SOME REMARKS ON THE BOSON MASS SPECTRUM IN A 3-3-1 GAUGE MODEL ADRIAN PALCU, SORIN NĂDĂBAN, ANDREA ȘANDRU Faculty of Exact Sciences – “Aurel Vlaicu”… [602641]

SOME REMARKS ON THE BOSON MASS SPECTRUM
IN A 3-3-1 GAUGE MODEL
ADRIAN PALCU, SORIN NĂDĂBAN, ANDREA ȘANDRU
Faculty of Exact Sciences – “Aurel Vlaicu” University Arad, Str . Elena Drăgoi 2, 310330 –
Arad, Romania
Received August 27, 2010
The bosson mass spectrum of a 3-3-1 gauge model with right-hand ed neutrinos
is investigated by tuning a unique free parameter within the ex act algebraical approach
of solving gauge models with high symmetries. The resulting mas ses and the possible
breaking scale of the model are discussed assuming certain phen omenological
constrains. A critical point can occur for a particular value o f the free parameter, so that
the masses in the neutral boson sector and the charged boson se ctor respectively,
become degenerate.
PACS numbers: 12.60.Cn, 14.70.Pw.
Key words: 3-3-1 models, boson mass spectrum.
1. INTRODUCTION
In this paper we analyze the boson mass spectrum in a particula r 3-3-1 gauge
model and emphasize the fact that a critical point could well o ccur at a not very
high breaking scale. All the Particle Data [1] suggest that the neutral boson 'Zof
whatever extension of the Standard Model (SM) has to be heavier than the neutral
boson Z of the SM in order to keep consistency with low energy phenomenology.
A very suitable manner to investigate the whole mass spectrum i s supplied by the
exact algebraical approach for solving gauge theories with high symmetries
proposed a decade ago by Cotăescu [2] and developed by one of t he authors in a
recent series of papers [3] – [7] on the 3-3-1 model with right -handed neutrinos.
For some interesting details of the rich phenomenology in such models, the reader
is referred also to Refs. [8] – [39] where the “orthodox method ” of treating gauge
models is involved.
The paper is organized as follows. Section 2 briefly reviews th e main results
of the theoretical method employed to solve the particular 3-3- 1 model with right-
handed neutrinos, so that the bos on mass spectrum – depending o n the unique free
parameter a – is obtained. Section 3 deals with the restriction s imposed on the
resulting masses from a phenomenol ogical viewpoint and the circ umstances under
Rom. Journ. Phys., Vol. 56, N os. 5–6, P. 673–681, Bucharest, 20 11

Adrian Palcu, Sorin Nădăban, Andrea Șandru 2 674
which the critical point can occur. That is, for a
0.6 the neutral bosons of the
theory Z, 'Z and X gain the same mass! Moreover, at the same time the charg ed
bosons W± and Y± become degenerate. Last section is devoted to sketching our
conclusions.
2. BOSON MASS SPECTRUM
The anomaly-free particle content of the ()()() 33 1cL YSU SU U⊗⊗ gauge
model with right-handed neutrinos , under consideration here [3] –[39], reads:
Lepton families
() ( )1, 3, –1 / 3 1,1, –1c
LR
Lfe
eα
αα α
αν
=ν∼∼ ( 1 )
Quark families
() ()3 –3 , 3 , 0 3 , 3 , – 1 / 3i
iL i L
i LLDT
Qd Q t
ub∗  
  ==      ∼∼ ( 2 )
() () ,3 , 1 , – 1 / 3 ,3 , 1 , 2 / 3Ri R Ri Rbd tu + ∼∼ ( 3 )
() () 3,1, 2 / 3 3,1, –1/ 3Ri RTD+∼∼ ( 4 )
with i = 1,2. The numbers in paranthesis denote – in a self-explanato ry notation –
the representations and the characters of each fermionic triple t with respect to the
gauge group of the theory.
With these representations the particular 3-3-1 model under con sideration
here is anomaly-free, as one can easily check out by using litt le algebra. Note that,
although all the anomalies cancel by an interplay between famil ies, each family
still remains anomalous by itself. This could be a hint to the generation number
issue which in turn must be a multiple of 3. Assuming the QCD c ondition that the
number of families must be upper bounded by 5 in order to have quark
confinement, one is led to the c onclusion that the number of ge nerations is exactly 3.
These representations can be achieved starting with the general method [2] of
exactly solving gauge models with high symmetries by just choos ing an
appropriate set of parameters (fo r certain details of dealing w ith the algebraical
procedure and the special parameterization involved here, the r eader is referred to
Ref. [7]). They are:

3 Some remarks on th e boson mass spectrum in a 3-3-1 gauge mode l 675
012 e, , 0, 0, 1Wθ ν =ν =ν = ( 5 )
being imposed by experimental arguments () e,Wθ [1] or by internal reasons of the
general method ()iν [2].
Along with the above parameters, one must add some new ones – a s they
determine the Higgs sector of the model – grouped in a paramete r matrix which
reads:
() ()()22
0111– 1– , , –22Di a g a ab ab η= η +   ( 6 )
where, initially, a and b are arbitrary non-vanishing real parameters that ensure the
condition ()()22
0 1– Trη=η . At the same time, [] 0,0 , 1aη∈ . All the details of the
Higgs sector and its involvement in the spontaneous symmetry br eakdown (SSB)
of the model are explained in Ref. [2]. We do not insist over i t, since we are here
interested only in the final results – mass spectrum and curren ts – allowed by the
method. They were already obtained in Ref. [7].
We must mention here that these parameters will determine, afte r the SSB –
which takes place (like in the SM) up to the universal residual ()1emU o n e – a
plausible non-degenerate boson mass. The exact expressions of t he boson masses
are given by the Eqs. (53) – (55) in Ref. [2], namely
()()()()22 1
2j ij
iMg =φη + η    ( 7 )
for the non-diagonal gauge bosons which usually are charged but – a s o n e c a n
easily observe in the 3-3-1 mode l under consideration here – on e of them comes
out neutral, and
()()22
ij ijM Tr B B=φ ( 8 )
with
()1–c o s
cosii iBg D Dθ  =+ ν ν η θ   ( 9 )
for the diagonal bosons of the model. The angle θ is the rotation angle around the
versor ν orthogonal to the electromagnetic direction in the parameter s pace [2].
The versor condition holds 1i
iνν = .
Since the electro-weak sector of the model is described now by the chiral
gauge group () ()31LYSU U⊗ , the two diagonal generators D 1 and D 2 (Ds – stands
for the Hermitian diagonal genera tors of the Cartan subalgebra) in the fundamental

Adrian Palcu, Sorin Nădăban, Andrea Șandru 4 676
representation of ()3LSU are: D 1=T3 and D 2=T8 – connected to the Gell-Mann
matrices in the manner / 2aaT=λ – and D 0=I for the chiral new hypercharge.
In our 3-3-1 model, the relation between θ in the general method [2] and the
Weinberg angle Wθ from SM was established [3, 7] and it is
2sin sin
3W θ=θ (10)
Boson mass spectrum
By using Eq. (7) one can express the masses of the non-diagonal bosons.
They are (according to the parameter order in the 2η matrix):
22
Wmm a= (11)
()22 11–2Ymm a b =+   (12)
()22 11– –2Xmm a b =   (13)
Throughout this paper we consider the following notation:
()222 2
0 1– /4 mg=φ η .
Evidently, W is the “old” charged boson of the SM which links p ositions 2–3
in the fermion triplet, namely the left-handed neutrino to its charged lepton partner
and, respectively, the “up” left-handed quarks to “down” left-h anded quarks. The
neutral Y boson couples the left- handed neutrino to the right-h anded one, and the
“classical” up (down) quarks to t he “exotic” up (down) quarks, that is positions 1–2
in fermion triplet are involved. The remaining X boson is respo nsible for the
charged weak current between positions 1–3 in each triplet.
The “pure” neutral bosons (diagona l ones) get their mass eigens tates by
diagonalizing the resulting matrix:
2
22
2 211 1 311– – 1– –22 22 3–4
13 1 1 3 3–1 – – – 1 –22 3 – 4 22 3–4ab ab
sMm
ab abs s +   =+     ( 1 4 )
after combining Eqs. (8), (9) and (6), where the notation sinWsθ= has been made
for simplicity.

5 Some remarks on th e boson mass spectrum in a 3-3-1 gauge mode l 677
One of the two diagonal bosons has to be identical to the neutr al boson Z
from SM. Therefore, the latter should be an eigenvector of this m a s s m a t r i x
corresponding to the eigenvalue 22 2/c o sZ WW mm= θ firmly established in the SM
[1]. The eigenvalue problem reads:
2
2
21–maMZZs>= > (15)
That is, one computes ()22 2–/ 1 – 0 Det M m a s = which leads to the constraint
upon the parameters: 2tanW ba=θ . Consequently, the parameter matrix (6)
becomes:
() ()22 2
021– d i a g 1– , , 1–t a n2c o s 2W
Waaa η= η θ  θ   (16)
Under these circumstances, the boson mass spectrum yields:
22
Wmm a= (17)
22
21–2co sY
Wamm=θ  (18)
()22 21– 1–t a n2XWamm =θ   (19)
2
2
2cosZ
Wmam=θ (20)
2
22
22tan 11– 13–4s i n 3–4s i nW
Z
WWmm a′  θ=+ + θθ    (21)
since ()22 2
Z Z Tr M m m=+ holds.
We obtained a mass spectrum depending on a single free paramete r a to be
tuned. One can observe that, alt hough the fermion representatio ns and even the
order in the parameter matrix 2η are not the same with those chosen in Ref. [3],
the resulting mass spectrum exhibits the same structure. That m eans there are
equivalent ways to choose the parameters in the general method in order to reach
the same particle content of the model and the same physics. In addition, by
inspecting the above mass spect rum one recovers the decoupling theorem.

Adrian Palcu, Sorin Nădăban, Andrea Șandru 6 678
3. PHENOMENOLOGICAL CONSEQUENCES
The method presented above relies heavily on the role played by t h e
parameter a. In fact, it determines the breaking scale and the structure o f the mass
spectrum in the model. For instance, from Eq.(17) it is obvious ly that assuming
()80.4 GeV mW , the smaller the parameter, the greater the mass scale m (and
consequently the breaki ng scale of the model).
When inspecting the boson mass spectrum – Eqs. (17) – (21) – on e can enforce
certain conditions on the parameter a as to obtain realistic values, in accordance
with the available experimental data. Furthermore, the neutrino phenomenology
was investigated [4] and, because a very high breaking scale φ was required, the
method suggested a natural see-saw mechanism [5], embedded in o rder to keep
consistency with the tiny observe d masses in the neutrino secto r.
However, a special and unexplored yet opportunity is offered by our method.
It was for the first time mentioned by Cotăescu in a communicat ion [40] on the
well known Pisano-Pleitez-Frampton 3-3-1 model [41, 42] and con sequently
developed in a regular paper devoted to exactly solving this mo del [43].
As long as the exact masses of the new bosons have not been exp erimentally
determined to date, one is entitle d to ask if there is no scree ning between them or –
more precisely – if the new ne utral boson does not “cover” the old one. Are their
masses degenerate? And if so, what kind of consequences has suc h a hypothesis?
What kind of hidden symmetry can unfold?
From Eqs. (20) and (21) it results that the free parameter has to be
2
22co s
3–2s i nW
Waθ=θ (22)
in order to achieve Z Z mm′= . That is 0.6a if we consider 2sinWθ [1].
Furthermore, what are the values gained by the masses of the re maining
bosons? Embedding (22) in (17), ( 18) and (19) respectively, one obtains the
amazing results:
2
22 2
22co s
3–2s i nW
WY
Wmm mθ==θ (23)
and simultaneously
22 2 2
22
3–2s i nZZ X
Wmm mm′===θ. (24)
These are the well-known values predicted by SM, namely 91.2 Ge V for the
neutral bosons, and 80.4 GeV for the charged ones. Assuming tha t in the SM

7 Some remarks on th e boson mass spectrum in a 3-3-1 gauge mode l 679
2W SMgm=φ (25)
holds, one can estimate the required breaking scale φ of the 3-3-1 model under
consideration here, by comparing it to (17). That is /SMa φ≥φ . This leads to
320 GeV.φ≥ However, we present below a plot with the mass spectrum
depending on the parameter a. The most plausible region of interest lies arround
the 1TeVm≥ , corresponding to 0.0065a≥ .

Fig. 1 – The masses of the bosons X, Y, Z’.
4. CONCLUDING REMARKS
We have proven in this brief report that the exact algebraical approach for
solving gauge models with high symmetries offers – when it is a pplied to a 3-3-1
model with right-handed neutrinos – a plausible way to investig ate the boson mass
spectrum. It allows one to make predictions regarding the resul ting masses of the
“new” bosons, while keeping the “old” boson ones at their estab lished values from
SM.

Adrian Palcu, Sorin Nădăban, Andrea Șandru 8 680
The critical point of the model can be seen as a point where th e free
parameter of the model enforces the breakdow n of the gauge grou p of the model up
to the SM's one. This can occur at a not very high breaking sca le 320 GeVφ≥ .
Since this result exactly holds at tree level, one is entitled to ask if the radiative
corrections do not significantly alter it. Of course, a detaile d analysis has to be
done in a future work [44] regarding the way the oblique parame ters S, T, U
(computed already for this particular 3-3-1 model in Ref. [13]) can influence these
results.
Therefore, we consider that the strange coincidence that simult aneously
occurs – namely, WXmm= and Z ZY mm m′== – for a particular value of the free
parameter seems more than a simple “fit”. It suggests a possibl e deeper identity
between the “same charge” bosons. This hypothesis must not be r uled out a priori ,
since more accurate results regarding the decays of the “new” b osons and high-
energy scatterings involving their couplings to fermions – expe rimental details
which can reveal some new restrictions on the parameter a – have to be more
exactly investigated at LHC.
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