A Trigger for Displaced Muon Pairs Following the CMS Phase II Upgrades

We show that the phase II upgrade of the CMS tracking detector could enable the experiment to trigger on very low mass $\mathcal{O}(1\,\text{GeV})$ displaced muon pairs with minimal $p_T$ cuts. As a result, CMS can be competitive with LHCb when searching for low mass displaced exotics originating from heavy flavor decays. The method can also be applied to signatures without muons but with a moderate amount of MET, $H_T$ or multiple displaced vertices in the event.


I. INTRODUCTION
With the LHC now gearing up for its high luminosity phase, all detectors will undergo substantial upgrades to cope with the expected high levels of pile-up. The tracking detectors especially are an important component of this effort. ATLAS [1], CMS [2] and LHCb [3] are all aiming to incorporate tracking information as early on in the trigger decisions as possible, for the phase II upgrade or sooner. For CMS in particular, the modules in the outer tracker will consist out of two closely spaced sensors, such that a rudimentary momentum measurement will be possible on a single module using the pairs of hits (stubs) [4]. A momentum preselection on these track stubs then greatly reduces the occupancy, rendering preliminary track finding feasible under the stringent time constraints of the L1 trigger.
It has been shown recently that this approach can in fact reconstruct tracks with a transverse impact parameter up to roughly 10 cm, opening up the exciting possibility of a dedicated L1 trigger for long-lived particles (LLPs) [5]. Concretely, [6] showed that demanding a number of displaced tracks combined with a moderate H T requirement can greatly increase the trigger efficiency for Higgs bosons decaying to a pair of displaced dijets. In this work we build on [5,6] by exploring the feasibility of an L1 trigger for a low mass displaced dimuon vertex with minimum muon p T thresholds. This would give CMS greatly improved access to e.g. exotic B-meson decays to light hidden sector particles, which can arise for instance from a light scalar mixing with the Higgs boson. In particular, we will show in Fig. 7 that this capability could make CMS competitive with LHCb [7,8] for this class of signatures. Dark photon models are another application, and have been studied extensively already in the context of the (upgraded) LHCb detector [9][10][11].
Our strategy can be summarized as follows: We use the simulation framework described in [5] to reconstruct tracks from the stubs retained by the future CMS outer tracker at the L1 trigger, and hereby include resolution smearing and multiple scattering. We subsequently find the most compatible vertex and require that the reconstructed mother particle trajectory points back to the interaction point. The latter is needed to suppress the large rate of vertices formed by random crossings of fake tracks. Once a suitable vertex is identified, its tracks can be matched to activity in the muon detectors, which should suffice to suppress the rate to O(kHz).
This paper is organized as follows: We describe simulation framework in Sec. II and our analysis and results in Sec. III. We conclude and present an outlook in Sec. IV. In Appendix A we provide some additional figures which we hope could be useful to recast our results for different models.

A. Signal and background simulation
The example signal we consider is that of a light particle (ϕ) produced through the B → X s ϕ decay with ϕ decaying to muons (ϕ → µ + µ − ) through a displaced vertex. Such a signal can arise for instance from a light scalar mixing with the Higgs boson, which itself is a feature in many UV complete models of beyond the Standard Model physics. A model of this kind typically predicts correlations between the lifetime, production rate and branching ratios of the new particle, however in this work we simply parametrize the yield in terms of the lifetime (cτ ) of ϕ and its production rate, expressed in terms of is the inclusive B-meson branching ratio. The signal was generated with pythia 8 [12,13] and we assumed an inclusive b-b cross section of 500 µbn [14].
Two sources of background are considered. First, and largest, is the random crossings of fake (i.e. comprised of the unrelated stubs) tracks. We assume the number of fake tracks averages 30 per event, with all five track parameters (see e.g. [15]) (q/p T , φ, d 0 , tan λ, and z 0 ) uniformly distributed. Not all combinations of parameters produce tracks with observable stubs, mostly due to the anti-correlation between the q/p T and d 0 (See Fig. 8 in Appendix A). We throw uniformly distributed track parameters (see Table I) until we get a track that is expected to have at least four observable stubs.
The second source of background is the the production of known long-lived particles, dominated by the K S . We used inclusive QCD events generated with pythia 8 [12,13]  We neglect the effects related to interactions of prompt particles with the material of the detector. Material interactions are relatively rare and are not expected to produce candidates that point back to the primary vertex (PV). High energy photon conversions point back to the PV, but are trivial to remove based on the smallness of the opening angle. In fact, it is likely that the vertex selection cuts in Sec. III remove most of them.

B. Track reconstruction
The toy Monte Carlo to estimate track reconstruction and resolution is similar to the one in [5], with several important additions.
The toy tracker has six perfectly cylindrical double layers covering |η| < 2.4. Hits in the double layers produce a stub if the azimuthal offset between them is consistent with one from a prompt track with p T > 2 GeV. The hits are smeared according to the expected resolutions [4]. We also make a simple estimate of multiple scattering in the detector, assuming tracks change direction of travel when crossing a tracking layer by a random angle with mean of zero and Gaussian sigma of 4 · 10 −4 /p T [4,16]. To account for scattering in the pixel detector, we add scattering layers corresponding to approximate positions of the pixel layers.
The resulting hits are then fit to a 5-parameter helical trajectory. Fig. 1 shows the impact of the multiple scattering on the track impact parameter resolution for a signal sample with m ϕ = 0.5 GeV.

C. Vertex reconstruction
The fitted tracks are the inputs to the vertex finding algorithm, which is deliberately kept as simple as possible, as proper vertexing is computationally prohibitive at the L1 trigger level. Our simplified vertex finder proceeds as follows: First we calculate the intersections between the two helices in the transverse plane. If no intersection exist, we identify the transverse location of the candidate vertex with the point on the line defined by the centers of the circles for which the distance to the circle boundaries is equal. The distance between both circle boundaries measured along this line is a measure of the quality of the vertex in the transverse plane, and is labeled by ∆ T . We Effect of multiple scattering on impact parameter resolution, for a signal sample with mϕ = 0.5 GeV and cτϕ = 1 cm. subsequently compute the distance between the tracks in the z-direction for this candidate vertex, which we denote by ∆ z . If two intersections are found, we calculate the distance in the z-direction between both tracks (∆ z ) for each of the two solutions, and select the solution for which |∆ z | is smallest. The ∆ T variable is set to zero in this case. Fig. 2 shows the resolution of the reconstructed vertex radius, with and without multiple scattering, for m ϕ = 0.5 GeV.
Once a candidate vertex is found, we use its coordinates and the reconstructed track momenta to compute the impact parameter of the mother particle's trajectory in the transverse plane with respect to the origin (d T ). We further compute the angle (α T ) in the transverse plane between the mother's reconstructed trajectory and the line connecting the vertex location with the origin. Finally, we record the distance of the vertex from the origin, in the transverse plane (R T ). The above variables are summarized in the diagram in Fig. 3. Note that we use the origin of the CMS coordinate system as a reference point, rather than the primary vertex, since we do not assume we identified the primary vertex at the trigger level.

III. ANALYSIS AND RESULTS
Figs. 4, 5 and 6 show the distributions of ∆ z , cos α T and d T respectively, for the signal and for fake vertices. Based on these distributions, we define the following set of cuts: i) d 0 > 1 mm for each track ii) at least 5 stubs for one of the tracks, and at least 4 stubs for the other.
The cut on the track impact parameter d 0 in i) prevents prompt tracks from contributing to the rate. Cuts ii) to vi) are intended to suppress the contribution from fake tracks to the trigger rate. Of these iv) and v) provide roughly 10 −2 background suppression each, as can be seen from Figs. 4 and 5. On the other hand, the background reduction from ii) and iii) combined is only ∼ 50% and these cuts could in principle be omitted. The cuts on d T and α T are rather strongly correlated, but some additional background suppression is nevertheless gained by imposing both, with virtually no loss in signal efficiency (see Fig. 6). Finally, the cut on the vertex radius in vii) reduces the rate from B-decays to the kHz level, though it is to a large degree degenerate with the cuts on d 0 .
The rates for finding a vertex satisfying the above criteria are given in Tab. II for various p T cuts on the tracks. Fake vertices and K S decay contribute roughly equally to the trigger rate. 1 As is, these rates are still too high for a realistic trigger and an additional selection is therefore needed to further reduce the rate with roughly two orders of magnitude. Depending on the signature of interest, this could for instance be an H T or MET requirement, or a second displaced vertex in the event. For our example, we require that both tracks are matched to a track candidate in the muon detectors, and assume that this provides the remaining necessary suppression of the fake rate. This will also greatly reduce the combinatorics for the muon detector matching, since only the tracks associated with a displaced vertex need to be checked.
The required fake track rejection from matching with the muon system corresponds to a factor of roughly 1/50 suppression per muon, which we consider conservative [17]. While it is possible that requiring a single muon may suffice, here we require both muons to be matched. The efficacy of the match is a strong function of the muon p T and is hard to estimate in the toy simulation. Therefore we present the yield for a few different choices of thresholds.
Both ATLAS and CMS are aiming to include a standalone dimuon trigger in their HL-LHC trigger menus, with p T 10 GeV thresholds for both muons [17,18]. As a point of comparison, we loosely model the acceptance of such a trigger by i) |η| < 2.4 for both muons ii) p T > 10 GeV for both muons iii) R T < 600 cm The distance along the z direction between the two helices at their intersection in the transverse plane, for a signal with mϕ = 0.5 GeV and cτ = 1 cm (red) and random crossings by fake tracks (blue), after cuts i) through iii). 1 Algorithms are being considered to identify the primary vertex already at the track trigger level. Given the relative softness of the b-b events, the primary vertex would still be misidentified a fraction of the time. If the associated loss in signal efficiency however proves to be acceptable, a cut on the longitudinal impact angle could be added (See Fig. 10 in Appendix A). This would render the rate from fake vertices negligible compared to that from K S decays.  Impact parameter of the LLP candidate in the transverse plane, for a signal with mϕ = 0.5 GeV and cτ = 1 cm (red) and random crossings by fake tracks (blue), after cuts i) through v).
where the R T cut is loosely based on geometry of the ATLAS muon system. We assume that this trigger is 100% efficient, which is conservative for the sake of our comparison. We further compare with the acceptance of LHCb, following the discussion in [11]: i) 2.0 < η < 5 for both tracks ii) p T > 0.5 GeV for both tracks iii) 3-momentum of both tracks larger than 6 GeV iv) R T < 2.2 cm  The cuts on the vertex location are motivated by the geometry of the VELO detector. We assumed a total integrated luminosity of 300 fb −1 for LHCb and 3 ab −1 for ATLAS and CMS. The resulting yield for all three strategies is shown in Fig. 7 for two benchmark mass points, shown in terms of the production rate and lifetime of ϕ. Given that the suppression factor from the matching with the muon system is unknown at this time, we show the results for the CMS displaced vertex trigger for a few different p T thresholds on the muons. With the assumptions above, we find that CMS has the potential to improve on the LHCb yield in the large cτ regime. In the low cτ regime LHCb clearly performs better due to the better vertex resolution of the VELO detector and LHCb's capability to do the full event reconstruction online.
Our proposed displaced vertex trigger is moreover highly complementary to the standalone dimuon trigger (Dashed purple curve in Fig. 7): At high cτ the standalone dimuon trigger has much larger geometric acceptance than our displaced vertex trigger, however for soft signals, such as heavy flavor decays, this is offset by our lower p T requirements. For high energy signals from e.g. supersymmetric particles, the standalone dimuon trigger will continue to have the best efficiency.
It should be noted that all curves are each somewhat optimistic in different ways: For LHCb and the standalone dimuon trigger no detector smearing is attempted and the reconstruction is assumed to be 100% efficient. For the displaced vertex trigger we have assumed negligible losses in signal efficiency at the high level trigger. Given that the backgrounds in all cases are highly non-trivial and are best estimated from data, we do not attempt to project exclusion limits at this stage.

IV. DISCUSSION
We have performed a toy simulation of the future CMS track trigger for the heavy flavor benchmark B → X s ϕ, where ϕ is a long-lived particle with an appreciable branching ratio to muons. The rate from random crossings of fake tracks and K S mesons can be brought under control by a combination of pointing cuts and by matching the tracks to activity in the muon chambers. The track trigger could allow CMS to compete with LHCb on soft, displaced signatures beyond the Standard Model physics, in particular in the context of exotic heavy flavor decays and dark photon models.
As mentioned above, at least two handles (pointing and muon matching) are needed to bring the rate from random track crossings down to an acceptable level for the high level trigger. There are therefore a number of possible variations on our strategy, by relaxing the pointing and/or muon requirements and instead demanding: • A dimuon vertex without pointing but with a mod- erate amount of MET. This may be powerful for inelastic dark matter models.
• Two reconstructed vertices in the event. This is of interest for e.g. hidden valley/dark shower models or models where the dark sector states are pair produced.
• A vertex with 4 or more tracks, which would capture hadronic decays or decays to τ + τ − .
• A single, non-pointing vertex with one muon with moderate p T , which can record decays of heavy neutral leptons.
We leave these and other possibilities for future work.  Expected number of stubs for a track given its curvature and impact parameter.
In Fig. 9 we show the trigger efficiency as a function of the truth-level R T , after imposing (truth-level) accep-tance cuts on the p T and η range for both muons. For higher masses and high R T , the higher opening angle between the muons implies that the stubs are more likely to fail the cut on the pitch, such that insufficient stubs are found to reconstruct the track. At low radii, the m ϕ = 0.5 GeV benchmark point looses efficiency faster due to the d 0 cut on the tracks.
Finally, with Fig. 10 we also include the distribution of the impact angle in the longitudinal direction. If the primary vertex can be identified, cut on in this variable can further reduce the fake background to the extend that is negligible compared to K S decays.