Finite-order Perturbation Theory Research Articles

Vibrational corrections to static electric properties of diatomics by Numerov–Cooley integration

We present a semi-numerical method based on Numerov–Cooley (NC) integration for calculating vibrational corrections to static electric properties of diatomic molecules. The approach exploits the Born–Oppenheimer energy and property functions, utilizing such information in the NC integration of the field-dependent vibrational equation, in a way which exposes separate contributions to the total electric property. The scheme is illustrated by computation of electronic and vibrational contributions to the dipole polarizability and first hyperpolarizability of the KLi dimer. A comparison with results obtained from finite-order perturbation theory reveals various sources of anharmonicity.

Infrared safe but infinite: soft-gluon divergences inside the physical region

We show that QCD observables defined as infrared- and collinear-safe, according to the usual Sterman-Weinberg criteria, can nevertheless be infinite at accessible points inside phase space, to any finite order of perturbation theory. Infrared finiteness is restored after resummation of divergent terms to all orders. The resulting characteristic structure, which we call a Sudakov shoulder, represents an interesting new class of QCD predictions.

Perturbative unification of soft supersymmetry-breaking terms

Perturbative unification of soft supersymmetry-breaking (SSB) parameters is proposed in gauge-Yukawa unified models. The method, which can be applied in any finite order in perturbation theory, consists in searching for renormalization group invariant relations among the SSB parameters, which are consistent with perturbative renormalizability. For the minimal gauge-Yukawa unified model based on SU(5) we find that the low energy SSB sector contains a single arbitrary parameter, the unified gaugino mass. Within a certain approximation we find that the model predicts a superpartner spectrum which is consistent with the experimental data.

Dyson summation without violating Ward identities and the Goldstone-boson equivalence theorem.

In contrast to the conventional treatment of gauge theories, in the background-field method the Ward identities for connected Green functions are not violated by Dyson summation of self-energies in finite orders of perturbation theory. Thus, Dyson summation does not spoil gauge cancelations at high energies which are ruled by the Goldstone-boson equivalence theorem. Moreover, in the background-field method the precise formulation of the equivalence theorem in higher orders (including questions of renormalization) is simplified rendering actual calculations easier. Finally, the equivalence theorem is also formulated for the Standard Model with a non-linearly realized scalar sector and for the gauged non-linear $\sigma$-model.

Renormalons

The calculation of higher twist (or dimension) corrections to physical quantities using operator product expansions is delicate. If dimensional regularization is used to regulate the ultra-violet divergences then there are ambiguities in the Wilson coefficient functions due to infra-red renormalon singularities. With a hard ultra violet cut-off, such as the inverse lattice spacing $a$, there are no renormalon ambiguities, as a result of cancellations between terms which in finite orders of perturbation theory diverge as inverse powers of $a$, and those which diverge at most logarithmically. In this lecture I review these questions, explaining the steps necessary to obtain predictions for physical quantities from lattice measurements of matrix elements of higher dimensional operators. The ideas are illustrated by considering quantities computed using the heavy quark effective theory beyond leading order in the heavy quark mass.

The algebra of effective Hamiltonians and operators: Truncated operators and computational aspects

We extend to finite orders of perturbation theory our previous analysis of effective Hamiltonians h and effective operators a which produce, respectively, exact energies and matrix elements of a time-independent operator A for a finite number of eigenstates of a time-independent Hamiltonian H. The validity of various properties is examined here for perturbatively truncated h and a, particularly, the preservation upon transformation to effective operators of commutation relations involving H and/or constants of the motion, of symmetries, and of the equivalence between dipole length and velocity transition moments. We compare formal and computational features of all a definitions and of the more limited Hellmann–Feynman theorem based ‘‘effective operators,’’ which provide only diagonal matrix elements of A in special cases. Norm-preserving transformations to effective operators are found to yield a simpler effective operator formalism from both formal and computational viewpoints.

A coupled-cluster study of inversion symmetry breaking in the F+2 molecular ion

Several coupled-cluster methods have been used to calculate equilibrium properties (re, ωe, and De ) of the ground state (2Πg) of the F+2 molecular ion. Two unrestricted Hartree–Fock reference determinants have been used. The first was made up of inversion symmetry constrained D2h orbitals, the second of inversion symmetry broken C2v orbitals. The results of the coupled-cluster calculations are rather insensitive to the choice of reference determinant, in contrast to what is observed for finite-order perturbation theory. It follows that for certain symmetry breaking problems, single reference coupled-cluster methods are sufficiently powerful to overcome a symmetry broken reference function, thus in principle obviating the need for either a multireference starting point or a symmetry constrained single reference starting point. Some extended basis set coupled-cluster calculations of equilibrium properties of F+2 and F2 were performed. Very good agreement with experiment was obtained for F2, suggesting that the results for F+2 are also reliable.

Asymptotic freedom and mass generation in the O(3) nonlinear σ-model

We complement a recently discovered collective Monte Carlo algorithm with a cluster variance reduction technique for the two-point function of σ-models. It is used to determine both the nonperturbative mass and the asymptotic freedom scale Λ MS from simulations of the O(3)-model with correlation lengths up to 121 on large lattices. We also estimate Λ MS in lattice units by studying the mass gap in physically small volumes. Asymptotic scaling with the bare coupling still does not occur. We propose an approximate lattice β-function that goes beyond finite-order perturbation theory and leads to better scaling behavior. The various methods give m/Λ MS = 2.5−3.0 for the O(3)-model.

Analysis of the lattice-chiral-fermion proposal of Aoki, Funakubo, and Kashiwa.

We examine the lattice-chiral-fermion scheme proposed by Aoki, Funakubo, and Kashiwa (AFK) in the context of chiral QED in two and four dimensions. In four dimensions, we find that, although the AFK scheme ensures gauge invariance by rendering amplitudes transverse, it does not improve their ultraviolet behavior; as in the Wilson theory, the effective vacuum polarization is quadratically divergent. In both two and four dimensions, we find that there are contributions to the effective gauge-field action beyond those analyzed by AFK. These contributions are not accessible through finite-order perturbation theory.

Variational evaluation of correlation functions for lattice electrons in high dimensions

We present a general formalism for the diagrammatic calculation of correlation functions for Hubbard-type models in terms of projected wave functions. It is shown that in the limit of high spatial dimensionsd only diagrams with bubble-structure remain. This causes correlation functions to have an overall RPA-type form ind→∞. Exact evaluations are performed for the Gutzwiller wave function. Nearest neighbor correlations are shown to be proportional to their value in the non-interacting case, i.e. are renormalized. However, their absolute value is only of order 1/d. Hence this wave function does not describe spin correlations adequately in high dimensions. The asymptotic behavior of the spin-correlation function is extracted and is found to have a scaling form similar tod=1. Assuming this form to hold in all dimensions we show that the Brinkman-Rice transition only occurs ind=∞. Finite orders of perturbation theory in 1/d around this singular point are not sufficient to remove the transition.

Integral quantum Hall effect for nonspecialists

An attempt is made to develop a description of the multielectron quantum state responsible for the integral quantum Hall effect. One goal is to provide intuitive support for the very powerful and general argument of Laughlin that the theoretical relationship is insensitive to complicating details in the interior of the sample. The model the author uses is somewhat more realistic than heretofore in that it is three dimensional, does not ignore the atomic structure of the bulk matter, and does not use an effective-mass approximation. In order to treat the problem quantum mechanically, the complete system, including circuitry external to the system of interest, is replaced by a model closed system consisting of a finite number of electrons. In this model, states with a finite Hall current and voltage are metastable against decay caused by interactions outside the model, such as those with bulk matter excitations. Such states describe the true situation well only in the conductivity plateaus; between plateaus, there would be current flow between the Hall voltage probes corresponding to decaying states. Experimental constraints replace this transverse current by a voltage drop along the direction of current flow. The interactions between the electrons are expressed in terms of a self-consistent potential which gives an independent-particle description as a starting point, and residual interactions which are treated by perturbation theory. The self-consistent potential is found to be important in understanding the properties of the quantum state of the system, such as the existence of the plateaus in conductivity and how the electrons in the (effective) two-dimensional region come to equilibrium with the different Fermi levels in the voltage probes. To all finite orders of perturbation theory, the residual interactions are found not to alter the quantized Hall conductivity.

In-plane next-nearest-neighbour exchange coupling in uniaxial metamagnets: its influence on the low-temperature self-energy

The ladder approximation is performed for metamagnets with uniaxial and exchange anisotropy and the following exchange couplings: in-plane nearest-neighbour ferromagnetic coupling J1, in-plane, next-nearest-neighbour antiferromagnetic coupling J2, and small interplane antiferromagnetic coupling J'. Taking J2 into account encounters non-trivial difficulties but the authors have tackled this problem because a suitable lattice-dependent value of J2=J2 causes the long-wavelength magnon dispersion curve to become a quartic power of the wavevector instead of a quadratic one, so one can expect dramatic consequences. In effect this is the case: the results the authors obtain for FeBr2, which has just the convenient ratio J2/J1 to fit the situation described above, give weak renormalisation and damping for the uniform mode, while for FeCl2 dramatic variances, at least from a quantitative point of view, do not arise with respect to previous approaches, where J2 was neglected. Such a result cannot be inferred on the basis of a finite-order perturbation theory which, on the contrary, would suggest a heavier renormalisation when the dispersion curve is more flat for small wavevectors, as in FeBr2.

Approximate Effective Action for Quantum Gravity

A new gauge-invariant effective action is proposed for quantum gravity, based on older results that go beyond finite-order perturbation theory. Expressed in coordinate space rather than momentum space it should find important applications in the theory of the early universe.

Gauge theory and effective lagrangians

We extend our previous methods to show how to find the effective lagrangian for non-abelian gauge theories quantized in the covariant gauge for arbitrary gauge parameter α. We explicitly calculate the effective lagrangian to the two-loop level. The inductive algorithm used to carry out this calculation is applicable to any quantum field theory, evaluated to any finite order of perturbation theory.

Graphical solutions of renormalization group equations

We show that when the β and γ functions in the Callan–Symanzik equation are calculated to a finite order in perturbation theory, the solution to the equation may be represented by an infinite series of Feynman diagrams obtained by repeated insertions of the lower order diagrams from which the β and γ functions are calculated. We demonstrate this assertion in two ways, by explicitly calculating these graphs in the leading logarithm approximation, and by proving that this set of diagrams satisfies exactly the Callan–Symanzik equation.

A multipole expansion and the Casimir-Polder effect in quantum chromodynamics

A multipole expansion for correlation functions in quantum chromodynamics in the presence of a static quark-antiquark pair is described. We discover a retardation effect, analogous to the Casimir-Polder effect in atomic physics, which is proven to persist to all finite orders of perturbation theory. The correlation functions studied may be used to gain some insight into the total rates for decays of the form ( Q Q ) n′l′ → ( Q Q ) nl + hadrons among states of a heavy quark Q. We also derive the Casimir-Polder potential between heavy mesons.

Field intensity and relativistic considerations in the choice of gauge in electrodynamics

The properties of the electric field gauge are compared with the radiation gauge and other simple gauges for problems where intense-field and/or relativistic effects are significant. Electromagnetic shifts in free-electron mass and momentum, which can be of major importance in both kinematics and dynamics when the field is intense, and which are clearly in evidence in second-order radiation-gauge formulations, do not appear in the electric field gauge in any finite order of perturbation theory. Intense-field expressions like the closed-form Volkov solution, and mass and momentum intensity parameters, are shown to be invariant in form in simple gauges, but find no natural expression in the electric field gauge. It is also shown that intensity effects introduce strongly spin-coupled terms even in nonrelativistic problems. For both charged and neutral bound-state systems, crossed vector-potential terms occur in electric field gauge which prevent separation of the Schr\odinger equation into center-of-mass and relative-coordinate equations when the field intensity is high. This difficulty does not arise in the radiation gauge. These considerations inhibit the use of electric field gauge in intense-field problems.

Elementary finite order perturbation theory for vertical ionization energies

The analysis of ionization energies in Rayleigh–Schrödinger perturbation theory and in propagator theory, previously known separately through third order in electron interaction, are compared in detail using elementary algebraic methods and their equivalence is explicitly shown. Relaxation terms are identified as the ΔESCF contributions and appropriate rules are described for their construction from the electron propagator diagrams of arbitrary order. Correlation terms are obtained separately. The transition operator method is analyzed and found to differ from ΔESCF in third order, contrary to earlier claims.

Spontaneous symmetry breaking in massless field theories, finite temperature and criticality

We study the behaviour at finite temperature of massless field theories exhibiting spontaneously broken solutions. We establish the occurence of a phase transition of the first kind at some critical point T c which can be calculated to any finite order in perturbation theory. Similarly, perturbative methods can be used for thermodynamic functions in all regions, including the critical region. For the case of a gauge theory, we demonstrate the gauge independence of the critical point, the thermodynamic potentials and the order parameter to all orders of perturbation theory.

Hermiticity of the superoperator Hamiltonian in propagator theory

Spurious terms are shown to appear in finite order perturbation theory treatment of the electron propagator in the so-called equation of motion approach. These terms, which cause the superoperator Hamiltonian not to be Hermitian, are canceled when the reference state of the propagator is improved to the next higher order in Rayleigh–Schrödinger perturbation theory.