Phenomenological Constraints on SUSY GUTs with Nonuniversal Gaugino Masses
Abstract
We study phenomenological aspects of supersymmetric grand unified theories with nonuniversal gaugino masses. For large , we investigate constraints from the requirement of successful electroweak symmetry breaking, the positivity of stau mass squared and the decay rate. In the allowed region, the nature of the lightest supersymmetric particle is determined. Examples of mass spectra are given. We also calculate loop corrections to the bottom mass due to superpartners.
pacs:
11.30.Pb, 12.10.DmI Introduction
Supersymmetric gauge field theories are among the most promising models for physics beyond the standard model. The lowenergy supersymmetry (SUSY) solves the socalled hierarchy problem, which basically follows from the tremendous scale differences in realistic models including gravity.
After SUSY breaking, SUSY models, e.g. the minimal supersymmetric standard model (MSSM) have over hundred free parameters in general. Most of these new parameters in the MSSM are in fact related to the SUSY breaking, i.e. gaugino masses , soft scalar masses , SUSY breaking trilinear couplings and SUSY breaking bilinear couplings. They are expected to be of the order of 1 TeV.
To probe the SUSY breaking mechanisms is very important in order to produce solid information on physics beyond the standard model. Two types of SUSY breaking mechanisms, gravitymediated SUSY breaking and gaugemediated SUSY breaking, have been actively studied in recent years. The signatures of gravity mediated and gauge mediated SUSY breaking are quite different. A specific SUSY breaking mechanism usually reduces the number of a priori free parameters from about one hundred to only a few by introducing solid relations among the SUSY breaking parameters. This makes the phenomenology of the MSSM more accessible for study.
For phenomenology of SUSY models, various aspects have been studied in several regions of the parameter space. Most phenomenological analyses have been done under the assumption that the soft SUSY breaking parameters are universal, i.e. for , for any scalar and at a certain energy scale, e.g. the Planck scale or the grand unified theory (GUT) scale. From the phenomenological viewpoint, the universality assumption is useful to simplify analysis. Actually, the universal parameters can be derived from a certain type of underlying theories, e.g. minimal supergravity.
However, the universality assumption may remove some interesting degrees of freedom. Indeed, there exist interesting classes of models in which nonuniversal soft SUSY breaking terms can be derived. For example, stringinspired supergravity can lead to nonuniversality for SUSY breaking parameters at the Planck scale [3, 4]. Also, gaugemediated SUSY breaking models, in general, lead to nonuniversality [5].
Recently phenomenological implications of nonuniversal SUSY breaking parameters have been investigated. For example, in Ref.[6, 8] phenomenological implications have been studied for nonuniversal gaugino masses derived from string models. GUTs without a singlet also lead to nonuniversal gaugino masses [9, 10, 11]. In Ref.[10] phenomenological aspects in the small scenario have been discussed, e.g. mass spectra and some decay modes. Some phenomenological constraints reduce the allowed region of the universal SUSY breaking parameters a lot. For example, in the large scenario, it is hard to fulfill the constraints due to the requirement of successful electroweak breaking, SUSY corrections to the bottom mass [12, 13] and the decay rate [14, 15]. These constraints can be relaxed in nonuniversal cases.
In this paper, we study phenomenological aspects of SUSY GUTs where the gaugino masses come from a condensation of component with a representation 24, 75 or 200. Each of them leads to a proper pattern of nonuniversal gaugino masses. We mostly concentrate on the large scenario. We take into account the full oneloop effective potential of the MSSM, in order to calculate the physical spectrum of the MSSM, given the initial conditions at the GUT scale. In particular, we investigate constraints from the requirement of successful electroweak symmetry breaking, the positivity of stau mass squared and the decay rate. We take SUSY corrections to the bottom quark mass carefully into account. We then find the allowed parameter space for each model and describe the particle spectrum.
This paper is organized as follows. In section 2 SUSY GUTs with nonuniversal gaugino masses are reviewed. In section 3 we study their phenomenological aspects, i.e. successful radiative breaking of the electroweak symmetry, the LSP mass, the stau mass, SUSY corrections to the bottom mass and the decay. We also give comments on small cases. Section 4 is devoted to conclusions.
Ii Susy GUTs with nonuniversal gaugino masses
We discuss the nonuniversality of soft SUSY breaking gaugino masses in SUSY GUT and the constraints on parameters at the GUT scale in our analysis. The gauge kinetic function is given by
(1)  
where are indices related to gauge generators, ’s are chiral superfields and is the gaugino field. The scalar and compontents of are denoted by and , respectively. The ’s are classified into two categories. One is a set of singlet supermultiplets and the other one is a set of nonsinglet ones . The gauge kinetic function is, in general, given by
(2) 
where and are functions of gauge singlets and is the reduced Planck mass defined by . Since the gauge multiplets are in adjoint representation, one finds the possible representations of with nonvanishing by decomposing the symmetric product as
(3) 
Thus, the representations of allowed as a linear term of in are 24, 75 and 200.
Here we make two basic assumptions. The first one is that SUSY is broken by nonzero VEVs of components , i.e., where is the gravitino mass. The second one is that the gauge symmetry is broken down to the standard model gauge symmetry by nonzero VEVs of nonsinglet scalar fields at the GUT scale .
After the breakdown of , the gauge couplings ’s of , are, in general, nonuniversal at the scale [16] as we see from the formula . The index represents generators as a whole. The gaugino field acquires soft SUSY breaking mass after SUSY breaking. The mass formula is given by
(4) 
Thus the ’s are also, in general, nonuniversal at the scale [9].
Next we will consider the constraints on the physical parameters used
at the scale for the analysis in this paper.
1) Gauge
couplings
We take a gauge coupling unification scenario within the
framework of the MSSM, that is,
(5) 
where and GeV.
The relation (5) leads to . We neglect the contribution of nonuniversality to the gauge
couplings. Such corrections of order have little effects on phenomenological aspects
which we will discuss in the next section, although such corrections
would be important for precision study on the gauge coupling
unification.
2) Gaugino masses
We assume that dominant component
of gaugino masses comes from one of nonsinglet components. The
VEV of the component of a singlet field whose scalar component
has a VEV of in is supposed to be
small enough such as dilaton
multiplet in modulidominant SUSY breaking in string models. In this
case, ratios of gaugino masses at are determined by group
theoretical factors and shown in table I. The patterns of gaugino
masses which stem from term condensation of 24, 75 and
200 are different from each other. The table also shows
corresponding ratios at the weak scale based on MSSM. In the
table, gaugino masses are shown in the normalization .
Note that the signs of are also fixed by group theory
up to an overall phase as shown in Table 1.
There is no direct experimental constraint on these signs.
For example, these signs affect radiative corrections of
terms and thus offdiagonal elements of sfermion matrices, that is,
radiative corrections of to are constructive
in the universal case, while
in the model 24 radiative corrections between and the others
interfere with each other leading to reduction.
In the other cases, the radiative corrections are larger by
than the universal case.
3) Scalar masses
For simplicity, we assume universal soft SUSY
breaking scalar masses at in our analysis in
order to clarify phenomenological implications of nonuniversal
gaugino masses. The magnitude of is supposed not
to be too large compared with that of ’s in order not to
overclose the universe with a huge amount of relic abundance of the
lightest neutralino [8].
The nonuniversal gaugino masses and scalar masses may have sizable SUSY threshold corrections for running of gauge couplings [17]. These threshold effects and nonuniversal contributions of in will be discussed elsewhere.
Iii Phenomenological constraints and mass spectra
In this section, we study several phenomenological aspects of SUSY GUTs with nonuniversal gaugino masses. The patterns of the gaugino masses in the models are different from each other as shown in table I. That leads to different phenomenology in these models. For example, in the model 24 we have a large gap between and , i.e. . In the model 75 gaugino masses are almost degenerate at the weak scale. In the model 200, is smallest. Some phenomenological aspects have been previously studied in the case with low [10]. We will study the case of a large value of , e.g. .
We take the trilinear scalar couplings, the socalled terms, to vanish at the GUTscale. Similarly, the case with nonvanishing terms can be studied, but the conclusions remain qualitatively unchanged. Also we ignore the supersymmetric violating phase of the bilinear scalar coupling of the two Higgs fields, the socalled term. Assuming vanishing terms and a real term, we have no SUSYCP problem. Ignoring the complex phases has no significant effect on the results of this work, although they would naturally be very relevant to the problem of CPviolation. We could fix magnitudes of the supersymmetric Higgs mixing mass and by assuming some generation mechanism for the term. However, we do not take such a procedure here. We will instead fix these magnitudes by use of the minimization conditions of the Higgs potential as shall be shown.
Given the quantum numbers of irreducible representation, one can characterize the models as a function of four parameters: , the gluino mass at the GUT scale, the universal mass of the scalar fields at the GUT scale and the sign of the term .
We will check the compatibility of the model with the experimental branching ratio . Since this branching ratio increases with , we will study the four models at the region of large , taking as a representative value and scanning over the gaugino mass and the scalar mass squared term. We require that the gauge coupling constants unify at the scale GeV.
Successful electroweak symmetry breaking is an important constraint. The oneloop effective potential written in terms of the VEVs, and , is
(6) 
where
(7) 
where and are respectively the spin and the number of degrees of freedom. Here and denote soft SUSY breaking Higgs masses.
We use the minimization conditions of the full oneloop effective potential,
(8) 
so that we can write and in terms of other parameters, that is, the soft scalar masses, the gaugino masses and . Here and denote the values determined only by use of the treelevel potential, and and denote the corrections due to the full oneloop potential, which are obtained
(9) 
Numerically, the most significant oneloop contribution to and comes from the (s)top and (s)bottom loops [18].
Successful electroweak symmetry breaking requires . Furthermore, we require the mass squared eigenvalues for all scalar fields to be nonnegative. In particular, in the large scenario the stau mass squared becomes easily negative due to large negative radiative corrections from the Yukawa coupling against positive radiative corrections from the gaugino masses.
These constraints are shown in Fig.1. In the model 1 with the universal gaugino mass, requirement of proper electroweak symmetry breaking excludes the region with very small ( GeV) scalar mass and gaugino masses. In the model the region where radiative symmetry breaking fails is considerably larger than in the model 1 because of negative for small . In the model and are quite small compared with . Such small values of and are not enough to push up against large negative radiative correction due to the Yukawa coupling. It is interesting to note that in the model 75 large gaugino masses drive the Higgs boson masssquared to very large positive values at the SUSY scale. This aspect combined with the contribution from the effective potential correction makes small and negative at large gaugino masses. As a result, in the model 75 there are no consistent solutions having large gluino mass GeV. Furthermore, around the border to the region with , i.e. GeV, the magnitude of is very small, and the lightest neutralino and the lighter chargino are almost higgsinos. Thus, the region around the border GeV is excluded by the experimental lower bound of the chargino mass, GeV. The region with GeV leads to large enough to predict GeV. In the model 200 radiative symmetry breaking works for all the scanned values.
From the experimental point of view, a crucial issue is the nature of the LSP, since it is a decisive factor in determing signals of the models in detectors. One candidate for the LSP is the lightest neutralino . In the large scenario, the lightest stau is another possibility ^{1}^{1}1In Ref.[19] cosmological implications of the stau LSP have been discussed.. Figs. 1 show what is the LSP for the four models. They also show the excluded region by the current experimental limit GeV [20]. The limit on the stau mass excludes the models 1 and 24 having small scalar masses. The models 75 and 200, on the other hand, always have relatively heavy stau, independent of the SUSY parameters, and in the latter two models the neutralino is always the LSP. For the models 1 and 24, the content of the LSP is similar and narrow regions lead to the stau LSP.
In our models the present experimental lower bound of the Higgs mass does not provide a strong constraint, because in the large scenario the Higgs mass is heavy.
We also consider the constraint due to the decay. The prediction of the decay branching ratio [14] should be within the current experimental bounds [22]
(10) 
Combined with the theoretical uncertainty in the SM prediction ( ) the branching ratio must be between 0.3 and 1.4 times the SM prediction.
As expected, the constraint is very strong for negative muterm ^{2}^{2}2We follow the conventional definition of the sign of [23]., because the supersymmetric contributions interfere constructively to the amplitude, causing the branching ratio to exceed the experimental bound. Figs. 1 show excluded regions due to for the four models with . We have taken into account squark mixing effects. These excluded regions are similar for the four models. The regions with small gluino masses GeV are ruled out due to too large branching ratio in the four models unless TeV. In addition, the model 75 has an excluded region with GeV due to the unsuccessful electroweak symmetry breaking. Thus, in the model 75 with only a narrow region for is allowed for TeV. In the case with positive muterm the constraints are much weaker, only some models with small gluino and soft scalar masses are ruled out due to the charged Higgs contribution.
The superpartnerloop corrections to the bottomquark Yukawa coupling become numerically sizable for large . These corrections are significant for precise prediction of the bottom mass. Thus, we also show how these SUSYcorrections to the bottom mass depend on our models with nonuniversal gaugino masses. The threshold effect can be expressed as [12]
(11) 
where are the bottom quark Yukawa couplings in the standard model and MSSM, respectively. The dominant part of the corrections is given by
(12) 
where is the top Yukawa coupling and
(13) 
The sign of is the same as the one of . Figs. 2 show for the four models the regions with , and . Most of the allowed regions in the models 1 and 24 lead to for , while most of the allowed region in the model 75 leads to . In the model 200, small leads to , while large leads to .
The large correction to the bottom mass affects the Yukawa coupling unification, which is one of interesting aspects in GUTs. We assume the Yukawa coupling unification at the GUT scale and use the experimental value GeV. Without the SUSY correction we would have GeV for . The present experimental value of the bottom mass contains large uncertainties: Ref. [24], for instance, gives
(14) 
while the analysis of the system [25] and the lattice result [26] give GeV and GeV, respectively, ^{3}^{3}3See also ref. [27]. which translate into
(15) 
Thus, the negative SUSY corrections, that is , with are favored for . Hence most of the region in the model leads to too small to fit the experimental value for . The SUSY correction is proportional to . Therefore, in the case with large , e.g. and 55, some parameter regions in the model 75 as well as the model 200 become more favorable. Because the prediction GeV without the SUSY correction is similar for , 50 and 55.
Finally we show sparticle spectra in the regions allowed by the electroweak breaking conditions and the constraint due to for and . The whole particle spectrum is fixed by gluino mass , the soft scalar mass and . The sign of the term has numerically insignificant effect to the mass spectrum. In the case of negative term the experimental upper bound to the decay branching ratio severely restricts the parameter space. As an example, we show mass spectra of the four models for and in table II. These parameters correspond to almost smallest mass parameters allowed by theoretical and experimental considerations common in the four models. Most of the nonSM degrees of freedom have masses around 1 TeV. Note that the model 75 with GeV and GeV predicts very small and the lightest chargino mass, which is actually excluded by the experimental lower bound. On the other hand, the model 24 for small predicts a very small mass of the lightest neutralino.
In the models 75 and 200 the lightest neutralino and the lightest chargino are almost degenerate. This would potentially create a very difficult experimental setup [28, 6, 29]. The charginos would be extremely difficult to detect, at least near the kinematical production threshold: as the charginos decay practically all of the reaction energy is deposited into the invisible LSP neutralinos. If the charginos decay very close to the interaction point, the photon background would quite effectively hide the signal. The chargino would be easy to detect only if it is sufficiently stable, having a decay length of at least millimeters.
In the models 1 and 24, the LSP is almost the bino. On the other hand, the winolike LSP or the higgsinolike LSP can be realised in the models 75 and 200. In particular, the model 75 has the region around GeV where the higgsino is very light. These different patterns of mass spectra also have cosmological implications, which will be discussed elsewhere [30].
We have assumed universal soft scalar mass at the GUT scale in order to concentrate on phenomenological implications of the nonuniversal gaugino masses, but we give some comments on nonuniversal soft scalar masses. Certain types of nonuniversalities can relax the given constraints. For example, the nonuniversality between the stau mass and the others is important for the constraint and obviously a large value of the stau mass at the GUT can remove the excluded region. For the electroweak symmetry breaking, the nonuniversality between the Higgs masses and is interesting and a large difference of enlarges the parameter region for the succesful electroweak symmetry breaking.
We give a comment for the small scenario. For small , the stau (mass) has no sizable and negative radiative corrections. Thus, the constraints and GeV are no longer serious. In addition most cases lead to the neutralino LSP. Furthermore, the SUSY contributions to is roughly proportional to . Hence, the constraint due to is also relaxed for small .
Iv Conclusions
We have studied the large scenario of the SUSY model in which the gaugino masses are not universal at the GUT scale. We find that the gluino mass at the electroweak scale is restricted to multiTeV values due to experimental limits on the decay for . In the model 75 the allowed region is narrow for . We find that in two of the models 1 and 24 we have neutralino LSP and stau NLSP, while in the models 75 and 200 the lightest neutralino and the lighter chargino are almost mass degenerate. This would provide for quite different kind of the first signature for the MSSM as is usually assumed within the minimal supergravity scenario. We have also calculated the SUSY correction to the bottom mass . The model 75, as well as the model 200 with large , leads to smaller than the others.
We have possibilities that gaugino fields acquire a different pattern of nonuniversal masses. For example, there is the case that some linear combination of components of 1, 24, 75 and 200 contributes to gaugino masses. It is pointed out that there exists a modelindependent contribution to gaugino masses from the conformal anomaly [11]. Furthermore, soft scalar masses and parameters at the GUT scale can, in general, be nonuniversal. We leave these types of extension to future work.
Note added:
After completion of this paper, Ref.[31] appears, where several signals of the GUTs with nonuniversal gaugino masses have been discussed for and 25.
Acknowledgments
The authors would like to thank K. Österberg for useful correspondence. This work was partially supported by the Academy of Finland under Project no. 44129. Y.K. acknowledges support by the Japanese GrantinAid for Scientific Research (10740111) from the Ministry of Education, Science and Culture.
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1  

24  
75  
200 
Model  

()  
(800, 400)  
(800, 400)  
(800, 400)  
(800, 400)  
(100, 1500)  
(100, 1500)  
(100, 1500)  
(100, 1500) 