One-Loop Corrections to Radiative Muon Decay

One-loop QED corrections to the differential width of radiative muon decay are considered. Results can be used to analyze high statistics data of modern and future experiments.


Introduction
Since the discovery of muon in 1937, the studies of its properties were always very important for the progress of the elementary particle physics. Nowadays, such precision observations like the muon life time and the muon anomalous magnetic moment are important for the checks of the Standard Model and searches for new physics. Besides many others, the process of radiative muon decay, is investigated in the modern experiments. In particular, the set of data from the PIBETA (πβ) experiment [1] at the Paul-Scherrer Institute contains a considerable Email address: arbuzov@thsun1.jinr.ru (A.B. Arbuzov). 1 A part of this work was performed in University of Alberta, Edmonton, Canada amount of these decays. Accurate measurements of the process provide interesting information about the structure of weak interactions.
In this paper we construct an advanced theoretical prediction for the differential distribution of process (1). Our calculations of radiative corrections (RC) allow to reduce the theoretical uncertainty. That makes it possible to perform precision comparisons with the experimental data and potentially look for new physics or rule out certain extensions of the Standard Model.
In the limit of small energy loss (carried away by the neutrinos), radiative corrections to the process were considered in Ref. [2]. In this limit the standard decay produces a background to the searches for the neutrinoless decay µ → eγ.
In this paper we will consider the general kinematics assuming that the energies of the final state electron and photon are above of a certain threshold and the angle between their momenta is not small (see Sect. 3.3). The tree-level distribution and the notation are introduced in the next Section. Then we consider different RC contributions. In Conclusions we present some numerical results and estimate the theoretical uncertainty in description of the radiative muon decay.

The tree-level distribution
Within the Fermi model of four-fermion interaction, the differential width of radiative muon decay was first considered in Refs. [3,4]. Accurate formulae including the terms suppressed by the factor (m e /m µ ) 2 were recently presented in Ref. [5]. We checked that their results coincide with the relevant contribution, which have appeared in calculations of exact one-loop radiative corrections to the muon decay spectrum [6]. At the Born level the differential distribution of the electrons and photons of the process (1) has the form where G F is the Fermi coupling constant; m e and m µ are the electron and muon masses, respectively; P µ is the muon polarization vector; x and y are the electron and photon energy fractions in the muon rest reference frame, x = 2E e /m µ and y = 2E γ /m µ ; bŷ p e andp γ we denote the unit vectors in the directions of motion of the electron and photon,p e = p e /| p e | andp γ = p γ /| p γ |; c = cos( p e p γ ). Functions F (x, y, d), G(x, y, d), and H(x, y, d) can be found in Appendix of Ref. [5].
In what follows we will concentrate on the case of unpolarized muon decay, since it is the one measured in the PIBETA experiment. In the unpolarized case only three variables are relevant and the tree-level distribution can be represented as Model independent parameterization of four-fermion interaction (see Particle Data Group [7]) leads to the appearance of two additional contributions. One of them is proportional to the difference (1 − 4ρ/3), which describes the deviation of the Michel parameter ρ from its value in the Standard Model. And the other one contains parameterη, which is a positive semi-definite quantity (see Ref. [8]) where g S,V,T RL,LR are the right-left (RL) and left-right (LR) coupling constants, which parameterize non-standard scalar (S), vector (V) and tensor (T) four-fermion interactions. In principle, one can look also for other exotic interactions, e.g., for the ones mediated by antisymmetric tensor fields [9]. Extraction ofη from the experimental data potentially can put strict limits on physics beyond the Standard Model.

Radiative corrections
New precision experiments call for an adequate level of accuracy in theoretical predictions within the Standard Model. Effects of higher orders of the perturbation theory become important. Here we will consider the first order QED radiative corrections. As usually, we separate them into three parts: i) emission of an additional soft photon; ii) effect due to one-loop virtual photonic correction; iii) emission of an additional hard photon. Note that all the relevant pure week corrections (like loop insertions into the W -propagator) are included into the G F coupling constant [10,11], which is measured directly from the muon lifetime. Effects of strong interactions in the process under consideration are negligible for the moment. They start to appear only at the order O (α 2 ) through hadronic vacuum polarization.

Soft Photon Contribution
We assume, that emission of an additional soft photon of energy below certain threshold is not distinguished by the experiment from the tree-level process (1). The energy of the soft photon, ω 2 , is limited by the parameter ∆: The corresponding correction can be factorized out in front of the tree-level differential distribution: dx dy dc , where λ is a fictitious photon mass; L is the so-called large logarithm, L = ln(m 2 µ /m 2 e ) ≈ 10.66 ; the dilogarithm and the Riemann zeta-function are defined as usual: Quantity δ Soft coincides with the corresponding factor, arising in the correction to the non-radiative muon decay (see e.g. Ref. [6]). Expression (6) takes into account the dependence on the electron mass exactly. Omitting small terms proportional to (m e /m µ ) 2 , we get

One-loop virtual correction
Here we will consider the effect of one-loop photonic corrections. Some representatives of the relevant Feynman diagrams are given in Fig. 1. There are two diagrams of class (a) with photon emission from an external leg (electron or muon line). In the same way the two box-type diagrams of class (b) describe real photon emission from virtual electron and muon propagators. Diagrams of classes (c) and (d) give corrections to photon radiation from a single leg. To get the corresponding correction to the muon decay spectrum we have to multiply the complete set of amplitudes of classes (a − d) by two tree-level amplitudes, describing single photon emission. In our calculations we followed the procedure which has been applied in Ref. [12]. The standard technique for one-loop integration was used. The list of relevant integrals is given in Appendix A. To eliminate the ultraviolet divergences we applied renormalization of the masses and wave functions of the electron and muon. Note that this is enough in the case of muon decay (see Ref. [13,14]), contrary to the general case of the Fermi four-fermion interaction. An analytical result for the virtual correction was obtained. We do not give the full formula here, since it is rather long.

Emission of an additional collinear hard photon
Events with registration of two hard photons are supposed to be rejected by the experimental event selection. But if the additional photon is emitted at a small angle with respect to the momentum of the outgoing electron (positron), the former is not recognized by a calorimetric detector as an independent particle (this can happen if there is no any considerable magnetic field in the detector volume). So, for the socalled collinear photon emission, one observes an effective electron with the energy and momentum composed by the sum of the corresponding quantities of the photon and the bare electron. Let us assume that this kind of calorimetric registration happens in the experiment, if the angle between the electron and photon momenta does not exceed a certain value θ 0 , which plays the role of a small parameter. We demand m e /m µ ≪ θ 0 ≪ 1. Typical experimental values for this parameter, a few degrees, satisfy our conditions. On the other hand, the angle between the observed photon and the electron should satisfy the condition θ = p e p γ ≥ θ 0 .
According to the general factorization procedure, we can represent the result for the contribution of collinear photon radiation as the product of two factors: The tree-level radiative muon decay (with photon emission at large angles with respect to the electron momentum) serves as a short-wave sub-process. Emission of a collinear photon by the outgoing electron serves as a long-wave sub-processes. The formula for the collinear radiation factor agrees with the one in Ref. [15].
Integration over the energy fraction of the collinear photon, z, gives Note that the lower limit of the collinear hard photon energy fraction is adjusted to the upper limit of soft photon emission.

Results and conclusions
Summing up the contributions of soft, virtual, and hard collinear photonic corrections we receive the final answer for the first order radiative correction to the process (1).
Here is our result for the corrected distribution, which substitutes function F (x, y, d) from Eq. (3): We presented explicitly only the factorized part of the correction. The remaining nonfactorizable part, B F (x, y, d), is rather long. We use it in a FORTRAN code for numerical estimates. Expressions for the radiatively corrected functions G Corr. It is worth to note that all the leading logarithm terms were factorized in each of the contributions, but they cancel out in the sum in accord with the Kinoshita-Lee-Nauenberg theorem [16,17]. Moreover, all the dependence on the parameters ∆ and θ 0 is contained in A(x, y, d).
In Fig. 2 we plotted the Born-level differential branching ratio of the radiative muon decay for a fixed value of c,  The relative contribution of radiative corrections is illustrated by Fig. 3, The dependence on c value of δ RC is rather weak. For given values of c and x the  maximal value of y is defined by the kinematics: For the given set of parameters, the factorized part of the correction dominates and gives about 4/5 of the total effect.
To illustrate also the case of 100% polarized muon decay we present in Fig. 4 a plot for the relative contribution of radiative corrections for a set of fixed variables. Namely, P µ p e = 30 • , P µ p γ = 60 • ; c, θ 0 and ∆ are the same as in Fig. 2. Quantity δ RC pol. is defined in analogy to Eq. (13) by adding the relevant contributions of G and H functions according to Eq. (2). Thus we presented the calculation of one-loop QED corrections to the differential distribution of unpolarized muon decay. Our FORTRAN code is available upon request from the authors. The results can be applied also for the decays τ → µν µ ν τ γ and τ → eν e ν τ γ. The theoretical uncertainty of the spectrum description is defined by higher order QED radiative corrections (EW and QCD effects are negligible compared to the QED ones). As a rough upper estimate we can consider the relative contribution of the omitted higher order terms to be about (δ RC ) 2 < ∼ 3·10 −3 , which is small compared to the present experimental precision. In principle, one can easily get the most important higher order terms with logarithms of ∆ and θ 0 by means of soft and collinear approximations.
Appendix A.

One-loop integrals
Here we give the list of integrals over the loop momentum k 1 , which are used for calculations of the virtual loop contribution. The notation for the loop integrals is as follows: where (i − l) are the denominators of the relevant propagators: Tensor and vector integrals are decomposed as We need different integrals with 4, 3, and 2 propagators. In the integrals below we dropped the dimension and put m µ = 1 (the dimension can be restored by multiplying by the required power of m µ ). In some cases we express tensor and vector integrals through a combination of more simple ones. These relations can be obtained by multiplying the tensor and vector integrals by certain particle momenta and subsequent cancellation of denominators, where possible.