Yukawa coupling unification in non-supersymmetric

SO(10) models with an intermediate scale

Abdelhak Djouadi1,2, Ruiwen Ouyang2, Martti Raidal2

1 Centro Andaluz de Fisica de Particulas Elementales (CAFPE)
and
Departamento de Física Teórica y del Cosmos,
Universidad de Granada, E–18071 Granada, Spain

2 Laboratory of High Energy and Computational Physics, NICPB, Rävala pst. 10,
10143 Tallinn, Estonia

###### Abstract

We discuss the possibility of unifying in a simple and economical manner the
Yukawa couplings of third generation fermions in a non-supersymmetric SO(10)
model with an intermediate symmetry breaking, focusing on two possible
patterns with intermediate Pati-Salam and minimal left-right groups. For this
purpose, we start with a two Higgs doublet model at the electroweak scale and
assume a minimal Yukawa sector at the high energy scales. We first enforce
gauge coupling unification at the two-loop level by including the threshold
corrections in the renormalisation group running which are generated by the
heavy fields that appear at the intermediate symmetry breaking scale. We then
study the running of the Yukawa couplings of the top quark, bottom quark and
tau lepton at two-loops in these two breaking schemes, when the appropriate
matching conditions are imposed. We find that the unification of the third
family Yukawa couplings can be achieved while retaining a viable spectrum,
provided that the ratio of the vacuum expectation values of the two Higgs
doublet fields is large, $\tan\beta\approx 60$.

## 1 Introduction

The paradigm of grand unification [1, 2] is at the heart of particle physics
as it exploits the power of symmetries to unify in a most elegant way the
electromagnetic, weak and strong interactions of the Standard Model (SM) into
a single force [3]. Grand unified theories (GUTs) provide natural solutions to
theoretical questions such as charge quantization and anomaly cancellation, in
addition to the explanation of the existence of three separate gauge symmetry
groups. GUTs can also successfully address most, if not all, of the important
issues that call for beyond the SM physics. This is particularly the case for
the problems of neutrino masses and mixing, the baryon asymmetry in the
universe and the nature of the dark matter. Hence, leaving aside the issue of
naturalness and the large hierarchy between the weak and Planck scales that
induces quadratic “divergences” to the observed Higgs boson mass (for which
one can, for instance, adopt an anthropic point of view just as in the case of
the cosmological constant), non-supersymmetric GUTs can be viewed as the royal
path to physics beyond the SM.

Unification in the context of SO(10) [4] is particularly interesting as this
symmetry group possesses a representation of dimension 16 in which, for each
generation, one can accommodate the 15 chiral fermions of the SM and an
additional Majorana neutrino. If the mass of this new state is very large,
somewhere at a scale of $10^{10}$ GeV, the see-saw mechanism [5] could explain
the present pattern in the neutrino sector, the baryon asymmetry could be
achieved through leptogenesis [6] and a suitable axion [7] could account for
dark matter; see Refs. [8, 9, 10] for reviews. This intermediate scale can
naturally be present in SO(10) as the group is of rank five, i.e., larger than
the rank of the SM group by one unit, so that the symmetry breaking may occur
in three steps, one at the GUT scale $M_{U}$, one at this intermediate scale
$M_{I}$ and a last one at the electroweak scale. This solves one of the main
drawbacks of non-supersymmetric GUTs, namely, the failure of the gauge
couplings to unify at the high energy scale. Indeed, threshold effects [11]
are generated by the contributions of the scalar multiplets that break the
intermediate symmetry down to the SM group at the energy $M_{I}$, and these
modify the renormalisation group evolution of the three coupling constants
such that they finally intersect at the scale $M_{U}$ [12, 13]. Hence, gauge
coupling unification can also be realized without the need of supersymmetry,
which was one of its main attractive points [14].

Another argument in favor of supersymmetry was the possibility of also
unifying the Yukawa couplings of third generation fermions [15]. Indeed, in
the minimal supersymmetric extension of the SM, the MSSM, two-Higgs doublets
fields are required in order to generate separately masses for the isospin up-
and down-type fermions and, in constrained scenarios with universal “soft”
SUSY–breaking parameters, these Yukawa couplings can be unified at the GUT
scale. This occurs at large values of the ratio of the vacuum expectation
values of the two Higgs fields, $\tan\beta=v_{u}/v_{d}$, which induces the
proper hierarchy for the starting top and bottom quark masses,
$\tan\beta\approx m_{t}/m_{b}\approx 60$.

In this letter, we show that the unification of the Yukawa couplings of third
generation fermions can also be achieved in a rather simple and most
economical way in a non-supersymmetric SO(10) scenario, taking as examples two
of the most interesting and widely discussed intermediate breaking patterns:
the Pati-Salam [2] and the minimal left-right symmetric [16] groups. As a
matter of fact, above the scale at which the intermediate breaking occurs,
only two Higgs bi-doublet fields will be necessary to describe the Yukawa
interactions of standard fermions, which reduce to two doublets below this
intermediate scale down to the electroweak scale. Hence, one would have an
effective two Higgs doublet model (2HDM) of type II [17] at low energies, just
as in the MSSM, with vacuum expectation values such that the parameter
$\tan\beta$ is large as to obtain the correct hierarchy for the top and bottom
quark masses. Using this minimal scalar sector, it is possible to make that
the renormalisation group running of third generation Yukawa couplings in
these two breaking schemes, with suitable matching conditions at the
intermediate scale for which gauge coupling unification occurs, leads to
Yukawa coupling unification at the GUT scale. This can be achieved while
reproducing the third family fermion and electroweak gauge boson masses and
preserving some important features such as ensuring the stability of the
electroweak vacuum up to the intermediate scale and keeping the Yukawa
couplings perturbative at all scales.

The paper is organized as follows. In the next section, we introduce our
theoretical framework and discuss the breaking of SO(10) with intermediate
steps. In section 3, we discuss the known issue of gauge couplings unification
in SO(10) when threshold corrections are added at an intermediate scale but
with a new ingredient, namely, the presence of an additional Higgs doublet
field at low energies. In section 4, we study the running of the third
generation Yukawa couplings and show that they can reach a common value at the
same scale that allows for gauge coupling unification, while keeping a viable
low energy spectrum. Our conclusions are given in section 5.

## 2 Theoretical framework

The SO(10) group has many attractive features [8, 9, 10] and most of them
follow from the fact that it possesses a fundamental representation of
dimension-16 in which, for each generation, the 15 SM chiral fermions as well
as one right-handed neutrino can be embedded. In this case, the Yukawa
couplings of the scalar bosons to pairs of these fermions belong to the direct
product of $\mathbf{16\otimes 16}$, which can be decomposed into

$\displaystyle\mathbf{16_{F}\otimes
16_{F}}=\mathbf{10}+\mathbf{120}+\mathbf{126}\,.$ (1)

Thus, the most general Yukawa interaction which is SO(10) invariant is given
by

$\displaystyle-{\cal
L}_{Y}=\mathbf{16_{F}}(Y_{10}\mathbf{10_{H}}+Y_{126}\mathbf{\overline{126}_{H}}+Y_{120}\mathbf{120_{H}})\mathbf{16_{F}}\,.$
(2)

The special case with only the first two Yukawa terms with the
$\mathbf{10_{H}}$ and $\mathbf{\overline{126}_{H}}$ representations, leading
to the so-called minimal SO(10) model, has been thoroughly discussed, see e.g.
Refs. [9, 18]. The extended SO(10) model including the $\mathbf{120_{H}}$
representation has been also explored [19, 20]. The first model usually
requires an extra U(1) symmetry to complexify the $\mathbf{10_{H}}$
representation to achieve the required splitting in the fermionic spectrum,
otherwise the ratio $m_{t}/m_{b}$ would be fixed to unity at the GUT scale
[9]. In turn, in the latter scenario, it has been shown that a realistic
fermion spectrum can be achieved with or without introducing such an extra
U(1) symmetry [20].

In this work, we will restrict to the minimal and most studied SO(10) scenario
in which only the $\mathbf{10_{H}}$ and $\mathbf{\overline{126}_{H}}$
representations are kept, but without an extra U(1) symmetry to complexify the
$\mathbf{10_{H}}$ representation. This will constrain the parameter space,
making the model more predictive, while allowing the possibility of neutrino
mass generation via a seesaw mechanism and being consistent with present data
[9, 18].

The breaking of SO(10) to the SM gauge group ${\cal G}_{\rm
SM}\\!\equiv\\!{\cal G}_{321}\\!=\\!{\rm SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}}$ can be triggered in several ways, but we will be only interested in
two patterns that involve one intermediate gauge group at a high scale
$M_{I}$: the Pati-Salam (PS) group [2] ${\cal G}_{422}\\!=\\!{\rm
SU(4)_{C}\times SU(2)_{L}\times SU(2)_{R}}$ and the minimal left-right (LR)
symmetry group [16] ${\cal G}_{3221}\\!=\\!{\rm SU(3)_{C}\times
SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}}$. To achieve the desired symmetry-
breaking in these two scenarios, one would necessarily need to introduce
scalar multiplets that acquire vacuum expectation values (vevs) at the
corresponding high scales.

For the Pati-Salam scenario, the breaking chain from SO(10) to the SM gauge
group is e.g. achieved by the ${\bf(15,1,1)}$ component of the scalar
representation $\mathbf{210_{H}}$ which acquires a vev at the GUT scale
$M_{U}$, and by the $\mathbf{\overline{126}_{H}}$ that acquires a vev at the
intermediate scale $M_{I}$. In turn, in the minimal left-right scenario, the
symmetry should be broken first by the $\mathbf{45_{H}}$ which acquires a vev
at the GUT scale and then by the $\mathbf{\overline{126}_{H}}$ that acquires
it at the intermediate scale [20]. One thus has

$\displaystyle\text{PS}:\quad\text{SO(10)}|_{M_{U}}\xrightarrow{\langle\mathbf{210_{H}}\rangle}{\cal
G}_{422}|_{M_{I}}\xrightarrow{\langle\mathbf{\overline{126}_{H}}\rangle}{\cal
G}_{321}|_{M_{Z}}\xrightarrow{\langle\mathbf{10_{H}}\rangle}{\cal G}_{31}\,;$
(3)
$\displaystyle\text{LR}:\quad\text{SO(10)}|_{M_{U}}\xrightarrow{\langle\mathbf{45_{H}}\rangle}{\cal
G}_{3221}|_{M_{I}}\xrightarrow{\langle\mathbf{\overline{126}_{H}}\rangle}{\cal
G}_{321}|_{M_{Z}}\xrightarrow{\langle\mathbf{10_{H}}\rangle}{\cal G}_{31}\,.$
(4)

According to the extended survival hypothesis [21], all the scalar fields that
do not participate in the symmetry breaking patterns above by acquiring vevs
will have masses of the order of the high scales $M_{U}$ and $M_{I}$. In these
two breaking chains, the scalar content that acquires vevs at the intermediate
scale $M_{I}$ or at the electroweak scale $M_{Z}$ consists of, respectively,
the $\mathbf{\overline{126}_{H}}$ and $\mathbf{10_{H}}$ representations. More
specifically, of the SO(10) scalar representations that can be decomposed
under the intermediate gauge groups, only certain scalar fields from
$\mathbf{10_{H}}$ and $\mathbf{\overline{126}_{H}}$ have masses below the GUT
scale and will contribute to the renormalisation group equations (RGEs)
between the two scales $M_{I}$ and $M_{U}$. These are, in the PS scenario,
${\bf(1,2,2)}$ ($\Phi_{10}$) from $\mathbf{10_{H}}$ and
${\bf(15,2,2)}\oplus{\bf(10,1,3)}$ ($\Sigma_{126}\oplus\Delta_{R}$) from
$\mathbf{\overline{126}_{H}}$ and, in the minimal LR scenario,
${\bf(1,2,2,0)}$ from $\mathbf{10_{H}}$ and the
${\bf(1,2,2,0)}\oplus{\bf(1,1,3,2)}$ from $\mathbf{\overline{126}_{H}}$. We
will thus only consider this restricted set of scalar fields between the GUT
and intermediate scales111This in contrast to most studies which are done in
this context as, generally, a very complicated scalar sector of the SO(10)
group is needed to fit the low energy spectrum, in particular the fermion
(including the light and sometimes even the heavy neutrino sector) masses and
mixings, by adjusting the numerous input parameters that are available..

At low energies, among the two Higgs bi-doublets that we had at a high energy
scale, only two Higgs doublets survive and develop vevs at the electroweak
scale. Thus, in our study, we will have in fact a model with two Higgs doublet
fields $H_{u}$ and $H_{d}$ that couple separately to isospin $+\frac{1}{2}$
and $-\frac{1}{2}$ fermions and acquire vevs $v_{u}$ and $v_{d}$

$\displaystyle\left\langle
H_{u}\right\rangle=\frac{1}{\sqrt{2}}\binom{0}{v_{u}}\quad,\quad\left\langle
H_{d}\right\rangle=\frac{1}{\sqrt{2}}\binom{0}{v_{d}}\,,$ (5)

to give masses to the $W,Z$ bosons implying the relation
$\sqrt{v_{u}^{2}+v_{d}^{2}}\\!=\\!v_{\rm SM}\\!\simeq\\!246\,\text{GeV}$; we
then define the ratio of these two vevs to be $\tan\beta=v_{u}/v_{d}$. The
most general renormalizable scalar potential of this two Higgs doublet model
can be found in Ref. [17] to which we refer for all details. The Yukawa
interactions of the fermions are those of a type-II 2HDM with a Lagrangian
given by

$\displaystyle-{\cal
L}_{Y}^{\text{2HDM}}=Y_{u}\bar{Q}_{L}H_{u}\;u_{R}+Y_{d}\bar{Q}_{L}H_{d}\;d_{R}+Y_{e}\bar{L}_{L}{H_{d}}\;e_{R}+{\rm
h.c.}\,,$ (6)

with $Q_{L}/L_{L}$ the quark/lepton left-handed doublets and $f_{R}$ the
right-handed singlets. In our study, only the third generation of fermions
will be considered and the small Yukawa couplings of the first two generations
will be neglected.

At the intermediate scale $M_{I}$, the minimal Yukawa interaction Lagrangian
is obtained when only two Higgs bi-doublets couple to fermions. One of them
should be from the $\mathbf{\overline{126}_{H}}$ which also has a triplet
field that breaks the left-right symmetry. The other can be chosen to be the
$\mathbf{10_{H}}$. Starting from eq. (2), the Yukawa Lagrangian for fermions
at the intermediate scale $M_{I}$ can be written in the considered two schemes
as

$\displaystyle-{\cal L}_{Y}^{PS}$ $\displaystyle=$
$\displaystyle\bar{F}_{L}(Y^{10}_{PS}\Phi_{10}+Y^{126}_{PS}\Sigma_{126})F_{R}+F_{R}^{T}Y^{R}_{PS}C\overline{\Delta_{R}}F_{R}+{\rm
h.c.}\,,$ (7) $\displaystyle-{\cal L}_{Y}^{LR}$ $\displaystyle=$
$\displaystyle\bar{Q}_{L}(Y^{10}_{LR}\Phi_{10}+Y^{126}_{LR}\Sigma_{126})Q_{R}+\bar{L}_{L}(Y^{10}_{LR}\Phi_{10}+Y^{126}_{LR}\Sigma_{126})L_{R}$
(8) $\displaystyle+$
$\displaystyle{\small\frac{1}{2}}L_{R}^{T}Y^{R}_{LR}i\sigma_{2}\Delta_{R}L_{R}+{\rm
h.c.}\,,$

where $F_{L,R}$ are generic left or right-handed quark/lepton fields and
$\sigma_{2}$ a Pauli matrix. In both cases, we have assumed that terms like
$\bar{F}_{L}^{T}\tilde{\phi}F_{R}$ with $\phi=\Phi$ or $\Sigma$ and
$\tilde{\phi}=\sigma_{2}^{T}\phi^{*}\sigma_{2}$ are forbidden by suitably
chosen ${\rm U(1)_{Y}}$ charges [22]. Below the intermediate scale, the PS and
LR models include, besides the triplet field $\Delta_{R}$ that gives masses to
the heavy neutrino species, four Higgs doublets: two doublets $\phi_{1}$ and
$\phi_{3}$ with opposite hypercharge from the
$(\mathbf{1},\mathbf{2},\mathbf{2})$ representation and the doublets
$\phi_{2}$ and $\phi_{4}$ again with opposite hypercharge from
$(\mathbf{15},\mathbf{2},\mathbf{2})$. The fields $\phi_{1}$ and $\phi_{2}$
couple to up-type quarks and heavy neutrinos, while $\phi_{3}$ and $\phi_{4}$
couple to down-type quarks and the light leptons.

While the triplet fields acquire a very large vev,
$\langle\Delta_{R}\rangle=v_{R}\sim{\cal O}(M_{I})$, the bi-doublet fields
acquire vevs of the order of the electroweak scale which implies that
$\sum_{i=1}^{4}v_{i}^{2}=v_{\rm SM}^{2}$. This ensures that the right-handed
gauge bosons are very heavy, $M_{W_{R}},M_{Z_{R}}\approx gv_{R}$, while the
${\rm SU(2)_{L}}$ $W$ and $Z$ bosons have weak scale masses,
$M_{W},M_{Z}\approx gv_{\rm SM}$. In fact, only two linear combinations of the
four scalar doublet fields $\phi_{1}\cdots\phi_{4}$ will have weak scale
masses, while the two other field combinations will have masses close to the
very high scale. One has thus to tune the scalar potentials of the two
scenarios to achieve this situation and discussions about the constraints to
which it leads can be found in Refs. [24, 23] for instance. The two fields
with weak scale masses will be ultimately identified with the doublets $H_{u}$
and $H_{d}$ of our low energy 2HDM. At the intermediate scale $M_{I}$, these
fields should match the $\Phi_{10}$ and $\Sigma_{126}$ fields, the
interactions of which have been given in eqs. (7,8) as will be discussed
shortly.

## 3 Gauge coupling unification

Assuming the 2HDM structure at low energies and the two breaking patterns of
SO(10) down to the SM group with the intermediate scale $M_{I}$ discussed
previously, namely PS and LR , we study the renormalisation group running of
the three SM gauge couplings $\alpha_{i}=g_{i}^{2}/(4\pi)$. We closely follow
Ref. [13] in which the standard case with only one electroweak Higgs doublet
was studied. The analytical expressions for the gauge coupling RGEs at the two
loop level, including the relevant $\beta$ functions can be found, e.g., in
Ref. [26] where the dependence of the number of Higgs doublets is explicitly
given. Naively, the more intermediate scale scalar particles are included in
the running of the couplings, the lower would be the resulting unification
scale.

At the intermediate scale, threshold effects [11] due to all the particles
that have masses in the vicinity of $M_{I}$, and in particular all the scalar
fields that develop vevs at this scale, will be active. These higher order
corrections will modify the matching conditions of the gauge couplings at the
scale of symmetry breaking, depending on the particle content. For a symmetry
breaking from a group ${\cal G}$ to a subgroup ${\cal H}$ at the scale $\mu$,
the matching conditions with the threshold corrections take the form

$\displaystyle\alpha_{i,{\cal H}}^{-1}(\mu)=\alpha_{i,{\cal
G}}^{-1}(\mu)-{\lambda_{i,{\cal H}}^{\cal G}}/{(12\pi)}\,,$ (9)

where $i=1,2,3,...$ refers to the particular gauge coupling $\alpha_{i}$ and
$\lambda_{i,{\cal H}}^{\cal G}$ are usually weighted by the parameters
$\eta_{i}=\ln({M_{i}}/{\mu})$ with $M_{i}$ being the masses of the heavy
particles integrated out at the low energy. The complete expressions for the
one-loop threshold corrections $\lambda_{i,{\cal H}}^{\cal G}$ are given in
Ref. [13] (see tables IV and VI of their Appendices B and C, respectively). As
a result, the intermediate and the unification scales $M_{I}$ and $M_{U}$
could be shifted by an order of magnitude or more even when only small
threshold corrections are included. In the following, we show an explicit
example of gauge coupling unification in the PS and LR breaking chains when
these thresholds are included.

We start with the following initial conditions for the SM gauge couplings
calculated in the $\overline{\text{MS}}$ renormalization scheme with two-loop
accuracy and first evaluated at the electroweak scale that we take to be the
$Z$ boson, mass $M_{Z}=91.2$ GeV [27],

$\displaystyle[g_{Y}(M_{Z}),g_{2}(M_{Z}),g_{3}(M_{Z})]=[0.3574,0.6517,1.2182]\,,$
(10)

where $g_{Y}$ should be normalized with the usual GUT condition leading to
$\alpha_{1}/\alpha_{Y}=5/3$. Using the two-loop RGEs in the case in which two
Higgs doublets are present at low energies and including the relevant
threshold corrections following Ref. [13], we determine the point at which the
couplings intersect when appropriately adjusting the intermediate scale
$M_{I}$. In the two symmetry breaking chains that we consider, the tree-level
matching conditions that determine the gauge couplings of the intermediate
scale models from the low energy ones read

$\displaystyle{\rm PS}:$
$\displaystyle\alpha_{4}^{-1}(M_{I})=\alpha_{3}^{-1}(M_{I})\,,\
\alpha_{2L}^{-1}(M_{I})=\alpha_{2}^{-1}(M_{I})\,,\
\alpha_{2R}^{-1}(M_{I})=\frac{5}{3}\alpha_{Y}^{-1}(M_{I})-\frac{2}{3}\alpha_{3}^{-1}(M_{I})\,,~{}~{}$
$\displaystyle{\rm LR}:$
$\displaystyle\\!\\!\alpha_{3}^{-1}(M_{I})=\alpha_{2L}^{-1}(M_{I})=\alpha_{3}^{-1}(M_{I})\,,\
\alpha_{B\\!-\\!L}^{-1}(M_{I})=\kappa\alpha_{2R}^{-1}(M_{I})=\big{(}\frac{2\kappa+3}{5}\big{)}^{-1}\alpha_{Y}^{-1}(M_{I})\,,$
(11)

where in LR we assume
$\alpha_{B\\!-\\!L}^{-1}(M_{I})=\kappa\alpha_{2R}^{-1}(M_{I})$ as we are
matching three couplings to four; this normalization factor $\kappa$ of ${\cal
O}(1)$ is to be solved with the scales $M_{I}$ and $M_{U}$.

Note that in eq. (10), we have ignored, for simplicity, the experimental
errors on the couplings constants (as well as the theoretical uncertainties)
and kept only the central values. These errors, the largest of which being the
one that affects the strong coupling constant $\alpha_{3}$ which is at the
percent level, will generate an uncertainty on the derived GUT an intermediate
scales of the order of a few percent only and, hence, do not affect our
discussion in a significant way as will be shown shortly.

In our analysis, the RGEs of the gauge couplings are solved up to two loop
order with the help of the program SARAH [29], and the inclusion of the
thresholds corrections is performed by randomly sampling parameters
$\eta_{i}=\ln(M_{i}/\mu)$ within the range of $\eta_{i}\in[-1,1]$. We then
impose the tree-level matching conditions of the gauge couplings at the
intermediate scale as in eq. (11) when the one-loop threshold corrections are
included as in eq. (9), and determine the values of the two scales $M_{I}$ and
$M_{U}$ for each sampling parameter set. More precisely, we take at least
10,000 points for the parameters $\eta_{i}$ within the range of
$\eta_{i}\in[-1,1]$ and determine the sets of all scales that allow for gauge
coupling unification.

In the two intermediate SO(10) scenarios that we consider, gauge coupling
unification with the inclusion of threshold corrections can, for instance, be
achieved for the following values of the unification and intermediate scales

$\displaystyle{\rm PS}:$ $\displaystyle M_{U}=7.5\times 10^{15}~{}{\rm
GeV~{}~{}and~{}~{}}M_{I}=6.6\times 10^{10}~{}\text{GeV}\,,$ $\displaystyle{\rm
LR}:$ $\displaystyle M_{U}=3.9\times 10^{15}~{}{\rm
GeV~{}~{}and~{}~{}}M_{I}=6.0\times 10^{9}~{}\text{GeV}\,.$ (12)

The evolution of the inverse of the coupling constants $\alpha_{i}^{-1}$ from
the scale $M_{U}$ down to $M_{I}$ and then down to $M_{Z}$ is shown in Fig. 1
as a function of the energy scale in the two breaking patterns PS (left panel)
and LR (right panel) when the two-loop (solid lines) and one-loop (dashed
lines) RGEs are used and the threshold effects are included at the
intermediate scale. While the three couplings are clearly different at the
scale $M_{I}$ of the order of a few times $10^{10}$ GeV (as required to
reproduce neutrino phenomenology), the slope are significantly modified at
this energy by the additional contributions so that the couplings meet at a
scale $M_{U}$ of the order of a few times $10^{15}$ GeV (which is high enough
to prevent fast proton decay). Both the two-loop corrections and the threshold
corrections have a noticeable impact and make the intermediate scale lower.
The small impact of the experimental errors on the couplings is illustrated by
the narrow red bands at the scales $M_{I}$ and $M_{U}$.

Figure 1: The evolution of the inverse of the gauge coupling constants
$\alpha_{i}=g_{i}^{2}/(4\pi)$ as a function of the energy scale $\mu$ in the
2HDM+${\cal G}_{422}$ Pati-Salam model (left) and 2HDM+${\cal G}_{3221}$
minimal LR model (right) at the one-loop (dashed lines) and two-loop (solid
lines) orders. The GUT and intermediate scales $M_{U}$ and $M_{I}$ are
indicated and the threshold effects are included. The red bands reveal the
effects of the experimental uncertainties in the measurement of the couplings

## 4 Yukawa coupling unification

We now turn to the Yukawa sector of the theory. As already mentioned, we will
ignore the very small Yukawa couplings of the first and second generation
fermions222As in the supersymmetric case, fermions with masses below a few GeV
cannot be realistically described in our approach as it will be plagued by
strong interaction uncertainties when running the RGEs down to the fermion
mass scale. and consider only those of the top quark, the bottom quark and the
tau lepton, neglecting all possible mixings. Below the intermediate scale
$M_{I}$, the Yukawa interactions of these fermions are those of a type-II 2HDM
with a Lagrangian given by eq. (6). It leads to the following relations
between the fermion masses and the Yukawa couplings

$m_{t}=\frac{1}{\sqrt{2}}Y_{t}v_{u}\,,\ \ \
m_{b}=\frac{1}{\sqrt{2}}Y_{b}v_{d}\,,\ \ \
m_{\tau}=\frac{1}{\sqrt{2}}Y_{\tau}v_{d}\,.$ (13)

In the region between the intermediate scale and the GUT scale, we assume the
Yukawa structure of eqs. (7) and (8) for the PS and minimal LR breaking
patterns, respectively. With a real $\mathbf{10_{H}}$ representation with its
vevs denoted by $v_{10}^{u}=v^{d\,*}_{10}=v_{10}$ and by adopting a phase
convention in which $v_{10}$ is real (this can be done via, e.g., an SU(2)
rotation) [20], and denoting by $v_{126}^{u,d}$ the vevs of the $\Sigma_{126}$
field, the fermion masses for the two considered breaking chains will be given
by

$\displaystyle
m_{t}\\!=\\!\frac{v_{10}Y_{10}^{PS}\\!+\\!v_{126}^{u}Y_{126}^{PS}}{\sqrt{2}},\,m_{b}\\!=\\!\frac{v_{10}Y_{10}^{PS}\\!+\\!v_{126}^{d}Y_{126}^{PS}}{\sqrt{2}},\,m_{\tau}\\!=\\!\frac{v_{10}Y_{10}^{PS}\\!-\\!3v_{126}^{d}Y_{126}^{PS}}{\sqrt{2}},~{}~{}$
$\displaystyle
m_{t}\\!=\\!\frac{v_{10}Y_{10,q}^{LR}\\!+\\!v_{126}^{u}Y_{126,q}^{LR}}{\sqrt{2}},\,m_{b}\\!=\\!\frac{v_{10}Y_{10,q}^{LR}\\!+\\!v_{126}^{d}Y_{126,q}^{LR}}{\sqrt{2}},\,m_{\tau}\\!=\\!\frac{v_{10}Y_{10,l}^{LR}\\!+\\!v_{126}^{d}Y_{126,l}^{LR}}{\sqrt{2}}\,.$
(14)

Finally, above the GUT scale $M_{U}$, the third generation Yukawa couplings
are unified as in eq. (2) and are given by

$\displaystyle m_{t}=v_{10}Y_{10}+v_{126}^{u}Y_{126}\,,\
m_{b}=v_{10}Y_{10}+v_{126}^{d}Y_{126}\,,\
m_{\tau}=v_{10}Y_{10}-3v_{126}^{d}Y_{126},$ (15)

(with the additional masses for neutrinos
$M_{\nu_{D}}\\!=\\!v_{10}Y_{10}\\!-\\!3v_{126}^{u}Y_{126}$ and
$M_{\nu_{R}}\\!=\\!v_{R}Y_{126}$). The normalization factors can be absorbed
into the redefinition of the Yukawa couplings at the GUT scale. The factors of
3 and the relative signs between the various terms are due to the Clebsh-
Gordan coefficients coming from the vev of the traceless adjoint 15 of $\rm
SU(4)$ in ${\bf(2,2,15)}$.

As the evolution of the couplings near the scale $M_{I}$ should be affected by
threshold corrections, one should expect a significant discontinuity of the
Yukawa couplings when the contributions of the numerous scalar and vector
fields are included in the RGEs. Nevertheless, as these Yukawa couplings are
directly related to the masses of the fermions, one can simply assume that the
physical fermion masses are continuous at the scale $M_{I}$ [25] when these
threshold corrections are included. This means that the masses calculated in
the low-energy 2HDM should coincide with those obtained from the intermediate
left-right or Pati–Salam models or the unified SO(10) model, up to their
running. One can then consider this relation as the matching conditions for
the Yukawa couplings at the intermediate and the GUT scales. For example, at
the scale $M_{I}$, equating eqs. (13-14) for the PS and LR breaking chains
leads to333For this exploratory work, we simply follow Ref. [25] and ignore
the small running of the vevs. This issue, together with other refinements,
will be postponed to a forthcoming publication.

$\displaystyle Y_{t}(M_{I})$
$\displaystyle\\!=Y_{10}^{PS}(M_{I})\frac{v_{10}}{v_{u}}\\!+\\!Y_{126}^{PS}(M_{I})\frac{v_{126}^{u}}{v_{u}}$
$\displaystyle{\rm
or}~{}~{}Y_{10,q}^{LR}(M_{I})\frac{v_{10}}{v_{u}}\\!+\\!Y_{126,q}^{LR}(M_{I})\frac{v_{126}^{u}}{v_{u}},\,$
$\displaystyle Y_{b}(M_{I})$
$\displaystyle\\!=Y_{10}^{PS}(M_{I})\frac{v_{10}}{v_{d}}\\!+\\!Y_{126}^{PS}(M_{I})\frac{v_{126}^{d}}{v_{d}}$
$\displaystyle{\rm
or}~{}~{}~{}Y_{10,q}^{LR}(M_{I})\frac{v_{10}}{v_{d}}\\!+\\!Y_{126,q}^{PS}(M_{I})\frac{v_{126}^{d}}{v_{d}},\,$
$\displaystyle Y_{\tau}(M_{I})$
$\displaystyle\\!=Y_{10}^{PS}(M_{I})\frac{v_{10}}{v_{d}}\\!-3Y_{126}^{PS}(M_{I})\frac{v_{126}^{d}}{v_{d}}$
$\displaystyle{\rm
or}~{}~{}~{}Y_{10,l}^{LR}(M_{I})\frac{v_{10}}{v_{d}}\\!+Y_{126,l}^{LR}(M_{I})\frac{v_{126}^{d}}{v_{d}}.\,$
(16)

As for the matching conditions at the GUT scale, one has to carefully take the
Clebsh-Gordan factors into account for the Yukawa interactions when the field
representations are embedded into the SO(10) group [30, 22]. One can then
enforce Yukawa coupling unification by requiring the matching conditions at
the GUT scale to be

$\displaystyle Y_{f}(M_{U})$ $\displaystyle\equiv$ $\displaystyle
Y_{10}^{PS}(M_{U})=\frac{1}{4}Y_{126}^{PS}(M_{U})\,,$ (17) $\displaystyle
Y_{f}(M_{U})$ $\displaystyle\equiv$ $\displaystyle
Y_{10,q}^{LR}(M_{U})=\frac{1}{4}Y_{126,q}^{LR}(M_{U})=Y_{10,l}^{LR}(M_{U})=-\frac{1}{12}Y_{126,l}^{LR}(M_{U})\,,$
(18)

where the unified Yukawa coupling $Y_{f}(M_{U})$ is taken to be a free
parameter of SO(10).

One has then to fit these parameters with the actual observables, namely the
top, bottom and tau masses using the relations in eq. (13) at the low energy
scale, chosen again to be $M_{Z}=91.2$ GeV. We use the following input
$\overline{\text{MS}}$ running fermion masses in the SM [27, 28] (we again
ignore the related experimental uncertainties for now),

$\displaystyle[m_{t}(M_{Z}),m_{b}(M_{Z}),m_{\tau}(M_{Z})]=[168.3,2.87,1.73]~{}{\rm
GeV}\,,$ (19)

and we then turn them into the corresponding input masses in the 2HDM by using
the appropriate RGEs in the evolution from the scale of the fermion masses to
$M_{Z}$.

In the PS model, the colored-quarks and leptons are charged under the same
local SU(4) symmetry so that all fermions can be unified into the same
representation $F_{L,R}$. When these fermions couple to the Higgs fields ${\bf
10_{H}}$, one cannot distinguish the bottom quark from the tau lepton and one
should have $m_{b}=m_{\tau}$ if the vev $v_{126}^{d}$ is small,
$v_{126}^{d}/v_{d}\ll 1$, as can be seen from eq. (14). If this mass equality
is still valid slightly below the intermediate scale, we should then have
$Y_{b}(M_{I})=Y_{\tau}(M_{I})$ in our low energy 2HDM by virtue of eqs. (16)
and (13). The scale at which the bottom and tau Yukawa couplings are equal,
that we denote by $M_{b\tau}$, is simply determined (within some accuracy) by
the point at which the curves for their RG running from the weak scale $M_{Z}$
upwards intersect, which critically depends on the value of the parameter
$\tan\beta$. In order to use the matching conditions given by the equations
above, the scale for $b$-$\tau$ unification should be identical to the
intermediate PS breaking scale, $M_{b\tau}=M_{I}$, and this can be achieved by
selecting the appropriate value of $\tan\beta$.

The RGEs for the Yukawa couplings from $M_{U}$ to the intermediate scale
$M_{I}$ and from $M_{I}$ to the electroweak scale $M_{Z}$ up to the two-loop
level have been given in Ref. [31] in the standard case with one electroweak
Higgs doublet only. In our case, we also include the additional contributions
of the extra Higgs doublet at the low scale. We solve the system using again
the program SARAH [29].

Figure 2: An example of the running of third generation fermion Yukawa
couplings from the weak to the high scales for the value $\tan\beta=58$ for
which the bottom and tau couplings unify at a scale $M_{b\tau}=6.6\times
10^{10}$ (left) and the dependence of this unification scale $M_{b\tau}$ on
the value of $\tan\beta$ (right).

In the PS model, the running of the third generation Yukawa couplings from the
low to the high energy scales are shown in the left panel of Fig. 2, for the
specific case where the input value $\tan\beta=58$ is chosen. One can see
that, indeed, the curves for $Y_{b}$ and $Y_{\tau}$ intersect at an energy
scale $M_{b\tau}\\!\simeq\\!7\\!\times\\!10^{10}$ GeV, which is very close to
the intermediate scale for which the gauge couplings unify in the PS scheme.

The right panel of the figure shows the dependence of the bottom-tau
unification scale $M_{b\tau}$ on the ratio of vevs $\tan\beta$ and, as can be
seen, intermediate scale values between $M_{I}=10^{9}$ GeV and $M_{I}=10^{11}$
GeV would imply high values of $\tan\beta$, in the range $\tan\beta\approx
50\\!-\\!60$. Note that the value of $\tan\beta$ cannot be arbitrary high in
order to avoid that the Yukawa couplings run to non-perturbative values at
these scales.

In the minimal LR model, the discussion above does not hold and the bottom and
tau Yukawa couplings do not unify at the intermediate scale. Nevertheless, one
should have close if not equal values for the Yukawa terms $Y_{10,q}^{LR}$ and
$Y_{10,l}^{LR}$ such that they can run to a common value at the scale $M_{U}$
where, according to eq. (18), one has $Y_{10,q}^{LR}=Y_{10,l}^{LR}$.

We come now to the unification of all Yukawa couplings. With the four Yukawa
couplings, the randomly chosen one $Y_{f}(M_{U})$ and the three weak scale
ones $Y_{t},Y_{b},Y_{\tau}$, the scales $M_{U}$ and $M_{I}$ to be determined
from gauge coupling unification, five parameters are needed to entirely
describe our Yukawa sector: the 2HDM vevs $v_{u},v_{d}$ at $M_{Z}$ and the
vevs $v_{126}^{u},v_{126}^{d}$ and $v_{10}$ of the bi-doublets at the scale
$M_{I}$. Nevertheless, we have many constraints at hand as, besides the
matching relations given in eqs. (16-18), one needs to reproduce the
experimental values of the standard particle masses.

Indeed, at both the scales $M_{Z}$ and $M_{I}$, the correct $W,Z$ masses
should be reproduced, giving $\sqrt{v_{u}^{2}\\!+\\!v_{d}^{2}}\\!=\\!v_{\rm
SM}\\!=\\!246\,{\rm
GeV}\\!=\\!v_{10}^{2}\\!+\\!v_{126}^{2}\\!+\\!v_{126d}^{2}$[24]. One needs
also to reproduce the heavy fermion masses at the weak scale $M_{Z}$, eq.
(19), using the relations of eq. (13). We will assume that there is an
uncertainty of the order of $2\%$ in reproducing all these masses, an
uncertainty that is not only due to the experimental errors (on $\alpha_{3}$
and the fermion masses for instance) but which is supposed to account for the
whole procedure, including the higher order effects in the RGEs among other
sources.

Figure 3: The renormalisation group running of the Yukawa couplings at two-
loop in the 2HDM+${\cal G}_{422}$ model (top) and 2HDM+${\cal G}_{3221}$ model
(bottom) including matching conditions and threshold effects at an
intermediate scale $M_{I}$ for a parameter set that is compatible with the
observed top, bottom, tau as well as gauge boson masses.

For completeness, we make sure in addition that the electroweak vacuum remains
stable up to the intermediate scale $M_{I}\approx 10^{10}$ GeV for the chosen
top quark and SM-like Higgs boson masses. To do so, we use the necessary and
sufficient conditions of Refs. [17, 32] on the 2HDM quartic scalar couplings
to ensure the scalar potential to be bounded from below, with the input values
for the relevant weak scale parameters given in Ref. [33]. As an additional
and final constraint, we force the three Yukawa couplings to remain
perturbative at all scales by imposing the conditions
$Y_{i}^{2}(\mu)/(4\pi)\leq 1$.

We then scan this constrained parameter space in order to find a viable
solution to the system of equations. Our main results are displayed in Fig. 3
and in Table 1.

Fig. 3 shows the evolution of the three Yukawa couplings as a function of the
energy scale in the PS (upper panel) and LR (lower panel) scenarios and it can
be seen that all of them reach a common value, $Y_{f}(M_{U})\approx 0.1$, at
the same GUT scale $M_{U}$ that leads to gauge coupling unification eq. (3
Gauge coupling unification). At the intermediate scale $M_{I}$ that is also
required by gauge coupling unification, eq. (3 Gauge coupling unification),
one notices the discontinuity for the Yukawa couplings which is due to the
matching conditions.

Finally, in Table 1, we show examples of points in the 2HDM+PS and 2HDM+LR
model parameter spaces that satisfy all the criteria discussed above and list
the sets of values for the three fermion couplings and all the relevant vevs
which lead to Yukawa coupling unification, with the GUT and intermediate
scales that allow for gauge coupling unification, eqs. (3 Gauge coupling
unification), and with all constraints implemented.

One can see that in both the PS and LR breaking schemes, one obtains
approximately the same unified Yukawa coupling $Y_{f}(M_{U})={\cal O}(0.1)$.
One can also see that the relations $\sum_{i}v_{i}^{2}\\!=\\!v_{\rm SM}^{2}$
are fulfilled at the relevant scales and that all Yukawa couplings are such
that their squares are smaller than $4\pi$ even at $M_{I}$.

In both cases, the obtained values of the input 2HDM parameter
$\tan\beta\\!=\\!v_{u}/v_{d}$ at the electroweak scale are large but still
reasonable, approximately $\tan\beta\\!=\\!58$ and $\tan\beta\\!=\\!70$ for
the PS and LR models respectively, as they ensure that the bottom quark Yukawa
coupling remains perturbative at all energy scales before $M_{U}$, and give
the correct hierarchy of quark masses at the electroweak scale,
$\tan\beta\\!\approx\\!m_{t}/m_{b}$.

scale |  | $M_{Z}$ |  |  | $M_{I}$ |  | $M_{U}$ |  | $M_{Z}\hskip 14.22636pt$ |  | $M_{I}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---
| $Y_{t}$ | $Y_{b}$ | $Y_{\tau}$ | $Y_{t}$ | $Y_{b}$ | $Y_{\tau}$ | $Y_{f}$ | $v_{d}$ | $v_{u}$ | $v_{10}$ | $v_{126}^{u}$ | $v_{126}^{d}$
PS | 0.97 | 1.09 | 0.58 | 0.55 | 0.76 | 0.75 | 0.10 | 4.21 | 246.2 | 23.4 | 244.7 | 0.004
LR | 0.97 | 1.44 | 0.68 | 0.62 | 2.76 | 2.01 | 0.14 | 3.50 | 246.2 | 53.2 | 241.0 | 0.079
Table 1: A set of the third generation fermion Yukawa couplings at the scales
$M_{Z},M_{I}$ and $M_{U}$, and the relevant vevs at the weak and intermediate
scales, which fit all observables within 2% accuracy at the two-loop level and
lead to both gauge coupling and Yukawa coupling unification in SO(10) with PS
and LR intermediate breaking.

Hence, Yukawa coupling unification can also be achieved in a simple manner in
a non-supersymmetric SO(10) scenario. One can arrange to achieve it for lower
values of $\tan\beta$ than above, at a minimal cost and without affecting the
simplicity of the approach, by complexifying the $\mathbf{10_{H}}$
representation. One still makes use of two Higgs bi-doublets above the scale
$M_{I}$ but there are four non-zero vevs instead of three as $v_{10}^{u}\neq
v_{10}^{d}$. This additional input can be adjusted to have more adequate
solutions to the system of Yukawa coupling RGEs. This possibility, as well as
other interesting extensions of the simple scheme proposed here, will be
addressed in a forthcoming publication.

## 5 Conclusions

We have analyzed the possibility of unifying the Yukawa couplings of third
generation fermions in the context of a non-supersymmetric SO(10) scenario
with intermediate breaking, focusing on the Pati-Salam and minimal left-right
breaking chains. The framework that we adopt is rather simple as the relevant
scalar sector of the theory consists of only two Higgs bi-doublets at the
intermediate breaking scale, $M_{I}={\cal O}(10^{10})$ GeV, reducing to a two
Higgs doublet model of type II at the electroweak scale.

We first discussed gauge coupling unification which can indeed be achieved at
a GUT scale close to $M_{U}\approx 10^{16}$ GeV, by including the threshold
effects of the scalar multiplets that appear at the intermediate scale. This
is somehow expected as the contribution of the additional electroweak Higgs
doublet (and all scalar fields in general) does not significantly modify the
running of the gauge coupling constants.

We have then studied the renormalisation group running of the Yukawa couplings
of the top and bottom quarks and the tau lepton in the Pati-Salam and minimal
left-right SO(10) scenarios, with the proper matching conditions at the
unification, intermediate and electroweak scales. We have performed a scan of
the parameter space of the two models, imposing that the phenomenology at low
energy and, in particular the third generation fermion and the electroweak
gauge boson masses, is correctly reproduced within 2% accuracy. We find that
the unification of the Yukawa couplings of third generation can be indeed
realized in regions of the parameter space in which the ratio of the two
electroweak Higgs doublet vevs is large, $\tan\beta\approx 60$.

Hence, similarly to the well known and widely studied supersymmetric case, not
only gauge but also Yukawa coupling unification can be achieved in SO(10)
while using a rather simple Higgs sector and retaining a viable particle
spectrum at the weak scale.

An interesting feature of this possibility is that while most of the
ingredients of the conventional SO(10) model are expected to be at a too high
scale, ${\cal O}(10^{10})$ GeV, to be probed effectively in collider
experiments, our scenario requires a second Higgs doublet at low energies. The
model thus predicts additional Higgs particles with weak scale masses which
could be searched for and eventually be observed at the Large Hadron Collider
or at the next generation of high-energy colliders.

Acknowledgement: We thank Paco del Aguila, Renato Fonseca, Luca Marzola and
Davide Meloni for discussions and comments on the manuscript, and Jérémie
Quevillon for collaboration at an early stage of the study. This work was
supported by the Estonian Research Council grants MOBTT86, PRG803 and by the
EU through the European Regional Development Fund CoE program TK133 “The Dark
Side of the Universe.” A.D. is also supported by the Junta de Andalucia
through the Talentia Senior program as well as by A-FQM-211-UGR18, P18-FR-4314
with ERDF.

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