# Helium-4 gravitational form factors:
exchange currents

Fangcheng He<EMAIL_ADDRESS>Center for Nuclear Theory,
Department of Physics and Astronomy, Stony Brook University, Stony Brook, New
York 11794‚Äì3800, USA  Ismail Zahed<EMAIL_ADDRESS>Center for
Nuclear Theory, Department of Physics and Astronomy, Stony Brook University,
Stony Brook, New York 11794‚Äì3800, USA

###### Abstract

We evaluate the leading exchange corrections to the Helium-4 gravitational
form factors (GFFs) upto momenta of the order of the nucleon mass. We use both
the K-harmonic method with simple pair nucleon potential, and a Jastrow trial
function using the Argonne $v_{14}$ potential, to evaluate the Helium-4 GFFs.
The exchange current contributions include the pair interaction, plus the
seagull and the pion exchange interactions, modulo the recoil corrections. To
estimate the off-shellness of the pion nucleon coupling in this momenta range,
we discuss the results using either the pseudo-scalar (PS) or pseudo-vector
(PV) pion-nucleon couplings. When the PV coupling is used, the pair diagram
contribution is higher order in the non-relativistic expansion. The results
for the Helium-4 A-GFF are comparable to those given by the impulse
approximation, especially for the PS coupling using both the K-Harmonic method
and variational method. The exchange current contributions with the PS
coupling for the charge form factor of Helium-4, yield better agreement with
the existing data over a broad range of momenta, especially when the Argonne
$v_{14}$ potential including the D-wave admixture is used.

##  I Introduction

The gravitational form factors (GFFs) of hadrons carry important information
on their mass and spin structure. Recent lattice QCD simulations of the pion
and nucleon GFFs have started to show how the quarks and gluons compose these
form factors Hackett  _et al._ (2023a, b); Wang  _et al._ (2024). This has led
to a better understanding of the internal properties of these hadrons Polyakov
and Schweitzer (2018); Burkert  _et al._ (2023), beyond their standard
electromagnetic properties Punjabi  _et al._ (2015).

Recent measurements of threshold photoproduction of charmonium off nucleons by
two collaborations at JLab Ali  _et al._ (2019); Duran  _et al._ (2023), have
given a first empirical estimate of nucleon gluonic GFFs, with the help of
model analyses Mamo and Zahed (2020); Guo  _et al._ (2021). These measurements
complement the quark GFFs extracted from deeply virtual Compton scattering
also at Jlab Jo  _et al._ (2015). The combined empirical and numerical
estimates of the nucleon GFFs, allow for a better understanding of the mass
and stress distribution inside the nucleon Abidin and Carlson (2009); Belitsky
and Ji (2002); Polyakov and Sun (2019); Won  _et al._ (2024); Hatta (2024);
Lorcé (2024). In particular, the mass radius of the nucleon was found to be
smaller than the electromagnetic radius, as suggested initially in dual
gravity Abidin and Carlson (2009); Mamo and Zahed (2022). Similar
understanding is being also reached for the pion Broniowski and Arriola
(2008); Frederico  _et al._ (2009); Fanelli  _et al._ (2016); Freese and Cloët
(2019); Krutov and Troitsky (2021); Raya  _et al._ (2022); Xu  _et al._
(2024); Li and Vary (2024); Broniowski and Ruiz Arriola (2024); Liu  _et al._
(2024a, b).

Nuclei are composed of nucleons bound by strong interactions. Most of our
understanding of nuclei follows from decades of electromagnetic scattering
Sick (2008), showing that the electric and magnetic form factors receive
sizable contributions from mesonic exchange currents in a wide range of
momenta Riska (1989). Furthermore, electron and muon scattering on nuclei have
also shown that the quark sub-constituents of the nucleon in nuclei get
modified through shadowing (large Bjorken-x) and anti-shadowing (small
Bjorken-x). However, it is less clear how the gluon sub-constituents are being
affected in nuclei. Coherent photoproduction of phi-mesons off the deuteron by
the CLAS collaboration Mibe  _et al._ (2007) and LEPS collaborations Chang
_et al._ (2008), have not added more insights beyond those expected from two
loosely bound nucleons.

Helium-4 is the lightest nucleus with strongly bound nucleons. It is a spin-
isospin singlet with a well measured electromagnetic form factor, mass and
charge radius. Recently we have shown that the Helium-4 GFFs in the impulse
approximation, deviate from those of independent nucleons in a range of
momenta within the nucleon mass He and Zahed (2024a). Similar observations
were made using the Skyrme model Garcia Martin-Caro  _et al._ (2023). However,
it is known that meson exchange-current contributions to the electromagnetic
form factors are important Hockert  _et al._ (1973); Chemtob  _et al._ (1974);
Kloet and Tjon (1974); Jackson  _et al._ (1975). The purpose of this work is
to address these exchanges for the Helium-4 GFFs. For completeness, we note
the light cone analysis of the deuteron GFFs in Freese and Cosyn (2022).

This paper is a follow up on our recent analysis of the EMT of light nuclei in
the impulse approximation He and Zahed (2024a, b). In section II we briefly
detail the ground state wavefunctions for the description of Helium-4 using
the K-harmonic method with no D-wave admixture, and the variational method
using the Argonne $v14$ potential with D-wave admixture. In section III we
describe the exchange current contributions to the GFFs of Helium-4, and their
generic matrix elements with both wavefunctions. In section IV we explicitly
evaluate all exchange matrix elements for Helium-4 using the wavefunction from
the K-harmonic method. Those from the variational method are multi-dimensional
and are evaluated numerically using Monte Carlo method. The numerical results
are discussed in section V. Our conclusions are in section VI. In the
Appendices we list the numerical results for the variational wavefunction for
future use, and apply the same analysis for the Helium-4 charge form factor
for comparison.

##  II Ground state of Helium-4

In this section we briefly outline the construction of the ground state
wavefunction of Helium-4, using two methods. The first method makes use of the
Reid nucleon pair potential and K-harmonics in the hyper-radius approximation,
without D-wave contribution. We recently used for the analysis of the GFFs of
Helium-4 in the impulse approximation He and Zahed (2024a). The second method,
makes use of the Argonne $v14$ nucleon pair potentials, retaining the D-wave
contribution Wiringa  _et al._ (1984).

###  II.1 K-Harmonic Method

The essential aspects of the radial part of the ground state of Helium-4 using
the K-harmonic method in the hyper-radius approximation was discussed recently
in He and Zahed (2024a) (and references therein). More specifically, the 4
nucleons in the $0^{++}$ spin-isospin configurations are described by

$\displaystyle\Phi_{H}[1,...,4]=\varphi[r_{1},r_{2},r_{3},r_{4}]\,\Phi[\sigma,\tau],$
(1)

with $\Phi$ refers to the properly symmetrized spin-isospin wavefunction. To
remove the spurious center of mass motion in Eq. (1), the nucleon coordinates
in (1) are split into three Jacobi coordinates $\vec{\xi}_{1,2,3}$ plus the
center of mass $\vec{R}_{C}$

$\displaystyle\vec{\xi}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\vec{r}_{2}-\vec{r}_{1}),$
$\displaystyle\vec{\xi}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}(\vec{r}_{1}+\vec{r}_{2}-2\vec{r}_{3}),$
$\displaystyle\vec{\xi}_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{3}}(\vec{r}_{1}+\vec{r}_{2}+\vec{r}_{3}-3\vec{r}_{4}),$
$\displaystyle\vec{R}_{C}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(\vec{r}_{1}+\vec{r}_{2}+\vec{r}_{3}+\vec{r}_{4}),$
(2)

with the radial hyperdistance

$\displaystyle R^{2}=\frac{1}{4}\sum_{i\neq
j}(\vec{r}_{i}-\vec{r}_{j})^{2}=\vec{\xi}_{1}^{2}+\vec{\xi}_{2}^{2}+\vec{\xi}_{3}^{2}.$
(3)

To keep this analysis simple as in He and Zahed (2024a), we will ignore the
D-wave contribution. Its role will be detailed in the second analysis to
follow, using the Argonne potential below. With this in mind, the radial
eigenstates carry constant hyper-spherical harmonic, with the spin-isospin
content Castilho Alcaras and Pimentel Escobar (1974)

$\displaystyle\Phi[\sigma,\tau]$ $\displaystyle=$
$\displaystyle\frac{\sqrt{105}}{8\pi^{2}}\bigg{(}([\sigma(1),\sigma(2)]_{1}[\sigma(3),\sigma(4)]_{1})_{00}([\tau(1),\tau(2)]_{0}[\tau(3),\tau(4)]_{0})_{00}$
(4)
$\displaystyle\qquad\qquad-([\sigma(1),\sigma(2)]_{0}[\sigma(3),\sigma(4)]_{0})_{00}([\tau(1),\tau(2)]_{1}[\tau(3),\tau(4)]_{1})_{00}\bigg{)}.$

The form of Eq. (1) in hyper-spherical coordinates can be expressed as

$\displaystyle\Phi_{H}[1,...,4]=\varphi[R]\,\Phi[\sigma,\tau].$ (5)

The radial part $\varphi[R]=u(R)/R^{4}$ where $u(R)$ is the reduced
wavefunction, solves

$\displaystyle
u{{}^{\prime\prime}}-\frac{12}{R^{2}}u-\frac{2m_{N}}{\hbar^{2}}(W(R)+V_{C}(R)-E)u=0,$

Here $W(R)$ and $V_{C}$ represent the pair nucleon and Coulomb potentials,
which can be written as Castilho Alcaras and Pimentel Escobar (1974); He and
Zahed (2024a)

$\displaystyle W(R)$ $\displaystyle=$
$\displaystyle\frac{315}{4}\int_{0}^{1}\,dx\,(1-x^{2})^{2}x^{2}\,V(\sqrt{2}Rx),$
$\displaystyle V_{C}(R)$ $\displaystyle=$ $\displaystyle\frac{2.23{\rm
MeV}{\rm fm}}{R},$ (7)

with

$\displaystyle V(r)$ $\displaystyle=$
$\displaystyle+1310.21\,e^{-(r/0.7)^{2}}-467.97\,e^{-(r/1.16)^{2}}.$

The radial solution to Eq. (II.1) is shown in the top panel in Fig. 1. The
ensuing wavefunction reproduces the correct binding energy and charge radius
of Helium-4. It also describes fairly well the empirical charge form factor of
Helium-4 up-to momentum of the order of half the nucleon mass He and Zahed
(2024a).

Figure 1: Top: The reduced wave function used in K-Harmonic Method; Bottom:
Helium-4 parial wave-functions following from the use of the Argonne $v14$
potential Wiringa (1991): central $f_{c}(r)$ (solid-bue), spin $u_{\sigma}(r)$
(solid-orange) and isospin-tensor $u_{t\tau}$ (solid-green).

###  II.2 Argonne $v14$ variational method

To include the D-wave contribution and more realistic nucleon pair
interactions in Helium-4, we will make use of the Argonne $v14$ symmetrized
($\cal S$) variational wavefunction Lomnitz-Adler and Pandharipande (1980);
Lomnitz-Adler  _et al._ (1981); Wiringa (1991)

$\displaystyle|\Psi_{v}\rangle={\cal
S}\prod\limits_{i<j}(1+U_{ij})|\Psi_{J}\rangle,$ (9)

with the Jastrow trial state

$|\Psi_{J}\rangle=\prod\limits_{i<j}f_{c}(r_{ij})|\Phi\rangle,$

where $f_{c}(r)$ is a central pair correlation function, and $\Phi$ the
antisymmetric spin-isospin wavefunction in Eq. (4). The pair correlation
functions $U_{ij}$ built in the variational state, is valued in the set of
operators making up the Argonne $v14$ potential

$\displaystyle U_{ij}=\sum_{p=2}^{m}u_{p}(r_{ij})O_{ij}^{p},$ (10)

with $O_{ij}^{p}$ making up the 14 spin-isospin pair operators in the $v14$
potential Lomnitz-Adler and Pandharipande (1980); Lomnitz-Adler  _et al._
(1981); Wiringa (1991)

$\displaystyle O_{ij}^{p=1,14}$ $\displaystyle=$ $\displaystyle
1,\tau_{i}\cdot\tau_{j},\sigma_{i}\cdot\sigma_{j},(\tau_{i}\cdot\tau_{j})(\sigma_{i}\cdot\sigma_{j}),S_{ij},$
(11) $\displaystyle
S_{ij}(\tau_{i}\tau_{j}),\vec{L}.\vec{S},\vec{L}.\vec{S}(\tau_{i}\tau_{j}),\vec{L}^{2},\vec{L}^{2}(\tau_{i}\cdot\tau_{j}),$
$\displaystyle\vec{L}^{2}(\sigma_{i}\cdot\sigma_{j}),\vec{L}^{2}(\sigma_{i}\cdot\sigma_{j})(\tau_{i}\cdot\tau_{j}),$
$\displaystyle(\vec{L}.\vec{S})^{2},(\vec{L}.\vec{S})^{2}(\tau_{i}\tau_{j}).$

Numerically the trial functions $u_{p}(r_{ij})$ will turn out to be small,
which allows us to simplify Eq. (9)

$\displaystyle|\Psi_{v}\rangle=(1+\sum_{i<j}U_{ij})|\Psi_{J}\rangle.$ (12)

Using the spin-isospin identities Lomnitz-Adler and Pandharipande (1980)

$\displaystyle\tau_{i}\cdot\tau_{j}|\Phi\rangle$ $\displaystyle=$
$\displaystyle-(2+\sigma_{i}\cdot\sigma_{j})|\Phi\rangle,$
$\displaystyle(\tau_{i}\cdot\tau_{j})(\sigma_{i}\cdot\sigma_{j})\Phi\rangle$
$\displaystyle=$ $\displaystyle-3|\Phi\rangle,$ $\displaystyle
S_{ij}(\tau_{i}\cdot\tau_{j})\Phi\rangle$ $\displaystyle=$
$\displaystyle-3S_{ij}|\Phi\rangle,$
$\displaystyle(\vec{L}.\vec{S})|\Phi\rangle$ $\displaystyle=$
$\displaystyle\vec{L}^{2}|\Phi\rangle=0.$ (13)

To include the central three body correlation, the pair correlation function
can be modified as follows Lomnitz-Adler and Pandharipande (1980); Lomnitz-
Adler  _et al._ (1981); Wiringa (1991).

$\displaystyle u_{p}(r_{ij})\rightarrow{\cal F}_{ij}u_{p}(r_{ij})=[\Pi_{k\neq
i,j}f_{ijk}]u_{p}(r_{ij}),$ (14)

where $f_{ijk}$ is chosen to be Wiringa (1991)

$\displaystyle f_{ijk}$ $\displaystyle=$ $\displaystyle
1-9\left(\frac{r_{ij}}{r_{ij}+r_{ik}+r_{jk}}\right)^{4}$ (15)
$\displaystyle\times$ $\displaystyle\textbf{exp}[-0.1(r_{ij}+r_{ik}+r_{jk})],$

we can rewrite Eq. (12) in the form

$\displaystyle|\Psi_{v}\rangle=\Big{\\{}1+\sum_{i<j}{\cal
F}_{ij}\big{[}u_{\sigma}(r_{ij})\sigma_{i}.\sigma_{j}$
$\displaystyle+u_{t\tau}(r_{ij})S_{ij}\tau_{i}.\tau_{j})\big{]}\Big{\\}}|\Psi_{J}\rangle.$
(16)

The Jastrow pair correlation function $f(r)$ and the operator correlation
functions $u_{\sigma,t\tau}$, can be repacked in Eq. (II.2), in terms of their
contributions to the singlet and triplet spin and isospin channels, using the
identity Wiringa (1991); Lomnitz-Adler  _et al._ (1981)

$\displaystyle
f_{c}(r_{ij})\left[1+u_{\sigma}(r_{ij})\sigma_{i}.\sigma_{j}+u_{t\tau}(r_{ij})\sigma_{i}.\sigma_{j}\right]|\Phi\rangle$
$\displaystyle=\bigg{(}\sum_{S,T}f_{S,T}(r_{ij})P_{S}P_{T}+f_{t,0}(r_{ij})S_{ij}P_{T=0}\bigg{)}|\Phi\rangle.$
(17)

Here $P_{S}$ and $P_{T}$ are the spin and isospin projection operators, i.e.

$\displaystyle P_{S=1}=(3+\sigma_{i}.\sigma_{j})/4,$ $\displaystyle
P_{S=0}=(1-\sigma_{i}.\sigma_{j})/4,$ $\displaystyle
P_{T=1}=(3+\tau_{i}.\tau_{j})/4,$ $\displaystyle
P_{T=0}=(1-\tau_{i}.\tau_{j})/4.$ (18)

The pair correlation functions $f_{c}$, $u_{\sigma}$ and $u_{t\tau}$ can be
related to the projected functions through

$\displaystyle f_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{4}(3f_{1,0}+f_{0,1}),$ $\displaystyle u_{\sigma}$
$\displaystyle=$ $\displaystyle\frac{1}{4}(f_{1,0}-f_{0,1})/f_{c},$
$\displaystyle u_{t\tau}$ $\displaystyle=$
$\displaystyle-\frac{1}{3}f_{t,0}/f_{c},$ (19)

where we made use of Eq. (II.2). In the ground state of Helium-4 with $L=0$,
the spin-isospin projected functions $f_{S,T}$ and spin-singlet-tensor
function $f_{t,0}$ can be obtained by solving the coupled Schrodinger
equations Wiringa (1991)

$\displaystyle-\frac{\hbar^{2}}{m}[r\,f_{0,1}(r)]^{\prime\prime}+(\bar{v}_{0,0}+\lambda_{0,T})f_{0,1}(r)r=0,$
$\displaystyle-\frac{\hbar^{2}}{m}[r\,f_{1,0}(r)]^{\prime\prime}+(\bar{v}_{1,0}+\lambda_{1,0})f_{1,0}(r)r+8(\bar{v}_{t,0}+\lambda_{t,0})f_{t,0}(r)r=0,$
$\displaystyle-\frac{\hbar^{2}}{m}[(r\,f_{t,0}(r))^{\prime\prime}-\frac{6}{r^{2}}(r\,f_{t,0}(r))]+(\bar{v}_{t,0}+\lambda_{t,0})f_{1,0}(r)r$
$\displaystyle~{}~{}~{}~{}~{}~{}+\left(\bar{v}_{t,0}+\lambda_{t,0}-2(\bar{v}_{t,0}+\lambda_{t,0})-3(\bar{v}_{b,0}+\lambda_{b,0})+6\bar{v}_{q1,0}+q\bar{v}_{bb,0}\right)f_{t,0}(r)r=0,$
(20)

here $\bar{v}_{S,T}$ refer to the explicit $v14$ pair nucleon potentials. The
eigenvalues $\lambda_{S,T}$ include the screening effects at short distances,
and their behavior at large distances is fixed by the asymptotic of the
correlation functions. We refer to Wiringa  _et al._ (1984); Wiringa (1991)
for their explicit form. In Fig 1 we show our numerical results for $f_{c}$,
$u_{\sigma}$ and $u_{t\tau}$, which are in overall agreement with those
reported in Wiringa (1991). In appendix. A, we have tabulated our results for
future comparison.

##  III Exchange currents in Helium-4

Helium-4 gravitational form factors in the impulse approximation were
discussed in He and Zahed (2024a). In this section we extend the analysis to
the exchange currents, following our recent discussion in the deuteron case He
and Zahed (2024b). More specifically, the gravitational exchange contributions
to Helium-4 in the single pion exchange approximation are illustrated in Fig.
2. We will refer to them generically by $T_{X}^{\mu\nu}$ with the pair
contribution (a), the seagull contribution (b), the pion exchange contribution
(c) and the recoil corrections plus wavefunction renormalization (d). For a
further assessment of the off-shellness effects between the pion-nucleon
pseudoscalar and pseudovector couplings, we will discuss both, where we note
that the seagull contribution (c) is absent for the former coupling. The
explicit operator forms for each of these contributions are quoted in section
IV below.

###  III.1 Matrix element: K-harmonic method

The generic structure of the matrix element in the K-harmonic method, can be
recast into a multi-dimensional integral over the Jacobi coordinates

$\displaystyle\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{X}^{\mu\nu}\bigg{|}-\frac{k}{2}m\right>$
$\displaystyle=$ $\displaystyle\int d^{3}\xi_{1}\int d^{3}\xi_{2}\int
d^{3}\xi_{3}\frac{d^{3}q_{2}}{(2\pi)^{3}}$ $\displaystyle\times$
$\displaystyle\textbf{exp}\Bigg{[}-i\sqrt{2}\vec{q}_{2}\vec{\xi}_{1}+i\vec{k}\left(\frac{\vec{\xi}_{1}}{\sqrt{2}}-\frac{\vec{\xi}_{2}}{\sqrt{6}}-\frac{\vec{\xi}_{3}}{2\sqrt{3}}\right)\Bigg{]}\langle\Phi_{H}[\xi]|T_{X}^{\mu\nu}(\sigma_{1},\sigma_{2})|\Phi_{H}[\xi]\rangle+{\rm
permutation}$

with $\vec{q}_{2}$ the momentum of exchanged pion in Fig. 2. The argument of
the Fourier transform refers to the struck nucleon relative to the center of
mass coordinate, i.e.

$\displaystyle\frac{\vec{\xi}_{1}}{\sqrt{2}}-\frac{\vec{\xi}_{2}}{\sqrt{6}}-\frac{\vec{\xi}_{3}}{2\sqrt{3}}=\vec{R}_{C}-\vec{r}_{1}$
(22)

$T_{X}^{\mu\nu}$ is short for the inserted EMT operators in Fig. 2. Their
explicit non-relativistic forms are listed in Eqs. (27), (33), (36) and (39)
for the pair contribution, the seagull contribution, the pion exchange
contribution and recoil contribution, respectively. To keep the hyperspherical
symmetric, the Jacobi coordinates are set to be

$\displaystyle\xi_{1}$ $\displaystyle=$ $\displaystyle
R\,cos\theta(sin\theta_{1}cos\phi_{1},sin\theta_{1}sin\phi_{1},cos\theta_{1}),$
$\displaystyle\xi_{2}$ $\displaystyle=$ $\displaystyle R\,sin\theta
cos\phi(sin\theta_{2}cos\phi_{2},sin\theta_{2}sin\phi_{2},cos\theta_{2}),$
$\displaystyle\xi_{3}$ $\displaystyle=$ $\displaystyle R\,sin\theta
sin\phi(sin\theta_{3}cos\phi_{3},sin\theta_{3}sin\phi_{3},cos\theta_{3}),$

The above integral can be reduced after integrating over the angular

$\displaystyle\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{X}^{\mu\nu}\bigg{|}-\frac{k}{2}m\right>$
(24) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dR\int_{0}^{\frac{\pi}{2}}d\theta\,sin\theta^{5}cos\theta^{2}\int_{0}^{\frac{\pi}{2}}d\phi\,sin\phi^{2}cos\phi^{2}\int
d\Omega_{2}\textbf{exp}\Bigg{[}-i\vec{k}\frac{\vec{\xi}_{2}}{\sqrt{6}}\Bigg{]}\int
d\Omega_{3}\textbf{exp}\Bigg{[}-i\vec{k}\frac{\vec{\xi}_{3}}{2\sqrt{3}}\Bigg{]}$
$\displaystyle\times$ $\displaystyle\int\frac{d^{3}q_{2}}{(2\pi)^{3}}\int
d\Omega_{1}\textbf{exp}\Bigg{[}-i\sqrt{2}\vec{q}_{2}\vec{\xi}_{1}+i\vec{k}\frac{\vec{\xi}_{1}}{\sqrt{2}}\Bigg{]}u(R)^{2}\langle\Phi|T_{X}^{\mu\nu}(\sigma_{1},\sigma_{2})|\Phi\rangle+{\rm
permutation}$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\int_{0}^{\infty}dR\int_{0}^{\frac{\pi}{2}}d\theta\,sin\theta^{5}cos\theta^{2}\int_{0}^{\frac{\pi}{2}}d\phi\,sin\phi^{2}cos\phi^{2}j_{0}\left(\frac{kR\,sin\theta
cos\phi}{\sqrt{6}}\right)j_{0}\left(\frac{kR\,sin\theta
sin\phi}{2\sqrt{3}}\right)$ $\displaystyle\times$
$\displaystyle\int\frac{d^{3}q_{2}}{(2\pi)^{3}}\int
d\Omega_{1}\textbf{exp}\Bigg{[}-i\sqrt{2}\vec{q}_{2}\vec{\xi}_{1}+i\vec{k}\frac{\vec{\xi}_{1}}{\sqrt{2}}\Bigg{]}u(R)^{2}\langle\Phi|T_{X}^{\mu\nu}(\sigma_{1},\sigma_{2})|\Phi\rangle+{\rm
permutation},$

Figure 2:  Exchange contributions to the EMT: (a) The pair contribution, (b)
seagull contribution, (c) pion exchange contribution, (d) recoil plus wave
function renormalization. The solid, dashed and wavy lines represent the
nucleon, pion and graviton, respectively. Note that there is no seagull
contribution for a pseudoscalar pion-nucleon coupling.

###  III.2 Matrix element: $v14$ potential

The generic form of the exchange corrections, using the Argonne potential with
the variational wave function (II.2) are of the form

$\displaystyle\left<+\frac{k}{2}m^{\prime}\bigg{|}T_{X\pi}^{\mu\nu}\bigg{|}-\frac{k}{2}m\right>$
$\displaystyle=$ $\displaystyle\frac{1}{Z}\int d^{3}\vec{r}_{12}\int
d^{3}\vec{r}_{13}\int d^{3}\vec{r}_{14}\frac{d^{3}q_{2}}{(2\pi)^{3}}$ (25)
$\displaystyle\times$
$\displaystyle\textbf{exp}\Bigg{[}i\vec{q}_{2}.\vec{r}_{12}-i\vec{k}.(\vec{r}_{1}-\vec{R}_{C})\Bigg{]}\langle\Psi_{v}|T_{X}^{\mu\nu}(\sigma_{1},\sigma_{2})\Psi_{v}\rangle+{\rm
permutation},$

where $\vec{r}_{ij}=\vec{r}_{i}-\vec{r}_{j}$. Here $Z$ enforces the
wavefunction normalization

$\displaystyle Z=\int d^{3}\vec{r}_{12}\int d^{3}\vec{r}_{13}\int
d^{3}\vec{r}_{14}\langle\Psi_{v}|\Psi_{v}\rangle.$ (26)

Note that we have switched back to initial coordinates $\vec{r}_{12}$,
$\vec{r}_{13}$ and $\vec{r}_{14}$ as integration variables, which tie to the
Jacobi coordinates (II.1) by translational invariance, modulo a constant
Jacobian. The latter can be eliminated in the ratio (25). To evaluate the
multi-dimensional integral in Eq. (25), we will use the Monte Carlo method
described in Lomnitz-Adler  _et al._ (1981); Wiringa (1991).

Figure 3: Helium-4 GFFs with PS coupling (left) and PV coupling (right) using
the K-Harmonics method: “imp" refers to the impulse approximation, “$X\pi$",
“$XS$", “$X\pi\pi$" and “XR" refer to the exchange contributions in (Fig. 2a),
seagull contribution in (Fig. 2b), pion exchange contribution in (Fig. 2c),
and recoil contribution in (Fig. 2d), respectively.

##  IV Matrix element of EMT in Helium-4

To extract the non-relativistic contributions to the exchange currents, we
will make use of the diagrams in Fig. 2, with the positive nucleon
contribution in (a) subtracted, and the recoil contribution and wavefunction
renormalization in (d) retained. The wiggly line refers to the insertion of
the EMT, and the dashed line to the pion exchange contribution using both
pseudo-vector coupling and pseudo-scalar coupling. Again, the explicit
operator form for the EMT insertion $T_{X}$ for nucleon pairs in the context
of the impulse approximation, is given in He and Zahed (2024a) and to which we
refer for completeness. Instead, we will give the explicit integral
expressions for the form factors for each contribution, using the K-Harmonic
Method only. Those from the Argonne $v14$ will be evaluated numerically using
Eq. (25) by Monte Carlo sampling, with only the final results given in section
V.

###  IV.1 Pair contribution

The EMT operator for the pair diagrams in Fig. 2a with pseudo-scalar coupling
and pseudo-vector coupling, in the leading non-relativistic reduction read He
and Zahed (2024b)

$\displaystyle T_{X\pi,PS}^{00}(k)$ $\displaystyle=$
$\displaystyle(A(k)+B(k))\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\frac{\vec{\sigma}_{1}.\vec{k}\vec{\sigma}_{2}.\vec{q_{2}}}{w_{q_{2}}^{2}}+(1\leftrightarrow
2)+{\cal O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$ (27a)
$\displaystyle T_{X\pi,PS}^{ij}(k)$ $\displaystyle=$
$\displaystyle-A(k)\tau_{1}\cdot\tau_{2}\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\frac{q_{2}^{i}\sigma_{1}^{j}+q_{2}^{j}\sigma_{1}^{i}}{2w_{q_{2}}^{2}}\vec{\sigma}_{2}.\vec{q}_{2}+(1\leftrightarrow
2)+{\cal O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)},$ (27b)
$\displaystyle T_{X\pi,PV}^{00,ij}(k)$ $\displaystyle=$ $\displaystyle{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}$ (27c)

with the pion energy $w_{q_{2}}=\sqrt{q_{2}^{2}+m_{\pi}^{2}}$. The GFFs $A$
and $D$ for Helium-4 are related to the matrix elements of $T^{00}$ and
$T^{ij}$ through

$\displaystyle\left<+\frac{k}{2}\bigg{|}T_{X\pi}^{00}\bigg{|}-\frac{k}{2}\right>$
$\displaystyle=$
$\displaystyle(m_{\alpha}+\frac{k^{2}}{4m_{\alpha}})A^{H}_{X\pi}+\frac{k^{2}}{4m_{\alpha}}D^{H}_{X\pi},$
$\displaystyle\left<+\frac{k}{2}\bigg{|}T_{X\pi}^{ij}\bigg{|}-\frac{k}{2}\right>$
$\displaystyle=$
$\displaystyle\frac{k_{i}k_{j}-\delta^{ij}k^{2}}{4m_{\alpha}}D^{H}_{X\pi}+\delta^{ij}D^{H}_{1,X\pi}.$

Note that the term including $D^{H}_{1,X\pi}$ breaks down the conservation law
$k^{i}T_{X\pi}^{ij}=0$. As noted in He and Zahed (2024b), all contributions in
Fig .2 need to be summed up, to uphold the conservation law. In particular,
the contributions $\delta^{ij}$ will be cancelled out. Therefore, we will not
keep such a contribution in our following calculation. With this in mind, the
corresponding GFFs for the pair diagrams with PS and PV coupling can be
expressed as

$\displaystyle A^{H}_{X\pi,PS}$ $\displaystyle=$
$\displaystyle-\frac{\vec{k}^{2}}{\vec{k}^{2}+4m_{\alpha}^{2}}D_{X\pi,PS}^{H}+6\frac{(4\pi)^{2}g_{\pi
N}^{2}}{(2m_{N})^{2}}m_{\pi}^{2}\int_{0}^{\infty}dR\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{\frac{\pi}{2}}d\phi\,{\mathbb{V}}\,j_{1}\left(\frac{kRcos\theta}{\sqrt{2}}\right)\frac{ku^{2}Y_{1}^{H}}{6m_{N}}$
(29a) $\displaystyle\times$
$\displaystyle(A(k)+B(k))\langle\Phi|\tau_{1}\cdot\tau_{2}\,\sigma_{1}\cdot\sigma_{2}|\Phi\rangle,$
$\displaystyle D^{H}_{X\pi,PS}$ $\displaystyle=$ $\displaystyle
6\frac{(4\pi)^{2}g_{\pi
N}^{2}}{(2m_{N})^{2}}m_{\pi}^{2}\int_{0}^{\infty}dR\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{\frac{\pi}{2}}d\phi\,{\mathbb{V}}\,j_{2}\left(\frac{kRcos\theta}{\sqrt{2}}\right)\frac{-8m_{\alpha}m_{\pi}u^{2}Y_{2}^{H}}{3k^{2}}A(k)$
(29b) $\displaystyle\times$
$\displaystyle\langle\Phi|\tau_{1}\cdot\tau_{2}\,\sigma_{1}\cdot\sigma_{2}|\Phi\rangle,$
$\displaystyle A^{H}_{X\pi,PV}$ $\displaystyle=$ $\displaystyle
D^{H}_{XS,PV}={\cal O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)},$ (29c)

where $u(R)$ is the reduced hyper-radius wavefunction in Eq. (II.1), and
factor ${\mathbb{V}}$ is defined as

$\displaystyle{\mathbb{V}}$ $\displaystyle=$ $\displaystyle
sin\theta^{5}cos\theta^{2}sin\phi^{2}cos\phi^{2}j_{0}\left(\frac{kR\,sin\theta
cos\phi}{\sqrt{6}}\right)j_{0}\left(\frac{kR\,sin\theta
sin\phi}{2\sqrt{3}}\right),$ (30)

with the shorthand Yukawa potentials,

$\displaystyle Y_{0}^{H}$ $\displaystyle=$ $\displaystyle
e^{-m_{\pi}\sqrt{2}R\,cos\theta}/(m_{\pi}\sqrt{2}R\,cos\theta),$
$\displaystyle\bar{Y}_{1}^{H}$ $\displaystyle=$
$\displaystyle(1+1/(m_{\pi}\sqrt{2}R\,cos\theta))Y_{0}^{H},$ $\displaystyle
Y_{2}^{H}$ $\displaystyle=$
$\displaystyle(3/(m_{\pi}\sqrt{2}R\,cos\theta)^{2}+3/(m_{\pi}\sqrt{2}R\,cos\theta)+1)Y_{0}^{H}.$
(31)

In the $0^{++}$ Helium-4 ground state, we can use Eq. (4) to obtain

$\displaystyle\langle\Phi|\tau_{1}\cdot\tau_{2}\,\sigma_{1}\cdot\sigma_{2}|\Phi\rangle=-\frac{315}{32\pi^{4}}.$
(32)

###  IV.2 Seagull contribution

For the Seagull in Fig .2(b), the EMT operator can be written as He and Zahed
(2024b)

$\displaystyle T_{XS,PV}^{00}$ $\displaystyle=$ $\displaystyle{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)},$ (33a) $\displaystyle
T_{XS,PV}^{ij}$ $\displaystyle=$
$\displaystyle-\tau_{1}\cdot\tau_{2}\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\frac{q_{2}^{i}\sigma_{1}^{j}+q_{2}^{j}\sigma_{1}^{i}}{2w_{q_{2}}^{2}}\vec{\sigma}_{2}.\vec{q}_{2}$
(33b) $\displaystyle\qquad+(1\leftrightarrow 2)+{\cal O}\bigg{(}\frac{g_{\pi
N}^{2}}{m_{N}^{4}}\bigg{)}.$

There is no contribution from the seagull diagram for the PS coupling case.
Also the contribution to $T^{ij}_{XS}$ from the PV coupling is equivalent to
the contribution to $T^{ij}_{X\pi}$ from the pair diagram for the PS
coupling,with the exception of the extra nucleon GFF A(k) in Eq. (27b). The
GFFs can be obtained using the following relations

$\displaystyle\left<+\frac{k}{2}\bigg{|}T_{XS}^{00}\bigg{|}-\frac{k}{2}\right>$
$\displaystyle=$
$\displaystyle(m_{\alpha}+\frac{k^{2}}{4m_{\alpha}})A^{H}_{XS}+\frac{k^{2}}{4m_{\alpha}}D^{H}_{XS}$
$\displaystyle=$ $\displaystyle{\cal O}\bigg{(}\frac{g_{\pi
N}^{2}}{m_{N}^{4}}\bigg{)},$
$\displaystyle\left<+\frac{k}{2}\bigg{|}T_{XS}^{ij}\bigg{|}-\frac{k}{2}\right>$
$\displaystyle=$
$\displaystyle\frac{k_{i}k_{j}-\delta^{ij}k^{2}}{4m_{\alpha}}D^{H}_{XS},$ (34)

where

$\displaystyle A^{H}_{XS,PS}$ $\displaystyle=$ $\displaystyle
D^{H}_{XS,PS}=0,$ (35a) $\displaystyle A^{H}_{XS,PV}$ $\displaystyle=$
$\displaystyle-\frac{\vec{k}^{2}}{\vec{k}^{2}+4m_{\alpha}^{2}}D_{XS,PV}^{H},$
(35b) $\displaystyle D^{H}_{XS,PV}$ $\displaystyle=$ $\displaystyle
6\frac{(4\pi)^{2}g_{\pi
N}^{2}}{(2m_{N})^{2}}m_{\pi}^{2}\int_{0}^{\infty}dR\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{\frac{\pi}{2}}d\phi{\mathbb{V}}\,j_{2}\left(\frac{kRcos\theta}{\sqrt{2}}\right)\frac{-8m_{\alpha}m_{\pi}u^{2}Y_{2}^{H}}{3k^{2}}$
(35c) $\displaystyle\times$
$\displaystyle\langle\Phi|\tau_{1}\cdot\tau_{2}\,\sigma_{1}\cdot\sigma_{2}|\Phi\rangle.$

Figure 4:  Helium-4 GFFs with PS coupling (left panel) and PV coupling (right
panel), using the Argonne $v14$ potential via a Monte Carlo method. The bands
represent the statistical uncertainty.

###  IV.3 Pion exchange contribution

The EMT exchange contributions stemming from the one-pion exchange in the non-
relativistic reduction, are equivalent whether we use the PS or PV couplings.
More specifically, the pertinent operators are He and Zahed (2024b)

$\displaystyle T_{X\pi\pi}^{00}$ $\displaystyle=$ $\displaystyle-\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\frac{\vec{k}^{2}\vec{\sigma_{1}}.\vec{q_{1}}\vec{\sigma_{2}}.\vec{q_{2}}}{2w_{q_{1}}^{2}w_{q_{2}}^{2}}T_{1\pi}(k)+\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\frac{\sigma_{1}.q_{1}\sigma_{2}.q_{2}}{2}\left(\frac{1}{w_{q_{2}}^{2}}+\frac{1}{w_{q_{1}}^{2}}\right)$
(36a) $\displaystyle+$ $\displaystyle{\cal O}\bigg{(}\frac{g_{\pi
N}^{2}}{m_{N}^{4}}\bigg{)},$ $\displaystyle T_{X\pi\pi}^{ij}$ $\displaystyle=$
$\displaystyle\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\left(\frac{(\vec{k}^{2}\delta^{ij}-k^{i}k^{j})\vec{\sigma_{1}}.\vec{q_{1}}\vec{\sigma_{2}}.\vec{q_{2}}}{2w_{q_{1}}^{2}w_{q_{2}}^{2}}T_{1\pi}(k)+\frac{(q_{1}^{i}+q_{2}^{i})(q_{1}^{j}+q_{2}^{j})\vec{\sigma_{1}}.\vec{q_{1}}\vec{\sigma_{2}}.\vec{q_{2}}}{2w_{q_{1}}^{2}w_{q_{2}}^{2}}T_{2\pi}(k)\right)$
(36b) $\displaystyle-$ $\displaystyle\delta^{ij}\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\frac{\sigma_{1}.q_{1}\sigma_{2}.q_{2}\left(\vec{q}_{1}^{2}+\vec{q}_{2}^{2}+2m_{\pi}^{2}\right)}{2w_{q_{1}}^{2}w_{q_{2}}^{2}}+{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)},$

with the identical matrix elements

$\displaystyle\left<+\frac{k}{2}\bigg{|}T_{X\pi\pi}^{00}\bigg{|}-\frac{k}{2}\right>$
$\displaystyle=$
$\displaystyle(m_{\alpha}+\frac{k^{2}}{4m_{\alpha}})A^{H}_{X\pi\pi}+\frac{k^{2}}{4m_{\alpha}}D^{H}_{X\pi\pi},$
$\displaystyle\left<+\frac{k}{2}\bigg{|}T_{X\pi\pi}^{ij}\bigg{|}-\frac{k}{2}\right>$
$\displaystyle=$
$\displaystyle\frac{k_{i}k_{j}-\delta^{ij}k^{2}}{4m_{\alpha}}D^{H}_{X\pi\pi},$
(37)

where

$\displaystyle A_{X\pi\pi}^{H}$ $\displaystyle=$
$\displaystyle-\frac{\vec{k}^{2}}{\vec{k}^{2}+4m_{\alpha}^{2}}D_{X\pi\pi}^{H}+6\frac{(4\pi)^{2}g_{\pi
N}^{2}}{(2m_{N})^{2}}T_{1\pi}(k)\int_{-\frac{1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}}dt\int_{0}^{\infty}dR\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{\frac{\pi}{2}}d\phi{\mathbb{V}}\Bigg{\\{}$
(38a)
$\displaystyle\frac{kL_{\pi}u^{2}\left(\sqrt{2}k^{2}Rt^{2}Y_{0}^{H,\pi\pi}cos\theta+3L_{\pi}\bar{Y}_{1}^{H,\pi\pi}\right)}{12t\,m_{\alpha}}j_{1}(ktR\,cos\theta)-\frac{k^{2}L_{\pi}^{2}Ru^{2}\bar{Y}_{1}^{H,\pi\pi}cos\theta
j_{2}(ktR\,cos\theta)}{12m_{\alpha}}$ $\displaystyle-$
$\displaystyle\frac{k^{2}u^{2}Y_{0}^{H,\pi\pi}\left(k^{2}R\left(2t^{2}-1\right)cos\theta+6\sqrt{2}L_{\pi}\right)j_{0}(ktRcos\theta)}{48m_{\alpha}}\Bigg{\\}}\langle\Phi|\tau_{1}\cdot\tau_{2}\,\sigma_{1}\cdot\sigma_{2}|\Phi\rangle$
$\displaystyle+$ $\displaystyle 6\frac{(4\pi)^{2}g_{\pi
N}^{2}}{(2m_{N})^{2}}\int
dR\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{\frac{\pi}{2}}d\phi
V\frac{m_{\pi}^{2}u^{2}}{12km_{\alpha}R}\Bigg{\\{}-2j_{1}\left(\frac{kRcos\theta}{\sqrt{2}}\right)\left(k^{2}r\bar{Y}^{H}_{1}+3\sqrt{2}m_{\pi}Y^{H}_{2}sec\theta\right)$
$\displaystyle+$ $\displaystyle
2km_{\pi}RY^{H}_{2}j_{2}\left(\frac{kRcos\theta}{\sqrt{2}}\right)+3\sqrt{2}k\bar{Y}^{H}_{1}sec\theta
j_{0}\left(\frac{krcos\theta}{\sqrt{2}}\right)\Bigg{\\}}\langle\Phi|\tau_{1}\cdot\tau_{2}\,\sigma_{1}\cdot\sigma_{2}|\Phi\rangle,$
$\displaystyle D_{X\pi\pi}^{H}$ $\displaystyle=$ $\displaystyle
6\frac{(4\pi)^{2}g_{\pi
N}^{2}}{(2m_{N})^{2}}T_{1\pi}(k)\int_{-\frac{1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}}dt\int_{0}^{\infty}dR\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{\frac{\pi}{2}}d\phi\,{\mathbb{V}}$
(38b) $\displaystyle\times$
$\displaystyle\Bigg{\\{}-\frac{m_{\alpha}u^{2}Y_{0}^{H,\pi\pi}\left[k^{2}R\left(2t^{2}-1\right)cos\theta+6\sqrt{2}L_{\pi}\right]j_{0}(ktR\,cos\theta)}{12}$
$\displaystyle+$
$\displaystyle\frac{m_{\alpha}L_{\pi}u^{2}j_{1}(ktR\,cos\theta)\left(\sqrt{2}k^{2}Rt^{2}Y_{0}^{H,\pi\pi}cos\theta+3L_{\pi}\bar{Y}_{1}^{H,\pi\pi}\right)}{3kt}$
$\displaystyle-$
$\displaystyle\frac{m_{\alpha}L_{\pi}^{2}Ru^{2}\bar{Y}_{1}^{H,\pi\pi}cos\theta
j_{2}(ktR\,cos\theta)}{3}\Bigg{\\}}\langle\Phi|\tau_{1}\cdot\tau_{2}\,\sigma_{1}\cdot\sigma_{2}|\Phi\rangle$
$\displaystyle+$ $\displaystyle 6\frac{(4\pi)^{2}g_{\pi
N}^{2}}{(2m_{N})^{2}}T_{2\pi}(k)\int_{-\frac{1}{\sqrt{2}}}^{\frac{1}{\sqrt{2}}}dt\int_{0}^{\infty}dR\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{\frac{\pi}{2}}d\phi{\mathbb{V}}\Bigg{\\{}$
$\displaystyle-$
$\displaystyle\frac{u^{2}m_{\alpha}j_{0}(ktRcos\theta)}{54k^{2}}[RY_{0}^{H,\pi\pi}cos\theta\left(9k^{4}t^{2}\left(1-2t^{2}\right)+40L_{\pi}^{4}\right)$
$\displaystyle+$ $\displaystyle
2\sqrt{2}L_{\pi}\left(10L_{\pi}^{2}(Y_{0}^{H,\pi\pi}+5Y_{2}^{H,\pi\pi})-63k^{2}t^{2}Y_{0}^{H,\pi\pi}\right)]$
$\displaystyle+$
$\displaystyle\frac{L_{\pi}tu^{2}m_{\alpha}j_{1}(ktRcos\theta)}{15k}[-\sqrt{2}Rcos\theta\left(5k^{2}\left(4t^{2}-1\right)Y_{0}^{H,\pi\pi}-28L_{\pi}^{2}Y_{2}^{H,\pi\pi}\right)-210L_{\pi}\bar{Y}_{1}^{H,\pi\pi}]$
$\displaystyle-$ $\displaystyle\frac{L_{\pi}^{2}u^{2}\sec^{2}\theta
m_{\alpha}j_{2}(ktRcos\theta)}{756k^{4}R^{2}t^{2}}\Big{[}452k^{2}L_{\pi}^{2}R^{3}t^{2}Y_{0}^{H,\pi\pi}cos(3\theta)+3Rcos\theta\Big{(}4L_{\pi}^{2}Y_{0}^{H,\pi\pi}\left(113k^{2}R^{2}t^{2}-700\right)$
$\displaystyle+$ $\displaystyle
63k^{4}R^{2}t^{2}\left(1-12t^{2}\right)\bar{Y}_{1}^{H,\pi\pi}\Big{)}+452\sqrt{2}k^{2}L_{\pi}R^{2}t^{2}Y_{0}^{H,\pi\pi}+4\sqrt{2}k^{2}L_{\pi}R^{2}t^{2}cos(2\theta)(113Y_{0}^{H,\pi\pi}+124Y_{2}^{H,\pi\pi})$
$\displaystyle+$ $\displaystyle
496\sqrt{2}k^{2}L_{\pi}R^{2}t^{2}Y_{2}^{H,\pi\pi}+63k^{4}R^{3}t^{2}\bar{Y}_{1}^{H,\pi\pi}cos(3\theta)-756k^{4}R^{3}t^{4}\bar{Y}_{1}^{H,\pi\pi}cos(3\theta)$
$\displaystyle-$ $\displaystyle
4200\sqrt{2}L_{\pi}Y_{0}^{H,\pi\pi}-21000\sqrt{2}L_{\pi}Y_{2}^{H,\pi\pi}\Big{]}$
$\displaystyle-$
$\displaystyle\frac{4\sqrt{2}L_{\pi}^{3}tu^{2}Y_{2}^{H,\pi\pi}R\,cos\theta
m_{\alpha}j_{3}(ktRcos\theta)}{5k}$ $\displaystyle-$
$\displaystyle\frac{10L_{\pi}^{3}u^{2}m_{\alpha}j_{4}(ktR\,cos\theta)\left(2L_{\pi}RY_{0}^{H,\pi\pi}cos\theta+\sqrt{2}(Y_{0}^{H,\pi\pi}+5Y_{2}^{H,\pi\pi})\right)}{63k^{2}}\Bigg{\\}}\langle\Phi|\tau_{1}\cdot\tau_{2}\,\sigma_{1}\cdot\sigma_{2}|\Phi\rangle.$
Here
$L_{\pi}=\left[m_{\pi}^{2}+(\frac{1}{4}-\frac{t^{2}}{2})\vec{k}^{2}\right]^{1/2}$
and the shorthand Yukawa potentials are expressed as $\displaystyle
Y_{0}^{H,\pi\pi}$ $\displaystyle=$ $\displaystyle
e^{-L_{\pi}\sqrt{2}R\,cos\theta}/(L_{\pi}\sqrt{2}R\,cos\theta),$
$\displaystyle\bar{Y}_{1}^{H,\pi\pi}$ $\displaystyle=$
$\displaystyle(1+1/(L_{\pi}\sqrt{2}R\,cos\theta))Y_{0}^{H,\pi\pi},$
$\displaystyle Y_{2}^{H,\pi\pi}$ $\displaystyle=$
$\displaystyle(3/(L_{\pi}\sqrt{2}R\,cos\theta)^{2}+3/(L_{\pi}\sqrt{2}R\,cos\theta)+1)Y_{0}^{H,\pi\pi}.$
(38c)

###  IV.4 Recoil contribution

The exchange operators stemming from the recoil terms after non-relativistic
reduction are also equivalent whether we use the PS or PV coupling, they can
be expressed as He and Zahed (2024b)

$\displaystyle T^{00}_{XR}$ $\displaystyle=$ $\displaystyle-\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}A(k)\frac{q_{2}.k\sigma_{1}.q_{2}\sigma_{2}.q_{2}}{2w_{q_{2}}^{4}}+(1\leftrightarrow
2)+{\cal O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)},$ (39a)
$\displaystyle T^{ij}_{XR}$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}.$ (39b)

with the corresponding matrix elements

$\displaystyle\left<+\frac{k}{2}\bigg{|}T_{XR}^{00}\bigg{|}-\frac{k}{2}\right>$
$\displaystyle=$
$\displaystyle(m_{\alpha}+\frac{k^{2}}{4m_{\alpha}})A^{H}_{XR}+\frac{k^{2}}{4m_{\alpha}}D^{H}_{XR},$
$\displaystyle\left<+\frac{k}{2}\bigg{|}T_{X}^{ij}\bigg{|}-\frac{k}{2}\right>$
$\displaystyle=$
$\displaystyle\frac{k_{i}k_{j}-\delta^{ij}k^{2}}{4m_{\alpha}}D^{H}_{XR}={\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)},$ (40)

and

$\displaystyle A^{H}_{XR}=$ $\displaystyle 6\frac{(4\pi)^{2}g_{\pi
N}^{2}}{(2m_{N})^{2}}A(k)$ (41)
$\displaystyle\times\int_{0}^{\infty}dR\int_{0}^{\frac{\pi}{2}}d\theta\int_{0}^{\frac{\pi}{2}}d\phi{\mathbb{V}}\frac{km_{\pi}^{2}u^{2}j_{1}\left(\frac{kRcos\theta}{\sqrt{2}}\right)(\sqrt{2}m_{\pi}RY_{2}^{H}cos\theta-5\bar{Y}_{1}^{H})}{6m_{\alpha}}\langle\Phi|\tau_{1}\cdot\tau_{2}\,\sigma_{1}\cdot\sigma_{2}|\Phi\rangle.$

##  V Numerical results

To summarise, the EMT exchange contributions to Helium-4 for the PS and PV
pion nucleon couplings are given by

$\displaystyle T^{00}_{DX,PS}$ $\displaystyle=$ $\displaystyle
T_{X\pi}^{00}+T_{X\pi\pi}^{00}+T^{00}_{XR},$ $\displaystyle T^{ij}_{DX,PS}$
$\displaystyle=$ $\displaystyle T_{X\pi}^{00}+T_{X\pi\pi}^{00},$
$\displaystyle T^{00}_{DX,PV}$ $\displaystyle=$ $\displaystyle
T_{X\pi\pi}^{00}+T^{00}_{XR},$ $\displaystyle T^{ij}_{DX,PV}$ $\displaystyle=$
$\displaystyle T_{XS}^{ij}+T^{ij}_{X\pi\pi}.$ (42)

The main difference between the PS and PV couplings, is the additional
contribution from the pair diagram to the $T^{00}$ component in the PS
coupling case. A similar addition is noted in the charge form factor as in
(B). We also note that the pair diagram contribution following from the PS
coupling to $T^{ij}$ , is very similar to that of the seagull diagram (absent
for the PS coupling) with PV coupling. The only difference is the extra
nucleon form factor $A(k)$ included in (27b) in comparison to (33b).

The corresponding GFFs from the exchange currents are

$\displaystyle A^{H}_{X,PS}$ $\displaystyle=$ $\displaystyle
A^{H}_{X\pi,PS}(\ref{eq:FF_AXpairPS})+A^{H}_{X\pi\pi}(\ref{eq:FF_A_pi})$
$\displaystyle+$ $\displaystyle A^{H}_{XR}(\ref{eq:FF_A_XR})$ $\displaystyle
D^{H}_{X,PS}$ $\displaystyle=$ $\displaystyle
D^{H}_{X\pi,PS}(\ref{eq:FF_DXpairPS})+D^{H}_{X\pi\pi}(\ref{eq:FF_D_pi})$
$\displaystyle A^{H}_{X,PV}$ $\displaystyle=$ $\displaystyle
A^{H}_{XS,PV}(\ref{eq:FF_AXSPV})+A^{H}_{X\pi\pi}(\ref{eq:FF_A_pi})$
$\displaystyle+$ $\displaystyle A_{XR}(\ref{eq:FF_A_XR})$ $\displaystyle
D^{H}_{X,PV}$ $\displaystyle=$ $\displaystyle
D^{H}_{XS,PV}(\ref{eq:FF_DXSPV})+D^{H}_{X\pi\pi}(\ref{eq:FF_D_pi})$ (43)

To assess the pion GFFs in the pion exchange current contribution, we borrow
their parameterization from the recent Lattice calculation in Hackett  _et
al._ (2023a),

$\displaystyle
T_{1\pi}^{g}(k)=\frac{0.596}{1+\frac{\vec{k}^{2}}{0.677~{}\text{GeV}^{2}}}$
$\displaystyle
T_{1\pi}^{q}(k)=\frac{0.304}{1+\frac{\vec{k}^{2}}{1.44~{}\text{GeV}^{2}}}$
$\displaystyle
T_{2\pi}^{g}(k)=\frac{0.546}{1+\frac{\vec{k}^{2}}{1.129~{}\text{GeV}^{2}}}$
$\displaystyle
T_{2\pi}^{q}(k)=\frac{0.481}{1+\frac{\vec{k}^{2}}{1.262~{}\text{GeV}^{2}}}$
(44)

In Fig. 3 we show the Helium-4 $A^{H},D^{H}$ GFFs using the pion-nucleon PS
coupling (left) and the pion-nucleon PV coupling (right) with the K-Harmonic
method. The results from the impulse approximation (blue-dots) are labeled by
"$Imp$" are from He and Zahed (2024a). The various contributions labeled by
“$X\pi$", “$XS$", “$X\pi\pi$", “$XR$" stem from the exchange contributions
illustrated in Fig. 2a for the pair contribution, in Fig. 2b for the seagull
contribution, in Fig. 2c for the pi-pi contribution and in Fig. 2d for the
recoil contribution, respectively. The total corrections from the exchange
currents using the PS coupling (red-triangle), do not yield qualitative
changes to the GFFs $A^{H}$ from the impulse approximation (blue-dot).
However, the contribution of the exchange currents with PV coupling yield lowe
results compared to the impulse approximation. The difference is due to the
pair contribution in Fig. 2a, whose contribution is higher order in the PV
coupling. The contributions of the exchange currents to $D^{H}$ are comparable
in the PS and PV case, since the only difference is the extra nucleon form
factor $A(k)$ included in $D^{H}_{X\pi,PS}$, but absent in $D^{H}_{XS,PV}$.
Both shift the diffractive minima to the left of the impulse approximation.

In Fig. 4 we show the Helium-4 $A^{H},D^{H}$ GFFs using the pion-nucleon PS
coupling (left) and the pion-nucleon PV coupling (right) with the Argonne
$v14$ potential. The statistical uncertainties following from the Monte-Carlo
sampling, are reflected in the colored-spreads when the various exchange
contributions are added to the impulse approximation labeled by "$Imp$" (blue-
spread) results. Again, different contributions labeled by “$X\pi$", “$XS$",
“$X\pi\pi$", “$XR$" stem from the exchange contributions for the pair
contribution, the seagull contribution, the pi-pi contribution and the recoil
contribution, respectively. Unlike the K-harmonic method, the use of the
Argonne $v14$ potential with D-wave admixture, shows sensitivity to the use of
the PS (left) versus PV (right) pion-nucleon coupling. The use of the PS
coupling (left) shows that the additional contributions stemming from the
exchange contributions when added up (read-spread), contribute net $A,D$ GFFS
that are comparable with the impulse approximation (blue-spread) within
statistics. This is not the case when using the PV coupling for the Helium-4
GFFs (right), with the net result of form factor $A^{H}$ (red band) is lower
than result with impulse approximation (blue-spread), the diffractive minima
of form factor $D^{H}$ (green-spread) is shifted to the left in comparison to
that for impulse approximation (blue-spread). Again, the difference of form
factor $A^{H}$ stems primarily from the $T^{00}$ component of pair diagram in
the PS coupling. The difference in $D^{H}$ is caused by the extra nucleon GFF
A(k) in (27b) with the PS coupling, when compared to (33b) with PV coupling.
In both cases, the GFF diffractive minima from the impulse approximation,
shift rightward in comparison to the K-harmonic method.

r(fm) | $f_{0,1}(r)$ | $f_{1,0}(r)$ | $f_{t,0}(r)$
---|---|---|---
0.05 | 0.1644 | 0.2291 | 0
0.25 | 0.2626 | 0.3448 | 0.0339
0.50 | 0.5690 | 0.6491 | 0.0859
0.75 | 0.9069 | 0.9480 | 0.1264
1.00 | 1.0586 | 1.0735 | 0.1361
1.25 | 1.0512 | 1.0525 | 0.1255
1.50 | 0.9812 | 0.9694 | 0.1084
1.75 | 0.8965 | 0.8719 | 0.0916
2.00 | 0.8135 | 0.7783 | 0.0770
2.25 | 0.7369 | 0.6942 | 0.0650
2.50 | 0.6677 | 0.6203 | 0.0550
2.75 | 0.6056 | 0.5560 | 0.0469
3.00 | 0.5500 | 0.4999 | 0.0403
3.25 | 0.5002 | 0.4508 | 0.0349
3.50 | 0.4555 | 0.4077 | 0.0305
3.75 | 0.4153 | 0.3694 | 0.0268
4.00 | 0.3791 | 0.3354 | 0.0238
4.25 | 0.3464 | 0.3049 | 0.0212
4.50 | 0.3168 | 0.2775 | 0.0189
4.75 | 0.2900 | 0.2528 | 0.0169
5.00 | 0.2657 | 0.2305 | 0.0151
5.25 | 0.2436 | 0.2103 | 0.0136
5.50 | 0.2235 | 0.1920 | 0.0122
5.75 | 0.2052 | 0.1754 | 0.0109
6.00 | 0.1885 | 0.1603 | 0.0099
6.25 | 0.1733 | 0.1466 | 0.0089
6.50 | 0.1593 | 0.1341 | 0.0080
6.75 | 0.1466 | 0.1227 | 0.0072
7.00 | 0.1349 | 0.1123 | 0.0065
7.25 | 0.1242 | 0.1029 | 0.0059
7.50 | 0.1144 | 0.0943 | 0.0054
7.75 | 0.1054 | 0.0864 | 0.0049
8.00 | 0.0972 | 0.0792 | 0.0044
8.25 | 0.0896 | 0.0726 | 0.0040
8.50 | 0.0826 | 0.0666 | 0.0036
8.75 | 0.0762 | 0.0611 | 0.0033
9.00 | 0.0704 | 0.0561 | 0.0030
9.25 | 0.0650 | 0.0515 | 0.0027
9.50 | 0.0600 | 0.0472 | 0.0025
9.75 | 0.0554 | 0.0434 | 0.0023
10.00 | 0.0512 | 0.0388 | 0.0020
Table 1: The wave functions $f_{0,1}$, $f_{1,0}$, $f_{t,0}$ solutions to Eq.
(II.2).

Figure 5: Charge form factor of Helium-4 with the PS coupling(left) and PV
coupling (right) using the K-Harmonics method. The subscript “imp" refers to
the impulse approximation, while the subscripts “$X\pi$" and “XR" refer to the
exchange contributions from the pair diagram shown in Fig. 2a and recoil
contribution shown in Fig. 2d, respectively. “Exp" are the experimental
results for the charge form factor in Garcon  _et al._ (1994).

Figure 6: The charge form factor of Helium-4 with PS coupling (left) and PV
coupling (right) using the Monte Carlo method. The bands represent the
statistic uncertainty. The orange data points are the experimental results in
Garcon  _et al._ (1994).

##  VI Conclusions

We have analyzed the exchange contributions to the GFFs $A,D$ for Helium-4.
For that, we used a wavefunction that is maximally symmetric in the hyper-
radius following from the K-harmonic method, with no D-wave admixture for
simplicity. The results for the form factor $A^{H}$ are overall consistent
with the one we recently reported using the impulse approximation in He and
Zahed (2024a), whether we use the pion-nucleon PS or PV couplings. The
diffractive minima of the form factor $D^{H}$ shift leftward when the exchange
currents are added.

For comparison, we have also used a more realistic wavefunction following from
the Argonne $v14$ potential, with D-wave admixture. This approach is
numerically intensive, and relies on Monte-Carlo sampling to generate the
GFFs, with inherent statistical uncertainties that can be improved with longer
runs. In this case, the exchange contributions were found to be overall
consistent with the impulse approximation within statistics, when the pion-
nucleon PS coupling is used. When the PV coupling is used, the exchange
contributions are significant.

Our analysis of the charge form factor for Helium-4 carried for comparison in
Appendix B. The exchange contributions from the PS coupling to the Helium-4
charge form factor, are larger in comparison to the impulse approximation when
using the Argonne $v14$ potential, significantly improving the agreement with
experiment, while the PV coupling contributions are smaller and in agreement
with the impulse approximation. For both cases, the diffractive GFF minima
following from the impulse approximation, shift rightward in comparison to the
K-harmonic result.

Overall, most of the exchange contributions are small in the range
$k\leq\frac{1}{2}m_{N}$, leaving unchanged the mass radii. While they are
separately sizable at higher momenta, fortunately their sum total is small,
with the exception of the GFFs with PV coupling used with the Argonne $v14$
potential. Since the Helium charge form factor is better described by the PS
coupling using the Argonne $v14$ potential, we expect the PS results for the
GFFs with this potential to be more reliable.

The nucleon GFFs have been recently probed at JLab Ali  _et al._ (2019); Duran
_et al._ (2023), through near threshold photoproduction of charmonium. It
would be useful to extend these experiments to Helium-4, for a better
understanding of the role of the exchange currents in understanding the mass
distribution in light nuclei, with a particular interest in extracting the
pion GFF.

Acknowledgements

We thank Zein-Eddine Meziani and Robert Wiringa for discussions. FH is
supported by the National Science Foundation under CAREER Award PHY- 1847893.
IZ is supported by the Office of Science, U.S. Department of Energy under
Contract No. DE-FG-88ER40388. This research is also supported in part within
the framework of the Quark-Gluon Tomography (QGT) Topical Collaboration, under
contract no. DE-SC0023646.

##  Appendix A Trial wave functions

In table 1, we list the numerical values of three wave-functions $f_{0,1}$,
$f_{1,0}$, $f_{t,0}$ solution to the coupled Schrodinger equations (II.2) in
the regime $r<10$ fm. In the regime $r>10$ fm, we use the asymptotic
expressions in Wiringa (1991),

$\displaystyle f_{0,1}(r>10fm)$ $\displaystyle=$ $\displaystyle
1.88\left(\frac{e^{-0.850961r}}{r}\right)^{\frac{1}{3}},$ $\displaystyle
f_{1,0}(r>10fm)$ $\displaystyle=$ $\displaystyle
1.75\left(\frac{e^{-0.912554r}}{r}\right)^{\frac{1}{3}},$ $\displaystyle
f_{t,0}(r>10fm)$ $\displaystyle=$ $\displaystyle
0.0665\left(1-e^{-\frac{r^{2}}{4}}\right)$ (45) $\displaystyle\times$
$\displaystyle\left(\frac{3.6025}{r^{2}}+\frac{3.28748}{r}+1\right)$
$\displaystyle\times$
$\displaystyle\left(\frac{e^{-0.912554r}}{r}\right)^{\frac{1}{3}}.$

##  Appendix B Electric form factor of Helium-4

In this appendix, we summarise the results for the charge form factor for
Helium-4 using the K-Harmonic method, and the trial state using the Argonne
$v14$ potential. More specifically, the leading non-relativistic form for the
isoscalar electric operators for the pair diagram with PS and PV couplings are

$\displaystyle J_{X\pi,PS}^{0}(k)$ $\displaystyle=$ $\displaystyle
G_{M}^{S}\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}\frac{\vec{\sigma}_{1}.\vec{k}\vec{\sigma}_{2}.\vec{q_{2}}}{m_{N}w_{q_{2}}^{2}}.$
$\displaystyle+$ $\displaystyle(1\leftrightarrow 2)+{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)},$ $\displaystyle
J_{X\pi,PV}^{0}(k)$ $\displaystyle=$ $\displaystyle{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)}.$ (46)

The operator for the recoil correction for both PS and PV is

$\displaystyle J^{0}_{XR}$ $\displaystyle=$ $\displaystyle-\frac{g_{\pi
N}^{2}}{(2m_{N})^{2}}\tau_{1}\cdot\tau_{2}G_{E}^{S}\frac{q_{2}.k\sigma_{1}.q_{2}\sigma_{2}.q_{2}}{2m_{N}w_{q_{2}}^{4}}$
(47) $\displaystyle+$ $\displaystyle(1\leftrightarrow 2)+{\cal
O}\bigg{(}\frac{g_{\pi N}^{2}}{m_{N}^{4}}\bigg{)},$

where $G_{M}^{S}$ and $G_{M}^{S}$ represent the iso-singlet combination of
nucleon electric and magnetic form factors. The seagull diagram and pion
exchange diagrams only contribute to the isospin vector form factor, not
discussed here. The results of electric form factors can be easily obtained
since the operator $J_{X\pi,PS}^{0}(k)$ and $J^{0}_{XR}$ are quite similar as
$T_{X\pi,PS}^{00}(k)$ and $T^{00}_{XR}$ defined in Eq. (27a) and Eq. (39a).

In Fig. 5 we show the results for Helium-4 charge form factor with the PS
pion-nucleon coupling (left) and PV pion-nucleon coupling (right) using the
K-Harmonics method with no D-wave admixture. The labels are “$Imp$" (impulse
approximation), “$X\pi$" (pair correction), “$XR$" (recoil correction) and
“$Exp$" (experiment) from Garcon  _et al._ (1994). For comparison, we show in
Fig. 6 the results of the Monte-Carlo simulation using the wavefunction
generated from the Argonne $v14$ potential with D-wave admixture, with the
same labelings and color coding. The colored-spreads reflect on the
statistical uncertainties. In the latter, the PS coupling yields a charge form
factor that is in reasonable agreement with the data.

## References

  * Hackett  _et al._ (2023a) Daniel C. Hackett, Patrick R. Oare, Dimitra A. Pefkou, and Phiala E. Shanahan, “Gravitational form factors of the pion from lattice QCD,” Phys. Rev. D 108, 114504 (2023a), arXiv:2307.11707 [hep-lat] .
  * Hackett  _et al._ (2023b) Daniel C. Hackett, Dimitra A. Pefkou, and Phiala E. Shanahan, “Gravitational form factors of the proton from lattice QCD,” (2023b), arXiv:2310.08484 [hep-lat] .
  * Wang  _et al._ (2024) Bigeng Wang, Fangcheng He, Gen Wang, Terrence Draper, Jian Liang, Keh-Fei Liu, and Yi-Bo Yang ($\chi$QCD), “Trace anomaly form factors from lattice QCD,” Phys. Rev. D 109, 094504 (2024), arXiv:2401.05496 [hep-lat] .
  * Polyakov and Schweitzer (2018) Maxim V. Polyakov and Peter Schweitzer, “Forces inside hadrons: pressure, surface tension, mechanical radius, and all that,” Int. J. Mod. Phys. A 33, 1830025 (2018), arXiv:1805.06596 [hep-ph] .
  * Burkert  _et al._ (2023) V. D. Burkert, L. Elouadrhiri, F. X. Girod, C. Lorcé, P. Schweitzer, and P. E. Shanahan, “Colloquium: Gravitational form factors of the proton,” Rev. Mod. Phys. 95, 041002 (2023), arXiv:2303.08347 [hep-ph] .
  * Punjabi  _et al._ (2015) V. Punjabi, C. F. Perdrisat, M. K. Jones, E. J. Brash, and C. E. Carlson, “The Structure of the Nucleon: Elastic Electromagnetic Form Factors,” Eur. Phys. J. A 51, 79 (2015), arXiv:1503.01452 [nucl-ex] .
  * Ali  _et al._ (2019) A. Ali _et al._ (GlueX), “First Measurement of Near-Threshold J/$\psi$ Exclusive Photoproduction off the Proton,” Phys. Rev. Lett. 123, 072001 (2019), arXiv:1905.10811 [nucl-ex] .
  * Duran  _et al._ (2023) B. Duran _et al._ , “Determining the gluonic gravitational form factors of the proton,” Nature 615, 813–816 (2023), arXiv:2207.05212 [nucl-ex] .
  * Mamo and Zahed (2020) Kiminad A. Mamo and Ismail Zahed, “Diffractive photoproduction of $J/\psi$ and $\Upsilon$ using holographic QCD: gravitational form factors and GPD of gluons in the proton,” Phys. Rev. D 101, 086003 (2020), arXiv:1910.04707 [hep-ph] .
  * Guo  _et al._ (2021) Yuxun Guo, Xiangdong Ji, and Yizhuang Liu, “QCD Analysis of Near-Threshold Photon-Proton Production of Heavy Quarkonium,” Phys. Rev. D 103, 096010 (2021), arXiv:2103.11506 [hep-ph] .
  * Jo  _et al._ (2015) H. S. Jo _et al._ (CLAS), “Cross sections for the exclusive photon electroproduction on the proton and Generalized Parton Distributions,” Phys. Rev. Lett. 115, 212003 (2015), arXiv:1504.02009 [hep-ex] .
  * Abidin and Carlson (2009) Zainul Abidin and Carl E. Carlson, “Nucleon electromagnetic and gravitational form factors from holography,” Phys. Rev. D 79, 115003 (2009), arXiv:0903.4818 [hep-ph] .
  * Belitsky and Ji (2002) Andrei V. Belitsky and X. Ji, “Chiral structure of nucleon gravitational form-factors,” Phys. Lett. B 538, 289–297 (2002), arXiv:hep-ph/0203276 .
  * Polyakov and Sun (2019) Maxim V. Polyakov and Bao-Dong Sun, “Gravitational form factors of a spin one particle,” Phys. Rev. D 100, 036003 (2019), arXiv:1903.02738 [hep-ph] .
  * Won  _et al._ (2024) Ho-Yeon Won, Hyun-Chul Kim, and June-Young Kim, “Flavor structure of the energy-momentum tensor form factors of the proton,” Phys. Lett. B 850, 138489 (2024), arXiv:2302.02974 [hep-ph] .
  * Hatta (2024) Yoshitaka Hatta, “Accessing the gravitational form factors of the nucleon and nuclei through a massive graviton,” Phys. Rev. D 109, L051502 (2024), arXiv:2311.14470 [hep-ph] .
  * Lorcé (2024) Cédric Lorcé, “Electromagnetic and gravitational form factors of the nucleon,” in  _25th International Spin Symposium_ (2024) arXiv:2402.00429 [hep-ph] .
  * Mamo and Zahed (2022) Kiminad A. Mamo and Ismail Zahed, “J/$\psi$ near threshold in holographic QCD: A and D gravitational form factors,” Phys. Rev. D 106, 086004 (2022), arXiv:2204.08857 [hep-ph] .
  * Broniowski and Arriola (2008) Wojciech Broniowski and Enrique Ruiz Arriola, “Gravitational and higher-order form factors of the pion in chiral quark models,” Physical Review D 78 (2008), 10.1103/physrevd.78.094011.
  * Frederico  _et al._ (2009) T. Frederico, E. Pace, B. Pasquini, and G. Salmè, “Pion generalized parton distributions with covariant and light-front constituent quark models,” Physical Review D 80 (2009), 10.1103/physrevd.80.054021.
  * Fanelli  _et al._ (2016) Cristiano Fanelli, Emanuele Pace, Giovanni Romanelli, Giovanni Salmè, and Marco Salmistraro, “Pion generalized parton distributions within a fully covariant constituent quark model,” The European Physical Journal C 76 (2016), 10.1140/epjc/s10052-016-4101-1.
  * Freese and Cloët (2019) Adam Freese and Ian C. Cloët, “Gravitational form factors of light mesons,” Physical Review C 100 (2019), 10.1103/physrevc.100.015201.
  * Krutov and Troitsky (2021) A. F. Krutov and V. E. Troitsky, “Pion gravitational form factors in a relativistic theory of composite particles,” Phys. Rev. D 103, 014029 (2021), arXiv:2010.11640 [hep-ph] .
  * Raya  _et al._ (2022) Khepani Raya, Zhu-Fang Cui, Lei Chang, Jose-Manuel Morgado, Craig D. Roberts, and Jose Rodriguez-Quintero, “Revealing pion and kaon structure via generalised parton distributions *,” Chin. Phys. C 46, 013105 (2022), arXiv:2109.11686 [hep-ph] .
  * Xu  _et al._ (2024) Yin-Zhen Xu, Minghui Ding, Khépani Raya, Craig D. Roberts, José Rodríguez-Quintero, and Sebastian M. Schmidt, “Pion and kaon electromagnetic and gravitational form factors,” Eur. Phys. J. C 84, 191 (2024), arXiv:2311.14832 [hep-ph] .
  * Li and Vary (2024) Yang Li and James P. Vary, “Stress inside the pion in holographic light-front qcd,” Physical Review D 109 (2024), 10.1103/physrevd.109.l051501.
  * Broniowski and Ruiz Arriola (2024) Wojciech Broniowski and Enrique Ruiz Arriola, “Gravitational form factors of the pion and meson dominance,” (2024), arXiv:2405.07815 [hep-ph] .
  * Liu  _et al._ (2024a) Wei-Yang Liu, Edward Shuryak, and Ismail Zahed, “Gluon contributions to nucleon matrix elements in the QCD instanton vacuum,” To Appear (2024a).
  * Liu  _et al._ (2024b) Wei-Yang Liu, Edward Shuryak, Christian Weiss, and Ismail Zahed, “Pion gravitational form factors in the QCD instanton vacuum I,” (2024b), arXiv:2405.14026 [hep-ph] .
  * Sick (2008) Ingo Sick, “Precise root-mean-square radius of He-4,” Phys. Rev. C 77, 041302 (2008).
  * Riska (1989) Dan Olof Riska, “EXCHANGE CURRENTS,” Phys. Rept. 181, 207 (1989).
  * Mibe  _et al._ (2007) T. Mibe _et al._ (CLAS), “First measurement of coherent phi-meson photoproduction on deuteron at low energies,” Phys. Rev. C 76, 052202 (2007), arXiv:nucl-ex/0703013 .
  * Chang  _et al._ (2008) W. C. Chang _et al._ , “Forward coherent phi-meson photoproduction from deuterons near threshold,” Phys. Lett. B 658, 209–215 (2008), arXiv:nucl-ex/0703034 .
  * He and Zahed (2024a) Fangcheng He and Ismail Zahed, “Gravitational form factors of light nuclei: Impulse approximation,” Phys. Rev. C 109, 045209 (2024a), arXiv:2310.12315 [nucl-th] .
  * Garcia Martin-Caro  _et al._ (2023) Alberto Garcia Martin-Caro, Miguel Huidobro, and Yoshitaka Hatta, “Gravitational form factors of nuclei in the Skyrme model,” Phys. Rev. D 108, 034014 (2023), arXiv:2304.05994 [nucl-th] .
  * Hockert  _et al._ (1973) J. Hockert, D. O. Riska, M. Gari, and A. Huffman, “Meson exchange currents in deuteron electrodisintegration,” Nucl. Phys. A 217, 14–28 (1973).
  * Chemtob  _et al._ (1974) M. Chemtob, E. J. Moniz, and Mannque Rho, “Deuteron Electromagnetic Structure at Large Momentum Transfer,” Phys. Rev. C 10, 344–352 (1974).
  * Kloet and Tjon (1974) W. M. Kloet and J. A. Tjon, “MESON EXCHANGE EFFECTS ON THE CHARGE FORM-FACTORS OF THE TRINUCLEON SYSTEM,” Phys. Lett. B 49, 419–422 (1974).
  * Jackson  _et al._ (1975) A. D. Jackson, A. Lande, and D. O. Riska, “Pion exchange currents in elastic electron deuteron scattering,” Phys. Lett. B 55, 23–27 (1975).
  * Freese and Cosyn (2022) Adam Freese and Wim Cosyn, “Spatial densities of momentum and forces in spin-one hadrons,” Phys. Rev. D 106, 114013 (2022), arXiv:2207.10787 [hep-ph] .
  * He and Zahed (2024b) Fangcheng He and Ismail Zahed, “Deuteron gravitational form factors: exchange currents,” (2024b), arXiv:2401.09318 [nucl-th] .
  * Wiringa  _et al._ (1984) Robert B. Wiringa, R. A. Smith, and T. L. Ainsworth, “Nucleon Nucleon Potentials with and Without Delta (1232) Degrees of Freedom,” Phys. Rev. C 29, 1207–1221 (1984).
  * Castilho Alcaras and Pimentel Escobar (1974) J. Castilho Alcaras and B. Pimentel Escobar, “Application of the Basic Approxirnation of the K-Harmonics Method to the Alpha Particle,” Revista Brasileira de Física 4, 83 (1974).
  * Wiringa (1991) Robert B. Wiringa, “Variational calculations of few-body nuclei,” Phys. Rev. C 43, 1585–1598 (1991).
  * Lomnitz-Adler and Pandharipande (1980) J. Lomnitz-Adler and V. R. Pandharipande, “A simple and realistic triton wave function,” Nucl. Phys. A 342, 404–420 (1980).
  * Lomnitz-Adler  _et al._ (1981) J. Lomnitz-Adler, V. R. Pandharipande, and R. A. Smith, “Monte Carlo calculations of triton and 4 He nuclei with the Reid potential,” Nucl. Phys. A 361, 399–411 (1981).
  * Garcon  _et al._ (1994) M. Garcon _et al._ , “Tensor polarization in elastic electron - deuteron scattering in the momentum transfer range 3.8-fm**-1 $<$= q $<$= 4.6-fm**-1,” Phys. Rev. C 49, 2516–2537 (1994).


17 |  Ye2016Face (Ye et al., 2016) | MM | 2016 | 1 | MT
18 |  Kucher2017Active (Kucher et al., 2017) | TIIS | 2017 | 1 | IL
19 |  Ratner2017Snorkel (Ratner et al., 2017) | VLDB | 2017 | 2 | MT
20 |  Bernard2018VIAL (Bernard et al., 2018b) | TVC | 2018 | 1 | IL, DOS, FE, DL, SA
21 |  Bernard2018VIAL (Bernard et al., 2018b) | TVC | 2018 | 2 | IL, DOS, MT, FE, DL
22 |  Felix2018Exploratory (Felix et al., 2018) | UIST | 2018 | 1 | IL, QA, LI
23 |  Zhang2018Collaborative (Zhang et al., 2018) | MM | 2018 | 2 | IL, DL
24 |  Shang2019Annotating (Shang et al., 2019) | ICMR | 2019 | 2 | IL
25 |  Xiang2019Interactive (Xiang et al., 2019) | VAST | 2019 | 2 | IL, DOS, DL
26 |  Liu2019Interactive (Liu et al., 2019) | TVCG | 2019 | 3 | IL, DOS, FE, DL, QA
27 |  Choi2019AILA (Choi et al., 2019) | CHI | 2019 | 6 | IL, DOS, FE
28 |  Wang2019DeepIGeoS (Wang et al., 2019) | TPAMI | 2019 | 1 | IL, DL, SA
29 |  Halter2019VIAN (Halter et al., 2019) | CGF | 2019 | 2 | IL, FE, DL
30 |  Evensen2020Ruler (Evensen et al., 2020) | EMNLP | 2020 | 2 | IL, DOS, MT
31 |  Baeuerle2020Classifier (Bäuerle et al., 2020) | CGF | 2020 | 1 | MT, DL, QA
32 |  Lekschas2020Peax (Lekschas et al., 2020) | CGF | 2020 | 3 | IL, DOS
33 |  Oelen2021Crowdsourcing (Oelen et al., 2021) | IUI | 2021 | 2 | IL, DOS, DL
34 |  Rietz2021Cody (Rietz and Maedche, 2021) | CHI | 2021 | 4 | IL, MT, DL
35 |  Zhang2021ChartNavigator (Zhang et al., 2021b) | TKDE | 2021 | 1 | DOS, FE
36 |  Zhang2021MI3 (Zhang et al., 2021a) | TIIS | 2021 | 3 | IL, DOS, MT, FE, DL, QA, SA

###  A.3. A Coding Example

Table 4.  Coding the flowchart in AILA (Choi et al., 2019). The final codes are abbreviated by initials. A slash (/) in the final code refers to the case that the theme is excluded from the final code.  Preliminary Code | Theme | Final Code
---|---|---
preprocessing - stemming | feature extraction | FE
preprocessing - bag of words | feature extraction | FE
preprocessing - term-document matrix | feature extraction | FE
preprocessing - word vector | feature extraction | FE
preprocessing - sentence vector | feature extraction | FE
document analysis - re-ordering - selecting | data object selection | DOS
document analysis - re-ordering - sorting | data object selection | DOS
document classifier - interactive attentive module - attention weight | preprocessing | /
document classifier - interactive attentive module - prediction score | preprocessing | /
labeling interface - document embedding | data object selection | DOS
labeling interface - document visualization | labeling interface | IL

AILA (Choi et al., 2019) supports labeling document classification. Figure 6
in its paper is a flowchart. The figure contains nested blocks, and we use a
hyphen to denote the nesting relation in the preliminary code. For example,
“preprocessing - stemming” refers to a block named “stemming” placed in a
block named “preprocessing”. We decompose the nested blocks to the lowest
level instead of treating the root block, such as “preprocessing”, as a single
phrase because different nested blocks may correspond to different themes. The
flowchart contains 11 phases, all categorized as preliminary codes for modules
because they describe actions. The 11 phrases are grouped into themes and
final codes as shown in Table 4. We categorize “attention weight” and
“prediction score” as preprocessing because they prepare the data visualized
in the labeling interface. “Labeling interface - document embedding” is
categorized as “data object selection” because it refers to an interactive
projection where the annotator can select data objects to label.

##  Appendix B Graph-Theoretic Constraints on Data Labeling Workflows

###  B.1. Notations

  * •

$G=(V,E)$: a directed graph modeling a data labeling tool.

  * •

$V$: a set of nodes, each corresponding to a software module, or
initialization, decision, and exit node.

  * •

$E$: a set of edges specifying the execution order.

  * •

$type(v)$: the type of a node $v$, which takes one of the following values:
initialization, process, decision, exit.

  * •

$fun(v)$: the function of a node $v$. When $type(v)$ takes value in $\\{$
initialization, decision, exit $\\}$, $fun(v)=type(v)$. When
$type(v)=process$, $fun(v)$ takes one of the following values:
interactiveLabeling, dataObjectSelection, modelTraining, featureExtraction,
defaultLabeling, qualityAssurance, stoppageAnalysis, labelIdeation.

  * •

$input(v)$: the set of input states to a node $v$.

  * •

$output(v)$: the output state of a node $v$.

  * •

$Exec(G)$: the set of all the directed walks on the graph from the
initialization node to the exit node.

###  B.2. Constraints on the Workflow Graph

Valid Flowchart:

  * •

The graph contains no parallel edges (i.e., $E$ is not a multi-set).

  * •

The graph contains one initialization node (denote as $v_{s}$).

$\exists!v(v\in V\wedge type(v)=initialization)$

  * •

The graph contains one exit node (denote as $v_{e}$).

$\exists!v(v\in V\wedge type(v)=exit)$

  * •

A process node has outdegree 1.

$\forall v(v\in V\wedge type(v)=process\rightarrow deg^{+}(v)=1)$

  * •

A decision node has outdegree 2.

$\forall v(v\in V\wedge type(v)=decision\rightarrow deg^{+}(v)=2)$

  * •

An initialization node has indegree 0 and outdegree 1.

$\forall v(v\in V\wedge type(v)=initialization\rightarrow deg^{+}(v)=1\wedge
deg^{-}(v)=0)$

  * •

An exit node has outdegree 0.

$\forall v(v\in V\wedge type(v)=exit\rightarrow deg^{+}(v)=0)$

  * •

All the nodes can be reached from the initialization node. The exit node can
be reached from all the nodes.

$\forall v(v\in V\rightarrow\exists exec(exec\in Exec(G)\wedge
exec=(v_{s},...,v,...,v_{e})))$

  * •

No self loops.

$\forall v(v\in V\rightarrow(v,v)\notin E)$

Input Initialized:

$\forall exec=(v_{s},...,v_{e})\in Exec(G)\forall v_{i}\in exec\forall
input\in input(v_{i})$ $\exists j(j<i\wedge output(v_{j})=input)$

No Redundancy:

  * •

After a module is visited, it should not be revisited until at least one of
its inputs’ value has been changed.

$\forall exec=(v_{1},...,v_{k})\in Exec(G)$ $((\exists 1\leq i<j\leq
k(v_{i}=v_{j}))\rightarrow$ $(\exists i<l<j(output(v_{l})\in input(v_{j}))))$

  * •

After a module is visited, its output(s) should be used by a module.

$\forall exec=(v_{1},...,v_{k})\in Exec(G)$ $\forall 1\leq i\leq k(output\in
output(v_{i})\rightarrow$ $(\exists i<j\leq k(output\in
input(v_{j})\wedge\nexists i<l<j(output\in output(v_{l})))))$

Involve Labeling:

$\forall exec=(v_{s},...,v_{e})\in Exec(G)\exists
i(fun(v_{i})=interactiveLabeling)$

##  Appendix C Customized Workflow Template

Below is the code snippet to declare a customized workflow template for
webpage classification.

⬇

1/** The declaration of a minimal webpage classification workflow. */

2export default {

3 nodes: [

4 {

5 label: ’initialization’,

6 type: ’Initialization’,

7 value: {

8 dataType: ’Webpage’,

9 labelTasks: [’Classification’],

10 },

11 layout: { x: 40, y: 40 },

12 },

13 {

14 label: ’random sampling’,

15 type: ’DataObjectSelection’,

16 value: [’Random’, { params: { nBatch: { value: 2 } } }],

17 layout: { x: 160, y: 40 },

18 },

19 {

20 label: ’grid matrix’,

21 type: ’InteractiveLabeling’,

22 value: [’GridMatrix’, {

23 params: {

24 nRows: { value: 1 },

25 nColumns: { value: 2 },

26 }

27 }],

28 layout: { x: 280, y: 40 },

29 },

30 {

31 label: ’check all labeled’,

32 type: ’StoppageAnalysis’,

33 value: ’AllLabeled’,

34 layout: { x: 400, y: 40 },

35 },

36 {

37 label: ’stop?’,

38 type: ’Decision’,

39 layout: { x: 400, y: 130 },

40 },

41 {

42 label: ’exit’,

43 type: ’Exit’,

44 layout: { x: 410, y: 220 },

45 },

46 ],

47 edges: [

48 { source: ’initialization’, target: ’random sampling’ },

49 { source: ’random sampling’, target: ’grid matrix’ },

50 { source: ’grid matrix’, target: ’check all labeled’ },

51 { source: ’check all labeled’, target: ’stop?’ },

52 { source: ’stop?’, target: ’exit’, condition: true },

53 { source: ’stop?’, target: ’random sampling’, condition: false },

54 ],

55}

##  Appendix D Customized Algorithm Module

Below is the code snippet to declare a customized default labeling module for
span annotation by detecting nouns. The module is implemented in the form of
an algorithm server.

⬇

1# A customized default labeling module

2# for span annotation by detecting nouns.

3import json

4import nltk

5import tornado.httpserver

6import tornado.ioloop

7import tornado.options

8import tornado.web

9import uuid

10from typing import List

11

12# Core function

13def predict(data_object: dict) -> List[dict]:

14 """ Create default spans by detecting nouns. """

15 sentence = data_object[’content’]

16 pos_tag = nltk.pos_tag(sentence.split())

17 start = 0

18 end = 0

19 spans = []

20 for i, (segment, tag) in enumerate(pos_tag):

21 start = end + 1 if i != 0 else end

22 end = start + len(segment)

23 # filter tags that are not nouns

24 if tag not in [’NN’, ’NNP’, ’NNS’]:

25 continue

26 spans.append({

27 ’text’: segment,

28 ’start’: start,

29 ’end’: end,

30 ’category’: ’subject’,

31 ’uuid’: str(uuid.uuid4()),

32 })

33 return spans

34

35class DefaultLabelingHandler(tornado.web.RequestHandler):

36 def post(self):

37 self.set_header(’Access-Control-Allow-Origin’, ’*’)

38 data_objects = json.loads(self.request.body)[’dataObjects’]

39 labels = [{ ’spans’: predict(d) } for d in data_objects]

40 self.write({ ’labels’: labels })

41

42def main():

43 tornado.options.parse_command_line()

44 http_server = tornado.httpserver.HTTPServer(

45 tornado.web.Application(

46 handlers=[(r’/defaultLabels’, DefaultLabelingHandler)],

47 debug=True,

48 )

49 )

50 http_server.listen(8007)

51 print(’Serve at http://localhost:8007/defaultLabels’)

52 tornado.ioloop.IOLoop.instance().start()

53

54if __name__ == "__main__":

55 main()

##  Appendix E User Study Instructions

The following are the instructions we gave the participants for the 4 labeling
tool development tasks.

###  E.1. Task 1 (Image Segmentation)

Please create a labeling tool for image segmentation according to the
following specification of the labeling tool’s functionality. Please do not
use OneLabeler’s built-in templates for this task.

Specification:

  1. (1)

In the image segmentation tool, data objects are iteratively selected for
annotators to label. 1 data object is randomly selected each time.

  2. (2)

The interface displays the 1 data object in a single object display.

  3. (3)

After annotating the 1 data object, the system checks whether all the data
objects are labeled.

  4. (4)

If all the data objects are labeled, the labeling STOPS.

  5. (5)

Otherwise, goto step 1.

Please tell the experimenter when you have finished.

###  E.2. Task 2 (Image Classification)

Based on the previous image segmentation workflow in Task 1, please modify the
workflow to create another labeling tool for image classification according to
the following specification of the labeling tool’s functionality.

Specification:

  1. (1)

In the image classification tool, data objects are iteratively selected for
annotators to label. 16 data objects are randomly selected each time.

  2. (2)

The interface displays the 16 data objects in a grid matrix with 4 rows and 4
columns.

  3. (3)

After annotating the 16 data objects, the system checks whether all the data
objects are labeled.

  4. (4)

If all the data objects are labeled, the labeling STOPS.

  5. (5)

Otherwise, goto step 1.

Please tell the experimenter when you have finished.

###  E.3. Task 3 (Machine-aided Image Classification)

Based on the previous image classification workflow in Task 2, please modify
the workflow to create another labeling tool for image classification
according to the following specification of the labeling tool’s functionality.

Specification:

  1. (1)

At the beginning, SVD features are extracted for the data objects.

  2. (2)

Then, data objects are iteratively selected for annotators to label. 16 data
objects are randomly selected each time.

  3. (3)

The selected data objects are then assigned default labels by a decision tree
classifier.

  4. (4)

The interface displays the 16 data objects in a grid matrix with 4 rows and 4
columns.

  5. (5)

After annotating the 16 data objects, the system checks whether all the data
objects are labeled.

  6. (6)

If all the data objects are labeled, the labeling STOPS.

  7. (7)

Otherwise, the decision tree classifier used for default labeling is updated
by retraining.

  8. (8)

Then, goto step 2.

Please tell the experimenter when you have finished.

###  E.4. Task 4 - Option 1 (Customize Data Type)

Based on the given image classification workflow template, please modify the
workflow to create another labeling tool for webpage classification according
to the following specification of the labeling tool’s functionality.

Specification:

  1. (1)

In the webpage classification tool, data objects are iteratively selected for
annotators to label. 2 data objects are randomly selected each time.

  2. (2)

The interface displays the 2 data objects in a grid matrix with 1 row and 2
columns.

  3. (3)

After annotating the 2 data objects, the system checks whether all the data
objects are labeled.

  4. (4)

If all the data objects are labeled, the labeling STOPS.

  5. (5)

Otherwise, goto step 1.

The webpage data type is not currently supported in OneLabeler. Before you
start to build the labeling tool in the visual programming interface, you will
need to first create the new webpage data type with some coding. To create
this new data type, please follow the instructions below.

  1. (1)

Enter the “./client” folder of OneLabeler’s source code.

  2. (2)

Execute “npm run customize” in the command-line, and choose suitable options
in the command-line interface to create template code files for the webpage
data type.

  3. (3)

Revise template code files to create the webpage data type.

The webpage data type should satisfy this specification:

When using a grid matrix to display webpage data objects, each grid should
display the webpage content. Each webpage data object should fill the grid
that contains it.

Additional Tips:

  * •

You may first read the “README.md” file in the created template code files,
which gives you an overview of what the template code files are.

  * •

When editing the code, you may refer to the HTML iframe tag API
(https://developer.mozilla.org/docs/Web/HTML/Element/iframe) if you are
unfamiliar with it.

  * •

In this task, you will not need to edit any code files other than the files
created by the “npm run customize” command.

  * •

You may refer to the “Customization” section of the documentation website.

  * •

If you do not have web development experience, you may ask the experimenter to
help you with the programming part.

Please tell the experimenter when you have finished.

###  E.5. Task 4 - Option 2 (Reproduce Your Tool)

Please use OneLabeler to reproduce your labeling tool. The reproduction should
have similar major functionalities as your labeling tool but does not need to
be the same. For this task, you may ask the experimenter for help.