Goldstone modes in renormalizable supersymmetric SO(10) model

We solve the Goldstone modes in the renormalizable SUSY SO(10) model with general couplings. The Goldstones are expressed by the Vacuum Expectation Values and the Clebsch–Gordan coefficients of relevant symmetries without explicit dependence on the parameters of the model.

Spontaneously Symmetry Breaking (SSB) generates the massless Goldstone bosons corresponding to the broken generators [16][17][18]. In gauge theories, these Goldstone bosons act as the longitudinal components of the gauge bosons. In model buildings, we need to check the spectra to verify the existence of these Goldstone bosons. When more than one Higgs multiplets contribute to the SSB, we need to check that there are massless eigenstates in the mass squared matrices with correct representations. a e-mail: 15012100098@pku.edu.cn b e-mail: dxzhang@pku.edu.cn In SUSY models, there are easier ways to get the Goldtone bosons. The Goldstinos, which are the fermionic partners of the Goldstone bosons, are also massless which mix with gauginos, the SUSY partners of the gauge bosons. Then, we can get the Goldstinos by solving the massless eigenstates of the mass matrices. Being the SUSY partners, the Goldstone bosons are determined accordingly.
In this work, we will study the Goldstone modes within SUSY. In [37] it has been realized that for the SSB of U (1) symmetries in SUSY models, in the Goldstone mode a component is proportional to the charge and to the Vacuum Expectation Value (VEV). Furthermore, it has been shown in the Non-Abelian cases [38], that in any Goldstone mode, the component from a representation is proportional to the VEV from the same representation and the CGCs determine the remainder dependence. Examples on the simplest SU (2) SSB are given in [38]. Here we will show test explicitly in the SUSY SO(10) model. We will give the Goldstone modes for the SSB of SO (10) into G 321 in the general renormalizable model.
In Sect. 2 we will give a general discussion on the method to solve the Goldstone modes. Then, in Sect. 3, we associate in SO (10) the Goldstones and the corresponding broken symmetries. In Sects. 4 and 5, Goldstones associated with the SSB of G 422 and SU (5) are solved, respectively, and relevant identities among the CGCs are examined. In Sect. 6, Goldstones associated with the SSB of the flipped SU (5) are solved. The relevant CGCs, which are not available in the literature, are presented and the identities are examined. The mass matrix for the Goldstones is given in the Appendix. Finally we summarize in Sect. 7.

General SSB in SUSY models
Following [38], we consider the SSB of G 1 → G 2 in SUSY models. For models with only real fields, the general Higgs superpotential can be written as where I, J, K denotes the superfields, i, j, k denote the representations of group G 1 . Under G 2 , the couplings in (1) are of the forms where a, b, c are the representations under G 2 , and C I J K i a j b k c is the Clebsch-Gordan coefficient (CGC). A singlet of G 2 , denoted by I i 1 etc., can have a VEVÎ i 1 in the SSB. SUSY requires the F-flatness conditions for the singlets, The Goldstinos in the SSB will be denoted by α, α under G 2 . A mass matrix element for the Goldstinos is For G I iᾱ denotes the element of the Goldstino corresponding toᾱ, the zero-eigenvalue equation is Eliminating M I in (3) and (4) gives It follows that if a representations of G 1 does not contain a G 2 singlet, even if it may contain representations of the Goldstinos, it does not contribute to the Goldstino modes since the M I term in (4) cannot be eliminated unless multiplied by zero.
The superpotential parameters can be arbitrary so that we can focus on a specified coupling λ I J K , and the summation over different λs is unnecessary. Furthermore, denoting The nonzero λ I J KĴ j 1K k 1 can be eliminated and we can reiterate the same operation for J, K , then Similar result is given for the Goldstino in α. (7) is followed by an identity that the determinant of the square matrix in (7) is zero, which must hold for any SSB. Some special cases need to be clarified now.
(1) If k does not contain α,ᾱ, (7) is simplified into which gives an identity that the determinant of the square matrix is zero, and a simple ratio between the two components. Taking all different couplings λs setup all relations among the Goldstino components. This determines the Goldstino content up to an overall normalization.
(2) For J = K and j = k, we have (3) For I = J = K and i = j = k, the result is without information on T . Generalization to the models with complex fields is straightforward with the general results given in [38]. We have also identities among the CGCs and equations for the Goldstinos involving the CGCs. Two types of special superpotentials might be relevant in the SO(10) study.
(4) For I, I , K contain Goldstones and K is real, if where X = C I I K Furthermore, in case that I i (I ) contains only α (ᾱ) but not α (α), we have We summarize in this section that the Goldstino and thus the Goldstone components are proportional to the VEVs, and the remaining determinations of these components need only the calculations of the CGCs. This is the approach which we will use to determine the Goldstones in the following sections, and the identities of the CGC determinants will be used to check the consistencies of these calculations.

SSB and Goldstones in SO(10)
We study the most general renormalizable couplings containing Higgs H (10), D(120), (126) + (126), A(45), E(54) and (210) in the SUSY SO(10) models. The most general renormalizable Higgs superpotential is [29] When we study the SSB of a SO(10) subgroup G 1 into G 2 , we firstly decompose the SO(10) representations into G 1 representations, and the couplings are now of the form where, for example, I i stands for the representations i of G 1 from SO(10) superfield I . Then, under G 2 , i, j, k are further decomposed, (15) is where I i a is a representations a of G 2 coming from I i , and C I J K i a j b k c is the SO(10) CGC. Most of these CGCs have been given [29,36] so that no separate calculations of C I J K i jk and C i jk abc are needed. A special kind of CGCs relevant for the SSB of the flipped SU (5) symmetry will be given later.
To study SSB of SO (10) into G 321 , it is necessary to decompose the Higgs representations of SO(10) under the G 321 subgroup. There are 45 − 12 = 33 Goldstone modes. We link the Goldstones of a SM representation with the SSB of a specific subgroup of SO(10), as is summarized in Table 1. In Table 1, the first three modes can be studied by the SO(10) CGCs using G 422 as the maximal subgroup [29], the (3, 2, − 5 6 ) + c.c. modes need the CGCs using SU (5) [36], and (3, 2, 1 6 ) + c.c. using G 51 CGCs which will be given below.

Goldstone modes for SSB of G 422
Under G 422 , the following representations contain the SM singlets (1, 1, 0) whose VEVs will be denoted in the same symbols. Obviously, E and 1 contain no Goldstones relevant for the G 422 breaking, A 1 contains Goldstone components relevant for the SU (2) R breaking, A 2 , 2 for the SU (4) C breaking, and v R , v R , 3 for both. The needed CGCs are taken from [29] and are summarized in Tables 2,  3 and 4.  Table 2 CGCs for the SM singlets (1, 1, 0) under G 422 SSB. These SM singlets also correspond to Goldstones (1, 1, 0) in the SSB of Hereafter, the CGCs for A are understood as those for 1 120 A [29]. Here, for example, A 1 stands for A and has been studied in [37]. In the renormalizable models, only (126) + (126) have the SM singlets with nonzero U (1) I 3R , U (1) B−L charges. Furthermore, if we include also fields in the spinor representations (16)+ (16) of SO (10), which are usually used in building non-renormalizable models and have halves of the charges of (126) + (126), the Goldstino and thus the Goldstone mode is found to be where N is a simple normalization factor. Equation (18) tell us two important conclusions in the U(1) symmetry break- ing. First, the component of a field in the Goldstone mode is proportional to its charge under the breaking U (1) and to its VEV. Second, there is no dependence on the superpotential parameters of the model, besides through the VEV determinations indirectly.
In the model (14), we have up to an obvious normalization.
Note that D has no VEV, the Goldstone corresponding to Here A 1 . . . are the VEVs, A, . . . are the fields and T s are what will be solved. We will first classify the couplings and discuss their results, both on solve the Goldstones and on the identities among the CGCs.
(1) The representations E, A 2 , 1 , 2 have no (1, 1, ±1) under G 321 , so that couplings of them with two same representations, such as A 2 1 E or 2 3 1 , do not give relations among the T s. These couplings lead to relations of CGCs such as which are trivial.
(2) Any coupling of E, A 2 , 1 , 2 with two different representations will set up a relation of the two T s. Following where the determinant is zero, so The coupling A 1 3 2 also gives couplings with all three representations contain the Goldstone modes. However, v R contains only α while v R contains only α. Following (13), the coupling 3 Altogether, we have consistently with an obvious normalization.
They are the Goldstones for SU (4)  Again, D does not have a VEV, the Goldstone mode is (1) According to (10), 3 2 leads to 0 = C 2 2 2 (2) A 1 , E, 1 does not contain α or α and are SU(4) singlets, their couplings with the same fields, A 2 2 E, 2 3 E, 2 3 1 lead to only trivial results. In the couplings of one of them with two different fields, according to (8), they give relations between two Goldstone components. A 2 3 A 1 gives (3) v R contains only α while v R contains only α. According to (13) (4) According to (9) or : Similarly, 2 A 2 2 gives or : gives  (24) etc.  All together we have

The Goldstone corresponding to
.

Summary
We have studied the Goldstone modes of SSB in the SUSY SO(10) model with general renormalizable couplings. VEVs and CGCs determine the contents of these Goldstone modes while the parameters of the model are irrelevant. Identities among the CGCs are examined to be in accord with the general conclusions in [38].
using G 51 as the maximal subgroup of SO (10). This matrix is transformed from [29] through (25)