130 lines
5.4 KiB
Markdown
130 lines
5.4 KiB
Markdown
---
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title: "06 – Radiative Corrections"
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date: 2026-04-26
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tags:
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- beta-spectrum/correction/radiative
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status: active
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aliases: [Outer Radiative, δ_R, Sirlin Function]
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cssclasses: []
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related_components: [[10-nuclear-structure]], [[07-recoil-effects]]
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---
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# Outer Radiative Corrections — $R(W,W_0) = 1 + \delta_R(W,W_0)$
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## Overview: Inner vs. Outer Corrections
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The radiatively corrected spectrum (Hayen et al., Eq. 46):
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$$\frac{d\Gamma}{dW} = \frac{d\Gamma_0}{dW}(1 + \Delta_R^{V/A})(1 + \delta_R(W,W_0))$$
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| Type | Symbol | Dependence | Treatment |
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|---|---|---|---|
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| **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 |
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| **Outer** | $\delta_R(W,W_0)$ | Energy-dependent, nucleus-dependent | This is the correction implemented in code ([[06-radiative-corrections]]) |
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## Total Spectral Influence (Eq. 47–48)
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$$R(W,W_0) = 1 + \frac{\alpha}{2\pi}\left[g(W_0,W) - 3\ln\left(\frac{m_p}{2W_0}\right)\right] + Z\alpha^2\sum_i \Delta_i(W) + Z^2\alpha^3 \delta_{3h} + ... \tag{48}$$
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### Order α — Sirlin's $g(W_0, W)$ Function (Eq. 50–52)
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The universal correction (same for electrons and positrons, Fermi and GT):
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$$g(W_0,W) = 3\ln(m_p) - \frac{3}{4} + \frac{4\beta}{1+\beta}\left[4\tanh^{-1}\beta - 1\right] $$
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$$+ \frac{W_0-W}{\beta}\left(-\frac{3}{3W} + \ln[2(W_0-W)]\right) $$
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$$- \frac{(W_0-W)^2}{6W^2} + 4\frac{\tanh^{-1}\beta}{\beta}(1+\beta^2) - 4\tanh^{-1}\beta$$
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where $\beta = p/W$.
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The Spence function (dilogarithm):
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$$L_s(x) = \int_0^x \frac{\ln(1-t)}{t} dt = -Li_2(x) = -\sum_{k=1}^\infty \frac{x^k}{k^2} \tag{51}$$
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Large $W_0$ limit:
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$$g(W_0 \to \infty, W) = 3\ln\left(\frac{M_N}{2W_0}\right) + \frac{81}{10} - \frac{4\pi^2}{3} \tag{52}$$
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> [!warning] Endpoint divergence
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> $g(W_0, W)$ has a **logarithmic divergence at $W = W_0$** related to soft real photon emission. This must be handled via resummation.
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### Soft Photon Resummation (Eq. 53–54)
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To handle the endpoint divergence:
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$$t(\beta)\ln(W_0 - W) \to (W_0-W)^{t(\beta)} - 1 \tag{53}$$
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where:
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$$t(\beta) = \frac{2\alpha}{\pi}\left[\frac{\tanh^{-1}\beta}{\beta} - 1\right] \tag{54}$$
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For tritium the effect is negligible (low endpoint). For higher energies, the $ft$ value shift can reach several parts in $10^4$.
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### Order Zα² Correction (Eq. 55–62)
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Three dominant Feynman diagrams: vacuum polarization of bremsstrahlung photon and two electron propagator renormalizations. The axial vector component contributes specifically to $\Delta_3$:
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$$\delta_2(Z,W) = Z\alpha^2 \sum_{i=1}^4 \Delta_i(W) \tag{55}$$
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The leading term $\Delta_1$ depends on nuclear form factor $F(q^2)$ and is split into:
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- **Nucleus-independent** part $\Delta_1^0$: energy-dependent terms combined with $\Delta_4$
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- **Nuclear-dependent** part $\Delta_1^F$: explicit integral forms for uniformly charged sphere
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For modified Gaussian model, differences between models for superallowed Fermi transitions ($^{10}$O to $^{54}$Co) are < $10^{-4}$.
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### Order Z²α³ — Sirlin's Heuristic Formula (Eq. 63–67)
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$$\delta_{3h} = Z^2\alpha^3 \left[a\ln\left(\frac{\Lambda}{W}\right) + bf(W) + \frac{4\pi}{3}g(W) - 0.649\ln(2W_0)\right]$$
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with $a = \frac{\pi}{3} - \frac{\sqrt{3}}{2}$, $b = \frac{4}{3\pi^4}(\pi^2 - \gamma_E) - \frac{\pi^2}{18}$.
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### Higher Order (Eq. 68–69)
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$$\delta_{Z^n\alpha^{n+1}} \approx Z^n\alpha^{n+1} K_{nm} \ln^{m-n}\left(\frac{\Lambda}{W}\right), \quad K_{nm} \approx 0.5 \tag{68}$$
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$$\sum_{n=3}^\infty Z^n\alpha^{n+1} = \frac{Z^3\alpha^4}{1-Z\alpha} \tag{69}$$
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Relevant for higher $Z$ nuclei only.
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## Neutrino Radiative Corrections (Eq. 70–71)
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Even though neutrinos have no direct electromagnetic interaction, they receive indirect corrections via virtual photon exchange and energy conservation from inner bremsstrahlung:
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$$R_\nu = 1 + \frac{\alpha}{2\pi}\left[3\ln\left(\frac{m_p}{m_e}\right) + \frac{23}{4} - Li_2\left(-\frac{8}{\hat{\beta}}\right) + ...\right]$$
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where $\hat{W} = W_0 - W_\nu$. Much smaller than electron spectrum correction — relevant for reactor neutrino oscillation studies.
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## Internal Bremsstrahlung (Eq. 72–74)
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An additional photon in the final state: $n \to p + e^- + \bar{\nu}_e + \gamma$
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Photon ejection probability per unit energy:
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$$\Phi(W,\omega) = \frac{\alpha p}{\pi\omega p'}\left[\frac{W+W'^2}{p^2}\ln\left(\frac{W+p}{W'-p}\right) - 2\right] \tag{73}$$
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Experimental branching ratios: $BR_{\beta\gamma} \sim 3\times 10^{-3}$ (neutron), $\sim 2\times 10^{-3}$ ($^{32}$P).
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## Calculator Implementation
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> [!tip] Code mapping
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> Module: `beta_spectrum/components/radiative.py` → `RadiativeCorrection(W0, use_endpoint_resummation=True)` class
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>
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> Implements outer radiative corrections per Hayen Eqs. (47)–(53):
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> - Sirlin's $g(W_0,W)$ with `scipy.special.spence` for dilogarithm
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> - Soft-photon resummation at endpoint via Eq. (53)
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> - Order Zα² terms Δ₁ through Δ₄
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> - Optional endpoint resummation flag
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### API Usage
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```python
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from beta_spectrum import RadiativeCorrection
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rad = RadiativeCorrection(
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W0=1.02, # endpoint in m_e units
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use_endpoint_resummation=True # handles endpoint divergence
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)
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values = rad(W_grid)
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```
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## Magnitude Summary
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| Term | Typical Size | Notes |
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|---|---|---|
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| $\alpha/2\pi \cdot g$ | ~1–3% | Universal, dominant contribution |
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| $Z\alpha^2 \Delta_i$ | ~0.01–0.1% | Nuclear-form-factor dependent |
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| $Z^2\alpha^3 \delta_{3h}$ | < 0.01% | Only for heavy nuclei |
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