Nuclear Electrodynamics Off-Shell Effects and Gauge Invariance

Electromagnetic properties of o -shell particles in the electrodynamics of the few-body systems are discussed in a connection with the fundamental requirements of gauge invariance. The de nition of the o -shell electromagnetic (EM) form factors for any virtual particles is not gauge invariant, depends on the representation of the full conserved current and cannot be investigated in practice. Any \modi cations" of the EM properties of the nucleons in ห้องสมุดไป่ตู้he nuclear media are unobservable.
2 O -Shell Form Fsctors and Gauge Invariance
2.1 Polarization states of the virtual photons
Here we will remind only few statements from standard QED which will help us to understand the problems connected with the conservation of the nuclear current in the case of nuclear electrodynamics. To introduce polarization states of the virtual photons, let us note that spin one massless eld has two physical polarization states only (q), with = 1; 2. However, these three 4-vectors: q , =1;2 (q) do not span 4-dimensional space. As a result standard Lorentz condition k = 0, which de ne polarization vector of the \on-shell" (real) photon completely, is not enough for spin one massive particles to determine the polarization vectors uniquely. To get a unique de nition of (q2 6= 0) for off-shell photons we have to sacri ce Lorentz covariance, and to choose an additional vector n : n q 6= 0 ; n = 0; (1) 2
Nuclear Electrodynamics: O -Shell E ects and Gauge Invariance
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S.I. Nagorny1 2 NIKHEF, P.O. Box 41882, 1009 DB Amsterdam, The Netherlands
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Frije Universiteit, de Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
Abstract
1 Introduction
Nuclear electrodynamics is a well known source of the \troubles" connected with gauge invariance and the couplig of the EM eld to the o -shell constituents. For a long time \media modi cation" of the EM properties of the bound nucleons attracted attention of the theorists and experimentalists. Electromagnetic properties of o -shell (virtual) particles have been the subject of intensive investigations1?4 for a long time. It was shown that the number of o -shell form factors (FF) in the general ? NN vertex with one (two)- o shell nucleon(s) increases2 to 4 (6) compared to two FF for the free nucleon as a spin-1/2 particle. Dispersion relations1 for o -shell FF were introduced by Bincer. Various one-loop calculations2?4 suggested their model-dependence and possible in uence the observables in the electron-nucleus scattering. It is well known that any o -mass-shell extension of a matrix element is not unique: various representations of the eld operators give rise to di erent o shell Green functions and o -shell matrix elements, while all of these lead to the same S-matrix. O -shell properties of Green functions and in particular their dependence on the representation of the Lagrangian have been discussed in the framework of eld theory for a long time5. Recently the method of e ective Lagrangians was applied to real Compton scattering where the representation dependence of o -shellness has been pointed out6 . The possibility to move o shell e ects from irreducible vertices of the Born current to the regular part of the amplitude was used6 to obtain a unique de nition for polarizabilities in Compton scattering on nucleons. Also in chiral perturbation theory applied to Real Compton scattering o the pion, it was shown7 that o -shell e ects 1
depend not only upon the model of Lagrangian, but also on its representation. The possibility to shift any explicit o -shell dependence of the irreducible threepoint Green function to the regular part of the amplitude has also been used to derive8;9 the low energy theorem for the virtual Compton scattering (VCS). From the general point of view it is clear that the de nition of an off-shell FF is directly connected with the possibility to introduce a one-body off-shell EM current independently of the full EM reaction amplitude. However, due to requirements of gauge invariance one- and many-body currents cannot be considered separately10 . Since the one-body o -shell current is not conserved itself, the de nition of o -shell FF will not be gauge invariant. Therefore, o shell e ects in the ? NN vertex should be considered in connection with the representation of the full conserved current. Here we will use a eld-theoretical approach10 formulated in terms of the Green functions, and explore the gauge nature of the EM eld. In the \minimal" scheme the insertion of t he EM eld into the strong n-point Green function (suppose to be known) provides the de nition of the (n+1)-point EM Green function independent of the details of the strong interaction. This phenomenological way is especially convenient for high-energy nuclear (or intermediate energy particle) physics when the microscopic lagrangian is not known and we have to deal often only with its \matrix elements" - nuclear vertex functions. The Ward Takahashi identities (WTI), which connect (n+1)-point EM Green functions (with EM eld) and strong n-point Green functions (without EM eld), are an important ingredient of the gauge invariant theory.