Top-bottom-tau Yukawa coupling unification in the MSSM+1VF and fermion masses as IR fixed points

In the MSSM extended by a complete vectorlike family, precise top, bottom and tau Yukawa coupling unification can be achieved assuming SUSY threshold corrections which are typical for comparable superpartner masses. Furthermore, the unification is possible with a large unified coupling, implying that all three fermion masses can be simultaneously close to their IR fixed points. Assuming unified Yukawa couplings of order one or larger, the preferred common scale of new physics (superpartners and vectorlike matter) is in the 3 TeV to 30 TeV range, with larger couplings favoring smaller scales. Splitting superpartner masses from masses of vectorlike fields, the preferred scales extend in both directions. The multi-TeV scale for superpartners is compatible with and independently suggested by the Higgs boson mass.


I. INTRODUCTION
Values of some of the free parameters in the standard model (SM) can be understood if they are related by additional symmetries to other parameters. Gauge coupling unification in the minimal supersymmetric extension of the standard model (MSSM) is a well known example that points to a larger symmetry of a grand unified theory (GUT) at the scale where gauge couplings meet. Similarly, embedding the particle content of the SM into GUT multiplets offers a possibility to understand Yukawa couplings and thus fermion masses from a unified Yukawa coupling at the same scale. There are indications that at least the masses of the third generation fermions (top quark, bottom quark and tau lepton) can be understood in this way as motivated by SO(10) symmetry .
However, the predictive power of Yukawa coupling unification is reduced because other (so far unknown) parameters also enter the determination of fermion masses. In the MSSM, the crucial parameter is the ratio of vacuum expectation values of the two Higgs doublets, tan β, that sets the required top, bottom and tau Yukawa couplings from their measured masses. Furthermore, there are significant supersymmetric (SUSY) threshold corrections [5][6][7]27] that, in the range of tan β favored by Yukawa coupling unification, can comprise up to about half of the bottom quark mass depending on superpartner masses. Without knowing tan β and at least basic features of SUSY spectrum, there is no sharp prediction for fermion masses. Nevertheless, we can instead require that the third generation of fermion masses originate from a single Yukawa coupling at the GUT scale and predict tan β and the SUSY spectrum consistent with this assumption. This has been done in a variety of scenarios [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] typically pointing to certain hierarchies or relations among SUSY parameters. does not work in the MSSM, 1 we will see that it works very well in the MSSM extended by 1 In the MSSM the top quark mass can be understood from the IR fixed point value of the top Yukawa coupling [28][29][30]. However, it requires small tan β precluding Yukawa coupling unification. For large tan β, the top Yukawa coupling is below the IR fixed point and approaches it very slowly in the RG evolution. a complete vectorlike family (an exact copy of a SM family: q,ū,d, l,ē and corresponding fields with conjugate quantum numbers).
We show that in the MSSM extended by a complete vectorlike family (MSSM+1VF), precise top, bottom and tau Yukawa coupling unification can be achieved with a large unified coupling, implying that all three fermion masses can be simultaneously close to their IR fixed points. All three Yukawa couplings approach IR fixed points rapidly from a large range of boundary conditions both above and below the IR fixed point values. Furthermore, the unification is possible assuming SUSY threshold corrections which are typical for comparable superpartner masses and thus no hierarchies or specific relations among SUSY parameters are required. Assuming unified Yukawa couplings of order one or larger, the preferred common scale of new physics (superpartners and vectorlike matter) is in the 3 TeV to 30 TeV range, with larger couplings favoring smaller scales. Splitting superpartner masses from masses of vectorlike fields, the preferred scales extend in both directions. The required scale of new physics is to a large extent driven by fitting the measure values of gauge couplings [31] with fermion masses further constraining the preferred range. However, due to the IR fixed point behavior it is highly non-trivial that Yukawa couplings point to a similar scale of new physics as gauge couplings. Furthermore, the multi-TeV scale for superpartners is compatible with and independently suggested by the Higgs boson mass.
Enlarging the particle content of the model also results in new parameters that cast a shadow on the predictivity of Yukawa coupling unification. The fields in a vectorlike family can have Yukawa couplings to Higgs doublets, they can mix with SM families (we will not consider this possibility) and they must have vectorlike masses to avoid detection. However We assume the common scale of new physics only for simplicity. The results do not differ much as long as superpartner masses and vectorlike masses are comparable. Nevertheless, after presenting the main results, we will also explore effects resulting from abandoning our simple assumptions. We will consider the scale of superpartners independent from vectorlike masses. Moreover, since the assumption of a common scale for superpartners has an impact on the predicted bottom quark mass, through SUSY threshold corrections, we will also consider splitting gaugino masses from scalar masses. Furthermore, we will present results in terms of the required SUSY correction to the bottom quark mass that could be used in a variety of scenarios that are not approximated well by our assumptions.
Extensions of the SM or the MSSM with vectorlike matter were previously explored in a variety of contexts. Examples include studies of their effects on gauge couplings [32][33][34][35][36][37][38][39], [31] and on electroweak symmetry breaking and the Higgs boson mass [40][41][42]. In addition, vectorlike fermions are often introduced on purely phenomenological grounds to explain various anomalies. Examples include discrepancies in precision Z-pole observables [43][44][45][46] and the muon g-2 anomaly [47,48] among many others. More related to our study, the fast approach of Yukawa couplings to the IR fixed points in asymptotically divergent models was observed in Refs. [49], [38,39], [31] and the b − τ Yukawa coupling unification in the MSSM with vectorlike matter was recently discussed in Ref. [50]. This paper is organized as follows. In Sec. II, we discuss model parameters and assumptions, provide approximate formulas for the RG equations of Yukawa couplings and SUSY threshold corrections and summarize details of the numerical analysis. The main results and their discussion are contained in Sec. III and we conclude in Sec. IV.

II. MODEL PARAMETERS, RG EQUATIONS AND PROCEDURE
We start exploring predictions for top, bottom and tau Yukawa couplings with the following set of model parameters: representing the GUT scale, the common mass of vectorlike matter and superpartners, M ≡ M V = M SU SY , and the ratio of vacuum expectation values of the two Higgs doublets, denoting the unified value of gauge couplings at the GUT scale, the GUT scale threshold correction to gauge couplings and the GUT scale boundary conditions for the common Yukawa coupling of top, bottom and tau, and the common Yukawa coupling of vectorlike matter. We neglect Yukawa couplings of first two SM generations.
We define the GUT scale as the scale where the α 1 and α 2 differ from α 3 by equal amounts and we identify α G with α 3 at this scale: where the third generation SM fields originate from 16 3 where t = ln Q/Q 0 , with Q being the RG scale, and We will see that the three parameters in Eq. However, the assumption of a common scale for superpartners has a significant impact, 2 The impact of the assumption of a common vectorlike mass at the EW scale versus the GUT scale on gauge couplings in this scenario was studied in Ref. [31]. The common mass at the GUT scale leads to an improvement in gauge coupling unification. However the difference is not dramatic.
especially on the predicted bottom quark mass, through finite SUSY threshold corrections [5][6][7]27]. We match the SM top, bottom and tau Yukawa couplings to those in the MSSM+1VF at the M SU SY scale: where t,b,τ are SUSY threshold corrections. Typically dominant contributions are from gluino-stop loops for the top quark, gluino-sbottom and chargino-stop loops for the bottom quark, and bino-stau loops for the tau lepton, where subscripts 1 and 2 label two mass eigenstates of corresponding scalars and The SUSY threshold corrections for the top Yukawa coupling are small in the large tan β region characteristic for Yukawa coupling unification. The corrections are also small for the tau Yukawa coupling since they are proportional to α 1 . However, for the bottom Yukawa coupling, they are of order 1%×tan β and typically in the 30%-40% range assuming comparable superpartner masses. In the limit where all superpartner masses are equal, given by M SU SY , the chargino correction is an order of magnitude smaller than the gluino correction for A-terms as large as M SU SY . 3 In addition, whether the chargino correction adds to or subtracts from the gluino correction depends on the relative sign of the A-term 3 For very large A-terms the chargino correction can be comparable to gluino correction or even dominate.
The region of the parameter space in the MSSM where gluino and chargino corrections almost cancel leading to successful Yukawa coupling unification was explored in Refs. [9][10][11][12].
and gluino mass and thus, for simplicity we assume zero A-terms when presenting main results. In the limit of degenerate superpartner masses the loop function also simplifies, Finally, electroweak symmetry breaking requires µ 2 −m 2 Hu and the typical result from the RG flow over few orders of magnitude in the energy scale [42]. Thus, the typical expectation is µ ± √ 2M SU SY with either sign. With these assumptions and simplifications the approximate formulas for the SUSY threshold corrections are: from which the typical sizes can be readily obtained. For any specific SUSY breaking scenario the SUSY corrections could be evaluated precisely. However, the above formulas should be a good approximation in large regions of the parameter space of scenarios with both high and low mediation scales of SUSY breaking. In addition to the main results assuming a common scale of superpartners we will explore the impact of splitting gaugino masses from scalar masses. Furthermore, we will also present results in terms of the required SUSY correction to the bottom quark mass that could be used in a variety of scenarios that are not approximated well by our assumptions.
In the numerical study we use 3-loop RG equations for gauge couplings and 2-loop RG equations for the third generation Yukawa couplings and Yukawa couplings of vectorlike fields [51][52][53][54][55][56], [34]. All the particles above the EW scale are integrated out at their corresponding mass scales. The complete set of SUSY threshold corrections to the third generation Yukawa couplings (for which the approximate formulas can be found above) is included at the M SU SY scale [5-7, 27, 57] with the assumption that µ = − √ 2M SU SY (we will see that only the negative sign is consistent with Yukawa coupling unification assuming comparable where α ≡ α 3 + 9 16 is the combination of gauge couplings, α i = g 2 i /4π, appearing in the RG equation for y t , Eq. (6). If we assume that all Yukawa couplings have the same boundary condition, then the only difference in the RG evolution of up-type (coupling to H u ) and down-type (coupling to H d ) couplings of quarks is due to hypercharge, see Eqs.
where s q is the number of y * y factors of large up-type quark Yukawa couplings. In our case s q = 13. Inserting the 1-loop RG equations for gauge couplings, dα i /dt = (b i /2π) α 2 i , with the beta function coefficients b i = (53/5, 5, 1) corresponding to the MSSM+1VF, we find For α G > 0.2 this approximation differs from the precise numerical value by about 2%.  Fig. 2 where . Taking the limit of y t (M G ) → ∞ we get the formula for the quasi fixed point As mentioned, in the SM or the MSSM the difference between Eq. (26) and Eq.
As anticipated, the ratio of y 2 τ to quark Yukawa couplings squared (or to gauge couplings) is decreasing for larger α G and further away from the GUT scale it is evaluated. 4 So far we have not included the threshold effects from superpartners or vectorlike matter, the same particle content is assumed all the way to the EW scale. As a result, the gauge couplings do not reproduce the measured values exactly. This is intentional since we want to infer the scale of superpartners and vectorlike matter from the IR fixed point values of Yukawa couplings. In order to do this, with dashed lines and shaded regions at low energies in Fig. 1, we plot the evolution of y t (black), y b (upper gray) and y τ (lower gray)    Fig. 4. We also show contours of the required Y V and a subset of other model parameters. Those not shown have similar values as in Fig. 3.
Perhaps more interesting than the Yukawa coupling unification itself is the fact that the unification is possible with large boundary conditions for both gauge and Yukawa couplings.
It means that the EW scale values are very insensitive to the boundary conditions due to the IR fixed point behavior discussed above. This can also be inferred from the large size of the parameter space leading to the top quark mass in 1% or 2% ranges around the central value in Fig. 3. Moreover, from the shaded range of the bottom quark mass in Fig. 4 we see that for Yukawa couplings larger than one, the preferred range of superpartners and vectorlike matter is 3 TeV to 30 TeV, with larger couplings favoring smaller scales of new physics. This mass range is also compatible with the Higgs boson mass.
Another interesting feature is that the required SUSY correction to m b in the whole plotted plane is in the range that is generically achieved with comparable values of SUSY parameters. Thus, no extreme regions of SUSY parameters are required to simultaneously obtain all three fermion masses correctly. This is important, since due to the IR fixed point nature, there are no other parameters that can affect the fermion masses significantly.
Splitting gaugino and scalar masses can actually be used to fit the bottom quark mass everywhere in the plane. DefiningR parameter as the ratio of gaugino masses and scalar masses and still, for simplicity, assuming that scalar masses are the same as vectorlike quark and lepton masses, this is illustrated in Fig. 5      decreasing the Yukawa couplings (or α G ) requires larger scales of new physics. However, in this limit, the understanding of the third generation fermion masses as IR fixed points gradually fades away. We should keep in mind, however, that these results assume the typical SUSY corrections resulting from comparable SUSY spectrum and different assumptions about soft SUSY breaking terms or the µ-term could shift the preferred range of model parameters as indicated (by b ) in Fig. 6(b) and previous figures.

IV. CONCLUSIONS
We have found that in the MSSM extended by a complete vectorlike family, precise top, bottom and tau Yukawa coupling unification can be achieved with a large unified coupling, implying that all three fermion masses can be simultaneously close to their IR fixed points.
All three Yukawa couplings approach IR fixed points rapidly from a large range of boundary Acknowledgments: This work was supported in part by the U.S. Department of Energy