Question

Now my concern is when I plug in what I have for Δ and keep only terms of order ϵ0, I am going to have a lot of i's running around. But, were the formulas as defined in appendix A of Peskin & Schroeder with or without the iδ prescription (where in P&S they use iϵ instead which can be confusing so I changed it to iδ). If this is the case then great, I'll just drop the iδ from my Δand proceed to simplify.

But if not, then throughout my calculation will I have to back-track and replace all of my loop momentum products, for example if I was using lμ, l2 with −l2E as told to in A.43 of P&S? And something I would want to clarify on this point is then the question of whether or not I would have to also replace the product of lμki,μ where ki,μ is the momentum of an external particle with a negative as well or just the square of the loop momentum?

Answer 1

The iδ (or iε) prescription is critical in quantum field theory calculations to ensure causality and avoid singularities, but usually does not alter external momenta. One should maintain consistency with signs and prescriptions throughout the calculation.

Step-by-step explanation:

The iδ prescription (sometimes represented as iε in certain texts like Peskin & Schroeder) is integral to the calculation of Feynman propagators in quantum field theory to ensure the correct causal structure of the theory. This term acts as a mathematical device to pick out the proper solutions to the equations of motion and to avoid singularities in the Feynman integrals. It is subtly different from the actual physical quantities you are trying to calculate, and it usually disappears from the final, physical results, particularly after renormalization and taking the real part of the amplitude.

When regarding loop calculations, and specifically when handling loop momenta (as in the product lμ lν), the replacement with negative energy components is done to maintain the causal structure. For instance, if lμ is the loop momentum and kμ is the external momentum, the products involving these would be affected by whether you are using the Feynman prescription or not. In most cases, the external momenta (like those of an external particle) would not be altered by the iδ prescription, but ensuring the consistency in signs and prescriptions throughout the calculation is critical.

Therefore, one does not typically replace the product of loop momentum with external momentum with a negative sign unless there is a specific reason dictated by the calculation procedure or the Feynman rules that have been established for the theory and situation in question.