Dark photons from charm mesons at LHCb

Dark photons from charm mesons at LHCb

Philip Ilten philten@cern.ch Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.    Jesse Thaler jthaler@mit.edu Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.    Mike Williams mwill@mit.edu Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.    Wei Xue weixue@mit.edu Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.
Abstract

We propose a search for dark photons at the LHCb experiment using the charm meson decay . At nominal luminosity, decays will be produced at about 700 kHz within the LHCb acceptance, yielding over 5 trillion such decays during Run 3 of the LHC. Replacing the photon with a kinetically-mixed dark photon, LHCb is then sensitive to dark photons that decay as . We pursue two search strategies in this paper. The displaced strategy takes advantage of the large Lorentz boost of the dark photon and the excellent vertex resolution of LHCb, yielding a nearly background-free search when the decay vertex is significantly displaced from the proton-proton primary vertex. The resonant strategy takes advantage of the large event rate for and the excellent invariant mass resolution of LHCb, yielding a background-limited search that nevertheless covers a significant portion of the parameter space. Both search strategies rely on the planned upgrade to a triggerless-readout system at LHCb in Run 3, which will permit identification of low-momentum electron-positron pairs online during data taking. For dark photon masses below about 100 MeV, LHCb can explore nearly all of the dark photon parameter space between existing prompt- and beam-dump limits.

preprint: MIT-CTP 4702

I Introduction

Rare decays of mesons are a powerful probe of physics beyond the standard model (SM). Precise measurements of branching fractions and decay kinematics indirectly constrain extensions of the SM by bounding symmetry-violating or higher-dimensional operators. More directly, non-SM particles could be produced in meson decays when kinematically allowed, and depending on their lifetimes, these particles could yield striking signals with displaced vertices. A well-motivated hypothetical particle is the dark photon which inherits a small coupling to the SM via kinetic mixing with the ordinary photon  Okun (1982); Galison and Manohar (1984); Holdom (1986); Pospelov et al. (2008); Arkani-Hamed et al. (2009); Bjorken et al. (2009). Indeed, some of the most stringent constraints on the properties of dark photons come from rare decays of mesons, including  Bernardi et al. (1986); Meijer Drees et al. (1992); Astier et al. (2001); Gninenko (2012a); Adlarson et al. (2013); Agakishiev et al. (2014); Adare et al. (2015); Batley et al. (2015),  Bergsma et al. (1986); Gninenko (2012b), and  Archilli et al. (2012); Babusci et al. (2013).

The minimal dark photon scenario involves a single broken gauge symmetry, along with mixing between the and SM hypercharge fields via the operator . After electroweak symmetry breaking and diagonalizing the gauge boson kinetic terms, the dark photon gains a suppressed coupling to the ordinary electromagnetic current , where the relevant terms in the Lagrangian are

(1)

This minimal scenario has two free parameters: the dark photon mass and the kinetic-mixing parameter (often reported in terms of ). The constraints placed on dark photons in the plane are shown in Fig.1 for , assuming that the dominantly decays into visible SM states (see LABEL:Essig:2013lka for a review).111There are also interesting searches where the dark photon decays invisibly to dark matter Aubert et al. (2008); Batell et al. (2009); deNiverville et al. (2011); Wojtsekhowski et al. (2012); Kahn and Thaler (2012); Izaguirre et al. (2013); Batell et al. (2014); Kahn et al. (2015); Izaguirre et al. (2015a, b). For , the most stringent bounds come from searches for prompt decays at collider and fixed-target experiments Batley et al. (2015); Merkel et al. (2011, 2014); Abrahamyan et al. (2011). As decreases, the lifetime increases, while as decreases, the lifetime and Lorentz boost factor both increase. Therefore, the constraints obtained from beam-dump experiments exclude wedge-shaped regions in the plane 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); Adler et al. (2004); Bjorken et al. (2009); Artamonov et al. (2009); Essig et al. (2010); Blumlein and Brunner (2011); Gninenko (2012b); Blumlein and Brunner (2014). Also shown in Fig.1 are electron bounds Hanneke et al. (2008); Giudice et al. (2012); Endo et al. (2012)222Since we follow the analysis in LABEL:Essig:2013lka, we obtain more conservative bounds from than shown in LABEL:Endo:2012hp., the preferred region to explain the muon anomaly Pospelov (2009), and supernova bounds from cooling Dent et al. (2012) and emissions Kazanas et al. (2014). Anticipated limits from other planned experiments are shown later in Fig.9.

Figure 1: Current bounds on dark photons with visible decays to SM states, adapted and updated from LABEL:Essig:2013lka. The upper bounds are from prompt- searches while the wedge-shaped bounds are from beam-dump searches and supernova considerations. The LHCb search region in Fig.2 covers most of the gap between these bounds for , with a reach extending to . Anticipated limits from other planned experiments are shown in Fig.9.
Figure 2: Potential bounds from LHCb after Run 3, for both the displaced (pre-module, solid blue) and resonant (dashed blue) searches. Also shown is an alternative displaced search strategy (post-module, dotted blue) that looks for vertices downstream of the first tracking module.

In this paper, we propose a search for dark photons through the rare charm meson decay

(2)

at the LHCb experiment during Run 3 of the LHC (scheduled for 2021–23).333Throughout this paper, and the inclusion of charge-conjugate processes is implied. The goal of this search is to explore the region between the prompt- and beam-dump limits for the range , which roughly includes . Reaching such small values of is only possible for decays where the yield of the corresponding SM process (i.e. replacing with ) is at least . Within the LHCb acceptance, over five trillion decays will be produced in proton-proton () collisions at 14 during Run 3, making this decay channel a suitable choice.

The range of values that is in principle accessible in this search is , where Olive et al. (2014)

(3)

The proximity of to leads to phase-space suppression of the decay , which results in a sizable branching fraction of about 38% for the decay .444This explains why we choose the decay instead of , since the corresponding branching fraction is only 1.6%. The small value of , however, also leads to typical electron momenta of within the LHCb acceptance. Therefore, the planned upgrade to a triggerless-readout system employing real-time calibration at LHCb in Run 3 LHC (2014a)—which will permit identification of relatively low-momentum pairs online during data taking—will be crucial for carrying out this search.

To cover the desired dark photon parameter space, we employ two different search strategies, shown in Fig.2. The displaced search, relevant at smaller values of , looks for an decay vertex that is significantly displaced from the collision. This search benefits from the sizable Lorentz boost factor of the produced dark photons and the excellent vertex resolution of LHCb. Our main displaced search looks for decays within the beam vacuum upstream of the first tracking module (i.e. pre-module), where the dominant background comes from misreconstructed prompt events.555We thank Natalia Toro for extensive discussions regarding this background. Because the gains a transverse momentum kick from collisions, the flight trajectory intersects the LHCb detector, making it possible to identify displaced pairs with smaller opening angles than the HPS experiment Moreno (2013). We also present an alternative displaced search for decays downstream of the first tracking module (i.e. post-module), where the dominant background comes from events with conversion within the LHCb material.

The resonant search, relevant at larger values of , looks for an resonance peak over the continuum SM background. This search benefits from the large yield of decays during LHC Run 3, which is larger than the yield in fixed-target experiments like MAMI/A1 Merkel et al. (2011, 2014) and APEX Abrahamyan et al. (2011). Furthermore, the narrow width of the meson, which is less than the detector invariant-mass resolution, provides kinematical constraints that can be used to improve the resolution on . This resonant search can also be employed for non-minimal dark photon scenarios where the might also decay invisibly into dark matter, shortening the lifetime. In that case, the anticipated limits in Fig.2 would roughly apply to the combination .

The remainder of this paper is organized as follows. In Sec.II, we estimate the signal and SM background cross sections, extracting the production rate and decay modes from an event generator and estimating the decay rates using a simple operator analysis. In Sec.III, we describe the LHCb detector and charged-particle tracking, provide the selection requirements applied to and meson candidates, and derive the mass resolution. We present the pre-module displaced search in Sec.IV, a post-module variant in Sec.V, and the resonant search in Sec.VI. Possible improvements are outlined in Sec.VII and a comparison to other experiments (especially HPS) is given in Sec.VIII. We summarize in Sec.IX and discuss how the LHCb dark photon search strategy might be extended above the threshold.

Ii Signal and Background Rates

Dark photon production in meson decays proceeds mainly via

(4)

though the low-energy photon in the latter decay is unlikely to be detected at LHCb. Here and throughout, we use the notation to mean in a subsequent decay. Because , is the only relevant visible decay channel.

The dominant backgrounds to the pre-module displaced search (Sec.IV) are and , where the pair is misreconstructed as being displaced due to a hard electron scatter in material. These backgrounds can be highly suppressed by requiring that the kinematics are consistent with a displaced vertex occurring in the proper decay plane. The dominant background to the post-module displaced search (Sec.V) is , where the converts into an pair during interactions with the detector material. This background can be highly suppressed by requiring that the vertex position is not consistent with the location of any detector material. The dominant backgrounds to the resonant search (Sec.VI) are again and , where the has been replaced by an off-shell . The first background is irreducible, making the resolution on the driving factor in the resonant search reach.

ii.1 Meson Production

We simulate production in collisions at a center-of-mass energy of 14 using Pythia 8.201 T. Sjöstrand et al. (2015) with the default settings. Since a large fraction of charm quarks are produced from gluon splitting and since we need to model forward physics at small transverse momentum , we run all soft QCD processes in Pythia (i.e. SoftQCD:all = on). While the production cross section is not yet known at 14, the result obtained using Pythia for the inclusive production cross section at 7 agrees with the measured value by LHCb Aaij et al. (2013a) to within about 5%.666During the final preparation of this article, LHCb presented the first prompt charm cross section measurement at 13 Aaij et al. (). Based on this result, we estimate that the relevant cross section for determining the dark photon reach should be about 20% higher than the one used in this paper. Since Pythia does not record the spin of the mesons, they are treated as unpolarized in this analysis.

To define the fiducial region, we require the meson to satisfy the following transverse momentum and pseudorapidity requirements:

(5)

Note that this requirement is placed on the meson, not on the , to suppress backgrounds to the component of the signal. The production cross section within this fiducial region is

(6)

excluding secondary production of mesons from -hadron decays. It may be possible to make use of some secondary decays; in this analysis, however, we require that the originates from the collision to suppress backgrounds (see Sec.IV.1).

The nominal instantaneous luminosity expected at LHCb during Run 3 is 2nb per second LHC (2014a), which will produce mesons at a rate of almost 2MHz (equivalently, at 0.7MHz). Assuming an integrated luminosity of 15 in Run 3,777The length of Run 3 is scheduled to be about the same as Run 1. LHCb collected a total of 3 fb in Run 1. The instantaneous luminosity will be five times higher in Run 3. Therefore, assuming the LHC performance is the same (including the slow ramp up), this gives an estimate of 15 fb in Run 3. this results in an estimated yield of 14 trillion mesons produced within this fiducial region, or

(7)

which we use as the baseline for our estimated reach.

ii.2 Meson Decays

The meson is an state with a mass of and a width less than . It decays promptly mainly into two final states with branching fractions of

(8)
(9)

where the meson is a state Olive et al. (2014). As mentioned above, is the dominant background to the pre-module displaced search as well as to the resonant search. To our knowledge, this branching fraction has not yet been measured; therefore, we will estimate the rate for this decay using an operator analysis. This same approach is used to determine the rate.

To calculate these transition amplitudes, we must first determine the matrix element. By parity, time reversal, and Lorentz invariance, this transition dipole matrix element can be written in the form

(10)

where is the four velocity of the meson, is the momentum flowing out of the current, and is the polarization of the meson. Here, is a -dependent effective dipole moment, whose value could be determined using a simple quark model (see, e.g., LABEL:Miller:1988tz) or using a more sophisticated treatment with heavy meson chiral perturbation theory (see, e.g., LABEL:Stewart:1998ke). For our purposes, we simply need to treat as being roughly constant over the range , which is a reasonable approximation given that . (Indeed, this relation is always satisfied in the heavy charm quark limit, where .) The precise value of is irrelevant for our analysis since it cancels out when taking ratios of partial widths.

Using Eq.(10), we estimate the decay rate for within the SM and in the limit to be

(11)

where . To calculate the decay rate, the off-shell photon propagator must be included. In the limit, the amplitude for this process is

(12)

where and are the electron and positron momenta. The ratio of partial widths is determined numerically to be

(13)

Since the dark photon also couples to , we use Eq.(10) to calculate the decay rate. The ratio of partial widths is

(14)

where we assume . This expression has the expected kinetic-mixing and phase-space suppressions. Since the meson is treated as unpolarized in Pythia, we ignore spin correlations in the subsequent decay.888As a technical note, to generate events, we reweight a sample of events from Pythia. In particular, we implement in the meson rest frame, boost to match the kinematics from Pythia, and then boost the decay products to account for the altered momentum. A similar strategy is employed for generating all other decays in our study.

ii.3 Rare Decays

To determine the decay rate in Eq.(4), we start by estimating the rate of the decay using the SM effective Lagrangian

(15)

where is the pion decay constant and the pion form factor is ignored. The dark photon is accounted for by making the replacement

(16)

which leads to the ratio of partial widths

(17)

The same effective Lagrangian can also be used for the SM decay . The amplitude is

(18)

The ratio of partial widths is obtained numerically to be

(19)

which agrees with the nominal value for this ratio Olive et al. (2014).

ii.4 Dark Photon Decays

Figure 3: Dark photon Lorentz boost factors for . These factors are independent of .

Figure 4: Flight distance distributions for showing (left) and (right) .

Assuming the only allowed decay mode is , the total width of the is

(20)

In the lab frame, the mean flight distance of the dark photon is approximately

(21)

where is the Lorentz boost factor. In Fig.3 we show some example spectra of boost factors from simulated decays, where both electrons are required to satisfy and so that they can be reconstructed by LHCb (see Sec.III.1 below). The inherits sizable momentum from the meson, leading to factors that reach . The corresponding spectra for the total () and transverse () flight distance of the are shown in Fig.4. For the displacement between the and -collision vertices is resolvable by LHCb.

Iii Baseline LHCb Selection

The LHCb detector is a single-arm spectrometer covering the forward region of  Alves Jr. et al. (2008); Aaij et al. (2015a). The detector, which was built to study the decays of hadrons containing and quarks, includes a high-precision tracking system capable of measuring charged-particle momenta with a resolution of about 0.5% in the region of interest for this search.999The precision of electron momentum measurements is limited by bremsstrahlung radiation; see Sec.III.4. The silicon-strip vertex locator (VELO) that surrounds the interaction region measures heavy-flavor hadron lifetimes with an uncertainty of about 50 fs Aaij et al. (2014a). Different types of particles are distinguished using information from two ring-imaging Cherenkov (RICH) detectors Adinolfi et al. (2013), a calorimeter system, and a system of muon chambers Alves Jr. et al. (2013). Both the momentum resolution and reconstruction efficiency are times worse for neutral particles than for charged ones. For this reason, the analysis strategy outlined below is based entirely on charged-particle information.

iii.1 Track Types

After exiting the VELO a distance of from the collision, charged particles next traverse the first RICH detector (RICH1) before reaching a large-area silicon-strip detector located just upstream of a dipole magnet with a bending power of about  Arink et al. (2014). Downstream of the magnet, there are three stations of silicon-strip detectors and straw drift tubes. All tracking systems will be upgraded for Run 3, though only the changes to the tracking systems upstream of the magnet are relevant here. The VELO has been redesigned to use pixels and is expected to have slightly better lifetime resolution and a lower material budget in Run 3 LHC (2013a). The tracking station just upstream of the magnet will also be replaced by a pixel detector and provide better coverage in than the current detector LHC (2014b). This tracking station is known as the upstream tracker (UT).

In LHCb jargon, there are two types of tracks relevant for this search:

  • LONG tracks that have hits in the VELO, the UT, and the stations downstream of the magnet. These tracks have excellent momentum resolution in both magnitude and direction.

  • UP tracks that have hits in the VELO and the UT, but not in the stations downstream of the magnet. These tracks have excellent directional resolution obtained from the VELO. Since the curvature measurement is based only on the fringe field in which the UT operates, however, the uncertainty on the magnitude of the momentum is about 12% LHC (2014b).

We also note that LHCb defines DOWN tracks which have hits in the UT and downstream of the magnet but no hits in the VELO. While DOWN tracks are not used in this search, they could be useful for other searches involving long-lived particles.

Charged particles may end up being reconstructed as UP tracks if they are swept out of the LHCb acceptance by the dipole magnetic field. This may occur if a particle is produced near the edge of the detector or if it is produced with low momentum. For simplicity, we take any charged particle with and to have 100% efficiency of being reconstructed as a LONG track. Any track that is not LONG, but satisifies and is assigned as an UP track. In reality, the reconstruction efficiency is not a step function—particles with may be reconstructed as LONG tracks, while particles with may produce UP tracks or not be reconstructed at all—but this simple choice reproduces well the overall tracking performance. The momentum resolution for each track type is derived in App.A.

iii.2 Reconstruction

The meson momentum must be reconstructed for this search, since the kinematic constraints imposed by the mass will be used to suppress backgrounds and to improve the resolution on . We consider two categories of reconstruction.

  • F-type: All of the children are charged particles so that the can be fully reconstructed. At least two of the decay products must be reconstructed as LONG tracks. This suppresses combinatorial backgrounds and provides excellent resolution on the location of the decay vertex and on the momentum . The remaining decay products are permitted to be reconstructed as either LONG or UP tracks.

  • P-type: At least two of the children are reconstructed as LONG tracks so that there is excellent resolution on the location of the decay vertex (there may be UP tracks as well). Requiring significant flight distance then permits reconstructing with good precision the direction of the momentum using the vector from the collision to the decay vertex. For the case where the invariant mass of the missing particle(s) is known, can be solved for as discussed below. In this way, the is pseudo-fully reconstructed.

The F-type decays considered in this search are given in Table 1. Each is of the form or , where or . We do not consider doubly Cabibbo-suppressed decays (e.g. ) since they have small branching fractions and can be difficult experimentally to separate from the related Cabibbo-favored decays. LHCb has already published results using most of the F-type decays listed here (see, e.g., Refs. Aaij et al. (2014b); Aaij (2014)), and each decay is expected to have minimal combinatorial background contamination even with only loose selection criteria applied. Here we assume a baseline F-type selection efficiency of 90%. The total efficiency is then , which is dominated by the requirement that all decay products are reconstructed by LHCb. As shown in App.A, the resolution on for F-type decays is excellent.

Decay
4.4% 2.5%
9.1% 4.5% 1.0%
6.8% 2.0%
22.0% 6.6%
1.5% 0.5%
8.5% 1.4%
Total 7.0% 11.5%
Table 1: Decays of mesons used in this search. The branching fraction and efficiency-corrected branching fraction are given for each decay, for both the F-type (fully reconstructed) and P-type (pseudo-fully reconstructed) selections. The notation denotes that is not reconstructed. Entries with an subscript count any decay where the invariant mass of the non-reconstructed system satisfies as signal.

In P-type decays, we can use the measured flight direction to pseudo-fully reconstruct . The direction is a unit-normalized vector from the collision to the decay vertex. The magnitude is , where is the reconstructed (visible) momentum and is the non-reconstructed (missing) momentum. Balancing the momentum transverse to the direction of flight, requires . Assuming that the invariant mass of the missing decay products is known, can be solved for in the rest frame using conservation of energy and the known meson mass. Since is invariant under boosts along , in the rest frame is easily obtained. Finally, can be determined in the lab frame up to a two-fold ambiguity that arises because the sign of in the rest frame is not known. However, once the is combined with an candidate to form a candidate, the vast majority of the time only the correct solution produces an invariant mass consistent with that of the meson. As described in App.A, we take the baseline selection efficiency for P-type decays to be 50%, since the flight distance must be large relative to the vertex resolution to obtain good resolution on .

The P-type decays considered in this search are given in Table 1. We again do not consider doubly Cabibbo-suppressed decays. Other decays that are ignored include those where the missing mass cannot be reliably predicted, such as , which dominantly has as the missing system. Note that solving for in P-type decays requires using the known missing mass as a constraint. That said, the resolution is only degraded slightly if the true missing mass differs from that used in the reconstruction by up to about . For example, when the visible part of the decay is , the most likely missing system is a single ; if the missing mass is taken to be , but the actual decay is , the resolution obtained on by applying the “wrong” kinematical constraints to the candidate is only worse by about 10%. In Table 1, we list the missing mass ranges considered as signal for each P-type decay. Candidates where the missing mass falls outside of these windows are ignored in this analysis, since they have worse resolution and anyways make up a small fraction of the P-type decays. A derivation of the P-type resolution is given in App.A. The resolution on is about an order of magnitude worse in P-type than F-type decays; however, the resolution after performing a mass-constrained fit is similar (as shown in Fig.5 below).

iii.3 Reconstruction

To reduce the background from unassociated combinations, we require that the reconstructed mass difference

(22)

satisfies

(23)

The looser requirement is placed on the lower edge due to bremsstrahlung by the electrons. This mass requirement highly suppresses the decay and its counterpart, except when is large (see Sec.IV.4 below). The efficiency of this requirement is about . Note that this cut can be tightened at the expense of signal efficiency if combinatorial backgrounds turn out to be problematic (see Sec.IV.3 below).

iii.4 Reconstruction

The reconstructed electrons produced in decays are a mixture of UP and LONG tracks. Only a few percent of the electrons have momenta large enough that equivalent-momenta non-electrons would be able to emit Cherenkov light in RICH1. Therefore, identification of the and should be highly efficient with a low hadron-misidentification rate. Furthermore, the signature of a maximum-Cherenkov-angle ring in coincidence with a track should suppress the fake-track background which can be sizable at low momenta.

Bremsstrahlung radiation and multiple scattering of the electrons significantly affect the resolution. We implement this numerically in our simulation following Refs. Olive et al. (2014); Koch and Motz (1959) and using the Run 3 LHCb VELO LHC (2013a), RICH1 LHC (2013b), and UT LHC (2014b) material budgets. Bremsstrahlung downstream of the magnet does not affect the momentum measurement and is ignored.

Figure 5: Distribution of with (solid, dashed) and without (dotted) incorporating the mass constraint for . The solid curve shows better performance than the dashed one because F-type candidates have better momentum resolution than P-type ones.

In Fig.5, we show the resolution on for several values of , where the candidates are constrained to originate from the collision. Bremsstrahlung creates large low-mass tails resulting in poor resolution on . Since the mass is known and its width is less than the detector resolution, though, we can correct the distribution once we identify the candidate and apply the cut. As a heuristic, one can rescale the value by a simple correction factor

(24)

A more sophisticated approach involves performing a mass-constrained fit to enforce energy-momentum conservation and the known mass using the covariance matrices of all reconstructed particles. Using this fit, we find 10–20% improvement in relative to the simple correction given in Eq.(24). As shown in Fig.5, the resolution on after the applying the kinematic fit is 2–3 using F-type candidates, and 2–5 using P-type candidates.

The key difference between the pre-module displaced, post-module displaced, and resonant searches are the requirements placed on the flight distance. These are described in more detail in the subsequent sections.

Iv Displaced Search (Pre-Module)

The typically has a large Lorentz boost factor, resulting in the decay vertex being significantly displaced from the collision for . The combined signature of a displaced decay vertex, a displaced vertex, consistent with , and a consistent decay topology will result in a nearly background-free search. This pre-module displaced search is aimed at decay vertices that occur within the beam vacuum upstream of the first VELO module intersected by the trajectory.

iv.1 Conversion and Misreconstruction Backgrounds

At LHCb, the first layer of material is the foil that separates the beam vacuum from the VELO vacuum. This foil is corrugated to accommodate the VELO modules, such that if the decays prior to the foil, it still effectively decays within the VELO tracking volume. The average transverse distance that the will travel before hitting a VELO module is LHC (2013a), which, because of the corrugated foil geometry, is roughly the average transverse flight distance to the foil as well.

To effectively eliminate backgrounds from conversions in the foil, we require the decay vertex to be reconstructed upstream of the foil. Furthermore, each reconstructed electron must have an associated hit in the first relevant VELO module given the location of the reconstructed decay vertex. These hits are required to have at least one vacant VELO pixel between them to avoid any charge-sharing issues, imposing an effective buffer distance between the decay vertex and the foil:

(25)

where is the electron-positron opening angle. In reality, the VELO pixels in Run 3 will be squares; the definition of is based on treating the pixels as circles with 0.123 mm being twice the effective diameter (the precise value used here has no impact on our search). The pre-module requirement can then be approximated by requiring the transverse flight distance to satisfy

(26)

where gives the flight direction. To remove trajectories that first intersect the foil far from a module, we require . We also impose to avoid possible contamination due to collisions that are not properly reconstructed.101010An candidate may be accidentally formed from a prompt pair produced in a collision if the event is not properly reconstructed. In particular, if a meson is produced in another collision upstream of that interaction point, the “displaced” would produce a consistent decay topology, albeit with .

Having suppressed conversion backgrounds, the dominant background comes from prompt events where the vertex is misreconstructed as being displaced because of multiple scattering of the electrons in the detector material. We estimate this background in a toy simulation of the Run 3 VELO, taking scattering angle distributions from a Geant simulation which includes non-Gaussian Molière scattering tails.111111It is likely that Geant overestimates the probability for large-angle scatterings (see Ref. HPS collaboration ()). If so, our results are conservative, since these scattering tails effectively define the reach for the pre-module search. Many of these fake vertices can be eliminated by requiring a consistent decay topology, in particular that the angle between and the vector formed from the collision to the decay vertex is consistent with zero, and the electrons travel within a consistent decay plane.

The remaining misreconstructed background events have a consistent topology, so a cut on transverse flight distance is required to ensure a significant displaced vertex. To avoid fake displaced vertices from one electron experiencing a large-angle scattering, we also require both the electron and positron to have a non-trivial impact parameter (IP) with respect to the collision. These requirements are summarized by

(27)

where the value of is adjusted to yield background event in each mass window, with ranging from 3 to 5 as a function of . The selection in Eq.(27) is meant to be simple and robust, and could certainly be optimized in a full analysis. See App.B for details on the and IP resolution.

iv.2 Event Selection

Figure 6: Pre-module displaced search signal efficiency for several values. The dashed line shows the efficiency of the track-reconstruction requirements placed on the electrons alone, while the solid lines show the total efficiency. This efficiency does not include the contribution from the or selection (i.e.  or ).

Summarizing, the event selection for the pre-module displaced search is:

  • F-type or P-type candidate;

  • and from LONG or UP tracks with hits in the first VELO module they intersect;

  • reconstructed candidate from the , , and ;

  • reconstructed satisfying the conversion veto (, );

  • reconstructed with significant displacement ().

In Fig.6, we show the resulting signal efficiency . For , which is near the low end of the reach, the efficiency is limited by the efficiency of the conversion veto. As increases, the requirement of a significant displacement ultimately limits the reach of the displaced search.

iv.3 Additional Backgrounds

Beyond the misreconstructed background, a full accounting of the potential backgrounds for the displaced search is difficult since all SM processes with large rates are highly suppressed. Therefore, any additional backgrounds will be dominated by extremely rare processes or highly unlikely coincidences.

One possible source of backgrounds would be decays, since the resulting vertex is truly displaced. Such decays can be suppressed by making the following requirements: the and momenta must intersect the -collision point when traced upstream from their respective decay vertices; the decay vertex is consistent with the -collision vertex; and there are no additional tracks consistent with originating from the decay vertex. Furthermore, one could require that the decay vertex is downstream of the decay vertex, which would be efficient for the smaller values probed in this search. Therefore, we do not expect a significant amount of background coming from decays.

The decays of other long-lived mesons could also be sources of displaced vertices. Decays of charged pions and kaons that produce an pair are rare, though, and the probability for these particles to decay in the VELO is small. A more likely source is the decay , where the is produced in the decay of a long-lived meson. All of the other meson-decay products must be neutral, of course, otherwise the presence of additional charged particles consistent with originating from the vertex could be used as a veto. For example, the decays and occur with huge rates within the LHCb VELO. Such decays, however, are unlikely to result in the candidate momentum intersecting the -collision point or to occur in coincidence with a meson such that is consistent with .

To see whether we could estimate displaced combinatorial backgrounds in Monte Carlo, we generated a sample of 30 million Pythia collisions at 14. We found that no combination of a true with two displaced tracks (not necessarily electrons, but assigned the electron mass) had an invariant mass within the mass window. In this simulated sample, there are only three candidates with using true electrons and none within 150 of our mass window. However, we are anticipating times more meson decays in the full LHCb data sample, so it appears that it is not feasible to use Monte Carlo to precisely estimate the displaced combinatorial background. This type of background will therefore need to be examined in data using the sidebands. If specific sources of combinatorial background are identified as problematic, then the selection will need to be adjusted to remove them.

iv.4 Contribution from Pion Decays

Thus far, we have ignored the channel , which is another potential source of signal events. Decays of this type are highly suppressed by the requirement in Eq.(23), though, unless is large. For most allowed values, one can choose whether or not to include such decays in the analysis by adjusting the requirement. After removing the requirement, the expected yields of and decays are comparable, but so is the expected background contamination from misreconstructed and . We choose not to include this channel when estimating the reach below, but note that such decays may prove useful in a complete analysis.

If one does try to use the channel, then one should be aware of an important subtlety when incorporating information. As described in Sec.III.4, a kinematic fit can be used to improve the resolution. For decays, however, the missing is not accounted for when enforcing energy-momentum conservation. We find that this results in the peak being shifted up in mass by about 20, with the resolution on degraded by about a factor of two. This results in two peaks in the reconstructed spectrum, coming from the and channels. If the background level is low, then this second peak could be used to boost the significance of an signal. Indeed, one narrow peak with a second, wider peak shifted in mass by a fixed amount would be a striking signature. If the background level is high, then this wider peak would largely be absorbed into the background and have no impact on the signal significance.

Finally, photon conversions arising from decays are also highly suppressed by the requirement, and in the absence of misreconstruction, can be eliminated by the pre-module requirement in Eq.(26). We expect such conversions to contribute less than those from decays.

iv.5 Reach

The expected signal yield for the displaced search as a function of and is given by

(28)

As discussed above, we adjusted the requirement in Eq.(27) to ensure background event in any given window. Assuming that all relevant backgrounds have been accounted for, the reach would be set at 95% confidence level by requiring . However, to allow for the possibility of extremely rare background sources, the reach is set by requiring . In this way, we account for either additional background candidates in the final data sample or for a lower selection efficiency due to the criteria required to suppress these additional backgrounds. The reach is shown in Fig.2 assuming 15 fb of data collected by LHCb in Run 3, which covers a significant part of the allowed parameter space for MeV.

V Displaced Search (Post-Module)

In order to capture more signal events, one can effectively reverse the pre-module requirements in Eq.(26) and search for post-module decays. Here, the dominant background is , where the on-shell converts into via interactions with the detector material. As we will see, this post-module search does not cover much additional parameter space compared to the pre-module search, but is important as a cross check of a possible discovery.

v.1 Misreconstruction and Conversion Backgrounds

The background considerations in the post-module case are reversed compared to the pre-module case in Sec.IV.1. Here, the background from misreconstructed events can be effectively eliminated by requiring no hits in the first VELO module intersected by the reconstructed electron trajectories.

The dominant background in the post-module search comes from with photon conversions. We simulate this background using the Run 3 LHCb VELO material as described in LABEL:LHCb-TDR-013 with the Bethe-Heitler spectrum as given in LABEL:PhysRev.89.1023.121212It is vital that all searches use the Bethe-Heitler spectrum, rather than the one produced by Geant. Geant vastly underestimates the fraction of conversions that produce large due to the usage of a less-CPU-intensive approximation of the Bethe-Heitler equation. We start with electron tracks that each have at least three hits in the VELO. This imposes an effective transverse flight distance requirement of

(29)

In reality, the electron hit requirement does not result in a step function for the efficiency. However, in the long-lifetime limit, this simple approximation produces the same integrated efficiency. We then require the reconstructed vertex to be significantly displaced from the VELO material. This can be well-approximated by treating the VELO as a stack of modules located at longitudinal distances , where is measured from the point where the has (i.e. the average position where the trajectory crosses a VELO module). From a given decay vertex at a location between modules and , one requires

(30)

where is the same buffer distance in Eq.(25) and IP is defined with respect to the location where the trajectory intersects the -th module (see App.B for the corresponding resolutions). We also impose the same requirement as in Sec.IV.1. Using our simulation, we adjust such that event will survive these criteria in each mass window.

v.2 Event Selection and Reach

Figure 7: Post-module displaced search signal efficiency for several values. The dashed line shows the efficiency of the track-reconstruction requirements placed on the electrons alone, while the solid lines show the total efficiency. This efficiency does not include the contribution from the or selection (i.e.  or ).

Summarizing, the event selection for the post-module displaced search is:

  • F-type or P-type candidate;

  • and from LONG or UP tracks with no hits in the first VELO module they intersect;

  • reconstructed candidate from the , , and ;

  • reconstructed satisfying the prompt veto (, );

  • reconstructed