Models of Dark Matter Coupled to Light Mediators

Matt Reece Harvard University

April 21, 2017

Based partly on: 1509.03628 “Continuum-mediated dark matter - baryon scattering” with A. Katz and A. Sajjad

1303.1521 “Double-disk dark matter” with J. Fan, A. Katz, and L. Randall

BSM Physics and LatticeWhat would I like to see from the lattice? A partial wish list:

- Clearer interpretation of hadronic EDM searches, especially given recent progress and expected future results.

- Anything that sheds light on B meson anomalies

- Boundary of the conformal window, spectroscopy in this region of QCD-like theories?

- But because we know our universe contains dark matter, interesting to think about dark sector as location for exotic types of physics. Strongly coupled dark sectors?

Aside: EDMs

h0|qgsGµ⌫�µ⌫q|0i = �(0.8± 0.2) GeV2h0|qq|0i

If you start trying to understand how searches for an EDM in mercury are turned into constraints on quark CEDMs you run into people quoting results like:

from papers by V.M. Belyaev and I.B. Ioffe in 1982 based on QCD sum rules. And it gets worse from there.(*)

So one thing I’d love to see is accurate translation of quark CEDMs into CP-violating pion-nucleon couplings. But for now the nuclear physics uncertainties are even larger, so maybe not the most urgent thing.

(* no offense to those who have made heroic efforts to do the calculations! but uncertainties are huge and approximations not well-controlled.)

DM with Long-Range Forces

A Dark Sector?

“Minimal” probably isn’t the first adjective that would come to mind to describe the Standard Model. We still don’t know “who ordered that?”

𝝌?

Why Light Mediators for Dark Matter?

Possible hint from data:Kaplinghat, Tulin, Yu 1508.03339

(gray points are simulated—others are data)

If drop in cross section at v <~ 1000 km/s taken seriously, could hint at break in propagator: m� ⇠ 10�3mDM

Light Mediators

• Naturalness: the Higgs puzzles us. If DM has a massive, but light, scalar or vector mediator (with associated Higgs mechanism), reinforces the hierarchy problem.

• If mediator is very light (compared to DM kinetic energy), can radiate away energy via mediator.

• Different signal shapes in direct detection.

• Interesting interplay with cosmology (e.g. dark radiation if massless, constraints from decays if not)

A few of the reasons why I think it’s interesting to think about scenarios where DM interacts through light force-carriers:

Cooling and StructureSpiral galaxies, like the Milky Way, have a flattened shape.

Scale radius ~ 3 kpc Scale height ~ 300 pc Why this anisotropic shape?

The answer lies in cooling processes. Particles (protons, electrons, atoms) scatter and emit photons that carry away kinetic energy but not (much) angular momentum. Shrink but keep spinning: forced to form a disk.

DM coupled to dark photonsIf dark matter interacts through long-range forces, it can cool through processes like bremsstrahlung, Compton scattering, recombination, collisional excitation of dark atoms, ….

Fan, Katz, Randall, MR 1303.1521 and followups

t coolbt Univ

t eqbt cool

Not Ionized

WDDDMthermal b 0.05 WDM

10-6 10-5 10-4 10-3 10-210-4

10-3

10-2

0.100

mC@GeVD

aD

e = 0.05, mX = 100 GeV, nX = nC = 3.3 ¥ 10-6 cm-3

t coolb tun

ivt eqbt cool

Not Ionized

WDDDMthermal b 0.05 WDM

10-6 10-5 10-4 10-3 10-210-4

10-3

10-2

0.100

mC@GeVD

aD

e = 0.05, mX = 1 GeV, nX = nC = 3.3 ¥ 10-4 cm-3

t eqbt cool

t coolbt univ

Not Ionized

WDDDMthermal b 0.05 WDM

10-6 10-5 10-4 10-3 10-210-4

10-3

10-2

0.100

mC@GeVD

aD

e = 0.05, mX = 100 GeV, nX = nC = 7.3 ¥ 10-5 cm-3

t coolb tun

iv

t eqbt cool

Not Ionized

WDDDMthermal b 0.05 WDM

10-6 10-5 10-4 10-3 10-210-4

10-3

10-2

0.100

mC@GeVD

aD

e = 0.05, mX = 1 GeV, nX = nC = 7.3 ¥ 10-3 cm-3

Figure 5: Cooling in the (mC ,↵D) plane. The purple shaded region is the allowed region that coolsadiabatically within the age of the universe. The light blue region cools, but with heavy and light particlesout of equilibrium. We take redshift z = 2 and TD = T

CMB

/2. The two plots on the left are for mX = 100

GeV; on the right, mX = 1 GeV. The upper plots are for a 110 kpc radius virial cluster; the lower plots,a 20 kpc NFW virial cluster. The solid purple curves show where the cooling time equals the age of theuniverse; they have a kink where Compton-dominated cooling (lower left) transitions to bremsstrahlung-dominated cooling (upper right). The dashed blue curve delineates fast equipartition of heavy and lightparticles. Below the dashed black curve, small ↵D leads to a thermal relic X, ¯X density in excess of the Oortlimit. To the upper right of the dashed green curve, BXC is high enough that dark atoms are not ionizedand bremsstrahlung and Compton cooling do not apply (but atomic processes might lead to cooling).

18

The next thing we have to worry about is a 3-loop kinetic mixing (Fig. 6). However, Ithink this should give zero just because Tr(Q) = 0.

�D

� �D

�D�

�D

�D

� �D�

Figure 8: Four-loop kinetic mixing

References

4

Figure 11: Possible 4-loop contributions to kinetic mixing when there are particles charged under bothU(1)s. At left: this diagram is proportional to Tr(Q3Q5

D). The condition that TrQD (Q3

) = 0, i.e. thatthe trace of visible charged cubed vanishes in the sector with any given dark charge, is sufficient to makethis diagram vanish. At right: this diagram is proportional to Tr(Q2Q2

D)Tr(QQ3

D). The condition thatTrQ(Q3

D) = 0 is sufficient for it to be zero.

two fermion loops, like the right-hand plot in Fig. 11, but these are set to zero by the same traceconstraints. These are anomaly-like constraints in the sense that they demand certain vanishingtraces, but they are much more restrictive than anomalies: they apply to scalars in the loop as wellas to fermions, and they restrict the trace of dark charges in the sector with fixed visible charge andvice versa. Although these conditions are restrictive, they could potentially be satisfied, and couldforbid kinetic mixing up to 5 loops. Such models would be consistent with the bound in Eq. 40.

Another possibility is that U(1)Y or U(1)D is embedded in a nonabelian group. For example,suppose that U(1)D arises from a group SU(2)D broken by an adjoint Higgs �D. If the lightestparticle charged under both SU(2)D and U(1)Y has mass M > h�Di, we expect that the mixingarises from a dark S-parameter operator [131]

gDg0

16⇡2

1

MTr(�

aDW a

Dµ⌫)Bµ⌫ . (41)

This is consistent with the bound <⇠ 10

�9 if, for example, SU(2)D is broken at the weak scale andall particles charged under both groups have masses above 10

9 GeV. In such a scenario, the threatthat GUT- or string-scale physics renders the model inconsistent with the bounds can be avoided.

Another distinctive scenario is to consider that an unbroken nonabelian dark force remains atlow energies. In this case kinetic mixing with the photon is completely impossible. One might worrythat such a force would confine and prevent long-range interactions. However, with the relativelysmall values of ↵D at which cooling can be effective, the temperature of our dark plasma in galaxieswill be too high for confinement to occur. Nonabelian sectors that don’t confine because they flowto infrared fixed points are also a possibility, but in this case we would need more fields with darkcharge and the BBN bounds on the number of light degrees of freedom would become more severe.

One final possibility is that the U(1)D is not exact and the gauge boson is not massless.

A.2 Other Interactions Between C, C̄ and the SM Plasma

In models with indirect detection signals, we assume that the heavy dark sector particles X and ¯Xcan annihilate to Standard Model particles; for instance, we can consider the process X ¯X ! ��.If the light particles C have couplings similar to those of X, then the inverse process �� ! C ¯Ccan produce relativistic C particles at late times. This may be in conflict with the bound on lightdegrees of freedom discussed in Sec. 3.1. Unlike kinetic mixing, this constraint would arise from a

30

“DDDM”: Dark QED. Theoretically, why not? Only challenge is suppressing kinetic mixing:

“Partially Interacting DM”: fraction of DM can do something different than the bulk of it.

(earlier work on dark plasmas: 0810.5126 by Ackerman, Buckley, Carroll, Kamionkowski; 0905.3039 by Feng, Kaplinghat, Tu, Yu)

Dark QED sectorCosmological constraints from dark acoustic oscillations (Cyr-Racine, de Putter, Raccanelli, Sigurdson 1310.3278)

Dark disk density bounds from Milky Way (MR, Randall 1403.0576; Kramer, Randall 1603.03058, 1604.01407)

Generic expectation if DM interacts with long range forces: dark matter-dark radiation interplay in early universe.

Allow ~2% of all DM to be dissipative

Is It Directly Detectable?We could detect this kind of dark matter through the dark disk. Could we detect it directly? There is a tension: a big coupling to the SM tends to thermalize, leading to too much dark radiation.

(Cyr-Racine, de Putter, Raccanelli, Sigurdson 1310.3278)

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‡

0.35 0.40 0.45 0.50 0.55 0.60 0.65

-56

-54

-52

-50

-48

-46

-44

x = TdarkêTSM

log 10@sê

cm2 D

Direct Detection Cross Section vs Dark Radiation Temperature

SM g*sHTLBSM g*sHTL

Best LUX BoundTdec = 1 GeV

Tdec = 10 GeV

Tdec = 100 GeV

Figure 1: The tension between the cosmological preference for a small dark radiation temperature ratio⇠ and a detectable dark matter–nucleon direct detection cross section � . Although physics beyond theStandard Model can modify the count of degrees of freedom and bend ⇠ toward smaller values, as shownby the dashed orange BSM curve, this can only happen at temperatures above about 100 GeV, which alreadycorrespond to undetectably small cross sections.

To make the tension clearer, we combine the equations 9 and 17 to obtain

� <⇠µ2

n

⇤4eff

⇡ 2.3 ⇥ 10�56 cm2✓

10 TeVTdec

◆3 ✓ g⇤(Tdec)352

◆1/2. (18)

For a given decoupling temperature, we can compare this cross section � with the dark radiationtemperature ratio ⇠ from eq. 3. We use a tabulation of g⇤(T) and g⇤s(T) in the Standard Modelfrom the DarkSUSY code [80], which is particularly useful for the regime around the QCD scale.The result is shown in Figure 1. The solid orange curve shows the attainable ⇠ including onlyStandard Model degrees of freedom. The dashed orange curve allows for new physics beyondthe Standard Model (BSM) to add to g⇤s(T). We have chosen to add a new degree of freedom forevery 10 GeV energy increase above 100 GeV:

gBSM⇤ (T) =

(gSM⇤ (T) if T 100 GeV

gSM⇤ (T) + T�100 GeV

10 GeV if T > 100 GeV.(19)

This is a rapid growth in the number of states, but we see from Fig. 1 that it is hopeless: already,when the decoupling temperature is 100 GeV, the largest attainable cross section is around 10�50

cm2. To reach smaller values of ⇠ at cross sections that are near the current bounds, we wouldhave to significantly alter the density of states already between 1 and 10 GeV. Given that colliderexperiments have thoroughly explored this energy regime, this does not seem possible.

8

(MR, T. Roxlo 1511.06768)

However, can reconcile with nonthermal cosmology: e.g. late-time modulus decay.

Dark-Charged DMAgrawal, Cyr-Racine, Randall, Scholtz 1610.04611, 1702.05482Argue that ellipticity bounds are weaker than previously thought: 100% of DM can be charged under U(1) with surprisingly big couplings.

Buen-Abad, Marques Tavares, Schmaltz 1505.05432; Lesgourgues, Marques Tavares, Schmaltz 1507.04351

DM with SM SU(2) charge and a much more weakly coupled dark SU(N) charge. Affects large-scale structure, possibly in good ways.

Continuum-Mediated DM

General DM-SM ScatteringWe can parametrize the scattering of dark matter on ordinary matter in terms of 3 ingredients:

An SM operator or current. NR limit, classified by Dobrescu & Mocioiu ’06; adapted to DM context by Fan, MR, Wang ’10; related nuclear physics worked out systematically by Fitzpatrick, Haxton, Katz, Lubbers, Xu ’12

A DM current, possibly involving nontrivial dark matter form factors: Chang, Pierce, Weiner ’09; Feldstein, Fitzpatrick, Katz ’09. (Potential for lattice input for composite dark matter models.)

A mediator (could be contact interaction) between DM and SM. Light mediators have received some attention, but the most general mediators went unexplored.

Continuum-Mediated Direct Detection

with Andrey Katz and Aqil Sajjad, 1509.03628 (following up on an earlier remark in 1008.1591 with Fan and Wang)

possible ways to model a mass gap. We show that the direct detection rates are almostnot a↵ected by our assumptions about the modeling of the mass gap. We also briefly ad-dress basic cosmological concerns related to our scenario. In Sec. 3 we study continuummediators coupling to the SM Higgs portal [44–48]. It is a simple illustration of the generalidea, but we find that there is not a viable parameter space for the sort of direct detectionsignal we are interested in. In Sec. 4 we analyze the DM direct detection and collider con-straints in the case when the DM-nucleus interaction is mediated via an antisymmetric tensoroperator which couples to the SM via the hypercharge portal [49, 50]. This case realizes in-teresting continuum-mediated phenomenology in a parameter space compatible with variousconstraints. Finally in the last section we conclude. Some technical details are relegated tothe appendix.

2 The scenario: mass scales and kinematics

2.1 The basic picture

The majority of models of dark matter–baryon scattering considered so far take one of twoforms:

Light particle mediator: = J�(p, p� q) 1

q2JSM

(k, k + q).

(2)

Pointlike interaction: = J�(p, p� q)JSM

(k, k + q).(3)

In position space, these correspond to potentials V (r) / 1/r and V (r) / �(3)(~r), respec-tively. (These are point-particle idealizations and should be appropriately convolved withthe nuclear form factor and, if it exists, dark matter form factor.) We could also consider thecase of a massive mediator with mass m ⇠ q, which would correspond to a Yukawa potentialinterpolating between these two extremes. In this paper, we consider a di↵erent scenario, inwhich we exchange a continuum of light modes. One way to think of this is as the result ofcoupling to states of multiple light particles:

Continuum mediator: = J�(p, p� q)⇣

1

q2

⌘↵

JSM

(k, k + q).

(4)

Here the scale-invariant factor of (1/q2)↵ is a stand-in for more general possible behaviorof the intermediate continuum. In the simplest case, we could consider just a loop of twolight, non-interacting particles. More generally, the continuum could consist of multipleparticles that are themselves interacting, as suggested by the shaded region in the figure.

3

We’re used to:

Can also have continuum (loops, “unparticle,” RS2…)

possible ways to model a mass gap. We show that the direct detection rates are almostnot a↵ected by our assumptions about the modeling of the mass gap. We also briefly ad-dress basic cosmological concerns related to our scenario. In Sec. 3 we study continuummediators coupling to the SM Higgs portal [44–48]. It is a simple illustration of the generalidea, but we find that there is not a viable parameter space for the sort of direct detectionsignal we are interested in. In Sec. 4 we analyze the DM direct detection and collider con-straints in the case when the DM-nucleus interaction is mediated via an antisymmetric tensoroperator which couples to the SM via the hypercharge portal [49, 50]. This case realizes in-teresting continuum-mediated phenomenology in a parameter space compatible with variousconstraints. Finally in the last section we conclude. Some technical details are relegated tothe appendix.

2 The scenario: mass scales and kinematics

2.1 The basic picture

The majority of models of dark matter–baryon scattering considered so far take one of twoforms:

Light particle mediator: = J�(p, p� q) 1

q2JSM

(k, k + q).

(2)

Pointlike interaction: = J�(p, p� q)JSM

(k, k + q).(3)

In position space, these correspond to potentials V (r) / 1/r and V (r) / �(3)(~r), respec-tively. (These are point-particle idealizations and should be appropriately convolved withthe nuclear form factor and, if it exists, dark matter form factor.) We could also consider thecase of a massive mediator with mass m ⇠ q, which would correspond to a Yukawa potentialinterpolating between these two extremes. In this paper, we consider a di↵erent scenario, inwhich we exchange a continuum of light modes. One way to think of this is as the result ofcoupling to states of multiple light particles:

Continuum mediator: = J�(p, p� q)⇣

1

q2

⌘↵

JSM

(k, k + q).

(4)

Here the scale-invariant factor of (1/q2)↵ is a stand-in for more general possible behaviorof the intermediate continuum. In the simplest case, we could consider just a loop of twolight, non-interacting particles. More generally, the continuum could consist of multipleparticles that are themselves interacting, as suggested by the shaded region in the figure.

3

Continuum Mediators?If we really had something like a conformal sector—spectral weight extending all the way down to zero mass—then we have extra radiation coupling to DM and ordinary matter in the early universe, and run into constraints.

If instead we have a continuum of stuff only above ~5 or 10 MeV, we avoid having radiation d.o.f. at BBN.

So imagine either lots of independent narrow states beginning at 5 MeV, or a smooth continuum beginning above such a gap.

Continuum in Direct Detection?

quite stringent. The safest way to avoid these bounds is if our continuum of particles doesnot extend all the way down to zero mass and develops a mass gap; if the particles becomenonrelativistic before BBN, they are no longer dark radiation from the viewpoint of cosmo-logical bounds. For instance, our approximately conformal sector could confine at a scale ofat least a few MeV. This mass means that during BBN, the continuum of modes behaves asdark matter rather than dark radiation. We can also arrange that, after acquiring a mass,the continuum states simply decay before BBN, dumping their energy into the StandardModel plasma. For instance, if they couple to the Standard Model through dimension-sixoperators suppressed by a scale ⇤, they can have a lifetime

⌧ ⇠ ⇤4

m5

⇠ 7⇥ 10�7 sec

✓⇤

100 GeV

◆4

✓10 MeV

m

◆5

, (5)

easily decaying before BBN (although stable on collider timescales). More refined estimatescan be done for the particular models we discuss. In the particular case of the hyperchargeportal, we will discuss such estimates further in Sec. 4.2. It is also possible that deviationsfrom thermal equilibrium just before BBN could assist in circumventing the dark radiationbounds even if there is no mass gap (or the mass gap is much smaller than the BBN scale) [61].

2.3 The direct detection scale meets the BBN scale

Direct detection experiments search for nuclear recoil events. If a nucleus of mass mN recoilswith kinetic energy ER, the momentum transfer is q =

p2mNER and the incoming dark

matter velocity must have been at least vmin

= q2µ

where µ is the dark matter–nucleus reducedmass. A typical dark matter velocity in the galactic halo is v̄ = 220km/s, which (takingµ = 100 GeV for reference) can impart at most a momentum transfer of q ⇡ 147 MeV. Alow-threshold nuclear recoil might be taken as, for instance, ER = 2 keV for a silicon nucleusof mass ⇡ 26 GeV [62], corresponding to about a 10 MeV momentum transfer. Thus,the range of momentum transfers that are relevant for direct detection might be roughlyconstrued as

10 MeV <⇠ q <⇠ 400 MeV, (6)

with the typical momentum transfer of interest in the middle of this range and the exactdetails depending on the particular experiment. LUX, for instance, studies nuclear recoilsin xenon between about 3 keV and 25 keV [63], corresponding to 27 MeV <⇠ q <⇠ 78 MeV.

From these estimates we see that, if the continuum of modes mediating scattering acquiresa mass gap m ⇡ 10 MeV in order to avoid dark radiation bounds during BBN, this masswill be a subdominant correction to the two-point function hO

med

(q)Omed

(�q)i at the valuesof q that are most relevant for direct detection. The “mass-gap” solution to the BBN boundsu↵ers from a coincidence problem: we have no explanation for why the gap should fall inthe relatively small interval between the temperatures relevant for BBN and the momentarelevant for direct detection. (Particular models may o↵er solutions, but no general solutionis apparent.) But if we assume that T

BBN

<⇠ m <⇠ qexp

, we have interesting direct detection

5

The momentum transfers involved in detecting a weak-scale DM particle are something like

above our mass gap—and in fact the physics can be not too sensitive to the details of the gap because direct detection probes spacelike q rather than timelike q.

-20 -15 -10 -5 0

-1

0

1

2

3

q2/m2

Π[q2]

Two-Point Functions at Spacelike Momentum Transfer

0 5 10 15 200

1

2

3

4

5

6

q2/m2

Π[q2]

Smeared Spectral Function ρΔ(q2), Δ = 1/4

Figure 1: Four toy models for a mass gap in the two-point function of a dimension-2 scalaroperator. In every case, ⇧(q2) ! log(�q2) in the spacelike region q2 ⌧ 0. The four modelsare similar at spacelike momentum (left) but di↵er greatly for timelike momentum (right).Solid blue: the simple replacement log(�q2+m2). Dashed orange: the result for a loop of freemassive scalars. Dotted green: the Randall-Sundrum or “hard-wall” ansatz, characteristic ofconfinement at large ’t Hooft coupling. Dot-dashed red: the digamma function, a toy modelof QCD-like confinement.

These four toy models are illustrated in Fig. 1. The right-hand panel shows the smearedspectral function, which varies from a step function (in the simple log(�q2+m2) ansatz) to asmooth curve that turns on (the loop of massive scalars) to wildly bumpy curves characteristicof confining large-N theories with many narrow resonances. Despite these dramaticallydi↵erent spectral functions, the behavior for spacelike momentum, as shown in the left-handplot, is qualitatively similar in every case. In particular, the corrections to the asymptoticconformal answer when �q2 � m2 are very small. This gives us confidence that the behaviorof the direct detection cross section can be approximated with the simplest toy ansatz,q2 ! q2 �m2, even without a detailed model for the origin of the mass gap. Such an ansatzwill certainly lead to the correct qualitative physics, and will be quantitatively reasonableunless we probe too close to |q2| ⇡ m2.

Although we have only given examples for the special case d = 2, we expect that similarresults would be obtained for other dimensions. (Hard and soft wall extra dimensionaltheories can produce results for arbitrary operator dimension by varying the bulk mass;generalizing the loop ansatz in a well-motivated way appears more di�cult.)

3 Higgs Portal

First, let us briefly analyze the higgs portal for the continuum-mediated Dirac fermion DM.Because it involves a scalar operator, this case is formally the simplest. Although we will

9

Higgs PortalIf we start looking for low-dimension operators in the SM to couple to, this is an obvious place to start.

The Higgs coupling to renormalizable O produces a mass gap in the hidden sector.

For BBN we want the lightest massive scalars sourced by O to decay early enough:

Higgs Portal: Doesn’t Work!In short, this model is too predictive. Without tuning, the coupling sets a minimum size of the mass gap.

The same coupling sets the decay width.

We want the mass gap to be small, or we lose our interesting phenomenology; but if the width is too small, we have BBN trouble.

Might be room to evade problems with nonstandard cosmology pre-BBN, but another example works better.

Hypercharge PortalThe other gauge-invariant dimension two operator in the SM is the hypercharge field strength. Can couple to an antisymmetric tensor in the hidden sector:

dim(Oµ⌫) � 2

L =cB

⇤d�2Bµ⌫Oµ⌫

Unitarity implies that , with the critical dimension arising for field strengths of gauge bosons.

So this generalizes “dark photon” models.

Question: are there field theories where the dimension of a 2-form is near, but significantly larger than, 2?

Dark Sector CouplingWe can have fermionic dark matter coupled to the antisymmetric tensor, e.g. via

one can think of this as a q2-dependent magnetic dipole moment for the dark matter,

Similarly for electric dipole moment or anapole moment couplings:

Hypercharge Portal

10 20 30 400.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

ER [keV]

dR/dER[arbitraryunits]

131Xe, EDM coupling, no mass gap, mDM = 100 GeV

d = 2.3

d = 2.7

d = 3.4

10 20 30 400

2

4

6

8

ER [keV]

dR/dER[arbitraryunits]

19F, EDM coupling, no mass gap, mDM = 100 GeV

d = 2.3

d = 2.7d = 3.4

10 20 30 400.0

0.5

1.0

1.5

2.0

2.5

ER [keV]

dR/dER[arbitraryunits]

131Xe, MDM coupling, no mass gap, mDM = 100 GeV

d = 2.3

d = 2.7

d = 3.4

0 10 20 30 40 500

2

4

6

8

ER [keV]

dR/dER[arbitraryunits]

19F, MDM coupling, no mass gap, mDM = 100 GeV

d = 2.3 d = 2.7

d = 3.4

Figure 2: Event rate in arbitrary units for electric-hypercharge (top panel) and magnetic-hypercharge (bottom panel) portals for di↵erent values of d. Green, orange and brown linesstand for d = 2.3, 2.7, 3.4 respectively. The curves have been artificially scaled to intersectat a single point, because the goal is to convey shape information only. The isotope 131Xe isshown at left and 19F at right. Notice that, due to di↵erent behavior of nuclear form factors,the shapes can be quite di↵erent for di↵erent atoms. To guide the reader’s intuition for howthese compare to more familiar scenarios, in the upper-left figure we also show the recoilspectrum arising from the standard contact operator coupling to mass (�̄�N̄N) as a dashedgray line and the same operator with a massless mediator (�̄�N̄N/q2) as a dotted gray line.

For the EDM and the anapole hypercharge portals we get exactly the same expression for thee↵ective moment with an obvious replacement c

2

! c̃2

for the EDM and c2

! c̄2

,⇤2d�3 !⇤2d�2 for the anapole moment.

Because all the e↵ective moments depend in a non-trivial way on the momentum transferq2, we expect that the di↵erential event rates dR/dER will be modified compared to the“standard” DM dipole/anapole scenario. One can refer to the factor (q2)d�2 as an e↵ectiveDM form factor. However this form factor arises because of the non-trivial dynamics inthe mediation sector, rather than non-trivial structure of the DM. Of course these e↵ectiveDM form-factors multiply the nuclear form-factor in the direct detection picture, yieldingdistinctive event distributions in the direct detection experiments.

It is also worth noticing that the e↵ective DM form-factors will be further modified

13

dashed: contact interaction dotted: massless mediator

10 20 30 400.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

ER [keV]

dR/dER[arbitraryunits]

131Xe, EDM coupling, no mass gap, mDM = 100 GeV

d = 2.3

d = 2.7

d = 3.4

10 20 30 400

2

4

6

8

ER [keV]

dR/dER[arbitraryunits]

19F, EDM coupling, no mass gap, mDM = 100 GeV

d = 2.3

d = 2.7d = 3.4

10 20 30 400.0

0.5

1.0

1.5

2.0

2.5

ER [keV]

dR/dER[arbitraryunits]

131Xe, MDM coupling, no mass gap, mDM = 100 GeV

d = 2.3

d = 2.7

d = 3.4

0 10 20 30 40 500

2

4

6

8

ER [keV]

dR/dER[arbitraryunits]

19F, MDM coupling, no mass gap, mDM = 100 GeV

d = 2.3 d = 2.7

d = 3.4

Figure 2: Event rate in arbitrary units for electric-hypercharge (top panel) and magnetic-hypercharge (bottom panel) portals for di↵erent values of d. Green, orange and brown linesstand for d = 2.3, 2.7, 3.4 respectively. The curves have been artificially scaled to intersectat a single point, because the goal is to convey shape information only. The isotope 131Xe isshown at left and 19F at right. Notice that, due to di↵erent behavior of nuclear form factors,the shapes can be quite di↵erent for di↵erent atoms. To guide the reader’s intuition for howthese compare to more familiar scenarios, in the upper-left figure we also show the recoilspectrum arising from the standard contact operator coupling to mass (�̄�N̄N) as a dashedgray line and the same operator with a massless mediator (�̄�N̄N/q2) as a dotted gray line.

For the EDM and the anapole hypercharge portals we get exactly the same expression for thee↵ective moment with an obvious replacement c

2

! c̃2

for the EDM and c2

! c̄2

,⇤2d�3 !⇤2d�2 for the anapole moment.

Because all the e↵ective moments depend in a non-trivial way on the momentum transferq2, we expect that the di↵erential event rates dR/dER will be modified compared to the“standard” DM dipole/anapole scenario. One can refer to the factor (q2)d�2 as an e↵ectiveDM form factor. However this form factor arises because of the non-trivial dynamics inthe mediation sector, rather than non-trivial structure of the DM. Of course these e↵ectiveDM form-factors multiply the nuclear form-factor in the direct detection picture, yieldingdistinctive event distributions in the direct detection experiments.

It is also worth noticing that the e↵ective DM form-factors will be further modified

13

Operator dimensions determine shape of recoil spectrum in direct detection:

Massive TowerThe mediating operator sources composite states, which effectively behave like a tower of dark photon mediators.

(Previous related work based on KK towers or confining gauge groups: Kristian McDonald with David Morrissey in 2010, with Benedict von Harling in 2012; Jaeckel, Roy, Wallace 2014)

Oµ⌫

BBN bound on lifetimes:

Hence effective mixing of states:

PhenomenologyWe have a tower of dark photon-like states, all of which kinetically mix with hypercharge. Many experimental constraints, including beam dumps and supernova cooling.

10. 100. 1000. 10000.2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.810. 100. 1000. 10000.

Λ [GeV]

d

MDM on 131Xe: max. σpeff[cm2], mDM = 100 GeV

Dark photon bounds strongest: ϵV = 10-3

Γ(Z→CFT) bound strongest2×10-4 < ϵV < 10-3

Excluded (beam dump / SN)

10-40 10-42 10-44

10-46

10-48

10. 100. 1000. 10000.2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.810. 100. 1000. 10000.

Λ [GeV]

d

EDM on 131Xe: max. σpeff[cm2], mDM = 100 GeV

Dark photon bounds strongest: ϵV = 10-3

Γ(Z→CFT) bound strongest2×10-4 < ϵV < 10-3

Excluded (beam dump / SN)

10-32.5

10-35 10-37.510-40

10-42.5

Figure 6: The largest e↵ective cross section �e↵

p for dark matter scattering on 131Xe allowedby the constraints from dark photon searches and the Z boson width. The left-hand plotis for magnetic dipole-type couplings of DM to Oµ⌫ and the right-hand plot for electricdipole-type coupling. We fix c

2

= 1 and a dark matter mass of 100 GeV. For a given ⇤and d, the largest allowed value of cB is chosen. At small d this is determined by the darkphoton constraint ✏V <⇠ 10�3. At somewhat larger d >⇠ 2.4 the Z width becomes the dominantconstraint. At the point that the Z width no longer allows ✏V >⇠ 2 ⇥ 10�4, the beam dumpand supernova constraints on dark photons force us all the way down to ✏V so small that thedark photons would decay after BBN, so no cross section is allowed above d ⇡ 2.6. Whatis plotted is the cross section �p in square centimeters associated to a model with scalarcontact interaction achieving the same integrated rate.

by v2 ⇠ 10�6 relative to that through the EDM operator.To summarize our results, despite the existence of a variety of stringent constraints on

the operator Oµ⌫ coupling to hypercharge, arising from both low-energy probes like beam-dump experiments and pion decay and high-energy probes like Z boson decays, there is arange of dimensions—roughly 2 <⇠ d <⇠ 2.6—for which sizable direct detection cross sectionscould occur. Of course, direct detection experiments themselves can constrain this region.However, due to the unusual spectral shapes that appear in continuum-mediated scattering,a new analysis of the direct detection data will be necessary to derive precise bounds. Weleave such analyses for future work.

5 Conclusions

As the simplest models of WIMPs come under strain from a variety of experiments (directdetection, indirect detection, and colliders), it becomes increasingly important to broadenour theoretical vision of what dark matter might be. In recent years, a large number of

22

There is viable parameter space with detectable scattering rates and unusual shapes in recoil energy.

Questions for LatticeWhat sort of low-dimension operators can be realized? For instance, do we have surprises in operator spectrum near the lower end of the conformal window? (Perhaps also a question for conformal bootstrap.)

For continuum-mediated DM-baryon scattering, interesting to find antisymmetric tensor operators with low dimension but still far from free photon limit.

Are there theories with non-obvious spectral “gaps” or large ratios between lighter and heavier states? Examples other than pseudo-Goldstone bosons?