# Isospectral Potentials and Quantum Mechanical Functions for Neutron-Neutron
Scattering

Anil Khachi1

1 Department of Physics
St. Bedes College, 171002,
Himachal Pradesh, Bharat (India)

###### Abstract

In this paper we have constructed inverse isospectral potentials for
${}^{1}S_{0}-nn$ state by fitting the experimental SPS using Variational
Monte-Carlo technique in tandem with PFM technique. The isospectral potentials
are obtained such that the cost measure i.e., mean absolute error (MAE)
between the obtained and experimental SPS are less than 1 and the parameters
give the low energy scattering parameters (‘$a$’ and ‘$r_{e}$’) very close to
the experimental values. The S-channel SPS for ${}^{1}S_{0}-nn$ have been
obtained, with a MAE with respect to experimental data for lab energies up to
350 MeV, to less than 1.

keywords: nn-scattering, Inverse potential, Phase function method, Variational
Monte-Carlo, Morse potential, Amplitude, wavefunction

##  1 Introduction

The experimental setup in neutron-neutron scattering consists of a neutron
beam generated by ${}^{3}H(d,n)^{4}He$ reaction. The target consists of heavy
water enriched in $D_{2}O$. In total the nd-breakup reaction i.e.,
$n+d(deuterium)\rightarrow p+n+n$ is the reaction leading to the appearance of
interacting two neutrons in the final state. In addition to n-d, another
reaction i.e., $\pi^{+}$ $\rightarrow$ $\gamma+n+n$ helps in determinining the
scattering length for nn reaction.

Neutron-neutron interaction is important because the reaction helps in the
investigation of charge idependence which has been fundamentally interesting
to the nuclear community. It is well known that to investigate the charge
independence in nuclear forces it is quite sufficient to compare the np and pp
interactions. It is also known that to get more accurate information about
these forces one needs to get precise low energy scattering data. Under such
circumstances only S- waves are important. The investigation of low energy
scattering parameters is directly related to test the hypothesis of charge
independence and charge symmetry. Both theoreticians and experimentalists are
keen to precisly determine the low energy parameters for np, pp and nn
interactions.

The nucleon-nucleon interactions at low energies are usually expressend in
terms of effective range ($r_{e}$) and scattering length (a). These parameters
provides information on the charge idependence of the nuclear forces. There
have been various arguments regarding the possibility of violation of charge
independence and charge symmentry some of them are discussed here:

  * •

Darewych et al. [1] found that there was substantial difference between the np
and nn potential curves with ${}^{1}S_{0}-nn$ ($V_{0}$=40.38 MeV) and for
${}^{1}S_{0}-np$ ($V_{0}$=61.99 MeV) hence observed “violation of charge
independence”. The violation was credited to the unavailability of high energy
nn data. They used high energy np and pp data for fitting the nn SPS curve.

  * •

Henley and others [2] have discussed about the small departure from charge
symmetry and charge indpendence. The departure can be originated by em forces
and is related to the basic understanding nuclear interaction and is
ultimately connected with the fundamental particle principles of elementary
physics. Since departures have been observed to be very small it is imperative
to acquire precise data on the nucleon nucleon interaction.

  * •

Babenko and Petrov [3] observed from a comparison of the low-energy parameters
for the np system and their counterparts pp and nn systems that the charge
dependence of nuclear forces is violated, which is associated with the mass
difference between the charged and neutral pions.

  * •

The values of scattering length and effective range are sensitive to any small
change in nn potentials. As stated by E.S. Konobeevski et al. [4]. A change of
7% in V(r) may lead to change of 20-30% in the scattering length calculation.

The above mentioned points are either related to the precise knowledge of data
or related to the fundamental knowledge of elementary physics. Our main aim in
this paper would be to use the adapted nn data by Wiringa et al. [5] and
obtain the interaction potentials by GOA and CDA techniques developed by our
group [6][7][8][9].

  * •

By GOA we will be running 3 parameters ($V_{0},r_{m}\&a_{m}$) for all 11 SPS
data points at once and obtain one set of globally optimized model parameters.
This is shown by solid blue line in Figure 2.

  * •

While by CDA we will be running 3 parameters for only 3 SPS data points. The
CDA results creates a set of isospectral potential shown in Figure 2 as yellow
bar.The details of the CDA procedure have been provided in the Appendix
section 6.

##  2 Methodology:

The Morse function [10] is given by:

$V_{\text{Morse}}(r)=V_{0}\left(e^{-2(r-r_{m})/a_{m}}-2e^{-(r-r_{m})/a_{m}}\right)$
(1)

where the model parameters $V_{0}$, $r_{m}$ and $a_{m}$ reflect strength of
interaction, equilibrium distance at which maximum attraction is felt
(equilibrium separation) and range parameter respectively. It has all the
interesting features that are observed in any typical scattering experiment,
such as, strong repulsion at short distances, maximum attraction at an
equilibrium distance $r_{m}$, followed by a quickly decaying tail at large
distances. It can also be observed that realistic N-N interaction potentials,
of Argonne v18 and Reid93 [8], for S-sates resemble that of a Morse function.
Also, the phenomenological Malfeit-Tjohn potential [11] has a similar shape.
Further, the analytical solution for the TISE with Morse potential interaction
has already been solved for bound state energies ($E<0$) [12]. Hence, it can
be considered as a good choice for modeling the interaction between any two
scattering particles.

###  2.1 Phase Function Method:

The Schr$\ddot{o}$dinger wave equation for a spinless particle with energy E
and orbital angular momentum $\ell$ undergoing scattering is given by

$\frac{\hbar^{2}}{2\mu}\bigg{[}\frac{d^{2}}{dr^{2}}+\big{(}k^{2}-\ell(\ell+1)/r^{2}\big{)}\bigg{]}u_{\ell}(k,r)=V(r)u_{\ell}(k,r)$
(2)

Where $k=\sqrt{E/(\hbar^{2}/2\mu)}$.

Second order differential equation Eq.2 has been transformed to the first
order non-homogeneous differential equation of Riccati type [13, 14] given by
following equation:

$\delta_{\ell}^{\prime}(k,r)=-\frac{V(r)}{k}\bigg{[}\cos(\delta_{\ell}(k,r))\hat{j}_{\ell}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\bigg{]}^{2}$
(3)

Here in Eq. 3 prime denotes differentiation of phase shift with respect to
distance and the Riccati Hankel function of first kind is related to
$\hat{j_{\ell}}(kr)$ and $\hat{\eta_{\ell}}(kr)$ by
$\hat{h}_{\ell}(r)=-\hat{\eta}_{\ell}(r)+\textit{i}~{}\hat{j}_{\ell}(r)$ . In
integral form the above equation can be simply written as:

$\delta(k,r)=\frac{-1}{k}\int_{0}^{r}{V(r)}\bigg{[}\cos(\delta_{\ell}(k,r))\hat{j_{\ell}}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta_{\ell}}(kr)\bigg{]}^{2}dr$
(4)

Eq.3 is numerically solved using Runge-Kutta 5th order (RK-5) method with
initial condition $\delta_{\ell}(0)=0$. For $\ell=0$, the Riccati-Bessel and
Riccati-Neumann functions $\hat{j}_{0}$ and $\hat{\eta}_{0}$ get simplified as
$\sin(kr)$ and $-\cos(kr)$, so Eq.3, for $\ell=0$ becomes

$\delta^{\prime}_{0}(k,r)=-\frac{V(r)}{k}\sin^{2}[kr+\delta_{0}(k,r)]$ (5)

The equation for amplitude function [15] with initial condition is obtained in
the form

$\displaystyle A_{\ell}^{\prime}(r)=$
$\displaystyle-\frac{A_{\ell}V(r)}{k}\left[\cos(\delta_{\ell}(k,r))\hat{j}_{\ell}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\right]$
(6)
$\displaystyle\times\left[\sin(\delta_{\ell}(k,r))(\hat{j}_{\ell}(kr)+\cos(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\right]$

also the equation to obtained wavefunction [15] is

$u_{\ell}(r)=A_{\ell}(r)\left[\cos(\delta_{\ell}(k,r))\hat{j}_{\ell}(kr)-\sin(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\right]$
(7)

In above equation the function $\delta_{0}(k,r)$ was termed “Phase function”
by Morse and Allis [16]. The significant advantage of PFM method is that, the
phase-shifts are directly expressed in terms of the potential and have no
relation to the wavefunction. This has been utilised in this paper to obtain
inverse potentials in an innovative way by implementing a modified VMC in
tandem with PFM. The technique optimizes the model parameters of the potential
to obtain the best match with respect to the experimental SPS values. Also,
rather than solving the second order Schr$\ddot{o}$dinger equation, we only
need to solve the first order non-homogeneous differential equation whose
asymptotic value gives directly the value of SPS.

###  2.2 Optimization of Morse function model parameters using VMC:

Typically, VMC is utilized for obtaining the ground state energy for a given
potential. The method initiates with a trial wavefunction, which is varied at
a random location by a random amount in the Monte-Carlo sense. Then, the
energy is determined using the newly obtained wavefunction and variational
principle is applied. This process is done iteratively till one converges to
the ground state. Here, we consider to vary the potential instead of
wavefunction and minimise variance with respect to experimental data, as
follows:
Initialisation step: To begin the optimisation procedure, Morse parameters
$V_{0}$, $r_{m}$ and $a_{m}$ are given some initial values. The phase equation
is integrated using RK-5 method for different values of $k$, a function of lab
energies E, to obtain the simulated SPS, say $\delta^{sim}_{k}$. The mean
percentage error (MPE) has been determined with respect to the SPS analysis
data of Wiringa et al., [5], say $\delta^{exp}_{k}$, as

$MPE=\frac{1}{N}\sum_{i=1}^{N}\frac{\lvert\delta^{exp}_{k}-\delta^{sim}_{k}\rvert}{\lvert\delta^{exp}_{k}\rvert}\times
100$ (8)

This is named as $MPE_{old}$ and is also assigned to $MPE_{min}$.
Monte-Carlo step: A random number $r$, generated in an interval [-I, I], is
added to one of the parameters, say $V_{0new}=V_{0}+r$.
PFM step: Again, the phase equation is integrated with new set of parameters
$V_{0new}$, $r_{m}$ and $a_{m}$ to obtain new set of simulated scattering
phase shifts (SPS), say $\delta^{sim-new}_{k}$, using which $MPE_{new}$ is
determined.
Variational step: If $MPE_{new}<MPE_{old}$, then $V_{0}=V_{0new}$,
$MPE_{min}=MPE_{new}$, else old values are retained.

The final three steps are repeated for each of the parameters to complete one
iteration. The size of interval is reduced after a certain number of
iterations, if there is no significant reduction in $MPE_{min}$. The process
is completed when $MPE_{min}$ does not change any further, that is,
convergence is reached.

##  3 Singlet Pseudo Bound S-wave Energy for nn

We know:

$k\cot(\delta)=-\frac{1}{a}+\frac{1}{2}kr_{e}^{2}$ (9)

where a and $r_{e}$ are scattering length and effective range. If use is made
of the approximation specified by Eq. 9, the S matrix can be written in the
form [3]:

$S(k)=\left(\frac{k+i\alpha}{k-i\alpha}\right)\left(\frac{k+i\beta}{k-i\beta}\right)$
(10)

Where S(k) is related to scattering length a and effective range $r_{e}$ by
following relations

$\alpha=\frac{1}{r_{e}}\bigg{[}1-\big{(}1-\frac{2r_{e}}{a}\big{)}^{1/2}\bigg{]}$
(11)
$\beta=\frac{1}{r_{e}}\bigg{[}1+\big{(}1-\frac{2r_{e}}{a}\big{)}^{1/2}\bigg{]}$
(12)

The S matrix in Eq. 10 has two poles in the complex plane of the wave number
k.

The first pole $i\beta$ ($\beta>0$), situated in the upper half-plane of k, is
the well-known redundant pole of the S matrix. The second pole $i\alpha$ ,
situated in the lower half-plane of k, corresponds to a virtual ($i\alpha<0$)
state of the two-nucleon system at the energy.

$\epsilon=\frac{\hbar^{2}\alpha^{2}}{2m}$ (13)

Using the obtained values for $a=-16.49(1.56)$ and $r_{e}=2.56(0.02)$ in Eq.
11 and then substituting the value in Eq. 13 the energy of the virtual
${}^{1}S_{0}-nn$ state comes out to be 0.135(22) MeV in comparison with
$0.1293(158)$ given in [3].

##  4 Results and Discussion

At laboratory energies of nn scattering larger that $E_{\ell ab}=250MeV$ the
measured scattering phase-shift in the ${}^{1}S_{0}$-wave interaction channel
becomes negative, i.e., the interaction becomes repulsive. The fits to SPS are
improved to reproduce the experimentally observed scattering length and
effective range. To reproduce the observed negative S-wave phase shifts at
higher energies with a static potential it is necessary to incorporate a
repulsion into the potential. Here the Morse potential serves as a far more
realistic potential in comparison to square well, exponential, Gaussian, or
Hulthen potential used in the earlier studies. In addition to SPS and pseudo
bound state energy for the nn system we have also obtained important quantum
mechanical functions like (i) SPS vs. r(fm) (ii) Amplitude A(r) vs r(fm) and
Wavefunction $u_{0}(r)$ for energy E=[5, 20, 50, 150, 250, 350]. These
functions are shown in figure 3 respectively.

Table 1: Optimized parameters for ${}^{1}S_{0}-nn$ states using GOA and CDA.
In later case, parameter values consisting of extreme depths are shown.
Scattering length ($a$ in fm) and effective range ($r_{e}$ in fm) obtained,
using SPS determined from these optimized parameters, are shown with
experimental values (bold) [5] in curly brackets. Virtual ${}^{1}S_{0}$ state
energy is taken from Petrov et al.

Analysis |  $[V_{0}$, $r_{m}$, $a_{m}]$ | MAE | $a(fm)$ | $r_{e}(fm)$ | $E_{nn}$
---|---|---|---|---|---
GOA | [67.523, 0.925, 0.376] | 0.5 | –17.64$\\{$-18.5(0.4)$\\}$ | 2.58$\\{$2.8(0.11)$\\}$ | 0.117{0.1293(158)}
CDA | [74.427,0.915,0.359] | 0.7 | -16.49(1.56) | 2.56(0.02) | 0.135(22){0.1293(158)}
[65.191,0.915,0.381] | 0.9

Figure 1: ${}^{1}S_{0}-nn$ potentilas obtained through GOA (solid blue line)
and CDA (yellow ribbon) analysis. Figure 2: ${}^{1}S_{0}-nn$ SPS obtained
through GOA (solid blue line) and CDA (yellow ribbon) analysis.

Figure 3: ${}^{1}S_{0}-nn$ SPS, amplitude and wavefunction function vs.
distance r(fm) for ${}^{1}S_{0}-nn$ state.

##  5 Conclusion

The outcomes of the paper is: nn data from Wiringa et al. have been fitted to
obtain the class of interaction potentials which further produces scattering
parameters ($r_{e}$ and a) and pseudo bound state energy for the nn system.
For this purpose we used VMC as an optimization procedure which was ran in
tandem with phase function method. CDA (computationally time taking procedure
$\approx$ 2,20,000 iterations) was performed to overcome the overfitting
nature of GOA (less computational time $\approx$ 500 iterations) such that the
class of potentials generated via CDA could give rise to the potentials which
are capable in obtaining the low energy scattering parameters and energy. In
addition we also obtained quantum mechanical functions like: amplitude and
wavefunctions for nn interaction. We will be communicating GOA plus CDA for
higher states of nn scattering in near future.

##  6 Appendix

###  6.1 Singlet State Analysis

  * •

Since, we have three parameters to be determined for ${}^{1}S_{0}-nn$ state, a
total of 165 combinations need to considered. All these are shown in Table
LABEL:165combi, where the data have been once again presented in ascending
order of overall MAE.

  * •

Out of these 165, only 112 of them have $MAE<2$ (Table LABEL:165combi). The
average value for $r_{m}$ from these 112 combinations is determined to be
0.915 fm.

  * •

Keeping $r_{m}=0.915$ fixed, one needs to vary only two parameter $V_{0}$ and
$a_{m}$. So, only ${}^{11}C_{2}$, that is 55 combinations need to be worked
out. These are given in Table LABEL:55combi.

  * •

A total of 21 combinations are having $MAE<1$. These have been considered for
determining energy scattering parameters ($a_{s}$ and $r_{s}$) and are shown
in Table LABEL:21combi.

  * •

The values of depth $V_{0}$ and width $a_{m}$, given in bold (Table
LABEL:21combi), are utilized for obtaining possible range of values for
scattering parameters in our calculations.

Table 2: ${}^{1}S_{0}$ state: Model parameters for 165 combinations, each with three lab energies and obtained by minimising MSE. The overall MAE is determined by obtaining SPS for remaining experimental data points. The data is sorted with respect to MAE in ascending order. Sr. No. |  $E_{1}$ (MeV) |  $E_{2}$ (MeV) |  $E_{3}$ (MeV) |  $V_{0}$ (MeV) |  $r_{m}$ (fm) |  $a_{m}$ (fm) |  Overall MAE
---|---|---|---|---|---|---|---
1 | 5 | 100 | 300 | 77.115479 | 0.903579 | 0.350886 | 0.662538
2 | 5 | 100 | 250 | 74.794074 | 0.905138 | 0.356135 | 0.664066
3 | 5 | 50 | 250 | 66.555057 | 0.923181 | 0.377144 | 0.667124
4 | 5 | 50 | 300 | 68.714993 | 0.923036 | 0.371343 | 0.667529
5 | 1 | 50 | 250 | 63.920909 | 0.932580 | 0.386325 | 0.683027
6 | 1 | 50 | 300 | 66.096318 | 0.932593 | 0.380109 | 0.685002
7 | 1 | 100 | 250 | 72.609854 | 0.912592 | 0.363168 | 0.689700
8 | 1 | 100 | 300 | 74.974928 | 0.911009 | 0.357580 | 0.689736
9 | 5 | 100 | 350 | 78.975352 | 0.902434 | 0.346849 | 0.710647
10 | 5 | 150 | 250 | 80.031139 | 0.895761 | 0.344532 | 0.724937
11 | 5 | 100 | 200 | 72.078202 | 0.907161 | 0.362599 | 0.725451
12 | 5 | 150 | 300 | 82.482578 | 0.893344 | 0.339510 | 0.734344
13 | 1 | 100 | 350 | 76.876625 | 0.909845 | 0.353275 | 0.744264
14 | 5 | 150 | 200 | 77.282940 | 0.898713 | 0.350445 | 0.744734
15 | 5 | 50 | 350 | 70.484158 | 0.922960 | 0.366796 | 0.752458
16 | 1 | 100 | 200 | 69.825095 | 0.914668 | 0.370109 | 0.756181
17 | 1 | 150 | 250 | 78.083707 | 0.902422 | 0.350603 | 0.765884
18 | 5 | 150 | 350 | 84.381371 | 0.891599 | 0.335772 | 0.770784
19 | 5 | 50 | 200 | 63.979704 | 0.923428 | 0.384456 | 0.777142
20 | 1 | 150 | 300 | 80.592060 | 0.899901 | 0.345280 | 0.778598
21 | 1 | 50 | 350 | 67.882315 | 0.932638 | 0.375236 | 0.781104
22 | 5 | 200 | 250 | 83.695427 | 0.889976 | 0.337076 | 0.792821
23 | 5 | 25 | 250 | 59.371657 | 0.943254 | 0.399038 | 0.799728
24 | 1 | 50 | 200 | 61.317014 | 0.932620 | 0.394214 | 0.805355
25 | 5 | 25 | 300 | 61.414424 | 0.944445 | 0.392522 | 0.813973
26 | 1 | 150 | 350 | 82.547036 | 0.898076 | 0.341299 | 0.820508
27 | 5 | 200 | 300 | 86.368368 | 0.886791 | 0.331951 | 0.822296
28 | 10 | 50 | 250 | 69.079223 | 0.914998 | 0.369006 | 0.826098
29 | 10 | 50 | 300 | 71.223951 | 0.914683 | 0.363555 | 0.830384
30 | 1 | 25 | 250 | 56.293894 | 0.955358 | 0.410959 | 0.840832
31 | 1 | 200 | 250 | 81.911322 | 0.896193 | 0.342571 | 0.847136
32 | 5 | 200 | 350 | 88.277749 | 0.884659 | 0.328433 | 0.858753
33 | 1 | 25 | 300 | 58.320941 | 0.956947 | 0.403929 | 0.859694
34 | 5 | 100 | 150 | 68.510238 | 0.910179 | 0.371682 | 0.867157
35 | 10 | 100 | 250 | 76.890502 | 0.898528 | 0.349814 | 0.867544
36 | 10 | 100 | 300 | 79.172683 | 0.896958 | 0.344850 | 0.872223
37 | 10 | 25 | 250 | 62.328957 | 0.932878 | 0.388595 | 0.878832
38 | 1 | 200 | 300 | 84.642341 | 0.892865 | 0.337174 | 0.882417
39 | 10 | 25 | 300 | 64.384332 | 0.933691 | 0.382510 | 0.892251
40 | 10 | 50 | 350 | 72.976295 | 0.914475 | 0.359286 | 0.909298
41 | 1 | 100 | 150 | 66.168294 | 0.917781 | 0.379889 | 0.912405
42 | 10 | 100 | 200 | 74.240851 | 0.900538 | 0.355870 | 0.914718
43 | 5 | 250 | 300 | 89.779662 | 0.881551 | 0.325726 | 0.917400
44 | 10 | 50 | 200 | 66.533663 | 0.915464 | 0.375827 | 0.918218
45 | 10 | 100 | 350 | 80.993022 | 0.895804 | 0.341044 | 0.920199
46 | 1 | 200 | 350 | 86.617050 | 0.890615 | 0.333431 | 0.924701
47 | 10 | 150 | 250 | 81.907492 | 0.889782 | 0.339023 | 0.942778
48 | 5 | 25 | 200 | 56.917854 | 0.941723 | 0.407363 | 0.944494
49 | 5 | 25 | 350 | 63.104300 | 0.945385 | 0.387386 | 0.945559
50 | 10 | 150 | 200 | 79.250014 | 0.892611 | 0.344558 | 0.952231
51 | 5 | 250 | 350 | 91.482999 | 0.879424 | 0.322753 | 0.955824
52 | 10 | 150 | 300 | 84.307897 | 0.887426 | 0.334255 | 0.958265
53 | 1 | 250 | 300 | 88.148662 | 0.887337 | 0.330608 | 0.991374
54 | 10 | 150 | 350 | 86.153677 | 0.885734 | 0.330728 | 0.996277
55 | 1 | 25 | 200 | 53.855529 | 0.953255 | 0.419992 | 1.002329
56 | 10 | 25 | 200 | 59.867095 | 0.931867 | 0.396320 | 1.004255
57 | 10 | 25 | 350 | 66.079670 | 0.934349 | 0.377716 | 1.009266
58 | 1 | 25 | 350 | 60.000686 | 0.958187 | 0.398395 | 1.010507
59 | 10 | 200 | 250 | 85.417081 | 0.884354 | 0.332064 | 1.016014
60 | 5 | 300 | 350 | 93.552354 | 0.876249 | 0.319243 | 1.027944
61 | 10 | 100 | 150 | 70.755154 | 0.903526 | 0.364365 | 1.029546
62 | 1 | 250 | 350 | 89.938622 | 0.885053 | 0.327404 | 1.036422
63 | 5 | 50 | 150 | 60.654246 | 0.923860 | 0.394616 | 1.046363
64 | 10 | 200 | 300 | 88.041188 | 0.881264 | 0.327161 | 1.050844
65 | 10 | 200 | 350 | 89.885755 | 0.879225 | 0.323846 | 1.088957
66 | 5 | 10 | 250 | 52.061359 | 0.969479 | 0.426049 | 1.092622
67 | 1 | 50 | 150 | 57.955534 | 0.932734 | 0.405222 | 1.108618
68 | 1 | 300 | 350 | 92.123306 | 0.881627 | 0.323616 | 1.120125
69 | 5 | 10 | 300 | 53.988044 | 0.972202 | 0.418513 | 1.130341
70 | 10 | 50 | 150 | 63.245032 | 0.916219 | 0.385275 | 1.147237
71 | 10 | 250 | 300 | 91.366002 | 0.876248 | 0.321223 | 1.148571
72 | 10 | 250 | 350 | 92.984555 | 0.874252 | 0.318459 | 1.187414
73 | 1 | 10 | 250 | 48.534912 | 0.985705 | 0.441887 | 1.226465
74 | 10 | 300 | 350 | 94.940727 | 0.871296 | 0.315198 | 1.257704
75 | 5 | 10 | 200 | 49.742061 | 0.965777 | 0.435794 | 1.269821
76 | 1 | 10 | 300 | 50.407365 | 0.989203 | 0.433686 | 1.275819
77 | 25 | 50 | 250 | 74.165885 | 0.900566 | 0.354256 | 1.287763
78 | 25 | 50 | 300 | 76.288280 | 0.899839 | 0.349376 | 1.308031
79 | 5 | 10 | 350 | 55.591458 | 0.974292 | 0.412578 | 1.323138
80 | 5 | 25 | 150 | 53.769108 | 0.939484 | 0.418971 | 1.325565
81 | 10 | 25 | 150 | 56.704177 | 0.930439 | 0.407046 | 1.333711
82 | 25 | 50 | 200 | 71.675838 | 0.901540 | 0.360268 | 1.336092
83 | 25 | 50 | 350 | 78.008622 | 0.899316 | 0.345574 | 1.382325
84 | 1 | 10 | 200 | 46.281762 | 0.980843 | 0.452557 | 1.422110
85 | 1 | 150 | 200 | 101.253374 | 0.880595 | 0.309167 | 1.424588
86 | 1 | 25 | 150 | 50.732258 | 0.950087 | 0.432647 | 1.434219
87 | 25 | 50 | 150 | 68.456491 | 0.903012 | 0.368552 | 1.476343
88 | 25 | 100 | 250 | 81.169693 | 0.886459 | 0.338039 | 1.499279
89 | 1 | 10 | 350 | 51.971163 | 0.991852 | 0.427233 | 1.500739
90 | 25 | 100 | 200 | 78.640776 | 0.888565 | 0.343416 | 1.503322
91 | 25 | 100 | 300 | 83.385047 | 0.884770 | 0.333542 | 1.529394
92 | 1 | 5 | 250 | 44.703470 | 1.004339 | 0.460146 | 1.531274
93 | 25 | 100 | 150 | 75.299148 | 0.891671 | 0.350961 | 1.544265
94 | 5 | 50 | 100 | 56.161277 | 0.924590 | 0.409879 | 1.566059
95 | 10 | 50 | 100 | 58.818554 | 0.917491 | 0.399325 | 1.586276
96 | 25 | 100 | 350 | 85.132502 | 0.883530 | 0.330126 | 1.587127
97 | 1 | 5 | 300 | 46.491365 | 1.008976 | 0.451198 | 1.600957
98 | 25 | 150 | 200 | 83.291825 | 0.881340 | 0.333479 | 1.622311
99 | 25 | 150 | 250 | 85.776189 | 0.878634 | 0.328588 | 1.651279
100 | 25 | 150 | 300 | 88.087350 | 0.876299 | 0.324237 | 1.695171
101 | 1 | 50 | 100 | 53.407413 | 0.932875 | 0.421910 | 1.701221
102 | 1 | 5 | 200 | 42.555702 | 0.997748 | 0.471868 | 1.742130
103 | 25 | 50 | 100 | 64.168570 | 0.905385 | 0.380620 | 1.746915
104 | 25 | 150 | 350 | 89.830087 | 0.874644 | 0.321074 | 1.747854
105 | 25 | 200 | 250 | 88.985575 | 0.873745 | 0.322482 | 1.765219
106 | 5 | 10 | 150 | 46.786039 | 0.960056 | 0.449510 | 1.772808
107 | 25 | 200 | 300 | 91.528935 | 0.870731 | 0.317930 | 1.830865
108 | 1 | 5 | 350 | 47.990655 | 1.012439 | 0.444169 | 1.870402
109 | 25 | 200 | 350 | 93.247202 | 0.868811 | 0.314968 | 1.884770
110 | 25 | 250 | 300 | 94.707257 | 0.866003 | 0.312440 | 1.963546
111 | 1 | 10 | 150 | 43.425162 | 0.973119 | 0.467661 | 1.991153
112 | 10 | 25 | 100 | 52.429141 | 0.928047 | 0.423295 | 1.997752
113 | 25 | 250 | 350 | 96.143870 | 0.864233 | 0.310072 | 2.013517
114 | 25 | 300 | 350 | 97.857572 | 0.861669 | 0.307287 | 2.094282
115 | 5 | 25 | 100 | 49.514747 | 0.935561 | 0.436721 | 2.101010
116 | 50 | 100 | 250 | 85.638400 | 0.875559 | 0.327097 | 2.166302
117 | 50 | 100 | 300 | 87.817526 | 0.873615 | 0.322932 | 2.232250
118 | 50 | 100 | 350 | 89.496261 | 0.872214 | 0.319838 | 2.306645
119 | 1 | 25 | 100 | 46.523119 | 0.944299 | 0.452180 | 2.322517
120 | 50 | 150 | 200 | 87.586643 | 0.870942 | 0.322978 | 2.354390
121 | 1 | 5 | 150 | 39.854431 | 0.986974 | 0.488576 | 2.381987
122 | 50 | 150 | 250 | 89.892435 | 0.868248 | 0.318643 | 2.433698
123 | 50 | 150 | 300 | 92.144370 | 0.865792 | 0.314587 | 2.520239
124 | 50 | 150 | 350 | 93.782322 | 0.864105 | 0.311741 | 2.595768
125 | 50 | 200 | 250 | 92.796838 | 0.863730 | 0.313268 | 2.609630
126 | 50 | 200 | 300 | 95.310954 | 0.860607 | 0.308912 | 2.726528
127 | 50 | 200 | 350 | 96.894458 | 0.858748 | 0.306268 | 2.804948
128 | 5 | 10 | 100 | 42.834193 | 0.949276 | 0.470912 | 2.835505
129 | 50 | 250 | 300 | 98.394826 | 0.855944 | 0.303696 | 2.923710
130 | 50 | 250 | 350 | 99.610614 | 0.854397 | 0.301742 | 2.988923
131 | 50 | 300 | 350 | 101.038510 | 0.852221 | 0.299448 | 3.086309
132 | 1 | 10 | 100 | 39.645644 | 0.957950 | 0.491504 | 3.204863
133 | 100 | 150 | 200 | 93.222892 | 0.859267 | 0.310799 | 3.215942
134 | 100 | 150 | 250 | 95.213565 | 0.856649 | 0.307229 | 3.350851
135 | 10 | 25 | 50 | 45.667704 | 0.921380 | 0.454834 | 3.443735
136 | 100 | 150 | 300 | 97.445560 | 0.853883 | 0.303385 | 3.501372
137 | 100 | 200 | 250 | 97.627765 | 0.852638 | 0.302836 | 3.597631
138 | 100 | 150 | 350 | 98.929260 | 0.852136 | 0.300916 | 3.605990
139 | 1 | 5 | 100 | 36.346508 | 0.964850 | 0.515314 | 3.763530
140 | 100 | 200 | 300 | 100.240006 | 0.849066 | 0.298428 | 3.805282
141 | 5 | 25 | 50 | 42.830270 | 0.924177 | 0.471778 | 3.810106
142 | 100 | 200 | 350 | 101.632354 | 0.847259 | 0.296164 | 3.915039
143 | 150 | 200 | 250 | 100.455374 | 0.846841 | 0.297276 | 4.110838
144 | 100 | 250 | 300 | 103.385872 | 0.844043 | 0.293166 | 4.124725
145 | 100 | 250 | 350 | 104.197584 | 0.842930 | 0.291885 | 4.193478
146 | 100 | 300 | 350 | 105.126331 | 0.841428 | 0.290387 | 4.291667
147 | 1 | 25 | 50 | 39.988895 | 0.926528 | 0.491485 | 4.297531
148 | 150 | 200 | 300 | 103.488161 | 0.842317 | 0.292178 | 4.430524
149 | 150 | 200 | 350 | 104.716717 | 0.840585 | 0.290197 | 4.554876
150 | 150 | 250 | 300 | 107.091983 | 0.836250 | 0.286146 | 4.904307
151 | 150 | 300 | 350 | 107.712611 | 0.835260 | 0.285151 | 4.981076
152 | 250 | 300 | 350 | 108.438873 | 0.833609 | 0.283742 | 5.165724
153 | 5 | 10 | 50 | 36.900274 | 0.914826 | 0.515154 | 5.246407
154 | 200 | 300 | 350 | 109.443045 | 0.831385 | 0.281837 | 5.412344
155 | 200 | 250 | 350 | 110.400050 | 0.829656 | 0.280247 | 5.569773
156 | 200 | 250 | 300 | 111.292402 | 0.828317 | 0.278881 | 5.680566
157 | 1 | 10 | 50 | 34.242821 | 0.905331 | 0.542148 | 5.964818
158 | 1 | 5 | 50 | 31.855840 | 0.880018 | 0.574156 | 6.904728
159 | 5 | 10 | 25 | 33.160455 | 0.827556 | 0.571797 | 7.980513
160 | 1 | 10 | 25 | 32.404745 | 0.749428 | 0.610644 | 9.053548
161 | 1 | 5 | 25 | 35.401342 | 0.554285 | 0.660232 | 10.338711
162 | 1 | 5 | 10 | 60.167715 | 0.010001 | 0.707299 | 11.760083
163 | 50 | 100 | 200 | 21.505459 | 2.000000 | 1.357916 | 19.074560
164 | 50 | 100 | 150 | 18.792839 | 2.000000 | 1.460051 | 19.352376
165 | 150 | 250 | 350 | 39.237156 | 2.000000 | 1.232217 | 25.195640
Table 3: ${}^{1}S_{0}$ state: Model parameters for 55 combinations for ${}^{1}S_{0}-nn$ state. After fixing $r_{m}=0.9155$ (average) fm from 112 combinations of previous table LABEL:165combi, two parameters produces ${}^{11}C_{2}$ i.e. 55 combinations. The data is sorted with respect to MAE in ascending order. Sr. No. |  $E_{1}$ (MeV) |  $E_{2}$ (MeV) |  $V_{0}$ (MeV) |  $a_{m}$ (fm) |  Overall MAE |
---|---|---|---|---|---|---
1 | 5 | 300 | 71.730802 | 0.363578 | 0.666437 |
2 | 5 | 250 | 69.812442 | 0.368384 | 0.666720 |
3 | 1 | 250 | 71.193511 | 0.366651 | 0.689214 |
4 | 1 | 300 | 72.930120 | 0.362403 | 0.689361 |
5 | 5 | 50 | 72.914193 | 0.360607 | 0.736652 |
6 | 5 | 350 | 73.343871 | 0.359686 | 0.738354 |
7 | 5 | 200 | 67.660861 | 0.374021 | 0.753482 |
8 | 1 | 350 | 74.427456 | 0.358861 | 0.754361 |
9 | 1 | 200 | 69.374385 | 0.371273 | 0.758771 |
10 | 50 | 300 | 70.968527 | 0.364323 | 0.814791 |
11 | 50 | 250 | 68.917538 | 0.369509 | 0.816551 |
12 | 100 | 200 | 69.587353 | 0.370934 | 0.817602 |
13 | 10 | 250 | 68.868276 | 0.369571 | 0.827726 |
14 | 10 | 300 | 70.898325 | 0.364391 | 0.833286 |
15 | 50 | 350 | 72.663694 | 0.360200 | 0.890809 |
16 | 100 | 150 | 66.847782 | 0.377443 | 0.900150 |
17 | 100 | 250 | 71.804840 | 0.365886 | 0.911899 |
18 | 10 | 350 | 72.583883 | 0.360260 | 0.914793 |
19 | 50 | 200 | 66.521411 | 0.375867 | 0.917600 |
20 | 10 | 200 | 66.517146 | 0.375874 | 0.918413 |
21 | 5 | 150 | 65.191362 | 0.380843 | 0.939942 |
22 | 100 | 300 | 73.759048 | 0.361589 | 1.044861 |
23 | 25 | 250 | 68.110429 | 0.370526 | 1.104946 |
24 | 25 | 300 | 70.216211 | 0.365056 | 1.122282 |
25 | 10 | 150 | 63.620993 | 0.384135 | 1.141107 |
26 | 5 | 100 | 63.240549 | 0.386527 | 1.142037 |
27 | 50 | 150 | 63.503218 | 0.384385 | 1.165667 |
28 | 25 | 200 | 65.625739 | 0.377329 | 1.190309 |
29 | 100 | 350 | 75.376753 | 0.358133 | 1.206715 |
30 | 25 | 350 | 71.951289 | 0.360735 | 1.217653 |
31 | 25 | 150 | 62.456573 | 0.386623 | 1.421515 |
32 | 50 | 100 | 59.621094 | 0.396317 | 1.614747 |
33 | 150 | 200 | 72.531742 | 0.366298 | 1.691748 |
34 | 25 | 100 | 58.241976 | 0.400252 | 1.876912 |
35 | 150 | 250 | 74.641661 | 0.362351 | 2.037006 |
36 | 150 | 300 | 76.578661 | 0.358811 | 2.376252 |
37 | 150 | 350 | 78.165101 | 0.355970 | 2.684166 |
38 | 10 | 50 | 101.859853 | 0.305723 | 2.942245 |
39 | 200 | 250 | 77.056751 | 0.359363 | 3.026641 |
40 | 200 | 300 | 79.074761 | 0.356343 | 3.612583 |
41 | 200 | 350 | 80.655030 | 0.354009 | 4.083227 |
42 | 10 | 25 | 42.587021 | 0.472826 | 4.263083 |
43 | 5 | 25 | 40.200809 | 0.489460 | 4.659452 |
44 | 250 | 300 | 81.441998 | 0.354000 | 4.800802 |
45 | 1 | 25 | 37.934492 | 0.507781 | 5.094683 |
46 | 5 | 10 | 36.968052 | 0.514500 | 5.213176 |
47 | 1 | 50 | 36.061249 | 0.523395 | 5.384020 |
48 | 250 | 350 | 82.929463 | 0.352199 | 5.389926 |
49 | 1 | 10 | 34.833315 | 0.534637 | 5.588524 |
50 | 1 | 5 | 32.866784 | 0.554584 | 5.952004 |
51 | 300 | 350 | 84.672221 | 0.350802 | 6.390542 |
52 | 25 | 50 | 30.513851 | 0.596846 | 7.700140 |
53 | 1 | 100 | 21.346168 | 0.752283 | 7.851272 |
54 | 10 | 100 | 18.349456 | 0.917733 | 9.809530 |
55 | 1 | 150 | 10.445956 | 1.341045 | 9.942812 |
Table 4: ${}^{1}S_{0}$ state: Model parameters for 21 combinations, each with three lab energies and obtained by minimising MSE. The overall MAE is determined by obtaining SPS for remaining experimental data points. The data is sorted with respect to $E_{1}$ in ascending order. Sr. No. |  $E_{1}$ (MeV) |  $E_{2}$ (MeV) |  $V_{0}$ (MeV) |  $a_{m}$ (MeV) |  Overall MAE |  $r_{e}$ (fm) |  $a$ (fm)
---|---|---|---|---|---|---|---
1 | 1 | 250 | 71.193511 | 0.366651 | 0.689214 | -18.380 | 2.539
2 | 1 | 300 | 72.930120 | 0.362403 | 0.689361 | -18.373 | 2.534
3 | 1 | 350 | 74.427456 | 0.358861 | 0.754361 | -18.368 | 2.530
4 | 1 | 200 | 69.374385 | 0.371273 | 0.758771 | -18.387 | 2.544
5 | 5 | 300 | 71.730802 | 0.363578 | 0.666437 | -16.173 | 2.556
6 | 5 | 250 | 69.812442 | 0.368384 | 0.666720 | -16.212 | 2.561
7 | 5 | 50 | 72.914193 | 0.360607 | 0.736652 | -16.076 | 2.554
8 | 5 | 350 | 73.343871 | 0.359686 | 0.738354 | -16.142 | 2.552
9 | 5 | 200 | 67.660861 | 0.374021 | 0.753482 | -16.258 | 2.566
10 | 5 | 150 | 65.191362 | 0.380843 | 0.939942 | -16.314 | 2.573
11 | 10 | 250 | 68.868276 | 0.369571 | 0.827726 | -14.962 | 2.576
12 | 10 | 300 | 70.898325 | 0.364391 | 0.833286 | -14.896 | 2.571
13 | 10 | 350 | 72.583883 | 0.360260 | 0.914793 | -14.844 | 2.568
14 | 10 | 200 | 66.517146 | 0.375874 | 0.918413 | -15.043 | 2.582
15 | 50 | 300 | 70.968527 | 0.364323 | 0.814791 | -14.997 | 2.570
16 | 50 | 250 | 68.917538 | 0.369509 | 0.816551 | -15.024 | 2.575
17 | 50 | 350 | 72.663694 | 0.360200 | 0.890809 | -14.972 | 2.566
18 | 50 | 200 | 66.521411 | 0.375867 | 0.917600 | -15.047 | 2.582
19 | 100 | 200 | 69.587353 | 0.370934 | 0.817602 | -18.681 | 2.541
20 | 100 | 150 | 66.847782 | 0.377443 | 0.900150 | -17.673 | 2.556
21 | 100 | 250 | 71.804840 | 0.365886 | 0.911899 | -19.499 | 2.529
|  | Avg. | 70.226561 | 0.367646 | 0.807472 | -16.492 | 2.558
|  | St. Dev. | 2.605663 | 0.006510 | 0.091322 | 1.557 | 0.017

## References

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# Heavy quarkonia spectroscopy at zero and finite temperature in bottom-up
AdS/QCD

Miguel Angel Martin Contreras<EMAIL_ADDRESS>Instituto de Física y
Astronomía,
Universidad de Valparaíso,
A. Gran Bretaña 1111, Valparaíso, Chile  Alfredo Vega<EMAIL_ADDRESS>Instituto de Física y Astronomía,
Universidad de Valparaíso,
A. Gran Bretaña 1111, Valparaíso, Chile  Saulo Diles<EMAIL_ADDRESS>Campus
Salinópolis,
Universidade Federal do Pará,
68721-000, Salinópolis, Pará, Brazil

###### Abstract

S-wave states of charmonium and bottomonium are described using bottom-up
AdS/QCD. We propose a holographic model that unifies the description of masses
and decay constants, leading to a precise match with experimental data on
heavy quarkonia. Finite temperature effects are considered by calculating the
current-current spectral functions of heavy vector mesons. The identification
of quasi-particle states as Breit-Wigner resonances in the holographic
spectral function was made. We develop a prescription to subtract background
contributions from the spectral function to isolate the Breit-Wigner peak. The
quasi-particle holographic thermal evolution is described, allowing us to
estimate the melting temperature for vector charmonia and bottomonia. Our
holographic model predicts that $J/\Psi$ melts at $415$ MeV $(\sim
2.92\leavevmode\nobreak\ T_{c})$ and $\Upsilon$ melts at $465$ MeV $(\sim
3.27\leavevmode\nobreak\ T_{c})$.

First keyword and Second keyword and More

##  I Introduction

Heavy quarkonia work as a probe of quark-gluon plasma formation in heavy-ion
collisions, where charmonium suppression seemed to play the fundamental role
Matsui and Satz (1986). It happens that $J/\Psi$ track is hard to reconstruct
due to physical effects such as nuclear absorption and recombination Chaudhuri
(2002); Liu et al. (2011); Abreu et al. (2018). This difficulty in tracking
back the charmonium trajectories made unfavorable $J/\Psi$ as a precise probe
of QGP. On the other hand, bottomonium production by recombination and
regeneration effects is small Song et al. (2012); Emerick et al. (2012); Reed
(2011). Bottomonium then emerges as a promising candidate to probe QGP
properties, but not invalidating the importance of charmonium in this context.
See Krouppa et al. (2019); Yao and Müller (2019).

Charmonium and bottomonium mesons were experimentally discovered, latter a
than its light cousins ($\rho,\phi$), due to its considerable threshold
energies imposed by the heavy $c,b$ quark masses. Curiously, current
experimental data about the spectrum of radial excitations is more extensive
and complete for the heavy vector than the light ones. The decay constants for
the excited S-wave states are entirely determined from experiments for the
heavy vector quarkonium Tanabashi et al. (2018). Decay constants of charmonium
and bottomonium are observed to be decreasing with excitation levels. For the
$\phi$ meson, the decay constants of excited states are estimated from
experimental data. These estimations predict they are also decreasing with
excitation level Pang (2019); Badalian and Bakker (2019).

Meson spectroscopy is a static low energy phenomenon. In this case, the color
interaction is strongly coupled and a non-perturbative approach for strong
interactions is required Gross and Wilczek (1973); Politzer (1973); van
Ritbergen et al. (1997). One important non-perturbative approach is the
holographic dual of QCD, referred as AdS/QCD Polchinski and Strassler (2002);
Boschi-Filho and Braga (2003); Erlich et al. (2005); Brodsky and de Teramond
(2008). AdS/QCD models are inspired in the exact duality between the conformal
and supersymmetric field theory $\mathcal{N}=4$ SYM in four space-time
dimensions, and the type IIB string theory in $AdS_{5}\times S^{5}$ Maldacena
(1999); Aharony et al. (2000). In top-down AdS/QCD models, the energy scales
are fixed by probe branes located in the bulk geometry. The presence of these
probe branes in the AdS bulk breaks conformal symmetry and set the energy
scales in the boundary theory Karch and Katz (2002); Sakai and Sugimoto
(2005a, b). On the other hand, bottom-up AdS/QCD models implement deformations
in the bulk geometry directly associated with observed phenomena in hadronic
physics. The most popular bottom-up AdS/QCD models are the hard wall
Polchinski and Strassler (2002); Boschi-Filho and Braga (2004, 2003) and the
soft wall Karch et al. (2006). The soft wall model is particularly interesting
for investigating the radial excitations of mesons since it predicts a linear
Regge trajectory for the hadron masses. Bottom-up AdS/QCD models have been
systematically applied in the description of the spectrum of mesons Karch et
al. (2006); Grigoryan and Radyushkin (2007); Erdmenger et al. (2008);
Colangelo et al. (2008); Ballon Bayona et al. (2010); Cotrone et al. (2011)
and in particular for heavy quarkonia Kim et al. (2007); Grigoryan et al.
(2010); Li et al. (2016); Braga et al. (2016a). Heavy quark potentials have
been analyzed for different botton-up AdS/QCD models, finding in all cases the
linear behaviour for large separation Boschi-Filho and Braga (2005); Boschi-
Filho et al. (2006); Andreev and Zakharov (2006, 2007); Colangelo et al.
(2011); Bruni et al. (2019); Diles (2020).

The observed decay constants of quarkonia S-wave states increase the
difficulty in obtaining an accurate description of their spectrum. The
challenge comes from the fact that decay constants decrease in a monotonic and
non-linear way with excitation level. The hard-wall model predicts decay
constants increasing with excitation level, while the soft-wall model
(quadratic dilaton) predicts completely degenerate decay constants. This poor
description of decay constants at zero temperature leads to bad results at
finite temperature, such as the disappearance of the spectral peaks of the
fundamental state at low temperatures Fujita et al. (2009, 2010); Mamani et
al. (2014). A good description of decay constants in the vacuum is needed to
get a consistent spectral analysis at finite temperature. Decay constant
defines the strength of the resonances fixing the zero-temperature limit of
the spectral function.

In Ref. Grigoryan et al. (2010) it is proposed an holographic description of
$c\bar{c}$ considering modifications in the holographic potential. These
modifications lead to an improvement in the description of masses and decay
constants of $J/\Psi,\Psi^{\prime}$. However, the holographic potential of
Grigoryan et al. (2010) does not capture the decrease in decay constants. An
alternative proposal is to set up an ultraviolet scale by calculating
correlation functions in an AdS slice at finite $z_{uv}$ Evans and Tedder
(2006); Afonin (2011, 2012); Braga et al. (2016b). This ultraviolet cut-off
results in decay constants that decrease with excitation level. However, this
model predicts a small decrease in the excitation level than experimental data
that shows a fast decrease. So, it captures the decrease in decay constants
but not the correct slope. The problem of the slope in decay constants was
circumvented in a different holographic model proposed in Ref. Braga et al.
(2017) and refined in Ref. Braga and Ferreira (2018). The holographic model of
Ref. Braga and Ferreira (2018) captures the correct observed spectrum of decay
constants of either charmonium and bottomonium with good precision. This
success in describing the decay constants does not extend to the mass
spectrum. An accurate description of the radial excitations of heavy quarkonia
involves either the masses and the decay constants. Here we propose a
holographic model that simultaneously describes the masses and decay constants
of the radial excitations of charmonium and bottomonium. The predictions of
our model agree with experimental data within an RMS error near to $6\%$ for
charmonium and $7,2\%$ for bottomonium, providing a precise description of
quarkonia spectroscopy at zero temperature. We consider the effects of hot
plasma on quarkonia states and use our model to compute in-medium spectral
functions. We propose a prescription for background subtraction, isolating the
contribution of the quasi-particle states in the spectral function from the
medium effects. The melting temperatures of
$J/\Psi,\Psi^{\prime},\Upsilon,\Upsilon^{\prime}$ are estimated and their
thermal masses analyzed.

The paper is organized as follows: in Section II, we motivate and present the
dilaton that defines our holographic model. In Section III, we describe
precisely the spectrum of masses and decay constants of charmonium and
bottomonium. In Section IV we consider our model at finite temperature: we
discuss the confinement/ deconfinement phase transition, compute finite
temperature spectral functions of $c\bar{c}$ and $b\bar{b}$ and analyse the
quasi-particle states associated with the resonance peaks. In section V we
perform the Breit-Wigner analysis to the holographic spectral densities
calculated for heavy quarkonia. Finally, we elaborate in Section VI the main
conclusions of this work.

##  II Holographic Model

In the context of the AdS/QCD bottom-up models, heavy vector quarkonium is
described as an abelian massless bulk gauge field. This affirmation follows
from the standard field/operator duality Aharony et al. (2000). Recall the
scaling dimension of the source operators creating mesons at the conformal
boundary defines the dual bulk field mass, according to the relation:

$M_{5}^{2}\,R^{2}=(\Delta-S)(\Delta+S-4),$ (1)

where $S$ is the meson spin, and $R$ is the AdS radius. This relation defines
a primitive notion of _hadronic identity_ since their corresponding bulk mass
will categorize the dual hadronic states defined by the boundary source
operator. In the case of _any_ $q\,\bar{q}$ vector meson state, it is
generated by structures with $\Delta=3$, implying $M_{5}^{2}\,R^{2}=0$. Thus,
the action in the bulk space is given by

$I_{\text{Vector
$Q\bar{Q}$}}=-\frac{1}{4\,g_{5}^{2}}\,\int{d^{5}x\,\sqrt{-g}\,e^{-\Phi(z)}}\,g^{mp}\,g^{nr}\,F_{mn}\,F_{pr},$
(2)

where $g_{5}$ is a constant that fixes units in the action given above and
$F_{mn}$ is the field strength. This coupling is calculated from the large
$q^{2}$ behavior of the holographic vector two-point functions Erlich et al.
(2005). The geometrical background is either AdS5 or AdS5 BH, depending on
whether we are at zero or finite temperature. We will postpone this discussion
to the next section. Independent of the geometry, the equations of motion for
the bulk gauge fields are

$\frac{1}{\sqrt{-g}}\,\partial_{n}\left[\sqrt{-g}\,e^{-\Phi(z)}\,g^{np}\,g^{mr}\,F_{pr}\right]=0.$
(3)

Confinement in this model is induced _via_ the static dilaton field $\Phi(z)$.
In the standard AdS/QCD softwall model, characterized by the static quadratic
dilaton, large $z$ behavior guaranteed the emergence of linear radial Regge
trajectories. However, it does not properly describe the meson decay constants
since they are expected to decrease with the radial excitation number $n$. The
softwall model calculation brings degenerate decay constants for $n$.

A lesson learned from Martin Contreras and Vega (2020a) was that decay
constants depend on the low $z$ limit behavior of the AdS/QCD model at hand.
We can modify this behavior by two possible forms: by deforming the background
Braga et al. (2016b, a) or by introducing terms in the dilaton that becomes
relevant at low $z$ Braga et al. (2017); Braga and Ferreira (2018). The
resulting Regge trajectories are still linear, and the decay constant behavior
is corrected.

On the experimental side, these sorts of linear Regge trajectories describe
better the light sector. Nevertheless, in the heavy one, the linear
approximation does not seem to fit the available experimental data. By looking
closely at the heavy quarkonium radial trajectories, we observed linearity in
the highly excited states. On the other side, the linear spectrum deviate due
to the heavy constituent quark mass effect in the meson. This picture can be
seen from the angular quantization of the string Afonin and Pusenkov (2014) or
the Bethe-Salpeter analysis Chen (2018) by writing the radial trajectory as

$(M_{n}-m_{Q_{1}}-m_{Q_{2}})^{2}=a(n+b),$ (4)

where $a$ is a universal slope and $b$ is related to the mesonic quantum
numbers. Therefore, nonlinearities emerge when the constituent quark mass
comes to play. The nonlinear trajectories can be written in general as

$M^{2}=a(n+b)^{\nu}.$ (5)

In a recent work Martin Contreras and Vega (2020b), these nonlinear Regge
trajectories were described in the context of bottom-up holographic QCD. The
main idea behind this model is that the inclusion of quark constituent masses
implies deviation from the quadratic behavior imposed on the static dilaton.
This model successfully described vector mesons in the light unflavored,
strange, heavy-light, and heavy sectors.

This nonquadratic and the softwall model dilaton share the same low $z$
behavior. Therefore, in the nonquadratic context, the decay constants do not
behave following the phenomenological constraints. An attempt to circumvent
this drawback is by adding the proper low $z$ behavior that captures the
expected decay constants phenomenology. Therefore we propose the following
nonquadratic dilaton

$\Phi(z)=\left(\kappa\,z\right)^{2-\alpha}+M\,z\,+\text{tanh}\left[\frac{1}{M\,z}-\frac{\kappa}{\sqrt{\Gamma}}\right],$
(6)

where the low $z$ contributions written above were read from Braga and
Ferreira (2018). The parameters $\kappa$, $M$ and $\sqrt{\Gamma}$ are energy
scales controlling the slope and the intercept, whereas $\alpha$ is
dimensionless and measures the constituent quark mass effect in the heavy
meson, as it was introduced in Martin Contreras and Vega (2020b).

In the later sections, we will discuss the application of this dilaton for
charmonium and bottomonium systems at zero and finite temperature.

##  III zero temperature

In the case of zero temperature, the geometrical background is given by the
Poincaré patch

$dS^{2}=g_{mn}\,dx^{m}\,dx^{n}=\frac{R^{2}}{z^{2}}\left[dz^{2}+\eta_{\mu\,\nu}\,dx^{\mu}\,dx^{\nu}\right],$
(7)

with the signature $\eta_{\mu\nu}=\text{diag}(-1,1,1,1)$ and $z\in(0,\infty)$.

Following the AdS/CFT methodology, we will write the Fourier transformed bulk
vector field in terms of the bulk modes $\psi(z,q)$ and the boundary sources
as

$A_{\mu}(z,q)=A_{\mu}(q)\,\psi(z,q),$ (8)

where we have implicitly imposed the gauge fixing $A_{z}=0$. We use the $z$
component of the equations of motion, $\partial_{z}(\partial_{\mu}A^{\mu})=0$,
and the Lorentz gauge in the boundary to set $\partial_{\mu}A^{\mu}=0$
everywhere. These definitions yield the following equations for the eigenmodes

$\partial_{z}\left[e^{-B(z)}\,\partial_{z}\,\psi_{n}(z,q)\right]+(-q^{2})\,e^{-B(z)}\,\psi_{n}(z,q)=0.$
(9)

where we have defined the background information $B(z)$ function as

$B(z)=\Phi(z)-\text{log}\left(\frac{R}{z}\right).$ (10)

Confinement emerges in this model by the effect of the dilaton field that
induces a holographic confining potential. We apply the Bogoliubov
transformation $\psi(z)=e^{B(z)/2}\,\phi(z)$ to the expression (9) obtaining a
Schrodinger-like equation defined as

$-\phi_{n}^{\prime\prime}(z)+U(z)\,\phi_{n}(z)=M_{n}^{2}\,\phi_{n}(z),$ (11)

where $M_{n}^{2}=-q^{2}$ defines the meson spectrum, and the holographic
potential is constructed in terms of the derivatives of the $\Phi(z)$ dilaton
field in eqn. (6) as follows

$U(z)=\frac{3}{4\,z^{2}}+\frac{\Phi^{\prime}(z)}{2\,z}+\frac{1}{4}\Phi^{\prime}(z)^{2}-\frac{1}{2}\Phi^{\prime\prime}(z).$
(12)

By solving the Schrodinger-like equation numerically, we obtain the associated
bulk modes and the holographic mass spectrum. The results for charmonium and
bottomonium, with the corresponding parameter fixing, are summarized in tables
1 and 2.

Charmonium States $I^{G}(J^{PC})=0^{+}(1^{--})$
---
Parameters: |  $\kappa=1.8$ GeV, $M=1.7$ GeV, $\sqrt{\Gamma}=0.53$ GeV and $\alpha=0.54$
$n$ | State |  $M_{\text{Exp}}$ (MeV) |  $M_{\text{Th}}$ (MeV) | %$M$ |  $f_{\text{Exp}}$ (MeV) |  $f_{\text{Th}}$ (MeV) | %$f$
$1$ | $J/\psi$ | $3096.916\pm 0.011$ | $3140.1$ | $1.42$ | $416.16\pm 5.25$ | $412.4$ | $1.4$
$2$ | $\psi(2S)$ | $3686.109\pm 0.012$ | $3656.9$ | $0.9$ | $296.08\pm 2.51$ | $272.7$ | $8.0$
$3$ | $\psi(4040)$ | $4039\pm 1$ | $4055.7$ | $0.4$ | $187.13\pm 7.61$ | $201.8$ | $7.8$
$4$ | $\psi(4415)$ | $4421\pm 4$ | $4376$ | $0.9$ | $160.78\pm 9.70$ | $164.1$ | $2.0$
Nonlinear Regge Trajectory: |  $M_{n}^{2}=8.097(0.39+n)^{0.58}$GeV2 with $R^{2}=0.999$
Table 1: Summary of results for charmonium states. $M_{Th}$ and $f_{Th}$ are
calculated with the parameters mentioned on header, and corresponding errors
appear in columns $\%M$ and $\%f$. Experimental results are read from PDG
Tanabashi et al. (2018) and total error is $\delta_{\text{RMS}}=6.0\,\%$. The
Regge trajectories are also presented. Bottomonium States
$I^{G}(J^{PC})=0^{+}(1^{--})$
---
Parameters: |  $\kappa=9.9$ GeV, $M=2.74$ GeV, $\sqrt{\Gamma}=1.92$ GeV and $\alpha=0.863$
$n$ | State |  $M_{\text{Exp}}$ (MeV) |  $M_{\text{Th}}$ (MeV) | %$M$ |  $f_{\text{Exp}}$ (MeV) |  $f_{\text{Th}}$ (MeV) | %$f$
$1$ | $\Upsilon(1S)$ | $9460.3\pm 0.26$ | $9506.5$ | $0.5$ | $714.99\pm 2.40$ | $718.8$ | $0.5$
$2$ | $\Upsilon(2S)$ | $10023.26\pm 0.32$ | $9892.9$ | $1.0$ | $497.37\pm 2.23$ | $575.7$ | $16$
$3$ | $\Upsilon(3S)$ | $10355.2\pm 0.5$ | $10227.2$ | $1.2$ | $430.11\pm 1.94$ | $413.0$ | $4.0$
$4$ | $\Upsilon(4S)$ | $10579.4\pm 1.2$ | $10497.5$ | $0.8$ | $340.65\pm 9.08$ | $324.3$ | $4.8$
$5$ | $\Upsilon(10860)$ | $10889.9^{+3.2}_{-2.6}$ | $10721.5$ | $1.5$ | – | – | –
$6$ | $\Upsilon(11020)$ | $10992.9^{+10.0}_{-3.1}$ | $10912.7$ | $0.7$ | – | – | –
Nonlinear Regge Trajectory: |  $M_{n}^{2}=7.376(1.31+n)^{0.24}$GeV2 with $R^{2}=0.999$
Table 2: Summary of results for bottomonium states. $M_{Th}$ and $f_{Th}$ are
calculated with the parameters mentioned on header, and corresponding errors
appear in columns $\%M$ and $\%f$. Experimental results are read from PDG
Tanabashi et al. (2018) and total error is $\delta_{\text{RMS}}=7.2\,\%$. The
Regge trajectories are also presented.

In the case of electromagnetic decay constants $f_{n}$, they arise as the
residues of the expansion in poles $-q^{2}=M_{n}^{2}$ of the two-point
function, defined from the correlator of two electromagnetic currents:

$\displaystyle\Pi_{\mu\nu}(q^{2})$ $\displaystyle=$ $\displaystyle
i\,\int{d^{4}x\,e^{i\,q\cdot x}\langle
0\left|\mathcal{T}\left\\{j_{\mu}(x)\,j_{\nu}(0)\right\\}\right|0\rangle}$
(13) $\displaystyle=$
$\displaystyle\left(q_{\mu}\,q_{\nu}-q^{2}\,\eta_{\mu\nu}\right)\,\Pi(-q^{2})$
$\displaystyle=$
$\displaystyle\left(q_{\mu}\,q_{\nu}-q^{2}\,\eta_{\mu\nu}\right)\,\sum_{n}{\frac{f_{n}^{2}}{-q^{2}-M_{n}^{2}+i\,\varepsilon}}.$

The tensor structure written in parentheses is nothing else than the
transverse projector, coming from the fulfillment of the Ward-Takahashi
identities.

The importance of the two-point function relies on the description of the
intermediate hadronic states that appear in scattering processes involving
hadrons. Decay constants measure the probability of finding such states in the
collision final products.

In the case of heavy quarks, the electromagnetic quark currents
$e\,\bar{c}\,\gamma_{\mu}\,c$ and $e\,\bar{b}\,\gamma_{\mu}\,b$ creates the
$J/\psi$ and $\Upsilon$ mesons respectively. At the physical level, these
mesonic vector states appear as observed resonances in the $e^{+}\,e^{-}$
annihilation process when the center of mass energy is around the mass of the
corresponding mesonic states. Therefore, these states are expected to be also
poles in the two-point function.

Experimentally, decay constants are measured from the vector meson decaying
process $V\to e^{+}\,e^{-}$, according to the expression:

$f_{n}^{2}=\frac{3\,M_{n}\,\Gamma_{V\to
e^{+}e^{-}}}{4\,\pi\,\alpha^{2}_{\text{em}}\,C_{V}^{2}},$ (14)

where $\Gamma_{V\to e^{+}\,e^{-}}$ is the heavy vector decay width, and
$C_{V}$ stands for the heavy quark electromagnetic charge contribution to the
meson, i.e., $C_{J/\psi}=2/3$ and $C_{\Upsilon}=1/3$.

The holographic dual of the two-point function is determined from the on-shell
boundary action Karch et al. (2006). Following the field/operator duality, the
holographic two-point is written as

$\Pi\left(-q^{2}\right)=-\left.\frac{e^{-B\left(z\right)}}{g_{5}^{2}\,(-q^{2})}\,\partial_{z}\,V\left(z,q\right)\right|_{z\to
0},$ (15)

where $V(z,q)$ is the bulk-to-boundary propagator. It is straightforward to
prove that this object can be written in terms of the normalizable modes
$\psi(z,q)$ by using the Green’s function associated with the equations of
motion (9). In work Martin Contreras and Vega (2020a), authors follow this
path deriving a general expression for the decay constants calculated for any
general AdS/QCD model depending only on the value of the quotient
$\psi(z,q)/z^{2}$ and the dilaton at the conformal boundary

$f_{n}^{2}=\frac{1}{M_{n}^{2}\,g_{5}^{2}}\,\lim_{z\to
0}{\,e^{-2\,\Phi(z)}\,\left|\frac{2\,\psi_{n}(z,q)}{z^{2}}\right|^{2}}.$ (16)

Let us stop here and see how the decay constants are calculated in the soft
wall model, i.e., static and quadratic dilaton. Following Karch et al. (2006),
we see that the mass spectrum has the linear structure
$M_{n}^{2}=4\,k^{2}(n+1)$, with $k$ being the dilaton slope. The
eigenfunctions are defined in terms of Laguerre associated polynomials

$\psi_{n}(z)=\sqrt{\frac{2\,k^{4}\,n!}{(n+1)!}}\,z^{2}\,L_{n}^{1}(k^{2}\,z^{2}),$
(17)

therefore, the decay constants follow from eqn. (16) yielding

$f_{n}^{2}=\frac{F_{n}^{2}}{M_{n}^{2}}=\frac{1}{4\,g_{5}^{2}\,k^{2}\,(n+1)}\times\,\frac{8\,k^{4}\,(n+1)!}{n!}=\frac{2\,k^{2}}{g_{5}^{2}},$
(18)

where we have used the asymptotic form of the Laguerre associated polynomials
when $z\to 0$. Therefore, we can conclude that decay constants are degenerate
in the softwall model. If we do similar computations in the hardware model
context Boschi-Filho and Braga (2003), they will lead to increasing decays
$f_{n}$ with the excitation number $n$. This drawback can be avoided by
deforming the low $z$ limit in the static dilaton, as it was first noticed by
Braga et al. Braga and Ferreira (2016). We will extend this idea in the
context of non-quadratic dilatons.

The numerical results for the charmonium and bottomonium decay constants,
calculated in the deformed non-quadratic dilaton context, are summarized in
tables 1 and 2. The deviations presented in the caption of tables 1 and 2
represent the difference between the theoretical prediction and the most
probable value of a given experimental measure. The total deviation
$\delta_{RMS}$ is defined as

$\delta_{\text{RMS}}=\sqrt{\frac{1}{N-N_{p}}\sum_{i}^{N}\left(\frac{\delta\,O_{i}}{O_{i}}\right)^{2}},$
(19)

where $O_{i}$ is a given experimental measure with $\delta\,O_{i}$ defining
the deviation of the theoretical value from the experimental one, $N_{p}$ is
the number of model parameters, and $N$ the total number of available
observables.

##  IV Finite temperature

For the finite-temperature extension, we will consider the heavy quarkonium
system living in a thermal bath, addressed by a colored plasma.
Holographically, we will deal with vector bulk field living in an AdS-
Schwarzschild black hole background, described by the metric

$dS_{\text{AdS-
Schw}}^{2}=\frac{R^{2}}{z^{2}}\left[\frac{dz^{2}}{f(z)}-f(z)\,dt^{2}+d\vec{x}\cdot
d\vec{x}\right],$ (20)

with the blackening factor defined as

$f(z)=1-\frac{z^{4}}{z_{h}^{4}}.$ (21)

where $z_{h}$ is the event horizon locus.

The description of heavy quarkonium at finite temperature in the context of
the softwall model was developed in Fujita et al. (2010). However, as it was
discussed in Vega and Cabrera (2016); Vega and Ibañez (2017); Vega and Martin
Contreras (2019), by analyzing the holographic potential in the context of
Bogoliubov transformations and tortoise coordinates, the mesonic melting
temperature appears to be too low as the ones expected from lattice QCD. This
bad behavior is attached to the holographic decay constant description in the
softwall model, where these objects are degenerate. This affirmation is
sustained by the thermal analysis of the hadronic part of the two-point
function Dominguez et al. (2010, 2013). For instance, the hadronic spectral
density calculated from thermal sum rules

$\left.\frac{1}{\pi}\mathbb{I}\text{m}\,\Pi(s,T)\right|_{\text{hadron}}=\frac{f_{n}^{2}\,M_{n}(T)^{3}\,\Gamma_{n}(T)}{[s-M_{n}^{2}(T)]^{2}+M_{n}^{2}(T)\,\Gamma_{n}(T)^{2}},$
(22)

establishes the formal dependence of the melting process with the decay
constant.

This softwall model issue was circumvented by introducing low $z$
modifications into the model, as it was done in Braga et al. (2016c).
Therefore, it is natural to suppose that this hybrid dilaton should exhibit
the expected raising in the melting temperatures in agreement with
phenomenology.

Let us focus on reviewing the holographic description of the heavy quarkonium.
Our starting point is the calculation of the hadronic spectral density. To do
so, we will follow the Minkowskian prescription given by Son and Starinets
(2002). Let us perform the variable change $z=z_{h}\,u$ in the metric (20) in
order to fix the horizon locus at $u=1$. We will also fix $-q^{2}=\omega^{2}$
in our analysis.

###  IV.1 Confinement/Deconfinement phase transition

In the boundary gauge theory, the formation of a deconfined plasma is
holographically described via the Hawking-Page phase transition in the dual
geometry Hawking and Page (1983); Herzog (2007). On the gauge theory side,
above the critical temperature, $T_{c}$, the fundamental quarks and gluons
inside the colorless matter are allowed to walk away from its partners,
forming a plasma of deconfined colored particles. It is usually considered
that the light vector meson dominates the deconfinement transitions. That is,
the medium is formed when the light quarks can escape from the hadrons.
Consequently, we use the light meson spectrum to fix the energy scales
governing the confinement/deconfinement transition.

The observed spectrum of radial excitations of the $\rho$ meson includes the
masses of the first five radial excitations, and the decay constant of the
ground state Tanabashi et al. (2018). It is important to mention that
additional scales in the model encode heavy quarkonia properties and bring no
particular advantages in describing the light meson spectrum. In particular,
for light mesons, the parameter $\alpha$ in eq.(6) is set to vanish. The
observed spectrum of the radial excitations of the $\rho$ meson are reasonable
fitted using the model parameters $\kappa=0.6$ GeV, $M=0.06$ GeV,
$\sqrt{\Gamma}=0.02$ GeV. Using these parameters to fix the dilaton profile,
we compute the gravitational on-shell action of the AdS-Schwarzschild black
hole geometry and the thermal AdS geometry. The normalized difference is then
obtained as

$\Delta
S=\int_{\epsilon}^{z_{h}}dz\frac{e^{-\Phi(z)}}{z^{5}}-\sqrt{f(\epsilon)}\int_{\epsilon}^{\infty}dz\frac{e^{-\Phi(z)}}{z^{5}}.$
(23)

We show in Figure 1 the difference in action as a function of temperature. In
the region where $\Delta S$ is positive, the thermal AdS is stable. In the
region with $\Delta S$ is negative, the black hole is stable. The condition
$\Delta S=0$ defines the critical temperature, and we obtain

$T_{c}=142\,\leavevmode\nobreak\ \textrm{MeV.}$ (24)

There are two important comments to make at this point. First, using the
$\rho$ meson spectrum to fix model parameters is a particular choice. As it
was recently pointed out in Afonin and Katanaeva (2020), the definition of
$T_{c}$ through a Hawking-Page transition is model depending. The same authors
performed a similar calculation considering the gluon condensate obtaining a
critical temperature of $156$ MeV Afonin (2020). Second, the phase transition
associated with QGP formation in heavy-ion collisions is more likely a
continuous crossing over than an abrupt transition Aoki et al. (2006).
However, the present computation of $T_{c}$ has no intention of dealing with
these subtleties. The critical temperature we obtain ($T_{c}=142$ MeV) is
consistent with the present holographic model and will be adopted from now on.

Figure 1: The difference between the on-shell gravitational action of AdS-
Schwarzschild and Thermal AdS geometries is depicted as a function of
temperature in GeV. The intersection with the horizontal axis gives the
critical temperature of the deconfinement transition.

###  IV.2 Spectral density

The holographic spectral density comes from the thermal Green’s function. We
define the bulk-to-boundary propagator in momentum space
$V_{\mu}(z,q)=V(z,q)V^{0}_{\mu}(q)$, where $V^{0}_{\mu}(q)$ is the source at
the boundary. According to the Minkowskian prescription, this correlator is
written in terms of the derivatives of the bulk-to-boundary propagator
$V(z,q)$ as

$G_{R}(\omega)=-\left.\frac{2}{z_{h}\,\mathcal{N}}\,\,e^{-B(u)}\,f(u)\,V(u,-\omega)\,\partial_{u}\,V(u,\omega)\right|_{u=0}.$
(25)

The spectral density, according to the Kubo relations, is written as the
imaginary part of the Green’s function

$\rho(\omega)=-\mathbb{I}\text{m}\,G_{R}(\omega).$ (26)

The bulk-to-boundary propagator obeys the bulk spatial vector equation of
motion

$\partial_{u}\left[e^{-B(u)}\,f(u)\,\partial_{u}\,V(u,\omega)\right]+\frac{z_{h}^{2}\,\omega^{2}}{f(u)}e^{-B(u)}\,V(u,\omega)=0.$
(27)

Although we are at finite temperature, the bulk-to-boundary propagator still
preserves its properties at the conformal boundary. If this is not guaranteed,
the field/operator duality does not hold anymore. Recall that at the conformal
boundary, we require that $V(u\to 0)\to 1$. On the other side, we also need
that $V(u,\omega)$ obeys the out-going boundary condition $\phi_{-}(u)$,
defined as

$\phi_{-}(u)=\left(1-u\right)^{-\,i\frac{\omega\,z_{h}}{4}}$ (28)

These conditions define the procedure to compute the spectral density. We will
follow the method depicted in Teaney (2006); Fujita et al. (2010); Miranda et
al. (2009); Fujita et al. (2009). Our starting point is writing a general
solution $v(u)$ for the Eqn. (27) in terms of the normalizable $\psi_{0}(u)$
and non-normalizable $\psi_{1}(u)$, that form a basis, in the following form

$v(u)=A\,\left[\psi_{1}(u)+\frac{B}{A}\,\psi_{0}(u)\right],$ (29)

such that the bulk-to-boundary propagator is written as
$V(\omega,u)=A^{-1}\,v(u)$, and satisfying the asymptotic solutions near the
conformal boundary

$\displaystyle\psi_{0}(u\,\omega)$ $\displaystyle=$
$\displaystyle\frac{2}{\omega\,z_{h}}\,u\,J_{1}(\omega\,z_{h}\,u)$ (30)
$\displaystyle\psi_{1}(u\,\omega)$ $\displaystyle=$
$\displaystyle-\frac{\pi\,\omega\,z_{h}}{2}\,u\,Y_{1}(\omega\,z_{h}\,u)$ (31)

After replacing this solution into the Green’s function definition we obtain

$G_{R}(\omega)=-\left.\frac{2\,R}{z_{h}\,\mathcal{N}}\left[\frac{B}{A}-\frac{\omega^{2}\,z_{h}^{2}}{2}\,\text{log}\,\left(\frac{e^{\gamma_{e}}\,\varepsilon\,\omega\,z_{h}}{2}\right)\,\varepsilon^{2}\right]\right|_{\varepsilon\to
0}$ (32)

Finally, the spectral density is written as the imaginary part of the Green’s
function

$\displaystyle\rho(\omega)$ $\displaystyle=$
$\displaystyle-\mathbb{I}\text{m}\,G_{R}(\omega)$ (33) $\displaystyle=$
$\displaystyle\frac{2\,R}{z_{h}\,\mathcal{N}}\,\mathbb{I}\text{m}\frac{B}{A}.$

Numerical results for the spectral density calculated for charmonium and
bottomonium system are shown in Fig. 2.

---
Figure 2: This figure describes the spectral density for charmonium (left
panel) and bottomonium (right panel) calculated using Eqn. (33), depicting the
melting process.Dashed lines corresponds to the melting temperature in each
case.

###  IV.3 Thermal holographic potential

Another essential quantity that carries valuable information about the heavy
quarkonium thermal picture is the thermal potential. At zero temperature case,
the potential translates the dilaton effect into the holographic confinement.
Holographic mesonic states appear as eigenfunctions of this potential.

The thermal dissociation of mesons is connected with the holographic
potential. In Vega and Martin Contreras (2019), this idea was discussed in the
context of softwall-like dilatons that vanish at the conformal boundary. In
this proposal, the melting is characterized by the disappearance of the
potential well. At zero temperature, the dilaton vanishes near the boundary,
and the potential holographic displays one single minimum that is global at
zero temperature. The disappearance of the global minimum of the holographic
potential encodes the information of meson dissociation.

In this work, we consider a dilaton that does not vanish near the boundary.
This dilaton field, given in Eqn. (6) interpolates between linear and the
deformed quadratic behavior, which induces a nonlinear spectrum. This dilaton
also changes the global structure of the potential by introducing local minima
near the UV at zero temperature. As argued in Grigoryan et al. (2010); Martin
Contreras and Vega (2020a), this UV deformation is needed in order to describe
the proper phenomenological behavior the decay constants of the excited
quarkonia states.

It is expected that, at finite temperature, the holographic potential also has
information about the melting process. To make a formal approach to this
phenomenology, we apply the Liouville (tortoise) transformation. It transforms
the equations of motion into a Schrödinger-like equation in terms of a
Liouville (tortoise) coordinate $r^{*}$. The potential exhibits a barrier that
decreases with the temperature, mimicking how the confinement starts to cease
when the temperature rises. Following Vega and Martin Contreras (2019), one
expect that the barrier disappears when all of the quarkonia states melt down
into the thermal medium. However, the appearance of a local minima near $z=0$
can sustain the state after the disappearance of the barrier.

The Liouville transformation appears in the core of the Liouville theory of
second-order differential equations. Given a differential equation, we can
associate it with a differential diagonalizable operator. As a consequence,
this operator will acquire a spectrum of eigenvalues and eigenfunctions. In
the holographic case at hand, the associated potential is defined _via_ the
transformation

$r^{*}(u)=z_{h}\,\int_{0}^{u}{\frac{d\,\xi}{1-\xi^{4}}}=\frac{z_{h}}{2}\left(\text{tan}^{-1}\,u+\text{tanh}^{-1}\,u\right).$
(34)

The equations of motion (27) transform into the following Schrodinger-like
equation

$-\frac{d^{2}\,\phi(r^{*})}{d\,r^{*2}}+U(r^{*})\,\phi(r^{*})=\omega^{2}\,z_{h}^{2}\,\phi(r^{*}),$
(35)

with the following definitions

$U(r*)=f(u)^{2}\left[\frac{3}{4\,u^{2}}+\frac{\Phi^{\prime}(u)}{2\,u}+\frac{\Phi^{\prime}(u)^{2}}{4}-\frac{\Phi^{\prime\prime}(u)}{2}\right.\\\
\left.-\frac{f^{\prime}(u)}{2\,u\,f(u)}-\frac{f^{\prime}(u)\,\Phi^{\prime}(u)}{2\,f(u)}\right]$
(36) $\displaystyle\phi(r^{*})$ $\displaystyle=$
$\displaystyle\psi(u)\,e^{\frac{1}{2}\,B(u)}$ (37) $\displaystyle u$
$\displaystyle=$ $\displaystyle u(r^{*}).$ (38)

where $u=u(r^{*})$ is obtained by inverting the Liouville coordinate defined
in Eqn. (34).

In figure 3, we depict the melting process from the Liouville potential for
the heavy quarkonia. In the zero temperature case, the potential reduces to
the holographic one described in Eqn. (12).

---
Figure 3: In this figure, we plot the holographic Liouville potential for
charmonium (left panel) and bottomonium (right) panel. Also, we plot the first
three masses calculated a zero temperature to illustrate the melting process.
When the barrier decreases below the mass, we can consider that such a state
had undergone a melting process.

The melting process in the present case is a two step process involving two
different energy scales. The first step is the disappearance of the infra-red
barrier when the temperature is increased above $T_{c}$ allowing for the bulk
modes to be absolved by the event horizon. At this step all the excited states
melts in the thermal medium. But this is not sufficient to state the melting
of the ground state. The appearance of a deep, narrow and persistent well near
$z=0$ produces a barrier greater them the mass of the ground state. The well
is separated from the event horizon by a barrier which narrows with the
raising of temperature. At the melting temperature the barrier is too narrow
to hold the bulk wave packet, that escapes from the well and is absolved by
the event horizon. A quantitative description of the tunneling process is not
performed here and the melting temperature depicted in Figure (3) are obtained
from the Breight-Wigner analysis performed in the next section.

##  V Breit-Wigner analysis

Once the spectral functions are calculated, we will perform the Breit-Wigner
analysis to discuss the thermal properties captured by the holographic model
described above. This analysis allows extracting information about the meson
melting process, as the temperature and the thermal mass shifting. Recall that
when a meson starts to melt, the resonance begins to broad (the width becomes
large), and the peak height, which is proportional to the decay constant,
decreases. In other words, the mesonic couplings tend to zero as the
temperature rises, implying these states ceased to be formed in the colored
medium.

Therefore, comparing the peak height and the width size will be the natural
form to define the meson melting temperature: the temperature at which the
width size overcomes the peak high is where the meson starts to melt. This
phenomenological landscape also comes in the context of pNRQCD at thermal
equilibrium.

The next thing to consider is the background. These background effects
observed in the spectral function come from continuum contribution, and they
should be subtracted in order to isolate the Breit-Wigner behavior. The
background subtraction methodology is not unique, and in general, is model
depending. However, most of the authors define interpolation polynomials in
terms o powers of $\omega^{2}$. See, for example, Colangelo et al. (2009); Cui
et al. (2016) in the light scalar sector and Fujita et al. (2010) for heavy
vector quarkonium. In these references, authors worked with quadratic-like
dilatons.

In our particular case, we will follow a different path: we will consider the
large $\omega^{2}$ behavior to deduce a background subtraction mechanism. As
ref. Grigoryan et al. (2010) pointed it out, in a conformal theory at short
distances, we could expect that

$\lim_{\omega^{2}\to\infty}\frac{\rho(\omega^{2})}{\omega^{2}}=\frac{\pi}{2\,g_{5}^{2}}\,\,\,\text{
i.e., a dimensionless constant},$ (39)

for the case of quadratic-like dilatons. The OPE-expansion of the 2-point
function dictates this behavior, allowing the match between the bulk and the
boundary theories. In the purely phenomenological sense, the existence of this
dimensional constant is a signature of asymptotic freedom. Thus, the spectral
function for these quadratic-like dilatons can be rescaled as

$\bar{\rho}(\omega^{2})=\frac{\rho}{\omega^{2}},$ (40)

in order to test the asymptotic freedom signature in the model. Therefore, if
the rescaled spectral function behavior does not match this criterion, the
model does not have a proper large $\omega^{2}$ limit compared with QCD. The
softwall model with quadratic dilaton perfectly matches this condition.

Then, what happens when the model does not have a quadratic dilaton? To answer
this question, we can go further by imposing the same asymptotic condition.
However, changing the quadratic structure on the dilaton will imply that the
asymptotic behavior of the spectral function is different: it is still linear
in $\omega^{2}$, but with a shifted value of the coupling $g_{5}$, defined at
zero temperature from the holographic 2-point function. Thus, we suggest the
following rescaling:

$\bar{\rho}(\omega^{2})=\frac{\rho(\omega^{2})}{\delta\,\omega^{2}},$ (41)

where $\delta$ is determined from the large $\omega^{2}$ behavior observed in
the spectral function $\rho(\omega^{2})$. From this rescaled spectral
function, we will subtract the background effects and construct the Breit-
Wigner analysis. For our practical purposes, we will write the Breit-Wigner
distribution as

$\bar{\rho}(\omega^{2})=\frac{1}{2}\frac{A_{0}\,\omega^{2}_{0}\,\Gamma_{0}\,\omega^{a_{0}}}{(\omega^{2}-\omega^{2}_{0})^{2}+\frac{\omega^{2}_{0}\,\Gamma^{2}_{0}}{4}},$
(42)

where $A_{0}$, $a_{0}$ are fitting parameters, $\omega_{0}$ is the mesonic
peak and $\Gamma_{0}$ is the decay width, proportional to the inverse of the
meson life-time.

###  V.1 Background substraction

In the thermal approach to heavy quarkonium, the colored medium is vital since
it strongly modifies the vacuum phenomenology. In particular, following the
Feynman-Hellman theorem analysis, it is expected that bound states energy
decrease when constituent mass is increased at zero temperature Quigg and
Rosner (1979). Consequently, zero temperature spectral peaks experience
shifting in their positions, color singlet excitations transform into other
singlet states by thermal fluctuations, or these singlet excitations transform
into another color octets. All of this intricated phenomenology is encoded in
the medium. Therefore, in order to isolate the thermal information regarding
the heavy quarkonium state melting process, a proper subtraction scheme is
needed. In our case, we will consider an interpolating polynomial in
$\omega^{2}$ that will be subtracted to the spectral density, allowing us to
obtain a Breit-Wigner distribution associated with the heavy quark state only.
In figure 4, we depict the subtraction process for the melting of $J/\psi$,
observed in our model at 415 MeV (2.92 $T_{c}$).

---
Figure 4: This figure depicts the subtraction procedure for $J/\psi$ at 400
MeV and 415 MeV, $\psi^{\prime}$ at 85 MeV and 90 MeV, and $\Upsilon$ at 465
MeV. Notice that the background polynomial appears as the orange function in
both cases. We plot the subtracted spectral density on the top right part of
each figure that we fit with the Breit-Wigner distribution (42). In the lower
panels, we plot the bottomonium case for the same temperature, 465 MeV, with
two different interpolating polynomials. In both situations, changing the
polynomial does not affect the melting criterium. Recall that, unless other
non-holographic effective models, the in-medium effects are encoded into the
metric tensor. Thus, any proper characteristic behavior, as heavy quarkonium
regeneration or Gluon radiation, is indistinguishable.

At this step, an important remark should be made. The interpolating polynomial
is not defined univocally. We can fix a criterium that these sorts of
polynomial should obey. In principle, since we do not have a proper
phenomenological tool at hand to split the behavior of the medium from the
hadronic state, we will ask for a _smooth subtraction_. In other words, the
region where the interpolating polynomial splits from the spectral function
should not display an abrupt change. Since the possible functions that could
match this condition are infinite, we can only bring a temperature interval
where the meson starts to melt. However, choosing similar polynomials will
lead to the same melting interval. See lower panels in figure 4.

###  V.2 Melting Temperature Criterium

As we observe in figure 2, mesonic states disappear progressively with
increasing temperature. In the holographic potential case, the melting
temperature is not connected with the disappearing of the confining barrier.
Since the potential has a depth well in the UV region, the thermal stability
would be associated with the tunneling of such a barrier.

In the holographic situation, the generated dual object is a colored medium at
thermal equilibrium, where the heavy quarkonium exists. In such a static
situation, mesonic states either exist or have melted down. Thus, the only
relevant information at the holographic level we have is the spectral function
and the background subtraction.

In order to find the interval where heavy mesons start to melt, we will follow
the standard criterium connecting the Breit-Wigner maximum with its graphical
width, defined as a product of the meson mass and the thermal width

$\frac{\bar{\rho}(\omega_{0}^{2})}{\omega_{0}\,\frac{\Gamma}{2}}<1.$ (43)

Notice that the definition depicted above is an alternative to the criteria
defined from the effective potential models and lattice QCD, defined where the
melting occurs when the in-medium binding energy equals the thermal decay
width Rothkopf (2020). In the holographic case, melting temperatures are
intrinsically connected to decay constants, proportional to the two-point
function residues at zero temperature. Recall the decay constants carry
information about how the mesonic states decay electromagnetically into
leptons. Thus, indirectly they measure the mesonic stability affected by
thermal changes: excited states with lower binding energy than the ground one
melt first. This connection with meson stability is supported by the
experimental fact that decay constants decrease with the excitation number.
Another possible form to explore the connection between the mesonic melting
process and stability is done in the context of configurational entropy,
discussed in refs. Braga et al. (2018); Braga and Ferreira (2018); Braga and
da Mata (2020); Braga and Junqueira (2020).

In the case of the charmonium, the $\psi^{\prime}$ state melts near 90 MeV or
$0.63\,T_{c}$. The ground state, the $J/\psi$ meson melts near to 415 MeV or
$2.92\,T_{c}$. If we compare with the pNRQCD results Burnier et al. (2015), we
obtain a lower temperature for the $2\,S$ charmonium state (lattice result:
$0.95\,T_{c}$) but higher for the ground state (lattice result:
$1.37\,T_{c}$). The main difference in both results is that in our holographic
case we are considering heavy quarkonium at rest, i.e., $|\vec{p}|=0$.

A similar situation is observed in the bottomonium case: the $\Upsilon(2\,S)$
melts near to $115$ MeV (or $0.81\,T_{c}$), compared with the pNRQCD result of
$1.25\,T_{c}$. For the ground state we have $465$ MeV (or $3.27\,T_{c}$),
compared with the lattice result of $2.66\,T_{c}$.

If we compare with holographic stringy models Andreev (2019), where the
melting temperature is estimated from the string tension in an AdS deformed
target space, we found bigger results for heavy quarkonium melting
temperature. They predict $1.05\,T_{c}$ and $2.52\,T_{c}$ for charmonium and
bottomonium respectively.

###  V.3 Thermal Mass

---
Figure 5: Resonance location as a function of the temperature. The shaded
region in each panel describes the increase of the thermal width until the
meson melting occurs. The left panels correspond to ground states, and the
right panels are the first excited states. Upper panels correspond to
charmonium, and lower panels correspond to bottomonium.

Other important quantities to discuss are the masses and widths of the
different hadronic states since these parameters have information about the
interaction with the colored medium. Figure 5 has summarized the mass thermal
behavior modeled for the first two charmonium and bottomonium excited states.
Comparing with other holographic models (see Fujita et al. (2010, 2009) for
heavy mesons; Colangelo et al. (2009, 2009) and Cui et al. (2016) for light
mesons), the mass for the ground state in our case tends to increase with
temperature until the meson melting takes place, as the upper (J/$\psi$) and
lower ($\Upsilon$) panels in figure 5 display. The same behavior is observed
for the charmonium first excited state, depicted in Figure 5 right upper
panel. However, this very same behavior is not observed for the first excited
state of the bottomonium. In the $\Upsilon(2S)$ meson case, the hadronic
resonance location decreases with the temperature.

The observed behavior for the thermal mass in our case seems to be quite
different from the one depicted in Fujita et al. (2009). In their case, the
thermal mass increases towards a maximum, where the authors claimed the
melting process starts, and then thermal mass decreases up to the last
charmonium meson is melted. In our case, such a concavity change occurs for
low temperatures compared with $T_{c}$, far from the melting temperatures,
around three times $T_{c}$. The monotonicity of the thermal mass appears to be
more consistent with lattice calculations Burnier et al. (2016); Rothkopf
(2020). In those approaches, writing the NRQCD heavy quark potential is done
in the soft scale, i.e., kinematical scale. In the case of hard scales, near
to the constituent quark masses, other approaches are necessary.

In the context of QCD sum rules Dominguez et al. (2010), following the Hilbert
moment mechanism, the thermal mass in the case of heavy quarks does not change
with the temperature until the system reaches the critical temperature, where
it drops. As an interesting observation, in this model, the decay constants go
to zero as the temperature comes closer to the critical one, indicating that
the melting has occurred.

##  VI Conclusions

By deforming the non-quadratic dilaton defined in Martin Contreras and Vega
(2020b) using the proposal given by Braga et al. in Braga et al. (2017), it
was possible to fit for the vector charmonium and bottomonium both the mass
spectra as non-linear Regge trajectories and their decreasing decay constants.
The precise holographic description of the heavy vector meson excited states
is reached by considering all the lessons learned in the last decade of
bottom-up AdS/QCD.

The precision of the fit is measured by the $\delta_{RMS}$, defined in
eq.(19), being $6\%$ for charmonium and $7,2\%$ for bottomonium. The dilaton
deformations are necessary for a precise description of the spectrum of masses
and decay constants. If we use the original quadratic dilaton to describe the
charmonium spectrum by fixing $k=1.55$ GeV, we find $\delta_{RMS}=74\%$. So,
the new parameters introduced in the dilaton do allow an accurate description
of the spectrum. Notice that the model has predictability even though we are
using four parameters to fit each heavy quarkonium family. As a matter of
fact, for the non-linear trajectory $M^{2}=a(n+b)^{\nu}$ we need three
parameters. If we assume that decay constants are functions of the excitation
number $n$ only, we can write them as $f(n)=c(-n+d)$, if we suppose linearity
as our first guest. The minus sign in the parametrization emphasizes the
decreasing behavior of the decays with $n$. Thus, if we count the maximum
number of parameters need for both decays and masses, we obtain five
parameters. If we assume non-linear behavior for decays, we have one extra
parameter, implying six instead of five maximum parameters per family. Thus,
in our case, we have four. Thus our model is predictable. Such precision is
essential to set the correct zero temperature behavior of the spectral
functions. If we think of the increasing temperature as an analog for time
evolution, zero-temperature properties play the role of initial conditions.

Spectral functions have been numerically computed for several representative
values of the temperature. As expected, pronounced resonance peaks around the
zero temperature masses of charmonium and bottomonium are observed near
$T_{c}$. To discuss the fate of the particle states when increasing
temperature, it is necessary to subtract background contributions from the
spectral functions. We provide a detailed discussion on this subject and
propose a numerical scheme to perform such a subtraction. The Breit-Wigner
peaks are analyzed. We obtain the melting temperature of $J/\Psi$ and
$\Upsilon$ to be, respectively, $T_{J/\Psi}=415$ MeV and $T_{\Upsilon}=465$
MeV. These high melting temperatures obtained are directly connected to the
correct description of the decay constants of the corresponding fundamental
states of $c\bar{c}$ and $b\bar{b}$. The excited states
$\Psi^{\prime},\Upsilon^{\prime}$ melts at temperatures smaller them $T_{c}$.
So, we consider smaller temperatures around $50-60$ MeV where we can see the
pronounced peaks associated with the states. Within this range of
temperatures, around $50-470$ MeV, we consider the thermal mass shifting of
$J/\Psi,\Psi^{\prime}$ and $\Upsilon,\Upsilon^{\prime}$. We observe a small
and monotonic increase in the masses of the ground states with temperature.

The specific form of the dilaton leads to a holographic potential that differs
from the one obtained in quadratic dilaton models. In the present case, there
is a narrow well in the ultra-violet region. The melting of the fundamental
state is no longer entirely governed by the disappearance of the infra-red
barrier. For this shape of holographic potential, the criteria for defining
the melting of the states established in Vega and Martin Contreras (2019) does
not apply. It is a task for future work to understand the melting process from
the thermal evolution of this class of holographic potentials.

###### Acknowledgements.

We wish to acknowledge the financial support provided by FONDECYT (Chile)
under Grants No. 1180753 (A. V.) and No. 3180592 (M. A. M. C.). Saulo Diles
thanks the Campus Salinopolis of the Universidade Federal do Pará for the
release of work hours for research.

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