# On the Cottingham formula and the electromagnetic contribution to the proton-neutron mass splitting

## TL;DR

This paper reviews the Cottingham Formula's role in calculating the electromagnetic contribution to the proton-neutron mass difference, discussing its development, challenges, and modeling approaches for accurate predictions.

## Contribution

It provides a comprehensive review of the Cottingham Formula, including its renormalization, the impact of QED and QCD mixing, and compares models for the subtraction function to improve uncertainty estimates.

## Key findings

- Analysis of the Cottingham Formula's development and challenges
- Comparison of models for the subtraction function
- Discussion on theoretical uncertainties in QED corrections

## Abstract

The excess mass of the neutron over the proton arises from two sources within the Standard Model, electromagnetism and the splitting of the down and up quark masses. The Cottingham Formula provides a means of determining the QED corrections from the forward Compton Amplitude, but this is challenged by the need for a subtraction function and the mixing of the QED and QCD (electro-weak) effects. I review the present understanding of the Cottingham Formula, including a discussion on the development of the formula, its renormalization which induces the mixing of QED and QCD effects, and the necessary modeling of the subtraction function that must be done to arrive a numerical prediction. I summarize the Regge Model originally proposed by Gasser and Leutwyler and I also review the proposed model by Walker-Loud, Carlson and Miller, which is an interpolation function between the low and high $Q^2$ regimes, both of which are anchored by rigorous theoretical underpinnings, for which I argue a more reliable theoretical uncertainty estimate can be obtained.