Dark matter annihilation with s-channel internal Higgsstrahlung

We study the scenario of fermionic dark matter that annihilates to standard model fermions through an s-channel axial vector mediator. We point out that the well-known chirality suppression of the annihilation cross section can be alleviated by s-channel internal Higgsstrahlung. The shapes of the cosmic ray spectra are identical to that of t-channel internal Higgsstrahlung in the limit of a heavy mediating particle. Unlike the general case of t-channel bremsstrahlung, s-channel Higgsstrahlung can be the dominant annihilation process even for Dirac dark matter. Since the s-channel mediator can be a standard model singlet, collider searches for the mediator are easily circumvented.


I. INTRODUCTION
A key strategy in the search for dark matter (DM) is indirect detection: the search for cosmic rays arising from dark matter annihilation in the cosmos. But it is well-known that dark matter annihilation to standard model (SM) fermion/anti-fermion pairs, a key signature, is suppressed if the dark matter is a real particle and flavor violation is minimal. In this broad scenario, which includes the constrained minimal supersymmetric standard model, one finds that the bremsstrahlung processes XX →f f Y (Y = γ, Z, h) can dominate over the XX →f f annihilation process.
Thus far, the focus of such studies has been on t-/u-channel annihilation, where the dominant contribution to bremsstrahlung arises from the coupling of a SM boson to a new charged scalar. These models yield predictable spectra which, remarkably, depend largely on the choice of final state and are independent of the details of the DM-SM interaction [8]. However, the allowed parameter space is tightly constrained by LHC searches for the charged mediator.
In this work, we point out that the Higgsstrahlung processes XX →f f h, can dominate over XX →f f in the case of s-channel annihilation, where the mediator is a SM gauge singlet. This scenario is far less constrained by LHC searches, but also yields predictions for cosmic ray spectra arising from dark matter annihilation which can by utilized in indirect searches. We focus on the case where the emitted boson is the SM Higgs boson, and the mediator is a new SM singlet boson. But our results also apply to the scenario in which the emitted boson is a new neutral scalar which may or may not decay to SM particles. Regardless of whether or not the scalar decays visibly, the associated f f spectrum will be unsuppressed, yielding an enhancement in the cosmic ray signal over the XX →f f annihilation process.
In Section II, we describe the general principles that underly the chirality suppression of s-wave dark matter annihilation, and describe a model which lifts this suppression through s-channel Higgsstrahlung. In Section III, we compute the cross sections and spectra, and compare them to the previously studied case of t-channel Higgsstrahlung [9]. We conclude with a discussion of our results in Section IV.

II. GENERAL PRINCIPLES
The suppression of the XX →f f process for the case of real dark matter and minimal flavor violation (MFV) can be understood from general principles. If the initial dark matter state consists of two identical particles, then it must be invariant under charge conjugation. Equivalently, the wave function must be totally symmetric (antisymmetric) if the particle is a boson (fermion). Since the two-particle initial state is multiplied by (−1) L+S under charge conjugation, an s-wave (L = 0) initial state must have S even; for either a spin-0 or spin-1/2 DM particle, this implies S = 0, and thus J = 0. The final state then must also have J = 0, implying that thef f pair travel back-to-back with the same helicity. The fermions must arise from different SM Weyl spinors, and the matrix element must be proportional to the mixing of the left-and right-handed spinors. Such mixing violates SM flavor symmetries; if flavor violation is minimal, then the s-wave annihilation matrix element must be suppressed by m f /m X . This suppression is no longer required if the final state isf f Y , where Y is a SM boson. Previous work has focussed on the case in which the boson is emitted from the virtual mediating particle, a process called virtual internal bremsstrahlung (VIB). (Of course, if Y is a gauge boson that couples to f ,f , it will be emitted from an external line as well.) This class of models is important because, if the boson cannot be emitted from an internal line or the initial particles, then the process of boson emission is essentially the same as final state radiation, which is dominated by soft/collinear emission for which the final state fermion propagator becomes nearly on-shell. As a result, the soft/collinear contribution is suppressed by a factor m f /m X , just as for the XX →f f process [10]. Moreover, if the mediator and dark matter particle are nearly degenerate in mass, then the VIB matrix element is enhanced in the region of phase space where one final state fermion is soft, and the propagator of the mediator is nearly on-shell.
If the boson emitted through VIB is a photon, then the mediator is charged under U (1) em , implying that it must be exchanged in the t-or u-channel. But if the emitted boson is a scalar, then it may be emitted from a SM singlet particle. In this case, VIB can occur even if dark matter annihilates in the s-channel through a SM singlet mediator.

A. Model
We consider the case where the mediator is a heavy real spin-1 particle, B µ , which couples to fermion dark matter (X) and SM matter through the following Lagrangian: where f is a SM fermion, H = H + h is the Higgs field and H = v EW ∼ 246 GeV. This interaction structure (fermion dark matter and a spin-1 mediator which couples to an axial vector dark matter current and a vector and/or axial vector SM current) is the only one that is suitable for our purpose. Higgsstrahlung is relevant only if the DMmediator interaction term has an unsuppressed matrix element with an s-wave initial state, and if the SM-mediator interaction is necessarily suppressed for the kinematics of a two-particle final state when the outgoing SM particles are relativistic. The appropriate suppression of the SM-mediator interaction for a two-body final state only occurs for the time-like component of a spin-1 mediator, coupling to either a vector or axial-vector SM fermion current [11] (in the axial vector current case, the interaction is suppressed by m f , and in the vector current case it vanishes identically). The mediator must then couple to a vector or axial vector dark matter current, such that only the time-like component of the dark matter current has an unsuppressed matrix element with an s-wave initial state. This requirement is only satisfied if the dark matter is spin-1/2 and couples to the mediator through an axial vector interaction [11]. Note, if dark matter is spin-1 and couples to the mediator through a vector interaction (X ν ∂ ν X µ B µ ), then the time-like component of the DM current does indeed have a non-trivial matrix element for an s-wave initial state, but this matrix element vanishes in the non-relativistic limit because it involves time-like polarizations of the DM particles [11]. The shapes of the energy spectra for the process XX →f f h, summed over final state spins, are independent of θ in the m f /m X → 0 limit. In this limit, θ only determines the relative branching fraction to final states with left-handed and right-handed f . For simplicity, we set θ = 0.
It is interesting to also note that in this scenario, the dark matter fermion X can be either Dirac or Majorana, while still exhibiting chirality suppression of theXX →f f cross section, which is lifted by s-channel Higgsstrahlung. This differs from the case of t-channel Higgsstrahlung, for the which the dark matter must be Majorana. This is because if dark matter interacts with SM matter through the t-or u-channel, one must use a Fierz transformation to construct the dark matter current which acts on the initial state. Generically, one gets a linear combination of all possible DM currents, including those which have a non-trivial matrix element with an L = 0, S = 1, J = 1 initial state. If the initial state is J = 1, then the final state is J = 1 as well, and the chirality suppression in the m f /m X → 0 limit no longer applies. In the t-channel case, it is thus necessary to assume that dark matter is Majorana in order to eliminate the J = 1 contribution. For the s-channel case, no such assumption is necessary because the choice of interaction Lagrangian picks out a particular dark matter current that couples to the s-channel mediator; if the DM current is axial vector, then it has a trivial matrix element with the L = 0, S = 1, J = 1 state. Such an interaction Lagrangian naturally arises for Dirac dark matter if the mediator is an axial vector, and if the DM-mediator interaction respects C and P .

III. CROSS SECTIONS AND SPECTRA
The XX →f f cross section is given by where N c is the color factor associated with f , and v rel is the relative velocity of the initial state particles. As expected, it vanishes in the limit m f /m X → 0. The amplitude for the process, X(k 1 )X(k 2 ) → f (p 1 )f (p 2 )h(k), can be written as The differential cross section in the limit m f → 0 is where r B ≡ m 2 B /m 2 X , r h ≡ m 2 h /m 2 X , and similar to the notation in Ref. [8], we define x 1 ≡ E f /m X , x 2 ≡ Ef /m X and x h ≡ E h /m X , so that in the static center of mass frame The energy distribution of f can be obtained by integrating over In the large r B limit, we have The Higgs spectrum can be obtained by integrating over We do not consider the regime 2m X ≥ m B + m h , as in this case the s-channel annihilation cross section would be dominated by the on-shell 2 → 2 process, XX → Bh. If 2m X > m B , then it is possible to produce an on-shell B and an off-shell h, which in turn couples to SM particles. Unless λ f is very small, B has a larger decay width than the Higgs, which implies that on-shell h production dominates over on-shell B production. Thus, in the entire mass range m h < 2m X < m B + m h , we are justified in considering only final states with an on-shell h.
The t-channel differential cross section can be written as where y DM is the coupling between X, the mediator, and SM matter. By a slight abuse of notation, we denote the t-channel scalar mediator by B. We successfully reproduced the primary t-channel Higgsstrahlung spectra of Ref. [9] (but not the secondary spectra, as we comment on below). Comparing Eqs. (4) and (8), we see that the main difference between the s-channel and t-channel annihilation is the propagator. In the limit of a heavy mediator, s-channel and t-channel Higgsstrahlung yield the same normalized primary spectra, making them impossible to distinguish; see the left panel of Fig. 1. From Fig. 2, we see that for m h ≪ m X , m B , the s-channel and t-channel spectra are distinguishable because r B no longer dominates the denominators of Eqs. (4) and (8).
The similarity of the spectra arising from s-channel and t-channel Higgsstrahlung in the heavy mediator limit is easily understood. In the heavy mediator limit, the mediator can be integrated out and the matrix element for the XX →f f h annihilation process can be derived from an effective contact operator. Since we and Ref. [9] have assumed MFV and taken the m f /m X → 0 limit, the operators relevant for either s-or t-channel bremsstrahlung cannot mix left-handed and right-handed f Weyl spinors. Moreover, because there is no mixing of SM Weyl spinors, and because X is a SM singlet, SU (2) L gauge-invariance requires an explicit insertion of a Higgs vev, v EW . The relevant contact operator must therefore be at least dimension 8. There are two dimension 8 contact operators which satisfy these constraints and have non-trivial matrix elements with an L = 0 dark matter initial state: Note that in the heavy mediator limit, one expects Λ ∝ m B , implying that the Higgsstrahlung cross section scales as r −4 B , as expected. In the heavy mediator limit, s-and t-channel higgsstrahlung are produced by different linear combinations of O AA and O AV . But in the m f /m X → 0 limit, these operators produce the same energy spectra. They differ only in the relative sign of the matrix element for coupling to left-handed and right-handed f , but this sign is unobservable in the chiral limit. Although this argument is only valid in the contact-interaction limit, we see that for r B > 4 the normalized spectra are already quite similar.  Finally, the total cross section can be expressed as As expected, there is a resonant enhancement as r B → 4. In the large r B limit, the total cross section becomes It is easy to verify that the above equation has the same form as Eq. (A.3) in Ref. [9], up to a normalization factor and a change of variable.

A. Secondary Spectra
In our model, the injected cosmic ray spectrum arises both from the direct injection off f pairs, and from the decay products of the Higgs boson. More generally, the mediator could couple to any real scalar φ via L φ = (λ φ /2)v EW φB µ B µ , where the factor of v EW ∼ 246 GeV is included as a convenient energy scale for the coupling. The primaryf f spectrum would be as in Eq. (5), with the emitted scalar boson mass a free parameter. The part of the cosmic ray spectrum arising from scalar decay would now depend on the branching fractions for φ to decay to various SM final states, and could be absent entirely if φ decayed invisibly. The features of these total spectra thus depend in detail on the choice of f , as well as on the visible decays of the scalar.
We use the cookbook of Ref. [13] to obtain the spectra of stable particles at the source (including decays, showering and hadronization) for a few special cases in which we assume the mediator couples equally to first generation leptons, and does not couple to other SM matter fields. From Fig. 3, we see that in each case, including the m X = 100 GeV and m B = 105 GeV case, the resultant s-channel and t-channel spectra are similar for the SM Higgs. 1 As an example of a new real scalar, we consider a Higgs-like boson with a mass that lies between 2m µ and 2m π , as may occur in models with Higgs portals. Such a boson decays dominantly to muons. As can be seen from Fig. 4, the Higgsstrahlung signatures can be very different from that for the SM Higgs.

IV. CONCLUSIONS
We calculated the differential cross section for the s-channel Higgsstrahlung process XX →f f h. This scenario arises when a spin-1 mediating particle has vector or axial vector couplings to a SM fermion f , axial vector coupling to a fermion dark matter particle X, and a coupling to the Higgs boson. The spectra reduce to the previously known t-channel Higgsstrahlung spectra in the contact-interaction limit. But there are differences in the viability of these scenarios, given data from the LHC. t-channel Higgsstrahlung necessarily involves a electroweak and/or QCD charged mediator, and there are tight constraints on the masses of such particles from current LHC data. Since an s-channel mediator may be a SM singlet, it can evade such bounds, opening up new regions of parameter space where Higgsstrahlung is relevant to dark matter annihilation.
Unlike the case of t-channel annihilation, s-channel annihilation can receive a chirality suppression which is lifted by Higgsstrahlung even if the dark matter is a Dirac fermion. This provides an interesting correlation between cosmic ray signatures of dark matter annihilation and the properties of dark matter, assuming that dark matter is stable. In particular, in the case of t-channel annihilation, the dominance of internal bremsstrahlung processes over chiralitysuppressed XX →f f annihilation processes would imply that dark matter must be a real particle. Since a real particle cannot be charged under an exact continuous symmetry, this would imply that dark matter was stabilized by a discrete symmetry. But if dark matter annihilates through the s-channel, then it may be stabilized by a continuous symmetry and still exhibit a chirality-suppressed XX →f f annihilation cross section; the chirality suppression can then be lifted by Higgsstrahlung.
Although we have focused on the Higgsstrahlung process XX →f f h, the Higgs boson may be replaced by any new scalar particle φ without altering the form of the primary fermion spectrum. In this case, both m X and m φ may be well below the electroweak scale. The annihilation of low mass dark matter to either b-quarks or τ -leptons has been considered as a possible source of the excess in GeV-scale photons observed from the Galactic Center (GC), and detailed fits of the observed photon spectrum from the GC to the spectra expected from the processes XX →bb,τ τ have been performed [14]. But these processes are relevant only if s-wave dark matter annihilation to fermions is not very chirality-suppressed; if it is suppressed, then scalar bremsstrahlung processes could dominate. The softening of the primary fermion injection spectrum arising from the process XX →f f φ would change the spectrum of photons produced at the GC. It would be interesting to reconsider the GC excess in this light.