# Transverse charge and current densities in the nucleon
from dispersively improved chiral effective field theory

J. M. Alarcón<EMAIL_ADDRESS>University of Alcalá, Nuclear and
Particle Physics Group, Department of Physics and Mathematics, 28805 Alcalá de
Henares (Madrid), Spain  C. Weiss<EMAIL_ADDRESS>Theory Center, Jefferson
Lab, Newport News, VA 23606, USA

###### Abstract

Background

The transverse densities $\rho_{1,2}(b)$ describe the distributions of
electric charge and magnetic moment at fixed light-front time and connect the
nucleon’s elastic form factors with its partonic structure. The dispersive
representation of the form factors $F_{1,2}(t)$ expresses the densities in
terms of exchanges of hadronic states in the $t$-channel and permits their
analysis using hadronic physics methods.

Purpose

Compute the densities at peripheral distances $b=\mathcal{O}(M_{\pi}^{-1})$,
where they are generated predominantly by the two-pion states in the
dispersive representation. Quantify the uncertainties.

Methods

Dispersively improved chiral effective field theory (DI$\chi$EFT) is used to
calculate the isovector spectral functions $\textrm{Im}\,F_{1,2}(t)$ on the
two-pion cut. The method includes $\pi\pi$ interactions ($\rho$ resonance)
through elastic unitarity and provides realistic spectral functions up to
$t\approx$ 1 GeV2. Higher-mass states are parametrized by effective poles and
constrained by sum rules (charges, radii, superconvergence relations). The
densities $\rho_{1,2}(b)$ are obtained from their dispersive representation.
Uncertainties are quantified by varying the spectral functions. The method
respects analyticity and ensures the correct $b\rightarrow\infty$ asymptotic
behavior of the densities.

Results

Accurate densities are obtained at all distances $b\gtrsim 0.5$ fm, with
correct behavior down to $b\rightarrow 0$. The region of distances is
quantified where transverse nucleon structure is governed by the two-pion
state. The light-front current distributions in the polarized nucleon are
computed and discussed.

Conclusions

Peripheral nucleon structure can be computed from first principles using
DI$\chi$EFT. The method can be extended to generalized parton distributions
and other nucleon form factors.

###### Contents

  1. I Introduction
  2. II Methods
    1. II.1 Transverse densities
    2. II.2 Dispersive representation
    3. II.3 Spectral functions
    4. II.4 Uncertainty estimates
  3. III Results
    1. III.1 Spectral functions
    2. III.2 Isovector densities
    3. III.3 Proton and neutron densities
    4. III.4 Current densities in polarized nucleon
  4. IV Discussion
  5. A Nucleon radii
  6. B $N$ functions
  7. C Isoscalar parametrization

##  I Introduction

Transverse densities have emerged as a key concept in nucleon structure
physics. The functions $\rho_{1,2}(b)$ describe the transverse coordinate
distributions of charge and current in the nucleon at fixed light-front time
$x^{+}=x^{0}+x^{3}$ and provide a spatial representation appropriate to the
relativistic nature of the dynamical system Soper (1977); Burkardt (2000,
2003); Miller (2007). They are defined as two-dimensional Fourier transforms
of the invariant form factors (FFs) $F_{1,2}(t)$ parametrizing the current
matrix element between nucleon states. At the same time, they represent a
projection of the generalized parton distributions (GPDs) describing the
distribution of partons in light-front longitudinal momentum and transverse
position Burkardt (2000, 2003). As such, the transverse densities connect the
FFs measured in low-energy electron-nucleon elastic scattering with the
partonic structure probed in high-energy processes such as deep-inelastic
scattering and hard exclusive processes.

The nucleon FFs $F_{1,2}(t)$ at spacelike momentum transfers $t<0$ can be
interpreted in terms of hadronic exchanges between the current and the
nucleon. The mathematical framework is provided by the dispersive
representation of the FFs based on analyticity in $t$. The FFs at $t<0$ are
expressed as integrals over their imaginary parts on the cut at $t>t_{\rm
thr}>0$, $\textrm{Im}\,F_{1,2}(t)$, the so-called spectral functions, which
correspond to $t$-channel states with definite hadronic composition and
quantum numbers. A similar dispersive representation can be derived for the
transverse densities Strikman and Weiss (2010). It establishes a
correspondence between the densities at a given distance $b$ and the hadronic
exchanges at various masses $t>0$ Miller  _et al._ (2011a). In particular, it
connects the densities at large distances $b\gtrsim$ 1 fm with the lowest-mass
$t$-channel states and permits a systematic study of “peripheral” nucleon
structure Strikman and Weiss (2010); Granados and Weiss (2014, 2015a, 2016).
Using this framework one can compute the peripheral densities from first
principles employing methods of hadronic physics. One can also explore the
duality between the hadronic exchanges in the $t$-channel and the partonic
structure in $s$-channel Miller  _et al._ (2011a).

The lowest-mass $t$-channel state in the nucleon electromagnetic FFs is the
two-pion ($\pi\pi$) state. It appears in the isovector channel and saturates
the isovector spectral functions up to $t\approx$ 1 GeV2. The $\pi\pi$ system
in this mass region interacts strongly and forms the $\rho$ resonance at
$t\approx 0.6$ GeV2. The picture of “vector dominance” abstracted from this
situation explains many observations in the phenomenology of the
electromagnetic form factors of the nucleon and other hadrons. The spectral
functions in the $\pi\pi$ channel have been constructed empirically using
methods of hadronic amplitude analysis, such as elastic unitarity with input
from $\pi N$ scattering data Hohler  _et al._ (1976); Belushkin  _et al._
(2006), or Roy-Steiner equations Hoferichter  _et al._ (2016). The transverse
densities have been studied using the dispersive representation with such
empirical spectral functions Miller  _et al._ (2011a).

It would be interesting if the transverse densities could be computed using
spectral functions derived from chiral effective field theory (chiral EFT).
This approach would open up several new possibilities. First, chiral EFT is
predictive and allows one to reduce the information content of the spectral
functions and densities to a few universal parameters (low-energy constants),
which are determined from independent measurements. Second, one can quantify
the uncertainties in the peripheral densities using the parametric expansion
of chiral EFT. Third, because chiral EFT is a point-particle field theory, one
can explore the duality between $t$-channel exchanges and $s$-channel
structure at a microscopic level and derive a partonic representation of the
chiral processes. Fourth, because of the universality of chiral EFT one can
relate the electromagnetic densities to other elements of peripheral nucleon
structure.

Traditional chiral EFT calculations of the spectral functions are limited to
the near-threshold region $t-4M_{\pi}^{2}\sim$ few $M_{\pi}^{2}$ and cannot
describe the $\rho$ resonance region, because the strong $\pi\pi$ interactions
amount to large higher-order corrections Gasser  _et al._ (1988); Bernard  _et
al._ (1996); Kubis and Meissner (2001); Kaiser (2003). When applied to the
transverse densities, this only allows one to compute the densities at very
large distances $b\gtrsim$ 2 fm, where they are extremely small and of little
practical interest Strikman and Weiss (2010); Granados and Weiss (2014, 2015a,
2016). In order to go to smaller distances, one needs an approach that takes
into account the $\pi\pi$ interactions in the $\rho$ region in a different
manner. Dispersively improved chiral EFT (DI$\chi$EFT) is an approach that
incorporates $\pi\pi$ interactions through elastic unitarity and enables EFT-
based calculations of the spectral functions in the $\rho$ meson region
Alarcón and Weiss (2017, 2018a, 2018b) (an equivalent alternative formulation
is described in Ref. Granados  _et al._ (2017)). Using the $N/D$ method, the
spectral function is separated into a part containing the nonperturbative
interactions in the $\pi\pi$ system, which is taken from the measured pion
timelike form factor, and a part describing the coupling of the $\pi\pi$
system to the nucleon, which can be computed with chiral EFT with good
convergence. The DI$\chi$EFT spectral functions have been used to compute the
electromagnetic form factors Alarcón and Weiss (2018a, b) and extract the
proton radii from elastic scattering data Alarcón  _et al._ (2019, 2020); the
approach has also been applied to the nucleon scalar FF Alarcón and Weiss
(2017). A first study of transverse densities has been performed in leading-
order (LO) accuracy Alarcón  _et al._ (2017).

In this work we use DI$\chi$EFT to compute the peripheral transverse charge
and magnetization densities $\rho_{1,2}(b)$ in the nucleon and study their
properties. We construct the $\pi\pi$ spectral functions
$\textrm{Im}\,F_{1,2}(t)$ in partial next-to-leading-order (N2LO) accuracy.
High-mass states are described by effective poles, whose parameters are fixed
by dispersive sum rules and superconvergence relations. We compute the
transverse densities in the dispersive representation and quantify the region
of distances where they are dominated by the $\pi\pi$ state. We quantify the
uncertainties of the densities resulting from the low-energy constants and the
high-mass poles. We also compute the transverse light-front current densities
and construct 2-dimensional images of the transversely polarized nucleon. We
discuss the interpretation of the results and possible applications to physics
studies with transverse densities.

Novel aspects of the present study of transverse densities are as follows: (a)
The dispersive representation respects the analytic properties of the FFs
(position and strength of singularities) and produces densities with the
correct asymptotic behavior at $b\rightarrow\infty$. This makes it possible to
reliably compute the peripheral densities and estimate their uncertainties.
Methods based on the Fourier transform of empirical FFs become unstable at
large $b$ and are not adequate for peripheral densities Venkat  _et al._
(2011). (b) The DI$\chi$EFT approach incorporates $\pi\pi$ interactions and
the $\rho$ resonance and produces realistic spectral functions up to $t\approx
1$ GeV2, which allows one to compute the densities down to distances
$b\lesssim 0.5$ fm, substantially smaller than possible with traditional
chiral EFT Strikman and Weiss (2010); Granados and Weiss (2014, 2015a, 2016).
This means that a large fraction of transverse nucleon structure is now
amenable to an EFT description and can be deconstructed in terms of effective
degrees of freedom and low-energy constants, representing a significant gain
of information. It also means that the EFT description at large distances can
be matched with a quark model-based description of the densities at distances
$b\lesssim 1$ fm, enabling studies of quark-hadron duality in the transverse
densities. (c) The uncertainties of the densities are estimated in the context
of the dispersive representation, by varying the elements of the spectral
functions. The unknown high-mass part of the spectral functions is
parametrized by a random ensemble of high-mass poles, whose distribution is
constrained by stability criteria imposed on the spacelike form factors. This
new formulation minimizes the model dependence in the description of the high-
mass states Alarcón and Weiss (2018a, b) and enables robust uncertainty
estimates for the densities.

The article is organized as follows. In Sec. II we describe the methods used
in the present study, including the properties of the transverse densities,
their dispersive representation, the construction of the DI$\chi$EFT spectral
functions, and the uncertainty estimates in the dispersive representation. In
Sec. III we describe the results, including the DI$\chi$EFT isovector spectral
functions, the isovector transverse densities, the proton and neutron
densities, and the light-front current densities in the polarized nucleon. In
Sec. IV we discuss the results and outline possible future applications to
quark-hadron duality and other structures. In Appendix A we collect the
nucleon radii and their uncertainties, which serve as input parameters in the
DI$\chi$EFT calculation. In Appendix B we summarize the formulas for the $N$
functions appearing in the calculation of the spectral functions with the
$N/D$ method. In Appendix C we describe the parametrization of the isoscalar
spectral functions used in the calculation of proton and neutron densities.

##  II Methods

###  II.1 Transverse densities

The transition matrix element of the electromagnetic current operator between
nucleon (proton, neutron) states with 4-momentum transfer $\Delta\equiv
p^{\prime}-p$ is described by the FFs $F_{1}(t)$ and $F_{2}(t)$ (Dirac and
Pauli FFs); see Ref. Alarcón  _et al._ (2017) for details. They are functions
of the invariant momentum transfer $t=\Delta^{2}$, with $t<0$ in the physical
region of elastic scattering. Their values at $t=0$ are given by the nucleon
charges and anomalous magnetic moments (in units of nuclear magnetons),

$\displaystyle F_{1}^{p,n}(0)\;$ $\displaystyle=\;Q^{p,n}\;=\;(1,0),$ (1)
$\displaystyle F_{2}^{p,n}(0)\;$
$\displaystyle=\;\kappa^{p,n}\;=\;(1.793,-1.913).$ (2)

The isovector and isoscalar components are defined as

$\displaystyle F_{i}^{V,S}\;\equiv\;{\textstyle\frac{1}{2}}(F_{i}^{p}\mp
F_{i}^{n})\hskip 20.00003pt(i=1,2);$ (3)

the same convention is used for other quantities (densities, radii; see
below).

The FFs are invariant functions and can be analyzed without specifying the
form of relativistic dynamics or choosing a reference frame. Their
interpretation in terms of spatial distributions requires specific choices. In
the light-front form of relativistic dynamics one follows the evolution of
strong interactions in light-front time $x^{+}\equiv x^{0}+x^{3}$ and
describes the structure of systems at fixed $x^{+}$ Dirac (1949); Leutwyler
and Stern (1978); Lepage and Brodsky (1980). In this context it is natural to
consider the FFs in a class of reference frames where the 4-momentum transfer
has only transverse components,
$\Delta^{0}=\Delta^{3}=0,\bm{\Delta}_{T}\equiv(\Delta^{1},\Delta^{2})\neq 0$,
with $|\bm{\Delta}_{T}|^{2}=-t$. The FFs can then be represented as
2-dimensional Fourier integrals over a transverse coordinate variable
$\bm{b}$, with $b\equiv|\bm{b}|$,

$\displaystyle F_{i}(t=-|\bm{\Delta}_{T}|^{2})$ $\displaystyle=\int
d^{2}b\;e^{i\bm{\Delta}_{T}\cdot\bm{b}}\;\rho_{i}(b)\hskip 20.00003pt(i=1,2)$
(4)

(the two-dimensional Fourier integrals can also be expressed as radial
Fourier-Bessel integrals) Soper (1977); Burkardt (2000, 2003); Miller (2007).
The functions $\rho_{1,2}(b)$ are the transverse densities. Their spatial
integrals reproduce the total charge and anomalous magnetic moment,

$\displaystyle\int d^{2}b\;\rho_{1}^{p,n}(b)$
$\displaystyle=F_{1}^{p,n}(0)=Q^{p,n},$ (5) $\displaystyle\int
d^{2}b\;\rho_{2}^{p,n}(b)$ $\displaystyle=F_{2}^{p,n}(0)=\kappa^{p,n}.$ (6)

The functions $\rho_{1,2}(b)$ describe the transverse spatial distributions of
electric charge and anomalous magnetic moment in the nucleon at fixed light-
front time. The distributions are frame-independent (they are invariant under
longitudinal light-front boosts and transform kinematically under transverse
boosts) and provide a spatial representation appropriate to the relativistic
nature of the dynamical system; see Refs. Burkardt (2000); Miller (2010);
Granados and Weiss (2014) for discussion.

Figure 1: Interpretation of the transverse densities in a proton state
localized at the transverse origin, $x=y=0$, Eq. (7). $\rho_{1}(b)$ describes
the spin-independent $J^{+}$ current at transverse radius $b$.
$\widetilde{\rho}_{2}(b)$ describes the spin-dependent distortion in a nucleon
polarized in the $y$-direction.

A simple interpretation of the densities can be provided in nucleon states
which are localized in transverse position (see Fig. 1; we use $x,y,z$ to
denote the $1,2,3$ spatial directions) Burkardt (2000); Granados and Weiss
(2014). In a state where the nucleon is localized at the transverse origin,
and its spin quantized along the $y$-direction, the expectation value of the
current $J^{+}$ at the transverse position $\bm{b}$ is

$\displaystyle\langle J^{+}(\bm{b})\rangle^{p}_{\textrm{\scriptsize
localized}}$
$\displaystyle=(...)\left[\rho_{1}^{p}(b)+(2S^{y})\cos\phi\,\widetilde{\rho}_{2}^{\,p}(b)\right],$
(7) $\displaystyle\widetilde{\rho}_{2}^{\,p}(b)$
$\displaystyle\equiv\frac{\partial}{\partial
b}\left[\frac{\rho_{2}^{p}(b)}{2m_{N}}\right]$ (8)

(same for $p\rightarrow n$). Here $(...)$ represents a factor resulting from
the normalization of states Granados and Weiss (2014), $\phi$ is the angle of
the vector $\bm{b}$ relative to the $x$–axis, and $S^{y}=\pm 1/2$ is the
expectation value of the spin in the $y$–direction. $m_{N}$ is the nucleon
mass (same for $p$ and $n$). Thus $\rho_{1}(b)$ describes the spin-independent
part of the plus current in the localized nucleon state, and
$\widetilde{\rho}_{2}(b)$ describes the spin- and angle-dependent part of the
current in a transversely polarized nucleon. Note that
$\widetilde{\rho}_{2}(b)$ satisfies the integral relation

$\displaystyle\int d^{2}b\;b\,\widetilde{\rho}_{2}^{\,p,n}(b)$
$\displaystyle=-\frac{\kappa^{\,p,n}}{m_{N}},$ (9)

which is obtained from Eq.(6) by integration by parts.

The derivatives of the FFs at $t=0$ are related to the average squared
transverse radii of the distributions,

$\displaystyle 4\frac{dF_{1}^{p,n}}{dt}(0)$ $\displaystyle=\langle
b^{2}\rangle_{1}^{p,n}\equiv\int d^{2}b\,b^{2}\,\rho_{1}^{p,n}(b),$ (10)
$\displaystyle\frac{4}{\kappa^{p,n}}\frac{dF_{2}^{p}}{dt}(0)$
$\displaystyle=\langle b^{2}\rangle_{2}^{p,n}\equiv\frac{1}{\kappa^{p,n}}\int
d^{2}b\,b^{2}\,\rho_{2}^{p,n}(b).$ (11)

The factor 4 results from the 2-dimensional distributions and replaces the
well-known factor 6 in the representation of the FF derivatives in terms of
conventional 3-dimensional radii. While the 3-dimensional radii have a
physical interpretation only in nonrelativistic systems, the transverse radii
here are averages of spatial distributions that have a well-defined meaning
for relativistic system. The relation between the transverse radii and the
3-dimensional Dirac and Pauli radii $\langle r^{2}\rangle_{1,2}$ is

$\displaystyle\langle b^{2}\rangle_{1}^{p,n}=\frac{2}{3}\langle
r^{2}\rangle_{1}^{p,n}\hskip 20.00003pt\langle
b^{2}\rangle_{2}^{p,n}=\frac{2}{3}\langle r^{2}\rangle_{2}^{p,n}.$ (12)

The nucleon radii are used as parameters in the dynamical calculations of the
transverse densities in this work. Because form factor phenomenology usually
quotes the 3-dimensional nucleon radii, we present our calculations such that
they use the 3-dimensional radii as input, keeping in mind that they are
related to the transverse radii by Eq. (12). The empirical values of the
3-dimensional radii and their uncertainties are summarized in Appendix A and
will be quoted in the following.

In studies of the nucleon’s partonic structure in QCD one considers the
transverse coordinate distributions of quarks and antiquarks with a given
light-cone momentum fraction $x$ in the proton, $f^{a}(x,b)$ and
$\bar{f}^{a}(x,b)$, where $a=u,d,...$ denotes the quark flavor. They are
defined as the Fourier representation of the GPDs $H^{a}(x,\xi=0,t)$, which
describe the form factors of partons with light-cone plus momentum fraction
$x$ in the proton, in the situation where the plus momentum difference between
the proton states is $\xi=0$ and the momentum transfer has only transverse
components Burkardt (2000, 2003),

$\displaystyle H^{a}(x,\xi=0,t=-|\bm{\Delta}_{T}|^{2})$ $\displaystyle=\int
d^{2}b\;e^{i\bm{\Delta}_{T}\cdot\bm{b}}\;f^{a}(x,b),$ (13)
$\displaystyle-H^{a}(-x,\xi=0,t=-|\bm{\Delta}_{T}|^{2})$ $\displaystyle=\int
d^{2}b\;e^{i\bm{\Delta}_{T}\cdot\bm{b}}\;\bar{f}^{a}(x,b).$ (14)

In this context the transverse charge density $\rho_{1}^{p}(b)$ represents the
integral over $x$ of the difference of the proton’s quark and antiquark
distributions at transverse distance $b$, weighted by the quark charges
$e_{a}$,

$\displaystyle\rho_{1}^{p}(b)=\sum_{a}e_{a}\int_{0}^{1}dx\;[f^{a}(x,b)-\bar{f}^{a}(x,b)],$
(15)

which can be interpreted as the cumulative charge of the partons in the proton
at the transverse radius $b$. Equivalently, $\rho_{1}^{p}(b)$ represents the
Fourier transform of the first moment of the charge-weighted GPDs,

$\displaystyle\int d^{2}b\;e^{i\bm{\Delta}_{T}\cdot\bm{b}}\;\rho_{1}^{p}(b)$
$\displaystyle=\sum_{a}e_{a}\int_{-1}^{1}dx$ $\displaystyle\times
H^{a}(x,\xi=0,t=-|\bm{\Delta}_{T}|^{2}).$ (16)

A similar relation connects the density $\rho_{2}^{p}(b)$ with the proton
helicity-flip GPDs $E^{a}(x,\xi=0,t)$ Burkardt (2000, 2003). The transverse
densities are thus directly related to the nucleon’s transverse partonic
structure in QCD.

The concepts of light-front quantization and partonic structure referenced
here are used only for the interpretation of the transverse densities but are
not needed for their computation. The densities are simple Fourier transforms
of the invariant FFs and can be computed using hadronic physics methods such
as dispersion theory and effective field theory. It is this “dual” character
that makes the transverse densities so useful for nucleon structure studies.

###  II.2 Dispersive representation

The nucleon FFs are analytic functions of $t$. The physical sheet has a
principal cut at positive real $t>t_{\rm thr}$; the threshold $t_{\rm thr}$
depends on the isospin channel (see below). The FFs satisfy unsubtracted
dispersion relations,

$\displaystyle F_{i}(t)\;$ $\displaystyle=\;\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt^{\prime}\,\frac{\textrm{Im}\,F_{i}(t^{\prime})}{t^{\prime}-t-i0}\hskip
20.00003pt(i=1,2),$ (17)

which express the functions at complex $t$ as integrals over their imaginary
parts on the cut. The real functions $\textrm{Im}\,F_{i}(t^{\prime})$ are
known as the spectral functions. They correspond to processes in which the
electromagnetic current with timelike momentum transfer $t>t_{\rm thr}$
couples to the nucleon through a hadronic state in the $t$-channel. These
processes occur in the unphysical region below the two-nucleon threshold,
$t_{\rm thr}<t<4m_{N}^{2}$, where the spectral functions cannot be measured
directly and have to be constructed using theoretical methods; see Ref. Lin
_et al._ (2021) for a review. In the isovector FFs the lowest-mass $t$-channel
state is two-pion state with $t_{\rm thr}=4M_{\pi}^{2}$; in the isoscalar FF
it is the three-pion state with $t_{\rm thr}=9M_{\pi}^{2}$.

The transverse densities Eqs. (4) can be computed as the Fourier transform of
the dispersive representation of FFs, Eq. (17) Strikman and Weiss (2010);
Miller  _et al._ (2011a). One obtains a dispersive (or spectral)
representation of the densities as

$\displaystyle\rho_{1}(b)=\phantom{-}\frac{1}{2\pi^{2}}\int_{t_{\rm
thr}}^{\infty}dt\ K_{0}(\sqrt{t}b)\ \textrm{Im}\,F_{1}(t),$ (18)
$\displaystyle\widetilde{\rho}_{2}(b)=-\frac{1}{2\pi^{2}}\int_{t_{\rm
thr}}^{\infty}dt\ \frac{\sqrt{t}}{2m_{N}}\ K_{1}(\sqrt{t}b)\
\textrm{Im}\,F_{2}(t),$ (19)

where $K_{0}$ and $K_{1}$ are the modified Bessel functions of the second
kind. Equations (18) and (19) express the densities at a given distance $b$ as
a superposition of contributions of $t$-channel states (or exchanges) with
squared mass $t$. The modified Bessel functions decay exponentially at large
arguments,

$\displaystyle K_{0,1}(\sqrt{t}b)$
$\displaystyle\sim\sqrt{\frac{\pi}{2\sqrt{t}b}}\;e^{-\sqrt{t}b}\hskip
20.00003pt(\sqrt{t}b\;\gg\;1).$ (20)

The dispersive integrals for the densities therefore converge exponentially at
large $t$, and the integration effectively extends over masses

$\displaystyle\sqrt{t}$ $\displaystyle\sim 1/b.$ (21)

This provides a mathematical formulation of the connection between the masses
of the exchanges and the ranges of the spatial distributions in the nucleon.
In particular, the peripheral densities at $b=\mathcal{O}(M_{\pi}^{-1})$ are
governed by lowest-mass states in the dispersive representation and can be
computed and analyzed accordingly. Note that the actual spectral composition
of the densities — how much the states with various $\sqrt{t}$ contribute to
the densities at given $b$ — depends on the distribution of strength in the
spectral functions and can be established only with specific models of the
latter.

The dispersive representation offers several theoretical and practical
advantages for studying the peripheral transverse densities compared to other
approaches. (a) The dispersive representation permits efficient calculation of
the peripheral densities. The exponential convergence of the dispersion
integrals reduces the contribution from the high-mass region, where the
spectral functions are poorly known. Calculations can focus on the low-mass
region — the two-pion part of the isovector spectral function, for which
dedicated theoretical methods are available. The high-mass region can be
parametrized by effective poles, whose coefficients are fixed by sum rules;
only the overall strength in this region is relevant to the peripheral
densities, not the details of the distribution. (b) The dispersive
representation automatically generates densities with the correct asymptotic
behavior at $b\rightarrow\infty$. The asymptotic behavior of the densities at
$b\rightarrow\infty$ is governed by the analytic properties of the FFs in $t$
(position and strength of singularities), which are explicitly realized in the
dispersive representation. The densities exhibit an exponential decay with a
range governed by lowest-mass exchanges, modified by a pre-exponential factor
resulting from the behavior of the spectral function near the threshold
Granados and Weiss (2014). The spectral integrals Eqs. (18) and (19) permit
stable numerical evaluation of the densities in the region where they are
exponentially small.111In Ref. Gramolin and Russell (2021) the low-$t$ nucleon
form factors were analyzed by expanding the transverse densities in a set of
basis functions (orthogonal polynomials). That method produces peripheral
densities with an oscillating behavior, which is in conflict with the smooth
exponential fall-off dictated by the dispersive representation; see Ref.
Miller  _et al._ (2011a) and the results of the present study, esp. Fig. 7.
The basis function expansion does not naturally describe the exponential
smallness of the densities, and correlations between many terms would be
required to express it correctly. In contrast, methods calculating the
densities as Fourier transforms of the FFs become numerically unstable at
large $b$, even if the FF parametrization have correct analytic
properties.222FF parametrizations with incorrect analytic properties, e.g.
rational functions with singularities at complex $t$ with $\textrm{Im}\,t\neq
0$ on the physical sheet, produce Fourier densities with qualitatively
incorrect asymptotic behavior at $b\rightarrow\infty$. Such FF
parametrizations are principally not adequate for evaluating densities at
distances above $b\gtrsim 2$ fm, even if they provide good fits to the
spacelike form factor data at small $t<0$; see Ref. Miller  _et al._ (2011a)
for a discussion. (c) The dispersive representation enables uncertainty
estimates of the peripheral densities. The densities generated by Eqs. (18)
and (19) depend smoothly on the parameters of spectral function, even at large
$b$ where they are exponentially small. Varying the parameters of the low-mass
spectral functions one can estimate the uncertainties of the peripheral
densities in a manner that respects analyticity and is numerically stable.
Methods using the frequency spectrum of the Fourier transform of FFs for
estimating the uncertainties of the densities are not appropriate for
distances significantly larger than 1 fm Venkat  _et al._ (2011).

The spectral functions obey certain integral relations (sum rules), which
result from the constraints on the nucleon FFs at $t=0$ and their derivatives
in the dispersive representation Eq. (17),

$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\frac{\textrm{Im}\,F_{1}(t)}{t}$ $\displaystyle=Q,$ (22)
$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\frac{\textrm{Im}\,F_{1}(t)}{t^{2}}$
$\displaystyle=\frac{1}{6}\langle r^{2}\rangle_{1}=\frac{1}{4}\langle
b^{2}\rangle_{1},$ (23) $\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\frac{\textrm{Im}\,F_{2}(t)}{t}$ $\displaystyle=\kappa,$
(24) $\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\frac{\textrm{Im}\,F_{2}(t)}{t^{2}}$
$\displaystyle=\frac{1}{6}\kappa\langle
r^{2}\rangle_{2}=\frac{1}{4}\kappa\langle b^{2}\rangle_{2}.$ (25)

Additional relations follow from the asymptotic behavior of the nucleon FFs at
large spacelike $|t|$,

$\displaystyle F_{1}(t)\sim|t|^{-2},\hskip
20.00003ptF_{2}(t)\sim|t|^{-3}\hskip 20.00003pt(|t|\rightarrow\infty).$ (26)

This behavior is predicted by the QCD hard scattering mechanism up to
logarithmic corrections (counting rules) Lepage and Brodsky (1980). In $F_{1}$
the predicted behavior is approximately observed in data at $|t|\gtrsim$ 1
GeV2; in $F_{2}$ the predicted behavior is not observed at presently available
momentum transfers Gayou  _et al._ (2002); see Ref. Belitsky  _et al._ (2003)
for a possible theoretical explanation in the context of QCD. When imposed on
the dispersive representation of the FFs, Eq. (26) implies the relations (so-
called superconvergence relations)

$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\textrm{Im}\,F_{1}(t)$ $\displaystyle=0,$ (27)
$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\textrm{Im}\,F_{2}(t)$ $\displaystyle=0,$ (28)
$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;t\;\textrm{Im}\,F_{2}(t)$ $\displaystyle=0.$ (29)

At the level of the densities, these relations constrain the behavior in the
limit $b\rightarrow 0$. This can be derived from the dispersive
representation, Eqs. (18) and (19), by using the limiting behavior of the
modified Bessel functions at small argument (here $z\equiv\sqrt{t}b$),

$\displaystyle K_{0}(z)$
$\displaystyle=-\left(1+\frac{z^{2}}{4}+...\right)\log z+\textrm{analytic},$
(30) $\displaystyle K_{1}(z)$
$\displaystyle=\frac{1}{z}+\left(\frac{z}{2}+...\right)\log
z+\textrm{analytic},$ (31)

where “analytic” denotes the parts that are analytic at $z=0$ (constants or
positive powers). Equation (27) implies that

$\displaystyle\rho_{1}(b)\;\rightarrow\;\textrm{finite}\hskip
20.00003pt\;\textrm{for}\;\;b\rightarrow 0.$ (32)

The kernel $K_{0}(\sqrt{t}b)$ in Eq. (18) diverges logarithmically for
$b\rightarrow 0$ according to Eq. (30); this would cause a logarithmic
divergence of $\rho_{1}(b)$; however, the coefficient of the divergent term is
zero because of Eq. (27).333Such a logarithmic divergence at $b\rightarrow 0$
is observed in the charge density in the pion, where the FF behaves as
$F_{\pi}(t)\sim|t|^{-1}$ for $|t|\rightarrow\infty$ and no relation like Eq.
(27) exists Miller (2009); Miller  _et al._ (2011b). The finiteness of the
charge density in the nucleon at $b=0$ appears natural in the parton picture,
as the nucleon is a more composite system than the pion. Similarly, Eqs. (28)
and (29) imply that

$\displaystyle\left.\begin{array}[]{rcl}\widetilde{\rho}_{2}(b)&\rightarrow&0\\\\[4.30554pt]
\displaystyle\frac{d\widetilde{\rho}_{2}}{db}(b)&\rightarrow&\textrm{finite}\end{array}\right\\}\;\;\textrm{for}\;\;b\rightarrow
0.$ (35)

###  II.3 Spectral functions

The spectral functions of the isovector nucleon FFs on the two-pion cut have
been computed using various theoretical approaches, such as analytic
continuation of pion-nucleon amplitudes Hohler  _et al._ (1976), chiral EFT
Gasser  _et al._ (1988); Bernard  _et al._ (1996); Kubis and Meissner (2001);
Kaiser (2003), and Roy-Steiner equations Hoferichter  _et al._ (2016). Here we
employ the method of DI$\chi$EFT, which combines general methods of dispersion
theory (elastic unitarity in the $\pi\pi$ channel, $N/D$ method) with specific
dynamical input from chiral EFT. The foundations of the method and its
applications to FFs are described in detail in Refs. Alarcón and Weiss (2017,
2018a, 2018b). Here we summarize only the main steps and the new features
arising in the present application to densities. The new features are: (a) We
now construct the spectral functions $\textrm{Im}\,F_{1,2}(t)$ as needed for
the transverse densities, whereas in Refs. Alarcón and Weiss (2018a, b) we
worked with $\textrm{Im}\,G_{E,M}(t)$. (b) We impose the superconvergence
relations resulting from the asymptotic behavior of $F_{1,2}(t)$, which
provide additional constraints on the high-mass part of the spectral
functions. (c) We implement a more flexible parametrization of the high-mass
part of the spectral function to enable more realistic uncertainty estimates.

The isovector spectral functions are organized as

$\displaystyle\textrm{Im}\,F_{i}^{V}(t)\;$
$\displaystyle=\;\textrm{Im}\,F_{i}^{V}(t)[\pi\pi]\;\Theta(4M_{\pi}^{2}<t<t_{\rm
max})$ $\displaystyle+\;\textrm{Im}\,F_{i}^{V}(t)[\textrm{high-
mass}]\;\Theta(t>t_{\rm max})$ $\displaystyle(i=1,2),$ (36)

where $\Theta$ is the step function (1 if the argument is true, otherwise 0).
The first term is the contribution of the two-pion cut that starts at
$4M_{\pi}^{2}$ and extends up to $t_{\rm max}\approx$ 1 GeV2 (see below); this
part is calculated theoretically. The second term represents the contribution
of high-mass states above $t_{\rm max}$ of unspecified hadronic composition;
this part is parametrized through effective poles.

Figure 2: Elastic unitarity relation for the two-pion part of the isovector
spectral functions $\textrm{Im}\,F_{i}^{V}(t)[\pi\pi]$ ($i=1,2$). The first
line shows the relation in the original form with the complex amplitudes, Eqs.
(37); the second line shows it in the explicitly real form obtained with the
$N/D$ method, Eq. (38).

The two-pion spectral functions are obtained from the elastic unitarity
relation in the two-pion channel (see Fig. 2). The relation is written in a
manifestly real form by applying the $N/D$ method Hohler and Pietarinen
(1975); Frazer and Fulco (1960a, b)

$\displaystyle\textrm{Im}\,F_{i}^{V}(t)[\pi\pi]$ $\displaystyle=\frac{k_{\rm
cm}^{3}}{\sqrt{t}}\;\Gamma_{i}(t)\;F_{\pi}^{*}(t)$ (37)
$\displaystyle=\frac{k_{\rm cm}^{3}}{\sqrt{t}}\;N_{i}(t)|F_{\pi}(t)|^{2},$
(38) $\displaystyle k_{\rm cm}$ $\displaystyle\equiv\sqrt{t/4-M_{\pi}^{2}},$
(39) $\displaystyle N_{i}(t)$
$\displaystyle\equiv\frac{\Gamma_{i}(t)}{F_{\pi}(t)}\hskip 20.00003pt(i=1,2).$
(40)

$k_{\rm cm}$ is the center-of-mass momentum of the $\pi\pi$ state in
$t$-channel, $\Gamma_{i}(t)(i=1,2)$ are the $\pi\pi\rightarrow N\bar{N}$
partial-wave amplitudes, and $F_{\pi}(t)$ is the pion timelike form factor.
The functions $\Gamma_{i}(t)$ and $F_{\pi}(t)$ in Eq. (37) are complex for
$t>4M_{\pi}^{2}$ because of $\pi\pi$ rescattering but have the same phase. The
ratios $N_{i}(t)(i=1,2)$ in Eq. (38) are real for $t>4M_{\pi}^{2}$ and possess
only left-handed singularities; they are free of $\pi\pi$ rescattering effects
and describe the coupling of the $\pi\pi$ system to the nucleon. These
function can be computed in $\chi$EFT with good convergence. The $\pi\pi$
rescattering effects in Eq. (38) are contained in the squared modulus of the
pion form factor $|F_{\pi}(t)|^{2}$. This function is measured in
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ exclusive annihilation experiments and
can be taken from a parametrization of the data Druzhinin  _et al._ (2011). In
this way Eq. (38) factorizes the rescattering effects in the $\pi\pi$ channel
and permits chiral EFT-based computation of the spectral functions. While Eq.
(37) and Eq. (38) are strictly valid only up to the 4-pion threshold
$t<16\,M_{\pi}^{2}$, the $e^{+}e^{-}$ exclusive annihilation data indicate
that 4-pion and other states are strongly suppressed up to $\sim$ 1 GeV2
Druzhinin  _et al._ (2011), so that the elastic unitarity relations can
practically be used up to $t_{\rm max}=1$ GeV2 Hohler and Pietarinen (1975);
Hohler  _et al._ (1976).

We have computed the functions $N_{i}(t)$ in relativistic $\chi$EFT with $N$
and $\Delta$ intermediate states at LO and NLO accuracy. Contributions at N2LO
accuracy have been estimated assuming that the loop corrections have the same
functional form as the tree-level result (partial N2LO, or pN2LO
approximation). The $\chi$EFT results for $N_{i}(t)$ can be obtained from
those of the functions $J^{1}_{\pm}(t)$, which appear in the $N/D$
representation of the unitarity relation for the $G_{E,M}(t)$ form factors and
were calculated in our earlier studies Alarcón and Weiss (2018a, b). The
explicit formulas are given in Appendix B.

The LO and NLO results for the functions $N_{i}(t)$ are given in terms of
known low-energy constants in the chiral Lagrangian. The pN2LO estimates
involve one unknown parameter, $\lambda_{i}$, describing the size of the loop
corrections relative to the tree-level result. The total results for the
function $N_{i}(t)$ are thus given by $(i=1,2)$

$\displaystyle
N_{i}(t)=N_{i}(t)[\textrm{LO}]+N_{i}(t)[\textrm{NLO}]+N_{i}(t)[\textrm{pN2LO}],$
(41) $\displaystyle
N_{i}(t)[\textrm{pN2LO}]\equiv\lambda_{i}\,N_{i}(t)[\textrm{N2LO-tree}].$ (42)

The explicit form of $N_{i}(t)[\textrm{N2LO-tree}]$ is given in Eq. (80). In
summary, at pN2LO accuracy the two-pion part of each of the spectral functions
in Eq. (36) involves one free parameter, $\lambda_{i}$ ($i=1,2$).

The high-mass part of the isovector spectral functions in Eq. (36) is
parametrized by effective poles. It is important to note that for our
calculation of peripheral densities we need only a summary description of the
high-mass strength of the spectral function, because of the strong numerical
suppression of large $t$ in the dispersion integrals Eqs. (18) and (19). (see
Sec. II.2). We construct an appropriate parametrization by making reasonable
assumptions about the positions of the poles, treating variations of the
position as part of the theoretical uncertainty, and and fixing the strength
of the poles through the dispersive sum rules Eq. (22) et seq. and Eq. (27) et
seq.

Several observations suggest that the main strength of the isovector spectral
functions beyond the two-pion region is located around $t\approx$ 2 GeV2, and
that higher values of $t$ are strongly suppressed. (a) The
$e^{+}e^{-}\rightarrow$ hadrons exclusive annihilation data show that the
cross section at $t>1$ GeV2 is dominated by the $4\pi$ channel and
concentrated around $t\approx 2$ GeV2 Druzhinin  _et al._ (2011). This
suggests similar behavior of the nucleon spectral functions, even if the
connection with the annihilation cross section is at the amplitude level and
cannot be made explicit. (b) Dispersive fits to the spacelike nucleon FFs with
flexible parametrizations of the high-mass states using multiple effective
poles find most of the strength in the region around $t\approx$ 2 GeV2
Belushkin  _et al._ (2007); Lin  _et al._ (2022). (c) The dual resonance model
describes the isovector spectral functions of the pion or nucleon FFs through
the exchange of vector resonances with masses $M^{2}_{n}=M_{\rho}^{2}(1+2n)$,
with $M_{0}^{2}\equiv M_{\rho}^{2}$. The first resonance after the $\rho$ has
mass $M_{1}^{2}=3\,M_{\rho}^{2}=$ 1.8 GeV2. If in the nucleon FFs the
resonance contributions decrease rapidly with $n$, the dominant contribution
beyond the $\rho$ should come from this resonance.

Based on these observations, we parametrize the high-mass part of the
isovector spectral function $\textrm{Im}\,F_{1}^{V}(t)$ as

$\displaystyle\textrm{Im}\,F_{1}^{V}(t)[\textrm{high-mass}]$
$\displaystyle=\pi a_{1}^{(V,0)}\delta(t-t_{1}^{(V,0)})$ $\displaystyle+\pi
a_{1}^{(V,1)}\delta^{\prime}(t-t_{1}^{(V,1)}).$ (43)

The first term is a delta function, the second term is the derivative of a
delta function. The pole masses have the nominal value

$\displaystyle
t_{1}^{(V,0)},t_{1}^{(V,1)}[\textrm{nominal}]\;=\;1.8\,\textrm{GeV}^{2}\;=\;3\,M_{\rho}^{2}.$
(44)

Their actual values are considered undetermined and will be allowed to vary
randomly in a plausible range; their distribution will be constrained by
further physical requirements, and the resulting variation in physical
quantities will be regarded as a theoretical uncertainty of the model (see
Sec. II.4). Note that the sum of the functions in Eq. (43) parametrizes both
the “strength” and the “shape” of the high-mass spectral function in region
around $t\approx 2$ GeV2 in an effective form.444In dispersive fits of the
nucleon isovector FFs, the high-mass states are traditionally parametrized as
a sum of simple delta functions at different positions. These parametrizations
give clusters of poles around $\sim 2$ GeV2 with varying signs and unnaturally
large coefficients $\gg 1$ Belushkin  _et al._ (2007), which can effectively
be combined to a sum of a single delta function and delta function
derivatives. In this sense our parametrization Eq. (43) is equivalent to the
traditional sum of simple delta functions. An advantage of our parametrization
is that all coefficients have natural size $\sim 1$, and that one can directly
implement the feature that all the “structures” arise from the same mass
region.

Combining the two-pion part given by Eqs. (38), (41), and (42), and the high-
mass part given by Eq. (43), the isovector spectral function
$\textrm{Im}\,F_{1}^{V}(t)$ contains three unknown parameters:

$\displaystyle\lambda_{1},\;a_{1}^{(V,0)},\;a_{1}^{(V,1)}.$ (45)

We fix these parameters through the sum rules for the isovector charge and
radius, see Eqs. (22) and (23), and the superconvergence relation for
$F_{1}^{V}$, see Eq. (27):

$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\frac{\textrm{Im}\,F_{1}^{V}(t)}{t}$ $\displaystyle=Q^{V},$
(46) $\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\frac{\textrm{Im}\,F_{1}^{V}(t)}{t^{2}}$
$\displaystyle=\frac{1}{6}\langle r^{2}\rangle_{1}^{V},$ (47)
$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\textrm{Im}\,F_{1}^{V}(t)$ $\displaystyle=0,$ (48)

where

$\displaystyle Q^{V}$
$\displaystyle\equiv\frac{1}{2}(Q^{p}-Q^{n})\;=\;\frac{1}{2},$ (49)
$\displaystyle\langle r^{2}\rangle_{1}^{V}$
$\displaystyle\equiv\frac{1}{2}\left(\langle r^{2}\rangle_{1}^{p}-\langle
r^{2}\rangle_{1}^{n}\right).$ (50)

The relations Eqs. (46)–(48) constrain weighted integrals of the total
isovector spectral function $\textrm{Im}\,F_{1}^{V}(t)$. The integrals extend
over the two-pion and the high-mass parts of the spectral functions (here
$n=0,1,2$),

$\displaystyle\int_{t_{\rm
thr}}^{\infty}dt\;t^{-n}\;\textrm{Im}\,F_{1}^{V}(t)$ $\displaystyle\equiv$
$\displaystyle\int_{t_{\rm thr}}^{t_{\rm
max}}dt\;t^{-n}\;\textrm{Im}\,F_{1}^{V}(t)[\pi\pi]$ $\displaystyle+$
$\displaystyle\int_{t_{\rm
max}}^{\infty}dt\;t^{-n}\;\textrm{Im}\,F_{1}^{V}(t)[\textrm{high-mass}].$ (51)

The integral over the two-pion part is a continuous integral and computed
numerically; the integral over the high-mass part is a sum of delta function
derivative integrals and computed exactly. Since the integrands depend
linearly on the parameters Eq. (45), one obtains a system of linear equations
for the parameters, which can easily be solved.

In the spectral function $\textrm{Im}\,F_{2}^{V}(t)$, we parametrize the high-
mass part as

$\displaystyle\textrm{Im}\,F_{2}^{V}(t)[\textrm{high-mass}]$
$\displaystyle=\pi a_{2}^{(V,0)}\delta(t-t_{2}^{(V,0)})$ $\displaystyle+\pi
a_{2}^{(V,1)}\delta^{\prime}(t-t_{2}^{(V,1)})$ $\displaystyle+\pi
a_{2}^{(V,2)}\delta^{\prime\prime}(t-t_{2}^{(V,2)}),$ (52)

which compared to Eq. (43) includes also a term with a second derivative of a
delta function. The three pole masses again have the nominal value

$\displaystyle
t_{2}^{(V,0)},t_{2}^{(V,1)},t_{2}^{(V,2)}[\textrm{nominal}]\;=\;1.8\,\textrm{GeV}^{2}\;=\;3\,M_{\rho}^{2},$
(53)

and will be allowed to vary randomly in an interval around this value (see
Sec. II.4). The combined spectral function now contains four unknown
parameters:

$\displaystyle\lambda_{2},\;a_{2}^{(V,0)},\;a_{2}^{(V,1)},\;a_{2}^{(V,2)}.$
(54)

They are fixed through the sum rules for the isovector magnetic moment and the
magnetic radius, see Eqs. (24) and (25), and by the two superconvergence
relations for $F_{2}$, see Eqs. (28) and (29):

$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\frac{\textrm{Im}\,F_{2}^{V}(t)}{t}$
$\displaystyle=\kappa^{V},$ (55) $\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\frac{\textrm{Im}\,F_{2}^{V}(t)}{t^{2}}$
$\displaystyle=\frac{1}{6}\langle r^{2}\rangle_{2}^{V},$ (56)
$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;\textrm{Im}\,F_{2}^{V}(t)$ $\displaystyle=0,$ (57)
$\displaystyle\frac{1}{\pi}\int_{t_{\rm
thr}}^{\infty}dt\;t\;\textrm{Im}\,F_{2}^{V}(t)$ $\displaystyle=0,$ (58)

where

$\displaystyle\kappa^{V}$
$\displaystyle\equiv\frac{1}{2}(\kappa^{p}-\kappa^{n}),$ (59)
$\displaystyle\langle r^{2}\rangle_{2}^{V}$
$\displaystyle\equiv\frac{1}{2}\left(\kappa^{p}\langle
r^{2}\rangle_{2}^{p}-\kappa^{n}\langle r^{2}\rangle_{2}^{n}\right).$ (60)

In our approach the unknown parameters in the spectral functions are fixed by
the dispersive sum rules and expressed in terms of the values and derivatives
of the FFs at $t=0$. Since values of the FFs (charges and magnetic moments)
are known, this leaves the derivatives (radii) as the effective parameters of
our model. With the spectral functions determined by the radii, our approach
can then predict the spacelike FFs and the densities in terms of the radii.
This particular “information flow” is made possible by the analytic properties
of the FFs, which relate integrals over the spectral functions to derivatives
of the FF at $t=0$.

Figure 3: Contributions of the two-pion and high-mass parts of the spectral
function $\textrm{Im}\,F_{1}^{V}(t)$, Eq. (36), to the dispersion integral of
the density $\rho_{1}^{V}(b)$ (nominal parameters). Upper panel: Absolute
contributions. Dashed line: $\rho_{1}^{V}(b)[\pi\pi]$. Dotted line:
$\rho_{1}^{V}(b)[\textrm{high-mass}]$. Solid line:
$\rho_{1}^{V}(b)[\textrm{total}]$. Lower panel: Relative contributions. Same
as in upper panel, but divided by total density
$\rho_{1}^{V}(b)[\textrm{total}]$.

When computing the densities in the dispersive representation, Eqs. (18) and
(19), the integrals receive contributions from the two-pion and the high-mass
parts of the spectral functions, Eq. (36). An important question is how the
relative contributions depend on the distance $b$, which represents an
external parameter in the integrals. Figure 3 shows the two-pion and high-mass
contributions to $\rho_{1}^{V}(b)$ obtained with our spectral functions (here,
for the the nominal parameter values). The top panel shows the absolute
contributions; the bottom panel shows the relative contributions, i.e., the
fractions of $\rho_{1}^{V}(b)$ due to the two-pion and high-mass parts. One
observes that in $\rho_{1}^{V}(b)$ the two-pion part accounts for $>80\%$ of
at $b>1$ fm, and $>60\%$ at $b>0.5$ fm. Similar relative contributions are
found in the dispersion integral for $\widetilde{\rho}_{2}^{V}(b)$ (not shown
in the figure). In $\widetilde{\rho}_{2}^{V}(b)$ the two-pion part accounts
for $>97\%$ of at $b>1$ fm, and $>50\%$ at $b>0.5$ fm. These findings are
central to our approach, as they quantify the dominance of the two-pion state
at large distances and justify the summary description of the high-mass states
for the purpose of computing the peripheral densities.

In the present study our focus is on the isovector channel, where the two-pion
state in the spectral functions generates the dominant contributions to the
peripheral nucleon densities. In order to compute the individual proton and
neutron densities in the dispersive representation, we need also the isoscalar
spectral function. A parametrization of the isoscalar spectral functions,
constructed along similar lines as for the isovector spectral function but
relying more on empirical information, is described in Appendix C.

###  II.4 Uncertainty estimates

Our dispersive approach allows us to estimate the uncertainties of the
spectral functions and the densities obtained from them. We consider two
sources of uncertainties:

(I) Uncertainties due to the parametrization of the high-mass part of the
spectral functions. The high-mass part of the isovector spectral function is
linked to the low-mass part through the dispersive sum rules, Eqs. (46)–(48)
and Eqs. (55)–(58). The parametrization of the high-mass part can therefore
influence the low-mass part of spectral functions and indirectly affect
observables sensitive to the low-mass part, such as the peripheral densities.
We estimate this uncertainty by varying the positions of the high-mass poles
in the isovector spectral functions, Eqs. (44) and Eqs. (53). As the plausible
range of variation we consider

$\displaystyle\left.\begin{array}[]{r}t_{1}^{(V,0)},t_{1}^{(V,1)}\\\\[4.30554pt]
t_{2}^{(V,0)},t_{2}^{(V,1)},t_{2}^{(V,2)}\end{array}\right\\}$
$\displaystyle=(1.2-2.4)\;\textrm{GeV}^{2}\,=\,(2-4)\,M_{\rho}^{2}.$ (63)

This range allows for variations of the pole masses with a maximum/minimum
ratio of 2, which is a very significant change. Eq. (63) covers the entire
region of the secondary peak of the $e^{+}e^{-}$ annihilation cross section
above the $\rho$ resonance Druzhinin  _et al._ (2011). In the context of the
dual resonance model, Eq. (63) corresponds to varying the pole position from
the $n=1$ resonance at $3M_{\rho}^{2}$ to values that are half way between
this one and the $n=0$ or 2 resonances. Note that we let the mass parameters
in the delta functions vary independently of each other over the given range,
so that the parametrization represents a wide range of “shapes” of the
spectral function.

We further constrain the set of mass parameters by requiring that the
variation of the spacelike form factor generated by the spectral function be
within a certain range around the nominal value. This is essentially a
stability condition, which eliminates extreme values of the mass parameters
that would lead to large excursions of the spacelike form factor and can be
ruled out on physical grounds. We implement this by requiring that (here
$i=1,2$)

$\displaystyle\frac{F_{i}^{V}(t_{\rm ref})[\textrm{varying pole
masses}]}{F_{i}^{V}(t_{\rm ref})[\textrm{nominal}]}\;-\;1\;<\;\epsilon,$ (64)

where $t_{\rm ref}<0$ is a spacelike $t$ value. In the following applications
we choose $t_{\rm ref}=-1$ GeV2 and $\epsilon=0.1$ for both $F_{1}^{V}$ and
$F_{2}^{V}$; the choice is justified in the following; other choices are
possible. The parameter variation Eq. (63), supplemented by the stability
condition Eq. (64), generates a functional variation in the high-mass part of
the spectral function which we regard as the theoretical uncertainty of our
model. Note that the stability condition Eq. (64) restricts the variation of
the theoretical FF prediction relative to the nominal value of the model, not
relative to an experimental value; no fitting to the FF data is performed
here. The parameters $t_{\rm ref}$ and $\epsilon$ are chosen such that the
resulting theoretical model uncertainty of the FFs is reasonable and covers
the experimental data. In this way the experimental FF data are used only in
estimating the theoretical uncertainty of the model, not in determining the
nominal model prediction.

Figure 4: Illustration of the procedure for estimating the uncertainty
resulting from the high-mass part of the spectral function
$\textrm{Im}\,F_{2}^{V}(t)$. A random ensemble of parameters for the high-mass
pole positions Eq. (63) is generated. With these parameters, the spectral
function and the spacelike FF are computed. The parameter ensemble is then
restricted to values for which the spacelike FF satisfies the stability
condition Eq. (64) (see upper panel, accept/reject). The restricted parameter
ensemble is then used to evaluate the uncertainty of the spectral function
(middle panel) and the density (lower panel). In all panels, the functions
with accepted parameters by Eq. (64) are shown as solid black lines, the ones
with rejected parameters as dot-dashed red lines. The functions with the
nominal parameters are shown by the dashed white line.

To map out the theoretical uncertainty in practice, we generate a random
ensemble of mass parameters in the range of Eq. (63) and retain those for
which the spacelike FFs satisfy the condition Eq. (64). We then use this
restricted ensemble to generate uncertainty bands in the spectral functions
and transverse densities (and possibly other quantities derived from the
spectral functions). Figure 4 illustrates the procedure in the case of
$F_{2}^{V}$. One observes that the procedure generates natural uncertainty
bands, which are approximately symmetric around the nominal value. The
resulting uncertainty will be quoted as “high-mass uncertainty” in the results
below.

We emphasize that the procedure respects analyticity and the dispersive sum
rules. Each instance in the ensemble corresponds to a form factor with correct
analyticity in $t$, and a density with correct asymptotic behavior at large
$b$. Each instance represents a spectral function that satisfies the sum rules
Eqs. (46)–(48) and Eqs. (55)–(58) and produces form factors and densities with
the correct normalization. The only differences between the instances are in
the form of the high-mass spectral function, and in the distribution of
strength between the low-mass and high-mass regions.

(II) Uncertainties due to the nucleon radii. The nucleon radii determine the
spectral function parameters through the dispersive sum rules Eqs. (46)–(48)
and Eqs. (55)–(58). We can estimate the resulting uncertainty by varying the
value of the radii. The empirical values of the radii and their uncertainties
are summarized in Appendix A. For our uncertainty estimate, we vary the the
radii in a range corresponding to their empirical uncertainty

$\displaystyle\langle r^{2}\rangle_{1}^{V}$ $\displaystyle\rightarrow\langle
r^{2}\rangle_{1}^{V}[\textrm{nominal}]\times(1\pm 0.03),$ (65)
$\displaystyle\langle r^{2}\rangle_{2}^{V}$ $\displaystyle\rightarrow\langle
r^{2}\rangle_{2}^{V}[\textrm{nominal}]\times(1\pm 0.02).$ (66)

We then follow the effect of this variation from the spectral function to the
form factors and densities calculated as dispersive integrals. The resulting
uncertainty will be quoted as “radius uncertainty” in the results below.

##  III Results

###  III.1 Spectral functions

|
---|---
Figure 5: Two-pion part of isovector spectral functions $\textrm{Im}\,F_{1}^{V}(t)$ (left column) and $\textrm{Im}\,F_{2}^{V}(t)$ (right column) obtained in DI$\chi$EFT. In each column: Upper panel: DI$\chi$EFT result with nominal parameters. Middle panel: Uncertainties resulting from the parametrization of high-mass states. Lower panel: Uncertainties resulting from the nucleon isovector radii. |
---|---
Figure 6: Isovector nucleon form factors $F_{1}^{V}(t)$ (left column) and
$F_{2}^{V}(t)$ (right column) at spacelike $t<0$ obtained from the DI$\chi$EFT
spectral functions. In each column: Upper panel, solid line: DI$\chi$EFT
result with nominal parameters. Middle panel: Uncertainty resulting from the
parametrization of high-mass states. Lower panel: Uncertainty resulting from
the nucleon isovector radii. Upper panel, dotted line: Empirical FFs Ye  _et
al._ (2018). All FFs are shown divided by the standard dipole FF.

Figure 5 shows our results for the two-pion part of the isovector spectral
functions $\textrm{Im}\,F_{1}^{V}(t)$ and $\textrm{Im}\,F_{2}^{V}(t)$ and
their uncertainties obtained with the methods of Sec. II.3 and II.4. The upper
panels show the spectral functions obtained with the nominal parameters for
the high-mass poles and the radii. One observes: (a) The spectral functions
show the characteristic peak from the $\rho$ resonance in the $\pi\pi$
channel. This essential feature arises through the pion timelike form factor
in the elastic unitarity relation Eq. (38). (b) Both spectral functions (with
the nominal parameters) change sign and become negative above the $\rho$
region.

The middle and lower panels of Fig. 5 show the uncertainties in the two-pion
part of the spectral functions resulting from the parametrization of the high-
mass part and from the nucleon radii, estimated with the procedure of Sec.
II.4. (Note that the figure shows only the variation of the two-pion part of
the spectral function; the high-mass part undergoes a corresponding variation
with the parameters, so that the sum rule are satisfied; this part is not
shown in the figure.) One observes: (a) The uncertainties of the spectral
functions are negligible in the region of $t$ from the threshold at
$4M_{\pi}^{2}$ to $\sim 0.4$ GeV2. In this region the chiral expansion of the
functions $N_{i}(t)$ is well convergent, and the spectral functions represent
genuine predictions of the theory. Notice that the enhancement of the spectral
functions through $\pi\pi$ rescattering is already very significant in this
region Alarcón  _et al._ (2017). (b) In the region of the $\rho$ resonance,
the spectral functions show significant uncertainties from the high-mass
states and from the radii. The behavior is this region is mainly constrained
by the sum rules, so that the spectral functions become sensitive to the
parameters, as expected. The relative uncertainties are of order unity and
approximately the same in $\textrm{Im}\,F_{1}^{V}(t)$ and
$\textrm{Im}\,F_{2}^{V}(t)$.

In the present study we use the spectral functions to compute the peripheral
densities. The dispersive integrals for the peripheral densities converge
rapidly and sample mostly the two-pion part of the spectral functions; the
contribution of high-mass states is strongly suppressed (see Fig. 3). Our
DI$\chi$EFT method and uncertainty estimates aim to provide a realistic
description of the two-pion part, while parametrizing the high-mass part in
summary form. The dispersive integrals for the spacelike FFs converge more
slowly and are more sensitive to the high-mass states. The computation of
spacelike FFs therefore generally places stronger demands on the description
of the high-mass states than are needed in the present study. Still, it is
instructive to see how our simple spectral functions perform in the
computation of the spacelike form factors, for which experimental data are
available.

Figure 6 shows the spacelike FFs $F_{1}(t)$ and $F_{2}(t)$ obtained with our
spectral functions. [The plots show the FFs divided by the standard dipole FF
$\textrm{SD}(t)\equiv(1-t/0.71\,\textrm{GeV}^{2})^{-2}$.] The top panels show
the predictions with the nominal values of the high-mass pole position and the
nucleon radii. One observes that the nominal predictions agree very well with
the empirical FFs extracted from experimental data Ye  _et al._ (2018). This
indicates that our assumptions made in parametrizing the high-mass part of the
spectral function (in particular, the rapid saturation at low masses $t\approx
3M_{\rho}^{2}$) are realistic at the quantitative level. The middle and lower
panels show the uncertainties of the predictions due to the position of the
high-mass poles and the values of the nucleon radii, estimated with the
procedure of Sec. II.4. One observes that the procedure gives a reasonable
uncertainty estimate of the spacelike FFs that is approximately symmetric
around the nominal value (by design of the procedure) and covers the
experimental values. We emphasize that our goal here is not to predict or
analyze the spacelike FFs, but just to validate that our spectral function
results are compatible with the spacelike FF data. It is clear that a much
more accurate description of the spacelike FFs could be achieved within our
framework if the value of the high-mass poles were used as fit parameters, as
is commonly done in dispersive fits Belushkin  _et al._ (2007); Lin  _et al._
(2022).

###  III.2 Isovector densities

|
---|---
Figure 7: Isovector densities $\rho_{1}^{V}(b)$ (left column) and $-\widetilde{\rho}_{2}^{V}(b)$ (right column) and their relative uncertainties obtained with the DI$\chi$EFT spectral functions. In each column: Upper panel, solid line: DI$\chi$EFT prediction with nominal parameters. Middle panel: Relative uncertainty from high-mass states. Lower panel: Relative uncertainty from nucleon isovector radii. Upper panel, dotted line: Density from empirical FFs Ye  _et al._ (2018). |
---|---
Figure 8: Same as Fig. 7, but showing the radial densities $2\pi
b\rho_{1}^{V}(b)$ (left column) and $-2\pi b\widetilde{\rho}_{2}^{V}(b)$
(right column) and their absolute uncertainties.

Figures 7 and 8 show our results for the isovector transverse densities
$\rho_{1}^{V}(b)$ and $\widetilde{\rho}_{2}^{V}(b)$, obtained by evaluating
the dispersive integrals Eqs. (18)–(19) with the spectral functions of Sec.
II.3, and quantifying the uncertainties with the procedure of Sec. II.4. These
densities are the principal objective of the present study. Figure 7 shows the
densities $\rho_{1}^{V}(b)$ and $-\widetilde{\rho}_{2}^{V}(b)$ on a
logarithmic scale and their relative uncertainties. Figure 8 shows the radial
densities $2\pi b\rho_{1}^{V}(b)$ and $-2\pi b\widetilde{\rho}_{2}^{V}(b)$ on
a linear scale and their absolute uncertainties.

One observes: (a) The densities exhibit an exponential decrease at $b\gtrsim
0.5$ fm, as dictated by the analytic properties of the form factor (see Sec.
II.2). This behavior is naturally obtained from the dispersive representation
and is the principal reason for the use of this method for the computation of
peripheral densities. (b) The uncertainties of the densities resulting from
the high-mass part of the spectral function and from the nucleon radii show a
characteristic dependence on $b$ (nodes, maxima). This dependence is explained
by the way in which these parameters influence the low-mass and high-mass
parts of the spectral functions through the dispersive sum rules (see Sec.
II.3). (c) The uncertainty bands are bounded at large distances and permit
stable estimates of the uncertainties of the peripheral densities. In both
$\rho_{1}^{V}(b)$ and $\widetilde{\rho}_{2}^{V}(b)$, the estimated relative
uncertainties from high-mass states and radii are $\lesssim 10\%$ at distances
$b>0.5$ fm. (d) At $b>0.5$ fm the relative uncertainties of $\rho_{1}^{V}$ and
$\widetilde{\rho}_{2}^{V}$ are comparable. At $b<0.5$ fm, the relative
uncertainty of $\widetilde{\rho}_{2}^{V}$ is larger than that of
$\rho_{1}^{V}$. This happens because at small $b$ the dispersion integral
samples the high-mass part of the spectral function, and our parametrization
of $\textrm{Im}\,F_{2}^{V}$ allows for more variation than that of
$\textrm{Im}\,F_{1}^{V}$.

We emphasize that our theoretical calculations are aimed only at the
“peripheral” densities, with the boundary in $b$ determined by the uncertainty
estimates. Our results for both the densities and uncertainty estimates are to
be understood in this sense. At distances $b\lesssim 0.3$ fm the densities
become sensitive to the details of the high-mass states in spectral function.
Our effective description is not expected to be accurate in this region but
still allows us to estimate the uncertainty and demonstrate its increase.

Figures 7 and 8 also show the empirical densities, computed as Fourier
transforms of the FF fit of Ref. Ye  _et al._ (2018), see Eqs. (4) and (8).
Because it is difficult to quantify the uncertainties of the Fourier transform
densities at large $b$ (see discussion in Sec. III.1), we show only their
central value. One observes that our theoretical results show excellent
agreement with the empirical densities at distances $b\gtrsim 0.5$ fm, where
our calculations are accurate according to our intrinsic uncertainty
estimates. Our result obtained with the nominal parameters follow the
empirical densities even down to $b\rightarrow 0$. This strongly indicates
that our assumptions made in parametrizing the high-mass part of the spectral
function are realistic regarding the nominal values, and that the uncertainty
of our predictions at small $b$ could be reduced by constraining the variation
of the high-mass part through further theoretical arguments or empirical
information.

###  III.3 Proton and neutron densities

|
---|---
Figure 9: Transverse charge density $\rho_{1}(b)$ in the proton (left column) and neutron (right column) obtained from the DI$\chi$EFT results (see text). In each column, the panels show the nominal DI$\chi$EFT results and their absolute uncertainties from high-mass states and from the nucleon radii. The empirical densities are obtained from the FF parametrizations of Ref. Ye  _et al._ (2018). |
---|---
Figure 10: Transverse magnetization density $\widetilde{\rho}_{2}(b)$ in the
proton (left column) and neutron (right column) obtained from the DI$\chi$EFT
results (see text). In each column, the panels show the nominal DI$\chi$EFT
results and their absolute uncertainties from high-mass states and from the
nucleon radii. The empirical densities are obtained from the FF
parametrizations of Ref. Ye  _et al._ (2018).

Figure 11: The $J^{+}$ current density on the $x$-axis, Eq. (67), in the
transversely polarized proton (left column) and neutron (right column) with
$S^{y}=+1/2$. In each column: Upper panel: Nominal DI$\chi$EFT result. Middle
panel: Absolute uncertainty from high-mass states. Lower panel: Absolute
uncertainty from nucleon radii.

The DI$\chi$EFT method allows us to predict the isovector densities, which
contain the effect of the two-pion states and dominate peripheral nucleon
structure. In order to compute the individual proton and neutron densities in
the dispersive representation we need also the isoscalar spectral functions.
For the purposes of this study we use the parametrization of Appendix C, which
describes the isoscalar spectral function through effective poles and
implements the dispersive sum rules in a similar manner as in the isovector
sector. The parameters are fixed in terms of the isoscalar nucleon radii and
their uncertainties. The proton and neutron densities and their uncertainties
are obtained by combining the isovector and isoscalar densities. As the high-
mass uncertainty of the proton and neutron densities we take only the one
resulting from the isovector densities estimated in Sec. III.2, which is
expected to be dominant. As the radius uncertainty of the proton and neutron
densities we quote the change of the density under variation of the nucleon’s
“own” radius, i.e., $\langle r^{2}\rangle_{1,2}^{p}$ for the proton and
$\langle r^{2}\rangle_{1,2}^{n}$ for the neutron, corresponding to a
simultaneous change of the isovector and isoscalar radii. This is the dominant
radius uncertainty in most of the densities. The only exception is the neutron
charge density, where the uncertainty resulting from the change of the neutron
radius is smaller than that resulting from the proton radius, through its
combined effect on the isovector and isoscalar densities.

Figures 9 and 10 show the densities $\rho_{1}^{p,n}(b)$ and
$\widetilde{\rho}_{2}^{p,n}(b)$ densities and their uncertainties obtained in
this way. One observes: (a) The calculation predicts the peripheral nucleon
densities with good accuracy. In the proton charge density $\rho_{1}^{p}$, and
the proton and neutron magnetization densities $\widetilde{\rho}_{2}^{p,n}$,
the relative uncertainties are estimated at $\lesssim 10\%$ at $b>0.5$ fm
(these densities are uniformly positive or negative, so that one can sensibly
quote the relative uncertainty). (b) In the neutron charge density
$\rho_{1}^{n}$, the absolute uncertainty is estimated at $>50\%$ near the
positive maximum at $b=0.65$ fm, and rapidly decreasing at larger $b$ (this
density has different sign in different regions). (c) The nominal DI$\chi$EFT
predictions agree well with the empirical densities at $b\gtrsim 0.5$ fm. In
particular, the theoretical calculation reproduces the behavior of the neutron
density, which changes from negative values at $b\gtrsim 1.5$ fm to positive
values at $1.5\gtrsim b\gtrsim 0.35$ fm to negative values at $b\lesssim 0.35$
fm Miller (2007, 2010). This behavior arises as the result of a delicate
cancellation of isovector and isoscalar densities in the different regions of
$b$.

###  III.4 Current densities in polarized nucleon

Combining our results for the densities $\rho_{1}(b)$ and
$\widetilde{\rho}_{2}(b)$, we can compute the $J^{+}$ current density in the
transversely polarized nucleon, Eq. (7). In order to display the theoretical
uncertainties it is useful to show a one-dimensional projection of the two-
dimensional current density. We consider the current density Eq. (7) in the
nucleon with $S^{y}=+1/2$ on the transverse $x$ axis, where $\bm{b}=(b^{x},0)$
with $b^{x}<0$ or $>0$, which is given by

$\displaystyle\langle J^{+}(b^{x})\rangle_{\textrm{\scriptsize localized}}\;$
$\displaystyle=\;(...)\;[\rho_{1}(|b^{x}|)\,+\,\textrm{sign}(b^{x})\,\widetilde{\rho}_{2}(|b^{x}|)].$
(67)

This function describes the current density to the “left” and “right” when
looking at the nucleon from $z=+\infty$ (see Fig. 1). Notwithstanding its
piecewise definition in Eq. (67), it is a smooth function of $b^{x}$ because
$\widetilde{\rho}_{2}(|b^{x}|=0)=0$.

Figure 11 shows the $J^{+}$ current density Eq. (67) in the proton and neutron
obtained from our DI$\chi$EFT results, including its theoretical uncertainty.
[The plot shows the expression in the square bracket in Eq. (67) without the
normalization factor denoted by (…).] In the high-mass uncertainties we have
added the uncertainty bands in $\rho_{1}(b)$ and $\widetilde{\rho}_{2}(b)$
assuming no correlation between the two (the positions of the effective high-
mass poles in $\textrm{Im}\,F_{1}^{V}$ and $\textrm{Im}\,F_{2}^{V}$ are not
related, and their variation in Sec. II.4 is performed independently). In the
radius uncertainty we show separately the variations of the density under the
changes of the nucleon’s radii $\langle r^{2}\rangle_{1}$ and $\langle
r^{2}\rangle_{2}$. One observes: (a) The numerical densities behave smoothly
at $b^{x}=0$, as they should. (b) The $J^{+}$ current densities in the proton
and the neutron exhibit a strong left-right asymmetry. In the context of the
parton picture, this shows that the internal motion due to the nucleon spin
causes a significant distortion of the plus momentum distribution and attests
to the essentially relativistic character of the system. A “mechanical”
interpretation of the left-right asymmetry of the peripheral densities in
traditional chiral EFT, as arising from the motion of a soft pion in the
nucleon’s periphery, has been developed in Refs. Granados and Weiss (2015b, a,
2016, 2019).

##  IV Discussion

In the present work we have computed the peripheral transverse charge and
magnetization densities in the nucleon using the DI$\chi$EFT method and
quantified their uncertainties. The main findings are: (a) The dispersive
representation permits stable calculation of the peripheral densities. The
densities exhibit the exponential decrease implied by analyticity of the form
factor and depend smoothly on the parameters of the spectral function. (b)
Uncertainties can be estimated by allowing for variation of the spectral
function (functional form, parameters) and following its effect on the
densities. The procedure makes use of the particular “information flow”
implied by analyticity and relates the peripheral densities to the spacelike
form factor in a controlled manner. (c) Using a minimal parametrization of the
high-mass part of the isovector spectral function, the isovector densities are
computed with an estimated accuracy of $\sim\pm 10\%$ at $b\gtrsim 0.3$ fm.

In the present calculation we have not used any spacelike form factor data
beyond the nucleon radii (FF derivatives at $t=0$) to constrain the isovector
spectral functions. In particular, we do not fit the high-mass part of the
spectral function to the spacelike form factor data, as is done is dispersive
fits. [The stability condition Eq. (64), controlling the variation of the
high-mass spectral function, applies to the variation relative to the nominal
theoretical prediction, not relative to the FF data.] Our results represent
theoretical predictions based on a minimal parametrization of the high-mass
spectral function, and our uncertainty estimates should be understood in this
sense. It is clear that a much more accurate description could be achieved if
spacelike FF data were used to constrain the high-mass part of the isovector
spectral functions. Our estimates of the high-mass uncertainty therefore
should not be regarded as “final,” but rather as showing how far one can go
without fitting spacelike FF data.

The methods developed here enable an EFT-based computation of the transverse
densities down to distances $b\gtrsim 0.5$ fm. At such distances the
transverse densities can be described in approaches using other effective
degrees of freedom, e.g. quark models. This makes it possible to match the
EFT-based description with quark model predictions of the transverse densities
at “intermediate” distances and explore quark-hadron duality in new ways.
While quark models may not be able to accurately reproduce the absolute
densities, they can predict qualitative features such as the spin/isospin
dependence and flavor decomposition of the densities, which can lead to
interesting conclusions when matched with the EFT description. We plan to
explore the use of transverse densities for quark-hadron duality studies in a
separate work.

Some comments are in order regarding the interpretation of our results in
terms of a “pion cloud” of the nucleon. It is true that the isovector
densities at distances $\gtrsim$ 0.5 fm are generated mostly by the two-pion
states in the dispersive representation (see Fig. 3). One might be tempted to
explain this in a picture where a bare nucleon fluctuates into a pion-nucleon
state, and the peripheral structure arises from the propagation of that pion.
Such a picture is indeed obtained in traditional chiral EFT, where the pion
and nucleon are pointlike, and the peripheral densities emerge from the
propagation of the pointlike pions. However, in our unitarity-based approach
the pion is not pointlike and has an extended structure of the same range as
the nucleon, as required by unitarity. The results of our approach should
therefore not be interpreted in terms of the traditional pion could picture.
The space-time interpretation of the densities in the unitarity-based approach
is an interesting question which we plan to investigate in a future study.

In the present study we have applied our unitarity-based approach to the
peripheral densities associated with the nucleon matrix element of the
electromagnetic current operator. The approach could be extended to compute
the peripheral densities of other operators whose form factors possess a two-
pion cut, such as the QCD energy momentum tensor (spin-2 operator) or the
leading-twist QCD operators whose matrix elements determine the moments of the
GPDs (twist-2, spin-$n$ operators, $n\geq$ 1). This would allow one to
“deconstruct” not only the nucleon’s electromagnetic current but also its
peripheral partonic structure in terms of EFT degrees of freedom. One
difference between the electromagnetic and the generalized form factors is
that for the latter the “radii” (derivatives at $t=0$) are generally not known
from independent measurements, so that one has to adjust the procedure of
fixing the parameters of the spectral functions and recruit new sources of
information.

##  Appendix A Nucleon radii

In this appendix we list the values of the nucleon radii used as parameters in
the DI$\chi$EFT calculation of the spectral functions. The Dirac and Pauli
radii of the proton and neutron are defined in terms of the FF derivatives

$\displaystyle\frac{dF_{1}^{p}}{dt}(0)=\frac{1}{6}\,\langle
r^{2}\rangle_{1}^{p},\hskip
20.00003pt\frac{dF_{2}^{p}}{dt}(0)=\frac{1}{6}\,\kappa^{p}\,\langle
r^{2}\rangle_{2}^{p}$ (68)

(same for $p\rightarrow n$). They are related to the conventional electric and
magnetic radii by

$\displaystyle\langle r^{2}\rangle_{1}^{p}\;=$ $\displaystyle\;\langle
r^{2}\rangle_{E}^{p}-\frac{3\kappa^{p}}{2m_{N}^{2}},$ (69)
$\displaystyle\kappa^{p}\langle r^{2}\rangle_{2}^{p}\;=$
$\displaystyle\;-\langle r^{2}\rangle_{E}^{p}+\mu^{p}\,\langle
r^{2}\rangle_{M}^{p}+\frac{3\kappa^{p}}{2m_{N}^{2}}$ (70)

(same for $p\rightarrow n$). Here $\kappa^{p,n}$ are the anomalous magnetic
moments, and $\mu^{p,n}=Q^{p,n}+\kappa^{p,n}=(2.793,-1.913)$ are the ordinary
magnetic moments of the nucleons.

We estimate the values of $\langle r^{2}\rangle_{1,2}$ and their uncertainties
from the empirical values of $\langle r^{2}\rangle_{E,M}$ and their
uncertainties, neglecting correlations between the uncertainties of $\langle
r^{2}\rangle_{E}$ and $\langle r^{2}\rangle_{M}$. We use the following numbers
and sources: $\langle r^{2}\rangle_{E}^{p}=(0.7090\pm 0.0168)$ fm2 Alarcón
_et al._ (2020), $\langle r^{2}\rangle_{E}^{n}=(-0.1161\pm 0.0022)$ fm2
Tanabashi  _et al._ (2018), $\langle r^{2}\rangle_{M}^{p}=(0.7225\pm 0.0170)$
fm2 Alarcón  _et al._ (2020), $\langle r^{2}\rangle_{M}^{n}=(0.7465\pm
0.0156)$ fm2 Tanabashi  _et al._ (2018). The proton and neutron radii $\langle
r^{2}\rangle_{1,2}$ thus obtained are summarized in Table 1.

We also list in Table 1 the isovector and isoscalar combinations of the radii,
defined by Eqs. (49)–(50) and (59)–(60), and Eqs. (86)–(87) and (88)–(89),
which enter in the sum rules for the isovector and isoscalar spectral
functions. In calculating the uncertainties we neglect correlations between
the uncertainties of the proton and neutron radii. Note that the
isovector/isoscalar combinations of $\langle r^{2}\rangle_{2}$ defined by Eqs.
(59)–(60) and (88)–(89) involve the nucleon anomalous magnetic moments and
cannot directly be interpreted as nucleon radii.

Type |  $\langle r^{2}\rangle$ [fm2] |  $\delta\langle r^{2}\rangle$ [fm2] | $\delta\langle r^{2}\rangle/\langle r^{2}\rangle$
---|---|---|---
$\langle r^{2}\rangle_{1}^{p}$ | 0.5906 | 0.0168 | 0.0285
$\langle r^{2}\rangle_{1}^{n}$ | 0.0102 | 0.0022 | 0.2157
$\langle r^{2}\rangle_{2}^{p}$ | 0.7961 | 0.0281 | 0.0353
$\langle r^{2}\rangle_{2}^{n}$ | 0.7518 | 0.0156 | 0.0207
$\langle r^{2}\rangle_{1}^{V}$ | 0.2902 | 0.0085 | 0.0293
$\langle r^{2}\rangle_{1}^{S}$ | 0.3004 | 0.0085 | 0.0283
$\langle r^{2}\rangle_{2}^{V}$ | 1.4328 | 0.0293 | 0.0204
$\langle r^{2}\rangle_{2}^{S}$ |  $-$0.0054 | 0.0293 |  $-$5.3759
Table 1: Nucleon Dirac and Pauli radii and their uncertainties used as input
in DI$\chi$EFT calculation.

##  Appendix B $N$ functions

In this appendix we give the expressions for the $N_{i}(t)$ functions
appearing in the unitarity relations for the isovector spectral functions,
$\textrm{Im}\,F_{1,2}^{V}(t)$ Eqs. (38) and (40). We do not compute these
functions explicitly but express them in terms of our earlier results for the
$J^{1}_{\pm}(t)$ functions appearing in the unitarity relations for the
$\textrm{Im}\,G_{E,M}^{V}(t)$ spectral functions Alarcón and Weiss (2018a).

The Dirac/Pauli FFs $F_{1,2}(t)$ and the electric/magnetic FFs $G_{E,M}(t)$
are related by

$\displaystyle G_{E}(t)$
$\displaystyle=F_{1}(t)+\frac{t}{4m_{N}^{2}}F_{2}(t),$ (71) $\displaystyle
G_{M}(t)$ $\displaystyle=F_{1}(t)+F_{2}(t),$ (72)

or, inversely,

$\displaystyle F_{1}(t)$
$\displaystyle=\left[G_{E}(t)-\frac{t}{4m_{N}^{2}}G_{M}(t)\right]\left/\left(1-\frac{t}{4m_{N}^{2}}\right),\right.$
(73) $\displaystyle F_{2}(t)$
$\displaystyle=\left[-G_{E}(t)+G_{M}(t)\phantom{\frac{0}{0}}\right]\left/\left(1-\frac{t}{4m_{N}^{2}}\right),\right.$
(74)

which hold for any complex $t$. The elastic unitarity relation for
$\textrm{Im}\,G_{E,M}(t)$, in its manifestly real form analogous to Eq. (38),
is written as (here $t>4M_{\pi}^{2}$)

$\displaystyle\textrm{Im}\,G_{E}^{V}(t)[\pi\pi]$ $\displaystyle=\frac{k_{\rm
cm}^{3}}{m_{N}\sqrt{t}}\,J_{+}^{1}(t)\,|F_{\pi}(t)|^{2},$ (75)
$\displaystyle\textrm{Im}\,G_{M}^{V}(t)[\pi\pi]$ $\displaystyle=\frac{k_{\rm
cm}^{3}}{\sqrt{2t}}\,J_{-}^{1}(t)\,|F_{\pi}(t)|^{2},$ (76)

where $k_{\rm cm}$ is given in Eq. (39). The relation between the functions
$N_{1,2}(t)$ and $J_{\pm}^{1}(t)$ is

$\displaystyle N_{1}(t)$
$\displaystyle=\left[\frac{J_{+}^{1}(t)}{m_{N}}-\frac{t}{4m_{N}^{2}}\,J_{-}^{1}(t)\right]\left/\left(1-\frac{t}{4m_{N}^{2}}\right),\right.$
(77) $\displaystyle N_{2}(t)$
$\displaystyle=\left[-\frac{J_{+}^{1}(t)}{m_{N}}+\frac{J_{-}^{1}(t)}{\sqrt{2}}\right]\left/\left(1-\frac{t}{4m_{N}^{2}}\right).\right.$
(78)

Note that

$\displaystyle 1-\frac{t}{4m_{N}^{2}}\;$
$\displaystyle=\;\frac{\widetilde{p}_{\rm cm}^{2}}{m_{N}^{2}}$ (79)

where $\widetilde{p}_{\rm cm}={\textstyle\sqrt{m_{N}^{2}-t/4}}$ is the
unphysical nucleon center-of-mass momentum in the $\pi\pi\rightarrow N\bar{N}$
process in the $t$-channel.

The explicit expressions for the $J_{\pm}^{1}(t)$ functions in chiral EFT at
LO and NLO accuracy are given in Ref. Alarcón and Weiss (2018a). The
expressions for the functions $N_{1,2}(t)$ at this order can be obtained from
those results using Eqs. (77) and (78). Note that in this procedure Eqs. (77)
and (78) are evaluated only in the region of elastic unitarity in the two-pion
channel, $t<t_{\rm max}\approx$ 1 GeV2, far away from the singularity at
$t=4m_{N}^{2}$.

In our calculation at pN2LO accuracy we approximate the full N2LO corrections
to the $N_{1,2}(t)$ functions by rescaling the tree-level N2LO result, see Eq.
(42). For the N2LO tree-level result we use the simplified form

$\displaystyle N_{i}(t)[\textrm{N2LO-tree}]\;\propto\;t,$ (80)

the overall coefficients are irrelevant since they are absorbed by the
parameters $\lambda_{i}$ in Eq. (42). Equation (80) differs from the exact
N2LO tree-level result [as obtained from Eqs. (77) and (78) and the N2LO tree
level result for the $J_{\pm}^{1}(t)$] only by additive terms $\propto
M_{\pi}^{2}$ with coefficient of order unity, which are numerically small at
$t=$ 20–50 $M_{\pi}^{2}$ where the pN2LO term Eq. (42) is significant. The
simplified form Eq. (80) ensures that each of the functions $N_{1,2}(t)$ gets
an pN2LO term with its own parameter $\lambda_{1,2}$, which can then be fixed
through the dispersive sum rules.

##  Appendix C Isoscalar parametrization

In this appendix we present a simple parametrization of the isoscalar spectral
functions, which is used in the dispersive calculations of the individual
proton and neutron densities. The isoscalar spectral functions are constructed
along similar lines as the isovector ones in Sec. II.3, but relying more on
empirical information.

The isoscalar spectral function starts with the $3\pi$ channel and can be
organized in a similar way as Eq. (36) (here $i=1,2$),

$\displaystyle\textrm{Im}\,F_{i}^{S}(t)=\textrm{Im}\,F_{i}^{S}(t)[\pi\pi\pi]+\textrm{Im}\,F_{i}^{S}(t)[\textrm{high-
mass}].$ (81)

The strength in the $3\pi$ channel is overwhelmingly concentrated in the
$\omega$ resonance at $t=M_{\omega}^{2}=0.61$ GeV2; for an estimate of non-
resonant $3\pi$ contributions in chiral EFT, see Ref. Kaiser and Passemar
(2019). We parametrize the $3\pi$ part of the spectral function as ($i=1,2$)

$\displaystyle\textrm{Im}\,F_{i}^{S}(t)[\pi\pi\pi]$ $\displaystyle=\pi
a_{i}^{\omega}\delta(t-M_{\omega}^{2}).$ (82)

High-mass strength appears at $t\gtrsim 1$ GeV2 through the $K\bar{K}$ channel
and the $\phi$ resonance, as well as through other hadronic states such as
$\pi\rho$ Meissner  _et al._ (1997); Ünal and Meißner (2019). We assume that
the high-mass part of the isoscalar spectral function is approximately
exhausted by these states at $t\sim 1$ GeV2 and parametrize the spectral
functions as

$\displaystyle\textrm{Im}\,F_{1}^{S}(t)[\textrm{high-mass}]$
$\displaystyle=\pi a_{1}^{(S,0)}\delta(t-t^{(S)})$ $\displaystyle+\pi
a_{1}^{(S,1)}\delta^{\prime}(t-t^{(S)})$ (83)

and

$\displaystyle\textrm{Im}\,F_{2}^{S}(t)[\textrm{high-mass}]$
$\displaystyle=\pi a_{2}^{(S,0)}\delta(t-t^{(S)})$ $\displaystyle+\pi
a_{2}^{(S,1)}\delta^{\prime}(t-t^{(S)})$ $\displaystyle+\pi
a_{2}^{(S,2)}\delta^{\prime\prime}(t-t^{(S)}),$ (84)

where the effective pole mass is chosen as

$\displaystyle t^{(S)}=M_{\phi}^{2}=1.04\,\textrm{GeV}^{2}$ (85)

The pole strengths $a_{1}^{\omega},a_{1}^{(S,0)}$, and $a_{1}^{(S,1)}$ in
$\textrm{Im}\,F_{1}^{S}$, and $a_{2}^{\omega},a_{2}^{(S,0)},a_{2}^{(S,1)}$,
and $a_{2}^{(S,2)}$ in $\textrm{Im}\,F_{2}^{S}$, are then determined by
imposing the dispersive sum rules for the isoscalar spectral functions. These
sum rules are given by the formulas analogous to Eqs. (46)–(48) and Eqs.
(55)–(58) with $V\rightarrow S$, in which the quantities on the right-hand
side are now the isoscalar charge and radius,

$\displaystyle Q^{S}$
$\displaystyle\equiv\frac{1}{2}(Q^{p}+Q^{n})\;=\;\frac{1}{2},$ (86)
$\displaystyle\langle r^{2}\rangle_{1}^{S}$
$\displaystyle\equiv\frac{1}{2}\left(\langle r^{2}\rangle_{1}^{p}+\langle
r^{2}\rangle_{1}^{n}\right),$ (87)

and the isoscalar anomalous magnetic moment and radius

$\displaystyle\kappa^{S}$
$\displaystyle\equiv\frac{1}{2}(\kappa^{p}+\kappa^{n}),$ (88)
$\displaystyle\langle r^{2}\rangle_{2}^{S}$
$\displaystyle\equiv\frac{1}{2}\left(\kappa^{p}\langle
r^{2}\rangle_{2}^{p}+\kappa^{n}\langle r^{2}\rangle_{2}^{n}\right).$ (89)

The isoscalar spectral functions defined by Eq. (81) et seq. generate
remarkably accurate isoscalar nucleon form factors (see Fig. 12) and provide a
sufficient description of the isoscalar sector for the purposes of our study.

Figure 12: Isoscalar form factors $F_{1}^{S}(t)$ and $F_{2}^{S}(t)$ obtained
with our parametrization of the isoscalar spectral functions Eqs. (C) and (C).
Solid lines: Nominal parameters. Bands: Uncertainty resulting from the
variation of the isoscalar radii $\langle r^{2}\rangle_{1}^{S}$ and $\langle
r^{2}\rangle_{2}^{S}$ in the range given in Table 1. Dashed lines: Empirical
form factors of Ref. Ye  _et al._ (2018). All FFs are shown divided by the
standard dipole FF.

In the applications in Sec. III, we take into account the uncertainty in the
isoscalar spectral function resulting from the nucleon radii. We do not
attempt to assign a theoretical uncertainty to the high-mass part of the
isoscalar parametrization; this uncertainty could be estimated empirically
from dispersive fits to the data Belushkin  _et al._ (2007); Lin  _et al._
(2022).

## Acknowledgments

J.M.A. acknowledges support from the Spanish MECD grant FPA2016-77313-P. This
material is based upon work supported by the U.S. Department of Energy, Office
of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.

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