#  $J/\psi$ near threshold in holographic QCD:
A and D gravitational form factors

Kiminad A. Mamo<EMAIL_ADDRESS>Physics Division, Argonne National Laboratory,
Argonne, Illinois 60439, USA  Ismail Zahed<EMAIL_ADDRESS>Center
for Nuclear Theory, Department of Physics and Astronomy, Stony Brook
University, Stony Brook, New York 11794-3800, USA

###### Abstract

The diffractive photoproduction of $J/\Psi$ on a nucleon, is mostly due to
gluonic exchanges at all $\sqrt{s}$. In holographic QCD (large number of
colors and strong ′t Hooft coupling), these exchanges are captured by
gravitons near threshold, and their Reggeized Pomeron form asymptotically. We
revisit our holographic analysis of the A and D gravitational form factors in
light of the new lattice data, and use them to refine our predictions for the
photoproduction of $J/\Psi$ near threshold, and the comparison to the GlueX
data. We use these results to estimate the scalar and mass radii of the
nucleon, and describe the gravitational pressure and shear across a nucleon.

##  I Introduction

The breaking of conformal and chiral symmetries in QCD are at the origin of
most of the visible mass in the Universe, a remarkable quantum feat, starting
from a classical field theory without explicit mass Wilczek (2012); Roberts
(2021) (and references therein). At low resolution, these phenomena are tied
semi-classically to topologically active gauge fluctuations: eternal tunneling
events between gauge vacuua with different winding numbers, also known as
instantons and anti-instantons Shuryak (2018); Zahed (2021) (and references
therein), with strong evidence from numerical lattice simulations using
cooling procedures Leinweber (1999); Biddle et al. (2020).

An empirical understanding of these mechanisms can be made using dedicated
electron machines Hafidi et al. (2017); Ali et al. (2019a); Meziani and
Joosten (2020); Anderle et al. (2021). Recently, the GlueX collaboration Ali
et al. (2019a) at JLAB has reported threshold data for photo-production of
charmonium $J/\Psi$ that may start to lift the lid on some of these
fundamental questions. Indeed, near threshold the diffractive production of
charmonia and bottomia is dominated by gluons or glueballs, and currently
measurable.

In a recent analysis of the GlueX data using a holographic construction, we
have shown Mamo and Zahed (2020a) that the threshold differential cross
section is only sensitive to the tensor gravitational form factor, and
suggested that this tensor form factor or A-term is extractable from the
current data under a minimal but universal set of holographic assumptions.
This allows for a first extraction of the tensor mass radius among other
things. Remarkably, the holographic construction ties the A- and
D-gravitational form factors, thereby allowing for the extraction of the
gluonic pressure and shear inside the proton. Similar ideas were also explored
in a hybrid form of holography in Hatta and Yang (2018), and in general in
Kharzeev (2021); Ji (2021); Hatta and Strikman (2021); Guo et al. (2021); Sun
et al. (2021); Wang et al. (2022).

In this work, we revisit our holographic analysis of the A and D gravitational
form factors Mamo and Zahed (2020a, 2021a), by improving our determination of
$A(0)$ and $D(0)$ from a comparison to the newly released lattice data Pefkou
et al. (2022). We will use these improved results, to reanalyze the threshold
photoproduction of charmonium as reported by the GlueX collaboration, and the
interplay of the A and D form factors in these results.

In section II, we briefly review the holographic arguments for the A and D
form factors for the proton, and compare them to the newly released lattice
results for the gravitational form factors of the proton Pefkou et al. (2022).
In section III, we show that the A and D form factors control the electro-
production amplitude of charmonium near threshold, with the D contribution
suppressed by the squared skewness of the process. We use these form factors
to characterize the scalar and mass radii, and the gravitational pressure and
shear across a nucleon. We reduce the electroproduction process to the photon
point, and revisit our comparison to the GlueX data. Our conclusions are in
section IV. Some of the details regarding parts of the derivations are
Appended.

##  II Gravitational form factors

The standard decomposition of the energy-momentum form factor in a nucleon
state is Kobzarev and Okun (1962); Pagels (1966); Carruthers (1970); Polyakov
and Schweitzer (2018)

$\displaystyle\left<p_{2}|T^{\mu\nu}(0)|p_{1}\right>=\overline{u}(p_{2})\left(A(k)\gamma^{(\mu}p^{\nu)}+B(k)\frac{ip^{(\mu}\sigma^{\nu)\alpha}k_{\alpha}}{2m_{N}}+C(k)\frac{k^{\mu}k^{\nu}-\eta^{\mu\nu}k^{2}}{m_{N}}\right)u(p_{1})\,,$
(II.1)

with $a^{(\mu}b^{\nu)}=\frac{1}{2}(a^{\mu}b^{\nu}+a^{\nu}b^{\mu})$,
$k^{2}=(p_{2}-p_{1})^{2}=t$, $p=(p_{1}+p_{2})/2$ and the normalization
$\overline{u}u=2m_{N}$. (II.1) is conserved and tracefull. Throughout,
$D(k)=4C(k)$ will be used interchangeably. In holography, (II.1) sources the
metric fluctuations in bulk, $g_{MN}(z)\rightarrow g_{MN}(z)+h_{MN}(x,z)$,
with line element $ds^{2}=g_{MN}(z)dx^{M}dx^{N}$ in a 5-dimensional anti-
deSitter space or AdS5, in the double limit of large $N_{c}$ and strong gauge
coupling Nastase (2007) (and references therein). The form factors in (II.1)
follow from the coupling of the irreducible representations of the metric
fluctuations $h_{\mu\nu}$, to a bulk Dirac fermion with chiral components
$\psi_{L,R}$. The bulk metric fluctuations can be decomposed as Kanitscheider
et al. (2008).

$\displaystyle
h_{\mu\nu}(k,z)\sim\epsilon_{\mu\nu}^{TT}h(k,z)+\frac{1}{3}\eta_{\mu\nu}f(k,z)$
(II.2)

following the $2\oplus 1\oplus 0$ invariant decomposition with the spin-1 part
omitted, as it drops from the threshold production analysis by parity.

###  II.1 A-term

To determine the A-term, we contract the energy-momentum form factor (II.1)
with a spin-2 transverse-traceless polarization tensor
$\epsilon_{\mu\nu}^{TT}$,

$\displaystyle\left<p_{2}|\epsilon_{\mu\nu}^{TT}T^{\mu\nu}(0)|p_{1}\right>$
$\displaystyle=$
$\displaystyle\overline{u}(p_{2})\left(A(k,\kappa_{T})\epsilon_{\mu\nu}^{TT}\gamma^{\mu}p^{\nu}\right)u(p_{1})$

and use a Witten diagram and the holographic dictionary in the soft wall
construction to evaluate it Nastase (2007); Abidin and Carlson (2009); Mamo
and Zahed (2020a). (II.1) sources in bulk the transverse and traceless part of
the metric or a genuine spin-2 graviton (dual to $2^{++}$ tensor glueballs)
coupled to a bulk Dirac fermion, with the result Abidin and Carlson (2009);
Mamo and Zahed (2020a)

$\displaystyle A(k,\kappa_{T})=A(0)\times
6\times\frac{\Gamma\left(2+\frac{a_{K}}{2}\right)}{\Gamma\left(4+\frac{a_{K}}{2}\right)}\times\,_{2}F_{1}\left(3,\frac{a_{K}}{2};\frac{a_{K}}{2}+4;-1\right)\,,$
(II.4)

with $a_{K}=K^{2}/4\kappa_{T}^{2}$ and $k^{2}=-K^{2}$ space-like. The scale
$\kappa_{T}$ determines the dilaton profile in bulk, and its value fixes most
hadron and glueball trajectories (see below). The hypergeometric function
${}_{2}F_{1}$ is related to the LerchPhi function. In the large $N_{c}$ limit,
$A(0)=1+\mathcal{O}(1/N_{c})$ as the nucleon mass is totally glue dominated at
low resolution. At finite $N_{c}$, only a fraction is glue dominated at the
same resolution, so $A(0)$ is a free parameter that can be fixed by comparison
to lattice results or experimental data.

(a)  (b)

Figure 1: (a) The $A$ form factor from the recent lattice QCD result Pefkou et
al. (2022) (red lattice data points), and our holographic fit using (II.4)
with $\kappa_{T}=0.388~{}\text{GeV}$, and $A(0)=0.430$ (solid-blue curve). The
solid-green curve is the tripole lattice fit (II.3). (b) The $D$ form factor
from the recent lattice QCD result Pefkou et al. (2022) (red lattice data
points), and our holographic fit using (II.7) with
$\kappa_{T}=0.388~{}\text{GeV}$, $\kappa_{S}=0.217~{}\text{GeV}$, and
$A(0)=0.430$ (lower blue-solid-curve). The blue shaded curve is determined by
varying $\kappa_{S}$ between $\kappa_{S}=0.217~{}\text{GeV}$ (lower blue-
solid-curve) with $D(0)=-1.275$, and
$\kappa_{S}=\kappa_{T}=0.388~{}\text{GeV}$ (upper blue-solid-curve) with
$D(0)=0$ (large $N_{c}$ limit). The solid-green curve is the tripole lattice
fit (II.3). See also Table 1.

###  II.2 C- or D-term

To determine the C-term or D-term ($D=4C$), we contract the energy-momentum
form factor (II.1) with $\frac{1}{3}\eta_{\mu\nu}$,

$\displaystyle\frac{1}{3}\left<p_{2}|\eta_{\mu\nu}T^{\mu\nu}(0)|p_{1}\right>=\overline{u}(p_{2})\left(A_{S}(k,\kappa_{S})\frac{m_{N}}{3}\right)u(p_{1})$

and use again a Witten diagram for its holographic evaluation. (II.2) sources
in bulk the trace part of the metric (dual to $0^{++}$ tensor glueballs)
coupled to a bulk Dirac fermion, with the result Mamo and Zahed (2021a)

$\displaystyle
A_{S}(k,\kappa_{S})=A(k,\kappa_{T})+\frac{k^{2}}{4m^{2}_{N}}B(k)-\frac{3k^{2}}{m^{2}_{N}}C(k,\kappa_{T},\kappa_{S})$

and

$\displaystyle A_{S}(K,\kappa_{S})$ $\displaystyle=$ $\displaystyle A(0)\times
6\times\frac{\Gamma\left(2+\frac{\tilde{a}_{K}}{2}\right)}{\Gamma\left(4+\frac{\tilde{a}_{K}}{2}\right)}\times\,_{2}F_{1}\left(3,\frac{\tilde{a}_{K}}{2};\frac{\tilde{a}_{K}}{2}+4;-1\right)\,,$
(II.7)

with $\tilde{a}_{K}=K^{2}/4\kappa_{S}^{2}$. The B-term is kinematically
excluded in holography, giving for $D=4C$

$\displaystyle
D(k,\kappa_{T},\kappa_{S})=\frac{4}{3}\frac{m_{N}^{2}}{k^{2}}\bigg{(}A(k,\kappa_{T})-A_{S}(k,\kappa_{S})\bigg{)}$
(II.8)

with $A_{S}(0)/A(0)=1$ by Poincare symmetry Mamo and Zahed (2021a). By
expanding (II.8) near $k^{2}=0$, we can determine $D(0)$ as

$\displaystyle D(0)$ $\displaystyle\equiv$ $\displaystyle
D(0,\kappa_{T},\kappa_{S})=-0.231\times A(0)\times
m_{N}^{2}\times\left(\frac{1}{\kappa_{S}^{2}}-\frac{1}{\kappa_{T}^{2}}\right)\,.$
(II.9)

(II.8) is the difference between the $2^{++}$ glueball Regge trajectory and
the $0^{++}$ glueball Regge trajectory, both of which are degenerate in
leading order in $1/N_{c}$ (same anomalous dimension) with
$D(k,\kappa_{T},\kappa_{S})={\cal O}(1/N_{c})$. The degeneracy is lifted at
finite $1/N_{c}$ through different assignments for the anomalous dimensions
Gubser et al. (2008). Since the leading scalar glueballs are lighter than the
tensor glueballs, $D(k,\kappa_{T},\kappa_{S})$ is in general negative, but
small in magnitude Mamo and Zahed (2021a)!

|  $\tau$ |  $\kappa_{T}(\text{GeV})$ |  $\kappa_{S}(\text{GeV})$ |  $m_{TT}(\text{GeV})$ |  $m_{SS}(\text{GeV})$ |  $A(0)$ |  $D(0)$ |  $\mathcal{N}e\,(\text{nb}\,{\rm GeV^{-2}})$
---|---|---|---|---|---|---|---|---
Holographic QCD $(This~{}work)$ |  3 |  0.388 |  0.217 (or 0.388) |  NA |  NA |  0.430 |  $-1.275$ (or 0) |  2.032 (or 2.311 with $D(0)=0$)
Holographic QCD $(tripole~{}approx.)$ |  NA |  NA |  NA |  1.612 |  0.963 |  0.430 |  $-1.275$ (or 0) |  2.032 (or 2.311 with $D(0)=0$)
Lattice QCD $(tripole~{}fit)$ Pefkou et al. (2022) |  NA |  NA |  NA |  1.641 |  1.070 |  0.429 |  $-1.930$ |  2.549
Lattice QCD $(dipole~{}fit)$ Shanahan and Detmold (2019) |  NA |  NA |  NA |  1.130 |  0.480 |  0.580 |  $-10.000$ |  3.244
Table 1: Summary of the values we have used for the holographic QCD parameters
of the gravitational form factors, and the corresponding normalization
constant of the differential cross section (III.31) (which are $\tau$,
$\kappa_{T}$, $\kappa_{S}$, $A(0)$, $\mathcal{N}e$, as D(0) is totally fixed),
and the lattice QCD tripole (II.3) Pefkou et al. (2022) and dipole fit (II.3)
Shanahan and Detmold (2019) parameters of the gravitational form factors, and
the corresponding normalization constant of the differential cross section
(III.31) (which are $m_{TT},m_{SS}$, $A(0)$, $D(0)$, and $\mathcal{N}e$). Note
that NA is a short hand for not available (i.e., the parameter is not part of
the corresponding model fit).

###  II.3 Comparison to the new lattice results

The parameter $\kappa_{T}$ is related to the dilaton profile in the soft wall
model, and its value is fixed by the rho meson Regge trajectory
$\kappa_{T}=m_{\rho}/2=0.388~{}\text{GeV}$. Note that this value fixes rather
well most of the electromagnetic radii of the nucleon Mamo and Zahed (2021b).
The value of the tensor gravitational form factor $A(0)=1+{\cal O}(1/N_{c})$
is only fixed at asymptotic $N_{c}$. At finite $N_{c}$, a comparison with the
recent lattice results Pefkou et al. (2022) suggests $A(0)=0.430$. We fix the
parameter $\kappa_{S}=\kappa_{T}(1+{\cal O}(1/N_{c}))$ in two ways: 1/ At
large $N_{c}$ with $\kappa_{S}=\kappa_{T}$, for which the D form factor is
null; 2/ At finite $N_{c}$ with $\kappa_{S}=0.217$ GeV, so that the
holographic $D$ form factor (III.26) is comparable to the recent lattice $D$
form factor Pefkou et al. (2022). For numerical convenience, we can also
approximate the holographic A and D form factors (II.4 and II.8) by a tripole
form as

$\displaystyle A(K,m_{TT})$ $\displaystyle=$
$\displaystyle\frac{A(0)}{\left(1+\frac{K^{2}}{m_{TT}^{2}}\right)^{3}}\,,$
$\displaystyle D(K,m_{SS})$ $\displaystyle=$
$\displaystyle\frac{D(0)}{\left(1+\frac{K^{2}}{m_{SS}^{2}}\right)^{3}}\,,$
(II.10)

with $m_{TT}=1.612~{}\text{GeV}$, $A(0)=0.430$ and
$m_{SS}=0.963~{}\text{GeV}$, $D(0)=-1.275$, respectively, see also Table 1.

For comparison, we quote the tripole fit to the lattice data for the A and D
form factors given in Pefkou et al. (2022)

$\displaystyle A(K,m_{TT})$ $\displaystyle=$
$\displaystyle\frac{A(0)}{\left(1+\frac{K^{2}}{m_{TT}^{2}}\right)^{3}}\,,$
$\displaystyle D(K,m_{SS})$ $\displaystyle=$
$\displaystyle\frac{D(0)}{\left(1+\frac{K^{2}}{m_{SS}^{2}}\right)^{3}}\,,$
(II.11)

with $m_{TT}=1.641~{}\text{GeV}$, $A(0)=0.429$ and
$m_{SS}=1.070~{}\text{GeV}$, $D(0)=-1.930$ respectively. Also and for
completeness, the dipole fit to the lattice data for the A and D form factors
given in Shanahan and Detmold (2019)

$\displaystyle A(K,m_{TT})$ $\displaystyle=$
$\displaystyle\frac{A(0)}{\left(1+\frac{K^{2}}{m_{TT}^{2}}\right)^{2}}\,,$
$\displaystyle D(K,m_{SS})$ $\displaystyle=$
$\displaystyle\frac{D(0)}{\left(1+\frac{K^{2}}{m_{SS}^{2}}\right)^{2}}\,,$
(II.12)

with $m_{TT}=1.130~{}\text{GeV}$, $A(0)=0.580$, and
$m_{SS}=0.480~{}\text{GeV}$, and $D(0)=-10$ respectively.

In Fig. 1a, we show the A gravitational form factor versus $K^{2}$. The solid-
blue line is our holographic prediction with the newly adjusted value
$A(0)=0.430$, the green-solid line is the tripole lattice fit (II.3). The red
lattice data points are from Pefkou et al. (2022). The holographic results is
a total match. In Fig. 1b, we show the D gravitational form factor versus
$K^{2}$. The holographic result is indicated by the blue-band which is
obtained by varying $\kappa_{S}$ between $\kappa_{S}=0.217~{}\text{GeV}$
(lower blue-solid-curve) with $D(0)=-1.275$, and
$\kappa_{S}=\kappa_{T}=0.388~{}\text{GeV}$ (upper blue-solid-curve) with
$D(0)=0$ (large $N_{c}$ limit). The solid-green curve is the tripole lattice
fit (II.3), and the lattice data are from Pefkou et al. (2022). A comparison
of the parameters entering the holographic approach used in this work, and the
ones we used in an earlier work Mamo and Zahed (2021a) are recorded in Table
1. We have also noted the changes in the lattice parameters reported in Pefkou
et al. (2022) from those reported earlier in Shanahan and Detmold (2019).

|  $\langle r^{2}_{GS}\rangle~{}(\text{fm}^{2})$ |  $\langle r^{2}_{GM}\rangle~{}(\text{fm}^{2})$
---|---|---
Holographic QCD $(This~{}work)$ |  $\frac{1.04}{\kappa_{S}^{2}}$ = $0.926^{2}$ ($or~{}0.518^{2}~{}with$ $\kappa_{S}=\kappa_{T}$ ) |  $\frac{0.693}{\kappa_{T}^{2}}$ \+ $\frac{0.347}{\kappa_{S}^{2}}$ = $0.682^{2}$ ($or~{}0.518^{2}~{}with$ $\kappa_{S}=\kappa_{T}$ )
Holographic QCD $(tripole~{}approx.)$ |  $-\frac{D(0)}{A(0)}\times\frac{4.5}{m_{N}^{2}}+\frac{18}{m_{TT}^{2}}=0.926^{2}$ |  $-\frac{D(0)}{A(0)}\times\frac{1.5}{m_{N}^{2}}+\frac{18}{m_{TT}^{2}}=0.682^{2}$
Lattice QCD $(tripolefit)$ Pefkou et al. (2022) |  $-\frac{D(0)}{A(0)}\times\frac{4.5}{m_{N}^{2}}+\frac{18}{m_{TT}^{2}}=1.073^{2}$ |  $-\frac{D(0)}{A(0)}\times\frac{1.5}{m_{N}^{2}}+\frac{18}{m_{TT}^{2}}=0.747^{2}$
Lattice QCD $(dipolefit)$ Shanahan and Detmold (2019) |  $-\frac{D(0)}{A(0)}\times\frac{4.5}{m_{N}^{2}}+\frac{12}{m_{TT}^{2}}=1.945^{2}$ |  $-\frac{D(0)}{A(0)}\times\frac{1.5}{m_{N}^{2}}+\frac{12}{m_{TT}^{2}}=1.227^{2}$
Table 2: Summary of the gluonic scalar (GS), and gluonic mass (GM) radii
(II.13 and II.15) with the holographic gravitational form factors (II.4 and
II.8) (including their tripole approximation (II.3)), and the lattice QCD
tripole (II.3) Pefkou et al. (2022) and dipole fits (II.3) Shanahan and
Detmold (2019) of the lattice gravitational form factor data.

(a)  (b)

Figure 2: (a) The gluonic mass radius $r_{GM}$ defined in (II.15). The blue
data point at $r_{GM}=0.682\pm 0.012~{}\rm fm$ is determined by using our
holographic A and D form factors (II.4 and II.8), and the green data point at
$r_{GM}=0.747\pm 0.033~{}\rm fm$ is determined using the lattice tripole fit
to the lattice A and D form factor data (II.3) Pefkou et al. (2022). (b) The
gluonic scalar radius $r_{GS}$ defined in (II.13). The blue data point at
$r_{GS}=0.926\pm 0.008~{}\rm fm$ is determined by using our holographic A and
D form factors (II.4 and II.8), and the green data point at $r_{GS}=1.073\pm
0.033~{}\rm fm$ is determined by using the lattice tripole fit to the lattice
A and D form factor data (II.3) Pefkou et al. (2022). For comparison, we have
also shown the charge radius of the proton (the orange band) centered around
the recent PRad measurement of $r_{c}=0.831~{}\rm fm$ Xiong et al. (2019). The
black data point is our holographic prediction of the charge radius of proton
$r_{c}=0.831\pm 0.008~{}\rm fm$ Mamo and Zahed (2021b). The other charge
radius data points, within the orange band, are from Pohl 2010 ($\mu$ H
spect.) Pohl et al. (2010), Antognini 2013 ($\mu$ H spect.) Antognini et al.
(2013), Beyer 2017 (H spect.) Beyer et al. (2017), CODATA 2018 Mohr et al.
(2019), and Bezignov 2019 (H spect.) Bezginov et al. (2019).

###  II.4 Radii, pressure, and shear distributions

####  II.4.1 Radii

Given the gravitational A and D form factors, a number of radii can be
defined. The simplest are those directly given by A and D respectively, which
characterize the tensor and scalar glueball range as discussed in Mamo and
Zahed (2021a). Alternatively, we may follow Ji (2021) and define the gluonic
scalar mass radius $r_{GS}$ of the proton (derived from the trace of its
energy-momentum tensor (II.1) with $k^{2}=-K^{2}$) as

$\displaystyle\langle
r^{2}_{GS}\rangle=-\frac{6}{A_{S}(0)}\bigg{(}\frac{dA_{S}(K)}{dK^{2}}\bigg{)}_{0}\,\hbar^{2}c^{2}\,,$
(II.13)

where

$\displaystyle A_{S}(K)\equiv
A(K)-\frac{K^{2}}{4m^{2}_{N}}B(K)+\frac{3K^{2}}{4m^{2}_{N}}D(K)\,.$

Similarly, the gluonic mass radius $r_{GM}$ of the proton (derived from the
$00$ component of its energy-momentum tensor (II.1) with $k^{2}=-K^{2}$) is
defined as Ji (2021)

$\displaystyle\langle
r^{2}_{GM}\rangle=-\frac{6}{A_{M}(0)}\bigg{(}\frac{dA_{M}(K)}{dK^{2}}\bigg{)}_{0}\,\hbar^{2}c^{2}\,,$
(II.15)

where

$\displaystyle A_{M}(K)\equiv
A(K)-\frac{K^{2}}{4m^{2}_{N}}B(K)+\frac{K^{2}}{4m^{2}_{N}}D(K)\,.$

Throughout, we use $\hbar c=0.197\,\text{GeV}\,\rm fm$. See Table 2, for
comparison of the gluonic radii of the proton that we have found by using the
holographic (II.4 and II.8) and lattice tripole or dipole fits (II.3 or II.3)
of the $A$ and $D$ gravitational form factors in (II.13) and (II.15). Note
that all the radii in Table 2 are computed with $B(K)=0$.

In Fig. 2a, we show the gluonic mass radius $r_{GM}$ defined in (II.15). The
blue data point at $r_{GM}=0.682\pm 0.012~{}\rm fm$ is determined by using our
holographic A and D form factors (II.4 and II.8), and the green data point at
$r_{GM}=0.747\pm 0.033~{}\rm fm$ is determined using the lattice tripole fit
to the lattice A and D form factor data (II.3) Pefkou et al. (2022). In Fig.
2b, we show the gluonic scalar radius $r_{GS}$ defined in (II.13). The blue
data point at $r_{GS}=0.926\pm 0.008~{}\rm fm$ is determined by using our
holographic A and D form factors (II.4 and II.8), and the green data point at
$r_{GS}=1.073\pm 0.033~{}\rm fm$ is determined by using the lattice tripole
fit to the lattice A and D form factor data (II.3) Pefkou et al. (2022). Note
that the gluonic scalar radius is smaller for $D(0)=0$ (strict large $N_{c}$
limit with $\kappa_{S}=\kappa_{T}$) with $r_{GS}=r_{GM}=0.518\pm 0.018~{}\rm
fm$, determined using our holographic A form factor (II.4). The theoretical
uncertainty for the holographic results are determined by varying
$\kappa_{T}=0.388~{}\text{GeV}$ to $\kappa_{T}=0.402~{}\text{GeV}$ (which
gives a charge radius of the proton to be $0.831~{}\rm fm$ Mamo and Zahed
(2021b) (in perfect agreement with the recent PRad measurement Xiong et al.
(2019))). We have also estimated the error in the lattice results to be $\pm
0.033$, see Table X in Pefkou et al. (2022). See also Table 2.

####  II.4.2 Pressure and shear distributions

In QCD, the proton, as an extended quantum object, is composed of interacting
partons in different proportions of quarks and gluons, at different
resolutions. This composition at the nucleon mass resolution, is clearly non-
perturbative. At large $N_{c}$ and fixed ′t Hooft coupling
$\lambda=g^{2}N_{c}$, this nonperturbative description is captured by
interacting and topological pion and vector meson fields, in the form of a
Skyrmion Zahed and Brown (1986). The meson mediated force field is repulsive
at the core, and attractive at the periphery, setting the pressure force
across the nucleon Jung et al. (2014).

Holography is yet another dual and nonperturbative description at large
$N_{c}$ and large ′t Hooft coupling $\lambda=g^{2}N_{c}$, whereby the nucleon
hologram is held together by gravitons (dual of tensor glueballs) and possibly
scalars (dual of scalar glueballs) emerging from bulk string fluctuations. The
short range tensor gravitons are dominant at the core and repulsive, while the
long range scalars are attractive at the periphery Mamo and Zahed (2021a).
Using the analytic Fourier transform of a tripole D form factor (with
$-t=K^{2}={\mathbf{\Delta}}^{2}$) Lorcé et al. (2019); Pefkou et al. (2022)

$\displaystyle\tilde{D}(r)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{\mathbf{\Delta}}}{(2\pi)^{3}}\,e^{-i{\mathbf{\Delta}}\cdot{\mathbf{r}}}\,D({\mathbf{\Delta}},m_{SS})=\int\frac{d^{3}{\mathbf{\Delta}}}{(2\pi)^{3}}\,e^{-i{\mathbf{\Delta}}\cdot{\mathbf{r}}}\,\frac{D(0)}{\left(1+\frac{{\mathbf{\Delta}}^{2}}{m_{SS}^{2}}\right)^{3}}=D(0)\times\frac{m_{SS}^{3}}{32\pi}\times(1+m_{SS}r)\times
e^{-m_{SS}r}\,,$

of the holographic tripole approximation (II.3), and lattice tripole fits
(II.3), we can determine the gluonic pressure and shear distributions inside
the proton (in the Breit frame with ${\mathbf{p}}=0$) Polyakov (2003);
Polyakov and Schweitzer (2018)

$\displaystyle p(r)=$
$\displaystyle\frac{1}{6m_{N}}\frac{1}{r^{2}}\frac{d}{dr}\bigg{(}r^{2}\frac{d}{dr}{\tilde{D}(r)}\bigg{)}\,,$
(II.18) $\displaystyle s(r)=$
$\displaystyle-\frac{1}{4m_{N}}r\frac{d}{dr}\bigg{(}\frac{1}{r}\frac{d}{dr}{\tilde{D}(r)}\bigg{)}\,.$
(II.19)

The pressure and shear distributions capture the anisotropic spatial content
of the energy momentum tensor as

$\displaystyle
T^{ij}({\mathbf{r}})=\frac{1}{3}\delta^{ij}p(r)+\bigg{(}\hat{r}^{i}\hat{r}^{j}-\frac{1}{3}\delta^{ij}\bigg{)}s(r)\,.$
(II.20)

In Fig. 3a, we show the pressure distribution $\hbar^{2}c^{2}\times r^{2}p(r)$
inside proton (II.18 with II.4.2) where $m_{ss}=0.963~{}\text{GeV}/\hbar c$
and $D(0)=-1.275$ (solid-blue curve (holographic QCD)), and
$m_{ss}=1.070~{}\text{GeV}/\hbar c$ and $D(0)=-1.930$ (solid-green curve
(lattice QCD)). In Fig. 3b, we show the shear distribution
$\hbar^{2}c^{2}\times r^{2}s(r)$ inside proton (II.19 with II.4.2) where
$m_{ss}=0.963~{}\text{GeV}/\hbar c$ (solid-blue curve (holographic QCD)), and
$m_{ss}=1.070~{}\text{GeV}/\hbar c$ (solid-green curve (lattice QCD)). The
shaded blue curves in (a) and (b) correspond to varying the holographic $D(0)$
value of the form factor between $-1.275$ and $0$. For the latter, the scalar
and tensor glueball Regge trajectories become degenerate, balancing
identically each other, with zero pressure across the proton!

(a)  (b)

Figure 3: (a) The pressure distribution $\hbar^{2}c^{2}\times r^{2}p(r)$
inside the proton (II.18 with II.4.2) where $m_{ss}=0.963~{}\text{GeV}/\hbar
c$ and $D(0)=-1.275$ (solid-blue curve (holographic QCD)), and
$m_{ss}=1.070~{}\text{GeV}/\hbar c$ and $D(0)=-1.930$ (solid-green curve
(lattice QCD)). (b) The shear distribution $\hbar^{2}c^{2}\times r^{2}s(r)$
inside proton (II.19 with II.4.2) where $m_{ss}=0.963~{}\text{GeV}/\hbar c$
(solid-blue curve (holographic QCD)), and $m_{ss}=1.070~{}\text{GeV}/\hbar c$
(solid-green curve (lattice QCD)). The shaded blue curves in (a) and (b)
correspond to varying the holographic $D(0)$ between $-1.275$ and $0$.

##  III Holographic $J/\psi$ production near-threshold

The A and to a lesser extent D gravitational form factors, are the key
elements in the electromagnetic production of heavy mesons in the threshold
region. The holographic approach allows not only for their determination as we
briefly reviewed, but fixes the entire heavy vector meson production process
time-like from threshold to asymptotia, all within one consolidated framework.
We now briefly detail the electroproduction process for charmonia near
threshold and close to the photon point, and use that to reanalyze the GlueX
data, for a better understanding of the role of the D term.

###  III.1 Holographic electroproduction

The t-channel $2^{++}$ glueball exchange contribution to the electroproduction
amplitude of vector mesons is described in bulk by the exchange of a graviton
Mamo and Zahed (2021c). The Witten amplitude for the transverse and
longitudinal ($T,L$) electroproduction of a heavy meson ($V$) off a nucleon,
are

$\displaystyle i{\cal A}^{h}_{\gamma^{*}_{T/L}p\rightarrow
Vp}(s,t)=\sum_{n}i\tilde{{\cal A}}^{h}_{\gamma^{*}_{T/L}p\rightarrow
Vp}(m_{n},s,t)$ $\displaystyle i\tilde{{\cal
A}}^{h}_{\gamma_{T/L}^{*}p\rightarrow
Vp}(m_{n},s,t)=\frac{1}{g_{5}}\times(-i)V_{h\gamma^{*}_{T/L}V}^{\mu\nu}(q,q^{\prime},k,m_{n})\times\tilde{G}_{\mu\nu\alpha\beta}(m_{n},k)\times(-i)V_{h\bar{\Psi}\Psi}^{\alpha\beta}(p_{1},p_{2},k,m_{n})\,,$

with the bulk vertices (defining $t=k^{2}=\Delta^{2}$, and
$k=\Delta=p_{2}-p_{1}=q-q^{\prime}$)

$\displaystyle
V_{h\gamma^{*}_{T/L}V}^{\mu\nu}(q,q^{\prime},k,m_{n})\equiv\left(\frac{\delta
S_{h\gamma^{*}_{T/L}V}^{k}}{\delta(\epsilon_{\mu\nu}h(k,z))}\right)\,J_{h}(m_{n},z)=\sqrt{2\kappa^{2}}\times\frac{1}{2}\int
dz\sqrt{g}\,e^{-\phi}z^{4}K_{T/L}^{\mu\nu}(q,q^{\prime},\epsilon,\epsilon^{\prime},z)J_{h}(m_{n},z)\,,$
$\displaystyle
V_{h\bar{\Psi}\Psi}^{\alpha\beta}(p_{1},p_{2},k,m_{n})\equiv\left(\frac{\delta
S_{h\bar{\Psi}\Psi}^{k}}{\delta(\epsilon_{\alpha\beta}h(k,z))}\right)\,J_{h}(m_{n},z)=-\sqrt{2\kappa^{2}}\times\frac{1}{2}\int
dz\sqrt{g}\,e^{-\phi}z\bar{\Psi}(p_{2},z)\gamma^{\alpha}p^{\beta}\Psi(p_{1},z)J_{h}(m_{n},z)\,.$
(III.22)

The graviton bulk-to-bulk propagator $\tilde{G}$, and the U(1) vector meson
bulk-to-boundary transverse and longitudinal propagators $K_{T/L}$ are given
in Appendix A. The holographic wavefunction for the spin-2 glueball
$J_{h}(m_{n},z)$ is given in Appendix A.2 for the soft wall model, and the
holographic wavefunction $\Psi(p,z)$ refers to the bulk Dirac fermion. The
details regarding the reduction of (III.1-III.1) can be found in Appendix A,
with the reduced transverse and longitudinal amplitudes

$\displaystyle{\cal A}^{h}_{\gamma_{T}^{*}p\rightarrow Vp}(s,t)$
$\displaystyle=$ $\displaystyle-\frac{1}{g_{5}}\times
2\kappa^{2}\times\mathcal{V}_{h\gamma_{T}^{*}V}(Q,M_{V})\times\frac{1}{m_{N}}\times\bar{u}(p_{2})u(p_{1})$
$\displaystyle\times$ $\displaystyle\left(p_{1}\cdot
q\right)^{2}\times\left[A_{T}(K,\kappa_{T})+\eta^{2}D(K,\kappa_{T},\kappa_{S})\right]\,,$
$\displaystyle{\cal A}^{h}_{\gamma_{L}^{*}p\rightarrow Vp}(s,t)$
$\displaystyle=$ $\displaystyle-\frac{1}{g_{5}}\times
2\kappa^{2}\times\frac{1}{QM_{V}}\times\mathcal{V}_{h\gamma_{L}^{*}V}(Q,M_{V})\times\frac{1}{m_{N}}\times\bar{u}(p_{2})u(p_{1})$
(III.23) $\displaystyle\times$ $\displaystyle\left(p_{1}\cdot
q\right)^{2}\times\left[A_{T}(K,\kappa_{T})+\eta^{2}D(K,\kappa_{T},\kappa_{S})\right]\,,$

with the A and D gravitational form factors following from

$\displaystyle A(K,\kappa_{T})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\int
dz\,\sqrt{g}\,e^{-\phi}z\,\left(\psi_{R}^{2}(z)+\psi_{L}^{2}(z)\right)\times\sum_{n=0}^{\infty}\frac{\sqrt{2}\kappa
F_{n}\psi_{n}(z)}{K^{2}+m_{n}^{2}}\,,$ (III.24) $\displaystyle=$
$\displaystyle\frac{1}{2}\int
dz\,\sqrt{g}\,e^{-\phi}z\,\left(\psi_{R}^{2}(z)+\psi_{L}^{2}(z)\right)\times\mathcal{H}(K,z)\,,$
$\displaystyle=$ $\displaystyle A(0)\times
6\times\frac{\Gamma\left(2+\frac{a_{K}}{2}\right)}{\Gamma\left(4+\frac{a_{K}}{2}\right)}\times\,_{2}F_{1}\left(3,\frac{a_{K}}{2};\frac{a_{K}}{2}+4;-1\right)\,,$

and

$\displaystyle D(K,\kappa_{T})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\int
dz\,\sqrt{g}\,e^{-\phi}z\,\left(\psi_{R}^{2}(z)+\psi_{L}^{2}(z)\right)\times\sum_{n=0}^{\infty}\,\frac{4}{3}\frac{p_{1}^{2}}{m_{n}^{2}}\times\frac{\sqrt{2}\kappa
F_{n}\psi_{n}(z)}{K^{2}+m_{n}^{2}}\,,$ (III.25) $\displaystyle=$
$\displaystyle\frac{1}{2}\int
dz\,\sqrt{g}\,e^{-\phi}z\,\left(\psi_{R}^{2}(z)+\psi_{L}^{2}(z)\right)\times\sum_{n=0}^{\infty}\,\frac{4}{3}\frac{p_{1}^{2}}{k^{2}}\times\frac{\sqrt{2}\kappa
F_{n}\psi_{n}(z)}{-k^{2}+m_{n}^{2}}\,,$ $\displaystyle=$
$\displaystyle-\frac{4}{3}\frac{m_{N}^{2}}{K^{2}}\times A(K,\kappa_{T})\,,$

with the dilaton profile $\phi(z)=\kappa_{T}^{2}z^{2}$. In deriving the last
two relations in (III.25), we have assumed that the D form factor is analytic
in $K^{2}$, and traded $m_{n}^{2}\rightarrow k^{2}$ at the pole time-like,
before switching back space-like. Note that only the A form factor
contribution appears in (III.25), as expected from a graviton exchange. The
complementary trace part contribution due to the exchange of a dilaton in
bulk, fails to couple to a heavy vector meson described by a U(1) bulk gauge
field Mamo and Zahed (2021a). Since we cannot rule out a holographic
formulation with a heavy vector meson in bulk with a tracefull coupling at
finite $N_{c}$, we will assume Mamo and Zahed (2021a)

$\displaystyle D(K,\kappa_{T})\rightarrow D(K,\kappa_{T},\kappa_{S})$
$\displaystyle=$
$\displaystyle-\frac{4}{3}\frac{m_{N}^{2}}{K^{2}}\times\left[A(K,\kappa_{T})-A_{S}(K,\kappa_{S})\right]\,.$
(III.26)

###  III.2 Holographic photoproduction

The coupling between $2^{++}$ glueballs (denoted by $h$), vector mesons with
mass $M_{V}$ (denoted by $V$), and transverse virtual photons with $Q^{2}\neq
0$ (denoted by $\gamma_{T}^{*}$) or transverse real photons with $Q^{2}=0$
(denoted by $\gamma_{T}$) is given by (A.53)

$\displaystyle\mathcal{V}_{h\gamma_{T}V}$ $\displaystyle\equiv$
$\displaystyle\mathcal{V}_{h\gamma^{*}_{T}V}(Q=0,M_{V})=g_{5}\times\bigg{(}\frac{f_{V}}{M_{V}}\bigg{)}\times{\mathbb{V}}_{h\gamma
V}\,,$

where $f_{V}/M_{V}$ is the dimensionless ratio of the decay constant $f_{V}$
of $V=J/\psi$, and its mass $M_{V}$. Here we have also defined

$\displaystyle{\mathbb{V}}_{h\gamma
V}=\frac{1}{2}\frac{1}{4\tilde{\kappa}_{V}^{4}}\int_{0}^{\infty}d\tilde{\xi}\,e^{-\tilde{\xi}}\,\tilde{\xi}^{2}\times
L_{0}^{1}(\tilde{\xi})=\frac{1}{4\tilde{\kappa}_{V}^{4}}\,.$

The pertinent differential cross section at the photon point, is of the form

$\displaystyle\left(\frac{d\sigma}{dt}\right)=\frac{e^{2}}{16\pi(s-m_{N}^{2})^{2}}\,\frac{1}{2}\sum_{{\rm
T=1,2}}\frac{1}{2}\sum_{{\rm spin}}\Bigg{|}{\cal
A}^{h}_{\gamma_{T}p\rightarrow J/\Psi p}(s,t)\Bigg{|}^{2}\,,$

which is dominated by the TT-part of the graviton or $2^{++}$ glueball
exchange as we noted earlier. The first sum is over the photon and $J/\Psi$
polarizations $T=1,2$. The second sum is over the initial and final bulk Dirac
fermion (proton) spin, evaluated explicitely as

$\displaystyle\sum_{s,s^{\prime}}\bar{u}_{s^{\prime}}(p_{2})u_{s}(p_{1})\bar{u}_{s}(p_{1})u_{s^{\prime}}(p_{2})$
$\displaystyle=$ $\displaystyle{\rm
Tr}\Big{(}\sum_{s.s^{\prime}}u_{s^{\prime}}(p_{2})\bar{u}_{s^{\prime}}(p_{2})u_{s}(p_{1})\bar{u}_{s}(p_{1})\Big{)}$
$\displaystyle=$ $\displaystyle\frac{1}{4}{\rm
Tr}\Big{(}\big{(}\gamma_{\mu}p_{2}^{\mu}+m_{N}\big{)}\big{(}\gamma_{\mu}p_{1}^{\mu}+m_{N}\big{)}\Big{)}=8m_{N}^{2}\times\left(1+\frac{K^{2}}{4m_{N}^{2}}\right)\approx
8m_{N}^{2}\,,$

for $\frac{K^{2}}{4m_{N}^{2}}\ll 1$ or $p_{1}\sim p_{2}$ which we are assuming
in the high energy limit $s\gg-t$.

Carrying explicitly these summations yield the differential cross section for
photoproduction of heavy meson in the spin $j=2$ exchange approximation as

$\displaystyle\left(\frac{d\sigma}{dt}\right)$
$\displaystyle=2\times\frac{e^{2}}{64\pi(s-m_{N}^{2})^{2}}\times\bigg{(}\frac{f_{V}}{M_{V}}\bigg{)}^{2}\times{\mathbb{V}}_{h\gamma
V}\times
2\kappa^{2}\times\left[A(K,\kappa_{T})+\eta^{2}D(K,\kappa_{T},\kappa_{S})\right]^{2}\times\tilde{F}(s)\times\frac{8m_{N}^{2}}{m_{N}^{2}}\,,$
(III.31)
$\displaystyle=\mathcal{N}^{2}\times\frac{e^{2}}{64\pi(s-m_{N}^{2})^{2}}\times\frac{\left[A(K,\kappa_{T})+\eta^{2}D(K,\kappa_{T},\kappa_{S})\right]^{2}}{A^{2}(0)}\times\tilde{F}(s)\times
8\,,$

with all vertex insertions shown explicitly and, in the last line, we have
defined the normalization factor $\mathcal{N}$ as

$\displaystyle\mathcal{N}^{2}\times e^{2}\equiv
2\times\bigg{(}\frac{f_{V}}{M_{V}}\bigg{)}^{2}\times{\mathbb{V}}_{h\gamma
V}\times 2\kappa^{2}\times A^{2}(0)$

for $V=J/\psi$, and ${\mathbb{V}}_{h\gamma
V}=\frac{1}{4\tilde{\kappa}_{V}^{4}}$ as defined in (III.2). We have also
defined the kinematic factor

$\tilde{F}(s)=(q\cdot p_{1})^{4}\approx\frac{1}{16}\times s^{4}\,,$

for $s\gg-t,m_{N}^{2}$, a typical signature of a bulk graviton (or spin-2
glueball) exchange.

Figure 4: The skewness parameter $\eta$ (defined in (III.33)) for
photoproduction of $J/\Psi$ at $E_{\gamma}=10.72$ GeV (and with
$t_{min}=-0.438~{}\rm GeV$).

(a)  (b)

Figure 5: (a) Differential cross section for $V=J/\Psi$ photoproduction for
$E_{\gamma}=10.72$ GeV (and with $t_{min}=-0.438~{}\rm GeV$). The solid-blue
curve is our soft-wall holographic QCD result (III.31) with $\mathcal{N}\times
e=2.311\,\text{nb}\,{\rm GeV^{-2}}$,
$\kappa_{T}=\kappa_{S}=0.388~{}\text{GeV}$, and $D(0)=-0.231\times A(0)\times
m_{N}^{2}\times\left({1}/{\kappa_{S}^{2}}-{1}/{\kappa_{T}^{2}}\right)=0$. The
dashed-blue curve is our soft-wall holographic QCD result (III.31) with
$\mathcal{N}\times e=2.032\,\text{nb}\,{\rm GeV^{-2}}$,
$\kappa_{T}=0.388~{}\text{GeV}$, $\kappa_{S}=0.217~{}\text{GeV}$, and
$D(0)=-0.231\times A(0)\times
m_{N}^{2}\times\left({1}/{\kappa_{S}^{2}}-{1}/{\kappa_{T}^{2}}\right)=-1.275$.
The data is from GlueX Ali et al. (2019b). (b) The dashed-green curve is
(III.31) with the tripole fits (II.3) to the recent lattice A and D form
factors data Pefkou et al. (2022). The dashed-purple curve is (III.31) with
the dipole fits (II.3) to the previous lattice A and D form factors data
Shanahan and Detmold (2019). See also Table 1.

We note that the kinematic factor (the skewness parameter) $\eta$ is

$\eta=\frac{M_{V}^{2}}{4p_{1}\cdot q-M_{V}^{2}+t}\,,$ (III.33)

where $2p_{1}\cdot q=s-m_{N}^{2}$, see Fig. 4. In general, $\eta\neq\xi$, with
$\xi$ is given by (see Eq.5.12 in Belitsky and Radyushkin (2005) for a very
general definition of $\eta$ and $\xi$)

$\xi=\frac{-M_{V}^{2}+t/2}{4p_{1}\cdot q-M_{V}^{2}+t}\,.$ (III.34)

In Fig. 4, we show the behavior of the skewness parameter $\eta(t)$ versus
$-(t-t_{\rm min})$ at the photon-point. From (III.31) we conclude that the
A-contribution dwarfs the D-contribution by about 10 to 1.

###  III.3 Comparison to GlueX

In Fig. 5a we show the holographic differential cross section for
phtoproduction of $J/\psi$ (III.31) versus $-(t-t_{\rm min})$ near threshold
for a photon energy $E_{\gamma}=10.72$ GeV. The solid-blue curve is the result
for $\kappa_{T}=\kappa_{S}$ = 0.338 GeV for which the D form factor is null or
$D=0$. The overall normalization factor (III.31) is fixed to
$\mathcal{N}\times e=2.311\,\text{nb}\,{\rm GeV^{-2}}$. The dashed-blue curve
is the result for $\kappa_{T}=0.388~{}\text{GeV}$,
$\kappa_{S}=0.217~{}\text{GeV}$, and

$D(0)=-0.231\times A(0)\times
m_{N}^{2}\times\left({1}/{\kappa_{S}^{2}}-{1}/{\kappa_{T}^{2}}\right)=-1.275\,.$

The overall normalization factor is now adjusted to $\mathcal{N}\times
e=2.032\,\text{nb}\,{\rm GeV^{-2}}$. The data are from GlueX Ali et al.
(2019b).

For comparison, we show in Fig. 5b the same differential cross section
(III.31), but now using the lattice A and D gravitational form factors Pefkou
et al. (2022) instead. The dashed-green curve follows from the lattice A and D
using the tripole fits in (II.3), while the dashed-purple curve follows from
the lattice A and D using the dipole fits in (II.3).

##  IV Conclusion

The holographic construction using the soft wall model offers the most
economical way of describing the gravitational form factors of the nucleon as
a Dirac fermion in bulk. In the double limit of large $N_{c}$ and large ′t
Hooft coupling, the A form factor follows from the exchange of a bulk-to-
boundary graviton, which is dual to the exchange of Reggeized and massive
$2^{++}$ glueballs at the boundary. Similarly, the D form factor follows from
the difference between the exchange of a bulk-to-boundary graviton and
dilaton, dual to Reggeized and massive $2^{++}$ and $0^{++}$ tensor and scalar
glueballs respectively. The holographic B form factor is null.

The holographic gravitational form factor depends on the dilaton profile
parameter $\kappa_{T}=m_{\rho}/2=0.388$ GeV, which is fixed by the rho meson
trajectory in bulk (so it is not really a parameter), and the value of
$A(0)=1+{\cal O}(1/N_{c})$ by Poincare symmetry. In the strict large $N_{c}$
limit, the D form factor vanishes since the Reggeized $2^{++}$ and $0^{++}$
trajectories are degenerate. At finite $N_{c}$ they are not, and the D form
factor follows by fixing $\kappa_{S}=0.217$ GeV to reproduce the newly
released lattice QCD results. Remarkably, holography predicts the D form
factor to be small, negative and tripole-like, while the A form-factor to be
large and tripole-like.

By refining the parameter choice of $A(0)$ to the newly released lattice data,
we have also sharpened our holographic predictions for the mass and scalar
radii of the proton. A better estimate of the D form factor yields also a
sharper description of the pressure and shear across the nucleon. The
comparison to the lattice estimates are in qualitative agreement, although
better lattice results for the D form factor at low momentum, would be
welcome.

The current GlueX data appears to support the holographic construction in
three ways: 1/ the threshold photo-production of $J/\psi$ is dominated by the
exchange of a tensor glueball, as captured by the A form factor; 2/ the
contribution of the D form factor appears to be very small, as it is
suppressed by the squared skewness (about $\frac{1}{10}$) and its inherent
smallness (it is null in the strict large $N_{c}$ limit); The holographic B
form factor is null.

Holography provides a nonperturbative calculational framework based on
$1/N_{c}$ and $1/g^{2}N_{c}$ counting, which captures the stringy character of
the gluon interactions both in the conformal and confined regimes. The QCD
string is a reality, that dominates most gluon exchanges in the IR. Detailed
lattice simulations show this string to be surprisingly close to the Nambu-
Goto string. The gravitational excitations in bulk at the origin of the A and
D form factors space-like, are the massless modes of this string in a warped
space, which allows the interpolation between the conformal UV limit and the
confining IR limit. The time-like photoproduction of charmonia and bottomia
appears to be dominated by the closed string spin-2 radial Regge trajectory,
as captured by the bulk graviton.

Holography is the most economical way of enforcing QCD symmetries, duality and
crossing symmetries among others, all usually sought by dispersive analysis,
and yet within a well defined and minimal organisational principle using a
stringy field theory. The present analysis and results are an illustration of
that.

However, holography provides much more. It allows for the assessment of any
n-point function on the boundary, using field theoretical methods through dual
Witten diagrams. It is the QCD string made user friendly. It also provides for
novel physics for processes at low parton-x Mamo and Zahed (2020a), and DIS on
nuclei as dual to black holes Mamo and Zahed (2020b, c).

Acknowledgements

We thank Zein-Eddine Meziani for discussions. K.M. is supported by the U.S.
Department of Energy, Office of Science, Office of Nuclear Physics, contract
no. DE-AC02-06CH11357, and an LDRD initiative at Argonne National Laboratory
under Project No. 2020-0020. I.Z. is supported by the Office of Science, U.S.
Department of Energy under Contract No. DE-FG-88ER40388.

##  Appendix A Details of the electroproduction amplitude

###  A.1 Wave functions and propagators of bulk virtual photons and vector
mesons

The transverse and longitudinal bulk-to-boundary U(1) propagators $K_{T,L}$ in
(III.1) are given by

$\displaystyle
K_{T}^{\mu\nu}(q,q^{\prime},\epsilon_{T},\epsilon^{\prime}_{T},z)$
$\displaystyle\approx$
$\displaystyle\epsilon_{T}\cdot\epsilon_{T}^{\prime}\times
q^{\mu}q^{\nu}\times\mathcal{V}_{\gamma^{*}}(Q,z)\mathcal{V}_{V}(M_{V},z)=-q^{\mu}q^{\nu}\times\mathcal{V}_{\gamma^{*}}(Q,z)\mathcal{V}_{V}(M_{V},z)\,$
$\displaystyle\equiv$ $\displaystyle q^{\mu}q^{\nu}\times K_{T}(Q,M_{V},z)\,,$
$\displaystyle
K_{L}^{\mu\nu}(q,q^{\prime},\epsilon_{L},\epsilon^{\prime}_{L},z)$
$\displaystyle\approx$
$\displaystyle-\epsilon_{L}^{\mu}\epsilon_{L}^{\prime\nu}\times\partial_{z}\mathcal{V}_{\gamma^{*}}(Q,z)\partial_{z}\mathcal{V}_{V}(M_{V},z)=-q^{\mu}q^{\nu}\times\frac{1}{Q\,M_{V}}\times\partial_{z}\mathcal{V}_{\gamma^{*}}(Q,z)\partial_{z}\mathcal{V}_{V}(M_{V},z)\,$
$\displaystyle\equiv$ $\displaystyle q^{\mu}q^{\nu}\times K_{L}(Q,M_{V},z)\,,$

where we have defined

$\displaystyle K_{T}(Q,M_{V},z)$ $\displaystyle\equiv$
$\displaystyle-\mathcal{V}_{\gamma^{*}}(Q,z)\mathcal{V}_{V}(M_{V},z)\,$
$\displaystyle K_{L}(Q,M_{V},z)$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{Q\,M_{V}}\times\partial_{z}\mathcal{V}_{\gamma^{*}}(Q,z)\partial_{z}\mathcal{V}_{V}(M_{V},z)$
(A.36)

The bulk-to-boundary propagator of the incoming virtual photon
$\mathcal{V}_{\gamma^{*}}(Q,z)$, and the normalized wave function of the
outgoing $V=J/\psi$ denoted as $\mathcal{V}_{V}(M_{V},z)$ are given by

$\displaystyle\mathcal{V}_{\gamma^{*}}(Q,z)$ $\displaystyle=$
$\displaystyle\tilde{\xi}\,\Gamma(1+a_{Q})\,\,{\cal
U}(1+a_{Q};2;\tilde{\xi})=\tilde{\xi}\int_{0}^{1}\frac{dx}{(1-x)^{2}}x^{a_{Q}}{\rm
exp}\Big{(}-\frac{x\,\tilde{\xi}}{1-x}\Big{)}\,,$
$\displaystyle\mathcal{V}_{V}(M_{V},z)$ $\displaystyle\equiv$
$\displaystyle\phi_{0}(z)=g_{5}\times\frac{f_{V}}{M_{V}}\times
2\tilde{\xi}L_{0}^{1}(\tilde{\xi})=g_{5}\times\frac{f_{V}}{M_{V}}\times
2\tilde{\xi}\,,$ (A.37)

with $a_{Q}=Q^{2}/(4\tilde{\kappa}^{2})$ and
$\tilde{\xi}=\tilde{\kappa}_{V}^{2}z^{2}$. The kinematical analysis of the
holographic photoproduction amplitude is simplified using $s\gg-t$ even at
threshold (which is satisfied even for charmonium), and specifically
$p_{1}^{\mu}\approx p_{2}^{\mu}$, $q^{\mu}\approx q^{\prime\mu}$,
$\epsilon_{T}\cdot\epsilon_{T}^{\prime}\approx-1$,
$\epsilon_{L}^{\mu}\approx\frac{q^{\mu}}{Q}$,
$\epsilon_{L}^{\prime\mu}\approx\frac{q^{\prime\mu}}{M_{V}}$,
$\epsilon_{T}\cdot A=\epsilon^{\prime}_{T}\cdot A\approx 0$ for
$A=q,q^{\prime},p,p_{1},p_{2}$.

###  A.2 Spin-2 glueball wavefunctions and propagators

In the soft-wall holographic QCD, the normalized wave function for spin-2
glueballs is given by Mamo and Zahed (2020a)

$\displaystyle
J_{h}(m_{n},z)\equiv\psi_{n}(z)=c_{n}\,z^{4}L_{n}^{2}(2\tilde{\xi}_{T})\,,$

with $\tilde{\xi}_{T}=\kappa_{T}^{2}z^{2}$, and normalization

$\displaystyle
c_{n}=\Bigg{(}\frac{2^{4}\kappa_{T}^{6}\Gamma(n+1)}{\Gamma(n+3)}\Bigg{)}^{\frac{1}{2}}=\frac{4\kappa_{T}^{3}}{\sqrt{(n+2)(n+1)}}\,,$
(A.39)

which is determined from the normalization condition (for soft-wall model with
background dilaton $\phi=\kappa_{T}^{2}z^{2}$)

$\displaystyle\int
dz\,\sqrt{g}e^{-\phi}\,{\left|g^{xx}\right|}\,\psi_{n}(z)\psi_{m}(z)=\delta_{nm}\,.$

Therefore, defining

$\displaystyle F_{n}=\frac{1}{\sqrt{2}\kappa}\bigg{(}-\frac{1}{z^{\prime
3}}\partial_{z^{\prime}}\psi_{n}(z^{\prime})\bigg{)}_{z^{\prime}=\epsilon}=-\frac{4}{\sqrt{2}\kappa}c_{n}L_{n}^{2}(0)\,,$

with $\psi_{n}(z\rightarrow 0)\approx
c_{n}\,z^{4}L_{n}^{2}(0)=c_{n}\,z^{4}\binom{2+n}{n}$, we can re-write the
normalized wave function of glueballs (A.2) in terms of $F_{n}$ as

$\displaystyle\psi_{n}(z)=-\sqrt{2}\kappa\times
F_{n}\times\frac{1}{4}\times\frac{1}{\kappa_{T}^{4}}\times\frac{1}{L_{n}^{2}(0)}\times\tilde{\xi}^{2}L_{n}^{2}(2\tilde{\xi})\,,$
(A.42)

with $L_{n}^{2}(0)=\binom{2+n}{n}$. Also note that
$m_{n}^{2}=8\kappa_{T}^{2}(n+2)$,

$\displaystyle
F_{n}^{2}=\frac{16}{2\kappa^{2}}\times\frac{16\kappa_{T}^{6}}{(n+2)(n+1)}\times\left(L_{n}^{2}(0)\right)^{2}\,.$
(A.43)

For space-like momenta ($t=k^{2}=-K^{2}$), we also have the bulk-to-bulk
propagator near the boundary Mamo and Zahed (2020a)

$\displaystyle G(z\rightarrow
0,z^{\prime})\approx\frac{z^{4}}{4}\sum_{n}\frac{\sqrt{2}\kappa
F_{n}\psi_{n}(z^{\prime})}{K^{2}+m_{n}^{2}}=\frac{z^{4}}{4}\mathcal{H}(K,z^{\prime}),$
(A.44)

where, for the soft-wall model Mamo and Zahed (2020a)

$\displaystyle\mathcal{H}(K,z)$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{\sqrt{2}\kappa
F_{n}\psi_{n}(z^{\prime})}{K^{2}+m_{n}^{2}}\,,$ (A.45) $\displaystyle=$
$\displaystyle
4z^{4}\Gamma(\frac{a_{K}}{2}+2)U\Big{(}\frac{a_{K}}{2}+2,3;2\tilde{\xi}_{T}\Big{)}=\Gamma(\frac{a_{K}}{2}+2)U\Big{(}\frac{a_{K}}{2},-1;2\tilde{\xi}_{T}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{\Gamma(\frac{a_{K}}{2}+2)}{\Gamma(\frac{a_{K}}{2})}\int_{0}^{1}dx\,x^{\frac{a_{K}}{2}-1}(1-x){\rm
exp}\Big{(}-\frac{x}{1-x}(2\tilde{\xi}_{T})\Big{)}\,,$

with $\tilde{\xi}_{T}=\kappa_{T}^{2}z^{2}$, $a_{K}={K^{2}}/{4\kappa_{T}^{2}}$,
and we have used the transformation $U(m,n;y)=y^{1-n}U(1+m-n,2-n,y)$. (A.45)
satisfies the normalization condition ${\cal H}(0,z)={\cal H}(K,0)=1$.

The full bulk-to-bulk graviton propagator
$G_{\mu\nu\alpha\beta}(m_{n},k,z,z^{\prime})$ is given by Raju (2011); D’Hoker
et al. (1999)

$\displaystyle
G_{\mu\nu\alpha\beta}(m_{n},k,z,z^{\prime})=J_{h}(m_{n},z)\times\tilde{G}_{\mu\nu\alpha\beta}(m_{n},k)\times
J_{h}(m_{n},z^{\prime})\,,$

where the massive spin-2 boundary propagator
$\tilde{G}_{\mu\nu\alpha\beta}(m_{n},k)$ is given by

$\displaystyle\tilde{G}_{\mu\nu\alpha\beta}(m_{n},k)=P_{\mu\nu;\alpha\beta}(m_{n},k)\times\frac{-i}{k^{2}-m_{n}^{2}+i\epsilon}\,,$
(A.47)

with the massive spin-2 projection operator $P_{\mu\nu;\alpha\beta}(m_{n},k)$
defined as

$\displaystyle P_{\mu\nu;\alpha\beta}(m_{n},k)={1\over
2}\left(P_{\mu;\alpha}P_{\nu;\beta}+P_{\mu;\beta}P_{\nu;\alpha}-\frac{2}{3}P_{\mu;\nu}P_{\alpha;\beta}\right)\,,$
(A.48)

which is written in terms of the massive spin-1 projection operator

$\displaystyle
P_{\mu;\alpha}(m_{n},k)=-\eta_{\mu\alpha}+k_{\mu}k_{\alpha}/m_{n}^{2}\,.$
(A.49)

For $z\rightarrow 0$, $t=-K^{2}$, and focusing on the transverse part of the
boundary propagator, i.e.,

$\tilde{G}_{\mu\nu\alpha\beta}(m_{n},k)\approx\frac{1}{2}\eta_{\mu\alpha}\eta_{\nu\beta}\times\frac{-i}{k^{2}-m_{n}^{2}+i\epsilon}$

(as we did in Mamo and Zahed (2020a)), we can simplify (III.1) as

$\displaystyle i{\cal A}^{h}_{\gamma^{*}_{T/L}p\rightarrow
Vp}(s,t)\approx\frac{1}{g_{5}}\times(-i)\mathcal{V}^{\mu\nu}_{h\gamma^{*}_{T/L}V}(q,q^{\prime},k)\times\bigg{(}\frac{-i}{2}\eta_{\mu\alpha}\eta_{\nu\beta}\bigg{)}\times(-i)\mathcal{V}^{\alpha\beta}_{h\bar{\Psi}\Psi}(p_{1},p_{2},k)\,,$

with

$\displaystyle\mathcal{V}^{\mu\nu}_{h\gamma^{*}_{T/L}V}(q,q^{\prime},k_{)}=\sqrt{2\kappa^{2}}\times\frac{1}{2}\int
dz\sqrt{g}\,e^{-\phi}z^{4}K_{T/L}^{\mu\nu}(q,q^{\prime},\epsilon,\epsilon^{\prime},z)\times\frac{z^{4}}{4}\,,$
$\displaystyle\mathcal{V}^{\alpha\beta}_{h\bar{\Psi}\Psi}(p_{1},p_{2},k_{z})=-\sqrt{2\kappa^{2}}\times\frac{1}{2}\int
dz\,\sqrt{g}\,e^{-\phi}z\,\bar{\Psi}(p_{2},z)\gamma^{\mu}p^{\nu}\,\Psi(p_{1},z)\times\mathcal{H}(K,z)\,.$
(A.51)

However, if we instead use the full massive spin-2 boundary propagator
$\tilde{G}_{\mu\nu\alpha\beta}(m_{n},k)$ (A.47), we find

$\displaystyle{\cal A}^{h}_{\gamma_{T}^{*}p\rightarrow Vp}(s,t)$
$\displaystyle=$ $\displaystyle\frac{1}{g_{5}}\times
2\kappa^{2}\times\mathcal{V}_{h\gamma_{T}^{*}V}(Q,M_{V})\times\frac{1}{m_{N}}\times\bar{u}(p_{2})u(p_{1})$
$\displaystyle\times$
$\displaystyle\sum_{n=0}^{\infty}\,\left[q^{\mu}q^{\nu}\,P_{\mu\nu;\alpha\beta}(m_{n},k)\,p_{1}^{\alpha}p_{1}^{\beta}\right]\times\mathcal{V}_{h\bar{\Psi}\Psi}(K,m_{n})\,,$
$\displaystyle{\cal A}^{h}_{\gamma_{L}^{*}p\rightarrow Vp}(s,t)$
$\displaystyle=$ $\displaystyle\frac{1}{g_{5}}\times
2\kappa^{2}\times\frac{1}{QM_{V}}\times\mathcal{V}_{h\gamma_{L}^{*}V}(Q,M_{V})\times\frac{1}{m_{N}}\times\bar{u}(p_{2})u(p_{1})$
$\displaystyle\times$
$\displaystyle\sum_{n=0}^{\infty}\,\left[q^{\mu}q^{\nu}\,P_{\mu\nu;\alpha\beta}(m_{n},k)\,p_{1}^{\alpha}p_{1}^{\beta}\right]\times\mathcal{V}_{h\bar{\Psi}\Psi}(K,m_{n})\,,$

with

$\displaystyle\mathcal{V}_{h\gamma^{*}_{T}V}(Q,M_{V})=-\frac{1}{2}\int
dz\sqrt{g}\,e^{-\phi}z^{4}\mathcal{V}_{\gamma^{*}}(Q,z)\mathcal{V}_{V}(M_{V},z)\times\frac{z^{4}}{4}\,,$
(A.53) $\displaystyle\mathcal{V}_{h\gamma^{*}_{L}V}(Q,M_{V})=-\frac{1}{2}\int
dz\sqrt{g}\,e^{-\phi}z^{4}\partial_{z}\mathcal{V}_{\gamma^{*}}(Q,z)\partial_{z}\mathcal{V}_{V}(M_{V},z)\times\frac{z^{4}}{4}\,,$
(A.54) $\displaystyle\mathcal{V}_{h\bar{\Psi}\Psi}(K,m_{n})=-\frac{1}{2}\int
dz\,\sqrt{g}\,e^{-\phi}z\,\left(\psi_{R}^{2}(z)+\psi_{L}^{2}(z)\right)\times\frac{\sqrt{2}\kappa
F_{n}\psi_{n}(z)}{K^{2}+m_{n}^{2}}\,.$ (A.55)

Following Nishio and Watari (2014), we can evaluate

$\displaystyle
q^{\mu}q^{\nu}\,P_{\mu\nu;\alpha\beta}(m_{n},k)\,p_{1}^{\alpha}p_{1}^{\beta}=\left(p_{1}\cdot
q\right)^{2}\times\left(1-\frac{4}{3}\frac{p^{2}}{m_{n}^{2}}\times\eta^{2}\right)\,,$
(A.57)

using the general result (see Eq.279 in Nishio and Watari (2014))

$\displaystyle
q^{\mu_{1}}q^{\mu_{2}}...q^{\mu_{j}}\,P_{\mu_{1}\mu_{2}...\mu_{j};\nu_{1}\nu_{2}...\nu_{j}}(m_{n},k)\,p_{1}^{\nu_{1}}p_{1}^{\nu_{2}}...p_{1}^{\nu_{j}}=\left(p_{1}\cdot
q\right)^{j}\times\hat{d}_{j}(\eta,m_{n}^{2})$ (A.58)

where $\hat{d}_{j}(\eta,m_{n}^{2})$, for even $j=2,4,...$, is a polynomial of
skewness $\eta$ of degree $j$ which can be written explicitly in terms of the
hypergeometric function ${}_{2}F_{1}(a,b,c,d)$ as

$\displaystyle\hat{d}_{j}(\eta,m_{n}^{2})=\,_{2}F_{1}\left(-\frac{j}{2},\frac{1-j}{2};\frac{1}{2}-j;-\frac{4p_{1}^{2}}{m_{n}^{2}}\times\eta^{2}\right)\,.$
(A.59)

Note that we have also replaced $p^{2}\sim p_{1}^{2}$ by $-p_{1}^{2}$, and
$-\Delta^{2}$ by $m_{n}^{2}$ in Eq.279 of Nishio and Watari (2014), since we
are using the massive spin-1 projection operator
$P_{\mu;\alpha}(m_{n},k)=-\eta_{\mu\alpha}+k_{\mu}k_{\alpha}/m_{n}^{2}$
instead of the massless one
$P_{\mu;\alpha}(\Delta=k)=\eta_{\mu\alpha}-k_{\mu}k_{\alpha}/\Delta^{2}$ used
in Mamo and Zahed (2020a) for the conformal case with
$\eta_{\mu\alpha}=(-,+,+,+)$ signature. With this in mind and using (A.57) in
(A.2), we obtain (III.1).

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