# Emission of Microwave Photon Pairs by a Tunnel Junction

Jean-Charles Forgues<EMAIL_ADDRESS>Département de
Physique, Université de Sherbrooke Sherbrooke, Québec, Canada, J1K 2R1
Christian Lupien<EMAIL_ADDRESS>Département de Physique,
Université de Sherbrooke Sherbrooke, Québec, Canada, J1K 2R1  Bertrand Reulet
<EMAIL_ADDRESS>Département de Physique, Université de
Sherbrooke Sherbrooke, Québec, Canada, J1K 2R1

Generation and control of non-classical electromagnetic fields is of crucial
importance for quantum information physics. While usual methods for the
production of such fields rely on a non-linearity (of a crystal, a Josephson
junction, etc.), a recent experiment performed on a normal conductor, a tunnel
junction under microwave irradiation, has unveiled an alternative: the use of
electron shot noise in a quantum conductor[1]. Here we show that such a device
can emit _pairs of microwave photons_ of different frequencies with a rate as
high as that of superconducting Josephson junctions[2]. This results in
intensity fluctuations of the photon field at two different frequencies being
correlated below the photon shot noise _i.e._ two-mode amplitude squeezing.
Our experiment constitutes a fundamental step towards the understanding of
electronic noise in terms of quantum optics, and shows that even a normal
conductor could be used as a resource for quantum information processing.

While a classical current generates a classical field [3], also known as a
coherent state of light, it appears that when electron transport requires a
quantum mechanical description, the random electromagnetic field that
corresponds to current noise is non-classical. This was shown in [1] by the
observation of vacuum noise squeezing. Squeezed electromagnetic fields are
usually the result of non-linearities which appear in the Hamiltonian
describing the electromagnetic field by terms such as $a^{2}$ and $a^{\dagger
2}$, with $a$ and $a^{\dagger}$ the photon annihilation and creation
operators, _i.e._ by emission/absorption of _pairs_ of photons [4]. Since a
tunnel junction under appropriate dc and ac bias can emit a squeezed
electromagnetic field despite its linearity, it is natural to consider whether
the field it generates contains pairs of photons. It is precisely the goal of
the present letter to address this question experimentally.

Another recent experiment has demonstrated that photo-assisted noise may
exhibit correlations between the power fluctuations measured at two different
frequencies for an adequate choice of the excitation frequency [5]. Since this
experiment was performed at a relatively high temperature $T=3\mathrm{~{}K}$
with regard to the frequency range, $4-8\mathrm{~{}GHz}$, _i.e._ $k_{B}T\gg
hf$, the power fluctuations were classical, meaning that the measured
correlations corresponded to fluctuations of photon fluxes with many ($\sim
40$) photons emitted within an experimental detection window. Here we report
the observation of similar correlations at very low temperature
$T=20\mathrm{~{}mK}$ and under weak excitation. We demonstrate that there
still are correlations between power fluctuations measured at two frequencies,
$f_{1}$ and $f_{2}$, even when in the quantum regime $k_{B}T\ll hf_{1,2}$ and
when the average number of photons observed within a detection window is
smaller than one. This is strong evidence for photons of frequencies $f_{1}$
and $f_{2}$ being emitted as a pair when the junction is irradiated at
frequency $f_{1}+f_{2}$. Our data are in very good agreement with the
theoretical predictions based on the fourth cumulant of current fluctuations
in the quantum regime [5].

_Principle of the experiment_

Figure 1: Experimental setup. The sample is a $23.6\mathrm{~{}\Omega}$
Al/Al2O3/Al tunnel junction in the presence of a magnetic field to insure that
the aluminum remains a normal metal at all temperatures. A triplexer seperates
the frequency spectrum in three bands corresponding to the dc bias
($<4\mathrm{~{}GHz}$), the ac bias ($>8\mathrm{~{}GHz}$) and the detection
($4-8\mathrm{~{}GHz}$). The noise generated by the junction in the
$4-8\mathrm{~{}GHz}$ range is amplified then separated in two frequency bands
centered at $f_{1}=4.4\mathrm{~{}GHz}$ and $f_{2}=7.2\mathrm{~{}GHz}$ with
bandwidths $\Delta f_{1}=0.65\mathrm{~{}GHz}$ and $\Delta
f_{2}=0.38\mathrm{~{}GHz}$. The powers $P_{1,2}$ were measured with fast power
detectors (diode symbols) with a $1\mathrm{~{}ns}$ response time and digitized
at a rate of $400\mathrm{~{}MS/s}$ to compute
$G_{2}=\left<P_{1}P_{2}\right>-\left<P_{1}\right>\left<P_{2}\right>$ in real
time.

The power $P(t)$ of the electromagnetic field radiated by a tunnel junction in
a given frequency band centered on frequency $f$ fluctuates in time:
$P(t)=\left<P\right>+\delta P(t)$. The average power $\left<P\right>$ is
related to the spectral density $S(f)$ of the current fluctuations in the
sample at frequency $f$ by
$\left<P\right>=G\left(f\right)\left[S\left(f\right)+S_{a}\left(f\right)\right]\Delta
f$, where $G\left(f\right)$ is the gain of the setup illustrated in Fig. 1,
$\Delta f$ the detection bandwidth and $S_{a}\left(f\right)$ the noise
spectral density of the HEMT amplifier placed at the $3\mathrm{~{}K}$ stage.
We measure the correlation $G_{2}=\left<\delta P_{1}\delta P_{2}\right>$
between the power fluctuations $\delta P_{1,2}$ in two separate, non-
overlapping frequency bands centered on $f_{1}$ and $f_{2}$ as a function of
the dc and ac bias of the junction. The detection frequencies are chosen in
the 4-8 GHz range, such that $hf_{1,2}\gg k_{B}T$, _i.e._ where quantum
properties of the radiated field are prominent. Details on the calibration
procedure are given in the Method section.

_Results_

As in [5], we observe $G_{2}\neq 0$ only for excitation frequencies $f_{0}$
which respect $f_{0}=\left(f_{1}\pm f_{2}\right)/p$ with $p$, an integer. In
the following we will focus only on $f_{0}=f_{1}+f_{2}=11.6\mathrm{~{}GHz}$,
which best corresponds to the quantum regime.

Figure 2: Reduced power-power correlator. Plot of $G_{2}$ vs dc bias voltage
for various ac excitation amplitudes at frequency $f_{0}=f_{1}+f_{2}=11.6$
GHz. Symbols are experimental data and solid lines theoretical expectations of
Eqs. (1,2). The overall gain was obtained by fitting the value of $G_{2}$ at
large ac and dc bias (classical limit).

We show on Fig. 2 the result for $G_{2}$, expressed in units of
$\mathrm{K}^{2}$, as a function of the dc bias voltage $V_{\mathrm{dc}}$
(coloured symbols), for various ac voltages $V_{\mathrm{ac}}$. The data
exhibits kinks at integer multiples of $V_{\mathrm{dc}}=hf_{2}/e\approx
30\mathrm{~{}\mu V}$, which are characteristic of quantum effects related to
emission or absorption of photons [6]. Solid lines are the theoretical
predictions for $G_{2}$ in terms of the fourth cumulant of current
fluctuations $C_{4}$ given by [5]:

$C_{4}=\left|\left<i\left(f_{1}\right)i\left(f_{2}\right)\right>\right|^{2}$
(1)

The two-frequency current-current correlator, which describes the noise
dynamics under ac excitation, is given by [7, 8, 9]:

$\left<i\left(f\right)i\left(f_{0}-f\right)\right>=\sum_{n}\frac{\alpha_{n}}{2}[S_{0}\left(f_{n+}\right)-S_{0}\left(f_{n-}\right)]$
(2)

with $S_{0}\left(f\right)=\left(hf/R\right)\coth\left(hf/2k_{B}T\right)$, the
equilibrium noise spectral density at frequency $f$ in a tunnel junction of
resistance $R$, $f_{n\pm}=f+nf_{0}\pm eV_{\mathrm{dc}}/h$ and
$\alpha_{n}=J_{n}\left(eV_{\mathrm{ac}}/hf_{0}\right)J_{n+1}\left(eV_{\mathrm{ac}}/hf_{0}\right)$,
where $J_{n}$ are the Bessel functions of the first kind.

As evidenced by Fig. 2, our data match very well with the theoretical
expectation, thereby validating the proportionality between $G_{2}$ and
$C_{4}$ in the quantum regime. The only difference between theory and
experiment is that we observe a small extra contribution to $G_{2}$, which is
most noticeable at very low ac bias. In particular, the photo-assisted $G_{2}$
should be zero in the absence of ac excitation while we measure a tiny
contribution of magnitude $\sim 2\times 10^{-4}\mathrm{~{}K^{2}}$. This might
be due to an imperfect calibration of the setup, but could also correspond to
a real signal. Indeed, the current fluctuations generated by the junction are
not Gaussian, which causes the existence of an intrinsic fourth cumulant,
given by $e^{3}V\Delta f/\left(k_{B}^{2}G\right)\sim 1.5\times
10^{-5}\mathrm{~{}K^{2}}$ at $V_{\mathrm{dc}}=100\mathrm{~{}\mu V}$, _i.e._
smaller than the observed signal by more than an order of magnitude. As is the
case for the third moment[10, 11, 12], environmental effects can contribute to
the fourth cumulant. In particular, a contribution
$\sim\left(\frac{\mathrm{d}S}{\mathrm{d}V}\right)^{2}\left<\delta
V^{2}\right>$ is expected. Here,
$\left(\frac{\mathrm{d}S}{\mathrm{d}V}\right)$ stands for the noise
susceptibility [7, 8, 9] and $\left<\delta V^{2}\right>$ the voltage noise
experienced by the sample within the relevant bandwidth. Considering low-
frequency $\left(\leq 400\mathrm{~{}MHz}\right)$ fluctuations of a
$50\mathrm{~{}\Omega}$ environment, a noise temperature of $5\mathrm{~{}K}$
would be required to explain the observed signal. Further experimental study
is required in order to fully explore this phenomenon.

_Photon-Photon Correlations._

The power detectors used in our experiment are not photo-detectors, but are
sensitive to the total electric field generated by the sample. The detected
power thus contains the contributions of photons emitted and absorbed by the
junction as well as vacuum fluctuations. The amplifier noise adds a large
contribution to this, which contributes to the average power but not to
$G_{2}$. We will now evaluate $G_{2}$ in terms of photons emitted by the
junction.

The total noise spectral density we detect can be decomposed into
$S=S_{em}+Ghf$ where $S_{em}=2Ghf\left<n(f)\right>$ is the emission noise [13,
14, 15] and $\left<n(f)\right>$ is the average number of photons emitted at
frequency $f$ per unit time per unit bandwidth. At equilibrium, one obtains
the average number of photons emitted from $S_{0}\left(f\right)$:
$n_{0}\left(f\right)=[\exp(hf/k_{B}T)-1]^{-1}$, here $\sim 10^{-8}$, which
corresponds to the Bose-Einstein distribution, as expected for thermal
radiation.

Similarly, we can express the power-power correlator $G_{2}$ in terms of a
photon-photon correlator. Using $\left<\delta P_{1}\delta
P_{2}\right>\propto\left<\delta n_{1}\delta
n_{2}\right>=\left<n_{1}n_{2}\right>-\left<n_{1}\right>\left<n_{2}\right>$
with $n_{1,2}=n(f_{1,2})$, we define the normalized correlator[4, 16, 17]:

$g_{2}=\frac{\left<n_{1}n_{2}\right>}{\left<n_{1}\right>\left<n_{2}\right>}=1+\frac{G_{2}}{S_{em}\left(f_{1}\right)S_{em}\left(f_{2}\right)}$
(3)

which is shown by the red circles in Fig. 3, left scale.

While $g_{2}=1$ corresponds to photons at frequencies $f_{1}$ and $f_{2}$
being emitted independently, as for chaotic light, a value above 1 like the
one observed here indicates the existence of correlations in the emission of
photons [4]. When the number of emitted photons is large, correlations can be
of classical origin, as in [5]. However, $g_{2}>1$ for
$\left<n_{1,2}\right>\ll 1$, which can be seen in the large peak we observe on
Fig. 3, implies the existence of correlations at the single photon level,
_i.e._ photon pairs. The average photon numbers $\left<n_{1,2}\right>$ are
plotted on the right scale of Fig. 3. Indeed, the peak in $g_{2}$ coincides
with small photon numbers $\left<n_{1}\right>\simeq 0.11$ and
$\left<n_{2}\right>\simeq 0.03$. If photons of frequency $f_{2}$ were always
part of a pair, one would have $\left<n_{1}n_{2}\right>=\left<n_{2}\right>$
and $g_{2}=1/\left<n_{1}\right>=9.1$. As we discuss below, this is almost the
case.

In order to quantify the probability that a photon detected by our setup was
emitted as part of a pair, we adopt a simple model valid in the
$\left<n_{1,2}\right>\ll 1$ limit: we neglect the possibility of two photons
reaching the same detector within a detection window. In that case,
probabilities of detecting one photon at frequencies $f_{1}$ and $f_{2}$
within a detection window are respectively
$P\left(1\right)=\left<n_{1}\right>$ and $P\left(2\right)=\left<n_{2}\right>$
while that of detecting a pair of photons is given by
$P\left(1,2\right)=\left<n_{1}n_{2}\right>$. The probability of detecting a
pair of photons when a photon is detected at $f_{2}$ is therefore
$P\left(1|2\right)=\left<n_{1}n_{2}\right>/\left<n_{2}\right>$. The data in
Fig. 3 yields a probability of 94% at $V_{\mathrm{dc}}=16\mathrm{~{}\mu V}$,
$V_{\mathrm{ac}}=22\mathrm{~{}\mu V}$ that a photon detected at $f_{2}$ was
emitted as part of a pair. The creation of photon pairs of different
frequencies in the microwave domain has been achieved recently with the help
of superconducting circuits [2, 18, 19]. Specifically, [2] reported a yield of
$6\times 10^{6}$ pairs per second, while our junction generates the same
amount of pairs for a bandwidth of $200\mathrm{~{}MHz}$. Since the two photons
have different frequencies, a (purely dispersive) diplexer can be used to
separate them spatially without loss.

Figure 3: Photon-photon correlator. _Left scale_ : $g_{2}$ vs dc bias voltage
at $V_{\mathrm{ac}}=22\mathrm{~{}\mu V}$ for an excitation frequency. Red
circles are experimental data and the full line represents theoretical
expectations from Eq. (3). _Right scale_ (dashed lines): Number of emitted
photons per unit bandwidth per unit time $\left<n_{1,2}\right>$ in branches 1
and 2. They are calculated from the noise spectral density under ac excitation
using
$S\left(f\right)=\frac{1}{2}\Sigma_{n}J_{n}^{2}(eV_{\mathrm{ac}}/hf_{0})\left[S_{0}\left(f_{n+}\right)+S_{0}\left(f_{n-}\right)\right]$.
The number of photons detected in each measurement, $\left<n\right>\Delta
f\tau$, is close to $\left<n\right>$ given that $\Delta f_{1}\tau=1.65$ and
$\Delta f_{2}\tau=0.95$.

_Noise reduction factor_

Figure 4: Noise reduction factor. Measured NRF vs dc bias voltage at
$V_{\mathrm{ac}}=22\mathrm{~{}\mu V}$ for an excitation frequency
$f_{0}=f_{1}+f_{2}=11.6\mathrm{~{}GHz}$ (red circles). The full line
represents theoretical expectations.

The previous description corresponds to coincidence measurements in optics.
Our observations can also be viewed as evidence for two-mode amplitude
squeezing, _i.e._ the ability for two light beams to have relative intensity
fluctuations below the classical limit [16]. Classically, the variance of
$n_{1}-n_{2}$ is limited by the photon shot noise of the two beams, given by
$\left<n_{1}\right>+\left<n_{2}\right>$. Thus, one usually defines the noise
reduction factor NRF as [20, 21, 17]:

$NRF=\frac{\langle(\delta n_{1}-\delta n_{2})^{2}\rangle}{\langle
n_{1}\rangle+\langle n_{2}\rangle}$ (4)

which is greater than 1 for classical light. To calculate the NRF for our
experiment, one needs to know the variance of the photon number fluctuations
at each frequency $\left<\delta n_{1,2}^{2}\right>$.

This quantity is difficult to ascertain experimentally given the large
contribution of the amplifier. However, the voltage noise measured in each
frequency band is almost Gaussian, _i.e._ identical to that of thermal noise
of a macroscopic resistor, whose photon statistics is that of chaotic light.
Thus it obeys $\left<\delta
n^{2}\right>=\left<n\right>\left(\left<n\right>+1\right)$ [22, 23]. Using this
result, we can calculate the NRF, which clearly goes below 1, as represented
on Fig. 4 where we observe a minimum value of 0.71. This proves the existence
of two-mode vacuum amplitude squeezing in the noise emitted by the junction, a
direct consequence of the presence of photon pairs. In terms of the current-
current correlator, our result is another example of violation by electronic
quantum noise of a Cauchy-Schwartz inequality[24].

_Methods_

$G\left(f\right)$, $S_{a}\left(f\right)$, the electron temperature $T$ and the
attenuation of the excitation line were all ascertained by measuring and
fitting the photo-assisted noise at relatively high ac excitations. This
yields $T=20\mathrm{~{}mK}$ and a setup noise temperature of $5\mathrm{~{}K}$.
Since the signal measured here is 1000 times weaker than that measured in [5],
the experiment was first performed by replacing the sample with a macroscopic
$50\mathrm{~{}\Omega}$ resistor heated with a dc current. For such a sample
one expects $G_{2}=0$. The observed signal thus allowed us to calibrate out
the spurious signal of the detection, mainly due to the cross-talk between the
digitizer channels.

_Acknoledgements_

We acknowledge fruitful discussions with A. Bednorz, W. Belzig, A. Blais, J.
Gabelli, G. Gasse, P. Grangier and F. Qassemi. We thank L. Spietz for
providing us with the tunnel junction and G. Laliberté for technical help.
This work was supported by the Canada Excellence Research Chairs program, the
NSERC, the MDEIE, the FRQNT, the INTRIQ and the Canada Foundation for
Innovation.

## References

  * [1] Gasse, G., Lupien, C. & Reulet, B.  Observation of squeezing in the electron quantum shot noise of a tunnel junction.  _Phys. Rev. Lett._ 111, 136601 (2013).
  * [2] Flurin, E., Roch, N., Mallet, F., Devoret, M. H. & Huard, B.  Generating entangled microwave radiation over two transmission lines.  _Phys. Rev. Lett._ 109, 183901 (2012).
  * [3] Glauber, R. J.  Some notes on multiple-boson processes.  _Phys. Rev._ 84, 395–400 (1951).
  * [4] Loudon, R.  _The quantum theory of light_ (Oxford university press, Oxford, 2000), 3rd edn.
  * [5] Forgues, J.-C. _et al._ Noise intensity-intensity correlations and the fourth cumulant of photo-assisted shot noise.  _Sci. Rep._ 3, 2869 (2013).
  * [6] Lesovik, G. B. & Levitov, L. S.  Noise in an ac biased junction: Nonstationary aharonov-bohm effect.  _Phys. Rev. Lett._ 72, 538–541 (1994).
  * [7] Gabelli, J. & Reulet, B.  Dynamics of quantum noise in a tunnel junction under ac excitation.  _Phys. Rev. Lett._ 100, 026601 (2008).
  * [8] Gabelli, J. & Reulet, B.  The noise susceptibility of a photo-excited coherent conductor. (2008).  ArXiv:0801.1432.
  * [9] _The noise susceptibility of a coherent conductor_ , vol. 6600 of _The Proccedings Of SPIE_.
  * [10] Reulet, B., Senzier, J. & Prober, D. E.  Environmental effects in the third moment of voltage fluctuations in a tunnel junction.  _Phys. Rev. Lett._ 91, 196601 (2003).
  * [11] Beenakker, C. W. J., Kindermann, M. & Nazarov, Y. V.  Temperature-dependent third cumulant of tunneling noise.  _Phys. Rev. Lett._ 90, 176802 (2003).
  * [12] Kindermann, M., Nazarov, Y. V. & Beenakker, C. W. J.  Feedback of the electromagnetic environment on current and voltage fluctuations out of equilibrium.  _Phys. Rev. B_ 69, 035336 (2004).
  * [13] Lesovik, G. & Loosen, R.  On the detection of finite-frequency current fluctuations.  _Journal of Experimental and Theoretical Physics Letters_ 65, 295–299 (1997).
  * [14] Gavish, U., Levinson, Y. & Imry, Y.  Detection of quantum noise.  _Phys. Rev. B_ 62, R10637–R10640 (2000).
  * [15] Aguado, R. & Kouwenhoven, L. P.  Double quantum dots as detectors of high-frequency quantum noise in mesoscopic conductors.  _Phys. Rev. Lett._ 84, 1986–1989 (2000).
  * [16] Iskhakov, T., Lopaeva, E., Penin, A., Rytikov, G. & Chekhova, M.  Two methods for detecting nonclassical correlations in parametric scattering of light.  _JETP Letters_ 88, 660–664 (2008).
  * [17] Agafonov, I. N., Chekhova, M. V. & Leuchs, G.  Two-color bright squeezed vacuum.  _Phys. Rev. A_ 82, 011801 (2010).
  * [18] Eichler, C. _et al._ Observation of two-mode squeezing in the microwave frequency domain.  _Phys. Rev. Lett._ 107, 113601 (2011).
  * [19] Nguyen, F. m. c., Zakka-Bajjani, E., Simmonds, R. W. & Aumentado, J.  Quantum interference between two single photons of different microwave frequencies.  _Phys. Rev. Lett._ 108, 163602 (2012).
  * [20] Heidmann, A. _et al._ Observation of quantum noise reduction on twin laser beams.  _Phys. Rev. Lett._ 59, 2555–2557 (1987).
  * [21] Debuisschert, T., Reynaud, S., Heidmann, A., Giacobino, E. & Fabre, C.  Observation of large quantum noise reduction using an optical parametric oscillator.  _Quantum Optics: Journal of the European Optical Society Part B_ 1, 3 (1989).
  * [22] Beenakker, C. W. J. & Schomerus, H.  Counting statistics of photons produced by electronic shot noise.  _Phys. Rev. Lett._ 86, 700–703 (2001).
  * [23] Gabelli, J. _et al._ Hanbury brown״wiss correlations to probe the population statistics of ghz photons emitted by conductors.  _Phys. Rev. Lett._ 93, 056801 (2004).
  * [24] Bednorz, A., Bruder, C., Reulet, B. & Belzig, W.  Nonsymmetrized correlations in quantum noninvasive measurements.  _Phys. Rev. Lett._ 110, 250404 (2013).


# Deploying heavier $\eta$ meson states: configurational entropy hybridizing
AdS/QCD

R. da Rocha ID<EMAIL_ADDRESS>Federal University of ABC, Center of
Mathematics, Santo André, 09580-210, Brazil

###### Abstract

The meson family of $\eta$ pseudoscalars is studied in the context of the
AdS/QCD correspondence and the differential configurational entropy (DCE). For
it, two forms of configurational-entropic Regge-like trajectories are
engendered, relating the $\eta$ mesonic states excitation number to both their
experimental mass spectrum in the Particle Data Group (PDG) and the DCE as
well. Hence, the mass spectrum of $\eta$ pseudoscalar mesonic states, beyond
the already detected states $\eta(550)$, $\eta(1295)$, $\eta(1405)$,
$\eta(1475)$, $\eta(1760)$, $\eta(2225)$, and $\eta(2320)$, is derived for any
excitation number. The three first ulterior members of this family are then
analyzed and also compared to existing candidates in PDG.

###### pacs:

89.70.Cf, 11.25.Tq, 14.40.Aq

##  I Introduction

Shannon’s information entropy paradigm resides in converting information into
a coded form in random systems. Compressed messages do present the same amount
of information as the original ones. Nevertheless, they are less redundant
Shannon:1948zz . In his seminal work, Shannon replaced information, from an
unestablished concept, into a precise theory that has underlain the digital
revolution. It has inspired the development of all modern data-compression
algorithms and error-correcting codes as well. Information entropy is the core
of formulating the configurational entropy (CE) and consists of the part of
the entropy of a system that is related to representative correlations among
its finite number of integrants. The CE of any system evaluates the rate at
which information is condensed into wave modes. For the continuous limit, the
differential configurational entropy (DCE) takes place and can be applied to a
considerable variety of problems when one introduces some distribution
portraying the system that is related to physical quantities under scrutiny
Gleiser:2018kbq ; Gleiser:2011di ; Bernardini:2016hvx .

The DCE merges informational and dynamical constituents of physical systems
Gleiser:2012tu . The DCE logarithmically evaluates the number of bits needed
to designate the organization of the studied system. Employing the DCE to
approach a system requires a scalar field that is spatially localized. One
typically uses the energy density, as the $T_{00}$ component of the energy-
momentum tensor. Other choices, like scattering amplitudes, are also suitable,
according to the system to be approached Karapetyan:2018oye ;
Karapetyan:2018yhm . The DCE was also employed to distinguish dynamical
systems among regular, entirely random, and also chaotic evolution in QCD,
providing observables and techniques to study physical phenomena Ma:2018wtw .
The DCE setup additionally refined the comprehension of phenomenological
AdS/QCD, playing the color glass condensate an important role on studying
meson states Karapetyan:2016fai ; Karapetyan:2017edu ; Karapetyan:2019fst .
With the DCE tools, heavy-ion interactions were else investigated Ma:2018wtw .
The DCE has been systematically shown to be a very important criterion to
examine some of the most essential features of AdS/QCD.

The DCE estimated new features of light-flavor mesonic states of several
families in AdS/QCD Bernardini:2016hvx ; Bernardini:2018uuy ; Ferreira:2019inu
, corroborated by several experiments reported in Particle Data Group (PDG)
pdg . Besides, charmonia and bottomonia, tensor mesons, scalar glueballs,
odderons, pomerons, and baryons, amongst other particle states, were
scrutinized in the DCE and AdS/QCD contexts, also incorporating finite
temperature Ferreira:2019nkz ; Colangelo:2018mrt ; Ferreira:2020iry ;
Braga:2018fyc ; Braga:2020myi ; Braga:2020hhs . A variety of leading
applications of the DCE have been also investigated in Refs. Correa:2016pgr ;
Cruz:2019kwh ; Lee:2019tod ; Bazeia:2018uyg ; Alves:2014ksa ; Alves:2017ljt ;
Gleiser:2018jpd . Besides, the DCE is decisive for investigating the gravity
point of view in AdS/QCD duality. Black branes were studied in Ref.
Braga:2019jqg  in the context of the DCE, whereas black holes and thermal AdS
backgrounds yielded a temperature-dependent DCE in Ref. Braga:2020opg . Also,
Bose-Einstein condensates in AdS/CFT, modeling self-interacting gravitons,
were explored under the DCE apparatus Casadio:2016aum .

The AdS/QCD framework is constituted by an AdS bulk wherein gravity, which is
weakly coupled, emulates the dual setup to the four-dimensional (conformal)
field theory (CFT). In the duality dictionary, physical fields in the AdS bulk
are dual objects to boundary operators of (strongly coupled) QCD. The bulk
fifth dimension is nothing more than the energy scale of QCD. Confinement can
be, hence, achieved by either by a Heaviside cut-off in the bulk, the hard
wall, or by a dilatonic field, that accomplishes a smooth cut-off along the
AdS bulk – the soft wall model Csaki ; Karch:2006pv . The soft wall model
asserts that Regge trajectories naturally emerge from confinement when one
employs a quadratic dilaton in the action that governs the bulk fields.
Although the soft wall model succeeds in some aspects, it does not address
other relevant aspects of QCD. Those aspects that are not encompassed by the
soft wall include the derivation of mesonic decay constants, the chiral
symmetry breaking (CSB) and the quarkonium mass spectrum, among others
Contreras:2018hbi . A decisive answer to these issues yielded a ultraviolet
(UV) cut-off to be considered Braga:2015jca , permitting an energy scale
$z_{0}$ sitting along the bulk. Although pseudoscalar mesons were scrutinized
within a non-perturbative setup, in AdS/CFT Katz:2007tf , here a modified
AdS/QCD model will be employed, where $\eta$ pseudoscalar mesons are dual
objects to free bulk fields, having an appropriate anomalous dimension that
gauge the bulk field mass Braga:2016wkm .

In this work, the DCE for the $\eta$ pseudoscalar mesons family will be
computed, as a function of both the $\eta$ meson family excitation number and
mass spectrum. The interpolation curves to these data engender two forms of
DCE Regge-like trajectories. Hence, the mass spectrum of $\eta$ meson states,
with higher excitation numbers, can be then obtained, consisting of the next
generation of $\eta$ mesons to be experimentally detected. This article is
displayed as follows: Sec. II devotes to review the AdS/QCD duality. From the
CSB and the introduction of the anomalous dimension, the mass spectrum of
$\eta$ pseudoscalar mesons family is derived, with the aid of bulk-to-boundary
propagators and correlators. In Sec. III, the fundamentals of the DCE are
introduced and discussed, being the DCE computed for the $\eta$ pseudoscalar
meson family. The DCE is then expressed as a function of the $\eta$ mesons $n$
excitation number and their experimental mass spectrum as well. Therefore, two
forms of DCE Regge-like trajectories of $\eta$ mesons consist of the resulting
interpolation curves for these two functions, predicting the mass spectrum of
heavier $\eta$ mesonic states, which do correspond to higher $n$ mesonic
$\eta$ states. Sec. IV encompasses an overall analysis of the results and
conclusions.

##  II AdS/QCD essential framework

The starting point is the model introduced in Braga:2015jca . One defines the
AdS Poincaré patch with UV cut-off as

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle g_{AB}\,dx^{A}\,dx^{B}$
$\displaystyle=$
$\displaystyle\frac{R^{2}}{z^{2}}\left(dz^{2}+\upeta_{\mu\nu}\,dx^{\mu}\,dx^{\nu}\right)\Uptheta\left(z-z_{0}\right),$

for $\Uptheta\left(x\right)$ being the Heaviside step function, $R$ is the AdS
radius, and $z_{0}$ is the UV cut-off defined by the geometrical locus of a
D-brane. Choosing the AdS boundary yields the conformal invariance to be
naturally broken by the energy scale $z_{0}$ Braga:2015jca . This model has
been consistently employed in studying, in particular, meson families in
AdS/QCD Colangelo:2008us . Making the conformal boundary $z_{0}\to 0$ limit,
the mass spectrum of mesons satisfies the standard Regge trajectory for the
dilaton $\Upphi(z)=k^{2}z^{2}$.

Pseudoscalar mesons can be described by a function $P(x^{\mu},z)$, and the
respective action reads

$\displaystyle S_{P}=-\frac{1}{2g_{P}^{2}}\int
d^{5}x\,\sqrt{-g}{\scalebox{1.0}{$L$}},$ (2)

where the Lagrangian is given by

$\displaystyle{\scalebox{1.0}{$L$}}=e^{-\Upphi(z)}\left[g^{AB}\,\partial_{A}P\partial_{B}P+\mathcal{M}_{5}^{2}P^{2}\right],$
(3)

The constant $g_{P}$ fixes the units of the action with respect to the number
of colors, $N_{c}$, of the CFT, whereas $\mathcal{M}_{5}^{2}$ stands for the
bulk mass of pseudoscalar mesons. The action (3) yields the bulk fields
equations of motion. According to the gauge/gravity setup, scalar bulk fields
solutions scale as $z^{\Updelta-4}$ in the UV regime, being dual to an
$\mathcal{O}$ operator in the boundary CFT. Its associated 2-point correlator
reads Witten:1998qj

$\left\langle\mathcal{O}(x)\,\mathcal{O}(\mathbf{0})\right\rangle\propto\left|x\right|^{-2\Updelta},$
(4)

where $\Updelta$ denotes the scaling dimension of $\mathcal{O}$. The latter
$\Updelta$ term is also related to the operators that create hadrons. Mesons
can be thus implemented when one employs $q\bar{q}$ operators, corresponding
to $\Updelta=3$. As bulk mass terms in the action affect the UV limit of the
solutions, therefore $\Updelta$, the term $\mathcal{M}_{5}^{2}$, and the spin
$s$ are coupled to as Witten:1998qj

$\mathcal{M}_{5}^{2}\,R^{2}=\Updelta\,\left(\Updelta-4\right)-s\left(s-4\right).$
(5)

Scalar mesons correspond to $\mathcal{M}_{5}^{2}\,R^{2}=-3$, whereas vector
mesons satisfy $\mathcal{M}_{5}^{2}\,R^{2}=0$. Eq. (5) holds for the $s$-wave,
$\ell=0$, approach of mesons. For $\ell\neq 0$ states, a twist-like operator,
$\uptau$, must modify the conformal dimension as
$\Updelta\mapsto\Updelta+\uptau$ Vega:2008te ; BoschiFilho:2012xr ;
Branz:2010ub .

The introduction of an anomalous dimension yields the exploration of other
features of mesons. Indeed, mesons happen to be non-degenerate states, after
CSB. For distinguishing them, the anomalous dimension, $\Updelta_{P}$,
identifies the parity of the mesonic state, here to be considered the $\eta$
pseudoscalar family. It promotes, hence, a CSB mechanism, by adjusting the
bulk mass as

$\displaystyle\\!\\!\\!\\!\\!\\!\mathcal{M}_{5}^{2}=\frac{1}{R^{2}}\left(\Updelta_{0}\\!+\\!\Updelta_{P}\right)\left(\Updelta_{0}\\!+\\!\Updelta_{P}-4\right)\\!-\\!s\left(s-4\right),$
(6)

where $\Updelta_{0}$ emulates an effective anomalous dimension, as proposed in
details in Ref. Contreras:2018hbi . Employing Eqs. (2, 3), one can derive the
mass spectrum of the $\eta$ mesonic states. Under parity, $J^{PC}=0^{-+}$, the
$\eta$ mesons Regge trajectory is not invariant. Ref. Contreras:2018hbi  used
the Breitenholner–Freeman limit to show that $\Updelta_{P}=-1$, for the $\eta$
meson family. Besides, the inequality $\mathcal{M}_{5}^{2}R^{2}\geq-4$
generates stable pseudoscalar mesonic states.

The equation of motion that rules the $\eta$ pseudoscalar meson family is
obtained by varying the action (2), yielding

$\displaystyle z^{5}\frac{\partial}{\partial
z}\left(\frac{e^{-k^{2}z^{2}}}{z}\frac{\partial P}{\partial
z}\right)+{e^{-k^{2}z^{2}}}\left(3P-z^{2}\Box P\right)=0.$ (7)

One implements the Fourier transform with respect to the $q^{\mu}$-momentum
space,

$\displaystyle
P(q,z)=\frac{1}{(2\pi)^{2}}\iiiint_{\scalebox{0.6}{Boundary}}\,d^{4}q\,e^{ix^{\mu}q_{\mu}}P(x^{\mu},z),$
(8)

denoting $P(q,z)=\tilde{P}(q)p(q,z),$ where $\tilde{P}(q)$ stands for the
sources and $p(q,z)$ is the pseudoscalar bulk-to-boundary propagator, with
condition $P(q,z_{0})=1$. Therefore, Eq. (7) yields

$\displaystyle z^{2}\frac{\partial^{2}p}{\partial
z^{2}}-(3+2k^{2}z^{2})p\frac{\partial p}{\partial
z}+\left(z^{2}q^{2}+4\right)p=0.$ (9)

Following well established holographic procedures for mesons Karch:2006pv ;
Branz:2010ub ; Vega:2009zb , the mass spectrum can be derived. Eq. (9) is a
Kummer differential equation, whose solutions are expressed by the Kummer
confluent hypergeometric function, $F(a,b,x)$, as

$\displaystyle
p\left(q,z\right)=\frac{z^{2}F\left(1-\mathfrak{q},1,k^{2}\,z^{2}\right)}{z_{0}^{2}F\left(1-\mathfrak{q},1,k^{2}\,z_{0}^{2}\right)},$
(10)

where $\mathfrak{q}\equiv\frac{q^{2}}{4k^{2}}.$

The on-shell boundary action in the pseudoscalar case is obtained by
evaluating the solution (10) into Eq. (3):

$\\!\\!\\!\\!\\!\\!\\!I_{\scalebox{0.6}{Boundary}}^{\eta}=\frac{R^{3}}{g^{2}_{S}}\\!\iiiint_{\scalebox{0.6}{Boundary}}\frac{d^{4}q}{\left(2\pi\right)^{4}}\left[\frac{e^{-k^{2}\,z^{2}}}{z^{3}}\tilde{P}\left(q\right)\tilde{P}\left(-q\right)\right.\\\
\left.p\left(q,z\right)\frac{\partial p\left(z,-q\right)}{\partial
z}\Big{|}_{z=z_{0}}\right].$ (11)

Therefore the correlator,
$\Pi_{\eta}(q^{2})=\frac{\delta^{2}I_{\scalebox{0.6}{Boundary}}^{\eta}}{\delta\tilde{P}\left(-q\right)\delta\tilde{P}\left(q\right)}$,
reads Braga:2015jca

$\Pi_{\eta}\left(q^{2}\right)=\frac{R^{3}e^{-k^{2}z^{2}}}{g^{2}_{S}z^{3}}\frac{\partial
p\left(q,z\right)}{\partial z}\Big{|}_{z=z_{0}}.$ (12)

Replacing Eq. (10) into (12) yields

$\\!\\!\\!\\!\\!\\!\Pi_{\eta}\left(q^{2}\right)\\!=\\!-\left[\frac{2}{z_{0}}\\!+\\!\left(1-\mathfrak{q}\right)\frac{2k^{2}z_{0}F\left(2-\mathfrak{q},2,k^{2}z^{2}_{0}\right)}{F\left(\mathfrak{q},1,k^{2}z^{2}_{0}\right)}\right]\\\
\times\frac{R^{3}e^{-k^{2}z^{2}_{0}}}{g_{S}^{2}z_{0}^{3}}.$ (13)

As one regards the on-shell mass condition, $q^{2}=M_{n}^{2}$, the $\eta$
mesons family mass spectrum can be read of the poles of (13), corresponding to
the roots $\xi_{n}\equiv M_{n}^{2}/4k^{2}$ of the Kummer function in the
denominator of Eq. (13), namely Cortes:2017lgz ,
$F\left(1-\xi_{n},1,k^{2}z^{2}_{0}\right)=0.$ Hence the $\eta$ mesons mass
spectrum reads

$M_{n}^{2}=4k^{2}\xi_{n}\left(k,z_{0},\Updelta_{P}\right).$ (14)

The numerical results are presented in the fourth column of Table 1, when
taking $k=0.450$ GeV and $z_{0}=5.0$ GeV-1.

Figure 1: Experimental squared mass spectrum of the $\eta$ pseudoscalar meson
family, for $n=1,\ldots,9$, with error bars, with states $\eta(550)$,
$\eta^{\prime}(958)$, $\eta(1295)$, $\eta(1405)\sim\eta(1475)$, $\eta(1760)$,
$\eta(2225)$, $\eta(2320)$ in PDG pdg .

———– $\eta$ pseudoscalar mesons mass spectrum ————

$n$ | State |  $M_{\scalebox{0.67}{{Experimental}}}$ (MeV) |  $M_{\scalebox{0.67}{{Theory}}}$ (MeV)
---|---|---|---
1 |  $\eta(550)\;$ | $547.862\pm 0.017$ | 975.2
2 |  $\eta^{\prime}(958)\;$ | $957.78\pm 0.06$ | 1011.1
3 |  $\eta(1295)\;$ | $1294\pm 4$ | 1234.0
4 |  $\eta(1405)/\eta(1475)\;$ | $1408.8\pm 1.8$ | 1455.4
5 |  $\eta(1760)\;$ | $1751\pm 15$ | 1829.1
6 |  $\eta(\Box)\;$ |  | 1992.6
7 |  $\eta(\Box)\;$ |  | 2087.3
8 |  $\eta(2225)\;$ | $2221^{+13}_{-10}$ | 2193.6
9 |  $\eta(2320)\;$ | $2320\pm 15$ | 2289.4

Table 1: Both the experimental and the AdS/QCD-predicted mass spectrum of the
$\eta$ pseudoscalar meson family. The identification of the $\eta$ particle
states to their respective $n$ excitation numbers follow the seminal work
Rinaldi:2021dxh .

As discussed in Sec. 63 of PDG 2020 pdg , the $\eta(1405)$ and the
$\eta(1475)$ states might be the same particle amsler ; masoni and here this
identification is implemented. The $\eta(\Box)$ state with $n=6$, in Table 1,
might be identified to the $\eta(2010)$, with experimental mass
$2010^{+35}_{-60}$ MeV, whereas the $\eta(\Box)$ state with $n=7$ might be
identified to the $\eta(2100)$, with experimental mass
$2100^{+30+75}_{-24-26}$ MeV pdg . The $\eta(2100)$ is argued to be
alternatively interpreted as a pseudoscalar glueball or just a Regge
excitation of $\eta^{\prime}$. For more details see, e. g., Refs. He:2009sb ;
Godfrey:1998pd ; Amsler:2004ps ; Klempt:2007cp ; Bugg:1999jc .

##  III DCE Regge-like trajectories of $\eta$ mesons

The DCE evaluates correlations amongst the fluctuations of the energy
configurations into the physical system to be studied. To portray the system,
the energy density – the time component of the stress-energy-momentum tensor
$T_{00}({\bf r})$ – is the essential ingredient, where ${\bf
r}\in\mathbb{R}^{m}$, for any finite dimension $m$. The correlator

$\displaystyle\Pi({\bf r})=\int_{\mathbb{R}^{m}}\,d^{m}x\,T_{00}(\mathring{\bf
r}+{\bf r})T_{00}(\mathring{\bf r})$ (15)

defines the probability distribution that determines the DCE as the Shannon’s
(information) entropy of correlations Braga:2018fyc . For constructing the
DCE, one first calculates the Fourier transform

$\displaystyle T_{00}({\bf
k})=\frac{1}{(2\pi)^{m/2}}\int_{\mathbb{R}^{m}}\,d^{m}x\,T_{00}({\bf
r})e^{-i{\bf k}\cdot{\bf r}}.$ (16)

A detector identifies a wave mode, within a volume $d^{m}{k}$ centered at
${\bf k}$, with a probability that is proportional to the power spectrum in
that mode, $p({\bf k}|d^{m}{k})\sim{\left|T_{00}({\bf k})\right|^{2}}d^{m}{k}$
Gleiser:2018kbq . Eq. (16) can be also seen as a probability distribution in
momentum space for different wavelengths that impart the generation of
correlations across the system. This probability distribution engenders the
modal fraction Gleiser:2012tu ,

$\displaystyle\mathcal{T}_{00}({\bf k})=\frac{\left|T_{00}({\bf
k})\right|^{2}}{\int_{\mathbb{R}^{m}}\,d^{m}{k}\,\left|T_{00}({\bf
k})\right|^{2}}.$ (17)

The amount of information to describe $T_{00}$, with respect to wave modes, is
computed by the DCE,

$\displaystyle{\rm
DCE}_{T_{00}}=-\int_{\mathbb{R}^{m}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\mathcal{T}_{00}^{\scalebox{0.93}{$\star$}}}({\bf
k})\ln{\mathcal{T}_{00}^{\scalebox{0.93}{$\star$}}}}({\bf k})\,d^{m}k\,,$ (18)

where $\mathcal{T}_{{00}}^{\scalebox{0.93}{$\star$}}({\bf
k})=\mathcal{T}_{00}({\bf k})/\mathcal{T}_{{00}}^{\scalebox{0.58}{max}}({\bf
k})$, and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathcal{T}_{{00}}^{\scalebox{0.58}{max}}({\bf
k})}$ is the supremum of $\mathcal{T}_{00}$ in $\mathbb{R}^{m}$ wherein the
power spectrum peaks. The DCE has units of nat/unit volume111Nat denotes the
natural unit of information. One nat equals $1/\ln 2$ bits, representing the
information that underlies a uniform distribution defined on the real range
$[0,e]$.. The DCE encodes a scale information, as the power spectrum is
represented the Fourier transform of the 2-point correlator Gleiser:2018kbq .

As we are here scrutinizing the $\eta$ pseudoscalar meson family, the value
$m=1$ is taken into account in Eqs. (16 – 18), representing the scale energy,
$z$, along the AdS bulk. Replacing the Lagrangian (3) into the time component
of the energy-momentum tensor,

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\!\\!\\!\\!\\!\\!\\!\\!T_{00}\\!=\\!\frac{2}{\sqrt{-g}}\\!\\!\left[\frac{\partial(\sqrt{-g}{L})}{\partial{g^{00}}}\\!-\\!\frac{\partial}{\partial{x^{\gamma}}}\frac{\partial(\sqrt{-g}{L})}{\partial\left(\frac{\partial
g^{00}}{\partial x^{\gamma}}\right)}\right]},$ (19)

the expectation value of the energy density, to be used in Eqs. (16 – 18),
reads

$\displaystyle\left\langle
T_{00}(z)\right\rangle=\frac{1}{z^{4}}\,\left[P^{\prime
2}(z)+\mathcal{M}_{5}^{2}P^{2}(z)\right]\,.$ (20)

With the energy density (20) of the $\eta$ pseudoscalar meson family in hands,
an alternative procedure for deriving their mass spectrum is put into action,
solely using the DCE. It is worth to emphasize that this method, that
hybridizes AdS/QCD, can be realized as a more accurate one, when compared to
pure AdS/QCD predictions. In fact, it is based on the interpolation of the
$\eta$ meson family experimental mass spectrum in PDG pdg .

The DCE, using the protocol (16 – 18), for the $\eta$ pseudoscalar meson
family, can be numerically computed and the results are shown in Table 2.

$n$ |  State |  CE
---|---|---
1 |  $\eta(550)$ |  3.994
2 |  $\eta^{\prime}(958)\;$ |  6.212
3 |  $\eta(1295)$ | 8.248
4 |  $\eta(1405)\sim\eta(1475)$ | 9.943
5 |  $\eta(1760)$ |  14.540
6 |  $\eta(\Box)$ | 23.834
7 |  $\eta(\Box)$ |  28.821
8 |  $\eta(2225)$ |  35.561
9 |  $\eta(2320)$ |  42.035
10* |  $\eta_{10}$ |  48.474
11* |  $\eta_{11}$ |  54.007
12* |  $\eta_{12}$ |  58.464
Table 2: The DCE of the $\eta$ pseudoscalar meson family. States with an
asterisk represent the members of the $\eta$ meson family whose mass is
obtained by the DCE Regge-like trajectories (21, 22), respectively for the
excitation numbers $n=10,11,12$.

Standard Regge trajectories AdS/QCD show the light-flavor meson spin to be
proportional to the square of the mass spectrum. One can then emulate them,
using the intrinsic DCE of the $\eta$ pseudoscalar meson family. For it, the
DCE of the $\eta$ pseudoscalar mesonic states can be regarded as a function of
the $\eta$ mesons mass spectrum that have been already detected in experiments
pdg . The interpolation curve of this data consists of the second form of DCE
Regge-like trajectories, whereas the first form takes into account the DCE as
a function of the $n$ excitation modes of $\eta$ meson states. Fig. 2 shows
the corresponding results, wherein numerical interpolation yields the first
form of DCE Regge-like trajectory. The explicit expression is numerically
obtained by interpolation of data in Table 2:

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\\!\\!\\!\\!{\rm
DCE}_{\eta}(n)}\\!$ $\displaystyle=$
$\displaystyle\\!{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-0.021n^{4}\\!+\\!0.377n^{3}\\!-\\!1.604n^{2}}\\!$
(21)
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+4.155n+1.297},$

within $\sim 1.5\%$ standard deviation.

Figure 2: DCE of the $\eta$ pseudoscalar meson family, for $n=1,\ldots,9$
(corresponding to the $\eta(550)$, $\eta^{\prime}(958)$, $\eta(1295)$,
$\eta(1405)\sim\eta(1475)$, $\eta(1760)$, $\eta(2225)$, $\eta(2320)$ in PDG
pdg  and to the two states $\eta(\Box)$) as a function of $n$. The first form
of DCE Regge-like trajectory is plotted as the gray dashed line. Figure 3: DCE
of the $\eta$ pseudoscalar meson family, for $n=1,\ldots,9$ (corresponding to
the $\eta(550)$, $\eta^{\prime}(958)$, $\eta(1295)$,
$\eta(1405)\sim\eta(1475)$, $\eta(1760)$, $\eta(2225)$, $\eta(2320)$ in PDG
pdg  and to the two states $\eta(\Box)$) as a function of their (squared)
mass. The second form of DCE Regge-like trajectory is plotted as the gray
dashed line.

As there is no known experimental data associated with the $\eta(\Box)$
states, the hybrid interpolation in Fig. 3 takes the experimental masses of
$\eta(550)$, $\eta^{\prime}(958)$, $\eta(1295)$, $\eta(1405)\sim\eta(1475)$,
$\eta(1760)$, $\eta(2225)$, $\eta(2320)$ in PDG pdg  and the AdS/QCD-predicted
masses of the two states $\eta(\Box)$, as in Table 1. As already pointed out,
one might try to identify the $\eta(\Box)$ state with $n=6$ to $\eta(2010)$,
with mass $2010^{+35}_{-60}$ MeV, and the $\eta(\Box)$, $n=7$, state to
$\eta(2100)$, whose mass equals $2100^{+30+75}_{-24-26}$ MeV pdg . However, as
the $\eta(2100)$ may be a pseudoscalar glueball or a Regge excitation of
$\eta^{\prime}$, taking the $\eta(2100)$ mass in PDG for deriving the DCE
Regge-like trajectories is too speculative. Anyway, either using the hybrid
interpolation or the interpolation with the experimental masses of
$\eta(2010)$ and $\eta(2100)$ yields a tiny difference of $0.1\%$ in the
derived masses of $\eta_{10},\eta_{11},\eta_{12}$. Hence, as both methods
provide similar masses of the $\eta_{10},\eta_{11},\eta_{12}$ states up to
$0.1\%$, only the hybrid interpolation in Fig. 3 will be taken into account in
what follows. The DCE Regge-like trajectory, as a function of the $\eta$
mesons mass, $m$ (MeV), is listed as follows:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\rm
DCE}_{\eta}(m)}$ $\displaystyle=$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}1.476\times\\!10^{-25}m^{8}+1.808\times\\!10^{-18}m^{6}}$
(22)
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+7.798\times\\!10^{-12}m^{4}-8.320\times\\!10^{-6}m^{2}}$
$\displaystyle\qquad{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+5.823},$

within $\sim 0.17\%$ standard deviation. The DCE Regge-like trajectories in
Eqs. (21, 22) illustrate several important features about the $\eta$
pseudoscalar meson family. Using (21, 22) makes one to compute the DCE of
members of the $\eta$ pseudoscalar meson family, for
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}n\geq
10}$. They consist of the first next states of the $\eta$ pseudoscalar meson
family to be detected in experiments.

As illustrated in Table 2, replacing $n=10$ in Eq. (21) yields the DCE equal
to 48.474. Then, substituting this value in the left-hand side of Eq. (22),
and solving for $m$, one obtains for the tenth member, $\eta_{10}$, of the
$\eta$ pseudoscalar meson family, the mass $m_{\eta_{10}}=2428.6$ MeV.
Implementing a similar procedure for $n=11$, one derives the mass of the
eleventh member, $\eta_{11}$, as $m_{\eta_{11}}=2533.1$ MeV. Looking at PDG
pdg , one can attempt to identify the $X(2632)$ meson, having mass
$m=2635.2\pm 3.3\text{MeV}$, to the $\eta_{11}$. Besides, replacing $n=12$ in
Eq. (21) yields the DCE equals to 58.464. Therefore, putting back this value
into the left-hand side of Eq. (22), solving for $m$ yields the mass
$m_{\eta_{12}}=2695.9$ MeV, for the $\eta_{12}$ member of the $\eta$
pseudoscalar meson family. One can speculate that the $\eta_{12}$ mesonic
state corresponds to the already detected $X(2680)$, with mass $2676\pm 27$
MeV pdg .

##  IV Concluding remarks

We regarded a useful technique, hybridizing DCE to AdS/QCD, to derive the mass
spectrum of $\eta$ pseudoscalar meson states. with higher excitation number.
For it, the quadratic dilaton model with anomalous dimension was employed. The
compactness of DCE methods, used in Sec. III, is here advantageous for at
least two main reasons. The first one is the avoidance of computational issues
that are common in approaching AdS/QCD phenomenology. The second one resides
in hybridizing DCE to AdS/QCD. In this case, obtaining the mass spectrum of
the next generation of $\eta$ mesons to be detected takes into account the
nine first $\eta$ mesons masses that were experimentally detected pdg , as
illustrated in Fig. 3 and analytically approximated by Eq. (22) with a very
good accuracy. With the aid of data in Fig. 2 and Table 2, another form of DCE
Regge-like trajectory (21) completes the necessary ingredients for deriving
the mass spectrum of $\eta$ pseudoscalar meson states. The derived mass
spectrum agrees with meson states candidates, recently listed in PDG 2020 pdg
, as already comprehensively discussed in the last paragraph of Sec. III.

According to Table 2 and Fig. 2, $\eta$ pseudoscalar meson states with lower
$n$ excitation number have higher configurational stability, from the DCE
point of view. Lower values of the DCE can point to more predominant $\eta$
meson states. Similarly, the more massive the $\eta$ pseudoscalar meson state,
the higher its configurational instability is. Hence, more massive $\eta$
pseudoscalar meson states are more unstable from the point of view of the DCE,
being also less predominant and prevalent $\eta$ meson states. This
interpretation also corroborates with the number of events, that characterizes
each $\eta$ meson, already detected by experiments pdg . Besides the
peculiarities of producing each $\eta$ meson as a by-product of precise
interactions and reactions, the DCE might also suggest the phenomenological
prevalence of states in the $\eta$ pseudoscalar meson family.

### Acknowledgments:

RdR is grateful to FAPESP (Grant No. 2017/18897-8) and to the National Council
for Scientific and Technological Development – CNPq (Grants No. 303390/2019-0
and No. 406134/2018-9), for partial financial support.

## References

  * (1) C. E. Shannon, Bell Syst. Tech. J. 27 (1948) 379.
  * (2) M. Gleiser, M. Stephens and D. Sowinski, Phys. Rev. D 97 (2018) 096007 [arXiv:1803.08550 [hep-th]].
  * (3) M. Gleiser and N. Stamatopoulos, Phys. Lett. B 713 (2012) 304 [arXiv:1111.5597 [hep-th]].
  * (4) A. E. Bernardini and R. da Rocha, Phys. Lett. B 762 (2016) 107 [arXiv:1605.00294 [hep-th]].
  * (5) M. Gleiser and N. Stamatopoulos, Phys. Rev. D 86 (2012) 045004 [arXiv:1205.3061 [hep-th]].
  * (6) G. Karapetyan, Phys. Lett. B 781 (2018) 201 [arXiv:1802.09105 [nucl-th]].
  * (7) G. Karapetyan, Phys. Lett. B 786 (2018) 418 [arXiv:1807.04540 [nucl-th]].
  * (8) C. W. Ma and Y. G. Ma, Prog. Part. Nucl. Phys. 99 (2018) 120 [arXiv:1801.02192 [nucl-th]].
  * (9) G. Karapetyan, EPL 117 (2017) 18001 [arXiv:1612.09564 [hep-ph]].
  * (10) G. Karapetyan, EPL 118 (2017) 38001 [arXiv:1705.10617 [hep-ph]]. [arXiv:1912.10071 [hep-ph]].
  * (11) G. Karapetyan, EPL 125 (2019) 58001 [arXiv:1901.05349 [hep-ph]].
  * (12) A. E. Bernardini and R. da Rocha, Phys. Rev. D 98 (2018) 126011 [arXiv:1809.10055 [hep-th]].
  * (13) L. F. Ferreira and R. da Rocha, Phys. Rev. D 99 (2019) no.8, 086001 [arXiv:1902.04534 [hep-th]].
  * (14) P. A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020 (2020) 083C01.
  * (15) L. F. Ferreira and R. da Rocha, Phys. Rev. D 101 (2020) 106002 [arXiv:1907.11809 [hep-th]].
  * (16) N. R. F. Braga, L. F. Ferreira and R. da Rocha, Phys. Lett. B 787 (2018) 16 [arXiv:1808.10499 [hep-ph]].
  * (17) N. R. Braga and R. da Mata, Phys. Rev. D 101 (2020) 105016 [arXiv:2002.09413 [hep-th]].
  * (18) N. R. F. Braga, R. da Mata, Phys. Lett. B 811 (2018) 135918 [arXiv:2008.10457 [hep-th]].
  * (19) P. Colangelo and F. Loparco, Phys. Lett. B 788 (2019) 500 [arXiv:1811.05272 [hep-ph]].
  * (20) L. F. Ferreira and R. da Rocha, Phys. Rev. D 101 (2020) 106002 [arXiv:2004.04551 [hep-th]].
  * (21) R. A. C. Correa, D. M. Dantas, C. A. S. Almeida and R. da Rocha, Phys. Lett. B 755 (2016) 358 [arXiv:1601.00076 [hep-th]].
  * (22) W. T. Cruz, D. M. Dantas, R. V. Maluf and C. A. S. Almeida, Annalen Phys. 531 (2019) 1970035 [1810.03991 [gr-qc]].
  * (23) C. O. Lee, Phys. Lett. B 800 (2020) 135030 [arXiv:1908.06074 [hep-th]].
  * (24) D. Bazeia, D. C. Moreira and E. I. B. Rodrigues, J. Magn. Magn. Mater. 475 (2019) 734.
  * (25) A. Alves, A. G. Dias, R. da Silva, Physica 420 (2015) 1 [arXiv:1408.0827 [hep-ph]].
  * (26) A. Alves, A. G. Dias, R. da Silva, Nucl. Phys. B 959 (2020) 115137 [arXiv:2004.08407 [hep-ph]].
  * (27) M. Gleiser and D. Sowinski, Phys. Rev. D 98 (2018) 056026 [arXiv:1807.07588 [hep-th]].
  * (28) N. R. Braga, Phys. Lett. B 797 (2019) 134919 [arXiv:1907.05756 [hep-th]].
  * (29) N. F. Braga, O. Junqueira, [arXiv:2010.00714 [hep-th]].
  * (30) R. Casadio and R. da Rocha, Phys. Lett. B 763 (2016) 434 [arXiv:1610.01572 [hep-th]].
  * (31) A. Karch, E. Katz, D. T. Son and M. A. Stephanov, Phys. Rev. D 74 (2006) 015005 [hep-ph/0602229].
  * (32) C. Csaki, M. Reece, JHEP 05 (2007) 062 [arXiv:hep-ph/0608266].
  * (33) M. Á. Martín Contreras, A. Vega and S. Cortés, Chin. J. Phys. 66 (2020) 715 [arXiv:1811.10731 [hep-ph]].
  * (34) N. R. F. Braga, M. A. Martin Contreras, S. Diles, Phys. Lett. B 763 (2016) 203 [arXiv:1507.04708 [hep-th]].
  * (35) E. Katz and M. D. Schwartz, JHEP 08 (2007) 077 [arXiv:0705.0534 [hep-ph]].
  * (36) N. R. F. Braga, M. A. Martin Contreras, S. Diles, Eur. Phys. J. C 76 (2016) 598 [arXiv:1604.08296 [hep-ph]].
  * (37) P. Colangelo, F. De Fazio, F. Giannuzzi, F. Jugeau and S. Nicotri, Phys. Rev. D 78 (2008) 055009 [arXiv:0807.1054 [hep-ph]].
  * (38) E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150].
  * (39) A. Vega and I. Schmidt, Phys. Rev. D 79 (2009) 055003 [arXiv:0811.4638 [hep-ph]].
  * (40) H. Boschi-Filho, N. R. F. Braga, F. Jugeau and M. A. C. Torres, Eur. Phys. J. C 73 (2013) 2540 [arXiv:1208.2291 [hep-th]].
  * (41) T. Branz, T. Gutsche, V. E. Lyubovitskij, I. Schmidt and A. Vega, Phys. Rev. D 82 (2010), 074022 [arXiv:1008.0268 [hep-ph]].
  * (42) A. Vega, I. Schmidt, T. Branz, T. Gutsche and V. E. Lyubovitskij, Phys. Rev. D 80 (2009) 055014 [arXiv:0906.1220 [hep-ph]].
  * (43) S. Cortés, M. Á. Martín Contreras, J. R. Roldán, Phys. Rev. D 96 (2017) 106002 [arXiv:1706.09502 [hep-ph]].
  * (44) M. Rinaldi and V. Vento, [arXiv:2101.02616 [hep-ph]].
  * (45) C. Amsler et al., Eur. Phys. J. C 33 (2004) 23.
  * (46) A. Masoni, C. Cicalo, G. L. Usai, J. Phys. G 32 (2006) R293.
  * (47) S. He, M. Huang and Q. S. Yan, Phys. Rev. D 81 (2010) 014003 [arXiv:0903.5032 [hep-ph]].
  * (48) S. Godfrey and J. Napolitano, Rev. Mod. Phys. 71 (1999) 1411 [arXiv:hep-ph/9811410 [hep-ph]].
  * (49) C. Amsler, N. A. Tornqvist, Phys. Rept. 389 (2004) 61.
  * (50) E. Klempt and A. Zaitsev, Phys. Rept. 454 (2007) 1 [arXiv:0708.4016 [hep-ph]].
  * (51) D. V. Bugg, L. Y. Dong and B. S. Zou, Phys. Lett. B 458 (1999) 511.