Electroweak and Higgs boson internal bremsstrahlung. General considerations for Majorana dark matter annihilation and application to MSSM neutralinos

Journal of High Energy Physics, Sep 2017

It is well known that the annihilation of Majorana dark matter into fermions is helicity suppressed. Here, we point out that the underlying mechanism is a subtle combination of two distinct effects, and we present a comprehensive analysis of how the suppression can be partially or fully lifted by the internal bremsstrahlung of an additional boson in the final state. As a concrete illustration, we compute analytically the full amplitudes and annihilation rates of supersymmetric neutralinos to final states that contain any combination of two standard model fermions, plus one electroweak gauge boson or one of the five physical Higgs bosons that appear in the minimal supersymmetric standard model. We classify the various ways in which these three-body rates can be large compared to the two-body rates, identifying cases that have not been pointed out before. In our analysis, we put special emphasis on how to avoid the double counting of identical kinematic situations that appear for two-body and three-body final states, in particular on how to correctly treat differential rates and the spectrum of the resulting stable particles that is relevant for indirect dark matter searches. We find that both the total annihilation rates and the yields can be significantly enhanced when taking into account the corrections computed here, in particular for models with somewhat small annihilation rates at tree-level which otherwise would not be testable with indirect dark matter searches. Even more importantly, however, we find that the resulting annihilation spectra of positrons, neutrinos, gamma-rays and antiprotons differ in general substantially from the model-independent spectra that are commonly adopted, for these final states, when constraining particle dark matter with indirect detection experiments.

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Electroweak and Higgs boson internal bremsstrahlung. General considerations for Majorana dark matter annihilation and application to MSSM neutralinos

Received: May Electroweak and Higgs boson internal bremsstrahlung. General considerations for Majorana dark matter annihilation and application to MSSM neutralinos Torsten Bringmann 1 3 Francesca Calore 0 1 Ahmad Galea 1 3 Mathias Garny 1 2 0 Universite Savoie Mont Blanc , 27 Rue Marcoz, 73000, Chambery , France 1 9 Chemin de Bellevue , BP-110, Annecy-le-Vieux, 74941, Annecy Cedex , France 2 Technical University Munich , James-Franck-Str. 1, D-85748 Garching , Germany 3 Department of Physics, University of Oslo , Box 1048, NO-0371 Oslo , Norway It is well known that the annihilation of Majorana dark matter into fermions is helicity suppressed. Here, we point out that the underlying mechanism is a subtle combination of two distinct e ects, and we present a comprehensive analysis of how the suppression can be partially or fully lifted by the internal bremsstrahlung of an additional boson in the nal state. As a concrete illustration, we compute analytically the full amplitudes and annihilation rates of supersymmetric neutralinos to nation of two standard model fermions, plus one electroweak gauge boson or one of the ve physical Higgs bosons that appear in the minimal supersymmetric standard model. Beyond Standard Model; Cosmology of Theories beyond the SM; Supersym- 1 Introduction 2 Majorana dark matter and relevant symmetries Chiral symmetry, gauge symmetry and helicity suppression Lifting of Yukawa and isospin suppression Gauge invariance in IB processes 3 Neutralino annihilation to f f and an additional nal-state particle 4 5 2.1 2.2 2.3 3.1 3.2 3.3 3.4 3.5 4.1 4.2 5.1 5.2 5.3 Full analytic amplitudes and gauge-invariant subsets Helicity suppression in the MSSM 3.2.1 3.2.2 Yukawa suppression Isospin suppression Yukawa and isospin suppression lifting Heavy propagator suppression Threshold e ects Narrow width approximation and total cross section Spectrum of stable particles Double counting issues Results for the MSSM Theoretical benchmark models Total annihilation rate Yield enhancement 5.4 Indirect detection spectra 6 Conclusions A Neutralino annihilation amplitudes A.1 Expansion of amplitudes in the helicity basis A.2 Results for expanded amplitudes A.3 Suppression lifting from individual diagrams B Numerical implementation C Spin correlations of decaying resonances { 1 { Introduction The prime hypothesis for the cosmologically observed dark matter (DM) [1] is a new type of elementary particle [2]. Among theoretically well-motivated candidates, weakly interacting massive particles (WIMPs) play a prominent role. This is because such WIMPs very often appear in theories that attempt to cure the ne-tuning problems in the Higgs sector of the standard model of particle physics (SM), and because thermal relics with weak masses and cross sections at the electroweak scale are typically produced with the correct abundance to account for the DM density today [ 3, 4 ]. Another advantage is that the WIMP hypothesis can be tested in multiple ways: at colliders, where the signature consists in missing energy, in direct detection experiments aiming to observe DM particles recoiling o the nuclei of deep underground detectors, and in indirect searches for the debris of DM annihilation in cosmic regions with large DM densities. Direct detection experiments have become extremely competitive in constraining smaller and smaller scattering rates [5, 6], and collider searches have pushed the scale of new physics to TeV energies in many popular models [ 7, 8 ]. It is worth stressing, however, that only `indirect' searches would eventually allow to test the WIMP DM hypothesis in situ, i.e. in places that are relevant for the cosmological evidence for DM. Also indirect searches have become highly competitive during the last decade, now probing the `thermal cross section' (the one that is needed to produce the observed DM abundance) up to WIMP masses of the order of 100 GeV [9, 10]. A key quantity for both thermal production of WIMPs and indirect searches is the total annihilation cross section. Multiplied by the relative velocity v of the incoming DM particles, it can in the non- relativistic limit be expanded as v = a + bv2 + O(v4) : (1.1) It was noted early [11, 12] that radiative corrections to v can be huge because of symmetries of the annihilating DM pair in the v ! 0 limit. For indirect DM searches, changes in either the partial cross section, for a given annihilation channel, or the di erential cross section, d v=dE, may be phenomenologically even more important. The reason is that an additional photon in the nal state can give rise to pronounced spectral features in the DM signal in both gamma [13] and charged cosmic rays [14]. For electroweak corrections, the situation is in some sense even more interesting because, on top of the just mentioned e ects, completely new indirect detection channels may open up. In this way, antiproton data can for example e ciently constrain DM annihilation to light leptons when considering the associated emission of W or Z bosons [15]. In the presence of point-like interactions, such as described by e ective operators, the resulting spectra can be computed in a model-independent way by using splitting functions inspired by a parton picture [16]. This approach is very useful for generic DM phenomenology and is, for example, the one implemented in the `cookbook' for indirect detection [17]. One of the main results of this article (see also [18]) is that the resulting cosmic ray spectra from DM annihilation can di er substantially from the actual spectra, calculated in a fully consistent way from the underlying particle framework. { 2 { Here we revisit in detail one of the most often discussed examples where radiative corrections can be large, namely the case of a Majorana DM particle . The tree-level annihilation rate into light fermions f is then on general grounds `helicity suppressed', for v ! 0, as a consequence of the conserved quantum numbers of the initial state [19]. The resulting suppression by a factor of mf2 =m2 can be lifted by allowing for an additional vector [11] or scalar [20] boson in the nal state, implying that for DM masses at the electroweak scale the radiative `corrections' can be several orders of magnitude larger than the result from lowest order in perturbation theory.1 Here, we revisit these arguments and point out that the e ect commonly referred to as helicity suppression is in fact the culmination of two distinct suppression mechanisms, in the sense that they can be lifted independently. This results, in general, in a rather rich phenomenology of such radiative corrections. As an application, we consider electroweak corrections to the annihilation cross section of the lightest supersymmetric neutralino | one of the most often discussed DM prototypes [28] and still a leading candidate despite null searches for supersymmetry at ever higher energies and luminosities at the LHC [ 7, 8 ] | though our main ndings can be extended in an analogous way to other DM candidates that couple to the SM via the electroweak or Higgs sector. Concretely, we provide a comprehensive analysis, both analytically and numerically, of all three-body nal states from neutralino annihilation that contain a fermion pair and either an electroweak gauge boson or one of the ve Higgs bosons contained in the minimal supersymmetric standard model (MSSM), for a neutralino that can be an arbitrary admixture of Wino, Bino and Higgsino.2 We nd large parameter regions where these three-body nal states signi cantly enhance the DM annihilation rate, with the impact on the shape of the cosmic-ray spectra relevant for indirect detection being even more signi cant. One of the technically most involved aspects, apart from the shear number of diagrams to be considered, is how to avoid `double counting' the on-shell parts of the three-body amplitudes that are already, implicitly, included in the corresponding two-body results. We provide an in-depth treatment of this issue and demonstrate how to accurately treat not only the total cross section but also the resulting cosmic-ray spectra. We again nd signi cant e ects on the latter, indicating the need to correctly adopt this method also for other DM candidates. In fact, in order to reliably test the underlying particle models, our ndings suggest that at least for fermionic nal states it is in general not su cient to use the model-independent spectra traditionally provided by numerical packages. The numerical routines that implement our results for the neutralino case will be fully available with the next public release of DarkSUSY [39, 40]. 1The lifting of helicity suppression via three-body nal states is also relevant for real scalar dark matter [21{25] and, under certain conditions, for vector dark matter [26]. The case in which the additional boson is a Z0 has been considered in [27]. 2For neutralino annihilation, so far only the cases of photon [13] and gluon [29] internal bremsstrahlung for pure binos in [30{35], for Higgsinos in [36], and for pure Winos in [37]. A rst study for a general neutralino has been performed in [18]. Finally, nal states involving the SM-like Higgs boson have been considered in [20] for a toy model encompassing a pure Bino (see also [38]). { 3 { This article is organized as follows. We start in section 2 with a general discussion of Majorana DM annihilating to fermions and the relevant symmetries that arise for v ! 0, revisiting in particular the often invoked `helicity suppression' arguments and how this suppression can be lifted fully or partially. In section 3 we then consider the concrete case of neutralino DM, and discuss the various possibilities of how the presence of an additional nal state boson can add sizeable contributions to, or even signi cantly enhance, the two-body annihilation rates. The double counting issues mentioned above are then addressed in detail in a separate section 4. We scan the parameter space of several MSSM versions and demonstrate the e ect of these newly implemented corrections to neutralino annihilation in section 5, both for the annihilation rates and the cosmic-ray spectra relevant for indirect DM searches, and present our conclusions in section 6. In a more technical appendix, we describe the details of our analytical calculations to obtain the three-body matrix elements for fully general neutralino annihilation in the MSSM (appendix A), the numerical implementation of these results in DarkSUSY (appendix B), and how to correctly treat spin correlations of decaying resonances (appendix C). 2 Majorana dark matter and relevant symmetries For DM annihilation in the Milky Way halo, where DM particles have typical velocities of order 10 3, only the rst term in eq. (1.1) gives a sizeable contribution. In the following we therefore neglect p and higher partial wave contributions, and it is understood that all (di erential) cross sections are e ectively evaluated in the zero-velocity limit. For an s-wave, the relative angular momentum in the initial state is L = 0. Due to the Majorana nature the initial particles are identical, but because we consider fermions the total wave function still needs to be antisymmetric with respect to exchanging the incoming particles. The orbital wave function for L = 0 is symmetric, so in order to get an anti-symmetric total wave function the spins must couple into an antisymmetric state. This is only possible for the singlet state, with S = 0, resulting in the following quantum numbers: J = 0; C = ( 1)L+S = 1; P = ( 1)L+1 = 1 : (2.1) Here, the general expressions for C and P apply because we have a system of two fermions. Assuming no signi cant sources of CP violation in the theory, which generally are highly constrained by measurements of the electric dipole moment and other precision experiments, the symmetry of the nal state is hence also restricted to be JCP = 0 . This implies the well-known `helicity' suppression of the annihilation rate into light fermions, similar to the case of charged pion decay. In the following we rst brie y review the origin of this suppression, and then argue that it can in fact be related to a combination of two rather independent suppression mechanisms. 2.1 Chiral symmetry, gauge symmetry and helicity suppression We want to study Dirac fermions f as possible nal states from the annihilation of Majorana mf )f , which is invariant under Lorentz transformations, i.e. invariant under SU(2)L+R. In the massless limit, mf ! 0, { 4 { where the arrows indicate the spin direction along the z-axis (the rst entry refers to the HJEP09(217)4 antifermion, the second to the fermion). If the two fermion momenta are (anti-)parallel | e.g. because they are emitted back-to-back as the nal states of a DM annihilation process | the z-axis can be chosen to be aligned in the same direction as the momenta, and the above spin projections on the z axis are directly related to the helicities of the two particles. Choosing pf (pf ) to point along the positive (negative) z-axis, the helicity con gurations h = Szpz= jpzj of the singlet and triplet state are then given by (2.2) (2.3) (2.4) (2.5) and 1 p 2 + ; 1 2 1 2 ; 1 2 1 2 1 ; p 2 1 2 1 2 + ; + 1 2 ; 1 2 1 2 1 2 1 2 + + ; + ; 1 2 ! p fR; fL fL; fR 1 2 ; + ; 1 2 this symmetry is upgraded to a chiral symmetry SU(2)L SU(2)R, in which the left and right handed Weyl states transform independently of each other, and helicity and chiral eigenstates unify. For a fermion pair f f , the spins can combine to either a singlet (S = 0), or a triplet (S = 1) spin state, (j"#i j#"i) =p2; n j##i ; (j"#i + j#"i) =p2; o j""i ; ! fL; fL ; p fR; fL + fL; fR ; fR; fR where the arrows indicate the chiral states in the left/right decoupling limit, i.e. for mf ! 0. The momentum con guration thus restricts which helicity states can be associated to the spin states. Angular momentum and the assumed CP invariance, on the other hand, restrict which spin state can be realized. Since CP = ( 1)L+S ( 1)L+1 = ( 1)S+1, for example, only the singlet state with S = 0 is compatible with the odd CP parity of the initial state. Eq. (2.4) then tells us that both fermion and antifermion in this momentum con guration must have the same helicity. In any chirally symmetric theory, however, the antifermion must necessarily have the opposite helicity of the fermion. We note that angular momentum conservation alone leads to the same conclusion: since L = r p, we must have Lz = 0 and hence Sz = Jz = 0. Eqs. (2.4), (2.5) then imply that fermion and antifermion must, independently of the value of S, have the same helicity. The annihillation process ! f f is therefore only possible if chiral symmetry is broken in the Lagrangian, for example through an explicit fermionic mass term mf fLfR or through the coupling of the fermion f to a scalar eld fLfR. It follows that the amplitude of the annihilation process must be proportional to the chiral symmetry breaking parameters mf or . In addition, it is instructive to consider also the isospin of the involved particles. Since left-and right-handed SM fermions transform under di erent representations of SU(2)L, the nal states fLfR and fRfL have total isospin I = 1=2. The initial state , on the other { 5 { hand, has necessarily integer isospin, implying I 6= 0. The annihilation rate thus has to vanish for an unbroken SU(2)L, and therefore has to be proportional to at least one power of the Higgs vacuum expectation value (VEV) vEW . For heavy DM the ratio v vEW =m becomes small, and processes with I 6= 0 will be suppressed by some power of v. In total, this implies that the amplitude for ! f f has to involve (at least) one parameter that breaks chiral symmetry, and one power of vEW that controls breaking of the SU(2)L symmetry. For the SM fermions that receive their mass from the Higgs mechanism, both of these conditions are ful lled for the usual helicity suppression factor mf . Depending on the model, however, there can be further possibilities, as we will discuss in detail for the case of the MSSM below, and it is useful to discriminate between the two suppression mechanisms. In the following, we therefore refer to the suppression related to chiral symmetry breaking as Yukawa suppression, and to the one related to electroweak symmetry breaking as isospin suppression. While isospin suppression is controlled by only one parameter, v = vEW =m , there can in principle be several sources of chiral symmetry breaking, for example in models with more complicated Higgs sectors. Nevertheless, as we discuss in detail in section 3, all terms that break chiral invariance in the MSSM are accompanied by Yukawa couplings yf / mf =vEW . Even though the following discussion of suppression lifting is completely model independent we will thus continue to assume, for concreteness, that chiral symmetry breaking is controlled by yf . 2.2 Lifting of Yukawa and isospin suppression With the above discussion in mind, the only way to avoid the suppression of non-relativistic Majorana DM annihilation is to allow for an additional nal state particle. Lorentz invariance requires this additional particle to be a boson, such that the leading process we are interested in is of the form ! BF f (2.6) where B is a scalar or vector boson, and F = f if B is electrically neutral. The additional boson can be either a SM particle, in particular a photon (refered to as electromagnetic IB), a gluon, a weak gauge boson (W , Z) or the Higgs boson h, or it can be a new particle beyond the SM (for example a heavy Higgs boson within the MSSM). For the moment we want to keep the discussion model-independent, and therefore focus on the former case. A frequently used approximation is to restrict the discussion to B being radiated o a fermion line in the nal state, as described by soft and/or collinear splitting functions [16, 17]. We emphasize that this approach does not capture the (partial) lifting of helicity suppression, and therefore is inadequate for the case of heavy Majorana DM annihilation to fermions. Taking the gauge restoration limit vEW ! 0, it becomes straight-forward to exhibit the scaling of a given process with yf and vEW . (We emphasize that we consider this limit only in order to discuss the possible mechanisms of Yukawa and isospin suppression lifting, while all our numerical results later on take the full dependence on vEW and yf into account). In this limit, the left- and right-handed components of the fermions in the nal state and in internal lines can not only be considered as gauge interaction eigenstates but as independently propagating degrees of freedom. The fermion mass is treated perturbatively in the mass in{ 6 { FRfL or FLfR nal state boson B and fermion combinations. Entries / 1 correspond to processes that potentially can lift both Yukawa and isospin suppression of the two-body process. Entries / yf can lift isospin suppression but are still suppressed by the Yukawa coupling, while those / vEW can lift Yukawa suppression but are still suppressed by v = vEW =m for large m . sertion approximation, and is associated with a chirality ip along with a suppression factor mf / yf vEW . In addition, longitudinally polarized gauge bosons WL=ZL can be replaced by the corresponding Goldstone bosons G ; G0 by virtue of the Goldstone boson equivalence theorem, cf. eqs. (2.7){(2.8) below. All nal states thus have de nite SU(2)L quantum numbers (i.e. I = 1=2 for G ; G0; h; fL, I = 0 for fR, and I = 1 for WT ), except for the ZT , which is a mixture of I = 0; 1 states (even in the gauge restauration limit, we nd it convenient to express our results in terms of the Z boson instead of the neutral SU(2)L boson). The amplitude of the generic three-body process indicated in eq. (2.6) can be non-zero for vEW ! 0 only if I = 0, i.e. if isospin is conserved. Furthermore, the amplitude must vanish for yf ! 0 unless both fermions have the same chirality. Note that this is possible for three-body processes because the kinematics does not force the fermions to be emitted back-to-back in the center-of-mass (CMS) frame, and therefore the arguments discussed in section 2.1 do not apply.3 These two observations immediately determine which annihilation processes can lift either Yukawa or isospin suppression (or both). In these considerations, for various combinations of fermion chiralities and nal state bosons (where the longitudinal gauge bosons represent the corresponding Goldstone bosons). Both suppression factors can be lifted only in processes where a transverse gauge boson (ZT , WT , , or gluon g) is emitted and the nal state fermions are described by spinors of equal chirality (FRfR or FLfL). For longitudinal gauge bosons (ZL or WL) or the Higgs boson h, only one of the suppression factors can (potentially) be avoided for three-body nal states: isospin suppression can be lifted if the fermions are of opposite chirality, and Yukawa suppression can be lifted if the fermions are of equal chirality. 3In the extreme case where both fermions are emitted in the same direction, e.g., one simply has to exchange fR $ fL in eqs. (2.4), (2.5), which allows equal chiralities of the fermions in both the singlet and triplet spin state. In this kinematical con guration, it is easy to visualize how the fermion momentum can be balanced by the emitted boson B, and how their spin can combine with SB and L to the required J = 0 for both SB = 0 and SB = 1. In general, the spin singlet and triplet states will be linear combinations of all chiral states, with expectation values that depend on the angle between the fermion momenta, thus rendering the above argument essentially independent of the speci c kinematical con guration. Also the requirement of CP conservation is much less restrictive for three-body than for two-body nal states. A general discussion is somewhat complicated by the fact that e.g. F f is not necessarily a CP eigenstate that could be analysed individually, but in principle straight-forward by classifying all possible e ective operators that Let us stress that the symmetry arguments presented above simply guarantee that the amplitude must vanish for yf ! 0 and vEW ! 0, respectively, and the same applies to any gauge invariant sub-sets of diagrams. The actual suppression can thus be stronger than indicated by table 1, i.e. by additional powers of vEW or yf . At the same time, we caution that single diagrams can scale in a di erent way, depending on the gauge choice, such that the vanishing for yf ! 0 or vEW ! 0 is in general not guaranteed. Following up on the last comment, let us for convenience brie y recall how to verify gauge independence and identify gauge invariant subsets of diagrams. While for photon emission a good test is to check whether a given set of diagrams satis es the Ward identity M ( ! f f )k = 0, where k is the momentum of the photon, this does not work for electroweak IB because SU(2)L U(1)Y has been spontaneously broken. Indeed the question of gauge invariance changes in general, as weak hypercharge and isospin are no longer conserved in their original form. For the spontaneously broken Glashow-Weinberg-Salam theory the correct way to de ne gauge invariance is in terms of the preserved BRST symmetry [42, 43], under which SM eld transformations involve ghost elds which arise from the electroweak gauge xing procedure. This implies a new set of Ward identities, which in general depend on the choice of gauge. Using the standard R class of gauges [44], we arrive at the Ward identities for electroweak IB as expected from the Goldstone equivalence theorem: M ( M ( ! f f Z)k ! F f W )k = imZ M( = mW M( ! f f G0) ; ! F f G ) : (2.7) (2.8) We reiterate that eqs. (2.7) and (2.8) in general apply to (subsets of) the full amplitude, not individual diagrams, and are a valuable test for the results outlined in the next section. 3 Neutralino annihilation to f f and an additional nal-state particle In this section we apply the general discussion of helicity suppression lifting in Majorana DM annihilation to the lightest supersymmetric neutralino as DM candidate, and additional nal state bosons charged under SU(2)L. For photon or gluon IB we refer to the references listed in the introduction. Concerning the choice of DM candidate, we note that much of the following discussion is still rather generic and can thus be extended in a straight-forward way to any theory with an extended Higgs sector or where the DM particles belong to a di erent electroweak multiplet. We will introduce the relevant three-body processes and Feynman diagrams in section 3.1, re-visit the discussion of the helicity suppression in light of the speci c situation encountered in the MSSM (section 3.2) and then demonstrate in detail how these suppressions can be lifted, fully or partially, in section 3.3. In sections 3.4 and 3.5, nally, we discuss two mechanisms by which three-body cross sections can be enhanced which are not related to the helicity suppression of two-body nal states. { 8 { F¯ V/S f V/S F¯ F¯ f F¯ V/S V/S χ˜01 χ˜01 χ˜01 f˜i f˜i V/S f F¯ f F¯ V/S χ˜01 χ˜01 χ˜01 χ˜01 f˜i χ˜0n/χ˜±n χ˜0n/χ˜±n F¯ V/S f f F¯ χ˜01 χ˜01 ). See text for more details on how the individual topologies are referred to in this article. HJEP09(217)4 3.1 Full analytic amplitudes and gauge-invariant subsets From now on, we thus assume DM to be composed of the lightest neutralino, which is a superposition of Wino, Bino and Higgsino states, f F¯ f V/S F¯ V/S ~01, (3.1) (3.2) (3.3) obtained by diagonalizing the neutralino mass matrix = N11B~ + N12W~ 3 + N13H~10 + N14H~20 ; M1 0 0 B B B B B p 2 2 0 M2 2 pgv2 2 pg0v1 gv1 p Higgsino mass parameter; v1 and v2 are the VEVs of the two Higgs doublets, with vEW = v1=v2, and g and g0 are the SU(2)L and U(1)Y couplings, respectively. We follow the conventions of ref. [45], as implemented in DarkSUSY, and take all mass eigenvalues to be positive, while the diagonalization matrix N can be complex. We want to consider here all three-body nal states that contains a fermion pair and a boson that is charged under SU(2)L. Assuming CP -violating terms to be small, the full list of processes of interest is thus ! W +F f; Zf f; H+F f; Af f; Hf f; hf f : Here, A denotes the CP -odd Higgs, H+ the charged Higgs, and H and h the heavy and light CP -even Higgs bosons, respectively. For charged boson nal states, f denotes any fermion doublet component with isospin +1=2, and F the corresponding one with isospin 1=2; for neutral bosons, f can be any SM fermion. A Z H± F¯ W± F¯ f W± f W± f W± F¯ A Z and a W boson, mediated by s-channel bosons with a mass at the scale of the CP -odd Higgs A. f F¯ W± f F¯ W± f F¯ χ˜01 χ˜01 χ˜01 χ˜01 χ˜±n H± χ˜±n W± χ˜01 f˜¯i χ˜01 χ˜±n f W± F¯ f W± F¯ f W± F¯ χ˜01 χ˜01 χ˜01 χ˜01 χ˜01 A Z f˜¯ i In gure 1, we show all contributing Feynman diagrams in a condensed form (note that some of these diagrams may vanish for speci c combinations of internal and external particles). For future reference, we follow ref. [18] and refer to the top row of diagrams as (derived from two-body) s-channel processes, and to the bottom row of diagrams as t=u-channel processes (noting that t- and u-channel amplitudes are identical in the v ! 0 limit). Likewise, we denote diagrams of the type that appear in the rst column as virtual internal bremsstrahlung (VIB), diagrams of the type that appear in the second and third column as nal state radiation (FSR),4 and diagrams of the type that appear in the last two columns as initial state radiation (ISR). We explicitly calculate the full analytical expressions for all these processes in the limit of vanishing relative velocity of the annihilating neutralino pair, see appendix A.1 for technical details. We then use the Ward identities in eqs. (2.7) and (2.8) to group diagrams into gauge invariant sets for the case of vector boson nal states. In general we identi ed only two of such invariant sets: those diagrams that are derived from two-body s-channel processes and those that are derived from two-body t-channel processes. In the limit mA m | which is phenomenologically particularly relevant because the observed 4We stress that this distinction between VIB and FSR, while useful for the speci c purpose of our discussion, is not gauge invariant and exclusively refers to the topology of the involved diagrams. In particular, it should not be confused with an often used gauge invariant alternative set of de nitions where FSR refers exclusively to the soft or collinear photons radiated from the nal legs [13, 16, 17], while VIB is de ned as the di erence between the full amplitude squared and the FSR contribution [13]. f F¯ W± f F¯ W± f F¯ W± Higgs is very SM like | the s-channel diagrams however split into two gauge-invariant subsets. All diagrams then fall quite neatly into 3 categories: heavy Higgs s-channel, which are the set of diagrams with (at least one) mediator at the mass scale MA (see gure 2), weak-scale s-channel, which are the set of diagrams with s-channel mediators at the weak scale (see gure 3), and t-channel, which are the set of diagrams with sfermion mediators (see gure 4).5 For Zf f and hf f nal states the three sets of diagrams can be obtained analogously: t-channel contributions involve at least one sfermion line, while the remaining diagrams belong to the s-channel category (which can be further split into subsets involving at least one mediator at scale MA, or none, respectively). lation rate by a factor of mf2 =m2 is indeed the combination of in principle independent Yukawa and isospin suppressions. Let us now turn back to this observation and discuss it in more detail in light of the MSSM, where both mechanisms are still intrinsically linked because of the connection between gauge symmetry and chiral structure in the MSSM Lagrangian. 3.2.1 Yukawa suppression The chiral symmetry of the MSSM Lagrangian is broken by terms proportional to Yukawa couplings (in order to avoid avour-changing neutral currents, we assume as usual that the A-terms are proportional to the Yukawa coupling matrices). Following the general arguments of section 2.1, any amplitude contributing to ! f f must therefore be proportional to yf . Within the MSSM the values of yf are functions of tan but, except for the top quark, in general so small that this can lead to a suppression of the two-body amplitudes by many orders of magnitude. From the point of view of the broken theory, this Yukawa suppression appears to arise from rather di erent types of contributions to the Lagrangian: i) fermion mass terms ii) couplings of any of the ve physical Higgs elds to fermions iii) couplings of fermions to sfermion mass eigenstates (which mix the left- and righthanded elds). For example, the rst case is relevant for annihilation into fermions via t-channel sfermion exchange if the sfermion mixing is small (otherwise, the third contribution can dominate the amplitude), and the second for annihilation via s-channel pseudoscalar mediation. We note that all three interaction types couple left- and right-handed states and hence can ` ip' the helicity of one of the nal state fermions. The helicity combinations that 5We note that for v ! 0 the two s-channel ISR diagrams are actually identical, but for clarity we still include them separately in gures 2 and 3. For v ! 0 and mF ! 0, also the two t-channel ISR diagrams are important cross-check of our nal amplitudes, we con rmed analytically that these identities indeed hold. would result in a chirally symmetric theory, fR;LfR;L, can thus be transformed into those compatible with the global symmetry requirements outlined in section 2.1, fR;LfL;R. Traditionally, the notion of this helicity ip is sometimes taken to refer speci cally to the case (i), in which it is the (kinematic) fermion mass that breaks chiral symmetry in the Lagrangian. Instead, we associate the e ect directly with the Yukawa couplings in the MSSM Lagrangian (which of course give rise to the SM fermion masses). As also discussed in section 2.1, the annihilation process ! f f furthermore violates I 6= 0, and therefore its amplitude has to vanish in the gauge restoration ! 0. The resulting isospin suppression by a factor v vEW =m , for heavy neutralinos, can arise from di erent terms in the Lagrangian of the broken theory: a) fermion mass terms b) mixing of di erent gauge multiplets (Bino, Higgsino, Wino) that contribute to the lightest neutralino mass eigenstate given by eq. (3.1) c) mixing of left- and right-handed sfermion eigenstates. The structure of the neutralino mass matrix (3.2) indeed con rms that neutralino mixings vanish for vEW ! 0, as required by SU(2)L invariance. Note that case (a) and (c) are intrinsically linked to an accompanying chirality violation, since mf / yf vEW and the o diagonal terms in the sfermion mass matrix are also proportional to yf within the MSSM. Let us consider as an illustration the t- and s-channel contributions to ! f f . The kinematical helicity suppression due to the fermion mass mf is relevant for the t-channel (sfermion exchange). In this case Yukawa and isospin suppression simply arise from the two factors in mf / yf vEW (case (a) and (i), respectively). In addition, the Yukawa and isospin violation can be due to the sfermion mixing (case (c) and (iii)). Indeed, due to the mixing, a given sfermion mass eigenstate can couple to both left- and right-handed fermions, which then gives rise to the required chirality ip. For s-channel annihilation, on the other hand, the situation is more interesting in the sense that Yukawa and isospin suppression cannot simply be traced back to the same origin. For a pseudoscalar Higgs boson A as mediator, e.g., the Yukawa suppression stems directly from the Yukawa coupling / yf Af f (case (ii)), while the isospin suppression arises from the neutralino mixing (case b): for pure gauge multiplets the coupling A would be forbidden by SU(2)L invariance, and therefore vanishes for vEW ! 0. For a Z-boson in the s-channel, the discussion of the limit vEW ! 0 is a bit more involved (see appendix A.3), but is essentially analogous to the case of an A mediator. 3.3 Yukawa and isospin suppression lifting In section 2.2, we discussed which three-body nal states the Yukawa- and/or isospin suppression of the process B is a SM gauge boson or a Higgs boson. This general discussion based on isospin and ! BF f can potentially lift ! f f , for the case in which FRfL or FLfR FLfL FRfR t=u; s vEW vEW yf t=u; s vEW t=u vEW h=H yf t=u; s vEW t=u; s vEW t=u; s We also indicated whether the process can be realized with the maximal enhancement allowed by chiral and isospin symmetry in t + u and s-channel annihilation processes, respectively. For the rst two columns we also specify for which neutralino composition (B~ = bino-like, W~ =winolike, H~ =Higgsino-like) the maximal enhancement occurs. For the last three columns t + u-channel processes are possible for B~- or W~ -like neutralino as well as mixed H~ =B~ or H~ =W~ , and s-channel processes are possible for mixed H~ =B~ or H~ =W~ . Entries with a dash do not contribute to the order we are working in (see appendices A.2 and A.3 for details). chiral symmetry in the limit vEW ! 0 can be extended to the MSSM, as shown in table 2, by noting that all physical Higgs bosons h; H; A; H have isospin I = 1=2. Compared to the gauge restoration limit ZL is given by the Goldstone boson G0 (and hence transforms in a similar way as the pseudoscalar A). Similar arguments apply to the other Higgs bosons. In appendix A.2, we consider the full analytic expressions for six di erent mass hierarchies of particular phenomenological interest and determine for each of the previously discussed gauge-invariant subsets of diagrams the leading order in vEW and yf . The result of this exercise is collected in tables 9{11, where we present the ratio of the leading term for the three-body amplitude and the corresponding two-body amplitude. This allows us, as also indicated in table 2, to identify which contributions to the three-body amplitudes actually realize the suppression lifting that we can maximally expect on the basis of our general symmetry arguments; the `missing' cases, for which we did not nd a contribution within the MSSM, are marked by a `-'. For a detailed technical discussion of the various lifting mechanisms, and how they are realized at the level of individual diagrams, we refer to appendix A.3. We provide a graphical summary in table 3, where we show representative diagrams that realize the lifting of isospin and/or Yukawa suppression, for the sets of gauge invariant classes of diagrams that can be discriminated in the gauge restoration limit (in addition to the three sets discussed before, the t-channel can be split into contributions that remain non-zero in the limit of pure neutralino states (I), and those that require neutralino mixing (II)). Isospin suppression can be lifted in all cases by the emission of longitudinal gauge bosons (here represented by the Goldstone bosons) or a Higgs boson. Lifting of Yukawa suppression, as well as lifting of both suppression factors, is more restricted. This can be traced back to basic properties of the unbroken MSSM Lagrangian and the conservation of JCP = 0 (see appendix A.3 for details), explaining the `missing' ZT , WT FLfL, FRfR χ˜01 χ˜01 χ˜01 χ˜01 χ˜01 χ˜01 gv χ˜01 χ˜01 χ˜01 χ˜01 g gv g gv f˜ f˜ f˜ g g g g g Yf G0 A0 Yfv Yfv Yf Yf f f¯ f f¯ f f¯ f f¯ f f¯ χ˜01 χ˜01 χ˜01 χ˜01 f˜ g g χ˜01 f˜ Yukawa + Isospin Isospin Yukawa ! 0. The rows correspond to the four gauge-invariant subsets of diagrams that can be discriminated in this limit (see appendix A.3 for details). The rst column corresponds to the two-body process, and the other columns show various three-body processes. The diagrams shown in the second column lift both Yukawa and isospin suppression. The diagrams in the third column lift only isospin suppression, and in the fourth column only Yukawa suppression. We show only one representative diagram for each topology (ISR/FSR/VIB) and suppression mechanism. The coupling factors attached to vertices and mass/mixing insertions give the scaling with yf , vEW and g of each diagram (for Bino- or Wino-like neutralinos; modi cations for Higgsino-like neutralinos are described in appendix A.3). Note that contributions with WT emitted via ISR (second column, rst and third row) exist for Wino- or Higgsino-like neutralinos; those with ZT emitted via ISR occur only for a Higgsino-like neutralino. entries in table 2. Let us also highlight that the classi cation procedure revealed ways to lift the two-body suppression that have not been pointed out for the MSSM before (in particular Higgsstrahlung via t-channel ISR and a speci c s-channel VIB process, shown in the last column and second/third row in table 3, respectively). An additional form of suppression, unrelated to the discussion so far, arises in diagrams that rely on mixing between neutralinos or contain heavy propagators. This mass suppression takes the form X m =MX, where X is the heavy state in question. In particular, both HJEP09(217)4 s-channel contributions to ! f f and a subset of t-channel contributions | those of type (II), see appendix A.2 | rely on mixing the Bino/Wino with the Higgsino. For example, for a Bino- or Wino-like neutralino, the two-body amplitude in the s-channel is suppressed by a factor 2 = m2 = 2 if j j m . For a Higgsino-like neutralino, on the other hand, it is suppressed by Mi = m =Mi for Mi m , where Mi = min(M1; M2) (see table 12). These suppression factors of the s-channel annihilation can be lifted for the case of a Wino- or Higgsino-like neutralino by the emission of a (transverse) W or Z from one of the initial neutralino lines (ISR). (The corresponding diagram is illustrated in the third row, second column of table 3.) Additionally, this three-body process simultaneously lifts both isospin- and Yukawa suppression. It is particularly relevant if the two-body nal states W W and ZZ are kinematically forbidden, such that the internal gauge boson is o -shell. This is a special case of the threshold e ects that we turn to next. 3.5 Threshold e ects A given two-body channel ! AB is strongly phase-space suppressed if the CMS energy is close to the mass of the nal-state particles, and for 2m mA + mB the corresponding partial cross section vanishes completely in the v ! 0 limit. If either A or B are o -shell and decay into much lighter states, however, the phase-space opens up again and thereby potentially increases even the total two-body annihilation rate signi cantly. For the MSSM, this is particularly relevant for the W +W and tt channels, which has previously been studied for speci c neutralino compositions [46, 47] (for an approximate numerical implementation in the context of relic density calculations, see [48]). For the processes we are interested in here, threshold e ects can in general appear for any two-boson nal states (or tt). For a more detailed discussion of this e ect, it is useful to rewrite the three-body cross section as (see e.g. [49, 50]) v2!3 = = S S 4E 1 E 2 4E 1 E 2 Z Z jM2!3j2 d 3(P ; p1; p2; p3) jM2!3j2 d 2(P ; p1; q) dq2 2 d 2(q; p2; p3) ; (3.4) where d n(P ; p1; : : : ; pn) = (2 )4 (4)(P P pi) Q d3pi i (2 )32Ei is the n-body phase space element, P = p 1 + p 2 the sum of the 4-momenta of the annihilating neutralinos and E i their energy; the pi denote the nal-state momenta. Since q2 = (p2 + p3)2 is time-like, we will in the following often use the notation q2 m223 instead. For the processes considered here, cf. eq. (3.3), the symmetry factor S is always 1.6 Furthermore, jMj2 denotes the usual squared matrix element, averaged over initial spins and summed over nal spins/helicities. We now assume that the amplitude is dominated by a resonant, almost on-shell internal propagator that decays into particles 2 and 3, and hence carries momentum q. For a resonance R with mass M , width , and spin 1, 1=2, or 0, respectively, we then have M2!3 = 1 m223 M 2 + iM < : > > 8 >> M(2q!) 2( g M(2q!) 2(=q + M )M1!2 (q) M2!2M1!2 (q) + q q =M 2)M1!2 vector fermion scalar (3.5) where M2!2 (M1!2) is the matrix element for ! p1q (R ! p2p3), up to polarization vectors or spinors for the `external' particle R (as indicated by the superscript q). The decisive observation is now that R 2 jM1!2j d 2(q; p2; p3) must be independent of the polarization state of R once all the nal state polarizations are summed over. This is familiar from on-shell momenta q | the total (but not di erential) decay rate of a particle is independent of its polarization state | but holds more generally for time-like initial momenta q [50]. As long as the full phase-space integral is performed (see section 4.2 for how to treat di erential cross sections), one may thus conveniently replace the correlated polarization or spin structure of eq. (3.5) with an unpolarized sum: M(2q!) 2( g + q q =M 2)M1!2 (q) 2 X (q) M2!2 M(2q!) 2(=q + M )M1!2 (q) 2 M(2q!) 2ususM1!2 (q) 2 = 1 = In this way, we can independently of the spin of R replace jM2!3j2 ! (m223 2 in eq. (3.4) which, for v ! 0, leads to v2re!s 3 = S Z (2m m1)2 dm223 (m2+m3)2 (m223 m23 M 2)2 + M 2 2 R!23 fv ~ !1R : 6In general, if some of the nal state particles are of identical type, con gurations that di er only by exchanging these particles should be counted only once in the phase space integration. Since this will be convenient later on, we thus use a convention where one integrates over all of the phase space as if all particles were distinct, and then correct for the corresponding over-counting by a symmetry factor S. It is S = 1 if all nal-state particles are distinct, and S = 1=2 (S = 1=6) if two (all three) of them are identical. (3.6) (3.7) (3.8) (3.9) Here, the decay rate of the o -shell resonance in the frame where q = (m23; 0) is given by and the cross section for the annihilation into an o -shell resonance is given by ~ R!23 v f !1R S23 Z 2m23 S1R Z P 2 2 2 16 S1R 128 jM1!2j d 2(q; p2; p3) = In the last step we performed the phase-space integral explicitly by using the fact that for v ! 0 the annihilation process is kinematically the same as a pseudo-scalar decay, implying that jMj cannot have any angular dependence.7 Eq. (3.9) will thus continue to hold for general s-wave annihilation, provided one replaces 4m ! s in eq. (3.11). The squared matrix elements are here again summed (averaged) over nal (initial) spins/helicities, leading to an overall symmetry factor of S = S=(S1RS23) (with S1R; S23 de ned in accordance with footnote 6). We note that eq. (3.9) can be signi cantly simpli ed by a few well-motivated assumptions. Concretely, let us assume the o -shell particle to decay to massless nal states, m2 = m3 = 0, and jM1!2j2 / M 2 close to the threshold; this implies ~R!23 = (m23=M ) R!23. We also introduce a reduced cross section ( v)red ( v) !1R= n+1=2(1; 1; R) ; (3.13) with R m223=s and 1 m21=s, allowing for the two-body cross section close to threshold to be suppressed not only by a phase-space factor (n = 0), but by an additional such factor from the matrix element itself (as e.g. in the example of Higgsino annihilation below, for which we have n = 1). By de nition, ( v)red thus remains nite both above and below the threshold. Assuming ( v)red to be independent of m23 close to threshold, eq. (3.9) simpli es to v !1R ' S( v)red Z max d 0 ( 1)2 + 2 n+1=2(1; 1; R) ; (3.14) where max = ( s p m1)2=m2R and the sense that the threshold correction can be directly estimated for any given two-body cross section (i.e. without rst having to compute ~v or ~). R=M . This expression is model-independent in As an illustrative and concrete example, let us consider the process the limit of pure Higgsino DM. For simplicity, we assume that sleptons are much heavier 7For this reason, the result takes the same form as for o -shell decays [51, 52], suggesting a straightforward generalization to 4-body nal states dominated by the annihilation into two o -shell particles: v2re!s 4 = S Z dm122 dm324 m12 m23 (m212 MR21 )2 +MR21 2R1 (m223 MR22 )2 +MR22 2R2 ~R1!12 ~R2!34 fv !R1R2 : ! W e+ e in (3.12) mχ±=200 GeV W Γ ± W m χ [ ! W e+ . For the latter process, we show the cross section divided by the branching fraction W !e = W ' 1=9 (solid lines). For comparison, we also include the model-independent estimate of (W +) (dotted lines). For m . mW , the three-body cross section is clearly larger than the lowest-order result; above the threshold, on the other hand, the two agree exactly. ! W +W , compared to than the neutralino, such that the only contributing diagrams are of the V = W ISR type, with a virtual Higgsino-like chargino and a resonance R = W + . In this limit, we nd v f !W W 16m4 8m2 (m2W + m223) + (m2W 2m2 + 2m2~+ m2W m223 We calculate the full three-body cross section as derived in appendix A.1, in the pure Higgsino limit, and then compare it to the result given in eq. (3.9). As shown in gure 5, we obtain excellent agreement even though both the directly involved amplitudes and the numerical phase-space integrations are very di erent in nature (the two results for the three-body cross section, shown as solid lines, lie exactly on top of each other). This should of course be expected for a process which by construction only receives contributions from an o -shell nal-state particle, but we stress that eq. (3.9) is in general much simpler to calculate in praxis for such cases. For comparison, we also indicate (with dotted lines) the model-independent result given in eq. (3.14); as one can see, even this simpli ed expression provides an excellent approximation to the full result. Most importantly, our example illustrates the much more general point that a threebody process around or below the kinematic threshold of a large two-body process can be signi cantly enhanced over the total annihilation rate at lowest order. Above the threshold and rescaled to the relevant branching ratio for the decay of the resonance R, on the other (3.15) (3.16) hand, the three-body cross section for a process ! 1R; R ! 23 equals almost exactly the two-body result | an e ect which we will discuss in detail in the next section. 4 Double counting issues We now turn to double counting issues related to unstable nal-state particles. If the nal state of a two-body annihilation process undergoes a subsequent 1 ! 2 decay, in particular, this can also be viewed as a three-body process with the unstable particle (the resonance, in our wording) as an intermediate state. While we discussed the situation below the kinematic threshold for the production of the unstable particle in section 3.5 as a way of enhancing the total cross section, we are here interested in the kinematic region above the threshold. As before, this is relevant for all massive diboson as well as tt nal states considered here. One possibility to avoid over-counting identical kinematic con gurations when adding two-body and three-body processes would be to altogether disregard the former for massive diboson or tt nal states. Interferences between (nearly) on- and o -shell contributions to the amplitude would then be correctly accounted for, as well as the impact of the spin of the resonance. However, this procedure has several drawbacks on a practical level, and furthermore turns out to be incorrect for two-body processes with identical particles in the nal state (such as e.g. ! ZZ), as will be discussed in more detail below. We therefore prefer to explicitly subtract on-shell contributions to the three-body processes, which allows us to keep most of the advantages of the full three-body computation while correctly taking into account all symmetry factors. In the following we describe this procedure in more detail for both the total cross section and the di erential yield of e.g. gamma rays. 4.1 Narrow width approximation and total cross section For three-body processes dominated by an on- or o -shell resonance, the total cross section can be written as in eq. (3.9). If the intermediate particle corresponds to a nearly on-shell resonance with M , furthermore, the Breit-Wigner propagator can be approximated as 1 (m223 M 2)2 + M 2 2 ! M (m223 M 2) : This narrow-width approximation (NWA) yields the on-shell contribution of the resonance R, and we denote the corresponding, approximated cross section by vNW A. Strictly speaking, for the approximation to work well, the kinematic boundaries have to be su ciently far away from the pole, jm223 M 2j phase-space factors apart from the Breit-Wigner propagator should be smooth functions of m223 in the vicinity of the pole, which we assume in the following. With this replacement in eq. (3.9), we immediately recover the well-known result M , and all contributions from the matrix element and v2N!W3A = S v !1;R BRR!23 ; S2!2=P 2 R d 2jMpj2 is the two-body cross section. cles 2 and 3, R!23 = S1!2=(2M ) R d 2jMdj2 is the partial decay width, and v where BRR!23 = R!23= is the branching ratio for the resonance R to decay into parti!1R = (4.1) (4.2) For Higgs nal states, the summation over polarizations is absent, and we de ne the corresponding ratios RLR=RL and RLL=RR analogously, corresponding to hfRfL + hfLfR and hfLfL + hfRfR nal states, respectively. We then start from our full result for the helicity amplitudes, using the explicit representations of the generic couplings and mass matrices that appear there, and expand them up to O( v2). Note that the limit v ! 0 implies in particular that we expand in the fermion mass mf / yf vEW and in gauge boson masses mW=Z / gvEW . In order to simplify the resulting analytic expressions, we set all sfermion masses equal to the neutralino mass, noting that larger sfermion masses would suppress t=u-channel rates relatively strongly because v2t!c3hannel / mf~8(mf~4) for Bino- (Higgsino/Wino-)like neutralinos, as discussed previously [ 31, 33, 36, 37, 92 ]. Furthermore, we use the notation BF f where for neutral bosons (B = Z; h) the nal state fermion types are identical, F = f , while for charged bosons we adopt in the following the convention that f denotes the up-type fermion (e.g. the top quark in ! W bt). In these cases, we keep for simplicity only the dependence on the Yukawa coupling of the up-type fermion, and set the other one to zero. We furthermore consider six distinct scenarios describing the dominant neutralino composition, which result from di erent assumptions about the involved mass hierarchies and which are of particular phenomenological interest: Higgsino DM, with small Bino admixture ( Higgsino DM, with small Wino admixture ( Bino DM, with small Wino admixture (M1 M2; Bino DM, with small Higgsino admixture (M1 Wino DM, with small Bino admixture (M2 M1; Wino DM, with small Higgsino admixture (M2 M2; M1 ! 1) M1; M2 ! 1) ! 1) ! 1) ; M2 ! 1) ; M1 ! 1) From the Bino-, Wino- and Higgsino mass parameters M1, M2 and , we de ne the dimensionless mass suppression factors M1 m =M1, M2 m =M2 and m = . For all six scenarios listed above, we expand the amplitude ratios to leading order in these mass suppression factors. E ectively, the neutralino mixing between either Bino or Wino and Higgsino then becomes a perturbative `mass insertion' / gvEW represented by the respective o -diagonal entries in the mass matrix of eq. (3.2). Furthermore, for de niteness, we also expand to linear order in A m =MA, i.e. we work in the decoupling limit where the heavy Higgs states are much heavier than the neutralino or SM-like Higgs boson. We note that it is straightforward to generalize these results, and our numerical results anyway include all MSSM Higgs bosons and are valid for arbitrary mass hierarchies. To lowest order in the expansion parameters de ned above, isospin and fermion chirality have to be conserved in all interaction vertices (assuming that the mass splitting between M1, M2 and j j is large compared to gvEW ). One of the implications, as it turns out, is that the gauge-invariant subset of t-channel diagrams discussed in section 3.1 can be further split into two separate gauge-invariant sets. The rst, which we will denote by (I), does not contain any neutralino mixing insertion / gvEW , and would hence contribute even in the limit of a pure neutralino state. The set of diagrams that contain at least one such insertion (denoted by (II)), on the other hand, require a mixing in the neutralino sector (just like is the case for all s-channel diagrams). In tables 9{11, we show the results of this expansion for the helicity-summed ratios R that we have introduced in eq. (A.16), where the di erent tables correspond to the three types of nal states (W F f , Zf f , and hf f , respectively). Each table contains the results for all six mass hierarchy scenarios speci ed above, broken down to contributions from each set of gauge-invariant diagrams.19 For the sake of the presentation, we keep only contributions that lift the isospin- or Yukawa suppression of the corresponding two-body process (or both). In particular, as apparent from table 2, the ratio RLTR=RL cannot lift any of these suppressions, and is therefore not included in tables 9{11. Furthermore, for each of the gauge-invariant sets of diagrams, we include only those amplitude ratios that actually do lift at least one of the suppression factors. For the remaining entries, a `0' indicates three-body amplitudes that vanish to the order we consider, while for entries containing a ` ' both 2- and three-body amplitudes vanish.20 The two-body amplitudes, nally, are for convenience summarized in table 12. 19We checked explicitly (up to O( v2)) that the Ward identities, eqs. (2.7), (2.8) are satis ed for each set separately. Note that we use Breit-Wigner widths in the amplitudes, and while they break gauge invariance at O( v), they do not contribute to the amplitudes as v ! 0. 20While in this case the ratio would be formally ill-de ned, we only identi ed one example where the two-body amplitude vanishes while the three-body amplitude does not, marked with a ( ). We note that the relevant process, ! W W ! W F f for a Wino-like neutralino, is phenomenologically not important because for m > MW it is largely captured by annihilation into W W , while Wino-like neutralinos with m < MW are practically excluded. ) I I r o t a i d e l e n Z n m a h c s f f h ! + 2 0 0 L R f f L ) R L 2 f f ;x v L R 1 x 2 f f ( h h E ) 2 ;x v 1 x 2 ( ~ L 2 R J x 2 x ) 1 2 x + 1 x + 1 x ( f y 2 ) 1 2 x v 2 1 x x + v x1 2 p 2 )J +1 x2 )(x 1 (x x2 g + ) 1 2 x 1 x ( f y 4 ) 1 L ) 2 x 2 +1 x1 x x 2twp4 2 g 2 ;x v 1 x 2 ( ~ ) ) E E E ) 2 Y o R L 2 1 f f x x L R + v fh fh x1 2 p 2 L R ( f fR 2J )2 L f f g h h 2 x 2 r o t a i d e A R L f f L f f m h h a h c u + t R L f f 2 x v (x p v + 1 1 1 2 1 2 v p 2 2 p p v v v v R 1 2 1 2 1 2 p 2 p p 2 2 1 2 | p 2 p | s o 1 1 1 1 1 1 ! ! ! ! ! 1 2 2 1 ! f o i t M f i M M ; ; ; 2 1 M M M ;2 - 2g Y t HJEP09(217)4 a w a k u n i p s o s i n i p s o s i a w a k u n i p s o s i n i s i g n L d F e d l o a c b a e t s h s a l T u o e . i h g v t i in ilft e e n r p h t in t 1= sp in so sa !i r d t o e e a n c w e r la a d p e k u re Y a s t o e s t h n o i d o t e t a t en te o e u N n ib . o r ly e t ;` on iv b c t ; c n e p s ; o d s s o e = b r , f Z 1 r a 2 F th x . i w + ;t e x1 ; s c ; u t o p h = = l L y R = n R L f o r ,r R o o f t d a n e i t d a a e ) t s 1 m l an lea 2 c x f -k f a h e s )(1 2.) 2fR + w 1 Y r o x fo r ( t t c u le p= 2fL b e Y R ( R 4 = 6 ,0 th 1 i e lb s a L w L + R2) R m s t a a f s rg d Y A ia ez + i . d l L 1 l a f 1 e e nn rom (4Y n l a b h a -c re 2 T s a L f f y h c r a r e i H s s a M t = ) ) 2 s t = + v 1 ( ( 2w 2 A v t v t a i d e m Z r o t a i d e 2 M f y 2 g 1 2 2 s + 2 M m 1 A 2 A f y 2 f y g 2 4 g 4 t t = = ) ) t ) 1 I ( I 2 ( M 2 g f y 1 2 2 s ) = I v ( 2 s = v 3 f 3 f y r o v 2 2 c c y 2w t f v c2 v 1 M 2 ) ) 2 fR 2 fR Y Y + + 2w t f y 2 g 2 fL Y ( v y 2 g 2 2w t f y 2 g 2w t f y 2 g 8 t 1 ( 2w t 1 M v 2 g f y 1 2 2w t f y 2 g 1 2 t = 1 t = f y 2 g 2 2 c v 2 f y 2 g t = v Y ( v 2 2 fL v v f f y y 2 g g 2w t f y 2 g 2 | | | | ;c r o u f = te f o r N o . F s k . r a ;;t qu = 3 . y l f = e r c iv o N tc f , r e p n o s e t v c e r i a g f , e r ra l n o o i o s c s e a rp s t l u s y p e b u r s s u i o l d n e i p s i v i re l p u i m d p t o s e n h e a t b h o a ic t w h s a w ah ku Y o t d e o e r t v i au sd t a q n l s e r t o p n s e ,2 e !m rr q 22j lee c o x ta jMir h t t a , s m v t n e e h d n t a e m s w o d o w an ep t - n to t o - = ) o 1 t s a !le th t t a f e ( o c a e e l v lu ep l o a r v n d u t le en il p b e a n m T o a 1 1 1 1 1 1 w t ! ! ! ! ! ! e 1 2 2 1 M M ; 2 ; 1 M M M ;2 ; m T a | 2 A | le . f e f y 2 g 8 o $ r x i r V o i 1 1 2 2 . t s 2 s e M M M M 1 ed In order to assess the parametric enhancement of three-body over two-body processes, it is su cient to consider the amplitude ratios just presented, and we will continue with a more detailed discussion of the various lifting mechanisms at the level of individual diagrams in the following subsection A.3. Before doing so, let us brie y remark that the corresponding cross section ratio for obtained by ! BF f , normalized to the one for ! f f , is 1 v2!2 d( v)2!3 = 1 are the dimension-less fermion energies of the three-body nal state. Using the results from tables 9{11, one can thus obtain the contribution to this ratio from each of the gauge-invariant subsets of diagrams separately. In the limit of massless nal state particles, the integration ranges are 0 < x1 < 1 and 1 x1 < x2 < 1, implying that some of these integrations become logarithmically divergent. This is an expected artefact of the expansion in v and, in practice, the corresponding infrared divergent contributions are cut o by the non-zero mass of the vector boson. Throughout this work, we assume that the resulting logarithmic enhancement O( ln2(EB=ps)) can be treated perturbatively down to the infrared cuto mass of roughly m EB mB gvEW . This imposes an upper limit on the neutralino O(gvEW e =g) O(10) TeV. If one is interested in higher masses, it would be interesting to apply the resummation methods discussed e.g. in refs. [55{57]. On the other hand, we stress that the logarithmic sensitivity to ln2(g v) does not spoil the power counting arguments related to lifting of isospin suppression factors, since the latter is described by powers vn of v. In our numerical results, we fully take into account the masses of all annihilation products. A.3 Suppression lifting from individual diagrams It is rather illustrative to re ect the results of the previous subsection at the level of individual diagrams. In table 3, displayed for clarity already in the main text (see section 3.3), we therefore organize all relevant amplitudes in a large table, with the four rows corresponding to the four gauge-invariant subsets. For each type of diagram, and assuming a Bino- or Wino-like neutralino, we furthermore explicitly indicate the scaling with the gauge coupling g, the Yukawa coupling yf , and the vev vEW (we comment on the Higgsino-like case below). Let us start our discussion with the rst column, which contains the diagrams contributing to the two-body process / g2yf vEW , but the origin di ers: ! f f . As expected, all these amplitudes scale as t-channel I. The factor yf vEW enters either via the chirality ip of one of the nal-state fermions, or via a L=R mixing insertion of the sfermion (for brevity, we show only one representative diagram in table 3 for each of these cases). t-channel II. The factor vEW enters via the gaugino/Higgsino mixing insertion on one of the initial lines, and the Yukawa suppression enters via the Higgsino-sfermion-fermion coupling. s-channel EW. The s-channel with electroweak-scale mediator corresponds to the Zexchange diagram mentioned earlier. In the s-wave limit, and from the perspective of the unbroken theory, this diagram is represented by the exchange of the pseudoscalar Goldstone boson G0. The factor vEW arises from the gaugino/Higgsino mixing, and the Yukawa coupling from the Yukawa interaction G0f f . s-channel MA. This case is similar to the previous one, except that the mediator is replaced by the (physical) heavy pseudoscalar Higgs A. Let us now turn our discussion to the remaining columns of table 3, which contain all relevant three-body processes. Here, the second column shows representative Feynman diagrams that lead to a lifting of both isospin and Yukawa suppression, while the third and fourth column show diagrams that lift only one of them, respectively: HJEP09(217)4 Lifting of Yukawa and isospin suppression. Both suppression factors can be lifted only for two of the gauge-invariant sets of diagrams (t-I and s-EW). In the former case, a transverse ZT or WT is emitted from either fermion line in the nal state, from the sfermion line, or from the initial lines (this last case cannot occur in the Bino-like case). We remark that FSR can only lift the helicity suppression if the virtual fermion is strongly o -shell, i.e. not for soft and collinear photons (which are sometimes dened as FSR, see footnote 4). In the s-channel case, the diagrams can be thought of as an annihilation ! W W , with subsequent decay of W (see section 3.5 for a discussion of such o -shell internal states). It is impossible to lift both suppression factors for the other two classes: for t-II, this would require a gaugino-Higgsino-W=Z vertex, which is absent for vEW ! 0. The same applies for s-MA, noting in addition that the Af f coupling requires also the presence of a Yukawa coupling. Lifting of only isospin suppression. The isospin suppression can be lifted for all four subsets, by replacing the insertion of vEW within the two-body amplitude by the emission of a Higgs boson or a Goldstone boson, respectively. Note that for the set t-I this amounts to replacing the fermion mass insertion by a fermion-fermion-Higgs/Goldstone coupling (or replacing the sfermion L=R mixing insertion by a sfermion-sfermion-Higgs/Goldstone coupling, respectively). For all other sets one replaces the gaugino-Higgsino mixing insertion in the initial line by a gaugino-Higgsino-Higgs/Goldstone vertex. For the s-channel, the diagrams can also be thought of as an annihilation into a pair of scalars, with subsequent decay of one of them. This mechanism of suppression lifting is very general, and appears for all gauge invariant subsets of diagrams as well as for all nal states (involving W=Z or a Higgs boson). We expect it to be relevant especially for heavy neutralino masses. Lifting of only Yukawa suppression. This case is in some sense the most di cult to realize. The reason is that it requires a Higgs (or Goldstone) boson in the nal state, and therefore only diagrams where the nal-state boson does not couple directly to the nal-state fermions can potentially contribute in the limit yf ! 0. We identi ed three such processes, shown in the last column in table 3: for t-I, the Higgs (or charged Goldstone boson; note that there is no sfermion-sfermion-G0 vertex for yf ! 0) can be emitted from the sfermion line in the t-channel, i.e. via VIB. The corresponding vertex is derived from a four-scalar sfermion-sfermion-Higgs-Higgs interaction, involving the full Higgs doublets. This coupling leads to the required vertices at O(vEW ), and scales with g2 for yf ! 0 within the MSSM (see refs. [33] and [20] for a discussion within a toy model for the Goldstone- and Higgs-emission, respectively). In addition, for t-II, the Higgs can be emitted via ISR (second row, last column of table 3). While this contribution lifts Yukawa suppression, it is suppressed compared to the two-body process for a large mass hierarchy between gaugino and Higgsino mass parameters; we nevertheless kept this contribution, because the former e ect can easily compensate for the latter. Finally, for the s-EW case, the Higgs can be emitted from the s-channel mediator via a Goldstone-Higgs-Z coupling (third row, last column in table 3). Note that this mechanism is distinct from the one discussed in [38], and that the toy-model discussed there cannot be realized within the MSSM. To the best of our knowledge, both the t-channel ISR and the s-channel Higgstrahlung processes that we identi ed within the MSSM have not been discussed before. One can understand the diagrams that lift Yukawa or isospin suppression as shown in table 3 based on basic properties of the unbroken MSSM Lagrangian, as well as the symmetry requirement JCP = 0 of the s-wave initial state. For example, mixing insertions / gvEW of the neutralino line can turn a Bino into a Higgsino, but not into a Wino. In addition, the Higgsino coupling to fermion/sfermion pairs is proportional to the Yukawa coupling, while the corresponding coupling for Bino- and Wino-like neutralinos involves a gauge coupling and is therefore generally much less suppressed (except for the top quark). One slightly more involved example is the diagram in the last column of the rst row. For nal states involving a longitudinal WL, the corresponding sfermion vertex derives from the interaction term / g2(f~Ly H)(Hyf~L) present for sfermion doublet under SU(2)L. After inserting the decomposition H = (G+; (vEW + h + iG0=p2)) elds that transform as of the SM-like Higgs doublet one easily veri es that at linear order in vEW one obtains a sfermion coupling to G and h, but not to G0, which explains why no longitudinal ZL boson can be produced in this case. The Higgs nal state also receives a further contribution from the interaction term / HyHf~yf~, which exists for all (left and right) sfermion elds. Furthermore, for the s-channel processes of the type ! hB ! hf f that give a non-zero contribution in the s-wave limit, the mediator B is a pseudoscalar or transverse vector (i.e. G0; A0, ZT ), while for ! G0B ! G0f f , B is a scalar (i.e. h; H0). This is consistent with the odd CP parity of the initial state. Note that the above arguments are only valid when expanding around the unbroken theory, and representing longitudinal degrees of freedom by Goldstone bosons. In fact, within the broken theory, analogous arguments would be hampered by large cancellations that occur among individual diagrams, and that make the power counting less transparent. Nevertheless, we carefully cross checked that all these arguments can indeed be reproduced when using the full matrix elements within the broken theory, and expanding the sum of all diagrams within a gauge invariant subset for heavy neutralino mass. While the discussion above assumed a gaugino-like neutralino, the case of a Higgsinolike neutralino is very similar. For the third and fourth row in table 3, in particular, nothing changes except that the incoming neutralino is now a Higgsino in the limit vEW ! 0, and the insertion / gvEW denotes mixing with either a Bino or Wino (in addition, both ZT and WT ISR is possible, while only WT ISR is possible for Wino-like neutralinos). The same applies to the second line, after interchanging the label of g and yf on the vertices involving a sfermion in all diagrams in the rst and second column (this does not a ect the overall scaling of the amplitude), while the diagram in the last column would receive an additional yf2 suppression. For the rst row, the two neutralino-sfermion-fermion vertices scale with yf instead of g in all diagrams. Thus, this class is additionally suppressed by a factor yf2 compared to the other subsets. Nevertheless, for completeness, we kept this case because the three-body processes can lift the additional suppression factors yf vEW of the two-body amplitude in the same way as for a gaugino-like neutralino. In summary, we con rmed the general symmetry arguments outlined in section 2.2 for the MSSM and explicitly identi ed the contributions to the three-body amplitudes that realize the suppression lifting, focussing on nal states containing (tranverse or longitudinal) gauge bosons as well as the SM-like Higgs boson. By expanding the full amplitudes in various limits that correspond to Bino-, Wino- or Higgsino-like neutralino, respectively, we nd that (almost) all of the possibilities allowed by symmetries are realized. The cases for which we did not nd a contribution within the MSSM are marked by a `-' in table 2. For processes involving W bosons and purely right-handed fermions an additional suppression arises that can be traced back to the chiral structure of the SU(2)L interaction. For processes involving ZL (represented by G0) or A, and fermions of equal chirality, on the other hand, lifting of Yukawa suppression would require that the amplitude does not contain Yukawa interaction vertices. In addition, vertices such as sfermion-sfermion-G0=A are absent for yf ! 0 (as required by CP -invariance), such that a t-channel process analogous to the one in the rst row, last column of gure 3 does not exist. For the s-channel, the symmetries of the initial state would require a CP-even mediator if the G0 or A was emitted via ISR. Within the MSSM, only the Higgs bosons are available. However, their coupling to fermions necessarily involve a Yukawa coupling, such that Yukawa suppression cannot be lifted in this speci c process. Similarly, one can convince oneself that the s-channel VIB process (3rd row, 4th column of gure 3) as well as the remaining t-channel process (2nd row, 4th column) cannot occur when replacing h ! G0; A. B Numerical implementation For each Feynman diagram, we have implemented the full analytical expressions for the helicity amplitudes in DarkSUSY [40]. We numerically sum over these contributions to obtain the total amplitude for a given helicity con guration, M appendix A.1. Di erential and partial cross sections are computed according to eq. (A.13), by numerically integrating over the energies of the nal state particles; for consistency checks, this can be done for any pair of energies and in any speci ed order. In order to (h; ) !F fX , as introduced in HJEP09(217)4 improve convergence and accuracy of the numerical integrations, we use taylored integration routines that make use of the known locations of kinematic resonances [87]. For the total cross sections, we have explicitly implemented the NWA approximations contained in eqs. (4.3){(4.9). We have extensively checked our code, and hence also the prescription of subtracting the NWA contribution detailed above, by comparing the total cross section de ned in eq. (4.10), on a channel-by-channel basis and for various SUSY models, with numerical results obtained with CalcHEP [93].21 For all models, and all annihilation channels, we nd remarkable agreement. We also checked agreement for individual classes of diagrams (s=tu-channel, ISR/FSR/VIB) as classi ed in section 3.1. Let us stress that in terms of computation time the implementation via helicity amplitudes, together with HJEP09(217)4 the taylored integration routines, is less expensive compared to the evaluation of squared matrix elements via Monte Carlo integration as implemented in CalcHEP. This is especially signi cant for the three-body processes to which a large number of diagrams contribute, and for which the di erence in computation times amounts to several orders of magnitude in the speci c kinematic limit we are interested in here. For the yields of stable particles, we have implemented the procedure described in section 4.2, using unpolarized yields for decaying particles given that these are the only ones that are currently available in DarkSUSY [95]. As discussed, as long as the total yields (i.e. summed over all channels) are concerned, our prescription still captures any double counting. We note that extending our implementation to fully polarized yields will be straight-forward for future work, given the results provided in section 4.2 and the helicity amplitudes reported in appendix A. Let us mention a few of the extensive numerical checks that we performed to test the yield implementation. We considered, in particular, models for which the three-body annihilation is dominated by an almost on-shell intermediate resonance. In this case, the 21We compared our implementation of three-body cross sections based on DarkSUSY 5.1 with CalcHEP 3.4 [93]. In particular, we adapted the ewsbMSSM implementation of CalcHEP to compute the spectrum from a given set of pMSSM input parameters at scale Q = MZ (except for MA which is the pole mass) using SoftSusy 3.4 [94]. The Susy les Houches output le written by SoftSusy is then used as input for DarkSUSY via the slha interface. In order to be able to directly compare the output it is necessary to adapt various routines in order to match the conventions. Apart from making sure that all SM input parameters agree (we used mb = 4:92 GeV, sin( W ) = 0:47162, W = 2:07 GeV, t = 2:0 GeV), we made the following changes for the purpose of cross checking: for CalcHEP, we switched o the running bottom mass (dMbOn=0) and used unitary gauge (for the comparison on a diagram-by-diagram basis; only the sum is gauge-independent). For DarkSUSY , the Yukawa couplings are by default read in from the blocks YU, YE and YD in the slha le. For the purpose of comparison, it is convenient to x the Yukawa couplings at yi = mi=v, especially for the top. Therefore, we commented out the corresponding lines in dsfromslha.f. Additionally, in su/dssuconst yukawa running.f, we commented out the running Yukawas, such that the default Yukawa couplings, which are simply related to the (on-shell) masses, are used. In addition, the call to dshigwid() was commented out in dsfomslha.f in order to avoid a rescaling of Higgs couplings that takes certain NLO corrections into account. For the purpose of comparison, it is more convenient to have tree-level couplings. In addition, we then set the Higgs widths to a common value in both programs. Finally, we set the rst and second generation quark masses to zero and the CKM mixing matrix to unity in order to match the conventions of the ewsbMSSM model implemented in CalcHEP. We veri ed that the conventions agree by comparing also the two-body cross sections for all channels allowed at s-wave, for which we nd perfect agreement after the changes described above. subtraction procedure described in section 4.2 is expected to lead to a large cancellation between the full three-body contribution and the NWA term. We explicitly veri ed this cancellation for all yields of stable particles, and over the full energy range. The cancellation amounts to several orders of magnitude in speci c cases, and therefore provides a robust check of the implementation. In addition, we also veri ed that the yields obtained from all of the models contained in our MSSM scan results pass a number of checks (e.g. yields within an expected range at E > m =2 and E > m =10). Finally, we also considered three-body nal states that contain directly one or more stable particles (such as e.g. ! W e ). In this case, we veri ed that the neutrino and positron spectra match the analytical result discussed in appendix C for speci c models for which this nal state is dominantly produced by an intermediate W resonance. C Spin correlations of decaying resonances In section 4, we discussed how to subtract double counting due to on-shell intermediate states (`resonances') contributing to three-body annihilation processes. If the resonance carries a spin, the spectrum of nal state particles depends on how much the various helicity states of the resonance contribute. In section 4 we argued that for annihilation of Majorana fermions in the s-wave limit, CP and angular momentum conservation uniquely determine the helicity of all possible intermediate states that can contribute to the threebody processes considered here. Here we present a formal derivation of this result, based on a description that would in principle allow us to treat also more general cases. In full generality, several helicity states of the resonance contribute to the amplitude, and can also interfere with each other when taking the absolute value squared. As a starting point we consider the example ! HW ! Hf F . We are interested in the contribution from the on-shell intermediate W boson. The full matrix element squared can then be written in the form (Ms1;s2)1!2 = us1 (p1)(gPL )vs2 (p2) jMresj 2 X s1;s2 X M2!2 (Ms1;s2)1!2 2 where we indicated explicitly the summation over the and the polarization states of the internal W . To extract the on-shell contribution in the narrow-width limit we assume that the momentum q of the W is (almost) on-shell, q2 ' M W2 . This implies that the kinematics of the H and W momenta is identical to the two-body annihilation. The rst term inside the square contains the helicity amplitude for the two-body part, M2!2 M2!2 : q For concreteness we can take the momentum of the W to be along the z-axis, q = (Eq; 0; 0; jqj), where Eq = jqj2 + M W2 and jqj is determined by the neutralino, W and Higgs mass via the two-body kinematics (identical to jp3j, see (C.22) below). The decay W ! f F gives (fermion momenta p1 and p2) (C.1) (C.2) (C.3) Inserting these into the resonant matrix element, and writing out the square gives after some renaming of indices jMresj = 2 X (Ms1s2 )1!2(Ms1s2 )1!2 where X ; 0 X ; 0 0 0 M2!2M2!2 M2!2M2!2 s1;s2 s1;s2 g2 tr((p=2 2g2 (p2 p1 mF ) PR(p=1 + mf )PL p2 p1 ) (C.4) (C.5) (C.6) (C.7) (C.8) (C.9) (C.11) (C.12) (C.13) The decorrelation-approximation (i.e. the replacement of the matrix element by the product of two matrix elements with independent summation over spin states, (3.6)) would be obtained by replacing . To compute it, one can use the D 0 ! D 0 decorrelated 1 3 X = (0; 1; i; 0)=p2; 0 as well as P 0 = (jqj; 0; 0; Eq)=MW ; g + q q =M W2 . helicity basis which ful ll 0 axis of the W ), A convenient frame to evaluate it is the rest frame of the W , obtained by boosting along the z direction. In this frame 0 = (0; 0; 0; 1). The momenta of the fermions can be parameterized by the angle w.r.t. to the z-axis (which is singled out as the polarization p1 = (Ep1 ; 0; jp1j sin ; jp1j cos ); p2 = (MW Ep1 ; 0; jp1j sin ; jp1j cos ) (C.10) where jp1j and Ep1 = qjp1j2 + mf2 are the momentum and energy of f in the W rest frame, see (C.22). Inserting this and evaluating the trace yields an explicit expression for D 0 = D 0 ( ) in terms of . Using that the polarization vectors have zero temporal component in the basis we are working in, and that p~2 = p~1, D 0 ( ) = 2g2 ( 2(~ 0 p~1) (~ p~1) + 0 p1 p2 + iMW (~ 0 ~ ) p~1) X (Ms1s2 )1!2(Md;s1s2 )1!2 HJEP09(217)4 Now one can use p1 p2 = (M W2 m2 f m2F )=2 and ~0p~1 = jp1j cos ; ~+ = +i~0; ~ p~1 = i jpp1j sin 2 form where d( vHfF )NW A = (2 )3 16(2M )2 jMresj MW W 2MW 1 jp1jjp3jd cos To obtain the di . cross section, we use the representation of the phase space in the d( vHfF ) = d j4MMj2 = 2 (2 )5 16(2M )2 jMj jp1jjp3jdm12d 1 d 3 1 where p3 is the Higgs momentum, and m212 = q2 the resonant momentum. Now we can do an approximation where we replace 2 2 jMj ! jMresj MW W M W2 ) but keep the fully correlated matrix element jMresj . By integrating over dm12 = dm212=(2m12) = dq2=(2MW ), and doing the trivial Higgs angle d 3 and d 1 integrals, one obtains the di . cross section w.r.t. to the angle of the fermion f and the polarization axis of the W boson in the back-to-back system, D D D 0 = D00 = 2g2(p1 p2 = 2g2(p1 p2 = 2g2jp1j2 sin2 2jp1j2 cos2 ) jp1j2 sin2 MW jp1j cos ) ip2g2jp1j sin (MW 2jp1j cos ) = D0 : One can check that the average over the diagonal contributions corresponds to the usual unpolarized decay matrix element, jM1!2j2 = 3 D 1 g2(6p1 p2 4jp1j2) = g2 M W2 mf2 m2F 4 3 jp1j2 jp1j = jp3j = [(M W2 (mf + mF )2)(M W2 (mf mF )2)]1=2 2MW One can rewrite this expression, using that the two-to-two and W decay rate are given by vHW = W !fF = jp3j X 1 8 8 (2M )2 1 jp1j M W2 jM1!2j jM2!2j 2 2 where M2!2 is the helicity amplitude and jM1!2j2 is the usual summed/averaged decay matrix element. Expressed in terms of the matrix introduced above, jM1!2j = 13 P D 2 2 2 1 1 2 4M (C.14) (C.15) (C.16) (C.17) (C.18) (C.19) (C.20) (C.21) (C.22) (C.23) (C.24) (C.25) HJEP09(217)4 The dependence on the angle cancels in this sum. Then, d( vHfF )NW A d cos vHW W !fF W vHW BR(W ! f F ) 2(P 2 jM2!2j2)( 13 P F !HW !HfF ( ) D ) (C.26) (C.27) (C.28) (C.29) where we have de ned the function F which characterizes the angular dependence. Using also (C.4) for Mres, one can write it as F !HW !HfF ( ) = P ; 0 M2!2M2!2 0 2(P jM2!2j2)( 13 P ) If one would replace the matrix D 0 by the decorrelated approximation (C.8), the last term becomes constant F !HW !HfF ( ) D!Ddecorrelated = 1 2 D : o -diagonal entries. for D 0 , this imples that F !HW !HfF ( ) = Integrating over the angle d cos (which yields a factor 2), one then recovers the familiar relation for the NWA of the total cross section. However, in general the matrix D 0 di ers from the decorrelated approximation, and has a non-trivial angular dependence as well as For Majorana DM annihilation into a scalar and a vector, only the longitudinal polar= 1 ization contributes to the s-wave, i.e. M2!2 ! 0 for v ! 0. Using the explicit expression D00( ) 2 13 P D M W2 m2 m2F 2(M W2 m2 m2F 4jp1j2 cos2 3 sin2 4 ; (C.30) where the last expression applies for massless fermions. This corresponds to the decay spectrum of a longitudinally polarized W boson. The integral of this expression over d cos coincides with the decorrelated case. Therefore, the result for the total cross section in the NWA is nevertheless accurate, with error governed by W =MW , as expected. Instead of the angle one can use the energy Ef of the fermion in the rest frame of the annihilating particles, qjp1j2 + mf2 + jp1j cos dEf = jp1jd cos (C.31) d vHfF d vW +fF Ef = q where = jp3j= jp3j2 + M W2 and in the narrow-width limit, = (1 2) 1=2. This nally yields the fermion spectrum vHW BR(W ! f F ) jp1j F !HW !HfF ( ) cos = Ef r jp1j2+mf2 (C.32) jp1j This procedure can be generalized to other three-body nal states in a straightforward way. For example, for ! W f F , the contribution from the W resonance is R=W vW W BR(W ! f F ) jp1j F !W W !W +fF ( ) cos = Ef r jp1j jp1j2+mf2 (C.33) where F !W W !W +fF ( ) = P ; 0; 3 M2!2 M2!32 3 0 2(P ; 3 jM2!32j2)( 13 P ) ; and M2!32 = 3 (p3) (q)M2!2 is the helicity amlitude for the ! W W annihilation process. In comparison to before, we have to sum in addition over the polarizations of the W + that appears in the three-body nal state. The matrix D 0 ( ) is the same as before. For s-wave annihilation the pair of vector bosons is in a state with S = L = 1, J = 0, and Lz = Sz = 0, when choosing the z-axis along the momentum of one of the nal state particles. The possible spin projections m1 and m2 of the vector bosons are then determined by the Clebsch-Gordon coe cients for coupling two spin-1 states (S1 = S2 = 1) to a total spin S = 1 state with m Sz = 0, hS1S2; m1m2jS1S2; Smi = < 8 p1=2 m1 = 1; m2 = m1 = 0; m2 = 0 p1=2 m1 = 1; m2 = 1 1 Since the spatial momenta of the vectors are opposite, this means they can only be in equal helicity states, and additionally have to be transverse, more precisely Mp which implies (C.34) (C.35) (C.36) (C.37) F s wave !W W !W +fF ( ) = M W2 3 4 2(M W2 1 m2 1 2 m2 sin2 m2F 2jp1j2 sin2 m2F This corresponds to the decay spectrum of transversely polasized W bosons, and the last line applies to massless fermions. It is straightforward to derive the corresponding matrix D 0 for Z decay and to generalize the procedure to a fermionic (top) resonance. One nds similarly that due to CP and angular momentum conservation only a de nite helicity state can contribute, which then determines the decay spectrum. 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Torsten Bringmann, Francesca Calore, Ahmad Galea, Mathias Garny. Electroweak and Higgs boson internal bremsstrahlung. General considerations for Majorana dark matter annihilation and application to MSSM neutralinos, Journal of High Energy Physics, 2017, 41, DOI: 10.1007/JHEP09(2017)041