A Gauge Invariant Unitary Theory for Pion Photoproduction

Abstract

A covariant, unitary and gauge invariant theory for pion photoproduction on a single nucleon is presented. To achieve gauge invariance at the operator level one needs to include both the πN and γπN thresholds. The final amplitude can be written in terms of a distorted wave in the final πN channel provided one includes additional diagrams to the standard Born term in which the photon is coupled to the final state pion and nucleon. These additional diagrams are required in order to satisfy gauge invariance.Most calculations to date for pion photoproduction on a single nucleon have included two ingredients: (i) A Born term which is taken to be gauge invariant. (ii) A final state interaction or distortion that is needed to satisfy the Watson theorem or unitarity.However, the inclusion of the pionic degrees of freedom into the problem changes the current and charge distribution and thus the coupling of the photon to the nucleon.Hence any consistent theory of pion photoproduction has to include the effect of the pionic degrees of freedom on the charge and current distribution. In other words one needs to satisfy unitarity and gauge invariance at the same time. Clearly, the inclusion of both of these symmetries to all orders requires a full solution to the field theory. Here, we

Photon– and pion–nucleon interactions in a unitary and causal effective field theory based on the chiral Lagrangian

“…While the tree-level contributions to the K-matrix and the generalized potential are identical, the one-loop contributions differ. The K-matrix does not receive a contribution of the second order rescattering term (35) nor the CDD pole correction term (17). Only the real part of the one-loop diagrams contribute to the K-matrix.…”

“…In (35) the reference to angular momentum and parity is omitted. The integral ( 35) is finite since the leading-order potential is asymptotically bounded.…”

“…We neglect this imaginary part as higher order and decompose the residual loop-expression according to (21). The truncation of the sum in ( 21) is uniform for all chiral moments of the potential, in particular for the leading-order term specifying the subtraction integral (35).…”

“…Gauge invariant approximations for the Bethe-Salpeter equation are much more difficult to construct[35]. This is because the scattering kernel is required for offshell kinematics and therefore is gauge dependent in this case.…”

Electromagnetic currents of the pion-nucleon system

“…To gauge the bare πN N vertices, we use the method of minimal substitution. The application of this procedure to πN N vertices with cutoff factors is discussed at length in the PhD thesis of van Antwerpen [29] (an abridged version of which was published in [30]). By taking a combination of his results for pseudoscalar and pseudovector coupling, we can immediately write,…”