How dark are Majorana WIMPs? Signals from magnetic inelastic dark matterand Rayleigh dark matter

Neal Weiner1,* and Itay Yavin2,3,†

1Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, New York 10003, USA2Department of Physics and Astronomy, McMaster University, 1280 Main Street, West Hamilton, Ontario, Canada L8S 4L8

3Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5(Received 3 August 2012; published 16 October 2012)

The effective interactions of dark matter with photons are fairly restricted. Yet both direct detection as

well as monochromatic �-ray signatures depend sensitively on the presence of such interactions. For a

Dirac fermion, electromagnetic dipoles are possible but are very constrained. For Majorana fermions, no

such terms are allowed. We consider signals of an effective theory with a Majorana dark matter particle

and its couplings to photons. In the presence of a nearby excited state, there is the possibility of a magnetic

dipole transition (magnetic inelastic dark matter), which yields both direct and indirect detection signals

and, intriguingly, yields essentially the same size over a wide range of dipole strengths. Absent an excited

state, the leading interaction of weakly interacting massive particles is similar to the Rayleigh scattering of

low-energy photons from neutral atoms, which may be captured by an effective operator of dimension 7 of

the form ���F��F��. While it can be thought of as a phase of the magnetic inelastic dark matter scenario

where the excited state is much heavier than the ground state, it can arise from other theories as well. We

study the resulting phenomenology of this scenario: gamma-ray lines from the annihilation of weakly

interacting massive particles; nuclear recoils in direct detection; and direct production of the weakly

interacting massive particle pair in high-energy colliders. Considering recent evidence in particular for a

130 GeV line from the Galactic center, we discuss the detection prospects at upcoming experiments.

DOI: 10.1103/PhysRevD.86.075021 PACS numbers: 12.60.Jv, 12.60.Cn, 12.60.Fr

I. INTRODUCTION

Weakly interacting massive particles (WIMPs) havelong been studied as potential candidates for the colddark matter observed in the Universe. The most well-motivated and deservedly most well-studied WIMPs arethose that emerge in extensions of the Standard Modelassociated with the seemingly unrelated problems of theelectroweak scale, such as supersymmetric extensions. Anorthogonal line of inquiry is motivated by the deceptivelyelementary question of ‘‘how dark is dark matter?’’Namely, what are the strongest constraints on the interac-tion of dark matter with the electromagnetic field?Numerous studies already exist and in particular the ideaof electric and magnetic dipole interactions have recentlyattracted considerable attention [1–11]. In these models,single-photon exchange provides a possible direct detec-tion signal, while annihilation into two photons mightprovide an indirect detection signal (see e.g., Ref. [12]).

However, if dark matter is a Majorana fermion, thenthese single-photon couplings through electromagnetic di-poles do not exist—the dipole operator vanishes identicallyfor Majorana fermions. For Dirac fermions, it is naturallyoff-diagonal [13,14]. For pseudo-Dirac fermions, in whichcase the ground state is the dark matter candidate, thedipole interaction mediates transitions between this ground

state � and an excited state ��. The authors of Ref. [9]exploited this possibility to build a model, dubbed mag-netic inelastic dark matter (MiDM), to explain the DAMAresults through dipole-dipole dominated scattering. Theinteraction Lagrangian of MiDM is

L ¼���

2

�������B

���þ c:c:; (1)

where �� is the dipole strength, B�� is the hypercharge

field-strength tensor, and ��� ¼ i½��; ���=2. This cou-

pling contains within it the interaction with the electro-magnetic field. We study the signatures of this model in thefirst part of this paper.In the limit that we take the excited state heavy, we are

left with a Majorana fermion, and we can again ask thequestion ‘‘how dark is dark Matter?’’ Starting with theMiDM Lagrangian above if the excited state �� is muchheavier than the energy available, then it can be integratedout to yield the interactions

L ¼ �2�

2m��ð ���B��B

�� þ i ���5�B��~B��Þ; (2)

where ~B�� ¼ 12 �

����B�� and ����� is the Levi-Civita

symbol. Motivated by this form, in the second part ofthis paper we will concentrate on the slightly more generalcase for the interaction of DM with the electroweak fieldstrengths

*neal.weiner@nyu.edu†iyavin@perimeterinstitute.ca

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1550-7998=2012=86(7)=075021(16) 075021-1 � 2012 American Physical Society

L ¼ 1

4�3R

f ���ðcos�B��B�� þ sin� TrW��W

��Þ

þ i ���5�ðcos�B��~B�� þ sin� TrW��

~W��Þg: (3)

Here � quantifies the relative coupling to the field strength

of hypercharge in comparison to that of SUWð2Þ and �R issome high scale related to the cutoff scale of the theory. Wewill discuss UV realizations in a later section, but simplescenarios can arise either as a limit of MiDM or, forinstance, integrating out a dilaton (or axidilaton). Theinteractions of Eq. (3) are akin to the familiar interactionsof photons with neutral atoms at long wavelengths that leadto Rayleigh scattering. Hence we dub this scenarioRayleigh dark matter (RayDM). This could be the entiretyof the DM interaction with the Standard Model, but it alsoserves as a reasonable form of the effective operatorsresponsible for � lines in many models (even when theyfreeze out dominantly through other channels). The specialform of this interaction, which necessitates at least twoforce mediators, requires a reconsideration of the basicprocesses by which we hope to detect dark matter andthis constitutes a part of the current work.

In this paper we set to explore these different possibil-ities for the interaction of Majorana WIMPS with light.The paper is organized as follows: In Sec. II we discuss indetail the MiDM scenario including its signatures ingamma rays as well as the prospects for seeing it in directdetection experiments; in Sec. III we explore the phenome-nology of RayDM; Sec. IV is devoted to the prospects ofcollider searches for both MiDM as well as RayDM;finally, the main findings of this work are summarized inthe conclusions, Sec. V.We caution the reader that the clearseparation between MiDM and RayDM is not alwaysappropriate. As we shall discuss and emphasize below,there are certain aspects of the phenomenology where thetwo scenarios and the operators involved cannot be logi-cally separated.

II. MIDM

In this section we concentrate on the MiDM scenario butconsider a slightly more general form of the magneticdipole interactions

L¼���

2

�������F

���þ��Z

2

�������Z

���þ c:c:; (4)

where F�� and Z�� are, respectively, the field strength of

the photon and Z boson, �� and �Z are the corresponding

dipole strength, and � and �� are twoWeyl fermions. If theinteraction above the electroweak scale is entirely with thefield strength of hypercharge, then �Z=�� ¼ � tanW . In

what follows we explore the more general possibility sinceone can entertain additional operators with the fieldstrength of SUWð2Þ that would result in a different ratio.An example of such an operator is the dimension 7 operator

ð��

2 Þ �������TrhyW��h. However, considering that such

operators are generically further suppressed we expectthe deviation away from the relation �Z=�� ¼ � tanWto be small.The phenomenology of this theory depends crucially on

the mass splitting between the two states � and ��. Whenthe mass splitting is large m�� �m� * m� the phenome-

nology is similar to that of RayDM, which is described inthe next section. More generally, even for a smaller split-ting such that m�� * m�=20 the heavier state �� is not

present during the early Universe freeze-out of the lighterstate �. In this case, unless other channels are availablethrough new interactions, the relic abundance is deter-mined by the annihilation into photons and vector bosons,which typically requires larger dipole strength. The pros-pects for direct detection in this case are fairly gloomy, butcollider constraints provide strong and interesting boundson this possibility as we describe in Sec. IV.As a consequence, in this section we focus on the case

when the mass splitting is small. In particular, when themass splitting to the excited state vanishes or is smallerthan the kinetic energy of the WIMP in the halo �M ¼m�� �m� � 100 keV the cross section of scattering on the

nucleus is much larger and the corresponding rates in directdetection experiments are phenomenologically interesting.Before moving onto the details, it is important to make

the phenomenological point: ‘‘for thermally producedMiDM, the direct and gamma-ray line indirect signaturesare roughly independent of the size of the dipole.1’’ That isto say, even if MiDM is a subdominant component ofthe dark matter, the amplitude of these signals would beunchanged.This arises quite simply. The annihilation rate is gov-

erned by the annihilation of WIMPs into fermion pairs [15]which scales as�2

�. This implies that the number density of

WIMPs scales as n� ���2� , i.e.,

MiDM ¼ 0 ��2thermal

�2�

; (5)

where�thermal is the dipole needed to achieve the appropri-ate relic abundance that arises from a cross section of6� 10�26 cm3 s�1. For a WIMP of mass m� ¼ 130 GeV

we find �thermal � 1:2� 10�3 �N (2:0� 10�3 �N) for adipole ratio of�Z=��¼�tanW (�Z=��¼0). Here�N ¼0:16 GeV�1 is the nuclear magneton. This dipole strengthcorresponds to an annihilation rate into gamma rays of�ð�� ! ��Þ ¼ 2:5� 10�29 cm3=s (1:6� 10�28 cm3=s).The collision rate in direct detection experiments scales

as n���N; thus,

1Since the breaking of hypercharge makes the phenomenologydepend in principle on both �� as well as �Z, this statementassumes the absence of any unexpectedly large difference inscale between �Z and ��.

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RDD / n��2� ¼ 0

m�

�2thermal

�2�

��2� ¼ 0

m�

�2thermal; (6)

where 0 � 0:4 GeV=cm3 is the local density of a WIMPcomprising all of the darkmatter. Such a scaling phenomenonis well known in many WIMP models, that when s-channelannihilation diagrams are directly linked to t-channel scat-tering, the lower relic abundance is compensated by thehigher scattering cross section (see, e.g., Ref. [16]).2

A possibly more remarkable scaling is associated withthe �� and �Z signatures. The cross section for theseprocesses is proportional to �4

�. Thus, the indirect �-ray

rate scales as

R�� / n2��4� ¼ 2

0

m2�

�4thermal

�4�

��4� ¼ 2

0

m2�

�4thermal: (7)

Again, if the WIMP is thermal, the �� signal is indepen-dent of the size of the dipole, even if the fraction of thedark matter is much smaller. There are important caveats tothis, as we shall discuss below, but they do not change thefact that even for very large dipoles (yielding underabun-dant dark matter) the signals are at the same level as that ofa thermal dominant WIMP.3

However, such a scenario is excluded unless � ¼ m�� �m� * 1=2m�v

2 and the direct detection scattering is either

inelastic (if the excited state is accessible) or not present(if it is not). Intriguingly, the size of the signals would beappropriate for DAMA [17] (via the MiDM scenario [9])and approximately for the recently reported 130 GeV sig-nal, given the astrophysical uncertainties [18–21].

A. Annihilation rate and relic abundance

To understand these points in detail, let us consider theprecise values realized. When the excited state �� ispresent, the annihilation into StandardModel fermion pairsthrough �=Z dominates. The annihilation cross section toleading order in velocity is given by

�ð��� ! f �fÞv ¼ �q2f�2�

�1þ 2vf

�Z

��

�ð4m2�Þ

þ ðv2f þ a2fÞ

�2Z

�2�

�2ð4m2�Þ�: (8)

Here qf is the fermion’s electric charge, vf (af) is the ratio

of its vector (axial-vector) coupling to the Z boson to its

electromagnetic coupling, and �ðsÞ ¼ s=ðs�m2ZÞ. In the

above expression we took m�� ¼ m� since it is only when

the mass splitting is not too large that the heavier state isrelevant. The annihilation rates of the lighter state intovector bosons are

�ð��� ! ��Þv ¼ m4�

4

2�2

�

m��

!2 1

ð1þm2�=m

2�� Þ2 ; (9)

�ð��� !�ZÞv¼ 2m4

�

4

2�2

�

m��

2�2Z

m��

! 1� m2

Z

4m2�

!3

� 1þ m2

Z

4m�m��

!2�

1þ2m2��m2

Z

2m2��

!2

;

(10)

�ð��� ! ZZÞv¼m4�

4

2�2

Z

m��

!2 1�m2

Z

m2�

!3=2

� 1þ m2

Z

2m�m��

!2�

1þm2� �m2

Z

m2��

!2

:

(11)

In the case of large mass splittings, we are effectively leftwith a single species at freeze-out and its annihilation intovector bosons must therefore be sufficiently large so as toyield�totv � 3� 10�26 cm3=s. This implies a rather largedipole strength which is in tension with collider searchesfor monophotons discussed in Sec. IV.In the case of intermediate and small mass splittings, the

cosmological history is slightly different and the necessarytotal annihilation rate is consequently altered. Since theheavier state decays to the lighter state through the dipoletransition only after freeze-out, we effectively have twospecies during freeze-out, each of which can only annihi-late on the other. The relic abundance necessary at freeze-out is therefore only half its usual value.4 This impliesthat the total annihilation rate is larger: �totv �6 � 10�26 cm3=s. Interestingly, since this requires a

dipole strength larger by a factor offfiffiffi2

p, this leads to an

increase of the annihilation rate into �� and �Z of a factorby 4.In two recent papers, Refs. [18,19] reported on a tenta-

tive gamma-ray line in the Fermi/LAT data at E� ¼130 GeV, which has recently been confirmed by Tempel,Hektor, and Raidal [20] and Su and Finkbeiner [21].This result can be accommodated within the MiDM sce-nario with a WIMP mass m� ¼ 130 GeV leading to the

gamma-ray line at E� ¼ 130 GeV through �� ! ��. It

will generically also result in an additional line of compa-rable strength at E� ¼ m� �m2

Z=4m� ¼ 114 GeV from

2We do not consider the implications of a CP-violatinginelastic electric dipole moment here. Due to a velocity-unsuppressed dipole-charge scattering (see e.g., Ref. [2]), theconstraints in Ref. [15] constrain the dipole �� & 10�8 �N . Atthese levels the indirect signals would be negligible, unless theexcited state is inaccessible.

3Once �� is large enough this scaling ceases because theannihilation to gauge bosons dominates. For 130 GeV this occursat �� * 0:05 �N . At this size, we shall see that collider con-straints would exclude the scenario already.

4This is true only as long as the heavier state’s lifetime issufficiently short so that present day DM is entirely composed ofthe lighter state. Form�� �m� * 100 keV the lifetime is shorterthan about a microsecond.

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�� ! �Z, which is consistent with the data (see also thediscussion in Ref. [22]).5 In Fig. 1 we plot the annihilationrate into fermions as a function of the dipole ratio �Z=��

for a WIMP mass of m� ¼ 130 GeV in the case of small

splitting m� � m�� .

Surprisingly, when the dipole strength is normalized toyield an annihilation rate into �� (�Z) in the range re-cently reported in Refs. [19,21], 0:3–1:3� 10�27 cm3=s,the annihilation rate at freeze-out into f �f pairs near theexpected ratio of�Z=�� ¼ � tanW is only a factor of 3–7

(5)–(10) larger than the needed value of 6� 10�26 cm3=s.As we shall see in the next subsection this also yields a ratein direct detection experiments which can be probed withexisting experiments and is surprisingly close to that re-ported by the DAMA Collaboration for m� �m�� �100 keV. Evidently, this surprising concordance is numeri-cally not perfect but is sufficiently interesting given thelarge astrophysical uncertainties.

B. Direct detection of MiDM: Constraining cosmicray gammas underground

A WIMP with a magnetic dipole can scatter against anucleus’ own magnetic dipole as well as its charge. Inthe case of elastic scattering the current limits from directdetection experiments on a WIMP of massm� � 130 GeV

and local mass density of 0:3 GeV=cm3 are at the level

of �X & 6� 10�5 �N (see e.g., Ref. [15]). Thisexcludes annihilation rates into diphotons at the level of�ð�� ! ��Þv & 10�34 cm3=s, far below anything wecan hope to measure anytime soon. Thus, current directdetection limits robustly exclude the possibility of observ-ing gamma-ray lines from magnetic dipole annihilationsindependently of the dipole strength.A more promising possibility is offered by MiDM [9]

where the WIMP couples via a magnetic dipole to anexcited state ��. If the mass splitting is of order the kineticenergy of the WIMP in the halo �M ¼ m�� �m� �100 keV, then it may undergo inelastic scattering againstthe nucleus, but the corresponding rates in direct detectionexperiments are much reduced compared to the elasticscattering discussed in the last paragraph. Thus, the dipolestrength can be larger and the annihilation rates of WIMPsinto gamma rays considerably enhanced. This scenariothen offers an interesting connection between observationsin gamma-ray lines and direct detection efforts that can beprobed with current experiments. In particular, as dis-cussed in Ref. [9] it may explain the signal claimed bythe DAMACollaboration6 [17] for WIMP masses ofm� �100 GeV and magnetic dipole strength in the range�� ¼ 10�2–10�3 �N . Interestingly, for m� ¼ 130 GeV

and �� ¼ 3� 10�3 �N the annihilation rate of WIMPs

into gamma rays is �vð�� ! ��Þ � 10�27 cm3=s whichcan accommodate the excess recently reported inRefs. [19,21].

C. Variations on a theme: Model dependencesin indirect signals

As we laid out in Eq. (7), the �-ray line signals shouldbe independent of the size of the dipole. The questionarises as to how robust the size of the gamma-ray signalwill be to changes in the underlying model. Indeed, wehave already seen examples of this: an MiDM model thatonly coupled to � would be a thermal relic with �� ¼�thermal � 2� 10�3 �N , which leads to a �� signal for asignal size of h�vi ’ 1:6� 10�28 cm3 s�1 (normalized tothe case with�� ¼ �DM). In contrast, in the presence of a

dipole with the hypercharge gauge boson, the needed

dipole for a thermal relic is roughlyffiffiffi3

psmaller, leading

to an effective signal size of h�vi ’ 2:5� 10�29 cm3 s�1.While the pure � dipole is close enough to explain thesignal at 130 GeV, a dipole of hypercharge seems too smallexcept in very cuspy halos. Thus, we should inquire

FIG. 1 (color online). The total annihilation rate for m� ¼m�� ¼ 130 GeV to fermion pairs, �ð�� ! ��Þ in the MiDM

scenario as a function of the relative strength of the photondipole to Z dipole. The dipole itself is normalized to yield theannihilation rate into �� (blue/dark band) and �Z (green/lightband). The bands span the range shown in units of 10�27 cm3=s.The red vertical line shows the relevant ratio of dipoles in thecase of MiDM interaction with hypercharge only.

5Alternatively, m� ¼ 144 GeV may give rise to the line atE� ¼ 130 GeV through the annihilation �� ! �Z. In this casethe annihilation to two photons would have to be somewhatsuppressed since no feature is observed at E� ¼ 144 GeV. Sucha suppression is more natural in the RayDM scenario discussedin the next section.

6Recently, the KIMS Collaboration has claimed to exclude thepossibility by Oð1Þ at 90% confidence [23]. However, this was aparticular range of energies arising for a specific choice ofquenching factors, both on NaI and CsI. When combined withthe absence of a thorough discussion of energy resolutions (giventhat signal may leak into surrounding bins), the MiDM scenarioappears intact, although likely requires an Oð1Þ modulationfraction.

NEAL WEINER AND ITAY YAVIN PHYSICAL REVIEW D 86, 075021 (2012)

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whether there are any effects that modify this. As it turnsout, there are at least two simple elements without enlarg-ing the effective theory that can affect this.

The first effect is the presence of form factors. Since theappropriate dipole scale is �� * e=TeV, it should be

resolved near the WIMP mass scale. The annihilationinto �� samples a form factor with spacelike q2 ¼ �m2

�,

while the annihilation into charged pairs samples a formfactor with timelike q2 ¼ 4m2

�. [The �Z signal samples

form factors with q2 ¼ �ðm2� �m2

z=2Þ.] If the form factor

is being resolved at these momentum transfers, then treat-ing it as a contact interaction is clearly wrong. Since q2 forthe s-channel diagram is 4 times larger than the t-channel,it is reasonable that it could pick up a larger suppressionfactor. In this case the dipole would have to be increased by�2

� ¼ �2thermal=F

2ð4m2�Þ. The resulting signal in �� would

be increased by ðF2ð�m2�Þ=F2ð4m2

�ÞÞ2. We note that in this

case, if the scale � of new physics is not much higher thanm�, the Rayleigh operator we discuss in Sec. III may be

generated as well, which could constructively interfere,enhancing the �� rate.

It is worth dwelling on this last point for a moment.Generally, we should in fact expect the presence of theRayleigh operator whenever the dipole operator is present.If the dipole is generated by a loop process as shown inFig. 2(a), then assuming a coupling ��X ~X, the natural sizefor the dipole operator is

gY�2

16 2

1

MX

: (12)

By attaching a second external photon, the Rayleigh op-erator is also generated through the diagram shown inFig. 2(b) with a natural size

g2Y�2

16 2

1

M3X

: (13)

Thus, annihilation to �� through the RayDM process is aone-loop process, while annihilation through the MiDM2

process is effectively two-loop. Thus RayDM annihilationis relatively enhanced over MiDM2 by a factor of�

�2

16 2

��2 m2�

M2X

: (14)

Even for m� �MX, if the theory is at all perturbative, the

Rayleigh contribution should dominate. If the Rayleigh

contribution to the amplitude is even a few times largerthan the MiDM2 contribution, the size of the signal shouldbe easily large enough to explain the 130 GeV signal. Thus,not only is it reasonable to believe that the Rayleighoperator could contribute, it should be a likely expectation.A second possibility is the presence of CP violation.

While both CP-conserving magnetic and CP-violatingelectro dipole moments (EDMs) produce s-wave �� sig-nals, only the s-channel diagram via a magnetic dipoleyields an s-wave annihilation to charged fermions [15].Thus, with EDMs one can increase the present day ��signal while producing only a p-wave suppressed annihi-lation at freeze-out into f �f. With only EDMs, assuminga freeze-out at T � m�=20, the annihilation into gauge

bosons dominates, and one has a signal cross section of� 6� 10�26 cm3 s�1 (as it is a Dirac fermion at freeze-out), exceeding the Fermi limits. Thus, while a pure EDMis excluded, a combination of EDM and magnetic dipolemoment could produce the signal consistent with con-straints. However, as we have stated, in the presence ofsuch a large EDM, the direct detection limits would haveexcluded it unless the excited state is completely inacces-sible. So while the first possibility still offers the prospectof discovery at upcoming direct detection experiments, thissecond case seems unlikely to be found underground.

III. RAYDM

A. Searching for RayDM in gamma rays

The nonrelativistic annihilation cross section of RayDMinto the different electroweak vector bosons is sensitiveonly to the axial ���5� components to leading order inthe velocity expansion. The differential cross section isgiven by

�ð�� ! VVÞv ¼ g2VV4

m4�

�6R

KVV; (15)

with the kinematic functions KVV and couplings gVVdefined as

K �� ¼ 1; g�� ¼ c�cW2 þ s�sW

2; (16)

K �Z ¼ 2

1� m2

Z

4m2�

!3

; g�Z ¼ ðs�� c�ÞsWcW; (17)

FIG. 2 (color online). (Left) Loop diagram contributing to dipole operator for MiDM model. (Right) Comparable diagramcontributing the Rayleigh operator.

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K ZZ ¼ 1�m2

Z

m2�

!3=2

; gZZ ¼ c�sW2 þ s�cW

2; (18)

KWW ¼ 2

1�m2

W

m2�

!3=2

; gWW ¼ s�: (19)

Here c� ¼ cos� and s� ¼ sin� and cW and sW are

similarly defined with respect to the Weinberg angles.When m� is not too much smaller than �R we expect

some form-factor suppression to soften the behavior ofthis cross section.

In Fig. 3 we plot the annihilation cross section of �� !�� as a function of the WIMP mass for several values ofcos�, the relative coupling to the field strengths in Eq. (3).

Requiring the right relic abundance, which for a Majoranafermion is obtained when the total annihilation cross sec-tion at freeze-out is 3� 10�26 cm3=s, we can normalizethe Rayleigh scale �R. For m� ¼ 130 GeV this results in

�R ¼ 440 GeV (�R ¼ 490 GeV) in the case of cos� ¼1 ( cos� ¼ 0). For a Dirac fermion the necessary annihi-

lation cross section is 6� 10�26 cm3=s and the Rayleigh

scale is correspondingly a factor of 21=6 smaller. Theresulting gamma rays are monochromatic with E� ¼ m�.

To qualitatively understand these results, we consider thelimit where the WIMP mass is much heavier than thevector boson’s. Then the expression for the total crosssection is particularly simple and by equating it to therequired cross section from relic abundance we can solvefor �R in terms of the WIMP mass and the angle �:

XVV0

�ð��!VV 0Þv¼ 3�10�26 cm3=s)�R

¼ 600 GeV

�m�

200 GeV

�2=3ð2� cos2�Þ16: (20)

With this value of the Rayleigh scale the annihilation ratesinto �� and �Z are

�ð�� ! ��Þv3� 10�26 cm3 s�1

¼ ðcW2c� þ sW2s�Þ2

2� cos2�

�th

R

�R

!6

; (21)

12�ð�� ! �ZÞv

3� 10�26 cm3 s�1¼ cW

2sW2ðc� � s�Þ2

2� cos2�

�th

R

�R

!6

: (22)

Here �thR is the value of the Rayleigh scale that leads to the

correct relic abundance. As can be expected when theRayleigh operator is mostly associated with hyperchargethe total cross section is very close to the cross section forannihilation into a photon pair�tot � �ð�� ! ��Þ, whichwould result in too large a signal.7 When the Rayleighoperator is mostly associated with the non-AbelianSUWð2Þ part, the annihilation into photons is suppressedcompared with the total cross section due to the Weinbergangle and the �� ! WþW� channel. More quantita-tively, in the pure SUWð2Þ case, �gg=�tot � 1=28 and

1=2� ��Z=�tot � 1=12. The photon-to-hadron ratio

ð��� þ 1=2��ZÞ=ð�tot � ��� � 1=2��ZÞ � 1=7:6. In the

case of hypercharge RayDM the equivalent numbers are�gg=�tot � 1=1:4, 1=2� ��Z=�tot � 1=7. The photon-to-

hadron ratio is � 5, so no significant hadronic emission ispresent in this case.An additional process contributing to the monochro-

matic gamma-ray signal is of course �� ! �Z with alower energy of E� ¼ m� �m2

Z=4m�. In Fig. 4 we plot

the annihilation rate associated with this channel aswell as its ratio to the diphoton rate. Depending on theDM halo profile and the angle cos�, both the �� and

�Z rates are in the interesting range reported in Ref. [19],3� 10�28–2� 10�27 cm3=s. This points to a fairly lowRayleigh scale of �R � 500 GeV.One might worry about the validity of this picture given

that the Rayleigh scale is rather low. We come back to thispoint in Sec. IV where this issue is particularly important;however, for the purpose of nonrelativistic annihilation it isonly necessary for the Rayleigh scale to be somewhatlarger than the WIMP mass. Nevertheless, since theWIMP mass is not much lower than the Rayleigh scale,it may be appropriate to include a form factor. Thinkingabout RayDM asMiDM2 allows one to resolve the 4-pointinteraction with the exchange of the excited state ��.Consulting the corresponding annihilation rates in

FIG. 3 (color online). The annihilation rate of WIMPs todiphotons, �ð�� ! ��Þ, as a function of the angle � for

different choices of the WIMP mass. For each mass choice theRayleigh scale �R is chosen so that the total annihilation crosssection yields the correct relic abundance. Shown are m� ¼100 GeV (solid black curve) and m� ¼ 130 GeV (dashed red

curve). The dotted blue curve depicts the asymptotic formulaEq. (21) which is independent of mass.

7Although, as we discuss in the conclusions, a subdominantDM component would plausibly give the right signal.

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MiDM, Eqs. (9)–(11), we see for example that the process�� ! �� is diminished by a factor of ð1þm2

�=m2�� �2.

B. Direct detection of RayDM

The scattering of RayDM against matter is complicatedby the fact that at least two force mediators have to beexchanged. The exchange of two photons leads to the leastamount of suppression and so we concentrate on this caseand consider the Lagrangian

L ¼ g��

4�3R

���F��F��: (23)

Two distinct processes are possible: �N ! �N elasticscattering through the loop shown in Fig. 5 and �N !�N þ � tree-level scattering. The latter channel is ex-tremely suppressed due to phase space. The total crosssection for the elastic channel was previously calculatedin the thorough work of Ref. [2] in the approximation thatthe nucleus is much heavier than the WIMP by consideringthe scattering of the WIMP off the external electric fieldgenerated by the nucleus. In Appendix A we provide adifferent derivation which leads to a more exact result thatis valid when the WIMP mass cannot be neglected relative

to that of the nucleus. At leading order in the velocityexpansion the amplitude for this process is given by

iM ¼ i�Z2g��4

Q0

�3R

F

jqj2Q2

0

!�uðk0ÞuðkÞ �uðp0ÞuðpÞ; (24)

where Z is the nucleus charge, � ¼ 137�1 is the fine-structure constant, Q0 is the nuclear coherence scale, andthe momentum transfer is related to the relative velocitybetween the WIMP and the nucleus and the angle ofscattering in the center-of-mass frame through jqj2 ¼2�2v2ð1� cosÞ. The function F ðxÞ decreases exponen-tially for high momentum transfers and is of order unitynear the origin, F ð0Þ ¼ 2=

ffiffiffiffi

p. The spin-independent dif-

ferential cross section in the center-of-mass frame is then

d�

d cos¼ �2

�N

2

���������Z2g��4

Q0

�3R

F�jqj2Q2

0

���������2

: (25)

Here ��N is the nucleus-WIMP reduced mass. To a good

approximation we can use F ðjq2j=Q20Þ � F ð0Þ and so

the total spin-independent cross section per nucleon isgiven by

�SIp � �2Z4g2��

4 2A4

m2NQ

20

�6R

; (26)

where A is the nucleon number. This is an extremelysmall cross section for an electroweak scale WIMP. For

example, taking the nuclear coherence scale Q0¼ffiffiffi6

p ð0:3þ0:89A1=3Þ�1, the Rayleigh scale �R¼500GeV,and setting g�� ¼ 1 yields �SI

p � 10�49 cm2 for scattering

on xenon. If the Rayleigh scale is considerably lower,�R & 100 GeV, then the scattering rates become appre-ciable. That requires much lighter WIMPs than what we setto explore in this work and we leave it for a future study toelucidate the prospects associated with this part of parame-ter space.

IV. COLLIDER PHENOMENOLOGY

In MiDM, the production mode in colliders is simplyf �f ! ��� followed by the prompt decay of the heavier

FIG. 4 (color online). The top pane is similar to Fig. 3 above,but for half the annihilation rate into a photon and a Z boson,12�ð�� ! �ZÞ. In the bottom pane we plot the ratio of the

annihilation rate into �Z to that into �� for m� ¼ 130 GeV as

a function of the angle �.

FIG. 5. Elastic scattering of RayDM against the nucleusthrough two-photon exchange.

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state to a photon or a Z boson, as shown in Fig. 6(a). As weemphasized throughout this paper, the phenomenology ofMiDM depends crucially on the mass difference betweenthe WIMP � and the heavier state �� and its colliderphenomenology is no exception. When the mass splittingis small searching for MiDM in collider proceeds in thesame fashion as searching for other WIMPs, namely bylooking for monojet or monophoton events from an unbal-anced initial state radiation (see for example the excellentrecent works of Refs. [24–26]). When the mass splitting islarge the collider signatures of MiDM are more similar tothat of RayDM.

In RayDM, since the coupling to WIMPs requires atleast two vector bosons, the collider signatures are some-what different than usual. There are two different processesthat may be searched for: f �f ! f �f�� through a vectorboson or photon fusion; and f �f ! ��V, through an inter-mediate vector boson, where V ¼ �, Z, or W�. This lastprocess, shown in Fig. 6(b), enjoys a larger cross sectionand results in the production of a photon or an electroweakvector boson in association with large missing energy. Wetherefore concentrate on this possibility below. We beginby discussing the MiDM scenario, followed by RayDM,and finally discuss the actual constraints. General formulasfor the differential cross sections in the different cases aregiven in Appendix C.

A. MiDM

In MiDM the dominant mode is the production of theheavier state in association with the WIMP throughf �f ! �=Z ! ���. After production, the heavier statesubsequently decays to the WIMP through the emission ofa photon or a vector boson as shown in Fig. 6(a).The differential cross section for this process is given inEq. (C8) in Appendix C.

When the splitting between the excited state and theground state is much smaller than the mass �M � m� the

resulting photon or vector boson is too soft to be searchedfor directly. Reference [27] proposed some interestingideas for looking for iDM in colliders when the masssplitting is in the GeV range and the heavier state decaysinto pions. More generally and without reliance on suchspecialized techniques, the collider phenomenology in thiscase is identical to the usual case of WIMP pair production,but through a dipole operator. It can be searched for in ageneral way by tagging on initial state radiation. This wasnicely worked out in Ref. [15] for monojet searches in thecase of degenerate states (m�� ¼ m�) for both magnetic as

well as electric dipoles. When the splitting is of order themass and larger, the emitted photon or vector boson may besufficiently energetic to be searched for directly. This canbe searched for without reliance on initial state radiationhence enjoying a larger cross section. In this case collidersearches for monophotons place strong constraints on thisscenario as discussed below in the final part of this section.

In Fig. 7 we plot the production cross section for MiDMas a function of the WIMP mass for several choices of theparameters and the mass splittings. We recall that in thecase of small splitting where the relic abundance is deter-mined by the annihilation into fermions the �� signal isindependent of the dipole strength whereas the collidercross section scales as the square of the dipole. Thus, thecross section shown in Fig. 7 should be interpreted as theminimal cross sections when the dipole strength is normal-ized to yield the correct relic abundance � ¼ �thermal. For

the same reason, the ratio �ðpp!�=Z!���Þ2�vð��!��Þ is independent of

the dipole strength for a given mass and choice of �Z=��.

FIG. 6. The production of WIMPs at colliders is shown in(b) for the RayDM scenario. When the Rayleigh scale is com-parable to or lower than the energies involved in the collision, theRayleigh operator must be resolved. In (a) we show the corre-sponding process in the case where RayDM is the result ofintegrating out a heavy excited state in MiDM. In (c) we showthe process in the case where the Rayleigh operator is resolved interms of a new scalar. More details on the UV completions areprovided in the text.

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It is given in Table I and allows for a straightforwardcomparison between the rates in astrophysical processesand the cross sections in colliders.

B. RayDM

Collider searches for dark matter are inevitably tied upin the embedding of dark matter into a complete theory andRayDM is no exception. In this subsection we discuss thepossibility of directly producing WIMPs in colliders in theRayDM scenario. As we shall see, the phenomenology issensitive to the UV physics that resolves the Rayleigh

operator and as a result is more model dependent. This isin contrast to the phenomenology associated with directand indirect detection efforts discussed in the previoussections which is insensitive to the details of the UVphysics. It is important to keep this contrast in mindwhen considering the impact of collider constraints onthe previous sections (this issue also arises in other modelsof DM; see Ref. [28] and references therein for somerelated discussion).In the case of RayDM, the differential cross section for

f �f ! ���� at center-of-mass energyffiffiffis

p m� through an

intermediate �=Z is given by

1

�tot

d�

dpT

¼ 20pT

s

�1� 4p2

T

s

�3=2

; (27)

where pT is the transverse momentum of the photon in thecenter-of-mass frame, and

�totðf �f ! ����Þ ¼ �q2f

3840 2

s2

�6R

� ðg2�� þ 2g��g�Zvf�ðsÞ

þ g2�Zðv2f þ a2fÞ�2ðsÞÞ: (28)

Here qf is the fermion’s electric charge, vf (af) is the ratio

of its vector (axial-vector) coupling the Z boson to itselectromagnetic coupling, and �ðsÞ ¼ s=ðs�m2

ZÞ. Wenote that the transverse momentum distribution is suchthat most photons are fairly central, which is importantfor the monophoton searches. Similar relations can beobtained for the production cross section for f �f ! ���W.In the limit where the WIMP mass and the W-boson massare both much smaller than the incoming center of massenergy an identical distribution in pT results. This moti-vates mono-W searches, looking for the final state W�produced in association with the invisible �� pair.At LEP for example, where

ffiffiffis

p � 200 GeV, one obtains�tot � 7� 10�3 fb ð500 GeV=�RÞ6 for � ¼ 0. This

cross section is much too low unless the Rayleigh scaleis brought down considerably. This is in good qualitativeagreement with the very thorough investigation ofRef. [29] where bounds on unparticle production at LEPwere presented.8

At Tevatron and LHC, one must convolve the aboveexpressions against the parton luminosity functions. Theresulting cross section is larger, but for those values of theRayleigh scale where the cross section is sufficiently largeto be interesting the theory requires a UV completion. Onepossible UV completion of RayDM is of course MiDM inthe case when the mass splitting is very large. Integrating

FIG. 7 (color online). Minimal production cross section forMiDM, pp ! ���, through �=Z in the s-channel against themass of the WIMP. The lower (black) curves show the case ofsmall splitting m� � m�� for two choices of the dipoles’ ratio

�Z=�� ¼ � tanW (solid curve) and �Z=� ¼ 0 (dotted curve).

The dipole strengths in this case are normalized to yield thecorrect relic abundance when the annihilation is dominantly intofermion pairs, Eq. (8). While the �� signal remains invariant asthe dipole strength increases, the collider cross section willincrease as �2

�. The uppermost curve (solid blue) depicts the

case when the mass splitting is large m�� �m� ¼ 100 GeV for

�Z=�� ¼ � tanW . The dipole strength is normalized to yield

the correct relic abundance when the annihilation is dominantlyinto vector-boson pairs, Eqs. (9)–(11).

TABLE I. For a given WIMP mass m� and splitting �M ¼m�� �m� the ratio �ðpp!�=Z!���Þ2

�vð��!��Þ is independent of the dipole

strength and is displayed in this table for several choices of themasses and couplings. The production cross section is calculatedat leading order for the LHC with

ffiffiffis

p ¼ 7 TeV.

�Z=�� ¼ � tanW �Z=�� ¼ cotW

ðm�;�MÞ(130 GeV, 0) ð28 fbÞ2

10�28 cm3 s�1ð0:4 pbÞ2

10�28 cm3 s�1

(130 GeV, 100 GeV) ð16 fbÞ210�28 cm3 s�1

ð0:2 pbÞ210�27 cm3 s�1

(300 GeV, 0) ð1:6 fbÞ210�28 cm3 s�1

ð23 fbÞ210�28 cm3 s�1

(300 GeV, 100 GeV) ð1:1 fbÞ210�28 cm3 s�1

ð15 fbÞ210�28 cm3 s�1

8The Rayleigh operator has a scaling dimension of � ¼ 3 inthe notation of Ref. [29], which was not considered by theauthors for good reasons. For � ¼ 2 they find that the unparticlescale can be as low as 190 GeV. Attempting to extrapolate to� ¼ 3 is not very useful since the energy available is greaterthan the cutoff scale and some UV completion is needed toresolve the nonrenormalizable Rayleigh operator.

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out the excited state �� one recovers the RayDM interac-tions. So schematically

MiDM2 !m��ERayDM: (29)

In this case the Rayleigh scale �R is connected with themagnetic dipole �� through �3

R ¼ m��=2�2� and one can

easily translate the results for MiDM in the previoussubsection to the case of RayDM.

Another possible UV completion involves a scalar s anda pseudoscalar a, which couple directly to the WIMP aswell as to the field strengths of UYð1Þ and SUWð2Þ. Weparametrize this theory with

L ¼ 1

2@�s@

�s� 1

2m2

ss2 þ 1

2@�a@

�a� 1

2m2

aa2 þ s ���

þ a ���5�þ cos�4�UV

sB��B�� þ cos�

4�UV

aB��~B��

þ sin�4�UV

sTrW��W�� þ sin�

4�UV

aTrW��~W��: (30)

Here we have taken the Yukawa couplings of the scalars tothe WIMP to be order unity and used �UV to denote thescale of the dimension-5 operators. Integrating out thescalars s and a we generate the different operators ofRayDM. When the scalars are light enough to be producedin colliders the dominant process shown in Fig. 6(c) isf �f ! VsðaÞ followed by the decay of s (a) to aWIMP pair.We note that this example in fact results in a more generalversion of RayDMwhere the relative coupling of the scalar��� and pseudoscalar ���5� to the field strengths is arbi-trary. This is important in the case of comparing directdetection rates (which are sensitive to the scalar piece) toindirect detection rates (which are sensitive to the pseudo-scalar piece). This distinction does not play an importantrole for the purpose of collider phenomenology [24–26].

In this case, the distribution of the transverse momentumof the photon in the center-of-mass frame is given by

1

�tot

d�

dpT

¼ 3pT

s

ðð1� m2s

s Þ2 þ 2p2T

s Þð1� m2

s

s Þ3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� m2

s

s Þ2 � 4p2T

s

q : (31)

We note that this function is strongly peaked towards the

kinematical limit pðmaxÞT ¼

ffiffis

p2 ð1�m2

s=sÞ. This is in sharp

contrast to typical monophoton signatures of dark matterproduction where the photon originates from initial stateradiation and hence its pT is dominantly soft. Here the totalcross section is given by

�totðf �f ! ����Þ ¼ �q2f

24�2UV

�1�m2

s

s

�3

� ðg2�� þ 2g��g�Zvf�ðsÞþ g2�Zðv2

f þ afÞ2�2ðsÞÞ: (32)

The couplings and the function �ðsÞ are defined afterEq. (28) above. Similar expressions hold for the axialscalar a. In Fig. 8 we plot the production cross section ofa photon in association with one of the scalars against theannihilation rate of WIMP into two photons.

C. Limits from colliders

Limits on RayDM from colliders come primarily fromeither monojet or monophoton searches. In the case ofMiDM with a small splitting the production in collidersis observable only through the emission of a gluon orphoton from the initial state partons. Thus the most con-straining limits on this scenario come from monojetsearches. The most recent search from CMS [30] place alimit of a few picobarns in the range m� & 103 GeV (see

Fig. 9). This is not quite sufficient to exclude the interest-ing production cross sections in the MiDM scenario (seeFig. 7), but it comes close. Consulting Table I we see forexample that in the case of hypercharge dominated inter-actions and a WIMP mass of m� � 100 GeV monojet

searches at the LHC can begin probing annihilation ratesinto diphotons of the order of 10�26–10�27 cm3=s.On the other hand, when the splitting is large (�M *

100 GeV) the photon emitted from the excited state’sdecay is hard enough to be searched for directly. In thatcase the relevant limits are the limits on �ðpp ! 6E�Þcoming from monophoton searches. These are muchmore constraining and are at the level of �ðpp ! ���Þ &14 fb for a WIMP mass below the TeV range [31]. As canbe seen from Fig. 7 this bound all but excludes MiDMwith

FIG. 8 (color online). A plot of the production cross section forthe heavy scalars in association with a photon, pp ! �sðaÞagainst the annihilation rate of WIMPs to two photons. Themass of the scalar is allowed to vary between 2m� and �UV. The

cutoff scale is set to �UV ¼ TeV, but since both the productioncross section as well as the annihilation rate scale as ��2

UV any

other choice can be obtained with a simple rescaling. The solidcurves correspond to m� ¼ 130 GeV with the blue (left) curve

corresponding to cos� ¼ 1 and the black (right) curve corre-

sponding to cos� ¼ 0. Similarly, the dashed curves correspond

to m� ¼ 300 GeV.

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large splittings. In the case of heavy scalar production inassociation with a photon one can again compare theproduction rate directly to the limits on �ðpp ! 6E�Þ sincethe resulting photon is hard. Since the production crosssection in the case of scalars is generically lower (seeFig. 8) the current constraints are not quite strong enoughto exclude this scenario, but they are now probing the mostinteresting parts of parameter space.

V. CONCLUSIONS

The effective theory describing the interactions of aMajorana WIMP with photons is of critical importance,given that our best indirect detection searches comethrough monoenergetic �-ray lines, and direct detectionis clearly sensitive to scattering through a photon ex-change. Interestingly, this effective theory is quite re-stricted: in the presence of a nearby excited state, there isthe possibility of an interaction with electromagnetism viaa dipole transition to the excited state (or MiDM); in theabsence of a nearby state, the leading operator comes in theform ��W��W

�� or ��B��B�� or its the pseudoscalar

and CP-violating equivalents. These two scenarios haverelated but distinct phenomenology.

Remarkably, in the case of MiDM both the size of thesignal in direct detection and ��þ �Z signatures areindependent of the size of the dipole, with the relic abun-dance suppression precisely canceling out against the en-hanced scattering and annihilation cross sections. Thisoffers a surprising concordance whereby the annihilationrates into �� are in the range to explain the tentative excessin gamma rays at around 130 GeVand possibly explain theDAMA annual modulation. MiDM predicts a secondaryline at around 114 GeV from �Z with a relative rate of

about 1:3 compared with the �� line at 130. Fermi shouldbe able to test the �-ray signature and, for small masssplitting m�� �m� � 100 keV, the MiDM scenario also

predicts collision rates with nuclei that can now be testedat direct detection experiments. The production rates incolliders are below the current sensitivity of the LHC forthermal cross sections but can exclude some regions ofparameter space where this particle constitutes only afraction of the total dark matter. The case of the thermalWIMP constituting all of the dark matter should be ob-servable in the near future. We showed that, given thescaling of the different quantities involved, the concord-ance is in fact independent of the dipole strength and ismaintained even with an increased dipole strength whereMiDM forms only a fraction of the total DM. Model-dependent corrections outside of the effective theory canchange this result, however. Moreover, constraints fromcolliders place an ultimate limit on such an increase in thedipole to be no more than Oð10Þ.In the case of RayDM it is also possible to simultaneously

achieve the right relic abundance as well as rates in the rangenow explored by gamma-ray observations. But, in contrastwith MiDM, it favors stronger coupling to the SUWð2Þvector bosons than to hypercharge. If RayDM is describingonly the interactions with photons, however, and freezes outthrough some other channel, coupling to hypercharge alonegives a good description of the data. RayDM predicts a ratioof �Z to �� of 1:5 when coupling to hypercharge domi-nates, or about 5:2 in the more likely case of dominantcoupling to SUWð2Þ. Unfortunately, the direct detectionprospects in this case are gloomy as the collision rateswith nuclei due to two-photon exchange are much too small.In contrast, this scenario offers interesting phenomenologyin colliders including monophoton, mono-Z, and mono-Wsignatures with rates that can now be probed at the LHC.There are a few interesting variations on the scenarios

we have discussed. A particularly natural scenario isMiDMþ RayDM, where the �� signal is naturallyboosted in an MiDM model by the presence of an addi-tional hypercharge Rayleigh operator. Such an operator isgenerally present and would be expected to often dominatethe �� signal from these models.An alternative possibility is that some amount of

hypercharge-dominated RayDM is just a subdominantcomponent of the dark matter. Since the density scales as� h�vi�1

ann, the overall rate scales as 2h�viann �h�vi�1

ann. Thus, rather than having all dark matter annihilateto �� with a cross section h�viann � 3� 10�27 cm3 s�1,we could have a cross section �10� h�vithermal and yieldthe claimed �� signal from a subdominant component ofdark matter.While a number of opportunities exist to distinguish the

MiDM scenario from a RayDM scenario, there is anotherimportant difference: in RayDM, in particular when thedominant operator is ��W��W

��, there is a sizable

FIG. 9 (color online). A plot of the collider constraints on theproduction cross section of a WIMP pair coming from CMSmonojet search [30] (upper solid line—red) and CMS mono-photon search [31] (dashed red line). The constraints from CDFmonojet search [44] are only slightly weaker compared with theCMS results. In the lower solid line (black) we plot the expectedproduction cross section in the case of MiDM with a small masssplitting and with a dipole strength 10 times larger than �thermal.

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hadronic annihilation channel (via W’s and Z’s) comparedto ��. In contrast, for MiDM, the f �f channel is not presentin the late Universe as it is only present for ��� annihila-tions rather than ��. Limits on the continuum photonemissions such as those from dwarf galaxies [32,33] orthe Galactic center [34–36] could potentially distinguishthese scenarios.

Ultimately, if a Majorana dark matter interacts signifi-cantly with light, there are a number of conclusions thatcan be drawn right away. While direct detection signalsrequire a nearby state, collider signatures do not. The era ofdata—approaching dark matter with direct, indirect andcollider experiments—may be on the verge of revealingits nature.

ACKNOWLEDGMENTS

We would like to thank Maxim Pospelov for veryuseful discussions. N.W. is supported by NSF GrantNo. 0947827. I. Y. is supported in part by funds from theNatural Sciences and Engineering Research Council(NSERC) of Canada. Research at the Perimeter Instituteis supported in part by the Government of Canada throughIndustry Canada and by the Province of Ontario throughthe Ministry of Research and Information (MRI).

APPENDIX A: ELASTIC SCATTERING DUE TOTWO-PHOTON EXCHANGE

In this Appendix we derive the amplitude for the elasticscattering of the WIMP on the nucleus due to two-photonexchange, Eq. (24). We begin by noting that there areseveral separate scales in the problem: �R, the high scaleassociated with the Rayleigh operator; mN and m�, the

masses of the nucleus and the WIMP, respectively; q2,the momentum exchange, approximately 100 MeV; RN ,the nuclear coherence size approximately 100 MeV; q0,ER, and �v2, the kinetic energies involved, approximately10 KeV. By itself this diagram is logarithmically divergent,but inclusion of the charge form factor provides a naturalcutoff at a momentum scale around R�1

N . Therefore, themomenta running in the loop are nonrelativistic and weevaluate this diagram using known techniques from heavyquark effective theory [37]. In Appendix B we show thatusing this technique one can recover the results obtained inRef. [38] where the second-order Born cross section wasused to calculate the elastic channel of usual iDM [39]. Webegin by writing the momentum of the intermediate nu-cleus in the usual velocity expansion,

P ¼ mNvþ ~P; (A1)

where v ¼ ð1; ~vÞ is the 4-velocity of the nucleus. Thepropagator for the nucleus can be approximated as

6PþmN

P2 �m2N

� 1~P vþ i0

1þ 6v2

; (A2)

where ð1þ 6vÞ=2 is the projector onto the two large com-ponents of the 4-spinor.9 The projector causes the QEDvertex of the nucleus to simplify to iZev� instead of theusual iZe��. This embodies the fact that, in the nonrela-tivistic limit, the polarization of charged particles does notchange under the exchange of a photon. Given the above,the amplitude associated with this diagram at leading orderin the velocity is given by

iM ¼ ð �uðk0ÞuðkÞÞI��ðq2Þð �uðp0Þ���ðvÞuðpÞÞ; (A4)

with

���ðvÞ ¼�v�2

�1þ 6v2

�v�2

�; (A5)

and

I�2�2ðq2Þ ¼ e2Z2g��

�3R

Z d4l

ð2 Þ4� �g�1�2

ðlþ q=2Þ2�

�� �g�1�2

ð�lþ q=2Þ2�ðð�l2 þ q2=4Þg�1�1

� ðlþ q=2Þ�1ð�lþ q=2Þ�1Þ� 1

ð~pþ lþ q=2Þ vþ i0: (A6)

Here I�2�2ðq2Þ has dimensions of inverse mass square.

Since we are interested only in the leading order in thevelocity only the timelike indices are important and weobtain

I00ðq2Þ ¼e2Z2g��

�3R

Z d4l

ð2 Þ4l2 � q2=4

D

� F

���������lþ q

2

���������F

����������lþ q

2

���������: (A7)

Here, boldface letters denote 3-vectors, and we includedthe charge form factor FðjqjÞ by hand. The denominator is

D¼ðlþq=2Þ2ð�lþq=2Þ2ð~p0þ l0þq0=2þ i0Þ: (A8)

In order to allow for exact evaluation of this integral wechoose to work with the Helm form factor, which is afunction of the momentum exchange q:

FðjqjÞ ¼ e�q2=Q20 : (A9)

Here Q0 ¼ffiffiffi6

pR�1N , and the nuclear radius is (see for

example the excellent review by Salati [40])

RN ¼ fm� ð0:3þ 0:89A1=3Þ: (A10)

9In the diagram above, P ¼ pþ lþ q=2 and since ~P� l� qwe can write

6pþ6 lþ 6q=2þmN

ðpþ lþ q=2Þ2 �m2N

� 1

ðpþ lþ q=2Þ vþ i0

1þ 6v2

: (A3)

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Neglecting the overall constant in front, the integral ofEq. (A7) can now be written as

I00ðq2Þ/Z d4l

ð2 Þ4l2�q2=4

D�exp

��l2þq2=4

Q20

�: (A11)

The denominatorD, given in Eq. (A8), is a factor of severalseparate propagators that can be combined together usingthe usual Feynman parameter together with an heavy quarkeffective theory (HQET) parameter with the dimensions ofenergy

1

D¼Z 1

0dxZ 1

0dE

2

ðEð~p02 þ l0 þ q0=2Þ þ l2 þ q2=4þ ð1� 2xÞl qþ i�Þ3 : (A12)

Shifting the momentum variable l� ! l�þ12 ðð1� 2xÞq� þ Eg0�Þ we can write the integral as

ZdxdE

d3l

ð2 Þ3Z dl0

2

N

ððl0Þ2 � �þ i�Þ3

� exp

�� l2 � ð1� 2xÞl qþ ð2� 4xþ 4x2Þq2=4

Q20

�;

where

N ¼ 2ðl2 � ð1� 2xÞl qþ xðx� 1Þq2Þ; (A13)

� ¼ l2 þ E2

4þ xðx� 1Þq2 � Eð~p0

2 þ xq0Þ: (A14)

The mixing term in� can be neglected as it is always muchsmaller than the other two terms. Either way, the integralover l0 can be done exactly and yields

Z 1

�1dl0

2

1

ððl0Þ2 ��þ i�Þ3 ¼ � 3i

16�5=2: (A15)

The dependence of the form factor on l q ¼ jljjqj coscauses the integration over d3l to be slightly more compli-cated than usual. We proceed by first doing the integralover cos followed by the integral over the dimensionalFeynman parameter E. We are left with two integrals, oneover the Feynman parameter x and the other over the radialcomponent of the spatial momentum jlj. Defining thedimensionless variables ~l ¼ jlj=Q0 and ~q ¼ jqj=Q0 wearrive at the result quoted in the text in Eq. (24):

iM ¼ i�Z2g��4

Q0

�3R

F�jq2jQ2

0

��uðk0ÞuðkÞ �uðp0ÞuðpÞ; (A16)

where

F ð~qÞ ¼ 4

Z 1

0dxZ 1

0d~l

~l2

ð~l2 þ ð1� xÞx~q2Þ2

� exp

��~l2 � ~q2

�1

2� xþ x2

���coshðð1� 2xÞ~l ~qÞ

�~l2 � ð1� xÞx~q2 þ 1

ð1� 2xÞ~l ~q sinðð1� 2xÞ~l ~qÞ�: (A17)

APPENDIX B: DERIVATION OF THE ELASTICCHANNEL OF IDM USING HQET

The inelastic dark matter scenario of Ref. [39] involves adark matter state � that interacts with normal matter onlythrough a transition involving an excited state �� separatedin mass by �M. Reference [38] considered the particularcase where the interaction with the standard model isthrough a new massive U’(1) vector boson that kineticallymixes with hypercharge, a Holdom boson [41]. They com-puted for the first time the contribution to the elasticchannel from the second-order diagram shown in Fig. 10in the case where the mass splitting is much larger than thekinetic energy available, �M * MeV. In what follows, wereproduce this result using the HQETmethods discussed inthe text and Appendix A. For simplicity we ignore thecharge form factor.

Writing the four-momentum of the WIMP as k ¼m�uþ ~k with u ¼ ð1; ~uÞ being the 4-velocity, we can

express the momentum of the excited state as

K ¼ k� l� q=2 ¼ m�uþ ~k� l� q=2: (B1)

Since the mass splitting,�M is greater than the kineticenergy, the fermionic propagator of the excited state canbe written as

6K þm��

K2 �m2��

� 1

�M

1þ 6u2

: (B2)

The propagator for the nucleus is as given in Eq. (A2).Replacing �� by 4-velocities v� in the vertices and con-sidering the leading order term in the velocity expansionthe amplitude is given by

iM ¼ ð �uðk0ÞuðkÞÞIðq2Þð �uuðpÞÞ; (B3)

with

Iðq2Þ ¼ 2� 16 2�2Z2��0

�M

Z d4l

ð2 Þ41

D: (B4)

Here the factor of 2 arises from a second diagramwhere theA0 lines cross and which contributes equally at this order.Here � is the kinetic mixing parameter, and �0 ¼ e02=4 isthe U’(1) charge. D is the same denominator as previouslyconsidered and presented in Eq. (A8) except that themassless photon propagators are replaced by a massive

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propagator ðq2 þ i�Þ�1 ! ðq2 �m2A0 þ i�Þ�1. Following

the same steps as in Appendix A, we introduce theFeynman parameters x and E to combine the denominator,shift the loop momentum l� ! l� þ 1

2 ðð1� 2xÞq� þEg0�Þ, and then integrate over dl0:

Iðq2Þ ¼ � 3i

4

�16 2�2Z2��0

�M

�Z 1

0dxZ 1

0dE

Z d3l

ð2 Þ3

� 1

ðl2 þ �0Þ5=2 : (B5)

Here �0 ¼ E2=4 þ xðx � 1Þq2 þ m2A0 � Eð~p0 þ xq0Þ.

Neglecting the term linear in E as before, all the integralscan easily be done to yield

IðjqjÞ ¼ � i

4

�16 2�2Z2��0

�M

�

� Arctanðjqj=2mA0 Þjqj !mA0!0 2 2Z2��0�2

jqj�M : (B6)

The differential cross section in the center-of-mass frame issimply

d�

d�¼ �2

4 2I2ðjqjÞ

¼4�2�02Z4�4�2

q2�MArctan

� jqj2mA0

�2 !mA0!0 2��0Z2�2�2

q2�M;

(B7)

which reproduces the result obtained in Ref. [38] in theappropriate limit.

APPENDIX C: DIFFERENTIAL CROSS-SECTIONFORMULAS FOR MIDM AND RAYDM

PRODUCTION AT COLLIDERS

In this Appendix we give the formulas for the productioncross section and distributions for the process f �f ! ���Vwhere V ¼ �, Z, W� through the Rayleigh operators ofEq. (3) as well as related processes in UV completions ofRayDM. A convenient way of presenting the results isobtained by treating the ��� system as having four-momentum p4 and mass p2

4. The 2 ! 3 phase-space factorcan then be written as

dPS3ðpfp �f!pVþp ��þp�Þ

¼dPS2ðpfp �f!pVþp4Þ�dp24

2 �dPS2ðp4!p ��þp�Þ:

(C1)

Averaging (summing) over initial (final) polarization forthe process shown in Fig. 6(b), and neglecting the incom-ing particles’ masses, the matrix element squared is

1

4

Xpol

jMj2 ¼ ðg2v þ g2aÞg2VVsp24

�6R

1

2ðs�m2VÞ2

� ½a2Sð1

� 4m2�=p

24Þðm4

V þ ðs� p24Þ2 þ 2m2

Vðs� tÞþ 2ðs� p2

4Þtþ 2t2Þ þ a2Aðm4V þ ðs� p2

4Þ2þ 2ðs� p2

4Þtþ 2t2 � 2m2Vðsþ tÞÞ�; (C2)

where gv;a are the vector and axial couplings of V to �ff,respectively, gVV are defined in Eq. (16), and theMandelstam variables s ¼ ðpf þ p �fÞ2, t ¼ ðpf � pVÞ2,and u ¼ ðpf � p4Þ2 are defined as usual [42]. Here we

kept explicit the separate contributions from the scalar���FF (axial ���5�F ~F) piece by preceding it with aS(aA). This separation will prove useful below when discus-sing the corresponding formulas in the UV completionof RayDM with heavy scalars. The integral overdPS2ðp4 ! p �� þ p�Þ is straightforward and can be done

in its entirety since the squared amplitude in Eq. (C2)contains no dependence on the ��� system’s angular dis-tribution. The integral over the azimuthal angle ofdPS2ðpfp �f ! pV þ p4Þ can also be done and the differ-

ential cross section is then given by

d2�ðf �f ! ���VÞd cosdp2

4

¼ ðg2v þ g2aÞg2VVp24

2048 3�6Rðs�m2

VÞ2ð1þ cos2Þ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4m2

�

p24

vuut ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

�1;m2

V

s;p24

s

�s �a2Sð1� 4m2

�=p24Þ�m4

V þ ðp24 � sÞ2

þ 2m2V

��3� cos2

1þ cos2

�s� p2

4

��þ a2Aðm4

V þ ðs� p24Þ2 � 2m2

Vðp24 þ sÞÞ

�; (C3)

FIG. 10. The second Born amplitude for the process �N !�N through an intermediate excited state ��. A second diagramwith the A0 lines crossed contributes the same as this one.

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where is the angle between the incoming fermion f andthe vector V in the center-of-mass frame, and �ðx; y; zÞ ¼x2 þ y2 þ z2 � 2xy� 2yz� 2zx is the usual kinematicfunction. The angular region is �1 � cos < 1 andthe mass of the ��� system is in the range 4m2

� < p24 <ð ffiffiffi

sp �m2

VÞ2. The photon case is particularly simple andyields

d2�ðf �f! ����Þdcosdp2

4

¼ �g2��

512 2

p24ð1þ cos2Þ

�6R

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4m2

�

p24

vuut ða2A þa2Sð1� 4m2�=p

24ÞÞ�1�p2

4

s

�3=2

:

(C4)

This can be integrated exactly to yield the total crosssection that is quoted in Eq. (28) for the case where

ffiffiffis

p m�. What is often of more interest is the transverse mo-mentum distribution of the vector boson. The transversemomentum in the center-of-mass frame is given by pT ¼pV sin and the momentum of the vector in the center-of-mass frame is pV ¼

ffiffis

p2 �1=2ð1; m2

V

s ;p24

s Þ. This can be usedto obtain the differential distribution

d�

dpT

¼Z

dp24d cos

d2�

d cosdp24

�ðpTðcos; p24Þ � pTÞ:

(C5)

Needless to say, for hadronic colliders such as the LHC andthe Tevatron these expressions have to be convolvedagainst the appropriate parton distribution functions. Wehave verified the formulas above against the MADGRAPH 4package [43].

Similarly, in the case of heavy scalars, by putting the ���momentum on-shell p2

4 ¼ m2s we obtain the matrix-

element squared for the process f �f ! �s through �=Z:

1

4

Xpol

jMj2 ¼ �q2f

�2UV

t2 þ u2

s� ðg2�� þ 2g��g�Zvf�ðsÞ

þ g2�Zðv2f þ afÞ2�2ðsÞÞ: (C6)

Here s, t, and u are the usual Mandelstam variables. Use ofthe relation

d�

dpT

¼0B@pT

p

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 � p2

T

q1CA d�

d cos; (C7)

leads to the differential cross section given by Eq. (31).

Here p ¼ffiffis

p2 ð1� m2

s

s Þ is the momentum of the photon in the

center-of-mass frame. Similar expressions hold for theother distributions involving Z and W�.Finally, in the case of MiDM, the production cross

section is given by

d�ðf �f ! ���Þd cos

¼ �q2f�2�

8

�1þ 2vf

�Z

��

�ðsÞ

þ ðv2f þ a2fÞ �

�2Z

�2�

�2ðsÞ�

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�

�1;m2

�

s;m2

��

s

�vuut ��1� �M2

s

�

��sin2þ ðm� þm�� Þ2

sð1þ cos2Þ

�;

(C8)

where is the scattering angle in the center-of-mass frame,and v�;Z (a�;Z) is the vector (axial) coupling of the corre-

sponding vector boson to the incoming fermions.

[1] J. Bagnasco, M. Dine, and S. D. Thomas, Phys. Lett. B320, 99 (1994).

[2] M. Pospelov and T. ter Veldhuis, Phys. Lett. B 480, 181(2000).

[3] K. Sigurdson, M. Doran, A. Kurylov, R. R. Caldwell, andM. Kamionkowski, Phys. Rev. D 70, 083501 (2004).

[4] S. Gardner, Phys. Rev. D 79, 055007 (2009).[5] E. Masso, S. Mohanty, and S. Rao, Phys. Rev. D 80,

036009 (2009).[6] W. S. Cho, J.-H. Huh, I.-W. Kim, J. E. Kim, and B. Kyae,

Phys. Lett. B 687, 6 (2010).[7] H. An, S.-L. Chen, R.N. Mohapatra, S. Nussinov, and

Y. Zhang, Phys. Rev. D 82, 023533 (2010).

[8] S. D. McDermott, H.-B. Yu, and K.M. Zurek, Phys. Rev.D 83, 063509 (2011).

[9] S. Chang, N. Weiner, and I. Yavin, Phys. Rev. D 82,125011 (2010).

[10] T. Banks, J.-F. Fortin, and S. Thomas, arXiv:1007.5515.[11] E. Del Nobile, C. Kouvaris, P. Panci, F. Sannino, and

J. Virkajarvi, J. Cosmol. Astropart. Phys. 08 (2012) 010.[12] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd, T.M.

Tait, and H.-B. Yu, Nucl. Phys. B844, 55 (2011).[13] H. K. Dreiner, H. E. Haber, and S. P. Martin, Phys. Rep.

494, 1 (2010).[14] J. Kopp, T. Schwetz, and J. Zupan, J. Cosmol. Astropart.

Phys. 02 (2010) 014.

HOW DARK ARE MAJORANAWIMPs? SIGNALS FROM . . . PHYSICAL REVIEW D 86, 075021 (2012)

075021-15

[15] J.-F. Fortin and T.M. Tait, Phys. Rev. D 85, 063506(2012).

[16] G. Duda, G. Gelmini, and P. Gondolo, Phys. Lett. B 529,187 (2002).

[17] R. Bernabei et al., Eur. Phys. J. C 67, 39 (2010)10.1140/epjc/s10052-010-1303-9.

[18] T. Bringmann, X. Huang, A. Ibarra, S. Vogl, and C.Weniger, J. Cosmol. Astropart. Phys. 07 (2012) 054.

[19] C. Weniger, J. Cosmol. Astropart. Phys. 08 (2012) 007.[20] E. Tempel, A. Hektor, and M. Raidal, arXiv:1205.1045.[21] M. Su and D. P. Finkbeiner, arXiv:1206.1616.[22] A. Rajaraman, T.M. Tait, and D. Whiteson, J. Cosmol.

Astropart. Phys. 09 (2012) 003.[23] S. Kim et al., Phys. Rev. Lett. 108, 181301 (2012).[24] Y. Bai, P. J. Fox, and R. Harnik, J. High Energy Phys. 12

(2010) 048.[25] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd,

T.M. Tait, and H.-B. Yu, Phys. Lett. B 695, 185(2011).

[26] J. Goodman, M. Ibe, A. Rajaraman, W. Shepherd,T.M. Tait, and H.-B. Yu, Phys. Rev. D 82, 116010(2010).

[27] Y. Bai and T.M. Tait, Phys. Lett. B 710, 335 (2012).[28] I.M. Shoemaker and L. Vecchi, Phys. Rev. D 86, 015023

(2012).[29] S. Kathrein, S. Knapen, and M. J. Strassler, Phys. Rev. D

84, 015010 (2011).

[30] S. A. Malik and f. t. C. Collaboration (Collaboration forthe CMS), arXiv:1206.0753.

[31] S. Chatrchyan et al. (CMS Collaboration), Phys. Rev. Lett.108, 261803 (2012).

[32] A. Geringer-Sameth and S.M. Koushiappas, Phys. Rev.Lett. 107, 241303 (2011).

[33] M. Ackermann et al. (Fermi-LAT Collaboration), Phys.Rev. Lett. 107, 241302 (2011).

[34] W. Buchmuller and M. Garny, J. Cosmol. Astropart. Phys.08 (2012) 035.

[35] T. Cohen, M. Lisanti, T. R. Slatyer, and J. G. Wacker,arXiv:1207.0800.

[36] I. Cholis, M. Tavakoli, and P. Ullio, arXiv:1207.1468[Phys. Rev. D (to be published)].

[37] A. Grozin, arXiv:hep-ph/0008300.[38] B. Batell, M. Pospelov, and A. Ritz, Phys. Rev. D 79,

115019 (2009).[39] D. Tucker-Smith and N. Weiner, Phys. Rev. D 64, 043502

(2001).[40] P. Salati, Proc. Sci., CARGESE2007 (2007) 009.[41] B. Holdom, Phys. Lett. 166B, 196 (1986).[42] M. E. Peskin and D.V. Schroeder (to be published).[43] J. Alwall, P. Demin, S. de Visscher, R. Frederix, M.

Herquet, F. Maltoni, T. Plehn, D. L. Rainwater, and T.Stelzer, J. High Energy Phys. 09 (2007) 028.

[44] T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett.108, 211804 (2012).

NEAL WEINER AND ITAY YAVIN PHYSICAL REVIEW D 86, 075021 (2012)

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