# Radiative Meson and Glueball Decays in the Witten-Sakai-Sugimoto Model

Florian Hechenberger  Josef Leutgeb  Anton Rebhan  Institut für Theoretische
Physik, Technische Universität Wien, Wiedner Hauptstrasse 8-10, A-1040 Vienna,
Austria

###### Abstract

We calculate radiative decay rates of mesons and glueballs in the top-down
holographic Witten-Sakai-Sugimoto model with finite quark masses. After
assessing to what extent this model agrees or disagrees with experimental
data, we present its predictions for so far undetermined decay channels.
Contrary to widespread expectations, we obtain sizeable two-photon widths of
scalar, tensor, and pseudoscalar glueballs, suggesting in particular that the
observed two-photon rate of the glueball candidate $f_{0}(1710)$ is not too
large to permit a glueball interpretation, but could be even much higher. We
also discuss the so-called exotic scalar glueball, which in the Witten-Sakai-
Sugimoto model is too broad to match either of the main glueball candidates
$f_{0}(1500)$ and $f_{0}(1710)$, but might be of interest with regard to the
alternative scenario of the so-called fragmented scalar glueball. Employing
the exotic scalar glueball for the latter, much smaller two-photon rates are
predicted for the ground-state glueball despite a larger total width;
relatively large two-photon rates would then apply to the excited scalar
glueball described by the predominantly dilatonic scalar glueball. In either
case, the resulting contributions to the muon $g-2$ from hadronic light-by-
light scattering involving glueball exchanges are small compared to other
single meson exchanges, of the order of $\lesssim 10^{-12}$.

###### Contents

  1. I Introduction and Summary
  2. II The Witten-Sakai-Sugimoto model augmented by quark masses
    1. II.1 Pseudoscalar masses
    2. II.2 Hadronic vector and axial vector meson decays
  3. III Radiative Meson Decays
    1. III.1 Vector meson dominance
    2. III.2 Radiative decays of pseudoscalars and vector mesons
      1. III.2.1 Vector meson $1\gamma$-decays
      2. III.2.2 Pseudoscalar meson $2\gamma$-decays
    3. III.3 Radiative axial-vector decays
      1. III.3.1 Axial-vector $1\gamma$-decays
      2. III.3.2 Axial-vector $2\gamma$-decays
  4. IV Glueballs in the Witten-Sakai-Sugimoto model
    1. IV.1 Exotic and dilatonic scalar glueballs
    2. IV.2 Extrapolations to realistic glueball masses
  5. V Radiative Glueball Decays
    1. V.1 Dilatonic Scalar Glueball Decays
      1. V.1.1 Dilatonic scalar glueball $1\gamma$-decays
      2. V.1.2 Dilatonic scalar glueball $2\gamma$-decays
    2. V.2 Tensor Glueball Decays
      1. V.2.1 Tensor glueball $1\gamma$-decays
      2. V.2.2 Tensor glueball $2\gamma$-decays
    3. V.3 Pseudoscalar Glueball Decays
  6. VI Glueball contributions to hadronic light-by-light scattering and the muon $g-2$
  7. A Hadronic decays of the scalar and tensor glueballs
    1. A.1 Dilatonic scalar glueball
    2. A.2 Exotic scalar glueball
    3. A.3 Tensor glueball
  8. B Radiative Decays of the Exotic Scalar Glueball
    1. B.0.1 Exotic scalar glueball $1\gamma$-decays
    2. B.0.2 Exotic scalar glueball $2\gamma$-decays

##  I Introduction and Summary

Glueballs, bound states of gluons without valence quarks, have been proposed
as a consequence of QCD from the start Fritzsch:1972jv ; Fritzsch:1973pi ;
Fritzsch:1975tx ; Jaffe:1975fd , but it is still a widely open question how
they manifest themselves in the hadron spectrum Klempt:2007cp ; Crede:2008vw ;
Ochs:2013gi ; Klempt:2021nuf ; Chen:2022asf . Lattice QCD Bali:1993fb ;
Morningstar:1999rf ; Chen:2005mg ; Gregory:2012hu ; Chen:2021dvn , mostly in
the quenched approximation, provides more or less clear predictions for the
spectrum, with a lightest glueball being a scalar, followed by a tensor
glueball with an important role as the lightest state associated with the
pomeron Donnachie:2002en , a pseudoscalar glueball participating in the
manifestation of the U(1)A anomaly responsible for the large mass of the
$\eta^{\prime}$ meson Rosenzweig:1981cu , and towers of states with arbitrary
integer spin as well as parity. However, it has turned out to be difficult to
discriminate glueball states from bound states of quarks with the same quantum
numbers with which they can mix, since the various available phenomenological
models give strongly divergent pictures, in particular for the lightest
glueballs. For the ground-state scalar glueball, the initially favored
scenario that the isoscalar meson $f_{0}(1500)$ contains the most glue content
while being strongly mixed with quarkonia Amsler:1995td ; Close:2001ga ;
Close:2005vf  is contested by models which identify the $f_{0}(1710)$ as a
glueball candidate Lee:1999kv ; Janowski:2014ppa ; Cheng:2015iaa  with more
dominant glue content. The latter also appears favored by its larger
production rate in supposedly gluon-rich radiative $J/\psi$ decays Gui:2012gx
, but there it was proposed that the glue content might rather be distributed
over several scalars involving a new meson $f_{0}(1770)$ previously lumped
together with the established $f_{0}(1710)$ Sarantsev:2021ein ; Klempt:2021nuf
; Klempt:2021wpg .

In order to clarify the situation, dynamical information on decay patterns is
required from first principles, which is difficult to extract from Euclidean
lattice QCD. Analytical approaches always involve uncontrollable
approximations, albeit recently interesting progress has been made using
Schwinger-Dyson equations Huber:2020ngt .

In this work we continue the analytical explorations made using gauge/gravity
duality, which has been employed for studying glueball spectra in strongly
coupled nonabelian theories shortly after the discovery of the AdS/CFT
correspondence Gross:1998gk ; Csaki:1998qr ; deMelloKoch:1998vqw ;
Hashimoto:1998if ; Csaki:1999vb , inspiring phenomenological “bottom-up” model
building for glueball physics Boschi-Filho:2002wdj ; Colangelo:2007pt ;
Forkel:2007ru ; Li:2013oda ; FolcoCapossoli:2015jnm ; Ballon-Bayona:2017sxa ;
Rinaldi:2021dxh . Of particular interest here is the top-down construction of
a dual to low-energy QCD in the large-$N_{c}$ limit from type-IIA string
theory by Witten Witten:1998zw , where the glueball spectrum has been obtained
in Constable:1999gb ; Brower:2000rp . Sakai and Sugimoto Sakai:2004cn ;
Sakai:2005yt  have extended this model by a D-brane construction introducing
$N_{f}$ chiral quarks in the ’t Hooft limit $N_{c}\gg N_{f}$, which turns out
to reproduce many features of low-energy QCD and chiral effective theory, not
only qualitatively, but often semi-quantitatively, while having a minimal set
of free parameters.

Glueball decay patterns have been first studied in the Witten-Sakai-Sugimoto
(WSS) model for the scalar glueball in Hashimoto:2007ze  and revisited and
extended in Brunner:2015oqa . This involves a so-called exotic scalar glueball
Constable:1999gb  for which it is unclear whether it should be identified with
the ground-state glueball in QCD or instead be discarded together with the
other states that more evidently do not relate to states in QCD.

Assuming that the ground-state scalar glueball corresponds to the
predominantly dilatonic bulk metric fluctuations which do not involve
polarizations in the extra Kaluza-Klein dimension employed for supersymmetry
breaking, Brunner:2015yha ; Brunner:2015oga  found that the resulting decay
pattern could match remarkably well the one of the $f_{0}(1710)$ meson when
effects of finite quark masses are included (or $f_{0}(1770)$ when this is
split off from a tetraquark $f_{0}(1710)$ Sarantsev:2021ein ). Instead of the
chiral suppression postulated for flavor asymmetries of scalar glueball decay
Chanowitz:2005du , a nonchiral enhancement of decays into heavier
pseudoscalars was obtained, which is correlated with a reduction of the
$\eta\eta^{\prime}$ decay mode Brunner:2015oga . This mechanism of flavor
symmetry violation is absent for the tensor glueball, whose hadronic decays
have been worked out also in Brunner:2015oqa ; hadronic decays of pseudoscalar
and pseudovector glueballs have been studied in Brunner:2016ygk ;
Leutgeb:2019lqu ; Brunner:2018wbv .

In the present paper, we revisit and extend the study of glueball decay
patterns of Brunner:2015oqa ; Brunner:2015yha ; Brunner:2015oga  to also
include radiative decays. As discussed already in Sakai:2005yt , the WSS model
naturally incorporates vector meson dominance (VMD), crucially involving an
infinite tower of vector mesons. After assessing the predictions of the WSS
model with regard to radiative decays of ordinary pseudoscalar and (axial)
vector mesons, we analyze its corresponding results for glueballs.

Contrary to widespread expectations, the WSS model predicts that glueballs can
have sizeable radiative decay widths in the keV range, exceeding even the
claimed observation of two-photon rates for $f_{0}(1710)$ by the BESIII
collaboration Belle:2013eck , which was taken as evidence against its glueball
nature.

In this context we also reconsider the exotic scalar glueball, which differs
from the dilatonic one in that it has smaller couplings to vector mesons as
well as photons, while having a total width in excess of the one of either
$f_{0}(1500)$ or $f_{0}(1710)$, when its mass is suitably adjusted. As such it
may instead be a candidate for the so-called fragmented scalar glueball
proposed in Sarantsev:2021ein ; Klempt:2021nuf ; Klempt:2021wpg , which is a
wider resonance distributed over $f_{0}(1710)$, a novel $f_{0}(1770)$,
$f_{0}(2020)$, and $f_{0}(2100)$, without showing up as an identifiable meson
on its own.

In the case of the tensor glueball, where the WSS model is unequivocal in
identifying the ground state, even though its mass also needs correction, we
find again two-photon rates in the keV region, larger than the old predictions
of Kada et al. Kada:1988rs , but comparable to those obtained by Cotanch and
Williams Cotanch:2005ja  using VMD. (The latter have obtained even larger two-
photon rates for the scalar glueball, which are an order of magnitude above
the WSS results.)

The next heavier glueball, the pseudoscalar glueball, which plays an important
rule in the realization of the U(1)A anomaly Leutgeb:2019lqu , is also found
to have two-photon rates in the keV region.

Because of their sizeable two-photon coupling in the WSS model, we consider
also the effect the lightest three glueballs may have as single-meson
contributions to hadronic light-by-light scattering, which is an important
ingredient of the Standard Model prediction of the anomalous magnetic moment
of the muon Aoyama:2020ynm  $a_{\upmu}=(g-2)_{\upmu}/2$. With the dilatonic
scalar glueball as ground state, we find results of $a_{\upmu}^{G}=-(1\dots
1.6)\times 10^{-12}$, and one order of magnitude smaller when the exotic
scalar glueball is used instead with mass raised to the value of the
fragmented glueball of Sarantsev:2021ein . With its larger mass and comparable
two-photon rate, the tensor glueball is bound to contribute less than the
dilatonic scalar glueball. The pseudoscalar glueball, which contributes with a
different sign, yields $a_{\upmu}^{G_{PS}}=+(0.2\dots 0.4)\times 10^{-12}$
depending on its actual mass. All these results are thus safely smaller than
the current uncertainties in the hadronic light-by-light scattering
contributions to $a_{\upmu}$.

##  II The Witten-Sakai-Sugimoto model augmented by quark masses

The Witten-Sakai-Sugimoto (WSS) model Sakai:2004cn ; Sakai:2005yt  is
constructed by placing a stack of $N_{f}$ flavor probe D8 and
$\overline{\text{D8}}$-branes into the near-horizon double Wick rotated black
D4-brane background proposed in Witten:1998zw  as a supergravity dual of four-
dimensional $\mathrm{U}(N_{c}\to\infty)$ Yang-Mills (YM) theory at low
energies, where supersymmetry and conformal symmetry are broken by
compactifications. It thus serves as a model for the low-energy limit of large
$N_{c}$ QCD with $N_{f}\ll N_{c}$, corresponding to a quenched approximation
when extrapolated to $N_{f}=N_{c}=3$. The background geometry is given by the
metric

$\displaystyle\text{d}s^{2}=\left(\frac{U}{R_{\mathrm{D4}}}\right)^{3/2}\left[\eta_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}+f(U)\text{d}\tau^{2}\right]+\left(\frac{R_{\mathrm{D4}}}{U}\right)^{3/2}\left[\frac{\text{d}U^{2}}{f(U)}+U^{2}\text{d}\Omega_{4}^{2}\right],$
$\displaystyle
e^{\phi}=g_{s}\left(\frac{U}{R_{\mathrm{D4}}}\right)^{3/4},\qquad
F_{4}=\text{d}C_{3}=\frac{(2\pi l_{s})^{3}N_{c}}{V_{4}}\epsilon_{4},\qquad
f(U)=1-\frac{U_{\text{KK}}^{3}}{U^{3}},$ (1)

with dilaton $\phi$ and Ramond-Ramond three-form field $C_{3}$, a solution of
type IIA supergravity, whose bosonic part of the action reads

$S_{\text{grav}}=\frac{1}{2\kappa_{10}^{2}}\int\text{d}^{10}x\sqrt{-g}\left[e^{-2\phi}\left(R+4\left(\nabla\phi\right)^{2}\right)-\frac{1}{2}|F_{4}|^{2}\right].$
(2)

The $N_{c}$ D4-branes extend along the directions parametrized by the
coordinates $x^{\mu}$, $\mu=0,1,2,3$ and another spatial dimension with
coordinate $\tau$, while $U$ corresponds to the radial (holographic) direction
transverse to the D4-brane. The remaining four transverse coordinates span a
unit $S^{4}$ with line element $\text{d}\Omega_{4}^{2}$, volume form
$\epsilon_{4}$ and volume $V_{4}=8\pi^{2}/3$. The $\tau$-direction is
compactified to a supersymmetry breaking $S^{1}$, whose period is chosen as

$\displaystyle\tau\simeq\tau+\delta\tau=\tau+2\pi M_{\text{KK}}^{-1},\quad$
$\displaystyle
M_{\text{KK}}=\frac{3}{2}\frac{U_{\text{KK}}^{1/2}}{R_{\mathrm{D4}}^{3/2}},$
(3)

to avoid a conical singularity at $U=U_{\text{KK}}$. The radius
$R_{\mathrm{D4}}$ is related to the string coupling $g_{s}$ and the string
length $l_{s}$ through $R_{\mathrm{D4}}^{3}=\pi g_{s}N_{c}l_{s}^{3},$ and the
’t Hooft coupling of the dual four-dimensional Yang-Mills theory is given by

$\lambda=g_{\mathrm{YM}}^{2}N_{c}=\frac{g_{5}^{2}}{\delta\tau}N_{c}=2\pi
g_{s}l_{s}M_{\mathrm{KK}}N_{c}.$ (4)

The flavor D8 and $\overline{\text{D8}}$-branes extend along $x^{\mu}$, $U$,
and the $S^{4}$. They are placed antipodally on the $\tau$-circle to join at
$U_{\text{KK}}$. In adopting the probe approximation, i.e. $N_{c}\gg N_{f}$
for the $N_{f}$ D8 branes, one can ignore backreactions from the D8-branes to
the D4-brane background. The gauge fields on the D8-branes, which are dual to
left and right chiral quark currents separated in the Kaluza-Klein ($\tau$)
direction, are governed at leading order by a Dirac-Born-Infeld (DBI) plus
Chern-Simons (CS) action

$\displaystyle S_{\text{DBI}}=$ $\displaystyle-
T_{8}\int\text{d}^{9}xe^{-\phi}\,\text{Tr}\,\sqrt{-\det\left(g_{MN}+2\pi\alpha^{\prime}F_{MN}\right)},$
$\displaystyle S_{\text{CS}}=$ $\displaystyle
T_{8}\int_{D8}C\wedge\,\text{Tr}\,\left[\exp\left\\{\frac{F}{2\pi}\right\\}\right]\sqrt{\hat{A}(\mathcal{R})},$
(5)

where $\hat{A}(\mathcal{R})$ is the so-called A-roof genus Green:1996dd ;
Polchinski1998 .

Considering only SO(5)-invariant excitations and restricting to terms
quadratic in the field strength, the nine-dimensional DBI action can be
reduced to a five-dimensional Yang-Mills theory with action Sakai:2004cn ;
Sakai:2005yt 111Note that in (6) one uses the Minkowski metric
$\eta_{\mu\nu}$, in the mostly plus convention, to contract the four-
dimensional spacetime indices.

$S_{\text{D8}}^{\text{DBI}}=-\kappa\int\text{d}^{4}x\,\text{d}z\,{\,\text{Tr}\,}\left[\frac{1}{2}K^{-1/3}F_{\mu\nu}^{2}+M_{\text{KK}}^{2}KF_{\mu
z}^{2}\right],$ (6)

with

$\kappa\equiv\frac{\lambda N_{c}}{216\pi^{3}},\qquad K(z)\equiv
1+z^{2}={U^{3}}/{U_{\text{KK}}^{3}}.$ (7)

To identify the four-dimensional meson fields, we make the ansatz

$\displaystyle
A_{\mu}(x^{\mu},z)=\sum_{n=1}^{\infty}B_{\mu}^{\left(n\right)}(x^{\mu})\psi_{n}(z)$
$\displaystyle
A_{z}(x^{\mu},z)=\sum_{n=0}^{\infty}\varphi^{\left(n\right)}(x^{\mu})\phi_{n}(z)$
(8)

for the five-dimensional gauge field using the complete sets
$\left\\{\psi_{n}(z)\right\\}_{n\geq 1}$ and
$\left\\{\phi_{n}(z)\right\\}_{n\geq 0}$ of normalizable functions of $z$ with
normalization conditions

$\begin{split}&\kappa\int\text{d}zK^{-1/3}\psi_{m}\psi_{n}=\delta_{mn},\\\
&\kappa\int\text{d}zK\phi_{m}\phi_{n}=\delta_{mn},\end{split}$ (9)

satisfying the completeness relations

$\begin{split}&\kappa\sum_{n}K^{-1/3}\psi_{n}(z)\psi_{n}(z^{\prime})=\delta(z-z^{\prime}),\\\
&\kappa\sum_{n}K\phi_{n}(z)\phi_{n}(z^{\prime})=\delta(z-z^{\prime}).\end{split}$
(10)

With this ansatz, the fields $B_{\mu}^{\left(n\right)}$ and
$\varphi^{\left(n\right)}$ have canonical kinetic terms; the eigenvalue
equation

$\displaystyle-K^{-1/3}\partial_{z}\left(K\partial_{z}\psi_{n}\right)=$
$\displaystyle\lambda_{n}\psi_{n},$ (11)

which can be used to relate the two complete sets via
$\phi_{n}(z)\propto\partial_{z}\psi_{n}(z)$ for $\left(n\geq 1\right)$, yields
a mass term for $B_{\mu}^{\left(n\right)}$. The remaining massless mode is
given by $\phi_{0}(z)=1/\left(\sqrt{\pi\kappa}M_{\text{KK}}K(z)\right)$.

Inserting the separation ansatz (8) into the DBI action (6) and integrating
over $z$, we obtain

$\displaystyle S_{\text{D8}}^{\text{DBI}}$ $\displaystyle=$
$\displaystyle-\,\text{Tr}\,\int\text{d}^{4}x\left[\left(\partial_{\mu}\varphi^{\left(0\right)}\right)^{2}+\sum_{n=1}^{\infty}\left(\frac{1}{2}\left(\partial_{\mu}B_{\nu}^{\left(n\right)}-\partial_{\nu}B_{\mu}^{\left(n\right)}\right)^{2}+m_{n}^{2}\left(B_{\mu}^{\left(n\right)}-m_{n}^{-1}\partial_{\mu}\varphi^{\left(n\right)}\right)^{2}\right)\right]$
(12) $\displaystyle+\left(\text{interaction terms}\right).$

The scalar fields $\varphi^{\left(n\right)}$ with $\left(n\geq 1\right)$ can
be absorbed by the fields $B_{\mu}^{\left(n\right)}$, which are interpreted as
(axial) vector meson fields, with masses
$m_{n}=\sqrt{\lambda_{n}}M_{\mathrm{KK}}$ determined by the eigenvalue
equation for the normalizable modes (11). The lightest vector mesons,
identified with the rho and omega mesons, have
$m_{\rho}=m_{1}=\sqrt{0.669314}M_{\mathrm{KK}}$, with the traditional value
Sakai:2004cn ; Sakai:2005yt  of $M_{\mathrm{KK}}=949$ MeV corresponding to
$m_{\rho}=776.4$ MeV.

The remaining field $\varphi^{\left(0\right)}$ is identified as the multiplet
of massless pion fields produced by chiral symmetry breaking, which is
realized geometrically by D8 and $\overline{\text{D8}}$-branes joining at
$z=0$, with the $\mathrm{U}(N_{f})$-valued Goldstone boson field given by the
holonomy

$U(x)=e^{i\Pi^{a}(x)\lambda^{a}/f_{\pi}}=\mathrm{P}\,\exp
i\int_{-\infty}^{\infty}\text{d}z\,A_{z}(z,x),$ (13)

where $\lambda^{a}=2T^{a}$ are Gell-Mann matrices including
$\lambda^{0}=\sqrt{2/N_{f}}\mathbf{1}$. For $N_{f}=3$ we have

$\Pi(x)\equiv\Pi^{a}(x)T^{a}=\frac{1}{2}\begin{pmatrix}\pi^{0}+\eta^{8}/\sqrt{3}+\eta^{0}\sqrt{2/3}&\sqrt{2}\pi^{+}&\sqrt{2}K^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}+\eta^{8}/\sqrt{3}+\eta^{0}\sqrt{2/3}&\sqrt{2}K^{0}\\\
\sqrt{2}K^{-}&\sqrt{2}\bar{K}^{0}&-2\eta^{8}/\sqrt{3}+\eta^{0}\sqrt{2/3}\end{pmatrix}.$
(14)

The pion decay constant is determined by

$f_{\pi}^{2}=\frac{\lambda N_{c}M_{\mathrm{KK}}^{2}}{54\pi^{4}};$ (15)

with the choice $f_{\pi}\approx 92.4$ MeV one obtains $\lambda\approx 16.63$.
Following Brunner:2015oqa , we shall also consider the smaller value
$\lambda\approx 12.55$ obtained by matching the large-$N_{c}$ lattice result
for the string tension obtained in Ref. Bali:2013kia  (resulting in
$f_{\pi}\approx 80.3$ MeV). A smaller ’t Hooft coupling has also been argued
for in Ref. Imoto:2010ef  from studies of the spectrum of higher-spin mesons
in the WSS model. The downward variation of $\lambda\approx 16.63\dots 12.55$
will thus be used as an estimate of the variability of the predictions of this
model.

###  II.1 Pseudoscalar masses

In the WSS model, the U(1)A flavor symmetry is broken by an anomalous
contribution of order $1/N_{c}$ due to the $C_{1}$ Ramond-Ramond field, which
gives rise to a Witten-Veneziano Witten:1979vv ; Veneziano:1979ec  mass term
for the singlet $\eta_{0}$ pseudoscalar with Sakai:2004cn

$m_{0}^{2}=\frac{2N_{f}}{f_{\pi}^{2}}\chi_{g}=\frac{N_{f}}{27\pi^{2}N_{c}}\lambda^{2}M_{\mathrm{KK}}^{2},$
(16)

where $\chi_{g}$ is the topological susceptibility.

For $N_{f}=N_{c}=3$, one has $m_{0}=967\dots 730$ MeV for $\lambda=16.63\dots
12.55$, which is indeed a phenomenologically interesting ballpark when finite
quark masses are added to the model by the addition of an effective Lagrangian

$\begin{split}&\mathcal{L}_{m}^{\mathcal{M}}\propto\,\text{Tr}\,\left(\mathcal{M}\,U(x)+h.c.\right),\\\
&\mathcal{M}={\rm diag}(m_{u},m_{d},m_{s}).\end{split}$ (17)

This deformation can be generated by either worldsheet instantons
Aharony:2008an ; Hashimoto:2008sr  or nonnormalizable modes of bifundamental
fields corresponding to open-string tachyons 0708.2839 ; Dhar:2008um ;
McNees:2008km ; Niarchos:2010ki . Assuming for simplicity isospin symmetry,
$m_{u}=m_{d}=\hat{m}$, this leads to masses Brunner:2015oga

$m^{2}_{\eta,\eta^{\prime}}=\frac{1}{2}m_{0}^{2}+m_{K}^{2}\mp\sqrt{\frac{m_{0}^{4}}{4}-\frac{1}{3}m_{0}^{2}(m_{K}^{2}-m_{\pi}^{2})+(m_{K}^{2}-m_{\pi}^{2})^{2}}$
(18)

for the mass eigenstates

$\displaystyle\eta$ $\displaystyle=$
$\displaystyle\eta_{8}\cos\theta_{P}-\eta_{0}\sin\theta_{P}$
$\displaystyle\eta^{\prime}$ $\displaystyle=$
$\displaystyle\eta_{8}\sin\theta_{P}+\eta_{0}\cos\theta_{P},$ (19)

with mixing angle

$\theta_{P}=\frac{1}{2}\arctan\frac{2\sqrt{2}}{1-\frac{3}{2}{m_{0}^{2}}/({m_{K}^{2}-m_{\pi}^{2}})}.$
(20)

Using $m_{\pi}^{2}=m_{\pi_{0}}^{2}\approx(135{\rm MeV})^{2}$ and

$m_{K}^{2}=\frac{1}{2}(m_{K_{\pm}}^{2}+m_{K_{0}}^{2})-\frac{1}{2}(m_{\pi_{\pm}}^{2}-m_{\pi_{0}}^{2})\approx(495{\rm
MeV})^{2}$ (21)

as isospin symmetric parameters, the WSS result $m_{0}\approx 967\dots 730$
MeV for $\lambda=16.63\dots 12.55$ leads to
$\theta_{P}\approx-14^{\circ}\dots-24^{\circ}$ and $m_{\eta}\approx 520\dots
470$, $m_{\eta^{\prime}}\approx 1080\dots 890$ MeV. In the following we shall
consider this range of mixing angles in conjunction with the variation of
$\lambda$, but we shall fix $m_{\eta}$ and $m_{\eta^{\prime}}$ to their
experimental values when evaluating phase space integrals. In the radiative
decay rates considered below, the explicit quark masses will not modify the
(chiral) results for the couplings; they only appear in phase space factors.

###  II.2 Hadronic vector and axial vector meson decays

Vertices for the hadronic decays of vector and axial vector meson involving
pseudoscalar mesons are contained in the second term of the DBI action (6).
For the $\rho$ meson, this contains the term (with indices restricted to the
first two quark flavors)

$\displaystyle\mathcal{L}_{\rho\pi\pi}=-g_{\rho\pi\pi}\varepsilon_{abc}(\partial_{\mu}\pi^{a})\rho^{b\mu}\pi^{c},$
$\displaystyle g_{\rho\pi\pi}=\int\text{d}z\frac{1}{\pi
K}\psi_{1}=33.98\,\lambda^{-\frac{1}{2}}N_{c}^{-\frac{1}{2}},$ (22)

yielding $\Gamma_{\rho\to\pi\pi}=98.0\dots 130$ MeV for $\lambda=16.63\dots
12.55$, which somewhat underestimates the experimental result of $\approx 150$
MeV.

There is also a vertex involving one vector, one axial vector, and one
pseudoscalar meson, which for the ground-state isotriplet mesons reads

$\displaystyle\mathcal{L}_{a_{1}\rho\pi}=g_{a_{1}\rho\pi}\varepsilon_{abc}a_{\mu}^{a}\rho^{b\mu}\pi^{c},$
$\displaystyle
g_{a_{1}\rho\pi}=2M_{\mathrm{KK}}\sqrt{\frac{\kappa}{\pi}}\int\text{d}z\psi_{2}^{\prime}\psi_{1}={-34.43}\,\lambda^{-\frac{1}{2}}N_{c}^{-\frac{1}{2}}M_{\mathrm{KK}}.$
(23)

In the WSS model, the predicted mass of the $a_{1}$ meson, 1186.5 MeV, is
rather close to the experimental result Workman:2022ynf  of 1230(40) MeV. The
predicted width for $a_{1}\to\rho\pi$ (already studied in Sakai:2005yt ) is
$425\dots 563$ MeV, which is within the experimental result for the total
width of 250…600 MeV (average value 420(35) MeV), but according to
CLEO:1999rzk  only 60% of the three-pion decays are due to S-wave $\rho\pi$
decays, whereas the latter saturate the hadronic decays in the WSS model.

For the light quark flavors, these results for the decay rates of $\rho$ and
$a_{1}$ seem to indicate that the WSS model is working quite well. When the
mass of the strange quark is included, a shortcoming of the model, which is
shared by many bottom-up holographic QCD models (see e.g. Abidin:2009aj ), is
that the $\phi$ meson remains degenerate with $\rho$ and $\omega$. In the
following we shall nevertheless also consider $K^{*}$ and $\phi$ mesons by
simply raising their masses in the resulting phase space factors while keeping
their vertices such as $g_{K^{*}K\pi}=g_{\phi KK}=g_{\rho\pi\pi}$ unchanged.
The resulting widths, $\Gamma(K^{*}\to K\pi)=28\ldots 37$ MeV and
$\Gamma(\phi\to K\bar{K})=2.12\ldots 2.82$ MeV, are between 40 and 20% too
small. These deviations are at least not dramatically larger than the one for
the $\rho$ width, which amounts to 33…12%; all appear to remain in the range
to be expected for a large-$N$ approach.

##  III Radiative Meson Decays

Before considering radiative decays of the experimentally elusive glueballs,
we shall evaluate the predictions of the WSS model with nonzero quark masses
for radiative decay widths of regular mesons and compare with experimental
data as far as available. As discussed extensively in the second paper of
Sakai and Sugimoto Sakai:2005yt , holographic QCD models naturally provide a
realization of vector meson dominance (VMD) Gell-Mann:1961jim ; Kroll:1967it ;
Sakurai:1969ss ; Sakurai:1972wk  involving an infinite tower of vector mesons.
There it was already observed that the chiral WSS model yields a result for
$\Gamma(\omega\to\pi^{0}\gamma)$ which is roughly consistent with the
experimental value. In the following we shall recapitulate the results of
Sakai:2005yt  and extend them to the WSS model including quark masses and the
Witten-Veneziano mass term.

###  III.1 Vector meson dominance

According to the holographic principle, non-normalizable modes are interpreted
as external sources. This permits to study electromagnetic interactions to
leading order by setting asymptotic values of the gauge field $A_{\mu}$ on the
D8 branes according to Sakai:2005yt

$\lim_{z\to\pm\infty}A_{\mu}(x,z)=A_{L,R\mu}(x)=eQA_{\mu}^{\text{em}}(x),$
(24)

where $e$ is the electromagnetic coupling constant and $Q$ is the electric
charge matrix, given as

$\displaystyle Q$ $\displaystyle=\frac{1}{3}\left(\begin{array}[]{ccc}2\\\
&-1\\\ &&-1\end{array}\right)$ (28)

for the $N_{f}=3$ case. The ansatz (8) changes to

$\begin{split}A_{\mu}(x^{\mu},z)=&A_{L\mu}(x^{\mu})\psi_{+}(z)+A_{R\mu}(x^{\mu})\psi_{-}(z)\\\
&+\sum_{n=1}^{\infty}v_{\mu}^{n}(x^{\mu})\psi_{n}(z),\end{split}$ (29)

with the functions $\psi_{\pm}(z)$ defined as

$\psi_{\pm}(z)\equiv\frac{1}{2}\left(1\pm\psi_{0}(z)\right),\qquad\psi_{0}(z)\equiv\frac{2}{\pi}\arctan
z,$ (30)

They satisfy (11) as non-normalizable zero modes, because
$\partial_{z}\psi_{\pm}(z)\propto\phi_{0}(z)\propto 1/K(z)$.

To distinguish between vector and axial-vector fields we introduce the
notation

$\begin{split}&\mathcal{V}_{\mu}\equiv\frac{1}{2}\left(A_{L\mu}+A_{R\mu}\right),\qquad\mathcal{A}_{\mu}\equiv\frac{1}{2}\left(A_{L\mu}-A_{R\mu}\right),\\\
&v_{\mu}^{n}\equiv B_{\mu}^{\left(2n-1\right)},\qquad a_{\mu}^{n}\equiv
B_{\mu}^{\left(2n\right)},\end{split}$ (31)

so that

$\begin{split}A_{\mu}(x^{\mu},z)=&\mathcal{V}_{\mu}(x^{\mu})+\mathcal{A}_{\mu}(x^{\mu})\psi_{0}(z)\\\
&+\sum_{n=1}^{\infty}v_{\mu}^{n}(x^{\mu})\psi_{2n-1}(z)+\sum_{n=1}^{\infty}a_{\mu}^{n}(x^{\mu})\psi_{2n}(z).\end{split}$
(32)

The first term in (6) can then be expanded as

$\displaystyle\frac{\kappa}{2}\int\text{d}z$
$\displaystyle\\!\\!\\!\\!\\!\\!K^{-1/3}F_{\mu\nu}^{2}$ (33) $\displaystyle=$
$\displaystyle\frac{a_{\mathcal{V}\mathcal{V}}}{2}\text{tr}\left(\partial_{\mu}\mathcal{V}_{\nu}-\partial_{\nu}\mathcal{V}_{\mu}\right)^{2}$
$\displaystyle+\frac{a_{\mathcal{A}\mathcal{A}}}{2}\text{tr}\left(\partial_{\mu}\mathcal{A}_{\nu}-\partial_{\nu}\mathcal{A}_{\mu}\right)^{2}$
$\displaystyle+\frac{1}{2}\text{tr}\left(\partial_{\mu}v_{\nu}^{n}-\partial_{\nu}v_{\mu}^{n}\right)^{2}+\frac{1}{2}\text{tr}\left(\partial_{\mu}a_{\nu}^{n}-\partial_{\nu}a_{\mu}^{n}\right)^{2}$
$\displaystyle+a_{\mathcal{V}v^{n}}\text{tr}\left(\left(\partial_{\mu}\mathcal{V}_{\nu}-\partial_{\nu}\mathcal{V}_{\mu}\right)\left(\partial_{\mu}v_{\nu}^{n}-\partial_{\nu}v_{\mu}^{n}\right)\right)$
$\displaystyle+a_{\mathcal{A}a^{n}}\left(\left(\partial_{\mu}\mathcal{A}_{\nu}-\partial_{\nu}\mathcal{A}_{\mu}\right)\left(\partial_{\mu}a_{\nu}^{n}-\partial_{\nu}a_{\mu}^{n}\right)\right)$
$\displaystyle+\left(\text{interaction terms}\right),$

with coupling constants

$\begin{split}&a_{\mathcal{V}v^{n}}=\kappa\int dzK^{-1/3}\psi_{2n-1},\quad
a_{\mathcal{V}\mathcal{V}}=\kappa\int dzK^{-1/3},\\\
&a_{\mathcal{A}a^{n}}=\kappa\int dzK^{-1/3}\psi_{2n}\psi_{0},\quad
a_{\mathcal{A}A}=\kappa\int dzK^{-1/3}\psi_{0}^{2},\end{split}$ (34)

mixing the photon field $\mathcal{V}$ with every vector meson $v^{n}$. The
coefficients $a_{\mathcal{V}\mathcal{V}}$ and $a_{\mathcal{A}\mathcal{A}}$ are
divergent, since the external fields correspond to non-normalizable modes in
the radial direction, and need to be renormalized to canonical values. The
photon field $\mathcal{V}$ does not appear in the interaction terms of this
model and can only couple via the mixing (34), fully realizing VMD.
Alternatively, it is possible to perform a field redefinition to diagonalize
the action and to get rid of the mixing terms, thus producing new interaction
terms coupling mesons to photons.

###  III.2 Radiative decays of pseudoscalars and vector mesons

The relevant vertices for radiative decays of pseudoscalars and (axial) vector
mesons come from the Chern-Simons term

$\displaystyle S_{CS}\supset$
$\displaystyle\,T_{8}\int\,\text{tr}\,\left(\exp\left(2\pi\alpha^{\prime}F_{2}+B_{2}\right)\wedge
C_{3}\right)$ $\displaystyle\supset\,$
$\displaystyle\frac{N_{c}}{96\pi^{2}}\epsilon^{\mu\nu\rho\sigma
z}\int\,\text{tr}\,\left(3A_{z}F_{\mu\nu}F_{\rho\sigma}-4A_{\mu}\partial_{z}A_{\nu}F_{\rho\sigma}\right),$
(35)

where we have used partial integration.

Inserting the mode expansion (32) and integrating over the radial coordinate
we obtain for the interaction term involving two vectors and one pseudoscalar

$\displaystyle\mathcal{L}_{\Pi v^{m}v^{n}}=$
$\displaystyle\frac{N_{c}}{4\pi^{2}f_{\pi}}c_{v^{n}v^{m}}\epsilon^{\mu\nu\rho\sigma}\,\text{tr}\,\left(\Pi\partial_{\mu}v_{\nu}^{n}\partial_{\rho}v_{\sigma}^{m}\right),$
(36)

with coupling constants

$c_{v^{n}v^{m}}=\frac{1}{\pi}\int\text{d}zK^{-1}\psi_{2n-1}\psi_{2m-1}=\left\\{\frac{1350.83}{\lambda
N_{c}},\ldots\right\\}$ (37)

as studied in Sakai:2005yt , where numerical results for the coefficients
beyond $c_{v^{1}v^{1}}$ given above can be found.

####  III.2.1 Vector meson $1\gamma$-decays

Using VMD, we can calculate the interaction term for the radiative decay of a
vector meson into a pseudoscalar and one photon as

$\mathcal{L}_{\Pi\mathcal{V}v^{n}}=\frac{N_{c}}{4\pi^{2}f_{\pi}}c_{\mathcal{V}v^{n}}\epsilon^{\mu\nu\rho\sigma}\,\text{tr}\,\left(\Pi\partial_{\mu}v_{\nu}^{n}\partial_{\rho}\mathcal{V}_{\sigma}+\Pi\partial_{\mu}\mathcal{V}_{\nu}\partial_{\rho}v_{\sigma}^{n}\right),$
(38)

with coupling

$\begin{split}c_{\mathcal{V}v^{n}}=&\sum_{m}c_{v^{n}v^{m}}a_{\mathcal{V}v^{m}}=\frac{1}{\pi}\int\text{d}zK^{-1}\psi_{2n-1}\\\
&=\\{33.9839,\dots\\}(N_{c}\lambda)^{-1/2},\end{split}$ (39)

where we have used the completeness relation (10) to eliminate the summed-over
modes.

Performing the polarization sums we get

$\begin{split}&\left|\mathcal{M}_{\left(v^{n}\rightarrow\Pi\mathcal{V}\right)}\right|^{2}=\sum_{\left(v^{n}\right)}\sum_{\left(\mathcal{V}\right)}\frac{1}{3}\epsilon_{\mu}^{\left(v^{n}\right)}\epsilon_{\nu}^{\left(v^{n}\right)*}\epsilon_{\rho}^{\left(\mathcal{V}\right)}\epsilon_{\sigma}^{\left(\mathcal{V}\right)*}\mathcal{M}_{\left(\Pi
v^{n}\mathcal{V}\right)}^{\mu\rho}\mathcal{M}_{\left(\Pi
v^{n}\mathcal{V}\right)}^{\nu\sigma*}\\\
&=\frac{c_{\mathcal{V}v^{n}}^{2}e^{2}N_{c}^{2}}{96\pi^{4}f_{\pi}^{2}}\left(\,\text{tr}\,\left(T_{\Pi}T_{v^{n}}Q\right)+\,\text{tr}\,\left(T_{\Pi}QT_{v^{n}}\right)\right)^{2}\left(m_{\Pi}^{2}-m_{v^{n}}^{2}\right)^{2}.\end{split}$

The partial width then reads

$\displaystyle\Gamma_{v^{n}\rightarrow\Pi\gamma}=$
$\displaystyle\frac{1}{8\pi}\left|\mathcal{M}_{\left(v^{n}\rightarrow\Pi\mathcal{V}\right)}\right|^{2}\frac{\left|\mathbf{p}_{v}\right|}{m_{v}^{2}}.$
(40)

|  $\Gamma^{{\rm exp.}}$[keV] |  $\Gamma^{{\rm WSS}}$[keV]
---|---|---
$\pi^{0}\rightarrow 2\gamma$ | 0.00780(12) | 0.00773…0.0102
$\eta\rightarrow 2\gamma$ | 0.515(18) | 0.480…0.978
$\eta^{\prime}\rightarrow 2\gamma$ | 4.34(14) | 5.72…5.87…5.75
$\rho^{0}\to\pi^{0}\gamma$ | 70(12) | 56.2…98.6
$\rho^{\pm}\to\pi^{\pm}\gamma$ | 68(7) | 56.2…98.6
$\rho^{0}\to\eta\gamma$ | 45(3) | 40.3…90.5
$\omega\to\pi^{0}\gamma$ | 725(34) |  $521$…915
$\omega\to\eta\gamma$ | 3.9(4) | 4.87…10.9
$\eta^{\prime}\to\rho^{0}\gamma$ | 55.4(1.9)fit,68(7)av. | 54.1…59.2…58.5
$\eta^{\prime}\to\omega\gamma$ | 4.74(20)fit,5.8(7)av. | 5.37…5.89…5.81
$\phi\to\pi^{0}\gamma$ | 5.6(2) | 0
$\phi\to\eta\gamma$ | 55.3(1.2) | 84.7…92.8…91.6
$\phi\to\eta^{\prime}\gamma$ | 0.264(10) | 0.525…1.18
$K^{*0}\to K^{0}\gamma$ | 116(10) | 124…218
$K^{*\pm}\to K^{\pm}\gamma$ | 50(5) | 31.0…54.5
Table 1: Results for various radiative decay widths of pseudoscalar and vector
mesons involving vector and pseudoscalar mesons, with ’t Hooft coupling
$\lambda=16.63\dots 12.55$ ($\lambda=16.63$ is the traditional Sakai:2004cn ;
Sakai:2005yt  value matching $f_{\pi}=92.4$ MeV; $\lambda=12.55$ an
alternative choice matching the large-$N$ string tension at the expense of
$f_{\pi}$). For nonmonotonic dependence on $\lambda$ intermediate extremal
values are also given. Ideal mixing is assumed for $\omega$ and $\phi$, fixing
the WSS result for $\phi\to\pi^{0}\gamma$ to zero. Experimental results are
from the PDG Workman:2022ynf  except for the $\pi^{0}$ width, which is from
PrimEx-II:2020jwd .

####  III.2.2 Pseudoscalar meson $2\gamma$-decays

Employing VMD a second time, we can derive the interaction term for a decay of
a pseudoscalar meson in two photons

$\displaystyle\mathcal{L}_{\Pi\mathcal{V}\mathcal{V}}=$
$\displaystyle-\frac{N_{c}}{4\pi^{2}f_{\pi}}c_{\mathcal{V}\mathcal{V}}\epsilon^{\mu\nu\rho\sigma}\,\text{tr}\,\left(\Pi\partial_{\mu}\mathcal{V}_{\nu}\partial_{\rho}\mathcal{V}_{\sigma}\right),$
(41)

where the sum over the entire tower of vector mesons yields

$\displaystyle c_{\mathcal{V}\mathcal{V}}=$
$\displaystyle\sum_{m}c_{\mathcal{V}v^{m}}a_{\mathcal{V}v^{m}}=\frac{1}{\pi}\int\text{d}zK^{-1}=1,$
(42)

leading to the standard result

$\displaystyle\Gamma_{\Pi\rightarrow\gamma\gamma}=$
$\displaystyle\frac{1}{8\pi}\left|\mathcal{M}_{\left(\Pi\rightarrow\mathcal{V}\mathcal{V}\right)}\right|^{2}\frac{\left|\mathbf{p}_{\gamma}\right|}{m_{\Pi}^{2}}\frac{1}{2}$
(43)

with

$\displaystyle\left|\mathcal{M}_{\left(\Pi\rightarrow\mathcal{V}\mathcal{V}\right)}\right|^{2}=$
$\displaystyle\frac{e^{4}N_{c}^{2}}{4\pi^{4}f_{\pi}^{2}}\left(\,\text{tr}\,\left(T_{\Pi}Q^{2}\right)\right)^{2}m_{\Pi}^{4}.$
(44)

The numeric results for the various radiative decays involving one
pseudoscalar and two vector particles are summarized in Table 1 for
$\lambda=16.63\dots 12.55$. As mentioned above, $\lambda=16.63$ is the
traditional Sakai:2004cn ; Sakai:2005yt  value matching $f_{\pi}=92.4$ MeV,
whereas $\lambda=12.55$ is an alternative choice matching the large-$N$ string
tension at the expense of $f_{\pi}$. The decay rate for $\pi^{0}$ is therefore
close to the experimental value only for the first value of $\lambda$, but the
partial widths of the decays $\rho$ and $\omega$ into $\pi\gamma$ are
reproduced by an intermediate value of $\lambda$.

In processes involving $\eta$ and $\eta^{\prime}$, we have used the
pseudoscalar mixing angle following from (20), which varies as
$\theta_{P}\approx-14^{\circ}\dots-24^{\circ}$ when $\lambda=16.63\dots
12.55$, which enters the flavor matrix $T_{\Pi}$ in (44). Here the dependence
on $\lambda$ is nonmonotonic, because also $f_{\pi}$ in the prefactor depends
on $\lambda$; Table 1 also gives the extremal values attained at intermediate
values of $\lambda$.

The vector couplings in the WSS model augmented by quark masses according to
(17) are flavor-symmetric, but we distinguish $\phi$ and $\omega$ mesons
through their experimental masses. The undetermined mixing of $\phi$ and
$\omega$ could be fixed by fitting for example the small ratio of the widths
for their decays into $\pi^{0}\gamma$, 5.6/725, which yields a mixing angle
close to ideal mixing, $\theta_{V}=\theta_{V}^{\mathrm{ideal}}+3.32^{\circ}$,
as in Ambrosino:2009sc . However, here and in the following we shall assume
completely ideal mixing for simplicity, which eliminates $\phi\to\pi\gamma$
but does not change the other partial widths of $\phi$ significantly. This
gives generally good results for decays involving $\omega$, but larger
discrepancies with experiment for $\phi$ mesons irrespective of the precise
value of $\theta_{V}$.222With a $\phi$-$\omega$ mixing angle of $3.32^{\circ}$
above ideal mixing Ambrosino:2009sc , we would have
$\Gamma^{\mathrm{WSS}}(\phi\to\pi\gamma)=4\ldots 7$ keV, consistent with
experiment; the result for $\omega\to\eta\gamma$ would be somewhat closer to
the experimental value, but the one for $\eta^{\prime}\to\omega\gamma$ further
off. Note that the standard value of $M_{\mathrm{KK}}=949$ MeV, which we are
using, is chosen such that the $\rho$ mass is reproduced, which is rather
close to the mass of the $\omega$ meson, but less suitable for the $\phi$
meson.

###  III.3 Radiative axial-vector decays

From the 5-dimensional CS-term (35) we can also extract a term including two
vector mesons and one axial-vector meson

$\displaystyle\mathcal{L}_{v^{m}v^{n}a^{p}}=$
$\displaystyle-\frac{N_{c}}{4\pi^{2}}d_{v^{m}v^{n}a^{p}}\epsilon^{\mu\nu\rho\sigma}\,\text{tr}\,\left(v_{\mu}^{m}a_{\nu}^{p}\partial_{\rho}v_{\sigma}^{n}\right),$
(45)

with

$\displaystyle d_{v^{m}v^{n}a^{p}}=$
$\displaystyle\int\text{d}z\psi_{2m-1}\psi_{2n-1}^{\prime}\psi_{2p},$ (46)

where we again made use of partial integration.

As noted already in Sakai:2005yt  and observed before in other holographic
models Son:2003et ; DaRold:2005mxj ; Hirn:2005nr  as well as in the hidden
local symmetry approach of Bando:1984ej , the vertex for the decay of an axial
vector meson into a pseudoscalar and a photon, which would have to come from
the DBI part of the action, vanishes,333In the hidden local symmetry approach,
$a_{1}\to\pi\gamma$ has been included by adding higher-derivative terms to the
action Bando:1987ym . even though there is a nonvanishing vertex for
$a_{1}^{\pm}\to\pi^{\pm}\rho^{0}$, see (II.2). But the corresponding coupling
for an on-shell photon is obtained by replacing $\psi_{1}$ therein by a unity,
leading to

$g_{a_{1}\pi\mathcal{V}}=2M_{\mathrm{KK}}\sqrt{\frac{\kappa}{\pi}}\int\text{d}z\psi_{2}^{\prime}=0,$
(47)

implying a cancellation between the contribution from the lowest vector meson
and the remaining tower. Indeed, the experimental result for
$a_{1}^{\pm}\to\pi^{\pm}\gamma$ is much smaller than expected from naive VMD
Zielinski:1984au .

####  III.3.1 Axial-vector $1\gamma$-decays

Employing VMD once we obtain for the interaction between one axial vector
meson, one vector meson and one photon

$\displaystyle\mathcal{L}_{\mathcal{V}v^{n}a^{p}}=$
$\displaystyle-\frac{N_{c}}{4\pi^{2}}d_{\mathcal{V}v^{n}a^{p}}\epsilon^{\mu\nu\rho\sigma}\,\text{tr}\,\left(v_{\mu}^{n}a_{\nu}^{p}\partial_{\rho}\mathcal{V}_{\sigma}\right),$
(48)

with

$\displaystyle d_{\mathcal{V}v^{n}a^{p}}=$
$\displaystyle\int\text{d}z\psi_{2n-1}^{\prime}\psi_{2p}=\\{-2497.14,\dots\\}N_{c}^{-1}\lambda^{-1},$
(49)

where we had to sum over the radial mode without the derivative to get a non-
vanishing result since the bulk-to-boundary propagator associated to an on-
shell photon is constant. The amplitudes for the decay $a\rightarrow
v\mathcal{V}$, for the different combinations of polarizations read

$\displaystyle\left|\mathcal{M}_{-101}^{a^{p}\rightarrow
v^{n}\mathcal{V}}\right|=$
$\displaystyle\frac{d_{\mathcal{V}v^{n}a^{p}}\left(m_{a^{p}}^{2}-m_{v^{n}}^{2}\right)N_{c}}{8m_{v^{n}}\pi^{2}}\,\text{tr}\,\left(eQT_{a^{p}}T_{v^{n}}\right)$
$\displaystyle\left|\mathcal{M}_{-110}^{a^{p}\rightarrow
v^{n}\mathcal{V}}\right|=$
$\displaystyle\frac{d_{\mathcal{V}v^{n}a^{p}}\left(m_{a^{p}}^{2}-m_{v^{n}}^{2}\right)N_{c}}{8m_{a^{p}}\pi^{2}}\,\text{tr}\,\left(eQT_{a^{p}}T_{v^{n}}\right),$
(50)

which yields

$\displaystyle\left|\mathcal{M}_{a^{p}\rightarrow
v^{n}\mathcal{V}}\right|^{2}=$
$\displaystyle\frac{1}{3}\left(2\left|\mathcal{M}_{-101}^{a^{p}\rightarrow
v^{n}\mathcal{V}}\right|^{2}+2\left|\mathcal{M}_{-110}^{a^{p}\rightarrow
v^{n}\mathcal{V}}\right|^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{d_{\mathcal{V}v^{n}a^{p}}^{2}\left(m_{a^{p}}^{2}-m_{v^{n}}^{2}\right)^{2}\left(m_{a^{p}}^{2}+m_{v^{n}}^{2}\right)N_{c}^{2}}{96\pi^{4}m_{a^{p}}^{2}m_{v^{n}}^{2}}\left(\,\text{tr}\,\left(eQT_{a^{p}}T_{v^{n}}\right)\right)^{2}.$
(51)

The decay width is given by

$\displaystyle\Gamma_{a^{p}\to v^{n}\gamma}=$
$\displaystyle\frac{1}{8\pi}\left|\mathcal{M}_{a^{p}\rightarrow
v^{n}\mathcal{V}}\right|^{2}\frac{\left|\mathbf{p}_{\mathcal{V}}\right|}{m_{a^{p}}^{2}},$
(52)

and the numerical results are listed in Table 2.

|  $\Gamma^{\text{exp}}$[keV] |  $\Gamma^{\text{WSS}}$[keV]
---|---|---
$a_{1}(1260)\rightarrow\rho\gamma$ |  | 28.9…50.8
$a_{1}(1260)\rightarrow\omega\gamma$ |  | 247…434
$f_{1}(1285)\rightarrow\rho\gamma$ | 1380(300)…640(240) | 295…518$|$270…473
$f_{1}(1285)\rightarrow\omega\gamma$ |  | 31.3…54.9$|$28.6…50.2
$f_{1}(1285)\rightarrow\phi\gamma$ | 17(7) | 2.44…4.29$|$3.97…6.98
$f_{1}(1420)\rightarrow\rho\gamma$ |  | 73.0…128$|$119…209
$f_{1}(1420)\rightarrow\omega\gamma$ |  | 7.80…13.7$|$12.7…22.3
$f_{1}(1420)\rightarrow\phi\gamma$ | 164(55) | 52.9…92.9$|$48.3…84.8
Table 2: Radiative axial-vector meson decay with $\lambda=16.63\dots 12.55$
and two values of the $f_{1}$ mixing angle
$\theta_{f}=20.4^{\circ}|26.4^{\circ}$. Experimental values are from the PDG
Workman:2022ynf  with the exception of the lower values for
$f_{1}(1285)\to\rho\gamma$, which are from VES Amelin:1994ii ; Zanke et al.
Zanke:2021wiq  propose here as experimental average 950(280) keV.

The PDG Workman:2022ynf  gives experimental results only for the $f_{1}$
mesons, which in the WSS model have the same mass as the $a_{1}$ meson.
Besides extrapolating to their experimental masses we consider also two
possible values (motivated below) for the mixing angle for the $f_{1}$ and
$f_{1}^{\prime}$ mesons using the convention

$\displaystyle|f_{1}(1285)\rangle$
$\displaystyle=\cos\theta_{f}|\bar{n}n\rangle-\sin\theta_{f}|\bar{s}s\rangle$
$\displaystyle|f_{1}(1420)\rangle$
$\displaystyle=\sin\theta_{f}|\bar{n}n\rangle+\cos\theta_{f}|\bar{s}s\rangle$
(53)

so that ideal mixing corresponds to $\theta_{f}=0$.

In Table 2, the $\phi$-$\omega$ mixing is again assumed to be ideal. A value a
bit above ideal mixing increases somewhat the branching ratio of $\phi\gamma$
over $\omega\gamma$ for $f_{1}(1285)$, while decreasing it for $f_{1}(1420)$.

####  III.3.2 Axial-vector $2\gamma$-decays

As mentioned above, the radial derivative of the bulk-to-boundary propagator
for a photon vanishes for on-shell photons, which implies that in accordance
with the Landau-Yang theorem at least one photon in the two-photon-decay of an
axial vector meson has to be off-shell. Denoting the virtual photon by $v^{*}$
we have

$\displaystyle d_{\mathcal{V}v^{*}a^{p}}=$
$\displaystyle\int\text{d}z\mathcal{J}^{\prime}\psi_{a^{p}},$ (54)

where we have introduced the (off-shell) bulk-to-boundary propagator
$\mathcal{J}$ defined by

$\displaystyle\left(1+z^{2}\right)^{1/3}\partial_{z}\left[\left(1+z^{2}\right)\partial_{z}\mathcal{J}\right]$
$\displaystyle=\frac{Q^{2}}{M_{\text{KK}}^{2}}\mathcal{J}.$ (55)

Since we are only interested in the low $Q$ regime we make the ansatz

$\displaystyle\mathcal{J}(Q,z)$
$\displaystyle=1+\frac{Q^{2}}{M_{\text{KK}}^{2}}\alpha(z)+\mathcal{O}(Q^{4})$
(56)

satisfying

$\displaystyle\left(1+z^{2}\right)^{1/3}\partial_{z}\left[\left(1+z^{2}\right)\partial_{z}\alpha\right]$
$\displaystyle=1.$ (57)

With the solution

$\displaystyle\partial_{z}\alpha=$
$\displaystyle\frac{z}{\left(1+z^{2}\right)}\;{}_{2}F_{1}\left(\frac{1}{3},\frac{1}{2},\frac{3}{2},-z^{2}\right)$
(58)

we obtain for the relevant coupling constant

$\displaystyle d_{\mathcal{V}v^{*}a^{p}}=$
$\displaystyle\frac{Q^{2}}{M_{\text{KK}}^{2}}\int\text{d}z\alpha^{\prime}\psi_{a^{p}}+\mathcal{O}(Q^{4})$
$\displaystyle=$
$\displaystyle\frac{Q^{2}}{M_{\text{KK}}^{2}}c_{\mathcal{V}v^{*}a^{p}}+\mathcal{O}(Q^{4})$
(59)

with

$\displaystyle c_{\mathcal{V}v^{*}a^{p}}=$ $\displaystyle
101.309N_{c}^{-1/2}\lambda^{-1/2}.$ (60)

The decay widths then read

$\displaystyle\Gamma(f(1285)\rightarrow\gamma_{L}^{*}\gamma_{T})=$
$\displaystyle\frac{2}{3}\left(\frac{c_{\mathcal{V}v^{*}a}m_{a}^{2}N_{c}}{8\pi^{2}M_{\text{KK}}^{2}}\right)^{2}\frac{1}{8\pi}\frac{|\mathbf{p}|}{m_{a}^{2}}\left(\frac{5e^{2}}{18}\cos\theta_{f}-\frac{e^{2}}{9\sqrt{2}}\sin\theta_{f}\right)^{2}Q^{2}+\mathcal{O}(Q^{4})$
$\displaystyle\Gamma(f(1285)\rightarrow\gamma_{T}^{*}\gamma_{T})=$
$\displaystyle\mathcal{O}\left(Q^{6}\right).$ (61)

and

$\displaystyle\Gamma(f(1420)\rightarrow\gamma_{L}^{*}\gamma_{T})=$
$\displaystyle\frac{2}{3}\left(\frac{c_{\mathcal{V}v^{*}a}m_{a}^{2}N_{c}}{8\pi^{2}M_{\text{KK}}^{2}}\right)^{2}\frac{1}{8\pi}\frac{|\mathbf{p}|}{m_{a}^{2}}\left(\frac{5e^{2}}{18}\sin\theta_{f}+\frac{e^{2}}{9\sqrt{2}}\cos\theta_{f}\right)^{2}Q^{2}+\mathcal{O}(Q^{4}).$
(62)

In the literature one usually finds the values for the so-called equivalent
photon rate

$\displaystyle\tilde{\Gamma}_{\gamma\gamma}=$
$\displaystyle\lim_{Q^{2}\rightarrow
0}\frac{m_{a}^{2}}{Q^{2}}\frac{1}{2}\Gamma\left(a\rightarrow\gamma_{L}^{*}\gamma_{T}\right),$
(63)

which are listed in 3.

|  $\tilde{\Gamma}_{\gamma\gamma}^{\text{exp}}$[keV] |  $\tilde{\Gamma}_{\gamma\gamma}^{\text{WSS}}$[keV]
---|---|---
$a_{1}\left(1260\right)$ | - | 1.60…2.12$|$1.39…1.85
$f_{1}\left(1285\right)$ | 3.5(8) | 3.84…5.09$|$2.39…3.17
$f_{1}\left(1420\right)$ | 3.2(9) | 3.50…4.64$|$2.19…2.90
Table 3: Equivalent photon rates of axial vector mesons for two values of the
$f_{1}$ mixing angle $\theta_{f}=20.4^{\circ}|26.4^{\circ}$ (in the latter
case with $M_{\mathrm{KK}}$ rescaled such that $m_{a}$ is raised to the
experimental value which reduces $\xi$ in (64) to zero); the range denoted by
dots corresponds again to $\lambda=16.63\dots 12.55$, where only the first
value is matching the axial anomaly exactly. Experimental values from L3
Achard:2001uu ; Achard:2007hm , see also Zanke:2021wiq .

The mixing angle is inferred from

$\tan^{2}\left(\theta_{f}-\arctan\frac{\sqrt{2}}{5}\right)=\left(\frac{m_{f_{1}}}{m_{f_{1}^{\prime}}}\right)^{1+\xi}\frac{\tilde{\Gamma}_{\gamma\gamma}^{f_{1}^{\prime}\text{exp}}}{\tilde{\Gamma}_{\gamma\gamma}^{f_{1}\text{exp}}},$
(64)

where the usual assumption of $\xi=0$ leads to $\theta_{f}=26.4^{\circ}$,
corresponding to the central value of $\theta_{A}=62(5)^{\circ}$ in
Zanke:2021wiq . However, in the WSS the coupling $d_{\mathcal{V}v^{*}a^{p}}$
is proportional to $1/M_{\text{KK}}^{2}$, which leads to a scaling of
$\tilde{\Gamma}_{\gamma\gamma}$ with four additional powers of $m_{a}$, i.e.
$\xi=4$, resulting in $\theta_{f}=20.4^{\circ}$.

In Tables 2 and 3 we consider two possible extrapolations to axial vector
mesons with realistic masses. In the first we keep the parameters of the
theory unchanged in the expressions for the couplings and use the measured
masses only in kinematical factors, which leads to $\xi=4$ and
$\theta_{f}=20.4^{\circ}$; in the second we rescale $M_{\text{KK}}$
proportional to $m_{a}^{\mathrm{exp}}/m_{a}^{\mathrm{WSS}}$ such that $\xi=0$
and $\theta_{f}=26.4^{\circ}$.

While the predictions for the equivalent photon rate for the $f_{1}$ mesons
(shown in Table 3) agree well with the experimental result for the standard
choice of $\lambda=16.63$ and $\theta_{f}=20.4^{\circ}$, the 1-$\gamma$ decay
rates are significantly underestimated. In contrast to the radiative decays of
vector mesons, lowering $\lambda$ does not increase the rates sufficiently to
cover the experimental results. Unfortunately no experimental results are
available for isotriplet axial vector mesons, where the WSS model is generally
performing best.

##  IV Glueballs in the Witten-Sakai-Sugimoto model

Glueballs are realized in the WSS model as fluctuations of the background in
which the probe D8 branes are placed, where certain superselection rules are
applied. In particular states with odd parity in the extra circle along $\tau$
are discarded, as well as Kaluza-Klein modes of the compact $S^{1}$ and
$S^{4}$ subspaces. The resulting glueball spectrum was discussed in
Brower:2000rp , where the lift of (1) to 11-dimensional supergravity is used.
In the following we shall consider scalar, tensor, and pseudoscalar glueballs,
for which hadronic decays have been worked out in the WSS model in
Hashimoto:2007ze ; Brunner:2015oqa ; Brunner:2015yha ; Brunner:2015oga ;
Leutgeb:2019lqu  and which we review and update in Appendix A in some detail
for the scalar and tensor glueballs.

The lift of a type IIA string-frame metric to 11-dimensional supergravity is
given by the relation

$\begin{split}\text{d}s^{2}&=G_{MN}\text{d}x^{M}\text{d}x^{N}\\\
&=e^{-2\phi/3}g_{AB}\text{d}x^{A}\text{d}x^{B}+e^{4\phi/3}\left(\text{d}x^{11}+A_{B}\text{d}x^{B}\right)^{2},\end{split}$
(65)

with $M,N=0,\dots 10$ and $A,B=0,\dots 9$, omitting the 11th index. By
introducing the radial coordinate $r$ related to $U$ by $U=\frac{r^{2}}{2L}$,
we get the lifted metric

$\text{d}s^{2}=\frac{r^{2}}{L^{2}}\left[f(r)\text{d}x_{4}^{2}+\eta_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}+\text{d}x_{11}^{2}\right]+\frac{L^{2}}{r^{2}}\frac{\text{d}r^{2}}{f\left(r\right)}+\frac{L^{2}}{4}\text{d}\Omega_{4}^{2},$
(66)

and the field strength
$F_{\alpha\beta\gamma\delta}=\frac{6}{L}\sqrt{g_{S^{4}}}\epsilon_{\alpha\beta\gamma\delta}$,
which are solutions to the equations of motion following from the unique
supergravity action

$2\kappa_{11}^{2}S_{11}=\int\mathrm{d}^{11}x\sqrt{-G}\left(R-\frac{1}{2}|F_{4}|^{2}\right)-\frac{1}{3!}\int
A_{3}\wedge F_{4}\wedge F_{4}.$ (67)

Scalar and tensor glueball modes appear as normalizable modes of metric
fluctuations $\delta G$, which translate to perturbations of the type-IIA
string metric and dilaton through

$\displaystyle g_{\mu\nu}=$
$\displaystyle\frac{r^{3}}{L^{3}}\left[\left(1+\frac{L^{2}}{2r^{2}}\delta
G_{11,11}\right)\eta_{\mu\nu}+\frac{L^{2}}{r^{2}}\delta G_{\mu\nu}\right]$
$\displaystyle g_{44}=$
$\displaystyle\frac{r^{3}f}{L^{3}}\left[1+\frac{L^{2}}{2r^{2}}\delta
G_{11,11}+\frac{L^{2}}{r^{2}f}\delta G_{44}\right]$ $\displaystyle g_{rr}=$
$\displaystyle\frac{L}{rf}\left[1+\frac{L^{2}}{2r^{2}}\delta
G_{11,11}+\frac{r^{2}f}{L^{2}}\delta G_{rr}\right]$ $\displaystyle g_{r\mu}=$
$\displaystyle\frac{r}{L}\delta G_{r\mu}$ $\displaystyle g_{\Omega\Omega}=$
$\displaystyle\frac{r}{L}\left(\frac{L}{2}\right)^{2}\left(1+\frac{L^{2}}{2r^{2}}\delta
G_{11,11}\right)$ $\displaystyle e^{4\phi/3}=$
$\displaystyle\frac{r^{2}}{L^{2}}\left(1+\frac{L^{2}}{r^{2}}\delta
G_{11,11}\right).$ (68)

Inducing these metric fluctuations to the world volume of the D8-brane system
described by the action (5), Hashimoto:2007ze  calculated interaction vertices
of the lightest scalar glueball with mesons, which was revisited and extended
in Brunner:2015oqa .

Pseudoscalar, vector, and pseudovector glueballs appear as fluctuations of the
type-IIA form fields; glueballs with higher spin would need a stringy
description beyond the supergravity approximation Sonnenschein:2018fph .

###  IV.1 Exotic and dilatonic scalar glueballs

Superficially, the emerging glueball spectrum resembles the one found in
lattice calculations (see Fig. 1 in Brunner:2015oga ), containing a lightest
scalar glueball with a mass below that of the tensor glueball, whereas most
other holographic models have the scalar glueball degenerate with the tensor.
This is achieved by an “exotic” polarization of the bulk metric involving the
extra compact dimension $(\tau)$ separating the D8-branes,

$\displaystyle\delta G_{\tau\tau}$
$\displaystyle=-\frac{r^{2}}{\mathcal{N}_{E}\,L^{2}}f(r)S_{4}(r)G_{E}(x^{\sigma}),$
$\displaystyle\delta G_{\mu\nu}$
$\displaystyle=\frac{r^{2}}{\mathcal{N}_{E}\,L^{2}}S_{4}(r)\left[\frac{1}{4}\eta_{\mu\nu}-\left(\frac{1}{4}+\frac{3r_{\text{KK}}^{6}}{5r^{6}-2r_{\text{KK}}^{6}}\right)\frac{\partial_{\mu}\partial_{\nu}}{M^{2}}\right]G_{E}(x^{\sigma}),$
$\displaystyle\delta G_{11,11}$
$\displaystyle=\frac{r^{2}}{\mathcal{N}_{E}\,4L^{2}}S_{4}(r)G_{E}(x^{\sigma}),$
$\displaystyle\delta G_{rr}$
$\displaystyle=-\frac{L^{2}}{\mathcal{N}_{E}\,r^{2}f(r)}\frac{3r_{\text{KK}}^{6}}{5r^{6}-2r_{\text{KK}}^{6}}S_{4}(r)G_{E}(x^{\sigma}),$
$\displaystyle\delta G_{r\mu}=\delta G_{\mu r}$
$\displaystyle=\frac{90r^{7}r_{\text{KK}}^{6}}{\mathcal{N}_{E}\,M^{2}L^{2}\left(5r^{6}-2r_{\text{KK}}^{6}\right)^{2}}S_{4}(r)\partial_{\mu}G_{E}(x^{\sigma}),$
(69)

with eigenvalue equation Brower:2000rp

$\frac{\text{d}}{\text{d}r}\left(r^{7}-r\,r_{KK}^{6}\right)\frac{\text{d}}{\text{d}r}S_{4}(r)+\left(L^{4}M_{E}^{2}r^{3}+\frac{432r^{5}r_{\text{KK}}^{12}}{\left(5r^{6}-2r_{KK}^{6}\right)^{2}}\right)S_{4}(r)=0.$
(70)

However, with $M_{E}=855$ MeV its mass is only a bit higher than that of the
$\rho$ meson, whereas the predominantly dilatonic mode that is the ground
state of another tower of scalar modes with respect to 3+1 dimensions is only
a little lighter than the traditional glueball candidates $f_{0}(1500)$ and
$f_{0}(1710)$. This mode is degenerate with the tensor mode and involves only
metric fluctuations $\delta G_{11,11}$ and $\delta G_{\mu\nu}$, see (108).

The exotic scalar glueball, denoted by $G_{E}$ in the following, turns out
Brunner:2015oqa  to have a relative width $\Gamma/M$ that is much higher than
that of the predominantly dilatonic scalar glueball ($G_{D}$), but only the
latter has a $\Gamma/M$ in the ballpark of $f_{0}(1500)$ and $f_{0}(1710)$.

It was therefore proposed in Brunner:2015oqa  to discard $G_{E}$ from the
spectrum of glueballs of the WSS model as a spurious mode that perhaps
disappears in the inaccessible limit $M_{\mathrm{KK}}\to\infty$, where the
supergravity approximation breaks down. Already in Constable:1999gb  it was
speculated that only one of the two scalar glueball towers might correspond to
the glueballs in QCD. Since it appears somewhat unnatural that an excited
scalar glueball should have a smaller width than the ground-state scalar
glueball, Brunner:2015oqa  preferred the dilatonic scalar glueball as
candidate for the actual ground state.

Indeed, the dilatonic scalar glueball turns out to have a decay pattern that
can match surprisingly well the glueball candidate $f_{0}(1710)$, in
particular when including additional couplings associated with the quark mass
term Brunner:2015yha ; Brunner:2015oga . This may actually apply instead to
$f_{0}(1770)$, which was proposed originally in BES:2004twe  as an additional
$f_{0}$ resonance between 1700 and 1800 MeV and more recently in
Sarantsev:2021ein  in radiative $J/\psi$ decays, where it appears dominantly
as the most glue-rich resonance.444The next (2023) update of the PDG
Workman:2022ynf  will in fact include $f_{0}(1770)$ as a separate resonance
(C. Amsler, private communication).

The fact that the ratio $\Gamma_{f_{0}\to K\bar{K}}/\Gamma_{f_{0}\to\pi\pi}$
is significantly higher for $f_{0}(1710)$ Workman:2022ynf  (or for
$f_{0}(1770)$ according to Sarantsev:2021ein ) than expected from a flavor-
symmetric glueball coupling can be attributed to the fact that dilaton
fluctuations couple naturally to quark mass terms, similar to, but more
pronounced than, in a model by Ellis and Lanik Ellis:1984jv . There is
therefore no need to invoke the previous conjecture of chiral suppression of
scalar glueball decay Carlson:1980kh ; Sexton:1995kd ; Chanowitz:2005du ,
which was questioned in Frere:2015xxa .

In the following we shall mainly explore the consequences of this
identification of the scalar glueball. In the radiative decay rates considered
here, the explicit quark masses will however not modify the (chiral) results
for the couplings; they are only included in phase space factors.

We shall however need to make assumptions on how to extrapolate to realistic
glueball masses, which we describe in more detail below. While the mass of
$f_{0}(1710)$ is not too much above the original mass of $G_{D}$ in the WSS
model, larger extrapolations are required for the tensor and pseudoscalar
glueballs when comparing to the various glueball candidates or lattice
results.

$M_{E}$ |  $\Gamma_{G_{E}}^{x=0}$[MeV] |  $\Gamma_{G_{E}}^{x=1}$[MeV]
---|---|---
855 | 72…96 | 85…113
1506 | 286…383 | 430…570
1712 | 351…469 | 483…640
1865 | 398…530 | 521…691

$M_{D}$ |  $\Gamma_{G_{D}}^{x=0}$[MeV] |  $\Gamma_{G_{D}}^{x=1}$[MeV]
---|---|---
1487 | 19…26 | 80…106
1506 | 19…27 | 80…106
1712 | 88…113 | 139…180
1865 | 151…197 | 198…259

Table 4: Total decay widths of the exotic and the dilatonic scalar glueball
$G_{E}$ and $G_{D}$ with original masses of 855 MeV and 1487 MeV,
respectively, and also with extrapolations to the masses of the glueball
candidates $f_{0}(1510)$ or $f_{0}(1710)$ and the fragmented glueball of
Sarantsev:2021ein ; Klempt:2021nuf ; Klempt:2021wpg , for two choices of the
extra coupling parameter $x$ associated with the quark mass term as defined in
Brunner:2015oga . The range of results corresponds again to
$\lambda=16.63\dots 12.55$. In addition to the two-body decays reviewed in
Appendix A, the decays $G_{D}\to\rho\pi\pi\to 4\pi$ which interfere
destructively with $G_{D}\to\rho\rho\to 4\pi$ have been taken into account
here.

As an alternative scenario, we shall also consider the option of keeping the
exotic scalar glueball mode $G_{E}$, whose relative decay width $\Gamma/M$ is
much too large to be identified with the traditional glueball candidates
$f_{0}(1510)$ or $f_{0}(1710)$ with total width 112(9) MeV and 128(18) MeV,
respectively, see Table 4. It would in fact fit better to the proposal in
Sarantsev:2021ein ; Klempt:2021nuf ; Klempt:2021wpg  of a relatively broad
fragmented glueball of mass 1865 MeV and a width of 370 MeV that does not show
up as a separate meson but only as admixture in the mesons $f_{0}(1710)$, a
novel $f_{0}(1770)$, $f_{0}(2020)$, and $f_{0}(2100)$. Of course, this
requires a drastic rise of the original mass of $G_{E}$ by a factor of over 2,
but also the mass of the tensor mode $G_{T}$ would have to be raised by a
factor of 1.6 to match the expectation of $m_{T}\sim 2400$ MeV from lattice
QCD; the mass of $G_{D}$, which would then be identified with the first
excited scalar glueball, would need to be raised somewhat more, as lattice
results point to a mass above the tensor glueball, from around 2670
MeVMorningstar:1999rf  to around 3760 MeV Gregory:2012hu .

###  IV.2 Extrapolations to realistic glueball masses

In the WSS model, the masses of glueballs are given by pure numbers times
$M_{\mathrm{KK}}$, which is also the case for the (axial) vector mesons.
However, when $M_{\mathrm{KK}}$ has been fixed by the mass of the $\rho$
meson, the glueball masses appear to be too small compared to lattice QCD
results.

In order to predict decay rates for different glueball candidates we manually
change the masses of glueball modes in amplitudes and phase space integrals,
which could be viewed as assuming a different scale $M_{\mathrm{KK}}$ for the
glueball sector. The coupling constants involving glueballs are all inversely
proportional to $M_{\mathrm{KK}}$ and we interpret this appearance of
$M_{\mathrm{KK}}$ to be tied to the mass scale of glueballs, which shows up
also in their normalization factors $\mathcal{N}$, whereas explicit
appearances of $M_{\mathrm{KK}}$ in the DBI action of the D branes are
considered as being fixed like the mass of the $\rho$ meson. When upscaling
glueball masses, we have therefore correspondingly reduced the dimensionful
glueball-meson/photon coupling constants. [Without such a rescaling, the
results for all glueball decay rates and the glueball contributions to
$a_{\upmu}$ presented in Sect. V extrapolated to some mass $M_{G}$ would be
simply larger by a factor $(M_{G}/M_{G}^{\mathrm{WSS}})^{2}$.]

We consider this rescaling plausible in that the overlap integrals of glueball
and meson holographic profiles should become smaller when glueball and meson
modes are separated further in energy. It may well be, however, that this
reduction is only insufficiently accounted for by the overall change of the
mass scale in the glueball coupling constants; thus our numerical results
should be considered as somewhat rough estimates.

##  V Radiative Glueball Decays

In the following we shall concentrate on glueball interactions involving
vector mesons which through VMD also give rise to glueball-photon vertices.
Other hadronic interactions of glueballs are reviewed in Appendix A.

We shall consider the first three lightest glueball states, scalar, tensor,
and pseudoscalar in turn, choosing the dominantly dilatonic scalar glueball
over the exotic scalar glueball, since the former has been found to match
remarkably well to the decay pattern of the glueball candidate $f_{0}(1710)$.
The more unwieldy results for the exotic scalar glueball are worked out in
Appendix A and B.

###  V.1 Dilatonic Scalar Glueball Decays

Inducing the fluctuation (108) in the D8 brane action (5) we obtain the
interaction terms of the dilatonic scalar glueball with two vector mesons as

$\displaystyle\mathcal{L}_{G_{D}v^{m}v^{n}}=$
$\displaystyle\,\text{tr}\,\int\text{d}^{4}x\,\left(d_{3}^{mn}\eta^{\rho\sigma}F_{\mu\rho}^{m}F_{\nu\sigma}^{n}+d_{2}^{mn}M_{\text{KK}}^{2}v_{\mu}^{m}v_{\nu}^{n}\right)\left(\eta^{\mu\nu}-\frac{\partial^{\mu}\partial^{\nu}}{\Box}\right)G_{D},$
(71)

where the coupling constants are given by

$\begin{split}&d_{2}^{mn}=\kappa\int\text{d}z\,K\psi_{2n-1}^{\prime}\psi_{2m-1}^{\prime}H_{D}=\\{4.3714,\dots\\}\frac{1}{\lambda^{\frac{1}{2}}N_{c}M_{\text{KK}}},\\\
&d_{3}^{mn}=\kappa\int\text{d}z\,K^{-1/3}\psi_{2n-1}\psi_{2m-1}H_{D}=\\{18.873,\dots\\}\frac{1}{\lambda^{\frac{1}{2}}N_{c}M_{\text{KK}}}.\end{split}$
(72)

Restricting to the ground-state vector mesons ($m=n=1$), the amplitudes for
the decay of the dilatonic scalar glueball into vector mesons with transverse
and longitudinal polarizations read

$\displaystyle\left|\mathcal{M}_{T}^{\left(G_{D}\rightarrow
v^{1}v^{1}\right)}\right|$
$\displaystyle=\left[d_{3}^{11}\left(2m_{v^{1}}^{2}-\frac{3M_{D}^{2}}{4}\right)-d_{2}^{11}M_{\text{KK}}^{2}\right],$
$\displaystyle\left|\mathcal{M}_{L}^{\left(G_{D}\rightarrow
v^{1}v^{1}\right)}\right|$
$\displaystyle=\left[\frac{d_{2}^{11}M_{D}^{2}M_{\text{KK}}^{2}}{4m_{v^{1}}^{2}}+d_{3}^{11}m_{v^{1}}^{2}\right],$
(73)

in terms of which the partial decay width is given by

$\displaystyle\Gamma_{G_{D}\rightarrow v^{1,a}v^{1,b}}$
$\displaystyle=\frac{1}{S}\left(2\left|\mathcal{M}_{T}^{\left(G_{D}\rightarrow
v^{1}v^{1}\right)}\right|^{2}+\left|\mathcal{M}_{L}^{\left(G_{D}\rightarrow
v^{1}v^{1}\right)}\right|^{2}\right)\frac{\left|\mathbf{p}_{v^{1}}\right|}{8\pi
M_{D}^{2}},$ (74)

where $S$ equals 2 for identical particles ($a=b$) and 1 otherwise.

In the narrow-resonance approximation, this vanishes for the WSS model mass
$M_{D}=1487$ MeV, which is below the threshold of two $\rho$ mesons. However,
when $M_{D}$ is manually adjusted to the mass of $f_{0}(1710)$, which we
assume as 1712 MeV (the average of the $T$-matrix pole results of
WA102:1999fqy  and WA102:2000lao ), the decay $G_{D}\to\rho\rho$ becomes the
largest channel, exceeding even the dominant pseudoscalar channel $G_{D}\to
KK$ (see Appendix A, Table 8).

As discussed in Brunner:2015oqa , the holographic prediction for the total
rate $G_{D}\to 4\pi$ is somewhat reduced by a destructive interference from
$G_{D}\to\rho\pi\pi$, rendering the partial width of $G_{D}\to 4\pi$ similar
to and slightly less than $G_{D}\to KK$ Brunner:2015yha . Remarkably, data
from radiative $J/\psi$ decays BES:1999dmf  for $f_{0}(1740)$ (or
$f_{0}(1770)$ in Sarantsev:2021ein ) seem to be fairly consistent with this
result.

####  V.1.1 Dilatonic scalar glueball $1\gamma$-decays

From the interaction terms (71) we can also derive the interactions including
photons by using VMD. Replacing one vector meson by a photon we find

$\displaystyle\mathcal{L}_{G_{D}\mathcal{V}v^{m}}=$ $\displaystyle
2d_{3}^{m\mathcal{V}}\eta^{\rho\sigma}\,\text{tr}\,\left(F_{\mu\rho}^{m}F_{\nu\sigma}^{\mathcal{V}}\right)\left(\eta^{\mu\nu}-\frac{\partial^{\mu}\partial^{\nu}}{\Box}\right)G_{D},$
(75)

with

$\displaystyle d_{3}^{m\mathcal{V}}\equiv$
$\displaystyle\kappa\int\text{d}z\,K^{-1/3}\psi_{2m-1}H_{D}$
$\displaystyle=\left\\{0.46895,\dots\right\\}\frac{1}{M_{\text{KK}}\sqrt{N_{c}}}.$
(76)

The other coupling $d_{2}^{m\mathcal{V}}$ vanishes for an on-shell photon,
since at zero virtuality its radial mode is constant and drops out in the
replacement $\psi^{\prime}\to\mathcal{J}^{\prime}=0$.

In radiative decays, only the transverse amplitude remains, which reads

$\displaystyle\left|\mathcal{M}_{T}^{\left(G_{D}\rightarrow\mathcal{V}v^{m}\right)}\right|$
$\displaystyle=\frac{d_{3}^{m\mathcal{V}}\left(m^{4}_{v}-4m^{2}_{v}M_{D}^{2}+3M_{D}^{4}\right)}{2M_{D}^{2}}\,\text{tr}\,\left(eQT_{v}\right),$
(77)

yielding

$\displaystyle\Gamma_{G_{D}\rightarrow v^{m}\gamma}$
$\displaystyle=2\left|\mathcal{M}_{T}^{\left(G_{D}\rightarrow\mathcal{V}v^{m}\right)}\right|^{2}\frac{\left|\mathbf{p}_{\mathcal{V}}\right|}{8\pi
M_{D}^{2}}.$ (78)

The results are displayed in Table 5 for two mass parameters corresponding to
$f_{0}(1500)$ and $f_{0}(1710)$, where ideal mixing was assumed for the
$\omega$ and $\phi$ mesons. The latter implies that $\rho\gamma$ and
$\omega\gamma$ decay rates are very close to the ratio $9:1$. The ratio of
decay rates $\phi\gamma$ and $\omega\gamma$, which would be 2:1 with equal
masses, is, however, significantly reduced by the larger $\phi$ mass.555A more
realistic value for the $\phi$-$\omega$ mixing angle of 3.32∘ above ideal
mixing Ambrosino:2009sc  increases the partial width for $\omega\gamma$ by
about 17% and decreases the one for $\phi\gamma$ by about 8.5%. This also
holds true for all the other glueball decay widths below.

####  V.1.2 Dilatonic scalar glueball $2\gamma$-decays

Replacing the second vector meson by a photon by means of VMD, we obtain the
$2\gamma$ interactions

$\displaystyle\mathcal{L}_{G_{D}\mathcal{V}\mathcal{V}}=$ $\displaystyle
d_{3}^{\mathcal{V}\mathcal{V}}\eta^{\rho\sigma}\,\text{tr}\,\left(F_{\mu\rho}^{\mathcal{V}}F_{\nu\sigma}^{\mathcal{V}}\right)\left(\eta^{\mu\nu}-\frac{\partial^{\mu}\partial^{\nu}}{\Box}\right)G_{D},$
(79)

with

$\displaystyle d_{3}^{\mathcal{V}\mathcal{V}}\equiv$
$\displaystyle\kappa\int\text{d}z\,K^{-1/3}H_{D}=0.0130195\lambda^{1/2}M_{\text{KK}}^{-1}$
(80)

which gives

$\displaystyle\left|\mathcal{M}_{T}^{\left(G_{D}\rightarrow\mathcal{V}\mathcal{V}\right)}\right|$
$\displaystyle=\frac{3}{2}d_{3}^{\mathcal{V}\mathcal{V}}M_{D}^{2}\,\text{tr}\,(e^{2}Q^{2})$
(81)

The resulting width

$\displaystyle\Gamma_{G_{D}\rightarrow\gamma\gamma}$
$\displaystyle=\frac{1}{8\pi}\left|\mathcal{M}_{T}^{\left(G_{D}\rightarrow\mathcal{V}\mathcal{V}\right)}\right|^{2}\frac{\left|\mathbf{p}_{\mathcal{V}}\right|}{M_{D}^{2}}$
(82)

is again displayed in Table 5 for the two mass parameters corresponding to
$f_{0}(1500)$ and $f_{0}(1710)$, which in both cases is above 1 keV.

This is larger than the old prediction by Kada et al. Kada:1988rs , but an
order of magnitude smaller than the VMD based result of Cotanch and Williams
Cotanch:2005ja , who obtained 15.1 keV for a scalar glueball with mass 1700
MeV after correcting their previous result of 2.6 keV in Cotanch:2004py  (note
that the corresponding preprint has erroneously 2.6 eV instead). Also all
other radiative decay rates obtained in Kada:1988rs  are about an order of
magnitude larger than ours (not uniformly so, however, but varying between a
factor of 7 to 26, thereby deviating from the ratios discussed at the end of
Sect. V.1.1).

|  $\Gamma_{G_{D}}$[keV]
---|---
$f_{0}\left(1500\right)\rightarrow\rho\gamma$ | 184
$f_{0}\left(1500\right)\rightarrow\omega\gamma$ | 19.9
$f_{0}\left(1500\right)\rightarrow\phi\gamma$ | 14.1
$f_{0}\left(1500\right)\rightarrow\gamma\gamma$ | 1.74…1.32
$f_{0}\left(1710\right)\rightarrow\rho\rho$ | (53.5…71.0)$\cdot 10^{3}$
$f_{0}\left(1710\right)\rightarrow\omega\omega$ | (16.6…22.0)$\cdot 10^{3}$
$f_{0}\left(1710\right)\rightarrow\rho\gamma$ | 276
$f_{0}\left(1710\right)\rightarrow\omega\gamma$ | 30.1
$f_{0}\left(1710\right)\rightarrow\phi\gamma$ | 29.4
$f_{0}\left(1710\right)\rightarrow\gamma\gamma$ | 1.98…1.50
Table 5: Radiative scalar glueball decay with $G_{D}$ identified alternatively
with $f_{0}(1500)$ and $f_{0}(1710)$ with masses 1506 MeV and 1712 MeV,
respectively, for $\lambda=16.63\dots 12.55$.

On the other hand, the two-vector meson decay rates obtained in Cotanch:2004py
(44.4 MeV for $\rho\rho$ and 34.6 MeV for $\omega\omega$) are not very far
from our results. In fact, our holographic prediction for
$f_{0}(1710)\to\omega\omega$ with $f_{0}(1710)$ as a (predominantly dilaton)
glueball appears to be in the right ballpark considering the measured
branching ratios of radiative $J/\psi$ decays in $\gamma f_{0}(1710)\to\gamma
K\bar{K}$ and $\gamma f_{0}(1710)\to\gamma\omega\omega$ Workman:2022ynf
(which according to Sarantsev:2021ein  may be instead $f_{0}(1770)$). The PDG
Workman:2022ynf  quotes two results for $\mathcal{B}(K\bar{K})$: a BNL
measurement Longacre:1986fh  from 1986 with
$\mathcal{B}(K\bar{K})=0.38^{+0.09}_{-0.19}$ and a phenomenological analysis
Albaladejo:2008qa  concluding $0.36(12)$, which both are consistent with the
WSS result obtained in Brunner:2015yha  as approximately 0.35. Using
$\mathcal{B}(K\bar{K})=0.36(12)$ and the total decay width of $f_{0}(1710)$
Workman:2022ynf  of 123(18) MeV lead to a partial decay width for
$f_{0}(1710)\to\omega\omega$ of about 15(8) MeV, for which the holographic
prediction from $G_{D}$ amounts to $16.6\dots 22.0$ MeV.

No experimental results for single-photon decays of $f_{0}(1710)$ appear to be
available, but in Belle:2013eck  the BELLE collaboration reports a measurement
of $f_{0}(1710)\rightarrow\gamma\gamma$ with the result
$\Gamma_{\gamma\gamma}\mathcal{B}(K\bar{K})=12_{-2-8}^{+3+227}\;\mathrm{eV}$,
with the stated conclusion that the $f_{0}(1710)$ meson was unlikely to be a
glueball because of a width larger than that expected (“much less than 1 eV”)
for a pure glueball state. However the holographic prediction for
$\Gamma_{\gamma\gamma}\mathcal{B}(K\bar{K})\approx 690\dots 520\;\mathrm{eV}$
is 3-2 $\sigma$ above the upper limit of the BELLE result.666Older upper
limits for $\Gamma_{\gamma\gamma}\mathcal{B}(K\bar{K})$ are 480 eV from ARGUS
ARGUS:1989ird , 200 eV from CELLO CELLO:1988xbx , and 560 eV from TASSO
TASSO:1985tme . (The latter two are quoted by the PDG Workman:2022ynf  with
lower values, 110 eV and 280 eV, respectively, corresponding however to the
assumption of helicity 2 which leads to smaller upper limits.) Ironically, the
BELLE result for the two-photon rate appears to be rather too small for a pure
(predominantly dilaton) glueball interpretation of $f_{0}(1710)$ within the
WSS model.777Assuming a tensor glueball $f_{2}(1720)$, Kada:1988rs  predicted
$\Gamma_{\gamma\gamma}\mathcal{B}(K\bar{K})\approx 95$ eV. The central value
of the BELLE result for $\Gamma_{\gamma\gamma}$ of only a few tens of eV would
thus seem to indicate that VMD does not apply for radiative decays of
$f_{0}(1710)$.

In Appendix B we also evaluate radiative decays of the exotic glueball of the
WSS model. The two-photon decay width of $G_{E}$ is considerably smaller than
that of $G_{D}$, 87…65 eV, when the mass of $G_{E}$ is extrapolated to that of
$f_{0}(1710)$. However, the decay pattern of $G_{E}$ does not fit to either
$f_{0}(1500)$ or $f_{0}(1710)$ when extrapolating to their masses.

###  V.2 Tensor Glueball Decays

The holographic mode functions associated with tensor glueballs are reviewed
in Appendix A.3 together with the results of hadronic two-body decays.

Radiative decays of tensor glueballs can be derived from the interaction terms
with two vector mesons, which are given by

$\displaystyle\mathcal{L}_{G_{T}v^{m}v^{n}}=$
$\displaystyle\,\text{tr}\,\left[t_{2}M_{\text{KK}}^{2}v_{\mu}^{m}v_{\nu}^{n}G_{T}^{\mu\nu}+t_{3}F_{\mu\rho}^{m}F_{\nu}^{n\rho}G_{T}^{\mu\nu}\right],$
(83)

with

$\displaystyle t_{2}^{mn}=$
$\displaystyle\int\text{d}zK\psi_{2m-1}^{\prime}\psi_{2n-1}^{\prime}T=2\sqrt{3}d_{2}^{mn}.$
(84) $\displaystyle t_{3}^{mn}=$
$\displaystyle\int\text{d}zK^{-1/3}\psi_{2m-1}\psi_{2n-1}T=2\sqrt{3}d_{3}^{mn}$
(85)

and $d_{2,3}^{mn}$ as given in (LABEL:d2d3). (Note that due to a different
normalization of the tensor field, the tensor coupling constants differ from
those in Brunner:2015oqa  by a factor $\sqrt{2}$; all other glueball coupling
constants are defined as in Brunner:2015oqa .)

The decay rate of a tensor glueball into two vector mesons reads

$\displaystyle\Gamma_{G_{T}\to vv}=\frac{1}{S}\biggl{\\{}$
$\displaystyle\frac{t_{2}^{2}}{120}\frac{M_{\text{KK}}^{4}}{m_{v}^{4}}(M_{G}^{4}+12m_{v}^{2}M_{G}^{2}+56m_{v}^{4})$
$\displaystyle+\frac{2}{3}t_{2}t_{3}M_{\text{KK}}^{2}(M_{G}^{2}-m_{v}^{2})$
$\displaystyle+\frac{t_{3}^{2}}{10}(M_{G}^{4}-3m_{v}^{2}M_{G}^{2}+6m_{v}^{4})\biggr{\\}}\frac{\left|\mathbf{p}_{v}\right|}{8\pi
M_{G}^{2}},$ (86)

where $S$ is again the symmetry factor for identical particles.

####  V.2.1 Tensor glueball $1\gamma$-decays

Through VMD (83) leads to a coupling of the tensor glueball with one photon
and one vector meson with interaction Lagrangian

$\displaystyle\mathcal{L}_{G_{T}v^{n}\mathcal{V}}=$ $\displaystyle
2t_{3}^{\mathcal{V}n}G_{T}^{\mu\nu}\eta^{\rho\sigma}\,\text{tr}\,\left(F_{\mu\rho}^{\mathcal{V}}F_{\nu\sigma}^{n}\right),$
(87)

with

$\displaystyle t_{3}^{\mathcal{V}n}=$
$\displaystyle\int\text{d}zK^{-1/3}\psi_{2n-1}T=2\sqrt{3}d_{3}^{\mathcal{V}n}$
(88)

and $d_{3}^{\mathcal{V}n}$ as given in (V.1.1).

This yields

$\Gamma_{G_{T}\to
v\gamma}=\frac{1}{15M_{G}^{4}}\left(\,\text{tr}\,\left(eQT_{v}\right)\right)^{2}\left(M_{G}^{2}-m_{v}^{2}\right)^{2}\left(6M_{G}^{4}+3M_{G}^{2}m_{v}^{2}+2m_{v}^{4}\right)\frac{|\mathbf{p}_{v}|}{8\pi
M_{G}^{2}}.$ (89)

####  V.2.2 Tensor glueball $2\gamma$-decays

Similarly (83) leads to

$\displaystyle\mathcal{L}_{G_{T}\mathcal{V}\mathcal{V}}=$ $\displaystyle
t_{3}^{\mathcal{V}\mathcal{V}}G_{T}^{\mu\nu}\eta^{\rho\sigma}\,\text{tr}\,\left(F_{\mu\rho}^{\mathcal{V}}F_{\nu\sigma}^{\mathcal{V}}\right),$
(90)

with

$\displaystyle t_{3}^{\mathcal{V}\mathcal{V}}=$
$\displaystyle\int\text{d}zK^{-1/3}T=2\sqrt{3}d_{3}^{\mathcal{V}\mathcal{V}}.$
(91)

and $d_{3}^{\mathcal{V}\mathcal{V}}$ as given in (80).

The resulting two-photon decay width of the tensor glueball is given by

$\Gamma_{G_{T}\to\gamma\gamma}=\frac{1}{5}\left[t_{3}^{\mathcal{V}\mathcal{V}}M_{G}^{2}\,\text{tr}\,\left(e^{2}Q^{2}\right)\right]^{2}\frac{|\mathbf{p}_{\gamma}|}{8\pi
M_{G}^{2}}.$ (92)

The resulting partial widths are listed in Table 6 for three values of the
mass of the tensor glueball, the unrealistically small WSS model mass value
1487 MeV as well as two higher values motivated by pomeron physics
Donnachie:2002en 888A candidate for a tensor glueball around 2000 MeV is the
broad resonance $f_{2}(1950)$, which has recently also been argued for in
Vereijken:2023jor  on the basis of a chiral hadronic model. The latter turns
out to yield a dominance of the decay modes into two vector mesons, in
qualitative agreement with the WSS model, which in fact predicts a very broad
tensor glueball (see Appendix A.3). and QCD lattice studies Morningstar:1999rf
, respectively, assuming ideal mixing of $\omega$ and $\phi$ mesons. With
increasing mass of the glueball, the partial decay widths for $\rho\gamma$,
$\omega\gamma$, and $\phi\gamma$ gradually approach the ratios $9:1:2$ for
degenerate vector meson masses; again, a more realistic value of $\theta_{V}$
changes the $\omega\gamma$ and $\phi\gamma$ results only slightly (cf.
footnote 5).

The radiative decay widths obtained for the tensor glueball turn out to be
comparable with those for the dilatonic scalar glueball for equal glueball
mass, rising approximately linear with glueball mass (due to the rescaling
described in Sect. IV.2).

Our prediction of the two-photon width of $\sim$ 2-3 keV is significantly
larger than the old prediction of Kada et al. Kada:1988rs  who have values in
the range of hundreds of eV, and also higher than the more recent prediction
in Godizov:2016vuw , where $\Gamma_{f_{2}(1950)\to\gamma\gamma}=960(50)$ eV
was obtained. Cotanch and Williams Cotanch:2005ja , on the other hand, have
also results above 1 keV, $\Gamma_{G_{T}(2010)\to\gamma\gamma}=1.72$ keV and
$\Gamma_{G_{T}(2300)\to\gamma\gamma}=1.96$ keV, by using VMD. Also their
results for single-photon decays are comparable with ours, even though their
results for decays into two vector mesons are significantly smaller than ours.
A particular point of disagreement is their result for a relatively large
$\omega\phi$ decay mode, which in the WSS model is absent. As noted in
Giacosa:2005bw , this is possible only by allowing for a rather strong
deviation from the large-$N_{c}$ limit.

|  $\Gamma_{G_{T}^{\mathrm{WSS}}}$[keV] |  $\Gamma_{G_{T}(2000)}$[keV] |  $\Gamma_{G_{T}(2400)}$[keV]
---|---|---|---
$G_{T}\rightarrow\rho\rho$ | - | (270…358)$\cdot 10^{3}$ | (382…507)$\cdot 10^{3}$
$G_{T}\rightarrow\omega\omega$ | - | (88.2…117)$\cdot 10^{3}$ | (127…169)$\cdot 10^{3}$
$G_{T}\rightarrow K^{*}K^{*}$ | - | (240…318)$\cdot 10^{3}$ | (417…552)$\cdot 10^{3}$
$G_{T}\rightarrow\phi\phi$ | - | - | (76.7…102)$\cdot 10^{3}$
$G_{T}\rightarrow\rho\gamma$ | 260 | 522 | 716
$G_{T}\rightarrow\omega\gamma$ | 28.3 | 57.5 | 79.1
$G_{T}\rightarrow\phi\gamma$ | 24.7 | 81.1 | 127
$G_{T}\rightarrow\gamma\gamma$ | 1.84…1.39 | 2.47…1.86 | 2.97…2.24
Table 6: Radiative tensor glueball decays and decays into two vector mesons
for $\lambda=16.63\dots 12.55$. Besides the pristine results for the WSS model
mass of 1487 MeV, their extrapolations to glueball masses of 2000 and 2400 MeV
are given.

###  V.3 Pseudoscalar Glueball Decays

In the WSS model, the pseudoscalar glueball is represented by a Ramond-Ramond
1-form field $C_{1}$, which has a kinetic mixing with the singlet $\eta_{0}$
given by Leutgeb:2019lqu

$\eta_{0}\to\eta_{0}+\zeta_{2}\,G_{PS}=\eta_{0}+0.01118\sqrt{N_{f}/N_{c}}\,\lambda\,G_{PS},$
(93)

with $G_{PS}$ remaining unchanged to leading order in $\sqrt{N_{f}/N_{c}}$
(formally treated as a small quantity because of the probe brane
approximation). In contrast to the conventional mixing scenarios of Ref.
Rosenzweig:1981cu ; Mathieu:2009sg  mass mixing is absent here, while the mass
of the pseudoscalar glueball is raised by a factor $(1+\zeta_{2}^{2})$ from
1789 MeV to (1819.7…1806.5) MeV for $\lambda=16.63\dots 12.55$. Lattice QCD
(in the quenched approximation), however, typically finds values around 2600
MeV, so we also consider the latter in our extrapolations.999Note that
historically the pseudoscalar glueball was expected to be the lightest
glueball, with $\eta(1405)$ a prominent candidate after $\iota(1440)$
Edwards:1982nc  was split into $\eta(1405)$ and $\eta(1475)$. This is still
occasionally considered a possibility, see for example Masoni:2006rz  and
Cheng:2008ss .

Through (93) the pseudoscalar glueball acquires the same interactions as
$\eta_{0}$, and the same form of transition form factors, only with
correspondingly modified coupling constants. Thus the formulae given in III.2
for the decays of pseudoscalars in vector mesons or photons remain essentially
unchanged, but the higher mass of the pseudoscalar glueball permits also
decays into pairs of vector mesons.

The resulting interaction Lagrangian reads

$\mathcal{L}_{G_{PS}vv/v\mathcal{V}/\mathcal{V}\mathcal{V}}=G_{PS}\epsilon^{\mu\nu\rho\sigma}\,\text{tr}\,\left[k_{1}^{vv}\partial_{\mu}v_{\nu}\partial_{\rho}v_{\sigma}+2k_{1}^{v\mathcal{V}}\partial_{\mu}v_{\nu}\partial_{\rho}\mathcal{V}_{\sigma}+k_{1}^{\mathcal{V}\mathcal{V}}\partial_{\mu}\mathcal{V}_{\nu}\partial_{\rho}\mathcal{V}_{\sigma}\right]$
(94)

with101010The couplings differ by a factor of 2 from Leutgeb:2019lqu  since we
use SU(3) generators $T^{a}=\lambda^{a}/2$.

$\displaystyle k_{1}^{v^{1}v^{1}}$
$\displaystyle=19.6184N_{c}^{-1}\lambda^{-1/2}M_{\text{KK}}^{-1},$ (95)
$\displaystyle k_{1}^{v^{1}\mathcal{V}}$
$\displaystyle=0.493557N_{c}^{-1/2}M_{\text{KK}}^{-1},$ (96) $\displaystyle
k_{1}^{\mathcal{V}\mathcal{V}}$
$\displaystyle=0.0145232\lambda^{1/2}M_{\text{KK}}^{-1}.$ (97)

The various resulting partial widths are listed in Table 7.

In the WSS model, all other hadronic decay channels of the pseudoscalar
glueball, such as those considered in Eshraim:2012jv ; Eshraim:2016mds , turn
out to be very weak compared to two-vector-meson decays Leutgeb:2019lqu . The
relative strength of the latter entails correspondingly important radiative
decay modes, and a two-photon partial width in the keV range. Note, however,
that these results have been obtained from the first term in a formal
expansion in $\sqrt{N_{f}/N_{c}}$, which is not a small parameter in real QCD.
It might nevertheless be meaningful, since the parameter $\zeta_{2}$ in (93)
is reasonably small, 0.19…0.14 for $\lambda=16.63\dots 12.55$.

|  $\Gamma_{G_{PS}^{\mathrm{WSS}}}$[keV] |  $\Gamma_{G_{PS}(2600)}$[keV]
---|---|---
$G_{PS}\rightarrow\rho\rho$ | (36.8…45.0)$\cdot 10^{3}$ | (190…248)$\cdot 10^{3}$
$G_{PS}\rightarrow\omega\omega$ | (11.3…13.8)$\cdot 10^{3}$ | (62.2…81.3)$\cdot 10^{3}$
$G_{PS}\rightarrow\phi\phi$ | - | (29.2…38.2)$\cdot 10^{3}$
$G_{PS}\rightarrow K^{*}K^{*}$ | (2.69…1.81)$\cdot 10^{3}$ | (188…246)$\cdot 10^{3}$
$G_{PS}\rightarrow\rho\gamma$ | 272…263 | 536…528
$G_{PS}\rightarrow\omega\gamma$ | 29.8…28.9 | 59.2…58.3
$G_{PS}\rightarrow\phi\gamma$ | 35.6…34.1 | 95.4…94.0
$G_{PS}\rightarrow\gamma\gamma$ | 1.75…1.30 | 2.49…1.86
Table 7: Radiative pseudoscalar glueball decay and decays into two vector
mesons $\lambda=16.63\dots 12.55$. Besides the WSS model result for the
pseudoscalar mass, $M_{G}=1813\pm 7\text{MeV}$, an extrapolation to 2600 MeV
(motivated by lattice results) is considered.

##  VI Glueball contributions to hadronic light-by-light scattering and the
muon $g-2$

In order to calculate the contribution of the glueball exchange diagram in the
light-by-light scattering amplitude, which enters the muon-photon vertex at
two loop order, the above results for the vertices of a glueball with two on-
shell photons need to be generalized to nonzero photon virtualities.

In the case of the dilatonic scalar glueball $G_{D}$, this involves two
interaction terms that are obtained by replacing $v_{\mu}$ in (71) by
$eQA_{\mu}^{e.m.}$ and the holographic profile functions $\psi(z)$ in
(LABEL:d2d3) by the bulk-to-boundary propagator $\mathcal{J}(Q,z)$ defined in
(55), yielding two form factors,

$\displaystyle d_{2}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})$
$\displaystyle\equiv\kappa\int
dz\,K\partial_{z}\mathcal{J}(Q_{1},z)\partial_{z}\mathcal{J}(Q_{2},z)H_{D}(z),$
$\displaystyle d_{3}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})$
$\displaystyle\equiv\kappa\int
dz\,K^{-1/3}\mathcal{J}(Q_{1},z)\mathcal{J}(Q_{2},z)H_{D}(z),$ (98)

in place of the coupling constants $d_{2}$ and $d_{3}$.

The exotic scalar glueball $G_{E}$ has more complicated interactions with two
vector fields, written out in (B), with five coupling constants (B). The
latter are generalized in a completely analogous manner to form factors
$c_{i}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})$ with $i=2,3,4$, and
$\breve{c}_{j}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})$ with $j=2,3$.

Following the notation of Danilkin:2021icn , the result for the matrix element
of a scalar glueball with two electromagnetic currents
$j_{\text{em}}^{\mu}(x)$ can be written in terms of two transition form
factors $\mathcal{F}_{1,2}^{S}$ defined by

$\displaystyle\mathcal{M}^{\mu\nu}(p\to q_{1},q_{2})$ $\displaystyle=$
$\displaystyle i\int d^{4}xe^{iq_{1}\cdot x}\langle
0\left|j_{\text{em}}^{\mu}(x)j_{\text{em}}^{\nu}(0)\right|G_{S}(p)\rangle$
(99) $\displaystyle=$
$\displaystyle\frac{\mathcal{F}_{1}^{S}(q_{1}^{2},q_{2}^{2})}{M_{S}}T_{1}^{\mu\nu}+\frac{\mathcal{F}_{2}^{S}(q_{1}^{2},q_{2}^{2})}{M_{S}^{3}}T_{2}^{\mu\nu}$

with

$\displaystyle T_{1}^{\mu\nu}$ $\displaystyle=$ $\displaystyle q_{1}\cdot
q_{2}g^{\mu\nu}-q_{2}^{\mu}q_{1}^{\nu},$ $\displaystyle T_{2}^{\mu\nu}$
$\displaystyle=$ $\displaystyle q_{1}^{2}q_{2}^{2}g^{\mu\nu}+q_{1}\cdot
q_{2}q_{1}^{\mu}q_{2}^{\nu}-q_{1}^{2}q_{2}^{\mu}q_{2}^{\nu}-q_{2}^{2}q_{1}^{\mu}q_{1}^{\nu}.$
(100)

For the dilatonic scalar glueball we obtain

$\displaystyle\mathcal{F}_{1}^{D}$
$\displaystyle=-2\frac{d_{3}^{\mathcal{VV}}(Q_{1}^{2},Q_{2}^{2})\,\text{tr}\,Q^{2}}{M_{D}}\left[(q_{1}^{2}+q_{2}^{2})+(q_{1}\cdot
q_{2})+2M_{D}^{2})\right]-\frac{d_{2}^{\mathcal{VV}}(Q_{1}^{2},Q_{2}^{2})M_{\mathrm{KK}}^{2}\,\text{tr}\,Q^{2}}{M_{D}},$
(101) $\displaystyle\mathcal{F}_{2}^{D}$
$\displaystyle=-2d_{3}^{\mathcal{VV}}(Q_{1}^{2},Q_{2}^{2})\,\text{tr}\,Q^{2}M_{D}+\frac{d_{2}^{\mathcal{VV}}(Q_{1}^{2},Q_{2}^{2})M_{\mathrm{KK}}^{2}\,\text{tr}\,Q^{2}M_{D}}{q_{1}^{2}q_{2}^{2}}\left[(q_{1}\cdot
q_{2})+M_{D}^{2})\right],$ (102)

and for the exotic scalar glueball

$\displaystyle\mathcal{F}_{1}^{E}$
$\displaystyle=-2\frac{\,\text{tr}\,Q^{2}}{M_{E}}\left[c_{3}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})((q_{1}^{2}+q_{2}^{2})+(q_{1}\cdot
q_{2})+M_{E}^{2})-\breve{c}_{3}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})M_{E}^{2}\right.$
$\displaystyle\left.+c_{2}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})M_{\mathrm{KK}}^{2}-\frac{3}{2}\left(c_{4}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})+c_{4}^{\mathcal{V}\mathcal{V}}(Q_{2}^{2},Q_{1}^{2})\right)\right],$
(103) $\displaystyle\mathcal{F}_{2}^{E}$
$\displaystyle=-2\,\text{tr}\,Q^{2}M_{E}\left[c_{3}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})-c_{2}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})M_{\mathrm{KK}}^{2}\frac{(q_{1}\cdot
q_{2})}{q_{1}^{2}q_{2}^{2}}\right.$
$\displaystyle\left.+\breve{c}_{2}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})\frac{M_{E}^{2}M_{\mathrm{KK}}^{2}}{q_{1}^{2}q_{2}^{2}}-\frac{3}{2}M_{\mathrm{KK}}^{2}\frac{c_{4}^{\mathcal{V}\mathcal{V}}(Q_{1}^{2},Q_{2}^{2})q_{1}^{2}+c_{4}^{\mathcal{V}\mathcal{V}}(Q_{2}^{2},Q_{1}^{2})q_{2}^{2}}{q_{1}^{2}q_{2}^{2}}\right],$
(104)

where $q_{1}\cdot q_{2}=-\frac{1}{2}(q_{1}^{2}+q_{2}^{2}+M_{D/E})$ and
$\,\text{tr}\,Q^{2}=2/3$ for $N_{f}=3$.

We have used these results to estimate the glueball contribution to the muon
anomalous magnetic moment $a_{\upmu}=(g-2)_{\upmu}/2$ in a narrow-width
approximation by inserting the above expressions in the two-loop expression
for the muon-photon vertex.

In the scenario where the exotic scalar glueball is discarded from the
spectrum and $G_{D}$ is identified with the ground-state scalar glueball, we
obtain for $M_{D}=1506$ MeV and $M_{D}=1712$ MeV corresponding to the glueball
candidates $f_{0}(1500)$ and $f_{0}(1710)$

$\displaystyle a_{\upmu}^{G_{D}(1506)}$ $\displaystyle=-1.62\times 10^{-12},$
$\displaystyle a_{\upmu}^{G_{D}(1712)}$ $\displaystyle=-1.01\times 10^{-12}.$
(105)

While the former result is approximately identical to the unmodified WSS
result, since $M_{D}^{\mathrm{WSS}}=1487$ MeV, the latter depends on the
specific extrapolations laid out in Sect. IV.2. Had we only raised the mass,
it would have been somewhat larger, $-1.35\times 10^{-12}$, but in this case
the rather good agreement of the hadronic decay pattern obtained for
$G_{D}(1712)$ with the experimental results for the glueball candidate
$f_{0}(1710)$ (or $f_{0}(1770)$ according to Sarantsev:2021ein ) would have
deteriorated.

If the exotic scalar glueball is not discarded from the spectrum but
identified with the ground-state scalar glueball, its mass needs to be raised
substantially to match the predictions from lattice QCD. Its decay pattern and
in particular its large width then does not fit to either $f_{0}(1500)$ and
$f_{0}(1710)$; it might instead be identified with the broad “fragmented”
glueball $G(1865)$ proposed in Sarantsev:2021ein ; Klempt:2021nuf ;
Klempt:2021wpg . Raising the mass of $G_{E}$ artificially to this glueball, we
obtain for its $a_{\upmu}$ contribution

$a_{\upmu}^{G_{E}(1865)}=-0.10\times 10^{-12},$ (106)

which is an order of magnitude smaller in accordance with the much smaller
two-photon rate of $G_{E}$. Since in this case the narrow-width approximation
is rather questionable, we have also considered the space-like Breit-Wigner
function proposed in Knecht:2018sci . However, this changes the result (106)
only by about 2%.

In Knecht:2018sci  the authors consider scalar resonances including
$f_{0}(1500)$, which is assumed to have a sizeable photon coupling while being
a glueball-like state, with a coupling constant similar to the one obtained
for $f_{0}(980)$, leading to $\Gamma^{f_{0}(1500)\to\gamma\gamma}\approx
0.79\,\text{keV}$. The assumed transition form factors therein yield
$a_{\upmu}=-(1.3\dots 2)\times 10^{-12}$. This is comparable to our results,
even though the two-photon rate obtained with $G_{D}$ is about twice as large.

In the WSS model, tensor glueballs have two-photon decay rates comparable to
$G_{D}$ with similar values of $\Gamma_{\gamma\gamma}/M_{G}$. We have not
evaluated their contribution to $a_{\upmu}$, but we expect that they will be
smaller than those of $G_{D}$ by some power of the ratio
$M_{G_{T}}/M_{G_{D}}$.

We have however evaluated the contribution of pseudoscalar glueballs, which
contribute with a positive sign. With the WSS model mass of $1789$ MeV we find
$a_{\upmu}^{G_{PS}^{\mathrm{WSS}}}=0.39\times 10^{-12}$, and when extrapolated
to a value typically found in quenched lattice QCD calculations of $2600$ MeV
this reduces to

$a_{\upmu}^{G_{PS}(2600)}=0.19\times 10^{-12}.$ (107)

This is about an order of magnitude smaller than the pseudoscalar contribution
called $G/\eta^{\prime\prime}$ in the bottom-up holographic model of
Leutgeb:2022lqw , $a_{\upmu}^{\eta^{\prime\prime}}\approx 2\times 10^{-12}$.
In this more realistic model, the pseudoscalar glueball mixes not only with
$\eta_{0}$ but also with excited $\eta(^{\prime})$ mesons (which are absent in
our simple extension of the WSS model to massive pseudoscalars).

###### Acknowledgements.

We would like to thank Claude Amsler for useful discussions. We are also
indebted to Jonas Mager for his assistance in the numerical evaluation of the
contributions to the anomalous magnetic moment of the muon. F. H. and J. L.
have been supported by the Austrian Science Fund FWF, project no. P 33655-N
and the FWF doctoral program Particles & Interactions, project no. W1252-N27.

##  Appendix A Hadronic decays of the scalar and tensor glueballs

In the following we review the hadronic decays of scalar and tensor glueballs
in the WSS model as worked out in Brunner:2015oqa ; Brunner:2015yha ;
Brunner:2015oga , including additional subdominant decay channels neglected
therein, in particular $G\to a_{1}\pi$. The latter has been emphasized in the
phenomenological analysis of Burakovsky:1998zg , where it was providing the
largest partial decay width of a pure glueball (177 MeV for a glueball mass of
1600 MeV). While their results for decays of a scalar glueball into two vector
mesons are remarkably compatible with the WSS result for $G_{D}$ when the mass
is raised to 1500-1700 MeV, the WSS prediction for $G\to a_{1}\pi$ turns out
to be fairly small, $\lesssim 1$ MeV, in stark contrast to the model of
Burakovsky:1998zg .111111For $f_{0}(1500)$ the experimental value from
CRYSTALBARREL:2001ldq  is 12(5)% of $\Gamma_{4\pi}$, i.e., $\sim 7$ MeV; for
$f_{0}(1710)$ no corresponding experimental results seem to be available.

We also review the dependence on the so far unconstrained extra coupling to be
associated with the quark mass term that we have added to the chiral WSS model
(parametrized by $x$ in Table 4). As discussed in Brunner:2015oga , this
correlates the flavor asymmetries in the decay pattern in two pseudoscalars
with the $\eta\eta^{\prime}$ partial width. Good agreement of the decay
pattern of $G_{D}$ with $f_{0}(1710)$ (or $f_{0}(1770)$) is obtained only for
small or vanishing $\eta\eta^{\prime}$ decay rates. Here a new experimental
result has been published in BESIII:2022iwi :
$\mathcal{B}(f_{0}(1710)\to\eta\eta^{\prime})/\mathcal{B}(f_{0}(1710)\to\pi\pi)<1.61\times
10^{-3}$, contradicting Sarantsev:2021ein ; Klempt:2021wpg  where this ratio
is $\sim 1$ for $f_{0}(1710)$ and $\sim 0.1$ for $f_{0}(1770)$.

###  A.1 Dilatonic scalar glueball

The scalar glueball fluctuation which in Brunner:2015oqa  is referred to as
(predominantly) dilatonic scalar glueball, reads

$\displaystyle\delta G_{\mu\nu}$
$\displaystyle=\frac{r^{2}}{\mathcal{N}_{D}\,L^{2}}T_{4}(r)\left(\eta_{\mu\nu}-\frac{\partial_{\mu}\partial_{\nu}}{\Box}\right)G_{D}(x^{\sigma}),$
$\displaystyle\delta G_{11,11}$
$\displaystyle=-3\frac{r^{2}}{\mathcal{N}_{D}\,L^{2}}T_{4}(r)G_{D}(x^{\sigma}),$
(108)

with an undetermined normalization parameter $\mathcal{N}_{D}$. To be a
solution of the Einstein equations, the radial function $T_{4}(r)$ has to
satisfy the differential equation

$\frac{\text{d}}{\text{d}r}\left(r^{7}-r\,r_{KK}^{6}\right)\frac{\text{d}}{\text{d}r}T_{4}(r)+L^{4}M_{D}^{2}r^{3}T_{4}(r)=0,$
(109)

with boundary conditions $T_{4}(r_{\text{KK}})=1$ and
$T_{4}^{\prime}(r_{\text{KK}})=0$, and therefore is normalizable for a
discrete set of mass eigenvalues $M_{D}$. In the following, we will only
consider the lightest mode with $M_{D}=1.567M_{\mathrm{KK}}=1487\,\text{MeV}$.

The kinetic and mass term for $G_{D}$ reads

$\displaystyle\left.\mathcal{L}_{4}\right|_{G_{D}^{2}}$
$\displaystyle=\mathcal{C}\int\text{d}r\frac{3r^{3}T_{4}(r)^{2}}{L^{3}\mathcal{N}_{D}^{2}}G_{D}\left(\Box-
M_{D}^{2}\right)G_{D}$ (110)

with the constant

$\displaystyle\mathcal{C}=$
$\displaystyle\left(\frac{L}{2}\right)^{4}\Omega_{4}\frac{1}{2\kappa_{11}^{2}}\left(2\pi\right)^{2}R_{4}R_{11}.$
(111)

The radial integration for the lightest mode yields the constant

$\displaystyle\int\text{d}r\frac{r^{3}T_{4}(r)^{2}}{L^{3}}=0.22547\left[T_{4}(r_{\text{KK}})\right]^{2}\frac{r_{\text{KK}}^{4}}{L^{3}}.$
(112)

To get a canonically normalized kinetic term

$\displaystyle\left.\mathcal{L}_{4}\right|_{G_{D}^{2}}$
$\displaystyle=\frac{1}{2}G_{D}\left(\Box-M_{D}^{2}\right)G_{D},$ (113)

we have to set

$\mathcal{N}_{D}=0.0335879\lambda^{\frac{1}{2}}N_{C}M_{\text{KK}}.$ (114)

Inducing the fluctuation (108) in the D8 brane action (5) we obtain the
derivative coupling of two pseudoscalar mesons to $G_{D}$ as

$\mathcal{L}_{G_{D}\Pi\Pi}=d_{1}\,\text{tr}\,\partial_{\mu}\Pi\partial_{\nu}\Pi\left(\eta^{\mu\nu}-\frac{\partial^{\mu}\partial^{\nu}}{\Box}\right)G_{D}$
(115)

where

$d_{1}=\frac{17.2261}{\sqrt{\lambda}M_{\text{KK}}N_{c}}$ (116)

(see Brunner:2015oqa  for further couplings).

Already in the chiral WSS model, a mass term arises for the singlet component
of $\Pi$ through the $U(1)_{A}$ anomaly Sakai:2004cn . The latter requires a
redefinition of the Ramond-Ramond 2-form field strength $F_{2}$ which is
associated with a $\theta$ term. The bulk action is thus given by

$S_{C_{1}}=-\frac{1}{4\pi(2\pi
l_{s})^{6}}\int\text{d}^{10}x\sqrt{-g}|\tilde{F}_{2}|^{2},$ (117)

where

$\tilde{F}_{2}=\frac{6\pi
U_{\text{KK}}}{U^{4}M_{\text{KK}}}\left(\theta+\frac{\sqrt{2N_{f}}}{f_{\pi}}\eta_{0}\right)\text{d}U\wedge
dx^{4},$ (118)

from which one obtains the Witten-Veneziano mass as Sakai:2004cn

$m_{0}^{2}=\frac{N_{f}}{27\pi^{2}N_{c}}\lambda^{2}M_{\text{KK}}.$ (119)

Inducing the metric fluctuations gives rise to an additional coupling between
the scalar glueballs and $\eta_{0}$. For the dilatonic glueball it is given by
Brunner:2015yha ; Brunner:2015oga

$\mathcal{L}_{\eta_{0}}\supset\frac{3}{2}m_{0}^{2}\eta_{0}^{2}d_{0}G_{D},$
(120)

with ($H_{D}\equiv T_{4}/\mathcal{N}_{D}$)

$d_{0}=3U^{3}_{\text{KK}}\int_{U_{\text{KK}}}^{\infty}\text{d}UH_{D}(U)U^{-4}\approx\frac{17.915}{\sqrt{\lambda}N_{c}M_{\text{KK}}}.$
(121)

Massive quarks can be introduced by worldsheet instantons Aharony:2008an ;
Hashimoto:2008sr ; Bergman:2007pm  or tachyon condensation Dhar:2007bz ;
Dhar:2008um ; McNees:2008km , which give

$\mathcal{L}_{m}^{\mathcal{M}}\propto\int\text{d}^{4}x\mathrm{Tr}\left(\mathcal{M}U(x)+h.c.\right),$
(122)

where

$U(x)=\mathrm{P}\exp
i\int\text{d}zA_{z}(z,x)=e^{i\Pi^{a}\lambda^{a}/f_{\pi}}.$ (123)

Expanding the mass term with
$\mathcal{M}=\mathrm{diag}(\hat{m},\hat{m},m_{s})$ leads to

$\displaystyle\mathcal{L}_{m}^{\mathcal{M}}=-\frac{1}{2}m_{\pi}^{2}\pi_{0}^{2}-m_{\pi}^{2}\pi^{+}\pi^{-}-m_{K}^{2}(K_{0}\bar{K}_{0}+K_{+}K_{-})$
$\displaystyle-\frac{1}{2}m_{1}^{2}\eta_{0}^{2}-\frac{1}{2}m_{8}^{2}\eta_{8}+\frac{2\sqrt{2}}{3}(m_{K}^{2}-m_{\pi}^{2})\eta_{0}\eta_{8},$
(124)

with

$\displaystyle m_{\pi}^{2}=2\hat{m}\mu,\quad m_{K}^{2}=(\hat{m}+m_{s})\mu,$
$\displaystyle m_{1}^{2}=\frac{2}{3}m_{K}^{2}+\frac{1}{3}m_{\pi}^{2},\quad
m_{8}^{2}=\frac{4}{3}m_{K}^{2}-\frac{1}{3}m_{\pi}^{2},$ (125)

and $\mu$ being the overall scale. We also note a sign error in the
$\eta_{0}\eta_{8}$ mixing term in Brunner:2015yha . With

$\eta=\eta_{8}\cos\theta_{P}-\eta_{0}\sin\theta_{P},\quad\eta^{\prime}=\eta_{8}\sin\theta_{P}+\eta_{0}\cos\theta_{P}$
(126)

the mass term is diagonalized by

$\theta_{P}=\frac{1}{2}\arctan\frac{2\sqrt{2}}{1-\frac{3}{2}m_{0}^{2}/(m_{K}^{2}-m_{\pi}^{2})}$
(127)

leading to

$m_{\eta,\eta^{\prime}}^{2}=\frac{1}{2}m_{0}^{2}+m_{K}^{2}\mp\sqrt{\frac{m_{0}^{4}}{4}-\frac{1}{3}m_{0}^{2}(m_{K}^{2}-m_{\pi}^{2})+(m_{K}^{2}-m_{\pi}^{2})^{2}}$
(128)

for the $\eta$ and $\eta^{\prime}$ meson, respectively.

As in Brunner:2015yha ; Brunner:2015oga , we assume a scalar glueball coupling
to the quark mass terms of the form (correcting a typo in Brunner:2015oga )

$\mathcal{L}_{G_{D}q\bar{q}}=-3d_{m}G_{D}\mathcal{L}_{m}^{\mathcal{M}}$ (129)

with $d_{m}$ being of the same order as $d_{0}$, i.e.

$d_{m}=xd_{0},\quad x=\mathcal{O}(1).$ (130)

This leads to a $G_{D}\eta\eta^{\prime}$ interaction given by

$\mathcal{L}_{G_{D}\eta\eta^{\prime}}=-\frac{3}{2}(1-x)d_{0}\sin(2\theta_{P})m_{0}^{2}G_{D}\eta\eta^{\prime}.$
(131)

With these modifications we obtain the coupling of the dilaton glueball to
$\eta\eta$ as

$\mathcal{L}_{G_{D}\eta\eta}=\frac{3}{2}d_{0}m_{0}^{2}(1-x)\sin\theta^{2}_{P}G_{D}\eta\eta+\frac{3}{2}d_{0}xm_{\eta}^{2}G_{D}\eta\eta+\frac{d_{1}}{2}\partial_{\mu}\eta\partial_{\nu}\eta\left(\eta^{\mu\nu}-\frac{\partial^{\mu}\partial^{\mu}}{\Box}\right)G_{D}.$
(132)

For the coupling to the $\eta^{\prime}$ meson we get $\cos\theta_{P}^{2}$
instead of $\sin\theta_{P}^{2}$.

The partial decay width for $G_{D}$ decaying into two identical pseudoscalar
mesons becomes

$\Gamma_{G_{D}\to
PP}=\frac{n_{P}d_{1}^{2}M_{D}^{3}}{512\pi}\left(1-4\frac{m_{P}^{2}}{M_{D}^{2}}\right)^{1/2}\left(1+\alpha\frac{m_{P}^{2}}{M_{D}^{2}}\right)^{2},$
(133)

where $P$ refers to pions ($n_{P}=3$), kaons ($n_{P}=4$) or
$\eta^{(^{\prime})}$ ($n_{P}=1)$ mesons, and

$\alpha=4\left(3\frac{d_{0}}{d_{1}}x-1\right)$ (134)

for pions and kaons, and

$\alpha=4\left[3\frac{d_{0}}{d_{1}}\left(x+\frac{m_{0}^{2}}{m_{P}^{2}}\sin^{2}\theta_{P}(1-x)\right)-1\right].$
(135)

for $\eta\eta$, and with the replacement $\sin\theta_{P}\to\cos\theta_{P}$ for
$\eta^{\prime}\eta^{\prime}$.

There is also a trilinear coupling of a dilatonic scalar glueball with one
axial vector and one pseudoscalar meson, which has been neglected in
Brunner:2015oqa , given by

$\displaystyle\mathcal{L}_{G_{D}\Pi a^{m}}=$
$\displaystyle-2d_{6}^{m}M_{\text{KK}}\,\text{tr}\,\left(\partial_{\mu}\Pi
a_{\nu}^{m}\right)\left(\eta^{\mu\nu}-\frac{\partial^{\mu}\partial^{\nu}}{\Box}\right)G_{D},$
(136)

with

$\displaystyle d_{6}^{m}$
$\displaystyle\equiv\sqrt{\frac{\kappa}{\pi}}\int\text{d}z\,\psi_{2m}^{\prime}H_{D}$
$\displaystyle=\left\\{11.768,7.809,2.350,\ldots\right\\}\frac{1}{M_{\text{KK}}N_{c}\sqrt{\lambda}}.$
(137)

Restricting ourselves to two-body decays, for which the relevant vertices for
vector mesons are given in Sect. V.1, the resulting partial decay widths are
collected in Table 8.

|  $\Gamma_{G_{D}}^{WSS}$[MeV] |  $\Gamma_{G_{D}(1506)}$[MeV] |  $\Gamma_{G_{D}(1712)}$[MeV] |  $\Gamma_{G_{D}(1865)}$[MeV]
---|---|---|---|---
$G_{D}\rightarrow\pi\pi$ | 12.4…16.5$|$15.2…20.1 | 12.6…16.7$|$15.4…20.4 | 14.6…19.3$|$17.0…22.5 | 16.1…21.3$|$18.3…24.2
$G_{D}\rightarrow KK$ | 4.16…5.51$|$50.5…67.0 | 4.43…5.87$|$50.4…66.8 | 7.49…9.93$|$49.4…65.4 | 9.87…13.1$|$48.8…64.7
$G_{D}\rightarrow\eta\eta$ | 1.85…3.71$|$14.1…18.7 | 1.93…3.82$|$14.1…18.7 | 2.77…4.96$|$13.9…18.4 | 3.38…5.75$|$13.7…18.1
$G_{D}\rightarrow\eta\eta^{\prime}$ | - | 0.29…0.30$|$0.00…0.00 | 4.35…4.54$|$0.00…0.00 | 4.19…4.38$|$0.00…0.00
$G_{D}\rightarrow a_{1}\pi$ | 0.14…0.18 | 0.17…0.23 | 0.66…0.87 | 1.08…1.43
$G_{D}\rightarrow\rho\rho$ | - | - | 53.5…71.0 | 90.1…119
$G_{D}\rightarrow\omega\omega$ | - | - | 16.6…22.0 | 28.7…38.1
$G_{D}\rightarrow K^{*}K^{*}$ | - | - | - | 42.6…56.4
Sum | 18.6…25.9$|$79.9…106 | 19.4…26.9$|$80.0…106 | 100…133$|$151…200 | 196…260$|$243…322
Table 8: Hadronic two-body decays of the dilatonic scalar glueball $G_{D}$
with WSS model mass and extrapolated to the masses of $f_{0}(1500)$,
$f_{0}(1710)$, and $M=1865$ MeV, for $\lambda=16.63\dots 12.55$. In decays
into two pseudoscalar mesons, the two sets of values correspond to $x=0$ and
$x=1$ in the coupling to the quark mass term (129).

###  A.2 Exotic scalar glueball

The lighter exotic scalar glueball fluctuation with mass
$M_{E}=0.901M_{\mathrm{KK}}=855$ MeV, which we have discarded from the
spectrum when identifying the dilatonic scalar glueball with the ground-state
glueball of QCD, is given by (IV.1) with eigenvalue equation (70). This mode
involves the metric component $h_{\tau\tau}$, which has no analogues in other
holographic QCD models, and has therefore been termed “exotic” in
Constable:1999gb . Its canonical normalization is obtained from

$\displaystyle\left.\mathcal{L}_{4}\right|_{G_{E}^{2}}$
$\displaystyle=\mathcal{C}\int\text{d}r\frac{r^{3}S_{4}(r){}^{2}}{2L^{3}\mathcal{N}_{E}^{2}}\frac{5}{8}G_{E}\left(\Box-
M_{E}^{2}\right)G_{E}$ $\displaystyle=\frac{1}{2}G_{E}\left(\Box-
M_{E}^{2}\right)G_{E},$ (138)

with

$\displaystyle\int\text{d}r\frac{r^{3}S_{4}(r)^{2}}{L^{3}}=0.09183\left[S_{4}(r_{\text{KK}})\right]^{2}\frac{r_{\text{KK}}^{4}}{L^{3}}$
(139)

and

$\mathcal{N}_{E}=0.008751\lambda^{\frac{1}{2}}N_{C}M_{\text{KK}}.$ (140)

Derivative couplings of pseudoscalars to $G_{E}$ are given by

$\mathcal{L}_{G_{E}}\supset-\,\text{tr}\,\left\\{c_{1}\left[\partial_{\mu}\Pi\partial_{\nu}\Pi\frac{\partial^{\mu}\partial^{\nu}}{M_{E}^{2}}G_{E}+\frac{1}{2}\left(\partial_{\mu}\Pi\right)^{2}\left(1-\frac{\Box}{M_{E}^{2}}\right)G_{E}\right]+\breve{c}_{1}\partial_{\mu}\Pi\partial^{\mu}\Pi
G_{E}\right\\}$ (141)

with $c_{1}$ and $\breve{c}_{1}$ as in Brunner:2015oqa .

In the Witten-Veneziano mass term for $\eta_{0}^{2}$, inducing the metric
fluctuations leads to additional couplings between the scalar glueballs and
$\eta_{0}$. For the exotic scalar glueball it is given by

$\mathcal{L}_{\eta_{0}}\supset-\frac{5}{2}m_{0}^{2}\eta_{0}^{2}\breve{c}_{0}G_{E},$
(142)

with ($H_{E}\equiv S_{4}/\mathcal{N}_{E}$)

$\breve{c}_{0}=\frac{3}{4}U^{3}_{\text{KK}}\int_{U_{\text{KK}}}^{\infty}\text{d}UH_{E}(U)U^{-4}\approx\frac{15.829}{\sqrt{\lambda}N_{c}M_{\text{KK}}}$
(143)

as previously studied in Brunner:2015oga .

Assuming the coupling of the exotic scalar glueball to quark masses to be of
the form

$\mathcal{L}_{G_{E}q\bar{q}}=5\breve{c}_{m}G_{E}\mathcal{L}_{m}^{\mathcal{M}}$
(144)

with $\breve{c}_{m}$ being of the same order as $\breve{c}_{0}$, i.e.

$\breve{c}_{m}=x\breve{c}_{0},\quad x=\mathcal{O}(1),$ (145)

we get

$\mathcal{L}_{G_{E}\eta\eta^{\prime}}=\frac{5}{2}(1-x)\breve{c}_{0}\sin(2\theta_{P})m_{0}^{2}G_{E}\eta\eta^{\prime}.$
(146)

All together we obtain the coupling of the exotic scalar glueball to
$\eta\eta$ as

$\displaystyle\mathcal{L}_{G_{E}\eta\eta}=\frac{5}{2}\breve{c}_{0}m_{0}^{2}(x-1)\sin\theta^{2}_{P}G_{E}\eta\eta-\frac{5}{2}\breve{c}_{0}xm_{\eta}^{2}G_{E}\eta\eta$
$\displaystyle-\frac{c_{1}}{2}\partial_{\mu}\eta\partial_{\nu}\eta\left(\frac{1}{2}\eta^{\mu\nu}\left(1-\frac{\Box}{M_{E}^{2}}\right)+\frac{\partial^{\mu}\partial^{\nu}}{M_{E}^{2}}\right)G_{E}-\frac{\breve{c}_{1}}{2}\partial_{\mu}\eta\partial^{\mu}\eta
G_{E}.$ (147)

For pions and kaons we have

$\left|\mathcal{M}_{G_{E}\to
PP}\right|=\frac{1}{4}\left|20\breve{c}_{0}m_{P}^{2}x+2\breve{c}_{1}(M_{E}^{2}-2m_{P}^{2})+c_{1}M_{E}^{2}\right|$
(148)

and for $\eta$

$\left|\mathcal{M}_{G_{E}\to\eta\eta}\right|=\frac{1}{4}\left|-20\breve{c}_{0}m_{0}^{2}(x-1)\sin\theta_{P}^{2}+20\breve{c}_{0}m_{P}^{2}x+2\breve{c}_{1}(M_{E}^{2}-2m_{P}^{2})+c_{1}M_{E}^{2}\right|,$
(149)

from which the $\eta^{\prime}$ amplitude is obtained by the replacement
$\sin\theta_{P}\to\cos\theta_{P}$.

In both cases the decay width is given by

$\Gamma_{G_{E}\to PP}=\frac{n_{P}}{2}\frac{1}{8\pi}\left|\mathcal{M}_{G_{E}\to
PP}\right|^{2}\frac{\left|\mathbf{p}_{P}\right|}{M_{E}^{2}}.$ (150)

The coupling of the exotic scalar glueball to one axial vector meson and one
pseudoscalar meson is given by

$\displaystyle\mathcal{L}_{G_{E}\Pi a^{m}}=$ $\displaystyle
2c_{6}^{m}M_{\text{KK}}\,\text{tr}\,\left(\partial_{\mu}\Pi
a_{\nu}^{m}\right)\frac{\partial^{\mu}\partial^{\nu}}{M_{E}^{2}}G_{E},$ (151)

with

$\displaystyle c_{6}^{m}$
$\displaystyle=\sqrt{\frac{\kappa}{\pi}}\int\text{d}z\,\psi_{2m}^{\prime}\left[\frac{1}{4}+\frac{3}{5K-2}\right]H_{E}$
$\displaystyle=\left\\{57.659,72.057,65.190,\ldots\right\\}\frac{1}{M_{\text{KK}}N_{c}\sqrt{\lambda}}.$
(152)

Restricting ourselves to two-body decays, for which the relevant vertices for
vector mesons are given separately in Appendix B, the resulting partial decay
widths are collected in Table 9.

|  $\Gamma_{G_{E}}^{WSS}$[MeV] |  $\Gamma_{G_{E}(1506)}$[MeV] |  $\Gamma_{G_{E}(1712)}$[MeV] |  $\Gamma_{G_{E}(1865)}$[MeV]
---|---|---|---|---
$G_{E}\rightarrow\pi\pi$ | 72.2…95.7$|$84.9…113 | 135…179$|$142…189 | 154…205$|$161…213 | 169…224$|$175…231
$G_{E}\rightarrow KK$ | - | 120…158$|$229…304 | 152…202$|$255…338 | 176…233$|$273…362
$G_{E}\rightarrow\eta\eta$ | - | 31.3…45.4$|$57.7…76.4 | 40.0…56.9$|$65.1…86.3 | 45.9…64.6$|$69.8…92.5
$G_{E}\rightarrow\eta\eta^{\prime}$ | - | 0.21…0.22$|$0.00…0.00 | 3.12…3.26$|$0.00…0.00 | 3.01…3.14$|$0.00…0.00
$G_{E}\rightarrow a_{1}\pi$ | - | 0.06…0.08 | 0.55…0.73 | 1.36…1.80
$G_{E}\rightarrow\rho\rho$ | - | - | 0.77…1.02 | 2.91…3.86
$G_{E}\rightarrow\omega\omega$ | - | - | 0.19…0.26 | 0.84…1.12
$G_{E}\rightarrow K^{*}K^{*}$ | - | - | - | 0.15…0.20
Sum | 72.2…95.7$|$84.9…113 | 286…383$|$430…570 | 351…469$|$483…640 | 398…530$|$521…691
Table 9: Hadronic two-body decays of the exotic scalar glueball $G_{E}$ with
WSS model mass 855 MeV and extrapolated to the masses of $f_{0}(1500)$,
$f_{0}(1710)$, and the scalar glueball at 1865 MeV proposed in
Sarantsev:2021ein , for $\lambda=16.63\dots 12.55$. In decays into two
pseudoscalar mesons, the two sets of values correspond to $x=0$ and $x=1$ in
the coupling to the quark mass term (144).

###  A.3 Tensor glueball

The tensor glueball fluctuations read

$h_{\mu\nu}=q{}_{\mu\nu}\frac{r^{2}}{L^{2}\,\mathcal{N}_{T}}T_{4}(r)G_{T}(x^{\sigma}),$
(153)

where $q_{\mu\nu}$ is a symmetric, transverse, and traceless polarization
tensor, which we normalize such that $q_{\mu\nu}q^{\mu\nu}=1$, differing from
Brunner:2015oqa .

$T_{4}(r)$ satisfies the same eigenvalue equation as in the case of the
dilatonic scalar glueball, (109), but it acquires a different normalization.
The Lagrangian reads

$\displaystyle\left.\mathcal{L}_{4}\right|_{G_{T}^{2}}$
$\displaystyle=\mathcal{C}\int\text{d}r\frac{r^{3}T_{4}(r)^{2}}{4L^{3}\mathcal{N}_{T}^{2}}G_{T}\left(\Box-M^{2}\right)G_{T}$
$\displaystyle=\frac{1}{2}G_{T}\left(\Box-M^{2}\right)G_{T},$ (154)

with

$\displaystyle\int\text{d}r\frac{r^{3}T_{4}(r)^{2}}{2L^{3}}=0.112735[T_{4}(r_{\text{KK}})]^{2}\frac{r_{\text{KK}}^{4}}{L^{3}}$
(155)

and

$\mathcal{N}_{T}=0.00969598\lambda^{\frac{1}{2}}N_{C}M_{\text{KK}}=\frac{1}{2\sqrt{3}}\mathcal{N}_{D}.$
(156)

This leads to

$\mathcal{L}_{G_{T}\Pi\Pi}=t_{1}\,\text{tr}\,\left(\partial_{\mu}\Pi\partial_{\nu}\Pi\right)G^{\mu\nu}_{T}$
(157)

with ($T\equiv T_{4}/\mathcal{N}_{T}$)

$t_{1}=\frac{1}{\pi}\int\text{d}zK^{-1}T=\frac{59.6729}{\sqrt{\lambda}M_{\text{KK}}N_{c}}=2\sqrt{3}d_{1}.$
(158)

Here no additional couplings arise from the mass terms of the pseudoscalars,
because the tensor glueball fluctuations are traceless.

There is also a coupling of the tensor glueball to one axial vector and one
pseudoscalar meson,

$\mathcal{L}_{G_{T}\Pi
a^{m}}=-2t_{6}M_{\text{KK}}\,\text{tr}\,\left(\partial_{\mu}\Pi
a_{\nu}^{m}\right)G_{T}^{\mu\nu}$ (159)

with

$t_{6}=\sqrt{\frac{\kappa}{\pi}}\int\text{d}z\psi_{2m}^{\prime}T=\left\\{40.764,27.050,8.140,\ldots\right\\}\frac{1}{M_{\text{KK}}N_{c}\sqrt{\lambda}}.$
(160)

Restricting ourselves to two-body decays, for which the relevant vertices for
vector mesons are given in Sect. V.2, the resulting partial decay widths are
collected in Table 10.

|  $\Gamma_{G_{T}}^{WSS}$[MeV] |  $\Gamma_{G_{T}(2000)}$[MeV] |  $\Gamma_{G_{T}(2400)}$[MeV]
---|---|---|---
$G_{T}\rightarrow\pi\pi$ | 19.9…26.3 | 27.7…36.8 | 33.8…44.7
$G_{T}\rightarrow KK$ | 6.66…8.83 | 19.2…25.4 | 29.2…38.6
$G_{T}\rightarrow\eta\eta$ | 1.02…1.35 | 3.97…5.26 | 6.48…8.58
$G_{T}\rightarrow a_{1}\pi$ | 0.53…0.71 | 5.12…6.78 | 8.00…10.6
$G_{T}\rightarrow\rho\rho$ | - | 270…358 | 382…507
$G_{T}\rightarrow\omega\omega$ | - | 88.2…117 | 127…169
$G_{T}\rightarrow K^{*}K^{*}$ | - | 240…318 | 417…552
$G_{T}\rightarrow f_{1}\eta$ | - | 0.98…1.71 | 3.97…6.89
$G_{T}\rightarrow\eta^{\prime}\eta^{\prime}$ | - | - | 0.92…1.22
$G_{T}\rightarrow\phi\phi$ | - | - | 76.7…102
Total | 28.1…37.2 | 655…869 | 1084…1437
Table 10: Hadronic two-body decays of the tensor glueball $G_{T}$ with WSS
model 1487 MeV mass and extrapolated to masses of 2000 and 2400 MeV, for
$\lambda=16.63\dots 12.55$. In decays involving $f_{1}$ we additionally vary
$\theta_{f}=20.4^{\circ}\dots 26.4^{\circ}$. Partial decay widths much smaller
than 1 MeV are left out.

Recently, Ref. Vereijken:2023jor  calculated branching ratios of tensor
glueball decays in a chiral hadronic model, the so-called extended linear
sigma model, where the ratios of all the decay modes of Table 10 can be
obtained, although not their absolute magnitudes. In that model a similar
dominance of decays into two vector mesons (when kinematically allowed) has
been obtained, which is numerically even more pronounced.121212For example,
while in the WSS model the branching ratio $\rho\rho:\pi\pi$ is around 10-11
for a tensor glueball mass between 2000 and 2400 MeV, in Ref.
Vereijken:2023jor  it varies between 60 and 50. Also the branching ratio
$\rho\rho:a_{1}\pi$ is 6 to 5 times larger there for this mass range. The
authors of Ref. Vereijken:2023jor  also gave a rough estimate of
$\Gamma(G_{T}\to\pi\pi)\sim 15$ MeV, which turns out to be comparable with the
WSS result.

##  Appendix B Radiative Decays of the Exotic Scalar Glueball

The exotic glueball interactions contain the vertices

$\displaystyle\mathcal{L}_{G_{E}v^{m}v^{n}}=$
$\displaystyle-\,\text{tr}\,\left\\{c_{2}^{mn}M_{\text{KK}}^{2}\left[v_{\mu}^{m}v_{\nu}^{n}\frac{\partial^{\mu}\partial^{\nu}}{M_{E}^{2}}G_{E}+\frac{1}{2}v_{\mu}^{m}v^{n\mu}\left(1-\frac{\Box}{M_{E}^{2}}\right)G_{E}\right]\right.$
$\displaystyle+c_{3}^{mn}\left[\eta^{\rho\sigma}F_{\mu\rho}^{m}F_{\nu\sigma}^{n}\frac{\partial^{\mu}\partial^{\nu}}{M_{E}^{2}}G_{E}-\frac{1}{4}F_{\mu\nu}^{m}F^{n\mu\nu}\left(1+\frac{\Box}{M_{E}^{2}}\right)G_{E}\right]+3c_{4}^{mn}\frac{M_{\text{KK}}^{2}}{M_{E}^{2}}v_{\mu}^{n}F^{m\mu\nu}\partial_{\nu}G_{E}$
$\displaystyle\left.+\breve{c}_{2}^{mn}M_{\text{KK}}^{2}v_{\mu}^{m}v^{n\mu}G_{E}+\frac{1}{2}\breve{c}_{3}^{mn}F_{\mu\nu}^{m}F^{n\mu\nu}G_{E}\vphantom{\left(\frac{\Box}{M_{E}^{2}}\right)}\right\\},$
(161)

with coupling constants

$\displaystyle
c_{2}^{mn}=\kappa\int\text{d}zK\psi_{2m-1}^{\prime}\psi_{2n-1}^{\prime}\overline{H}_{E}=\frac{\\{7.116,\ldots\\}}{M_{\mathrm{KK}}N_{c}\sqrt{\lambda}},$
$\displaystyle
c_{3}^{mn}=\kappa\int\text{d}zK^{-1/3}\psi_{2m-1}\psi_{2n-1}\overline{H}_{E}=\frac{\\{69.769,\ldots\\}}{M_{\mathrm{KK}}N_{c}\sqrt{\lambda}}$
$\displaystyle
c_{4}^{mn}=\kappa\int\text{d}z\,\frac{20zK}{\left(5K-2\right)^{2}}\psi_{2m-1}\psi_{2n-1}^{\prime}H_{E}=\frac{\\{-10.5798,\ldots\\}}{M_{\text{KK}}N_{c}\sqrt{\lambda}},$
$\displaystyle\breve{c}_{2}^{mn}=\frac{\kappa}{4}\int\text{d}zK\psi_{2m-1}^{\prime}\psi_{2n-1}^{\prime}H_{E}=\frac{\\{2.966,\ldots\\}}{M_{\mathrm{KK}}N_{c}\sqrt{\lambda}}$
$\displaystyle\breve{c}_{3}^{mn}=\frac{\kappa}{4}\int\text{d}zK^{-1/3}\psi_{2m-1}\psi_{2n-1}H_{E}=\frac{\\{18.122,\ldots\\}}{M_{\mathrm{KK}}N_{c}\sqrt{\lambda}},$
(162)

where $\overline{H}_{E}=\left[\frac{1}{4}+\frac{3}{5K-2}\right]H_{E}$.

Calculating the amplitude for different polarizations we get

$\displaystyle\left|\mathcal{M}_{T}^{\left(G_{E}\rightarrow
v^{1}v^{1}\right)}\right|$
$\displaystyle=\frac{1}{2}\left[c_{3}\left(M_{E}^{2}-4m_{v}^{2}\right)-6c_{4}M_{\mathrm{KK}}^{2}-4\breve{c}_{2}M_{\text{KK}}^{2}-2\breve{c}_{3}\left(M_{E}^{2}-2m_{v}^{2}\right)\right]$
$\displaystyle\left|\mathcal{M}_{L}^{\left(G_{E}\rightarrow
v^{1}v^{1}\right)}\right|$
$\displaystyle=\frac{c_{2}M_{\text{KK}}^{2}\left(M_{E}^{2}-4m_{v}^{2}\right)+2\breve{c}_{2}M_{\text{KK}}^{2}\left(M_{E}^{2}-2m_{v}^{2}\right)+6c_{4}M_{\mathrm{KK}}^{2}m_{v}^{2}+4\breve{c}_{3}m_{v}^{4}}{2m_{v}^{2}}.$
(163)

####  B.0.1 Exotic scalar glueball $1\gamma$-decays

For the decay in one vector meson and one photon, we use

$\displaystyle\mathcal{L}_{G_{E}\mathcal{V}v^{m}}=-\,\text{tr}\,$
$\displaystyle\left\\{c_{3}^{m\mathcal{V}}\left[2\eta^{\rho\sigma}F_{\mu\rho}^{m}F_{\nu\sigma}^{\mathcal{V}}\frac{\partial^{\mu}\partial^{\nu}}{M_{E}^{2}}G_{E}-\frac{1}{2}F_{\mu\nu}^{m}F^{\mathcal{V}\mu\nu}\left(1+\frac{\Box}{M_{E}^{2}}\right)G_{E}\right]\right.$
$\displaystyle+3c_{4}^{\mathcal{V}n}\frac{M_{\text{KK}}^{2}}{M_{E}^{2}}v_{\mu}^{n}F^{\mathcal{V}\mu\nu}\partial_{\nu}G_{E}\left.+\breve{c}_{3}^{m\mathcal{V}}F_{\mu\nu}^{m}F^{\mathcal{V}\mu\nu}G_{E}\vphantom{\left(\frac{\Box}{M_{E}^{2}}\right)}\right\\},$
(164)

with

$\displaystyle c_{3}^{m\mathcal{V}}=$
$\displaystyle\kappa\int\text{d}zK^{-1/3}\psi_{2m-1}\overline{H}_{E}=\frac{\\{1.551,\ldots\\}}{M_{\text{KK}}N_{c}^{\frac{1}{2}}}$
$\displaystyle c_{4}^{\mathcal{V}m}=$
$\displaystyle\kappa\int\text{d}z\,\frac{20ZK}{\left(5K-2\right)^{2}}\psi_{2m-1}^{\prime}H_{E}=\frac{\\{-0.262,\ldots\\}}{M_{\text{KK}}N_{c}^{\frac{1}{2}}}$
$\displaystyle\breve{c}_{3}^{m\mathcal{V}}=$
$\displaystyle\frac{\kappa}{4}\int\text{d}zK^{-1/3}\psi_{2m-1}H_{E},=\frac{\\{0.425,\ldots\\}}{M_{\text{KK}}N_{c}^{\frac{1}{2}}}$

to obtain

$\displaystyle\left|\mathcal{M}_{T}^{\left(G_{E}\rightarrow
v^{m}\mathcal{V}\right)}\right|$
$\displaystyle=\frac{\left(M_{E}^{2}-m_{v}^{2}\right)}{2M_{E}^{2}}\left|3c_{4}^{\mathcal{V}n}M_{\mathrm{KK}}^{2}+2\breve{c}_{3}^{m\mathcal{V}}M_{E}^{2}+c_{3}^{m\mathcal{V}}\left(m_{v}^{2}-M_{E}^{2}\right)\right|\,\text{tr}\,\left(eQT_{v^{m}}\right).$
(165)

####  B.0.2 Exotic scalar glueball $2\gamma$-decays

The two-photon decay rate is obtained from

$\displaystyle\mathcal{L}_{G_{E}\mathcal{V}\mathcal{V}}=$
$\displaystyle-\,\text{tr}\,\left\\{c_{3}^{\mathcal{V}\mathcal{V}}\left[F_{\mu\rho}^{\mathcal{\mathcal{V}}}F_{\nu}^{\mathcal{V}\rho}\frac{\partial^{\mu}\partial^{\nu}}{M_{E}^{2}}G_{E}-\frac{1}{4}F_{\mu\nu}^{\mathcal{V}}F^{\mathcal{V}\mu\nu}\left(1+\frac{\Box}{M_{E}^{2}}\right)G_{E}\right]\right.$
$\displaystyle\left.+\frac{1}{2}\breve{c}_{3}^{\mathcal{V}\mathcal{V}}F_{\mu\nu}^{\mathcal{V}}F^{\mathcal{V}\mu\nu}G_{E}\vphantom{\left(\frac{\Box}{M_{E}^{2}}\right)}\right\\}$
(166)

with

$\displaystyle c_{3}^{\mathcal{V}\mathcal{V}}$
$\displaystyle=\kappa\int\text{d}zK^{-1/3}\overline{H}_{E}=\frac{237.587\kappa}{M_{\text{KK}}N_{c}\lambda^{1/2}}=0.0355\frac{\lambda^{\frac{1}{2}}}{M_{\text{KK}}},$
(167) $\displaystyle\breve{c}_{3}^{\mathcal{V}\mathcal{V}}$
$\displaystyle=\frac{\kappa}{4}\int\text{d}zK^{-1/3}H_{E}=\frac{71.18\kappa}{M_{\text{KK}}N_{c}\lambda^{1/2}}=0.0106\frac{\lambda^{\frac{1}{2}}}{M_{\text{KK}}},$
(168)

yielding

$\left|\mathcal{M}_{T}^{\left(G_{E}\rightarrow\mathcal{V}\mathcal{V}\right)}\right|=\frac{M_{E}^{2}}{2}\left(c_{3}^{\mathcal{V}\mathcal{V}}-2\breve{c}_{3}^{\mathcal{V}\mathcal{V}}\right)\,\text{tr}\,\left(e^{2}Q^{2}\right).$
(169)

In Table 11 the results for the partial widths for the radiative and two-
vector decays of the exotic scalar glueball are given when the above
amplitudes are substituted in the respective formulae for the dilaton scalar
glueball, (74), (78), and (82). Again, these are evaluate for the WSS model
mass, which is only 855 MeV for the exotic scalar glueball, as well as for
three higher masses, corresponding to the glueball candidates $f_{0}(1500)$,
$f_{0}(1710)$, and the one proposed in Sarantsev:2021ein . While the total
decay width of $G_{E}$ is much larger than that of $G_{D}$ at equal mass, see
Table 4, the radiative and two-vector widths of $G_{E}$ are much smaller than
those of $G_{D}$, see 5.

|  $\Gamma_{G_{E}^{\mathrm{WSS}}}$[keV] |  $\Gamma_{G_{E}(1506)}$[keV] |  $\Gamma_{G_{E}(1712)}$[keV] |  $\Gamma_{G_{E}(1865)}$[keV]
---|---|---|---|---
$G_{E}\rightarrow\rho\rho$ | - | - | 771…1022 | 2910…3857
$G_{E}\rightarrow\omega\omega$ | - | - | 194…257 | 843…1117
$G_{E}\rightarrow K^{*}K^{*}$ | - | - | - | 149…197
$G_{E}\rightarrow\rho\gamma$ | 0.047 | 13.4 | 20.7 | 26.4
$G_{E}\rightarrow\omega\gamma$ | 0.003 | 1.4 | 2.23 | 2.86
$G_{E}\rightarrow\phi\gamma$ | - | 0.30 | 0.98 | 1.72
$G_{E}\rightarrow\gamma\gamma$ | 0.043…0.033 | 0.076…0.058 | 0.087…0.066 | 0.095…0.071
Table 11: Radiative and two-vector decays of the exotic scalar glueball
$G_{E}$ with WSS model mass 855 MeV and extrapolated to the masses of
$f_{0}(1500)$, $f_{0}(1710)$ and the scalar glueball at 1865 MeV proposed in
Sarantsev:2021ein .

## References

  * (1) H. Fritzsch and M. Gell-Mann, _Current algebra: Quarks and what else?_ , _eConf_ C720906V2 (1972) 135–165, [hep-ph/0208010].
  * (2) H. Fritzsch, M. Gell-Mann and H. Leutwyler, _Advantages of the Color Octet Gluon Picture_ , _Phys. Lett. B_ 47 (1973) 365–368.
  * (3) H. Fritzsch and P. Minkowski, _$\Psi$ Resonances, Gluons and the Zweig Rule_, _Nuovo Cim. A_ 30 (1975) 393.
  * (4) R. L. Jaffe and K. Johnson, _Unconventional States of Confined Quarks and Gluons_ , _Phys. Lett. B_ 60 (1976) 201–204.
  * (5) E. Klempt and A. Zaitsev, _Glueballs, Hybrids, Multiquarks. Experimental facts versus QCD inspired concepts_ , _Phys. Rept._ 454 (2007) 1–202, [0708.4016].
  * (6) V. Crede and C. A. Meyer, _The Experimental Status of Glueballs_ , _Prog. Part. Nucl. Phys._ 63 (2009) 74–116, [0812.0600].
  * (7) W. Ochs, _The Status of Glueballs_ , _J. Phys._ G40 (2013) 043001, [1301.5183].
  * (8) E. Klempt, _Scalar mesons and the fragmented glueball_ , _Phys. Lett. B_ 820 (2021) 136512, [2104.09922].
  * (9) H.-X. Chen, W. Chen, X. Liu, Y.-R. Liu and S.-L. Zhu, _An updated review of the new hadron states_ , _Rept. Prog. Phys._ 86 (2023) 026201, [2204.02649].
  * (10) UKQCD collaboration, G. S. Bali, K. Schilling, A. Hulsebos, A. C. Irving, C. Michael and P. W. Stephenson, _A Comprehensive lattice study of SU(3) glueballs_ , _Phys. Lett. B_ 309 (1993) 378–384, [hep-lat/9304012].
  * (11) C. J. Morningstar and M. J. Peardon, _The Glueball spectrum from an anisotropic lattice study_ , _Phys. Rev._ D60 (1999) 034509, [hep-lat/9901004].
  * (12) Y. Chen et al., _Glueball spectrum and matrix elements on anisotropic lattices_ , _Phys. Rev. D_ 73 (2006) 014516, [hep-lat/0510074].
  * (13) E. Gregory, A. Irving, B. Lucini, C. McNeile, A. Rago, C. Richards et al., _Towards the glueball spectrum from unquenched lattice QCD_ , _JHEP_ 10 (2012) 170, [1208.1858].
  * (14) F. Chen, X. Jiang, Y. Chen, K.-F. Liu, W. Sun and Y.-B. Yang, _Glueballs at Physical Pion Mass_ , 2111.11929.
  * (15) S. Donnachie, H. G. Dosch, O. Nachtmann and P. Landshoff, _Pomeron physics and QCD_ , vol. 19.  Cambridge University Press, 12, 2004.
  * (16) C. Rosenzweig, A. Salomone and J. Schechter, _A Pseudoscalar Glueball, the Axial Anomaly and the Mixing Problem for Pseudoscalar Mesons_ , _Phys. Rev. D_ 24 (1981) 2545–2548.
  * (17) C. Amsler and F. E. Close, _Is $f_{0}(1500)$ a scalar glueball?_, _Phys. Rev. D_ 53 (1996) 295–311, [hep-ph/9507326].
  * (18) F. E. Close and A. Kirk, _Scalar glueball q anti-q mixing above 1-GeV and implications for lattice QCD_ , _Eur. Phys. J. C_ 21 (2001) 531–543, [hep-ph/0103173].
  * (19) F. E. Close and Q. Zhao, _Production of $f_{0}(1710)$, $f_{0}(1500)$, and $f_{0}(1370)$ in $J/\psi$ hadronic decays_, _Phys. Rev. D_ 71 (2005) 094022, [hep-ph/0504043].
  * (20) W.-J. Lee and D. Weingarten, _Scalar quarkonium masses and mixing with the lightest scalar glueball_ , _Phys. Rev. D_ 61 (2000) 014015, [hep-lat/9910008].
  * (21) S. Janowski, F. Giacosa and D. H. Rischke, _Is $f_{0}(1710)$ a glueball?_, _Phys. Rev. D_ 90 (2014) 114005, [1408.4921].
  * (22) H.-Y. Cheng, C.-K. Chua and K.-F. Liu, _Revisiting Scalar Glueballs_ , _Phys. Rev. D_ 92 (2015) 094006, [1503.06827].
  * (23) CLQCD collaboration, L.-C. Gui, Y. Chen, G. Li, C. Liu, Y.-B. Liu, J.-P. Ma et al., _Scalar Glueball in Radiative $J/\psi$ Decay on the Lattice_, _Phys. Rev. Lett._ 110 (2013) 021601, [1206.0125].
  * (24) A. V. Sarantsev, I. Denisenko, U. Thoma and E. Klempt, _Scalar isoscalar mesons and the scalar glueball from radiative $J/\psi$ decays_, _Phys. Lett. B_ 816 (2021) 136227, [2103.09680].
  * (25) E. Klempt and A. V. Sarantsev, _Singlet-octet-glueball mixing of scalar mesons_ , _Phys. Lett. B_ 826 (2022) 136906, [2112.04348].
  * (26) M. Q. Huber, C. S. Fischer and H. Sanchis-Alepuz, _Spectrum of scalar and pseudoscalar glueballs from functional methods_ , _Eur. Phys. J. C_ 80 (2020) 1077, [2004.00415].
  * (27) D. J. Gross and H. Ooguri, _Aspects of large N gauge theory dynamics as seen by string theory_ , _Phys. Rev. D_ 58 (1998) 106002, [hep-th/9805129].
  * (28) C. Csaki, H. Ooguri, Y. Oz and J. Terning, _Glueball mass spectrum from supergravity_ , _JHEP_ 01 (1999) 017, [hep-th/9806021].
  * (29) R. de Mello Koch, A. Jevicki, M. Mihailescu and J. P. Nunes, _Evaluation of glueball masses from supergravity_ , _Phys. Rev. D_ 58 (1998) 105009, [hep-th/9806125].
  * (30) A. Hashimoto and Y. Oz, _Aspects of QCD dynamics from string theory_ , _Nucl. Phys. B_ 548 (1999) 167–179, [hep-th/9809106].
  * (31) C. Csaki, J. Russo, K. Sfetsos and J. Terning, _Supergravity models for (3+1)-dimensional QCD_ , _Phys. Rev. D_ 60 (1999) 044001, [hep-th/9902067].
  * (32) H. Boschi-Filho and N. R. F. Braga, _QCD / string holographic mapping and glueball mass spectrum_ , _Eur. Phys. J. C_ 32 (2004) 529–533, [hep-th/0209080].
  * (33) P. Colangelo, F. De Fazio, F. Jugeau and S. Nicotri, _On the light glueball spectrum in a holographic description of QCD_ , _Phys. Lett. B_ 652 (2007) 73–78, [hep-ph/0703316].
  * (34) H. Forkel, _Holographic glueball structure_ , _Phys. Rev. D_ 78 (2008) 025001, [0711.1179].
  * (35) D. Li and M. Huang, _Dynamical holographic QCD model for glueball and light meson spectra_ , _JHEP_ 11 (2013) 088, [1303.6929].
  * (36) E. Folco Capossoli and H. Boschi-Filho, _Glueball spectra and Regge trajectories from a modified holographic softwall model_ , _Phys. Lett. B_ 753 (2016) 419–423, [1510.03372].
  * (37) A. Ballon-Bayona, H. Boschi-Filho, L. A. H. Mamani, A. S. Miranda and V. T. Zanchin, _Effective holographic models for QCD: glueball spectrum and trace anomaly_ , _Phys. Rev. D_ 97 (2018) 046001, [1708.08968].
  * (38) M. Rinaldi and V. Vento, _Meson and glueball spectroscopy within the graviton soft wall model_ , _Phys. Rev. D_ 104 (2021) 034016, [2101.02616].
  * (39) E. Witten, _Anti-de Sitter space, thermal phase transition, and confinement in gauge theories_ , _Adv. Theor. Math. Phys._ 2 (1998) 505–532, [hep-th/9803131].
  * (40) N. R. Constable and R. C. Myers, _Spin two glueballs, positive energy theorems and the AdS / CFT correspondence_ , _JHEP_ 10 (1999) 037, [hep-th/9908175].
  * (41) R. C. Brower, S. D. Mathur and C.-I. Tan, _Glueball spectrum for QCD from AdS supergravity duality_ , _Nucl. Phys._ B587 (2000) 249–276, [hep-th/0003115].
  * (42) T. Sakai and S. Sugimoto, _Low energy hadron physics in holographic QCD_ , _Prog. Theor. Phys._ 113 (2005) 843–882, [hep-th/0412141].
  * (43) T. Sakai and S. Sugimoto, _More on a holographic dual of QCD_ , _Prog. Theor. Phys._ 114 (2005) 1083–1118, [hep-th/0507073].
  * (44) K. Hashimoto, C.-I. Tan and S. Terashima, _Glueball decay in holographic QCD_ , _Phys. Rev._ D77 (2008) 086001, [0709.2208].
  * (45) F. Brünner, D. Parganlija and A. Rebhan, _Glueball Decay Rates in the Witten-Sakai-Sugimoto Model_ , _Phys. Rev._ D91 (2015) 106002, [1501.07906].
  * (46) F. Brünner and A. Rebhan, _Nonchiral enhancement of scalar glueball decay in the Witten-Sakai-Sugimoto model_ , _Phys. Rev. Lett._ 115 (2015) 131601, [1504.05815].
  * (47) F. Brünner and A. Rebhan, _Constraints on the $\eta\eta^{\prime}$ decay rate of a scalar glueball from gauge/gravity duality_, _Phys. Rev. D_ 92 (2015) 121902, [1510.07605].
  * (48) M. Chanowitz, _Chiral suppression of scalar glueball decay_ , _Phys.Rev.Lett._ 95 (2005) 172001, [hep-ph/0506125].
  * (49) F. Brünner and A. Rebhan, _Holographic QCD predictions for production and decay of pseudoscalar glueballs_ , _Phys. Lett. B_ 770 (2017) 124–130, [1610.10034].
  * (50) J. Leutgeb and A. Rebhan, _Witten-Veneziano mechanism and pseudoscalar glueball-meson mixing in holographic QCD_ , _Phys. Rev. D_ 101 (2020) 014006, [1909.12352].
  * (51) F. Brünner, J. Leutgeb and A. Rebhan, _A broad pseudovector glueball from holographic QCD_ , _Phys. Lett. B_ 788 (2019) 431–435, [1807.10164].
  * (52) Belle collaboration, S. Uehara et al., _High-statistics study of $K^{0}_{S}$ pair production in two-photon collisions_, _PTEP_ 2013 (2013) 123C01, [1307.7457].
  * (53) E. H. Kada, P. Kessler and J. Parisi, _Two $\gamma$ Decay Widths of Glueballs_, _Phys. Rev. D_ 39 (1989) 2657.
  * (54) S. R. Cotanch and R. A. Williams, _Tensor glueball photoproduction and decay_ , _Phys. Lett. B_ 621 (2005) 269–275, [nucl-th/0505074].
  * (55) T. Aoyama et al., _The anomalous magnetic moment of the muon in the Standard Model_ , _Phys. Rept._ 887 (2020) 1–166, [2006.04822].
  * (56) M. B. Green, J. A. Harvey and G. W. Moore, _I-brane inflow and anomalous couplings on D-branes_ , _Class. Quant. Grav._ 14 (1997) 47–52, [hep-th/9605033].
  * (57) J. Polchinski, _String Theory: Volume 2, Superstring Theory and Beyond_.  Cambridge University Press, Cambridge, 1998.
  * (58) G. S. Bali, F. Bursa, L. Castagnini, S. Collins, L. Del Debbio et al., _Mesons in large-N QCD_ , _JHEP_ 1306 (2013) 071, [1304.4437].
  * (59) T. Imoto, T. Sakai and S. Sugimoto, _Mesons as Open Strings in a Holographic Dual of QCD_ , _Prog.Theor.Phys._ 124 (2010) 263–284, [1005.0655].