Calculation of size for bound-state constituents

TL;DR

This paper explores how the size of a proton, as inferred from bound-state energy measurements, depends on the lepton mass in lepton-proton systems using a quantum field theory approach, highlighting the need for higher-order calculations.

Contribution

It introduces a method to estimate proton size from bound-state observables within the RGPEP framework at second order, revealing mass-dependent size variations.

Findings

Proton size extraction depends on lepton mass.

Smaller lepton mass leads to larger inferred proton size.

Differences in proton size between hydrogen and muonic atoms could be a few percent.

Abstract

Elements are given of a calculation that identifies the size of a proton in the Schroedinger equation for lepton-proton bound states, using the renormalization group procedure for effective particles (RGPEP) in quantum field theory, executed only up to the second order of expansion in powers of the coupling constant. Already in this crude approximation, the extraction of size of a proton from bound-state observables is found to depend on the lepton mass, so that the smaller the lepton mass the larger the proton size extracted from the same observable bound-state energy splitting. In comparison of Hydrogen and muon-proton bound-state dynamics, the crude calculation suggests that the difference between extracted proton sizes in these two cases can be a few percent. Such values would match the order of magnitude of currently discussed proton-size differences in leptonic atoms. Calculations using the RGPEP of higher order than second are required for a precise interpretation of the energy splittings in terms of the proton size in the Schroedinger equation. Such calculations should resolve the conceptual discrepancy between two conditions: that the renormalization group scale required for high accuracy calculations based on the Schroedinger equation is much smaller than the proton mass (on the order of a root of the product of reduced and average masses of constituents) and that the energy splittings due to the physical proton size can be interpreted ignoring corrections due to the effective nature of constituents in the Schr\"odinger equation.