Towards Realistic SUSY Grand Unification for Extended MSSM

Low-energy supersymmetric models such as MSSM, NMSSM and MSSM with vectorlike fermion are consistent with perturbative unification. While the non-minimal extensions naturally explain Higgs mass and dark matter in the low energy region, it is unclear how seriously they are constrained in the ultraviolet region. Our study shows that $i)$, In the case of embedding MSSM into $\rm{SU}(5)$, the fit to SM fermion masses requires a singlet $S$, which leads to unviable embedding of NMSSM into $\rm{SU}(5)$ because such $S$ feeds singlet $N$ a mass of order unification scale as well. $ii)$, Similar result holds in the case of embedding NMSSM into $\rm{SO}(10)$, where $S$ is replaced by some Higgs fields responsible for $\rm{SO}(10)$ breaking. $iii)$, On the contrary, for the embedding of MSSM with $16$-dimensional vectorlike fermions into $\rm{SO}(10)$, the Higgs field responsible for the vectorlike mass of order TeV scale can evade those problems the singlet $N$ encounters because of an intermediate mass scale in the $126$-dimensional Higgs field.


I. INTRODUCTION
At the frontiers of new physics, beyond the standard model (SM) natural or tera electron volt-scale supersymmetry (SUSY) offers us a grand unification (GUT) of SM gauge coupling constants [1][2][3][4]. Such natural SUSY hosts a lot of SUSY particles, which can be directly detected at the particle collider LHC or dark matter direct detection facilities such as Xenon-1T. Meanwhile, embedding these tera electron volt-scale SUSY models into the ultraviolet completions-SUSY GUT-may solve the long-standing issues such as the SM flavor puzzle and neutrino masses.
Nowadays, experimental data seem to oppose the minimal supersymmetric standard model (MSSM) from either the bottom or top viewpoint. In the former one, the 125 GeV Higgs mass [5,6] requires either a large mixing effect or soft masses of order 10 TeV for the stop scalars [7][8][9]. When the mixing effects among generations are significant, the constraints from flavor violation tend to require the SUSY mass order far above the weak scale. Moreover, the direct detection limits of dark matter [10,11] impose rather strong pressure on the scenario of neutralino dark matter. In the latter perspective, the minimal SUSY SU (5) referring to the MSSM is significantly constrained by the proton decay [12]. It requires the color-triplet Higgs mass of order GUT scale, which together with unification leads to the MSSM mass spectrum at least of order 100 TeV [13][14][15].
Therefore, it is of great interest to explore the MSSM with rational extensions that can resurrect the natural SUSY once again. Along this direction, there are at least two simple examples: the next-to-minimal supersymmetric standard model (NMSSM) [16] and the MSSM with vectorlike (VL) fermions (VMSSM) [17], which are consistent with unification [18,19]. While these extensions provide natural explanations of Higgs mass and dark matter in the low-energy region, it is unclear what the statuses of them are in the ultraviolet energy region. This is the main focus of this study.
In this paper, we discuss the embeddings of the MSSM, NMSSM and VMSSM into realistic GUT [20][21][22]. In each case, both the SUSY SU(5) [23][24][25] and SUSY SO(10) [26] representations will be explored. In these SUSY GUTs, we discuss the GUT-scale superpotential with the following features: (i) W Y and W SB are both renormalizable.
(ii) All of vacuum expectation values (vevs) are dynamically generated from W SB . (iii) All of SM matters and extended matters obtain their masses via the Higgs mechanism in W Y . Since W Y is fixed by the SM and the extra matters such as the singlet N or VL fermions at the tera-electron-volt scale, it is crucial to find suitable content of W SB that achieves the breaking of gauge group G GUT → G SM . In Sec. II, we explore the embedding of the MSSM into realistic SUSY GUTs, and useful conventions and notation will be introduced. The analysis on the MSSM is of great use to guide us towards the embeddings of the NMSSM and VMSSM. Sections III and IV are devoted to studying the embeddings of the NMSSM and VMSSM into realistic SUSY GUTs, respectively. Finally, we conclude in Sec. V.

II. BENCHMARK MODEL: MSSM
In the minimal SU(5), SM fermions of each generation are assigned as 1 for right-hand neutrino N R ,5 (ψ) for L and down quark d, and 10 (Φ) for Q, up quark u and e, respectively, whereas in the SO(10) representation, the SM fermions of each generation are embedded into a 16dimensional representation, which decomposes as 16 ¼ 1 þ5 þ 10 under the SU(5).

A. SU(5)
The Yukawa superpotential W Y in Eq. (1) contains two parts, which refer to SM fermions without neutrinos and neutrinos, respectively. According to the product5 × 10 ¼ 5 þ 45 and 10 × 10 ¼5 s þ45 a þ50 s , where subscripts s and a refer to symmetric and antisymmetric, respectively. With the Higgs representations composed of 5,5, and45 [27], W where a, b, c, etc., denote the SU(5) indices, i and j are the generation indices, and Y u;d;45 are Yukawa matrices. Note that Φ ab is an antisymmetric field and H a bc ¼ −H a cb . The reason to include45 is clear in the SM fermion mass matrices as derived from Eq. (3), where υ 5 u and υ¯5 d is the vev of the Higgs doublet H u and H d in 5 and5, respectively, and υ 45 is the vev of doublet σ d in 45, which is defined as hH b5 a i ¼ υ 45 ð1; 1; 1; −3Þ diag for The GUT-scale mass relations in Eq. (5) strongly constrain the Yukawa matrices Y u;d; 45 . For example, some specific choices of Y u;d;45 in the Georgi-Jarlskog scheme [27] lead to a stable b quark. To solve the SM flavor issue, we choose the Fritzsch scheme [28], in which Y u;d;45 take the forms where there are small mass hierarchies C f ≫ B f ≫ A f with f ¼ fu; d; eg in Eq. (6) so as to address the SM flavor mass hierarchies. Substituting Eq. (6) into Eq. (5) implies that there is a fine-tuning between B d and D d , With this fine-tuning solution, the diagonalizations of matrices in Eq.
which is in good agreement with experimental data. For W Y ν responsible neutrino masses, we take a simple form, where S is a singlet of G SM ¼ SUð3Þ c × SUð2Þ L × Uð1Þ Y , with hSi of order GUT scale. In Eq. (9), one finds the neutrino Dirac mass M ν ¼ Y ij N υ 5 u and the right-handed neutrino mass M N R ¼ Y ij S hSi, which results in the left-hand neutrino masses in terms of the type-I [29][30][31][32] seesaw mechanism, Given υ 5 u ∼ 10 2 GeV, hSi ∼ 10 15 − 10 16 GeV, and Yukawa couplings of order unity, the neutrino mass is of order approximately 10 −2 − 10 −3 eV. Similar to Y u;d;e , Y S and Y N in Eq. (9) are also constrained by the fit to neutrino mixings as described by the Pontecorvo-Maki-Nakagawa- where U ν and U e are defined to diagonalize mass matrices m ν and M e , respectively. Now that we have established a benchmark solution 1 to the input parameters at the GUT scale that can explain the SM flavor issue and neutrino masses, we turn to the structure of W SB : 1) To obtain light neutrino masses, hSi of order GUT scale is required. 2). To break the SU(5), we introduce 75ðZÞ. With a 75, we can add a 50 and50 to achieve the doublet-triplet splitting [33] for 5 and5. 3). To gain nonzero vev υ4 5 d , we include another 75ðZ 0 Þ with a vev of the GUT scale. The reason for this is that neither 1 nor 24 with a large vev is favored by the product Hð5Þ ×Hð5Þ ¼ 1 þ 24. 4), Because of the singlet S, there is an unsafe operator, which must be eliminated. Shown in Table I is the Z 2 × Z 0 2 parity, which can eliminate the unsafe operator in Eq. (11). Under this parity, W SB reads as According to the F terms in Eq. (12), the nonzero singlet vevs hð1; 1; where and A few comments are in order regarding the parity assignments. First, the Z 2 parity eliminates the unsafe operator in Eq. (11). Second, without Z 0 2 , a large μ term for the doublets in 5 and5 would be induced by the mixings with 45 and45. Instead, imposing Z 0 2 forbids the operator Z 0 Hð5ÞHð45Þ, which then keeps the doublets in 5 and5 light. Finally, due to the last line in Eq. (12), which is consistent with Z 2 × Z 0 2 , the effective operator Hð5ÞHð45ÞZ 02 =M Z is produced after integrating Z. Thus, the effective superpotential for the doublets in 45 and45 at the leading order is given by where corrections due to those mixings among singlets of Z, Z 0 , and S have been neglected. From Eq. (15), we obtain the vev Given λ 3 ∼ 0.1 and υ d ∼ 10 GeV, we have σ d ∼ 100 MeV.
A rational Yukawa texture such as Y 45 ∼ ð0.01; 0.1; 1Þ for TABLE I. Z 2 × Z 0 2 parity assignments in the case of embedding the MSSM into SU (5), which are consistent with the superpotentials in Eqs. (2) and (12).
Field N R ψ Φ 5545 1ðSÞ 1ðS 0 Þ 75ðZÞ 75ðZ 0 Þ 45 5050 It is of special interest to examine whether there is a viable solution to the input parameters in the case in which all mass matrices such as M u;d;e;N R are assigned the Fritzsch form. the three generations then reproduces the mass relations in Eq. (5).

B. SO(10)
Unlike the case of SU(5), the input parameters in the SO(10) that control the SM fermions masses M u;d;e and neutrino masses m ν are tied to each other. The main reason for this is that the MSSM matters of each generation are contained in a single 16ðϕÞ. Here, we give a brief review of the embedding of the MSSM into SO (10).
We employ the solution of modifying W SB [48], in which the Higgs fields are composed of 210ðYÞ, Hð126Þ,Hð1 26Þ, and 54ðXÞ, and W SB takes the form 2 Here, a few comments about W SB are in order: 1). Under the notation of SUð4Þ × SUð2Þ L × SUð2Þ R , the SM singlet vevs

III. NMSSM
With the embedding of the MSSM into realistic SUSY GUTs as a benchmark, in this section, we analyze the NMSSM. According to the starting points in the Introduction, a viable embedding should satisfy two constraints: (i) The mass of N should be of order tera-electronvolt scale. (ii) The vev of N should be of order tera-electronvolt scale. Both of them may be spoiled by a few dangerous mixings between N and Higgs fields that contain a singlet vev of order GUT scale. The key point is whether there is suitable symmetry to avoid such mixings.

A. SU(5)
In this situation, W Y in Eq. (2) should be extended to include NH u H d þ κ 3 N 3 þ H:c: in the NNSSM, which means that 3 Equation (23) does not affect the fit to SM flavor masses and mixings in Sec. II A. Nevertheless, compared to the MSSM, W SB is allowed to contain superpotential terms These new terms in Eq. (24) yield corrections to the F terms of Z, Z 0 , and S such as F S ¼ F MSSM S þ 2λ N SN, which can be adjusted to the case of the MSSM by, e.g., hNi ¼ 0. Even so, the singlet vevs hZi, hZ 0 i, and hSi still lead to either large N mass or large mixing.
To avoid all of the mixing terms in Eq. (24), we need to impose new parity. The first observation is that an odd N under a Z 2 parity as shown in Table I excludes the first three terms in Eq. (24). But the last term therein still remains. 4 A similar result holds for Z N or an Abelian symmetry.

B. SO(10)
Similar to the embedding of the NMSSM into SU(5), W Y in Eq. (17) is modified by Instead of Eq. (24), W SB in Eq. (21) is allowed by gauge invariance to contain in which N mixes with the SM singlets of Y, 126,1 26, and X. Thus, all of Yukawa couplings in Eq. (27) have to be extremely small. What kind of parity allows Yukawa superpotential in Eq. (26) but eliminates that in Eq. (27) simultaneously? The first observation is that a Z 2 parity does not work, since Eq. (27) would imply that N is an odd field, which contradicts with the Yukawa superpotential in Eq. (26). Similar results hold for any Z N parity. Because the rational assignment n Y ¼ 0 as required by successful symmetry breaking implies that n 1 0 ¼ n 1 26 ¼ n1 26 ¼ N=2 in order to allow the Yukawa superpotential in Eq. (26). Accordingly, n N ¼ 0 from N10ðHÞ10ðHÞ, which implies that some of the terms in Eq. (27) are still allowed. To conclude, in our setup, embedding the NMSSM into the minimal SO(10) is not viable.

IV. VMSSM
Let us proceed to discuss the embedding of the VMSSM into SUSY SU(5) and SO (10). The VL fermions with mass of order TeV scale can be composed of 5 with5, 10 with10 in the SU(5), or 16 with16 in the SO(10) [18,19]. A realistic embedding should satisfy the following constraints: (i) The vev of the Higgs field ρ responsible for the VL fermion masses should be of order tera-electronvolt scale. (ii) The VL fermions are prevented from directly coupling to the Higgs fields which trigger high-scale gauge symmetry breaking. Violating the first constraint is likely to occur because either ρ ¼ f1; 24; 75g or ρ ¼ f1; 45; 210g may directly couple to S,S 0 Z, Z 0 in the case of SU(5) or X, and Y in the case of SO (10), which tends to yield hρi of GUT scale. In contrast, hρi of order tera-electron-volt scale can be only realized by the effective operator such as where A, B, Á Á Á, refer to Hð5Þ,Hð5Þ, S, S 0 , Z, and Z 0 in the SU(5) or Hð126Þ,Hð1 26Þ, X, and Y in the SO(10), with M U denoting the GUT scale.

A. SU(5)
For the VL fermions of 5ðΣÞ and5ðΣÞ, W Y in Eq. (2) is extended by where ρ ¼ f1; 24g of SU(5). The reason for adding ρ is that either singlet vev S or S 0 in Sec. II A is too large to provide a VL mass of order tera-electron-volts.  (5), which is consistent with the superpotential in Eq. (29). Yet, This parity is unable to exclude all the unsafe structures in Eq. (31).
Field N R ψ Φ ΣΣ 5545 ρ 1ðSÞ 1ðS 0 Þ 75ðZÞ 75ðZ 0 Þ 45 5050 Operator NHð5ÞHð5Þ contributes to Yukawa interaction NH cHc beyond the MSSM, with H c andH c being the colortriplet Higgs fields. However, it does not affect proton decay at all, as the singlet N mass is always far larger than proton mass. 4 An economic solution to keeping light N is adding another singlet S 0 ¼ 1 that is even under the Z 2 . With such an S 0 , the W SB is further extended by The two different contributions to the N mass cancel each other, leaving us a light N. Unfortunately, neither Abelian or Z N parity can ensure M S 0 ¼ m SS 0 .
In this case, the unsafe superpotential at least includes which can be excluded by imposing the Z 2 parity assignments as shown in Table II. W unsafe in Eq. (30) can also contain the following terms depending on ρ: Besides the unsafe operators in Eq. (31), there are also no suitable Feynman graphs to generate the desired effective operator with correct mass order in Eq. (28). In principle, the form of effective operator in Eq. (28)  B. SO (10) As shown in Ref. [19], the nonminimal extension through the 16-dimensional VL fermions remains consistent with the SO(10) unification. In this model, W Y is modified by  (10), which is critical to solving the problem.
We first consider ρ ¼ 210. We add Higgs fields 54ðVÞ and 54ðUÞ to W SB , with the Z 2 parity assignments as shown in Table III. The Z 2 parity excludes the unsafe operator and simultaneously allows Yukawa interactions In terms of Eqs. (34) and (21), the effective operator for ρ is given by For calculating an effective superpotential in the infrared region from those in the ultraviolet region, integrating out heavy chiral superfields in the Feynman graphs is equivalent to solving the nonlinear equations of F terms related to these heavy chiral superfields. The leading-order terms with coefficient λ 0 10 U þ λ 00 10 V ¼ δF Y in Eq. (35) are obtained after integrating out superfield Y for λ 1 and λ 10 less than unity. Similarly, the next-leading-order operators therein are induced by further integrating out X, referring to which the Feynman graph is shown in Fig. 1. Note that we have used the mass term in Eq. (35) representing those quadratic terms and neglected the higher-order terms.
Apart from the F-term contributions in Eq. (35), the potential for the singlet component in ρ also contains soft SUSY-breaking terms such as A ρ ρ 3 þ H:c: The scale of A ρ depends on the details of SUSY breaking. It can be neglected in gauge mediation but be of order approximately 1 TeV in the other scenarios. In the latter case, it is easily to verify that the F-term contribution still dominates over TABLE III. Z 2 parity assignments in the VMSSM model with ρ ¼ 210, which are consistent with the superpotentials in Eqs. (17), (21), (32), and (34) and simultaneously avoid the unsafe operator in Eq. (33).
Field ϕ ΔΔ 101 26 ρ X 126 54ðVÞ 54ðUÞ Y Super-Feynman graph for the generation of the higherdimensional effective operator in the case of ρ ¼ 210.
those soft terms. In what follows, we simply ignore those soft terms for the estimate of hρi.
The leading-order contribution in Eq. (35) has to be eliminated if one wants to obtain singlet vev hρi ∼ TeV scale. Alternatively, the coefficient δF Y must be suppressed without much fine-tuning. There is a dynamical realization for this purpose. From Eq. (34), one finds that the SUSY vacuum described by the vevs in Eq. (22) remains only if the constraints Given singlet vev υ 126 s ≃ υ1 26 s ∼ 10 13 GeV fixed by the fit to SM flavor masses [48] and M ρ ∼ M U ∼ hVi ∼ 10 16 GeV, we have hρi ∼ 1-10 TeV.
The analysis for ρ ¼ 210 can shed light on other cases such as ρ ¼ f1; 45g. For ρ ¼ 1, we can naively choose Z 2 odd fields U ¼ V ¼ 210 in Fig. 1. However, an unsafe operator VðUÞΔð16ÞΔð16Þ appears again. For ρ ¼ 45, one may choose U ¼ 45 and V ¼ 45 or 54, which is unfavored by an operator similar to ρ ¼ 1.

V. CONCLUSION
According to the observed Higgs mass at the LHC and the dark matter direct detection limits, the conventional MSSM-the simplest natural SUSY that is consistent with unification-is under more pressure than ever. Such stress can be greatly relaxed in the extended MSSM models such as the NMSSM and VMSSM, which retain the unification and are still simple. In this paper, following the assumptions that W Y is fixed by the SM matter content and its tera electron volt-scale extension and that they receive their masses from W SB through the Higgs mechanism, we have studied the embeddings of these three models into SUSY GUTs.
First of all, we discussed the MSSM, in which the realistic SU(5) and SO(10) realizations serve as benchmark solutions to the SM flavor issue and neutrino masses. Then, we utilized the benchmark MSSM as guidance for the embedding of the NMSSM and VMSSM. We found that the embedding of the NMSSM is not viable due to a large amount of mixings between the singlet N and the Higgs fields responsible for the GUT symmetry breaking. But the problem can be evaded in the VMSSM because the Higgs field ρ that provides 16-dimensional VL mass of order teraelectron-volt scale can avoid the same problems the singlet N encounters due to the intermediate mass scale in the 126dimensional Higgs.