Higher-order NLO radiative corrections to polarized muon decay spectrum

Higher-order QED radiative corrections to muon decay spectrum are evaluated within the QED structure function approach in the next-to-leading order logarithmic approximation. New analytical results are given in the O (cid:0) α 3 ln 2 ( m 2 µ /m 2 e ) (cid:1) order. Earlier results in O (cid:0) α 2 ln 1 ( m 2 µ /m 2 e ) (cid:1) and O (cid:0) α 3 ln 3 ( m 2 µ /m 2 e ) (cid:1) orders are partially corrected. Numerical estimates of different contributions are presented.

Higher-order QED radiative corrections to muon decay spectrum are evaluated within the QED structure function approach in the next-to-leading order logarithmic approximation.New analytical results are given in the O α 3 ln 2 (m 2 µ /m 2 e ) order.Earlier results in O α 2 ln 1 (m 2 µ /m 2 e ) and O α 3 ln 3 (m 2 µ /m 2 e ) orders are partially corrected.Numerical estimates of different contributions are presented.

I. INTRODUCTION
Studies of muon decay µ − −→ e − + νe + ν µ (1) are one of the cornerstones of modern particle physics.This process is almost a pure weak-interaction process with small QED, QCD, and possibly new physics additions.High-precision and high-sensitivity experiments with muons can test small deviations from the Standard Model (SM) predictions, which would be the traces of new physics.Differential distributions in muon decays allows studying properties of weak interactions, including even the Dirac or Majorana nature of neutrinos [1,2].Such experiments as TWIST [3,4], Mu2e [5], Mu3e [6] require accurate advanced theoretical predictions.Precision of the predictions can be increased by calculation of higher-order radiative corrections.QED corrections to the muon lifetime are known from the works [7][8][9][10][11][12][13] upto the O(α 2 ) order, and the O(α 3 ) corrections were also recently calculated [14].The TWIST experiment required corrections to the muon decay spectrum in at least the O(α 2 ) order.In ref. [15] radiative corrections to unpolarized muon decay spectrum to the order O(α 2 L) where L ≡ ln(m 2 µ /m 2 e ), and in ref. [16] radiative corrections to the polarized muon decay spectrum in the O(α 3 L 3 ) and O(α 2 L) orders were calculated analytically.Complete two-loop QED corrections to muon decay spectrum were calculated numerically [17] in a restricted kinematics domain.Recently, the results for the pure-photonic part of the higher-order QED corrections in the leading and next-to-leading logarithmic approximations were presented in [18].
Our aim here is to calculate O(α 3 L 2 ) order corrections to the electron energy spectrum in decays of polarized and unpolarized muons.The paper is organized as follows.In the next Sect.we describe application of the QED structure function formalism for calculation of corrections to the muon decay spectrum.Sect.III contains numerical results and a discussion about the factorization scale choice.Lengthy formulae with analytic results are shifted to Appendices.

II. CORRECTIONS TO ELECTRON ENERGY SPECTRUM
Analytic calculations of higher-order radiative corrections to muon decay spectrum as well as to differential distributions of other processes like Bhabha scattering and electron-positron annihilation are very difficult because of the presence of several energy scales.Only a few complete results for O(α 2 ) QED radiative corrections to differential distributions are known.
On the other hand, the bulk of QED radiative corrections typically comes from the terms enhanced by powers of the large logarithms ln Q 2 /m 2 , where Q 2 is the square of the characteristic energy scale and m is the mass of a light charged lepton, e.g., L = ln M 2 Z /m 2 e ≈ 24 for the process of e + e − annihilation into Z boson.
In the QED structure function approach [19], one can get corrections enhanced by the large logarithms by performing a convolution of a hard scattering cross section and the corresponding parton distribution or fragmentation functions which are independent of the process.In general, the large logarithm reads where µ F is the factorization scale and µ R is the renormalization scale.If µ F ≫ µ R , the corrections proportional to powers of L yield the most significant contributions.In the case of muon decay, we can calculate the electron energy spectrum in the following way [15]: where c is the cosine of the angle θ between muon polarization vector and electron momentum.Above, z is the energy fraction of the parton j produced in muon decay, x is the energy fraction of the resulting massive electron, d Γj /(dcdz) is the energy and angle distribution arXiv:2312.10778v1[hep-ph] 17 Dec 2023 of the massless parton j, D ej is the fragmentation function that describes the probability density for transformation of the massless parton j into the physical electron in the final state, µ F is the factorization scale.Here the standard MS subtraction scheme is used.We can take µ F = m µ and µ R = m e , so the large logarithm is The perturbative expansion of the kernel coefficient function d Γj /(dcdz) in powers of α reads where F ) is the renormalized fine structure constant taken at the factorization energy scale.
Here we used process-independent fragmentation functions, calculated by solving the QED evolution equation For the initial conditions and other details, see ref. [20].
Note that here and in what follows we apply the natural choice of the QED renormalization constant µ R = m e .In the unpolarized case, the relevant coefficient functions are U,e (z) = f (1)  e (z), In the polarized case, the corrections include the parts which are dependent on the polarization degree P µ and the cosine of the angle between the muon spin and electron's momentum c = cosθ, Â(1) P,e (z) = f (1)  e (z) + cP µ g (1)  e (z), ( 10) Â(1) The expression for the differential distribution of electrons (averaged over electron spin states) in a polarized muon decay reads [16] where "+" and "-" correspond to e + and e − respectively, G F is Fermi coupling constant, E e and z = 2E e /m µ are the energy and the energy fraction of the electron (or positron), respectively.The complete expression for the spectrum functions (H = F, G) up to the O(α 3 L 2 ) order reads where indices γ, N S, S and int correspond to the pure photonic contribution, the non-singlet fermion pair one, the singlet fermion pair one, and the interference of the singlet and non-singlet pair corrections.Here and in what follows we omit the arguments of the functions h i for convenience with h ≡ f or h ≡ g.
To get the contribution of the order O(α 3 L 2 ), we have to make convolutions of the fragmentation functions with functions h i e (z) and h i γ (z): and take only the terms proportional to α 3 L 2 from the result.
Expressions for the polarized part can be received from the equation ( 15) by substitution f . Index T in the above equation marks (timelike) fragmentation functions.
We recalculated the O(α 2 L) corrections and found that the term d eγ (see Eq. (A8)) has been missed in the electron fragmentation function used in Ref. [15] .The difference is A similar change with respect to the result given in Ref. [16] is found for function G 21 (z), the corrected expression for it is shown in Appendix A.
In our previous work [20] we also corrected a mistake in the result for the O(α 3 L 3 ) singlet contribution to structure and fragmentation functions obtained in Ref. [21].As the result, the singlet part of the NLO electron frag-mentation function should read Functions a and g a read [9]: The relevant fragmentation functions are shown in the Appendix B, see ref. [20] for details of notation and explicit expressions for these functions.
Convolutions were calculated using our own program in FORM [22] and crosschecked with the help of the HPL [23] and MT [24] Wolfram Mathematica packages.The results are presented in Appendix B. A part of the results for the unpolarized case were presented in [25], here we reproduce them for the sake of completeness.
We calculated separately the parts of F 21 , F 22 , F 43 , F 44 and G 21 , G 22 , G 43 , G 44 with pure-photonic contributions in order to compare with the results of Ref. [18].Our results completely agreed with the ones from this work in the orders O(α 3 L 2 ), O(α 4 L 4 ), and O(α 4 L 3 ).

III. FACTORIZATION SCALE CHOICE AND NUMERICAL RESULTS
The factorization scale choice allows some arbitrariness.Above and in earlier papers [15,16] the muon mass was taken as the factorization scale.This choice is certainly good for the leading logarithmic approximation, but can be optimized if one goes beyond it.We suggest choosing the factorization scale as So we expand on the powers of new large logarithm: With this choice, NLO contributions are shifted by an additional term and the same for the G part.Here indices a and b are powers of α and L, respectively; Fab is the NLO contribution for the new factorization scale choice and F ab is for the old one.The LO contributions do not change, The new factorization scale choice increases the difference between the NLO and LO contributions, thus improving the convergence of the expansion in the powers of the large logs.The results for the two factorization scale choices are shown on the plots for O(α 1 ) (Fig. 1), O(α 2 ) (Fig. 2), and O(α 3 ) (Fig. 3).We also can look at the relative values of the contributions of different orders and the relative value of the full correction F (in the unpolarized case) We can't show the same picture for G part because of the zero of the function g 0 (z) at z = 0.5, but the effect would be of the same order.

IV. CONCLUSIONS
In this way, we computed radiative corrections to polarized and non-polarized muon decay spectrum in the O(α 3 L 2 ) and O(α 4 L 4 ) orders.A new factorization scale is chosen to improve the convergence of the expansion in the powers of the large logarithm and thus suppress unknown NNLO effects.With this factorization scale we The next-to-leading order electron splitting function can be divided into four parts: x, (A2) x 2 , (A3) Note that we have removed the term 10 9 P (0) ij from the expressions for functions P (1,N S) ij given in refs.[15,16] because it naturally comes from the running coupling constant and can be kept there as discussed in [20].
The fragmentation function [D ee ] T can also be divided ee The D eγ T fragmentation function reads Appendix B: Analytic results Here we present the analytic formulae for the computed higher-order contributions which appear in Eq. ( 14).

FIG. 1 .
FIG. 1. Contributions in O(α 1 L 0 ) and O(α 1 L 1 ) orders for the old and new factorization scales.We present our results for the new factorization scale for F (Figs. 4) and G (Figs.5, 6) functions.We take corrections of the orders O(α) and O(αL) from[9], and contributions of the order O(α) are recalculated with the new factorization scale.