Momentum Dependent Dark Matter Scattering
MMCTP-09-42
Momentum Dependent Dark Matter Scattering
Spencer Chang, Aaron Pierce, and Neal Weiner Physics Department, University of California Davis, Davis, California 95616 Michigan Center for Theoretical Physics (MCTP),Department of Physics, University of Michigan, Ann Arbor, MI 48109 Center for Cosmology and Particle Physics,Department of Physics, New York University, New York, NY 10003 (Dated: October 26, 2018)It is usually assumed that WIMPs interact through spin-independent and spin-dependentinteractions. Interactions which carry additional powers of the momentum transfer, q , are assumedto be too small to be relevant. In theories with new particles at the ∼ GeV scale, however, these q -dependent interactions can be large, and, in some cases dominate over the standard interactions.This leads to new phenomenology in direct detection experiments. Recoil spectra peak at non-zero energies, and the relative strengths of different experiments can be significantly altered. Wepresent a simple parameterization for models of this type which captures much of the interestingphenomenology and allows a comparison between experiments. As an application, we find that darkmatter with momentum dependent interactions coupling to the spin of the proton can reconcile theDAMA annual modulation result with other experiments. INTRODUCTION
The presence of dark matter (DM) in our universeis now well established by a variety of astrophysicalmeasurements, over a wide range of scales from sub-kpc to Gpc. Its clustering and low interaction crosssections are supported by the success of the CDMframework and the ability of N-body simulationsto reproduce observed structures. In spite of thesegreat successes, we remain ignorant to its detailednature. A direct detection of DM via its recoilsoff of nuclei would confirm its particle nature andyield insight into its origin. An examination of therecoil spectrum would provide important informa-tion about its properties, and possibly the formationhistory of the galaxy.If the dark matter is a Majorana fermion χ - asupersymmetric neutralino being the most promi-nent example - the types of interactions availableare significantly limited. The dominant scatteringsare mediated via the operators: O SI = ( ¯ χχ )(¯ qq ) , (1) O SD = ( ¯ χγ µ γ χ )(¯ qγ µ γ q ) , (2)which give respectively spin-independent (SI) andspin-dependent (SD) scattering. As these operatorstypically dominate the interaction rate of WeaklyInteracting Massive Particles (WIMPs) with nucleartargets, direct detection experiments quote results asbounds on the spin-independent and spin-dependentcross section per nucleon. Nonetheless, there are more dimension-6 opera-tors that can contribute to the direct detection crosssection. Namely, O = ( ¯ χγ χ )(¯ qq ) , (3) O = ( ¯ χχ )(¯ qγ q ) , (4) O = ( ¯ χγ χ )(¯ qγ q ) , (5) O = ( ¯ χγ µ γ χ )(¯ qγ µ q ) . (6)The operators O , O , and O are not presentif parity is a good symmetry of the theory, butsince parity is badly broken in the Standard Modeland it could be badly broken in the dark mattersector, it is reasonable to include them. If χ isa Dirac fermion, instead of Majorana, additionaloperators are possible. In particular, there is thepossibility of a dipole or charge radius coupling todark matter and a vector coupling to quarks [1, 2].Such an operator is quantitatively similar to O ,with the principle difference that it typically couplesto atomic number Z rather than mass number A .These operators in Eqns. (3)–(6) are present evenin the context of the minimal supersymmetric Stan-dard Model (MSSM), but there the contributions toscattering are typically far subdominant, as they aresuppressed relative to O SI/SD by additional powersof momentum O ( q /M W ) ∼ − , or in the case of O also by velocity suppression v . Consequently,they are usually ignored [3], but see [4]. Moreover,even if the dominant operators are zero, because ofthis suppression, these new operators are typicallynegligible in the context of direct detection experi- a r X i v : . [ h e p - ph ] A ug ments. Thus, even neglecting O SI/SD it might seemunlikely that such interactions would be relevant forupcoming direct detection experiments.However, this reasoning ignores that these twofacts often go hand in hand. O SI/SD are typicallysmall when there is a symmetry reason for them tobe small, in particular, when the DM-nucleon forceis mediated by a pseudo-Goldstone boson (PGB).Because of their shift symmetry, PGBs have q suppressed interactions. At the same time, PGBsare also naturally much lighter than the weak scale.Thus, O − are no longer insignificant if the media-tor has mass (cid:46) O (GeV). When we combine this withthe recent interest in new GeV-scale particles, e.g.,[5], and in particular PGBs [6] arising from modelsto explain PAMELA, Fermi, ATIC and HESS, weare strongly motivated to consider these scenarios.In this paper, we explore a class of dark mat-ter models where the scattering is momentum de-pendent (MDDM), i.e., where the operators O − dominate and are large enough to be observablein upcoming direct detection experiments. As weshall see, these operators can have a significantimpact on the spectral shape and the sensitivity ofvarious experiments. As an example, we shall seethat these effects can improve the ability to explainthe DAMA annual modulation signal while beingconsistent with other direct detection exclusions. SIGNALS OF MDDM
The recoil rates at a direct detection experimentcan be written as dRdE R = N T m N ρ χ m χ µ σ ( q ) (cid:90) ∞ v min f ( v ) v dv, (7)where m N is the nucleus mass, N T is the number oftarget nuclei in the detector, ρ χ = 0 . is the WIMP density, µ is the reduced mass ofthe WIMP-nuclei system, and f ( v ) is the halovelocity distribution function in the lab frame. Theminimum velocity to scatter with energy E R is v min = (cid:112) m N E R / µ . The rest of the expressiondepends on the scattering’s q = 2 m N E R . For SIinteractions, we have σ ( q ) SI = 4 G F µ π [ Zf p + ( A − Z ) f n ] F ( q ) , (8)where f p , f n are respectively the couplings to theproton and neutron, and F factor is the form factor. We take the limit f p = f n . This expression is thenproportional to the nucleon scattering cross section σ p = π G F µ p f p , where µ p is the reduced mass of theWIMP-proton system. For SD interactions, we have σ ( q ) SD = 32 G F µ J + 1 [ a p S pp ( q ) + a p a n S pn ( q )+ a n S nn ( q )] , (9)where a p , a n are respectively the couplings to theproton and neutron, and the S factors are theform factors for SD scattering. The correspondingnucleon cross sections are σ ( p,n ) = π G F µ p,n ) a p,n ) .The effect of the new operators can be parameter-ized simply: dR MDDMi dE R = (cid:32) q q ref (cid:33) n (cid:32) q ref + m φ q + m φ (cid:33) dR i dE R , (10)where i indexes the interaction, i.e., SD-proton, SD-neutron or SI, and we have included the propagatordue to a light mediator φ with mass m φ . For thebenchmark cases, we will take m φ (cid:29) q , to arrive atthe simple form dR MDDMi dE R = (cid:32) q q ref (cid:33) n dR i dE R . (11)We have chosen to normalize the new factors outfront at a reference value q ref ≡ (100 MeV) ,a characteristic value for many direct detectionexperiments. For operators O , O the exponent n = 1, while for O , n = 2. For O , the interactionis spin-independent on the nucleus side, while forthe others it is spin-dependent. This form of therecoil rate defines the nucleon cross sections σ p,n formomentum dependent scattering. O ’s scatteringcannot be written in this form, since it has termsproportional to the DM velocity. However, we findthat its spectra is almost identical to standard SIscattering, so we neglect it for the rest of the paper.MDDM is characterized by a modification of itsnuclear recoil spectrum. Typically, direct detectionexperiments optimize their searches by going tolower energy thresholds, where standard WIMPsignatures are expected to peak. In contrast, thespectrum of MDDM vanishes at zero recoil energy,and then can be either peaked or fairly flat over therange in question.We show in Fig. 1 the spectra of MDDM scenariosfor the case of SI germanium scattering. As we a) NR b) NR FIG. 1: Germanium spectra plots with arbitrary normalization versus energy recoil for SI momentum dependentscattering of a 100 GeV dark matter mass. Plot a) displays the effect of additional powers of q with q , q , and q in solid, long dash and short dash. Plot b) illustrates the effect of m φ on the q suppressed scenario with m φ =(1 , . , .
01) GeV in solid, long dash and short dash. can see, the spectra differ dramatically from thoseexpected for conventional dark matter. The powersof q suppress the low energy events resulting ina peaked spectrum reminiscent of inelastic darkmatter (iDM) [7–9]. In contrast to inelastic darkmatter, here the peaking arises without needing acoincidence of parameters (specifically, δ in iDMmodels tuned to the WIMP kinetic energy). Thespectrum need not be sharply peaked, however, andcan be broadly spread over a large range of recoilenergies. The non-trivial propagator of Eqn. (10)allows the possibility that events can be suppressedfor q (cid:29) m φ , as can be seen in Fig. 1. Finally,increasing the dark matter mass shifts the spectrato higher energies. Given the possibilities, the lessonis that search strategies developed for the simplestdark matter candidates are by no means optimalfor every dark matter candidate. The true darkmatter candidate may not be one of these simplestpossibilities, and it is important to cast a wide net. EXISTING SEARCHES AND DAMA
To understand the effects of MDDM, we studyhow these q effects can modify the limits arisingfrom existing experiments. We show in Fig. 2 thelimits on interactions mediated by O comparedwith limits on standard SI interactions, and in Fig. 3the limits of O when compared with standard SD-proton interactions. We follow the procedure laidout in Ref. [10] for CDMS [11] and XENON10 [12]limits and Ref. [9] for KIMS [13] limits. AlthoughPICASSO [14] and COUPP [15] limits are compara-ble, we only discuss PICASSO limits in what follows.Our methods better reproduce their (momentum independent) result in the SD-proton case, makingus more confident that the MDDM limit is realistic.Inspecting the exclusion limits of these plots, wesee important changes with respect to the traditionalcases. First, consider the SI case with q dependence( O ). While CDMS-Ge and XENON10 remain thestrongest, KIMS becomes stronger than CDMS-Siover much of the parameter space. In the SD-protoncase, limits from PICASSO are significantly weaker,and XENON10 becomes stronger than KIMS in the15-25 GeV range.These results are easy to understand. Due tosuppression of low energy events in the MDDMscenario, experiments that rely upon low energythresholds (in particular, XENON10) are weakenedwhen compared with others with higher thresholds(such as CDMS), which is why CDMS improvesrelative to XENON10. On the other hand, since q = 2 M N E R , at a given recoil energy, heavier nucleiare preferred by momentum dependent scattering,which is why KIMS improves over CDMS Si andwhy PICASSO weakens relative to the other ex-periments. Another effect occurs for COUPP andPICASSO. In these bubble chamber experiments,operation at varying temperature or pressure es-sentially integrates the recoil spectrum above somethreshold. The background from alpha decays isknown to be a flat spectrum above some specifictemperature or pressure and is fit to in the data.For the broadest MDDM spectra (see Fig. 1), thedark matter signal looks similar to this alpha back-ground. Unfortunately, this complicates backgroundsubtraction and reduces the present sensitivity tothese models.Intriguingly, momentum dependent scattering canalso modify the interpretation of dark matter ex- a) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m Χ (cid:72) GeV (cid:76) l og (cid:64) Σ p (cid:68) (cid:72) c m (cid:76) b) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m Χ (cid:72) GeV (cid:76) l og (cid:64) Σ p (cid:68) (cid:72) c m (cid:76) FIG. 2: Plots of the SI nucleon cross section σ p vs DM mass m χ without (a) and with q suppression (b). The coloredregions show the 68, 90, and 99% CL regions for the best DAMA fit. The 90% exclusions limits are KIMS (orangedashed), CDMS Si (red solid), CDMS Ge (red dotted) and XENON10 (brown dot-dashed). We have taken f p = f n . planations of DAMA’s annual modulation signaland exclusions from other direct detection exper-iments. The annual modulation signal [16, 17],originally seen at the DAMA experiment [18], hasrecently been confirmed by DAMA/LIBRA [19]. Onthe other hand, limits from XENON and CDMSstrongly constrain the simplest dark matter interpre-tation of the DAMA experiment: a signal resultingfrom the SI scattering of a WIMP. Explanation ofthe DAMA signal with spin-dependent scatterings[20, 21] is now also strongly constrained by COUPPand PICASSO.Fig. 2 shows that adding momentum dependenceto the SI interactions can only weaken, but not elim-inate the limits other experiments put on DAMAexplanations at the 90% confidence level, at leastwithin a Maxwellian halo model. Employing thecaveats discussed in [10], alternative statistical tech-niques [21] or a non-Maxwellian halo [22] might allowa window at low mass when combined with thesenew effects.In light of this, we now focus discussion on thescenario with the weakest direct detection limits,SD-proton scattering, shown in Fig. 3. Theseplots show that the relative importance of differentexperiments can invert as one adds momentum de-pendence, for precisely the reasons described above.In fact, the normal SD-proton case [20, 21] whichis ruled out by PICASSO is allowed for the q scenario. Interestingly, these factors are also ableto improve the fit with DAMA’s spectral shape,so that there are new masses that can now fit the DAMA spectrum. In particular, the mass regionat ∼ −
60 GeV would have normally had ashape that was inconsistent with DAMA. Since thesemomentum factors suppress the low energy scat-tering, the constraint from DAMA’s unmodulatedevent rate [10] is also weakened, leading to betterconsistency with DAMA’s full data set. For theseplots, we assumed a mediator mass of 1 GeV and 100MeV. As we will discuss later in the next section, alighter mediator mass of O (100) MeV is more suitedto generate cross sections of this size. As seen inFig. 3(c), for this lighter choice of mass, the 10 GeVDM mass region survives, but the KIMS limit cutsinto about half of the higher mass region. We shouldnote that our approach to the KIMS limits is notaggressive, and does not yield as strong a limit asthat in [13]. Consequently, a more aggressive limitmight also be able to exclude this region as well.For mediator masses much less than an MeV, themomentum independent case is recovered since the q factors cancel in Eqn. (10), at least in the rangethat the first Born approximation is valid.One important point about these scenarios isthat the expected relationship between direct andindirect detection signals breaks down. Typically,one assumes that the annihilation proceeds intosome Standard Model final state. Here, since werely upon the light mediator for our interaction,it provides an annihilation channel. Then, if themediator is (cid:46) GeV in mass, it is natural for itto decay dominantly to e.g., electrons, muons andpions. The limits from Super–Kamiokande WIMP a) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m Χ (cid:72) GeV (cid:76) l og (cid:64) Σ p (cid:68) (cid:72) c m (cid:76) b) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m Χ (cid:72) GeV (cid:76) l og (cid:64) Σ p (cid:68) (cid:72) c m (cid:76) c) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) m Χ (cid:72) GeV (cid:76) l og (cid:64) Σ p (cid:68) (cid:72) c m (cid:76) FIG. 3: Plots of the SD-proton cross section σ p vs DM mass m χ without (a) and with q suppression (b) and (c),where the mediator mass is 1 GeV (b) and 100 MeV (c). The colored regions show the 68, 90, and 99% CL regionsfor the best DAMA fit. The 90% exclusions limits are PICASSO (gray solid), KIMS (orange dashed), XENON10(brown dot-dashed), and CDMS (red dotted). capture are then trivially evaded [23, 24]. MODEL BUILDING CONSTRAINTS
For the dark matter scattering to display thenovel phenomenology discussed here, the scatteringrate must be dominated by the new operators, andnot the typical SI or SD coupling. This is not atrivial requirement. For comparable coefficients, thescattering mediated by operators of Eqns. (3)–(6)are suppressed by powers of the velocity or momentarelative to Eqns. (1) and (2). It is possible, however,that the coefficients of these new operators are muchlarger than the coefficients of the other operators.We discuss this further below.If the signal is to be observable at near-futureexperiments, the q n suppression must be compen-sated by a large coefficient for the operator. Thiscould be due in part to particularly large couplingsof a mediator to the Dark Sector or a local over-density of the dark matter, but the simplest way toget an enhancement is just for the mediator massto be small: dR/dE R ∝ m − φ . The necessary m φ depends on the amount of q suppression. If thereis a single q , as in O and O , a mediator massof a m φ ∼ few GeV, O (1) couplings to the DMwhile having Yukawa suppression on quark side,one finds a ∼ − cm cross section. This isnear the interesting region for O . In the case ofcoherent scattering off of nuclei (as for O ), thiswould actually already be strongly ruled out byCDMS and XENON for a large mass range, a viable 10 − cm cross section would require something like m φ ∼
100 GeV. For q suppression, the mass has tobe O (100) MeV to get a ∼ − cm cross section.In specific cases, large contributions to O - O can be expected. If there is a light pseudoscalarpresent that couples both to dark matter and toquarks then O - O can be generated through itsexchange without generation of either O SI or O SD .If there is no parity violation, then the expectationis that O dominates. On the other hand, if parityviolation is present, then it is plausible that thecoefficients of O - O could all be comparable.In this case, it is likely that it would be easiestto probe O because of its coherent scattering offof nuclei. Alternatively, parity violation might beconfined to couplings in the dark matter sector. Inthis case, pseudoscalar exchange could dominantlyinduce O . We note that if the light pseudoscalaris naively realized as a pseudo-Goldstone, it isdifficult to sufficiently suppress the contributions to O SI . Contributions are induced by exchanging thescalar whose vacuum expectation f φ value breaksthe global symmetry and made the φ light.Interactions with only q suppression can con-ceivably still dominate over standard interactionswithout significant model-building efforts. Thesimplest example comes from charge-radius or dipolecouplings to a composite WIMP, whose constituentsare charged under a new, dark gauge group [1].This generates the phenomenology of O straight-forwardly, although typically with a coupling to Z instead of A . If the mediation arises through aPGB, O , can dominate over the scalar exchangeas only one vertex will be suppressed by f φ , whilethe scalar exchange is suppressed by f φ .The most challenging model-building comes inrealizing the q suppressed interaction, without in-ducing SI scattering from the accompanying scalarmediator. While this seems difficult from the per-spective of a standard PGB, it can arise fairly simplyin SUSY theories. While PGBs are a natural way torealize a shift symmetry, such a shift symmetry couldsimply be present in the theory from other origins.For instance, in theories with N = 2 SUSY in thegauge sector (e.g., [25]), there is a chiral superfieldpartner for every gauge boson. The pseudoscalarcontained in it possesses a shift symmetry which canbe thought of as a higher-dimensional gauge symme-try, compactified on an S /Z orbifold. SUSY break-ing will make the associated scalar massive. Thiscan arise either from F − term breaking, throughthe operator X † X ( φ + φ † ) (with X a spurionthat gets the non-zero F -term, and φ a superfieldcontaining the PGB), or from D -terms, through W α W (cid:48) α φ (with W α the U (1) Y supersymmetric fieldstrength, and W (cid:48) α the supersymmetric field strengththat gets a non-zero D -term). Even in the presenceof this supersymmetry breaking, the pseudoscalarremains massless, and will only pick up a massradiatively through diagrams violating both the shiftsymmetry and SUSY. Thus, the scalar contributioncan be effectively decoupled from the strength of thepseudoscalar-mediated q interactions.Finally, a sizable coefficient for O can be gener-ated in theories with a light gauge boson that couplesto the dark matter and mixes with the B µ gauge fieldof the Standard Model.There are model-dependent constraints on lightpseudoscalar mediators. In particular, searches foraxions can apply. For particles in the GeV range,the process Υ → γφ is relevant. These branchingratios are constrained to be in the range 10 − − − ;the precise bound depends on the final state of φ decay, φ → µ ¯ µ, τ ¯ τ or invisible, [26–28]. If the φ hascouplings comparable to Standard Model Yukawas,the branching ratio is a f ew × − for masses wellbelow the Υ mass. Thus, these bounds constrain the φ coupling to b quarks to be somewhat smaller thanthe Standard Model Yukawa coupling. For lightermediators, depending on the flavor structure of the φ couplings, K → ππφ may be relevant. The rate forthe potentially more stringent process K + → π + φ issuppressed – a pure pseudoscalar coupling does notmediate this process, see for e.g., [29]. The dominant contribution to this decay comes from π − φ mixing[30], which is model dependent. In cases where thepseudoscalar couples only to 3rd generation quarks,the Kaon decays are absent; however, the Upsilonconstraints still apply. Following the procedure in[31], we find that 3rd generation couplings alone cangenerate a detectable rate, as pseudoscalar couplingsto heavy quarks generate a coupling to G (cid:101) G [32].Incidentally, in general, experimental uncertainties(in particular, the light quark contribution to theproton spin ∆Σ) and parameters like tan β allowa proton dominated coupling to be generated. Inthe minimal case of 3rd generation couplings, the φ decays to two photons with a decay length cτ ∼ g − | δa µ | < × − enforces m φ >
300 MeV [29]. Finally with couplingsto electrons, it is possible to search for e + e − → φγ for either invisible or electron decays of φ [33].However, for pseudoscalar φ , suppression by theelectron yukawa coupling makes the production ratebelow the projected sensitivities [33]. CONCLUSIONS
As the sensitivity of new dark matter directdetection experiments continues to increase at arapid pace, the ability to test for new scenariosfor dark matter will grow simultaneously. Presentexperiments are optimized to search for WIMPs withsignals that peak at low nuclear recoil energies. Incontrast, models with momentum-suppressed inter-actions (MDDM), have spectra that peak at inter-mediate energies, thus changing the expected signalsand relative strengths of various direct detectionexperiments. While interesting scenarios, specifi-cally inelastic dark matter, have been proposed withspectra that peak at high recoil energy, we find thatthe scenarios with this feature are more ubiquitousthan previously thought. In models with new lightvectors or pseudoscalars, momentum dependent in-teractions can be large, and this phenomenology canbe present.A simple parameterization captures much of therelevant phenomenology. Specifically, one can re-place dR i /dE R with ( q ) n dR i /dE R , where i in-dexes the interaction type and q is the momentumtransfer in units of 100 MeV. While additionalfeatures can arise at low mediator masses, this pa-rameterization is sufficient to reproduce the peakingin the spectrum, and provides a convenient wayto compare different experiments. In analyzingthe presently allowed parameter space in this way,we find that momentum dependent couplings canopen allowed ranges of parameters for DAMA withdominantly spin-dependent proton couplings, if ac-companied by an additional q suppression.Whatever model of dark matter nature has chosento realize, it is important to be cognizant of the widerange of possible phenomenology, so that possiblesignals are not missed or attributed to backgrounds.The framework of MDDM provides motivation, anda prescription to study and constrain these modelsin the future.Note added: As this paper was being finished, webecame aware of [34], which appeared in the arXivand discusses momentum dependent interactionsarising from the couplings to one or more new gaugebosons, and their ability to explain DAMA fromspin-independent interactions. ACKNOWLEDGEMENTS
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