Arbitrary Order In Perturbation Theory Research Articles

A theorem on the power of supersymmetry in Matrix theory

For the so-called source-probe configuration in Matrix theory, we prove the following theorem concerning the power of supersymmetry (SUSY): let δ be a quantum-corrected effective SUSY transformation operator expandable in powers of the coupling constant g as δ=∑ n⩾0 g 2 n δ ( n) , where δ (0) is of the tree-level form. Then, apart from an overall constant, the SUSY Ward identity δΓ=0 determines the off-shell effective action Γ uniquely to arbitrary order of perturbation theory, provided that the SO(9) symmetry is preserved. Our proof depends only on the properties of the tree-level SUSY transformation laws and does not require the detailed knowledge of quantum corrections.

Renormalizability of the dynamical two-form

A proof of renormalizability of the theory of the dynamical non-Abelian two-form is given using the Zinn-Justin equation. Two previously unknown symmetries of the quantum action, different from the BRST symmetry, are needed for the proof. One of these is a gauge fermion dependent nilpotent symmetry, while the other mixes different fields with the same transformation properties. The BRST symmetry itself is extended to include a shift transformation by use of an anticommuting constant. These three symmetries restrict the form of the quantum action up to arbitrary order in perturbation theory. The results show that it is possible to have a renormalizable theory of massive vector bosons in four dimensions without a residual Higgs boson.

Qualitative treatment of nonrigid motions in the HOOH molecule

Internal motion in the hydrogen peroxide (HOOH) molecule is analyzed qualitatively with inclusion of trans- and cis-type transitions between its two independent, energetically equivalent equilibrium configurations and with the use of the chain concept of symmetry groups. The analysis involves the classification of stationary states and construction of effective operators of physical parameters which characterize the rotational spectrum of the molecule in the presence of nonrigid motions in an arbitrary order of perturbation theory. The set of necessary equilibrium configurations is specified using the coordinate spin operator. An effective Hamiltonian is used as an example to show that the simplicity of the description and its operator form for all types of motion under study are the important advantages of this approach.

Calculation of multiloop diagrams in high orders of perturbation theory

A number of scalar massless diagrams is rigorously calculated in higher orders of perturbation theory. A method of calculating a wide class of diagrams, which may be calculated in an arbitrary order of perturbation theory with the use of the uniqueness technique, is presented. A massless diagram with five loops, which has a special form and which does not belong to the mentioned class, is calculated analytically as well.

Operator regularization and massive Yang–Mills theory

Operator regularization has proved to be a practical method for computing finite, symmetry preserving Green's functions to arbitrary order in perturbation theory. This finiteness has been shown to persist in such nonrenormalizable theories as [Formula: see text] and quantum gravity; in this paper we apply the technique to massive gauge theories. The mass for the vector particle is inserted by hand using the Stueckleberg–Kunimasa–Goto mechanism; no degree of freedom associated with the Higgs scalar is present in the initial Lagrangian. Several one-loop calculations serve to illustrate that no divergences ever arise in this theory if one uses operator regularization. The one-loop effective action is examined in both 4 and n = (4 − ε) dimensions. The result is finite in 4 dimensions and in (4 − ε) dimensions has the pole structure at ε = 0 found by Kafiev. In four dimensions, we find that, in contrast to the case of renormalizable theories, radiatively induced interactions occur with coefficients depending on log (p2/μ2). Consequently μ2 is no longer an arbitrary parameter but must be fixed experimentally.

The high energy behavior of open string scattering

The high energy scattering of open strings is studied using semiclassical techniques. It is shown that the integrals for the open string amplitudes are dominated by saddlepoints. In the case of orientable strings the dominant saddlepoints, to arbitrary order in perturbation theory, can be constructed from the closed string saddlepoints obtained previously. A variant of this technique is used to construct the dominant saddles for nonorientable surfaces. It is found that certain topologies do not contain dominant saddles. The one loop open string amplitudes are studied in detail to illustrate these issues. The semiclassical trajectories corresponding to these saddlepoints are described. This analysis is then applied, following the analogous treatment of the closed string, to derive an infinite set of linear between open string scattering amplitudes at infinite energy that hold order by order in perturbation theory.

The high-energy behavior of string scattering amplitudes

The large-energy, fixed-angle, behavior of string scattering amplitudes is considered to arbitrary order in perturbation theory. It is shown that the path integal over surfaces is dominated, in this limit, by a saddle point. The dominant saddle point is identified and its contribution to the amplitude is evaluated.

Topologically massive chromodynamics in the perturbative regime.

Topologically massive SU(N) gauge theories are studied by using the loop expansion in the Landau gauge. Ward identities for infinitesimal and topologically nontrivial gauge transformations are derived, and checked to one-loop order. The renormalized propagators and vertices are shown to be well behaved about zero momentum to arbitrary order in perturbation theory. We also establish that only massive states contribute to the discontinuities of physical amplitudes.

Calculation of ladder diagrams in arbitrary order

A scalar ladder propagator-type diagram is rigorously calculated in an arbitrary order of perturbation theory.

Number of Feynman diagrams in arbitrary order of perturbation theory

Recurrence relations are established to determine the number of Feynman diagrams in arbitrary order of perturbation theory for four expansions: (i) the Green's function $\mathcal{G}$ expanded in the noninteracting Green's function ${\mathcal{G}}^{(0)}$ and the bare interaction $V$, (ii) the proper self-energy $\ensuremath{\Sigma}$ expanded in ${\mathcal{G}}^{(0)}$ and $V$, (iii) $\ensuremath{\Sigma}$ expanded in $\mathcal{G}$ and $V$, and (iv) $\ensuremath{\Sigma}$ expanded in $\mathcal{G}$ and the particle-particle (hole-hole) ladder sum $\ensuremath{\Gamma}$. In each case, the number of diagrams has the asymptotic behavior $\mathrm{const}\ifmmode\times\else\texttimes\fi{}(2n+1)$!! for large $n$.

Dielectric Function of Free Carriers with a Nonparabolic Dispersion Relation

AbstractGeneral expressions for the Bloch density matrix, the Dirac density matrix, and the generalized density matrix for free carriers with a spherical but nonparabolic dispersion relation are derived in arbitrary order of perturbation theory. The linear response function of a free carrier gas with Kane's dispersion law is considered in detail. The deviations from the case of parabolic bands are discussed.

On exponentiation of the singlet quark form factor in the leading-logarithm approximation

The leading-logarithm approximation of the singlet quark form factor is studied. In arbitrary order of perturbation theory the leading asymptotics of graphs involving no more than one three-gluon vertex is calculated. Cancellation of terms proportional to the Casimir operator CA is shown.

Second-order contributions to the structure functions in deep inelastic scattering (I). Theoretical calculations

In this paper we present a simple analytical expression for the next-to-leading corrections to the contributions to scaling violations of non-singlet, twist-two operators. Using these expression, we solve the point-like inversion problem to this order of perturbation theory in an explicit form. The formulas given are general enough to allow continuation of the procedure to arbitrary orders in perturbation theory. The method is furthermore extended to give also an analytic treatment of target mass corrections. To exemplify the method, a few phenomenological implications are discussed, a full comparison with experiment being left for a following paper.

Weak Turbulence Theory of Velocity Space Diffusion in a Magnetic Field

The particle-wave interaction to arbitrary order in perturbation theory is investigated for a weakly turbulent plasma in the presence of a uniform magnetic field. It is found that to every order in perturbation theory, the ensemble average distribution function obeys a diffusion equation in v⊥ vz space. By conservation of energy and momentum between waves and particles, the growth or damping rate to any order may be determined. The damping rate for nonlinear cyclotron damping, the particle energization rate from very low-frequency turbulence, and the growth rate for a nonlinear instability are calculated.

Weak Turbulence Theory of Velocity Space Diffusion and the Nonlinear Landau Damping of Waves

The particle-wave interaction to arbitrary order in perturbation theory is investigated. One finds that to every order the ensemble average distribution function obeys a diffusion equation. The diffusion constant specifies how the energy and momentum of particles change in time. By conservation of energy and momentum between waves and particles, the growth or damping rates of waves to any order can be determined. The growth rates for nonlinear instabilities and the damping rates for nonlinear Landau damping are explicitly calculated.

Soft-Photon Theorem for Bremsstrahlung in a Potential Model

The bremsstrahlung matrix element calculated from the Schr\"odinger equation for two particles interacting with each other via a local potential and with the electromagnetic field in the standard gauge-invariant manner is shown to satisfy the soft-photon theorem. That is, the first two terms of the expansion of the radiative matrix element in powers of the photon's energy are calculable from a knowledge of the nonradiative matrix element. The proof is given first without, then with, spin for arbitrary order of perturbation theory in the strong interaction. While the derivation is very different from the one which Low used for a relativistic theory, the final formulas are similar in appearance and agree in the nonrelativistic limit. For low particle momenta, it is found that there is a relation between the on- and off-energy-shell derivatives of the nonradiative $T$ matrix to the leading order in the momentum. Furthermore, the $p$-wave contribution to the internal-emission matrix element is of the same order in the momentum as the $s$-wave part. For particle momenta much less than the reciprocal of the $s$-wave scattering length, the second term of the expansion of the bremsstrahlung matrix element in powers of the photon's energy is negligible compared with the first term.

On infra-red divergence in quantum electrodynamics

The paper examines the cancellation of infra-red divergence in the perturbation theory of quantum electrodynamics. The prescription for the cancellation of the infra-red divergence, in the interaction energy of an electron and electromagnetic field is given to all orders of perturbation theory. It is shown that the magnetic moment of electron computed to an arbitrary order of perturbation theory is not dependent on the mode of regularization of divergent integrals in the infra-red region.