CALT-TH-2016-014

Strong CMB Constraint On P -Wave Annihilating Dark Matter

Haipeng An, Mark B. Wise, and Yue ZhangWalter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125

We consider a dark sector consisting of dark matter that is a Dirac fermion and a scalar mediator.This model has been extensively studied in the past. If the scalar couples to the dark matter ina parity conserving manner then dark matter annihilation to two mediators is dominated by theP -wave channel and hence is suppressed at very low momentum. The indirect detection constraintfrom the anisotropy of the Cosmic Microwave Background is usually thought to be absent in themodel because of this suppression. In this letter we show that dark matter annihilation to boundstates occurs through the S-wave and hence there is a constraint on the parameter space of themodel from the Cosmic Microwave Background.

Introduction. The Standard Model (SM) has no accept-able dark matter (DM) candidate. As its name impliesDM must be uncharged and various direct detection aswell as astrophysical and cosmological constraints existon its couplings to ordinary matter as well as its self in-teractions. These constraints motivate a class of verysimple extensions of the SM that contain a dark sectorwith particles that carry no SM gauge quantum numbers.For thermal DM the minimal dark sector model consistsof the DM and a mediator that the DM annihilates intoin the early universe. There are various possibilities forthe Lorentz quantum numbers of the DM and mediator.Two well studied examples are a Dirac fermion with a me-diator that is either a new massive U(1)D gauge boson(the dark photon) or a massive scalar. In the first casecommunication with the SM degrees of freedom occursthrough the vector portal (via kinetic mixing betweenthe U(1)D and U(1)Y field strength tensors) and in thelatter case through the Higgs portal.

Constraints on the parameter space of these modelsoccur from the so-called indirect detection signals. Anni-hilation of DM in the early universe at the time of recom-bination injects energy into the plasma of SM particleselongating the recombination process and changing ex-pectations for the cosmic microwave background (CMB)radiation anisotropy. Annihilation of DM today in ourgalaxy contributes to electromagnetic and charged par-ticle astrophysical spectra observed, for example, by theFermi satellite.

In a recent paper [1], we have highlighted the role thatDM bound state formation can play on indirect detec-tion signals from DM annihilation in our galaxy whenthe mediator is a dark photon (there bound state forma-tion was not important for the CMB constraint). In thisletter, we again consider the influence of DM bound stateformation on indirect signals but focus on the case wherethe mediator is a real scalar and on the CMB constraint.We impose a parity symmetry on the dark sector withthe real scalar mediator having even parity. Then, theLagrange density for the DM sector is,

L = iχγµ∂µχ−mDχχ−gχχφ+1

2∂µφ∂

µφ− 1

2m2φφ

2 , (1)

p-wave annihilation

capture HmonopoleL

capture HquadrupoleL

Coulomb HmonopoleLHulthen HmonopoleL

mD=5TeV, ΑD=0.27, mΦ=0.8GeV

10-8 10-7 10-6 10-5 10-4 0.001 0.0110-27

10-26

10-25

10-24

10-23

10-22

10-21

v

ΣvHcm

3 �secL

FIG. 1: DM relative velocity dependence in various cross sec-tions. The black curve is the p-wave direct annihilation crosssection for χχ → φφ. The red curve is the (χχ) bound stateformation cross section via monopole transition, evaluatednumerically using Eqs. (4) and (5). The blue curve standsfor quadrupole transition counterpart. The brown line is themonopole transition cross section in the Coulomb limit, whilethe green curve is based on the Hulthen potential which givesa quite good approximation to the realistic Yukawa potential.

where χ and φ are the DM and the dark mediator and theHiggs portal couplings are omitted. This model has beenwell studied for various reasons [2–18]. For DM heavierthan 5 − 10 GeV, direct detection experiments [19] andthe requirement that φ decays before BBN set the lowerbound, mφ > 2mµ ' 0.2 GeV. In our calculations below,we assume a thermal DM relic density, which fixes thevalue of αD = g2/(4π) as a function of the DM mass,mD.

The most often considered DM annihilation process inthis model is χχ→ φφ. The parity of a 2φ system mustbe even and so does the χχ system because parity is con-served by the Lagrange density in Eq. (1). Therefore thisannihilation is mostly P -wave for slow DM and anti-DMparticles. 1 With the P -wave Sommerfeld enhancement

1 If parity was not conserved S-wave annihilation would be possi-ble.

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factor [20] included, the cross section times velocity canbe written as

(σv)P -waveA =

3πα2Dv

2

8m2D

×∣∣∣∣√ 3

4πp2dRp1dr

(r = 0)

∣∣∣∣2 , (2)

where p = mDv/2, v is the relative velocity. Rp` is de-fined as the radial part of the initial scattering wave func-tion (with the relative momentum aligned along the z-axis), Ψ~p=pz(~r) =

∑`Rp`(r)Y`0(r), and Ψ~p(~r) is asymp-

totic to exp(i~p·~r) at infinity. A typical curve of (σv)P -waveA

as a function of v is the black curve in Fig. 1. As v getssmaller, (σv)P -wave

A first grows as 1/v due to the Som-merfeld enhancement, and then at around v ∼ mφ/mD,(σv)P -wave

A gets strongly suppressed. The drop-off is dueto the effective potential barrier at r ∼ m−1φ generatedby the sum of the attractive Yukawa potential and therepulsive centrifugal potential. The transmission coeffi-cient for tunneling through the barrier diminishes as v2

in the small v limit, as illustrated by Fig. 1.After thermal freeze out (chemical decoupling), DM

can still maintain kinetic equilibrium with the φ particlesin the universe. The DM velocity only red-shifts linearlywith the expansion after the kinetic decoupling. For DMmass in the TeV range, their relative velocity v during re-combination is extremely small, v �

√Trec/mD ∼ 10−6,

where Trec is the temperature of the universe at the re-combination era. Hence it has been thought that therewill be no CMB constraint for the P -wave annihilatingDM in this model. In this letter, we show that this isnot the case. In some regions of parameter space, a pairof free DM particles can capture into a DM bound statevia the emission of a φ particle, and then annihilate intoφ’s inside the bound state. The bound state formationprocess dominantly occurs in an S-wave and therefore isnot suppressed at low velocity due to the absence of thecentrifugal potential barrier. The mediator eventuallydecays to SM particles via the Higgs portal resulting in aCMB constraint on the region of the parameter space inthe model where the kinematics allows for bound stateformation.

Bound state formation cross section. The Hamiltonianfor a non-relativistic DM-anti-DM system interactingwith the mediator field is (in the center of mass frame)

Hint = g [φ (~r/2) + φ (−~r/2)]

−g [φ (~r/2)− φ (−~r/2)]∇2

2m, (3)

where g is the dark Yukawa coupling. ~r is the relativeposition of the DM-and-anti-DM particles, and φ is theSchrodinger picture mediator field. In the bound stateformation transition amplitude a mediator particle is cre-ated by the field φ . The mode expansion of the mediatorfield has exponential dependence on the wave-vector ~k

that can be expanded, e±i~k·~r = 1± i~k ·~r− (~k ·~r)2/2+ · · · .

In the first line of Eq. (3), due to the orthogonality be-tween the initial and final states, the leading order con-tribution vanishes. The contributions at the i~k · ~r orderfrom DM and anti-DM cancel with each other. The con-tribution from the (~k · ~r)2 order yields both monopoleand quadrupole transitions. The second line of Eq. (3)represents the leading relativistic correction, which con-tributes to the monopole transition at the zeroth orderin ~k · ~r.

The bound state formation cross section times the rel-ative velocity can be written as

σv =∑f

∑M=M,Q

∫d3~k

(2π)32k0(2π)δ(Ef + k0−Ei)|VM

fi |2 ,

(4)where Ei and Ef are the energies of the initial and finalstates of the DM-anti-DM system. The sum over f isover final bound state azimuthal, magnetic, and princi-pal quantum numbers, but because we have aligned thedark matter relative momentum along the z-axis only themagnetic quantum number m = 0 contributes. Here weare neglecting the spin degrees of freedom for the darkmatter. Including them would give a factor of 1/4 fromspin averaging and then for each f = n, l,m there wouldbe four final bound states; one with spin 0 and three withspin 1.

For the monopole (M) transition

|VMfi |2 =g2∣∣∣∣∫ drr2

[1

12k2r2 +

αDe−mφr

mDr

]Rn`(r)Rp`(r)

∣∣∣∣2 ,(5)

where k ≡ |~k|, Rk` and Rn` are the initial and final radialwave functions. For quadrupole transition,

|V Qfi |2 =

g2k4

120

[(`+ 1)(`+ 2)

(2`+ 1)(2`+ 3)

∣∣∣∣∫ drr4R∗n`(r)Rp`+2(r)

∣∣∣∣2+

2`(`+ 1)

3(2`− 1)(2`+ 3)

∣∣∣∣∫ drr4R∗n`(r)Rp`(r)

∣∣∣∣2+

`(`− 1)

(2`− 1)(2`+ 1)

∣∣∣∣∫ drr4R∗n`(r)Rp`−2(r)

∣∣∣∣2].(6)

During the time of recombination the DM and anti-DM particles have negligible kinetic energy, hence toemit an on-shell φ, mφ < α2

DmD/(4n2) is required in

the Coulomb limit. This indicates mφ � αDmD/(2n).Therefore, the relevant bound state wave functions can betreated as Coulombic for the computation of the boundstate formation cross section. On the other hand, wesolve for the scattering state wave functions numericallyusing the shooting method described in [1].

From numerical solutions, we find that after sum-ming over the azimuthal quantum number `, for boththe monopole and quadrupole transitions, (σv) ∼ n−2

roughly. For mD = 5.0 TeV, αD = 0.27, mφ = 0.8 GeV,

3

CMB excl.

mD=5 TeV, ΑD=0.27, v=10-8

0.2 0.5 1.0 2.0 5.0

10-23

10-22

mΦ HGeVL

HΣv

L CM

BHcm

3�se

cL

FIG. 2: Scalar mediator mass dependence in the bound stateformation cross section at very low DM velocity, v � mφ/mD.This is the cross section to be constrained by the CMB ob-servation. In this plot, the yellow shaded region is excluded.The magenta dot-dashed line is the appoximate envelop ofthe blue curves using Eq. (11).

the numerical solution of total cross sections times ve-locity for the monopole and quadrupole transitions areshown as the red and blue curves in Fig. 1 respectively.

For v > mφ/mD, σv goes like v−1 and agrees with theresult from the Coulomb potential scattering states whichis shown by the brown line in Fig. 1. For v � αD, theCoulomb scattering wavefunction takes the approximateform

Rp`(r) ' 4π

√2`+ 1

4prJ2`+1

(√4αDmDr

). (7)

In this limit, the monopole transition cross section timesvelocity can be written as

(σv)Mn` =24`+7(2`+ 1)n2`−2Γ(n− `)π2α5

D

9Γ(n+ `+ 1)e4nm2Dv

(L2`+1n−`−1(4n)

)2,

(8)where L is the associated Laguerre polynomial, and herewe have used Eq. (7.421 (4)) of [21]. For the ground state(n = 1, ` = 0) formation, it can be simplified to (σv)M10 =

128π2α5D/(9e

4m2Dv). The quadrupole piece is, (σv)Q10 =

512π2α5D/(45e4m2

Dv). The v−1 behavior originates fromthe Sommerfeld enhancement.

Due to the potential barrier, the contributions fromincoming partial waves with ` > 0 are suppressed whenv < mφ/mD. This causes the sharp drop-off in thesmaller v direction in the curve for quadrupole transi-tions in the region v . mφ/mD, which is roughly 10−4 inFig. 1. The only transition that does not get suppressedby the barrier is from ` = 0 to ` = 2, which causes theblue curve to plateau at the small v region. However, itsvalue is suppressed by the phase space since the ` = 2bound state starts from n = 3.

In the case of finite mφ, the Huthen potential canbe used as an approximation to the Yukawa potential.

This is a useful approximation for S-wave scattering. Formφ � αDmD, the incoming S-wave function can be ap-proximated as [20]

Rp0(r) =

√4π

αDmDr

∣∣∣∣Γ(a−)Γ(a+)

Γ(1 + 2iw)

∣∣∣∣ J1 (√4αDmDr),

(9)where w = mDv/(2mφ), a± = 1 + iw(1 ±

√1− x/w)

and x = 2αD/v. One can get an analytic solution of themonopole transition into the S-wave bound state. In thelimit v � mφ/mD

(σv)Mn0 =26πα4

D

9n3m2D

∣∣∣∣Γ(a−)Γ(a+)

Γ(1 + 2iw)

∣∣∣∣2 e−4n (L1n−1(4n)

)2,

(10)This simplifies to,

(σv)Mn0 =26π3α5

De−4n (L1

n−1(4n))2

9n3mDmφ sin2(π√αDmD/mφ

) , (11)

unless the value of√αDmD/mφ is very close to an in-

teger. The divergence one encounters in the cross sec-tion (σv)Mn0 using the expression above will be regularizedby the small imaginary parts of a±. For values of mφ,where an S-wave state crosses threshold a peak appearsin (σv)Mn0. This structure is depicted in Fig. 2, where theapproximate lower envelop corresponding to Eq. (11) bysending the sine square factor in the dominator to unityis also shown as the magenta dot-dashed curve.

Annihilation decay. In the simple model of Eq. (1), thereexist two ground states with quantum numbers JPC =1−− and 0−+. The 1−− state, once formed, is stable dueto the C-parity symmetry. It is part of the dark matter.The 0−+ state, on the other hand, can decay. It is easy toverify that systems made of 2 and 3 real scalars are parityeven. Because in this model, the Yukawa interaction alsopreserves parity, the leading decay channel of the 0−+

state is into 4φ’s, and the decay rate is,

Γ(0−+)→4φ =F |ΨB(0)|2α4

D

192π2m2D

, (12)

where F ' 0.01 has been determined numerically.

CMB constraint. The 0−+ bound state is spin-singletand 1−−, spin-triplet. Therefore, in this simple model(1) only 1/4 of the dark bound states can decay. In theCoulomb limit, the ground state wave function at theorigin is, ΨB(0) =

√α3Dm

3D/(8π). Thus, the lifetime of

the 0−+ state is very short compared to the cosmologi-cal time scale during the recombination era. As a result,during recombination, the total formation rate of the 0−+

bound states is equal to their overall decay rate. The en-ergy injection rate due to DM annihilation via the boundstate channel is proportional to the 0−+ bound state for-mation cross section [1]. (The 1−− state is stable andthese bound states are part of the DM today.)

4

LU

X+

BB

Nex

cl.

mΦ>1

4ΑD

2 mD

0.1 0.5 1.0 5.0 10.0 50.0 100.0

500

1000

2000

5000

0.05

0.1

0.15

0.2

0.3

mΦ HGeVL

mD

HGeV

L

Α D

FIG. 3: The blue region is the parameter space excluded bythe CMB due to bound state formation, which is the mainpoint of this work. The yellow region is known to be excludedjointly by BBN and direct detection experiments.

This cross section is bounded from above, in order notto distort the CMB spectrum, which is roughly [22],

limv→0

(σv) < 3× 10−24 cm3sec−1 ×(mD

TeV

). (13)

Based on the bound state formation cross section we de-rived in Eq. (4), the CMB constraint is shown by Fig. 3,where the blue region is excluded by current Planckdata [23]. The blue solid triangle region in the upper-left corner of Fig. 3 is fully excluded. The strips in thelarger mφ region are also ruled out due to the resonanceeffect.

Concluding Remark. In this letter we have shown thatfor a scalar mediator dark matter annihilation into boundstates can give rise to qualitatively different physical ef-fects. Without including bound state formation, at thetime of recombination, the constraints in Fig. 3 (whichare the main results of this letter) would be absent.

Additional Comments. The DM-anti-DM particles canalso annihilate into 4φ’s when they are in an S-wave ini-tial scattering state. With the Sommerfeld enhancement,the cross section times velocity for this channel is,

(σv)S-waveA =

1

4

|ΨS(0)|2

|ΨB(0)|2Γ(0−+)→4φ , (14)

where |ΨS(0)|2 = |Γ(a−)Γ(a+)/Γ(1 + 2iw)|2 is the scat-tering state wavefunction at the origin. In practice, wefind the ratio (σv)S-wave

A /(σv)M10 = 3e4F/(16384π3)� 1,so such a direct annihilation is numerically irrelevantthroughout our analysis.

In Eq. (1) the DM is assumed to be a Dirac fermion.If it is a Majorana fermion, its direct annihilation is stilldominated by the P -wave channel. In this case, due toits Majorana nature, the S-wave two-DM system, canonly be in a spin singlet. As a result only 1/4 of DM

annihilation can happen via the monopole transition intothe ground state. One expects a similar constraint as inthe case of Dirac DM.

Acknowledgement. We acknowledge an illuminating con-versation with Haibo Yu. This work is supported by theDOE Grant de-sc0011632, de-sc0010255, and by the Gor-don and Betty Moore Foundation through Grant No. 776to the Caltech Moore Center for Theoretical Cosmol-ogy and Physics. We are also grateful for the supportprovided by the Walter Burke Institute for TheoreticalPhysics.

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