# Production of Tensor Glueball in Ultraperipheral $pp$, $AuAu$ and $PbPb$
Collisions

Wei Kou<EMAIL_ADDRESS>Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  Xurong Chen<EMAIL_ADDRESS>(Corresponding author) Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  University of Chinese Academy of Sciences,
Beijing 100049, China  Guangdong Provincial Key Laboratory of Nuclear Science,
Institute of Quantum Matter, South China Normal University, Guangzhou 510006,
China

###### Abstract

In this work, we discuss the production of a tensor glueball in the proton-
proton and nuclei-nuclei ultraperipheral collisions (UPCs). The cross section
is calculated in the equivalent photon approximation (EPA). The total UPCs
cross section is related to the matrix elements of
$\gamma\gamma\to\pi^{0}G_{2}$ process, which can be described as in terms of
neutral pion and tensor glueball $G_{2}$’s distribution amplitudes. The
predictions for the rapidity distributions of UPCs cross sections and total
cross sections are presented. The theoretical uncertainty of the whole
calculation is considered. We consider the decay modes of the final-state
tensor glueball and estimate the corresponding number of events. We estimate
that there are about 10 tensor glueball $G_{2}\to\phi\phi$ signal events in
$PbPb$ collisions at the LHC. We also argue that the lead-lead UPC experiments
at future HE-LHC and FCC experiments will provide larger opportunities to
measure tensor glueballs.

###### pacs:

24.85.+p, 13.60.Hb, 13.85.Qk

##  I Introduction

The photon-induced processes at RHIC, Tevatron and LHC have motivated a series
of studies, which test the validity of the Standard Model and deepen our
understanding of the Beyond Standard Model (BSM) physics. The analysis of
different final states produced in $\gamma\gamma$ and $\gamma h$ interactions
at hadronic collisions are fully discussed e.g. in the review Akiba  _et al._
(2016). The hadronic states made up only from gluons are known as glueballs
Fritzsch and Gell-Mann (1972); Fritzsch  _et al._ (1973); Fritzsch and
Minkowski (1975). The first study to give the glueball mass came from bag
model DeGrand  _et al._ (1975). The spectrum of glueball states have been
determined by lattice QCD calculations, see Refs. Bali  _et al._ (1993);
Morningstar and Peardon (1999); Chen  _et al._ (2006); Richards  _et al._
(2010); Gregory  _et al._ (2012); Athenodorou and Teper (2020). Because of the
special nature of glueballs under SU(3) symmetry, which have the same quantum
number as a meson state composed of quark-antiquark, finding glueball in
experiments is a challenge. The analytical approximation of QCD predicts a
scalar glueball at 1850 $\sim$ 1980 MeV Szczepaniak and Swanson (2003); Huber
_et al._ (2020); Rinaldi and Vento (2021). Unlike scalar glueballs, which have
some scalar candidates Klempt and Zaitsev (2007); Crede and Meyer (2009); Ochs
(2013), much less is even known about tensor glueballs. The mass spectra of
the lowest $2^{++}$ tensor glueballs were analyzed from the lattice QCD, and
these works give the masses around 1900 $\sim$ 2600 MeV Klempt (2022). We note
there are some experimental evidences that such state have been seen in
various processes Etkin  _et al._ (1985); Uehara  _et al._ (2013); Ablikim
_et al._ (2016); Klempt (2022). In particular, the recent work has shown the
related analysis of scalar glueball as well as tensor glueball Klempt (2022).

In hadron-hadron collisions, ultraperipheral collisions (UPCs) are important
in the theoretical and experimental investigations Baur  _et al._ (2002);
Bertulani  _et al._ (2005); Baltz (2008). In hadron–hadron UPCs, the impact
parameter between the two hadrons is larger than the sum of the two hadron’s
radius. Consequently, strong interaction between the hadrons is suppressed for
the large distance. On the other hand, photon can be emitted from the hadrons
at high energy, thus, the photon can interact with the hadron or the photon
emitted from the other hadron. There has been a lot of published works on the
topic of glueball production by UPCs process Schramm (1999); Machado and da
Silva (2011); Klusek-Gawenda and Szczurek (2013). These studies include the
calculation of cross sections for scalar and tensor glueballs, and some of
these studies have estimated the production of glueballs in peripheral
collisions by strong double diffractive scattering, i.e. Pomeron-Pomeron (PP)
exchange Schramm (1999); Machado and da Silva (2011). In the present work, we
consider the glueball production in two-photon fusion which can be calculated
using the narrow-resonance approximation Schramm (1999); Machado and da Silva
(2011).

The production cross section in our method is computed using the QCD
factorization approach Lepage and Brodsky (1980); Brodsky and Lepage (1981);
Efremov and Radyushkin (1980); Kivel and Vanderhaeghen (2018). In Ref. Kivel
and Vanderhaeghen (2018), the authors studied the production of tensor
$2^{++}$ glueballs in two-photon collisions at high momentum transfer. One
considered that the coupling of quarks and gluons to the final mesonic states
is described by the distribution amplitudes (DAs) describing the momentum
fraction distribution of partons at zero transverse separation in a two-
particle Fock state Kivel and Vanderhaeghen (2018). We review the details of
this part in the following section.

Based on the above discussion, tensor glueballs can be considered to be
produced during the UPCs process and a significant number of events may be
observed in future heavy ions collision experiments. The calculation of
$\gamma\gamma\to\pi^{0}G_{2}$ process and the corresponding UPCs process are
reviewed in Sec. II. The results for the total cross sections and rapidity
distributions are shown in Sec. III. Summary and outlook are presented in the
final section.

##  II $\gamma\gamma\to\pi^{0}G_{2}$ to Ultraperipheral Collisions

As a starting point for the core, one can recall the two-photon collisions in
Ref. Kivel and Vanderhaeghen (2018) that produce $\pi^{0}$ and tensor glueball
$G_{2}$. The production amplitude for the $\gamma\gamma\to\pi^{0}G_{2}$
process should be described in terms of the helicity amplitudes Kivel and
Vanderhaeghen (2018)

$\displaystyle iA_{\pm\pm}=\varepsilon_{1\mu}(\pm)\varepsilon_{2\nu}(\pm)$ (1)
$\displaystyle\times\int d^{4}xe^{-i\left(q_{1}x\right)}\left\langle
G_{2}(p),\pi^{0}(k)\left|T\left\\{J_{\mathrm{em}}^{\mu}(x),J_{\mathrm{em}}^{\nu}(0)\right\\}\right|0\right\rangle,$

where $p$ and $k$ denote the momenta of glueball $G_{2}$ and $\pi^{0}$.
$\varepsilon_{1\mu}$ and $\varepsilon_{2\nu}$ are the polarization vectors of
initial photons. The electromagnetic currents are described as
$J_{\mathrm{em}}^{\mu}(x)$ and $J_{\mathrm{em}}^{\nu}(0)$ (see FIG. 1). From
the helicity amplitudes, the differential cross section is written as Budnev
_et al._ (1975)

$\frac{d\sigma_{\gamma\gamma}\left[\pi^{0}G_{2}\right]}{d\cos\theta}=\frac{1}{64\pi}\frac{W_{\gamma\gamma}+m^{2}}{W_{\gamma\gamma}^{2}}\left(\left|\overline{A_{++}}\right|^{2}+\left|\overline{A_{+-}}\right|^{2}\right),$
(2)

where $\theta$ is the scattering angle in the c.m.s., $W_{\gamma\gamma}$
denotes the $\gamma\gamma$ c.m.s. energy and one can set $|A_{++}|=|A_{--}|$
and, $|A_{-+}|=|A_{+-}|$ Kivel and Vanderhaeghen (2018). $m$ is the mass of
tensor glueball and we choose $m=2.3$ GeV ($f_{2}(2300)$ or $f_{2}(2340)$)
which have been recently observed by the Belle Uehara  _et al._ (2013) and
BESIII Ablikim  _et al._ (2016) collaborations are good candidates to be
tensor glueball. The sum over polarization of amplitudes are

$\left|\overline{A_{+\pm}}\right|^{2}=\sum_{\lambda=-2}^{2}A_{+\pm}(\lambda)A_{+\pm}^{*}(\lambda),$
(3)

where $\lambda$ is the polarization of the glueball. The authors in Ref. Kivel
and Vanderhaeghen (2018) also introduced light-like vectors and some
approximations. They considered two diagrams (see the internal parts in FIG.
1). The blobs in FIG. 1 denote the light-cone matrix elements which define the
DAs of the outgoing meson and glueball. One can finally get the polarized
amplitudes when the colliding photons have the same helicities. It can be
treated with the DAs (see eq. (15-19) in Ref. Kivel and Vanderhaeghen (2018)
for details). The total cross section of two-photons interactions is
$\sigma_{\gamma\gamma}(W_{\gamma\gamma})=\int
d\cos\theta\frac{d\sigma_{\gamma\gamma}(W_{\gamma\gamma})}{d\cos\theta}$.

Photon interactions lead to a wide variety of final states. They couple to all
charged particles, including leptons, quarks, and charged gauge bosons Hencken
_et al._ (1995); Baur  _et al._ (2002). The production of massive tensor
glueball $G_{2}$ in hadronic collisions is represented in FIG. 1. Both
incident hadrons can be considered as the source of photons.

Figure 1: Two typical diagrams of $h_{1}h_{2}\to h_{1}h_{2}\pi^{0}G_{2}$. The
blobs denote the distribution amplitudes discussed in Ref. Kivel and
Vanderhaeghen (2018). $h_{1},\ h_{2}$ represent protons or nucleus.

In UPCs, the total cross section of $h_{1}h_{2}\to h_{1}h_{2}\pi^{0}G_{2}$ can
be written in the well-known form form Baur and Ferreira Filho (1990)

$\displaystyle\sigma(h_{1}h_{2}\rightarrow
h_{1}h_{2}\pi^{0}G_{2};s_{h_{1}h_{2}})$ (4) $\displaystyle=\int
d^{2}\mathbf{b}_{1}d^{2}\mathbf{b}_{2}\theta(|\bm{b}_{1}-\mathbf{b}_{2}|-(R_{h_{1}}+R_{h_{2}}))\frac{d\omega_{1}}{\omega_{1}}\frac{d\omega_{2}}{\omega_{2}}$
$\displaystyle\times
N_{h_{1}}\left(\omega_{1},\bm{b}_{1}\right)N_{h_{2}}\left(\omega_{2},\mathbf{b}_{2}\right)\hat{\sigma}(\gamma\gamma\rightarrow\pi^{0}G_{2};4\omega_{1}\omega_{2}),$

where $\bm{b}_{1}$ and $\bm{b}_{2}$ represent the impact parameters,
$R_{h_{1(2)}}$ denotes the hadron or nucleus’ radius. For proton, the radius
$R_{p}=0.84$ fm, for nucleus we choose $R_{A}=r_{0}A^{\frac{1}{3}}$ with
$r_{0}=1.2$ fm. In particular, $N(\omega,\bm{b})$ is the equivalent photon
flux for a given photon energy $\omega$ and impact parameter $\bm{b}$, which
can be expressed in terms of the form factor $F(q^{2})$ for the equivalent
photon source as follows

$\displaystyle
N_{A}(\omega,b)=\frac{Z^{2}\alpha_{em}}{\pi^{2}}\frac{1}{b^{2}\omega}$ (5)
$\displaystyle{\left[\int
u^{2}J_{1}(u)F\left(\sqrt{\frac{(b\omega/\gamma_{L})^{2}+u^{2}}{b^{2}}}\right)\frac{1}{(b\omega/\gamma_{L})^{2}+u^{2}}\mathrm{~{}d}u\right]^{2}},$

where $Z$ is the proton number of nucleus and $\alpha_{em}$ denotes the fine
structure constant. $\gamma_{L}$ is the Lorentz factor which is discussed
below. $J_{n}(u)$ is the first kind Bessel function. Form factor is the
Fourier transform of charge distribution in the nucleus. If one assume
$\rho(r)$ is the spherical symmetric charge distribution, the form factor is a
function of photon virtuality $q^{2}$ Klusek-Gawenda and Szczurek (2010).

$F\left(q^{2}\right)=\int\frac{4\pi}{q}\rho(r)\sin(qr)rdr=1-\frac{q^{2}\left\langle
r^{2}\right\rangle}{3!}+\frac{q^{4}\left\langle
r^{4}\right\rangle}{5!}\cdots.$ (6)

When the form factor $F(q^{2})=1$, it means that the charge distribution is
point-like, the integration can be computed analytically. After analytical
integration, the equivalent photon flux is given as follows Bertulani  _et
al._ (2005)

$N(\omega,\bm{b})=\frac{Z^{2}\alpha_{\mathrm{em}}}{\pi^{2}}\frac{\omega}{\gamma_{L}^{2}}K_{1}^{2}(|b|\omega/\gamma_{L}).$
(7)

In the above formula, $K_{1}(x)$ is the second Bessel function. In this paper,
we denote the above equivalent photon flux as model-1.

Moreover, if the charge distribution is not point-like, the form factor can be
expressed in other functions. We refer Refs. Klusek-Gawenda and Szczurek
(2010); Hencken  _et al._ (1995) then the form factor function is given as

$F(q)=\frac{\Lambda^{2}}{\Lambda^{2}+q^{2}}$ (8)

with $\Lambda=0.088$ GeV for nucleus Goncalves (2013); Goncalves and Sauter
(2015); Gonçalves and Moreira (2018) and $\Lambda=0.71$ GeV for protons Xie
and Chen (2018). If the form factor function is taken as above formula, the
equivalent photon flux is written as

$N(\omega,\bm{b})=\frac{Z^{2}\alpha_{\mathrm{em}}}{\pi^{2}}\frac{1}{\omega}\left[\frac{\omega}{\gamma_{L}}K_{1}\left(b\frac{\omega}{\gamma_{L}}\right)-\sqrt{\frac{\omega^{2}}{\gamma_{L}^{2}}+\Lambda^{2}}K_{1}\left(b\sqrt{\frac{\omega^{2}}{\gamma_{L}^{2}}+\Lambda^{2}}\right)\right]^{2}.$
(9)

where $K_{1}$ is the modified Bessel function of the second kind. In this
paper, we denote the above equivalent photon flux as model-2. Lorentz factor
$\gamma_{L}$ is computed as $\gamma_{L}=\sqrt{s_{NN}}/2m_{N}$, $m_{N}$ is the
nucleon mass and the rapidity-type variable $Y$ defined through

$\displaystyle\omega_{1}=\frac{W_{\gamma\gamma}}{2}e^{Y},\quad\omega_{2}=\frac{W_{\gamma\gamma}}{2}e^{-Y},$
(10)
$\displaystyle\frac{d\omega_{1}}{\omega_{1}}\frac{d\omega_{2}}{\omega_{2}}=2\frac{dW_{\gamma\gamma}}{W_{\gamma\gamma}}dY.$

In next section, we consider the $pp$ and $PbPb$ collisions at LHC and $AuAu$
collisions at RHIC. We also predict the planned c.m.s. energies for the next
run of the LHC, as well for the future High-Energy LHC Abada  _et al._ (2019a)
and Future Circular Collider (FCC) Abada  _et al._ (2019b).

##  III Results and discussion

We first discuss the scattering matrix element of the $\gamma\gamma\to
G_{2}\pi^{0}$ process and the corresponding differential cross section
calculation. In Ref. Kivel and Vanderhaeghen (2018) the authors considered the
mesonic DAs of pion and tensor glueball. The DA of $2^{++}$ glueball is
treated as the tensor meson, see, e.g. Ref. Braun  _et al._ (2015). In general
case there are three light-cone matrix elements which define two gluon DAs and
one quark DA Kivel and Vanderhaeghen (2018). Constructing the formalization of
the glueball DA is more complicated, and it is not the focus of this work. We
give here only the simple conclusion in Ref. Kivel and Vanderhaeghen (2018) –
the DAs’ model of the glueball depends formally on several coupling constants,
$f_{g}^{S}$, $f_{g}^{T}$ and $f_{q}$. It is natural to assume that the
glueball state strongly overlaps with the gluon wave function and the value of
the gluon couplings are relatively large and can be of the same order as the
quark coupling $f_{q}\sim 100$ MeV for quark-antiquark mesons, i.e.
$f_{g}^{S}\sim f_{g}^{T}\sim 100$ MeV. After our tests, the helicity amplitude
$|A^{+-}|$ containing the coupling constants $f_{q}$ and $f_{g}^{S}$ is always
about two orders of magnitude smaller than the amplitude $|A^{++}|$ containing
$f_{g}^{T}$. One is therefore able to conclude that the contribution with
$|A^{+-}|$ does not provide significant numerical impact Kivel and
Vanderhaeghen (2018). Hence the cross section (2) is only sensitive to the
value of tensor coupling $f_{g}^{T}$. This can also be seen from the analysis
of the $G_{2}\to\phi\phi$ decay, which can be used for the identification of
the glueball state Ablikim  _et al._ (2016). Based on the above discussion, we
give an example of the calculation of the differential cross section (2). In
FIG. 2, we take the coupling $f_{g}^{T}=50$, 100, 150 MeV, respectively and
set $W_{\gamma\gamma}^{2}=16$ GeV2. The other parameters that appear in the
helicity amplitude we are consistent with Ref. Kivel and Vanderhaeghen (2018).

Figure 2: The example of cross section as a function of $\cos\theta$ at
$W_{\gamma\gamma}^{2}=16$ GeV2. The solid, dashed and dotted lines correspond
to $f_{g}^{T}=50$ MeV, 100 MeV and 150 MeV Kivel and Vanderhaeghen (2018).

Next we give the results for UPCs. Considering the parameter selection of
$\gamma\gamma\to G_{2}\pi^{0}$ cross section, we calculate the total cross
section of UPCs in two ways, model-1 and model-2 mentioned in the previous
section, respectively. We consider $AuAu$ collisions on RHIC and $pp$ and
$PbPb$ collisions on LHC, respectively. We go further and consider the planned
center-of-mass energies for the next run of the LHC, as well for the future
High-energy LHC Abada  _et al._ (2019a) and Future Circular Collider Abada
_et al._ (2019b). The results presented in FIGs. 3 to 5. It is seen that there
are only minor differences between model-1 and model-2. The selection of
different coupling constants in each sub-figure can be significantly different
and the trend is consistent with the FIG. 2. It is because the differential
cross section (2) is proportional to the square of $f_{g}^{T}$. Moreover, we
have that for larger energies the rapidity distribution increases and becomes
wider in rapidity. As expected from our previous discussions, the distribution
is symmetric for $pp$ and $PbPb$ collisions. If one compares the FIGs. 4 and
5, it is clear that the value of predictions for midrapidities ($Y\simeq 0$)
increase with the energy and are a factor $\simeq 10^{6}$ larger in $PbPb$
collisions than in $pp$ collisions. The enhancement in $PbPb$ collisions is
mainly associated to the $Z^{4}$ \- factor in the nuclear photon flux Coelho
and Gonçalves (2020).

In the Tables 1 and 2 we present our predictions for the total cross sections
and number of events, which includes the results of both model-1 and model-2
calculations (see Eqs. (7) and (9)). Here we set the previously mentioned
coupling constant $f_{g}^{T}=100$ MeV. Because of the low energy for RHIC, the
total cross section is small and not shown in text. For $pp$ and $PbPb$
collisions, the expected integrated luminosities we use to estimate the number
of events come from Refs. Abada  _et al._ (2019a, b) for the next planned run
of LHC, HE-LHC and FCC. Moreover, we will also take into account the tensor
glueball decay mode, with the associated branching ratio for the processes
$G_{2}\to\phi\phi$ being $1.74\%$ Yang  _et al._ (2013); Ablikim  _et al._
(2016). The uncertainty of the branching ratio is not considered. We need to
emphasize that there are fewer existing studies related to the branching ratio
of $G_{2}\to\phi\phi$, thus we try to combine the lattice calculations with
the experimental measurements presented by BESIII collaboration, which the
tensor glueball is treated as tensor meson $f_{2}(2340)$. In principle, with
fixed coupling $f_{g}^{T}$, both Tables 1 and 2 show the difference between
model-1 and model-2. Although the difference is small, one could see that the
calculation result of model-2 is smaller than that of model-1.

Some comments are in order. First, in our analysis we only consider one of the
possibilities of production of tensor glueball from two-photon collisions,
other photoproduction processes such as those mentioned above and the
“Pomeron-Pomeron exchange” are not considered in detail. These contributions
cannot be ignored and serve as background processes for UPCs. For a complete
discussion we suggest that readers refer to Refs. Schramm (1999); Machado and
da Silva (2011). Second, we provide some theoretical uncertainties about the
calculation of $\gamma\gamma\to G_{2}\pi^{0}$ matrix element and UPCs. In
addition to the uncertainties analysis already mentioned, we need to emphasize
other sources of theoretical uncertainties for double photon collisions Kivel
and Vanderhaeghen (2018), which are provided by various subleading
corrections. Besides, we also note the hadronic survival probabilities
mentioned in Ref. Shao and d’Enterria (2022), which corresponds to the
probability that both scattered protons do not dissociate due to secondary
soft hadronic interactions Dokshitzer  _et al._ (1987). These factors will
also modify the calculation of the UPCs cross section. Finally, we would like
to declare something about the number of events. We emphasize that this work
only gives results for the ideal case, the parameters and properties of the
corresponding detectors are not the focus of this work. As shown in Tables 1
and 2, the number of events produced by $pp$ collisions is generally greater
than that of $PbPb$ collisions. This is due to the integrated luminosities of
$pp$ beam, which is higher than that of heavy ions. However, there is likely
no possibility to observe the low-$p_{T}$ decay products of the very low-mass
$G_{2}$ glueball (and $\pi^{0}$) decays in the ATLAS/CMS detectors as those
experiments are designed to trigger on and to reconstruct particles at much
higher transverse momenta/masses (typically above $p_{T}$, $m\sim 5$ GeV). In
addition, higher luminosity implies larger pileup that will only make
impossible this measurement in $pp$ collisions (see the review article Soyez
(2019)). Thus the absence of pileup in $PbPb$ collisions may make this system
more beneficial to trigger on and reconstruct any UPCs compared to $pp$
collisions.

##  IV Summary and outlook

As a summary, in this work we have predicted the production of tensor glueball
$G_{2}$ in $AuAu$, $pp$ and $PbPb$ collisions at UPCs. The two-photons
interactions are used to construct the cross section of
$\gamma\gamma\to\pi^{0}G_{2}$ process, thus we use it in UPCs
($h_{1}h_{2}\rightarrow h_{1}h_{2}\pi^{0}G_{2}$). We also provide the rapidity
distributions and total cross sections in $pp$ and $PbPb$ collisions at UPCs
and the corresponding number of events. Although the $pp$-collision case has
higher integrated luminosities, the large pileup phenomenon previously
mentioned makes it more difficult to reconstruct the final state tensor
glueball. In contrast, $PbPb$ collisions have the opportunity to reconstruct
the production of tensor glueball. From our estimation, one can expect that
the $PbPb$ UPCs process could already produce about 10 signal events of
$G_{2}\to\phi\phi$ when the integrated luminosity is $10$ nb-1 at the LHC.
Likely, the ALICE/LHCb experiments have better possibilities, to reconstruct
the $G_{2}+\pi^{0}$ final state at low masses. Finally, we expect that future
$PbPb$ UPC experiments performed by HE-LHC and FCC will provide higher
statistics of tensor glueball production events.

###### Acknowledgements.

We are indebted to Ya-Ping Xie for discussions and comments. This work is
supported by the Strategic Priority Research Program of Chinese Academy of
Sciences under the Grant NO. XDB34030301 and Guangdong Major Project of Basic
and Applied Basic Research NO. 2020B0301030008.

Figure 3: Rapidity distributions for the UPC process $Au+Au\to
Au+Au+G_{2}+\pi^{0}$ at RHIC with model-1 (left panels) and model-2 (right
panels). The center of mass energies is $\sqrt{s}=200$ GeV. The solid, dashed
and dotted lines correspond to $f_{g}^{T}=50$ MeV, 100 MeV and 150 MeV Kivel
and Vanderhaeghen (2018).

Figure 4: Rapidity distributions for the UPC process $p+p\to
p+p+G_{2}+\pi^{0}$ with model-1 (left panels) and model-2 (right panels). From
top to bottom, it represents the predictions for the LHC, HE-LHC and FCC
energies, respectively. The solid, dashed and dotted lines correspond to
$f_{g}^{T}=50$ MeV, 100 MeV and 150 MeV Kivel and Vanderhaeghen (2018).

Figure 5: Rapidity distributions for the UPC process $Pb+Pb\to Pb+Pb+G_{2}+\pi^{0}$ with model-1 (left panels) and model-2 (right panels). From top to bottom, it represents the predictions for the LHC, HE-LHC and FCC energies, respectively. The solid, dashed and dotted lines correspond to $f_{g}^{T}=50$ MeV, 100 MeV and 150 MeV Kivel and Vanderhaeghen (2018).  Table 1: Cross sections and associated number of events in the tensor glueball decay mode $G_{2}\to\phi\phi$ for the tensor glueball production in $pp$ collisions. The integrated luminosities selection come from Abada  _et al._ (2019a, b). The table includes the results of both model-1 and model-2 calculations. Here we set the previously mentioned coupling constant $f_{g}^{T}=100$ MeV. |  |  $\sigma_{tot}$ (fb) |  $\\#$events ($G_{2}\to\phi\phi$)
---|---|---|---
$pp$ collisions | $\int\mathcal{L}$ | model-1 | model-2 | model-1 | model-2
LHC $\sqrt{s}=14$ TeV | 3 ab${}^{-1}/$20 years | $8.48$ | $8.25$ | 442.20 | 428.51
HE-LHC $\sqrt{s}=27$ TeV | 15 ab${}^{-1}/$20 years | $12.03$ | $11.72$ | 3127.08 | 3059.66
FCC $\sqrt{s}=100$ TeV | 25 ab${}^{-1}/$25 years | $20.57$ | $20.12$ | 8904.33 | 8734.73
Table 2: Cross sections and associated number of events in the tensor glueball decay mode $G_{2}\to\phi\phi$ for the tensor glueball production in $PbPb$ collisions. The integrated luminosities selection $\int\mathcal{L}$ come from Abada  _et al._ (2019a, b). The table includes the results of both model-1 and model-2 calculations. Here we set the previously mentioned coupling constant $f_{g}^{T}=100$ MeV. |  |  $\sigma_{tot}$ (nb) |  $\\#$events ($G_{2}\to\phi\phi$)
---|---|---|---
$PbPb$ collisions | $\int\mathcal{L}$ | model-1 | model-2 | model-1 | model-2
LHC $\sqrt{s}=5.5$ TeV | 10 nb${}^{-1}/$experiment | 44.50 | 42.46 | 7.8 | 7.4
HE-LHC $\sqrt{s}=10.6$ TeV | 10 nb${}^{-1}/$month | 88.23 | 84.35 | 15.79 | 14.63
FCC $\sqrt{s}=39$ TeV | 110 nb${}^{-1}/$month | 218.89 | 212.94 | 419.31 | 407.36

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