GeV-scale dark matter: production at the Main Injector

Assuming that dark matter particles interact with quarks via a GeV-scale mediator, we study dark matter production in fixed target collisions. The ensuing signal in a neutrino near detector consists of neutral-current events with an energy distribution peaked at higher values than the neutrino background. We find that for a $Z'$ boson of mass around a few GeV that decays to dark matter particles, the dark matter beam produced by the Main Injector at Fermilab allows the exploration of a range of values for the gauge coupling that currently satisfy all experimental constraints. The NO$\nu$A detector is well positioned for probing the presence of a dark matter beam, while future LBNF near-detectors would provide more sensitive probes.


Introduction
Dark matter (DM) provides solid evidence for physics beyond the Standard Model (SM), but its identity remains unknown. A central question is whether DM particles experience interactions with ordinary matter beyond gravity. If DM particles have weak-scale masses and order-one couplings, then their relic abundance given by the thermal freeze-out is consistent with the observed value [1]. An extensive experimental program has been carried out to explore this WIMP (weakly-interacting massive particles) paradigm, setting impressive bounds on the viable parameter space. Direct detection experiments [2], in particular, have imposed strong constraints on the interactions between nucleons and WIMPs of mass larger than about 5 GeV.
Here we consider the possibility that DM particles have mass below 5 GeV and have interactions with quarks mediated by a new boson. If the latter is relatively light, then the DM interactions with nucleons can be probed in experiments. A natural choice for the mediator is a leptophobic Z boson of mass near the GeV scale. The constraints on a boson of this type interacting with quarks are rather loose [3,4]. We will show that the constraints remain weak even when the Z interacts with DM.
A promising search method for such light DM particles is to test if they are produced in fixed target experiments, which benefit from large beam intensities. We will focus on the case where the Z boson decays into a pair of DM particles. Proton scattering off a fixed target may copiously produce Z bosons of mass around a few GeV as long as the proton energy is larger than a few tens of GeV. Thus, the Main Injector [5] at Fermilab, which accelerates protons at 120 GeV, is well suited to test this GeV-scale DM scenario. In the NuMi beam line, where the 120 GeV protons hit a Carbon target, if DM particles are produced, then they could be detected as neutral-current events in neutrino near-detectors such as NOνA [6,7] and MINOS [8,9]. The future LBNF [10] beam line would be even better suited for this type of search, provided a detector is placed within a few hundred meters from the target.
The possibility of searching for dark matter beams at neutrino detectors has been recently proposed and explored in [11][12][13][14], especially for DM much lighter than 1 GeV, which can be produced in the Fermilab Booster beam line where the protons have an energy of 8 GeV [15]. Hadron collider experiments are also sensitive to light DM that interacts with quarks, because a pair of DM particles could be produced in association with a jet or a photon or other particles [16][17][18][19][20][21][22][23]. Quarkonium decays that involve missing energy provide another probe of the GeV-scale DM scenario [24][25][26][27][28]. We are going to compare the existing limits from these classes of experiments, and show that DM beams produced at the Main Injector may lead to thousands of deep-inelastic neutral current events in existing and future neutrino detectors. This is an example of a broader capability of high-intensity fixed target experiments to probe the existence of light hidden particles [29][30][31][32][33][34][35].
2 Leptophobic Z as portal to hidden particles We are focusing on a vector boson Z of mass M Z in the 1 − 10 GeV range, so that it can be produced by an O(100) GeV proton beam scattering off a fixed target. To that end, we extend the SM gauge group by including an U (1) z group under which the quarks are charged while the leptons are neutral. The simplest charge assignment that allows quark masses and evades constraints from flavor-changing neutral currents (FCNC) is charges given by the baryon number, U (1) B [36][37][38][39][40]. Another possibility is to assign charge 0 to the left-handed quark doublets and to charge either the downor the up-type right-handed quarks; a simple choice is U (1) ds , where d R and s R have charges +1 and −1, while all other quarks are neutral [3]. Thus, the Z boson has the following couplings to the SM quarks q = u, d, s, c, b, t, Let us also include a very long lived particle, generically labelled by χ, of mass m χ < M Z /2, which is a color singlet, electrically neutral, but charged under U (1) z .
We focus on the cases where χ is either a Dirac fermion or a complex scalar (possible DM scenarios are discussed in Section 3). Occasionally we will use the notations ψ χ or φ χ when we need to emphasize the difference between the fermion and scalar χ. If χ is a Dirac fermion, then its left-and right-handed components may have different U (1) z charges; for simplicity we will ignore this possibility though, and label the U (1) z charge of χ by z χ whether it is a fermion or a complex scalar. The Z couplings to the long-lived particle χ are 2) The partial width for the Z decay into a pair of χ particles is while for larger M Z the width increases by a factor of up to 4/3 as decay channels in- volving c quarks open up, and above 2m b by another factor of up to 5/4. For illustration we use a benchmark set of values for the parameters: The branching fractions of Z into χ particles for this set of parameters, marked with an index 0, are given in the U (1) B model by for M Z in the 3 − 3.5 GeV range, and by somewhat smaller values as M Z increases above 2M D 0 .
The Z widths into hadrons in the U (1) ds model, for M Z 3 GeV (where the phase space suppression can be neglected for decays into mesons made up of s or d quarks) are given by The branching fractions of Z into χ particles for the above set of parameters in the U (1) ds model are for M Z 3 GeV, and decrease for smaller M Z .
We now turn to deriving the constraints on the Z in the 1 − 10 GeV mass range, in the U (1) B and U (1) ds models.

Limits from monojet searches
Hadron colliders set bounds on light Z via mono-jet and mono-photon searches. For M Z < 10 GeV the strongest constraint comes from the CDF search pp → j + / E T [41], and is given by [28]: (2.10) The limits from ATLAS [42,43] and CMS [44,45] are weaker due to stronger cuts imposed on missing energy and the jet p T . In Fig. 1 we show the regions in the (M Z , g z ) plane excluded by these constraints for the benchmark values Eq. (2.6).

Invisibile quarkonium decays
The searches for an invisible Υ decay constrain the U (1) B model, while for the U (1) ds model there is no such constraint since the Z does not couple to b quarks. The Z B exchange induces an Υ → χχ decay, with [27]: (2.11) The most stringent bound on the Υ invisible branching fraction has been set by the BaBar Collaboration [46], B(Υ → invisible) < 3 × 10 −4 at the 90% confidence level.
This implies that the shaded region labelled "Υ" in Fig. 1 is excluded. Similarly, a limit for M Z near 3 GeV arises from J/ψ decays, with the limit on invisible branching fraction given by B(J/ψ → invisible) < 7 × 10 −4 [47].

Monophoton limits from BaBar data
At tree level the Z does not couple to leptons, but at one loop a kinetic mixing, −( B /2)Z µν F µν , is generated. Therefore, bounds from dark photon searches apply also to a leptophobic Z . In the 1 GeV < M Z < 10 GeV mass range the strongest constraint comes from the BaBar monophoton search reinterpreted in terms of invisibly decaying Z produced along with a single photon in e + e − collisions [48]. In the U (1) B model the kinetic mixing at the BaBar center of mass energy (E ∼ 10 GeV) is [27] B (10 GeV) ∼ 10 −2 g z , (2.12) while for the U (1) ds model the mixing is generated only below the strange quark mass and it is negligible. In Fig. 1 we present the bounds on γZ production taken from Fig.   5 of [48] and interpreted as a bound on g z using Eq. (2.12).

Anomaly cancellation versus collider limits on fermions
The inclusion of a leptophobic gauge group U (1) z requires new electrically-charged fermions which are vector-like with respect to the SM gauge group in order to cancel the gauge anomalies. These fermions acquire a mass m f through a Yukawa coupling to a scalar ϕ whose VEV breaks U (1) z . The collider limits on m f then translate then into an upper bound on the gauge coupling [3]: where z ϕ is the U (1) z charge of ϕ, λ is the Yukawa coupling, and we imposed a perturbativity bound λ 3.8.

Viable DM scenarios
Let us outline some possible scenarios which give rise in our framework to a viable DM candidate. Since we are interested in DM of mass m χ below a few GeV, direct detection bounds are currently very mild [49][50][51].
The most stringent constraint is provided by the bounds on energy injection around redshifts z ∼ 100−1000, coming from observations of the cosmic microwave background (CMB) [52][53][54][55]. This constrains the annihilation of DM into charged SM particles during recombination, and in particular rules out DM lighter than about 10 GeV if it annihilates via s-wave processes. Therefore, CMB forces the dominant annihilation to be p-wave suppressed or to go into neutrinos. In our scenario a Dirac fermion ψ χ annihilates into quarks via s-wave processes, and the thermal averaged cross section times velocity is [55] σ (3.1) Therefore, the CMB bound implies that ψ χ can be a DM particle only if it is part of a hidden sector that is more complex than the minimal model of Eq. (2.1). One possibility is to interpret the CMB bound as an upper limit on the s-wave annihilation into SM particles, that is [55]: where the ionizing efficiency factor is f ≈ 0.2 for pions. Since the annihilation is suppressed, σv 1 pb, the minimal model leads to overabundant DM, and therefore needs to be extended. A simple extension, outlined in [14], includes a scalar η that has a Yukawa coupling, dominates, and gives the correct relic abundance, e.g for y 1 ∼ 0.05, m ψχ = 1 GeV and m η = 100 MeV. This annihilation mode is p-wave suppressed and therefore CMB safe.
The η scalar can then decay into SM particles via a small Higgs portal coupling. The condition in Eq.(3.2) is satisfied for values of g z below the dashed red curves in Fig. 2.
Another way to satisfy the CMB constraint is to suppress the ψ χ relic density rather than the annihilation cross section into SM charged particles. This possibility requires a substantially larger cross section than the one corresponding to the correct relic abundance [55]: This is self-consistent if ψ χ is a subdominant DM component. Alternatively the minimal model can be part of an asymmetric DM model [56], so that Eq. The shaded region in Fig. 2, between the two lines corresponding to the two scenarios just described, is not necessarily ruled out: if both ingredients discussed above are present then the CMB constraints can be accommodated. We see that the CMB constraints are very model dependent, and hence there is still a large region of the parameter space yet unexplored which leads to robust DM scenarios where Dirac fermions interact with SM quarks via s-wave processes.
In the case of scalar DM, φ χ , the annihilation cross section into quarks is [57]: where v ∼ 0.3 is the DM velocity at freeze out. This is p-wave suppressed, and hence CMB safe. Large gauge couplings are typically required in order to achieve the correct relic abundance (which requires at freeze out σv ∼ 1.5 pb for light dark matter [58]) and these could then be possibly already excluded by the current constraints. For m χ ≈ M Z /4 and z χ = 3 the correct relic abundance is obtained with g z ∼ 0.06 (M Z /1GeV) 1

DM production through proton scattering off nucleons
Having examined the bounds on GeV-scale leptophobic Z bosons decaying into DM particles, we now proceed to discuss the potential sensitivity of proton fixed target experiments to this scenario.
We assume that the Z boson is produced on-shell and then decays into DM particles (M Z > 2m χ ). This way the DM particles are produced resonantly: pN → Z → χχ, where N indicates the nucleon inside the target. The cross section for protonnucleon scattering, computed within the parton model, is where B(Z → χχ) is the branching fraction of the Z boson into DM particles. If the vector boson is produced on-shell, the tree-level partonic cross section iŝ In the U (1) B model z q L = z q R = 1/3, so that the proton and the neutron cross sections are the same. In the U (1) ds model only the right-handed d and s quarks have nonzero charges (z 2 q R = 1 for q = d, s), leading to different proton-neutron and proton-proton cross sections: σ(pn → Z ds ) 2σ(pp → Z ds ). As a result, the average proton-nucleon cross section is material dependent. For a target of atomic mass A T and atomic number Z T , the average pN cross section is Comparing the Z production rate with the total proton-proton cross section, σ(pp), which for a 120 GeV beam is given by σ(pp) ≈ 40 mb (see fig. 46.10 [1]), we find the number of DM particles produced in the target: where N POT is the number of protons on target.
Using MadGraph 5 [59] to compute the production cross section, and FeynRules [60] to implement the Z models, we find the number N T χ of produced DM particles shown in Fig. 3 for 10 21 protons on target. We focus on M Z > 2 GeV because the validity of the parton model is questionable in the case of lighter Z production. The Z ds line shown in Fig. 3 corresponds to an isospin-symmetric target (A T = 2Z T ). More generally, the number of DM particles produced has only a mild dependence on Z T /A T (and is material independent in the Z B model).
The value of the gauge coupling used in Fig. 3 is g z = 0.1; for other values, N χ scales as (g z /0.1) 2 . The branching fraction for Z → χχ used in Fig. 3 is 100%; more realistic choices, discussed in Section 2, depend on m χ , z χ and on whether χ is a fermion or a scalar (in the case where χ is a Dirac fermion and z χ = 3, the branching fraction is large, of about 87% for Z B and 75% for Z ds ).

DM flux through detectors
We now proceed to compute how many of the produced dark matter particles pass through detectors, as well as their energy distribution. We will discuss both off-axis and on-axis detectors, with examples given by the NOνA and MINOS near detectors.

Angular distribution of DM particles
Let us denote the polar angle in the lab frame (i.e., the angle between the direction of one of the DM particles produced in Z decays and the beam direction) by θ. Using the output of the MadGraph simulation, we obtain the polar angular distributions shown in Fig. 4 in the cases where the DM particle is a Dirac fermion or a complex scalar. The fermion angular distribution is more spread and peaked towards smaller angles than the scalar one. This different behavior can be understood by considering the scalar and    Table I we list the approximate angular cuts required for a DM particle produced in the absorber or the target to reach the MINOS or NOνA near detectors.
We compute the geometrical acceptance of the detector, det , by imposing angular cuts on the DM particles produced in the simulated events. In the case of a Dirac fermion, we find det O(10 −3 ) both for MINOS and NOνA near detectors for particles produced at the absorber, while for particles produced at the target the acceptance is smaller by an order of magnitude due to the larger distance. The values of the acceptance are given in Table I for M Z = 3 GeV. For a scalar, dσ/dθ vanishes faster for θ → 0, so that the acceptance of on-axis detectors is suppressed. Therefore, this offers a possibility to measure the spin of a discovered DM particle via a parallel MINOS and NOνA analysis.
Higher-order processes that include real radiation, pp → Z + jets, can potentially change the scalar angular distribution. As a crude approximation, we computed the tree-level production of Z together with one or two hard jets, imposing a jet-p t cut of 1 GeV, and we found that these processes are not large enough compared to pp → Z to change qualitatively the above result.

Energy distribution of DM particles
The where DM transfer a significant energy to nucleus, and imposing a cut on the energy of the hadronic activity.
The difference between energy distributions of DM and neutrinos is less pronounced for on-axis detectors such as MINOS, where the neutrinos have a wider energy distribution with a long tail [8,9,61]. However, a dedicated search for MINOS near detector would also be possible and highly interesting, particularly for reasons mentioned in section 5.1 related to the distinction between DM beams of scalars or fermions.
6 DM scattering inside the detector DM particles may scatter off the nucleons in the detector by exchanging a Z boson in the t channel, and producing neutral-current events. Let us study the interaction between the DM particles and nucleons inside the detector.
The cross section for DM interacting with nucleons can be much larger than the neutrino one since it is mediated by a lighter boson. For momentum transfers much smaller than M Z , the ratio of the DM to neutrino cross sections for neutral-current events is where j stands for any hadronic final state. C N (Z ) is a coefficient of order one that depends on whether the nucleon N is a proton (N = p) or neutron (N = n), as well as on the Z model; neglecting the interactions of sea quarks and nuclear form factors, this coefficient takes the values where s W ≡ sin θ W is evaluated at a scale of a few GeV (sin 2 θ W ≈ 0.235). For a DM charge under the new U (1) group of z χ = 3, the values of the Z gauge coupling and mass shown in Fig. 1 allow the ratio in Eq. (6.1) to be as large as 10 3 . However, even with such a large cross section for DM-nucleon scattering, the total number of DM events in the detector is much smaller than that of neutrino neutral-current events, because of the very large QCD production of pions and other mesons, whose decays generate the neutrino beam.
Before discussing selection cuts that reduce the neutrino background, let us compute the total number of DM events in the detector. The average DM-nucleon scattering cross section in the detector can be written as  Fig. 3, and the geometric acceptance T det is given in Table I. Multiplying this number of particles by the fraction of those that have energy between E χ and E χ + dE χ (shown in Fig. 5) gives For a detector of density ρ d and length L d , the number of signal events due to the beam produced in the target is given by Without imposing cuts there are O(10 7 ) neutral-current neutrino scattering events, which is a too large background for allowing sensitivity to our signal. Thus, it is necessary to find some selection cuts that reduce the neutrino background without reducing the DM signal too much. If we label the incoming and outgoing χ fourmomenta by k µ and k µ respectively, and the momentum transfer by q µ = k µ −k µ , then the DIS regime is realized for Q 2 > m 2 p . For lighter mediators, M Z < 1 GeV, the most relevant region for fixed target experiments is Q 2 = −q 2 < 1 GeV [62] . This is explained by the Q 2 dependence of the cross-section: 1/(M 2 Z + Q 2 ). For heavier Z bosons, of mass around a few GeV, we expect the DIS regime to dominate. Consequently, we expect that it is helpful to impose a cut on the energy E j of the hadronic activity produced by the DM particle in the detector.
The peak energy of the neutrinos that enter the NOνA detector is near 3 GeV, while the energy hadronic activity due to the neutrinos peaks at smaller values. We impose a cut E j > 2 GeV, as stronger cuts reduce the signal too much in some cases.
We expect this cut to not be sufficient to reduce enough the huge neutrino background.
We computed the σ(χN → χj) d cross section after this cut with MadGraph 5; in Fig. 6 we show the regions in the (M Z , g z ) plane (above the red lines) where the number of DM scattering events in the NOνA detector satisfies S > 10 n , n = 1, ..., 4, for z χ = 3. We also show there the regions (shaded) excluded by various experiments discussed in Section 2. We find that the number of signal events S can be as large as 10 3 for Z B and 10 4 for Z ds , with N POT = 10 21 . It appears that NOνA may be able to explore a viable region of parameter space.
The MINOS near detector may also be able to probe the case of fermionic χ, even though the background is larger because the neutrinos traveling closer to the axis are typically more energetic [61]. Therefore, we urge both the NOνA and MINOS Collaborations to perform dedicate analyses to probe the presence of a DM beam.

Outlook
It is important for DM searches to be as broad as possible in order to cover the wide range of allowed masses and the many potential portals to the visible sector. Proton fixed target experiments offer the possibility to explore a region of the parameter space left unconstrained by the existing searches involving direct detection, collider searches, or invisible quarkonium decays. The high beam intensity of these experiments makes them a promising ground for testing models with light DM particles.
In this paper we have studied the potential sensitivity of the neutrino near detectors to a DM beam produced at the Fermilab Main Injector, in the NuMI beam line. We have considered DM candidates, either a fermion or a scalar, charged under a new leptophobic gauge group, with the associated Z boson having a mass M Z in the 1 − 10 GeV range. Assuming that the DM mass satisfies m χ < M Z /2, pairs of DM particles may be resonantly produced when the 120 GeV proton beam scatters off the target or is dumped into the absorber.
We have found a potentially interesting reach for these experiments provided that an efficient way to reduce the neutrino background is found. We have outlined possible solutions in this sense, focusing on the NOνA near detector since its off-axis position is better suited than the MINOS one for reducing the neutrino background.
Even though the NOνA near detector is better positioned for our purpose, the MINOS near detector can also be sensitive to a DM beam, especially in the case of fermionic DM particles. A parallel analysis done by the MINOS and NOνA collaborations can reveal the spin of the particle potentially discovered. The MInerνa detector is on-axis, but about 10% closer to the absorber than MINOS, so it may provide further tests.
The neutrino detectors along the Booster beam line at Fermilab may also be used to probe the presence of DM beams generated at the NuMI beam line. The MiniBoone detector is 6.3 • off-axis when viewed from the NuMI target [63], and at more than 90 • off-axis when viewed from the NuMI absorber; it also happens to be closer to the absorber by a factor of 2 compared to the MINOS near detector. The MicroBoone detector [64] is ∼ 8 • off-axis when viewed from the NuMI target, and the proposed LAr1-ND detector [65] would be ∼ 30 • off-axis.
The proposed LBNF [66] beam line at Fermilab would have a substantially larger N POT than the NuMI beam line. Thus, if a near detector is built close enough to the LBNF target or absorber, then the expected number of DM events can be an order of magnitude higher than in NOνA. Furthermore, the steep downwards slope of the proposed LBNF beam, in conjuction with the shape of the DM beam (a conic shell originating at the target and another one originating at the absorber), offers interesting physics opportunities if two or more near detectors are placed off-axis.