Direct Detection Constraints on Dark Photon Dark Matter
aa r X i v : . [ h e p - ph ] J u l CALT-TH 2014-173
Direct Detection Constraints on Dark Photon Dark Matter
Haipeng An, Maxim Pospelov,
Josef Pradler, and Adam Ritz Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada Perimeter Institute for Theoretical Physics, Waterloo, ON N2J 2W9, Canada Institute of High Energy Physics, Austrian Academy of Sciences, Nikolsdorfergasse 18, 1050 Vienna, Austria (Dated: December 2014)Dark matter detectors built primarily to probe elastic scattering of WIMPs on nuclei are alsoprecise probes of light, weakly coupled, particles that may be absorbed by the detector material.In this paper, we derive constraints on the minimal model of dark matter comprised of long-livedvector states V (dark photons) in the 0 . −
100 keV mass range. The absence of an ionization signalin direct detection experiments such as XENON10 and XENON100 places a very strong constrainton the dark photon mixing angle, down to O (10 − ), assuming that dark photons comprise thedominant fraction of dark matter. This sensitivity to dark photon dark matter exceeds the indirectbounds derived from stellar energy loss considerations over a significant fraction of the availablemass range. We also revisit indirect constraints from V → γ decay and show that limits frommodifications to the cosmological ionization history are comparable to the updated limits from thediffuse γ -ray flux.
1. INTRODUCTION
The Standard Model of particle physics (SM) is knownto be incomplete, in that it needs to be augmented toinclude the effects of neutrino mass. Furthermore, cos-mology and astrophysics provide another strong motiva-tion to extend the SM, through the need for dark matter(DM). Evidence ranging in distance and time scales fromthe horizon during decoupling of the cosmic microwavebackground (CMB) to sub-galactic distances points tothe existence of ‘missing mass’ in the form of cold, non-baryonic DM. The particle (or, more generally, field the-oretic) identity of dark matter remains a mystery – onethat has occupied the physics community for many years.While the ‘theory-space’ for DM remains enormous,several model classes can be broadly identified. Shouldnew physics exist at or near the electroweak scale, aweakly interacting massive particle (WIMP) becomes aviable option. The WIMP paradigm assumes the exis-tence of a relatively heavy particle (typically with a massin the GeV to TeV range) having sizeable couplings tothe SM. The self-annihilation into the SM regulates theWIMP cosmic abundance according to thermal freeze-out, and the observed relic density requires a weak-scaleannihilation rate. The simplest models of this type alsopredict a significant scattering rate for WIMPs in thegalactic halo on nuclei, when up to 100 keV of WIMPkinetic energy can be transferred to atoms, offering a va-riety of pathways for detection. Direct detection, as it hasbecame known, is a rapidly growing field, with significantgains in sensitivity achieved in the last two decades, andwith a clear path forward [1].Alternatively, DM could be in the form of super-weaklyinteracting particles, with a negligible abundance in theearly Universe, and generated through a sub-Hubblethermal leakage rate (also known as the ‘freeze-in’ pro-cess). Dark matter of this type is harder to detect di-rectly, as the couplings to the SM are usually smaller than those of WIMPs by many orders of magnitude. Metasta-bility of such states offers a pathway for the indirect de-tection of photons in the decay products, as is the casefor metastable neutrino-like particles in the O (10 keV)mass range (see, e.g. [2]). It was also pointed out in [3]that WIMP direct detection experiments are sensitive tobosonic DM particles with couplings of O (10 − ) or be-low, that could be called super-WIMPs (referring to the‘super-weak’ strength of their SM interactions).Finally, a completely different and independent classof models for dark matter involves light bosonic fieldswith an abundance generated via the vacuum misalign-ment mechanism [4–6]. In this class of models, DM par-ticles emerge from a cold condensate-like state with verylarge particle occupation numbers, which can be well de-scribed by a classical field configuration. The mass andinitial amplitude of the DM field defines its present en-ergy density. The most prominent example in this class,the QCD axion, does have a non-vanishing interactionwith SM fields, although other forms of ‘super-cold’ DMdo not necessarily imply any significant coupling. Whileaxion dark matter has been the focus of many experimen-tal searches and proposals [7], other forms of super-colddark matter have received comparatively less attention(see e.g. [8–10, 13]). In the course of these latter in-vestigations, and subsequent work, several experimentalstrategies for detecting such dark matter scenarios havebeen suggested [14–16].Regarding the latter class of models, it is also possi-ble to generate a dark matter abundance not only froma pre-existing condensate (vacuum misalignment) butgravitationally, during inflation, through perturbationsin the field that carry finite wave number k [11]. Recentwork [12] investigates this possibility for vector parti-cles, reaching the conclusion that such mechanism avoidslarge-scale isocurvature constraints from CMB observa-tions, and allows light vectors to be generated in sufficientabundance as viable dark matter candidates. CMBdiffuse γ XMASSXENON100XENON10 m V (eV) k i n e t i c m i x i n g κ RGHBsun 10 − − − − − m V (eV) k i n e t i c m i x i n g κ − − − − − FIG. 1.
A summary of constraints on the dark photon kinetic mixing parameter κ as a function of vector mass m V (see Secs. 2 and 3for the details). The thick lines exclude the region above for dark photons with dark matter relic density. The solid (dashed) line is fromXENON10 (XENON100); the limit from XMASS is taken from [25]. The dash-dotted lines show our newly derived constraints on thediffuse γ -ray flux from V → γ decays, assuming that decays contribute 100% (thick line) or 10% (thin line) to the observed flux. Thethick dotted line is the corresponding constraint from CMB energy injection. Shaded regions depict (previously considered) astrophysicalconstraints that are independent of the dark photon relic density. The limits from anomalous energy loss in the sun (sun), horizontalbranch stars (HB), and red giant stars (RG) are labeled. The shaded region that is mostly inside the solar constraint is the XENON10limit derived from the solar flux [31]. In this paper, we consider ‘dark photon dark matter’generated through inflationary perturbations, or possiblyother non-thermal mechanisms. While existing proposalsto detect dark photons address the range of masses be-low O (meV), we will investigate the sensitivity of existingWIMP-search experiments to dark photon dark matterwith mass in the 10 eV - 100 keV window. As we willshow, the coupling constant of the dark photon to elec-trons, eκ , can be probed to exquisitely low values, downto mixing angles as low as κ ∼ O (10 − ). Furthermore,sensitivity to this mixing could be improved with carefulanalysis of the ‘ionization-only’ signal available to a va-riety of DM experiments. The sensitivity of liquid xenonexperiments to vector particles has already been exploredin [17] and many experiments have already reported rel-evant analyses [18–25]. While we concentrate on theStuckelberg-type mass for the vector field, our treatmentof direct detection of V will equally apply to the Higgsedversion of the model. Moreover, the existence of a Higgsfield charged under U (1) ′ opens up additional possibil-ities for achieving the required cosmological abundanceof V .The rest of this work is organized as follows. In Sec. 2we introduce the dark photon model in some more detail,describe existing constraints, and reconsider indirect lim-its. In Sec. 3 we compile the relevant formulæ for direct detection, confront the model with existing direct detec-tion results and derive constraints on the mixing angle κ . The results are summarized in Fig. 1, which showsthe new direct detection limits in comparison to variousastrophysical constraints. In Sec. 4, we provide a gen-eral discussion of super-weakly coupled DM, and possi-ble improvements in sensitivity to (sub-)keV-scale DMparticles.
2. DARK PHOTON DARK MATTER
It has been well-known since 1980s that the SM allowsfor a natural UV-complete extension by a new massive ormassless U (1) ′ field, coupled to the SM hypercharge U (1)via the kinetic mixing term [26]. Below the electroweakscale, the effective kinetic mixing of strength κ betweenthe dark photon ( V ) and photon ( A ) with respective fieldstrengths V µν and F µν is the most relevant, L = − F µν − V µν − κ F µν V µν + m V V µ V µ + eJ µ em A µ , (1)where J µ em is the electromagnetic current and m V is thedark photon mass. This model has been under signif-icant scrutiny over the last few years, as the minimalrealization of one the few UV-complete extensions of theSM (portals) that allows for the existence of light weaklycoupled particles [27]. For simplicity, we will considerthe St¨uckelberg version of this vector portal, in which m V can be added by hand, rather than being inducedvia the Higgs mechanism. Light vector particles with m V < m e have multi-ple contributions to their cosmological abundance, suchas (a) production through scattering or annihilation, γe ± → V e ± and e + e − → V γ , possibly with sub-Hubblerates, (b) resonant photon-dark photon conversion, or(c) production from an initial dark photon condensate,as could be seeded by inflationary perturbations. Noticethat if mechanisms (a) and (b) are the only sources thatpopulate the DM, they are not going to be compatiblewith cold dark matter when m V . keV.For mechanism (a), naive dimensional analysis sug-gests a dark photon interaction rate Γ int ∼ κ α n e /s ,where n e is the electron number density and √ s is thecentre-of-mass energy. At temperatures T ≫ m e , wherethe number density of charge carriers is maximal, n e ∼ T , this production rate scales linearly with temperature,whereas the Hubble rate is a quadratic function of T . Itfollows that for sub-MeV mass dark vectors, the ther-mal production of V is maximized at T ∼ m e . However,simple parametric estimates of this kind may require re-finement due to matter effects that alter the most naivepicture. At finite temperature T , the in-medium effectscan be cast into a modification of the mixing angle, κ T,L = κ × m V | m V − Π T,L | , (2)where Π T,L ( ω, | ~q | , T ) are the transverse (T) and longi-tudinal (L) polarization functions of the photon in theisotropic primordial plasma. They depend on photon en-ergy ω and momentum | ~q | and their temperature depen-dence is exposed by noting that Re Π T,L ∝ ω where ω P is the plasma frequency; for the cases of interestIm Π T,L ≪ Re Π
T,L .The consequences of these in-medium effects are two-fold. First, at high temperatures, they suppress themixing angle since ω ∼ αT (in the relativistic limit),thereby diminishing contributions to thermal productionfor T ≫ m V . Second, the presence of the mediumallows the production to proceed resonantly, wheneverRe Π T,L ( T r , ω ) = m V [process (b) above]. Indeed, res-onant conversion dominates the thermal dark photonabundance for m V < m e , but the constraints from di-rect detection experiments rule out the possibility of athermal dark photon origin for 10 eV . m V <
100 keValtogether. The values of κ that are required for the cor-rect thermal relic abundance, estimated in [3, 28], arelarger than the direct detection bounds discussed hereby several orders of magnitude. Dark photon dark matter remains a possibility whenthe relic density receives contributions from a vacuumcondensate and/or from inflationary perturbations, pro-cess (c). The displacement of any bosonic field from theminimum of its potential can be taken as an initial con-dition, and during inflation any non-conformal scalar orvector field receives an initial contribution to such dis-placements scaling as H inf / (2 π ), where H inf is the Hub-ble scale during inflation. Even in absence of initial mis-alignment, the inflationary production of vector bosonscan account for the observed dark matter density with aspectrum of density perturbations that is commensuratewith those observed in the CMB [12]. While the pro-duction of transverse modes is suppressed, longitudinalmodes can be produced in abundance [12],Ω V ∼ . r m V (cid:18) H inf GeV (cid:19) . (3)For our mass range of interest the correct relic densitywould then be attained with H inf in the 10 GeV ball-park.Undoubtedly, interactions between dark photons andthe plasma are present, and the evolution of any macro-scopic occupation number of vector particles is compli-cated by (resonant) dissipation processes [29]. For smallenough couplings, these processes may be made ineffi-cient, and most of the vector particles are preserved toform the present day DM. Equation (3) illustrates that—depending on the value H inf —a successful cosmologicalmodel amenable to direct detection phenomenology canalways be found, and in the remainder of this work weassume that Ω V h = 0 .
12, in accordance with the CMB-inferred cosmological cold dark matter density. Conse-quently, we also assume that the galactic dark matteris saturated by V -particles, and neglect any effects fromsubstructure. The latter is a possibility when inflation-ary perturbations produce excess power on very smallscales [12], and which will make the direct detection phe-nomenology ever more interesting. In this work, we re-strain ourselves to the smooth dark matter density andhence to the time-independent part of the absorption sig-nal. In vacuum, this theory is exceedingly simple, as it cor-responds to one new vector particle of mass m V with acoupling eκ to all charged particles. Some of this sim-plicity disappears once the matter effects for the SMphoton become important, and the effective mixing an-gle becomes suppressed. The subtleties of these calcula-tions, taking proper account of the role of the longitu-dinal modes of V , were fully accounted for only recently[30–33]. An understanding of these effects is importantbecause they determine the exclusion limits set by the en-ergy loss processes in the Sun, and other well-understoodstars [34]. In the limit of small m V (small comparedto the typical plasma frequency in the central region ofthe Sun), the energy loss into vector particles scales as ∝ κ m V , and is dominated by the production of longi-tudinal modes [30]. Although the resulting constraintsfrom energy loss processes turn out to be quite strongin the m V ∼
100 eV region, they weaken considerablyfor very small m V , opening a vast parameter space for avariety of laboratory detection methods.For m V >
10 eV, dark matter experiments are sensi-tive enough to compete with stellar energy loss boundsif dark photons contribute to a significant fraction of thedark matter cosmological abundance. Here we reviewthe most important aspects of stellar emission for theSt¨uckelberg case, whereby we also update our previouslyderived constraint on horizontal branch (HB) stars.Ordinary photons inside a star can be assumed to bein good local thermal equilibrium so that their distribu-tion function is time independent, ˙ f γ ( ω, T ) = 0. This al-lows one to relate photon production and absorption pro-cesses, d Γ prod γ /dωdV = ω | ~q | / (2 π ) e − ω/T Γ abs γ . In analogy,for the production rate of on-shell dark photons one has, d Γ prod T,L dωdV = κ T,L ω p ω − m V π e − ω/T Γ abs γ,T,L , (4)where d Γ prod T,L /dωdV is the rate of emission for a spin-1vector particle with mass m V and longitudinal ( L ) ortransverse ( T ) polarization, while κ T,L is defined in (2).Inside active stars like our sun, the rate is dominated bybremsstrahlung processes; for explicit formulae see [30]and [32]. The expression (4) is useful since the op-tical theorem (at finite temperature) relates Γ abs γ,T,L = − Im Π
T,L ( ω, ~q ) / [ ω (1 − e − ω/T )].Importantly, as alluded to above, emission can proceedresonantly when m V = Re Π T,L ; see (2). In the emis-sion of an on-shell dark photon, Re Π L = ω P m V /ω andRe Π T = ω P , up to corrections of O ( T /m e ). A resonanceinside a star occurs when either ω P ( r res ) = ω (longi-tudinal) or ω P ( r res ) = m V (transverse). The emissionthen proceeds from a spherical shell of radius r res and therates become independent of the details of the emissionprocess. One may then integrate over the stellar profileby using the narrow width approximation [30, 32], d Γ prod dω ≃ r e ω/T ( r ) − p ω − m V | ∂ω P ( r ) /∂r | ! r = r res × ( κ m V ω longitudinal ,κ m V transverse , (5)for each polarization of transverse V -bosons. This formnicely exhibits the different decoupling behavior with re-spect to m V . The bounds derived from stellar energyloss may qualitatively be understood on noting that thetypical plasma frequency at the center of the star is given by, Sun: ω P ( r = 0) ≃
300 eV , Horizontal Branch: ω P ( r = 0) ∼ . , Red Giant: ω P ( r = 0) ∼
200 keV , and both longitudinal and transverse resonant emissionstops once m V > ω P ( r = 0). In our numerical analysis,we employ the full expressions for emission that also coverthe case in which dark photons are emitted off-resonance.The shaded regions in Fig. 1 are a summary of the as-trophysical constraints on the mixing parameter κ thatare independent of the relic density of dark photon darkmatter. The thin solid (dotted) gray lines show the con-straints that are based solely on the emission of trans-verse (longitudinal) modes.For the sun, the limit on the anomalous energy lossrate is identical to the one in previous work [30, 32]. Asa criterion we require that the luminosity in dark photonscannot exceed 10% of the solar luminosity, L ⊙ = 3 . × W. The limit is derived from observations of the Bneutrino flux; for details we refer the reader to the abovereferences.For Horizontal Branch (HB) stars, we update our ownpreviously derived limit as follows (a similar limit hasalready been presented in [32]): as an HB representa-tive, we consider a 0 . M ⊙ solar mass star with stel-lar profiles as shown in in [34, 35]. The energy lossis then limited to 10% of the HB’s luminosity [34], forwhich we take L HB = 60 L ⊙ [35]. The transverse modesdominate the energy loss in HB stars. Since the cor-responding resonant emission originates from one shell r res , T for all energies, the derived constraint is sensi-tive to the stellar density profile in the resonance region m V < ω P ( r = 0) ≃ . m V ∼
150 eV originates fromentering the He-burning shell. Our result is in qualitativeagreement with [32]; quantitative differences may be as-signed to our use of full emission rate expressions [ratherthen (5)] that are integrated using Monte Carlo meth-ods over the assumed stellar profile. In either case, suchbounds are—by construction—only representative in na-ture and a detailed comparison of the derived limits willnot yield much further insight.Finally, the constraint that can be derived from RedGiant (RG) stars extends sensitivity to larger m V .We require a dark photon luminosity that is less then10 erg / g / s originating from the degenerate He core with ρ ∼ g / cm , T ≃ . m V = ω P (core) ∼
20 keV. Here we note that there isroom for improvement when deriving the limit from RGstars. For example, recent high-precision photometry forthe Galactic globular cluster M5 has allowed the authorsof [36] to derive constraints on axion-electric couplingsand neutrino dipole moments that are based on the ob-served brightness of the tip of the RG branch. In con-junction with an actual stellar model, however, the betterobservations do not yield a drastic improvement of limits,as there appears to be a slight preference for extra cool-ing [36]. Albeit such hints to new physics are tantalizing,we in turn expect only mild changes to our representativeRG constraint when a detailed stellar model is employedand/or better observational data is used; we leave suchstudy for the future. V → γ decay Next we consider constraints imposed by energy injec-tion from γ -rays originating from V → γ decays belowthe e + e − threshold, for which the one-photon inclusivedifferential rate was computed in [3]. It reads, d Γ dx = κ α π m V m e x (cid:20) − x + 29192 x (cid:21) , (6)where x = 2 E γ /m V with 0 ≤ x ≤
1; the total decaywidth is obtained by integration, Γ V → γ = R dx d Γ /dx ,and it sets the lifetime of dark photons for m V < m e .A limit from observations of the diffuse γ -ray back-ground was estimated in [28] by translating the re-sults for monochromatic photon injection obtained in[37] and assuming a photon injection energy of m V / dN/dE γ denote the differential spectrum such that R dE γ dN γ /dE γ = 3. It follows that E γ ( dN/dE γ ) =3Γ − V → γ x ( d Γ /dx ).There are then two contributions to the diffuse photonbackground from V → γ decays. For the flux from thedark matter density at cosmological distances we find, E γ dφ eg dE γ = Ω V ρ c Γ V → γ πm V Z z f dz E γ H ( z ) dN [(1 + z ) E γ ] dE γ , (7)where we have made the assumption that most of thedark matter has not yet decayed today, Γ V ≪ H , with H being the present day Hubble rate. H ( z ) is the Hub-ble rate at redshift z and we cut off the integral at the(blueshifted) kinematic boundary, z f = m V / (2 E γ ) − E γ →
0, at some maximal redshift that is numeri-cally inconsequential. In turn, the galactic diffuse flux isgiven by, E γ dφ gal dE γ = Γ V → γ πm V E γ dNdE γ ρ sol R sol J , (8)where J ( ψ ) is the ρ sol R sol –normalized line-of-sight in-tegral at an angle ψ from the galactic center; ρ sol ≃ . / cm is the dark matter density at the sun’s po-sition, R sol ≃ . ψ = π or π/ J ≃ . J = 1 as fiducialvalue in (8). galactic diffuseextragalacticdiffuse γE γ (keV) E γ d φ / d E γ ( k e V / c m / s ec / s r ) m V = 100 keV κ = 3 × − .1001010 − FIG. 2.
Representative diffuse gamma ray bolometric flux (thicksolid top line) together with computed extragalactic (galactic) pho-ton fluxes depicted by the dashed (dotted) line from V → γ decay.We constrain the sum of these fluxes (solid line) to not exceed theobserved one. Figure 2 depicts the representative diffuse gamma rayflux of photons (thick solid line) as taken from [37]. Theextragalactic and diffuse galactic fluxes originating fromdark photon decay with m V = 100 keV and κ = 3 × − are respectively shown by the dashed and dotted lines.We constrain the flux contribution from dark photon de-cay by requiring that their sum (solid line) does not ex-ceed 100% (10%) of the observed flux. The ensuing limitsin the ( m V , κ ) parameter space are shown in Fig. 1 andthey constrain the region m V >
100 keV. While the de-rived limit represents a conceptual improvement becauseuse of the differential photon spectrum has been made,quantitatively, the strength of the limits is comparableto the previous estimate [28].The final constraint discussed in this section is dueto precise measurements of the cosmic microwave back-ground (CMB) radiation, and its sensitivity to DM de-cay. Specifically, V → γ decays at redshift O (1000) alterthe ionization history, raising TE and EE amplitudes onlarge scales, and damping TT temperature fluctuationson small scales. An energy density of dEdV dt = 3 ζm p Γ V → γ e − Γ V → γ t (9)is injected into the plasma per unit time where ζ =( f / V / Ω b is related to the injected energy per baryon,which is equal to 3 ζm p ; m p is the mass of the proton and f denotes the overall efficiency with which the plasmais heated and ionized. In the case at hand f = 1. In[38] limits on the combination ( ζ, Γ V ) were derived fordecaying heavy dark photons with m V > m e , utilizingthe Planck 2013 and WMAP polarization data. (For ear-lier analyses, see e.g. [39, 40].) For lifetimes significantlylonger than the cosmic time of recombination, the limitamounts to ζ Γ V . × − eV / s or τ V & s. Weshow this constraint in Fig. 1 and it is very comparablein sensitivity to the one derived from diffuse γ -ray lines.
3. DM ABSORPTION SIGNALS IN DIRECTDETECTION EXPERIMENTS3.1. Dark vector-induced ionization
If the energy of dark vectors is above the photoelectricthreshold E V ≥ E th , atomic ionization becomes viable,for example in Xenon:Xe I+ V → Xe II+ e − ; Xe I+ V → Xe III+2 e − ; ... (10)Here I’s are used according to the usual atomic notation,and Xe I represents the neutral Xenon atom which ismost relevant for our discussion. Most of the DM is coldand non-relativistic, so that E V = m V with good accu-racy. The astrophysics bounds, on the other hand, areoften derived in the regime E V ≫ m V . We will addressthe E V ≃ m V case first, where the distinction between Land T modes all but disappears.When m V ≥ E th = 12 .
13 eV, matter effects are notvery important, and the problem reduces to the absorp-tion of a massive nonrelativistic particle with eκ cou-pling to electrons. The difference with the absorption ofa photon with ω = m V amounts to the following: thephoton carries momentum | ~q | = ω , whereas the nonrela-tivistic dark vector carries a negligibly small momentum, | ~q | = m V v DM ∼ O (10 − ) ω where v DM is the dark pho-ton velocity. Fortunately, this difference has little effecton the absorption rate for the following reason. Both thephoton wavelength and the DM Compton wavelength aremuch larger than the linear dimension of the atom, allow-ing for a multipole expansion in the interaction Hamil-tonian, ( ~p e ~ǫ ) exp( i~q~r e ) ≃ ( ~p e ~ǫ ) × (1 + i~q~r e + ... ), where ~ǫ is the (dark) photon polarization. The first term cor-responds to the E1 transition that dominates over othermultipole contributions, making the matix elements forabsorption of ‘normal’ and dark photons approximatelyequal. Accounting for the differences in flux, and aver-aging over polarization, gives the relation between theabsorption cross sections [3] σ V ( E V = m V ) v V ≃ κ σ γ ( ω = m V ) c, (11)where v V is the velocity of the incoming DM particle.This relation is not exact and receives corrections of order O ( ω r ) where r at is the size of corresponding electronicshell participating in the ionization process. Near ion-ization thresholds this factor varies from ∼ α for outershells to ∼ Z α for inner shells. We deem this accuracyto be sufficient, and point out that further improvementscan be achieved by directly calculating the absorptioncross section for dark photons using the tools of atomictheory. (Analogous calculations have already been per-formed for the case of axion-like DM [41].)Relation (11) is nearly independent of the DM veloc-ity, and results in complete insensitivity of the DM ab-sorption signal to the (possibly) intricate DM velocitydistribution in the galactic halo; this is in stark contrast to the case of WIMP elastic scattering. The resultingabsorption rate is given byRate per atom ≃ ρ DM m V c × κ σ γ ( ω = m V ) c, (12)where ρ DM is the local galactic DM energy density, andfactors of c are restored for completeness.The above formulae are sufficiently accurate providedall medium effects can be ignored. In general, however,the process of absorption of a dark photon must also ac-count for the modification of V − γ kinetic mixing dueto in-medium dispersion effects. While the absorptionof m V ≫ E th particles cannot be affected significantly,close to the lowest theshold such effects can be impor-tant. To account for in-medium effects, we follow ouroriginal derivation in [31]. The matrix element for pho-ton absorption q + p i → p f with photon four momentum q = ( ω, ~q ) and transverse ( T ) or longitudinal ( L ) polar-ization vectors ǫ T,Lµ is given by, M i → f + V T,L = − eκm V m V − Π T,L ( q ) h p f | J µem (0) | p i i ε T,Lµ ( q ) . (13)Squaring the matrix element and summing over finalstates f , one obtains the absorption rate of L or T pho-tons,Γ T,L = 12 ω (2 π ) δ (4) ( q + p i − p f ) e κ T,L ε ∗ µ ε ν × X f h p i | J µem (0) | p f ih p f | J νem (0) | p i i (14)= e ω Z d x e iq · x κ T,L ε ∗ µ ε ν h p i | [ J µem ( x ) , J νem (0)] | p i i , (15)where the in-medium effective V − γ mixing angle is givenin (2). The polarization functions Π T,L are obtained fromthe in-medium polarization tensor Π µν ,Π µν ( q ) = ie Z d x e iq · x h | T J µem ( x ) J νem (0) | i = − Π T X i =1 , ε T µi ε T νi − Π L ε Lµ ε Lν . (16)Noting that Z d x e iq · x h | [ J µem ( x ) , J νem (0)] | i = 2 Im (cid:20) i Z d x e iq · x h | T J µem ( x ) J νem (0) | i (cid:21) , (17)we can express the absorption rate in the lab-frame ofthe detector (14) as follows,Γ T,L = − κ T,L
Im Π
T,L ω . (18)
Im( n )Re( n )LXE refractive index. ω (eV) 1000010001001010 − − − − − − XENON100background model
30 keV10 keV5 keV1 keV κ = 10 − .S1(PE) e v e n t s / PE − FIG. 3.
Left:
Real and imaginary parts of the liquid xenon refractive index computed from tabulated atomic scattering factors and using theKronig-Kramers relation. Note that the maximum of the Im( n ) function corresponds to the photoelectric cross section σ γ ∼ × − cm . Right:
Simulated events in ‘xenon-units’ of photo-electrons (PE) for various dark photon masses as labeled. Also shown are the reportedevent counts and the background model as taken from [24].
This particular form is suitable for calculation, as wecan relate Π
T,L to tabulated optical properties of thematerial. For an isotropic and non-magnetic medium,Π L = ( ω − ~q )(1 − n ) , Π T = ω (1 − n ) , (19)where n refr is the (complex) index of refraction for elec-tromagentism. When | ~q | ≪ ω , Π L = Π T ≃ Π, and allformulae for the absorption of L and T modes becomeidential, as expected.As the final step, we obtain n refr from its relation to theforward scattering amplitude f (0) = f + if , where theatomic scattering factors f , are tabulated e.g. in [42].Close to the ionization threshold we make use of theKramers-Kronig dispersion relations to relate f and f in estimating n refr . Alternatively, one may establish anintegral equation relating the real and imaginary parts of ε ; see [31].When m V ≫ Π, κ L ( T ) ≃ κ , and the in-medium modi-fication of absorption can be negelected. In that case theabsorption rate per DM particle isΓ ≃ κ ω × Im n = κ σ γ × (cid:18) N at V (cid:19) , (20)leading to the same formula for the absorption rate peratom as before, Eq. (12). The XENON10 data set from 2011 exemplifies thepower of ionization-sensitive experiments when it comesto very low-energy absorption-type processes. With anionization threshold of ∼
12 eV, the absorption of a 300 eV dark photon already yields about 25 electrons,and the relatively small exposure of 15 kg-days is stillsufficient to provide the best limits on dark photons orig-inating from the solar interior [31]. The same type of sig-nature is used to provide important contraints on WIMP-electron scattering [43, 44].Despite significant uncertainties in electron yield, en-ergy calibration, and few-electron backgrounds, we wouldlike to emphasize the fact that robust and conservative limits can be derived which are independent of the abovesystematics. The procedure is straightforward, and fol-lows the one already outlined in [31]. First, we count allionization events (246) with up to 80 ionization electrons,or, equivalently, within 20 keV of equivalent nuclear re-coil. If we do not attempt to subtract backgrounds(which is conservative), this implies a 90% C.L. upperlimit of less than 19 . . m V .
200 eV the newlimit is stronger than the previously derived solar energyloss constraint.
The XENON100 collaboration has performed a low-threshold search using the scintillation signal S1 with anexposure of 224.6 live days and an active target massof 34 kg liquid xenon [24]. A very low background rateof ∼ × − / kg / day / keV has been achieved through acombination of xenon purification, usage of ultra-low ra-dioactivity materials, and through self-shielding by vol-ume fiducialization. In addition, with energy depositionin the keV range and above, the XENON100 experimentprovides a sufficient energy resolution, allowing for massreconstruction of a potential DM absorption signal.We derive the signal in the XENON100 detector asfollows. For the dark photon dark matter the kineticenergy is negligible with respect to its rest energy since( v/c ) ∼ − . Therefore, a mono-energetic peak at thedark photon mass is expected in the spectrum. To derivethe constraint, we first convert the absorbed energy m V into the number of photo-electrons (PE) using Fig. 2 ofRef. [24]. This may result in a 10% uncertainty due to thecorrections from binding energies of electrons at variousenergy levels as shown in Fig. 1 of Ref. [45]. We take intoaccount the Poissonian nature of the process, and includethe detector’s acceptance as a function of S1, shown inFig. 1 of Ref. [24]. The resulting S1 spectrum for variousdark photon masses together with the reported data isshown in Fig. 3.A likelihood analysis is used to constrain the kineticmixing κ . The likelihood function is defined as L ( κ, m V ) = Y i ≥ Poiss( N ( i ) | N ( i ) s ( κ, m V ) + N ( i ) b ) , (21)where i labels the bin number (which equals the numberof S1 for each event) as shown in Fig. 6 of Ref. [24], N ( i ) b and N ( i ) are the background and number of observedevents as presented in Ref. [24]. Following the latter ex-perimental work, we apply a cut S1 ≥
3. Here we neglectthe contribution from the uncertainty of n exp to the like-lihood function, since from Fig. 2 of Ref. [24] one can seethat after we apply the S1 ≥ κ is less than 10%. A standard likelihood anal-ysis then yields the resulting 2 σ limit on κ as a functionof m V . It is shown as the black dashed curve in Fig. 1.Again, we find the direct detection constraints to be verycompetitive with astrophysical bounds.
4. DISCUSSION AND CONCLUSION
With an array of direct detection experiments nowsearching for signatures of elastic nuclear recoil ofWIMPs on nuclei, and with sensitivity levels marchingtowards the neutrino background, it is important to keepin mind that other dark matter scenarios can also be sen-sitively probed with this technology. In particular, theexquisite sensitivity to ionization signatures at variousexperiments allows stringent constraints to be placed ongeneric models of super-weakly-interacting dark matter.In this paper, we have studied the sensitivity to the mini-mal model of dark photon dark matter, and obtained lim-its (summarized in Fig. 1) that exceed those from stellarphysics over a significant mass range.The sensitivity of current direct detection experimentsalready excludes dark photon dark matter with a ther-mally generated abundance. This is not a problem for the model, as the DM abundance may be determined by non-thermal mechanisms. For example, perturbations duringinflation may create the required relic abundance [12],and further constraints on such models may be achievedif an upper bound on H inf were to be established by ex-periments probing the CMB.Dark photon dark matter has certain advantages overaxion-like-particle dark matter with respect to direct de-tection. The absence of the dark photon decay to twophotons removes the constraint from monochromatic X-ray lines. This latter signature usually provides a morestringent constraint on axion-like keV-scale DM than di-rect detection. Furthermore, the cross section for darkphotons is significantly enhanced for small masses, rela-tive to the cross section for absorption of axion-like par-ticles.The analysis presented in this paper addresses themodel of a very light dark photon field, that is partic-ularly simple and well-motivated. In addition, one couldconstruct a whole family of ‘simplified’ models of verylight dark matter, with observational consequences fordirect detection [3]. The most relevant of these would in-volve couplings to electrons, and one could consider DMof different spin and parity:(pseudo)scalar g S S ¯ ψψ, g P P ¯ ψγ ψ, (pseudo)vector g V V µ ¯ ψγ µ ψ, g A A µ ¯ ψγ µ γ ψ, (22)tensor g T T µν ¯ ψσ µν ψ, · · · Here ψ stands for the electron field, g i parametrizes thedimensionless couplings, and V, A , S, P, T... are the fieldsof metastable but very long lived DM. The case consid-ered in this paper corresponds to g V = eκ , and the lightmass m V is protected by gauge invariance. However,even cases where the mass of DM is not protected by anysymmetry are of interest, and can be considered withineffective (or simplified) models. In this case, loop pro-cesses tend to induce a finite mass correction, which is atmost ∆ m DM i ∼ g i Λ UV . With the cutoff Λ UV at a TeV,it is natural to expect that, for a DM mass of ∼
100 eVfor example, one should have g i < − . As demon-strated by the analysis in this paper, DM experimentscan probe well into this naturalness-inspired regime, andset meaningful constraints on many variations of lightDM models.Finally, we would like to emphasize that furtherprogress can be achieved through the analysis of‘ionization-only’ signatures. For example, in noble gas-and liquid-based detectors one can improve the boundsfor E < keV by accounting for multiple ionization elec-trons (see Ref. [44]). The ionization of Xe atoms from thelowest electronic shells is likely accompanied by Augerprocesses, which generate further photo-electrons, andthe corresponding bounds can be tightened. Analysis ofthese complicated processes may require additional inputfrom atomic physics.
Acknowledgements
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