Search

The use of antithetic variates as a variance reduction method is investigated by numerical experimentation. It is found that if variance reduction is not carefully defined, it can go undetected. With this caveat in mind, it is shown that antithetic variates can reduce variance up to a point. The phenomenon of spontaneous correlation is identified as the cause of failure. The surprising result that it can pay to use non-scoring histories is demonstrated. A total energy deposition problem is studied.

An analytic technique for regulating the infinites of nonlinear equations of quantum gauge theories is described. The technique is ideally suited for dealing with high-order effects characterized by high powers of logarithms. The regulated equation, possessing high order poles with nonlogarithmic residues, is renormalizable.

Two integrals fundamental to the analysis of the integral transport equation in infinitely long, annular geometry are defined and represented as sums of Meijer’s G‐function. Transformations useful for thick or thin annuli are derived, limiting forms for voided annuli are recorded, and asymptotic series are investigated. Special results include the identification of probability integrals for infinite cylinders and generalized, associated Bickley‐Nayler functions as G‐functions with appropriate parameters and variables. The incomplete, generalized, associated Bickley‐Nayler functions are defined for annuli of finite length, and represented as sums of G‐functions for long and short annuli. Asymptotic forms are given for both integrals, one of which reduces to Sievert’s integral. Some illustrative examples important for numerical analysis, evaluation of multiple integrals, summation of series, and the study of the transport equation are reported.

It is shown that the principle of analytic continuation is well suited for regulating the spurious singularity peculiar to Feynman integrals in the axial gauges. This new, analytic regularization is much more powerful than the standard regularization based on the principal-value prescription, at the same time yielding identical results compared to the latter. An analytic representation based on a Meijer G-function is derived for the class of massless, two-point integrals ∫d2ωq[(p − q)2] κ × (q2)μ(q · n)v, where n is the vector defining the axial guage and ω, κ, μ and v are continuous variables. Several important aspects of the representation are discussed and examples of its application are given.

The general one-loop three-vertexГμeλabc(p, q, r) in the four-component formulation of the Yang-Mills theory is calculated in the light-cone gauge. The nonvanishing counter Lagrangian constructed from this three-vertex and the self-energy is proportional to the original Lagrangian, the single renormalization constant being -11g2CYMГ(2−ω)/48π2. Gauge dependent and nonlocal counterterms do not contribute to the renormalization constant, but are needed to verify the appropriate Slavnov-Taylor (ST) and Becchi-Rouet-Stora (BRS) identities.

A new algorithm is presented for determining whether or not a point lies inside a polygon. The method is particularly useful when the coordinates of point and polygon are given with respect to a multi-dimensional coordinate basis. The polygon must be convex and connected, but may be open. As a corollary, a method to determine the convexity of a polygon is revealed.

The one-loop vacuum-polarization tensor for pure Yang-Mills theory is examined using background-field quantization in the light-cone planar gauge, i.e. , with the gauge-fixing Lagrangian Wgf — ( 1 /2a )n Q'D '"( A )n Q, where n =0. The divergent part of the vacuum polarization, [H„,(p)]d;„ is found to depend on both a and n„, and hence is gauge dependent This result d.oes not comply with Kallosh's theorem, according to which the counterterms should be independent of gauge choice. We argue that the occurrence of nonlocal counterterms in the light-cone-type gauges violates one of the implicit assumptions of Kallosh's theorem. We also point out that a similar violation of Kallosh's theorem occurs also in the ordinary light-cone gauge, i.e., using the gaugefixing Lagrangian Wst= — (1/2a)(n. g), where n =0. The role of the light-cone gauge' in supersymmetric and superstring models has recently stimulated an examination of radiative processes in Yang-Mills theories quantized in this gauge. One definite advantage of this gauge choice is that it allows one to eliminate unphysical degrees of freedom from the analysis. In earlier studies of this gauge singularities of the form (k n) ' pres.ented a problem in Feynman integrals when n =0. The "principal value" prescription, first suggested for handling such singularities in the context of dimensional regularization, proved to be deficient as it led to integrals which do not obey naive power counting. Explicit calculations using this prescription also led to results inconsistent with the usual axial anomaly, " renormalizability, ' ' and the vanishing of the )33 function in the N=4 supersymmetric theory. Subsequently Mandelstam and Leibbrandt found a prescription that overcomes all the above-mentioned deficiencies of the principal-value prescription, is consistent with canonical quantization, ' and preserves gauge invariance.

By explicit calculation of radiative corrections to the self-energy and the three-vertex at one-loop level for Yang-Mills theories in the light-cone gauge, it is demonstrated that analytically regulated Feynman integrals defined by the principal value prescription satisfy the one, two, and three-point Ward identities. However, both the calculated self-energy and three-vertex have anomalous, unrenormalizable infinite parts, thus confirming the belief, based on previous calculations of only the self-energy, that the principal value prescription is seriously flawed in the light-cone gauge.

The Mandelstam-Leibbrandt prescription is used to study one-loop structures of the two-component (LCM2) and four-component (LCM4) formalisms of the same Yang-Mills theory in the light-cone gauge. The complete one-loop counter lagrangians are constructed by computing the one-loop two-, three- and four-point vertices. LCM2 is renormalizable order-by-order in g with δ L counter = (Z − 1) L , Z = 1 + 11g 2 C 2 /48π 2 ϵ . For LCM4, both the two- and three-vertices generate anomalous counterterms which, however, cancel upon summation so that the total δ L counter is the same as LCM2. Slavnov-Taylor identities are satisfied in LCM4; they do not exist in LCM2. The method of analytic regularization is used in computation; all invariant and tensor integrals are evaluated using a single representation for light-cone invariant two-point integrals. The calculation is exceedingly simple in LCM2, far less so in LCM4.

The generalized integro-exponential function is defined in terms of the exponential integral (incomplete gamma function) and its derivatives with respect to order. A compendium of analytic results is given in one section. Rational minimax approximations sufficient to permit the computation of the first six first-order functions are reported in another section.

Algorithms are given which permit the swift and accurate evaluation of the collision probabilities Pio and Poo as a function of two variables lying in the range 0 ⩽ κ ⩽ 1 and 0 ⩽ z. The method gives results which are correct to a fractional error of less than 2 × 10−5; six other probabilities may be evaluated simultaneously with no loss of significant figures. The method is designed to take advantage of characteristics of a large set of problems for which the probabilities are required. In these cases-nuclear transmutation or neutron slowing-down—the geometry remains invariant and only the cross section varies. Precalculation of geometrical quantities thus leads to extremely efficient evaluations because in a majority of practical cases the variable z is small and only a few terms in the series are needed. For practical applications however, the method is superior in both speed and accuracy in comparison to contemporary techniques of evaluation, and may have potential application to other functions of two variables (e.g. Sievert's integral).

The background field quantization procedure for pure YM theories is used in conjunction with non-covariant gauges characterized by the gauge fixing term ( 1 2 α)n·Q a f i ab n·Q b where fiab can asume the forms fI = δ ab (i.e. axial gauge), fIIab = (n·D(A))2ab/(n2)2 or fIII = D2(Aab/n2 (i.e. planar gauge) where n2 ≠ 0. Here Aμa and Qμa represent respectively the classical background field and the quantum field. It is noted that if fiab explicitly depends on the background field, then it is necessary to introduce Nielsen-Kallosh ghosts in addition to the expected Faddeev-Popov ghosts. Explicit calculations to one-loop order show that for fII and fIII, the divergent part of the vacuum polarization is [( i /16π 2 )C 2 δ ab g 2 /(2−ω)] 11 3 (p 2 δ μν −p μ p ν ) , while in the axial gauge the vacuum polarization is transverse but α- and n-dependent. The latter result - an apparent contradiction of Kallosh's theorem — is shown to arise due to the unconventional asymptotic behaviour of the vector propagator in the axial gauge.

A comprehensive set of two-point integrals for Yang-Mills theories in the light-cone gauge defined in the Mandelstam-Leibbrandt prescription is evaluated and presented in tabulated form.

A hybrid of dimensional and analytic regularization is used to regulate and uncover a Meijer's G -function representation for a class of massless, divergent Feynman integrals in an axial gauge. Integrals in the covariant gauge belong to a subclass and those in the light-cone gauge are reached by analytic continuation. The method decouples the physical ultraviolet and infrared singularities from the spurious axial gauge singularity but regulates all three simultaneously. For the axial gauge singularity, the new analytic method is more powerful and elegant than the old principal value prescription, but the two methods yield identical infinite as well as regular parts.

It is shown that the method of analytically regulating Yang–Mills theories in the axial gauge preserves gauge invariance. Two‐ and three‐point Ward identities are computed and verified at the one‐loop level. The method also permits a convenient and gauge‐invariant separation of infrared and ultraviolet singularities in the axial gauge. In the axial gauge the renormalization constants Z3=Z1=1+11g2C2/(48πe2), leading to a β function which is identical to that computed in the covariant ξ gauges.

A numerical study of the convergence of the R-function solution to the Helmholtz equation is presented in this paper. Several two-dimensional domains of simple polygonal shape have been considered. Calculations have been carried out by four types of trial functions derived from two different solution structures. In addition, a singular function series is applied for the purpose of comparison. In the case of convex domains, one of the presented approximations yields an accurate solution with a very low number of degrees of freedom. However, the accuracy is very poor for the reentrant region and a separate treatment of singularity seems to be necessary.