PHYSICAL REVIEW D 98, 053008 (2018) + − Discovering true muonium 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 à 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 electron and the off-shell photon invariant mass squared. This two-body muon observables should occur only due to their mass decay is the reach of rare kaon 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 meson decays [4] have shown hints of simple kinematics with a single, monochromatic photon ðμþμ−Þ þ − violations to this universality. The bound state 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 lepton universality in and beyond the standard model 2 2 electrons (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: particle 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 mesons [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 proton beam, DImeson Relativistic 2 þ − how small the Q bins can be made. In contrast to this, the Atom 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 FKLγγà 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 γγÃðQ ¼ 4MμÞ ∼ F γγÃð0Þ 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 leptons, and is derived from experiment for the hadronic constribution. à 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 → γÃγà → 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- FγÃγà 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 γγà ðzÞ=F γγà ð0Þ, and C0 is the sum of the arises from the direct weak vertex KL V which then KL KL à 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-quark model calculations assumed. → γÃð ÞþγÃð − 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 photons. 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 à Mann-Okubo relation [21,22]. In the nonet realization, of the double virtual photon contribution KL → γ ðkÞþ 0 à 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 à à 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 à 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 à à 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 hadrons, C is the vertex correction 0 ver the process KL → π γγ as well, and we use the unweighted term of [8],whileCγÃγà is the contribution from Fig. 1. average of the two most recent measurements of this A similar calculation for positronium, 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 à weak vector-vector interaction (KL → γ þ K → γ þ ρ; Z ω; ϕ → γ þ γà ΔI ¼ 1 ∞ Πð Þ ).