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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: [[corrections/06-recoil]]

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# 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)` class
>
> 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(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

```python
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 $W_0 - W < \\delta\_{\\text{cut}}$ (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

| 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            |