The Gerasimov-Drell-Hearn sum rule and nucleon structure

Abstract

The text between equations (16) and (19) (including equation (19)) should be replaced by the following: Neither equation (14) nor the commutator in equations (15) and (16) are in general covariant under Lorentz transformation. They contain Schwinger-type terms that depend on the spatial components of qµ. Therefore, equation (16) cannot be applied to any frame in general. However, for the physically interesting covariant parts of equations (14)-(16), equation (16) is correct in any frame. Equation (15) can, in fact, be generalized to the case in which q2 is fixed and is allowed to go to infinity [3] in an infinite momentum frame for the nucleon in which a sum rule without the seagull and Schwinger term contributions to the final results can be obtained for the commutator of the current (see [1] for a partial demonstration of such a cancellation of seagull and Schwinger terms in the context of one loop electroweak theory in the infinite momentum frame). Using the form of the EM current operator written in terms of quark fields, the naive commutator between the currents in a specific frame is where Q is the charge operator. The naive equal-time commutator in equation (17) is not Lorentz covariant. The canonical value of the commutator is defined to coincide with the naive one in equation (17) when its matrix elements are taken between nucleon states at rest. So, the matrix element of the canonical equal-time commutator between two EM current density operators can be obtained from the rest-frame matrix elements by a proper Lorentz transformation. It is of the following form: with pµ the nucleon 4-momentum and GNaxial the finite `charge square weighted axial charge' of a nucleon. Then the form of equation (16) in an infinite momentum frame of the nucleon takes the form References [1] Pantförder R, Rollnik H and Pfeil W 1998 Eur. Phys. J. C 1 585