Effects of supersymmetric threshold corrections on the Yukawa matrix unification

We present an updated analysis of the Yukawa matrix unification within the renormalizable Minimal Supersymmetric Standard Model. It is assumed that the soft terms are non-universal but flavour-diagonal in the super-CKM basis at the GUT scale. Trilinear Higgs-squark-squark A-terms can generate large threshold corrections to the Yukawa matrix $\mathbf{Y}^d$ at the superpartner decoupling scale. In effect, the SU(5) boundary condition $\mathbf{Y}^d=\mathbf{Y}^{e\,T}$ at the GUT scale can be satisfied. However, such large trilinear terms make the usual Higgs vacuum metastable (though long-lived). We broaden previous studies by including results from the first LHC phase, notably the measurement of the Higgs particle mass, as well as a quantitative investigation of flavour observables.


Introduction
Supersymmetric Grand Unified Theories (SUSY GUTs) have been a topic of multiple studies since the original formulation of the SU (5) model [1]. A successful unification of gauge couplings in the Minimal Supersymmetric Standard Model (MSSM) is a phenomenological triumph of this programme. GUT symmetries are decisively helpful as providers of boundary conditions at the high energy scale, which reduces the dimensionality of the MSSM huge parameter space. Despite a notable historical success of an approximate bottom-tau Yukawa unification, the absence of such a unification for the first two generations remains a long-standing issue. Modifications of the GUT field content that aimed at solving this problem have been applied already in the the nonsupersymmetric case [2]. The most exhaustively studied alterations to the boundary conditions at the GUT scale arise from assuming non-negligible effects from higher-dimensional operators in the SU (5) model Lagrangian. Different mass ratios obtained by such manipulations are reviewed, e.g., in Refs. [3,4].
An important problem of the minimal SU (5) model [1] is the proton decay triggered by the higgsino triplet exchange. Although it remains a non-trivial constraint, several analyses have shown that the theory has not yet been excluded. The proton lifetime can be enhanced by several orders of magnitude by contributions from higher-dimensional operators [5]. Moreover, it has been shown that the tension with experimental results becomes weaker when one uses three-loop Renormalization Group Equations (RGEs) and two-loop decoupling conditions [6]. In the present paper, we do not restrict ourselves to the minimal model. In particular, we allow the higgsino triplets to acquire superheavy masses from their couplings to additional 5 and5 fields that do not couple to ordinary matter. No proton decay problem occurs within such a setup, while the Yukawa unification constraint remains the same as in the minimal case.
It has been observed a long time ago that threshold corrections at the superpartner decoupling scale µ sp can significantly change or even generate the light fermion masses [7]. This mechanism was applied in the context of grand unification in Ref. [8]. However, in most of the contemporary phenomenological analyses, Yukawa unification has been exhaustively studied only in the third generation case.
A quantitative study that achieved Y s (M GUT ) = Y µ (M GUT ) within renormalizable MSSM was performed in Ref. [9]. It included only the threshold corrections coming from gluino and higgsino loops, and concluded that a tension arises between the Yukawa unification and flavour observables. That was likely to happen because flavour off-diagonal soft terms were used to generate the Cabibbo angle as well. This analysis was later broadened and simplified to the flavour-diagonal case in Ref. [10]. It provided examples of points in the MSSM parameter space where the SU (5) Yukawa unification was achieved for tan β ≤ 20. In another article [11] where the leading MSSM threshold corrections were investigated, the problem of proton decay was addressed by raising the Higgs soft masses above 30 TeV.
Our work updates the one of Ref. [10] with a broader range of tan β (reaching 40), inclusion of the contemporary experimental data, as well as a quantitative study of flavour observables. Results from the first phase of the LHC have constrained the superpartner masses and delivered the lightest Higgs mass measurement, thus calling for an up-to-date analysis of Yukawa unification.
We shall make use of a corrected account of chirally enhanced threshold corrections to fermion masses in the MSSM. It was summarized in Ref. [12] and earlier published as parts of other analyses [13][14][15]. However, the twoloop effects computed and described in Ref. [16] are not included in the present work. It might be interesting to study their effect in the future even though they are unlikely to affect our final conclusions.
The article is organised as follows. In Sec. 2, our phenomenological scenario within the MSSM is described. Sec. 3 is devoted to analysing threshold corrections to the Yukawa couplings, and to studying in what manner their unification depends on the most important variables. In Sec. 4, particular examples of points in the MSSM parameter space with Yukawa unification are given. The impact of large A-terms on flavour observables is scrutinized in Sec. 5, whereas a correlation with the vacuum metastability is explained in Sec. 6.

Relevant aspects of the SUSY SU(5) GUT
The SM gauge group is a subgroup of SU (5). A standard embedding of the MSSM superfields Q, U , D, L, E into the 5-and 10-dimensional representations of SU (5) is given by where the hypercharges have been displayed in the conventional SM normalization. We are going to consider SU (5) GUTs whose Yukawa terms in the superpotential read [1] where H5 and H 5 are the two Higgs superfields that couple to matter. Masses of the known fermions are thus determined by only two independent 3 × 3 matrices Y de and Y u . Below the gauge unification scale M GUT 2 × 10 16 GeV, the model reduces to the MSSM with the superpotential given by Thence, Y d and Y e T are equal at the matching scale M GUT , up to a basis redefinition, and up to one-loop threshold corrections at this scale. Unification constraints for Y d and Y e take the simplest form in the super-CKM basis where the superpotential flavour mixing has been entirely included in Y u , while Y d and Y e are real and diagonal. Then we just require equality of the diagonal entries Below M GUT , the relation between Y d and Y e is affected by the RGE and, most importantly, by the threshold corrections at µ sp that strongly depend on the soft supersymmetry breaking terms. These terms include the gaugino masses MB, MW , Mg, soft scalar masses m 2 q , m 2 u , m 2 d , m 2 l , m 2 e , m 2 h d , m 2 hu , as well as the bi-and trilinear interactions of the higgses and sfermions (squarks and sleptons) Our assumptions concerning the soft supersymmetry breaking terms are outlined in the next section.

Choice of the parameters
In order to approach the question of unifying Yukawa couplings by an appropriate choice of the MSSM parameters, we assume the validity of its RGEs up to M GUT where the SU (5) boundary conditions are imposed. The phenomenological motivation behind the discussed scenario within renormalizable MSSM is to achieve Yukawa unification and fulfil experimental conditions in the simplest manner, constraining as few parameters as possible. To independently influence the ratios Y d ii /Y e ii for all the three families, one needs to adjust at least three real parameters.
Diagonal entries of the trilinear A de -terms in the super-CKM basis (which we use throughout the article) can well serve this purpose, as they have a strong influence on the relevant threshold corrections. Moreover, to obtain a correct mass of the lightest Higgs boson for given sparticle masses, one has to adjust A u 33 that governs the stop mixing [17]. Both the Higgs soft mass terms and tan β = vu v d are unconstrained by the SU (5) unification conditions, and can serve other phenomenological purposes. As far as the gaugino and the soft sfermion masses at the GUT scale are concerned, we restrict ourselves here to the simplest choice of a common gaugino mass M 1/2 and a universal soft mass m 2 0 for all the sfermions (but not the Higgs doublets). Such a choice reduces the number of free parameters and makes the analysis transparent. However, it is by no means necessary for achieving the Yukawa matrix unification.
To parameterize the Higgs sector, we have chosen to specify tan β, the value of µ at the electroweak symmetry breaking scale, and the pole pseudoscalar mass m A 0 . This choice, in accordance with the SUSY Les Houches Accord [18,19], has proven superior to the usage of Higgs soft masses at M GUT , i.e. it facilitates finding regions with proper radiative electroweak symmetry breaking, especially for high tan β.

Tools
A standard numerical procedure that for a given parameter set leads to a full spectrum of the MSSM can be summarized as follows. The renormalization group equations of MSSM are solved by an iterative algorithm that interpolates between various scales at which the parameter values are assumed. The boundary with the SM (i.e. the scale µ sp ) is currently set by most of the public programs to be at M Z . Such a choice has considerable disadvantages, one of which is excluding too many parameter points from the analysis. For instance, some fields become formally tachyonic only well below their actual mass scale but above M Z , which is still acceptable, though most programs usually reject such points.
Minimization of the MSSM scalar potential is performed at the scale M SUSY = √ mt 1 mt 2 , where the scale dependence of the electroweak breaking conditions is relatively mild. A recent article [20] has shown that the contemporary spectrum generators find only one of the potentially many models corresponding to a given set of parameters that are specified at multiple energy scales. In particular, it affects the cMSSM dark matter analyses [21]. However, this fact hardly mat-ters for our present investigation because we only search for sample regions in the parameter space where the Yukawa unification constraint is satisfied.
For the purpose of the current analysis, we have modified Softsusy 3.3.8 [22] that distinguishes itself among other spectrum generators by possessing a technical documentation. We implemented threshold corrections to the first and second family Yukawa couplings as given in Ref. [12], which Softsusy was lacking at the moment of writing.
Flavour observables are calculated with the help of SUSY_FLAVOR v2.10 [23]. Our input values of the SM parameters are collected in Tab. 1  3 Analysis of threshold corrections to the Yukawa matrices at µ sp Given our choice of the GUT-scale parameters, the only source of flavour violation at this scale is the Yukawa matrix Y u . Since it affects the RGE for the remaining parameters, neither Y d nor the soft terms are going to remain strictly flavour-diagonal below M GUT . However, the corresponding flavour violation is going to be given by the CKM matrix and remain genuinely small. While such flavour violation is taken into account in our numerical study, we shall neglect it for simplicity in the following discussion where large corrections to the flavour-diagonal terms are of main interest. Within such an approximation, it is sufficient to consider only real diagonal Yukawa matrices corrections are needed for Y s , as illustrated in Fig. 1. Even in this case, the corrections are well in the perturbative regime because the corresponding leading term is small enough (Y s ∼ 10 −2 ).
As described in Ref. [12], in the SUSY-decoupling limit, the chiralityflipping parts of the quark (lepton) self energies Σ are linear functions of the Yukawa couplings, with a proportionality factor and an additive term Σ /
Threshold corrections to the down-quark Yukawa couplings can be enhanced by either tan β or large values of the A-terms. In such a case, a corrected relation between the MSSM Yukawa couplings and the quark masses has the following approximate form: where m d,SM i is the SM quark mass at the matching scale µ sp , mq, md are the respective family squark soft masses, and α s is the strong coupling constant.
The trilinear coupling A d ii controls the most significant contribution from a loop with the gluino -see the first diagram in Fig. 2. It can be used to adjust the threshold correction and to achieve the Yukawa unification for given values of other parameters. In the following, we shall illustrate how the threshold corrections to the Yukawa couplings at µ sp as well as the ratio Y d ii /Y e ii at the GUT scale depend on the most important parameters of the model. Using the point no. 3 in Tab. 2 (Sec. 4) as a reference, we have varied only two parameters at a time, which gives an estimate of the shape of the relation in the vicinity of the considered point. We concentrate on the cases of the second and third family, as the first and second ones are qualitatively similar.
In Figs. 3-6, we show only the points that fulfil all the necessary phenomenological requirements, in particular that the Higgs vacuum is a local minimum of the scalar potential 1 , and that no Landau poles arise below M GUT . White regions in the plots mean that either one of above conditions was not fulfilled, or that Softsusy rejected the point as its iterative algorithm had not converged.
Starting from the largest couplings, we notice that three parameters play a crucial role in the case of bottom-tau unification: A de 33 , µ and m 0 (which for given M 1/2 governs masses of the third family sfermions). Non-universal sfermion masses, independent for each family, could grant additional freedom to our model. Although they are not necessary to achieve Yukawa unification, relaxation of the universality could facilitate finding points with even higher tan β than presented in the next section. A non-trivial interplay among the parameters is illustrated in Fig. 3 where threshold corrections to the bottom and tau Yukawa couplings are shown. One observes that the effects in Y b are significantly larger than in Y τ , which is due to the absence of gluino-mediated corrections in the latter case. The corresponding values of the ratio Y b /Y τ at M GUT are presented in Fig. 4, again as functions of A de 33 , µ and m 0 . The equality of Y b and Y τ at this scale demands an adjustment of all the three above-mentioned parameters because excluded points tightly surround the allowed region.
In the second family case, unification of Y s and Y µ is usually possible by a manipulation of just one parameter, namely A de 22 , despite the fact that it influences both the Yukawa couplings. Its impact on the threshold corrections can be seen in detail in Fig. 5. For the second family, µ has little influence on the unification in the considered region because the higgsino loop gives a much smaller contribution, due to m c m t . The ratio Y s /Y µ at M GUT plotted in Fig. 6 against m 0 and A de 22 shows that a large value of A de 22 is required to achieve unification. The corresponding values of A d 22 /m 2 at M SUSY are shown in Fig. 6. Such ratios will be relevant for our discussion of the vacuum metastability in Sec. 6. Here,m i are defined bỹ (10)    Unification of the down-quark and electron Yukawa couplings is illustrated in Fig. 7. It is the simplest case, because the necessary adjustment of the respective A-term neither triggers any phenomenological problems nor influences any parameters that are relevant for other families.
4 Examples of points with successful SU(5) Yukawa matrix unification In Tab. 2, we present several examples of points in the parameter space where a proper Yukawa matrix unification has been achieved in our setup. We aimed at fulfilling the unification constraints and reproducing the lightest Higgs particle mass (up to the theoretical uncertainty of 3 GeV) for a broad range of tan β. We have chosen the sparticle masses so that the gluino is heavy enough to have evaded the current bounds, but could possibly be detected in the second LHC phase. Tab. 3 shows the corresponding threshold corrections to the Yukawa couplings, as defined in Eq. (9). In addition, we give the GUT-scale ratios which parametrize the unification quality. Their (small) deviations from unity determine sizes of the necessary GUT-scale threshold corrections. Finally, we also present the ratios A i /m i that will be relevant for our vacuum metastability considerations in Sec. 6, withm i at the scale M SUSY defined in Eq. (10).  Table 2: Examples of points with a successful Yukawa unification. They are given by their defining sets of parameters: tan β, common gaugino mass M 1/2 , common sfermion mass m 0 , the superpotential parameter µ(M SUSY ), soft trilinear couplings A ii (M GUT ), and the pseudoscalar mass m A 0 . Nr

Flavour observables
In this section, we discuss the impact of large A-terms on flavour observables. We shall plot how they change when the A-terms grow from 0 to 150% of the value that is necessary for the Yukawa unification Y d (M GUT ) = Y e T (M GUT ) to take place. Among the observables calculable with the help of SUSY_FLAVOR v2.10, only three turn out to be significantly altered: Moreover, the only A-term component they noticeably depend on is A de 33 . Another important parameter to which these observables are sensitive is tan β.
In Fig. 8 [27][28][29] and the branching ratio can be smaller by about 15% compared to the SM prediction B SM sµ = (3.65 ± 0.23) × 10 −9 [30]. As far as B dµ is concerned, it undergoes an almost identical alteration with respect to the SM. However, it remains in perfect agreement with the present   [27][28][29] within its large uncertainties. The experimental sensitivity would need to be improved by more than an order of magnitude to distinguish between the SM prediction B SM dµ = (1.06 ± 0.09) × 10 −10 [30] and the corresponding MSSM results for our sample points.
The three considered decays share the crucial property of being sensitive to supersymmetric contributions even if no sources of flavour violation beyond the CKM matrix are present. It follows from the fact that they are all chirally suppressed in the SM.

Electroweak symmetry breaking
The MSSM contains a large number of scalar fields. In a proper analysis of the electroweak symmetry breaking, one would need to prove that only the neutral Higgs fields acquire non-zero values in the global minimum of the MSSM scalar potential. However, it is well known that there exist large regions in the MSSM parameter space where other, deeper minima arise. At such minima, also sfermions develop non-vanishing vacuum expectation values.
In particular, along the direction in the MSSM scalar field space where However, the usual Higgs vacuum does not need to be absolutely stable. The standard viability condition is that the its lifetime must be longer than the age of the Universe. According to Ref. [32], such a condition is fulfilled when This requirement turns out to be satisfied in all our examples of Yukawa unification. One can verify this by inspecting Tab. 3 where the ratios Ã m have been presented for all the three generations.

Conclusions
Searches for supersymmetric particles during the first LHC phase have significantly constrained the MSSM parameter space. With heavier superpartners, the little hierarchy problem becomes more difficult, but the SUSY flavour problem is rendered less severe. Thus, if the MSSM is the proper low-energy theory, one should consider its possibly non-trivial flavour structure. To check its consistency with grand unification, we need to understand all the factors involved in fulfilling the GUT boundary conditions for the Yukawa matrices. Given only few existing analyses of the MSSM threshold correction impact on the Yukawa matrix unification, we performed an update that takes the recent experimental data into account.
Our article provided examples of successful SU (5) Yukawa unification that is consistent with the current experimental bounds in a scenario where all the soft terms are flavour-diagonal. It did come at a price. An adjustment of the down-quark Yukawa couplings governed by the A-terms of the sfermion mass size led to a conclusion that the usual Higgs vacuum becomes metastable. However, given its long enough lifetime, such a situation is still phenomenologically viable.
Different ways of explaining the Yukawa matrix unification are complementary. In a general case, both the MSSM threshold corrections and the GUT-scale higher-dimensional operators can be present. If a complete quantitative study of a specific GUT model were to be performed, all such options would need to be simultaneously taken into account.
We satisfied the Yukawa matrix unification constraint in possibly the simplest manner, by adjusting just three parameters. The remaining freedom in the choice of other soft parameters in this scenario gives it the advantage of modularity. Such a freedom is likely to facilitate satisfying additional phenomenological constraints (like the observed dark matter relic density) or fitting new observables that might prove relevant for future studies.