arXiv subjects
Publications and source records attributed to Michael Creutz.
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.
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.
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.
I respond to the Bernard et al. comment on my letter ``Chiral anomalies and rooted staggered fermions.''
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.
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.
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.
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.
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.
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.
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.
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.
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.
The admissibility condition usually used to define the topological charge in lattice gauge theory is incompatible with a positive transfer matrix.
The charges of magnetic monopoles are constrained to a multiple of $2π$ times the inverse of the elementary unit electric charge. In the standard model, quarks have fractional charge, raising the question of whether the basic magnetic monople unit is a multiple of $2 π/e$ or three times that. A simple lattice construction shows how a magnetic monopole of the lower strength is possible if it interacts with gluonic fields as well. Such a monopole is thus a hadron. This is consistent with the construction of magnetic monopoles in grand unified theories.
The Yang-Mills theory lies at the heart of our understanding of elementary particle interactions. For the strong nuclear forces, we must understand this theory 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 some of 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.
Some time ago Dashen pointed out that spontaneous CP violation can occur in the strong interactions. I show how a simple effective Lagrangian exposes the remarkably large domain of quark mass parameters for which this occurs. I close with some warnings for lattice simulations.
It has long been known that no physical singularity is encountered as up quark mass is adjusted from small positive to negative values as long as all other quarks remain massive. This is tied to an additive ambiguity in the definition of the quark mass. This calls into question the acceptability of attempts to solve the strong CP problem via a vanishing mass for the lightest quark.