Yukawa Unification with Four Higgs Doublets in Supersymmetric GUT

We discuss the Yukawa coupling unification, a very interesting prediction of the grand unified theory, in the context of scenarios with more than one pair of Higgs doublet since the current LHC constraint has become a problem for the Yukawa unification scenarios with just one pair of Higgs doublet. More than one pair of Higgs doublets can easily arise in missing partner mechanism which solves the doublet-triplet splitting problem. In such a scenario, the Yukawa unification occurs at a medium $\tan\beta$ value, e.g., $\sim$ 30, which corresponds to much smaller threshold corrections compared to usual large $\tan\beta$ scenario for $t-b-\tau$ unification in the context of SO(10) and $b-\tau$ unification in the context of SU(5) models. Further, we show that an additional Higgs doublet pair lowers the sensitivity of the radiative symmetry breaking of the electroweak vacuum.


Introduction
The Standard Model (SM) is well established to describe the physics below the weak scale, and the SM particle content is complete after the discovery of the Higgs boson whose mass is 125 GeV. However, 27% abundance of the universe, origin of the electroweak scale, neutrino masses etc. are not explained in the SM. The minimal supersymmetric standard model (MSSM), one of the most elegant extension of the SM, with origin in a grand unified model, e.g., SO(10) [1], has answers for all these puzzles.
However, there is no evidence of supersymmetry (SUSY) at the LHC so far and this has generated constraints on the colored SUSY masses, e.g., squarks, gluino masses need to be ≥ 2 TeV [2,3]. Similarly, the lower bounds on non-colored SUSY masses have been kept on increasing. This situation has impacts on scenarios with predictions for lighter SUSY masses. One such example is a scenario which possesses unification of third generation Yukawa couplings motivated by the grand unified theories [4,5,6]. This scenario is running into difficulty with the current LHC constraints on the sparticle masses [7,8]. Since the unification of third generation Yukawa coupling is a very interesting prediction in the context of minimal SO(10) unification scenarios [9], one wonders whether there is a way to circumvent this problem. In addition, the little hierarchy is becoming more fine-tuned with the non-observation of SUSY. Since the SUSY breaking scale (Q S ) associated with stop mass is moving up, it becomes closer to the symmetry breaking scale (Q 0 ) where the Higgs squared mass turns negative by renormalization group equation (RGE). Q S is a dimensionful parameter while the smallness of Q 0 compared to the Planck scale is due to dimensionless parameters. The closeness of these two unrelated scales defines the fine tuning of the little hierarchy which is elevating with the non-observation of SUSY particles [10,11,12,13,14,15]. Both problems seem to be ingrained in the choice of number of Higgs doublets in the low energy theory 1 . In the context of SO(10) or SU(5) GUT models, more than one pair of Higgs doublets may exist in the full theory. The light pair of doublet arises by choosing smaller mass for one of the higher dimensional Higgs representations in the missing partner doublet-triplet splitting mechanism. However, more than one pair can easily be made light as well. We consider such a scenario and show that both problems can be solved, i.e., (i) Yukawa unification and (ii) less fine tuning in little hierarchy can be achieved in the context of 4 Higgs doublet (4HD) SUSY models arising from SO(10) or SU (5). We show that the Yukawa coupling unification can be realized for lower tan β, for which the threshold corrections are quite small. This paper is organized as follows. In section 2, we discuss missing partner model to The matrices A T and A D are determined by the masses of ∆ and Φ and their GUT-scale vevs, which depend on the SO(10) symmetry breaking vacua. The matrices B T,D and C T,D depend on the Higgs coupling H a ∆Φ and the GUT scale vevs. The matrices D D and D T are obtained by the mass term of 10-dimensional Higgs fields. If the mass of 10 are suppressed by a discrete symmetry (as discussed for the µ term in Giudice-Masiero mechanism [18]), one linear combination of the Higgs doublets remain light (at weak scale) while all the other linear combinations of doublets and triplets are massive at the GUT scale.
The Yukawa interactions to generate the fermion masses are given as The charged fermion Yukawa matrices are given by a linear combination of h a since the mixing of∆ u,d in the light Higgs doublets are tiny under the assumption above. The left-and righthanded Majorana neutrino masses can be generated by the f coupling. By investigating M −1 T , one finds that the f coupling does not contribute to the proton decay amplitudes and the dimension-five operators C ijkl L,R are the linear combination of h a ij h b kl . Therefore, compared to the minimal SO(10) model, though the predictivity of the neutrino masses and mixings is lost, the proton decay suppression is easier to be realized (in type II seesaw) by choosing the hierarchy pattern in h a (e.g., h 1 is a nearly rank-1 matrix, which gives top, bottom and tau Yukawa couplings, and 1st and 2nd generation masses and CKM mixings are generated by the other h a ). Surely, in this naive choice, the Georgi-Jarskog relations are not obtained. Instead of requiring four 10 Higgs fields, by adopting one 120 representation (which contains two triplets and two pairs of doublets) and two 10 fields, one can realize the same situation where only one pair of doublets is light 2 and the Georgi-Jarskog relations can be realized.
In the context of a minimal SO(10) model, the doublet-triplet splitting arises just by cancellation in the determinant of the doublet mass matrix, and one of the linear combination is fine-tuned to be light. In the missing partner doublet realization of the doublet-triplet splitting, on the other hand, the lightness of one pair of doublets is realized by the smallness of the mass of the 10 (and 120) Higgs representations, and in principle, there is no strong reason that only one pair of doublets is light since it just depends on the number of 10-dimensional Higgs fields. It is possible that multi-pair of Higgs doublets can be light in this scenario, which is true in the missing partner mechanism also in SU(5) and flipped-SU(5).
Here, let us consider the possibility that two pairs of Higgs doublets (totally, four Higgs doublets) remain light. There are two possibility depending on the number of excess of the triplet (3, 1, −1/3) compared to (1, 2, 1/2): 1. Two pairs of doublets are light, and one triplet (and one anti-triplet) Higgs is light.

Two pairs of doublets are light, and no triplet Higgs is light.
In the case 1, to avoid rapid proton decay, the Yukawa interaction to the fermions of the Higgs triplet needs to be very tiny (by the discrete symmetry or anomalous U(1) symmetry). Then, the Yukawa coupling of one of the linear combination of the Higgs doublets is absent. In We consider the consequence of the case 1 3 . Denoting that the linear combination of the Higgs doublets which couples to fermions asĤ 1u andĤ 1d and the other combinations asĤ 2u andĤ 2d (we call this as Yukawa-basis), we obtain the Yukawa terms (below the GUT scale) : The µ-term and the SUSY breaking Higgs mass terms are given as and Via RGE (with a large Y 33 u ), m 2 Hu 11 becomes negative and the electroweak symmetry is broken. Not onlyĤ 0 1u,d but alsoĤ 0 2u,d acquires vevs (denote them as v iu and v id ). We define a new basis (called as vev-basis): where tan +Y ij e ℓ i e c j (cos ζ d H 1d − sin ζ d H 2d ), and the fermion mass matrices are We find that the RGE running of the top, bottom and tau Yukawa couplings (whose description is easier in Yukawa-basis) for cos ζ u ≃ 1, tan ζ d ∼ 1 and tan β ∼ 30 resembles the running in MSSM for tan β ≃ 50. In other words, for tan β < ∼ 35 in the MSSM, the bottom Yukawa coupling is small and the RGE running is governed by the loop diagram arising from the gauge  Fig.1 for different values of cos ζ u , assuming that the third generation Yukawa couplings are unified at M U = 2 × 10 16 GeV. In Fig.2, we plot the bottom-tau mass ratio at M Z leaving out the weak scale threshold corrections as a function of cos ζ u . In the MSSM, it has been discussed that the bottom and tau unification is possible if tan β ∼ 2 or tan β ∼ 50. For tan β ∼ 2, the top Yukawa coupling is large and it can contribute to the RGE running of bottom Yukawa coupling, but, at present tan β ∼ 2 is excluded due to the 125 GeV Higgs mass. For tan β ∼ 50, the finite corrections and the TeV scale threshold corrections are large and it is difficult to realize the bottom-tau unification for the current bounds on the SUSY mass spectrum. Actually, the finite correction of q 3 b c H * u is important for large tan β: Naively, for a stop mass ∼ 2 − 3 TeV, A t has to be large (for the 125 GeV Higgs) which makes the chargino contribution is large (Xχ ± ≃ ). In 4HD case, there are additional contributions to X compared to MSSM if µ 12 (in vevbasis) is not zero. In the chargino loop (Higgsino component), µ 12 can directly contribute in the Higgsino propagator. In the gluino loop, there is a term y b (µ 11 − µ 12 tan ζ d )q 3b c H * 1u in the F -term, |∂W/∂H id | 2 . Therefore, if µ 12 is small, the contribution to X is similar to MSSM and the finite correction is not sizable for tan β which is not so large.
In summary, in the context of MSSM with 2HD, RGE running is important for the bottomtau unification for large tan β but the large finite correction associated with the non-observation of SUSY masses destroys the realization of the bottom-tau unification. In 4HD, however, the suitable RGE running can be realized even for tan β which somewhere in the middle where the finite corrections are not sizable, and as a result, top, bottom and tau Yukawa unification is possible in a simple manner. In the missing partner mechanism for the doublet-triplet splitting, the existence of two pairs of Higgs doublets with masses around the weak scale is not at all unnatural.

Minimization of the Higgs potential
In 2HD case, the Higgs potential of the neutral Higgs vevs is where m 2 1 = m 2 H d + µ 2 and m 2 2 = m 2 Hu + µ 2 . The Z boson mass (at tree-level) is written as The other minimization condition gives The symmetry breaking condition (which is equivalent to M 2 Z > 0) is m 2 1 m 2 2 − m 4 3 < 0. For a large tan β, we obtain M 2 Z ≃ −2m 2 2 and a cancellation between µ 2 and −m 2 Hu is needed (if |m 2 Hu | is large for a given boundary condition of SUSY breaking). It is often said that a smaller µ is preferable for "Natural SUSY" due to the tree-level relation. However, the Higgs mass parameters run by RGEs, and it is still unnatural if the RGEs give a large logarithmic correction to the mass parameters near the minimization scale (where the 1-loop correction of the scalar potential ∆V gives small derivatives ∂∆V /∂v u ≈ ∂∆V /∂v d ≈ 0). For example, if m 2 Hu runs rapidly, the radiative electroweak symmetry breaking is still sensitive to the SUSY breaking parameters (even if one tunes |m 2 Hu | to be small at a scale). Such a situation can be expressed by equations as follows: By Tailor expansion around the scale Q 0 , the Z boson mass relation can be expressed as where Q 0 is the symmetry breaking scale satisfying m 2 1 m 2 2 −m 4 3 = 0, and Q S is the minimization scale, which is roughly same as the geometric average of the stop masses. The RGEs of m 2 2 , m 2 1 and m 2 3 are given as One can find that ln Q 0 /Q S to needed to be tuned to be small (irrespective of the smallness of µ) if the stop masses and A t are large. Let us examine the "naturalness" of the little hierarchy in the case of 4HD. The Higgs potential in 4HD (in the Yukawa-basis) is given as where and m 2 u11 = (m 2 Hu ) 11 +μ 2 11 +μ 2 12 , and so on. The minimization conditions of the tree-level potential can be written as We note that M 2 Z (− cos 2β)/2 is an eigenvalue of the matrix: diag.(−1, −1, 1, 1)M 2 0 , and the corresponding eigenvector is (v 1u , v 2u , v 1d , v 2d ). The Z boson mass can be written as The symmetry breaking condition is det M 2 0 < 0. In this case, a large tan β (and cos ζ u ∼ 1) can be obtained by small m 2 u12 ,b 11 andb 12 . The symmetry breaking can arise when the determinant of the sub-matrix (M 2 0 ) ij (i, j = 2, 3, 4) is negative, and it is not necessarily true that a cancellation in m 2 u11 between −(m 2 Hu ) 11 andμ 2 11 +μ 2 12 needs to occur to obtain the little hierarchy. Surely, a cancellation is needed to make the magnitude of the determinant of M 2 0 small for the little hierarchy. The cancellation happens radiatively at Q 0 (by definition) and the important tuning quantity is the size of ln Q 0 /Q S . In 4HD case, we obtain In the Yukawa-basis, d(M larger at the lower energy. This can make to keep det M 2 0 ≈ 0 for a wider range of Q S compared to 2HD case. Roughly speaking, for a lighter stop mass ∼ 2 − 3 TeV, if the heavier Higgsino mass is O(10) TeV, one finds that the sensitivity for ln Q 0 /Q S is relaxed, and the little hierarchy is much less fine-tuned compared to the 2HD case.
In order to illustrate the above statement, let us rewrite Eq.(29) using a bold approximation. We neglect the terms which depend on cos β, and gaugino masses M 1 and M 2 . We also neglect the terms which depends onμ 12 andμ 21 , assuming that the Higgs mixing ζ u is mainly generated by SUSY breaking term, (m 2 Hu ) 12 , and that the dominant contribution from 2HD case is proportional toμ 2 22 . Then, we can write approximately as For example, suppose that mt L = mt R = 2 TeV and A t = 5 TeV. In 2HD case (which corresponds to sin ζ u = 0), we obtain ln Q 0 /Q S ≃ 0.003, and it means that m 2 1 m 2 2 − m 2 3 ≈ 0 is satisfied only in a narrow range, and Q S needs to be fine-tuned and to be very close to Q 0 . The approximate relation tells us that det M 2 0 ≈ 0 can be satisfied in a wide range if the heavier Higgsino mass is chosen to beμ and electroweak symmetry breaking can happen "naturally". One can find thatμ 22 ∼ 20, 30, 50 TeV for cos ζ u = 0.92, 0.96, 0.98, respectively. We note that the wino, bino and one of the Higgsino (and one of the charged Higgs (as well as the CP-odd neutral Higgs)) can be light (∼ 1 TeV) in the 4HD scenario, while the other one needs to be heavy to relax the sensitivity which appears in the 2HD case.

Conclusion
The doublet-triplet splitting problem is one of the major issue in the grand unified models. The missing partner mechanism is known to provide a solution to the problem. In principle, the number of the pairs of Higgs doublets is a free parameter in this mechanism, though one pair of Higgs doublets is the minimal choice and it is preferable for the gauge coupling unification which can have additional contributions from GUT thresholds and intermediate scales. In this paper, we have investigated the possibility that two pairs of Higgs doublets (i.e., four Higgs doublets, 4HD) remain at the TeV scale. In 2HD, the bottom-tau unification, which is one of the major implication of GUTs, does not appear to be successful after including the current experimental constraints from LHC. In fact, for a suitable parameter region of tan β where the RGE runnings allows us to generate bottom-tau unification, large threshold corrections from SUSY breaking are generated which ruin this unification. In 4HD, on the other hand, we find that the threshold corrections can be small even if the tree-level Yukawa coupling associated with bottom and tau are unified by RGE. This happens due to the freedom of the Higgs mixing terms at the TeV scale. It is possible to choose two pairs of Higgs doublet to be light and a linear combination of these two pairs acquire the vacuum expectation values by the minimization of the Higgs potential. The top-bottom-tau and bottom-tau Yukawa unifications are also implied in the context of SO(10) and SU(5) models respectively in this scenario. We also discuss the merits of 4HD compared to the 2HD choice for the little hierarchy between the SUSY breaking masses and the Z boson mass. The additional Higgs pair at O(10) TeV appears to relax the sensitivity of the radiative electroweak symmetry breaking.