7.8 KiB
title: "06 – Radiative Corrections" date: 2026-04-26 tags:
- beta-spectrum/correction/radiative status: active aliases: [Outer Radiative, δ_R, Sirlin Function] cssclasses: [] related_components: 10-nuclear-structure, 07-recoil-effects
Outer Radiative Corrections — R(W,W_0) = 1 + \\delta_R(W,W_0)
Overview: Inner vs. Outer Corrections
The radiatively corrected spectrum (Hayen et al., Eq. 46):
\\frac{d\\Gamma}{dW} = \\frac{d\\Gamma_0}{dW}(1 + \\Delta_R^{V/A})(1 + \\delta_R(W,W_0))
| Type | Symbol | Dependence | Treatment |
|---|---|---|---|
| Inner | \\Delta_R |
Nuclear-independent, absorbed into effective coupling constants | \\Delta_V^R = (2.361 \\pm 0.038)% (Marciano & Sirlin 2006) — NOT part of spectrum shape calculation |
| Outer | \\delta_R(W,W_0) |
Energy-dependent, nucleus-dependent | This is the correction implemented in code (06-radiative-corrections) |
Total Spectral Influence (Eq. 47–48)
R(W,W_0) = 1 + \\frac{\\alpha}{2\\pi}\\left[g(W_0,W) - 3\\ln\\left(\\frac{m_p}{2W_0}\\right)\\right] + Z\\alpha^2\\sum_i \\Delta_i(W) + Z^2\\alpha^3 \\delta\_{3h} + ... \\tag{48}
Order α — Sirlin's g(W_0, W) Function (Eq. 50–52)
The universal correction (same for electrons and positrons, Fermi and GT):
g(W_0,W) = 3\\ln(m_p) - \\frac{3}{4} + \\frac{4\\beta}{1+\\beta}\\left[4\\tanh^{-1}\\beta - 1\\right]
+ \\frac{W_0-W}{\\beta}\\left(-\\frac{3}{3W} + \\ln[2(W_0-W)]\\right)
- \\frac{(W_0-W)^2}{6W^2} + 4\\frac{\\tanh^{-1}\\beta}{\\beta}(1+\\beta^2) - 4\\tanh^{-1}\\beta
where \\beta = p/W.
The Spence function (dilogarithm):
L_s(x) = \\int_0^x \\frac{\\ln(1-t)}{t} dt = -Li_2(x) = -\\sum\_{k=1}^\\infty \\frac{x^k}{k^2} \\tag{51}
Large W_0 limit:
g(W_0 \\to \\infty, W) = 3\\ln\\left(\\frac{M_N}{2W_0}\\right) + \\frac{81}{10} - \\frac{4\\pi^2}{3} \\tag{52}
[!warning] Endpoint divergence
g(W_0, W)has a logarithmic divergence at $W = W_0$ related to soft real photon emission. This must be handled via resummation.
Soft Photon Resummation (Eq. 53–54)
To handle the endpoint divergence, the leading logarithmic term C(\\beta)\\ln(W_0 - W) is replaced with an exponentiated form:
t(\\beta)\\ln(W_0 - W) \\to (W_0-W)^{t(\\beta)} - 1 \\tag{53}
where:
t(\\beta) = \\frac{2\\alpha}{\\pi}\\left[\\frac{\\tanh^{-1}\\beta}{\\beta} - 1\\right] \\tag{54}
C(\\beta) = 4\\left[\\frac{\\tanh^{-1}\\beta}{\\beta} - 1\\right]
Implementation note: The resummation is only applied when W_0 - W < \\delta\_{\\text{cut}} (default \\delta\_{\\text{cut}} = 10^{-3} in m_e units, ≈ 0.5 keV). For W_0 - W \\gg \\delta\_{\\text{cut}}, the standard logarithmic form is used. This ensures resummed and standard modes are identical far from the endpoint.
For tritium the effect is negligible (low endpoint). For higher energies, the ft value shift can reach several parts in 10^4.
Order Zα² Correction (Eq. 55–62)
Three dominant Feynman diagrams: vacuum polarization of bremsstrahlung photon and two electron propagator renormalizations. The axial vector component contributes specifically to \\Delta_3:
\\delta_2(Z,W) = Z\\alpha^2 \\sum\_{i=1}^4 \\Delta_i(W) \\tag{55}
The leading term \\Delta_1 depends on nuclear form factor F(q^2) and is split into:
- Nucleus-independent part
\\Delta_1^0: energy-dependent terms combined with\\Delta_4 - Nuclear-dependent part
\\Delta_1^F: explicit integral forms for uniformly charged sphere
For modified Gaussian model, differences between models for superallowed Fermi transitions ($^{10}$O to $^{54}$Co) are < 10^{-4}.
Order Z²α³ — Sirlin's Heuristic Formula (Eq. 63–67)
\\delta\_{3h} = Z^2\\alpha^3 \\left[a\\ln\\left(\\frac{\\Lambda}{W}\\right) + bf(W) + \\frac{4\\pi}{3}g(W) - 0.649\\ln(2W_0)\\right]
with a = \\frac{\\pi}{3} - \\frac{\\sqrt{3}}{2}, b = \\frac{4}{3\\pi^4}(\\pi^2 - \\gamma_E) - \\frac{\\pi^2}{18}.
Higher Order (Eq. 68–69)
\\delta\_{Z^n\\alpha^{n+1}} \\approx Z^n\\alpha^{n+1} K\_{nm} \\ln^{m-n}\\left(\\frac{\\Lambda}{W}\\right), \\quad K\_{nm} \\approx 0.5 \\tag{68}
\\sum\_{n=3}^\\infty Z^n\\alpha^{n+1} = \\frac{Z^3\\alpha^4}{1-Z\\alpha} \\tag{69}
Relevant for higher Z nuclei only.
Neutrino Radiative Corrections (Eq. 70–71)
Even though neutrinos have no direct electromagnetic interaction, they receive indirect corrections via virtual photon exchange and energy conservation from inner bremsstrahlung:
R\_\\nu = 1 + \\frac{\\alpha}{2\\pi}\\left[3\\ln\\left(\\frac{m_p}{m_e}\\right) + \\frac{23}{4} - Li_2\\left(-\\frac{8}{\\hat{\\beta}}\\right) + ...\\right]
where \\hat{W} = W_0 - W\_\\nu. Much smaller than electron spectrum correction — relevant for reactor neutrino oscillation studies.
Internal Bremsstrahlung (Eq. 72–74)
An additional photon in the final state: n \\to p + e^- + \\bar{\\nu}\_e + \\gamma
Photon ejection probability per unit energy:
\\Phi(W,\\omega) = \\frac{\\alpha p}{\\pi\\omega p'}\\left[\\frac{W+W'^2}{p^2}\\ln\\left(\\frac{W+p}{W'-p}\\right) - 2\\right] \\tag{73}
Experimental branching ratios: BR\_{\\beta\\gamma} \\sim 3\\times 10^{-3} (neutron), \\sim 2\\times 10^{-3} ($^{32}$P).
Calculator Implementation
[!tip] Code mapping Module:
beta_spectrum/components/radiative.py→RadiativeCorrection(W0, Z, A, use_endpoint_resummation, delta_cut)classImplements outer radiative corrections per Sirlin (1967) and Sirlin (1987):
- O(α): Universal Sirlin function
g(W,W_0)withscipy.special.spencefor dilogarithm- O(Zα²): Finite nuclear size correction
\\delta_2(Z,W)using modified Gaussian model- Soft-photon resummation near endpoint via Sirlin (1987) prescription (only when
W_0 - W < \\delta\_{\\text{cut}})- Small-beta Taylor expansion for numerical stability at threshold
- Optional endpoint resummation flag
API Usage
from beta_spectrum import RadiativeCorrection
# Light nucleus (tritium-like) — resummation has negligible effect at low W0
rad = RadiativeCorrection(
W0=1.02,
Z=1,
use_endpoint_resummation=True
)
# Heavy nucleus (uranium) with nuclear model correction
rad = RadiativeCorrection(
W0=5.0,
Z=92,
A=238, # affects Delta_F nuclear model term
use_endpoint_resummation=True,
delta_cut=1e-3 # only apply resummation within 1 keV of endpoint
)
# Standard (non-resummed) mode for comparison
rad_standard = RadiativeCorrection(
W0=5.0,
Z=92,
use_endpoint_resummation=False
)
values = rad(W_grid)
[!note] Resummation scope The soft-photon resummation (Eq. 53) is only applied when
W_0 - W < \\delta\_{\\text{cut}}(default10^{-3}inm_eunits ≈ 0.5 keV). Far from the endpoint, both resummed and standard modes produce identical results. This matches the Sirlin (1987, 2011) prescription where resummation replaces only the leading logarithmic divergence nearW \\to W_0.
Magnitude Summary
| Term | Typical Size | Notes |
|---|---|---|
\\alpha/2\\pi \\cdot g |
~1–3% | Universal, dominant contribution |
Z\\alpha^2 \\Delta_i |
~0.01–0.1% | Nuclear-form-factor dependent |
Z^2\\alpha^3 \\delta\_{3h} |
< 0.01% | Only for heavy nuclei |