Constraints on Millicharged Particles from Cosmic-Ray Production

PHYSICAL REVIEW D 102, 115032 (2020)

Constraints on millicharged particles from cosmic-ray production

† ‡ Ryan Plestid ,1,2,3,4,* Volodymyr Takhistov,5, Yu-Dai Tsai ,2,6, Torsten Bringmann ,7,§ ∥ Alexander Kusenko,5,8, and Maxim Pospelov9,10,¶ 1Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506, USA 2Theoretical Physics Department, Fermilab, Batavia, Illinois 60510, USA 3Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada 4Department of Physics and Astronomy, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4M1, Canada 5Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA 6Cosmic Physics Center, Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA 7Department of Physics, University of Oslo, Box 1048, N-0371 Oslo, Norway 8Kavli Institute for the Physics and Mathematics of the Universe (WPI), UTIAS The University of Tokyo, Kashiwa, Chiba 277-8583, Japan 9School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 10William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA

(Received 13 March 2020; accepted 7 December 2020; published 24 December 2020)

We study cosmic-ray-atmosphere collisions as a permanent production source of exotic millicharged particles (MCPs) for all terrestrial experiments. [MCPs are also known as charged massive particles (CHAMPs).] Based on data from Super-K, this allows us to derive new limits on MCPs that are competitive with, or improve, the currently leading bounds from accelerator-based searches for masses up to 1.5 GeV. In models where a subdominant component of dark matter (DM) is fractionally charged, these constraints probe parts of the parameter space that is inaccessible for conventional direct-detection DM experiments, independently of assumptions about the DM abundance.

DOI: 10.1103/PhysRevD.102.115032

I. INTRODUCTION [2–4]. At these energies, new physics can be efficiently probed by fixed-target experiments [5–9] with high inten- The remarkable success of the Standard Model (SM), sity electron [10,11] and proton beams [12–20] where dark along with null results at the LHC, suggests that if new sector particles can be produced either directly, or through physics exists below the TeV scale it must be weakly meson decays; collider experiments typically provide coupled to SM degrees of freedom. Such a dark sector the leading constraints at higher masses [21–23]. A less would likely leave its strongest imprint on SM degrees of explored opportunity is the production of dark sector freedom commensurate with its own dynamical energy particles in cosmic-ray interactions and their subsequent scales [1]. The MeV-GeV regime both contains many SM detection in large detectors [24–34]. particles and hosts a number of persistent anomalies, Historically, the discovery of particles in the MeV-GeV including the anomalous magnetic moment of the muon regime (e.g., pions [35] and muons [36]) often resulted from cosmic rays, mostly protons, bombarding the upper *[email protected] atmosphere. This essentially constitutes a fixed-target † [email protected] “ ” ‡ experiment, where the proton beam is always on. With [email protected] modern neutrino telescopes, the detectors located “down- §[email protected] ∥ ” [email protected] stream can be kton-Mton scale (e.g., IceCube [37], Super- ¶[email protected] K [38], Hyper-K [39], JUNO [40], DUNE [41]) and can serve as a powerful tool with which to probe the dark sector. Published by the American Physical Society under the terms of In this work, we calculate a lower bound on the flux of the Creative Commons Attribution 4.0 International license. – χ Further distribution of this work must maintain attribution to millicharged particles (MCPs) [42 48], , arising from the author(s) and the published article’s title, journal citation, meson decays in the upper atmosphere for mχ in the and DOI. Funded by SCOAP3. ∼100 MeV to few GeV regime. For this, we adopt a

2470-0010=2020=102(11)=115032(13) 115032-1 Published by the American Physical Society RYAN PLESTID et al. PHYS. REV. D 102, 115032 (2020) minimal model about the new particle that is based on only fact the leading, probe of MCPs in the 100 MeV–1.5 GeV two assumptions which are as follows: regime, free from cosmological assumptions. Our results (1) χ couples to the photon with strength Qχ ¼ ϵ × e; apply to MCP parameter space important to 21 cm we remain agnostic as to the origin of this charge. cosmology (potentially explaining the EDGES anomaly (2) χ is stable, which is a natural consequence if Qχ is [56–59]) and to the same regime that has motivated the the smallest (nonzero) charge in the dark sector. proposal of dedicated detectors such as MilliQan [22,23] Our analysis then serves as a benchmark for the sensitivity and FerMINI [18]. of neutrino observatories to stable dark sector particles that More broadly, we note that MCPs have connections to can be produced in the upper atmosphere. In particular, our charge quantization, which is itself connected to, but does constraints apply (possibly conservatively) to any model not necessarily preclude [65], the existence of magnetic that satisfies the above two assumptions. monopoles [66], grand unification [67–70], and quantum One immediate consequence of our calculation is that gravity [71]. Furthermore, they appear naturally in models existing constraints on an ambient ionizing MCP flux— of light dark matter (DM) [65,72–74], and a fast-moving ϵ Φ ðϵÞ— quoted as a function of and ion from MACRO flux of MCPs has recently been proposed to explain a [49,50], Kamiokande-II [51], LSD [52], CDMS [53], and reported excess in direct-detection experiments [75,76]. Majorana [54] can now for the first time be recast as direct ϵ constraints on as a function of mχ. Better yet, we find that II. STRONGLY INTERACTING DM neutrino observatories can set new competitive (and in some cases leading) bounds on MCP couplings in the Our results can be recast as limits on millicharged 100–500 MeV range based on existing data, surpassing the strongly interacting DM (SIDM) [77,78]. This refers to a reach of fixed-target experiments with neutrino detectors. class of models where the DM-SM particle (mostly nuclei We demonstrate this point explicitly by providing novel and electrons) cross section is so large that the DM flux constraints based on published analyses by the Super- reaching conventional underground direct-detection experi- Kamiokande (Super-K, SK) Collaboration searching for the ments would be significantly attenuated, thus preventing ϵ diffuse supernova neutrino background (DSNB) [55]. detection [77,79]. Therefore, above some critical value of Our results, summarized in Fig. 1, suggest that future fractionally charged DM remains unconstrained by such neutrino telescopes could be able to act as a robust, and in experiments [78], and there is a window of available parameter space where MCPs could constitute a subdominant component of DM (≲0.4% to avoid cosmological constraints [80–83]); we will refer to this as the millicharged SIDM window (see Fig. 2). New balloon and satellite experiments have been proposed [78] to further explore this window, which could accommodate DM models capable of explaining the EDGES anomaly [56–59,84]. Our results exclude parameter space for millicharged SIDM that is not accessible at terrestrial detectors because the typical velocity of MCPs produced by cosmic rays is much higher than the ambient dark matter wind, such that cosmic-ray-produced MCPs can reach subterranean detectors with enough kinetic energy to leave detectable sig- natures. In Fig. 2, we translate our results from Fig. 1 to constraints on a conventional “reference cross section” σ¯ ¼ 16πα2ϵ2μ2 4 1 e;ref χe=qd;ref for DM-electron scattering (which for light MCPs dominates over scattering with nuclei). Here, μχe is the reduced mass of the electron and χ χ − and qd;ref are the typical momentum transfer in e FIG. 1. Exclusion limits for MCPs from cosmic-ray interactions scatterings for semiconductor or noble-liquid targets, taken (SK, red solid), based on the diffuse supernova neutrino back- to be αme [78]. In the same figure, we compare to existing ground search in Super-K [55], as well as sensitivity projections for an upgrade with gadolinium doping (SK+, red dotted) and 1In Ref. [78], the DM millicharge is generated through a near-future Hyper-K (HK, red dashed). We also display new coupling to a massless dark photon that kinetically mixes with the limits (blue) from recasting data of MACRO [49,50] and SM Uð1Þ ; here we directly consider the DM to have a Majorana [54]. Previous limits from electron fixed-target (SLAC Y ð1Þ millicharge under U Y, with minimal theoretical assumptions MilliQ [10,11], MiniBooNE [16,60], ArgoNeuT [61]) and about the origin of this charge. The reference cross section we – α ¼ α 0 → 0 collider experiments [6,62 64] (as compiled in Ref. [61]) are consider corresponds to D and mA in Eq. (2.6) of shown for comparison. Ref. [78].

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experiments.2 We focus on the dominant η, light vector, and J=ψ mesons. While we have also quantitatively considered ϒ meson as well as direct Drell-Yan production, we found these contributions to be negligibly small (6 orders of magnitude smaller than J=ψ). We focus our attention on mχ ≳ 100 MeV due to strong existing constraints from a combination of SLAC’s milliQ experiment [10] and LSND’s search for electronlike scattering events [102]. This eliminates π0 as a potential source of MCPs. The resulting meson flux from cosmic-ray collisions in the upper atmosphere is then given by Z σ ðγ Þ Φ ðγ Þ¼Ω I ðγ Þ m cm ðγ jγ Þ γ ð Þ m m eff CR cm σ ðγ Þ P m cm d cm; 1 FIG. 2. Constraints and sensitivity reaches around the milli- in cm charged SIDM window, including our new bounds from recasting I ðγ Þ ≡ I ð Þ γ Ω ≈ 2π Super-K data [55] (red) and projections for a Super-K upgrade where CR cm CR Ep ×dEp=d cm and eff is (red, dashed) and Hyper-K (red, dotted). We also show new the effective solid angle from which MCPs can arrive at 3 ðγ jγ Þ bounds (blue) from recasting data of MACRO [49,50] and the detector, and P m cm represents the probability to Majorana [54], along with existing accelerator-based constraints get a meson with boost γm in the lab frame. We – σ σ [6,10,11,16,61 64], limits from the above-atmosphere detector emphasize that the primary yield per proton m= in (RRS) [85], a rocket experiment (XQC) [86,87], and ground- and the cosmic-ray intensity I are well measured based direct detection [78]. (see Appendix B) and determine the total integrated flux ðγ jγ Þ of mesons. In contrast, P m cm does not affect the constraints, but do not consider here bounds based on the normalization, only serving to redistribute the flux as a γ MCP acceleration from astrophysical sources that rely on function of m. This probability can be conveniently additional assumptions beyond the local DM density estimated (see Appendix B) from the differential produc- ≡ [26,47,48,88]. Our constraints are independent of the tion cross section with respect to xF pL=pmax, where fractional composition of DM [80–82], since our milli- pL is the longitudinal momentum and pmax is the charged particles are directly produced in the upper maximum possible momentum, atmosphere; for reference cross sections below approxi- ðαÞ −17 2 X 1 σ mately 10 cm , our results are insensitive to attenuation ðγ jγ Þ ≈ d m dxF ð Þ P m cm σ × × γ : 2 in the Earth. α m dxF d m

III. COSMIC-RAY MESON PRODUCTION Here α ¼ denotes the two different branches that appear when inverting γmðxFÞ,anddσm=dxF is a function We begin by discussing the production of mesons from γ ðγ Þ of cm and xF m . The meson-production energy spec- cosmic rays, which then subsequently decay to MCPs. trum thus depends on both the total meson cross section, Cosmic rays hitting the upper atmosphere dominantly σ ðγ Þ m cm , and the differential cross section with respect to consist of high-energy protons and are isotropically dis- ðγ jγ Þ xF, or equivalently on P m cm . Our parametrization of tributed. Their flux is typically quoted in terms of intensity, dσm=dxF is guided by empirical measurements, but, with ½I ¼GeV−1 cm−2 s−1 str−1 [89], which in our analysis CR the exception of J=ψ,dataonthexF distribution are we take from DarkSUSY [90] (based on Ref. [91]). limited to specific center-of-mass energies and we adopt a Taking into account that all incoming cosmic rays are prescription outlined in Appendix B. The normalization eventually absorbed by the atmosphere, we can treat it as of the flux is independent of this prescription, and the a “thick target.” In what follows, we (i) treat pA collisions shape of the resultant meson fluxes depends only weakly as an incoherent sum of pp collisions and (ii) we include on it. only primary production. With these two assumptions, we can calculate the yield of primary mesons per incident 2 σ σ σ → m For η, see Refs. [92–97]. For ρ, ω, and ϕ, see Refs. [94,95,97]. proton as m= in where m is the inclusive pp X ψ – σ Finally, for J= , see Refs. [98 101]. Details of the analysis are production cross section, and in is the total pp inelastic discussed in Appendix B. cross section. This provides a robust lower bound on 3By rescaling the muon’s stopping power [26,89], we estimate the flux of mesons produced by cosmic rays in the that the energy loss of MCPs in the Earth’s crust (standard rock) is σ ð Þ roughly 50 MeV=km for ϵ ∼ 10−2. While for the range of ϵ and upper atmosphere. We take in pp from Ref. [89], γ ¼ energies that we are interested in here, MCPs interact too strongly expressedpffiffiffi in terms of the center-of-mass boost cm to penetrate the entire Earth, they are not significantly impeded to 1 2 s=mp and σm from a number of accelerator-based reach the detector when originating from the upper hemisphere.

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I FIG. 3. Differential cosmic-ray intensity, CR, multiplied by the FIG. 4. Fast flux of MCPs due to meson decays, Φ ,asa σ σ γ cut yield per proton m= in as a function of cm. For the resulting function of the MCP mass, mχ, for three different choices of γcut. meson spectra, see Appendix E. The full integrated MCP flux is obtained for γcut ¼ 1. The meson mass thresholds are clearly visible, stemming from η, ω=ρ, ϕ, and ψ In Fig. 3, we display the differential cosmic-ray intensity finally J= (sequentially from left to right). ðγ Þ σ ðγ Þ ICR cm multiplied by yield per proton m cm for each γ γ ¼ 1 meson species. This plot shows us which parts of the several choices of cut: cut i.e., the full integrated MCP γ ¼ 6 primary cosmic-ray spectrum are responsible for each flux relevant for ionization experiments, cut as ’ ¼ 16 meson s (primary) production; we emphasize again that required for an electron recoil of Tmin MeV (relevant γ ¼ 12 this results in a robust lower bound on the flux of mesons for Super-K), and cut for illustration. from the upper atmosphere computed from microscopic considerations. The shape of these curves is determined by V. DETECTING MCPs IN LABORATORIES the competition between a rising inclusive cross section and a sharply falling cosmic-ray flux and illustrates which parts Our calculation of the flux as a function of ϵ and mχ of the cosmic-ray spectrum predominantly contribute to a allows us to derive constraints on ϵ as a function of mχ.For given meson species’ flux. the case of ionization experiments, we can directly compare γ ¼ 1 the cut curve in Fig. 4 to published bounds in terms IV. MCP FLUX FROM MESON DECAYS of the ambient MCP flux (see Appendix C for details), and our Φ results establish a quantitative baseline that can be used to Upon constructing m as outlined above, we find the estimate the impact of upcoming projects such as LEGEND associated flux of MCPs from meson decays by folding the −1 [103]. For MCPs with large charges of ϵ ≳ 10 , effects of meson flux with the unit-normalized spectrum of MCPs in attenuation when passing through Earth to reach typical the lab frame, PðγχjγmÞ, and weighting by the decay detector depths of ∼1 km of standard rock, i.e., few km water branching ratio, Meson kinematics are discussed in equivalent, become significant (see e.g., Ref. [26]). We do not Appendix A consider these effects here as this region is already well Z X constrained by collider searches (which are not sensitive to ΦχðγχÞ¼2 BRm→χχ¯ dγmΦmðγmÞPðγχjγmÞ; ð3Þ attenuation). m Electron scattering inside Cherenkov detectors, with ∼10 where the factor of 2 accounts for the contribution from recoils in the MeV range, can also probe MCPs −1 [15] (see also [26]). The detection of MCPs is dominated both χ¯ and χ. The quantity PðγχjγmÞ¼½Γ dΓ=dγχ can lab by soft scattering from electrons as can be readily under- be calculated from first principles, at leading order in ϵ, stood by considering the differential scattering cross where Γ is the decay rate for m → χχ¯ and dΓ=dγχ is the section which, being mediated by photon exchange, scales differential rate with respect to the MCP boost, both as dσ=dQ2 ∼ 1=Q4. For elastic scattering from a target evaluated in the lab frame, Appendix B. of mass M, the momentum transfer is given by We define the integrated “fast flux” of MCPs satisfying 2 0 0 γ ≥ γ Q ¼ 2MðE − MÞ, where E is the total recoil energy of χ cut as Z the target. The cross section is therefore maximized by ∞ dΦχ scattering off the lightest target possible with the lowest Φ ð γ Þ¼ γ ð Þ cut mχ; cut d χ ; 4 γ dγχ possible recoil energy. In practice, experimental consi- cut derations such as detection efficiency and background γ where cut is set by the relevant experimental threshold. In reduction will set a minimum electron recoil energy which Fig. 4, we display this quantity’s mass dependence for will, in turn, dictate the detection cross section for that

115032-4 CONSTRAINTS ON MILLICHARGED PARTICLES FROM COSMIC- … PHYS. REV. D 102, 115032 (2020) given experiment. We will therefore consider a windowed 0 cross section for electron recoils with kinetic energy, Te ¼ ð 0 − Þ Ee me between Tmin and Tmax, or equivalently with 2 2 momentum transfers between Qmin and Qmax,

Z 2 Qmax dσ χ σ˜ ¼ e 2 ð Þ eχ 2 dQ : 5 2 dQ Qmin Since the four-momentum transfer is directly related to 0 0 the recoil energy in the lab frame, Te ¼ Ee − me, via 2 0 2 0 2 0 Q ¼ −2ðpe − peÞ ¼ð2meEe − 2meÞ¼2meTe, this is 2 ≥ 2 equivalent to demanding that Q meTmin. In the FIG. 5. Dependence of windowed cross section σ˜ eχ on MCP γ 0 ¼ 16 0 ¼ 80 center-of-mass frame, the maximal momentum transfer boost factor χ for Tmin MeV and Tmax MeV as is given when the scattering is back-to-back such that compared to the approximation Eq. (8). 2 2 2 Q ≤ 4Pe − 2me where Pe is the electron’s momentum in the center-of-mass frame, In principle, the thresholds of large neutrino detectors can sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi be rather low, a few MeV in case of SK, and as low as 4 2 2 2 2 me − 2meðmχ þ sÞþðmχ − sÞ 200 keV for Borexino. We choose, however, a much higher P ¼ : ð6Þ ∼15–16 e 4s threshold of MeV, that removes all the events generated by solar neutrinos, so that background counting In terms of lab-frame variables, this implies that rates reduce to OðfewÞ per year. Super-Kamiokande has searched for inverse beta decay 2 2 0 mePχ 2 positrons [55]. This search looked at inverse beta decays T ≤ ≈ 2m ðβχγχÞ ; ð Þ e 2 þ 2 þ 2 e 7 þ meEχ me mχ ν¯ep → ne with a positron recoil energy 16 MeV

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ACKNOWLEDGMENTS We would like to thank Dr. T.-T. Yu for a stimulating discussion of the cosmic-ray-generated light dark matter flux. We also thank Professor M. Kachelrieß, Dr. M. Smy and Professor H. Sobel for discussions. We thank Roni Harnik and Zhen Liu for providing us with the ArgoNeuT exclusion curves directly. R. P. and Y. -D. T. thank the University of Washington and the Institute for Nuclear theory for its hospitality during the final portion of this work. R. P. also thanks the Fermilab theory group for their hospitality and support. Y. -D. T. would like to thank Kavli Institute of Cosmological Physics (KICP), University of Chicago, for the hospitality and support. The work of A. K. FIG. 6. Comparison of event shapes for MCP elastic scattering and V. T. was supported by the U.S. Department of Energy off electrons and inverse beta decay from supernova background (DOE) Grant No. DE-SC0009937. The work of A. K. was neutrinos. The MCP signal was obtained by folding the differ- also supported by the World Premier International Research σ ential scattering cross section d eχ =dTe against the cosmic- Center Initiative, MEXT Japan. The work of R. P. was ray- induced MCP flux. The supernova background curves supported by an the Government of Canada through an 2 ð Eν=Tν þ 1Þ ¼ þ 1 3 correspond to Eν= e where Eν Te . MeV; these NSERC PGS-D award and by the U.S. Department of correspond to the fixed temperature profiles used in Ref. [55] as Energy, Office of Science, Office of High Energy Physics, can be readily verified by reproducing their Fig. 19. under Award No. DE-SC0019095. This paper has been authored by Fermi Research Alliance, LLC under Contract volume of 190 kt, the near future Hyper-K water No. DE-AC02-07CH11359 with the U.S. Department of Cherenkov experiment [39], as also shown in this figure, Energy, Office of Science, Office of High Energy Physics. can further improve significantly on these projections. This work was partly performed at the Aspen Center for Other near future experiments with sizable fiducial vol- Physics, which is supported by National Science umes, such as DUNE (40 kton, liquid argon) [41] and Foundation Grant No. PHY-1607611. Research at JUNO (20 kton, liquid scintillator) [40], will complement the Perimeter Institute was supported in part by the water-based Cherenkov detectors as probes of the DSNB Government of Canada through NSERC and by the [105], and hence can also serve as probes of atmospheri- Province of Ontario through Ministry of Economic cally produced MCPs. Development, Job Creation and Trade (MEDT).

VI. SUMMARY APPENDIX A: COSMIC-RAY PRODUCTION We have considered MCP production from standard KINEMATICS cosmic rays interacting with the atmosphere, constituting a permanent MCP source for all terrestrial experiments. 1. Boost of produced mesons Explicitly calculating the flux of MCPs from cosmic rays The lab-frame energy of a meson produced in a collision, as a function of mχ and ϵ, in particular, closes a long- ¼ γ E þ γ β P Em, can be written as Em cm cm cm k, where curly standing gap in the MCP literature as it allows to directly script variables refer to center-of-mass frame quantities. We translate well-known bounds on an ambient MCP flux into ¼ P can rewrite this expression inp termsffiffiffi of xF k=pmax limits on these parameters. As a result, we show that large- 1 2 (“Feynman-x”), where p ¼ sð1 − mm=sÞ is the larg- scale underground neutrino experiments can potentially max 2 est possible longitudinal momentum allowed by kinematic serve as the leading probe of stable dark sector particles 1 constraints; xF therefore varies from −1 (backward point- with masses below mχ < m ψ ≃ 1.5 GeV. The new limits 2 J= ing) to þ1 (forward pointing). Written in terms of x , our on MCPs from Super-K are highly relevant, for example, in F formula is given by Em ¼ γ p ðE=p þ β x Þ,or scenarios where MCPs constitute an SIDM component cm max max cm F because they are (i) independent of the DM fraction made sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! of such MCPs and (ii) probe a part of the parameter space 2 2 that is inaccessible at ground-based direct-detection experi- γ ¼ γ pmax 2 þ pT þ mm þ β ð Þ m cm xF 2 2 cmxF : A1 ments. In addition to improvements from larger detectors, mm pmax pmax since we only consider primary meson production (providing a robust lower bound on meson production), our results ¼ P ≪ can likely be further strengthened by a more detailed Since pT T pmax, we can neglect it in our analysis. modeling of cosmic-ray showers. Therefore, we can obtain γmðxFÞ from

115032-6 CONSTRAINTS ON MILLICHARGED PARTICLES FROM COSMIC- … PHYS. REV. D 102, 115032 (2020) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 described above provided we integrate over all such Eχ, γ ≈ γ pmax 2 þ mm þ β ð Þ m cm xF 2 cmxF : A2 weighted by the differential decay rate. Therefore, the lab- mm p max frame distribution of MCPs from η decay is given by Z This equation can be inverted to yield two branches 1 dΓη 1 dΓη ðÞ ¼ dEχ ×BoxðEχjγmÞ; ðA7Þ x ðγmÞ, Γ Γ E F η dEχ lab η d χ rest ðÞ ¼ −γ γ ðβ β Þ mm ð Þ x cm m cm m ; A3 where Γη ¼ Γðη → χχγ¯ Þ. From Eq. (A7), PðγχjγmÞ is F p max readily obtained using corresponding to the two solutions of the quadratic 1 Γ equation. ðγ jγ Þ¼ d ð Þ P χ m Γ γ A8 d χ lab 2. Meson decay to millicharged particles η Most of the decay modes we consider involve two-body and the chain rule. We neglect the form factor and 1 ½ Γ E γγ final states. For example, in the case of the J=ψ, the compute Γ d =d using the Wess-Zumino P vertex, differential decay in the rest frame of the parent meson is with P as pseudoscalar meson [106,107]. 1 monoenergetic dΓ=dEχ ∝ δðEχ − 2 mJ=ψ Þ (in this subsec- tion, curly letters refer to meson rest-frame quantities). APPENDIX B: ATMOSPHERIC MESON Upon boosting to the lab frame, this becomes a box PRODUCTION RATE ðþÞ ð−Þ distribution, BoxðEχjγmÞ, of width Eχ − Eχ and height Our treatment of meson production in the upper atmos- ðþÞ ð−Þ 1=ðEχ − Eχ Þ, where phere is data driven and centers mostly around the ratio of σðpp → mXÞ=σ ðppÞ which varies as a function of ðÞ inel Eχ ¼ γJ=ψ ðEχ βJ=ψ PχÞ: ðA4Þ center-of-mass energy. Although we have tried to inform our fits using data across a wide range of center-of-mass Equivalently, in terms of the MCP’s lab-frame boosts, we γ energies (or equivalently cm), there is a limited window have of “important” center-of-mass boosts that is determined by the competition between a rising inclusive cross section ðÞ ˜ γχ ¼ γJ=ψ γ˜χð1 βJ=ψ βχÞ; ðA5Þ and a sharply falling cosmic-ray flux as a function of γ ∼ −2.7 cm (the typical ICR E scaling translates to roughly γ˜ β˜ I ∼ γ−4.5 where χ and χ are the boost and velocity of the MCP in CR cm ). This is illustrated in Fig. 3 of the main text ρ0 ϕ γ the meson rest frame. In the case of and , the dominant where one can see that the relevant ranges are cm between decay mode is also a two body final state (e.g., ρ0 → χχ¯). 1.5–5 for η mesons, between 1.5 and 10 for ρ (and ω) and For ω, the SM branching ratio for ω → π0lþl− is roughly ϕ, and between 3 and 25 for J=ψ. 10 times larger than ω → lþl− [89], but this decay mode is The rest of this section is devoted to our parametrization 1 only accessible for mχ ≤ 2 ðmω − mπÞ ≈ 325 MeV, as of the available inclusive cross section data, which we 1 separate into a discussion of σmðγ Þ and Pðγmjγ Þ.Itis opposed to mχ ≤ mω ≈ 390 MeV for the direct two body cm cm 2 ðγ jγ Þ decay. We therefore neglect this decay mode5 which will important to note that although P m cm is poorly con- underestimate the MCP flux by a factor of ∼ O(few) in the strained by the data, we were able to find that its impact on our sensitivity curves is marginal; this is because the total window 275 MeV ≳ mχ ≳ 325 MeV and focus instead on number of MCPs produced is independent of this quantity. ω → χχ¯. The branching ratio for MCPs can be obtained by In contrast, although it has a relatively comprehensive data a simple rescaling of the di-muon branching ratio set, the production cross section σmðγ Þ in the window of sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cm 2 2 maximal production (as shown in Fig. 3 of the main text) mm − 4mχ ðm → χχ¯Þ¼ϵ2 ðm → μþμ−Þ ð Þ can have a substantialpffiffiffiffiffiffiffiffiffiffiffiffi impact on the MCP signal (bounds BR 2 2BR ; A6 4 mm − 4mμ on ϵ scale as signal) because it alters the total number of MCPs produced. We therefore anticipate that the uncer- where BRðρ0 → μþμ−Þ¼4.55 × 10−5,BRðω→μþμ−Þ¼ tainties in the production cross section are the dominant 7.4×10−5, and BRðϕ → μþμ−Þ¼2.87 × 10−4 [89]. source of error in our analysis [at the level of ∼OðfewÞ]. For the Dalitz decay η → γχχ¯, the MCPs are not monoenergetic in the meson rest frame. Nevertheless, each 1. η mesons E infinitesimal rest frame energy χ can be treated as Eta meson production in pp collisions has been most extensively measured in the near-threshold regime for the 5Including ω → π0χχ¯ would involve a chiral perturbation exclusive process pp → ηpp [95,97]. Near threshold this is theory calculation analogous to the one performed for η → γχχ¯. the only available channel, such that this cross section can

115032-7 RYAN PLESTID et al. PHYS. REV. D 102, 115032 (2020) be taken as a reasonable estimate of the total inclusive cross section. Further away from threshold bona fide measurements of the inclusive cross section are scarcer, butpffiffiffi we have identified four measurements in the literature at s ¼ 3.17, 27.45, 38.8, and 53 GeV [92–94,96,97]. We split the available data into two subsets, near-threshold exclusivep productionffiffiffi (defined as pp → ppη measurements for s ≤ 3 GeV) and far-from-thresholdpffiffiffi inclusive data (defined as pp → ηX for pffiffiffis > 3 GeV). We fit σ ð Þ the near-threshold data for η spffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwith the function b c fðxÞ¼aðx − 2.42Þ x where x ≡ spp=GeV. For the far-from-threshold data, we instead use gðxÞ¼að1 þjbj= ð − 2 42Þ2Þ 2ð Þ → η x . log x . In both cases, a weighted linear FIG. 7.p Productionffiffiffi cross section for pp X as a function of 1 regression to the data was performed. Using the best fit γcm ¼ 2 s=mp. The data are taken from Refs. [92–97] and fitted values for both fits, and demanding that the function is using the piecewise procedure described in the text; the smooth continuous, we find curve is Eq. (B1). σ ðγ Þ¼Θðγ − γ0Þ ð1 876γ Þ η cm cm f . cm without weighted errors. We find the data to be reasonably þ Θðγ0 − γ Þgð1.876γ Þ; ðB1Þ cm cm well described by pwhereffiffiffi the numerical factor comes from the relationship ¼ 2 γ ¼ð1 876 Þγ ð Þ s mp cm . GeV cm. The functions f x and ð Þ 2 g x , with their best fit values, are given by σρðγ Þ ≈ ð1.35 mbÞlog ð1.876γ Þ cm cm −1 2 22 4 59 13.4 fðxÞ¼ð0.0176 mbÞ × ðx − 2.42Þ . x . ; ðB2Þ 1 þ : ð Þ × ð1 876γ − 2 61Þ2 B6 . cm . 0.356 −1 gðxÞ¼ð1.32 mbÞ log2ðxÞ × 1 þ ; ðB3Þ ðx − 2.42Þ2 A comparison between the available data and our smooth fit and γ0 ¼ 1.59 is chosen such that Eq. (B1) is continuous; is shown in Fig. 8. the fit is shown versus the data (with error bars when We found that the data for σðpp → ρXÞ had a much available) in Fig. 7. better coverage than the corresponding ω production cross For the differential cross section dσm=dxF, measure- section and where there are measurements of the ω cross ments at NA27 [94] strongly suggest an exponential section it is nearly identical to the ρ cross section. We distribution, therefore estimated the ω cross section σωðγωÞ ≃ σρðγρÞ. For the ϕ meson, we find that the functional form hðxÞ¼ dση cη=2 2 −1 c ¼ ση × exp ½−cηjxFj; ðB4Þ að1 þjbj=ðx − 2.896Þ Þ x gives a reasonable fit to the dx 1 − exp½−cη F data [93,95,97]. After an unweighted regression, we find γ that σϕ is well described, cf. Fig. 9,by wherepffiffiffi cη depends on cm. Measurements from NA27 at ¼ 27 5 γ ¼ 14 6 ≈ 9 5 s . GeV (corresponding to cm . )fixcη . [94]. One generally expects that cη will be a monotonically γ 0 increasing function of cm and that cη > . The simplest functional form that satisfies these expectations and agrees with the measurement of [94] is

ðγ Þ¼9 5 þð Þ ðγ − 14 6Þ ð Þ cη cm . slope × cm . : B5

1 We take slope≈ 2. We checked that our sensitivity to MCPs from experiments such as SK is relatively insensitive to the value of the slope parameter.

2. Light vector mesons The production cross section for the ρ0 meson is relatively well measured [93,95,97]. Like the η meson, FIG. 8. Compilation of pp → ρX cross sections as a function of ð Þ γ we perform a best fit analysis with the function g x ,but cm taken from [93,95,97].

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→ ϕ → ψ FIG. 9. Compilation of pp X cross sections as a function FIG. 10. Productionpffiffiffi cross section for pp J= X as a function γ – 1 of cm taken from [93 95]. of γcm ¼ 2 s=mp. The data points have been digitized from E-789’s compilation [98] and HERA-B’s compilation [101]. The σ ðγ Þ¼ð0 01 Þð1 876γ Þ1.23 ϕ cm . mb . cm solid curve is digitized from the fit presented in Ref. [101], while the dashed curve is taken from Ref. [98]. For our sensitivity 2.4 1 þ : ð Þ analysis, we use the solid curve from [101]. × ð1 876γ − 2 896Þ2 B7 . cm . relatively mild effect on the fast flux of MCPs. Data from Like the η meson, the longitudinal momentum distribu- ≈ experimentspffiffiffi at lower energies show a preference for cJ=ψ tions for the vector mesons were more difficult to find in 2 for s ≤ 15 GeV [100], whereas experiments at higher the literature, and we rely on a single measurement at pffiffiffi pffiffiffi energies find larger values such as c Ψ ≈ 6 for s ≈ s ¼ 27.5 GeV [94] which shows the x dependence to be J= F 40 GeV [99]; we did not find a robust set of measurements described by Eq. (B4) with cV ¼ cρ ¼ cω ¼ cϕ ≈ 7.7.We of c ψ spanning the entire range of γ relevant for cosmic- expect this value to be smaller at lower center-of-mass J= cm ray pp collisions. For simplicity, and because our final energies and so take results are relatively insensitive to the details of the xF 5 7 distribution, we take c ψ to vary linearly with γ , ¼ 7 7 þ . ðγ − 14 6Þ ð Þ J= cm cV . 13 cm . ; B8 1 ¼ 2 þ ðγ − 5Þ ð Þ which, just like the cη, should be viewed as a cartoon of the cJ=ψ 5 cm : B10 σ γ behavior of d =dxF as a function of cm rather than a faithful representation. In this case, there are data at lower center-of-mass energies that suggest this formula is a reasonable interpolation. 3. J=ψ mesons ψ For the J= mesons, we found two convenient summa- APPENDIX C: IONIZATION EXPERIMENTS ries of the available data: one from E-739 (Fig. 7 in Ref. [98]) and one from HERA-B (Fig. 8 of Ref. [101]). Ionization is a very low threshold process and so we use The HERA-B compilation includes measurements at sig- the full flux (integrated over all boosts γχ) of MCPs for γ ¼ 1 nificantly higher center-of-mass energies. For comparison, ionization experiments; this corresponds to cut as we plot both sets of data in Fig. 10 where we see that the shown in Fig. 4 of the main text; we denote this total flux HERA-B compiled data are roughly consistent with that by ΦðmχÞ. To translate existing bounds on an ambient MCP from the E739 paper, but suggests a steeper growth with flux in the literature, we demand that rising center-of-mass energy. We use the best fit to the σ ðγ Þ ϵ2 ðϵ−2Φð ÞÞ ¼ Φ ðϵÞ ð Þ former to calculate m cm . × mχ ion ; C1 For the differential distribution, we used the standard σ ϵ−2Φð Þ γ ¼ 1 parametrization of d J=ψ =dxF [100], where mχ corresponds to the cut curve in Fig. 4 of the main text (i.e., the integrated MCP flux generated in dσJ=ψ ðcJ=ψ þ 1Þ the upper atmosphere), and Φ ðϵÞ is the joint exclusion cJ=ψ ion ¼ σJ=ψ × ð1 − jxFjÞ : ðB9Þ dxF 2 curve obtained by combining data from MACRO [49,50] and Majorana [54] as shown in Fig. 7 of [54]. We then solve Like cη, the fit parameter cJ=ψ depends on the center-of- for ϵ for each value of mχ which determines a critical value mass energy, and like cη the precise value of cJ=ψ has a of, ϵcðmχÞ, above which MCPs are excluded.

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APPENDIX D: COSMOGENIC COINCIDENCE AND CUTS AT SUPER-KAMIOKANDE In this section, we discuss the possibility that cuts from the Super-Kamiokande Collaboration designed to limit cosmogenic activity could also remove MCP scattering events were they to occur. We conclude that this is extremely unlikely and can be neglected in our analysis. As is clearly seen in Fig. 3, the energy of the pp collision that sources an MCP can be (roughly) inferred given the meson species that was produced. Perhaps the most concern- ing possibility is a high-energy pp collision that produces J=ψ mesons (these dominate the flux for the heaviest MCPs we considered). A center-of-mass boost of γcm ¼ 25 corresponds to an incident cosmic-ray energy of E ≈ 75 GeV. FIG. 11. Mesons fluxes from primary pp collisions assuming a p longitudinal momentum distribution as described in Eqs. (B4), Consulting Table II-28 of [108] we find that muons with (B5), and (B8)–(B10). energy less than 75 GeV would penetrate only 200 m of water. Noting that the SKoverburden is 3 × 103 m water equivalent, we conclude that any SM cosmic-ray particles sourced from possible that a high-energy muon (i.e., with energy larger the proton-proton collisions that produce almost all of the than 1 TeV) produced independently of an MCP could be MCP flux would be stopped by the overburden; while it is time coincident with an MCP scattering event, this is also possible that higher energy cosmic rays also produce MCPs unlikely; the rate of muon activity in SK is 2 Hz such that and associated penetrating showers, this component of the the probability of 50 microsecond coincidence is roughly flux and its contribution to the expected MCP event rate at SK 10−4. We therefore conclude that the efficiency of the are totally negligible for our analysis. DSNB cuts for MCPs is a relatively small effect that can be In addition to the (rather robust) energetic argument safely neglected at the level of accuracy considered in presented above, we note that while cosmic rays are our study. collimated, they have a characteristic separation on the order of a few degrees or 0.03 radians, having two APPENDIX E: MESON FLUXES coincident events inside SK’s volume within the same shower requires an angular separation of 20 m (detector A useful biproduct of our research are the lab-frame −3 size)/5 km (travel length)∼4 × 10 radians and is hence spectra of mesons as a function of γm. Given any calculable extremely unlikely. m → dark sector decay, using the meson spectra as inputs, The final consideration that one may be worried about is a flux of dark sector particles originating from primary coincident cosmic-ray activity from a separate high-energy cosmic-ray collisions can be obtained. Our results, shown muon. In the DSNB analysis, no event is allowed to be in Fig. 11, rely only on simple parametrizations of the within 50 microseconds after a muon detection, requiring differential cross sections dσ=dxF and the measured pro- very strong time coincidence as well. Although it is duction cross sections as outlined above.

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