Dark Matter as a Solution to Muonic Puzzles
DDark Matter as a Solution to Muonic Puzzles
Maxim Perelstein, Yik Chuen San
Laboratory for Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA (Dated: November 17, 2020)We propose a simple model in which dark matter particle exchanges mediate a new quantumforce between muons and nucleons, resolving the proton charge radius puzzle. At the same time,the discrepancy between the measured anomalous magnetic moment of the muon and the StandardModel prediction can be accommodated, and thermal relic abundance of the dark matter candidateis consistent with observations. The dark matter particle mass is in the MeV range. We show thatthe model is consistent with a variety of experimental and observational constraints.
I. INTRODUCTION
Observational evidence for the existence of dark matter(DM) is overwhelming. While DM comprises most of thematter in today’s universe, and contributes about 20%of the total energy density, there is no known elementaryparticle that can account for it. Many candidate theorieshave been proposed, extending the Standard Model (SM)of particle physics to include one or more dark matterparticles. In many theories, DM particles have potentialexperimental or observational signatures going beyondthe purely gravitational effects that have been observed.However, no non-gravitational signature of DM has beenconclusively established so far.In this paper, we propose that dark matter particles aredirectly responsible for explaining a long-standing puzzlein particle physics, the proton charge radius anomaly.The value of the proton charge radius measured us-ing Lamb shift in muonic hydrogen [1, 2] does notagree, at about 5 σ level, with the value obtained fromelectron-proton scattering and electron hydrogen spec-troscopy [3, 4]. A similar discrepancy was observed inmuonic deuterium [5]. We note that some recent dataon ep scattering [6] and H spectroscopy [7, 8] are in bet-ter agreement with muonic hydrogen results, while oth-ers [9] confirm the discrepancy. At present, the sources ofdisagreement among electronic experiments are not un-derstood, and in this paper, we take the point of viewthat the anomaly is real and demands an explanation interms of Beyond-the-SM (BSM) phenomena. (For previ-ous works that proposed BSM explanations of the protoncharge radius puzzle, see e.g. [10–14].)The key ingredient of our proposal is the idea that loopdiagrams involving light dark matter states can inducea new “quantum” force between SM particles [15, 16].Since the flavor structure of DM couplings to the SM isunconstrained, it is plausible that this new force may befelt by muons but not electrons. As will be shown below,such flavor-dependent quantum force can account for theproton charge radius puzzle, without conflict with anyexisting experimental constraints. At the same time, thedark matter particle responsible for this force can be athermal relic consistent with the measured cosmologicalDM abundance, as well as with all known bounds on DMproperties. The DM particle is predicted to have a mass FIG. 1: One loop diagram involving exchange of dark mat-ter particle χ that induces a new force between muons andprotons. of about 10 MeV, an interesting range from the point ofview of direct detection experiments.Another prominent experimental anomaly involvingthe muon is the anomalous magnetic moment a µ , whosemeasured value differs from the SM prediction by about3 σ [17–19]. While non-perturbative SM contributionsmay account for some or all of this discrepancy, in thispaper we take the point of view that it is real and requiresa BSM explanation. It turns out that in our model, theeffect of DM loops on a µ is subdominant to the shift in-duced by the mediator particles which connect DM andSM sectors. We show that this shift can indeed accountfor the observed discrepancy, and demonstrate a set ofparameters for which a µ and proton charge puzzles aresimultaneously solved, DM particle is a thermal relic withcorrect relic density, and all experimental and observa-tional constraints are satisfied. It is remarkable that thesimple model presented here can account for such a di-verse set of data pointing to physics beyond the SM. II. MODEL
We introduce a DM field χ , a Dirac fermion with noSM gauge charges. DM is coupled to the SM via tworeal-scalar mediator fields, a leptophilic mediator X anda leptophobic mediator X (cid:48) . Both mediators are also SM a r X i v : . [ h e p - ph ] N ov m χ m X m X (cid:48) g g (cid:48) p g (cid:48) n y y (cid:48) λ . · − − TABLE I: Model parameters at the benchmark point. gauge singlets. At energies below the QCD confinementscale, where all physics relevant for this study takes place,the interaction Lagrangian is given by L int = − g ¯ µµX − ( g (cid:48) p ¯ pp + g (cid:48) n ¯ nn ) X (cid:48) − y ¯ χχX − y (cid:48) ¯ χχX (cid:48) , (1)where µ , p and n are muon, proton and neutron fields,respectively. Throughout this paper, we will study theregime in which both mediators are much heavier thanthe DM particle, m X , m X (cid:48) (cid:29) m χ , and can be integratedout, leading to an effective Lagrangian L eff = − ygm X ¯ χχ ¯ µµ − y (cid:48) g (cid:48) p m X (cid:48) ¯ χχ ¯ pp − y (cid:48) g (cid:48) n m X (cid:48) ¯ χχ ¯ nn + . . . (2)The quantum force between proton and muon, arisingfrom the diagrams in Fig. 1, provides a new contribu-tion to the Lamb shift in muonic hydrogen, resolving theproton charge radius puzzle.A few comments are in order. For simplicity, we as-sumed that the mediator X (and therefore the DM) cou-ples to muon but not to the electron. While the flavor-dependent nature of this coupling is crucial to resolvingthe proton charge radius puzzle, a non-zero value of elec-tron coupling (for example, a plausible scenario in which g (cid:96) ∝ m (cid:96) ) can be introduced without altering the basicpicture. Further, X (cid:48) is generically expected to couple topions and other mesons. We do not include such cou-plings since they would play no role in our analysis. Fi-nally, while the interactions in Eq. (1) are sufficient toexplain the proton charge radius and the a µ anomaly,requiring that χ be a thermal relic with observed cosmo-logical abundance necessitates an additional interactioninvolving neutrinos ν :∆ L int = − λ ¯ ννX, (3)if neutrinos are Dirac, or its Majorana counterpart. Thisinteraction can arise from the operator X ( HL ) abovethe weak scale, and its strength is a priori unrelated tothe coupling of X to charged muons which arises fromthe operator X ( HL ) µ R .We also note that if X and X (cid:48) were replaced with asingle mediator, coupled to both muons and quarks, thedominant new physics effect in muonic hydrogen wouldcome from a tree-level exchange of the mediator, ratherthan the DM-induced quantum force. To leading order,this model would in fact be identical to that already con-sidered in [14]. III. RESULTS
Using the above model, we performed a fit to rele-vant experimental data and observational constraints on
FIG. 2: Fit to experimental data indicating non-SM physics(green) in the plane of mediator particle masses, m X and m X (cid:48) ,with the other parameters fixed to the values listed in Table I.Relevant experimental and observational constraints on DMand mediator particles are also shown; the shaded areas areruled out.FIG. 3: Fit to experimental data indicating non-SM physicsin the plane of DM particle mass m χ and the leptophilic me-diator coupling to muons, g , with the other parameters fixedto the values listed in Table I. DM properties. The results of the fit are summarizedin Figs. 2, 3 and 4. The model can explain the protoncharge radius puzzle, the a µ anomaly, and the observedDM relic density, while maintaining consistency with allknown experimental and observational constraints. Asample benchmark point in the model parameter spacewhich satisfies these requirements is shown in Table I.Some details of the analysis are presented below. Proton Charge Radius:
The fact that the protonhas a finite size (with radius r p ) introduces shifts in en-ergy levels of hydrogen-like atoms [1]. In particular, therewould be a change in the energy difference between the2 S and 2 P levels, i.e. Lamb shift ∆ E Lamb . By measur-ing ∆ E Lamb , one is able to deduce the value of r p . Inour model, the extra non-SM contribution to the Lambshift in muonic hydrogen arises at one loop from the dia- FIG. 4: Fit to experimental data indicating non-SM physicsin the plane of DM particle mass m χ and the leptophobicmediator coupling to protons, g (cid:48) p , with the other parametersfixed to the values listed in Table I. Relevant experimentaland observational constraints on DM and mediator particlesare also shown. gram in Fig. 1. In the non-relativistic limit, this interac-tion can be captured by a potential between protons andmuons [16]: V ( r ) = − π r (cid:18) ygm X (cid:19) (cid:18) y (cid:48) g (cid:48) p m X (cid:48) (cid:19) m χ r K (2 m χ r ) , (4)where K is the modified Bessel function of the secondkind of order 2. As a result, there is a new contributionto ∆ E Lamb in muonic hydrogen, given by∆ E L = (cid:104) S | V | S (cid:105) − (cid:104) P | V | P (cid:105) = − π yy (cid:48) gg (cid:48) p m X m X (cid:48) m χ a J ( x , a ) . (5)Here J ( x , a ) = (cid:90) ∞ x d x − x ) + x x e − x K (2 m χ ax ) , (6) a is the Bohr radius of muonic hydrogen, and x ≈ ( a Λ QCD ) − is the short-distance cutoff corresponding tothe breakdown of the effective field theory descriptionin Eq. (1) at length scales below O (Λ QCD ). We do notinclude the additional contribution to the Lamb shiftfrom length scales below r , which can only be calculatedwithin a specific UV completion of Eq. (1). To estimatethe associated theoretical uncertainty, we vary the valueof the cutoff by a factor of two around the assumed cen-tral value ( a Λ QCD ) − . This uncertainty is reflected in thewidth of the “proton charge radius” band in Figs. 2—4.Our fit assumes that this effect fully accounts for the“extra” Lamb shift measured in the muonic hydrogen,compared to the baseline value inferred from electronichydrogen and ep scattering: ∆ E L = − . FIG. 5: Fit to muonic deuterium data, as well as constraintsfrom nuclear physics experiments, in the plan of the mediatorcoupling ratio g (cid:48) n /g (cid:48) p and the DM particle mass m χ . The otherparameters are fixed to the values listed in Table I. Muonic Deuterium:
After the proton charge radiuspuzzle was discovered, there has been interest in per-forming similar experiments with other muonic atoms, inparticular muonic deuterium µD . It was reported thatthe deuteron charge radius r d extracted from µD showssimilar discrepancy from world-averaged CODATA 2014value [5], [4]. When comparing against spectroscopic val-ues of r d that involve deuterium only ( i.e. independentof r p ), this discrepancy is reduced to 3 . σ [20], but isstill statistically significant. We therefore require thatour model produces an extra contribution to Lamb shiftin µD [5]: ∆ E µD L = − . µD is due to dark matter-mediated quantum force, andis given by∆ E µD L = − π ygm X y (cid:48) (cid:0) g (cid:48) p + g (cid:48) n (cid:1) m X (cid:48) m χ a µD J ( x D , a µD ) , (7)where a µD is the Bohr radius of the muonic deuteriumsystem, r D is the deuteron radius, and x D = r D /a µD .The shift depends on both g (cid:48) n and g (cid:48) p , while various com-binations of these couplings are constrained by nuclearphysics experiments (see below). As shown in Fig. 5,isospin-preserving coupling g (cid:48) p = g (cid:48) n is consistent withboth the deuteron charge radius and nuclear physics con-straints. Muon Anomalous Magnetic Moment:
The lead-ing new contribution to a µ is given by the one-loop di-agram shown in Fig. 6. Note that this contribution isindependent of the dark matter candidate χ itself, whichonly enters at the two-loop level. The shift in a µ is givenby ∆ a µ = 2 g (4 π ) (cid:90) d x (1 − x ) (1 + x )(1 − x ) + x ( m X /m µ ) . (8) FIG. 6: Contribution to muon anomalous magnetic momentdue to the mediator particle X .FIG. 7: Dominant annihilation channel for χ at the time offreeze-out, if coupling to neutrinos are absent. In our fit we assume that this effect fully accounts for theexperimental discrepancy ∆ a µ = 287(80) × − , within2 standard deviations. Relic Density:
We assume that χ is a thermal relicand that it accounts for all of the observed cosmologicalDM abundance, Ω h = 0 . ± .
001 [21]. With interac-tions in Eq. (1) and m χ ∼ O (MeV), the dominant DMannihilation channel at the time of freeze-out is χχ → γ ,see Fig. 7. The leading ( p -wave) contribution to the crosssection in the non-relativistic limit is given by σ = 3 e π (cid:12)(cid:12)(cid:12)(cid:12) m µ I ( τ µ ) ygm X + m p I ( τ p ) y (cid:48) g (cid:48) p m X (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) . (9)Here the loop function I ( τ f ) for a fermion f is definedby I ( τ f ) = (cid:90) d x (cid:90) − x d y − xyτ f − xy (10)with τ f = m f /s , where √ s is the center-of-mass energyof the scattering process. σ can be used to compute therelic density of χ [22] by solving the Boltzmann equationnumerically.We find that the χχ → γ cross section is too small toprovide the observed relic density for model paramatersrequired to fit the proton charge radius and a µ anomalies.A simple solution is to consider an additional annihilationchannel, χχ → νν , via the interaction in Eq. (3). Thecross section is given by σ = π (cid:16) yλm X (cid:17) m χ . Since thisfinal state arises at tree level, it naturally dominates overthe 2 γ channel. With this addition, all three constraintscan be satisfied simultaneously, see Figs. 2—4.In addition to fits to the data indicating deviationsfrom the SM, a number of constraints from data and observations consistent with the SM have to be takeninto account: Dark Matter Self-Scattering:
Tree-level exchangesof mediator particles X and X (cid:48) induce DM short-rangeself-interactions of the form L self = − (cid:18) y m X + y (cid:48) m X (cid:48) (cid:19) ( ¯ χχ ) . (11)DM self-scattering cross sections at low velocities arebounded by observations of halo shapes [23, 24]: σ T /m χ (cid:46) / g, where σ T is the momentum-transfercross section defined by σ T ≡ (cid:90) dΩ d σ χχ → χχ dΩ (1 − cos θ ) . (12)This constraint translates into an upper bound on theDM mass: m χ (cid:46) π (cid:18) y m X + y (cid:48) m X (cid:48) (cid:19) − × − . (13) Dark Matter Direct Detection:
Direct detection ofdark matter in the MeV mass range has been the subjectof much interest recently [25, 26]. Most techniques relyon detection of DM scattering on electrons. In our model,this channel is not available, since by construction DMdoes not couple to electrons. However, scattering on anucleon can occur, with cross section σ ( χN → χN ) = 1 π (cid:18) g (cid:48) N y (cid:48) m X (cid:48) (cid:19) (cid:18) m χ m N m χ + m N (cid:19) , (14)where N = n or p . For our benchmark point, this crosssection is about 6 × − cm . This is about half an orderof magnitude below the strongest current constraint fromnon-observation of signal due to energetic DM componentgenerated through collisions with cosmic rays [27, 28],shown by the XENON-1T curve in Fig. 4. MeV-scale darkmatter can also be detected using the Migdal effect [29–33]. However, the recent results from
SENSEI collabora-tion [34] are not yet sensitive enough to constrain ourmodel.
Early-Universe Cosmology:
Measurements of Cos-mic Microwave Backgound (CMB) place strong con-straints on possible reionization due to DM annihila-tions [35]. However in our model, χ annihilation proceedsin p -wave, and thus not subject to this constraint. CMBdata together with the Lyman-alpha forest flux powerspectrum from the Sloan Digital Sky Survey constrainsthe elastic scattering of DM on baryons [36, 37]. Thisconstraint is shown by the curve labeled "cosmology" on Fig. 2. In addition, a scenario where DM freeze-outoccurs after neutrinos decouple from the rest of the SMplasma is constrained by the CMB bound on ∆ N eff , sincein this case the neutrino temperature at recombinationwould be raised relative to T γ by the entropy transferredfrom the DM . This argument imposes a lower bound m χ > ∼ a few MeV [38, 39]. A similar bound is imposedby the success of Big-Bang Nucleosynthesis (BBN) [40–42]. Dark Matter Mediator Searches:
In addition todirect searches for dark matter particles, there are manyexperiments looking for mediator particles produced atcolliders or in a fixed-target setup [25, 26]. While mostanalyses present the results in terms of bounds on darkphotons, which couple to both leptons and quarks pro-portional to their electric charges, the interpretation ofinterest to us is in terms of leptophilic or leptophobic me-diators. Leptophilic mediator searches rely on their pro-duction via their interaction with electrons, making theminsensitive to our model. (An exception is the recentlyreported NA64 search for scalars produced through theircoupling to photons [43]. However, this search does notplace relevant bounds on our model, since mediator cou-plings to photons are loop-suppressed.) Thus we onlyconsider bounds from leptophobic mediator searches.With sub-MeV dark matter, the most stringent boundcurrently comes from the
MiniBooNE experiment [26, 44],which places a bound on the parameter Y related to darkmatter annihilation cross section (see referenced papersfor the precise definition). In our model, this parameteris given by Y = (cid:18) g (cid:48) p e (cid:19) g (cid:48) p π (cid:18) m χ m X (cid:48) (cid:19) . (15)Note that the mediator X (cid:48) in our model is a scalar, whilethe MiniBooNE bounds were derived using a spin-1 me-diator. An order-one correction to the bound may arisedue to the differing kinematic acceptances and spin fac-tors in the two cases, but we do not expect it to affectour conclusions.
Nuclear Interactions:
Leading non-SM contribu-tions to nucleon-nucleon potential are given by V N N = − g (cid:48) N g (cid:48) N πr e − m X (cid:48) r − π (cid:18) y (cid:48) m X (cid:48) (cid:19) g (cid:48) N g (cid:48) N m χ r K (2 m χ r ) . (16)This extra potential can be probed at various nuclearphysics experiments. It turns out that the second term in V N N , arising from the DM-loop exchange, is the mostrelevant for our analysis, since 1 /r exp (cid:28) m χ (cid:28) m X (cid:48) ,where r exp is the length scale probed by the experiments.The relevant constraints are summarized in Fig. 5.The binding energy difference between He and H hasbeen well-established to be caused by Coulomb force andcharge asymmetry of nuclear forces [45], [46], [47]. In or-der not to spoil this agreement, the non-SM contribution We are grateful to Gordan Krnjaic for bringing this constraintto our attention. is required to be less than 30 keV [14]. It is worth notingthat this contribution is proportional to (cid:0) g (cid:48) p − g (cid:48) n (cid:1) , andvanishes in the isospin limit, see Fig. 5.The charge-independence breaking (CIB) scatteringlength is defined as∆ a = 12 ( a nn + a pp − a np ) , (17)where a N N is the scattering length between two nucle-ons N and N . Experimental and theoretical values for∆ a are known to be 5 . ± .
60 fm [48] and 5 . ± . δa th = − m N π (cid:18) y (cid:48) m X (cid:48) (cid:19) m χ (cid:0) g (cid:48) p − g (cid:48) n (cid:1) log (cid:32) m χ Λ (cid:33) , (18)where m N is the nucleon mass and Λ is the cut-off scaleof the effective theory. We require δa th < . a at 2 σ level. This constraint again vanishes inthe isospin limit.Scattering lengths between cold neutrons and nucleican be measured by different methods, such as Braggdiffraction and the transmission method [50]. In ourmodel, this scattering length is given by Eq. (18) byreplacing (cid:0) g (cid:48) p − g (cid:48) n (cid:1) with g (cid:48) n or g (cid:48) n g (cid:48) p . By comparingthe scattering lengths measured by different methods,bounds can be placed on contributions from non-contactoperators. The detailed analysis can be found in [51].Neutron scattering places the most restrictive nuclear-physics bound on the model in the isospin-symmetriclimit [16]. IV. CONCLUSIONS
In this paper we presented a simple model of MeV-scale Dirac fermion dark matter χ coupled to nucleonsand muons, but not electrons. The quantum force dueto χ loops is responsible for resolving the proton (anddeutron) charge radius puzzles, while a scalar particle in-troduced to mediate DM-muon interactions can accountfor the discrepancy between the SM prediction for theanomalous magnetic moment of the muon and the mea-sured value. If an additional interaction between χ andneutrinos is postulated, the former can be a thermal relicresponsible for all of the observed DM abundance. Weverified the existence of a region in the paramater spaceof the model where all of the above features are obtainedsimultaneously, while all known experimental and obser-vational constraints on light DM and mediators are sat-isfied.New experimental data will soon be available that willtest our model on various fronts. Numerous efforts, e.g. MUSE experiment [52], are under way that will hopefullyclarify the status of the proton charge radius puzzle.Anomalous magnetic moment of the muon measurementwill be improved by the Fermilab Muon g − Acknowledgments
We are grateful to Gordan Krnjaic and Philip Tanedofor useful comments on the first version of this paper.This work is supported by the U.S. National ScienceFoundation grants PHY-1719877 and PHY-2014071. [1] R. Pohl et al.,
The size of the proton , Nature , 213(2010).[2] A. Antognini et al.,
Proton Structure from the Measurementof S − P Transition Frequencies of Muonic Hydrogen ,Science , 417 (2013).[3] P. J. Mohr, B. N. Taylor, and D. B. Newell,
CODATARecommended Values of the Fundamental PhysicalConstants: 2006 , Rev. Mod. Phys. , 633 (2008),0801.0028.[4] P. J. Mohr, D. B. Newell, and B. N. Taylor, CODATARecommended Values of the Fundamental PhysicalConstants: 2014 , Rev. Mod. Phys. , 035009 (2016),1507.07956.[5] R. Pohl et al. (CREMA), Laser spectroscopy of muonicdeuterium , Science , 669 (2016).[6] W. Xiong et al.,
A small proton charge radius from anelectron–proton scattering experiment , Nature , 147(2019).[7] A. Beyer et al.,
The Rydberg constant and proton size fromatomic hydrogen , Science , 79 (2017).[8] N. Bezginov, T. Valdez, M. Horbatsch, A. Marsman,A. Vutha, and E. Hessels,
A measurement of the atomichydrogen Lamb shift and the proton charge radius , Science , 1007 (2019).[9] H. Fleurbaey, S. Galtier, S. Thomas, M. Bonnaud, L. Julien,F. Biraben, F. Nez, M. Abgrall, and J. Guena,
NewMeasurement of the 1S-3S Transition Frequency ofHydrogen: Contribution to the Proton Charge Radius Puzzle ,Phys. Rev. Lett. , 183001 (2018), 1801.08816.[10] D. Tucker-Smith and I. Yavin,
Muonic hydrogen and MeVforces , Phys. Rev. D , 101702 (2011), 1011.4922.[11] B. Batell, D. McKeen, and M. Pospelov, NewParity-Violating Muonic Forces and the Proton ChargeRadius , Phys. Rev. Lett. , 011803 (2011), 1103.0721.[12] C. E. Carlson and B. C. Rislow,
New Physics and the ProtonRadius Problem , Phys. Rev. D , 035013 (2012), 1206.3587.[13] S. G. Karshenboim, D. McKeen, and M. Pospelov, Constraints on muon-specific dark forces , Phys. Rev. D ,073004 (2014), [Addendum: Phys.Rev.D 90, 079905 (2014)],1401.6154.[14] Y.-S. Liu, D. McKeen, and G. A. Miller, ElectrophobicScalar Boson and Muonic Puzzles , Phys. Rev. Lett. ,101801 (2016), 1605.04612.[15] S. Fichet,
Quantum Forces from Dark Matter and Where toFind Them , Phys. Rev. Lett. , 131801 (2018),1705.10331.[16] P. Brax, S. Fichet, and G. Pignol,
Bounding Quantum DarkForces , Phys. Rev.
D97 , 115034 (2018), 1710.00850.[17] G. Bennett et al. (Muon g-2),
Final Report of the MuonE821 Anomalous Magnetic Moment Measurement at BNL ,Phys. Rev. D , 072003 (2006), hep-ex/0602035. [18] M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Reevaluation of the Hadronic Contributions to the , Eur.Phys. J. C , 1515 (2011), [Erratum: Eur.Phys.J.C 72,1874 (2012)], 1010.4180.[19] K. Hagiwara, R. Liao, A. D. Martin, D. Nomura, andT. Teubner, re-evaluated using new precise data , J. Phys. G , 085003 (2011), 1105.3149.[20] R. Pohl, F. Nez, T. Udem, A. Antognini, A. Beyer,H. Fleurbaey, A. Grinin, T. W. Hansch, L. Julien,F. Kottmann, et al., Deuteron charge radius and rydbergconstant from spectroscopy data in atomic deuterium ,Metrologia , L1 (2017), URL https://doi.org/10.1088%2F1681-7575%2Faa4e59 .[21] N. Aghanim et al. (Planck), Planck 2018 results. VI.Cosmological parameters (2018), 1807.06209.[22] E. W. Kolb and M. S. Turner,
The Early Universe , Front.Phys. , 1 (1990).[23] A. Kusenko and L. J. Rosenberg, Snowmass-2013 cosmicfrontier 3 (cf3) working group summary: Non-wimp darkmatter (2013), 1310.8642.[24] J. Zavala, M. Vogelsberger, and M. G. Walker,
Constrainingself-interacting dark matter with the milky way?s dwarfspheroidals , Monthly Notices of the Royal AstronomicalSociety: Letters , L20?L24 (2013), ISSN 1745-3925, URL http://dx.doi.org/10.1093/mnrasl/sls053 .[25] J. Alexander et al., in
Dark Sectors 2016 Workshop:Community Report (2016), 1608.08632.[26] M. Battaglieri et al., in
U.S. Cosmic Visions: New Ideas inDark Matter (2017), 1707.04591.[27] T. Bringmann and M. Pospelov,
Novel direct detectionconstraints on light dark matter , Phys. Rev. Lett. ,171801 (2019), 1810.10543.[28] C. Cappiello and J. F. Beacom,
Strong New Limits on LightDark Matter from Neutrino Experiments , Phys. Rev. D ,103011 (2019), 1906.11283.[29] M. Ibe, W. Nakano, Y. Shoji, and K. Suzuki,
Migdal Effectin Dark Matter Direct Detection Experiments , JHEP , 194(2018), 1707.07258.[30] M. J. Dolan, F. Kahlhoefer, and C. McCabe, Directlydetecting sub-GeV dark matter with electrons from nuclearscattering , Phys. Rev. Lett. , 101801 (2018), 1711.09906.[31] N. F. Bell, J. B. Dent, J. L. Newstead, S. Sabharwal, andT. J. Weiler,
Migdal effect and photon bremsstrahlung ineffective field theories of dark matter direct detection andcoherent elastic neutrino-nucleus scattering , Phys. Rev. D , 015012 (2020), 1905.00046.[32] D. Baxter, Y. Kahn, and G. Krnjaic,
Electron Ionization viaDark Matter-Electron Scattering and the Migdal Effect ,Phys. Rev. D , 076014 (2020), 1908.00012.[33] R. Essig, J. Pradler, M. Sholapurkar, and T.-T. Yu,
Relationbetween the Migdal Effect and Dark Matter-Electron
Scattering in Isolated Atoms and Semiconductors , Phys.Rev. Lett. , 021801 (2020), 1908.10881.[34] L. Barak, I. M. Bloch, M. Cababie, G. Cancelo,L. Chaplinsky, F. Chierchie, M. Crisler, A. Drlica-Wagner,R. Essig, J. Estrada, et al.,
Sensei: Direct-detection resultson sub-gev dark matter from a new skipper-ccd (2020),2004.11378.[35] T. R. Slatyer,
Indirect dark matter signatures in the cosmicdark ages. I. Generalizing the bound on s-wave dark matterannihilation from Planck results , Phys. Rev. D , 023527(2016), 1506.03811.[36] K. K. Boddy and V. Gluscevic, First CosmologicalConstraint on the Effective Theory of Dark Matter-ProtonInteractions , Phys. Rev. D , 083510 (2018), 1801.08609.[37] W. L. Xu, C. Dvorkin, and A. Chael, Probing sub-GeV DarkMatter-Baryon Scattering with Cosmological Observables ,Phys. Rev. D , 103530 (2018), 1802.06788.[38] C. M. Ho and R. J. Scherrer, Limits on MeV Dark Matterfrom the Effective Number of Neutrinos , Phys. Rev. D ,023505 (2013), 1208.4347.[39] C. Boehm, M. J. Dolan, and C. McCabe, Increasing Neffwith particles in thermal equilibrium with neutrinos , JCAP , 027 (2012), 1207.0497.[40] E. W. Kolb, M. S. Turner, and T. P. Walker, The Effect ofInteracting Particles on Primordial Nucleosynthesis , Phys.Rev. D , 2197 (1986).[41] M. Hufnagel, K. Schmidt-Hoberg, and S. Wild, BBNconstraints on MeV-scale dark sectors. Part I. Sterile decays ,JCAP , 044 (2018), 1712.03972.[42] K. M. Nollett and G. Steigman, BBN And The CMBConstrain Neutrino Coupled Light WIMPs , Phys. Rev. D , 083505 (2015), 1411.6005.[43] D. Banerjee et al. (NA64), Search for Axionlike and ScalarParticles with the NA64 Experiment , Phys. Rev. Lett. ,081801 (2020), 2005.02710.[44] A. Aguilar-Arevalo et al. (MiniBooNE DM),
Dark MatterSearch in Nucleon, Pion, and Electron Channels from a Proton Beam Dump with MiniBooNE , Phys. Rev. D ,112004 (2018), 1807.06137.[45] J. L. Friar and B. F. Gibson, Coulomb energies in s -shellnuclei and hypernuclei , Phys. Rev. C , 908 (1978), URL https://link.aps.org/doi/10.1103/PhysRevC.18.908 .[46] V. Koch and G. A. Miller, Six quark cluster effects andbinding energy differences between mirror nuclei , Phys. Rev.C , 602 (1985), URL https://link.aps.org/doi/10.1103/PhysRevC.31.602 .[47] G. Miller, B. Nefkens, and I. ?laus, Charge symmetry, quarksand mesons , Physics Reports , 1 (1990), ISSN0370-1573, URL .[48] T. Ericson and G. Miller,
Charge dependence of nuclearforces , Physics Letters B , 32 (1983), ISSN 0370-2693,URL .[49] R. Machleidt and I. Slaus,
The nucleon-nucleon interaction ,Journal of Physics G: Nuclear and Particle Physics , R69(2001), URL https://doi.org/10.1088%2F0954-3899%2F27%2F5%2F201 .[50] L. Koester, H. Rauch, and E. Seymann, Neutron scatteringlengths: A survey of experimental data and methods , AtomicData and Nuclear Data Tables , 65 (1991), ISSN0092-640X, URL .[51] V. Nesvizhevsky, G. Pignol, and K. Protasov, Neutronscattering and extra short range interactions , Phys. Rev. D , 034020 (2008), 0711.2298.[52] R. Gilman et al. (MUSE), Technical Design Report for thePaul Scherrer Institute Experiment R-12-01.1: Studying theProton ”Radius” Puzzle with µp Elastic Scattering (2017),1709.09753.[53] J. Grange et al. (Muon g-2),