11institutetext: G. Krein 22institutetext: Instituto de Física Teórica,
Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II,
01140-070 São Paulo, SP, Brazil.
33institutetext: T. C. Peixoto 44institutetext: Instituto de Física Teórica,
Universidade Estadual Paulista Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II,
01140-070 São Paulo, SP, Brazil.
and
Instituto Federal de Educação, Ciência e Tecnologia de Sergipe, Rodovia
Juscelino Kubitschek, s/n,
49680-000 Nossa Senhora da Glória, SE, Brazil

G. Krein [Corresponding author]55institutetext: 55email<EMAIL_ADDRESS>

# Femtoscopy of the origin of the nucleon mass

G. Krein  T. C. Peixoto

(Received: )

###### Abstract

We study the prospects of using femtoscopic low-momentum correlation
measurements at the Large Hadron Collider to access properties of the
$J/\psi$-nucleon interaction. The QCD multipole expansion in terms of the
$J/\psi$ chromopolarizability relates the forward scattering amplitude to a
key matrix element to the origin of the nucleon mass problem, the average
chromoelectric gluon distribution in the nucleon. We use information on the
$J/\psi$-nucleon interaction provided by lattice QCD simulations and
phenomenological models to compute $J/\psi$-nucleon correlation functions. The
computed correlation functions show clear sensitivity to the interaction, in
particular to the $J/\psi$ chromopolarizability.

###### Keywords:

Quantum chromodynamics Trace anomaly Proton mass Femtoscopy Heavy-ion
collisions

††journal: Few-Body Systems

##  1 Introduction

What is the origin of the mass of protons and neutrons (nucleons) and,
therefore, most of the universe’s visible matter? Computer simulations of
quantum chromodynamics (QCD) have given an answer to the question, namely: the
mass comes mostly from the gluons and the nearly massless quarks. Yet, we are
still unsatisfied and want more; we want to understand, to quote Wilczek [1]:
“How did it happen?”. With this mindset, an ever-growing effort is underway to
find that kind of answer, both theoretically [2, 3] and experimentally [4,
5, 6]. The present work adds to this effort. We study the prospects of using
femtoscopy in high-energy proton-proton and heavy-ion collisions for learning
about the origin of the nucleon mass.

Femtoscopy [7] is a technique that makes it possible to obtain spatio-temporal
information on particle production sources at the femtometer scale. Two-hadron
momentum correlation functions carry such information [8, 9]. These
correlation functions, remarkably, also carry information on low-energy
hadron-hadron forces as final-state effects [10, 11]. Relevant to the origin
of the mass problem is the correlation function of a heavy quarkonium (such as
$J/\psi$, $\eta_{c}$, $\Upsilon$,$\eta_{b}$) and a nucleon, for it gives
direct access to the quarkonium-nucleon forward scattering amplitude. The QCD
multipole expansion relates this amplitude to a key matrix element to the mass
problem: the average chromoelectric gluon distribution in the nucleon [12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. It is key to the problem
because it relates to the trace of the QCD energy-momentum tensor in the
nucleon, which defines the nucleon mass [23, 24].

The quarkonium-nucleon scattering amplitude is also accessible with $J/\psi$
and $\Upsilon$ electro- and photo-production experiments [5]. However, the
kinematics of the production process forbids direct access to the forward
amplitude. In femtoscopy there are no such kinematics constraints. In
addition, the two-particle correlation functions are measurable, in principle,
down to zero relative momentum. Exemplar of femtoscopy’s capabilities are the
hyperon-proton and hyperon-hyperon correlation measurements in heavy-ion
($AA$), proton-ion ($pA$) and proton-proton ($pp$) collisions, ongoing for the
last 15 years [25, 26, 27, 28, 29, 30, 31, 32, 33]. Closer to our
interest in this paper is the recent feasibility study [34] of the
$\phi-$proton system using data from $pp$ collisions collected by LHC’s ALICE
detector. In this and the envisioned quarkonium-nucleon case, the theoretical
interpretation of the measurements profits from the absence of the Coulomb
interaction and quantum-statistics, features that allow us to link a
correlation signal to a strong-interaction effect. Experimentally, however,
the situation is not as clear-cut as in theory. Complications arise due to
non-femtoscopic correlations, momentum resolution, and other experimental
issues, which are always present in an experiment. Such effects need to be
accounted for to extract the genuine strong-interaction correlation. In
addition, knowledge of the particle source emission form and size are
important issues for the theoretical interpretation of the data.
Notwithstanding experimental issues, the subject’s relevance and successes
with the hyperon-proton system and preliminary results on the $\phi$-proton
system motivate our prospective study. We hope our results will motivate
experimental studies as well.

We focus on the $J/\psi$-nucleon ($J/\psi\,N$) system, having in mind
experiments at the LHC. The motivation for preferring $J/\psi$ over the other
heavy quarkonia is twofold. First, $J/\psi$ production yields and weak decay
rates are relatively high—the branching ratios of decays to $e^{+}e^{-}$ and
$\mu^{+}\mu^{-}$ is of the order of 6% each. Second, there is some theoretical
knowledge on the $J/\psi\,N$ system from lattice QCD studies [35, 36, 37,
38, 39, 40, 41, 42, 43]. These studies revealed an attractive and not
very strong interaction. One lattice study [44] found a bound $\eta_{c}\,N$
state with $20$ MeV binding, a binding energy much larger than
phenomenological expectations [22, 45]. For our femtoscopic study, we use the
lattice results of Refs. [35, 36, 37], which provide values for the $S-$wave
scattering length $a_{0}$ and effective range $r_{0}$ parameters, and of Ref.
[38] which gives in addition an $S-$wave $J/\psi\,N$ potential extracted with
the HALQCD method. These lattice results were obtained either with quenched
gluon configurations [35, 37] or large pion masses [36, 38] and, therefore,
require extrapolation to the physical mass. A recent study [46] performed such
an extrapolation with an effective field theory (EFT) specially set up for
studying the low-energy the quarkonium-nucleon interaction. The EFT, dubbed
QNEFT, obtained expressions for $a_{0}$ and $r_{0}$ at leading order (LO) and
next-to-leading order (NLO) in the pion mass. The extrapolated values preserve
the overall qualitative picture of a weakly attractive interaction. The QNEFT
also predicts a model-independent van der Waals type of potential of range
$1/2m_{\pi}\simeq 0.7$ fm, with a strength controlled by the $J/\psi$
chromopolarizability. We note that the $J/\psi\,N$ system can have spin 1/2 or
spin 3/2. The early latttice results of Refs. [35, 36, 37, 38] found no
significant hyperfine splitting in $a_{0}$ and $r_{0}$. Therefore, the QNEFT
extrapolations should be considered as spin-1/2 and spin-3/2 averages.
Interestingly, a recent lattice study [41] within the HALQCD method, but with
larger lattice volumes and a relativistic heavy quark action for the charm
quark, found a somewhat sizable hyperfine splitting, $a^{1/2}_{0}\simeq
1.7\,a^{3/2}_{0}$.

Further knowledge on the $J/\psi\,N$ interaction comes from phenomenological
models. We consider two models, those of Refs. [47, 48]. Both models address
the interaction of a charmonium with a nucleon within the hadro-charmonium
picture [49]. In this picture, a charmonium interacts as a compact object
within the volume of a light hadron. The model of Ref. [47] treats the nucleon
as a spherical finite well. This is a very simple but insightful model, as it
can be solved analytically. The model of Ref. [48] uses the chiral soliton
quark model ($\chi$QSM) of Ref. [50]. The model uses the QCD trace anomaly
[51, 52, 53, 54, 55] to obtain an effective $J/\psi\,N$ potential in terms
of the gluon energy-momentum density inside a nucleon calculated with the
$\chi$QSM.

We use the knowledge on the $J/\psi\,N$ system from lattice QCD simulations,
extrapolated to the physical pion mass with the QNEFT [46], and from the
phenomenological models of Refs. [47, 48] to make predictions for the
$J/\psi\,N$ correlation function. In the next section we review the basics of
femtoscopy. We discuss limits for which the correlation function can be linked
directly to the scattering amplitude and how this allows us to link the
correlation function to the average chromoelectric gluon distribution in the
nucleon. In Section 3 we present predictions for the $J/\psi\,N$ correlation
function. We study the dependence of the correlation function on source
emission size and on parameters of the interaction. Section 4 presents a
summary of our work.

##  2 Correlation function and scattering amplitude

The observable of interest in femtoscopy is a two-hadron correlation function
$C({\bm{p}}_{1},{\bm{p}}_{2})$ of measured hadron momenta ${\bm{p}}_{1}$ and
${\bm{p}}_{2}$ [8, 9]. The extraction of the experimental correlation
function involves computing the ratio of two yields, $A(k)/B(k)$, where
$k=|{\bm{k}}|$ with ${\bm{k}}={{\bm{p}}}_{1}=-{{\bm{p}}}_{2}$, the relative
momentum in the center of mass of the pair111The total momentum is
${\bm{P}}={\bm{p}}_{1}+{\bm{p}}_{2}$ and the relative momentum is
${\bm{k}}=(m_{2}{\bm{p}}_{1}-m_{1}{\bm{p}}_{2})/(m_{1}+m_{2})$, where
$m_{1},m_{2}$ are the particles’ masses. In the center of mass frame,
${\bm{P}}=0$ and ${\bm{p}}_{1}=-{\bm{p}}_{2}$; hence, the relative momentum is
${\bm{k}}={\bm{p}}_{1}=-{\bm{p}}_{2}$.. $A(k)$ is the coincidence yield (or
signal distribution), formed by pairs with a given $k$ produced in the same
collision event, and $B(k)$ is the uncorrelated yield (or background
distribution), formed by pairs with the same $k$ but collected from different
collision events. Corrections for non-femtoscopic correlations in $A(k)$ not
accounted for in $B(k)$, and other experimental effects, are taken into
account by a multiplicative factor $\xi(k)$, so that $C(k)=\xi(k)\times
A(k)/B(k)$. The theoretical interpretation of the experimental correlation
function is usually based on the Koonin-Pratt formula [10, 56]:

$C(k)=\xi(k)\,\frac{A(k)}{B(k)}=\int
d^{3}\\!r\;S_{12}({\bm{r}})\,|\psi({\bm{k}},{\bm{r}})|^{2}\ .$ (1)

Here $\psi({\bm{k}},{\bm{r}})$ is the relative wave function of the pair and
$S_{12}({\bm{r}})$ a static emission source, a pair’s relative distance
distribution in the pair frame. Refs. [8, 9, 57, 58] address in depth the
validity of the assumptions and approximations behind Eq. (1).

The current knowledge of the quarkonium-nucleon interaction lets us assume
that it affects only the wave function’s $S-$wave component. Therefore,
separating from $\psi({\bm{k}},{\bm{r}})$ the $l=0$ component,
$\psi_{0}(k,r)$, which contains the effects of the strong interaction, we can
write $\psi({\bm{k}},{\bm{r}})$ as

$\psi({\bm{k}},{\bm{r}})=e^{i{\bm{k}}\cdot{\bm{r}}}+\psi_{0}(k,r)-j_{0}(kr),$
(2)

where $j_{0}(kr)$ is the $l=0$ spherical Bessel function, the $S-$wave
component of the non-interacting wave function. Taking a one-parameter
Gaussian form for the emission source, $S_{12}(r)=(4\pi
R^{2})^{-3/2}\exp(-r^{2}/4R^{2})$, the Koonin-Pratt formula can be written as

$C(k)=1+\frac{4\pi}{(4\pi
R^{2})^{3/2}}\int^{\infty}_{0}dr\,r^{2}\,e^{-r^{2}/4R^{2}}\left[|\psi_{0}(k,r)|^{2}-|j_{0}(kr)|^{2}\right].$
(3)

The Gaussian form is a common choice as it simplifies the analysis [59, 60,
61, 62], but it represents an experimental issue, as we discuss in the next
section.

Two length scales in $C(k)$ are important for extracting properties of the
interaction: the emission source radius $R$ and the two-particle interaction
effective range. When these lengths are of comparable size, most of the
emitted particles are under the influence of the interaction. Therefore, one
needs the pair wave function $\psi_{0}(k,r)$ in the entire range $0\leq
r\leq\infty$ of integration in Eq. (3). One can use either the Schrödinger
equation or the Lippmann-Schwinger equation to obtain $\psi_{0}(k,r)$; the
latter is well suited for treating nonlocal potentials and coupled channels
[61, 62]. We deal with local $J/\psi\,N$ potentials in this paper and use the
Schrödinger equation to obtain $\psi_{0}(k,r)$.

On the other hand, for $R$ much larger than the effective range of the
interaction, most of the emitted pairs are not under the influence of the
interaction. Then, one can replace $\psi_{0}(k,r)$ with its asymptotic form

$\psi^{asy}_{0}(k,r)=\frac{\sin(kr+\delta_{0})}{kr}=e^{-i\delta_{0}}\left[j_{0}(kr)+f_{0}(k)\frac{e^{ikr}}{r}\right]\hskip
14.22636pt\text{with}\hskip
14.22636ptf_{0}(k)=\frac{e^{i\delta_{0}}\sin\delta_{0}}{k},$ (4)

where $f_{0}(k)$ is the scattering amplitude and $\delta_{0}$ the phase shift.
However, use of $\psi^{asy}_{0}(k,r)$ in place of $\psi_{0}(k,r)$ in Eq. (3)
for all values of $r$ incurs in error for pairs emitted from within the range
of the interaction. One way to account for this error uses [11] effective
range theory [63] to evaluate the correction $|\psi_{0}(k,r)-\psi^{\rm
asy}(k,r)|^{2}$ for $r\simeq 0$. Using $\psi^{asy}_{0}(k,r)$ in place of
$\psi_{0}(k,r)$ in Eq. (3) and including the short-range correction, one
obtains for $C(k)$ [11]:

$\displaystyle C(k)$ $\displaystyle=$ $\displaystyle
1+\frac{|f_{0}(k)|^{2}}{2R^{2}}\left(1-\frac{r_{0}}{2\sqrt{\pi}R}\right)+\frac{2{\rm
Re}f_{0}(k)}{\sqrt{\pi}R}F_{1}(2kR)-\frac{{\rm Im}f_{0}(k)}{R}F_{2}(2kR),$ (5)

where $r_{0}$ the effective range parameter that appears in the effective
range expansion (ERE) of $f_{0}(k)$:

$f_{0}(k)=\frac{1}{k\cot\delta_{0}-ik}\overset{k\approx
0}{\longrightarrow}\frac{1}{-\frac{1}{a_{0}}+\frac{1}{2}r_{0}\,k^{2}-ik},$ (6)

and $a_{0}$ is the scattering length, and

$F_{1}(x)=\frac{1}{x}\int^{x}_{0}dt\,e^{\,t-x},\hskip
28.45274ptF_{2}(x)=\frac{1}{x}\left(1-e^{-x^{2}}\right).$ (7)

The short-range correction is the term ${r_{0}}/{2\sqrt{\pi}R}$ in Eq. (5);
taking $r_{0}=0$ in that equation amounts to replacing $\psi_{0}(k,r)$ by
$\psi^{asy}_{0}(k,r)$ everywhere in Eq. (3). Of course, to make sense, the
correction must be small. Eq. (5) is known as the Lednicky-Lyuboshits (LL)
formula. For small values of $k$, one can use the ERE for $f_{0}(k)$ and then
Eq. (5) becomes universal, in the sense that it depends only on $a_{0}$,
$r_{0}$, and $R$; no further knowledge is required to compute the correlation
function.

If the LL formula is applicable, $C(k)$ at small $k$ gives direct access to
the matrix element of the average chromoelectric gluon distribution in the
nucleon, $\langle
N|(g{\bm{E}}^{a})\cdot(g{\bm{E}}^{a})|N\rangle\equiv\langle(g{\bm{E}})^{2}\rangle_{N}$,
where ${\bm{E}}^{a},a=1,\cdots,8,$ is the chromoelectric gluon field, $g$ the
strong coupling constant, and $|N\rangle$ the nucleon state. The average
$\langle(g{\bm{E}})^{2}\rangle_{N}$ appears in the low energy $J/\psi\,N$
forward scattering amplitude within the QCD multipole expansion framework, in
that the heavy quarkonium behaves like a small color dipole interacting with
soft gluon fields of the nucleon [12, 13, 16, 20, 21, 22]. For an
$S-$wave dominating interaction, as it seems to be the case for $J/\psi\,N$,
the forward amplitude at small $k$ is real and given by $f_{\rm
forw}(k)=f_{0}(k)\simeq-a_{0}$ [14, 20, 21], with

$a_{0}=\frac{\mu}{4\pi}\,\alpha_{J/\psi}\,\langle(g{\bm{E}})^{2}\rangle_{N},$
(8)

where $\mu$ is the $J/\psi\,N$ reduced mass, $\alpha_{J/\psi}$ the $J/\psi$
chromopolarizability, with $\langle(g{\bm{E}})^{2}\rangle_{N}$ evaluated with
the nucleon at rest. Therefore, if the value of $\alpha_{J/\psi}$ is known,
one can obtain $\langle(g{\bm{E}})^{2}\rangle_{N}$ from the $a_{0}$ extracted
from the measured $C(k)$ via the LL formula. Unfortunately, at present
$\alpha_{J/\psi}$ is not well constrained by data; as such, while the
situation persists, there will be an associated uncertainty in the extraction
of $\langle(g{\bm{E}})^{2}\rangle_{N}$. We note that this difficulty is not
restricted to femtoscopy; any experiment attempting to extract
$\langle(g{\bm{E}})^{2}\rangle_{N}$ from the scattering length via Eq. (8), as
for example $J/\psi$ electro- and photo-production experiments [5], share with
femtoscopy this difficulty.

If the LL formula is not applicable, the link between the measured $C(k)$ and
$\langle(g{\bm{E}})^{2}\rangle_{N}$ is more indirect, as one needs a model for
the interaction to extract the scattering length from the experimental $C(k)$.
The $\Sigma^{+}\Sigma^{+}$ interaction, discussed in Ref. [61], is an example
indicating failure of the LL formula (cf. Fig. [5] of that reference), the
failure being associated with the large value of the effective range parameter
$r_{0}$. In the next section, we show results for $C(k)$ computed with two
models for the $J/\psi\,N$ interaction for which, depending on the value of
$\alpha_{J/\psi}$, $r_{0}$ is large.

To conclude this section, we note that the connection between
$\langle(g{\bm{E}})^{2}\rangle_{N}$ and the trace of the QCD energy-momentum
tensor and the nucleon mass comes through the inequality [21]:

$\displaystyle\langle
N|\left[\left(g{\bm{E}}^{a}\right)^{2}-\left(g{{\bm{B}}^{a}}\right)^{2}\right]|N\rangle=-\frac{1}{2}\langle
N|g^{2}G^{a}_{\mu\nu}G^{a\mu\nu}|N\rangle=\frac{16\pi^{2}}{9}m_{N}\leq\langle(g{\bm{E}})^{2}\rangle,$
(9)

where ${{\bm{B}}^{a}}$ is the chromomagnetic field. Here we used the trace-
anomaly relationship [51, 52, 53, 54, 55]

$T^{\mu}_{\mu}(x)=-\dfrac{9}{32\pi^{2}}\,g^{2}G^{a}_{\mu\nu}(x)G^{a\mu\nu}(x),$
(10)

valid in the chiral limit. The last inequality in Eq. (9) follows from the
fact that $\langle N|\left(g{{\bm{B}}^{a}}\right)^{2}|N\rangle\geq 0$. The
normalization of the nucleon state we use is such that the expectation value
of $T^{00}$ is the energy [24, 22]. Away from the chiral limit, Eq. (10)
contains the contribution from the quark-mass term of the QCD Lagrangian,
whose contribution to $m_{N}$ seems to be small [22].

##  3 Numerical results and discussions

Here we present predictions for the $J/\psi\,N$ correlation function. We use
information on the $J/\psi\,N$ interaction from lattice QCD simulations,
extrapolated to the physical pion mass with QNEFT [46], and from the
phenomenological models of Refs. [47, 48]. Both QNEFT and the
phenomenological models consider spin-1/2 and spin-3/2 degeneracy. The radius
$R$ of the assumed one-parameter Gaussian form for the emission source is
treated as a free parameter; we present results for $R=1$ fm and $R=3$ fm. The
lower value of $R$ is thought appropriate for $pp$ collisions, and the highest
for $pA$ and $AA$ collisions. These are smaller source sizes as those employed
in hyperon-nucleon studies [61, 34, 62]. The use of smaller source sizes is
motivated by evidence that emission sources scale with the inverse of the
emitted particles masses [9]. We recall that the source in femtoscopy refers
to the “region of homogeneity”, the region from which particle pairs with a
certain momentum are most likely emitted, which is significantly smaller than
the size of the entire source emitting particles [8, 9].

We begin presenting results using the QNEFT-extrapolated lattice information
[46] for the $J/\psi\,N$ interaction. The interaction contains contact terms
and a long-range, model independent van der Waals potential:

$V_{\rm
vdW}(r)=\frac{3g^{2}_{A}}{128\pi^{2}F^{2}}\left\\{c_{di}\left[6+m_{\pi}r(2+m_{\pi}r)(6+m_{\pi}r(2+m_{\pi}r))\right]+c_{m}m^{2}_{\pi}r^{2}(1+m_{\pi}r)^{2}\right\\}\,\frac{e^{-2m_{\pi}r}}{r^{6}},$
(11)

where $g_{A}=1.27$ is the nucleon axial charge, $F=93$ MeV the pion decay
constant, and $c_{di}$ and $c_{m}$ are low-energy constants; $c_{di}$ reflects
the hadronization into two pions of the soft-gluon coupling $(E^{a})^{2}$
mediating the interaction between the nucleon and $J/\psi$, and $c_{m}$
relates to the light-quark masses. The value of $c_{di}$ can be determined by
using the QCD trace anomaly [64]. Explicit expressions for $c_{di}$ and
$c_{m}$ are given in Eq. (5) of Ref. [46]; they also depend on the $J/\psi$
chromopolarizability, $\alpha_{J/\psi}$ (denoted by $\beta$ in that
reference), the only parameter in Eq. (11) not well constrained by experiment.
Ref. [46] extracted the value $\alpha_{J/\psi}=0.24~{}{\rm GeV}^{-3}$ by
fitting $V_{\rm vdW}(r)$ to the $J/\psi\,N$ HALQCD potential [38]. Given the
uncertainties in the lattice values for $a_{0}$ and $r_{0}$ [35, 36, 37],
the corresponding QNEFT-extrapolated values are $-0.71~{}{\rm fm}\leq
a_{0}\leq-0.35~{}{\rm fm}$, and $1.29~{}{\rm fm}\leq r_{0}\leq 1.35~{}{\rm
fm}$. We note that a much smaller value for the scattering length,
$a_{0}=-0.05$ fm, was obtained in Ref. [46] when identifying the LO term of
the QNEFT scattering amplitude with the one in Ref. [20], which uses the
multipole expansion for quarkonium interaction. A low value for $a_{0}$ was
also obtained in an earlier study [65] using QCD sum rules, $a_{0}=-0.10$ fm.
Smaller values were also extracted from old [66] and recent [67, 68]
photoproduction data—it should be noted that the extraction of the scattering
length in such experiments rely on assumptions and extrapolations to zero
momentum transfer, as kinematics forbids direct access to the forward
amplitude.

The values of range of the van der Waals force, $1/2m_{\pi}$, strength at
$r=1/2m_{\pi}$, 3 MeV, and the values of $a_{0}$ and $r_{0}$, justify use of
Eq. (5) to compute $C(k)$ for $R=1$ and $R=3$. Since the value of the
effective range $r_{0}$ does not vary much within the uncertainties, we fix it
to $r_{0}=1.3$ fm and present results for the scattering length varying in the
range $-0.7~{}{\rm fm}\leq a_{0}\leq-0.05~{}{\rm fm}$. Fig. 1 displays results
for $C(k)$ for two values of the source radius, $R=1$ fm and $R=3$ fm.

Figure 1: $J/\psi\,N$ correlation function computed with Eq. (5) for two
values of the source radius $R$. The $a_{0}$ and $r_{0}$ values are from
lattice QCD simulations [35, 36, 37, 38] extrapolated to the physical pion
mass with QNEFT [46].

Figure 1 reveals the expected trend about correlation strength and source
size, in that stronger correlations happen for smaller sources. From this
perspective alone, $pp$ collisions are preferred over collisions with heavy
ions, although production yields are higher in the latter. Regarding the
predictions of the correlation strengths for the different values of the
scattering length $a_{0}$, they are small for $R=3$ fm for the smallest values
of $a_{0}$; namely $C(0)\simeq 1.04$ for $a_{0}=-0.10$ fm, $C(0)\simeq 1.02$
for $a_{0}=-0.05$ fm. On the other hand, for $R=1$, the correlation strengths
are comparable to those extracted for the $\phi\,N$ system in Ref. [34].

Next, we present results for $C(k)$ computed with scattering wave functions
obtained with the Schrödinger equation for the potentials of Refs. [47] and
[48]. The first uses a spherical finite well of radius $R_{N}$, given in Eq.
(4) of Ref. [47]:

$V(r)=\left\\{\begin{array}[]{ccc}-\frac{2\pi}{3}\left(\frac{\alpha_{J/\psi}}{R^{3}_{N}}\right)m_{N}&\mbox{for}&r<R_{N}\\\\[8.5359pt]
0&\mbox{for}&r>R_{N}\end{array}\right.\mbox{ }.$ (12)

The second potential is Eq. (11) of Ref. [48]:

$\displaystyle
V(r)=-\alpha_{J/\psi}\frac{4\pi^{2}}{b}\left(\frac{g^{2}}{g^{2}_{s}}\right)\left[\nu\rho_{E}(r)-3p(r)\right],$
(13)

with $b=(11N_{c}-3N_{f})/3=27/3$ and $g^{2}/g^{2}_{s}=1$ (adequate values for
$J/\psi$) and $\nu=1.5$. The energy density $\rho_{E}(r)$ and pressure $p(r)$,
given in terms of matrix elements of $T^{00}(x)$ and $T^{\mu}_{\mu}(x)$ in the
nucleon state, are computed with the $\chi$QSM in Ref. [50]—we scanned their
profiles from Fig. 1 of this reference. In both potentials, only
$\alpha_{J/\psi}$ is not well constrained by theory and experiment. Therefore,
we present results for $C(k)$ for four values of $\alpha_{J/\psi}$, covering a
wide range of values commonly used in the literature, namely:
$\alpha_{J/\psi}=2~{}{\rm GeV}^{-3}$ [20, 21], $1.60~{}{\rm GeV}^{-3}$ [69],
$0.54~{}{\rm GeV}^{-3}$ [65], $0.24~{}{\rm GeV}^{-3}$ [46]. We set the well
radius to $R_{N}=1$ fm, as the results for $C(k)$ with $R=3$ fm follow the
trend shown in Fig. 1.

Table 1 lists the scattering length and effective range parameters extracted
from the wave functions. One sees that the $a_{0}$ values corresponding to the
different $\alpha_{J/\psi}$, for both potentials, are in close correspondence
to those used in Fig. 1. Table 1 also reveals the well known fact that when
$|a_{0}|$ is much smaller than the actual range of the potential $R_{\rm
range}$, $r_{0}$ can be very different from $R_{\rm range}$. In this case, use
of the ERE for $f_{0}$ in the Lednicky model becomes problematic [61].

Table 1: Scattering length and effective range parameters (in fm) corresponding to the finite well with $R_{N}=1$ fm and $\chi$SQM potentials, Eqs. (12) and (13), for several values of $\alpha_{J/\psi}$ (in ${\rm GeV}^{-3}$).  | Finite well |  $\chi$SQM
---|---|---
$\alpha_{J/\psi}$ | $a_{0}$ | $r_{0}$ | $a_{0}$ | $r_{0}$
$2.00$ | $-0.68$ | $1.59$ | $-0.42$ | $1.86$
$1.60$ | $-0.47$ | $1.86$ | $-0.30$ | $2.25$
$0.54$ | $-0.12$ | $4.50$ | $-0.08$ | $6.00$
$0.24$ | $-0.05$ | $9.46$ | $-0.03$ | $13.05$

Figure 2: $J/\psi\,N$ correlation function calculated with pair wave functions
$\psi_{0}(k,r)$ computed with the Schrödinger equation for the entire range
$0\leq r\leq\infty$ of the integrand of Eq. (3). The $\psi_{0}(k,r)$
correspond to the potentials in Eqs. (12) and (13).

Figure 2 displays the results for $C(k)$ using the wave functions
corresponding to the two potentials. The results for the finite well are very
similar to those in Fig. 1. This is not surprising given the fact already
mentioned regarding the extracted values of $a_{0}$. The $C(k)$ values for the
$\chi$SQM are a little smaller than those for the finite well because the
$a_{0}$ values are a little smaller for the former.

Given the results in Figs. 1 and 2, one sees that if the $J/\psi\,N$
interaction turns out to be very weak, to extract information from femtoscopic
measurements will be very challenging. As already mentioned, there will always
be experimental errors related to source size, momentum resolution, etc that
will not allow to resolve weak correlations. In concrete terms, there seems to
be room for optimism if the value of the chromopolarizability
$\alpha_{J/\psi}$ turns out to be above $0.24~{}{\rm GeV}^{-3}$.

To conclude, we note that none of these potentials form bound states. A
potential well as in Eq. (12) with $R_{N}=1$ holds an $S-$wave bound state
when $\alpha_{J/\psi}>4.4~{}{\rm GeV}^{-3}$, a value twice as large as the
largest value commonly practiced in the literature. The situation is different
in the case of a nucleus. Even when the $J/\psi\,N$ interaction is too weak to
bind the two hadrons, nuclear many-body effects play an important role. Of
particular importance are the effects of the nuclear mean fields on sub-
threshold $DD-$states [22, 70, 71, 72].

##  4 Conclusions and perspectives

We studied the prospects of using femtoscopy in high-energy proton-proton and
heavy-ion collisions for learning about the low-momentum $J/\psi$-nucleon
interaction. Femtoscopic correlation measurements offer the opportunity to
access information on low-energy hadron-hadron forces inaccessible by other
means. Within the QCD multipole expansion framework, the forward
$J/\psi$-nucleon scattering amplitude is given in terms of the $J/\psi$
chromopolarizability and a key matrix element to the origin of mass problem,
the average chromoelectric gluon distribution in the nucleon, which in turn
relates to the nucleon mass via the QCD trace anomaly.

Although we focused our study on the $J/\psi$-nucleon system, femtoscopic
measurements can also be performed for other heavy quarkonia. Our choice of
the $J/\psi$ was motivated by two main facts: the relatively high $J/\psi$
production yields and weak decay rates; and the theoretical information
available from lattice QCD simulations and phenomenological models. We made
use of this information to compute $J/\psi\,N$ correlation functions. The
available information points towards a relatively short-ranged, weakly
attractive interaction. These features of the interaction, together with the
QCD multipole expansion framework, lead to a direct relationship between the
correlation function at small momenta and the average chromoelectric gluon
distribution in the nucleon when using the Lednicky-Lyuboshits (LL) model.
Away from the LL model, the link between the chromoelectric gluon distribution
in the nucleon and the femtoscopic momentum correlation function is less
direct. Although one would still have access to information on the
interaction, e.g. on scattering parameters, the theoretical interpretation of
the data would be more subtle. In any case, the strength of the correlation
depends on the $J/\psi$ chromoelectric polarizabilty, $\alpha_{J/\psi}$, a
$J/\psi$ property. This quantity, however, is poorly constrained by
experiment. If its value turns out to be $\alpha_{J/\psi}>0.24~{}{\rm
GeV}^{-3}$, our model calculations indicate a good likelihood for a
femtoscopic extraction of the average chromoelectric gluon distribution in the
nucleon; or else, one will have to find other ways to get it.

We have not addressed experimental issues that can impact the extraction of a
low-momentum $J/\psi$-nucleon correlation function. As already mentioned,
nontrivial issues include source form and size, momentum resolution, and non-
femtoscopic correlations. Notwithstanding these issues, we hope the positive
prospects of our theoretical study motivate an experimental study as well.

###### Acknowledgements.

Enlightening and useful discussions with Johann Haidenbauer are gratefully
acknowledged. G.K was partially supported by: Conselho Nacional de
Desenvolvimento Científico e Tecnológico - CNPq, Grants No. Grant. nos.
309262/2019-4, 464898/2014-5 (INCT Física Nuclear e Aplicações), Fundação de
Amparo à Pesquisa do Estado de São Paulo - FAPESP, Grant No. 2013/01907-0.
T.C.P was supported by a scholarship from Conselho Nacional de Desenvolvimento
Científico e Tecnológico - CNPq.

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