We show that the energy spectra from dark matter (DM) annihilation into $f^+f^-\gamma$ (where $f$ is a light fermion) via chirality preserving interactions are identical in 3 scenarios: (a) DM is a Majorana fermion and the particle exchanged is a scalar, (b) DM is a Majorana fermion and the particle exchanged is a vector and (c) DM is a scalar and the particle exchanged is a fermion. For cases (a) and (c), we also calculate the differential cross section to $\ell^+f^-V$, where $V=W,Z,\gamma$, and $\ell$ is a light fermion that may or not be the same as $f$. The form of the cross section depends on whether the DM is a Majorana fermion or a scalar but in both cases its form is independent of $V$.
Intensive searches for dark matter (DM) via their annihilation signatures [1] have spurred interest in the effects of electromagnetic and electroweak bremsstrahlung on the rates for particular final states and on the resulting spectra [2,3,4].
In this Letter, we show that for 3 different interactions that are chirality preserving and appear in popular scenarios, dark matter annihilation to f + f -γ in the static limit gives identical energy spectra. We then connect the common structure for the amplitude to an operator basis. Finally, for 2 of the 3 cases, we derive the differential cross section to ℓ + f -V , where V = W, Z, γ and find its form to be different for Majorana DM and scalar DM, but independent of V .
We consider the annihilation final state e + e -γ for the interaction terms in the table below. The exchanged particles S, W ′ and E (collectively denoted by X) are taken to be negatively charged and heavier than the corresponding DM particle D = χ, N and φ. We assume that D is a standard model singlet and its coupling to the electron to be chiral with the interaction involving e R . For e L , straightforward replacements can be made.
Model 2 Model 3
The supersymmetric case of neutralino DM with selectron exchange is a good example of Model 1. The case of a new heavy charged W ′ gauge boson that couples e R and a right-handed heavy Majorana neutrino N in variations of left-right theory falls into the class of Model 2 [5].
Scenarios of scalar DM which have been of interest recently are realizations of Model 3 [4].
For the annihilation process, DD → e(p 1 )ē(p 2 )γ(k, ǫ), in the nonrelativistic limit we find a unified description of the energy spectra of the decay products for all three models. In standard notation, the amplitude turns out to be
where the minus sign applies to Models 1 and 2 (with Majorana DM) and the plus sign applies to Model 3 (with scalar DM). Note that for the Majorana DM models only the initial state with vanishing total angular momentum participates in the annihilation. The overall constant
Remarkably, we find the same structure for all 3 Models even though Model 2 involves the trigauge boson vertex W ′ W ′ γ. The amplitude obeys QED gauge invariance and its two terms do not interfere when photon polarizations are summed in the limit m e → 0. The spin-averaged annihilation rate is
where r = m 2 X /m 2 D . The scaling variables
are defined in the static center of mass frame so that x 1 + x 2 + x 3 = 2. The spin-averaged factor 1 (2S D +1) 2 is 1 4 for Models 1, 2 and unity for Model 3. The photon energy distribution is obtained by integrating over
The unified formulas for the amplitude and the distribution for the 3 models are simple and interesting. We now study the structure of the amplitude. The numerator of the amplitude involves two pieces, each of which are QED gauge invariant. The first one is
where we have used the massless fermion on-shell condition and the last step identifies the stucture with an operator [4],
Similarly, the second piece is
where [4] O
is the conjugate of O. On the other hand, we can use the Chisholm identity
Similarly,
So we have the relations,
where
Thus, either O and O † , or F and F can be used as bases to describe the amplitude. Because the dual substitution interchanges E ↔ B, it also interchanges a right-polarized photon for a leftpolarized photon. If the energy distribution sums up polarizations, both F and F give the same result. Note that since F is CP even, it corresponds to Model 3 with scalar DM. On the contrary, the Majorana pair is CP odd, and the amplitude picks up F.
For the annihilation of Majorana DM χ via scalar exchange, we derive a univeral amplitude for vector boson V emission χ + χ → f (p 1 ) l(p 2 )V (k, ǫ) in the static limit. The vector boson can be W, Z or γ. As v rel → 0, only the composite of χχ with zero total angular momentum gives a contribution, and we get an amplitude of the form of Eq. ( 1) with the minus sign selected, and C replaced by C V (in which we collect the model-dependent couplings and coefficients). As usual, we assume that the chirality of the massless lepton field is conserved. Note that the two terms in the numerator do not interfere only when k 2 = 0 (as for the photon V = γ). QED-like gauge invariance (i.e. M • k = 0) is satisfied even for k 2 = m 2 V = 0. We calculate the annihilation cross section to be
where
In the case of W/Z emission, we assume a common mass m X for all SU (2) partners of the heavy exchanged particles. Note that
respectively. The energy distribution of V can be obtained by integrating over x 1 within the limits
, and the energy distribution of f can be obtained by integrating over x 3 from 1 -x 1 + δ V /(1 -x 1 ) to 1 + δ V . On setting δ V = 0 we recover the cross section for V = γ and C γ is simply C given in the photon case. The values of C W and C Z are given by
and
where T, T 3 , T ′ 3 are the weak-isospin numbers of the exchanged particles and θ
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