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CERN-TH-2019-049

Discovering at LHCb

Xabier Cid Vidal,1, ∗ Philip Ilten,2, † Jonathan Plews,2, ‡ Brian Shuve,3, 4, § 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 . We propose a search for the vector 1 S1 state, TM, which kinetically mixes with the 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 form bound states; by far, the most well TM known of these are the . Similar -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 (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 , 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 are massive spin-1 states that couple via triplet state. This vector muonium state, which we de- a kinetic mixing ε to the (SM) photon: note as , kinetically mixes with the photon resulting in a phenomenologyTM similar to the [10–15]. Dark photons have been the subject of much recent study, ε 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] †Electronic address: [email protected] ‡ Electronic address: [email protected] mTM = 2mµ BE 211 MeV , (2) § − ≈ Electronic address: [email protected] 2 −5 ¶ εTM = 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 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 ,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 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 (ε) using Darkcast [28], (made of AlMg3) and the silicon vertex locator (VELO) where corresponds to a single point given by the ε 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. tector during Run 3 [5], the boost distribution for TM 3

100 signal STM 3π mµ 3 20 MeV −6 signal w/out dis α 1.2 10 . (7) P BEM ≈ 16 σm ≈ σm × background ee ee 1 10− ]

1 The dominant source of off-shell photons in the mass − ∗ range mTM 211 MeV is from η γγ decays. We

[mm therefore focus≈ on searching for →produced from η dr dσ +TM− → 1 σ 2 γ decays with a e e final state. The sig- 10− nalTM can be fully normalizedTM → by the data using the pro- cedure outlined above. The ratio of Eq. (7) must be corrected by the different acceptance and efficiency fac- tors for a displaced e+e− signal relative to the prompt 3 10− 0 2 4 6 8 10 signal. Additionally, the signal rate should be corrected r (radial flight distance) [mm] by the expected dissociation factor, to account for that dissociate without decaying. TM FIG. 2: Normalized radial flight distance distributions for the The number of signal events can be estimated as fol- TM → e+e− signal (blue solid) with dissociation, (blue dot- lows: we simulate in Pythia 8.2 [40] both the pp total + − ted) without dissociation, and (red dashed) the e e back- cross section, σtot = 100 mb, and the average number ground from B- decays. of η mesons produced per collision within the LHCb ac- ceptance, Nη = 0.83. The former is in agreement with the LHCb inelastic cross-section measurement [41], while produced within LHCb acceptance, and dis, roughly the latter correctly predicts the low mass limit of the half of the produced are expected to dissociateP with- LHCb inclusive µ+µ− dark photon search [39]. Given out decayingTM into an e+e− final state. The radial flight that BR(η γ ) = 4.8 10−10 [22], which agrees distance distribution of the particles which do de- well with Eq.→ (7)TM using the differential× η γe+e− shape + − TM + − cay into e e , is compared to the expected e e back- from Pythia, the signal cross section in→ the fiducial vol- ground in Fig. 2. On average, has a higher boost ume is than the background, resultingTM in a flatter distribution that is abruptly truncated by dissociation. + − fid This dissociation gives rise to a signal of µ µ origi- σ = σtotNη BR(η γ ) 40 pb . (8) TM → TM ≈ nating from the regions of high material density at LHCb. While nearly half of the produced is a considerable In our analysis we consider two possible search strate- fraction of the total signal,TM the dissociated µ+µ− signal is gies: (i) inclusive search – the final state is e+e− and we difficult to reconstruct and suffers from large irreducible do not search for the photon, thus the η is not recon- backgrounds. The two muons will be nearly collinear structed (in principle this search is sensitive to any TM and will typically share hits within the VELO, resulting production mechanism); (ii) exclusive search – the final in poorly defined tracks. Additionally, since the disso- state is γ e+e− and the η is reconstructed. Each of these ciation occurs in material, the conversion background of methods has both advantages and disadvantages. The γ µ+µ− can no longer be eliminated with a material inclusive search is simpler and expected to have smaller veto→ without eliminating the signal itself. Therefore, for systematic uncertainties, while the background rates for the remainder of the paper we focus on the 50% of signal the exclusive analysis are smaller. Without a full detector events which decay via e+e−. simulation and data-driven background estimates with TM → their corresponding uncertainties, we cannot definitively state which of the two strategies is optimal; we therefore estimate the potential sensitivities of both. The details IV. PROPOSED LHCB SEARCH of our signal and background simulations are provided in Appendix A. We propose searching for as a displaced e+e− The LHCb experiment is a forward arm spectrometer . Since behavesTM like a dark photon, the which covers pseudorapidities between 2 and 5 [42, 43]. signal rate can beTM calculated directly from the off-shell This is a simplification of the coverage provided by the in- photon rate as given by the prompt e+e− spectrum dividual sub-systems, but provides an adequate descrip- data [15, 33, 39]. For any initial (Y ) and final (X) states, tion, given the evolving nature of the upgraded detector the ratio between the number of Y X X e+e− and the weak assumptions made on identifica- → TM+ →− events, STM, and the number of prompt e e events, tion efficiencies in this paper. While the exact perfor- ∗ + − + − Y Xγ Xe e , BEM, is fixed. For the e e in- mance of LHCb during Run 3 and 4 is yet to be fully un- → → variant mass within the range of mee mTM < 2σmee , derstood, we estimate the relevant quantities as follows, + − | − | + − where σmee is the e e invariant mass resolution, this with more details given in Appendix B. The e e invari- ratio is given by ant mass resolution around the mass is estimated to TM 4

0 + − + − (i) (i) (ii) (ii) be σmee 20 MeV, based on the KS e e e e LHCb requirement S B S B ≈ → TM tot TM tot study [44], while σmeeγ around the η mass is estimated 3 7 3 6 to be 50 MeV based on Refs. [43, 45, 46]. base 3.4 × 10 3.2 × 10 1.6 × 10 5.4 × 10 DOCA(trk, e) 3.0 × 103 8.5 × 106 1.3 × 103 1.1 × 106 We apply the following baseline selection criteria for θ 1.5 × 103 1.8 × 104 6.4 × 102 1.9 × 103 both cases (i) and (ii): efficiency 4.4 × 10−1 5.6 × 10−4 4.0 × 10−1 3.5 × 10−4 1. Two opposite-sign in the LHCb accep- ± ± tance and with p(e ) > 10 GeV, pT (e ) > 0.5 GeV, TABLE I: Expected signal and background yields for the and transverse impact parameter (IPT) which is ee (eeγ) final state label as i (ii), assuming 100 % reconstruc- ± tion efficiency for the final state and a collected Run 3 dataset not consistent with zero, IPT(e ) > 3σIP (e), T of 15 fb−1. where σIPT(e) is the IPT resolution; 2. A reconstructed e+e− candidate in the TM → B-mesons tend to decay to a high multiplicity of tracks LHCb acceptance and with pT( ) > 1.0 GeV, TM that originate from the same decay vertex. These events mee mTM < 2σm , and the distance of closest | − | ee are, in principle, readily suppressed by B-decay vetoes approach (DOCA) between the two electrons con- 0 + − used in the LHCb dark photon search [39] and Bs sistent with zero, DOCA(e , e ) < 3 σ + − DOCA(e ,e ) µ+µ− lifetime measurement [49]. As a simple proxy for→ (the details on DOCA resolution are given in the these vetoes, we apply the following additional selections: Appendix B). This ensures that the electron pair forms a high-quality vertex. 5. The candidate is isolated from other tracks TM in the LHCb acceptance: tracks with pT(trk) > For case (ii), in which we reconstruct the additional 0.5 GeV and IPT(trk) > 3σIP (trk) must satisfy photon from the η decay, there are two additional base- T DOCA(trk, e) > 3σDOCA(trk,e) for both electrons. line selections: 6. The opening angle, θ, between the flight and mo- 3. A photon in the LHCb acceptance and p(γ) > mentum vectors of the candidate is consistent TM 5 GeV, and pT(γ) > 0.65 GeV; with zero. The resolution on this opening angle de- pends upon the reconstructed flight distance and 4. A reconstructed η candidate within the LHCb ac- IPT resolution of the two electrons. ceptance and meeγ mη < 2σmeeγ . | − | The numbers of expected candidates are given in For both cases (i) and (ii), data is expected to be col- Table I for the signal and backgroundTM after the baseline lected using an e+e− trigger. During Run 1 and 2, only selection, as well as after each of the two additional re- a single electron trigger with tight kinematic cuts was quirements. Less than 0.1 % of the signal events pass the available in the first-level hardware trigger, which is not baseline selection, largely due to the inefficiency of the efficient for this signal. However, in Run 3 and 4 full on- pT requirements; however, the pT selections cannot be line reconstruction with triggerless readout will be avail- significantly loosened. The efficiencies of the additional able [7], which will allow the reconstruction of lower mo- selections beyond baseline, however, are of order one for mentum signals such as the electrons from decays. the signal and 10−3 10−4 for the background, allowing Because decays are displaced and insideTM a narrow for efficient background∼ − reduction. There is an additional invariantTM mass window, the candidates can be re- efficiency for reconstruction of all the final-state parti- TM constructed and recorded in Run 3 and 4 with a high cles, εf , which originates from the reconstruction of the efficiency. tracks, both online and offline, and from applying particle The dominant background after the baseline selection identification criteria. Because the expected electron and is from B-hadron decays, which are also displaced. De- photon efficiencies are not yet public for Runs 3 and 4, we cays of D- are a sub-dominant background since leave εf as an unspecified quantity in our expression for these rarely produce an e+e− pair which creates a recon- the significance and discuss the implications shortly. We structible vertex in the chosen kinematic regime. The note that final state reconstruction efficiencies can be es- background from photon conversions was also estimated timated based on current LHCb performance. From the ∗0 and found to be sub-dominant, using techniques from the B J/ψK analysis [50] we find that εe+e− > 10 %, ∗0 0 + − → proposed D D e e dark photon search [32] and a and from Ref. [45] we estimate εγe+e− 0.3 εe+e− > 3 %. material veto similar→ to that used in the LHCb inclusive For further details see Appendix B. ≈ µ+µ− dark photon search [47]. In the same regard, the Because the background rate in the signal region can background from η e+e−γ decays will be also sub- be estimated using the invariant mass sidebands, we dominant taking into→ account the expected displacement expect the significance to be limited by the statistical of the before decaying (see Fig. 2). Given the excel- uncertainty of the sample. The LHCb inclusive dark lent LHCbTM resolution for reconstructing the signal decay photon di- search [39] successfully used such a vertex [48], a moderate cut in this displacement would technique [51], although inclusion of known background be enough to reduce this background to negligible levels. structure helped improve significance. The shape of the 5

30 assumption is placed on the lifetime of the candi- date, and it does not appear to be necessaryTM to reach a 25 5 σstat discovery significance.

20 ]

1 V. DISCUSSION AND OUTLOOK − 15 [fb

L As outlined above, we project that LHCb will be able 10 to discover with a statistical significance exceeding 5 σ in RunTM 3. Ultimately, LHCb and other experi- (i) e+e stat 5 − ments can directly measure the mass, lifetime, and + (ii) e e−γ production rate (from η decaysTM or other mechanisms). 0 Since the properties are well predicted by the SM, 0.00 0.05 0.10 0.15 0.20 0.25 0.30 this will beTM a test of the SM predictions in Eqs. (2)– εf [unitless] (4), and any deviation from them is a clear sign of new

FIG. 3: The required integrated luminosity for a 5 σstat dis- physics coupled to muons. Examples include dark pho- covery of TM as function of the final reconstruction efficiency, tons, Lµ Lτ gauge , scalars, or -like parti- + − + − − εf for the proposed (blue) e e and (red) e e γ searches. cles. In the presence of any of these particles, the mass (via the binding energy), lifetime, branchingTM ra- tios and (see discussion in [52]) are mod- B-hadron background has been demonstrated to be well ified, and thus measurements can discover or con- TM modeled [39], and there is a similar expectation for this strain new muonic interactions. Such new forces are mo- analysis. Therefore, the signal significance is ap- tivated by several possible discrepancies with predictions proximately given by TM of the SM in other experiments, including measurements of the muon anomalous magnetic moment (g 2)µ [53], and the charge radius problem [54–57].− How- r STM εf ever, strong constraints on new physics exist from di- σstat L−1 , (9) ≈ √Btot 15 fb rect searches [58, 59], measurements of (g 2)µ, experiments [60–66], and eµ spectroscopy− [67–69]. In- where STM and Btot are the expected number of signal deed, these constraints are generally more powerful than and background events from Table I, ε is the final state the expected sensitivity of LHCb to , although some f TM reconstruction efficiency, and is the integrated lumi- exceptions exist (for example, (g 2)µ constraints can nosity of the dataset. Using theL expected Run 3 dataset be alleviated if there are other new− particles whose ef- −1 of 15 fb , can be discovered with σstat 5 when fects partially cancel). New muonic forces can also be TM + − + − ≥ εf > 20 % (12 %) for the e e (e e γ) final state. Given probed as in Refs. [70–75]. For a detailed analysis see the current LHCb performance, these efficiencies are real- Appendix C. istic; see the above discussion and Appendix B. In Fig. 3 In the context of this study, we also considered the we plot the required integrated luminosity for discovery possibility of an inclusive search for a τ +τ − bound state, of , e.g. σstat 5, as a function of εf . see e.g. Ref. [20]. In particular, ortho-tauonium, with InTM addition, Fig.≥ 4 shows the differential cross sections a significant branching fraction to µ+µ−, would appear with respect to the e+e− invariant mass for signal and to be the best candidate for an LHCb search. We find, combinatorial background at LHCb, assuming a global however, that the short lifetime of the tauonium (close efficiency to reconstruct the candidates of 20%, or to the τ itself), and the small signal yield compared to 6% when also considering theTM reconstruction of the addi- the background make the prospects very poor for being tional photon from the η decay. observed at LHCb. We conclude this section by commenting that we con- In summary, we have studied the potential for LHCb to sidered additional selection criteria that we found to be discover an as-yet-undiscovered long-lived particle in the sub-optimal and therefore did not include in our analysis. SM: the µ+µ− true muonium bound state. We have pro- 3 First, we can require that the IPT of the two electrons, posed a search for the vector 1 S1 true muonium state, projected onto the normal of the decay plane, is consis- , which kinetically mixes with the photon and de- tent with zero. The decay plane is defined by the first hit caysTM to e+e−. We have demonstrated that a search for + − of each electron track and the primary vertex. We found η γ , e e can exceed a 5 σstat statis- that this observable does not provide strong separation tical→ significanceTM TM using → a displaced vertex search, and after the above selection has been applied. Second, the we have presented two possible searches: an inclusive expected proper lifetime of the candidate is known, search for the e+e− vertex, as well as an exclusive search and so in principle the transverseTM flight distance can be where we reconstruct the additional photon and require used to select events that are most consistent with this m(γ, ) = mη. Since mixes kinetically with the hypothesis. However, the analysis is more robust if no photonTM and has a signatureTM similar to the dark photon, 6

4500 200 signal case (i) signal case (ii) 4000 background 175 background 3500 signal + background signal + background 150 3000 125 2500

[fb/GeV] [fb/GeV] 100 − −

e 2000 e + + e e dσ dσ 75 dm 1500 dm 50 1000

500 25

0 0 0.10 0.15 0.20 0.25 0.30 0.10 0.15 0.20 0.25 0.30

+ + me e− [GeV] me e− [GeV]

FIG. 4: The differential cross sections with respect to the e+e− invariant mass for the expected TM signal and combinatorial background at LHCb, assuming the normalization in Tab. I for case (i) and (ii). Global efficiencies of 20% and 6% are assumed to reconstruct the TM and TM plus photon candidates, respectively. In these conditions, a 5 σ observation would be possible with an integrated luminosity of 15 fb−1 in case (i) and of 30 fb−1 in case (ii). The invariant mass resolution of the signal is described in the text. The shift observed in the central position of the signal peak, due to the lack of reconstructed bremsstrahlung from the electrons, is compatible with that of Ref. [44]. For the combinatorial background, the resulting invariant mass distribution is obtained from simulation. this method could also have sensitivity to dark photons to generate a sufficiently large background sample. The in a similar mass window. results from this large sample were found to be in agree- ment with a smaller background sample generated us- ing the more inclusive SoftQCD configuration. Addition- Acknowledgments ally, including more sophisticated B-hadron decays using EvtGen [76] was found to have no noticeable effect on the final result. This is because Pythia already uses the We thank Gil Paz, Mike Williams and Jure Zupan branching fraction tables from EvtGen, and many of for useful discussions, and Johannes Albrecht, Matthew the inclusive EvtGen decays use Pythia for showering John Charles, Maxim Pospelov, Mike Williams and Jos´e and hadronization. The results from Pythia for both Zurita for comments on the manuscript. We also thank signal and background are demonstrated to be reliable, the organizers of the “New ideas in detecting long-lived with the Pythia study of Ref. [33] accurately predict- particles at the LHC” workshop at LBL for a stimulating ing the reach of the LHCb inclusive µ+µ− dark photon environment for discussions, along with other members search [39]. of our working group: Jeff Asaf Dror, Maxim Pospelov, Conversion backgrounds were estimated using the pho- Yuhsin Tsai and Jos´eZurita. ton flux generated from Pythia configured with the flag The work of XCV is supported by MINECO (Spain) SoftQCD:all = on, and modeling the expected conver- through the Ram´ony Cajal program RYC-2016-20073 sion rate within the material of the upgraded LHCb de- and by XuntaGal under the ED431F 2018/01 project. tector. The cross section for photon conversions was cal- PI is supported by a Birmingham Fellowship. JP is culated using a method [77], similar to that implemented supported by the UK Science and Technology Facilities in the material simulation package Geant [78]. The ap- Council. The work of BS is supported by the U.S. Na- proximation of the opening angle between the converted tional Science Foundation under Grant PHY-1820770. electron- pair is under-estimated at high masses by the Geant model [79], and so a correction was applied to produce an invariant mass spectrum of the converted Appendix A: Signal and Background Simulation pair that matches the full analytic expression [80].

All signal and background samples are simulated us- ing Pythia 8.240 [40]. The signal from η de- cays is generated using the flag SoftQCD:all = on, while the B-hadron background is generated using the flag HardQCD:bbbar = on. For the latter, the HardQCD flag in conjunction with repeated B-hadron decays was used 7

Appendix B: LHCb Performance nal state. One drawback of reconstructing electrons that are swept away by the magnet is the missing informa- 1. Invariant mass resolution and reconstruction tion from the PID detectors located after the magnet, efficiencies e.g. RICH 2, the calorimeters, and the muon system. However, the PID information from RICH 1, specially An upgraded version of the LHCb detector will record designed for low-momentum particles [81], would still be available. the result of proton-proton collisions at √s = 14 TeV + − during Runs 3 and 4 of the LHC. Similar, if not better, Concerning the e e invariant mass resolution for the performances of the detector are expected during that reconstruction, Ref. [44] claims an invariant mass resolutionTM of 8% to reconstruct K0 e+e−e+e− de- period [4]. The upgrade of the detector is currently tak- ∼ S → ing place. One important feature of this upgrade is the cays at LHCb. The kinematic cuts in that study are expected triggerless readout [7], removing the need for a softer with respect to this one, and therefore the mo- first-level hardware trigger that is present in other LHC mentum resolution for the electrons in this analysis is detectors. This will allow a dramatic increase in the effi- expected to be better, due to the smaller effect of multi- ciency to reconstruct low-momentum signatures, such as ple scattering. However, here we assume a similar in- variant mass resolution, taking σ 20 MeV with the decay products of . mee radiative tails based on the invariant mass∼ distribution An estimation of theTM efficiency to reconstruct the from K0 e+e−e+e− decays. This conservative ap- candidates can be achieved by comparing to other LHCbTM S proach can→ be confirmed by the σ distribution from analyses containing an e+e− final state. In the B0 mee B0 J/ψK∗0 decays, with the J/ψ decaying to an e+e− J/ψK∗0 analysis, with the J/ψ decaying to an e+e− pair,→ pair→ [82]. For these decays, using final state electrons in reconstruction and selection efficiencies at the level of 5% a kinematic range similar to this study, resolutions at the could be achieved during the first years of LHCb run- level of 2% can be achieved with LHCb. The kinematic ning [50]. This efficiency includes the reconstruction of constraint mentioned above, arising from the knowledge the accompanying K∗0 particles decaying to Kπ pairs as of the decay position and the pp collision point, could well as selection cuts on the mother B candidate. The also beTM used to improve the e+e− invariant mass resolu- kinematics of the selected signal electrons and those tion by 20%. from J/ψ decay have beenTM checked to be in reasonable The full≈ reconstruction of the η γ decay also re- agreement. Therefore, reconstruction and selection effi- → TM ciencies above 10% should be easy to achieve. Since the quires the determination of the reconstruction efficiency performance of the upgraded LHCb detector is still to be of the γ. To obtain this, Ref. [45] is used, aligning our γ determined, we chose to show the expected significance selection cuts with those in that analysis. In that study, as a function of the final state reconstruction efficiency, an efficiency of 10% is claimed to reconstruct the photon. rather than choosing a fixed value. This efficiency will This includes both the effect of the kinematic cuts applied also account for additional selection requirements to be and of the reconstruction in the LHCb ECAL. If the ef- applied in the experimental analysis. This includes the fect of the kinematic cuts is factored out, an efficiency use of particle identification cuts or more sophisticated of 30% is obtained. This is taken as a baseline for this≈ analysis. In order to estimate the η γ decay variables to discriminate against the combinatorial back- → TM ground. In the same regard, additional potential inef- invariant mass resolution, an estimate of the γ momen- ficiencies in the online reconstruction at the upgraded tum resolution is needed. This has two components, the detector can be factorized as part of that efficiency. It direction and energy resolution of the photons. The first should be remarked that the 5% efficiency, given as a depends on the ECAL cell size and on its distance to the baseline above, already includes this online reconstruc- pp collision point. Most of the signal photons are found tion in the current detector. to fall in the most inner region of the ECAL, where the cells have a size of 4 cm [43]. This provides an an- One of the main challenges to reconstruct low mo- gular resolution of ≈0.002. For the energy resolution, mentum electrons at LHCb is the fact that the mag- Ref. [43] reports δE/E≈ 9%pGeV/E 0.8%. Combin- net sweeps away an important fraction of these particles, ing both effects together,' an invariant mass⊕ resolution of which then only leave hits in the pre-magnet tracking σ 50 MeV is obtained. The methodology is vali- stations. Therefore, these electrons can be reconstructed, meeγ dated using≈ multiple LHCb analyses with γ in the final but their momenta are unknown. However, for the recon- states [45, 46]. struction of the mass, the knowledge of the pp colli- sion vertex (whereTM the was produced), the decay position, and the directionsTM and momenta ofTM the decay electrons is over-constrained. In this case, only the full 2. Impact parameter and DOCA resolution reconstruction of one of the final-state electrons is neces- sary. For the other electron, only the direction is needed, The description of the upgraded LHCb vertex locator such that hits in the pre-magnet tracking stations would (VELO) is taken from Ref. [5], using a nominal single be sufficient. The use of this technique could signifi- hit resolution of 12 µm in x and y. Multiple scattering is cantly increase the reconstruction efficiency of the fi- modeled [83] assuming an RF-foil thickness of 0.25 mm TM 8 and sensor thicknesses of 0.2 mm. This material descrip- tion is validated against the full LHCb upgrade simula- tion where the transverse impact parameter for a track Scalar (S): S =ySµ Sµµ¯ + ySe See¯ , (C1) L is parameterized by, 5 5 Pseudoscalar (a): a =yaµ aµγ¯ µ + yae aeγ¯ e L g + aγ aF F˜µν , (C2) 4 µν  1.3 GeV  ν −2 Vector (V ): V =gV µ µγ¯ µVν σIPT = 1.1 + 10 mm , (B1) pT × L ν + gV e eγ¯ eVν , (C3) ν 5 Axial Vector (A): A =gAµ µγ¯ γ µAν where the first term is determined by the detector ge- L ν 5 ometry and the second term arises from multiple scat- + gAe eγ¯ γ eAν , (C4) tering. The uncertainty on the distance of closest ap- µν µνρσ proach (DOCA) between two tracks is well approximated where F˜ = ε Fρσ/2. as,

(1) (2) 2. TM decay to a photon and a mediator σDOCA = σ σ , (B2) IPT ⊕ IPT If the mediator X = S, a, V or A couples to muons, given σ(1) and σ(2) are the IP uncertainties for the first IPT IPT T we can have decays as γX or XX. Since and second track, respectively. the decay to two mediatorsTM is → typicallyTM suppressed → by the square of the mediator coupling to muons, the decay to γX is the most important. Depending on the lifetime of X and its decay modes, the signature can be mono- Appendix C: Muonium and Physics Beyond the + − photon, or photon and e e . Assuming that ΓTM Standard Model ≈ ΓTM→e+e− , see Eq. (4), we find the following branching ratios Since the properties of are completely determined by the SM, the ability of LHCbTM to independently measure y2 the mass, production rate, and lifetime of provides Sµ 2  TM BR( γS) = 1 + 4xS + xS , (C5) the possibility of a precision test of the SM. New particles TM → 2πα(1 xS) − and forces coupled to muons, including dark photons, (1 x ) BR( γa) = a y2 Lµ Lτ gauge bosons, low-mass scalars, and axion-like − aµ − TM → 2πα particles, could potentially alter the muon binding energy 2  and decay rates by providing additional annihilation 2 2 (1 xa) + gaγ mTM − , (C6) channelsTM for the µ+µ− bound state. Such new muonic 16 forces have already been predicted in the context of the BR( γV ) =0 , (C7) persistent anomalous measurements of (g 2) and the TM → µ g2 1 + 10x + x2 proton charge radius problem, see e.g. [84].− BR( γA) = Aµ A A , (C8) TM → 2πα 1 xA Here, we focus on BSM contributions to the decay − TM rate, both to SM states mediated by new interactions but 2 2 where xX = mX /mTM and we neglect the relative mo- also the decay to hidden-sector states. Since the mentum of the muons in the state. This is reason- productionTM rate depends on the wavefunction at TM TM TM able because this kinetic energy is a small contribution the origin, a new force can only appreciably modify this to the energy released in the decay. if its structure constant is comparable to α. However, The limits on the coupling ofTM the mediator to muons is this structure constant is strongly constrained by (g − generally model dependent. However, the measurement 2)µ and other precision measurements. Therefore, the of (g 2)µ provides a sensitive probe of new physics cou- prospects for BSM modifications to the decay are − TM pled to muons. In principle, it is generally possible to more promising than for its production, although still evade these constraints by having another contribution challenging to observe. to (g 2)µ that almost cancels the one from the me- diator.− In Fig. 5 we plot the maximal branching ratio to final states in Eqs. (C5)–(C8) whichTM is allowed by measurements of (g 2)µ at the 5 σ level, i.e. ∆aµ = 1. Hidden-Sector Models 1 1 − −9 (g 2)µ(obs) (g 2)µ(SM) [ 1.1, 6.9] 10 [53]. 2 − − 2 − ∈ − × We do not include the effects of the coupling gaγ on the We consider the following scenarios, which give rise branching fraction to pseudoscalars because of the pow- to modifications of the decay rate and branching erful constraints on direct searches for axion-like particles fractions: TM from LEP data, which lead to a negligible contribution 9

∗ 100 The maximal allowed value of BR( V χχ¯ ) by TM → → 10 1 (g 2)µ for gV χ = 4π and mχ = 0 is plotted in Fig. 5. − − 2 For mV mTM, this gives a hidden-sector branching 10−  3 fraction at the level of 2%. While this is likely too small 10− 4 to be seen as a change in the lifetime or cross sec- 10− TM 5 tion, it could be detectable if the χ decays themselves 10− 6 are visible, which is challenging. If mV = 160 MeV, the 10− branching fraction is enhanced to 10%. If the states 7 10− γS become much more degenerate than∼ this, it is: (a) tuned; 8 TM → max(BR) [unitless] 10 γa, y − TM → aµ (b) would require some careful treatment of the width 9 10− γA TM → and mixing between the two states. This is especially 10 10 χχ, m = 0, g = 4π − TM → χ vχ true if the coupling gV χ is very large, because the width 11 10− would be large as well. If we instead take gV χ = 1, then 0.00 0.05 0.10 0.15 0.20 −4 −3 the branching fraction is 10 for mV = 0 and 10 mX [GeV] ∼ ∼ for mV = 170 MeV. FIG. 5: TM branching ratio to BSM final states in Eqs. (C5)– (C8) which are allowed by (g − 2)µ at the 5 σ. 4. Modifications to TM decay to e+e− to the branching fraction into pseudoscalars [85– In this section, we consider s-channel contributions of 87]; seeTM also a recent recast of PrimEx data [88, 89]. the mediator to the decay of e+e−. This is sim- TM → The expressions for NP contributions to ∆aµ are taken ilar to the decay from Section C 3, but we must include from [87, 90, 91]. As we can see the maximal branching interference with the contribution from the SM photon. ratios are typically below the 1 % level and require high We obtain (in the limit me mTM, mV ) precision measurements to exceed this sensitivity.  TM 3 + − α Γ( e e ) = 2 2 3. TM decay to hidden-sector particles TM → 192π (1 xV ) − 2 [gV µgV e + 4πα(1 xV )] , (C10) In this section, we calculate decay rates of to × − hidden-sector particles χ such as χχ¯ , whereTM χ which appropriately reduces to the V -only or photon- TM → is a hidden-sector particle. This results from muon an- only results in the limits α 0 and gV µ = gV e = 0, → nihilation via an s-channel mediator into the χ particles. respectively. For gV µ, gV e √4πα, the dominant cor- This final state dominates when the mediator has a much rection to the width from the SM value scales like larger coupling to hidden sector particles than SM parti- cles. These χ particles could be invisible, or in turn decay to lighter hidden-sector particles. We consider the same ∆Γ gV µgV e −5 = 2 10 . (C11) Γ 2πα(1 x ) . mediators as in Section C 2 and assume that mTM = mX ; V xV 1 × otherwise, we have to take into account mixing between6 − the states. We note that the SM rate of Z∗ νν¯ Note that we can apply this to a dark photon by simply is completely negligible. TM → → choosing gV e = gV µ = ε√4πα, where ε is the kinetic Let us assume for concreteness that χ is a Dirac mixing of the dark photon. . The coupling to χ has the same parity structure as to SM , e.g. we assume that a scalar couples to References χχ¯ , a pseudoscalar toχγ ¯ 5χ, etc. Because the state TM [1] M. Deutsch, “Evidence for the Formation of we are considering is a vector, the only contribution is Positronium in Gases,” Phys. Rev. 82 (1951) 455–456. via decay through a vector state. Then, we have [2] V. W. Hughes, D. W. McColm, K. Ziock, and R. Prepost, “Formation of Muonium and Observation of 2 2 its Larmor Precession,” Phys. Rev. Lett. 5 (1960) 63–65. ∗ gV µgV χ [3] V. W. Hughes and B. Maglic, “True muonium,” Bull. BR( V χχ¯ ) = 2 2 2 TM → → 16π α (1 xV ) Am. Phys. Soc. 16 (1971) 65. − p [4] LHCb Collaboration, “Framework TDR for the LHCb (1 + 2xχ) 1 4xχ . (C9) × − Upgrade: Technical Design Report,” Tech. Rep. CERN-LHCC-2012-007/LHCb-TDR-12, Apr, 2012. If we consider mV < mTM such that there is no sup- https://cds.cern.ch/record/1443882. pression of the V propagator and mχ mTM, we ob-  [5] LHCb Collaboration, “LHCb VELO Upgrade tain constraints on the coupling gV µ from (g 2)µ. The Technical Design Report,” Tech. Rep. − coupling to νµ leads to constraints on neutrino trident CERN-LHCC-2013-021. LHCB-TDR-013, Nov, 2013. rates, so for a vector coupling these also constrain gV µ . https://cds.cern.ch/record/1624070. 10

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