Reply to the "Comment on `Once more about the $K\bar{K}$ molecule approach to the light scalars' "

TL;DR

This paper clarifies that high-momentum kaon loops are significant in gauge-invariant regularization methods, countering the misconception that only low-momentum contributions matter in the $Kar{K}$ molecule approach.

Contribution

It demonstrates that regularization techniques like Pauli-Villars do not exclude high-momentum effects, challenging previous assumptions about nonrelativistic dominance.

Findings

High-momentum kaon loops are relevant in gauge-invariant regularizations.

Regularization methods can obscure the true momentum contributions.

Nonrelativistic approximations may overlook significant effects.

Abstract

The need to regularize loop integrals in a manner that preserves gauge invariance, for example, using the Pauli-Villars method, requires a subtraction that in the large mass limit hides its high momentum origin. This gives rise to the illusion that only nonrelativistic kaon loop momenta are relevant, when in fact this is not the case, as we show.