KIAS–P16059, UMN–TH–3601/16, FTPI–MINN–16/23
The Super-GUT CMSSM Revisited
John Ellis, Jason L. Evans, Azar Mustafayev,
and Keith A. Olive
Theoretical Physics and Cosmology Group, Department of Physics,
King’s College London, Strand, London WC2R 2LS, UK
Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland
School of Physics, KIAS, Seoul 130-722, Korea
William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA
We revisit minimal supersymmetric SU(5) grand unification (GUT) models in which the soft supersymmetry-breaking parameters of the minimal supersymmetric Standard Model (MSSM) are universal at some input scale, , above the supersymmetric gauge coupling unification scale, . As in the constrained MSSM (CMSSM), we assume that the scalar masses and gaugino masses have common values, and , respectively, at , as do the trilinear soft supersymmetry-breaking parameters . Going beyond previous studies of such a super-GUT CMSSM scenario, we explore the constraints imposed by the lower limit on the proton lifetime and the LHC measurement of the Higgs mass, . We find regions of , , and the parameters of the SU(5) superpotential that are compatible with these and other phenomenological constraints such as the density of cold dark matter, which we assume to be provided by the lightest neutralino. Typically, these allowed regions appear for and in the multi-TeV region, for suitable values of the unknown SU(5) GUT-scale phases and superpotential couplings, and with the ratio of supersymmetric Higgs vacuum expectation values .
There have been many phenomenological studies of the minimal supersymmetric (SUSY) extension of the Standard Model (MSSM) that assume some degree of universality for the soft supersymmetry-breaking scalar and gaugino masses, and , and the trilinear soft supersymmetry-breaking parameters . Scenarios in which these parameters are universal at the supersymmetric grand unification (GUT) scale, , called the constrained MSSM (CMSSM) [1, 2, 3, 4, 5, 6], have been particularly intensively studied, usually assuming that the lightest supersymmetric particle (LSP) is a neutralino, which is stable because of the conservation of -parity , and provides (all or some of) the cosmological cold dark matter. These and other GUT-universal models are under strong pressure from LHC data [5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], in particular, the notable absence of missing transverse energy signals at the LHC [19, 20], with the measurement of the Higgs mass [21, 22], , providing an additional important constraint.
Fewer studies have been performed for scenarios in which the soft supersymmetry-breaking parameters are universal at some other scale , which might be either below the GUT scale (so-called sub-GUT or GUT-less scenarios [23, 11, 18]) or above the GUT scale (so-called super-GUT scenarios [24, 25, 26]). For example, in our current state of confusion about the possible mechanism of supersymmetry breaking, and specifically in the absence of a convincing dynamical origin at , one could well imagine that the universality scale might lie closer to the Planck or string scale: .
When studying such super-GUT scenarios, there appear additional ambiguities beyond those in the conventional CMSSM. What is ? Which GUT model to study? What are its additional parameters? How much additional freedom do they introduce? In parallel, once one commits to a specific GUT model, one must also consider the constraint imposed by the absence (so far) of proton decay . In order to minimize the ambiguities and the number of additional GUT parameters, we study here the minimal supersymmetric SU(5) GUT .
It is well known that the length of the proton lifetime is a significant challenge for this model [29, 30], and one of the principal new ingredients in this paper, compared to previous studies of super-GUT CMSSM models, is the incorporation of this constraint in our exploration of the model parameter space. Another improvement on previous super-GUT CMSSM studies is the incorporation of LHC constraints, of which the measurement of the Higgs mass turns out to be the most relevant.
We find regions of the soft supersymmetry-breaking parameters , , and the unknown coefficients in the SU(5) superpotential that are compatible with these and other phenomenological constraints such as the density of cold dark matter. As usual, we assume that this is provided by the LSP, which we assume to be the lightest neutralino. The Higgs mass and proton lifetime constraints both favor and in the multi-TeV region, and proton stability favours a value for the ratio of supersymmetric Higgs vacuum expectation values (VEVs), . The cosmological constraint on the cold dark matter density typically favors narrow strips of parameter space where coannihilation with the lighter stop brings the LSP density into the cosmological range. All these constraints can be reconciled for suitable values of the unknown SU(5) superpotential couplings.
The layout of this paper is as follows. In Section 2 we review our set-up of the super-GUT CMSSM, with particular attention to the model parameters and the matching to the relevant parameters below the GUT scale. Section 3 then reviews our treatment of proton decay, paying particular attention to the potential implications of unknown GUT-scale phases. Our results are presented and explained in Section 4, and Section 5 then summarizes our conclusions. An Appendix reviews details of our nucleon decay calculations.
2 Super-GUT CMSSM Models
2.1 Minimal SUSY SU(5)
We first review briefly the minimal supersymmetric SU(5) GUT , specifying our notation. This model is the simplest supersymmetric extension of the original SU(5) GUT model due to Georgi and Glashow . In this model, the right-handed down-type quark and left-handed lepton chiral superfields, and , respectively, reside in representations, , while the left-handed quark doublet, right-handed up-type quark, and right-handed charged-lepton chiral superfields, , , and , respectively, are in representations, , where the index denotes the generations. The MSSM Higgs chiral superfields and are embedded into and representations, and , respectively, where they are accompanied by the and coloured Higgs superfields and , respectively.
The SU(5) GUT gauge symmetry is assumed to be spontaneously broken down to the Standard Model (SM) gauge group by the vacuum expectation value (vev) of a chiral superfield, , where the () are the generators of SU(5) normalized so that . The renormalizable superpotential for this model is then given by
where Greek sub- and superscripts denote SU(5) indices, and is the totally antisymmetric tensor with .
The adjoint Higgs is assumed to have a vev of the form
where . In this case, the GUT gauge bosons acquire masses , where is the SU(5) gauge coupling. In order to realize the doublet-triplet mass splitting in and , we need to impose the fine-tuning condition , which we discuss in Section 2.4. In this case, the masses of the color and weak adjoint components of are equal to , while the singlet component of acquires a mass . The color-triplet Higgs states have masses .
2.2 Planck-scale suppressed higher-dimensional operators
In supersymmetric GUTs, gauge-coupling unification predicts that the unification scale is GeV. Since the unification scale is fairly close to the reduced Planck mass GeV, interactions of gravitational strength may give rise to sizable effects. We accommodate these effects by considering higher-dimensional effective operators suppressed by powers of .
We may expect that such effective operators play significant roles in the minimal SUSY SU(5) GUT. For example, in minimal SU(5) GUTs fthe down-type Yukawa couplings are predicted to be equal to the corresponding lepton Yukawa couplings at the GUT scale, since they both originate from . Nevertheless, in most of the parameter space we consider, this Yukawa unification is imperfect. For the third generation, the deviation is typically at the % level. For the first two generations, on the other hand, there are differences. These less successful predictions can be rectified if one considers the following dimension-five effective operators that are suppressed by the Planck scale [32, 33]:
These operators induce non-universal contributions to the effective Yukawa couplings that are after the adjoint Higgs acquires a VEV 111There is another class of dimension-five operators of the form . However, they do not spoil Yukawa unification, but only modify the overall sizes of the down-type quark and charged-lepton Yukawa couplings by ., which is sufficient to account for the observed deviations 222One may also use higher-dimensional Higgs representations to explain the observed differences between down-type and lepton Yukawa couplings . However, in this paper we focus on the minimal SU(5) GUT, and do not consider this alternative..
There are several other dimension-five operators that one may consider. Among them is
where denotes the superfields corresponding to the field strengths of the SU(5) gauge vector bosons . The term (4) can have a significant effect, since it changes the matching conditions of the gauge coupling constants after develops a VEV [35, 36, 37]. This operator also modifies the matching conditions for gaugino masses, thereby modifying gaugino mass unification [35, 37, 38]. We discuss these effects in detail in Section 2.4.
We may also have terms of the form 
These operators can split the masses of the color and SU(2) adjoint components in , and by . This mass difference induces threshold corrections to gauge coupling constants of . This effect is negligible for but could be significant for very small . However, in order to simplify our analysis, we neglect the effects of these operators in this paper.
2.3 Soft supersymmetry-breaking mass parameters
The soft supersymmetry-breaking terms in the minimal supersymmetric SU(5) GUT are
where and are the scalar components of and , respectively, the are the SU(5) gauginos, and we use the same symbols for the scalar components of the Higgs fields as for the corresponding superfields.
In the super-GUT CMSSM model, we impose the following universality conditions for the soft-mass parameters at a soft supersymmetry-breaking mass input scale :
The bilinear soft SUSY-breaking therms and are determined from the other parameters, as we shall see in the following. Note that, if we set , the above conditions are equivalent to those in the CMSSM.
These parameters are evolved down to using the renormalization-group equations (RGEs) of the minimal supersymmetric SU(5) GUT, which can be found in [39, 40, 25], with appropriate changes of notation. During the evolution, the GUT parameters in Eq. (1) affect the running of the soft supersymmetry-breaking parameters, which results in non-universality in the soft parameters at . In particular, the coupling enters into the RGEs for the soft masses of the and Higgs fields, and can have significant effects on their evolution. These effects become particularly important in the vicinity of the focus-point region at large , since it is very close to the boundary of consistent electroweak symmetry breaking (EWSB). In addition, contributes to the running of the Yukawa couplings and the corresponding -terms. On the other hand, affects directly only the running of , , and (besides and ), and thus can affect the MSSM soft mass parameters only at higher-loop level. Both of and contribute to the RGEs of the soft masses of matter multiplets only at higher-loop level, and thus their effects on these parameters are rather small. Thus, the low-energy phenomenology is rather insensitive to the value of . The parameters and , as well as the corresponding bilinear parameters and , do not enter into RGEs of the rest of the parameters, and thus their values give no effects on the running of the parameters in Eq. (7). We note in passing that, if we set , we obtain the CMSSM and there is no effect from the running above the GUT scale on the low-energy spectrum 333However, we find that the GUT-scale matching condition on the parameter gives a constraint on the model parameter space even though , as we see below. .
2.4 GUT-scale matching conditions
At the unification scale , the SU(5) GUT parameters are matched onto the MSSM parameters. In this Section, we summarize these matching conditions and discuss the constraints on the parameters from the low-energy observables.
The matching conditions for the Standard Model gauge couplings at one-loop level in the scheme are given by
where , , and are the U(1), SU(2), and SU(3) gauge couplings, respectively, and is a renormalization scale taken in our analysis to be the unification scale: . The last terms in these equations represent the contribution of the dimension-five operator (4). Since , these terms can be comparable to the one-loop threshold corrections, and thus should be taken into account when discussing gauge-coupling unification . From these equations, we have
We note that there is no contribution to (12) from the dimension-five operator 444This feature can be understood as follows. The contributions of the color-triplet Higgs multiplets to the gauge coupling beta functions are given by . In this notation, the matching conditions may be rewritten as (14) Since and , neither nor appears in (12).. By running the gauge couplings up from their low-energy values, we can determine the combination via (12) [41, 42, 43]. Notice that without the dimension-five operator (), is also determined from the values of the gauge couplings at the GUT scale via Eq. (11). The contribution of this operator relaxes this constraint, and allows us to regard as a free parameter. The last matching condition, Eq. (13), will be used to determine and as will be discussed below.
For the Yukawa couplings, we use the tree-level matching conditions. However, we note here that there is an ambiguity in the determination of the GUT Yukawa couplings. As we mentioned in Section 2.2, Yukawa unification in the MSSM is imperfect in most of the parameter space. Although this is cured by the higher-dimensional operators in (3), they introduce additional contributions to the matching conditions for the Yukawa couplings. With this in mind, in this paper, we use
for the third-generation Yukawa couplings, where , , , , and are eigenvalues of , , the MSSM up-type Yukawa couplings, the MSSM down-type Yukawa couplings, and the MSSM lepton Yukawa couplings, respectively. This condition is the same as that used in Ref. . For the first- and second-generation Yukawa couplings, on the other hand, we use
We chose the down-type Yukawa couplings for the matching condition, rather than the lepton Yukawa couplings, since it results in longer proton decay lifetimes and thus gives a conservative bounds on the model parameter space [18, 44].
Next we obtain the matching conditions for the soft supersymmetry-breaking terms. To this end, we first note that in the presence of soft supersymmetry-breaking terms the VEV of deviates from by , where denotes the supersymmetry-breaking scale . In addition, develops a non-vanishing -term. We find that
We again find that the contribution of the dimension-five operator can be comparable to that of the one-loop threshold corrections.
The soft masses of the MSSM matter fields, as well as the -terms of the third-generation sfermions, are given by
Finally, for the and terms we have 
These equations display the amount of fine-tuning required to obtain values of and that are . Equation (23) shows that we need to tune to be . On the other hand, Eq. (24) indicates that should be , which requires . Therefore, we can neglect in the following calculations. Notice that the condition is stable against radiative corrections as shown in Ref. .
The and parameters are determined by using the electroweak vacuum conditions:
where and denote loop corrections .
We can determine the parameters in minimal SU(5) by solving the conditions (24) and 555We need to determine the parameters in order to obtain the MSSM gaugino masses via Eqs. (19–21).. However, we find that there is an additional condition that must be satisfied in order for these equations to be solvable. When eliminating from Eq. (24) using , we obtain an equation that is quadratic in . This equation has a real solution only if
This condition gives a non-trivial constraint on the input parameters, especially on the trilinear coupling . In particular, for , this constraint leads to .
From these equations, we obtain
We can then determine using Eq. (12). Eq. (13) can be reduced to an equation with undetermined parameters and using Eq. (29) and (31). Then once and are chosen, this equation plus Eq. (32) can be used to determine and . However, since is only logarithmical dependent on , it will remain fairly constant for a broad range of . As mentioned above, if we do not include the contribution of the dimension-five operator, Eq. (11) fixes . In this case, and are restricted via Eq. (32), and thus we cannot regard both of them as free parameters. The last term in Eq. (11) can relax this restriction, and enables us to take and as input parameters. In this case, is given by Eq. (32), and Eq. (11) determines the parameter . In the following analysis, we check that the coefficient has reasonable values, i.e., .
Using the above results, we see how the super-GUT CMSSM model is specified by the following set of input parameters:
where the trilinear superpotential Higgs couplings, , are specified at .
3 Proton Decay and GUT-Scale Phases
As is well known, in the minimal supersymmetric SU(5) GUT with weak-scale supersymmetry breaking, the dominant decay channel of proton is the mode , which is induced by the exchange of the color-triplet Higgs multiplets, and the model is severely restricted by the proton decay bound [29, 30]. The exchange of the GUT-scale gauge bosons can also induce proton decay, but this contribution is usually subdominant because of the large GUT scale in supersymmetric GUTs. The strong constraint from the decay may, however, be evaded if the masses of supersymmetric particles are well above the electroweak scale [51, 52, 53, 44, 18]. In addition, it turns out that the decay mode depends sensitively on the extra phases in the GUT Yukawa couplings , which can suppress the proton decay rate, as we discuss in this Section. For more details of the proton decay calculation, see Refs. [51, 53, 44, 18] and the Appendix.
In supersymmetric models, the largest contribution to the decay rate of the proton is determined by the dimension-five effective operators generated by integrating out the colored Higgs multiplets ,
with and defined by
where are generation indices, are SU(3) color indices, and is the totally antisymmetric three-index tensor. The Wilson coefficients are given by
where are the familiar CKM matrix elements, and the () are the new CP- violating phases in the GUT Yukawa couplings. These are subject to the constraint , so there are two independent degrees of freedom for these new CP-violating phases  666The number of extra degrees of freedom in the GUT Yukawa couplings can be counted as follows. Since is a symmetric complex matrix, it has 12 real degrees of freedom, while has 18. Field redefinitions of and span the transformation group, and thus 18 parameters are unphysical. Hence, we have 12 physical parameters. Among them, 6 are specified by quark masses, while 4 are for the CKM matrix elements. The remaining 2 are the extra CP phases, which we take to be and .. We take and as free input parameters in the following discussion. The coefficients in Eq. (36) are then run to the SUSY scale using the RGEs. At the SUSY scale, the sfermions associated with these Wilson coefficients are integrated out through a loop containing either a wino mass insertion or a Higgsino mass insertion, which are proportional to and , respectively. The wino contribution to the decay amplitude for the mode is given by the sum of the Wilson coefficients and multiplied by the corresponding matrix elements (see Eq. (A.12)). These coefficients are approximated by
where , , , and are the masses of the charm quark, top quark, boson, and down-type quarks, respectively, and . Since the ratio of Yukawa couplings and CKM matrix elements in the parenthesis in Eq. (37) is , this Wilson coefficient may be suppressed for certain ranges of the phases. On the other hand, the Higgsino exchange process contributes only to the mode, and gives no contribution to the modes. The relevant Wilson coefficients for the mode are and in Eq. (37), as well as and , which are approximately given by
where , , and are the masses of down quark, strange quark, and tau lepton, respectively. Contrary to the coefficients in Eq. (37), the absolute values of these coefficients do not change when the phases vary.
Equations (37) and (38) show that the proton decay rate receives a enhancement as well as a suppression by the sfermion mass scale . To evade the proton decay bound, therefore, a small and a high supersymmetry-breaking scale are favored as shown in the subsequent section. In addition, we note that the proton decay rate decreases as is taken to be large. From Eq. (32), we find , and thus the proton lifetime is proportional to . This indicates that larger values and smaller values help avoid the proton decay bound.
To show the phase dependence of these contributions more clearly, we show in Fig. 1(a) each contribution to the decay amplitude of the channel as a function of with fixed to be . The red dashed, green dash-dotted and black solid lines represent the absolute values of the wino, Higgsino, and total contributions, respectively. We take the parameter point indicated by the star in Fig. 4 below. This figure shows that the wino contribution can vary by almost an order of magnitude, while the size of the Higgsino contribution remains constant. These contributions are comparable, and thus a significant cancellation can occur. As a result, the total amplitude varies by more than an order of magnitude. The wino contribution is minimized at , while the total amplitude is minimized at . This mismatch is due to the Higgsino contribution.
In Fig. 1(b) we show the phase dependence of the lifetime of each decay mode with the same parameter set. The green dash-dotted, blue dotted and red dashed lines represent the first-, second-, and third-generation neutrino decay modes, respectively, while the black solid line shows the total lifetime. We see that the lifetimes of the and modes, which are induced by wino exchange only, are maximized at , which deviates from the point where is maximized. Due to this deviation, the phase dependence of the total lifetime is much smaller than that of each partial lifetime, but still it can change by an factor.
In Fig. 2(a), we show a contour plot for the proton decay lifetime in units of years in the – plane, using the same parameter set as in Fig. 1. We find that the proton lifetime exceeds the current experimental bound, yrs [27, 55], in a significant area of the phase space shown by the contour labeled 0.066. The peak lifetime is marked in the upper part of the figure by a spade.
Although the modes may be suppressed for certain values of the phases, other decay modes that depend on the same phases are not suppressed in the same way. The other decay modes that could restrict the parameter space are and . The Wilson coefficients for these proton decay modes are quite similar to those that generate , and depend on exactly the same combination of SUSY parameters. The differences in the calculations of their lifetimes come from their different dependences on CKM matrix elements. The and modes are suppressed relative to the modes by off-diagonal components of the CKM matrix. Moreover, the experimental constraints on these modes are weaker: yrs and yrs [27, 56], so these decay modes are less restrictive on the parameter space. To ensure that these modes are not problematic, in Fig. 2(b), we show the lifetimes of these decay modes as functions of for the same parameter set as in Fig. 1. We find that, although the mode can be dominant, it is still above the present experimental limit. The is always sub-dominant, and again exceeds the current bound. We also note that the and modes exhibit the same phase dependence, since they are related to each other through isospin symmetry.
In the following analysis, we choose the CP-violating phases so as to maximize the lifetime, thereby obtaining a conservative constraint on the super-GUT model parameter space. Although not shown in the figures below, we have verified that each allowed point also meet the experimental constraint coming from and .
To appreciate the effect of choosing , we begin by reviewing briefly some results for the CMSSM with . We note that we use here the FeynHiggs 2.11.3 code  to compute the Higgs mass. Previously we used FeynHiggs 2.10.0, and we note that due to a bug fix, the new version yields a significant change in at large positive 777Note that our sign convention for is opposite that found in many public codes such as SoftSusy .. A large value of is necessary to obtain the correct relic density along the stop-coannihilation strip [59, 14], where the lighter stop and neutralino LSP are nearly degenerate in mass. For , we find that FeynHiggs 2.11.3 results in a GeV drop in the value of relative to the previous result, necessitating a lower value of . However, for , the stop strip is no longer present. On the other hand, the effect of updating FeynHiggs on at large negative is less pronounced. We further note that our calculation of the proton lifetime here is also updated with bug-fixes.
4.1 CMSSM update
In view of the proton lifetime constraint, which favours larger sparticle masses, we consider here the possibilities that the correct relic density of neutralino dark matter is obtained either in the focus-point strip [60, 13] or the stop-coannihilation strip , updating the results found in . We use SSARD  to compute the particle mass spectrum, the dark matter relic density, and proton lifetimes. The discussion of the proton lifetime in Section 3 motivates us to focus on relatively small values of . For larger values of , the proton lifetime becomes smaller than the current experimental bound, and minimal supersymmetric SU(5) is not viable. For the CMSSM cases with , we have set and taken from Eq. (11).
In Fig. 3, we show four CMSSM planes displaying the focus-point (left) and stop-coannihilation (right) relic density strips for the two choices of the sign of . Higgs mass contours are shown as red dot-dashed curves labeled by in GeV in 1 GeV intervals starting at 122 GeV. In the left panels, we choose 888As we discussed in Section 2.4, if we assume the minimal SU(5) GUT with the universality condition (7), then the -term matching condition restricts via Eq. (28). This constraint can, however, be evaded if we relax the universality condition (7) (for in particular) or consider non-minimal Higgs content. With these possibilities in mind, we do not take the condition (28) into account in Section 4.1, which allows the choice . with (top) and (bottom). For this choice of , there is a relatively minor effect on due to the updated version of FeynHiggs. The light mauve shaded region in the parts of the left panels with large are excluded because there are no solutions to the EWSB conditions: along this boundary . Just below the regions where EWSB fails, there are narrow dark blue strips where the relic density falls within the range determined by CMB and other experiments  999Since the relic density of dark matter is now determined quite accurately (), for the purpose of visibility we display expanded strips for which the relic density lies in the range .. These strips are in the focus-point region [13, 60]. We note also that the brown shaded regions in the portions of the panels with low are excluded because there the LSP is the lighter charged stau lepton. The planes also feature stau-coannihilation strips  close to the boundaries of these brown shaded regions. They extend to TeV, but are very difficult to see on the scale of this plot, even with our enhancement of the relic density range. There are also ‘thunderbolt’-shaped brown shaded bands at intermediate where the chargino is the LSP. There are no accompanying chargino-coannihilation strips, as at these multi-TeV mass scales any such strip would lie within the shaded region and is therefore excluded.
Contours of the proton lifetime calculated using down-type Yukawa couplings (see the discussion in Section 2.4) are shown as solid black curves that are labeled in units of yrs. The current limit yrs [27, 55] would exclude the entire area below the curve labeled 0.066. For the nominal value of GeV, neglecting the theoretical uncertainties in the calculation of , we see that in the upper left plane of Fig. 3 the Higgs contour intersects the focus-point region where yrs, very close to the experimental limit. Much of the focus-point strip in this figure may be probed by future proton decay experiments. Changing the sign of has almost no effect on the proton lifetime, as seen in the lower left panel of Fig. 3, but the calculated Higgs mass is smaller by GeV, which is less than the uncertainty in the FeynHiggs calculation of .
In the right panels of Fig. 3, we have chosen large negative and , which allows a sufficiently heavy Higgs and a viable stop strip. There are now brown shaded regions in the upper left wedges of the planes where the stop is the LSP (or tachyonic). Though it is barely visible, there is a stop strip that tracks that boundary 101010In this case, and in the super-GUT cases to follow, we have further extended the range on to [0.01,2.0]. Otherwise the thickness of the strips which are typically 10–50 GeV would be pixel thin for the range of masses shown.. Since we have taken an enhanced range for the relic density the blue strip continues to the edge of the plot. In reality, however, the stop strip ends  at the position marked by the X in the figure. We see that, for , the stop strip ends when GeV, whereas for the strip ends when GeV, both of which are acceptable given the uncertainty in the calculation of . At the endpoint, which occurs at TeV, the proton lifetime is approximately yrs. Had we chosen a smaller value of , the stop strip would have extended to higher . For example, for , the stop strip extends to 125 GeV for