Uncovering quirk signal via energy loss inside tracker

The quirk particle carries Lorentz force and long-range infracolor force, while suffers relatively large ionization energy loss inside the detector. It can be indirectly constrained by mono-jet search or directly search through co-planar hits if the confinement scale is not too low ($\Lambda \gtrsim 100$ eV). Considering the ionization energy loss inside tracker, we improve the co-planar search. We also will solve the equation of motion for quirks numerically by including all of the important contributions. Based on our selection strategy, the $\sim 100$ fb$^{-1}$ dataset at the LHC will be able to probe the colored fermion/scalar quirks with masses up to {2.1/1.1 TeV}, and the color neutral fermion/scalar quirks with masses up to {450/150 GeV}, respectively.


I. INTRODUCTION
The Large Hadron Collider (LHC) has already collected tremendous data at its Run-1 and Run-2 with center of mass energy ranging from 7 TeV to 13 TeV. However, the traditional analyses failed to find any new physics signals beyond the standard model (BSM) from those data. Yet there still exists possibility that some new physics processes have been copiously produced at the LHC without being probed due to its nonconventional behavior. The long-lived exotic searches are receiving increased interests at the LHC [1,2] and some future facilities [3,4]. Many BSM models, which introduce extra gauge symmetries (such as hidden valley models [5,6]), predict exotic signals at the detector, for example, emerging jet [7], trackless jet [8], and soft bomb [9], etc. While the others that predict the new charged particles around the detector scale can lead to the kinked track [10] and disappearing track [11] at the detector. Even though those exotic signals were overlooked by traditional searches, it does not mean that they are difficult to probe. In fact, there are already specific searches designed for emerging jet [12] and disappearing track [13] by the CMS Collaboration. Very strong constraints on those signals were obtained because of their small backgrounds.
In this paper, we shall consider the quirk signals at the colliders [14,15]. Quirks are long lived exotic particles that are charged under both the Standard Model (SM) gauge group and a new confining gauge group. The mass of the lightest quirk is much larger than the confinement scale (Λ) of the new gauge group. So after the quirk pair is produced at collider through the SM charge, there will be confining force binding the quirk and anti-quirk pair with strength proportional to Λ 2 . In the center of mass energy frame, the typical amplitude of the oscillations can be estimated as Given a quirk with mass around electroweak scale, when the Λ O(10) MeV, the confining force will lead to intensive oscillations in quirk system. It will lose energy quickly via hidden glueball and photon radiations. Thus, annihilations into the SM particles are almost promptly happened inside detector. Searches for new resonances in the SM final states have been proposed to look for such quirk signals [16][17][18][19][20]. For Λ ∈ [10 keV, 10 MeV], the oscillation amplitude is microscopic while the glueball and photon radiation is not efficient. The quirk system will leave a straight line inside tracker. It will be reconstructed as an single boosted charged particle with high ionization energy loss and has been searched at Tevatron [21]. While for Λ O(10) eV, the confining force becomes negligible comparing to the Lorentz force. Each quirk propagates with helical trajectory and is found to be constrained by heavy stable charged particle searches at the LHC [22][23][24]. The case with Λ ∼ 100 eV-10 keV is the most interesting, since the oscillation amplitude is macroscopic and is visible by detectors. The quirk trajectory can not be simply reconstructed as helix. Moreover, we will show that its energy deposit in electromagnetic/hadronic calorimeter (ECal/HCal) is also usually small within the time scale of 25 ns (which is bunch crossing period at the LHC). So they will be totally missed by conventional reconstructions in collider searches and will just behave as missing energy. As a result, those quirks can only be constrained by mono-jet searches [24][25][26] if they are produced with recoiling against an energetic initial state radiated jet. On the other hand, if the quirk pair is produced with little kinetic energy, the ECal/HCal is able to capture the quirk system. The quirk pair will annihilate inside the calorimeter after long time oscillation which gives out-of-time decay with the active pp collisions [27].
However, the quirk signal could be more informative than just missing transverse energy. For a relatively boosted quirk pair system, the dominating confining force will lead to an almost co-planer trajectories of quirk pair. Searching for co-planer hits in the tracker can greatly suppress backgrounds while maintain very high signal efficiency [28]. We will further develop the quirk search along this strategy, with including the information of ionization energy loss (dE/dx) inside tracker. The dE/dx of each cluster generated by the charged particle throughout the detector can be derived from the cluster charge, average energy needed to create an electron-hole pair, the density of silicon and the thickness of each layer [29]. We will introduce a new method to tag the quirk track hits in the case when it leaves relatively large dE/dx hits in the tracker compared to the SM particles (including both background and pile-up events) [30]. We find the SM Z(→ νν)+ jets process turns out to be dominating over the Z + γ + j background that was used in Ref. [28]. And our reconstructing algorithm allows wider separation between quirk and anti-quirk in the plane thus smaller Λ can be detected. Moreover, we numerically solve the complete equation of motion for each quirk including the Lorentz force as well as the ionization energy loss according to the architecture of the CMS detector. Another advantage of the complete numerical study is that the variation of the direction of infracolor force in the center of mass frame due to external forces can be taken into account. This paper is organized as follows. In Sec. II, we briefly discuss the theoretical frameworks for quirk. Detailed study on the equation of motion for quirk will be given in Sec. III. In Sec. IV, based on monte carlo events, we propose the method to separate quirk signal from backgrounds. We draw the conclusion in Sec. V.

II. THE NATURE OF QUIRKS
The color neutral quirks are commonly present in many BSM model of neutral naturalness [31], which is proposed to partly solve the little hierarchy problem [32,33]. Such kind of models includes folded supersymmetry [34,35], quirky little Higgs [36], twin Higgs [37][38][39], minimal neutral naturalness model [40] and so on. While other more general BSMs predict quirks that carry strong interaction [41]. The quirks can be either a fermion or a scalar as well.
In this work, we will take the simplified model frameworks as benchmark. The quantum number assignments for the quirks of interest under SU (N IC ) × SU C (3) × SU L (2) × U Y (1) are given as where we will take N IC = 3 for the infracolor gauge group.D c andẼ c have spin zero, D c and E c are fermions. The electric charges ofD c /D c andẼ c /E c are 1 3 and -1, respectively. However, due to the color confinement, one can only observe hadrons of quirk-quark bound state for D c and D c . The probability for final state hadrons having charge ±1 is around 30% [28]. Since we are only interested in final state quirk with non-zero electric charge, in the following, we will simply refer to the charge ±1 quirk-quark bound state asD c or D c . This is justified because the mass of quirk is much larger than quark. The quirk pair can be produced at colliders through the SM gauge interaction. Depending on its gauge representation, there are several processes that could be important as given in Fig 1. The colored fermionic quirk production is given by the first two diagrams, while the colored scalar quirk has an extra contribution from the third diagram. The color neutral quirk productions are simply given by the Drell-Yan processes in the fourth diagram.
We show the production cross sections for these quirks at 13 TeV LHC in Fig. 2. Because our analysis is heavily rely on the co-planar feature of quirk trajectories as well as to trigger the quirk signal events efficiently, only quirk production associated with an energetic initial state radiated (ISR) jet will be concerned. A missing transverse energy trigger E miss T 100 GeV is found to be efficient in mono-jet search [25]. So we require the ISR jet to have p T > 100 GeV in generating the events. The scalar quirks have less degrees of freedom than fermionic quirks, as well as their couplings to gauge bosons are momentum suppressed, we can see from the figure that the cross sections of scalar quirks are typically more than one order of magnitude smaller than fermionic quirks with the same quantum number and mass.
FIG. 2. The production cross sections for different quirks at 13 TeV LHC. We have also required an ISR jet with pt(jet) > 100 GeV in production processes.

III. QUIRK EQUATION OF MOTION
The quirk equation of motion (EoM) inside detector is given by where v = v ·ŝ and v ⊥ = v − v ŝ withŝ being a unit vector along the string pointing outward at the endpoints. F s corresponds to the infracolor force and is described by the Nambu-Goto action [42]. F ext represents the external forces including Lorentz force and the effects of ionization energy loss for charged quirk propagating under magnetic field and through materials.
There are several subdominant effects that were not taken into account in the EoM. Colored quirks are surrounded by a cloud of non-perturbative QCD "brown muck" [15]. Due to the non-perturbative QCD interaction, a hadron with energy ∼ Λ QCD will be radiated at each time of quirk pair crossing. Similarly, at every quirk oscillation, the two quirks binding by infracolor string can emit a soft infracolor gullball with energy roughly of order Λ. And the energy loss due to Larmor radiation of charged quirk is even one order of magnitude smaller in each crossing ∼ √ αΛ.

A. The dE/dx inside the CMS detector
It is well-known that the ionization energy loss of charged particle as function of velocity in the Bethe-Bloch (BB) and Lindhard-Scharff (LS) region are well predicted by theory [43,44]. They can be expressed in simplified forms where the coefficients are given by with the A, Z and ρ correspond to the relative atomic mass, atomic number and density of material, respectively. The ionization energy loss function in the region between LS and BB are interpolated from experimental data.
The CMS detector chooses the silicon for the tracker, lead tungstate (PbWO 4 ) for the ECal, copper for the HCal and iron for the muon chamber. Taking parameters of materials from particle data group [45], we plot the ionization energy loss function for the quirk propagating through each of those materials in Fig. 3. It has to be note that each part of the detector is not fully occupied by corresponding material. The occupancy rate for each component can be described by a factor ζ.
They are 0.05, 0.33 and 0.88 for track, ECal and HCal, respectively. They should be multiplied on the ionization energy loss function in Eq. (III.4) and Eq. (III.5) when solving the quirk EoM.

B. Numerical solution of the EoM
The detector measures the quirk trajectory in the lab frame. However, the dynamic of quirk in lab frame is much more difficult to describe than that in the centre of mass (CoM) frame. In the CoM frame, the string is straight. So the direction of theŝ of one quirk is simply given by the relative displacement respect to the other quirk. However, the CoM frame itself is changing all the time due to the effects of F ext , which arises from the magnetic field B as well as ionization energy loss. Such that the changing of CoM is velocity dependent. It breaks the simultaneity and collineation of the infracolor force between two quirks in the lab frame.
We numerically solve the EoM by slowly increasing the time with small steps. In the lab frame, four momenta of the two quirks are denoted by (E i , P i ) and space-time position by (t i , r i ) with i = 1, 2. In order to ensure the simultaneity in the CoM frame, the following condition needs to be fulfilled . As a result, the time moving step i ( 1) in solving each quirk EoM should satisfy where F i = F si + F exti includes the infracolor force and external forces. Then, at the space time with relation given in Eq. (III.6), we can boost the infracolor force in the CoM frame to the lab frame with theŝ in Eq. III.2 given by the unit vector of for each quirk respectively. In solving the EoM, we require the time step to be smaller than 10 −4 ns, which should be smaller than T ∝ m Q /Λ 2 to ensure the infinitesimal. And time evolution stops when one of the following criteria are met • evolution time longer than 25 ns.

C. Thickness of the quirk hits plane
Not the full trajectory of quirk can be reconstructed by detectors. Charged quirks can only leave a number of hits in the silicon tracker and deposit a small fraction of its energy inside the calorimeter. The CMS detector is segmented into cylindrical barrels that surround the beam pipe and end caps on each side of the barrel. It consists of two subsystems of barrels and end caps: the pixel detector and the strip tracker. The details of the reconstruction parameters can be found in Ref. [46]. In the upper panel of Fig. 4, we plot the distributions of the hit numbers for the quirk-antiquirk system with varying confinement scale Λ = 200 eV, 600 eV, and 2 keV. The events adopted for plotting have quirk mass m Q = 100 GeV and the transverse momentum of ISR jet p T > 100 GeV. Since the CMS tracker consists of 13 barreled layers, the quirk system which is relatively transverse boosted by ISR jet recoiling is likely to leave 26 hits in the tracker. There is also great possibility that the quirk system leaves more than 26 hits inside tracker, mainly due to the end cap hits or backward propagation of quirks. This possibility increases for larger Λ as shown in the figure.
For quirk-antiquirk system propagates in the tracker, it leaves N hits located at h i (i = 1, 2, . . . , N ) when crosses the detector layers. We can define to describe the distance between the hits and a virtual plane n, which includes the origin. There will be a n giving the smallest d min , which we call the normal vector of the quirk plane. Then the thickness of the quirk plane arises from the n-component of the Lorentz forces, which can be estimated as [47]  where B xy = | B − ( B · n) n|, which induces the Lorentz forces perpendicular to the quirk plane.
On the other hand, the thickness of the quirk hits plane can also be calculated by the algorithm proposed in Ref. [28], which is simplified to an eigenvalue problem. We have verified that the thickness calculated from this method matches well with our estimation in Eq. (III.11). In the lower panel of Fig. 4, we plot the distributions of the quirk plane thickness, varying both the quirk mass and the confinement scale, where the event sample requires an ISR jet with p T > 100 GeV as before. There is no doubt that the dependence on the confinement scale is much stronger than quirk mass. We can also observe that for Λ = O(100) eV − O(1) keV, the hits of most of the events lie on a plane with thickness smaller than ∼ O(100)µm. In what follows, we mainly concern the quirk system with confinement scale ∼ keV.

IV. SIGNAL AND BACKGROUND ANALYSES AT 13 TEV LHC
To conduct Monte Carlo simulation, the simplified model for four different quirks are implemented in Feyn-Rules [48]. The general purpose event generation framework MG5_aMC@NLO [49] is able to simulate the quirk production processes with the model file provided by FeynRules. Then the built-in Pythia8 [50] is used for implementing the SM particle parton shower, hadronization and decay. However, the QCD parton shower as well as hadronization do not significantly change the kinetic energy of quirks, we will ignore those effects on quirks momenta for simplicity.
As have been discussed before, we are interested in events with quirk pair recoiling against an hard ISR jet.
The hits of the associating jets are obtained in a much easier way than that explained in previous section for quirk, since the trajectories of the SM particles are simply given by the helix. After the parton shower and hadronization of the ISR parton, we collect all of the charged hadrons inside the jet. And the trajectory for each of charged hadrons is obtained by fitting helix.
The full quirk pair trajectories inside the CMS tracker are shown in Fig. 5 for illustration, where a few different confinement scale has been chosen. In the plots, the quirk mass has been fixed to 100 GeV and three trajectories share the same initial quirk momenta.
The overall time of quirk traveling inside the detector is dominantly controlled by the transverse velocity v T of the quirk system, the distribution of which is provided in the upper panel of Fig. 6. Even though the single quirk can have typical velocity 0.5, the quirk system is moving relatively slow v T ∼ 0.1. As a result, the time for quirk system to propagate outside tracker is 10 − 20 ns. And the heavier quirk is, the longer time delay to reach ECal and HCal.
As have been discussed in the Sec. III A, the quirk pair is not totally invisible by calorimeters due to the relatively strong ionization energy loss there. According to our simulation, we find the total energy losses inside ECal and HCal are around 1 GeV and 4 GeV, respectively, for Λ ∼ O(1) keV and m Q ∼ O(100) GeV as shown in the lower panel of Fig. 6. Moreover, the quirk pair is flying along a similar direction due to the infracolor force. This leads to a significant transverse momentum imbalance in the final states of quirk production process. The feature is helpful in trigging the quirk signal at a realistic detector, i.e., missing transverse energy trigger. We note that the event samples are required to have ISR jet with p T > 100 GeV and the confinement scale Λ = 1 keV.
Taking the advantage of sizable missing transverse energy in the final state, the dominant background process for the quirk signal will be the SM Z(→ νν)+jets. The production cross section of this process at the LHC is huge. We requires the transverse momentum of the neutrino system to be p T > 200 GeV. This is conservative since a much stronger cut will be applied in the analysis. The event sample are matched up to two additional associated jets. We also consider the background process of Z(→ νν)e + e − + jet as pointed out in Ref. [28], with p T (νν) > 100 GeV and p T (e) > 1 GeV. The production cross section for Z(→ νν)+jets and Z(νν)e + e − + jet after applying preliminary cuts are 3.6 pb and 2.5 fb, respectively. To get the hits information for those two background processes, we collect all charged hadrons in the final state that are provided in the Pythia8 output and conduct helix fitting for them.

A. Pileup simulation
During a bunching crossing at the LHC, there will be multiple proton-proton collisions (referred as pile-up), which is dominated by the non-diffractive events (the sample is referred as the minimum-bias) with small transverse momenta transfer. Pythia8 [50] adopts perturbative parton shower, Lund-string hadronization, multiple parton interaction and colour reconnection models to simulate the minimum-bias. However, those models contain many parameters, which can only be deducted from experimental data. The set of chosen parameters are refered as Pythia tunes [51]. The A3 tune with phenomenological parameters provided in Refs. [52,53] is taken in our simulation of pile-up events, since it is found to provide a good agreement with the ATLAS charged particle distributions.
The number of pile-up events per bunch crossing at the LHC follows Poisson distribution with average value around µ ∼ 30 − 50. They will give O(10 4 ) hits inside the tracker, and thus become the main background in our analysis. In our simulation, each of signal and background events are mixed with µ = 50 pile-up events in order to draw a conservative conclusion. Moreover, due to the finite size of the beam spot, the interaction points of the pile-up events could be spread along the z−direction. We will assume the z−coordinate of the interaction points follows a Gaussian distribution with width of 45 mm.
The hits caused by pile-up events can be obtained by the same method as used for the Z+jets background, i.e., crossing the helix trajectories of the charged particles with the CMS tracker architecture, taking into account all charged particles in all events.

B. Quirk signal selections
Now we are ready to propose a dedicated algorithm to separate the quirk hits from SM particles hits. Without loss of generality, we will fix Λ = 1 keV in the following discussion. And the quirk below is referring to theD c otherwise specified.
In searching for the signals of quirk pairs, we should first find the plane that quirks move in based on recorded hits inside tracker.
a. First, we find a way to reduce the SM background hits utilizing high dE/dx of quirk hits. To do so, we remove all the hits with dE/dx smaller than 3.0 MeV/cm. For those remaining hits, we divide all of them into different classes based on their dE/dx. The a th class has an average value equal to where N a is the hit number of the a th class. If the next hit satisfies dE dx next − dE dx c aver < 1.0 MeV/cm, the hit is assigned to the c th class. If a hit does not belong to any existing classes, then we create a new class and put this point in. The iteration continues until all the hits have been assigned to a class. The classes, which contain less than 80 hits, are selected out and considered as quirk hits candidate. For other classes that have more than 80 hits, we remove the sets of hits which can be reconstructed as helix. This step is found to be very useful for suppressing pileup events. Probability We will try to reconstruct the circles from hits in the transverse plane instead of reconstructing a true helix. The location of the possible center of circles (COC) in transverse plane for any two hits in the class is given by (the origin is assumed to be on the circle) (IV.2) where k 1 = y 1 /x 1 , k 2 = y 2 /x 2 , as well as x 1 , y 1 and x 2 , y 2 are x-y coordinates of two hits in a given class. For those pairs of hits with COCs distance smaller than 0.5 cm, and the differences among the radii of those circles smaller than 0.1 cm, the hits are considered as background hits, and thus should be removed from the class. The reduction of hits for each class is repeated until less than 1500 hits are left.
The last column is the reduction efficiencies given at a reduction when N hit goes below 1500, which is ∼ 1500/Ntot.
In Tab. I, we summarize the reduction efficiencies for several full reductions (a full reduction means running over all of the hits in a class) as well as at a reduction when the number of hits in the class is smaller than 1500. We can see that for all signals with different quirk masses and backgrounds, an single full reduction is able to reduce the number of hits close to 1500.
b. Second, we should calculate the normal vector of the plane based on remaining hits from the above reduction. Since the origin is contained in the quirk plane, the plane normal vector can be determined for any two of the remaining hits: n = r 1 × r 2 , where r 1 and r 2 are the coordinates of the two hits. For each given plane (i.e. given two hits), the distance of other hits to the plane is simply d i = | r i × n|. According to our choice of Λ = 1 keV which lead to typical plane thickness ∼ O(10 −5 ) m, we account the total number of hits within the distance of 30 µm to the given plane. The plane with the largest number of close hits is kept and will be regarded as the quirk plane. In the upper panel of Fig. 7, we plot the distributions of the hits number within the quirk plane (d < 30 µm) for signal with different quirk mass as well as background. We can observe that the signals tend to have much larger number hits than background.
c. Third, based on the reconstructed quirk plane, we can define several important variables for further signal and background discrimination. The distance-weighted ionization energy loss is defined as where i runs over all remaining hits from step a and d 0 = 30 µm. Furthermore, the variation of the S p can also be useful for signal identification. S p corresponds to the maximal/minimal S p if we variate the direct of quirk plane within angle of π 12 . The distributions of S p for signals and background are shown in the lower panel of Fig. 7. Due to the slowness of quirk, it givens harder S p spectra than background.
Finally, as we have discussed in Sec. IV, the missing transverse energy (E miss T ) is also an important feature of quirk signal, so we propose the following cuts to select quirk signals at the LHC The cutflow of several quirk signals and backgrounds are given in Tab. II, where we can find that the E miss T and S p play the most important roles.
In Fig. 8, we plot the 95% Confidence Level (C.L.) exclusion limits for different quirks. The sensitivity is dominated by the production rate, while the effects of the quirk kinematics is relevant but not so important. And the sensitivity grows as √ L, where L is integrated luminosity. The LHC Run-2 provides dataset of ∼ 100 fb −1 , which means the colored fermion/scalar quirks with masses up to 2.1/1.1 TeV, and color neutral fermion/scalar quirks with masses up to 450/150 GeV can be possibly probed/excluded.

V. CONCLUSION
Quirk particles widely exist in new physics models with extra gauge symmetries. Especially, the color neutral quirk is well motivated from neutral naturalness model which can solve the little hierarchy problem. We considered a simplified model framework for quirks with different quantum numbers and studied the discovery potential at the LHC.
The quirk equation of motion inside the detector are mainly controlled by the Lorentz force, the long-range infracolor force as well as the ionization energy loss. Moreover, the infracolor force between two quirks are correlated, leads to a coupled partial differential equation system. It can be solved with assumption that the infracolor force is much larger than the other external forces, such that the string connecting two quirks is approximately straight in the center of mass frame. We numerically solved the full EoM for the quirk system, taking into account the architecture of the CMS detector. For the parameter space of interest (i.e. λ ∼ O(100 − 1000) eV, m Q ∼ O(100) GeV, p T (QQ) 100 GeV), most of the quirk pair can leave total number of ∼ 26 hits inside the tracker. Those hits are lie on the plane with thickness O(100) µm. Meanwhile, there will be O(10)% ( 50%) of quirks that can not reach the ECal (HCal) of the CMS detector within the bunch cross time, which is 25 ns. Due to the heaviness of the quirk, its energy losses in both ECal and HCal are found to be small ( 10 GeV).
The main background for the quirk signal will be abundant pileup events (which we take µ = 50) as well as the SM Z(→ νν)+jets process. We proposed a dedicated hit reduction algorithm to remove the SM particle hits while keep the quirk hits as much as possible. The quirk plane is reconstructed for each event based on the remaining hits. Subsequently, three discriminative variables can be defined: the number of remaining hits within the d < 30 µm of the quirk plane,distance-weighted ionization energy loss in the tracker and the variation of the S p . Together with the missing transverse energy, they can provide very sensitive probe for the quirk signal, we found the ∼ 100 fb −1 dataset at the LHC will be able to probe the colored fermion/scalar quirks with masses up to 2.1/1.1 TeV, and color neutral fermion/scalar quirks with masses up to 450/150 GeV, respectively.