Convergence Of Perturbation Theory Research Articles

SU(8) family unification with boson–fermion balance

We formulate an SU(8) family unification model motivated by requiring that the theory should incorporate the graviton, gravitinos, and the fermions and gauge fields of the standard model, with boson–fermion balance. Gauge field SU(8) anomalies cancel between the gravitinos and spin ½ fermions. The 56 of scalars breaks SU(8) to SU(3) family × SU(5) × U(1)/Z5, with the fermion representation content needed for "flipped" SU(5) with three families, and with residual scalars in the 10 and [Formula: see text] representations that break flipped SU(5) to the standard model. Dynamical symmetry breaking can account for the generation of 5 representation scalars needed to break the electroweak group. Yukawa couplings of the 56 scalars to the fermions are forbidden by chiral and gauge symmetries, so in the first stage of SU(8) breaking fermions remain massless. In the limit of vanishing gauge coupling, there are N = 1 and N = 8 supersymmetries relating the scalars to the fermions, which restrict the form of scalar self-couplings and should improve the convergence of perturbation theory, if not making the theory finite and "calculable." In an Appendix we give an analysis of symmetry breaking by a Higgs component, such as the (1, 1)(-15) of the SU(8) 56 under SU(8) ⊃ SU(3) × SU(5) × U(1), which has nonzero U(1) generator.

Heavy quark diffusion in QCD and 𝒩 = 4 SYM at next-to-leading order

We present the full details of a calculation at next-to-leading order of the momentum diffusion coefficient of a heavy quark in a hot, weakly coupled, QCD plasma. Corrections arise at O(g_s); physically they represent interference between overlapping scatterings, as well as soft, electric scale (p ~ gT) gauge field physics, which we treat using the hard thermal loop (HTL) effective theory. In 3-color, 3-flavor QCD, the momentum diffusion constant of a fundamental representation heavy quark at NLO is kappa = (16\pi/3) alpha_s^2 T^3 (log(1/g) + 0.07428 + 1.9026 g). We extend the computation to a heavy fundamental representation ``probe'' quark in large N_c, N=4 Super Yang-Mills theory, where the result is kappa^{SYM}= (lambda^2 T^3)(6\pi) (log(1/\sqrt{\lambda}) + 0.4304 + 0.8010 \sqrt{lambda}) (where lambda=g_s^2 N_c is the t'Hooft coupling). In the absence of some resummation technique, the convergence of perturbation theory is poor.

Fermionic sign cancellations in the continuous renormalization group equation

Abstract I derive a Wick-ordered continuous renormalization group equation for fermions and show a determinant bound for the solution. This reduces the combinatorial factor in the recursion for fermionic theories by a factorial, as compared to bosons, and thus suggests that convergence of perturbation theory can be proven in the absence of couplings that grow under the renormalization flow.

Gauge dependence of the high-temperature two-loop effective potential for the Higgs field.

The high-temperature limit of the 2-loop effective potential for the Higgs field is calculated from an effective 3d theory, in a general covariant gauge. It is shown explicitly that a gauge-independent result can be extracted for the equation of state from the gauge-dependent effective potential. The convergence of perturbation theory is estimated in the broken phase, utilizing the gauge dependence of the effective potential.

Polarization electron beams in high energy storage rings Part I. Convergence of perturbation theory

Abstract The spin resonances in high energy electron storage rings are calculated using perturbation theory, for various model storage rings, and the results are compared against analytical formulas (in the case of higher order synchrotron resonances), to elucidate the convergence of the perturbation series. A new analytical formula for synchrotron resonances centered on an integer is also derived, because it is shown that the older formula contains approximations not always valid in modern storage rings. The perturbation series is shown to converge and to agree with the analytical formulas, and the number of term required for convergence is estimated. Some other pedagogical results are also presented.

Comparison between the convergence of perturbation expansions in one-dimensional square and triangle-well fluids

The convergence of perturbation theory in the isobaric and canonical ensembles is investigated for one-dimensional square and triangle-well fluids. It is found that for the square-well fluid the canonical expansion is in much better agreement with the exact results than the isobaric expansion, but that both expansions are in very good agreement with the exact results for the triangle-well fluid.

A note on the convergence of perturbation theory in polymer problems

It is shown that the Fixman perturbation expansion for the number of configurations of a self-interacting polymer molecule (from which the entropy, end-to-end distance etc. can be calculated) is not convergent for any strength of interaction, but is asymptotic.

A numerical study of analyticity in the coupling constant

For two simple field theories, both of zero-space dimension, the Riemann surface of the energy as a function of the square of the coupling constant is studied. The convergence of perturbation theory is established and a comparison is made between the exact energies and their Padé approximations.

The convergence of perturbation expansions in classical statistical mechanics

The most successful theory of liquids is perturbation theory in which the attractive energy is treated as a perturbation on the hard-sphere potential and the canonical partition function expanded in powers of the perturbations. The convergence of perturbation theory in the constant pressure, canonical, and grand canonical ensembles is investigated for a one-dimensional square-well fluid, where the calculations are not too lengthy and where a comparison with exact results can be made. At high temperatures, the convergence is excellent in all three ensembles but, at lower temperatures, the convergence is excellent in the canonical ensemble but only fair in the constant pressure and grand canonical ensembles.

An effective interaction for nuclear hartree-fock calculations

An effective potential is presented which matches S, P, and D-wave nucleon-nucleon phase parameters up to 320 MeV without generating the usual strong, short-range correlations. A suitably defined set of separable potentials, including a noncentral force and having small off-energy-shell matrix elements, is employed to produce smooth two-body wave functions. As a preliminary theoretical test of this effective interaction and of the applicability of perturbation theory, the average energy per particle, the single particle potential, and the symmetry energy of infinite nuclear matter have been computed. A saturation of nuclear matter is found in first order at a Fermi momentum of k F = 1.6 F −1 with an average energy per particle of −8 MeV; second order contributions move the minimum to k F = 1.8 F −1 with an average energy per particle of −14.1 MeV. This saturation is characterized by significant P-wave contributions. A convergence of perturbation theory is indicated by a ratio of second to first order potential energy of <0.10 in all but the coupled 3S 1 + 3D 1 state where the second order potential energy per particle is −4 MeV at k F = 1.8 F −1. It is suggested that this smooth potential, which is related to the realistic interaction by a canonical transformation, can be used in nuclear Hartree-Fock calculations.

Green’s function formulation of the Feynman model of the polaron

The Feynman model of the polaron replaces the real polaron problem, in which the convergence of perturbation theory is rather slow, by a model problem in which perturbation theory converges much faster. The present paper demonstrates a relation between the true electronic Green’s functions and the model Green’s functions which enables one to calculate all the properties of the real polaron in terms of the model Green’s functions. This result is essentially similar to the earlier work of Schultz except for a small but significant difference in the handling of the fictitious field of the model. The Green’s function theory is used to calculate a transport equation for the polaron. The lowest-order transport equation is identical to that published earlier by Kadanoff at low temperatures. The derivation indicates, however, that Kadanoff’s equation is wrong at high temperatures.