On the effective operators for Dark Matter annihilations

0.71 MB·English

by  Andrea De Simone

CERN-PH-TH/2012-369
SISSA 01/2013/FISI
On the effective operators for
Dark Matter annihilations
3 Andrea De Simone1,2,3, Alexander Monin4, Andrea Thamm1,4, Alfredo Urbano2
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n 1 CERN, Theory Division, CH-1211 Geneva 23, Switzerland
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J 2 SISSA, via Bonomea 265, I-34136 Trieste, Italy

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3 INFN, sezione di Trieste, I-34136 Trieste, Italy

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h 4 Institut de Th´eorie des Ph´enom`enes Physiques,
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E´cole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland
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p
e
h
[

1
v Abstract
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8
WeconsidereffectiveoperatorsdescribingDarkMatter(DM)interactionswithStandard
4
1 Model fermions. In the non-relativistic limit of the DM field, the operators can be
.
1 organized according to their mass dimension and their velocity behaviour, i.e. whether
0
they describe s- or p-wave annihilations. The analysis is carried out for self-conjugate
3
1 DM (real scalar or Majorana fermion). In this case, the helicity suppression at work in
:
v the annihilation into fermions is lifted by electroweak bremsstrahlung. We construct and
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X study all dimension-8 operators encoding such an effect. These results are of interest in
r indirect DM searches.
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1 Introduction
SignificantexperimentalactivityiscurrentlydevotedtothesearchforDarkMatter(DM)bylooking
at the excesses in cosmic ray production through DM annihilations (or decays) in the galactic halo.
Detailed predictions for the fluxes largely depend on the particle physics model for the DM, e.g.
the mass, the annihilation channels etc. It is therefore desirable to describe DM annihilations and
theirproductswithinageneralandmodel-independentframeworkandEffectiveFieldTheory(EFT)
provides such a tool.
Of course, the EFT is only applicable whenever there is a separation of scales between the
process to describe (the annihilation of non-relativistic DM at a scale ∼ M ) and the underlying
DM
microscopic physics of the interactions (at a scale Λ). This may not always be the case, as, for
example, in the case of supersymmetry with a compressed spectrum.
If we want to describe the annihilation of two non-relativistic DM particles, whose relative
velocity is v ∼ 10−3 (in units of c) in our Galaxy today, it is convenient to expand the cross section
in powers of v
vσ = a+bv2+O(v4), (1.1)
where the first term corresponds to annihilation in the state of orbital angular momentum L = 0
(s-wave) while the second term describes L = 1 (p-wave). For the annihilation DM DM → ff¯of a
self-conjugate DM particle (real scalar or Majorana fermion) into SM fermions of mass m , helicity
f
arguments lead to a ∝ (m /M )2, and hence a very suppressed s-wave term for light final state
f DM
fermions (e.g. leptons), while the p-wave is suppressed by v2.
It is clear that the correct operator expansion must be done in terms of two parameters: mass
dimension of the operator and relative velocity. The effective lagrangian would be generically given
by an infinite series of non-renormalizable operators
(cid:88) 1 (cid:16) (cid:17)
L = c(d,s)O(d,s)+c(d,p)O(d,p) , (1.2)
eff Λd−4
d>4
where O(d,sorp) indicates that the operator of dimension d describes s-wave or p-wave annihilations,
and we neglect annihilations in waves higher than p. There is no obvious ordering of the importance
of the operators. An important role in this respect is played by ElectroWeak (EW) bremsstrahlung,
which has received significant attention recently [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] (for earlier studies
on the impact of gauge boson radiation on DM annihilations or cosmic ray physics, see [12, 13, 14]).
Taking into account processes with the inclusion of EW radiation eludes the helicity suppression
and opens up an s-wave contribution to the cross section [2, 3, 6, 7, 10, 12].
In this paper we will classify the operators (up to dimension 8) according to the v-behaviour of
the amplitude connecting two self-conjugate particles – real scalars or Majorana fermions – in the
initial state with the final state of two massless fermions and possibly a gauge boson.
The effects of lifting the helicity suppression by means of EW radiation has so far been studied
withinthecontextofexplicitmodels[2,3,6,7,10]. Theresultsofthepresentpaperprovideamodel-
independentapproachtothisproblemandcanbeusedtoplacerobustconstraintsonthenewphysics
responsible for the DM sector. Although several analyses in the literature place phenomenological
constraints on the coefficients of the dimension-six operators (see e.g. Refs. [15, 16, 17, 18]), the
important role of higher-dimensional operators is typically underestimated.
1
The remainder of the paper is organized as follows. In Section 2, we explain our methodology
and construct the effective operators contributing to s-wave DM annihilations. We compute the
differential and total annihilation cross sections for each operator in Section 3, and compare them to
thecontributionfromthetypicallowest-dimensionaloperator. WeconcludeinSection4, mentioning
phenomenological applications and possible analyses that could be carried out using these results.
Finally, we collect some useful relations and identities in the Appendices, for the convenience of the
reader.
2 Effective operators
We will carry out our analysis under the following set of assumptions on the DM sector:
1. the DM is either a Majorana fermion or a real scalar field;
2. the DM is neutral under the SM gauge group;
3. there exists a Z symmetry under which the DM is odd and the SM is even;
2
4. the DM couples only to the fermions in the SM spectrum, which are assumed to be massless.
Ofcourse,anyoftheseassumptionsmaynotholdinrealityandifthisisthecaseouranalysisrequires
modifications. Assumption1specifiestheconditionsunderwhichthehelicitysuppressioniseffective,
while assumption 2 is there to simplify the discussion, even though it is not strictly necessary and
relaxing this assumption can also lead to interesting effects (see Refs. [7, 19]). Assumption 3 is
commonly used in DM phenomenology to ensure the stability of the DM particle.
Moreclarificationsaboutassumption4areinorder. Consideringm (cid:54)= 0wouldintroduceanother
f
mass scale into the problem and would render the operator classification much less transparent.
Our analysis is still valid in the regime m /Λ (cid:28) v, which may not hold for the third-generation
f
quarks. The other piece of information in assumption 4 is that the DM particle only couples to
the fermion sector of the SM. This needs not to be true, of course. Allowing for DM interactions
with the other SM particles, namely gauge bosons or the Higgs, the possibility of having additional
operators contributing to DM annihilations in s-wave opens up. For instance for Majorana DM, χ,
coupling to the Higgs doublet H or a generic field-strength Fµν one can have (χ¯γ5χ)(H†H) and
(cid:0)χ¯γ5χ(cid:1)F Fµν, which are CP odd, and (cid:0)χ¯γ5χ(cid:1)F F˜µν, which is CP even. For a list of effective
µν µν
operators connecting DM to vector bosons and/or Higgs bosons see e.g. Refs.[20, 21]. We will not
consider these possibilities here.
We want to classify the operators of dimension d = 6,7,8 according to the v-behaviour of
the amplitude connecting two DM particles in the initial state with two massless SM fermions f
and a gauge boson in the final state. We look for operators which are hermitian, gauge invariant
(under SU(2) ⊗U(1) ) and giving a non-zero contribution to the amplitude for the annihilation
L Y
of DM into two fermions and one gauge boson. We will further classify the operators according to
their CP transformation properties (see Appendix A for the relevant transformation properties).
The operators containing D/f give no contribution to the process under consideration, due to the
Equation Of Motion (EOM). In order to ensure manifest gauge invariance, we first introduce the
2
following notation for the covariant derivative
→− (cid:16)→− (cid:17)
D f = ∂ +igTaWa+ig(cid:48)Y B f, (2.1)
µ µ µ f µ
←− (cid:16)←− (cid:17)
f¯D = f¯ ∂ −igTaWa−ig(cid:48)Y B , (2.2)
µ µ µ f µ
where Ta = σa/2, with σa being the usual Pauli matrices, and the charge Q of the fermion f is
f
related to its hypercharge Y by Q = T3 + Y . Furthermore, the field-strength W of the SM
f f f µν
gauge fields is related to the covariant derivatives as usual
(cid:104)→− →− (cid:105)
D ,D f = (igTa)Wa f, (2.3)
µ ν µν
(cid:104)←− ←− (cid:105)
f D ,D = f(igTa)Wa . (2.4)
µ ν µν
In the following, we will separately deal with the cases where the DM is a Majorana fermion or a
real scalar. For simplicity, we will consider only left-handed SM fermions f , but the analysis can
L
be applied to right-handed fermions straightforwardly.
2.1 Majorana fermion DM
LetusfirstsupposetheDMparticleisaMajoranafermionχ. Itispossibletobuildseveraloperators
containing two DM fields, two SM fermion fields and zero or one gauge bosons. They can be built
in full generality requiring gauge invariance and hermiticity, and further classified according to their
CP properties. The Majorana-flip properties and the chiralities of the SM fermions make several
structures identically zero. The only two Majorana fermion bilinears that are non-vanishing in the
limit v → 0 are χ¯γ5χ and χ¯γµγ5χ (see Appendix B). As we are interested in s-waves only, we will
limit our analysis to these two bilinears for the Majorana fermions. For the pseudo-scalar bilinear
χ¯γ5χ no contractions with SM fermions can be built, as they would vanish either by chirality or by
the EOM of fermions, so we are left with the axial-vector bilinear.
The lowest-dimensional terms that can be written out of two χ’s and two f ’s are of dimension
L
6. At this level, there is only one non-vanishing operator satisfying all criteria:
O = (cid:0)χ¯γ5γµχ(cid:1)(cid:2)f¯ γ f (cid:3) . (2.5)
M L µ L
The µ = 0 component of this operator could a priori give a v-independent contribution to the
scattering amplitude, but it actually vanishes because of the identity u†(p)v (−p) = 0. Therefore
f f
this dimension-6 operator only contributes to the p-wave, as noted in Ref. [3], due to the helicity
suppression, which cannot be removed by simply radiating a gauge boson from the external final
state leg. In order to look for s-wave terms, we need to consider higher-dimensional operators with
one EW gauge boson whose radiation in the annihilation process lifts the helicity suppression.
At dimenion 7, there are no operators contributing to the s-wave cross section. In fact, all
possible structures vanish either due to Majorana-flip properties or the chirality of the SM fermions,
or because of the EOM of the f s. A priori, the µ = 0 component of the operator (χ¯γ5∂ χ)[f γµf ]
L µ L L
would give a v-independent contribution to the scattering amplitude, but it vanishes again because
of the identity u†(p)v (−p) = 0.
f f
At the level of dimension 8, there are several structures that can be built requiring gauge invari-
ance and hermiticity. They contain two χ’s (in the bilinear χ¯γµγ5χ), two f ’s and two covariant
L
3
Name Operator CP
(cid:104)(cid:16) ←− (cid:17) (cid:16)→− (cid:17)(cid:105)
O (cid:0)χγ5γµχ(cid:1) f D γ Dρf +
M1 L ρ µ L
(cid:104) →− (cid:16)→− (cid:17) (cid:16) ←− (cid:17)←− (cid:105)
O i(cid:15) (cid:0)χγ5γµχ(cid:1) f γνDρ Dσf − f Dσ Dργνf +
M2 µνρσ L L L L
(cid:104)(cid:16) ←− (cid:17) (cid:16)→− (cid:17) (cid:16) ←− (cid:17) (cid:16)→− (cid:17)(cid:105)
O i(cid:15) (cid:0)χγ5γµχ(cid:1) f Dν γρ Dσf − f Dσ γρ Dνf +
M3 µνρσ L L L L
(cid:104) →− (cid:16)→− (cid:17) (cid:16) ←− (cid:17)←− (cid:105)
O i(cid:0)χγ5γµχ(cid:1) f D/ D f − f D D/f −
M4 L µ L L µ L
(cid:104) →− (cid:16)→− (cid:17) (cid:16) ←− (cid:17)←− (cid:105)
O i(cid:0)χγ5γµχ(cid:1) f γ D Dρf − f D Dργ f −
M5 L µ ρ L L ρ µ L
Table 1: List of dimension-8 operators contributing to the s-wave cross section for the annihilation
of Majorana DM into two fermions and a gauge boson.
derivatives. It is possible to reduce the number of independent operators, contributing to the cross
section for the process under consideration, by using EOM and the identities in Appendix C. In
addition, some structures can be related to each other by terms (like (cid:0)χ¯γ5γµχ(cid:1)∂2(cid:2)f¯ γ f(cid:3)), which
L µ
do not contribute to the s-wave annihilation into two fermions and a gauge boson. Therefore they
contribute in exactly the same way to the amplitude for the process we are interested in. There
remain only five independent operators of dimension 8 contributing to the s-wave annihilation of
DM into two SM fermions and a gauge boson, listed in Table 1 together with their CP conjugation
properties. We remain agnostic about the presence or absence of CP violation in the Dark Matter
sector, which can possibly induce CP violation in the SM at loop level and therefore be further
constrained.
All other operators have either a larger dimensionality or produce more powers of v2 in the
annihilation cross section. Notice that we chose to keep a Lorentz-covariant formalism, despite
looking at the non-relativistic limit. This implies that the same operator can lead to both v-
independent and v-dependent terms in the amplitudes; for example, the operator O also gives a
M1
contribution to the p-wave cross section, but we will not consider it as it is very suppressed.
To summarize, the dimension-8 operator contributing to the s-wave annihilation cross section
(8,s)
of Majorana DM into SM fermions is given by the sum of the operators in Table 1, O =
M
(cid:80)5 c(8,s)O ; the leading interactions are therefore described in terms of only a few operators
i=1 i Mi
5
L = 1 c(6,p)O + 1 (cid:88)c(8,s)O +higher-dim. (2.6)
eff Λ2 M Λ4 i Mi
i=1
In absence of CP-violation in the DM sector, only the first three operators in the sum need to be
considered.
2.2 Real scalar DM
Next, let us consider the case where the DM particle is a real scalar φ. By angular momentum
conservation, two real scalars cannot annihilate into two massless fermions in the configuration with
4
Name Operator CP
(cid:104) →−→− →− ←− ←− ←− (cid:105)
O iφ2 f D/Dν(D f )−(f D )DνD/f +
R1 L ν L L ν L
(cid:104) →− →− →− ←− ←−←− (cid:105)
O iφ2 f DνD/(D f )−(f D )D/Dνf +
R2 L ν L L ν L
(cid:104) →− →− →− ←− ←− ←− (cid:105)
O φ2(cid:15) f γµDνDρ(Dσf )+(f Dσ)DρDνγµf +
R3 µνρσ L L L L
(cid:104) →− ←− (cid:105)
O i(∂ φ∂ φ) f¯ γµDνf −f¯ Dνγµf +
R4 µ ν L L L L
(cid:104) ←− →− →− ←− ←− →− (cid:105)
O φ2 (f Dν)D/(D f )+(f D )D/(Dνf ) −
R5 L ν L L ν L
(cid:104) →− →− →− ←− ←− ←− (cid:105)
O iφ2(cid:15) f γµDνDρ(Dσf )−(f Dσ)DρDνγµf −
R6 µνρσ L L L L
(cid:104) ←− →− →− ←− ←− →− (cid:105)
O iφ2(cid:15) (f Dν)γµDρ(Dσf )−(f Dσ)Dργµ(Dνf ) −
R7 µνρσ L L L L
Table 2: List of dimension-8 operators contributing to the s-wave cross section for the annihilation
of real scalar DM into two fermions and a gauge boson.
L = 0; this process would only occur through a chirality flip induced by a mass term. For this
reason, in the limit m = 0 we are considering here, no s-wave annihilation φφ → f¯ f is possible.
f L L
One can recover this result in the language of effective operators. At dimension 5, only a single
operator can be constructed which, however, vanishes due to chirality, φ2(f f ) = 0. At dimension
L L
6, we have φ2∂ (f γµf ) = 0, by the EOM (cf. Eq. (C.2)). At dimension 7, there are no possible
µ L L
Lorentz contractions to construct an operator.
Nevertheless, atthelevelofdimension8, severalgaugeinvarianthermitianoperatorscanbebuilt
out of two φ’s, two f ’s and covariant derivatives. As discussed already for the Majorana case, it
L
is possible to reduce the number of independent operators, contributing to the cross section for the
process under consideration, by using the EOM and the identities in Appendix C. We are left with
four CP-even operators and three CP-odd operators, listed in Table 2. A v-dependent annihilation
φφ → f¯ f is mediated by the operator O , while the s-wave annihilation of two φ’s can proceed
L L R4
by switching on the emission of a gauge boson in the final state.
Otheroperatorsinvolvingφ∂ φoragaugefield-strengthFµν canbeobtainedfromthelistedones
µ
byintegrationbyparts, usingtheEOMoftheDMparticleortheidentities(2.3)-(2.4). Forinstance,
theoperatorφ2∂ (cid:2)f¯ γ f (cid:3)Fµν, consideredinRef.[22], isexpressedinthisbasisas1/g(O −O ).
ν L µ L R1 R2
To summarize, the dimension-8 operator contributing to the s-wave annihilation cross section
(8,s)
of real scalar DM into SM fermions is given by the sum of the operators in Table 2: O =
R
(cid:80)7 c(8,s)O ; the leading interactions are therefore described in terms of only a few operators
i=1 i Ri
7
L = 1 (cid:88)c(8,s)O +higher-dim. (2.7)
eff Λ4 i Ri
i=1
In absence of CP-violation in the DM sector, only the first four operators in the sum need to be
considered.
5
Figure 1: Diagrams for the annihilation process in Eq. (3.1).
3 Annihilation cross sections
In this section we show analytical results for the annihilation cross sections due to the operators
found above. For simplicity, we restrict ourselves to considering left-handed SM fermions only, but
the results can be easily adapted to account for annihilations into right-handed fermions as well. We
consider the process
DM(k )DM(k ) → f (p )f¯ (p )V(k), (3.1)
1 2 i,L 1 j,L 2
where light fermions in the final state - described here by the generic SU(2) doublet F = (f ,f )T
L 1 2
- can be both leptons and quarks, and where V = W±,Z,γ. Note that diagrams with gauge boson
emissionfromthefinalstatelegshavetobeincludedinordertocomputeagaugeinvariantamplitude
(see Fig. 1).
It is convenient to introduce the kinematical variables y,z defined by
√
p0 = (1−y) s/2, (3.2)
1
√
p0 = (1−z) s/2, (3.3)
2
√
k0 = (y+z) s/2, (3.4)
which are subject to the following phase space constraints
m2 m2 m2
V ≤ y ≤ 1, V ≤ z ≤ 1−y+ V , (3.5)
s sy s
where s = (k + k )2 = 4M2 /(1 − v2/4). The scattering amplitudes will be proportional to a
1 2 DM
coefficient A containing the correct gauge couplings according to the different possible final states
and their SU(2) ⊗U(1) quantum numbers; more explicitly,
L Y
gs gs
W W
A(f f γ) = − (1+y ), A(f f γ) = + (1−y ),
1 1 f 2 2 f
2 2
g g
A(f f Z) = − [1−(1+y )s2 ], A(f f Z) = + [1−(1−y )s2 ], (3.6)
1 1 2c f W 2 2 2c f W
W W
g g
A(f f W+) = −√ , A(f f W−) = −√ ,
1 2 2 1
2 2
where Y = y /2 (e.g. y = −1, y = 1/3), g is the gauge coupling and s ,c are the sin and cos
f f L Q W W
of the weak angle, respectively.
Thedouble-differentialannihilationcrosssectionisgivenintermsofthesquaredamplitude|M|2
(averaged over the initial spins and summer over the final ones) as
d2σ |M|2
v = , (3.7)
dydz 128π3
6
while the total cross section is obtained by integrating over the kinematical variables on the phase
space domain defined by Eq. (3.5). Let us now show the results of the computation of these cross
sections for each of the operators found in the previous section.
3.1 Majorana fermion DM
In the zero-velocity approximation, s = 4M2 and the double-differential cross sections for the
DM
annihilation process (3.1) mediated by the dimension-8 operators O ,...,O in Table 1 are
M1 M5
v d2σ (cid:12)(cid:12)(cid:12) = |c(8,s)|2A2MD6M (cid:20)1−y−z+ m2V (cid:21)(cid:20)y2+z2− m2V (cid:21) , (3.8)
dydz(cid:12) M1 2π3Λ8 4M2 2M2
OM1 DM DM
v d2σ (cid:12)(cid:12)(cid:12) = 4 |c(M8,2s)|2 v d2σ (cid:12)(cid:12)(cid:12) , (3.9)
dydz(cid:12)OM2 |c(M8,1s)|2 dydz(cid:12)OM1
v d2σ (cid:12)(cid:12)(cid:12) = 4 |c(M8,3s)|2 v d2σ (cid:12)(cid:12)(cid:12) , (3.10)
dydz(cid:12)OM3 |c(M8,1s)|2 dydz(cid:12)OM1
v d2σ (cid:12)(cid:12)(cid:12) = |c(8,s)|22A2MD6M (cid:20)(1−y−z)(y2+z2)+ m2V (cid:2)(y+z)2−2(y+z)+2(cid:3)(cid:21) ,
dydz(cid:12) M4 π3Λ8 4M2
OM4 DM
(3.11)
v d2σ (cid:12)(cid:12)(cid:12) = |c(M8,5s)|2 v d2σ (cid:12)(cid:12)(cid:12) . (3.12)
dydz(cid:12)OM5 |c(M8,4s)|2 dydz(cid:12)OM4
Notice that in the limit m /M → 0 the differential cross sections are the same for all operators,
V DM
up to an overall numerical factor.
IntheindirectsearchesforDM,theobservablesmeasuredexperimentallyarethefluxesofcosmic
rays, which are directly related to the energy spectra of particles generated by DM annihilations
at the production point. By integrating the double-differential cross sections listed above once, one
obtains the energy spectra of the SM fermions and of the gauge bosons at production. We consider
the distributions of the final fermion energy (E ) and of the final gauge boson energy (E ), defined
f V
as
dN 1 dσ(DMDM → ffV)
≡ , (3.13)
dlnx σ(DMDM → ffV) dlnx
where x ≡ E /M , and shown in Fig. 2.
f,V DM
The distributions originating from the set of operators O ,O ,O differ just by an overall
M1 M2 M3
factor, which cancels out by normalizing the spectra, and therefore produce the same curves. The
same argument applies to the other set O ,O . However, the operator O would instead give a
M4 M5 M1
differentresultwithrespecttoO ,butthedifferenceisnotvisibleinthefermionenergydistribution
M4
(left panel of Fig. 2) because of the smallness of m /M .
V DM
The dimension-6 operator O mediates the two-body annihilation χχ → f f in p-wave. The
M L L
processes where a gauge boson is radiated from the final state fermion are still in p-wave, as dis-
cussed in Ref. [3]. Therefore, the energy spectra of fermions and gauge bosons originating from
the dimension-6 operator are going to be much less important than those from the dimension-8
operators. The total cross section for the two-body process χχ → f f mediated by O is simply
L L M
vσ(χχ → fLfL)(cid:12)(cid:12)M = |c(6,p)|21M2πD2ΛM4v2, (3.14)
7
1
1
M1,M2,M3
M1,M2,M3,M4,M5 10(cid:45)1
x 10(cid:45)1 x
n n
l l
d d
(cid:144) (cid:144)
dN dN 10(cid:45)2
10(cid:45)2
M4,M5
10(cid:45)3
10(cid:45)3 MDM(cid:61)1TeV MDM(cid:61)1TeV
0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0
x (cid:61) Ef(cid:145)MDM x (cid:61) EV(cid:145)MDM
Figure 2: Majorana fermion DM. The energy distributions dN/dln(E/M ) of the final fermion
DM
(left panel) and of the final gauge boson (right panel), for the different dimenion-8 operators. We
set c(6,p) = c(8,s) = 1 and M = 1 TeV.
Mi DM
whilethetotalcrosssectionsofthethree-bodyprocesses(3.1)mediatedbythedimension-8operators
are obtained by integrating over the full phase space
A2M6 (cid:20) m2 m4 (cid:18) 2M (cid:19) (cid:18) m6 (cid:19)(cid:21)
vσ| = |c(8,s)|2 DM 4−15 V +5 V −4+6ln DM +O V ,(3.15)
M1 M1 240π3Λ8 M2 M4 m M6
DM DM V DM
|c(8,s)|2
vσ| = 4 M2 vσ| , (3.16)
M2 |c(8,s)|2 M1
M1
|c(8,s)|2
vσ| = 4 M3 vσ| , (3.17)
M3 |c(8,s)|2 M1
M1
A2M6 (cid:20) m2 m4 (cid:18) 2M (cid:19) (cid:18) m6 (cid:19)(cid:21)
vσ| = |c(8,s)|2 DM 8+15 V + V 40−60ln DM +O V ,(3.18)
M4 M4 120π3Λ8 M2 M4 m M6
DM DM V DM
|c(8,s)|2
vσ| = M5 vσ| . (3.19)
M5 |c(8,s)|2 M4
M4
Forsimplicity, wehaveonlyreportedheretheleadingtermsintheexpansioninpowersofm /M ,
V DM
but the complete analytical expressions are used in the plots. The sub-leading terms can be of the
same order as the contributions from higher-dimensional operators we are neglecting.
The relative importance of the s-wave three-body process due to dimension-8 operators with
respect to the p-wave two-body annihilation due to the dimenion-6 operator is captured by the
ratios of the total cross sections, plotted in Fig. 3. Three-body cross sections can be sizeably larger
thanthetwo-bodyones. Itisevidentthatthedimension-8operatorsdominatethetotalcrosssection,
provided that the effective operator scale Λ is not too large with respect to the DM mass M . It
DM
is clear that limiting an EFT analysis for DM annihilations to the dimension-six operators misses
the right result, as the cross section receives important contributions from operators of dimension
higher than six.
One could have expected this result by an order-of-magnitude estimate of the two-body and
8
(cid:76)(cid:61)5TeV MDM(cid:61)1TeV
102 M4,M5 103
M1
(cid:76)body (cid:76)body 10 body(cid:76) body(cid:76) 102
Σ(cid:72)3(cid:45)dim8 Σ(cid:72)2(cid:45)dim6 1 Σ3(cid:72)(cid:45)dim8 Σ2(cid:72)(cid:45)dim6 10 M2,M3,M4,M5
10(cid:45)1 1 M1
M2,M3
10(cid:45)2 10(cid:45)1
500 1000 1500 2000 2500 3000 2 4 6 8 10
MDM(cid:64)GeV(cid:68) (cid:76)(cid:145)MDM
Figure 3: Left panel: The ratio of the total cross sections vσ(χχ → (cid:96)+(cid:96)−Z)(cid:12)(cid:12) / vσ(χχ → (cid:96)+(cid:96)−)(cid:12)(cid:12)
L L Mi L L M
as a function of M . We set c(6,p) = c(8,s) = 1, and Λ = 5 TeV. The curves corresponding to
DM i
O ,O (in green) departs from those of O ,O at low M and are slightly higher. Right
M4 M5 M2 M3 DM
panel: The ratio of the total cross sections vσ(χχ → (cid:96)+(cid:96)−Z)(cid:12)(cid:12) / vσ(χχ → (cid:96)+(cid:96)−)(cid:12)(cid:12) as a function
L L Mi L L M
of Λ/M . We set c(6,p) = c(8,s) = 1 and M = 1 TeV.
DM i DM
three-body total cross sections originating from the operators above
M2
vσ(2 → 2)| ∼ |c(6,p)|2v2 DM (3.20)
O6 Λ4
α M6
vσ(2 → 3)| ∼ |c(8,s)|2 W DM , (3.21)
O8 4π Λ8
where α = g2/(4π), from which we learn that the three-body cross section due to the dimension-8
W
operator can be bigger than the two-body one due to dimension-6, provided that
|c(6,p)| Λ2 1(cid:114)α
(cid:46) W (cid:39) 50 (3.22)
|c(8,s)|M2 v 4π
DM
for v = 10−3. This estimate is confirmed by the numerical results in Fig. 3.
Let us conclude this subsection with a comment on the toy-model studied in Ref. [3], consisting
of a Majorana spinor χ with mass M and a scalar S with mass M > M , in addition to the
DM S DM
SM particle content. The added particles are a SM singlet and a SU(2) -doublet respectively. The
L
Lagrangian is of the form
1
L = L + χ¯(i∂/−M )χ+(D S)†(D S)−M2S†S +(y χ¯(Fiσ S)+h.c.). (3.23)
SM 2 DM µ µ S χ 2
In the limit M (cid:29) M , the interactions can be accurately described by the effective operator O
S DM M1
in Table 1. Indeed, in the limit v → 0, the amplitudes for the process χχ → f f¯ Z as due to the
L L
toy model Eq. (3.23) and the effective operator O match, provided that 4c(8,s)/Λ4 = −y2/M4.
M1 M1 χ S
3.2 Real scalar DM
As discussed in the previous section, there is no two-body annihilation φφ → f f in s-wave in
L L
the limit m = 0. So the EW bremsstrahlung opens up a three-body annihilation channel which
f
is otherwise absent. The first non-vanishing contribution to the s-wave annihilation cross section
9