Inclusive Dark Photon Search at LHCb
We propose an inclusive search for dark photons at the LHCb experiment based on both prompt and displaced di-muon resonances. Because the couplings of the dark photon are inherited from the photon via kinetic mixing, the dark photon rate can be directly inferred from the off-shell photon rate, making this a fully data-driven search. For Run 3 of the LHC, we estimate that LHCb will have sensitivity to large regions of the unexplored dark-photon parameter space, especially in the – and – mass ranges. This search leverages the excellent invariant-mass and vertex resolution of LHCb, along with its unique particle-identification and real-time data-analysis capabilities.
Dark matter—firmly established through its interactions with gravity—remains an enigma. Though there are increasingly stringent constraints on direct couplings between visible matter and dark matter, little is known about the dynamics within the dark sector itself. An intriguing possibility is that dark matter might interact via a new dark force, felt only feebly by standard model (SM) particles. This has motivated a worldwide effort to search for dark forces and other portals between the visible and dark sectors (see LABEL:Essig:2013lka for a review).
A particularly compelling dark-force scenario is that of a dark photon which has small SM couplings via kinetic mixing with the ordinary photon through the operator Okun (1982); Galison and Manohar (1984); Holdom (1986); Pospelov et al. (2008); Arkani-Hamed et al. (2009); Bjorken et al. (2009). Previous beam dump Bergsma et al. (1986); Konaka et al. (1986); Riordan et al. (1987); Bjorken et al. (1988); Bross et al. (1991); Davier and Nguyen Ngoc (1989); Athanassopoulos et al. (1998); Astier et al. (2001); Adler et al. (2004); Bjorken et al. (2009); Artamonov et al. (2009); Essig et al. (2010); Blumlein and Brunner (2011); Gninenko (2012a); BlÃ¼mlein and Brunner (2014), fixed target Abrahamyan et al. (2011); Merkel et al. (2014, 2011), collider Aubert et al. (2009); Curtin et al. (2014); Lees et al. (2014), and rare meson decay Bernardi et al. (1986); Meijer Drees et al. (1992); Archilli et al. (2012); Gninenko (2012b); Babusci et al. (2013); Adlarson et al. (2013); Agakishiev et al. (2014); Adare et al. (2015); Batley et al. (2015); Anastasi et al. (2016) experiments have already played a crucial role in constraining the dark photon mass and kinetic-mixing strength . Large regions of the – plane, however, are still unexplored (see Fig. 1). Looking to the future, a wide variety of innovative experiments have been proposed to further probe the dark photon parameter space Essig et al. (2011); Freytsis et al. (2010a); Balewski et al. (2013); Wojtsekhowski et al. (2012); Beranek et al. (2013); Echenard et al. (2015); Battaglieri et al. (2015); Curtin et al. (2015); Alekhin et al. (2015); Gardner et al. (2015); Ilten et al. (2015), though new ideas are needed to test and .
In this Letter, we propose a search for dark photons via the decay
at the LHCb experiment during LHC Run 3 (scheduled for 2021–2023). The potential of LHCb to discover dark photons was recently emphasized in LABEL:Ilten:2015hya, which exploits the exclusive charm decay mode with . Here, we consider an inclusive approach where the production mode of need not be specified. An important feature of this search is that it can be made fully data driven, since the signal rate can be inferred from measurements of the SM prompt spectrum. The excellent invariant-mass and vertex resolution of the LHCb detector, along with its unique particle-identification and real-time data-analysis capabilities Benson et al. (2015); Aaij et al. (2016), make it highly sensitive to . We derive the LHCb sensitivity for both prompt and displaced decays, and show that LHCb can probe otherwise inaccessible regions of the – plane.
The is a hypothetical massive spin-1 particle that, after electroweak symmetry breaking and diagonalizing the gauge kinetic terms, has a suppressed coupling to the electromagnetic (EM) current Okun (1982); Galison and Manohar (1984); Holdom (1986); Pospelov et al. (2008); Arkani-Hamed et al. (2009); Bjorken et al. (2009):
There is also a model-dependent coupling to the weak current (see e.g. LABEL:Barello:2015bhq), which appears at . We provide nearly model-independent sensitivity estimates for the mass range by ignoring the coupling to the . We include model-dependent -mixing effects for , adopting the parameters of Cassel et al. (2010); Cline et al. (2014).
The partial widths of to SM leptons are
where and . Because the couples to , the branching fraction of to SM hadrons can be extracted from the measured value of (taken from LABEL:Agashe:2014kda)
In particular, (4) already includes the effect of the mixing with the QCD vector mesons , , , etc. It is also possible for the to couple to non-SM particles with an invisible decay width , in which case the total width is
Below, we consider , though our analysis can be easily adapted to handle non-vanishing invisible decay modes.
To estimate the signal rate, we follow the strategy outlined in LABEL:Bjorken:2009mm. Consider the signal production process in proton-proton () collisions
where is any (multiparticle) final state. Ignoring and corrections, this process has the identical cross section to the prompt SM process which originates from the EM current
up to differences between the and propagators and the kinetic-mixing suppression. Interference between and is negligible for a narrow resonance. Therefore, for any selection criteria on , , and , the ratio between the differential cross sections is
where is the di-muon invariant mass, for the case . The factor arises because both the production and decay rates scale like .
To obtain a signal event count, we integrate over an invariant-mass range of , where is the detector resolution on . The ratio of signal events to prompt EM background events is
neglecting phase space factors for leptons lighter than . This expression already accounts for the branching-fraction suppression when is large. Despite the factor of in (8), the ratio in (9) is proportional to because of the scaling of .
We emphasize that (9) holds for any final state (and any kinematic selection) in the limit for tree-level single-photon processes. In particular, it already includes production from QCD vector mesons that mix with the photon. This allows us to perform a fully data-driven analysis, since the efficiency and acceptance for the (measured) prompt SM process is the same as for the (inferred) signal process, excluding lifetime-based effects. The dominant component of at small comes from meson decays , especially , and is denoted as (which includes feed-down contributions from heavier meson decays). There are also two other important components: final state radiation (FSR) and Drell-Yan (DY) production. Non-prompt production is small and only considered as a background.
Beyond , there are other important sources of backgrounds that contribute to the reconstructed prompt di-muon sample, ordered by their relative size:
: A fake di-muon pair can also arise from one real muon (primarily from charm or beauty decays) combined with one misID pion or kaon. This background can be subtracted similarly to .
: The Bethe-Heitler (BH) background played an important role in the analysis of LABEL:Bjorken:2009mm. This is a subdominant process at the LHC due in part to the small effective photon luminosity function. We verified that is small using a parton shower generator (see below), and it will be neglected in estimating the reach.
True displaced di-muon pairs, which arise from beauty decays, are rarely reconstructed as prompt at LHCb. Such backgrounds, however, are dominant in the displaced search discussed below.
Summarizing, the reconstructed prompt di-muon sample contains the following background components:
where for simplicity we ignore interference terms between the various components. After subtracting from Abelev et al. (2012); SUP (), we can use (9) to infer from for any and . Since both and are extracted from data, this strategy is fully data driven.
We now present an inclusive search strategy for dark photons at LHCb. The LHCb experiment will upgrade to a triggerless detector-readout system for Run 3 of the LHC LHC (2014), making it highly efficient at selecting decays in real time. Therefore, we focus on Run 3 and assume an integrated luminosity of (see LABEL:Ilten:2015hya)
The trigger system currently employed by LHCb is efficient for many decays included in our search. We estimate that the sensitivity in Run 2 will be equivalent to using about 10% of the data collected in Run 3. Therefore, inclusion of Run 2 data will not greatly impact the reach by the end of Run 3, though a Run 2 analysis could explore much of the same parameter space in the next few years.
The LHCb detector is a forward spectrometer covering the pseudorapidity range Alves et al. (2008); Aaij et al. (2015a). Within this acceptance, muons with three-momentum are reconstructed with near 100% efficiency with a momentum resolution of and a di-muon invariant mass resolution of Aaij et al. (2015a, 2013)
For the displaced search, the vertex resolution of LHCb depends on the Lorentz boost factor of the ; we therefore use an event-by-event selection criteria in the analysis below. That said, it is a reasonable approximation to use a fixed proper-lifetime resolution Aaij et al. (2015a)
except near the di-muon threshold where the opening angle between the muons is small.
To suppress fake muons, our strategy requires muon candidates have (transverse) momenta () , and are selected by a neural-network muon-identification algorithm Aaij et al. (2015b) with a muon efficiency of and a pion fake rate of SUP (). To a good approximation, the neural-network performance is independent of the kinematics. Such a low pion misID rate is a unique feature of LHCb and is vital for probing the low- region in decays.
To further suppress for , we require muons to satisfy an isolation criterion based on clustering the charged component of the final state with the anti- jet algorithm Cacciari et al. (2008) with in FastJet 3.1.2 Cacciari et al. (2012); muons with are rejected, excluding the contribution to from the other muon if it is contained in the same jet. By considering charged particles only, this isolation strategy is robust to pileup. The di-muon isolation efficiencies obtained from simulated LHCb data (see below) are 50% for FSR, DY, and BH, 25% for meson decays (dominantly from charmonium states), and 1% for fake pions ( and have similar efficiencies).
The baseline selection for the LHCb inclusive search is therefore:
two opposite-sign muons with , , and ;
a reconstructed candidate with , , and passing the isolation criterion for ;
Following a similar strategy to LABEL:Ilten:2015hya, we use the reconstructed muon impact parameter (IP) and transverse flight distance to define three non-overlapping search regions:
Displaced (pre-module): ;
Displaced (post-module): .
The resolution on IP and are taken from LHC (2013); SUP (). The displaced search is restricted to to ensure at least three hits per track in the vertex locator (VELO). We define two search regions based on the average to the first VELO module (i.e. 6 mm), where each VELO module is a planar silicon-pixel detector oriented perpendicular to the LHC beamline.
To estimate the reach for this search using the data-driven strategy in (9), we need to know with the above selection criteria applied. To our knowledge, LHCb has not published such a spectrum, so we use Pythia 8.212 SjÃ¶strand et al. (2015) for illustrative purposes to understand the various components of .111We caution the reader that the di-muon spectra published by ATLAS Aad et al. (2014) and CMS Chatrchyan et al. (2012) do not impose prompt selection criteria nor do they subtract fake di-muons. LHCb has published measurements of meson Aaij et al. (2011a), charmonium Aaij et al. (2011b), bottomonium Aaij et al. (2012), and DY LHCb (2012) production in 7 collisions, and we find that Pythia accurately reproduces these measurements. Therefore, we assume that Pythia also adequately predicts their production at 14. The ALICE collaboration has published the low-mass di-muon spectrum at in a similar kinematic region as proposed for this search Abelev et al. (2012). Within the kinematic region used by ALICE, we find that Pythia accurately describes the production of the mesons, but overestimates and production by factor of two; we therefore reduce the Pythia prediction for these mesons to match the observed ALICE spectrum.
Including our selection criteria and modifications, the prompt di-muon spectrum from Pythia is shown in Fig. 2. The background is dominated by meson decays like at low invariant mass, and transitions to DY production at larger , with FSR being subdominant throughout. Note the sharp change in the spectrum at due to the muon-isolation requirement. We also show in Fig. 2 the expected non-EM background contamination from and . The misidentification background is large and dominates for , though this is also the region where Pythia likely underestimates di-muon production from excited meson decays (e.g. ) SUP ().
We also use Pythia to understand the backgrounds for the displaced searches, where the dominant contribution comes from double semi-leptonic heavy-flavor decays of the form followed by . Such decays are highly suppressed by our consistent-decay-topology requirements SUP (), but they still contribute at a large rate because of the copious heavy-flavor production in high-energy collisions. Semi-leptonic decays of charm and beauty mesons, where one real muon and one fake muon arise from the same secondary vertex, also contribute but at a much lower rate. Decays of heavy-flavor hadrons with two misID pions or with are similarly subdominant.
For the pre-module displaced region, we find background events per mass bin. For the post-module displaced region, relevant for long-lived dark photons with , we estimate the background to be candidates per mass bin by scaling the observed combinatorial background in a published LHCb search Aaij et al. (2015b) by the increase in luminosity used in this analysis. In the post-module region, the heavy-flavor background is on the order of few events per bin, and the dominant contribution is from interactions with the detector material. This contribution can likely be reduced following a strategy similar to LABEL:Ilten:2015hya.
The estimated sensitivity of LHCb to inclusive production is shown in Fig. 1. For the prompt search, we estimate from using data in the neighboring sidebands and take as a rough criterion for the exclusion limit. This sideband method fails near narrow QCD resonances, which would need a dedicated analysis. Figure 1 shows that for one can probe down to – with the prompt search, improving on current limits. The reach is limited at higher masses due to , where the expected sensitivity is comparable to the present bound. Going to higher masses where the production rate depends on model-dependent mixing with the , LHCb can extend anticipated ATLAS and CMS limits Curtin et al. (2015) for .
For the displaced search, the spectrum of Lorentz boost factors can be inferred from the prompt spectrum observed in data in a given bin; the lifetime acceptance can then be obtained from simulation. Following the background discussion above, the exclusion criterion for the pre-module (post-module) search is (), yielding the regions shown in Fig. 1. A comparable reach is obtained by simply assuming the fixed proper-lifetime resolution in (13). Because of the large rate, the displaced search has the potential to probe with , a region that is challenging to access through other experiments.
There are a number of possible improvements and generalizations to this search. For example, dark photons can be searched for during LHC Run 2, by adapting our analysis to include di-muon hardware trigger requirements. Because the search is entirely data driven, di-muon triggers need not be fully efficient to be useful in such an analysis. The real-time analysis, event-selection, and multi-search-region Williams (2015) strategies employed by LHCb could be improved, and data collected in LHC Runs 4 and 5 would greatly improve the sensitivity SUP (). One could also pursue a semi-inclusive strategy, where an candidate is selected along with another required object; for most semi-inclusive modes, one can still use the data-driven method in (9). If the fake muon backgrounds could be controlled, a similar search could be performed at ATLAS and CMS. Beyond dark photons, these searches are sensitive to spin-0 di-muon resonances (see related work in Freytsis et al. (2010b); Aaij et al. (2015c); Haisch and Kamenik (2016)). An inclusive search in the electron channel could explore the mass region, though this is considerably more challenging due to Bremsstrahlung radiation and multiple scattering SUP ().
In summary, we proposed an inclusive search strategy for dark photons at the LHCb experiment using di-muon resonances. Since the coupling of the to the standard model is dictated by the kinetic-mixing parameter , the signal rate can be directly inferred from the off-shell photon rate, enabling a data-driven search. Through a combination of prompt and displaced searches, LHCb is sensitive to interesting regions in the – parameter space, some of which are difficult to probe with other proposed experiments. This search leverages the excellent invariant-mass and vertex resolution of LHCb, along with its unique particle-identification and real-time data-analysis capabilities. Provided that the appropriate real-time selections are employed starting this year, LHCb could probe much of this parameter space using data collected in Run 2 of the LHC. Given the simplicity of this proposed search strategy, it could easily be adapted to other experiments at the LHC and beyond.
Acknowledgements.We thank R. Essig, T. Gershon, J. Kamenik, Z. Ligeti, and V. Vagnoni for helpful feedback. Y.S., J.T., and W.X. are supported by the U.S. Department of Energy (DOE) under cooperative research agreement DE-SC-00012567. J.T. is also supported by the DOE Early Career research program DE-SC-0006389, and by a Sloan Research Fellowship from the Alfred P. Sloan Foundation. M.W. and P.I. are supported by the U.S. National Science Foundation grant PHY-1306550.
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Here, we provide a more detailed discussion on the following aspects of our proposed search at LHCb: how to subtract the misidentified di-muon background; how to estimate the LHCb muon-identification performance from LABEL:Aaij:2014azz; what the consistent-decay-topology requirements are in our analysis; how the sensitivity of LHCb to could be improved; and the reach plotted with an extended range.
.1 Fake Di-Muon Subtraction
Most fake muons come from misidentified pions, with a subdominant contribution from misidentified kaons and protons. For simplicity, we denote all misID particles as pions below, though the argument presented is completely general. The following logic will allow us to use data-driven methods to subtract the fake di-muon background.
We first consider the double-misID case where two pions are each misidentified as muons. The number of same-sign (SS) pairs from a collision is related to the number of pions that satisfy our kinematic requirements by
while the number of opposite-sign (OS) pairs is
In the limit, and assuming equal acceptance for SS and OS pairs with the same invariant mass, we obtain the simple relationship
where the right-most relationship assumes , which is a good approximation in collisions. Therefore, the total number of SS pion pairs in a data sample is
where and the sum is over all collisions in the sample.
Next, we consider the single-misID case where one true muon is combined with a misidentified prompt pion. This true muon dominantly comes from a displaced heavy-flavor decay which is mis-reconstructed as prompt. In this case, the combinatorics only enter for the pion, and in the full data sample we find
Combining the double- and single-misID cases together, one expects
where the lack of superscripts on denotes that we do not need to separate these into and categories experimentally. This simple estimate, based on taking the asymptotic limit and assuming charge-symmetric pion samples, could easily be improved in an actual analysis, since the true combinatorics can be determined from the data. The small correction required to account for the difference in acceptance between SS and OS pairs can be obtained reliably from simulation. We expect that (19) is accurate to and that a highly-accurate misID subtraction can be performed using the data.
Finally, we note that an alternative approach is also possible using OS samples directly with the pion misID rate measured in data, along with OS samples where the muon is displaced. The actual analysis could use both methods and check their consistency to assess the systematic uncertainty in the fake-muon background subtraction.
.2 Muon Identification
For small , most decays produce low- muons. Since high-energy collisions produce many low- pions, there are many possible pairs per collision that could result in a double misID of as . Furthermore, the decay-in-flight probability of is inversely proportional to momentum. Therefore, the low-mass signal is obscured by the enormous double-misID background if muon identification is based soley on whether the particle is a muon when it reaches the muon system. Our baseline selection requires , , and . By convolving the pion momentum spectrum obtained from Pythia with the decay-in-flight probability given by the pion lifetime, we obtain an estimate that of all pions satisfying these kinematic requirements will be identified as muons by the muon system. This results in in the low- region.
One way to reduce the double-misID background is to increase the muon threshold. At low-, however, the signal is predominantly produced via , so increasing the muon threshold greatly reduces the potential signal yield. For example, increasing the muon threshold from 0.5, as in our nominal proposed search, to 2 reduces the low- yield by a factor of . That said, such an approach may prove viable at ATLAS and CMS as they plan to collect 200 times more luminosity by the end of the HL-LHC era than LHCb will collect in Run 3, making it plausible that the low- reach estimate in this letter could be representative of the ultimate ATLAS/CMS sensitivity.
Instead of raising the muon threshold, here we take advantage of the unique particle-identification features of LHCb. The LHCb detector employs two ring-imaging Cherenkov (RICH) detectors to identify charged particles with momenta from . The primary motivation for incorporating such detector systems into LHCb was to provide hadronic particle-identification capabilities to facilitate studying Cabbibo-suppressed weak decays. For our purposes, these systems are also very powerful tools for lepton identification. For , the RICH detectors are capable of identifying electrons and muons without the need for additional information from the calorimeter or muon systems. By combining the information of the RICH detectors with all other LHCb subsystems into a neural network (NN), LHCb is able to greatly reduce the pion (and kaon) misID probabilities Aaij et al. (2015b).
To our knowledge, the only published performance of the LHCb muon-identification NN is from a search for the decay Aaij et al. (2015b). The muon kinematic requirements in that analysis are similar to ours, and so we estimate the NN performance directly from LABEL:Aaij:2014azz. Specifically, we assume the 2012 performance and a requirement that removes the lowest two bins in NN response (see Fig. 2d of LABEL:Aaij:2014azz). The efficiency to identify a true di-muon pair is taken to be , which we reduce to 50% to account for other selection criteria applied to candidates. Since the selection only used displaced tracks, the background sample dominantly contains at least one true muon. Assuming that all background candidates are gives ; this value includes the probability of decay-in-flight of per pion. Since the background likely contains a non-negligible fraction of candidates, this is an underestimate of the single-pion rejection from the NN. Therefore, we obtain a conservative estimate of the reach in regions where is important. Finally, we note that using a simple likelihood-based approach, as in LABEL:LHCb-DP-2012-003, LHCb obtains a per-pion misID rate that is only a factor of two higher for the same muon efficiency than the NN value used in our analysis.
.3 Decay-Topology Criteria
One of the key ingredients in our analysis is to enforce a consistent decay topology. These requirements are the same as in LABEL:Ilten:2015hya, but with electrons replaced by muons; we repeat them here for the convenience of the reader. We also apply one additional criterion here that is not used in LABEL:Ilten:2015hya: displaced di-muon vertices are rejected if a third displaced particle within LHCb acceptance and satisfying has a distance of closest approach (DOCA) less than 1 mm relative to either muon; this reduces the heavy-flavor background.
The one-dimensional track IP resolution expected in Run 3 is well approximated by LHC (2013)
while the resolution on is
where is the decay opening angle and the is constrained to originate from the collision. As stated in the main text, we require in the prompt search.
For the displaced searches, we require the reconstructed candidate to satisfy the following consistency requirements:
the decay vertex is downstream of the collision;
the DOCA between the two muon tracks is consistent with zero;
the angle between and the spatial vector formed from the collision to the decay vertex is consistent with zero;
the IP out of the decay plane for each muon is consistent with zero, where the decay plane is defined by the collision point and the first hits on the and tracks.
In each case, we define “consistent with zero” as having a -value greater than 1%. Therefore, the efficiency on a true displaced decay is close to 100%. We also apply the DOCA requirement in the prompt search.
.4 Possible Improvements
The following improvements could result in an increased sensitivity at LHCb to decays:
Excited mesons: When estimating , Pythia does not produce known heavy mesons (e.g. ) through Lund string fragmentation (though they can be produced through heavy-flavor decays). These excited mesons have been observed to decay to di-electrons, so one would assume that they also decay to di-muons with the same rate up to phase space effects. The fact that such mesons are not included in our study likely means that we underestimate the reach for . In fact, it is plausible that such mesons provide the dominant source of potential production in this mass region. If so, then one would likely want to shift the isolation criteria to apply only for (instead of as in the text).
Event selection: In this study, we used a simple and robust cut-based selection strategy. A multivariate approach would likely perform better, especially in the displaced searches where correlations between various features used in the consistent-decay-topology requirements could be exploited. Furthermore, the and requirements discard about 70% of the signal decays that could be fully reconstructed in LHCb. If the low- misID background can be suppressed in the real-time data-analysis system, then the yield could be increased by up to a factor of 3 for the same luminosity.
Search strategy: Here, we considered the reach assuming three distinct search regions: prompt, pre-module, post-module. One could optimally combine these regions following LABEL:Williams:2015xfa which should improve the reach in the low-mass region.
Semi-inclusive search: Instead of using the inclusive di-muon spectrum, a similar search could be done in semi-inclusive hadron decays such as , more in the spirit of LABEL:Ilten:2015hya. Depending on the channel, one could use the invariant mass of the or system as a constraint to help control fake muon backgrounds.
Di-electron search. To cover the mass range , one could pursue a similar inclusive search strategy for the di-electron final state. That said, the di-electron mass resolution is significantly degraded by Bremsstrahlung radiation and multiple scattering Ilten et al. (2015). In LABEL:Ilten:2015hya, the resolution could be improved by imposing the kinematic constraints from charm meson decays, which is not an option in an inclusive search. For the displaced search, these same effects degrade the vertex resolution, and pairs from photon conversion are a challenging background in the post-module region. For these reasons, we suspect that is best probed using an exclusive (or semi-inclusive) strategy, but it would be worth testing the fully inclusive approach on LHCb data.
Luminosity: Our study is based on 15 fb of data collected by LHCb, which is a conservative estimate of what is expected in Run 3. LHCb expects to collect at least 50 fb of data in Runs 3 and 4 combined, and may eventually collect 10–30 times more data than considered in this study. The impact on the dark photon reach from scaling up the LHCb luminosity is shown in Fig. 4.