Contribution to "Sense of beauty in physics," 26-27 Jan. 2006, Univ. Pisa. Revised version adds references and makes minor clarifications

The bosonization process elegantly shows the equivalence of massless scalar and fermion fields in two space-time dimensions. However, with multiple fermions the technique often obscures global symmetries. Witten's non-Abelian bosonization makes these symmetries explicit, but at the expense of a somewhat complicated bosonic action. Frenkel and Kac have presented an intricate mathematical formalism relating the various approaches. Here I reduce these arguments to the simplest case of a single massless scalar field. In particular, using only elementary quantum field theory concepts, I expose a hidden $SU(2)\times SU(2)$ chiral symmetry in this trivial theory. I then discuss in what sense this field should be interpreted as a Goldstone boson., Comment: 15 pages. Revision adds numerous references and puts things in better historical context

With two degenerate quarks, the chiral condensate exhibits a jump as the quark masses pass through zero. I discuss how this single transition splits into two Ising like transitions when the quarks are made non-degenerate. The order parameter is the expectation of the neutral pion field. The transitions represent long distance coherent phenomena occuring without the Dirac operator having vanishingly small eigenvalues., Comment: 6 pages, 3 figures; contribution to the XXIIIrd International Symposium on Lattice Field Theory, 25-30 July 2005, Trinity College, Dublin, Ireland

I explore the regions of quark masses where CP will be spontaneously broken in the strong interactions. The boundaries of these regions are controlled by the chiral anomaly, which manifests itself in ambiguities in the definition of non-degenerate quark masses. In particular, the concept of a single massless quark is ill defined., Comment: 9 pages, 5 figures, contribution to the 6th Conference on Quark Confinement and the Hadron Spectrum