Nucleon Structure, Duality and Elliptic Theta Functions

Abstract

Nucleon structure functions are shown to have a qualitative (or `formal') relation to the classical elliptic theta functions. In particular, theta_1'/2 shows a clear resemblance to a non-singlet structure function like xF_3. In the appropriate range the Q^2-dependence of the moments of theta_1'/2 is in near-quantitative agreement with QCD, and at low-Q^2 the moments converge to a common value, as observed experimentally for xF_3. At very high Q^2 (Q^2 -> M_PL), theta_1'/2 -> eta^3(x) (where eta(x) is the Dedekind eta function), while xF_3 in the same limit appears closer to eta^5(x). A comparison of the theta function identity theta_2^4(0)=theta_3^4(0)-theta_4^4(0) with the relation xF_3=Q(x)-Q^-(x) suggests that singlet structure functions have more in common with theta_3 and theta_4. The possibility of interesting large-x/small-x `duality' relations for structure functions emerges naturally from the analysis.