Possible Explanation for the Rapid Approach to "Universality" of the Inelastic Electron-Scattering Structure Functions

We have determined the single variable analyticity in the complex x-plane of the inelastic electron scattering structure functions, with s kept fixed and real, to all orders of Feynman perturbation theory. We find that its Landau singularities, which move as a function of s, rapidly approach their asymptotic s-independent position once s is large. We discuss how this observation offers a possible explanation for a rapid approach to “universality” of the inelastic electron scattering structure functions and shows that sW2(s,q2/s) should “scale” faster than vW2(s,x) 0 Recent experimental data’ on inelastic electron scattering indicate that for fixed x the structure functions W1 and v W2 become approximately independent2 of s once s is far above the resonance production region (s > 4 GeV2). We will call this region the “deep inelastic region”. This fact has been referred to as “scaling” of the structure functions. 334 We consider two forms of “scaling’*. The first is the s independence of the magnitude of the structurwfunctions for fixed x=q2/(2P*q), which we call “universality of magnitude”, The second is the s-independence of the shape of the curve of the structure functions versus x, which we call the “universality of shape”. *work supported by the U. S. Atomic Energy Commission, (Submitted for publication) We propose that a rapid approach to a universal shape for the uW2 (or WI) curve for s>> 4 GeV2 can be understood as a consequence of the rapid approach of its physical x-sheet singularities to their s-independent asymptotic position once s is large enough. This is provided the %trengths** of these singularities (i, e. : residues of poles and discontinuities across cuts) are slowly varying functions of s for large s. We ignore spin and other quantum numbers since they affect only the **strengths** of these singularities and not their position. We analyze the Landau singularities of the Feynman integrals contributing to the nonBorn term part of the inelastic structure function W,(s, x) for fixed real s. In a more detailed publication we will show to all orders of Feynman perturbation theory that the only Landau singularities on the physical sheet of the complex x-plane are the s independent normal q2 threshold branch points (for real time like q2) and the set of anomalous singularities like x,(s), which move with s, and correspond to the single loop box or triangle reduced graphs shown in Fig. 1. ’ . Their equation is given by5 G(s) = T+K (M2-P-f9(s-~-p)/(W) f (1/2p)[A(M2,P, K) A@, T, p)]1’2 h(x,y,z) = [x-(Jy+ Jz,2J[x-(JyJz)2]= h(z,x,y), etc. x&4 = ‘I 2 /(IqM2*) = -l/w&) where qa(s) = c&s-a) Mf (1)