beta_shape_pnpi/docs/06-radiative-corrections.md

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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: 10-nuclear-structure, 07-recoil-effects

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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:

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}

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) 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 at endpoint via Sirlin (1987) prescription
• Small-beta Taylor expansion for numerical stability at threshold
• Optional endpoint resummation flag

API Usage

from beta_spectrum import RadiativeCorrection

# Light nucleus (tritium-like)
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
)

values = rad(W_grid)

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