Operator product expansions near the light cone

Abstract

The behaviour of products A( x) B(0) of local field operators near the light cone x 2 = 0 is studied in renormalized perturbation theory. Operator product expansions are shown to exist and describe this behaviour in each order. This result is an extensive generalization of known results about short distance ( x μ → 0) behaviour in quantum field theory. For example, the local scalar current j( x) ≡ : ϕ( x) ϕ( x) : of dimension two in ϕ 4 theory, ignoring logs and c-numbers, has the short distance behaviour j(x)j(0) → x→0 c 0 1 x2 j(0)+c 1 1 x2 xα:φ∂αφ. It is shown that the light cone behaviour is j(x)j(0) → x 2→0 1 x2 ∑ n=0 ∞ x αn…x αn0 (n) α 1…α n (0) for some necessarily infinite set 0 (n) α1…αn of local field operators satisfying dim O ( n) α1… α n = n+ 2. One must have O ( o) = c 0 j and O (1) α 1 = c 1 : ϕ∂ α 1 ϕ:, and the ifO (n) for n ⩾ 2 do not contribute for x μ → 0 but they are necessary to describe the x 2 → 0 limit. Similar expansions are derived for the product j μ ( x) j ν (0) of vector currents and for other interesting products in ϕ 4 theory, the gluon model, all other renormalizable models, and in soluble field theoretic models. Properties of the expansions are discussed in detail. The usefulness of such expansions arises from the fact that they describe, for example, the configuration space limit corresponding to the physical momentum space limit in which a very massive current interacts with a hadronic system in a high energy inelastic collision. The operator expansions predict the strength of the light cone singularities and thereby provide a means of measuring the dimensions of inteacting fields, they determine properties of amplitudes in several variables, and they provide relations between form factors describing different experiments corresponding to different matrix elements of the current products. Several such applications are described and it is concluded that the present experimental results are in good agreement with the naive field dimensions and canonical singularity structure of renormalizable field theories.