Summary Talk at Chiral '99
A brief summary of talks relating to massless lattice fermions is presented. This summary is not a review and reading it certainly is no substitute to reading the various original contributions.
hep-lat↗
arXiv subjects
H. Neuberger
Publications and source records attributed to H. Neuberger.
Overlap
The overlap formulation of regulated vectorial and chiral gauge theories is reviewed. Ostensibly new constructions, based on the Ginsparg-Wilson relation are essentially just overlap with new notation. At present there exists no satisfactory realization of chiral symmetries outside perturbation theory which is structurally different from the overlap.
hep-lat↗
The Overlap Dirac Operator
This introductory presentation describes the Overlap Dirac Operator, why it could be useful in numerical QCD, and how it can be implemented.
hep-lat↗
Chiral fermions on the lattice
Chiral fermions resisted being put on the lattice for twenty years. This raised the suspicion that asymptotically free chiral gauge theories were not renormalizable outside perturbation theory and therefore could not be mathematically extended to infinite energies. During the last several years the situation has reversed itself. Today we believe that all the essential ingredients for a full lattice definition of non-anomalous chiral gauge theories are in place within the overlap construction. This construction is based on earlier work by Callan and Harvey, by Kaplan and by Frolov and Slavnov. It can be reinterpreted as coming from the Ginsparg-Wilson relation, but, at the moment, it is a unique construction and therefore might also be the way by which Nature itself regularizes chiral fermions. This is yet another instance in which lattice field theory makes a potentially important contribution to fundamental particle physics.
hep-lat↗
Tricks to implement the overlap Dirac operator
I present several tricks to help implement the overlap Dirac operator numerically.
hep-lat↗
Chirality on the Lattice
During the last several years a non-perturbative formulation of exact chiral symmetry on the lattice has been developed. I shall outline the main ideas of these developments and discuss prospects for the future. The focus will be on the basic concepts enciphered in a new jargon consisting of terms like ``infinite number of fermions'', ``domain wall fermions'', ``the overlap'', ``the Ginsparg-Wilson relation''. Technical details will be omitted.
hep-lat↗
Geometrical aspects of chiral anomalies in the overlap
The set of one dimensional lowest energy eigenspaces used to construct the overlap induces a two form on gauge orbit space which is the locally exact curl of Berry's connection. If anomalies do not cancel, examples of two dimensional closed sub-manifolds of orbit space are produced over which the integral of the above two form does not vanish. Based on these observations, a natural definition of covariant currents is obtained, a simple way to calculate chiral anomalies on the lattice is found, and indications for how to construct an ideal regularization of chiral gauge theories are seen to emerge.
hep-lat↗
Remark on lattice BRST invariance
A recently claimed resolution to the lattice Gribov problem in the context of chiral lattice gauge theories is examined. Unfortunately, I find that the old problem remains.
hep-lat↗
More about exactly massless quarks on the lattice
In a previous publication [hep-lat/9707022] I showed that the fermion determinant for strictly massless quarks can be written on the lattice as $\det D$, where $D$ is a certain finite square matrix explicitly constructed from the lattice gauge fields. Here I show that $D$ obeys the Ginsparg-Wilson relation $Dγ_5 D = Dγ_5 +γ_5 D$.
hep-lat↗
Exponential suppression of radiatively induced mass in the truncated overlap
A certain truncation of the overlap (domain wall fermions) contains $k$ flavors of Wilson-Dirac fermions. We show that for sufficiently weak lattice gauge fields the effective mass of the lightest Dirac particle is exponentially suppressed in $k$. This suppression is seen to disappear when lattice topology is non-trivial. We check explicitly that the suppression holds to one loop in perturbation theory. We also provide a new expression for the free fermion propagator with an arbitrary additional mass term.
hep-lat↗
Overlap in odd dimensions
In odd dimensions the lattice overlap formalism is simpler than in even dimensions. Masslessness of fermions can still be preserved without fine tuning and gauge invariance without gauge averaging can be maintained, although, sometimes, only at the expense of parity invariance. When parity invariance is enforced invariance under small gauge transformations can be maintained and continuum global gauge anomalies are reproduced.
hep-lat↗
Finite size corrections in two dimensional gauge theories and a quantitative chiral test of the overlap
An argument is presented for a certain universality of finite size corrections in two dimensional gauge theories. In the abelian case a direct calculation is carried out for a particular chiral model. The analytical result confirms the above universality and that the 't Hooft vertex previously measured using the overlap smoothly approaches the correct continuum limit within statistical errors.
hep-th↗
Massless Composite Fermions in Two Dimensions and the Overlap
There exist chiral gauge models in two dimensions that have massless composite fermions. Two examples are presented and it is suggested that they be accepted as benchmark test-cases for generic proposals of non-perturbatively regulating chiral gauge theories in any dimension. We apply the overlap to the simpler of the two benchmarks and present the results of a numerical simulation of modest size.
hep-lat↗
Progress in lattice chiral gauge theories
Some key features of continuum chiral fermions are shown to be satisfied by the overlap.
hep-lat↗
Some Applications of the Overlap Formalism
The massless vector Schwinger model with $N_f=1,2,3,4$ number of flavors is studied on the lattice using the overlap formalism. A full Monte Carlo simulation yields values for the bilinear fermion condensate that are in agreement with the exact solution of the continuum Schwinger model with an added Thirring interaction.
hep-lat↗
Chiral determinant on the lattice -- Anomalies and Instantons
An expression for the lattice effective action induced by chiral fermions in any even dimensions in terms of an overlap of two states is shown to have promising properties in two and four dimensions: The correct abelian anomaly is reproduced and gauge field configurations with non-zero topological charge are completely suppressed.
hep-lat↗
Numerical analysis of the Higgs mass triviality bound
Previous large $N$ calculations are combined with numerical work at $N=4$ to show that the Minimal Standard Model will describe physics to an accuracy of a few percent up to energies of the order 2 to 4 times the Higgs mass, $M_H$, only if $M_H \le 710\pm60 ~ GeV$. This bound is the result of a systematic search in the space of dimension six operators and is expected to hold in the {\it continuum}. Given that studying the scalar sector in isolation is already an approximation, we believe that our result is sufficiently accurate and that further refinements would be of progressively diminishing interest to particle physics. Note: The full ps file of this preprint is also available via anonymous ftp to ftp.scri.fsu.edu. To get the ps file, ftp to this address and use for username "anonymous" and for password your name. The file is in the directory pub/vranas (to go to that directory type: cd pub/vranas) and is called num_hig.ps (to get it type: get num_hig.ps)
hep-ph↗
The triviality bound on the Higgs mass; its value and what it means
Older lattice work exploring the Higgs mass triviality bound is briefly reviewed. It indicates that a strongly interacting scalar sector in the minimal standard model cannot exist; on the other hand low energy QCD phenomenology might be interpreted as an indication that it could. We attack this puzzle using the $1/N$ expansion and discover a simple criterion for selecting a lattice action that is more likely to produce a heavy Higgs particle. Depending on the precise form of the limitation put on the cutoff effects, our large $N$ calculations, when combined with old numerical data, suggest that the Higgs mass bound might be around 750 $GeV$, which is higher than the $\sim 650~GeV$ previously obtained. Preliminary numerical work indicates that an increase of at least 19\% takes place at $N=4$ on the $F_4$ lattice when the old simple action is replaced with a new action (still containing only nearest neighbor interactions) if one uses the lattice spacing as the physical cutoff for both actions. It appears that, while a QCD like theory could produce $M_H / F ~ \sim 6$, a meaningful ``minimal elementary Higgs'' theory cannot have $M_H/ F~ \gtapprox 3$. Still, even at 750 $GeV$, the Higgs particle is so wide ($\sim 290~$GeV), that one cannot argue any more that the scalar sector is weakly coupled.
hep-lat↗