###### Abstract

We analyse the phenomenology of orbifold scenarios from the heterotic superstring, and the resulting theoretical predictions for the direct detection of neutralino dark matter. In particular, we study the parameter space of these constructions, computing the low-energy spectrum and taking into account the most recent experimental and astrophysical constraints, as well as imposing the absence of dangerous charge and colour breaking minima. In the remaining allowed regions the spin-independent part of the neutralino-proton cross section is calculated and compared with the sensitivity of dark matter detectors. In addition to the usual non universalities of the soft terms in orbifold scenarios due to the modular weight dependence, we also consider D-term contributions to scalar masses. These are generated by the presence of an anomalous , providing more flexibility in the resulting soft terms, and are crucial in order to avoid charge and colour breaking minima. Thanks to the D-term contribution, large neutralino detection cross sections can be found, within the reach of projected dark matter detectors.

FTUAM 07/02

IFT-UAM/CSIC-07-09

KUNS-2091

arXiv:yymm.nnnn

6 September 2007

Prospects for the direct detection of neutralino dark matter in orbifold scenarios

David G. Cerdeño , Tatsuo Kobayashi , Carlos Muñoz

[0.2cm]

Departamento de Física Teórica C-XI, Universidad Autónoma de Madrid,

[0pt] Cantoblanco, E-28049 Madrid, Spain

[0pt] Instituto de Física Teórica C-XVI, Universidad Autónoma de Madrid,

[0pt] Cantoblanco, E-28049 Madrid, Spain

[0pt] Department of Physics, Kyoto University, Kyoto 606-8502, Japan.

[0pt]

PACS: 11.25.Wx, 95.35.+d

Key words: String phenomenology, Dark matter

## 1 Introduction

One of the most interesting candidates for the dark matter in the
Universe is a
Weakly Interacting Massive Particle (WIMP), and in fact
many underground
experiments are being carried out around the world in order to detect
its flux on the Earth [1].
These try to observe the elastic scattering of
WIMPs on target nuclei through nuclear recoils.
Although one of the experiments, the DAMA collaboration [2],
reported data favouring the existence of a signal with
WIMP-proton cross section pb
for a WIMP mass smaller than GeV [2, 3],
other collaborations such as
CDMS Soudan [4], EDELWEISS [5], and
ZEPLIN I [6]
claim to have excluded important regions of the DAMA
parameter space^{1}^{1}1
For attempts to show that DAMA and these experiments
might not be in conflict, see Ref. [7]..
Recently, the
XENON10 experiment at the Gran Sasso National Laboratory
[8] has set the strongest upper limit for the WIMP-proton
cross section, further disfavouring the DAMA result.
This controversy will be solved in the future since many
experiments are running or in preparation around the world.
For example, LIBRA [9] and ANAIS [10] will probe the
region compatible with DAMA result.
Moreover, CDMS Soudan will be able to explore a WIMP-proton cross
section pb, and
planned 1 tonne Ge/Xe detectors are expected to reach cross
sections as low as
pb [11].

The leading candidate within the class of WIMPs is the lightest neutralino, , a particle predicted by supersymmetric (SUSY) extensions of the standard model. Given the experimental situation, and assuming that the dark matter is a neutralino, it is natural to wonder how big the cross section for its direct detection can be. This analysis is crucial in order to know the possibility of detecting dark matter in the experiments. In fact, the analysis of the neutralino-proton cross section has been carried out by many authors and during many years [1]. The most recent studies take into account the present experimental and astrophysical constraints on the parameter space. Concerning the former, the lower bound on the Higgs mass, the and branching ratios, and the muon anomalous magnetic moment have been considered. The astrophysical bounds on the dark matter density,

In the usual minimal supergravity (mSUGRA) scenario, where the soft terms of the minimal supersymmetric standard model (MSSM) are assumed to be universal at the unification scale, GeV, and radiative electroweak symmetry breaking is imposed, the neutralino-proton cross section turns out to be constrained by pb. Clearly, in this case, present experiments are not sufficient and only the planned 1 tonne Ge/Xe detectors would be able to test part of the parameter space. However, in the presence of non-universal soft scalar and gaugino masses [14] the cross section can be increased significantly [15] in some regions with respect to the universal scenario (see, e.g., the discussion in [16], and references therein). Although the current upper limit on the decay seriously affects these results, as was pointed out in [17], regions of the parameter space can still be found where the neutralino detection cross section can be within the reach of experiments such as CDMS Soudan. An analysis, summarizing all these results in the context of SUGRA, can be found in [18].

On the other hand, the low-energy limit of superstring theory is SUGRA, and therefore the neutralino is also a candidate for dark matter in superstring constructions. Let us recall that, in the late eighties, working in the context of the heterotic superstring, a number of interesting four-dimensional vacua with particle content not far from that of the SUSY standard model were found (see, e.g., the discussion in the introduction of [19], and references therein). Such constructions have a natural hidden sector built-in: the complex dilaton field arising from the gravitational sector of the theory, and the complex moduli fields parametrizing the size and shape of the compactified space. The auxiliary fields of those gauge singlets can be the seed of SUSY breaking, solving the arbitrariness of SUGRA where the hidden sector is not constrained. In addition, in superstrings the gauge kinetic function, , and the Kähler potential, , can be computed explicitly, leading to interesting predictions for the soft parameters [20]. More specifically, in orbifold constructions they show a lack of universality due to the modular weight dependence. From these resulting SUGRA models one can also obtain predictions for the value of the neutralino-proton cross section. In fact, analyses of the detection cross section in these constructions were carried out in the past in [21, 22, 23].

Our aim in this work is to study in detail the phenomenology of these orbifold models, including the most recent experimental constraints on low-energy observables, as well as those coming from charge and colour breaking minima, and to determine how large the cross section for the direct detection of neutralino dark matter can be. We therefore calculate the theoretical predictions for the spin-independent part of the neutralino-nucleon cross section, , and compare it with the sensitivities of present and projected experiments. Since the soft terms in superstring scenarios are a subset of the general soft terms studied in SUGRA theories we make use of previous results on departures from the mSUGRA scenario to look for values of the orbifold soft terms giving rise to a large cross section accessible for experiments.

In addition, we introduce a new ingredient in the analysis, namely the modification produced in the soft parameters by the presence of an anomalous . Let us recall that in string theory, and in particular in orbifold constructions [24, 25] of the heterotic superstring [26], the gauge groups obtained after compactification are larger than the standard model gauge group, and contain generically extra symmetries, [27]. One of these ’s is usually anomalous, and although its anomaly is cancelled by the four-dimensional Green-Schwarz (GS) mechanism, it generates a Fayet-Iliopoulos (FI) contribution to the D-term [28]. This effect is crucial for model building [29] since some scalars acquire large vacuum expectation values (VEVs) in order to cancel the FI contribution, thereby breaking the extra gauge symmetries, and allowing the construction of realistic standard-like models in the context of the orbifold [30, 31, 32] (see also [33]). Recently other interesting models in the context of the orbifold [34, 35, 36], and orbifold [37, 38], have been analysed. Due to the FI breaking, also D-term contributions to the soft scalar masses are generated [39, 40, 41, 42, 43, 44]. This allows more flexibility in the soft terms and, consequently, in the computation of the associated neutralino-proton cross section.

The paper is organised as follows. In Section 2 we briefly review the departures from mSUGRA which give rise to large values of the neutralino detection cross section. Then, in the next sections, we use this analysis to study several orbifold scenarios where such departures may be present. Special emphasis is put on the effect of the various experimental constraints on the SUSY spectrum and low-energy observables. We start in Section 3 with the simplest (but not the most common) possibility, where an anomalous is not present. Then, in Section 4, we discuss the important modifications produced in the soft terms by the presence of an anomalous , and their effects on the computation of the neutralino-proton cross section, considering the effect of D-term contributions to soft scalar masses. The conclusions are left for Section 5.

## 2 Neutralino-proton cross section and departures from mSUGRA

In this section we review possible departures from the mSUGRA scenario, and their impact on the neutralino-proton cross section. This will allow us to discuss orbifold scenarios more easily. Let us first recall that in mSUGRA one has only four free parameters defined at the GUT scale: the soft scalar mass, , the soft gaugino mass, , the soft trilinear coupling, , and the ratio of the Higgs vacuum expectation values, . In addition, the sign of the Higgsino mass parameter, , remains undetermined. Using these inputs the neutralino-proton cross section has been analysed exhaustively in the literature, as mentioned in the Introduction. Taking into account all kind of experimental and astrophysical constraints, the resulting scalar cross section is bounded to be pb.

Departures from the universal structure of the soft parameters in mSUGRA allow to increase the neutralino-proton cross section significantly. As it was shown in the literature, it is possible to enhance the scattering channels involving exchange of CP-even neutral Higgses by reducing the Higgs masses, and also by increasing the Higgsino components of the lightest neutralino. A brief analysis based on the Higgs mass parameters, and , at the electroweak scale can clearly show how these effects can be achieved.

First, a decrease in the values of the Higgs masses can be obtained by increasing at the electroweak scale (i.e., making it less negative) and/or decreasing . More specifically, the value of the mass of the heaviest CP-even Higgs, , can be very efficiently lowered under these circumstances. This is easily understood by analysing the (tree-level) mass of the CP-odd Higgs , which for reasonably large values of can be approximated as . Since the heaviest CP-even Higgs, , is almost degenerate in mass with , lowering we obtain a decrease in which leads to an increase in the scattering channels through Higgs exchange

Second, through the increase in the value of an increase in the Higgsino components of the lightest neutralino can also be achieved. Making less negative, its positive contribution to in the minimization of the Higgs potential would be smaller. Eventually will be of the order of , and will then be a mixed Higgsino-gaugino state. Thus scattering channels through Higgs exchange become more important than in mSUGRA, where is large and is mainly bino. It is worth emphasizing however that the effect of lowering the Higgs masses is typically more important, since it can provide large values for the neutralino-nucleon cross section even in the case of bino-like neutralinos.

Non-universal soft parameters can produce the above mentioned effects. Let us consider in particular the non-universality in the scalar masses, which will be the most interesting possibility in orbifold scenarios. We can parametrize these in the Higgs sector, at the high-energy scale, as follows:

(2.1) |

Concerning squarks and sleptons we will assume that the three generations have the same mass structure:

(2.2) |

Such a structure avoids potential problems with flavour changing
neutral currents^{2}^{2}2
Another possibility would be to assume that the first
two generations have the common scalar mass , and
that non-universalities are allowed only for the third
generation (as it occurs for the models analysed in
Ref. [45]).
This would not modify our analysis since, as we will see below,
only the third generation is relevant in our
discussion.
(FCNC), and arises naturally e.g. in orbifold constructions with
two Wilson lines, where realistic models have been obtained.
Note also that whereas all ’s in (2.2) have to
satisfy in order to avoid an unbounded from below
(UFB) direction breaking charge and colour^{3}^{3}3
If we allow metastability of our vacuum, tachyonic masses
for some sfermions, , at the high-energy scale might be allowed.
However, we do not consider such a possibility.,
in (2.1) is possible as long as
and
are fulfilled.

An increase in at the electroweak scale can be obviously achieved by increasing its value at the high-energy scale, i.e., with the choice . In addition, this is also produced when and at the high-energy scale decrease, i.e. taking , due to their (negative) contribution proportional to the top Yukawa coupling in the renormalization group equation (RGE) of .

Similarly, a decrease in the value of at the electroweak scale can be obtained by decreasing it at the high-energy scale with . The same effect is obtained when and increase at the high-energy scale, due to their (negative) contribution proportional to the bottom Yukawa coupling in the RGE of . Thus one can deduce that will also be reduced by choosing .

In fact non-universality in the Higgs sector gives the most important effect, and including the one in the sfermion sector the cross section only increases slightly.

Taking into account this analysis, several scenarios were discussed in Ref. [13], obtaining that large values for the cross section are possible. For example, with , regions of the parameter space are found which are accessible for experiments such as CDMS Soudan [18]. Interestingly, it was also realised that these choices of parameters were helpful in order to prevent the appearance of UFB minima in the Higgs potential.

The different UFB directions were classified in Ref. [46]. Among these, the one labelled as UFB-3, which involves VEVs for the fields with , yields the strongest bound. After an analytical minimization of the relevant terms of the scalar potential the value of the fields can be written in terms of . Then, for any value of satisfying

(2.3) |

the potential along the UFB–3 direction reads

(2.4) |

Otherwise

(2.5) |

In these expressions denotes the leptonic Yukawa coupling of the th generation, the deepest direction corresponding to . The UFB-3 condition is then , where , with , is the value of the potential at the realistic minimum. is evaluated at the typical scale of SUSY masses, , and at the renormalization scale, , which is chosen to be , in order to minimize the effect of one-loop corrections to the scalar potential.

As we see from Eqs. (2.4) and (2.5), the potential along this direction can be lifted when increases (becomes less negative) and for large values of the stau mass parameters, thereby making the UFB-3 condition less restrictive. In this sense, non-universal soft terms, like the ones discussed above, can be very helpful.

The question now is whether it is possible to find explicit realisations of these scenarios within orbifold models. In the following sections we will study this issue in detail.

## 3 Orbifold scenarios

Let us recall first the structure of the SUGRA theory in four-dimensional constructions from the heterotic superstring. The tree-level gauge kinetic function is independent of the moduli sector and is simply given by

(3.6) |

where is the Kac-Moody level of the gauge factor. Usually (level one case) one takes for the MSSM. In any case, the values are irrelevant for the tree-level computation since they do not contribute to the soft parameters. On the other hand, the Kähler potential has been computed for six-dimensional Abelian orbifolds, where three moduli are generically present. For this class of models the Kähler potential has the form

(3.7) |

Here are (zero or negative) fractional numbers usually called ‘modular weights’ of the matter fields .

In order to determine the pattern of soft parameters it is crucial to know which fields, either or , play the predominant role in the process of SUSY breaking. Thus one can introduce a parametrization for the VEVs of dilaton and moduli auxiliary fields [47]. A convenient one is given by [47, 48, 49]

(3.8) |

where labels the three complex compact dimensions, is the gravitino mass, and the angles and , with , parametrize the Goldstino direction in the , field space. Here we are neglecting phases and the cosmological constant vanishes by construction.

Using this parametrization and Eqs. (3.6) and (3.7) one obtains the following results for the soft terms [47, 48, 49]:

(3.9) | |||||

Although in the case of the parameter an explicit -dependence may appear in the term proportional to , where are the Yukawa couplings and , it disappears in several interesting cases [47, 49]. For example, the -term which is relevant to electroweak symmetry-breaking is the one associated to the top-quark Yukawa coupling. Thus, in order to obtain the largest possible value of the coupling, the fields should be untwisted or twisted associated to the same fixed point. In both cases , and we will only consider this possibility here.

Using the above information, one can analyse the structure of soft parameters available in Abelian orbifolds. In the dilaton-dominated SUSY-breaking case () the soft parameters are universal, and fulfil [50, 51]

(3.10) |

where the positive (negative) sign for corresponds to (). Of course, these are a subset of the parameter space of mSUGRA, and as a consequence one should expect small dark matter detection cross sections, as discussed in the previous Section.

However, in general, the soft terms (scalar masses and trilinear parameters) given in Eq. (3.9) show a lack of universality due to the modular weight dependence. For example, assuming an overall modulus (i.e., and ), one obtains

(3.11) | |||||

(3.12) |

where we have defined the overall modular weights . In the case of Abelian orbifolds, these can take the values . Fields belonging to the untwisted sector of the orbifold have . Fields in the twisted sector but without oscillators have usually modular weight , and those with oscillators have . Of course, if all modular weights of the standard model fields are equal, one recovers the universal scenario. For example, taking all one has [47] , , .

Using notation (2.1) and (2.2), the degree of non-universality in the scalar masses is therefore given by

(3.13) |

It is worth noticing here that as a consequence of the negativeness of the modular weights. As we will see, this has important phenomenological implications.

On the other hand, the apparent success of the joining of gauge coupling constants at, approximately, GeV in the MSSM is not automatic in the heterotic superstring, where the natural unification scale is GeV, where is the unified gauge coupling. Therefore unification takes place at energies around a factor smaller than expected in the heterotic superstring. This problem might be solved with the presence of large string threshold corrections which explain the mismatch between both scales [52, 53]. In a sense, what would happen is that the gauge coupling constants cross at the MSSM unification scale and diverge towards different values at the heterotic string unification scale. These different values appear due to large one-loop string threshold corrections.

It was found that these corrections can be obtained for restricted values of the modular weights of the fields [53]. In fact, assuming generation independence for the as well as , the simplest possibility corresponds to taking the following values for the standard model fields:

(3.14) |

where, e.g., denotes the three family squarks , , . The above values together with Re lead to good agreement for and [53]. The associated soft sfermion masses are given by [47]:

(3.15) |

whereas for the soft Higgs masses, choosing , one obtains:

(3.16) |

For convenience, this set of modular weights is summarised in Table 1 and labelled as case A).

For example, with , using notation (2.1) and (2.2), the non-universalities in the Higgs and sfermion sectors correspond to , , , , and .

A) | -1 | -2 | -1 | -3 | -3 | -3 | -1 |
---|---|---|---|---|---|---|---|

B) | -1 | -2 | -1 | -3 | -3 | -3 | -2 |

C) | -1 | -2 | -2 | -1 | -1 | -2 | -1 |

D) | -2 | -1 | -1 | -2 | -1 | -2 | -2 |

E) | -1 | -1, -3, -3 | -1 | -3 | -1, -3, -3 | -2 | -3 |

Concerning the soft gaugino masses, they are given by:

(3.17) |

The small departure from universality is due to the effect of the string threshold corrections on the gauge kinetic function [47].

Finally, for the above modular weights, and using (3.12), the expressions for the trilinear parameters read

(3.18) |

The -term which is relevant to radiative symmetry breaking is the one associated to the top-quark Yukawa coupling .

These soft terms serve as an explicit model for the study of the neutralino detection cross section. Since they are completely determined in terms of just the gravitino mass and the Goldstino angle, we are left with three free parameters, namely , , and , plus the sign of . Note, however, that the absence of negative mass-squared of the sleptons at the GUT scale implies the constraint . Besides, the shift implies in the above equations , and . This fact makes it unnecessary to consider both signs of the parameter. The reason is that the RGEs are symmetric under the change . Notice in this sense that we will always have for whereas for . This will have important implications, as we will soon see, on the effect of the experimental constraints on the rare decays and , and on the SUSY contribution to the muon anomalous magnetic moment, . . Consequently, in the remainder of this paper we will assume

The resulting structure of the soft parameters for case A), given at the GUT scale, is represented in Fig. 1 as a function of the Goldstino angle in units of the gravitino mass. Two generic features of this kind of orbifold constructions are evidenced by the plot, namely, the fact that scalar masses are always smaller than gaugino masses, and the presence of regions which are excluded because some scalar masses-squared become negative. In the present example, as already mentioned, the strongest bound is set by slepton masses, for which (3.15) implies . The ruled areas correspond to those where this bound is not fulfilled. This reduces the allowed parameter space to two strips in , around the dilaton-dominated case, .

With this information, the RGEs are numerically solved and the low-energy supersymmetric spectrum is calculated. Fig. 2 shows the resulting particle spectrum as a function of the Goldstino angle for GeV and and . As we can see, although slepton masses-squared are positive at the GUT scale for , the RGEs can still drive them negative, or lead to tachyonic mass eigenstates. This is typically the case of the lightest stau, , and lightest sneutrino, (the latter only for low values of the gravitino mass), due to their small mass parameters (3.15). This is more likely to happen for large , since the lepton Yukawas (which are proportional to ) increase and induce a larger negative contribution to the slepton RGEs. In such a case, the lightest stau can be the lightest SUSY particle (LSP) in larger regions of the parameter space, thus potentially reducing the allowed areas for neutralino dark matter, as we see in the example with . The supersymmetric spectrum also displays a heavy squark sector, due to the gluino contribution on the running of their mass parameters. Similarly, the heavy Higgs masses (represented here only with the pseudoscalar, ) are also sizable. For reference, the value of the term is also displayed and found to be large.

Notice at this point that there are regions of the parameter space where the lightest neutralino is the LSP and the stau, being the next-to-lightest SUSY particle (NLSP), has a very similar mass. As we will soon see, this allows reproducing the correct dark matter relic density by means of a coannihilation effect. On the other hand, one can readily see that in these examples and therefore there is no enhancement in the annihilation of neutralinos mediated by the CP-odd Higgs.

Finally, it is worth emphasizing that in these scenarios the gravitino is never the LSP. Despite the bino mass being larger than at the string scale, its RGE always leads to at the electroweak scale (even in the dilaton-dominated limit for which is at its maximum) so that, at least, the neutralino mass is always lighter than .

Having extracted the supersymmetric spectrum, we are ready to determine the implications for low-energy observables and study how the associated bounds further restrict the allowed parameter space. In our analysis the most recent experimental and astrophysical constraints have been included. In particular, the lower bounds on the masses of the supersymmetric particles and on the lightest Higgs have been implemented, as well as the experimental bound on the branching ratio of the process, B(). The latter has been calculated taking into account the most recent experimental world average for the branching ratio reported by the Heavy Flavour Averaging Group [54, 55, 56], as well as the new re-evaluation of the SM value [57], with errors combined in quadrature. We also take into account the improved experimental constraint on the branching ratio, B, obtained from a combination of the results of CDF and D0, [58, 59, 60]. The evaluation of the neutralino relic density is carried out with the program micrOMEGAs [61], and, due to its relevance, the effect of the WMAP constraint will be shown explicitly. Finally, dangerous charge and colour breaking minima of the Higgs potential will be avoided by excluding unbounded from below directions.

Concerning , we have taken into account the experimental result for the muon anomalous magnetic moment [62], as well as the most recent theoretical evaluations of the Standard Model contributions [63, 64, 65]. It is found that when data are used the experimental excess in would constrain a possible supersymmetric contribution to be , where theoretical and experimental errors have been combined in quadrature. However, when tau data are used, a smaller discrepancy with the experimental measurement is found. Due to this reason, in our analysis we will not impose this constraint, but only indicate the regions compatible with it at the level, this is, .

For a better understanding of all these constraints, we have represented in Fig. 3 their effect on the plane for . For comparison, the cases with are also shown in Fig. 4.

The first thing to notice is that extensive regions are excluded due to the occurrence of tachyonic masses for sleptons. As already discussed, the area excluded for this reason becomes larger when increases, an effect which is clearly displayed in Figs. 3 and 4. This implies an increase in the lower bound of the gravitino mass. Whereas for the smallest allowed value is GeV, in the case with one needs GeV.

The above mentioned smallness of the slepton mass parameters, together
with the fact that gaugino masses are always larger than scalar masses
(), also imply that the areas in the parameter
space where the lightest neutralino is the LSP are not very extensive.
These regions occur for small values of (they are
centered around ), since the
ratio increases with^{4}^{4}4
The lack of a complete mirror symmetry at and
is due to the trilinear terms (3.12) being a
combination of and .
. Note that such values of the Goldstino angle mean that
the breaking of SUSY is mainly due to the dilaton auxiliary term.
Once more, the allowed areas shrink for large values of
and eventually disappear for . In the rest of the
parameter space the role of the LSP
is mainly played by the lightest stau. Although, as already mentioned,
for small values of the sneutrino can also be the LSP in a
very narrow band for small gravitino masses, this area is always
excluded by experimental bounds.

The relevance of the experimental constraints is also evidenced by Figs. 3 and 4. Reproducing the experimental result of the branching ratio of is much easier in the region around , since it has . On the contrary, it poses a stringent lower bound on the value of for the region around , for which . As expected, the area excluded for this reason also increases for larger values of . Thus, whereas this constraint implies GeV for in the area around , . GeV is necessary for

Having , the whole region around also fails to fulfil the experimental constraint on , and is therefore further disfavoured.

The bound on the lightest Higgs mass also rules out some regions for small gravitino masses. This is only relevant for small values of and in the region around . Already for or constraints. The areas not fulfilling the experimental constraints on sparticle masses are always contained within those already excluded by other bounds and are therefore not shown explicitly. this bound becomes less important than the

The allowed parameter space is further reduced when the constraint on the relic density is imposed. The WMAP result is only reproduced along the narrow regions close to the area where the stau becomes the LSP. This is due to the well known coannihilation effect that takes place when the neutralino mass is close to the stau mass. The equivalent of the “bulk region” in the mSUGRA parameter space is here excluded by the experimental constraints. Finally, no regions are found where , and consequently resonant annihilation of neutralinos does not play any role in this case.

Having shown that there are regions with viable neutralino dark matter, let us now turn our attention to its possible direct detection. Following the discussion of Section 2, the Higgs modular weights giving rise to the soft masses (3.16), could induce an increase of the neutralino detection cross section with respect to the universal case. In order to investigate this possibility, the theoretical predictions for the spin-independent part of the neutralino-nucleon cross section have been calculated in the accepted regions of the parameter space. They are represented versus the neutralino mass in Fig. 5 for and , where the sensitivities of present and projected dark matter experiments are also shown. These results resemble those of mSUGRA, as no high values are obtained. As in mSUGRA, in this scenario the parameter and the heavy Higgs masses are sizable (see Fig. 2), thus implying bino-like neutralinos and a suppressed contribution to from Higgs-exchanging processes. This is illustrated in Fig. 6, where the resulting values of the parameter are plotted as a function of the pseudoscalar Higgs mass. After analysing the range to we found that pb, the maximum values corresponding to . These results are therefore beyond the present sensitivities of dark matter detectors and would only be partly within the reach of the projected 1 tonne detectors.

So far we have not commented on the bounds
imposed by the UFB-3 constraint
to avoid dangerous charge and colour breaking minima of the Higgs
potential. This turns out to play a crucial role in disfavouring this
scenario. Indeed, most of the parameter space is excluded on these
grounds^{5}^{5}5This is consistent with previous analyses of
charge and colour breaking minima in different superstring
and M-theory scenarios [66]..
Only for small values of and heavy gravitinos do allowed
regions appear (see for instance Fig. 3, where
GeV is necessary), but these always correspond to
areas where the neutralino relic density is too large and
exceeds the WMAP constraint. For GeV. Once more, the reason for this is the low value of
the slepton masses, and more specifically, of the stau mass.
Let us recall that the smaller this value, the more negative
in (2.4) or (2.5) is,
and thus the stronger the UFB-3 bound becomes.
Moreover, the fact that in this scenario the value of is
not very large (since is negative)
also contributes in driving the potential deeper along this
direction. the UFB-3
constraint already excludes the complete region with

Let us finally remark that other examples with different choices of modular weights for the Higgs parameters satisfying (3.14) have been investigated, such as case B) in Table 1, and lead to qualitatively similar results.

The previous analysis suggests how to modify the model to ‘optimise’ its behaviour under the UFB-3 constraint [66], increasing also the regions in the parameter space where the lightest neutralino is the LSP. The most favorable case would correspond to slepton masses as large as possible, i.e.,

(3.19) |

For squark and Higgs mass parameters we will continue using the
modular weights of case A)^{6}^{6}6
Of course, with such a choice of modular weights we know that
the string threshold corrections cannot account for the joining
of gauge couplings at the MSSM unification scale.
Thus we will be tacitly assuming that there is some other
effect (e.g., the existence of further chiral fields in the spectrum
below the heterotic string scale [69, 33, 19]) which
appropriately produces the correct low-energy experimental values
for gauge couplings..
Note that now the bound on is less constraining,
since we only need to impose , thus allowing
a larger degree of
non-universality. For example, with , we get
and for the Higgs
masses.

The resulting supersymmetric spectrum is shown in Fig. 7 as a function of the Goldstino angle for GeV with and . Notice that now the whole region with is free from tachyons at the GUT scale. For small the increase in the slepton mass-squared parameters leaves extensive allowed regions where the neutralino can be the LSP. As expected, larger values of lead to a reduction in the stau mass, which now easily becomes the LSP, and gives rise to tachyons in some regions.

These features are evidenced in Fig. 8, where the corresponding parameter space is depicted for . Notice that, due to the increase in the slepton mass terms, the stau only becomes the LSP on narrow bands on the right-hand side of the allowed areas for (for smaller values of the neutralino is always the LSP). As expected, these areas with stau LSP become more sizable for and eventually dominate the whole parameter space for the sneutrino can also be the LSP on a narrow region for very light gravitinos, although this is always excluded by experimental constraints. . Also, for

The decrease of the stau mass towards the right-hand side of the allowed areas can be understood by analysing the expressions for the trilinear soft terms. The trilinear terms associated to the top, bottom and tau Yukawa coupling read in this example

(3.20) |

It can be checked that for all of them the ratio increases towards the right-hand side of both allowed areas. In particular, for and becomes for . The increase in leads to a larger negative correction in the RGE for the slepton mass terms, implying lighter staus. Large values of increase the corresponding Yukawas thus further decreasing the stau mass.

The variation in the stau mass affects the area excluded by the UFB-3 constraint, which becomes more stringent towards the right-hand side of the allowed regions. On the left, as expected, the effect of the UFB constraints is less severe than in the previous examples and regions with are allowed. Interestingly, for part of these areas can also reproduce the correct value for the neutralino relic density. GeV for

The corresponding predictions for the neutralino-nucleon cross section are depicted in Fig. 9. Although regions with the correct relic density can appear with pb for , the points fulfilling the UFB-3 constraints only correspond to those with pb for 10, where the resulting parameter is represented versus the CP odd Higgs mass, this is due to the large values of and the heavy Higgs masses. . Once more, as evidenced in Fig.

Notice, finally, that in this example the non-universality of the Higgs masses, given by (3.16), was chosen to be the maximal allowed by the modular weights ( and ). Also, the stau mass, for which we have , cannot be further increased and therefore the behaviour under the UFB-3 constraint cannot be improved. Consequently, this optimised scenario represents a good estimate of how large the neutralino detection cross section can be in heterotic orbifolds with overall modulus, where soft masses are given by (3.11). We therefore conclude that in this class of models pb. The neutralino in these scenarios would escape detection in all present experiments and only tonne detectors would be able to explore some small areas of the allowed parameter space.

For completeness, we have also analysed other three scenarios, described in [50], which also give rise to gauge coupling unification with an overall modulus. Their corresponding modular weights are summarised in Table 1.

For example, in case C) unification is possible with Re , but extra massless chiral fields (one octet, one triplet, and two multiplets transforming like right-handed electrons), with modular weight equal to , are needed. This scenario seems promising, since the modular weights for sleptons are less negative (). In fact, although squarks become tachyonic at the GUT scale for , sleptons have a positive mass squared in the whole remaining area . As we have learned from the optimised example, this might be helpful in order to avoid the UFB constraints. The presence of extra matter alters the running of the gauge coupling constants, which are now dictated by the following beta functions, , , and . As a consequence, the running of the soft masses is also modified. In particular, all the gaugino masses become smaller at the EW scale, as compared to the usual running within the MSSM. Notice in particular that the gluino mass does not run (at tree level) from the GUT to the EW scales.

The decrease in and (especially) affect the running of the scalar mass parameters, rendering them smaller at the EW scale. This is enough to offset the increase in due to the smaller modular weights. Similarly, the important decrease in the gluino mass implies a very light squark sector. This leads to a qualitatively different structure of the SUSY spectrum in which squarks and sleptons have a similar mass. This is illustrated in Fig. 11, where we have represented the resulting spectrum for GeV and and . Unlike the previous cases, this example displays very light gluinos and squarks. There are even regions where the stop is the LSP (especially for low values of for which the top Yukawa is larger).

The regions allowed by experimental constraints become larger in this example, as we can see in Fig. 12, where the plane is depicted for and . It is important to mention that, due to the resulting light squarks, the supersymmetric contribution to B() becomes sizable. Unlike in the previous examples, the experimental bound on this observable becomes the most stringent constraint, even for small values of . There are also areas which reproduce the correct dark matter relic density through coannihilation effects with the stop (for small values of ) and the stau (for ) imply smaller values for the Higgs mass parameters. In particular, is more negative, making it more difficult to avoid the UFB-3 constraint. Only for heavy gravitinos do allowed regions occur ( GeV is necessary for , whether for larger gravitinos heavier than 1 TeV are needed). As already observed in case A), these regions never correspond to those with the correct neutralino relic density. ). Noticeably, in spite of the less negative modular weights for sleptons, the modifications in the RGEs (especially the decrease in

Finally, it is worth mentioning that, since the Higgs mass parameters have a smaller departure from universality, we do not expect large neutralino detection cross sections in this example. The results for are represented in Fig. 13 for and and, clearly lie beyond the reach of current and projected direct dark matter searches.

Another potentially interesting scenario is case D), once more due to the reduced modular weights for sleptons. As in the previous example, the region allowed at the GUT scale is , where . In this scenario one needs four extra multiplets, transforming like , and Re for unification to take place, thus implying , , and . The absence of running for and therefore for the gluino mass parameters has the same consequences as in case C), leading to a light squark spectrum. This is illustrated in Fig. 14, where the sparticle masses are plotted as a function of the Goldstino angle for GeV and and . Interestingly, the running of the wino mass parameter is slightly enhanced and is found at the electroweak scale. This makes the lightest neutralino a mixed bino-wino state and almost degenerate in mass with the lightest chargino. Although this can lead to an increase of the resulting neutralino direct detection cross section, it also implies a more efficient neutralino annihilation and, consequently, a relic density which is too small to account for the dark matter of the Universe. In particular, one obtains for the whole region with gravitinos lighter than TeV, independently of the value of . Thus, although the area allowed by experimental constraint, represented in Fig. 15 for and is sizable, the astrophysical constraint on the relic density is never fulfilled. As in the previous examples, the presence of light squarks induce larger contributions to B() and the experimental constraint on it excludes extensive regions of the parameter space, even at low . Furthermore, the steeper running of renders more negative, which makes the UFB constraints even more restrictive. All the area represented in Fig. 15 becomes excluded for this reason.

Finally, case E) in Table 1 corresponds to a orbifold with a universal modulus, in which case Re is needed. Due to the small modular weights for sleptons this example yields similar results regarding the UFB constraints as case A), with allowed regions appearing only for very massive gravitinos and incompatible with the astrophysical bound on the dark matter relic density. Moreover, since the non-universality in the Higgs mass parameters () is not the optimal to increase the neutralino detection cross section, the theoretical predictions for are even smaller than those represented in Fig. 5.

The presence of different moduli can provide some extra freedom to the non-universalities of the soft scalar masses (3.9), which are then parametrized by new Goldstino angles, . We have analysed two scenarios of this kind, whose modular weights are described in Table 2, and which have also been shown to reproduce gauge coupling unification [50]. Scenario F) corresponds to a orbifold with a rotated plane for which unification is achieved with Re Re . In scenario G), a orbifold was taken, again non-isotropic, with .

F) | |||||||
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G) |

In both examples, some combinations of the Goldstino angles lead to potentially interesting non-universalities in the soft scalar masses. In particular, it is always possible to enlarge the stau mass, thus avoiding UFB constraints. For instance, in case F) this can be done by choosing (hence ) obtaining the following expressions for the soft terms,

(3.21) |

Alternatively, in case G) one can take (and therefore ) and obtain

(3.22) |

Notice, however, that the resulting Higgs mass parameters are not adequate to obtain large neutralino detection cross sections. On the one hand, in case F) these are universal by construction since they have the same modular weights. On the other hand, in case G), the Higgs soft masses are related by