CERN-TH-2019-049 Discovering True Muonium at LHCb Xabier Cid Vidal,1, ∗ Philip Ilten,2, y Jonathan Plews,2, z Brian Shuve,3, 4, x and Yotam Soreq5, 6, { 1Instituto Galego de Fısica de Altas Enerxıas (IGFAE), Universidade de Santiago de Compostela, Santiago de Compostela, Spain 2School of Physics and Astronomy, University of Birmingham, Birmingham, B152 2TT, UK 3Harvey Mudd College, 301 Platt Blvd., Claremont, CA 91711, USA 4University of California, 900 University Ave., Riverside, CA 92521, USA 5Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland 6Department of Physics, Technion, Haifa 32000, Israel We study the potential of the LHCb experiment to discover, for the first time, the µ+µ− true 3 muonium bound state. We propose a search for the vector 1 S1 state, TM, which kinetically mixes with the photon and dominantly decays to e+e−. We demonstrate that a search for η ! γTM, TM! e+e− in a displaced vertex can exceed a significance of 5 standard deviations assuming statistical uncertainties. We present two possible searches: an inclusive search for the e+e− vertex, and an exclusive search which requires an additional photon and a reconstruction of the η mass. I. INTRODUCTION cay are to γγ, which is challenging to reconstruct with the LHCb detector. Therefore, we concentrate on the discovery of , the spin-triplet true muonium state. Electromagnetic (EM) interactions between oppositely Other possibleTM search avenues for are with the cur- charged particles form bound states; by far, the most well TM known of these are the atoms. Similar atom-like bound rently running HPS experiment [19] or via rare B decays states of elementary particles have since been discovered, into leptonium at LHCb [20]. However, both of these including positronium (a bound state of e+e−) [1] and methods are statistically limited with potentially large muonium (a bound state of µ+e−) [2]. The properties of backgrounds and are not expected to have discovery po- these bound states are predicted by quantum electrody- tential. The proposed RedTop [21, 22] experiment at namics (QED), and measurements of the mass and spec- Fermilab is designed to produce a large flux of η mesons, tra provide precision tests of QED. and using the methods outlined in this work, might also be sensitive to . Searching for a γ final state However, there remain heavier QED bound states that from e+e− collisionsTM has also been proposedTM [23], which have not yet been experimentally observed which can pro- may be accessible to Belle II. However, discovery is vide unique probes that are sensitive to beyond the stan- not expected given the Belle II dark photonTM reach [24]. dard model (BSM) physics. In particular, the hypoth- + − The rest of the paper is organized as follows. In Sec- esized bound state known as true muonium (µ µ ) [3] tions II and III we describe the analogy between and has yet to be discovered. In this work, we explore the dark-photon and highlight the differences. SectionTM IV potential of the LHCb experiment to discover the lowest contains the details of the proposed LHCb search. We spin-1 state of true muonium via its displaced decays to + − conclude in Section V. The appendices contains techni- e e pairs. We show that true muonium can be observed cal details and a discussion about and new physics. with a statistical significance exceeding 5 standard devi- TM ations using the expected 15 fb−1 of LHC Run 3 data to be collected with the upgraded LHCb detector [4{9]. II. TRUE MUONIUM SIGNAL AS A DARK The most promising true muonium state for discov- PHOTON 3 ery is the 1 S1 state, which in the non-relativistic limit has zero orbital angular momentum and is in the spin- arXiv:1904.08458v2 [hep-ph] 13 Sep 2019 Dark photons are massive spin-1 states that couple via triplet state. This vector muonium state, which we de- a kinetic mixing " to the standard model (SM) photon: note as , kinetically mixes with the photon resulting in a phenomenologyTM similar to the dark photon [10{15]. Dark photons have been the subject of much recent study, " 0µν Fµν F ; (1) e.g. [16{18], allowing us to use these latest developments L ⊃ 2 in the discovery of at LHCb. Note that spin-singlet µν 0µν true muonium statesTM also exist, but their dominant de- where F and F are the dark photon and SM pho- ton field strengths, respectively. The phenomenology of is similar to that of a dark photon, and the mass and kineticTM mixing are predicted by QED at leading order: ∗Electronic address: [email protected] yElectronic address: [email protected] z Electronic address: [email protected] mTM = 2mµ BE 211 MeV ; (2) x − ≈ Electronic address: [email protected] 2 −5 { "TM = α =2 2:66 10 ; (3) Electronic address: [email protected] ≈ × 2 2 2 where BE mµα =4 = 1:41 keV is the binding 10− ≈ TM energy, estimated in the non-relativistic limit. Our result Belle II FASER is in agreement with Ref. [25], where the kinetic mixing of HPS SeaQuest hidden sector onium states was calculated. We emphasise 10 3 + − LHCb µ µ− SHiP that the above analogy between and the dark photon 0 LHCb D∗ is valid only at energies close to theTM mass, as relevant TM 4 to our study. 10− [unitless] As noted earlier, decays through the same kinetic ε mixing to an e+e− finalTM state with a branching fraction of TM 5 BR( e+e−) 98 %, while the sub-dominant decay 10− modeTM! has BR( ≈ 3γ) 1:7 % . The lifetime at leading order isTM! ≈ TM 6 10− 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 m [GeV] 6 −12 FIG. 1: Dark photon parameter space in dark photon mass τTM 5 1:8 10 sec : (4) ≈ α mµ ≈ × and kinetic mixing with (gray) previous limits and future reach from (magenta) Belle II, (purple) FASER, (cyan) HPS, and (green/yellow) LHCb. TM corresponds to the marked Because of the forward coverage of LHCb, light par- point, using Eqs. (2) and (3). ticles produced within LHCb acceptance typically have large boosts. Given the expected boost of within LHCb and the relatively long proper lifetimeTM of 0:53 mm, III. DISSOCIATION OF TRUE MUONIUM the decay of into e+e− within LHCb will typically TM produce a resolvable displaced vertex. While searches Because is a bound state rather than an elemen- for long-lived particles typically focus on new BSM tary particle,TM there are significant differences between states [26], is an example of a SM long-lived par- and dark photon phenomenology. Most importantly, TM ticle that can be searched for at LHCb. Predictions of TM can dissociate when the constituent muons of the the mass and lifetime at higher order than those derived boundTM state interact with the detector material, result- here are available [22, 27]; however, it is unlikely that ing in a separated µ+ and µ− with an invariant mass just LHCb will be sensitive to these higher order corrections. above the mass of , mTM. TM Since and dark photon phenomenology are sim- The dissociation cross section is estimated to TM 2 ilar, excludingTM dissociation detailed in Section III, be [34{37] σTM!µµ 13Z b , where Z is the atomic ≈ projected darkTM photon reaches from future experiments number of the material inducing the dissociation. The can provide a rough guide to sensitivity. In Fig. 1 the bulk of the material traversed by within LHCb prior TM dark photon parameter spaceTM is plotted in dark photon to its decay is the aluminum radio frequencey (RF)-foil mass (m) and kinetic mixing (") using Darkcast [28], (made of AlMg3) and the silicon vertex locator (VELO) where corresponds to a single point given by the " sensors. Since both aluminum and silicon have similar and mTMof Eqs. (2) and (3). The gray regions correspond Z and number densities, the mean free path for TM to already excluded parameter space, while the colored traversing the material of the detector is, regions represent possible reach from relevant future ex- periments. Dashed lines indicate experiments where dis- −1 −1 λ = σTM!µµna 13 mm ; (5) sociation will be an issue. These include searches by ≈ FASER [29], SeaQuest [30], and SHiP [31] where where the number density is na 6:0 (5:0) will dissociate as it passes through the shielding. TM 1019 atoms=mm3 [38] and Z = 13≈ (14) for alu-× ∗0 0 0 + minum (silicon). Thus, the probability of dissoci- Both the proposed LHCb D D A ( e e) [32] TM and inclusive A0( µ+µ−) [33] searches! are! shown, to ating is given by demonstrate how! dark photon searches based on this study could be used to fill the gap between the two −x/λ dis = 1 e ; (6) searches. The dashed regions for these LHCb searches P − correspond to post-module search strategies where the where x is the distance of the material traversed. The will dissociate. The expected displaced reach of RF-foil will have a nominal width of 0:25 mm in Run 3 HPSTM [19] does not cover the parameter space point, and the VELO sensors a nominal width of 0:2 mm. Con- and will also suffer from someTM dissociation. Additionally, sequently, every encounter of with material in the the expected prompt Belle II reach [24] does not extend VELO results in a minimum dissociationTM probability of to large enough lifetimes to discover , and the nom- dis & 90%. inal Belle II lifetime resolution will notTM be sufficient for P Given the expected material budget of the LHCb de- effective displaced searches.