A couplet from flavored dark matter

Abstract

Abstract

We show that a couplet, a pair of closely spaced photon lines, in the X-ray spectrum is a distinctive feature of lepton flavored dark matter models for which the mass spectrum is dictated by Minimal Flavor Violation. In such a scenario, mass splittings between different dark matter flavors are determined by Standard Model Yukawa couplings and can naturally be small, allowing all three flavors to be long-lived and contribute to the observed abundance. Then, in the presence of a tiny source of flavor violation, heavier dark matter flavors can decay via a dipole transition on cosmological timescales, giving rise to three photon lines. Two of these lines are closely spaced, and constitute the couplet. Provided the flavor violation is sufficiently small, the ratios of the line energies are determined in terms of the charged lepton masses, and constitute a prediction of this framework. For dark matter masses of order the weak scale, the couplet lies in the keV-MeV region, with a much weaker line in the eV-keV region. This scenario constitutes a potential explanation for the recent claim of the observation of a 3.5 keV line. The next generation of X-ray telescopes may have the necessary resolution to resolve the double line structure of such a couplet.

Published for SISSA by Springer

Received: April 3, 2015 Accepted: July 18, 2015 Published: August 17, 2015

Prateek Agrawal,a Zackaria Chacko,b Can Kilicc and Christopher B. Verhaarenb

aFermilab,

P.O. Box 500, Batavia, IL, 60510 U.S.A.

bMaryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD, 20742-4111 U.S.A.

cTheory Group, Department of Physics and Texas Cosmology Center,The University of Texas at Austin, 2515 Speedway Stop C1608, Austin, TX, 78712-1197 U.S.A.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , [email protected]

Abstract: We show that a couplet, a pair of closely spaced photon lines, in the X-ray spectrum is a distinctive feature of lepton avored dark matter models for which the mass spectrum is dictated by Minimal Flavor Violation. In such a scenario, mass splittings between di erent dark matter avors are determined by Standard Model Yukawa couplings and can naturally be small, allowing all three avors to be long-lived and contribute to the observed abundance. Then, in the presence of a tiny source of avor violation, heavier dark matter avors can decay via a dipole transition on cosmological timescales, giving rise to three photon lines. Two of these lines are closely spaced, and constitute the couplet. Provided the avor violation is su ciently small, the ratios of the line energies are determined in terms of the charged lepton masses, and constitute a prediction of this framework. For dark matter masses of order the weak scale, the couplet lies in the keV-MeV region, with a much weaker line in the eV-keV region. This scenario constitutes a potential explanation for the recent claim of the observation of a 3.5 keV line. The next generation of X-ray telescopes may have the necessary resolution to resolve the double line structure of such a couplet.

Keywords: Beyond Standard Model, Cosmology of Theories beyond the SM, Global Symmetries

ArXiv ePrint: 1503.03057

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP08(2015)072

Web End =10.1007/JHEP08(2015)072

A couplet from avored dark matter

JHEP08(2015)072

Contents

1 Introduction 1

2 The framework 3

3 A benchmark model 53.1 Mass splittings 53.2 Relic abundance 73.3 Direct detection 93.4 Indirect detection 93.5 The couplet 103.6 The 3.5 keV line 11

4 Conclusions 13

1 Introduction

Increasingly precise cosmological measurements indicate that about 80% of the matter density of the universe is composed of particles that are non-baryonic, and neutral under both color and electromagnetic interactions. However, the precise nature of this dark matter (DM) remains a mystery. One theoretically appealing possibility is that DM is composed of Weakly Interacting Massive Particles (WIMPs), particles with mass of order the weak scale that have interactions of weak scale strength with the standard model (SM) elds. This scenario is compelling because, provided the WIMPs were in thermal equilibrium with the SM at early times, just enough of them survive today as thermal relics to account for the observed DM density.

While the WIMP framework requires that DM have interactions of weak scale strength with the SM elds, e orts to produce it at high energy colliders have, so far, proven fruitless. Likewise, e orts to directly detect DM in the laboratory through its scattering o nucleons, in spite of the increased sensitivity of current experiments, have also been unsuccessful. There are some tentative hints from indirect detection of DM annihilation to SM today, but there is no conclusive signal. In the wake of these null results, DM scenarios that retain the cosmological success of the WIMP framework while satisfying the current experimental bounds have become increasingly compelling.

The matter elds of the SM (Q, Uc, Dc, L and Ec) are known to come in three copies, or avors. Di erent avors carry the same charges under the SM gauge groups, but have di erent couplings to the Higgs, and so di er in their masses. One interesting possibility, which has been receiving increased attention, is that DM, like the SM matter elds, also comes in three avors [111] or has avor-specic couplings to the SM [1215]. Specic

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DM candidates of this type include sneutrino DM in supersymmetric extensions of the SM [16], and Kaluza-Klein neutrino DM in theories with a universal extra dimension.

In [2], the simplest theories of avored DM (FDM) were classied, and labelled as lepton avored, quark avored or internal avored, based on the form of the interactions of the DM candidate with the SM elds. Models in which the DM has tree level interactions with the SM leptons but not with the quarks as in lepton FDM can naturally account for the observed abundance of DM while remaining consistent with all experimental bounds [1719]. The reason is that strong production at a hadron collider or scattering o a nucleus rely on DM-quark interactions, which are loop suppressed in this scenario. In addition, because the average number of photons generated in DM annihilation to hadrons is much larger than in the case of DM annihilation to leptons, the limits from indirect DM searches using gamma rays are also weaker. Some other indirect signals of DM annihilation, such as the positron ux, are enhanced for lepton FDM, but regions of parameter space for which the DM is a thermal relic remain viable.

In general the couplings of lepton FDM violate the avor symmetries of the SM. In order to avoid conict with the very stringent bounds on avor violating processes such as [notdef] ! e , while giving rise to an annihilation cross section of weak scale strength, the

couplings of DM to the SM leptons must be aligned with the SM Yukawa interactions. In theories where the avor structure is consistent with Minimal Flavor Violation (MFV), so that the only sources of avor violation are associated with the SM Yukawa couplings, this condition is automatically satised. Then each DM avor is associated with a corresponding lepton avor. It is this class of theories that we shall focus on.

In realizations of FDM that respect MFV, the mass splitting between a pair of di erent DM avors is dictated by the corresponding SM Yukawa couplings. In simple models, this splitting is proportional to the di erence in the squares of the Yukawa couplings, so that for Dirac fermions we obtain,

m[notdef],i m[notdef],j [similarequal] (y2i y2j) . (1.1)

In this expression the index i = 1, 2, 3 runs over the three lepton avors e, [notdef], . While m[notdef],i

represents the mass of the ith DM avor, and yi represents the Yukawa coupling of the ith lepton avor. The constant has the dimensions of mass and depends on the dynamics which UV completes avor at some high scale . If threshold e ects at this scale generate mass splittings at tree level, can naturally be of order m[notdef], where m[notdef] is the average DM mass. In this case the largest splitting, that between the e and avors, is expected to be in the MeV-GeV range for weak scale DM. If tree-level contributions at the threshold are absent, the leading e ects are then loop suppressed. The largest splitting is then much smaller, in the keV-MeV range.

Since the Yukawa couplings are small, the splittings between the di erent DM avors are suppressed relative to the mass of each avor. If the splittings are smaller than the electron mass, the dominant avor-conserving decay mode,

[notdef]i ! [notdef]j + i +

j , (1.2)

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2

is slow on cosmological timescales, so that the lifetimes of the heavier avors are much longer than the age of the universe. Then all three avors are expected to contribute to the observed DM abundance.

Now, suppose a tiny additional source of explicit avor breaking is present in the theory, so that the avor violating decays,

[notdef]i ! [notdef]j + , (1.3)

can occur on cosmological timescales and dominate over the avor-conserving decay. The monochromatic photons produced in such decays then constitute a striking signal of DM. Provided this new source of avor violation is too small to contribute signicantly to the splittings between the di erent DM avors, the frequencies of the resulting gamma ray lines depend on the SM Yukawa couplings as in eq. (1.1). The constant of proportionality in eq. (1.1) cancels out when ratios of frequencies are considered. For example, if the avor of DM is the heaviest and [notdef]e the lightest, we have,

! ([notdef] ! [notdef][notdef]) ! ([notdef] ! [notdef]e)

! ([notdef][notdef] ! [notdef]e) ! ([notdef] ! [notdef]e)

m2 = 3.5 [notdef] 103 . (1.5)

We see that this scenario predicts a pair of very closely spaced lines in the keV-MeV region corresponding to the [notdef] ! [notdef]e and [notdef] ! [notdef][notdef] transitions (the couplet), as well as an

isolated line in the eV-keV region. Remarkably, we see that in this limit of negligible avor violation, the ratios of these lines frequencies are a prediction of this scenario.

The fact that decaying DM particles will, in general, be in motion, leads to some broadening of these lines. The size of this e ect scales with the velocity of DM. For typical astrophysical sources, the DM velocity v ranges from (13)[notdef]103, and a line of frequency !

is smeared by order v!. Comparing to eq. (1.5), we see that whether the double line feature gets washed out by this e ect depends on the astrophysical source. While this splitting is not currently measurable, it is within the design resolution of upcoming experiments such as ASTRO-H [20, 21].

In the next section we review the MFV framework for models of lepton avored DM. In section 3 we choose a simple benchmark to illustrate the phenomenology of this scenario, focusing on constraints from various DM experiments and potential signals. We conclude in section 4.

2 The framework

In this section we study the restrictions that MFV places on the parameters of theories of lepton FDM. The SM has an approximate U(3)5 avor symmetry that acts on the three generations of fermions Q, Uc, Dc, L and Ec. This symmetry is explicitly broken by the SM Yukawa couplings. In extensions of the SM that respect MFV, any new parameter

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m2[notdef]

= m2 m2[notdef] m2 m2e

1

m2 , (1.4)

and,

m2[notdef]

= m2[notdef] m2e m2 m2e

that violates the SM avor symmetries must be aligned with the SM Yukawa couplings. Specically, the Yukawa couplings of the SM are promoted to spurions that transform under the U(3)5 avor symmetry. Any new interactions are then required to be invariant under this spurious symmetry.

In the lepton sector of the SM there is an approximate U(3)L [notdef]U(3)E avor symmetry

that acts on the left-handed SU(2) doublets LA and SU(2) singlets Eci. We denote U(3)L indices by capital Latin letters and U(3)E indices by lowercase Latin letters. This symmetry is violated by the SM Yukawa interactions,

LY = Y i

A LAHEci + h.c. (2.1)

We can, however, make the Yukawa interactions formally invariant under the avor symmetry by promoting the matrix Y i

A to be a spurion that transforms as (3, 3) under SU(3)L [notdef] SU(3)E subgroup of U(3)L [notdef] U(3)E, and has appropriate charges under the

U(1) factors.

Theories of FDM posit a U(3)[notdef] avor symmetry that acts on the DM eld [notdef] . We use lowercase Greek indices to denote the DM avor. We focus on the case where the DM eld is a fermion that transforms as a singlet under the SM gauge interactions, and has renormalizable couplings to the to the SU(2) singlet leptons Eci through a scalar mediator . The mediator does not transform under the SM and DM avor groups. The relevant interaction takes the form

L = i

[notdef] Eci + h.c. (2.2)

This interaction explicitly violates the U(3)E [notdef]U(3)[notdef] symmetry. In general, it will give rise

to lepton avor violating processes at one loop. However, when MFV is imposed, the form of this interaction is restricted, with the result that avor violating processes are forbidden.

We impose MFV by identifying the DM avor symmetry U(3)[notdef] with the U(3)E avor symmetry that acts on the SU(2) singlet leptons in the SM,1 and requiring that the new interaction be invariant under the spurious U(3)5 symmetry. This leads to a restriction on the form of j

ji. To leading order in Y

i

A we then have,

e Y AiY

j

A . (2.3) MFV also governs the form of the DM mass matrix. If [notdef] is a Dirac fermion we can write the mass term in the Lagrangian as

LM = m

[notdef] [notdef] m ji[notdef]j[notdef]i . (2.4)

MFV restricts the mass matrix to be of the form

m ji =

bm ji + emY AiY

j

A . (2.5)

Then, in a basis where the lepton mass matrix is diagonal, we see that the splittings between the di erent DM mass eigenstates are governed by the SM Yukawa couplings in

1One could also identify U(3)~ with the U(3)L symmetry that acts on the SU(2) doublet leptons (see [2]), but the main results do not depend on this choice.

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ji =

b ji +

accordance with eq. (1.1). Our interest is in the scenario where the parameter

em, which controls the mass splittings, is about a loop factor smaller than the parameter m, which sets the scale of the DM mass. Then, for DM masses of order the weak scale, the splittings can be less than an MeV, allowing all three DM avors to be stable on cosmological time scales.

3 A benchmark model

In this section we construct a simplied benchmark model which exhibits the features described above and consider its phenomenology in detail. We choose the DM candidate [notdef] to be a Dirac fermion which transforms as a fundamental under the SM avor group SU(3)E. The only additional terms in the Lagrangian beyond the SM are given by eq. (2.2)

and (2.4). We assume that the structure of the coupling and mass terms is restricted by MFV, and follows eq. (2.3) and (2.5). For this benchmark we will further restrict to the special case where the DM mass terms and couplings generated at the high avor scale are universal, so that ~m and ~

are zero at this scale.2 The mass di erences between the di erent DM avors are then radiatively generated at one loop. The largest splittings will lie in the keV-MeV range for WIMP DM, allowing all three avors to contribute to the observed abundance. The mass and interaction terms for the DM in four-component notation at scale are then given by,

L = m[notdef][notdef]i[notdef]i + [bracketleftbigg]

. (3.1)

Here ei is the four-component spinor corresponding to the charged leptons of the SM. The only free parameters in this model are the masses of the DM and the mediator, and a coupling . As we shall see, the mass splittings generated in this case are nite at one loop. Later we will introduce a tiny source of avor violation.

In what follows we determine the splittings between the di erent DM avors, and show that all three avors are stable on cosmological time scales. We then determine the range of parameters consistent with the observed relic abundance of DM. This model is constrained by a number of experiments. Constraints from g 2 of the muon and monophoton searches

tend to be subdominant to direct detection constraints [19, 22]. LHC constraints on the production of the mediator and indirect detection constraints from DM annihilations into positrons and photons can also be relevant. We study these bounds in turn. Finally, we introduce a small source of avor violation, obtain expressions for the lifetimes of the heavier avors, and show that this naturally leads to the couplet feature.

3.1 Mass splittings

The breaking of the lepton avor symmetry U(3) ! U(1)3 by the SM lepton Yukawa

couplings is communicated to the DM sector through the FDM interaction, eq. (2.2). Even

2Note that allowing non-vanishing ~m and ~

at the scale would not qualitatively a ect our results, provided that the parameter ~m is small enough that the mass splittings between the di erent DM avors are less than an MeV. In particular, the predictions for the ratios of the mass splittings, eqs. (1.4) and (1.5), do not depend on this assumption.

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[notdef]i(1 + 5)ei + h.c.

[bracketrightbigg]

i

i

i

Figure 1. One-loop correction to the two-point function of [notdef].

though they are assumed to be degenerate at tree level, mass splittings between the three [notdef] avors will be induced at loop level. In particular, let us consider the 2-point function for the three avors of [notdef] (see gure 1). The one-loop contributions are identical in the limit of massless leptons, but at order O(m2[lscript]) they begin to di er, giving rise to di erences in the

wavefunction renormalizations for the [notdef]i. Due to the chiral nature of the FDM coupling, it is easy to see that there is no direct mass renormalization. Once the elds [notdef]i are brought back to canonical normalization, however, a mass splitting is induced between them,

m[notdef],i m[notdef],j mij =

m[notdef] 2

322

[integraldisplay]

1

0 dx x log[parenleftBigg][parenleftBigg]

x m2 + (1 x)m2[lscript],i x(1 x)m2[notdef]

x m2 + (1 x)m2[lscript],j x(1 x)m2[notdef] [parenrightBigg]

JHEP08(2015)072

. (3.2)

To leading order in the Yukawa couplings this yields,

mij

m[notdef] [similarequal]

2(y2i y2j) 642

v2 m2

F (m2[notdef]/m2), (3.3)

where as before, yi denote the Yukawa couplings of lepton i, and v = 246 GeV is the Higgs vacuum expectation value. The loop function F (x) is given by,

F (x) =

x + log(1 x)

x2 [similarequal]

1

2 +

x

3 + O(x2) . (3.4)

We see that [notdef] is split signicantly further from [notdef]e and [notdef][notdef] than these two states are split from each other. For an overall mass scale m[notdef] in the GeV regime and m O(100) GeV, m[notdef], m[notdef],[notdef] [similarequal] m[notdef], m[notdef],e m keV, (3.5)

m[notdef],[notdef] m[notdef],e m eV . (3.6)

It is interesting to note that the one-loop splitting calculated above is a nite e ect, suppressed by v2/m2 for large masses. Note that the sign of m is not arbitrary, and as a consequence [notdef] is the heaviest DM avor. At two loops there arises a logarithmically divergent contribution to the mass splitting, where the log divergence is cut o at the UV avor scale, . This two-loop e ect can become important for very large m and . We estimate that it is subleading to the nite one-loop calculation for the range of m we are interested in, provided the new physics scale is less than 100 TeV.

6

i

i

i

W

i

j

i

j

j

j

Figure 2. a) Flavor preserving decay [notdef]i ! [notdef]j i

j. b) A Feynman diagram contributing to the

avor violating decay [notdef]i ! [notdef]j .

Once the [notdef] masses are split by these loop corrections, only the lightest [notdef] is truly stable. This is true even in the exact MFV limit where the U(1)3 avor symmetry is preserved. For this benchmark, the splittings are smaller than the mass of the electron. Then, the leading contribution for avor-preserving [notdef] decays arises at one loop and is illustrated in the left panel of gure 2. Note that this contribution is very suppressed due to the following three factors.

With a [notdef] mass splitting of order keV, the kinematically available phase space is

extremely small. This results in a signicant suppression for the 1 ! 3 process.

The loop amplitude is suppressed by the momentum-exchange scale, or more con

cretely by ( m/m).

The lepton propagators in the loop couple to on one end and to W on the other.

However, the former couples to right-chirality leptons while the latter couples to left-chirality leptons. Therefore both lepton propagators need a mass insertion to obtain a nonzero amplitude, so the decay rate is further suppressed by m2[lscript]im2[lscript]j.

As a consequence of these e ects, heavier avors are long-lived on cosmological time scales.

3.2 Relic abundance

The DM annihilation rate in each channel ([notdef]i[notdef]j ! [lscript]i[lscript]j) is given by,

hv[angbracketright] =

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4m2[notdef]

32(m2[notdef] + m2)2

. (3.7)

Since the DM candidate is a Dirac fermion, there is no p-wave or chirality suppression, and thus the annihilation cross section today is the same as in the early universe to a good approximation.

Since all avor combinations of DM co-annihilate with one another with the same cross section, the cross section that gives rise to the correct relic abundance today is the same as for a single species of Dirac fermion DM, given by

hv[angbracketright] = 2 [notdef] (2.2 [notdef] 1026 cm3/s). (3.8)

The factor of two relative to the canonical quoted value (for Majorana DM) arises due to the Dirac nature of the DM particle. The region of parameter space leading to the correct relic density is shown in gure 3 as a red band.

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Figure 3. For mediator masses m = 100, 200 and 400 GeV, we plot the position of the X-ray signal (in keV, gray dashed contours) as well as a number of constraints. Direct detection constraints from LUX are shown as the blue-shaded region, while the indirect detection constraints from positrons and photons are shown as the purple and green-shaded regions, respectively. The red band shows the region where correct relic abundance is obtained.

8

Note that the above calculation applies only if we assume that the interaction with leptons alone is responsible for the DM thermal relic abundance. Any coupling to additional non-SM states will alter these numbers. This constraint can also be relaxed if the DM relic density is set by an asymmetry, which can arise rather naturally in these models [9].

3.3 Direct detection

As discussed in detail in [2], lepton avored DM can scatter o nuclei via a one-loop photon exchange. These constraints can be severe in the region where the DM is a thermal relic. The dominant contribution to the WIMP-nucleon cross section is avor diagonal and for each avor of FDM it is given by,

n = [notdef]2nZ2 A2

X[lscript]

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"1 + 23 log 2[lscript]

m2

[bracketrightBigg][parenrightBigg]2. (3.9)

Here [notdef]n is the reduced mass of the DM-nucleon system and [lscript] represents the infrared cuto in the loop calculation for the e ective DM-photon coupling. This cuto is m[lscript], the mass of the corresponding lepton, unless m[lscript] is smaller than the momentum exchange in the process, of the order of 10 MeV. We therefore use = m and [notdef] = m[notdef], but set e = 10 MeV. In extracting constraints from the null results of direct detection experiments such as LUX, we use the total rate, summed over all three FDM avors. The region of parameter space excluded by LUX is shown in gure 3 as the blue-shaded region.

3.4 Indirect detection

In the limit m[notdef] m, ( m)2 and [angbracketleft]v[angbracketright] both scale approximately as m2[notdef] 4/m4, and

therefore choosing a xed mass splitting or requiring thermal relic abundance leads to potentially observable signals for indirect detection searches in photons and positrons (with the caveats mentioned at the end of the previous paragraph). The constraints from both indirect detection channels are more stringent for lower mass DM, since the signal rate scales as the square of the [notdef] number density, which itself scales as m1[notdef], while the background ux as a function of energy does not change as rapidly. Therefore, for a given m or [angbracketleft]v[angbracketright],

these constraints can be weakened by increasing the DM mass and either decreasing the coupling or increasing m.

For the positron constraint from the AMS-02 experiment [23], the signal has contributions both from the prompt positrons produced when one of the annihilating particles is [notdef]e, and also from secondary positrons from the decays of [notdef]+ and + that are produced when one of the annihilating particles is [notdef][notdef] or [notdef]. The spectrum of the secondary positrons is shifted towards lower energies compared to the prompt positrons (not to mention that the branching ratio of ! e + X is rather low), and therefore the bound from AMS-02 comes

mostly from the prompt positrons. The bound is shown in gure 3 as the purple-shaded region. Note that the positron constraint is signicantly weaker than the constraint from direct detection across the parameter space, and therefore the inclusion or non-inclusion of secondary positrons in determining the bound turns out to be academic. Owing to the relative factor of two between Dirac and Majorana DM (eq. (3.8)), the latter being relevant

9

2e2

642m2

for SUSY for which the AMS bounds are calculated, the bound on the FDM annihilation cross section leading from prompt positrons (any one of the three [notdef][lscript] avors annihilating with [notdef]e) is related to the total annihilation cross section (all nine annihilation channels) as

hv[angbracketright] 6[angbracketleft]v[angbracketright]bound,e+ , (3.10)

where [angbracketleft]v[angbracketright]bound,e

+ is the experimental bound quoted in [23] for a Majorana DM annihilating to e+e with 100% branching fraction.

Similar to the case of positron constraints being most sensitive to prompt positrons in the nal state, indirect detection in photons is most sensitive to s in the nal state, since more photons are produced from s than from es or [notdef]s. One can therefore formulate the bound from indirect detection in the photon nal state [24] in terms of the e ective annihilation cross section leading to the production of s,

hv[angbracketright] 6[angbracketleft]v[angbracketright]bound, , (3.11)

where [angbracketleft]v[angbracketright]bound, is the experimental bound quote in [24] for a Majorana DM annihilating

to + with 100% branching ratio. The constraint from indirect detection in photons is shown in gure 3 as the green-shaded region.

DM annihilating to leptons can potentially have signicant constraints from the CMB [25, 26]. However, the annihilation into muons and taus has a low e ciency to inject energy into the CMB [27], and the constraints are subdominant to the other constraints considered above.

3.5 The couplet

Since the rates of avor-preserving [notdef] decays are so extremely small, even a very small avor-violating contribution can easily be the dominant channel for the decays of the heavier [notdef]. Such decays could, for example, arise from the avor violating dipole operator,

[notdef]i[notdef] [notdef]jF [notdef] , which generates the process [notdef]i ! [notdef]j + . The monochromatic X-ray photons

from these transitions would then constitute a striking signal of this scenario, and exhibit the couplet feature. The dipole operator is, however, non-renormalizable. We can obtain the same e ect at the renormalizable level by adding to the Lagrangian a tiny avor violating contribution to the DM-visible matter interaction term,

LFV = 12 ij

[notdef]i(1 + 5)ej + h.c. (3.12)

This radiatively generates the avor violating dipole transition [notdef]i ! [notdef]j + as illustrated

in the right panel of gure 2. Note that unlike the avor-preserving decays, these are two-body decays, and there are no suppressions due to lepton masses. The rate for these avor-violating decays can be calculated in a straightforward manner. If we assume that all o -diagonal couplings in ij are of the same size , then to leading order

, (3.13)

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ij

[notdef]i![notdef]j =

e2 2 2

1024 5

( mij)3m2[notdef]

m4

10

where we have neglected higher order terms proportional to lepton masses. We see that

ij/ ( mij)3. Then, in the absence of any hierarchy in the , it follows that the rates

for the [notdef] ! [notdef][notdef] and [notdef] ! [notdef]e transitions are comparable. However, transitions between

[notdef][notdef] and [notdef]e will be many orders of magnitude slower than this, and are not expected to be be observable.

The parameters allow the heaviest avor to decay on cosmological timescales. The current sensitivity for dark matter decay lifetimes from X-ray observations in the keV-MeV range is roughly at the level of 10271028 sec [28, 29]. Having restricted the parameters

, m[notdef] and m to satisfy the relic abundance condition eq. (3.7), this can be translated into a bound on the avor vioating coupling , expressed in terms of the splitting m between [notdef] and the two lighter avors,

[lessorsimilar] 107 [parenleftbigg]

m[notdef]100 GeV. (3.14)

It follows that, in the allowed range, is much too small to a ect the relic abundance calculation, or any of the bounds on this scenario considered in the previous sections. It remains to verify that the contribution of the to the DM mass splittings is subdominant to the previously calculated splittings associated with the SM Yukawa couplings, eq. (3.3). The o -diagonal entries in the coupling matrix responsible for the avor-violating decay only a ect the mass eigenvalues at quadratic order in . Consequently, once the bound eq. (3.14) is imposed, this e ect is much smaller than the splitting associated with the SM Yukawa couplings calculated in eq. (3.3). However, in the absence of a symmetry that restricts the form of , we expect that the diagonal elements of this matrix will be parametrically of the same size as the o -diagonal avor-violating entries. These diagonal terms will, in general, not be avor universal, and will contribute to the mass eigenvalues at linear order in . Therefore, it is important to understand whether this e ect can dominate over the splitting calculated in eq. (3.3). At one loop there is a logarithmically enhanced contribution from that is cuto at the UV avor scale ,

m[notdef] m[notdef] =

3

7

5

JHEP08(2015)072

keV m

2

[radicalbigg]

. (3.15)

For satisfying the bound, eq. (3.14), this e ect is always much smaller than the splitting between [notdef] and the other avors, showing that the existence of the couplet is a robust prediction of this framework. However, for / [greaterorsimilar] 108, corresponding to m [lessorsimilar] 10 keV, this splitting can become comparable to or larger than the splitting between [notdef]e and [notdef][notdef] from eq. (3.3). Hence the prediction for the splitting between the lines in the couplet, eq. (1.5), is only valid if the mass di erence between [notdef] and the other avors is larger than about 10 keV.

3.6 The 3.5 keV line

An X-ray line signal has been observed at 3.5 keV [30, 31]. We note that there is currently no consensus about the interpretation of this observation as arising from a DM signal [32 43]. Nevertheless, a large number of DM models have been proposed to explain this signal.

11

[integraldisplay]

1

0 dx xlog 2

6

4

xm2 x(1 x)m2[notdef][parenrightBig]

2

Figure 4. Constraints on the mass of DM, m~ and the mediator, m when the X-ray line is at 3.5 keV. The contours show the value of coupling , and the red band shows the region where correct relic abundance is obtained. The blue and purple-shaded regions show the exclusion from LUX and from AMS respectively.

The ideas that have been proposed include sterile neutrinos [4469], axions [7085], supersymmetry [8692] and a number of other mechanisms [93124]. We now show that the 3.5 keV line, if conrmed, can be easily accommodated within the framework of the minimal model. We show in gure 3 the region of parameter space that provides the right relic abundance for DM, consistent with a 3.5 keV line. Fixing the splitting to be 3.5 keV, we show the direct detection and the AMS positron constraint in gure 4 along with the region of parameter space consistent with the requirement of correct relic abundance.

Our scenario further predicts that closer inspection of this line will reveal two closely spaced lines corresponding to the [notdef] ! [notdef][notdef] and [notdef] ! [notdef]e transitions. This couplet consti

tutes a characteristic indirect detection signal of lepton FDM scenarios. However, because the frequencies of these lines is less than 10 keV, in general the prediction for their splitting, eq. (1.5), is not expected to apply. To understand this in greater detail, note that the lifetime for a decaying DM candidate to be consistent with the observed signal is given by (see for instance [99]),

DM [similarequal] (1028 sec)

7 keVm[notdef] . (3.16)

Then, for m[notdef] = 150 GeV, one obtains DM 1020 s. For m = 500 GeV and [similarequal] 1, this

would require [similarequal] 108. Then the contribution to the splittings from the avor diagonal

elements of the matrix , given by eq. (3.15), can comparable to or larger than the small mass splitting between the two light DM avors that arises from the SM Yukawa couplings, eq. (3.3). Therefore, eq. (1.5) is not expected to apply. However, the contribution from is still much smaller than the large 3.5 keV splitting between [notdef] and the other two avors.

Consequently, the characteristic couplet feature survives.

12

JHEP08(2015)072

4 Conclusions

We have studied models of lepton FDM within the MFV framework. In this scenario, mass splittings between di erent DM avors can naturally be small enough that tree level decays of heavy avors are kinematically forbidden. Then all three avors of DM can be long-lived on cosmological time scales (the lightest avor being exactly stable). The ratios of the mass splittings between the three possible pairings among the DM avors are predicted.

When even a very small source of avor violation is present in the dark sector, a new decay channel becomes available for the decay of heavier DM avors through a dipole transition. While the lifetime associated with this decay may be orders of magnitude longer than the age of the universe, it can still be the dominant decay channel and gives rise to a very distinct nal state. These decays result in two very closely separated photon lines, the couplet. For weak scale DM, the overall energy of the couplet is naturally in the keV-MeV range, with the splitting of the two lines in the eV-keV range.

We have focused on the detailed phenomenology of a specic model of lepton avored DM. In this model, the mass splittings are radiatively generated by nite one loop e ects arising from the breaking of the lepton avor symmetry by Yukawa couplings. The sign of the contribution is xed, with the result that [notdef] is the heaviest and [notdef]e the lightest state.

This scenario is a potential explanation for the claimed observation of a 3.5 keV line in the X-ray spectrum, and there exist regions in parameter space of the model where such an explanation is entirely consistent with the observed DM relic density as well as with experimental constraints set by a number of direct and indirect detection experiments. While a DM explanation of this line is in dispute, the characteristic double-line structure predicted by the couplet can be directly tested experimentally. In particular, the next generation experiments might be able to resolve any such feature in the X-ray spectrum.

Acknowledgments

We thank Ilias Cholis and Dan Hooper for useful discussions. CK would also like to thank the Aspen Center for Physics (supported by the National Science Foundation under Grant No. PHYS-1066293) as well as the Perimeter Institute for Theoretical Physics (supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation), where part of this work was completed, for their hospitality. ZC and CV are supported by NSF under grant PHY-1315155. CK is supported by NSF grant numbers PHY-1315983 and PHY-1316033. Fermilab is operated by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

13

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