inspired SUSY models with exact custodial symmetry
[8mm] Roman Nevzorov111E-mail: email@example.com
[3mm] Department of Physics and Astronomy, University of Hawaii,
Honolulu, HI 96822, USA
[2mm] Institute for Theoretical and Experimental Physics,
Moscow, 117218, Russia
The breakdown of gauge symmetry at high energies may lead to supersymmetric (SUSY) models based on the Standard Model (SM) gauge group together with extra and gauge symmetries. To ensure anomaly cancellation the particle content of these inspired models involves extra exotic states that generically give rise to non-diagonal flavour transitions and rapid proton decay. We argue that a single discrete symmetry can be used to forbid tree-level flavor-changing transitions, as well as the most dangerous baryon and lepton number violating operators. We present and orbifold GUT constructions that lead to the inspired SUSY models of this type. The breakdown of and gauge symmetries that preserves matter parity assignment guarantees that ordinary quarks and leptons and their superpartners, as well as the exotic states which originate from representations of survive to low energies. These inspired models contain two dark-matter candidates and must also include additional TeV scale vectorlike lepton or vectorlike down type quark states to render the lightest exotic quark unstable. We examine gauge coupling unification in these models and discuss their implications for collider phenomenology and cosmology.
inspired models are well motivated extensions of the Standard Model (SM). Indeed, supersymmetric (SUSY) models based on the gauge symmetry or its subgroup can originate from the ten–dimensional heterotic superstring theory . Within this framework gauge and gravitational anomaly cancellation was found to occur for the gauge groups or . However only can contain the SM since it allows for chiral fermions while does not. Compactification of the extra dimensions results in the breakdown of up to or one of its subgroups in the observable sector . The remaining couples to the usual matter representations of the only by virtue of gravitational interactions and comprises a hidden sector that is thought to be responsible for the spontaneous breakdown of local SUSY (supergravity). At low energies the hidden sector decouples from the observable sector of quarks and leptons, the gauge and Higgs bosons and their superpartners. Its only manifest effect is a set of soft SUSY breaking terms which spoil the degeneracy between bosons and fermions within one supermultiplet . The scale of soft SUSY breaking terms is set by the gravitino mass, . In the simplest SUSY extensions of the SM these terms also determine the electroweak (EW) scale. A large mass hierarchy between and Planck scale can be caused by the non–perturbative effects in the hidden sector that may trigger the breakdown of supergravity (SUGRA) .
Since is a rank - 6 group the breakdown of symmetry may result in low energy models based on rank - 5 or rank - 6 gauge groups, with one or two additional gauge group factors in comparison to the SM. Indeed, contains the maximal subgroup while can be decomposed in terms of the subgroup –. By means of the Hosotani mechanism  can be broken directly to
which has rank–6. This rank–6 model may be reduced further to an effective rank–5 model with only one extra gauge symmetry which is a linear combination of and :
In the models based on rank - 6 or rank - 5 subgroups of the anomalies are automatically cancelled if the low energy particle spectrum consists of a complete representations of . Consequently, in -inspired SUSY models one is forced to augment the minimal particle spectrum by a number of exotics which, together with ordinary quarks and leptons, form complete fundamental representations of . Thus we will assume that the particle content of these models includes at least three fundamental representations of at low energies. These multiplets decompose under the subgroup of as follows:
The first, second and third quantities in brackets are the representation and extra and charges respectively, while is a family index that runs from 1 to 3. An ordinary SM family, which contains the doublets of left–handed quarks and leptons , right-handed up– and down–quarks ( and ) as well as right–handed charged leptons , is assigned to + . Right-handed neutrinos are associated with the last term in Eq. (2), . The next-to-last term, , represents new SM-singlet fields , with non-zero charges that therefore survive down to the EW scale. The pair of –doublets ( and ) that are contained in and have the quantum numbers of Higgs doublets. They form either Higgs or Inert Higgs multiplets. 222We use the terminology “Inert Higgs” to denote Higgs–like doublets that do not develop VEVs. Other components of these multiplets form colour triplets of exotic quarks and with electric charges and respectively. These exotic quark states carry a charge twice larger than that of ordinary ones. In phenomenologically viable inspired models they can be either diquarks or leptoquarks.
The presence of the bosons associated with extra gauge symmetries and exotic matter in the low-energy spectrum stimulated the extensive studies of the inspired SUSY models over the years , . Recently, the latest Tevatron and early LHC mass limits in these models have been discussed in  while different aspects of phenomenology of exotic quarks and squarks have been considered in . Also the implications of the inspired SUSY models have been studied for EW symmetry breaking (EWSB) –, neutrino physics –, leptogenesis –, EW baryogenesis , muon anomalous magnetic moment , electric dipole moment of electron  and tau lepton , lepton flavour violating processes like  and CP-violation in the Higgs sector . The neutralino sector in inspired SUSY models was analysed previously in , –, –. Such models have also been proposed as the solution to the tachyon problems of anomaly mediated SUSY breaking, via D-term contributions , and used in combination with a generation symmetry to construct a model explaining fermion mass hierarchy and mixing . An important feature of inspired SUSY models is that the mass of the lightest Higgs particle can be substantially larger in these models than in the minimal supersymmetric standard model (MSSM) and next-to-minimal supersymmetric standard model (NMSSM) , –. The Higgs sector in these models was examined recently in , , .
Within the class of rank - 5 inspired SUSY models, there is a unique choice of Abelian gauge symmetry that allows zero charges for right-handed neutrinos and thus a high scale see-saw mechanism. This corresponds to . Only in this Exceptional Supersymmetric Standard Model (ESSM) – right–handed neutrinos may be superheavy, shedding light on the origin of the mass hierarchy in the lepton sector and providing a mechanism for the generation of the baryon asymmetry in the Universe via leptogenesis –. Indeed, the heavy Majorana right-handed neutrinos may decay into final states with lepton number , thereby creating a lepton asymmetry in the early universe. Since in the ESSM the Yukawa couplings of the new exotic particles are not constrained by neutrino oscillation data, substantial values of the CP–asymmetries can be induced even for a relatively small mass of the lightest right–handed neutrino () so that successful thermal leptogenesis may be achieved without encountering a gravitino problem .
Supersymmetric models with an additional gauge symmetry have been studied in  in the context of non–standard neutrino models with extra singlets, in  from the point of view of mixing, in  and – where the neutralino sector was explored, in ,  where the renormalisation group (RG) flow of couplings was examined and in – where EWSB was studied. The presence of a boson and of exotic quarks predicted by the Exceptional SUSY model provides spectacular new physics signals at the LHC which were analysed in –, . The presence of light exotic particles in the ESSM spectrum also lead to the nonstandard decays of the SM–like Higgs boson that were discussed in details in . Recently the particle spectrum and collider signatures associated with it were studied within the constrained version of the ESSM .
Although the presence of TeV scale exotic matter in inspired SUSY models gives rise to specatucular collider signatures, it also causes some serious problems. In particular, light exotic states generically lead to non–diagonal flavour transitions and rapid proton decay. To suppress flavour changing processes as well as baryon and lepton number violating operators one can impose a set of discrete symmetries. For example, one can impose an approximate symmetry, under which all superfields except one pair of and (say and ) and one SM-type singlet field () are odd –. When all symmetry violating couplings are small this discrete symmetry allows to suppress flavour changing processes. If the Lagrangian of the inspired SUSY models is invariant with respect to either a symmetry, under which all superfields except leptons are even (Model I), or a discrete symmetry that implies that exotic quark and lepton superfields are odd whereas the others remain even (Model II), then the most dangerous baryon and lepton number violating operators get forbidden and proton is sufficiently longlived –. The symmetries , and obviously do not commute with because different components of fundamental representations of transform differently under these symmetries.
The necessity of introducing multiple discrete symmetries to ameliorate phenomenological problems that generically arise due to the presence of low mass exotics is an undesirable feature of these models. In this paper we consider rank - 6 inspired SUSY models in which a single discrete symmetry serves to simultaneously forbid tree–level flavor–changing transitions and the most dangerous baryon and lepton number violating operators. We consider models where the and gauge symmetries are spontaneously broken at some intermediate scale so that the matter parity,
is preserved. As a consequence the low-energy spectrum of the models will include two stable weakly interacting particles that potentially contribute to the dark matter density of our Universe. The invariance of the Lagrangian with respect to and symmetries leads to unusual collider signatures associated with exotic states that originate from –plets. These signatures have not been studied in details before. In addition to the exotic matter multiplets that stem from the fundamental representations of the considered models predict the existence of a set of vector-like supermultiplets. In particular the low-energy spectrum of the models involves either a doublet of vector-like leptons or a triplet of vector-like down type quarks. If these extra states are relatively light, they will manifest themselves at the LHC in the near future.
The layout of this paper is as follows. In Section 2 we specify the rank–6 inspired SUSY models with exact custodial symmetry. In Section 3 we present five–dimensional () and six–dimensional () orbifold Grand Unified theories (GUTs) that lead to the rank–6 inspired SUSY models that we propose. In Sections 4 and 5 the RG flow of gauge couplings and implications for collider phenomenology and cosmology are discussed. Our results are summarized in Section 6.
2 inspired SUSY models with exact custodial symmetry
In our analysis we concentrate on the rank–6 inspired SUSY models with two extra gauge symmetries — and . In other words we assume that near the GUT or string scale or its subgroup is broken down to . In the next section we argue that this breakdown can be achieved within orbifold GUT models. We also allow three copies of 27-plets to survive to low energies so that anomalies get cancelled generation by generation within each complete representation of . In models the renormalisable part of the superpotential comes from the decomposition of the fundamental representation and can be written as
Here the summation over repeated family indexes () is implied. In the considered models number is conserved automatically since the corresponding global symmetry is a linear superposition of and . At the same time if terms in and are simultaneously present in the superpotential then baryon and lepton numbers are violated. In other words one cannot define the baryon and lepton numbers of the exotic quarks and so that the complete Lagrangian is invariant separately under and global symmetries. In this case the Yukawa interactions in and give rise to rapid proton decay.
Another problem is associated with the presence of three families of and . All these Higgs–like doublets can couple to ordinary quarks and charged leptons of different generations resulting in the phenomenologically unwanted flavor changing transitions. For example, non–diagonal flavor interactions contribute to the amplitude of oscillations and give rise to new channels of muon decay like . In order to avoid the appearance of flavor changing neutral currents (FCNCs) at the tree level and forbid the most dangerous baryon and lepton number violating operators one can try to impose a single discrete symmetry. One should note that the imposition of additional discrete symmetry to stabilize the proton is a generic feature of many phenomenologically viable SUSY models.
In our model building strategy we use SUSY GUT as a guideline. Indeed, the low–energy spectrum of the MSSM, in addition to the complete multiplets, contains an extra pair of doublets from and fundamental representations, that play a role of the Higgs fields which break EW symmetry. In the MSSM the potentially dangerous operators, that lead to the rapid proton decay, are forbidden by the matter parity under which Higgs doublets are even while all matter superfields, that fill in complete representations, are odd. Following this inspirational example we augment three 27-plets of by a number of components and from extra and below the GUT scale. Because additional pairs of multiplets and have opposite , and charges their contributions to the anomalies get cancelled identically. As in the case of the MSSM we allow the set of multiplets to be used for the breakdown of gauge symmetry. If the corresponding set includes , , and then the symmetry can be broken down to associated with electromagnetism. The VEVs of and break and entirely while the symmetry remains intact. When the neutral components of and acquire non–zero VEVs then symmetry gets broken to and the masses of all quarks and charged leptons are generated.
As in the case of the MSSM we assume that all multiplets are even under symmetry while three copies of the complete fundamental representations of are odd. This forbids couplings in the superpotential that come from . On the other hand the symmetry allows the Yukawa interactions that stem from , and The multiplets have to be even under symmetry because some of them are expected to get VEVs. Otherwise the VEVs of the corresponding fields lead to the breakdown of the discrete symmetry giving rise to the baryon and lepton number violating operators in general. If the set of multiplets includes only one pair of doublets and the symmetry defined above permits to suppress unwanted FCNC processes at the tree level since down-type quarks and charged leptons couple to just one Higgs doublet , whereas the up-type quarks couple to only.
The superfields can be either odd or even under this symmetry. Depending on whether these fields are even or odd under a subset of terms in the most general renormalizable superpotential can be written as
where and are Yukawa couplings and and are mass parameters. Also one should keep in mind that only and components of and appear below the GUT scale. If is odd under symmetry then the term and are forbidden while can have non-zero values. When is even vanish whereas and are allowed by symmetry. In general mass parameters and are expected to be of the order of GUT scale. In order to allow some of the multiplets to survive to low energies we assume that the corresponding mass terms are forbidden at high energies and get induced at some intermediate scale which is much lower than .
The VEVs of the superfields and (that originate from and ) can be used not only for the breakdown of and gauge symmetries, but also to generate Majorana masses for the right–handed neutrinos that can be induced through interactions
The non–renormalizable operators (6) give rise to the right–handed neutrino masses which are substantially lower than the VEVs of and . Because the observed pattern of the left–handed neutrino masses and mixings can be naturally reproduced by means of seesaw mechanism if the right–handed neutrinos are superheavy, the and are expected to acquire VEVs . This implies that symmetry is broken down to near the GUT scale, where symmetry is a linear superposition of and , i.e.
under which right-handed neutrinos have zero charges. Since and acquire VEVs both supermultiplets must be even under symmetry.
At the same time the VEVs of and may break symmetry. In particular, as follows from Eq. (4) the VEV of can induce the bilinear terms and in the superpotential. Although such breakdown of gauge symmetry might be possible the extra particles tend to be rather heavy in the considered case and thus irrelevant for collider phenomenology. Therefore we shall assume further that the couplings of to are forbidden. This, for example, can be achieved by imposing an extra discrete symmetry . Although this symmetry can forbid the interactions of with three complete representations of it should allow non–renormalizable interactions (6) that induce the large Majorana masses for right-handed neutrinos. These requirements are fulfilled if Lagrangian is invariant under symmetry transformations and . Alternatively, one can impose symmetry () under which only transforms. The invariance of the Lagrangian with respect to symmetry () under which only transforms implies that the mass term in the superpotential (5) is forbidden. On the other hand this symmetry allows non–renormalizable term in the superpotential
In this case and can develop VEVs along the –flat direction so that
where is a low–energy supersymmetry breaking scale. This mechanism permits to generate resulting in right-handed neutrino masses of order of
The mechanism of the gauge symmetry breaking discussed above ensures that the low–energy effective Lagrangian is automatically invariant under the matter parity . Such spontaneous breakdown of the gauge symmetry can occur because is a discrete subgroup of and . This follows from the and charge assignments presented in Eq. (2). Thus in the considered case the VEVs of and break gauge symmetry down to . As a consequence the low–energy effective Lagrangian is invariant under both and discrete symmetries. Moreover the symmetry is a product of
where is associated with most of the exotic states. In other words all exotic quarks and squarks, Inert Higgs and Higgsino multiplets as well as SM singlet and singlino states that do not get VEV are odd under symmetry. The transformation properties of different components of , and multiplets under the , and symmetries are summarized in Table 1. Since the Lagrangian of the considered inspired models is invariant under and symmetries it is also invariant under the transformations of symmetry. Because is conserved the lightest exotic state, which is odd under this symmetry, is absolutely stable and contributes to the relic density of dark matter.
It is also well known that in SUSY models the lightest supersymmetric particle (LSP), i.e. the lightest –parity odd particle (), must be stable. If in the considered models the lightest exotic state (i.e. state with ) has even –parity then the lightest –parity odd state cannot decay as usual. When the lightest exotic state is –parity odd particle either the lightest –parity even exotic state or the next-to-lightest –parity odd state with must be absolutely stable. Thus the considered inspired SUSY models contain at least two dark-matter candidates.
The residual extra gauge symmetry gets broken by the VEV of the SM–singlet superfield (and possibly ). The VEV of the field induces the mass of the associated with symmetry as well as the masses of all exotic quarks and inert Higgsinos. If acquires VEV of order (or even lower) the lightest exotic particles can be produced at the LHC. This is the most interesting scenario that we are going to focus on here. In some cases the superfield may also acquire non–zero VEV breaking symmetry as we will discuss later. If this is a case then should be even under the symmetry. Otherwise the superfield can be odd.
The above consideration indicate that the set of multiplets has to contain at least , , and in order to guarantee the appropriate breakdown of the gauge symmetry in the rank–6 inspired SUSY models. However if the set of even supermultiplets involve only , , and then the lightest exotic quarks are extremely long–lived particles. Indeed, in the considered case the symmetry forbids all Yukawa interactions in and that allow the lightest exotic quarks to decay. Moreover the Lagrangian of such model is invariant not only with respect to and but also under symmetry transformations
The invariance ensures that the lightest exotic quark is very long–lived. The , and global symmetries are expected to be broken by a set of non–renormalizable operators which are suppressed by inverse power of the GUT scale or . These operators give rise to the decays of the exotic quarks but do not lead to the rapid proton decay. Since the extended gauge symmetry in the considered rank–6 inspired SUSY models forbids any dimension five operators that break global symmetry the lifetime of the lightest exotic quarks is expected to be of order of
where is the mass of the lightest exotic quark. When the lifetime of the lightest exotic quarks , i.e. considerably larger than the age of the Universe.
The long–lived exotic quarks would have been copiously produced during the very early epochs of the Big Bang. Those lightest exotic quarks which survive annihilation would subsequently have been confined in heavy hadrons which would annihilate further. The remaining heavy hadrons originating from the Big Bang should be present in terrestrial matter. There are very strong upper limits on the abundances of nuclear isotopes which contain such stable relics in the mass range from to . Different experiments set limits on their relative concentrations from to per nucleon . At the same time various theoretical estimations  show that if remnant particles would exist in nature today their concentration is expected to be at the level of per nucleon. Therefore inspired models with very long–lived exotic quarks are ruled out.
To ensure that the lightest exotic quarks decay within a reasonable time the set of even supermultiplets needs to be supplemented by some components of -plet that carry color or lepton number. In this context we consider two scenarios that lead to different collider signatures associated with the exotic quarks. In the simplest case (scenario A) the set of even supermultiplets involves lepton superfields and/or that survive to low energies. This implies that and can interact with leptons and quarks only while the couplings of these exotic quarks to a pair of quarks are forbidden by the postulated symmetry. Then baryon number is conserved and exotic quarks are leptoquarks.
In this paper we restrict our consideration to the inspired SUSY models that lead to the approximate unification of the , and gauge couplings at some high energy scale . This requirement implies that in the one–loop approximation the gauge coupling unification is expected to be almost exact. On the other hand it is well known that the one–loop gauge coupling unification in SUSY models remains intact if the MSSM particle content is supplemented by the complete representations of (see for example ). Thus we require that the extra matter beyond the MSSM fill in complete representations. In the scenario A this requirement can be fulfilled if and are odd under the symmetry while is even supermultiplet. Then and from the can get combined with the superposition of the corresponding components from so that the resulting vectorlike states gain masses of order of . The supermultiplets and are also expected to form vectorlike states. However these states are required to be light enough to ensure that the lightest exotic quarks decay sufficiently fast333Note that the superfields and are not allowed to survive to low energies because they spoil the one–loop gauge coupling unification.. The appropriate mass term in the superpotential can be induced within SUGRA models just after the breakdown of local SUSY if the Kähler potential contains an extra term .
The presence of the bosonic and fermionic components of at low energies is not constrained by the unification of the , and gauge couplings since is the SM singlet superfield. If is odd under the symmetry then it can get combined with the superposition of the appropriate components of . The corresponding vectorlike states may be either superheavy () or gain TeV scale masses. When is even superfield then its scalar component is expected to acquire a non-zero VEV breaking gauge symmetry.
Thus scenario A implies that in the simplest case the low energy matter content of the considered inspired SUSY models involves:
where the right–handed neutrinos are expected to gain masses at some intermediate scale, while the remaining matter survives down to the EW scale. In Eq. (13) and . Integrating out , and as well as neglecting all suppressed non-renormalisable interactions one gets an explicit expression for the superpotential in the considered case
A second scenario, that allows the lightest exotic quarks to decay within a reasonable time and prevents rapid proton decay, is realized when the set of multiplets together with , , and contains an extra superfield (instead of ) from . If the even supermultiplet survives to low energies then exotic quarks are allowed to have non-zero Yukawa couplings with pair of quarks which permit their decays. They can also interact with and right-handed neutrinos. However if Majorana right-handed neutrinos are very heavy () then the interactions of exotic quarks with leptons are extremely suppressed. As a consequence in this scenario B and manifest themselves in the Yukawa interactions as superfields with baryon number .
Although in the scenario B the baryon and lepton number violating operators are expected to be suppressed by inverse powers of the masses of the right–handed neutrinos they can still lead to the rapid proton decay. The Yukawa interactions of the even superfield with other supermultiplets of ordinary and exotic matter can be written in the following form
Integrating out Majorana right-handed neutrinos one obtains in the leading approximation
where is an effective seesaw scale which is determined by the masses and couplings of and . In the considered case the baryon and lepton number violation takes place only when all three terms in Eqs. (15)–(16) are present in the superpotential. If () or the baryon and lepton number conservation requires exotic quarks to be either diquarks or leptoquarks respectively. When vanish the conservation of the baryon and lepton numbers implies that the superfields , and have the following and charges and . This consideration indicates that in the case when all three terms are present in Eqs. (15)–(16) the and global symmetries can not be preserved. It means that in the leading approximation the proton decay rate is caused by all three types of the corresponding Yukawa couplings and has to go to zero when the Yukawa couplings of at least one type of Yukawa interactions vanish. In practice, the proton lifetime is determined by the one–loop box diagram that leads to the dimension seven operator
where and . In Eq. (17) Greek indices denote the color degrees of freedom while indices are suppressed. Here we assume that all particles propagating in the loop have masses of the order of . For and the appropriate suppression of the proton decay rate can be achieved if the corresponding Yukawa couplings are less than .
Once again, the requirement of the approximate unification of the , and gauge couplings constrains the low energy matter content in the scenario B. The concept of gauge coupling unification implies that the perturbation theory method provides an adequate description of the RG flow of gauge couplings up to the GUT scale at least. The requirement of the validity of perturbation theory up to the scale sets stringent constraint on the number of extra and supermultiplets that can survive to low energies in addition to three complete fundamental representations of . For example, the applicability of perturbation theory up to the high energies permits only one extra pair of triplet superfields to have mass of the order of TeV scale. The same requirement limits the number of pairs of doublets to two.
Because in the scenario B the even supermultiplets and are expected to form vectorlike states which have to have TeV scale masses the limit caused by the validity of perturbation theory up to the scale is saturated. Then in order to ensure that the extra matter beyond the MSSM fill in complete representations and should survive to the TeV scale as well. As before we assume that these supermultiplets are odd under the symmetry so that they can get combined with the superposition of the corresponding components from at low energies forming vectorlike states. Again the superfield may or may not survive to the TeV scale. It can be either even or odd under the symmetry. If is even, it should survive to low energies and its scalar component is expected to get a VEV.
Following the above discussion the low energy matter content in the simplest case of the scenario B may be summarized as:
All states in Eq. (18) are expected to be considerably lighter than the GUT scale . Assuming that , and