#
UNification
With GAugino

And SFermion
MAss Non-UNiversality

###### Abstract

In the context of a SUSY GUT inspired MSSM version, the
low energy consequences of the asymptotic Yukawa coupling
Unification are examined, under the assumption of universal or
non-universal boundary conditions for the gaugino and sfermion
masses. Gaugino non-universality is applied, so that the SUSY
corrections to -quark mass can be reconciled with the present
experimental data on muon anomalous magnetic moment. Restrictions
on the parameter space, originating from the cold dark matter
abundance in the universe, the inclusive branching ratio of
and the accelerator data are, also,
investigated and the scalar neutralino-proton cross section is
calculated. In the case of a bino-like LSP and universal boundary
conditions for the sfermion masses, the constraints, arising from
the cold dark matter and can be
simultaneously satisfied, mainly thanks to the -pole effect or
the neutralino-stau coannihilations. In addition, sfermion mass
non-universality provides the possibility of new coannihilation
phenomena (neutralino-sbottom or neutralino-tau sneutrino-stau),
which facilitate the simultaneous satisfaction of all the other
requirements. In both cases above, the neutralino abundance can
essentially decrease for a W-ino or higgsino like LSP creating
regions of parameter space with additional neutralino-chargino
and/or heavier neutralino coannihilations. The neutralino-sbottom
mass proximity significantly ameliorates the detectability of LSP.

Keywords: Supersymmetric Models, Dark Matter

PACS codes: 12.60.Jv, 95.35.+d

^{†}

^{†}preprint: SISSA-27/2003/EP \JPub

Nucl. Phys. B678, 398 (2004)

[\fancyplain 1]\fancyplain UNIFICATION WITH GAUGINO AND SFERMION MASS NON-UNIVERSALITY \lhead[\fancyplain ]\fancyplain1 \cfoot

## 1 INtroduction

In some recent papers [1, 2], the early prediction in the context of Grand Unified Theory (GUT) [3] of the asymptotic Yukawa coupling Unification (YU) was beautifully combined with the present experimental data on neutrino physics within models. However, applying this scheme in the framework of the Constrained Minimal Supersymmetric (SUSY) Standard Model (CMSSM) [15] and given the top and tau experimental masses, the parameter is restricted to negative values [5, 6]. This is due to the fact that the tree level -quark mass receives sizeable SUSY corrections [7], which can drive the corrected -quark mass within its experimental range only for . On the other hand, the case is in conflict with the present experimental data [8] on the muon anomalous magnetic moment [5, 6]. Indeed, the deviation, , of the measured value of the muon anomalous magnetic moment from its predicted value in the Standard Model (SM), seems to favor the regime [9, 10]. In addition, the negative sign of is severely restricted by the recent experimental results [11] on the inclusive branching ratio [5, 6], which bounds below the SUSY spectrum to rather high values.

At the same time, the SUSY spectrum can be bounded above by the requirement that the relic density, of the lightest SUSY particle (LSP) in the universe does not exceed the upper limit on the Cold Dark Matter (CDM) abundance implied by cosmological considerations. After the recent experimental results of WMAP [12], this limit turns out to be as much stringent as the previous one (see, e.g. Ref. [13]) with a significantly better accuracy (see sec. 2.1). Since increases with the LSP mass , this limit imposes a very strong upper bound on . However, this can significantly weaken in regions of the parameter space, where a substantial reduction of can be achieved, mainly thanks to the pole effect (PE) and/or the coannihilation mechanism (CAM). In the first case, which is applicable for large [14, 15], the reduction is caused by the presence of a resonance () in the mediated LSP’ s annihilation channel. On the other hand, CAM is activated for any , when a mass proximity occurs between the LSP and the next to LSP (NLSP). In the context of CMSSM, NLSP can be the lightest slepton [16, 17] and particularly stau [18, 15] for large or stop [19, 20] for very large trilinear coupling, or also, chargino [21, 22]. Possible non-universality in the Higgs sector [23] gives rise to CAMs between LSP and /-sneutrino and/or chargino-sleptons.

As induced from the previous considerations, the viability of the YU hypothesis in the context of CMSSM becomes rather dubious. On the other hand, the embedding of MSSM into a or SUSY GUT leads, also, to a variety of possibilities [24] beyond the CMSSM universality. In this paper we will employ these scenaria in order to obtain SUSY spectra compatible with all these Cosmo-Phenomenological requirements. Namely non universal gaugino masses (UGMs) [25, 26] are applied, so that the inconsistency between the constraint and the -quark mass experimental data is removed. This effect can produce gaugino inspired CAMs (neutralino-chargino and/or heavier neutralino, ). On the other hand, new sfermionic CAMs (neutralino-sbottom, and neutralino-tau sneutrino-stau, ) can be caused by applying non universal sfermion masses (USFMs) [27, 28]. Both phenomena by themselves or in conjunction can drastically reduce to an acceptably low level, increasing the upper bound on almost up to . As a bonus, in both latter cases, the satisfaction of the BR() criterion is facilitated, allowing viable parameter space consistent even with the optimistic upper bound on from constraint (see sec. 2.5). Consequently the neutralino-proton cross sections are sensitively increased, especially in the case of CAM.

Combination of YU and non UGMs were previously considered in Refs. [25, 26], where, contrary to our case, a single dominant direction in gauge kinetic function has been assumed (see sec. 3). Additional presence of non USFMs were not studied until now, while the CAM was just noticed in Ref. [27]. Our main improvements are the consideration of the constraint, the reproduction of the CAM in a much more restrictive framework, the finding of the CAM and the study of the co-existence of the gaugino inspired CAMs.

The framework of our analysis is described in some detail in sec. 2. The basic features of our model are established in sec. 3. Our numerical approach and results are exhibited in secs. 4 and 5. We end up with our conclusions and some open issues in sec. 6. Throughout the text and the formulas, brackets are used by applying disjunctive correspondence.

## 2 COsmo-PHenomenological FRamework

We briefly describe the operation of the various Cosmo-Phenomenological criteria that we will use in our investigation. In the following formulas, gaugino masses, , top trilinear coupling , parameter and the various SUSY corrections, to -quark [-lepton] mass are calculated, at a SUSY breaking scale, specified in sec. 4.

### 2.1 COld DArk MAtter COnsiderations

According to WMAP results [12], the total (M) and the baryonic (B) matter abundance in the universe is respectively:

(1) |

We, thus, deduce the confidence level (c.l.) range for the CDM abundance [29]:

(2) |

In the context of MSSM, the lightest neutralino, can be the LSP. It consists the most natural candidate for solving the CDM problem, being neutral, weakly interacting and stable in the context of SUSY theories with -Parity [30] conservation. Hence, in our analysis, we require the LSP relic density not to exceed the upper bound of Eq. (2):

(3) |

We calculate , using micrOMEGAs [31], which is one of the most complete publicly available codes. This includes accurately thermally averaged exact tree-level cross sections of all possible (co)annihilation processes, treats poles properly and uses one loop QCD corrections to the Higgs decay widths and couplings into fermions [32].

### 2.2 SCalar NEutralino-PRoton CRoss SEction

Neutralinos could be detected via their elastic scattering with nuclei [33]. The quantity which is being conventionally used in the literature (e.g. [34, 35]) to compare experimental [36, 37] and theoretical results is the scalar neutralino-proton () cross section,

(4) |

and is the scalar contribution to the effective coupling. We calculate it, using the full one loop treatment of Ref. [35] (with some typos fixed [38]). This is indispensable for a reliable result in the region , where the tree level approximation (which, indeed works well elsewhere) fails. For the involved renormalization-invariant functions were adopted the values [34] (in the notation of Ref. [35]):

(5) |

Combining the sensitivities of the recent [36] and planned [37] experiments, we end in the following phenomenologically interesting region, for :

(6) |

For SUSY spectra of our models consistent with all the other constraints of sec. 2, the obtained lies beyond the claimed by DAMA preferred range, , which however, has mostly been excluded by other collaborations (e.g. EDELWEISS, ZEPLIN I).

### 2.3 Susy COrrections To -QUark And -LEpton MAss

In the large and intermediate regime, the tree level -quark mass, receives sizeable SUSY corrections [7, 39, 40], which arise from sbottom-gluino, (mainly) and stop-chargino, loops. interferes destructively (see Eq. 12)) to which, normally (an exception is constructed in Ref. [41]) dominates . Consequently:

(7) |

using the standard sign convention of Ref. [43]. Hence, for , the corrected -quark mass at a low scale ,

(8) |

turns out to be above [below] its tree level value . The result is to be compared with the c.l. experimental range for . This is derived by appropriately [44] evolving the corresponding range for the pole -quark mass, up to scale with , in accord with the analysis in Ref. [45]:

(9) |

Less important but not negligible (almost 5) are the SUSY corrections to -lepton, originated from [39] sneutrino-chargino, (mainly) and stau-neutralino, loops, with the following signs:

(10) |

### 2.4 BRanching RAtio Of

Taking into account the recent experimental results [11] on this ratio, , and combining appropriately the experimental and theoretical involved errors [44], we obtain the following c.l. range:

(11) |

We compute by using an updated version of the relevant calculation contained in the micrOMEGAs package [31]. In this code, the SM contribution is calculated using the formalism of Ref. [46] including the improvements of Ref. [47]. The contribution is evaluated by including the next-to-leading order (NLO) QCD corrections from Ref. [48] and enhanced contributions from Ref. [49]. The dominant SUSY contribution, , includes resummed NLO SUSY QCD corrections from Ref. [49], which hold for large . The contribution interferes constructively with the SM contribution, whereas interferes de[con]-structively with the other two contributions for , since [42], in general:

(12) |

However, the SM plus contributions and the decrease as increases and so, a lower bound on can be derived from Eq. (11a [11b]) for with the latter being much more restrictive. It is obvious from Eqs. (12) and (7) that simultaneous combination of negative correction to -quark mass and destructive contribution of is impossible [42].

### 2.5 MUon ANomalous MAgnetic MOment

The deviation, of the measured value from its predicted value in the SM, can be attributed to SUSY contributions, arising from chargino-sneutrino and neutralino-smuon loops. is calculated by using micrOMEGAs routine based on the formulæof Ref. [50]. The absolute value of the result decreases as increases and its sign is:

(13) |

On the other hand, the calculation is not yet stabilized mainly due to the instability of the hadronic vacuum polarization contribution. According to the evaluation of this contribution in Ref. [9], the findings based on annihilation and -decay data are inconsistent with each other. Taking into account these results and the recently announced experimental measurements [8] on , we impose the following c.l. ranges:

(14) | |||||

(15) |

A lower bound on can be derived for from Eq. (14b [15a]) and an optimistic upper bound for from Eq. (14a) which, however is not imposed as an absolute constraint due to the former computational instabilities. Although the case can not be excluded [51], it is considered as quite disfavored [10], because of the poor -decay data. For this reason, following the common practice [23, 29], we adopt the restrictions to parameters induced from Eq. (14).

### 2.6 COllider BOunds

## 3 PArticle MOdel

The embedding of MSSM in a SUSY GUT enriches the model with extra constraints and opens new possibilities beyond the CMSSM universality [24]. We below select some of them, constructing the Yukawa (sec. 3.1), gaugino (sec. 3.2) and scalar (sec. 3.3) sector of our theory. Actually, it is a variant of the models proposed in secs. II and III of Ref. [24]. The phenomenological reasons which push us in the introduction of the non-universalities of sec. 3.2 and 3.3 will become obvious during the presentation of our results in sec. 5. However, for clarity, let us outline them shortly. Consistency of YU with Eq. (14a) requires a proper application of non UGMs. The resulting model has still two shortcomings: (i) Uninteresting due to large minimal , since this is derived from Eq. (11b) (ii) Inability for the satisfaction of Eq. (14b). Non USFMs assist us to alleviate both disadvantages.

### 3.1 UNification

In the minimal SUSY GUT [24, 6], the third generation left handed superfields , belong to the representation (reps), while , , belong to the reps. Assuming that the electroweak Higgs superfields , are contained in and reps, respectively, the model predicts YU at GUT scale, ( is determined by the requirement of gauge coupling unification):

(17) |

since, the Yukawa coupling terms of the resulting version of MSSM:

(18) |

originate from an unique term, of the underlying GUT.

The asymptotic relation of Eq. (17) can also arise in the context of SUSY GUT. In this case, a family of fermions is incorporated in the spinorial reps. Assuming that , are contained in two different Higgses [24, 1] in and , the terms in Eq. (18) can be again derived from an unique term, . Alternatively, if , are expressed as a combination of and , large atmospheric neutrino mixing, produced through a non canonical see-saw mechanism, requires [2] YU.

Assuming exact YU at and given the top and tau masses, and can not be both free parameters (see, also, sec. 4). We choose as input parameter in our presentation (in contrast with the usual in the literature strategy, see, e.g. Refs. [5, 6, 25, 26]). As a consequence, a prediction can be made for . Furthermore, the sign of has to be negative. This is, because close to the complete YU (), we obtain close to the upper edge of the range of Eq. (9). As decreases, increases [55] and so, a negative can drive for some values of (see, e.g. Fig. 4 of Ref. [56]) within the above range. Combining this result with Eq. (7), we conclude that YU can become viable only for (in accordance with the findings of Refs. [5, 6, 25, 26]).

### 3.2 GAugino SEctor

From the discussion of the previous section, we can induce that YU in the context of CMSSM [15], is viable only for . Consequently the parameter space of the model can be restricted through Eq. (15), which, however, is rather oracular. To liberate the model from this ugly feature, we invoke a departure from the UGMs. The importance of non UGMs in addressing the former inconsistency has already been stressed in Refs. [25, 26]. Indeed, from Eqs. (7) and (13), we can infer that negativity of and positivity of , can be reconciled with the following arrangement:

(19) |

and . Such a condition can arise, by employing a moderate deviation from the minimal Supergravity (mSUGRA) scenario [58, 59, 60, 61, 62] (for an other approach see, e.g. Refs. [57]) as follows.

According to the gravity-mediated SUSY breaking mechanism [64], gaugino masses are generated by a left handed chiral superfield , which appears linearly in the gauge kinetic function ( run over the GUT gauge group generators). During the spontaneous breaking of the GUT symmetry, auxiliary fields , components of , coupled to gauginos, acquire vacuum expectation values (vevs), which can be considered as the asymptotic . In mSUGRA models, the fields are treated as singlets under the underlying GUT and therefore, UGMs result. However, can belong to any reps in the symmetric product () of two adjoints [58]. Thus, after the breaking of the GUT symmetry to the SM one, acquire vevs in the SM neutral direction , where are group theoretical factors. Therefore, the gaugino masses at can be parameterized as follows ():

(20) |

where is a gaugino mass parameter and the characteristic numbers of every reps have been worked out in Ref. [63] for the and in Ref. [65] for the GUT. Also, are the relative weight of the reps in the sum. They can be treated as free parameters, while there is no direct experimental constraint on the resulting signs [60]. As emphasized in Refs. [59, 24], all the cases are compatible with the gauge coupling unification with the assumption that the scalar component of develops negligible vev. One can next show, by solving the resulting 3x3 systems (we ignore for simplicity the large reps 220 [770] for ), that there is a wide and natural set of ’s in Eq. (20) for , so that the ratio of Eq. (19) can be realized. To keep our investigation as general as possible, we will not assume (as usually [25, 59, 65]) dominance of a specific direction in Eq. (20). Nevertheless, we will comment on these more restrictive but certainly more predictive cases, in the conclusions.

We will close this section, quoting several important comments:

i. In both cases of Eq. (19), interferes constructively to SM+ contribution and therefore, the satisfaction of Eq. (11b) has to be attained.

ii. The renormalization group running is affected very little by the specific choice one of the two possibilities in Eq. (19). However, we choose to work with , since in this case the resulting value of is slightly diminished. This is due to the fact that for , increases (due to the additive correlation of the two contributions in Eq. (10) which does not exist for with from Eq. (19)). As a consequence, with given the tau yukawa coupling , a larger (or smaller ) for is needed so as a successful is obtained (see, also, sec. 4).

iii. For and , LSP is mainly a pure B-ino. However for and/or , LSP can become W-ino or Higgsino like and the mass of charginos and/or gluinos decrease so as they can coannihilate with LSP reducing even lower than the expectations [72]. Despite the fact that much lower than the bound of Eq. (2) can not be characterized as a fatal disadvantage of the theory (since other production mechanisms of LSP may be activated [73] and/or other CDM candidates may contribute to ) we will keep our investigation in regions of , where turns out to be close to the bound of Eq. (2) for .

### 3.3 SCalar SEctor

In the minimal SUSY GUT [67, 24], the soft SUSY breaking terms for the sfermions in reps (, , ), and in reps (, ), can be different. Consequently, the soft SUSY breaking masses for the sfermions of the resulting MSSM at can be written as:

(21) | |||||

(22) |

where we have adopted the parameterization and the range for of Refs. [68, 27]. Similar splitting between sfermion masses can also occur in the context of [69, 24, 28] although in the presence of some -terms. To suppress dangerous flavor changing neutral currents [70], we maintain the universality among generations. Remarkably, such a non-universality among sleptons can be probed at colliders [71]. As we will see in sec. 5, can decrease considerably the resulting masses of and so as to have a chance to play the rôle of coannihilator during the LSP freeze out in the Early Universe, reducing efficiently .

As regards the soft masses of the two higgses , included in , they can be in general arbitrary (even in the case of SUSY GUT since we assumed two higgses in sec. 3.1)

(23) |

However, trying to isolate the non USFMs and reduce the number of the free parameters, we will restrict ourselves to the simplificative case . Nevertheless, we will comment on the conclusion how this assumption can influence our results.

## 4 NUmerical CAlculation

In our numerical calculation, we closely follow the notation as well as the renormalization group and radiative electroweak symmetry breaking (RESB) analysis of Refs. [44]. We integrate the 2-loop renormalization group equations (RGEs) for the gauge and Yukawa coupling constants and 1-loop for the soft SUSY breaking terms between and a common SUSY threshold ( are the stop mass eigenstates) determined in consistency with the SUSY spectrum. At we impose RESB, evaluate the SUSY spectrum and incorporate the SUSY corrections to and masses [39, 40]. Between and , the running of gauge and Yukawa couplings is continued using the SM RGEs.

We use fixed values for the running top quark mass and tau lepton mass . Using an iterative up-down approach, and are determined for each at , while is derived from the YU assumption, Eq. (17). Equivalently, turning the procedure around, can be adjusted so, that the derived corresponds to a desired . Fixing it, also, to its central experimental value of Eq. (9), , a prediction for can be made, as already mentioned in sec. 3.1. Finally, we impose the boundary conditions given by Eqs. (19) with for the gaugino masses, by Eqs. (21)-(23) for the scalar masses and we also, assume a universal trilinear scalar coupling, .

In summary, our effective theory below depends on the parameters :

To further reduce the parameter space of the model, we fix (as usually [15, 23, 29]) . is not expected to change dramatically our results. Also, for presentation purposes, and can be replaced by and a relative mass splitting, , defined as follows:

(24) |

The choice of this parameter is convenient, since it determines, for given , the strength of the PE for P : or of the CAM for P : . It, thus, essentially unifies the description of both reduction “procedures” (see sec. 5). Note that although can be defined through a relation similar to this in the second line of Eq. (24) with P : , these can not be used in order to determine the spectrum, since they depend crucially only on and not on . So, they vary very slowly, once or have been chosen. Consequently, our final set of the considered free parameters is:

## 5 REsults

We proceed, now in the delineation of the parameter space of our model. For the sake of illustration, we divide this section in subsections devoted to each applied reduction “procedure”. The CAMs are classified to 3 main categories based to the sfermionic ones with or without the presence of CAMs. In this case, the allowed ranges of the basic parameters and are listed comparatively in the Tables 1, 2, 3 together with the relative contributions beyond a threshold value of the (co)annihilation processes to the calculation as and vary in their allowed (or, indicative in some cases) ranges. The allowed regions on the plane from the various absolute constraints of sec. 2 are shaded, while the regions favored by the optimistic upper bound of Eq. (14a) are ruled. For simplicity, we do not show bounds from constraints less restrictive than those which are crucial.

Let us introductionary explain the reasons for which we will focus on some specific and . Initially, in order to make contact with the highly predictive and well-investigated parameter space of CMSSM (see, e.g. Refs. [19, 21, 22, 17, 29]) we will consider . Indeed, for , the resulting low energy values of the soft SUSY breaking terms turn out to be quite similar to those that we would have obtained, if we had imposed UGMs with . The gaugino running (see, e.g. Ref. [66]) and essentially, the LSP gaugino purity, (in the notation of Ref. [66]), remain unaltered. The scalar running is altered by a few percent due to the resulting lower values of the trilinear couplings. This is, because their running crucially depends on the relative sign of and . The latter difference has the following remarkable consequences. In our case: (i) turns out to be larger. This is because, anti-correlates more weakly with due to lower (sec. 2.3). (ii) is significantly decreased (especially in the case of universal sfermion masses), since the tree level has to be larger (sec. 3.1), so that after the subtraction of the larger , the resulting is within its experimental margin of Eq. (9). (iii) is lower, since and the contribution is diminished, mainly due [49] to the larger denominator of the resummation and lower enhanced contributions, respectively.

For and , we will present the mass parameters and the allowed regions on the plane. Possible variation of is not expected to change the general characteristics of the mass parameters. Also, we checked that and/or do not create new CAMs and so, do not essentially deform the allowed regions. However, for and/or , additional CAMs can further enlarge them. A first example will be given for and . With this choice, is established, creating a background of useful (not very drastic) CAMs (in accord with Ref. [72]) which can be combined with the sfemionic ones. However, essential reduction of is obtained only for . For and , new situation for the calculation emerges for . Then, and CAMs reduce lower than the expectations. Instead, we will insist on the choice, and or 0.6, for which CAMs can be kept under control.

Finally, let the hadronic inputs of Eq. (5) vary within their ranges we will derive the corresponding bands on the plane for various and , while possible improvement for and will be illustrated, too. The findings will be collectively presented in Fig. 5, but the explanations will be given separately in each subsection.

### 5.1 -POle EFfect (With or Without COannihilations)

For and , reduction of caused by the PE is possible. Especially, for , there are two different combinations of and , which support this possibility. In Figs. 1-(a) and 1-(c), we present the mass parameters , , and versus for in these two cases. We observe that the main difference between them is related to the value of , which turns out to be relatively high [low] in Fig. 1-(a [c]). In these, the various lines terminate at low [high] ’s due to improper RESB () [].

The corresponding allowed areas on the plane are displayed in Figs. 1-(b) and 1-(d). In both cases, the left (almost vertical) boundary of the allowed (shaded) region comes from Eq. (11b) while the lower and upper curved boundaries correspond to the saturation of Eq. (3). A simultaneous satisfaction of Eq. (14a) is impossible. More explicitly, we find the following allowed ranges:

(25) |

with in the high case (Fig. 1-(b)). Saturation of the optimistic bound from Eq. (14a) is not possible in the overall investigated parameter space.

(26) |

with in the low case (Fig. 1-(d)). The bound of Eq. (14a) implies .

Comparing Figs. 1-(b) and 1-(d), we observe that the allowed area for low can be included in this for high , with the lower and upper curved boundaries being almost identical. The main difference is that the bound from Eq. (11b) is more restrictive in the low case, due to the lighter stop spectrum. This difference is also shown in Fig. 5-(a), where we depict versus for and low [high] (grey [light grey]) band. Obviously the high is phenomenologically more attractive.

Coexistence of PE and CAM can further enlarge the allowed areas. Indeed, analyzing two cases, we find

(27) |

with and high for and (Fig. 2-(a)) without possibility of saturation of the optimistic bound from Eq. (14a). The obtained turns out to be quite similar to this of light grey band in Fig. 5-(a).

(28) |

with and low for and (Fig. 2-(b)). The bound of Eq. (14a) can be satisfied for due to low . The corresponding is also increased due to stronger higgsino component of the LSP as is shown in Fig. 5-(a), cyan band.

Due to the existence of the CAM, in both latter (i, ii) cases, there is no upper [lower] bound on , for , contrary to the former cases (i, ii). Evident is, also, in any case that the reduction, because of the PE, is more efficient for than for , in accord with the findings of Ref. [74].

In both cases, the relative contributions beyond 5 of the (co)annihilation processes to as and vary in the ranges of Eq. (25) or (26) [(27) or (28)], are:

If we had imposed UGMs with and , the high case would not have survived, due to larger which would have invalidated the RESB, while for low , we could have found allowed area similar to this in Fig. 2-(b) with and .

### 5.2 COannihilations

The most usual and easily achieved (for , also) CAM is the CAM. The mass parameters , , and which support this situation, are plotted versus for and in Fig. 3-(a). We observe that , unlike all the other cases.

The corresponding allowed area on the plane, for the same ’s is depicted in Fig. 3-(b) and it turns out to be disconnected from these of subsec. 5.1. The left (almost vertical) [right curved] boundary of the allowed (shaded) region is derived from Eq. (11b [3]). It is obvious that strong degeneracy among LSP and NLSP is needed in order the criteria of Eqs. (3) and (11b) to be simultaneously fulfilled without a possible achievement of Eq. (14a), since it implies (Table 1, left column). Consequently, the corresponding lies also well below the range of Eq. (6b), as shown in Fig. 5-(b), dark grey band.

MODEL PARAMETERS | |||
---|---|---|---|

ALLOWED RANGES | |||

PROCESSES WHICH CONTRIBUTE MORE THAN | |||

PROCESS | CONTRIBUTION () | ||