#  The LHC Higgs Boson Discovery: Updated implications for Finite Unified
Theories and the SUSY breaking scale

S. Heinemeyer1,2,3 email<EMAIL_ADDRESS>M. Mondragón4 email:
<EMAIL_ADDRESS>G. Patellis5 email<EMAIL_ADDRESS>N.
Tracas5 and G. Zoupanos5,6
1Instituto de Física Teórica, Universidad Autónoma de Madrid Cantoblanco,
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,
México 01000
5 Physics Department, Nat. Technical University, 157 80 Zografou, Athens,
Greece
6 Max-Planck Institut für Physik, Föhringer Ring 6, D-80805 München, Germany
email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Finite Unified Theories (FUTs) are $N=1$ supersymmetric Grand Unified Theories
which can be made finite to all orders in perturbation theory, based on the
principle of reduction of couplings. The latter consists in searching for
renormalization group invariant relations among parameters of a renormalizable
theory holding to all orders in perturbation theory. FUTs have proven very
successful so far. In particular, they predicted the top quark mass one and
half years before its experimental discovery, while around five years before
the Higgs boson discovery a particular FUT was predicting the light Higgs
boson in the mass range $\sim 121-126~{}{\rm GeV}$, in striking agreement with
the discovery at LHC. Here we review the basic properties of the
supersymmetric theories and in particular finite theories resulting from the
application of the method of reduction of couplings in their dimensionless and
dimensionful sectors. Then we analyse the phenomenologically favoured FUT,
based on SU(5). This particular FUT leads to a finiteness constrained version
of the MSSM, which naturally predicts a relatively heavy spectrum with
coloured supersymmetric particles above 2.7 TeV, consistent with the non-
observation of those particles at the LHC. The electroweak supersymmetric
spectrum starts below 1 TeV and large parts of the allowed spectrum of the
lighter might be accessible at CLIC. The FCC-hh will be able to fully test the
predicted spectrum.

IFT-UAM/CSIC-18-013

##  1 Introduction

In 2012 the discovery of a new particle at the LHC was announced [1, 2].
Within theoretical and experimental uncertainties the new particle is
compatible with predictions for the Higgs boson of the Standard Model (SM) [3,
4], constituting a milestone in the quest for understanding the physics of
electroweak symmetry breaking (EWSB). However, taking the experimental results
and the respective uncertainties into account, also many models beyond the SM
can accomodate the data. Furthermore, the hierarchy problem, the neutrino
masses, the Dark Matter, the over twenty free parameters of the model, just to
name some questions, ask for a more fundamental theory to answer some, if not
all, of those.

Therefore, one of the main aims of this fundamental theory is to relate these
free parameters, or rephrasing it, to achieve a reduction of these parameters
in favour of a smaller number (or ideally only one). This reduction is usually
based in the introduction of a larger symmetry, rendering the theory more
predictive. Very good examples are the Grand Unified Theories (GUTs) [5, 6, 7,
9, 8] and their supersymmetric extensions [10, 11]. The case of minimal
$SU(5)$ is one example, where the number of couplings is reduced to one due to
the corresponding unification. Data from LEP [12] suggested that a $N=1$
global supersymmetry (SUSY) [10, 11] is required in order for the prediction
to be viable. Relations among the Yukawa couplings are also suggested in GUTs.
For example, the $SU(5)$ predicts the ratio of the tau to the bottom mass
$M_{\tau}/M_{b}$ [13] in the SM. GUTs introduce, however, new complications
such as the different ways of breaking this larger symmetry as well as new
degrees of freedom.

A way to relate the Yukawa and the gauge sector, in other words achieving
Gauge - Yukawa Unification (GYU) [14, 15, 16] seems to be a natural extension
of the GUTs. The possibility that $N=2$ SUSY [17] plays such a role is highly
limited due to the prediction of light mirror fermions. Other phenomenological
drawbacks appear in composite models and superstring theories.

A complementary approach is to search for all-loop Renormalization Group
Invariant (RGI) relations [18, 19] which hold below the Planck scale and are
preserved down to the unification scale [20, 24, 21, 22, 23, 14, 25, 15, 16].
With this approach Gauge - Yukawa unification (GYU) is possible. A remarkable
point is that, assuming finiteness at one-loop in $N=1$ gauge theories, RGI
relations that guarantee finiteness to all orders in perturbation theory can
be found [28, 27, 29].

The above approach seems to need SUSY as an essential ingredient. However the
breaking of SUSY has to be understood too, since it provides the SM with
several predictions for its free parameters. Actually, the RGI relation
searches have been extended to the soft SUSY breaking (SSB) sector [25, 30,
31, 32] relating parameters of mass dimension one and two. This is indeed
possible to be done on the RGI surface which is defined by the solution of the
reduction equations.

Applying the reduction of couplings method to $N=1$ SUSY theories has led to
very interesting phenomenological developments. Previously, an appealing
“universal” set of soft scalar masses was assumed in the SSB sector of SUSY
theories, given that, apart from economy and simplicity, (1) they are part of
the constraints that preserve finiteness up to two loops [34, 35], (2) they
are RGI up to two loops in more general SUSY gauge theories, subject to the
condition known as $P=1/3Q$ [30] and (3) they appear in the attractive dilaton
dominated SUSY breaking superstring scenarios [36, 37, 38]. However, further
studies exhibited problems all due to the restrictive nature of the
“universality” assumption for the scalar masses. For instance, (a) in Finite
Unified Theories (FUTs) the universality predicts that the lightest SUSY
particle is a charged particle, namely the superpartner of the $\tau$ lepton
$\tilde{\tau}$, (b) the MSSM with universal soft scalar masses is inconsistent
with the attractive radiative electroweak symmetry breaking and, worst of all,
(c) the universal soft scalar masses lead to charge and/or colour breaking
minima deeper than the standard vacuum [39]. Therefore, there have been
attempts to relax this constraint without loosing its attractive features.
First, an interesting observation was made that in $N=1$ GYU theories there
exists a RGI sum rule for the soft scalar masses at lower orders; at one loop
for the non-finite case [40] and at two loops for the finite case [42]. The
sum rule manages to overcome the above unpleasant phenomenological
consequences. Moreover, it was proven [41] that the sum rule for the soft
scalar masses is RGI to all orders for both the general and the finite case.
Finally, the exact $\beta$-function for the soft scalar masses in the Novikov-
Shifman-Vainstein-Zakharov (NSVZ) scheme [43, 44, 45] for the softly broken
SUSY QCD has been obtained [41]. The use of RGI both in the dimensionful and
dimensionless sector, together with the above mentioned sum rule, allows for
the construction of realistic and predictive $N=1$ all-loop finite $SU(5)$
SUSY GUTs, also with interesting predictions, as was shown in Refs. [20] and
[14, 22, 26, 33, 32, 46, 47, 48].

This paper is organised as follows. In Section 2 we review the theoretical
basis of the method of the reduction of couplings, which is extended in the
subsection 2.1 to the dimensionful parameters. Section 3 is devoted to
Finiteness in the dimensionless sector of a SUSY theory in some detail. In
Section 4 we discuss the implications of the method of reduction of couplings
in the SUSY breaking sector of an $N=1$ SUSY theory including the finite case.
Then in Section 5 we review the best Finite Unified Model selected previously
on the basis of agreement with the known experimental data at the time [33].
The current set-up of experimental constraints and predictions is briefly
reviewed in Section 6 and applied to our best Finite Unified Model in Section
7, including in particular the latest improvements in the prediction of the
light Higgs boson mass (as implemented in FeynHiggs). Our conclusions can be
found in Section 8.

##  2 Theoretical basis

In this section we outline the idea of reduction of couplings. Any RGI
relation among couplings (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)

$\mu\,\frac{d\Phi}{d\mu}={\vec{\nabla}}\cdot{\vec{\beta}}~{}=~{}\sum_{a=1}^{A}\,\beta_{a}\,\frac{\partial\Phi}{\partial
g_{a}}~{}=~{}0~{},$ (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) [18, 19, 49],

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

$g_{a}=\sum_{n}\rho_{a}^{(n)}\,g^{2n+1}~{},$ (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 [18, 19, 49]. To
illustrate this, let us assume that the $\beta$-functions have the form

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

$\displaystyle M(r)_{a}^{d}$ $\displaystyle=3\sum_{b,c\neq
g}\,\beta^{(1)\,bcd}_{a}\,\rho_{b}^{(1)}\,\rho_{c}^{(1)}+\beta^{(1)\,d}_{a}-(2r+1)\,\beta^{(1)}_{g}\,\delta_{a}^{d}~{},$
(5) $\displaystyle 0$ $\displaystyle=\sum_{b,c,d\neq
g}\,\beta^{(1)\,bcd}_{a}\,\rho_{b}^{(1)}\,\rho_{c}^{(1)}\,\rho_{d}^{(1)}+\sum_{d\neq
g}\beta^{(1)\,d}_{a}\,\rho_{d}^{(1)}-\beta^{(1)}_{g}\,\rho_{a}^{(1)}~{},$ (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 SUSY theories have the same asymptotic behaviour. Therefore,
searching for a power series solution of the form (3) to the REs (2) is
justified. This is not the case in non-SUSY theories, although the deeper
reason for this fact is not fully understood.

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 [50, 51].

###  2.1 Reduction of dimension one and two parameters

The reduction of couplings was originally formulated for massless theories on
the basis of the Callan-Symanzik equation [18, 19]. 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 [52]. There has been a lot
of progress in this direction starting from ref. [25], 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 [53, 31]
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. [25] and we, too, use 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

$\mathcal{D}\Gamma\left[\Phi^{\prime}s;\hat{g}_{0},\hat{g}_{1},...,\hat{g}_{N};\hat{h}_{1},...,\hat{h}_{L};\hat{m}_{1}^{2},...,\hat{m}_{M}^{2};\mu\right]=0~{},$
(7)

where

$\mathcal{D}=\mu\frac{\partial}{\partial\mu}+\sum_{i=0}^{N}\beta_{i}\frac{\partial}{\partial\hat{g}_{i}}+\sum_{a=1}^{L}\gamma_{a}^{h}\frac{\partial}{\partial\hat{h}_{a}}+\sum_{\alpha=1}^{M}\gamma_{\alpha}^{m^{2}}\frac{\partial}{\partial\hat{m}_{\alpha}^{2}}+\sum_{J}\Phi_{I}\gamma^{\phi
I}_{\,\,\,\,J}\,\frac{\delta}{\delta\Phi_{J}}~{},$ (8)

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

$\begin{split}\gamma^{h}_{a}&=\sum_{b=1}^{L}\gamma_{a}^{h,b}(g_{0},g_{1},...,g_{N})\hat{h}_{b},\\\
\gamma_{\alpha}^{m^{2}}&=\sum_{\beta=1}^{M}\gamma_{\alpha}^{m^{2},\beta}(g_{0},g_{1},...,g_{N})\hat{m}_{\beta}^{2}+\sum_{a,b=1}^{L}\gamma_{\alpha}^{m^{2},ab}(g_{0},g_{1},...,g_{N})\hat{h}_{a}\hat{h}_{b},\end{split}$
(9)

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 parameters left

$\begin{split}\hat{g}_{i}&=\hat{g}_{i}(g),\qquad(i=1,...,N),\\\
\hat{h}_{a}&=\sum_{b=1}^{P}f_{a}^{b}(g)h_{b},\qquad(a=P+1,...,L),\\\
\hat{m}^{2}_{\alpha}&=\sum_{\beta=1}^{Q}e^{\beta}_{\alpha}(g)m^{2}_{\beta}+\sum_{a,b=1}^{P}k^{ab}_{\alpha}(g)h_{a}h_{b},\qquad(\alpha=Q+1,...,M)\end{split}$
(10)

is consistent with the RG equations (7,8). It turns out that the following
relations should be satisfied

$\begin{split}\beta_{g}\,\frac{\partial\hat{g}_{i}}{\partial
g}&=\beta_{i},\qquad(i=1,...,N),\\\
\beta_{g}\,\frac{\partial\hat{h}_{a}}{\partial
g}+\sum_{b=1}^{P}\gamma^{h}_{b}\,\frac{\partial\hat{h}_{a}}{\partial
h_{b}}&=\gamma^{h}_{a},\qquad(a=P+1,...,L),\\\
\beta_{g}\,\frac{\partial\hat{m}^{2}_{\alpha}}{\partial
g}+\sum_{a=1}^{P}\gamma_{a}^{h}\,\frac{\partial\hat{m}^{2}_{\alpha}}{\partial
h_{a}}+\sum_{\beta=1}^{Q}\gamma_{\beta}^{m^{2}}\,\frac{\partial\hat{m}_{\alpha}^{2}}{\partial
m_{\beta}^{2}}&=\gamma_{\alpha}^{m^{2}},\qquad(\alpha=Q+1,...,M).\end{split}$
(11)

Using Eqs. (9) and (10), the above relations reduce to

$\begin{split}&\beta_{g}\,\frac{df^{b}_{a}}{dg}+\sum_{c=1}^{P}f^{c}_{a}\left[\gamma^{h,b}_{c}+\sum_{d=P+1}^{L}\gamma^{h,d}_{c}f^{b}_{d}\right]-\gamma^{h,b}_{a}-\sum_{d=P+1}^{L}\gamma^{h,d}_{a}f^{b}_{d}=0,\\\
&(a=P+1,...,L;\,b=1,...,P),\\\
&\beta_{g}\,\frac{de^{\beta}_{\alpha}}{dg}+\sum_{\gamma=1}^{Q}e^{\gamma}_{\alpha}\left[\gamma_{\gamma}^{m^{2},\beta}+\sum_{\delta=Q+1}^{M}\gamma_{\gamma}^{m^{2},\delta}e^{\beta}_{\delta}\right]-\gamma_{\alpha}^{m^{2},\beta}-\sum_{\delta=Q+1}^{M}\gamma_{\alpha}^{m^{2},d}e^{\beta}_{\delta}=0,\\\
&(\alpha=Q+1,...,Mq\,\beta=1,...,Q),\\\
&\beta_{g}\,\frac{dk_{\alpha}^{ab}}{dg}+2\sum_{c=1}^{P}\left(\gamma_{c}^{h,a}+\sum_{d=P+1}^{L}\gamma_{c}^{h,d}f_{d}^{a}\right)k_{\alpha}^{cb}+\sum_{\beta=1}^{Q}e^{\beta}_{\alpha}\left[\gamma_{\beta}^{m^{2},ab}+\sum_{c,d=P+1}^{L}\gamma_{\beta}^{m^{2},cd}f^{a}_{c}f^{b}_{d}\right.\\\
&\left.+2\sum_{c=P+1}^{L}\gamma_{\beta}^{m^{2},cb}f^{a}_{c}+\sum_{\delta=Q+1}^{M}\gamma_{\beta}^{m^{2},d}k_{\delta}^{ab}\right]-\left[\gamma_{\alpha}^{m^{2},ab}+\sum_{c,d=P+1}^{L}\gamma_{\alpha}^{m^{2},cd}f^{a}_{c}f^{b}_{d}\right.\\\
&\left.+2\sum_{c=P+1}^{L}\gamma_{\alpha}^{m^{2},cb}f^{a}_{c}+\sum_{\delta=Q+1}^{M}\gamma_{\alpha}^{m^{2},\delta}k_{\delta}^{ab}\right]=0,\\\
&(\alpha=Q+1,...,M;\,a,b=1,...,P)~{}.\end{split}$ (12)

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

$\begin{split}\Gamma_{R}&\left[\Phi\textrm{'s};g;h_{1},...,h_{P};m_{1}^{2},...,m_{Q}^{2};\mu\right]\equiv\\\
&\Gamma\left[\Phi\textrm{'s};g,\hat{g}_{1}(g)...,\hat{g}_{N}(g);h_{1},...,h_{P},\hat{h}_{P+1}(g,h),...,\hat{h}_{L}(g,h);\right.\\\
&\left.\qquad\qquad\qquad
m_{1}^{2},...,m^{2}_{Q},\hat{m}^{2}_{Q+1}(g,h,m^{2}),...,\hat{m}^{2}_{M}(g,h,m^{2});\mu\right]\end{split}$
(13)

has the same renormalization group flow as the original one.

The assumptions 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 (10), should be expressed as a power series in
the primary coupling $g$:

$\begin{split}\hat{g}_{i}&=g\sum_{n=0}^{\infty}\rho_{i}^{(n)}g^{n},\qquad
f_{a}^{b}=g\sum_{n=0}^{\infty}\eta_{a}^{b(n)}g^{n}\\\
e^{\beta}_{\alpha}&=\sum_{n=0}^{\infty}\xi^{\beta(n)}_{\alpha}g^{n},\qquad
k_{\alpha}^{ab}=\sum_{n=0}^{\infty}\chi_{\alpha}^{ab(n)}g^{n}.\end{split}$
(14)

The above expansion coefficients can be found by inserting these power series
into Eqs. (11), (12) 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 one 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 one parameter which could be a gaugino mass
which corresponds to the independent (gauge) coupling. In case, on top of
that, there are no independent mass-dimension two parameters ($\hat{m}^{2}$),
the corresponding reduction takes analogous form

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

##  3 Finiteness in N = 1 Supersymmetric Gauge Theories

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

$W=\frac{1}{2}\,m_{ij}\,\phi_{i}\,\phi_{j}+\frac{1}{6}\,C_{ijk}\,\phi_{i}\,\phi_{j}\,\phi_{k}~{},$
(15)

where $m_{ij}$ and $C_{ijk}$ are gauge invariant tensors and the matter field
$\phi_{i}$ transforms according to the irreducible representation $R_{i}$ of
the gauge group $G$. The renormalization constants associated with the
superpotential (15), assuming that SUSY is preserved, are

$\displaystyle\phi_{i}^{0}$
$\displaystyle=(Z^{j}_{i})^{(1/2)}\,\phi_{j}~{},~{}$ (16) $\displaystyle
m_{ij}^{0}$
$\displaystyle=Z^{i^{\prime}j^{\prime}}_{ij}\,m_{i^{\prime}j^{\prime}}~{},~{}$
(17) $\displaystyle C_{ijk}^{0}$
$\displaystyle=Z^{i^{\prime}j^{\prime}k^{\prime}}_{ijk}\,C_{i^{\prime}j^{\prime}k^{\prime}}~{}.$
(18)

The $N=1$ non-renormalization theorem [54, 55, 56] ensures that there are no
mass and cubic-interaction-term infinities and therefore

$\begin{split}Z_{ijk}^{i^{\prime}j^{\prime}k^{\prime}}\,Z^{1/2\,i^{\prime\prime}}_{i^{\prime}}\,Z^{1/2\,j^{\prime\prime}}_{j^{\prime}}\,Z^{1/2\,k^{\prime\prime}}_{k^{\prime}}&=\delta_{(i}^{i^{\prime\prime}}\,\delta_{j}^{j^{\prime\prime}}\delta_{k)}^{k^{\prime\prime}}~{},\\\
Z_{ij}^{i^{\prime}j^{\prime}}\,Z^{1/2\,i^{\prime\prime}}_{i^{\prime}}\,Z^{1/2\,j^{\prime\prime}}_{j^{\prime}}&=\delta_{(i}^{i^{\prime\prime}}\,\delta_{j)}^{j^{\prime\prime}}~{}.\end{split}$
(19)

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 [57]

$\beta^{(1)}_{g}=\frac{dg}{dt}=\frac{g^{3}}{16\pi^{2}}\,[\,\sum_{i}\,l(R_{i})-3\,C_{2}(G)\,]~{},$
(20)

where $l(R_{i})$ is the Dynkin index of $R_{i}$ and $C_{2}(G)$ is the
quadratic Casimir invariant of the adjoint representation of the gauge group
$G$. 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:

$\beta_{ijk}=\frac{dC_{ijk}}{dt}~{}=~{}C_{ijl}\,\gamma^{l}_{k}+C_{ikl}\,\gamma^{l}_{j}+C_{jkl}\,\gamma^{l}_{i}~{}.$
(21)

At one-loop level $\gamma_{ij}$ is [57]

$\gamma^{i(1)}_{j}=\frac{1}{32\pi^{2}}\,[\,C^{ikl}\,C_{jkl}-2\,g^{2}\,C_{2}(R)\delta_{j}^{i}\,],$
(22)

where $C_{2}(R)$ is the quadratic Casimir invariant of the representation
$R_{i}$, and $C^{ijk}=C_{ijk}^{*}$. Since dimensional coupling parameters such
as masses and couplings of cubic scalar field 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 SUSY 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 $C_{ijk}^{2}$ are always
positive numbers.

As one can see from Eqs. (20) and (22), all the one-loop $\beta$-functions of
the theory vanish if $\beta_{g}^{(1)}$ and $\gamma_{ij}^{(1)}$ vanish, i.e.

$\sum_{i}\ell(R_{i})=3C_{2}(G)\,,$ (23)

$C^{ikl}C_{jkl}=2\delta^{i}_{j}g^{2}C_{2}(R_{i})\,,$ (24)

The conditions for finiteness for $N=1$ field theories with $SU(N)$ gauge
symmetry are discussed in [58], and the analysis of the anomaly-free and no-
charge renormalization requirements for these theories can be found in [59]. A
very interesting result is that the conditions (23,24) are necessary and
sufficient for finiteness at the two-loop level [57, 60, 62, 61, 63].

In case SUSY is broken by soft terms, the requirement of finiteness in the
one-loop soft breaking terms imposes further constraints among themselves
[34]. In addition, the same set of conditions that are sufficient for one-loop
finiteness of the soft breaking terms renders the soft sector of the theory
two-loop finite[35].

The one- and two-loop finiteness conditions (23,24) 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 (15).
Note in particular that the finiteness conditions cannot be applied to the
minimal SUSY standard model (MSSM), since the presence of a $U(1)$ gauge group
is incompatible with the condition (23), 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
[64] terms are incompatible with finiteness, as well as D-type [65]
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 [28, 66] 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,
the necessary and also sufficient condition for this to happen is to require
that such relations are solutions to the REs

$\beta_{g}\frac{dC_{ijk}}{dg}=\beta_{ijk}$ (25)

and hold at all orders. Remarkably, the existence of all-order power series
solutions to (25) can be decided at one-loop level, as already mentioned.

Let us now turn to the all-order finiteness theorem [28, 66], which states the
conditions under which an $N=1$ SUSY gauge theory can become finite to all
orders in the sense of vanishing $\beta$-functions, that is of physical scale
invariance. It is based on (a) the structure of the supercurrent in $N=1$ SUSY
gauge theory [67, 68, 69], and on (b) the non-renormalization properties of
$N=1$ chiral anomalies [28, 66, 27, 70, 71]. Details on the proof can be found
in refs. [28, 66] and further discussion in Refs. [27, 70, 71, 29, 72]. Here,
following mostly Ref. [72] we present a comprehensible sketch of the proof.

Consider an $N=1$ SUSY 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 $N=1$ SUSY theory is invariant under a $U(1)$ chiral
transformation $R$ under which the various fields transform as follows

$\begin{split}A^{\prime}_{\mu}&=A_{\mu},~{}~{}\lambda^{\prime}_{\alpha}=\exp({-i\theta})\lambda_{\alpha}\\\
\phi^{\prime}&=\exp({-i\frac{2}{3}\theta})\phi,~{}~{}\psi_{\alpha}^{\prime}=\exp({-i\frac{1}{3}\theta})\psi_{\alpha},~{}\cdots\end{split}$
(26)

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

$J^{\mu}_{R}(x)=\bar{\lambda}\gamma^{\mu}\gamma^{5}\lambda+\cdots$ (27)

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

$\partial_{\mu}J^{\mu}_{R}=r(\epsilon^{\mu\nu\sigma\rho}F_{\mu\nu}F_{\sigma\rho}+\cdots).$
(28)

From its known topological origin in ordinary gauge theories [73, 74, 75], 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 SUSY
theories [27, 70, 71]. Therefore

$r=\hbar\beta_{g}^{(1)}.$ (29)

(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

$T^{\mu}_{\mu}=\beta_{g}F^{\mu\nu}F_{\mu\nu}+\cdots$ (30)

(3) Massless, $N=1$ SUSY gauge theories are classically invariant under the
SUSY 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 $J$,

$J\equiv\\{J^{\prime\mu}_{R},~{}Q^{\mu}_{\alpha},~{}T^{\mu}_{\nu},...\\},$
(31)

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

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

where $T^{\mu}_{\mu}$ is given in Eq.(30) and

$\displaystyle\partial_{\mu}J^{\prime\mu}_{R}$
$\displaystyle~{}=~{}\beta_{g}\epsilon^{\mu\nu\sigma\rho}F_{\mu\nu}F_{\sigma\rho}+\cdots$
(32)
$\displaystyle\sigma^{\mu}_{\alpha\dot{\beta}}\bar{Q}^{\dot{\beta}}_{\mu}$
$\displaystyle~{}=~{}\beta_{g}\lambda^{\beta}\sigma^{\mu\nu}_{\alpha\beta}F_{\mu\nu}+\cdots$
(33)

(4) It is very important to note that the Noether current defined in (27) is
not the same as the current associated to R invariance that appears in the
supercurrent $J$ in (31), 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 (28) and the current
$J^{\prime\mu}_{R}$ belonging to the supercurrent multiplet $J$ are not the
same, there is a relation [28, 66] between quantities associated with them

$r=\beta_{g}(1+x_{g})+\beta_{ijk}x^{ijk}-\gamma_{A}r^{A}$ (34)

where $r$ was given in Eq.(29). 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 equality (34) is independent of the
renormalization scheme.

One-loop finiteness, i.e. vanishing of the $\beta$-functions at one loop,
implies that the Yukawa couplings $C_{ijk}$ 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 (25).

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

Theorem:

Consider an $N=1$ SUSY 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

$\beta^{(1)}_{g}=0=\sum_{i}l(R_{i})-3\,C_{2}(G).$ (35)

  3. 3.

There exist solutions of the form

$C_{ijk}=\rho_{ijk}g,~{}\qquad\rho_{ijk}\in{{\rm I}\kern-5.89996pt{\rm C}}$
(36)

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

$\begin{split}&\gamma^{i~{}(1)}_{j}~{}=~{}0\\\
&=\frac{1}{32\pi^{2}}~{}[~{}C^{ikl}\,C_{jkl}-2~{}g^{2}~{}C_{2}(R)\delta_{j}^{i}].\end{split}$
(37)

  4. 4.

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

$\beta_{ijk}=0.$ (38)

Then, each of the solutions (36) 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 (28). 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 [76].

Proof:

Insert $\beta_{ijk}$ as given by the REs into the relationship (34) between
the axial anomalies coefficients and the $\beta$-functions. Since these chiral
anomalies vanish, we get for $\beta_{g}$ an homogeneous equation of the form

$0=\beta_{g}(1+O(\hbar)).$ (39)

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 (25),
$\beta_{ijk}=0$ too.

Thus we see that finiteness and reduction of couplings are intimately related.
Since an equation like Eq.(34) is lacking in non-SUSY theories, one cannot
extend the validity of a similar theorem in such theories.

##  4 The SSB sector of reduced $N=1$ SUSY and Finite Theories

As we have seen in subsection 2.1, the method of reducing the dimensionless
couplings has been extended[25], to the soft SUSY breaking (SSB) dimensionful
parameters of $N=1$ SUSY theories. In addition it was found [40] that RGI SSB
scalar masses in GYU models satisfy a universal sum rule.

Consider the superpotential given by (15) along with the Lagrangian for SSB
terms

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

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

We assume that the reduction equations admit power series solutions of the
form

$C^{ijk}=g\,\sum_{n}\,\rho^{ijk}_{(n)}g^{2n}~{}.$ (40)

If we knew higher-loop $\beta$-functions explicitly, we could follow the same
procedure and find higher-loop RGI relations among SSB terms. However, the
$\beta$-functions of the soft scalar masses are explicitly known only up to
two loops. In order to obtain higher-loop results some relations among
$\beta$-functions are needed.

In the case of finite theories we assume that the gauge group is a simple
group and the one-loop $\beta$-function of the gauge coupling $g$ vanishes.
According to the finiteness theorem of Refs. [66, 28] and the assumption given
in (40), the theory is then finite to all orders in perturbation theory, if,
among others, the one-loop anomalous dimensions $\gamma_{i}^{j(1)}$ vanish.
The one- and two-loop finiteness for $h^{ijk}$ can be achieved by [35]

$h^{ijk}=-MC^{ijk}+\dots=-M\rho^{ijk}_{(0)}\,g+O(g^{5})~{},$ (41)

where $\dots$ stand for higher order terms.

Now, to obtain the two-loop sum rule for soft scalar masses, we assume that
the lowest order coefficients $\rho^{ijk}_{(0)}$ and also $(m^{2})^{i}_{j}$
satisfy the diagonality relations

$\rho_{ipq(0)}\rho^{jpq}_{(0)}\propto\delta_{i}^{j}~{}\mbox{for
all}~{}p~{}\mbox{and}~{}q~{}~{}\mbox{and}~{}~{}(m^{2})^{i}_{j}=m^{2}_{j}\delta^{i}_{j}~{},$
(42)

respectively. Then we find the following soft scalar-mass sum rule [42, 16,
77]

$(~{}m_{i}^{2}+m_{j}^{2}+m_{k}^{2}~{})/MM^{{\dagger}}=1+\frac{g^{2}}{16\pi^{2}}\,\Delta^{(2)}+O(g^{4})~{}$
(43)

for i, j, k with $\rho^{ijk}_{(0)}\neq 0$, where $\Delta^{(2)}$ is the two-
loop correction

$\Delta^{(2)}=-2\sum_{l}[(m^{2}_{l}/MM^{{\dagger}})-(1/3)]~{}T(R_{l}),$ (44)

which vanishes for the universal choice in accordance with the previous
findings of Ref. [35] (in the above relation $T(R_{l})$ is the Dynkin index of
the $R_{l}$ irrep).

Making use of the spurion technique [78, 79, 56, 80, 81], it is possible to
find the following all-loop relations among SSB $\beta$-functions, [82, 83,
84, 85, 86, 87]

$\displaystyle\beta_{M}$ $\displaystyle=2{\cal
O}\left(\frac{\beta_{g}}{g}\right)~{},$ (45) $\displaystyle\beta_{h}^{ijk}$
$\displaystyle=\gamma^{i}{}_{l}h^{ljk}+\gamma^{j}{}_{l}h^{ilk}+\gamma^{k}{}_{l}h^{ijl}$
$\displaystyle-2\gamma_{1}^{i}{}_{l}C^{ljk}-2\gamma_{1}^{j}{}_{l}C^{ilk}-2\gamma_{1}^{k}{}_{l}C^{ijl}~{},$
(46) $\displaystyle(\beta_{m^{2}})^{i}{}_{j}$
$\displaystyle=\left[\Delta+X\frac{\partial}{\partial
g}\right]\gamma^{i}{}_{j}~{},$ (47) $\displaystyle{\cal O}$
$\displaystyle=\left(Mg^{2}\frac{\partial}{\partial
g^{2}}-h^{lmn}\frac{\partial}{\partial C^{lmn}}\right)~{},$ (48)
$\displaystyle\Delta$ $\displaystyle=2{\cal O}{\cal
O}^{*}+2|M|^{2}g^{2}\frac{\partial}{\partial
g^{2}}+\tilde{C}_{lmn}\frac{\partial}{\partial
C_{lmn}}+\tilde{C}^{lmn}\frac{\partial}{\partial C^{lmn}}~{},$ (49)

where $(\gamma_{1})^{i}{}_{j}={\cal O}\gamma^{i}{}_{j}$,
$C_{lmn}=(C^{lmn})^{*}$, and

$\tilde{C}^{ijk}=(m^{2})^{i}{}_{l}C^{ljk}+(m^{2})^{j}{}_{l}C^{ilk}+(m^{2})^{k}{}_{l}C^{ijl}~{}.$
(50)

Assuming, following [83], that the relation

$h^{ijk}=-M(C^{ijk})^{\prime}\equiv-M\frac{dC^{ijk}(g)}{d\ln g}~{},$ (51)

among couplings is all-loop RGI and using the all-loop gauge $\beta$-function
of Novikov et al. [43, 44, 45] given by

$\beta_{g}^{\rm
NSVZ}=\frac{g^{3}}{16\pi^{2}}\left[\frac{\sum_{l}T(R_{l})(1-\gamma_{l}/2)-3C(G)}{1-g^{2}C(G)/8\pi^{2}}\right]~{},$
(52)

the all-loop RGI sum rule [41] was found:

$\begin{split}m^{2}_{i}+m^{2}_{j}+m^{2}_{k}&=|M|^{2}\\{~{}\frac{1}{1-g^{2}C(G)/(8\pi^{2})}\frac{d\ln
C^{ijk}}{d\ln g}+\frac{1}{2}\frac{d^{2}\ln C^{ijk}}{d(\ln g)^{2}}~{}\\}\\\
&+\sum_{l}\frac{m^{2}_{l}T(R_{l})}{C(G)-8\pi^{2}/g^{2}}\frac{d\ln
C^{ijk}}{d\ln g}~{}.\end{split}$ (53)

In addition the exact-$\beta$-function for $m^{2}$ in the NSVZ scheme has been
obtained [41] for the first time and is given by

$\begin{split}\beta_{m^{2}_{i}}^{\rm
NSVZ}&=\left[~{}|M|^{2}\\{~{}\frac{1}{1-g^{2}C(G)/(8\pi^{2})}\frac{d}{d\ln
g}+\frac{1}{2}\frac{d^{2}}{d(\ln g)^{2}}~{}\\}\right.\\\
&\left.+\sum_{l}\frac{m^{2}_{l}T(R_{l})}{C(G)-8\pi^{2}/g^{2}}\frac{d}{d\ln
g}~{}\right]~{}\gamma_{i}^{\rm NSVZ}~{}.\end{split}$ (54)

Surprisingly enough, the all-loop result (53) coincides with the superstring
result for the finite case in a certain class of orbifold models [42] if $d\ln
C^{ijk}/{d\ln g}=1$.

###  4.1 All-loop RGI Relations in the SSB Sector

Let us now see how the all-loop results on the SSB $\beta$-functions, Eqs.
(45)-(50), lead to all-loop RGI relations. We make two assumptions:
(a) the existence of an RGI surface on which $C=C(g)$, or equivalently that

$\frac{dC^{ijk}}{dg}=\frac{\beta_{C}^{ijk}}{\beta_{g}}$ (55)

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

$h^{ijk}=-M\frac{dC(g)^{ijk}}{d\ln{g}}$ (56)

holds, too, in all orders.

Then one can prove, [88, 89], 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)

$\displaystyle M$ $\displaystyle=M_{0}\frac{\beta_{g}}{g}~{},$ (57)
$\displaystyle h^{ijk}$ $\displaystyle=-M_{0}\beta_{C}^{ijk}~{},$ (58)
$\displaystyle b^{ij}$ $\displaystyle=-M_{0}\beta_{\mu}^{ij}~{},$ (59)
$\displaystyle(m^{2})^{i}_{\phantom{a}j}$
$\displaystyle=\frac{1}{2}|M_{0}|^{2}\mu\frac{d\gamma^{i}_{\phantom{a}j}}{d\mu}$
(60)

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

$C_{\alpha}\frac{\partial}{\partial
C_{\alpha}}=C_{\alpha}^{*}\frac{\partial}{\partial C_{\alpha}}$ (61)

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

$\frac{d}{dg}=\Big{(}\frac{\partial}{\partial g}+2\frac{\partial}{\partial
C}\frac{dC}{dg}\Big{)}=\Big{(}\frac{\partial}{\partial
g}+2\frac{\beta_{C}}{\beta_{g}}\frac{\partial}{\partial C}\Big{)}~{},$ (62)

where Eq.(55) has been used. Now let us consider the partial differential
operator $\cal O$ in Eq.(48) which, assuming Eq.(51) becomes

${\cal O}=\frac{1}{2}M\frac{d}{d\ln{g}}~{}.$ (63)

In turn, $\beta_{M}$ given in Eq.(45) becomes

$\beta_{M}=M\frac{d}{d\ln{g}}(\frac{\beta_{g}}{g})~{},$ (64)

which by integration provides us [88, 90] with the generalized, i.e. including
Yukawa couplings, all-loop RGI Hisano - Shifman relation [82]

$M=\frac{\beta_{g}}{g}M_{0}~{},$ (65)

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.(65) becomes the all-loop RGE Eq.(57).
Note that $\beta_{M}$ using Eqs. (64) and (65) can be written as

$\beta_{M}=M_{0}\frac{d}{dt}(\beta_{g}/g)~{}.$ (66)

Similarly

$(\gamma_{1})^{i}_{\phantom{a}j}={\cal
O}\gamma^{i}_{\phantom{a}j}=\frac{1}{2}M_{0}\frac{d\gamma^{i}_{\phantom{a}j}}{dt}~{}.$
(67)

Next, form Eq.(51) and (65) we obtain

$h^{ijk}=-M_{0}\beta_{C}^{ijk}~{},$ (68)

while $\beta_{h}^{ijk}$, given in Eq.(46) and using Eq.(67), becomes [88]

$\beta_{h}^{ijk}=M_{0}\frac{d}{dt}\beta_{C}^{ijk}$ (69)

which shows that Eq.(68) is all-loop RGI. In a similar way Eq.(59) 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.(53) has been proven [41] to be all-loop RGI,
which (using Eq.(65)) gives us a generalization of Eq.(60) to be applied in
considerations of non-universal soft scalar masses, which are necessary in
many cases. Moreover, the sum rule holds also in the more general cases,
discussed in subsection 2.1, according to which exact relations among the
squared scalar and gaugino masses can be found.

##  5 A successful Finite Unified Theory

We review an all-loop FUT with $SU(5)$ as gauge group, where the reduction of
couplings has been applied to the third generation of quarks and leptons. This
FUT was selected previously on the basis of agreement with the known
experimental data at the time [33] and was predicting the Higgs mass to be in
the range $121-126$ GeV almost five years before the discovery. The particle
content of the model we will study, which we denote $SU(5)$-FUT 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 [20, 21, 22, 23, 15, 24].

A predictive GYU $SU(5)$ model which is finite to all orders, in addition to
the requirements mentioned already, should also have the following properties:

  1. 1.

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

  2. 2.

Three fermion generations in the irreducible representations $\overline{\bf
5}_{i},{\bf 10}_{i}~{}(i=1,2,3)$ which obviously 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.

After the reduction of couplings the symmetry is enhanced, leading to the
following superpotential [42, 91]

$\displaystyle W$
$\displaystyle=\sum_{i=1}^{3}\,[~{}\frac{1}{2}g_{i}^{u}\,{\bf 10}_{i}{\bf
10}_{i}H_{i}+g_{i}^{d}\,{\bf 10}_{i}\overline{\bf
5}_{i}\,\overline{H}_{i}~{}]+g_{23}^{u}\,{\bf 10}_{2}{\bf 10}_{3}H_{4}$ (70)
$\displaystyle+g_{23}^{d}\,{\bf 10}_{2}\overline{\bf
5}_{3}\,\overline{H}_{4}+g_{32}^{d}\,{\bf 10}_{3}\overline{\bf
5}_{2}\,\overline{H}_{4}+g_{2}^{f}\,H_{2}\,{\bf
24}\,\overline{H}_{2}+g_{3}^{f}\,H_{3}\,{\bf
24}\,\overline{H}_{3}+\frac{g^{\lambda}}{3}\,({\bf 24})^{3}~{}.$

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

$\displaystyle(g_{1}^{u})^{2}=\frac{8}{5}~{}g^{2}~{},~{}(g_{1}^{d})^{2}=\frac{6}{5}~{}g^{2}~{},~{}(g_{2}^{u})^{2}=(g_{3}^{u})^{2}=\frac{4}{5}~{}g^{2}~{},$
(71)
$\displaystyle(g_{2}^{d})^{2}=(g_{3}^{d})^{2}=\frac{3}{5}~{}g^{2}~{},~{}(g_{23}^{u})^{2}=\frac{4}{5}~{}g^{2}~{},~{}(g_{23}^{d})^{2}=(g_{32}^{d})^{2}=\frac{3}{5}~{}g^{2}~{},$
$\displaystyle(g^{\lambda})^{2}=\frac{15}{7}g^{2}~{},~{}(g_{2}^{f})^{2}=(g_{3}^{f})^{2}=\frac{1}{2}~{}g^{2}~{},~{}(g_{1}^{f})^{2}=0~{},~{}(g_{4}^{f})^{2}=0~{},$

and from the sum rule we obtain:

$m^{2}_{H_{u}}+2m^{2}_{{\bf 10}}=M^{2}~{},~{}m^{2}_{H_{d}}-2m^{2}_{{\bf
10}}=-\frac{M^{2}}{3}~{},~{}m^{2}_{\overline{{\bf 5}}}+3m^{2}_{{\bf
10}}=\frac{4M^{2}}{3}~{},$ (72)

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

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 [94, 20, 24, 92, 93], in such a way that only one pair of Higgs
doublets, coupled mostly to the third family, remains light and acquires
vacuum expectation values. To avoid fast proton decay the usual fine tuning to
achieve doublet-triplet splitting is performed, although 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 not restricted.

##  6 Phenomenological constraints

In this section we briefly review the relevant experimental constraints that
we apply in our phenomenological analysis.

###  6.1 Flavour Constraints

We consider four types of flavour constraints to apply to the SU(5)-FUT, where
SUSY is known to have significant impact. More specifically, we consider the
flavour observables ${\rm BR}(b\to s\gamma)$, ${\rm
BR}(B_{s}\to\mu^{+}\mu^{-})$, ${\rm BR}(B_{u}\to\tau\nu)$ and $\Delta
M_{B_{s}}$.222We do not use the latest experimental and theoretical values
here. However, this has a minor impact on the general form of our results. The
uncertainties are the linear combination of the experimental error and twice
the theoretical uncertainty in the MSSM.333We include the MSSM uncertainty
also into the ratios of exp. data and SM prediction, to apply it readily to
our prediction of the ratio of our MSSM and SM calculation. In the case that
no specific estimate is available, we use the SM uncertainty.

For the branching ratio ${\rm BR}(b\to s\gamma)$, we take a value from the
Heavy Flavour Averaging Group (HFAG) [95, 96]

$\frac{{\rm BR}(b\to s\gamma)^{\rm exp}}{{\rm BR}(b\to s\gamma)^{\rm
SM}}=1.089\pm 0.27~{}.$ (73)

For the branching ratio ${\rm BR}(B_{s}\to\mu^{+}\mu^{-})$, a combination of
CMS and LHCb data [98, 97, 99, 100, 101] is used

${\rm BR}(B_{s}\to\mu^{+}\mu^{-})=(2.9\pm 1.4)\times 10^{-9}~{}.$ (74)

For the $B_{u}$ decay to $\tau\nu$ we use the limit [102, 96, 103]

$\frac{{\rm BR}(B_{u}\to\tau\nu)^{\rm exp}}{{\rm BR}(B_{u}\to\tau\nu)^{\rm
SM}}=1.39\pm 0.69~{},$ (75)

while for $\Delta M_{B_{s}}$ [104, 105]

$\frac{\Delta M_{B_{s}}^{\rm exp}}{\Delta M_{B_{s}}^{\rm SM}}=0.97\pm 0.2$
(76)

At the end of the phenomenological discussion we also comment on the cold dark
matter (CDM) density. It is well known that the lightest neutralino, being the
lightest SUSY particle (LSP), is an excellent candidate for CDM [106].
Consequently one can in principle demand that the lightest neutralino is
indeed the LSP and parameters leading to a different LSP could be discarded.

The current bound, favoured by a joint analysis of WMAP/Planck and other
astrophysical and cosmological data, is at the $2\,\sigma$ level given by the
range [107, 108],

$\Omega_{\rm CDM}h^{2}=0.1120\pm 0.0112~{}.$ (77)

###  6.2 The light Higgs boson mass

The quartic couplings in the Higgs potential are given by the SM gauge
couplings. As a consequence, the lightest Higgs mass is not a free parameter,
but rather predicted in terms of the other parameters of the model. Higher
order corrections are crucial for a precise predictions of $M_{h}$, see Refs.
[109, 111, 110] for reviews.

The discovery of a Higgs boson at ATLAS and CMS in July 2012 [112, 113] can be
interpreted as the discovery of the light $\cal CP$-even Higgs boson of the
MSSM Higgs spectrum [114, 115, 116]. The experimental average for the (SM)
Higgs boson mass is taken to be [117]

$M_{H}^{\rm exp}=125.1\pm 0.3~{}{\rm GeV}~{},$ (78)

and adding in quadrature a 3 (2) GeV theory uncertainty [118, 119, 120] for
the Higgs mass caclulation in the MSSM we arrive at an allowing range of

$M_{h}=125.1\pm 3.1~{}(2.1)~{}{\rm GeV}~{}.$ (79)

We used the code FeynHiggs [118, 120, 121] (version 2.14.0 beta) to predict
the lightest Higgs boson mass. The evaluation of the Higgs masses with
FeynHiggs is based on the combination of a Feynman-diagrammatic calculation
and a resummation of the (sub)leading and logarithms contributions of the
(general) type of $\log{(m_{\tilde{t}}/m_{t})}$ in all orders of perturbation
theory. This combination ensures a reliable evaluation of $M_{h}$ also for
large SUSY scales. Several refinements in the combination of the fixed order
log resummed calculation have been included w.r.t. previous versions, see Ref.
[120]. They resulted not only in a more precise $M_{h}$ evaluation for high
SUSY mass scales, but in particular in a downward shift of $M_{h}$ at the
level of ${\cal O}(2~{}{\rm GeV})$ for large SUSY masses.

##  7 Numerical analysis

In this section we will analyse the particle spectrum predicted in the
$SU(5)$-FUT . 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 (71), the $h=-MC$ (41) relation, and the soft
scalar-mass sum rule at $M_{\rm GUT}$.

In Fig.1 we show the $SU(5)$-FUT predictions for $m_{t}$ and $m_{b}(M_{Z})$ as
a function of the unified gaugino mass $M$, for the two cases $\mu<0$ and
$\mu>0$. We use the experimental value of the top quark pole mass as [103]444
We did not include the latest LHC/Tevatron combination, leading to $m_{t}^{\rm
exp}=(173.34\pm 0.76)\,\,\mathrm{GeV}$ [122], which would have a negligible
impact on our analysis.

$m_{t}^{\rm exp}=(173.2\pm 0.9)\,\,\mathrm{GeV}~{}.$ (80)

The bottom mass is calculated at $M_{Z}$ to avoid uncertainties that come from
running down to the pole mass; the leading SUSY radiative corrections to the
bottom and tau masses have been taken into account [123]. We use the following
value for the bottom mass at $M_{Z}$ [103],

$m_{b}(M_{Z})=(2.83\pm 0.10)\,\,\mathrm{GeV}.$ (81)

The bounds on the $m_{b}(M_{Z})$ and the $m_{t}$ mass clearly single out
$\mu<0$, as the solution most compatible with these experimental constraints.

Figure 1: The bottom quark mass at the $Z$ boson scale (left) and top quark
pole mass (right) are shown as function of $M$ for both signs of $\mu$.

As was already mentioned, for the lightest Higgs boson mass we used the code
FeynHiggs (2.14.0 beta). The prediction for $M_{h}$ of $SU(5)$-FUT with
$\mu<0$ is shown in Fig.2, in a range where the unified gaugino mass varies
from $0.5~{}{\rm TeV}\lesssim M\lesssim 9~{}{\rm TeV}$. The green points
include the $B$-physics constraints. One should keep in mind that these
predictions are subject to a theory uncertainty of 3(2) GeV [118]. Older
analysis, including in particular less refined evaluations of the light Higgs
boson mass, are given in Refs. [46, 124, 125].

Figure 2: The lightest Higgs mass, $M_{h}$, as a function of $M$ for the FUT
model with $\mu<0$. The green points are the ones that satisfy the $B$-physics
constraints.

The allowed values of the Higgs mas put a limit on the allowed values of the
SUSY masses, as can be seen in Fig.3. In the left (right) plot we impose
$M_{h}=125.1\pm 3.1(2.1)~{}{\rm GeV}$ as discussed above. In particular very
heavy coloured SUSY particles are favoured (nearly independent of the $M_{h}$
uncertainty), in agreement with the non-observation of those particles at the
LHC [126]. Overall, the allowed coloured SUSY masses would remain unobservable
at the (HL-)LHC, the ILC or CLIC. However, the coloured spectrum would be
accessible at the FCC-hh [127], as could the full heavy Higgs boson spectrum.
On the other hand, the lightest observable SUSY particle (LSOP) is the scalar
tau. Some parts of the allowed spectrum of the lighter scalar tau or the
lighter charginos/neutralinos might be accessible at CLIC with
$\sqrt{s}=3~{}{\rm TeV}$.

Figure 3: The left (right) plot shows the spectrum of the $SU(5)$-FUT (with
$\mu<0$) model after imposing the constraint $M_{h}=125.1\pm 3.1(2.1)~{}{\rm
GeV}$. The light (green) points are the various Higgs boson masses, the dark
(blue) points following are the two scalar top and bottom masses, the gray
ones are the gluino masses, then come the scalar tau masses in orange (light
gray), the darker (red) points to the right are the two chargino masses
followed by the lighter shaded (pink) points indicating the neutralino masses.

$\delta M_{h}=2.1$ | $M_{h}$ | $M_{H}$ | $M_{A}$ | $M_{H^{\pm}}$ | $m_{\tilde{t}_{1}}$ | $m_{\tilde{t}_{2}}$ | $m_{\tilde{b}_{1}}$ | $m_{\tilde{b}_{2}}$ | $m_{\tilde{g}}$
---|---|---|---|---|---|---|---|---|---
lightest | 123.1 | 1533 | 1528 | 1527 | 2800 | 3161 | 2745 | 3219 | 4077
heaviest | 127.2 | 4765 | 4737 | 4726 | 10328 | 11569 | 10243 | 11808 | 15268
| $m_{\tilde{\tau}_{1}}$ | $m_{\tilde{\tau}_{2}}$ | $m_{\tilde{\chi}^{\pm}_{1}}$ | $m_{\tilde{\chi}^{\pm}_{2}}$ | $m_{\tilde{\chi}^{0}_{1}}$ | $m_{\tilde{\chi}^{0}_{2}}$ | $m_{\tilde{\chi}^{0}_{3}}$ | $m_{\tilde{\chi}^{0}_{4}}$ | $\tan\beta$
lightest | 983 | 1163 | 1650 | 2414 | 900 | 1650 | 2410 | 2414 | 45
heaviest | 4070 | 5141 | 6927 | 8237 | 3920 | 6927 | 8235 | 8237 | 46

$\delta M_{h}=3.1$ | $M_{h}$ | $M_{H}$ | $M_{A}$ | $M_{H^{\pm}}$ | $m_{\tilde{t}_{1}}$ | $m_{\tilde{t}_{2}}$ | $m_{\tilde{b}_{1}}$ | $m_{\tilde{b}_{2}}$ | $m_{\tilde{g}}$
---|---|---|---|---|---|---|---|---|---
lightest | 122.8 | 1497 | 1491 | 1490 | 2795 | 3153 | 2747 | 3211 | 4070
heaviest | 127.9 | 4147 | 4113 | 4103 | 10734 | 12049 | 11077 | 12296 | 16046
| $m_{\tilde{\tau}_{1}}$ | $m_{\tilde{\tau}_{2}}$ | $m_{\tilde{\chi}^{\pm}_{1}}$ | $m_{\tilde{\chi}^{\pm}_{2}}$ | $m_{\tilde{\chi}^{0}_{1}}$ | $m_{\tilde{\chi}^{0}_{2}}$ | $m_{\tilde{\chi}^{0}_{3}}$ | $m_{\tilde{\chi}^{0}_{4}}$ | $\tan\beta$
lightest | 1001 | 1172 | 1647 | 2399 | 899 | 647 | 2395 | 2399 | 44
heaviest | 4039 | 6085 | 7300 | 8409 | 4136 | 7300 | 8406 | 8409 | 45

Table 1:  Two example spectra of the $SU(5)$-FUT (with $\mu<0$) . All masses
are in GeV and rounded to 1 (0.1) GeV (for the light Higgs mass).

In Table 1 we show two example spectra of the $SU(5)$-FUT (with $\mu<0$) which
span the mass range of the parameter space that is in agreement with the
$B$-physics observables and the Higgs-boson mass measurement. We give the
lightest and the heaviest spectrum for $\delta M_{h}=2.1$ and $\delta
M_{h}=3.1$, respectively. The four Higgs boson masses are denoted as $M_{h}$,
$M_{H}$, $M_{A}$ and $M_{H^{\pm}}$. $m_{\tilde{t}_{1,2}}$,
$m_{\tilde{t}_{1,2}}$, $m_{\tilde{g}}$, $m_{\tilde{\tau}_{1,2}}$, are the
scalar top, scalar bottom, gluino and scalar tau masses, respectively.
$m_{\tilde{\chi}^{\pm}_{1,2}}$ and $m_{\tilde{\chi}^{0}_{1,2,3,4}}$ denote the
chargino and neutralino masses.

We find that no point of $SU(5)$-FUT (with $\mu<0$) fulfills the strict bound
of Eq. (77). (For our evaluation we have used the code MicroMegas [128, 129,
130].) Consequently, on a more general basis a mechanism is needed in our
model to reduce the CDM abundance in the early universe. This issue could, for
instance, be related to another problem, that of neutrino masses. This type of
masses cannot be generated naturally within the class of finite unified
theories that we are considering in this paper, although a non-zero value for
neutrino masses has clearly been established [103]. However, the class of FUTs
discussed here can, in principle, be easily extended by introducing bilinear
R-parity violating terms that preserve finiteness and introduce the desired
neutrino masses [131, 132, 133]. R-parity violation [134] would have a small
impact on the collider phenomenology presented here (apart from the fact that
SUSY search strategies could not rely on a ‘missing energy’ signature), but
remove the CDM bound of Eq. (77) completely. The details of such a possibility
in the present framework attempting to provide the models with realistic
neutrino masses will be discussed elsewhere. Other mechanisms, not involving
R-parity violation (and keeping the ‘missing energy’ signature), that could be
invoked if the amount of CDM appears to be too large, concern the cosmology of
the early universe. For instance, “thermal inflation” [135] or “late time
entropy injection” [136] could bring the CDM density into agreement with the
WMAP measurements. This kind of modifications of the physics scenario neither
concerns the theory basis nor the collider phenomenology, but could have a
strong impact on the CDM bounds. (Lower values than those permitted by Eq.
(77) are naturally allowed if another particle than the lightest neutralino
constitutes CDM.)

##  8 Conclusions

The MSSM is considered a very attractive candidate for describing physics
beyond the SM. However, the serious problem of the SM having too many free
parameters is further proliferated in the MSSM. Assuming a GUT beyond the
scale of gauge coupling unification, based on the idea that a Particle Physics
Theory should be more symmetric at higher scales, seems to fit to the MSSM. On
the other hand, the unification scenario seems to be unable to further reduce
the number of free parameters.

Attempting to reduce the free parameters of a theory, a new approach was
proposed in Refs. [18, 19] based on the possible existence of RGI relations
among couplings. Although this approach could uncover further symmetries, its
application opens new horizons too. At least the Finite Unified Theories seem
to be a very promising field for applying the reduction approach. In the FUT
case, the discovery of RGI relations among couplings above the unification
scale ensures at the same time finiteness to all orders.

The discussion in the previous sections of this paper shows that the
predictions of the particular FUT discussed here are impressive. In addition,
one could add some comments on a successful FUT from the theoretical side,
too. The developments on treating the problem of divergencies include string
and non-commutative theories, as well as $N=4$ SUSY theories [137, 138], $N=8$
supergravity [139, 140, 141, 142, 143] and the AdS/CFT correspondence [144].
It is very interesting that the $N=1$ FUT discussed here includes many ideas
which survived phenomenological and theoretical tests as well as the
ultraviolet divergence problem. It is actually solving that problem in a
minimal way.

We concentrated our examination on the predictions of one particular $SU(5)$
Finite Unified Theory, including the restrictions of third generation quark
masses and $B$-physics observables. The model, $SU(5)$-FUT (with $\mu<0$), is
consistent with all the phenomenological constraints. Compared to our previous
analyses [46, 47, 124, 125], the improved evaluation of $M_{h}$ prefers a
heavier (Higgs) spectrum and thus in general allows only a very heavy SUSY
spectrum. The coloured spectrum could easily escape the (HL-)LHC searches, but
can likely be tested at the FCC-hh. The lower part of the electroweak spectrum
could be accessible at CLIC.

### Acknowledgements

We thank H. Bahl, T. Hahn, W. Hollik, D. Lüst and E. Seiler for helpful
discussions. The work of S.H. is supported in part by the MEINCOP Spain under
contract FPA2016-78022-P, in part by the Spanish Agencia Estatal de
Investigación (AEI) and the EU Fondo Europeo de Desarrollo Regional (FEDER)
through the project FPA2016-78645-P, and in part by the AEI through the grant
IFT Centro de Excelencia Severo Ochoa SEV-2016-0597. The work of M.M. is
partly supported by UNAM PAPIIT through grant IN111518. The work of N.T. and
G.Z. are supported by the COST actions CA15108 and CA16201. G.Z. thanks the
MPI Munich for hospitality and the A.v.Humboldt Foundation for support.

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