Discretization Errors and Rotational Symmetry: The Laplacian Operator on Non-Hypercubical Lattices

Discretizations of the Laplacian operator on non-hypercubical lattices are discussed in a systematic approach. It is shown that order $a^2$ errors always exist for discretizations involving only nearest neighbors. Among all lattices with the same density of lattice sites, the hypercubical lattices always have errors smaller than other lattices with the same number of spacetime dimensions. On the other hand, the four dimensional checkerboard lattice (also known as the Celmaster lattice) is the only lattice which is isotropic at order $a^2$. Explicit forms of the discretized Laplacian operators on root lattices are presented, and different ways of eliminating order $a^2$ errors are discussed.Comment: 30 pages in REVTe

Liquid-gas and other unusual thermal phase transitions in some large-N magnets

Much insight into the low temperature properties of quantum magnets has been gained by generalizing them to symmetry groups of order N, and then studying the large N limit. In this paper we consider an unusual aspect of their finite temperature behavior--their exhibiting a phase transition between a perfectly paramagetic state and a paramagnetic state with a finite correlation length at N = \infty. We analyze this phenomenon in some detail in the large ``spin'' (classical) limit of the SU(N) ferromagnet which is also a lattice discretization of the CP^{N-1} model. We show that at N = \infty the order of the transition is governed by lattice connectivity. At finite values of N, the transition goes away in one or less dimension but survives on many lattices in two dimensions and higher, for sufficiently large N. The latter conclusion contradicts a recent conjecture of Sokal and Starinets, yet is consistent with the known finite temperature behavior of the SU(2) case. We also report closely related first order paramagnet-ferromagnet transitions at large N and shed light on a violation of Elitzur's theorem at infinite N via the large q limit of the q-state Potts model, reformulated as an Ising gauge theory.Comment: 27 pages, 7 figures. Added clarifications requested by a refere

Two-loop QCD corrections of the massive fermion propagator

The off-shell two-loop correction to the massive quark propagator in an arbitrary covariant gauge is calculated and results for the bare and renormalized propagator are presented. The calculations were performed by means of a set of new generalized recurrence relations proposed recently by one of the authors. From the position of the pole of the renormalized propagator we obtain the relationship between the pole mass and the \bar{MS} mass. This relation confirms the known result by Gray et al.. The bare amplitudes are given for an arbitrary gauge group and for arbitrary space-time dimensions.Comment: 18 pages LaTeX, misprints in formula (12) are correcte

Complete Renormalization Group Improvement-Avoiding Factorization and Renormalization Scale Dependence in QCD Predictions

For moments of leptoproduction structure functions we show that all dependence on the renormalization and factorization scales disappears, provided that all the ultraviolet logarithms involving the physical energy scale Q are completely resummed. The approach is closely related to Grunberg's method of Effective Charges. A direct and simple method for extracting the universal dimensional transmutation parameter of QCD from experimental data is advocated.Comment: 16 pages, no figure

qˉq{\bar {q}}q condensate for light quarks beyond the chiral limit

We determine the ${\bar{q}}q$ condensate for quark masses from zero up to that of the strange quark within a phenomenologically successful modelling of continuum QCD by solving the quark Schwinger-Dyson equation. The existence of multiple solutions to this equation is the key to an accurate and reliable extraction of this condensate using the operator product expansion. We explain why alternative definitions fail to give the physical condensate.Comment: 13 pages, 8 figure

Perturbative Strong Interaction Corrections to the Heavy Quark Semileptonic Decay Rate

We calculate the part of the order $\alpha_s^2$ correction to the semileptonic heavy quark decay rate proportional to the number of light quark flavors, and use our result to set the scale for evaluating the strong coupling in the order $\alpha_s$ term according to the scheme of Brodsky, Lepage and Mackenzie. Expressing the decay rate in terms of the heavy quark pole mass $m_Q$, we find the scale for the $\overline{MS}$ strong coupling to be $0.07\, m_Q$. If the decay rate is expressed in terms of the $\overline{MS}$ heavy quark mass $\overline m_Q(m_Q)$ then the scale is $0.12\, m_Q$. We use these results along with the existing calculations for hadronic $\tau$ decay to calculate the BLM scale for the nonleptonic decay width and the semileptonic branching ratio. The implications for the value of $|V_{bc}|$ extracted from the inclusive semileptonic $B$ meson decay rate are discussed.Comment: 7 pages in Latex plus 1 uuencoded figure, uses epsf, UTPT-94-24, CMU-HEP 94-29, CALT-68-1950 (previous results unchanged; we add a short discussion of nonleptonic decays

Exact mass dependent two--loop αˉs(Q2)\bar{\alpha}_s(Q^2) in the background MOM renormalization scheme

A two-loop calculation of the renormalization group $\beta$--function in a momentum subtraction scheme with massive quarks is presented using the background field formalism. The results have been obtained by using a set of new generalized recurrence relations proposed recently by one of the authors (O.V.T.). The behavior of the mass dependent effective coupling constant is investigated in detail. Compact analytic results are presented.Comment: 20 pages, 5 figures, LaTeX, uses axodraw.sty, revised version, Sec. 5 (numerical results) changed (quark masses were not set properly) and enhance

Quark-gluon vertex in arbitrary kinematics

We compute the quark-gluon vertex in quenched lattice QCD, in the Landau gauge using an off-shell mean-field O(a)-improved fermion action. The complete vertex is computed in two specific kinematical limits, while the Dirac-vector part is computed for arbitrary kinematics. We find a nontrivial and rich tensor structure, including a substantial infrared enhancement of the interaction strength regardless of kinematics.Comment: 6 pages, 8 figures, talk by JIS at QCD Down Under, Adelaide, 10-19 March 200

Dynamical mass generation by source inversion: Calculating the mass gap of the Gross-Neveu model

We probe the U(N) Gross-Neveu model with a source-term $J\bar{\Psi}\Psi$. We find an expression for the renormalization scheme and scale invariant source $\hat{J}$, as a function of the generated mass gap. The expansion of this function is organized in such a way that all scheme and scale dependence is reduced to one single parameter d. We get a non-perturbative mass gap as the solution of $\hat{J}=0$. In one loop we find that any physical choice for d gives good results for high values of N. In two loops we can determine d self-consistently by the principle of minimal sensitivity and find remarkably accurate results for N>2.Comment: 13 pages, 3 figures, added referenc

Gauge parameter dependence in the background field gauge and the construction of an invariant charge

By using the enlarged BRS transformations we control the gauge parameter dependence of Green functions in the background field gauge. We show that it is unavoidable -- also if we consider the local Ward identity -- to introduce the normalization gauge parameter $\xi_o$, which enters the Green functions of higher orders similarly to the normalization point $\kappa$. The dependence of Green functions on $\xi_o$ is governed by a further partial differential equation. By modifying the Ward identity we are able to construct in 1-loop order a gauge parameter independent combination of 2-point vector and background vector functions. By explicit construction of the next orders we show that this combination can be used to construct a gauge parameter independent RG-invariant charge. However, it is seen that this RG-invariant charge does not satisfy the differential equation of the normalization gauge parameter $\xi_o$, and, hence, is not $\xi_o$-independent as required.Comment: 29 pages, LaTe