Momentum-cutoff regularization and gauge invariance in QED

Symmetry preserving regularization with a cutoff

“…Various conditions were derived to maintain gauge symmetry, to have the freedom of momentum routing or shifting the loop momentum. These conditions were known by several authors [11,13,20,21]. Our new proposal is that these conditions will be satisfied for the regularization process if terms proportional to loop momenta with an even number of free Lorentz indices (e.g.…”

“…This example shows that the Ward identity is fulfilled only if the shift of the loop momentum does not change the value of the integral, as in (11). In [21] based on the general diagrammatic proof of gauge invariance it is shown that the Ward identity is fulfilled if the difference of a general n-point loop and its shifted version vanishes…”

“…We show at the one loop level that gauge invariance of the vacuum polarization function is equivalent to invariance of a special loop integrand to shifting the loop momentum (11). Consider Π µν defined in (6), performing the trace we obtain…”

“…It holds in dimensional regularization and in the momentum cutoff based on DREG of Section 2. In [11,13] a similar relation defined the finite or infinite Pauli-Villars terms to maintain gauge invariance. So far the integrals have not been performed.…”

“…Various authors formulated consistency conditions to maintain gauge invariance in the evaluation of divergent loop integrals. When finite [11] or infinite [12,13] number of new regulator terms added to the propagators a'la Pauli-Villars, the integrals have at most logarithmic singularities and become tractable. The Pauli-Villars regularization technique was applied with subtractions to gauge invariant and chiral models [14][15][16][17].…”