Light Pseudo-Goldstone Higgs Boson from SO(10) GUT with Realistic Phenomenology

Within the supersymmetric SO(10) grand unified theory (GUT), a new mechanism, giving the light Higgs doublet as a pseudo-Goldstone mode, is suggested. Realizing this mechanism, we present an explicit model with fully realistic phenomenology. In particular, desirable symmetry breaking and natural all-order hierarchy are achieved. The constructed model allows one to have a realistic fermion pattern, nucleon stability, and successful gauge coupling unification. The suggested mechanism opens prospects in the field for a novel $SO(10)$ GUT model building and for further investigations.

symmetry breaking is achieved, and the MSSM Higgs doublets are light pseudo-Goldstones, even after taking into account all allowed high-order operators. The model I construct offers a realistic fermion pattern, nucleon stability, and successful gauge coupling unification. Because of these issues (usually turning out to be severe problems for various GUTs), I find the presented SO(10) model to be quite successful.
The mechanism proposed in this paper opens the door for future work to build varieties of PGB SO(10) models and investigate their phenomenological implications.

II. THE SETUP AND THE MECHANISM
First, I describe the PGB mechanism within the SUSY SO(10) GUT and then, on a concrete model, demonstrate its natural realization. The minimal superfield content, ensuring SO(10) → SU (3) c ×SU (2) L ×U (1) Y ≡ G SM breaking, is {45 H , 16 H , 16 H } [and possibly some additional SO(10) singlet states]. Thus, the symmetrybreaking scalar superpotential W H consists of three parts: and ν c H , the breaking U (16) → U (15) happens and pseudo-Goldstone states emerge. Clearly, those modes (at least some of those) would gain masses by including the terms of W (45, 16) H . Our goal is to see if it is possible to arrange the couplings such that MSSM Higgs doublets emerge as light pseudo-Goldstones while all remaining states acquire large masses. It turns out that in this setup this is indeed possible.
Assume that the 45 H has a VEV towards the With this breaking, the 30 states (from 45 H ) absorbed by superheavy gauge fields, become genuine Goldstones. These are the fragment (6,2,2) [given in (2) (3), the second and last lines possess U (4) and U (12) symmetries, respectively. The U (4) global symmetry, which has leptonic states, is spontaneously broken down to U (3) by the VEVs ν C H = ν C H = v R . Because of U (4) → U (3) breaking the seven Goldsones emerge. Among them, three -one SM singlet and the pair e c H ,ē c H -are genuine Goldstones absorbed by the coset [SU (2) R × U (1) B−L ]/U (1) Y gauge fields. The states l H andl H , having the quantum numbers of the MSSM Higgs superfields, emerge as light (as desired) pseudo-Goldstone modes. The remaining fragments, given in the last line in (3), will be heavy.
However, for this mechanism to be realized, some care needs to be exercised. The λ coupling term, with nonzero W 45H 16H 16H S S1 10H 10 ′ H 16i i.e., smallness or absence of any coupling will follow from the symmetries (we will invoke).

III. MODEL WITH ALL-ORDER NATURAL HIERARCHY
Realizing the mechanism described above, we augment the SUSY SO(10) GUT by U(1) A × Z 4 symmetry, where U(1) A is an anomalous gauge symmetry and Z 4 is discrete R symmetry. For symmetry breaking and all-order DT hierarchy, the anomalous U(1) A was first applied in Ref. [17] [within the PGB SU (6) GUT] and was proven to be very efficient for realistic SO(10) model building [9,12]. Besides the states 45 H , 16 H and 16 H , we introduce two SO(10) singlet superfields S and S 1 , which will be used in superpotential W H of Eq. (1). In Table I, I display the U(1) A × Z 4 transformation properties of these fields together with other states (introduced and discussed later on). Note that, under Z 4 symmetry, the whole superpotential transforms as W → −W . Thus, the relevant superpotential couplings are With anomalous U(1) A , having the string origin, the Fayet-Iliopoulos term d 4 θξV A is always generated [18], and the corresponding D-term potential is At the last stage in (5), we have taken into account that ν c H = ν c H ≡ v R , which ensures the vanishing of all D terms of SO (10). With ξ > 0 and looking for the solution of S 1 ≪ v R , the condition D A = 0 from (5) gives Next, we investigate the symmetry-breaking pattern from the superpotential couplings given in (4) and imposing F φi = 0, from the superpotential couplings (4), taking into account that v R is fixed as (6), we obtain From (8)- (10), one can see that the desirable VEV configuration is obtained. Indeed, from (8), with v R ∼ 10 17 GeV [fixed from (6)] and Λ = (10 13 − 10 14 ) GeV, we obtain S1 M * ≃ 10 −8 −10 −6 . Thus, an effective, and suppressed, λ coupling (ensuring suppressed Higgs mass) is generated. Small S 1 also ensures naturally suppressed V R (in a limit S 1 → 0, one has the solution V R → 0). At the leading order, in powers of S1 M * , from (8)-(10) we get For the masses of l H andl H , coming from the 16 H and 16 H , we have (in the Appendix, I present the decomposition helping to compute the masses of this and remaining fragments) The latter's value, i.e., the µ term, with v R ∼ V B−L and for Λ vR ∼ (1 − 2)·10 −4 will be ∼ 1 − 10 TeVof desirable magnitude. Allowed higher-order operator The masses of colored modes (from 16 H , 16 H ) are around ∼ 10 8 GeV. Below, I show how MSSM Higgs doublets can have desirable couplings to the MSSM matter and how nucleon stability is ensured [19]. Before discussing these, I give the scalar sector's extension, which ensures desirable Yukawa couplings and realistic phenomenology. I introduce two scalar superfields in the fundamental representation (10-plets) of SO (10) In (15), the O(TeV) stand for possible corrections of the order of (1−10)TeV (I will comment shortly) and are harmless. From (15), we see that one doublet pair, identified with MSSM Higgs superfields h u and h d , is light. The remaining doublets (denoted as D 1,2 andD 1,2 ) are heavy. Analyzing the matrix (15), the distribution of h u and h d in original states can be found: the composition crucial for building a realistic fermion pattern (especially for large top Yukawa coupling λ t ).
As far as other operators, allowed by symmetries, are concerned, the couplings SS145H , will contribute to the µ term by an amount of ∼ few TeV and therefore are harmless [20].

IV. YUKAWA COUPLINGS AND PROTON STABILITY
All MSSM matter is embedded in SO(10)'s 16 i -plets (i = 1, 2, 3). Their U(1) A ×Z 4 charges are given in Table  I. The effective operators, generating up-and down-type quark and charged lepton masses, are The first term is responsible for the up-type quark Yukawa couplings. According to (17), the 10 H -plet includes the h u by the weight |x|, which with |λ ′ V R | ∼ |M | ∼ |M ′ | can naturally be |x| ≈ 0.5 − 1. Therefore, the λ t coupling can naturally have a desirable value.
On the other hand, the h d entirely resides in the 16 Hplet, and down quark and charged lepton masses are emerging from the second term of (18) generates a term with y iα coupling above. I will not discuss here more details (should be pursued elsewhere), but emphasize that after the GUT symmetry breaking some fragments from 10 f , 16 f and 16 f can have masses below the GUT scale and their contribution, as thresholds, may play an important role for the gauge coupling unification (discussed at the end).