beta_shape_pnpi/docs/corrections/06-recoil.md

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title: "07 – Recoil Effects" date: 2026-04-26 tags:

• beta-spectrum/correction/recoil status: todo aliases: [Nuclear Recoil, R_N, Weak Magnetism] cssclasses: [] related_components: 02-phase-space, corrections/05-radiative

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Nuclear Recoil Effects — R_N, Q, and Induced Currents

Status in Calculator

[!todo] Not yet implemented Module: beta_spectrum/components/recoil.py exists but is empty (stub only). This correction needs to be added.

Why Recoil Matters

The finite nuclear mass turns beta decay from a two-body into a three-body process, introducing recoil corrections that become significant at the 10^{-4} precision level. Additionally:

• Weak magnetism b_{wm} must be included as experimental precision increases — otherwise it limits BSM sensitivity
• The oft-neglected induced pseudoscalar interaction may be significant

Kinematic Recoil Correction R_N(W,W_0,M) (Eq. 41–43)

R_N = 1 + \frac{r_0}{W} + \frac{r_1}{W^2} + r_2 W + \frac{r_3}{W^2} \tag{41}

For Fermi (vector) transitions:

r_0^V = W_0^2/(2M^2) - 11/(6M^2), \quad r_1^V = W_0/(3M) r_2^V = 2/M - 4W_0^2/(3M), \quad r_3^V = 16/(3M^2)

For Gamow-Teller (axial) transitions:

r_0^A = -2W_0/(3M) - W_0^2/(6M^2) - 77/(18M^2), \quad r_1^A = -2/(3M) + 7W_0/(9M^2) r_2^A = 10/(3M) - 28W_0/(9M^2), \quad r_3^A = 88/(9M^2)

Mixed transitions:

Terms are weighted by 1/(1+\rho^2) and 1/(1+\rho^{-2}) where \rho = \mathcal{M}_{GT}/\mathcal{M}_F.

[!tip] Magnitude Order 10^{-5} to 10^{-3}. See Wilkinson (1990) Fig. 1 for magnitude plot.

Electromagnetic Recoil Correction Q(Z,W,M) (Eq. 45)

The recoiling nucleus's Coulomb field differs from a static one:

Q \simeq 1 \mp \frac{\pi\alpha Z}{M}\left(1 + a\right)\frac{W_0 - W}{3W} \tag{45}

where a = (1-\rho^2/3)/(1+\rho^2) is the β-ν correlation coefficient (a=1 for Fermi, a=-1/3 for GT).

[!note] Typically negligible The effect amounts to at most a few percent of the typical error in phase space factor f due to Q-value uncertainty — usually not limiting.

Weak Magnetism and Induced Currents (Section VI)

The weak vertex receives QCD-induced form factors beyond pure V-A:

Behrens-Bühring Form (Eq. 82):

J_\mu^{BB} = i\langle \bar{u}_p | C_V\gamma_\mu - f_M\sigma_{\mu\nu}q^\nu + if_S q_\mu - \frac{C_A}{C_V}\left(\gamma_\mu\gamma^5 - f_T\sigma_{\mu\nu}\gamma^5 q^\nu\right) + i\frac{f_P}{C_V}\gamma^5 q_\mu | u_n \rangle

Holstein Form (Eq. 83):

J_\mu^{HS} = i\langle \bar{u}_p | g_V\gamma_\mu - \frac{g_M-g_V}{2M}\sigma_{\mu\nu}q^\nu + i\frac{g_S}{2M}q_\mu + g_A\gamma^5\gamma_\mu - \frac{g_{II}}{2M}\sigma_{\mu\nu}\gamma^5 q^\nu + i\frac{g_P}{2M}\gamma^5 q_\mu | u_n \rangle

Form Factor	Name	CVC/PCAC Status
C_V = g_V	Vector (Fermi)	Conserved — from EM data
f_M/C_V	Weak magnetism	Determined by CVC from M1 transitions
f_S/g_S	Induced scalar	Excluded by CVC for Fermi; Second-class current
C_A = g_A	Axial-vector (GT)	Measured in neutron decay (g_A \approx 1.276)
f_T	Tensor	First-class if exists
g_{II}	Induced tensor	Second-class current → experimental null results
f_P/g_P	Induced pseudoscalar	PCAC prediction: g_P(0) \approx -229, but quenched ~80% in nuclei

CVC Constraint on Weak Magnetism (Eq. 87–91)

For mirror transitions and isovector M1 decays, weak magnetism is determined from electromagnetic data — no free parameters:

\mu_p - \mu_n = g_M(0)/\mu_N \quad (\text{from CVC})

PCAC / Goldberger-Treiman (Eq. 92)

g_P(0) \approx -229 \quad (\text{free nucleon})

With ~5% higher-order corrections. In nuclear medium, g_P is quenched by up to 80% — great care needed when evaluating pseudoscalar currents for nuclear decays.

Spectral Shape Modification (Eq. 3)

Weak magnetism and Fierz term modify the spectrum shape:

N(W)\,dW \propto p W (W_0 - W)^2 \left[1 + \frac{\gamma m_e}{W} b_{\text{Fierz}} \pm \frac{4W}{3M} b_{wm}\right] dW \tag{3}

The weak magnetism term \propto W/M grows with energy — opposite to Fierz term which falls as 1/W. This distinct energy dependence allows experimental separation.

Implementation Plan for Calculator

To implement recoil in recoil.py:

1. Create RecoilCorrection class inheriting from SpectrumComponent
2. Implement R_N(W,W_0,M) per Eqs. (41)–(44) with transition-type parameter (Fermi/GT/mixed)
3. Optionally implement Q(Z,W,M) for electromagnetic recoil
4. Accept weak magnetism form factor from nuclear data or CVC calculation