#  IFT-UAM/CSIC-19-020
LAPTH-004/19
Reduction of Couplings and its application
in Particle Physics

S. Heinemeyer1,2,3, email<EMAIL_ADDRESS>M. Mondragón4, email:
<EMAIL_ADDRESS>N. Tracas5, and G. Zoupanos5,6,7,8,
1Instituto de Física Teórica (UAM/CSIC), Universidad Autonóma de Madrid, 28049
Madrid, Spain
2Campus of International Excellence UAM+CSIC, Cantoblanco, 28049, Madrid,
Spain
3Instituto de Física de Cantabria (CSIC-UC), E-39005 Santander, Spain
4Instituto de Física, Universidad Nacional Autónoma de México, A.P. 20-364,
CDMX 01000, México
5Physics Department, Nat. Technical University, 157 80 Zografou, Athens,
Greece
6Institute of Theoretical Physics, D-69120 Heidelberg, Germany
7Max-Planck Institut für Physik, Föhringer Ring 6, D-80805 München, Germany
8Laboratoire d’Annecy-le-Vieux de Physique Théorique, Annecy, France
Dedicated to the memory of Wolfhart Zimmermann,
the brilliant theoretical physicist,
who initiated the subject of reduction of couplings email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The idea of reduction of couplings in renormalizable theories will be
presented and then will be applied in Particle Physics models. Reduced
couplings appeared as functions of a primary one, compatible with the
renormalization group equation and thus solutions of a specific set of
ordinary differential equations. If these functions have the form of power
series the respective theories resemble standard renormalizable ones and thus
widen considerably the area covered until then by symmetries as a tool for
constraining the number of couplings consistently. Still on the more abstract
level reducing couplings enabled one to construct theories with beta-functions
vanishing to all orders of perturbation theory. Reduction of couplings became
physics-wise truly interesting and phenomenologically important when applied
to the standard model and its possible extensions. In particular in the
context of supersymmetric theories it became the most powerful tool known
today once it was learned how to apply it also to couplings having dimension
of mass and to mass parameters. Technically this all relies on the basic
property that reducing couplings is a renormalization scheme independent
procedure. Predictions of top and Higgs mass prior to their experimental
finding highlight the fundamental physical significance of this notion.

Prologue and Synopsis

In spite of their limitations, perturbative local field theories are still of
prominent practical value.

It is remarkable that the intrinsic ambiguities connected with locality and
causality - most of the time associated with ultraviolet infinities - can be
summarized in terms of a formal group which acts in the space of the coupling
constants or coupling functions attached to each type of local interaction.

It is therefore natural to look systematically for stable submanifolds. Some
such have been known for a long time: e.g., spaces of renormalizable
interactions and subspaces characterized by system of Ward identities mostly
related to symmetries.

A systematic search for such stable submanifolds has been initiated by W.
Zimmermann in the early eighties.

Disappointing for some time, this program has attracted several other active
researchers and recently produced physically interesting results.

It looks at the moment as the only theoretically founded algorithm potentially
able to decrease the number of parameters within the physically favoured
perturbative models aaa The above text has appeared, as Geleitwort (preface),
in the book “Reduction of Couplings and its Application in Particle Physics
Finite Theories Higgs and Top Mass Predictions”, Ed. Klaus Sibold, Authors:
Jisuke Kubo, Sven Heinemeyer, Myriam Mondragon, Olivier Piguet, Klaus Sibold,
Wolfhart Zimmermann, George Zoupanos. Published in PoS (Higgs & top)001. .

Raymond Stora, CERN (Switzerland), December 16, 2013

###### Contents

  1. 1 Introduction: The Basic Ideas
  2. 2 Theoretical Basis
    1. 2.1 Reduction of Dimensionless Parameters
    2. 2.2 Reduction of Couplings in N = 1 Supersymmetric Gauge Theories. Partial Reduction
    3. 2.3 Reduction of Dimension-1 and -2 Parameters
    4. 2.4 Reduction of Couplings of Soft Breaking Terms in $N=1$ Suspersymmetric Theories
  3. 3 Reduction of Couplings in the Standard Model and Predictions
  4. 4 Finiteness
    1. 4.1 The idea behind finiteness
    2. 4.2 Finiteness in N=1 Supersymmetric Gauge Theories
  5. 5 Reduction of Couplings in Phenomenologically Viable Models
    1. 5.1 Finite Unified Models
      1. 5.1.1 FUT A
      2. 5.1.2 FUT B
      3. 5.1.3 Predictions for Quark Masses
    2. 5.2 Reduction of Couplings in the Minimal Supersymmetric $SU(5)$ GUT
    3. 5.3 Gauge-Yukawa Unification in other Supersymmetric GUTs by Reducing the Couplings
      1. 5.3.1 Asymptotically Non-Free Pati-Salam Model
      2. 5.3.2 Asymptotically Non-Free $SO(10)$ Model
    4. 5.4 Finite $SU(N)^{3}$ Unification
    5. 5.5 Reduction of Couplings in the MSSM
    6. 5.6 Comments on the Gauge-Yukawa unification
  6. 6 Low Energy Phenomenology of the Finite Unified Model and the Reduced MSSM
    1. 6.1 Phenomenological Constraints
    2. 6.2 Numerical Analysis of the FUT
      1. 6.2.1 FUT Predictions for Future Colliders
      2. 6.2.2 FUT Conclusions
    3. 6.3 Numerical Analysis of the Reduced MSSM
      1. 6.3.1 Free Parameters of the Reduced MSSM
      2. 6.3.2 Constraints from Fermion Masses
      3. 6.3.3 Predictions of the reduced MSSM for future colliders
      4. 6.3.4 Conclusions about the Reduced MSSM
  7. 7 Conclusions

##  Chapter 1 Introduction: The Basic Ideas

In the recent years the theoretical endeavours that attempt to achieve a
deeper understanding of Nature have presented a series of successes in
developing frameworks such as String Theories and Noncommutativity that aim to
describe the fundamental theory at the Planck scale. However, the essence of
all theoretical efforts in Elementary Particle Physics (EPP) is to understand
the present day free parameters of the Standard Model (SM) in terms of few
fundamental ones, i.e. to achieve reductions of couplings[1]. Unfortunately,
despite the several successes in the above frameworks they do not offer
anything in the understanding of the free paramaters of the SM. The pathology
of the plethora of free parameters is deeply connected to the presence of
infinities at the quantum level. The renormalization program can remove the
infinities by introducing counterterms, but only at the cost of leaving the
corresponding terms as free parameters.

Although the Standard Model (SM) has been very successful in describing
elementary particles and its interactions, it has been known for some time
that it must be the low energy limit of a more fundamental theory. This quest
for a theory beyond the Standard Model (BSM) has expanded in various
directions. The usual, and very efficient, way of reducing the number of free
parameters of a theory to render it more predictive, is to introduce a
symmetry. Grand Unified Theories (GUTs) are very good examples of such a
procedure [2, 3, 4, 5, 6, 7]. First in the case of minimal $SU(5)$, because of
the (approximate) gauge coupling unification, it was possible to reduce the
gauge couplings of the SM and give a prediction for one of them. By adding a
further symmetry, namely $N=1$ global supersymmetry [8, 9, 10] it was possible
to make the prediction viable. GUTs can also relate the Yukawa couplings among
themselves, again $SU(5)$ provided an example of this by predicting the ratio
$M_{\tau}/M_{b}$ [11] in the SM. Unfortunately, requiring more gauge symmetry
does not seem to help, since additional complications are introduced due to
new degrees of freedom, for instance in the ways and channels of breaking the
symmetry.

A possible way to look for relations among unrelated parameters is the method
of reduction of couplings [12, 13, 14]; see also refs [15, 16, 17]. This
method, as its name proclaims, reduces the number of couplings in a theory by
relating either all or a number of couplings to a single coupling denoted as
the “primary coupling”. This method might help to identify hidden symmetries
in a system, but it is also possible to have reduction of couplings in systems
where there is no apparent symmetry. The reduction of couplings is based on
the assumption that both the original and the reduced theory are
renormalizable and that there exist renormalization group invariant (RGI)
relations among parameters.

A natural extension of the GUT idea and successful application of the method
of reduction of couplings is to find a way to relate the gauge and Yukawa
sectors of a theory, that is to achieve gauge-Yukawa Unification (GYU). This
will be presented in Chapter 5. Following the original suggestion for reducing
the couplings within the framework of GUTs we were hunting for renormalization
group invariant (RGI) relations holding below the Planck scale, which in turn
are preserved down to the GUT scale. It is indeed an impressive observation
that one can guarantee the validity of the RGI relations to all-orders in
perturbation theory by studying the uniqueness of the resulting relations at
one-loop. Even more remarkable is the fact that it is possible to find RGI
relations among couplings that guarantee finiteness to all-orders in
perturbation theory. The above principles have only been applied in $N=1$
supersymmetric GUTs for reasons that will be transparent in the following
sections, here we should only note that the use of $N=1$ supersymmetric GUTs
comprises the demand of the cancellation of quadratic divergencies in the SM.
The above GYU program applied in the dimensionless couplings of supersymmetric
GUTs had a great success by predicting correctly, among others, the top quark
mass in the finite [18, 19] and in the minimal $N=1$ supersymmetric $SU(5)$
[20] before its discovery [21].

Although supersymmetry seems to be an essential feature for a successful
realization of the above program, its breaking has to be understood too, since
it has the ambition to supply the SM with predictions for several of its free
parameters. Indeed, the search for RGI relations has been extended to the soft
supersymmetry breaking sector (SSB) of these theories, which involves
parameters of dimension one and two. In addition, there was important progress
concerning the renormalization properties of the SSB parameters, based on the
powerful supergraph method for studying supersymmetric theories, and it was
applied to the softly broken ones by using the “spurion” external space-time
independent superfields. According to this method a softly broken
supersymmetric gauge theory is considered as a supersymmetric one in which the
various parameters, such as couplings and masses, have been promoted to
external superfields. Then, relations among the soft term renormalization and
that of an unbroken supersymmetric theory have been derived. In particular the
$\beta$-functions of the parameters of the softly broken theory are expressed
in terms of partial differential operators involving the dimensionless
parameters of the unbroken theory. The key point in solving the set of coupled
differential equations so as to be able to express all parameters in a RGI
way, was to transform the partial differential operators involved to total
derivative operators. It is indeed possible to do this by choosing a suitable
RGI surface.

On the phenomenological side the application on the reduction of coupling
method to $N=1$ supersymmetric theories has led to very interesting
developments too. Previously an appealing “universal” set of soft scalar
masses was assumed in the SSB sector of supersymmetric theories, given that
apart from economy and simplicity (1) they are part of the constraints that
preserve finiteness up to two-loops, (2) they appear in the attractive dilaton
dominated supersymmetry breaking superstring scenarios. However, further
studies have exhibited a number of problems, all due to the restrictive nature
of the “universality” assumption for the soft scalar masses. Therefore, there
were attempts to relax this constraint without loosing its attractive
features. Indeed an interesting observation on $N=1$ GYU theories is that
there exists a RGI sum rule for the soft scalar masses at lower orders in
perturbation theory, which was later extended to all-orders, and manages to
overcome all the unpleasant phenomenological consequences. Armed with the
above tools and results we were in a position to study the spectrum of the
full finite models in terms of few free parameters, with emphasis on the
predictions of supersymmetric particles and the lightest Higgs mass.

The result was indeed very impressive since it led to a prediction of the
Higgs mass which coincided with the results of the LHC for the Higgs mass by
ATLAS [22, 23] and CMS [24, 25], and predicted a relatively heavy spectrum
consistent with the non-observation of supersymmetric particles at the LHC.
The coloured supersymmetric particles are predicted to be above 2.7 TeV, while
the electroweak supersymmetric spectrum starts below 1 TeV. These successes
will be presented in Chapter 6.

Last but certainly not least, the above machinery has been recently applied in
the MSSM with impressive results concerning the predictivity of the top,
bottom and Higgs masses, being at the same time consistent with the non-
observation of supersymmeric particles at the LHC. More specifically the
electroweak supersymmetric spectrum starts at 1.3 TeV and the coloured at
$\sim 4$ TeV. These results will be presented too in Chapter 6.

##  Chapter 2 Theoretical Basis

###  2.1 Reduction of Dimensionless Parameters

In this section we outline the idea of reduction of couplings. Any RGI
relation among couplings (i.e. which does not depend on the renormalization
scale $\mu$ explicitly) can be expressed, in the implicit form
$\Phi(g_{1},\cdots,g_{A})~{}=~{}\mbox{const.}$, which has to satisfy the
partial differential equation (PDE)

$\mu\,\frac{d\Phi}{d\mu}={\vec{\nabla}}\Phi\cdot{\vec{\beta}}~{}=~{}\sum_{a=1}^{A}\,\beta_{a}\,\frac{\partial\Phi}{\partial
g_{a}}~{}=~{}0~{},$ (2.1)

where $\beta_{a}$ is the $\beta$-function of $g_{a}$. This PDE is equivalent
to a set of ordinary differential equations, the so-called reduction equations
(REs) [12, 13, 14],

$\beta_{g}\,\frac{dg_{a}}{dg}=\beta_{a}~{},~{}a=1,\cdots,A~{},$ (2.2)

where $g$ and $\beta_{g}$ are the primary coupling and its $\beta$-function,
and the counting on $a$ does not include $g$. Since maximally ($A-1$)
independent RGI “constraints” in the $A$-dimensional space of couplings can be
imposed by the $\Phi_{a}$’s, one could in principle express all the couplings
in terms of a single coupling $g$. However, a closer look to the set of Eqs.
(2.2) reveals that their general solutions contain as many integration
constants as the number of equations themselves. Thus, using such integration
constants we have just traded an integration constant for each ordinary
renormalized coupling, and consequently, these general solutions cannot be
considered as reduced ones. The crucial requirement in the search for RGE
relations is to demand power series solutions to the REs,

$g_{a}=\sum_{n}\rho_{a}^{(n)}\,g^{2n+1}~{},$ (2.3)

which preserve perturbative renormalizability. Such an ansatz fixes the
corresponding integration constant in each of the REs and picks up a special
solution out of the general one. Remarkably, the uniqueness of such power
series solutions can be decided already at the one-loop level [12, 13, 14]. To
illustrate this, let us assume that the $\beta$-functions have the form

$\begin{split}\beta_{a}&=\frac{1}{16\pi^{2}}\left[\sum_{b,c,d\neq
g}\beta^{(1)\,bcd}_{a}g_{b}g_{c}g_{d}+\sum_{b\neq
g}\beta^{(1)\,b}_{a}g_{b}g^{2}\right]+\cdots~{},\\\
\beta_{g}&=\frac{1}{16\pi^{2}}\beta^{(1)}_{g}g^{3}+\cdots~{},\end{split}$
(2.4)

where $\cdots$ stands for higher order terms, and $\beta^{(1)\,bcd}_{a}$’s are
symmetric in $b,c,d$. We then assume that the $\rho_{a}^{(n)}$’s with $n\leq
r$ have been uniquely determined. To obtain $\rho_{a}^{(r+1)}$’s, we insert
the power series (2.3) into the REs (2.2) and collect terms of ${\cal
O}(g^{2r+3})$ and find

$\sum_{d\neq g}M(r)_{a}^{d}\,\rho_{d}^{(r+1)}=\mbox{lower order
quantities}~{},$

where the r.h.s. is known by assumption, and

$\displaystyle M(r)_{a}^{d}$ $\displaystyle=3\sum_{b,c\neq
g}\,\beta^{(1)\,bcd}_{a}\,\rho_{b}^{(1)}\,\rho_{c}^{(1)}+\beta^{(1)\,d}_{a}-(2r+1)\,\beta^{(1)}_{g}\,\delta_{a}^{d}~{},$
(2.5) $\displaystyle 0$ $\displaystyle=\sum_{b,c,d\neq
g}\,\beta^{(1)\,bcd}_{a}\,\rho_{b}^{(1)}\,\rho_{c}^{(1)}\,\rho_{d}^{(1)}+\sum_{d\neq
g}\beta^{(1)\,d}_{a}\,\rho_{d}^{(1)}-\beta^{(1)}_{g}\,\rho_{a}^{(1)}~{}.$
(2.6)

Therefore, the $\rho_{a}^{(n)}$’s for all $n>1$ for a given set of
$\rho_{a}^{(1)}$’s can be uniquely determined if $\det M(n)_{a}^{d}\neq 0$ for
all $n\geq 0$.

As it will be clear later by examining specific examples, the various
couplings in supersymmetric theories have the same asymptotic behaviour.
Therefore searching for a power series solution of the form (2.3) to the REs
(2.2) is justified.

The possibility of coupling unification described in this section is without
any doubt attractive because the “completely reduced” theory contains only one
independent coupling, but it can be unrealistic. Therefore, one often would
like to impose fewer RGI constraints, and this is the idea of partial
reduction [26, 27].

The above facts lead us to suspect that there is and intimate connection among
the requirement of reduction of couplings and supersymmetry which still waits
to be uncovered. The connection becomes more clear by examining the following
example.

Consider an $SU(N)$ gauge theory with the following matter content:
$\phi^{i}({\bf N})$ and $\hat{\phi}_{i}(\overline{\bf N})$ are complex
scalars, $\psi^{i}({\bf N})$ and $\hat{\psi}_{i}(\overline{\bf N})$ are left-
handed Weyl spinor, and $\lambda^{a}(a=1,\dots,N^{2}-1)$ is a right-handed
Weyl spinor in the adjoint representation of $SU(N)$.

The Lagrangian, omitting kinetic terms, includes:

${\cal L}\supset
i\sqrt{2}\\{~{}g_{Y}\overline{\psi}\lambda^{a}T^{a}\phi-\hat{g}_{Y}\overline{\hat{\psi}}\lambda^{a}T^{a}\hat{\phi}+\mbox{h.c.}~{}\\}-V(\phi,\overline{\phi}),$
(2.7)

where

$V(\phi,\overline{\phi})=\frac{1}{4}\lambda_{1}(\phi^{i}\phi^{*}_{i})^{2}+\frac{1}{4}\lambda_{2}(\hat{\phi}_{i}\hat{\phi}^{*~{}i})^{2}+\lambda_{3}(\phi^{i}\phi^{*}_{i})(\hat{\phi}_{j}\hat{\phi}^{*~{}j})+\lambda_{4}(\phi^{i}\phi^{*}_{j})(\hat{\phi}_{i}\hat{\phi}^{*~{}j}),$
(2.8)

which is the most general renormalizable form of dimension four, consistent
with the $SU(N)\times SU(N)$ global symmetry.

Searching for a solution of the form of Eq. (2.3) for the REs (2.2,) we find
in lowest order the following one ($g$ is the gauge coupling):

$\begin{split}g_{Y}&=\hat{g}_{Y}=g~{},\\\
\lambda_{1}&=\lambda_{2}=\frac{N-1}{N}g^{2}~{},\\\
\lambda_{3}&=\frac{1}{2N}g^{2}~{},~{}\lambda_{4}=-\frac{1}{2}g^{2}~{},\end{split}$
(2.9)

which corresponds to an $N=1$ supersymmetric gauge theory. Clearly the above
remarks do not answer the question of the relation among reduction of
couplings and supersymmetry but rather try to trigger the interest for further
investigation.

###  2.2 Reduction of Couplings in N = 1 Supersymmetric Gauge Theories.
Partial Reduction

Let us consider a chiral, anomaly free, $N=1$ globally supersymmetric gauge
theory based on a group G with gauge coupling constant $g$. The superpotential
of the theory is given by

$\displaystyle W$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,m_{ij}\,\phi_{i}\,\phi_{j}+\frac{1}{6}\,C_{ijk}\,\phi_{i}\,\phi_{j}\,\phi_{k}~{},$
(2.10)

where $m_{ij}$ and $C_{ijk}$ are gauge invariant tensors and the matter field
(chiral superfield) $\phi_{i}$ transforms according to the irreducible
representation $R_{i}$ of the gauge group $G$. The renormalization constants
associated with the superpotential (2.10), assuming that supersymmetry is
preserved, are

$\displaystyle\phi_{i}^{0}$
$\displaystyle=\left(Z^{j}_{i}\right)^{(1/2)}\,\phi_{j}~{},~{}$ (2.11)
$\displaystyle m_{ij}^{0}$
$\displaystyle=Z^{i^{\prime}j^{\prime}}_{ij}\,m_{i^{\prime}j^{\prime}}~{},~{}$
(2.12) $\displaystyle C_{ijk}^{0}$
$\displaystyle=Z^{i^{\prime}j^{\prime}k^{\prime}}_{ijk}\,C_{i^{\prime}j^{\prime}k^{\prime}}~{}.$
(2.13)

The $N=1$ non-renormalization theorem [28, 29, 30, 31] ensures that there are
no mass and cubic-interaction-term infinities and therefore

$\begin{split}Z_{ij}^{i^{\prime}j^{\prime}}\left(Z^{i^{\prime\prime}}_{i^{\prime}}\right)^{(1/2)}\left(Z^{j^{\prime\prime}}_{j^{\prime}}\right)^{(1/2)}&=\delta_{(i}^{i^{\prime\prime}}\,\delta_{j)}^{j^{\prime\prime}}~{},\\\
Z_{ijk}^{i^{\prime}j^{\prime}k^{\prime}}\left(Z^{i^{\prime\prime}}_{i^{\prime}}\right)^{(1/2)}\left(Z^{j^{\prime\prime}}_{j^{\prime}}\right)^{(1/2)}\left(Z^{k^{\prime\prime}}_{k^{\prime}}\right)^{(1/2)}&=\delta_{(i}^{i^{\prime\prime}}\,\delta_{j}^{j^{\prime\prime}}\delta_{k)}^{k^{\prime\prime}}~{}.\end{split}$
(2.14)

As a result the only surviving possible infinities are the wave-function
renormalization constants $Z^{j}_{i}$, i.e., one infinity for each field. The
one-loop $\beta$-function of the gauge coupling $g$ is given by [32, 33, 34,
35, 36]

$\beta^{(1)}_{g}=\frac{dg}{dt}=\frac{g^{3}}{16\pi^{2}}\left[\,\sum_{i}\,T(R_{i})-3\,C_{2}(G)\right]~{},$
(2.15)

where, as usual, $t$ is the logarithm of the ratio of the energy scale over a
reference scale, $C_{2}(G)$ is the quadratic Casimir of the adjoint
representation of the associated gauge group $G$ and $T(R)$ is given by the
relation $\textrm{Tr}[T^{a}T^{b}]=T(R)\delta^{ab}$ while $T^{a}$ is the
generators of the group in the appropriate representation. The
$\beta$-functions of $C_{ijk}$, by virtue of the non-renormalization theorem,
are related to the anomalous dimension matrix $\gamma_{ij}$ of the matter
fields $\phi_{i}$ as:

$\beta_{ijk}=\frac{dC_{ijk}}{dt}~{}=~{}C_{ijl}\,\gamma^{l}_{k}+C_{ikl}\,\gamma^{l}_{j}+C_{jkl}\,\gamma^{l}_{i}~{}.$
(2.16)

At one-loop level $\gamma^{i}_{j}$ is given by [32]

$\gamma^{(1)}{}_{j}^{i}=\frac{1}{32\pi^{2}}\,[\,C^{ikl}\,C_{jkl}-2\,g^{2}\,C_{2}(R_{i})\delta^{i}_{j}\,],$
(2.17)

where $C_{2}(R_{i})$ is the quadratic Casimir of the representation $R_{i}$,
and $C^{ijk}=C_{ijk}^{*}$. Since dimensional coupling parameters such as
masses and couplings of scalar field cubic terms do not influence the
asymptotic properties of a theory on which we are interested here, it is
sufficient to take into account only the dimensionless supersymmetric
couplings such as $g$ and $C_{ijk}$. So we neglect the existence of
dimensional parameters, and assume furthermore that $C_{ijk}$ are real so that
$C_{ijk}^{2}$ always are positive numbers. For our purposes, it is convenient
to work with the square of the couplings and to arrange $C_{ijk}$ in such a
way that they are covered by a single index $i~{}(i=1,\cdots,n)$:

$\alpha=\frac{g^{2}}{4\pi}~{},~{}\alpha_{i}~{}=~{}\frac{g_{i}^{2}}{4\pi}~{}.$
(2.18)

The evolution equations of $\alpha$’s in perturbation theory then take the
form

$\begin{split}\frac{d\alpha}{dt}&=\beta~{}=~{}-\beta^{(1)}\alpha^{2}+\cdots~{},\\\
\frac{d\alpha_{i}}{dt}&=\beta_{i}~{}=~{}-\beta^{(1)}_{i}\,\alpha_{i}\,\alpha+\sum_{j,k}\,\beta^{(1)}_{i,jk}\,\alpha_{j}\,\alpha_{k}+\cdots~{},\end{split}$
(2.19)

where $\cdots$ denotes the contributions from higher orders, and
$\beta^{(1)}_{i,jk}=\beta^{(1)}_{i,kj}$.

Given the set of the evolution equations (2.19), we investigate the asymptotic
properties, as follows. First we define [12, 14, 37, 38, 16]

$\tilde{\alpha}_{i}\equiv\frac{\alpha_{i}}{\alpha}~{},~{}i=1,\cdots,n~{},$
(2.20)

and derive from Eq. (2.19)

$\begin{split}\alpha\frac{d\tilde{\alpha}_{i}}{d\alpha}&=-\tilde{\alpha}_{i}+\frac{\beta_{i}}{\beta}=\left(-1+\frac{\beta^{(1)}_{i}}{\beta^{(1)}}\,\right)\tilde{\alpha}_{i}\\\
&-\sum_{j,k}\,\frac{\beta^{(1)}_{i,jk}}{\beta^{(1)}}\,\tilde{\alpha}_{j}\,\tilde{\alpha}_{k}+\sum_{r=2}\,\left(\frac{\alpha}{\pi}\right)^{r-1}\,\tilde{\beta}^{(r)}_{i}(\tilde{\alpha})~{},\end{split}$
(2.21)

where $\tilde{\beta}^{(r)}_{i}(\tilde{\alpha})~{}(r=2,\cdots)$ are power
series of $\tilde{\alpha}$’s and can be computed from the $r$-th loop
$\beta$-functions. Next we search for fixed points $\rho_{i}$ of Eq. (2.20) at
$\alpha=0$. To this end, we have to solve

$\left(-1+\frac{\beta^{(1)}_{i}}{\beta^{(1)}}\right)\rho_{i}-\sum_{j,k}\frac{\beta^{(1)}_{i,jk}}{\beta^{(1)}}\,\rho_{j}\,\rho_{k}=0~{},$
(2.22)

and assume that the fixed points have the form

$\rho_{i}=0~{}\mbox{for}~{}i=1,\cdots,n^{\prime}~{};~{}\rho_{i}~{}>0~{}\mbox{for}~{}i=n^{\prime}+1,\cdots,n~{}.$
(2.23)

We then regard $\tilde{\alpha}_{i}$ with $i\leq n^{\prime}$ as small
perturbations to the undisturbed system which is defined by setting
$\tilde{\alpha}_{i}$ with $i\leq n^{\prime}$ equal to zero. As we have seen,
it is possible to verify at the one-loop level [12, 13, 14, 37] the existence
of the unique power series solution

$\tilde{\alpha}_{i}=\rho_{i}+\sum_{r=2}\rho^{(r)}_{i}\,\alpha^{r-1}~{},~{}i=n^{\prime}+1,\cdots,n~{}$
(2.24)

of the reduction equations (2.21) to all orders in the undisturbed system.
These are RGI relations among couplings and keep formally perturbative
renormalizability of the undisturbed system. So in the undisturbed system
there is only one independent coupling, the primary coupling $\alpha$.

The small perturbations caused by nonvanishing $\tilde{\alpha}_{i}$ with
$i\leq n^{\prime}$ enter in such a way that the reduced couplings, i.e.
$\tilde{\alpha}_{i}$ with $i>n^{\prime}$, become functions not only of
$\alpha$ but also of $\tilde{\alpha}_{i}$ with $i\leq n^{\prime}$. It turned
out that, to investigate such partially reduced systems, it is most convenient
to work with the partial differential equations

$\begin{split}\left\\{\tilde{\beta}\,\frac{\partial}{\partial\alpha}+\sum_{a=1}^{n^{\prime}}\,\tilde{\beta_{a}}\,\frac{\partial}{\partial\tilde{\alpha}_{a}}\right\\}~{}\tilde{\alpha}_{i}(\alpha,\tilde{\alpha})&=\tilde{\beta}_{i}(\alpha,\tilde{\alpha})~{},\\\
\tilde{\beta}_{i(a)}~{}=~{}\frac{\beta_{i(a)}}{\alpha^{2}}-\frac{\beta}{\alpha^{2}}~{}\tilde{\alpha}_{i(a)}&,\qquad\tilde{\beta}~{}\equiv~{}\frac{\beta}{\alpha}~{},\end{split}$
(2.25)

which are equivalent to the reduction equations (2.21), where we let $a,b$ run
from $1$ to $n^{\prime}$ and $i,j$ from $n^{\prime}+1$ to $n$ in order to
avoid confusion. We then look for solutions of the form

$\tilde{\alpha}_{i}=\rho_{i}+\sum_{r=2}\,\left(\frac{\alpha}{\pi}\right)^{r-1}\,f^{(r)}_{i}(\tilde{\alpha}_{a})~{},~{}i=n^{\prime}+1,\cdots,n~{},$
(2.26)

where $f^{(r)}_{i}(\tilde{\alpha}_{a})$ are supposed to be power series of
$\tilde{\alpha}_{a}$. This particular type of solution can be motivated by
requiring that in the limit of vanishing perturbations we obtain the
undisturbed solutions (2.24) [27, 39]. Again it is possible to obtain the
sufficient conditions for the uniqueness of $f^{(r)}_{i}$ in terms of the
lowest order coefficients.

###  2.3 Reduction of Dimension-1 and -2 Parameters

The reduction of couplings was originally formulated for massless theories on
the basis of the Callan-Symanzik equation [12, 13]. The extension to theories
with massive parameters is not straightforward if one wants to keep the
generality and the rigor on the same level as for the massless case; one has
to fulfill a set of requirements coming from the renormalization group
equations, the Callan-Symanzik equations, etc. along with the normalization
conditions imposed on irreducible Green’s functions [40]. There has been a lot
of progress in this direction starting from ref. [41], as it is already
mentioned in the Introduction, where it was assumed that a mass-independent
renormalization scheme could be employed so that all the RG functions have
only trivial dependencies on dimensional parameters and then the mass
parameters were introduced similarly to couplings (i.e. as a power series in
the couplings). This choice was justified later in [42, 43] where the scheme
independence of the reduction principle has been proven generally, i.e it was
shown that apart from dimensionless couplings, pole masses and gauge
parameters, the model may also involve coupling parameters carrying a
dimension and masses. Therefore here, to simplify the analysis, we follow Ref.
[41] and make use also of a mass-independent renormalization scheme.

We start by considering a renormalizable theory which contain a set of $(N+1)$
dimension-zero couplings,
$\left(\hat{g}_{0},\hat{g}_{1},...,\hat{g}_{N}\right)$, a set of $L$
parameters with mass-dimension one,
$\left(\hat{h}_{1},...,\hat{h}_{L}\right)$, and a set of $M$ parameters with
mass-dimension two, $\left(\hat{m}_{1}^{2},...,\hat{m}_{M}^{2}\right)$. The
renormalized irreducible vertex function $\Gamma$ satisfies the RG equation

$\mathcal{D}\Gamma\left[\Phi^{\prime}s;\hat{g}_{0},\hat{g}_{1},...,\hat{g}_{N};\hat{h}_{1},...,\hat{h}_{L};\hat{m}_{1}^{2},...,\hat{m}_{M}^{2};\mu\right]=0~{},$
(2.27)

where

$\mathcal{D}=\mu\frac{\partial}{\partial\mu}+\sum_{i=0}^{N}\beta_{i}\frac{\partial}{\partial\hat{g}_{i}}+\sum_{a=1}^{L}\gamma_{a}^{h}\frac{\partial}{\partial\hat{h}_{a}}+\sum_{\alpha=1}^{M}\gamma_{\alpha}^{m^{2}}\frac{\partial}{\partial\hat{m}_{\alpha}^{2}}+\sum_{J}\Phi_{I}\gamma^{\phi
I}_{\,\,\,\,J}\,\frac{\delta}{\delta\Phi_{J}}~{},$ (2.28)

where $\mu$ is the energy scale, while $\beta_{i}$ are the $\beta$-functions
of the various dimensionless couplings $g_{i}$, $\Phi_{I}$ are the various
matter fields and $\gamma_{\alpha}^{m^{2}}$, $\gamma_{a}^{h}$ and
$\gamma^{\phi I}_{\,\,\,\,J}$ are the mass, trilinear coupling and wave
function anomalous dimensions, respectively (where $I$ enumerates the matter
fields). In a mass independent renormalization scheme, the $\gamma$’s are
given by

$\begin{split}\gamma^{h}_{a}&=\sum_{b=1}^{L}\gamma_{a}^{h,b}(g_{0},g_{1},...,g_{N})\hat{h}_{b},\\\
\gamma_{\alpha}^{m^{2}}&=\sum_{\beta=1}^{M}\gamma_{\alpha}^{m^{2},\beta}(g_{0},g_{1},...,g_{N})\hat{m}_{\beta}^{2}+\sum_{a,b=1}^{L}\gamma_{\alpha}^{m^{2},ab}(g_{0},g_{1},...,g_{N})\hat{h}_{a}\hat{h}_{b},\end{split}$
(2.29)

where $\gamma_{a}^{h,b}$, $\gamma_{\alpha}^{m^{2},\beta}$ and
$\gamma_{\alpha}^{m^{2},ab}$ are power series of the $g$’s (which are
dimensionless) in perturbation theory.

We look for a reduced theory where

$g\equiv g_{0},\qquad h_{a}\equiv\hat{h}_{a}\quad\textrm{for $1\leq a\leq
P$},\qquad m^{2}_{\alpha}\equiv\hat{m}^{2}_{\alpha}\quad\textrm{for
$1\leq\alpha\leq Q$}$

are independent parameters and the reduction of the remaining parameters

$\begin{split}\hat{g}_{i}&=\hat{g}_{i}(g),\qquad(i=1,...,N),\\\
\hat{h}_{a}&=\sum_{b=1}^{P}f_{a}^{b}(g)h_{b},\qquad(a=P+1,...,L),\\\
\hat{m}^{2}_{\alpha}&=\sum_{\beta=1}^{Q}e^{\beta}_{\alpha}(g)m^{2}_{\beta}+\sum_{a,b=1}^{P}k^{ab}_{\alpha}(g)h_{a}h_{b},\qquad(\alpha=Q+1,...,M)\end{split}$
(2.30)

is consistent with the RG equations (2.27,2.28). It turns out that the
following relations should be satisfied

$\begin{split}\beta_{g}\,\frac{\partial\hat{g}_{i}}{\partial
g}&=\beta_{i},\qquad(i=1,...,N),\\\
\beta_{g}\,\frac{\partial\hat{h}_{a}}{\partial
g}+\sum_{b=1}^{P}\gamma^{h}_{b}\,\frac{\partial\hat{h}_{a}}{\partial
h_{b}}&=\gamma^{h}_{a},\qquad(a=P+1,...,L),\\\
\beta_{g}\,\frac{\partial\hat{m}^{2}_{\alpha}}{\partial
g}+\sum_{a=1}^{P}\gamma_{a}^{h}\,\frac{\partial\hat{m}^{2}_{\alpha}}{\partial
h_{a}}+\sum_{\beta=1}^{Q}\gamma_{\beta}^{m^{2}}\,\frac{\partial\hat{m}_{\alpha}^{2}}{\partial
m_{\beta}^{2}}&=\gamma_{\alpha}^{m^{2}},\qquad(\alpha=Q+1,...,M).\end{split}$
(2.31)

Using Eqs. (2.29) and (2.30), the above relations reduce to

$\begin{split}&\beta_{g}\,\frac{df^{b}_{a}}{dg}+\sum_{c=1}^{P}f^{c}_{a}\left[\gamma^{h,b}_{c}+\sum_{d=P+1}^{L}\gamma^{h,d}_{c}f^{b}_{d}\right]-\gamma^{h,b}_{a}-\sum_{d=P+1}^{L}\gamma^{h,d}_{a}f^{b}_{d}=0,\\\
&\hskip 244.69386pt(a=P+1,...,L;\,b=1,...,P),\\\
&\beta_{g}\,\frac{de^{\beta}_{\alpha}}{dg}+\sum_{\gamma=1}^{Q}e^{\gamma}_{\alpha}\left[\gamma_{\gamma}^{m^{2},\beta}+\sum_{\delta=Q+1}^{M}\gamma_{\gamma}^{m^{2},\delta}e^{\beta}_{\delta}\right]-\gamma_{\alpha}^{m^{2},\beta}-\sum_{\delta=Q+1}^{M}\gamma_{\alpha}^{m^{2},d}e^{\beta}_{\delta}=0,\\\
&\hskip 236.15796pt(\alpha=Q+1,...,M;\,\beta=1,...,Q),\\\
&\beta_{g}\,\frac{dk_{\alpha}^{ab}}{dg}+2\sum_{c=1}^{P}\left(\gamma_{c}^{h,a}+\sum_{d=P+1}^{L}\gamma_{c}^{h,d}f_{d}^{a}\right)k_{\alpha}^{cb}+\sum_{\beta=1}^{Q}e^{\beta}_{\alpha}\left[\gamma_{\beta}^{m^{2},ab}+\sum_{c,d=P+1}^{L}\gamma_{\beta}^{m^{2},cd}f^{a}_{c}f^{b}_{d}\right.\\\
&\left.+2\sum_{c=P+1}^{L}\gamma_{\beta}^{m^{2},cb}f^{a}_{c}+\sum_{\delta=Q+1}^{M}\gamma_{\beta}^{m^{2},d}k_{\delta}^{ab}\right]-\left[\gamma_{\alpha}^{m^{2},ab}+\sum_{c,d=P+1}^{L}\gamma_{\alpha}^{m^{2},cd}f^{a}_{c}f^{b}_{d}\right.\\\
&\left.+2\sum_{c=P+1}^{L}\gamma_{\alpha}^{m^{2},cb}f^{a}_{c}+\sum_{\delta=Q+1}^{M}\gamma_{\alpha}^{m^{2},\delta}k_{\delta}^{ab}\right]=0,\\\
&\hskip 227.62204pt(\alpha=Q+1,...,M;\,a,b=1,...,P)~{}.\end{split}$ (2.32)

The above relations ensure that the irreducible vertex function of the reduced
theory

$\begin{split}\Gamma_{R}&\left[\Phi\textrm{'s};g;h_{1},...,h_{P};m_{1}^{2},...,m_{Q}^{2};\mu\right]\equiv\\\
&\Gamma\left[\Phi\textrm{'s};g,\hat{g}_{1}(g)...,\hat{g}_{N}(g);h_{1},...,h_{P},\hat{h}_{P+1}(g,h),...,\hat{h}_{L}(g,h);\right.\\\
&\left.\qquad\qquad\qquad
m_{1}^{2},...,m^{2}_{Q},\hat{m}^{2}_{Q+1}(g,h,m^{2}),...,\hat{m}^{2}_{M}(g,h,m^{2});\mu\right]\end{split}$
(2.33)

has the same renormalization group flow as the original one.

The assumption that the reduced theory is perturbatively renormalizable means
that the functions $\hat{g}_{i}$, $f^{b}_{a}$, $e^{\beta}_{\alpha}$ and
$k_{\alpha}^{ab}$, defined in Eq. (2.30), should be expressed as a power
series in the primary coupling $g$:

$\begin{split}\hat{g}_{i}&=g\sum_{n=0}^{\infty}\rho_{i}^{(n)}g^{n},\qquad
f_{a}^{b}=g\sum_{n=0}^{\infty}\eta_{a}^{b(n)}g^{n}\\\
e^{\beta}_{\alpha}&=\sum_{n=0}^{\infty}\xi^{\beta(n)}_{\alpha}g^{n},\qquad
k_{\alpha}^{ab}=\sum_{n=0}^{\infty}\chi_{\alpha}^{ab(n)}g^{n}.\end{split}$
(2.34)

The above expansion coefficients can be found by inserting these power series
into Eqs. (2.31), (2.32) and requiring the equations to be satisfied at each
order of $g$. It should be noted that the existence of a unique power series
solution is a non-trivial matter: It depends on the theory as well as on the
choice of the set of independent parameters.

It should also be noted that in the case that there are no independent mass-
dimension 1 parameters ($\hat{h}$) the reduction of these terms take naturally
the form

$\hat{h}_{a}=\sum_{b=1}^{L}f_{a}^{b}(g)M,$

where $M$ is a mass-dimension 1 parameter which could be a gaugino mass that
corresponds to the independent (gauge) coupling. Furthermore, if there are no
independent mass-dimension 2 parameters ($\hat{m}^{2}$), the corresponding
reduction takes the analogous form

$\hat{m}^{2}_{a}=\sum_{b=1}^{M}e_{a}^{b}(g)M^{2}.$

###  2.4 Reduction of Couplings of Soft Breaking Terms in $N=1$
Suspersymmetric Theories

The method of reducing the dimensionless couplings was extended[41, 44], as we
have discussed in the introduction, to the soft supersymmetry breaking (SSB)
dimensionful parameters of $N=1$ supersymmetric theories. In addition it was
found [45, 46] that RGI SSB scalar masses in Gauge-Yukawa unified models
satisfy a universal sum rule.

Consider the superpotential given by

$W=\frac{1}{2}\,\mu^{ij}\,\Phi_{i}\,\Phi_{j}+\frac{1}{6}\,C^{ijk}\,\Phi_{i}\,\Phi_{j}\,\Phi_{k}~{},$
(2.35)

along with the Lagrangian for SSB terms

$-{\cal L}_{\rm
SSB}=\frac{1}{6}\,h^{ijk}\,\phi_{i}\phi_{j}\phi_{k}+\frac{1}{2}\,b^{ij}\,\phi_{i}\phi_{j}+\frac{1}{2}\,(m^{2})^{j}_{i}\,\phi^{*\,i}\phi_{j}+\frac{1}{2}\,M\,\lambda\lambda+\mbox{H.c.},$
(2.36)

where the $\phi_{i}$ are the scalar parts of the chiral superfields
$\Phi_{i}$, $\lambda$ are the gauginos and $M$ their unified mass.

Let us recall (see Eqs.(2.15-2.17)) that the one-loop $\beta$-function of the
gauge coupling $g$ is given by [32, 33, 34, 35, 36]

$\displaystyle\beta^{(1)}_{g}=\frac{dg}{dt}=\frac{g^{3}}{16\pi^{2}}\,\left[\,\sum_{i}\,T(R_{i})-3\,C_{2}(G)\,\right]~{},$
(2.37)

the $\beta$-function of $C_{ijk}$ is given by

$\beta_{C}^{ijk}=\frac{dC_{ijk}}{dt}~{}=~{}C_{ijl}\,\gamma^{l}_{k}+C_{ikl}\,\gamma^{l}_{j}+C_{jkl}\,\gamma^{l}_{i}~{},$
(2.38)

and, at one-loop level, the anomalous dimension $\gamma^{(1)}\,{}^{i}_{j}$ of
the chiral superfield is

$\gamma^{(1)}\,{}^{i}_{j}=\frac{1}{32\pi^{2}}\,\left[\,C^{ikl}\,C_{jkl}-2\,g^{2}\,C_{2}(R_{i})\delta^{i}_{j}\,\right].$
(2.39)

Then, the $N=1$ non-renormalization theorem [28, 29, 31] ensures there are no
extra mass and cubic-interaction-term renormalizations, implying that the
$\beta$-functions of $C_{ijk}$ can be expressed as linear combinations of the
anomalous dimensions $\gamma^{i}_{j}$.

Here we assume that the reduction equations admit power series solutions of
the form

$C^{ijk}=g\,\sum_{n=0}\,\rho^{ijk}_{(n)}g^{2n}~{}.$ (2.40)

In order to obtain higher-loop results instead of knowledge of explicit
$\beta$-functions, which anyway are known only up to two-loops, relations
among $\beta$-functions are required.

Judicious use of the spurion technique, [31, 47, 48, 49, 50] leads to the
following all-loop relations among SSB $\beta$-functions (in an obvious
notation), [51, 111, 55, 54, 53, 56, 57]

$\displaystyle\beta_{M}$ $\displaystyle=2{\cal
O}\left(\frac{\beta_{g}}{g}\right)~{},$ (2.41) $\displaystyle\beta_{h}^{ijk}$
$\displaystyle=\gamma^{i}_{l}h^{ljk}+\gamma^{j}_{l}h^{ilk}+\gamma^{k}_{l}h^{ijl}$
$\displaystyle\,-2\left(\gamma_{1}\right)^{i}_{l}C^{ljk}-2\left(\gamma_{1}\right)^{j}_{l}C^{ilk}-2\left(\gamma_{1}\right)^{k}_{l}C^{ijl}~{},$
(2.42) $\displaystyle(\beta_{m^{2}})^{i}_{j}$
$\displaystyle=\left[\Delta+X\frac{\partial}{\partial
g}\right]\gamma^{i}_{j}~{},$ (2.43)

where

$\displaystyle{\cal O}$ $\displaystyle=\left(Mg^{2}\frac{\partial}{\partial
g^{2}}-h^{lmn}\frac{\partial}{\partial C^{lmn}}\right)~{},$ (2.44)
$\displaystyle\Delta$ $\displaystyle=2{\cal O}{\cal
O}^{*}+2|M|^{2}g^{2}\frac{\partial}{\partial
g^{2}}+\tilde{C}_{lmn}\frac{\partial}{\partial
C_{lmn}}+\tilde{C}^{lmn}\frac{\partial}{\partial C^{lmn}}~{},$ (2.45)
$\displaystyle(\gamma_{1})^{i}_{j}$ $\displaystyle={\cal O}\gamma^{i}_{j},$
(2.46) $\displaystyle\tilde{C}^{ijk}$
$\displaystyle=(m^{2})^{i}_{l}C^{ljk}+(m^{2})^{j}_{l}C^{ilk}+(m^{2})^{k}_{l}C^{ijl}~{}.$
(2.47)

The assumption, following [55], that the relation among couplings

$h^{ijk}=-M(C^{ijk})^{\prime}\equiv-M\frac{dC^{ijk}(g)}{d\ln g}~{},$ (2.48)

is RGI and furthermore, the use of the all-loop gauge $\beta$-function of
Novikov et al. [58, 59, 60] given by

$\beta_{g}^{\rm
NSVZ}=\frac{g^{3}}{16\pi^{2}}\left[\frac{\sum_{l}T(R_{l})(1-\gamma_{l}/2)-3C_{2}(G)}{1-g^{2}C_{2}(G)/8\pi^{2}}\right]~{},$
(2.49)

lead to the all-loop RGI sum rule [61] (assuming
$(m^{2})^{i}_{j}=m^{2}_{j}\delta^{i}_{j}$),

$\begin{split}m^{2}_{i}+m^{2}_{j}+m^{2}_{k}&=|M|^{2}\left\\{~{}\frac{1}{1-g^{2}C_{2}(G)/(8\pi^{2})}\frac{d\ln
C^{ijk}}{d\ln g}+\frac{1}{2}\frac{d^{2}\ln C^{ijk}}{d(\ln
g)^{2}}~{}\right\\}\\\
&\qquad\qquad+\sum_{l}\frac{m^{2}_{l}T(R_{l})}{C_{2}(G)-8\pi^{2}/g^{2}}\frac{d\ln
C^{ijk}}{d\ln g}~{}.\end{split}$ (2.50)

Surprisingly enough, the all-loop result of Eq.(2.50) coincides with the
superstring result for the finite case in a certain class of orbifold models
[62, 63, 46] if

$\frac{d\ln C^{ijk}}{d\ln g}=1~{},$

as discussed in ref. [19].

Let us now see how the all-loop results on the SSB $\beta$-functions,
Eqs.(2.41)-(2.47), lead to all-loop RGI relations. We assume:
(a) the existence of a RGI surfaces on which $C=C(g)$, or equivalently that
the expression

$\frac{dC^{ijk}}{dg}=\frac{\beta^{ijk}_{C}}{\beta_{g}}$ (2.51)

holds, i.e. reduction of couplings is possible, and
(b) the existence of a RGI surface on which

$h^{ijk}=-M\frac{dC(g)^{ijk}}{d\ln g}$ (2.52)

holds too in all-orders.
Then one can prove [64, 65], that the following relations are RGI to all-loops
(note that in both (a) and (b) assumptions above we do not rely on specific
solutions of these equations)

$\displaystyle M$ $\displaystyle=M_{0}~{}\frac{\beta_{g}}{g},$ (2.53)
$\displaystyle h^{ijk}$ $\displaystyle=-M_{0}~{}\beta_{C}^{ijk},$ (2.54)
$\displaystyle b^{ij}$ $\displaystyle=-M_{0}~{}\beta_{\mu}^{ij},$ (2.55)
$\displaystyle(m^{2})^{i}_{j}$
$\displaystyle=\frac{1}{2}~{}|M_{0}|^{2}~{}\mu\frac{d\gamma^{i}{}_{j}}{d\mu},$
(2.56)

where $M_{0}$ is an arbitrary reference mass scale to be specified shortly.
The assumption that

$C_{a}\frac{\partial}{\partial C_{a}}=C_{a}^{*}\frac{\partial}{\partial
C_{a}^{*}}$ (2.57)

for a RGI surface $F(g,C^{ijk},C^{*ijk})$ leads to

$\frac{d}{dg}=\left(\frac{\partial}{\partial g}+2\frac{\partial}{\partial
C}\,\frac{dC}{dg}\right)=\left(\frac{\partial}{\partial
g}+2\frac{\beta_{C}}{\beta_{g}}\frac{\partial}{\partial C}\right)\,,$ (2.58)

where Eq.(2.51) has been used. Now let us consider the partial differential
operator ${\cal O}$ in Eq.(2.44) which, assuming Eq.(2.48), becomes

${\cal O}=\frac{1}{2}M\frac{d}{d\ln g}\,.$ (2.59)

In turn, $\beta_{M}$ given in Eq.(2.41), becomes

$\beta_{M}=M\frac{d}{d\ln g}\big{(}\frac{\beta_{g}}{g}\big{)}~{},$ (2.60)

which by integration provides us [66, 64] with the generalized, i.e. including
Yukawa couplings, all-loop RGI Hisano - Shifman relation [54]

$M=\frac{\beta_{g}}{g}M_{0}~{},$ (2.61)

where $M_{0}$ is the integration constant and can be associated to the
unification scale $M_{U}$ in GUTs or to the gravitino mass $m_{3/2}$ in a
supergravity framework. Therefore, Eq.(2.61) becomes the all-loop RGI
Eq.(2.53). Note that $\beta_{M}$ using Eqs.(2.60) and (2.61) can be written as

$\beta_{M}=M_{0}\frac{d}{dt}(\beta_{g}/g)~{}.$ (2.62)

Similarly

$(\gamma_{1})^{i}_{j}={\cal
O}\gamma^{i}_{j}=\frac{1}{2}~{}M_{0}~{}\frac{d\gamma^{i}_{j}}{dt}~{}.$ (2.63)

Next, from Eq.(2.48) and Eq.(2.61) we obtain

$h^{ijk}=-M_{0}~{}\beta_{C}^{ijk}~{},$ (2.64)

while $\beta^{ijk}_{h}$, given in Eq.(2.42) and using Eq.(2.63), becomes [64]

$\beta_{h}^{ijk}=-M_{0}~{}\frac{d}{dt}\beta_{C}^{ijk},$ (2.65)

which shows that Eq.(2.64) is all-loop RGI. In a similar way Eq.(2.55) can be
shown to be all-loop RGI.

Finally we would like to emphasize that under the same assumptions (a) and (b)
the sum rule given in Eq.(2.50) has been proven [61] to be all-loop RGI, which
(using Eq.(2.61)) gives us a generalization of Eq.(2.56) to be applied in
considerations of non-universal soft scalar masses, which are necessary in
many cases including the MSSM.

Having obtained the Eqs.(2.53)-(2.56) from Eqs.(2.41)-(2.47) with the
assumptions (a) and (b), we would like to conclude the present section with
some remarks. First it is worth noting the difference, say in first order in
$g$, among the possibilities to consider specific solution of the reduction
equations or just assume the existence of a RGI surface, which is a weaker
assumption. So in the case we consider the reduction equation (2.51) without
relying on a specific solution, the sum rule of Eq.(2.50) reads

$m^{2}_{i}+m^{2}_{j}+m^{2}_{k}=|M|^{2}\frac{d\ln C^{ijk}}{d\ln g},$ (2.66)

and we find that

$\frac{d\ln C^{ijk}}{d\ln
g}=\frac{g}{C^{ijk}}\frac{dC^{ijk}}{dg}=\frac{g}{C^{ijk}}\frac{\beta^{ijk}_{C}}{\beta_{g}},$
(2.67)

which is clearly model dependent. However assuming a specific power series
solution of the reduction equation, as in Eq.(2.3), which in first order in
$g$ is just a linear relation among $C^{ijk}$ and $g$, we obtain that

$\frac{d\ln C^{ijk}}{d\ln g}=1$ (2.68)

and therefore the sum rule of Eq.(2.66) becomes model independent. We should
also emphasize that in order to show [55] that the relation

$(m^{2})^{i}_{j}=\frac{1}{2}\frac{g^{2}}{\beta_{g}}|M|^{2}\frac{d\gamma^{i}_{j}}{dg},$
(2.69)

which using Eq.(2.61) becomes Eq.(2.56), is RGI to all-loops a specific
solution of the reduction equations has to be required. As it has already been
pointed out above such a requirement is not necessary in order to obtain the
all-loop RG invariance of the sum rule of Eq.(2.50).

As it was emphasized in ref. [64] the set of the all-loop RGI relations
(2.53)-(2.56) is the one obtained in the Anomaly Mediated SB Scenario [67, 68,
69, 70, 71, 72], by fixing the $M_{0}$ to be $m_{3/2}$, which is the natural
scale in the supergravity framework.

A final remark concerns the resolution of the fatal problem of the anomaly
induced scenario in the supergravity framework, i.e. the negative mass squared
for sleptons, leading to tachyonic sleptons. Here, the problem is solved
thanks to the sum rule of Eq.(2.50), as it will become clear in the next
section. Other solutions have been provided by introducing Fayet-Iliopoulos
terms [73].

##  Chapter 3 Reduction of Couplings in the Standard Model and Predictions

The first application of the idea of reduction of couplings in realistic
models was presented in the celebrated paper [26]. We encourage the reader to
study the original article and here we limit ourselves to an introduction,
comments and updated remarks of the authors presented in the book “Reduction
of couplings and its application in particle physics, Finite theories, Higgs
and top mass predictions” [74].

Even today, more than twenty years after the first paper on reduction of
couplings in the Standard Model the original motivation for applying this
method to this Model has not become obsolete, neither by time nor by new
insight. The theoretical predictions originating from the Standard Model are
in extremely good agreement with experiment. Two decades of precision
measurement and precision calculation yielded essentially on all available
observables a truly astonishing coincidence[75]. And, yet there is no
convincing explanation why the number of families is three; why the mass
scales –the Planck mass and the electroweak breaking scale– differ so much in
magnitude, why the Higgs mass is so small compared to the Planck scale. And,
quite generally, there is also no explanation for the mixing of the families.

Reduction of couplings offers a way to understand at least to some degree
masses and mixings of charged leptons and quarks and the mass of the Higgs
particle. It extends the well known case of closed renormalization orbits due
to symmetry to other, more general ones. Which structure these orbits have had
to be learned, i.e. deduced from the relevant renormalization group equation
in the specific model. In particular, one had to take into account the
different behaviour of abelian versus non-abelian gauge groups and of the
Higgs self-coupling, say in the ultraviolet region. If asymptotic expansions
should make sense in the transition from a non-perturbative theory to a
perturbative version it should be possible to rely on common ultraviolet
asymptotic freedom. One also has to respect gross features coming from
phenomenology. In mathematical terms this is the problem of integrating
partial differential equations by imposing suitable boundary conditions
(originating from physical requirements): partial reduction.

Perhaps the most important and not obvious result of the entire analysis is
the fact that reduction of couplings (even the version of “partial reduction”)
is extremely sensitive to the model. If one accepts the integration “paths” as
derived in the relevant papers, the ordinary Standard Model can neither
support a mass of the top quark nor of the Higgs particle as large as they
have been found experimentally. There is an apparent mismatch among the the
reduced Standard Model predictions and the experimental findings of the top
and Higgs masses. Renormalization group improvements of the original
theoretical predictions were concerning essentially the QCD sector, which was
taken into account in the reduction. Whereas the differences originating from
the other couplings turned out to be negligibly small. Hence it became clear
that other model classes are to be studied and further constraining principles
had to be found. This will be the subject of Chapters 4 and 5.

In ref. [26] within the context of the Standard Model with one Higgs doublet
and n families the principle of reduction of couplings was applied. For
simplicity mixing of the families was assumed to be absent: the Yukawa
couplings are diagonal and real. For the massless model reduction solutions
can be found to all orders of perturbation theory as power series in the
“primary” coupling, thus superseding fixed point considerations based on one-
loop approximations. Due to the different asymptotic behaviour of the SU(3),
SU(2) and U(1) couplings the space of solutions is clearly structured and
permits reduction in very distinct ways only. Since reducing the gauge
couplings relative to each other is either inconsistent or phenomenologically
not acceptable, $\alpha_{s}$ (the largest coupling) has been chosen as the
expansion parameter –the primary coupling– and thus UV-asymptotic freedom as
the relevant regime. This allows to neglect in the lowest order approximation
the other gauge couplings and to take their effect into account as
corrections.

In the matter sector (leptons, quarks, Higgs) discrete solutions emerge for
the reduced couplings which permit essentially only the Higgs self-coupling
and the Yukawa coupling to the top quark to be non-vanishing. Stability
considerations (Liapunov’s theory) show how the power series solutions are
embedded in the set of the general solutions. Couplings of the massless model
were converted into masses in the tree approximation of the spontaneously
broken model. For three generations one finds $m_{H}=61$ GeV, $m_{t}=81$ GeV
with an error of about 10-15%.

Reduction of couplings is based on the requirement that all reduced couplings
vanish simultaneously upon reduction of the primary coupling. This is clearly
only possible if the couplings considered have the same asymptotic behavior or
have vanishing $\beta$-functions. Hence in the Standard model, based on
$SU(3)\times SU(2)\times U(1)$ straightforward reduction cannot be realized.
Since however the strong coupling $\alpha_{s}$ is, say at the $W$-mass,
considerably larger than the weak and electromagnetic coupling one may put
those equal to zero, reduce within the system of quantum chromodynamics
including the Higgs and the Yukawa couplings and subsequently take into
account electroweak corrections as a kind of perturbation. This is called
“partial reduction”. In [27] a new perturbation method was developed and then
applied using the updated experimental values of the strong coupling and the
Weinberg angle at the time.

In asymptotically free theories the $\beta$-functions usually go to zero with
some power of the couplings involved. Thus, reduction equations are singular
for vanishing coupling and require a case by case study at this singular
point. In particular this is true for the reduction equations of Yukawa and
Higgs couplings when reducing to $\alpha_{s}$. It was shown in the paper that
for the non-trivial reduction solution (i.e. only the top Yukawa coupling and
the Higgs coupling do not vanish) one can de-singularize the system by a
variable transformation and thereafter go over to a partial differential
equation which is easier to solve than the ordinary differential equations one
started with. The reduction solutions of the perturbed system are then in one-
to-one correspondence with the unperturbed ones.

In terms of mass values the non-trivial reduction yields $m_{t}=91.3$ GeV,
$m_{H}=64.3$ GeV. These mass values are at the same time the upper bound for
the trivial reduction, where the Higgs mass is a function of the top mass.
Here is used as definition for “trivial” that the ratios of top-Yukawa
coupling and Higgs coupling with respect to $\alpha_{s}$ go to zero for the
weak coupling limit $\alpha_{s}$ going to zero.

Still there are corrections to the above values:
1\. The above mass values depend on the SM parameters, in particular the
strong coupling constant $\alpha_{s}$ and $\sin\theta_{W}$. Since the values
of $\alpha_{s}$ and $\sin\theta_{W}$ were updated, the above predictions had
to be updated, too.
2\. Two-loop corrections could be important.
3\. In ref. [26] the difference of the physical mass (pole mass) and the mass
defined in the $\overline{\textrm{MS}}$ scheme has been ignored. In ref. [76]
all these corrections are included. It was found that the correction coming
from the $\overline{\textrm{MS}}$ to the pole mass transition increases
$m_{t}$ by about 4%, while $m_{H}$ is increased by about 1%. The two-loop
effect is non-negligible especially for $m_{t}:+2$% for $m_{t}$ and 0.2% for
$m_{H}$. Taking into account all these corrections it was found

$m_{t}=98.6\pm 9.2\textrm{ GeV},m_{H}=64.5\pm 1.5\textrm{ GeV},$ (3.1)

where the 1991 values of $M_{Z}$, $\alpha_{s}(M_{Z})$,
$\sin^{2}\theta_{W}(M_{Z})$ and $\alpha_{em}(M_{Z})$ were used. Even with
updated values it was found [74] that the change of the prediction is
negligible. Obviously, this prediction is inconsistent with the experimental
observations.

An optimistic point of view is that the failure of the reduction of couplings
programme in the SM shows that it is not the final theory but only a very
interesting part of it and therefore we have to search further for the
ultimate theory.

##  Chapter 4 Finiteness

The principle of finiteness requires perhaps some more motivation to be
considered and generally accepted these days than when it was first envisaged.
It is however interesting to note that in the early days of field theory the
feeling was quite different. Probably the well known Dirac’s phrase that
“…divergencies are hidden under the carpet” is representative of the views of
that time. In recent years we have a more relaxed attitude towards
divergencies. Most theorists believe that the divergencies are signals of the
existence of a higher scale, where new degrees of freedom are excited. Even
accepting this dogma, we are naturally led to the conclusion that beyond the
unification scale, i.e. when all interactions have been taken into account in
a unified scheme, the theory should be completely finite. In fact, this is one
of the main motivations and aims of string, non-commutative geometry, and
quantum group theories, which include also gravity in the unification of the
interactions. In our work on reduction of couplings and finiteness we
restricted ourselves to unifying only the known gauge interactions, based on a
lesson of the history of Elementary Particle Physics (EPP) that if a nice idea
works in physics, usually it is realised in its simplest form.

###  4.1 The idea behind finiteness

Finiteness is based on the fact that it is possible to find renormalization
group invariant (RGI) relations among couplings that keep finiteness in
perturbation theory, even to all orders. Accepting finiteness as a guiding
principle in constructing realistic theories of EPP, the first thing that
comes to mind is to look for an $N=4$ supersymmetric unified gauge theory,
since any ultraviolet (UV) divergencies are absent in these theories. However
nobody has managed so far to produce realistic models in the framework of
$N=4$ SUSY. In the best of cases one could try to do a drastic truncation of
the theory like the orbifold projection of refs. [77, 82], but this is already
a different theory than the original one. The next possibility is to consider
an $N=2$ supersymmetric gauge theory, whose beta-function receives corrections
only at one-loop. Then it is not hard to select a spectrum to make the theory
all-loop finite. However a serious obstacle in these theories is their mirror
spectrum, which in the absence of a mechanism to make it heavy, does not
permit the construction of realistic models. Therefore, we are naturally led
to consider $N=1$ supersymmetric gauge theories, which can be chiral and in
principle realistic.

Let us be clear at this point and state that in our approach (ultra violet,
UV) finiteness means the vanishing of all the $\beta$-functions, i.e. the non-
renormalization of the coupling constants, in contrast to a complete (UV)
finiteness where even field amplitude renormalization is absent. Before our
work the studies on $N=1$ finite theories were following two directions: (a)
construction of finite theories up to two-loops examining various
possibilities to make them phenomenologically viable, (b) construction of all-
loop finite models without particular emphasis on the phenomenological
consequences. The success of our work was that we constructed the first
realistic all-loop finite model, based on the theorem presented in the Sect.
4.2, realising in this way an old theoretical dream of field theorists.
Equally important was the correct prediction of the top quark mass one and
half year before the experimental discovery. It was the combination of these
two facts that motivated us to continue with the study of $N=1$ finite
theories. It is worth noting that nobody expected at the time such a heavy
mass for the top quark. Given that the analysis of the experimental data
changes over time, the comparison of our original prediction with the updated
analyses will be discussed later.

###  4.2 Finiteness in N=1 Supersymmetric Gauge Theories

Let us, once more, consider a chiral, anomaly free, $N=1$ globally
supersymmetric gauge theory based on a group G with gauge coupling constant
$g$. The superpotential of the theory is given by (see Eq.(2.10))

$W=\frac{1}{2}\,m_{ij}\,\phi_{i}\,\phi_{j}+\frac{1}{6}\,C_{ijk}\,\phi_{i}\,\phi_{j}\,\phi_{k}~{}.$
(4.1)

The $N=1$ non-renormalization theorem, ensuring the absence of mass and cubic-
interaction-term infinities, leads to wave-function infinities. The one-loop
$\beta$-function is given by (see Eq.(2.15)

$\beta^{(1)}_{g}=\frac{dg}{dt}=\frac{g^{3}}{16\pi^{2}}\left[\,\sum_{i}\,T(R_{i})-3\,C_{2}(G)\right]~{},$
(4.2)

the $\beta$-function of $C_{ijk}$ by (see Eq. (2.16))

$\beta_{ijk}=\frac{dC_{ijk}}{dt}~{}=~{}C_{ijl}\,\gamma^{l}_{k}+C_{ikl}\,\gamma^{l}_{j}+C_{jkl}\,\gamma^{l}_{i}~{}$
(4.3)

and the one-loop wave function anomalous dimensions by (see Eq. (2.17 ))

$\gamma^{(1)}{}_{j}^{i}=\frac{1}{32\pi^{2}}\,[\,C^{ikl}\,C_{jkl}-2\,g^{2}\,C_{2}(R)\delta_{j}^{i}\,]~{}.$
(4.4)

As one can see from Eqs. (4.2) and (4.4), all the one-loop $\beta$-functions
of the theory vanish if $\beta_{g}^{(1)}$ and $\gamma^{(1)}{}_{j}^{i}$ vanish,
i.e.

$\displaystyle\sum_{i}T(R_{i})$ $\displaystyle=3C_{2}(G)\,,$ (4.5)
$\displaystyle C^{ikl}C_{jkl}$
$\displaystyle=2\delta^{i}_{j}g^{2}C_{2}(R_{i})\,,$ (4.6)

The conditions for finiteness for $N=1$ field theories with $SU(N)$ gauge
symmetry are discussed in [86], and the analysis of the anomaly-free and no-
charge renormalization requirements for these theories can be found in [87]. A
very interesting result is that the conditions (4.5,4.6) are necessary and
sufficient for finiteness at the two-loop level [32, 33, 34, 35, 36].

In case SUSY is broken by soft terms, the requirement of finiteness in the
one-loop soft breaking terms imposes further constraints among themselves
[88]. In addition, the same set of conditions that are sufficient for one-loop
finiteness of the soft breaking terms render the soft sector of the theory
two-loop finite[89].

The one- and two-loop finiteness conditions of Eqs. (4.5,4.6) restrict
considerably the possible choices of the irreducible representations (irreps)
$R_{i}$ for a given group $G$ as well as the Yukawa couplings in the
superpotential (4.1). Note in particular that the finiteness conditions cannot
be applied to the minimal supersymmetric standard model (MSSM), since the
presence of a $U(1)$ gauge group is incompatible with the condition (4.5), due
to $C_{2}[U(1)]=0$. This naturally leads to the expectation that finiteness
should be attained at the grand unified level only, the MSSM being just the
corresponding, low-energy, effective theory.

Another important consequence of one- and two-loop finiteness is that SUSY
(most probably) can only be broken due to the soft breaking terms. Indeed, due
to the unacceptability of gauge singlets, F-type spontaneous symmetry breaking
[90] terms are incompatible with finiteness, as well as D-type [91]
spontaneous breaking which requires the existence of a $U(1)$ gauge group.

A natural question to ask is what happens at higher loop orders. The answer is
contained in a theorem [93, 92] which states the necessary and sufficient
conditions to achieve finiteness at all orders. Before we discuss the theorem
let us make some introductory remarks. The finiteness conditions impose
relations between gauge and Yukawa couplings. To require such relations which
render the couplings mutually dependent at a given renormalization point is
trivial. What is not trivial is to guarantee that relations leading to a
reduction of the couplings hold at any renormalization point. As we have seen
(see Eq. (2.51)), the necessary and also sufficient, condition for this to
happen is to require that such relations are solutions to the REs

$\beta_{g}\frac{dC_{ijk}}{dg}=\beta_{ijk}$ (4.7)

and hold at all orders. Remarkably, the existence of all-order power series
solutions to (4.7) can be decided at one-loop level, as already mentioned.

Let us now turn to the all-order finiteness theorem [93, 92], which states
under which conditions an $N=1$ supersymmetric gauge theory can become finite
to all orders in perturbation theory, that is attain physical scale
invariance. It is based on (a) the structure of the supercurrent in $N=1$
supersymmetric gauge theory [94, 95, 96], and on (b) the non-renormalization
properties of $N=1$ chiral anomalies [93, 92, 97, 98, 99]. Details of the
proof can be found in refs. [93, 92] and further discussion in Refs. [97, 98,
99, 100, 101]. Here, following mostly Ref. [101] we present a comprehensible
sketch of the proof.

Consider an $N=1$ supersymmetric gauge theory, with simple Lie group $G$. The
content of this theory is given at the classical level by the matter
supermultiplets $S_{i}$, which contain a scalar field $\phi_{i}$ and a Weyl
spinor $\psi_{ia}$, and the vector supermultiplet $V_{a}$, which contains a
gauge vector field $A_{\mu}^{a}$ and a gaugino Weyl spinor
$\lambda^{a}_{\alpha}$.

Let us first recall certain facts about the theory:

(1) A massless $N=1$ supersymmetric theory is invariant under a $U(1)$ chiral
transformation $R$ under which the various fields transform as follows

$\begin{split}A^{\prime}_{\mu}&=A_{\mu},~{}~{}\lambda^{\prime}_{\alpha}=\exp({-i\theta})\lambda_{\alpha}\\\
\phi^{\prime}&=\exp({-i\frac{2}{3}\theta})\phi,~{}~{}\psi_{\alpha}^{\prime}=\exp({-i\frac{1}{3}\theta})\psi_{\alpha},~{}\cdots\end{split}$
(4.8)

The corresponding axial Noether current $J^{\mu}_{R}(x)$ is

$J^{\mu}_{R}(x)=\bar{\lambda}\gamma^{\mu}\gamma^{5}\lambda+\cdots$ (4.9)

is conserved classically, while in the quantum case is violated by the axial
anomaly

$\partial_{\mu}J^{\mu}_{R}=r\left(\epsilon^{\mu\nu\sigma\rho}F_{\mu\nu}F_{\sigma\rho}+\cdots\right).$
(4.10)

From its known topological origin in ordinary gauge theories [102, 103, 104],
one would expect the axial vector current $J^{\mu}_{R}$ to satisfy the Adler-
Bardeen theorem and receive corrections only at the one-loop level. Indeed it
has been shown that the same non-renormalization theorem holds also in
supersymmetric theories [97, 98, 99]. Therefore

$r=\hbar\beta_{g}^{(1)}.$ (4.11)

(2) The massless theory we consider is scale invariant at the classical level
and, in general, there is a scale anomaly due to radiative corrections. The
scale anomaly appears in the trace of the energy momentum tensor $T_{\mu\nu}$,
which is traceless classically. It has the form

$T^{\mu}_{\mu}=\beta_{g}F^{\mu\nu}F_{\mu\nu}+\cdots$ (4.12)

(3) Massless, $N=1$ supersymmetric gauge theories are classically invariant
under the supersymmetric extension of the conformal group – the superconformal
group. Examining the superconformal algebra, it can be seen that the subset of
superconformal transformations consisting of translations, SUSY
transformations, and axial $R$ transformations is closed under SUSY, i.e.
these transformations form a representation of SUSY. It follows that the
conserved currents corresponding to these transformations make up a
supermultiplet represented by an axial vector superfield called the
supercurrent $J$,

$J\equiv\left\\{J^{\prime\mu}_{R},~{}Q^{\mu}_{\alpha},~{}T^{\mu}_{\nu},...\right\\},$
(4.13)

where $J^{\prime\mu}_{R}$ is the current associated to R invariance,
$Q^{\mu}_{\alpha}$ is the one associated to SUSY invariance, and
$T^{\mu}_{\nu}$ the one associated to translational invariance (energy-
momentum tensor).

The anomalies of the R current $J^{\prime\mu}_{R}$, the trace anomalies of the
SUSY current, and the energy-momentum tensor, form also a second
supermultiplet, called the supertrace anomaly

$S=\left\\{Re~{}S,~{}Im~{}S,~{}S_{\alpha}\right\\}=\left\\{T^{\mu}_{\mu},~{}\partial_{\mu}J^{\prime\mu}_{R},~{}\sigma^{\mu}_{\alpha\dot{\beta}}\bar{Q}^{\dot{\beta}}_{\mu}~{}+~{}\cdots\right\\}$

where $T^{\mu}_{\mu}$ is given in Eq.(4.12) and

$\displaystyle\partial_{\mu}J^{\prime\mu}_{R}$
$\displaystyle~{}=~{}\beta_{g}\epsilon^{\mu\nu\sigma\rho}F_{\mu\nu}F_{\sigma\rho}+\cdots$
(4.14)
$\displaystyle\sigma^{\mu}_{\alpha\dot{\beta}}\bar{Q}^{\dot{\beta}}_{\mu}$
$\displaystyle~{}=~{}\beta_{g}\lambda^{\beta}\sigma^{\mu\nu}_{\alpha\beta}F_{\mu\nu}+\cdots$
(4.15)

(4) It is very important to note that the Noether current defined in (4.9) is
not the same as the current associated to R invariance that appears in the
supercurrent $J$ in (4.13), but they coincide in the tree approximation. So
starting from a unique classical Noether current $J^{\mu}_{R(class)}$, the
Noether current $J^{\mu}_{R}$ is defined as the quantum extension of
$J^{\mu}_{R(class)}$ which allows for the validity of the non-renormalization
theorem. On the other hand $J^{\prime\mu}_{R}$, is defined to belong to the
supercurrent $J$, together with the energy-momentum tensor. The two
requirements cannot be fulfilled by a single current operator at the same
time.

Although the Noether current $J^{\mu}_{R}$ which obeys (4.10) and the current
$J^{\prime\mu}_{R}$ belonging to the supercurrent multiplet $J$ are not the
same, there is a relation [93, 92] between quantities associated with them

$r=\beta_{g}(1+x_{g})+\beta_{ijk}x^{ijk}-\gamma_{A}r^{A}$ (4.16)

where $r$ was given in Eq. (4.11). The $r^{A}$ are the non-renormalized
coefficients of the anomalies of the Noether currents associated to the chiral
invariances of the superpotential, and –like $r$– are strictly one-loop
quantities. The $\gamma_{A}$’s are linear combinations of the anomalous
dimensions of the matter fields, and $x_{g}$, and $x^{ijk}$ are radiative
correction quantities. The structure of Eq. (4.16) is independent of the
renormalization scheme.

One-loop finiteness, i.e. vanishing of the $\beta$-functions at one-loop,
implies that the Yukawa couplings $\lambda_{ijk}$ must be functions of the
gauge coupling $g$. To find a similar condition to all orders it is necessary
and sufficient for the Yukawa couplings to be a formal power series in $g$,
which is solution of the REs (4.7).

We can now state the theorem for all-order vanishing $\beta$-functions [93].

Theorem:

Consider an $N=1$ supersymmetric Yang-Mills theory, with simple gauge group.
If the following conditions are satisfied

  1. 1.

There is no gauge anomaly.

  2. 2.

The gauge $\beta$-function vanishes at one-loop

$\beta^{(1)}_{g}=0=\sum_{i}T(R_{i})-3\,C_{2}(G).$ (4.17)

  3. 3.

There exist solutions of the form

$C_{ijk}=\rho_{ijk}g,~{}\qquad\rho_{ijk}\in{{\rm I}\kern-5.89996pt{\rm C}}$
(4.18)

to the conditions of vanishing one-loop matter fields anomalous dimensions

$\gamma^{(1)}{}_{j}^{i}~{}=~{}0=\frac{1}{32\pi^{2}}~{}[~{}C^{ikl}\,C_{jkl}-2~{}g^{2}~{}C_{2}(R)\delta_{j}^{i}].$
(4.19)

  4. 4.

These solutions are isolated and non-degenerate when considered as solutions
of vanishing one-loop Yukawa $\beta$-functions:

$\beta_{ijk}=0.$ (4.20)

Then, each of the solutions (4.18) can be uniquely extended to a formal power
series in $g$, and the associated super Yang-Mills models depend on the single
coupling constant $g$ with a $\beta$-function which vanishes at all-orders.

It is important to note a few things: The requirement of isolated and non-
degenerate solutions guarantees the existence of a unique formal power series
solution to the reduction equations. The vanishing of the gauge $\beta$
function at one-loop, $\beta_{g}^{(1)}$, is equivalent to the vanishing of the
R current anomaly (4.10). The vanishing of the anomalous dimensions at one-
loop implies the vanishing of the Yukawa couplings $\beta$ functions at that
order. It also implies the vanishing of the chiral anomaly coefficients
$r^{A}$. This last property is a necessary condition for having $\beta$
functions vanishing at all orders.111There is an alternative way to find
finite theories [105, 106, 107, 110].

Proof:

Insert $\beta_{ijk}$ as given by the REs into the relationship (4.16). Since
these chiral anomalies vanish, we get for $\beta_{g}$ an homogeneous equation
of the form

$0=\beta_{g}(1+O(\hbar)).$ (4.21)

The solution of this equation in the sense of a formal power series in $\hbar$
is $\beta_{g}=0$, order by order. Therefore, due to the REs (4.7),
$\beta_{ijk}=0$ too.

Thus we see that finiteness and reduction of couplings are intimately related.
Since an equation like eq. (4.16) is lacking in non-supersymmetric theories,
one cannot extend the validity of a similar theorem in such theories.

A very interesting development was done in ref [111]. Based on the all-loop
relations among the beta-functions of the soft supersymmetry breaking terms
and those of the rigid supersymmetric theory with the help of the differential
operators, discussed in Sect. 2.4, it was shown that certain RGI surfaces can
be chosen, so as to reach all-loop finiteness of the full theory. More
specifically it was shown that on certain RGI surfaces the partial
differential operators appearing in Eqs. (2.41,2.42) acting on the beta- and
gamma-functions of the rigid theory can be transformed to total derivatives.
Then the all-loop finiteness of the beta and gamma functions of the rigid
theory can be transferred to the beta functions of the soft supersymmetry
breaking terms. Therefore a totally all-loop finite $N=1$ SUSY gauge theory
can be constructed, including the soft supersymmetry breaking terms.

##  Chapter 5 Reduction of Couplings in Phenomenologically Viable Models

In this chapter we apply the idea of reduction of couplings to
phenomenologically viable supersymmetric models. These models make clear
predictions for the top and bottom quark masses. Confronting the models with
the experimental values allows to restrict the parameter space and to single
out the viable models. The full set of experimental constraints for a subset
of these models will then be discussed in Chapter 6.

###  5.1 Finite Unified Models

From the classification of theories with vanishing one-loop gauge
$\beta$-function [112], one can easily see that there exist only two candidate
possibilities to construct $SU(5)$ GUTs with three generations. These
possibilities require that the theory should contain as matter fields the
chiral supermultiplets ${\bf 5},~{}\overline{\bf 5},~{}{\bf
10},~{}\overline{\bf 10},~{}{\bf 24}$ with the multiplicities $(6,9,4,1,0)$ or
$(4,7,3,0,1)$, respectively. Only the second one contains a ${\bf 24}$-plet
which can be used to provide the spontaneous symmetry breaking (SB) of $SU(5)$
down to $SU(3)\times SU(2)\times U(1)$. For the first model one has to
incorporate another way, such as the Wilson flux breaking mechanism in higher
dimensional theories, to achieve the desired SB of $SU(5)$ [18, 19].
Therefore, for a self-consistent field theory discussion we would like to
concentrate only on the second possibility.

The particle content of the models we will study consists of the following
supermultiplets: three ($\overline{\bf 5}+{\bf 10}$), needed for each of the
three generations of quarks and leptons, four ($\overline{\bf 5}+{\bf 5}$) and
one ${\bf 24}$ considered as Higgs supermultiplets. When the gauge group of
the finite GUT is broken the theory is no longer finite, and we will assume
that we are left with the MSSM.

Therefore, a predictive Gauge-Yukawa unified $SU(5)$ model which is finite to
all orders, in addition to the requirements mentioned already, should also
have the following properties:

  1. 1.

The one-loop anomalous dimensions are diagonal, i.e.,
$\gamma^{(1)}{}_{j}^{i}\propto\delta^{j}_{i}$.

  2. 2.

The three fermion generations, in the irreducible representations
$\overline{\bf 5}_{i},{\bf 10}_{i}$ $(i=1,2,3)$, should not couple to the
adjoint ${\bf 24}$.

  3. 3.

The two Higgs doublets of the MSSM should mostly be made out of a pair of
Higgs quintet and anti-quintet, which couple to the third generation.

In the following we discuss two versions of the all-order finite model: the
model of Ref. [18, 19], which will be labeled ${\bf A}$, and a slight
variation of this model (labeled ${\bf B}$), which can also be obtained from
the class of the models suggested in Ref. [56, 57] with a modification to
suppress non-diagonal anomalous dimensions [46].

The superpotential which describes the two models, before the reduction of
couplings takes place, is of the form [18, 19, 46, 113, 114]

$\displaystyle W$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\,[~{}\frac{1}{2}g_{i}^{u}\,{\bf 10}_{i}{\bf
10}_{i}H_{i}+g_{i}^{d}\,{\bf 10}_{i}\overline{\bf
5}_{i}\,\overline{H}_{i}~{}]+g_{23}^{u}\,{\bf 10}_{2}{\bf 10}_{3}H_{4}$
$\displaystyle+g_{23}^{d}\,{\bf 10}_{2}\overline{\bf
5}_{3}\,\overline{H}_{4}+g_{32}^{d}\,{\bf 10}_{3}\overline{\bf
5}_{2}\,\overline{H}_{4}+\sum_{a=1}^{4}g_{a}^{f}\,H_{a}\,{\bf
24}\,\overline{H}_{a}+\frac{g^{\lambda}}{3}\,({\bf 24})^{3}~{},$

where $H_{a}$ and $\overline{H}_{a}~{}~{}(a=1,\dots,4)$ stand for the Higgs
quintets and anti-quintets.

The main difference between model ${\bf A}$ and model ${\bf B}$ is that two
pairs of Higgs quintets and anti-quintets couple to the ${\bf 24}$ in ${\bf
B}$, so that it is not necessary to mix them with $H_{4}$ and
$\overline{H}_{4}$ in order to achieve the triplet-doublet splitting after the
symmetry breaking of $SU(5)$ [46]. Thus, although the particle content is the
same, the solutions to Eqs. (4.3,4.4) and the sum rules are different, which
will be reflected in the phenomenology, discussed in Sect. 6.2.

####  5.1.1 FUT A

| $\overline{{\bf 5}}_{1}$ | $\overline{{\bf 5}}_{2}$ | $\overline{{\bf 5}}_{3}$ | ${\bf 10}_{1}$ | ${\bf 10}_{2}$ | ${\bf 10}_{3}$ | $H_{1}$ | $H_{2}$ | $H_{3}$ | $H_{4}$ | $\overline{H}_{1}$ | $\overline{H}_{2}$ | $\overline{H}_{3}$ | $\overline{H}_{4}$ | ${\bf 24}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$Z_{7}$ | 4 | 1 | 2 | 1 | 2 | 4 | 5 | 3 | 6 | -5 | -3 | -6 | 0 | 0 | 0
$Z_{3}$ | 0 | 0 | 0 | 1 | 2 | 0 | 1 | 2 | 0 | -1 | -2 | 0 | 0 | 0 | 0
$Z_{2}$ | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Table 5.1: Charges of the $Z_{7}\times Z_{3}\times Z_{2}$ symmetry for Model
FUT A.

This model was introduced and examined first in refs [18, 19]. After the
reduction of couplings the symmetry of the superpotential $W$ (Eq. (5.1)), is
enhanced. For model ${\bf A}$ one finds that the superpotential has the
$Z_{7}\times Z_{3}\times Z_{2}$ discrete symmetry with the charge assignment
shown in Tab. 5.1, and with the following superpotential

$W_{A}=\sum_{i=1}^{3}\,[~{}\frac{1}{2}g_{i}^{u}\,{\bf 10}_{i}{\bf
10}_{i}H_{i}+g_{i}^{d}\,{\bf 10}_{i}\overline{\bf
5}_{i}\,\overline{H}_{i}~{}]+g_{4}^{f}\,H_{4}\,{\bf
24}\,\overline{H}_{4}+\frac{g^{\lambda}}{3}\,({\bf 24})^{3}~{}.$ (5.2)

The non-degenerate and isolated solutions to $\gamma^{(1)}_{i}=0$ for model
FUT A , which are the boundary conditions for the Yukawa couplings at the GUT
scale, are:

$\displaystyle(g_{1}^{u})^{2}=\frac{8}{5}~{}g^{2}~{},~{}(g_{1}^{d})^{2}=\frac{6}{5}~{}g^{2}~{},~{}(g_{2}^{u})^{2}=(g_{3}^{u})^{2}=\frac{8}{5}~{}g^{2}~{},$
(5.3)
$\displaystyle(g^{\lambda})^{2}=\frac{15}{7}g^{2}~{},~{}(g_{2}^{d})^{2}=(g_{3}^{d})^{2}=\frac{6}{5}~{}g^{2}~{},~{}(g_{4}^{f})^{2}=g^{2}$
$\displaystyle(g_{23}^{u})^{2}=(g_{23}^{d})^{2}=(g_{32}^{d})^{2}=(g_{2}^{f})^{2}=(g_{3}^{f})^{2}=(g_{1}^{f})^{2}=0~{}.~{}$

In the dimensionful sector, the sum rule gives us the following boundary
conditions at the GUT scale for this model [46]:

$m^{2}_{H_{u}}+2m^{2}_{{\bf 10}}=m^{2}_{H_{d}}+m^{2}_{\overline{{\bf
5}}}+m^{2}_{{\bf 10}}=M^{2}~{}~{},$ (5.4)

and thus we are left with only three free parameters, namely
$m_{\overline{{\bf 5}}}\equiv m_{\overline{{\bf 5}}_{3}}$, $m_{{\bf 10}}\equiv
m_{{\bf 10}_{3}}$ and $M$.

####  5.1.2 FUT B

This model was introduced and was presented its first study in ref [46]. Also
in the case of FUT B the symmetry is enhanced after the reduction of
couplings. The superpotential has now a $Z_{4}\times Z_{4}\times Z_{4}$
symmetry with charges shown in Tab. 5.2 and with the following superpotential

$W_{B}=\sum_{i=1}^{3}\,[~{}\frac{1}{2}g_{i}^{u}\,{\bf 10}_{i}{\bf
10}_{i}H_{i}+g_{i}^{d}\,{\bf 10}_{i}\overline{\bf
5}_{i}\,\overline{H}_{i}~{}]+g_{23}^{u}\,{\bf 10}_{2}{\bf 10}_{3}H_{4}\\\
+g_{23}^{d}\,{\bf 10}_{2}\overline{\bf
5}_{3}\,\overline{H}_{4}+g_{32}^{d}\,{\bf 10}_{3}\overline{\bf
5}_{2}\,\overline{H}_{4}+g_{2}^{f}\,H_{2}\,{\bf
24}\,\overline{H}_{2}+g_{3}^{f}\,H_{3}\,{\bf
24}\,\overline{H}_{3}+\frac{g^{\lambda}}{3}\,({\bf 24})^{3}~{},$ (5.5)

For this model the non-degenerate and isolated solutions to
$\gamma^{(1)}_{i}=0$ give us:

$\displaystyle(g_{1}^{u})^{2}=\frac{8}{5}~{}g^{2}~{},~{}(g_{1}^{d})^{2}=\frac{6}{5}~{}g^{2}~{},~{}(g_{2}^{u})^{2}=(g_{3}^{u})^{2}=(g_{23}^{u})^{2}=\frac{4}{5}~{}g^{2}~{},$
$\displaystyle(g_{2}^{d})^{2}=(g_{3}^{d})^{2}=(g_{23}^{d})^{2}=(g_{32}^{d})^{2}=\frac{3}{5}~{}g^{2}~{},$
(5.6)
$\displaystyle(g^{\lambda})^{2}=\frac{15}{7}g^{2}~{},~{}(g_{2}^{f})^{2}=(g_{3}^{f})^{2}=\frac{1}{2}~{}g^{2}~{},~{}(g_{1}^{f})^{2}=(g_{4}^{f})^{2}=0~{},$

and from the sum rule we obtain [46]:

$m^{2}_{H_{u}}+2m^{2}_{{\bf 10}}=M^{2}~{},\quad m^{2}_{H_{d}}-2m^{2}_{{\bf
10}}=-\frac{M^{2}}{3}~{},\quad m^{2}_{\overline{{\bf 5}}}+3m^{2}_{{\bf
10}}=\frac{4M^{2}}{3}~{},$ (5.7)

i.e., in this case we have only two free parameters $m_{{\bf 10}}\equiv
m_{{\bf 10}_{3}}$ and $M$ for the dimensionful sector.

| $\overline{{\bf 5}}_{1}$ | $\overline{{\bf 5}}_{2}$ | $\overline{{\bf 5}}_{3}$ | ${\bf 10}_{1}$ | ${\bf 10}_{2}$ | ${\bf 10}_{3}$ | $H_{1}$ | $H_{2}$ | $H_{3}$ | $H_{4}$ | $\overline{H}_{1}$ | $\overline{H}_{2}$ | $\overline{H}_{3}$ | $\overline{H}_{4}$ | ${\bf 24}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$Z_{4}$ | 1 | 0 | 0 | 1 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0
$Z_{4}$ | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 2 | 0 | 3 | 0 | -2 | 0 | -3 | 0
$Z_{4}$ | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 2 | 3 | 0 | 0 | -2 | -3 | 0
Table 5.2: Charges of the $Z_{4}\times Z_{4}\times Z_{4}$ symmetry for Model
FUT B.

As already mentioned, after the $SU(5)$ gauge symmetry breaking we assume we
have the MSSM, i.e. only two Higgs doublets. This can be achieved by
introducing appropriate mass terms that allow to perform a rotation of the
Higgs sector [114, 18, 19, 115, 113], in such a way that only one pair of
Higgs doublets, coupled mostly to the third family, remains light and acquire
vacuum expectation values. To avoid fast proton decay the usual fine tuning to
achieve doublet-triplet splitting is performed. Notice that, although similar,
the mechanism is not identical to minimal $SU(5)$, since we have an extended
Higgs sector.

Thus, after the gauge symmetry of the GUT theory is broken we are left with
the MSSM, with the boundary conditions for the third family given by the
finiteness conditions, while the other two families are basically decoupled.

####  5.1.3 Predictions for Quark Masses

We will now examine the prediction of such all-loop Finite Unified theories
with $SU(5)$ gauge group for the third generation quark masses (for the
reasons expressed above). An extension to three families, and the generation
of quark mixing angles and masses in Finite Unified Theories has been
addressed in [116], where several examples are given. These extensions are not
discussed here.

Since the gauge symmetry is spontaneously broken below $M_{\rm GUT}$, the
finiteness conditions do not restrict the renormalization properties at low
energies, and all it remains are boundary conditions on the gauge and Yukawa
couplings (5.3) or (5.1.2), the relation

$h^{ijk}=-MC^{ijk}+...=-M\rho_{(0)}^{ijk}g+O(g^{5})~{},$

which follow from Eq.(2.52) and the power series solution Eq.(2.40) and the
soft scalar-mass sum rule [117, 46, 118]

$(~{}m_{i}^{2}+m_{j}^{2}+m_{k}^{2}~{})/MM^{{\dagger}}=1+\frac{g^{2}}{16\pi^{2}}\,\Delta^{(2)}+O(g^{4})~{},$
(5.8)

where the $g^{2}$ term is given by

$\Delta^{(2)}=-2\sum_{l}\left[\frac{m^{2}_{l}}{MM^{\dagger}}-\frac{1}{3}\right]T(R_{l})~{},$

all taken at $M_{\rm GUT}$, as applied in the two models. Thus we examine the
evolution of these parameters according to their RGEs up to two-loops for
dimensionless parameters and at one-loop for dimensionful ones with the
relevant boundary conditions. Below $M_{\rm GUT}$ their evolution is assumed
to be governed by the MSSM. We further assume a unique SUSY breaking scale
$M_{\rm SUSY}$ (which we define as the geometrical average of the stop masses)
and therefore below that scale the effective theory is just the SM. This
allows to evaluate observables at or below the electroweak scale.

In the following, we review the derivation of the prediction for the third
generation of quark masses that allows for a direct comparison with
experimental data and to determine the models that are in good agreement with
the observed quark mass values [121, 120, 119].

Figure 5.1: The bottom quark mass at the $Z$ boson scale (upper) and top quark
pole mass (lower plot) are shown as function of $M$ for both models and both
signs of $\mu$.

In Fig. 5.1 we show the FUT A and FUT B predictions for the top pole mass,
$M_{\rm top}$, and the running bottom mass at the scale $M_{Z}$, $m_{\rm
bot}(M_{Z})$, as a function of the unified gaugino mass $M$, for the two cases
$\mu<0$ and $\mu>0$. The running bottom mass is used to avoid the large QCD
uncertainties inherent to the pole mass. In the evaluation of the bottom mass
$m_{\rm bot}$, we have included the corrections coming from bottom squark-
gluino loops and top squark-chargino loops [122]. The prediction is compared
to the experimental values [123]111 These values correspond to the
experimental measurements at the time of the original evaluation. However, the
small change to the current values would not change the phenomenological
analysis in a relevant way.

$m_{b}(M_{Z})=2.83\pm 0.10\,\,\mathrm{GeV}~{}.$ (5.9)

and

$m_{t}^{\rm exp}=(173.2\pm 0.9)\,\,\mathrm{GeV}~{}.$ (5.10)

One can see that the value of $m_{b}$ depends strongly on the sign of $\mu$
due to the above mentioned radiative corrections involving SUSY particles. For
both models ${\bf A}$ and ${\bf B}$ the values for $\mu>0$ are above the
central experimental value, with $m_{b}(M_{Z})\sim 4.0-5.0$ GeV. For $\mu<0$,
on the other hand, model ${\bf B}$ shows overlap with the experimentally
measured values, $m_{b}(M_{Z})\sim 2.5-2.8$ GeV. For model ${\bf A}$ we find
$m_{b}(M_{Z})\sim 1.5-2.6$ GeV, and there is only a small region of allowed
parameter space at large $M$ where we find agreement with the experimental
value at the two $\sigma$ level. Therefore, the experimental determination of
$m_{b}(M_{Z})$ clearly selects the negative sign of $\mu$.

Now we turn to the top quark mass. The predictions for the top quark mass
$M_{t}$ are $\sim 183$ GeV and $\sim 172$ GeV in the models ${\bf A}$ and
${\bf B}$ respectively, as shown in the lower plot of Fig. 5.1. (Here it
should be kept in mind that theoretical values for $M_{t}$ may suffer from a
correction of $\sim 4\%$ [125, 124, 118]). One can see clearly that model
${\bf B}$ is singled out. In addition the value of $\tan\beta$ is found to be
$\tan\beta\sim 54$ and $\sim 48$ for models ${\bf A}$ and ${\bf B}$,
respectively. Thus from the comparison of the predictions of the two models
with experimental data only FUT B with $\mu<0$ survives.

###  5.2 Reduction of Couplings in the Minimal Supersymmetric $SU(5)$ GUT

In this section we consider the partial reduction of couplings in the minimal
$N=1$ supersymmetric gauge model based on the group SU(5) according to refs
[20, 41] The three generations of quarks and leptons are accommodated by three
chiral superfields in $\Psi^{I}({\bf 10})$ and $\Phi^{I}(\overline{\bf 5})$,
where $I$ runs over the three generations. A $\Sigma({\bf 24})$ is used to
break $SU(5)$ down to $SU(3)_{\rm C}\times SU(2)_{\rm L}\times U(1)_{\rm Y}$,
and $H({\bf 5})$ and $\overline{H}({\overline{\bf 5}})$ to describe the two
Higgs superfields appropriate for electroweak symmetry breaking [126, 127].
Note that the Finite Unified Models discussed in Sect. 5.1 contain four $({\bf
5}+{\bf\bar{5}})$ to describe the Higgs superfields appropriate for
electroweak symmetry breaking instead of one set of $({\bf 5}+{\bf\bar{5}})$
used here in the minimal $N=1$ supersymmetric $SU(5)$ version. This minimality
makes the present version asymptotically free (negative $\beta_{g}$) instead
of finite at one loop, which was the case of the models in Sect. 5.1. The
superpotential of the model is [126, 127]

$\begin{split}W&=\frac{g_{t}}{4}\,\epsilon^{\alpha\beta\gamma\delta\tau}\,\Psi^{(3)}_{\alpha\beta}\Psi^{(3)}_{\gamma\delta}H_{\tau}+\sqrt{2}g_{b}\,\Phi^{(3)\alpha}\Psi^{(3)}_{\alpha\beta}\overline{H}^{\beta}+\frac{g_{\lambda}}{3}\,\Sigma_{\alpha}^{\beta}\Sigma_{\beta}^{\gamma}\Sigma_{\gamma}^{\alpha}+g_{f}\,\overline{H}^{\alpha}\Sigma_{\alpha}^{\beta}H_{\beta}\\\
&+\frac{\mu_{\Sigma}}{2}\,\Sigma_{\alpha}^{\gamma}\Sigma_{\gamma}^{\alpha}++\mu_{H}\,\overline{H}^{\alpha}H_{\alpha}~{},\end{split}$
(5.11)

where $\alpha,\beta,\ldots$ are the $SU(5)$ indices, and we have suppressed
the Yukawa couplings of the first two generations. The Lagrangian containing
the SSB terms is

$\begin{split}-{\cal L}_{\rm
soft}&=m_{H_{u}}^{2}{\hat{H}}^{*\alpha}{\hat{H}}_{\alpha}+m_{H_{d}}^{2}\hat{\overline{H}}^{*}_{\alpha}\hat{\overline{H}}^{\alpha}+m_{\Sigma}^{2}{\hat{\Sigma}}^{{\dagger}~{}\alpha}_{\beta}{\hat{\Sigma}}_{\alpha}^{\beta}+\sum_{I=1,2,3}\,[\,m_{\Phi^{I}}^{2}{\hat{\Phi}}^{*~{}(I)}_{\alpha}{\hat{\Phi}}^{(I)\alpha}\\\
&+\,m_{\Psi^{I}}^{2}{\hat{\Psi}}^{{\dagger}~{}(I)\alpha\beta}{\hat{\Psi}}^{(I)}_{\beta\alpha}\,]+\\{\,\frac{1}{2}M\lambda\lambda+B_{H}\hat{\overline{H}}^{\alpha}{\hat{H}}_{\alpha}+B_{\Sigma}{\hat{\Sigma}}^{\alpha}_{\beta}{\hat{\Sigma}}_{\alpha}^{\beta}+h_{f}\,\hat{\overline{H}}^{\alpha}{\hat{\Sigma}}_{\alpha}^{\beta}{\hat{H}}_{\beta}\\\
&+\frac{h_{\lambda}}{3}\,{\hat{\Sigma}}_{\alpha}^{\beta}{\hat{\Sigma}}_{\beta}^{\gamma}{\hat{\Sigma}}_{\gamma}^{\alpha}+\frac{h_{t}}{4}\,\epsilon^{\alpha\beta\gamma\delta\tau}\,{\hat{\Psi}}^{(3)}_{\alpha\beta}{\hat{\Psi}}^{(3)}_{\gamma\delta}{\hat{H}}_{\tau}+\sqrt{2}h_{b}\,{\hat{\Phi}}^{(3)\alpha}{\hat{\Psi}^{(3)}}_{\alpha\beta}\hat{\overline{H}}^{\beta}+\mbox{h.c.}\,\\}~{},\end{split}$
(5.12)

where a hat is used to denote the scalar component of each chiral superfield.

The RG functions of this model may be found in refs. [128, 129, 109], and we
employ the usual normalization of the RG functions, $d{\rm
A}/d\ln\mu~{}=~{}[\beta^{(1)}(A)~{}~{}\mbox{or}~{}~{}\gamma^{(1)}(A)]/16\pi^{2}+\dots$,
where $\dots$ are higher orders, and $\mu$ is the renormalization scale:

$\displaystyle\beta^{(1)}(g)$
$\displaystyle=-3g^{3}~{},~{}\beta^{(1)}(g_{t})=[\,-\frac{96}{5}\,g^{2}+9\,g_{t}^{2}+\frac{24}{5}\,g_{f}^{2}+4\,g_{b}^{2}\,]\,g_{t}~{},$
$\displaystyle\beta^{(1)}(g_{b})$
$\displaystyle=[\,-\frac{84}{5}\,g^{2}+3\,g_{t}^{2}+\frac{24}{5}\,g_{f}^{2}+10\,g_{b}^{2}\,]\,g_{b}~{},$
$\displaystyle\beta^{(1)}(g_{\lambda})$
$\displaystyle=[\,-30\,g^{2}+\frac{63}{5}\,g_{\lambda}^{2}+3\,g_{f}^{2}\,]\,g_{\lambda}~{},$
$\displaystyle\beta^{(1)}(g_{f})$
$\displaystyle=[\,-\frac{98}{5}\,g^{2}+3\,g_{t}^{2}+4\,g_{b}^{2}+\frac{53}{5}\,g_{f}^{2}+\frac{21}{5}\,g_{\lambda}^{2}\,]\,g_{f}~{},~{}\gamma^{(1)}(M)=-6g^{2}\,M~{},$
$\displaystyle\gamma^{(1)}(\mu_{\Sigma})$
$\displaystyle=[\,-20g^{2}+2g_{f}^{2}+\frac{42}{5}g_{\lambda}^{2}\,]\,\mu_{\Sigma}~{},~{}\gamma^{(1)}(\mu_{H})=[\,-\frac{48}{5}g^{2}+\frac{48}{5}g_{f}^{2}+4g_{b}^{2}+3g_{t}^{2}\,]\,\mu_{H}~{},$
$\displaystyle\gamma^{(1)}(B_{H})$
$\displaystyle=[\,-\frac{48}{5}g^{2}+\frac{48}{5}g_{f}^{2}+4g_{b}^{2}+3g_{t}^{2}\,]\,B_{H}$
$\displaystyle+[\,\frac{96}{5}g^{2}M+\frac{96}{5}h_{f}g_{f}+8g_{b}h_{b}+6g_{t}h_{t}]\,\mu_{H}~{},$
$\displaystyle\gamma^{(1)}(B_{\Sigma})$
$\displaystyle=[\,-20g^{2}+2g_{f}^{2}+\frac{42}{5}g_{\lambda}^{2}\,]\,B_{\Sigma}+[\,40g^{2}M+4h_{f}g_{f}+\frac{84}{5}g_{\lambda}h_{\lambda}]\,\mu_{\Sigma}~{},$
$\displaystyle\gamma^{(1)}(h_{t})$
$\displaystyle=[\,-\frac{96}{5}\,g^{2}+9\,g_{t}^{2}+\frac{24}{5}\,g_{f}^{2}+4\,g_{b}^{2}\,]\,h_{t}$
$\displaystyle+[\,\frac{192}{5}Mg^{2}+18h_{t}g_{t}+8h_{b}g_{b}+\frac{48}{5}h_{f}g_{f}\,]\,g_{t}~{},$
$\displaystyle\gamma^{(1)}(h_{b})$
$\displaystyle=[\,-\frac{84}{5}\,g^{2}+3\,g_{t}^{2}+\frac{24}{5}\,g_{f}^{2}+10\,g_{b}^{2}\,]\,h_{b}$
$\displaystyle+[\,\frac{168}{5}Mg^{2}+6h_{t}g_{t}+20h_{b}g_{b}+\frac{48}{5}h_{f}g_{f}\,]\,g_{b}~{}~{},$
$\displaystyle\gamma^{(1)}(h_{\lambda})$
$\displaystyle=[\,-30\,g^{2}+\frac{63}{5}\,g_{\lambda}^{2}+3\,g_{f}^{2}\,]\,h_{\lambda}+[\,60Mg^{2}+\frac{126}{5}h_{\lambda}g_{\lambda}+6h_{f}g_{f}\,]\,g_{\lambda}~{},$
$\displaystyle\gamma^{(1)}(h_{f})$
$\displaystyle=[\,-\frac{98}{5}\,g^{2}+3\,g_{t}^{2}+4\,g_{b}^{2}+\frac{53}{5}\,g_{f}^{2}+\frac{21}{5}\,g_{\lambda}^{2}\,]\,h_{f}$
$\displaystyle~{}+[\,\frac{196}{5}Mg^{2}+6h_{t}g_{t}+8h_{b}g_{b}+\frac{42}{5}h_{\lambda}g_{\lambda}+\frac{106}{5}h_{f}g_{f}\,]\,g_{f}~{},$
$\displaystyle\gamma^{(1)}(m_{H_{d}}^{2})$
$\displaystyle=-\frac{96}{5}g^{2}M^{2}+\frac{48}{5}g_{f}^{2}(m_{H_{u}}^{2}+m_{H_{d}}^{2}+m_{\Sigma}^{2})$
$\displaystyle+8g_{b}^{2}(m_{H_{d}}^{2}+m_{\Psi^{3}}^{2}+m_{\Phi^{3}}^{2})+\frac{48}{5}h_{f}^{2}+8h_{b}^{2}~{},$
$\displaystyle\gamma^{(1)}(m_{H_{u}}^{2})$
$\displaystyle=-\frac{96}{5}g^{2}M^{2}+\frac{48}{5}g_{f}^{2}(m_{H_{u}}^{2}+m_{H_{d}}^{2}+m_{\Sigma^{3}}^{2})+6g_{t}^{2}(m_{H_{u}}^{2}+2m_{\Psi^{3}}^{2})+\frac{48}{5}h_{f}^{2}+6h_{t}^{2}~{},$
$\displaystyle\gamma^{(1)}(m_{\Sigma}^{2})$
$\displaystyle=-40g^{2}M^{2}+2g_{f}^{2}(m_{H_{u}}^{2}+m_{H_{d}}^{2}+m_{\Sigma}^{2})+\frac{126}{5}g_{\lambda}^{2}m_{\Sigma}^{2}+2h_{f}^{2}+\frac{42}{5}h_{\lambda}^{2}~{},$
$\displaystyle\gamma^{(1)}(m_{\Phi^{3}}^{2})$
$\displaystyle=-\frac{96}{5}g^{2}M^{2}+8g_{b}^{2}(m_{H_{d}}^{2}+m_{\Psi^{3}}^{2}+m_{\Phi^{3}}^{2})+8h_{b}^{2}~{},$
$\displaystyle\gamma^{(1)}(m_{\Psi^{3}}^{2})$
$\displaystyle=-\frac{144}{5}g^{2}M^{2}+6g_{t}^{2}(m_{H_{u}}^{2}+2m_{\Psi^{3}}^{2})+4g_{b}^{2}(m_{H_{d}}^{2}+m_{\Psi^{3}}^{2}+m_{\Phi^{3}}^{2})+6h_{t}^{2}+4h_{b}^{2}~{},$
$\displaystyle\gamma^{(1)}(m_{\Phi^{1,2}}^{2})$
$\displaystyle=-\frac{96}{5}g^{2}M^{2}~{},~{}\gamma^{(1)}(m_{\Psi^{1,2}}^{2})=-\frac{144}{5}g^{2}M^{2}~{},$
(5.13)

where $g$ stands for the gauge coupling.

The reduction solution is found as follows. We require that the reduced theory
should contain the minimal number of the SSB parameters that are consistent
with perturbative renormalizability. We will find that the set of the
perturbatively unified SSB parameters significantly differ from the so-called
universal SSB parameters.

Without loss of generality, one can assume that the gauge coupling $g$ is the
primary coupling. Note that the reduction solutions in the dimension-zero
sector is independent of the dimensionful sector (under the assumption of a
mass independent renormalization scheme). It has been found [128] that there
exist two asymptotically free (AF) solutions that make a Gauge-Yukawa
Unification possible in the present model:

$\begin{split}a&:g_{t}=\sqrt{\frac{2533}{2605}}g+{\cal
O}(g^{3})~{},~{}g_{b}=\sqrt{\frac{1491}{2605}}g+{\cal
O}(g^{3})~{},~{}g_{\lambda}=0~{},~{}g_{f}=\sqrt{\frac{560}{521}}g+{\cal
O}(g^{3})~{},\\\ b&:g_{t}=\sqrt{\frac{89}{65}}g+{\cal
O}(g^{3})~{},~{}g_{b}=\sqrt{\frac{63}{65}}g+{\cal
O}(g^{3})~{},~{}g_{\lambda}=0~{},~{}g_{f}=0~{},\end{split}$ (5.14)

where the higher order terms denote uniquely computable power series in $g$.
It has been also found that the two solutions in (5.14) describe the
boundaries of an asymptotically free RG-invariant surface in the space of the
couplings, on which $g_{\lambda}$ and $g_{f}$ can be different from zero. This
observation has enabled us to obtain a partial reduction of couplings for
which the $g_{\lambda}$ and $g_{f}$ can be treated as (non-vanishing)
independent parameters without loosing AF. Later we have found [130] that the
region on the AF surface consistent with the proton decay constraint has to be
very close to the solution $a$. Therefore, we assume in the following
discussion that we are exactly at the boundary defined by the solution $a$ 222
Note that $g_{\lambda}=0$ is inconsistent, but $g_{\lambda}<\sim 0.005$ has to
be fulfilled to satisfy the proton decay constraint [130]. We expect that the
inclusion of a small $g_{\lambda}$ will not affect the prediction of the
perturbative unification of the SSB parameters..

In the dimensionful sector, we seek the reduction of the parameters in the
form of Eqs. (2.30). First, one can realize that the supersymmetric mass
parameters, $\mu_{\Sigma}$ and $\mu_{H}$, and the gaugino mass parameter $M$
cannot be reduced; that is, there is no solution in the desired form.
Therefore, they should be treated as independent parameters. We find the
following lowest order reduction solution:

$B_{H}=\frac{1029}{521}\,\mu_{H}M~{},~{}B_{\Sigma}=-\frac{3100}{521}\,\mu_{\Sigma}M~{},$
(5.15)
$\begin{split}h_{t}&=-g_{t}\,M~{},~{}h_{b}=-g_{b}\,M~{},~{}h_{f}=-g_{f}\,M~{},~{}h_{\lambda}=0~{},\\\
m_{H_{u}}^{2}&=-\frac{569}{521}M^{2}~{},~{}m_{H_{d}}^{2}=-\frac{460}{521}M^{2}~{},~{}m_{\Sigma}^{2}=\frac{1550}{521}M^{2}~{},\\\
m_{\Phi^{3}}^{2}&=\frac{436}{521}M^{2}~{},~{}m_{\Phi^{1,2}}^{2}=\frac{8}{5}M^{2}~{},~{}m_{\Psi^{3}}^{2}=\frac{545}{521}M^{2}~{},~{}m_{\Psi^{1,2}}^{2}=\frac{12}{5}M^{2}~{}.\end{split}$
(5.16)

So, the gaugino mass parameter $M$ plays a similar role as the gravitino mass
$m_{2/3}$ in supergravity coupled GUTs and characterizes the scale of the
supersymmetry–breaking.

In addition to the $\mu_{\Sigma}$, $\mu_{H}$ and $M$, it is possible to
include also $B_{H}$ and $B_{\Sigma}$ as independent parameters without
changing the one-loop reduction solution (5.16).

The prediction of the minimal supersymmetric $SU(5)$ GUT, following the Gauge-
Yukawa Unification of the solution a in Eqs (5.14) is:

$\begin{split}M_{t}&\simeq 1.8\times 10^{2}~{}\mbox{GeV}~{},~{}M_{b}\simeq
5.4~{}\mbox{GeV}~{},~{}\alpha_{3}(M_{Z})\simeq 0.12~{},\\\ M_{\rm GUT}&\simeq
1.7\times 10^{16}~{}\mbox{GeV}~{},\alpha_{\rm GUT}\simeq
0.040~{},~{}\tan\beta(M_{\rm SUSY})\simeq 48~{},\end{split}$

where $M_{t}$ and $M_{b}$ are the physical top and bottom quark masses. These
values suffer from corrections coming from different sources such as threshold
effects, which are partly taken into account and estimated in Ref. [130]. In
Ref. [41], Tab. 1, also the prediction of the specific model for several SSB
parameters can be found. Just for completeness we mention that the input
parameters for the above prediction were:

$\alpha_{1}(M_{Z})=0.0169~{},~{}\alpha_{2}(M_{Z})=0.0337~{},~{}\alpha_{\tau}(M_{Z})=8.005\times
10^{-6}~{}$

while the SUSY scale was fixed at $500$ GeV.

The present model has very good chances to survive the recent experimental
constraints. A more detailed examination is in order to determine its
viability.

###  5.3 Gauge-Yukawa Unification in other Supersymmetric GUTs by Reducing the
Couplings

####  5.3.1 Asymptotically Non-Free Pati-Salam Model

In this section a model is discussed, where a partial reduction of couplings
is achieved, which however is not based on a single gauge group, but on a
product of simple groups. In order for the RGI method for the gauge coupling
unification to work, the gauge couplings should have the same asymptotic
behavior. Recall that this common behavior is absent in the standard model
with three families. A way to achieve a common asymptotic behavior of all the
different gauge couplings is to embed $SU(3)_{C}\times SU(2)_{L}\times
U(1)_{Y}$ to some non-abelian gauge group, as it was done in the previous
sections. However, in this case still a major rôle in the GYU is due to the
group theoretical aspects of the covering GUT. Here we would like to examine
the power of RGI method by considering theories without covering GUTs. We
found [131] that the minimal phenomenologically viable model is based on the
gauge group of Pati and Salam [132]– ${\cal G}_{\rm PS}\equiv SU(4)\times
SU(2)_{R}\times SU(2)_{L}$. We recall that $N=1$ supersymmetric models based
on this gauge group have been studied with renewed interest because they could
in principle be derived from superstrings [133, 134].

In our supersymmetric, Gauge-Yukawa unified model based on ${\cal G}_{\rm PS}$
[131], three generations of quarks and leptons are accommodated by six chiral
supermultiplets, three in $({\bf 4},{\bf 2},{\bf 1})$ and three
$({\bf\overline{4}},{\bf 1},{\bf 2})$, which we denote by
$\Psi^{(I)\mu~{}i_{R}}$ and $\overline{\Psi}_{\mu}^{(I)i_{L}}$, ($I$ runs over
the three generations, and $\mu,\nu~{}(=1,2,3,4)$ are the $SU(4)$ indices
while $i_{R}~{},~{}i_{L}~{}(=1,2)$ stand for the $SU(2)_{L,R}$ indices.) The
Higgs supermultiplets in $({\bf 4},{\bf 2},{\bf 1})$, $({\bf\overline{4}},{\bf
2},{\bf 1})$ and $({\bf 15},{\bf 1},{\bf 1})$ are denoted by
$H^{\mu~{}i_{R}}~{},~{}\overline{H}_{\mu~{}i_{R}}$ and $\Sigma^{\mu}_{\nu}$,
respectively. They are responsible for the spontaneous symmetry breaking (SSB)
of $SU(4)\times SU(2)_{R}$ down to $SU(3)_{C}\times U(1)_{Y}$. The SSB of
$U(1)_{Y}\times SU(2)_{L}$ is then achieved by the nonzero VEV of
$h_{i_{R}i_{L}}$ which is in $({\bf 1},{\bf 2},{\bf 2})$. In addition to these
Higgs supermultiplets, we introduce $G^{\mu}_{\nu~{}i_{R}i_{L}}~{}({\bf
15},{\bf 2},{\bf 2})$, $\phi~{}({\bf 1},{\bf 1},{\bf 1})$ and
$\Sigma^{{}^{\prime}\mu}_{\nu}~{}({\bf 15},{\bf 1},{\bf 1})$. The
$G^{\mu}_{\nu~{}i_{R}i_{L}}$ is introduced to realize the $SU(4)\times
SU(2)_{R}\times SU(2)_{L}$ version of the Georgi-Jarlskog type ansatz [135]
for the mass matrix of leptons and quarks while $\phi$ is supposed to mix with
the right-handed neutrino supermultiplets at a high energy scale. With these
things in mind, we write down the superpotential of the model $W$, which is
the sum of the following superpotentials:

$\begin{split}W_{Y}&=\sum_{I,J=1}^{3}g_{IJ}\,\overline{\Psi}^{(I)i_{R}}_{\mu}\,\Psi^{(J)\mu~{}i_{L}}~{}h_{i_{R}i_{L}}~{},~{}\\\
W_{GJ}&=g_{GJ}\,\overline{\Psi}^{(2)i_{R}}_{\mu}\,G^{\mu}_{\nu~{}i_{R}j_{L}}\,\Psi^{(2)\nu~{}j_{L}}~{},\\\
W_{NM}&=\sum_{I=1,2,3}\,g_{I\phi}~{}\epsilon_{i_{R}j_{R}}\,\overline{\Psi}^{(I)i_{R}}_{\mu}~{}H^{\mu~{}j_{R}}\,\phi~{},\\\
W_{SB}&=g_{H}\,\overline{H}_{\mu~{}i_{R}}\,\Sigma^{\mu}_{\nu}\,H^{\nu~{}i_{R}}+\frac{g_{\Sigma}}{3}\,\mbox{Tr}~{}[~{}\Sigma^{3}~{}]+\frac{g_{\Sigma^{\prime}}}{2}\,\mbox{Tr}~{}[~{}(\Sigma^{\prime})^{2}\,\Sigma~{}]~{},\\\
W_{TDS}&=\frac{g_{G}}{2}\,\epsilon^{i_{R}j_{R}}\epsilon^{i_{L}j_{L}}\,\mbox{Tr}~{}[~{}G_{i_{R}i_{L}}\,\Sigma\,G_{j_{R}j_{L}}~{}]~{},\\\
W_{M}&=m_{h}\,h^{2}+m_{G}\,G^{2}+m_{\phi}\,\phi^{2}+m_{H}\,\overline{H}\,H+m_{\Sigma}\,\Sigma^{2}+m_{\Sigma^{\prime}}\,(\Sigma^{\prime})^{2}~{}.\end{split}$

Although $W$ has the parity, $\phi\to-\phi$ and
$\Sigma^{\prime}\to-\Sigma^{\prime}$, it is not the most general potential,
but, as we already mentioned, this is not a problem in N = 1 SUSY theories.

We denote the gauge couplings of $SU(4)\times SU(2)_{R}\times SU(2)_{L}$ by
$\alpha_{4}~{},~{}\alpha_{2R}$ and $\alpha_{2L}$, respectively. The gauge
coupling for $U(1)_{Y}$, $\alpha_{1}$, normalized in the usual GUT inspired
manner, is given by $1/\alpha_{1}~{}=~{}2/5\alpha_{4}+3/5\alpha_{2R}~{}$. In
principle, the primary coupling can be any one of the couplings. But it is
more convenient to choose a gauge coupling as the primary one because the one-
loop $\beta$ functions for a gauge coupling depends only on its own gauge
coupling. For the present model, we use $\alpha_{2L}$ as the primary one.
Since the gauge sector for the one-loop $\beta$ functions is closed, the
solutions of the fixed point equations (2.22) are independent on the Yukawa
and Higgs couplings. One easily obtains
$\rho_{4}^{(1)}=8/9~{},~{}\rho_{2R}^{(1)}~{}=~{}4/5$, so that the RGI
relations (2.26) at the one-loop level become

$\displaystyle\tilde{\alpha}_{4}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{4}}{\alpha_{2L}}~{}=~{}\frac{8}{9}~{},~{}\tilde{\alpha}_{1}~{}=~{}\frac{\alpha_{1}}{\alpha_{2L}}~{}=~{}\frac{5}{6}~{}.$
(5.17)

The solutions in the Yukawa-Higgs sector strongly depend on the result of the
gauge sector. After slightly involved algebraic computations, one finds that
most predictive solutions contain at least three vanishing $\rho_{i}^{(1)}$’s.
Out of these solutions, there are two that exhibit the most predictive power
and moreover they satisfy the neutrino mass relation
$m_{\nu_{\tau}}>m_{\nu_{\mu}}~{},~{}m_{\nu_{e}}$. For the first solution we
have $\rho_{1\phi}^{(1)}=\rho_{2\phi}^{(1)}=\rho_{\Sigma}^{(1)}=0$, while for
the second solution, $\rho_{1\phi}^{(1)}=\rho_{2\phi}^{(1)}=\rho_{G}^{(1)}=0$,
and one finds that for the cases above the power series solutions (2.26) take
the form

$\begin{split}\tilde{\alpha}_{GJ}&\simeq\left\\{\begin{array}[]{l}1.67-0.05\tilde{\alpha}_{1\phi}+0.004\tilde{\alpha}_{2\phi}-0.90\tilde{\alpha}_{\Sigma}+\cdots\\\
2.20-0.08\tilde{\alpha}_{2\phi}-0.05\tilde{\alpha}_{G}+\cdots\end{array}\right.~{}~{},\\\
\tilde{\alpha}_{33}&\simeq\left\\{\begin{array}[]{l}3.33+0.05\tilde{\alpha}_{1\phi}+0.21\tilde{\alpha}_{2\phi}-0.02\tilde{\alpha}_{\Sigma}+\cdots\\\
3.40+0.05\tilde{\alpha}_{1\phi}-1.63\tilde{\alpha}_{2\phi}-0.001\tilde{\alpha}_{G}+\cdots\end{array}\right.~{}~{},\\\
\tilde{\alpha}_{3\phi}&\simeq\left\\{\begin{array}[]{l}1.43-0.58\tilde{\alpha}_{1\phi}-1.43\tilde{\alpha}_{2\phi}-0.03\tilde{\alpha}_{\Sigma}+\cdots\\\
0.88-0.48\tilde{\alpha}_{1\phi}+8.83\tilde{\alpha}_{2\phi}+0.01\tilde{\alpha}_{G}+\cdots\end{array}\right.~{}~{},\\\
\tilde{\alpha}_{H}&\simeq\left\\{\begin{array}[]{l}1.08-0.03\tilde{\alpha}_{1\phi}+0.10\tilde{\alpha}_{2\phi}-0.07\tilde{\alpha}_{\Sigma}+\cdots\\\
2.51-0.04\tilde{\alpha}_{1\phi}-1.68\tilde{\alpha}_{2\phi}-0.12\tilde{\alpha}_{G}+\cdots\end{array}\right.~{}~{},~{}~{}\\\
\tilde{\alpha}_{\Sigma}&\simeq\left\\{\begin{array}[]{l}---\\\
0.40+0.01\tilde{\alpha}_{1\phi}-0.45\tilde{\alpha}_{2\phi}-0.10\tilde{\alpha}_{G}+\cdots\end{array}\right.~{},\\\
\tilde{\alpha}_{\Sigma^{\prime}}&\simeq\left\\{\begin{array}[]{ll}4.91-0.001\tilde{\alpha}_{1\phi}-0.03\tilde{\alpha}_{2\phi}-0.46\tilde{\alpha}_{\Sigma}+\cdots\\\
8.30+0.01\tilde{\alpha}_{1\phi}+1.72\tilde{\alpha}_{2\phi}-0.36\tilde{\alpha}_{G}+\cdots\end{array}\right.~{}~{},\\\
\tilde{\alpha}_{G}&\simeq\left\\{\begin{array}[]{ll}5.59+0.02\tilde{\alpha}_{1\phi}-0.04\tilde{\alpha}_{2\phi}-1.33\tilde{\alpha}_{\Sigma}+\cdots\\\
\---\end{array}\right.~{}~{}.\end{split}$ (5.18)

We have assumed that the Yukawa couplings $g_{IJ}$ except for $g_{33}$ vanish.
They can be included into RGI relations as small perturbations, but their
numerical effects will be rather small.

The number $N_{H}$ of the Higgses lighter than $M_{\rm SUSY}$ could vary from
one to four while the number of those to be taken into account above $M_{\rm
SUSY}$ is fixed at four. We have assumed here that $N_{H}=1$. The dependence
of the top mass on $M_{\rm SUSY}$ in this model is shown in Fig. 5.2. One can
see that for any reasonable supersymmetry breaking scale in the TeV region the
experimentally found top quark mass cannot be reproduced within this model.

Figure 5.2: The values for $M_{t}$ predicted by the Pati-Salam model for
different $M_{\rm SUSY}$ scales.

####  5.3.2 Asymptotically Non-Free $SO(10)$ Model

In this section a model based on SO(10) is discussed, which also admits a
partial reduction of couplings [136]. We denote the hermitean $SO(10)$-gamma
matrices by $\Gamma_{\alpha}~{},~{}\alpha=1,\cdots,10$. The charge conjugation
matrix $C$ satisfies
$C=C^{-1}~{},~{}C^{-1}\,\Gamma_{\alpha}^{T}\,C~{}=~{}-\Gamma_{\alpha}$, and
the $\Gamma_{11}$ is defined as
$\Gamma_{11}\equiv(-i)^{5}\,\Pi_{\alpha=1}^{10}\Gamma_{\alpha}~{}~{}\mbox{with}~{}~{}(\Gamma_{11})^{2}~{}=~{}1$.
The chiral projection operators are given by ${\cal
P}_{\pm}=\frac{1}{2}(\,1\pm\Gamma_{11})$.

In $SO(10)$ GUTs [137, 138, 139], three generations of quarks and leptons are
accommodated by three chiral supermultiplets in ${\bf 16}$ which we denote by

$\Psi^{I}({\bf 16})~{}~{}\mbox{with}~{}~{}{\cal
P}_{+}\,\Psi^{I}~{}=~{}\Psi^{I}~{},$ (5.19)

where $I$ runs over the three generations and the spinor index is suppressed.
To break $SO(10)$ down to $SU(3)_{\rm C}\times SU(2)_{\rm L}\times U(1)_{\rm
Y}$, we use the following set of chiral superfields:

$S_{\\{\alpha\beta\\}}({\bf 54})~{},~{}A_{[\alpha\beta]}({\bf
45})~{},~{}\phi({\bf 16})~{},~{}\overline{\phi}({\overline{\bf 16}})~{}.$
(5.20)

The two $SU(2)_{\rm L}$ doublets which are responsible for the spontaneous
symmetry breaking (SSB) of $SU(2)_{\rm L}\times U(1)_{\rm Y}$ down to
$U(1)_{\rm EM}$ are contained in $H_{\alpha}({\bf 10})$. We further introduce
a singlet $\varphi$ which after the SSB of $SO(10)$ will mix with the right-
handed neutrinos so that they will become superheavy.

The superpotential of the model is given by

$\displaystyle W$ $\displaystyle=$ $\displaystyle
W_{Y}+W_{SB}+W_{HS}+W_{NM}+W_{M}~{},$ (5.21)

where

$\displaystyle W_{Y}$
$\displaystyle=\frac{1}{2}\sum_{I,J=1}^{3}g_{IJ}\,\Psi^{I}\,C\Gamma_{\alpha}\,\Psi^{J}~{}H_{\alpha}~{},$
$\displaystyle W_{SB}$
$\displaystyle=\frac{g_{\phi}}{2}\,\overline{\phi}\,\Gamma_{[\alpha\beta]}\,\phi~{}A_{[\alpha\beta]}+\frac{g_{S}}{3!}\,\mbox{Tr}~{}S^{3}+\frac{g_{A}}{2}\,\mbox{Tr}~{}A^{2}\,S~{},$
$\displaystyle W_{HS}$ $\displaystyle=\frac{g_{HS}}{2}\,\
H_{\alpha}\,S_{\\{\alpha\beta\\}}\,H_{\beta}~{},~{}W_{NM}^{I}~{}=~{}\sum_{I=1}^{3}\,g_{INM}\,\Psi^{I}\,\overline{\phi}\,\varphi~{},$
$\displaystyle W_{M}$
$\displaystyle=\frac{m_{H}}{2}\,H^{2}+m_{\varphi}\,\varphi^{2}+m_{\phi}\,\overline{\phi}\phi+\frac{m_{S}}{2}\,S^{2}+\frac{m_{A}}{2}\,A^{2}~{},$
(5.22)

and
$\Gamma_{[\alpha\beta]}=i(\Gamma_{\alpha}\Gamma_{\beta}-\Gamma_{\beta}\Gamma_{\alpha})/2$.
As in the case of the $SU(5)$ minimal model, the superpotential is not the
most general one, but this does not contradict the philosophy of the coupling
unification by the reduction method. $W_{SB}$ is responsible for the SSB of
$SO(10)$ down to $SU(3)_{C}\times SU(2)_{W}\times U(1)_{Y}$, and this can be
achieved without breaking supersymmetry, while $W_{HS}$ is responsible for the
triplet-doublet splitting of $H$. The right-handed neutrinos obtain a
superheavy mass through $W_{NM}$ after the SSB, and the Yukawa couplings for
the leptons and quarks are contained in $W_{Y}$. We assume that there exists a
choice of soft supersymmetry breaking terms so that all the vacuum expectation
values necessary for the desired SSB corresponds to the minimum of the
potential.

Given the supermultiplet content and the superpotential $W$, we can compute
the $\beta$ functions of the model. The gauge coupling of $SO(10)$ is denoted
by $g$, and our normalization of the $\beta$ functions is as usual, i.e.,
$dg_{i}/d\ln\mu~{}=~{}\beta^{(1)}_{i}/16\pi^{2}+O(g^{5})$, where $\mu$ is the
renormalization scale. We find:

$\begin{split}\beta^{(1)}_{g}&=7\,g^{3}~{},\\\
\beta^{\left(1\right)}_{g_{T}}&=g_{T}\,\left(\,14g_{T}^{2}+\frac{27}{5}g_{HS}^{2}+g_{3NM}^{2}-\frac{63}{2}g^{2}\,\right)~{},\\\
\beta^{\left(1\right)}_{g_{\phi}}&=g_{\phi}\left(\,53g_{\phi^{2}}+\frac{48}{5}g_{A}^{2}+\frac{1}{2}g_{1NM}^{2}+\frac{1}{2}g_{2NM}^{2}+\frac{1}{2}g_{3NM}^{2}-\frac{77}{2}g^{2}\,\right),\\\
\beta^{\left(1\right)}_{S}&=g_{S}\left(\,\frac{84}{5}g_{S}^{2}+12g_{A}^{2}+\frac{3}{2}g_{HS}^{2}-60g^{2}\,\right)~{},\\\
\beta^{\left(1\right)}_{A}&=g_{A}\left(\,16g_{\phi}^{2}+\frac{28}{5}g_{S}^{2}+\frac{116}{5}g_{A}^{2}+\frac{1}{2}g_{HS}^{2}-52g^{2}\,\right)~{},\\\
\beta^{\left(1\right)}_{HS}&=g_{HS}\left(\,8g_{T}^{2}+\frac{28}{5}g_{S}^{2}+4g_{A}^{2}+\frac{113}{10}g_{HS}^{2}-38g^{2}\,\right)~{},\\\
\beta^{\left(1\right)}_{1NM}&=g_{1NM}\left(\,\frac{45}{2}g_{\phi}^{2}+9g_{1NM}^{2}+\frac{17}{2}g_{2NM}^{2}+\frac{17}{2}g_{3NM}^{2}-\frac{45}{2}g^{2}\,\right)~{},\\\
\beta^{\left(1\right)}_{2NM}&=g_{2NM}\left(\,\frac{45}{2}g_{\phi}^{2}+\frac{17}{2}g_{1NM}^{2}+9g_{2NM}^{2}+\frac{17}{2}g_{3NM}^{2}-\frac{45}{2}g^{2}\,\right)~{},\\\
\beta^{\left(1\right)}_{3NM}&=g_{3NM}\left(\,5g_{T}^{2}+\frac{45}{2}g_{\phi}^{2}+\frac{17}{2}g_{1NM}^{2}+\frac{17}{2}g_{2NM}^{2}+9g_{3NM}^{2}-\frac{45}{2}g^{2}\,\right)~{}.\end{split}$
(5.23)

We have assumed that the Yukawa couplings $g_{IJ}$ except for $g_{T}\equiv
g_{33}$ vanish. They can be included as small perturbations. Needless to say
that the soft susy breaking terms do not alter the $\beta$ functions above.

We find that there exist two independent solutions, $A$ and $B$, that have the
most predictive power, where we have chosen the $SO(10)$ gauge coupling as the
primary coupling:

$\displaystyle\rho_{T}$
$\displaystyle=\left\\{\begin{array}[]{ll}163/60&\simeq 2.717\\\
0&\end{array}\right.~{}~{},~{}~{}\rho_{\phi}~{}=~{}\left\\{\begin{array}[]{ll}5351/9180&\simeq
0.583\\\ 1589/2727&\simeq 0.583\end{array}\right.~{},$ (5.28)
$\displaystyle\rho_{S}$
$\displaystyle=\left\\{\begin{array}[]{ll}152335/51408&\simeq 2.963\\\
850135/305424&\simeq
2.783\end{array}\right.,\rho_{A}=\left\\{\begin{array}[]{ll}31373/22032&\simeq
1.424\\\ 186415/130896&\simeq 1.424\end{array}\right.,$ (5.33)
$\displaystyle\rho_{HS}$ $\displaystyle=\left\\{\begin{array}[]{ll}7/81&\simeq
0.086\\\ 170/81&\simeq
2.099\end{array}\right.~{},~{}\rho_{1NM}=\rho_{2NM}=\left\\{\begin{array}[]{ll}191/204&\simeq
0.936\\\ 191/303&\simeq 0.630\end{array}\right.~{}~{},$ (5.38)
$\displaystyle\rho_{3NM}$ $\displaystyle=\left\\{\begin{array}[]{ll}0&\\\
191/303&\simeq
0.630\end{array}\right.~{}~{}~{}\mbox{for}~{}~{}\left\\{\begin{array}[]{l}A\\\
B\end{array}\right.~{}.$ (5.43)

Clearly, the solution B has less predictive power because $\rho_{T}=0$. So, we
consider below only the solution A, in which the coupling $\alpha_{3NM}$
should be regarded as a small perturbation because $\rho_{3NM}=0$.

Given this solution it is possible to show, as in the case of $SU(5)$, that
the $\rho$’s can be uniquely computed in any finite order in perturbation
theory.

The corrections to the reduced couplings coming from the small perturbations
up to and including terms of $O(\tilde{\alpha}_{3NM}^{2}$) are:

$\begin{split}\tilde{\alpha}_{T}&=(\,163/60-0.108\cdots\tilde{\alpha}_{3NM}+0.482\cdots\tilde{\alpha}_{3NM}^{2}+\cdots\,)+\cdots~{},\\\
\tilde{\alpha}_{\phi}&=(\,5351/9180+0.316\cdots\tilde{\alpha}_{3NM}+0.857\cdots\tilde{\alpha}_{3NM}^{2}+\cdots\,)+\cdots~{},\\\
\tilde{\alpha}_{S}&=(\,152335/51408+0.573\cdots\tilde{\alpha}_{3NM}+5.7504\cdots\tilde{\alpha}_{3NM}^{2}+\cdots\,)+\cdots~{},\\\
\tilde{\alpha}_{A}&=(\,31373/22032-0.591\cdots\tilde{\alpha}_{3NM}-4.832\cdots\tilde{\alpha}_{3NM}^{2}+\cdots\,)+\cdots~{},\\\
\tilde{\alpha}_{HS}&=(7/81-0.00017\cdots\tilde{\alpha}_{3NM}+0.056\cdots\tilde{\alpha}_{3NM}^{2}+\cdots\,)+\cdots~{},\\\
\tilde{\alpha}_{1NM}&=\tilde{\alpha}_{2NM}=(\,191/204-4.473\cdots\tilde{\alpha}_{3NM}+2.831\cdots\tilde{\alpha}_{3NM}^{2}+\cdots\,)+\cdots,\end{split}$
(5.44)

where $\cdots$ indicates higher order terms which can be uniquely computed. In
the partially reduced theory defined above, we have two independent couplings,
$\alpha$ and $\alpha_{3NM}$ (along with the Yukawa couplings
$\alpha_{IJ}~{},~{}I,J\neq T$).

At the one-loop level, Eq. (5.44) defines a line parametrized by
$\tilde{\alpha}_{3NM}$ in the $7$ dimensional space of couplings. A numerical
analysis shows that this line blows up in the direction of
$\tilde{\alpha}_{S}$ at a finite value of $\tilde{\alpha}_{3NM}$ [136]. So if
we require $\tilde{\alpha}_{S}$ to remain within the perturbative regime
(i.e., $g_{S}\leq 2$, which means $\tilde{\alpha}_{S}\leq 8$ because
$\alpha_{\rm GUT}\sim 0.04$), the $\tilde{\alpha}_{3NM}$ should be restricted
to be below $\sim 0.067$. As a consequence, the value of $\tilde{\alpha}_{T}$
is also bounded

$2.714\leq\tilde{\alpha}_{T}\leq 2.736~{}.$ (5.45)

This defines GYU boundary conditions holding at the unification scale $M_{\rm
GUT}$ in addition to the group theoretical one,
$\alpha_{T}=\alpha_{t}~{}=~{}\alpha_{b}~{}=~{}\alpha_{\tau}$. The value of
$\tilde{\alpha}_{T}$ is practically fixed so that we may assume that
$\tilde{\alpha}_{T}=163/60\simeq 2.72$, which is the unperturbed value.

Figure 5.3: Top quark mass prediction versus $M_{\rm SUSY}$ for
$\tilde{\alpha}_{T}=2.717$.

Fig. 5.3 shows the prediction for the top quark mass in this model for
different values of the supersymmetry breaking scale $M_{\rm SUSY}$. While the
value for the top quark mass predicted is below its infrared value ($\sim
189\,\,\mathrm{GeV}$) [136], it is above the experimental value [140].
Consequently, also this particular model has difficulties to meet the
experimental data on the top quark mass, despite the theoretical uncertainties
involved.

###  5.4 Finite $SU(N)^{3}$ Unification

We continue examining the possibility of constructing realistic FUTs based on
product gauge groups. Consider an $N=1$ supersymmetric theory, with gauge
group $SU(N)_{1}\times SU(N)_{2}\times\cdots\times SU(N)_{k}$, with $n_{f}$
copies (number of families) of the supersymmetric multiplets
$(N,N^{*},1,\dots,1)+(1,N,N^{*},\dots,1)+\cdots+(N^{*},1,1,\dots,N)$. The one-
loop $\beta$-function coefficient in the renormalization-group equation of
each $SU(N)$ gauge coupling is simply given by

$b=\left(-\frac{11}{3}+\frac{2}{3}\right)N+n_{f}\left(\frac{2}{3}+\frac{1}{3}\right)\left(\frac{1}{2}\right)2N=-3N+n_{f}N\,.$
(5.46)

This means that $n_{f}=3$ is the only solution of Eq.(5.46) that yields $b=0$.
Since $b=0$ is a necessary condition for a finite field theory, the existence
of three families of quarks and leptons is natural in such models, provided
the matter content is exactly as given above.

The model of this type with best phenomenology is the $SU(3)^{3}$ model
discussed in Ref. [141], where the details of the model are given. It
corresponds to the well-known example of $SU(3)_{C}\times SU(3)_{L}\times
SU(3)_{R}$ [142, 143, 144, 145], with quarks transforming as

$q=\begin{pmatrix}d&u&h\\\ d&u&h\\\
d&u&h\end{pmatrix}\sim(3,3^{*},1),~{}~{}~{}q^{c}=\begin{pmatrix}d^{c}&d^{c}&d^{c}\\\
u^{c}&u^{c}&u^{c}\\\ h^{c}&h^{c}&h^{c}\end{pmatrix}\sim(3^{*},1,3),$ (5.47)

and leptons transforming as

$\lambda=\begin{pmatrix}N&E^{c}&\nu\\\ E&N^{c}&e\\\
\nu^{c}&e^{c}&S\end{pmatrix}\sim(1,3,3^{*}).$ (5.48)

Switching the first and third rows of $q^{c}$ together with the first and
third columns of $\lambda$, we obtain the alternative left-right model first
proposed in Ref. [145] in the context of superstring-inspired $E_{6}$.

In order for all the gauge couplings to be equal at an energy scale, $M_{\rm
GUT}$, the cyclic symmetry $Z_{3}$ must be imposed, i.e.

$q\to\lambda\to q^{c}\to q,$ (5.49)

where $q$ and $q^{c}$ are given in eq. (5.47) and $\lambda$ in eq. (5.48).
Then, the first of the finiteness conditions (4.5) for one-loop finiteness,
namely the vanishing of the gauge $\beta$-function is satisfied.

Next let us consider the second condition, i.e. the vanishing of the anomalous
dimensions of all superfields, eq. (4.6). To do that first we have to write
down the superpotential. If there is just one family, then there are only two
trilinear invariants, which can be constructed respecting the symmetries of
the theory, and therefore can be used in the superpotential as follows

$f~{}Tr(\lambda
q^{c}q)+\frac{1}{6}f^{\prime}~{}\epsilon_{ijk}\epsilon_{abc}(\lambda_{ia}\lambda_{jb}\lambda_{kc}+q^{c}_{ia}q^{c}_{jb}q^{c}_{kc}+q_{ia}q_{jb}q_{kc}),$
(5.50)

where $f$ and $f^{\prime}$ are the Yukawa couplings associated to each
invariant. Quark and leptons obtain masses when the scalar parts of the
superfields $(\tilde{N},\tilde{N}^{c})$ obtain vacuum expectation values
(vevs),

$m_{d}=f\langle\tilde{N}\rangle,~{}~{}m_{u}=f\langle\tilde{N}^{c}\rangle,~{}~{}m_{e}=f^{\prime}\langle\tilde{N}\rangle,~{}~{}m_{\nu}=f^{\prime}\langle\tilde{N}^{c}\rangle.$
(5.51)

With three families, the most general superpotential contains 11 $f$
couplings, and 10 $f^{\prime}$ couplings, subject to 9 conditions, due to the
vanishing of the anomalous dimensions of each superfield. The conditions are
the following

$\sum_{j,k}f_{ijk}(f_{ljk})^{*}+\frac{2}{3}\sum_{j,k}f^{\prime}_{ijk}(f^{\prime}_{ljk})^{*}=\frac{16}{9}g^{2}\delta_{il}\,,$
(5.52)

where

$\displaystyle f_{ijk}=f_{jki}=f_{kij},$ (5.53) $\displaystyle
f^{\prime}_{ijk}=f^{\prime}_{jki}=f^{\prime}_{kij}=f^{\prime}_{ikj}=f^{\prime}_{kji}=f^{\prime}_{jik}.$
(5.54)

Quarks and leptons receive masses when the scalar part of the superfields
$\tilde{N}_{1,2,3}$ and $\tilde{N}^{c}_{1,2,3}$ obtain vevs as follows

$\displaystyle({\cal
M}_{d})_{ij}=\sum_{k}f_{kij}\langle\tilde{N}_{k}\rangle,~{}~{}~{}({\cal
M}_{u})_{ij}=\sum_{k}f_{kij}\langle\tilde{N}^{c}_{k}\rangle,$ (5.55)
$\displaystyle({\cal
M}_{e})_{ij}=\sum_{k}f^{\prime}_{kij}\langle\tilde{N}_{k}\rangle,~{}~{}~{}({\cal
M}_{\nu})_{ij}=\sum_{k}f^{\prime}_{kij}\langle\tilde{N}^{c}_{k}\rangle.$
(5.56)

We will assume that the below $M_{\rm GUT}$ we have the usual MSSM 333For
details of how the spontaneous breaking of $SU(3)^{3}$ to MSSM can be achieved
see refs [146, 147] and refs therein., with the two Higgs doublets coupled
maximally to the third generation. Therefore we have to choose the linear
combinations $\tilde{N}^{c}=\sum_{i}a_{i}\tilde{N}^{c}_{i}$ and
$\tilde{N}=\sum_{i}b_{i}\tilde{N}_{i}$ to play the role of the two Higgs
doublets, which will be responsible for the electroweak symmetry breaking.
This can be done by choosing appropriately the masses in the superpotential
[114], since they are not constrained by the finiteness conditions. We choose
that the two Higgs doublets are predominately coupled to the third generation.
Then these two Higgs doublets couple to the three families differently, thus
providing the freedom to understand their different masses and mixings. The
remnants of the $SU(3)^{3}$ FUT are the boundary conditions on the gauge and
Yukawa couplings, i.e. Eq.(5.52), the $h=-MC$ relation, and the soft scalar-
mass sum rule eq. (5.8) at $M_{\rm GUT}$, which, when applied to the present
model, takes the form

$\displaystyle
m^{2}_{H_{u}}+m^{2}_{\tilde{t}^{c}}+m^{2}_{\tilde{q}}=M^{2}=m^{2}_{H_{d}}+m^{2}_{\tilde{b}^{c}}+m^{2}_{\tilde{q}}~{},$
(5.57)

where ${\tilde{t}^{c}},~{}{\tilde{b}^{c}}$, and ${\tilde{q}}$ are the scalar
parts of the corresponding superfields.

Concerning the solution to Eq.(5.52) we consider two versions of the model:
I) An all-loop finite model with a unique and isolated solution, in which
$f^{\prime}$ vanishes, which leads to the following relations

$f^{2}=f^{2}_{111}=f^{2}_{222}=f^{2}_{333}=\frac{16}{9}g^{2}\,.$ (5.58)

As for the lepton masses, since all $f^{\prime}$ couplings have been fixed to
be zero at this order, in principle they would be expected to appear
radiatively induced by the scalar lepton masses appearing in the SSB sector of
the theory. However, due to the finiteness conditions they cannot appear
radiatively and remain as a problem for further study.
II) A two-loop finite solution, in which we keep $f^{\prime}$ non-vanishing
and we use it to introduce the lepton masses. The model in turn becomes finite
only up to two-loops since the corresponding solution of Eq.(5.52) is not an
isolated one any more, i.e. it is a parametric one. In this case we have the
following boundary conditions for the Yukawa couplings

$\displaystyle f^{2}=r\left(\frac{16}{9}\right)g^{2}\,,\quad f^{\prime
2}=(1-r)\left(\frac{8}{3}\right)g^{2}\,,$ (5.59)

where $r$ is a free parameter which parametrizes the different solutions to
the finiteness conditions. As for the boundary conditions of the soft scalars,
we have the universal case.

Below $M_{\rm GUT}$ all couplings and masses of the theory run according to
the RGEs of the MSSM. Thus we examine the evolution of these parameters
according to their RGEs up to two-loops for dimensionless parameters and at
one-loop for dimensionful ones imposing the corresponding boundary conditions.
We further assume a unique SUSY breaking scale $M_{\rm SUSY}$ and below that
scale the effective theory is just the SM.

We compare our predictions with the experimental value of $m_{t}^{\rm exp}$
444 As before, these values correspond to the experimental measurements at the
time of the original evaluation. Again, the small change to the current values
would not change the phenomenological analysis in a relevant way. and recall
that the theoretical values for $m_{t}$ suffer from a correction of $\sim 4\%$
[125, 124, 118]. In the case of the bottom quark, we take again the value
evaluated at $M_{Z}$, see Eq. (5.9). In the case of model I, the predictions
for the top quark mass (in this case $m_{b}$ is an input) $m_{t}$ are $\sim
183\,{\rm~{}GeV}$ for $\mu<0$, which is above the experimental value, and
there are no solutions for $\mu>0$.

Figure 5.4: The figures show the values for the top and bottom quark masses
for the FUT model $SU(3)^{3}$, with $\mu<0$, vs the parameter $r$. The thicker
horizontal line is the experimental central value, and the lighter green and
orange ones are the one and two sigma limits respectively. The red points are
the ones that satisfy the $B$-physics constraints, as discussed in Chapter 6.

For the two-loop model II, we look for the values of the parameter $r$ which
comply with the experimental limits given above for top and bottom quarks
masses. In the case of $\mu>0$, for the bottom quark, the values of $r$ lie in
the range $0.15\lesssim r\lesssim 0.32$. For the top mass, the range of values
for r is $0.35\lesssim r\lesssim 0.6$. From these values we can see that there
is a very small region where both top and bottom quark masses are in the
experimental range for the same value of $r$. In the case of $\mu<0$ the
situation is similar, although slightly better, with the range of values
$0.62\lesssim r\lesssim 0.77$ for the bottom mass, and $0.4\lesssim r\lesssim
0.62$ for the top quark mass. In the above mentioned analysis, the masses of
the new particles $h$’s and $E$’s of all families were taken to be at the
$M_{\rm GUT}$ scale.

Taking into account new thresholds for these exotic particles below $M_{\rm
GUT}$ we find a wider phenomenologically viable parameter space [148]. This
can be seen in Fig. 5.4, where we took only one down-like exotic particle
decoupling at $10^{14}$ GeV, below than the usual MSSM.

In this case, for $r\sim 0.5\sim 0.62$ we have reasonable agreement with
experimental data for both top and bottom quark masses, where the red points
in the figure are the ones that satisfy the $B$-physics constraints (at the
time of the analysis) [149]. The above analysis shows that it is worth
returning with a fresh examination of this model taking into account all new
experimental constraints.

###  5.5 Reduction of Couplings in the MSSM

In this section we are working in the framework of MSSM, assuming though the
existence of a covering GUT.

The analysis of the partial reduction of couplings in this framework was first
done in refs [150, 151].

The superpotential of the MSSM (where again we restrict ourselves to the third
generation of fermions) is defined by

$W=Y_{t}H_{2}Qt^{c}+Y_{b}H_{1}Qb^{c}+Y_{\tau}H_{1}L\tau^{c}+\mu H_{1}H_{2}\,,$
(5.60)

where $Q,L,t,b,\tau,H_{1},H_{2}$ are the usual superfields of MSSM, while the
SSB Lagrangian is given by

$\begin{split}-\mathcal{L}_{\rm
SSB}&=\sum_{\phi}m^{2}_{\phi}\hat{\phi^{*}}\hat{\phi}+\left[m^{2}_{3}\hat{H_{1}}\hat{H_{2}}+\sum_{i=1}^{3}\frac{1}{2}M_{i}\lambda_{i}\lambda_{i}+\textrm{h.c}\right]\\\
&+\left[h_{t}\hat{H_{2}}\hat{Q}\hat{t^{c}}+h_{b}\hat{H_{1}}\hat{Q}\hat{b^{c}}+h_{\tau}\hat{H_{1}}\hat{L}\hat{\tau^{c}}+\textrm{h.c.}\right],\end{split}$
(5.61)

where $\hat{\phi}$ represents the scalar component of all superfields,
$\lambda$ refers to the gaugino fields while all hatted fields refer to the
scalar components of the corresponding superfield. The Yukawa $Y_{t,b,\tau}$
and the trilinear $h_{t,b,\tau}$ couplings refer to the third generator only,
neglecting the first two generations.

Let us start with the dimensionless couplings, i.e. gauge and Yukawa. As a
first step we consider only the strong coupling and the top and bottom Yukawa
couplings, while the other two gauge couplings and the tau Yukawa will be
treated as corrections. Following the above line, we reduce the Yukawa
couplings in favour of the strong coupling $\alpha_{3}$

$\frac{Y^{2}_{i}}{4\pi}\equiv\alpha_{i}=G_{i}^{2}\alpha_{3},\qquad i=t,b,$

and using the RGE for the Yukawa, we get

$G_{i}^{2}=\frac{1}{3},\qquad i=t,b.$

This system of the top and bottom Yukawa couplings reduced with the strong one
is dictated by (i) the different running behaviour of the $SU(2)$ and $U(1)$
coupling compared to the strong one [26] and (ii) the incompatibility of
applying the above reduction for the tau Yukawa since the corresponding
$G^{2}$ turns negative [152]. Adding now the two other gauge couplings and the
tau Yukawa in the RGE as corrections, we obtain

$G_{t}^{2}=\frac{1}{3}+\frac{71}{525}\rho_{1}+\frac{3}{7}\rho_{2}+\frac{1}{35}\rho_{\tau},\qquad
G_{b}^{2}=\frac{1}{3}+\frac{29}{525}\rho_{1}+\frac{3}{7}\rho_{2}-\frac{6}{35}\rho_{\tau}$
(5.62)

where

$\rho_{1,2}=\frac{g_{1,2}^{2}}{g_{3}^{2}}=\frac{\alpha_{1,2}}{\alpha_{3}},\qquad\rho_{\tau}=\frac{g_{\tau}^{2}}{g_{3}^{2}}=\frac{\displaystyle{\frac{Y^{2}_{\tau}}{4\pi}}}{\alpha_{3}}$
(5.63)

Note that the corrections in Eq.(5.62) are taken at the GUT scale and under
the assumption that

$\frac{d}{dg_{3}}\left(\frac{Y_{t,b}^{2}}{g_{3}^{2}}\right)=0.$

Let us comment on our assumption above, which led to the Eq.(5.62). In
practice we assume that even including the corrections from the rest of the
gauge as well as the tau Yukawa couplings, at the GUT scale the ratio of the
top and bottom couplings $\alpha_{t,b}$ over the strong coupling are still
constant, i.e. their scale dependence is negligible. Or, rephrasing it, our
assumption can be understood as a requirement that in the ultraviolet (close
to the GUT scale) the ratios of the top and bottom Yukawa couplings over the
strong coupling become least sensitive against the change of the
renormalization scale. This requirement sets the boundary condition at the GUT
scale, given in Eq.(5.62). Alternatively one could follow the systematic
method to include the corrections to a non-trivially reduced system developed
in ref. [27], but considering two reduced systems: the first one consisting of
the “top, bottom” couplings and the second of the “strong, bottom” ones.

In the next order the corrections are assumed to be in the form

$\alpha_{i}=G_{i}^{2}\alpha_{3}+J_{i}^{2}\alpha_{3}^{2},\qquad i=t,b.$

Then, the coefficients $J_{i}$ are given by

$J_{i}^{2}=\frac{1}{4\pi}\,\frac{17}{24},\qquad i=t,b$

for the case where only the strong gauge and the top and bottom Yukawa
couplings are active, while for the case where the other two gauge and the tau
Yukawa couplings are added as corrections we obtain

$J_{t}^{2}=\frac{1}{4\pi}\frac{N_{t}}{D},\quad
J_{b}^{2}=\frac{1}{4\pi}\frac{N_{b}}{5D},$

where