A Selection Rule for Enhanced Dark Matter Annihilation · TIFR/TH/16-43 A Selection Rule for Enhanced Dark Matter Annihilation Anirban Das and Basudeb Dasguptay Tata Institute of

TIFR/TH/16-43

Selection Rule for Enhanced Dark Matter Annihilation

Anirban Das∗ and Basudeb Dasgupta†

Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India.(Dated: June 29, 2017)

We point out a selection rule for enhancement (suppression) of odd (even) partial waves of darkmatter coannihilation or annihilation using Sommerfeld effect. Using this, the usually velocity-suppressed p-wave annihilation can dominate the annihilation signals in the present Universe. Theselection mechanism is a manifestation of the exchange symmetry of identical incoming particles,and generic for multi-state DM with off-diagonal long-range interactions. As a consequence, therelic and late-time annihilation rates are parametrically different and a distinctive phenomenology,with large but strongly velocity-dependent annihilation rates, is predicted.

PACS numbers: 95.35.+d

1. Introduction.– If DM is a thermal relic of the earlyUniverse, then its cosmological abundance provides ameasure of its “annihilation rate” 〈σv〉 [1–3]. This anni-hilation rate has contributions from various partial wavesof the scattering amplitude, each with its characteristicdependence on the relative velocity v of the colliding par-ticles,

〈σv〉 = a︸︷︷︸〈σv〉s

+ bv2︸︷︷︸〈σv〉p

+ . . . . (1)

The first term on the right, 〈σv〉s, represents the velocity-independent s-wave contribution and the second term,〈σv〉p, which scales as v2, has the p-wave contribution.Omitted terms appear with higher powers of v2, andfor nonrelativistic DM, the contribution of these higherpartial waves are small. In the simplest models, thes-wave contribution dominates and the annihilation rateis 〈σv〉relic ' 2.2 × 10−26 cm3s−1 [4], to produce the ob-served DM abundance, practically independent of v.

Detection of a non-s-wave DM annihilation rate, e.g.,〈σv〉 ∝ v2, would reveal a crucial clue to the nature ofDM. However, it is believed to be highly challenging. Tothe best of our knowledge, annihilations of very dense orvery fast DM are the only avenues that have interestingsensitivity to p-wave annihilations [5–8]. Unfortunately,even these become inefficient for heavy DM.

Sommerfeld effect induces further nontrivial velocity-dependence of the annihilation rate [9, 10]. Long-rangeinteractions of DM distort the wave-functions of incomingparticles and change the annihilation rate, 〈σv〉 → S〈σv〉,by the velocity-dependent Sommerfeld factor S. This ef-fect has been studied extensively in recent years [11–20],after it was initially invoked [21–23] to explain the cosmic-ray positron excesses [24, 25] using a large DM annihila-tion rate. As this enhancement occurs for small v, againmodels with dominantly s-wave annihilations are popu-lar. Therein, the enhancement is always larger for smallerv and as a result a large annihilation rate is predictedaround recombination, which leaves an imprint on theCosmic Microwave Background (CMB) [26, 27].

In this Letter, we point out a selection mechanismthat allows enhanced p-wave DM annihilation, with noenhancement but rather a possible suppression of thes-wave rate. Models employing the mechanism aretestable and predict a distinctive, large but stronglyvelocity-dependent, annihilation rate: highest at inter-mediate velocities, e.g., v ' 10−3–10−4 in galaxies, whilebeing lower at both larger and smaller velocities, e.g., ingalaxy clusters and at recombination, respectively. In thefollowing, we explain this mechanism, provide a concretemodel, discuss the main signatures and constraints, andfinally conclude.2.Mechanism.– The basic idea is that, for coannihi-

lations or annihilations of multi-level DM, the effectiveone-level interaction potential can be attractive or re-pulsive, depending on the angular momentum of the in-coming state, and leads to enhancement or suppression,respectively. We now explain this selection mechanismin more detail, for coannihilations or annihilations of twoDM fermions A and B.

Let Ψi be the wave-function of an incoming two-bodystate, i.e., |AB〉 or |BA〉 for co-annihilation and |AA〉 or|BB〉 for annihilation, with the state labeled by i ∈ {1, 2}in each case. Its long-distance distortion is governed bythe two-level Schrodinger equation

− 1

2µi

dΨi2

dr2+`(`+ 1)

2µir2Ψi + Vij(r)Ψj =

k2i

2µiΨi , (2)

where ki = µiv, with µi being the reduced mass of thei-th two-body state, ` is the angular momentum, and

V =

(V11 V12

V21 V22

), (3)

the potential energy matrix dependent on interactions.The Sommerfeld factor for coannihilation or annihilationchannel i and partial wave ` is given by [14]

S(i)` =

((2`− 1)!!

k`i

)2 (T †Γ`T

)ii

(Γ`)ii(no sum) . (4)

arX

iv:1

611.

0460

6v3

[he

p-ph

] 2

8 Ju

n 20

17

2

The matrix Tij = Ψ∗iΨje−ikir|r→∞ consists of the am-

plitudes of asymptotic wavelike solutions to eq. (2). TheΓ`-matrix contains the a, b, . . . coefficients of coannihila-tion or annihilation rates, calculable in the framework ofnon-relativistic effective theory of the DM model [28–30].

Equivalent One-level Problem.– For co-annihilation,physically there is no distinction between the two states,|AB〉 and |BA〉, as one is obtained from the other by amere exchange of particles. Therefore, one expects thesestates to be identical up to an overall phase,

|BA〉 = (−1)`+s|AB〉 , (5)

where `, s are the angular momentum and spin of thetwo-body state [28]. A factor of (−1)` comes from thechange in relative momentum, (−1)s+1 from exchange ofspins, and a (−1) from the Wick exchange of fermionfields. Clearly, the potentials must satisfy V11 = V22 andV12 = V21. Plugging eq. (5) in eq. (2), reduces eq. (2) toits one-level-equivalent with the effective potential

Veff = V11 + (−1)`+sV12 +`(`+ 1)

2µir2. (6)

The effective potential Veff leads to selective Sommer-feld enhancement of odd or even partial waves. Consider,for example, the potentials Vij are attractive and that theincoming state has s = 1. For even-integer values of `,e.g., s-wave, the effective interaction V11 − V12 may van-ish if V12 ' V11 or become repulsive if |V12| & |V11|.Thus one expects no enhancement or perhaps even asuppression of the s-wave rate. On the other hand,for odd-integer values of `, e.g., p-wave, the potential isVeff = V11 + V12 + `(`+ 1)/(2µir

2), which can be attrac-tive if V11 + V12 falls off slower than 1/r2 in the relevantrange. A minimum in the potential then develops at finitenonzero r, where higher ` wave-functions peak, and leadsto an enhancement. As a result, one has S`=even . 1 andS`=odd � 1. If |V12| � |V11|, this mechanism is notas effective, Veff being dominated by the diagonal po-tential that does not switch its sign. The general lessonhere is that a strong off-diagonal long-range interaction ofmulti-level DM can enforce a spin/angular-momentum-dependent selection rule on Sommerfeld enhancement.

Figure 1 shows a typical manifestation of this selectionmechanism. At high velocities, v ' 1, the Sommerfeldfactors are close to 1, not appreciably affecting relic an-nihilation. At smaller velocities, s-wave rates are sup-pressed but the p-wave rates are enhanced, i.e., Ss . 1and Sp � 1. Specifically for v . 10−3−10−4, the p-wave

Sommerfeld factors Sco/annp saturate to large constant

values, rising roughly as ∼ 1/v3 in the intermediate re-gion. This stronger velocity dependence for intermediatev can overcome the v2 suppression in 〈σv〉p and producesa unique phenomenology.

S = 1

~1/ 3

p-wave

s-wave

CMB dSph MW cluster relic

Ssco-ann

Spco-ann

Spann

10-8 10-7 10-4 10-3 10-2 10-1 10010-6

10-4

10-2

100

102

104

106

108

1010

SS = 1

~1/υ3

p-wave

s-wave

υCMB υdSph υMW υcluster υrelic

Ssco- ann

Spco- ann

Spann

10-8 10-7 10-4 10-3 10-2 10-1 10010-610-410-21001021041061081010

υ

S

≈≈

s-wave

p-wave

S = 1

S = 1

~1/υ3

p-wave

s-wave

υCMB υdSph υMW υcluster υrelic

Ssco- ann

Spco- ann

Spann

10-8 10-7 10-4 10-3 10-2 10-1 10010-610-410-21001021041061081010

υ

S

≈≈

s-wave

p-wave

S = 1

FIG. 1. Sommerfeld factors for s-wave and p-wave coannihila-tion or annihilation processes at velocities v. At smaller veloc-ities the p-wave coannihilation or annihilation processes arestrongly enhanced but s-wave co-annihilation is suppressed.The ∼1/v3-rise of Sp at intermediate velocities predicts that〈σv〉 ' Sp〈σv〉p peaks for v at the edge of the satura-tion plateau at low v. Typical DM velocities in differentsources/epochs are annotated. See text for details of themodel.

For annihilation, where the states |AA〉 and |BB〉 arenot obviously related, the one-level-equivalent does notexist. Yet, as we see in Fig. 1, the p-wave annihilationalso shows a large enhancement. We will show that thisis a consequence of an approximate |AA〉 ↔ |BB〉 ex-change symmetry, which when exact makes |AA〉 and|BB〉 identical to each other. Then, the preceding argu-ment applies for annihilation as well, with small correc-tions proportional to the breaking of this symmetry.

3.Model.– The above selection mechanism or its vari-ants will crop up in many existing DM models. For exam-ple, multiple DM fermions universally coupled to a bosonin the Standard Model (SM) naturally exhibit the selec-tion mechanism. Here, we discuss a simple model thatpresents an interesting version of the selection mecha-nism, where the late-time signal can be due to a purelyp-wave process.

Consider a Dirac fermion χ and a complex scalar φ,with charges +1 and −2, respectively, under an explicitlybroken global dark U(1) symmetry [31–33],

L ⊃ ∂µφ†∂µφ+ µ2|φ|2 − λ|φ|4 + LU(1)−breaking

+ iχ/∂χ−Mχχ−(f√2φχχc + h.c.

). (7)

φ develops a vacuum expectation value vφ, to give φ =(vφ + ρ+ iη)/

√2 and splits χ into two pseudo-Dirac DM

particles χ1 = (χ−χc)/(√

2i) and χ2 = (χ+χc)/√

2 withmasses M∓∆/2. Taking LU(1)−breaking = − 1

2m2ηη

2 keepsthe residual Z2-symmetry, which stabilizes the lighter χ1

of mass mχ and makes it a good DM candidate while ηbecomes a pseudo-Nambu-Goldstone boson of mass mη.

3

The fermion interactions are − f2ρ (χ1χ1 − χ2χ2) −f2 η (χ1χ2 + χ2χ1), i.e., η only mediates between differ-ent fermions, while ρ mediates between alike fermions.The interaction potentials are then given by V11 =−αe−mρr/r, V12 = V21 = −αe−mηr/r, V22 = −αe−mρr/rfor co-annihilation and V22 = −αe−mρr/r + 2∆ for an-nihilation, with the dark fine-structure constant α ≡f2/(4π). A chiral fermion χL instead of χ in eq. (7) [34],would have led to a spin-dependent singular potentialmediated by η and the Sommerfeld effects would bevery different [35, 36]. This problem does not arisehere. We will be interested in the parameter space wheremη,mρ,∆� mχ.4.Methods & Results.– The co-annihilation process

has both s-wave and p-wave amplitudes, while for annihi-lation the s-wave process is forbidden by having identicalMajorana fermions in the initial state [31–34]. To com-pute the Sommerfeld factors for 〈σv〉co-ann

s,p and 〈σv〉annp ,

using eq. (4), following refs. [28–30] we first computed Γ`:

Γco-anns =

πα2

3m2χ

(+1 −1−1 +1

), (8)

Γco-annp =

πα2m4ρ

4m4χ∆2

(+1 +1+1 +1

), (9)

Γannp =

6πα2

m2χ

(+1 +1+1 +1

), (10)

to leading order in m2ρ/(mχ∆) � 1, i.e., when χ1,2 are

not overly degenerate [37]. We then computed T by nu-merically solving the corresponding Schrodinger equa-tions [eq. (2)] using two methods: (i) directly solving thetwo-level equations [14], and (ii) using the variable phasemethod with an ansatz for the wave-functions in terms ofBessel functions [30, 38]. The second method is especiallyuseful in cases with exponentially growing solutions. Forco-annihilation, where the two-level system can be ex-actly mapped into a one-level system, we computed thefactor using the one-level equation as well [36, 39]. Allmethods gave identical results.

Figure 1 shows the dependence of S on velocity, formρ = 10−3mχ, mη = 0.9mρ, ∆ = 10−3mχ, and α = 0.1as representative values. The main feature, i.e., the en-hancement of p-wave rates and suppression of the s-waverate, is understood in terms of the effective potential.

One-level Interpretation.– The co-annihilating |χ1χ2〉or |χ2χ1〉 states have total spin s = 1. The equivalentone-level problem then has the effective potential,

Veff = −α e−mρr

r+ (−1)`

α e−mηr

r+`(`+ 1)

2µir2. (11)

The η gives a Yukawa potential that being stronger than1/r2 at r∼m−1

η obviously satisfies the condition for se-lective enhancement. As expected from the general se-lection rule, the `-dependent sign of the second Yukawapotential leads to Sco-ann

p � 1 and Sco-anns . 1.

↵

m⇢/m�

↵

m⇢/m�

FIG. 2. Sommerfeld enhancement of p-wave annihilation, forDM of mass mχ interacting via a mediator of mass mρ withdark fine-structure constant α. The Sommerfeld factor is largefor α & few×10−3 and mρ/mχ . few×10−2, using v = 10−3

and ∆/mχ = 10−3 as representative values.

The annihilating |χ1χ1〉 or |χ2χ2〉 states have spins = 1, with ` + s being an even-integer due to antisym-metry. Thus, ` is odd. This two-level problem does notreduce to a one-level problem directly. However, in thelimit ∆ � mχ one has V11 = V22 and |χ1χ1〉 ↔ |χ2χ2〉,and only the linear combination |χ1χ1〉 − (−1)`|χ2χ2〉 isphysically relevant. For this linear combination, Veff isthe same as in eq. (11) and one gets Sann

p � 1, i.e., p-wave annihilation is also strongly Sommerfeld-enhanced;the physical origin of the enhancement being the approx-imate exchange symmetry at ∆→ 0.

As χ2 may decay to χ1, we focus on annihilations atlate time. Figure 2 illustrates the dependence of Sann

p onthe strength and range of the interaction. Here, Sann

p �1, when α & 10−3 and mρ/mχ . few × 10−2. At smallα there is no significant enhancement, whereas larger en-hancements are possible when the momentum in the firstBohr orbit, ∼αmχ, becomes larger than the relative mo-mentum of incoming particles ∼mχv [40]. For large mρ,the Yukawa potential is negligible and Sann

p → 1, whereasin the small mρ limit, the potential and the solutionbecome independent of mρ. For intermediate values ofmρ ' 6αmχ/(π

2(n+ 2)2), with n = 0, 1, 2, . . . [13, 14],the particles form zero-energy bound states and exhibitresonances. The ∆-dependence is weak.

5. Signatures & Constraints.– What are the ro-bust signatures of models employing this selection mech-anism? The primary signal is a velocity-dependent anni-hilation rate, but that in itself is not unique to this mech-anism. The smoking-gun is that the velocity-dependenceis non-monotonic: growing as 1/v at intermediate v,through the competition of v2-suppression of the barep-wave rate and the ∼ 1/v3 Sommerfeld enhancement,and falling off as v2 elsewhere. This means that the con-straints from reionization of the CMB should be easily

4

Dark Matter Mass m� [TeV]

Annih

ilat

ion

Rat

eh�

vi[

cm3s�

1]

H.E.S.S. @MW

IceCube @ Virgo

Fermi-LAT @ dSph

Thermal Relic

Clusters , υ ~ 10-2

Galaxies , υ ~ 10-3100%BRruled out

dSph , υ ~ 10-4

0.3 0.5 1 3 5 1010-32

10-30

10-28

10-26

10-24

10-22

10-20

0.02 0.03 0.05 0.1 0.2 0.3

mχ [TeV]

AnnihilationRate[cm3 /s]

α↵

FIG. 3. p-wave DM annihilation rates in different astrophys-ical sources and source-specific constraints from indirect de-tection searches. The annihilation rates have a signature non-monotonic v-dependence over and above the resonances, e.g.,for mχ > 4 TeV the galactic annihilation rate (solid line) ex-ceeds that in clusters (dashed line) and dwarf galaxies (dot-dashed line). In DM mass-ranges shown by gray verticalbands a 100% branching ratio to µ+µ− is ruled out.

evaded as v is too small, and the signals from galaxy clus-ters, where v is larger, may be small. The signal may pri-marily come from intermediate-sized objects such as theMilky Way (MW), nearby galaxies, and dwarf spheroidalgalaxies (dSph). Interestingly, these late-time p-wave an-nihilation rates may be significantly larger than the relicannihilation rate.

Figure 3 illustrates the signatures and constraints forthis mechanism, through the model described here. Wechoose a representative mediator mass mρ = 30 GeV andmass-gap ∆ = 10 GeV. The relic density constraint issatisfied everywhere; the dark fine-structure constant α(shown on the upper abscissa) is determined for a givenmχ (shown on the lower abscissa) by the s-wave co-annihilation rate πα2/(3m2

χ) ' 〈σv〉relic. At small mχ

the late-time annihilation rate scales as v2, as expectedfor p-wave annihilations when S is not too large. 〈σv〉is the largest in galaxy clusters (dashed line), followedby galaxies (solid line) and dwarf galaxies (dot-dashedline). However, with stronger selective Sommerfeld en-hancement, at larger mχ the rate is the largest in galax-ies. At resonant values of mχ, it may be the largest indwarf galaxies. The Sommerfeld factor at recombina-tion is similar to that in dwarfs, both being in the sat-urated S regime, but the annihilation rate is suppressedby v2

CMB/v2dSph ' 10−8.

Observation of DM annihilation requires a connectionbetween the dark sector and the visible sector. Thisis model-dependent and parametrized in the branch-

ing ratio BR of the annihilation rate to the specificSM particles. As always, indirect detection constrainsBR× 〈σv〉. For these models, constraints obtained usingone source do not directly apply to another, thanks to thenon-monotonic velocity-dependence. Naturally, velocity-resolved multi-source indirect detection of the annihila-tion signal is of key importance here [41–45]. We comparethe predicted rate in each source-class with the limitsobtained for that source-class. In Fig. 3, a 100% branch-ing ratio to µ+µ− is ruled out within the gray verticalbands, due to H.E.S.S. observations of the Milky Way(red shaded region) [46]. Constraints from observationsof dwarf galaxies, e.g., by Fermi-LAT [47] (blue shaded re-gion) and AMS-02 [48, 49], also independently constrainresonant slivers within these bands. Improvements in Ice-Cube observations of the Virgo cluster (green shaded re-gion) [50–52] and Fermi-LAT observations of the Fornaxcluster may be interesting for mχ ' (2-4) TeV [53]. CMBdata are significantly less constraining (not shown), thanfor s-wave models. Surprisingly, a purely p-wave late-time annihilation rate can be larger than 〈σv〉relic and iseminently detectable.

DM has long-range interactions in these models, andconstraints on small-scale structure, e.g., from BulletCluster, may apply [54–59]. For the model parametersconsidered here, they happen to be weak. Specific mod-els may also be constrained using collider limits on dark-visible mixing [60, 61]. A rather generic prediction ofthese models is dark radiation ∆Neff & 0.13 [33], due tothe presence of light mediators or their decay into lightSM particles, that will be detectable via future CMB ob-servations [62, 63].

6. Summary & Outlook.– We have pointed out aselection mechanism that leads to large and possibly ob-servable p-wave annihilation rates in the present Uni-verse, without enhancing s-wave rates. The smoking gunof this mechanism is the signature velocity-dependenceand source-dependence of 〈σv〉, with the possibility of itexceeding 〈σv〉relic. These features are distinctive of largep-wave annihilation of degenerate multi-level DM.

We then discussed a concrete model implementing theselection mechanism and showed that large portions of itsparameter space are already probed by existing experi-ments. The exact constraints are model-dependent, butin general multi-source indirect DM detection, cosmo-logical searches for dark radiation, and small-scale DMstructure are the main avenues for testing this mecha-nism. Collider searches can pin down the dark-to-visiblesector connection.

This mechanism opens a new area for model-buildingand phenomenology, allowing enhanced DM annihila-tions in specific sources where DM has velocities in anoptimal range. As further work, one may also considerthe several variations on this theme: more than two DMparticles in the dark sector, even-s incoming states, re-pulsive interactions, multiple mediators, etc. Some of

5

these possibilities may also turn out to be theoreticallyinteresting and find phenomenological application.

Acknowledgements.– This work was partiallyfunded through a Ramanujan Fellowship of the Dept. ofScience and Technology, Government of India, and theMax-Planck-Partnergroup “Astroparticle Physics” of theMax-Planck-Gesellschaft awarded to B.D. We acknowl-edge use of the FeynCalc package [64] and invaluablehelp from Vladyslav Shtabovenko. We thank RanjanLaha and Kenny C. Y. Ng for their many useful commentson the manuscript.

∗ [email protected]† [email protected]

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