Dirac operator index and topology of lattice gauge fields

The fermionic topological charge of lattice gauge fields, given in terms of a spectral flow of the Hermitian Wilson--Dirac operator, or equivalently, as the index of Neuberger's lattice Dirac operator, is shown to have analogous properties to L\"uscher's geometrical lattice topological charge. The main new result is that it reduces to the continuum topological charge in the classical continuum limit. (This is sketched here; the full proof will be given in a sequel to this paper.) A potential application of the ideas behind fermionic lattice topological charge to deriving a combinatorial construction of the signature invariant of a 4-manifold is also discussed.Comment: 16 pages, based on talk at Chiral'99 (Sept. 13-18, 1999, Taipei), to be published in the Proceeding

Relation between bare lattice coupling and MSbar coupling at one loop with general lattice fermions

A compact general integral formula is derived from which the fermionic contribution to the one-loop coefficient in the perturbative expansion of the MSbar coupling in powers of the bare lattice coupling can be extracted. It is seen to reproduce the known results for unimproved naive, staggered and Wilson fermions, and has advantageous features which facilitate the evaluation in the case of improved lattice fermion formulations. This is illustrated in the case of Wilson clover fermions, and an expression in terms of known lattice integrals is obtained in this case which gives the coefficient to much greater numerical accuracy than in the previous literature.Comment: 26 pages, 1 figure, to appear in Phys.Rev.D. Completely rewritten with new title and new material added (see abstract). Some material from the previous version has been removed since it was superceded by arXiv:0709.078

Index and overlap construction for staggered fermions

Recent developments regarding index and overlap construction for staggered fermions are reviewed, highlighting the surprising and unexpected aspects.Comment: proceedings contribution for 28th International Symposium on Lattice Field Theory, Lattice2010, June 14-19, 2010, Villasimius, Italy (slightly extended version, 8 pages

Axial anomaly and topological charge in lattice gauge theory with Overlap Dirac operator

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, given by the index of the overlap Dirac operator, is then established for a class of topologically non-trivial fields in the aforementioned finite volume setting. Problematic issues concerning the index in the infinite volume setting are also discussed.Comment: Latex, 33 pages. v6: shortened and (hopefully) more succinct version, to appear in Ann.Phy

Theoretical foundation for the Index Theorem on the lattice with staggered fermions

A way to identify the would-be zero-modes of staggered lattice fermions away from the continuum limit is presented. Our approach also identifies the chiralities of these modes, and their index is seen to be determined by gauge field topology in accordance with the Index Theorem. The key idea is to consider the spectral flow of a certain hermitian version of the staggered Dirac operator. The staggered fermion index thus obtained can be used as a new way to assign the topological charge of lattice gauge fields. In a numerical study in U(1) backgrounds in 2 dimensions it is found to perform as well as the Wilson index while being computationally more efficient. It can also be expressed as the index of an overlap Dirac operator with a new staggered fermion kernel.Comment: 4 revtex pages. v3: slightly shortened and revised, to appear in Phys.Rev.Lett

Renormalisation group evolution for the ΔS=1\Delta S = 1 effective Hamiltonian with Nf=2+1N_f=2+1

We discuss the renormalisation group (RG) evolution for the $\Delta S = 1$ operators in unquenched QCD with $N_f = 3$ ($m_u=m_d=m_s$) or, more generally, $N_f = 2+1$ ($m_u=m_d \ne m_s$) flavors. In particular, we focus on the specific problem of how to treat the singularities which show up only for $N_f=3$ or $N_f = 2+1$ in the original solution of Buras {\it et al.} for the RG evolution matrix at next-to-leading order. On top of Buras {\it et al.}'s original treatment, we use a new method of analytic continuation to obtain the correct solution in this case. It is free of singularities and can therefore be used in numerical analysis of data sets calculated in lattice QCD.Comment: 7 pages, minor revisions, to appear in PR