Beta Decay Spectrum — Full Expression

The Master Equation

The fully differential allowed β spectrum shape, including all corrections needed for 10^{-4} precision (Hayen et al., Eq. 4):

N(W)\,dW = \frac{G_V^2 V_{ud}^2}{2\pi^3} \; K(Z,W,W_0,M) \; A(Z,W) \; C'(Z,W) \; p W (W_0 - W)^2 dW

or expanded:

N(W)\,dW = \frac{G_V^2 V_{ud}^2}{2\pi^3} F_0 L_0 U DFS R R_N Q S X r C D_C \; p W (W_0 - W)^2 dW \tag{4}

Correction Factors

Symbol	Name	Module / Note	Status
`F_0(Z,W)`	Point charge Fermi function	01-fermi-function	✓
`L_0(Z,W)`	Electrostatic finite size (uniform sphere)	03-finite-size	✓
`U(Z,W)`	Charge distribution correction	03-finite-size	✓
`DFS(Z,W,\beta^2)`	Nuclear deformation correction	03-finite-size	✓
`R(W,W_0) = 1 + \delta_R`	Outer radiative corrections	06-radiative-corrections	✓
`R_N(W,W_0,M)`	Recoil kinematic correction	07-recoil-effects	✗
`Q(Z,W,M)`	Recoil electromagnetic correction	07-recoil-effects	—
`S(Z,W)`	Atomic electron screening	04-screening-correction	✓
`X(Z,W)`	Atomic exchange effect	05-exchange-correction	✓
`r(Z,W)`	Atomic mismatch / overlap	12-atomic-overlap	—
`A(Z,W) = S \cdot X \cdot r`	Combined atomic factors	12-atomic-overlap	—
`C(Z,W)`	Nuclear structure shape factor	10-nuclear-structure	—
`D_C(Z,W,\beta^2)`	Nuclear deformation (convolution)	10-nuclear-structure	—

Baseline: Phase Space

The uncorrected spectral shape is pure phase space (02-phase-space):

\frac{dN}{dW}\bigg|_{\text{PS}} = p W (W_0 - W)^2

with p = \sqrt{W^2-1}, W total energy in units of m_e, and W_0 the endpoint.

Precision Goal

The paper achieves accurate to a few parts in 10^{-4} down to 1 keV for low-to-medium Z nuclei — extending previous work by nearly an order of magnitude Hayen2017_summary.

[!key] Why this matters A precision of ~0.5% is required when determining the Fierz term b_{\text{Fierz}} to improve limits on BSM scalar/tensor couplings. Weak magnetism b_{wm} must be included as experimental precision increases, otherwise it limits BSM sensitivity.

Three Formalisms Compared

Approach	Style	Strengths
Behrens-Bühring (BB)	Rigorous, numerical Dirac solutions	Most precise results
Holstein (HS)	Analytical, manifestly covariant	Clean symmetry properties
Wilkinson	Analytical parametrizations	Practical fits for dominant effects

The Hayen paper combines BB's accuracy with HS's transparency.

Conventions

Natural units: c = \hbar = m_e = 1 throughout
Z is always positive; \beta^- and \beta^+ distinguished via upper/lower signs (\mp)
W = E/m_ec^2 + 1, p = \sqrt{W^2 - 1}
G_V: vector coupling strength; V_{ud} = \cos^2\theta_C

[!note] Two types of "finite size" The authors distinguish two corrections conflated under this term:

Electrostatic (L_0, U, DFS): Potential difference from point charge → realistic nuclear shape

Convolution: Integration of leptonic wave functions over the nuclear volume (appears in C(Z,W))

Energy Conversion

W = 1 + \frac{T}{m_ec^2}, \quad m_ec^2 = 510.998950\text{ keV}

In code: see beta_spectrum/utils.py → T_to_W(), W_to_T() functions.

Calculator Implementation

Each correction factor C_i(W) is a callable class inheriting from SpectrumComponent. The total spectrum is the product of all enabled components:

N(W) = PhaseSpace(W0) × FermiFunction(Z) × FiniteSizeL0(Z,A) × ...

Configuration toggles are defined in SpectrumConfig (spectrum.py). See README for architecture details.

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