PHYSICAL REVIEW D 98, 053008 (2018)

+ − Discovering true in KL → ðμ μ Þγ

Yao Ji * Institut für Theoretische Physik, Universität Regensburg, Regensburg 93040, Germany † Henry Lamm Department of Physics, University of Maryland, College Park, Maryland 20742, USA

(Received 15 June 2017; published 25 September 2018)

þ − −13 Theoretical and phenomenological predictions of BRðKL → ðμ μ ÞγÞ ∼ 7 × 10 are presented for 2 ð Þ different model form factors FKLγγ Q . These rates are comparable to existing and near-term rare KL decay searches at J-PARC and CERN, indicating a discovery of true muonium is possible. Further discussion of potential backgrounds is made.

DOI: 10.1103/PhysRevD.98.053008

Lepton universality predicts differences in and the off-shell invariant mass squared. This two-body observables should occur only due to their mass decay is the reach of rare decay searches and is an þ − difference. Measurements of ðg − 2Þl [1], nuclear charge attractive process for discovering ðμ μ Þ. The decay has radii [2,3], and rare decays [4] have shown hints of simple kinematics with a single, monochromatic photon ðμþμ−Þ þ − violations to this universality. The of , (of Eγ ¼ 203.6 MeV if the KL is at rest) plus ðμ μ Þ which true muonium, presents a unique opportunity to study could undergo a two-body dissociate or decay into two universality in and beyond the 2 2 (with Mll ∼ 4Mμ). [5]. To facilitate these studies, efforts are ongoing to Another outcome of this search is its unique dependence improve theoretical predictions [6]. Alas, true muonium on the form factor, which provides complimentary infor- remains undetected today. mation for determining model parameters. Previous extrac- ðμþμ−Þ Since the late 1960s, two broad categories of tions of the form factor relied upon radiative Dalitz decays, production methods have been discussed: colli- þ − KL → l l γ, the most recent being from the KTEV sions (fixed-target and collider) [7], or through rare decays collaboration [16]. In these analyses, the phenomenological of [8,9]. Until recently, none have been attempted form factor is integrated over 10’sofMeVQ2 bins, and fit ∝ α4 due to the low production rate ( ). Currently, the Heavy to differential cross section data. Although any measure- Photon Search (HPS) [10] experiment is searching for true ment of BRðK → ðμþμ−ÞγÞ would be accompanied by − → ðμþμ−Þ L muonium [11] via e Z X. Another fixed-target larger statistics of the radiative Dalitz decay, it is unclear experiment, but with a beam, DImeson Relativistic 2 þ − how small the Q bins can be made. In contrast to this, the Complex (DIRAC) [12] studies the ðπ π Þ bound ðμþμ−Þ þ − branching ratio gives the form factor at an effec- state and could look for ðμ μ Þ in a upgraded run [13]. 2 tively keV sized Q bin, tightening the correlation between In recent years, a strong focus on rare kaon decays has any parameters in the model form factors with cleaner developed in the search for new physics. The existing KOTO systematic uncertainties. experiment at J-PARC [14] and proposed NA62-KLEVER at BRð → ðμþμ−ÞγÞ BR ∼ 10−13 In this paper, we present the KL CERN [15] hope to achieve sensitivities of including full OðαÞ radiative corrections and four different BRð → π0ννÞ ∼ 10−11 allowing a 1% measurement of KL . ð 2Þ þ − treatments of the form factor FKLγγ Q , thereby avoiding Malenfant was the first to propose KL as a source of ðμ μ Þ þ − −13 Malenfant’s approximation. It is shown that the approxima- [9]. He estimated BRðKL → ðμ μ ÞγÞ ∼ 5 × 10 by 2 2 2 tion underestimates the branching ratio by a model- approximating F γγðQ ¼ 4MμÞ ∼ F γγð0Þ where Q is KL KL dependent 15–60%. Possible discovery channels are discussed and brief comments on important backgrounds are made. Following previous calculations for atomic decays of *[email protected] † [email protected] mesons [8,9,17], the branching ratio can be computed þ − BRðKL → ðμ μ ÞγÞ Published by the American Physical Society under the terms of BRðK → γγÞ the Creative Commons Attribution 4.0 International license. L Further distribution of this work must maintain attribution to α4ζð3Þ α ’ ¼ ð1 − Þ3 1 þ j ð Þj2 ð Þ the author(s) and the published article s title, journal citation, zTM C0 f zTM ; 1 and DOI. Funded by SCOAP3. 2 π

2470-0010=2018=98(5)=053008(5) 053008-1 Published by the American Physical Society YAO JI and HENRY LAMM PHYS. REV. D 98, 053008 (2018)

from the spectral functions ImΠðtÞ. This function is known to leading order analytically for the , and is derived from experiment for the hadronic constribution. ð0Þ BR FKLγγ is fixed to the experimental value of −4 ðKL → γγÞ¼5.47ð4Þ × 10 [19]. Evaluating Eq. (1), BRð →ðμþμ−ÞγÞ¼5 13ð4Þ 10−13j ð Þj2 we find KL . × f zTM , þ − where the dominant error is from BRðKL → ðμ μ ÞγÞ, preventing the measurement of these radiative corrections þ − from this ratio. An improved value of BRðKL → ðμ 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.Itis FIG. 1. Feynman diagram of K → γγ → ðμþμ−Þγ which L typically decomposed into b ¼ bV þ bD. bV arises from a contributes to the branching ratio at Oðα5Þ and is proportional to weak transition from KL → P followed by a strong- Fγγ ðz1;z2Þ. interaction vector interchange P → Vγ and concluding P with the vector meson mixing with the off-shell photon. ζð3Þ¼ 1 3 P ðπ0; η; η0Þ where n =n arising from the sum over all Here, we denote with the pseudoscalars and ðμþμ−Þ ¼ 2 2 ≈ 4 2 2 ρ ω ϕ allowed states, zTM M =M Mμ=M , with V the vector mesons ( ; ; ). The second term, bD, TM K K → γ fðzÞ¼F γγ ðzÞ=F γγ ð0Þ, and C0 is the sum of the arises from the direct weak vertex KL V which then KL KL leading order corrections to the branching ratio. mixes with γ þ γ which requires modeling. Following Previous computations of radiative corrections considered [20], the predictions of bV and bD are divided into the vacuum polarization from the flavor found in the whether nonet or octet symmetry in the light mesons is final state [8] and constituent- model calculations assumed. → γð Þþγð − Þ → [17] of the QED process KL k PKL k To compute bV , one integrates out the vector mesons γ þ → γ TM demonstrated by Fig. 1 where PKL is the four- from the P V vertex and assuming a particular pseu- momentum of the KL,andk is the four-momentum of one doscalar symmetry, the effective Lagrangian is derived and of the virtual . We have computed the full ðμþμ−Þ octet ¼ 0 low energy constants can be used. bV at leading results including the electronic, muonic, and hadronic order due to the cancellation between π0 and η in the Gell- vacuum polarization [6] as well as an improved calculation Mann-Okubo relation [21,22]. In the nonet realization, of the double virtual photon contribution KL → γ ðkÞþ 0 η bnonet ¼ γ ðP − kÞ → γ þ TM. For this contribution, one should a nonzero contribution coming from yields V KL 2 2 ∼ 0 46 take the convolution of the QED amplitude with double- rVMK=Mρ . [23], where rV is a model-independent 2 2 2 ð ð − Þ parameter depending on the couplings of each decomposed virtual-photon form factor FKLγ γ k =MK; PKL k = 2 Þ meson fields in the effective Lagrangian and are ultimately MK . For our purpose, however, taking the form factor ð0 Þ to be a constant equal to Fγγ ;zTM and factoring it from determined by experimental data. the integral is a sufficient approximation as shown in [18]. For bD, the derivation is more complicated and relies on We find models. In the naive factorization model (FM) [24], the dominant contribution to the weak vertex is assumed to be ¼ þ þ þ þ C0 CeVP CμVP ChVP Cver Cγ γ factorized current × current operators which neglect the 62 4 16 1 754ð4Þ 36 12 6 chiral structure of QCD. A free parameter, k , is introduced ¼ . − − . − − . ð Þ F 9 9 9 9 9 : 2 that is related to goodness of the factorized current approximation. If this factorization was exact, kF ¼ 1.In where the C indicate vacuum polarization contributions nonet ¼ 2 octet ¼ 1 41 iVP this scheme, bD bD . kF. This model predicts from i ¼ e, μ, and , C is the vertex correction 0 ver the process KL → π γγ as well, and we use the unweighted term of [8],whileCγγ is the contribution from Fig. 1. average of the two most recent measurements of this A similar calculation for , where other lepton process to fix kF ¼ 0.55ð6Þ [25,26]. flavors and hadronic loop corrections are negligible, finds In the Bergström-Massó-Singer (BMS) model [27], the α ¼ þ ¼ −52 9 the π coefficient is C0 CVP Cver = [8]. CiVP are direct transition is instead assumed to be dominated by a found by computing weak vector-vector interaction (KL → γ þ K → γ þ ρ; Z ω; ϕ → γ þ γ ΔI ¼ 1 ∞ Πð Þ ). BMS further assumes that no 2 ¼ 4 2 Im t ð Þ CiVP mμ dt 2 3 enhancement occurs. This model produces a complete 4m2 tð4mμ − tÞ i form factor:

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1 CαK is derived from the differential cross sections of these fγ ðzÞ¼ 2 þ 2 ;BMS ð α Þ ¼ −0 517ð30Þ ð22Þ MK MK processes; yielding C K . [16] 1 − 2 z 1 − 2 z e stat sys Mρ M K ð α Þ ¼ −0 37ð7Þ 0 1 and C K μ . [16], which are each input into our prediction for ðμþμ−Þ. B4 1 1 1 2 1 C × @ − − − A: ð4Þ We also consider the D’Ambrosio-Isidori-Portol´es (DIP) M2 M2 M2 3 K 9 K 9 K ð Þ 1 − 2 z 1 − 2 z 1 − 2 z phenomenological Fγ γ z1;z2 [29]: Mρ Mω Mϕ 0 1

The two terms correspond to the vector interchange and B z1 z2 C ð Þ¼1 þ α þ fγ γ ;DIP z1;z2 DIP@ 2 2 A direct transition, respectively. Expanding this expression in Mρ Mρ z1 − 2 z2 − 2 powers of z, we find the BMS model predicts MK MK 2 1 2 2 2 þ β z1z2 ð Þ MK MK MK MK 2 2 : 6 ¼ − α 9 þ 2 þ DIP bBMS 2 C K 2 2 2 − Mρ − Mρ Mρ 9 Mρ M Mω z1 2 z2 2 ϕ MK MK ¼ 0 41205 − 0 509926 α ¼ ¼ 0 ðμþμ−Þ . . C K where z1 zTM;z2 for production. To set α → þ −γ α ¼ ¼ b þ b ð5Þ DIP, we take the values from KL e e , DIP;e V;BMS D;BMS −1 729ð43Þ ð28Þ →μþμ−γ α ¼ . stat sys [16], and from KL , DIP;μ −α Under the model assumptions, K is theoretically esti- −1.54ð10Þ [16]. Our phenomenological results are com- ∼0 2–0 3 ¼ 2 7ð4Þ mated to be . . [27]. C . depends on a piled in Table II. Comparing the phenomenological form number of other mesonic decay rates [16,28], and we used factors, they are indistinguishable within uncertainty in the modern values [19]. The error comes from the exper- ðμþμ−Þ production. This is perhaps unsurprising because imental uncertainty which is dominated by the two K they arise from the same underlying data, but the difference 0 measurements. BRðK → K γÞ contributes ΔC ∼ 13% and in functional forms could be discriminated by higher Γ Δ ∼ 4% K ;tot contributes C due to a disagreement between precision data. 2 2 α ≈ 4 decay modes. This choice of C and K is consistent with Due to the small value of zPs Me=MK, the branching → lþl−γ þ − −13 the measured rates for KL . ratio to positronium, BRðK →ðe e ÞγÞ¼9.31ð5Þ×10 , ’ L D Ambrosio et al. advocates the view that bD;BMS is one is independent of the form factor within the error of þ − of a series of contributions to bD, which should be summed BRðKL → γγÞ and slightly larger than ðμ μ Þ. While this together with the model-independent bV [20]. They con- branching ratio also has not been measured, one can struct another contribution by factorizing the vector cou- construct a ratio pling (FMV) similar to FM but first restricting the Lagrangian to left-handed currents. For the different BRð → ðμþμ−ÞγÞ ¼ KL symmetry realizations, bnonet ¼ 3.14η ∼ 0.66 and boctet ¼ R þ − D D BRðKL → ðe e ÞγÞ 2.42η ∼ 0.51 where η is a coefficient multiplying the naive ð1 − z Þ3ð1 − 0.439 αÞjfðz Þj2 weak coupling G8 and like k is related to the quality of the ¼ TM π TM F 3 52 α 2 η ¼ Wilson ð1 − z Þ ð1 − Þjfðz Þj factorization assumption. We use their value of g8 = Ps 9 π Ps jg8j ¼ 0.21. Our theoretical results are compiled in ð Þ2 K→ππ;LO ¼ 0 55767ð2Þ f zTM ð Þ Table I. These values disagree outside their error, and a . ; 7 fðzPsÞ 10% precision measurement would be able to discriminate between them. This is in contrast to the radiative Dalitz which is independent of the BRðKL → γγÞ uncertainty and decays, where the theoretical values are consistent. directly measures lepton universality without an uncer- The BMS form factor also has been used to phenom- 2 þ − tainty due to Q binning. By taking the largest and smallest → l l γ l ¼ μ α enologically fit KL for both e, , and C K theoretical values of b to give a gross range, we predict

TABLE I. Theoretical values of b and BRðK → ðμþμ−ÞγÞ for L j ð Þj BRð → ðμþμ−ÞγÞ the models considered in this paper. TABLE II. Values of f zTM and KL com- puted using the phenomenological form factors with parameters 13 set by either radiative K decay to e or μ. Model btheory BRTM × 10 L ð Þoctet a 13 FM 0.40(4) 5.90(9) Model jfðzTMÞj BRTM × 10 ð Þnonet a FM 1.24(6) 7.68(15) a ðBMSÞnonet 0.76(9) 6.63(20) BMSeeγ 1.134(6) 6.60(10) ð þ Þoctet BMSμμγ 1.119(8) 6.42(11) BMS FMV 0.85(10) 6.82(22) a ðBMS þ FMVÞnonet 1.45(10) 8.16(25) DIPeeγ 1.139(6) 6.66(10) a 0 DIPμμγ 1.124(9) 6.48(12) Using value of kF ¼ 0.55ð6Þ derived from KL → π γγ [25,26]. aThe systematic and statistical errors have been summed.

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¼ 0 76ð14Þ þ − R . . Applying the same procedure to the phe- KL → l l γ. We compute the branching ratio for this by nomenological form factors yields R ¼ 0.707ð9Þ. integrating the differential cross section in an invariant mass ðμþμ−Þ We now focus upon the experimental situation. bin, Mbin,aroundthe peak to obtain a background Throughout, we assume a 10% acceptance. The largest estimate. In the case of electrons, the bin is centered around previous experimental data set that could be used to the ðμþμ−Þ peak; for muon final states it is defined as BRð → ðμþμ−ÞγÞ study KL is KTEV. We estimate from ½2mμ; 2mμ þ M . This difference in binning reflects that BRð → lþl−γÞ bin the number of events reported for KL [16] the are above threshold. For bin size similar to KTEV, that at least 1000 times the luminosity would be required BRð → þ −γÞ ¼ 1 2 10−8 þ − the values are KL e e bin . × Mbin, and for just one ðμ μ Þ event. From the existing data, þ − −9 BRðK → μ μ γÞ ¼ 5.0 × 10 M where M is in one might expect to place a limit on the order of BR L bin bin bin MeV. This large raw background (∼105× the signal) will ðK → ðμþμ−ÞγÞ ≲ 10−9. L have to be reduced, but it has distinct features compared to The KOTO experiment at J-PARC has reported true muonium decays which can be leveraged. 3.560ð0.013Þ × 107 K per 2 × 1014 on target L The smoothness of the background differential cross (POT) [30]. Their 2013 physics run accumulated 1.6 × þ − section around the ðμ μ Þ peak should allow accurate 1018 POT [14] which would correspond to 0.015 ðμþμ−Þ modeling from the sidebands. Reconstruction of the KL events. Through their 2015 physics run, 20 times the KL allows the energy of the KL to be used to cut on the γ and decays have been recorded [14], indicating 0.3 produced leptonic energies. The two two-body decay topology ðμþμ−Þ ≲10−11 events and a limit of . Unfortunately, the suggests cuts on momenta and angular distribution would KOTO experiment is designed to detect only photons, and be powerful in background suppression. As an example, for detecting purely photon decay products of ðμþμ−Þ would radiative Dalitz decay the angle θe between the electrons be difficult. The J-PARC kaon beam hopes to run into the can be arbitrary, but from the true muonium decay e will 2020s with an additional flux upgrade so a discovery is θ ∼ ∼ 50o GeV have e mTM=ETM × E . This suggests the higher quite possible in an experiment with lepton identification. KL The NA62-KLEVER proposal [15] for a rare KL beam at energy of the proposed CERN beamline would be desir- CERN hopes to start by 2026 and accumulate 3 × 1013 K able. Additionally, vertex cuts can be made using the proper L 3 over 5 years, which would also be nearly sufficient for lifetime of true muonium cτ ¼ 0.5n mm, where n is the single-event sensitivity. principal quantum number. A more rigorous study of A few channels are available to measure the branching backgrounds is planned for the future. ratio of true muonium: dissociated μþμ− with or without γ, þ − γ lγ decayed e e with or without ,or similar to SUSY ACKNOWLEDGMENTS searches with invisible decays [31]. The decay to π0γ is suppressed by 10−5 but KOTO can search for it without H. L. is supported by the U.S. Department of Energy modification [32]. under Contract No. DE-FG02-93ER-40762. Y. J. acknowl- For each channel, different backgrounds matter. The edges the Deutsche Forschungsgemeinschaft for support dominant backgrounds will arise from the free decays under Grant No. BR 2021/7-1.

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