Dark matter annihilations into two light fermions and one gauge boson:
general analysis and antiproton constraints
We study in this paper the scenario where the dark matter is constituted by Majorana particles which couple to a light Standard Model fermion and an extra scalar via a Yukawa coupling. In this scenario, the annihilation rate into the light fermions with the mediation of the scalar particle is strongly suppressed by the mass of the fermion. Nevertheless, the helicity suppression is lifted by the associated emission of a gauge boson, yielding annihilation rates which could be large enough to allow the indirect detection of the dark matter particles. We perform a general analysis of this scenario, calculating the annihilation cross section of the processes when the dark matter particle is a singlet or doublet, is a lepton or a quark, and is a photon, a weak gauge boson or a gluon. We point out that the annihilation rate is particularly enhanced when the dark matter particle is degenerate in mass to the intermediate scalar particle, which is a scenario barely constrained by collider searches of exotic charged or colored particles. Lastly, we derive upper limits on the relevant cross sections from the non-observation of an excess in the cosmic antiproton-to-proton ratio measured by PAMELA.
Among the various proposals to characterize the dark matter in our Universe, the scenario where the dark matter is constituted by weakly interacting Majorana particles stands as the most promising one. In this scenario, thermal scatterings of Standard Model particles in the early Universe can produce a relic density of dark matter particles which is of the correct order of magnitude, when the interaction strength is of the order of the weak interaction strength and the dark matter mass is about 1 TeV. Furthermore, this scenario has the appealing feature that the dark matter particle might be directly detected in underground detectors, indirectly detected in cosmic ray detectors, gamma-ray and neutrino telescopes, and directly produced at the LHC.
In this paper we will focus on the possibility of indirectly detecting Majorana dark matter particles via their self-annihilation in the Milky Way dark matter halo. Concretely we will study the annihilation process into two fermions and one gauge boson which, under some circumstances, can have a non-negligible or even a larger cross section than the lowest order annihilation process into two fermions. More specifically, the -wave contribution to the thermally averaged cross section for the process is helicity suppressed by the mass of the final fermion, while the -wave contribution is suppressed by the small velocity of the dark matter particles in the Milky Way halo. In contrast, for the process the -wave contribution is no longer suppressed, due to the associated emission of a vector in the final state. As a result, the processes can even have a larger cross section than the processes as the lifting of the helicity suppression can compensate the suppression due to the additional coupling , provided the mediating scalar particles are not too heavy.
Dark matter annihilations into two fermions and one photon with the mediation of a heavy scalar particle were first studied in the framework of the Minimal Supersymmetric Standard Model in neutralino annihilations [1, 2], and explored in a number of subsequent papers . It was also pointed out that this process not only could have a sizable cross-section but also produces a gamma-ray with a very peculiar spectral shape which, if detected in gamma-ray telescopes, could be unequivocally identified as being originated in dark matter annihilations [4, 5]. If the mediating scalar particle is electrically charged it must necessarily carry hypercharge and therefore the annihilation process must also produce weak gauge bosons and, in turn, antiprotons. This process has been studied in [6, 7] employing a toy model where the dark matter particle is a singlet under the Standard Model gauge group and the mediating particle is a doublet. The constraints on the annihilation cross section in this model from the non-observation of an excess in the cosmic antiproton-to-proton fraction measured by PAMELA were derived in . Lastly, annihilations into quarks, so that the emission of a gluon is allowed in the final state were discussed in [2, 8].
In this paper we aim to extend this analysis, considering various toy models where the dark matter particle is a singlet or doublet, which couples to the left-handed or right-handed leptons or quarks of the first generation. For each case, we will calculate the cross sections for the different processes and we will calculate the constraints on the cross sections from the PAMELA measurements of the antiproton-to-proton fraction .
The paper is organized as follows. In section 2, we discuss in detail the cross sections for the various electromagnetic and electroweak internal bremsstrahlung processes occurring for singlet as well as doublet dark matter particle coupling to leptons. The corresponding cases, when assuming a coupling to quarks, are discussed in section 3. In section 4 we present the constraints on the cross sections from the PAMELA measurements of the antiproton-to-proton fraction, and translate them into upper limits on an astrophysical boost factor for the various toy models. Finally, we conclude in section 5. Our full analytical results for the cross sections of all processes considered in this work can be found in the Appendix.
2 Dark matter coupling to leptons
We consider an extension of the Standard Model by one Majorana fermion, , which we assume to constitute the dominant component of dark matter in the Universe, and one scalar particle, , which mediates the annihilation process into light fermions. The Lagrangian is
Here, is the Standard Model Lagrangian which includes a potential for the Higgs doublet , . On the other hand and are the parts of the Lagrangian involving just the Majorana fermion and the scalar particle , respectively, and which are given by
where denotes the covariant derivative. Lastly, and denote the fermionic and scalar interaction terms of the new particles to the leptons and to the Higgs doublet. These terms depend on the details of the model and will be discussed case by case below.
The quantum numbers of the relevant Standard Model particles under the gauge group are: , , . On the other hand, the quantum numbers of the dark matter particle and the scalar are constrained in our setup by the requirement that the dark matter particle is colorless and electrically neutral, and by the requirement that a Yukawa coupling to the leptons (either left-handed or right-handed) is invariant under the Standard Model gauge group. We will assume in the following that the dark matter particle only couples to the first generation of leptons, which can be ensured by postulating that the extra scalar particle carries electron lepton number , while the dark matter particle does not carry lepton number. Lastly, in order to guarantee the stability of the dark matter particle, we impose a discrete symmetry under which and are odd while the Standard Model particles are even.
In the scenarios of interest for this paper, the intermediate scalar is electrically charged and could possibly lead to experimental signatures in collider experiments. Precise measurements of the invisible decay width of the boson at LEP set the upper bound MeV , which rules out the existence of exotic charged scalar particles with mass below 40 GeV . Furthermore, the OPAL collaboration searched for an excess with respect to the Standard Model expectations of dilepton events with missing energy induced by the production of exotic scalar charged particles which decay into an electron and an invisible particle (in the framework of supersymmetry, the production of selectrons which decay into an electron and the lightest neutralino). The non-observation of an excess in a sample of 680 pb of collisions at center-of-mass energy between 192 GeV and 209 GeV, leads to the lower bound GeV, assuming GeV . A similar search was undertaken by the L3 collaboration using a sample of 450 pb collisions at GeV, resulting in the lower bound GeV assuming GeV , by the ALEPH collaboration using a sample of 207 pb collisions at GeV, resulting in GeV assuming GeV  and by the DELPHI collaboration using a sample of 609 pb collisions at GeV, resulting in GeV assuming GeV, and GeV assuming GeV and GeV . For smaller mass splittings the detection efficiency is significantly reduced and the lower bounds derived by the LEP experiments can be avoided.
In the remainder of this section we present a classification of models, according to the charge of the dark matter particle under .
2.1 singlet dark matter
When the dark matter particle is a singlet, its hypercharge must be zero in order to render an electrically neutral particle, hence the gauge quantum numbers must be . On the other hand, the gauge quantum numbers of the scalar depend on whether the dark matter particle couples to the right-handed electron singlet or to the left-handed electron doublet.
If the dark matter has a Yukawa coupling to the right-handed electron singlet and a scalar field , then gauge invariance requires . With these quantum numbers the only interaction terms in the Lagrangian are:
In the Minimal Supersymmetric Standard Model (MSSM) this possibility is realized if is the bino and is the right-handed selectron, .
On the other hand, if the dark matter only couples to the left-handed electron doublet then . The interaction terms are then:
In the MSSM, this possibility is realized if is the bino and the left-handed selectron doublet, .
After the electroweak symmetry breaking, the mass of the electrically charged scalar is given by . If the dark matter couples to the left-handed electron doublet, there exists also a neutral scalar with mass given by . We will assume that , such that the Majorana fermion is stable and can constitute the dark matter. The interactions lead to a thermal production of in the Early Universe that is compatible with the WMAP value if the coupling is of order one and the masses lie between the weak and the TeV scale [16, 7].
The annihilation of dark matter in the Milky Way today can proceed via the annihilation channels , as well as . However, the cross-sections are highly suppressed because the -wave contribution is helicity suppressed, while the -wave contribution is suppressed by the dark matter velocity in the Milky Way halo. The helicity suppression can be lifted by emitting an additional spin-1 particle in the final state [1, 2, 3, 6, 7]. The lifting of the helicity suppression can compensate the suppression due to the additional coupling provided the mediating scalar particles are not too heavy, typically . Therefore, the dominant annihilation processes are channels, like , or .
Note that one could similarly consider a coupling of the dark matter particle to the leptons of the second or third generation. The cross-sections for the processes and the constraints from the antiproton flux that will be discussed later are independent of the lepton flavor to a good accuracy. Due to the larger masses of the and leptons, the helicity suppression of the annihilation cross-sections is less pronounced than for a coupling to electrons, while the velocity suppressed contributions to the annihilation cross-sections are flavor-independent.
The annihilation mode leads to a gamma ray signal with a pronounced peak at the dark matter mass, that is potentially observable by the Fermi-LAT (see  for a recent discussion) and by current and future IACTs [4, 5]. On the other hand, the annihilation channels involving weak gauge bosons yield a primary contribution to the cosmic flux of antiprotons. In the following, we will analyze the relative strength of these channels, and in Section 4 we will derive upper limits on the cross-section from the PAMELA measurement of the antiproton to proton ratio .
Coupling to right-handed electrons
Let us first consider the possibility that the dark matter particle couples to right-handed electrons. Since the mediating particle carries hypercharge and electric charge, the annihilation channels and are possible [6, 7]. The branching ratios compared to the annihilation cross section are shown in the upper panel of Fig. 1. For dark matter masses , far above the -threshold, the ratio of the cross sections for the electromagnetic and the electroweak bremsstrahlung processes approach constant values given by the ratio of the respective coupling constants:
The general formulas for the double differential cross sections and for arbitrary dark matter masses are given in the Appendix. In order to obtain a gauge-invariant result it is important to take into account the diagrams for which the gauge boson is emitted off the internal line and off the final state particles (later on, when considering doublet dark matter, also contributions from initial state radiation have to be included). The corresponding Feynman diagrams are also shown in the Appendix. In the following, we will refer to all these processes as internal bremsstrahlung (IB). As noted before, these comprise also the contributions from final state radiation in general. Note that the contributions from soft and collinear emission, which are in principle logarithmically enhanced, are typically negligible in this context because they are helicity suppressed, like the processes. Instead, the dominant contribution arises from the diagrams where the gauge boson is emitted either off the internal line, or from a final state particle with an off-shell intermediate state (see Ref.  for a detailed discussion).
The dependence of the cross sections on the mass is shown in the upper right panel of Fig. 1. It is apparent from the figure that the branching ratio of the processes is largest when is close to . Furthermore, for , the cross section of the processes fall of as , while the cross sections scale like . The former remain dominant as long as the dark matter mass and the mass of the mediating particle are of comparable size, roughly . The qualitative properties discussed here are common also to most other cases considered below. However, there are some quantitative and also qualitative differences which we will stress in the following.
Coupling to left-handed electrons
If the dark matter particle couples to the left-handed electron doublet, it can give rise to annihilations into final states involving or bosons. Note that this case has been discussed in detail in Ref. . We will briefly review it here for completeness. The branching ratios compared to the annihilation cross section are shown in the lower part of Fig. 1. For , and assuming that , the asymptotic values are again given by the ratios of the appropriate couplings:
Generically, one expects a non-zero mass splitting of the neutral and charged components of induced by the breaking of the electroweak symmetry, . Compared to the degenerate limit, the branching ratios get modified due to two effects. First, the masses in the t-channel propagators of the mediating particles corresponding to the charged and the neutral component of differ from each other. Second, the mass splitting opens up a new channel, namely the annihilation into longitudinally polarized -bosons. In the limit , the branching ratios are approximately given by
where and . The ratio , in contrast, is not affected by the mass splitting. The emission of longitudinal bosons also leads to a spectrum that is harder compared to the case . Analytical expressions for the double differential cross sections, from which the spectra can be easily obtained, are given in the Appendix.
2.2 doublet dark matter
In this case the dark matter doublet must have one electrically neutral component, which is achieved by postulating that the hypercharge is . This matter content, however, leads to gauge anomalies which can be canceled by introducing another doublet with opposite hypercharge. Then, the minimal model with doublet fermionic dark matter must contain the new fermions , , both charged under the discrete symmetry. In the case of the MSSM these two particles can be identified with the two higgsinos.
Under these assumptions, the only gauge invariant and invariant fermionic mass term in the Lagrangian is , which generates identical tree level masses for , , , . Quantum corrections induced by the Standard Model gauge bosons generate a mass splitting between the charged and the neutral component of the multiplet GeV , inducing the decay of the former into the latter. Therefore, this toy model predicts the existence of two stable particles, candidates of dark matter, , which will annihilate, among other channels, , with a vector. Since we are interested in the general features from annihilations of doublet dark matter particles and not in constructing a fully realistic model, we will assume in what follows that only one of these, or , is present in our Universe today. This can be achieved by postulating a mass splitting between them, so that one of them decays into the other at very early times, for instance by introducing the dimension five operators
with a mass scale larger than the Higgs vacuum expectation value and coefficients of order one. Let us denote the dark matter mass eigenstate as , the heavier neutral state as and the charged component by . The mass splittings induced by the dimension five operator are given by,
up to corrections of order . Up to the same order, the mass eigenstates are related to the two doublet fields by and , where , and are Majorana fields and is a Dirac field. We will assume in the following that the radiative corrections to the mass splittings can be neglected compared to the ones induced by the dimension five operators.
By decomposing the gauge interactions of and into mass eigenstates one obtains
Note that this coincides with the interactions of the neutralino in the higgsino limit within the MSSM. These interactions give rise to the dark matter annihilation channels into a pair of weak bosons with cross-sections given by
As is well-known, these cross-sections can be altered substantially for dark matter masses in the TeV range, and if the mass splittings are of order GeV or below, by the multiple exchange of weak bosons among the fermions in the initial state, analogous to Sommerfeld enhancement in electrodynamics . We will assume here that the mass splittings are large enough, such that the effect of Sommerfeld enhancement is in a perturbative regime, and comment on its impact below. As we will see this requires TeV.
In the present work, we are motivated by the observation that internal bremsstrahlung can lift the helicity suppression of fermionic final states. In fact, as we will discuss below, the annihilation can be under certain conditions as important as the gauge processes . Let us start by discussing the fermionic interactions analogous to the singlet case. In particular, the dark matter can couple to the right-handed electron singlet and a scalar field , yielding the two following interaction terms in the Lagrangian:
Alternatively, the dark matter particle can couple to the left-handed electron doublet and a scalar field ,
In a supersymmetric context, the scalars can be identified with and , respectively. Note that one could in principle consider additional scalar particles that lead to analogous couplings involving . We do not consider this possibility in the following.
The relic abundance produced by the thermal freeze-out is determined by the cross-sections arising from gauge and Yukawa interactions. If the latter are subdominant, an abundance in accordance with the WMAP value can be achieved for dark matter masses of order TeV . By adding the Yukawa interactions, and adjusting the coupling , it is in principle possible to obtain the WMAP value also for even higher dark matter masses. However, in the following we will not restrict the range of the dark matter mass or the coupling in order to determine the constraints arising from the measurements of the antiproton flux in a way that is independent of the production mechanism.
We will first discuss the relative size of the electromagnetic and electroweak bremsstrahlung in analogy to the case of singlet dark matter, and then the relative importance of fermionic and diboson final states. Throughout, we will assume that , and use the notation in analogy to the singlet case.
Coupling to right-handed electrons
If the mediating scalar has quantum numbers , it leads to annihilations of dark matter into right-handed electrons. For and , and taking only annihilations mediated by the scalar into account, the branching ratio of electromagnetic to electroweak bremsstrahlung is given approximately by
where , and and are the sine and cosine of the weak mixing angle. More general analytical expressions are given in the Appendix. The neutral component plays no role because it could only lead to the production of right-handed neutrinos, which are absent in the SM. Note that the branching ratio increases
with the mass of the mediating particle to the fourth power, . The reason for this behaviour is that the dark matter particle, being an doublet, couples also to the boson. Therefore, annihilation to can occur also via initial state radiation. The latter leads to a non-zero contribution to the s-wave cross-section already at the -level, while electromagnetic bremsstrahlung occurs at order . A similar result has been obtained within an effective operator approach for Wino-like dark matter in . Note that, for very large values of , eventually the p-wave contribution to will dominate over the s-wave contribution to . However, the former also scales like . This means the ratio of electroweak and electromagnetic cross sections saturates for very large values of , which can be estimated roughly as for and GeV. However, since a strong gamma signal with spectrum peaked at high energies is produced only for , we will not discuss this case in further detail. The full dependence of the branching ratios on the dark matter mass and the mass of the mediating scalar particle is shown in the upper part of Fig. 2. In the right part, it is also shown that the p-wave contribution to the annihilation into becomes important when is large.
Since the electroweak bremsstrahlung is strongly enhanced compared to electromagnetic bremsstrahlung even for moderate values , it is important to investigate whether higher-order contributions can lift the suppression of electromagnetic bremsstrahlung. It turns out that this is indeed the case, when considering the corrections arising from Sommerfeld enhancement. Here, we will estimate the leading effect when the enhancement is perturbatively small, following Ref. [21, 23]. In general the s-wave amplitude for annihilation into some SM final state can be written as
where the amplitudes denote the tree-level amplitudes for annihilations of the various components of the doublet, and are enhancement factors. They are related to the wave-functions for radially symmetric (s-wave) two-fermion initial states, where is a dimensionless variable related to the spatial separation of the fermions. The wave-functions are solutions of a set of coupled Schrödinger equations in the presence of a Yukawa potential that is generated by the exchange of vector bosons among the fermion pair [21, 22]. For the dark matter masses we are interested in we can safely apply the low-velocity limit and assume that . The latter condition implies that the charged and heavier neutral components of the doublet cannot be produced on-shell, such that their wave-functions decay exponentially at large separations. At leading order in the gauge couplings one then finds the approximate solution
The approximation can be expected to hold if . For the range of dark matter masses we are interested in, this is safely the case if GeVTeV (see e.g. ). Here, we will assume that this inequality holds and thus that the tree-level cross sections yield a reliable estimate. Nevertheless, for the annihilation involving electromagnetic bremsstrahlung, the annihilation proceeding via an intermediate charged fermion pair, , can be important. Concretely, we find that for
Here the Sommerfeld enhanced cross section is normalized to the leading order cross-section, and the cross section for should be evaluated for an initial state with total spin zero  (note that, for the same reason, the channel is helicity suppressed although is a Dirac particle). Thus, as expected, we find that for the annihilation via a charged intermediate state the cross section scales like instead of as for the tree-level contribution. Therefore, if is large enough, it may compensate the suppression factor , and yield a significant contribution to the annihilation via electromagnetic bremsstrahlung. The influence of Sommerfeld enhancement is shown also in Fig. 2 for mass splittings GeV. Note that analogous corrections exist also for electroweak bremsstrahlung. However, since their leading order cross sections scale already like , these will be small corrections even when is large.
Coupling to left-handed electrons
If the mediating scalar has quantum numbers , it leads to annihilations of dark matter into left-handed electrons. For and , we find the following ratios
Annihilation into does not occur at tree level, because the quantum numbers of imply that it couples the dark matter particle only to the charged lepton. Also, since is a singlet, only transversally polarized bosons are produced via initial as well as final state radiation provided that . The reason for the -dependence of the ratios of cross sections is due to initial state radiation, as discussed above. The full dependence of the branching ratios on the dark matter mass and the mass of the mediating scalar particle is shown in the lower part of Fig. 2.
When including Sommerfeld corrections, the annihilation channels and appear. In particular, the latter yields a contribution that scales like instead of as for . Its cross section is given by
The Sommerfeld corrections shown in the lower panel of Fig. 2 refer to the sum of the electromagnetic annihilation cross sections .
The relative size of the cross sections and depend on the ratio of the Yukawa coupling to the gauge coupling as well as the ratio of the mass of the mediating particle and the dark matter mass. In general, due to the lifting of helicity suppression, the processes are much less suppressed than the corresponding annihilations into fermions. However, the annihilations mediated by gauge interactions are also not helicity suppressed and therefore generically dominate over the channels. Nevertheless, there can be some cases when the latter are important. For example, this can be the case if the dark matter mass is in the range , such that the diboson states are kinematically disfavored. Another possibility is a rather large value for the coupling . In Fig. 3 the various cross sections are shown for . In this case, the cross sections of the channels arising from electroweak bremsstrahlung are the dominant annihilation channels provided that . For even larger , the latter restriction can be relaxed. We note that couplings of that size are required for dark matter masses in the multi-TeV range, when imposing the relic density constraint from thermal freeze out. In order to determine constraints from the antiproton flux, we will consider both the case that the annihilation channels or dominate.
3 Dark matter coupling to quarks
The analysis of the annihilation of dark matter particles into quarks is completely parallel to the annihilation into leptons discussed in the previous section, the main difference being the inclusion of a color quantum number in the final fermion states and in the intermediate scalar state. As a result, in addition to the electromagnetic and electroweak bremsstrahlung processes, a new annihilation channel arises where a gluon can be radiated off the final quark states or off the internal colored scalar state. Being both the gluon and the photon massless gauge bosons, the resulting spectra will be identical . However, the cross section for the gluon internal bremsstrahlung will be enhanced compared to the electromagnetic internal bremsstrahlung by the larger coupling constant and by the color factor, therefore we expect a larger impact of the processes in scenarios where the dark matter couples to quarks compared to scenarios where the dark matter couples to leptons, and in particular a larger impact of the present measurements of the antiproton-to-proton fraction on the constraints on the couplings of the model.
Scenarios where the dark matter particle couples to quarks and to a colored scalar particle are strongly constrained by experiments searching for exotic colored particles. The LEP constraint on the invisible width of the boson, MeV , allows to set the absolute lower limits GeV if or and GeV if , corresponding to the searches for left-handed quark doublets, right-handed up quarks and right-handed down quarks, respectively .
Recently, the ATLAS collaboration has reported in  the results for the search of squark and gluinos using final states with jets and missing transverse momentum using 1.04 fb of data taken in proton-proton collisions with TeV at the Large Hadron Collider. In this analysis it is considered a simplified supersymmetric model with R-parity conserved containing only squarks of the first two generations, , a gluino octet, , and the lightest neutralino, , while all other supersymmetric particles are assumed to be very heavy. In this simplified scenario, the supersymmetric particles are produced in pairs and the final states produce more than 2, 3 or 4 jets plus missing energy. Concretely, the production of two squarks is followed by the decay , thus producing at least two jets plus missing energy. In contrast the production of two gluinos is followed by the decay , which yields in the final state at least four jets plus missing energy. Lastly, the associated production of one squark and one gluino yields at least three jets plus missing energy. The non observation of an excess in any of these channels above the Standard Model background can be translated into constraints on the plane, assuming .
Our toy model for dark matter corresponds to the simplified SUSY model considered by the ATLAS collaboration in the limit where the gluino is also very heavy and is not kinematically accessible to the LHC with TeV. Hence, we conclude that the non-observation of an excess over the Standard Model background of dijet events with missing energy in the present LHC data translates into a lower bound on the colored scalar state of our toy model GeV at 95% c.l. It is important to note that this lower bound assumes that the mass splitting between and is large enough to produce jets passing the requirements for the transverse momenta ( GeV for the leading jet and GeV for the second jet). Then, this stringent lower bound on the colored scalar mass can be avoided if and present a degenerate mass spectrum, as discussed in , concretely when GeV, so that the dijet event does not pass all the cuts required by the ATLAS analysis.
Searches for colored scalar states were also undertaken at the Tevatron and at LEP. The searches at the Tevatron by the CDF  and D0  collaborations employ similar cuts as the ATLAS analysis described above, and have by now been superseded. On the other hand, the searches at LEP, despite limited by the smaller center of mass energy and by the smaller luminosity, employed a smaller cut for the jet transverse momentum and are relevant for our analysis. Concretely, the L3 collaboration has presented limits on the squark masses searching for an excess in dijet events with missing energy in collisions at center of mass energies between 192 GeV and 209 GeV with an integrated luminosity of 450.5 pb . The non-observation of an excess with respect to the expected Standard Model background translates in our toy model into the lower bound GeV for GeV.
As a summary, we conclude that the toy model with a Majorana dark matter particle which couples to the quarks and a colored scalar, , via a Yukawa coupling is in agreement with the present searches of new physics if
for any .
, if .
, if .
Following the same scheme as in the case of annihilations into leptons, we will analyze the features of various dark matter scenarios coupling to quarks according to the charge of the dark matter particle under .
3.1 singlet dark matter
This choice of the charge requires that the gauge quantum numbers of the dark matter particle must be in order to render an electrically neutral particle, while the quantum numbers of the intermediate colored scalar depend on whether the dark matter particle couples to the right-handed up quarks, , the right-handed down quarks, , or to the quark doublet, .
The fermionic interaction term when the dark matter couples to the right-handed up quark reads:
which requires quantum numbers for the intermediate scalar particle , while the scalar interaction term is given by Eq.(2.3). Similarly, a Yukawa interaction between the dark matter and the down quark requires . In the MSSM these particles correspond to the right-handed up and down squarks, respectively.
On the other hand, the fermionic interaction Lagrangian of the dark matter particle to the quark doublet reads:
with , which in the MSSM corresponds to the left-handed squark doublet. The scalar interaction Lagrangian is given in Eq.(2.4).
In this case, the helicity suppression of the annihilation channels , can be lifted by the associated emission of photons, weak gauge bosons or gluons together with the light quarks. Let us analyze the relative strength of these channels for each of the scenarios.
Coupling to right-handed up quarks
In this case the particle mediating the dark matter annihilations carries hypercharge, electric and color charge, therefore the annihilation channels , , are possible. We show in Fig. 4, upper plot, the corresponding cross sections relative to the helicity suppressed cross section for . In the left plot we present the ratio of cross sections for dark matter masses between 50 GeV and 5 TeV and different values of the mass splitting GeV, 50 GeV and 10 GeV, to study the impact of the constraints from collider searches of exotic colored particles. It is apparent from the plot that the smaller the mass splitting, the larger is the relative cross section of the processes, especially for light dark matter particles. On the other hand, in the limit the ratios are fairly independent of the mass splitting and take the values:
Exact formulae for the various cross sections can be found in the Appendix. For the numerical values given here and below, we have evaluated the strong coupling constant at a scale GeV for illustration.
Moreover, we show in the upper-right plot the dependence of the ratio of cross sections on the mass of the intermediate colored scalar particle for a fixed dark matter mass GeV. The maximal value of the ratio of cross sections, which is as large as for , is reached when . We note that this is precisely the region of the parameter space which is most difficult to constrain at colliders, as the jet produced in the decay of the colored scalar particle is too soft to be triggered. As in the leptonic case, for , the cross sections of the processes scale as , while that of the process scale as .
Coupling to right-handed down quarks
The results for the annihilations into right-handed down quarks are completely analogous to the results for the annihilations into right-handed up quarks presented above, the only difference being the different hypercharge (and electric charge) of the intermediate scalar. As a consequence, the cross sections for the annihilations are a factor of 1/4 smaller than the corresponding cross sections for . In particular, when the cross sections for annihilations into right-handed down quarks satisfy the relations:
Coupling to left-handed quarks
When the dark matter particle couples to the left-handed quarks, a new annihilation channel is open, involving bosons which can be radiated off the internal colored scalar or off the final fermions legs. The ratios are shown in Fig. 4, lower-left plot, in the limiting case for dark matter masses between 50 GeV and 5 TeV and different mass splittings between the dark matter mass and the intermediate scalar mass, 10, 50 and 100 GeV. Compared to the coupling into right-handed quarks, it is noticeable the enhancement of the branching ratio into electroweak gauge bosons, arising from the additional channel involving bosons and both up and down quarks, as well as a stronger coupling of left-handed compared to right-handed quarks to the -boson. On the other hand, the branching ratio into gluons is identical to the right-handed case, while the branching ratio for differs by a factor of 8/5 (2/5) with respect to () due to the different electric charges of the particles involved and due to the doubling of diagrams. In the limit the cross sections satisfy the relations:
The dependence of the branching ratios with the mass of the intermediate scalar is shown in Fig. 4, lower-right plot.
In a more realistic scenario, the two weak isospin components of the intermediate scalar particle will not be degenerate in mass, but will have a mass splitting proportional to the order parameter of the electroweak symmetry breaking: