Renormalization Group and Grand Unification with 331 Models
ABSTRACT By making a renormalization group analysis we explore the possibility of having a 331 model as the only intermediate gauge group between the standard model and the scale of unification of the three coupling constants. We shall assume that there is no necessarily a group of grand unification at the scale of convergence of the couplings. With this scenario, different 331 models and their corresponding supersymmetric versions are considered, and we find the versions that allow the symmetry breaking described above. Besides, the allowed interval for the 331 symmetry breaking scale, and the behavior of the running coupling constants are obtained. It worths saying that some of the supersymmetric scenarios could be natural frameworks for split supersymmetry. Finally, we look for possible 331 models with a simple group at the grand unification scale, that could fit the symmetry breaking scheme described above.
arXiv:hep-ph/0505096v3 15 Aug 2007
Renormalization Group and Grand Unification with 331 Models
Rodolfo A. Diaz(1)∗, D. Gallego(1,2), R. Martinez(1)†
(1)Universidad Nacional de Colombia, Departamento de F´ ısica, Bogot´ a, Colombia.
(2)The Abdus Salam International Center for Theoretical Physics, Trieste, Italy.
By making a renormalization group analysis we explore the possibility of having a 331 model as the only
intermediate gauge group between the standard model and the scale of unification of the three coupling con-
stants. We shall assume that there is no necessarily a group of grand unification at the scale of convergence
of the couplings. With this scenario, different 331 models and their corresponding supersymmetric versions are
considered, and we find the versions that allow the symmetry breaking described above. Besides, the allowed
interval for the 331 symmetry breaking scale, and the behavior of the running coupling constants are obtained.
It worths saying that some of the supersymmetric scenarios could be natural frameworks for split supersymme-
try. Finally, we look for possible 331 models with a simple group at the grand unification scale, that could fit
the symmetry breaking scheme described above.
PACS: 11.10Hi, 12.10.-g, 12.60.Jv, 11.15.Ex, 11.30.Ly
Keywords: renormalization group equations, grand unification, 331 models, supersymmetric unification.
Since the birth of the Standard Model (SM) many attempts have been done to go beyond it, and solve some of
the problems of the model such as the charge quantization and the unification of the gauge couplings. In some
cases the unification is done by taking a simple group of grand unification, arising the so called Grand Unification
Theories (GUT), where the three interactions described by SM are treated as only one -, the most common
GUT’s are SO(10) and E6. The first condition for these kind of theories is an equal value for the three couplings at
certain scale of energy, MU. This condition cannot be fulfilled by the simplest grand unification schemes with the
minimal SM particle content and taking the precision low-energy data. However, the minimal supersymmetric SM
can achieve this scenario for the coupling constants -. Other possibilities for unification are the introduction
of more degrees of freedom like fermions and scalar fields that lead the three couplings to converge at a high energy
scale . Polychromatic extensions of the SM i.e. SU(N)C⊗ SU(2)L⊗ U(1)1/N have been considered where
the unification of gauge couplings is achieved with N = 7, 5 and with two, three Higgs doublets, respectively
. Alternative proposals of unification of quarks and leptons at TeV scale were considered too . Finally,
another interesting alternative consists of enlarging the electroweak sector of the SM gauge group such that the
renormalization group equations (RGE) could lead to unification of the three gauge couplings at certain scale
MU, in which there is no necessarily a group of grand unification at the scale of convergence of the couplings. In
particular, the model based on the SU(3)C⊗SU(3)L⊗U(1)Xgauge group (hereafter 331 models) is an interesting
choice that could address problems like the charge quantization - and the existence of three families based
on cancellation of anomalies -.
One way to look for new Physics in 331 models is to check for tT production with t denoting the ordinary
top quark and T being an exotic quark with charge 2/3 or 4/3 according to the model considered. In LHC the
production channels would be pp → X0→ tT, when the charge of T is 2/3, and pp → X++→ tT, in the case
in which T has an exotic charge 4/3. In the first case, in order to identify the signals, the decay of t is known
and identified by the energy spectrum and angular distribution of the final fermions -, then this signal is
correlated with the one of an exotic particle with the same charge of the top, but with a totally different decay in
the final state. The decays of T would be of the form T → X0t → νEt, T → K+b → νEb, which could de easily
identifyable if exotic leptons have already been produced at LHC. In the second case in which T has an exotic
Rodolfo A. Diaz, D. Gallego, R. Martinez
charge 4/3, it should be looked for decays of the type T → X++t where X++is a doubly charged gauge field that
should be easily identified if this model is correct. The other possible channel is T → K+b → νEb.
In the scenario of SUSY 331 models, different channels can be searched. A good perspective is pp → gg → g →
? g? g where ? g →?tT,t?T,T?T in the case in which T posseses an ordinary charge 2/3. Another production mechanism
In looking for unification of the coupling constant by passing through a 331 model, we shall assume that 1) The
331 gauge group is the only extension of the SM before the unification of the running coupling constants. 2) The
hypercharge associated with the 331 gauge group is adequately normalized such that the three gauge couplings
unify at certain scale MU. and 3) There is no necessarily a unified gauge group at the scale of convergence of the
couplings MU. In the absence of a grand unified group, there are no restriction on MUcoming from proton decay1.
Under our scheme, we have three characteristic energy scales: MU where the three gauge couplings converge,
MXwhere the 331 symmetry is broken, and MZwhere the SM breaking occurs. We are going to consider different
scenarios for 331 models with one and three families2, i. e., one family models. We also introduce Supersymmetric
versions of the 331 models with different scalar Higgs multiplets. As for the SUSY breaking, we shall consider two
different scenarios: when SUSY is broken at the electroweak scale, or when SUSY breaks at the scale of MX.
In our scheme we have four parameters to take into account, and to look for a possible unification of the coupling
constants (UCC). They are the scales MX, MU, the value of the coupling contants at the unification convergence
point, αU, and the parameter associated with the normalization of the hypercharge (denoted by a). Since we
are interested mostly in possible phenomenological scenarios, the relevant parameter will be the gauge symmetry
breaking scale MX; and the parameters MU, a can be viewed as functions of this one.
If the unification came from a grand unified symmetry group G, the normalization of the hypercharge Y would
be determined by the group structure. However, under our assumptions, this normalization factor is free and the
problem could be addressed the opposite way, since the values obtained for a could in turn suggest possible groups
of grand unification in which the 331 group is embedded, we shall explore this possibility as well.
In the present work, we study six different versions of 331 models with non-SUSY and SUSY particle content,
and find which models could lead to a unification at certain scale MUwith only one symmetry breaking between MZ
and MUscales. Some of the SUSY versions studied, could provide a quite natural scenario for split supersymmetry.
Finally, we also consider the possibility of embedding those 331 versions into a grand unified theory (GUT) in which
a gauge group at the unification scale appears.
in the gaugino sector is could be pp → X++→ χ++χ0, pp → χ++→ X++χ0, pp → Z,Z′++χ−−-
2Running Coupling Constants
The evolution for the Running Coupling Constants (RCC) at one-loop order is ruled by the solution of the Renor-
malization Group Equations (RGE), which can be written in the form :
where αi= g2
i/4π, and the coefficients biare given by :
The summations run over Weyl fermions and scalars, respectively. The coefficient TRis the Dynkin index
with the generators in the representation R. The last term is the quadratic Casimir for the adjoint representation
with C2(G) = N for SU(N). On the other hand, the respective supersymmetric versions are
where the usual non-supersymmetric degrees of freedom are counted.
TRi(s) − 3C2i(G).(2.5)
1Notwithstanding, proton decay could be induced even in the absence of a group of grand unification.
dimensional 331 invariant effective operators, that violates barionic and leptonic numbers .
2331 models with three identical fermion multiplets are usually call one family models.
It may occurs via six
Renormalization group in 331 models
The general expression for the electromagnetic charge operator will be a linear combination of diagonal generators
for the gauge group 331:
Q = T3+ Y = T3+
√3bT8+ X (2.6)
with Tithe Gell-Mann matrices normalized as Tr(TiTj) =1
proportional to the identity matrix 3 × 3. The hypercharge will be given by
2δij. The X operator for the abelian group U(1)X is
√3bT8+ X. (2.7)
b is a known parameter that determines the class of 331 models to be considered .
The renormalization group analysis compares the couplings for different gauge groups at given energy scales,
and models with symmetry breakings need relations for couplings at different energy regions which are called the
matching conditions; they are extracted from the way in which the unbroken group is embedded into a bigger
broken group. Also, in order to have all couplings in the same ground, all the generators should be normalized in
the same way, and well normalized couplings are those which will converge in an unification point.
Calling˜Y the well-normalized hypercharge operator, it will be proportional to the original one,
Y = a˜Y ,(2.8)
where a is the normalizing parameter so that the convergence of the running coupling constants at certain scale
MU is guaranteed. In the same way the operator X has a well-normalized˜ X, i.e. X = c˜ X, and the normalizing
parameter for it, is given by Eq. (2.7) requiring the same normalization for˜Y , T8 which satisfies the following
3b2+ c2. (2.9)
Therefore, the parameter a should be such that
Then, the well-normalized hypercharge operator can be written as a function of the unknown parameter a, as
And from this equation, we obtain the following matching condition for the corresponding couplings :
where ˜ αY, ˜ αX, α3Lare related with U(1)˜Y, U(1)˜
The following relations must also be satisfied
Xand SU(3)L, respectively.
=a2αY,˜ αX= (a2−4b2
where αX, αY and α2Lare related with U (1)X, U(1)Y and SU(2)L, respectively. The third relation corresponds
to the strong interaction where αsis associated with the Standard Model and α3Cis related with the color part in
the 331 model. Finally, the last relation corresponds to the embedding of SU(2)Linto SU(3)L.
Rodolfo A. Diaz, D. Gallego, R. Martinez
By replacing the relations described by Eqs. (2.12, 2.13) into Eq. (2.1), we can write the evolution for the RCC
from the Z boson-pole MZpassing through a 331 symmetry breaking scale MX, up to a certain Scale of unification
where the coefficients bs, b2Land bY are related with SU(3)C,SU(2)L,U(1)Y, respectively and they are calculated
at energies in the range MZ ≤ µ ≤ MX. The coefficients b3C, b3L, and bX, are related with SU(3)C,SU(3)L,
and U(1)X respectively; they are calculated for energies in the range MX ≤ µ ≤ MU. For our study, we need bi
coefficients for energy scales below (and above) the symmetry breaking scale MX, which will be given by SM (and
331) degrees of freedom. Then for different models, we have different bi’s for the intervals of energy scales which
can change the running of the coupling constants. The input parameters from precision measurements are 
The MU scale, where all the well-normalized couplings have the same value, can be calculated from (2.15) and
(2.16) as a function of the symmetry breaking scale MX
The hierarchy condition MX≤ MU ≤ MPlanck, must be satisfied. We shall however impose a stronger condition
of MU? 1017GeV, in order to avoid gravitational effects. Hence, the hierarchy condition becomes
MX≤ MU≤ 1017GeV(2.19)
Such condition can establish an allowed range for the symmetry breaking scale MX in order to obtain grand
unification for a given normalizing parameter a.
With a similar procedure, the expression for a2is found, and is given by
In order to analyze the possibility of having unification at certain scale MU with 331 as the only intermediate
gauge group between the SM and MU scales, we could distinguish three scenarios of unification pattern
Renormalization group in 331 models
The scenario with b3C ?= b3L and (b3C− b3L) ?= (bs− b2L). We shall call it the first unification pattern
(1UP). In that case, we obtain an allowed region for the scale MX. It is carried out by combining the results
of Section 2.2. The procedure to get the allowed interval for MX is described in detail in Section 3.1 for the
so called model A.
The scenario with b3C = b3L. We call it (2UP). In such a case Eq. (2.18) is not valid anymore, and we
should go back to Eqs. (2.15, 2.16). For an arbitrary value of MXthe couplings α3Cand α3Lgo parallel each
other for energies larger than MX. Therefore, the only possible way to still obtain unification is by setting
α3C= α3Lat the scale MX such that both couplings go together for scales above MX. Unification with the
third coupling could occur at any scale bigger than MX. By equating Eqs (2.15, 2.16), we find a single value
of MXthat makes the couplings α3Cand α3Lto converge. This convergence occurs at the scale
It worths emphasizing that this scenario leads to a unique value of MXand not to an allowed range. Finally,
Eq. (2.20) for a2must also be recalculated to find
The 3UP with (b3C− b3L) = (bs− b2L) ?= 0; according to Eq. (2.18), the unification scale MU becomes
independent of MX.
The case (b3C− b3L) = (bs− b2L) = 0, does not lead to unification as can be seen by trying to equate Eqs.
(2.15) and (2.16). Since the first scenario is the most commom one, we shall only indicate when the other two
scenarios appear. We will study non-SUSY and SUSY versions of the 331 models. In the case of SUSY models we
shall consider two scenarios for the SUSY breaking pattern
1. The SUSY Breaking Scenario at the Z−pole (ZSBS), in which the SUSY breaking scale is taken as ΛSUSY ∼
MZ. Although this is not a very realistic scenario, numerical results do not change significantly with respect
to the more realistic scenario with ΛSUSY lying at some few TeV’s.
2. The SUSY Breaking Scenario at the MX scale (XSBS), with ΛSUSY ∼ MX i.e. SUSY breaking at the 331
Analogously to the SM, fermions will transform as singlets or in the fundamental representation of SU(3)L, and
gauge fields in the adjoint representation. Assignment of U(1)X quantum numbers should be done ensuring a
model free of anomalies. There are models with three different families necessary to cancel out the anomalies, and
there are models with only one family and the other two are a copy of the first. We will take into account six
different versions of the 331 model for the analysis of the unification scheme.
The minimal spectrum necessary for symmetry breaking and generation of masses is given by 
Where the quantum numbers are associated with SU (3)C, SU (3)L, and U (1)Xrespectively. The first multiplet
acquires a vacuum expectation value (VEV) at MXscale, breaking the symmetry as 331 → 321; the other two will