Update docs for radiative correction
Added summaries for Sirlin's articles and spec file based on them.
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# General Properties of the Electromagnetic Corrections to the Beta Decay of a Physical Nucleon
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A. Sirlin
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*Physical Review*, Volume 164, Number 5, 25 December 1967
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Department of Physics, New York University, New York, New York
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(Received 26 June 1967)
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---
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## Abstract
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Some general properties of the electromagnetic corrections to the \(\beta\) decay of a physical nucleon (a nucleon in the presence of strong interactions) are discussed. The aim is to isolate and determine general properties independent of the details of strong interactions and assumptions about the existence of the intermediate boson. The method separates out in a finite and gauge-invariant manner all terms of order \(1/k\) in the hadronic covariants, then examines properties of other contributions. Under general and plausible mathematical assumptions, all terms of order \(\alpha\) in the correction factor to the electron spectrum that explicitly depend on the electron energy \(E\) and electron mass \(m\) can be rigorously computed, despite strong interaction complications, provided contributions of relative order \(\alpha q/M\), \(\alpha(E/M)\ln(M/E)\), and \(\alpha \nu_N/c\) are neglected. The electromagnetic correction of order \(\alpha\) to the shape of the allowed electron spectrum is given by a single universal function \(g(E, E_m, m)\), independent of strong interaction details, intermediate boson assumptions, and the ratio \(M_{GT}/M_F\). This function is physically significant, particularly for nuclear \(\beta\) decays with \(E_m/m \gg 1\). Model-dependent terms that contribute to the lifetime are discussed but not evaluated. Other observables for which corrections can be evaluated model-independently are illustrated by the virtual radiative corrections to the longitudinal polarization of the electron in allowed transitions. The paper implicitly assumes the validity of the vector and axial-vector theory of weak interactions.
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---
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## I. INTRODUCTION
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The radiative corrections to \(\beta\) decay of a nucleon have been discussed extensively.\(^{1-7}\) They are of interest for their relevance to universality of weak interactions and because they alter the shape of the allowed electron spectrum in \(\beta\) decay, with significant effects when the end-point energy is large compared to the electron mass.\(^{8}\)
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A key theoretical aspect is that these corrections are logarithmically divergent in the \(V-A\) theory when evaluated for a bare nucleon, while the corresponding quantities for muon decay are finite and well defined. Two main approaches to save universality have been suggested:
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1. Weak interactions are mediated by an intermediate boson.\(^{9-11}\)
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2. Strong interactions themselves may provide a natural ultraviolet cutoff.\(^{12-14}\)
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More recently, Bjorken\(^{15}\) and Abers, Norton, and Dicus,\(^{16}\) using current-algebra techniques, have asserted that part of the electromagnetic corrections to the vector decay coupling constant contains a divergent part independent of strong interactions. However, this does not give a complete answer because: (a) there may be contributions from the axial current that are also divergent and model-dependent, and (b) possible intrinsic structure effects of weak interactions from intermediate boson calculations may alter convergence properties.\(^{11}\)
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This paper bypasses the convergence problem, taking the hopeful view that the weak, electromagnetic, and strong interactions conspire to give a finite value for the relative renormalization of \(\beta\) and \(\mu\) decay coupling constants. The aim is to isolate and determine general properties of the corrections independent of strong interaction details and intermediate boson assumptions.
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---
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## II. THE SEPARATION
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The virtual radiative corrections of order \(\alpha\) are separated into two parts. Part one is: (i) independent of strong interaction details; (ii) contains all terms of order \(1/k\) in hadronic covariants; (iii) finite (in the uv region) and gauge-invariant. Part two depends on strong interaction details and involves three- and four-point functions.
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### A. Photon Exchange Between Hadronic and Electron Lines
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Diagrams where a photon is exchanged between hadronic and electron lines are divided into three classes: photon emitted from an electromagnetic vertex attached to the external proton line, external neutron line, or from internal structure of the weak vertex. The contribution of the first class [Fig. 1(a)] is:\(^{18}\)
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\[M^{(p)} = -\frac{G_{\nu}^{0}\alpha}{\sqrt{2}4\pi^{2}i}\int d^{4}k D_{\mu \nu}(k)[\bar{u}_{e}(2l_{r} - \gamma_{r}k)O_{\lambda}\nu], \quad (1)\]
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where \(p_1\), \(p_2\), \(l\), and \(k\) are the neutron, proton, electron, and photon four-momenta; \(\lambda_{\min}\) is the photon mass; \(O_{\lambda} \equiv \gamma_{\lambda}(1 + \gamma_{5})\); \(G_{\nu}^{0}\) is the bare coupling constant of the weak-vector current; \(W_{\lambda}\) and \(\bar{\Gamma}_{\mu}^{(p)}\) are the fully dressed renormalized proper vertices; and \(i\bar{S}_{p}^{\prime}\) is the fully dressed renormalized propagator of the proton. The photon propagator is chosen in the Landau gauge:
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\[D_{\mu \nu}(k) = \frac{(g_{\mu\nu} - k_{\mu}k_{\nu} / k^2 + i\epsilon)}{k^2 - \lambda_{\min}k^2 + i\epsilon}. \quad (2)\]
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The "proton electromagnetic form factor" \(F_{\mu}^{(p)}(p_2, p_2 + k)\) is introduced for the case where \(p_2\) corresponds to an on-shell proton but \(p_2 + k\) is off-shell:
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\[\begin{array}{rl} & {\bar{u}_p\bar{\Gamma}_{\mu}^{(p)}(p_2,p_2 + k)\bar{S}_p^{\prime}(p_2 + k)}\\ & {\qquad = \bar{u}_pF_{\mu}^{(p)}(p_2,p_2 + k)[\bar{p}_2 + k - m_p]^{-1},} \end{array} \quad (3)\]
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where \(m_p\) is the proton mass. Using the generalized Ward identity:
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\[k_{\mu}F_{\mu}^{(p)}(p_{2},p_{2} + k) = k, \quad (4a)\]
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\[F_{\mu}^{(p)}(p_{2},p_{2}) = \gamma_{\mu}. \quad (4b)\]
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Equations (4) suggest writing:\(^{19}\)
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\[F_{\mu}^{(p)}(p_{2},p_{2} + k) = \gamma_{\mu} + X_{\mu}^{(p)}(p_{2},p_{2} + k), \quad (5a)\]
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where
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\[k_{\mu}X_{\mu}^{(p)}(p_{2},p_{2} + k) = 0, \quad (5b)\]
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\[X_{\mu}^{(p)}(p_{2},p_{2}) = 0. \quad (5c)\]
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The terms of order \(1/k\) in the hadronic covariant are separated:
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\[\begin{array}{rl} & {H_{\mu \lambda}^{(p)}(p_{2},p_{1},k)\equiv \bar{u}_p\bar{\Gamma}_{\mu}^{(p)}(p_{2},p_{2} + k)\bar{S}_p^{\prime}(p_{2} + k)}\\ & {\qquad \times W_{\lambda}(p_{2} + k,p_{1})u_n = (k^2 +2p_{2}\cdot k + i\epsilon)^{-1}}\\ & {\times \bar{u}_p[2(p_{2} + k_\mu)W_\lambda (p_{2},p_{1}) + R_{\mu \lambda}^{(p)}(p_{2},p_{1},k)]u_n,} \end{array} \quad (6a)\]
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where
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\[R_{\mu \lambda}^{(p)}(p_{2},p_{1},k) = (2p_{2\mu} + k_{\mu})[W_{\lambda}(p_{2} + k,p_{1}) - W_{\lambda}(p_{2},p_{1})] + X_{\mu}^{(p)}(p_{2},p_{2} + k)(p_{2} + k + m_{p})W_{\lambda}(p_{2} + k,p_{1}) + [\gamma_{\mu},k]W_{\lambda}(p_{2} + k,p_{1}) / 2. \quad (6b)\]
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Assuming \(\partial W_{\lambda} / \partial k_{\mu}(p_{2} + k,p_{1})|_{k_{\mu} = 0}\) exists, \(R_{\mu \lambda}^{(p)}\to 0\) linearly with \(k_{\mu}\) as \(k_{\mu}\to 0\).
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A similar analysis applies to diagrams with photon exchange between the electron and the electromagnetic vertex attached to the external neutron line [Fig. 1(b)]. Because the neutron has zero charge:
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\[k_{\mu}F_{\mu}^{(n)}(p_{1} - k,p_{1}) = 0, \quad (7a)\]
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\[F_{\mu}^{(n)}(p_{1} - k,p_{1}) = 0. \quad (7b)\]
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The matrix element for this contribution has the hadronic covariant:
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\[H_{\mu \lambda}^{(n)}(p_{2},p_{1},k) = [k^{2} - 2p_{1}\cdot k + i\epsilon ]^{-1} [\bar{u}_{\mu}W_{\lambda}(p_{2},p_{1} - k)(p_{1} - k + m_{n}) F_{\mu}^{(n)}(p_{1} - k,p_{1})u_{n}]. \quad (7c)\]
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\(H_{\mu \lambda}^{(n)}\) is "regular" as \(k_{\mu}\to 0\).
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Let \(G_{\mu \lambda}(p_{2},p_{1},k)\) be the matrix element for photon emission from any internal line of the proper vertex of the weak-interaction current [Fig. 1(c)]. Using the generalized Ward identity for electromagnetic vertex functions:\(^{20}\)
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\[k_{\mu}G_{\mu \lambda}(p_{2},p_{1},k) = W_{\lambda}(p_{2},p_{1}) - W_{\lambda}(p_{2} + k,p_{1}). \quad (8a)\]
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### B. Combined Matrix Element \(M_1\)
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The matrix element \(M_1\) (in the Landau gauge) of all diagrams with photon exchange between hadronic and electron lines [Figs. 1(a)-(c)] is:
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\[M_{1} = -\frac{G_{\nu}^{0}}{\sqrt{2}}\frac{\alpha}{4\pi^{2}i}\int \frac{d^{4}k D_{n\mu}(k)[\bar{u}_{e}(2l_{r} - \gamma_{r}k)O_{\lambda}\nu_{\tau}]}{(k^{2} - 2l\cdot k + i\epsilon)} \times \left[\bar{u}_{p}\left[\frac{(2p_{2\mu} + k_{\mu})W_{\lambda}(p_{2\mu}p_{1})}{(k^{2} + 2p_{2}\cdot k + i\epsilon)} +T_{\mu \lambda}(p_{2\mu}p_{1}k)\right]u_{n}\right], \quad (9a)\]
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where
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\[T_{\mu \lambda}(p_{2},p_{1},k) = \frac{R_{\mu\lambda}^{(p)}(p_{2},p_{1},k)}{k^{2} + 2p_{2}\cdot k + i\epsilon} +G_{\mu \lambda}(p_{2},p_{1},k) +\frac{R_{\mu\lambda}^{(n)}(p_{2},p_{1},k)}{k^{2} - 2p_{1}\cdot k + i\epsilon}, \quad (9b)\]
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\[R_{\mu \lambda}^{(n)}(p_{2},p_{1},k) = W_{\lambda}(p_{2},p_{1} - k)(p_{1} - k + m_{n}) F_{\mu \lambda}^{(n)}(p_{1} - k,p_{1}). \quad (9c)\]
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The tensor-pseudotensor \(T_{\mu \lambda}\) has two basic properties: (a) it is "regular" as \(k\rightarrow 0\); (b) it is transversal:
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\[k_{\mu}T_{\mu \lambda} = 0. \quad (10)\]
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Equation (10) follows from Eqs. (4a), (7a), (8a) and definitions (9b) and (9c).
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Thus, Eq. (9a) achieves a separation of Figs. 1(a)-(c) into two parts: the first part is finite (in the Landau gauge), contains all terms of order \(1/k\) in the hadronic covariant, and can be evaluated independently of strong interaction details; the second part depends on strong interaction details through \(T_{\mu \lambda}\) and is regular as \(k_{\mu}\rightarrow 0\).
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### C. Gauge Invariance and Additional Diagrams
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To obtain gauge-invariant matrix elements, one must add diagrams where the photon is emitted and absorbed by the electron line (electron wave function renormalization, Fig. 2) and by the hadronic line (Fig. 3).
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In the Landau gauge, after mass renormalization, the electron wave function renormalization to order \(\alpha\) is:
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\[\begin{array}{l}{M_{2} = -\frac{\alpha}{8\pi^{2}i}\frac{G_{\nu}^{0}}{\sqrt{2}} [\bar{u}_{p}W_{\lambda}(p_{2},p_{1})u_{n}]}\\ {\times \int \frac{d^{4}k D_{n\mu}(k)[\bar{u}_{e}(2l_{r} - \gamma_{r}k)I(2l_{r} - k\gamma_{r})(l + m)O_{\lambda}v_{\tau}]}{(k^{2} - 2l\cdot k + i\epsilon)^{2}2m^{2}}.} \end{array} \quad (11)\]
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The integral in Eq. (11) is finite in the uv region and \(M_2\) is proportional to the zeroth-order matrix element:
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\[M_{0} = \frac{G_{\nu}^{0}}{\sqrt{2}} [\bar{u}_{e}O_{\lambda}v_{\tau}][\bar{u}_{p}W_{\lambda}(p_{2},p_{1})u_{n}]. \quad (12)\]
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The matrix element for diagrams where a photon is emitted and absorbed by the hadronic line (Fig. 3) must have the structure:
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\[M_{3} = \frac{\alpha}{2\pi} [\bar{u}_{e}O_{\lambda}v_{\tau}][V_{\lambda} + A_{\lambda}], \quad (13a)\]
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where
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\[V_{\lambda} = \bar{u}_{p}[f_{1}(q^{2})\gamma_{\lambda} + if_{2}(q^{2})\sigma_{\lambda \nu}q_{\nu} + f_{3}(q^{2})q_{\lambda}]u_{n}, \quad (13b)\]
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and \(q\) is the total momentum transfer to the leptons. Neglecting terms proportional to \(\alpha q\) and the \(q^{2}\) dependence of form factors, only \(f_{1}(0)\gamma_{\lambda}\) and \(g_{1}(0)\gamma_{\lambda}\gamma_{5}\) survive.
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All infrared terms from Fig. 3 arise from graphs where the photon is emitted and absorbed by the outgoing proton and are contained in the convection-convection contribution:
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\[M_{3}^{(c)} = -\frac{\alpha}{8\pi^{2}i} M_{0}\int \frac{d^{4}k D_{\mu \lambda}(k)(2p_{2} - k)_{\mu}(2p_{2} - k)_{\nu}}{(k^{2} - 2p_{2}\cdot k + i\epsilon)^{2}}. \quad (14)\]
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The contributions of Eqs. (11), (14), and the first term in Eq. (9a) are each finite in the uv region, and their sum is invariant under gauge transformations of the photon propagator.
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### D. Extension to Intermediate Boson Case
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The separation and all equations extend immediately to the case of an intermediate boson by: replacing \(W_{\lambda}\) everywhere by \(W_{\mu}D_{\mu \lambda}^{a}\) (where \(D_{\mu \lambda}^{a}\) is the boson propagator), and redefining \(G_{\mu \lambda}\) to include photon emission by the intermediate boson. Graphs with photon exchange between the boson and electron are included in the \(T_{\mu \lambda}\) contribution in Eq. (9a); all other graphs with photon attached to the intermediate boson are included in Eq. (13a).
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---
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## III. GENERAL PROPERTIES OF MODEL-DEPENDENT CONTRIBUTIONS
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The term involving \(T_{\mu \lambda}\) in Eq. (9a) is discussed. Aside from constant factors, it is given by:
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\[I = -\frac{\alpha}{2\pi}\int \frac{d^{4}k[\bar{u}_{e}(2l_{\mu} - \gamma_{\mu}k)O_{\lambda}v_{\nu}][\bar{u}_{p}T_{\mu\lambda}(p_{2},p_{1},k)u_{n}]}{(k^{2} + i\epsilon)(k^{2} - 2l\cdot k + i\epsilon)}, \quad (15)\]
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where the transversality relation of Eq. (10) has been used and \(\lambda_{\min} = 0\) because the integral is infrared-convergent.
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Consider the contribution \(K\) involving the factor \(\gamma_{\mu}k\) in the leptonic covariant:
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\[K = -\frac{\alpha}{2\pi} [\bar{u}_{e}\gamma_{\mu}\gamma_{\rho}O_{\lambda}v_{\nu}]\int \frac{d^{4}k k_{\rho}[\bar{u}_{p}T_{\mu\lambda}(p_{2},p_{1},k)u_{n}]}{(k^{2} + i\epsilon)(k^{2} - 2l\cdot k + i\epsilon)}. \quad (16)\]
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Using the separation:
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\[[k^{2} - 2l\cdot k + i\epsilon]^{-1} = [k^{2} + i\epsilon]^{-1} + 2l\cdot k[k^{2} + i\epsilon]^{-1}[k^{2} - 2l\cdot k + i\epsilon]^{-1}, \quad (17a)\]
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\(K\) can be written as:
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\[K = K_{1} + K_{2}, \quad (17b)\]
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\[K_{1} = -\frac{\alpha}{2\pi} [\bar{u}_{e}\gamma_{\mu}\gamma_{\rho}O_{\lambda}v_{\nu}]\int \frac{d^{4}k k_{\rho}[\bar{u}_{p}T_{\mu\lambda}(p_{2},p_{1},k)u_{n}]}{(k^{2} + i\epsilon)^{2}}, \quad (17c)\]
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\[K_{2} = -\frac{\alpha}{\pi} [\bar{u}_{e}\gamma_{\mu}\gamma_{\rho}O_{\lambda}v_{\nu}]\int \frac{d^{4}k k_{\rho}k_{\sigma}[\bar{u}_{p}T_{\mu\lambda}(p_{2},p_{1},k)u_{n}]}{(k^{2} + i\epsilon)^{2}(k^{2} - 2l\cdot k + i\epsilon)}. \quad (17d)\]
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Because \(T_{\mu \lambda}\) is "regular" as \(k_{\mu}\rightarrow 0\), \(K_1\) and \(K_2\) converge in the infrared region.
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The integral in Eq. (17c) does not involve the electron four-momentum \(l_{\mu}\) explicitly. If the total momentum transfer \(q\) is neglected in \(T_{\mu \lambda}\) so that \(T_{\mu \lambda}(p_{2},p_{1},k) \approx T_{\mu \lambda}(p_{1},p_{1},k)\), after performing the \(k\) integration \(K_1\) is of the form (see Appendix A):
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\[K_{1} = (\alpha /2\pi)[\bar{u}_{e}O_{\lambda}v_{\nu}][\bar{u}_{p}(a\gamma_{\lambda} + b\gamma_{\lambda}\gamma_{5})u_{n}], \quad (17e)\]
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where \(a\) and \(b\) are constants independent of \(l_{\mu}\), whose precise values depend on \(T_{\mu \lambda}\) and therefore on strong interaction details and intermediate boson assumptions.
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For the contribution \(K_2\), considering the \(R_{\mu \lambda}^{(p)}\) term in \(T_{\mu \lambda}\), one expands:
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\[R_{\mu \lambda}^{(p)}(p_{2},p_{1},k) = k_{\alpha}S_{\alpha \mu \lambda}^{(a)}(p_{2},p_{1}) + k_{\alpha}k_{\beta}S_{\alpha \beta \mu \lambda}^{(b)}(p_{2},p_{1},k), \quad (18a)\]
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where \(S_{\alpha \mu \lambda}^{(a)}(p_{2},p_{1})\) is independent of \(k\) and \(S_{\alpha \beta \mu \lambda}^{(b)}(p_{2},p_{1},k)\) is regular as \(k\rightarrow 0\). The contribution from \(R_{\mu \lambda}^{(p)}\) is:
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\[K_{2}^{(p)} = (\alpha /\pi)l_{\sigma}[\bar{u}_{e}\gamma_{\mu}\gamma_{\rho}O_{\lambda}v_{\nu}] \int \frac{d^{4}k k_{\sigma}k_{\rho}[\bar{u}_{p}(k_{\alpha}S_{\alpha \mu \lambda}^{(a)}(p_{2},p_{1}) + k_{\alpha}k_{\beta}S_{\alpha \beta \mu \lambda}^{(b)}(p_{2},p_{1},k))u_{n}]}{[k^{2} + i\epsilon][k^{2} - 2l\cdot k + i\epsilon][k^{2} + 2p_{2}\cdot k + i\epsilon]}. \quad (18b)\]
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The integral involving \(S_{\alpha \mu \lambda}^{(a)}\) contains terms of logarithmic order in \(E\) and \(m\). As it is multiplied by \(l_{\sigma}\), the first term in Eq. (18b) gives contributions of order \(\alpha(E/m_{p})\ln(m_{p}/E)\), \(\alpha(E/m_{p})\), and smaller—"terms of higher order in the lepton four-momentum."
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For the term involving \(S_{\alpha \beta \mu \lambda}^{(b)}\), defining:
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\[L^{(b)}(\xi) = \int \frac{d^{4}k k_{\sigma}k_{\rho}k_{\alpha}k_{\beta}S_{\alpha \beta \mu \lambda}^{(b)}(p_{2},p_{1},k)}{[k^{2} + i\epsilon][k^{2} - 2\xi l\cdot k + i\epsilon][k^{2} + 2p_{2}\cdot k + i\epsilon]}, \quad (18c)\]
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where \(\xi\) is a numerical variable. For \(\xi = 0\), \(L^{(b)}(0)\) is well defined (convergent in the infrared, same uv convergence as \(L^{(b)}(1)\)). Assuming \(L^{(b)}(\xi)\) is continuous in \(-1 \leq \xi \leq 1\) or at least remains bounded, the second term in Eq. (18b) is also of higher order in \(l\), of order \(\alpha E/M\) where \(M\) may be \(m_p\), the intermediate boson mass, or hadron masses characterizing \(S^{(b)}\).
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**Conclusion**: The contributions of \(T_{\mu \lambda}\) in Eq. (9a) are of the form (17e), provided terms of order \(\alpha(E/M)\ln(M/E)\) and \(\alpha q/M\) are neglected.
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---
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## IV. GENERAL RESULTS
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All "model-dependent terms" can be written as:
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\[\begin{array}{r}S = \frac{G_{\nu}^{0}}{\sqrt{2}}\frac{\alpha}{2\pi} [\bar{u}_{e}O_{\lambda}v_{\nu}][\bar{u}_{p}(c\gamma_{\lambda} + \rho d\gamma_{\lambda}\gamma_{5})u_{n}],\\ \rho = -G_{A}^{0}/G_{\nu}^{0}, \end{array} \quad (19a)\]
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where \(c\) and \(d\) are constants independent of \(E\) and \(m\). Equation (19) contributes to the lifetime corrections. However, neglecting terms of order \(\alpha^{2}\), these terms can be formally "absorbed" in the zeroth-order matrix elements by redefining the vector and axial-vector coupling constants. Therefore, up to terms of order \(\alpha\), Eq. (19) will not alter any observable which to zero order in \(\alpha\) is independent of \(G_{\nu}^{0}\) and \(G_{A}^{0}\). For an unpolarized initial neutron, two such observables are: (i) the electron spectrum when neutrino direction and lepton polarizations are undetected; (ii) the longitudinal polarization of the electron when neutrino direction and polarization are undetected.
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### A. Universal Correction Function \(g(E, E_m, m)\)
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Combining the first term of Eq. (9a) with Eqs. (11), (14), and (19), and the inner bremsstrahlung diagrams, the decay probability for a neutron (after averaging over initial nucleon polarization, integrating over neutrino direction, and summing over lepton polarizations) is:
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\[Pd\rho = \frac{\tilde{\xi}[E_m - E]^{2}p^{2}}{2\pi^{3}} d\rho\left[1 + \frac{\alpha}{2\pi} g(E,E_m,m)\right], \quad (20a)\]
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where:
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\[\begin{array}{rl} g(E,E_m,m) = & 3\ln \left(\frac{m_p}{m}\right) - \frac{3}{4} + \frac{4}{6}\left[\frac{\tanh^{-1}\beta}{\beta} -1\right] \\ & \times \left[\frac{(E_m - E)}{3E} -\frac{3}{2} +\ln \left[\frac{2(E_m - E)}{m}\right]\right] \\ & + \frac{4}{\beta} L\left(\frac{2\beta}{1 + \beta}\right) \\ & +\frac{1}{\beta}\tanh^{-1}\beta \left[2(1 + \beta^{2}) + \frac{(E_m - E)^{2}}{6E^{2}} -4\tanh^{-1}\beta \right], \end{array} \quad (20b)\]
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and:
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\[\begin{array}{r}\tilde{\xi} = |G_{V}^{0}|^{2}\left(1 + \frac{\alpha}{\pi}\mathrm{Re}\,c\right)|M_F|^{2} \\ +|G_{A}^{0}|^{2}\left(1 + \frac{\alpha}{\pi}\mathrm{Re}\,d\right)|M_{GT}|^{2}. \end{array} \quad (20c)\]
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In these equations, \(E_m\) is the end-point energy, \(p\) is the electron momentum, \(\beta = p/E\), \(M_F\) and \(M_{GT}\) are the Fermi and Gamow-Teller matrix elements, and \(L(x)\) is the Spence function:
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\[L(x) = \int_{0}^{x}\frac{dt}{t}\ln(1 - t). \quad (20d)\]
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For free neutron decay, \(|M_F|^{2} = 1\), \(|M_{GT}|^{2} = 3\).
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|
||||
The model-dependent constants \(c\) and \(d\) are absorbed in \(\tilde{\xi}\). Equations (20) show that these terms contribute to the lifetime but do not alter the shape of the electron spectrum (up to terms of order \(\alpha\)). The Coulomb term of order \(\alpha\) is not included in Eq. (20a), as it is already incorporated in the Fermi function \(F(Z,E)\).\(^{3}\) Terms of relative order \(\alpha(E/M)\ln(M/E)\), \(\alpha q/M\), and smaller have been neglected.
|
||||
|
||||
Equations (20) can be applied to allowed nuclear \(\beta\) decay using an independent particle model of the nucleus.\(^{7}\) The function \(g(E, E_m, m)\) is:
|
||||
|
||||
- Independent of strong interaction details and intermediate boson assumptions
|
||||
- Independent of \(M_{GT}/M_F\)
|
||||
- The same for electron and positron decay
|
||||
|
||||
Thus, \(g(E, E_m, m)\) is a **universal function** describing deviations from the allowed electron spectrum arising from radiative corrections of order \(\alpha\). It differs from previous calculations by a numerical constant independent of \(E\), \(E_m\), and \(m\).\(^{28}\) The corrections to the spectral shape are significant, particularly in allowed nuclear \(\beta\) decays with \(E_m \gg m\).\(^{3}\)
|
||||
|
||||
For given \(G_{V}^{0}\) and \(G_{A}^{0}\), the function \(g\) increases the decay probability for neutron \(\beta\) decay by \(1.5\%\) and that of \(^{14}\mathrm{O}\) by \(1.3\%\). This is not a complete lifetime calculation because model-dependent contributions of Eq. (20c) are not considered. However, the effect of \(g\) on the lifetime is rather large as it gives rise to the term \(3(\alpha/2\pi)\ln(m_p/E_m)\), discussed in detail in Ref. 4.
|
||||
|
||||
### B. Longitudinal Polarization
|
||||
|
||||
Aside from the electron spectrum, there are other observables for which radiative corrections can be computed model-independently, provided terms of order \(\alpha q/M\) and \(\alpha(E/M)\ln(M/E)\) are neglected.\(^{24}\) For the longitudinal polarization of the electron (initial nucleus unpolarized, neutrino direction undetected), the virtual radiative corrections give:\(^{25,26}\)
|
||||
|
||||
\[P = -\beta \left[1 + \frac{\alpha}{2\pi} \frac{(1 - \beta^{2})}{\beta}\ln \left(\frac{1 + \beta}{1 - \beta}\right)\right]. \quad (21)\]
|
||||
|
||||
This result is valid for an arbitrary value of \(M_{GT}/M_F\). The absence of model-dependent terms \(c\) and \(d\) indicates that Eq. (21) is independent of strong interaction details and intermediate boson assumptions.
|
||||
|
||||
---
|
||||
|
||||
## APPENDIX A
|
||||
|
||||
In Eq. (17c), writing \(O_{\lambda} \equiv \gamma_{\lambda}(1 + \gamma_{5})\) and using the identity:
|
||||
|
||||
\[\gamma_{\mu}\gamma_{\rho}\gamma_{\lambda} = g_{\mu \rho}\gamma_{\lambda} - g_{\mu \lambda}\gamma_{\rho} + g_{\rho \lambda}\gamma_{\mu} - i\epsilon_{\mu \rho \lambda \alpha}\gamma_{\alpha}\gamma_{5}, \quad (A1)\]
|
||||
|
||||
with convention \(\epsilon_{0123} = 1\), the contribution of the first term to \(K_1\) is:
|
||||
|
||||
\[\frac{\alpha}{2\pi} [\bar{u}_{e}\gamma_{\lambda}(1 + \gamma_{5})v_{\nu}] \int \frac{d^{4}k[\bar{u}_{p}k_{\mu}T_{\mu\lambda}(p_{2},p_{1},k)u_{n}]}{(k^{2} + i\epsilon)^{2}}. \quad (A2)\]
|
||||
|
||||
Neglecting the momentum transfer to the leptons \(q = p_1 - p_2\) so that \(T_{\mu \lambda}(p_{2},p_{1},k) \approx T_{\mu \lambda}(p_{1},p_{1},k)\), and noting that after integration only the four-vectors \(p_{2\lambda}\), \(\gamma_{\lambda}\), and \(\gamma_{\lambda}\gamma_{5}\) are available, (A2) reduces to an expression of the form (17e), provided terms of order \(q\) are neglected. Identical conclusions hold for the other contributions of (A1).
|
||||
|
||||
---
|
||||
|
||||
## APPENDIX B
|
||||
|
||||
Continuity properties of Feynman integrals with reference to Eq. (18c) are discussed. As an illustration, consider the special form:
|
||||
|
||||
\[S_{\alpha \beta \mu \lambda}^{(b)} \to g_{\alpha \beta} t_{\mu \lambda} \Lambda^{2} / (\Lambda^{2} - k^{2}), \quad (B1)\]
|
||||
|
||||
where \(t_{\mu \lambda}\) is independent of \(k\) and \(\Lambda \gg E\). The integral \(L^{(b)}(\xi)\) is then proportional to:
|
||||
|
||||
\[\begin{array}{rl} \mathcal{J}_{3\sigma \tau}(\xi) = & 8i(2\pi)^{-2} \\ & \times \int \frac{d^{4}k k_{\sigma}k_{\tau}[\Lambda^{2} / (\Lambda^{2} - k^{2})]}{[k^{2} + i\epsilon][k^{2} - 2\xi l\cdot k + i\epsilon][k^{2} + 2p_{2}\cdot k + i\epsilon]}. \end{array} \quad (B2)\]
|
||||
|
||||
This integral is a continuous function of \(\xi\) in \(-1 \leq \xi \leq 1\). If \(\Lambda \gg m_p\) and terms of order \(m_p/\Lambda\) are neglected:
|
||||
|
||||
\[\lim_{\xi \to 0}J_{3\sigma \tau}(\xi) = J_{3\sigma \tau}(0) = \frac{p_{2\sigma}p_{2\tau}}{m_{p}^{2}} - g_{\sigma \tau}\left[\frac{3}{4} + \ln\left(\frac{\Lambda}{m_{p}}\right)\right]. \quad (B3)\]
|
||||
|
||||
This agrees with the general discussion after Eq. (18c).
|
||||
|
||||
However, \(J_{3\sigma \tau}\) is **not** a continuous function of the four variables \(l_{\mu}\) in the neighborhood of \(l_{\mu} = 0\). The cone \(l^{2} = 0\) ("the light cone of the massless electron") is a surface of logarithmic singularities—these are the mass singularities discussed in Ref. 3 and elsewhere.\(^{27,28}\) In the proton rest frame, one contribution to (B2) can be written approximately as \(-[(l_{\sigma}l_{\tau})/(2m_{p}p)]\ln[(E+p)/(E-p)]\). Replacing \(l_{\sigma} \to \xi l_{\sigma}\), \(p \to \xi p\), \(E \to \xi E\), this is continuous in \(\xi\) and tends to 0 as \(\xi = 0\), but is not continuous in \(E\) and \(p\) as \(E = p\) is a line of logarithmic singularities. By examining continuity with respect to \(\xi\), one avoids the light cone of the massless electron. For the general argument of Sec. III, it is not actually necessary to assume continuity or boundedness—the minimal required property is that such contributions should not diverge faster than \(\ln \xi\) as \(\xi \to 0\).
|
||||
|
||||
---
|
||||
|
||||
## ACKNOWLEDGMENTS
|
||||
|
||||
The author is greatly indebted to Professor W. Zimmermann and Dr. J. Abate and Dr. R. Rabitz for many helpful and stimulating discussions. He also wishes to thank Professor G. Kallen and Professor R. E. Norton and Mrs. H. R. Quinn for very interesting communications.
|
||||
|
||||
---
|
||||
|
||||
## References
|
||||
|
||||
1. R. E. Behrends, R. J. Finkelstein, and A. Sirlin, Phys. Rev. 101, 866 (1956).
|
||||
2. S. M. Berman, Phys. Rev. 112, 267 (1958).
|
||||
3. T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959).
|
||||
4. S. M. Berman and A. Sirlin, Ann. Phys. (N.Y.) 20, 20 (1962).
|
||||
5. L. Durand III, L. F. Landovitz, and R. B. Marr, Phys. Rev. 130, 1188 (1963).
|
||||
6. G. Källén, Nucl. Phys. B1, 225 (1967).
|
||||
7. For nuclear \(\beta\) decay, the independent-particle model calculations of the electromagnetic corrections of order \(Z\alpha^{2}\) have been discussed by W. Jaus and G. Rasche, Nucl. Phys. A143, 202 (1970) and W. Jaus, Phys. Letters 40B, 616 (1972).
|
||||
8. See, for example, the discussion in Sec. 4 of Ref. 3.
|
||||
9. T. D. Lee, Phys. Rev. 128, 899 (1962).
|
||||
10. M. Veltman, Physica 29, 186 (1963).
|
||||
11. R. A. Shaffer, Phys. Rev. 131, 2203 (1963); 135, B187 (1964).
|
||||
12. R. P. Feynman, unpublished.
|
||||
13. S. M. Berman and A. Sirlin, Ref. 4.
|
||||
14. G. Källén, Ref. 6.
|
||||
15. J. D. Bjorken, Phys. Rev. 148, 1467 (1966).
|
||||
16. E. S. Abers, R. E. Norton, and D. A. Dicus, Phys. Rev. Letters 18, 676 (1967).
|
||||
17. D. R. Yennie, S. C. Frautschi, and H. Suura, Ann. Phys. (N.Y.) 13, 379 (1961); N. Meister and D. R. Yenn
|
||||
@@ -0,0 +1,209 @@
|
||||
# Accurate Verification of the Conserved-Vector-Current and Standard-Model Predictions
|
||||
|
||||
A. Sirlin and R. Zucchini
|
||||
|
||||
*Physical Review Letters*, 20 October 1986
|
||||
|
||||
Department of Physics, New York University, New York, New York 10003
|
||||
(Received 11 June 1986)
|
||||
|
||||
PACS numbers: 23.40.Bw, 12.15.Ji, 13.40.Ks
|
||||
|
||||
---
|
||||
|
||||
## Abstract
|
||||
|
||||
An approximate analytic calculation of \(O(Z\alpha^{2})\) corrections to Fermi decays is presented. When the analysis of Koslowsky et al. is modified to take into account the new results, it is found that each of the eight accurately studied \(\mathcal{F}t\) values differs from the average by \(\leq 1\sigma\), thus significantly improving the comparison of experiments with conserved-vector-current predictions. The new \(\mathcal{F}t\) values are lower than before, which also brings experiments into very good agreement with the three-generation standard model, at the level of its quantum corrections.
|
||||
|
||||
---
|
||||
|
||||
## 1. Introduction and Motivation
|
||||
|
||||
The analysis of superallowed Fermi transitions has long been a cornerstone in the study of weak interactions and plays a basic role in verification of the standard model (SM) at the level of its quantum corrections.\(^{1,2}\) In recent years, experiments have attained great precision (\(\approx 0.1\%\)), and at this level a sharp discrepancy has been found between the four low-\(Z\) and the four high-\(Z\) decays.\(^{3,4}\)
|
||||
|
||||
Motivated by this experimental situation and compelling theoretical arguments, the authors have reexamined the \(O(Z\alpha^{2})\) corrections to Fermi decays. These are defined as the residual \(O(Z\alpha^{2})\) corrections not contained in the product \(F(Z,E)(1 + \delta_{1})\), where \(F(Z,E)\) is the Fermi function and \(\delta_{1}\) the \(O(\alpha)\) correction.
|
||||
|
||||
The problem of evaluating \(O(Z\alpha^{2})\) and \(O(Z^{2}\alpha^{3})\) corrections was tackled in the influential papers of Jaus and Rasche\(^{5}\) (called I) and Jaus\(^{6}\) (called II). These authors developed a systematic perturbative method in the framework of the independent-particle model. However, the results obtained in these pioneering papers face severe difficulties when confronted with theoretical expectations based on general theorems of perturbative quantum field theory.
|
||||
|
||||
---
|
||||
|
||||
## 2. General Theoretical Considerations
|
||||
|
||||
### 2.1 Diagram Classification
|
||||
|
||||
The crucial diagrams are depicted in two figures:
|
||||
|
||||
- **Figure 1**: Leading corrections of \(O(Z\alpha^{2})\). Here \(q\) stands for the momentum of the Coulombic photon interacting with the daughter nucleus, while \(k\) represents the momentum of the usual four-dimensional photon. In diagram (a), one subtracts from the diagram on the left the same diagram with \(q\) set equal to zero in the indicated propagator.
|
||||
- **Figure 2**: Nonleading corrections of \(O(Z\alpha^{2})\).
|
||||
|
||||
The combination of graphs in Fig. 1 is finite in both the ultraviolet and infrared domains. The same is true of the diagrams of Fig. 2 after charge renormalization [which affects Fig. 2(a)] and mass renormalization [which must be included in Fig. 2(b)]. Because all relevant mass scales are \(<< m_{W}\), these \(O(Z\alpha^{2})\) contributions may be treated in the local Fermi theory.
|
||||
|
||||
### 2.2 Logarithmic Structure and Mass Singularities
|
||||
|
||||
According to I, when the daughter nucleus is regarded as a point particle, the graphs in Fig. 1 lead to a large contribution to the decay rate, namely \(Z\alpha^{2}\ln(M/m)\), where \(M\) and \(m\) are the proton and positron masses, respectively. After the finite nuclear size is taken into account (II), \(M\) is essentially replaced by \(\Lambda \equiv \sqrt{6}/a\), where \(a\) is the rms radius of the charge distribution of the daughter nucleus.
|
||||
|
||||
A key theoretical question is posed: Knowing the coefficient of \(\ln M\) in the corrections to the total decay rate, can one determine on the basis of general principles the coefficients of the other possible logarithms, \(\ln E_m\) and \(\ln m\), in the limit \(m/E_m \to 0\)? (Here \(E\) and \(E_m\) are the relativistic energy of the positron and its maximum value, respectively.)
|
||||
|
||||
The answer is yes, by invoking the theorems on cancellation of mass singularities for total decay rates.\(^{7}\) If \(F(Z,E)\) is expanded in powers of the bare charge \(e_0\), there are no \(m\) singularities in the corrections to the total decay rate. Thus, the only \(m\) singularities are induced by renormalization. To obtain the coefficient of \(\ln m\), one expands: \(F(Z,E) = 1 - \pi Z a_0 / \beta + \ldots\) (with \(a_0 = e_0^2/4\pi\)) and inserts \(a_0 = a[1 + (2a/3\pi)\ln(1/m) + \ldots]\). The \(m\) singularity is associated with the renormalization of the vacuum-polarization graph of Fig. 2(a).
|
||||
|
||||
With knowledge of the coefficients of \(\ln M\) and \(\ln m\), the coefficient of \(\ln E_m\) is determined, leading to the conclusion that the corrections to the total decay rate must be of the form:
|
||||
|
||||
\[\frac{\overline{\Delta P}}{\overline{P}_0} = Z a^2 \left[ \ln\left(\frac{M}{2E_m}\right) - \ln\left(\frac{2E_m}{m}\right) + c + \ldots \right] \quad (1)\]
|
||||
|
||||
for a point daughter nucleus, and a similar expression with \(M \to \Lambda\) when finite nuclear size is considered. Here the bars denote integration over the positron spectrum, \(c\) is a constant, and the ellipses stand for terms that vanish as \(m/E_m \to 0\) as well as very small energy-dependent contributions of \(O(E/M, E_m/\Lambda)\).
|
||||
|
||||
Neglecting terms that vanish as \(m/E_m \to 0\) corresponds to treating the positrons in the extreme relativistic approximation (ERA). Because in all accurately measured superallowed decays the positron is relativistic throughout most of the spectrum, this is expected to be a good approximation, especially for high-\(Z\) nuclei.
|
||||
|
||||
### 2.3 Discrepancy with Previous Results
|
||||
|
||||
It is very difficult to reconcile the results of I and II with Eq. (1). The contribution from Fig. 1 was expressed in those papers as \(Z\alpha^2[\ln(M/m) + \Delta^0(E_0)]\), where \(E_0 = E_m/m\). The term \(\Delta^0(E_0)\) was evaluated numerically and found to be a positive and monotonically increasing function of \(E_0\). Instead, from Eq. (1) one expects contributions from Fig. 1 to be of the form:
|
||||
|
||||
\[Z\alpha^2[\ln(M/2E_m) + c' + \ldots] = Z\alpha^2[\ln(M/m) - \ln(2E_0) + c' + \ldots]\]
|
||||
|
||||
so that \(\Delta^0(E_0)\) should be a monotonically decreasing function of \(E_0\). Moreover, unless \(c'\) is a large positive constant, one expects \(\Delta^0(E_0)\) to be negative, especially for the high-\(Z\) cases.
|
||||
|
||||
---
|
||||
|
||||
## 3. Detailed Calculation
|
||||
|
||||
### 3.1 Point Nucleus Contribution from Figure 1
|
||||
|
||||
The starting point is the consideration of Fig. 1 for a point daughter nucleus. Only terms of leading order in \(M\) are retained because other contributions involve additional powers of \(k\) and \(q\) and, as explained in II, are suppressed by small factors of \(O(\Lambda/M)\) when finite nuclear size is considered.
|
||||
|
||||
After evaluating the two-loop integrals over \(k\) and \(q\) and performing the appropriate traces, the contribution of Fig. 1 at the level of the positron spectrum is:
|
||||
|
||||
\[\frac{\Delta P_1}{P_0} = \frac{2Z\alpha^2 M}{\pi} \int_0^1 \frac{u du}{(1-u)^{1/2}} \int_0^1 dv \int_0^1 dz \frac{1}{A^2} \left[ Lv \tan^{-1}\left(\frac{Lv}{X}\right) + \frac{9(L^2 v^2 - A^2)}{X} \frac{\pi}{2} + \ldots \right], \quad (2)\]
|
||||
|
||||
where \(L\) is the three-momentum of the positron, \(A^2 \equiv a_0^2 - L^2(1-uv)\), \(X \equiv (A^2 - L^2 v^2)^{1/2}\), \(a_0 = Muv(1-z) + E(1-uv)\), and the ellipsis represents subdominant contributions.
|
||||
|
||||
In the ERA, including subdominant contributions:
|
||||
|
||||
\[\Delta P_1 = P_0 Z\alpha^2[\ln(M/2E) + 5/2 - \pi^2/6]. \quad (3)\]
|
||||
|
||||
In the opposite (nonrelativistic) limit, setting \(L = 0\) in Eq. (2):
|
||||
|
||||
\[\Delta P_1 = P_0 Z\alpha^2[\ln(M/m) + 4 - 6\ln 2].\]
|
||||
|
||||
Writing \(\Delta P_1/P_0 = Z\alpha^2[\ln(M/E) + f(E)]\), one finds that \(f(E)\) varies from a small positive constant (\(\sim -0.16\)) in the extreme relativistic domain to a small negative one (\(\sim -0.16\)) in the NR limit, indicating that \(\ln(M/E)\) is by far the dominant contribution.
|
||||
|
||||
### 3.2 Figure 2 Contributions
|
||||
|
||||
The diagrams from Fig. 2 are much less important numerically. In the ERA:
|
||||
|
||||
\[\Delta P_2 = P_0 Z\alpha^2\left[-\frac{5}{12}\ln(2E/m) + \frac{1}{9}\pi^2 - \frac{7}{18}\right]. \quad (4)\]
|
||||
|
||||
### 3.3 Combined Result
|
||||
|
||||
Combining Eqs. (3) and (4):
|
||||
|
||||
\[\frac{\Delta P}{P_0} = Z\alpha^2\left[\ln\left(\frac{M}{2E}\right) - \frac{5}{12}\ln\left(\frac{2E}{m}\right) + \frac{43}{18}\right]. \quad (5)\]
|
||||
|
||||
Averaging over the positron spectrum\(^{8}\):
|
||||
|
||||
\[\frac{\overline{\Delta P}}{\overline{P}_0} = Z\alpha^2\left[\ln\left(\frac{M}{2E_m}\right) - \frac{5}{12}\ln\left(\frac{2E_m}{m}\right) + \frac{133}{36}\right], \quad (6)\]
|
||||
|
||||
or, recalling \(E_0 = E_m/m\):
|
||||
|
||||
\[\frac{\overline{\Delta P}}{\overline{P}_0} = Z\alpha^2\left[\ln\left(\frac{M}{m}\right) - \frac{17}{12}\ln(2E_0) + \frac{133}{36}\right]. \quad (7)\]
|
||||
|
||||
Equation (6) is in agreement with the theoretically expected Eq. (1) and determines \(c = 133/36\). The constant \(c'\) is obtained by averaging Eq. (3): \(c' = \frac{5}{2} - \pi^2/6 + \frac{47}{60} \approx 1.64\). Thus \(-\ln(2E_0) + c'\) is negative for the eight decays, confirming the expectation that \(\Delta^0(E_0)\) should be negative.
|
||||
|
||||
---
|
||||
|
||||
## 4. Finite Nuclear Size Effects
|
||||
|
||||
Neglecting terms of \(O(\Lambda/M)\), the fractional correction is:
|
||||
|
||||
\[\Delta = Z\alpha^2\left[1 - \gamma - \ln(Ma) - 4\pi \int_0^\infty \rho(r)r^2 \ln(r/a)dr\right],\]
|
||||
|
||||
where \(\gamma = 0.5772\) and \(\rho(r)\) is the nuclear charge density of the daughter nucleus.
|
||||
|
||||
For the modified Gaussian distribution\(^{9,10}\): \(\rho(r) = \mathrm{const}\times (1 + \alpha k^2 r^2/a^2)e^{-r^2 k^2/a^2}\) with \(k^2 = 3(2+5\alpha)/2(2+3\alpha)\) and \(\alpha = (Z-2)/3\):
|
||||
|
||||
\[\Delta = -Z\alpha^2[\ln(M/\Lambda) + \kappa_1(Z)], \quad (8)\]
|
||||
|
||||
where \(\kappa_1(Z) \equiv \frac{1}{2}[\gamma + \ln(3/2k^2) + 2\alpha/(2+3\alpha)]\) is a small constant varying from 0.33 for \(Z=7\) to 0.36 for \(Z=26\).
|
||||
|
||||
For a uniformly charged sphere of radius \(R\) [\(a = (\frac{3}{5})^{1/2}R\)], \(\Delta\) is analogous with \(\kappa_1(Z)\) replaced by \(\kappa_2(Z) \equiv \gamma - \frac{4}{3} + \frac{1}{2}\ln 10 = 0.40\).
|
||||
|
||||
The variation in \(\kappa\) is negligible; the authors use Eq. (8) with \(\kappa_1(Z) = 0.35\) and \(a = (\frac{3}{5})^{1/2}r_0 A^{1/3}\) fm (equivalently, \(M/\Lambda = r_0 A^{1/3}/0.665\)), with \(r_0\) given by Wilkinson.\(^{11}\)
|
||||
|
||||
The sum \(\langle\delta_2(E)\rangle\) of Eqs. (7) and (8) is the final expression for the \(O(Z\alpha^2)\) corrections.\(^{12}\)
|
||||
|
||||
---
|
||||
|
||||
## 5. Estimate of \(O(Z^2\alpha^3)\) Corrections
|
||||
|
||||
An estimate of \(O(Z^2\alpha^3)\) corrections (residual contributions not contained in \(F(Z,E)(1+\delta_1+\delta_2)\)) is given. Jaus\(^{6}\) evaluated the coefficient of the ultraviolet logarithm from diagrams analogous to Fig. 1 (with one additional Coulombic photon attached to the positron). Based on the earlier discussion, this is identified as \(\ln(\Lambda/E)\). A heuristic estimate that includes \(m\) singularities is:\(^{14}\)
|
||||
|
||||
\[\langle\delta_1^{\mathrm{HE}}(E)\rangle = Z^2 a^3 \left[ a\ln(\Lambda/E_m) + b\ln(2E_m/m) + d f(E)\ln(2E_m/m) + \ldots \right]\]
|
||||
|
||||
where \(\alpha = 1.232\), \(b = (4/3\pi)(\frac{11}{4} - \gamma - \pi^2/6)\), \(d = 4/3\pi\), \(f(E) \equiv \ln(2E_m/m) - 5/6\), \(g(E) \equiv \frac{1}{2}[\ln^2(Rm) - \ln^2(2E/m)] + (\frac{5}{4})\ln(2RE)\), \(R = (\frac{5}{4})^{1/2}a\), and \(h = -0.649\) is a constant adjusted to cancel spurious \(m\) singularities. This will be used as an estimate of the error made in stopping the calculation at \(O(Z\alpha^2)\).
|
||||
|
||||
---
|
||||
|
||||
## 6. Results: \(\mathcal{F}t\) Values and CVC Test
|
||||
|
||||
**Table I** presents fractional radiative corrections (in percent) and \(\mathcal{F}t\) values:
|
||||
|
||||
| Decay | \(\langle\delta_1^{\mathrm{qu}}\rangle\) | \(\langle\delta_2\rangle\) | \(\langle\delta_1^{\mathrm{HE}}\rangle\) | \(\mathcal{F}t\) (old)\(^a\) (s) | \(\mathcal{F}t\) (new)\(^b\) (s) |
|
||||
|-------|---------------------|----------------|------------------------|---------------------------|---------------------------|
|
||||
| \(^{14}\mathrm{O}\) | 1.29 | 0.21 | 0.02 | 3075.5 ± 3.9 | 3073.4 ± 3.9 |
|
||||
| \(^{26}\mathrm{Al}^{m}\) | 1.11 | 0.30 | 0.04 | 3072.9 ± 3.7 | 3066.9 ± 3.9 |
|
||||
| \(^{34}\mathrm{Cl}\) | 1.00 | 0.35 | 0.06 | 3076.9 ± 4.7 | 3066.9 ± 5.0 |
|
||||
| \(^{38}\mathrm{K}\) | 0.96 | 0.37 | 0.07 | 3076.6 ± 4.6 | 3064.2 ± 5.1 |
|
||||
| \(^{42}\mathrm{Sc}\) | 0.94 | 0.40 | 0.08 | 3089.3 ± 7.5 | 3074.7 ± 7.9 |
|
||||
| \(^{46}\mathrm{V}\) | 0.90 | 0.41 | 0.10 | 3088.6 ± 4.3 | 3071.3 ± 5.2 |
|
||||
| \(^{50}\mathrm{Mn}\) | 0.87 | 0.43 | 0.11 | 3085.9 ± 5.7 | 3066.2 ± 6.5 |
|
||||
| \(^{54}\mathrm{Co}\) | 0.84 | 0.44 | 0.12 | 3087.5 ± 4.4 | 3065.4 ± 5.7 |
|
||||
| **Average** | | | | **3080.1 ± 2.4** | **3068.6 ± 1.8** |
|
||||
| **CL** | | | | **3%** | **80%** |
|
||||
|
||||
\(^a\) Taken from Ref. 4.
|
||||
\(^b\) Modified values obtained in this paper on the basis of \((\delta_1^{\mathrm{qu}} + \delta_2)\); \(\langle\delta_1^{\mathrm{HE}}\rangle\) has been included as a theoretical error.
|
||||
|
||||
The \(\langle\delta_2(E)\rangle\) correction ranges from 0.21% for \(^{14}\mathrm{O}\) to 0.44% for \(^{54}\mathrm{Co}\), while the corresponding correction reported in II varies from 0.26% to 0.93%. The main difference can be traced to the evaluation of \(\Delta^0(E_0)\) arising from Fig. 1.\(^{13}\) A more physical characterization: while papers I and II state that the leading contribution is \(Z\alpha^2\ln(M/m)\), this analysis indicates the dominant contribution to the spectrum is roughly \(Z\alpha^2\ln(\Lambda/E)\), a significantly different logarithm.
|
||||
|
||||
The overall pattern is much better than before. Seven of the \(\mathcal{F}t\) values differ from the new weighted average by \(< 1\sigma\), while \(^{14}\mathrm{O}\) differs by 1.2\(\sigma\). The \(\chi^2/\nu\) for the old and new fits are 2.2 and 0.55, respectively (\(\nu = 7\)), corresponding to confidence levels of 3% and 80%. If \(\langle\delta_1^{\mathrm{HE}}\rangle\) is not included in the error, the CL is slightly reduced to 72%.
|
||||
|
||||
The weighted averages of the four low-\(Z\) and four high-\(Z\) values are now \(3068.4\pm 2.2\) s and \(3069.0\pm 3.1\) s, respectively. Thus, the sharp discrepancy pointed out in Refs. 3 and 4 has disappeared.
|
||||
|
||||
---
|
||||
|
||||
## 7. Standard Model Test and \(V_{ud}\) Determination
|
||||
|
||||
**Table II** lists values of \(V_{ud}\) and \(V_{ud}^2 + V_{us}^2 + V_{ub}^2\) based on various combinations of the new \(\mathcal{F}t\) values and \(V_{us} = 0.220\pm 0.002\):\(^{17}\)
|
||||
|
||||
| Combination | \(V_{ud}\) | \(V_{ud}^2 + V_{us}^2 + V_{ub}^2\) |
|
||||
|-------------|--------|-------------------------------|
|
||||
| (a) \(^{14}\mathrm{O}\) | 0.9740 ± 0.0015 | 0.9971 ± 0.0031 |
|
||||
| (b) Average of \(^{14}\mathrm{O}\) and \(^{26}\mathrm{Al}^{m}\) | 0.9745 ± 0.0013 | 0.9981 ± 0.0027 |
|
||||
| (c) Average of four low-\(Z\) decays | 0.9748 ± 0.0011 | 0.9986 ± 0.0023 |
|
||||
| (d) Average of 8 decays | 0.9747 ± 0.0011 | 0.9984 ± 0.0023 |
|
||||
|
||||
The treatment of inner corrections and calculation of errors is the same as in Ref. 2.\(^{16}\) There is now very good agreement between a large number of experiments and the three-generation SM, at the level of its quantum corrections. If the latter were not included, the unitarity bound would be exceeded by \(\approx 3.7\%\) and the SM would not be tenable.
|
||||
|
||||
In spite of small errors, the possibility of a fourth generation with substantial mixing is still open. Using the last entry in Table II, one finds \(|V_{ub}| \leq 0.064\) (90% CL). These calculations can put sharp constraints on various types of new physics, as competing theories must match the SM at a high level of precision. On the experimental side, accurate measurements of \(^{10}\mathrm{C}\) (Ref. 18) and \(\pi\beta\) decay would be important to verify the consistent trend.
|
||||
|
||||
---
|
||||
|
||||
## References
|
||||
|
||||
1. A. Sirlin, Nucl. Phys. B71, 29 (1974); Rev. Mod. Phys. 50, 573 (1978); Ann. N.Y. Acad. Sci. 461, 521 (1986).
|
||||
2. W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 56, 22 (1986).
|
||||
3. I. S. Towner and J. C. Hardy, in *Proceedings of the Seventh International Conference on Atomic Masses and Fundamental Constants*, edited by O. Klepper (Gesellschaft fur Schwerionenforschung, Darmstadt, 1984), p. 564.
|
||||
4. V. T. Koslowsky et al., in Ref. 3, p. 572.
|
||||
5. W. Jaus and G. Rasche, Nucl. Phys. A143, 202 (1970).
|
||||
6. W. Jaus, Phys. Lett. 40, 616 (1972).
|
||||
7. T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959); T. Kinoshita, J. Math. Phys. (N.Y.) 3, 650 (1962); T. D. Lee and M. Nauenberg, Phys. Rev. B 133, 1549 (1964).
|
||||
8. The positron spectrum \(S(E)\) in the ERA: \(S(E)dE = E_0^2 x^2(1-x)^2 dx\), \(x \equiv E/E_m\), which leads to \(\langle\ln x\rangle = -\frac{1}{20}\).
|
||||
9. R. Hofstadter, Annu. Rev. Nucl. Sci. 7, 231 (1957).
|
||||
10. W. Jaus, Nucl. Phys. A177, 70 (1971).
|
||||
11. D. H. Wilkinson, in *Nuclear Physics with Heavy Ions and Mesons*, edited by R. Balian, M. Rho, and G. Ripka (North-Holland, Amsterdam, 1978), p. 877.
|
||||
12. All reference to the nucleon (in contraposition to the nucleus) has disappeared in \(\delta_1(E)\), which strongly suggests that the leading contributions do not depend on the independent-particle model.
|
||||
13. Note that an additional contribution of the form \(\mathrm{const}/E_0\) cannot explain the difference for all the nuclei.
|
||||
14. A. Sirlin, unpublished.
|
||||
15. A. Sirlin, Phys. Rev. 164, 1767 (1967).
|
||||
16. The \(\mathcal{F}t\) values from Ref. 4 include the Coulomb correction evaluated by I. S. Towner, J. C. Hardy, and M. Harvey, Nucl. Phys. A284, 296 (1977), with an allowance for theoretical uncertainties. In extracting \(V_{ud}\), the total errors in the \(\mathcal{F}t\) values have been further doubled (Ref. 2).
|
||||
17. H. Leutwyler and M. Roos, Z. Phys. C 25, 91 (1984); J. Donoghue, B. Holstein, and S. Klimt, to be published.
|
||||
18. The ERA is less reliable for low-\(Z\) decays such as \(^{10}\mathrm{C}\) but the \(O(Z\alpha^{2})\) corrections are less important for such cases. It would be desirable to perform new computer-based calculations that retain the small terms neglected in the ERA as well as those of \(O(\Lambda/M)\).
|
||||
@@ -0,0 +1,210 @@
|
||||
# Remarks concerning the \(O(Za^2)\) corrections to Fermi decays, conserved-vector-current predictions, and universality
|
||||
|
||||
A. Sirlin
|
||||
|
||||
*Physical Review D*, 1 June 1987
|
||||
|
||||
Department of Physics, New York University, New York, New York 10003
|
||||
(Received 2 December 1986; revised manuscript received 18 February 1987)
|
||||
|
||||
---
|
||||
|
||||
## Abstract
|
||||
|
||||
This article studies finite-nuclear-size contributions to the \(O(Za^2)\) corrections to Fermi decays using realistic nuclear-charge distributions. Combined with results from Koslowsky et al. and recent work by the author with Zucchini and by Jaus and Rasche, these refinements yield an average \(\mathcal{F}_t = 3070.6 \pm 1.6\) s for accurately measured superallowed Fermi transitions. Correspondingly, \(V_{ud} = 0.9744 \pm 0.0010\) and \(V_{ud}^2 + V_{us}^2 + V_{ub}^2 = 0.9979 \pm 0.0021\), in good agreement with the three-generation standard model at the level of its quantum corrections. Agreement with conserved-vector-current (CVC) predictions is very good, with each of the eight transitions differing from the average by \(< 1\sigma\). The consequences of using two other calculations of the nuclear mismatch correction \(\delta_c\) (Wilkinson's microscopic analysis and the recent results of Ormand and Brown) are briefly discussed. A useful upper bound on \(\mathcal{F}_t\), independent of the \(\delta_c\) calculation, is given.
|
||||
|
||||
---
|
||||
|
||||
## 1. Introduction and Background
|
||||
|
||||
After a 14-year hiatus, significant progress has been achieved in the analysis of the \(O(Za^2)\) corrections to Fermi decays, bringing highly accurate experiments into close agreement with CVC predictions and the standard model (SM) at the level of quantum corrections.\(^{1,2}\)
|
||||
|
||||
The \(O(Za^2)\) corrections are defined as contributions of this order to the positron-nucleus interaction not contained in the product \(F(Z,E)(1 + \delta_1)\), where \(F(Z,E)\) is the Fermi function and \(\delta_1\) the \(O(\alpha)\) correction. An ingenious perturbative formalism to treat these residual corrections in the independent-particle model of \(\beta\) decay was developed by Jaus and Rasche in the early 1970s.\(^{3,4}\) Originally, these corrections relied heavily on numerical, computer-based calculations.
|
||||
|
||||
When experiments on superallowed Fermi transitions reached \(\sim 0.1\%\) accuracy, a sharp discrepancy between low-\(Z\) and high-\(Z\) decays was found.\(^{5,6}\) The first recent theoretical progress occurred when an approximate analytic calculation, supported by general theorems of perturbative quantum field theory, revealed a sharp difference with the older numerical results. Reexamination of \(\mathcal{F}t\) values in light of new \(O(Za^2)\) corrections showed very good agreement with CVC and SM predictions.\(^{1}\) Subsequently, Jaus and Rasche reanalyzed the complicated two-loop calculations, found an error in the older work, and obtained answers completely consistent with the analytical work of Ref. 1.\(^{7}\)
|
||||
|
||||
Some relatively small differences remain between Refs. 1 and 7 (referred to as I and II, respectively), concerning the treatment of finite-nuclear-size effects and small \(O(Z^2 a^3)\) corrections. This article provides explicit expressions for finite-nuclear-size effects in terms of realistic nuclear-charge distributions, compares results of I and II, modifies the treatment of \(O(Z^2 a^3)\) corrections, and discusses the effect of these refinements on CVC verification, SM tests, and determination of \(V_{ud}\).
|
||||
|
||||
---
|
||||
|
||||
## 2. Classification of \(O(Za^2)\) Corrections
|
||||
|
||||
The \(O(Za^2)\) corrections involve the exchange of a photon of momentum \(k\) and a Coulombic photon of momentum \(q\). The diagrams fall into two classes:
|
||||
|
||||
- **Class (a)**: The \(k\) photon is interchanged between the decaying nucleons and the positron.
|
||||
- **Class (b)**: The \(k\) photon is fully attached to the positron (pure QED correction).
|
||||
|
||||
The sum of diagrams in each class is ultraviolet and infrared convergent.
|
||||
|
||||
Class (a) contributions are further classified according to dependence on the nucleon mass \(M\):
|
||||
- Terms of leading order in \(M\), called \(Za^2 \Delta_1\) in II, are complicated. Writing the charge form factor of the daughter nucleus as \(G(Q^2) = 1 + G(Q^2) - 1\) (with \(Q^2 \equiv -q^2 \geq 0\)), \(\Delta_1\) splits into:
|
||||
- \(\Delta_1^0(E)\): corresponds to a point nucleus, depends on positron energy \(E\).
|
||||
- \(\Delta_1^F\): arises from \(G(Q^2) - 1\), is energy independent (neglecting \(O(E/\Lambda)\) terms, where \(\Lambda \equiv \sqrt{6}/a\) and \(a\) is the rms nuclear charge radius), depends on the charge distribution.
|
||||
- Terms not of leading order in \(M\): constants of first and higher orders in \(\Lambda/M\). Vector and axial-vector contributions are called \(Za^2 \Delta_2\) and \(Za^2 \Delta_3\), respectively.
|
||||
- Class (b) diagrams are denoted by \(Z\alpha^{2}\Lambda_{4}(E)\) in II.
|
||||
|
||||
---
|
||||
|
||||
## 3. Comparison of Analytic and Computer-Based Calculations
|
||||
|
||||
Analytic expressions for \(\Delta_{1}^{0}(E)\) and \(\Delta_{4}(E)\) obtained in the extreme relativistic approximation (ERA) are given in paper I. In the ERA, the sum is:
|
||||
|
||||
\[\Delta_{1}^{0}(E) + \Delta_{4}(E)\equiv \ln (M / m) - \frac{5}{12}\ln (2E / m) + \frac{43}{18}, \quad (1)\]
|
||||
|
||||
where \(m\) is the positron mass.
|
||||
|
||||
**Table I** compares Eq. (1) with the new computer-based calculations of paper II:
|
||||
|
||||
| \(p/m\) | \(\Delta_1+\Delta_4\) [Eq. (1)] | \(\Delta_1+\Delta_4\) (II) | Diff. (%) | \(\delta P(E)\) (Z=26) (%) |
|
||||
|--------|---------------------------|----------------------|------------|------------------------|
|
||||
| 0.1 | 8.74 | 8.99 | -2.8 | -0.03 |
|
||||
| 0.5 | 8.56 | 8.48 | +0.9 | +0.01 |
|
||||
| 1.0 | 8.17 | 7.98 | +2.4 | +0.03 |
|
||||
| 5.0 | 6.03 | 6.01 | +0.3 | 0.00 |
|
||||
| 10.0 | 4.90 | 4.93 | -0.6 | 0.00 |
|
||||
| 16.0 | 4.12 | 4.18 | -1.4 | -0.01 |
|
||||
|
||||
The numerical calculation agrees remarkably well with the analytic expression, and the difference has a very small effect on the decay probability and \(\mathcal{F}t\) values.
|
||||
|
||||
---
|
||||
|
||||
## 4. Finite-Nuclear-Size Effects
|
||||
|
||||
### 4.1 General Expressions
|
||||
|
||||
After neglecting terms of \(O(E/\Lambda)\) and performing integrations, \(\Delta_{1}^{F}\) can be written as:
|
||||
|
||||
\[\Delta_{1}^{F} = \frac{4}{\pi^{2}}\int_{0}^{1}\left[\frac{u}{1 - u}\right]^{1 / 2}du\int_{0}^{1}dv\int_{0}^{\infty}\frac{dQ}{Q} [G(Q^{2}) - 1]\arctan \left[\frac{Mu^{1 / 2}v}{Q(1 - u)^{1 / 2}}\right]. \quad (2)\]
|
||||
|
||||
Evaluating this for a spherically symmetric charge distribution \(\rho(r)\) (normalized such that \(\int \rho(r)d^{3}r = 1\)) yields:
|
||||
|
||||
\[\Delta_{1}^{F} = 1 - \gamma -4\pi \int_{0}^{\infty}\rho (r)r^{2}\ln (Mr)d r\] \[-(8 / M)\int_{0}^{\infty}\rho (r)r[1 + \gamma +\ln (Mr)]d r~, \quad (4)\]
|
||||
|
||||
where \(\gamma = 0.5772\ldots\), and terms of \(O((\Lambda/M)^{3})\) are neglected.
|
||||
|
||||
The contributions from \(\Delta_{2}\) and \(\Delta_{3}\) are:
|
||||
|
||||
\[\Delta_{2} = (4 / M)\int_{0}^{\infty}\rho (r)r dr(1 - \pi /4Mr), \quad (5)\]
|
||||
|
||||
\[\Delta_{3} = \frac{8g_{A}(1 + \mu\nu)}{M}\int_{0}^{\infty}\rho (r)r\left[\gamma +\ln (Mr) - \frac{1}{2} +\frac{\pi}{8Mr}\right]dr, \quad (6)\]
|
||||
|
||||
where \(g_{A} = 1.25\), \(1 + \mu \nu = 4.70\) arises from the isovector magnetic moment of the nucleon, and terms of \(O((\Lambda/M)^3)\) are neglected.
|
||||
|
||||
### 4.2 Specific Nuclear Models
|
||||
|
||||
**Uniformly charged sphere** of radius \(R = (\frac{2}{3})^{1 / 2}a = 10^{1 / 2} / \Lambda\):
|
||||
|
||||
\[\begin{array}{l}{\Delta_{1}^{F} = \ln \left[\frac{\Lambda}{M}\right] - \kappa_{2} - \frac{3}{10^{1 / 2}\pi}\frac{\Lambda}{M}\left[\frac{1}{2} +\gamma +\ln 10^{1 / 2} + \ln \left[\frac{M}{\Lambda}\right]\right],}\\ {\Delta_{2} = \frac{3}{2\times 10^{1 / 2}\pi}\frac{\Lambda}{M}\left[1 - \frac{\pi}{2\times 10^{1 / 2}}\frac{\Lambda}{M}\right],}\\ {\Delta_{3} = \frac{3}{10^{1 / 2}\pi}g_{A}(1 + \mu \nu)\frac{\Lambda}{M}\left[\gamma -1 + \ln 10^{1 / 2} + \ln \left[\frac{M}{\Lambda}\right] + \frac{\pi}{4\times 10^{1 / 2}}\frac{\Lambda}{M}\right],} \end{array} \quad (7)\]
|
||||
|
||||
where \(\kappa_{2}\equiv \gamma -\frac{4}{3} +\ln 10^{1 / 2} = 0.395\).
|
||||
|
||||
**Modified Gaussian model** \(\rho(r) = \mathrm{const}\times (1 + \alpha k^{2}r^{2} / a^{2})e^{- r^{2}k^{2} / a^{2}}\) with \(\alpha = (Z - 2) / 3\), \(k^{2} = (\frac{3}{2})(2 + 5\alpha) / (2 + 3\alpha)\) (Ref. 8):
|
||||
|
||||
\[\begin{array}{l}{\Delta_{1}^{F} = \ln \left[\frac{\Lambda}{M}\right] - \kappa_{1}(Z) - \frac{8}{(6\pi^{3})^{1 / 2}}\frac{k}{2 + 3\alpha}\frac{\Lambda}{M}\left[\left[1 + \frac{\gamma}{2} +\ln \left[\frac{6^{1 / 2}}{k}\frac{M}{\Lambda}\right]\right](1 + \alpha) + \frac{\alpha}{2}\right],}\\ {\Delta_{2} = \frac{4}{(6\pi^{3})^{1 / 2}}\frac{k}{2 + 3\alpha}\frac{\Lambda}{M}\left[1 + \alpha -\left[\frac{\pi^{3}}{96}\right]^{1 / 2}\frac{k}{M}\left[1 + \frac{\alpha}{2}\right]\right],}\\ {\Delta_{3} = \frac{8}{(6\pi^{3})^{1 / 2}}g_{A}(1 + \mu \nu)\frac{k}{2 + 3\alpha}\frac{\Lambda}{M}\left[\left[\frac{\gamma - 1}{2} +\ln \left[\frac{6^{1 / 2}}{k}\frac{M}{\Lambda}\right]\right](1 + \alpha) + \frac{\alpha}{2} +\left[\frac{\pi^{3}}{6}\right]^{1 / 2}\frac{k}{8}\frac{\Lambda}{M}\right],} \end{array} \quad (10)\]
|
||||
|
||||
where \(\kappa_{1}(Z)\equiv \frac{1}{2} [\gamma +\ln (3 / 2k^{2}) + 2\alpha /(2 + 3\alpha)]\).
|
||||
|
||||
---
|
||||
|
||||
## 5. Results for Finite-Nuclear-Size Contributions
|
||||
|
||||
**Table II** lists values for \(\Delta^{F}\equiv \Delta_{1}^{F} + \Delta_{2} + \Delta_{3}\) for the uniformly charged sphere (UCS), modified Gaussian (MG), and Yukawa (Y) models, and the corresponding \(\langle \delta_{2}(E)\rangle\) values:
|
||||
|
||||
| Nucleus | \(\Delta^{F}\) (Y) | \(\Delta^{F}\) (UCS) | \(\Delta^{F}\) (MG) | \(\langle\delta_{2}\rangle\) (Y) (%) | \(\langle\delta_{2}\rangle\) (MG) (%) |
|
||||
|---------|---------------|----------------|----------------|--------------------------|---------------------------|
|
||||
| \(^{14}\mathrm{O}\) | -0.68 | -1.39 | -1.31 | 0.24 | 0.22 |
|
||||
| \(^{26}\mathrm{Al}^{m}\) | -0.99 | -1.60 | -1.53 | 0.35 | 0.32 |
|
||||
| \(^{34}\mathrm{Cl}\) | -1.12 | -1.69 | -1.64 | 0.43 | 0.39 |
|
||||
| \(^{38}\mathrm{K}^{m}\) | -1.17 | -1.73 | -1.73 | 0.46 | 0.41 |
|
||||
| \(^{42}\mathrm{Sc}\) | -1.22 | -1.77 | -1.72 | 0.50 | 0.45 |
|
||||
| \(^{46}\mathrm{V}\) | -1.26 | -1.80 | -1.75 | 0.53 | 0.47 |
|
||||
| \(^{50}\mathrm{Mn}\) | -1.30 | -1.83 | -1.79 | 0.55 | 0.49 |
|
||||
| \(^{54}\mathrm{Co}\) | -1.33 | -1.86 | -1.82 | 0.57 | 0.50 |
|
||||
|
||||
The UCS and MG models give very similar answers; both differ significantly from the Yukawa results because \(\Lambda^{2}/(\Lambda^{2}+Q^{2})\) is not a realistic representation of nuclear charge form factors (it decreases too slowly as \(Q^{2}\) increases and has a singularity at \(r=0\)).
|
||||
|
||||
---
|
||||
|
||||
## 6. \(\mathcal{F}t\) Values and CVC Test
|
||||
|
||||
**Table III** presents the fractional radiative corrections and \(\mathcal{F}t\) values:
|
||||
|
||||
| Decay | \(\langle\delta_{1}^{m}\rangle\) (%) | \(\langle\delta_{2}\rangle^{m}\) (%) | \(\langle\delta_{3}^{m}\rangle\) (%) | \(\mathcal{F}t\) (s) |
|
||||
|-------|--------------------------|------------------------------|-------------------------------|----------------|
|
||||
| \(^{14}\mathrm{O}\) | 1.29 | 0.22 | 0.01 | 3074.0 ± 3.9 |
|
||||
| \(^{26}\mathrm{Al}^{m}\) | 1.11 | 0.32 | 0.02 | 3068.1 ± 3.7 |
|
||||
| \(^{34}\mathrm{Cl}\) | 1.00 | 0.39 | 0.03 | 3069.0 ± 4.7 |
|
||||
| \(^{38}\mathrm{K}^{m}\) | 0.96 | 0.41 | 0.04 | 3066.6 ± 4.6 |
|
||||
| \(^{42}\mathrm{Sc}\) | 0.94 | 0.45 | 0.04 | 3077.5 ± 7.5 |
|
||||
| \(^{46}\mathrm{V}\) | 0.90 | 0.47 | 0.05 | 3074.7 ± 4.3 |
|
||||
| \(^{50}\mathrm{Mn}\) | 0.87 | 0.49 | 0.05 | 3069.6 ± 5.7 |
|
||||
| \(^{54}\mathrm{Co}\) | 0.84 | 0.50 | 0.06 | 3069.0 ± 4.4 |
|
||||
| **Average** | | | | **3070.6 ± 1.6** |
|
||||
|
||||
The \(\chi^{2}/\nu\) is 0.57, corresponding to a 78% confidence level. Each of the eight accurately measured Fermi transitions stands within \(1\sigma\) of the average, in very good agreement with CVC predictions.
|
||||
|
||||
The difference of 2 sec between the average \(\mathcal{F}t = 3070.6\pm 1.6\) sec and the result of paper I is due to inclusion of \(O(\Lambda/M)\) terms (≈0.9 sec) and \(O(Z^{2}\alpha^{3})\) terms (≈1.1 sec). The difference of 1.8 sec with paper II is mainly due to use of realistic nuclear charge distributions.
|
||||
|
||||
---
|
||||
|
||||
## 7. Determination of \(V_{ud}\) and Standard Model Test
|
||||
|
||||
**Table IV** lists values of \(V_{ud}\) and \(V_{ud}^{2} + V_{us}^{2} + V_{ub}^{2}\) obtained from various combinations of \(\mathcal{F}t\) values, using \(V_{us} = 0.220\pm 0.002\) (Ref. 13):
|
||||
|
||||
| Combination | \(V_{ud}\) | \(V_{ud}^{2}+V_{us}^{2}+V_{ub}^{2}\) |
|
||||
|-------------|---------|-------------------------------|
|
||||
| \(^{14}\mathrm{O}\) | 0.9739 ± 0.0015 | 0.9969 ± 0.0031 |
|
||||
| Average of \(^{14}\mathrm{O}\) and \(^{26}\mathrm{Al}^{m}\) | 0.9744 ± 0.0012 | 0.9979 ± 0.0025 |
|
||||
| Average of 4 low-Z decays | 0.9746 ± 0.0011 | 0.9982 ± 0.0023 |
|
||||
| Average of 8 decays | 0.9744 ± 0.0010 | 0.9979 ± 0.0021 |
|
||||
|
||||
The results are in very good agreement with the third-generation SM at the level of its quantum corrections. Such accurate agreement can be used to derive interesting constraints on new physics.\(^{14}\)
|
||||
|
||||
---
|
||||
|
||||
## 8. Dependence on Nuclear Mismatch Corrections \(\delta_c\)
|
||||
|
||||
Using the same radiative corrections and errors as in Table III, results based on alternative \(\delta_c\) calculations are:
|
||||
|
||||
- **Wilkinson (W)** (Ref. 9): \((\mathcal{F}t)_{\mathrm{ave}} = 3069.8\pm 1.6\) s (\(\chi^{2}/\nu = 0.97\))
|
||||
- **Ormand and Brown (OB)** (Ref. 15): \((\mathcal{F}t)_{\mathrm{ave}} = 3076.1\pm 1.6\) s (\(\chi^{2}/\nu = 1.01\))
|
||||
- **Towner-Hardy-Harvey (THH)** (Ref. 12): 3070.6 ± 1.6 s (\(\chi^{2}/\nu = 0.57\)) [Table III]
|
||||
|
||||
All fits with the older \(O(Z\alpha^{2})\) corrections were very poor, so the new results are highly significant. The THH and W calculations lead to similar averages, while the OB result differs from THH by +0.18%, implying \(V_{ud} = 0.9735\pm 0.0010\) and \(V_{ud}^{2} + V_{us}^{2} + V_{ub}^{2} = 0.9962 \pm 0.0021\).
|
||||
|
||||
### Model-Independent Upper Bound
|
||||
|
||||
Assuming all corrections except \(\delta_c\) are well understood and \(\delta_c > 0\), the lowest \(\mathcal{F}t\) value calculated with \(\delta_c = 0\) provides an upper bound. From \(^{26}\mathrm{Al}^{m}\) data, \(\mathcal{F}t \leq 3078.6\pm 1.4\) s. The THH and W \(\delta_c\) values essentially obey this bound, but the central values of \(^{14}\mathrm{O}\), \(^{34}\mathrm{Cl}\), \(^{42}\mathrm{Sc}\), and \(^{46}\mathrm{V}\) \(\mathcal{F}t\) values calculated with OB \(\delta_c\) values are somewhat larger.
|
||||
|
||||
### Alternative Determination
|
||||
|
||||
Using the \(\mathcal{F}t\) value for \(^{26}\mathrm{Al}^{m}\) calculated with \(\delta_c = (0.34\pm 0.08)\%\) (the average of THH, W, and OB values, with error covering the range) leads to \((\mathcal{F}t)_{\mathrm{Al}} = 3068.1\pm 2.8\) s, corresponding to \(V_{ud} = 0.9748\pm 0.0010\) and \(V_{ud}^{2} + V_{us}^{2} + V_{ub}^{2} = 0.9986 \pm 0.0021\). This is compatible with the Table IV determination.
|
||||
|
||||
---
|
||||
|
||||
## 9. Discussion and Conclusions
|
||||
|
||||
The claim by Drukarev and Strikman\(^{16}\) that screening due to atomic electrons is considerably larger than given by conventional theory\(^{9}\) is noted: inclusion of their additional screening correction would worsen the very good fit of Table III.
|
||||
|
||||
The discrepancies mentioned are at the 0.1–0.2% level—accuracies seldom attained in particle physics—and are much smaller than the overall quantum corrections of ∼4%. It is hoped that nuclear and atomic theorists and \(\beta\)-decay experimentalists will reexamine these small corrections in light of the new \(O(Za^{2})\) calculations. Accurate measurements of \(C\) and \(\pi \beta\) decays would be most welcome.
|
||||
|
||||
---
|
||||
|
||||
## References
|
||||
|
||||
1. A. Sirlin and R. Zucchini, Phys. Rev. Lett. 57, 1994 (1986) (called I).
|
||||
2. A. Sirlin, Nucl. Phys. B71, 29 (1974); Rev. Mod. Phys. 50, 573 (1978); Ann. N.Y. Acad. Sci. 461, 521 (1986).
|
||||
3. W. Jaus and G. Rasche, Nucl. Phys. A143, 202 (1970); Aust. J. Phys. 39, 1 (1986).
|
||||
4. W. Jaus, Phys. Lett. 40, 616 (1972).
|
||||
5. I. S. Towner and J. C. Hardy, in *Proceedings of the Seventh International Conference on Atomic Masses and Fundamental Constants*, edited by O. Klepper (Gesselschaft fur Schwerionenforschung, Darmstadt, 1984), p. 564.
|
||||
6. V. T. Koslowsky et al., in *Proceedings of the Seventh International Conference on Atomic Masses and Fundamental Constants* (Ref. 5), p. 572.
|
||||
7. W. Jaus and G. Rasche, Phys. Rev. D 35, 3420 (1987) (called II).
|
||||
8. R. Hofstadter, Annu. Rev. Nucl. Sci. 7, 231 (1957).
|
||||
9. D. H. Wilkinson, in *Nuclear Physics with Heavy Ions and Mesons*, edited by R. Balian, M. Rho, and G. Ripka (North-Holland, Amsterdam, 1978), p. 877.
|
||||
10. In paper I the factor \(a\) in the term \(Z^{2}a^{3}a\ln (A / E)\) must be changed to \(a = (\pi^{2} / 3 - \frac{1}{2}) / \pi = 0.5697\) to reflect the new result of Jaus and Rasche (II) concerning the leading logarithm of \(O(Z^{2}a^{3})\) and, in the definition of \(f(E)\), \(E_{m}\) should be replaced by \(E\) to correct a misprint.
|
||||
11. W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 56, 22 (1986).
|
||||
12. I. S. Towner, J. C. Hardy, and M. Harvey, Nucl. Phys. A284, 296 (1977).
|
||||
13. H. Leutwyler and M. Roos, Z. Phys. C 25, 91 (1984); J. Donoghue, B. Holstein, and S. Klimt (unpublished).
|
||||
14. W. J. Marciano and A. Sirlin, Phys. Rev. D 35, 1672 (1987).
|
||||
15. W. E. Ormand and B. A. Brown, in *Weak and Electromagnetic Interactions in Nuclei*, proceedings of the International Symposium Heidelberg, July 1986, edited by H. V. Klapdor (Springer, Berlin, 1986), p. 545.
|
||||
16. E. G. Drukarev and M. I. Strikman, Leningrad Nuclear Physics Institute Report No. 1221, 1986 (unpublished).
|
||||
@@ -0,0 +1,207 @@
|
||||
# Radiative correction to the \(\bar{\nu}_{e}(\nu_{e})\) spectrum in \(\beta\) decay
|
||||
|
||||
A. Sirlin
|
||||
|
||||
*Physical Review D*, 84, 014021 (2011)
|
||||
|
||||
Department of Physics, New York University, 4 Washington Place, New York, New York 10003, USA
|
||||
(Received 18 May 2011; published 19 July 2011)
|
||||
|
||||
DOI: 10.1103/PhysRevD.84.014021
|
||||
PACS numbers: 13.30.Ce, 13.40.Ks
|
||||
|
||||
---
|
||||
|
||||
## I. INTRODUCTION
|
||||
|
||||
The \(\mathcal{O}(\alpha)\) radiative correction (RC) to the \(e^{-}(e^{+})\) spectrum has played an important role in precision studies of \(\beta\) decay. It is described by a RC \((\alpha/2\pi)g(E,E_{0})\), where \(E\) and \(E_{0}\) are the energy and end-point energy of the \(e^{-}(e^{+})\) [1,2]. This RC is not contained in Fermi's Coulomb function \(F(Z,E)\), which is separately included. The effect of \((\alpha/2\pi)g(E,E_{0})\) on the spectrum shape is significant, particularly for large values of \(E_{0}\) [1,3]. It also gives important contributions to theoretical calculations of the \(\mathcal{F}t\)-values of superallowed \((0^{+}\rightarrow 0^{+})\) Fermi transitions [4] and the neutron lifetime [5].
|
||||
|
||||
The superallowed Fermi transitions play a crucial role in the precise verification of the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The current analysis, which includes large short-distance effects [4], the RC \((\alpha/2\pi)g(E,E_{0})\), and a number of more recent refinements [5,6], leads to the conclusion [7]:
|
||||
|
||||
\[|V_{ud}|^2 +|V_{us}|^2 +|V_{ub}|^2 = 0.99995(61), \quad (1)\]
|
||||
|
||||
an impressive test of the three-generation standard theory at the level of its quantum corrections. The overall electroweak corrections to Eq. (1) are of \(\mathcal{O}(4\%)\), i.e., 66 times larger than the \(0.061\%\) error.
|
||||
|
||||
The correction \((\alpha/2\pi)g\), in conjunction with short-distance effects and refinements discussed in Refs. [4-6], has also been recently applied to the calculation of the muon capture rate in the singlet state of \(\mathrm{H}\) [8].
|
||||
|
||||
The aim of this paper is to derive the corresponding RC to the \(\bar{\nu}_{e}(\nu_{e})\) spectrum in allowed \(\beta\) decay, which is of considerable current interest for reactor studies of neutrino oscillations [9-11].
|
||||
|
||||
---
|
||||
|
||||
## II. RADIATIVE CORRECTION TO THE \(\bar{\nu}_{e}\) (\(\nu_{e}\)) SPECTRUM
|
||||
|
||||
As in Ref. [1], the basic process of neutron \(\beta\) decay, \(n\rightarrow p + e^{-} + \bar{\nu}_{e}\), is considered, neglecting terms of \(\mathcal{O}(\alpha E/M)\) where \(M\) is the nucleon mass.
|
||||
|
||||
### A. Virtual Corrections and Zeroth Order Spectrum
|
||||
|
||||
The \(\mathcal{O}(\alpha)\) virtual RC, as well as the zeroth order transition probability, are obtained from those in the \(e^{-}\) spectrum by simply replacing \(E\rightarrow \hat{E}, p\rightarrow \hat{p}, \beta\rightarrow \hat{\beta}\), where \(E\), \(p\), and \(m\) are the \(e^{-}\) energy, momentum, and mass, \(\beta = p/E\), \(K\) is the \(\bar{\nu}_{e}\) energy, and \(\hat{E}\), \(\hat{p}\), and \(\hat{\beta}\) are defined by:
|
||||
|
||||
\[\hat{E}\equiv E_0 - K;\qquad \hat{p}\equiv (\hat{E}^2 - m^2)^{1/2};\qquad \hat{\beta}\equiv \hat{p}/\hat{E}. \quad (2)\]
|
||||
|
||||
The reason is that in deriving the \(\bar{\nu}_{e}\) spectrum, one integrates over the \(e^{-}\) phase space factor \(d^3 p / 2E\) rather than the \(\bar{\nu}_{e}\)'s \(d^3 K / 2K\), with energy conservation expressed by \(\delta(E_0 - K - E)\).
|
||||
|
||||
### B. Inner Bremsstrahlung Contributions
|
||||
|
||||
The inner bremsstrahlung (IB) contributions present more subtle and theoretically interesting differences with the \(e^{-}\) spectrum.
|
||||
|
||||
Calling \(\epsilon_{\gamma}\) the radiated photon energy, the IB amplitude involves two terms: the first behaves as \(1/\epsilon_{\gamma}\) as \(\epsilon_{\gamma}\to 0\); the second is of \(\mathcal{O}(\epsilon_{\gamma}^{0})\). The square of the absolute value of the first term, combined with the \(d^{3}k / 2\epsilon_{\gamma}\) photon phase space factor (\(k\) is the photon momentum), leads to the infrared (IR) divergent contribution. The interference of the two terms and the square of the absolute value of the second are IR convergent and are evaluated separately.
|
||||
|
||||
As in [1], the IR divergence is regularized by introducing an infinitesimal photon mass \(\lambda_{\mathrm{min}}\) so that \(\epsilon_{\gamma} = (k^{2} + \lambda_{\mathrm{min}}^{2})^{1/2}\). After integrating over the proton and electron momenta, as well as the \(\bar{\nu}_{e}\) direction, the IR contribution is proportional to \(\int_{0}^{k_{m}}(d^{3}k / 2\epsilon_{\gamma}^{3})f(\epsilon_{\gamma},\nu)\), where:
|
||||
|
||||
\[\begin{array}{r}f(\epsilon_{\gamma},\nu) = pE\{(1 / \beta \nu)\ln [(1 + \beta \nu) / (1 - \beta \nu)]\\ -(1 - \beta^{2}) / (1 - \beta^{2}\nu^{2}) - 1\}, \end{array} \quad (3)\]
|
||||
|
||||
\(\nu = k / \epsilon_{\gamma}\), \(k_{m} = \hat{E} - m\) is the maximum photon energy for given \(K\), and it is understood that:
|
||||
|
||||
\[E = \hat{E} - \epsilon_{\gamma}, \quad (4)\]
|
||||
|
||||
which follows from energy conservation.
|
||||
|
||||
The complicated dependence of \(f(\epsilon_{\gamma},\nu)\) on \(\epsilon_{\gamma}\) and \(\nu\) hinders the subsequent integration over photon momenta. A convenient strategy is to split \(f(\epsilon_{\gamma},\nu)\) according to:
|
||||
|
||||
\[f(\epsilon_{\gamma},\nu) = f(0,\nu) + [f(\epsilon_{\gamma},\nu) - f(0,\nu)]. \quad (5)\]
|
||||
|
||||
The IR divergence arises from \(f(0,\nu)\), a function that only involves photon momenta through its dependence on \(\nu\). Changing the integration variable from \(k\) to \(\nu\), the integral \(\int_{0}^{k_{m}}(d^{3}k / 2\epsilon_{\gamma}^{3})f(0,\nu)\) can be performed analytically and leads to a function of \(\hat{\beta}\) and \(\ln(k_{m}/\lambda_{\mathrm{min}})\). Its combination with the virtual \(\mathcal{O}(\alpha)\) RC is well defined since the IR divergences cancel exactly.
|
||||
|
||||
The integration of the second term in Eq. (5) is IR convergent, so we set \(\lambda_{\mathrm{min}} = 0\) (\(\nu = 1\)) and replace \(f(\epsilon_{\gamma},\nu) - f(0,\nu) \rightarrow f(k,1) - f(0,1)\). In this latter expression, the \(\epsilon_{\gamma} = k\) dependence is contained in \(E\) and, therefore, also in \(p\), \(\beta\), and \(f(k,1)\) (cf. Equations 3 and 4).
|
||||
|
||||
Changing the variable of integration from \(k\) to \(E\) via Eq. (4), the integral \(\int d^{3}k / 2k^{3}[f(k,1) - f(0,1)]\) can also be performed analytically. This includes the integral:
|
||||
|
||||
\[I = \int_{m / \hat{E}}^{1}d t\ln (1 - t) / (t^{2} - m^{2} / \hat{E}^{2})^{1/2}, \quad (6)\]
|
||||
|
||||
for which a compact analytic representation has been derived:
|
||||
|
||||
\[I = -\mathrm{Li}_{2}(2\hat{\beta} /(1 + \hat{\beta})) + \mathrm{tanh}^{-1}\hat{\beta}\ln (2\hat{\beta}^{2} / (1 + \hat{\beta})), \quad (7)\]
|
||||
|
||||
where:
|
||||
|
||||
\[\mathrm{Li}_{2}(z) = -\int_{0}^{z}(d t / t)\ln (1 - t) \quad (8)\]
|
||||
|
||||
is the dilogarithm.
|
||||
|
||||
The \(d^{3}k\) integration of the two remaining contributions—the interference of the two terms in the IB amplitude and the absolute value square of the second—are IR convergent. Changing again the variable of integration from \(k\) to \(E\) via Eq. (4), they can also be performed analytically.
|
||||
|
||||
### C. Final Expression for \(h(\hat{E},E_0)\)
|
||||
|
||||
Combining all the \(\mathcal{O}(\alpha)\) contributions, the radiatively corrected expression for the \(\bar{\nu}_{e}\) spectrum is:
|
||||
|
||||
\[dP_{\nu} = dP_{\nu}^{0}[1 + (\alpha /2\pi)h(\hat{E},E_0)], \quad (9)\]
|
||||
|
||||
where:
|
||||
|
||||
\[dP_{\nu}^{0} = A\hat{p}\hat{E}F(Z,\hat{E})K^{2}dK \quad (10)\]
|
||||
|
||||
is the zeroth order spectrum, and:
|
||||
|
||||
\[\begin{array}{l}{h(\hat{E},E_0) = 3\ln \left(\frac{m_{p}}{m}\right) + \frac{23}{4} -\frac{8}{\beta}\mathrm{Li}_{2}\left(\frac{2\hat{\beta}}{\mathrm{1} + \hat{\beta}}\right)}\\ {+8\left(\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}} -1\right)\ln \left(\frac{2\hat{E}\hat{\beta}}{m}\right)}\\ {+4\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}}\left[\frac{7 + 3\hat{\beta}^{2}}{8} -2\tanh^{-1}\hat{\beta}\right],} \end{array} \quad (11)\]
|
||||
|
||||
where \(m_{p}\) is the proton mass. In Eq. (10), \(F(Z,\hat{E})\) is the Fermi Coulomb function evaluated at \(E = \hat{E}\), and \(A\) is a constant independent of \(K\). In neutron \(\beta\) decay, for example, \(A = (g_{V}^{2} + 3g_{A}^{2}) / 2\pi^{3}\), where \(g_{V} = G_{F}V_{ud}\) and \(g_{A}\) are the vector and axial vector coupling constants.
|
||||
|
||||
In analogy with the \(e^{-}\) spectrum case [1], Eqs. (9) and (11) can be applied to allowed nuclear \(\beta\) decay using an independent particle model of the nucleus. In that case, \(g_{V}^{2} + 3g_{A}^{2}\) is replaced by \(g_{V}^{2}|M_{F}|^{2} + g_{A}^{2}|M_{\mathrm{GT}}|^{2}\), where \(M_{\mathrm{F}}\) and \(M_{\mathrm{GT}}\) are the Fermi and Gamow-Teller matrix elements. Equation (11) is valid for both \(\bar{\nu}_{e}\) and \(\nu_{e}\) transitions.
|
||||
|
||||
Large additional electroweak corrections in the standard theory give very important contributions to the \(\mathcal{F}t\)-values of superallowed Fermi transitions and the neutron lifetime. However, they are constants independent of \(K\) and \(E\), so they do not affect the shape of the \(\bar{\nu}_{e}\) and \(e^{-}\) spectra and are not included in this discussion.
|
||||
|
||||
### D. The \(m \rightarrow 0\) Limit
|
||||
|
||||
Equation (11) simplifies considerably in the \(m\rightarrow 0\) (\(\hat{\beta}\rightarrow 1\)) limit. Noting that \(\tanh^{-1}\hat{\beta} = \ln[\hat{E}(1+\hat{\beta})/m]\) and \(\mathrm{Li}_{2}(1) = \pi^{2}/6\):
|
||||
|
||||
\[\lim_{m\to 0}h(\hat{E},E_0) = 3\ln \left(\frac{m_p}{2E_0}\right) + \frac{23}{4} -\frac{4\pi^2}{3} -3\ln (1 - y), \quad (12)\]
|
||||
|
||||
where \(y = K / E_0\). Thus, the \(m\rightarrow 0\) limit of \(h(\hat{E},E_0)\) is **convergent** and has a very simple form.
|
||||
|
||||
### E. Comparison with the Electron Spectrum Correction
|
||||
|
||||
This is in sharp contrast with the behavior of \(g(E,E_0)\), the function that describes the RC to the \(e^{-}(e^{+})\) spectrum. In the relativistic \(m^{2}\ll E^{2}\) limit:
|
||||
|
||||
\[\begin{array}{rl} & {g(E,E_0)}\\ & {\quad \rightarrow 3\ln \left(\frac{m_{p}}{2E_0}\right) - \frac{3}{4} -\frac{2\pi^{2}}{3} +4(\ln x - 1)}\\ & {\quad \times \left[\frac{1 - x}{3x} -\frac{3}{2} +\ln \left(\frac{1 - x}{x}\right)\right] + \frac{(1 - x)^2}{6x^{2}}\ln x}\\ & {\quad +\ln \left(\frac{2E_0}{m}\right)\left[\frac{4(1 - x)}{3x} -3 + \frac{(1 - x)^2}{6x^{2}} +4\ln \left(\frac{1 - x}{x}\right)\right],} \end{array} \quad (13)\]
|
||||
|
||||
where \(x = E / E_0\). Equation (13) can be gleaned from the expression between curly brackets in Eq. (4.4) of Ref. [3], after subtracting \(6\ln(\lambda/m_p) + 9/4\) (cf. footnote 23 in Ref. [1]). Thus, the correction to the electron spectrum **diverges logarithmically** in the \(m\rightarrow 0\) limit.
|
||||
|
||||
If one multiplies Eq. (13) by \(x^{2}(1-x)^{2}\) (the energy dependence of the zeroth order probability in the \(m\rightarrow 0\) limit) and integrates over \(0\leq x\leq 1\), the cofactor of \(\ln(2E_0/m)\) vanishes, so the correction to the inverse lifetime is indeed free from the electron "mass singularity" [3,12,13].
|
||||
|
||||
### F. Theoretical Explanation
|
||||
|
||||
The fact that \(h(\hat{E},E_0)\) is convergent in the \(m\rightarrow 0\) limit while \(g(E,E_0)\) is not can be explained theoretically: For given \(K\), in the \(m\rightarrow 0\) limit, all collinear \(e-\gamma\) configurations become energy degenerate with energy \(E_0 - K\) and generally give rise to mass singularities. An elementary but powerful theorem in quantum mechanics leads to the conclusion that these singularities are cancelled in the power series expansions of transition probabilities if the latter are summed over an appropriate ensemble of such degenerate states [13]. In deriving the corrections to the \(\bar{\nu}_{e}(\nu_{e})\) spectrum, the \(d^{3}p\) and \(d^{3}k\) integrations are performed, so indeed one sums over the set of collinear \(e-\gamma\) configurations that become energy degenerate in the \(m\rightarrow 0\) limit. Thus, \(h(\hat{E},E_0)\) should be free of \(\ln m\) singularities, as found. In contrast, this is not the case in the derivation of the electron spectrum, since the \(d^{3}p\) integration is not carried out.
|
||||
|
||||
### G. Consistency Check and Illustrative Examples
|
||||
|
||||
A useful consistency check: Multiplying Eq. (12) by \(y^{2}(1-y)^{2}\) and Eq. (13) by \(x^{2}(1-x)^{2}\) and integrating over \(x\) and \(y\) from 0 to 1 yields equal results, since both lead to the correction to the inverse lifetime. This has been verified to high accuracy.
|
||||
|
||||
As an illustrative example, for \(E_0 = 8\mathrm{MeV}\):
|
||||
|
||||
- The RC \((\alpha/2\pi)h(\hat{E},E_0)\) to the \(\bar{\nu}_{e}(\nu_{e})\) spectrum **increases** the transition probability by \(1.17\%\) at \(y = 0.8\) (\(K = 6.4\mathrm{MeV}\)) and by \(0.64\%\) at \(y = 0.2\) (\(K = 1.6\mathrm{MeV}\)). This behavior is reflected in Eq. (12), which shows a positive, monotonically increasing function of \(y\).
|
||||
- The RC \((\alpha/2\pi)g(E,E_0)\) to the \(e^{-}(e^{+})\) spectrum **decreases** the transition probability by \(1.13\%\) at \(x = 0.8\) (\(E = 6.4\mathrm{MeV}\)) and **increases** it by \(2.90\%\) at \(x = 0.2\) (\(E = 1.6\mathrm{MeV}\)). This can be qualitatively understood by noting that the cofactor of \(\ln(2E_0/m)\) in Eq. (13) is negative for \(x\geq 0.44\) and positive for \(x\leq 0.44\).
|
||||
|
||||
Equations (12) and (13) provide very good approximations to the exact expressions when \((E_0-K)^{2}\gg m^{2}\) and \(E^{2}\gg m^{2}\), respectively. As shown in Ref. [1], the \(\mathcal{O}(\alpha)\) RC to the shape of the \(e^{-}(e^{+})\) spectrum is not affected by strong interactions if terms of \(\mathcal{O}(\alpha E/M)\) are neglected; the same conclusion applies to the \(\mathcal{O}(\alpha)\) RC to the \(\bar{\nu}_{e}(\nu_{e})\) spectrum.
|
||||
|
||||
---
|
||||
|
||||
## III. RADIATIVE CORRECTION TO THE CONVERSION FROM THE \(e^{-}\) SPECTRUM TO THE \(\bar{\nu}_{e}\) SPECTRUM
|
||||
|
||||
Including the \(\mathcal{O}(\alpha)\) RC, the theoretical expressions for the \(e^{-}\) and \(\bar{\nu}_{e}\) spectra in allowed \(\beta\) decay can be cast in the form:
|
||||
|
||||
\[\frac{dP_e}{dE} = f_e(E,E_0), \quad (14)\]
|
||||
|
||||
\[\frac{dP_\nu}{dK} = f_\nu (K,E_0), \quad (15)\]
|
||||
|
||||
where:
|
||||
|
||||
\[\begin{array}{r}{f_{e}(E,E_{0}) = A p E(E_{0} - E)^{2}F(Z,E)\Big[1 + \frac{\alpha}{2\pi} g(E,E_{0})\Big],} \end{array} \quad (16)\]
|
||||
|
||||
\[\begin{array}{r}{f_{\nu}(K,E_{0}) = A\hat{p}\hat{E} K^{2}F(Z,\hat{E})\Big[1 + \frac{\alpha}{2\pi} h(\hat{E},E_{0})\Big],} \end{array} \quad (17)\]
|
||||
|
||||
and \(\hat{E}\) and \(\hat{p}\) are defined in Eq. (2).
|
||||
|
||||
Comparing Eqs. (16) and (17) and recalling that \(\hat{E} = E_{0} - K\):
|
||||
|
||||
\[\begin{array}{r}{f_{\nu}(K,E_{0}) = f_{e}(\hat{E},E_{0})\Big[1 + \frac{\alpha}{2\pi} h(\hat{E},E_{0})\Big] / } \end{array} \begin{array}{r}{\left[1 + \frac{\alpha}{2\pi} g(\hat{E},E_{0})\right],} \end{array} \quad (18)\]
|
||||
|
||||
or, neglecting terms of order \(\mathcal{O}(\alpha^{2})\):
|
||||
|
||||
\[\begin{array}{r}{f_{\nu}(K,E_{0}) = f_{e}(\hat{E},E_{0})\Big[1 + \frac{\alpha}{2\pi} (h(\hat{E},E_{0}) - g(\hat{E},E_{0}))\Big].} \end{array} \quad (19)\]
|
||||
|
||||
Using Eq. (11) and the exact expression for \(g(E,E_{0})\) (Eq. (20b) of Ref. [1]):
|
||||
|
||||
\[\begin{array}{r}{h(\hat{E},E_{0}) - g(\hat{E},E_{0})}\\ {= \frac{13}{2} -\frac{4}{\hat{\beta}}\mathrm{Li}_{2}\Big(\frac{2\hat{\beta}}{1 + \hat{\beta}}\Big) + 4\Big(\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}} - 1\Big)}\\ {\times \Big[\frac{11}{6} +\ln \Big(\frac{2E_{0}}{m}\Big) + 2\ln \hat{\beta} +2\ln (1 - y) - \ln y - \frac{1}{3(1 - y)}\Big]}\\ {+\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}}\Big[\frac{3 - \hat{\beta}^{2}}{2} - 4\tanh^{-1}\hat{\beta} -\frac{y^{2}}{6(1 - y)^{2}}\Big],} \end{array} \quad (20)\]
|
||||
|
||||
where \(\hat{\beta}\) is defined in Eq. (2) and \(y\equiv K / E_{0}\).
|
||||
|
||||
Equations (19) and (20) describe the conversion from the \(e^{-}\) spectrum in a specific decay into the corresponding \(\bar{\nu}_{e}\) spectrum when \(\mathcal{O}(\alpha)\) RC are included.
|
||||
|
||||
### Illustrative Example
|
||||
|
||||
For \(E_{0} = 8\mathrm{MeV}\), using Eqs. (19) and (20), the RC \((\alpha/2\pi)[h(\hat{E},E_{0}) - g(\hat{E},E_{0})]\):
|
||||
|
||||
- **Decreases** \(f_{\nu}(K,E_{0})\) by \(1.74\%\) at \(y = 0.8\) (\(K = 6.4\mathrm{MeV}\))
|
||||
- **Increases** \(f_{\nu}(K,E_{0})\) by \(1.77\%\) at \(y = 0.2\) (\(K = 1.6\mathrm{MeV}\))
|
||||
|
||||
Thus, the correction factor \([1 + (\alpha/2\pi)(h(\hat{E},E_{0}) - g(\hat{E},E_{0}))]\) in Eq. (19) is \(3.57\%\) larger at \(K = 1.6\mathrm{MeV}\) than at \(K = 6.4\mathrm{MeV}\). This behavior reflects the dominant contribution of \(-g(\hat{E},E_{0})\). As a consequence, the RC decreases (increases) \(f_{\nu}(K,E_{0})\) for high (low) \(K\), an effect that becomes sharper as \(E_{0}\) increases.
|
||||
|
||||
---
|
||||
|
||||
## IV. CONCLUSIONS
|
||||
|
||||
In summary, an analytic expression for the \(\mathcal{O}(\alpha)\) RC to the \(\bar{\nu}_{e}(\nu_{e})\) spectrum in allowed \(\beta\) decay has been derived (Eqs. (9) and (11)). The \(m\rightarrow 0\) limit of this correction is convergent and leads to a very simple expression (Eq. (12)). This is in sharp contrast to the \(\mathcal{O}(\alpha)\) RC to the \(e^{-}(e^{+})\) spectrum, which diverges as \(m\rightarrow 0\), an important difference that has been explained on theoretical grounds. These results have been applied to derive the \(\mathcal{O}(\alpha)\) RC to the conversion from the \(e^{-}\) spectrum to the \(\bar{\nu}_{e}\) spectrum (Eqs. (19) and (20)), a relation that plays an important role in reactor studies of neutrino oscillations [9-11]. A recent study concludes that the conversion procedure can be implemented with an error \(\lesssim 1\%\), provided some conditions are met [10]. If this accuracy is achieved, it is likely that the RC will play a significant role in precision studies of the \(\bar{\nu}_{e}\) spectrum.
|
||||
|
||||
---
|
||||
|
||||
## ACKNOWLEDGMENTS
|
||||
|
||||
The author is indebted to W. J. Marciano for calling his attention to the conversion from the \(e^{-}\) spectrum to the \(\bar{\nu}_{e}\) spectrum, and for very useful observations, and to A. Ferroglia and G. Ossola for very interesting and helpful discussions. This work was supported in part by NSF Grant No. PHY-0758032.
|
||||
|
||||
---
|
||||
|
||||
## References
|
||||
|
||||
1. A. Sirlin, Phys. Rev. 164, 1767 (1967).
|
||||
2. A. Sirlin, Acta Physica Austriaca, Supplement V, edited by P. Urban, (Springer-Verlag, New York, 1968), p. 353-390.
|
||||
3. T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959).
|
||||
4. A. Sirlin, Rev. Mod. Phys. 50, 573 (1978); 50, 905(E) (1978).
|
||||
5. A. Czarnecki, W. J. Marciano, and A. Sirlin, Phys. Rev. D 70, 093006 (2004).
|
||||
6. W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 96, 032002 (2006).
|
||||
7. J. C. Hardy and I. S. Towner, Phys. Rev. C 79, 055502 (2009).
|
||||
8. A. Czarnecki, W. J. Marciano, and A. Sirlin, Phys. Rev. Lett. 99, 032003 (2007).
|
||||
9. C. Bemporad, G. Gratta, and P. Vogel, Rev. Mod. Phys. 74, 297 (2002), and references cited therein.
|
||||
10. P. Vogel, Phys. Rev. C 76, 025504 (2007), and references cited therein.
|
||||
11. Th. A. Mueller et al., Phys. Rev. C 83, 054615 (2011), and references cited therein.
|
||||
12. T. Kinoshita, J. Math. Phys. (N.Y.) 3, 650 (1962).
|
||||
13. T. D. Lee and M. Nauenberg, Phys. Rev. 133, B1549 (1964).
|
||||
@@ -0,0 +1,363 @@
|
||||
# Algorithm for Radiative Corrections to Beta-Decay Spectra
|
||||
|
||||
## 1. Physical Overview
|
||||
|
||||
The radiatively corrected beta-decay spectrum takes the form:
|
||||
|
||||
\[\frac{d\Gamma}{dW} = \frac{d\Gamma_0}{dW} \cdot R(W, W_0)\]
|
||||
|
||||
where \(d\Gamma_0/dW\) is the uncorrected spectrum and \(R(W, W_0)\) is a multiplicative correction factor. The correction is organized by powers of the fine-structure constant \(\alpha\) and the nuclear charge \(Z\):
|
||||
|
||||
\[R(W, W_0) = 1 + \delta^{(1)}(W, W_0) + \delta^{(2)}(Z, W) + \delta^{(3)}(Z, W) + \cdots\]
|
||||
|
||||
**Important distinction:** Inner radiative corrections (\(\Delta_R^V\), \(\Delta_R^A\)) are energy-independent constants absorbed into effective coupling constants. They are **not** part of the spectrum shape calculation. Only the outer (energy-dependent) correction \(\delta_R(W, W_0)\) modulates the spectral shape.
|
||||
|
||||
---
|
||||
|
||||
## 2. Input Parameters
|
||||
|
||||
| Symbol | Description | Units |
|
||||
|--------|-------------|-------|
|
||||
| \(W\) | Total electron energy | \(m_e c^2\) |
|
||||
| \(W_0\) | Endpoint energy | \(m_e c^2\) |
|
||||
| \(Z\) | Nuclear charge of daughter | dimensionless |
|
||||
| \(m_p\) | Proton mass | \(m_e c^2\) (\(\approx 1836.15\)) |
|
||||
| \(\alpha\) | Fine-structure constant | \(\approx 1/137.036\) |
|
||||
|
||||
Derived quantities:
|
||||
- Momentum: \(p = \sqrt{W^2 - 1}\)
|
||||
- Velocity: \(\beta = p / W\)
|
||||
- Antineutrino energy (for neutrino spectrum corrections): \(\hat{W} = W_0 - W\)
|
||||
|
||||
---
|
||||
|
||||
## 3. Order \(\alpha\): The Universal Sirlin Function \(g(W, W_0)\)
|
||||
|
||||
### 3.1 Mathematical Form
|
||||
|
||||
The dominant correction, derived in Sirlin (1967) and refined in Sirlin & Zucchini (1986) and Sirlin (1987), is universal—identical for Fermi and Gamow-Teller transitions, electrons and positrons:
|
||||
|
||||
\[g(W, W_0) = 3\ln\left(\frac{m_p}{m_e}\right) - \frac{3}{4} + 4\left(\frac{\tanh^{-1}\beta}{\beta} - 1\right)\left[\frac{W_0 - W}{3W} - \frac{3}{2} + \ln\left(\frac{2(W_0 - W)}{m_e}\right)\right] + \frac{4}{\beta}L\left(\frac{2\beta}{1+\beta}\right) + \frac{\tanh^{-1}\beta}{\beta}\left[2(1+\beta^2) + \frac{(W_0-W)^2}{6W^2} - 4\tanh^{-1}\beta\right]\]
|
||||
|
||||
The correction to the spectrum is:
|
||||
|
||||
\[\delta^{(1)}(W, W_0) = \frac{\alpha}{2\pi} \, g(W, W_0)\]
|
||||
|
||||
### 3.2 The Dilogarithm (Spence Function)
|
||||
|
||||
The function \(L(x)\) appearing above is the Spence function (dilogarithm):
|
||||
|
||||
\[L(x) = \int_0^x \frac{\ln(1-t)}{t} dt = -\text{Li}_2(-x/(1-x)) \quad \text{or equivalently} \quad L(x) = -\text{Li}_2(x)\]
|
||||
|
||||
Using the standard dilogarithm \(\text{Li}_2(x) = -\int_0^x \frac{\ln(1-t)}{t} dt = \sum_{k=1}^\infty \frac{x^k}{k^2}\):
|
||||
|
||||
\[L\left(\frac{2\beta}{1+\beta}\right) = -\text{Li}_2\left(\frac{2\beta}{1+\beta}\right)\]
|
||||
|
||||
### 3.3 Numerical Stability Considerations
|
||||
|
||||
**Inverse hyperbolic tangent:**
|
||||
|
||||
\[\tanh^{-1}\beta = \frac{1}{2}\ln\left(\frac{1+\beta}{1-\beta}\right)\]
|
||||
|
||||
When \(\beta \to 1\) (ultra-relativistic limit), compute using the identity:
|
||||
|
||||
\[\tanh^{-1}\beta = \ln\left(\frac{W+p}{m_e}\right) = \ln(W + p) \quad (\text{in } m_e \text{ units where } m_e = 1)\]
|
||||
|
||||
For \(\beta \to 0\) (non-relativistic limit), use the Taylor expansion:
|
||||
|
||||
\[\frac{\tanh^{-1}\beta}{\beta} = 1 + \frac{\beta^2}{3} + \frac{\beta^4}{5} + \cdots\]
|
||||
|
||||
Avoid computing \(\tanh^{-1}\beta\) directly when \(\beta\) is very small.
|
||||
|
||||
**Terms containing \(1/\beta\):** When \(\beta < \epsilon\) (e.g., \(\epsilon = 10^{-6}\)), evaluate the expression using a small-\(\beta\) series expansion to avoid division by zero.
|
||||
|
||||
**Dilogarithm evaluation:** Use a robust numerical implementation. For \(x > 1\), apply the identity:
|
||||
|
||||
\[\text{Li}_2(x) = \frac{\pi^2}{3} - \frac{1}{2}\ln^2(x) - \text{Li}_2(1/x) - i\pi\ln(x) \quad (x > 1)\]
|
||||
|
||||
The imaginary part cancels in the real-valued total correction.
|
||||
|
||||
### 3.4 Endpoint Behavior and Soft-Photon Resummation
|
||||
|
||||
**Critical observation:** The function \(g(W, W_0)\) contains a term proportional to \(\ln(W_0 - W)\), which diverges as \(W \to W_0\). This divergence is physical—arising from the emission of arbitrarily soft real photons—and the total decay rate integrated over all photon energies remains finite.
|
||||
|
||||
For the differential spectrum shape, the leading logarithmic behavior near the endpoint is:
|
||||
|
||||
\[g(W, W_0) \sim 4\left(\frac{\tanh^{-1}\beta}{\beta} - 1\right) \ln(W_0 - W) + \text{regular terms}\]
|
||||
|
||||
**Resummation prescription (Sirlin 1987, 2011):** Replace the logarithmically divergent term with an exponentiated form:
|
||||
|
||||
\[\ln(W_0 - W) \cdot t(\beta) \quad \longrightarrow \quad (W_0 - W)^{t(\beta)} - 1\]
|
||||
|
||||
where the exponent is:
|
||||
|
||||
\[t(\beta) = \frac{2\alpha}{\pi}\left[\frac{\tanh^{-1}\beta}{\beta} - 1\right]\]
|
||||
|
||||
**Implementation strategy:**
|
||||
|
||||
1. Compute the full \(g(W, W_0)\) including the \(\ln(W_0 - W)\) term.
|
||||
2. For \(W_0 - W < \Delta_{\text{cut}}\) (suggested \(\Delta_{\text{cut}} \sim 10^{-3}\) in units of \(m_e c^2\)), extract the coefficient of \(\ln(W_0 - W)\):
|
||||
|
||||
\[C(\beta) = 4\left(\frac{\tanh^{-1}\beta}{\beta} - 1\right)\]
|
||||
|
||||
3. Replace \(C(\beta) \cdot \ln(W_0 - W)\) with \(C(\beta) \cdot \frac{(W_0-W)^{t(\beta)} - 1}{t(\beta)}\) using:
|
||||
|
||||
\[\lim_{t \to 0} \frac{x^t - 1}{t} = \ln x\]
|
||||
|
||||
4. For energies well below the endpoint (\(W_0 - W \gg \Delta_{\text{cut}}\)), no resummation is needed and the original logarithmic form suffices.
|
||||
|
||||
**For low endpoint energies** (e.g., tritium with \(W_0 \approx 1.02\)), the endpoint resummation has negligible effect and can be omitted for simplicity.
|
||||
|
||||
---
|
||||
|
||||
## 4. Order \(Z\alpha^2\): Finite Nuclear Size Corrections
|
||||
|
||||
### 4.1 Structure
|
||||
|
||||
The \(O(Z\alpha^2)\) correction, derived in Sirlin (1987), depends on the nuclear charge distribution:
|
||||
|
||||
\[\delta^{(2)}(Z, W) = Z\alpha^2 \sum_{i=1}^4 \Delta_i(W)\]
|
||||
|
||||
The four contributions \(\Delta_i\) arise from distinct classes of Feynman diagrams (see Sirlin 1987, Sec. 2). The dominant energy dependence comes from \(\Delta_1^0(W) + \Delta_4(W)\), for which the relativistic limit gives:
|
||||
|
||||
\[\Delta_1^0(W) + \Delta_4(W) = \ln\left(\frac{M}{m_e}\right) - \frac{5}{12}\ln\left(\frac{2W}{m_e}\right) + \frac{43}{18}\]
|
||||
|
||||
where \(M\) is the nucleon mass (use proton mass \(m_p\)).
|
||||
|
||||
### 4.2 Nuclear-Model-Dependent Part
|
||||
|
||||
The finite-nuclear-size contribution \(\Delta^F = \Delta_1^F + \Delta_2 + \Delta_3\) depends on the nuclear charge distribution. For the **modified Gaussian model** (Sirlin 1987, Eq. 10):
|
||||
|
||||
\[\Delta^F(Z) = \ln\left(\frac{\Lambda}{M}\right) - \kappa_1(Z) + O(\Lambda/M)\]
|
||||
|
||||
where:
|
||||
- \(\Lambda = \sqrt{6}/a\), with \(a\) the rms charge radius
|
||||
- \(M/\Lambda = r_0 A^{1/3} / 0.665\) (using Wilkinson's \(r_0\))
|
||||
- \(\kappa_1(Z) = \frac{1}{2}\left[\gamma + \ln(3/2k^2) + 2\alpha/(2+3\alpha)\right]\), with \(\gamma = 0.5772\), \(\alpha = (Z-2)/3\), and \(k^2 = \frac{3}{2}\frac{2+5\alpha}{2+3\alpha}\)
|
||||
|
||||
### 4.3 Practical Simplification
|
||||
|
||||
For light and medium nuclei (\(Z \lesssim 30\)), the total \(O(Z\alpha^2)\) correction can be tabulated once per isotope as an energy-independent shift (the energy dependence from \(\Delta_1^0 + \Delta_4\) is small compared to the order \(\alpha\) term). **For spectrum shape calculations:**
|
||||
|
||||
1. Calculate the energy-dependent part \(\Delta_1^0(W) + \Delta_4(W)\) using the relativistic formula above.
|
||||
2. Add the energy-independent nuclear-structure part \(\Delta^F(Z)\) from precomputed values (see Sirlin 1987, Table II).
|
||||
3. If high precision is required for low-energy electrons \((p \ll m_e)\), use the full non-relativistic expressions from Sirlin 1987.
|
||||
|
||||
For a **uniformly charged sphere**, \(\kappa_2 = \gamma - 4/3 + \ln(\sqrt{10}) = 0.395\), giving results within ~2% of the modified Gaussian model.
|
||||
|
||||
---
|
||||
|
||||
## 5. Order \(Z^2\alpha^3\): Higher-Order Corrections
|
||||
|
||||
The heuristic estimate from Sirlin (1986, 1987) is:
|
||||
|
||||
\[\delta^{(3)}(Z, W) \approx Z^2\alpha^3 \left[a\ln\left(\frac{\Lambda}{W}\right) + b f(W) + \frac{4\pi}{3}g(W) - 0.649\ln(2W_0)\right]\]
|
||||
|
||||
with:
|
||||
\[a = \frac{\pi}{3} - \frac{\sqrt{3}}{2}, \quad b = \frac{4}{3\pi^4}(\pi^2 - \gamma_E) - \frac{\pi^2}{18}\]
|
||||
|
||||
And:
|
||||
\[f(W) = \ln(2W_0/m_e) - 5/6\]
|
||||
\[g(W) = \frac{1}{2}\left[\ln^2(R m_e) - \ln^2(2W/m_e)\right] + \frac{5}{4}\ln(2RW)\]
|
||||
|
||||
where \(R = \sqrt{5/4}\,a\) and \(a\) is the rms charge radius.
|
||||
|
||||
**Practical note:** For \(Z \lesssim 30\), \(\delta^{(3)} < 0.01\%\) and can typically be neglected for spectrum shape. For superallowed Fermi transitions with heavy nuclei, this correction contributes ~0.05–0.12% to the total correction (Sirlin 1987, Table III).
|
||||
|
||||
---
|
||||
|
||||
## 6. Neutrino Spectrum Correction
|
||||
|
||||
For the antineutrino/neutrino spectrum (Sirlin 2011, Eq. 11), define:
|
||||
|
||||
\[\hat{W} \equiv W_0 - W_\nu, \quad \hat{p} \equiv \sqrt{\hat{W}^2 - 1}, \quad \hat{\beta} \equiv \hat{p}/\hat{W}\]
|
||||
|
||||
The correction function \(h(\hat{W}, W_0)\) is:
|
||||
|
||||
\[h(\hat{W}, W_0) = 3\ln\left(\frac{m_p}{m_e}\right) + \frac{23}{4} - \frac{8}{\hat{\beta}}\text{Li}_2\left(\frac{2\hat{\beta}}{1+\hat{\beta}}\right) + 8\left(\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}} - 1\right)\ln\left(\frac{2\hat{W}\hat{\beta}}{m_e}\right) + 4\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}}\left[\frac{7+3\hat{\beta}^2}{8} - 2\tanh^{-1}\hat{\beta}\right]\]
|
||||
|
||||
The neutrino spectrum correction is:
|
||||
|
||||
\[\delta_\nu(\hat{W}, W_0) = \frac{\alpha}{2\pi} \, h(\hat{W}, W_0)\]
|
||||
|
||||
**Key difference from the electron correction:** The neutrino correction \(h\) is finite in the \(m_e \to 0\) limit, while the electron correction \(g\) diverges logarithmically. The \(m_e \to 0\) limit of \(h\) is particularly simple:
|
||||
|
||||
\[h(\hat{W}, W_0) \xrightarrow{m_e \to 0} 3\ln\left(\frac{m_p}{2W_0}\right) + \frac{23}{4} - \frac{4\pi^2}{3} - 3\ln\left(\frac{W_\nu}{W_0}\right)\]
|
||||
|
||||
This finite limit is explained theoretically by the Kinoshita-Lee-Nauenberg theorem: summing over the degenerate collinear \(e^-\gamma\) final states (as is done when integrating over electron momenta to obtain the neutrino spectrum) cancels the mass singularities (Sirlin 2011, Sec. II.F).
|
||||
|
||||
---
|
||||
|
||||
## 7. Conversion Between Electron and Neutrino Spectra
|
||||
|
||||
Given the electron spectrum \(f_e(W, W_0)\) for a specific decay, the corresponding neutrino spectrum including radiative corrections is (Sirlin 2011, Eq. 19):
|
||||
|
||||
\[f_\nu(W_\nu, W_0) = f_e(\hat{W}, W_0) \left[1 + \frac{\alpha}{2\pi}\left(h(\hat{W}, W_0) - g(\hat{W}, W_0)\right)\right]\]
|
||||
|
||||
where \(\hat{W} = W_0 - W_\nu\).
|
||||
|
||||
The difference function is given explicitly in Sirlin 2011, Eq. 20:
|
||||
|
||||
\[h(\hat{W}, W_0) - g(\hat{W}, W_0) = \frac{13}{2} - \frac{4}{\hat{\beta}}\text{Li}_2\left(\frac{2\hat{\beta}}{1+\hat{\beta}}\right) + 4\left(\frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}} - 1\right) \times \Bigg[\frac{11}{6} + \ln\left(\frac{2W_0}{m_e}\right) + 2\ln\hat{\beta} + 2\ln\left(\frac{W_\nu}{W_0}\right) - \ln\left(\frac{W_\nu}{W_0}\right) - \frac{1}{3W_\nu/W_0}\Bigg] + \frac{\tanh^{-1}\hat{\beta}}{\hat{\beta}}\left[\frac{3-\hat{\beta}^2}{2} - 4\tanh^{-1}\hat{\beta} - \frac{(W_\nu/W_0)^2}{6(1-W_\nu/W_0)^2}\right]\]
|
||||
|
||||
This conversion is important for reactor neutrino oscillation studies.
|
||||
|
||||
---
|
||||
|
||||
## 8. Complete Calculation Algorithm
|
||||
|
||||
### 8.1 Full Correction Factor for Electron Spectrum
|
||||
|
||||
```
|
||||
Algorithm: RadiativeCorrection(W, W0, Z)
|
||||
Input: W — array of electron total energies [m_e c^2]
|
||||
W0 — endpoint energy [m_e c^2]
|
||||
Z — nuclear charge of daughter nucleus
|
||||
Output: R — multiplicative correction factor at each W
|
||||
|
||||
Constants:
|
||||
alpha = 1/137.035999084
|
||||
m_p = 938.272 / 0.51099895 # proton mass in m_e units
|
||||
gamma_E = 0.5772156649
|
||||
pi = 3.141592653589793
|
||||
|
||||
For each W_i in W:
|
||||
1. Compute derived quantities:
|
||||
p = sqrt(W_i^2 - 1)
|
||||
beta = p / W_i
|
||||
|
||||
2. Order α correction (Section 3):
|
||||
a. Calculate tanh^{-1}(beta) / beta
|
||||
If beta < 1e-6: use Taylor series 1 + beta^2/3 + beta^4/5
|
||||
Else: tanh_inv_beta_beta = 0.5 * ln((1+beta)/(1-beta)) / beta
|
||||
|
||||
b. Calculate dilogarithm L(2*beta/(1+beta)) = -Li_2(2*beta/(1+beta))
|
||||
Use robust numerical implementation
|
||||
|
||||
c. Compute g(W_i, W0) per Eq. (20b) of Sirlin 1967
|
||||
|
||||
d. If use_endpoint_resummation and (W0 - W_i) < delta_cut:
|
||||
t_beta = (2*alpha/pi) * (tanh_inv_beta_beta - 1)
|
||||
C_beta = 4 * (tanh_inv_beta_beta - 1)
|
||||
Replace: C_beta * ln(W0 - W_i)
|
||||
-> C_beta * ((W0 - W_i)^t_beta - 1) / t_beta
|
||||
|
||||
e. delta_1 = (alpha / (2*pi)) * g(W_i, W0)
|
||||
|
||||
3. Order Zα² correction (Section 4):
|
||||
a. Energy-dependent part:
|
||||
Delta_1_4 = ln(m_p) - (5/12)*ln(2*W_i) + 43/18
|
||||
|
||||
b. Energy-independent nuclear part:
|
||||
Look up Delta_F for the given (Z, A) from precomputed table
|
||||
(Use Sirlin 1987 Table II or compute from nuclear model)
|
||||
|
||||
c. delta_2 = Z * alpha^2 * (Delta_1_4 + Delta_F)
|
||||
|
||||
4. Order Z²α³ correction (Section 5, optional):
|
||||
For Z >= 20 or precision better than 0.01% required:
|
||||
Compute delta_3 from Sirlin heuristic formula
|
||||
Else:
|
||||
delta_3 = 0
|
||||
|
||||
5. Combine corrections:
|
||||
R_i = 1 + delta_1 + delta_2 + delta_3
|
||||
|
||||
Return R array
|
||||
```
|
||||
|
||||
### 8.2 Neutrino Spectrum Correction
|
||||
|
||||
```
|
||||
Algorithm: NeutrinoRadiativeCorrection(W_nu, W0, Z)
|
||||
Input: W_nu — array of neutrino energies [m_e c^2]
|
||||
W0 — endpoint energy [m_e c^2]
|
||||
Z — nuclear charge
|
||||
Output: R_nu — multiplicative correction factor
|
||||
|
||||
For each W_nu in array:
|
||||
W_hat = W0 - W_nu
|
||||
p_hat = sqrt(W_hat^2 - 1)
|
||||
beta_hat = p_hat / W_hat
|
||||
|
||||
Compute h(W_hat, W0) per Sirlin 2011 Eq. (11)
|
||||
(Use same numerical stability measures as for g-function)
|
||||
|
||||
R_nu_i = 1 + (alpha/(2*pi)) * h(W_hat, W0)
|
||||
```
|
||||
|
||||
### 8.3 Conversion Algorithm (Electron → Neutrino Spectrum)
|
||||
|
||||
```
|
||||
Algorithm: ElectronToNeutrinoConversion(W_nu, W0, Z, f_e)
|
||||
Input: W_nu — neutrino energies
|
||||
W0 — endpoint
|
||||
Z — nuclear charge
|
||||
f_e — function returning electron spectrum values
|
||||
Output: f_nu — neutrino spectrum values
|
||||
|
||||
For each W_nu in array:
|
||||
W_hat = W0 - W_nu
|
||||
|
||||
# Evaluate electron spectrum at W_hat
|
||||
f_e_val = f_e(W_hat)
|
||||
|
||||
# Compute difference correction
|
||||
Compute delta_diff = (alpha/(2*pi)) * [h(W_hat, W0) - g(W_hat, W0)]
|
||||
per Sirlin 2011 Eq. (20)
|
||||
|
||||
f_nu_i = f_e_val * (1 + delta_diff)
|
||||
```
|
||||
|
||||
---
|
||||
|
||||
## 9. Physical Interpretation and Validation
|
||||
|
||||
### 9.1 Dominant Terms
|
||||
|
||||
The leading \(O(\alpha)\) correction is the **universal Sirlin function** \(g(W, W_0)\). Its dominant contribution comes from the logarithm:
|
||||
|
||||
\[3\ln\left(\frac{m_p}{m_e}\right) \approx 15.4\]
|
||||
|
||||
multiplied by \(\alpha/2\pi \approx 1.16 \times 10^{-3}\), giving a typical ~1.8% correction. The energy-dependent terms modulate this by ~±1% across the spectrum.
|
||||
|
||||
### 9.2 Sign Conventions
|
||||
|
||||
- The correction factor \(R(W, W_0)\) **multiplies** the uncorrected spectrum.
|
||||
- \(R > 1\) means the correction increases the transition probability at that energy.
|
||||
- For the electron spectrum: the correction typically **decreases** the probability at high \(W\) (\(W/W_0 \gtrsim 0.44\)) and **increases** it at low \(W\) (Sirlin 2011, Sec. III).
|
||||
- For the neutrino spectrum: the correction is **positive and monotonically increasing** with \(W_\nu\) (Sirlin 2011, Sec. II.G).
|
||||
|
||||
### 9.3 Consistency Checks
|
||||
|
||||
1. **Lifetime check:** Integrate the corrected and uncorrected spectra:
|
||||
|
||||
\[\int_1^{W_0} \frac{d\Gamma_0}{dW} R(W, W_0) dW\]
|
||||
|
||||
The relative correction to the lifetime should be finite (the logarithmic divergence at the endpoint integrable over the spectrum).
|
||||
|
||||
2. **Electron-neutrino consistency:** The lifetime calculated from the corrected electron spectrum must equal that from the corrected neutrino spectrum.
|
||||
|
||||
3. **Zero-charge limit:** For \(Z = 0\), \(R(W, W_0) \to 1 + \frac{\alpha}{2\pi}g(W, W_0)\).
|
||||
|
||||
4. **Low-energy limit:** As \(W \to 1\) (non-relativistic), the correction should approach the values tabulated in Sirlin 1967.
|
||||
|
||||
---
|
||||
|
||||
## 10. Summary of Required Special Functions
|
||||
|
||||
| Function | Definition | Numerical Source |
|
||||
|----------|------------|------------------|
|
||||
| \(\text{Li}_2(x)\) | \(\sum_{k=1}^\infty x^k/k^2\) | Standard library (e.g., `scipy.special.spence` in Python, `gsl_sf_dilog` in C) |
|
||||
| \(\tanh^{-1}(x)\) | \(\frac{1}{2}\ln\frac{1+x}{1-x}\) | Built-in or \(\text{arctanh}(x)\) |
|
||||
| \(\gamma_E\) | Euler-Mascheroni constant | 0.57721566490153286 |
|
||||
|
||||
---
|
||||
|
||||
## References
|
||||
|
||||
1. A. Sirlin, Phys. Rev. 164, 1767 (1967) — Universal correction function \(g(E, E_m, m)\) and proof of model independence.
|
||||
2. A. Sirlin and R. Zucchini, Phys. Rev. Lett. 57, 1994 (1986) — Analytic \(O(Z\alpha^2)\) calculation resolving CVC discrepancy.
|
||||
3. A. Sirlin, Phys. Rev. D 35, 3420 (1987) — Refined finite-nuclear-size effects with realistic charge distributions.
|
||||
4. A. Sirlin, Phys. Rev. D 84, 014021 (2011) — Neutrino spectrum radiative correction and conversion formula.
|
||||
5. T. Kinoshita and A. Sirlin, Phys. Rev. 113, 1652 (1959) — Mass singularity cancellation theorems.
|
||||
Reference in New Issue
Block a user