Radiative corrections to inverse muon decay for accelerator neutrinos

Inverse muon decay ($\nu_\mu e^- \to \nu_e \mu^-$) is a promising tool to constrain neutrino fluxes with energies $E_{\nu} \ge 10.9~\mathrm{GeV}$. Radiative corrections introduce percent-level distortions to energy spectra of outgoing muons and depend on experimental details. In this paper, we calculate radiative corrections to the scattering processes $\nu_\mu e^- \to \nu_e \mu^-$ and $\bar{\nu}_e e^- \to \bar{\nu}_\mu \mu^-$. We present the muon energy spectrum for both channels, double-differential distributions in muon energy and muon scattering angle and in photon energy and photon scattering angle, and the photon energy spectrum for the dominant $\nu_\mu e^- \to \nu_e \mu^-$ process. Our results clarify and extend the region of applicability of previous results in the literature for the double differential distribution in muon energy and photon energy, and in the muon energy spectrum with a radiated photon above a threshold energy. We provide analytic expressions for single, double and triple differential cross sections, and discuss how radiative corrections modify experimentally interesting observable distributions.


Introduction
interaction with scale-independent Fermi coupling constant G F [34,[36][37][38][39][40] L eff = −2 √ 2G Fνe γ λ P L ν µμ γ λ P L e + h.c.. (1) The reaction is kinematically allowed only for sufficiently high energies of the incoming neutrino E νµ , Eν e ≥ E thr ν , where In radiation-free kinematics, the muon goes predominantly in the forward direction with a scattering angle θ µ : with the momentum-space photon propagator Π µν : where the photon mass λ regulates the infrared divergence, ξ γ is the photon gauge-fixing parameter, and a is an arbitrary constant. The corresponding field renormalization factors for the external charged leptons are evaluated from the one-loop self energies in the MS renormalization scheme as [10,[41][42][43] with the renormalization scale in dimensional regularization µ, where the number of dimensions is d = 4−2ε. Neglecting Lorentz structures whose contractions with the (anti)neutrino current vanish at m ν = 0 and denoting the ratio of lepton masses as r = m e /m µ , the resulting correction to the charged lepton current is expressed as where the form factors g M , g 5 M , f 2 , and f 5 2 are [34]: M (η, r, β) = −1 + 1 β f (5) 2 (η, r, β) = Here β is the velocity of the muon in the electron rest frame, η = 1 for the form factors g M , f 2 and η = −1 for the form factors g 5 M , f 5 2 .

Real radiation
Inverse muon decay with single photon emission is described by the Bremsstrahlung contribution T 1γ : e p e · k γ μγ µ P L e + 1 2μ γ ν k / γ γ µ p µ · k γ + γ µ k / γ γ ν p e · k γ P L e ε ν , (14) with the photon polarization four-vector ε ν . Let us consider separately the cases of soft and hard photon emission.

Soft-photon Bremsstrahlung
The inverse muon decay with radiation of photons of arbitrary small energy cannot be experimentally distinguished from the decay without radiation. All events with photons below some energy cutoff k γ ≤ ∆E (in the electron rest frame) must be included in measured observables. The corresponding scattering cross section factorizes in terms of the tree-level result of Eqs. (6) and (7) as with the universal correction δ s (∆E) [10,14,18,26,44]: This region of the phase space with low-energy photons cancels the infrared-divergent contributions from virtual diagrams. As a result, soft and virtual contributions multiply the tree-level cross sections of Eqs. (6) and (7) with infrared-finite factor, i.e., independent of the fictitious photon mass λ [45][46][47][48], as where we can obtain the contributions in theν e e − →ν µ µ − reaction by replacing the momenta of neutrinos with the momenta of antineutrinos.

Contribution of hard photons
Here we evaluate the contribution of photons above the energy cutoff k γ ≥ ∆E to the muon energy spectrum. Squaring the matrix element of Eq. (14) for the inverse muon decay ν µ e − → ν e µ − , we obtain the result in terms of Lorentz invariants as while the result forν e e − →ν µ µ − is given by the replacement of neutrino momenta by corresponding antineutrino momenta. We perform the integration following the technique that was introduced in [11] and further developed in [9,10,34,35]. For the inverse muon decay, the implementation is slightly more involved by having two mass scales: the electron mass and the muon mass.
First, we introduce the four-vector l: l = p e + k ν − p µ = l 0 , f . Working in the rest frame of atomic electrons, we have for the components of l: Accounting for the conservation of energy and momentum, we obtain where γ denotes the angle between the photon direction and the vector f . Using energy and momentum conservation to perform the integration over the final-state neutrino momentum components and the photon energy, we obtain the muon energy spectrum as It is convenient to split the phase space into two regions with distinct ranges of integration. There are no restrictions on the photon phase space in the region I: l 2 ≥ 2∆E (l 0 − f cos γ). In this region, the range of kinematic variables is given by In the complementary region II: l 2 ≤ 2∆E (l 0 − f cos γ) that is close to the kinematics of the radiation-free process, the angle between the photon momentum and the vector f is restricted as This region contributes a factorizable contribution δ II , which adds linearly to δ s (∆E) of Eq. (17) [10]: where the angle δ 0 is given by and ρ = 1 − β 2 . Only the first term from Eq. (18) contributes in this region. The same term generates the ∆E-dependence after integrating over region I. For the muon energy spectrum including both soft and hard photons, this dependence cancels with the contribution of soft photons from Eq. (15). For other terms, we can safely set ∆E = 0 starting from Eq. (18).

Muon energy spectrum and integrated cross section
Adding virtual and real corrections, we obtain the ∆E-and λ-independent result for the muon energy spectrum in the inverse muon decay ν µ e − → ν e µ − with photons of arbitrarily large energy allowed by kinematics. In the limit E ν m e (i.e., neglecting order m e /E ν power corrections) our results are in agreement with the calculation of Bardin and Dokuchaeva [31]: Comparing the double-differential distribution in photon energy and muon energy in Ref. [31] to our numerical evaluation, we do not find agreement; in particular, the double-differential distribution in the calculation of Bardin and Dokuchaeva is not positive-definite in the kinematically allowed region of inverse muon decay. However, we reproduce the contribution to the muon energy spectrum from the events with photons above some energy cutoff in the limit 2m e ∆E m 2 µ for neutrino energies of modern accelerator experiments, not much larger than the threshold region in the inverse muon decay reaction.
At the fixed neutrino energy of 15 GeV, Fig. 1 shows muon energy spectra for the tree-level processes ν µ e − → ν e µ − andν e e − →ν µ µ − as well as the O (α) contribution to ν µ e − → ν e µ − from Eq. (29). We show the latter with an opposite sign for convenience. The radiative corrections reduce the cross section by 3 − 4 %. They have the largest relative size for backward scattering and increase going to forward angles.
Integrating the muon energy spectrum over the kinematically allowed range in Eq. (5), we obtain the O (α) contribution to the unpolarized inverse muon decay cross section σ. For illustration, we present two limits of interest. The leading term in m e /E ν expansion is given by while the high-energy limit, x 1, is described with where the leading terms coincide with the well-known expression in Ref. [25]. We provide the total cross section, double-differential distribution in muon energy and muon scattering angle, double-differential distribution in muon energy, and triple-differential distribution in muon energy, muon scattering angle, and photon energy in Appendices A, B and in Supplemental material.

Distortion of experimentally-accessed distributions
Experimentally, inverse muon decay events are distinguished from other reactions by looking for highenergy muons, above the E min µ of Eq. (23) with no other particles in the final state, and which are along the direction of the incoming neutrino due to the kinematics of elastic scattering from electrons. Radiative corrections cause events with real photons in the final state and with a different distribution of muon energies and angles than in the tree-level process. This section explores those changes from the tree-level predictions.
Experiments will need to reject events from ν µ quasielastic scattering on nucleons in nuclei which may appear to be consistent with elastic kinematics, but which will have a recoiling proton in the final state. Similarly, inelastic processes can produce high-energy forward muons with other particles in the final state. Because there are many possible elastic and inelastic reactions, a common experimental strategy is to remove events with any other visible energy than the muon in the final state. An energetic real photon from radiative processes, even one nearly collinear with the muon, may produce visible energy that will veto the event due to this requirement.
While this experimental strategy will be common to all measurements, the details of the effect will be particular to each experimental setup. In its analysis [5], MINERvA predicted the relative acceptance as a function of photon energy, and that prediction is shown in Fig. 2.
Averaging over high-energy tails of the expected flux in the DUNE experiment [49] and medium-energy "neutrino" (forward horn current) mode for the MINERvA experiment [2,3,50], we provide the effect of O (α) on muon energy spectra for two representative examples of neutrino experiments that do or will use IMD to constrain its high-energy flux tails in Figs. 3. For MINERvA and DUNE predictions, we average over the (anti)neutrino energy above the threshold value E thr ν but below 30 and 80 GeV respectively. We illustrate this averaging on the left panel of Fig. 3 and compare it to fluxes averaged over the same region in both experiment. The average over the flux decreases the resulting cross section compared to the fixed energy E ν = 15 GeV result shown in Fig. 1, since the monotonically falling with (anti)neutrino energy flux is convoluted with slower rising cross sections. Distortions of the muon energy spectrum increase as the neutrino energy approaches the threshold of the inverse muon decay from above. The effect on the measurable cross section from the removal of some events with real photons is also shown in Fig. 3 for comparison with the O (α) correction, and it is less than a 1% reduction in the observed rate, with a larger effect for higher muon energies.
The kinematics of elastic scattering from electrons produces a relationship between the muon energy and angle with respect to the incoming neutrino direction. A useful combination is When E ν E µ and E µ E min µ , F can approach its upper limit of 2m e . In measurements of elastic neutrino-electron scattering by the MINERvA experiment [1,3,4], the same quantity was used to select events that were due to elastic scattering from electrons. In this case E e E min e for all of the selected events. However, for IMD for the experimental fluxes considered above from DUNE and MINERvA, neither condition above is true for most events, and therefore typically F 2m e . In particular because the factor 1 −

Eµ Eν
is usually small, one might want to consider an "idealized" version of F , However, this quantity is not accessible since the neutrino energy is not known on an event-by-event basis.
MINERvA, E thr ν ≤E ν ≤30 GeV MINERvA, E thr ν ≤E ν ≤80 GeV DUNE, E thr ν ≤E ν ≤80 GeV  In the measurement by the MINERvA experiment [5], the analysis enforced elastic kinematics for a "maximum" energy of likely candidate events in its beam. The variable F MINERvA , with E max = 35 GeV, was used for the selection of signal events by placing a cut on F MINERvA (E µ , θ µ ).
To illustrate various definitions for the variable F , we present all three variants as a function of the final-state muon energy E µ for the fixed neutrino energy E ν = 15 GeV in Fig. 4. The size of this variable in the inverse muon decay is below 10 − 100 keV. F vanishes both in forward and backward directions for definitions in Eqs. (33) and (35) contrary to the forward scattering only for the definition in Eq. (34). In Fig. 5, we also present the tree-level distributions of the variable F for the same neutrino energy in the region, which is allowed kinematically for all three definitions. We observe a significant redistribution of events moving from one definition of the variable F to another.
To illustrate the effect of radiative corrections on the distribution of the F variables, we keep MIN-ERvA's definition in Eq. (35) for applications to MINERvA's study [2,50]. However for a general experiment, including the application to the DUNE flux [49] in this paper, we don't wish to enforce a maximum neutrino energy above which we would drop the constraint, so we study instead the original F of Eq. (33). We present in Fig. 6 the distribution of the variable F MINERvA at tree level, and compare it to the O (α) contribution of radiative corrections by integrating the double-differential distribution in muon energy and muon scattering angle, and by providing the naive estimate assuming the kinematics of the radiation-free process and Eq. (29). O (α) contributions shift the distribution of F MINERvA variable by a percent-level correction. Note also that all inverse muon decay events from neutrinos of energy E ν ≤ 30 GeV belong to the first bin in the variable F MINERvA , i.e. 0 ≤ F MINERvA ≤ 250 keV, with the current experimental resolution [5].

Conclusions and Outlook
The goal of this paper is to enable percent-level constraints on the incoming (anti)neutrino fluxes by measuring the inverse muon decay reaction on the atomic electrons. Thus, we performed a study of radiative corrections and various cross sections in the inverse muon decay. We confirmed analytical expression for the muon energy spectrum and have provided triple-differential distribution in muon energy, muon scattering angle, and photon energy; double-differential distribution in muon energy and muon scattering angle; double-differential distribution in muon energy and photon energy, and total cross section. We investigated the effects of O (α) radiative corrections on the muon energy spectrum and experimentallyaccessed distribution of the variable F . In both cases, the corresponding distortions have the percent-level size.
We have clarified the definition for the experimentally-acessed variable F . Inverse muon decay events contribute to the lowest bins of the variable F . The size of this variable in the inverse muon decay is much smaller than in the elastic (anti)neutrino-electron scattering. The F distribution provides a reasonable discriminant of keV to tens of keV size for (anti)neutrino energies well below the parameter E max ν .
Providing radiative corrections to the inverse muon decay paves the way for percent-level constraints of high-energy tails in incoming neutrino flux of modern and future neutrino oscillation and cross-section experiments.

F [keV]
< l a t e x i t s h a 1 _ b a s e 6 4 = " w J T q w h B f 8 I  with the kinematic notations