[a,b]Andrew W. Jackura

# Connecting Matrix Elements to Multi-Hadron Form-Factors

###### Abstract

We discuss developments in calculating multi-hadron form-factors and
transition processes via lattice QCD. Our primary tools are finite-volume
scaling relations, which map spectra and matrix elements to the corresponding
multi-hadron infinite-volume amplitudes. We focus on two hadron processes
probed by an external current, and provide various checks on the finite-volume
formalism in the limiting cases of perturbative interactions and systems
forming a bound state. By studying model-independent properties of the
infinite-volume amplitudes, we are able to rigorously define form-factors of
resonances.

Preprint number: JLAB-THY-21-3522

##  1 Introduction

Understanding the composition and structure of the excited states of Quantum
Chromodynamics (QCD) requires not only constructing model independent
observables which can be rigorously attached to an interpretation, but also
connecting the excited states via their decays to multi-hadron systems.
Lattice QCD, combined with principles of scattering theory and general
features of effective field theories (EFTs), offers a systematic pathway to
compute excited states from QCD. Since we are interested in capturing the few-
body effects of such states, we require a non-perturbative framework to map
finite volume observables computed with lattice QCD to the infinite volume
scattering amplitudes, from which we can determine the spectrum of the excited
states of such reaction processes.

The Lüscher methodology provides such a framework, where one exploits the fact
that poles of the correlation function, corresponding to the finite volume
spectrum, can be related to scattering amplitudes when writing the correlation
function to all-orders in the strong coupling [1, 2, 3, 4, 5, 6, 7].
Applications of the Lüscher framework to the scattering of two particles have
been extremely successful, including cases of particles with spin and multiple
coupled channels (see Ref. [8] for a review). Generalizations have been
studied for transitions from a single hadron to two hadron states [9, 10, 11,
12, 13], including applications to cases such as $\pi+\gamma^{\star}\to\pi\pi$
in the $\rho$-resonance channel [14, 15, 16].

Recent extensions [17, 18, 19, 20] have opened up the possibility of studying
two-hadron bound and resonant states under the influence of some external
current [21]. The simplest structural observable we investigate are form-
factors, which provide the first quantitative information on the size of these
excited states, e.g. electromagnetic charge radii of resonances such as
$\rho+\gamma^{\star}\to\rho$. Figure 1 shows the workflow in determining
properties of two-body hadronic states. Here we first define our observables
of interest, and then review the connection to finite-volume matrix elements
and various consistency checks. Then we connect the corresponding infinite-
volume observables to an on-shell representation which allow us to perform
global analyses using lattice QCD data.

Figure 1: Schematic work flow for connecting finite volume spectra and matrix
elements of multiple hadrons computed with lattice QCD to infinite volume
bound and resonant state properties.

##  2 Bound and resonant states of two-hadron systems

Figure 2: Diagrammatic representations of the $\mathbf{2}\to\mathbf{2}$ and
$\mathbf{2}+\mathcal{J}\to\mathbf{2}$ amplitudes, defined in Eqs. (1) and (2),
respectively, and representations near bound or resonant state poles as
defined in Eqs. (3) and (4).

We are interested in the spectrum of multi-hadron systems, which is encoded in
their scattering amplitudes. For a $\mathbf{2}\to\mathbf{2}$ scattering system
with initial state having four-momentum $P=(E,\mathbf{P})$ and final state
having $P^{\prime}=(E^{\prime},\mathbf{P}^{\prime})$, the scattering amplitude
$\mathcal{M}$ is defined as the asymptotic matrix element

$\displaystyle\langle P^{\prime},\mathrm{out}\lvert
P,\mathrm{in}\rangle_{\mathrm{conn.}}\equiv(2\pi)^{4}\delta^{(4)}(P^{\prime}-P)\,i\mathcal{M}(P)\,,$
(1)

where the delta function enforces total momentum conservation, and we have
implicitly assumed states in a definite angular momentum. The
“$\mathrm{conn}.$” reminds us that only the connected pieces contribute to the
amplitude. Since we are interested in the form-factors of scattering states,
we also introduce the $\mathbf{2}+\mathcal{J}\to\mathbf{2}$ amplitude
$\mathcal{W}$, where $\mathcal{J}$ is some external current with an arbitrary
Lorentz structure. We define this amplitude via the matrix element

$\displaystyle\langle P^{\prime},\mathrm{out}\rvert\mathcal{J}(0)\lvert
P,\mathrm{in}\rangle_{\mathrm{conn.}}\equiv\mathcal{W}(P^{\prime},P)\,.$ (2)

Since the current can inject a momentum $P^{\prime}-P$, $P\neq P^{\prime}$ in
general and the angular momentum between the initial and final state need not
be conserved as well. Although the framework present holds for arbitrary
currents, we will focus on the case of an external vector current unless
otherwise stated. Figure 2 illustrates these amplitudes diagrammatically.

States of multi-hadron systems are characterized by pole singularities of
scattering amplitudes. Bound states, such as the deuteron in $NN$, are poles
of the amplitude below the scattering threshold, while resonant or virtual
states are poles on the analytically continued amplitude to the second Riemann
sheet 111This holds for single channel scattering. For coupled channels,
resonances lie on the nearest unphysical sheet to the real energy region under
study.

$\displaystyle\mathcal{M}(P)\sim\frac{(ig_{p})^{2}}{s-s_{p}}\quad\mathrm{as\,\,}s\to
s_{p}\,,$ (3)

where $s\equiv P^{2}$, $s_{p}$ is the pole position, and $g_{p}$ is the
coupling to the channel. In the same manner, we can define a form factor of a
multi-hadron state by continuing the $\mathcal{W}$ amplitude to the
appropriate Riemann sheet, and isolating the pole contributions,

$\displaystyle\mathcal{W}^{\mu}(P^{\prime},P)\sim\mathcal{W}_{\textrm{df}}^{\mu}(P^{\prime},P)\sim\frac{ig_{p^{\prime}}}{s^{\prime}-s_{p^{\prime}}}\,\sum_{j}K_{j}^{\mu}\,f_{p\to
p^{\prime}}^{(j)}(Q^{2})\,\frac{ig_{p}}{s-s_{p}}\quad\mathrm{as\,\,}s^{\prime}\to
s_{p^{\prime}}\,\,\mathrm{and\,\,}s\to s_{p}\,,$ (4)

where $Q^{2}=-(P^{\prime}-P)^{2}$ is the invariant momentum transfer to the
system via the external current, and $\mathcal{W}_{\textrm{df}}$ removes the
contributions where the external current couples to one of the external legs
infinite volume $\mathbf{2}+\mathcal{J}\to\mathbf{2}$ amplitude $\mathcal{W}$,
as discussed in Refs. [19, 20, 22] and shown in Fig. 3, Note that the terms
which contain the current insertion on one of the external legs do not
contribute as one of the two spectral poles will cancel against this term,
meaning the spectral poles can be determined from $\mathcal{W}_{\textrm{df}}$
alone. Here, $p$ and $p^{\prime}$ are the states in the spectrum, $K_{j}$ is a
kinematic function determining the Lorentz structure, and $f_{p\to
p^{\prime}}^{(j)}$ are scalar form factors. Elastic form factors are those
where the state is identical, $p\to p$, and transitions are for $p\neq
p^{\prime}$. This allows us to provide rigorous definitions for elastic
resonance form factors $f_{R\to R}$, for example $\rho+\gamma^{\star}\to\rho$
which in turn can give us the electromagnetic charge radius and provide first
insight to a structural observable of a resonance from QCD. See Fig. 2 for a
pictorial representation of Eqs. (3) and (4).

Figure 3: Definition of the divergence free $\mathcal{W}_{\textrm{df}}$, where
contributions with current insertions on one of the external legs are removed.
For details, we refer to Refs. [19, 20, 22]

##  3 Two-hadron matrix element finite volume formalism

We consider a cubic volume $L^{3}$ subject to periodic boundary conditions
with an infinite temporal extent $T\to\infty$. The primary objects we consider
are the 2- and 3-point correlation functions, which can be related to the
finite volume spectrum and matrix elements. For some interpolating operator
$\mathcal{O}$, the 2-point correlator is given by

$\displaystyle C_{2,L}(P)=\int_{L}\\!\textrm{d}^{4}x\,e^{iP\cdot
x}\langle\mathcal{O}(x)\mathcal{O}^{{\dagger}}(0)\rangle=L^{3}\sum_{\mathfrak{n}}\frac{iZ_{\mathfrak{n}}Z_{\mathfrak{n}}^{{\dagger}}}{E-E_{\mathfrak{n}}+i\epsilon}\,,$
(5)

where $x=(t,\mathbf{x})$ is a Minkowski spacetime point and we assume a time
ordering $t>0$ and $Z_{\mathfrak{n}}\equiv\langle 0\rvert\mathcal{O}(0)\rvert
P_{\mathfrak{n}},L\rangle$ is an overlap factor for the interpolating operator
onto some finite volume state $\mathfrak{n}$ with
$P_{\mathfrak{n}}=(E_{\mathfrak{n}},\mathbf{P})$. The second equality is the
spectral decomposition of the correlator, which shows that the spectrum
$E_{\mathfrak{n}}$ for the system arises as pole singularities on the real
energy axis of the function. In practice, one computes these correlation
functions in Euclidean time $t\to-i\tau$. Here, we simply work with Minkowski
time $t$ since the spectrum and matrix elements considered here are
independent of the time signature. Similarly, the $3$-point function is
defined by

$\displaystyle C_{3,L}(P^{\prime},P)$
$\displaystyle=\int_{L}\\!\textrm{d}^{4}x^{\prime}\int_{L}\\!\textrm{d}^{4}x\,e^{iP^{\prime}\cdot
x^{\prime}}e^{-i(P^{\prime}-P)\cdot
x}\langle\mathcal{O}(x^{\prime})\mathcal{J}(x)\mathcal{O}^{{\dagger}}(0)\rangle\,,$
$\displaystyle=L^{6}\sum_{\mathfrak{m},\mathfrak{n}}\frac{iZ_{\mathfrak{m}}}{E^{\prime}-E_{\mathfrak{m}}+i\epsilon}\,\langle
P_{\mathfrak{m}}^{\prime},L\rvert\mathcal{J}(0)\lvert
P_{\mathfrak{n}},L\rangle\,\frac{iZ_{\mathfrak{n}}^{{\dagger}}}{E-E_{\mathfrak{n}}+i\epsilon}\,,$
(6)

where it is assumed that $t^{\prime}>t>0$ and $\langle
P_{\mathfrak{m}}^{\prime},L\rvert\mathcal{J}(0)\lvert
P_{\mathfrak{n}},L\rangle$ is the matrix element between two finite volume
states $\mathfrak{n}$ and $\mathfrak{m}$. For some particular set of
interpolating operators with the desired quantum numbers, one computes these
correlation functions and extracts the finite volume spectrum and matrix
elements.

In addition to the spectral representations, we can write expressions for the
2- and 3-point correlation functions to all-orders within some generalized
EFT. Since we are interested in the infrared limit of QCD, that is hadronic
processes, the generalized EFT provides an intermediary tool to encapsulate
the hadronic degrees of freedom. We do not specify any particular interaction,
and allow any and all interactions which satisfy the symmetries of the system
under consideration. Within this framework, one can write diagrammatic
representations of the correlators in terms of the fully dressed propagators
of the hadrons, and one- and two-particle irreducible kernels that encode all
the physics which does not allow two-particle states to go on-shell. Figure 4
illustrates an example of the 2-point correlator for two-particle systems.

Figure 4: (a) Diagrammatic representation for the 2-point correlator in terms
of the fully dressed propagator (line with black dot), the Bethe-Salpeter two-
particle irreducible kernel (gray circle with four legs), and the end-cap
kernels. Each loop labeled with ‘$\mathrm{V}$’ is an integral over the
temporal component of momentum and a sum over the spatial. (b) The infinite-
volume $\mathbf{2}\to\mathbf{2}$ amplitude in terms of the Bethe-Salpeter
kernel.

Identifying the poles of the all-orders expression with the spectral
representation allows us to connect finite-volume and infinite-volume
quantities. For example, if we consider a system such that the ground state of
some function is associated with the finite volume mass of the hadron,
$M_{L}=E_{\mathfrak{0}}$, then when we express all finite-volume sums in the
loops as integrals plus a correction determined from the Poisson summation
formula and find that the finite volume mass of a hadron is equal to its
infinite volume mass $M$ up to some exponentially small correction,

$\displaystyle M_{L}=M+\mathcal{O}(e^{-mL})\,,$ (7)

where $m$ is the smallest mass scale of the theory, e.g. the $\pi$ for QCD. We
assume we work in a regime such that $mL\gg 1$, thus this correction is
negligible compared to other volume effects which we consider shortly. Besides
matching pole positions, we may also match the residues at the poles between
the spectral and all-orders representations. For the 3-point function, this
allows us to link matrix elements with form factors, e.g. for the elastic form
factor of a stable hadron, one finds that the form factors $f_{L}$ are equal
to their infinite-volume counterparts $f$ up to some exponentially suppressed
correction

$\displaystyle f_{L}(Q^{2})=f(Q^{2})+\mathcal{O}(e^{-mL})\,,$ (8)

where $Q^{2}=-(P^{\prime}-P)^{2}$ is the momentum transfer.

Finite volume corrections for multi-hadron systems scale as a power law of the
inverse volume, since on-shell intermediate states lead to singularities in
the correlation function which enhance the volume dependence. Within a
kinematic region where only two particles can go on their mass-shell, i.e.
$2m\leq E^{\star}<E_{\mathrm{inel.}}^{\star}$ where
$E^{\star}\equiv\sqrt{P^{2}}$ is the center-of-momentum (CM) energy of the
system and $E_{\mathrm{inel.}}^{\star}$ is the first inelastic threshold, the
finite volume correction to the correlator leads to the Lüscher quantization
condition, which relates the poles of the correlator to the scattering
amplitude $\mathcal{M}$,

$\displaystyle\det\left[\,\mathcal{M}^{-1}(P_{\mathfrak{n}})+F_{L}(P_{\mathfrak{n}})\,\right]=0\,,$
(9)

where the determinant is over angular momentum space and
$P_{\mathfrak{n}}=(E_{\mathfrak{n}},\mathbf{P})$ is the four-momentum of the
two-particle system in a frame with momentum $\mathbf{P}$ and finite volume
energy $E_{\mathfrak{n}}$. Here, $F_{L}$ is a known function characterizing
the geometry of the volume, and is a dense matrix in angular momentum space.
For example, considering identical particles in $S$ wave, $F_{L}$ takes the
form

$\displaystyle
F_{L}(P)=\xi\left[\,\frac{1}{L^{3}}\sum_{\mathbf{k}}-\int\\!\frac{\textrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\,\right]\,\frac{1}{2\omega_{k}}\frac{1}{(P-k)^{2}-m^{2}+i\epsilon}\bigg{\rvert}_{k^{0}=\omega_{k}}\,,$
(10)

where $\xi=1/2$ for identical particles and the sum over $\mathbf{k}$ is over
all $\mathbf{k}=2\pi\mathbf{n}/L$ where $\mathbf{n}\in\mathbb{Z}^{3}$.
Equation (9) can be subduced into irreducible representation of the Octahedral
group or its little groups for systems at rest or moving, respectively.

Turning to $\mathbf{2}+\mathcal{J}\to\mathbf{2}$ processes, cf. Fig. 5,
examining simultaneous residues of the finite volume correction, one can
identify a relation between the matrix element and the corresponding infinite
volume amplitude [19, 20]

$\displaystyle\left\lvert\,\langle
P_{\mathfrak{m}}^{\prime},L\rvert\mathcal{J}^{\mu}(0)\lvert
P_{\mathfrak{n}},L\rangle\,\right\rvert^{2}=\frac{1}{L^{6}}\operatorname{Tr}\left[\,\mathcal{W}_{L,\textrm{df}}^{\mu}(P_{\mathfrak{n}},P_{\mathfrak{m}}^{\prime})\cdot\mathcal{R}_{L}(P_{\mathfrak{m}}^{\prime})\cdot\mathcal{W}_{L,\textrm{df}}^{\mu}(P_{\mathfrak{m}}^{\prime},P_{\mathfrak{n}})\cdot\mathcal{R}_{L}(P_{\mathfrak{n}})\,\right]$
(11)

where $\mathcal{R}_{L}$ is the generalized Lellouch-Lüscher factor [10, 11],

$\displaystyle\mathcal{R}_{L}(P_{\mathfrak{n}})=\lim_{E\to
E_{\mathfrak{n}}}\,\frac{(E-E_{\mathfrak{n}})}{\mathcal{M}^{-1}(P)+F_{L}(P)}\,,$
(12)

and $\mathcal{W}_{L,\textrm{df}}$ is defined as

$\displaystyle\mathcal{W}_{L,\textrm{df}}^{\mu}(P^{\prime},P)=\mathcal{W}_{\textrm{df}}^{\mu}(P^{\prime},P)+\mathcal{M}(P^{\prime})\cdot\left[\,\sum_{j}f_{j}(Q^{2})\,\mathcal{G}_{L,j}^{\mu}(P^{\prime},P)\,\right]\cdot\mathcal{M}(P)\,,$
(13)

with $f_{j}$ are the single hadron form factors of the constituent system, and
$\mathcal{G}_{L}$ is a geometric function which characterizes the finite-
volume corrections for this object. Consider a system with a vector current,
with total $S$ wave, then there is a single form factor $f$ and function
$\mathcal{G}_{L}$, which takes the form

$\displaystyle\mathcal{G}_{L}^{\mu}(P^{\prime},P)=\left[\,\frac{1}{L^{3}}\sum_{\mathbf{k}}-\int\\!\frac{\textrm{d}^{3}\mathbf{k}}{(2\pi)^{3}}\,\right]\,\frac{1}{2\omega_{k}}\frac{(P^{\prime}+P)^{\mu}-2k^{\mu}}{\left[(P^{\prime}-k)^{2}-m^{2}+i\epsilon\right]\left[(P-k)^{2}-m^{2}+i\epsilon\right]}\bigg{\rvert}_{k^{0}=\omega_{k}}\,.$
(14)

A diagrammatic representation of the infinite-volume portion is shown in Fig.
6.

Figure 5: (a) Diagrammatic expansion for the 3-point correlator, with short-
distance kernels for the single current insertions on single particles (cross)
and on two-particle irreducible kernels (gray circle with four legs). (b) The
all-orders representation for the infinite-volume
$\mathbf{2}+\mathcal{J}\to\mathbf{2}$ amplitude in terms of its kernels and
the $\mathbf{2}\to\mathbf{2}$ amplitude.

###  3.1 Volume dependence of conserved vector charge

As part of a series of consistency checks, we examine some special cases of
the finite-volume formalism for the $\mathbf{2}+\mathcal{J}\to\mathbf{2}$
processes. First, we consider a conserved vector current
$\partial_{\mu}\mathcal{J}^{\mu}=0$. Current conservation places restrictions
in the form of the Ward-Takahashi identity, leading to a constraint on the
forward direction of the $\mathcal{W}_{\textrm{df}}$ amplitude. For $S$ wave
scattering of a system composed of a neutral and charged particle, e.g.
$\pi^{0}\pi^{+}$, the relation is given by

$\displaystyle\mathcal{W}_{\textrm{df}}^{\mu}(P,P)=\mathrm{Q}\,\frac{\partial}{\partial
P_{\mu}}\mathcal{M}(P)\,,$ (15)

where $\mathrm{Q}=f(0)$ is the charge of the single particle. This relation
ensures that the charge of the matrix element is protected from finite-volume
effects [23],

$\displaystyle\langle P_{\mathfrak{n}},L\rvert\mathcal{Q}\lvert
P_{\mathfrak{n}},L\rangle\equiv L^{3}\langle
P_{\mathfrak{n}},L\rvert\mathcal{J}^{0}(0)\lvert
P_{\mathfrak{n}},L\rangle=\mathrm{Q}\,.$ (16)

###  3.2 Volume dependence for matrix elements of bound states

If we consider energies below the two-body production threshold, then the
power-law finite-volume corrections become exponentially suppressed via the
Poisson summation formula. If there is a bound state in this vicinity, then
the bound state mass will gain exponentially suppressed corrections [24]

$\displaystyle M_{B,L}=M_{B}+\mathcal{O}\left(e^{-\kappa L}\right)\,,$ (17)

where $\kappa$ is the binding momentum. If we examine Eq. (11) in this region,
we find that a bound state form-factor will gain similar exponentially
suppressed corrections. Consider a scalar current, $\mathcal{J}_{S}$, we
investigate the volume dependence of the scalar charge and find [23]

$\displaystyle\mathrm{g}_{B,L}$ $\displaystyle\equiv 2E_{B,L}\,\langle
P_{B},L\rvert\mathcal{J}_{S}(0)\lvert P_{B},L\rangle\,,$
$\displaystyle=\mathrm{g}_{B}+\mathcal{O}\left(e^{-\kappa L}\right)\,,$ (18)

where $\mathrm{g}_{B}$ is the infinite-volume scalar charge.

###  3.3 Threshold expansion of matrix elements

As a final check of the formalism, we explore the near threshold behavior of
matrix elements as a series in the inverse volume. For identical scalars, the
ground state CM energy follows the well-known expansion [1]

$\displaystyle E_{0}(L)=2m+\frac{4\pi
a}{mL^{3}}+\mathcal{O}\left(\frac{1}{L^{4}}\right)\,,$ (19)

where $a$ is the $S$ wave scattering length. One arrives at Eq. (19) by
looking at the non-relativistic limit of Eq. (9), and expand to the desired
order. If we now consider a scalar current in the forward direction, we find
that following a similar expansion of Eq. (11) leads to the expression [25]

$\displaystyle
L^{3}\langle\mathfrak{n}=0,L\rvert\mathcal{J}_{S}(0)\lvert\mathfrak{n}=0,L\rangle=\frac{\mathrm{g}}{m}\left[1-\frac{2\pi
a}{m^{2}L^{3}}+\mathcal{O}\left(\frac{1}{L^{4}}\right)\right]\,,$ (20)

where $\mathrm{g}$ is the scalar charge of a single particle. Alternatively,
one can arrive at this expansion by using time-ordered perturbation theory on
the 3-point correlation function. This expansion agrees with the Feynman-
Hellman theorem,

$\displaystyle
L^{3}\langle\mathfrak{n}=0,L\rvert\mathcal{J}_{S}(0)\lvert\mathfrak{n}=0,L\rangle=\mathrm{g}\,\frac{\textrm{d}E_{0}(L)}{\textrm{d}m^{2}}\,,$
(21)

where we expect the scaling of the correction to the energy matches that of
the matrix element.

##  4 Resonant form-factors from on-shell amplitudes

Given a set of lattice QCD spectra and matrix elements for some given channel,
we use Eq. (9), along with (11) to constrain the infinite volume on-shell
$\mathcal{M}$ and $\mathcal{W}_{\textrm{df}}$. Within a limited region of the
CM energy, below inelastic thresholds, the infinite volume amplitudes admit an
on-shell representation which isolates the non-analyticities fixed from
unitarity, i.e. branch point singularities associated with on-shell
intermediate states are separated from short-distance physics which is
unknown. The well-known $K$ matrix form for the $\mathbf{2}\to\mathbf{2}$
partial wave amplitude is one such representation,

$\displaystyle\mathcal{M}(P)=\mathcal{K}(P)\cdot\frac{1}{1-i\rho(P)\cdot\mathcal{K}(P)}$
(22)

where $\rho$ is the two-body phase space factor characterizing the kinematics
of on-shell two-particle states

$\displaystyle\rho(P)=\frac{\xi q^{\star}}{8\pi E^{\star}}\,,$ (23)

with $q^{\star}=\sqrt{P^{2}/4-m^{2}}$. The function $\mathcal{K}$ captures all
short-distance (off-shell) physics, and is a real function in the kinematic
region of consideration. Unitarity alone does not fix the form of
$\mathcal{K}$, however using Eq. (9) with some spectrum determined from
lattice QCD allows us to determine $\mathcal{K}$ in a global analysis with
some suitable parameterization respecting the on-shell structure.

Figure 6: Triangle topology $\sum_{j}f_{j}\mathcal{G}_{j}^{\mu}$ for the
$\mathcal{W}_{\textrm{df}}$ amplitude shown in Eq. (24).

In a similar manner to $\mathcal{M}$, the $\mathcal{W}$ amplitude can be
written in an on-shell form, allowing us to parameterize a smooth function and
ensure the analytic structure of the function is respected. When we project
the all-orders self-consistent equation [Fig. 5] to an on-shell form, we find
that $\mathcal{W}_{\mathrm{df}}$, cf. Fig. 3, takes the form [22]

$\displaystyle\mathcal{W}_{\mathrm{df}}^{\mu}(P^{\prime},P)=\mathcal{M}(P^{\prime})\cdot\left[\,\mathcal{A}^{\mu}(P^{\prime},P)+\sum_{j}f_{j}(Q^{2})\,\mathcal{G}_{j}^{\mu}(P^{\prime},P)\,\right]\cdot\mathcal{M}(P)\,,$
(24)

where $f_{j}$ are the on-shell single particle form factors, $\mathcal{G}$ is
a known kinematic function shown in Fig. 6, the infinite volume triangle
function characterizing on-shell singularities, and $\mathcal{A}$ is a smooth
real function which encapsulates the short-distance physics of two-particle
scattering under the influence of an external current. The
$\mathbf{2}\to\mathbf{2}$ amplitudes in the initial and final states describe
the initial and final state interactions between the two particles. Again, we
have presented the form of Eq. (24) with a vector Lorentz index, but the on-
shell form is identical for any Lorentz structure.

The path to bound or resonant state form factors is as follows: First, one
computes the two-particle finite volume spectrum for some channel and
determines the $K$ matrix and resulting $\mathbf{2}\to\mathbf{2}$ amplitude
for a given channel using Eq. (9) and (22). The single particle form factors
$f_{j}$ of the hadrons in the channel must also be determined for the desired
current. Next, the matrix elements for the two hadron system subject to the
external current must be calculated, and the $\mathcal{W}_{\textrm{df}}$
amplitude can be constrained by fitting parameterizations of $\mathcal{A}$ in
Eq. (24) such that (11) is satisfied. Once an analytic parameterization for
$\mathcal{A}$ can be determined, the elastic or transition form factors for a
bound or resonant state can be determined by inverting Eq. (4) for $f_{p\to
p^{\prime}}$. In terms of the on-shell representation for
$\mathcal{W}_{\textrm{df}}$, the form factors of a state in the two-particle
spectrum is given by

$\displaystyle\sum_{j}K_{j}^{\mu}\,f_{p\to
p^{\prime}}^{(j)}(Q^{2})=\lim_{\begin{subarray}{c}s^{\prime}\to
s_{p^{\prime}}\\\ s\to
s_{p}\end{subarray}}g_{p^{\prime}}g_{p}\,\left[\,\mathcal{A}^{\mu}(P^{\prime},P)+\sum_{j}f_{j}(Q^{2})\,\mathcal{G}_{j}^{\mu}(P^{\prime},P)\,\right]\,,$
(25)

where for resonances the expression is defined with $\mathcal{G}$ on the
second Riemann sheet as detailed in Ref. [22]. At zero-momentum transfer, the
charge of the resonance must be the sum of the constituent hadrons. For a
charge-neutral system, e.g. $\rho^{+}\to\pi^{+}\pi^{0}$, we recover $f_{p\to
p}(0)=\mathrm{Q}$.

##  5 Summary

Scattering calculations from lattice QCD have matured over the past decade,
and the push for developing the framework to compute more complicated
observables has evolved in tandem. Including the effects of external current
sources interacting with two-body systems has been developed both for the
finite volume formalism and the infinite volume amplitude. Non-trivial checks
on the formalisms have been made in various cases. Coupled with lattice QCD
data, the finite- and infinite-volume formalisms provide a pathway to compute
resonance form-factors, providing the first structural observables to study
the composition of excited QCD states.

##  6 Acknowledgements

I would like to thank my collaborators Raúl A. Briceño, Maxwell T. Hansen,
Felipe G. Ortega-Gama, and Keegan H. Sherman for their contributions in
completing this work and comments on these proceedings. This work is supported
in part by USDOE grant No. DE-AC05-06OR23177, under which Jefferson Science
Associates, LLC, manages and operates Jefferson Lab. This work is also
supported from the USDOE Early Career award, contract de-sc0019229.

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