The Minimal SUSY Model: From the Unification Scale to the LHC
ABSTRACT: This paper introduces a random statistical scan over the high-energy initial parameter space of the minimal SUSY model–denoted as the MSSM. Each initial set of points is renormalization group evolved to the electroweak scale–being subjected, sequentially, to the requirement of radiative and electroweak symmetry breaking, the present experimental lower bounds on the vector boson and sparticle masses, as well as the lightest neutral Higgs mass of 125 GeV. The subspace of initial parameters that satisfies all such constraints is presented, shown to be robust and to contain a wide range of different configurations of soft supersymmetry breaking masses. The low-energy predictions of each such “valid” point–such as the sparticle mass spectrum and, in particular, the LSP–are computed and then statistically analyzed over the full subspace of valid points. Finally, the amount of fine-tuning required is quantified and compared to the MSSM computed using an identical random scan. The MSSM is shown to generically require less fine-tuninng.
- I Introduction
- II The TeV Scale Model
- III Journey From the Unification Scale
- IV The Framework
- V The Parameter Space and Scan
- VI Results
- VII Fine-Tuning
- VIII Conclusion
- IX Acknowledgments
- A Renormalization Group Equations
- B Physical Masses
- C Application of the Checks and Iterative Procedure
The minimal supersymmetric standard model (MSSM) is the simplest possible supersymmetric extension of the standard model of particle physics. It has the gauge group , replaces each gauge field by a vector supermultiplet, each matter fermion by a chiral supermultiplet, has a conjugate pair of Higgs chiral superfields and has no right-handed neutrino multiplets. The MSSM was introduced in various contexts in Dimopoulos:1981zb (); Nappi:1982hm () and has been extensively reviewed in Martin:1997ns (). However, without further modification, the most general superpotential for the MSSM contains cubic interactions that explicitly violate both lepton and baryon number and lead, among other things, to rapid proton decay–which is unobserved. The traditional solution to this problem is to demand that, in addition to the gauge group, the MSSM be invariant under an ad hoc finite symmetry–-parity–which acts on individual component fields as . Although there have been many attempts to explain how -parity can arise in the MSSM Aulakh:1999cd (); Aulakh:2000sn (); Babu:2008ep (); Feldman:2011ms (); FileviezPerez:2011dg () or be spontaneously broken Aulakh:1982yn (); Hayashi:1984rd (); Masiero:1990uj (). These are all “non-minimal” in the sense, for example, that they require additional matter multiplets, have other ad hoc assumptions and so on. Is there a more natural and minimal approach to -parity in the MSSM?
One begins by noting that, within the context of supersymmetry, -parity is a finite subgroup of –see for example Mohapatra:1986su (). It follows that -parity might arise as a consequence of demanding that the MSSM be invariant under a global symmetry. This would be consistent with current bounds on baryon and lepton number violation, and can also be imposed on the MSSM extended to include three families of right-handed neutrino chiral supermultiplets . However, one of the implications of the standard model is that a global continuous symmetry group is likely to appear in its local form–that is, as a gauge symmetry. With this in mind, one might ask if -parity could arise as the consequence of extending the MSSM gauge group to include a new gauged symmetry. It has long been known that the standard MSSM is anomalous with respect to gauged symmetry, whereas the MSSM extended by three families of right-handed neutrino supermultiplets is anomaly free. Furthermore, this is the minimal such extension of the MSSM. We will call this anomaly free, minimal content theory the MSSM, and propose that this is a more “natural” way in which -parity can arise in low energy supersymmetric particle physics.
The MSSM was identified from a low-energy “bottom-up” point of view in FileviezPerez:2008sx (); Barger:2008wn (); Everett:2009vy ()††See Mohapatra:1986aw () for a similar idea in the context of .. In several papers FileviezPerez:2012mj (); Perez:2013kla (), these authors explored its structure and some phenomenological consequences. In addition, these ideas, and other directions, have recently been reviewed in Perez:2015rza (). Interestingly, the MSSM was also discovered from a high-energy “top-down” viewpoint in Braun:2005ux (); Braun:2005nv (); Braun:2013wr (), where it was shown that this model arises within the context of heterotic superstring theory Lukas:1998yy (); Lukas:1998tt (). More specifically, the MSSM is the low-energy effective theory associated with compactifying the heterotic string on a Schoen Calabi-Yau threefold Braun:2004xv () with a specific class of vector bundles Braun:2005zv (). An important aspect of this high-energy point of view is that the parameters of the theory are specified near the gauge coupling unification scale, and then run down to the electroweak scale using the renormalization group (RG). This allows one to explore fundamental aspects of the theory–such as and electroweak symmetry breaking. First steps in this direction were taken in Ambroso:2009jd (); Ambroso:2009sc (); Ambroso:2010pe (), where it was shown–for a restrictive set of initial parameters–that radiative breaking of both of these symmetries can indeed occur. A further study of the Wilson lines, spectra, mass scales and the unification of gauge couplings from the high-energy superstring point of view was presented in Ovrut:2012wg (). Combining the bottom-up and top-down approaches to the MSSM, various aspects of both LHC and neutrino phenomenology were studied in the special case where a stop or a sbottom sparticle is the lightest supersymmetric particle (LSP) Marshall:2014kea (); Marshall:2014cwa ().
However, all of the previous analyses involved specific assumptions–either about high-energy initial conditions or the low-energy structure of the theory. For the MSSM to be a realistic contender for the low-energy theory of particle physics, it is essential that its initial parameter space be explored in a generic way, and that its low-energy predictions be compared with all present experimental data. This will be carried out in detail in this paper. We perform a statistical scan over a well-defined and wide range of initial parameters and, for each fixed set of such parameters, scale the theory down to low energy using the RGEs with specified threshold conditions. The results will be examined to determine the subset of the parameter space that, sequentially, 1) breaks symmetry at a scale consistent with experiment, 2) breaks electroweak symmetry, 3) has all sparticle masses above their present experimental lower bounds and 4) predicts the mass of the lightest neutral Higgs scalar to be within 2 standard deviation from the ATLAS measured value of 125.36 GeV. A small subset of important results from this statistical scan were given in Ovrut:2014rba (). Here, we give present the details of the method use, as well as a wide array of new results and experimental predictions.
This paper is organized as follows. In Section II, the MSSM model at the TeV scale is described in detail. This includes a discussion of both and electroweak symmetry breaking and much of our notation. The connection between the UV picture at the scale of gauge coupling unification and the TeV scale is outlined in Section III. Specifically, the important mass scales and energy regimes are fully described and the relationships between the different mass scales are presented. The content of these two sections is expanded upon in Section IV. Here, the details of the renormalization group equation (RGE) evolution of the parameters of the theory between the UV and TeV scales are discussed. This includes the appropriate input values of the parameters and the relevant equations–with reference to Appendix A when necessary. Special attention is given to the right-handed sneutrino RGE which drives radiative symmetry breaking. The relationship between the different running parameters and scales–introduced in Section III–is further discussed. The latter part of this section outlines the experimental bounds used in our analysis. First, collider bounds are discussed. These correspond to lower bounds on the physical sparticle masses–which are closely related to the running mass parameters. The section finishes by describing–and implementing–the well-known bounds from flavor changing neutral currents and CP violation.
This paper approaches the connection between UV and TeV physics in a novel way. Specifically, instead of assuming some universal conditions or relationships between UV soft SUSY breaking parameters, we simply allow all such parameters to be within about an order of magnitude of some chosen SUSY breaking scale. Our approach will then be to scan all relevant SUSY breaking parameters, at the high scale, over this possible range. These parameters will then be RG evolved to the TeV scale, following the discussion in Section IV. This very general approach, as well as other details of our scan, are described in Section V. Such an approach is especially applicable to the string realization of the MSSM model, since, in that case, each chiral supermultiplet arises from a different representation of . Therefore, their soft SUSY breaking masses do not obey boundary conditions at the high scale. Our analysis is also valid for a wide range of pure GUTs, which can impose disparate boundary conditions at the unification scale–including none at all. Finally a “meta” scan is conducted to choose the optimal value for the range of SUSY breaking mass scales.
The optimal range, arrived at in Section V, is used to generate all subsequent results in this paper–the bulk of which are presented in Section VI. Relating these results to the twenty-four phenomenologically relevant scanned parameters is daunting at best. Fortunately, a cohesive picture can be presented in terms of two so-called -parameters. These -parameters, which are each the sum of the squares of SUSY breaking mass parameters, play an important role in radiative symmetry breaking. Their role in this capacity is presented in Fig. 5. The results for all other experimental constraints can also be expressed in terms of these two parameters–see Fig. 6 and Fig. 7. A central result of this work is given is Fig. 8, which displays the frequency at which particles appear as LSPs in our scan. The section closes with histograms of the spectra, as well as some spectrum plots to help characterize specific features of our results.
Fine-tuning in our approach is addressed in Section VII. While fine-tuning in the MSSM is not drastically different than in the MSSM, there are several key issues to highlight. First, while one might expect that the scale associated with symmetry breaking could introduce new contributions to fine-tuning, it is shown that this is not the case. Second, the MSSM, analyzed using the same methods as in this paper, typically yields equal or more fine-tuning than in the our model for similar initial points. Finally, an LSP analysis similar to Fig. 8, but with fine-tuning constrained to be better than one part in a thousand, is presented in Fig. 17. We conclude in Section VIII.
In addition to the main sections of this paper, three Appendices are included to help elucidate various topics. Appendix A contains all one-loop RGEs for this model in the different regimes. The first part of Appendix B specifies how to relate the running soft mass parameters to the physical masses of the SUSY particles. The second part of Appendix B describes the procedure used to calculate the SM-like Higgs mass. Those readers interested in the details of how the random scan in this paper was conducted, are directed to Appendix C.
Ii The TeV Scale Model
Motivated by both phenomenological considerations and string theory, we analyze the minimal anomaly free extension of the MSSM with gauge group
As discussed in Ovrut:2012wg (), we prefer to work with the Abelian factors rather than – although they are physically equivalent. This is motivated by the fact that the former is the unique choice that does not introduce kinetic mixing between the associated field strengths at any scale in their renormalization group equation (RGE) evolution. The gauge covariant derivative can be written as
where is the charge and the factor of is introduced in the last term by a redefinition of the gauge coupling – thus simplifying many equations. As discussed in Ovrut:2012wg () and throughout this paper, a radiatively induced vacuum expectation value (VEV) for a right-handed sneutrino will spontaneously break the Abelian factors to , in analogy with the way that the Higgs fields break to in the SM. For simplicity, we will refer to this as “” symmetry breaking–even though it is technically the breaking of a linear combination of the and generators, leaving the hypercharge group generated by
invariant. The particle content of the minimal model is simply that of the MSSM plus three right-handed neutrino chiral multiplets. That is, three generations of matter superfields
along with two Higgs supermultiplets
We refer to this model throughout the remainder of this paper as the MSSM.
The superpotential of the MSSM is given by
where flavor and gauge indices have been suppressed and the Yukawa couplings are three-by-three matrices in flavor space. In principle, the Yukawa matrices are arbitrary complex matrices. However, the observed smallness of the three CKM mixing angles and the CP-violating phase dictate that the quark Yukawa matrices be taken to be nearly diagonal and real. The lepton Yukawa coupling matrix can also be chosen to be diagonal and real. This is accomplished by moving the rotation angles and phases into the neutrino Yukawa couplings which, henceforth, must be complex matrices. Furthermore, the smallness of the first and second family fermion masses implies that all components of the up, down, and lepton Yukawa couplings–with the exception of the (3,3) components–can be neglected for the purposes of this paper. Similarly, the very light neutrino masses imply that the neutrino Yukawa couplings are sufficiently small so as to be neglected for the purposes of this paper. The -parameter can be chosen to be real, but not necessarily positive, without loss of generality. The soft supersymmetry breaking Lagrangian is then given by
The parameter can be chosen to be real and positive without loss of generality. The gaugino soft masses can, in principle, be complex. This, however, could lead to CP-violating effects that are not observed. Therefore, we proceed by assuming they all are real. The -parameters and scalar soft mass can, in general, be Hermitian matrices in family space. Again, however, this could lead to unobserved flavor and CP violation. Therefore, we will assume they all are diagonal and real. Furthermore, we assume that only the (3,3) components of the up, down, and lepton -parameters are significant and that the neutrino parameters are negligible. For more explanation of these assumptions, see Section IV.2.
Spontaneous breaking of symmetry results from a right-handed sneutrino developing a non-vanishing VEV, since it carries the appropriate and charges. However, since sneutrinos are singlets under the gauge group, it does not break any of the SM symmetries. To acquire a VEV, a right-handed sneutrino must develop a tachyonic mass††Here and throughout this paper we use the term “tachyon” to describe a scalar particle whose parameter is negative. Although all parameters at high scale will be chosen positive, one or more can be driven negative at lower energy by radiative corrections. This signals dynamical instability at the origin–although a stable VEV may, or may not, develop.. As discussed in Mohapatra:1986aw (); Ghosh:2010hy (); Barger:2010iv (), a VEV can only be generated in one linear combination of the right-handed sneutrinos. Furthermore, beyond the fact that its VEV breaks symmetry, in which combination it occurs has no further observable effect. This is because there is no right-handed charged current to link the right-handed neutrinos to a corresponding right-handed charged lepton. Therefore, without loss of generality, one can assume that it is the third generation right-handed sneutrino that acquires a VEV. At a lower mass scale, electroweak symmetry is spontaneously broken by the neutral components of both the up and down Higgs multiplets acquiring non-zero VEV’s. In combination with the right-handed sneutrino VEV, this also induces a VEV in each of the three generations of left-handed sneutrinos. The notation for the relevant VEVs is
where is the generation index.
The neutral gauge boson that becomes massive due to symmetry breaking, , has a mass at leading order, in the relevant limit that , of
The second term in the parenthesis is a small effect due to mixing in the neutral gauge boson sector. The hypercharge gauge coupling is given by
Since the neutrino masses are roughly proportional to the and parameters, it follows that and . In this phenomenologically relevant limit, the minimization conditions of the potential are simple and worthwhile to note. They are
Here, the first two equations correspond to the sneutrino VEVs. The third and fourth equations are of the same form as in the MSSM, but new scale contribution to and shift their values significantly compared to the MSSM. Eq. (13) can be used to re-express the mass as
This makes it clear that, to leading order, the mass is determined by the soft SUSY breaking mass of the third family right-handed sneutrino. The term proportional to is insignificant in comparison and, henceforth, neglected in our calculations.
A direct consequence of generating a VEV for the third family sneutrino is the spontaneous breaking of -parity. The induced operators in the superpotential are
This general pattern of -parity violation is referred to as bilinear -parity breaking and has been discussed in many different contexts, especially in reference to neutrino masses– see references Mukhopadhyaya:1998xj (); Chun:1998ub (); Chun:1999bq (); Hirsch:2000ef () for early works. In addition, the Lagrangian contains additional bilinear terms generated by and from the super-covariant derivative. These are
The consequences of spontaneous -parity violation are quite interesting, and have been discussed in a variety of papers. For LHC studies, see FileviezPerez:2012mj (); Perez:2013kla () as well as recent work on stop and sbottom LSP’s in this context and the connection between their decays and the neutrino sector Marshall:2014kea (); Marshall:2014cwa (). Predictions for the neutrino sector were discussed in Mohapatra:1986aw (); Ghosh:2010hy (); Barger:2010iv (). It was shown that the lightest left-handed, or active, neutrino is massless and that the model contains two right-handed neutrinos, referred to as sterile neutrinos, that are lighter than the remaining two active neutrinos. Sterile neutrinos can influence the cosmological evolution of the universe due to their role as dark radiation. This effect was studied in Perez:2013kla ().
In this section, we have focussed on the TeV scale manifestation of the MSSM. However, the main content of this paper will be to study the connection between this low energy theory and its possible origins in heterotic string theory, thereby linking some of the MSSM phenomenology to high scale physics. In this context, our model is the remnant of an unified symmetry broken by two Wilson lines. Although the details of the various physical regimes of this theory, and the renormalization group scaling between them, were given in Ovrut:2012wg (), we review them in the next section for completeness.
Iii Journey From the Unification Scale
The goal of this section is to review the physics associated with the string construction of the MSSM–from unification to the electroweak scale. After compactification to four-dimensions, the unified gauge group is . This is then further broken to the MSSM gauge group by the turning on of two Abelian Wilson lines, denoted by and respectively. The energy scales associated with these Wilson lines need not be the same. In fact, exact gauge coupling unification at one-loop, which we will assume throughout this paper, requires that the scales be different– implying there is a two-step symmetry breaking process from to the gauge group of the MSSM. This leads to an intermediate regime between the two scales associated with the Wilson lines. The particle content and gauge group in this regime depends on which Wilson line turns on first. Defining the mass scales of and as and respectively, we find the following two initial symmetry breaking patterns.
: , the “left-right” model
: , a modified version of the “Pati-Salam” model
In each case, the subsequent turning on of the second Wilson line breaks the intermediate model to the MSSM.
Reference Ovrut:2012wg () studied these two cases and found that gauge coupling unification dictates that the Wilson line scales should be separated by less than an order of magnitude; that is, the intermediate regime is not very large. It follows that the TeV scale physics has little dependence on which of the above two models inhabit the intermediate regime. For simplicity, we will carry out our analysis under the assumption that it is the first of these symmetry breaking patterns that occurs. Hence, the intermediate regime contains the left-right model. We then make the identifications
These scales will be further discussed below.
In the intermediate regime, the particle content of the left-right model consists of nine copies of the matter family
two copies of a Higgs bi-doublet, which contains the MSSM Higgs fields,
and a pair of color triplets
Once the second Wilson line turns on, the extra particle content integrates out and one is left with exactly the spectrum of the MSSM.
At this point, it is important to make a quick note on notation for the gauge coupling. Thus far, we have discussed the gauge parameter , which couples to charge. As is well known, this gauge coupling has to be properly normalized so as to unify with the other gauge parameters. We use defined by
to denote the properly unifying coupling. The parameter couples to charge and will appear in the RGEs. For quantities of physical interest, such as physical masses, will be used.
To fully understand the evolution of this model from unification to the electroweak scale, it should be noted that there are five relevant mass scales of interest, two of which were mentioned briefly above. All five are described in the following:
: The unification mass and the scale of the first Wilson line. We assume that all gauge couplings unify at this scale. That is, .
: The intermediate scale associated with the second Wilson line and the symmetry breaking – that is, the right-handed isospin breaks into its third component. Since the gauge coupling of slightly above is equal to the gauge coupling slightly below , we use for both and . All gauge couplings have trivial thresholds at this scale.
: The scale is the mass at which the right-handed sneutrino VEV triggers . Physically, this corresponds to the mass of the neutral gauge boson of the broken symmetry and, therefore, the scale of decoupling. Specifically
where depends on parameters evaluated at –see Eq. (9). Substituting Eq. (9) into this relation yields a transcendental equation that must be solved using iterative numerical methods to obtain the correct value for .
At this scale, we also evaluate the hypercharge gauge coupling using its relationship to the gauge parameters of and the third component of right-handed isospin. This is given by
: The soft SUSY breaking scale. This is the scale at which all sparticles are integrated out with the exception of the right-handed sneutrinos. The right-handed sneutrinos are associated with breaking and, therefore, are integrated out at the scale. While there is obviously no single scale associated with the masses of all the SUSY partners, we use the scale of stop decoupling given by
This scale is useful because when the stops decouple, the parameter that controls electroweak symmetry breaking, that is, the soft mass parameter, effectively stops running– see Gamberini:1989jw () for more details. Like the scale, the SUSY scale must be determined using iterative numerical methods because the physical stop masses in Eq. (32) depend implicitly on the SUSY scale.
: The electroweak scale. This is the well-known scale associated with the and gauge bosons of the SM. We will make the identification
For correct electroweak breaking, one must satisfy the conditions
The first constraint guarantees that the Higgs potential is bounded from below while the second indicates that the trivial vacuum is not stable.
With the relevant mass scales appropriately defined, we can now discuss the physical regimes that exist in between them. To begin with, we will be interested in the evolution of the gauge couplings–since our assumption that they unify will help relate these disparate scales to each other. We present below, for each regime, the slope factors appearing in the gauge RGE’s
where indexes the associated gauge groups. Note that while , the hierarchy between the SUSY and scales depends on the point chosen in the initial parameter space. Each of the two possibilities will be addressed below.
: This regime is populated by the left-right model discussed above. In this interval, the factors are
We will refer to this scaling interval as the “left-right regime” and, when required, denote the associated -coefficients by .
: This regime is populated by the MSSM model. The factors in this case are
We will refer to this scaling interval as the “B-L MSSM regime” and, when required, denote its -coefficients by .
The remaining two regimes depend on which of the following two cases occurs: –the “right-side-up” hierarchy–and –the “upside-down” hierarchy.
: In this case has been broken but SUSY is still a good symmetry, thereby giving an MSSM-like theory–that is, the MSSM plus two light right-handed neutrino chiral multiplets. Another possible deviation might occur in the composition of the bino–more about this late. In general, however, this is the MSSM. Specifically, the gauge couplings in this regime evolve like the well-known MSSM gauge couplings with coefficients
We refer to this interval as the “MSSM” regime and denote the associated -coefficients by .
: In this regime, one simply has the SM with two sterile neutrinos. It has the well-known slope factors
We refer to this as the “SM” regime and denote the -coefficients by .
: Now remains a good symmetry below the average stop mass, where we effectively integrated out the SUSY partners. The resulting theory is simply a non-SUSY model, which also includes three generations of right-handed sneutrinos–the third of which acts as the Higgs. The slope factors are
: Here, again, we have the SM with two sterile neutrinos and the slope factors given in Eq. (40).
Given the above information, and the demand that all gauge couplings unify, we can solve for a given mass scale in terms of the others. First consider the unification mass–corresponding to the scale at which the four gauge couplings become equal to each other. Practically, it is derived as the energy-momenta at which . As is well-known, this will not be influenced by any scale that acts as a threshold for complete multiplets of a minimal group that unifies and –for example, . The and intermediate scales are both such thresholds. The scale is a threshold for singlets of , that is, the right-handed neutrinos, while is a threshold for six new matter generations, a pair of Higgs doublets and their color partners. All of these particles fit into the 1, 5, and 10 of – see Eqs. (23)-(26). Working through the algebra of setting yields
where the superscripts on the slope factors indicate their regime of relevance and the take their experimental values at PDG ():
Inserting all the coefficients, the unification scale becomes
Similarly, the intermediate scale can be solved for by setting and using the relationship between the gauge couplings of hypercharge , and the third component of right-handed isospin given in Eq. (29). The intermediate scale is found to be
Substituting for using Eq. (44) gives
These relationships are displayed in Fig. 1.
Iv The Framework
The purpose of this paper is twofold: (i) to analyze how the high scale parameter space is associated with the TeV scale phenomenology of the MSSM and (ii) to introduce a new way of analyzing this question. More specifically, (i) is an attempt to understand the kind of general UV conditions that yield valid, experimentally consistent points at the TeV scale–especially conditions that lead to breaking and a viable /electroweak hierarchy, a topic which has not been widely studied before. Such a study requires a framework for evolving parameters through the relevant regimes. The skeleton of this framework was discussed in the previous section. This section will continue this discussion and review the well-known experimental constraints on the SUSY parameters. Utilizing these constraints, the next section presents the motivation and the strategy for our scan. This addresses (ii).
The approach to the RG evolution of the parameters is similar to other such work, with several deviations that will be highlighted below. The RGEs of interest are calculated using reference Martin:1993zk () and are presented in Appendix A. Gauge couplings and gaugino masses are evolved up to the unification scale. The remaining parameters, Yukawa couplings, sfermion mass parameters and -terms, are only evaluated in the scaling regimes below the intermediate scale. This is because in the string construction considered here, the scaling regime between the unification scale and contains six additional copies of matter fields as well as an additional copy of Higgs fields.We note that each component field of a given generation of matter originates from a different 16 of . This is important and will be discussed later. Since these new Yukawa couplings are unknown, RG running them through this regime would not contribute to the predictability of this study. In practice, we implement these calculations piecewise starting with the analytically tractable equations first. These are the gauge couplings, gaugino mass parameters and the first and second generation sfermion mass parameters, as well as all sneutrino mass parameters. We then numerically calculate the evolution of the remaining parameters.
As is traditional, we begin by inputting the experimentally determined parameters–that is, the gauge couplings and Yukawa couplings derived from fermion masses–at the electroweak scale. The initial values of the gauge couplings were given above in Eq. (43). For the purposes of this paper, the SM Yukawa couplings, which are three-by-three matrices in flavor space, can all be approximated to be zero except for the three-three elements which give mass to the third generation SM fermions. We use the initial conditions
For details on relating fermion masses to Yukawa couplings, see Djouadi:2005gi (). Here the lower case represents Yukawa couplings in the non-SUSY regime. These can be evolved to the SUSY scale, both in the right-side-up hierarchy, Eqs. (90) - (92), and up-side-down hierarchy, Eqs. (93) - (95). At the SUSY scale, one has the non-trivial boundary conditions
The gauge couplings in the various regimes were discussed in the previous section. With those solutions in hand, the RGE evolution of the gauginos can be easily derived. Gaugino masses are inputted at the unification scale and evolved down. Naively, one might expect gaugino mass unification. However, this is not always the case–as has been discussed in a number of contexts, see for example Ellis:1985jn (); Choi:2005ge (). Therefore, and to be as general as possible, we impose no relationship between the different gaugino masses at the unification scale. The general RGE for a gaugino mass parameter is
where indexes the gauge groups. These equations can be solved analytically. For the gauginos associated with , , and one has
The bino, however, is a treated somewhat differently for each of the two possible hierarchies between the and SUSY scales. For the right-side-up hierarchy, at the scale, we have three neutral fermions that mix: the third generation right-handed neutrino , the gaugino (blino) and the gaugino (rino). This is a direct consequence of -parity violation in the MSSM. As we will see, it is possible for a neutralino LSP mass eigenstate to have a significant component. The mixing between the third-family right-handed neutrino and the gauginos is described in the basis by the mass matrix†† This mass matrix neglects mixing with the Higgsinos through the electroweak breaking Higgs VEV. This is a safe approximation since the lower bound on the mass implies that the electroweak Higgs VEV will be negligible compared to the third-family right-handed sneutrino VEV.
Due to the RGEs monotonically pushing the values of and down, they will typically be significantly lighter than . It is, therefore, instructive to perturbatively diagonalize this mass matrix in the limit . At zeroth order, the mass eigenstates are
At first order, the effect of adding and back into the mass matrix is to give the bino a mass of
This shows that, in the right-side-up hierarchy, between the scales and we have the gauge group and particle content of the MSSM plus two right-handed neutrino supermultiplets–that is, the two sneutrino generations that do not acquire a VEV.††At some points in parameter space, it is possible that the required limit will not be satisfied and there will not be a mass eigenstate that can clearly be identified as the bino. However, since the scaling regime between and is always small, the errors introduced by assuming the existence of a bino are insignificant. Below the scale, the bino mass is
In the upside-down case, all neutralinos are diagonalized at the SUSY mass scale.
The running of the tri-linear -terms is straightforward. Their initial values are randomly generated at the intermediate scale. The -term RGEs in the MSSM regime are given in Eqs. (102) - (104), while those for the MSSM are in Eqs. (105) - (107). All relevant threshold conditions are trivial.
The RGEs for the square of the soft sfermion mass parameters can be broken into two categories: 1) those with simple analytic solutions–given in Eqs. (114) - (119) and Eqs. (120) - (125) for the MSSM and MSSM regimes respectively–and 2) those requiring numerical solutions–given in Eqs. (127) - (133) and Eqs. (134) - (140) for the MSSM and MSSM regimes. These parameters are all inputted at the intermediate scale. The third generation right-handed sneutrino is then evolved to the scale–while all other sfermion mass squared parameters are RG evolved to the SUSY scale. The third generation right-handed sneutrino mass squared plays an important role here since, when it runs negative, it triggers breaking as was discussed in detail in Ambroso:2009jd (); Ambroso:2009sc (). The right-handed sneutrino mass RGE is
Despite the lack of a large Yukawa coupling, the right-handed sneutrino mass can still be driven tachyonic by appropriate signs and magnitudes of the -terms defined in Eqns (59, 60). To emphasize this, the analytic solution to the sneutrino mass RGE is presented here. It is
Recall that the value of any Abelian gauge couplings grows larger at higher scale. Therefore, we see that a tachyonic sneutrino is only possible when is negative and/or is positive. This demonstrates the central role played by the -terms in the breaking of symmetry. Note that in typical unification scenarios all soft masses are “universal” and, hence, both -terms vanish. However, it was mentioned earlier that, in this string construction, different elements of a given generation arise from different 16 representations of . Therefore, the soft masses of a given generation are generically non-degenerate. Hence, the -terms can be non-zero.
As mentioned above, and the relationship
is used to iteratively solve for the scale. The SUSY mass scale must also be solved for iteratively using the equation
where are the physical stop masses. The relationships between the soft mass parameters and the physical masses are given in Appendix B.1. The soft mass squared parameter for the up-type Higgs is driven tachyonic, as usual, by the large top Yukawa coupling. Furthermore, the decoupled values of the soft Higgs mass squared parameters are used to calculate the - and -terms using Eqs. (15) and (16).
The soft mass parameters have non-trivial boundary conditions at the scale due to the effects of the and -terms:
where and indicate a scale slightly below and slightly above the scale respectively, and and are the third component of right-handed isospin and hypercharge of a generic scalar .
Once the mass parameters have been properly evolved to their appropriate scales, the physical masses can be evaluated. For much of the spectrum, this has been discussed in the literature, see for example Martin:1997ns (), and has been included in Appendix B.1. The new element here is the mass of the scalar associated with the third generation right-handed sneutrino–the Higgs–degenerate with the mass. In addition, the calculation for the SM-like Higgs mass is crucial since the experimentally measured value of 125 GeV requires substantial radiative corrections from the stop sector in the MSSM. In this paper we follow the approach of references ArkaniHamed:2004fb (); Cabrera:2011bi (); Giudice:2011cg ()–taking into account the decoupling scales of the two stops, matching the quartic Higgs coupling at those scales and RGE evolving the quartic coupling to the electroweak scale to calculate the Higgs mass. Full details are given in Appendix B.2.
Once a given physical mass is calculated, it is compared to current lower bounds or, in the case of the SM Higgs, the experimentally measured value. If a given initial set of parameters predicts a physical mass that is inconsistent with current bounds, it is rejected as being an invalid point. These lower bounds are discussed in the next subsection.
iv.1 Collider Constraints
The bounds placed by collider data on SUSY masses are, in general, model dependent. That is, they depend on the spectrum and decay modes. Despite the much larger energy of the LHC, LEP 2 still has competitive bounds on colorless particles that couple to the and/or the photon–including sleptons in scenarios with both -parity conservation LEPSUSYWG/04-01.1 (); LEPSUSYWG/02-09.2 () and violation LEPSUSYWG/02-10.1 (), bounds on charginos LEPSUSYWG/01-03.1 (); LEPSUSYWG/02-04.1 () and bounds on sneutrinos in the case of -parity violation LEPSUSYWG/02-10.1 (). As one may expect, due to the relatively clean environment at LEP, these bounds are close to one half the center of mass energy of LEP 2. Therefore, for simplicity, we proceed with the bound that all colorless fields that couple to the photon must be heavier than 100 GeV. That is,
where is any charged slepton. Colorless states that couple to the , the left-handed sneutrino, must be heavier than half the mass:
Colorless states that do not couple to the Z, such as right-handed sneutrinos/neutrinos and the bino, have such small collider production cross-sections that they do not have collider-based lower bounds. Wino and Higgsino neutralinos are degenerate with their chargino partner, thereby effectively putting a lower bound of 100 GeV on those states as well.
The bounds from the LHC are much more dependent on the parameters. For example, if one investigated the bound on, for example, degenerate squarks in this model with a neutralino LSP, those bounds could be significantly different than in the case of a sneutrino, or some other, LSP. Allowing the squark masses to split would further alter the lower bounds. In fact, a full treatment would involve calculating the signatures of a given point in parameter space, comparing the number of events to the most recent LHC bounds on such events, and determining if the parameter point is valid. We do not expect the details of these lower bounds to heavily affect our results. We will, therefore, simply use the naive bounds
which are based on recent CMS CMS:2014ksa () and ATLAS Aad:2014wea () studies of the -parity conserving MSSM. In these studies, the colored states decay into jets and missing energy–possible final states in our model whenever the LSP decays into neutrinos. In this paper, we impose these bounds except in the case of a stop or sbottom LSP. These two cases were explicitly studied in Marshall:2014kea (); Marshall:2014cwa () and yielded the following lower bounds:
where () denotes the lightest stop (sbottom). Here, right-handed refers to a stop that is almost completely right-handed–that is, a stop mixing angle, or, equivalently, a state composed of 99 right-handed stop–while admixture stop refers to all other stops. This distinction is based on the phenomenology of the stops; right-handed stops have significant decays into a top quark and neutrinos while admixture stops decay almost exclusively to a bottom quark and a charged lepton.
The lower bound on the mass from LHC searches is 2.5 TeV ATLAS:2013jma (); CMS:2013qca (). Finally, we require that the Higgs mass be within the allowed range from the value measured at the ATLAS experiment at the LHC. We naively obtain the two sigma range by adding in quadrature the systematic and statistical uncertainties from Aad:2014aba (), and multiplying the result by two:
|Left-handed sneutrinos||45.6 GeV|
|Charginos, sleptons||100 GeV|
|Squarks, except for stop or sbottom LSP’s||1000 GeV|
|Stop LSP (admixture)||450 GeV|
|Stop LSP (right-handed)||400 GeV|
|Sbottom LSP||500 GeV|
iv.2 Constraints from Flavor and CP-Violation
A large number of low-energy experiments exist which place constraints on the SUSY parameter space. Some of the oldest and most well-known are the constraints placed on flavor changing neutral currents from the analyses in references Ellis:1981ts (); Barbieri:1981gn (); Campbell:1983bw ()–for example, those arising from oscillation–and on CP violation Ellis:1982tk (); Buchmuller:1982ye (); Polchinski:1983zd (); del Aguila:1983kc (); Nanopoulos:1983xd ()–for example, from electric dipole moment measurements. Generically, the implication of these constraints are, approximately, as follows:
Soft sfermion mass matrices are diagonal.
The first two generations of squarks are degenerate in mass.
The trilinear -terms are diagonal.
The gaugino masses and trilinear -terms are real.
In addition, it is typically assumed that the soft trilinear -terms are proportional to the Yukawa couplings, that is, generically for each fermions species. Each is a dimensionful real number on the order of a TeV, while each factor is a dimensionless matrix in flavor space. This condition effectively makes all non-third generation trilinear terms insignificant. Note that this assumption does not immediately follow from the above experimental constraints. However, significant radiative corrections to fermion masses, proportional to the -term, can arise in SUSY, as first discussed for fermions in references Hall:1993gn (); Carena:1994bv (); ArkaniHamed:1995fq (). For example, a down quark mass is modified by gluino exchange, through the diagram in Fig. 2, as follows:
and is the gluino mass. If is on the order of a TeV, this radiative correction can be quite large, possibly larger than the down quark mass. If this were the case, the radiative correction would have to be fine-tuned against the tree-level contribution to reproduce the correct down quark mass. This motivates allowing only the third generation -terms to be significant. Hence, our assumption that