Complementarity between dark matter direct searches and CE$\nu$NS experiments in $U(1)'$ models

We explore the possibility of having a fermionic dark matter candidate within $U(1)'$ models for CE$\nu$NS experiments in light of the latest COHERENT data and the current and future dark matter direct detection experiments. A vector-like fermionic dark matter has been introduced which is charged under $U(1)'$ symmetry, naturally stable after spontaneous symmetry breaking. We perform a complementary investigation using CE$\nu$NS experiments and dark matter direct detection searches to explore dark matter as well as $Z^{\prime}$ boson parameter space. Depending on numerous other constraints arising from the beam dump, LHCb, BABAR, and the forthcoming reactor experiment proposed by the SBC collaboration, we explore the allowed region of $Z^{\prime}$ portal dark matter.

However, here we are interested to investigate dark matter (DM) using these neutrino-nucleus scattering experiments. It has been known in literature that there exists numerous amount of astrophysical and cosmological evidence for the existence of DM in the universe, but so far there is no experimental result, neither from direct/indirect detection nor from accelerator-based experiments, that supports its existence.
Among the many dark matter candidates, the thermally produced Weakly Interacting Massive Particle (WIMP) has received much theoretical and experimental scrutiny. Many dark matter direct detection experiments, such as the Xenon1T [25] or PandaX-II [26] experiments for example, have also set stringent limits on the DM -nucleus cross-section. In particular, there exists some interesting light mediators models connecting dark photons with the light dark matter, as pointed in [27][28][29][30][31][32][33]. Moreover, the COHERENT collaboration has recently studied sub-GeV dark matter models using detector sensitive to CEνNS processes in Ref. [34].
In this work, we exploit an extension of the SM consisting of an additional anomaly-free U (1) gauge symmetry with a very light gauge boson, Z , and a dark matter candidate, stable due to a residual symmetry of U (1) . There exists very well-motivated studies, where a new U (1) gauge symmetry arises, namely, in the context of supersymmetry [35][36][37], grand unified theories [38][39][40], and string-motivated models [41,42]. We will study different scenarios for the U (1) where dark matter is charged under this symmetry, quarks have flavor independent charges, while leptons are allowed to have generation-dependent charge, namely U (1) B−L , U (1) B−2Lα−L β , and U (1) B−3Lα .
In this way the new gauge boson couples to quarks, leptons, and dark matter. If the interaction mediates both processes, namely dark matter -nucleus scattering and an extra contribution to neutrino -nucleus scattering, there will be a correlation between both cross-sections. In particular, if an extended gauge symmetry is considered to mediate both processes, the couplings are also correlated through the gauge coupling and the vector boson mediator mass. The light gauge boson will mediate the interaction between dark matter and nuclei and it will also contribute to the interaction of a neutrino with nuclei. We will focus on the complementarity of the searches for light gauge bosons and dark matter from colliders and CEνNS experiments.
Furthermore, considering the combined effect of different experimental constraints coming from the COHERENT collaboration, beam-dump experiments, the LHCb and the BABAR dark photon searches, we examine the allowed parameter space to constrain Z boson and dark matter for each U (1) models. In addition, we also explore the potential of upcoming reactor-based CEνNS experiment proposed by the Scintillating Bubble Chamber (SBC) collaboration [43,44]. Bounds arising from astrophysical observations such as Big Bang Nucleosynthesis (BBN) and Cosmic Microwave Background (CMB) have also been shown for the comparison.
We organize this work as follows. Sec. II is dedicated to a brief description of CEνNS experiments. Dark matter in U (1) models has been discussed in Sec. III. We discuss the DM relic density and direct detection process in Sec. IV. Our principal results are illustrated in Sec. V.
Summary of this work and concluding remarks are presented in Sec. VI.

II. CEνNS PROCESSES AND Z BOSON
The coherent elastic neutrino-nucleus scattering (CEνNS) was measured by the COHERENT experiment [1] using neutrinos from a stopped-pion source at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory. The COHERENT collaboration used two different detectors, one made of CsI [1] and the other one made of LAr [45].
Furthermore, there are several experiments trying to measure CEνNS using reactor antineutrinos, but the need of sub-keV thresholds and great control over background has made this task difficult. One of these experimental proposals is a liquid argon scintillating bubble chamber, currently under construction by the SBC collaboration [43,44]. With an expected energy threshold of 100 eV, a 10-kg chamber placed near a 1-MW th nuclear reactor will be able to measure SM parameters with high precision, such as the weak mixing angle, in addition to set competitive limits on some beyond SM scenarios [44].
The SM differential cross-section for CEνNS process is given by [46][47][48] where T is the nuclear recoil energy, E ν is the incoming neutrino energy, and M N is the nuclear mass. The weak nuclear charge, Q w , is given by 1 where g V p = 1/2 − 2 sin 2 θ W and g V n = −1/2 are the SM weak couplings, Q is the transferred momentum, Z(N ) is the proton (neutron) number of the nucleus, and F Z(N ) (Q 2 ) the nuclear form factor. Since g V p ∼ 0.02, the cross-section depends highly on the number of neutrons N . In presence of a new vector interaction, the cross-section for CEνNS is affected through the weak nuclear charge (see Eq. (2)) in the following way: where g and M Z are the coupling and mass of the new gauge boson, respectively. Therefore, by comparing this effective Lagrangian with the one from an effective field theory approach, we can relate the parameters with the Z interaction parameters as .
In order to extract limits on the Z parameters from the measurement by the COHERENT collaboration with the CsI detector, the following χ 2 function can be used [1] where N i meas and N i th are the measured and theoretical predicted number of events per energy bin, respectively, σ i stat = N i meas + B i on + 2B i ss is the statistical uncertainty of the measurement, and B i on (B i ss ) is the beam-on (steady-state) background. The systematic uncertainties of signal and background normalization are encoded in σ α = 0.28 and σ β = 0.25, respectively. The function in Eq. (6) has to be marginalized over α and β, which are nuisance parameters. Recently it has been shown that the limits for a light vector mediator obtained from the COHERENT measurements with the LAr detector are similar to the ones obtained with the CsI data [23,49,50]. Therefore, in this work we will present bounds only from the CsI measurement.
For the case where there is no current measurement, such as with the SBC-CEνNS detector, one can assume the measured signal as the SM expectation plus background. Hence, the projected sensitivities on the Z parameters can be obtained with the χ 2 function where B reac is the background due to the nuclear reactor. Here, the statistical uncertainty is defined as σ stat = √ N meas + 4B cosm , with B cosm the background from muon-induced and cosmogenic neutrons. The nuclear recoil threshold is set to (1+γ)·100 eV, where γ is an additional nuisance parameter, while the systematic uncertainties associated to the signal, background, and energy threshold are set to σ α = 0.024, σ β = 0.1, and σ γ = 0.05, respectively.   Table I. Scalar singlets φ i are added in order to spontaneously break U (1) symmetry as given by Table II It is well known that if we extend the SM with a U (1) symmetry spontaneously broken by a scalar field with a integer charge, it is possible to have a residual symmetry Z N . This happens if there is a scalar or fermion field with fractional U (1) charge [56]. This mechanism can be responsible for the DM stability and depending on the symmetry it can also tell us if neutrinos are Dirac or Majorana particles [57,58]. Consider introducing a SM singlet fermion χ which can be Dirac or Majorana, depending only on its U (1) charge. To avoid spoiling the anomaly-free nature of the model a vector-like pair of χ, χ L and χ R , is needed. If χ L/R transform as 1/2, they will be a pair of Majorana fields and their mass will be provided once the U (1) is broken by a flavon field φ transforming as 1 under U (1) . On the other hand, if χ L/R transform as 1/3 they will form a Dirac fermion. Thus, we end up with a residual Z 3 symmetry, which stabilizes the DM.
For the choice of Dirac fermion charge 1/3 the resulting dark matter mass eigenstate is given We can write the dark sector Lagrangian as follows The only coupling of χ is therefore to the Z gauge boson. The relic density of dark matter may be determined by this coupling in the freeze-out regime, through the Z mediated χχ → f f channel, where f is a SM fermion, as well as the χχ → Z Z channel. The Z channel also provides a tree-level spin-independent direct detection signature. It is well known that these two constraints We implemented the U (1) models in LanHEP [67] and micrOMEGAs [68] to calculate the dark matter observables, scanning over the ranges ( For the different models we will need different singlet scalar fields φ i transforming as i under the U (1) , as shown in

IV. RELIC DENSITY AND DIRECT DETECTION
In this work we investigate the possibility of χ being a thermally produced WIMP dark matter candidate, taking into account experimental Z constraints to its possible couplings to the SM fields. The relic density of χ is determined by the thermal freeze-out with resonant enhancement of the Z mediated annihilation cross-section. The kinematically allowed processes are χχ → f f for SM fermions f with masses M f > M χ , as shown in Fig. 1 (see left panel). The t-channel χχ → Z Z process is kinematically forbidden by the resonance condition 2M χ ∼ M Z . In the parameter space considered here, the leading contributions to the relic density determination are the annihilation into neutrinos, followed by charged leptons, while quarks contribute O(1 − 10%).
The resonant condition allows lower values of g compared to the non-resonant case down to g ∼ 10 −6 for a 50 MeV Z mass, for example. We filter the data to reproduce the observed relic density Ω CDM h 2 = 0.1198 [69].
The gauge coupling of DM to the Z leads to tree level spin-independent (SI) scattering of DM with nucleons. The Feynman diagram corresponding to this process is shown in Fig. 1  The dark matter-nucleus SI cross-section per nucleon, in the small momentum transfer limit, is given by [70] σ SI ≈ µ 2 where, µ χn is the WIMP-nucleon reduced mass, Z, A are the atomic number, atomic mass of the target nucleus, respectively. Also, f p and f n represent proton and neutron scattering functions and are given by In this study, . Therefore, the spin-independent cross-section reduces to Due to the vector-like nature of χ, axial couplings to Z of the form g A χγ µ γ 5 Z µ χ are not present  Fig. (2). The constraints arsing from the COHERENT-CsI data has been presented using the light-purple shaded region. The exclusion regions arising from the future reactor-based CEνNS experiment SBC-CEνNS [43,44] are shown using the black long-dashed line. In order to present results for these CEνNS experiments in the (M Z , g ) plane, we perform a χ 2 test using the latest COHERENT-CsI 10 -2. 10 -1. 10 0. 10 1. 10 -6.
II, and final exclusion limits are presented at 95% confidence level. Regarding the limits from dark matter direct detection experiments, we show these bounds using the light-orange, light-cyan, and light-yellow region for the SBC-DM, XENON1T and PandaX-II experiments, respectively.
Moreover, the light-red regions represent the exclusion limit of the relic density calculation for the given model, due to an overabundance of dark matter.
Bounds arising from the calculation of ∆N eff by the BBN + CMB [71] measurements are shown using the vertical red band. It is to be noted that different electron beam dump experiments like E141 [74], E137 [73], E774 [75], KEK [76], Orsay [80], and NA64 [78], which put bounds on dark photon searches, can also put bounds on masses and couplings of Z boson. Moreover, proton beam dump experiments like ν-CAL I [77], proton bremsstrahlung [82], CHARM [72], NOMAD [79], and PS191 [81] can also set bounds on Z boson searches. We consider these bounds from the literature. In order to recast these limits, Darkcast [86] code has been utilized to our specific model, and the combined electron and proton beam dump limits are presented using light-green region at 90% confidence level.
We consider limits set by the proton-proton collider LHCb [84], where Z arising from U (1) sym-metry decays to µ + µ − . Similarly, we also consider Z production in the electron-positron collider BABAR [83]. For both the LHCb and BABAR experiments, we have utilized Darkcast [86] code to recast their results as shown using the sky-blue and light-brown regions at 90% confidence level, respectively. Notice that for the BABAR, one could also have bound on µ + µ − production [87], i.e., for scenarios where we have x e = 0, x µ = 0. However, it has been observed that those bounds are much weaker compared to our COHERENT-CsI analysis or LHCb bounds [84]. Hence, throughout this work we only entertain bounds arising from the BABAR for x e = 0.
We notice from the left panel of Fig. (2) that the forthcoming CEνNS experiment SBC-CEνNS will be able to produce the most stringent constraint for Z masses between (0.02 − 1. 3) GeV and couplings in the range (10 −5 − 4 × 10 −4 ). The reason for this is the high antineutrino flux (10 5 times higher than the one from the SNS) given the considered 1 MW th power reactor and 3 m baseline [51,52], the very low energy threshold achieved by the detector, and the fact that the background is greatly reduced due to the detector insensitivity to electron recoils, for more details see [44]. It can be observed that the SBC-CEνNS will be able to constrain (M Z , g ) parameter space almost an O(1) stronger than the latest constraint provided by the COHERENT-CsI data.
For the mass range between (0.2 − 1.3) GeV the future SBC-CEνNS bounds will be competitive with the current LHCb exclusion limits. However, constraint coming from the BABAR collaboration puts the most stringent bounds in the range (0.5 − 10) GeV of M Z , whereas above 10 GeV the LHCb again shows the best exclusion limits. Interestingly, for coupling of Z boson ∼ 10 −4 or smaller are concerned, it can be seen that the BBN+CMB calculations as well as beam dump bounds are able to ruled out a significant amount of the parameter space for M Z less than 0.1 GeV (see shaded green and red regions). On the other hand, for M Z greater than 0.1 GeV, it is the relic density calculations that can rule out most of the parameter space for smaller g as shown by the light-red regions.
We observe further that the forthcoming SBC-DM dark matter constraint will surpass the present bounds provided by XENON1T and PandaX-II, respectively. It is to be noted further that in this scenario, the right-handed neutrino mass matrix is generated dynamically, once the the scalar field φ 2 takes it's vev by breaking of U (1) B−L symmetry. For the same model we also show the corresponding plot in the (M χ − σ SI ) plane in the right panel of Fig. (2), showing the same experimental constraints. The CEνNS process from solar, atmospheric and diffuse supernova neutrinos constitute an unavoidable background for dark matter direct detection searches. We have indicated this background [85] with a dotted magenta curve in the results, as a discovery limit, not an exclusion region. We notice that the bounds from 10 -2. 10 -1. 10 0. 10 1. 10 -6.

PandaX
FIG. 4. Same as Fig. (2) but for the MIII U (1) models as given by Table (II). BABAR and LHCb constrain the parameter space down to the neutrino floor of direct detection experiments, for dark matter masses below M χ ∼ 8 GeV.
Having discussed the flavor-independent B-L model, we now proceed to explore parameter space for the flavor-dependent B-L models, which are presented in Fig. (3). The four scenarios investigated here correspond to the model MII as given by Table (II) and depending on U (1) charges (see Table (I respectively. It is important to point out here that these four scenarios have very sharp predictions for the leptonic sector. All these cases lead to two-zero textures in the neutrino mass matrix, and are consistent with the latest global-fit of neutrino oscillation data, detailed phenomenology of these cases for the leptonic sector have been carried-out in [23]. Notice that beam dump experiments, BABAR as well as reactor experiment SBC-CEνNS only provide bounds for the electron channel, therefore we find their impact only for the cases where one has non-zero x e as can be seen from the first and second row of the Fig. (3). Similarly, the LHCb dark-photon searches are relevant for non-zero x µ , and hence one sees its presence for three cases In Fig. (4), we discuss our numerical results for U (1) models corresponding to the model MIII as given by Table (II). All scenarios of models MIII have correlations for the neutrino masses and mixing parameters, see appendix A, and are compatible with the latest neutrino data.
As per the constrained parameter space is concerned, Fig. (4) shows almost similar pattern like

VI. CONCLUSIONS
We have discussed scenarios for a light gauge boson Z , where the SM fermions and the DM interact with such a gauge boson. In these scenarios, the DM stability is due to a residual symmetry from the spontaneous breaking of U (1) → Z N . We have found that in order to have the correct thermal DM relic density, the DM annihilation in the early Universe must occur resonantly, i.e. DM mass should satisfy M χ ≈ M Z /2. Firstly, the flavor-independent U (1) model i.e., B − L scenario has been studied, where all leptons have the same charge. Later, different flavorful scenarios have also been analyzed, where different lepton flavors carry different U (1) charges. Since the quarks also transform under the extra gauge symmetry, the Z couples to the nucleons, and therefore, we have investigated constraints arising from CEνNS (when the electron and/or muon flavors are charged) as well as from DM direct detection experiments. However, it goes without saying that any new physics scenario beyond the SM undergoes various phenomenological constraints coming from numerous particle physics experiments. Using various other limits, coming from the beam dump, LHCb, and BABAR, we have first constrained M Z − g allowed parameter regions. Later, we have translated all these limits in the M χ − σ SI plane using the correlation between the DM mass (M χ ) , spin-independent DM direct detection cross-section (σ SI ) together with gauge coupling (g ).
Our noteworthy results are summarized in Figs. (2, 3, 4, and 5). Investigating all scenarios, it has been observed that the most stringent parameter space is obtained for the U (1) model with x e = 0. For the particular scenario, we have bounds from the CEνNS , beam dump, and BABAR together with DM direct detection experiments. As an example, from the first panel of GeV. Also, we have observed that the future SBC-CEνNS can put almost an order of magnitude stronger constraint compared to the latest COHERENT-CsI bound. On the other hand, the most unexplored regions are seen for scenarios with x e = 0, x µ = 0, as there are no bounds from the CEνNS , beam dump, LHCb, or BABAR. In this case, in the Z mass range (5.3 × 10 −3 − 0.1) GeV parameter space remains unconstrained. From the DM relic density, there is a lower bound in the M Z − g plane due to kinematics. All these constraints can be translated to the DM direct detection cross-section setting strong constraints for DM masses below 10 GeV for PandaX-II, 6 GeV for XENON1T, whereas the stringent constraint is observed for the SBC-DM searches, which puts a limit around 1.3 GeV. Finally, it is worth mentioning that for DM mass below 1. In the models B − 2L e − L µ and B − 2L e − L τ , the charged lepton and Dirac neutrino mass matrices are diagonal by construction. While it is not enough to include the φ 1 and φ 2 to reproduce neutrino masses and mixings [23]. This can be alleviated by the inclusion of a third flavon field φ 4 .
In this way the right-handed neutrino mass matrix takes the form for U B−2Le−Lµ , U B−2Le−Lτ models, respectively. Given the Dirac and Majorana neutrino mass matrices, one can construct the low-energy active neutrino mass matrices using type-I seesaw mechanism. We write the active neutrino mass matrix as