Abstract
Dark sector may couple to the Standard Model via one or more mediator particles. We discuss two types of mediators: the dark photon and the dark scalar mediator . The total cross-sections and various differential distributions of the processes and ( and b quarks) are discussed. We focus on the study of the invisible due to the cleaner background at future colliders. It is found that the kinematic distributions of the two-jet system could be used to identify (or exclude) the dark photon and the dark scalar mediator, as well as to distinguish between them. We further study the possibility of a search for dark photons at a future CEPC experiment with GeV and 240 GeV. With CEPC running at 91.2 GeV, it would be possible to perform a decisive measurement of the dark photon (20 GeV 60 GeV) in less than one operating year. The lower limits of the integrated luminosity for the significance 2, 3 and 5 are presented.
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1. Introduction
Signals of non-baryonic dark matter (DM) in the Universe have been identified in a number of astrophysical and cosmological observations, such as the Cosmic Microwave Background anisotropy measurements, galactic rotation curves, large scale structure surveys, X-ray observations and gravitational lensing [1-11]. The contribution of DM is nearly 75% of the total matter in the Universe. Specifically, the Planck data give the value of the relic density of DM of [1]. DM influences the dynamical effects from the scale of a galaxy up to the cosmic scale, and plays a crucial role in the galaxy rotation curve and the formation of structures in the Universe. However, the nature of the DM particles remains a mystery and has become one of the most important challenges of modern science. The underlying physics of DM particles is explored by various worldwide projects, such as the direct and indirect searches, collider experiments and astrophysical signatures arising from DM self-interactions [12-15].
Given the intricate structure of the Standard Model (SM), which describes only a sub-dominant component of the Universe, it would not be surprising if the dark sector contains itself a rich structure, with DM making only a part of it. In the dark sector, the DM particles do not interact directly with the known forces, except with the gravitational force. However, there are typically one or more mediator particles which are coupled with SM and act as a "portal" [16-21]. Such extended interactions associating the dark sector and SM depend on the spin and parity: the mediators can be vector , scalar , pseudoscalar a, axial-vector and even fermions N.
A new force mediated by dark photons has been a subject of considerable interest in high energy physics. The existence of the dark photon [22-24], associated to a hidden gauge interaction, has been the subject of many investigations, both theoretical and experimental. Substantial effort has been invested in the search for dark photons using various processes including bremsstrahlung [25-28], meson decays , , and [29-31], the Drell-Yan process [32,33], annihilation [34-38], etc. Stringent limits for the kinetic mixing parameter for a given dark photon mass have been obtained [17,18,24,39,40]. For GeV, only limited values of are allowed. However, for a heavy dark photon, a wide range of mixing parameter values has not been excluded by the current experiments.
Future high-energy electron-positron colliders provide an opportunity to search for the dark sector mediators. These colliders include CEPC [41], ILC [42], FCC-ee [43] and CLIC [44], with the center-of-mass energy varying from 91.2 GeV to 1 TeV. Assuming that dark mediators interact only with quarks, we investigate in this work the production of dark photon and dark scalar mediator at electron-positron colliders with 91.2 GeV, 240 GeV, 500 GeV and 1 TeV. We analyze the cross-sections and the normalized kinematic distributions of the processes and , and focus on the invisible due to a cleaner background. The corresponding background processes are also simulated.
The paper is organized as follows. In Sec. 2, we present a simple theoretical framework for the dark photon and dark scalar mediator. In Sec. 3, we investigate the production of dark photon and dark scalar mediator at future colliders, and discuss how to distinguish between them. In Sec. 4, we study the discovery potential of dark photons at a CEPC experiment. Finally, a short summary is given.
2. Dark photon and dark scalar mediator
In a simple extension of SM, one can introduce a as an extra gauge group. The gauge boson arises from the extra gauge group, which can be coupled weakly to electrically charged particles by "kinetic mixing" with the photon [22-24]. Kinetic mixing produces an effective parity-conserving interaction of with the electromagnetic current , suppressed relative to the electron charge by the parameter [18]. The gauge boson or dark photon play the role of the "vector portal" connecting the SM and DM particles. We assume that the dark photon only interacts with the DM particles and SM quarks. After diagonalization of the kinetic mixing term, the Lagrangian of the dark photon model is [19-21]
where , and denote the masses of SM quarks, DM particle and dark photon, respectively. is the charge of the quarks. and are the field strengths of the ordinary photon A and the dark photon , is the kinetic mixing parameter in the physical basis, is the coupling parameter between the dark photon and the dark sector, and is the dark fine structure constant.
A number of experiments have proposed restrictions on the mixing parameter [17,18,24,39]. However, for the dark photon mass 1 GeV, a wide range of mixing parameter values has still not been excluded by the current experiments. We can extract the maximum value of from the direct DM detection experiments. The differential cross-sections for DM particle-nucleon scattering in the non-relativistic limit can be written as [24,45,46]
where is the nuclear recoil energy, is the velocity of the DM particle in the nucleon rest frame, is the electromagnetic fine structure constant, is the mass of the target nucleus, is the number of protons in the target nuclei, and is the Helm form factor [47,48]. The dark fine structure constant can be determined from the relic abundance of DM. When is determined, the combined coupling parameter can be constrained from the experimental data by evaluating the function , where is the likelihood function [49,50]. Fig. 1 shows the 90% C.L. upper limits of the combined parameter with 8.6 GeV (CDMS-II-Si favors a DM mass of 8.6 GeV [51]), and 100 GeV constrained by the CDEX-10 [52], PandaX-II [53], DarkSide-50 [54] and XENON-1T [55] data.
Alternatively, in the dark scalar mediator model, the DM particles can interact with the SM particles through the "Higgs portal" [19,20]. The corresponding Lagrangian can be written as,
where H is the SM Higgs doublet, is the corresponding vacuum expectation value, and , are three parameters. In the case of and 0, after electroweak symmetry breaking, the relevant DM and mediator Lagrangian takes the following form,
where the interaction between SM particles and DM particles are mediated by Higgs-singlet mixing, i.e., the scalar exchange. We assume that the dark scalar mediator directly couples to the SM quarks q. The dark scalar mediator plays a crucial role in the "scalar portal". The mixing term can be written as . We choose for simplicity.
3. Production of dark photon and dark scalar mediator
In this section, we investigate the production of the massive dark photon and of the massive dark scalar mediator via the processes and ( and b) at the center-of-mass energies 91.2 GeV, 240 GeV, 500 GeV and 1 TeV, with different values of and . The Feynman diagrams for the production of and associated with two jets at colliders are shown in Fig. 2.
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Standard imageIn order to obtain the analytical amplitudes, we use FeynArts [56] and FeynCalc [57] to generate the Feynman diagrams and perform the calculations. We use the multidimensional numerical integration package Cuba [58] to analyze the kinematic distributions. The cross-sections of the processes ( ) are suppressed by factors of ( ). In order to see the general trend, we show the reduced cross-sections of the two processes as function of and or in Fig. 3. Fig. 3 (a) and (b) exhibit peaks due to the contribution from the resonant boson production. Taking GeV as an example, the cross-section decreases by about three orders of magnitude when increases from 91.2 GeV to 1 TeV. Fig. 3 (c) and (d) show that the reduced cross-sections become smaller as the mass becomes bigger. It is worth noting that since the value of the coupling parameter ) varies with the mass , the shape of the cross-section changes with ( ) when the mass dependent ( ) is used. As we focus on the production of invisible dark photons and dark scalar mediators at colliders, one can identify them by reconstructing the missing momentum, i.e. the recoil of two final jets. The four-momentum of the two-jet system is used to infer the characteristics of the two processes. Fig. 4 shows the normalized , , and distributions of the two-jet system for the processes (left panels) and (middle panels) for several and ( ) without any kinematic cuts. Here, is the transverse momentum of the two-jet system and is the invariant mass, is the angle between the momentum of the two-jet system and the particle beam axis, and is the rapidity of the two-jet system. For comparison, we use MadGraph [59] to analyze the kinematic distributions of the dominant background processes ( ), which is shown in Fig. 4 (right panels). For , the distributions of the background exhibit two peaks around 91 GeV and 125 GeV due to the contributions of the resonant and the Higgs boson. However, for , the peak is not obvious, because we have set the minimum transverse momentum of the jets to 0.5 GeV. From Fig. 4, one can see that the kinematic distributions of the two processes are somewhat different. We further investigate these distributions as function of and in Fig. 5 for several and ( ) values. Compared with the scalar mediator, the distributions for the dark photon are restricted to a smaller area. For example, for 91.2 GeV, the dominant area for is and (0.9, 1) with , while the area for is comparatively broader. For higher center-of-mass energies , this trend is even more obvious.
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Standard imageDownload figure:
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Standard imageAs mentioned above, the kinematic distributions of the dark photon and dark scalar mediator are different. The difference can be enhanced by imposing appropriate kinematic cuts on the and distributions. As we show below, there are significant differences between the production of the dark photon and dark scalar mediator at colliders.
In order to show the difference in , we impose a cut on such that , presented in Fig. 6. In the case of 91.2 GeV and = 20 GeV and for , the distributions of are attenuated as increases, while the distributions of are substantially flat in this region. In the case of 1 TeV and = 50 GeV, the differences of the distributions of the two processes become much easier to identify. For , the distributions of are monotonically attenuated as increases. However, the distributions of first increase quickly, and then decrease slowly in the same region. We also display the transverse momentum distributions of the background with the same cuts, which show quite a different shape for .
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Standard imageAs examples of distributions, we present in Fig. 7 the differential cross-sections of the two processes for = 91.2 GeV, 240 GeV, 500 GeV, 1 TeV and = 20 GeV or 50 GeV. It can be seen that the cuts of can enhance the difference between the dark photon and dark scalar mediator. Imposing the cuts 20, 50,100 and 240 GeV for the above center-of-mass energies, we find that the differential distributions of reach a maximum around , with the inverted "U" shape. However, for the scalar mediator, the maximum of the peak lies around , and the shape looks like the letter "M". The shape of the angular distribution of the background with the same cuts varies dramatically for typical values.
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Standard image4. Identifying the dark photon signal against the background
Future colliders are expected to play a crucial role in discovering the nature of DM (dark sector) particles since they have a cleaner background. In this section, we focus on how to identify the heavy dark photon signal against the expected background at a future CEPC experiment. The analysis is similar for the dark scalar mediator . In the dark photon model of Eq. (1), can decay into a pair of SM quarks and a DM pair. The related decay widths are defined as
where is the charge of the quarks. The branching ratio of can be written as
which is related to and , while the combined parameter can be obtained from Fig. 1. Here we choose = 8.6 GeV, , we extract from the XENON-1T curve in Fig. 1, and obtain the branching ratios of listed in Table 1. In the following, we study the process with due to its cleaner background. The dominant background process is ( , and ). In the final states of both the signal and background processes, we observe only two jets. The background process is simulated by MadGraph [59]. The invariant mass of the dark photon can be reconstructed from the recoil four-momentum of the two-jet system, where is defined as,
Table 1. The mixing parameter and the branching ratios of as function of the dark photon mass , for of 8.6 GeV and of 0.032.
20 GeV | 30 GeV | 40 GeV | 50 GeV | 60 GeV | |
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0.0030 | 0.0067 | 0.012 | 0.019 | 0.027 | |
Br | 0.996 | 0.985 | 0.955 | 0.898 | 0.809 |
where , , and are the four-momenta of the incoming electron, positron and the two jets in the final states, respectively. We focus on the light quark jets ( u, d, s, c and b) since the top quark decays quickly.
Theoretically, the on-shell dark photon events can be reconstructed precisely at in the invariant mass spectrum. However, the detector has a finite energy resolution, which results in bump structures in the spectrum. To make our estimate more realistic, we simulate this effect by smearing the jet energies assuming a Gaussian resolution,
where is the energy resolution, A is the sampling term, B a constant term, and denotes the sum in quadrature. According to the CEPC CDR [41], the energy resolution for light jets ranges from 6% at E=20 GeV to 3.6% at 100 GeV. We adopt the parameters A = 25.7% and B = 2.4%. The smearing effect is introduced in the same way in the reconstruction of the background events.
In order to identify the dark photon signal against the background, we need to impose proper kinematic cuts. The cuts are based on the kinematic distributions of the signal and background processes. We set the basic transverse momentum cut at 10 GeV and the rapidity cut at 4. In order to identify an isolated jet, the angular distribution between jets i and j is defined by
where ( ) denotes the azimuthal angle (rapidity) difference between the two jets. In the two-jet system, we set the basic cut at for both the signal and background processes.
In Fig. 8, we show the differential cross-section as function of the invariant mass of the dark photon for 20, 30, 40, 50 and 60 GeV, with the smearing and the above kinematic cuts. The reconstructed signal has a shape that complies with a Gaussian distribution with the expectation of and the standard deviation of the energy resolution of . In contrast to the case of 91.2 GeV, the signal at 240 GeV has a wider spread since is larger.
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Standard imageIn order to identify the dark photon signal against the background, the significance of the signal-to-noise ratio needs to be explored. To enhance the significance, we impose the following cuts on the invariant mass spectrum: 6 GeV at 91.2 GeV, and 12 GeV at 240 GeV. For 91.2 GeV and with the CEPC integrated luminosity of and for several values, we estimate the number of events for the signal ( ) and background ( ) processes, as well as the significance , as listed in Table 2. It can be seen that for 20, 30, 40 and 50 GeV, the significance is greater than 3 .
Table 2. Number of events for the signal ( ) and background ( ) processes and the significance for the integrated luminosity at 91.2 GeV, with the smearing and proper kinematic cuts.
20 GeV | 30 GeV | 40 GeV | 50 GeV | 60 GeV | |
---|---|---|---|---|---|
191 | 368 | 372 | 206 | 46 | |
2503 | 3697 | 3636 | 2304 | 799 | |
3.82 | 6.05 | 6.17 | 4.29 | 1.63 |
In the case of the CEPC operating energy of 240 GeV, we adopt a higher integrated luminosity of . The number of events for the signal and background processes and the significance are given in Table 3. In comparison with Table 2, we obtain a much smaller number of dark photon events. This is understandable since for , the cross-section decreases with the center-of-mass energy for 91.2 GeV, as demonstrated in Fig. 3 (a) and (c). In addition, we obtain many more background events for 240 GeV than for 91.2 GeV. This is due to the new topology of Feynman diagram for the background process shown in Fig. 9, whose contribution increases with . This topology is excluded in the signal since we assumed that the dark photon interacts only with quarks.
Table 3. The same as Table 2, but for , 240 GeV and 12 GeV.
20 GeV | 30 GeV | 40 GeV | 50 GeV | 60 GeV | |
---|---|---|---|---|---|
2 | 10 | 23 | 39 | 53 | |
60252 | 114953 | 210674 | 380295 | 682870 | |
0.00815 | 0.0295 | 0.0501 | 0.0632 | 0.0641 |
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Standard imageAs an additional element relevant for a future CEPC experiment, we present the significance versus the integrated luminosity for 91.2 GeV and 240 GeV in Fig. 10. In the case of 91.2 GeV, the minimum integrated luminosities for the 3 discovery of the dark photon with 20, 30, 40, 50 and 60 GeV are 1.23, 0.490, 0.473, 0.971 and 6.67 , respectively. Hence, it is understandable why the dark photon signal was not found at the Large Electron-Positron (LEP) collider, since the total luminosity of the LEP experiments [60] did not reach the minimum integrated luminosity for the 3 discovery of the dark photon with 20 GeV 60 GeV. At CEPC with 91.2 GeV, the yearly luminosity is expected to be for a single interaction point (CEPC will have two interaction points), and it would be possible for a CEPC experiment to perform a decisive measurement of the dark photon (20 GeV 60 GeV) in less than one operating year. In the case of 240 GeV, the minimum integrated luminosities required for one signal event with the above values are 7.06, 1.91, 0.853, 0.508 and 0.374 , respectively. Therefore, with CEPC running at 240 GeV and a luminosity of for a single interaction point, it would be hardly possible to get any signal of the dark photon (20 GeV 60 GeV) in one operating year.
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Standard image5. Summary
The dark sector may consist of not only DM but also of one or more new force-carrying mediators which couple to the SM particles. We discussed the vector dark photon and the scalar mediator which could be produced in the processes and at future colliders. The production cross-sections of these processes were predicted for = 91.2 GeV, 240 GeV, 500 GeV and 1 TeV. We further studied the kinematic distributions of the two-jet system in the final state, and found that they could be used to identify (or exclude) the dark photon and the dark scalar mediator, as well as to distinguish between them. In this work, we only considered the interaction between the dark photon and quarks, and with the process as an example, we investigated the discovery potential of the dark photon at CEPC with = 91.2 GeV and 240 GeV. It was shown that the dark photon with ranging from 20 GeV to 60 GeV might be discovered in the process at colliders, e.g. at the super-Z factory or CEPC, with the minimum required integrated luminosity for the 3 discovery of about 0.473~6.67 . If the interaction between the dark mediator and leptons is also considered, and could be the other interesting processes to study, where the production would be the background. The method proposed in this work could also be used to search for any other invisible particles in annihilation.
Footnotes
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This work was supported in part by the National Natural Science Foundation of China (11875179, 11325525, 11635009, 11775130, 11905112), the Natural Science Foundation of Shandong Province (ZR2017MA002, ZR2019QA012) and the Fundamental Research Funds of Shandong University (2019GN038)