Discovering true muonium in KL → ð μ + μ − Þ γ

Lepton universality predicts differences in electron and muon observables should occur only due to their mass difference. Measurements of ðg − 2Þl [1], nuclear charge radii [2,3], and rare meson decays [4] have shown hints of violations to this universality. The bound state of ðμþμ−Þ, true muonium, presents a unique opportunity to study lepton universality in and beyond the standard model [5]. To facilitate these studies, efforts are ongoing to improve theoretical predictions [6]. Alas, true muonium remains undetected today. Since the late 1960s, two broad categories of ðμþμ−Þ production methods have been discussed: particle collisions (fixed-target and collider) [7], or through rare decays of mesons [8,9]. Until recently, none have been attempted due to the low production rate (∝ α). Currently, the Heavy Photon Search (HPS) [10] experiment is searching for true muonium [11] via e−Z → ðμþμ−ÞX. Another fixed-target experiment, but with a proton beam, DImeson Relativistic Atom Complex (DIRAC) [12] studies the ðπþπ−Þ bound state and could look for ðμþμ−Þ in a upgraded run [13]. In recent years, a strong focus on rare kaon decays has developed in the search for new physics. The existingKOTO experiment at J-PARC [14] and proposedNA62-KLEVERat CERN [15] hope to achieve sensitivities of BR ∼ 10−13 allowing a 1% measurement of BRðKL → π0ννÞ ∼ 10−11. Malenfant was the first to proposeKL as a source of ðμþμ−Þ [9]. He estimated BRðKL → ðμþμ−ÞγÞ ∼ 5 × 10−13 by approximatingFKLγγ ðQ2 1⁄4 4MμÞ ∼ FKLγγ ð0ÞwhereQ2 is the off-shell photon invariant mass squared. This two-body decay is the reach of rare kaon decay searches and is an attractive process for discovering ðμþμ−Þ. The decay has simple kinematics with a single, monochromatic photon (of Eγ 1⁄4 203.6 MeV if the KL is at rest) plus ðμþμ−Þ which could undergo a two-body dissociate or decay into two electrons (with Mll ∼ 4Mμ). Another outcome of this search is its unique dependence on the form factor, which provides complimentary information for determining model parameters. Previous extractions of the form factor relied upon radiative Dalitz decays, KL → lþl−γ, the most recent being from the KTEV collaboration [16]. In these analyses, the phenomenological form factor is integrated over 10’s of MeV Q bins, and fit to differential cross section data. Although any measurement of BRðKL → ðμþμ−ÞγÞ would be accompanied by larger statistics of the radiative Dalitz decay, it is unclear how small the Q bins can be made. In contrast to this, the ðμþμ−Þ branching ratio gives the form factor at an effectively keV sized Q bin, tightening the correlation between any parameters in the model form factors with cleaner systematic uncertainties. In this paper, we present the BRðKL → ðμþμ−ÞγÞ including full OðαÞ radiative corrections and four different treatments of the form factor FKLγγ ðQ2Þ, thereby avoiding Malenfant’s approximation. It is shown that the approximation underestimates the branching ratio by a modeldependent 15–60%.Possible discoverychannels are discussed and brief comments on important backgrounds are made. Following previous calculations for atomic decays of mesons [8,9,17], the branching ratio can be computed BRðKL → ðμþμ−ÞγÞ BRðKL → γγÞ 1⁄4 α 4ζð3Þ

the off-shell photon invariant mass squared.This two-body decay is the reach of rare kaon decay searches and is an attractive process for discovering ðμ þ μ − Þ.The decay has simple kinematics with a single, monochromatic photon (of E γ ¼ 203.6 MeV if the K L is at rest) plus ðμ þ μ − Þ which could undergo a two-body dissociate or decay into two electrons (with M 2 ll ∼ 4M 2 μ ).Another outcome of this search is its unique dependence on the form factor, which provides complimentary information for determining model parameters.Previous extractions of the form factor relied upon radiative Dalitz decays, K L → l þ l − γ, the most recent being from the KTEV collaboration [16].In these analyses, the phenomenological form factor is integrated over 10's of MeV Q 2 bins, and fit to differential cross section data.Although any measurement of BRðK L → ðμ þ μ − ÞγÞ would be accompanied by larger statistics of the radiative Dalitz decay, it is unclear how small the Q 2 bins can be made.In contrast to this, the ðμ þ μ − Þ branching ratio gives the form factor at an effectively keV sized Q 2 bin, tightening the correlation between any parameters in the model form factors with cleaner systematic uncertainties.
In this paper, we present the BRðK L → ðμ þ μ − ÞγÞ including full OðαÞ radiative corrections and four different treatments of the form factor F K L γγÃ ðQ 2 Þ, thereby avoiding Malenfant's approximation.It is shown that the approximation underestimates the branching ratio by a modeldependent 15-60%.Possible discovery channels are discussed and brief comments on important backgrounds are made.
Following previous calculations for atomic decays of mesons [8,9,17], the branching ratio can be computed where ζð3Þ ¼ P n 1=n 3 arising from the sum over all , and C 0 is the sum of the leading order corrections to the branching ratio.Previous computations of radiative corrections considered the vacuum polarization from the flavor found in the final state [8] and constituent-quark model calculations [17] of the QED process 1 where P K L is the fourmomentum of the K L , and k is the four-momentum of one of the virtual photons.We have computed the full ðμ þ μ − Þ results including the electronic, muonic, and hadronic vacuum polarization [6] as well as an improved calculation of the double virtual photon contribution K L → γ Ã ðkÞþ γ Ã ðP K L − kÞ → γ þ TM.For this contribution, one should take the convolution of the QED amplitude with doublevirtual-photon form factor For our purpose, however, taking the form factor to be a constant equal to F γγ Ã ð0; z TM Þ and factoring it from the integral is a sufficient approximation as shown in [18].We find where the C iVP indicate vacuum polarization contributions from i ¼ e, μ, and hadrons, C ver is the vertex correction term of [8], while C γ Ã γ Ã is the contribution from Fig. 1.
A similar calculation for positronium, where other lepton flavors and hadronic loop corrections are negligible, finds the α π coefficient is [8].C iVP are found by computing from the spectral functions ImΠðtÞ.This function is known to leading order analytically for the leptons, and is derived from experiment for the hadronic constribution.F K L γγ Ã ð0Þ is fixed to the experimental value of BR ðK L → γγÞ ¼ 5.47ð4Þ × 10 −4 [19].Evaluating Eq. ( 1), we find BRðK L →ðμ þ μ − ÞγÞ¼5.13ð4Þ×10−13 jfðz TM Þj 2 , where the dominant error is from BRðK L → ðμ þ μ − ÞγÞ, preventing the measurement of these radiative corrections from this ratio.An improved value of BRðK L → ðμ þ μ − ÞγÞ or constructing a different ratio, as we do below, can allow sensitivity to these corrections.
The theoretical predictions for fðzÞ are computed as a series expansion to first order in z with slope b.
arises from a weak transition from K L → P followed by a stronginteraction vector interchange P → Vγ and concluding with the vector meson mixing with the off-shell photon.Here, we denote with P the pseudoscalars ðπ 0 ; η; η 0 Þ and with V the vector mesons (ρ; ω; ϕ).The second term, b D , arises from the direct weak vertex K L → Vγ which then mixes with γ þ γ Ã which requires modeling.Following [20], the predictions of b V and b D are divided into whether nonet or octet symmetry in the light mesons is assumed.
To compute b V , one integrates out the vector mesons from the P → Vγ vertex and assuming a particular pseudoscalar symmetry, the effective Lagrangian is derived and low energy constants can be used.b octet V ¼ 0 at leading order due to the cancellation between π 0 and η in the Gell-Mann-Okubo relation [21,22].In the nonet realization, a nonzero contribution coming from η 0 yields b nonet [23], where r V is a model-independent parameter depending on the couplings of each decomposed meson fields in the effective Lagrangian and are ultimately determined by experimental data.
For b D , the derivation is more complicated and relies on models.In the naive factorization model (FM) [24], the dominant contribution to the weak vertex is assumed to be factorized current × current operators which neglect the chiral structure of QCD.A free parameter, k F , is introduced that is related to goodness of the factorized current approximation.If this factorization was exact, k F ¼ 1.In this scheme, b nonet 41k F .This model predicts the process K L → π 0 γγ as well, and we use the unweighted average of the two most recent measurements of this process to fix k F ¼ 0.55ð6Þ [25,26].
In the Bergström-Massó-Singer (BMS) model [27], the direct transition is instead assumed to be dominated by a weak vector-vector interaction enhancement occurs.This model produces a complete form factor: which contributes to the branching ratio at Oðα 5 Þ and is proportional to The two terms correspond to the vector interchange and direct transition, respectively.Expanding this expression in powers of z, we find the BMS model predicts Under the model assumptions, −α K Ã is theoretically estimated to be ∼0.2-0.3 [27].C ¼ 2.7ð4Þ depends on a number of other mesonic decay rates [16,28], and we used the modern values [19].The error comes from the experimental uncertainty which is dominated by the two K Ã measurements.BRðK Ã → K 0 γÞ contributes ΔC ∼ 13% and Γ K Ã ;tot contributes ΔC ∼ 4% due to a disagreement between decay modes.This choice of C and α K Ã is consistent with the measured rates for  I.These values disagree outside their error, and a 10% precision measurement would be able to discriminate between them.This is in contrast to the radiative Dalitz decays, where the theoretical values are consistent.
We also consider the D'Ambrosio-Isidori-Portolés (DIP) phenomenological where To set α DIP , we take the values from K L → e þ e − γ, α DIP;e ¼ −1.729ð43Þ stat ð28Þ sys [16], and from K L →μ þ μ − γ, α DIP;μ ¼ −1.54ð10Þ [16].Our phenomenological results are compiled in Table II.Comparing the phenomenological form factors, they are indistinguishable within uncertainty in ðμ þ μ − Þ production.This is perhaps unsurprising because they arise from the same underlying data, but the difference in functional forms could be discriminated by higher precision data.Due to the small value of z Ps ≈ 4M 2 e =M 2 K , the branching ratio to positronium, BRðK L →ðe þ e − ÞγÞ¼9.31ð5Þ×10−13 , is independent of the form factor within the error of BRðK L → γγÞ and slightly larger than ðμ þ μ − Þ.While this branching ratio also has not been measured, one can construct a ratio which is independent of the BRðK L → γγÞ uncertainty and directly measures lepton universality without an uncertainty due to Q 2 binning.By taking the largest and smallest theoretical values of b to give a gross range, we predict  The systematic and statistical errors have been summed.R ¼ 0.76ð14Þ.Applying the same procedure to the phenomenological form factors yields R ¼ 0.707ð9Þ.We now focus upon the experimental situation.Throughout, we assume a 10% acceptance.The largest previous experimental data set that could be used to study BRðK L → ðμ þ μ − ÞγÞ is KTEV.We estimate from the number of events reported for BRðK L → l þ l − γÞ [16] that at least 1000 times the luminosity would be required for just one ðμ þ μ − Þ event.From the existing data, one might expect to place a limit on the order of BR ðK L → ðμ þ μ − ÞγÞ ≲ 10 −9 .
The KOTO experiment at J-PARC has reported 3.560ð0.013Þ× 10 7 K L per 2 × 10 14 protons on target (POT) [30].Their 2013 physics run accumulated 1.6 × 10 18 POT [14] which would correspond to 0.015 ðμ þ μ − Þ events.Through their 2015 physics run, 20 times the K L decays have been recorded [14], indicating 0.3 produced ðμ þ μ − Þ events and a limit of ≲10 −11 .Unfortunately, the KOTO experiment is designed to detect only photons, and detecting purely photon decay products of ðμ þ μ − Þ would be difficult.The J-PARC kaon beam hopes to run into the 2020s with an additional flux upgrade so a discovery is quite possible in an experiment with lepton identification.The NA62-KLEVER proposal [15] for a rare K L beam at CERN hopes to start by 2026 and accumulate 3 × 10 13 K L over 5 years, which would also be nearly sufficient for single-event sensitivity.
A few channels are available to measure the branching ratio of true muonium: dissociated μ þ μ − with or without γ, decayed e þ e − with or without γ, or l AE γ similar to SUSY searches with invisible decays [31].The decay to π 0 γ is suppressed by 10 −5 but KOTO can search for it without modification [32].
For each channel, different backgrounds matter.The dominant backgrounds will arise from the free decays K L → l þ l − γ.We compute the branching ratio for this by integrating the differential cross section in an invariant mass bin, M bin , around the ðμ þ μ − Þ peak to obtain a background estimate.In the case of electrons, the bin is centered around the ðμ þ μ − Þ peak; for muon final states it is defined as ½2m μ ; 2m μ þ M bin .This difference in binning reflects that the muons are above threshold.For bin size similar to KTEV, the values are BRðK L → e þ e − γÞ bin ¼ 1.2 × 10 −8 M bin , and BRðK L → μ þ μ − γÞ bin ¼ 5.0 × 10 −9 M bin where M bin is in MeV.This large raw background (∼10 5 × the signal) will have to be reduced, but it has distinct features compared to true muonium decays which can be leveraged.
The smoothness of the background differential cross section around the ðμ þ μ − Þ peak should allow accurate modeling from the sidebands.Reconstruction of the K L allows the energy of the K L to be used to cut on the γ and leptonic energies.The two two-body decay topology suggests cuts on momenta and angular distribution would be powerful in background suppression.As an example, for radiative Dalitz decay the angle θ e between the electrons can be arbitrary, but from the true muonium decay e will have θ e ∼ m TM =E TM ∼ 50 o × GeV E K L .This suggests the higher energy of the proposed CERN beamline would be desirable.Additionally, vertex cuts can be made using the proper lifetime of true muonium cτ ¼ 0.5n 3 mm, where n is the principal quantum number.A more rigorous study of backgrounds is planned for the future.

8 = jg 8 j
Ambrosio et al. advocates the view that b D;BMS is one of a series of contributions to b D , which should be summed together with the model-independent b V [20].They construct another by factorizing the vector coupling (FMV) similar to FM but first restricting the Lagrangian to left-handed currents.For the different symmetry realizations, b nonet D ¼ 3.14η ∼ 0.66 and b octet D ¼ 2.42η ∼ 0.51 where η is a coefficient multiplying the naive weak coupling G 8 and like k F is related to the quality of the factorization assumption.We use their value of η ¼ g Wilson K→ππ;LO ¼ 0.21.Our theoretical results are compiled in Table

TABLE I .
Theoretical values of b and BRðK L → ðμ þ μ − ÞγÞ for the models considered in this paper.

TABLE II .
Values of jfðz TM Þj and BRðK L → ðμ þ μ − ÞγÞ computed using the phenomenological form factors with parameters set by either radiative K L decay to e or μ. a