Initial Tests Of The Pion Beta Decay Detector

2.3.3 Radiative corrections

The empirical value of the [lambda]( p ⁺ -> e⁺ n ) / [lambda]( p ⁺ -> µ⁺ n ) branching ratio ((1.230 ± 0.004) × 10^-4) differs from the first order calculation outlined in section 2.3.1 (1.28 × 10^-4) by 3.9%. Most of this discrepancy can be accounted for by including the radiative corrections to the calculation.

Following the early works by Berman [Ber 58] and Kinoshita [Kin 59], corrections to the transition rates of the p ->e n and p ->µ n decay channels can be separated into three parts. These parts are (1) the inner bremsstrahlung, (2) the virtual photon emission and reabsorption (referred to as correction A), and (3) the lepton mass renormalization (referred to as correction B) corrections.

The inner bremsstrahlung process is one in which there is a real photon in the final state p ⁺ -> l⁺ n g . The lepton energies then occupy a range of values due to the three body final state. The photon energy can be infinitesimally small making the inner bremsstrahlung process with soft photons experimentally indistinguishable from a p -> l n process. The total probability of inner bremsstrahlung [Kin 59] is

where

and

with the infinitesimal energy of the low energy photons is denoted by [lambda]_min.

Virtual photon emission is really at the root of both correction A and correction B. The separation is made as such: The emission and reabsorption of virtual photons modify the interaction operator so that it is proportional to the bare mass of the lepton m_l⁰(this is correction A). The mass renormalization that must be done in order to express the bare mass m_l⁰ in terms of the observed mass m_lintroduces an additional correction (this is correction B).

Correction A takes the form

for the p -> l n decay rate. The values [lambda] and [lambda]_min are the upper and lower photon energy cutoffs respectively.

The [lambda]( p ⁺ -> e⁺ n ( g )) / [lambda]( p ⁺ -> µ⁺ n ( g )) branching ratio can be written as

where R₀ is the raw branching ratio calculated in section 2.3.1, [epsilon] is the combination of correction A and the inner bremsstrahlung correction, and d comes from correction B. These corrections reduce to d = -16.0 ( a / p ) and [epsilon] = -0.92 ( a / p ).

The "mass correction" term d reduces the raw branching ratio by 3.7% while the correction [epsilon] reduces it by another 0.2% bringing the calculation in good agreement with experiment.