Successful Supersymmetric Dark Matter with Thermal Over/UnderAbundance from Late Decay of a Visible Sector Scalar
Abstract
We present an explicit model where the decay of an parity even scalar with mass is the origin of nonthermal dark matter. The correct relic abundance can be produced for both large and small annihilation rates in accordance with the Fermi constraints on the annihilation crosssection. This scenario has advantages over that of nonthermal dark matter from modulus decay. First, branching ratio for production of parity odd particles can be made quite small by a combination of couplings to matter fields and kinematic suppression, enabling us to obtain the observed dark matter relic density in cases of thermal underproduction as well as overproduction. Second, gravitino production is naturally suppressed by the virtue of decaying scalar belonging to the visible sector. The decaying scalar can also successfully generate baryon asymmetry of the universe, and may provide an explanation for the baryondark matter coincidence puzzle.
MIFPA 1246
December, 2012
I Introduction
Weakly interacting massive particles (WIMPs) are promising dark matter (DM) candidates that can explain the DM relic abundance, as precisely measured by cosmic microwave background experiments WMAP (), via thermal freezeout of annihilation in the early universe. The nominal DM annihilation rate for this scenario, called the WIMP miracle, is cm s.
WIMPs typically arise in models of particle physics beyond the standard model (SM). In supersymmetric (SUSY) models with conserved parity, the lightest supersymmetric particle (LSP) is the DM candidate. The lightest neutralino is the most suitable candidate with prospects for detection in various direct and indirect searches. However, in large regions of SUSY parameter space the thermal relic abundance of the LSP is different from the observed value. Typically, if the lightest neutralino is a Higgsino or Wino, the annihilation crosssection is large compared to the nominal value, while for a Bino the annihilation crosssection is small.
A mainly Higgsino LSP is motivated by considerations of naturalness Papucci:2011wy (); Hall:2011aa (); Baer:2012uy (); Allahverdi:2012wb (); Gogoladze:2012yf (); Baer:2012cf (). If the Higgsino mass is in the subTeV region, the annihilation rate is typically larger than cm s, thus resulting in an insufficient thermal relic abundance. The FermiLAT data GeringerSameth:2011iw () constrains the DM annihilation rate at the present time. These bounds do not allow much room for too large an annihilation crosssection. Further, for lower values of the LSP mass, FermiLAT data appears to prefer smaller annihilation crosssections. This prefers Bino type LSP, which will have a thermally overproduced abundance.
Therefore, since the presently motivated DM annihilation crosssections are either larger or smaller than the nominal thermal freezeout value, one is naturally led to consider nonthermal scenarios to obtain the correct DM relic density. A latedecaying field that reheats the universe below the freezeout temperature ( denoting the DM particle) can provide a nonthermal origin for DM. If the annihilation rate is larger than the nominal value, the LSPs produced from decay undergo residual annihilation before reaching their final abundance. Models that prefer Higgsino DM can be accommodated in such a scenario. If the number density of LSPs produced from decay is very small and/or the annihilation rate is smaller than the nominal value, annihilation will be inefficient and the final DM relic density will the same as that from decay. This scenario can accommodate both types of models with Higgsino and Bino DM.
A gravitationally coupled modulus field, typically arising in SUSY and superstringinspired models BBN1 (), serves as a standard candidate for the field . Moduli that are heavier than 50 TeV decay before the onset of big bang nucleosynthesis (BBN), and hence will not ruin its successful predictions on the primordial abundance of light elements. The decay also provides a nonthermal origin for DM production Moroi:1999zb (); Dutta:2009uf (). However, in such a scenario, the branching ratio for gravitino production from late decay (denoted by ) must be sufficiently suppressed. Otherwise, the decay of gravitinos will lead to DM overproduction or, for gravitino mass below TeV, will ruin the success of BBN. It is not easy to satisfy this requirement in scenarios where modulus decay is the source of DM production. For example, in simple models such as KKLT Kachru:2003aw () turns out to be too large by a factor of Allahverdi:2010rh (). Furthermore, it is difficult to obtain a sufficiently small DM abundance from moduli decay without residual annihilation. Even if one suppresses LSP production from twobody decays of moduli, the threebody decays will still be unacceptably large Allahverdi:2010rh (). Suppressing the total decay rate of the modulus can help the situation, but it involves nontrivial conditions on the Kähler geometry of the underlying effective supergravity theory.
In this paper we explore the late decay of a visible sector scalar field that serves as the origin of nonthermal DM. It has the following advantages: Gravitino production from is naturally suppressed for gravitino mass as low as by virtue of the fact that is a visible sector field and Due to kinematic suppression, parity odd particles are produced at oneloop or twoloop level, which can yield . This immediately opens up the option of obtaining nonthermal DM purely from branchings of the scalar field, without undergoing further annihilation. As we discussed, for Bino LSP, this is the only option. For Higgsinos, this allows both options of a relic density set by pure branching ratios, as well as by annihilation.
We present an explicit model where a visible sector scalar with a mass and parity charge has direct couplings to new colored fields . If is lighter than particles, as well as all colored superparticles, it dominantly decays into two gluons at oneloop level. decay also produces DM particles via loops, but this mode is suppressed by powers of and , with being the gauge couplings respectively. As we will see, this model can yield the correct DM relic abundance for of the thermal underproduction and overproduction cases, and also lead to successful latetime baryogenesis.
The paper is organized as follows. In Section II, we briefly review nonthermal DM form modulus decay and the associated problems. In Section III, we introduce a model with late decaying scalar field and discuss its properties. In Section IV, we show that this model can successfully explain the DM content for both large and small annihilation cross sections. In Section V, we discuss baryogenesis in this model and point out that the presented nonthermal DM scenario can also address the baryonDM coincidence puzzle. We conclude the paper in Section VI.
Ii Nonthermal Dark Matter from a Late Decaying Scalar Field
In this section, we discuss the various possibilities for nonthermal production of DM from a late decay.
ii.1 Reheating by late decay
We consider a scalar field with mass and decay width . Assuming that has acquired a large vacuum expectation value during inflation, it will start oscillating about the minimum of its potential with an initial amplitude when the Hubble expansion rate is . Oscillations of behave like matter, with an initial energy density . The energy density of the universe at this time, dominated by thermal bath, is .
The quantity is redshifted , with being the scale factor of the universe. After using the fact that is redshifted for a radiationdominated universe, we find the necessary condition for to be dominant at the time of decay
(1) 
Decay of reheats the universe to a temperature . As a numerical example, for and MeV (in order to be compatible with BBN), Eq. (1) implies dominance for GeV.
If dominates the universe at a temperature , we will have
(2) 
where “before” and “after” are in reference to the epoch of decay, and we have used the fact that .
It is seen from Eq. (II.1) that decay releases a large entropy that dilutes any preexisting quantity in the thermal bath. For the above numerical example where and MeV, the entropy release factor can be as large as .
ii.2 Dark matter from late decay
Provided that , decay of will dilute any thermally produced DM by a large factor as mentioned above. However, decay itself produces DM particles. The abundance of nonthermally produced DM is given by
(3) 
Here , is the branching fraction for production parity odd particles from decay, and denotes DM abundance obtained via thermal freezeout that is related to the observed DM relic abundance through:
(5)  
The abundance of DM particles immediately after their production from decay is given by . If , DM annihilation will be inefficient at temperature . In this case, the final DM relic abundance will be given by the first inside the brackets in Eq. (3). On the other hand, if , annihilation will be efficient right after decay. This will somewhat reduce the abundance of DM particles produced from decay, in which case the final relic density will be given by the second term inside the brackets in Eq. (3).
There are therefore two possible scenarios for obtaining the correct DM relic density from decay:

Annihilation Scenario: If , then (hence “thermal underproduction”). The large annihilation cross section can reduce the abundance of DM particles produced from decay to an acceptable level, provided that:
(6) The final DM abundance will then be given:
(7) This scenario can work well in the case of Higgsino DM, for which ) as mentioned before, provided that the reheat temperature from decay satisfies Eq. (6).

Branching Scenario: If Eq. (6) is not satisfied, then annihilation will be rendered ineffective. This happens if is too low and/or is too small.
The first possibility is that , but is lower than that given in Eq. (6). In this case nonthermal Higgsino DM must be produced via “Branching Scenario”.
On the other hand, we note that Eq. (6) can never be satisfied if . It is seen from Eq. (5) that this results in (hence “thermal overproduction”). This leaves “Branching Scenario” as the only possibility for nonthermal DM production in this case. Bino DM provides a prime example of this case.
The final DM abundance will be the same as that produced from decay, which follows
(8)
ii.3 Challenges for nonthermal dark matter from modulus decay
Modulus decay provides a natural scenario for nonthermal DM Moroi:1999zb (). Moduli heavier than 50 TeV decay before the onset of BBN, which allows their utilization as the source of DM production. The modulus decay rate is given by Allahverdi:2010rh (). For TeV, this results in .
“Annihilation Scenario” can be realized in this framework. We have recently discussed nonthermal Higgsino DM from modulus decay in scenarios with mixed anomalymodulus mediation of supersymmetry breaking Allahverdi:2012wb (). For a typical modulus mass TeV in this scenario, one finds . Eq. (6) can be satisfied for Higgsino annihilation cross sections that are compatible with Fermi bounds GeringerSameth:2011iw () with a mass GeV Allahverdi:2012wb ().
“Branching Scenario”, however, is not easily realizable. The fact that requires for GeV. Such a small may be obtained for twobody decays of the modulus Moroi:1999zb (), but threebody decays will inevitably set a lower bound . Obtaining the correct DM relic abundance within “Branching Scenario” then requires that be further lowered, which may happen by means of geometric suppression Allahverdi:2010rh ().
An additional challenge is posed by gravitino production from modulus decay, whose abundance follows , where is the branching fraction for gravitino production from modulus decay. Gravitinos decay much later than the modulus. For example, in the scenario discussed in Allahverdi:2012wb () TeV, which implies they decay just before the onset of BBN. Gravitinos produce DM particles upon decay, which requires that
(9) 
For GeV, this results in the bound . Since , we then need . However, in the simples example based on KKLT model Kachru:2003aw () we have . One may lower by modifying the Kähler potential, but a successful implementation that does not affect other aspects of the nonthermal DM scenario is a nontrivial task.
To summarize, a completely successful nonthermal DM scenario from modulus decay is challenging, in particular in models with thermal overproduction (notably Bino DM). The reason being that and are typically too large in this case.
Iii Late Decay of a Visible Sector Scalar
In this section, we present an explicit model of nonthermal DM from late decay of a visible sector field. We describe the model and its field content, and show how it can address the abovementioned issue with .
iii.1 The Model
The visible sector consists of the minimal supersymmetric standard model (MSSM) augmented with extra superfields:
A singlet whose decay is the origin of DM.
Two flavors of isosinglet color triplets with hypercharges respectively.
Two flavors of singlets .
Multiple flavors of are introduced to accommodate latetime baryogenesis (which we will discuss in Section V).
We choose charge assignments under parity such that parity is conserved, hence LSP is stable. This implies that and are parity even, while their SUSY partners are parity odd. We also assume that the lightest MSSM neutralino is the DM candidate.
The superpotential of the visible sector is , where
(10) 
and
(11) 
For simplicity, we have omitted the flavor and color indices. Henceforth, we use the same symbol for superfields and their corresponding parity even component fields, while parity odd fields are distinguished by a .
We note that some terms that are gaugeinvariant are absent in , namely . The first two terms are forbidden by parity, while invoking some other discrete or continuous symmetry (like an symmetry) may help forbid or suppress the last two terms.
Soft SUSY breaking terms typically make scalar components of chiral superfields heavier than their fermionic counterparts. In the case of superfield, one has , where denotes the soft mass of and is the term associated with the superpotential mass term for superfield . We assume the following mass condition
(12) 
It is also reasonable to assume that all parity odd colored particles have a mass larger than . The reason being that soft breaking masses of these particles are driven toward large values at low energies by radiative corrections with gauge interactions while is a singlet.
These conditions imply that cannot decay to either of or fields. Moreover, it cannot decay to any parity odd colored fields. Similarly, decay to parity odd scalars is kinematically blocked since these masses are governed by the soft terms, which supposedly have the smallest value for a singlet field. can however decay to DM particles, which is an essential part of this model.
As a consequence, dominantly decays into two gluons through the oneloop diagram shown in Figure 1.^{1}^{1}1 can also decay into fourbody final states containing through mediation of offshell . However, these decays are suppressed compared to the twobody decays by phase space factors and additional powers of , and hence can be neglected. There are also subdominant decay modes of that proceed via loop diagrams and are important for DM production and baryogenesis. We will discuss these modes later. The corresponding decay is given by
(13) 
Here is the gauge coupling constant and the two flavors of are taken into account. The precise expression for is given in the Appendix.
The fact that decay is loop suppressed combined with help us obtain a sufficiently low reheat temperature . For example, for TeV and TeV, we find if .
The magnitude of needed is similar to the electron Yukawa coupling, and also comparable to typical values of neutrino Dirac Yukawa couplings in TeV scale seesaw models. Its smallness may be explained in different ways. There may exist some symmetry, broken at the scale of grand unified theories (GUT), under which is charged. Then the term arises from a higherorder suppressed operator after symmetry breaking, and its strength will be suppressed by powers of . Moreover, see Eq. (13), we notice that the combination appears in , which acts like an effective coupling. Therefore one can make larger by simultaneously increasing with the same factor. For example, we find if GeV.
We also note that for a gravitationally interacting field the decay rate will be . It is seen from Eq. (13) that decay to gluons occurs with a strength much larger than that for a gravitational decay as long as . This is clearly the case for and MeV, which justifies belonging to the visible sector rather than a hidden sector with gravitationally suppressed coupling to matter.
iii.2 Gravitino Production
Gravitinos can be produced from decay if . Considering that TeV, this is kinematically possible only if . This implies that there will be no gravitino production form decay for a very wide mass range .
When kinematically possible, the important decay mode is , whose decay width is given by Moroi:1995fs ()
(14) 
Gravitinos with mass decay long after BBN. Their abundance should be low enough in order not to ruin successful predictions of BBN. Since , and is assumed to be lighter than all colored superparticles, gravitino decay modes can only be radiative. Successful BBN in this case requires that BBN2 ()
(15) 
This also insures that gravitino decay will not overproduce DM particles with a mass GeV.
The abundance of gravitinos produced from decay follows
(16) 
For the limit in Eq. (15) is satisfied provided that MeV. We see that gravitino production will not be a problem for the entire range of allowed by BBN (i.e., MeV). Moreover, this condition will be irrelevant altogether if , in which case decay to the gravitino will be forbidden kinematically.
The fact that that gravitino overproduction from decay is avoided comes as a direct consequence of mainly decaying into visible sector fields (i.e., gluons), while its decay to gravitinos is suppressed. The situation is very different when is a modulus field, as pointed out earlier, since all decay modes are gravitationally suppressed in that case.
Iv Dark Matter from Visible Sector Decay
In this section we show how the model presented in the previous section can produce the observed DM relic abundance via the “Branching Scenario” and “Annihilation scenario” both. We also discuss successful nonthermal production of Higgsino and Bino DM from visible sector decay. Recently, Higgsino type LSP has attracted significant attention from the natural SUSY perspective Papucci:2011wy (); Hall:2011aa (); Baer:2012uy (); Allahverdi:2012wb (); Gogoladze:2012yf (); Baer:2012cf (). However, if the Higgsino mass is in the subTeV region, the annihilation rate is larger than the nominal value cm s, which yields insufficient thermal relic abundance. The concern with this scenario is that the current constraint from the FermiLAT data does not allow much room for the annihilation crosssection to be too large. These results appear to prefer smaller annihilation crosssections for smaller values of LSP mass. This prefers Bino type LSP, which will have a thermally overproduced abundance.
iv.1 Branching Fraction for Decay to Dark Matter Particles
The decaying scalar has parity charge +1, which implies that it can only decay to an even number of parity odd particles. If it was parity odd instead, all of the decay modes would produce parity odd particles, thus resulting in . However, one can now obtain a small if decay to parity odd particles is suppressed relative to that of decay to gluons.
The dominant mode for producing parity odd particles is that proceeds through the oneloop diagram shown in Figure 2 (top diagram). This mode is kinematically allowed if Bino is the DM particle, see Eq. (III.1). The corresponding decay width is given by (for more details see the Appendix)
(17) 
where is the gauge coupling constant. Both flavors of have been taken. After using Eq. (13), this results in the following branching fraction
(18) 
Inserting numerical values for the parameters TeV, GeV, , , and , we find . This is much smaller than what one finds from modulus decay Allahverdi:2010rh ().
If the LSP is not purely Bino but rather mixed Higgsino/Bino or Wino/Bino, the branching fraction will be
(19) 
where is the Bino fraction of the DM.
For a Higgsino LSP the branching fraction in Eq. (19) will be vanishingly small since . However, Higgsinos can be directly produced from decay at twoloops through diagrams shown in Figure 2 (middle and bottom diagrams). The decay widths for the twoloop diagrams are
(20)  
where is the top Yukawa coupling, is the coefficient of term in Eq. (III.1), and is the Higgs hypercharge.
Unless is very small, the first expression in Eq. (IV.1) will be dominant. Then the branching fraction for Higgsino DM will be
(21) 
Much smaller values of can be obtained in the case of Higgsino DM. For example, we can find for TeV, GeV, and .
We note that the second expression in Eq. (IV.1) takes over for , which provides a lower bound on regardless of how small is.
iv.2 Higgsino Dark Matter
In Table 1, we show the model parameters and typical mass scales considered in this work. In Table 2, we give a summary of the various possibilities for Higgsino and Bino DM.
For a purely Higgsino DM, is given by Eq. (21). Since Higgsino LSP is thermally underproduced, the correct relic abundance can be obtained via both of the “Annihilation” and “Branching” scenarios discussed in Section II. The important factor in determining which of the scenarios can work is the size of coupling .
To illustrate this, we first consider the case with . Then Eq. (21) results in for GeV. Combined with the condition for not overproducing gravitinos from decay, see Eq. (16), this leads to . Therefore the observed relic density cannot be obtained in the “Branching Scenario”. This leaves the “Annihilation Scenario” as the only possibility when is large. This is shown in Table 2. For , “Annihilation Scenario” is the only option even for at the BBN bound. The correct relic density may be obtained in “Annihilation Scenario” for GeV Allahverdi:2012wb ().
Next, we consider the case with . Then we see from Eq. (21) that for GeV. This implies that the correct relic abundance via the “Branching Scenario” can be obtained for . A typical example of this scenario is shown in Table 2.
The flexibility that one can make either of the “Annihilation” and “Branching” scenarios work by dialing is also important from another point of view. “Annihilation Scenario” relies on a large annihilation rate . Therefore, tightening of the Fermi bounds on Hooper:2012sr (), will put increasing pressure on this scenario. It is therefore important that one can have a viable “Branching Scenario”, regardless of the improving constraints, by lowering .
Parameter  Value 

subTeV  
DM  Mass  Scenario  


Annihilation ()  
Branching ()  

Branching () 
iv.3 Bino Dark Matter
Bino DM is thermally overproduced. This, as pointed before, leaves “Branching Scenario” as the only possibility for producing the correct DM density. In this case , where for TeV. Obtaining the observed relic density then requires that . We see from Eq. (18) that this can be found for a Bino on the lighter side GeV when the reheat temperature is close to its lower value from BBN bound MeV.
Having a viable scenario for nonthermal Bino DM is important. With the Fermi bounds on the annihilation rate improving GeringerSameth:2011iw (), Higgsino DM may be ruled out specially at the lower end of the subTeV mass range. This will motivate Bino DM, for which one can find a successful nonthermal scenario based on decay.
V Baryogenesis
The entropy released in decay dilutes any previously produced baryon asymmetry, see Eq. (II.1), thus necessitating mechanisms of baryogenesis at temperatures far below the electroweak symmetry breaking scale. This can be achieved by introducing baryon number and violating operators in a suitable extension of the SM Allahverdi:2010im (); Babu:2006wz (). Late decay of naturally provides the necessary departure from thermal equilibrium.
The model given in Eq. (III.1) can give rise to lowtemperature baryogenesis via decay Allahverdi:2010im (). The decay of produces quanta through the oneloop diagram shown in Fig. 3 (top diagram). The rate for decay to the heavier of the two flavors, denoted by , is
(22) 
where is the coefficient of term. We have chosen , so that the loop containing dominates the process. We note that decay to and quarks is subdominant due to the helicity suppression of fermionic final states, while decay to scalar and squarks is kinematically blocked, see Eq. (III.1) and the subsequent discussion.
decay via the diagram in Fig. 3 generates baryon asymmetry, whose density is given by
(23) 
where is the branching fraction for producing from decay, and is the asymmetry parameter in decay. is given by
(24) 
For , we find .
Taking , similar masses for fermions and scalars, for all flavors and colors (in accordance with Table I), and violating phases of , the asymmetry parameter is given by
(25) 
The factor comes from summing over three flavors of quarks plus including diagrams where are switched . Then, for , we get .
One comment is in order at this point. fermions produced in decay themselves to three quarks. This must happen well before the onset of BBN in order not to ruin its success. The decay rate of is given by Babu:2006wz ()
(26) 
where is the term associated with mass term in Eq. (III.1), and is a color multiplicity factor. One can see that decay well before 0.1 s for GeV, , and the parameter values given in Table I.
Finally, we note that the “Branching Scenario” for nonthermal DM production (like in the case of Bino LSP) provides a visible sector realization of “Cladogenesis” mechanism proposed in Allahverdi:2010rh () to address the baryonDM coincidence puzzle. It is clear from our discussions that this scenario is a considerably more flexible than that when modulus decay is the source of nonthermal DM Allahverdi:2010rh (); Allahverdi:2012wb ().
Vi Conclusion
In large regions of SUSY parameter space, the DM annihilation crosssection is larger (as for Higgsino or Wino LSP) or smaller (as for Bino LSP) than the nominal value of cm s in the thermal WIMP scenario. Nonthermal scenarios of DM production will be needed in these cases to yield the correct relic abundance, while being compatible with the bounds from indirect DM searches on the DM annihilation cross section particularly the FermiLAT results.
Motivated by these considerations, in this paper, we have presented a nonthermal scenario that relies on the visible sector of a SUSY model. In this model the late decay of an parity even scalar field that is a SM singlet (but may be charged under a higher rank gauge group) produces DM. is coupled to new colored fields with a mass relation . Assuming that all parity odd colored fields are heavier than , it will dominantly decay into gluons at the oneloop level. The combination of a small coupling between and , the mass relation , and the oneloop factor can naturally lead to a late decay of that yields a low reheat temperature . As we saw above, one can obtain for TeV, TeV, and .
The branching fraction for decay to DM particles depends on the nature of the LSP. If the LSP is Bino, then can decay to a pair of DM particles at the oneloop level, where for GeV. For a Higgsinos LSP, the same decay occurs at the twoloop level yielding (the exact value depending on the model parameters) for GeV. As a consequence, one can obtain the observed relic abundance in both larger and smaller annihilation crosssection regions of SUSY parameter space.
For Higgsino DM, the correct relic density can be found directly from decay if is sufficiently small. The DM annihilation crosssection will be irrelevant in this case (hence “Branching Scenario”). We showed a benchmark point for this scenario where GeV and . For larger values of , Higgsinos will undergo residual annihilation upon production from decay (hence “Annihilation Scenario”). This scenario can yield the correct relic abundance if the annihilation rate is larger than by a factor of .
In the case of Bino DM, where the annihilation crosssection is small, “Branching Scenario” is the only option. The correct relic density is obtained for a lighter Bino GeV and smaller values of reheat temperature MeV. One can reduce by lowering the coupling and/or raising . A viable scenario for nonthermal Bino DM is particularly important in light the FermiLAT result, which appear to prefer smaller annihilation crosssections for smaller values of DM mass.
The above scenarios are summarized in Table 2.
The scenario presented in this paper has two main advantages over that using moduli decay. First, gravitino production from decay is naturally suppressed by the virtue of belonging to the visible sector. Second, can be made sufficiently small by dialing the model parameters, which is essential for a successful realization of the “Branching Scenario” (the only possibility in the case of Bino DM). To attain these virtues in scenarios of nonthermal DM from moduli decay, one needs to satisfy nontrivial conditions on the Kähler geometry of the underlying effective supergravity theory.
Finally, our model can also successfully generate the baryon asymmetry of the universe. decay also produces the SM singlets at the oneloop level, whose subsequent decay via baryon number and violating couplings to the heavy colored states can yield the desired asymmetry . Moreover, the baryonDM coincidence puzzle can be addressed in the context of the “Branching Scenario”, since both baryon asymmetry and DM abundance are directly produced from decay in this case.
Vii Acknowledgement
This work is supported in part by the DOE grant DEFG0295ER40917.
Appendix A Calculations of Decay Width
We first discuss the decay of into two gluons. There is a contribution from the loop of fermions shown in Fig. 1, as well as a loop of sfermions. The total decay width is given by Djouadi:2001kba ()
(27)  
where
(28) 
and
(29) 
Here, is the mass of the particle in the loop. The amplitudes and refer to the fermion and sfermion loops respectively.
With the assumed mass conditions, one obtains the decay width given in Eq. (13), which further accounts for the two flavors of .
The decay width into Binos is given by
(30) 
where
In the above equation, and are VeltmanPassarino functions Passarino:1978jh (). With the assumed mass conditions, this expression is reduces to Eq. (17).
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