††thanks: Now at the Bureau of Meteorology, Adelaide, SA, Australia.

# Odd-Parity Nucleon Electromagnetic Transitions in Lattice QCD

Finn M. Stokes  Special Research Centre for the Subatomic Structure of Matter,
Department of Physics, University of Adelaide, South Australia 5005, Australia
Jülich Supercomputing Centre, Institute for Advanced Simulation,
Forschungszentrum Jülich, Jülich D-52425, Germany  Benjamin J. Owen  Special
Research Centre for the Subatomic Structure of Matter,
Department of Physics, University of Adelaide, South Australia 5005, Australia
Waseem Kamleh  Special Research Centre for the Subatomic Structure of Matter,
Department of Physics, University of Adelaide, South Australia 5005, Australia
Derek B. Leinweber  Special Research Centre for the Subatomic Structure of
Matter,
Department of Physics, University of Adelaide, South Australia 5005, Australia

###### Abstract

The parity-expanded variational analysis (PEVA) technique enables the
isolation of opposite-parity eigenstates at finite momentum. The approach has
been used to perform the first lattice QCD calculations of excited-baryon form
factors. In particular, these calculations show that the low-lying odd-parity
nucleon excitations are described well by constituent quark models at moderate
$u$ and $d$ quark masses approaching the strange quark mass. Herein, we extend
the PEVA technique to establish a formalism for the determination of odd-
parity nucleon electromagnetic transition form factors in lattice QCD. The
formalism is implemented in the first calculation of the helicity amplitudes
for transitions from the ground state nucleon to the first two odd-parity
excitations. Through a comparison with constituent quark model calculations of
these amplitudes, these new results give important insight into the structure
of these excitations. This work is a critical step towards confronting
experimental electroproduction amplitudes for the $N^{*}(1535)$ and
$N^{*}(1650)$ resonances with ab-initio lattice QCD calculations.

††preprint: ADP-24-06/T1245

##  I Introduction

The study of nucleon resonances is a challenging and rich field. The structure
of such states can be probed experimentally through meson electroproduction.
Such measurements of nucleon resonance electrocouplings have existed for over
a decade, but have yet to be confronted with ab-initio calculations.

Lattice QCD studies of resonance electrocouplings provide not only the
opportunity to confront experiment, but can also provide important insight
into the nature of baryon excitations and the manner in which their structure
changes as the quark masses take values away from the physical regime. Through
a comparison with model calculations, one can learn the veracity of the model
and gain insight into the manner in which QCD gives rise to the observed
phenomena.

This approach has already emphasised how low-lying odd-parity nucleons are
described well by constituent quark models when the light-quark masses in the
simulations take moderate values approaching the strange quark mass Stokes
_et al._ (2020); Chiang  _et al._ (2003); Liu  _et al._ (2005); Sharma  _et
al._ (2013). This finding is driving new extensions of Hamiltonian effective
field theory (HEFT) calculations Hall  _et al._ (2013, 2015); Liu  _et al._
(2016, 2017a, 2017b); Wu  _et al._ (2018); Abell  _et al._ (2022) where two
bare-baryon basis states are incorporated in a coupled-channel analysis of
$S_{11}$ pion-nucleon scattering Abell  _et al._ (2023a, b).

Computing the electrocouplings of physical resonances from lattice QCD is a
challenging and multifaceted endeavour that requires a number of problems to
be solved. First, a formalism for the determination of odd-parity nucleon
electromagnetic transition form factors in lattice QCD needs to be
established. This is the focus of the present investigation. Here the parity-
expanded variational analysis (PEVA) technique Stokes  _et al._ (2015, 2019,
2020) is essential to the isolation of baryon eigenstates at finite momentum.

Subsequently, contributions from multi-particle scattering states need to be
understood. This will require the use of non-local momentum-projected meson-
baryon operators and excellent progress is being made for two-point functions
relevant to the mass spectrum Lang and Verduci (2013); Andersen  _et al._
(2018); Silvi  _et al._ (2021); Morningstar  _et al._ (2022); Bulava  _et al._
(2023a, 2024a, 2024b, b). The calculation of scattering-state contributions to
baryon three-point functions relevant to matrix elements has yet to be
considered. Once the matrix elements are determined on the lattice, one must
draw on a formalism to connect the finite-volume calculations to the infinite-
volume resonances in nature. Here scattering state contributions will play a
nontrivial role.

In this light, the study presented here focuses on the matrix elements of low-
lying energy eigenstates at larger quark masses where the states dominated by
single-particle operators are the lowest-lying states in the finite-volume
spectrum. Our aim is to establish a formalism in lattice QCD for the
determination of odd-parity nucleon electromagnetic transition form factors
and illustrate the formalism in practice through first calculations of the
helicity amplitudes for transitions from the ground state nucleon to the first
two odd-parity excitations.

As such, our comparison is not with experiment, but rather with constituent
quark-model calculations of these amplitudes to gain insight into the veracity
of these early model calculations Capstick and Keister (1995). We are keen to
learn if the success of the quark model in capturing the essence of the
physics for both ground state and odd-parity excited-state magnetic moments of
the nucleon is realised for electromagnetic transitions between these states.
Here, nontrivial momentum dependencies introduce new challenges for the quark
model.

Thus, this investigation focuses on the problem of accessing the relevant
matrix elements in lattice QCD and the need for careful regard to state
isolation and opposite-parity contaminations. Establishing this formalism is a
critical step towards confronting experimental electroproduction amplitudes
for the $N^{*}(1535)$ and $N^{*}(1650)$ resonances with ab-initio lattice QCD
calculations.

It has been well-established that variational analysis techniques provide a
powerful tool for isolating individual states in lattice QCD Michael (1985);
Luscher and Wolff (1990). However, conventional variational analyses have
relied upon a naive zero-momentum parity projection which fails at finite
momentum. The recent introduction and application of the parity-expanded
variational analysis (PEVA) technique Stokes  _et al._ (2015, 2019, 2020)
offers a solution to this problem. By extending a conventional variational
analysis, both odd and even-parity baryon eigenstates can be simultaneously
isolated at finite momentum. In this paper, we extend this technique to the
calculation of transition form factors of baryons.

Section II commences with a review of the covariant vertex functions for both
positive and negative parity transition matrix elements and outlines the
relationship between the associated form factors and the helicity amplitudes
of experimental interest. Here we introduce a new formalism and discuss its
relative merits. Section III briefly reviews the PEVA technique and outlines
its application to odd-parity electromagnetic transitions between the nucleon
ground state and odd-parity excitations of the nucleon. Finally, the formalism
is applied to calculate the odd-parity nucleon resonance electrocouplings and
the results are compared with legacy quark-model calculations in Sec. IV.
Section V concludes our presentation with a summary.

##  II Covariant Vertex Functions and Helicity Amplitudes

In our previous work Stokes  _et al._ (2020), we made use of variational
techniques to evaluate the elastic electromagnetic form factors for the
lightest negative-parity nucleon states. While such quantities can provide
important insight into the underlying structure of these states and inform
model calculations, defining and measuring such quantities experimentally
present significant challenges. Experimentally the quantities of interest are
the transition elements for the electroproduction processes
$\gamma^{*}N\rightarrow N^{*}$, specifically the transverse and longitudinal
helicity amplitudes ${\cal A}_{\frac{1}{2}}$ and ${\cal S}_{\frac{1}{2}}$.

Previous lattice calculations have focused primarily on the $\Delta\rightarrow
N\gamma$ transition. The framework for such a calculation was established in
Ref. Leinweber  _et al._ (1993) and subsequently examined comprehensively both
in quenched Alexandrou  _et al._ (2004, 2005) and full QCD Alexandrou  _et
al._ (2008, 2011). For all of these studies, due to the choice of lattice
ensemble parameters, the $\Delta$ baryon remains the lightest state in this
channel and so this process can be examined using standard techniques. The
results of Alexandrou  _et al._ (2011) show good qualitative agreement with
the experimental data, especially in the value extracted for $E2/M1$ ratio,
however the authors outline that discrepancies observed in the exact behaviour
of these amplitudes, particularly the $M1$ amplitude, highlight important
chiral dynamics to this process.

The only other nucleon transition considered to date has been the
$N^{*}(1440)\rightarrow N\gamma$ transition Lin  _et al._ (2008); Lin and
Cohen (2012). However, as yet it is unlikely any group has properly isolated
the $N^{*}(1440)$ on the lattice. The current understanding is that the
$N^{*}(1440)$ Roper resonance is dynamically generated through rescattering in
the $\pi N$, $\pi\Delta$, and $\sigma N$ channels Liu  _et al._ (2017a); Wu
_et al._ (2018); Leinweber  _et al._ (2024), and significant advances on
accessing multiparticle states in lattice QCD are being made Morningstar  _et
al._ (2022); Bulava  _et al._ (2023a). Experimental determinations indicate
that the sign of the $\mathcal{A}_{\frac{1}{2}}$ transition form factor is
negative at low $Q^{2}$. This contrasts constituent quark models which predict
a positive sign, lending further evidence the Roper resonance is not
associated with a quark-model state. The need for at least three coupled
meson-baryon channels to generate this resonance Wu  _et al._ (2018) presents
a formidable challenge to first-principles lattice QCD calculations. However,
evaluation of this amplitude will assist in understanding the underlying
dynamics of the Roper resonance and the make up of this state, both on the
lattice and in phenomenology.

A notably absent calculation is the evaluation of the odd-parity
$N^{*}(1535)\rightarrow N\gamma$ transition. The focus of this work is the
extraction of the transition form factors for such states. We begin with the
presentation of a general framework through which one can determine
$N^{*}\rightarrow N\gamma$ transition amplitudes for all choices and
combinations of parity, all from the same baryon correlator. Then in the
following section, we utilise this framework to examine the ${\cal
A}_{\frac{1}{2}}$ and ${\cal S}_{\frac{1}{2}}$ form factors of the first two
negative parity nucleon eigenstates.

Experimentally the quantities of interest are the transverse and longitudinal
helicity amplitudes ${\cal A}_{\frac{1}{2}}$ and ${\cal S}_{\frac{1}{2}}$

$\displaystyle{\cal A}_{\frac{1}{2}}$
$\displaystyle=\sqrt{\frac{2\pi\alpha}{K}}\frac{1}{e}\langle\,N^{*},s^{\prime}_{z}=\nicefrac{{1}}{{2}}^{+}\,|\,\epsilon^{(+)}_{\mu}J^{\mu}\,|\,N,s_{z}=\nicefrac{{1}}{{2}}^{-}\,\rangle\,,$
(1) $\displaystyle{\cal S}_{\frac{1}{2}}$
$\displaystyle=\sqrt{\frac{2\pi\alpha}{K}}\frac{1}{e}\frac{|\vec{q}|}{Q}\langle\,N^{*},s_{z}^{\prime}=\nicefrac{{1}}{{2}}^{+}\,|\,\epsilon^{(0)}_{\mu}J^{\mu}\,|\,N,s_{z}=\nicefrac{{1}}{{2}}^{+}\,\rangle\,,$
(2)

where $e$ is the magnitude of the electron’s charge, $\alpha$ the
electromagnetic fine structure constant and $Q=\sqrt{Q^{2}}$. Here
$\epsilon_{\mu}$ represents the polarisation of the incoming virtual photon,
with $|\vec{q}|$ the photon’s 3-momentum in the $N^{*}$ rest frame and

$K=\frac{M^{2}-m^{2}}{2M}\,.$ (3)

We identify $M$ with the resonance $N^{*}$ and $m$ with the nucleon $N$. These
amplitudes can in turn be related to matrix elements with the familiar form

$\displaystyle\langle N^{*},p^{\prime},s^{\prime}|\,J^{\mu}\,|N,p,s\rangle\,=$
$\displaystyle\hskip
7.0pte\,\left(\frac{M\,m}{E^{\prime}(\vec{p}\;\\!^{\prime})\,E(\vec{p})}\right)^{\nicefrac{{1}}{{2}}}\bar{u}_{(M)}(p^{\prime},s^{\prime})\,\Gamma(p^{\prime},p)\,u_{(m)}(p,s)\,,$
(4)

which parametrises the interaction in terms of Lorentz covariant structures.
We note that all spinors in this and subsequent expressions are regular Dirac
spinors. Here the subscript, $u_{(\xi)}$, labels the mass of the state for
which the spinor describes. A convenient choice of parametrisation is that
first presented in Ref. Devenish  _et al._ (1976) and nicely summarised in
Refs. Aznauryan  _et al._ (2008); Aznauryan and Burkert (2012), which allows
us to express the normal-parity
($\nicefrac{{1}}{{2}}^{+}\rightarrow\nicefrac{{1}}{{2}}^{+}$) transition as

$\displaystyle\langle N^{*},p^{\prime},s^{\prime}|J^{\mu}|N,p,s\rangle=$
$\displaystyle\hskip
10.0pte\,\left(\frac{M\,m}{E^{\prime}(\vec{p}\;\\!^{\prime})\,E(\vec{p})}\right)^{\nicefrac{{1}}{{2}}}\,\bar{u}_{(M)}(p^{\prime},s^{\prime})\,\tilde{J}^{\mu}\,u_{(m)}(p,s)\,,$
(5)

and abnormal-parity
($\nicefrac{{1}}{{2}}^{+}\rightarrow\nicefrac{{1}}{{2}}^{-}$) transition

$\displaystyle\langle N^{*},p^{\prime},s^{\prime}|J^{\mu}|N,p,s\rangle=$
$\displaystyle\hskip
10.0pte\,\left(\frac{M\,m}{E^{\prime}(\vec{p}\;\\!^{\prime})\,E(\vec{p})}\right)^{\nicefrac{{1}}{{2}}}\,\bar{u}_{(M)}(p^{\prime},s^{\prime})\,\tilde{J}^{\mu}\,\gamma_{5}\,u_{(m)}(p,s)\,,$
(6)

with

$\displaystyle\tilde{J}^{\mu}=\,$
$\displaystyle-\left[q^{2}\,\gamma^{\mu}-\not{q}\,q^{\mu}\right]\,G_{1}(Q^{2})$
$\displaystyle-\left[\left(P\cdot
q\right)\,\gamma^{\mu}-\not{q}\,P^{\mu}\right]\,G_{2}(Q^{2})\,,$ (7)

and $P=\frac{1}{2}\left(p^{\prime}+p\right)$.

Using this decomposition, the helicity amplitudes can be expressed in terms of
$G_{1}$ and $G_{2}$. For normal transitions

$\displaystyle\mathcal{A}_{\frac{1}{2}}^{+}(Q^{2})=$ $\displaystyle\hskip
10.0ptb_{+}\,\left[2Q^{2}\,G_{1}(Q^{2})-(M^{2}-m^{2})\,G_{2}(Q^{2})\right]\,,$
(8) $\displaystyle\mathcal{S}_{\frac{1}{2}}^{+}(Q^{2})=$ $\displaystyle\hskip
10.0ptb_{+}\,\frac{|\vec{q}\,|}{\sqrt{2}}\,\left[2\,(M+m)\,G_{1}(Q^{2})+(M-m)\,G_{2}(Q^{2})\right]\,,$
(9)

and for abnormal transitions

$\displaystyle\mathcal{A}_{\frac{1}{2}}^{-}(Q^{2})=$ $\displaystyle\hskip
10.0ptb_{-}\,\left[2Q^{2}\,G_{1}(Q^{2})-(M^{2}-m^{2})\,G_{2}(Q^{2})\right]\,,$
(10) $\displaystyle\mathcal{S}_{\frac{1}{2}}^{-}(Q^{2})=$ $\displaystyle
b_{-}\,\frac{-|\vec{q}\,|}{\sqrt{2}}\left[2\,(M-m)\,G_{1}(Q^{2})+(M+m)\,G_{2}(Q^{2})\right]\,,$
(11)

with

$b_{\pm}=\sqrt{\frac{Q^{2}+(M\mp m)^{2}}{8\,m\,(M^{2}-m^{2})}}\,,$ (12)

Another popular choice of parametrisation is to express the vertex in terms of
Pauli-Dirac-like form factors $F_{1}^{*}(Q^{2})$ and $F_{2}^{*}(Q^{2})$.
Taking

$\displaystyle F_{1}^{*}(Q^{2})$ $\displaystyle=Q^{2}\,G_{1}(Q^{2})\,,$ (13)
$\displaystyle F_{2}^{*}(Q^{2})$
$\displaystyle=-\frac{(M^{2}-m^{2})}{2}\,G_{2}(Q^{2})\,,$ (14)

one can re-express the transition matrix elements Aznauryan and Burkert (2012)
as

$\langle
N^{*}_{i},p^{\prime},s^{\prime}|J^{\mu}|N,p,s\rangle\,=e\,\left(\frac{M\,m}{E^{\prime}(\vec{p}\;\\!^{\prime})\,E(\vec{p})}\right)^{\nicefrac{{1}}{{2}}}\,\bar{u}_{(M)}(p^{\prime},s^{\prime})\,\Gamma_{i}^{\mu}(p^{\prime},p)\,u_{(m)}(p,s)\,,$
(15)

where $i$ denotes normal, $n$, or abnormal, $a$, transitions. For normal
transitions

$\Gamma^{\mu}_{\mathrm{n}}(p^{\prime},p)=\left(\delta^{\mu}_{\nu}-\frac{q_{\nu}\,q^{\mu}}{q^{2}}\right)\gamma^{\nu}\,F_{1}^{*}(Q^{2})+\frac{i\sigma^{\mu\nu}\,q_{\nu}}{M+m}\,F_{2}^{*}(Q^{2})\,,$
(16)

and for abnormal transitions

$\Gamma^{\mu}_{\mathrm{a}}(p^{\prime},p)=\left(\delta^{\mu}_{\nu}-\frac{q_{\nu}\,q^{\mu}}{q^{2}}\right)\,\gamma^{\nu}\,\gamma_{5}\,F_{1}^{*}(Q^{2})+\frac{i\sigma^{\mu\nu}\,q_{\nu}}{M-m}\,\gamma_{5}\,F_{2}^{*}(Q^{2})\,.$
(17)

We note that our choice of normalisation for the $F_{2}^{*}$ form factor in
the abnormal transition vertex differs from the common choice Ramalho and Pena
(2011); Ramalho and Tsushima (2011) by a factor of
$\left(\frac{M-m}{M+m}\right)$. However, the absence of parity doubling in the
low-lying energy eigenstates of QCD admit this formalism. The advantage of our
normalisation is apparent if we consider the expressions for the helicity
amplitudes in terms of the Dirac- and Pauli-like form factors, namely

$\displaystyle\mathcal{A}^{\pm}_{\frac{1}{2}}(Q^{2})=2\,b_{\pm}\,\left[F_{1}^{*}(Q^{2})+F_{2}^{*}(Q^{2})\right]\,,$
(18) $\displaystyle\mathcal{S}^{\pm}_{\frac{1}{2}}(Q^{2})=$
$\displaystyle\hskip 10.0pt\pm\frac{\sqrt{2}\,b_{\pm}\,(M\pm
m)\,|\vec{q}\,|}{Q^{2}}\,\left[F_{1}^{*}(Q^{2})-\tau_{\pm}\,F_{2}^{*}(Q^{2})\right]\,,$
(19)

where $b_{\pm}$ is defined in Eq. (12) and

$\tau_{\pm}=\frac{Q^{2}}{(M\pm m)^{2}}\,,$ (20)

with the $\pm$ label indicating the parity of the resonant nucleon state.
Hence, the helicity amplitudes are proportional to generalisations of the
Sachs electric and magnetic form factors

$\displaystyle G_{E}^{*}(Q^{2})$
$\displaystyle=F_{1}^{*}(Q^{2})-\frac{Q^{2}}{(M\pm
m)^{2}}\,F_{2}^{*}(Q^{2})\,,$ (21) $\displaystyle G_{M}^{*}(Q^{2})$
$\displaystyle=F_{1}^{*}(Q^{2})+F_{2}^{*}(Q^{2})\,.$ (22)

Compare this to the choice of Refs. Ramalho and Pena (2011); Ramalho and
Tsushima (2011); Ramalho and Peña (2024) where

$F_{2}(Q^{2})=\frac{M+m}{M-m}\,F_{2}^{*}(Q^{2})\,,$ (23)

for the abnormal transition such that

$\displaystyle\mathcal{A}^{-}_{\frac{1}{2}}(Q^{2})=2\,b_{-}\,\left[F_{1}^{*}(Q^{2})+\frac{M-m}{M+m}\,F_{2}(Q^{2})\right]\,,$
(24) $\displaystyle\mathcal{S}^{-}_{\frac{1}{2}}(Q^{2})=$
$\displaystyle-\frac{\sqrt{2}\,b_{-}\,(M+m)\,|\vec{q}\,|}{Q^{2}}\,\left[\frac{M-m}{M+m}\,F_{1}^{*}(Q^{2})-\tau_{+}\,F_{2}(Q^{2})\right]\,.$
(25)

In this case, the above identification with generalisations of the Sachs form
factors is not possible due to a relative factor of
$\left(\frac{M+m}{M-m}\right)$ between $F_{2}^{*}$ and $F_{2}$. In our
analysis we shall use the decompositions given by Eqs. (16) and (17).

##  III Parity Expanded Variational Analysis

###  III.1 Two-point functions

The process of extracting transition from factors of baryonic excited states
via the PEVA technique follows similarly to the elastic case presented in Ref
Stokes  _et al._ (2019). We provide here a brief summary of this process and
detail the generalisations required to handle transition matrix elements.

In this section we focus on a lattice QCD implementation of the formalism. We
adopt the common Euclidean metric ${\delta^{{\mu}{\nu}}}{}$ and use the
Hermitian Pauli representation for the Dirac matrices. With a Euclidean
metric, there is no need to distinguish between contravariant and covariant
indices.

We begin with a basis of $n$ conventional spin-$\nicefrac{{1}}{{2}}$ operators
$\left\\{{\chi_{{i}}(x)}\right\\}$ that couple to the states of interest. We
introduce the PEVA projector Stokes  _et al._ (2015)

${\Gamma_{\\!{\mp\bm{p}}}}\equiv\frac{1}{4}\left(\mathbb{I}+{{\gamma^{{4}}}}\right)\left(\mathbb{I}\mp
i{{\gamma^{{5}}}}{{\gamma^{{k}}}}{{\smash{\widehat{\bm{p}}}\;\\!\vphantom{p}}^{{k}}}\right)\,,$
(26)

where there is an arbitrary sign choice in whether the unit vector
$\smash{\widehat{\bm{p}}}\;\\!\vphantom{p}$ is chosen to be parallel ($+$) or
antiparallel ($-$) to the momentum direction. This choice changes the sign of
the cross-parity terms, but otherwise has no effect on the analysis of the
two-point functions. However, as discussed later, it plays an important role
in optimising the extraction of transition form factors for general
kinematics. With this projector, a set of basis operators is constructed

$\displaystyle{\chi_{{\mp\bm{p}}\,{i}}(x)}$
$\displaystyle\equiv{\Gamma_{\\!{\mp\bm{p}}}}\,{\chi_{{i}}(x)}\,,$ (27a)
$\displaystyle{\chi_{{\mp\bm{p}}\,{i^{\prime}}}(x)}$
$\displaystyle\equiv\mp{\Gamma_{\\!{\mp\bm{p}}}}\,{\gamma^{{5}}}{}\,{\chi_{{i}}(x)}\,.$
(27b)

We then seek an optimised set of operators
${\phi^{{\alpha}\,{}}_{{\mp\bm{p}}}(x)}$ that each couple strongly to a single
energy eigenstate ${\alpha}$. These optimised operators are constructed as
linear combinations of the basis operators. The optimum linear combinations
are found by solving a generalised eigenvalue problem (GEVP) with
${G(\bm{p}\,;t+\Delta{}t)}$ and ${G(\bm{p}\,;t)}$, where the correlation
matrix

$\displaystyle{G_{{i}{j}}(\bm{p}\,;t)}$
$\displaystyle\qquad\equiv\operatorname{Tr}\left(\sum_{\bm{x}}\mathrm{e}^{-i\bm{p}\cdot\bm{x}}\braket{\Omega}{\,{\chi_{{\mp\bm{p}}\,{i}}(x)}\,{\smash{\overline{\chi}}\vphantom{\chi}_{{\mp\bm{p}}\,{j}}(0)}\,}{\Omega}\right)\,,$
(28)

with $i$ and $j$ ranging over both the primed and unprimed operators. This
process is described in detail in Ref. Stokes  _et al._ (2015).

Using the optimised operators, we can construct the eigenstate-projected two-
point correlation function

$\displaystyle{G(\bm{p}\,;t\,;{\alpha}{})}$
$\displaystyle\qquad\equiv\operatorname{Tr}\left(\sum_{\bm{x}}\mathrm{e}^{-i\bm{p}\cdot\bm{x}}\braket{\Omega}{\,{\phi^{{\alpha}\,{}}_{{\mp\bm{p}}}(x)}\,{\smash{\overline{\phi}}\vphantom{\phi}^{{\alpha}\,{}}_{{\mp\bm{p}}}(0)}\,}{\Omega}\right)$
$\displaystyle\qquad={v^{{\alpha}}_{{i}}}(\bm{p})\,{G_{{i}{j}}(\bm{p}\,;t)}\,{u^{{\alpha}}_{{j}}}(\bm{p})\,.$
(29)

###  III.2 Three-point functions

The same approach is applied to lattice three-point correlation functions
drawing on the eigenvectors obtained in solving the two-point GEVP

$\displaystyle{\mathcal{G}^{{3}}_{{\mp}}(\mathcal{J}\\!\,;\bm{p}^{\prime}\\!\,,\bm{p}\,;t_{2}\,,t_{1}\,;{\alpha}{}\xrightarrow{}{\beta}{})}$
$\displaystyle\qquad\equiv\sum_{\bm{x}_{1},\bm{x}_{2}}\mathrm{e}^{-i\bm{p}^{\prime}\cdot\bm{x}_{2}}\,\mathrm{e}^{i(\bm{p}^{\prime}-\bm{p})\cdot\bm{x}_{1}}$
$\displaystyle\qquad\qquad\quad\times\braket{\Omega}{\,{\phi^{{\beta}\,{}}_{{\mp\bm{p}^{\prime}}}(x_{2})}\,\mathcal{J}(x_{1})\,{\smash{\overline{\phi}}\vphantom{\phi}^{{\alpha}\,{}}_{{-\bm{p}}}(0)}\,}{\Omega}\,,$
(30)

where $\mathcal{J}(x)$ is some current operator, which is inserted with a
momentum transfer $\bm{q}=\bm{p}^{\prime}-\bm{p}$. The consideration of
${\mathcal{G}^{{3}}_{{+}}(\mathcal{J}\\!\,;\bm{p}^{\prime}\\!\,,\bm{p}\,;t_{2}\,,t_{1}\,;{\alpha}{}\xrightarrow{}{\beta}{})}$
(where the sink operator uses the opposite PEVA projector sign convention to
the source operator) is required to optimise the extraction of the form
factors for general kinematics. We note that it is sufficient to consider this
change of projector for the sink operator alone, leaving the source operator
as ${\smash{\overline{\phi}}\vphantom{\phi}^{{\alpha}\,{}}_{{-\bm{p}}}(0)}$ in
all cases considered.

In this paper, we investigate electromagnetic transitions of the proton and
neutron by choosing the current operator $\mathcal{J}(x)$ to be the vector
current. In particular, we use the $O(a)$-improved Martinelli  _et al._ (1991)
conserved vector current used in Ref. Boinepalli  _et al._ (2006),

${j^{{\mu}}_{{CI}}}{}(x)\equiv{j^{{\mu}}_{{C}}}{}(x)+\frac{r}{2}\,a\,{\smash{\overline{q}}\vphantom{q}}{}(x)\left({\overleftarrow{\nabla}^{{\rho}}}{}+{\overrightarrow{\nabla}^{{\rho}}}{}\right){\sigma^{{\rho}{\mu}}}{}\,{q}{}(x)\,,$
(31)

where $r$ is the Wilson parameter, and ${j^{{\mu}}_{{C}}}{}(x)$ is the
standard conserved vector current for the Wilson action, symmetrised about a
lattice site.

As discussed in Section II, this choice of current operator gives the parity-
conserving transition matrix element

$\displaystyle\braket{{\beta}^{\pm}\,;p^{\prime}\,;s^{\prime}}{\,{j^{{\mu}}_{{CI}}}{}\,}{{\alpha}^{\pm}\,;p\,;s}=\sqrt{\frac{{m^{{\alpha}}}}{{E^{{\alpha}}(\bm{p})}}}\,\sqrt{\frac{{m^{{\beta}}}}{{E^{{\beta}}(\bm{p}^{\prime})}}}\,{\smash{\overline{u}}\vphantom{u}^{{\beta}}{(p^{\prime},s^{\prime})}}{}\,\Gamma_{n}^{\mu}(p^{\prime},p)\,{u^{{\alpha}}{(p,s)}}{}$
$\displaystyle\qquad=\sqrt{\frac{{m^{{\alpha}}}}{{E^{{\alpha}}(\bm{p})}}}\,\sqrt{\frac{{m^{{\beta}}}}{{E^{{\beta}}(\bm{p}^{\prime})}}}\;{\smash{\overline{u}}\vphantom{u}^{{\beta}}{(p^{\prime},s^{\prime})}}{}\left(\left({\delta^{{\mu}{\nu}}}{}-\frac{{{q}^{{\mu}}}{{q}^{{\nu}}}}{q^{2}}\right){\gamma^{{\nu}}}{}\,{F^{{*}}_{{1}}(Q^{2})}-\frac{{\sigma^{{\mu}{\nu}}}{}\,{{q}^{{\nu}}}}{{m^{{\beta}}}+{m^{{\alpha}}}{}}\,{F^{{*}}_{{2}}(Q^{2})}\right){u^{{\alpha}}{(p,s)}}{}\,,$
(32)

where ${\alpha}$ and ${\beta}$ have the same parity, and the parity-changing
transition matrix element

$\displaystyle\braket{{\beta}^{\mp}\,;p^{\prime}\,;s^{\prime}}{\,{j^{{\mu}}_{{CI}}}{}\,}{{\alpha}^{\pm}\,;p\,;s}=\sqrt{\frac{{m^{{\alpha}}}}{{E^{{\alpha}}(\bm{p})}}}\,\sqrt{\frac{{m^{{\beta}}}}{{E^{{\beta}}(\bm{p}^{\prime})}}}\,{\smash{\overline{u}}\vphantom{u}^{{\beta}}{(p^{\prime},s^{\prime})}}{}\,\Gamma_{a}^{\mu}(p^{\prime},p)\,{u^{{\alpha}}{(p,s)}}{}$
$\displaystyle\qquad=\sqrt{\frac{{m^{{\alpha}}}}{{E^{{\alpha}}(\bm{p})}}}\,\sqrt{\frac{{m^{{\beta}}}}{{E^{{\beta}}(\bm{p}^{\prime})}}}\;{\smash{\overline{u}}\vphantom{u}^{{\beta}}{(p^{\prime},s^{\prime})}}{}\left(\left({\delta^{{\mu}{\nu}}}{}-\frac{{{q}^{{\mu}}}{{q}^{{\nu}}}}{q^{2}}\right){\gamma^{{\nu}}}{}{\gamma^{{5}}}{}\,{F^{{*}}_{{1}}(Q^{2})}-\frac{{\sigma^{{\mu}{\nu}}}{}\,{{q}^{{\nu}}}}{{m^{{\beta}}}{}-{m^{{\alpha}}}{}}\,{\gamma^{{5}}}\,{F^{{*}}_{{2}}(Q^{2})}\right){u^{{\alpha}}{(p,s)}}{}\,,$
(33)

where ${\alpha}$ and ${\beta}$ have opposite parity. Note that these
expressions differ from those in Section II due to conversion to the Pauli
representation used in the current section. Here
$Q^{2}=\bm{q}^{2}-{\left({E^{{\alpha}}(\bm{p}^{\prime})}-{E^{{\alpha}}(\bm{p})}\right)}^{2}$
is the squared four-momentum with the conventional sign, and the invariant
scalar functions ${F^{{*}}_{{1}}(Q^{2})}$ and ${F^{{*}}_{{2}}(Q^{2})}$ are
respectively the Dirac- and Pauli-like transition form factors defined in
Equations 13 and 14.

To extract our desired signal from this spinor structure, we can take the
spinor trace with some spin-structure projector ${\Gamma_{\\!{S}}}$. This
trace is then called the spinor-projected three-point correlation function

$\displaystyle{G^{{3}}_{{\mp}}({\Gamma_{\\!{S}}}\,;{j^{{\mu}}_{{CI}}}{}\,;\bm{p}^{\prime}\\!\,,\bm{p}\,;t_{2}\,,t_{1}\,;{\alpha}{}\xrightarrow{}{\beta}{})}\equiv\operatorname{Tr}\\!\left[\,{\Gamma_{\\!{S}}}\,{\mathcal{G}^{{3}}_{{\mp}}({j^{{\mu}}_{{CI}}}{}\,;\bm{p}^{\prime}\\!\,,\bm{p}\,;t_{2}\,,t_{1}\,;{\alpha}{}\xrightarrow{}{\beta}{})}\,\right]$
$\displaystyle\qquad=e^{-{E^{{\beta}}(\bm{p}^{\prime})}(t_{2}-t_{1})}\,e^{-{E^{{\alpha}}(\bm{p})}t_{1}}\,\frac{\bar{\mathcal{Z}}^{\alpha}(\bm{p})}{2{E^{{\alpha}}(\bm{p})}}\,\frac{\mathcal{Z}^{\beta}(\bm{p}^{\prime})}{2{E^{{\beta}}(\bm{p}^{\prime})}}\,\operatorname{Tr}\\!\left[\,{\Gamma_{\\!{S}}}\,\Gamma_{\mp\bm{p}^{\prime}}\left(\not{p}+m^{\alpha}\right)\,\Gamma^{\mu}_{i}(p,p^{\prime})\,\left(\not{p}^{\prime}+m^{\beta}\right)\,\Gamma_{-\bm{p}}\,\right]\,,$
(34)

where the coefficients $\mathcal{Z}^{\alpha}(\bm{p})$ parametrise the operator
overlaps

$\langle\Omega|\,{\phi^{{\alpha}\,{}}_{{\mp\bm{p}}}(0)}|{\alpha}\,;p\,;s\rangle=\mathcal{Z}^{\alpha}(\bm{p})\,\sqrt{\frac{{m^{{\alpha}}}}{{E^{{\alpha}}(\bm{p})}}}\,\Gamma_{\mp\bm{p}}\,{u^{{\alpha}}{(p,s)}}{}\,.$
(35)

These spinor-projected correlation functions have a nontrivial time
dependence, which can be removed by constructing the ratio Leinweber  _et al._
(1991)

$\displaystyle
R^{{\mu}{\nu}}_{\mp}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})$
$\displaystyle\equiv\,\sqrt{\left|\frac{{G^{{3}}_{{\mp}}({\Gamma^{{\nu}}}{}\,;{j^{{\mu}}_{{CI}}}{}\,;\bm{p}^{\prime}\\!\,,\bm{p}\,;t_{2}\,,t_{1}\,;{\alpha}{}\xrightarrow{}{\beta}{})}\,{G^{{3}}_{{\mp}}({\Gamma^{{\nu}}}{}\,;{j^{{\mu}}_{{CI}}}{}\,;\bm{p}\,,\bm{p}^{\prime}\\!\,;t_{2}\,,t_{1}\,;{\beta}{}\xrightarrow{}{\alpha}{})}}{{G(\bm{p}^{\prime}\,;t_{2}\,;{\beta}{})}\,{G(\bm{p}\,;t_{2}\,;{\alpha}{})}}\right|}$
$\displaystyle\quad\times\operatorname{sign}\\!\left({G^{{3}}_{{\mp}}({\Gamma^{{\nu}}}{}\,;{j^{{\mu}}_{{CI}}}{}\,;\bm{p}^{\prime}\\!\,,\bm{p}\,;t_{2}\,,t_{1}\,;{\alpha}{}\xrightarrow{}{\beta}{})}\right)\,,$
(36)

where ${\Gamma^{{4}}}{}=(\mathbb{I}+{\gamma^{{4}}}{})/2$ and
${\Gamma^{{k}}}{}=(\mathbb{I}+{\gamma^{{4}}}{})(i\,{\gamma^{{5}}}{}\,{\gamma^{{k}}}{})/2$
form the basis for the spin projectors we use.

We can then define the reduced ratio

$\displaystyle\smash{\overline{R}}\vphantom{R}^{{\mu}{\nu}}_{\mp}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})$
$\displaystyle\equiv\sqrt{\frac{2{E^{{\alpha}}(\bm{p})}}{{E^{{\alpha}}(\bm{p})}+{m^{{\alpha}}}{}}}\,\sqrt{\frac{2{E^{{\beta}}(\bm{p}^{\prime})}}{{E^{{\beta}}(\bm{p}^{\prime})}+{m^{{\alpha}}}{}}}R^{{\mu}{\nu}}_{\mp}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})$
$\displaystyle=\operatorname{Tr}\\!\left[\,{\Gamma^{{\nu}}}{}\,\Gamma_{\mp\bm{p}^{\prime}}\frac{\not{p}+{m^{{\alpha}}}}{{E^{{\alpha}}(\bm{p})}+{m^{{\alpha}}}}\,\Gamma^{\mu}_{i}(p,p^{\prime})\,\frac{\not{p}^{\prime}+{m^{{\beta}}}}{{E^{{\beta}}(\bm{p}^{\prime})}+{m^{{\beta}}}}\,\Gamma_{+\bm{p}}\,\right]\,.$
(37)

In order to isolate the form factors, we can express this trace (which is
itself a linear combination of the form factors) as a matrix product

$\displaystyle\smash{\overline{R}}\vphantom{R}^{{\mu}{\nu}}_{\mp}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})=\begin{bmatrix}K_{1,\mp}^{\mu\nu}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})&K_{2,\mp}^{\mu\nu}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})\end{bmatrix}\begin{bmatrix}F^{*}_{1}(Q^{2})\\\
F^{*}_{2}(Q^{2})\end{bmatrix},$ (38)

expressed here in terms of the Pauli-Dirac decomposition.

The kinematic weights
$K_{i,\mp}^{\mu\nu}(\bm{p}^{\prime},\bm{p};{\beta},{\alpha})$ are determined
by substituting the explicit form for the vertex function in the trace and
reducing the product of $\gamma$-matrices into expressions involving the
incoming and outgoing energies and momenta, the eigenstate masses and the
hadron spins via Pauli matrices. The evaluation of these expressions is
presented in the Appendix. By expressing each independent determination of
$\smash{\overline{R}}\vphantom{R}^{{\mu}{\nu}}_{\mp}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})$
in this way, we can combine the results into a single vector equation

$\mathbf{R}=\mathbb{K}\,\mathbf{F}\,,$ (39)

where $\mathbf{R}$ is a vector containing $n$ independent ratio
determinations, $\mathbb{K}$ is an $n\times 2$ matrix of kinematic factors and
$\mathbf{F}$ is a vector containing the $2$ form factors. Provided we have at
least two linearly independent determinations of
$\overline{R}^{\mu}_{\alpha^{i}\rightarrow\beta^{j}}(p^{\prime},p;\Gamma^{\prime},\Gamma;j,i)$,
we can solve the linear system by taking the pseudo-inverse of $\mathbb{K}$
via the Singular Value Decomposition (SVD) of $\mathbb{K}$, to give

$\mathbf{F}=\mathbb{K}^{+}\,\mathbf{R}\,.$ (40)

To express this solution in terms of another choice of form factor
decomposition, it is a simple matter of applying the relevant basis
transformation matrix relating the two parametrisations. Consequently we can
determine the the generalised Sachs form factors and the helicity amplitudes
through the following expressions,

$\displaystyle\mathbf{G}$ $\displaystyle=\begin{pmatrix}G^{*}_{E}(Q^{2})\\\
G^{*}_{M}(Q^{2})\end{pmatrix}=\begin{pmatrix}1&-\frac{Q^{2}}{(M\pm m)^{2}}\\\
1&1\end{pmatrix}\mathbb{K}^{+}\,\mathbf{R}\,$ (41) $\displaystyle\mathbf{H}$
$\displaystyle=\begin{pmatrix}{\cal S}_{\frac{1}{2}}(Q^{2})\\\ {\cal
A}_{\frac{1}{2}}(Q^{2})\end{pmatrix}=b_{\pm}\begin{pmatrix}\pm\frac{\sqrt{2}(M\pm
m)|\vec{q}|}{Q^{2}}&0\\\ 0&2\end{pmatrix}\begin{pmatrix}1&-\frac{Q^{2}}{(M\pm
m)^{2}}\\\ 1&1\end{pmatrix}\mathbb{K}^{+}\,\mathbf{R}\,.$ (42)

##  IV Results

The results presented in this paper are calculated on the PACS-CS
$(2+1)$-flavour full-QCD ensembles Aoki  _et al._ (2009), made available
through the ILDG Beckett  _et al._ (2011). These ensembles use a $32^{3}\times
64$ lattice, and employ a renormalisation-group improved Iwasaki gauge action
with $\beta=1.90$ and non-perturbatively $O(a)$-improved Wilson quarks, with
$C_{SW}=1.715$.

Our current study focuses on the matrix elements of low-lying energy
eigenstates at larger quark masses where the states dominated by single-
particle operators are the lowest-lying states in the finite-volume spectrum.
Our aim is to implement the formalism established in the previous section and
report the first ab initio calculation of the helicity amplitudes for
electromagnetic transitions from the ground state nucleon to the first two
odd-parity excitations.

The recent HEFT calculation of Ref. Abell  _et al._ (2023b) indicates that the
two odd-parity states observed on the lattice are the lowest states in the
spectrum for the heaviest PACS-CS quark mass ensemble. Moreover, the HEFT
analysis indicates scattering state contaminations in the two-point
correlation functions are well below $10\text{\,}\mathrm{\char 37\relax}$ for
this mass. Thus of the five PACS-CS ensembles, we select the ensemble with
$m_{\pi}=$702\text{\,}\mathrm{MeV}$$ Aoki  _et al._ (2009). Given our
phenomenological focus is on the quark model, we will consider this particular
quark mass where the assumptions of the non-relativistic constituent quark
model are best satisfied. We set the scale using the Sommer parameter with
$r_{0}=$0.4921(64)\text{\,}\mathrm{fm}$$ Aoki  _et al._ (2009) providing
$a=$0.1022(15)\text{\,}\mathrm{fm}$$ and
$m_{\pi}^{2}=$0.3884(113)\text{\,}\mathrm{GeV}$^{2}$ based on 399
configurations.

For the variational analyses in this paper, we begin with the same eight-
interpolator basis as in Refs. Stokes  _et al._ (2019, 2020), in which we
studied the elastic form factors of the ground-state nucleon and three
localised excitations. This basis is formed from the conventional
spin-$\nicefrac{{1}}{{2}}$ nucleon interpolators

$\displaystyle{\chi_{{1}}}=$
$\displaystyle{\epsilon^{{a}{b}{c}}}\,[{{u^{{a}}}}^{\top}\,(C{\gamma^{{5}}})\>{d^{{b}}}]\,{u^{{c}}}\,,\
\text{and}$ $\displaystyle{\chi_{{2}}}=$
$\displaystyle{\epsilon^{{a}{b}{c}}}\,[{{u^{{a}}}}^{\top}\,(C)\>{d^{{b}}}]\,{\gamma^{{5}}}{u^{{c}}}\,,$
(43)

with $16$, $35$, $100$, or $200$ sweeps of gauge-invariant Gaussian smearing
Gusken (1990) with a smearing fraction of $\alpha=0.7$, applied at the quark
source and sinks in creating the propagators. For the PEVA analyses, this
basis is expanded to sixteen operators as described in Section III. The gauge
links used in the Gaussian smearing are first smoothed by applying four sweeps
of three-dimensional isotropic stout-link smearing Morningstar and Peardon
(2004) with $\rho=0.1$. Using this basis, the electromagnetic transitions
between the ground state nucleon and the first two negative-parity excitations
are examined.

To extract the transition form factors, we fix the source at time slice
$N_{t}/4=$16$$ relative to a fixed boundary condition in time. Utilising the
sequential source technique (SST) Bernard  _et al._ (1985), we invert through
the current, fixing the current insertion at time slice $21$. We choose time
slice $21$ by inspecting the two point correlation functions associated with
each state and observing that excited-state contaminations in the eigenstate-
projected correlators are suppressed by time slice $21$. This is evaluated by
fitting the effective mass in this region to a single-state ansatz verifying
that the full correlated $\chi^{2}/\text{dof}$ is satisfactory. We then
extract the transition form factors as outlined in Section III for every
possible sink time and once again look for a plateau consistent with the
single-state ansatz.

When fitting correlators, the $\chi^{2}/\mathrm{dof}$ is calculated with the
full covariance matrix, and the $\chi^{2}$ values of all fits are consistent
with an appropriate $\chi^{2}$ distribution. To avoid significant multi-
particle contamination, we demand that our two- and three-point correlators
are consistent with a single-state ansatz within the Euclidean time regions
considered. As one proceeds to precision calculations of these transition form
factors, it will be essential to examine the scattering-state contributions to
the form factors in detail.

In performing the SST through the electromagnetic current insertion, one needs
to commit to specific momentum insertions, $\vec{q}$. In quanta of $2\pi/L$
where $L$ is the length of a side of the cubic volume, we select three values
for $\vec{q}$ including $(1,0,0)$, $(1,1,0)$ and $(2,0,0)$ for the three-
momentum components. In the traditional fashion the final momentum of the
state is projected by the Fourier transform of the three-point function, and
the initial state momentum is inferred via momentum conservation. We consider
several initial, $\vec{p}$, and final ${\vec{p}\,}^{\prime}$ momenta, with the
intention of using boosts to access several momentum transfers Stokes  _et
al._ (2019, 2020). For each $\vec{q}$, we draw from permutations of
$\vec{p}=(0,0,0)$, $(1,0,0)$, $(1,1,0)$, $(2,0,0)$, $(2,1,0)$, and $(3,0,0)$
with several momentum sign combinations as we aim to access space-like
momentum transfers where $Q^{2}=\vec{q}^{\,2}-q_{0}^{2}$ is positive. To do
so, we ensure the lower-lying nucleon ground state has the largest momentum
between $\vec{p}$ and $\vec{p}\,^{\prime}$ to minimise the difference
$q_{0}=p_{0}^{\prime}-p_{0}=E^{\prime}-E$ as the baryons are on-shell.

Of the space-like momentum-transfer combinations considered, four were found
to generate good signal-to-noise correlation functions with excellent plateaus
available for fitting. These momentum combinations are summarised in Table 1.
Using the masses of the ground-state nucleon, $m_{0}=1.429(09)$ GeV, the first
excitation $m_{1}=1.996(25)$ GeV, and the second excitation $m_{2}=2.010(39)$
GeV, and ensuring the lower-lying nucleon is assigned the larger momentum
magnitudes, the associated $Q^{2}$ values for the transitions from states
$0\to 1$ and $0\to 2$ are also reported in Table 1.

Table 1: The momenta considered in our three-point function analysis producing good signal-to-noise in the correlation functions at space-like momentum transfers. The initial three-momentum components, $\vec{p}$, the SST current momentum, $\vec{q}$, and the final momentum, $\vec{p}\,^{\prime}$, are indicated in quanta of $2\pi/L$. To obtain space-like momentum transfers, the lower-lying nucleon is assigned the larger momentum magnitudes of $\vec{p}\,^{\prime}$. Using the state masses provided in the text, $Q^{2}$ values are reported for the transitions from states $0\to 1$ and $0\to 2$.  $-$$\vec{p}$ | $\vec{q}$ | $\vec{p}\,^{\prime}$ |  $Q^{2}_{0\to 1}/$ GeV |  $Q^{2}_{0\to 2}/$ GeV
---|---|---|---|---
$-$0,0,0 | 1,1,0 | 1,1,0 | 0.066(24) | 0.053(38)
$-$1,0,0 | 1,1,0 | 2,1,0 | 0.150(18) | 0.140(30)
$-$1,0,0 | 2,0,0 | 1,0,0 | 0.268(28) | 0.253(44)
$-$0,0,0 | 2,0,0 | 2,0,0 | 0.431(20) | 0.420(31)

It’s interesting to note that $\vec{q}=(1,0,0)$ is insufficient to generate
space-like momentum transfers. The mass splitting between the states is
significant and larger momentum differences are required to bring the state
energies closer together.

As one might expect, when the excited state takes zero momentum, one obtains
good signal to noise in the three-point correlation function. Another
favourable momentum combination is when the momenta of the incoming and
outgoing states differ by a sign. It’s well known this type of Breit frame
kinematics is favourable for form factors, and we now see it to be favourable
in the case of transition form factors. The fourth favourable momentum
combination in the second row of Table 1 is perhaps somewhat of a surprise.
However, the three-momentum of the current insertion is the smallest available
to generate space-like momentum transfer, and the magnitude of $Q^{2}$ is the
second smallest of those considered.

In the next two subsections we examine the Pauli-Dirac form factors
$F_{1}^{*}(Q^{2})$ and $F_{2}^{*}(Q^{2})$ for electromagnetic transitions from
the ground-state nucleon to the first and second negative parity excitations.
These form factors are defined in the covariant vertex function of Eq. (17)
and used in Eq. (33). Subsequently, they will be related to the helicity
amplitudes via the kinematic factors of Eq. (42). However, the fits to our
correlation functions are performed for $F_{1}^{*}(Q^{2})$ and
$F_{2}^{*}(Q^{2})$.

The form factors are obtained in the solution of Eq. (40) where the vector
$\mathbf{R}$ is constructed from the nontrivial components of the $4\times
4\times 2$ elements of
$\smash{\overline{R}}\vphantom{R}^{{\mu}{\nu}}_{\mp}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})$
defined in Eq. (37). We report contributions from the doubly-represented
$u$-sector of the proton, $u_{p}$, and the singly-represented $d$-sector of
the proton, $d_{p}$, individually. Each sector is reported for single quarks
of unit charge such that the physical states are obtained through simple quark
counting and electric charge assignments. Schematically, the physical proton
and neutron transition amplitudes are obtained via

$\displaystyle p$
$\displaystyle=\phantom{-}\frac{2}{3}\,2\,u_{p}-\frac{1}{3}\,d_{p}\,,$ (44a)
$\displaystyle n$ $\displaystyle=-\frac{1}{3}\,2\,u_{p}+\frac{2}{3}\,d_{p}\,.$
(44b)

###  IV.1 Transitions to the first negative parity excitation

Figure 1:  Quark-sector contributions to the three-point correlation function
for the covariant vertex function $F_{1}^{*}(Q^{2})$ at $Q^{2}=0.066(24)$ GeV2
for the odd-parity transition between the ground state proton and the first
negative-parity excitation. Sector contributions are normalised to represent
single quarks of unit charge.

Figure 2:  Quark-sector contributions to the three-point correlation function
for the covariant vertex function $F_{2}^{*}(Q^{2})$ at $Q^{2}=0.066(24)$ GeV2
for the odd-parity transition between the ground state proton and the first
negative-parity excitation. Sector contributions are normalised to represent
single quarks of unit charge.

Figure 3:  Zoomed-in presentation of the lattice QCD results presented in Fig.
1 for the transition between the ground state proton and the first negative-
parity excitation. The onset of plateau behaviour is observed at two time
steps following the current insertion at $t=21$ and statistical uncertainties
remain stable throughout the fit regime.

Results for $F_{1}^{*}(Q^{2})$ and $F_{2}^{*}(Q^{2})$ in the odd parity
electromagnetic transition for the ground-state proton to the first negative
parity excitation are presented in Figs. 1 and 2 respectively. Here $Q^{2}$
takes the value 0.066(24) GeV2, the lowest space-like momentum transfer
available in this analysis. Values for $F_{1}^{*}$ are small and a zoomed-in
presentation of the fit is provided in Fig. 3.

The use of correlation matrix eigenvectors to aid in isolating the energy
eigenstates leads to a rapid onset of plateau behaviour following the current
insertion at $t=21$. By two-time slices later, a persistent plateau is
identified in the full covariance matrix fit.

It’s interesting that for both form factors, the doubly-represented $u$ sector
of the proton contributes with a negative sign, while the singly-represented
sector contributes with an opposite positive sign. In the dominant $F_{2}^{*}$
form factor, the singly-represented $d$ sector of the proton makes the largest
contribution. Nevertheless, upon including quark-counting and charge factors,
the doubly-represented $u$-quark sector of the proton will dominate, whereas
the singly represented $u$-quark sector in the neutron will dominate.

Figure 4: Transverse helicity amplitudes for transitions from the ground state
nucleon to the first negative-parity excitation at
$m_{\pi}=$702\text{\,}\mathrm{MeV}$$ (filled symbols) are presented in the
context of results from a constituent quark model incorporating relativistic
effects in a light-front framework Capstick and Keister (1995) (lines) and
experiment (open symbols).

Figure 5: Longitudinal helicity amplitudes for transitions from the ground
state nucleon to the first negative-parity excitation at
$m_{\pi}=$702\text{\,}\mathrm{MeV}$$ (filled symbols) are presented in the
context of results from a relativised constituent quark model Capstick and
Keister (1995) (lines) and experiment (open symbols).

Figure 6: Ratio of proton to neutron transverse helicity amplitudes for
transitions from the ground state to the first negative-parity excitation at
$m_{\pi}=$702\text{\,}\mathrm{MeV}$$ (filled symbols) are compared with the
relativised constituent quark model Capstick and Keister (1995).

With $F_{1}^{*}(Q^{2})$ and $F_{2}^{*}(Q^{2})$ determined in the correlation
function ratio fits, we proceed to calculate the helicity amplitudes for the
ground to first odd-parity excitation. Quark sectors are combined to create
physical proton and neutron states via Eqs. (44) and the helicity amplitudes
are obtained via Eq. (42). Figures 4 and 5 present the transverse and
longitudinal helicity amplitudes respectively, as a function of the momentum
transfer, $Q^{2}$. Here our lattice QCD results at
$m_{\pi}=$702\text{\,}\mathrm{MeV}$$ are presented in the context of
constituent quark model predictions for the $N^{*}(1535)$ and experimental
measurements. The quark model Capstick and Keister (1995) incorporates
relativistic effects in a light-front framework.

Our first observation for the transverse helicity amplitude is that the
$Q^{2}$ dependence is very mild, a result reflected in both the constituent
quark model and experiment. And while the sign of the proton and neutron
helicity amplitudes are in accord with the model and available experimental
results, the magnitude of our lattice results exceed both the model and
experiment. It will be interesting to learn the relevant dynamics as one
reduces the quark masses of the lattice QCD simulation toward the physical
point. Assuming the lattice results approach the experimental measurements, it
will be interesting to understand the relative roles of a simple quark mass
dependence for each state contributing to the resonance amplitude and the
resonance-related role of several finite-volume states contributing to the
amplitude.

The longitudinal helicity amplitudes presented in Fig. 5 are relatively small.
Here good agreement with the model results is observed for the excited proton
at small $Q^{2}$ but the lattice QCD results do not show the $Q^{2}$
invariance of the model, instead dropping towards zero and even changing sign
in accord with the experimental measurements. On the other hand, the lattice
QCD results agree very well with the model results for the transition of the
neutron to the $n^{*}(1535)$. Together, these results suggest that the quark
mass dependence of this observable is likely to be mild.

It’s interesting to examine the extent to which the lattice QCD and model
predictions of the helicity amplitudes agree after the ratio of proton to
neutron amplitudes are taken. In the ratio, quark mass effects have an
opportunity to compensate, leaving a prediction more robust to quark mass
variation. Figure 6 presents this ratio for the transverse helicity amplitude
as a function of $Q^{2}$. Again the trends match and the magnitudes are now
quite similar.

###  IV.2 Transitions to the second negative parity excitation

We now turn our attention to the electromagnetic transitions to the second
low-lying negative-parity nucleon excitation observed on the lattice at
$m_{\pi}=$702\text{\,}\mathrm{MeV}$$ and compare our results with quark model
calculations of the $N^{*}(1650)$ and associated experimental measurements. It
is here that the correlation matrix based eigenstate projection gains
significant importance as we examine electromagnetic transitions to the second
eigenstate in the odd-parity spectrum.

Figure 7:  Quark-sector contributions to the three-point correlation function
for the covariant vertex function $F_{1}^{*}(Q^{2})$ at $Q^{2}=0.053(38)$ GeV2
for the odd-parity transition between the ground state proton and the second
negative-parity excitation. Sector contributions are normalised to represent
single quarks of unit charge.

Figure 8:  Quark-sector contributions to the three-point correlation function
for the covariant vertex function $F_{2}^{*}(Q^{2})$ at $Q^{2}=0.053(38)$ GeV2
for the odd-parity transition between the ground state proton and the second
negative-parity excitation. Sector contributions are normalised to represent
single quarks of unit charge.

Figure 9:  Zoomed-in presentation of the lattice QCD results presented in Fig.
7 for the transition between the ground state proton and the second negative-
parity excitation. Statistical uncertainties are larger for this transition
relative to the transition to the first negative-parity excitation illustrated
in Fig. 3.

Results for $F_{1}^{*}(Q^{2})$ and $F_{2}^{*}(Q^{2})$ in the electromagnetic
transition from the ground-state proton to the second negative parity
excitation are presented in Figs. 7 and 8 respectively. Here the minimum
space-like $Q^{2}$ takes a slightly smaller value of 0.053(38) GeV2 due to the
change in the mass of the excitation. Once again, values for $F_{1}^{*}$ are
small and a zoomed-in presentation of the fit is provided in Fig. 9. Here we
observe slightly larger statistical uncertainties for the transition to the
second odd-parity excitation.

The use of correlation-matrix eigenvector projection for state isolation leads
to a rapid onset of plateau behaviour for three of the four correlators
presented. Some fluctuation in the proton’s singly-represented $d$-quark
sector contribution to $F_{2}^{*}(Q^{2})$ prevents an early fit due to
uncomfortably large values for the full covariance matrix $\chi^{2}/$dof.

This time there is a clear sign change between the contributions to
$F_{1}^{*}(Q^{2})$ and $F_{2}^{*}(Q^{2})$ for the singly-represented $d$-quark
sector in the proton. However, the doubly-represented $u$ sector of the proton
continues to uniformly contribute with a negative sign. In the dominant
$F_{2}^{*}$ form factor, both quark sectors make a contribution that is
similar in magnitude. Upon including quark-counting and charge factors, the
$u$-quark sector of the proton will dominate, whereas the sectors are more
balanced for the neutron.

Figure 10: Transverse helicity amplitudes for transitions from the ground
state nucleon to the second negative-parity excitation at
$m_{\pi}=$702\text{\,}\mathrm{MeV}$$ (filled symbols) are presented in the
context of results from a constituent quark model incorporating relativistic
effects in a light-front framework Capstick and Keister (1995) (lines) and
experiment (open symbols).

Figure 11: Longitudinal helicity amplitudes for transitions from the ground
state nucleon to the second negative-parity excitation at
$m_{\pi}=$702\text{\,}\mathrm{MeV}$$ (filled symbols) are presented in the
context of results from a relativised constituent quark model Capstick and
Keister (1995) (lines) and experiment (open symbols).

Figure 12: Ratio of proton to neutron transverse helicity amplitudes for
transitions from the ground state to the second negative-parity excitation at
$m_{\pi}=$702\text{\,}\mathrm{MeV}$$ (filled symbols) are compared with the
relativised constituent quark model Capstick and Keister (1995).

With $F_{1}^{*}(Q^{2})$ and $F_{2}^{*}(Q^{2})$ determined, we proceed to the
helicity amplitudes for the ground to second odd-parity excitation. Figures 10
and 11 present the transverse and longitudinal helicity amplitudes
respectively. Our lattice QCD results are presented in the context of the
relativised constituent quark model Capstick and Keister (1995) and
experimental measurements where available.

A similar pattern of results for the transverse helicity amplitude is
observed. Again, the $Q^{2}$ dependence is very mild in both the lattice and
constituent quark model results. The signs of the proton and neutron helicity
amplitudes from lattice QCD are in accord with the model and experiment.
Again, the magnitude of our lattice results exceed both the model and
experiment, suggesting an important quark mass dependence.

The longitudinal helicity amplitudes presented in Fig. 11 are relatively
small. Reasonable agreement with the model results are observed. However, the
sign of the experimental result at larger $Q^{2}$ differs. A high-statistics
analysis would be required to access this $Q^{2}$, but such an analysis would
be able to discern if the sign change is $Q^{2}$ related or quark-mass
related.

On a final note, we examine the extent to which the lattice QCD and model
predictions of the transverse helicity amplitudes agree after the ratio of
proton to neutron amplitudes are taken to make a comparison more robust to
quark mass variation. Figure 12 illustrates matching $Q^{2}$ trends with
comparable magnitudes in the ratio.

##  V Conclusion

A formalism for the determination of spin-1/2 electromagnetic transition form
factors in lattice QCD has been established. Both normal parity-conserving
transitions and abnormal parity-changing transitions have been included in our
considerations. After working with the standard covariant vertex functions, we
introduce a new set of covariant vertex functions which more closely resemble
the Pauli-Dirac elastic form factors.

Using the parity-expanded variational analysis (PEVA) technique Stokes  _et
al._ (2015, 2019, 2020), the new formalism is implemented in the first lattice
QCD calculation of the helicity amplitudes for transitions from the ground
state nucleon to both the first and the second odd-parity excitations observed
in lattice QCD at $m_{\pi}=$702\text{\,}\mathrm{MeV}$$. The selection of a
large pion mass avoids significant complications associated with the mixing
with two-particle states to become scattering states.

Statistical uncertainties in the transition form factors are found to be
similar for both states, with only slightly larger fluctuations in the Pauli-
Dirac form factor $F_{1}^{*}(Q^{2})$ and slightly larger statistical
uncertainties in $F_{2}^{*}(Q^{2})$ for the second odd-parity excitation of
the nucleon.

Our lattice QCD results are compared comprehensively with a relativised
constituent quark model using a light-front formalism Capstick and Keister
(1995). In every case, the lattice QCD results for the transverse and
longitudinal helicity amplitudes agree with the quark model in sign. This
includes both proton and neutron amplitudes and electromagnetic transitions to
both the first and the second excitation. Moreover, when a ratio of proton to
neutron results is considered to suppress quark-mass related effects, the
results compare favourably.

Combined with our earlier results favourably comparing lattice QCD and the
quark model for the electromagnetic form factors of these odd-parity
excitations, the results presented here provide further credence that the two
low-lying odd-parity excitations of the nucleon observed in experiment are
associated with quark-model like basis states which are dressed to form
resonances.

The magnitude of the lattice QCD results for the transverse helicity
amplitudes are larger than that predicted by the quark model as it was tuned
to describe experiment. Indeed, the experimental measurements are even smaller
than that predicted by the quark model. Together, these hint at a quark mass
effect that is significant for the transverse helicity amplitude. It will be
interesting to learn the relevant dynamics as one reduces the quark masses of
the lattice QCD simulation toward the physical point. While the predominant
effect may be a simple quark-mass dependence for each state contributing to
the resonance amplitude, effects from combining the contributions from several
finite-volume states to form the resonance are a novel possibility.

Thus working towards the physical quark-mass requires the introduction of
multi-particle interpolating fields that will mix with the single-particle
interpolators considered herein Abell  _et al._ (2023b). In the three-point
function calculation, one will need to perform current insertions on all five
of the quark propagators, further increasing the demands of the calculation.
As the bare quark-model-like basis state mixes with the nearby two-particle
states its contribution will be spread over several finite-volume states. It
will be fascinating to learn the transition amplitudes to each of these
finite-volume states and how they differ. Finally, it will be important to
probe larger space-like $Q^{2}$ values to make direct confrontations with
experimental measurements.

*

##  Appendix A Kinematic factors

As discussed in Section III.2, the reduced ratio
$\smash{\overline{R}}\vphantom{R}^{{\mu}{\nu}}_{\mp}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})$
can be expressed as a matrix product

$\displaystyle\smash{\overline{R}}\vphantom{R}^{{\mu}{\nu}}_{\mp}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})=\begin{bmatrix}K_{1,\mp}^{\mu\nu}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})&K_{2,\mp}^{\mu\nu}(\bm{p}^{\prime},\bm{p}\,;{\beta}{},{\alpha}{})\end{bmatrix}\begin{bmatrix}F^{*}_{1}(Q^{2})\\\
F^{*}_{2}(Q^{2})\end{bmatrix},$ (45)

of some kinematic weights
$K_{i,\pm}^{\mu\nu}(\bm{p}^{\prime},\bm{p};{\beta},{\alpha})$ and the Pauli-
Dirac-like form factors, $F^{*}_{1}(Q^{2})$ and $F^{*}_{2}(Q^{2})$. These
kinematic factors can be obtained by evaluating traces of the form

$F^{\prime}_{\mp}\left({\Gamma_{\\!{}}},\mathcal{J}\right)\equiv
8EE^{\prime}\operatorname{Tr}\\!\left({\Gamma_{\\!{}}}\,{\Gamma_{\\!{\mp\bm{p}^{\prime}}}}\,\frac{-i\gamma\cdot
p^{\prime}+\mathbb{I}M^{\prime}}{2E^{\prime}}\,\mathcal{J}\,\frac{-i\gamma\cdot
p+\mathbb{I}M}{2E}\,{\Gamma_{\\!{-\bm{p}}}}\right)\,.$ (46)

Specifically, in the Pauli representation, for normal, parity-conserving
transitions,

$\displaystyle K_{1,\mp,n}^{\mu\nu}(\bm{p}^{\prime},\bm{p};{\beta},{\alpha})$
$\displaystyle=\left(\delta^{\mu\rho}{}-\frac{{{q}^{{\mu}}}q^{\rho}}{q^{2}}\right)F^{\prime}_{\mp}({\Gamma^{{\nu}}}{},\gamma^{\rho})\,\text{
and}$ (47) $\displaystyle
K_{2,\mp,n}^{\mu\nu}(\bm{p}^{\prime},\bm{p};{\beta},{\alpha})$
$\displaystyle=-\frac{q^{\rho}}{{m^{{\beta}}}+{m^{{\alpha}}}{}}\,F^{\prime}_{\mp}({\Gamma^{{\nu}}}{},\sigma^{\mu\rho})\,,$
(48)

and for abnormal, parity-changing transitions

$\displaystyle K_{1,\mp,a}^{\mu\nu}(\bm{p}^{\prime},\bm{p};{\beta},{\alpha})$
$\displaystyle=\left(\delta^{\mu\rho}{}-\frac{{{q}^{{\mu}}}q^{\rho}}{q^{2}}\right)F^{\prime}_{\mp}({\Gamma^{{\nu}}}{},\gamma^{\rho}\gamma^{5})\,\text{
and}$ (49) $\displaystyle
K_{2,\mp,a}^{\mu\nu}(\bm{p}^{\prime},\bm{p};{\beta},{\alpha})$
$\displaystyle=-\frac{q^{\rho}}{{m^{{\beta}}}-{m^{{\alpha}}}{}}\,F^{\prime}_{\mp}({\Gamma^{{\nu}}}{},\sigma^{\mu\rho}\gamma^{5})\,.$
(50)

These traces were evaluated in the Pauli representation in Ref. Stokes (2019),
and are reproduced below for completeness:

$\displaystyle
F^{\prime}_{\mp}\left({{\Gamma^{{4}}}{}},{\mathbb{I}{{}}}\right)$
$\displaystyle=\left(\left(E+m\right)\left(E^{\prime}+m^{\prime}\right)\mp|\bm{p}||\bm{p}^{\prime}|\right)\left(1\pm\hat{p}^{i}\hat{p}^{\prime\,i}\right)$
$\displaystyle F^{\prime}_{\mp}\left({{\Gamma^{{4}}}{}},{\gamma^{i}}\right)$
$\displaystyle=-\left(|\bm{p}|\left(E^{\prime}+m^{\prime}\right)\pm|\bm{p}^{\prime}|\left(E+m\right)\right)\left(\pm\epsilon^{ijk}\hat{p}^{j}\hat{p}^{\prime\,k}+{i}\left(\hat{p}^{i}\pm\hat{p}^{\prime\,i}\right)\right)$
$\displaystyle F^{\prime}_{\mp}\left({{\Gamma^{{4}}}{}},{\gamma^{4}}\right)$
$\displaystyle=\left(\left(E+m\right)\left(E^{\prime}+m^{\prime}\right)\pm|\bm{p}||\bm{p}^{\prime}|\right)\left(1\pm\hat{p}^{i}\hat{p}^{\prime\,i}\right)$
$\displaystyle F^{\prime}_{\mp}\left({{\Gamma^{{4}}}{}},{\gamma^{5}}\right)$
$\displaystyle=\left(|\bm{p}|\left(E^{\prime}+m^{\prime}\right)\mp|\bm{p}^{\prime}|\left(E+m\right)\right)\left(1\pm\hat{p}^{i}\hat{p}^{\prime\,i}\right)$
$\displaystyle
F^{\prime}_{\mp}\left({{\Gamma^{{4}}}{}},{\gamma^{i}\gamma^{5}}\right)$
$\displaystyle=-\left(\left(E+m\right)\left(E^{\prime}+m^{\prime}\right)\pm|\bm{p}||\bm{p}^{\prime}|\right)\left(\pm\epsilon^{ijk}\hat{p}^{j}\hat{p}^{\prime\,k}+{i}\left(\hat{p}^{i}\pm\hat{p}^{\prime\,i}\right)\right)$
$\displaystyle
F^{\prime}_{\mp}\left({{\Gamma^{{4}}}{}},{\gamma^{4}\gamma^{5}}\right)$
$\displaystyle=\left(|\bm{p}|\left(E^{\prime}+m^{\prime}\right)\pm|\bm{p}^{\prime}|\left(E+m\right)\right)\left(1\pm\hat{p}^{i}\hat{p}^{\prime\,i}\right)$
$\displaystyle F^{\prime}_{\mp}\left({{\Gamma^{{4}}}{}},{\sigma^{4i}}\right)$
$\displaystyle={i}\left(|\bm{p}|\left(E^{\prime}+m^{\prime}\right)\mp|\bm{p}^{\prime}|\left(E+m\right)\right)\left(\pm\epsilon^{ijk}\hat{p}^{j}\hat{p}^{\prime\,k}+{i}\left(\hat{p}^{i}\pm\hat{p}^{\prime\,i}\right)\right)$
$\displaystyle F^{\prime}_{\mp}\left({{\Gamma^{{4}}}{}},{\sigma^{ij}}\right)$
$\displaystyle=-\left(\left(E+m\right)\left(E^{\prime}+m^{\prime}\right)\mp|\bm{p}||\bm{p}^{\prime}|\right)\left(\epsilon^{ijk}\left(\hat{p}^{k}\pm\hat{p}^{\prime\,k}\right)\mp{i}\left(\hat{p}^{i}\hat{p}^{\prime\,j}-\hat{p}^{j}\hat{p}^{\prime\,i}\right)\right)$
$\displaystyle
F^{\prime}_{\mp}\left({{\Gamma^{{k}}}{}},{\mathbb{I}{{}}}\right)$
$\displaystyle={-i}\left(\left(E+m\right)\left(E^{\prime}+m^{\prime}\right)\mp|\bm{p}||\bm{p}^{\prime}|\right)\left(\pm\epsilon^{ijk}\hat{p}^{j}\hat{p}^{\prime\,k}-{i}\left(\hat{p}^{i}\pm\hat{p}^{\prime\,i}\right)\right)$
$\displaystyle F^{\prime}_{\mp}\left({{\Gamma^{{k}}}{}},{\gamma^{j}}\right)$
$\displaystyle={i}\left(|\bm{p}|\left(E^{\prime}+m^{\prime}\right)\pm|\bm{p}^{\prime}|\left(E+m\right)\right)$
$\displaystyle\qquad\times\left(\pm\hat{p}^{i}\hat{p}^{\prime\,j}\pm\hat{p}^{j}\hat{p}^{\prime\,i}-{i}\epsilon^{ijk}\left(\hat{p}^{k}\mp\hat{p}^{\prime\,k}\right)+\delta^{ij}\left(1\mp\hat{p}^{k}\hat{p}^{\prime\,k}\right)\right)$
$\displaystyle F^{\prime}_{\mp}\left({{\Gamma^{{k}}}{}},{\gamma^{4}}\right)$
$\displaystyle={-i}\left(\left(E+m\right)\left(E^{\prime}+m^{\prime}\right)\pm|\bm{p}||\bm{p}^{\prime}|\right)\left(\pm\epsilon^{ijk}\hat{p}^{j}\hat{p}^{\prime\,k}-{i}\left(\hat{p}^{i}\pm\hat{p}^{\prime\,i}\right)\right)$
$\displaystyle F^{\prime}_{\mp}\left({{\Gamma^{{k}}}{}},{\gamma^{5}}\right)$
$\displaystyle={-i}\left(|\bm{p}|\left(E^{\prime}+m^{\prime}\right)\mp|\bm{p}^{\prime}|\left(E+m\right)\right)\left(\pm\epsilon^{ijk}\hat{p}^{j}\hat{p}^{\prime\,k}-{i}\left(\hat{p}^{i}\pm\hat{p}^{\prime\,i}\right)\right)$
$\displaystyle
F^{\prime}_{\mp}\left({{\Gamma^{{k}}}{}},{\gamma^{j}\gamma^{5}}\right)$
$\displaystyle={i}\left(\left(E+m\right)\left(E^{\prime}+m^{\prime}\right)\pm|\bm{p}||\bm{p}^{\prime}|\right)$
$\displaystyle\qquad\times\left(\pm\hat{p}^{i}\hat{p}^{\prime\,j}\pm\hat{p}^{j}\hat{p}^{\prime\,i}-{i}\epsilon^{ijk}\left(\hat{p}^{k}\mp\hat{p}^{\prime\,k}\right)+\delta^{ij}\left(1\mp\hat{p}^{k}\hat{p}^{\prime\,k}\right)\right)$
$\displaystyle
F^{\prime}_{\mp}\left({{\Gamma^{{k}}}{}},{\gamma^{4}\gamma^{5}}\right)$
$\displaystyle={-i}\left(|\bm{p}|\left(E^{\prime}+m^{\prime}\right)\pm|\bm{p}^{\prime}|\left(E+m\right)\right)\left(\pm\epsilon^{ijk}\hat{p}^{j}\hat{p}^{\prime\,k}-{i}\left(\hat{p}^{i}\pm\hat{p}^{\prime\,i}\right)\right)$
$\displaystyle F^{\prime}_{\mp}\left({{\Gamma^{{k}}}{}},{\sigma^{4j}}\right)$
$\displaystyle=\left(|\bm{p}|\left(E^{\prime}+m^{\prime}\right)\mp|\bm{p}^{\prime}|\left(E+m\right)\right)$
$\displaystyle\qquad\times\left(\pm\hat{p}^{i}\hat{p}^{\prime\,j}\pm\hat{p}^{j}\hat{p}^{\prime\,i}-{i}\epsilon^{ijk}\left(\hat{p}^{k}\mp\hat{p}^{\prime\,k}\right)+\delta^{ij}\left(1\mp\hat{p}^{k}\hat{p}^{\prime\,k}\right)\right)$
$\displaystyle F^{\prime}_{\mp}\left({{\Gamma^{{k}}}{}},{\sigma^{jk}}\right)$
$\displaystyle=\left(\left(E+m\right)\left(E^{\prime}+m^{\prime}\right)\mp|\bm{p}||\bm{p}^{\prime}|\right)\epsilon^{jkl}$
$\displaystyle\qquad\times\left(\pm\hat{p}^{i}\hat{p}^{\prime\,l}\pm\hat{p}^{l}\hat{p}^{\prime\,i}-{i}\epsilon^{ilm}\left(\hat{p}^{m}\mp\hat{p}^{\prime\,m}\right)+\delta^{il}\left(1\mp\hat{p}^{m}\hat{p}^{\prime\,m}\right)\right)$

###### Acknowledgements.

We thank the PACS-CS Collaboration for making their configurations available
via the International Lattice Data Grid (ILDG). This research was undertaken
with the assistance of resources from the Phoenix HPC service at the
University of Adelaide, the National Computational Infrastructure (NCI), which
is supported by the Australian Government, and by resources provided by the
Pawsey Supercomputing Centre with funding from the Australian Government and
the Government of Western Australia. These resources were provided through the
National Computational Merit Allocation Scheme and the University of Adelaide
partner share. FS is supported by a Ramsay Fellowship from the University of
Adelaide. WK was supported by the Pawsey Supercomputing Centre through the
Pawsey Centre for Extreme Scale Readiness (PaCER) program. This research is
supported by the Australian Research Council through grants no. DP140103067,
DP150103164, LE160100051, LE190100021, DP190102215, and DP210103706.

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