C. S. Lam

Structure function relations at large transverse momenta in Lepton-pair production processes

Abstract Features of the lepton angular distributions due to hard parton sub-processes in lepton-pair production at relatively large q⊥ are studied. Structure function relations which characterize these mechanisms are derived. Comparisons of these features with those derived from the Drell-Yan mechanism are made.

Decomposition of time ordered products and path ordered exponentials

We present a decomposition formula for Un, an integral of time-ordered products of operators, in terms of sums of products of the more primitive quantities Cm, which are the integrals of time-ordered commutators of the same operators. The resulting factorization enables a summation over n to be carried out to yield an explicit expression for the time-ordered exponential, an expression which turns out to be an exponential function of Cm. The Campbell–Baker–Hausdorff formula and the non-Abelian eikonal formula obtained previously are both special cases of this result.

Invariant integration over the unitary group

Integrals for the product of unitary-matrix elements over the U(n) group will be discussed. A group-theoretical formula is available to convert them into a multiple sum, but unfortunately the sums are often tedious to compute. In this article, we develop an alternative method in which these sums are avoided, and group theory is rendered unnecessary. Only unitarity and the invariance of the Haar measure are required for the computation. The method can also be used to get a closed expression for the simpler integral of monomials over a hypersphere.

GLUING AND SHIFTING LATTICE CONSTRUCTIONS AND RATIONAL EQUIVALENCE

This paper deals with the construction and classification of lattices. Two methods of constructing new lattices from old one — gluing and shifting — are discussed. A fairly exhaustive analysis of shifting is carried out, and its close relationships to gluing and to rational equivalence are investigated. This exploration enables rational equivalence to be seen in a more geometric light than has usually been the case. A related classification of lattices, called similarity, is introduced and studied. New geometric proofs of some old theorems are obtained, as well as a number of new results.

Lattices and θ-function identities. II : Theta series

The gluing construction of lattices is used to generate and study a number of theta function densities. It is shown, for example, that Riemann’s formula, a fundamental degree 4 identity, can be derived from a degree 2 identity. It is proved that all in a large class of identities can be derived from these lattice techniques. A list of 24 independent quadratic identities in the Jacobi functions ϑ1 and ϑ3, conjectured to be complete, with all but two of them seeming to be new, is given. The theta series of glue classes is also investigated, and it is shown that all of them, as well as the functions ϑa, b, satisfy polynomials whose coefficients are linear combinations of theta series of lattices; these polynomials have some interesting properties.

Invariant and group theoretical integrations over the U(n) group

In a previous article, an “invariant method” to calculate monomial integrals over the U(n) group was introduced. In this paper, we study the more traditional group-theoretical method, and compare its strengths and weaknesses with those of the invariant method. As a result, we are able to introduce a “hybrid method” which combines the respective strengths of the other two methods. There are many examples in the paper illustrating how each of these methods works.

Evaluation of multiloop diagrams via lightcone integration

We present a systematic method to determine the dominant regions of internal momenta contributing to any two-body high-energy near-forward scattering diagram. Such a knowledge is used to evaluate leading high-energy dependences of loop diagrams. It also gives a good idea where dominant multiparticle cross sections occur.

Can a lattice string have a vanishing cosmological constant

Given a modular-invariant partition function Q that integrates to a zero cosmological constant, there exists a whole class of functions C Q ={cQ+I} which does this as well; here I is an arbitrary imaginary modular-invariant function and c is an arbitrary scaling constant. The question of whether a nonsupersymmetric lattice string can be constructed to yield any of the partition functions in C Q is addressed. Two methods are devised to sidestep the arbitrariness of the function I, and lattice techniques are used to find conditions necessary for the existence of such a string

Lattices and Θ-function identities. I : Theta constants

The gluing construction of lattices is used to generate and study a number of theta constant identities. It is shown, for example, that the Jacobi identity, a well‐known degree 4 identity, can be derived from a degree 2 identity. A list of all quadratic identities in θ3 derivable from this lattice method, containing over 30 algebraically independent identities, and conjectured to yield all quadratic identities in θ3 derivable by any means, is included. In contrast, only two independent quadratic identities in θ3 have been found elsewhere in the mathematical literature. The theta constants of lattices and their glue classes are investigated, and the fact that the theta constant of each glue class is a root of a polynomial whose coefficients are linear combinations of theta constants of lattices is shown.

Time ordering, energy ordering, and factorization

Relations between integrals of time-ordered product of operators, and their representation in terms of energy-ordered products are studied. Both can be decomposed into irreducible factors and these relations are discussed as well. The energy-ordered representation was invented to separate various infrared contributions in gauge theories. It is shown that the irreducible time-ordered expressions can be used to accomplish the same purpose. Besides, it has the added advantage of being factorizable.