PHYSICAL REVIEW D 102, 075026 (2020)

New production channels for light dark matter in hadronic showers

A. Celentano ,1 L. Darm´e,2 L. Marsicano,1 and E. Nardi 2 1Istituto Nazionale di Fisica Nucleare, Sezione di Genova, 16146 Genova, Italy 2Istituto Nazionale di Fisica Nucleare, Laboratori Nazionali di Frascati, C.P. 13, 00044 Frascati, Italy

(Received 25 June 2020; accepted 11 August 2020; published 21 October 2020)

Hadronic showers transfer a relevant amount of their energy to electromagnetic subshowers. We show that the generation of “secondary” dark photons in these subshowers is significant and typically dominates the production at low dark photon masses. The resulting dark photons are however substantially less energetic than the ones originating from mesons decay. We illustrate this point both semianalytically and through Monte Carlo simulations. Existing limits on vector-mediator scenarios for light dark matter are updated with the inclusion of the new production processes.

DOI: 10.1103/PhysRevD.102.075026

I. INTRODUCTION sector blind to all SM interactions is not particularly exotic. In addition, given that in the SM there is no shortage of One of the most compelling empirical arguments to 1 =c2 search for extensions of the Standard Model (SM) of states with mass below, say, GeV , it is also rather natural to hypothesize that the same could be true for dark elementary particles is the need to explain the nature of sector particles, the lightest of which would be stable, thus dark matter (DM). In years past, theoretical and exper- providing a light dark matter (LDM) candidate. On the imental efforts mainly catalysed around the hypothesis that other hand, the dark sector could well come equipped with DM corresponds to a weakly interacting massive particle its own set of interactions (to which SM particles should (WIMP) with electroweak scale mass (for a recent review, clearly be blind) and if this set also contains the simplest see e.g., Ref. [1]). Such a hypothesis is certainly well type of gauge force, corresponding to a Uð1Þ gauge factor, grounded, given that in the early Universe WIMPs would then mixing between the dark spin-1 boson (often referred be produced via thermal processes, and their subsequent to as “dark photon” and denoted as V in this work) and the annihilation with typical weak interaction rates would photon would naturally occur [3]. This would provide a leave, almost independently of other details, a relic density portal though which the SM and the dark sector could of the correct size to match the observed cosmological communicate. amount of DM. However, null results of an extensive and Recent years have witnessed a steadily growing interest long lasting search program that combined direct, indirect, toward LDM and its possible detection through the vector and collider probes are presently triggering a waning of the portal, and many studies have appeared deepening our WIMP paradigm [2]. While WIMP searches should cer- understanding of the theoretical models and of their phe- tainly continue until all experimentally accessible corners nomenology, see for example Refs. [4–10]. Interestingly, of the parameter space are thoroughly probed, it is now besides promoting new experimental programs aiming to timely and important to put no lesser vigor in exploring also search both for the V and for LDM particles [11–13], the other pathways. LDM paradigm also stimulated the reanalysis and reinter- One alternative scenario, well motivated in first place by pretation of old data originally collected to search for other the evidence that DM is reluctant to interact with ordinary types of particles [5,14–16]. Accelerator-based thick-target matter, conjectures the existence of a new class of relatively experiments at moderate beam energy (∼10 ÷ 100 GeV) are light elementary particles not charged under the SM the ideal tool to probe the new hypothesis, since they SUð3Þ × SUð2Þ × Uð1Þ gauge group. After all, even C L Y have a very large discovery potential in a wide area of in the SM some particles are uncharged under one or more parameters space. Within this context, the main experi- gauge group factors, so that an extension to include a new mental techniques that have been considered so far are (1) missing energy/momentum/mass searches with electron and/or positron beams [16–19], (2) electron and proton Published by the American Physical Society under the terms of thick-target experiments searching for light new particles the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to via their scattering in a downstream detector [4,20], the author(s) and the published article’s title, journal citation, and (3) decay of long-lived dark sector fields into SM and DOI. Funded by SCOAP3. particles [21–29].

2470-0010=2020=102(7)=075026(21) 075026-1 Published by the American Physical Society CELENTANO, DARME,´ MARSICANO, and NARDI PHYS. REV. D 102, 075026 (2020)

Proton beam-dump experiments show an enhanced that we developed to compute the enhanced sensitivity of sensitivity to the dark sector. Thanks to the large beam proton beam-dump experiments, which results from taking energy and accumulated charge, typically higher than those into account the new production processes. Finally, in in electrons and positrons counterparts, a large LDM signal Sec. IV, after briefly reviewing the main features of a yield is expected, usually at the price of a larger back- representative set of proton beam-dump experiments, we ground [15,30,31]. The experimental intensity frontier is present the corresponding exclusion limits and sensitivity currently extremely active, and many new experiments will curves, updated by including the new LDM production start to take data during the course of the next decade channels. [11,32,33], making the accurate estimation of their poten- tial reaches an important issue. This is particularly true for II. LDM PRODUCTION BY SECONDARY e dark sector searches carried out at proton beam-dump IN PROTON BEAM-DUMP EXPERIMENTS experiments designed for neutrino physics [20,34], such as MiniBooNE [35], SBND [36], ICARUS [37] or DUNE The production of dark-sector particles in a proton beam- [38], and for lower energy COHERENT [39,40], where the dump experiment is a multistep process involving the irreducible neutrino background calls for an even more secondary particles produced in the thick target by the careful evaluation of the expected LDM signal. impinging hadron. Due to the variety of secondary particles In the aforementioned vector portal scenario in which a being part of the developing hadronic shower, a large light dark photon interacts with the SM sector via feeble number of production mechanisms is possible. In this work, gauge interactions, the main LDM production mechanism we include for the first time electron- and positron-induced involved in a proton beam-dump experiment is the two- processes in the computation of the LDM yield of proton photons decay of light mesons (π0 and η), where dark sector beam-dump experiments. In order to do so, we decouple the particles are produced thanks to the γ − V mixing. This problem into two separate parts: the development of the production mechanism has been widely studied in the last EM shower which is controlled by SM physics, and the new decades. However, a proton-induced hadronic shower is physics processes generating the dark photon (which we always accompanied by an electromagnetic counterpart, assume to be strongly subdominant compared to the which carries a significant fraction of the primary beam former). More in detail, we will first revisit the typical energy. This allows for a rich variety of electron- and structure of the EM component of a proton-induced hadronic shower, for a primary beam energy in the positron-induced LDM production processes, incrementing 10 100 the flux of LDM particles from the thick target, and thus the ÷ GeV range. We will next discuss the main experimental sensitivities. An early attempt to consider this processes responsible for LDM production by electrons effect was presented in [41], considering only the V visible and positrons in this regime. Finally, we will focus on the decay. In this work, for the first time we estimate the LDM production and detection of LDM in a typical proton-beam, production rate from the electromagnetic components of thick-target experiment. proton beam-dump experiments. We show that, in some cases, this is the dominant LDM production mechanism for A. Production of e in proton-induced a non-negligible region of the dark sector parameter space. hadronic showers Furthermore, we demonstrate that thanks to these new When a high-energy proton impinges on a thick target, a production processes proton beam-dump experiments can cascade of secondary hadrons with progressively degrading also probe nonminimal dark sector scenarios that were, so energy is produced, mostly containing protons, neutrons, far, considered to be an unique prerogative of lepton-beam and pions. Due to the isospin symmetry of hadron-induced efforts, such as protophobic models [11] in which the dark reactions, approximately 1=3 of the latter are π0. These photon coupling to quarks is strongly suppressed. The immediately decay to high-energy γγ pairs, which in turn recently proposed protophobic fifth-force interpretation initiate an EM shower accompanying the hadronic one. A [42,43] of the observed anomalies in internal e pair similar argument applies for η and η0 mesons, although their creation in 8Be and 4He nuclear transitions [44,45] is an contribution to the EM shower is reduced, both because of example of a particularly intriguing new physics scenario the smaller production cross section, and because of the that the new production mechanisms allow to test also in lower branching fraction for the γγ decay. On average, the proton beam thick-target experiments. fraction of the primary proton energy transferred to the EM The paper is organized as follows. In Sec. II we describe component is of the order 50% for a 100 GeV impinging the phenomenology of LDM production by secondary proton [46]. electrons and positrons in a ∼100 GeV proton beam-dump While a complete treatment based on numerical simu- experiment, discussing both the main properties of proton- lations will be presented in the next sections, a relatively induced electromagnetic showers and the dominant LDM good approximation of the energy distributions can be processes induced by eþ and e− at this energy scale. In obtained from a semi-analytical approach. Starting from the Sec. III we present the details of the numerical procedure typical differential number density of secondary neutral

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ð Þ dNγ mesons nM0 E from a pN collision at the beam energy, neutral mesons decay can be obtained, accounting for with N a nucleus of the target material, the differential yield dEγ the corresponding branching fraction. of mesons in the hadronic shower, per POT, can be While a thorough description of the EM shower develop- estimated approximately by just considering the first ment requires a complete Monte Carlo calculation, an interaction of the proton: approximate evaluation of the electrons and positrons track length can still be obtained with an analytical approach. We N ρ ð Þ ð Þ dNM0 A dnM0 E L dnM0 E introduce the dimensionless shower depth parameter in unit ¼ Lσ × ≡ × ; ð1Þ ≡ dE A pN dE λ dE of radiation length t d=X0 and the shower age as function T of the energy E and t [50,51]: where σ is the inelastic proton-nucleon cross section, A is 3 pN E ≃ t ð Þ ρ s ;t E ; 2 the atomic mass of the target material, the density, Eγ t − 2 ln 23 Eγ N A ¼ 6.022 × 10 , L is the length of the active part of the target, and the second equality follows from the where Eγ is the energy of the photon inducing the shower. λ definition of the nuclear interaction length T. If the target The differential distribution n of electron/positron grows ≳ λ e is thick enough, L T, and in the approximation of only exponentially with s, corresponding to the power law considering the first generation of secondary particles in the scaling hadronic shower, we can set L ≃ λT, which results in the dN dn simplified expression M0 ≃ M0 . Clearly, this approxima- 1 dE dE n ∼ : ð3Þ tion is expected to be more accurate for energies close to the e Esþ1 beam energy, while for lower energies the actual number of neutral mesons would be underestimated. This effect is Ultimately, the lowest energy electrons/positrons (resp. 0 photons) in the shower start interacting with the medium clearly visible in Fig. 1, where we show the differential π mostly via ionisation (resp. Compton scattering) and the yield from a 120 GeV proton beam impinging on a thick shower stops developing. The critical energy ϵ for which graphite target, comparing the approximate result from c this happens is roughly defined as the energy for which the Eq. (1) (red curve) with that obtained from a full simulation electron bremsstrahlung and ionisation rates are equal (in of the hadronic shower made with GEANT4 [47] (blue fact the energy at which the ionisation loss per X0 is equal curve). We used the QGSPJETII software [48] to compute dn to the electron/positron energy). Denoting with Z the M0 — dE a similar calculation with the EPOS-LHC software atomic number of the medium, the critical energy can be [49] yielded the same conclusion. From the knowledge of approximated by [52]: dN M0 , the differential distribution of primary photons from dE 610 MeV ϵ ∼ : ð4Þ c Z þ 1.24 10 A direct consequence of the two energy loss mechanisms described above is that one typically expects that in a thick 1 target the differential density distribution should be domi- nated by electron/positrons around the critical energy. − 10 1 In order to describe more quantitatively the electromag- netic shower, we will broadly follow the approach of Rossi and Griesen [50] as reported in [51]. We refer to −

/ dE (1/GeV) 2 0 10 Appendix A for details. The first step is to obtain the π

dN differential energy spectra of both electrons and positrons in the photon-induced electromagnetic shower. In this − 10 3 formalism eþ and e− are treated on equal footing, neglect- ing initially the influence of ionization or Compton

− scattering on the shower. This reads: 10 4   2 0 −1−s 1 10 10 dneðE; Eγ;tÞ 1 Gγ→eðsÞ E ¼ pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi λ1ðsÞt ð Þ E (GeV) 00 e ; 5 dEdt Eγ 2π λ1ðsÞt Eγ FIG. 1. Comparison of the differential π0 yield per proton on λ ð Þ ð Þ target from a 120 GeV proton impinging on a thick graphite where the functions 1 s and Gγ→e s are defined in target. Red curve: result obtained from Eq. (1) based on EPOS- Appendix A, s is the shower age defined in Eq. (2), and LHC [49]. Blue curve: results of a full GEANT4 -based simulation. Eγ is the energy of the primary photon. The two auxiliary

075026-3 CELENTANO, DARME,´ MARSICANO, and NARDI PHYS. REV. D 102, 075026 (2020) function are constructed from the cross sections for pair- 40 400 Gev SHiP Analytical production and bremstrahlung (along with several of their 400 Gev SHiP G4 momenta) and thus include the details of the underlying 30 physical processes leading to photons and positrons/ electrons creation in the shower following the approach of Rossi and Griesen [50]. 20 In a second step, an approximate solution including the cutoff effect from ionization/Compton scattering can be 0 10 obtained by multiplying ne by a cut-off function p1: ð Þ 0ð Þ dne E; Eγ;t dne E; Eγ;t ϵc E ¼ ð Þ 0 p1 s ;t ; ; 6 0.5 1 5 10 50 100 dEdt dEdt Eγ ϵc (a) and we will approximate the function p1 by its value at the ¼ 1 5 maximum of the shower (s ) [51]. 120 Gev DUNE Analytical Finally, the electrons and positrons differential track- 120 Gev DUNE G4 dT 4 length dE is obtained by integrating over the depth of the full shower and over the energy distribution of primary 3 photons (with Eini the energy of the primary proton initiating the shower). More precisely, Z Z 2 ∞ dT 1 Eini dneðE; Eγ;tÞ dNγðEγÞ ¼ dEγ dt ; ð7Þ dE 2 0 0 dEdt dEγ 1

where the factor 1=2 comes from the fact that the analytical 0 0.5 1 5 10 50 approach does not distinguish between electrons and posi- dT −1 trons. Note that dE has dimension of GeV . Intuitively, the (b) dT quantity dE dE represents the total path length in the dump, 0.30 þ − in radiation length units, taken by e =e with energy in the 8Gev MiniBooNE Analytical 8Gev MiniBooNE G4 interval between E and E þ dE. This acts as an effective 0.25 target length for LDM production, allowing the complicated EM shower to be condensed down to an effective fixed target 0.20 experiment, as discussed in the following. 0.15 We validate this approach in Fig. 2. In particular we show for reference the full result obtained from a GEANT4 0.10 simulation [47], as is described in the next sections. We present the positrons track-length times energy squared 0.05 distribution as function of the energy of the positrons. The 0.00 semianalytical approach carries an important uncertainty in 0.01 0.05 0.10 0.50 1 5 that it does not account for the full dynamics of the hadronic shower, and it assumes instead that the initial (c) proton interacts only once. Accordingly, the number of nuclei targets is set in Eq. (1) by what is assumed to be the FIG. 2. Differential track length times energy squared in GeV “active” part of the target. We can either set L to the nuclear for the positrons in the showers generated in the SHiP, DUNE and interaction length, or we can make the more conservative MiniBooNE targets in arbitrary units. The yellow lines represent choice of setting L to the nuclear collision length, thus the results from complete GEANT4 simulations. The blue regions represent the results obtained from the semianalytical approach ensuring that the incoming proton would not lose energy described in the text, with the upper lines obtained by fixing in before generating the shower. The results obtained for these Eq. (1) L ¼ λ (the nuclear interaction length) and the lower two choices delimit the blue region in Fig. 2, which can be T dashed lines corresponding to L ¼ λc (the nuclear collision taken as a proxy for the typical uncertainty associated to the length) which is a more conservative choice. semianalytical procedure. In any case, we find a very good agreement with the full numerical approach for the experi- the significant simplifications involved in the analytical ments with the lower beam energies. For the high-energy approach which does not include the effects of secondaries, case of SHiP, we still obtain an acceptable agreement given whose relevance increases with increasing beam energy.

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1 1 ε We further observe that the semianalytical approach L ⊃− 0μν 0 − 0μν − 0 J μ ð Þ F Fμν BμνF VμgD D; 8 becomes more conservative with increasing proton beam 4 2 cos θw energy (in SHiP for instance) as secondary mesons carry enough energy to generate sizeable subshowers of their own. where the parameter ε weights the kinetic mixing, Bμν the J μ Note that our final results will in any case be based on the hypercharge field strength, and D is the dark gauge complete numerical simulation shown in orange in Fig. 2. current, which depends on the details of the dark sector. One important comment is that this approach does not After electroweak symmetry breaking, and after performing incorporate the angular distribution of the produced a standard redefinition of the photon field γ → γ − εV0 to electrons/positrons. As can be readily inferred from the diagonalize the kinetic term, the dark photon also acquires a relative low energy of the peak of the spectrum in Fig. 2, ε-suppressed interaction with the SM electromagnetic the electrons/positrons angular distribution has a non- current: negligible width. Depending on the geometry of the 0 μ experiment (detector size and detector-dump distance), this L ⊃−VμeεJ em: ð9Þ effect can be critical, since it affects the angular distribution of the LDM particles produced, and thus the signal yield. In Note that the dark photon mass mV can originate either this work, we accounted for it by evaluating the double- from the Stueckelberg mechanism or from the VEV of a dTðE;ΩÞ differential track length dEdΩ . As an example, Fig. 3 dark Higgs boson. The latter typically constitutes an shows the angular distribution of positrons produced in the important part of the phenomenology if it has the same DUNE target by the 120 GeV Fermilab proton beam. mass as the dark matter candidate [23,26,53]; on the contrary, it can basically decouple if it is heavier than the dark photon. Here we will consider explicitly the B. LDM production channels second scenario. Finally, specifying the precise nature of χ 1. LDM model building and Lagrangian the dark matter candidate is not critical for the scope of this work. In order to compare our result with the recent The procedure described above is completely general limits from the MiniBooNE collaboration, we considered a and can be applied to any light new particle coupling to the complex scalar dark matter candidate, although our con- electrons/positrons or to the light quarks (for instance clusions also apply for other standard choices (Majorana axionlike particles and millicharged particles). For con- dark matter, pseudo-Dirac dark matter with a small mass creteness, in this work we focused on the case of a LDM splitting, etc...) since their production and detection mech- scenario where sub-GeV DM particles interact with the SM μ 0μν anisms are similar. For the case of a complex scalar the dark via a dark photon mediator V (with field strength F and current is given by: dark gauge coupling gD). The corresponding Lagrangian contains the following terms: J μ ¼ ðχ∂μχ − χ∂μχÞ ð Þ D i : 10

As long as mV > 2mχ, the interaction in Eq. (8) leads to 108 rapid dark photon decay into dark matter particles: this is the so-called invisible decay scenario on which we focus. 107 Note that often in the literature an extra factor of 1=2 is

6 included in the normalization of the dark gauge current in 10 Eq. (10). Thus, when relevant to carry out proper compar-

5 A.U. 10 isons, we have rescaled the existing limits on the dark gauge coupling in Eq. (8) to account for the choice 104 of normalization.1

103 2. Main production channels and cross sections 2 10 For low mass dark sectors, the main production mech- 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 anisms for dark photon from the hadronic development θ cos( ) of the shower are from the decay of light unflavored FIG. 3. Angular distribution of positrons produced by the mesons. Depending on the mass of the dark photon, the π0 → γ η η0 → γ 120 GeV proton beam from the Fermilab accelerator in the DUNE dominant meson decay process are V, ; V or target. The black, red and blue lines refer, respectively, to positrons with a 1 GeV, 3 GeV, and 8 GeV energy threshold. The 1Most notably, the recent works using the convention with an normalization of each curve is proportional to the total positron extra factor 1=2 include the prospects for the SHiP collaboration yield applying the corresponding energy threshold. The angular as reported in, e.g., [54,55], as well as the study of the projected distribution of electrons, not displayed, features a similar behavior. sensitivity of the NOνA near detector in [34].

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ρ; ω → V → χχ (in case the dark photon decays into dark for dark photon production by secondary eþ=e−, illustrated sector particles). The typical branching ratio is given by in Fig. 4, are the following: (i) Bremsstrahlung of electrons and positrons off 2 3 → m nuclei, e N e NV with typical cross section: BRðπ0 → VγÞ¼2ε2 1 − V ; ð11Þ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mπ0 4ε2α3 2 ξ 1 σ ≃ em 1 − mV brem log 2 2 ; α 3 2 2 m m In particular, note that there is no em insertion so that this E mV ð e V Þ Max m2 ; E2 process is only mildly suppressed. V On the other hand, hadronic showers develop a large ð12Þ electromagnetic component from the radiative decays of 0 light neutral mesons π ; η. All relevant processes here where ξ ∼ Z2 is the effective flux of photon from the depend on the density of the relevant targets (either nuclei accelerated nuclei in the incoming electron/positron σ for bremsstrahlung or atomic electrons for positron/photon frame [57]. We observe that brem increases quad- processes). While the dominant production mechanism for ratically for small dark photon mass, but is, however, α3 vector mediators in electron beam dumps is mostly via severely suppressed by em. Also, the emitted dark electron bremsstrahlung, it was recently realized that photons are typically very energetic, since they carry production mechanisms based on secondary positrons most of the energy of the initial eþ=e−, with the can dominate in the low mass ranges [56]. Since in median value for EV given by: hadronic showers the yield of secondary electrons and positrons is almost the same, positron-related processes 2 2 h i¼ 1 − me mV ð Þ dominate the LDM production rate. EV E Max 2 ; : 13 m E Denoting E the energy of the incoming positron/ V electron in the lab frame, the main processes responsible This mechanism dominates the dark photon produc- tion in electron beam-dump experiments, due to the fact that it is enhanced for very energetic primary electrons (see the comparison with the resonant production mode in [56,58]). In the proton-shower induced environment, both the electrons and the positrons are secondary particles, and therefore they contribute equally to the bremsstrahlung production rate. We review in more detail this production mechanism and our numerical approach for this (a) process in Appendix B. (ii) Direct positrons annihilation on target atomic elec- trons eþe− → V → χχ [59]. This process can be divided in two main regimes: resonant and off-shell. In the resonant regime, the dark photon is produced on-shell and the cross section is given by: 2π2ε2α 2 σ ¼ em δ − mV ð Þ res Eþ : 14 me 2me

(b) (c) While this process can only occurs around the resonant energy (depending on the width of the dark photon, which is here relatively large due to the dark decay V → χχ), it is still important because α it is only suppressed by em [59]. Furthermore, given the restricted kinematics, the energies of the incom- ing positron and of the outgoing dark photon are related by (d) m2 FIG. 4. Dominant processes for dark photon production during res ¼ V ð Þ EV 2 : 15 the electromagnetic development of a shower. me

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≫ 2 This allows us to estimate the range of the accessible production channel. In the limit where Eþme mV, the dark photon masses as: cross section becomes pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 th ≳ 2 ð Þ σ πε α 2Eþ mV meEth; 16 σ ≃ assoc ≃ em ð Þ Compton log ; 19 2 meEþ me where E is the experimental detection threshold. th α2 Note that the above expression is very conservative which is also em suppressed. Note that this cross section in that it assumes that the dark photon energy is falls much faster than that for associated production at entirely transmitted to the detector. larger dark photon mass. We present a thorough description In the off-shell regime, χχ pairs are produced via of the impact of this channel, comparing it to the associated exchange of an off-shell V, and this process can be and bremsstrahlung production channels, in Appendix C. relevant especially when considering a large dark α ∼ 0 1 gauge coupling D . . Accounting for off-shell 3. Comparison and total production rates χχ production requires including in the resonant In all the experiments we have considered the associated positrons annihilation the finite V width, and con- production process is often subdominant compared to the sidering the full four-particle s-channel reaction þ − bremsstrahlung or to the resonant production mechanism. e e → V → χ χ. More precisely, in the limit 2 pffiffiffi Indeed, the latter strongly dominates due to its ϵ α where the center-of-mass (CM) energy s is much em scaling when enough positrons with adequate energy larger than m (particularly relevant for small, MeV- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V 2 ð2 Þ scale dark photon), the off-shell contribution can be mV= me are produced in the showers. On the other estimated as: hand, as can be seen in Fig. 5, the bremsstrahlung cross section saturates at high incoming energy, while both πα ε2α associated and Compton-like process decrease due to their σ ¼ em D ð Þ 1 off−shell 6 : 17 =s dependence. This implies that, even for very small dark meEþ 1 2 photon masses where there is a =mV enhancement, bremsstrahlung production gets contributions from posi- (iii) Associated production from positrons in the shower, þ − 2 trons in the full range of energies available in the shower. e e → γV Eþm ≫ m . In the limit where e V, the Furthermore, in the opposite limit of large dark photon cross section becomes res masses, where the resonant dark photon energy EV , see Eq. (15), is larger than the beam energy and resonant 2πε2α2 2 σ ≃ em Eþ ð Þ production cannot occur, both associated and Compton-like assoc log ; 18 meEþ me processes are also forbidden. In this case, the bremsstrah- lung process has access to a larger CM energy since it α2 which is typically em suppressed but is enhanced by corresponds to an interaction with the nucleus, and can be 1 a =me factor. Note that, due to the presence of an effective up to E ∼ mV. additional photon in the final state, in this case the In order to illustrate the respective importance of energy of the emitted dark photon can differ from mesons decay process with respect to shower-induced m2 V . In fact, as shown in Appendix C, around half of 2me the dark photons from associated production retain most of the energy of the incoming positron 0.001 ∼ þ − → → EV Eþ. Compared with the direct e e V 4 χχ off-shell regime in Eq. (17), the log-enhanced 10 α ð2 Þ em log Eþ=me term is replaced by the dark gauge 10 5 coupling term αD. Therefore, in case αD ≳ 0.1, σ σ 6 assoc is negligible with respect to off−shell. On the 10 other hand, the experimental energy threshold tends Assoc. production mV = 10 MeV to suppress both these processes with respect to 10 7 Compton production mV = 10 MeV Brem. production m = 10 MeV bremsstrahlung. V Note that for the processes leading to an on-shell dark 0.01 0.10 1 10 photon, V decays to a χχ pair with near 100% branching ratio for sizeable dark gauge coupling, thus allowing to FIG. 5. Production cross section for the associated eþe− → γV easily derive the LDM production yield. and Compton-like process γe− → e−V as function of the energy Finally, the electromagnetic shower further contain a of the incoming particle (either a photon Eγ, a positron or an significant number of photons, making the Compton-like electron with energy E ) in the laboratory frame. We chose − − e scattering process γe → e V also a potentially relevant ε ¼ 0.001;mV ¼ 10 MeV.

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In all the experimental setups considered in this paper, the distance between the target and the detector is much larger than the length of the target, so that the entire shower can be approximated as starting from the initial vertex. Therefore, the number of LDM particles emitted through a process characterized by a cross section σ can be computed as: Z N X0ρ Eini dTðEÞ N ¼ A dE σðEÞ; ð20Þ A 0 dE

where the X0 is the radiation length of the material, ρ its 23 2 mass density, A its atomic mass and N A ¼ 6.022 × 10 . Depending on the production process being considered, dT− FIG. 6. Dark photon production rate per proton-on-target for dE dTþ the MiniBooNE experiment as function of the dark photon mass and/or dE should be used. Similarly, the differential yield dN dσ mV . The shower-induced leptonic production processes are can be obtained by replacing σ → . dEχ dEχ shown in green: electron/positron bremsstrahlung (dashed line) þ − As discussed before, this approach does not incorporate and resonant e e → V;V → χχ (solid line), The blue line corresponds to the rate for standard hadronic production proc- the angular distribution of the produced electrons/positrons. esses. We have applied basic cuts on the GEANT4 objects: A rough estimate of the detector geometric acceptance is 2 their angle θ with respect to the beam axis is selected such εT ∼ S=ðθχDÞ , with θχ being the average LDM emission that sin θ < 0.2, and their kinetic energy should be larger angle. This has to be computed by convolving the different than 10 MeV. processes that are ultimately resulting to LDM production in the thick target: the production of primary neutral ones, we present in Fig. 6 the corresponding dark photon mesons in the hadronic shower, their decay to photons, production rates for the 8 GeV proton beam servicing the the development of the EM shower, and the LDM pro- MiniBooNE experiment. We used the full GEANT4 simu- duction by electrons and positrons. Furthermore, the lation described in the next section to obtain both the angular shape of each of these processes has its own distribution of light mesons and the track length of energy dependency. Therefore, a numerical approach is secondary positrons. Interestingly, the secondary produc- here unavoidable. χ tion strongly dominates in the lower mass regimes. This is If the couples diagonally with the V, the two main both due to the fact that the meson production saturates in processes responsible for the interaction with the detector this regime and that the showers provide an abundant are the elastic scattering off electrons and the quasielastic number of positrons and electrons with enough energy to scattering off nucleons. In the electron case, since ≪ χ produce such light dark photons. Both hadronic and me mV, the electron carries most of the impinging shower-based processes have the same production rate energy and gives rise to an electromagnetic shower in the ∼ 16 detector. In the nucleon case, instead, due to the nucleon for a dark photon mass around mcross MeV. This “crossing” mass depends more generally on the energy larger mass, the recoil energy is typically lower, making the available in the initial proton beam as well as on the signal corresponding to this process more difficult to material of the target. For instance, for the 120 GeV beam identify. For this reason, in this work we focus on the χ − from Fermilab’s main injector, which will be used by the e scattering process only. The differential cross section ∼ 20 for χe → χe scattering with respect to the electron recoil DUNE experiment, mcross MeV, while for the pro- posed SHiP experiment with access to the SPS 400 GeV energy Ef in the laboratory frame is [15]: ∼ 30 beam and a high-Z material target, mcross MeV. dσ 2m E2 − f ðE ÞðE − m Þ f;s ¼ 4πε2αα e f;s f f e ð Þ D ð 2 − 2Þð 2 þ 2 − 2 2Þ2 21 C. Experimental LDM production and detection dEf E mχ mV meEf me The typical setup of a proton beam-dump experiment is where E is the incoming χ energy and f and s stand for shown in Fig. 7. The primary proton beam impinges on a fermion and scalar χ respectively; ffðEfÞ¼2meE − thick target, where LDM particles are produced. These þ 2 þ 2 2 ð Þ¼2 þ 2 propagate straight toward a detector with cross size S placed meEf mχ me, fs Ef meE mχ. The total signal at distance D downstream that reveals them. A sizeable yield can then be obtained analytically, convolving amount of shielding material is placed between the dump the differential cross section with the incoming LDM and the detector to range out all other particles produced by the primary beam, except neutrinos. 2Note that the cross section has to be expressed in cm−2.

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D Detector Dump χ p

Shielding

Production Interaction χ e + χ χ e e Beam direction χ

FIG. 7. Typical setup of a proton beam-dump experiment. The proton beam impinges on a thick target, where LDM particles are produced by secondaries—the inset shows the production of a LDM particle pairs from eþe− annihilation. LDM particles then propagate straight toward a downstream detector at distance D, where they are revealed via the scattering on atomic electrons and nuclei. distribution and the cut efficiency for electron recoil the subsequent propagation and detector interaction as detection. Note that while in this paper we consider the described below. detection of LDM via its scattering in the detector, the main All the necessary numerical ingredients, including in idea of secondary dark photon production is relevant also particular the track length distributions used to describe the for other types of dark sector searches. electrons and positrons from the subshowers are available We finally observe that, while in this work we focused on on the Zenodo online repository [60]. the case of LDM detection through the elastic scattering on atomic electrons, our idea also applies to LDM models A. LDM production predicting similar interaction mechanisms in the detector. The evaluation of the LDM production in the dump For example, in inelastic dark matter scenarios (iDM) [21], χ þ − was further factorized into two independent steps: (i) the if the splitting between the two dark 1e e states is small calculation of the electrons and positrons track-length in the with respect to the beam energy scale, the leptons-induced target, and (ii) the computation of the LDM differential LDM yield in the beam dump would not change signifi- yield from eþ interactions. cantly. At the same time, provided mχ >mχ þ 2m , the 2 1 e For each of the detector setups that we have considered expected signature in the detector would be either the direct þ − in this work, and that are described in the next section, we decay χ2 → χ1e e within the detector when the χ2 state is TðE;ΩÞ have computed dEdΩ , i.e., the electrons/positrons differ- sufficiently long-lived, or the nondiagonal scattering ential track-length distribution as a function of the particle χ → χ 1N 2N, with N an atomic nucleus, followed by the energy and angle, by means of a GEANT4 simulation. χ → χ þ − decay 2 1e e . In both cases, the result is a significant We have used the standard G4EmStandardPhysics energy deposition in the detector. In particular, we note that physics list to describe EM interactions, and the χ in the limit where the heavy state 2 has a decay length FTFP_BERT_HP physics list to parameterize hadronic much larger than the distance to the detector, the lower reactions. We have developed a custom class, inheriting boost factor of secondary production events will enhance from G4SteppingAction, that records, for each elec- the detection prospects. We will investigate the effect of tron and positron step in the target, the corresponding shower-induced iDM production in a future work (see e.g., particle energy and direction. The output of the simulation – [22 29] for recent works discussing the iDM physics case). TðE;ΩÞ is the distribution dEdΩ for discrete bins of the two observables. For the energy, we have used a bin width III. NUMERICAL EVALUATION ΔE corresponding to ∼0.1% of the primary proton beam To reevaluate the exclusion limits implied by existing energy. Since in the simulation, with default physics lists proton beam-dump results when the lepton-induced sec- settings, the typical energy loss for each positron step inside ondary production processes are properly included, and the dump volume is already much smaller than ΔE, we did to estimate the sensitivity of planned experiments, the not include any explicit step limiter. Finally, to speed-up expected number of signal events within the detector has to the calculation, we introduced for all particles an energy be computed as a function of the model parameters, and threshold equivalent to the detection threshold, discarding compared with the background yield. We performed the from the simulation all particles falling below this value. calculation of the signal yield numerically, decoupling the In order to make a fair comparison between the electron- evaluation of the LDM production in the beam-dump from and positron-induced production mechanisms with the

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“traditional” processes usually considered for proton beam- precisely, we propagated the LDM particles to the detector dump experiments involving neutral mesons decays, in the and estimated their intersection with the detectors using the simulations we have also sampled the differential distri- internal BdNMC routines. The scattering probability as a dn 0 ðE;ΩÞ M 0 ¼ π0 η function of the dark matter nature (complex scalar or Dirac- bution dEdΩ for M ; . The LDM yield in the target was then computed fermion) was estimated using Eq. (21) (note that the complex scalar case was already present in the original using the MadDump software [54] and a modified version BdNMC code). In order to simulate accurately the detector of the Monte Carlo generator BdNMC [20] depending on the production process. The former is a plugin for the response, we added at the generator-level the selection cuts from the experiments. To speed up the calculation, basic MadGraph5_aMC@NLO program [61,62] that allows to com- pute the differential yield of LDM particles in the target energy cuts were included directly in the cross section ð ΩÞ from the knowledge of dT E; . In particular, we evaluation, while the more advanced ones (such as that on dEdΩ E θ2) were applied after the scattering events had been used MadDump to generate a list of outgoing dark matter e e momenta from all the leptonic production channels, includ- simulated. Finally, starting from the knowledge of the sensitivity of ing the s-channel eþe− → VðÞ → χχ, the associated a given experiment in terms of signal yield, the corre- production and the bremsstrahlung processes. For the latter, sponding reach curve was sampled as follows. To reduce we adopted the nuclear form-factor parametrization the number of free parameters, we adopted the standard described in Ref. [57]. On the other hand the hadronic choice mV ¼ 3mχ and αD ¼ 0.1. Observing that in the production processes were handled by BdNMC. For the scenario considered in this work all LDM particles are production via light meson decays, we used the light neutral produced promptly in the beam dump, we can expect that meson distributions including secondary mesons as given for a given set of reduced model parameters the foreseen by GEANT4 (instead of the build-in empirical distributions) signal yield in the detector will scale as: and we have simulated their decay to dark matter via the vector portal. For completeness, we have further included 4 0 ε N ð χ εÞ¼ ð χÞ ð Þ the proton bremsstrahlung process and dark photon pro- S m ; NS m · ε ; 22 duction via resonant vector meson mixing as it is imple- 0 mented in BdNMC [20] (in particular, the timelike form 0 where N ðmχÞ is the signal yield corresponding to the factor used in the production rate is derived from [63] and S kinetic mixing parameter ε0. We can thus obtain the limit hence incorporates the effect of ρ=ω meson production). for ε by inverting the previous relation. We observe that, in the current version, both MadDump and BdNMC assume that all LDM particles are produced at the beginning of the target, neglecting the development of the IV. APPLICATIONS AND EXAMPLES EM shower in the corresponding volume. However, as In this section, we present the revised exclusion limits already mentioned, this approximation is well justified by and we discuss the estimates of the sensitivities that we the much larger distance between the target and the have obtained for a representative selection of existing and detector. planned proton beam-dump experiments, after the new The advantage of this dual approach, rather than han- positrons annihilation production mechanism is included in dling together the description of the EM shower develop- the evaluation of the LDM yield. For each case we briefly ment and the production of LDM in a single simulation, is discuss the relevant experimental details, and the assump- the fact that, for each considered experiment, the differ- tions made in carrying out the analysis(see also Table I). ential track length and the neutral mesons distribution have Our results are summarized in Sec. IV E. to be computed only once, thus saving a significant amount of computation time. Only the evaluation of the LDM yield A. MiniBooNE has to be repeated for different values of mV and mχ. Finally, to account for the different materials in the target MiniBooNE is a proton beam-dump experiment at geometry, the procedure we adopted was to compute Fermilab, originally designed to measure short-baseline separately for each of them the eþ=e− differential track neutrino oscillations [67]. The MiniBooNE detector is a length and the LDM yield using the procedure described 6 m radius spherical tank, filled with 818 tons of mineral oil before, summing the obtained results. To speed-up the [35]. It is installed approximately 540 m downstream of a calculation, only the materials with a non-negligible track- beryllium neutrino production target, where the 8 GeV length relative weight (≳1%) were further considered. proton beam from the Fermilab Booster impinges on. Recently, a dedicated LDM measurement was performed by the MiniBooNE-DM collaboration using data corre- B. Detector interaction and normalization sponding to 1.86 × 1020 protons on target [64]. Since We have used BdNMC to simulate the propagation and neutrino interactions in the detector represent an irreducible interaction of light dark matter with the detector. More background for the LDM measurement, the experiment was

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TABLE I. Beam, target, and detector main characteristics for the experiments considered in this work, along with the total number of protons on target (PoT) and the typical lower energy cut. The distance (to the center of the experiment) D and the typical detector dimensions (length L and cross-area S) are also indicated. Note that the SHiP design is not final. We list the number of events (NoE in the table) corresponding to a 90% confidence level exclusion limit. In the DUNE-PRISM case, we considered both an on-axis and an off- axis configuration (see text for details). The references in the first column refer either to a published analysis in the case of existing constraints, or to projected bounds in the case of planned experiments.

2 Experiment Ebeam Target PoT D (m) L/S ðm=m Þ Ecut (scat) NoE 90% MiniBooNE [64] 8 GeV Steel 1.86 × 1020 490 12=36 75 MeV 2.3 NOνA [64] 120 GeV C 2.97 × 1020 990 12.67=3.9 × 3.9 500 MeV 16.4 SHiP [65] 400 GeV W=Mo=Fe 2 × 1020 38 3.2=0.75 × 3.2 1 GeV 38 DUNE-PRISM [66] 120 GeV C 7.7 × 1020 574 5=3 × 4 50 MeV 350 (54) performed by steering the primary proton beam in an “off- distribution for the oscillation measurement. The detector is target” configuration, to avoid neutrino production in the a large volume of plastic (PVC) extrusions filled with liquid target, and to impinge directly on the steel beam-dump scintillator (active volume), followed by a muon detector installed 50 m downstream. This resulted in a neutrino made of alternating steel planes and scintillator planes. The background reduction of a factor ≃30. The experiment active volume is a high-granularity sampling calorimeter, considered both the nucleon and the electron scattering characterized by enhanced PID and tracking capabilities. channel to detect LDM, with the latter providing the most The corresponding mass is approximately 193 × 103 kg, stringent limits. After employing a sophisticated set of for a total volume of 3.9 × 3.9 × 12.67 m3 [70]. selection cuts to discriminate between the LDM signal and A first estimate of the NOνA-ND sensitivity to LDM was the residual backgrounds, zero events were observed in the discussed in [34] where, however, only the χ − e− scatter- signal region. This allowed the collaboration to set a ing channel was considered. This result was based on a 90% CL limit on the LDM parameters space, correspond- preliminary report of the ν − e elastic scattering analysis ing to 2.3 expected signal events. performed by the collaboration [71], for a total exposure of To compute the LDM flux in MiniBoone, we described 2.97 × 1020 POT. Both the elastic neutrino scattering signal the beam dump in GEANT4 as a 4 m long steel block. Since (120 expected events) and the corresponding backgrounds this correspond to approximately 24 hadronic interaction (40 expected events) were treated as an irreducible back- lengths, we ignored any further downstream material. Also, ground for the LDM search, for a 90% CL exclusion limit we did not include any material upstream the thick target. of ≃16.4 LDM events. In this work, we only considered the χ − e− scattering process. We reproduced the MiniBooNE-DM analysis following the same strategy adopted in Ref. [20].We parametrized the MiniBooNE-DM response with the fol- lowing selection cuts, Ee > 75 MeV and cosðθeÞ > 0.99, where Ee and θe are, respectively, the scattered electron energy, and the angle measured with respect to the primary beam direction. The validity of this parametrization can be assessed from Fig. 8, where we compare the sensitivity for the “traditional” LDM production as reported by the MiniBooNE-DM collaboration (dashed orange line) with that obtained applying the aforementioned selection cuts (solid rust line) observing a very good agreement.

B. NOνA NOνA is a neutrino experiment at Fermilab studying the oscillation of muon neutrinos to electron neutrinos [68]. The experiment measures neutrinos produced in the NuMI FIG. 8. Limits for the MiniBooNE experiments. The grey region represents the exclusion bounds from the BABAR [16] target facility by the 120 GeV proton beam from the FNAL ν ν and NA64 [17] collaborations. The dashed orange line corre- Main Injector [69]. The NO A near detector (NO A-ND) sponds to the sensitivity as extracted from [64], the rust solid line is located 990 m downstream from the target, at 14.6 mrad is our estimate based on hadronic processes only, the solid green angle from the primary beam direction. Such off-axis line is our estimate based on secondary production processes configuration was chosen to optimize the neutrino energy only, and the thick black line is the combination of the two.

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In this work, we computed the NOνA-ND sensitivity to studies [72]. DUNE will consist of a near detector, that will LDM by simulating electrons- and positrons-induced record interactions near the source of the beam, and of a much production processes in the NUMI target. We implemented larger far detector, located underground 1,300 km down- the official GEANT4 description of the target geometry and stream of the source. DUNE will detect neutrinos produced by materials, as was used to measure fundamental neutrino the primary 120 GeV proton beam of the Fermilab accelerator properties [68], and that was provided to us by the NOνA complex impinging on a graphite target. collaboration. We considered the NOνA-ND active volume In a recent work it was shown that, despite the abundant described before, with an average electron number density neutrino background, a dedicated analysis with the DUNE 23 −3 ne ≃ 3 × 10 cm . Finally, we parametrized the detector near detector data will be able to explore unknown response to the scattered electron with the following territories in the LDM parameters space, exploiting the 2 χ − − selection cuts: Ee > 500 MeV, Ee · θe < 5 MeV, where e scattering channel [66]. In this work, we adopted the θ is measured with respect to the impinging particle same description for the DUNE near detector geometry e 3 direction. used in Ref. [66], considering a 3 × 4 × 5 m liquid argon detector located 574 m downstream from the target. C. SHiP We described the target as a thin, 220-cm long graphite cylinder [73]. We parametrized the detector response with SHiP is a proposed beam-dump experiment at CERN the following cuts on the scattered electron kinematics: SPS to search for weakly interacting long lived particles 2 Eeθe < 2me, Ee > 50 MeV, with θe measured with respect [65]. The SHiP detector, currently being designed, foresees to the impinging χ direction. two complementary apparatus, to investigate the hidden To derive the DUNE near detector exclusion limits sector exploiting both the visible decay signature of hidden for LDM, we considered a total accumulated charge of particles and the recoil signal from the scattering on atomic 1.1 × 1021 POT/year, and a 7-years long measurement. We electrons and nuclei. In particular, the SHiP Scattering and observe that, as discussed in Ref. [66], the DUNE near Neutrino Detector (SND) is a hybrid apparatus consisting detector sensitivity to LDM can be significantly enhanced of alternating layers of an absorber, nuclear emulsion films by performing multiple measurements at different off-axis and fast electronic trackers, characterized by a very low locations, to exploit the different angular spectra of the detection threshold and enhanced PID capability. The LDM signal and the neutrino background (DUNE-PRISM detector is located approximately 40 m from the production detector concept). In this work, for simplicity we performed target where the 400 GeV proton beam impinges on. a first estimate of the DUNE sensitivity to LDM produced A first estimate of the SHiP experiment sensitivity to by secondary eþ considering both a single on-axis and a χ − − LDM was discussed in [20] considering both the e single off-axis measurement (at the maximum transverse χ − and the N scattering processes. More recently, the SHiP distance of 36m), leaving a more comprehensive evaluation χ − − collaboration presented an updated limit for the e for the future. We estimated the irreducible neutrino channel, based on a robust evaluation of the irreducible background for the on-axis (off-axis) measurement to be neutrino background and on a realistic parametrization ≃71 × 103 (≃1500) events, assuming an equal experiment of the foreseen detector response, for a total exposure of run time in neutrino and antineutrino mode [66]. This 2 1020 × POT [55]. corresponds to a 90% CL exclusion limit of 350 (54) signal In this work, we evaluated the SHiP sensitivity to LDM events. as follows. We computed the LDM flux due to positrons annihilation in the beam dump with GEANT4, implementing E. Results the current target geometry and material composition that were provided to us by the collaboration. We parametrized In this section we present our results for the limits and for the SND active volume as a 90 × 75 × 320 cm3 volume, the projected sensitivities of the four experiments described located 38 m from the beam dump, with a fiducial mass of above, assuming that LDM is a complex scalar particle, 10 ton. The following selection cuts were applied to the that is for the model discussed in Sec. II B. In order to consistently compare the dark matter production via meson scattered electron kinematics, 1 GeV

075026-12 NEW PRODUCTION CHANNELS FOR LIGHT DARK MATTER IN … PHYS. REV. D 102, 075026 (2020) scenarios, for which proton beam-dump experiments are usually believed to have no sensitivity. For the coupling ge of a dark photon interacting dominantly with the leptons and with suppressed couplings to hadrons, the limits on the couplings are given by the simple relation:

lim ¼ εe ð Þ ge e lim: 23 “ ” εe In the following figures, this lepton-only limit lim is represented as a solid green line. Note that being electron- based experiments, the limits from NA64 and BABAR also apply in this case. We first considered the reach of the MiniBooNE experi- ment. As can be seen in Fig. 8, we find excellent agreement between our simulation using light meson production (orange dashed line) and the original limit from the FIG. 9. Projected reach of the SHiP experiment. The grey collaboration [64] (rust solid line). This confirms the robust- region represents the exclusion bounds from the BABAR [16] and ness of our calculations. The dotted and dashed green lines NA64 [17] collaborations. The dashed orange line is the limit correspond, respectively, to the limits from bremsstrahlung, extracted from [74], the rust line our estimate based on hadronic processes only, the solid green line our estimate based on the and from positron-induced production, including both þ − → → χχ þ − → γ → secondary production processes only, and the thick black line is resonant e e V and associated e e V the combination of both. γχχ processes. They contribute significantly to the total ∼ 20 number of expected events for mV MeV, thus signifi- estimate). Even if the experiment will use the 400 GeV SPS cantly enhancing the full MiniBooNE exclusion limits proton beam, leading in principle to high-energy electro- compared with those from the NA64 collaboration. The magnetic showers, due to the high detection threshold mass range where the pure resonant process is active is (∼1 GeV) electrons- and positrons-induced processes re- clearly visible in the plot. In particular, the lower bound at present only a small fraction of the final events. Mχ ∼ 3 m ∼ 10 1 MeV ( V MeV) is due to the fact that, We show in more detail in Fig. 10 the LDM energy following Eq. (16), a dark photon resonantly produced at distribution for the different production mechanisms, for this low mass does not transfer enough energy to the LDM the specific choice mV ¼ 30 MeV. The energy distribution particle (and ultimately to the scattered electron) to pass the for the leading mesons decay channel peaks as the highest Eth selection cut. For dark photon masses below this energies, as expected since it originates from mesons from threshold, the dominant production processes are thus the the primary hadronic shower. The secondary production dark bremsstrahlung from electrons and positrons and the from electrons/positrons bremsstrahlung retains a signifi- associated dark photon production. Note that the limit from cant fraction of the energy of the shower and peaks just secondary production is conservative in that we do not include dark photon production via the Compton-like process γe− → Ve−.3 For the lowest dark photon mass, as discussed in Appendix B and C, the cross-section for bremsstrahlung increases quadratically with the inverse of the dark photon mass, while associated production saturates. The impact of the energy threshold on the limit is further visible in Fig. 9, where we plot the expected sensitivity of the SHiP experiment. Also in this case, the comparison between our calculation (rust dashed line) and the results of the collaboration (orange dashed line) for light mesons LDM production show a relatively good agreement (notice that we did not include possible detection efficiencies in our

3As shown in Appendix C, based on the similarities with the associated production differential cross section it is possible to estimate the typical size of the complete secondary production rate by multiplying by ∼3 the associated production rate. FIG. 10. Energy distribution of LDM particles impinging on Such modification, however, improves only marginally the limits the SHiP detector for different production mechanisms: produc- presented here. We thus leave a complete study of the Compton- tion from mesons decay (blue), positrons resonant annihilation like process for a future work. (orange), electrons and positrons bremsstrahlung (green).

075026-13 CELENTANO, DARME,´ MARSICANO, and NARDI PHYS. REV. D 102, 075026 (2020) above the GeV. As shown in Appendix B, this is due to both the fact that bremsstrahlung dark photons typically retain all the energy of the incoming eþ=e− and that the bremsstrahlung process itself is effective at large center- of-mass energy. Finally, LDM production through resonant positrons annihilation is peaked at a lower energy below the GeV, around half the energy of the outgoing dark pho- ¼ 2 ð2 Þ ∼ 0 9 ton EV mV= me . GeV. In the case of the NOνA experiment, the large energy ¼ 0 5 threshold Eth . GeV also limits significantly the con- tribution of electromagnetic shower-induced processes, ∼ with a corresponding lower mass threshold around Mχ1 10 MeV (mV ∼ 30 MeV), as seen in Fig. 11. Note that the relatively large energy threshold as well as the large distance between the beam dump and the experiment tends FIG. 12. Projected reach of the NOνA (green lines) and the to reduce the contribution from the shower-generated SHiP (red lines) experiments, with reduced energy thresholds at events, since they are typically both less collimated and 125 MeV for NOνA and 250 MeV for SHiP. Sensitivity estimates less energetic than their hadronic-generated counterparts. are based on 16.4 (38) signal events for NOνA (SHiP). The grey We illustrate the effect of lowering the energy threshold for region represents the exclusion bounds from the BABAR [16] and the NOνA and SHiP experiments in Fig. 12. In this case, we NA64 [17] collaborations. The solid lines are our estimate based did not combine the hadronic and leptonic limits as for the on hadronic processes only, while the dashed lines are based on other plots, to illustrate that the background level are likely secondary production processes. to be significantly modified, so that the proposed reaches should also be rescaled accordingly. On the other hand, it is the leptonic-induced events play an important role in the clear that the ratios between both production modes is not final production rates, particularly at small dark matter significantly modified by this change. In particular, in the masses. A particularity of the proposed DUNE-PRISM ν case of the NO A experiment, the small geometric accep- near detector concept is that it can be physically moved off- tance of the experiment suppresses naturally the shower- axis up to 36 m to reduce the overall background. While we induced events. did not performed a complete analysis like the one carried Finally, we present in Fig. 13 the long term prospect out in Ref. [66], we present in Fig. 14 the possible reach of based on the near detector of the DUNE experiment. This the DUNE near detector in case it will be moved at the experiment will adopt a much lower energy threshold maximal off-axis distance, considering the same run ν than NO A and SHiP. Consequently, we observe that parameters as the nominal on-axis mode. Interestingly,

FIG. 11. Projected reach of the NOνA experiment. The grey FIG. 13. Projected reach of DUNE near detector. The grey region represents the exclusion bounds from the BABAR [16] and region represents the exclusion bounds from the BABAR [16] and NA64 [17] collaborations. The rust line is our estimate based on NA64 [17] collaborations. The rust line represents our estimate hadronic processes only, the solid green line our estimate based based on hadronic processes only, the solid green line our on secondary production processes only, and the thick black line estimate based on secondary production only, and the thick is the combination of both. black line is the combination of both.

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mesons decays, this effect is more important for experi- ments characterized by low detection threshold on the scattered electron. Before concluding, it should be emphasized that, while we focused on the case of a dark photon mediator, our analysis can be easily extended to any other LDM model. Given that the increase in the LDM particle yield that we obtain only depends on the inclusion of new production channels, our results can be relevant also for LDM searches based on detection strategies different from the simple χe− → χe− scattering considered here, as for example measurements of energy deposition in the detector from visible decays of long-lived dark sector states. Finally, while we concentrated on proton beam-dump experiments, it would also be important to properly account for the new FIG. 14. Projected reach of DUNE near detector, if moved by processes analysed in this work for projected LHC-based 36 m off the beam axis. Same color coding as the previous intensity frontier experiments, such as FASER(ν) [75,76], DUNE plot. MATHUSLA [77], Codex-b [78], ANUBIS [33] or MilliQan [79]. The extremely high energy available at LHC interaction points may actually lead to an even the wide emission cone of the leptons-induced dark matter stronger production of dark sector particles from processes candidate enhances their importance with respect to the induced by electromagnetic showers. We thus believe that it standard mesons decay processes. would be particularly important for these experiments to consider carefully also shower-based dark sector produc- tions, and not only to estimate correctly their sensitivity V. CONCLUSIONS AND OUTLOOKS reach, but also to optimize the choice of the detection When a high-energy proton beam impinges on a thick energy thresholds for the physics run. target, a large fraction of the primary energy is transferred to the electromagnetic component of the developed par- ACKNOWLEDGMENTS ticles shower, resulting into an abundant production of photons, electrons, and positrons. In this work, starting A. C. and L. D. warmly thank L. Buonocore for very from this observation, we have discussed for the first time helpful discussions on MadDump. A. C. thanks P. Snopok the role of electrons- and positrons-induced processes in and A. Habig for their support in implementing the official proton beam-dump experiments in relation to LDM NUMI target geometry in our Monte Carlo simulation, searches. We have shown that LDM production from W. Bonivento and T. Ruf for their support in implementing shower induced electromagnetic processes, that was so the official SHIP target geometry in our Monte Carlo far overlooked, must be accounted for to properly assess the simulation, and K. Kelly for his help concerning the sensitivity of forthcoming proton-beam dump experiments, neutrino background expected in the DUNE-PRISM meas- and to derive limits on the LDM parameter space from the urement. The authors thank S. Trojanowski for his help on analysis of existing data. analtyical expression for the bremsstrahlung process. L. D. “ ” A numerical procedure, based on the MaddDump and and E. N. are supported by the INFN Iniziativa Specifica BdNMC simulations codes was developed to generate Theoretical Astroparticle Physics (TAsP-LNF). LDM particles, and, starting from the eþ=e− differential track length in the target computed with a GEANT4-based Note added.—Simultaneously with our paper, Ref. [80] simulation, to propagate them into a downstream detector. appeared which dealt with neutrino experiments based on We considered a representative set of proton thick-target high-intensity proton beam with ∼GeV energy (such as the experiments (MiniBooNE, NOνA, SHiP, and DUNE), COHERENT experiment) and investigated in details the finding that for each of them the new production mecha- use of timing and energy cuts to reduce the neutrino nism results into a non-negligible increment of the sensi- background. We point out that it would be interesting to tivity to LDM. For some regions of the parameters space, include the complete shower productions modes (in par- the eþ=e−-induced processes actually represent the dom- ticular resonant production) when estimating the efficiency inant production mechanism for LDM, and can lead to of this approach. Indeed, the kinematic distribution of these signal rates on par with the standard results. Due to the events is likely to be significantly different from the meson- typically softer spectrum of LDM particles generated from induced production, potentially leading to new ways of eþ=e− secondaries with respect to those originating from optimizing the selection cuts.

075026-15 CELENTANO, DARME,´ MARSICANO, and NARDI PHYS. REV. D 102, 075026 (2020)   0 −1−s APPENDIX A: ANALYTICAL TREATMENT dneðE; Eγ;tÞ 1 Gγ→eðsÞ E ¼ pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi λ1ðsÞt ð Þ OF EM SHOWERS 00 e ; A6 dEdt Eγ 2π λ1ðsÞt E0 In this Appendix we describe the technical details of the analytical shower modeling. Our treatment is based on the where we have used the primed notation for the derivatives ð Þ study of the development of high energy cosmic ray with respect to s. The auxiliary function Gγ→e s is defined showers in the atmosphere presented in Ref. [51], which as: in turn is based on the Rossi and Griesen approach [50]. p p The idea is to solve first the equations coupling the 1 ½σ þ λ1ðsÞ½σ þ λ2ðsÞ Gγ→eðsÞ¼− ; ðA7Þ differential density of electrons/positrons neðE; tÞ and of C λ1ðsÞ − λ2ðsÞ 0 photons nγ ðE; tÞ as function of the depth parameter t λ (expressed in unit of radiation length), that read: while the two functions 1;2 read: Z  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 b 0 1 1 ∂n ðEÞ 1 dσ n ðE=ð1 − xÞÞ p p 2 e ¼ − 0ð Þ − e λ1 2ðsÞ¼− ðA þ σ Þ ðA − σ Þ þ 4BC: ðA8Þ dx ne E ; 2 2 ∂t 0 dx 1 − x  p 2 dσ 0 We have used the following cross-sections momenta: − nγ ðE=xÞ ðA1Þ x dx Z 1 σb Z d s ∂ 0ð Þ 1 0ð Þ σp AðsÞ¼ dx ð1 − ð1 − xÞ Þ; ðA9Þ nγ E p 0 ne E=x d 0 dx ¼ −σ nγ þ dx ; ðA2Þ ∂t 0 x dx Z 1 σp d s dσb dσp BðsÞ¼2 dx x ; ðA10Þ where dx and dx are respectively the differential cross 0 dx section for bremsstrahlungR photon production and for e Z p b pair production, and σp ¼ 1 dx dσ is the integrated pair 1 dσ 0 dx CðsÞ¼ dx xs; ðA11Þ production cross section. The two differential cross sections 0 dx are given by:   which can also be straightforwardly expressed as (lengthy) dσb 1 2 expressions involving polylogarithm functions [51]. ðxÞ¼ 1 − − 2b ð1 − xÞþð1 − xÞ2 ðA3Þ dx x 3 Z Once this uncut distribution is estimated, the approach of Rossi and Griesen is to add a “loss” term in Eq. (A1) by p dσ 2 2 2 replacing ðxÞ¼ð1 − xÞ þ − 2b ð1 − xÞx þ x : ðA4Þ dx 3 Z ∂n ðEÞ ∂n ðEÞ ∂n ðEÞ e → e − ϵ e ; ðA12Þ The first two terms in Eq. (A1) represent respectively the ∂t ∂t c ∂E fraction of e of energy E which lose energy by brems- ϵ strahlung, and the fraction of higher energy e which end where c is the critical energy defined in Eq. (4). up with energy E following a bremsstrahlung. The last term Approximate solutions to the new system of equations accounts for e produced via photon conversion. The two can be searched for in the form: terms in Eq. (A2) represent, respectively, the photons lost to ð Þ¼ 0ð Þ ð ϵ Þ ð Þ pair-production and the photons produced via bremsstrah- ne E; s ne E; s × p1 E= c;s : A13 lung. The parameter x represents the energy ratio Ee=Eγ 0ð Þ between the incident e and the outgoing photon for In general ne E; s on the right-hand-side of this equation bremsstrahlung, while it represents the opposite ratio for should be multiplied by a cutoff function p that can in principle be obtained by replacing n0 × p in the system of e pair production. The effective parameter bZ can be e expressed as function of the atomic number Z of the differential equations. In our paper, we are using for medium as simplicity the interpolation of p, estimated at the shower maximum, that is p → p1ðx ¼ E=ϵc;s¼ 1Þ as given in 1 [51]. Note that a good analytical interpolation in x ¼ E=ϵ b ≃ : ðA5Þ c Z 18 logð183Z−1=3Þ is given by:

0.18 18 As was worked out long ago by Rossi and Greisen [50],itis p1ðx; 1Þ¼tanhð1.8x Þ : ðA14Þ possible to obtain an analytical solution for the above set of coupled equations valid for the later stage of shower Finally, since the original hadronic shower produces a development, i.e., when t; E=Eγ ≪ 1. For a shower induced large number of photons with different energy, the resulting by a photon of energy Eγ the solution reads: track-length distribution for the full electromagnetic

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el in shower is obtained by integrating over the initial differential with G2ðtÞ¼G2 þ G2 defined by distribution of photons, as shown in Eq. (7). a2t 2 1 2 Gel ¼ Z2; APPENDIX B: NUMERICAL APPROACH 2 1 þ a2t 1 þ t=d TO BREMSSTRAHLUNG PROCESSES 2 02 2 1 þ t ðμ − 1Þ a t 4m2 p in ¼ p ð Þ Bremsstrahlung production of dark photons is G2 02 t 4 Z; B4 1 þ a t ð1 þ 2Þ traditionally the dominant production mechanism consid- 0.71 GeV ered in electron beam-dump experiments. We give in this μ ¼ 2 79 ¼ 0 938 4 Appendix a few details about our estimation of this process with p . and the proton mass mp . GeV. via MadGraph5_aMC@NLO, starting from a brief summary of Interestingly, we see that the form factors disfavor very soft the analytical approach based on the Weizsacker-Williams or very hard photon exchanges due to either the screening approximation [57,81,82]. We present the result for the from the electrons in the atomic cloud when case of an incoming electron, but note that it also applies for 1 1 the case of an incoming positron. 2 02 ≪1 ≡111 0 ≡773 ð Þ a t; a t ;a 1=3 ;a 2=3 B5 We consider the process meZ meZ

− − 0 ðÞ − or from the finite nuclear size in the other limit e ðpÞNðPiÞ → e ðp ÞNðPfÞV ðkÞ → e Nχ χ; ðB1Þ

2 −2 3 where N is a nucleus with atomic number Z. For simplicity, dt ≪ 1;d¼ 0.164 GeV A = : ðB6Þ we focus on the case of a monochromatic impinging beam (the extension to the realistic case through a track length As pointed out by [57], all values of t contribute equally to approach is straightforward). We follow the notations and the integral—in particular, the integral it is not dominated ð Þ ∼ summarizing the discussion of [57]. We define as E0 EV by t tmin. Indeed, while the virtual photon propagator 1 2 ¼ the energy of the incoming electron (outgoing dark photon) squared, =t , is maximum at t tmin, the phase-space in the lab frame, and we introduce the ratio x ≡ EV=E0.As numerator balances it in the integral. The minimum value was noted in [81], the photons mediating the process are of t is given by only very mildly virtual so that their interaction with the electron are dominated by their transverse polarization. It is 2 2 2 ¼ − 2 ≈ U ∼ MV ð Þ then possible to decompose the cross section into a real tmin qmin ; B7 2ð1 − xÞ 2E0 photon-electron scattering, eðpÞγðqÞ → eðp0ÞVðkÞ where ≡ − the photon has the (small) virtual momentum q Pi Pf, where and a form factor for the emission of the photon from the 2 nucleus. Let us define t ≡ −q (not to be confused with the 1 − x U ≡ Uðx; θ Þ¼E2θ2 x þ m2 þ m2x: ðB8Þ depth parameter t introduced in the previous Appendix) and V 0 V V x e call θV the angle of the outgoing dark photon with respect ∼ 2 → 2 to the incoming electron in the lab frame. The full cross at t tmin Following [57], the cross section for the ∼ 2 section can be written [81]: process at t tmin can be written up to terms in me as:  dσð2 → 3Þ σ σ ð1 − Þ d ¼ 2 d ¼ð4πα2 ϵ2Þ x 1 þð1 − Þ2 dxd cos θ em 2 x V dðp · kÞ dt2 U 0  α F β σð þ → þ Þ 2 2 ¼ em E0x V d p q p k 2ð1 − xÞ m Ux E0 × ; þ V m2 − : ðB9Þ π ð1 − xÞ dðp · kÞ ¼ 2 V t tmin U 1 − x ðB2Þ θ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Putting everything together and neglecting the V depend- β ≡ 1 − 2 2 ence in F, the cross section can be integrated once yielding with V mV =E0. Importantly, the cross section for 2 → 2 the process is estimated at the minimum virtuality −1 2 α F dσ3→2 1 − x x ¼ em ¼ 4α3 ϵ2Fβ 2 þ 2 1 − þ t tmin. The term π describes the effective photon flux em V mV mex x ; ¼ dx x 3 integrated from t tmin to the total center of mass (CM) ¼ energy tmax s. It can be obtained by integrating the ðB10Þ nuclear and atomic form factors over the virtuality:

Z 4 t − Note that the last term of the inelastic form factor is not F ≡ max t tmin ð Þ ð Þ dt 2 G2 t ; B3 squared, following the original expression of [81] (see also [29]) tmin t compared to the expression in [57].

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(note that the original expression from [57] missed a factor frame (although the associated production process is also of 1=2 [82]). It is clear that this differential cross section has enhanced in the oppositepffiffiffi direction, θ ∼ π). For small ∼ 1 an approximate singularity for x , regulated by the angles and in the limit s ≫ mV;me, the following similar 2 ð1 − Þ ¼ me expressions hold: electron mass at x c1 2 , where the subscript c1 mV labels a first cutoff point. As remarked in [57], the σ dσ 4πε2α2 d assoc ¼ Compton ≃ em ð Þ approximation also breaks down if the virtuality is too θ θ 2 2 ; C1 2 d cos d cos sθ þ 4me mV large, yielding a second cutoff ð1 − xÞ ¼ 2 . The total c2 E0 cross section finally reads: In particular, both differential cross sections saturate at very θ2 4 2 small angle, when s < me. The total cross sections are 4 α3 ϵ2Fβ 1 also equivalent, with σ ≈ em V log ; ðB11Þ 3 m2 ð1 − xÞ V c 2πε2α2 2 σ ≃ em Eþ − 1 ð Þ assoc log C2 2 m2 ð1 − Þ ¼ ðme V Þ meEþ me where x c max m2 ; E2 . V 0 An important feature that can be read out from this πε2α2 2 1 σ ≃ em Eþ þ ð Þ formula is that the cross section is actually only mildly Compton log ; C3 m Eþ m 2 dependent on the incoming electron energy, either via the e e logarithm term (which saturates when the me contribution mV where the factor of 2 is compensated by the fact that the dominates), or via the form-factor contribution, which also associated production also generate efficiently events with saturates at high energy due to the atomic electrons a very forward photon, with the same rate as in the forward screening. dark photon region. Hence both processes lead to similar We have simulated this process in MadGraph5_aMC@NLO production rates of energetic dark photons, and since the using an effective NNγ interaction with form factor G2. cross section does not depend on mV, we expect these rates This implies that we did not use the Weizsacker-Williams to saturate in the light dark photon limit. Finally, note that approximation for the cross section, but we directly we have considered for both processes the atomic electrons estimated the 2 → 4 process with dark matter final states. to be free (i.e., described by a plane wave function) and in Furthermore, in order to regulate the numerical divergence particular we neglected the target electron motion [59]. which arises for large electron energies when the Furthermore, we observe that in an electromagnetic exchanged photon is very soft, we have modified the form shower, the distribution of photons actually follows factor G2ðtÞ. In particular, due to the screening effects relatively closely the one of the positron/electron as long 2 occurring when a t ≪ 1, we know that this part of the as the energy is above the critical energy. One has phase space is sub-dominant in the final production rate. Tγ ∼ ð1.3 − 1.5Þ · ðTeþ þ Te− Þ in most of the shower We therefore implemented a regularization cut by setting development—see for example the discussion in Ref. [51]. the form factor to 0 in the “screened” region: All in all, we therefore expect the production of very  forward dark photons in the electromagnetic subshower to ð Þ 2 1 3 r G2 t for a t> = be a factor of 2 larger for the Compton-like production than G2ðtÞ¼ ðB12Þ 0 for a2t<1=3: for the associated production, albeit with very similar kinematics. We have explicitly checked that the value of the final cross We have simulated the associated production process in 2 section is not modified by varying the cut between a t<1 MadGraph5_aMC@NLO using the positron track length esti- and a2t<0.05, and agrees with the analytical expression mated via GEANT4. As can be seen from the differential developed above. Furthermore, we have verified that the cross section Eq. (C1), the process has an approximate differential distribution in angles and energy are also not collinear divergence regulated by the electron mass which affected by this regularization procedure. leads to a logarithmic enhancement of the total cross section. We numerically-regulated this divergence in MadGraph5_aMC@NLO by adding a generator-level cut on APPENDIX C: ASSOCIATED AND θ as θ > 10−5 rad. Since this value is safely below the COMPTON-LIKE PROCESS pffiffiffi saturation value for the differential cross section 2me= s in We give in this Appendix more details about the the whole range of energies considered in this work, the associated production and Compton-like scattering which effect of this cut on the magnitude of the cross section is complement the pure resonant production of light dark negligible. Furthermore, the associated cross section also matter. presentspffiffiffi an infrared divergence from soft photon emission ∼ The differential cross section for both processes peaks when s mV, whichpffiffiffi is not present in the above formula forward at θ ∼ 0, with θ the V production angle in the CM since we assumed s ≫ mV. This second divergence

075026-18 NEW PRODUCTION CHANNELS FOR LIGHT DARK MATTER IN … PHYS. REV. D 102, 075026 (2020) formally cancels against the infrared divergence of the We have included in our numerical evaluation the virtual 1-loop correction to the resonant production proc- associated production rate, while we leave for future ess, and represents therefore an higher order effect. That is, refinements the estimation of the LDM signal arising from formally the events with a soft photon represent a QED Compton-like dark photon production. As pointed out in radiative correction to the resonantly produced dark pho- the main text, we expect this process to be sizeable only in ton. Since we are alreadypffiffiffi simulating the tree-level resonant the limited region where the dark photons are massive process, we imposed s >mV=0.95 at the generator-level, enough to suppress bremsstrahlung, but light enough so independently of the emission angle θ, to ensure that only that resonant production is not available due to the events with sufficiently hard photons are simulated. experimental energy thresholds.

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