beta_shape_pnpi/docs/corrections/05-radiative.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: 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

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