Extended MSSM in Supersymmetric $\rm{SO}(10)$ Grand Unification

We apply the perturbative grand unification due to renormalization to distinguish TeV-scale relics of supersymmetric $\rm{SO}(10)$ scenarios. With rational theoretical constraints taken into account, we find that for the breaking pattern of either $\rm{SU}(5)$ or Pati-Salam only extra matter $\mathbf{16}$ supermultiplet of $SO(10)$ can appear at TeV scale, apart from MSSM spectrum.


I. INTRODUCTION
The discovery of standard model (SM)-like Higgs [1,2] provides a new portal to TeV-scale new physics at the LHC in the forthcoming years. Among other things, such new physics models may reveal the "nature" of SM-like Higgs, and offer a novel mechanism to stabilize divergence involving SM Higgs. For those interesting scenarios in the literature, in this paper we are restricted to the idea of supersymmetry (SUSY). Specifically, we will utilize the grand unification (GUT) [3], which is one of the most beautiful features delivered by SUSY, to distinguish TeV-scale relics of SUSY GUT models. For reviews on this subject see e.g. [4,5].
In the viewpoint of unification, the minimal supersymmetric standard model (MSSM) can be embedded into conventional SU (5) [6][7][8], SO(10) [9,10] or other GUT models with gauge groups of higher ranks. In the light of our previous study on SU (5) [11], we will continue to explore the TeV-scale relics of SUSY SO(10) unification. Comparing with SU (5), the low-energy effective theories of SO(10) are more complex. The first major reason is that there may be multiple intermediate scales between the weak and GUT scale. The second reason is that since a lot of higher-dimensional representations of SO(10) trivially satisfy gauge anomaly free condition, the constraint imposed by this condition is much weaker in SO (10). Earlier studies on low-energy effective theory which is consistent with perturbative SUSY SO(10) unification are based on specific motivations such as Higgs mass [12] and neutrino physics [13][14][15].
Instead of particular phenomenological concerns, we will take a systematic analysis on the low-energy effective theory. In order to simplify the analysis on extra matter beyond the MSSM spectrum, we will explore SO(10) scenarios with the following theoretical features.
• In the chain of gauge symmetry breaking where G SM refers to the SM gauge group. When the Higgs component fields responsible for two * Electronic address: sibozheng.zju@gmail.com nearby steps of gauge symmetry breaking can be contained in a single Higgs supermultiplet, these two Higgs supermultiplets will be identified as the same one. Otherwise, they differ from each other 1 .
• Theoretical constraints above have been partially imposed in the literature to our knowledge. However, they have never been combined together to derive a systematic analysis on the low-energy effective theory. The paper is organized as follows. In Sec.II, we discuss the extra matter supermultiplets which are consistent with our starting points in two well known patterns of gauge symmetry breaking. In Sec.III, we examine the perturbative unification with these representations. Finally, we conclude in Sec.IV.

II. REPRESENTATIONS
According to our starting points, in this section we investigate the representation of extra matter which can directly couple to SM Higgs 10 H in the following two patterns of gauge symmetry breaking, Pattern A [16][17][18][19] is a two-step breaking with SU (5) subgroup, and pattern B [20][21][22] is a three-step breaking referred to Pati-Salam model [21].
Note, in the MSSM the SM fermion matters are described by 16 i of SO(10) with index i = 1 − 3, and the SM Higgs is contained in the 10 H of SO (10). In particular, 16 i contain three-generation right-hand neutrinos, whereas 10 H is composed of 5 H and5 H of SU (5) which contain the two Higgs doublets of MSSM and two colortriplets.
A. SU (5) In this pattern of symmetry breaking H 1 should contain an SU (5) singlet, there are two candidates H 1 = {16, 126}. The second Higgs H 2 should contain a 24 of SU (5), which corresponds to three potential choices H 2 = {45, 54, 210}. Since H 1 = H 2 , we take the rational that the splitting between these two broken scales is large.
With potential assignments on H 1 and H 2 above, there are six sets of combinations. In each case, there may exist four types of dangerous gauge-invariant superpotentials which violate the last two starting points in the Sec.I: In compared with breaking pattern A, there is another pattern of two-step breaking 3 SO(10)  [21,23,24] can be close to TeV scale (see, e.g. [25][26][27][28][29][30]).

III. PERTURBATIVE UNIFICATION
With the theoretical constraints in the Introduction, we have clarified that a single or two 16 supermultiplets are allowed in pattern A, whereas two 16s, a 10 with 54, a 16 with 144 or a pair of vector-like 144 may appear in the pattern B. Now, we examine whether any of them are consistent with the first constraint -perturbative unification.
We start with the one-loop renormalization group equations (RGEs) for SM gauge coupling constant, where RG scale t = lnµ and coefficients b i = (b U(1)Y , b SU(2)L , b SU(3)c ) are determined by [31,32], Here, C 2 (G) is the quadratic Casimir invariant, and T (r) refers to dynkin index that depends on details of the representation [5].
A. SU (5) In the case of SU (5) subgroup there are two intermediate scales Λ SUSY and Λ 5 between M Z and Λ 10 , corresponding to SUSY and SU (5) breaking, respectively. The b i coefficients are given by 4  Fig.1 shows the plots of RG running of SM gauge coupling constants according to Eq. (7). It reveals that for Λ SUSY = 1 TeV the SU (5) unification occurs at Λ 5 ≃ 10 16.3 GeV. Moreover, the SO(10) unification in both cases can occur at Λ 10 large than 10 18 GeV. Comparing Λ 5 with Λ 10 , one finds that there is indeed sufficient splitting between them, which verifies previous arguments.

B. Pati-Salam
In the case of Pati-Salam model there are two intermediate scales Λ SUSY and Λ R between M Z and Λ 10 , which denotes SUSY and SU (2) R × U (1) B−L breaking scale, respectively. In this case the coefficients b i are given by, for the RG scale between M Z and Λ R , and for the RG scale between Λ R and Λ 10 . Above the RG scale Λ R , MSSM matters and Higgs field H 3 = 16 con- tributes to δb i = (6, 0, 0, −3) and δb i = (2, 2, 2, 2) in Eq. (9), respectively. Regardless of what extra matter appears above Λ SUSY and what kind of Higgs H 1,2,3 above Λ R , SO(10) unification at the one-loop level yields ln(Λ 10 /Λ R ) ≃ 2.53 [33,34], which uniquely determines Λ R once the content of extra matters is identified.

IV. CONCLUSION
In the forthcoming years we will enter into a new era of precise Higgs physics, which means that studying new physics through the Higgs portal will become very interesting. In this paper, we have utilized perturbative unification due to renormalization to explore the low energy effective theory of SUSY SO(10) scenarios. With the rational theoretical constraints taken into account, we find that for the breaking pattern of either SU(5) or Pati-Salam only 16 supermultiplet can appear at TeV scale apart from the MSSM spectrum.
The quarks or leptons in the 16 supermultiplet(s) can be either chiral or vector-like. Note, vector-like fermion mass requires addition of SM singlet (with vev of order TeV) which does not affect our discussions. While the chiral case has been excluded, the vector-like quarks or leptons are smoking guns in these SUSY SO(10) scenarios. Moreover, the neutral fermions of singlet or doublets of the 16 supermultiplet can serve as dark matter totally, or partially with the neutralinos of the MSSM.