Electroweak fragmentation functions for dark matter annihilation

Electroweak corrections can play a crucial role in dark matter annihilation. The emission of gauge bosons, in particular, leads to a secondary flux consisting of all Standard Model particles, and may be described by electroweak fragmentation functions. To assess the quality of the fragmentation function approximation to electroweak radiation in dark matter annihilation, we have calculated the flux of secondary particles from gauge-boson emission in models with Majorana fermion and vector dark matter, respectively. For both models, we have compared cross sections and energy spectra of positrons and antiprotons after propagation through the galactic halo in the fragmentation function approximation and in the full calculation. Fragmentation functions fail to describe the particle fluxes in the case of Majorana fermion annihilation into light fermions: the helicity suppression of the lowest-order cross section in such models cannot be lifted by the leading logarithmic contributions included in the fragmentation function approach. However, for other classes of models like vector dark matter, where the lowest-order cross section is not suppressed, electroweak fragmentation functions provide a simple, model-independent and accurate description of secondary particle fluxes.


Introduction
The existence of dark matter [1] provides strong evidence for physics beyond the Standard Model (SM). A leading candidate for dark matter are weakly interacting massive particles, which may be produced at colliders or detected through direct and indirect detection experiments. In indirect detection, in particular, one searches for dark matter annihilation products, including antimatter particles like positrons and antiprotons, which propagate through the galactic halo and which can be detected in astrophysical experiments at the earth.
Electroweak (EW) corrections may be important for dark matter annihilation for two reasons . First, the radiation of a vector boson can lift the helicity suppression of cross sections for Majorana fermion dark matter annihilating into light fermions [13][14][15][16][25][26][27][28][29]. Moreover, the emission and decay of electroweak gauge bosons from the primary annihilation products alter the spectrum and composition of the secondary flux. In particular, the gauge boson decay will lead to a secondary flux which includes all stable SM particles (e + , e − , ν,ν, γ, p,p), irrespective of the model-specific composition of the primary annihilation products.
Many models provide dark matter candidates with masses in the TeV-range, see e.g. Ref. [1,[30][31][32]. For such heavy dark matter, soft and collinear electroweak gauge boson emission from the relativistic final-state particles is enhanced by Sudakov logarithms ln 2 (M 2 DM /M 2 EW ) [33][34][35], where M DM and M EW are the mass of the dark matter candidate and of the electroweak gauge boson, respectively. The leading logarithmic contributions to the annihilation cross section can be described in a model-independent way by electroweak fragmentation functions [12,36,37], where improved splitting functions are introduced in order to take into account the masses of the emitted gauge bosons.
The purpose of the present article is to examine the quality of the fragmentation function approximation. To this end we have compared the predictions obtained for the secondary flux after propagation using the fragmentation function approximation against those obtained from an exact calculation. To perform the comparison we have chosen two specific dark matter models, a simplified version of the minimal supersymmetric model (MSSM) [30,38] with neutralino dark matter, and a model with universal extra dimensions (UED) where the first Kaluza-Klein excitation of the photon provides a dark matter candidate [39,40]. Both models are generic for the annihilation of Majorana fermion and vector dark matter, respectively. To assess the quality of the fragmentation function approach we focus on the particular case where the dark matter particles annihilate at lowest order into electron-positron pairs only. We show that the fragmentation function approach reproduces well the exact result in the case of UED with vector dark matter, while the approximation does not work for the MSSM with Majorana fermion annihilation into electron-positron pairs. This is due to the fact that the annihilation of Majorana fermions into a light lepton pair is helicity suppressed and that the emission of soft and collinear gauge bosons from the final-state particles, included in the fragmentation function approximation, is not sufficient to lift this helicity suppression [16]. Hence, the fragmentation function approximation provides a simple and model-independent way to obtain realistic predictions for dark matter indirect detection for those models where the annihilation cross section is not suppressed at the lowest order.
This article is organised as follows: in Section 2 we briefly review the fragmentation function approximation to electroweak radiation. In Section 3 we present the calculation of EW gauge boson emission in two specific models with Majorana fermion and vector dark matter, respectively, and compare the fragmentation function approximation against the exact result for the primary flux. In Section 4 we perform the comparison for the secondary flux after propagation through the galactic halo. We conclude in Section 5.

Electroweak corrections
Electroweak corrections in the context of dark matter annihilation have been widely discussed in the literature . Therefore we only summarize the main ideas and describe the fragmentation function approach to describe electroweak gauge boson emissions.

The fragmentation function approach
We assume heavy dark matter i.e. M DM M EW , which annihilates into electron-positron pairs only. The radiation of W and Z bosons off the high energy final-state leptons is enhanced by logarithms of the form ln(M 2 DM /M 2 EW ) and ln 2 (M 2 DM /M 2 EW ) corresponding to collinear and soft/collinear emission, respectively. No such enhancement exists for radiation off the nonrelativistic initial-state dark matter particles, or radiation off intermediate particles. Thus, in the limit M DM M EW the electroweak gauge boson emission is dominated by soft and collinear final-state radiation, which is independent of the specific dark matter model or the form of the annihilation cross section.
We are interested in the energy spectrum of a given SM final state f = {e ± , γ, p,p, ν,ν}, resulting from the annihilation process DM DM → e + e − + (Z → f ), including the decay of the Z boson and the fragmentation and hadronization of the decay products, see The energy spectrum is given by with x = 2E f / √ s, and σv cm is the thermally averaged cross section for the process DM DM → e + e − (Z → f ). The centre-of-mass energy is , and E f denotes the energy of the final state SM particle of type f . By v cm we denote the velocity of the dark matter particle in the centre-of-mass frame. More specifically, as the dark matter particles are non-relativistic, we have x E f /M DM . In contrast to some conventions in the literature, the energy spectrum, Eq. (1), is normalised to one, dx dN f /dx = 1.
The logarithmically enhanced contributions to the emission of an electroweak boson can be described by fragmentation functions, D EW , that obey evolution equations similar to the where I denotes the final state of the (2 → 2) annihilation process, i.e. an electron-positron pair in the case at hand. The summation in J includes the final state particles after the radiation of an EW gauge boson. In our case, J = {e + , e − , Z}. The fragmentation functions D EW I→J (z) describe the probability that a particle I turns into a particle J, with a fraction z of the energy of the emitting particle, via the emission of a EW boson. Clearly at the lowest order we have D EW I→J (z) = δ IJ δ(1 − z). The further decay, fragmentation and hadronisation of the primary annihilation products, J → f , can be described by standard Monte Carlo event generators.
The fragmentation functions D EW I→J (z) can be obtained by computing the partonic splitting functions. They differ from those of QCD or QED as the emitted EW gauge bosons are massive. 2 For our purposes, we need the splitting functions of a massless fermion (F) splitting into a massless fermion and a massive vector particle (V): with The corresponding fragmentation functions are where m Z is the mass of the emitted Z boson, α 2 the SU(2) coupling and θ w the weak angle. The factor g f = T f 3 −sin 2 θ w Q f accounts for the coupling of the fermions to the Z boson, where T f 3 and Q f are the isospin and the charge of the fermion, in our case e ± L or e ± R . The function L, Eq. (4), respects the correct kinematic limits and vanishes below threshold for x < 2m Z / √ s. Note that the fragmentation functions depend on the chirality of the fermion through the coupling g f , so that one should sum over e + L e − L and e + R e − R pairs to obtain the annihilation cross section. For the implementation of the fragmentation function approach, we use a Sudakov parametrisation of the phase space [12]. The four-momentum of the initial state is denoted by The momentum of the electron/positron that radiates the Z boson can be parametrised as where x corresponds to the fraction of energy carried away by the emitted Z boson. The fourmomentum of the emitted Z boson takes the form Finally, the four-momentum of the electron/positron which does not radiate is with This parametrisation ensures the conservation of the four-momentum.

Majorana fermion and vector dark matter
In order to quantify the accuracy of the fragmentation function approximation, we have calculated dark matter annihilation into an e + e − -pair plus a Z boson, DM DM → e + e − Z, in the minimal supersymmetric model (MSSM) with neutralino dark matter and in a model with universal extra dimensions (UED), where the dark matter is provided by the first Kaluza-Klein excitation of the photon. For both models, we have compared the energy spectrum of the Z boson within the fragmentation function approximation with the exact calculation of the (2 → 3) process, including the radiation of a Z boson from the intermediate t-and u-channel particles. We have used FeynArts and FormCalc [41][42][43] for the MSSM calculation and CalcHep [44,45] for UED, and checked our results against MadGraph 5 [46]. The flux of the annihilation products is determined by the thermally averaged cross section, σv cm , which can be expanded in powers of the dark matter velocity [47]: Given that v 2 cm ≈ 10 −6 for the annihilation of non-relativistic dark matter, we only kept the first term of the expansion.

A supersymmetric model with Majorana fermion dark matter
As a specific model with Majorana fermion dark matter we have considered the MSSM and calculated the annihilation of neutralinos into an electron-positron pair,χ 0χ0 → e + e − . We consider a pure bino, which does not couple to a Z boson, so that the annihilation proceeds through the exchange of selectrons in the t-and u-channel only. To derive analytic results, we have furthermore assumed that there is no neutralino and selectron mixing, and that the masses of the left-and right-chiral selectrons are degenerate. Without mixing, the vertices drastically simplify to −ie √ 2P L /cos θ w for the left-chiral selectron,ẽ L , and ie √ 2P R /(2 cos θ w ) for the rightchiral selectron,ẽ R , respectively [48,49]. The Feynman diagrams for this process are displayed in Figure 2. We have reproduced the result for the matrix element squared obtained in [50]  and performed a low-velocity expansion of the Mandelstam variables to obtain the thermally averaged cross section [51,52]. Retaining only the first coefficient of this expansion i.e. the zeroth order in the velocity, we obtain where m e and mẽ = mẽ L = mẽ R are the mass of the electron and of the selectrons, respectively, and α is the QED coupling. In the following we will always set mẽ = M DM for simplicity. Note that because of the well-known helicity suppression [25,26] due to the Majorana nature of the neutralino the cross section vanishes in the limit of zero electron mass m e → 0. The (2 → 3) tree-level process,χ 0χ0 → e + e − Z, is finite, and we do not include virtual corrections as we are mainly interested in the shape of the secondary flux induced by the Zboson decay. The helicity suppression of the (2 → 2) process, Eq. (12), is lifted by Z-boson radiation only if the emission from both the final-and intermediate state particles, i.e. from the electron-positron pair and from the t-and u-channel selectron, is included [16]. The fragmentation function approximation which only includes soft/collinear radiation off the electron/positron pair is thus expected not to work in this particular case. This can be seen explicitly from the s-wave contribution to the neutralino annihilation cross section,χ 0χ0 → e + e − Z, obtained in Ref. [16]: where x is the energy fraction of the emitted Z boson. In Fig. 3 we compare the Z energy distribution of the full (2 → 3) calculation for bino annihilation into an electron-positron pair plus a Z boson with the fragmentation function approximation It is manifest from Fig. 3 that the fragmentation function approximation does not describe the energy distribution of the full (2 → 3) process. In general, the fragmentation function approach is not supposed to work if the lowest-order cross section is helicity suppressed, as in the case of Majorana fermion annihilation into light fermions.

A universal extra dimension model with vector dark matter
We now discuss a model with vector dark matter, where the lowest-order annihilation cross section to electron-positron pairs is not helicity suppressed. Specifically, we consider a model with universal extra dimensions where the first Kaluza-Klein excitation of the U(1) Y hypercharge gauge field, B (1) , provides a dark matter candidate. The Feynman diagrams for the leadingorder annihilation process, B (1) B (1) → e + e − , mediated by t-and u-channel exchange of the first Kaluza-Klein excitation of the electron, e L,R , are displayed in Fig. 4. For completeness we give the s-wave contribution of the thermally averaged annihilation cross section at leading-order [40,53]: The     ) only, and is thus expected become more and more accurate with increasing dark matter mass. This is born out by the explicit numerical comparison shown in Table 1. While the fragmentation function approximation underestimates the cross section by about a factor 3 for M DM = 150 GeV, at M DM = 500 GeV it already reproduces the normalisation of the full calculation within 10% accuracy.
The energy distributions of the emitted Z boson, dN Z /dx, is shown in Fig. 5 for different masses of the dark matter particle. One can see that the fragmentation function approximation   provides a very accurate description of the shape of the energy distributions for heavy dark matter with M DM /m Z 5. The quality of the fragmentation function approximation can be deduced from the analytical expression that we have obtained for the exact (2 → 3) process B (1) B (1) → e + L e − L Z. Integrating over the angles and considering the centre of mass of the annihilating particles, the differential cross section can be written as where |M| 2 is the matrix element squared and x 1 and x 2 parametrise the energies of the finalstate particles. In particular with the four momenta k, p 1 and p 2 corresponding to the Z boson, the positron and the electron, respectively. The integration limits of the phase space for the variable x 1 are with For x 2 the integration boundaries are Expanding in the velocity, integrating over x 1 and neglecting terms vanishing in the limit of vanishing Z boson mass, Eq. (16) becomes The function F is defined as . and coefficients A, B and C given by Given the Born cross section (see Eq. (15)), Eq. (23) can be recast in the form This is exactly the form expected within the fragmentation function approach, see Eqs.
(3), (4) and (5). Indeed in the limit m Z → 0, which is exactly Eq. (3) in this limit. For completeness we have computed the coefficients A, B and C for an intermediate mass m i different from the dark matter mass. In this general case they read with w = m i /m DM . Furthermore, in Eq. (24) ln(x 2 ) has to be replaced by ln w 2 −1+2x 2

The secondary flux after propagation
In this section we compare the positron and antiproton fluxes obtained with the fragmentation function approximation against the exact result after propagation through the galactic halo. We briefly explain how we obtain the energy spectra of the stable Standard Model particles at the production point and how we implement the propagation of positrons and antiprotons. In order to obtain the primary 3 stable SM particles we use Pythia 8 [54][55][56] to describe the Z boson decay, the evolution of its decay products (further decays and hadronisation) and the QCD and QED radiation. The fluxes produced by Pythia 8 are then propagated through the galactic halo to obtain predictions for positrons and antiprotons at the earth.
As we are not aiming at a detailed study of the different dark matter profiles, and the uncertainty due to propagation, we have chosen the Einasto model [57,58] as a specific example: where α = 0.17, r s = 28.44 kpc and ρ s = 0.033 GeV/cm 3 . These values correspond to a dark matter density ρ = 0.3 GeV/cm 3 at the location of the sun (r = 8.33 kpc). In order to calculate the flux of positrons/electrons at the location of the earth, we have used the Green function formalism [59]. The expression for the positron/electron flux after propagation is [59] with with the energy expressed in GeV, I = {e + , e − , Z} and v e ± is the velocity of the electron. The energy spectrum is normalized to the total cross section. The normalisation of the flux is given by b T ( ) = 2 /τ where τ = GeV/b (1 GeV, r ) = 5.7 × 10 15 sec. The functions λ D and I are defined as and with = log 10 (λ D /kpc). The values of the astrophysical parameters K 0 , δ, a 0 , a 1 , a 2 , a 3 , b 1 and b 2 are taken from Ref. [59] and correspond to the medium (MED) Einasto profile: (a 0 , a 1 , a 2 , The flux of antiprotons from dark matter annihilation at the earth is given by [59] dΦp dK where K = E − m p and m p are the kinetic energy and the mass of the proton respectively. The function R is defined as with κ = log 10 (K/ GeV) and the coefficients again taken from Ref.

Comparison and results
We consider the spectra for positrons and antiprotons from the annihilation of vector dark matter in the universal extra dimension model after propagation through the galactic halo. We compare the results from the full (2 → 3) calculation and the fragmentation function approximation. In Fig. 6 the positron spectra after parton shower and propagation are displayed. The fragmentation function approximation provides an accurate description of the positrons from the Z-boson decay. However, as the cross section of the (2 → 3) process is a genuine electroweak higher-order contribution of O(α ln 2 (M 2 DM /m 2 Z )), and thus highly suppressed by comparison to the leadingorder annihilation, the amount of additional positrons is small compared to those produced in the (2 → 2) process. The small dip in the fragmentation function prediction at high energies is a remnant of the kinematics of the 2 → 2 process and is disappearing as M DM /m Z increases.
In our simple leptophilic model set-up, antiprotons are generated exclusively from Z-boson decay. As the fragmentation function provides a good approximation to the Z-boson spectrum of the exact calculation, the flux of antiprotons is also expected to be reproduced well. This is indeed born out by the explicit calculation presented in Fig. 7. We find that the exact (2 → 3) calculation and the fragmentation function approach agree within 10% for M DM 500 GeV.     For heavy dark matter, M DM m Z , the emission of electroweak gauge bosons is enhanced by Sudakov logarithms ln 2 (M 2 DM /m 2 Z ). The leading logarithmic contributions from soft and collinear gauge boson emission can be described by electroweak fragmentation functions. We have quantified the quality of the fragmentation function approximation by comparing the fragmentation function prediction with the exact calculation for dark matter annihilation into an electron-positron pair plus a Z boson, DM + DM → e + e − + Z. Specifically, we have investigated a supersymmetric model and a universal extra dimension model with Majorana fermion and vector dark matter, respectively. We provide predictions for the energy distribution of the Z boson and the secondary flux of positrons and antiprotons after propagation through the galactic halo in both the fragmentation function approximation and the exact calculation of the (2 → 3) process.
We find that the fragmentation function approach fails for the supersymmetric model with Majorana fermion annihilation into electron-positron pairs. The emission of soft/collinear vector bosons, as included in the fragmentation function approximation, is not sufficient to lift the helicity suppression of the lowest-order annihilation cross section. For vector dark matter annihilation, on the other hand, the fragmentation function approach works very well and can describe the shape and normalisation of the secondary flux accurately. Specifically, we find that the particle fluxes obtained from the exact (2 → 3) calculation and the fragmentation function approach agree to better than 10% for M DM ≈ 500 GeV and to better than 2% for M DM ≈ 1 TeV.
Thus, the fragmentation function approximation provides a simple and model-independent way to obtain realistic predictions for particle fluxes from dark matter annihilation for those models where the annihilation cross section is not suppressed at the lowest order.