---
abstract: 'The supersymmetric grand unified theory where the $SU(5)$ gauge symmetry is broken by the Hosotani mechanism provides a natural solution to the so-called doublet-triplet splitting problem. At the same time, this model derives a general and distinctive prediction that is testable at TeV scale collider experiments. To be more concrete, adjoint chiral supermultiplets with masses around TeV scale appear. Since these additional fields originate from a higher-dimensional gauge supermultiplet, our model is highly predictive. We study especially the Higgs sector and show that our model is discriminative from the others by precision measurements of the couplings and masses. Namely, we may get a hint of the breaking mechanism of the grand unification at future collider experiments.'
author:
- 'T. Yamashita'
title: Higgs as a probe of supersymmetric grand unification with the Hosotani mechanism
---

Introduction
============

By the discovery of the standard model (SM) like Higgs boson whose mass is around 125 GeV, as reported in 2012 by the ATLAS and CMS collaborations of the CERN Large Hadron Collider (LHC) [@LHC], the SM has been established at least as a low energy effective theory. It is, however, not the end of the story, as the SM can not explain the neutrino oscillations, existence of dark matter and baryon asymmetry of the universe. In addition, there are several unsatisfactory points from the theoretical viewpoint, such as the so-called hierarchy problem and charge quantization problem. These two theoretical problems motivate the supersymmetry (SUSY) and the grand unified theory (GUT) [@GUT; @SUSY-GUT]. In particular, when these two ideas are assumed simultaneously, the three running gauge couplings meet with each others at a very high scale around $10^{16}$ GeV, called the GUT scale, in the minimal model. Thus, the combination, the SUSY-GUT, is worth examining seriously as the physics beyond the SM. Then, the so-called doublet-triplet (DT) splitting problem is the biggest one to be solved. In the minimal model, the mass splitting is realized by fine-tuning, but it dismisses the above motivation of the SUSY. Beyond the minimal model, many ideas to solve the DT splitting problem have been proposed [@DTS]. In extended SUSY-GUT models, however, the successful gauge coupling unification (GCU) is spoiled in many cases and the GCU becomes a constraint instead of a prediction. In addition, there is an unfavorable point that the typical scale of the SUSY-GUTs is too high to address directly at any practical collider experiments.

In these respects, especially the latter one, the model proposed in Ref. [@gGHU-DTS] by supersymmetrizing the grand gauge-Higgs unification (gGHU) [@gGHU] is attractive. The gGHU is a kind of GUTs where the $SU(5)$ gauge symmetry is broken by the so-called Hosotani mechanism [@hosotani]. In the SUSY version of the gGHU (SGGHU), the DT splitting problem is naturally solved and, as a by-product, existence of chiral adjoint supermultiplets as light as the SUSY-breaking scale is generally predicted. These additional fields may be tested at future collider experiments, if the SUSY-breaking scale is around TeV scale which is required to solve the hierarchy problem. In Ref. [@gGHU-pheno], we study the phenomenology of the SGGHU, especially focusing on its Higgs sector. In this article, we review our model in Sec. \[sec:gGHU\] and briefly describe some features of its low energy effective theory in Sec. \[sec:LEET\]. Then, we focus on its Higgs sector in Sec. \[sec:HiggsSector\]. Sec. \[sec:summary\] is devoted to the summary.

grand gauge-Higgs unification {#sec:gGHU}
=============================

In Ref. [@gGHU], a possibility to apply the Hosotani mechanism [@hosotani] to the GUT breaking is revisited.

The Hosotani mechanism is one of the mechanisms for gauge symmetry breaking. It works on a higher-dimensional gauge theory with the extra-dimensions compactified (to reproduce our four dimensions in the phenomenological applications). In view of the four-dimensional (4D) effective theory, the extra-dimensional components of the gauge fields behaves as scalar fields and may develop non-vanishing vacuum expectation values (VEVs) to break the gauge symmetry. In other words, the gauge field and Higgs fields are unified, and thus often called the gauge-Higgs unification (GHU) [@GHU] especially when it is applied to the electroweak symmetry breaking. We name our case where it is applied to the GUT breaking as the grand GHU [@gGHU][^1].

One feature of this mechanism is that the order parameter is a continuous Wilson loop defined by exponentiating the gauge field. This quantity is an element of the gauge group instead of the algebra. Thus, it is constrained to be special, ${\rm det}W=1$, instead of the traceless condition and, for instance, the form $\left< W\right>={\rm diag.}(1,1,1,-1,-1)\equiv P_W$ in the $SU(5)$ case is allowed. Since the entry $1$ corresponds to the vanishing exponent and thus to the trivial vacuum, the VEV is effectively written as $P_W\sim{\rm diag.}(0,0,0,v,v)$ which is a kind of those called [*missing VEVs*]{} that has vanishing eigenvalues. The missing VEVs can be used to [*split*]{} the masses of different components in a single multiplet, especially of the doublet and the triplet. Usually, the missing VEV can not be realized in the $SU(5)$ model due to the traceless condition and larger GUT symmetry, such as $SO(10)$, is considered [@DW]. Although the naively expected pattern of the mass splitting is opposite to the desired one, the VEV $P_W$ results in the desired one for anti-periodic bulk fields with vanishing mass terms [@gGHU-DTS]. Thus, when the SUSY version of this scenario is considered to protect the mass terms from the quantum corrections, the DT splitting problem can be solved, assuming the VEV $P_W$ is obtained.

In this way, the possibility of the GUT breaking via the Hosotani mechanism is interesting enough to be seriously examined. Once we start to try to construct a model on this line, however, we immediately face a difficulty. Since the higher-dimensional theories are essentially vector-like in view of the 4D theory, the extra-dimensions should be compactified on a so-called orbifold. The orbifold boundary conditions (BCs) that project out one chirality, however, tend to project out also the desired massless adjoint scalar fields. In Ref. [@gGHU], it is shown that the so-called diagonal embedding method [@DiagonalEmbedding] invented in context of the string theory can be applied also in field-theoretical setups, as demonstrated below. An advantage of the field-theoretical setup is that it is much easier to calculate the quantum corrections to the scalar potential which is calculable, [*i.e.*]{} finite [@finiteness], and determines the position of the vacuum. In the simplest case, an 5D $SU(5)$ model compactified on an $S^1/{{\mathbb Z}_2}$ orbifold, the VEV $P_W$ can be realized as a minimum for appropriate matter contents [@gGHU], with no need of fine-tuning as a ${{\mathbb Z}_2}$ symmetry is recovered on this vacuum [@gGHU-DTS].

In order to apply the diagonal embedding method to the above simplest case, we impose two copies of the $SU(5)$ gauge symmetry with a ${{\mathbb Z}_2}$ symmetry that exchanges the two $SU(5)$, and set the BCs around the endpoints of the $S^1/{{\mathbb Z}_2}$ as the two gauge fields are exchanged. It is straightforward to see that, with these BCs, the gauge symmetry remaining unbroken in the 4D effective theory is the diagonal part of the $SU(5)\times SU(5)$ (or our GUT symmetry is [*embedded*]{} into the [*diagonal*]{} part) and an adjoint scalar field is actually realized. These become apparent when the 5th dimension is put on a lattice which is very similar to the one of $S^1$ model, as shown in FIG. 1 in Ref. [@gGHU]. A difference from the $S^1$ model is that there is a brane at each endpoint of the $S^1/{{\mathbb Z}_2}$ on which chiral fermions can be put. Namely, our setup can be seen as a way to [*introduce*]{} chiral fermions in $S^1$ models. An interesting feature is that the bulk fields are similar to those in $S^1$ models and couple to the VEV $P_W$, and thus the zero modes appear as vector-like $SU(5)$ incomplete multiplets as the two Higgs doublets in the minimal SUSY SM (MSSM). On the other hand, the boundary fields are essentially 4D fields and do not couple to the VEV, and thus appear as chiral $SU(5)$ complete multiplets as the MSSM matter multiplets.

low energy effective theory {#sec:LEET}
===========================

Now, we examine the properties of the low energy effective theory of the SGGHU. By definition, in this scenario, there is an adjoint scalar field that originates from the gauge field, which is massless at the tree level. Its mass is generated by the quantum corrections which vanish in the limit of the exact SUSY. This means that the loop-induced mass is at most of order of the SUSY-breaking scale. The masses of its SUSY partners can differ from its mass again at most by the SUSY-breaking scale. As a result, adjoint chiral multiplets, in concrete an color octet, a weak triplet and a singlet all with vanishing hypercharge, with mass around the SUSY-breaking scale generally exist in the SGGHU [@gGHU-DTS]. Existence of these adjoints can be tested in future collider experiments if the SUSY-breaking scale lies around the TeV scale.

An immediate consequence of the existence is, however, that the success of the GCU is dismissed, although the unification can be recovered by introducing additional bulk matter fields whose zero modes consist of $SU(5)$ incomplete multiplets. An example we consider is that the additional zero modes are two vector-like pairs of $(\bar{L},L)$ $(({\bf1},{\bf2})_{-1/2})$, one of $(\bar{U},U)$ $((\bar
{\bf3},{\bf1})_{-2/3})$ and one of $(\bar{E},E)$ $(({\bf1},{\bf1})_{1})$ [@gGHU-DTS]. With these matter content, the three gauge couplings meet with each others at the GUT scale with a unified gauge coupling $\alpha_G\sim 0.3$. This value is still perturbative, but is rather strong. In particular, the $SU(3)$ is not asymptotic free and the quantum corrections to the colored particles are enhanced, and for instance the $\mu$ parameter of the octet becomes 70 times larger than the one of the triplet at the low energy [@gGHU-pheno].

Higgs sector {#sec:HiggsSector}
============

Since the colored particles tend to be too heavy to study at the LHC as mentioned above, below we focus on the colorless adjoints, [*i.e.*]{} the triplet $\Delta$ and the singlet $S$ which enlarge the Higgs sector. Now, this sector is composed of the MSSM two Higgs doublets $H_u$ and $H_d$, $\Delta$ and $S$. The superpotential is given by $$\begin{aligned}
 \label{eq:WHiggs}
  W=\mu H_u \cdot H_d+\mu_\Delta{\rm tr}(\Delta^2)+\frac{\mu_S}2S^2
  +\lambda_\Delta H_u \cdot \Delta H_d + \lambda_SSH_u \cdot H_d.\end{aligned}$$ Notice that there are no self-interactions among $S$ and $\Delta$, although such couplings are not forbidden by the symmetry of the effective theory, as $S$ and $\Delta$ originate from the gauge supermultiplet.

The two new couplings $\lambda_\Delta$ and $\lambda_S$ push up the SM-like Higgs boson mass via the $F$-term contributions as in the next-to-MSSM (NMSSM) [@NMSSM] and cause mixing between the MSSM doublets and the adjoints which results in modification of the SM-like Higgs coupling constants. Interestingly, these couplings are related with the unified gauge coupling $g_{\rm GUT}$ as $\lambda_\Delta =2\sqrt{5/3}\lambda_S (\equiv\lambda_S')=g_{\rm GUT}$ at the GUT scale. Thus, this model is very predictive, up to the SUSY-breaking parameters. To be more concrete, for the above example of the additional chiral matter multiplets to recover the GCU, we obtain $\lambda_\Delta=1.1$ and $\lambda_S=0.25$ at the TeV scale [@gGHU-pheno].

As for the SUSY-breaking parameters, since the unified gauge coupling is strong, the unified gaugino mass must be large, say 3 TeV. As a result, soft masses of the colorless fields at the TeV scale are typically $1$-$2$ TeV. In addition, as in the MSSM, the soft term of $H_u$ receives a large contribution via the large top Yukawa interaction. Therefore, some tuning is unfortunately needed to realize electroweak symmetry breaking, and the higgsino mass parameter $\mu$ and the CP-odd Higgs boson mass $m_A$ also tend to be $1$-$4$ TeV. In order to realize scenarios where some of the extra Higgs boson masses are of the order of ${\cal O}(100)$ GeV, further tuning is required among the input parameters. Therefore, we will show some benchmark points that reproduce the mass of the SM-like Higgs boson, instead of scanning the parameter space. We focus on the following three different cases:

-   All the Higgs bosons other than the SM-like Higgs boson are heavy.

-   The new Higgs bosons other than the MSSM-like Higgs bosons are heavy.

-   The new Higgs bosons affect the SM-like Higgs boson couplings.

Bearing the fact that there are a few GeV uncertainties in the numerical computation of the SM-like Higgs boson mass, we take the range of $122~{\rm GeV} < m_h < 129~{\rm GeV}$ as its allowed region. Values of parameters of the TeV-scale effective theory are obtained after RG running and shown in Table \[tab:benchmark-TeV\].

   Case        $\mu$              $B \mu$         $\widetilde{m}_{\Delta}$   $\widetilde{m}_{S}$   $\lambda_\Delta A_\Delta$   $\lambda'_S A_S$   $B_\Delta \mu_\Delta$        $B_S \mu_S$             $m_h$
  ------ ----------------- --------------------- -------------------------- --------------------- --------------------------- ------------------ ----------------------- ----------------------- -----------------
   (A)    $177~{\rm GeV}$   $42100~{\rm GeV}^2$       $585~{\rm GeV}$          $195~{\rm GeV}$          $284~{\rm GeV}$        $446~{\rm GeV}$    $288000~{\rm GeV}^2$     $-5750~{\rm GeV}^2$    $123~{\rm GeV}$
   (B)    $177~{\rm GeV}$   $40800~{\rm GeV}^2$       $784~{\rm GeV}$          $612~{\rm GeV}$         $1340~{\rm GeV}$        $1110~{\rm GeV}$    $30700~{\rm GeV}^2$    $-110000~{\rm GeV}^2$   $123~{\rm GeV}$
   (C)    $175~{\rm GeV}$   $41800~{\rm GeV}^2$       $548~{\rm GeV}$          $216~{\rm GeV}$          $284~{\rm GeV}$        $446~{\rm GeV}$    $207000~{\rm GeV}^2$    $-33600~{\rm GeV}^2$    $122~{\rm GeV}$

  : Examples of parameters of the TeV-scale effective theory obtained after RG running. For all the cases, $\tan\beta=3$, the gaugino masses are $(M_1,M_2,M_3)=(194, 388, 1390)~{\rm GeV}$ and the adjoint $\mu$ terms are $(\mu_\Delta,\mu_S)=(-252, -85.8)~{\rm GeV}$.

\[tab:benchmark-TeV\]

For these benchmark points, we calculate the deviation parameters $\kappa_X = \frac{g_{hXX}}{g_{hXX}|_{\rm SM}}$ where $X$ denotes SM particles, as shown in Figure \[Fig:HiggsCouplings\]. There the predictions of the three benchmark points (A), (B) and (C) in the SGGHU, in addition to those of the MSSM and NMSSM for comparison, are shown.

![The deviations in the SM-like Higgs couplings to a SM field $X$ from the SM predictions $\kappa_X$ are plotted. The predictions of the three benchmark points (A), (B) and (C) in the SGGHU are shown with green blobs. The MSSM and NMSSM predictions are shown with red and blue lines, respectively, for $\tan \beta=10$ (thick line) and $\tan \beta=3$ (dashed). Three lines for the NMSSM predictions indicate mixings between the SM-like and singlet-like Higgs bosons of 10%, 20% and 30% from the right to the left. For the purpose of illustration, the NMSSM line is slightly displaced from $\kappa_\tau = \kappa_b$ in the left figure. []{data-label="Fig:HiggsCouplings"}](taub.eps "fig:"){width="45mm"} ![The deviations in the SM-like Higgs couplings to a SM field $X$ from the SM predictions $\kappa_X$ are plotted. The predictions of the three benchmark points (A), (B) and (C) in the SGGHU are shown with green blobs. The MSSM and NMSSM predictions are shown with red and blue lines, respectively, for $\tan \beta=10$ (thick line) and $\tan \beta=3$ (dashed). Three lines for the NMSSM predictions indicate mixings between the SM-like and singlet-like Higgs bosons of 10%, 20% and 30% from the right to the left. For the purpose of illustration, the NMSSM line is slightly displaced from $\kappa_\tau = \kappa_b$ in the left figure. []{data-label="Fig:HiggsCouplings"}](vb.eps "fig:"){width="45mm"} ![The deviations in the SM-like Higgs couplings to a SM field $X$ from the SM predictions $\kappa_X$ are plotted. The predictions of the three benchmark points (A), (B) and (C) in the SGGHU are shown with green blobs. The MSSM and NMSSM predictions are shown with red and blue lines, respectively, for $\tan \beta=10$ (thick line) and $\tan \beta=3$ (dashed). Three lines for the NMSSM predictions indicate mixings between the SM-like and singlet-like Higgs bosons of 10%, 20% and 30% from the right to the left. For the purpose of illustration, the NMSSM line is slightly displaced from $\kappa_\tau = \kappa_b$ in the left figure. []{data-label="Fig:HiggsCouplings"}](cb.eps "fig:"){width="45mm"}

Since the triplet mass has to be rather heavy to make its VEV of the neutral component smaller than $10~{\rm GeV}$ in order to satisfy the rho parameter constraint, the predictions of the SGGHU are not different from those of the NMSSM. In order to discriminate the SGGHU from the NMSSM, the mass difference among the MSSM-like additional Higgs bosons is useful. The MSSM-like charged Higgs boson mass $m_{H^\pm}^{}$ is given by $$  m_{H^\pm}^2
  = m_{H^\pm}^2|_{\rm MSSM}^{} (1 +\delta_{H^\pm}^{})^2 
  \simeq m_A^2 +m_W^2 +\frac{1}{8}\lambda_\Delta^2 v^2
  -\frac{1}{2}\lambda_S^2v^2\, ,
\label{Eq:mHpm}$$ where $\delta_{H^\pm}$ is the deviation in $m_{H^\pm}^{}$ from the MSSM and $m_A^{}$ is the MSSM-like CP-odd Higgs boson mass. The sign of the singlet contribution is opposite to the triplet one due to the group theory. For the couplings $\lambda_\Delta = 1.1$ and $\lambda_S = 0.25$ obtained above, the triplet contribution dominates and the deviation parameter $\delta_{H^\pm}$ change the sign compared with the NMSSM case [@gGHU-pheno]. As shown in FIG. 6 in Ref. [@gGHU-pheno], this parameter can be a few percent for $m_A\lesssim500~{\rm GeV}$. Since the charged Higgs boson mass can be determined with an accuracy of a few percent at the LHC given such small masses [@HiggsWG], we can test our model.

When the masses of the adjoint-like scalars are below $500~{\rm GeV}$, the International Linear Collider (ILC) has capability to directly produce these new particles. For example, the benchmark point (C) gives the mass of the lighter triplet-like scalar $\Delta^\pm$ is less than $500~{\rm GeV}$ [@gGHU-pheno], and we can probe $\Delta^\pm$ using the channel $e^+ e^- \rightarrow
\Delta^+ \Delta^- \rightarrow tb \bar{t} \bar{b}$, which proceeds via the mixing between the MSSM-like and triplet-like charged Higgs bosons.

summary {#sec:summary}
=======

In the gGHU, the adjoint scalar field is realized in the 4D effective theory via the diagonal embedding method. Its VEV is determined by the calculable (finite) loop-induce scalar potential, and the VEV $P_W$ can be obtained without fine tuning in $SU(5)$ model with appropriate matter content. Assuming this [*missing*]{} VEV, the DT splitting problem can be solved in the SUSY version, and thus the SGGHU is theoretically well-motivated. In order to suppress the nucleon decay via the exchange of the gauge fields, the scale of the VEV should be at least the usual GUT scale, beyond the reach of direct search at any feasible experiments. The SGGHU, however, generally derives a testable prediction that adjoint chiral superfields with masses around TeV scale exist, and is attractive also from a viewpoint of the phenomenology.

Since the colored particles become rather heavy in the SGGHU, we study its Higgs sector. We show that the the SM-like Higgs mass is enhanced by the $F$-term contributions and that the deviations of the couplings from the SM predictions are a few percent, which is a good target of future electron-positron colliders, when the adjoint masses are below $1~{\rm TeV}$. The mass gap between the MSSM-like charged Higgs boson and CP-odd Higgs boson differs from that of the MSSM by a few percent, which is within the scope of the LHC, when their masses are below $500~{\rm GeV}$. Since the direction of the deviation is opposite to that in the NMSSM, the SGGHU can be discriminated from the other models by these measurements.

[99]{} G. Aad [*et al.*]{} \[ATLAS Collaboration\], Phys. Lett. B [**716**]{}, 1 (2012); S. Chatrchyan [*et al.*]{} \[CMS Collaboration\], Phys. Lett. B [**716**]{}, 30 (2012).

H. Georgi and S. L. Glashow, Phys. Rev. Lett.  [**32**]{} (1974) 438. E. Witten, Nucl. Phys.  B [**188**]{} (1981) 513; S. Dimopoulos, S. Raby and F. Wilczek, Phys. Rev.  D [**24**]{} (1981) 1681; S. Dimopoulos and H. Georgi, Nucl. Phys.  B [**193**]{} (1981) 150; N. Sakai, Z. Phys.  C [**11**]{} (1981) 153. See, for example, the references in Ref. [@gGHU-DTS]. T. Yamashita, Phys. Rev. D [**84**]{}, 115016 (2011). K. Kojima, K. Takenaga and T. Yamashita, Phys. Rev. D [**84**]{} (2011) 051701. Y. Hosotani, Phys. Lett.  B [**126**]{}, 309 (1983); ibid [**129**]{}, 193 (1983); M. Kakizaki, S. Kanemura, H. Taniguchi and T. Yamashita, Phys. Rev. D [**89**]{} (2014) 7, 075013. See, for example, the references in Ref. [@gGHU]. C. S. Lim and N. Maru, Phys. Lett.  B [**653**]{}, 320 (2007). S. Dimopoulos and F. Wilczek, Print-81-0600 (SANTA BARBARA), NSF-ITP-82-07; N. Maekawa, Prog. Theor. Phys.  [**106**]{} (2001) 401; N. Maekawa and T. Yamashita, Prog. Theor. Phys.  [**107**]{} (2002) 1201; ibid  [**110**]{} (2003) 93. K. R. Dienes and J. March-Russell, Nucl. Phys.  B [**479**]{} (1996) 113. N. V. Krasnikov, Phys. Lett. B [**273**]{}, 246 (1991); H. Hatanaka, T. Inami and C. S. Lim, Mod. Phys. Lett. A [**13**]{}, 2601 (1998); N. Maru and T. Yamashita, Nucl. Phys.  B [**754**]{}, 127 (2006); Y. Hosotani, N. Maru, K. Takenaga and T. Yamashita, Prog. Theor. Phys.  [**118**]{}, 1053 (2007). For a review, see, for example, U. Ellwanger, C. Hugonie and A. M. Teixeira, Phys. Rept.  [**496**]{}, 1 (2010). D. Cavalli [*et al.*]{} \[Higgs Working Group Collaboration\], hep-ph/0203056.

[^1]: This name is already used in Ref. [@LimMaru] in a slightly different context which we carelessly overlooked.
