# On the definition of electromagnetic local spatial densities for composite
spin-$1/2$ systems

J. Yu. Panteleeva  Institut für Theoretische Physik II, Ruhr-Universität
Bochum, D-44780 Bochum, 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  High
Energy Physics Institute, 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

###### Abstract

An unambiguous definition of the electromagnetic spatial densities for a
spin-1/2 system is proposed and worked out in the zero average momentum frame
and in moving frames. The obtained results are compared with the traditional
definition of the densities in terms of the three-dimensional Fourier
transforms of the electromagnetic form factors in the Breit frame.

##  I Introduction

The three-dimensional Fourier transform of the charge form factor in the Breit
frame is often interpreted as the electric charge density of the corresponding
hadron, owing to the seminal papers on electron-proton scattering by
Hofstadter, Sachs and others in the 60s of the last century Hofstadter:1958 ;
Ernst:1960zza ; Sachs:1962zzc . Similar interpretations have also been
proposed for the Fourier transforms of the gravitational form factors and for
various local distributions Polyakov:2002wz ; Polyakov:2002yz ;
Polyakov:2018zvc .

Despite common belief, the identification of spatial density distributions
with the Fourier transform of the corresponding form factors in the Breit
frame suffers from conceptual issues as repeatedly pointed out in the
literature Burkardt:2000za ; Miller:2007uy ; Miller:2009qu ; Miller:2010nz ;
Jaffe:2020ebz ; Miller:2018ybm ; Freese:2021czn . In Ref. Jaffe:2020ebz , it
was shown on the example of a spin-$0$ system that the above-mentioned
traditional expression for the charge density in terms of the Breit frame
distribution follows only in the static limit of an infinitely heavy particle.
On the other hand, doubt has been raised that local density distributions can
even be defined unambiguously, i.e. independently of the form of the wave
packet, for systems whose Compton wavelength is of the order of or larger than
the charge radius (defined via the derivative of the form factor at zero
momentum transfer) Jaffe:2020ebz .

The issue of a proper definition of the spatial distributions of matrix
elements of local operators has attracted much attention in the last few years
Lorce:2020onh ; Lorce:2022cle ; Guo:2021aik ; Lorce:2018egm ; Freese:2021mzg .
In the recent work of Ref. Epelbaum:2022fjc , the proper definition of the
charge density has been revisited for a spin-$0$ system.111We thank Cedric
Lorcé for pointing out that similar results have been published long ago in
Ref. Fleming:1974af .  Closely following the logic of Ref. Jaffe:2020ebz , the
charge density has been defined unambiguously in the zero average momentum
frame (ZAMF) of the system by using spherically symmetric sharply localized
wave packets without invoking any approximations.222Under ZAMF we mean a
Lorentz frame in which the expectation value of the three-momentum for the
state, specified by the given packet, is zero. For wave packets with a sharp
localization around an eigenstate of the four-momentum operator, the ZAMF
coincides with the rest-frame of the system. The definition has also been
generalized to moving frames, and it was shown that in the infinite-momentum
frame (IMF), the charge density turns into the well-known two-dimensional
distribution in the transverse plane times the delta-function in the
longitudinal direction.

In the current paper we work out the details of our new definition of the
electromagnetic densities for spin-$1/2$ systems. Analogously to the spin-$0$
case, we consider sharply localized wave packets and obtain local spatial
distributions for the ZAMF as well as for moving frames.

Our work is organized as follows, in Section II we consider the spatial
distributions of a spin-1/2 system in the ZAMF. Section III is devoted to the
moving frames. In Section IV, we discuss the interpretation of the novel
density distributions and comment on the proton radius controversy. We end
with a summary in Section V.

##  II The electromagnetic densities of a spin-1/2 system in the ZAMF

We start with considering the electromagnetic densities of a spin-$1/2$ system
in its ZAMF. We choose the four-momentum eigenstates $|p,s\rangle$
characterizing our system to be normalized as

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

where $p=(E,{\bf p})$, with $E=\sqrt{m^{2}+{\bf p}^{2}}$ and $m$ the
particle’s mass. These states are also eigenstates of the charge operator
given by $\hat{Q}=\int d^{3}{r}\,\hat{j}^{0}({\bf r},0)$, where
$\hat{j}^{\mu}({\bf r},0)$ is the electromagnetic current operator at $t=0$ in
the Heisenberg picture.

The matrix elements of the electromagnetic current operator between momentum
eigenstates of a spin-$1/2$ system can be parameterized in terms of two form
factors:

$\displaystyle\langle p^{\prime},s^{\prime}|\hat{j}^{\mu}({\bf
r},0)|p,s\rangle$ $\displaystyle=$ $\displaystyle e^{-i({\bf p^{\prime}}-{\bf
p})\cdot{\bf
r}}\,\bar{u}(p^{\prime},s^{\prime})\left[\gamma^{\mu}F_{1}(q^{2})+\frac{1}{2}i\sigma^{\mu\nu}q_{\nu}\,F_{2}(q^{2})\right]u(p,s),$
(2)

with $p$ and $s$ ($p^{\prime}$ and $s^{\prime}$) denoting the momentum and the
polarization of the initial (final) states, respectively. The Dirac spinors
are normalized as $\bar{u}(p,s^{\prime})u(p,s)=2m\delta_{s^{\prime}s}$. The
momentum transfer is given by $q=p^{\prime}-p$, and $F_{1}(q^{2})$ and
$F_{2}(q^{2})$ are the Dirac and the Pauli form factors, respectively. These
are normalized as $F_{1}(0)=1$ and $F_{2}(0)=\kappa/m$, with $\kappa$ being
the anomalous magnetic moment of the spin-$1/2$ particle. We point out the
difference in the normalization of the Pauli form factor compared to the
literature.

To define local electromagnetic densities we calculate the matrix element of
the current operator in a state localized in coordinate space and take the
size of the localization much smaller than all length scales characterizing
the system. A normalizable Heisenberg-picture state with the center-of-mass
position ${\bf X}$ can be written in terms of a wave packet as

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

where from the normalization condition it follows that

$\int d^{3}{p}\,|\phi(s,{\bf p})|^{2}=1\,.$ (4)

We define the density distributions in the ZAMF of the system by employing
spherically symmetric wave packets with spin-independent profile functions
$\phi(s,{\bf p})=\phi({\bf p})=\phi(|{\bf p}|)$. For later convenience, we
introduce a dimensionless profile function $\tilde{\phi}$ via

$\phi({\bf p})=R^{3/2}\,\tilde{\phi}(R{\bf p})\,,$ (5)

where $R$ characterizes the size of the wave packet such that sharp
localization corresponds to small values of $R$. The current density
distribution for the state defined in Eq. (3) takes the form

$\langle\Phi,{\bf X},s^{\prime}|\hat{j}^{\mu}({\bf r},0)|\Phi,{\bf
X},s\rangle=\int\frac{d^{3}{p}\,d^{3}{p}^{\prime}}{(2\pi)^{3}\sqrt{4EE^{\prime}}}\bar{u}(p^{\prime},s^{\prime})\left[\gamma^{\mu}F_{1}(q^{2})+\frac{i\sigma^{\mu\nu}q_{\nu}}{2}F_{2}(q^{2})\right]u(p,s)\phi^{\star}({\bf
p^{\prime}})\phi({\bf p})e^{i{\bf q}\cdot({\bf X}-{\bf r})}.$ (6)

Without loss of generality we can take ${\bf X}=0$. In terms of the momenta
${\bf P}=({\bf p}^{\prime}+{\bf p})/2$ and ${\bf q}$ the current density
distribution takes the form

$\displaystyle j^{\mu}_{\phi}({\bf r})$ $\displaystyle\equiv$
$\displaystyle\langle\Phi,{\bf 0},s^{\prime}|\hat{j}^{\mu}({\bf
r},0)|\Phi,{\bf 0},s\rangle$ (7) $\displaystyle=$
$\displaystyle\int\frac{d^{3}{P}\,d^{3}{q}}{(2\pi)^{3}\sqrt{4EE^{\prime}}}\,\,\bar{u}(p^{\prime},s^{\prime})\left[\gamma^{\mu}F_{1}((E-E^{\prime})^{2}-{\bf
q}^{2})+\frac{i\sigma^{\mu\nu}q_{\nu}}{2}\,F_{2}((E-E^{\prime})^{2}-{\bf
q}^{2})\right]u(p,s)$ $\displaystyle\times$ $\displaystyle\phi\bigg{(}{\bf
P}-\frac{\bf q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf
q}{2}\bigg{)}\,e^{-i{\bf q}\cdot{\bf r}},$

where $E=\sqrt{m^{2}+{\bf P}^{2}-{\bf P}\cdot{\bf q}+{\bf q}^{2}/4}$ and
$E^{\prime}=\sqrt{m^{2}+{\bf P}^{2}+{\bf P}\cdot{\bf q}+{\bf q}^{2}/4}$.

The standard definition of the current density in terms of the form factors in
the Breit frame, $F_{i}(q^{2})=F_{i}(-{\bf q}^{2})$, which we will refer to as
“naive” following the terminology of Ref. Jaffe:2020ebz , is obtained by first
approximating the integrand in Eq. (7) by the two leading terms in the
$1/m$-expansion and subsequently localizing the wave packet by taking the
limit $R\to 0$ Miller:2018ybm ; Jaffe:2020ebz . The $1/m$-expansion of the
integrand leads to the following expressions:

$\displaystyle J^{0}_{\phi,\,\rm naive}({\bf r})$ $\displaystyle\equiv$
$\displaystyle\int\frac{d^{3}{P}\,d^{3}{q}}{(2\pi)^{3}}\,\left\\{F_{1}(-{\bf
q}^{2})-F_{2}(-{\bf q}^{2})\left[\frac{{\bf q}^{2}}{4m}-\frac{i}{2m}\,{\bf
q}\cdot({\bf P}\times{\bm{\sigma}})\right]\right\\}\phi\bigg{(}{\bf
P}-\frac{\bf q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf
q}{2}\bigg{)}\,e^{-i{\bf q}\cdot{\bf r}},$ $\displaystyle{\bf J}_{\phi,\,\rm
naive}({\bf r})$ $\displaystyle\equiv$
$\displaystyle\frac{1}{m}\int\frac{d^{3}{P}\,d^{3}{q}}{(2\pi)^{3}}\,\left[{\bf
P}\,F_{1}(-{\bf q}^{2})-\frac{F_{1}(-{\bf q}^{2})+mF_{2}(-{\bf
q}^{2})}{2}\,i\,{\bf q}\times{\bm{\sigma}}\right]\phi\bigg{(}{\bf P}-\frac{\bf
q}{2}\bigg{)}\,\phi^{\star}\bigg{(}{\bf P}+\frac{\bf q}{2}\bigg{)}\,e^{-i{\bf
q}\cdot{\bf r}}\,.$ (8)

Here and in what follows, we use $J^{\mu}$ instead of $j^{\mu}$ to indicate
that these densities are written as operators in spin space rather than the
corresponding matrix elements. The $R\to 0$ limit of the expressions in Eq.
(8) can be calculated without specifying the form factors and the profile
function $\phi({\bf p})$ using the methods developed to analyze loop integrals
in quantum field theory Gegelia:1994zz ; Beneke:1997zp . For
$F_{1}\left(q^{2}\right)$ and $F_{2}\left(q^{2}\right)$ decreasing at large
$q^{2}$ faster than $1/q^{2}$ and $1/(q^{2})^{2}$, respectively, by using the
method of dimensional counting Gegelia:1994zz , the only non-vanishing
contribution to $J^{\mu}_{\phi,\,\rm naive}({\bf r})$ in the $R\to 0$ limit is
obtained by substituting ${\bf P}=\tilde{\bf P}/R$, expanding the integrands
in Eq. (8) in $R$ around $R=0$ and keeping up to the zeroth order terms. Doing
so we obtain

$\displaystyle{J}^{0}_{\rm naive}({\bf r})\;\equiv\;\lim_{R\to
0}J^{0}_{\phi,\,\rm naive}({\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\,\left\\{F_{1}(-{\bf
q}^{2})-F_{2}(-{\bf q}^{2})\left[\frac{{\bf
q}^{2}}{4m}-\frac{1}{R}\,\frac{i}{2m}\,{\bf
q}\cdot({\bf\tilde{P}}\times{\bm{\sigma}})\right]\right\\}|\tilde{\phi}({{\bf\tilde{P}}})|^{2}\,e^{-i{\bf
q}\cdot{\bf r}}$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf
r}}\,G_{E}(-{\bf q}^{2})$ $\displaystyle\equiv$ $\displaystyle\rho_{\rm
naive}^{\rm ch}(r)\,,$ $\displaystyle{{\bf J}}_{\rm naive}({\bf
r})\;\equiv\;\lim_{R\to 0}{\bf J}_{\phi,\,\rm naive}({\bf r})$
$\displaystyle=$
$\displaystyle\int\frac{d^{3}{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\,\,\left\\{\frac{{\bf\tilde{P}}}{mR}\,F_{1}(-{\bf
q}^{2})-\frac{i\,{\bf q}\times\bm{\sigma}}{2m}\,\left[F_{1}(-{\bf
q}^{2})+mF_{2}(-{\bf
q}^{2})\right]\right\\}|\tilde{\phi}({{\bf\tilde{P}}})|^{2}\,e^{-i{\bf
q}\cdot{\bf r}}$ (9) $\displaystyle=$ $\displaystyle\frac{{\bm{\nabla}_{\bf
r}}\times{\bm{\sigma}}}{2m}\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf r}}\,G_{M}(-{\bf q}^{2})$ $\displaystyle\equiv$
$\displaystyle{\bm{\nabla}_{\bf r}}\times{\bf M}_{\rm naive}(r)\,,$

where we employed the condition imposed on $\tilde{\phi}$ in Eq. (4) and used
the fact that the integral over ${\bf\tilde{P}}$ of an odd function vanishes.
Here, $G_{E}(q^{2})$ and $G_{M}(q^{2})$ refer to the Sachs electric and
magnetic form factors Ernst:1960zza  (note again the unconventional
normalization of the Pauli form factor):

$G_{E}(q^{2})=F_{1}(q^{2})+\frac{q^{2}}{4m}F_{2}(q^{2}),\quad\quad\quad
G_{M}(q^{2})=F_{1}(q^{2})+mF_{2}(q^{2})\,.$ (10)

Owing to the classical expression for the magnetization current ${\bf J}_{\rm
mag}({\bf r})={\bm{\nabla}}\times{\bf M}({\bf r})$, the vector-valued function
${\bf M}_{\rm naive}(r)$ (an operator in the spin space) is interpreted as the
magnetization distribution of the spin-$1/2$ particle. In the ZAMF of a
spherically symmetric system, the spin operator is the only available vector,
and the function ${\bf M}_{\rm naive}(r)$ is thus interpreted as the density
of magnetic dipoles. It is usually expressed in terms of a scalar
magnetization density $\rho_{\rm naive}^{\rm mag}(r)$, which corresponds to
the Fourier transform of the magnetic form factor $G_{M}$ in the Breit frame

${\bf M}_{\rm naive}(r)=\frac{1}{2m}{\bm{\sigma}}\rho_{\rm naive}^{\rm
mag}(r)\,,\quad\quad\quad\rho_{\rm naive}^{\rm
mag}(r)=\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf
r}}\,G_{M}(-{\bf q}^{2})\,.$ (11)

In the notation we employ, the charge and scalar magnetization densities
$\rho_{\rm naive}^{\rm ch}(r)$ and $\rho_{\rm naive}^{\rm mag}(r)$ are
normalized to $1$ and to the magnetic moment $\mu=1+\kappa$ of the particle,
respectively. These two scalar densities incorporate the complete information
about the internal structure of a composite spin-$1/2$ particle encoded in the
electromagnetic form factors.

Notice that the final expressions in Eq. (9) do not depend on the shape of the
wave packet. They correspond to the leading approximation to the spatial
densities for packets localized with $R$ much bigger than the Compton
wavelength $1/m$ while much smaller than any other length scale characterizing
the system. Such a definition, however, becomes doubtful for systems like
light hadrons, whose characteristic length scales are comparable to or smaller
than the Compton wavelength, see Ref. Jaffe:2020ebz  and references therein.

An alternative definition of spatial densities, applicable to any systems, is
obtained by localizing the wave packet describing the particle under
consideration without performing a nonrelativistic expansion for the integrand
in Eq. (7). Using the method of dimensional counting, the $R\to 0$ limit in
Eq. (7) can, in fact, be taken for arbitrary (nonvanishing) values of the
particle’s mass as discussed in Ref. Epelbaum:2022fjc . Doing so we obtain

$\displaystyle J^{0}_{\phi}({\bf r})$
$\displaystyle=\int\frac{d^{3}{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\left\\{F_{1}\bigg{[}\Big{(}{\bf{\hat{\tilde{P}}}\cdot{\bf
q}}\Big{)}^{2}-{\bf q}^{2}\bigg{]}+\frac{i}{2}\,{\bf
q}\cdot\left({\bf\hat{\tilde{P}}}\times{\bm{\sigma}}\right)F_{2}\bigg{[}\Big{(}{\bf{\hat{\tilde{P}}}\cdot{\bf
q}}\Big{)}^{2}-{\bf
q}^{2}\bigg{]}\right\\}\,|\tilde{\phi}({{\bf\tilde{P}}})|^{2}\,e^{-i{\bf
q}\cdot{\bf r}}\,,$ $\displaystyle{\bf J}_{\phi}({\bf r})$
$\displaystyle=\int\frac{d^{3}{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\left\\{{\bf\hat{\tilde{P}}}\,F_{1}\bigg{[}\Big{(}{\bf{\hat{\tilde{P}}}\cdot{\bf
q}}\Big{)}^{2}-{\bf q}^{2}\bigg{]}-\frac{i}{2}\,\left({\bf
q}\times{\bm{\sigma}}-{\bf
q}\times{\bf\hat{\tilde{P}}}\,{\bm{\sigma}}\cdot{\bf\hat{\tilde{P}}}\right)F_{2}\bigg{[}\Big{(}{\bf{\hat{\tilde{P}}}\cdot{\bf
q}}\Big{)}^{2}-{\bf
q}^{2}\bigg{]}\right\\}|\tilde{\phi}({\bf{\tilde{P}}})|^{2}e^{-i{\bf
q}\cdot{\bf r}}\,,$ (12)

where ${\bf\hat{\tilde{P}}}={\bf{\tilde{P}}}/\tilde{P}$, with
$\tilde{P}\equiv|{\bf\tilde{P}}|$. For spherically symmetric wave packets with
$\tilde{\phi}({\bf{\tilde{P}}})=\tilde{\phi}(|{\bf{\tilde{P}}}|)$, the
integration over ${\bf\tilde{P}}$ can be easily performed, and the current
densities become independent of $\tilde{\phi}$. Denoting $\alpha=\cos\theta$
with $\theta$ being the angle between the vectors ${\bf{\tilde{P}}}$ and
$\bf{q}$, we obtain

$\displaystyle J^{0}({\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf q}\cdot{\bf
r}}\int_{-1}^{+1}d\alpha\,\frac{1}{2}\,F_{1}\Big{[}(\alpha^{2}-1)\,{\bf
q}^{2}\Big{]}\;\equiv\;\rho_{1}(r)\,,$ $\displaystyle{\bf J}({\bf r})$
$\displaystyle=$ $\displaystyle\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf
r}}\int_{-1}^{+1}\frac{d\alpha}{2}\,\left\\{\alpha\,{\bf\hat{q}}F_{1}\Big{[}(\alpha^{2}-1){\bf
q}^{2}\Big{]}-\frac{i}{4}\,{\bf
q}\times{\bm{\sigma}}\left(1+\alpha^{2}\right)F_{2}\Big{[}(\alpha^{2}-1)\,{\bf
q}^{2}\Big{]}\right\\}$ (13) $\displaystyle=$
$\displaystyle\frac{{\bm{\nabla}_{\bf
r}}\times{\bm{\sigma}}}{2m}\int\frac{d^{3}{q}}{(2\pi)^{3}}\,e^{-i{\bf
q}\cdot{\bf
r}}\int_{-1}^{+1}d\alpha\,\frac{1}{4}\left(1+\alpha^{2}\right)mF_{2}\Big{[}(\alpha^{2}-1)\,{\bf
q}^{2}\Big{]}$ $\displaystyle\equiv$ $\displaystyle{\bm{\nabla}_{\bf
r}}\times{\bf M}(r)\;\equiv\;\frac{{\bm{\nabla}_{\bf
r}}\times{\bm{\sigma}}}{2m}\,\rho_{2}(r)\,.$

Notice that the first term in the curly brackets, being an odd function in
$\alpha$, does not contribute to the density after integrating over $\alpha$.
The spatial densities $J^{\mu}({\bf r})$ defined above do not depend on the
shape of the wave packet and are unambiguously expressed in terms of the
experimentally measurable form factors $F_{1}(q^{2})$ and $F_{2}(q^{2})$.
However, in contrast to the naive definition in Eq. (9), the validity of Eq.
(13) does not depend on the relationship between the Compton wavelength $1/m$
and other length scales characterizing the system. In particular, Eq. (13) can
be also used to define the spatial densities of light handrons.

It is striking that the obtained result for $J^{\mu}({\bf r})$, expressed in
terms of the form factors $F_{1}(q^{2})$ and $F_{2}(q^{2})$, does not depend
on the particle’s mass, similarly to the charge density for a spinless system
introduced in Ref. Epelbaum:2022fjc . This implies that the traditional
expression for the density, $J^{\mu}_{\rm naive}({\bf r})$, does _not_ emerge
from $J^{\mu}({\bf r})$ by expanding about the static limit. As explained in
Ref. Epelbaum:2022fjc  for the case of a spin-$0$ particle, this
counterintuitive feature originates from the non-commutativity of the $R\to 0$
and $m\to\infty$ limits of $J^{\mu}_{\phi}({\bf r})$ in Eq. (7). Notice that
while a finite-order approximation in the $1/m$-expansion is valid when
calculating the form factors in Eq. (2) provided $-q^{2}\ll m^{2}$, its
validity is violated in certain momentum regions when performing the
integration in Eq. (7) if $R$ is taken of the order of the Compton wavelength
or smaller.

The ZAMF expression for the charge density $\rho_{1}(r)$ in Eq. (13) coincides
with that of a scalar particle discussed in Ref. Epelbaum:2022fjc . Given that
the state $|\Phi,{\bf 0},s\rangle$ is an eigenstate of the charge operator
with an eigenvalue $1$, both $\rho_{1}(r)$ and the naive charge density
$\rho_{\rm naive}^{\rm ch}(r)$ are normalized to $1$. On the other hand,
$\rho_{2}(r)$ is normalized to $\frac{2}{3}\kappa$ rather than to the magnetic
moment $1+\kappa$ as it is the case for the naive magnetization density
$\rho^{\rm mag}_{\rm naive}(r)$. We will come back to this point in section
III, where a geometrical interpretation of the obtained rest-frame densities
will be given in terms of the densities defined in the infinite-momentum
frame. To facilitate such an interpretation, it is instructive to rewrite
$J^{\mu}({\bf r})$, specified in Eq. (13), as

$J^{\mu}({\bf r})=\frac{1}{4\pi}\int
d{\bf\hat{n}}\,J^{\mu}_{{\bf\hat{n}}}({\bf r})\,,$ (14)

where ${\bf\hat{n}}\equiv{\bf n}/|{\bf n}|$ is a unit vector along the
direction of the vector ${\bf\tilde{P}}$ in Eq. (12) and

$J^{0}_{{\bf\hat{n}}}({\bf r})=\rho_{1,{\bf\hat{n}}}({\bf
r})\,,\quad\quad\quad{\bf J}_{{\bf\hat{n}}}({\bf
r})=\frac{1}{2m}{\bm{\nabla}}_{\bf
r}\times{\bm{\sigma}}_{\perp}\,\rho_{2,{\bf\hat{n}}}({\bf r})\,,$ (15)

where

$\rho_{i,{\bf\hat{n}}}({\bf
r})=\rho_{i}({r_{\perp}})\,\delta(r_{\parallel})\,,$ (16)

and the two-dimensional auxiliary densities $\rho_{i}({r_{\perp}})$ are
defined in terms of the form factors $F_{i}(q^{2})$ via

$\rho_{1}({r_{\perp}})=\int\frac{d^{2}{q}_{\perp}}{(2\pi)^{2}}\,e^{-i{\bf
q}_{\perp}\cdot{\bf r}_{\perp}}\,F_{1}\left(-{\bf
q}_{\perp}^{2}\right)\,,\quad\quad\quad\rho_{2}({r_{\perp}})=\int\frac{d^{2}{q}_{\perp}}{(2\pi)^{2}}\,e^{-i{\bf
q}_{\perp}\cdot{\bf r}_{\perp}}\,mF_{2}\left(-{\bf q}_{\perp}^{2}\right)\,.$
(17)

Here and in what follows, ${\bf a}_{\parallel}\equiv{\bf
a}\cdot{\bf\hat{n}}\,{\bf\hat{n}}$ (${\bf a}_{\perp}\equiv{\bf a}-{\bf
a}\cdot{\bf\hat{n}}\,{\bf\hat{n}}={\bf\hat{n}}\times({\bf
a}\times{\bf\hat{n}})$) denote the component of a vector ${\bf a}$ parallel
(perpendicular) to the vector $\bf n$, and $a_{\parallel}\equiv|{\bf
a}_{\parallel}|$, $a_{\perp}\equiv|{\bf a}_{\perp}|$. Notice that the
auxiliary densities $\rho_{i,{\bf\hat{n}}}({\bf r})$ depend on the direction
${\bf\hat{n}}$ only through the arguments $r_{\perp}$ and $r_{\parallel}$ of
the corresponding two-dimensional densities and the delta-function.

##  III The electromagnetic densities of a spin-1/2 system in moving frames

To generalize the expressions in Eqs. (13) and (14) to moving frames, we
follow the procedure of Ref. Epelbaum:2022fjc  and replace the packet in Eq.
(7) with its boosted expression. Using Eq. (3) and the transformation
properties of the momentum eigenstates under a boost $\Lambda_{\bf v}$ with
velocity ${\bf v}$,

$|p,s\rangle\;\stackrel{{\scriptstyle\Lambda_{\bf
v}}}{{\longrightarrow}}\;U(\Lambda_{\bf
v})|p,s\rangle=\sum_{s_{1}}D_{s_{1}s}\bigg{[}W\bigg{(}\Lambda_{\bf
v},\frac{{\bf p}}{m}\bigg{)}\bigg{]}|\Lambda_{\bf v}p,s_{1}\rangle\,,$ (18)

where $D_{s_{1}s}\left[W\right]$ is a spin-1/2 representation of the
corresponding Wigner rotation $W$ Weinberg:1995mt , we express a normalizable
Heisenberg-picture state located at the origin of a moving frame in terms of
the spherically symmetric ZAMF quantity $\phi({\bf p})$ as (see also Ref.
Hoffmann:2018edo ):

$|\Phi,{\bf 0},s\rangle_{\bf
v}=\int\frac{d^{3}{p}}{\sqrt{2E(2\pi)^{3}}}\,\sqrt{\gamma\Big{(}1-\frac{{\bf
v}\cdot{\bf p}}{E}\Big{)}}\,\phi\Big{(}\Lambda_{\bf v}^{-1}{\bf
p}\Big{)}\sum_{s_{1}}D_{s_{1}s}\bigg{[}W\bigg{(}\Lambda_{\bf
v},\frac{\Lambda_{\bf v}^{-1}{\bf p}}{m}\bigg{)}\bigg{]}|p,s_{1}\rangle,$ (19)

where $\gamma=(1-v^{2})^{-1/2}$, $E=\sqrt{m^{2}+{\bf p}^{2}}$, $\Lambda_{\bf
v}$ is the Lorentz boost from the ZAMF to the moving frame and $\Lambda_{\bf
v}^{-1}{\bf p}={\bf{\hat{v}}}\times({\bf p}\times{\bf{\hat{v}}})+\gamma({\bf
p}\cdot{\bf{\hat{v}}}-vE){\bf{\hat{v}}}$.

Analogously to the case of the ZAMF, we use the method of dimensional counting
by substituting ${\bf P}={\bf\tilde{P}}/R$ and obtain in the $R\to 0$ limit:

$\displaystyle J^{0}_{\phi,{\bf v}}({\bf r})$
$\displaystyle=\int\frac{d^{3}{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\,\gamma\left(1-{\bf
v}\cdot{\bf\hat{\tilde{P}}}\right)\left\\{F_{1}\bigg{[}\Big{(}{{\bf\hat{\tilde{P}}}\cdot{\bf
q}}\Big{)}^{2}-{\bf q}^{2}\bigg{]}\right.$
$\displaystyle\left.+\frac{1}{4}\left({\bf q}\cdot{\bm{\Sigma}}_{{\bf
v,{\bf\hat{m}}}}\,{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}-{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\,{\bf
q}\cdot{\bm{\Sigma}}_{{\bf v,\hat{\bf
m}}}\right)F_{2}\bigg{[}\Big{(}{{\bf\hat{\tilde{P}}}\cdot{\bf
q}}\Big{)}^{2}-{\bf
q}^{2}\bigg{]}\right\\}\,\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,e^{-i{\bf
q}\cdot{\bf r}}\,,$ $\displaystyle{\bf J}_{\phi,{\bf v}}({\bf r})$
$\displaystyle=\int\frac{d^{3}{\tilde{P}}\,d^{3}{q}}{(2\pi)^{3}}\,\gamma\left(1-{\bf
v}\cdot{\bf\hat{\tilde{P}}}\right)\left\\{\frac{1}{2}\left({\bm{\Sigma}}_{{\bf
v,\hat{\bf
m}}}\,{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}+{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\,{\bm{\Sigma}}_{{\bf
v,\hat{\bf m}}}\right)F_{1}\bigg{[}\Big{(}{\bf{\hat{\tilde{P}}}\cdot{\bf
q}}\Big{)}^{2}-{\bf q}^{2}\bigg{]}\right.$
$\displaystyle\left.-\frac{i}{4}\,{\bm{q}}\times\left({\bm{\Sigma}}_{{\bf
v,{\bf\hat{m}}}}-{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\,{\bm{\Sigma}}_{{\bf
v,\hat{\bf
m}}}\,{\bf\hat{\tilde{P}}}\cdot{\bm{\sigma}}\right)F_{2}\bigg{[}\Big{(}{\bf{\hat{\tilde{P}}}\cdot{\bf
q}}\Big{)}^{2}-{\bf
q}^{2}\bigg{]}\right\\}\,\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,e^{-i{\bf
q}\cdot{\bf r}}\,,$ (20)

where we have introduced
${\bf\tilde{P}}^{\prime}={{\bf\hat{v}}}\times\big{(}{\bf\tilde{P}}\times{{\bf\hat{v}}}\big{)}+\gamma\big{(}{\bf\tilde{P}}\cdot{{\bf\hat{v}}}-v\tilde{P}\big{)}{{\bf\hat{v}}}$
and a unit vector ${{\bf\hat{m}}}\equiv{\bf{\hat{\tilde{P}}}^{\prime}}$.
Furthermore, ${\bm{\Sigma}}_{{\bf v,\hat{\bf m}}}$ refers to the Wigner
rotated spin-operator

${\bm{\Sigma}}_{{\bf v,{\bf\hat{m}}}}\equiv
D^{\dagger}\big{[}W\big{(}\Lambda_{\bf
v},{{\bf\hat{m}}}\big{)}\big{]}{\bm{\sigma}}D\big{[}W\big{(}\Lambda_{\bf
v},{{\bf\hat{m}}}\big{)}\big{]}\,,$ (21)

with

$D\big{[}W\big{(}\Lambda_{\bf v},{{\bf\hat{m}}}\big{)}\big{]}=\lim_{R\to
0}D\bigg{[}W\bigg{(}\Lambda_{\bf v},\frac{\Lambda_{\bf
v}^{-1}({\bf\tilde{P}}/R)}{m}\bigg{)}\bigg{]}\,.$ (22)

Next, we change the integration variable
${\bf\tilde{P}}\to{\bf\tilde{P}}^{\prime}$ and define a vector valued function

${\bf n}\big{(}{\bf
v},{{\bf\hat{m}}}\big{)}={{\bf\hat{v}}}\times\big{(}{\bf\hat{m}}\times{{\bf\hat{v}}}\big{)}+\gamma\big{(}{\bf\hat{m}}\cdot{{\bf\hat{v}}}+v){{\bf\hat{v}}}\,.$
(23)

Given that
${\bf\tilde{P}}={{\bf\hat{v}}}\times\big{(}{\bf\tilde{P}}^{\prime}\times{{\bf\hat{v}}}\big{)}+\gamma\big{(}{\bf\tilde{P}}^{\prime}\cdot{{\bf\hat{v}}}+v\tilde{P}^{\prime}\big{)}{{\bf\hat{v}}}$,
it follows that ${{\bf\hat{n}}}={\bf{\hat{\tilde{P}}}}$. It is easy to verify
that the Jacobian of the change of variables
${\bf\tilde{P}}\to{\bf\tilde{P}}^{\prime}$ cancels the first factor in the
integrands in Eq. (20), leading to

$\displaystyle J^{0}_{\phi,{\bf v}}({\bf r})$ $\displaystyle=$
$\displaystyle\int\frac{d{{\bf\hat{m}}}\,d\tilde{P}^{\prime}\,d^{3}{q}}{(2\pi)^{3}}\;\big{|}\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}\big{|}^{2}\,e^{-i{\bf
q}\cdot{\bf r}}\bigg{\\{}F_{1}\Big{[}({\bf\hat{n}}\cdot{\bf q})^{2}-{\bf
q}^{2}\Big{]}+\frac{1}{4}\left({\bf{q}}\cdot{\bm{\Sigma}}_{{\bf v,\hat{\bf
m}}}\,{\bf\hat{n}}\cdot{\bm{\sigma}}-{\bf\hat{n}}\cdot{\bm{\sigma}}\,{\bf{q}}\cdot{\bm{\Sigma}}_{{\bf
v,\hat{\bf m}}}\right)F_{2}\Big{[}({\bf\hat{n}}\cdot{\bf q})^{2}-{\bf
q}^{2}\Big{]}\bigg{\\}},$ $\displaystyle{\bf J}_{\phi,{\bf v}}({\bf r})$
$\displaystyle=$
$\displaystyle\int\frac{d{{\bf\hat{m}}}\,d\tilde{P}^{\prime}\,d^{3}{q}}{(2\pi)^{3}}\;\big{|}\tilde{\phi}\big{(}\tilde{\bf
P}^{\prime}\big{)}\big{|}^{2}\,e^{-i{\bf q}\cdot{\bf
r}}\bigg{\\{}\frac{1}{2}\left({\bm{\Sigma}}_{{\bf
v,{\bf\hat{m}}}}\,{\bf\hat{n}}\cdot{\bm{\sigma}}+{\bf\hat{n}}\cdot{\bm{\sigma}}\,{\bm{\Sigma}}_{{\bf
v,\hat{\bf m}}}\right)F_{1}\Big{[}\left({{\bf\hat{n}}}\cdot{\bf
q}\right)^{2}-{\bf q}^{2}\Big{]}$ $\displaystyle-$
$\displaystyle\frac{i}{4}\,{\bm{q}}\times\left({\bm{\Sigma}}_{{\bf v,\hat{\bf
m}}}-{\bf\hat{n}}\cdot{\bm{\sigma}}\,{\bm{\Sigma}}_{{\bf v,\hat{\bf
m}}}\,{\bf\hat{n}}\cdot{\bm{\sigma}}\right)F_{2}\Big{[}\left({{\bf\hat{n}}}\cdot{\bf
q}\right)^{2}-{\bf q}^{2}\Big{]}\bigg{\\}}.$

Using the spherical symmetry of
$\tilde{\phi}\big{(}{\bf\tilde{P}}^{\prime}\big{)}=\tilde{\phi}\big{(}|{\bf\tilde{P}}^{\prime}|\big{)}$
and Eq. (4), the integration over $\tilde{P}^{\prime}$ becomes trivial,
yielding the densities which are independent of the wave packet form. To keep
the notation compact, we express the resulting densities in the form similar
to Eq. (14):

$J^{\mu}_{\bf v}({\bf r})=\frac{1}{4\pi}\int d{\bf\hat{m}}\,J^{\mu}_{{\bf
v},{\bf\hat{m}}}({\bf r})\,,$ (24)

where

$\displaystyle J^{0}_{{\bf v},{\bf\hat{m}}}({\bf r})$ $\displaystyle=$
$\displaystyle\rho_{1,{\bf\hat{n}}}({\bf r})+\frac{i}{4m}{\bm{\nabla}}_{\bf
r}\cdot\left[{\bm{\Sigma}}_{{\bf
v,{\bf\hat{m}}}},\;{\bf\hat{n}}\cdot{\bm{\sigma}}\right]_{-}\rho_{2,{\bf\hat{n}}}({\bf
r}),$ $\displaystyle{\bf J}_{{\bf v},{\bf\hat{m}}}({\bf r})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[{\bm{\Sigma}}_{{\bf
v,{\bf\hat{m}}}},\;{\bf\hat{n}}\cdot{\bm{\sigma}}\right]_{+}\rho_{1,{\bf\hat{n}}}({\bf
r})+\frac{1}{4m}\,{\bm{\nabla}}_{\bf r}\times\left(\left[{\bm{\Sigma}}_{{\bf
v,{\bf\hat{m}}}},\;{\bf\hat{n}}\cdot{\bm{\sigma}}\right]_{-}\,{\bf\hat{n}}\cdot{\bm{\sigma}}\right)\rho_{2,{\bf\hat{n}}}({\bf
r})\,,$ (25)

with $[A,\;B]_{\pm}\equiv AB\pm BA$ and the auxiliary densities
$\rho_{i,{\bf\hat{n}}}({\bf r})$ being defined in Eq. (16) with the
superscripts $\perp$ and $\parallel$ of various vectors referring, as before,
to the direction of ${\bf n}$.

The above expressions simplify considerably in two extreme cases. First, in
the particle’s ZAMF with $v=0$ and $\gamma=1$, we have ${\bf n}\big{(}{\bf
v},{{\bf\hat{m}}}\big{)}={\bf\hat{m}}$ and ${\bm{\Sigma}}_{{\bf
v,{\bf\hat{m}}}}={\bm{\sigma}}$. Thus, one can simply replace in Eq. (24)
${\bf\hat{m}}$ with ${\bf\hat{n}}$. Then, given Eq. (16), the integration over
${\bf\hat{n}}$ of the second (first) term on the right-hand side of the first
(second) equality in Eq. (III) yields a vanishing result, so that one
encounters the expressions in Eqs. (14)-(17).

The second interesting case corresponds to the infinite-momentum frame (IMF)
with $v\to 1$ and $\gamma\to\infty$, for which the vector-valued function
${\bf\hat{n}}$ turns to ${\bf\hat{v}}$. Then, the integrand in Eq. (24)
depends on ${\bf\hat{m}}$ only through the Wigner rotation matrices.
Performing the integration over ${\bf\hat{m}}$, we obtain for the spatial
densities in the IMF

$\displaystyle J^{0}_{\rm IMF}({\bf r})$ $\displaystyle\equiv$ $\displaystyle
J^{0}_{{\bf\hat{v}}}({\bf r})\;=\;\rho_{1,{\bf\hat{v}}}({\bf
r})\;+\;{\bf\hat{v}}\cdot\,\frac{{\bm{\nabla}}_{\bf
r}\times{\bm{\sigma}}_{\perp}}{4m}\,\rho_{2,{\bf\hat{v}}}({\bf r})\,,$
$\displaystyle{\bf J}_{\rm IMF}({\bf r})$ $\displaystyle\equiv$
$\displaystyle{\bf J}_{{\bf\hat{v}}}({\bf r})\;=\;\frac{{\bm{\nabla}}_{\bf
r}\times{\bm{\sigma}}_{\perp}}{4m}\,\rho_{2,{\bf\hat{v}}}({\bf r})\,,$ (26)

where the densities $\rho_{i,{\bf\hat{v}}}({\bf r})$ are given in Eqs. (16)
and (17) with ${\bf\hat{n}}={\bf\hat{v}}$. The appearance of
$\delta(r_{\parallel})$, see Eq. (16), reflects the fact that in the IMF, the
system is Lorentz-contracted to a two-dimensional object perpendicular to the
velocity ${\bf\hat{v}}$ of the moving frame. It thus makes sense to introduce
two-dimensional densities in the impact parameter space spanned by ${\bf
r}_{\perp}$ by integrating the three-dimensional densities in Eq. (III) over
$r_{\parallel}$:

$\displaystyle\int dr_{\parallel}\,J^{0}_{{\bf\hat{v}}}({\bf r})$
$\displaystyle=$
$\displaystyle\rho_{1}(r_{\perp})\;+\;{\bf\hat{v}}\cdot\,\frac{{\bm{\nabla}}_{\bf
r}\times{\bm{\sigma}}_{\perp}}{4m}\,\rho_{2}(r_{\perp})\,,$ $\displaystyle\int
dr_{\parallel}\,{\bf J}_{{\bf\hat{v}}}({\bf r})$ $\displaystyle=$
$\displaystyle\frac{{\bm{\nabla}}_{\bf
r}\times{\bm{\sigma}}_{\perp}}{4m}\,\rho_{2}(r_{\perp})\,.$ (27)

Notice that the two-dimensional transverse charge and magnetization densities
of spin-1/2 particles in the IMF have been extensively discussed in the
literature, see e.g. Burkardt:2002hr ; Miller:2007kt ; Carlson:2007xd . As for
the expression in the last line of Eq. (III), the quantity
$\frac{1}{2m}\rho_{2}(r_{\perp})$ has been interpreted in Ref. Miller:2007kt
as a true magnetization density of the system, which generates the anomalous
magnetic moment. On the other hand, Ref. Miller:2010nz  argued that a more
natural interpretation of the anomalous magnetization density is provided by
the distribution

$\tilde{\rho}_{2}({\bf
r}_{\perp})=-\frac{1}{2m}r_{y}\frac{\partial\rho_{2}(r_{\perp})}{\partial
r_{y}}\,,$ (28)

where $r_{y}$ is a component of ${\bf r}$ that is perpendicular to both ${\bf
v}$ and ${\bm{\sigma}}_{\perp}$. Clearly, both densities
$\frac{1}{2m}\rho_{2}(r_{\perp})$ and $\tilde{\rho}_{2}({\bf r}_{\perp})$ are
normalized to the anomalous magnetic moment of the particle.

##  IV Discussion and interpretation

Before discussing the interpretation of the obtained results, it is useful to
briefly summarize what has been achieved so far. Starting from a general
matrix element of the electromagnetic current operator $\hat{j}^{\mu}({\bf
r},0)$ in a wave-packet state $|\Phi,{\bf 0},s\rangle$ of a composite
spin-$1/2$ system placed at the origin of the ZAMF, we have taken the limit of
a sharp localization, $R\to 0$, in order to remove the information about the
shape of the wave packet (for a spherically symmetric $\phi(|{\bf p}|)$),
thereby obtaining the corresponding spatial density $J^{\mu}({\bf r})$. The
resulting expression in the first line of Eq. (13) relates the charge density
$J^{0}(r)\equiv\rho_{1}(r)$ to the Dirac form factor $F_{1}(q^{2})$, while the
current distribution ${\bf J}({\bf r})$ is expressed in terms of the scalar
magnetization density $\rho_{2}(r)$ related to the Pauli form factor
$F_{2}(q^{2})$. These expressions for the charge and current densities differ
from the conventional ones given in terms of the Fourier transform of the
Sachs form factors in the Breit frame, see Eqs. (9) and (11). This raises
questions about the uniqueness of our results, their relation to the
conventional densities and physical interpretation, which will be addressed
below. To keep the discussion as transparent as possible, we focus in the
following on a spin-$0$ system considered in Ref. Epelbaum:2022fjc  (but the
arguments hold for spin-$1/2$ particles as well). The charge densities
$\rho(r)$ and $\rho_{\rm naive}(r)$ of Ref. Epelbaum:2022fjc  for a spinless
system coincide with $\rho_{1}(r)$ and $\rho_{\rm naive}^{\rm ch}(r)$ of this
paper if one sets $F_{1}(q^{2})\equiv F(q^{2})$ and $F_{2}(q^{2})=0$.

To uncover the meaning of the charge distribution we need to identify the way
it can be probed experimentally, and we usually think of elastic lepton
scattering. For an infinitely heavy system described by a static charge
distribution $\rho_{\rm naive}(r)$, there are no recoil effects, and the
differential cross section in the single-photon approximation is given by the
Mott cross section for scattering off a point-like charge times $|F(-{\bf
q}^{2})|$, see e.g. Halzen:1984mc  for a textbook discussion. This exact
result provides the means for directly accessing $\rho_{\rm naive}(r)$
experimentally, and it runs deep in our way of visualizing and interpreting
the charge distribution of a composite system in nonrelativistic, quantum
mechanical settings as common in atomic, nuclear and solid-states physics.
Notice that the static limit actually corresponds to the limit of
$c\to\infty$, in which the Lorentz group reduces to the Galilean one, so that
the charge distribution $\rho_{\rm naive}(r)$ becomes frame independent.

Clearly, the static limit is merely an idealization that can be imposed to
approximate low-energy dynamics of composite systems. Contrary to what is
sometimes claimed in the literature, see e.g. Halzen:1984mc , the Breit
distribution $\rho_{\rm naive}(r)$ can not be interpreted as the intrinsic
charge density of the system beyond the strict static limit Epelbaum:2022fjc .
While it is possible to perturbatively (i.e., based on the $1/m$-expansion)
take into account corrections beyond the static limit using an alternative
definition of the charge density with the wave packet localized at distances
well above its Compton wavelength Jaffe:2020ebz , the resulting spatial
distribution does not entirely reflect the internal structure of the system,
being dependent on the wave packet. The usefulness of such a definition of the
charge density thus relies on the Compton wave length being much smaller than
the size of the system as determined by the charge radius Jaffe:2020ebz .

In contrast, the charge density defined using sharply localized wave packets
as done here and in Ref. Epelbaum:2022fjc  is completely determined by
intrinsic properties of the system, regardless of the relationship between the
Compton wave length and the characteristic scale of the system related to the
charge radius. Moreover, the resulting charge distribution $\rho(r)$ comes out
to be independent of the particle’s mass, which implies that $\rho_{\rm
naive}(r)\neq\lim_{m\to\infty}\rho(r)$ and makes the static picture described
above inappropriate for the interpretation of $\rho(r)$. More precisely, the
large momentum components of the wave packet play a crucial role in the
definition of the charge density $\rho(r)$, which thus represents an
intrinsically relativistic quantity. Remarkably, imposing the sharp
localization limit in the definition of the charge density forces one to think
of $\rho(r)$ in a holographic-like picture in terms of a continuous
superposition of images taken in all possible IMF Epelbaum:2022fjc . The usual
non-relativistic interpretation of the rest-frame charge distribution of a
heavy system characterized by the density $\rho(r)$ can only be obtained by
reconstructing $\rho_{\rm naive}(r)$ from $\rho(r)$.

The above-mentioned holographic-like relationship between the ZAMF and IMF
distributions also holds for the electromagnetic densities of a spin-1/2
particle considered in this paper. Comparing Eqs. (14) and (15) to Eq. (III),
one observes that the spatial current densities $J^{\mu}({\bf r})$ in the ZAMF
are given by integrating the three-dimensional IMF densities
$J^{\mu}_{\hat{\bf v}}({\bf r})$ over all possible directions:

$J^{0}({\bf r})=\frac{1}{4\pi}\int d{\bf\hat{v}}\,J^{0}_{{\bf\hat{v}}}({\bf
r})\,,\quad\quad{\bf J}({\bf r})=2\times\frac{1}{4\pi}\int d{\bf\hat{v}}\,{\bf
J}_{{\bf\hat{v}}}({\bf r})\,.$ (29)

This feature can be understood already by looking at the defining expressions
for the densities in Eq. (7): taking the limit $R\to 0$ to remove the
information about the wave packet profile function from the definition of the
densities brings the integrands in Eq. (7) to the kinematics with the total
momentum $|{\bf P}|$ being larger than all other momentum scales, which in
turn corresponds to the IMF (up to the Wigner rotations). In the IMF, the
system is Lorentz-contracted to essentially a two-dimensional object
perpendicular to the velocity of the moving frame with the densities given in
Eq. (III). While only such two-dimensional images of the system are observed
in the IMF, the expression in the ZAMF reconstructs the full three-dimensional
structure by putting together all possible two-dimensional “images”. In
general, the full image of a $d$-dimensional object can be reconstructed by
putting together its all possible $d-1$ dimensional “images”. This kind of
representation is possible for all positive integers $d>1$. We further
emphasize that the extra factor of $2$ that appears in the second equality in
Eq. (29) has its origin in the Wigner rotations of spin states when performing
Lorentz boosts to the IMF.

It is worth mentioning that the second moment of the charge distribution
$\rho(r)$, the quantity that should be interpreted as the mean square charge
radius of the system, is related to the form factor slope via

$\langle r^{2}\rangle=4F_{1}^{\prime}(0)~{},$ (30)

where the prime denotes differentiation with respect to $q^{2}$. This is in
contrast to the usual relationship

$\langle r^{2}\rangle_{\rm
naive}=6\left(F_{1}^{\prime}(0)+\frac{F_{2}(0)}{4m}\right)~{},$ (31)

motivated by the static definition $\rho^{\rm ch}_{\rm naive}(r)$. Thus, the
size of e.g. the proton measured by the new charge distribution $\rho_{1}(r)$
is $\sqrt{\langle r_{\rm p}^{2}\rangle}=0.62649$ fm, which differs from
$\sqrt{\langle r_{\rm p}^{2}\rangle_{\rm naive}}=0.8409(4)$ fm
ParticleDataGroup:2020ssz ; Lin:2021xrc . This discrepancy, however, has no
practical implications since the radius extracted from electron-proton
scattering as well as from electronic and muonic hydrogen is defined based on
the expansion of the electric Sachs form factor around $q^{2}=0$,

$G_{E}(q^{2})=G_{E}(0)\left[1+q^{2}\frac{\langle
r^{2}\rangle}{6}+\ldots\right]\,,$ (32)

which is consistently used in the theory underlying these processes. What we
have shown is that the radius directly related to the internal charge density,
defined via sharply localized packets, is indeed smaller due to the squeezing
of the density already explained in Ref. Epelbaum:2022fjc . This also applies
to the magnetic radius related to the slope of the Pauli form factor at zero
momentum transfer.

##  V Summary and conclusions

Using the prescription introduced in Ref. Epelbaum:2022fjc , we worked out the
details of an unambiguous definition of spatial distributions of the
electromagnetic current for composite spin-$1/2$ systems. We obtained
relationships between the form factors and the local densities in the ZAMF and
moving frames. The resulting spatial densities are independent of the specific
form of the wave packet in which the state was prepared, provided that it is
spherically symmetric and independent of the spin polarization of the state in
the ZAMF of the system. Our definition is applicable to any system,
irrespective of the relation between the Compton wavelength and other length
scales characterizing the system. We have also worked out the expressions for
the electromagnetic densities in moving frames including IMF. Similarly to the
charge density of a spin-0 system studied in Ref. Epelbaum:2022fjc , the
obtained electromagnetic spatial distributions possess a holographic-like
interpretation in terms of the two-dimensional “images” made in all possible
IMF.

We also explored an alternative way to define the spatial densities by
employing the static approximation as suggested in Ref. Jaffe:2020ebz ,
thereby recovering the conventional expressions in terms of the Fourier
transform of the Sachs form factors in the Breit frame. Comparing the two
definitions, we find that the static distributions can not be obtained as a
systematic approximation to our exact expressions. This is due to non-
commutativity of the limits of an infinitely heavy system and a sharply
localized packet.

###### Acknowledgements.

We thank Cédric Lorcé for useful comments on the manuscript. This work was
supported in part by BMBF (Grant No. 05P18PCFP1), 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 ERC NuclearTheory (grant No. 885150) and ERC EXOTIC (grant No.
101018170), by CAS through a President’s International Fellowship Initiative
(PIFI) (Grant No. 2018DM0034), by the VolkswagenStiftung (Grant No. 93562),
and by the EU Horizon 2020 research and innovation programme (STRONG-2020,
grant agreement No. 824093).

## References

  * (1) R. Hofstadter, F. Bumiller, and M. R. Yearian, Rev. Mod. Phys. 30, 482 (1958).
  * (2) F. J. Ernst, R. G. Sachs and K. C. Wali, Phys. Rev. 119, 1105-1114 (1960).
  * (3) R. G. Sachs, Phys. Rev. 126, 2256-2260 (1962).
  * (4) M. V. Polyakov and A. G. Shuvaev, [arXiv:hep-ph/0207153 [hep-ph]].
  * (5) M. V. Polyakov, Phys. Lett. B 555, 57 (2003).
  * (6) M. V. Polyakov and P. Schweitzer, Int. J. Mod. Phys. A 33 (2018) no.26, 1830025.
  * (7) M. Burkardt, Phys. Rev. D 62 (2000), 071503(R), [erratum: Phys. Rev. D 66 (2002), 119903(E)].
  * (8) G. A. Miller, Phys. Rev. Lett. 99, 112001 (2007).
  * (9) G. A. Miller, Phys. Rev. C 79, 055204 (2009).
  * (10) G. A. Miller, Ann. Rev. Nucl. Part. Sci. 60 (2010), 1-25.
  * (11) R. L. Jaffe, Phys. Rev. D 103 (2021) no.1, 016017.
  * (12) G. A. Miller, Phys. Rev. C 99, no.3, 035202 (2019).
  * (13) A. Freese and G. A. Miller, Phys. Rev. D 103, 094023 (2021).
  * (14) C. Lorcé, Phys. Rev. Lett. 125, no.23, 232002 (2020), [arXiv:2007.05318 [hep-ph]].
  * (15) C. Lorcé, P. Schweitzer and K. Tezgin, [arXiv:2202.01192 [hep-ph]].
  * (16) Y. Guo, X. Ji and K. Shiells, Nucl. Phys. B 969, 115440 (2021), [arXiv:2101.05243 [hep-ph]].
  * (17) C. Lorcé, H. Moutarde and A. P. Trawiński, Eur. Phys. J. C 79, no.1, 89 (2019), [arXiv:1810.09837 [hep-ph]].
  * (18) A. Freese and G. A. Miller, Phys. Rev. D 105, no.1, 014003 (2022), [arXiv:2108.03301 [hep-ph]].
  * (19) G. N. Fleming, Physical Reality & Math. Descrip., 357 (1974).
  * (20) E. Epelbaum, J. Gegelia, N. Lange, U.-G. Meißner and M. V. Polyakov, [arXiv:2201.02565 [hep-ph]].
  * (21) J. Gegelia, G. S. Japaridze and K. S. Turashvili, Theor. Math. Phys. 101, 1313-1319 (1994).
  * (22) M. Beneke and V. A. Smirnov, Nucl. Phys. B 522, 321-344 (1998).
  * (23) S. Weinberg, “The Quantum theory of fields. Vol. 1, : Foundations,” Cambridge University Press (2005-06-02).
  * (24) S. E. Hoffmann, [arXiv:1804.00548 [quant-ph]].
  * (25) M. Burkardt, Int. J. Mod. Phys. A 18, 173-208 (2003), [arXiv:hep-ph/0207047 [hep-ph]].
  * (26) G. A. Miller, E. Piasetzky and G. Ron, Phys. Rev. Lett. 101, 082002 (2008), [arXiv:0711.0972 [nucl-th]].
  * (27) C. E. Carlson and M. Vanderhaeghen, Phys. Rev. Lett. 100, 032004 (2008), [arXiv:0710.0835 [hep-ph]].
  * (28) F. Halzen and A. D. Martin, Quarks and Leptons: An Introductory Cource in Modern Particle Physics. John Wiley & Sons, USA, 1 1984. ISBN 978-0-471-88741-6.
  * (29) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020).
  * (30) Y. H. Lin, H.-W. Hammer and U.-G. Meißner, Phys. Rev. Lett. 128, no.5, 052002 (2022) [arXiv:2109.12961 [hep-ph]].