On the controversy concerning the definition of quark and gluon angular momentum

A major controversy has arisen in QCD as to how to split the total angular momentum into separate quark and gluon contributions, and as to whether the gluon angular momentum can itself be split, in a gauge invariant way, into a spin and orbital part. Several authors have proposed various answers to these questions and offered a variety of different expressions for the relevant operators. I argue that none of these is acceptable and suggest that the canonical expression for the momentum and angular momentum operators is the correct and physically meaningful one. It is then an inescapable fact that the gluon angular momentum operator cannot, in general, be split in a gauge invariant way into a spin and orbital part. However, the projection of the gluon spin onto its direction of motion i.e. its helicity is gauge invariant and is measured in deep inelastic scattering on nucleons. The Ji sum rule, relating the quark angular momentum to generalized parton distributions, though not based on the canonical operators, is shown to be correct, if interpreted with due care. I also draw attention to several interesting aspects of QED and QCD, which, to the best of my knowledge, are not commented upon in the standard textbooks on Field Theory.Comment: 41 pages; Some incorrect statements have been rectified and a detailed discussion has been added concerning the momentum carried by quarks and the Ji sum rule for the angular momentu

Two-Loop Bethe Logarithms for Higher Excited S Levels

Processes mediated by two virtual low-energy photons contribute quite significantly to the energy of hydrogenic S states. The corresponding level shift is of the order of (alpha/pi)^2 (Zalpha)^6 m_e c^2 and may be ascribed to a two-loop generalization of the Bethe logarithm. For 1S and 2S states, the correction has recently been evaluated by Pachucki and Jentschura [Phys. Rev. Lett. vol. 91, 113005 (2003)]. Here, we generalize the approach to higher excited S states, which in contrast to the 1S and 2S states can decay to P states via the electric-dipole (E1) channel. The more complex structure of the excited-state wave functions and the necessity to subtract P-state poles lead to additional calculational problems. In addition to the calculation of the excited-state two-loop energy shift, we investigate the ambiguity in the energy level definition due to squared decay rates.Comment: 14 pages, RevTeX, to appear in Phys. Rev.

Multiplying unitary random matrices - universality and spectral properties

In this paper we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random hermitian matrix. This is equivalent to solving unitary diffusion generated by a hamiltonian random in time. We find that the result is universal and depends only on the second moment of the generator of the stochastic evolution. We find indications of critical behavior (eigenvalue spacing scaling like $1/N^{3/4}$) close to $\theta=\pi$ for a specific critical evolution time $t_c$.Comment: 12 pages, 2 figure

Improved α4\alpha^4 Term of the Electron Anomalous Magnetic Moment

We report a new value of electron $g-2$, or $a_e$, from 891 Feynman diagrams of order $\alpha^4$. The FORTRAN codes of 373 diagrams containing closed electron loops have been verified by at least two independent formulations. For the remaining 518 diagrams, which have no closed lepton loop, verification by a second formulation is not yet attempted because of the enormous amount of additional work required. However, these integrals have structures that allow extensive cross-checking as well as detailed comparison with lower-order diagrams through the renormalization procedure. No algebraic error has been uncovered for them. The numerical evaluation of the entire $\alpha^4$ term by the integration routine VEGAS gives $-1.7283 (35) (\alpha/\pi)^4$, where the uncertainty is obtained by careful examination of error estimates by VEGAS. This leads to $a_e = 1 159 652 175.86 (0.10) (0.26) (8.48) \times 10^{-12}$, where the uncertainties come from the $\alpha^4$ term, the estimated uncertainty of $\alpha^5$ term, and the inverse fine structure constant, $\alpha^{-1} = 137.036 000 3 (10)$, measured by atom interferometry combined with a frequency comb technique, respectively. The inverse fine structure constant $\alpha^{-1} (a_e)$ derived from the theory and the Seattle measurement of $a_e$ is $137.035 998 83 (51)$.Comment: 64 pages and 10 figures. Eq.(16) is corrected. Comments are added after Eq.(40

Constraints on the mass of the superlight gravitino from the muon anomaly

We reexamine the limits on the gravitino mass supplied by the muon anomaly in the frame of supergravity models with a superlight gravitino and a superlight scalar S and a superlight pseudoscalar P.Comment: 7 pages, REVTeX with 5 figures. Uses epsfi

Hadronic Contributions to the Muon Anomaly in the Constituent Chiral Quark Model

The hadronic contributions to the anomalous magnetic moment of the muon which are relevant for the confrontation between theory and experiment at the present level of accuracy, are evaluated within the same framework: the constituent chiral quark model. This includes the contributions from the dominant hadronic vacuum polarization as well as from the next--to--leading order hadronic vacuum polarization, the contributions from the hadronic light-by-light scattering, and the contributions from the electroweak hadronic $Z\gamma\gamma$ vertex. They are all evaluated as a function of only one free parameter: the constituent quark mass. We also comment on the comparison between our results and other phenomenological evaluations.Comment: Several misprints corrected and a clarifying sentence added. Three figures superposed and two references added. Version to appear in JHE

Non-Gaussian fixed point in four-dimensional pure compact U(1) gauge theory on the lattice

The line of phase transitions, separating the confinement and the Coulomb phases in the four-dimensional pure compact U(1) gauge theory with extended Wilson action, is reconsidered. We present new numerical evidence that a part of this line, including the original Wilson action, is of second order. By means of a high precision simulation on homogeneous lattices on a sphere we find that along this line the scaling behavior is determined by one fixed point with distinctly non-Gaussian critical exponent nu = 0.365(8). This makes the existence of a nontrivial and nonasymptotically free four-dimensional pure U(1) gauge theory in the continuum very probable. The universality and duality arguments suggest that this conclusion holds also for the monopole loop gas, for the noncompact abelian Higgs model at large negative squared bare mass, and for the corresponding effective string theory.Comment: 11 pages, LaTeX, 2 figure

High-statistics finite size scaling analysis of U(1) lattice gauge theory with Wilson action

We describe the results of a systematic high-statistics Monte-Carlo study of finite-size effects at the phase transition of compact U(1) lattice gauge theory with Wilson action on a hypercubic lattice with periodic boundary conditions. We find unambiguously that the critical exponent nu is lattice-size dependent for volumes ranging from 4^4 to 12^4. Asymptotic scaling formulas yield values decreasing from nu(L >= 4) = 0.33 to nu(L >= 9) = 0.29. Our statistics are sufficient to allow the study of different phenomenological scenarios for the corrections to asymptotic scaling. We find evidence that corrections to a first-order transition with nu=0.25 provide the most accurate description of the data. However the corrections do not follow always the expected first-order pattern of a series expansion in the inverse lattice volume V^{-1}. Reaching the asymptotic regime will require lattice sizes greater than L=12. Our conclusions are supported by the study of many cumulants which all yield consistent results after proper interpretation.Comment: revtex, 12 pages, 9 figure

The Pole Mass of The Heavy Quark. Perturbation Theory and Beyond

The key quantity of the heavy quark theory is the quark mass $m_Q$. Since quarks are unobservable one can suggest different definitions of $m_Q$. One of the most popular choices is the pole quark mass routinely used in perturbative calculations and in some analyses based on heavy quark expansions. We show that no precise definition of the pole mass can be given in the full theory once non-perturbative effects are included. Any definition of this quantity suffers from an intrinsic uncertainty of order \Lam /m_Q. This fact is succinctly described by the existence of an infrared renormalon generating a factorial divergence in the high-order coefficients of the $\alpha_s$ series; the corresponding singularity in the Borel plane is situated at $2\pi /b$. A peculiar feature is that this renormalon is not associated with the matrix element of a local operator. The difference \La \equiv M_{H_Q}-m_Q^{pole} can still be defined in Heavy Quark Effective Theory, but only at the price of introducing an explicit dependence on a normalization point $\mu$: \La (\mu ). Fortunately the pole mass $m_Q(0)$ {\em per se} does not appear in calculable observable quantities.Comment: 22 pages, Latex, 6 figures (available upon request), TPI-MINN-94/4-T, CERN-TH.7171/94, UND-HEP-94-BI