[a]Cédric Lorcé

# Electromagnetic and gravitational form factors of the nucleon

###### Abstract

Form factors are Lorentz invariant functions describing the internal structure
of a system. In particular, they encode how physical properties like, e.g.,
charge, energy, momentum, and pressure are spatially distributed. While
nucleon electromagnetic form factors have been studied for a long time, the
first extraction of nucleon gravitational form factors from experimental data
was reported in 2018, triggering a lot of enthusiasm and attention in the
hadronic community. In this contribution we review some theoretical bases,
discuss recent developments regarding the physical interpretation of these
form factors, and give a glimpse of what can be learned about the mass and
spin structure of the nucleon.

##  1 Introduction

Exclusive scattering experiments allow one to probe how physical properties of
a microscopic system are distributed in space. In particular, the
electromagnetic structure of hadrons has been measured with extremely high
precision over the past decades, see e.g. [1, 2] for recent reviews.
Interestingly, it turns out that the mechanical structure of hadrons can also
be measured, albeit indirectly [3].

The study of elastic lepton-hadron scatterings provides information on the
matrix elements of the electromagnetic current $\langle
p^{\prime},s^{\prime}|\hat{j}^{\mu}|p,s\rangle$, where $p$ is the four-
momentum and $s$ is the polarization of the hadron. These matrix elements can
be parametrized in terms of Lorentz-invariant functions known as
electromagnetic form factors (FFs), and are usually interpreted after Fourier
transform in the Breit frame (BF) in terms of charge and magnetization
densities [4, 5]. However, because of relativistic recoil corrections, these
cannot be considered as probabilistic distributions [6, 7, 8]. Genuine charge
and magnetization densities can alternatively be defined on the light front
(LF) [9, 10, 11], but they display distortions that are hard to conciliate
with the picture of a system at rest [12, 13].

Various attempts to clarify the concept of relativistic spatial distribution
have recently appeared in the literature, see e.g. [14, 15, 16, 17, 18, 19,
20, 21], but the only way that reconciles BF and LF distributions while
explaining at the same time the origin of LF distortions is to adopt a phase-
space approach [22, 23, 24]. The latter allows one to embrace the frame
dependence of relativistic spatial distributions by relaxing the density
interpretation to a quasi-probabilistic one. Some of the results obtained
within this approach are summarized in the rest of this contribution.

##  2 Phase-space formalism

The physical meaning of LF distributions can usually be better understood in
terms of the closely related distributions in the infinite-momentum frame
(IMF) [25], where the target moves with almost the speed of light relative to
the inertial observer. To interpolate between the BF and IMF descriptions, the
concept of elastic frame (EF) distribution was first introduced in [26] and
further motivated by the phase-space approach [22, 23].

The expectation value of a generic operator $O$ in a physical state
$|\psi\rangle$ can be represented in phase-space as follows

$\langle\psi|O|\psi\rangle=\int\frac{\mathrm{d}^{3}P}{(2\pi)^{3}}\,\mathrm{d}^{3}R\,\rho_{\psi}(\vec{R},\vec{P})\,\langle
O\rangle_{\vec{R},\vec{P}},$ (1)

where

$\rho_{\psi}(\vec{R},\vec{P})=\int\frac{\mathrm{d}^{3}q}{(2\pi)^{3}}\,e^{-i\vec{q}\cdot\vec{R}}\,\tilde{\psi}^{*}(\vec{P}+\tfrac{\vec{q}}{2})\tilde{\psi}(\vec{P}-\tfrac{\vec{q}}{2})$
(2)

with $\tilde{\psi}(\vec{p})=\langle p|\psi\rangle/2\sqrt{p^{0}}$ is the Wigner
distribution of the system [27]. It is then natural to interpret the amplitude

$\langle
O\rangle_{\vec{R},\vec{P}}=\int\frac{\mathrm{d}^{3}\Delta}{(2\pi)^{3}}\,e^{i\vec{\Delta}\cdot\vec{R}}\,\frac{\langle
p^{\prime}|O|p\rangle}{2\sqrt{p^{0}p^{\prime 0}}}$ (3)

as the expectation value of $O$ for a system localized in average around the
position $\vec{R}$ with average momentum
$\vec{P}=(\vec{p}^{\prime}+\vec{p})/2$. In the case of a local operator
$O(x)$, one obtains a static distribution when there is no energy transfer to
the system. Since the mass shell constraints imply
$\Delta^{0}=\vec{P}\cdot\vec{\Delta}/P^{0}$, the elastic condition
$\Delta^{0}=p^{\prime 0}-p^{0}=0$ is automatically satisfied in the BF defined
by $|\vec{P}|=0$. For a non-vanishing average momentum, one can choose the
$z$-axis such that $\vec{P}=P_{z}\vec{e}_{z}$. The elastic condition is then
satisfied provided one integrates over the longitudinal coordinate, leading to
the two-dimensional (2D) EF distribution

$O_{\text{EF}}(\vec{b}_{\perp};P_{z})=\int\mathrm{d}z\,\langle
O\rangle_{\vec{R},P_{z}\vec{e}_{z}}(\vec{x})=\int\frac{\mathrm{d}^{2}\Delta_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\,\frac{\langle
p^{\prime}|O(0)|p\rangle}{2P^{0}}\bigg{|}_{\Delta_{z}=|\vec{P}_{\perp}|=0}$
(4)

which depends on the relative transverse position
$\vec{b}_{\perp}=\vec{x}_{\perp}-\vec{R}_{\perp}$ owing to translation
symmetry.

The $P_{z}$-dependence of the EF distribution is essentially determined by the
amplitude in momentum space. For a general tensor operator and spin-$j$ state,
one expects that [28, 29]

$\displaystyle\langle
p^{\prime},s^{\prime}|O^{\mu_{1}\cdots\mu_{n}}(0)|p,s\rangle=$ (5)
$\displaystyle\qquad\sum_{s_{B}^{\prime},s_{B}}{D}^{*(j)}_{s_{B}^{\prime}s^{\prime}}(p_{B}^{\prime},\Lambda){D}^{(j)}_{s_{B}s}(p_{B},\Lambda)\,\Lambda^{\mu_{1}}_{\phantom{\mu_{1}}\nu_{1}}\cdots\Lambda^{\mu_{n}}_{\phantom{\mu_{n}}\nu_{n}}\,\langle
p_{B}^{\prime},s_{B}^{\prime}|O^{\nu_{1}\cdots\nu_{n}}(0)|p_{B},s_{B}\rangle,$

where $\langle
p_{B}^{\prime},s_{B}^{\prime}|O^{\nu_{1}\cdots\nu_{n}}(0)|p_{B},s_{B}\rangle$
is the BF matrix element, $\Lambda^{\mu_{i}}_{\phantom{\mu_{i}}\nu_{i}}$ is
the Lorentz boost matrix from the BF to a generic Lorentz frame, and
${D}^{(j)}_{s_{B}s}(p_{B},\Lambda)$ is the corresponding Wigner spin rotation
matrix. The latter contribution is the least familiar but is crucial for
understanding the distortions induced by the boost [24, 30, 31, 32]. In the
following we will only consider a nucleon target, i.e. $j=\frac{1}{2}$.

##  3 Electromagnetic current

According to the phase-space approach, the 3D BF distributions of electric
charge and current are given by

$J^{\mu}_{B}(\vec{r})=\langle
j^{\mu}\rangle_{\vec{R},\vec{0}}(\vec{x})=\int\frac{\mathrm{d}^{3}\Delta}{(2\pi)^{3}}\,e^{-i\vec{\Delta}\cdot\vec{r}}\,\frac{\langle
p^{\prime}_{B},s^{\prime}_{B}|j^{\mu}(0)|p_{B},s_{B}\rangle}{2P^{0}_{B}}$ (6)

with $p^{\mu}_{B}=(P^{0}_{B},-\vec{\Delta}/2)$ and
$p^{\prime\mu}_{B}=(P^{0}_{B},\vec{\Delta}/2)$ the initial and final momenta
in the BF, and $\vec{r}=\vec{x}-\vec{R}$ the relative position. The difference
with the conventional Sachs distributions [4, 5] lies in the normalization
factor $2P^{0}_{B}$ instead of $2M$. Similarly, the corresponding 2D EF
distributions are defined as

$J^{\mu}_{\text{EF}}(\vec{b}_{\perp};P_{z})=\int\mathrm{d}z\,\langle
j^{\mu}\rangle_{\vec{R},P_{z}\vec{e}_{z}}(\vec{x})=\int\frac{\mathrm{d}^{2}\Delta_{\perp}}{(2\pi)^{2}}\,e^{-i\vec{\Delta}_{\perp}\cdot\vec{b}_{\perp}}\,\frac{\langle
p^{\prime},s^{\prime}|j^{\mu}(0)|p,s\rangle}{2P^{0}}\bigg{|}_{\Delta_{z}=|\vec{P}_{\perp}|=0}.$
(7)

In the limit $P_{z}\to 0$, one obtains the projection of the BF distributions
onto the transverse 2D plane

$J^{\mu}_{\text{EF}}(\vec{b}_{\perp};0)=\int\mathrm{d}z\,J^{\mu}_{B}(\vec{r}).$
(8)

Moreover, the total electric charge

$\mathcal{Q}=\int\mathrm{d}^{2}b_{\perp}\,J^{0}_{\text{EF}}(\vec{b}_{\perp};P_{z})\big{|}_{s^{\prime}=s}=\frac{\langle
p,s|j^{0}(0)|p,s\rangle}{2p^{0}}$ (9)

does not depend on $P_{z}$, showing that the amplitudes in momentum space are
correctly normalized.

Using the traditional parametrization in terms of the Dirac and Pauli FFs

$\langle
p^{\prime},s^{\prime}|j^{\mu}(0)|p,s\rangle=e\,\overline{u}(p^{\prime},s^{\prime})\left[\gamma^{\mu}\,F_{1}(Q^{2})+\frac{i\sigma^{\mu\nu}\Delta_{\nu}}{2M}\,F_{2}(Q^{2})\right]u(p,s),$
(10)

with $Q^{2}=-\Delta^{2}$, one finds in the BF the same spin structure as in
the non-relativistic case [6, 4, 5]

$\displaystyle\langle
p^{\prime}_{B},s^{\prime}_{B}|j^{0}(0)|p_{B},s_{B}\rangle$
$\displaystyle=e\,2M\,\delta_{s^{\prime}_{B}s_{B}}\,G_{E}(Q^{2}),$ (11)
$\displaystyle\langle
p^{\prime}_{B},s^{\prime}_{B}|\vec{j}(0)|p_{B},s_{B}\rangle$
$\displaystyle=e\,(\vec{\sigma}_{s^{\prime}_{B}s_{B}}\times
i\vec{\Delta})\,G_{M}(Q^{2}),$

where $G_{E}=F_{1}-\tau F_{2}$ with $\tau=Q^{2}/(4M^{2})$ and
$G_{M}=F_{1}+F_{2}$ are the electric and magnetic Sachs FFs, and
$\vec{\sigma}$ are the three Pauli matrices.

The spin structure in a generic EF is more complicated [24, 15, 31] but can be
massaged in the following form

$\displaystyle\langle
p^{\prime},s^{\prime}|j^{0}(0)|p,s\rangle\big{|}_{\text{EF}}$
$\displaystyle=e\,2M\,\gamma\,\Bigg{[}\left(\delta_{s^{\prime}s}\,\cos\theta+\frac{(\vec{\sigma}_{s^{\prime}s}\times
i\vec{\Delta}_{\perp})_{z}}{|\vec{\Delta}_{\perp}|}\,\sin\theta\right)G_{E}(Q^{2})$
(12)
$\displaystyle\qquad\qquad\quad+\beta\left(-\delta_{s^{\prime}s}\,\sin\theta+\frac{(\vec{\sigma}_{s^{\prime}s}\times
i\vec{\Delta}_{\perp})_{z}}{|\vec{\Delta}_{\perp}|}\,\cos\theta\right)\sqrt{\tau}\,G_{M}(Q^{2})\Bigg{]},$
$\displaystyle\langle
p^{\prime},s^{\prime}|j^{3}(0)|p,s\rangle\big{|}_{\text{EF}}$
$\displaystyle=e\,2M\,\gamma\,\Bigg{[}\beta\left(\delta_{s^{\prime}s}\,\cos\theta+\frac{(\vec{\sigma}_{s^{\prime}s}\times
i\vec{\Delta}_{\perp})_{z}}{|\vec{\Delta}_{\perp}|}\,\sin\theta\right)G_{E}(Q^{2})$
$\displaystyle\qquad\qquad\quad+\left(-\delta_{s^{\prime}s}\,\sin\theta+\frac{(\vec{\sigma}_{s^{\prime}s}\times
i\vec{\Delta}_{\perp})_{z}}{|\vec{\Delta}_{\perp}|}\,\cos\theta\right)\sqrt{\tau}\,G_{M}(Q^{2})\Bigg{]},$
$\displaystyle\langle
p^{\prime},s^{\prime}|\vec{j}_{\perp}(0)|p,s\rangle\big{|}_{\text{EF}}$
$\displaystyle=e\,(\sigma_{z})_{s^{\prime}s}\,(\vec{e}_{z}\times
i\vec{\Delta}_{\perp})\,G_{M}(Q^{2}),$

in agreement with the general expectation (LABEL:EFLorentzTrans-Spinj). The
Lorentz boost parameters are

$\gamma=\frac{P^{0}}{\sqrt{P^{2}}}=\frac{P^{0}}{P^{0}_{B}}=\frac{P^{0}}{M\sqrt{1+\tau}},\qquad\beta=\frac{P_{z}}{P^{0}},$
(13)

and the Wigner rotation angle $\theta$ satisfies

$\cos\theta=\frac{P^{0}+M(1+\tau)}{(P^{0}+M)\sqrt{1+\tau}},\qquad\sin\theta=-\frac{\sqrt{\tau}P_{z}}{(P^{0}+M)\sqrt{1+\tau}}.$
(14)

In the BF limit $P_{z}\to 0$, one recovers (11) restricted to $\Delta_{z}=0$.
In the IMF limit $P_{z}\to\infty$, one obtains using the LF coordinates
$a^{\pm}=(a^{0}\pm a^{3})/\sqrt{2}$

$\displaystyle\langle
p^{\prime},s^{\prime}|j^{+}(0)|p,s\rangle\big{|}_{\text{IMF}}$
$\displaystyle=e\,2P^{+}\,\bigg{[}\delta_{s^{\prime}s}\,F_{1}(Q^{2})+\frac{(\vec{\sigma}_{s^{\prime}s}\times
i\vec{\Delta})_{z}}{2M}\,F_{2}(Q^{2})\bigg{]},$ (15) $\displaystyle\langle
p^{\prime},s^{\prime}|j^{-}(0)|p,s\rangle\big{|}_{\text{IMF}}$
$\displaystyle=e\,2P^{-}\,\bigg{[}\delta_{s^{\prime}s}\,G_{1}(Q^{2})+\frac{(\vec{\sigma}_{s^{\prime}s}\times
i\vec{\Delta})_{z}}{2M}\,G_{2}(Q^{2})\bigg{]},$ $\displaystyle\langle
p^{\prime},s^{\prime}|\vec{j}_{\perp}(0)|p,s\rangle\big{|}_{\text{IMF}}$
$\displaystyle=e\,(\sigma_{z})_{s^{\prime}s}\,(\vec{e}_{z}\times
i\vec{\Delta}_{\perp})\,G_{M}(Q^{2}),$

where $G_{1}=(G_{E}-\tau G_{M})/(1+\tau)$ and $G_{2}=-(G_{E}+G_{M})/(1+\tau)$.
These amplitudes agree with those defining the 2D distributions within the LF
formalism [9, 12, 13]. In particular, the canonical polarization in the IMF
coincides with the LF helicity.

Clearly, the effect of a boost is not limited to a mere mixing of the $j^{0}$
and $j^{3}$ components of the electromagnetic current, but induces also a
distortion of the spin structure. As a result, both the electric and magnetic
terms contribute to the unpolarized charge distribution as soon as
$\vec{P}\neq\vec{0}$. The fact that the induced magnetic contribution in the
neutron is large and opposite to the electric contribution explains in
particular why the center of the LF charge distribution of the neutron appears
to be negative, see Fig. 1. Because of these kinematical distortions one
cannot interpret LF distributions as intrinsic densities, although they are
probabilistic.

Figure 1: Unpolarized EF charge distributions $\rho^{EF}_{E}\equiv
J^{0}_{\text{EF}}(\vec{b}_{\perp};P_{z})$ in the proton and the neutron for
different values of the target average momentum. The small panels on the right
show the electric (or convective) and magnetic (or magnetization)
contributions. Figure adapted from [24].

##  4 Energy-momentum tensor

The same formalism can be applied to the energy-momentum tensor (EMT) to
discuss nucleon mechanical properties. The concept of EMT distributions in the
BF was introduced in [33] and generalized to both the EF and the LF in [23].

The spin-$\frac{1}{2}$ matrix elements of the quark EMT can conveniently be
parametrized as follows

$\displaystyle\langle p^{\prime},s^{\prime}|T^{\mu\nu}_{q}(0)|p,s\rangle$
$\displaystyle=\overline{u}(p^{\prime},s^{\prime})\left[\frac{P^{\mu}P^{\nu}}{M}\,A_{q}(Q^{2})+\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{4M}\,D_{q}(Q^{2})+Mg^{\mu\nu}\bar{C}_{q}(Q^{2})\right.$
(16)
$\displaystyle\qquad\qquad\quad+\left.\frac{P^{\\{\mu}i\sigma^{\nu\\}\lambda}\Delta_{\lambda}}{M}\,J_{q}(Q^{2})-\frac{P^{[\mu}i\sigma^{\nu]\lambda}\Delta_{\lambda}}{M}\,S_{q}(Q^{2})\right]u(p,s),$

and similarly for the gluon EMT. The first four terms describe the symmetric
part of the EMT $T^{\\{\mu\nu\\}}=(T^{\mu\nu}+T^{\nu\mu})/2$ while the last
term accounts for the antisymmetric part
$T^{[\mu\nu]}=(T^{\mu\nu}-T^{\nu\mu})/2$. Poincaré symmetry implies some
constraints on the so-called gravitational FFs

$A_{q}(0)+A_{g}(0)=1,\qquad
J_{q}(0)+J_{g}(0)=\tfrac{1}{2},\qquad\bar{C}_{q}(Q^{2})+\bar{C}_{g}(Q^{2})=0,$
(17)

which can be understood in terms of linear momentum conservation, angular
momentum conservation, and mechanical equilibrium, respectively.

To keep the presentation simple, the following discussion is restricted to the
BF. One finds

$\displaystyle\langle
p^{\prime}_{B},s^{\prime}_{B}|T^{00}_{q}(0)|p_{B},s_{B}\rangle$
$\displaystyle=2MP^{0}_{B}\,\delta_{s^{\prime}_{B}s_{B}}\left\\{A_{q}(Q^{2})+\bar{C}_{q}(Q^{2})+\tau[D_{q}(Q^{2})-B_{q}(Q^{2})]\right\\},$
(18) $\displaystyle\langle
p^{\prime}_{B},s^{\prime}_{B}|T^{\\{0k\\}}_{q}(0)|p_{B},s_{B}\rangle$
$\displaystyle=2P^{0}_{B}\,(\vec{\sigma}_{s^{\prime}_{B}s_{B}}\times
i\vec{\Delta})^{k}\,J_{q}(Q^{2}),$ $\displaystyle\langle
p^{\prime}_{B},s^{\prime}_{B}|T^{[0k]}_{q}(0)|p_{B},s_{B}\rangle$
$\displaystyle=-2P^{0}_{B}\,(\vec{\sigma}_{s^{\prime}_{B}s_{B}}\times
i\vec{\Delta})^{k}\,S_{q}(Q^{2}),$ $\displaystyle\langle
p^{\prime}_{B},s^{\prime}_{B}|T^{ij}_{q}(0)|p_{B},s_{B}\rangle$
$\displaystyle=2MP^{0}_{B}\,\delta_{s^{\prime}_{B}s_{B}}\left\\{\frac{\Delta^{i}\Delta^{j}}{4M^{2}}\,D_{q}(Q^{2})-\delta^{ij}[\bar{C}_{q}(Q^{2})+\tau
D_{q}(Q^{2})]\right\\},$

where $B_{q}=2J_{q}-A_{q}$. These matrix elements are key to the physics
program of the future Electron-Ion Collider [34] for they encode a lot of
information about a large number of fundamental properties of the nucleons.
Here are a few examples:

  * •

The normalized amplitude $\langle
p^{\prime}_{B},s^{\prime}_{B}|T^{00}_{q}(0)|p_{B},s_{B}\rangle/(2P^{0}_{B})$
describes after Fourier transform the spatial distribution of the quark energy
$T^{00}_{q,B}(\vec{r})$, and reduces in the forward limit $\Delta\to 0$ to the
quark contribution to the nucleon mass [35, 36].

  * •

The spatial distribution of quark Belinfante angular momentum is defined as

$\mathcal{J}^{i}_{q,B}(\vec{r})=\epsilon^{ijk}r^{j}T^{\\{0k\\}}_{q,B}(\vec{r})$
(19)

and differs from the spatial distribution of quark kinetic angular momentum
[37, 26]

$J^{i}_{q,B}(\vec{r})=L^{i}_{q,B}(\vec{r})+S^{i}_{q,B}(\vec{r}),$ (20)

where orbital and spin contributions read

$\displaystyle
L^{i}_{q,B}(\vec{r})=\epsilon^{ijk}r^{j}T^{0k}_{q,B}(\vec{r}),\qquad
S^{i}_{q,B}(\vec{r})=\frac{1}{2}\langle\overline{\psi}\gamma^{i}\gamma_{5}\psi\rangle_{\vec{0},\vec{0}}(\vec{r}).$
(21)

  * •

The stress tensor is of particular interest [33, 38, 23]

$T^{ij}_{q,B}(\vec{r})=\delta^{ij}\,p_{q}(r)+\left(\frac{r^{i}r^{j}}{r^{2}}-\frac{1}{3}\,\delta^{ij}\right)s_{q}(r).$
(22)

Once summed over quark and gluon contributions, isotropic pressure $p(r)$ and
pressure anisotropy (or shear forces) $s(r)$ are related via the EMT
conservation

$\nabla^{i}T^{ij}_{B}(\vec{r})=0\qquad\Rightarrow\qquad\frac{\mathrm{d}}{\mathrm{d}r}\bigg{(}p+\frac{2}{3}\,s\bigg{)}+\frac{2}{r}\,s=0.$
(23)

As a corollary of the virial theorem, the von Laue condition
$\int\mathrm{d}^{3}r\,p(r)=0$ is satisfied to ensure mechanical equilibrium of
the system. It has also been suggested that the quantity
$D_{q}(0)+D_{g}(0)=M\int\mathrm{d}^{3}r\,r^{2}\,p(r)$ should be negative for
stability reasons [39].

The first experimental extraction of the pressure distribution inside the
proton from deeply virtual Compton scattering appeared in [40], see Fig. 2.
More conservative analyses were later reported in [41, 42]. Gluon
gravitational FFs have also been extracted from the photoproduction of
$J/\psi$ in the threshold region [43]. On the Lattice QCD side, recent
estimates of the gravitational FFs can be found e.g. in [44, 45, 46]. For more
details and references, see the review [3].

Figure 2: Radial distribution of the quark pressure $r^{2}p_{q}(r)$ in the
proton based on JLab data [40]. Note that the contribution from
$\bar{C}_{q}(Q^{2})$ is currently not known and was set to zero. Figure
adapted from [3].

##  5 Conclusions

Electromagnetic and gravitational form factors are Lorentz-invariant functions
encoding the information about the spatial structure of a system. In a
relativistic regime, recoil effects cannot in general be neglected. As a
result, spatial distributions are usually frame-dependent. The phase-space
formalism is therefore the natural language to formulate the relativistic
notion of spatial distribution, albeit in a quasi-probabilistic picture. This
approach interpolates between the Breit frame ($P_{z}=0$) and light-front
($P_{z}\to\infty$) representations, and clearly shows that the spin of the
target plays a key role when it comes to understanding how spatial
distributions are affected by a Lorentz boost.

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