Electroweak and Higgs Boson Internal Bremsstrahlung: General considerations for Majorana dark matter annihilation and application to MSSM neutralinos
Torsten Bringmann, Francesca Calore, Ahmad Galea, Mathias Garny
TTUM-HEP 1083/17LAPTH-013/17
Prepared for submission to JHEP
Electroweak and Higgs Boson InternalBremsstrahlung
General considerations for Majorana dark matter annihilation andapplication to MSSM neutralinos
Torsten Bringmann, a Francesca Calore, b Ahmad Galea a and Mathias Garny c a Department of Physics, University of Oslo,Box 1048, NO-0371 Oslo, Norway b LAPTh, CNRS, 9 Chemin de Bellevue, BP-110, Annecy-le-Vieux, 74941, Annecy Cedex, France c Technical University Munich, James-Franck-Str. 1, D-85748 Garching, Germany
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
It is well known that the annihilation of Majorana dark matter into fermions ishelicity suppressed. Here, we point out that the underlying mechanism is a subtle combina-tion of two distinct effects, and we present a comprehensive analysis of how the suppressioncan be partially or fully lifted by the internal bremsstrahlung of an additional boson inthe final state. As a concrete illustration, we compute analytically the full amplitudes andannihilation rates of supersymmetric neutralinos to final states that contain any combina-tion of two standard model fermions, plus one electroweak gauge boson or one of the fivephysical Higgs bosons that appear in the minimal supersymmetric standard model. Weclassify the various ways in which these three-body rates can be large compared to thetwo-body rates, identifying cases that have not been pointed out before. In our analysis, weput special emphasis on how to avoid the double counting of identical kinematic situationsthat appear for two-body and three-body final states, in particular on how to correctlytreat differential rates and the spectrum of the resulting stable particles that is relevant forindirect dark matter searches. We find that both the total annihilation rates and the yieldscan be significantly enhanced when taking into account the corrections computed here, inparticular for models with somewhat small annihilation rates at tree-level which otherwisewould not be testable with indirect dark matter searches. Even more importantly, how-ever, we find that the resulting annihilation spectra of positrons, neutrinos, gamma-raysand antiprotons differ in general substantially from the model-independent spectra thatare commonly adopted, for these final states, when constraining particle dark matter withindirect detection experiments. a r X i v : . [ h e p - ph ] S e p ontents ¯ f f and an additional final-state particle 8 A.1 Expansion of Amplitudes in the Helicity Basis 40A.2 Results for expanded amplitudes 42A.3 Suppression lifting from individual diagrams 49
B Numerical implementation 52C Spin correlations of decaying resonances 54 – 1 –
Introduction
The prime hypothesis for the cosmologically observed dark matter (DM) [1] is a new type ofelementary particle [2]. Among theoretically well-motivated candidates, weakly interactingmassive particles (WIMPs) play a prominent role. This is because such WIMPs very oftenappear in theories that attempt to cure the fine-tuning problems in the Higgs sector of thestandard model of particle physics (SM), and because thermal relics with weak masses andcross sections at the electroweak scale are typically produced with the correct abundance toaccount for the DM density today [3, 4]. Another advantage is that the WIMP hypothesiscan be tested in multiple ways: at colliders , where the signature consists in missing energy,in direct detection experiments aiming to observe DM particles recoiling off the nuclei of deepunderground detectors, and in indirect searches for the debris of DM annihilation in cosmicregions with large DM densities. Direct detection experiments have become extremelycompetitive in constraining smaller and smaller scattering rates [5, 6], and collider searcheshave pushed the scale of new physics to TeV energies in many popular models [7, 8]. It isworth stressing, however, that only ‘indirect’ searches would eventually allow to test theWIMP DM hypothesis in situ , i.e. in places that are relevant for the cosmological evidencefor DM. Also indirect searches have become highly competitive during the last decade, nowprobing the ‘thermal cross section’ (the one that is needed to produce the observed DMabundance) 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 thetotal annihilation cross section. Multiplied by the relative velocity v of the incoming DMparticles, it can in the non- relativistic limit be expanded as σv = a + bv + O ( v ) . (1.1)It was noted early [11, 12] that radiative corrections to σv can be huge because of symme-tries of the annihilating DM pair in the v → limit. For indirect DM searches, changesin either the partial cross section, for a given annihilation channel, or the differential crosssection, dσv/dE , may be phenomenologically even more important. The reason is thatan additional photon in the final state can give rise to pronounced spectral features inthe DM signal in both gamma [13] and charged cosmic rays [14]. For electroweak cor-rections, the situation is in some sense even more interesting because, on top of the justmentioned effects, completely new indirect detection channels may open up. In this way,antiproton data can for example efficiently constrain DM annihilation to light leptons whenconsidering the associated emission of W or Z bosons [15]. In the presence of point-likeinteractions, such as described by effective operators, the resulting spectra can be computedin 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 oneimplemented in the ‘cookbook’ for indirect detection [17]. One of the main results of thisarticle (see also [18]) is that the resulting cosmic ray spectra from DM annihilation candiffer substantially from the actual spectra, calculated in a fully consistent way from theunderlying particle framework. – 2 –ere we revisit in detail one of the most often discussed examples where radiativecorrections can be large, namely the case of a Majorana DM particle χ . The tree-levelannihilation rate into light fermions f is then on general grounds ‘helicity suppressed’, for v → , as a consequence of the conserved quantum numbers of the initial state [19]. Theresulting suppression by a factor of m f /m χ can be lifted by allowing for an additional vector[11] or scalar [20] boson in the final state, implying that for DM masses at the electroweakscale the radiative ‘corrections’ can be several orders of magnitude larger than the resultfrom lowest order in perturbation theory . Here, we revisit these arguments and point outthat the effect commonly referred to as helicity suppression is in fact the culmination oftwo 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 sectionof 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 higherenergies and luminosities at the LHC [7, 8] – though our main findings can be extendedin an analogous way to other DM candidates that couple to the SM via the electroweakor Higgs sector. Concretely, we provide a comprehensive analysis, both analytically andnumerically, of all 3-body final states from neutralino annihilation that contain a fermionpair and either an electroweak gauge boson or one of the five Higgs bosons contained in theminimal supersymmetric standard model (MSSM), for a neutralino that can be an arbitraryadmixture of Wino, Bino and Higgsino. We find large parameter regions where these 3-body final states significantly enhance the DM annihilation rate, with the impact on the shape of the cosmic-ray spectra relevant for indirect detection being even more significant.One of the technically most involved aspects, apart from the shear number of diagramsto be considered, is how to avoid ‘double counting’ the on-shell parts of the 3-body ampli-tudes that are already, implicitly, included in the corresponding 2-body results. We providean in-depth treatment of this issue and demonstrate how to accurately treat not only thetotal cross section but also the resulting cosmic-ray spectra. We again find significant ef-fects on the latter, indicating the need to correctly adopt this method also for other DMcandidates. In fact, in order to reliably test the underlying particle models, our findingssuggest that at least for fermionic final states it is in general not sufficient to use the model-independent spectra traditionally provided by numerical packages. The numerical routinesthat implement our results for the neutralino case will be fully available with the next publicrelease of
DarkSUSY [39, 40].This article is organized as follows. We start in Section 2 with a general discussion ofMajorana DM annihilating to fermions and the relevant symmetries that arise for v → , The lifting of helicity suppression via three-body final 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 bosonis a Z (cid:48) has been considered in [27]. For neutralino annihilation, so far only the cases of photon [13] and gluon [29] internal bremsstrahlung(IB) have been considered in full generality. Final states with electroweak gauge bosons have been consideredfor pure binos in [30–35], for Higgsinos in [36], and for pure Winos in [37]. A first study for a generalneutralino has been performed in [18]. Finally, final states involving the SM-like Higgs boson have beenconsidered in [20] for a toy model encompassing a pure Bino (see also [38]). – 3 –evisiting in particular the often invoked ‘helicity suppression’ arguments and how this sup-pression can be lifted fully or partially. In Section 3 we then consider the concrete case ofneutralino DM, and discuss the various possibilities of how the presence of an additionalfinal state boson can add sizeable contributions to, or even significantly enhance, the 2-body annihilation rates. The double counting issues mentioned above are then addressed indetail in a separate Section 4. We scan the parameter space of several MSSM versions anddemonstrate the effect of these newly implemented corrections to neutralino annihilation inSection 5, both for the annihilation rates and the cosmic-ray spectra relevant for indirectDM searches, and present our conclusions in Section 6. In a more technical Appendix,we describe the details of our analytical calculations to obtain the 3-body matrix elementsfor fully general neutralino annihilation in the MSSM (Appendix A), the numerical im-plementation of these results in
DarkSUSY (Appendix B), and how to correctly treat spincorrelations of decaying resonances (Appendix C).
For DM annihilation in the Milky Way halo, where DM particles have typical velocities oforder − , only the first term in Eq. (1.1) gives a sizeable contribution. In the followingwe therefore neglect p and higher partial wave contributions, and it is understood thatall (differential) cross sections are effectively evaluated in the zero-velocity limit. For an s -wave, the relative angular momentum in the initial state is L = 0 . Due to the Majorananature the initial particles are identical, but because we consider fermions the total wavefunction 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 totalwave function the spins must couple into an antisymmetric state. This is only possible forthe singlet state, with S = 0 , resulting in the following quantum numbers: J = 0 , C = ( − L + S = 1 , P = ( − L +1 = − . (2.1)Here, the general expressions for C and P apply because we have a system of two fermions .Assuming no significant sources of CP violation in the theory, which generally are highlyconstrained by measurements of the electric dipole moment and other precision experiments,the symmetry of the final state is hence also restricted to be J CP = 0 − . This implies thewell-known ‘helicity’ suppression of the annihilation rate into light fermions, similar tothe case of charged pion decay. In the following we first briefly review the origin of thissuppression, and then argue that it can in fact be related to a combination of two ratherindependent suppression mechanisms. We want to study Dirac fermions f as possible final states from the annihilation of Majoranaparticles χ . Their free Lagrangian is given by L = ¯ f ( i /∂ − m f ) f , which is invariant underLorentz transformations, i.e. invariant under SU (2) L + R . In the massless limit, m f → ,this symmetry is upgraded to a chiral symmetry SU (2) L × SU (2) R , in which the left and– 4 –ight handed Weyl states transform independently of each other, and helicity and chiraleigenstates unify. For a fermion pair ¯ f f , the spins can combine to either a singlet ( S = 0 ), ( |↑↓(cid:105) − |↓↑(cid:105) ) / √ , (2.2)or a triplet ( S = 1 ) spin state, (cid:110) |↓↓(cid:105) , ( |↑↓(cid:105) + |↓↑(cid:105) ) / √ , |↑↑(cid:105) (cid:111) , (2.3)where the arrows indicate the spin direction along the z -axis (the first entry refers to theantifermion, the second to the fermion). If the two fermion momenta are (anti-)parallel –e.g. because they are emitted back-to-back as the final states of a DM annihilation process– the z -axis can be chosen to be aligned in the same direction as the momenta, and theabove spin projections on the z axis are directly related to the helicities of the two particles.Choosing p f ( p ¯ f ) to point along the positive (negative) z -axis, the helicity configurations h = S z p z / | p z | of the singlet and triplet state are then given by √ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) − , − (cid:29) − (cid:12)(cid:12)(cid:12)(cid:12) + 12 , + 12 (cid:29)(cid:19) → √ (cid:0)(cid:12)(cid:12) ¯ f R , f L (cid:11) − (cid:12)(cid:12) ¯ f L , f R (cid:11)(cid:1) (2.4)and (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) + 12 , − (cid:29) , √ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) − , − (cid:29) + (cid:12)(cid:12)(cid:12)(cid:12) + 12 , + 12 (cid:29)(cid:19) , (cid:12)(cid:12)(cid:12)(cid:12) − , + 12 (cid:29)(cid:27) → (cid:26)(cid:12)(cid:12) ¯ f L , f L (cid:11) , √ (cid:0)(cid:12)(cid:12) ¯ f R , f L (cid:11) + (cid:12)(cid:12) ¯ f L , f R (cid:11)(cid:1) , (cid:12)(cid:12) ¯ f R , f R (cid:11)(cid:27) , (2.5)where the arrows indicate the chiral states in the left/right decoupling limit, i.e. for m f → .The momentum configuration thus restricts which helicity states can be associated tothe spin states. Angular momentum and the assumed CP invariance, on the other hand,restrict which spin state can be realized. Since CP = ( − L + S × ( − L +1 = ( − S +1 , forexample, only the singlet state with S = 0 is compatible with the odd CP parity of theinitial state. Eq. (2.4) then tells us that both fermion and antifermion in this momentumconfiguration must have the same helicity. In any chirally symmetric theory, however, theantifermion must necessarily have the opposite helicity of the fermion. We note that angularmomentum conservation alone leads to the same conclusion: since L = r × p , we must have L z = 0 and hence S z = J z = 0 . Eqns. (2.4, 2.5) then imply that fermion and antifermionmust, independently of the value of S , have the same helicity. The annihillation process χχ → ¯ f f is therefore only possible if chiral symmetry is broken in the Lagrangian, forexample through an explicit fermionic mass term m f ¯ f L f R or through the coupling of thefermion f to a scalar field λφ ¯ f L f R . It follows that the amplitude of the annihilation processmust be proportional to the chiral symmetry breaking parameters m f or λ .In addition, it is instructive to consider also the isospin of the involved particles. Sinceleft-and right-handed SM fermions transform under different representations of SU (2) L , thefinal states ¯ f L f R and ¯ f R f L have total isospin I = 1 / . The initial state χχ , on the otherhand, has necessarily integer isospin, implying ∆ I (cid:54) = 0 . The annihilation rate thus has to– 5 –anish for an unbroken SU (2) L , and therefore has to be proportional to at least one powerof the Higgs vacuum expectation value (VEV) v EW . For heavy DM the ratio δ v ≡ v EW /m χ becomes small, and processes with ∆ I (cid:54) = 0 will be suppressed by some power of δ v .In total, this implies that the amplitude for χχ → ¯ f f has to involve (at least) oneparameter that breaks chiral symmetry, and one power of v EW that controls breakingof the SU (2) L symmetry. For the SM fermions that receive their mass from the Higgsmechanism, both of these conditions are fulfilled for the usual helicity suppression factor m f . Depending on the model, however, there can be further possibilities, as we will discussin detail for the case of the MSSM below, and it is useful to discriminate between the twosuppression mechanisms. In the following, we therefore refer to the suppression related tochiral symmetry breaking as Yukawa suppression , and to the one related to electroweaksymmetry breaking as isospin suppression . While isospin suppression is controlled by onlyone parameter, δ v = v EW /m χ , there can in principle be several sources of chiral symmetrybreaking, for example in models with more complicated Higgs sectors. Nevertheless, aswe discuss in detail in Section 3, all terms that break chiral invariance in the MSSM areaccompanied by Yukawa couplings y f ∝ m f /v EW . Even though the following discussionof suppression lifting is completely model independent we will thus continue to assume, forconcreteness, that chiral symmetry breaking is controlled by y f . With the above discussion in mind, the only way to avoid the suppression of non-relativisticMajorana DM annihilation is to allow for an additional final state particle. Lorentz invari-ance requires this additional particle to be a boson, such that the leading process we areinterested in is of the form χχ → B ¯ F f (2.6)where B is a scalar or vector boson, and F = f if B is electrically neutral. The additionalboson can be either a SM particle, in particular a photon (refered to as electromagneticIB), a gluon, a weak gauge boson ( W ± , Z ) or the Higgs boson h , or it can be a new particlebeyond the SM (for example a heavy Higgs boson within the MSSM). For the moment wewant to keep the discussion model-independent, and therefore focus on the former case. Afrequently used approximation is to restrict the discussion to B being radiated off a fermionline in the final state, as described by soft and/or collinear splitting functions [16, 17]. Weemphasize 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 v EW → , it becomes straight-forward to exhibitthe scaling of a given process with y f and v EW . (We emphasize that we consider thislimit only in order to discuss the possible mechanisms of Yukawa and isospin suppressionlifting, while all our numerical results later on take the full dependence on v EW and y f intoaccount). In this limit, the left- and right-handed components of the fermions in the finalstate and in internal lines can not only be considered as gauge interaction eigenstates but asindependently propagating degrees of freedom. The fermion mass is treated perturbativelyin the mass insertion approximation, and is associated with a chirality flip along with a– 6 – Z T /γ W T Z L W L h ¯ F R f L or ¯ F L f R y f v EW y f v EW y f v EW y f y f y f ¯ F L f L v EW v EW v EW ¯ F R f R v EW v EW v EW Table 1 . Summary of Yukawa and isospin suppression(-lifting) in 3-body annihilation processes χχ → B ¯ F f for various final state boson B and fermion combinations. Entries ∝ correspondto processes that potentially can lift both Yukawa and isospin suppression of the 2-body process.Entries ∝ y f can lift isospin suppression but are still suppressed by the Yukawa coupling, whilethose ∝ v EW can lift Yukawa suppression but are still suppressed by δ v = v EW /m χ for large m χ . suppression factor m f ∝ y f v EW . In addition, longitudinally polarized gauge bosons W L /Z L can be replaced by the corresponding Goldstone bosons G ± , G by virtue of the Goldstoneboson equivalence theorem, cf. Eqs. (2.7-2.8) below. All final states thus have definite SU (2) L quantum numbers (i.e. I = 1 / for G ± , G , h, f L , I = 0 for f R , and I = 1 for W T ),except for the Z T , which is a mixture of I = 0 , states (even in the gauge restaurationlimit, we find it convenient to express our results in terms of the Z boson instead of theneutral SU (2) L boson).The amplitude of the generic 3-body process indicated in Eq. (2.6) can be non-zerofor v EW → only if ∆ I = 0 , i.e. if isospin is conserved. Furthermore, the amplitudemust vanish for y f → unless both fermions have the same chirality. Note that thisis possible for 3-body processes because the kinematics does not force the fermions tobe emitted back-to-back in the center-of-mass (CMS) frame, and therefore the argumentsdiscussed in Section 2.1 do not apply. These two observations immediately determinewhich annihilation processes can lift either Yukawa or isospin suppression (or both). InTable 1, we show schematically the required scaling of the amplitude that results fromthese considerations, for various combinations of fermion chiralities and final state bosons(where the longitudinal gauge bosons represent the corresponding Goldstone bosons). Bothsuppression factors can be lifted only in processes where a transverse gauge boson ( Z T , W T , γ , or gluon g ) is emitted and the final state fermions are described by spinors ofequal chirality ( ¯ F R f R or ¯ F L f L ). For longitudinal gauge bosons ( Z L or W L ) or the Higgsboson h , only one of the suppression factors can (potentially) be avoided for 3-body finalstates: isospin suppression can be lifted if the fermions are of opposite chirality, and Yukawa In the extreme case where both fermions are emitted in the same direction, e.g., one simply has toexchange f R ↔ f L in Eqs. (2.4, 2.5), which allows equal chiralities of the fermions in both the singlet andtriplet spin state. In this kinematical configuration, it is easy to visualize how the fermion momentum canbe balanced by the emitted boson B , and how their spin can combine with S B and L to the required J = 0 for both S B = 0 and S B = 1 . In general, the spin singlet and triplet states will be linear combinationsof all chiral states, with expectation values that depend on the angle between the fermion momenta, thusrendering the above argument essentially independent of the specific kinematical configuration. Also therequirement of CP conservation is much less restrictive for 3-body than for 2-body final states. A generaldiscussion is somewhat complicated by the fact that e.g. ¯ F f is not necessarily a CP eigenstate that couldbe analysed individually, but in principle straight-forward by classifying all possible effective operators thatconnect initial and final states (similar in spirit to the analysis of 2-body final states presented in Ref. [41]). – 7 –uppression can be lifted if the fermions are of equal chirality.Let us stress that the symmetry arguments presented above simply guarantee that theamplitude must vanish for y f → and v EW → , respectively, and the same applies to any gauge invariant sub-sets of diagrams. The actual suppression can thus be stronger thanindicated by Table 1, i.e. by additional powers of v EW or y f . At the same time, we cautionthat single diagrams can scale in a different way, depending on the gauge choice, such thatthe vanishing for y f → or v EW → is in general not guaranteed. Following up on the last comment, let us for convenience briefly recall how to verify gaugeindependence and identify gauge invariant subsets of diagrams. While for photon emission agood test is to check whether a given set of diagrams satisfies the Ward identity M µ ( χχ → ¯ f f γ ) k µ = 0 , where k µ is the momentum of the photon, this does not work for electroweakIB because SU (2) L × U (1) Y has been spontaneously broken. Indeed the question of gaugeinvariance changes in general, as weak hypercharge and isospin are no longer conservedin their original form. For the spontaneously broken Glashow-Weinberg-Salam theory thecorrect way to define gauge invariance is in terms of the preserved BRST symmetry [42, 43],under which SM field transformations involve ghost fields which arise from the electroweakgauge fixing procedure. This implies a new set of Ward identities, which in general dependon the choice of gauge. Using the standard R ξ class of gauges [44], we arrive at the Wardidentities for electroweak IB as expected from the Goldstone equivalence theorem: M µ ( χχ → ¯ f f Z ) k µ = im Z M ( χχ → ¯ f f G ) , (2.7) M µ ( χχ → ¯ F f W ± ) k µ = m W M ( χχ → ¯ F f G ± ) . (2.8)We reiterate that Eqns. (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. ¯ f f and an additional final-state particle In this section we apply the general discussion of helicity suppression lifting in MajoranaDM annihilation to the lightest supersymmetric neutralino as DM candidate, and additionalfinal state bosons charged under SU (2) L . For photon or gluon IB we refer to the referenceslisted in the introduction. Concerning the choice of DM candidate, we note that muchof 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 belongto a different electroweak multiplet. We will introduce the relevant 3-body processes andFeynman diagrams in Section 3.1, re-visit the discussion of the helicity suppression in lightof the specific situation encountered in the MSSM (Section 3.2) and then demonstrate indetail how these suppressions can be lifted, fully or partially, in Section 3.3. In Sections 3.4and 3.5, finally, we discuss two mechanisms by which 3-body cross sections can be enhancedwhich are not related to the helicity suppression of 2-body final states.– 8 – χ ˜ χ ˜ χ n / ˜ χ ± n ˜ f i ˜ χ ˜ χ ˜ χ n / ˜ χ ± n ¯ Ff ¯ FfV/SV/SV/S ˜ χ ˜ χ f ¯ F ˜ χ n / ˜ χ ± n ¯ F ˜ χ ˜ χ V/SffV/S ˜ χ ˜ χ ¯ F ˜ f i ˜ f j ˜ f i V/Sf ¯ F ˜ χ ˜ χ ˜ f i fV/S ¯ F ˜ χ ˜ χ ˜ f i ˜ χ n / ˜ χ ± n ˜ χ ˜ χ ¯ FfV/S ¯ FV/S ˜ χ ˜ χ f ˜ χ ˜ χ ¯ FV/Sf
Figure 1 . Condensed representation of all Feynman diagrams for neutralino annihilation into ¯ F f V or ¯ F f S , where dotted lines indicate scalar ( S = A, h, H, H ± ) or vector ( V = Z, W ± ) mediators,depending on the final state configuration. Fermion final states are identical, F = f , for neutralboson emission ( h , H , A or Z ), while ( f, F ) constitute the two components of an SU (2) L doubletfor charged boson emission ( H ± or W ± ). See text for more details on how the individual topologiesare referred to in this article. From now on, we thus assume DM to be composed of the lightest neutralino, χ ≡ ˜ χ , whichis a superposition of Wino, Bino and Higgsino states, χ = N ˜ B + N ˜ W + N ˜ H + N ˜ H , (3.1)obtained by diagonalizing the neutralino mass matrix M − g (cid:48) v √ g (cid:48) v √ M gv √ − gv √ − g (cid:48) v √ gv √ − µ g (cid:48) v √ − gv √ − µ . (3.2)Here, M and M are the Bino and Wino mass parameters, respectively, and µ is theHiggsino mass parameter; v and v are the VEVs of the two Higgs doublets, with v EW = (cid:112) v + v and tan β ≡ v /v , and g and g (cid:48) 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 masseigenvalues to be positive, while the diagonalization matrix N can be complex.We want to consider here all 3-body final states that contains a fermion pair and aboson that is charged under SU (2) L . Assuming CP -violating terms to be small, the fulllist of processes of interest is thus χχ → W + ¯ F f, Z ¯ f f, H + ¯ F f, A ¯ f f, H ¯ f f, h ¯ f f . (3.3)Here, A denotes the CP -odd Higgs, H + the charged Higgs, and H and h the heavy andlight CP -even Higgs bosons, respectively. For charged boson final states, f denotes anyfermion doublet component with isospin +1 / , and F the corresponding one with isospin − / ; for neutral bosons, f can be any SM fermion.– 9 – χ ˜ χ ¯ FW ± f ¯ F ˜ χ ˜ χ W ± f W ± ˜ χ ˜ χ ˜ χ ˜ χ ¯ FW ± ˜ χ ˜ χ f ˜ χ ± n ˜ χ ± n A H ± f ¯ F f ¯ FW ± A A H ± H ± Figure 2 . Gauge invariant set of amplitudes for neutralino DM annihilation into a fermion pairand a W boson, mediated by s -channel bosons with a mass at the scale of the CP -odd Higgs A . ˜ χ ˜ χ ¯ FW ± f ¯ F ˜ χ ˜ χ W ± f W ± ˜ χ ˜ χ ˜ χ ˜ χ ¯ FW ± ˜ χ ˜ χ f ˜ χ ± n ˜ χ ± n Z W ± f ¯ F f ¯ FW ± Z Z W ± W ± Figure 3 . Same as Fig. 2, but mediated by s -channel bosons with a mass at the electroweak scale. fW ± ¯ F ˜ χ ˜ χ W ± ˜ χ ˜ χ f ¯ F W ± f ¯ F ˜ χ ˜ χ fW ± ¯ F ˜ χ ˜ χ ˜ χ ˜ χ ¯ FfW ± ˜¯ f i ˜¯ f j ˜¯ f i ˜¯ f i ˜¯ f i ˜¯ f i ˜ χ ± n ˜ χ ± n Figure 4 . Same as Fig. 2, but mediated by t -channel sfermions. In Fig. 1, we show all contributing Feynman diagrams in a condensed form (notethat some of these diagrams may vanish for specific combinations of internal and externalparticles). For future reference, we follow Ref. [18] and refer to the top row of diagramsas (derived from 2-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 → limit).Likewise, we denote diagrams of the type that appear in the first column as virtual internalbremsstrahlung (VIB), diagrams of the type that appear in the second and third column as final state radiation (FSR), and diagrams of the type that appear in the last two columnsas initial state radiation (ISR).We explicitly calculate the full analytical expressions for all these processes in thelimit of vanishing relative velocity of the annihilating neutralino pair, see Appendix A.1for technical details. We then use the Ward identities in Eqns. (2.7) and (2.8) to groupdiagrams into gauge invariant sets for the case of vector boson final states. In general we We stress that this distinction between VIB and FSR, while useful for the specific purpose of our dis-cussion, 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 definitions where FSR refersexclusively to the soft or collinear photons radiated from the final legs [13, 16, 17], while VIB is defined asthe difference between the full amplitude squared and the FSR contribution [13] . – 10 –dentified only two of such invariant sets: those diagrams that are derived from 2-body s -channel processes and those that are derived from 2-body t -channel processes. In the limit m A (cid:29) m χ – which is phenomenologically particularly relevant because the observed Higgsis 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 theset of diagrams with (at least one) mediator at the mass scale M A (see Fig. 2), weak-scale s -channel , which are the set of diagrams with s -channel mediators at the weak scale (seeFig. 3), and t -channel , which are the set of diagrams with sfermion mediators (see Fig. 4). For Z ¯ f f and h ¯ f f final states the three sets of diagrams can be obtained analogously: t -channel contributions involve at least one sfermion line, while the remaining diagramsbelong to the s -channel category (which can be further split into subsets involving at leastone mediator at scale M A , or none, respectively). As established in the previous Section, the ‘helicity suppression’ of the 2-body annihilationrate by a factor of m f /m χ is indeed the combination of in principle independent Yukawaand isospin suppressions. Let us now turn back to this observation and discuss it in moredetail in light of the MSSM, where both mechanisms are still intrinsically linked because ofthe connection between gauge symmetry and chiral structure in the MSSM Lagrangian. The chiral symmetry of the MSSM Lagrangian is broken by terms proportional to Yukawacouplings (in order to avoid flavour-changing neutral currents, we assume as usual thatthe A -terms are proportional to the Yukawa coupling matrices). Following the generalarguments of Section 2.1, any amplitude contributing to χχ → ¯ f f must therefore be pro-portional to y f . Within the MSSM the values of y f are functions of tan β but, except for thetop quark, in general so small that this can lead to a suppression of the 2-body amplitudesby many orders of magnitude. From the point of view of the broken theory, this Yukawasuppression appears to arise from rather different types of contributions to the Lagrangian: i) fermion mass terms ii) couplings of any of the five physical Higgs fields to fermions iii) couplings of fermions to sfermion mass eigenstates (which mix the left- and right-handed fields).For example, the first case is relevant for annihilation into fermions via t -channel sfermionexchange if the sfermion mixing is small (otherwise, the third contribution can dominatethe amplitude), and the second for annihilation via s -channel pseudoscalar mediation. We note that for v → the two s -channel ISR diagrams are actually identical , but for clarity we stillinclude them separately in figures 2 and 3. For v → and m F → , also the two t -channel ISR diagrams areidentical; in practice, the difference only matters for final states containing top and bottom quarks. As animportant cross-check of our final amplitudes, we confirmed analytically that these identities indeed hold. – 11 –e note that all three interaction types couple left- and right-handed states and hencecan ‘flip’ the helicity of one of the final state fermions. The helicity combinations thatwould result in a chirally symmetric theory, ¯ f R,L f R,L , can thus be transformed into thosecompatible with the global symmetry requirements outlined in Section 2.1, ¯ f R,L f L,R . Tra-ditionally, the notion of this helicity flip is sometimes taken to refer specifically to the case (i) , in which it is the (kinematic) fermion mass that breaks chiral symmetry in the La-grangian. Instead, we associate the effect directly with the Yukawa couplings in the MSSMLagrangian (which of course give rise to the SM fermion masses).
As also discussed in Section 2.1, the annihilation process χχ → ¯ f f furthermore violatesweak isospin, ∆ I (cid:54) = 0 , and therefore its amplitude has to vanish in the gauge restorationlimit v EW → . The resulting isospin suppression by a factor δ v ≡ v EW /m χ , for heavyneutralinos, can arise from different terms in the Lagrangian of the broken theory: a) fermion mass terms b) mixing of different gauge multiplets (Bino, Higgsino, Wino) that contribute to thelightest 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 confirms that neutralino mixingsvanish for v EW → , as required by SU (2) L invariance. Note that case ( a ) and ( c ) areintrinsically linked to an accompanying chirality violation, since m f ∝ y f v EW and the off-diagonal terms in the sfermion mass matrix are also proportional to y f within the MSSM.Let us consider as an illustration the t - and s -channel contributions to χχ → ¯ f f . Thekinematical helicity suppression due to the fermion mass m f is relevant for the t -channel(sfermion exchange). In this case Yukawa and isospin suppression simply arise from the twofactors in m f ∝ y f v EW (case (a) and (i) , respectively). In addition, the Yukawa and isospinviolation 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, whichthen gives rise to the required chirality flip.For s -channel annihilation, on the other hand, the situation is more interesting inthe sense that Yukawa and isospin suppression cannot simply be traced back to the sameorigin. For a pseudoscalar Higgs boson A as mediator, e.g., the Yukawa suppression stemsdirectly from the Yukawa coupling ∝ y f A ¯ f f (case (ii) ), while the isospin suppression arisesfrom the neutralino mixing (case b ): for pure gauge multiplets the coupling A ¯ χχ would beforbidden by SU (2) L invariance, and therefore vanishes for v EW → . For a Z -boson in the s -channel, the discussion of the limit v EW → is a bit more involved (see Appendix A.3),but is essentially analogous to the case of an A mediator. In Section 2.2, we discussed which 3-body final states χχ → B ¯ F f can potentially lift theYukawa- and/or isospin suppression of the process χχ → ¯ f f , for the case in which B is– 12 – T W T Z L /A W L /H ± h/H ¯ F R f L or ¯ F L f R y f v EW y f v EW y f y f y f no enhancement no enhancement t/u, s t/u, s t/u, s ¯ F L f L v EW v EW v EWt/u ( ˜ B, ˜ W ) , s ( ˜ H ) t/u ( ˜ B, ˜ W ) , s ( ˜ H, ˜ W ) − t/u t/u, s ¯ F R f R v EW v EW v EWt/u ( ˜ B ) , s ( ˜ H ) − − − t/u, s Table 2 . As Table 1, but applied to weak gauge and Higgs boson final states within the MSSM.We also indicated whether the process can be realized with the maximal enhancement allowedby chiral and isospin symmetry in t + u and s -channel annihilation processes, respectively. Forthe first two columns we also specify for which neutralino composition ( ˜ B = bino-like, ˜ W =wino-like, ˜ H =Higgsino-like) the maximal enhancement occurs. For the last three columns t + u -channelprocesses are possible for ˜ B - or ˜ W -like neutralino as well as mixed ˜ H/ ˜ B or ˜ H/ ˜ W , and s -channelprocesses are possible for mixed ˜ H/ ˜ B or ˜ H/ ˜ W . Entries with a dash do not contribute to the orderwe are working in (see Appendices A.2 and A.3 for details). a SM gauge boson or a Higgs boson. This general discussion based on isospin and chiralsymmetry in the limit v EW → can be extended to the MSSM, as shown in Table 2, bynoting that all physical Higgs bosons h, H, A, H ± have isospin I = 1 / . Compared to Table1, the amplitudes for A ¯ f f scale as expected in the same way as Z L ¯ f f , noting that in thegauge restoration limit Z L is given by the Goldstone boson G (and hence transforms in asimilar 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 different mass hi-erarchies of particular phenomenological interest and determine for each of the previouslydiscussed gauge-invariant subsets of diagrams the leading order in v EW and y f . The resultof this exercise is collected in Tables 9 – 11, where we present the ratio of the leading termfor the 3-body amplitude and the corresponding 2-body amplitude. This allows us, as alsoindicated in Table 2, to identify which contributions to the 3-body amplitudes actuallyrealize the suppression lifting that we can maximally expect on the basis of our generalsymmetry arguments; the ‘missing’ cases, for which we did not find a contribution withinthe MSSM, are marked by a ‘-’. For a detailed technical discussion of the various liftingmechanisms, and how they are realized at the level of individual diagrams, we refer toAppendix A.3. We provide a graphical summary in Table 3, where we show representativediagrams that realize the lifting of isospin and/or Yukawa suppression, for the sets of gaugeinvariant classes of diagrams that can be discriminated in the gauge restoration limit (inaddition to the three sets discussed before, the t -channel can be split into contributions thatremain non-zero in the limit of pure neutralino states (I), and those that require neutralinomixing (II)). Isospin suppression can be lifted in all cases by the emission of longitudi-nal gauge bosons (here represented by the Goldstone bosons) or a Higgs boson. Liftingof 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 theconservation of J CP = 0 − (see Appendix A.3 for details), explaining the ‘missing’ entries– 13 – → → Z T , W T Z L , W L , h ¯ F L f L , ¯ F R f R ¯ F L f R , ¯ F R f L ¯ F L f L , ¯ F R f R t -channel I ˜ χ ˜ χ f ¯ f ˜ f gg Y f v ˜ χ ˜ χ f ¯ f ˜ f gg Y f v W T ,Z T f ¯ F ˜ χ ˜ χ ˜ f ˜ χ ˜ χ ¯ FfW T ,Z T ˜ f W T ,Z T ˜ χ ˜ χ f ¯ F ˜ f ˜ f gg gggg ggg G ,G ± ,hf ¯ F ˜ χ ˜ χ ˜ f G ,G ± ,h ˜ χ ˜ χ f ¯ F ˜ f ˜ f gg Y f ggY f G ± ,h ˜ χ ˜ χ f ¯ F ˜ f ˜ f ggg v t -channel II ˜ χ ˜ χ f ¯ f ˜ f gY f gv ˜ χ ˜ χ ¯ FfG ,G ± ,h ˜ f ggY f ˜ χ ˜ χ ¯ ffh ˜ f ggggv s -channel EW ˜ χ ˜ χ ¯ ff G g Y f gv W T ,Z T ˜ χ ˜ χ f ¯ F gg g G ,G ± ,h ˜ χ ˜ χ h,G ± ,G f ¯ F gg Y f ¯ fh ˜ χ ˜ χ fG g g ggv s -channel M A ˜ χ ˜ χ ¯ ff A g Y f gv G ,G ± ,h ˜ χ ˜ χ H ,H ± ,A f ¯ F gg Y f Lifting of Yukawa + Isospin Isospin Yukawa
Table 3 . Diagrams for annihilation into fermions ¯ f f and B ¯ F f (for B = W, Z, h ) in the gaugerestoration limit v EW → . The rows correspond to the four gauge-invariant subsets of diagramsthat can be discriminated in this limit (see Appendix A.3 for details). The first column correspondsto the 2-body process, and the other columns show various 3-body processes. The diagrams shownin the second column lift both Yukawa and isospin suppression. The diagrams in the third columnlift only isospin suppression, and in the fourth column only Yukawa suppression. We show onlyone representative diagram for each topology (ISR/FSR/VIB) and suppression mechanism. Thecoupling factors attached to vertices and mass/mixing insertions give the scaling with y f , v EW and g of each diagram (for Bino- or Wino-like neutralinos; modifications for Higgsino-like neutralinosare described in Appendix A.3). Note that contributions with W T emitted via ISR (second column,first and third row) exist for Wino- or Higgsino-like neutralinos; those with Z T emitted via ISRoccur only for a Higgsino-like neutralino. in Table 2. Let us also highlight that the classification procedure revealed ways to lift the2-body suppression that have not been pointed out for the MSSM before (in particularHiggsstrahlung via t -channel ISR and a specific s -channel VIB process, shown in the lastcolumn and second/third row in Table 3, respectively).– 14 – .4 Heavy propagator suppression An additional form of suppression, unrelated to the discussion so far, arises in diagrams thatrely on mixing between neutralinos or contain heavy propagators. This mass suppression takes the form δ X ≡ m χ /M X , where X is the heavy state in question. In particular, both s -channel contributions to χχ → ¯ 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 2-body amplitude in the s -channel is suppressed bya factor δµ = m χ /µ if | µ | (cid:29) m χ . For a Higgsino-like neutralino, on the other hand, it issuppressed by δ M i = m χ /M i for M i (cid:29) m χ , where M i = min( M , M ) (see Table 12).These suppression factors of the s -channel annihilation can be lifted for the case of aWino- or Higgsino-like neutralino by the emission of a (transverse) W or Z from one ofthe initial neutralino lines (ISR). (The corresponding diagram is illustrated in the thirdrow, second column of Table 3.) Additionally, this 3-body process simultaneously lifts bothisospin- and Yukawa suppression. It is particularly relevant if the 2-body final states W W and ZZ are kinematically forbidden, such that the internal gauge boson is off-shell. Thisis a special case of the threshold effects that we turn to next. A given 2-body channel χχ → AB is strongly phase-space suppressed if the CMS energyis close to the mass of the final-state particles, and for m χ ≤ m A + m B the correspondingpartial cross section vanishes completely in the v → limit. If either A or B are off-shell and decay into much lighter states, however, the phase-space opens up again andthereby potentially increases even the total 2-body annihilation rate significantly. For theMSSM, this is particularly relevant for the W + W − and ¯ tt channels, which has previouslybeen studied for specific neutralino compositions [46, 47] (for an approximate numericalimplementation in the context of relic density calculations, see [48]). For the processes weare interested in here, threshold effects can in general appear for any two-boson final states(or ¯ tt ).For a more detailed discussion of this effect, it is useful to rewrite the 3-body crosssection as (see e.g. [49, 50]) σv → = S E χ E χ (cid:90) |M → | d Φ ( P ; p , p , p )= S E χ E χ (cid:90) |M → | d Φ ( P ; p , q ) × dq π × d Φ ( q ; p , p ) , (3.4)where d Φ n ( P ; p , . . . , p n ) = (2 π ) δ (4) ( P − (cid:80) p i ) (cid:81) i d p i (2 π ) E i is the n -body phase space ele-ment, P = p χ + p χ the sum of the -momenta of the annihilating neutralinos and E χ i theirenergy; the p i denote the final-state momenta. Since q = ( p + p ) is time-like, we will inthe following often use the notation q ≡ m instead. For the processes considered here,c.f. Eq. (3.3), the symmetry factor S is always . Furthermore, |M| denotes the usualsquared matrix element, averaged over initial spins and summed over final spins/helicities. In general, if some of the final state particles are of identical type, configurations that differ only by – 15 –e now assume that the amplitude is dominated by a resonant, almost on-shell internalpropagator that decays into particles 2 and 3, and hence carries momentum q . For aresonance R with mass M , width Γ , and spin , / , or , respectively, we then have M → = 1 m − M + iM Γ × M ( q ) µ → ( − g µν + q µ q ν /M ) M ( q ) ν → vector M ( q )2 → ( /q + M ) M ( q )1 → fermion M → M → scalar (3.5)where M → ( M → ) is the matrix element for χχ → p q ( R ∗ → p p ), up to polarizationvectors or spinors for the ‘external’ particle R (as indicated by the superscript q ).The decisive observation is now that (cid:82) |M → | d Φ ( q ; p , p ) must be independent ofthe polarization state of R once all the final state polarizations are summed over. This isfamiliar from on-shell momenta q – the total (but not differential) decay rate of a particle isindependent 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 treatdifferential cross sections), one may thus conveniently replace the correlated polarization orspin structure of Eq. (3.5) with an unpolarized sum: (cid:12)(cid:12)(cid:12) M ( q ) µ → ( − g µν + q µ q ν /M ) M ( q ) ν → (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:88) λ M ( q ) µ → (cid:15) ∗ λµ (cid:15) λν M ( q ) ν → (cid:12)(cid:12)(cid:12) → (cid:88) λ ,λ (cid:12)(cid:12)(cid:12) M ( q ) µ → (cid:15) ∗ λ µ (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) (cid:15) λ ν M ( q ) ν → (cid:12)(cid:12)(cid:12) , (3.6) (cid:12)(cid:12)(cid:12) M ( q )2 → ( /q + M ) M ( q )1 → (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:88) s M ( q )2 → u s ¯ u s M ( q )1 → (cid:12)(cid:12)(cid:12) → (cid:88) s ,s |M ( q )2 → u s | | ¯ u s M ( q )1 → | , (3.7)In this way, we can independently of the spin of R replace |M → | → |M → | |M → | ( m − M ) + M Γ (3.8)in Eq. (3.4) which, for v → , leads to σv res2 → = S (cid:90) (2 m χ − m ) ( m + m ) dm π m ( m − M ) + M Γ ˜Γ R → (cid:102) σv χχ → R . (3.9)Here, the decay rate of the off-shell resonance in the frame where q = ( m , ) is given by ˜Γ R → ≡ S m (cid:90) |M → | d Φ ( q ; p , p ) = S π λ ( m , m , m ) m |M → | , (3.10) exchanging these particles should be counted only once in the phase space integration. Since this will beconvenient later on, we thus use a convention where one integrates over all of the phase space as if allparticles were distinct, and then correct for the corresponding over-counting by a symmetry factor S . It is S = 1 if all final-state particles are distinct, and S = 1 / ( S = 1 / ) if two (all three) of them are identical. – 16 –nd the cross section for the annihilation into an off-shell resonance is given by (cid:102) σv χχ → R ≡ S R P (cid:90) |M → | d Φ ( P ; p , q ) = S R π λ (4 m χ , m , m ) m χ |M → | . (3.11)In the last step we performed the phase-space integral explicitly by using the fact that for v → the annihilation process is kinematically the same as a pseudo-scalar decay, implyingthat |M| cannot have any angular dependence. Eq. (3.9) will thus continue to hold forgeneral s -wave annihilation, provided one replaces m χ → s in Eq. (3.11). The squaredmatrix elements are here again summed (averaged) over final (initial) spins/helicities, lead-ing to an overall symmetry factor of S = S/ ( S R S ) (with S R , S defined in accordancewith footnote 6).We note that Eq. (3.9) can be significantly simplified by a few well-motivated assump-tions. Concretely, let us assume the off-shell particle to decay to massless final states, m = m = 0 , and |M → | ∝ M close to the threshold; this implies ˜Γ R → = ( m /M )Γ R → .We also introduce a reduced cross section ( σv ) red ≡ ( σv ) χχ → R /λ n +1 / (1 , µ , µ R ) , (3.13)with µ R ≡ m /s and µ ≡ m /s , allowing for the 2-body cross section close to thresholdto be suppressed not only by a phase-space factor ( n = 0 ), but by an additional such factorfrom the matrix element itself (as e.g. in the example of Higgsino annihilation below, forwhich we have n = 1 ). By definition, ( σv ) red thus remains finite both above and belowthe threshold. Assuming ( σv ) red to be independent of m close to threshold, Eq. (3.9)simplifies to σv χχ → R ∗ (cid:39) S ( σv ) red (cid:90) µ max dµπ γµ ( µ − + γ λ n +1 / (1 , µ , µµ R ) , (3.14)where µ max = ( √ s − m ) /m R and γ ≡ Γ R /M . This expression is model-independent inthe sense that the threshold correction can be directly estimated for any given 2-body crosssection (i.e. without first having to compute ˜ σv or ˜Γ ).As an illustrative and concrete example, let us consider the process χχ → W − e + ν e inthe limit of pure Higgsino DM. For simplicity, we assume that sleptons are much heavierthan 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 find (cid:102) σv χχ → W W ∗ = g π (cid:0) m χ − m χ ( m W + m ) + ( m W − m ) (cid:1) (cid:16) m χ + 2 m χ + − m W − m (cid:17) (3.15) For this reason, the result takes the same form as for off-shell decays [51, 52], suggesting a straight-forward generalization to 4-body final states dominated by the annihilation into two off-shell particles: σv res2 → = S (cid:90) dm π dm π m ( m − M R ) + M R Γ R m ( m − M R ) + M R Γ R ˜Γ R → ˜Γ R → (cid:102) σv χχ → R R . (3.12) – 17 – � �� ��� ��� ����� - �� �� - �� �� - �� �� - �� �� - �� �� - �� �� - �� � χ [ ��� ] σ � [ � � � � - � ] m W ± Γ W χχ ⟶ W - W + χχ ⟶ W - e + ν / BR W → e ν χχ ⟶ W - ( W + ) * pure Higgsino m χ ± =
200 GeV
Figure 5 . Cross section for pure Higgsinos annihilating to W bosons, χχ → W + W − , compared to χχ → W − e + ν . For the latter process, we show the cross section divided by the branching fraction Γ W → eν / Γ W (cid:39) / (solid lines). For comparison, we also include the model-independent estimateof Eq. (3.14) for χχ → W − ( W + ) ∗ (dotted lines). For m χ (cid:46) m W , the 3-body cross section is clearlylarger than the lowest-order result; above the threshold, on the other hand, the two agree exactly. and ˜Γ W ∗ → eν = g m π = m m W Γ W → eν . (3.16)We calculate the full 3-body cross section as derived in Appendix A.1, in the pure Higgsinolimit, and then compare it to the result given in Eq. (3.9). As shown in Fig. 5, we obtainexcellent agreement even though both the directly involved amplitudes and the numericalphase-space integrations are very different in nature (the two results for the 3-body crosssection, shown as solid lines, lie exactly on top of each other). This should of course beexpected for a process which by construction only receives contributions from an off-shellfinal-state particle, but we stress that Eq. (3.9) is in general much simpler to calculatein 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 simplified expressionprovides an excellent approximation to the full result.Most importantly, our example illustrates the much more general point that a 3-bodyprocess around or below the kinematic threshold of a large 2-body process can be signifi-cantly enhanced over the total annihilation rate at lowest order. Above the threshold andrescaled to the relevant branching ratio for the decay of the resonance R , on the other hand,the 3-body cross section for a process χχ → R, R → equals almost exactly the 2-bodyresult – an effect which we will discuss in detail in the next Section. We now turn to double counting issues related to unstable final-state particles. If the finalstate of a 2-body annihilation process undergoes a subsequent → decay, in particular,– 18 –his can also be viewed as a 3-body process with the unstable particle (the resonance, in ourwording) as an intermediate state. While we discussed the situation below the kinematicthreshold for the production of the unstable particle in Section 3.5 as a way of enhancingthe 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 final states considered here.One possibility to avoid over-counting identical kinematic configurations when adding 2-body and 3-body processes would be to altogether disregard the former for massive dibosonor ¯ tt final states. Interferences between (nearly) on- and off-shell contributions to theamplitude would then be correctly accounted for, as well as the impact of the spin ofthe resonance. However, this procedure has several drawbacks on a practical level, andfurthermore turns out to be incorrect for 2-body processes with identical particles in thefinal state (such as e.g. χχ → ZZ ), as will be discussed in more detail below. We thereforeprefer to explicitly subtract on-shell contributions to the 3-body processes, which allows usto keep most of the advantages of the full 3-body computation while correctly taking intoaccount all symmetry factors. In the following we describe this procedure in more detailfor both the total cross section and the differential yield of e.g. gamma rays. For 3-body processes dominated by an on- or off-shell resonance, the total cross section canbe written as in Eq. (3.9). If the intermediate particle corresponds to a nearly on-shellresonance with Γ (cid:28) M , furthermore, the Breit-Wigner propagator can be approximated as m − M ) + M Γ → πM Γ δ ( m − M ) . (4.1)This narrow-width approximation (NWA) yields the on-shell contribution of the resonance R , and we denote the corresponding, approximated cross section by σv NW A . Strictly speak-ing, for the approximation to work well, the kinematic boundaries have to be sufficiently faraway from the pole, | m − M | (cid:29) M Γ , and all contributions from the matrix element andphase-space factors apart from the Breit-Wigner propagator should be smooth functions of m in the vicinity of the pole, which we assume in the following. With this replacementin Eq. (3.9), we immediately recover the well-known result σv NW A → = S × σv χχ → ,R × BR R → , (4.2)where BR R → = Γ R → / Γ is the branching ratio for the resonance R to decay into particles and , Γ R → = S → / (2 M ) (cid:82) d Φ |M d | is the partial decay width, and σv χχ → R = S → /P (cid:82) d Φ |M p | is the 2-body cross section.In general, more than one resonance can contribute to a given 3-body process, and onehas to sum over all those contributions (in principle there can be interference effects foroverlapping resonances with | M − M | (cid:46) Γ + Γ ; we will assume this is not the case). The– 19 –arrow-width limit for the processes in Eq. (3.3) is thus given by σv NW AW + f ¯ F = σv W W BR ( W → f ¯ F ) + σv W + H − BR ( H − → f ¯ F ) (4.3) σv NW AW + b ¯ t = σv W + H − BR ( H − → b ¯ t ) + σv t ¯ t BR ( t → W + b ) (4.4) σv NW AZf ¯ f = 2 σv ZZ BR ( Z → f ¯ f ) + σv ZH BR ( H → f ¯ f )+ σv Zh BR ( h → f ¯ f ) (4.5) σv NW AH + f ¯ F = σv W − H + BR ( W → f ¯ F ) (4.6) σv NW AAf ¯ f = σv Ah BR ( h → f ¯ f ) + σv AH BR ( H → f ¯ f ) (4.7) σv NW AHf ¯ f = σv AH BR ( A → f ¯ f ) + σv ZH BR ( Z → f ¯ f ) (4.8) σv NW Ahf ¯ f = σv Ah BR ( A → f ¯ f ) + σv Zh BR ( Z → f ¯ f ) , (4.9)where f denotes any SM fermion, and F its SU (2) L doublet partner. The branching frac-tions BR are given by the tree-level decay widths, divided by the total width appearing inthe corresponding Breit-Wigner propagators. As stated in the second line, third-generationquarks have to be treated separately because of the contribution from top decay. Note thatthese results justify why the interference effects mentioned above can indeed be neglected:those would be potentially relevant only in small regions of the MSSM parameter space,where the charged Higgs is degenerate in mass with the W boson or the top quark (or,instead, one of the heavy neutral Higgses close in mass to the Z boson or the SM Higgs h ).The total annihilation cross section is then given by σv = σv → + σv → − σv NW A → , (4.10)where each term corresponds to the sum over all possible 2- and 3-body final states, re-spectively. In the following, we refer to the difference σv sub2 → ≡ σv → − σv NW A → as the(NWA-subtracted) contribution from 3-body final states, with a similar definition for in-dividual 3-body final states. We note that σv sub2 → can be negative (although σv > , ofcourse). To match our conventions for the computation of 3-body cross sections, the 2-body cross sections appearing above should be evaluated in s -wave approximation. Finally,we stress that, even when summing over all possible 3-body final states, σv NW A → is notequal to the sum over 2-body cross sections with diboson and t ¯ t final states, as one mayhave naively expected. This is partially because the Higgs resonances can also decay intopairs of bosons, and partially due to a mismatch in the combinatorial factors, which can betraced back to ambiguities in the narrow-width limit. For example, for χχ → ZZ ∗ → Zf ¯ f , We neglect loop corrections to σv → because only 3-body final states can lift the m f /m χ suppression.For very heavy neutralinos, however, enhancements ∝ απ ln ( m W /m χ ) from both soft/collinear IB and one-loop corrections to σv → can become sizeable. For EW corrections these logarithmic terms will in generalgive a non-zero contribution, unless the initial state is a singlet under SU (2) L × U (1) Y , such as for a pureBino [53, 54] (the latter also applies to U (1) and SU (3) IB; see Ref. [29] for an efficient model-independentway of taking the relevant loop contributions into account). Resummation of logarithmically enhancedcontributions has been discussed e.g. in [55–57] for pure Winos and Higgsinos. In addition, for neutralinoswith a significant Wino fraction and TeV mass, Sommerfeld enhancement can play an important role [58–60]. A joint treatment of all these effects for the general MSSM is beyond the scope of this work, but wouldbe desirable in view of future indirect detection probes. – 20 –ach of the Z bosons could act as intermediate resonance, which intuitively explains thefactor S = 2 encountered in this case. These ambiguities would disappear if one were totreat both Z bosons on equal footing, i.e. consider -body final states, which however isimpractical for a general MSSM computation. In general, our final state particles p will fragment and decay into potentially observableparticles P , such as gamma rays or antiprotons. For a given 3-body annihilation channel,and a conventional normalization to the corresponding yield from the 2-body rate, thespectrum can be written as dN P dE P = 1 σv tot2 → (cid:88) p (cid:90) E max p E min p dN p → P + ...P dE P dσv → dE p dE p . (4.11)Here, dN p → P + ...P /dE P describes the number of stable particles P , per energy bin, that resultfrom the inclusive process p → P + ... of a particle p decaying in flight (with energy E p ), andthe sum has in principle to be performed over all helicity states separately (because dN/dE P can differ for different helicities of p ). Assuming CP conservation and that the decayingparticles have very narrow widths, a very useful approximation in practice consists in consid-ering instead unpolarized cross sections and replace dN p → P + ...P /dE P → dN ¯ pp → P + ...P /dE P ,where the inclusive process ¯ pp → P + ... is evaluated for a CMS energy of E p (see e.g. [18]).This quantity can easily be obtained from event generators like Pythia [61] and, unlike dN p → P + ...P /dE P , has the further advantage of being manifestly color neutral. The above expression depends on the differential cross section dσ/dE p , rather than thetotal cross section discussed in the previous subsection, implying that we need to re-discusshow to correctly take into account double-counting issues. Consider for example the process χχ → H + f ¯ F . We want to remove the contribution already contained in χχ → H + W − ,say in the differential cross section for the fermion f , (cid:18) dσv H + f ¯ F dE f (cid:19) sub = dσv H + f ¯ F dE f − (cid:18) dσv H + f ¯ F dE f (cid:19) NW A . (4.13) Note that for quark final states Eq. (4.11) is not correct beyond leading order, where partons fragmentindependently, because it ignores flux tubes. In general, for a final state consisting of a (color-neutral)boson B and a quark pair ¯ qq , which may have different masses, one should instead consider σv tot2 → dN P dE P = (cid:90) E max B E min B dN B → P + ...P dE P dσv → dE B dE B + (cid:90) E max¯ q E min¯ q (cid:90) E max q E min q dN ¯ qq → P + ...P dE P dσv → dE ¯ q dE q dE ¯ q dE q . (4.12)The fragmentation function dN ¯ qq → P + ...P /dE P that appears here can be obtained by boosting to the back-to-back system of the quarks, defined as p bb q = − p bb¯ q , then evaluate the fragmentation function supplied by,e.g., Pythia for a CMS energy of E bb q + E bb¯ q , and finally boosting back to the DM frame. As we only considerleading order effects here, and the expected difference in dN P /dE P is anyway very small, we restrict ournumerical implementation to Eq. (4.11) also for quark final states. – 21 –he question is, what to use for the NWA term. The simplest assumption would be toreplace the branching ratio in Eq. (4.2) by the differential spectrum, i.e. (cid:18) dσv H + f ¯ F dE f (cid:19) NW A = σv H + W − dN W − → f ¯ F dE f (4.14)where the last factor is the spectrum of f per decay of the W , as seen in the CMS frame andnormalized such that (cid:82) dE f dN W /dE f = BR ( W → f ¯ F ) . Obviously, it is straightforwardto generalize Eqs. (4.13,4.14) to other final states, analogous to Eqs. (4.3–4.9).Unlike for the total cross section [50], however, the replacements (3.6) for vector andfermion resonances are in general not correct for the differential cross section. Instead, thelatter can be affected by the correlation of the helicities/spins of the resonance betweenthe production and decay processes, even in the limit Γ /M → . Fortunately, conservationof CP and total angular momentum uniquely fixes the polarization states of the vectorresonances for the case of Majorana pair annihilation in the s -wave limit (see, e.g., [62]):vector bosons in W W and ZZ final states are necessarily transversely polarized, and thosein H ± W ∓ , HZ and hZ final states longitudinal (while AZ final states are not possible,for the same reason). Therefore, spin correlations can fully be taken into account by usingthe appropriate energy spectra for polarized vector bosons in Eq. (4.14), noting that thebranching fractions are in fact independent of polarization (see Appendix C for a more de-tailed discussion). For the example above, this implies that one should use dN W L → f ¯ F /dE f instead of dN W → f ¯ F /dE f ; for e.g. χχ → W W ∗ → W f ¯ F , on the other hand, the correspond-ing narrow-width contribution contains dN W T → f ¯ F /dE f . For the decay of a top resonance, χχ → t ∗ ¯ t → W + b ¯ t , conservation of angular momentum and CP requires t ∗ and ¯ t to havethe same helicity. We note that the decay spectrum for polarized tops, dN t h → W + b /dE b ,differs for the two helicities h = ± / , due to the parity-violating W coupling. Neverthe-less, since both polarizations are produced with equal cross section, as a consequence of CP conservation, the relevant contribution to the total narrow-width spectrum is given by (cid:88) h = ± / σv t h ¯ t h dN t h → W + b dE b = σv t ¯ t (cid:32) dN t / → W + b dE b + dN t − / → W + b dE b (cid:33) = σv t ¯ t dN t → W + b dE b , (4.15)where σv t ¯ t / is the usual cross section summed over all final-state helicities. Thus, also fortop resonances, no polarization effects occur in the CP conserving MSSM.In summary, the differential 3-body cross section to be used in Eq. (4.11) is given by (cid:18) dσv → dE p (cid:19) sub ≡ dσv → dE p − (cid:18) dσv → dE p (cid:19) NWA , (4.16)where for fermionic final state particles ( p = f, f , F, F ) we have (cid:18) dσv W + f ¯ F dE p (cid:19) NWA = σv W W dN W T → f ¯ F dE p + σv W + H − dN H − → f ¯ F dE p (4.17) (cid:18) dσv Zf ¯ f dE p (cid:19) NWA = 2 σv ZZ dN Z T → f ¯ f dE p + σv ZH dN H → f ¯ f dE p + σv Zh dN h → f ¯ f dE p (4.18)– 22 – dσv H + f ¯ F dE p (cid:19) NWA = σv W − H + dN W L → f ¯ F dE p (4.19) (cid:18) dσv Af ¯ f dE p (cid:19) NWA = σv Ah dN h → f ¯ f dE p + σv AH dN H → f ¯ f dE p (4.20) (cid:18) dσv Hf ¯ f dE p (cid:19) NWA = σv AH dN A → f ¯ f dE p + σv ZH dN Z L → f ¯ f dE p (4.21) (cid:18) dσv hf ¯ f dE p (cid:19) NWA = σv Ah dN A → f ¯ f dE p + σv Zh dN Z L → f ¯ f dE p . (4.22)Here, the decay spectra are normalized to fermionic branching ratios, e.g. (cid:82) dE f dN W λ → f ¯ F /dE f = BR ( W → f ¯ F ) . For scalars, or when neglecting correlations, these spectra are flat, e.g. dN h → b ¯ b dE b = BR ( h → b ¯ b ) E max b − E min b , (4.23)where E max / min b are the maximal and minimal allowed energy of the b in the decay of the(boosted) Higgs. This is the usual box-shaped spectrum in a cascade decay. The non-trivialspectra are in principle straight-forward to derive (see Appendix C), but not needed in ournumerical implementation (see below). For bosonic final states ( p = Z, W ± , h, H, H ± , A ),on the other hand, there is no polarization effect and only the energy allowed by the 2-bodykinematics contributes. This implies that one has to replace σv pX dN X ( h ) → f ¯ F dE p −→ σv pX BR ( X → f ¯ F ) δ ( E p − E χχ → pXp ) (4.24)in Eqs. (4.17 – 4.22), where E χχ → pXp = m χ + (cid:0) m p − m X (cid:1) / m χ . Annihilation channelsinvolving top quarks , finally, are slightly special: (cid:18) dσv W + b ¯ t dE b (cid:19) NWA = σv W + H − dN H − → t ¯ b dE b + σv ¯ tt dN t → W + b dE b (4.25) (cid:18) dσv W + b ¯ t dE ¯ t (cid:19) NWA = σv W + H − dN H − → t ¯ b dE ¯ t + σv ¯ tt BR ( t → bW ) δ ( E ¯ t − m χ ) (4.26) (cid:18) dσv W + b ¯ t dE W (cid:19) NWA = σv ¯ tt dN t → W + b dE W + σv W + H − BR ( H − → ¯ tb ) δ ( E W − E χχ → W + H − W ) (4.27)Now let us consider these NWA corrections to the energy distribution of final stateparticles p in the context of Eq. (4.11), i.e. the spectrum of stable particles P . From eachof the terms on the r.h.s. of Eqs. (4.17 – 4.22), and a given channel χχ → Y f ¯ F , we pick upa contribution of the form σv Y X σv tot2 → (cid:88) p = f, ¯ F (cid:90) dN p → P + ...P dE P dN X ( h ) → f ¯ F dE p dE p + BR ( X → f ¯ F ) dN Y → P + ...P dE P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E Y = E χχ → Y XY = BR ( χχ → Y X ) dN X ( h ) → f ¯ F → P + ...P dE P + BR ( X → f ¯ F ) dN Y → P + ...P dE P E X,Y = E χχ → Y XX,Y (4.28)– 23 – BR ( χχ → Y X ) BR ( X → f ¯ F ) (cid:40) d ˆ N X ( h ) → P + ...P dE P + dN Y → P + ...P dE P (cid:41) E X,Y = E χχ → Y XX,Y (4.29) = BR ( χχ → Y X ) BR ( X → f ¯ F ) d ˆ N XY → P + ...P dE P . (4.30)Here, we can replace X ( h ) → X in the last step because, as stressed before, the helicity of X in χχ → XY is uniquely fixed by conservation of angular momentum and CP (for Y ,on the other hand, the helicity has been fixed that way right from the start). Furthermore,we introduced the notation d ˆ N X ( h ) → P + ...P /dE P for the yield of species P considering onlyfermionic decays of X . Similarly, for computing d ˆ N XY → P + ...P /dE P we take into account onlyfermionic decays of X (while for Y all decay modes are included). Note that the second step( → ) is then only valid under the assumption that dN/dE P has the same shape for a singledecay channel X ( h ) → f ¯ F as for the sum over all fermionic final states, which can be a ratherpoor approximation for a given channel (involving, e.g., neutrinos). The total spectrum of agiven stable state particle P , however, is correctly recovered when summing Eq. (4.30) overall possible 3-body channels involving the boson Y and a pair of fermions. Numerically, weimplement this sum for all fermionic final states for decays of X = Z, W, h, H, H ± , A . Forcases where a nearly on-shell t ∗ quark gives a large contribution, even the single-channelyield is well approximated because the decay t → W b dominates.Let us conclude this Section with two comments regarding the correct use of final statehelicities for the determination of yields of stable particles. i) Eq. (4.11) has indeed to besummed over all helicities of p that contribute to the cross section σ → ; it is thus in general not sufficient to determine only the yields dN ¯ pp → P + ...P /dE P that are required for 2-bodyprocesses (for which the helicities of p and ¯ p are fixed by the requirement J P = 0 − ). ii) The yields from the NWA subtraction, on the other hand, do result from final states withhelicities fixed by the same symmetry argument as in the 2-body case. This implies thatdouble counting is fully avoided in our prescription even if the yields dN XY → P + ...P /dE P are throughout approximated by using unpolarized final state particles X and Y : In thatcase, the procedure described above consistently removes the double counting related tothe yields produced from decay and fragmentation of Y . For X , on the other hand, the full3-body matrix element automatically takes into account polarization effects in the decay X → f ¯ F , while the NWA term subtracts the yield for unpolarized decays. This means that,in this case, the NWA-subtracted 3-body contribution accounts precisely for the difference,and correctly replaces the unpolarized by the polarized yield for the X decay after addingtwo- and 3-body contributions. In order to demonstrate the impact of our results on realistic models, we work in theframework of simplified phenomenological MSSM versions, introduced in Section 5.1, thatare however generic enough to capture the relevant phenomenology discussed in this work.We assess in turn the consequences for the overall annihilation cross section (Section 5.2)and yields (Section 5.3), before discussing in detail selected example spectra in Section 5.4.– 24 –SSM-91 µ M M M A tan β A t A b M ˜ q M ˜ (cid:96) MSSM-92 µ M M M A tan β A t A b M sf , . M sf , . MSSM-93 µ M M M A tan β A tb M ˜ q M ˜ (cid:96), . M ˜ (cid:96), . MSSM-94 µ M M M A tan β A tb M ˜ t R M ˜ t L / ˜ b L M sf , rest Table 4 . Free parameters for the four types of MSSM models considered in this work. Note thatfor the last two models we assume A t = A b . As we will see, some of the most relevant part of the parameter space involves SUSY spectrawith degenerate particle states, that are typically more difficult to test in proton colliderexperiments.
We introduce four phenomenological “pMSSM-9” realisations, each defined by 9 parametersat the electroweak scale as specified below (see also Table 4):
MSSM-91
The Higgsino mass parameter µ , the Bino and Wino masses M and M ,the CP -odd Higgs boson mass M A , the ratio of Higgs vacuum expectation values tan β , the squark and slepton mass terms M ˜q , and M ˜ (cid:96) , and the third generationtrilinear couplings, A t and A b (note that M and M are not constrained by the GUTunification relation). MSSM-92
Here, instead of distinguishing squark and slepton masses, we decouple the rd sfermion generation from the st and nd unified generations. MSSM-93
Squark and slepton mass terms are decoupled as in MSSM-91. Here, we allowfor a separate rd generation slepton mass, and a common slepton mass for the st and nd generation, respectively. In this case we assume A t = A b . MSSM-94
Adding more freedom to the squark sector, we allow here for an independentright-handed stop (UR) mass and left-handed rd generation squark mass (L), whileadopting a universal sfermion mass for all other cases.In addition, we used in all cases a fixed value of M = 5 TeV for the gluino massparameter. We performed Bayesian scans over the parameter space of these models byusing
MultiNest [63], which we interface to
DarkSUSY to compute all relevant quantitiesthat enter in the likelihood evaluations. The joint likelihood that we adopt takes the form ln L Joint = ln L Ω χ h + ln L m Higgs + ln L SUSY + ln L LUX , (5.1)where L Ω χ h refers to the constraint on the cold DM relic abundance from cosmic microwavebackground (CMB) observations; L m Higgs imposes the mass of the lightest SUSY Higgs toagree with the Higgs boson mass; L SUSY includes constraints from sparticles searches atcolliders; and L LUX accounts for the constraints on DM-nucleon interactions from the LUXdirect detection experiment. – 25 –odel parameters Low-mass High-mass µ [GeV] (70 , , M [GeV] (70 , , M [GeV] (70 , , M ˜ q , M ˜ (cid:96) ( M sf ,i ) [GeV] (70 , , A t /M ˜ t R , A b /M ˜ t R ( − ,
3) ( − , M A [GeV] (70 , , β (5 ,
40) (5 , Table 5 . Parameter ranges of the scans performed. The range for M sf ,i applies to all combinationsof independent sfermion mass parameters for the four models. The relic abundance is computed including co-annihilations [64, 65], using a centralvalue of Ω χ h = 0 . [1] and a generous error of % to account for both experimentaland, more importantly, theoretical uncertainties in the Ω χ h prediction. We impose the pre-dicted mass of the lightest Higgs to match the measured Higgs mass, m h = 125 . ± . GeV[66]. Since we are mainly interested in suppression lifting mechanisms, rather than a de-tailed phenomenological analysis of the MSSM parameter space, we adopt a conservativeset of further constraints to roughly indicate some of the more relevant experimental con-straints. Apart from LEP and TeVatron constraints as implemented in
DarkSUSY [39], weimpose LHC constraints on stop, sbottom, light squark and slepton masses assuming directproduction [67–77], as well as null results from direct DM detection [78].For each model, we performed scans with two different parameter ranges: one forlow-mass neutralinos, roughly ∼
50 – 2000 GeV, and one for high masses, ∼
500 – 3000GeV, adopting logarithmic priors on the parameters, except for the trilinear couplings A . Table 5 contains the parameters and the ranges of the “low-mass” and “high-mass”scans. We emphasise that we do not perform global scans of the MSSM in light of themost recent results from various experiments (for this, see instead Ref. [79]), nor is it ourpurpose to do so here. In particular, we note that important additional constraints canarise from electroweakino and MSSM Higgs searches, that are however highly dependent onthe specific configuration of masses and decay channels, and therefore beyond the scope ofthis work. Rather than identifying the most probable parameter regions of our pMSSM-9models, taking into account all experimental constraints, our focus is simply to providea phenomenological proof of concept concerning the impact of 3-body final states on theannihilation of DM particles and on the resulting cosmic-ray fluxes. In our extensive scans, we identified many MSSM models where the total annihilation crosssection is significantly enhanced by including the 3-body final states we consider here. Asan illustration, we show in Fig. 6 the ratio between the (NWA subtracted) 3-body and total2-body cross section, in function of the neutralino mass. In the left panel, we furthermoreindicate the neutralino composition, while in the right panel we indicate the models where– 26 – ààààààà à àààààààààààààààààààààààààààààààààààààààààààààààààààààààà à ôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôô ôô ô ôôôôôôôôôôôôôôôôôôôô ôôô ô ôôôôôôôôôôôôôôôôôôôô ôôôôôô ôôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôô ôôôôôôôôôôôôôô ôôôôôôôôôôôôôôô ô ô ôô ôôôôôôôôôôôôôôô ôôôôôôôôôôôôôô ô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ô ôôôôôôôôôôôôôôôôôô ôôôôôôôôô ô ôôôôôôôôôôôôôôôôôô ôôô ôôôôôôô ô ôô ôôôôôôôôôô ôô ô ôôô ô ôô ôôôôôôôôô ô ôôô ôôô ôôô ôô ôôôôô æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ ææ æ æ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ ææ ææ ææ ææ ææææ òò òòò ò ò òò òòòòòò òòòò òò òòòòòòòòò ò òòòòòòòòòòòòòòòò òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ò òòòò ò ò òòòòòòòòòòòòòòòòòòòòòòòòòòòò
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Left panel : Ratio between the (NWA subtracted) 3-body and total 2-body cross sectionfor the MSSM models defined in Section 5.1, as a function of the lightest neutralino mass m χ . Dif-ferent neutralino compositions are indicated as “Bino” ( | N | > . ) in red, “Wino” ( | N | > . )in blue, “Higgsino” ( | N | + | N | > . ) in cyan, and “mixed” (otherwise) in green. Right panel :Same, but now broken down to models that are either (a) close to a threshold X = W, Z, t (with − X < m χ − m X < Γ X , where Γ X is the total width of X ) or (b) with a degenerate sfermion ˜ f in the mass spectrum ( m χ /m ˜ f > . ) or (c) Bino-like neutralinos with Wino coannihilation( m χ /M > . and | N | > . , without degenerate sfermions). the neutralino is either close to a threshold or there exists an almost mass-degenerateSUSY particle. As anticipated, models with large enhancements indeed broadly fall intothose two classes. In particular, cross section enhancements by O (10) factors are possiblefor models with neutralino masses close to the kinematical threshold of producing W + W − , ZZ and ¯ tt final states; on the other hand, we did not find any models where the totalcross section is significantly enhanced below the Zh threshold. For models with neutralinomasses close to the threshold for annihilation into a heavy Higgs boson and a SM particle( ZH , A h, W ± H ∓ ) we identified up to ∼ enhancements.The second class of enhancement mechanisms are models with small mass splittingsbetween the neutralino and other SUSY particles. For neutralino masses m χ (cid:38) GeV,degenerate sleptons can significantly increase σv , while for even heavier neutralinos, m χ (cid:38) GeV, this also becomes possible for degenerate squarks. A final, somewhat less expectedclass of models where σv → can be of at least the same size as σv → are heavy Bino-likeneutralinos with almost mass-degenerate Winos. Here, the relic density is set by Winocoannihilation, while the DM annihilation rate is highly suppressed due to either the smallneutralino mixing (for W W or ZZ final states) or helicity factors m f /m χ (for fermionicfinal states). The lifting of the latter by 3-body annihilation processes is thus very relevantin this case, even for models that do not feature degenerate sfermions.One of the main new additions of this work is the full inclusion of final states containinga Higgs boson. To demonstrate their impact, we show in Fig. 7 the annihilation cross sectionto H ¯ f f /h ¯ f f /A ¯ f f /H ± ¯ f F as compared to the one for Z ¯ f f and W ± ¯ f F final states. Wefind that Higgs boson IB alone can increase the lowest-order cross section by up to a factor– 27 – ààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààà ààààà àààààà ààà ààààààààààààààààààààààààààààààààààà ààà ààà ààààààààààà àààààààààààààààààààààààààààààààààààààààààààààààààààà àà àààààààààààààààààààààààààààààààààààààààààààààààààà àà ààààààààààààààààààààààààààààààààààààààààààààààààà ààààààààààààààààààààààààààààà àà àààà àà ààààà ààààààà 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Cross section for annihilation into 3-body final states containing a gauge boson versusthat containing any of the Higgs bosons in the final state, normalized to the total 2-body crosssection. Different neutralino compositions are indicated as in Fig. 6. of ∼ , but in general the relative importance of the Higgs and gauge boson channels is verymodel dependent. Models where the former dominates (bottom right corner of the plot)feature neutralino masses in the TeV range, and a stop not much heavier than the neutralino(within a few hundred GeV). The dominant annihilation channels are then Htt, Att, H ± tb ,in particular via ˜ t exchange in the t/u channel, where the large Yukawa coupling to topquarks explains why the corresponding gauge boson final states are not equally pronounced.We note that these cases are examples where lifting of isospin suppression is more importantthan lifting of Yukawa suppression (see Section 3.3).A further class of models with large contributions of Higgs final states is character-ized by large values of µ and tan β , which lead to an enhancement of the Higgs coupling tosfermions. However, this enhances also the mixing between left- and right-handed sfermionsand hence implies only a mild Yukawa suppression; nevertheless the three-body processescan be particularly important for the antiproton yield as discussed in more detail for bench-mark model D3 further down.For masses below the SM thresholds, annihilation into Higgs plus top final states iskinematically forbidden. For Binos, as clearly seen in the top left part of Fig. 7, thiscan result in large IB enhancements without Higgs contribution. For neutralinos with asignificant Higgsino fraction, on the other hand, there is a correlation between gauge boson(mostly W ¯ f F ) and Higgs (mostly h ¯ f f ) final states (top middle part of Fig. 7). Thesemodels correspond to neutralino masses below the W W threshold, and the 3-body finalstates arise dominantly from virtual W or Z boson decays, W W ∗ → W ¯ f F and hZ ∗ → h ¯ f f ,respectively. The latter are suppressed compared to the former by the gaugino fraction,but otherwise of comparable size, which leads to the correlation seen in Fig. 7. Note that this also explains the observed separation: the upper left part in the figure corresponds tothreshold models, for which the Higgs contribution can be arbitrarily small in the pure Bino limit. Themiddle part corresponds to models where t -channel sfermion exchange is important. In this case, thereis a h ¯ ff contribution even in the pure Bino limit, that arises from VIB emission of the Higgs boson, as – 28 – àààààààààààààààààààà ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôô ô ô ôôôôôô ô ôôôôôôôôôôôôôôôôôôô ô ô ôô ôôôôô ô ô ôôôôôôôôôôôôôôôôôôôô ô ôô ôôôôôô ôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ô ôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôô ôôô ôôôôôôôô ôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ô ôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ô ô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ô ô ô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôô ôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôôô ôô ôôôôôôôôôôôôôôôôôôôô ôôôô ôô ôôôôôôôôôôôôôôôôôôôôô ôôôô ô ô ô ôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôô ôô ôô ôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôô ô ô ôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôô ôôôôôôôôôôôôôô ôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôô ôôôô ôôôôôô ô ôô ô ôôôôô ôôôô ô ôôô æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ ææ æ æ æ ææ æ æ æ ææ æ æ æ ææ æ æ æ ææ æ ææ æ æ ææ æ æ æ æ æ ææ æ ææ æ ææ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ ææ ææ ææ ææ ææ ææ æææ ò òòòòòòòòòòòòòòòòòòòòòòòòòòòò òò òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò Bino È Mixed È Higgsino È Wino ôô ææ òò àà - - m Χ @ GeV D N p t o3 (cid:144) N p t o2 p ààààààààààààààààààààà ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôô ôôôôôôô ôôôôôôôôô ô ôô ô ôôôôô ôô ôô ôôôôôôôôôôôôôôôôôôô ô ôô ôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôô ôôô ôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôô ôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôô ôôôôôôôô ô ôôôôôôôô ôôôôôôôô ôôôôôôôôôô ôôôôôôô ôôôôô ô ôôôôôôô ôôôôô ôôôôô ô ôôôôôôôôôô æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ ææ æ æ æ ææ æ æ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ æ æ ææ æ æ ææ æ ææ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ ææ ææ ææ ææ ææ ææææ ò òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò òòòòòòòòòòòòòòòòòòòòòòò Bino È Mixed È Higgsino È Wino ôô ææ òò àà - - m Χ @ GeV D N Γ t o3 (cid:144) N Γ t o2 Γ ààààààààààààààààààààà ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ô ô ôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôô ô ôôôôô ôôôôôôôôôôôôôôôôôô ôôô ô ôôôôô ôô ô ô ôôôôôôô ôô ôôôôôôôôôôôô ôôô ôô ôôôôôôô ô ô ô ôôôôôôôôôôôôôôôôôô ôôôô ôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôô ôôôôôôô ô ôôôôôôôôôôôôôôô ôôôôôôôôôôôô ôô ôôô ôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôô ôôôôôôôôôôôôôôôôôôôôô ô ôôôô ô ô ôôôôôôôôôôôôôôôôô ô ôô ô ô ôôôôôôôôôôôôôôôôô ô ôôôô ôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôô ôôô ôôô ô ôôôôôôô ôôô ô ô ôô ôôôôôô ôô ôôôôôô ôô ôô ôô ô ô ôô ô ô ô ôô ô ô ôôô ôô ô ôôô æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ æ æ ææ æ æ æ ææ æ æ æ æ æ ææ æ æ æ ææ æ æ ææ æ æ æ ææ æ æ ææ æ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ æ ææ ææ ææ ææ ææ ææ ææ ææ ææ ò òòòòòòòòòòòòòòòòòòòòòòòòòòò ò òòò òò òòòòòòò òòòòò òòòòò òò òò òòòòòòòòòòòòòòòòòòòòòòòòòòòòò Bino È Mixed È Higgsino È Wino ôô ææ òò àà - - m Χ @ GeV D N Ν Μ t o3 (cid:144) N Ν Μ t o2 Ν Μ Figure 8 . Ratio of integrated yields above E X > m χ / from (NWA-subtracted) 3-body finalstates and the 2-body result, for species X = ¯ p, γ, ν µ , plotted against the neutralino mass m χ .The other light lepton yields ( ν τ and e + ) are qualitatively very similar to the ν µ case. Differentneutralino compositions are indicated as in Fig. 6. ààààààààà ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôô ôôôôôô ôôô ô ôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôô ôôôôôôôôôôôôôôô ôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôô ô ôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôô ô ôô ô ôôôôôôôôôôôô ôôôôôôô ô ôô ôôôô ôôôôôôôôôôôôô ôôôôôôôô ôô ô ôô ôôôôôôôôôôôô ôôôôôôôô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôô ôôôôô ôô ôôôôô ôôôôôôôôôôôôôô ô ô ôô ôôôôôôôôôôôôôôôôôôôô ô ôô ôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôô ô ô ôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôô ô ô ôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôô ô ô ôôô ô ôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôô ôôô ôôôôôôôôôôôôôôôôôôôôô ô ô ôôôôôôôôôôôôôôôôôô ôô ô ôôôôôôôôôôôôôôôôô ô ô ôô ôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôô ôô ô ôôôôôôôôôôôôôôôô ô ô ôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôô ô ôô ôôô ôôôôôôôôôô ôô ôô ôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ô ôôôôôô ôô ôô ôôô ô ôô ôô ôô ôôô ôô æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ ææ æ æ æ ææ æ æ æ ææ æ æ ææ æ ææ ææ æ òòòòòòòòòòòòòòòò òòòò òò òòòòòò òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò òòòòò Bino È Mixed È Higgsino È Wino ôô ææ òò àà - - - - - - - - - Σ v @ cm (cid:144) s D N p t o3 (cid:144) N p t o2 p ààààààààà ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôô ôôô ôôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôô ôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ô ôô ôôôôôôôôôôôôôôôôôôôôôôôô ôôôôô ô ôôôôôôôôôôôôôôôôôôôôô ôôô ô ôô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôô ô ô ôôôôôôôôôôôôôôôôôôô ôô ôô ôô ôôôôôôôôôôôôôôôôôôôôôôô ô ôô ôôôôôôôôôôôôôôôôôôô ô ô ôôô ôôôôôôôôôôôôôôôôôôô ô ôô ôôôôôôôôôôôôôôô ô ô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ô ôôôôôôôôôôôôôôôô ôô ôô ôô ô ôô ôô æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ ææ æ æ æ æ ææ æ æ æ ææ æ æ ææ æ ææ ææ òòòòòòòòòòòòòòòòòòò òò ò òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò Bino È Mixed È Higgsino È Wino ôô ææ òò àà - - - - - - - - - Σ v @ cm (cid:144) s D N Γ t o3 (cid:144) N Γ t o2 Γ ààààààààà ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôô ô ô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôô ôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôô ôôôôôôôôôôô ôôôôôôô ôôôôôô ô ôô ô ôô ôô ôôôôôôôôôôôôôôôôôôôôô ôôôôô ô ô ôôôôôôôôôôôôôôôôôôô ôôô ôôôô ôôôôôôôôôôôôôôôôô ôôôô ô ô ôôô ôôôôôôôôôôôôôôôôôô ôôô ôô ôôôôôôôôôôôôôôôô ôôô ô ôô ôôôôôôôôôôôôôôôôôôô ô ôôôô ô ôôôôôôôôôôôôôôôô ô ôôô ôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôô ôôôô ôôôôôôôôôôôôôôô ôôô ôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôôôôôôôôôôôôôôôô ôôôôôôôô ôôôôôôôôôô ôôôôôôôô ôôôôôôô ôôôôôôôôôô ôô ô ôôôôô ôôôôô ô ôôôô ôô æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ æ ææ æ æ æ æ ææ æ æ æ æ ææ æ æ æ ææ æ æ æ æ ææ æ æ ææ æ æ ææ æ ææ æ ææ ææ æ òòòòòòòòòòòòòòòò òò òòòòòòò òòòò òòòòòòòò òò ò òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò Bino È Mixed È Higgsino È Wino ôô ææ òò àà - - - - - - - - - Σ v @ cm (cid:144) s D N Ν Μ t o3 (cid:144) N Ν Μ t o2 Ν Μ Figure 9 . Same as Fig. 8, but now showing the integrated yields, for E X > m χ / , against thetotal 2-body cross section instead of the neutralino mass. The zero-velocity annihilation cross section is already a good indicator for the reach ofindirect detection experiments, but observationally more relevant is the resulting numberof stable particles. Astrophysical background spectra are generally rather soft, i.e. theyfall quickly with energy, such that from the point of view of indirect DM searches mostlyannihilation products with relatively large energies are relevant (with the notable exceptionof CMB constraints that are mostly sensitive to the total energy deposition, see e.g. [80]).In Figs. 8–10, we therefore consider the integrated yield of all relevant stable particlesX = ¯ p, γ, ν µ , ν τ , e + above some threshold energy. Fig. 8, in particular, shows the ratioof the 3-body yield to the typically considered yield expected from 2-body final states, for E X > m χ / , as a function of the neutralino mass and for different neutralino compositions. discussed above. On the other hand, the gap between the pure Bino models on the very left part and themixed and Higgsino-like models in the top middle part is due to (large, but still limited) statistics of oursample. The reason is that the ratio of h ¯ ff and W/Z ¯ F f cross section is extremely sensitive to the Higgsinofraction (roughly ∝ ( Z + Z ) ), and therefore varies very rapidly with the input parameters. – 29 – ààààààààààààààààààààààààààà àààààààààààààààààààààààààààà àààààààààà àààààààà àààààààààààààààààààààààààààààààààààààààààà ààààà àààààààààààààààààààààààààààààààààààààààààààààààààààà àààààààààààààààààààààààààààààààààààààààààààààààààà ààààààààààààààààààààààààààààààààààààààààààààààààààà ààààààààààààààààààààààààààààààààààààààààààààààààà àààà ààààààààààààààààààààààààààààààààààààààààààààààà ààààààààààààààààààààààààààà àààààààààààààààààààààà àà àà àààààààààààààààààààààààààààààààààààààààààààààààà àà ààààààààààààààààààààààààààààààààààààààààààààààà ààà àààààààààààààààààààààààà ààààààààààààààààààààààà ààà àààààààààààààààààààààààà àààà ààààààààààààààààààà ààà àààààààààààààààààààààààà àààààà àààààààààààààààààà àà àààààààààààààààààààààààààà àààààà ààààààààààààààààààà àààààààààààààààààààààààààààààà 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Same yield ratios as in Fig. 8, but now for all species X = ¯ p, γ, ν µ , ν τ , e + and plottedagainst the ratio of cross sections. Left panel : Integrated yields above an energy threshold of E X > m χ / . Right panel : Integrated yields above an energy threshold of E X > m χ / . In Fig. 9, we show the same quantity, but now as a function of the total 2-body annihilationrate. This immediately allows us to make the interesting observation that the yields aremost strongly enhanced for models with cross sections somewhat below the ‘thermal’ crosssection of ∼ − cm s − ; including the effect of electroweak corrections, as already pointedout in Ref. [18], may thus turn out to be the decisive ingredient to make these modelsaccessible by current and near-future indirect detection experiments. We complement thesetwo sets of figures by Fig. 10, which demonstrates how the yield enhancement correlateswith the cross section enhancement discussed in the previous subsection. This is done bothfor the yield enhancement at low energies (left) and at high energies close to the kinematicthreshold (right), in order to get a first qualitative indication of how the spectral shape isaffected. In the following, we discuss the various relevant stable particles in turn. Antiprotons.
For the bulk of the models considered here, the enhancement in the ¯ p yieldscales as expected linearly with the enhancement in σv . For τ + τ − final states, how-ever, antiproton production is not kinematically allowed. Including the τ + τ − Z and τ ± ν τ W ∓ channels can therefore drastically enhance the ¯ p yield even if the correspond-ing enhancement of the cross section is at most moderate. This effect, clearly seen inFig. 10, is responsible for the largest enhancements (for Bino-like models) in the leftpanel of Fig. 8. It is also reflected in the left panel of Fig. 9, which shows that theenhancement is most pronounced for models with small 2-body annihilation rates.Antiprotons are only produced in the fragmentation and decay of the annihilationproducts, and therefore cannot obtain energies E ¯ p ∼ m χ . The additionally emittedgauge or Higgs boson must first decay to quarks, inducing an additional step in the ¯ p production chain compared to 2-body quark final states. The largest enhancementof the antiproton yield from 3-body final states will thus on average occur mostly atsmall energies, an effect clearly seen when comparing the two panels of Fig. 10. Photons.
Compared to antiprotons, gamma-ray yields lack the “atypical” enhancement– 30 –rom τ + τ − final states. Consequently, the enhancement in the yield is generallyweaker, and more strongly correlated with the one in σv . As for antiprotons, thespectrum is only enhanced at somewhat lower energies because the additional photonsonly result from steps further down in the decay chain. Unlike for the ¯ p case, on theother hand, our improved treatment of the NWA prescription for differential ratesdetailed in Section 4.2 can actually decrease the photon yield at large energies (seeSection 5.4 for example spectra). The main reason for the large difference between thetwo panels of Fig. 10, however, are rather the monochromatic photon final states γZ and γγ . While the annihilation rate into these states is loop-suppressed, they can stilldominate the differential photon yield at the highest energies. The models wherethe photon-yield enhancement due to the inclusion of 3-body final states is largest –which, from Fig. 8 are those close to the W threshold – thus at the same time featureparticularly large annihilation rates into γZ and γγ , too. Leptons.
Unlike photons and antiprotons, leptons can appear directly in the final statesconsidered here. This leads to a yield enhancement in particular at high energies, E (cid:96) (cid:38) m χ / , and with a strong correlation with the σv enhancement. Observationally,the resulting characteristic spectral features are especially relevant for positrons [10]in view of the excellent energy resolution of the AMS experiment [81], but also theneutrino spectra can be striking signatures to look for [82–84] if neutralino annihilationin the sun is sizeable (for more details, see the benchmark spectra discussed below).As expected, e + and ν e yields are in general very similar, and the same holds for ν µ . Large yield enhancements are in particular found (i) for models below the W threshold, dominated by χχ → W ∗ W → (cid:96)νW , and (ii) for TeV models with almostdegenerate sleptons and dominant 3-body rates Z(cid:96)(cid:96) and
W (cid:96)ν . Compared to theother leptons, tau neutrinos can receive a somewhat larger yield enhancement, andgenerally feature a spectrum that is less pronounced at small energies. The reasonfor both effects is that the decay of the copiously produced pions, both from 2-bodyfinal states and due to the additional final state boson, results in many low-energy 1 st and 2 nd generation leptons, but hardly any 3 rd generation leptons. The additional final state boson may not only enhance the yields significantly, as describedabove, but also change the shape of the resulting cosmic-ray spectra in a characteristic way.The maybe most striking examples that we identified are sharp spectral features near thekinematic endpoint of neutrino and positron spectra, resembling in fact the often discussedcases of positron [14] and gamma-ray [13] spectra for photon VIB. In this Section, wediscuss in more detail a few example models where the spectrum of at least one type ofstable particles changes significantly once 3-body processes are taken into account. Note that, throughout the manuscript, we include final states that vanish at tree-level when referringto two-body final states. This is not only the usual convention adopted in
DarkSUSY , but serves to stressour emphasis on the differences between 2-body and 3-body final states (rather than between tree-level andnext-to-leading order results). – 31 – ench Type µ (GeV) M (GeV) M (TeV) M A (GeV) tan β A ( M ˜ q ) M ˜ q (GeV) M ˜ (cid:96) (GeV)mark A t A b st /2 nd rd st /2 nd rd D1
93 -2022 1257 1125 2841 12.4 -2.00 2.00 1715 1715 1180 1147 D2
94 3531 3458 3396 2562 22.5 1.94 1.94 3599 3665 R L D3
93 3586 411.9 380.2 1766 11.7 -1.43 -1.43 3714 3714 485.3 470.1 T1
91 -102.0 601.7 119.8 521.1 12.5 -2.00 -1.81 1350 1350 1072 1072 T2
91 -586.4 189.0 172.6 981.4 8.66 -1.56 2.34 1733 1733 698.7 698.7 W
93 -3717 1346 1319 3160 9.01 -1.91 -1.91 2872 2872 1748 1885 H
93 3492 3976 3371 1418 9.47 1.57 1.57 3629 3629 3405 3783
Table 6 . Benchmark models for which we show the resulting spectra of stable particles in Figs. 11–14, with model parameters for the various pMSSM-9 types introduced in Section 5.1. A t/b are givenin units of M ˜ t R . Model D2 takes as input the right-handed stop mass (R), the left-handed thirdgeneration squarks mass (L), and otherwise assumes a common mass scale for all other sfermionmass terms. See Table 7 for some phenomenological properties, and main text for more details. Bench m χ ˜ χ m ˜ q ( ˜ q ) m ˜ l ( ˜ l ) m ˜ χ i σv b σv b σv b,h σv b,W/Z N b,X N b,X X mark [GeV] [GeV] [GeV] [GeV] D1 ˜ B ˜ t ) 1129 ( ˜ τ ) 1254 0.58 0.12 61 ν µ D2 ˜ B ˜ b ) 3397 ( ˜ τ ) 3458 2.7 0.64 8.9 ν τ D3 ˜ B ˜ t ) 385.3 ( ˜ τ ) 411.4 .
17 0 . ¯ p T1 ˜ B/ ˜ W ˜ t ) 1070 ( ˜ ν e ) 113.1 25 < − e + T2 ˜ B ˜ t ) 693.7 ( ˜ τ ) 188.8 3.0 < − ν τ W ˜ B ˜ t ) 1746 ( ˜ ν e ) 1345 0.39 0.12 26 ν µ H ˜ B ˜ b ) 3400 ( ˜ µ ) 3976 3.1 4.9 2.8 ν τ Bench → → mark D1 gg (48%) bb (42%) W ν(cid:96) (37%)
Z(cid:96)(cid:96) (33%)
Zνν (9%)
W tb (5%) D2 HW/Z, hA (54%) gg (25%) W ν(cid:96) (30%) h(cid:96)(cid:96) (17%)
Z(cid:96)(cid:96) (16%)
Htb (10%) D3 τ τ (99%) γγ (0.3%) W ν(cid:96) (42%) h(cid:96)(cid:96) (37%)
Z(cid:96)(cid:96) (20%) T1 cc (37%) gg (31%) W Qq (63%)
W ν(cid:96) (33%)
Zqq (2.6%) T2 W W (59%) gg (21%) W tb (99%) W gg (56%) bb (28%) W ν(cid:96) (31%)
Z(cid:96)(cid:96) (28%)
W tb (14%)
Zνν (7.5%) H HW/Z, hA (62%) gg (25%) Htb (42%)
Htt (19%)
Att (17%)
W tb (6.8%)
Table 7 . Characteristic properties of the benchmark models defined in Table 6. The upper tableshows neutralino mass and composition ( ˜ B = Bino-like, ˜ B/ ˜ W = mixed Bino/Wino), identity andmass of the lightest squark and slepton, next-to lightest neutralino, ratio of 3-body to 2-body crosssection, ratio of 3-body cross sections involving Higgs bosons to that involving weak gauge bosons,and the ratio of 3-body to 2-body yields for various species X (integrated above E X > m χ / forD3, T1, T2 and above E X > m χ / for D1, D2, W, H). The lower table shows the dominant two-and 3-body annihilation channels (for leptonic channels we sum over all three generations, whilefor quarks we quote separately the final states involving top quarks). For this purpose, we define seven pMSSM-9 benchmark models in Table 6, and collectin Table 7 the phenomenological properties that are most relevant for our discussion. Inparticular, we include two threshold models, T1 and T2, with neutralino masses just belowthe W and t mass, respectively. Three of the benchmark models, D1 to D3, show a mass– 32 –pectrum where at least one of the sfermions is degenerate in mass with the neutralino,while model W describes a TeV-scale Bino DM candidate where the correct relic densityis obtained due to coannihilations with an almost degenerate Wino. Model H, finally, isa model example with a particularly large rate to 3-body final states containing a Higgsboson. Degenerate mass spectra.
Models with all sleptons degenerate in mass with the neu-tralino show a significant overall enhancement of the yield in leptonic channels, (cid:96) = e ± , ν µ , ν τ , caused by sharp spectral features at the kinematic end-point of thosespectra. In full analogy to the positron spectrum from VIB e + e − γ final states [14],the annihilation in these models is dominated by t -channel diagrams with (cid:96) appear-ing directly in the final state (and additionally in the decay of the W or Z bosonin the three-body final state); these diagrams lift the helicity suppression, and aremaximized when the corresponding sleptons are degenerate with the neutralino. Asan example of this type of models, we show in Fig. 11 the spectrum of benchmark D1.For leptonic final states (left panel) we can clearly see these sharp spectral features,leading to a yield enhancement as large as O (100) at high energies for all leptons. SU (2) corrections thus further enhance the e + feature associated to photon IB, in-dicated separately with dotted lines, which the AMS experiment is highly sensitiveto [10]. In addition, similar features appear also in neutrino final states, giving riseto a potential smoking-gun signature for annihilating Majorana DM at neutrino tele-scopes [82, 83, 85]. In the right panel of Fig. 11, we show instead the impact ofradiative corrections on the gamma-ray and antiproton spectra. The impact of pho-ton IB on the former is as expected large [13], while SU (2) corrections lead to muchless significant, though still noticeable, spectral distortions. The antiproton spectrumonly receives an overall enhancement directly related to the total σv enhancement.If, on the other hand, only squarks are degenerate, then the lepton spectrum doesnot show any significant distortion but just an overall enhancement proportional tothe one in σv , while the photon spectrum again hardens slightly. In the left panelof Fig. 12 we show for illustration the case of benchmark model D2, which features both degenerate squarks and sleptons and hence, as expected qualitatively very similarspectra compared to those of D1. The spectra of benchmark model D3 (right panel of Fig. 12) are again qualitativelyvery different and feature a significant enhancement only in the antiproton channel.The reason, as already discussed in Section 5.3, is that the 2-body annihilation in D3is largely dominated by τ τ final states. Therefore, the 3-body final states, specifically If only first and second generation sleptons were degenerate in mass with the neutralino, a correspondingfeature for ν τ (but not for e ± and ν µ ) would be absent. The (non-)detection of such features can thus inaddition be a powerful tool to robustly discriminate between such scenarios. One noticeable feature is the step-like behaviour of lepton yields from 2-body annihilation (dashedlines). This drop by more than an order of magnitude, at x (cid:38) . , is due the channels H ± W ∓ , hA and HZ which contribute about to the 2-body cross section: the W / Z decays constrain the resulting leptonenergy to E (cid:96) < E W/Z = m χ (1 − ( m H − m W/Z ) / (4 m χ )) ∼ . m χ (for m H = 2 . TeV as in D2). The 3-bodyyields (solid lines) smear out the abrupt step (for ν µ ) or lead to a pronounced bump at x ∼ (for ν τ ). – 33 – (cid:43) Ν Μ Ν Τ (cid:43) IB U (cid:72) (cid:76) total (cid:45) x (cid:61) E (cid:144) m Χ x d N (cid:144) d x D1 Γ p2to2 2to2 (cid:43) IB U (cid:72) (cid:76) total (cid:45) x (cid:61) E (cid:144) m Χ x d N (cid:144) d x D1 Figure 11 . Spectral energy distribution of a model where all sleptons are degenerate (D1 in Table6). In the left panel , we display leptonic final states e ± , ν µ , and ν τ (green, blue and cyan linerespectively) and, in the right panel photon (red) and antiproton (orange) spectra. Solid linesindicate the total (NWA-corrected 2-body plus 3-body) contribution, the dashed lines the 2-bodyresult. The dotted lines represent the contribution from 2-body final states plus that from photon bremsstrahlung alone. Shaded areas thus highlight the effect of including SU (2) corrections. Γ e (cid:43) p Ν Μ Ν Τ (cid:43) IB U (cid:72) (cid:76) total x (cid:61) E (cid:144) m Χ x d N (cid:144) d x D2 Γ e (cid:43) p Ν Μ Ν Τ (cid:43) IB U (cid:72) (cid:76) total (cid:45) (cid:45) (cid:45) x (cid:61) E (cid:144) m Χ x d N (cid:144) d x D3 Figure 12 . Same as Fig. 11, but now for models where both squarks and sleptons are degeneratein mass with the neutralino (model D2, left panel) or with a degenerate stau that is a mixture of ˜ τ L and ˜ τ R (model D3, right panel). hτ τ and W τ ν , lead to a very large enhancement of the ¯ p flux – even though this onlyleads to a relatively small ( ∼ ) correction in σv . Threshold models.
In Fig. 13, we display the spectra for two models just below the W and top quark threshold, corresponding to models T1 and T2 from Tab. 6, respectively.Models with m χ ∼ m W show a strong enhancement in all channels and at all energies.In particular, the 3-body contributions induce pronounced bump-like features for ν µ , ν τ , e ± spectra at x ∼ . , which result from the decay W ( ∗ ) → (cid:96)ν (cid:96) . As expected fromthe discussion in Section 3.5, these enhancements are most significant for m χ slightly This model features a very large µ -term ( ∼ . TeV) and tan( β ) ∼ , leading to a large stau mixingand hence only a mild helicity suppression for χχ → τ + τ − . Corrections to leptonic channels are thussmall in this specific case, despite an almost degenerate stau. We also note that hτ τ final states, via HiggsVIB, are enhanced with respect to gauge boson IB in this type of model. The reason is that the mixingcontribution ∝ gµv EW ˜ τ L ˜ τ R to the stau mass is directly linked to a large ˜ τ L ˜ τ R h coupling to the Higgs boson(since the VEV and the Higgs field appear in the combination v EW + h in the Lagrangian). – 34 – (cid:43) Ν Μ Ν Τ (cid:43) IB U (cid:72) (cid:76) total (cid:45) x (cid:61) E (cid:144) m Χ x d N (cid:144) d x T1 Γ p2to2 2to2 (cid:43) IB U (cid:72) (cid:76) total (cid:45) x (cid:61) E (cid:144) m Χ x d N (cid:144) d x T1 e (cid:43) Ν Μ Ν Τ (cid:43) IB U (cid:72) (cid:76) total (cid:45) x (cid:61) E (cid:144) m Χ x d N (cid:144) d x T2 Γ p2to2 2to2 (cid:43) IB U (cid:72) (cid:76) total (cid:45) x (cid:61) E (cid:144) m Χ x d N (cid:144) d x T2 Figure 13 . Same as figure 11, but for a W threshold model (T1, top) and a top-quark thresholdmodel (T2, bottom). The features in the shape of the lepton and gamma ray spectra are due to aninterplay of various effects as discussed in detail in the text. below m W . Photon and antiproton spectra, on the other hand, are somewhat softerthan the 2-body spectra, but can be greatly enhanced at all energies.We also find enhancements in all channels when m χ ∼ m t , and observe a peculiardouble structure in the spectrum of leptonic channels for T2: a bump-like feature at x ∼ . and a sharp spectral feature at higher energies. The first is directly relatedto the dominant off-shell top decay close to the threshold, χχ → t ¯ t ( ∗ ) → W tb . Theline-like feature close to x = 1 , on the other hand, arises from leptonic decays of thetransversely polarized W -bosons in the process χχ → W ¯ W → W (cid:96)ν (cid:96) . For the antiproton and gamma spectra, finally, the inclusion of the 3-body resultcauses relatively large deviations as the 2-body yields can both increase and decrease,depending on the energy. Special cases.
In Fig. 14, we show two interesting cases that do not fall in either of Note that the NWA-subtracted cross section for
W (cid:96)ν (cid:96) is much smaller than for the kinematically acces-sible χχ → W + W − . The enhancement at x ∼ originates thus exclusively from our 3-body computationtaking the (transverse) W polarization into account (see Appendix C, specifically Eq. (C.37)). This typeof correction can occur whenever m χ (cid:29) m W ( m Z ) and χχ → W + W − , ZZ proceeds with a significant rate. The peculiar shape of the gamma-ray spectrum, in particular, can be explained as follows: for lowenergies x (cid:46) . there is a strong enhancement from W tb final states, while for intermediated energies x ∼ . − . , the spectrum is slightly suppressed compared to the 2-body case. At high energies x (cid:38) . , W W γ and ¯ (cid:96)(cid:96)γ , i.e. photon IB final states dominate; the sharp drop around x ∼ . is due to the kinematicendpoint of the W W γ contribution, while photons from ¯ (cid:96)(cid:96)γ dominate for x > . . – 35 – e (cid:43) p Ν Μ Ν Τ (cid:43) IB U (cid:72) (cid:76) total (cid:45) x (cid:61) E (cid:144) m Χ x d N (cid:144) d x W Γ e (cid:43) p Ν Μ Ν Τ (cid:43) IB U (cid:72) (cid:76) total (cid:45) x (cid:61) E (cid:144) m Χ x d N (cid:144) d x H Figure 14 . Same as figure 11, but for a Bino-like neutralino with almost degenerate Wino (W,left) and a model with large Hf ¯ f contribution (H, right). The features in the shape of the leptonand gamma ray spectra are due to an interplay of various effects as discussed in detail in the text. Left panel : Bino-like neutralino with almost degenerate Wino (benchmark model W). Final statechannels: photons (red), antiprotons (orange), positrons (green), ν µ (blue), and ν τ (cyan). Solidlines indicate the total (2-body and 3-body) contribution, the dashed lines the 2-body process. Right panel : Large Hf ¯ f contribution (benchmark model H). the categories above. In the left panel, we present benchmark model W, a Bino-like neutralino degenerate with the Wino. The (small) 2-body annihilation rate isdominated by gg final states, followed by ¯ f f . The 3-body process thus lifts thehelicity suppression of the latter and can be important even if the sfermions arenot highly degenerate in mass with the neutralino. Because the contribution to theneutrino and positron spectra still come dominantly from W ν(cid:96) final states, they showsharp spectral features like in models with even more degenerate sleptons. The rightpanel of Fig. 14, instead, corresponds to a model with a large ( ∼ ) contribution tothe cross section from channels that involve the MSSM Higgs bosons and top quarks(benchmark model H). The neutralino mass is rather heavy ( ∼ . TeV) such thateven t ¯ t final states suffer from a certain amount of helicity suppression. Due to thelarge top Yukawa coupling, the suppression is lifted preferably via Higgsstrahlung. Forthis model, leptons are dominantly produced indirectly, and correspondingly leptonspectra are enhanced broadly at all energies. The small additional spike at very highenergies results from the W/Z decay from W ¯ F f ( ) and Z ¯ f f ( ) final states.In Fig. 15, finally, we show for a subset of our benchmark models the ratios of 3-body to2-body yields, illustrating some of the features discussed above on a model-by-model basisfrom a slightly different angle. We note in particular the strong enhancement of high-energylepton spectra for model H, which is explained – similar to the situation for model D2 – bya sharp drop in the 2-body yield from W ± H ∓ and ZH due to the maximal lepton energyfrom W/Z decays that is kinematically possible.A widely used phenomenological approach to take into account electroweak correctionsto DM annihilation spectra, often referred to as ‘model-independent’ in the literature,is based on splitting functions inspired by a parton picture [16, 17]. These effectivelyresult from assuming point-like interactions being responsible for the 2-body annihilation– 36 – (cid:43) Ν Μ Ν Τ IB U (cid:72) (cid:76) SU (cid:72) (cid:76) (cid:43) IB U (cid:72) (cid:76) PPPC4DMID x (cid:61) E (cid:144) m Χ x d N (cid:144) d x D1 Γ pIB U (cid:72) (cid:76) SU (cid:72) (cid:76) (cid:43) IB U (cid:72) (cid:76) PPPC4DMID x (cid:61) E (cid:144) m Χ x d N (cid:144) d x D3 Γ pIB U (cid:72) (cid:76) SU (cid:72) (cid:76) (cid:43) IB U (cid:72) (cid:76) PPPC4DMID x (cid:61) E (cid:144) m Χ x d N (cid:144) d x T2 e (cid:43) Ν Μ Ν Τ IB U (cid:72) (cid:76) SU (cid:72) (cid:76) (cid:43) IB U (cid:72) (cid:76) PPPC4DMID x (cid:61) E (cid:144) m Χ x d N (cid:144) d x W e (cid:43) Ν Μ Ν Τ IB U (cid:72) (cid:76) SU (cid:72) (cid:76) (cid:43) IB U (cid:72) (cid:76) PPPC4DMID x (cid:61) E (cid:144) m Χ x d N (cid:144) d x H Γ pIB U (cid:72) (cid:76) SU (cid:72) (cid:76) (cid:43) IB U (cid:72) (cid:76) PPPC4DMID x (cid:61) E (cid:144) m Χ x d N (cid:144) d x H Figure 15 . Ratio of 3-body to 2-body yields, for some of the benchmark models from Tab. 6.Solid lines show the full ratios, while dashed lines indicate the contribution from photon IB alone.Dotted lines show the result from the method implemented in the ‘Poor Particle Physicist Cookbookfor Dark Matter Indirect Detection’ (PPPC4DMID) [17]; note that the relatively small resultingcorrections are not restricted to these benchmark models but generic for neutralino annihilation.We cut the ¯ p spectrum at x ∼ . in order to avoid numerical artefacts from very small, and hencepoorly sampled, 2-body yields provided by DarkSUSY . See text for more details. channels, such that 3-body final states are dominated by gauge (or Higgs) bosons thatare soft or collinear with the final state particle they are radiated from. In Fig. 15, wetherefore also indicate the ratios that result from this approach. As one can see, theresulting changes in the yield differ at times drastically. In fact, for the pMSSM modelsstudied here, we find much more generally that the full annihilation spectra from final Since only the SM Higgs boson is implemented in [17], we have approximated the missing yields byreplacing the heavy MSSM Higgs bosons with Goldstone bosons, i.e. longitudinal gauge boson components.
For the sake of Fig. 15 we thus implemented, e.g., hA final states as 50% hh and 50% Z L Z L final states. – 37 – tates containing fermions deviate substantially from the model-independent approximationwhenever electroweak corrections induce even O (10%) changes to the 2-body rates. Also inthe case of TeV DM models, the difference between 2-body and corrected yields is muchlarger than what one would expect when adopting the PPPC [17] implementation.For neutralino annihilation, this is quite straight-forward to understand, and for thisreason we also expect these conclusions to hold in even more generic MSSM models. Inthe ‘model-independent’ prescription, in particular, none of the enhancement mechanismsdescribed in detail in Section 3 is captured. Final states containing leptons are thus nec-essarily still suppressed by m (cid:96) /m χ , while quark and gluon final states are hardly affectedby electroweak corrections due to soft or collinear radiation [16] given the strong dynamicsleading to essentially immediate fragmentation. Furthermore, MSSM specific final states(heavy Higgs bosons) are not included, and even for the final states that are included theenergy distribution of the final state particles can differ substantially when taking into ac-count the full matrix element or just assuming a point-like interaction, respectively. Thelatter is for example visible in the antiproton spectrum of model D3: as the 2-body anni-hilation is dominated by τ lepton final states, the main contribution must come from thefinal state boson; in the PPPC case the softer antiproton spectrum can then be traced backto the fact that the emitted boson has a much softer spectrum. In this article, we have studied in detail the annihilation of Majorana DM particles intoa pair of fermions and an electroweak gauge or Higgs boson in the final state. We haverevisited the arguments why the annihilation to fermionic 2-body final states is helicitysuppressed, and pointed out that this can be traced back to a combination of two funda-mentally distinct effects, dubbed Yukawa- and isospin suppression, which can independentlybe lifted if the final state boson carries isospin. Furthermore, we have consistently gener-alized a standard way of avoiding ‘double-counting’ of contributions from two- and 3-bodyfinal states, which consists in subtracting the latter in the narrow-width approximation, todifferential cross sections and the yield of stable particles relevant for indirect DM searches.The latter constitutes one of our main results, which we believe will prove useful also inother contexts.As a concrete application, we have performed the first full analytical calculation ofall differential cross sections for the internal bremsstrahlung of electroweak gauge bosons,as well as any MSSM Higgs boson, for fermion final states from neutralino annihilation.We have performed a detailed analysis of these results in light of our general discussion,recovering specific examples pointed out previously in the literature, and extending them,but also pointing out qualitatively new processes. In order to estimate the size of thecorrections reported here, we have performed dedicated scans over the parameter space ofvarious MSSM realizations.We find that both lifting of the Yukawa and lifting of the isospin suppression cansignificantly increase the total neutralino annihilation rate. Even more importantly, how-ever, the resulting spectra of positrons, neutrinos, antiprotons and gamma rays can differ– 38 –ubstantially from those obtained for 2-body final states – even when including the ‘model-independent’ electroweak corrections implemented in PPPC [17]. We stress that this isa generic result which is not restricted to specific cases but holds whenever electroweakcorrections to fermionic channels are (at least) comparable to the 2-body results, e.g. forTeV DM models. Given that the supersymmetric neutralino still is the prototype WIMPDM candidate, our results thus underline the importance of performing full computationsconsistent with the model that is being studied. In other words, such radiative correctionsare intrinsically highly model-dependent, and even the often adopted ‘model-independent’approach of Ref. [17] can be argued to simply rest on one rather specific model realization(in the sense that the underlying assumptions essentially describe a point like interaction,which is a good approximation under roughly the same conditions under which an effectiveoperator analysis is valid to leading order in perturbation theory. )To conclude, we have shown that the way electroweak corrections to DM annihilationare commonly estimated can lead to rather misleading results for a given DM model. Theconsistently computed spectra of stable particles from DM annihilation can be much largeror offer striking spectral features, either of which may significantly help to indirectly detectDM in forthcoming experiments. We stress that this holds for all yields relevant for indirectDM detection, i.e. both gamma rays, charged cosmic rays ( ¯ p and e + ) and neutrinos. Theroutines needed to compute all relevant rates and particle yields for neutralino annihilationin the MSSM will be included in the shortly upcoming public release 6.0 of the DarkSUSY package [40].
Acknowledgments
We thank Maria Eugenia Cabrera Catalan, Marco Cirelli, Feng Luo, Are Raklev, RobertoRuiz de Austri Bazan and Filippo Sala for useful discussions. Note that while some of the effects of helicity suppression lifting could in principle be described viaadditional higher-dimensional effective operators in specific regions of parameter space [86], it turns outthat within the MSSM the part of parameter space in which this is possible typically does not coincide withthe one where the corrections are most relevant. – 39 –
Neutralino annihilation amplitudes
In this Appendix, we review our analytical approach of calculating the matrix elementsby means of an expansion in helicity amplitudes (A.1). For illustration of our full results,we then consider a number of phenomenologically interesting limiting cases concerning thecomposition of the lightest neutralino (A.2).
A.1 Expansion of Amplitudes in the Helicity Basis
For the analytical calculation of amplitudes we closely follow the procedure of Ref. [18],presented in detail in chapter 4 and corresponding appendices of Ref. [87]. We thus modifythe generic MSSM model file shipped with the
FeynArts mathematica package [88] such as toagree with the conventions adopted in
DarkSUSY [40, 45]. We then use
FeynArts to generateall possible Feynman diagrams for neutralino annihilation into 3-body final states containinga fermion, an anti-fermion and a Boson. In the next step, given that we want to restrictourselves to the v → limit, we project out the singlet state ( J P = 0 − ) of the annihilatingneutralino pair with total momentum p by replacing the two external Majorana spinorsin the amplitude with P S ≡ γ √ ( m χ − /p/ . Finally, we expand the amplitude for eachdiagram in terms of helicity amplitudes, applying a method used originally for neutralinoannihilation to 2-body final states [89] and extended to 3-body final states in [18, 87].Let us review those final steps in a bit more detail. By applying the P S projector, inparticular, we can reduce any of the matrix elements considered here to the generic form M ∝ ¯ u r ( k )Γ rs v s ( k ) (cid:15) ∗ µ ( k ) , (A.1)where only the final state spinors appear and Γ is a 4 × Feyncalc [90] todecompose Γ into the standard basis of matrices where the corresponding Dirac bilinearsare real and have definite transformation properties under the Lorentz group (i.e. scalar,vector, tensor, pseudo-vector and pseudo-scalar, respectively). In order to assign helicities,we work in the back-to-back frame of the outgoing fermion-antifermion pair, which we defineby k = − k . For states of definite helicity h = ± / , this implies that we can use [91] χ h (ˆ k , ) = χ − h ( − ˆ k , ) (A.2)for the two-component spinors that appear in the explicit Dirac representations of both u and v . Choosing the fermion momentum k to be aligned with the z -axis, we thus obtain u + = η + k η − k , u − = η + k − η − k , v + = η − k − η + k , v − = η − k η + k , (A.3)where we have introduced η ± k , ≡ (cid:0) k , ± m , (cid:1) / . In this frame, we can furthermore choosethe momentum of the final state boson, k , to lie in the y − z plane, spanning an angle θ with the z -axis. For the case of a massive vector boson, the 3 possible polarization states (cid:15) λ of definite helicity are thus given by (cid:15) ± = 1 √ , ∓ , − i cos θ, i sin θ ) , (cid:15) = 1 m (cid:0) | k | , , k sin θ, k cos θ (cid:1) . (A.4)– 40 – ¯ u Γ v ) h h = (0 , h = (1 , − h = (1 , h = (1 , uv p + ¯ uγ µ v E + e µ + p − e µu − E − e µ E + e µ − ¯ uσ µν v ip + ( e µ − e ν + − e µ + e ν − ) ip − ( e µ + e ν − e µ e ν + ) iE + ( e µ e νu − e µu e ν ) ip − ( e µ − e ν − e µ e ν − )+ iE − ( e µu e ν + − e µ + e νu ) + iE − ( e µu e ν − − e µ − e νu )¯ uγ µ γ v p − e µ − E − e µu − p + e µ + p + e µ − ¯ uγ v − E + Table 8 . Decomposition of basis Dirac bilinears into helicity eigenstates of the two final statefermions. For a definition of the quantities appearing here, see Eqs. (A.9–A.11).
Singlet and triplet spin states of the two-fermion system can now be constructed fromthe individual helicity states as in Eqs. (2.4, 2.5), i.e. we can decompose each bilinear as (¯ u Γ v ) (0 , = (¯ u + Γ v + − ¯ u − Γ v − ) / √ , (A.5) (¯ u Γ v ) (1 , − = ¯ u − Γ v + , (A.6) (¯ u Γ v ) (1 , = (¯ u + Γ v + + ¯ u − Γ v − ) / √ , (A.7) (¯ u Γ v ) (1 , +1) = ¯ u + Γ v − . (A.8)In Table 8, we show the result of this decomposition for each of the 16 basis Dirac bilinears,where for ease of notation we have introduced the following kinematic quantities (note thedifferent normalization convention with respect to [87]): E ± ≡ √ (cid:16) η + k η + k ± η − k η − k (cid:17) , (A.9) p ± ≡ √ (cid:16) η + k η − k ± η − k η + k (cid:17) . (A.10)Four-vectors and tensors, furthermore, are more conveniently expressed in the helicity basis , (cid:8) ˜ e ( µ ) (cid:9) = { ˜ e u , ˜ e + , ˜ e − , ˜ e } ≡ (cid:26) (1 , , , , (cid:18) , − √ , − i √ , (cid:19) , (cid:18) , √ , − i √ , (cid:19) , (0 , , , (cid:27) , (A.11)which is an orthonormal basis choice just like the canonical coordinate basis (cid:8) e ( µ ) (cid:9) with (cid:0) e ( µ ) (cid:1) ν = δ νµ . This implies, e.g., that the components of a four-vector V for these basischoices are related by V µ = A µν ˜ V ν , where A µν ≡ e ( µ ) · ˜ e ( ν ) .With the above decompositions of fermion and vector boson polarizations even thefull analytic expressions for the amplitudes turn out to be relatively easily manageable.We evaluate the amplitude for every helicity configuration, for each diagram separately,and simplify it further by explicitly contracting the remaining polarisation vectors (whenapplicable), basis vectors and four-momenta. We then sum over all diagrams to obtain thetotal helicity amplitudes M ( h,λ ) , where h is the helicity of the fermion-antifermion pair inthe back-to-back system and λ the polarisation state of the emitted vector boson. Finally,we obtain the total amplitude squared by averaging over initial ( r, s ) spins and summingover final ( r (cid:48) , s (cid:48) ) degrees of freedom: |M| ≡ (cid:88) r,s,r (cid:48) ,s (cid:48) ,λ (cid:12)(cid:12)(cid:12) M χχ → ¯ F fX (cid:12)(cid:12)(cid:12) ≡ (cid:88) h,λ (cid:12)(cid:12)(cid:12) M ( h,λ ) χχ → ¯ F fX (cid:12)(cid:12)(cid:12) , (A.12)– 41 –here X is either a vector boson ( W/Z ) or a scalar Higgs (in which case no polarisation ispresent). For convenience, we then transform back to the CMS frame. This allows us tocompute the total 3-body cross section by integrating over the phase space, d ( σv → ) dE dE = 116 m χ π ) |M| , (A.13)where E and E are the CMS energies of any two final state particles. For details concerningthe numerical implementation, we refer to Appendix B. A.2 Results for expanded amplitudes
Let us consider our analytical results in the limit of heavy neutralino masses, which amountsto taking the ratio of the electroweak VEV and the neutralino mass, δ v ≡ v EW /m χ , andexpanding the full results for the amplitude around δ v = 0 . Besides allowing for compactanalytic expressions, this limit is particularly useful for deriving the scaling behaviour ofthe amplitudes not only with δ v ≡ v EW /m χ , but also with the Yukawa couplings y f andthe gauge coupling g . We express the results of this procedure in terms of the ratio of the3-body to the corresponding 2-body amplitudes, the latter of which are suppressed by afactor m f = y f v EW . If the 3-body process lifts Yukawa suppression, the amplitude ratiowill thus scale as ∝ /y f , and if it lifts isospin suppression it scales as ∝ /δ v .We therefore introduce the dimensionless ratio of the helicity amplitudes to the spin-summed/-averaged matrix element for the corresponding 2-body process, R λ,h ≡ m χ M λ,h (cid:113) |M| → . (A.14)For the total amplitude squared, the individual helicity contributions have to be summedover, c.f. Eq. (A.12). For the sake of our discussion here, we organize this sum in a slightlydifferent way and split it into contributions from final state fermions with equal or oppositechirality (rather than singlet and triplet states), as well as longitudinal and transversepolarizations (like before). We note that, since the external fermions are massless in the limitthat we are considering here, helicity coincides with chirality and hence becomes Lorentzinvariant. In the back-to-back frame introduced in Appendix A.1, the helicity components h = + − and h = − + then correspond to chiralities f R ¯ f R or f L ¯ f L , respectively, andcoincide with the spin-triplet components (1 , +1) and (1 , − discussed there. The helicitycombinations h = ++ and h = −− of the fermion pair, on the other hand, correspond to f R ¯ f L and f L ¯ f R states, respectively. The sum over these latter two contributions is thenequivalent to the sum over the singlet and the remaining triplet states, (0 , and (1 , inthe notation from above. Altogether, this yields the decomposition (cid:88) λ,h (cid:12)(cid:12)(cid:12) R λ,h (cid:12)(cid:12)(cid:12) = (cid:16) R LLR/RL (cid:17) + (cid:16) R LLL/RR (cid:17) + (cid:16) R TLR/RL (cid:17) + (cid:16) R TLL/RR (cid:17) , (A.15)– 42 –here (cid:16) R LLR/RL (cid:17) ≡ (cid:88) λ =0 h =++ , −− (cid:12)(cid:12)(cid:12) R λ,h (cid:12)(cid:12)(cid:12) , (cid:16) R LLL/RR (cid:17) ≡ (cid:88) λ =0 h =+ − , − + (cid:12)(cid:12)(cid:12) R λ,h (cid:12)(cid:12)(cid:12) , (cid:16) R TLR/RL (cid:17) ≡ (cid:88) λ = ± h =++ , −− (cid:12)(cid:12)(cid:12) R λ,h (cid:12)(cid:12)(cid:12) , (cid:16) R TLL/RR (cid:17) ≡ (cid:88) λ = ± h =+ − , − + (cid:12)(cid:12)(cid:12) R λ,h (cid:12)(cid:12)(cid:12) . (A.16)For Higgs final states, the summation over polarizations is absent, and we define the cor-responding ratios R LR/RL and R LL/RR analogously, corresponding to h ¯ f R f L + h ¯ f L f R and h ¯ f L f L + h ¯ f R f R final states, respectively.We then start from our full result for the helicity amplitudes, using the explicit repre-sentations of the generic couplings and mass matrices that appear there, and expand themup to O ( δ v ) . Note that the limit δ v → implies in particular that we expand in the fermionmass m f ∝ y f v EW and in gauge boson masses m W/Z ∝ gv EW . In order to simplify the re-sulting analytic expressions, we set all sfermion masses equal to the neutralino mass, notingthat larger sfermion masses would suppress t/u -channel rates relatively strongly because σv t − channel2 → ∝ m − f ( m − f ) for Bino- (Higgsino/Wino-)like neutralinos, as discussed previ-ously [31, 33, 36, 37, 92]. Furthermore, we use the notation B ¯ F f where for neutral bosons( B = Z, h ) the final state fermion types are identical, ¯ F = ¯ f , while for charged bosonswe adopt in the following the convention that f denotes the up-type fermion (e.g. the topquark in χχ → W ¯ bt ). In these cases, we keep for simplicity only the dependence on theYukawa coupling of the up-type fermion, and set the other one to zero.We furthermore consider six distinct scenarios describing the dominant neutralino com-position, which result from different assumptions about the involved mass hierarchies andwhich are of particular phenomenological interest: • Higgsino DM, with small Bino admixture ( µ (cid:28) M , M → ∞ ) • Higgsino DM, with small Wino admixture ( µ (cid:28) M , M → ∞ ) • Bino DM, with small Wino admixture ( M (cid:28) M , µ → ∞ ) • Bino DM, with small Higgsino admixture ( M (cid:28) µ, M → ∞ ) • Wino DM, with small Bino admixture ( M (cid:28) M , µ → ∞ ) • Wino DM, with small Higgsino admixture ( M (cid:28) µ, M → ∞ )From the Bino-, Wino- and Higgsino mass parameters M , M and µ , we define the dimen-sionless mass suppression factors δ M ≡ m χ /M , δ M ≡ m χ /M and δ µ ≡ m χ /µ . For allsix scenarios listed above, we expand the amplitude ratios to leading order in these masssuppression factors. Effectively, the neutralino mixing between either Bino or Wino andHiggsino then becomes a perturbative ‘mass insertion’ ∝ gv EW represented by the respec-tive off-diagonal entries in the mass matrix of Eq. (3.2). Furthermore, for definiteness, wealso expand to linear order in δ A ≡ m χ /M A , i.e. we work in the decoupling limit wherethe heavy Higgs states are much heavier than the neutralino or SM-like Higgs boson. We– 43 –ote that it is straightforward to generalize these results, and our numerical results anywayinclude all MSSM Higgs bosons and are valid for arbitrary mass hierarchies.To lowest order in the expansion parameters defined above, isospin and fermion chiralityhave to be conserved in all interaction vertices (assuming that the mass splitting between M , M and | µ | is large compared to gv EW ). One of the implications, as it turns out,is that the gauge-invariant subset of t -channel diagrams discussed in Section 3.1 can befurther split into two separate gauge-invariant sets. The first, which we will denote by ( I ) ,does not contain any neutralino mixing insertion ∝ gv EW , and would hence contribute evenin the limit of a pure neutralino state. The set of diagrams that contain at least one suchinsertion (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 different tables correspond to thethree types of final states ( W ¯ F f , Z ¯ f f , and h ¯ f f , respectively). Each table contains theresults for all six mass hierarchy scenarios specified above, broken down to contributionsfrom each set of gauge-invariant diagrams. For the sake of the presentation, we keeponly contributions that lift the isospin- or Yukawa suppression of the corresponding 2-bodyprocess (or both). In particular, as apparent from Table 2, the ratio R TLR/RL cannot liftany 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 ratiosthat actually do lift at least one of the suppression factors. For the remaining entries,a ‘ ’ indicates 3-body amplitudes that vanish to the order we consider, while for entriescontaining a ‘ − ’ both 2- and 3-body amplitudes vanish . The 2-body amplitudes, finally,are for convenience summarized in Table 12. We checked explicitly (up to O ( δ v ) ) that the Ward identities, Eqs. (2.7, 2.8) are satisfied for each setseparately. Note that we use Breit-Wigner widths in the amplitudes, and while they break gauge invarianceat O ( δ v ) , they do not contribute to the amplitudes as δ v → While in this case the ratio would be formally ill-defined, we only identified one example where the2-body amplitude vanishes while the 3-body amplitude does not, marked with a ( ∗ ) . We note that therelevant process, χχ → W W ∗ → W ¯ F f for a Wino-like neutralino, is phenomenologically not importantbecause for m χ > M W it is largely captured by annihilation into W W , while Wino-like neutralinos with m χ < M W are practically excluded. – 44 – χ → W − ¯ F f M a ss H i e r a r c h y s - c h a nn e l t + u - c h a nn e l H ± m e d . Z / h / W m e d i a t o r ( I )( II ) W L ¯ F L f R W L ¯ F L f R W T ¯ F L f L W L ¯ F L f R W L ¯ F L f L W T ¯ F L f L W L ¯ F L f R W L ¯ F R f L W L ¯ F R f L W L ¯ F R f L W L ¯ F R f L µ (cid:28) M , M →∞ √ δ v ( + s β ) g C ( x , x ) √ y f δ v δ M c β ( x + x − ) √ δ v x g C ( x , x ) √ y f δ v x √ δ v ( + t β ) x µ (cid:28) M , M →∞ g C ( x , x ) √ y f δ v δ M t w c β ( x + x − ) √ δ v x g C ( x , x ) √ y f δ v x M (cid:28) µ , M →∞ √ δ v √ δ v A ( x , x ) √ δ v g c β K √ y f x x g D ( x , x ) K √ y f δ v B ( x , x ) √ δ v M (cid:28) M , µ →∞ -- A ( x , x ) √ δ v g c β K √ y f x x g D ( x , x ) K √ y f δ v - M (cid:28) µ , M →∞ √ δ v √ δ v g √ C ( x , x ) y f δ v δ µ c β ( x + x − ) √ δ v x g c β √ y f x x g D ( x , x ) √ y f δ v √ δ v x M (cid:28) M , µ →∞ -- g √ C ( x , x ) y f δ v ( x + x − ) ( ∗ ) √ δ v x g c β √ y f x x g D ( x , x ) √ y f δ v - L i f t i n go f i s o s p i n i s o s p i n Y u k a w aa nd i s o s p i n i s o s p i n Y u k a w a Y u k a nd i s o i s o s p i n T a b l e . A m p li t ud e r a t i o s f o r - b o d y p r o ce ss e s χχ → W − T ( L ) ¯ F X f Y ( w i t h X Y = LL / RR ; L R / R L )t o2 - b o d y p r o ce ss e s χχ → ¯ ff ,i n t h e li m i t δ v = v E W / m χ → a nd f o r d i ff e r e n t S U S Y m a ss h i e r a r c h i e s . C o m p a r e d t o t h e d e fin i t i o n s i n t r o du ce d i n E q . ( A . ) , w e s h o w h e r e R T LL / RR / √ x + x − , R LL R / R L / √ x + x − a nd R LLL / RR / (cid:112) ( − x )( − x ) , r e s p ec t i v e l y . W ee x p r e ss e v e r y t h i n g i nd i m e n s i o n l e ss q u a n t i t i e s , w h e r e x ( ) = E ( ) / m χ a r e t h e C M S e n e r g i e s o f t h e fin a l s t a t e f e r m i o n s a nd x X ≡ m χ / M X m a sss upp r e ss i o n f a c t o r s . W e a l s o d e fin e y f ≡ m f / v E W , u s i n g t hu s a c o n v e n t i o n w i t h o u t f a c t o r s o f t a n β , a ndn e g l ec t y ¯ F ≡ m F / v E W f o r F = d , s , b , (cid:96) . F u r t h e r m o r e , t w ≡ t a n ( θ W ) , s w ≡ s i n ( θ W ) , c w ≡ c o s ( θ w ) , s β ≡ s i n ( β ) , c β ≡ c o s ( β ) , s β ≡ s i n ( β ) a nd c β ≡ c o s ( β ) . q f i s t h e f e r m i o n c h a r g e , a nd Y f L ≡ q f − t f , Y f R ≡ q f a r e t h e h y p e r c h a r g e s ( w i t h t f =+ / f o r f = u , c , t , ν ) . F o r m o r ec o m p a c t e x p r e ss i o n s , w e a l s o i n t r o du ce d K ≡ Y f L / ( Y f L + Y f R ) , A ≡ K / x + ( − K ) / x , B ≡ Y f R / x − Y f L / x , C ≡ (cid:112) ( x − ) + ( x − ) / ( x + x − ) , D ≡ (cid:112) ( x − ) + ( x − ) / ( x x ) . A ‘ ’i nd i c a t e s - b o d y a m p li t ud e s t h a t v a n i s h t o t h e o r d e r w e c o n s i d e r , w h il e f o r e n t r i e s c o n t a i n i n ga ‘ − ’ b o t h - a nd - b o d y a m p li t ud e s v a n i s h . F o r t h e fi e l d m a r k e db y ( ∗ ) , fin a ll y , w e n o r m a li ze b y t h e a m p li t ud e f o r t h e t - c h a nn e l p r o ce ss a s t h ec o rr e s p o nd i n g2 - b o d y a m p li t ud e v a n i s h e s i n t h i s c a s e . – 45 – χ → Z ¯ ff M a ss H i e r a r c h y s - c h a nn e l t + u - c h a nn e l H m e d i a t o r Z / h m e d i a t o r ( I )( II ) Z L ¯ f L f R Z L ¯ f L f R Z T ¯ f L f L Z L ¯ f L f R Z T ¯ f L f L Z L ¯ f L f R Z L ¯ f R f L Z L ¯ f R f L Z T ¯ f R f R Z L ¯ f R f L Z T ¯ f R f R Z L ¯ f R f L µ (cid:28) M , M →∞ − s β √ δ v ( + s β ) √ δ v ( x + x − ) g C ( x , x ) J y f δ v δ M c w c β ( x + x − )( x + x ) √ δ v x x g F ( x , x ) δ v y f c w ( t β − ) E ( x , x ) δ v ( t β + ) µ (cid:28) M , M →∞ − s β √ δ v ( + s β ) √ δ v ( x + x − ) g C ( x , x ) J y f δ v δ M s w t w c β ( x + x − )( x + x ) √ δ v x x g F ( x , x ) δ v y f c w ( t β − ) E ( x , x ) δ v t w ( t β + ) M (cid:28) µ , M →∞ √ δ v √ δ v G ( x , x ) δ v g D ( x , x ) H y f δ v ˜ E ( x , x ) δ v M (cid:28) M , µ →∞ -- G ( x , x ) δ v g D ( x , x ) H y f δ v - M (cid:28) µ , M →∞ √ δ v √ δ v E ( x , x ) δ v g D ( x , x ) Z f L y f δ v c w E ( x , x ) δ v M (cid:28) M , µ →∞ -- E ( x , x ) δ v g D ( x , x ) Z f L y f δ v c w - L i f t i n go f i s o s p i n i s o s p i n Y u k a w aa nd i s o s p i n i s o s p i n Y u k a w aa nd i s o s p i n i s o s p i n T a b l e . A s T a b l e , bu t f o r Z ¯ ff fin a l s t a t e . N o t e t h a t i n t h e t h i r d c o l u m n t h e l o n g i t ud i n a l c o n t r i bu t i o n ( l e f t) o r i g i n a t e s f r o m Z / h - e x c h a n g e F S R a nd V I B , w h il e t h e t r a n s v e r s e ( r i g h t) o r i g i n a t e s f r o m I S R c o n t r i bu t i o n s t o Z e x c h a n g e (t h i s i m p li e s t h a tt h e s e t w o c l a ss e s o f d i ag r a m s a r e ga u g e - i nd e p e nd e n t s e p a r a t e l y , w h i c h w ec h ec k e d e x p li c i t l y ) . N o t a t i o n s a r e d e fin e d a s i n t a b l e , a nd y f = y ¯ f ≡ m f / v E W f o r f = q , ν , (cid:96) . I n a dd i t i o n , w e d e fin e t h e Z - c o up li n g s Z f L = q f s w − t f , Z f R = q f s w , a nd E ≡ (cid:112) x + x / ( x x ) , ˜ E ≡ (cid:112) ( Y f L + Y f R ) ( x − x ) + ( x + x ) / ( √ x x ) , F ≡ C ( x , x ) × (cid:112) ( Z f L ( x + x ) − Z f R ) + ( Z f R ( x + x ) − Z f L ) / ( x x ) , G ≡ (cid:114) ( Y f L + Y f R ) ( Y f L + Y f R ) ( x − x ) + ( x + x ) / ( √ x x ) , H ≡ (cid:113) Y f R Z f R + Y f L Z f L / ( Y f L + Y f R ) , J ≡ (cid:113) ( Z f L + Z f R ) / c w . – 46 – χ → h ¯ ff M a ss H i e r a r c h y s - c h a nn e l t + u - c h a nn e l A m e d i a t o r Z m e d i a t o r ( I )( II ) h ¯ f L f R h ¯ f L f R h ¯ f L f L h ¯ f L f R h ¯ f L f L h ¯ f L f R h ¯ f L f L h ¯ f R f L h ¯ f R f L h ¯ f R f R h ¯ f R f L h ¯ f R f R h ¯ f R f L h ¯ f R f R µ (cid:28) M , M →∞ √ δ v √ δ v ( x + x − ) g J y f ( x + x − )( x + x − ) x + x √ δ v x x g c β J y f x x E ( x , x ) δ v µ (cid:28) M , M →∞ √ δ v √ δ v ( x + x − ) g ( x + x ) J y f ( x + x − )( x + x − ) x + x √ δ v x x g c β J y f x x ˜ E ( x , x ) δ v M (cid:28) µ , M →∞ √ δ v √ δ v G ( x , x ) δ v g c β H y f c w x x ˜ E ( x , x ) δ v g δ µ t w t β ( x + x ) L √ y f x x M (cid:28) M , µ →∞ -- G ( x , x ) δ v g c β H y f c w x x - M (cid:28) µ , M →∞ √ δ v √ δ v E ( x , x ) δ v g c β Z f L y f c w x x E ( x , x ) δ v g δ µ t β ( x + x ) y f x x M (cid:28) M , µ →∞ -- E ( x , x ) δ v g c β Z f L y f c w x x - L i f t i n go f i s o s p i n i s o s p i n Y u k a w a i s o s p i n Y u k a w a i s o s p i n Y u k a w a T a b l e . A s T a b l e , bu t f o r h ¯ ff fin a l s t a t e f o r f = u , c , t , ν . F o r f = d , s , b , (cid:96) o n e n ee d s t o r e p l a ce t β → / t β i n t h e l a s t c o l u m n . F o r t h e s - c h a nn e l d i ag r a m s w i t h e l ec t r o w e a k - s c a l e m e d i a t o r , o n l y t h o s e w i t h a Z b o s o n c o n t r i bu t e t o t h e Y u k a w ao r i s o s p i n li f t i n g . T h e a m p li t ud e r a t i o s a r e n o r m a li ze d a s R LL / RR / (cid:112) ( x − )( x − ) a nd R L R / R L / √ x + x − , r e s p ec t i v e l y . N o t a t i o n s a r e d e fin e d a s i n t h e p r e v i o u s t a b l e s ,i n a dd i t i o n L ≡ ( Y f L + Y f R ) + ( Y f L + Y f R ) . – 47 – χ → ¯ ff M a ss H i e r a r c h y s - c h a nn e l t + u - c h a nn e l A m e d i a t o r Z m e d i a t o r ( I )( II ) µ (cid:28) M , M →∞ g y f δ A δ M δ v ( + s β ) / t β g y f δ M δ v c β y f δ v / s β y f g δ v δ M ( + t β ) / t β µ (cid:28) M , M →∞ g y f δ A δ M δ v t w ( + s β ) / t β g y f t w δ M δ v c β y f δ v / s β y f g δ v δ M t w ( + t β ) / t β M (cid:28) µ , M →∞ g y f t w δ A δ µ δ v / t β g y f t w δ µ δ v c β g y f t w δ v ( Y f L + Y f R ) g y f t w δ v δ µ / t β M (cid:28) M , µ →∞ -- g y f t w δ v ( Y f L + Y f R ) - M (cid:28) µ , M →∞ g y f δ A δ µ δ v / t β g y f δ µ δ v c β g y f δ v g y f δ v δ µ / t β M (cid:28) M , µ →∞ -- g y f δ v - T a b l e . V a l u e o f t h e t w o - t o - t w o m a t r i x e l e m e n t s (cid:113) | M | → , r e l a t i v e t o w h i c h t h e p r e v i o u s r e s u l t s a r e g i v e n , f o r f = u , c , t , ν . F o r f = d , s , b , (cid:96) o n e n ee d s t o r e p l a ce t β → / t β a nd s β ↔ c β . T h e m a t r i x e l e m e n t s q u a r e dh a s t o b e m u l t i p li e db y a c o l o r f a c t o r N c = f o r q u a r k s . N o t e t h a t a ll a m p li t ud e s i n v o l v e ( a t l e a s t) o n e p o w e r o f y f a nd δ v , t h a t c o rr e s p o nd s t o Y u k a w a - a nd i s o s p i n s upp r e ss i o n , r e s p ec t i v e l y . – 48 –n order to assess the parametric enhancement of 3-body over 2-body processes, it issufficient to consider the amplitude ratios just presented, and we will continue with a moredetailed discussion of the various lifting mechanisms at the level of individual diagrams inthe following subsection A.3. Before doing so, let us briefly remark that the corresponding cross section ratio for χχ → B ¯ F f , normalized to the one for χχ → ¯ f f , is obtained by σv → d ( σv ) → = 14 π (cid:113) − m f /m χ (cid:88) λ,h (cid:12)(cid:12)(cid:12) R λ,h (cid:12)(cid:12)(cid:12) dx dx (A.17)where x i = E i /m χ are the dimension-less fermion energies of the 3-body final state. Usingthe results from Tables 9 – 11, one can thus obtain the contribution to this ratio from eachof the gauge-invariant subsets of diagrams separately. In the limit of massless final stateparticles, the integration ranges are < x < and − x < x < , implying that someof these integrations become logarithmically divergent. This is an expected artefact of theexpansion in δ v and, in practice, the corresponding infrared divergent contributions are cutoff by the non-zero mass of the vector boson. Throughout this work, we assume that theresulting logarithmic enhancement O ( απ ln ( E B / √ s )) can be treated perturbatively downto the infrared cutoff E B ∼ m B ∼ gv EW . This imposes an upper limit on the neutralinomass of roughly m χ (cid:28) O ( gv EW e π/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 ln ( gδ v ) does not spoil thepower counting arguments related to lifting of isospin suppression factors, since the latteris described by powers δ nv of δ v . In our numerical results, we fully take into account themasses of all annihilation products. A.3 Suppression lifting from individual diagrams
It is rather illustrative to reflect the results of the previous subsection at the level of in-dividual diagrams. In Table 3, displayed for clarity already in the main text (see Section3.3), we therefore organize all relevant amplitudes in a large table, with the four rows cor-responding to the four gauge-invariant subsets. For each type of diagram, and assuming aBino- or Wino-like neutralino, we furthermore explicitly indicate the scaling with the gaugecoupling g , the Yukawa coupling y f , and the vev v EW (we comment on the Higgsino-likecase below). Let us start our discussion with the first column, which contains the diagramscontributing to the 2-body process χχ → ¯ f f . As expected, all these amplitudes scale as ∝ g y f v EW , but the origin differs: t -channel I : The factor y f v EW enters either via the chirality flip of one of the final-statefermions, or via a L/R mixing insertion of the sfermion (for brevity, we show onlyone representative diagram in Table 3 for each of these cases). t -channel II : The factor v EW enters via the gaugino/Higgsino mixing insertion on one ofthe initial lines, and the Yukawa suppression enters via the Higgsino-sfermion-fermioncoupling. – 49 – -channel EW: The s -channel with electroweak-scale mediator corresponds to the Z -exchange diagram mentioned earlier. In the s -wave limit, and from the perspective ofthe unbroken theory, this diagram is represented by the exchange of the pseudoscalarGoldstone boson G . The factor v EW arises from the gaugino/Higgsino mixing, andthe Yukawa coupling from the Yukawa interaction G ¯ f f . s -channel M A : This case is similar to the previous one, except that the mediator is re-placed by the (physical) heavy pseudoscalar Higgs A .Let us now turn our discussion to the remaining columns of Table 3, which containall relevant 3-body processes. Here, the second column shows representative Feynmandiagrams that lead to a lifting of both isospin and Yukawa suppression, while the third andfourth column show diagrams that lift only one of them, respectively: Lifting of Yukawa and isospin suppression:
Both suppression factors can be lifted onlyfor two of the gauge-invariant sets of diagrams ( t - I and s -EW). In the former case,a transverse Z T or W T is emitted from either fermion line in the final state, fromthe sfermion line, or from the initial lines (this last case cannot occur in the Bino-likecase). We remark that FSR can only lift the helicity suppression if the virtual fermionis strongly off-shell, i.e. not for soft and collinear photons (which are sometimes de-fined as FSR, see footnote 4). In the s -channel case, the diagrams can be thought ofas an annihilation χχ → W W ∗ , with subsequent decay of W ∗ (see Section 3.5 for adiscussion of such off-shell internal states). It is impossible to lift both suppressionfactors for the other two classes: for t - II , this would require a gaugino-Higgsino- W/Z vertex, which is absent for v EW → . The same applies for s - M A , noting in additionthat the A ¯ f f coupling requires also the presence of a Yukawa coupling. Lifting of only isospin suppression:
The isospin suppression can be lifted for all foursubsets, by replacing the insertion of v EW within the 2-body amplitude by the emis-sion of a Higgs boson or a Goldstone boson, respectively. Note that for the set t - I thisamounts to replacing the fermion mass insertion by a fermion-fermion-Higgs/Goldstonecoupling (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/Goldstonevertex. For the s -channel, the diagrams can also be thought of as an annihilation intoa pair of scalars, with subsequent decay of one of them. This mechanism of suppres-sion lifting is very general, and appears for all gauge invariant subsets of diagramsas well as for all final states (involving W/Z or a Higgs boson). We expect it to berelevant especially for heavy neutralino masses.
Lifting of only
Yukawa suppression:
This case is in some sense the most difficult torealize. The reason is that it requires a Higgs (or Goldstone) boson in the final state,and therefore only diagrams where the final-state boson does not couple directly to thefinal-state fermions can potentially contribute in the limit y f → . We identified threesuch processes, shown in the last column in Table 3: For t - I , the Higgs (or charged– 50 –oldstone boson; note that there is no sfermion-sfermion- G vertex for y f → ) can beemitted from the sfermion line in the t -channel, i.e. via VIB. The corresponding vertexis derived from a four-scalar sfermion-sfermion-Higgs-Higgs interaction, involving thefull Higgs doublets. This coupling leads to the required vertices at O ( v EW ) , and scaleswith g for y f → within the MSSM (see Refs. [33] and [20] for a discussion withina 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). Whilethis contribution lifts Yukawa suppression, it is suppressed compared to the 2-bodyprocess for a large mass hierarchy between gaugino and Higgsino mass parameters; wenevertheless kept this contribution, because the former effect can easily compensatefor the latter. Finally, for the s -EW case, the Higgs can be emitted from the s -channelmediator via a Goldstone-Higgs- Z coupling (third row, last column in Table 3). Notethat this mechanism is distinct from the one discussed in [38], and that the toy-modeldiscussed 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 identifiedwithin the MSSM have not been discussed before.One can understand the diagrams that lift Yukawa or isospin suppression as shownin Table 3 based on basic properties of the unbroken MSSM Lagrangian, as well as thesymmetry requirement J CP = 0 − of the s -wave initial state. For example, mixing insertions ∝ gv EW of the neutralino line can turn a Bino into a Higgsino, but not into a Wino. Inaddition, the Higgsino coupling to fermion/sfermion pairs is proportional to the Yukawacoupling, while the corresponding coupling for Bino- and Wino-like neutralinos involves agauge 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 first row.For final states involving a longitudinal W L , the corresponding sfermion vertex derivesfrom the interaction term ∝ g ( ˜ f † L H )( H † ˜ f L ) present for sfermion fields that transform asdoublet under SU (2) L . After inserting the decomposition H = ( G + , ( v EW + h + iG / √ of the SM-like Higgs doublet one easily verifies that at linear order in v EW one obtainsa sfermion coupling to G ± and h , but not to G , which explains why no longitudinal Z L boson can be produced in this case. The Higgs final state also receives a further contributionfrom the interaction term ∝ H † H ˜ f † ˜ f , which exists for all (left and right) sfermion fields.Furthermore, for the s -channel processes of the type χχ → hB ∗ → h ¯ f f that give a non-zerocontribution in the s -wave limit, the mediator B is a pseudoscalar or transverse vector (i.e. G , A , Z T ), while for χχ → G B ∗ → G ¯ f f , B is a scalar (i.e. h, H ). This is consistentwith the odd CP parity of the initial state.Note that the above arguments are only valid when expanding around the unbrokentheory, and representing longitudinal degrees of freedom by Goldstone bosons. In fact,within the broken theory, analogous arguments would be hampered by large cancellationsthat occur among individual diagrams, and that make the power counting less transparent.Nevertheless, we carefully cross checked that all these arguments can indeed be reproducedwhen using the full matrix elements within the broken theory, and expanding the sum ofall diagrams within a gauge invariant subset for heavy neutralino mass.– 51 –hile the discussion above assumed a gaugino-like neutralino, the case of a Higgsino-like neutralino is very similar. For the third and fourth row in Table 3, in particular, nothingchanges except that the incoming neutralino is now a Higgsino in the limit v EW → , andthe insertion ∝ gv EW denotes mixing with either a Bino or Wino (in addition, both Z T and W T ISR is possible, while only W T ISR is possible for Wino-like neutralinos). Thesame applies to the second line, after interchanging the label of g and y f on the verticesinvolving a sfermion in all diagrams in the first and second column (this does not affect theoverall scaling of the amplitude), while the diagram in the last column would receive anadditional y f suppression. For the first row, the two neutralino-sfermion-fermion verticesscale with y f instead of g in all diagrams. Thus, this class is additionally suppressed bya factor y f compared to the other subsets. Nevertheless, for completeness, we kept thiscase because the 3-body processes can lift the additional suppression factors y f v EW of the2-body amplitude in the same way as for a gaugino-like neutralino.In summary, we confirmed the general symmetry arguments outlined in Section 2.2 forthe MSSM and explicitly identified the contributions to the 3-body amplitudes that realizethe suppression lifting, focussing on final states containing (tranverse or longitudinal) gaugebosons as well as the SM-like Higgs boson. By expanding the full amplitudes in variouslimits that correspond to Bino-, Wino- or Higgsino-like neutralino, respectively, we find that(almost) all of the possibilities allowed by symmetries are realized. The cases for which wedid not find a contribution within the MSSM are marked by a ‘-’ in Table 2. For processesinvolving W bosons and purely right-handed fermions an additional suppression arises thatcan be traced back to the chiral structure of the SU (2) L interaction. For processes involving Z L (represented by G ) or A , and fermions of equal chirality, on the other hand, lifting ofYukawa suppression would require that the amplitude does not contain Yukawa interactionvertices. In addition, vertices such as sfermion-sfermion- G /A are absent for y f → (asrequired by CP -invariance), such that a t -channel process analogous to the one in the firstrow, last column of Fig. 3 does not exist. For the s -channel, the symmetries of the initialstate would require a CP-even mediator if the G or A was emitted via ISR. Within theMSSM, only the Higgs bosons are available. However, their coupling to fermions necessarilyinvolve a Yukawa coupling, such that Yukawa suppression cannot be lifted in this specificprocess. Similarly, one can convince oneself that the s -channel VIB process (3rd row, 4thcolumn of Fig. 3) as well as the remaining t -channel process (2nd row, 4th column) cannotoccur when replacing h → G , A . B Numerical implementation
For each Feynman diagram, we have implemented the full analytical expressions for thehelicity amplitudes in
DarkSUSY [40]. We numerically sum over these contributions toobtain the total amplitude for a given helicity configuration, M ( h,λ ) χχ → ¯ F fX , as introduced inAppendix A.1. Differential and partial cross sections are computed according to Eq. (A.13),by numerically integrating over the energies of the final state particles; for consistencychecks, this can be done for any pair of energies and in any specified order. In order to– 52 –mprove convergence and accuracy of the numerical integrations, we use taylored integrationroutines that make use of the known locations of kinematic resonances [87].For the total cross sections , we have explicitly implemented the NWA approximationscontained in Eqs. (4.3–4.9). We have extensively checked our code, and hence also the pre-scription of subtracting the NWA contribution detailed above, by comparing the total crosssection defined in Eq. (4.10), on a channel-by-channel basis and for various SUSY models,with numerical results obtained with
CalcHEP [93] . For all models, and all annihilationchannels, we find remarkable agreement. We also checked agreement for individual classesof diagrams ( s/tu -channel, ISR/FSR/VIB) as classified in Section 3.1. Let us stress thatin terms of computation time the implementation via helicity amplitudes, together withthe taylored integration routines, is less expensive compared to the evaluation of squaredmatrix elements via Monte Carlo integration as implemented in CalcHEP . This is especiallysignificant for the 3-body processes to which a large number of diagrams contribute, andfor which the difference in computation times amounts to several orders of magnitude inthe specific kinematic limit we are interested in here.For the yields of stable particles , we have implemented the procedure described inSection 4.2, using unpolarized yields for decaying particles given that these are the onlyones 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 doublecounting. We note that extending our implementation to fully polarized yields will bestraight-forward for future work, given the results provided in Section 4.2 and the helicityamplitudes reported in Appendix A.Let us mention a few of the extensive numerical checks that we performed to test theyield implementation. We considered, in particular, models for which the 3-body annihi-lation is dominated by an almost on-shell intermediate resonance. In this case, the sub- We compared our implementation of 3-body cross sections based on
DarkSUSY
CalcHEP
CalcHEP to compute the spectrum froma given set of pMSSM input parameters at scale Q = M Z (except for M A which is the pole mass) usingSoftSusy 3.4 [94]. The Susy les Houches output file 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 variousroutines in order to match the conventions. Apart from making sure that all SM input parameters agree (weused m b = 4 . GeV, sin( θ W ) = 0 . , Γ W = 2 . GeV, Γ t = 2 . GeV), we made the following changes forthe purpose of cross checking: For
CalcHEP , we switched off the running bottom mass ( dMbOn=0 ) and usedunitary 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 file. For the purpose of comparison, it is convenient to fix the Yukawa couplings at y i = m i /v , especiallyfor 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 Yukawacouplings, 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 NLOcorrections 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 firstand second generation quark masses to zero and the CKM mixing matrix to unity in order to match theconventions of the ewsbMSSM model implemented in CalcHEP . We verified that the conventions agreeby comparing also the 2-body cross sections for all channels allowed at s -wave, for which we find perfectagreement after the changes described above. – 53 –raction procedure described in Sec. 4.2 is expected to lead to a large cancellation betweenthe full 3-body contribution and the NWA term. We explicitly verified this cancellationfor all yields of stable particles, and over the full energy range. The cancellation amountsto several orders of magnitude in specific cases, and therefore provides a robust check ofthe implementation. In addition, we also verified that the yields obtained from all of themodels contained in our MSSM scan results pass a number of checks (e.g. yields within anexpected range at E > m χ / and E > m χ / ). Finally, we also considered 3-body finalstates that contain directly one or more stable particles (such as e.g. χχ → W eν ). Inthis case, we verified that the neutrino and positron spectra match the analytical resultdiscussed in App. C for specific models for which this final state is dominantly producedby an intermediate W resonance. C Spin correlations of decaying resonances
In Section 4, we discussed how to subtract double counting due to on-shell intermediatestates (‘resonances’) contributing to 3-body annihilation processes. If the resonance carriesa spin, the spectrum of final state particles depends on how much the various helicity statesof the resonance contribute. In Section 4 we argued that for annihilation of Majoranafermions in the s -wave limit, CP and angular momentum conservation uniquely determinethe helicity of all possible intermediate states that can contribute to the 3-body processesconsidered here. Here we present a formal derivation of this result, based on a descriptionthat 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 startingpoint we consider the example χχ → HW → Hf ¯ F . We are interested in the contributionfrom the on-shell intermediate W boson. The full matrix element squared can then bewritten in the form |M res | ≡ (cid:88) s ,s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) λ M µ → (cid:15) ∗ λµ (cid:15) λν ( M νs ,s ) → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (C.1)where we indicated explicitly the summation over the final-state spins of the fermions,and the polarization states of the internal W . To extract the on-shell contribution inthe narrow-width limit we assume that the momentum q µ of the W is (almost) on-shell, q (cid:39) M W . This implies that the kinematics of the H and W momenta is identical to the2-body annihilation. The first term inside the square contains the helicity amplitude forthe 2-body part, M λ → ≡ M µ → (cid:15) ∗ λµ . (C.2)For concreteness we can take the momentum of the W to be along the z-axis, q µ =( E q , , , | q | ) , where E q = (cid:113) | q | + M W and | q | is determined by the neutralino, W andHiggs mass via the 2-body kinematics (identical to | p | , see (C.22) below). The decay W → f ¯ F gives (fermion momenta p and p ) ( M νs ,s ) → = ¯ u s ( p )( gP L γ ν ) v s ( p ) (C.3)– 54 –nserting these into the resonant matrix element, and writing out the square gives aftersome renaming of indices |M res | = (cid:88) λ,λ (cid:48) M ∗ λ (cid:48) → M λ → (cid:15) ∗ λ (cid:48) µ (cid:15) λν (cid:88) s ,s ( M ∗ µs s ) → ( M νs s ) → = (cid:88) λ,λ (cid:48) M ∗ λ (cid:48) → M λ → × D λλ (cid:48) (C.4)where D λλ (cid:48) ≡ (cid:15) ∗ λ (cid:48) µ (cid:15) λν (cid:88) s ,s ( M ∗ µs s ) → ( M νd,s s ) → (C.5) = (cid:15) ∗ λ (cid:48) µ (cid:15) λν g tr(( /p − m F ) γ µ P R ( /p + m f ) P L γ ν ) (C.6) = (cid:15) ∗ λ (cid:48) µ (cid:15) λν g ( p µ p ν − g µν p · p + p ν p µ + i(cid:15) µνκρ p κ p ρ ) (C.7)The decorrelation-approximation (i.e. the replacement of the matrix element by the productof two matrix elements with independent summation over spin states, (3.6)) would beobtained by replacing D λλ (cid:48) → D decorrelated λλ (cid:48) ≡ δ λλ (cid:48) (cid:88) λ (cid:48)(cid:48) D λ (cid:48)(cid:48) λ (cid:48)(cid:48) (C.8)Instead, one can also try to use the full matrix D λλ (cid:48) . To compute it, one can use the helicitybasis (cid:15) ± µ = (0 , , ± i, / √ , (cid:15) µ = ( | q | , , , E q ) /M W , (C.9)which fulfill (cid:15) ∗ λ (cid:48) µ (cid:15) µλ = − δ λλ (cid:48) as well as (cid:80) λ (cid:15) ∗ λµ (cid:15) λν = − g µν + q µ q ν /M W .A convenient frame to evaluate it is the rest frame of the W , obtained by boostingalong the z direction. In this frame (cid:15) µ = (0 , , , . The momenta of the fermions can beparameterized by the angle w.r.t to the z-axis (which is singled out as the polarization axisof the W ), p = ( E ∗ p , , | p ∗ | sin θ, | p ∗ | cos θ ) , p = ( M W − E ∗ p , , −| p ∗ | sin θ, −| p ∗ | cos θ ) (C.10)where | p ∗ | and E ∗ p = (cid:113) | p ∗ | + m f are the momentum and energy of f in the W restframe, see (C.22). Inserting this and evaluating the trace yields an explicit expression for D λλ (cid:48) = D λλ (cid:48) ( θ ) in terms of θ . Using that the polarization vectors have zero temporalcomponent in the basis we are working in, and that (cid:126)p = − (cid:126)p , D λλ (cid:48) ( θ ) = 2 g ( − (cid:126)(cid:15) λ (cid:48) (cid:126)p ) ∗ ( (cid:126)(cid:15) λ (cid:126)p ) + δ λλ (cid:48) p · p + iM W ( (cid:126)(cid:15) ∗ λ (cid:48) × (cid:126)(cid:15) λ ) · (cid:126)p ) (C.11)Now one can use p · p = ( M W − m f − m F ) / and (cid:126)(cid:15) (cid:126)p = | p ∗ | cos θ, (cid:126)(cid:15) ± (cid:126)p = ± i | p ∗ |√ θ (C.12) (cid:126)(cid:15) ∗ + × (cid:126)(cid:15) + = + i(cid:126)(cid:15) , (cid:126)(cid:15) ∗− × (cid:126)(cid:15) − = − i(cid:126)(cid:15) , (cid:126)(cid:15) ∗ × (cid:126)(cid:15) ± = ∓ i(cid:126)(cid:15) ± , (cid:126)(cid:15) ∗± × (cid:126)(cid:15) = ∓ i(cid:126)(cid:15) ∓ (C.13)– 55 –he result is D = 2 g ( p · p − | p ∗ | cos θ ) (C.14) D ±± = 2 g ( p · p − | p ∗ | sin θ ∓ M W | p ∗ | cos θ ) (C.15) D ±∓ = 2 g | p ∗ | sin θ (C.16) D ± = − i √ g | p ∗ | sin θ ( M W ∓ | p ∗ | cos θ ) = D ∗ ± . (C.17)One can check that the average over the diagonal contributions corresponds to the usualunpolarized decay matrix element, |M → | = 13 (cid:88) λ D λλ = 13 g (6 p · p − | p ∗ | ) = g ( M W − m f − m F − | p ∗ | ) (C.18)To obtain the diff. cross section, we use the representation of the phase space in theform d ( σv Hf ¯ F ) = d Φ |M| M χ = 1(2 π ) M χ ) |M| | p ∗ || p | dm d Ω ∗ d Ω (C.19)where p is the Higgs momentum, and m = q the resonant momentum. Now we can doan approximation where we replace |M| → |M res | πM W Γ W δ ( q − M W ) (C.20)but keep the fully correlated matrix element |M res | . By integrating over dm = dm / (2 m ) = dq / (2 M W ) , and doing the trivial Higgs angle d Ω and dφ ∗ integrals, one obtains the diff.cross section w.r.t to the angle θ of the fermion f and the polarization axis of the W bosonin the back-to-back system, d ( σv Hf ¯ F ) NW A = 2(2 π ) M χ ) |M res | πM W Γ W M W | p ∗ || p | d cos θ (C.21)where | p ∗ | = [( M W − ( m f + m F ) )( M W − ( m f − m F ) )] / M W (C.22) | p | = [((2 M χ ) − ( m W + m H ) )((2 M χ ) − ( m W − m H ) )] / M χ (C.23)One can rewrite this expression, using that the two-to-two and W decay rate are given by σv HW = 18 π | p | (2 M χ ) (cid:88) λ |M λ → | (C.24) Γ W → f ¯ F = 18 π | p ∗ | M W |M → | (C.25)where M λ → is the helicity amplitude and |M → | is the usual summed/averaged decaymatrix element. Expressed in terms of the matrix introduced above, |M → | = (cid:80) λ D λλ .– 56 –he dependence on the angle cancels in this sum. Then, d ( σv Hf ¯ F ) NW A d cos θ = σv HW Γ W → f ¯ F Γ W × |M res | (cid:80) λ |M λ → | )( (cid:80) λ D λλ ) (C.26) = σv HW BR ( W → f ¯ F ) × F χχ → HW → Hf ¯ F ( θ ) (C.27)where we have defined the function F which characterizes the angular dependence. Usingalso (C.4) for M res , one can write it as F χχ → HW → Hf ¯ F ( θ ) = (cid:80) λ,λ (cid:48) M ∗ λ (cid:48) → M λ → × D λλ (cid:48) ( θ )2( (cid:80) λ |M λ → | )( (cid:80) λ D λλ ) . (C.28)If one would replace the matrix D λλ (cid:48) by the decorrelated approximation (C.8), the last termbecomes constant F χχ → HW → Hf ¯ F ( θ ) (cid:12)(cid:12) D→D decorrelated = 12 . (C.29)Integrating over the angle d cos θ (which yields a factor ), one then recovers the familiarrelation for the NWA of the total cross section. However, in general the matrix D λλ (cid:48) differsfrom the decorrelated approximation, and has a non-trivial angular dependence as well asoff-diagonal entries.For Majorana DM annihilation into a scalar and a vector, only the longitudinal polar-ization contributes to the s -wave, i.e. M λ = ± → → for v → . Using the explicit expressionfor D λλ (cid:48) , this imples that F s − wave χχ → HW → Hf ¯ F ( θ ) = D ( θ )2 (cid:80) λ D λλ = M W − m f − m F − | p ∗ | cos θ M W − m f − m F − | p ∗ | ) ≈ θ , (C.30)where the last expression applies for massless fermions. This corresponds to the decayspectrum of a longitudinally polarized W boson. The integral of this expression over d cos θ coincides with the decorrelated case. Therefore, the result for the total cross section in theNWA is nevertheless accurate, with error governed by Γ W /M W , as expected.Instead of the angle θ one can use the energy E f of the fermion in the rest frame ofthe annihilating particles, E f = γ (cid:16)(cid:113) | p ∗ | + m f + | p ∗ | β cos θ (cid:17) , dE f = γβ | p ∗ | d cos θ (C.31)where β = | p | / (cid:113) | p | + M W and γ = (1 − β ) − / . This finally yields the fermion spectrumin the narrow-width limit, (cid:18) dσv Hf ¯ F dE f (cid:19) NW A = σv HW BR ( W → f ¯ F ) γβ | p ∗ | × F χχ → HW → Hf ¯ F ( θ ) (cid:12)(cid:12)(cid:12) cos θ = Ef − γ (cid:114) | p ∗ | m f | p ∗ | βγ (C.32)This procedure can be generalized to other 3-body final states in a straightforward way.For example, for χχ → W f ¯ F , the contribution from the W resonance is (cid:18) dσv W + f ¯ F dE f (cid:19) NW A (cid:12)(cid:12)(cid:12) R = W = σv W W BR ( W − → f ¯ F ) γβ | p ∗ | ×F χχ → W W → W + f ¯ F ( θ ) (cid:12)(cid:12)(cid:12) cos θ = Ef − γ (cid:114) | p ∗ | m f | p ∗ | βγ (C.33)– 57 –here F χχ → W W → W + f ¯ F ( θ ) = (cid:80) λ,λ (cid:48) ,λ M ∗ λ λ (cid:48) → M λ λ → × D λλ (cid:48) ( θ )2( (cid:80) λ,λ |M λ λ → | )( (cid:80) λ D λλ ) , (C.34)and M λ λ → = (cid:15) λ µ ( p ) (cid:15) λµ ( q ) M µν → is the helicity amlitude for the χχ → W W annihilationprocess. In comparison to before, we have to sum in addition over the polarizations of the W + that appears in the 3-body final state. The matrix D λλ (cid:48) ( θ ) 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 L z = S z = 0 , when choosing the z -axis along the momentum of one of the final stateparticles. The possible spin projections m and m of the vector bosons are then determinedby the Clebsch-Gordon coefficients for coupling two spin-1 states ( S = S = 1 ) to a totalspin S = 1 state with m ≡ S z = 0 , (cid:104) S S ; m m | S S ; Sm (cid:105) = (cid:112) / m = 1 , m = − m = 0 , m = 0 − (cid:112) / m = 1 , m = − (C.35)Since the spatial momenta of the vectors are opposite, this means they can only be in equalhelicity states, and additionally have to be transverse, more precisely M λλ (cid:48) p ∝ (cid:104) λ ( − λ (cid:48) ) | (cid:105) ∝ diag(1 , , − , (C.36)which implies F s − wave χχ → W W → W + f ¯ F ( θ ) = M W − m f − m F − | p ∗ | sin θ M W − m f − m F − | p ∗ | ) (C.37) ≈ (cid:18) −
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