# Local spatial densities for composite spin-3/2 systems

H. Alharazin  Institut für Theoretische Physik II, Ruhr-Universität Bochum,
D-44780 Bochum, Germany  B.-D. Sun  Guangdong Provincial Key Laboratory of
Nuclear Science, Institute of Quantum Matter,
South China Normal University, Guangzhou 510006, China  Guangdong-Hong Kong
Joint Laboratory of Quantum Matter,
Southern Nuclear Science Computing Center,
South China Normal University, Guangzhou 510006, China  Helmholtz-Institut für
Strahlen- und Kernphysik and Bethe Center for Theoretical Physics, Universität
Bonn, D-53115 Bonn, Germany  E. Epelbaum  Institut für Theoretische Physik II,
Ruhr-Universität Bochum, D-44780 Bochum, Germany  J. Gegelia  Institut für
Theoretische Physik II, Ruhr-Universität Bochum, D-44780 Bochum, Germany
Tbilisi State University, 0186 Tbilisi, Georgia  U.-G. Meißner  Helmholtz-
Institut für Strahlen- und Kernphysik and Bethe Center for Theoretical
Physics, Universität Bonn, D-53115 Bonn, Germany  Institute for Advanced
Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics,
Forschungszentrum Jülich, D-52425 Jülich, Germany  Tbilisi State University,
0186 Tbilisi, Georgia

(22 December 2022)

###### Abstract

The definition of local spatial densities by using sharply localized one-
particle states is applied to spin-3/2 systems. Matrix elements of the
electromagnetic current and the energy-momentum tensor are considered and
integral expressions of associated spatial distributions in terms of form
factors are derived.

##  I Introduction

In close analogy with the electric charge density of hadrons Hofstadter:1958 ;
Ernst:1960zza ; Sachs:1962zzc  it has been suggested to interpret the Fourier
transforms of the gravitational form factors in the Breit frame as local
densities of physical quantities characterizing various composite systems
Polyakov:2002wz ; Polyakov:2002yz ; Polyakov:2018zvc . The identification of
the spatial densities with the Fourier transforms of the electromagnetic and
gravitational form factors in the Breit frame, especially for systems with
intrinsic sizes comparable to their Compton wavelengths, has been questioned
in Refs. Burkardt:2000za ; Miller:2007uy ; Miller:2009qu ; Miller:2010nz ;
Jaffe:2020ebz ; Miller:2018ybm ; Freese:2021czn . This issue has raised much
interest recently Lorce:2020onh ; Lorce:2022cle ; Lorce:2018egm ; Chen:2022smg
; Guo:2021aik ; Panteleeva:2021iip ; Epelbaum:2022fjc ; Panteleeva:2022khw ;
Kim:2021kum ; Kim:2021jjf ; Kim:2022bia ; Kim:2022wkc ; Freese:2021mzg ;
Freese:2022fat ; Carlson:2022eps . On the one hand, the formalism of Wigner
phase space distributions is utilized in Ref. Lorce:2018egm  and in subsequent
publications. On the other hand, two-dimensional densities in the transverse
plane obtained in the formalism of light front dynamics by integrating over
the $x^{-}$ coordinate have been considered as the only possible true internal
densities Freese:2021czn ; Freese:2022fat . Note that two-dimensional
densities have been also considered earlier in Ref. Lorce:2018egm . A
definition of spatial densities of local operators using sharply localized
wave packet states, applicable to systems with arbitrary Compton wavelengths,
has been suggested in Ref. Epelbaum:2022fjc , see also Ref. Fleming:1974af
for a related earlier study. Specifying the one-particle state of a spin-0
system by a spherically symmetric wave packet localized in space and taking
the size of the packet much smaller than all internal characteristic scales of
the considered system, spatial charge distributions have been defined in the
zero average momentum frame (ZAMF). The new definition has been also
generalized to moving Lorentz frames. Recently this definition has been also
applied to electromagnetic densities of spin-1/2 systems Panteleeva:2022khw ,
and to gravitational densities of spin-0 and spin-1/2 systems, defined via the
matrix elements of the energy-momentum tensor (EMT) Panteleeva:2022uii .

In the current work we consider spatial densities corresponding to the
electromagnetic current and the EMT for spin-3/2 systems. We obtain the
corresponding expressions in terms of form factors and discuss their physical
interpretation.

Our work is organized as follows. In Sec. II we define spatial densities
corresponding to the electromagnetic current for spin-3/2 systems. Section III
deals with the one-particle matrix elements of the EMT. In Sec. IV we discuss
the large-distance behavior of various distributions of the delta resonance,
which is the most studied composite spin-3/2 system, in chiral EFT. We
summarize our results in Sec. V. The appendix contains the lengthy expressions
of various quantities.

##  II Electromagnetic densities in the zero average momentum frame

We choose the four-momentum eigenstates $|p,s\rangle$ characterizing our
spin-3/2 system to be normalized as

$\langle
p_{f},s^{\prime}|p_{i},s\rangle=2E(2\pi)^{3}\delta_{s^{\prime}s}\delta^{(3)}({\bf
p}_{f}-{\bf p}_{i})\,,$ (1)

where $(p_{i},s)$ and $(p_{f},s^{\prime})$ are the momentum and polarization
of the initial and final state, respectively. Further, $p=(E,{\bf p})$ with
$E=\sqrt{m^{2}+{\bf p}^{2}}$, where $m$ is the particle’s mass.

To define the spatial densities via the matrix elements of local operators we
use normalizable Heisenberg-picture states written in terms of wave packets as
follows:

$|\Phi,{\bf X},s\rangle=\int\frac{d^{3}{p}}{\sqrt{2E(2\pi)^{3}}}\,\phi(s,{\bf
p})\,e^{-i{\bf p}\cdot{\bf X}}|p,s\rangle,$ (2)

where the parameters ${\bf X}$ are interpreted as the coordinates of the
center of the charge or mass distribution, corresponding to the operator under
consideration, and the profile function satisfies the normalization condition

$\int d^{3}{p}\,|\phi(s,{\bf p})|^{2}=1\,.$ (3)

To define the density distributions of the system we use spherically symmetric
wave packets and profile functions of which $\phi(s,{\bf p})=\phi({\bf
p})=\phi(|{\bf p}|)$ are also spin-independent. The average of the three-
momentum of the system vanishes in states corresponding to such packets, thus
they describe the system in the ZAMF. For our calculations it is convenient to
define dimensionless profile functions

$\phi({\bf p})=R^{3/2}\,\tilde{\phi}(R{\bf p})\,,$ (4)

where $R$ specifies the size of the wave packet. Small values of $R$
correspond to sharp localization of the packet.

The matrix elements of the electromagnetic current operator between momentum
eigenstates of a spin-3/2 system can be parameterized in terms of four form
factors, see Ref. Pascalutsa:2006up  for a review. We use here the notation of
Ref. Cotogno:2019vjb :

$\displaystyle\langle p_{f},s^{\prime}|J_{\mu}|p_{i},s\rangle$
$\displaystyle=$
$\displaystyle-\bar{u}^{\beta}(p_{f},s^{\prime})\Biggl{[}\frac{P_{\mu}}{m}\left(g_{\alpha\beta}F_{1,0}^{V}(q^{2})-\frac{q_{\alpha}q_{\beta}}{2m^{2}}\,F_{1,1}^{V}(q^{2})\right)$
(5) $\displaystyle+$
$\displaystyle\frac{i}{2m}\sigma_{\mu\rho}q^{\rho}\left(g_{\alpha\beta}F^{V}_{2,0}(q^{2})-\frac{q_{\alpha}q_{\beta}}{2m^{2}}\,F^{V}_{2,1}(q^{2})\right)\Biggr{]}u^{\alpha}(p_{i},s)\,,$

where $P=(p_{i}+p_{f})/2$, $q=p_{f}-p_{i}$. In terms of these variables, the
energies are given as $E=(m^{2}+{\bf P}^{2}-{\bf P}\cdot{\bf q}+{\bf
q}^{2}/4)^{1/2}$ and $E^{\prime}=(m^{2}+{\bf P}^{2}+{\bf P}\cdot{\bf q}+{\bf
q}^{2}/4)^{1/2}$. The spinors of the spin-3/2 states are defined as follows:

$\displaystyle u^{\mu}(p,s)$ $\displaystyle=$
$\displaystyle\sum_{\lambda,\sigma}\langle
1\lambda,\frac{1}{2}\sigma|\frac{3}{2}s\rangle
e^{\mu}(p,\lambda)u(p,\sigma)\,,$ $\displaystyle e^{\mu}(p,\lambda)$
$\displaystyle=$ $\displaystyle\left(\frac{\hat{\bf e}_{\lambda}\cdot{\bf
p}}{m},\hat{\bf e}_{\lambda}+\frac{{\bf p}(\hat{\bf e}_{\lambda}\cdot{\bf
p})}{m(p_{0}+m)}\right)\,,$ $\displaystyle u(p,\sigma)$ $\displaystyle=$
$\displaystyle\sqrt{p_{0}+m}\left(\chi_{\sigma},\frac{{\bm{\sigma}}\cdot{\bf
p}}{p_{0}+m}\,\chi_{\sigma}\right)^{T}\,,$ (6)

where $\langle 1\lambda,\frac{1}{2}\sigma|\frac{3}{2}s\rangle$ are the
pertinent Clebsh-Gordon coefficients and

$\hat{\bf e}_{+}=-\frac{1}{\sqrt{2}}\left(1,i,0\right),\qquad\hat{\bf
e}_{0}=\left(0,0,1\right),\qquad\hat{\bf
e}_{-}=\frac{1}{\sqrt{2}}\left(1,-i,0\right).$ (7)

The Dirac spinors are normalized as
$\bar{u}(p,s^{\prime})u(p,s)=2m\,\delta_{s^{\prime}s}$. The matrix element of
the electromagnetic current operator in localized states is given by

$\displaystyle j^{\mu}_{\phi}(s^{\prime},s,{\bf r})$ $\displaystyle\equiv$
$\displaystyle\langle\Phi,{\bf X},s^{\prime}|\hat{J}^{\mu}({\bf
x},0)|\Phi,{\bf X},s\rangle$ (8) $\displaystyle=$
$\displaystyle-\int\frac{d^{3}{P}\,d^{3}{q}}{(2\pi)^{3}\sqrt{4EE^{\prime}}}\,\bar{u}^{\beta}\left(P+\frac{q}{2},\sigma^{\prime}\right)\Biggl{[}\frac{P_{\mu}}{m}\left(g_{\alpha\beta}F_{1,0}^{V}(q^{2})-\frac{q_{\alpha}q_{\beta}}{2m^{2}}\,F_{1,1}^{V}(q^{2})\right)$
$\displaystyle+$
$\displaystyle\frac{i}{2m}\sigma_{\mu\rho}q^{\rho}\left(g_{\alpha\beta}F^{V}_{2,0}(q^{2})-\frac{q_{\alpha}q_{\beta}}{2m^{2}}\,F^{V}_{2,1}(q^{2})\right)\Biggr{]}u^{\alpha}\left(P-\frac{q}{2},\sigma\right)\phi\bigg{(}{\bf
P}-\frac{\bf q}{2}\bigg{)}\phi^{\star}\bigg{(}{\bf P}+\frac{\bf
q}{2}\bigg{)}e^{-i{\bf q}\cdot{\bf r}},$

where ${\bf r}={\bf x}-{\bf X}$. By applying the method of dimensional
counting of Ref. Gegelia:1994zz , the leading contributions in Eq. (8) for
$R\to 0$ can be obtained without specifying the form of the form-factors and
the profile function $\phi(|\bf{p}|)$. Provided that the form factors
$F_{1,0}^{V}(q^{2})$, $F_{1,1}^{V}(q^{2})$, $F_{2,0}^{V}(q^{2})$ and
$F_{2,1}^{V}(q^{2})$ decay for large $q^{2}$ as $1/q^{2}$, $1/q^{4}$,
$1/q^{3}$, $1/q^{5}$ (or faster), respectively, the only non-vanishing
contribution for $R\to 0$ is generated from the region of integration where
${\bf P}$ is large. It can be obtained by substituting ${\bf P}={\bf Q}/R$,
expanding the resulting integrand in Eq. (8) in powers of $R$ around $R=0$ and
keeping the leading order term for each component separately. Introducing
$\hat{n}={\bf Q}/|{\bf Q}|$ and using the spherical symmetry of the wave
packet, the integration over $|{\bf Q}|$ can be carried out without specifying
the radial profile function. The result of the integration is given below in
terms of irreducible tensors and the multipole operators which are defined as
follows. The $n$-th rank irreducible tensors in coordinate and momentum
spaces, respectively, are given by ($r\neq 0$)

$Y_{n}^{i_{1}i_{2}\cdots
i_{n}}(\Omega_{r})=\frac{(-1)^{n}}{(2n-1)!!}\,r^{n+1}\partial^{i_{1}}\partial^{i_{2}}\cdots\partial^{i_{n}}\frac{1}{r}\,,\qquad
Y_{n}^{i_{1}i_{2}\cdots
i_{n}}(\Omega_{p})=\frac{(-1)^{n}}{(2n-1)!!}\,p^{n+1}\partial^{i_{1}}\partial^{i_{2}}\cdots\partial^{i_{n}}\frac{1}{p}\,.$
(9)

The quadrupole- and octupole-operators, $\hat{Q}^{ij}$ and $\hat{O}^{ijk}$,
for a spin-3/2 system are given in terms of the spin operator $\hat{S}^{i}$
via

$\displaystyle\hat{Q}^{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\hat{S}^{i}\hat{S}^{j}+\hat{S}^{j}\hat{S}^{i}-\frac{2}{3}S(S+1)\delta^{ij}\right)\,,$
$\displaystyle\hat{O}^{ijk}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left(\hat{S}^{i}\hat{S}^{j}\hat{S}^{k}+\hat{S}^{j}\hat{S}^{i}\hat{S}^{k}+\hat{S}^{k}\hat{S}^{j}\hat{S}^{i}+\hat{S}^{j}\hat{S}^{k}\hat{S}^{i}+\hat{S}^{i}\hat{S}^{k}\hat{S}^{j}+\hat{S}^{k}\hat{S}^{i}\hat{S}^{j}\right.$
(10) $\displaystyle-$
$\displaystyle\left.\frac{6S(S+1)-2}{5}\left(\delta^{ij}\hat{S}^{k}+\delta^{ik}\hat{S}^{j}+\delta^{kj}\hat{S}^{i}\right)\right)\,,$

with $i,j,k=1,2,3$.

The results for the matrix elements after integration over $|{\bf Q}|$ in Eq.
(8) have the form:

$\displaystyle j^{0}_{\phi}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf
r}}\frac{1}{4\pi}\int d^{2}\hat{n}\left\\{
\mathcal{Z}_{0}(-q_{\perp}^{2})~{}\delta_{s^{\prime}s}+\left[\mathcal{Z}_{1}(-q_{\perp}^{2})~{}\hat{n}^{k}\hat{n}^{l}+\mathcal{Z}_{2}(-q_{\perp}^{2})~{}\frac{q^{k}_{\perp}q^{l}_{\perp}}{{m^{2}}}\right]{\hat{Q}^{kl}_{s^{\prime}s}}\right\\}\,$
(11) $\displaystyle=$
$\displaystyle\rho^{C}_{0}\left(r\right)\delta_{s^{\prime}s}+\rho^{C}_{2}\left(r\right)Y_{2}^{kl}\left(\Omega_{r}\right)\hat{Q}_{s^{\prime}s}^{kl},$
(12) $\displaystyle j^{i}_{\phi}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf
r}}\frac{i}{4\pi}\int
d^{2}\hat{n}\left\\{\left[\mathcal{A}_{0}(-q_{\perp}^{2})~{}\hat{n}^{i}\hat{n}^{l}\epsilon^{kln}+\mathcal{A}_{1}(-q_{\perp}^{2})\left(~{}\delta^{kl}-\hat{n}^{k}\hat{n}^{l}\right)\epsilon^{iln}\right]\frac{q_{\perp}^{n}}{m}{\hat{S}^{k}_{s^{\prime}s}}\right.$
(13) $\displaystyle+$
$\displaystyle\bigg{[}\left(\mathcal{A}_{2}(-q_{\perp}^{2})~{}\hat{n}^{t}\hat{n}^{z}+\mathcal{A}_{3}(-q_{\perp}^{2})~{}\frac{q_{\perp}^{t}q_{\perp}^{z}}{{m^{2}}}\right)\hat{n}^{i}\hat{n}^{l}\epsilon^{kln}$
$\displaystyle+$
$\displaystyle\left(\mathcal{A}_{4}(-q_{\perp}^{2})\hat{n}^{t}\hat{n}^{z}+\mathcal{A}_{5}(-q_{\perp}^{2})~{}\frac{q_{\perp}^{t}q_{\perp}^{z}}{{m^{2}}}\right)\epsilon^{iln}\left(\delta^{kl}-\hat{n}^{k}\hat{n}^{l}\right)\bigg{]}\frac{q_{\perp}^{n}}{m}{\hat{O}^{ktz}_{s^{\prime}s}}\Biggl{\\}}\,$
$\displaystyle=$
$\displaystyle{i\epsilon^{ikn}\hat{S}_{s^{\prime}s}^{k}Y_{1}^{n}\frac{1}{m}\frac{d}{dr}\rho^{M}_{1}\left(r\right)\text{
}+i\epsilon^{ikn}\hat{O}_{s^{\prime}s}^{ktz}Y_{3}^{ntz}\,\frac{r^{3}}{m^{3}}\left(\frac{1}{r}\frac{d}{dr}\right)^{3}\rho^{M}_{3}\left(r\right),}$
(14)

where $q_{\perp}^{2}={\bf q}^{2}-({\bf q}\cdot\hat{n})^{2}$ and the
coefficient functions $\mathcal{Z}_{i}$ and $\mathcal{A}_{i}$ are given in the
appendix. The quantities $\rho^{C}_{0}(r)$ and $\rho^{C}_{2}(r)$ are the
monopole and quadrupole charge densities, respectively, whereas
$\rho^{M}_{1}(r)$ and $\rho^{M}_{3}(r)$ are the dipole and octupole scalar
magnetization densities, respectively. These quantities are given by

$\displaystyle\rho^{C}_{0}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf r}}\int
d^{2}\hat{n}\Biggl{\\{}F_{1,0}^{V}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{6m^{2}}\left[-2F_{1,0}^{V}(-q_{\perp}^{2})+F_{1,1}^{V}(-q_{\perp}^{2})+F_{2,0}^{V}(-q_{\perp}^{2})\right]$
(15) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{4}}{24m^{4}}\left[-2F_{1,1}^{V}(-q_{\perp}^{2})+F_{2,1}^{V}(-q_{\perp}^{2})\right]\Biggl{\\}}~{},$
$\displaystyle\rho^{C}_{2}(r)$ $\displaystyle=$
$\displaystyle-\frac{r}{4\pi}\frac{d}{dr}\frac{1}{r}\frac{d}{dr}\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf r}}\int
d^{2}\hat{n}\,\frac{q_{\perp}^{2}}{12m^{2}q^{2}}\Biggl{\\{}4F_{1,0}^{V}(-q_{\perp}^{2})+F_{1,1}^{V}(-q_{\perp}^{2})-2F_{2,0}^{V}(-q_{\perp}^{2})$
(16) $\displaystyle-$
$\displaystyle\frac{3q_{\perp}^{2}}{q^{2}}\left[F_{1,0}^{V}(-q_{\perp}^{2})+F_{1,1}^{V}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{2}}{2m^{2}}\left[-2F_{1,1}^{V}(-q_{\perp}^{2})+F_{2,1}^{V}(-q_{\perp}^{2})\right]-\frac{3q_{\perp}^{4}}{4m^{2}q^{2}}F_{1,1}^{V}(-q_{\perp}^{2})\Biggl{\\}},$
$\displaystyle\rho^{M}_{1}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf r}}\int
d^{2}\hat{n}\frac{q_{\perp}^{2}}{2q^{2}}\bigg{\\{}F_{1,0}^{V}(-q_{\perp}^{2})-\frac{1}{3}F_{2,0}^{V}(-q_{\perp}^{2})$
(17) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{2}}{30m^{2}}\left[-2F_{1,0}^{V}(-q_{\perp}^{2})+7F_{1,1}^{V}(-q_{\perp}^{2})\right.+\left.2F_{2,0}^{V}(-q_{\perp}^{2})-2F_{2,1}^{V}(-q_{\perp}^{2})\right]$
$\displaystyle+$
$\displaystyle\frac{q_{\perp}^{4}}{60m^{4}}\left[F_{2,1}^{V}(-q_{\perp}^{2})-F_{1,1}^{V}(-q_{\perp}^{2})\right]\bigg{\\}},$
$\displaystyle\rho^{M}_{3}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf r}}\int
d^{2}\hat{n}\frac{q_{\perp}^{4}}{48q^{4}}\bigg{\\{}-4F_{1,0}^{V}(-q_{\perp}^{2})-F_{1,1}^{V}(-q_{\perp}^{2})+4F_{2,0}^{V}(-q_{\perp}^{2})+F_{2,1}^{V}(-q_{\perp}^{2})$
(18) $\displaystyle+$
$\displaystyle\frac{5q_{\perp}^{2}}{q^{2}}\left[F_{1,0}^{V}(-q_{\perp}^{2})+F_{1,1}^{V}(-q_{\perp}^{2})-F_{2,0}^{V}(-q_{\perp}^{2})-F_{2,1}^{V}(-q_{\perp}^{2})\right]+\frac{q_{\perp}^{2}}{m^{2}}\left[F_{2,1}^{V}(-q_{\perp}^{2})-F_{1,1}^{V}(-q_{\perp}^{2})\right]$
$\displaystyle+$
$\displaystyle\frac{5q_{\perp}^{4}}{4m^{2}q^{2}}\left[F_{1,1}^{V}(-q_{\perp}^{2})-F_{2,1}^{V}(-q_{\perp}^{2})\right]\bigg{\\}}.$

The standard expressions of the densities in terms of the form factors in the
Breit frame, $F_{i,j}(q^{2})=F_{i,j}(-{\bf{q}}^{2})$, which we will refer to
as ”naive”, are obtained by first approximating the integrand in Eq. (8) by
the two leading terms in the $1/m-$expansion111Factors of $m$ introduced in
Eqs. (5) and (20) for dimensional reasons in the parametrization of the matrix
elements in terms of form factors are not counted when expanding in $1/m$
neither here nor below in the case of EMT matrix elements. and subsequently
localizing the wave packet by taking the limit $R\mapsto 0$. The resulting
expressions have the form:

$\displaystyle j^{0}_{\text{naive}}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}e^{-i{\bf q}\cdot{\bf
r}}\Biggl{\\{}\left[F^{V}_{1,0}\left(-{\bf q}^{2}\right)+\frac{{\bf
q}^{2}}{6m^{2}}F^{V}_{1,1}\left(-{\bf
q}^{2}\right)\right]\delta_{s^{\prime}s}-F^{V}_{1,1}\left(-{\bf
q}^{2}\right)\frac{q^{k}q^{l}}{6m^{2}}{\hat{Q}^{kl}_{s^{\prime}s}}\Biggl{\\}},$
$\displaystyle j^{i}_{\text{naive}}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}e^{-i{\bf q}\cdot{\bf
r}}i\epsilon^{ikn}\frac{q^{n}}{3m}\Biggl{\\{}\left[F^{V}_{2,0}\left(-{\bf
q}^{2}\right)+\frac{{\bf q}^{2}}{5m^{2}}F^{V}_{2,1}\left(-{\bf
q}^{2}\right)\right]{\hat{S}^{k}_{s^{\prime}s}}-F^{V}_{2,1}\left(-{\bf
q}^{2}\right)\frac{q^{l}q^{z}}{2m^{2}}{\hat{O}^{klz}_{s^{\prime}s}}\Biggl{\\}}.$
(19)

##  III Gravitational densities in the zero average momentum frame

One is often interested in matrix elements of the quark and gluon
contributions to the EMT. As these are not separately conserved, we
parameterize the matrix element of a symmetric EMT for spin-3/2 states in
terms of ten form factors as follows Cotogno:2019vjb ; Kim:2020lrs :

$\displaystyle\langle
p_{f},s^{\prime}|{T_{\mu\nu}\left(\vec{x}\right)}|p_{i},s\rangle$
$\displaystyle=$
$\displaystyle-\bar{u}^{\beta}(p_{f},s^{\prime})\Biggl{[}\frac{P_{\mu}P_{\nu}}{m}\left(g_{\alpha\beta}F_{1,0}(q^{2})-\frac{q_{\alpha}q_{\beta}}{2m^{2}}\,F_{1,1}(q^{2})\right)$
(20) $\displaystyle+$
$\displaystyle\frac{q_{\mu}q_{\nu}-\eta_{\mu\nu}q^{2}}{4m}\left(g_{\alpha\beta}F_{2,0}(q^{2})-\frac{q_{\alpha}q_{\beta}}{2m^{2}}\,F_{2,1}(q^{2})\right)$
$\displaystyle+$ $\displaystyle
mg^{\mu\nu}\left(g_{\alpha^{\prime}\alpha}F_{3,0}(t)-\frac{\Delta_{\alpha^{\prime}}\Delta_{\alpha}}{2m^{2}}F_{3,1}(t)\right)$
$\displaystyle+$
$\displaystyle\frac{i}{2}\frac{\left(P_{\mu}\sigma_{\nu\rho}+P_{\nu}\sigma_{\mu\rho}\right)q^{\rho}}{m}\left(g_{\alpha\beta}F_{4,0}(q^{2})-\frac{q_{\alpha}q_{\beta}}{2m^{2}}\,F_{4,1}(q^{2})\right)$
$\displaystyle-$
$\displaystyle\frac{1}{m}\,\left(g_{\nu\beta}q_{\mu}q_{\alpha}+g_{\mu\beta}q_{\nu}q_{\alpha}+g_{\nu\alpha}q_{\mu}q_{\beta}+g_{\mu\alpha}q_{\nu}q_{\beta}-2g_{\mu\nu}q_{\alpha}q_{\beta}\right.$
$\displaystyle-$
$\displaystyle\left.g_{\mu\beta}g_{\nu\alpha}q^{2}-g_{\nu\beta}g_{\mu\alpha}q^{2}\right)F_{5,0}(q^{2})$
$\displaystyle+$ $\displaystyle
m(g^{\mu}_{\alpha^{\prime}}g^{\nu}_{\alpha}+g^{\nu}_{\alpha^{\prime}}g^{\mu}_{\alpha})F_{6,0}(t)\Biggr{]}u^{\alpha}(p_{i},s)e^{-i{\bf
q}\cdot{\bf r}}\,,$

where in case of a conserved EMT the form factors $F_{3,0}(t)$, $F_{3,1}(t)$
and $F_{6,0}(t)$ vanish. The matrix element of the EMT in localized states is
written as

$\displaystyle t^{\mu\nu}_{\phi}({\bf r})$ $\displaystyle\equiv$
$\displaystyle\langle\Phi,{\bf X},s^{\prime}|\hat{T}^{\mu\nu}({\bf
x},0)|\Phi,{\bf X},s\rangle$ (21) $\displaystyle=$
$\displaystyle-\int\frac{d^{3}{P}\,d^{3}{q}}{(2\pi)^{3}\sqrt{4EE^{\prime}}}\,\bar{u}^{\beta}\left(P+\frac{q}{2},\sigma^{\prime}\right)\Biggl{[}\frac{P_{\mu}P_{\nu}}{m}\left(g_{\alpha\beta}F_{1,0}(q^{2})-\frac{q_{\alpha}q_{\beta}}{2m^{2}}\,F_{1,1}(q^{2})\right)$
$\displaystyle+$
$\displaystyle\frac{q_{\mu}q_{\nu}-\eta_{\mu\nu}q^{2}}{4m}\left(g_{\alpha\beta}F_{2,0}(q^{2})-\frac{q_{\alpha}q_{\beta}}{2m^{2}}\,F_{2,1}(q^{2})\right)+\frac{i}{2}\frac{\left(P_{\mu}\sigma_{\nu\rho}+P_{\nu}\sigma_{\mu\rho}\right)q^{\rho}}{m}\left(g_{\alpha\beta}F_{4,0}(q^{2})-\frac{q_{\alpha}q_{\beta}}{2m^{2}}\,F_{4,1}(q^{2})\right)$
$\displaystyle-$
$\displaystyle\frac{1}{m}\,\left(g_{\nu\beta}q_{\mu}q_{\alpha}+g_{\mu\beta}q_{\nu}q_{\alpha}+g_{\nu\alpha}q_{\mu}q_{\beta}+g_{\mu\alpha}q_{\nu}q_{\beta}-2g_{\mu\nu}q_{\alpha}q_{\beta}\right.$
$\displaystyle-$
$\displaystyle\left.g_{\mu\beta}g_{\nu\alpha}q^{2}-g_{\nu\beta}g_{\mu\alpha}q^{2}\right)F_{5,0}(q^{2})\Biggr{]}u^{\alpha}\left(P-\frac{q}{2},\sigma\right)\phi\bigg{(}{\bf
P}-\frac{\bf q}{2}\bigg{)}\phi^{\star}\bigg{(}{\bf P}+\frac{\bf
q}{2}\bigg{)}e^{-i{\bf q}\cdot{\bf r}}.$

The matrix elements of the EMT in the localized states with $R\to 0$ can be
obtained analogously to the electromagnetic case. As we will see below, the
leading order contributions to $t^{00}_{\phi}({\bf r})$ and
$t^{0i}_{\phi}({\bf r})$ are of the order of $1/R$, and the
$t^{ij}_{\phi}({\bf r})$ terms need to be treated differently from the others,
when expanding in $R$. The reason for that is that, the components of
$t^{ij}_{\phi}({\bf r})$, unlike $t^{00}_{\phi}({\bf r})$ and
$t^{0i}_{\phi}({\bf r})$, which contain only information about the energy and
spin densities, respectively, encode information about the internal pressure
and shear forces as well as about the motion of the system Freese:2021mzg ;
Panteleeva:2022uii . That is, $t^{ij}_{\phi}({\bf r})$ needs to be decomposed
to a component $t^{ij}_{\phi,0}({\bf r})$ that describes the motion of the
system as whole, and a component that encodes information about pressure and
shear forces $t^{ij}_{\phi,2}({\bf r})$. Therefore, after expanding in $R$, we
keep the leading order contribution of each of these terms. The resulting
expressions have the form:

$\displaystyle t^{00}_{\phi}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle N_{\phi,R}\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf r}}\int d^{2}\hat{n}\Biggl{\\{}
\mathcal{E}_{0}(q_{\perp}^{2})~{}\delta_{s^{\prime}s}+\left[\mathcal{E}_{1}(q_{\perp}^{2})~{}\hat{n}^{k}\hat{n}^{l}+\mathcal{E}_{2}(q_{\perp}^{2})~{}\frac{q^{k}_{\perp}q^{l}_{\perp}}{{m^{2}}}\right]{\hat{Q}^{kl}_{s^{\prime}s}}\Biggl{\\}},$
(22a) $\displaystyle t^{0i}_{\phi}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle i\,N_{\phi,R}\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf r}}\int d^{2}\hat{n}
\Biggl{\\{}~{}\left[\mathcal{C}_{0}(q_{\perp}^{2})\,\epsilon^{kln}\hat{n}^{l}\hat{n}^{i}+\mathcal{C}_{1}(q_{\perp}^{2})\,\epsilon^{iln}\left(\delta^{kl}-\hat{n}^{k}\hat{n}^{l}\right)\right]\,\frac{q^{n}_{\perp}}{m}\,{\hat{S}^{k}_{s^{\prime}s}}$
(22b) $\displaystyle+$
$\displaystyle\Biggl{[}\left(\mathcal{C}_{2}(q_{\perp}^{2})\hat{n}^{t}\hat{n}^{z}+\mathcal{C}_{3}(q_{\perp}^{2})\frac{q_{\perp}^{t}q_{\perp}^{z}}{{m^{2}}}\right)~{}\epsilon^{kln}\hat{n}^{l}\hat{n}^{i}$
$\displaystyle+$
$\displaystyle\left(\mathcal{C}_{4}(q_{\perp}^{2})\hat{n}^{t}\hat{n}^{z}+\mathcal{C}_{5}(q_{\perp}^{2})s\frac{q_{\perp}^{t}q_{\perp}^{z}}{{m^{2}}}\right)\epsilon^{iln}\left(\delta^{kl}-\hat{n}^{k}\hat{n}^{l}\right)\Biggl{]}\frac{q^{n}_{\perp}}{m}\,{\hat{O}^{ktz}_{s^{\prime}s}}\Biggl{\\}},$
$\displaystyle t^{ij}_{\phi}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle t^{ij}_{\phi,0}(s^{\prime},s,{\bf
r})+t^{ij}_{\phi,2}(s^{\prime},s,{\bf r}),$ (22c)

where

$\displaystyle t^{ij}_{\phi,0}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle N_{\phi,R}\int\frac{d^{3}{q}}{(2\pi)^{3}}~{}e^{-i{\bf
q}\cdot{\bf r}}\int d^{2}\hat{n}\,\hat{n}^{i}\hat{n}^{j} \Biggl{\\{}
\mathcal{E}_{0}(q_{\perp}^{2})~{}\delta_{s^{\prime}s}+\left[\mathcal{E}_{1}(q_{\perp}^{2})~{}\hat{n}^{k}\hat{n}^{l}+\mathcal{E}_{2}(q_{\perp}^{2})~{}\frac{q^{k}_{\perp}q^{l}_{\perp}}{{m^{2}}}\right]{\hat{Q}^{kl}_{s^{\prime}s}}\Biggl{\\}}\,$
(23a) $\displaystyle t^{ij}_{\phi,2}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle N_{\phi,R,2}\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf r}}\int d^{2}\hat{n}$ (23b) $\displaystyle\times$
$\displaystyle\Biggl{\\{}\frac{1}{2m^{2}}\left(q^{i}q^{j}-{q}_{\perp}^{2}\delta^{ij}\right)\Biggl{[}\mathcal{W}_{0}(q_{\perp}^{2})\delta_{s^{\prime}s}+\Biggl{[}\mathcal{W}_{1}(q_{\perp}^{2})\hat{n}^{k}\hat{n}^{l}+\mathcal{W}_{2}(q_{\perp}^{2})\frac{q_{\perp}^{k}q_{\perp}^{l}}{{m^{2}}}\Biggl{]}{\hat{Q}^{kl}_{s^{\prime}s}}\Biggl{]}$
$\displaystyle-$ $\displaystyle
2\delta^{ij}\Biggl{[}\mathcal{U}_{0}(q_{\perp}^{2})\delta_{s^{\prime}s}+\left[\mathcal{U}_{1}(q_{\perp}^{2})\hat{n}^{k}\hat{n}^{l}+\mathcal{U}_{2}(q_{\perp}^{2})\frac{q_{\perp}^{k}q_{\perp}^{l}}{{m^{2}}}\right]{\hat{Q}^{kl}_{s^{\prime}s}}\Biggl{]}\Biggl{\\}}\
,$

and the coefficient functions
$\mathcal{E}_{i},\mathcal{C}_{i},\mathcal{W}_{i}$ and $\mathcal{U}_{i}$ are
given in the appendix and further,

$\displaystyle N_{\phi,R}$ $\displaystyle=$
$\displaystyle\frac{1}{R}\int_{0}^{\infty}\,d{Q}\,{Q}^{3}|\tilde{\phi}(|{\bf
Q}|)|^{2}\,,$ $\displaystyle N_{\phi,R,2}$ $\displaystyle=$
$\displaystyle\frac{m^{2}R}{2}\int_{0}^{\infty}\,dQ\,Q|\tilde{\phi}(|{\bf
Q}|)|^{2}\,.$ (24)

Below, we specify the multipole expansion of
$t^{\mu\nu}_{\phi}(s^{\prime},s,{\bf r})$. For the $00$th component the result
reads

$\displaystyle t^{00}_{\phi}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle
N_{\phi,R}\Biggl{\\{}\rho^{E}_{0}\left(r\right)\delta_{s^{\prime}s}+\rho^{E}_{2}\left(r\right)Y_{2}^{kl}\left(\Omega_{r}\right)\hat{Q}_{s^{\prime}s}^{kl}\Biggl{\\}},$
(25)

where the monopole and quadrupole energy densities are identified as
$\rho^{E}_{0}$ and $\rho^{E}_{2}$, respectively. Their expressions have the
form:

$\displaystyle\rho^{E}_{0}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf r}}\int
d^{2}\hat{n}\bigg{\\{}F_{1,0}(-q_{\perp}^{2})-\frac{2}{3}F_{6,0}(-q_{\perp}^{2})$
(26) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{2}}{3m^{2}}\left[-F_{1,0}(-q_{\perp}^{2})+\frac{1}{2}F_{1,1}(-q_{\perp}^{2})+F_{4,0}(-q_{\perp}^{2})+2F_{5,0}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{4}}{12m^{4}}\left[F_{1,1}(-q_{\perp}^{2})-F_{4,1}(-q_{\perp}^{2})\right]\bigg{\\}},$
$\displaystyle\rho^{E}_{2}\left(r\right)$ $\displaystyle=$
$\displaystyle-r\frac{d}{dr}\frac{1}{r}\frac{d}{dr}\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf r}}\int
d^{2}\hat{n}\frac{1}{q^{2}}\bigg{\\{}\frac{2}{3}F_{6,0}(-q_{\perp}^{2})-\frac{q_{\perp}^{2}}{q^{2}}F_{6,0}(-q_{\perp}^{2})$
(27) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{2}}{12m^{2}}\left[4F_{1,0}(-q_{\perp}^{2})+F_{1,1}(-q_{\perp}^{2})-4F_{4,0}(-q_{\perp}^{2})-8F_{5,0}(-q_{\perp}^{2})\right]+\frac{q_{\perp}^{4}}{12m^{4}}\left[F_{1,1}(-q_{\perp}^{2})-F_{4,1}(-q_{\perp}^{2})\right]$
$\displaystyle+$
$\displaystyle\frac{q_{\perp}^{4}}{4m^{2}q^{2}}\left[4F_{5,0}(-q_{\perp}^{2})-F_{1,0}(-q_{\perp}^{2})-F_{1,1}(-q_{\perp}^{2})\right]
-\frac{q_{\perp}^{6}}{16m^{4}q^{2}}F_{1,1}(-q_{\perp}^{2})\ \bigg{\\}}\,.$

For the $0i$th components we have

$\displaystyle t^{0i}_{\phi}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle
N_{\phi,R}\biggl{[}~{}\epsilon^{ikn}\hat{S}_{s^{\prime}s}^{k}Y_{1}^{n}\frac{1}{r}\rho^{J}_{1}\left(r\right)\text{
}+~{}\epsilon^{ikn}\hat{O}_{s^{\prime}s}^{ktz}Y_{3}^{ntz}\,\frac{1}{r}\rho^{J}_{3}\left(r\right)\biggl{]},$
(28)

where

$\displaystyle\rho^{J}_{1}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{r}{m}\frac{d}{dr}\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf r}}\int
d^{2}\hat{n}\frac{q_{\perp}^{2}}{2q^{2}}\bigg{\\{}F_{1,0}(-q_{\perp}^{2})-\frac{2}{3}F_{4,0}(-q_{\perp}^{2})-\frac{4}{15}F_{6,0}(-q_{\perp}^{2})$
(29) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{2}}{15m^{2}}\left[-F_{1,0}(-q_{\perp}^{2})+\frac{7}{2}F_{1,1}(-q_{\perp}^{2})+2F_{4,0}(-q_{\perp}^{2})-2F_{4,1}(-q_{\perp}^{2})+4F_{5,0}(-q_{\perp}^{2})\right]$
$\displaystyle+$
$\displaystyle\frac{q_{\perp}^{4}}{60m^{4}}\left[-F_{1,1}(-q_{\perp}^{2})+F_{4,0}(-q_{\perp}^{2})+F_{4,1}(-q_{\perp}^{2})\right]
\bigg{\\}},$ $\displaystyle\rho^{J}_{3}\left(r\right)$ $\displaystyle=$
$\displaystyle\frac{r^{4}}{m}\left(\frac{1}{r}\frac{d}{dr}\right)^{3}\int\frac{d^{3}q}{(2\pi)^{3}}e^{-i{\bf
q}\cdot{\bf r}}\int
d^{2}\hat{n}\frac{q_{\perp}^{2}}{12q^{4}}\bigg{\\{}-4F_{6,0}^{a}(-q_{\perp}^{2})+5\frac{q_{\perp}^{2}}{q^{2}}F_{6,0}^{a}(-q_{\perp}^{2})$
(30) $\displaystyle-$
$\displaystyle\frac{q_{\perp}^{2}}{4m^{2}}\left[4F_{1,0}^{a}(-q_{\perp}^{2})+F_{1,1}^{a}(-q_{\perp}^{2})-8F_{4,0}^{a}(-q_{\perp}^{2})-2F_{4,1}^{a}(-q_{\perp}^{2})-16F_{5,0}^{a}(-q_{\perp}^{2})\right]$
$\displaystyle+$
$\displaystyle\frac{5q_{\perp}^{4}}{4m^{2}q^{2}}\left[F_{1,0}^{a}(-q_{\perp}^{2})+F_{1,1}^{a}(-q_{\perp}^{2})-2F_{4,0}^{a}(-q_{\perp}^{2})-2F_{4,1}^{a}(-q_{\perp}^{2})-4F_{5,0}^{a}(-q_{\perp}^{2})\right]$
$\displaystyle-$
$\displaystyle\frac{q_{\perp}^{4}}{4m^{4}}\left[F_{1,1}^{a}(-q_{\perp}^{2})-2F_{4,1}^{a}(-q_{\perp}^{2})\right]+\frac{5q_{\perp}^{6}}{16m^{4}q^{2}}\left[F_{1,1}^{a}(-q_{\perp}^{2})-2F_{4,1}^{a}(-q_{\perp}^{2})\right]\bigg{\\}}.$

The spin density is given by

$\displaystyle J^{i}(\bm{r},s^{\prime},s)$ $\displaystyle=$
$\displaystyle\epsilon^{ijk}r^{j}t^{0k}_{\phi}(s^{\prime},s,{\bf
r})=N_{\phi,R}\,\Biggl{\\{}\left(\frac{2}{3}\delta^{il}Y_{0}-Y_{2}^{il}\right)\rho^{J}_{1}\left(r\right)\text{
}\hat{S}_{s^{\prime}s}^{l}$ (31)
$\displaystyle+\left[-Y_{4}^{iltz}+\frac{2}{35}\left(8\delta^{il}Y_{2}^{tz}+\delta^{it}Y_{2}^{lz}+\delta^{iz}Y_{2}^{lt}\right)\right]\rho^{J}_{3}\left(r\right)\,\hat{O}_{s^{\prime}s}^{ltz}\Biggl{\\}}.$

We identify the monopole angular momentum density $(J_{\text{mono}}^{i})$ as
the term with the structure $Y_{0}\hat{S}_{s^{\prime}s}^{i}$, i.e.

$\displaystyle J_{\text{mono}}^{i}\left({\bf r},s^{\prime},s\right)$
$\displaystyle=$
$\displaystyle\frac{2}{3}N_{\phi,R}\;\hat{S}_{s^{\prime}s}^{i}\rho^{J}_{1}\left(r\right),$
(32)

and, following Ref. Kim:2020lrs , the averaged angular momentum density is
given by

$\displaystyle\rho_{J}\left(r\right)$ $\displaystyle\equiv$
$\displaystyle\frac{S}{\text{Tr}\left[\hat{S}^{2}\right]}\sum_{s^{\prime},s,i}S_{ss^{\prime}}^{i}J_{\text{mono}}^{i}\left(\vec{r},s^{\prime},s\right)=N_{\phi,R}\,\rho^{J}_{1}\left(r\right)\
,$ (33)

where $S=3/2$ is the spin of the system.

Finally, for the $ij$th components we obtain

$\displaystyle t^{ij}_{\phi,0}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle
N_{\phi,R}\bigg{\\{}\left[{a}_{1}\left(r\right)\delta^{ij}-\left(\frac{\delta^{ij}}{3}\partial^{2}+Y_{2}^{ij}r\frac{d}{dr}\frac{1}{r}\frac{d}{dr}\right){a}_{2}\left(r\right)\right]\delta_{s^{\prime}s}+\hat{Q}_{s^{\prime}s}^{ij}{a}_{3}\left(r\right)$
(34)
$\displaystyle+\hat{Q}_{s^{\prime}s}^{kl}\partial_{k}\partial_{l}\left(\frac{\delta^{ij}}{3}\partial^{2}+Y_{2}^{ij}r\frac{d}{dr}\frac{1}{r}\frac{d}{dr}\right){a}_{4}\left(r\right)-\delta^{ij}\hat{Q}_{s^{\prime}s}^{kl}\partial_{k}\partial_{l}{a}_{5}\left(r\right)$
$\displaystyle-\left[\frac{2}{3}\hat{Q}_{s^{\prime}s}^{ij}\partial^{2}+\left(\hat{Q}_{s^{\prime}s}^{iv}Y_{2}^{jv}+\hat{Q}_{s^{\prime}s}^{jv}Y_{2}^{iv}\right)r\frac{d}{dr}\frac{1}{r}\frac{d}{dr}\right]{a}_{6}\left(r\right)\bigg{\\}},$
$\displaystyle t^{ij}_{\phi,2}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle
N_{\phi,R,2}\bigg{\\{}-\frac{1}{2m^{2}}\left(\frac{\delta^{ij}}{3}\partial^{2}+Y_{2}^{ij}r\frac{d}{dr}\frac{1}{r}\frac{d}{dr}\right){w}_{0}\left(r\right)\,\delta_{s^{\prime}s}$
(35)
$\displaystyle+\frac{1}{2m^{2}}\hat{Q}_{s^{\prime}s}^{kl}\partial_{k}\partial_{l}\left[\left(\frac{\delta^{ij}}{3}\partial^{2}+Y_{2}^{ij}r\frac{d}{dr}\frac{1}{r}\frac{d}{dr}\right){w}_{1}\left(r\right)\right]$
$\displaystyle+\delta^{ij}\left[{v}_{0}\left(r\right)~{}\delta_{s^{\prime}s}-\hat{Q}_{s^{\prime}s}^{kl}\partial_{k}\partial_{l}\,{v}_{1}\left(r\right)\right]\bigg{\\}},$

where the coefficient functions ${a}(r)$, $w(r)$ and $v(r)$ are given in the
appendix.

To obtain the pressure and shear force densities we consider a conserved EMT
and take the part of ${\tilde{t}}^{ij}_{\phi,2}(s^{\prime},s,{\bf r})$ linear
in $R$ (where the tilde means only conserved EMTs are considered), which we
parametrize as follows Polyakov:2019lbq :

$\displaystyle{\tilde{t}}^{ij}_{\phi,2}(s^{\prime},s,{\bf r})$
$\displaystyle=$ $\displaystyle
N_{\phi,R,2}\Biggl{\\{}p_{0}(r)\delta^{ij}\delta_{s^{\prime}s}+s_{0}(r)Y_{2}^{ij}\delta_{s^{\prime}s}+p_{2}(r)\hat{Q}_{s^{\prime}s}^{ij}+2s_{2}(r)\left[\hat{Q}_{s^{\prime}s}^{ik}Y_{2}^{kj}+\hat{Q}_{s^{\prime}s}^{jk}Y_{2}^{ki}-\delta^{ij}\hat{Q}_{s^{\prime}s}^{kl}Y_{2}^{kl}\right]$
(36) $\displaystyle-$
$\displaystyle\frac{1}{m^{2}}\hat{Q}_{s^{\prime}s}^{kl}\partial_{k}\partial_{l}\left[p_{3}(r)\delta^{ij}+s_{3}(r)Y_{2}^{ij}\right]\Biggl{\\}},$

where $p_{0}(r)$ and $s_{0}(r)$ are the pressure and shear force densities
also appearing in the spherically symmetric hadrons, respectively, $p_{2}(r)$,
$p_{3}(r)$ correspond to the quadrupole pressure densities, and $s_{2}(r)$,
$s_{3}(r)$ are the quadrupole shear force densities.222Another equivalent
parametrization is given in Ref. Panteleeva:2020ejw , where the normal and
tangential forces can be defined in a compact way. However, it has been shown
that the parametrization of Ref. Polyakov:2019lbq  has advantages in studying
the mechanical structure, whenever performing an Abel transformation is
involved Kim:2022wkc . Comparing Eqs. (36) and (35) we obtain for the pressure
and shear forces the following results:

$\displaystyle p_{0}(r)$ $\displaystyle=$
$\displaystyle\tilde{v}_{0}\left(r\right)-\frac{1}{6m^{2}}\partial^{2}{w}_{0}\left(r\right),\quad
s_{0}(r)=-\frac{1}{2m^{2}}r\frac{d}{dr}\frac{1}{r}\frac{d}{dr}{w}_{0}\left(r\right),$
$\displaystyle p_{2}(r)$ $\displaystyle=$ $\displaystyle
0,\qquad\qquad\qquad\qquad\qquad s_{2}(r)=0,$ $\displaystyle p_{3}(r)$
$\displaystyle=$ $\displaystyle
m^{2}\tilde{v}_{1}\left(r\right)-\frac{1}{6}\partial^{2}{w}_{1}\left(r\right),\quad
s_{3}(r)=-\frac{1}{2}r\frac{d}{dr}\frac{1}{r}\frac{d}{dr}{w}_{1}\left(r\right),$
(37)

where the coefficient functions ${w}(r)$ and ${\tilde{v}}(r)$ are given as

$\displaystyle{w}_{0}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\hat{n}d^{3}q}{\left(2\pi\right)^{3}}e^{-i\vec{q}\cdot\vec{r}}\Biggl{[}F_{2,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{6m^{2}}\left[-2F_{2,0}(-q_{\perp}^{2})+F_{2,1}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{4}}{12m^{4}}F_{2,1}(-q_{\perp}^{2})\Biggl{]},$
$\displaystyle\tilde{v}_{0}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\hat{n}d^{3}q}{\left(2\pi\right)^{3}}e^{-i\vec{q}\cdot\vec{r}}\left(-\frac{q_{\perp}^{2}}{2m^{2}}\right)\left[F_{2,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{6m^{2}}\left[-2F_{2,0}(-q_{\perp}^{2})+F_{2,1}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{4}}{12m^{4}}F_{2,1}(-q_{\perp}^{2})\right],$
$\displaystyle{w}_{1}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\hat{n}d^{3}q}{\left(2\pi\right)^{3}}e^{-i\vec{q}\cdot\vec{r}}\frac{q_{\perp}^{2}}{2m^{2}q^{2}}\left[\left(\frac{2}{3}-\frac{q_{\perp}^{2}}{2q^{2}}\right)F_{2,0}(-q_{\perp}^{2})+\frac{1}{2}\left(\frac{1}{3}+\frac{q_{\perp}^{2}}{3m^{2}}-\frac{q_{\perp}^{2}}{q^{2}}-\frac{q_{\perp}^{4}}{4m^{2}q^{2}}\right)F_{2,1}(-q_{\perp}^{2})\right],$
$\displaystyle\tilde{v}_{1}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\hat{n}d^{3}q}{\left(2\pi\right)^{3}}e^{-i\vec{q}\cdot\vec{r}}\left(-\frac{q_{\perp}^{4}}{4m^{4}q^{2}}\right)\left[\left(\frac{2}{3}-\frac{q_{\perp}^{2}}{2q^{2}}\right)F_{2,0}(-q_{\perp}^{2})+\frac{1}{2}\left(\frac{1}{3}+\frac{q_{\perp}^{2}}{3m^{2}}-\frac{q_{\perp}^{2}}{q^{2}}-\frac{q_{\perp}^{4}}{4m^{2}q^{2}}\right)F_{2,1}(-q_{\perp}^{2})\right].$
(38)

It is clear from Eqs. (37) and (38) that the pressure and shear forces are
expressed in terms of $F_{2,0}$ and $F_{2,1}$ only.333Notice that the feature
$p_{2}=s_{2}=0$ is not identical with the result of the large $N_{c}$ limit
for baryons in the chiral soliton model as obtained in Ref. Panteleeva:2020ejw
, where another parameterization of the pressure and shear forces is used.
This can be easily checked by converting the pressure and shear forces in Eq.
(37) to the notations used in Ref. Panteleeva:2020ejw .

Below we obtain the differential equation for the pressure and shear forces
that follows from the conservation of the EMT. In that case, we have to make
our matrix element in Eq. (20) time-dependent, i.e. we substitute $e^{-i{\bf
q}\cdot{\bf r}}\mapsto e^{iq_{0}t-i{\bf q}\cdot{\bf r}}$. The conservation of
the EMT leads to

$\displaystyle\partial_{\mu}t^{\mu\nu}_{\phi}(s^{\prime},s,{\bf
r},t)|_{t=0}=\partial_{0}t^{0\nu}_{\phi}(s^{\prime},s,{\bf
r},t)|_{t=0}+\partial_{i}t^{i\nu}_{\phi}(s^{\prime},s,{\bf r},t)|_{t=0}=0.$
(39)

Notice that $\partial_{0}$ and $|_{t=0}$ do not commute, i.e. to obtain
$\partial_{0}t^{0\nu}_{\phi}(s^{\prime},s,{\bf r},t)|_{t=0}$ we first take the
derivative of $t^{0\nu}_{\phi}(s^{\prime},s,{r})$, then put $t=0$ and after
that perform the expansion around $R=0$. Since we are interested in the
pressure and shear forces, we consider the case with $\nu=j$ and keep all
contributions linear in $R$. We obtain the following differential equation,
adhering to the notation of Ref. Polyakov:2019lbq ,

$\displaystyle
p_{n}^{\prime}(r)+\frac{2}{3}s_{n}^{\prime}(r)+\frac{2}{r}s_{n}(r)={h}_{n}^{\prime}\left(r\right),\
\ \ \text{with}\;n=0,2,3,$ (40)

where

$\displaystyle{h}_{0}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{\left(2\pi\right)^{3}}e^{-i{\bf q}\cdot{\bf
r}}\int d^{2}\hat{n}\frac{\left({\bf
q}\cdot\hat{n}\right)^{2}}{2m^{2}}\mathcal{W}_{0}\left(q_{\perp}^{2}\right),\quad{h}_{2}\left(r\right)=0,$
$\displaystyle{h}_{3}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{\left(2\pi\right)^{3}}e^{-i{\bf q}\cdot{\bf
r}}\int d^{2}\hat{n}\frac{\left({\bf
q}\cdot\hat{n}\right)^{2}}{2q^{2}}\left[\mathcal{W}_{1}\left(q_{\perp}^{2}\right)\;\left(1-\frac{3q_{\perp}^{2}}{2q^{2}}\right)+\mathcal{W}_{2}\left(q_{\perp}^{2}\right)~{}\frac{q_{\perp}^{2}}{2m^{2}}\left(\frac{3q_{\perp}^{2}}{q^{2}}-1\right)\right],$
(41)

with $r=|{\bf r}|$, and the coefficient functions
$\mathcal{W}_{0,1,2}(q^{2}_{\perp})$ are given in the Appendix. In contrast,
there are no $h_{n}(r)$ terms in Eq. (40) in the case of the Breit-frame
because of the absence of the temporal dependence in $t^{0\nu}$ component (due
to $q^{0}=0$).

The pressure densities $p_{n}(r)$ comply with the von Laue stability condition

$\displaystyle\int d^{3}r\,p_{n}(r)$ $\displaystyle=$ $\displaystyle 0,\ \ \
\mathrm{with}\ \ n=0,2,3,$ (42)

as long as $\text{lim}_{q^{2}_{\perp}\rightarrow
0}\left(q_{\perp}^{2}\right)^{\delta}F_{2,0}\left(-q^{2}_{\perp}\right)=0$ and
$\text{lim}_{q^{2}_{\perp}\rightarrow
0}\left(q_{\perp}^{2}\right)^{\delta}F_{2,1}\left(-q^{2}_{\perp}\right)=0$,
for $\delta>0$.

The dimensionless constants (generalized D-terms) are defined by

$\displaystyle\mathcal{D}_{n}$ $\displaystyle=$
$\displaystyle{-\frac{4}{15}\,m^{2}\int{\rm d}^{3}r\,r^{2}s_{n}(r)}=m^{2}\int
d^{3}r\,r^{2}\left[p_{n}\left(r\right)-{h}_{n}\left(r\right)\right],\ \ \
\mathrm{with}\ \ n=0,2,3.$ (43)

Note that the above definition differs from that of the Breit-frame case
Panteleeva:2020ejw  by the ${h}_{n}\left(r\right)$ terms.

The spherical components of the internal forces ($dF_{r},dF_{\theta}$ and
$dF_{\varphi}$) acting on the radial area element
($d\bm{S}=dS_{r}\hat{\bm{e}}_{r}+dS_{\theta}\hat{\bm{e}}_{\theta}+dS_{\varphi}\hat{\bm{e}}_{\varphi}$)
are expressed as follows

$\displaystyle\frac{dF_{r}}{dS_{r}}$ $\displaystyle=$ $\displaystyle
N_{\phi,R,2}\Bigg{[}\left(p_{0}(r)+\frac{2}{3}s_{0}(r)\right)\delta_{s^{\prime}s}+\left(p_{2}(r)+\frac{2}{3}s_{2}(r)\right)\hat{Q}_{s^{\prime}s}^{rr}$
(45)
$\displaystyle-\frac{1}{m^{2}}\hat{Q}_{s^{\prime}s}^{rr}\left(r\frac{d}{dr}\frac{1}{r}\frac{d}{dr}\left(p_{3}(r)+\frac{2}{3}s_{3}(r)\right)+s_{3}(r)\frac{2}{r^{2}}\right)\Bigg{]},$
$\displaystyle\frac{dF_{\theta}}{dS_{r}}$ $\displaystyle=$ $\displaystyle
N_{\phi,R,2}\bigg{[}\left(p_{2}(r)+\frac{2}{3}s_{2}(r)\right)\hat{Q}_{s^{\prime}s}^{\theta
r}-\frac{2}{m^{2}}\hat{Q}_{s^{\prime}s}^{\theta
r}\frac{d}{dr}\frac{s_{3}(r)}{r}\bigg{]},$ (46)
$\displaystyle\frac{dF_{\varphi}}{dS_{r}}$ $\displaystyle=$ $\displaystyle
N_{\phi,R,2}\bigg{[}\left(p_{2}(r)+\frac{2}{3}s_{2}(r)\right)\hat{Q}_{s^{\prime}s}^{\varphi
r}-\frac{2}{m^{2}}\hat{Q}_{s^{\prime}s}^{\varphi
r}\frac{d}{dr}\frac{s_{3}(r)}{r}\bigg{]}.$ (47)

Notice that for an unpolarized spin-3/2 hadron, the normal force acting on the
radial area element (${dF_{r}}/{dS_{r}}$) is solely due to
$p_{0}(r)+\frac{2}{3}s_{0}(r)$.

As defined in Ref. Polyakov:2018zvc , the mechanical radius is given by

$\displaystyle\langle r_{n}^{2}\rangle_{\mathrm{mech}}=\frac{\int
d^{3}r\,r^{2}\left[p_{n}(r)+\frac{2}{3}s_{n}(r)\right]}{\int
d^{3}r\,\left[p_{n}(r)+\frac{2}{3}s_{n}(r)\right]}\,.$ (48)

Notice that Eqs. (37), (38) and (48) lead to expressions for the radii which
differ from those of the Breit frame.

Analogously to the case of the electromagnetic current, the static
approximation is obtained by first expanding the integrand in $1/m$ and after
that taking the limit of a sharply localized wave packet. The resulting naive
densities read:

$\displaystyle t^{00}_{\text{naive}}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}e^{-i{\bf q}\cdot{\bf
r}}\Biggl{\\{}\left[mF_{1,0}\left(-{\bf q}^{2}\right)+\frac{{\bf
q}^{2}}{6m}F_{1,1}\left(-{\bf
q}^{2}\right)\right]\delta_{s^{\prime}s}-\frac{1}{6m}F_{1,1}\left(-{\bf
q}^{2}\right)q^{k}q^{n}{\hat{Q}^{kn}_{s^{\prime}s}}\Biggl{\\}}\,,$
$\displaystyle t^{0i}_{\text{naive}}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle-i\int\frac{d^{3}q}{(2\pi)^{3}}e^{-i{\bf q}\cdot{\bf
r}}\epsilon^{ilk}q^{l}\Biggl{\\{}\frac{1}{3}\left[F_{4,0}\left(-{\bf
q}^{2}\right)+\frac{{\bf q}^{2}}{5m^{2}}F_{4,1}\left(-{\bf
q}^{2}\right)\right]{\hat{S}^{k}_{s^{\prime}s}}-\frac{F_{4,1}\left(-{\bf
q}^{2}\right)}{6m^{2}}q^{n}q^{t}{\hat{O}^{knt}_{s^{\prime}s}}\Biggl{\\}},$
$\displaystyle t^{ij}_{\text{naive}}(s^{\prime},s,{\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}n}{R^{2}}|\tilde{\phi}({|{\bf
n}|})|^{2}n^{i}n^{j}\int\frac{d^{3}q}{(2\pi)^{3}}e^{-i{\bf q}\cdot{\bf
r}}\bigg{\\{}\left[\frac{1}{m}F_{1,0}\left(-{\bf q}^{2}\right)+\frac{{\bf
q}^{2}}{6m^{3}}F_{1,1}\left(-{\bf q}^{2}\right)\right]\delta_{s^{\prime}s}$
(49) $\displaystyle-\frac{1}{6m^{3}}F_{1,1}\left(-{\bf
q}^{2}\right)q^{k}q^{n}{\hat{Q}^{kn}_{s^{\prime}s}}\bigg{\\}}$
$\displaystyle+$ $\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}e^{-i{\bf
q}\cdot{\bf r}}\bigg{\\{}\left(q^{i}q^{j}-{\bf
q}^{2}\delta^{ij}\right)\bigg{[}\left(\frac{1}{4m}F_{2,0}\left(-{\bf
q}^{2}\right)+\frac{{\bf q}^{2}}{24m^{3}}F_{2,1}\left(-{\bf
q}^{2}\right)\right)\delta_{s^{\prime}s}$
$\displaystyle-\frac{1}{24m^{3}}F_{2,1}\left(-{\bf
q}^{2}\right)q^{k}q^{n}{\hat{Q}^{kn}_{s^{\prime}s}}\bigg{]}-\frac{4}{3m}F_{5,0}\bigg{[}\left(q^{i}q^{j}-{\bf
q}^{2}\delta^{ij}\right)\delta_{s^{\prime}s}$
$\displaystyle+\frac{1}{2}\left(q^{2}{\hat{Q}^{ij}_{s^{\prime}s}}+\delta^{ij}q^{k}q^{n}{\hat{Q}^{kn}_{s^{\prime}s}}-q^{k}q^{i}{\hat{Q}^{kj}_{s^{\prime}s}}-q^{k}q^{j}{\hat{Q}^{ki}_{s^{\prime}s}}\right)\bigg{]}+\frac{F_{3,1}\left(-{\bf
q}^{2}\right)}{6m}\delta^{ij}q^{k}q^{n}{\hat{Q}^{kn}_{s^{\prime}s}}$
$\displaystyle-\left(mF_{3,0}\left(-{\bf q}^{2}\right)+\frac{{\bf
q}^{2}}{6m}F_{3,1}\left(-{\bf q}^{2}\right)+\frac{2m}{3}F_{6,0}\left(-{\bf
q}^{2}\right)\right)\delta^{ij}\delta_{s^{\prime}s}+\frac{2m}{3}F_{6,0}\left(-{\bf
q}^{2}\right){\hat{Q}^{ij}_{s^{\prime}s}}\bigg{\\}}.$

The above results for $t^{00}_{\text{naive}}(s^{\prime},s,{\bf r})$ and
$t^{0i}_{\text{naive}}(s^{\prime},s,{\bf r})$ agree with the corresponding
Breit-frame expressions of Ref. Kim:2020lrs  modulo terms of higher-orders in
the $1/m$ expansion, contained in the Breit-frame expressions. On the other
hand, for the $ij$th component, Ref. Kim:2020lrs  only quotes the result
corresponding to the second term of $t^{ij}_{\text{naive}}(s^{\prime},s,{\bf
r})$, contained in the last four lines of Eq. (49), which is to be interpreted
as characterizing the internal forces of the considered system Freese:2021mzg
; Panteleeva:2022uii . Again, the two expressions agree modulo terms of
higher-orders in the $1/m$ expansion, contained in the Breit-frame expression
of Ref. Kim:2020lrs .

##  IV Large distance behavior of the energy, spin, pressure and shear forces
distributions of the delta resonance in chiral EFT

In Ref. Alharazin:2022wjj  the GFFs of the delta resonance have been
calculated up to third chiral order. Using these results and restricting
ourselves to the region of distances $1/\Lambda_{\text{strong}}\ll r\ll
1/M_{\pi}$ we obtained approximate asymptotic behavior of the spatial
densities from the singularities of GFFs at $t=0$ in the chiral limit. The
third order one-loop calculations of Ref. Alharazin:2022wjj  led to the
following expressions for corresponding leading non-analytical parts of the
GFFs:

$\displaystyle F_{\rm
1,0}(t)=-\frac{25g_{1}^{2}}{9216F^{2}m_{\Delta}}~{}t\,\sqrt{-t}\,,$
$\displaystyle F_{\rm
1,1}(t)=-\frac{5g_{1}^{2}m_{\Delta}}{1536F^{2}}\,\sqrt{-t}\,,$ $\displaystyle
F_{\rm 2,0}(t)=\frac{5g_{1}^{2}m_{\Delta}}{768F^{2}}\,\sqrt{-t}\,,$
$\displaystyle F_{\rm
2,1}(t)=\frac{5g_{1}^{2}m_{\Delta}^{3}}{384F^{2}}\,\frac{\sqrt{-t}}{t}\,,$
$\displaystyle F_{\rm
4,0}(t)=-\frac{5g_{1}^{2}}{1728\pi^{2}F^{2}}~{}t\ln(-t/m_{N}^{2})\,,$
$\displaystyle F_{\rm 4,1}(t)=0\,,$ $\displaystyle F_{\rm
5,0}(t)=-\frac{5g_{1}^{2}m_{\Delta}}{9216F^{2}}\,\sqrt{-t}\,,$ (50)

where $t=q^{2}$. Using the above results we obtain the following long-range
behavior for the densities derived in the previous section

$\displaystyle\rho^{E}_{0}(r)$ $\displaystyle=$
$\displaystyle\frac{25g_{1}^{2}}{1536F^{2}m_{\Delta}}\frac{1}{r^{6}}-\frac{10g_{1}^{2}}{81\pi^{2}F^{2}m_{\Delta}^{2}}\frac{1}{r^{7}}+{\cal
O}\left(\frac{1}{r^{8}}\right)\,,$ (51) $\displaystyle\rho^{E}_{2}(r)$
$\displaystyle=$
$\displaystyle\frac{35g_{1}^{2}}{6144F^{2}m_{\Delta}}\frac{1}{r^{6}}+\frac{35g_{1}^{2}}{162\pi^{2}F^{2}m_{\Delta}^{2}}\frac{1}{r^{7}}+{\cal
O}\left(\frac{1}{r^{8}}\right)\,,$ (52) $\displaystyle\rho^{J}_{1}(r)$
$\displaystyle=$
$\displaystyle\frac{5g_{1}^{2}}{162\pi^{2}F^{2}m_{\Delta}}\frac{1}{r^{5}}-\frac{125g_{1}^{2}}{3072F^{2}m_{\Delta}^{2}}\frac{1}{r^{6}}+{\cal
O}\left(\frac{1}{r^{7}}\right)\,,$ (53) $\displaystyle\rho^{J}_{3}(r)$
$\displaystyle=$
$\displaystyle-\frac{625g_{1}^{2}}{24576F^{2}m_{\Delta}^{2}}\frac{1}{r^{6}}+\frac{5g_{1}^{2}}{54\pi^{2}F^{2}m_{\Delta}^{3}}\frac{1}{r^{7}}+{\cal
O}\left(\frac{1}{r^{8}}\right)\,,$ (54) $\displaystyle p_{0}(r)$
$\displaystyle=$
$\displaystyle-\frac{25g_{1}^{2}}{2304F^{2}m_{\Delta}}\frac{1}{r^{6}}-\frac{75g_{1}^{2}}{1024F^{2}m_{\Delta}^{3}}\frac{1}{r^{8}}+{\cal
O}\left(\frac{1}{r^{9}}\right)\,,$ (55) $\displaystyle s_{0}(r)$
$\displaystyle=$
$\displaystyle\frac{5g_{1}^{2}}{96F^{2}m_{\Delta}}\frac{1}{r^{6}}+\frac{15g_{1}^{2}}{64F^{2}m_{\Delta}^{3}}\frac{1}{r^{8}}+{\cal
O}\left(\frac{1}{r^{8}}\right)\,,$ (56) $\displaystyle p_{3}(r)$
$\displaystyle=$
$\displaystyle\frac{85g_{1}^{2}m_{\Delta}}{221184F^{2}}\frac{1}{r^{4}}-\frac{155g_{1}^{2}}{196608F^{2}m_{\Delta}}\frac{1}{r^{6}}+{\cal
O}\left(\frac{1}{r^{8}}\right)\,,$ (57) $\displaystyle s_{3}(r)$
$\displaystyle=$
$\displaystyle-\frac{25g_{1}^{2}m_{\Delta}}{9216F^{2}}\frac{1}{r^{4}}+\frac{15g_{1}^{2}}{4096F^{2}m_{\Delta}}\frac{1}{r^{6}}+{\cal
O}\left(\frac{1}{r^{8}}\right)\,.$ (58)

Notice that while the delta resonances are unstable particles, our expressions
satisfy the general stability conditions of Ref. Polyakov:2018zvc , i.e.
$\rho^{E}_{0}(r)>0$ and $\frac{2}{3}s_{0}(r)+p_{0}(r)>0$. This result is in
agreement with the observation of Ref. Polyakov:2018zvc  that the general
stability conditions are necessary but not sufficient for a system to be
stable.

##  V Summary and conclusions

In this work we applied the novel definition of local spatial densities using
sharply localized wave packets Epelbaum:2022fjc  to spin-3/2 systems. Matrix
elements of the electromagnetic current and the energy-momentum tensor in the
ZAMF were considered and integral representations of associated spatial
densities in terms of form factors were derived. Following Ref. Jaffe:2020ebz
, the corresponding expressions in the Breit-frame were obtained by first
expanding the integrands in inverse powers of the mass of the system and then
taking the limit of sharply localized wave packets. This corresponds to
considering packet sizes that are much larger than the Compton wavelength of
the system. To apply the new definition as well as the Breit-frame formulas
one needs to take the packet sizes much smaller than any length scales
characterizing internal structure of the system. This makes clear that the
Breit-frame spatial densities cannot be used for systems whose Compton
wavelengths and the radii have comparable sizes Jaffe:2020ebz . However, the
novel definition used here does not impose any lower bound on the size of the
wave packet and therefore can be applied to any systems.

Considering the spatial components of the matrix elements of the EMT we
obtained the expressions of the pressure and the shear forces inside the spin
3/2-systems. We also obtained a differential equation satisfied by these
quantities due to the conservation of the EMT.

The formalism can be extended to the $\Delta\to N$ transition form factors,
however, this is not straightforward and requires a separate investigation.
Work along such lines is under way.

###### Acknowledgements.

We thank J. Panteleeva for numerous discussions and for checking some
formulas. This work was supported in part by BMBF (Grant No. 05P21PCFP1), by
DFG and NSFC through funds provided to the Sino-German CRC 110 “Symmetries and
the Emergence of Structure in QCD” (NSFC Grant No. 11621131001, DFG Project-ID
196253076 - TRR 110), by CAS through a President’s International Fellowship
Initiative (PIFI) (Grant No. 2018DM0034), by the VolkswagenStiftung (Grant No.
93562), by the EU Horizon 2020 research and innovation program (STRONG-2020,
grant agreement No. 824093), and by the Heisenberg-Landau Program 2021, by
Guangdong Provincial funding with Grant No. 2019QN01X172, the National Natural
Science Foundation of China with Grant No. 12035007 and No. 11947228,
Guangdong Major Project of Basic and Applied Basic Research No.
2020B0301030008, and the Department of Science and Technology of Guangdong
Province with Grant No. 2022A0505030010.

##  Appendix A Coefficient functions

Coefficient functions for $j^{0}_{\phi}(s^{\prime},s,{\bf r})$ and
$j^{i}_{\phi}(s^{\prime},s,{\bf r})$:

$\displaystyle\mathcal{Z}_{0}(-q_{\perp}^{2})$ $\displaystyle=$ $\displaystyle
F_{1,0}^{V}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{6m^{2}}\left[-2F_{1,0}^{V}(-q_{\perp}^{2})+F_{1,1}^{V}(-q_{\perp}^{2})+F_{2,0}^{V}(-q_{\perp}^{2})\right]$
(59) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{4}}{24m^{4}}\left[-2F_{1,1}^{V}(-q_{\perp}^{2})+F_{2,1}^{V}(-q_{\perp}^{2})\right]\,,$
$\displaystyle\mathcal{Z}_{1}(-q_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{q_{\perp}^{2}}{6m^{2}}\left[3F_{1,0}^{V}(-q_{\perp}^{2})-2F_{2,0}^{V}(-q_{\perp}^{2})\right]+\frac{q_{\perp}^{4}}{24m^{4}}\left[3F_{1,1}^{V}(-q_{\perp}^{2})-2F_{2,1}^{V}(-q_{\perp}^{2})\right]\,,$
(60) $\displaystyle\mathcal{Z}_{2}(-q_{\perp}^{2})$ $\displaystyle=$
$\displaystyle-{\frac{1}{6}}\left[-2F_{1,0}^{V}(-q_{\perp}^{2})+F_{1,1}^{V}(-q_{\perp}^{2})+2F_{2,0}^{V}(-q_{\perp}^{2})\right]+{\frac{q_{\perp}^{2}}{12m^{2}}}\left[F_{1,1}^{V}(-q_{\perp}^{2})-F_{2,1}^{V}(-q_{\perp}^{2})\right]\,,$
(61) $\displaystyle\mathcal{A}_{0}(-q_{\perp}^{2})$ $\displaystyle=$
$\displaystyle
F_{1,0}^{V}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{30m^{2}}\left[-2F_{1,0}^{V}(-q_{\perp}^{2})+7F_{1,1}^{V}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{4}}{60m^{4}}F_{1,1}^{V}(-q_{\perp}^{2})\,,$
(62) $\displaystyle\mathcal{A}_{1}(-q_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{3}F_{2,0}^{V}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{15m^{2}}\left[-F_{2,0}^{V}(-q_{\perp}^{2})+F_{2,1}^{V}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{4}}{60m^{4}}F_{2,1}^{V}(-q_{\perp}^{2})\,,$
(63) $\displaystyle\mathcal{A}_{2}(-q_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{q_{\perp}^{2}}{6m^{2}}F_{1,0}^{V}(-q_{\perp}^{2})+\frac{q_{\perp}^{4}}{24m^{4}}F_{1,1}^{V}(-q_{\perp}^{2})\,,$
(64) $\displaystyle\mathcal{A}_{3}(-q_{\perp}^{2})$ $\displaystyle=$
$\displaystyle-{\frac{1}{6}}F_{1,1}^{V}(-q_{\perp}^{2})\,,$ (65)
$\displaystyle\mathcal{A}_{4}(-q_{\perp}^{2})$ $\displaystyle=$
$\displaystyle\frac{q_{\perp}^{2}}{6m^{2}}F_{2,0}^{V}(-q_{\perp}^{2})+\frac{q_{\perp}^{4}}{24m^{4}}F_{2,1}^{V}(-q_{\perp}^{2})\,,$
(66) $\displaystyle\mathcal{A}_{5}(-q_{\perp}^{2})$ $\displaystyle=$
$\displaystyle-{\frac{1}{6}}F_{2,1}^{V}(-q_{\perp}^{2})\,.$ (67)

Coefficient functions for $t^{00}_{\phi}(s^{\prime},s,{\bf r})$ and
$t^{ij}_{\phi,0}(s^{\prime},s,{\bf r})$:

$\displaystyle\mathcal{E}_{0}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle
F_{1,0}(-q_{\perp}^{2})-\frac{2}{3}F_{6,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{3m^{2}}\left[-F_{1,0}(-q_{\perp}^{2})+\frac{1}{2}F_{1,1}(-q_{\perp}^{2})+F_{4,0}(-q_{\perp}^{2})+2F_{5,0}(-q_{\perp}^{2})\right]$
(68)
$\displaystyle+\frac{q_{\perp}^{4}}{12m^{4}}\left[-F_{1,1}(-q_{\perp}^{2})+F_{4,1}(-q_{\perp}^{2})\right],$
$\displaystyle\mathcal{E}_{1}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{2}{3}F_{6,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{m^{2}}\left[\frac{1}{2}F_{1,0}(-q_{\perp}^{2})-\frac{2}{3}F_{4,0}(-q_{\perp}^{2})-\frac{2}{3}F_{5,0}(-q_{\perp}^{2})\right]$
(69) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{4}}{m^{4}}\left[\frac{1}{8}F_{1,1}(-q_{\perp}^{2})-\frac{1}{6}F_{4,1}(-q_{\perp}^{2})\right],$
$\displaystyle\mathcal{E}_{2}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle{\frac{1}{3}}\left[F_{1,0}(-q_{\perp}^{2})-\frac{1}{2}F_{1,1}(-q_{\perp}^{2})-2F_{4,0}(-q_{\perp}^{2})\right]+{\frac{q_{\perp}^{2}}{12m^{2}}}\left[F_{1,1}(-q_{\perp}^{2})-2F_{4,1}(-q_{\perp}^{2})\right],$
(70) $\displaystyle{a}_{1}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf r}}\int
d^{2}\hat{n}\,\frac{q_{\perp}^{2}}{2q^{2}}\bigg{\\{}F_{1,0}(-q_{\perp}^{2})-\frac{2}{3}F_{6,0}(-q_{\perp}^{2})$
(71) $\displaystyle-$
$\displaystyle\frac{q_{\perp}^{2}}{3m^{2}}\left[F_{1,0}(-q_{\perp}^{2})-\frac{1}{2}F_{1,1}(-q_{\perp}^{2})-F_{4,0}(-q_{\perp}^{2})-2F_{5,0}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{4}}{12m^{4}}\left[F_{1,1}(-q_{\perp}^{2})-F_{4,1}(-q_{\perp}^{2})\right]\bigg{\\}},\
\ $ $\displaystyle{a}_{2}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf r}}\int
d^{2}\hat{n}\,\frac{1}{q^{2}}\left(1-\frac{3q_{\perp}^{2}}{2q^{2}}\right)\bigg{\\{}F_{1,0}(-q_{\perp}^{2})-\frac{2}{3}F_{6,0}(-q_{\perp}^{2})$
(72) $\displaystyle-$
$\displaystyle\frac{q_{\perp}^{2}}{3m^{2}}\left[F_{1,0}(-q_{\perp}^{2})-\frac{1}{2}F_{1,1}(-q_{\perp}^{2})-F_{4,0}(-q_{\perp}^{2})-2F_{5,0}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{4}}{12m^{4}}\left[F_{1,1}(-q_{\perp}^{2})-F_{4,1}(-q_{\perp}^{2})\right]\bigg{\\}},\
\ $ $\displaystyle{a}_{3}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf r}}\int
d^{2}\hat{n}\;\frac{q_{\perp}^{4}}{4q^{4}}\bigg{\\{}\frac{2}{3}F_{6,0}\left(-q_{\perp}^{2}\right)+\frac{q^{2}}{6m^{2}}\left[2F_{1,0}\left(-q_{\perp}^{2}\right)-F_{1,1}\left(-q_{\perp}^{2}\right)-4F_{4,0}\left(-q_{\perp}^{2}\right)\right]$
(73) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{2}}{6m^{2}}\left[F_{1,0}\left(-q_{\perp}^{2}\right)+F_{1,1}\left(-q_{\perp}^{2}\right)-4F_{5,0}\left(-q_{\perp}^{2}\right)\right]+\frac{q^{2}q_{\perp}^{2}}{12m^{4}}\left[F_{1,1}\left(-q_{\perp}^{2}\right)-2F_{4,1}\left(-q_{\perp}^{2}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{q_{\perp}^{4}}{24m^{4}}F_{1,1}\left(-q_{\perp}^{2}\right)\bigg{\\}},$
$\displaystyle{a}_{4}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf r}}\int
d^{2}\hat{n}\;\frac{1}{3q^{4}}\bigg{\\{}2F_{6,0}\left(-q_{\perp}^{2}\right)-\frac{10q_{\perp}^{2}}{q^{2}}F_{6,0}\left(-q_{\perp}^{2}\right)+\frac{35q_{\perp}^{4}}{4q^{4}}F_{6,0}\left(-q_{\perp}^{2}\right)$
(74) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{2}}{4m^{2}}\left[4F_{1,0}\left(-q_{\perp}^{2}\right)+F_{1,1}\left(-q_{\perp}^{2}\right)-4F_{4,0}\left(-q_{\perp}^{2}\right)-8F_{5,0}\left(-q_{\perp}^{2}\right)\right]$
$\displaystyle-$
$\displaystyle\frac{5q_{\perp}^{4}}{8m^{2}q^{2}}\left[5F_{1,0}\left(-q_{\perp}^{2}\right)+\frac{7}{2}F_{1,1}\left(-q_{\perp}^{2}\right)-2F_{4,0}\left(-q_{\perp}^{2}\right)-16F_{5,0}\left(-q_{\perp}^{2}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{q_{\perp}^{4}}{4m^{4}}\left[F_{1,1}\left(-q_{\perp}^{2}\right)-F_{4,1}\left(-q_{\perp}^{2}\right)\right]+\frac{35q_{\perp}^{6}}{16m^{2}q^{4}}\left[F_{1,0}\left(-q_{\perp}^{2}\right)+F_{1,1}\left(-q_{\perp}^{2}\right)-4F_{5,0}\left(-q_{\perp}^{2}\right)\right]$
$\displaystyle+$
$\displaystyle\frac{5q_{\perp}^{6}}{32m^{4}q^{2}}\left[-5F_{1,1}\left(-q_{\perp}^{2}\right)+2F_{4,1}\left(-q_{\perp}^{2}\right)\right]+\frac{35q_{\perp}^{8}}{64m^{4}q^{4}}F_{1,1}\left(-q_{\perp}^{2}\right)\bigg{\\}},$
$\displaystyle{a}_{5}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf r}}\int
d^{2}\hat{n}\;\frac{q_{\perp}^{2}}{12q^{4}}\bigg{\\{}4F_{6,0}\left(-q_{\perp}^{2}\right)-\frac{5q_{\perp}^{2}}{q^{2}}F_{6,0}\left(-q_{\perp}^{2}\right)$
(75) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{2}}{2m^{2}}\left[5F_{1,0}\left(-q_{\perp}^{2}\right)+\frac{1}{2}F_{1,1}\left(-q_{\perp}^{2}\right)-6F_{4,0}\left(-q_{\perp}^{2}\right)-8F_{5,0}\left(-q_{\perp}^{2}\right)\right]$
$\displaystyle-$
$\displaystyle\frac{5q_{\perp}^{4}}{4m^{2}q^{2}}\left[F_{1,0}\left(-q_{\perp}^{2}\right)+F_{1,1}\left(-q_{\perp}^{2}\right)-4F_{5,0}\left(-q_{\perp}^{2}\right)\right]+\frac{q_{\perp}^{4}}{8m^{4}}\left[5F_{1,1}\left(-q_{\perp}^{2}\right)-6F_{4,1}\left(-q_{\perp}^{2}\right)\right]$
$\displaystyle-$
$\displaystyle\frac{5q_{\perp}^{6}}{16m^{4}q^{2}}F_{1,1}\left(-q_{\perp}^{2}\right)\bigg{\\}}\,,$
$\displaystyle{a}_{6}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf r}}\int
d^{2}\hat{n}\;\frac{q_{\perp}^{2}}{3q^{4}}\bigg{\\{}2F_{6,0}\left(-q_{\perp}^{2}\right)-\frac{5q_{\perp}^{2}}{2q^{2}}F_{6,0}\left(-q_{\perp}^{2}\right)$
(76) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{2}}{8m^{2}}\left[2F_{1,0}\left(-q_{\perp}^{2}\right)+5F_{1,1}\left(-q_{\perp}^{2}\right)+4F_{4,0}\left(-q_{\perp}^{2}\right)-16F_{5,0}\left(-q_{\perp}^{2}\right)\right]$
$\displaystyle-$
$\displaystyle\frac{5q_{\perp}^{4}}{8m^{2}q^{2}}\left[F_{1,0}\left(-q_{\perp}^{2}\right)+F_{1,1}\left(-q_{\perp}^{2}\right)-4F_{5,0}\left(-q_{\perp}^{2}\right)\right]+\frac{q_{\perp}^{4}}{16m^{4}}\left[F_{1,1}\left(-q_{\perp}^{2}\right)+2F_{4,1}\left(-q_{\perp}^{2}\right)\right]$
$\displaystyle-$
$\displaystyle\frac{5q_{\perp}^{6}}{32m^{4}q^{2}}F_{1,1}\left(-q_{\perp}^{2}\right)\bigg{\\}}.$

Coefficient functions for $t^{ij}_{\phi,2}(s^{\prime},s,{\bf r})$:

$\displaystyle\mathcal{W}_{0}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle
F_{2,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{6m^{2}}\left[-2F_{2,0}(-q_{\perp}^{2})+F_{2,1}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{4}}{12m^{4}}F_{2,1}(-q_{\perp}^{2}),$
(77) $\displaystyle\mathcal{W}_{1}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{q_{\perp}^{2}}{2m^{2}}F_{2,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{4}}{8m^{4}}F_{2,1}(-q_{\perp}^{2}),$
(78) $\displaystyle\mathcal{W}_{2}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle{\frac{1}{6}}\left[2F_{2,0}(-q_{\perp}^{2})-F_{2,1}(-q_{\perp}^{2})\right]+{\frac{q_{\perp}^{2}}{12m^{2}}}F_{2,1}(-q_{\perp}^{2}),$
(79) $\displaystyle\mathcal{U}_{0}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle
F_{3,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{6m^{2}}\left[-2F_{3,0}(-q_{\perp}^{2})+F_{3,1}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{4}}{12m^{4}}F_{3,1}(-q_{\perp}^{2}),$
(80) $\displaystyle\mathcal{U}_{1}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{q_{\perp}^{2}}{2m^{2}}F_{3,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{4}}{8m^{4}}F_{3,1}(-q_{\perp}^{2}),$
(81) $\displaystyle\mathcal{U}_{2}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle{\frac{1}{6}}\left[2F_{3,0}(-q_{\perp}^{2})-F_{3,1}(-q_{\perp}^{2})\right]+{\frac{q_{\perp}^{2}}{12m^{2}}}F_{3,1}(-q_{\perp}^{2}),$
(82) $\displaystyle{v}_{0}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}q}{\left(2\pi\right)^{3}}e^{-i\vec{q}\cdot\vec{r}}\int
d^{2}\hat{n}\Biggl{\\{}-2F_{3,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{3m^{2}}\left[-\frac{3}{2}F_{2,0}(-q_{\perp}^{2})+2F_{3,0}(-q_{\perp}^{2})-F_{3,1}(-q_{\perp}^{2})\right]$
(83) $\displaystyle+$
$\displaystyle\frac{q_{\perp}^{4}}{6m^{4}}\left[F_{2,0}(-q_{\perp}^{2})-\frac{1}{2}F_{2,1}(-q_{\perp}^{2})+F_{3,1}(-q_{\perp}^{2})\right]+\frac{q_{\perp}^{6}}{24m^{6}}F_{2,1}(-q_{\perp}^{2})\Biggl{\\}},$
$\displaystyle{v}_{1}\left(r\right)$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}\hat{n}d^{3}q}{\left(2\pi\right)^{3}}e^{-i\vec{q}\cdot\vec{r}}\frac{q_{\perp}^{2}}{2m^{2}q^{2}}\Bigg{\\{}\left(\frac{2}{3}-\frac{q_{\perp}^{2}}{2q^{2}}\right)\left(-\frac{q_{\perp}^{2}}{2m^{2}}F_{2,0}(-q_{\perp}^{2})-2F_{3,0}(-q_{\perp}^{2})\right),$
(84)
$\displaystyle+\frac{1}{2}\left(\frac{1}{3}+\frac{q_{\perp}^{2}}{3m^{2}}-\frac{q_{\perp}^{2}}{q^{2}}-\frac{q_{\perp}^{4}}{4m^{2}q^{2}}\right)\left(-\frac{q_{\perp}^{2}}{2m^{2}}F_{2,1}(-q_{\perp}^{2})-2F_{3,1}(-q_{\perp}^{2})\right)\Bigg{\\}}\,.$

Coefficient functions for $t^{0i}_{\phi}(s^{\prime},s,{\bf r})$:

$\displaystyle\mathcal{C}_{0}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle
F_{1,0}(-q_{\perp}^{2})-\frac{1}{3}F_{4,0}(-q_{\perp}^{2})-\frac{4}{15}F_{6,0}(-q_{\perp}^{2})$
(85)
$\displaystyle+\frac{q_{\perp}^{2}}{15m^{2}}\left[-F_{1,0}(-q_{\perp}^{2})+\frac{7}{2}F_{1,1}(-q_{\perp}^{2})+F_{4,0}(-q_{\perp}^{2})-F_{4,1}(-q_{\perp}^{2})+4F_{5,0}(-q_{\perp}^{2})\right]$
$\displaystyle+\frac{q_{\perp}^{4}}{60m^{4}}\left[-F_{1,1}(-q_{\perp}^{2})+F_{4,0}(-q_{\perp}^{2})\right],$
$\displaystyle\mathcal{C}_{1}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{3}F_{4,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{15m^{2}}\left[-F_{4,0}(-q_{\perp}^{2})+F_{4,1}(-q_{\perp}^{2})\right]-\frac{q_{\perp}^{4}}{60m^{4}}F_{4,1}(-q_{\perp}^{2}),$
(86) $\displaystyle\mathcal{C}_{2}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{2}{3}F_{6,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{2}}{6m^{2}}\left[F_{1,0}(-q_{\perp}^{2})-F_{4,0}(-q_{\perp}^{2})-4F_{5,0}(-q_{\perp}^{2})\right]$
(87)
$\displaystyle+\frac{q_{\perp}^{4}}{24m^{4}}\left[F_{1,1}(-q_{\perp}^{2})-F_{4,1}(-q_{\perp}^{2})\right],$
$\displaystyle\mathcal{C}_{3}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle{\frac{1}{6}}\left[-F_{1,1}(-q_{\perp}^{2})+F_{4,1}(-q_{\perp}^{2})\right],$
(88) $\displaystyle\mathcal{C}_{4}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle\frac{q_{\perp}^{2}}{6m^{2}}F_{4,0}(-q_{\perp}^{2})+\frac{q_{\perp}^{4}}{24m^{4}}F_{4,1}(-q_{\perp}^{2}),$
(89) $\displaystyle\mathcal{C}_{5}\left(q_{\perp}^{2}\right)$ $\displaystyle=$
$\displaystyle-{\frac{1}{6}}F_{4,1}(-q_{\perp}^{2}).$ (90)

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