Yukawa Unification Predictions for the LHC
This paper is divided into two parts. In the first part we analyze the consequences, for the LHC, of gauge and third family Yukawa coupling unification with a particular set of boundary conditions defined at the GUT scale. We perform a global analysis including the observables and . The fit is performed in the MSSM in terms of 9 GUT scale parameters, while and are fixed at the weak scale. Good fits suggest an upper bound on the gluino mass, TeV. This constraint comes predominantly from fitting the bottom quark and Higgs masses (assuming a 125 GeV Higgs). Gluinos should be visible at the LHC in the 14 TeV run but they cannot be described by the typical simplified models. This is because the branching ratios for , , , , are comparable. Stops and sbottoms may also be visible. Charginos and neutralinos can be light with the LSP predominantly bino-like. In the second part of the paper we analyze a complete three family model and discuss the quality of the global fits and the differences between the third family analysis and the full three family analysis for overlapping observables. We note that the light Higgs in our model couples to matter like the Standard Model Higgs. Any deviation from this would rule out this model.
Gauge coupling unification in supersymmetric grand unified theories (SUSY GUTs) [1, 2, 3, 4, 5, 6] provides an experimental hint for low energy SUSY. However, it does not significantly constrain the spectrum of supersymmetric particles. On the other hand, it has been observed that Yukawa coupling unification for the third generation of quarks and leptons in models, such as or , can place significant constraints on the SUSY spectrum in order to fit the top, bottom and tau masses [7, 8, 9, 10, 11]. These constraints depend on the particular boundary conditions for sparticle masses chosen at the GUT scale (see for example, [9, 12, 13], which consider different GUT scale boundary conditions). In light of the present success of the LHC with the possible observation of the Higgs boson with mass of order 125 GeV and significant lower bounds on gluino and squark masses, it is a perfect time to review the viability of the constraints on the sparticle spectrum resulting from gauge and third generation Yukawa coupling unification.111For other analyses in this direction, see [14, 15]. This is what we do in this paper. In part one of the paper, we perform a global analysis assuming boundary conditions for sparticle masses and non-universal Higgs masses, which we have called “just so Higgs splitting.” We fit the observables, and in terms of 11 arbitrary parameters. These fits then place significant constraints on the gluino mass.
In the second part of the paper we study a complete three family model of quark and lepton Yukawa couplings at the GUT scale [16, 17] which is based on an GUT with a family symmetry. This model was shown to give good fits to precision electroweak data, including quark, charged lepton and neutrino masses and mixing angles (see most recently the global analysis in ). In light of the observation of it is again a perfect time to re-analyze this model. We are also able to compare the third family Yukawa unification analysis with the three family analysis which now includes hierarchical Yukawa matrices with unification of the (3,3) element of the Yukawa matrices. Hence, off-diagonal elements in the Yukawa matrices give small corrections to exact Yukawa unification.
The paper is organized as follows. In 2, we present the model. In 3, we present the procedure used in the paper for analyzing the model. In 4, we consider a model with gauge coupling unification and only the Yukawa couplings for the third family, which are assumed to unify at the GUT scale. We perform a global analysis fitting the relevant low energy observables. In 5, we extend the analysis to all three families of quarks and leptons using a particular GUT model. In this case, we look for the minimum values of for five different choices of the universal squark and slepton mass, , defined at the GUT scale, . Finally, the summary and conclusions are given in 6.
2 The Model
2.1 Third family model
Fermion masses and quark mixing angles are manifestly hierarchical. The simplest way to describe this hierarchy is with Yukawa matrices which are also hierarchical. Moreover the most natural way to obtain the hierarchy is in terms of effective higher dimension operators of the form
This version of models has the nice features that it only requires small representations of , has many predictions and can, in principle, find an UV completion in string theory. The only renormalizable term in is which gives Yukawa coupling unification
So instead we use Yukawa unification to predict the soft SUSY breaking masses. In order to fit the data, we need
Given the following GUT scale boundary conditions, namely universal squark and slepton masses, , universal cubic scalar parameter, , universal gaugino masses, , and non-universal Higgs masses [NUHM] or “just so” Higgs splitting, or , we find that fitting the top, bottom and tau mass forces us into the region of SUSY breaking parameter space with
In addition, radiative electroweak symmetry breaking requires , with roughly half of this coming naturally from the renormalization group running of neutrino Yukawa couplings from to GeV .
It is very interesting that the above region in SUSY parameter space results in an inverted scalar mass hierarchy at the weak scale with the third family scalars significantly lighter than the first two families . This has the nice property of suppressing flavor changing neutral current and CP violating processes. These results depend solely on Yukawa unification for the third family. In order to demonstrate this, we perform a separate analysis with only third family observables (4) and then a complete three family analysis (5).222The large Yukawa coupling for the third family is the driving force for the inverted scalar mass hierarchy. However, the particular boundary conditions of 5 were shown to maximize the effect.
2.2 Full Three Family Model
We now consider a complete three family model for fermion masses and mixing, including neutrinos [16, 17, 18]. The model also includes a family symmetry which is necessary to obtain a predictive theory of fermion masses by reducing the number of arbitrary parameters in the Yukawa matrices. Consider the superpotential generating the effective fermion Yukawa couplings:
where is an adjoint field which is assumed to obtain a VEV in the B – L direction, is a linear combination of an singlet and adjoint, and the index . Its VEV gives mass to Froggatt-Nielsen states . Here, and are elements of the Lie algebra of with in the direction of the which commutes with and the standard weak hypercharge, and , are arbitrary constants which are fit to the data. is an invariant mass scale which in principle could be obtained by integrating out additional Froggatt-Nielsen states. Note that both and are assumed to be above the GUT scale. , , are singlet “flavon” fields, is a non-trivial one dimensional representation under , and , are a pair of Froggatt-Nielsen states transforming as a and under . The so-called flavon fields are assumed to obtain VEVs of the form
After integrating out the Froggatt-Nielsen states one obtains the effective fermion mass operators in 1.
Inserting the flavon VEVs, one then obtains Yukawa matrices for up-quarks, down-quarks, charged leptons and neutrinos given by
with and a Dirac neutrino mass matrix given by
From 9 and 11 one can see that the flavor hierarchies in the Yukawa couplings are encoded in terms of the four complex parameters and the additional real ones . These matrices contain 7 real parameters and 4 arbitrary phases. Note, the superpotential (7) has many arbitrary parameters. However, at the end of the day the effective Yukawa matrices have much fewer parameters. This is good, because we then obtain a very predictive theory. Also, the quark mass matrices accommodate the Georgi-Jarlskog mechanism, such that . This is a result of the VEV in the direction.
We then add 3 real Majorana mass parameters for the neutrino see-saw mechanism. The anti-neutrinos obtain GUT scale masses by mixing with three singlets ( for and ) transforming as a doublet and singlet, respectively. The full superpotential is given by with
where the fields are additional flavon fields whose VEVs provide Majorana masses for the states . We assume obtains a VEV, , in the right-handed neutrino direction, and for and . The effective neutrino mass terms are given by
all assumed to be real. Finally, upon integrating out the heavy Majorana neutrinos we obtain the Majorana mass matrix for the light neutrinos in the lepton flavor basis given by
where the effective right-handed neutrino Majorana mass matrix is given by:
|Sector||Third Family Analysis||#||Full three family Analysis||#|
|gauge||, ,||3||, ,||3|
|SUSY (GUT scale)||, , , ,||5||, , , ,||5|
|textures||1||, , , , , ,||11|
|SUSY (EW scale)||,||2||,||2|
Renormalization Group Equations
The model parameters, summarized in 1, are defined at the grand unification scale with the exception of and that are defined at the electroweak scale. At the GUT scale, and , where is the GUT scale threshold correction333Without presenting a complete GUT we leave as a free parameter. In this way, our analysis will also apply to orbifold GUTs or string compactifications with a scale of order . necessary to fit the strong coupling to experimental data at the electroweak scale, . These 3 gauge parameters, the 11 Yukawa textures (described in 2.2), 5 SUSY boundary conditions, and 3 real neutrino mass parameters allow us to completely define the model at the GUT scale and derive all the parameters of the minimal supersymmetric standard model (MSSM).
First, the GUT scale parameters are RGE evolved to the right-handed neutrino scale where the RH neutrinos are integrated out (see 2). The right-handed neutrinos have three different scales associated with them, and the most relevant one is the third-family RHN that is mostly responsible for splitting the up and down type Higgs masses. We therefore choose to integrate out all the right-handed neutrinos at one single scale, .
Below the scale of the RHNs, we use the 2-loop MSSM RGEs for both dimensionful and dimensionless parameters. Ideally, one should evolve all parameters to the scale of the heavy scalars ( in this case, as shown in 2) and integrate them out and proceed to evolve to the weak scale using an effective theory without the first two generation scalars. We choose an alternative approach and use the 2-loop MSSM RGE444In scenarios with heavy scalars, it has been shown that the 2-loop contributions to the third generation scalars can lead to dramatic consequences, like driving the stop mass squared negative  and thus it is important to include the 2-loop RGEs in scenarios such as discussed here. evolution down to the weak scale and correct for the additional running by including 1-loop threshold corrections to the relevant observables555For the calculation of Higgs mass, we define an effective theory at the scale and interface our calculation with the code by authors in Ref.. This approximation eliminates the need to define multiple effective theories. In our analysis, we have been careful to take into account the corresponding threshold corrections for all observables.
|Observable||Exp. Value||Ref.||Program||Th. Error|
At the weak scale, we calculate the SUSY spectrum and the SUSY threshold corrections to the fermion masses and CKM matrix elements. Especially in the large regime, these SUSY threshold corrections are very important for the down type quarks and charged leptons and can be at the percent level in Yukawa-unified SUSY models . We then use the threshold corrected fermion masses to determine the tree level masses for the squarks and sleptons. In addition, we also determine the one-loop pole mass for gluino and the CP-odd Higgs mass. The precision electroweak observables , , , , are calculated including 1-loop threshold corrections, using the procedure described in [35, 36]. Following the prescription in , the condition for consistent radiative electroweak symmetry breaking is also imposed at the weak scale, and for this, we use the physical pole mass. The parameter is fixed by this procedure via a separate minimization, and in the process, we fit the mass precisely to the physical pole mass. In the calculation of and , we only include the 1-loop corrections from the third family scalars, since the first two generation scalars are integrated out at . We assign a theoretical uncertainty of 0.5% to our calculation of the electroweak observables (except for ) due to the approximate treatment of thresholds described above. We also assign a 1% theoretical uncertainty to our calculation of , since we neglect the SUSY vertex and box diagrams. Finally, to compare to experiment, is evolved to zero momentum transfer.
Charged Fermion masses and mixing angles
Below , we integrate out all SUSY partners and electroweak gauge bosons to obtain an effective low energy theory. We use 1-loop QED and 3-loop QCD RGEs to renormalize to the appropriate scales and calculate the low energy observables. We fit the top quark pole mass, and the bottom and charm quark masses are calculated at their respective masses. All the other light quark masses are calculated at the scale of 2 GeV. We fit 7 observables relevant to quark masses, 3 charged lepton masses, and 6 CKM observables. The theoretical uncertainty in their calculation is again estimated to be 0.5 %. Since the light quark masses are not measured to very high precision, we choose to fit multiple correlated observables. These include the strange quark mass, the mass ratio and the mass ratio Q defined in the PDG  as
The CKM matrix is calculated from the left and right mixing matrices by diagonalizing the Yukawa matrices and including the SUSY threshold corrections. 6 CKM observables (, , , , and ) are included in our global fit analysis. To account for the inconsistencies in the inclusive and exclusive measurements of and , we allow our result to be within the experimental error from both the inclusive and the exclusive measurement. The pole masses in the lepton sector are calculated with 1-loop electromagnetic threshold corrections.
To execute the steps elaborated so far, we use a code maton, originally developed by Radovan Dermíšek to study Yukawa unification in the model with family symmetry . maton has been restructured and extended appropriately to adapt to the current analysis.
The recent observation of the Higgs boson at the LHC [37, 30] will allow us to highly constrain the parameter space of the model. Flavor constraints have already pushed the first two generation scalars of Yukawa-unified SUSY models 10 TeV . In contrast, the third family scalars have mass about a few TeV, purely by the effects of RGE running. The hierarchy between the first two and the third generations alleviates the constraints from flavor physics and CP violating observables, and at the same time eases the large fine-tuning in models with heavy scalars. In addition to the TeV range scalars, the large -terms make it easy to obtain a Higgs mass of about 125 GeV. We integrate out all the scalars (including the third generation squarks and sleptons) below the scale , and calculate the Higgs boson mass using the dedicated code by the authors of Ref. , that is best suited to our case where the sfermions are very heavy. Given the boundary conditions,
at the scale , (where are the gaugino masses at the scale ), the routine  determines the Higgs mass by calculating the corrections to the Higgs quartic coupling:
are the contributions from chargino and neutralino diagrams. The quartic coupling is given by:
We have to point out an important difference in our approach. The conventional method is to use the SM inputs of , , , , , , to determine the gauge and the Yukawa couplings at the scales and further constrain the GUT scale parameters. We instead like to predict these low energy observables and constrain the GUT scale parameter space based on a global fit to the data. In our calculation of the Higgs mass, we take the gauge and Yukawa couplings as input at the scale , obtained from RGE evolution using maton and calculate the Higgs mass using these inputs. The approach we adopt here is purely top-down. We have adapted the routine  to suit this line of analysis. Nevertheless, we have compared the spectrum we obtain from maton with that from softsusy666Without making significant changes to softsusy or other publicly available codes, we find that we can only make rough comparisons of the spectra. This is because to the best of our knowledge, most of the currently available codes do not handle complex parameters. In addition, many do not include right-handed neutrinos, and do not offer an easy way to implement the particular GUT scale Yukawa texture of the model.  and find good agreement.
We are fitting 5 observables in the neutrino sector: the mixing angles , , , and the mass-squared differences and (cf. 2). The most dramatic change in the experimental determination of the neutrino parameters in recent years comes from the Daya Bay and Reno collaborations [40, 41] that have confirmed that is indeed large. Moreover, there are tentative hints that is not maximal [42, 43]. Whereas Ref.  sees a preference at for the first octant, i.e. , Ref.  finds an equal probability for being larger or smaller than . In the following, we will be using the best-fit values and the uncertainties quoted by the NuFIT collaboration  which are in agreement with Ref.  at .
The strongest constraints on the model come from B-physics. For calculating the flavor observables, we use two publicly available codes, namely susy_flavor  and SuperIso [45, 46]. Since the boundary conditions that we impose at the GUT scale may generate large off-diagonal and in general complex entries at the low scale, susy_flavor is better adapted to our needs. Note that susy_flavor, in contrast to comparable programs that calculate similar processes, does not assume minimal flavor violation (MFV), and allows for general, full three family, complex soft parameters. This is particularly important in our case, since we are calculating several CP violating observables and need to take into account777We calculate the particle spectrum using maton, see comments on p. 3. To the best of our knowledge, there is currently no publicly available spectrum generator that fully takes into account all the complex phases of the MSSM. the complex phases in the soft parameters. Hence, susy_flavor is our default choice for all flavor observables with the following exceptions. For , we use SuperIso, since susy_flavor does not include the NNLO SM corrections. We have verified that the discrepancy between susy_flavor and SuperIso in the parameter space that is of interest to us is at most 10% and typically less than 7%. Also, we use SuperIso for the observables connected to the decay process , since susy_flavor does not provide them. It is important to note that SuperIso has some built-in assumptions that prove to be too restrictive in our case. E.g. SuperIso assumes all soft parameters to be real, and only takes the diagonal entries of the third-family trilinear couplings into account. As a consequence, we have assigned larger theoretical uncertainties to the values calculated by SuperIso (see 2). Additional sources of uncertainties in the flavor observables derive from the theoretical determination of the B meson decay constant and from the experimental measurements of the CKM matrix elements.
LHCb has recently measured  the which is in good agreement with the SM prediction. This pushes the CP-odd Higgs mass to a few TeV and hence leads to the Higgs decoupling limit. Thus the light Higgs is predicted to be SM-like. The recent observation of zero-crossing in the forward-backward asymmetry of constrains the Wilson coefficient to be of the same sign as that in the SM. This imposes the additional constraint for the model that if , in order to satisfy the branching fraction observed in the process the first two generation scalars have to be heavier than at least 10 TeV.
In the last step of our calculation, we construct a function in terms of the 36 calculated observables (see 2).
and are the theoretical prediction and experimental measurement, respectively, for each observable. is the error on each observable, the theoretical and experimental errors added in quadrature. In the general case, we vary 23 parameters (see 1 and note that is fixed in all the analyses) in order to fit 36 observables, which amounts to 12 (or 13 counting the separate fit to the Z pole mass) degrees of freedom (d.o.f.). We will consider the per d.o.f. for the model as a qualitative measure of the goodness of fit. We will look at the pulls from the individual observables to assess the goodness of fit of the model.
Finding the global minimum for a model with 23 parameters is a formidable task. In the present analysis, we minimize the function using the Minuit package maintained by CERN . Note that Minuit is not guaranteed to find the global minimum, but will in most cases converge on a local one. For that reason, we iterate (100) times the minimization procedure for each set of input parameters, and in each step we take a different initial guess for the minimum (required by Minuit) so that we have a fair chance of finding the true minimum. This, of course, requires large computing resources, and to that end we have used the Ohio Supercomputer Center in Columbus and the “Centre de Calcul de l’Institut National de Physique Nucléaire et Physique des Particules” in Lyon.
4 Third family analysis
In this section we analyze the consequences of Yukawa unification for the the third family in the context of minimal supersymmetric grand unification defined by the superpotential term, W . The aim of this analysis is to study the SUSY spectrum, and we argue that the constraints on the SUSY spectrum come predominantly from the third family, the lightest Higgs mass and the branching ratio . There are 24 parameters in total in the DR model [16, 17], and in this section we focus on 11 parameters (summarized in 1) that are used to evaluate 11 low energy observables, , , , , , , , , , , and the lightest Higgs mass, . We specify the model with the full 24 parameters, but we only vary 11 in the minimization procedure to fit the 11 observables listed above. The irrelevant parameters for this analysis, namely, the neutrino parameters and the off-diagonal Yukawa textures, are set to constant values and do not enter into the minimization procedure.888In this section, when calculating flavor violating observables, we use susy_flavor with the experimental input values for the light fermion masses and mixing angles. Similarly, the low energy observables connected to the first two families do not enter the function. The effects of the off-diagonal Yukawa textures will be discussed in 5.
Consider first the SUSY spectrum in our analysis. The first and second family squarks and sleptons have mass of order , while stops, sbottoms and staus are all significantly lighter. This is the inverted scalar mass hierarchy which is a direct result of RG running. Nevertheless, gluinos are always lighter than the third family squarks and sleptons, and the lightest charginos and neutralinos are even lighter. Note the states and are approximately degenerate. A detailed spectrum is given in 3. Recent results from CMS and ATLAS give lower bounds on the gluino mass. These bounds are given in terms of the CMSSM or simplified models. The simplified models which are most relevant for our analysis are those in which (a) the third family of squarks and sleptons are lighter than the first two, and (b) the gluino is lighter than the stops and sbottoms. In this case, the lower bound on the gluino mass is now of order 1 - 1.2 TeV, assuming the branching ratio or [48, 49]. Although neither simplified model is appropriate for our model, we nevertheless impose a lower bound on the gluino pole mass in order to be roughly consistent with the latest LHC results.
In 3(a) we present the best fits as a function of for two values of the lower bound that we impose on the gluino pole mass, i.e. 850 and 1000 GeV. We note that is relatively insensitive to these lower bounds on the gluino mass, although lower values of are slightly favored. The minimum is found for TeV, and increases as either decreases or increases. Features of the model like the large A-terms and large are favorable to obtain a Higgs mass in the range of 122 - 127 GeV as observed at the LHC. However, the largest contribution to for lower values of comes from the Higgs mass constraint (see 4).
As the lower bound on the gluino mass is increased to 2 or 3 TeV, we find that dramatically increases (see 3(b)). Note, this is predominantly due to the constraint from the light Higgs mass (4). The simple explanation for this fact is that as the gluino mass increases the magnitude of at also increases, due to the infra-red fixed point. This has the effect of decreasing the light Higgs mass because now which goes beyond maximal mixing. As a consequence, there appears to be an upper bound on the gluino mass of order 2 TeV, which makes gluinos inevitably observable at the LHC 14 TeV. However, as discussed earlier, the usual simplified models do not apply since gluinos decay with branching ratios , , , , which are all significant.
In 5 we give the best fits for the third family analysis as a function of the lower bound on the gluino mass and the value of . In 5 we give the contours of constant gluino masses (roughly horizontal lines) and stop masses (vertical lines)999In a recent analysis , the authors found an upper bound on the stop mass for good Yukawa unification. Their result is a consequence of the constraint GeV, in order to satisfy dark matter bounds. We do not make any such assumption and do not find an upper bound on the stop mass for TeV.. Note, for values of TeV, the best fit gluino pole mass is always much larger than the lower bound imposed.
5 Full Three family analysis
In this section we present the global analysis for three families including all 24 arbitrary parameters. The function includes 36 observables. We present our results for fixed values of in 6 and in 5 to 9.
In 6 we give the best fits for two different values of the lower bound on the gluino pole mass imposed in the analysis. 3 and 6 have similar behavior. The value of increases dramatically for values of TeV. For larger values of TeV the increase is much slower. In the three family analysis, the minimum occurs around TeV, just as in the third family analysis. Moreover, the input parameters which minimize in the third family analysis also minimize for the full three family analysis.
In 5 to 9 we present the best fits for values of TeV, respectively. The best fit overall comes for TeV with /d.o.f (see 7). Let us just comment on a few of the initial values of the parameters for this point. We find . The magnitude of the Yukawa couplings are hierarchical. As expected, we have , and . The average Higgs mass parameter and the relative splitting are given by TeV and , respectively. The gravitino mass for this model is expected to be of order the largest scalar mass, i.e. TeV. We used micrOmegas to calculate the relic abundance for the benchmark points considered and found (10 TeV), (15 TeV), (20 TeV), (25 TeV), (30 TeV). This is a consequence of a purely bino-like LSP. In this case, a non-thermal process would be necessary to accommodate the observed dark matter abundance. Assuming the correct dark matter abundance, a bino LSP would not have been observed yet by direct detection methods, but should be observable by future detectors .
Let us now focus on the fit. Consider the observables with the largest pulls. Roughly half the contribution to at this point comes from just two observables, namely and . Our value of is larger than the experimental value, and this implies that our value of (see 19). We have allowed to range over values consistent with both exclusive and inclusive measurements. We find that our fit is more consistent with exclusive measurements. Moreover, our fit value of is at the lower bound allowed by the experiments. Otherwise we are able to fit an amazing array of experimental observables. The light Higgs mass is fit to within the 3 GeV theoretical uncertainty we have assigned. As for the neutrino mixing angle we obtain a value closer to 6, rather than the present experimental value of approximately 9. This may be a problem, however, it has been noticed recently that flavor violating corrections to the Kähler potential can have a significant effect on without affecting the other larger mixing angles . Our neutrino spectrum corresponds to the normal hierarchy. Note that the two large mixing angles are a consequence of a hierarchy in the right-handed neutrino masses.
|10 TeV||15 TeV||20 TeV||25 TeV||30 TeV|
|Current Limit||10 TeV||15 Te V||20 TeV||25 TeV||30 TeV|
In 3 we summarize the predictions for the SUSY spectrum given values of TeV, respectively. We give the spectrum of the lightest squark, slepton and gaugino masses, and the CP odd Higgs mass . The first and second generation squarks and sleptons all have mass of order . Note, in order to fit the branching ratio with large , we have . Thus we are in the decoupling limit where the light Higgs is predicted to couple to matter just like the Standard Model Higgs. Therefore, any deviation from this prediction would rule out our model. Finally, in 4 we present results for yet to be observed quantities such as electric dipole moments of charged leptons, flavor violating processes such as and the CP violating angle in the lepton sector, . The value of is close to zero and is thus consistent with tentative emerging hints for . We also find that the may in fact be observable by the MEG experiment in a few years .
6 Summary and Conclusions
We have performed a global analysis of an SUSY GUT times a family symmetry. The model fits all fermion masses and mixing angles, as well as many flavor observables, quite well. The model has 24 arbitrary parameters which we use to fit 36 low energy observables. Five of these parameters include the soft SUSY breaking masses, a universal squark and slepton mass, ; a universal cubic scalar coupling, ; a universal gaugino mass,