# A new analysis on matching conditions and the 331 perturbative regime

Mario W. Barela<EMAIL_ADDRESS>Instituto de Física Teórica,
Universidade Estadual Paulista,
R. Dr. Bento Teobaldo Ferraz 271, Barra Funda
São Paulo - SP, 01140-070, Brazil

###### Abstract

The Minimal 331 Model offers a compelling extension of the Standard Model of
Electroweak Interactions, with a rich particle content that allow exciting
phenomenological possibilities. The theory’s parameter space, however, remains
notably unconstrained – a fact unexpected to substantially change without new
strong experimental input. With regard to its effective range of validity, it
is of general knowledge that the model’s $U(1)_{X}$ coupling diverges at
around $4\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$. In this work, we review
this affirmation, supposedly valid for the 331 as an extension of the Standard
Model, stating it more precisely. Additionally, we investigate whether a
contrasting equally acceptable standpoint which poses the 331 as alternative,
rather than extension, may soften this perturbativity bound. We pay special
attention to the matter of symmetry matching conditions, revising the origin
and legitimacy of some relations between couplings regularly invoked within
the 331 context.

##  I Introduction

When the Standard Model (SM) issues, both theoretical inadequacies and
inabilities of accounting for important physical phenomena, became unavoidable
within its context, a pressing necessity of new fundamental theories emerged.
The elegant and revolutionary idea of extending the spacetime algebra by an
anti-commuting sector – with desirable consequences on the particle physics
model building side – as implemented by Supersymmetry appeared as an exciting
possibility [1], soon to be frustrated by phenomenology and high energy data
[2, 3].

A contrasting species of theories, candidates to replace the SM, is more
closely related to it, and features the property of being built around a non-
simple gauge group, to present lower and more immediately experimentable
characteristic energies. Known examples of this are the SM plus a $Z^{\prime}$
[4, 5, 6]; an $SU(4)_{L}\times U(1)_{X}$ [7] electroweak model; and left-right
symmetric electroweak sectors [8, 9, 10]. Another representative of this class
of theories is the 331 Model [11, 12], which stands out by presenting
solutions to one or various of the SM issues in each of its versions, such as
motivating an explanation for the number of families; Being capable of
accomodating Dark Model candidates [13, 14, 15]; Correctly fitting the
Neutrino Masses; And allowing for coupling constant unification even without
supersymmetrization [16, 17].

Besides exploratory or data-driven phenomenology, the evolution of the model’s
couplings under the Renormalization Group may be considered in order to
elucidate the theory’s characteristics and consequences while the experimental
desert paradigm does not change. Among the lessons this analysis might bring
is the observation of the possibility of unification of the couplings at some
scale – studies focusing on this aspect of the 331 have already been performed
and we do not revisit it [16, 17]. Further insights that may be gained from a
similar investigation regard the model’s perturbativity regime, as the
existence of arbitrary growth (sometimes called a Landau Pole) in the
behaviour of a coupling evolution poses a threat to the model’s unitarity,
indicating the necessity of newer physics or degrees of freedom.

The model’s Gauge Group, $SU(3)_{c}\times SU(3)_{L}\times U(1)_{X}$, extends
the electroweak sector and introduces five exotic physical vector bosons,
including a doubly charged one, another interesting feature. Initially
proposed as an extension of the SM, the 331 symmetry is, within this
perspective, only fully realized at some unknown (parametrically free) high
energy scale, below which it is broken to $SU(3)_{c}\times SU(2)_{L}\times
U(1)_{Y}$. An alternative rarely considered possibility, however, is that of
the model replacing the SM as a fundamental theory altogether, with a direct
breaking pattern that never crosses the SM symmetry:

$SU(3)_{c}\times SU(3)_{L}\times U(1)_{X}\to SU(3)_{c}\times
U(1)_{\text{EM}}.$ (1)

The difference between such group-theoretical constructions can be purely
semantic for a deal of the effective phenomenological consequences and,
therefore, to distinguish between them can be a delicate issue. This matter
raises the motivation for the study of the 331 Renormalization Group Equation
(RGE), as it may cast light over the benefits and burdens of each of the
Extension and Alternative point of views.

In this work, we attempt to achieve new information from the evolution of the
model’s electroweak couplings, while expanding on the traditional formalism,
didactically demystifying a few usual misconceptions and focusing on the
aspects of the 331 status as a fundamental theory mentioned above. We improve
earlier analysis in that we integrate exotic particles out by hand in a
diagramatic fashion and pay special attention to the matter of the symmetry
matching conditions. We focus solely on the minimal version of the model
(m331), and similar analysis for other versions are left for subsequent works.

The paper is organized as follows. In Section II we present a didactic
introduction to symmetry matching conditions and discuss some possible
equivocations; In Section III we perform a short review of the necessary
aspects of the m331; In Sections IV and V we carefully describe the
theoretical construction and formalism utilized in the analysis of the
Extension and Alternative scenarios, respectively, followed by the
corresponding results; and our conclusions and perspective are discussed in
Section VI.

##  II Symmetry Matching Conditions and Symmetric RGE

To motivate this section, consider the 331 Model as an extension of the SM. In
this scenario, the ocurring breaking pattern is

$SU(3)_{c}\times SU(3)_{L}\times
U(1)_{X}\xrightarrow[E_{\text{high}}]{\langle\chi\rangle}SU(3)_{c}\times
SU(2)_{L}\times
U(1)_{Y}\xrightarrow[E_{\text{low}}]{\langle\eta\rangle,\langle\rho\rangle}SU(3)_{c}\times
U(1)_{\text{EM}}.$ (2)

At the unknown scale $E_{\text{high}}$, the 331 symmetry is broken down to the
SM symmetry, after which, at $E_{\text{low}}$, which is to be identified with
the electroweak scale, the usual breaking occurs. To run the coupling constant
with the Renormalization Group Equation (RGE), we need to match the theory
across the symmetry thresholds. Since this sort of matching conditions can be
a source of confusion, we discuss them in some detail.

At $E_{\text{high}}$, $g_{2L}$ and $g_{Y}$ are to be matched to $g_{3L}$ and
$g_{X}$ above. As will be seen, the relations implied by this construction
didactically illustrate the general pattern, which can be subdivided into two
rules depending on the embedding structure of the groups. Consider the
breaking pattern $G_{1}\times G_{2}\times\cdots\times G_{n}\to G$, with
respective coupling constants $g_{1},g_{2},\cdots g_{n}$ and $g$, where all
factors are simple. In this case, at the breaking scale, the matching
conditions for $g$ may fall into two useful categories [18]:

  1. 1.

$G\in G_{1}$

An important special case occurs when the embedding is such that the lower
group is entirely contained within a simple higher group factor, in which
situation the condition becomes simply

$g=g_{1}.$ (3)

  2. 2.

$G\in G_{1}\times G_{2}\times\cdots G_{n}$

Suppose, in the more general case, that $G$ is contained within the non-simple
product and specialize to the case in which $G=U(1)$. Write the embedding
as111Clearly, the product of simple groups in which $G$ is embedded could be a
proper subset of the complete symmetry above the threshold, in which case
$p_{i}=0$ for some $i$.

$Z=\sum_{i=1}^{n}p_{i}T_{i},$ (4)

where $Z$ is the $U(1)$ generator and $T_{i}$ collectively denote every
generator of $G_{1}\times G_{2}\times\cdots G_{n}$. In this case, the matching
condition reads

$\frac{1}{g^{2}}=\sum_{i=1}^{n}\frac{p_{i}^{2}}{g_{i}^{2}}.$ (5)

Notice that, in the sum above, every distinct $i$-term which corresponds to
generators of a same group exhibits the same coupling factor, i.e.,
$g_{i}=g_{j}$ if $T_{i},T_{j}$ belong to the algebra of the same simple group.

These conditions are not to be practically viewed as similar to the finding of
Wilson coefficients when matching effective theories at thresholds where
massive particles are integrated out. They are, rather, simply a consequence
of requiring that the theory can be described by Lagrangians with the expected
symmetry and corresponding couplings at each energy range, and that at the
breaking scale the theories coincide, as obviously required by continuity. A
known example of the second case is the SM relation

$\frac{1}{e^{2}}=\frac{1}{g_{2L}^{2}}+\frac{1}{g_{Y}^{2}},$ (6)

which is a consequence of the definition of electric charge and that may, at
the level of the lagrangian, be readily read from

$Q=T_{3}+Y.$ (7)

Once the different couplings, defined to exist at distinct energy ranges, are
correctly matched at the symmetry transition scales, they may be evolved
through the RGE between thresholds or towards infinity according to

$\beta(g)=\mu\frac{dg}{d\mu}.$ (8)

The $\beta$-function may be written as

$\beta(g)=-\frac{g^{3}}{(4\pi)^{2}}b_{1},$ (9)

where the $\beta$-function coefficient at one loop, in an energy range where
the theory is fully symmetric, may be readily found for its symmetry group and
representation content as [19]

$b_{1}=\frac{11}{3}C_{2}(\text{Gauge})-\frac{4}{3}\kappa
S_{2}(\text{Fermion})-\frac{1}{6}\eta S_{2}(\text{Scalar}),$ (10)

where $C_{2}(R)$ and $S_{2}(R)$ are, respectively, the Casimir and Dynkin
Index invariants of the representation $R$, and $\kappa=1/2$ for Weyl
components and $\eta=2$ for complex scalars.

##  III 331 Review

For our purposes, a sufficient review of the m331 is required. The model’s
symmetry group is $SU(3)_{c}\times SU(3)_{L}\times U(1)_{X}$, with the
electric charge operator given by222The 331 general charge operator can be
written as $Q=T_{3}-\beta T_{8}+X$, with the $\beta$ parameter changing with
the model’s version. The minimal model presents $\beta=\sqrt{3}$.

$Q=T_{3}-\sqrt{3}T_{8}+X.$ (11)

The lepton sector representation content is

$L_{\ell}\equiv\left(\nu_{\ell},\ell,\ell^{c}\right)^{T}_{L}\thicksim(\mathbf{1},\mathbf{3},0),\;\;\;\;\ell=e,\mu,\tau.$
(12)

It may be realized that every degree of freedom of any generation is contained
within a single triplet.

The color sector is composed by

$\displaystyle
Q_{iL}\equiv\left(d_{i},u_{i},j_{i}\right)^{T}_{L}\thicksim(\mathbf{3},\mathbf{\overline{3}},-1/3),\;\;\;\;\;Q_{3L}\equiv\left(u_{3},d_{3},J\right)^{T}_{L}\thicksim(\mathbf{3},\mathbf{3},2/3),\;\;\;\;i=1,2;$
$\displaystyle u_{\alpha
R}\thicksim\left(\mathbf{3},\mathbf{1},2/3\right),\;\;\;\;d_{\alpha
R}\thicksim\left(\mathbf{3},\mathbf{1},-1/3\right),\;\;\;\;\alpha=1,2,3;$ (13)
$\displaystyle
J_{R}\thicksim\left(\mathbf{3},\mathbf{1},5/3\right),\;\;\;\;j_{iR}\thicksim\left(\mathbf{3},\mathbf{1},-4/3\right).$

Notice that $J^{+5/3}$ and $j_{1,2}^{-4/3}$ are exotic quarks.

The scalar sector is comprised, in the usual minimal model, by three triplets

$\eta\equiv\left(\eta^{0},\eta_{1}^{-},\eta_{2}^{+}\right)^{T}_{L}\thicksim(\mathbf{1},\mathbf{3},0),\;\;\;\;\;\rho\equiv\left(\rho^{+},\rho^{0},\rho^{++}\right)^{T}_{L}\thicksim(\mathbf{1},\mathbf{3},1),\;\;\;\;\;\chi\equiv\left(\chi^{-},\chi^{--},\chi^{0}\right)^{T}_{L}\thicksim(\mathbf{1},\mathbf{3},-1),$
(14)

and a sextet

$S\equiv\begin{pmatrix}\sigma_{1}^{0}&\frac{h_{2}^{+}}{\sqrt{2}}&\frac{h_{1}^{-}}{\sqrt{2}}\\\
\frac{h_{2}^{+}}{\sqrt{2}}&H_{1}^{++}&\frac{\sigma_{2}^{0}}{\sqrt{2}}\\\
\frac{h_{1}^{-}}{\sqrt{2}}&\frac{\sigma_{2}^{0}}{\sqrt{2}}&H_{1}^{--}\end{pmatrix}\thicksim(\mathbf{1},\mathbf{\overline{6}},0).$
(15)

This scalar sector is deemed minimal for being the smallest construction
sufficient to fit neutrino masses.

This representation content is projected onto the SM symmetry as follows:

$\displaystyle
L^{(2)}_{\ell}\equiv\left(\nu_{\ell},\ell\right)^{T}_{L}\thicksim(\mathbf{1},\mathbf{2},-1/2),\;\;\;\;\ell^{(2)}_{R}\thicksim\left(\mathbf{3},\mathbf{1},-1\right),$
$\displaystyle
Q^{(2)}_{iL}\equiv\left(d_{i},u_{i}\right)^{T}_{L}\thicksim(\mathbf{3},\mathbf{\overline{2}},1/6),\;\;\;\;\;Q^{(2)}_{3L}\equiv\left(u_{3},d_{3}\right)^{T}_{L}\thicksim(\mathbf{3},\mathbf{2},1/6)$
$\displaystyle u^{(2)}_{\alpha
R}\thicksim\left(\mathbf{3},\mathbf{1},2/3\right),\;\;\;\;d^{(2)}_{\alpha
R}\thicksim\left(\mathbf{3},\mathbf{1},-1/3\right),\;\;\;\;$ $\displaystyle
J^{(2)}_{R}\thicksim\left(\mathbf{3},\mathbf{1},5/3\right),\;\;\;\;J^{(2)}_{L}\thicksim\left(\mathbf{3},\mathbf{1},5/3\right),$
$\displaystyle
j^{(2)}_{iR}\thicksim\left(\mathbf{3},\mathbf{1},-4/3\right),\;\;\;\;j^{(2)}_{iL}\thicksim\left(\mathbf{3},\mathbf{1},-4/3\right).$
$\displaystyle\Phi_{\eta}\equiv\left(\eta^{0},\eta_{1}^{-}\right)^{T}_{L}\thicksim(\mathbf{1},\mathbf{2},-1/2),\;\;\;\;\;\eta^{+(2)}_{2}\thicksim(\mathbf{1},\mathbf{1},1),$
(16)
$\displaystyle\Phi_{\rho}\equiv\left(\rho^{+},\rho^{0}\right)^{T}_{L}\thicksim(\mathbf{1},\mathbf{2},1/2),\;\;\;\;\;\rho^{++(2)}\thicksim(\mathbf{1},\mathbf{1},2),$
$\displaystyle\chi^{(2)}\equiv\left(\chi^{-},\chi^{--}\right)^{T}_{L}\thicksim(\mathbf{1},\mathbf{2},-3/2),\;\;\;\;\;\chi^{0(2)}\thicksim(\mathbf{1},\mathbf{1},0),$
$\displaystyle\phi^{(2)}\equiv\left(h_{1}^{-},\sigma_{2}^{0}\right)^{T}_{L}\thicksim(\mathbf{1},\mathbf{\overline{2}},-1/2),\;\;\;\;\;H^{--(2)}\thicksim(\mathbf{1},\mathbf{1},-2),$
$\displaystyle\xi^{(2)}\equiv\begin{pmatrix}\sigma_{1}^{0}&\frac{h_{2}^{+}}{\sqrt{2}}\\\
\frac{h_{2}^{+}}{\sqrt{2}}&H_{1}^{++}\end{pmatrix}\thicksim(\mathbf{1},\mathbf{\overline{3}},1).$

Lastly in the representation content are the physical vector bosons, including
five exotic ones, implied by the extended symmetry. The electrically charged
sector is comprised by new singly and doubly charged bosons, written in terms
of the symmetry eigenstates, along with the $W$-boson, simply as

$\begin{split}W_{\mu}^{\pm}&=(W_{\mu}^{1}\mp iW_{\mu}^{2})/\sqrt{2},\\\
V_{\mu}^{\pm}&=(W_{\mu}^{4}\pm iW_{\mu}^{5})/\sqrt{2},\\\
U_{\mu}^{\pm\pm}&=(W_{\mu}^{6}\pm iW_{\mu}^{7})/\sqrt{2}.\\\ \end{split}$ (17)

The last addition is an extra neutral vector boson $Z^{\prime}$. The
diagonalization of the neutral sector is, however, immensely more complicated,
given, in general, by

$\begin{split}W^{3}_{\mu}&=\frac{t}{\sqrt{4t^{2}+1}}A_{\mu}-N_{1}\left(3m_{2}^{2}+\bar{v}_{\rho}^{2}-2\bar{v}_{W}^{2}\right)Z_{1\mu}-N_{2}\left(3m_{1}^{2}+\bar{v}_{\rho}^{2}-2\bar{v}_{W}^{2}\right)Z_{2\mu},\\\
\frac{W^{8}_{\mu}}{\sqrt{3}}&=-\frac{t}{\sqrt{4t^{2}+1}}A_{\mu}-N_{1}\left(m_{2}^{2}+\bar{v}_{\rho}^{2}-\frac{2}{3}\bar{v}_{W}^{2}-\frac{2}{3}\right)Z_{1\mu}-N_{2}\left(m_{1}^{2}+\bar{v}_{\rho}^{2}-\frac{2}{3}\bar{v}_{W}^{2}-\frac{2}{3}\right)Z_{2\mu},\\\
B_{\mu}&=\frac{1}{\sqrt{4t^{2}+1}}A_{\mu}+2t\left(1-\bar{v}_{\rho}^{2}\right)N_{1}Z_{1\mu}+2t(1-\bar{v}_{\rho}^{2})N_{2}Z_{2\mu},\end{split}$
(18)

where the overbar indicates the ratio by $v_{\chi}$ as
$\bar{v}_{\alpha}\equiv\frac{v_{\alpha}}{v_{\chi}}$, and
$t\equiv\tan\theta_{X}\equiv\frac{g_{X}}{g_{3L}}$. We have defined the factors

$\begin{split}A&=\frac{1}{3}\left[3t^{2}\left(\bar{v}_{\rho}^{2}+1\right)+\bar{v}_{W}^{2}+1\right],\\\
R&=\left\\{1-\frac{1}{3A^{2}}\left(4t^{2}+1\right)\left[\bar{v}_{W}^{2}\left(\bar{v}_{\rho}^{2}+1\right)-\bar{v}_{\rho}^{4}\right]\right\\}^{1/2},\end{split}$
(19)

and the dimensionless masses

$\begin{split}m_{1}^{2}&=\frac{2M^{2}_{Z_{1}}}{g^{2}v_{\chi}^{2}}=A(1-R),\\\
m_{2}^{2}&=\frac{2M^{2}_{Z_{2}}}{g^{2}v_{\chi}^{2}}=A(1+R),\end{split}$ (20)

The normalization factors are given by

$\begin{split}N_{1}^{-2}&=3\left(2m_{2}^{2}+\bar{v}_{\rho}^{2}-\frac{4}{3}\bar{v}_{W}^{2}-\frac{1}{3}\right)+(\bar{v}_{\rho}^{2}-1)^{2}(4t^{2}+1),\\\
N_{2}^{-2}&=3\left(2m_{1}^{2}+\bar{v}_{\rho}^{2}-\frac{4}{3}\bar{v}_{W}^{2}-\frac{1}{3}\right)+(\bar{v}_{\rho}^{2}-1)^{2}(4t^{2}+1).\end{split}$
(21)

This rotation is manifestly highly dependent on the vacuum expectation values
(VEV) $v_{\rho},v_{\chi}$, and the electroweak one, $v_{W}$, which, to
numerically fit the Higgs mass, comes up from the sum
$v_{\eta}^{2}+v_{\rho}^{2}+2v_{s}^{2}=v_{W}^{2}$ (considering the two sextet
VEVs to be equal). These VEVs are free parameters of the theory and their
skeptical analysis is out of the scope of this work. Here, instead, when
needed, we adopt a benchmark resulting from demanding that the 331 Symmetry
closes to the SM one at the electroweak scale.

The solution to the closure we adopt was found in [20], where the authors show
how the requirement that physically measurable parameters, such as neutral
boson masses and neutral current parameters, agree with their experimental
values, is automatically satisfied if an implicit relation between the
otherwise free VEVs is obeyed. Explicitly, this condition dictates

$v_{\rho}^{2}=\frac{1-4s_{W}^{2}}{2c_{W}^{2}}v_{W}^{2},$ (22)

where $s_{W}$ is the sine of the electroweak angle. If imposed, both neutral
current parameters, known vector boson masses and the custodial symmetry
$\rho$-parameter collapse to their expected values [21].

##  IV 331 as Extension

###  IV.1 Matching and strategy

We begin assessing the traditional construction of the m331 proposed as an
extension of the SM. In this scenario, we have, as already mentioned, two
couplings to match at each of two scales. Referring, again, to the breaking
pattern

$SU(3)_{c}\times SU(3)_{L}\times
U(1)_{X}\xrightarrow[E_{\text{high}}]{\langle\chi\rangle}SU(3)_{c}\times
SU(2)_{L}\times
U(1)_{Y}\xrightarrow[E_{\text{low}}]{\langle\eta\rangle,\langle\rho\rangle}SU(3)_{c}\times
U(1)_{\text{EM}},$ (23)

the entire process may be understood as: (i) Nature is assumed to be well
described by the $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$ symmetry
immediately above a $E_{\text{low}}$ scale, below which the theory is supposed
to be broken to the conserved $SU(3)_{c}\times U(1)_{\text{EM}}$; At this
threshold, the couplings are fixed to their numerically known values; (ii)
$g_{2L},g_{Y}$ are evolved, according to the appropriate RGE ansatz and
particle content (see below and in the next section), with increasing energy,
up to $E_{\text{high}}$; (iii) The second matching is performed to replace
$g_{2L},g_{Y}$ by the emergent $g_{3L},g_{X}$; (iv) The couplings are evolved
again, now up to the pole, according to a second ansatz corresponding to the
higher symmetry and/or larger set of particles suitable to the new energy
range, once $E_{\text{high}}$ is also set as the scale in which the exotic
particles’ effects become important.

Using the formalism reviewed above, the matching conditions may be found. At
$E_{\text{low}}$, generically taken to be at the $Z$-pole,
$E_{\text{low}}=M_{Z}=$91.1876\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, the
known SM couplings are simply fixed to their well measured numerical values
[22]:

$g_{2L}(E_{\text{low}})=0.62977,\;\;\;\;\;g_{Y}(E_{\text{low}})=0.34537.$ (24)

Table 1: $\beta$-function coefficients of the 331 couplings in the two energy ranges. We consider two different representation contents: ’All’ refers to the entire 331 content and; ’SM’ ignores the exotic 331 particles. |  $g_{Y(X)}$: All |  $g_{2L(3L)}$: All |  $g_{Y}$: SM |  $g_{2L}$: SM
---|---|---|---|---
$E_{\text{low}}<\mu<E_{\text{high}}$ | $38$ | $-2$ | 41/6 | -19/6
$\mu>E_{\text{high}}$ | $22$ | $-17/3$ |  |

At $E_{\text{high}}$, the two exotic couplings must be matched to the SM ones
according to the formalism of Sec. II. To deal with $g_{3L}$, it is enough to
realize that the relevant embedding structure obeys $SU(2)_{L}\in SU(3)_{L}$,
causing the necessary condition to simply be:

$g_{3L}(E_{\text{high}})=g_{2L}(E_{\text{high}}).$ (25)

Finally, for $g_{X}$, we recall

$Y=-\sqrt{3}T_{8}+X,$ (26)

which is a consequence of Eq. (11). Of inserting this information into formula
(5), results

$\frac{1}{g_{Y}(E_{\text{high}})^{2}}=\frac{3}{g_{3L}(E_{\text{high}})^{2}}+\frac{1}{g_{X}(E_{\text{high}})^{2}},$
(27)

into which, plugging the condition for $g_{3L}$, we obtain the matching
condition

$g_{X}(E_{\text{high}})=\frac{g_{2L}(E_{\text{high}})\,g_{Y}(E_{\text{high}})}{\sqrt{g_{2L}(E_{\text{high}})^{2}-3g_{Y}(E_{\text{high}})^{2}}}.$
(28)

Once more, it must be emphasized that these boundary conditions on the
breaking scales are merely consistency requirements to guarantee the supposed
symmetry structure and continuity. In fact, they obviously could be found –
and usually are – by explicit brute force comparison of the two lagrangians
with correct symmetry properties around the threshold.

The $\beta$-function coefficients needed to run the couplings between or above
thresholds are calculated for the entire representation content and for the
light, SM, subset, and are shown in Table 1.

###  IV.2 Results

The explicit dependence of $g_{X}$ with energy is given by

$\displaystyle g_{X}^{\text{naive}}(\mu)$
$\displaystyle=\frac{2\pi}{\sqrt{30.1-11.0\log{E_{\text{high}}}+22.0\log{E_{\text{low}}}-11.0\log{\mu}}},$
(29) $\displaystyle g_{X}^{\text{eff}}(\mu)$
$\displaystyle=\frac{2\pi}{\sqrt{30.1+2.83\log{E_{\text{high}}}+8.17\log{E_{\text{low}}}-11.0\log{\mu}}},$
(30)

where $\mu>E_{\text{high}}$ (there is no $g_{X}$ below it). Above,
$g_{X}^{\text{eff}}$ corresponds to a construction in which the heavy
particles are integrated out below their supposed scale of importance; and
$g_{X}^{\text{naive}}$ is resulting of one which fails to do so. Recall that
at the $E_{\text{high}}$ scale, the theory is altered in two ways: the
symmetry is enlarged and the set of exotic particles becomes relevant.

An immediate result to be noted is that for

$E_{\text{high}}\gtrsim$370\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,$ (31)

$g_{X}^{\text{naive}}$ becomes imaginary in the entire $\mu$-range. This
signifies that, for
$E_{\text{high}}\gtrsim$370\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, the
mandatory matching conditions are impossible to fulfil, rendering the theory
senseless in the entire higher symmetry regime.

In the other hand, if we integrate the heavy particles out at energies
$\mu<E_{\text{high}}$, we see, from Eq. (30), that for energies obeying

$E_{\text{high}}<$3770\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$,$ (32)

there exists a region within the higher symmetry regime where the theory can
be perturbative. This can be verified in Figure 1. For breaking scales greater
than $3770\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$, the matching is only
possible with $g_{X}$ originating larger than $4\pi$ and rapidly diverging.

We observe, however, that even in the most favourable case, the model’s
perturbativity is quickly harmed at the low scale of
$\sim$4500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.

Figure 1: Running, in the Extension Scenario, of $g_{Y}$ (solid), followed by
the corresponding running of $g_{X}$ (dashed) for five different benchmarks
for $E_{\text{high}}$.

##  V 331 as Alternative

###  V.1 Matching and strategy

A second possibility rarely explored within new fundamental theories is that
of omitting the SM from the start and considering the alternative model as the
nature descriptive already at low energies. Looking at the 331 through this
lens, the following breaking pattern must be assumed

$SU(3)_{c}\times SU(3)_{L}\times
U(1)_{X}\xrightarrow[E_{\text{low}}]{\langle\eta\rangle,\langle\rho\rangle,\langle\chi\rangle}SU(3)_{c}\times
U(1)_{\text{EM}}.$ (33)

For which, using Eqs. (5,11), the matching conditions may be found to be

$\displaystyle\frac{1}{e(E_{\text{low}})^{2}}=\frac{4}{g_{3L}(E_{\text{low}})^{2}}+\frac{1}{g_{X}(E_{\text{low}})^{2}}.$
(34)

The equation above is, at times, inconsistently claimed and used to derive the
following additional relation between 331 and SM couplings

$t^{2}=\frac{s_{W}^{2}}{1-4s_{W}^{2}}.$ (35)

There are several conceptual issues with this process. To start, it has always
been carried, without exception known to the author, while the Extension
Scenario is being assumed, in which case Eq. (34) does not follow; In fact, in
that context there is no scale in which $e$ and $g_{3L},g_{X}$ cohexist, which
the attempts to derive it from Eq. (28) fail to note. In the other hand, in
the Alternative Scenario, the SM couplings $g_{2L},g_{Y}$ are formally non-
existent, and the above relation for $t^{2}$ makes no strict sense. In any
case, it is, in general, derived mixing disjoint scenarios (in the jargon
employed here) and/or scales.

Although the strategies of investigation which ‘run’ the matching conditions
with energy [23, 24, 25] could be useful in some approximate or effective
situations, we advocate that the most precise ansatz is the one presented
here, in which the breaking scales are single energy points theoretically
fixed a priori (which does not prevent them to be scanned over), at which the
matching conditions are imposed to assure the theory’s continuity and, between
which, only the couplings contained within the set corresponding to the active
symmetry are effectively mentioned.

In the last section, the known fact that heavy particles must be integrated
out with decreasing energy has been touched upon. The origin of this is the
fact that, in $\overline{\text{MS}}$, the decoupling of heavy particles is not
automatic as happens in physical Renormalization Schemes, and must be
introduced by hand [26]. Therefore, in order for the effects of the heavy
particles contributions to be accurately considered, the RGE must be manually
altered.

This must be performed in one of two ways, depending on the status of the
effective low energy theory. If the theory that remains after eliminating the
heavy particles from the representation content is still symmetric, it is
enough to modify the $\beta$-function coefficient to encompass only the
remaining multiplets. This case is exemplified by the removal of the third
generation of quarks of the SM and, specially, corresponds to the calculation
carried in the context of the Extension Scenario discussed above: the SM
projection of the m331 contains every exotic degree of freedom as a singlet,
which may be removed without spoiling the symmetry.

In the other hand, if the gauge symmetry is broken by disregarding a few
particles, some brute force method becomes necessary. This corresponds to the
case of integrating out only the top quark of the SM. Another instance of this
setting is the current Alternative Scenario, in which the symmetry stays
$SU(3)_{L}\times U(1)_{X}$ above the electroweak scale, and the $SU(3)_{L}$
triplets and octet are broken once the heavy particles are removed. As a
consequence, below the threshold the theory is no longer effectively
symmetric, and the heavy particles must be manually removed within that energy
range.

The procedure amounts to the subtracting, from the complete, symmetric $\delta
Z_{g_{3L}},\delta Z_{g_{X}}$ (or, rather, from their resulting
$\beta$-functions, calculated through Eq. (9)), the 1-loop contributions of
the exotic particles, $\delta Z^{\mathscr{H}}_{g_{3L}},\delta
Z^{\mathscr{H}}_{g_{X}}$, only activated above the threshold. To perform it,
we construct a completely determined system of equations from which the
couplings’ counterterms may be obtained; for that, the counterterms for at
least two vertex functions are required, and are conveniently chosen to be
correspondent to the $W^{+}e^{-}\overline{\nu}$ and $\gamma e^{+}e^{-}$
interactions. The dependence of the 1-loop counterterms on the individual
contributions is additive, hence, the needed quantities,
$\\{\delta\Gamma^{\mathscr{H}}_{We\overline{\nu}},\delta\Gamma^{\mathscr{H}}_{\gamma
e\overline{e}},\delta Z^{\mathscr{H}}_{e},\delta
Z^{\mathscr{H}}_{\nu_{e}},\delta Z^{\mathscr{H}}_{W},\delta
Z^{\mathscr{H}}_{A}\\}$, are calculated referring only to the exotic diagrams,
shown in Fig. 4, and there are no crossed effects between those and the
entirely usual contributions.

The relations among the vertex function counterterms
$\delta\Gamma^{\mathscr{H}}$ and the other ones read

$\displaystyle\delta\Gamma^{\mathscr{H}}_{We\overline{\nu}}=i\gamma_{\mu}P_{L}\frac{g_{3L}}{4\sqrt{2}}\left(\delta
Z^{\mathscr{H}}_{\nu_{e}}+\delta Z^{\mathscr{H}}_{e}+\delta
Z^{\mathscr{H}}_{W}+2\delta Z^{\mathscr{H}}_{g_{3L}}\right),$ (36)
$\displaystyle\delta\Gamma^{\mathscr{H}}_{\gamma
e\overline{e}}=i\gamma_{\mu}\frac{g_{3L}g_{X}}{2\sqrt{g_{3L}^{2}+4g_{3L}^{2}}}\left[\delta
Z^{\mathscr{H}}_{A}+2\delta Z^{\mathscr{H}}_{e}+2\delta
Z^{\mathscr{H}}_{g_{3L}}\left(1-\frac{g_{3L}^{2}}{g_{3L}^{2}+4g_{X}^{2}}\right)+2\delta
Z^{\mathscr{H}}_{g_{X}}\left(1-\frac{4g_{X}^{2}}{g_{3L}^{2}+4g_{X}^{2}}\right)\right].$
(37)

The right-hand sides of the equations above are found by inspection of the
exact 331 Lagrangian after renormalization and linearization of the relevant
interactions, with $Z_{x}\equiv 1+\delta Z_{x}$.

The solution of the system above is then used to find the
$\beta^{\mathscr{H}}$-function by demanding perturbative consistency from the
RGE $\partial g_{0}/\partial\ln\mu=0$ as usual [27]. The final result of the
running in the alternative scenario, in the non-symmetric regime below the
heavy particles threshold, is finally obtained by numerically solving the
coupled system of differential equations

$\displaystyle\frac{\partial
g_{3L}(\mu)}{\partial\ln\mu}-\frac{g_{3L}(\mu)^{3}}{(4\pi)^{2}}b_{g_{3L}}$
$\displaystyle-\left[-\beta^{\mathscr{H}}_{g_{3L}}\left(g_{3L}(\mu),g_{X}(\mu)\right)\right]=0,$
(38) $\displaystyle\frac{\partial
g_{X}(\mu)}{\partial\ln\mu}-\frac{g_{X}(\mu)^{3}}{(4\pi)^{2}}b_{g_{X}}$
$\displaystyle-\left[-\beta^{\mathscr{H}}_{g_{X}}\left(g_{3L}(\mu),g_{X}(\mu)\right)\right]=0$
(39)

The exotic scalar degrees of freedom and their corresponding counterterms have
not been included in the calculations above because, since they come in fully
exotic triplets, their elimination can be correctly effected through
modification of the $b$ above. The exception is the SM scalar doublet, which
must be kept. To accomplish this, we consider it to be the one projected by
the $\rho$-triplet, $\rho^{(2)}$ of Eq. (16). The $\beta$-function
coefficients to be plugged into the equations above, already accounting for
the removal of the scalars other than $\rho^{(2)}$, are

$b_{g_{X}}=\frac{62}{3},\;\;\;\;\;b_{g_{3L}}=-\frac{41}{6}.$ (40)

To summarize, the evolution process within the current construction follows
as: (i) We assume that, above $E_{\text{low}}$, nature is accurately described
by the 331 Gauge Symmetry and match $g_{3L},g_{X}$ to the electric charge $e$
according to Eq. (34) and to the requirement that the $W$-mass, $Wee$\- and
$\gamma ee$-couplings all coincide with their numerically known values at this
scale; (ii) We initially evolve the couplings with increasing energy according
to the $SU(3)_{L}\times U(1)_{X}$ RGE and with an effective representation
content where every exotic particle has been integrated out through some
procedure; (iii) At a parametrically free scale $E_{\text{high}}$, referring
now to the scale where their effects become important, we include the heavy
particles into the theory, match to the model below, and start evolving the
couplings according to the new RGE ansatz up to the pole.

###  V.2 Results

Figure 2: Evolution of $g_{X}$ comparing the three considered ansatz in the
Alternative Scenario. The symmetric case contains every particles’ effects,
while the others never include them. The curves highlight how correctly
excluding the heavy particles from the theory below their scale of importance
is detrimental to keep the model perturbative.

The objective is to investigate whether the second way of treating the m331
can alleviate the initial stress put onto the model by the previous Scenario
analysis. The operational method implies several steps, and many amplitudes of
Figure 4 are derived from rules which originate from Lagrangians with explicit
charge conjugation, which turns their obtaining into a subtle matter [28, 29].
We cross check them using FeynRules [30] paired with FeynArts [31]. The loop
calculations are performed with the help of Package-X [32] connected to
FeynCalc through FeynHelpers [33]. We employ the Feynman-’t Hooft gauge [34],
and validate the results through verification of theoretical expectations such
as that the longitudinal vector propagators are not corrected.

To verify again the importance of correctly accounting for the heavy particles
contributions, three different ansatz are compared, as seen in Figure 2. The
Symmetric case presents a theory with an untouched gauge structure above
$E_{\text{low}}$, and the complete representation content active from the
beginning. As expected by the knowledge of the big $\beta$-function
coefficient $b_{1X}=22$, the respective running is steep and heavily limits
the theory’s perturbativity, approaching $4\pi$ and diverging at
$\sim$2000\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$.

The remaining two curves represent ansatz with exotic particles integrated out
and never included again. The Projection curve is relative to removing the
heavy particles by calculating the $b_{1X}$ of the m331 projected onto the SM,
but with gauge-theoretical factors of the 331 Group. Although this approach
fails to consider that the low theory is not symmetric, it is more realistic
while it integrates out the high energy content of the model below their
energy scale, and already saves unitarity allowing the theory to exist up to
the high multi-TeV scale.

Finally, the third curve relates to the more appropriate method of integrating
out the particles by explicit evaluation of their contributions to the
runnings, as described in the last section, which we called Refined. The fact
that it appears close to the Projection approximation is non-trivial and
provides further validation to the calculations. Furthermore, the more
sophisticated analysis delivers a coupling evolution even more permissive of
high energies. This reiterates the lesson that the elimination of the heavy
particles’ effects is crucial.

The analysis is completed by the inclusion of the exotic particles at the
threshold, $E_{\text{high}}$, and the matching of the effective theory below
it with the full, symmetric m331 above. The results are shown in Fig. 3 for
three benchmark $E_{\text{high}}$. As expected, the necessary inclusion of the
heavy particles at some scale immensely strengthens the growth of the
coupling, causing $g_{X}$, for
$E_{\text{high}}=$1000\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, for example,
to break perturbativity at around $10\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$.
By completeness, a plot of the $g_{3L}$ running is also displayed.

In the other hand, these two more realistic constructions are allowing of a
higher effective range for the m331 when compared to the Extension scenario,
and, in particular, the refined calculation, which integrates out the exact
heavy physical eigenstates, keeps the theory perturbative up to higher
energies, allowing, for
$E_{\text{high}}=$5000\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, the model
with the original representation content to be physical up to a
phenomenological characteristic scale of around
$62.5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$.

Figure 3: Left: Alternative Scenario comparison of the running of $g_{X}$ in
the two ansatz for the removal of heavy particles, which are included again
above a scale for which three benchmarks are presented. The plot attests how
the refined calculation favours the model perturbativity relative to the
Projection approximation. Right: Evolution of the left-handed coupling (to
avoid cluttering in the image, we omit the
$E_{\text{high}}=$3500\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$ curve).

##  VI Conclusions

The 331 Model is a well motivated extension of the SM, with naturally rich
phenomenology is that has not yet been exhaustively explored: Skeptical
analysis of the several VEVs of the model; Studies of the scalar potential and
its symmetries; Or loop-level phenomenology and its influence on fermion
masses are examples of topics still wide open for discussion. There are,
however, additional purely theoretical ways to examine the theory’s
usefulness, which complement the phenomenology discoveries.

The m331 is, as common for analogous models, traditionally seen as an
extension of the SM. In this case, the Spontaneous Symmetry Breaking (SSB)
occurs in two steps: at a scale $E_{\text{high}}$, expected to be near
$v_{\chi}$ (a free parameter directly related to the exotic charged vector
boson masses), the $\chi$-triplet neutral component condenses and the 331
Symmetry is broken down to the SM one. At the electroweak scale, then, the
usual SM breaking occurs through the remaining VEVs. The second possibility is
to treat the model as an entire alternative to the SM, in which case the SSB
occurs in a single step, and the SM couplings $g_{Y},g_{2L}$ formally do not
exist.

We test the model’s possibilities on perturbativity in the extension scenario
and confirm the value of integrating heavy particles out with decreasing
energy in keeping results sensible. This, however, does not improve the
situation sufficiently, keeping the model perturbative, in the most favourable
case, only up to $4600\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$.

The alternative perspective delivers a more exciting possibility. As before,
the elimination of the heavy degrees of freedom at low energies is crucial:
failing to implement it causes the model to, in principle, break down around
$2000\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$. In light of that, two
approaches for removing the heavy particles are assessed. In the first one,
called Effective, the $\beta$-function coefficients are modified, below the
threshold, precisely to disregard the exotic $U(1)_{X}$ content. This is an
approximation in that the $SU(3)_{L}$ content is considered untouched although
the triplets are broken. The result shows a safer model, perturbative,
depending on the $E_{\text{high}}$, up to very high scales.

In the Refined ansatz, in which the exact status of the broken symmetry is
taken into account by evaluation of the exotic loops, the resulting $g_{X}$
evolution is even more tolerant of high energies, achieving, for
$E_{\text{high}}=$5000\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$$, energies of
around $62.5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$.

These findings are interesting in that they give insight into the model’s
usefulness, besides expanding its understanding. Apart from assessing
alternative versions of the model, left for posterior works, there are a few
points in the analysis, ignored by simplicity, which could be addressed. A
first one is the possibility of splitting the threshold: we have considered a
single high energy one. In the Extension case, this implies that the scale in
which the symmetry is enlarged coincides with the scale of importance of heavy
particles. A more general analysis would consider two distinct energy points,
one for each phenomenon. Furthermore, a different scale for inclusion could be
considered for each species of heavy particle, since their masses could differ
by up to a few TeV – this would evidently play a role in the Alternative
scenario as well.

Besides that, it must be noted that a benchmark for the Higgs doublet within
the Alternative Scenario was utilized. The study could be improved by a more
thorough analysis of the physical parametrization of the Higgs boson –
considering it, for instance, to be projected out of the $\eta$-triplet or
from an arbitrary rotation. It could also be interesting to focus on the
numerical structure of the scalar free parameters, VEVs in particular. The
benchmark we adopted, motivated by the numerical fitting of the neutral
current parameters at $E_{\text{low}}$, is not unique, and a large theory
space may exist in assuring that the model is numerically coherent. Relaxing
the relation between VEVs or testing other hypotheses in this sense could
yield interesting conclusions.

Such new analysis are, however, intrinsically tied to experimental and
phenomenological advances and should either not immensely vary the paradigm
unveiled here or be tied to distinct and complementary premises. It must be
understood that the Landau Poles, by themselves, do not condemn the model, but
attest that, at the corresponding energy ranges, new degrees of freedom or
theoretical mechanism must arise to protect the theory.

## ACKNOWLEDGEMENTS

The author is immensely grateful to Vicente Pleitez for initial discussions
which originated this project and for the critical reading of the paper. MB is
also grateful to CNPq for the financial support.

##  Appendix A Heavy particles’ contributions to the $\beta$-function

Figure 4: The complete set of exotic diagrams of the 331 Model with relation
to the SM, relevant for the RGE analysis of the $\gamma ee$ and $W\nu e$
vertices. Time flows from left to right. Diagrams with $N$ labels on internal
lines should be read as $N$ diagrams, each with a combination of labels that
should be paired as they are ordered. To allow for a sanity check, we note
that the number of diagrams counts 35.

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