Field transformations and simple models illustrating the impossibility of measuring off-shell effects

Abstract

In the context of simple models illustrating field transformations in Lagrangian field theories we discuss the impossibility of measuring off-shell effects in nucleon-nucleon bremsstrahlung, Compton scattering, and related processes. To that end we introduce a simple phenomenological Lagrangian describing nucleon-nucleon bremsstrahlung and perform an appropriate change of variables leading to different off-shell behavior in the nucleonnucleon amplitude as well as the photon-nucleon vertex. As a result we obtain a class of equivalent Lagrangians, generating identical S-matrix elements, of which the original Lagrangian is but one representative. We make use of this property in order to show that what appears as an off-shell effect in an Smatrix element for one Lagrangian may originate in a contact term from an equivalent Lagrangian. By explicit calculation we demonstrate for the case of nucleon-nucleon bremsstrahlung as well as nucleon Compton scattering the equivalence of observables from which we conclude that off-shell effects cannot in any unambiguous way be extracted from an S-matrix element. Finally, we also discuss some implications of introducing off-shell effects on a phenomenological basis, resulting from the requirement that the description of one process be consistent with that of other processes described by the same Lagrangian.

The nucleon as a test case to calculate vector-isovector form factors at low energies

“…One might dub this "offshell ambiguity". In an effective field theory this ambiguity is compensated by the appearance of contact interactions [37].…”

Chiral behavior of vector meson self-energies

“…As far as we are aware, there is no model-independent or natural way to determine the coefficientsd n≥2 . Moreover, one has to be aware of the fact that the off-shell behavior of an amplitude like Π T (s) will in general depend on the chosen parameterization of the fields, see e. g. [56].…”

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“…(2). Although the half-off-shell FFs are not measurable by themselves [18], they can be an important part of various models for physical processes. For example, a two-photon exchange contribution to electron-proton scattering contains γNN vertices with an off-shell nucleon leg.…”