Full One-Loop Radiative Corrections to the Asymmetry Parameter of Polarised Neutron Beta Decay

We present a calculation of full one-loop radiative corrections, including the constant term, to the asymmetry parameter of polarised neutron beta decay. This gives the radiative correction to the axial coupling constant $g_A$ extracted from the beta asymmetry so that it ties to $g_A$ that appears in neutron decay lifetime in a consistent renormalisation scheme. We find that the ratio of axial-vector to vector couplings determined from the beta asymmetry, after taking account of the outer radiative correction, is related to the bare value as $G_A/G_V=1.0012 G_A^0/G_V^0$.

Beta decay asymmetry of the polarised neutron has been used to determine the axial vector coupling constant g A of the nucleon. One-loop radiative corrections to the asymmetry parameter has been calculated by several authors [1]- [4], but their calculations do not include the constant term, or so-called inner corrections, which require a special care in the treatment of the UV divergence of the radiative correction. This makes the identification of g A extracted from β decay asymmetry with that which appears in the nucleon beta decay rate ambiguous 1 .
The radiative correction to beta decay is UV divergent and it is rendered finite only with the use of Weinberg-Salam's theory of electroweak interaction. The complication arises from the fact that one must deal with quarks in electroweak theory, and one must continue to calculations with hadrons at low energies [5].
In this paper we calculate one-loop radiative corrections including the inner correction: hereby the coupling constant g A that appears in the β asymmetry parameter is unambiguously tied to that in the decay rate in a consistent renormalisation scheme. The key point of the calculation is the clarification of the universal and non-universal UV divergent parts by using the current algebra technique and the proof that the same combinations of the renormalisation factors appear in the beta asymmetry as in the beta decay rate. The separation of the UV divergences into universal and non-universal parts was done first by Abers et al. [6] for the Fermi transition of nuclear beta decay, and then used by Sirlin [7,5] to develop a practical scheme of one-loop radiative corrections for the 0 + → 0 + transition. The scheme was extended to the Gamow-Teller transition by our recent publication (Paper I) [8]. The present work is an application of the formalism developed in Paper I. We are content with the outline of the calculations in this paper, since the bulk of lengthy calculations are parallel to the ones presented in Paper I. We refer the readers who are interested in technical details of calculations to Paper I.
The tree amplitude for beta decay of the polarised neutron is given by where G V = G F cosθ C with the universal Fermi coupling G F and the Gell-Mann-Lévy-Cabibbo angle θ C , n µ is the polarisation vector of the neutron with n 2 = −1, n · p 1 = 0, and s = ±1, and W λ (p 2 , p 1 ) is a general form of the weak vertex of hadrons and reads 1 It is included in Sirlin's proof [7] that the inner radiative corrections can be factored out for the processes concerning the unpolarised neutron. As far as the authors know there is no proof which shows that the inner radiative correction for the polarised neutron and that for the unpolarised neutron should agree.
at low energies. We retain f V = 1 to trace the vector coupling in the calculation. The spinors of the neutron, proton, electron and antineutrino are denoted by u n , u p , u e , and v ν , respectively, with the momenta specified in parentheses. After spin summation and integration over p ν the amplitude square reads where E and E ν are energies of the electron and the antineutrino. Therefore, the asymmetry parameter is given by the electron velocity factor β = | ℓ|/E being removed as a convention.
To evaluate full one loop corrections, we divide the integration region of the virtual gauge bosons into long-and short-distance parts [5]: where k is the momentum of the virtual gauge bosons, and the mass scale M , introduced by hand, divides the low-and high-energy regimes and is supposed to lie between the protonneutron masses (m p and m n ) and the W and Z boson masses, m W and m Z . Old-fashioned four-Fermi interactions are applied to the proton and neutron in region (i), and the mass scale M is regarded as the ultraviolet cutoff of the QED (i.e., purely photonic) correction.
In region (ii), electroweak theory is used for quarks and leptons, and M is the mass scale that describes the onset of the asymptotic behaviour. The concern is to connect the results in (i) and (ii) smoothly. Abers et al. [6] proved that the logarithmic divergences that are proportional to f 2 V are universal for the Fermi transition on the basis of the conserved vector current with the use of the current algebra technique. The same was proven for the g 2 A terms for the Gamow-Teller transition for which current conservation is broken only with soft operators [9,8]. This guarantees smooth connection of the logarithmic divergence for the corrections of f 2 V and g 2 A . There appear, however, intereference terms of the order f V g A , for which logarithmic divergences depend on the model of hadrons. Marciano & Sirlin [10] proposed the prescription to evaluate the high and low energy contributions separately by rendering the UV divergence in the low-energy contribution milder by taking account of form factors of hadrons. We follow the same prescription [10,11] in the present calculation of the asymmetry parameter.
The diagrams of QED one-loop corrections are depicted in Figure 1, where (v) is the vertex correction, (s) is the self energy correction and (b) is bremsstrahlung. We write the one-loop amplitude The bremsstrahlung contribution is added separately. We consider the static limit for nucle- Our calculation is done in the Feynman gauge. We start with the vertex correction, which is given by where λ is the photon mass to regulate the infrared divergence. Using identities, we decompose (7) into three parts, Here M (v1) picks up the product of (2ℓ − k) µ in (8) and (2p 2 + k) µ in (9), and at the same time W λ (p 2 + k, p 1 ) is replaced by W λ (p 2 , p 1 ). It has the same gamma matrix structure as the Born term (1), and is then written as a multiplicative correction factor. This correction has both UV and IR divergences and depends on the electron velocity. The UV divergence in M (v1) is cancelled by that in the self energy correction of M (s) , and the IR divergence, along with that arising from M (s) , is cancelled when the contribution The term M (v2) represents the combination of iσ µν k ν in (8) and (2p 2 + k) µ in (9), and W λ (p 2 + k, p 1 ) is again replaced with W λ (p 2 , p 1 ). This term is UV and IR finite, but gives an electron-velocity dependent factor. In the static limit of the nucleon the correction is also a multiplication on the tree amplitude.
A straightforward calculation yields where g(E, E 0 ) is the conventional g function that appears in the radiative correction for the beta decay rate and is defined with an additional constant 3/4 [12,7].
where E 0 is the end point energy of the electron, andĝ(E, E 0 ) is a similar function for the spin-dependent term, Here is the Spence function. We also defineĝ(E, E 0 ) with an additional constant 3/4 as a convention. These are the outer corrections, which we may write and agree with the formulae given by Shann [1] and by García and Maya [3]. They also agree with the expression derived by Yokoo et al. [2] up to a constant.
We now consider the remaining term M (v3) , where We use the current algebra technique to classify logarithmic divergences into those that Adding the tree term and after integration over p ν spin where The short distance correction from the integration region (ii) in (5) is evaluated using electroweak theory [5]. When we consider corrections relative to muon decay, we only need to consider the box diagrams of photon (or Z) and W exchanges (see Fig. 2 of Paper I). In order to connect the quark-level amplitudes with hadronic ones, we assume that the ratio of the tree and loop amplitudes for beta decays of the assembly of quarks is the same as that for neutron beta decay [5]. This is justified at least for the universal logarithmic divergent part. With this prescription the correction amounts to a multiplication factor whereQ = 1/6 is the mean charge of the isodoublet of quarks, and the second line is the correction that appears in muon decay and thus subtracted when we consider the radiative correction relative to muon decay that determines G F .
This electroweak one-loop correction amounts to adding to δ F in ′ extra terms, where the terms proportional to g A /f V are collected in C F , and similarly for δ GT in ′ , where C GT is for point nucleons.
We observe that the M dependence (upper cutoff) that appears in the first term of (22) is cancelled by the first term in the braces in (24), which demonstrates a smooth connection from electroweak theory to effective hadronic theory for the Fermi transition. The UV divergence in the term proportional to g A /f V , however, fails to cancel against the divergence with the coefficient 3Q, unlessQ has a specific (and unrealistic) value of charge. Marciano and Sirlin [10] proposed to evaluate the model-dependent long-distance divergence of the Fermi transition by rendering it softer introducing nucleon form factors, and leave the term 3Q log(m Z /M ) as it is taking M as the mass scale of the onset of the asymptotic behaviour [10]. The same procedure was followed in Paper I for the GT part, while it was noted that the inclusion of the weak magnetism is important to evaluate the long-distance integral, especially for the GT transition.
The calculation of the long distance contribution is made replacing for the electromagnetic vertex, and in eq. (7). From the form we observed in eq. (21) we expect that the calculation incorporating form factors would give rise to a result summarised in the same form, while C F and C GT are modified exactly as in [8]. Since we have not found an immediate proof that it should, we repeated a long calculation as we did in Paper I including the spin projection operator, and confirmed the anticipated result. In fact, we obtained C F and C GT exactly those that appear in the spin independent part. So we take the result of numerical integral of Paper I, C GT = 0.727 + 2.554 = 3.281, where the two parts of numbers represent contributions from the (V,A) interaction and weak magnetism. The first number in C F was evaluated by Marciano and Sirlin [10] and by Towner [11], and agrees with their results up to slight differences in the input parameters.
In conclusion the radiative correction to polarised neutron beta decay to order O(α) is summarised as in the static nucleon approximation. Hence the asymmetry parameter is written as The denominator of eq. (34) is the combination that appears in the neutron decay rate.
The energy dependent prefactor C(E), is plotted in Figure 2 as a function of the kinetic energy T = E − m e . The magnitude of (α/2π)g(E, E 0 ) and (α/2π)ĝ(E, E 0 ) is about 2%, but the two corrections nearly cancel in where E is in units of MeV. The fit, also displayed in Figure 2, overlays nearly top on the true function of C(E).
The cancellation also takes place for the inner correction. After correcting for C(E), the axial-vector to vector coupling ratio extracted from the tree level formula is related to its tree-level value asḡ The dominant part of the inner correction, including log m Z /m p cancels in δ F in − δ GT in , and the net correction is of the order of 0.1% for g A /f V (which is usually denoted as λ ≡ G A /G V = −g A /f V ).
The Particle Data Group [13] gives a value g A = 1.2670 ± 0.0030. This is obtained by averaging 5 values reported in the literature, one of which [14] is obtained including the outer radiative correction (with the inner correction discarded), and others are results that do not include radiative corrections. The outer radiative correction reduces the value of |g A | by about 0.0007, but the scatter among the data from different authors is 0.005 (rms), so systematic errors other than the radiative correction dominate the uncertainty of g A .
As we have shown that the inner radiative corrections can be included into g A in common irrespective of quantities measured for beta decay, it is a matter of definition whether they are included in g A or not. If we define the tree-level axial coupling constant it is related with the value including the radiative correction by eq. (38).