Yukawa-unified natural supersymmetry
Dept. of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA
[2mm] Laboratoire de Physique Subatomique et de Cosmologie, UJF Grenoble 1, CNRS/IN2P3, INPG, 53 Avenue des Martyrs, F-38026 Grenoble, France
Previous work on Yukawa-unified supersymmetry, as expected from SUSY GUT theories based on the gauge group , tended to have exceedingly large electroweak fine-tuning (EWFT). Here, we examine supersymmetric models where we simultaneously require low EWFT (“natural SUSY”) and a high degree of Yukawa coupling unification, along with a light Higgs scalar with GeV. As Yukawa unification requires large , while EWFT requires rather light third generation squarks and low GeV, -physics constraints from and can be severe. We are able to find models with EWFT (better than 1–2% EWFT) and with Yukawa unification as low as (30% unification) if -physics constraints are imposed. This may be improved to if additional small flavor violating terms conspire to improve accord with -constraints. We present several Yukawa-unified natural SUSY (YUNS) benchmark points. LHC searches will be able to access gluinos in the lower TeV portion of their predicted mass range although much of YUNS parameter space may lie beyond LHC14 reach. If heavy Higgs bosons can be accessed at a high rate, then the rare decay might allow a determination of as predicted by YUNS models. Finally, the predicted light higgsinos should be accessible to a linear collider with TeV.
Keywords: Supersymmetry Phenomenology, Supersymmetric Standard Model, Large Hadron Collider
A striking feature in nature is that all the fermions of each generation fill out a complete 16-dimensional spinor multiplet of the gauge group . While ordinary grand unified theories (GUTs) suffer from the notorious gauge hierarchy problem, supersymmetric (SUSY) GUTs not only tame this hierarchy problem , but they also receive support from the well-known unification of gauge couplings . In the simplest SUSY GUT theories, where both MSSM Higgs doublets and occupy the same 10-dimensional representation, one also expects unification of third generation Yukawa couplings , and at GeV [4, 5]. The Yukawa coupling unification is highly sensitive to both 2-loop renormalization group running (RGEs) and to threshold corrections when transitioning between MSSM and SM effective theories at the SUSY particle mass scale. Thus, the entire SUSY mass spectrum enters into a precise computation of Yukawa coupling unification.
Many groups have explored Yukawa unification (YU) in SUSY theories [6, 7, 8, 9, 10, 11, 12, 13]. It has been found that, for , YU can occur at the few percent level in either the Higgs splitting (HS) model or the DR3 model (D-term splitting, right-hand neutrino effects and third generation splitting), provided that the GUT scale soft SUSY breaking (SSB) terms are related as
with . Moreover, the GUT scale Higgs splitting is needed to allow for an appropriate
radiative breakdown of electroweak symmetry. In these models, first generation squarks and sleptons
are required to be in the multi-TeV range [8, 10] while
third generation sfermions are driven to much lighter TeV-scale masses.
A benefit of these models is that the light SUSY Higgs boson mass tends naturally to be in the
125 GeV range,
In this work, we examine to what extent it is possible to reconcile YU with low EWFT. Minimization of the SUSY scalar potential allows one to relate the mass scale to the superpartner mass scale via the well-known relation
The radiative corrections and are given in the 1-loop approximation of the Higgs effective potential by:
where is the one-loop correction to the tree-level potential, and the derivative is evaluated in the physical vacuum: i.e. the fields are set to their vacuum expectation values after evaluating the derivative. At the one-loop level and in the limit of setting first/second generation Yukawa couplings to zero, contains 18 and contains 19 separate contributions from various particles/sparticles . We include contributions from , , , , , , , , and , , and . We adopt a scale choice to minimize the largest of the logarithms. The dominant contribution to the terms arise from superpotential Yukawa interactions of third generation squarks involving the top quark Yukawa coupling. For instance, the dominant contribution to is given by
where , and and . This expression thus grows quadratically with the stop mass.
We adopt the fine-tuning measure from , which requires that each of the 40 terms on the right-hand-side (RHS) of Eq. (2) should be of order . Labeling each term as (with ), we may require , where GeV, depending on how much EWFT one is willing to tolerate. This measure of fine-tuning is similar to (but not exactly the same as) Kitano–Nomura  but different from Barbieri–Giudice  beyond the tree-level. In the following, we will use the fine-tuning parameter
where lower values of correspond to less fine-tuning, and e.g. would correspond to fine-tuning.
Our goal in this paper is to search for parameter choices which
Maximize the degree of Yukawa coupling unification, i.e. minimize
with each Yukawa coupling evaluated at the GUT scale. Thus, a value of would give perfect Yukawa coupling unification.
Have as low EWFT as possible (in practice, we will require , or better than 1% EWFT).
Have GeV in accord with the recent LHC discovery of a Higgs-like resonance. In practice, we will require GeV to allow for a roughly 2–3 GeV error in the RG-improved one-loop effective potential calculation of the Higgs mass .
We recognize that in addition to EWFT, there also exists a fine-tuning associated with
generating particular weak scale SUSY spectra from distinct GUT scale parameters ,
(see also [23, 24] for related discussions).
Here we adopt the less restrictive weak scale fine-tuning condition, which nonetheless turns out to be
indeed very restrictive. In this vein, we regard particular GUT scale parameters as merely a parametrization
of our ignorance of the mechanism of SUSY breaking and soft term generation.
For the remainder of this paper, in Section 2 we present details of our scan over SUSY parameter space, and which constraints are invoked in our analysis. In Section 3, we present the results of our parameter space scans. We will find that requiring and GeV only allows for as low as . While this degree of Yukawa unification is not optimal, we feel it is still useful in that it might guide model builders towards models including additional GUT scale threshold corrections or extra matter or above-GUT-scale running which may ameliorate the situation. In Section 5, we discuss observable consequences of YUNS for LHC, ILC and dark matter searches. We pay some attention to methods which might allow one to distinguish YUNS from generic NS models at lower values. In Section 6 we present a summary and conclusions.
2 Parameter space and Yukawa unification
For our calculations, we adopt the ISAJET 7.83  SUSY spectrum generator ISASUGRA . ISASUGRA begins the calculation of the sparticle mass spectrum with input gauge couplings and , Yukawa couplings at the scale ( running begins at ) and evolves the 6 couplings up in energy to scale (defined as the value where ) using two-loop RGEs. At , we input the soft SUSY breaking parameters as boundary conditions, and evolve the set of 26 coupled MSSM RGEs  back down in scale to . Full two-loop MSSM RGEs are used for soft term evolution, while the gauge and Yukawa coupling evolution includes threshold effects in the one-loop beta-functions, so the gauge and Yukawa couplings transition smoothly from the MSSM to SM effective theories as different mass thresholds are passed. In ISASUGRA, the values of SSB terms of sparticles which mix are frozen out at the scale , while non-mixing SSB terms are frozen out at their own mass scale . The scalar potential is minimized using the RG-improved one-loop MSSM effective potential evaluated at an optimized scale which accounts for leading two-loop effects . Once the tree-level sparticle mass spectrum is computed, full one-loop radiative corrections are calculated for all sparticle and Higgs boson masses, including complete one-loop weak scale threshold corrections for the top, bottom and tau masses at scale . Since the GUT scale Yukawa couplings are modified by the threshold corrections, the ISAJET RGE solution must be imposed iteratively with successive up-down running until a convergent sparticle mass solution is found. Since ISASUGRA uses a “tower of effective theories” approach to RG evolution, we expect a more accurate evaluation of the sparticle mass spectrum for models with split spectra (this procedure sums the logarithms of potentially large ratios of sparticle masses) than with programs which make an all-at-once transition from the MSSM to SM effective theories. The fine-tuning measure described in Sec. 1 has been implemented in ISAJET 7.83 .
In models of “natural SUSY” (NS) [32, 21, 33, 34, 35, 36, 37, 20], the first requirement to gain a low EWFT is that the parameter be of the order of , while in Yukawa-unified SUSY, it is necessary to invoke some manner of Higgs soft term splitting at the GUT scale in order to obtain raditive EWSB. For these reasons, the model parameter space chosen in this study is the two-parameter non-universal Higgs model (NUHM2) where the weak scale values of and are input in lieu of the GUT scale values of and . In addition, to allow for (sub)TeV-scale third generation masses (as required by EWFT from Eq. (2) along with at least a partial decoupling solution to the SUSY flavor and CP problems, we allow for split first/second and third generations at the GUT scale. Thus, the parameter space we choose is given by
Here, and are the first/second and third generation sfermion soft masses, respectively; is the universal gaugino mass parameter; and is the universal trilinear coupling. These parameters are defined at , while , , and are defined at the weak scale. The top quark mass is set to GeV.
We search for mass spectra with low EWFT and low by performing a vast random scan over the following parameter ranges (masses in GeV units):
electroweak symmetry be radiatively broken (REWSB),
the neutralino is the lightest MSSM particle,
the light chargino mass obeys the rather model independent LEP2 limit that GeV , and
the light Higgs mass falls within the window GeV, where we adopt GeV as theoretical error on the Higgs mass calculation.
Regarding -physics constraints, we consider , where experimental and theoretical uncertainties have been added in squares, and at 95% CL. In the following we will impose the constraint only at the level (the reason for this is that calculating with both IsaTools and SuperISO, we observe deviations of the order of 5% in the relevant region of parameter space). For the constraint, we assume a theoretical uncertainty of 20%. This leads to the the following limits
which we will use throughout the numerical analysis.
Regarding neutralino relic density, we remark here that models of natural SUSY contain a higgsino-like lightest neutralino with thermal abundance of typically . This thermal under-abundance can be regarded as a positive feature of NS models in the sense that if one invokes the axion solution to the strong CP problem, then one expects mixed axion-higgsino dark matter, where the higgsino portion is typically enhanced by thermal axino production and decay to higgsinos in the early universe. Thus, a thermal under-abundance leaves room for additional non-thermal higgsino production plus an axion component to the dark matter .
3 Scan results
As our first result, we show in Fig. 1 points from our parameter space scan in the plane. All points have GeV and obey the current LEP and LHC SUSY mass limits. The dark blue triangles however violate the -physics constraints of Eq. (9). These points may still be valid if additional small flavor violating terms are allowed in the theory. The light blue triangles obey the constraint, but deviate from the measured by more than . Finally, the pink squares satisfy both the and constraints. This color scheme is used throughout the remainder of the paper. We see already from this plot that flavor physics constraints significantly affect the parameter space of Yukawa-unified natural SUSY. This is to be expected, since naturalness requires lighter third generation squarks while Yukawa unification requires : both these effects bolster SUSY contributions to -physics observables. (Analogous observations were made in [10, 42] in the context of generic YU models.) Without flavor constrains, we can obtain as low as . The constraint pushes this up to , and the constraint to . Aside from the -physics constraints, it is intriguing that points with lowest also have lowest values of . This is because low requires light third generation squark masses, while at the same time Yukawa unification requires large SUSY threshold corrections which also require lighter third generation squarks.
In Fig. 2, we show the value of versus various input parameters for (less than 1% fine-tuning). Here, the behavior deviates considerably from unified models with large and a spectrum derived from the radiatively driven inverted scalar mass hierarchy [7, 8, 10]. From frame a), we see that YUNS models actually prefer large whereas generic Yukawa unified SUSY models (with but arbitrary EWFT), YUS for short, prefer low . The preference for large helps to avoid LHC constraints on the gluino mass. If we impose the -physics constraints, then the distribution flattens out with some preference for lower values. In frame b), where is plotted vs. , we see that lowest values prefer TeV, which leads to rather light third generation squarks and typically violation of -physics constraints. If we respect -constraints, then larger values of TeV are preferred, at the cost of larger values of . In frame c), we plot versus . For YUS models, there is a strong preference for , while for YUNS, the lowest values are obtained for smaller . Imposing -constraints, the preference moves to . Frame d) shows . As expected, is preferred.
The dependence of on the input parameters is shown in Fig. 3 for points with . In frame a), we see a mild preference by for low , i.e. the gluino mass can’t be too heavy, lest it pushes the stop masses too high, leading to large . From frame b), we see low prefers TeV. If is much higher, then third generation squarks are too heavy to give low EWFT, while if is too ight, we generate tachyonic spectra: the optimal corresponds to TeV. For TeV, then sub-TeV top squarks are generated leading to violation of -constraints. In frame c), we see that low allows a wide range of unless -constraints are imposed, in which case is again preferred. The large negative values lead to larger values and also can lower the EWFT . In frame d), we see that there is only mild preference for values unless -constraints are respected, in which case is preferred.
In Fig. 4 we show versus various sparticle and Higgs masses. In frame a), the distribution versus is shown, and we see that low prefers the lower range of . This is understandable since low EWFT prefers lower top and bottom squark masses, which may not feed a sufficient radiative correction into the computation. These low third generation squark mass solutions also tend to give large contributions to -constraint. If we impose -constraints, then can live in the GeV range, at the cost of larger . In frame b), we show the distribution versus . While all solutions—especially low ones—favor the heavier range of ( TeV, likely beyond LHC reach), the solutions obeying -constraints tend to slightly favor lower , possibly within range of LHC14 searches. In frame c), the distribution versus is shown. Here, we see a clear demarcation: -constraints favor TeV to suppress SUSY loop contributions to . This constraint forces the minimum to move form to about . Additional small flavor-violating contributions to the MSSM Lagrangian could alter the predicted and/or rates and thus allow the lower solutions. In frame d), we show the distribution versus pseudoscalar. Higgs mass . We see low favors the lower range of , although values up to and beyond 5 TeV are also possible (at the cost of , however). Since , LHC searches for will access a significant range of in this case. It is also possible for some range of TeV for LHC to access production .
In Fig. 5, we show scatter plots of YUNS points with and in a) and b) space. From frame a), we see as expected that and are correlated, with some solutions reaching well below GeV, which should be accessible to LHC searches. However, in this case these points all violate -constraints, so that requiring -constraints within measured range requires instead TeV, likely beyond the 14 TeV LHC reach. In frame b), we see also that and are correlated. This is different from usual NS, where can be far above and . The reason here is that is large and so there is also a non-negligible contribution to from . Imposing -constraints, we find and both TeV (the former aids in lifting into its measured range).
4 Benchmark points
For HSb, , which is nearly perfect Yukawa coupling unification. The Higgs mass GeV is also sufficiently heavy. Unfortunately, the point is now excluded by LHC7 searches for multi-jet plus one -tag searches, since is only 351 GeV. We also list in Table 1 the EWFT measure; for HSb we have , indicating exceptionally high level of fine-tuning. Much of this comes from the parameter which turns out to be of order 3 TeV, and thus requires a large value of at the weak scale to cancel against.
In contrast, point YUNS1 in column 3 has low fine-tuning of , at the cost of relaxing to . For YUNS1, TeV, with first/secnd generation squarks at TeV. The point is likely beyond LHC8 reach, but should be accessible to LHC14 with 10–100 fb. In column 4, we list YUNS2 with as low as allowed by -constraints, but with . This point has TeV and TeV, so it is likely beyond LHC reach, including a high-luminosity upgrade. While the higgsino-like chargino is only GeV, it decays via 3-body mode into a higgsino-like with GeV, so that visible decay products are very soft, and likely impossible to observe above SM backgrounds at the LHC. Point YUNS3, listed in column 5, features as low as 1.19, with . This point has , and , so it falls out of the -physics allowed range. While gluinos and first/second generation squarks are beyond LHC reach, the rather light top and bottom squarks may be accessible to LHC searches. All these points have , leaving room for non-thermal higgsino production and axions. This contrasts point HSb, which has a much too thermal abundance and so would need an extremely light axino or huge late-time entropy production to tame this over-abundance .
In Fig. 6, we show the Yukawa coupling evolution of , and versus renormalization group scale , from to for benchmark points YUNS1 and YUNS3. These can be compared to similar plots for YUS, as in e.g. Fig. 6 of Ref. . The SUSY threshold corrections implemented at the scale show up as jumps in the curves. In the case of YUS models, the threshold correction is positive due to a large loop contribution which goes like , where both and are extremely large. For YUNS models, with rather low , these loops are suppressed and in the case of YUNS1, the loops actually dominate, and are of opposite sign to the loops, leading to the slight downward jump of and thus bad Yukawa coupling unification. For YUNS3, the loops are larger, and the jump goes upwards, thus providing better Yukawa coupling unification.
5 Yukawa-unified natural SUSY: LHC, ILC and DM searches
In this Section, we discuss the observable consequences of YUNS for LHC, ILC and direct and indirect WIMP and also axion searches.
5.1 YUNS at LHC
In natural SUSY models, it is favorable to have multi-TeV first/second generation squarks and sleptons because 1. they are safely beyond current LHC searches, 2. they provide at least a partial solution to the SUSY flavor and CP problems and 3. they provide additional suppression of third generation scalar masses via large 2-loop RGE effects. However, this means they are likely beyond any conceivable LHC reach. Third generation squarks may be much lighter, and in generic NS models are naively expected to be below the TeV scale (but see Ref.  where 1–4 TeV third generation squarks work just fine, and lift the value of into its measured range). In the case of YUNS, with , the combined light squarks and large usually imply violation of -constraints, and if these are imposed, then top and bottom squarks are beyond 1.5 TeV, and likely inaccessible to LHC searches. If additional sources of flavor violation are invoked, then the -constraints may be invalid, and then the solutions with much lower are accessible, along with much lighter top and bottom squarks, potentially accessible to LHC searches.
For YUNS, the gluino mass may lie anywhere in the 1–5 TeV range. It has been estimated in Ref.  that LHC14 with 100 fb should be able to access gluino pair production in the case of heavy squarks for up to 1.8 TeV. In the case of YUNS with heavier top and bottom squarks, the is expected to dominantly decay via 3-body modes into and final states. The gluino pair events will thus contain multi-jets plus missing energy plus isolated leptons plus several identifiable -jets . While the higgsino-like chargino and neutralino production cross sections can be large, their decays to soft visible particles, arising from the small energy release in their 3-body decays, will be difficult to detect at LHC above SM backgrounds .
In the case where TeV, then it may be possible to detect , especially if these are produced in association with -jets, e.g. production. It may also be possible to detect production with , since the production and decay are enhanced at large . In this case, the mass and width may be determined by reconstructing . At large , this width is typically in the tends of GeV range and is very sensitive to . This reaction offers a method to distinguish YUNS from NS, in that the former is expected to occur at .
5.2 YUNS at ILC
A linear collider operating at GeV would in many ways be an optimal discovery machine for YUNS. The reason is that by construction GeV, so chargino and neutralino pair production should always be available. While the small energy release in and decay is problematic at LHC, it should be much more easily observable in the clean environment of an collider. In this sense, an collider operating at GeV would be a higgsino in addition to a Higgs factory. It is also possible that some lighter third generation squarks are accessible to ILC with TeV or CLIC with TeV, depending if one avoids -constraints and accepts the low mass, low solutions.
5.3 Higgsino-like WIMPs
A generic prediction of both NS and YUNS models is that the LSP is a higgsino-like WIMP with a typical under-abundance of thermally produced (TP) neutralinos . However, in cases where is as low as GeV and is as large as GeV, then there can be substantial bino–higgsino mixing, boosting up to or even beyond. The situation is illustrated in Fig. 7, where we plot versus and versus the higgsino fraction from YUNS models satisfying and .
The typical under-abundance is an appealing feature if one invokes the Peccei-Quinn solution to the strong CP problem, in which case one must introduce an axion superfield which contains a pseudoscalar axion as well as a spin-1/2 axino and a spin-0 saxion . In such models, one expects both saxion and axino masses at or around the SUSY breaking scale, so that dark matter is comprised of an axion-WIMP admixture. In this case, thermal production of axinos and subsequent decay to states such a bolster the WIMP abundance beyond its TP-value. In addition, axions are produced as usual via coherent oscillations. It is also possible to suppress the WIMP abundance in the mixed cosmology via late-time entropy production from saxion production and decay, although this case tends to be highly constrained by maintaining successful Big Bang nucleosynthesis (BBN). The upshot is that in the mixed dark matter scenario, it may be possible to detect both a WIMP and an axion.
In the case of the YUNS model, the higgsino-like neutralinos have a substantial spin-independent (SI) direct detection cross section, as illustrated in Table 1, where pb. While this level of direct detection cross section is now highly constrained by recent XENON100  results, one must bear in mind that the higgsino-like WIMPs would constitute only a portion of the dark matter, so their local abundance might be up to a factor of 30 lower than is commonly assumed. In Table 1, we also list (relevant for WIMP detection at IceCube) and , relevant for detection of dark matter annihilation into gamma rays or anti-matter throughout the cosmos. While these cross sections are also at potentially observable levels, again one must take into account that the overall WIMP abundance may be up to a factor of about 30 below what is commonly assumed. Plots of , and have been presented in the case of higgsino-like WIMPs in Ref’s  and  and so similar plots will not be reproduced here.
6 Summary and conclusions
Previous analyses of Yukawa-unified models suffer from two problems: 1. they tend to predict a light gluino GeV (alhough solutions are possible for much heavier gluinos) which is now excluded by LHC searches, and 2. they suffer from extreme fine-tuning in the electroweak sector. In this paper, we examined how well the Yukawa couplings could unify in the natural SUSY context, where GeV, while at the same time requiring the light Higgs mass GeV. The small value of suppresses the large loop contributions to the -quark Yukawa coupling which seem to be needed for precision Yukawa coupling unification. Nonetheless, by scanning over NUHM2 parameters with split third generation, we are able to find solutions with as low as 1.18. These solutions, with very light third generations squarks and tend to violate -physics constraints. If -physics constraints are imposed, then only can be achieved. However, the -physics calculations can be modified if additional small flavor-violating terms are allowed in the MSSM Lagrangian, so it is not clear how seriously should be taken.
The Yukawa-unified natural SUSY spectra have important differences from previous YU spectra. The gluino mass can easily be in the 1–4 TeV range, thus avoiding LHC constraints from SUSY searches. The light higgsino-like charginos and neutralinos decay to soft particles, also avoiding LHC searches. However, light higgsinos should be easily accessible to an ILC with TeV, as is typical of all NS models. In addition, as in all NS models, the lightest neutralino is higgsino-like with a typical thermal underabundance of WIMP dark matter. We regard this as a positive feature in that the WIMP abundance is typically increased in non-standard (but more attractive) cosmologies such as those conatining mixed axion-neutralino cold dark matter.
The question arises as to how to distinguish YUNS from ordinary NS. The YUNS model requires , which leads to large production cross sections for heavy Higgs bosons and at LHC, and large widths for these particles. If the rare decays can be identified with suficiently high statistics (perhaps at a luminosity upgraded LHC), then the widths may be measured with precision, allowing one to highly constrain , and perhaps verify that it is consistent with YUNS models.
This work has been supported in part by the Office of Science, US Department of Energy and by IN2P3 under contract PICS FR–USA No. 5872. HB and SK acknowledge the hospitality of the Aspen Center for Physics which is supported by the National Science Foundation Grant No. PHY-1066293.
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