Bounds on the Wilson Dirac Operator

New exact upper and lower bounds are derived on the spectrum of the square of the hermitian Wilson Dirac operator. It is hoped that the derivations and the results will be of help in the search for ways to reduce the cost of simulations using the overlap Dirac operator. The bounds also apply to the Wilson Dirac operator in odd dimensions and are therefore relevant to domain wall fermions as well.Comment: 16 pages, TeX, 3 eps figures, small corrections and improvement

Noncompact chiral U(1) gauge theories on the lattice

A new, adiabatic phase choice is adopted for the overlap in the case of an infinite volume, noncompact abelian chiral gauge theory. This gauge choice obeys the same symmetries as the Brillouin-Wigner (BW) phase choice, and, in addition, produces a Wess-Zumino functional that is linear in the gauge variables on the lattice. As a result, there are no gauge violations on the trivial orbit in all theories, consistent and covariant anomalies are simply related and Berry's curvature now appears as a Schwinger term. The adiabatic phase choice can be further improved to produce a perfect phase choice, with a lattice Wess-Zumino functional that is just as simple as the one in continuum. When perturbative anomalies cancel, gauge invariance in the fermionic sector is fully restored. The lattice effective action describing an anomalous abelian gauge theory has an explicit form, close to one analyzed in the past in a perturbative continuum framework.Comment: 35 pages, one figure, plain TeX; minor typos corrected; to appear in PR

An alternative to domain wall fermions

We define a sparse hermitian lattice Dirac matrix, $H$, coupling $2n+1$ Dirac fermions. When $2n$ fermions are integrated out the induced action for the last fermion is a rational approximation to the hermitian overlap Dirac operator. We provide rigorous bounds on the condition number of $H$ and compare them to bounds for the higher dimensional Dirac operator of domain wall fermions. Our main conclusion is that overlap fermions should be taken seriously as a practical alternative to domain wall fermions in the context of numerical QCD.Comment: Revtex Latex, 26 pages, 1 figure, a few minor change

Another determination of the quark condensate from an overlap action

I use the technique of Hernandez, et al (hep-lat/0106011) to convert a recent calculation of the lattice-regulated quark condensate from an overlap action to a continuum-regulated number. I find Sigma(MSbar)(mu = 2 GeV) = (282(6) MeV)-cubed times (a-inverse/1766 MeV)-cubed from a calculation with the Wilson gauge action at beta=5.9.Comment: 3 pages, Revtex, 1 postscript figure. References added. COLO-HEP-47

Chiral Logs in Quenched QCD

The quenched chiral logs are examined on a $16^3 \times 28$ lattice with Iwasaki gauge action and overlap fermions. The pion decay constant $f_{\pi}$ is used to set the lattice spacing, $a = 0.200(3) {\rm fm}$. With pion mass as low as $\sim 180 {\rm MeV}$, we see the quenched chiral logs clearly in $m_{\pi}^2/m$ and $f_P$, the pseudoscalar decay constant. We analyze the data to determine how low the pion mass needs to be in order for the quenched one-loop chiral perturbation theory ($\chi$PT) to apply. With the constrained curve-fitting method, we are able to extract the quenched chiral log parameter $\delta$ together with other low-energy parameters. Only for $m_{\pi} \leq 300 {\rm MeV}$ do we obtain a consistent and stable fit with a constant $\delta$ which we determine to be 0.24(3)(4) (at the chiral scale $\Lambda_{\chi}=0.8 {\rm GeV}$). By comparing to the $12^3 \times 28$ lattice, we estimate the finite volume effect to be about 2.7% for the smallest pion mass. We also fitted the pion mass to the form for the re-summed cactus diagrams and found that its applicable region is extended farther than the range for the one-loop formula, perhaps up to $m_{\pi} \sim 500-600$ MeV. The scale independent $\delta$ is determined to be 0.20(3) in this case. We study the quenched non-analytic terms in the nucleon mass and find that the coefficient $C_{1/2}$ in the nucleon mass is consistent with the prediction of one-loop $\chi$PT\@. We also obtain the low energy constant $L_5$ from $f_{\pi}$. We conclude from this study that it is imperative to cover only the range of data with the pion mass less than $\sim 300 {\rm MeV}$ in order to examine the chiral behavior of the hadron masses and decay constants in quenched QCD and match them with quenched one-loop $\chi$PT\@.Comment: 37 pages and 24 figures, pion masses are fitted to the form for the re-summed cactus diagrams, figures added, to appear in PR

Proposal for the numerical solution of planar QCD

Using quenched reduction, we propose a method for the numerical calculation of meson correlation functions in the planar limit of QCD. General features of the approach are outlined, and an example is given in the context of two-dimensional QCD.Comment: 31 pages, 10 figures, uses axodraw.sty, To appear in Physical Review

Large N reduction in the continuum three dimensional Yang-Mills theory

Numerical and theoretical evidence leads us to propose the following: Three dimensional Euclidean Yang-Mills theory in the planar limit undergoes a phase transition on a torus of side $l=l_c$. For $l>l_c$ the planar limit is $l$-independent, as expected of a non-interacting string theory. We expect the situation in four dimensions to be similar.Comment: 4 pages, latex file, two figures, version to appear in Phys. Rev. Let

A practical implementation of the Overlap-Dirac operator

A practical implementation of the Overlap-Dirac operator ${{1+\gamma_5\epsilon(H)}\over 2}$ is presented. The implementation exploits the sparseness of $H$ and does not require full storage. A simple application to parity invariant three dimensional SU(2) gauge theory is carried out to establish that zero modes related to topology are exactly reproduced on the lattice.Comment: Y-axis label in figure correcte

Chiral symmetry restoration and axial vector renormalization for Wilson fermions

Lattice gauge theories with Wilson fermions break chiral symmetry. In the U(1) axial vector current this manifests itself in the anomaly. On the other hand it is generally expected that the axial vector flavour mixing current is non-anomalous. We give a short, but strict proof of this to all orders of perturbation theory, and show that chiral symmetry restauration implies a unique multiplicative renormalization constant for the current. This constant is determined entirely from an irrelevant operator in the Ward identity. The basic ingredients going into the proof are the lattice Ward identity, charge conjugation symmetry and the power counting theorem. We compute the renormalization constant to one loop order. It is largely independent of the particular lattice realization of the current.Comment: 11 pages, Latex2

One loop calculation in lattice QCD with domain-wall quarks

One loop corrections to the domain-wall quark propagator are calculated in massless QCD. It is shown that no additative counter term to the current quark mass is generated in this theory, and the wave function renormalization factor of the massless quark is explicitly evaluated. We also show that an analysis with a simple mean-field approximation can explain properties of the massless quark in numerical simulations of QCD with domain-wall quarks.Comment: 24 pages, REVTeX, with 3 epsf figure