# Two-pion interferometry for partially coherent sources in relativistic
heavy-ion collisions in a multiphase transport model

Shi-Yao Wang, Jun-Ting Ye, Wei-Ning<EMAIL_ADDRESS>School of
Physics, Dalian University of Technology, Dalian, Liaoning 116024, China

###### Abstract

We perform two-pion Hanbury Brown-Twiss (HBT) interferometry for the partially
coherent pion-emitting sources in relativistic heavy-ion collisions, using a
multi-phase transport (AMPT) model. A longitudinal coherent emission length,
as well as a transverse coherent emission length, are introduced to the pion
generation coordinates in calculating the HBT correlation functions of the
partially coherent sources. We compare the model results with and without
coherent emission conditions with experimental data in Au-Au collisions at
center-of-mass energy $\sqrt{s_{NN}}=$200 GeV, and in Pb-Pb collisions at
center-of-mass energy $\sqrt{s_{NN}}=$2.76 TeV, and find that the HBT results
of the partially coherent sources are closer to the experimental data than
those of chaotic sources.
Keywords: two-pion HBT correlations, partially coherent sources, AMPT model,
relativistic heavy ion collisions

###### pacs:

25.75.Gz, 25.75.-q, 21.65.jk

##  I Introduction

Two-pion Hanbury Brown-Twiss (HBT) correlation is an important observable to
detect the space-time structure of particle-emitting sources produced in
relativistic heavy-ion collisions Gyu79 ; Wongbook94 ; Wie99 ; Wei00 ;
Csorgo02 ; Lisa05 . Because the two pions emitted coherently have no HBT
correlation, this observable can also be used to study the source coherence
Gyu79 ; Wongbook94 ; Wie99 ; Wei00 ; Csorgo02 ; Lisa05 . Recent measurements
of two-pion HBT correlations in Pb-Pb collisions at the Large Hadron Collider
(LHC) ALICE_PRC89_2014 ; ALICE_PRC92_2015 ; ALICE_PRC93_2016a  and
measurements in Au-Au collisions at the Relativistic Heavy Ion Collider (RHIC)
STAR_PRL87_2001 ; PHENIX_PRL93_2004 ; STAR_PRC71_2005 ; STAR_PRC81_2010
showed that the two-pion correlation functions near zero relative momentum are
substantially less than 2, corresponding to the magnitude for completely
chaotic sources Gyu79 ; Wongbook94 ; Wie99 ; Wei00 ; Csorgo02 ; Lisa05 . These
observations and the suppression of the multi-pion correlation functions
observed in Pb-Pb collisions at the LHC ALICE_PRC89_2014 ; ALICE_PRC93_2016
indicate that the pion-emitting sources are perhaps partially coherent.
However, there is no widely accepted explanation for these observations.

Because of the complexity of relativistic heavy-ion collision systems, the
calculations for experimental observables based on models play an important
role in understanding the system properties. A multi-phase transport model
(AMPT) has been extensively used in relativistic heavy-ion collisions
LinKo_PRC65_2002 ; LinKo_JPG30_2004 ; Lin_PRC72_2005 ; Nasim_PRC82_2010 ;
XJunCMKo_PRC83_2011 ; DSolanki_PLB720_2013 ; YLXie_NPA920_2013 ;
BzdakGLMa_PRL113_2014 ; GLMaZWLin_PRC93_2016 ; HLi_PRC96_2017 ;
Haque_JPG46_2019 ; MDordevic_PRC101_2020 ; TShao_PRC102_2020 ;
KJSunCMKo_PRC103_2021 ; MagdtLacey_PRC104_2021 ; SBasu_PRC104_2021 ;
MagEvdLac_JPG48_2021 ; ZZhangNYuHXu_EPJA58_2022 . In Ref.
SYWangWNZhang_FP_2022 , the authors studied the two-pion HBT correlation
functions for the partially coherent pion-emitting sources constructed with
the AMPT model by introducing a coherent emission length $L_{\rm C}$ to the
pion longitudinal freeze-out coordinates. It was assumed that the emission of
the two pions with a longitudinal difference of freeze-out coordinates smaller
than $L_{\rm C}$ is coherent, and otherwise it is completely chaotic
SYWangWNZhang_FP_2022 . They found that the intercepts of the two-pion
correlation functions of the partially coherent sources with finite coherent
emission lengths are less than those of the completely chaotic sources in Au-
Au collisions at center-of-mass energy $\sqrt{s_{NN}}=$200 GeV, and in Pb-Pb
collisions at center-of-mass energy $\sqrt{s_{NN}}=$2.76 TeV.

As a microscopic transport model, the AMPT model can provide the coordinates
and momenta of the freeze-out particles at each time step in the source after
they are generated. In this paper, we introduce a transverse coherent emission
length and a longitudinal coherent emission length, which are momentum
dependent according to particle DeBroglie wavelength, to identical pion
generation coordinates to construct the partially coherent pion-emitting
sources. We calculate the two-pion correlation functions for the partially
coherent sources in the AMPT model, and compare the model results with
experimental data in Au-Au collisions at $\sqrt{s_{NN}}=$200 GeV
STAR_PRC71_2005  and in Pb-Pb collisions at $\sqrt{s_{NN}}=$2.76 TeV
ALICE_PRC93_2016a . We find that the HBT results of partially coherent sources
are closer to experimental data than those of chaotic sources. The results of
coherent fraction for the partially coherent sources in Pb-Pb collisions is
consistent with the values extracted from the experimental measurements of
multi-pion HBT correlations ALICE_PRC89_2014 ; ALICE_PRC93_2016 .

The rest of this paper is organized as follows. Section II presents a brief
introduction to the AMPT model, and discusses the generation and freeze-out
coordinates of identical pions generated by quark coalescence and by particle
scattering and decay. Section III gives the two-pion correlation function
calculations for the partially coherent and completely chaotic sources. Model
HBT interferometry results in Au-Au collisions at $\sqrt{s_{NN}}=$200 GeV, and
in Pb-Pb collisions at $\sqrt{s_{NN}}=$2.76 TeV, are presented and compared
with experimental data at the RHIC and LHC. We summarize and discuss this work
in section IV.

##  II Pion generation and freeze-out in AMPT model

The AMPT model is a hybrid composed of initialization, parton transport,
hadronization, and hadron transport Lin_PRC72_2005 . It has been successfully
used to describe the observables in high-energy heavy-ion collisions at the
RHIC and LHC LinKo_PRC65_2002 ; LinKo_JPG30_2004 ; Lin_PRC72_2005 ;
Nasim_PRC82_2010 ; XJunCMKo_PRC83_2011 ; DSolanki_PLB720_2013 ;
YLXie_NPA920_2013 ; BzdakGLMa_PRL113_2014 ; GLMaZWLin_PRC93_2016 ;
HLi_PRC96_2017 ; Haque_JPG46_2019 ; MDordevic_PRC101_2020 ; TShao_PRC102_2020
; KJSunCMKo_PRC103_2021 ; MagdtLacey_PRC104_2021 ; SBasu_PRC104_2021 ;
MagEvdLac_JPG48_2021 ; ZZhangNYuHXu_EPJA58_2022 . The initialization of
collisions in the AMPT model is performed using the HIJING model HIJING . The
parton and hadron transport are described by the ZPC parton cascade model ZPC
and ART model ART , respectively. In this study, we use the string melting
version of the AMPT model in which partons are hadronized by the quark
coalescence mechanism LinKo_PRC65_2002 ; Lin_PRC72_2005 . We take the model
parameter $\mu$ of parton screening mass to be 2.2814 fm-1, and the strong
coupling constant $\alpha_{s}$ to be 0.47, corresponding to a parton-
scattering cross section of 6 mb Lin_PRC72_2005 .

Figure 1: (Color online) Pion average transverse and longitudinal freeze-out
coordinates (dashed lines) and generation coordinates (solid lines) in Au-Au
collisions at $\sqrt{s_{NN}}=$200 GeV. (a) all identical (negative) pions; (b)
the pions generated by quark coalescence; (c) the pions generated by particle
scattering and decay.

After hadronization, the generated hadrons may interact with other particles
in the source, and may be absorbed or remain until freeze-out. Some hadrons
may also be generated by particle interactions in the source and hadron
decays. With the AMPT model, we can trace back to the origin of a freeze-out
particle, its generation coordinates, momentum, and parent. Figure 1 shows the
average transverse and longitudinal freeze-out coordinates (dashed lines) and
generation coordinates (solid lines) of identical pions versus the particle
transverse momentum in Au-Au collisions at $\sqrt{s_{NN}}=$200 GeV, where
panels (a) – (c) are for all identical (negative) pions, the pions generated
by quark coalescence, and the pions generated by particle scattering and
hadron decays, respectively. In our calculations, the cutoff time for hadron
cascade is taken to be 200 fm$/c$, as in previous HBT studies Lin_PRC72_2005 .
The impact parameter is taken to be 0–5 fm. For Au-Au collisions at
$\sqrt{s_{NN}}=$200 GeV, we take a rapidity cut, $|y|<0.5$, as in experimental
data analyses STAR_PRC71_2005 . It can be seen that the average freeze-out
coordinates are larger than the average generation coordinates. The average
values for quark coalescence generation are the smallest, and the average
values for the pions generated by particle scattering and decay are the
largest. This is because pions generated by quark coalescence occur earlier on
average than those generated by particle scattering and decay. One can also
see that the relative difference between the average freeze-out coordinate and
generation coordinate for quark coalescence is larger than that for particle
scattering and decay.

Figure 2: (Color online) Pion average transverse and longitudinal freeze-out
coordinates (dashed lines) and generation coordinates (solid lines) in Pb-Pb
collisions at $\sqrt{s_{NN}}=$2.76 TeV. (a) all the identical (negative)
pions; (b) the pions generated by quark coalescence; (c) the pions generated
by particle scattering and decay.

Figure 2 shows the average transverse and longitudinal freeze-out coordinates
(dashed lines) and generation coordinates (solid lines) of identical pions
versus the particle transverse momentum in Pb-Pb collisions at
$\sqrt{s_{NN}}=$2.76 TeV, where panel (a) is for all of the identical
(negative) pions, (b) is for the pions generated by quark coalescence, and (c)
is for the pions generated by particle scattering and hadron decays. In the
calculations for Pb-Pb collisions, we take a pseudorapidity cut, $|\eta|<0.8$,
as in experimental data analyses ALICE_PRC93_2016a . One can see that the
average coordinates are larger in Pb-Pb collisions at $\sqrt{s_{NN}}=$2.76 TeV
than in Au-Au collisions at $\sqrt{s_{NN}}=$200 GeV.

##  III Two-pion HBT analyses

###  III.1 Calculation of two-pion correlation function in AMPT model

The two-pion HBT correlation function is defined as:

$C({\mathbf{p}}_{1},{\mathbf{p}}_{2})=\frac{P({\mathbf{p}}_{1},{\mathbf{p}}_{2})}{P({\mathbf{p}}_{1})P({\mathbf{p}}_{2})},$
(1)

where $P({\mathbf{p}}_{i})~{}(i=1,2)$ is the distribution of single-pion
momentum ${\mathbf{p}}_{i}$, and $P({\mathbf{p}}_{1},{\mathbf{p}}_{2})$ is the
two identical pion momentum distribution in an event.

For a chaotic pion-emitting source, the denominator and numerator in Eq. (1)
can be expressed respectively as Wongbook94

$P({\mathbf{p}}_{1})P({\mathbf{p}}_{2})=\sum_{X_{1},X_{2}}A^{2}({\mathbf{p}}_{1},X_{1})A^{2}({\mathbf{p}}_{2},X_{2}),$
(2)
$P({\mathbf{p}}_{1},{\mathbf{p}}_{2})=\sum_{X_{1},X_{2}}\left|\Phi({\mathbf{p}}_{1},{\mathbf{p}}_{2};X_{1},X_{2})\right|^{2},$
(3)

where $A({\mathbf{p}}_{i},X_{i})$ is the magnitude of the amplitude for
emitting a pion with momentum ${\mathbf{p}}_{i}$ at four-coordinate $X_{i}$
(freeze-out coordinates), and

$\Phi({\mathbf{p}}_{1},{\mathbf{p}}_{2};X_{1},X_{2})=\frac{1}{\sqrt{2}}\Big{[}A({\mathbf{p}}_{1},X_{1})A({\mathbf{p}}_{2},X_{2})e^{ip_{1}\cdot
X_{1}}e^{ip_{2}\cdot
X_{2}}+A({\mathbf{p}}_{1},X_{2})A({\mathbf{p}}_{2},X_{1})e^{ip_{1}\cdot
X_{2}}e^{ip_{2}\cdot X_{1}}\Big{]},$ (4)

where $p_{i}$ is the four-momentum.

Because the HBT effect works on the identical pion pair with small relative
momentum ${\mathbf{q}}={\mathbf{p}}_{1}\\!-\\!{\mathbf{p}}_{2}$, we can
substitute $A({\mathbf{p}}_{j},X_{i})$ for $A({\mathbf{K}},X_{i})$ $(i,j=1,2)$
[smoothed approximation,
${\mathbf{K}}=({\mathbf{p}}_{1}+{\mathbf{p}}_{2})/2$]. Then we have

$\displaystyle P({\mathbf{p}}_{1},{\mathbf{p}}_{2})$ $\displaystyle=$
$\displaystyle\sum_{X_{1},X_{2}}A^{2}({\mathbf{K}},X_{1})A^{2}({\mathbf{K}},X_{2})\left[1+\cos\left[(p_{1}-p_{2})\cdot(X_{1}-X_{2})\right]\right]$
(5) $\displaystyle=$
$\displaystyle\sum_{X_{1},X_{2}}A^{2}({\mathbf{p}}_{1},X_{1})A^{2}({\mathbf{p}}_{2},X_{2})\left[1+\cos\left[(p_{1}-p_{2})\cdot(X_{1}-X_{2})\right]\right].$

In relativistic heavy-ion collisions at the RHIC and LHC, the systems are
initially compressed in the beam direction (longitudinal or $z$ direction),
and then expand longitudinally and transversely. A partially coherent pion-
emitting source was constructed in the AMPT model by introducing a coherent
emission length $L_{\rm C}$ to the longitudinal freeze-out coordinates of
identical pions SYWangWNZhang_FP_2022 . It is assumed that the emissions of
the two pions are coherent if they have a longitudinal distance of freeze-out
coordinates less than $L_{\rm C}$. However, a constant coherent emission
length is too simple. Pions, as the lightest hadron, have wide DeBroglie
wavelengths, $(h/|{\mathbf{p}}|)$. Considering the systems produced in
relativistic heavy-ion collisions are anisotropic in longitudinal and
transverse directions, we respectively introduce longitudinal and transverse
coherent emission lengths as:

$L_{\rm CZ}=a_{Z}\left[\frac{h}{k_{1Z}}+\frac{h}{k_{2Z}}\right],$ (6)

and

$L_{\rm CT}=a_{T}\left[\frac{h}{k_{1T}}+\frac{h}{k_{2T}}\right],$ (7)

where $a_{Z}$ and $a_{T}$ are two coherent-length parameters determined by
comparing model HBT results with experimental data, $k_{iZ}$ and $k_{iT}$
($i=1,2$) are pion longitudinal and transverse generation momenta. We assume
that the emission of two pions with the differences of longitudinal and
transverse generation coordinates respectively smaller than $L_{\rm CZ}$ and
$L_{\rm CT}$ is coherent. For this partially coherent source, we have

$P({\mathbf{p}}_{1},{\mathbf{p}}_{2})=\sum_{X_{1},X_{2}}A^{2}({\mathbf{p}}_{1},X_{1})A^{2}({\mathbf{p}}_{2},X_{2})R({\mathbf{p}}_{1},{\mathbf{p}}_{2};X_{1},X_{2}),$
(8)

where

$R({\mathbf{p}}_{1},{\mathbf{p}}_{2};X_{1},X_{2})=1,\hskip 102.42992pt{\rm
for}~{}\Delta z<L_{\rm CZ}~{}{\rm and}~{}\Delta r_{\rm T}<L_{\rm CT},$ (9)

and

$R({\mathbf{p}}_{1},{\mathbf{p}}_{2};X_{1},X_{2})=1+\cos\left[(p_{1}-p_{2})\cdot(X_{1}-X_{2})\right],~{}~{}~{}~{}~{}~{}~{}~{}{\rm
otherwise},\hskip 34.1433pt$ (10)

where $\Delta z=|z_{1}-z_{2}|$, $\Delta r_{\rm
T}=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}$, and
$(x_{i},y_{i},z_{i})~{}(i=1,2)$ are the pion generation coordinates.

Figure 3: (Color online) Two-pion correlation functions $C(q)$ for pion-
emitting sources with different coherent-length parameters, ($a_{\rm Z}=0$,
$a_{\rm T}=0$), ($a_{\rm Z}=0.3$, $a_{\rm T}=1$), ($a_{\rm Z}=0.3$, $a_{\rm
T}=3$), and ($a_{\rm Z}=1$, $a_{\rm T}=3$), in Au-Au collisions at
$\sqrt{s_{NN}}=200$ GeV, with impact parameter $0<b<5$ fm in the AMPT model.
Results calculated with (a) all of the identical (negative) pions; (b) the
pions generated by quark coalescence; (c) the pions generated by particle
scattering and decay. Figure 4: (Color online) Two-pion correlation functions
$C(q)$ for pion-emitting sources with different coherent-length parameters,
($a_{\rm Z}=0$, $a_{\rm T}=0$), ($a_{\rm Z}=0.3$, $a_{\rm T}=1$), ($a_{\rm
Z}=0.3$, $a_{\rm T}=3$), and ($a_{\rm Z}=1$, $a_{\rm T}=3$), in Pb-Pb
collisions at $\sqrt{s_{NN}}=2.76$ TeV, with impact parameter $0<b<5$ fm in
the AMPT model. Results calculated with (a) all of the identical (negative)
pions; (b) the pions generated by quark coalescence; (c) the pions generated
by particle scattering and decay.

Figures 3(a)–(c) show the two-pion correlation functions $C(q)$ calculated
with all of the identical (negative) pions, the pions generated by quark
coalescence, and by particle scattering and decay, respectively, in Au-Au
collisions at $\sqrt{s_{NN}}=$200 GeV and with impact parameter $0<b<5$ fm in
the AMPT model. The correlation functions for the pions generated by quark
coalescence are wider than those for the pions generated by particle
scattering and decay, because the former have smaller average freeze-out
coordinates (see Fig. 1). In Figs. 3(a)–(c), the circle, up-triangle, down-
triangle, and square symbols denote the coherent-length parameters ($a_{\rm
Z}=0$, $a_{\rm T}=0$), ($a_{\rm Z}=0.3$, $a_{\rm T}=1$), ($a_{\rm Z}=0.3$,
$a_{\rm T}=3$), and ($a_{\rm Z}=1$, $a_{\rm T}=3$), respectively. One can see
that the correlation function values for the partially coherent sources are
less than those for completely chaotic sources ($a_{\rm Z}=a_{\rm T}=0$) at
small relative momenta. Also, one can see that the difference between the
correlation functions of partially coherent source and chaotic source at small
$q$ is greater for the quark coalescence generation than that for the
generation of particle scattering and decay. This is because that the smaller
average generation coordinates for quark coalescence generation lead to a
greater effect than that for the generation of particle scattering and decay,
with the same coherent-length parameters.

Figures 4(a)–(c) respectively show the two-pion correlation functions $C(q)$
calculated with all of the identical (negative) pions, the pions generated by
quark coalescence, and the pions generated by particle scattering and decay,
in Pb-Pb collisions at $\sqrt{s_{NN}}=$2.76 TeV and with impact parameter
$0<b<5$ fm in the AMPT model, where circle, up-triangle, down-triangle, and
square symbols mean the same as in Fig. 3. Compared to the correlation
functions in Au-Au collisions, the two-pion correlation functions in Pb-Pb
collisions are narrower because the average source sizes in Pb-Pb collisions
are larger.

###  III.2 Interferometry results

In two-pion HBT analyses, the fitting correlation function is usually the
Gaussian form,

$C(q_{\rm out},q_{\rm side},q_{\rm long})=\kappa(1+\lambda\,e^{-q^{2}_{\rm
out}R^{2}_{\rm out}-q^{2}_{\rm side}R^{2}_{\rm side}-q^{2}_{\rm
long}R^{2}_{\rm long}}),$ (11)

where $q_{\rm out}$, $q_{\rm side}$, and $q_{\rm long}$ are the Bertsch-Pratt
variables BerGonToh_PRC37_NPA498 ; PraCsoZim_PRC42 , which respectively denote
the components of the relative momentum ${\mathbf{q}}$ in the transverse “out”
(parallel to the transverse momentum of the pion pair ${\mathbf{K}}_{\rm T}$),
transverse “side” (in the transverse plane and perpendicular to
${\mathbf{K}}_{\rm T}$), and longitudinal (“long”) directions in the
longitudinally co-moving system (LCMS) frame Lisa05 ; $\kappa$ is a
normalization parameter, $\lambda$ is the chaoticity parameter for the
partially coherent source; and $R_{\rm out}$, $R_{\rm side}$, and $R_{\rm
long}$ are the HBT radii in the respective out, side, and long directions.

Figure 5: (Color online) Two-pion correlation functions with respect to
relative momenta $q_{i,j,k}$ ($i,j,k={\rm out,side,long}$) for chaotic and
partially coherent sources in Au-Au collisions at $\sqrt{s_{NN}}=200$ GeV,
with impact parameter $0<b<5$ fm in the AMPT model. Top: all of the pions;
middle: pions generated by quark coalescence; bottom: pions generated by
particle scattering and decay.

Figure 5 shows projections of the two-pion correlation functions $C(q_{\rm
out},q_{\rm side},q_{\rm long})$ in the out, side, and long directions for
chaotic and partially coherent sources in Au-Au collisions at
$\sqrt{s_{NN}}=200$ GeV, with impact parameter $0<b<5$ fm in the AMPT model,
where, the top panels are for all of the identical pions, middle panels are
for the pions generated by quark coalescence, and bottom panels are for the
pions generated by particle scattering and decay. For each relative momentum
direction, the projections of the correlation functions are obtained from the
three-dimensional correlation functions by averaging the relative momenta in
the other two directions over 0 to 20 MeV/$c$. Dashed lines in the figure are
the fitting curves of Eq. (11). In the calculations, the longitudinal and
transverse coherent-length parameters for the partially coherent sources are
taken as $a_{\rm Z}=0.5$ and $a_{\rm T}=1.8$, respectively, according to the
comparison of the fitted HBT results in the AMPT model with experimental data
STAR_PRC71_2005  in different intervals of transverse momentum of pion pair.
The total number of events in the model calculations is $3\times 10^{4}$.

Figure 6: (Color online) Two-pion correlation functions with respect to
relative momenta $q_{i,j,k}$ ($i,j,k={\rm out,side,long}$) for chaotic and
partially coherent sources in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV,
with impact parameter $0<b<5$ fm in the AMPT model. Top: all of the pions;
middle: pions generated by quark coalescence; bottom: pions generated by
particle scattering and decay.

Figure 6 shows projections of the two-pion correlation functions $C(q_{\rm
out},q_{\rm side},q_{\rm long})$ in the out, side, and long directions for
chaotic and partially coherent sources in Pb-Pb collisions at
$\sqrt{s_{NN}}=2.76$ TeV, with impact parameter $0<b<5$ fm in the AMPT model.
In calculations in Pb-Pb collisions, the longitudinal and transverse coherent-
length parameters for the partially coherent sources are taken to be $a_{\rm
Z}=0.8$ fm and $a_{\rm T}=2.5$, respectively, according to the comparison of
the fitted HBT results in the AMPT model with experimental data
ALICE_PRC93_2016a  in different intervals of transverse momentum of pion pair.
The total number of events in the mode calculations in Pb-Pb collisions is
$6\times 10^{3}$.

One can see that the intercepts of the correlation functions of partially
coherent sources at zero relative momentum are lower than those of chaotic
sources. This effect is most significant for the pions generated by quark
coalescence and the least significant for the pions generated by particle
scattering and decay. This is because the average generation coordinate of
pions generated by quark coalescence is the smallest, and that of pions
generated by particle scattering and decay is the largest.

Figure 7: (Color online) Two-pion HBT results ($R_{\rm out}$, $R_{\rm side}$,
$R_{\rm long}$, $R_{\rm out}/R_{\rm side}$, $\lambda$) for the chaotic and
partially coherent source and the coherent fraction ($C\\!F$) of partially
coherent source, with respect to transverse momentum of pion pair $K_{\rm T}$,
in Au-Au collisions at $\sqrt{s_{NN}}=200$ GeV with impact parameter $0<b<5$
fm in the AMPT model. Experimental data STAR_PRC71_2005  in Au-Au collisions
at $\sqrt{s_{NN}}=200$ GeV with centralities 0-5% and 10-20% are presented for
comparison.

Figure 7 shows the results of the fitted HBT radii $R_{\rm out}$, $R_{\rm
side}$, and $R_{\rm long}$; ratio $R_{\rm out}/R_{\rm side}$; and chaoticity
parameter $\lambda$ for chaotic and partially coherent sources in AMPT Au-Au
collisions at $\sqrt{s_{NN}}=200$ GeV, with impact parameter $0<b<5$ fm, in
the transverse momentum intervals of pion pair $K_{\rm T}<200$ MeV$\\!/c$,
$200\leq K_{\rm T}<300$ MeV$\\!/c$, $300\leq K_{\rm T}<400$ MeV$\\!/c$,
$400\leq K_{\rm T}<500$ MeV$\\!/c$, $500\leq K_{\rm T}<600$ MeV$\\!/c$, and
$600\leq K_{\rm T}<1000$ MeV$\\!/c$. The experimental data in Au-Au collisions
at $\sqrt{s_{NN}}=200$ GeV with centralities 0-5% and 10-20% STAR_PRC71_2005
are also shown in Figs. 7(a)–(e) for comparison. We found that increasing the
longitudinal coherent-length parameter $a_{\rm Z}$ leads to a decrease of
$\lambda$ and increase of $R_{\rm long}$, while increasing the transverse
coherent-length parameter $a_{\rm T}$ decreases $\lambda$ and increases
$R_{\rm side}$ and $R_{\rm out}$. However, the increase of $a_{\rm T}$ does
not improve the results of $R_{\rm out}/R_{\rm side}$. Finally, we took the
longitudinal and transverse coherent-length parameters $a_{\rm Z}$ and $a_{\rm
T}$ to be 0.5 and 1.8, respectively, for the partially coherent source. One
can see that the model HBT radii and chaoticity parameter of the partially
coherent source are closer to the experimental data compared to those of the
chaotic source. The middle and right panels in Fig. 7 show the results for
chaotic and partially coherent sources with the pions generated by quark
coalescence and particle scattering and decay, respectively. One can see that
the results of $\lambda$ for quark coalescence show great differences between
chaotic and partially coherent sources. However, the corresponding differences
of $\lambda$ values are small in the case of particle scattering and decay.

With the numbers of coherent-emission pion-pairs $n_{c}=N_{c}(N_{c}-1)/2$ and
total pion-pairs $n_{t}=N_{t}(N_{t}-1)/2$, where $N_{c}$ and $N_{t}$ are the
corresponding pion numbers, we can obtain the source coherent fraction
$C\\!F=N_{c}/N_{t}$. In Figs. 7(e), 7(e′), and 7(e′′), the square symbols give
the results of coherent fraction for the partially coherent source in Au-Au
collisions at $\sqrt{s_{NN}}=200$ GeV in the AMPT model. At small transverse
momenta of pion pair, the values of coherent fraction are mainly determined by
pion longitudinal momentum. However, the decrease of coherent fraction with
increasing $K_{\rm T}$ at large transverse momenta of pion pair is mainly
determined by the increase of pion transverse momentum at large transverse
momenta of pion pair, which leads to a decrease of $L_{\rm CT}$ [see Eq. (7)].

Figure 8: (Color online) Two-pion HBT results ($R_{\rm out}$, $R_{\rm side}$,
$R_{\rm long}$, $R_{\rm out}/R_{\rm side}$, $\lambda$) for the chaotic and
partially coherent source and the coherent fraction ($C\\!F$) of partially
coherent source, with respect to transverse momentum of pion pair $K_{\rm T}$,
in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV with impact parameter $0<b<5$
fm in the AMPT model. Experimental data ALICE_PRC93_2016a  in Pb-Pb collisions
at $\sqrt{s_{NN}}=2.76$ TeV with centralities 0-5% and 10-20% are presented
for comparison.

Figure 8 shows the results of the fitted HBT radii $R_{\rm out}$, $R_{\rm
side}$, and $R_{\rm long}$; ratio $R_{\rm out}/R_{\rm side}$; and chaoticity
parameter $\lambda$ for chaotic and partially coherent sources in AMPT Pb-Pb
collisions at $\sqrt{s_{NN}}=2.76$ GeV, with impact parameter $0<b<5$ fm, in
the transverse momentum intervals of pion pair $K_{\rm T}<250$ MeV$\\!/c$,
$250\leq K_{\rm T}<350$ MeV$\\!/c$, $350\leq K_{\rm T}<450$ MeV$\\!/c$,
$450\leq K_{\rm T}<550$ MeV $\\!/c$, $550\leq K_{\rm T}<650$ MeV$\\!/c$,
$650\leq K_{\rm T}<750$ MeV$\\!/c$, $750\leq K_{\rm T}<850$ MeV$\\!/c$, and
$850\leq K_{\rm T}<1200$ MeV$\\!/c$. Figure 8 also shows the coherent fraction
$C\\!F$ of partially coherent source in the AMPT model, and the experimental
data of HBT radii in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ GeV with
centralities 0-5% and 10-20% ALICE_PRC93_2016a  for comparison. In
comprehensively comparing the model HBT radii $R_{\rm out}$, $R_{\rm side}$,
and $R_{\rm long}$ with the experimental data, and the $C\\!F$ results with
the values extracted from experimental measurements of multi-pion HBT
correlations ALICE_PRC89_2014 ; ALICE_PRC93_2016 , we took the longitudinal
and transverse coherent-length parameters $a_{\rm Z}$ and $a_{\rm T}$ to be
0.8 and 2.5, respectively, for the partially coherent source. One can see from
Figs. 8(a)–8(e) that the model HBT radii of the partially coherent source are
closer to the experimental data than those of the chaotic source. Also, the
results of coherent fraction of the partially coherent source are consistent
with the values 0.23$\pm$0.08 extracted from the experimental measurement of
three-pion HBT correlation functions ALICE_PRC89_2014  and
0.32$\pm$0.03(stat)$\pm$0.09(syst) extracted from the experimental measurement
of four-pion HBT correlation functions ALICE_PRC93_2016 .

The middle and right panels in Fig. 8 show the results for the chaotic and
partially coherent sources with the pions generated by quark coalescence, and
particle scattering and decay, respectively. One can see that the differences
between the $\lambda$ values for chaotic and partially coherent sources are
great in the case of quark coalescence. However, the differences of $\lambda$
values between the chaotic and partially coherent sources are slight in the
case of particle scattering and decay.

##  IV Summary and Discussion

We performed two-pion interferometry for pion-emitting sources in relativistic
heavy-ion collisions in the AMPT model. A partially coherent source was
constructed by introducing momentum dependent longitudinal and transverse
coherent emission lengths $L_{\rm CZ}$ and $L_{\rm CT}$, based on pion
DeBroglie wavelength, in the calculations of two-pion HBT correlation
functions. The two pions generated with longitudinal distance smaller than
$L_{\rm CZ}$ and with transverse distance smaller than $L_{\rm CT}$ were
assumed to be emitted coherently and without HBT correlation. We compared the
HBT results in the AMPT model with experimental data in Au-Au collisions at
$\sqrt{s_{NN}}=200$ GeV, and in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV.
We found that the HBT chaoticity parameter $\lambda$ decreases with increasing
$L_{\rm CZ}$ and $L_{\rm CT}$. The results of HBT radii $R_{\rm out}$, $R_{\rm
side}$, $R_{\rm long}$, and chaoticity parameter $\lambda$ of partially
coherent sources in the AMPT model are closer to the experimental data in Au-
Au collisions at $\sqrt{s_{NN}}=200$ GeV at the RHIC, compared to those of
chaotic sources. Also, the results of $R_{\rm out}$, $R_{\rm side}$, and
$R_{\rm long}$ of the partially coherent sources in the AMPT model are closer
to the experimental data in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV at
the LHC, compared to those of the chaotic sources. The results of coherent
fraction of partially coherent source in Pb-Pb collisions in the AMPT model
are consistent with the values extracted by experimental measurements of
multi-pion HBT correlations. We investigated the two-pion HBT results for
chaotic and partially coherent sources constructed with the pions generated by
quark coalescence and by particle scattering and decay, and found that the
chaoticity parameter values of partially coherent sources are significantly
less than those of chaotic sources for quark coalescence pions.

With the AMPT model, one can trace back a freeze-out particle to its
generation coordinates, momentum, and parent. It is useful for studying the
underlying physics of experimental observables. The work presented in this
paper provides a possible relationship between pion generation and the HBT
observable, the two-pion correlation function. However, it does not affect the
explanations that the AMPT model provides to other observables, such as
single-particle spectra and collective flows. Pions, as the lightest hadron,
may involve quantum effects in their production and propagation in the
sources. It is possible that some of these effects remain and affect two- and
multi-pion HBT correlations, the observables from quantum statistics. In
relativistic heavy-ion collisions, the suppressions of two- and multi-pion HBT
correlation functions at small relative momenta may indicate the pion-emitting
sources are partially coherent. More detailed studies of the suppressions of
pion HBT correlations in relativistic heavy-ion collisions will be of
interest.

###### Acknowledgements.

We thank Zi-Wei Lin for useful discussions and suggestions. This research was
supported by the National Natural Science Foundation of China under Grant No.
12175031.

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# Constraining 3-3-1 Models at the LHC and Future Hadron Colliders

A. Alves1<EMAIL_ADDRESS>L. Duarte7<EMAIL_ADDRESS>S. Kovalenko2,3,4,5
<EMAIL_ADDRESS>Y. M. Oviedo-Torres6,7<EMAIL_ADDRESS>F. S.
Queiroz3,7,8<EMAIL_ADDRESS>Y. S. Villamizar7,8<EMAIL_ADDRESS>1Departamento de Física, Universidade Federal de São Paulo, UNIFESP, Diadema,
São Paulo, Brazil  2Departamento de Ciencias Fisicas, Universidad Andres
Bello, Sazie 2212, Piso 7, Santiago, Chile  3 Millennium Institute for
Subatomic Physics at High-Energy Frontier (SAPHIR), Fernandez Concha 700,
Santiago, Chile  4 Centro Científico-Tecnológico de Valparaíso, Casilla 110-V,
Valparaíso, Chile  5 Bogoliubov Laboratory of Theoretical Physics, JINR,
141980 Dubna, Russia  6 Departamento de Fisica, Universidade Federal da
Paraiba, Caixa Postal 5008, 58051-970, Joao Pessoa, PB, Brazil  7
International Institute of Physics, Universidade Federal do Rio Grande do
Norte, Campus Universitario, Lagoa Nova, Natal-RN 59078-970, Brazil  8
Departamento de Física, Universidade Federal do Rio Grande do Norte,
59078-970, Natal, RN, Brasil

###### Abstract

In this work, we derive lower mass bounds on the $Z^{\prime}$ gauge boson
based on the dilepton data from LHC with 13 TeV of center-of-mass energy, and
forecast the sensitivity of the High-Luminosity-LHC with $L=3000fb^{-1}$, the
High-Energy LHC with $\sqrt{s}=27$ TeV, and also at the Future Circular
Collider with $\sqrt{s}=100$ TeV. We take into account the presence of exotic
and invisible decays of the $Z^{\prime}$ gauge boson to find a more
conservative and robust limit, different from previous studies. We investigate
the impact of these new decays channels for several benchmark models in the
scope of two different 3-3-1 models. We found that in the most constraining
cases, LHC with $139fb^{-1}$ can impose $m_{Z^{\prime}}>4$ TeV. Moreover, we
forecast HL-LHC, HE-LHC, and FCC bounds that yield $m_{Z^{\prime}}>5.8$ TeV,
$m_{Z^{\prime}}>9.9$ TeV, and $m_{Z^{\prime}}>27$ TeV, respectively. Lastly,
put our findings into perspective with dark matter searches to show the region
of parameter space where a dark matter candidate with the right relic density
is possible.

Collider Physics, High-Luminosity (HL-LHC), High-Energy LHC (HE-LHC), 3-3-1
Models, Dark Matter, Future Circular Collider (FCC)

##  I  Introduction

Neutral resonances decaying to lepton pairs occur in several beyond the
Standard Model (SM) theories that are motivated to explain open problems such
as dark matter (DM), neutrino masses, parity violation and grand-unification,
for example. Many of such models predict neutral gauge bosons which can be
produced at current and future colliders. In some dark matter models Arcadi
_et al._ (2018a), $Z^{\prime}$ gauge bosons mediate interactions with the SM
spectrum and are key to the dark matter phenomenology, as they can drive both
the relic density and direct detection signals. A dark matter particle at the
LHC is inferred from missing energy events, but the LHC can also indirectly
contribute to the dark matter hunting by observing decays of those gauge
bosons that mediate the DM-SM interactions. The dilepton channel is
particularly interesting once it is much cleaner than the dijet one, offering
a better signal-over-background ratio.

From a collider physics perspective, obtaining mass bounds on a new vector
boson is a promising strategy to assess which new physics models could be
observed at the LHC. For instance, if a dilepton signal is observed with
invariant mass at $4$ TeV, one could conclude from our findings that such
signal would not come from a model based on the $SU(3)_{C}\otimes
SU(3)_{L}\otimes U(1)_{X}$ gauge symmetry, 3-3-1 for short. Models based on
this symmetry are phenomenologicaly compelling because may solve some open
problems such as neutrino masses Montero  _et al._ (2001); Tully and Joshi
(2001); Montero  _et al._ (2002); Cortez and Tonasse (2005); Cogollo  _et al._
(2009, 2010, 2008); Okada  _et al._ (2016); Vien  _et al._ (2019); Cárcamo
Hernández  _et al._ (2018); Nguyen  _et al._ (2018); de Sousa Pires  _et al._
(2019); Cárcamo Hernández  _et al._ (2019a, b); Hernández  _et al._ (2021),
dark matter Fregolente and Tonasse (2003); Long and Lan (2003); de S. Pires
and Rodrigues da Silva (2007); Mizukoshi  _et al._ (2011); Ruiz-Alvarez  _et
al._ (2012); Profumo and Queiroz (2014a); Dong  _et al._ (2013a, b); Cogollo
_et al._ (2014a); Dong  _et al._ (2014a, b); Kelso  _et al._ (2014); Dong  _et
al._ (2014b); Mambrini  _et al._ (2016); Dong  _et al._ (2015); de S. Pires
_et al._ (2016); Alves  _et al._ (2017); Rodrigues da Silva (2016); Carvajal
_et al._ (2017); Dong  _et al._ (2018); Arcadi  _et al._ (2018b); Montero  _et
al._ (2018); Huong  _et al._ (2019); Alvarez-Salazar and Peres (2021); Van Loi
_et al._ (2021); Dutra  _et al._ (2021); Oliveira and Pires (2021), meson
anomalies Cogollo  _et al._ (2012, 2014b); Buras  _et al._ (2014); Buras and
De Fazio (2016a); Queiroz  _et al._ (2016); de Melo  _et al._ (2021); Buras
and De Fazio (2016b); Buras  _et al._ (2021), flavor violation Cabarcas  _et
al._ (2012); Hue  _et al._ (2018), among others Montero and Sanchez-Vega
(2011); Santos and Vasconcelos (2018); Barreto  _et al._ (2018); De Conto  _et
al._ (2018).

An important feature of these models is that the mass of the $Z^{\prime}$
gauge boson is determined by the energy scale at which the 3-3-1 symmetry
breaks down to the 3-2-1. Furthermore, masses of the particles belonging to
the 3-3-1 spectrum are also proportional to this energy scale. In other words,
a bound on the $Z^{\prime}$ mass is seen as a limit on the entire 3-3-1
spectrum. For concreteness, we will focus our analysis on two popular models
based on the 3-3-1 symmetry, namely 3-3-1 RHN and 3-3-1 LHN. As the symmetry
suggests, the fermion content is arranged in triplets under $SU(3)_{L}$. In
the 3-3-1RHN, the lepton triplet contains a right-handed neutrino, whereas the
3-3-1 LHN has a heavy neutral fermion. The key difference between these models
is the presence of a viable dark matter candidate. In the latter, such heavy
neutral fermion can reproduce the correct dark matter relic density and yield
signals at direct detection experiments via interactions mediated by the
$Z^{\prime}$ field. Therefore, a limit on the $Z^{\prime}$ boson translates
into a constraint on possible dark matter signals. Lastly, 3-3-1 models have
$W^{\prime}$ bosons, whose mass is also set by the energy scale of symmetry
breaking, i.e., by the $Z^{\prime}$ mass. These new gauge boson can induce
lepton flavor violation signals Lindner  _et al._ (2018); Arcadi  _et al._
(2018b), which again can be indirectly constrained by lower mass bounds on the
$Z^{\prime}$ mass.

Motivated by the importance of the $Z^{\prime}$ gauge boson to 3-3-1
constructions, we derive bounds for both 3-3-1RHN and 3-3-1LHN models using
LHC data taking into account overlooked exotic decays, such as decay into
exotic quarks and dark matter. These new decay channels might significantly
impact the lower bound obtained considering only decays into SM fermions as
previously assumed Ramirez Barreto  _et al._ (2007, 2010, 2013); Coutinho  _et
al._ (2013); Corcella  _et al._ (2017); Nepomuceno and Meirose (2020). We
assess the relevance of these new decay channels for several benchmark models.
Furthermore, we forecast limits based on the High-Luminosity (HL), High-Energy
(HE), and Future Circular Collider (FCC) LHC setups. Lastly, in the context of
the 3-3-1LHN model, we investigate if one can host a viable dark matter
candidate in light of those bounds.

This work is organized as follows: in Section II we review the 3-3-1 model and
the relevant $Z^{\prime}$ decay channels; in Section III we discuss the
benchmark models and kinematic cuts used in the production of the $Z^{\prime}$
signal; in Section IV we present our collider findings; in Section V we
connect our collider results with dark matter phenomenology; and finally in
Section VI we draw our conclusions.

##  II  3-3-1 RHN and LHN Models

In our work we analyze two models based on the local symmetry group
$\textbf{SU(3)}_{\textbf{C}}\times\textbf{SU(3)}_{\textbf{L}}\times\textbf{U(1)}_{\textbf{X}}$,
namely 3-3-1 with right-handed neutrinos (3-3-1 RHN), Long (1996a, b), and
3-3-1 with neutral left-handed fermion (3-3-1 LHN) Mizukoshi  _et al._ (2011);
Catano  _et al._ (2012), where C represents color charge, L Left-handed, and X
is the $\text{U(1)}_{X}$ quantum number. The electric charge operator in these
two models is the same,

$\frac{Q}{e}=\frac{1}{2}\left(\lambda_{3}-\frac{1}{\sqrt{3}}\lambda_{8}\right)+\text{X}\cdot\hat{\rm
I},$ (1)

where $\lambda_{3,8}$, are the diagonal generators of
$\text{SU(3)}_{\text{L}}$ and $\hat{\rm I}$ is the identity matrix that acts
as generator of $\text{U(1)}_{\text{X}}$ group. Regarding the 3-3-1 RHN model,
it contains triplet and singlet fermionic fields given by,

$\displaystyle f_{aL}=\left(\begin{array}[]{c}\nu_{L}^{a}\\\ \ell_{L}^{a}\\\
\nu_{R}^{ac}\end{array}\right)\sim(1,3,-1/3),\ \ \ell_{aR}\sim(1,1,-1),$ (5)
$\displaystyle Q_{iL}=\left(\begin{array}[]{c}d_{i}\\\ -u_{i}\\\
d_{i}^{\prime}\end{array}\right)_{L}\sim(3,\overline{3},0),\text{
}u_{iR}\sim(3,1,2/3),$ (9) $\displaystyle\hskip
22.76228ptd_{iR}\sim(3,1,-1/3),\text{ }d_{iR}^{\prime}\sim(3,1,-1/3),$
$\displaystyle Q_{3L}=\left(\begin{array}[]{c}u_{3}\\\ d_{3}\\\
T\end{array}\right)_{L}\sim(3,3,1/3),\text{ }u_{3R}\sim(3,1,2/3),$ (13)
$\displaystyle\hskip 22.76228ptd_{3R}\sim(3,1,-1/3),\text{
}T_{R}\sim(3,1,2/3),$

with $a=1,2,3$ and $i=1,2$ indicating the generation indices. Notice that we
have three new exotic quarks ${q}^{\prime}$ ($d_{i}^{\prime}$ and $T$). In the
3-3-1 LHN a new heavy neutral lepton $N^{a}_{L}$ replaces the
$\left({\nu}_{R}^{a}\right)^{c}$ in the lepton triplet. Besides, a right-
handed neutral fermion ${N}^{a}_{R}$ is introduced, transforming as a singlet
under ${SU(3)}_{L}$,

$N^{a}_{R}\sim(1,1,0).$ (14)

$Z^{\prime}$ Interactions in the 3-3-1 model
---
Interaction | $g^{\prime}_{V}$ | $g^{\prime}_{A}$
$Z^{\prime}\ \bar{u}u,\bar{c}c$ | $\displaystyle{\frac{3-8\sin^{2}\theta_{W}}{{6\sqrt{3-4\sin^{2}\theta_{W}}}}}$ | $\displaystyle{-\frac{1}{2\sqrt{3-4\sin^{2}\theta_{W}}}}$
$Z^{\prime}\ \bar{t}t$ | $\displaystyle{\frac{3+2\sin^{2}\theta_{W}}{{6\sqrt{3-4\sin^{2}\theta_{W}}}}}$ | $\displaystyle{-\frac{1-2\sin^{2}\theta_{W}}{2\sqrt{3-4\sin^{2}\theta_{W}}}}$
$Z^{\prime}\ \bar{d}d,\bar{s}s$ | $\displaystyle{\frac{3-2\sin^{2}\theta_{W}}{6\sqrt{3-4\sin^{2}\theta_{W}}}}$ | $\displaystyle{-\frac{{3-6\sin^{2}\theta_{W}}}{6\sqrt{3-4\sin^{2}\theta_{W}}}}$
$Z^{\prime}\ \bar{b}b$ | $\displaystyle{\frac{3-4\sin^{2}\theta_{W}}{{6\sqrt{3-4\sin^{2}\theta_{W}}}}}$ | $\displaystyle{-\frac{1}{2\sqrt{3-4\sin^{2}\theta_{W}}}}$
$Z^{\prime}\ \bar{\ell}\ell$ | $\displaystyle{\frac{-1+4\sin^{2}\theta_{W}}{2\sqrt{3-4\sin^{2}\theta_{W}}}}$ | $\displaystyle{\frac{1}{2\sqrt{3-4\sin^{2}\theta_{W}}}}$
$Z^{\prime}\overline{N}N$ | $\displaystyle{\frac{4\sqrt{3-4\sin^{2}\theta_{W}}}{9}}$ | $\displaystyle{-\frac{4\sqrt{3-4\sin^{2}\theta_{W}}}{9}}$
$Z^{\prime}\ \overline{\nu_{\ell}}\nu_{\ell}$ | $\displaystyle{\frac{\sqrt{3-4\sin^{2}\theta_{W}}}{18}}$ | $\displaystyle{-\frac{\sqrt{3-4\sin^{2}\theta_{W}}}{18}}$
$Z^{\prime}\overline{d^{i}_{i}}d^{i}_{i}$ | $-\frac{3-5\sin^{2}\theta_{W}}{3\sqrt{3-4\sin^{2}\theta_{W}}}$ | $\frac{1-\sin^{2}\theta_{W}}{\sqrt{3-4\sin^{2}\theta_{W}}}$
$Z^{\prime}\overline{T}T$ | $\frac{3-7\sin^{2}\theta_{W}}{3\sqrt{3-4\sin^{2}\theta_{W}}}$ |  -$\frac{1-\sin^{2}\theta_{W}}{\sqrt{3-4\sin^{2}\theta_{W}}}$
Table 1: Vector and Axial couplings of the ${Z^{\prime}}$ boson with fermions
in the 3-3-1 model RHN and LHN. In the 3-3-1 RHN model there are no
interactions with the heavy fermions N. Apart from that, the $Z^{\prime}$
interactions are precisely the same as the 3-3-1 LHN model.

The quark sector remains unchanged. Fermion masses are obtained through the
introduction of three scalar triplets, which we denote as $\chi$, $\rho$ and
$\eta$,

$\displaystyle\quad\quad\chi=\left(\begin{array}[]{c}\chi^{0}\\\ \chi^{-}\\\
\chi^{\prime
0}\end{array}\right)\sim(1,3,-1/3),\quad\langle\chi\rangle=\left(\begin{array}[]{c}0\\\
0\\\ v_{\chi}\end{array}\right),$ (21)
$\displaystyle\rho=\left(\begin{array}[]{c}\rho^{+}\\\ \rho^{0}\\\
\rho^{\prime+}\end{array}\right)\sim(1,3,2/3),\quad\quad\langle\rho\rangle=\left(\begin{array}[]{c}0\\\
v_{\rho}\\\ 0\end{array}\right),$ (28)
$\displaystyle\eta=\left(\begin{array}[]{c}\eta^{0}\\\ \eta^{-}\\\
\eta^{\prime
0}\end{array}\right)\sim(1,3,-1/3),\quad\langle\eta\rangle=\left(\begin{array}[]{c}v_{\eta}\\\
0\\\ 0\end{array}\right).$ (35)

where $v_{\chi}$, $v_{\rho}$ and $v_{\eta}$ correspond to the vacuum
expectation values (VEVs) defining a two-step spontaneous symmetry breaking
(SSB)

$\displaystyle\textbf{SU(3)}_{\textbf{L}}\times\textbf{U(1)}_{\textbf{X}}\xrightarrow{\langle\chi\rangle}\textbf{SU(2)}_{\textbf{L}}\times\textbf{U(1)}_{\textbf{Y}}\xrightarrow{\langle\eta\rangle,\langle\rho\rangle}\textbf{U(1)}_{\textbf{Q}}.$

Table 2: Implemented benchmark sets (BMs) corresponding to mass values of the heavy exotic quarks $q^{\prime}$ and the heavy neutral lepton $N$ in the 3-3-1 RHN and LHN models. Model | 3-3-1 LHN | 3-3-1 RHN
---|---|---
Mass |  $M_{q^{\prime}}$ [TeV] |  $M_{N}$ [TeV] |  $M_{q^{\prime}}$ [TeV]
BM1 | $10$ | $10$ | $10$
BM2 | $1$ | $10$ | $1$
BM3 | $1.5$ | $10$ | $1.5$
BM4 | $2$ | $10$ | $2$
BM5 | $2$ | $2$ | N/A
BM6 | $2$ | $2.5$ | N/A
BM7 | $2$ | $4$ | N/A
BM8 | $1$ | $1$ | N/A
BM9 | $0.5$ | $10$ | N/A
BM10 | $10$ | $0.5$ | N/A

Throughout, we adopt the decoupling limit where the energy scale of SSB of the
3-3-1 symmetry is much larger than the electroweak one, i.e., $v_{\chi}\gg
v_{\eta},v_{\rho}$. As a result of the large gauge group new gauge bosons
arise $W^{\prime\pm}$, $U^{0}$, and a $Z^{\prime}$). Moreover, new neutral and
charged scalars also appear Mizukoshi  _et al._ (2011). We will discuss the
importance of scalars to our findings soon, but we anticipate that they are
not relevant. Hence, we concentrate on the gauge bosons, whose masses are
found to be,

$\displaystyle m_{Z^{\prime}}^{2}$
$\displaystyle=\frac{g^{2}}{\left(3-4s_{W}^{2}\right)}\left(c_{W}^{2}v_{\chi}^{2}+\frac{v_{\rho}^{2}+v_{\eta}^{2}\left(1-2s_{W}^{2}\right)^{2}}{4c_{W}^{2}}\right),$
(36) $\displaystyle m_{W^{\prime}}^{2}$
$\displaystyle=\frac{g^{2}}{4}\left(v_{\eta}^{2}+v_{\chi}^{2}\right),\quad
m_{U^{0}}^{2}=\frac{g^{2}}{4}\left(v_{\rho}^{2}+v_{\chi}^{2}\right),$

where $v^{2}=v_{\eta}^{2}+v_{\rho}^{2}\simeq 246$ GeV, $g$ is the $SU(2)_{L}$
gauge coupling, ${c}_{W}\equiv cos\theta_{W}$, ${s}_{W}\equiv sin\theta_{W}$,
with $\theta_{W}$ being the Weinberg angle. From Eq.(36), one clearly sees
that the gauge boson masses are determined by $v_{\chi}$. Hence, once we set a
bound on the $Z^{\prime}$ mass, it can be translated into a constraint on the
$W^{\prime}$ and $U^{0}$ masses as well.

We derive this limit using the high-mass dilepton resonances searches at the
ATLAS detector with $\sqrt{s}=13$ TeV center-of-mass-energy, and later
estimate the future collider bounds. In that regard, the main ingredient is
the neutral current that reads,

$\mathcal{L}^{NC}_{Z^{\prime}ff}=\frac{g}{2c_{W}}\bar{f}\gamma^{\mu}\left[\text{g}_{\text{V}}^{(f)}+\gamma_{5}\text{g}_{\text{A}}^{(f)}\right]fZ_{\mu}^{\prime},$
(37)

where $\text{g}_{\text{V}}^{(f)}$ and ($\text{g}_{\text{A}}^{(f)}$) are the
vector (axial) coupling constant of fermions $f=\ell,{N},{q}^{\prime}$ with
$Z^{\prime}$ (see Table 1). The branching ratio of the $Z^{\prime}$ boson in
two leptons is defined as,

$\operatorname{Br}\left(Z^{\prime}\rightarrow\ell\bar{\ell}\right)=\frac{\Gamma\left(Z^{\prime}\rightarrow\ell\bar{\ell}\right)}{\Gamma_{Z^{\prime}}},$
(38)

where $\Gamma_{Z^{\prime}}$ is the total width,

$\displaystyle\Gamma_{Z^{\prime}}=\sum_{X}\Gamma\left(Z^{\prime}\rightarrow
2X\right),$ (39)

being $X$ the SM particles and new particles in 3-3-1 models, while
$\Gamma\left(Z^{\prime}\rightarrow\ell\bar{\ell}\right)$ is the partial decay
width into dileptons, $\ell=e,\mu$. Following Group  _et al._ (2020) the decay
into tau pairs is not considered for presenting hadronic decays.

$\Gamma\left(Z^{\prime}\rightarrow\ell\overline{\ell}\right)\simeq\cfrac{g^{2}}{{c_{W}}^{2}}\cfrac{\text{m}_{\text{Z}^{\prime}}}{24\pi}\left[\left(\text{g}_{\text{V}}^{\ell}\right)^{2}+\left(\text{g}_{\text{A}}^{\ell}\right)^{2}\right],$
(40)

being $\text{g}_{\text{V}}$ and $\text{g}_{\text{A}}$ the coupling constants
defined in Eq. (37), $g$ the $SU(2)_{L}$ gauge coupling, and $m_{Z^{\prime}}$
the $Z^{\prime}$ mass. For the calculation of the total width, we take into
account all possible decays, including new gauge bosons, scalars and exotic
quarks. The analytical expressions are presented, but we obtained the decay
widths using Calchep.

##  III  Data and signal output

We carry out our collider simulation with Madgraph5 Alwall  _et al._ (2014);
Frederix  _et al._ (2018), and compute the decay with CalcHEP Belyaev  _et
al._ (2013); Pukhov  _et al._ (9 20). We compute the $pp\to
Z^{\prime}\to\ell\bar{\ell}$ at $\sqrt{s}=13$ TeV, with $\ell=e,\mu$ and
compare our findings with the public results from ATLAS Collaboration
Collaboration  _et al._ (2019). We generate the Monte Carlo events to simulate
the cross-section of the Drell-Yan process using the parton distribution
function (PDF) NNPDF23LO Carrazza  _et al._ (2013). We require two opposite
charge leptons in the event and the following kinematic cuts in order to
compare our results with the ATLAS Collaboration data: $p_{\mathrm{T}}>30$
GeV111Transverse momentum is the component of the momentum that is
perpendicular to the beam axis. and $|\eta|<2.5$ 222The pseudo-rapidity $\eta$
is defined as, $\eta=-\ln\tan\frac{\theta}{2}$ , where $\theta$ is the polar
angle between the particle’s linear momentum and the positive direction of the
beam axis..

Instead of considering only $Z^{\prime}$ interactions to fermions as done in
previous works, we fully implemented the model in LANHEP Semenov (1998, 2009,
2016), SARAH-HEP Staub (3 20); Vicente (2017); Staub (2015) and Feynrules
Alloul  _et al._ (2014, 9 20) to generate the output files for CalcHeP and
Madgraph5, respectively. This is important because additional exotic decays of
the $Z^{\prime}$ gauge boson can significantly weaken the lower mass bounds
based on dilepton data. The more sizeable decay channels are added to the
total width, the smaller is the branching ratio into dileptons. Consequently,
weaker limits are found. We investigate the importance of each of the possible
new decay channels by considering several benchmark models.

###  III.1 Importance of exotic $Z^{\prime}$ decays and benchmarks models

In the 3-3-1 models explored here, the $Z^{\prime}$ might decay into SM
fermions, new scalars, new gauge bosons ($W^{\prime\pm}$,
$U^{0},U^{0\dagger}$), and invisibly ($N_{i}$). One of the three heavy
fermions is cosmologically stable, and is rendered as dark matter candidate.
The other two are long-lived. Suppose $N_{1}$ is the lightest one, for the
sake of the argument. Because of a $Z_{2}$ symmetry where $N_{i}\rightarrow-
N_{i}$, $N_{2}$ might decay into $N_{1}$ via the $W^{\prime}$ gauge boson, but
this decay width is suppressed for two reasons: the $W^{\prime}$ is heavy, and
second, the entries of the mixing matrix involving $N_{1}-N_{2}-N_{3}$ should
also be small otherwise one could observe lepton flavor violation processes as
explored in Arcadi  _et al._ (2018b). As far as collider searches are
concerned, without worrying about particular details of the masses and mixing
matrices, we can safely take the neutral fermions as stable particle, and thus
considered as invisible decay. We have checked that decays into new scalars
and exotic gauge bosons are either very suppressed or kinematically
prohibited. Hence, the only relevant new decay channels beyond the SM, are
those involving exotic quarks, and neutral fermions ($N_{i}$). That said, we
investigate several benchmark models varying the masses of the decay products
to quantify their importance in the derivation of lower mass bounds.

###  III.2 Procedure

We compute the $\sigma\times\operatorname{BR}(\ell\overline{\ell})$ at the LHC
using the high-energy physics tools aforementioned for each benchmark model
(BM) in Table 2, and later we compare with ATLAS data. Furthermore, we use
these results to obtain, through Collider Reach $(\beta)$ Salam and Weiler (9
21), new bounds for HL-HE, and FCC LHC. Collider Reach ($\beta$) takes the
input bound of $m_{Z^{\prime}}$ obtained in the first step at a certain
center-of-mass-energy and luminosity, and forecast new bounds for a different
collider configurations including center-of-mass-energy and luminosity. In our
work, we are interested in the High-Luminosity, High-Energy Cepeda  _et al._
(2019), and the FCC proton-proton setups Abada  _et al._ (2019).

##  IV Results and discussions

####  Branching ratios.

The partial width of $Z^{\prime}$ into leptons is the same for both 3-3-1
models, as they have identical interactions (see Eq. (37), Eq. (40) and Table
1), but the branching ratio into charged leptons can be quite different from
each other. We highlight that the key difference of the 3-3-1 RHN is the
presence of heavy neutral fermions, $N_{i}$. Therefore, when decays into
$N_{i}$ pairs are inaccessible, both models are indistinguishable as far as
$Z^{\prime}$ searches are concerned. We have shown in Figs. 1, 2, 3, and 4 the
branching ratio $BR(Z^{\prime}\to\ell\bar{\ell})$ as a function of
$m_{Z^{\prime}}$ for several BM in the 3-3-1 RHN and LHN models. We notice
that in both models the value of the branching ratio is less than 2% and 1.7%,
respectively. In Fig. 1, for the 3-3-1 LHN, the BM 2-3-4 lead to a drop near
$2000$ GeV, $3000$ GeV, and $4000$ GeV, respectively. This behavior is caused
by $Z^{\prime}$ decay into exotic quark. In contrast, the BM1 does not
experience such behavior since the new exotic quarks masses are fixed at 10
TeV.

Figure 1:  Branching ratio for the $Z^{\prime}$ decay into dilepton channel as a function to $m_{Z^{\prime}}$ for the benchmark sets BM1, BM2, BM3, and BM4 of the 3-3-1 LHN model.  Figure 2:  Branching ratio for the $Z^{\prime}$ decay into dilepton channel as a function of $m_{Z^{\prime}}$ for the benchmark sets BM 5, 6 and 7 of the the 3-3-1 LHN model. Figure 3: Branching ratio for the $Z^{\prime}$ decay into dilepton channel as a function to $m_{Z^{\prime}}$ for the benchmark sets BM 8, 9, and 10 the 3-3-1 LHN model. Figure 4: Branching ratio for the $Z^{\prime}$ decay into dilepton channel as a function to $m_{Z^{\prime}}$ for the 3-3-1 RHN model. Table 3: $m_{Z^{\prime}}$ lower bounds taking into account the dilepton signal data at the LHC Collaboration  _et al._ (2019) and the theoretical signal production from Fig. 6 for the 3-3-1 RHN and LHN models.  Model | BM |  $m_{Z^{\prime}}$[GeV]
---|---|---
3-3-1 RHN | BM 1 333The lower bounds of BM 1 for the 3-3-1 RHN model are equivalent to those of BM 10 in the 3-3-1 LHN model. | $4052$
BM 2 | $3960$
BM 3 444The lower bound of BM 3 for the 3-3-1 RHN model i equivalent to those of BM 9 in the 3-3-1 LHN model. | $3989$
BM 4 | $4040$
3-3-1 LHN | BM 1 | $4132$
BM 2 | $4013$
BM 3 | $4060$
BM 4, 6 and 7 | $4118$
BM 5 | $4094$
BM 8 | $3950$

,

Figure 5: Parameter space $m_{Z^{\prime}}\times m_{N_{1}}$ plane that explains
the thermal relic density. BM models are indicated in the figure, as well the
regions that lead to overabundant and underabundant dark matter. In the gray
region, $N_{1}$ is not stable.

(a) $\sigma_{\text{fid.}}\times\operatorname{BR}(\ell\bar{\ell})$ vs
$m_{Z^{\prime}}$ for BM sets 1, 2, and 3 in the 3-3-1 RHN model.  (b)
$\sigma_{\text{fid.}}\times\operatorname{BR}(\ell\bar{\ell})$ vs
$m_{Z^{\prime}}$ for BM sets 1, 2, and 3 in the 3-3-1 LHN model.  (c)
$\sigma_{\text{fid.}}\times\operatorname{BR}(\ell\bar{\ell})$ vs
$m_{Z^{\prime}}$ for BM sets 5, 6 and 7 in the 3-3-1 LHN model.  (d)
$\sigma_{\text{fid.}}\times\operatorname{BR}(\ell\bar{\ell})$ vs
$m_{Z^{\prime}}$ for BM sets 8, 9 and 10 in the 3-3-1 LHN model.

Figure 6: Solid red and dashed black lines symbolize
$\sigma_{\text{fid.}}\times\operatorname{BR}(\ell\bar{\ell})$ upper limits
observed and expected at $95\%\mathrm{CL}$ as a function of $Z^{\prime}$ mass
for the $10\%$ width signals for the dilepton channel
$Z^{\prime}\rightarrow\ell\bar{\ell}$ in the ATLAS experiment at a center of
mass energy 13 TeV (ATLAS Collaboration Collaboration  _et al._ (2019)). The
solid yellowgreen, dash-dot blue, and black dotted lines represent the
theoretical production $\sigma{(pp\to Z^{\prime})}\times
BR{(Z^{\prime}\to\ell\bar{\ell})}$ generated using MadGraph5 and CalcHEP for
several benchmark sets for the 3-3-1 RHN and LHN models. We assume different
masses for the new exotic quarks and heavy neutral lepton (see Table 2). The
lower mass bounds on the $Z^{\prime}$ obtained can be seen in Table 3.

In the same way, for the 3-3-1 LHN model, we observed some substantial
deacrease in the branching ratio into charged leptons for the BM sets 5, 6,
and 7 when $m_{Z^{\prime}}=4000$ GeV (see Fig. 2). It has to do with the
exotic quarks at $m_{q^{\prime}}=2000$ GeV. The importance of decays into
heavy neutral fermions can be seen in BM 6, which leads to a significant
decrease in the branching ratio when $m_{Z^{\prime}}\sim 5$ TeV, as we fixed
$m_{Ni}=2.5$ TeV. In a similar vein, the behavior seen in Fig. 3 can be
explained.

For the 3-3-1 RHN model, we observe that the behavior for the branching ratio
is similar to the 3-3-1 LHN model since by performing the same variations on
the exotic quark masses, we have the same decrease in the branching ratio (see
Fig. 4). However, the size of the branching ratio into charged leptons is
smaller because the $Z^{\prime}$ can always decay into right-handed neutrinos,
which are assumed to have keV masses Dias  _et al._ (2005).

####  Signal production

As explained above, the theoretical production of an $Z^{\prime}$ at the LHC
decaying into dileptons was generated using MadGraph5 and CalcHEP. To compare
the theoretical signal for the dilepton channel $Z^{\prime}$ with ATLAS
collaboration data presented in Fig. 3(a) in Collaboration  _et al._ (2019),
we plot $\sigma{(pp\to Z^{\prime})}\times BR{(Z^{\prime}\to\ell\bar{\ell})}$
as a function of $m_{Z^{\prime}}$ for the 3-3-1 RHN and LHN models, as seen in
Fig. 6. For the $Z^{\prime}$ mass we take different values in the interval of
200 GeV$<m_{Z^{\prime}}<$6000 GeV with steps of $40$ GeV. The lower mass
bounds on the $Z^{\prime}$ are obtained by considering the intersection of the
solid yellowgreen, dash-dot blue, and black dotted lines with the red solid
curve in Figs. 6(a), 6(b), 6(c) and 6(d), and these results are summarized in
Table 3.

####  HE-HL and FCC LHC.

After obtaining the lower bounds of ${m}_{{Z}^{\prime}}$ for the LHC (Table
3), we use these results as input for Collider Reach (${\beta}$) with the PDF
MMHTMMHT2014nnlo68cl Harland-Lang  _et al._ (2015), and forecast HL-LHC, HE-
LHC and FCC LHC limits. These new bounds are displayed in Table 4, for which
we use different values for the center-of-mass energy $\sqrt{s}=13$ TeV, $14$
TeV, $27$ TeV, and $100$ TeV, and for the integral luminosity $L_{int}=$
139$fb^{-1}$, 300$fb^{-1}$, 500$fb^{-1}$, and 3000$fb^{-1}$, respectively. In
summary, BM 1 and 3, in the 3-3-1 RHN model, coincide with BM 10 and 9 in the
3-3-1 LHN model, respectively. Moreover, BM 6-7 present a similar bound to BM
4 in the 3-3-1 LHN model, and these results are easily justified by the
presence or not of exotic $Z^{\prime}$ decays as discussed previously.

Having in mind that 3-3-1 LHN features heavy neutral fermions that can be dark
matter candidates, one may wonder if the current and projected collider bounds
derived in our work preclude the existence of a plausible dark matter
candidate in the model. We address this concern below.

##  V  dark matter

####  Thermal Production

The lightest of the neutral fermions, let us say $N_{1}$, can be a viable dark
matter candidate due to a $Z_{2}$ symmetry Mizukoshi  _et al._ (2011); Profumo
and Queiroz (2014a). The dark matter abundance is governed by s-channel
annihilations into SM fermions mediated by the $Z^{\prime}$ gauge field. The
$Z^{\prime}$ interactions with SM fermions are set by the gauge symmetry,
which features a fixed gauge coupling. In other words, the dark matter
abundance is governed by two parameters only, the dark matter and the
$Z^{\prime}$ masses Mizukoshi  _et al._ (2011); Oliveira and Pires (2021).
Furthermore, the dark matter scattering off nucleon occurs through a t-channel
$Z^{\prime}$ exchange. Hence, the dark matter phenomenology is quite
predictable once we fix the $Z^{\prime}$ mass.

In the context of thermal production, where the production of dark matter
occurs in the usual standard freeze-out, the curve that yields the correct
relic density, $\Omega h^{2}=0.11$ Aghanim  _et al._ (2020), is shown in Fig.
5 with solid blue curves. We also exhibit the region where the dark matter is
overabundant and underabundant. Considering current LHC bound, we observe that
for the BM 5, $m_{N}=2$ TeV, we can reproduce the correct relic density for
$m_{Z^{\prime}}\sim 4.6$ TeV, which safely obeys LHC constraints. However,
with $500fb^{-1}$ of data, LHC will already be able to probe this scenario
(see Table 4). For the BM 6, the right relic density is found for
$m_{Z^{\prime}}\sim 4.7$ and $5.7$ TeV, whereas for BM7 a $5.8$ TeV, $7.8$ TeV
and $8.8$ TeV $Z^{\prime}$ could yield the correct relic density. In
particular, for the BM 1, only HE-LHC has the potential to fully probe this
scenario, as it can exclude $Z^{\prime}$ masses up to $9.8$ TeV. There are
many interesting scenarios to be explored, but one can solidly see the
importance of orthogonal and complementary searches for new physics. Our
study, clearly shows that setting aside one’s theoretical prejudice for multi-
TeV mediators, vanilla thermal dark matter models, such as the one present in
the 3-3-1 LHN model, is fully consistent with current LHC data, and only the
next generation of colliders will be able to close the few TeV dark matter
particle window.

####  Direct Detection

The neutral fermion can indeed leave signals at direct detection experiments
through t-channel $Z^{\prime}$ exchange, but the limits are weak compared to
those stemming from collider searches. This is a typical feature of vector
mediator models that have sizeable couplings to fermions Alves  _et al._
(2014, 2015). The current and projected direct detection bounds, from XENON or
PANDAX collaborations Aprile  _et al._ (2018); Meng  _et al._ (2021); Aprile
_et al._ (2020) are significantly surpassed by current LHC data, mainly when
are put into perspective with future colliders. For this reason, we decided
not to show them in Fig. 5.

Table 4: $m_{Z^{\prime}}$ lower bounds for all benchmark sets considered in this work at HE-HL and FCC LHC by increasing the center-of-mass energy ($\sqrt{s}$) from 13TeV until 100TeV, and integral luminosity ($L_{int}$) from 139 $fb^{-1}$ to 3000 $fb^{-1}$, for the 3-3-1 RHN and LHN models. Values of $m_{Z^{\prime}}$ for HE-HL LHC appear between the fourth and sixth columns of the table, whereas for the FCC LHC, the $m_{Z^{\prime}}$ bounds are shown in the seventh column, when increasing the luminosity (column three) . Model | Benchmark (BM) sets | $L_{int}[fb^{-1}]$ |  $m_{Z^{\prime}}$[TeV]-13TeV |  $m_{Z^{\prime}}$[TeV]-14TeV |  $m_{Z^{\prime}}$[TeV]- 27TeV |  $m_{Z^{\prime}}$[TeV]-100TeV
---|---|---|---|---|---|---
3-3-1 RHN |  | 139 | $4.052$ | $4.288$ | $6.987$ | $17.180$
| 300 | $4.390$ | $4.651$ | $7.675$ | $19.447$
BM 1 555The lower bounds of BM 1 for the 3-3-1 RHN model are equivalent to those of BM 10 in the 3-3-1 LHN model. | 500 | $4.613$ | $4.892$ | $8.136$ | $21.006$
| 1000 | $4.916$ | $5.217$ | $8.763$ | $23.175$
| 3000 | $5.388$ | $5.727$ | $9.755$ | $26.711$
| 139 | $3.960$ | $4.189$ | $6.801$ | $16.548$
| 300 | $4.298$ | $4.552$ | $7.487$ | $18.821$
BM 2 | 500 | $4.521$ | $4.793$ | $7.947$ | $20.363$
| 1000 | $4.825$ | $5.119$ | $8.574$ | $22.514$
| 3000 | $5.298$ | $4.699$ | $9.566$ | $26.030$
| 139 | $3.989$ | $4.220$ | $6.860$ | $16.769$
| 300 | $4.327$ | $4.583$ | $7.547$ | $19.016$
BM 3 666The lower bounds of BM 3 for the 3-3-1 RHN model are equivalent to those of BM 9 in the 3-3-1 LHN model. | 500 | $4.550$ | $4.824$ | $8.006$ | $20.564$
| 1000 | $4.853$ | $5.149$ | $8.633$ | $22.721$
| 3000 | $5.326$ | $5.661$ | $9.626$ | $26.244$
| 139 | $4.040$ | $4.275$ | $6.963$ | $17.101$
| 300 | $4.378$ | $4.638$ | $7.651$ | $19.364$
BM 4 | 500 | $4.601$ | $4.879$ | $8.111$ | $20.921$
| 1000 | $4.904$ | $5.204$ | $8.739$ | $23.089$
| 3000 | $5.377$ | $5.715$ | $9.731$ | $26.652$
3-3-1 LHN |  | 139 | $4.132$ | $4.374$ | $7.149$ | $17.709$
| 300 | $4.470$ | $4.737$ | $7.839$ | $19.990$
BM 1 | 500 | $4.693$ | $4.978$ | $8.301$ | $21.571$
| 1000 | $4.995$ | $5.303$ | $8.928$ | $23.755$
| 3000 | $5.467$ | $5.812$ | $9.920$ | $27.306$
| 139 | $4.013$ | $4.246$ | $6.908$ | $16.924$
| 300 | $4.351$ | $4.609$ | $7.596$ | $19.197$
BM 2 | 500 | $4.574$ | $4.850$ | $8.056$ | $20.731$
| 1000 | $4.877$ | $5.175$ | $8.683$ | $22.894$
| 3000 | $5.350$ | $5.686$ | $9.675$ | $26.421$
| 139 | $4.060$ | $4.297$ | $7.003$ | $17.233$
| 300 | $4.398$ | $4.660$ | $7.692$ | $19.502$
BM 3 | 500 | $4.621$ | $4.901$ | $8.153$ | $21.062$
| 1000 | $4.924$ | $5.225$ | $8.780$ | $23.233$
| 3000 | $5.396$ | $5.736$ | $9.772$ | $26.770$
| 139 | $4.118$ | $4.359$ | $7.121$ | $17.616$
| 300 | $4.456$ | $4.722$ | $7.811$ | $19.902$
BM 4, 6, and 7 | 500 | $4.679$ | $4.963$ | $8.272$ | $21.472$
| 1000 | $4.981$ | $5.288$ | $8.900$ | $23.654$
| 3000 | $5.453$ | $5.797$ | $9.891$ | $27.202$
| 139 | $4.094$ | $4.333$ | $7.072$ | $17.457$
| 300 | $4.432$ | $4.696$ | $7.761$ | $19.736$
BM 5 | 500 | $4.655$ | $4.937$ | $8.223$ | $21.302$
| 1000 | $4.958$ | $5.262$ | $8.850$ | $23.479$
| 3000 | $5.430$ | $5.772$ | $9.842$ | $27.023$
| 139 | $3.950$ | $4.178$ | $6.781$ | $16.520$
| 300 | $4.288$ | $4.541$ | $7.467$ | $18.753$
BM 8 | 500 | $4.511$ | $4.782$ | $7.926$ | $20.294$
| 1000 | $4.815$ | $5.108$ | $8.553$ | $22.443$
| 3000 | $5.289$ | $5.620$ | $9.546$ | $25.956$

##  VI  conclusions

In this work, we derived LHC bounds on two different 3-3-1 models, namely
3-3-1 RHN and 3-3-1 LHN. We assessed the impact of overlooked exotic
$Z^{\prime}$ decays in the derivation of lower mass limits using dilepton
data. Later, we obtained solid lower mass bounds that range from $3.9$ TeV to
$4.1$ TeV, significantly weaker than previous studies. We also forecasted HL-
LHC, HE-LHC and FCC limits, and put our results into perspective with dark
matter phenomenology to conclude that one could successfully accommodate a few
TeV thermal dark matter candidate in agreement with direct detection and
collider bounds. Our main results are summarized in Table 4.

###### Acknowledgements.

FSQ thanks Universidad Tecnica Frederico Santa Maria for the hospitality
during the final states of this work. We thank Alfonso Zerwekh, Antonio
Carcamo, and Carlos Pires for discussions. Y.S.V acknowledges the financial
support of CAPES under Grant No. 88882.375870/2019-01. Y.M.O.T. acknowledges
financial support from CAPES under grants 88887.485509 / 2020-00. FSQ is
supported by ICTP-SAIFR FAPESP grant 2016/01343-7, CNPq grants 303817/2018-6
and 421952/2018 – 0, and the Serrapilheira Foundation (grant number Serra -
1912 – 31613). LCD thanks Simons Foundation (Award Number: 884966, AF) for the
financial support. S.K. is supported by ANID PIA/APOYO AFB180002 (Chile) and
by ANID FONDECYT (Chile) No. 1190845. F.S.Q. and S.K. acknowledge support from
ANID$-$Millennium Program$-$ICN2019_044 (Chile). AA is supported by CNPq
(307317/2021-8). AA and FSQ also acknowledge support from FAPESP
(2021/01089-1) grant.

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