1. Introduction

The asymptotic freedom of quantum chromodynamics (QCD) can be used to predict the behaviour of deep inelastic structure functions for large . These predictions involve matrix elements of certain local operators, or equivalently, effective quark and gluon momentum distributions (structure functions) at a fixed ``starting'' . We are at present unable to calculate these distributions: some features may be determined from experiment while others must simply be guessed. It is evident, however, that physically acceptable structure functions obtained from asymptotic freedom formulae must remain positive throughout the range of in which the formulae should apply. In this paper we investigate the constraints that this places on the starting momentum distributions.

The asymptotic freedom formulae give directly the variation of the moments and of the gluon and (anti)quark momentum distributions of each flavour , where

In sect. 2 we collect these predictions, and state Hausdorff's theorem, which provides the conditions that the must satisfy in order that be non-negative (for ). Equivalent conditions have previously been used by Nachtmann [1] to derive constraints on the dependence of anomalous dimensions. Nachtmann considered moments which are controlled by a single local operator in the operator product expansion. His analysis therefore applies to the singlet operator with the smallest anomalous dimension, which dominates the singlet structure functions at infinite , but does not address the question of whether the structure functions remain positive at accessible values of (non-singlet combinations of structure functions are controlled by a single local operator but can be negative even at ).

In sect. 3 we show that the moments predicted from any set of positive starting distributions are positive for . We argue that the further conditions which must be satisfied if the complete momentum distributions are to be positive also hold. For , however, one of any set of structure functions eventually becomes negative. This would be irrelevant if it happened only at very small , since asymptotic freedom formulae (1) should not be expected to hold there, but we find that unless the starting distributions are chosen carefully it occurs for close to . Advocates of particular quark and gluon distributions should therefore check that their distributions remain positive in the region where the use of asymptotic freedom formulae is not completely unreliable.

In sect. 4 we derive some analytic constraints on the starting distributions. Assuming that as

we show that for the moments become negative at (where is an arbitrarily small positive number) unless

In fact, the asymptotic freedom formulae cease to apply as at fixed , so that eq. (3) need not be obeyed exactly. However, the constraints found numerically for finite reflect this result. Note that if the formulae of asymptotic freedom are used to generate distributions for , their behaviour as will necessarily satisfy eq. (3).

We also present analytic results for the moments (total momenta carried by each species). We find that the requirement that distributions do not rapidly become negative for places constraints on the division of momentum between quarks, antiquarks and gluons. For example, unless gluons carry at least 20% of the total momentum at , the predicted gluon momentum distribution will become negative at a above . There is also an upper bound on the gluon momentum, and a bound on the antiquark momenta.

In sect. 5, we analyse the numerical consequences of our bounds for various forms of the starting distributions. A typical result is that if the are those deduced from studies of present deep inelastic scattering data, then must fall less steeply with x than about .

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