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--- ______________________________________________________________________
title: "06 – Radiative Corrections" title: "06 – Radiative Corrections"
date: 2026-04-26 date: 2026-04-26
tags: tags:
- beta-spectrum/correction/radiative - beta-spectrum/correction/radiative
status: active status: active
aliases: [Outer Radiative, δ_R, Sirlin Function] aliases: [Outer Radiative, δ_R, Sirlin Function]
cssclasses: [] cssclasses: []
related_components: [[10-nuclear-structure]], [[07-recoil-effects]] related_components: \[[10-nuclear-structure]\], \[[07-recoil-effects]\]
---
# Outer Radiative Corrections — $R(W,W_0) = 1 + \delta_R(W,W_0)$ ______________________________________________________________________
# Outer Radiative Corrections — $R(W,W_0) = 1 + \\delta_R(W,W_0)$
## Overview: Inner vs. Outer Corrections ## Overview: Inner vs. Outer Corrections
The radiatively corrected spectrum (Hayen et al., Eq. 46): The radiatively corrected spectrum (Hayen et al., Eq. 46):
$$\frac{d\Gamma}{dW} = \frac{d\Gamma_0}{dW}(1 + \Delta_R^{V/A})(1 + \delta_R(W,W_0))$$ $$\\frac{d\\Gamma}{dW} = \\frac{d\\Gamma_0}{dW}(1 + \\Delta_R^{V/A})(1 + \\delta_R(W,W_0))$$
| Type | Symbol | Dependence | Treatment | | 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 | | **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]]) | | **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) ## Total Spectral Influence (Eq. 47–48)
$$R(W,W_0) = 1 + \frac{\alpha}{2\pi}\left[g(W_0,W) - 3\ln\left(\frac{m_p}{2W_0}\right)\right] + Z\alpha^2\sum_i \Delta_i(W) + Z^2\alpha^3 \delta_{3h} + ... \tag{48}$$ $$R(W,W_0) = 1 + \\frac{\\alpha}{2\\pi}\\left[g(W_0,W) - 3\\ln\\left(\\frac{m_p}{2W_0}\\right)\\right] + Z\\alpha^2\\sum_i \\Delta_i(W) + Z^2\\alpha^3 \\delta\_{3h} + ... \\tag{48}$$
### Order α — Sirlin's $g(W_0, W)$ Function (Eq. 50–52) ### Order α — Sirlin's $g(W_0, W)$ Function (Eq. 50–52)
The universal correction (same for electrons and positrons, Fermi and GT): The universal correction (same for electrons and positrons, Fermi and GT):
$$g(W_0,W) = 3\ln(m_p) - \frac{3}{4} + \frac{4\beta}{1+\beta}\left[4\tanh^{-1}\beta - 1\right] $$ $$g(W_0,W) = 3\\ln(m_p) - \\frac{3}{4} + \\frac{4\\beta}{1+\\beta}\\left[4\\tanh^{-1}\\beta - 1\\right] $$
$$+ \frac{W_0-W}{\beta}\left(-\frac{3}{3W} + \ln[2(W_0-W)]\right) $$ $$+ \\frac{W_0-W}{\\beta}\\left(-\\frac{3}{3W} + \\ln[2(W_0-W)]\\right) $$
$$- \frac{(W_0-W)^2}{6W^2} + 4\frac{\tanh^{-1}\beta}{\beta}(1+\beta^2) - 4\tanh^{-1}\beta$$ $$- \\frac{(W_0-W)^2}{6W^2} + 4\\frac{\\tanh^{-1}\\beta}{\\beta}(1+\\beta^2) - 4\\tanh^{-1}\\beta$$
where $\beta = p/W$. where $\\beta = p/W$.
The Spence function (dilogarithm): The Spence function (dilogarithm):
$$L_s(x) = \int_0^x \frac{\ln(1-t)}{t} dt = -Li_2(x) = -\sum_{k=1}^\infty \frac{x^k}{k^2} \tag{51}$$ $$L_s(x) = \\int_0^x \\frac{\\ln(1-t)}{t} dt = -Li_2(x) = -\\sum\_{k=1}^\\infty \\frac{x^k}{k^2} \\tag{51}$$
Large $W_0$ limit: Large $W_0$ limit:
$$g(W_0 \to \infty, W) = 3\ln\left(\frac{M_N}{2W_0}\right) + \frac{81}{10} - \frac{4\pi^2}{3} \tag{52}$$ $$g(W_0 \\to \\infty, W) = 3\\ln\\left(\\frac{M_N}{2W_0}\\right) + \\frac{81}{10} - \\frac{4\\pi^2}{3} \\tag{52}$$
> [!warning] Endpoint divergence > [!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. > $g(W_0, W)$ 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) ### Soft Photon Resummation (Eq. 53–54)
To handle the endpoint divergence: To handle the endpoint divergence, the leading logarithmic term $C(\\beta)\\ln(W_0 - W)$ is replaced with an exponentiated form:
$$t(\beta)\ln(W_0 - W) \to (W_0-W)^{t(\beta)} - 1 \tag{53}$$ $$t(\\beta)\\ln(W_0 - W) \\to (W_0-W)^{t(\\beta)} - 1 \\tag{53}$$
where: where:
$$t(\beta) = \frac{2\alpha}{\pi}\left[\frac{\tanh^{-1}\beta}{\beta} - 1\right] \tag{54}$$ $$t(\\beta) = \\frac{2\\alpha}{\\pi}\\left[\\frac{\\tanh^{-1}\\beta}{\\beta} - 1\\right] \\tag{54}$$
$$C(\\beta) = 4\\left[\\frac{\\tanh^{-1}\\beta}{\\beta} - 1\\right]$$
**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 $10^4$. 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) ### 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 $\Delta_3$: Three dominant Feynman diagrams: vacuum polarization of bremsstrahlung photon and two electron propagator renormalizations. The axial vector component contributes specifically to $\\Delta_3$:
$$\delta_2(Z,W) = Z\alpha^2 \sum_{i=1}^4 \Delta_i(W) \tag{55}$$ $$\\delta_2(Z,W) = Z\\alpha^2 \\sum\_{i=1}^4 \\Delta_i(W) \\tag{55}$$
The leading term $\Delta_1$ depends on nuclear form factor $F(q^2)$ and is split into: 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 - **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 < $10^{-4}$. 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) ### Order Z²α³ — Sirlin's Heuristic Formula (Eq. 63–67)
$$\delta_{3h} = Z^2\alpha^3 \left[a\ln\left(\frac{\Lambda}{W}\right) + bf(W) + \frac{4\pi}{3}g(W) - 0.649\ln(2W_0)\right]$$ $$\\delta\_{3h} = Z^2\\alpha^3 \\left[a\\ln\\left(\\frac{\\Lambda}{W}\\right) + bf(W) + \\frac{4\\pi}{3}g(W) - 0.649\\ln(2W_0)\\right]$$
with $a = \frac{\pi}{3} - \frac{\sqrt{3}}{2}$, $b = \frac{4}{3\pi^4}(\pi^2 - \gamma_E) - \frac{\pi^2}{18}$. 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) ### Higher Order (Eq. 68–69)
$$\delta_{Z^n\alpha^{n+1}} \approx Z^n\alpha^{n+1} K_{nm} \ln^{m-n}\left(\frac{\Lambda}{W}\right), \quad K_{nm} \approx 0.5 \tag{68}$$ $$\\delta\_{Z^n\\alpha^{n+1}} \\approx Z^n\\alpha^{n+1} K\_{nm} \\ln^{m-n}\\left(\\frac{\\Lambda}{W}\\right), \\quad K\_{nm} \\approx 0.5 \\tag{68}$$
$$\sum_{n=3}^\infty Z^n\alpha^{n+1} = \frac{Z^3\alpha^4}{1-Z\alpha} \tag{69}$$ $$\\sum\_{n=3}^\\infty Z^n\\alpha^{n+1} = \\frac{Z^3\\alpha^4}{1-Z\\alpha} \\tag{69}$$
Relevant for higher $Z$ nuclei only. Relevant for higher $Z$ nuclei only.
@@ -84,28 +91,29 @@ Relevant for higher $Z$ nuclei only.
Even though neutrinos have no direct electromagnetic interaction, they receive indirect corrections via virtual photon exchange and energy conservation from inner bremsstrahlung: Even though neutrinos have no direct electromagnetic interaction, they receive indirect corrections via virtual photon exchange and energy conservation from inner bremsstrahlung:
$$R_\nu = 1 + \frac{\alpha}{2\pi}\left[3\ln\left(\frac{m_p}{m_e}\right) + \frac{23}{4} - Li_2\left(-\frac{8}{\hat{\beta}}\right) + ...\right]$$ $$R\_\\nu = 1 + \\frac{\\alpha}{2\\pi}\\left[3\\ln\\left(\\frac{m_p}{m_e}\\right) + \\frac{23}{4} - Li_2\\left(-\\frac{8}{\\hat{\\beta}}\\right) + ...\\right]$$
where $\hat{W} = W_0 - W_\nu$. Much smaller than electron spectrum correction — relevant for reactor neutrino oscillation studies. where $\\hat{W} = W_0 - W\_\\nu$. Much smaller than electron spectrum correction — relevant for reactor neutrino oscillation studies.
## Internal Bremsstrahlung (Eq. 72–74) ## Internal Bremsstrahlung (Eq. 72–74)
An additional photon in the final state: $n \to p + e^- + \bar{\nu}_e + \gamma$ An additional photon in the final state: $n \\to p + e^- + \\bar{\\nu}\_e + \\gamma$
Photon ejection probability per unit energy: Photon ejection probability per unit energy:
$$\Phi(W,\omega) = \frac{\alpha p}{\pi\omega p'}\left[\frac{W+W'^2}{p^2}\ln\left(\frac{W+p}{W'-p}\right) - 2\right] \tag{73}$$ $$\\Phi(W,\\omega) = \\frac{\\alpha p}{\\pi\\omega p'}\\left[\\frac{W+W'^2}{p^2}\\ln\\left(\\frac{W+p}{W'-p}\\right) - 2\\right] \\tag{73}$$
Experimental branching ratios: $BR_{\beta\gamma} \sim 3\times 10^{-3}$ (neutron), $\sim 2\times 10^{-3}$ ($^{32}$P). Experimental branching ratios: $BR\_{\\beta\\gamma} \\sim 3\\times 10^{-3}$ (neutron), $\\sim 2\\times 10^{-3}$ ($^{32}$P).
## Calculator Implementation ## Calculator Implementation
> [!tip] Code mapping > [!tip] Code mapping
> Module: `beta_spectrum/components/radiative.py` → `RadiativeCorrection(W0, Z, A, use_endpoint_resummation)` class > Module: `beta_spectrum/components/radiative.py` → `RadiativeCorrection(W0, Z, A, use_endpoint_resummation, delta_cut)` class
> >
> Implements outer radiative corrections per Sirlin (1967) and Sirlin (1987): > Implements outer radiative corrections per Sirlin (1967) and Sirlin (1987):
>
> - **O(α)**: Universal Sirlin function $g(W,W_0)$ with `scipy.special.spence` for dilogarithm > - **O(α)**: Universal Sirlin function $g(W,W_0)$ with `scipy.special.spence` for dilogarithm
> - **O(Zα²)**: Finite nuclear size correction $\delta_2(Z,W)$ using modified Gaussian model > - **O(Zα²)**: Finite nuclear size correction $\\delta_2(Z,W)$ using modified Gaussian model
> - Soft-photon resummation at endpoint via Sirlin (1987) prescription > - 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 > - Small-beta Taylor expansion for numerical stability at threshold
> - Optional endpoint resummation flag > - Optional endpoint resummation flag
@@ -114,7 +122,7 @@ Experimental branching ratios: $BR_{\beta\gamma} \sim 3\times 10^{-3}$ (neutron)
```python ```python
from beta_spectrum import RadiativeCorrection from beta_spectrum import RadiativeCorrection
# Light nucleus (tritium-like) # Light nucleus (tritium-like) — resummation has negligible effect at low W0
rad = RadiativeCorrection( rad = RadiativeCorrection(
W0=1.02, W0=1.02,
Z=1, Z=1,
@@ -126,16 +134,27 @@ rad = RadiativeCorrection(
W0=5.0, W0=5.0,
Z=92, Z=92,
A=238, # affects Delta_F nuclear model term A=238, # affects Delta_F nuclear model term
use_endpoint_resummation=True 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) values = rad(W_grid)
``` ```
> [!note] Resummation scope
> The soft-photon resummation (Eq. 53) is only applied when $W_0 - W < \\delta\_{\\text{cut}}$ (default $10^{-3}$ in $m_e$ units ≈ 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 near $W \\to W_0$.
## Magnitude Summary ## Magnitude Summary
| Term | Typical Size | Notes | | Term | Typical Size | Notes |
|---|---|---| | ---------------------------- | ------------ | -------------------------------- |
| $\alpha/2\pi \cdot g$ | ~1–3% | Universal, dominant contribution | | $\\alpha/2\\pi \\cdot g$ | ~1–3% | Universal, dominant contribution |
| $Z\alpha^2 \Delta_i$ | ~0.01–0.1% | Nuclear-form-factor dependent | | $Z\\alpha^2 \\Delta_i$ | ~0.01–0.1% | Nuclear-form-factor dependent |
| $Z^2\alpha^3 \delta_{3h}$ | < 0.01% | Only for heavy nuclei | | $Z^2\\alpha^3 \\delta\_{3h}$ | < 0.01% | Only for heavy nuclei |