# Understanding the nature of the $\mathbf{\Delta(1600)}$ resonance

Liam Hockley<EMAIL_ADDRESS>Curtis Abell  Derek Leinweber
Anthony Thomas  ARC Special Research Centre for the Subatomic Structure of
Matter (CSSM), Department of Physics, The University of Adelaide, SA, 5005,
Australia.

###### Abstract

We present a coupled-channel analysis of the $J^{P}=3/2^{+}\ \Delta$-baryon
spectrum, based in the framework of Hamiltonian Effective Field Theory (HEFT).
We construct a Hamiltonian which mixes quark model-like single-particle states
and two-particle meson-baryon channels, and constrain this via experimentally
measured $\pi N\to\pi N$ scattering observables. In the same vein as Lüscher’s
approach, we then connect this infinite-volume inspired Hamiltonian with
finite-volume lattice QCD results. Drawing on lattice correlation-matrix
eigenvectors identifying the $1s$ and $2s$ states in the finite-volume
$\Delta(3/2^{+})$ spectrum, and utilising the HEFT eigenvectors describing the
composition of the energy eigenstates, we resolve the structure of these
states and their relation to the $\Delta(1600)$ resonance. We find the
dominant contributions to this resonance come from strong rescattering in the
$\pi N$ and $\pi\Delta$ channels. This contrasts the long-held view of a
dominant quark model-like core for the $\Delta(1600)$. Further discussion of
other contemporary lattice results for the $\Delta$ spectrum and $\pi N$
scattering states is also presented.

††preprint: APS/123-QED

##  I Introduction

The strong nuclear force presents many unique challenges in particle physics,
one of which being short-lived states known as resonances. While the lowest-
lying hadrons can be arranged based on isospin, charge and strangeness
symmetries in a quark model for mesons and baryons [1, 2], the complex
dynamics of resonances makes modelling and analysis of these excited states
non-trivial.

The foremost example of this difficulty is the $N^{*}(1440)$ Roper resonance
[3, 4, 5], with similar concerns lying in the $\Delta$ baryon spectrum’s
$\Delta(1600)$ state [6, 7]. Essentially, the issue with these states is that
they have masses which disagree with the most simple quark model predictions
without any fine-tuning. For example, a naive quark model predicts that states
in the $\Delta(J=3/2)$ spectrum should alternate in the signs of their
internal parities as one climbs in energy. However, considering the
experimentally observed resonances, one immediately notices the apparent
mismatch in the ordering of states since two positive parity states (the
$\Delta(1232)$ and $\Delta(1600)$ resonances) lie below the first odd parity
state, the $\Delta(1700)$. This tension between theory and experiment
represents a fundamental gap in the collective understanding of QCD.

The only currently tenable method of resolving the properties of QCD exactly
in the hadronic region is lattice QCD. Lattice QCD is well established as a
non-perturbative framework within which many physical quantities can be
extracted such as particle masses, decay constants, form factors, etc [8].
Studying scattering processes and resonances has historically proven
challenging, with no way of instituting asymptotic scattering states on the
finite-volume of the lattice.

A breakthrough methodology was first realised by Lüscher [9, 10, 11], allowing
for scattering properties to be extracted from finite-volume spectra.
Initially derived for the case of single-channel 2-body decays, the formalism
has been extended for dealing with cases such as that of multiple channels
[12] as well as 3-body decays [13, 14, 15]. The development of this approach
towards ever-more generalised systems is ongoing. Useful reviews of the
Lüscher approach and other methods for studying hadron resonances are provided
in Refs. [16, 17].

While the Lüscher formalism provides rigourous quantisation conditions
relating finite-volume spectra to infinite-volume scattering observables, for
high energy resonances, the opening of multiple relevant channels and the
proximity of various 2- and 3-body thresholds rapidly complicates the
calculations. More recently, Hamiltonian Effective Field Theory (HEFT) has
emerged as a Lüscher based formalism which seamlessly incorporates multiple
scattering channels and 3-quark bare states into its description of resonances
[18]. Using infinite-volume scattering data, one is able to place initial
constraints on the Hamiltonian and check for validity through extraction of
the T-matrix poles. HEFT acts as an alternative for the Lüscher formalism in
the sense that this infinite-volume inspired Hamiltonian can then be confined
to a finite-size box and thus the energy eigenstates and eigenvalues become
discretised. This allows for direct comparison with results from lattice QCD
through the Hamiltonian eigenvalues, with additional structural information
coming from the eigenvectors.

The combination of lattice QCD and HEFT has been shown to yield insightful
results for the historically enigmatic Roper resonance $N^{*}(1440)$ [19, 20,
21, 22, 23, 24, 25, 26, 27] and the $\Lambda(1405)$ [28, 29, 30, 31], allowing
for new understanding of these complicated resonances. Similarly, an initial
case study of the $\Delta(1600)$ [32] suggested there is more to this
resonance than a predominant 3-quark radial excitation of the $\Delta(1232)$,
contrasting a mainstay in the literature as suggested, for example, by work
using Dyson-Schwinger models [5, 6].

The current work looks to build on recent lattice QCD results for the
$\Delta$-spectrum which identify the energy of the $2s$ excitation of the
ground state at $\sim 2.15$ GeV [33]. This is remarkably reminiscent of the
findings for the nucleon spectrum where the first radial excitation of the
ground state was found to lie at approximately $1.9$ GeV, about $500$ MeV
above the physical Roper resonance [22, 23, 20, 21, 24, 25]. The key results
of the recent lattice QCD $\Delta$ spectrum [33] are given in Fig. 1. The
current study seeks to use HEFT to understand this $2s$ state in the context
of the $\Delta(1600)$ resonance.

Figure 1: A summary of key lattice QCD results from Ref. [33]. Importantly,
the first excitation in the CSSM results was identified as a $2s$ excitation
of the ground state based on nodes of the wave function used to excite the
state. This state is far above the energy region one typically associates with
the $\Delta(1600)$ resonance. Results from other groups using 3-quark
operators are given for comparison. Discussion of the HSC [34] and Khan et al.
[35] results is given in Section V.

This paper is organised in the following way. Section II describes the HEFT
formalism and how it can be applied for an arbitrary number of single- and
two-particle basis states. We outline how one can constrain the various
Hamiltonian parameters by fitting phase shifts and inelasticities to $\pi N$
scattering data, and then extend the Hamiltonian to finite volumes at
unphysical quark masses. This allows for direct contact with the lattice
results of Ref. [33] through the Hamiltonian eigenvalues, with additional
information about quark-model-like and meson-baryon-like components coming
from the eigenvectors. In Section III this approach is taken to investigate
the low energy region around the mass of the $\Delta(1232)$ resonance before
being extended in Section IV to include additional bare states and channels
which couple to the $\Delta(1600)$ resonance. We obtain a model which
simultaneously describes both the infinite-volume phase shifts and
inelasticities, and the finite-volume lattice results of Ref. [33]. The key
information about the structure of the states in the finite-volume spectrum
comes from the Hamiltonian eigenvectors, and these are discussed in detail.

We show the diverse applicability of HEFT in Section V by recomputing the
finite-volume spectrum at a range of lattice sizes and comparing with recent
$\pi N$ energy level results from Refs. [36, 37, 38], as well as other lattice
QCD results using single-hadron operators in Refs. [34, 35]. This affords
another way of constraining our Hamiltonian and understanding the lattice
results of other groups. This also demonstrates the powerful insight offered
by HEFT in guiding future lattice QCD calculations.

##  II Hamiltonian Effective Field Theory

Hamiltonian Effective Field Theory (HEFT) has been shown to be a useful tool
in exploring the nucleon spectrum to understand the nature of the Roper
resonance [39, 26, 30, 27] and more recently in the $\Delta$-baryon spectrum
[32, 40]. Here, we briefly review the formalism for constructing the
Hamiltonian and constraining it using infinite-volume scattering data. We then
present the finite-volume, unphysical-quark-mass extension of the model and
describe how this can be used to make direct contact with results from lattice
QCD.

###  II.1 Hamiltonian Model

In general, the Hamiltonian for an interacting system can be decomposed into
two pieces

$H=H_{0}+H_{I}\,,$ (1)

where $H_{0}$ is the free (non-interacting) part, and $H_{I}$ governs the non-
trivial dynamics of the system through various interactions between its basis
states. In our HEFT formalism we consider two types of basis states: single-
particle bare baryon states (one can interpret these as being quark model-like
states [41, 42, 43]) and two-particle meson-baryon states. The two-particle
basis states are non-interacting.

We introduce the general notation $\ket{B_{0}}$ to describe our bare basis
states, and $\ket*{\alpha(\bm{k})}$ to denote our meson-baryon pairs in the
center of mass frame with back-to-back 3-momentum $\bm{k}$. The free
Hamiltonian is then given by

$\displaystyle H_{0}$
$\displaystyle=\sum_{B_{0}}\,\ket{B_{0}}\,m_{B_{0}}\,\bra{B_{0}}+\sum_{\alpha}\int
d^{3}k$
$\displaystyle\times\ket*{\alpha(\bm{k})}\quantity{\sqrt{m_{\alpha_{B}}^{2}+k^{2}}+\sqrt{m_{\alpha_{M}}^{2}+k^{2}}}\,\bra*{\alpha(\bm{k})}$
(2)

where $m_{B_{0}}$ is the mass of the bare state $B_{0}$, $m_{\alpha_{B}}$ is
the mass of the baryon in the two-particle channel $\alpha$, and likewise
$m_{\alpha_{M}}$ is the mass of the meson in channel $\alpha$.

Including interactions in our model, we decompose the interacting Hamiltonian
into two parts

$H_{I}=g+v\,,$ (3)

where $g$ describes interactions between bare states and two-particle basis
states, and $v$ describes the coupling of the channels through two-particle
states interacting with other two-particle states. These are represented
graphically by the processes shown in Fig. 2.

Figure 2: Interaction vertices for bare to two-particle state processes (top),
and 2-to-2 processes (bottom). Here $B_{0}$ is a bare state while $\alpha$
denotes a channel consisting of meson $\alpha_{M}$ and baryon $\alpha_{B}$.

Mathematically, we describe these vertex interactions by

$g=\sum_{\alpha,\,B_{0}}\int
d^{3}{k}\,\quantity{\,\ket*{\alpha(\bm{k})}\,G^{\dagger}_{\alpha,B_{0}}(k)\,\bra*{B_{0}}+h.c.\,}\,,$
(4)

where $h.c.$ denotes the Hermitian conjugate of the first term, and

$v=\sum_{\alpha,\beta}\int
d^{3}{k}\,d^{3}{k}^{\prime}\,\ket*{\alpha(\bm{k})}\,V_{\alpha\beta}(k,k^{\prime})\,\bra*{\beta(\bm{k}^{\prime})}\,.$
(5)

The momentum-dependent forms of the various $G_{\alpha,B_{0}}(k)$ are
motivated by $\chi$PT. We use the form

$\tilde{G}_{B_{0},\alpha}(k)=\frac{g^{B_{0}}_{\alpha}}{2\pi}\,\quantity(\frac{k}{f_{\pi}})^{l_{\alpha}}\,\frac{1}{\sqrt{\omega_{\pi}(k)}}\,,$
(6)

where the tilde denotes an unregulated function, and we’ve introduced some as
yet unknown coupling strength $g^{B_{0}}_{\alpha}$, as well as the pion decay
constant $f_{\pi}=92.4$ MeV. The angular momentum in channel $\alpha$ is
denoted $l_{\alpha}$. In general, this function will diverge as we perform
integrals over all values of 3-momentum in the two-particle states, so we
adopt a regularisation scheme to keep the results convergent and finite.

To regulate the so-called “1-to-2” interactions contained in the $g$ term of
$H_{I}$, we introduce a regulator

$u^{g}_{l_{\alpha}}(k,\Lambda_{B_{0},\alpha})=\frac{1}{\quantity(1+k^{2}/\Lambda^{2}_{B_{0},\alpha})^{(l_{\alpha}+3)/2}}\,.$
(7)

The exponent of $(l_{\alpha}+3)/2$ is motivated by the overall dimensionful
factor of $k^{{l_{\alpha}}+2}$ in Eq. (4) after converting the integral to
spherical coordinates.

For the case of a channel which is in $p$-wave, this has the usual form of a
dipole with $(l_{\alpha}+3)/2=2$, while for $f$-wave, we get
$(l_{\alpha}+3)/2=3$, in other words, a tripole form factor. These two cases
will be the relevant angular momenta for this study.

We can thus compute a fully regularised and finite contribution for each
interaction by taking

$G_{B_{0},\alpha}(k)=\tilde{G}_{B_{0},\alpha}(k)\,u^{g}_{l_{\alpha}}(k,\Lambda_{B_{0},\alpha})\,.$
(8)

Similarly, we define couplings for the 2-to-2 interactions by the separable
potentials

$\tilde{V}_{\alpha\beta}(k,k^{\prime})=\frac{v_{\alpha\beta}}{4\pi^{2}}\,\quantity(\frac{k}{f_{\pi}})^{\l_{\alpha}}\,\quantity(\frac{k^{\prime}}{f_{\pi}})^{l_{\beta}}\,\frac{1}{\omega_{\pi}(k)}\,\frac{1}{\omega_{\pi}(k^{\prime})}\,,$
(9)

where we’ve introduced a parameter $v_{\alpha\beta}$ which determines how
strongly the channels $\alpha$ and $\beta$ couple to one another.

To regulate these interactions, we introduce regulators $u^{v}_{l_{\alpha}}$.
These are of the same form as Eq. (7)

$u^{v}_{l_{\alpha}}(k,\Lambda^{v}_{\alpha})=\frac{1}{\quantity(1+k^{2}/\quantity(\Lambda^{v}_{\alpha})^{2})^{(l_{\alpha}+3)/2}}\,,$
(10)

but we make the distinction here that the regulator parameters
$\Lambda^{v}_{\alpha}$ may in principle be different from the
$\Lambda_{B_{0},\alpha}$.

Hence, we can construct fully regularised 2-to-2 couplings by

$V_{\alpha\beta}(k,k^{\prime})=\tilde{V}_{\alpha\beta}(k,k^{\prime})\,u^{v}_{l_{\alpha}}(k,\Lambda^{v}_{\alpha})\,u^{v}_{l_{\beta}}(k^{\prime},\Lambda^{v}_{\beta})\,.$
(11)

In this study, we find dipole and tripole regulators to be sufficient to
describe the experimental data. Previous HEFT studies have explored
alternative regulators [32].

###  II.2 Constraints from Scattering Observables

In order to constrain the various parameters entering into the model, we look
to construct the scattering T-matrix and from this, extract channel phase
shifts and inelasticities. This is achieved by first constructing a coupled-
channel potential

$V^{\text{coup}}_{\alpha\beta}(k,k^{\prime},E)=\sum_{B_{0}}\frac{G^{B_{0}\dagger}_{\alpha}(k)G^{B_{0}}_{\beta}(k^{\prime})}{E-m_{B_{0}}}+V_{\alpha\beta}(k,k^{\prime})\,,$
(12)

and then solving a 3-dimensional reduction of the Bethe-Salpeter equations
using this potential,

$\displaystyle T_{\alpha\beta}(k,k^{\prime};E)$
$\displaystyle=V^{\text{coup}}_{\alpha\beta}(k,k^{\prime};E)+\sum_{\gamma}\int
dq\,q^{2}\times$ $\displaystyle
V^{\text{coup}}_{\alpha\gamma}(k,q,E)\frac{1}{E-\omega_{\gamma}(q)+i\epsilon}T_{\gamma\beta}(q,k^{\prime};E)\,.$
(13)

The corresponding S-matrix elements are obtained from

$\displaystyle S_{\alpha\beta}(E)$
$\displaystyle=\delta_{\alpha\beta}-2i\pi\sqrt{\rho_{\alpha}\rho_{\beta}}\,T_{\alpha\beta}(k_{\text{on},\alpha},\,k_{\text{on},\beta};\,E)$
(14)

where the functions $\rho_{\alpha}$ are defined by

$\rho_{\alpha}=\frac{\sqrt{k^{2}_{\text{on},\,\alpha}+m^{2}_{\alpha_{M}}}\,\sqrt{k^{2}_{\text{on},\,\alpha}+m^{2}_{\alpha_{B}}}}{E}\,k_{\text{on},\,\alpha}\,.$
(15)

Here, $k_{\text{on},\,\alpha}$ is the on-shell back-to-back momentum carried
by the meson and baryon in the two-particle channel $\alpha$.

The diagonal elements of the S-matrix are readily related to the scattering
phase shift $\delta_{\alpha}$ and inelasticity $\eta_{\alpha}$ by

$S_{\alpha\alpha}(E)=\eta_{\alpha}\,\exp(2i\delta_{\alpha})\,.$ (16)

For the case of $\pi N\to\pi N$ scattering, which is of most importance to us,
these observables have been derived using partial wave analysis methods
applied to experimental scattering data. The results are publicly available on
the George Washington University (GWU) SAID partial wave analysis database
[44]. Hence, one can perform fits to this data and so constrain the
Hamiltonian model.

An additional point of comparison between our results and those reported by
experiment lies in the extraction of poles of the $T$-matrix. These are
complex energies $E_{\text{pole}}$ for which
$T_{\alpha\beta}(k,k^{\prime};E_{\text{pole}})^{-1}=0$.

###  II.3 Finite-volume Hamiltonian

In order to compare our Hamiltonian model with lattice QCD results, we need to
consider that lattice QCD results are calculated on a finite volume at
unphysical quark masses. Neither of these facets has been incorporated into
our infinite-volume Hamiltonian.

To address the first of these, consider taking our Hamiltonian and
constraining it to describe a finite-volume system with periodic boundary
conditions. The basis states interact within the volume to create scattering
states. The primary difference this induces is that our momenta obey the
quantisation condition

$\bm{k_{n}}=\frac{2\pi}{L}\,\bm{n}\,,$ (17)

where $\bm{n}\in\mathbb{Z}_{3}$ and $L$ describes the length of a side of the
cubic volume. Thus, the continuous integrals over momenta encountered
throughout Section II.1 are converted to discrete sums over momentum
$3$-indices $\bm{n}$ via

$4\pi\int k^{2}dk=\int
d^{3}k\to\sum_{\bm{n}\in\mathbb{Z}^{3}}\quantity(\frac{2\pi}{L})^{3}\,.$ (18)

The entries of $\bm{n}$ can take on any integer value, however when
constructing the $\pi N$ and $\pi\Delta$ states required for studying the
$\Delta$, one ignores the pure $S$-wave contribution ($\bm{n}=\bm{0}$).

We can reduce this 3-dimensional sum down to a 1-dimensional sum by
identifying degenerate momenta magnitudes

$k_{n}=\frac{2\pi}{L}\sqrt{n}\,,$ (19)

where

$n=n_{x}^{2}+n_{y}^{2}+n_{z}^{2}\,.$ (20)

The degeneracy of the magnitudes is captured by introducing the function
$C_{3}(n)$ which counts the number of ways to produce a given integer $n$ from
the sum of three squared integers. Some examples are $C_{3}(2)~{}=~{}12$ since
there are 12 ways to sum three squared integers and obtain a result of 2
taking into account the four allowed combinations of $\pm 1$ and the three
allowed positions for 0. As another example, $C_{3}(7)~{}=~{}0$, since $n=7$
is not a solution of Eq. (20) $\text{for any }n_{i}\in\mathbb{Z}$.

Finally then, the $3$-dimensional sums in Eq. (18) become

$\sum_{\bm{n}\in\mathbb{Z}^{3}}\quantity(\frac{2\pi}{L})^{3}\to\sum_{n\in\mathbb{Z}}\quantity(\frac{2\pi}{L})^{3}\,C_{3}(n)\,.$
(21)

Thus we need only label our momentum states by a single index $n$ and all
3-dimensional symmetries are carried in the degeneracy factors $C_{3}(n)$.

The discussion above leads to various finite-volume corrections to our
couplings and potentials $G_{B_{0},\alpha}$ and $V_{\alpha\beta}$. Before
elaborating on this however, we note that our Hamiltonian is regularised by
the parameters $\Lambda_{B_{0},\alpha},\ \Lambda^{v}_{\alpha}$ which allow for
the introduction of momentum cutoffs. Considering the coupling suppression of
high-momentum Hamiltonian basis states, we can impose a maximum magnitude
allowed for the momentum and thus a maximum number of basis states in the
Hamiltonian. Importantly then, the regulator parameters influence the size of
the Hamiltonian matrix; note that the choice of $L$ also plays a role since
with larger volumes we can fit more momentum states in the box. For fixed
values of $L$, we can thus determine a maximum momentum in the box by imposing
a minimum value on the regulators $u$, for example for a generic regulator
with $l_{\alpha}=1$

$u(k,\Lambda)=\frac{1}{\quantity(1+k^{2}/\Lambda^{2})^{2}}\,.$ (22)

Rearranging for the maximum value of momentum $k_{\text{max}}$ we have

$k_{\text{max}}=\Lambda\sqrt{u^{-1/2}_{\text{min}}-1}\,.$ (23)

Setting $u_{\text{min}}=0.01$ as in previous HEFT analyses [40, 32, 39, 26,
30, 27] this fixes a maximum momentum index by

$n_{\text{max}}=\quantity(\frac{k_{\text{max}}L}{2\pi})^{2}\,.$ (24)

In principle this maximum index can be computed for each channel, taking into
account the different values of angular momenta, and so determine a momentum
cutoff for each channel in the box. To keep the Hamiltonian matrix in a
suitable form we work with a single value for $k_{\text{max}}$.

Thus we see explicitly that the two-particle states (of which we had an
infinite number previously), are now of the form $\ket{\alpha(k_{n})}$ for
some single index $n$ which runs from 1 to $n_{\text{max}}$. Hence we are now
in a position to consider our Hamiltonian as a matrix, and solve for its
eigenvalues and eigenvectors.

The final consideration for the finite-volume Hamiltonian is that the coupling
functions $G_{B_{0},\,\alpha}$ and $V_{\alpha\beta}$ receive finite-volume
factors associated with momentum quantisation and cubic symmetry in Eqs. (18)
and (21) respectively. The finite-volume coupling functions $\bar{G}$ and
$\bar{V}$ are

$\bar{G}^{B_{0}}_{\alpha}(k_{n})=\sqrt{\frac{C_{3}(n)}{4\pi}}\quantity(\frac{2\pi}{L})^{3/2}G^{B_{0}}_{\alpha}(k_{n})\,,$
(25)

$\bar{V}_{\alpha\beta}(k_{n},k_{m})=\sqrt{\frac{C_{3}(n)}{4\pi}}\sqrt{\frac{C_{3}(m)}{4\pi}}\quantity(\frac{2\pi}{L})^{3}V_{\alpha\beta}(k_{n},k_{m})\,.$
(26)

These finite-volume corrected functions replace their corresponding infinite-
volume counterparts in Eqs. (4) and (5) in the calculation of the Hamiltonian
matrix.

###  II.4 Hamiltonian Matrix

Following the preceding discussion, we can simply fill in the entries of the
Hamiltonian matrix, now using the finite-volume coupling functions of Eqs.
(25) and (26). For ease of presentation, we will only show the specific case
of $n_{b}=2$ bare basis states and $n_{c}=2$ species of meson-baryon basis
states. We adopt the labels $\Delta_{1}$ and $\Delta_{2}$ for the bare states,
with the channel labels being $\alpha\in\\{\pi N,\,\pi\Delta\\}$, though the
following is easily generalised to any number of channels and bare states.
Recalling $H=H_{0}+H_{I}$, then the Hamiltonian matrix is given by

$H_{0}=\text{diag}\quantity(m_{\Delta_{1}},\,m_{\Delta_{2}},\,\omega_{\pi
N}(k_{1}),\,\omega_{\pi\Delta}(k_{1}),\,\dots,\omega_{\pi
N}(k_{n_{\text{max}}}),\,\omega_{\pi\Delta}(k_{n_{\text{max}}}))\,,$ (27)

$H_{I}=\begin{pmatrix}0&0&\bar{G}^{\Delta_{1}}_{\pi
N}(k_{1})&\bar{G}^{\Delta_{1}}_{\pi\Delta}(k_{1})&\bar{G}^{\Delta_{1}}_{\pi
N}(k_{2})&\bar{G}^{\Delta_{1}}_{\pi\Delta}(k_{2})&\dots&\bar{G}^{\Delta_{1}}_{\pi
N}(k_{n_{\text{max}}})&\bar{G}^{\Delta_{1}}_{\pi\Delta}(k_{n_{\text{max}}})\\\
0&0&\bar{G}^{\Delta_{2}}_{\pi
N}(k_{1})&\bar{G}^{\Delta_{2}}_{\pi\Delta}(k_{1})&\bar{G}^{\Delta_{2}}_{\pi
N}(k_{2})&\bar{G}^{\Delta_{2}}_{\pi\Delta}(k_{2})&\dots&\bar{G}^{\Delta_{2}}_{\pi
N}(k_{n_{\text{max}}})&\bar{G}^{\Delta_{2}}_{\pi\Delta}(k_{n_{\text{max}}})\\\
\bar{G}^{\Delta_{1}}_{\pi N}(k_{1})&\bar{G}^{\Delta_{2}}_{\pi N}(k_{1})\\\
\bar{G}^{\Delta_{1}}_{\pi\Delta}(k_{1})&\bar{G}^{\Delta_{2}}_{\pi\Delta}(k_{1})\\\
\bar{G}^{\Delta_{1}}_{\pi N}(k_{2})&\bar{G}^{\Delta_{2}}_{\pi N}(k_{2})\\\
\bar{G}^{\Delta_{1}}_{\pi\Delta}(k_{2})&\bar{G}^{\Delta_{2}}_{\pi\Delta}(k_{2})&&&&\tilde{\tilde{V}}\\\
\vdots&\vdots\\\ \bar{G}^{\Delta_{1}}_{\pi
N}(k_{n_{\text{max}}})&\bar{G}^{\Delta_{2}}_{\pi N}(k_{n_{\text{max}}})\\\
\bar{G}^{\Delta_{1}}_{\pi\Delta}(k_{n_{\text{max}}})&\bar{G}^{\Delta_{2}}_{\pi\Delta}(k_{n_{\text{max}}})\\\
\end{pmatrix}\,,$ (28)

where the $(n_{c}\times n_{\text{max}})^{2}$ symmetric matrix
$\tilde{\tilde{V}}$ is given by

$\tilde{\tilde{V}}=\begin{pmatrix}\bar{V}_{\pi N,\pi
N}(k_{1},k_{1})&\bar{V}_{\pi N,\pi\Delta}(k_{1},k_{1})&\bar{V}_{\pi N,\pi
N}(k_{1},k_{2})&\bar{V}_{\pi N,\pi\Delta}(k_{1},k_{2})&\dots&\bar{V}_{\pi
N,\pi\Delta}(k_{1},k_{n_{\text{max}}})\\\ \bar{V}_{\pi\Delta,\pi
N}(k_{1},k_{1})&\bar{V}_{\pi\Delta,\pi\Delta}(k_{1},k_{1})&\bar{V}_{\pi\Delta,\pi
N}(k_{1},k_{2})&\bar{V}_{\pi\Delta,\pi\Delta}(k_{1},k_{2})&\dots&\bar{V}_{\pi\Delta,\pi\Delta}(k_{1},k_{n_{\text{max}}})\\\
\bar{V}_{\pi N,\pi N}(k_{2},k_{1})&\bar{V}_{\pi
N,\pi\Delta}(k_{2},k_{1})&\bar{V}_{\pi N,\pi N}(k_{2},k_{2})&\bar{V}_{\pi
N,\pi\Delta}(k_{2},k_{2})&\dots&\bar{V}_{\pi
N,\pi\Delta}(k_{2},k_{n_{\text{max}}})\\\ \bar{V}_{\pi\Delta,\pi
N}(k_{2},k_{1})&\bar{V}_{\pi\Delta,\pi\Delta}(k_{2},k_{1})&\bar{V}_{\pi\Delta,\pi
N}(k_{2},k_{2})&\bar{V}_{\pi\Delta,\pi\Delta}(k_{2},k_{2})&\dots&\bar{V}_{\pi\Delta,\pi\Delta}(k_{2},k_{n_{\text{max}}})\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\ \bar{V}_{\pi N,\pi
N}(k_{n_{\text{max}}},k_{1})&\bar{V}_{\pi
N,\pi\Delta}(k_{n_{\text{max}}},k_{1})&\bar{V}_{\pi N,\pi
N}(k_{n_{\text{max}}},k_{2})&\bar{V}_{\pi
N,\pi\Delta}(k_{n_{\text{max}}},k_{2})&\dots&\bar{V}_{\pi
N,\pi\Delta}(k_{n_{\text{max}}},k_{n_{\text{max}}})\\\ \bar{V}_{\pi\Delta,\pi
N}(k_{n_{\text{max}}},k_{1})&\bar{V}_{\pi\Delta,\pi\Delta}(k_{n_{\text{max}}},k_{1})&\bar{V}_{\pi\Delta,\pi
N}(k_{n_{\text{max}}},k_{2})&\bar{V}_{\pi\Delta,\pi\Delta}(k_{n_{\text{max}}},k_{2})&\dots&\bar{V}_{\pi\Delta,\pi\Delta}(k_{n_{\text{max}}},k_{n_{\text{max}}})\par\end{pmatrix}\,.$
(29)

With the matrices above defined we can solve this symmetric Hamiltonian matrix
for its eigenvalues and eigenvectors. The final step required to compare with
results from lattice QCD is to extend our Hamiltonian model to unphysical
quark/pion masses.

###  II.5 Pion Mass Dependence

Thus far, all calculations have been carried out at the physical point, where
the hadron masses take their measured values. As noted in the introduction,
state-of-the-art lattice QCD calculations can now be carried out at the
physical quark-mass point. Nevertheless, comparison with lattice QCD results
computed over a broad range of quark masses introduces strong constraints on
the Hamiltonian vital to obtaining the composition of states in the
Hamiltonian eigenvectors. To extend our calculations at the physical point
into the heavier quark-mass regime, we seek a simple interpolation of the
finite-volume PACS-CS hadron masses used in Ref. [33]. Using the linear
dependence of $m_{\pi}^{2}$ on the quark mass, we find a simple linear
relationship between the baryon masses and $m_{\pi}^{2}$ to be adequate in
reproducing results from lattice QCD. While one anticipates important chiral
curvature, this is suppressed in the finite volume and at pion masses far from
the physical point where hadron masses follow a linear trend in $m_{\pi}^{2}$
[45, 46]. Thus, for the $N$ and $\Delta$ masses appearing in the two-particle
basis-state channels, we consider

$m_{B}(m^{2}_{\pi})=m^{*}_{B}+\alpha_{B}\,\quantity(m_{\pi}^{2}-m_{\pi}|_{\text{phys}}^{2})\,,$
(30)

where $m^{*}_{B}$ is the interpolated finite-volume mass of baryon $B$ at the
physical quark/pion mass point $m_{\pi}|_{\text{phys}}$. This and the slope
parameter $\alpha_{B}$ are obtained in a fit to lattice QCD results from the
PACS-CS collaboration [46]. Note that we allow for finite-volume effects by
allowing $m^{*}_{B}\neq m_{B}|_{\text{phys}}$ on the lattice.

One aims to repeat the finite-volume calculation at various choices of pion
mass and so extend to regimes compatible with lattice QCD calculations. This
builds up a finite-volume energy spectrum where each of the energy values is
an eigenvalue of the Hamiltonian at the given pion mass and lattice size. In
addition to interpolating the masses appearing in the two-particle channels,
one also needs to extend the bare masses from the physical point. This time
there is an additional complication as one needs to know which eigenstates of
the HEFT spectrum should be fit to the available lattice QCD results. Thus we
turn our attention to the Hamiltonian eigenvectors describing the composition
of the energy eigenstates.

###  II.6 Eigenvector Analysis

At the outset of this HEFT approach, the number of bare basis states $n_{b}$
is chosen to match the number of 3-quark interpolated lattice states to which
we aim to compare. For example, suppose one is comparing to only a set of
ground state lattice masses, then the number of bare states is set to
$n_{b}=1$. This essentially allows for the case where the bare state
corresponds to the observed lattice state. To see if there really is a
correspondence between the two, we inspect the content of the energy
eigenvectors.

Considering the form of the Hamiltonian in Eqs. (27)-(29), a general
eigenvector can be decomposed in terms of bare basis states and two-particle
basis states of different momenta as

$\ket{E_{n}}=\sum_{i=1}^{n_{b}}b_{n}^{i}\ket{B^{i}_{0}}+\sum_{i=1}^{n_{c}}\sum_{j=1}^{n_{\text{max}}}\,c_{n,j}^{\alpha_{i}}\ket{\alpha_{i}(k_{j})}\,,$
(31)

where, for the energy eigenstate with energy-level index $n$, the $b_{n}^{i}$
denote eigenvector components for each of the bare basis states $B^{i}_{0}$,
and the $c^{\alpha_{i}}_{n,j}$ are the eigenvector components for each two-
particle basis state $\alpha_{i}$ with discrete back-to-back momentum $k_{j}$.

Continuing with the $n_{b}=n_{c}=2$ case as an example, we write

$\displaystyle\ket{E_{n}}$
$\displaystyle=b_{n}^{1}\ket{\Delta_{1}}+b_{n}^{2}\ket{\Delta_{2}}$
$\displaystyle+\sum_{j=1}^{n_{\text{max}}}\,c_{n,j}^{\pi N}\ket{\pi
N(k_{j})}+\sum_{j=1}^{n_{\text{max}}}\,c_{n,j}^{\pi\Delta}\ket{\pi\Delta(k_{j})}\,.$
(32)

One can then consider the overlap of these eigenvectors with each of the basis
states, and in particular the bare basis states $\ket{\Delta_{1}}$ and
$\ket{\Delta_{2}}$. Since the lattice results of Ref. [33] were generated
using 3-quark interpolators, they should have significant overlap with the
bare basis states in our Hamiltonian. Thus we expect the bare basis state
content to be significant for energy eigenvalues near the lattice results and
we tune the parameters of the Hamiltonian to ensure this is the case. We can
test if our Hamiltonian allows this by simply computing the overlap
probabilities

$\absolutevalue{b^{n}_{i}}^{2}=\absolutevalue{\innerproduct{E_{n}}{B_{i}}}^{2}\,,$
(33)

and optimising the bare masses, $m_{B_{i}}$, and the bare mass slope
parameters, $\alpha_{B_{i}}$, such that the states with highest overlap with
the bare states are closest to the lattice QCD results. This optimisation is
controlled by a $\chi^{2}/{\text{dof}}$ measure. This is done with the aim of
achieving a single Hamiltonian which can describe both infinite-volume
scattering data, and the lattice data on finite volumes.

###  II.7 Model (in)dependence in HEFT

Understanding the model-dependent and model-independent aspects of HEFT is
important. HEFT incorporates the Lüscher formalism [47, 18], and therefore
there are aspects of the calculation that share the same level of model
independence as the Lüscher formalism itself.

####  II.7.1 Model independence

The Lüscher formalism provides a rigorous relationship between the finite-
volume energy spectrum and the scattering amplitudes of infinite-volume
experiment. In HEFT, this relationship is mediated by a Hamiltonian. In the
traditional approach, the parameters of the Hamiltonian are tuned to describe
lattice QCD results. When the fit provides a high-quality description of
lattice QCD results, the associated scattering-amplitude predictions are of
high quality. The key is to have a sufficient number of tunable parameters
within the Hamiltonian to accurately describe the lattice QCD results.

However, in the baryon sector, high-quality precise lattice QCD results are
scarce and HEFT is usually fit to experimental data first. The HEFT formalism
then describes the finite-volume dependence of the baryon spectrum, indicating
where high-precision lattice QCD results will reside. This is the approach
adopted herein. We will show high-quality fits to the experimental scattering
observables such that HEFT provides rigorous predictions of the finite-volume
lattice QCD spectrum with model independence at the level of the Lüscher
formalism.

Of course, this model independence is restricted to the case of matched quark
masses in finite-volume and infinite-volume. The Lüscher formalism provides no
avenue for changing the quark mass. In other words, direct contact with
lattice QCD results is only possible when the quark masses used in the lattice
QCD simulations are physical.

On the other hand, $\chi$PT is renowned for describing the quark mass
dependence of hadron properties in a model-independent manner, provided one
employs the truncated expansion in the power-counting regime, where higher-
order terms not considered in the expansion are small by definition. Given
that finite-volume HEFT reproduces finite-volume $\chi$PT in the perturbative
limit by construction [18, 32], it is reasonable to explore the extent to
which this model independence persists in the full nonperturbative calculation
of HEFT.

This has been explored in Ref. [32]. In the one-channel case where a single
particle basis state (e.g. a quark-model like $\Delta$) couples to one two-
particle channel (e.g. $\pi N$), the independence of the results on the form
of regularisation is reminiscent of that realised in $\chi$PT. Any change in
the regulator is absorbed by the low-energy coefficients such that the
renormalised coefficients are physical, independent of the renormalisation
scheme.

However, in the more complicated two-channel case with a $\pi\Delta$ channel
added, the same is not observed. The form of the Hamiltonian becomes
constrained, describing experimental data accurately for only a limited range
of parameters with specific regulator shapes. This property is also observed
in the calculations performed herein. The Hamiltonian becomes a model in this
case, with regulator-function scales and shapes governed by the experimental
data. The principles of $\chi$PT no longer apply in this nonperturbative
calculation. However, for fit parameters that describe the data well, the
model independence of the Lüscher formalism remains intact. The Hamiltonian is
only mediary.

####  II.7.2 Quark mass variation

The consideration of variation of the quark masses away from the physical
point provides further constraints on the Hamiltonian. In particular, lattice
QCD results away from the physical point provide new constraints on the form
of the Hamiltonian. In the two-channel case, we find the Hamiltonian becomes
tightly constrained when considering experimental scattering data and lattice
QCD results together. This was also observed in the analysis of Ref. [32].

With the Hamiltonian determined by one set of lattice results, one can then
make predictions of the finite-volume spectrum considered by other lattice
groups at different volumes and different quark masses. This is a central aim
of the current investigation where we will confront legacy results from the
HSC [34], as well as contemporary results from Khan et al. [35] and the CLS
Consortium [37], and very recent lattice QCD predictions for the $\Delta$
spectrum from the Cyprus collaboration at a physical pion mass of 139.4(1) MeV
[38].

Our Hamiltonian will be constrained by lattice QCD results from Ref. [33] for
the $1s$ and $2s$ radial excitations of the $\Delta$, such that the
confrontation with contemporary lattice QCD results is parameter free and
predictive.

For the cases previously considered in the baryon spectrum the predictions of
HEFT are in agreement with lattice QCD spectrum predictions. For example, in
the $N(1/2^{+})$ channel, HEFT reproduces the lattice QCD results from Lang et
al. [27, 48]. In the $N(1/2^{-})$ channel, HEFT successfully predicts spectra
from the CLS consortium [49, 50], the HSC [49, 51, 52] and Lang & Verducci
[49, 53]. In the odd-parity $\Lambda(1/2^{-})$ spectrum, HEFT accurately
predicts the finite-volume spectrum of the BaSc collaboration [31, 54]. In the
present analysis we will show successful predictions for the finite-volume
$\Delta$ spectrum of the CLS consortium [37]. Thus one concludes that the
systematic errors of the HEFT approach to quark-mass variation are small on
the scale of contemporary lattice QCD uncertainties. As the Hamiltonian is
constrained by model-independent scattering data and lattice QCD results, we
expect this success to be realised in general.

Variation in the quark mass is conducted in the same spirit as for $\chi$PT.
The couplings are held constant and the hadron masses participating in the
theory take values as determined in lattice QCD. The single-particle bare
basis states acquire a quark mass dependence and this is done in the usual
fashion by drawing on terms analytic in the quark mass. In most cases, lattice
QCD results are only able to constrain a term linear in $m_{\pi}^{2}$, as is
the case here, e.g. in Eq. (30).

The model independence associated with the movement of quark masses away from
the physical point is largely governed by the distance one chooses to move
from the physical quark-mass point. The HEFT approach is systematically
improvable, reliant on high-quality lattice QCD results to constrain the
higher-order terms that one can introduce. For example, one could include an
additional analytic $m_{\pi}^{4}$ term or higher-order interaction terms from
the chiral Lagrangian. However, this increased level of precision is not yet
demanded by current experimental measurements nor contemporary lattice QCD
results.

####  II.7.3 Model dependence

Now that the Hamiltonian has become a tightly constrained model, the
eigenvectors describing the manner in which the non-interacting basis states
come together to compose the eigenstates of the spectrum are model dependent.
At the same time, there is little freedom in the model parameters of the
Hamiltonian such that the predictions of the Hamiltonian are well defined.

The information contained in the Hamiltonian eigenvectors describing the
basis-state composition of finite-volume energy eigenstates is analogous to
the information contained within the eigenvectors of lattice QCD correlation
matrices describing the linear combination of interpolating fields isolating
energy eigenstates on the lattice. These too are model dependent, governed by
the nature of the interpolating fields used to construct the correlation
matrix.

What is remarkable is that, as shown in Refs. [27, 49], with a suitable
renormalisation scheme on the lattice (e.g. interpolators are normalised to
set diagonal correlators equal to 1 at one time slice after the source), the
composition of the states drawn from the lattice correlation matrix is very
similar to the description provided by HEFT. While both eigenvector sets are
model dependent, their similarity does indeed provide some relevant insight
into hadron structure. And because regularisation in the Hamiltonian is
tightly constrained, one can begin to separate out the contributions of bare
versus two-particle channels.

####  II.7.4 Summary

In summary, there is a direct model-independent link between the scattering
observables of experiment and the finite-volume spectrum calculated in HEFT at
physical quark masses. This model independence is founded on the Lüscher
formalism embedded within HEFT. Similarly, variations of the quark masses away
from the physical quark mass have systematic uncertainties that are small
relative to contemporary lattice QCD spectral uncertainties. Finally, the
Hamiltonian eigenvectors describing the basis-state composition of finite-
volume energy eigenstates are model dependent. They are analogous to the
interpolator dependent eigenvectors of lattice QCD correlation matrices
describing the linear combination of interpolating fields isolating energy
eigenstates on the lattice. The similarity displayed by these two different
sets of eigenvectors suggests that they do indeed provide insight into hadron
structure.

##  III Preliminary Calculation: $L=3\ \text{fm}$

To begin our analysis of the $\Delta$-baryon spectrum, we choose our basis
states based on the following:

  * •

The number of single-particle basis states should match the number of 3-quark-
interpolated states identified on the lattice.

  * •

The meson-baryon pairs should hold significant branching fractions in the
decays of the relevant $\Delta$ resonances.

Another consideration to take into account is the limited availability of
$S$-matrix scattering data. By this, we mean that while $\pi N\to\pi N$
scattering data is readily available to fit our infinite-volume Hamiltonian
to, the same cannot be said for other relevant transitions such as $\pi
N\to\pi\Delta$ and $\pi\Delta\to\pi\Delta$. This means that including many
channels will make it difficult to constrain the Hamiltonian as the number of
parameters increases rapidly with additional channels in which there is little
experimental support. It is here that lattice QCD results can provide
important constraints.

With these considerations in mind, we start by only looking to analyse the
lowest state in the lattice spectrum, that is, the $1s$ state identified in
Ref. [33] as corresponding to the $\Delta(1232)$ resonance. Hence, we take
$n_{b}=1$, with $\Delta_{1}$ denoting the single bare basis state. The most
relevant channels at this energy are $\pi N$ and $\pi\Delta$, both in
$p$-wave, based on branching fractions in the PDG [55]. Thus the relevant
basis states are categorised as:

  * •

Single-particle bare basis states:

$\ket{\Delta_{1}}$.

  * •

Two-particle basis states:

$\ket{\pi N(\bm{k})},\,\ket{\pi\Delta(\bm{k})}.$

Following the procedure outlined in Section II, the $S$-matrix can be computed
for a number of parameters, and the resulting $\pi N\to\pi N$ scattering phase
shifts $\delta$ and inelasticities $\eta$ can be extracted. These have then
been fit to scattering data from Ref. [44] as shown in Fig. 3, using a
$\chi^{2}$ per degree of freedom ($\chi^{2}/\text{dof}$) to control the
optimisation. We observe excellent agreement in both scattering observables
and the results of the fits are given in Table 1.

Figure 3: Plot of fits to experimental data for phase shifts $\delta$ (Left) and inelasticities $\eta$ (Right) in $\pi N\to\pi N$ scattering, up to a region close to the $\Delta(1600)$. The data we perform our fits to is shown by the red points with error bars, and the resulting fits are shown by blue solid lines. The phase shift passes through $90^{\circ}$ at the position of the $\Delta(1232)$. Table 1: Table of HEFT parameters from the fit to SAID partial wave analysis data. The measures of how well we fit the data are shown as the $\chi^{2}$ and $\chi^{2}/{\text{dof}}$, where dof is the degrees of freedom. Parameter |  Value
---|---
$m_{\Delta_{1}}/\text{GeV}$ |  $1.3952$
$g^{\Delta_{1}}_{\pi N}$ |  $0.5320$
$g^{\Delta_{1}}_{\pi\Delta}$ |  $0.6826$
$v_{\pi N,\pi N}$ |  $0.0324$
$v_{\pi N,\pi\Delta}$ |  $-1.5425$
$v_{\pi\Delta,\pi\Delta}$ |  $-1.6157$
$\Lambda^{\Delta_{1}}_{\pi N}/\text{GeV}$ |  $0.6377$
$\Lambda^{\Delta_{1}}_{\pi\Delta}/\text{GeV}$ |  $0.8011$
$\Lambda^{v}_{\pi N}/\text{GeV}$ |  $0.6764$
$\Lambda^{v}_{\pi\Delta}/\text{GeV}$ |  $0.7792$
dof |  $56$
$\chi^{2}$ |  $212.61$
$\chi^{2}/{\text{dof}}$ |  $3.80$

As was discussed in Ref. [32], the data we perform our fits to has neglected
systematic errors, making it difficult to interpret the $\chi^{2}/\text{dof}$
value. One can address this as in Ref. [56] by considering a small additional
contribution to the errors to account for neglected systematics [32, 56].
Taking this approach, we find that our $\chi^{2}/\text{dof}$ reduces to $0.57$
after assigning a $3\%$ uncertainty to the phase shift data. In addition to
this and the good visual agreement with the W108 solution from Ref. [44, 57],
we can also consider the T-matrix poles of our model, of which we can extract
two. These may be compared with those provided by the Particle Data Group
(PDG) for the $\Delta(1232)$ and $\Delta(1600)$, given in Table 2. We find
exact agreement with the lower-lying resonance pole position, and reasonable
agreement with the next highest resonance.

Table 2: Table of pole positions in GeV extracted from the T-matrix corresponding to our Hamiltonian model. Particle Data Group results [55] for the first two poles of the $\Delta(3/2^{+})$ spectrum are included for comparison. Pole # | This Work | PDG
---|---|---
1 | $1.210-0.050i$ | $(1.210\pm 0.001)-(0.050\pm 0.001)i$
2 | $1.475-0.141i$ | $(1.52^{+0.07}_{-0.05})-(0.14^{+0.02}_{-0.07})i$

Following these infinite-volume considerations, we perform the corresponding
finite-volume calculations at unphysical pion masses, using the approach
outlined in Sections II.3, II.4 and II.5. The various Sommer-scale lattice
spacings and sizes from the PACS-CS ensembles are given in Table 3 and we take
$L$ to vary linearly between these values as we interpolate to build up the
spectrum.

Table 3: Table of parameters for the PACS-CS lattice ensembles used by the CSSM, characterised by the Hopping parameter $\kappa$. The configurations are calculated on lattices with spacings $a$ and spatial lengths $L$, and pion masses $m_{\pi}$, as determined using the Sommer parameter to set the scale on an ensemble-by-ensemble basis [33]. $\kappa$ |  $m_{\pi}$ (MeV) |  $a$ (fm) |  $L$ (fm)
---|---|---|---
0.13781 | 169 | 0.0933(13) | 2.9856(416)
0.13770 | 280 | 0.0951(13) | 3.0432(416)
0.13754 | 391 | 0.0961(13) | 3.0752(416)
0.13727 | 515 | 0.1009(15) | 3.2288(480)
0.13700 | 623 | 0.1023(15) | 3.2736(480)

The values for the finite-volume mass and slope parameters are given in Table
4 where we note that the values for $B=N,\,\Delta$ are obtained by fitting to
results from the PACS-CS collaboration.

Table 4: Table of finite-volume baryon masses and slope parameters for the single bare state, $n_{b}=1$, and two channel, $n_{c}=2$, case. The Basis State column describes how the mass and slope parameters are used. $B$ | Basis State |  $m_{B}^{*}$ (GeV) |  $\alpha_{B}$ (GeV-1)
---|---|---|---
$N$ | two-particle | 1.002 | 1.097
$\Delta$ | two-particle | 1.263 | 0.972
$\Delta_{1}$ | single-particle | 1.395 | 0.896
Figure 4: Plot of the $L\sim 3$ fm finite volume energy spectrum. Solid lines
are fully interacting energy eigenvalues while dashed blue lines are non-
interacting energies. The blue dot-dashed line is the non-interacting energy
associated with the bare basis state. Coloured dressings on interacting energy
level lines show states with the largest, second largest and third largest
bare basis state components, as described in the text. The vertical grey
dashed line is at the physical quark mass point.

With the two-particle baryon mass parameters in hand, the full energy spectrum
is calculated across a broad range of pion masses, at lattice sizes $\sim 3$
fm. Identifying the Hamiltonian eigenstate dominated by the bare basis state,
the bare mass, $m^{*}_{\Delta_{1}}$, and slope parameter,
$\alpha_{\Delta_{1}}$, are constrained by recent lattice QCD results from the
CSSM [33]. These results are displayed in Fig. 4. The energy levels at each
value of $m_{\pi}^{2}$ are shown by the solid black lines. The states in the
spectrum with largest overlap with the bare basis state are shown by the solid
red dressings, while blue and green dressed lines represent states with the
2nd and 3rd largest $\ket{\Delta_{1}}$ components respectively. Importantly,
the state with the largest overlap with the bare basis state is constrained to
match the lattice QCD state excited by 3-quark operators. The excellent
agreement between the red solid dressed line and the lattice results follows
from this constraint.

In summary then, we have produced a Hamiltonian model which simultaneously
describes experiment and the lattice results, capturing the resonance physics
in both infinite and finite volumes. With this model however, we are unable to
comment on the structure of the $2s$ excitation observed in lattice QCD [33]
at $\sim 2.15$ GeV.

With this proof of concept established, we turn our attention to a more
sophisticated analysis which incorporates a second bare state and an
additional channel, to explore the higher energy region.

##  IV Results: $L=3\,\text{fm}$

In order to investigate the $2s$ state excited on the lattice, we extend our
Hamiltonian model to incorporate a second bare basis state. To capture the
physics involved at the higher energy scale near the $2s$ excitation, we
include a third meson-baryon channel in the form of $\pi\Delta$ coupled in
$f$-wave. This has strong coupling to the $\Delta(1920)$ and should be
significant in the $\sim 2$ GeV region. We distinguish between our $p$\- and
$f$-wave $\pi\Delta$ channels with a subscript $p$ or $f$. Our Hamiltonian
basis states for this extended model are then

  * •

Single-particle bare basis states:

$\ket{\Delta_{1}},\,\ket{\Delta_{2}}$.

  * •

Two-particle basis states:

$\ket{\pi
N(\bm{k})},\,\ket{\pi\Delta_{p}(\bm{k})},\,\ket{\pi\Delta_{f}(\bm{k})}.$

We note that while our Hamiltonian formalism doesn’t allow us to include a
direct 3-body $\pi\pi N$ threshold, the resonant contributions for this
process are incorporated in the 2-body channels chosen above. Thus we proceed
with the assumption that our 2-body channels are suitable for capturing the
relevant physics.

The Hamiltonian is then constructed from the various couplings between the
basis states given by Eqs. (8) and (11), and the masses of the two bare basis
states. We then constrain the Hamiltonian using a fit to available scattering
data up to $2.3$ GeV, ensuring that we produce the relevant resonance at
$1.920$ GeV. The fit to the scattering data is displayed in Fig. 5 with the
Hamiltonian parameters given in Table 5.

Figure 5: Plot of fits to experimental data for phase shifts $\delta$ (Left) and inelasticities $\eta$ (Right) in $\pi N\to\pi N$ scattering, for centre-of-mass energies up to $\sim 2.2$ GeV. The data we perform our fits to is shown by the red points with error bars, and the best fit of our HEFT analysis is shown by blue solid lines. Table 5: Table of HEFT parameters constrained by scattering phase shifts and inelasticities at 61 SAID partial wave analysis data points. Parameter | Value | Parameter | Value
---|---|---|---
$m_{\Delta_{1}}/\text{GeV}$ | $1.3894$ | $m_{\Delta_{2}}/\text{GeV}$ | $2.3177$
$g^{\Delta_{1}}_{\pi N}$ | $0.4974$ | $g^{\Delta_{2}}_{\pi N}$ | $0.2914$
$g^{\Delta_{1}}_{\pi\Delta_{p}}$ | $0.5300$ | $g^{\Delta_{2}}_{\pi\Delta_{p}}$ | $0.2289$
$g^{\Delta_{1}}_{\pi\Delta_{f}}$ | $0.0004$ | $g^{\Delta_{2}}_{\pi\Delta_{f}}$ | $0.0075$
$\Lambda^{\Delta_{1}}_{\pi N}/\text{GeV}$ | $0.8246$ | $\Lambda^{\Delta_{2}}_{\pi N}/\text{GeV}$ | $1.3384$
$\Lambda^{\Delta_{1}}_{\pi\Delta_{p}}/\text{GeV}$ | $0.8376$ | $\Lambda^{\Delta_{2}}_{\pi\Delta_{p}}/\text{GeV}$ | $0.5428$
$\Lambda^{\Delta_{1}}_{\pi\Delta_{f}}/\text{GeV}$ | $0.5776$ | $\Lambda^{\Delta_{2}}_{\pi\Delta_{f}}/\text{GeV}$ | $1.0549$
$v_{\pi N,\pi N}$ | $0.0454$ | $v_{\pi N,\pi\Delta_{f}}$ | $-0.0030$
$v_{\pi N,\pi\Delta_{p}}$ | $-1.5545$ | $v_{\pi\Delta_{p},\pi\Delta_{f}}$ | $-0.0053$
$v_{\pi\Delta_{p},\pi\Delta_{p}}$ | $-0.9694$ | $v_{\pi\Delta_{f},\pi\Delta_{f}}$ | $-0.0001$
$\Lambda^{v}_{\pi N}/\text{GeV}$ | $0.6032$ | $\Lambda^{v}_{\pi\Delta_{f}}/\text{GeV}$ | $1.3289$
$\Lambda^{v}_{\pi\Delta_{p}}/\text{GeV}$ | $0.8058$ |  |
dof | $99$ |  |
$\chi^{2}$ | $581.81$ |  |
$\chi^{2}/{\text{dof}}$ | $5.88$ |  |

Again, there is excellent visual agreement between our fit result and the
reported partial wave analysis data. While the $\chi^{2}/\text{dof}$ is once
again quite large, we treat the uncertainties in the same way as in Section
III and achieve a reduction to $1.07$. The poles achieved with this fit are
given in Table 6 and once again agree remarkably well with the PDG, though
both the real and imaginary parts of the second pole sit just beyond the
bounds of the corresponding PDG pole.

Table 6: Table of pole positions in GeV extracted from the T-matrix corresponding to our full Hamiltonian model. Particle Data Group results [55] for the first three poles of the $\Delta(3/2^{+})$ spectrum are included for comparison. Pole # | Our Value | PDG Value
---|---|---
1 | $1.211-0.049i$ | $(1.210\pm 0.001)-(0.050\pm 0.001)i$
2 | $1.46-0.17i$ | $(1.52^{+0.07}_{-0.05})-(0.14^{+0.02}_{-0.07})i$
3 | $1.91-0.19i$ | $(1.90\pm 0.05)-(0.15\pm 0.10)i$

The finite-volume spectrum is generated in the same manner as discussed in
Section III using baryon masses given in Table 7 and is shown in Fig. 6.
Considering the ground state first, we see that we have retained an excellent
quark model dominated description of the $\Delta(1232)$.

However this is not to say that the ground state has negligible mixing with
the other basis states. To see this, one can consider plots of the eigenvector
components for each state in the spectrum, such as those shown in Fig. 7. Note
that the “Overlap” in these eigenvector plots is a measure of the modulus-
square of the coefficient for each basis state, as in Eq. (33) for the bare
basis states. The coefficients for a given energy eigenstate are normalised
such that the sum of the overlap probabilities over all basis states is unity.

At the physical pion mass, the first eigenvector, corresponding to the ground
state of the spectrum, has $\sim 75\%$ component coming from the first bare
basis state $\Delta_{1}$. The remaining $\sim 25\%$ is composed of $\pi N$ and
$\pi\Delta$ in $p$-wave. Additionally, we note that further separating the
eigenvectors into their individual contributions from each basis state
(essentially separating each of the two-particle species into their different
back-to-back momentum components) as in the right-hand column of Fig. 7 yields
a further breakdown of the structural information.

Table 7: Table of finite-volume baryon mass and slope parameters for the two bare state, $n_{b}=2$, and three channel, $n_{c}=3$ case. The Basis State column describes how the mass and slope parameters are used. $B$ | Basis State |  $m_{B}^{*}$ (GeV) |  $\alpha_{B}$ (GeV-1)
---|---|---|---
$N$ | two-particle | 1.002 | 1.097
$\Delta$ | two-particle | 1.263 | 0.972
$\Delta_{1}$ | single-particle | 1.389 | 0.751
$\Delta_{2}$ | single-particle | 2.318 | 0.203

Considering the lattice QCD results at $\sim 2.15$ GeV corresponding to the
$2s$ state, we see that there is good overall agreement between these and the
blue-dressed line indicating the HEFT energy eigenstate with largest
$\ket{\Delta_{2}}$ basis state component. Thus this dressing indicates where
the second quark model-like state in the spectrum is to be found on the
lattice.

However, not all of the $2s$ excitation lattice points lie on a blue solid or
dashed line. The two points at $m_{\pi}^{2}\sim 0.15$ and $0.27$ GeV2 sit
between the blue lines instead of selecting one of the two. This suggests that
the lattice QCD analysis has reported an energy associated with a
superposition of two bare-basis-state dominated energy eigenstates.

Remarkably, our HEFT analysis actually predicts this superposition of energy
eigenstates. In both cases, the lattice QCD results sit at the precise point
where the bare-basis-state dominated energy eigenstate jumps from one
eigenstate energy level to another. Thus the lattice QCD points are at the
location where the 3-quark interpolating field has an approximately equal
overlap with two nearby energy eigenstates. Thus a superposition of nearby
states is excited and an average mass is reported in the lattice analysis.

Note how this situation contrasts the other three lattice QCD points. These
points are located away from the bare-basis-state cross-overs and sit well on
the bare-basis-state dominated HEFT eigenstate. Of the three points, the
lightest point is closest to a cross over and may have a small contamination
from the higher-energy second bare-basis-state dominated energy eigenstate.

In this light, we have succeeded in describing the finite-volume results from
lattice QCD, using our infinite-volume inspired Hamiltonian.

Figure 6: Plot of the finite-volume spectrum for $L\sim 3$ fm using 3 channels
and 2 bare states. Solid colours denote states with the largest overlaps with
the first and second bare basis states (red for the lower lying and blue for
the higher energy bare state). Dashed dressings show the states with second
largest overlap. Dot-dashed blue lines show the non-interacting states in the
spectrum. The vertical grey dashed line is at the physical quark mass point.

###  IV.1 Eigenvectors 1, 2, 3

At this point we can begin identifying key structural features of the energy
eigenstates using the superposition of basis states provided in the
Hamiltonian eigenvectors. Fig. 7 presents results for the composition of the
three lowest energy eigenstates. The left-hand column shows a sum over all
momentum basis states for a given channel, while the right-hand column reports
individual momentum components.

The dominant component for the lowest state (described by eigenvector 1) is
$\Delta_{1}$, with the overall $\pi N$ and $\pi\Delta_{p}$ contributions
coming from $n=1$ and $n=2$ momenta. So we can conclude that the state in the
HEFT spectrum most closely associated with the $\Delta(1232)$ is mostly
composed of a quark model-like bare basis state, with a relatively small
admixture of low-lying two-particle states. This explains why local 3-quark
operators were sufficient to excite the ground state on the lattice.

Notice also that there is a significant mass gap between the first non-
interacting energy level and the ground state in Fig. 6. We can now understand
this based on the eigenvector components. The significant $\Delta_{1}$
component and the large coupling strengths $g^{\Delta_{1}}_{\pi N}$ and
$g^{\Delta_{1}}_{\pi\Delta_{p}}$ lead to significant dressing of the bare
basis state mass of $m_{\Delta_{1}}=1.3894$ GeV at the physical point,
shifting it down to the interacting energy level observed.

Interestingly, these mass gaps are not present for all states in the spectrum.
Indeed, the very next state (2nd solid line in Fig. 6) lies almost exactly on
the corresponding non-interacting line. This is reflected in the eigenvector
components in the second row of Fig. 7. Here, there is clear dominance of the
$\pi N$ basis states (predominantly composed of the $n=1$ momentum state) with
only a small contribution from the first bare state. This leads to a much
smaller mass gap, as the smaller $\pi N$ and $\pi\Delta$ channel coupling
strengths are insufficient to drive the mass away from the non-interacting
energy level.

Climbing to the next state in the spectrum, described by eigenvector 3, the
first bare state contribution is significantly diminished and we see the
second bare state starting to become more involved. The larger couplings
between these bare states and the meson-baryon channels again drives the mass
away from the non-interacting level. Indeed, the composition of this state at
$\sim 1.7$ GeV is highly mixed, with the dominant contributions coming from
all three two-particle channels.

###  IV.2 Eigenvectors 4, 5, 6

The 4th and 5th eigenvectors have no significant bare state components and we
see that the interacting energy levels lie close to the non-interacting
levels. Interestingly, the 4th eigenvector is initially coincident with a non-
interacting energy level at the physical point and gradually diverges as we
increase in pion mass. This is also reflected in the eigenvector component
plots (top panels of Fig. 8) where less trivial two-particle mixing and also
mixing with the second bare state enter at larger pion masses. The evolution
of eigenvector 5 in $m_{\pi}^{2}$ is almost an inverse of this with the
initial mixture of $\pi N$ and $\pi\Delta$ basis states becoming dominated by
$\pi\Delta$ beyond $m_{\pi}^{2}\sim 0.15$ GeV2. This explains why the line
corresponding to eigenvector 5 begins to overlap with the nearby non-
interacting state in Fig. 6.

The 6th eigenvector is then the next most interesting where we again observe
some non-trivial bare state component (now almost exclusively from the second
bare state $\Delta_{2}$) along with the indicative mass gap in Fig. 6.

###  IV.3 Eigenvectors 7–12

Similar analysis of the 7th and 8th eigenvectors in Fig. 9 can be used to
understand their near degeneracy with the corresponding non-interacting levels
and the large mass gap for the 9th eigenvector is similarly associated with
the clear presence of the second bare state. The 10th, 11th and 12th
eigenvectors follow this same pattern.

###  IV.4 Momentum Modes

Comparing the eigenvector components between the left and right columns of
Fig. 7, and likewise for Fig. 8, it appears that typically only a single
momentum mode in each two-particle channel contributes in any significant way.
The exceptions to this appear to be the 3rd and 6th eigenvectors where
$n=1,\,2$ and $3$ momentum modes of the same two-particle species are mixed
(for example the $\pi N(n=1,2)$ basis states for the 3rd eigenvector). This
dominance by the lowest lying momentum modes is partly due to the inverse
proportionality of the lattice size $L$ and the discretised momentum $k_{n}$
in Eq. (19). This indicates that larger lattice sizes will allow for more
densely packed momentum states, and in particular, larger volumes allow more
momentum basis states to be kinematically allowed participants for low-lying
energy levels. This will be further explored in Section V.

###  IV.5 HEFT as a guide for lattice QCD

The eigenvector components for each energy eigenstate observed in HEFT can be
used to guide lattice QCD studies in their choice of interpolating fields. For
example, the ground state HEFT energy aligns very well with the observed
$\Delta(1232)$ resonance and has significant overlap with the first bare
state. This explains why the lattice calculation of the $\Delta(1232)$ mass in
Ref. [33] was effective when using 3-quark interpolators.

When it comes to the HEFT eigenstates nearby to the $\Delta(1600)$, namely
eigenvectors 2, 3, 4 and 5, we observe smaller bare state components than for
those eigenstates near the $2s$ excitation around $\sim 2.15$ GeV. This
explains why the lattice study of Ref. [33] was unable to yield eigenstates in
the vicinity of the $\Delta(1600)$; using 3-quark operators alone means their
interpolators couple more strongly to the states in the $\sim 2.15$ GeV region
which have largest $\Delta_{2}$ component.

This indicates that future lattice studies looking to extract the
$\Delta(1600)$ resonance properties require momentum projected two-particle
interpolators in the $\pi N$ and $\pi\Delta$ channels with momenta as
indicated in the right-hand columns of Figs. 7 and 8 for eigenstates 2 through
5. Here, the $\pi\Delta$ channel will capture the most important resonant
contributions of the three-body $\pi\pi N$ channel.

Figure 7: Plots of the first three eigenvectors describing the basis state
components of the energy eigenstates. The “Overlap” with an individual basis
state is equal to the modulus-square of the coefficient of that basis state in
the superposition describing the eigenstate, normalised to unity. Note that
the two-particle components have their back-to-back momenta $k_{n}$ summed
over in the left column of plots, while the right column shows the composition
in terms of separate two-particle basis state contributions with momenta
$k_{n}$ up to $n=3$. The vertical grey dotted line is at the physical quark
mass point.

Figure 8: Same as in Fig. 7 but for eigenvectors 4, 5 and 6.

Figure 9: Plots of the eigenvector components describing the basis state
composition of the energy eigenstates. Here the momenta, $k_{n}$, are summed
over for each two-particle channel. Eigenvectors 7–12 are arranged in a
column-ordered manner as in Figs. 7 and 8.

##  V Comparison with other lattice QCD results

In this final section of the paper, we use the HEFT Hamiltonian, constrained
with lattice QCD results from Ref. [33], to recalculate the finite-volume
eigenvalues and eigenvectors at various lattice sizes in order to make contact
with the results of Refs. [34, 35, 38, 37, 36]. To ensure direct comparisons
are valid, we match the masses and slope parameters of the two-particle basis
state participants ($N,\Delta$) to reproduce the corresponding non-interacting
energy levels of these studies. As in Refs. [32, 49, 40] we hold the single-
particle bare baryon masses and slope parameters fixed as the finite-volume
effects are contained in the interactions. In doing so, we find general
agreement between our HEFT predictions and the results of the studies
considered herein.

###  V.1 CLS Consortium Comparison

The studies in Refs. [37, 36] yield values for $\pi N$ energy levels using
two-particle momentum-projected operators on two gauge-field ensembles from
the CLS Consortium. The first ensemble has $m_{\pi}=200$ MeV and $L=4.16$ fm
and taking $m_{N}$ and $\alpha_{N}$ so as to match the non-interacting energy
levels of Ref. [37], we generate the corresponding HEFT spectrum. This is
shown in the left-hand plot of Fig. 10 with the three lowest-lying energy
levels of Ref. [37] overlaid. Likewise, the right-hand plot in Fig. 10 shows
the comparison with the ensemble used in Ref. [36], with $m_{\pi}=280$ MeV,
$L=3.7$ fm. By extending the baryon masses beyond the results of Ref. [37], we
generate the HEFT finite-volume spectrum across a range of $m_{\pi}^{2}$
values and this is given in Fig. 11.

Figure 10: $\pi N$ energy levels from Ref. [37] (Left) and Ref. [36] (Right).
Lattice results are shown by the data points. Our HEFT predictions are shown
as solid lines, with red solid lines showing states dominated by the bare
basis state $\Delta_{1}$, and green solid lines corresponding to $\pi N$
dominated states. Blue dashed lines denote the non-interacting energy levels.
Figure 11: Finite-volume spectrum for $L=4.16$ fm and with $\pi N$ non-
interacting energies matched to the non-interacting energy levels of the CLS
Consortium [37].

The HEFT energy levels are in excellent agreement with the lattice results,
particularly for the $m_{\pi}=200$ MeV ensemble. While the $m_{\pi}=200$ MeV
plot in Fig. 10 and the plot in Fig. 11 show four lattice results, only one of
the two highest energies in the cubic group $H_{g}$ has $J=3/2$ [58]. In Fig.
11, note that while the states dominated by $\Delta_{2}$ appear to shift
higher at low pion masses, there are still significant mixtures of this basis
state in the $\sim 2.0-2.2$ GeV region, as revealed by considering the
eigenvector components of each energy eigenstate illustrated in Figs. 12, 13
and 14. Fig. 12 gives a snapshot of the eigenvector components at the CLS pion
mass of $m_{\pi}=200$ MeV, while the plots of Figs. 13 and Figs. 14 show the
evolution of the eigenstate components in $m_{\pi}^{2}$.

In Fig. 12, we find that the first three finite-volume eigenstates matching
the CLS lattice points have significant overlap with the $\pi N$ channel and
the lower-lying bare basis state $\Delta_{1}$. With the CLS Consortium’s
interpolator basis including $\pi N$ interpolators with projected momenta as
per the right-hand column of Fig. 13, this explains the excellent agreement
between their results and our HEFT prediction.

Considering those energy eigenstates with energies near the mass of the
$\Delta(1600)$ resonance, in particular the eigenstates with eigenvalues in
the range $1.633-1.759$ GeV in Fig. 12, we find that they are either dominated
by a single scattering channel (e.g. the fifth eigenstate) or composed of a
mixture of basis states with only a small admixture of the bare basis states.
This again indicates that in order to excite energies in the finite-volume
spectrum corresponding to the $\Delta(1600)$ on the lattice, one must include
interpolators which capture the rich resonant properties of the
$\Delta(1600)$. Taking into account the dominant momentum basis states in the
right-hand columns of Figs. 13 and 14, we expect such excitations to respond
well to a basis containing single-hadron interpolators in combination with
both $\pi N$ and $\pi\Delta$ multi-hadron interpolators with momenta up to and
including $n=3$. The plots for eigenvector 7 show that $\pi N(k_{n})$ basis
states beyond $n=3$ become important for exciting the 7th energy eigenstate
(and higher) on the lattice.

In terms of the physical $\Delta(1600)$ resonance, this reinforces our key
finding that the $\Delta(1600)$ is not strongly associated with a 3-quark
core. Rather, it relies heavily on strong rescattering in the $\pi N$ and
$\pi\Delta$ channels.

As a final note, the interacting energy levels in Fig. 11 give a clear example
of how the density of states increases with the lattice size. This is readily
apparent by comparing with Fig. 6, where the smaller lattice volume leads to
sparser energy eigenstates. This will become especially relevant when
comparing with the HSC lattice QCD results on very small lattices, with $L\sim
2$ fm.

Figure 12: Eigenvectors calculated at the CLS pion mass of $m_{\pi}=200$ MeV
and decomposed into their basis state content. All two-particle channels have
had their momenta summed over.

Figure 13: Eigenvector components for the first three fully interacting HEFT
eigenstates in the $L=4.16$ fm calculation. The left-hand column shows the
basis state content with momenta summed over in each 2-particle channel, while
the righ-hand column gives the full basis state decomposition without
summation.

Figure 14: Same as for Fig. 13 but for eigenvectors 4-7, with energy
eigenvalues in the region of the $\Delta(1600)$ mass.

Figure 15: Eigenvector components for the 4th-11th fully interacting HEFT
eigenstates in the $L=4.16$ fm calculation.

###  V.2 Cyprus Collaboration Comparison

Recently, Ref. [38] provided lattice QCD results for $\pi N$ energies at the
physical quark mass point. To make contact with these results via HEFT, we use
$m_{\pi}=139.43$ MeV, $L=5.1$ fm and $m_{N}=944$ MeV. Assuming a similar
increase from the finite volume for the $\Delta$, we take $m_{\Delta}=1238$
MeV. The resulting HEFT energy level predictions at the physical pion mass
point are given in Fig. 16 along with the results from the four methods used
in Ref. [38] for determining the energy levels of $\pi N$ scattering states,
those being the Generalised Eigenvalue Problem (GEVP), Prony Generalised
Eigenvalue Method (PGEVM), Athens Model Independent Analysis Scheme (AMIAS)
and the Ratio method (RATIO).

There is fair agreement between the HEFT energy levels and the lattice QCD
results for the standard GEVP analysis. The largest discrepancy occurs for the
highest energy results reported by the Cyprus group. All four analysis methods
produce results consistent with the nearby non-interacting state. Future
calculations should explore additional interpolating fields and more extensive
Euclidean time evolution to examine the possible evolution of the eigenstate
energy away from the noninteracting basis-state energy.

Figure 16: Plot comparing the energy levels between our HEFT model and those
reported by the Cyprus group. Note that this comparison is made at the single
value of $m_{\pi}=139.43$ used in Ref. [38]. Non-interacting energies are
shown as blue dashed lines, while fully interacting HEFT levels are shown as
solid red and green lines indicating the predominant basis state component.
The lattice QCD results from the Cyprus group are shown by the various symbols
for each of the different methods used in extracting energies; GEVP (filled
circles), PGEVM (squares), AMIAS (triangles) and Ratio (stars).

###  V.3 HSC Comparison

Next, we compute the HEFT spectrum using a lattice size of $L=1.96$ fm, as in
Ref. [34]. Taking their values of $m_{\pi}=392,\,438$ and $521$ MeV, and
taking $m^{*}_{N}$, $\alpha_{N}$ and $m^{*}_{\Delta},\alpha_{\Delta}$ so as to
reproduce the non-interacting thresholds in Table VI of Ref. [34] yields the
finite-volume spectrum given in Fig. 17. Additionally, we provide the
eigenvector component plots for the four lowest energy levels in Fig. 18.

Notably, we see clear evidence that while the HSC results for the lowest- and
highest-energy states show fair agreement with the HEFT predictions, the first
excited state has no nearby interacting energy level with which it can be
associated. Indeed, on a lattice of around just 2 fm, the finite-volume
spectrum levels are very sparse at low energies. The first $\pi N$ dominated
state is at $\sim 2$ GeV. In this light, the intermediate states remain
unexplained. They cannot be associated with the $\Delta(1600)$ (as had been
suggested in Ref. [34]).

Disregarding the first excitation then, the remaining results are in fair
agreement with HEFT; the ground state is approximately where HEFT predicts it
to be, while the second excitation is likely a superposition of energy
eigenstates having overlap with the bare basis state $\Delta_{2}$.

Figure 17: Finite-volume spectrum at a lattice size of $L=1.96$ fm. Solid
lines show interacting energy levels, while dashed lines show non-interacting
levels. The lattice data from Ref. [34] are shown as data points with errors.

Figure 18: Eigenvector components for the first four fully interacting HEFT
eigenstates in the $L=1.96$ fm calculation.

###  V.4 Khan et al. Comparison

As our final comparison with contemporary lattice QCD results, we compute the
HEFT spectrum using a lattice size of $L=3.01$ fm, as in Ref. [35]. The
resulting finite-volume spectrum is given in Fig. 19. Additionally, we provide
the eigenvector component plots for the eight lowest energy levels in Fig. 20.

Here, the reported lattice points align fairly well with the finite-volume
spectrum generated via HEFT, with every point lying on or nearby a bare-basis
state dominated interacting energy level with the exception of the lowest-
lying state at the lighter quark mass. If we assume a systematic error in the
lower quark mass results associated with scale setting, then all these masses
move into positions which are in better agreement with the HEFT model (given
by the “scaled” points in Fig. 19).

The heavier quark-mass ensemble results are interesting as well; remarkably,
the lowest-lying energy at the heavier quark mass sits on the lowest-lying
HEFT energy level, despite the miniscule uncertainty. Additionally, the
highest energy level reported is reminiscent of the $m_{\pi}^{2}\sim 0.15$ and
$0.27$ GeV2 from Fig. 6 in that it lies in a superposition of two energy
levels dominated by the second bare state. Thus this result is also supported
by HEFT.

Turning our attention to the eigenvector component plots, we consider the
plots for eigenvectors 4, 5 and 6 as these correspond to the eigenvalues near
the first excitation results of Ref. [35], and are approximately in the region
associated with the $\Delta(1600)$ resonance. Notably, these lattice QCD
results were identified as being consistent with hybrid states excited by
local lattice operators. The quantum numbers of the operators used are not
exotic so these states can also be described by mixtures of two-particle
channels.

Eigenvectors 4 and 5 possess very little bare state content at the pion masses
used, instead being mostly comprised of two-particle basis states. On the
other hand, eigenvector 6 shows a greater mixture of bare and two-particle
states at the lighter quark mass. Importantly though, at the physical point,
the bare basis state content of eigenvector 6 is greatly diminished. This
provides further support for the interpretation of the $\Delta(1600)$ as being
a dynamically generated resonance, rather than a resonance with a dominant
3-quark core component.

Figure 19: Finite-volume spectrum at a lattice size of $L=3.01$ fm. Solid
lines show interacting energy levels, while dashed lines show non-interacting
levels. The lattice QCD results from Ref. [35] are shown as black circles with
errors. The orange squares show the points from the lighter ensemble scaled up
such that the ground state matches the prediction from HEFT.

Figure 20: Eigenvector components for the first 8 fully interacting HEFT
eigenstates in the $L=3.01$ calculation.

##  VI Conclusion

We have presented an analysis of the low-lying $\Delta$-baryon spectrum
through a combination of lattice QCD and Hamiltonian Effective Field Theory
(HEFT).

Using HEFT, we are able to simultaneously describe experimental data from Ref.
[44] and lattice QCD results obtained in Ref. [33]. Starting with a simple
description of the low-energy region around the $\Delta(1232)$ resonance, we
then extended this analysis to higher energies seeking to confront the
$\Delta(1600)$ and the $2s$ excitation observed in lattice QCD.

We have also validated our results through comparison with spectrum results
obtained using lattice QCD on an array of volumes, at several pion masses.
Excellent agreement was found with the results of the CLS Consortium and we
saw agreement with a comparison to state-of-the-art results from the Cyprus
Collaboration for the first two states. Our analysis was in fair agreement
with the results from Khan et al., with the main discrepancies accounted for
by a possible systematic error in the scale of their lightest-ensemble
results. We find that the legacy results of the HSC collaboration are not
fully supported by HEFT, with some of these results unable to be associated
with any nearby finite-volume energy levels.

Based on our HEFT analysis, the ground state $\Delta(1232)$ corresponds to a
$1s$ state, and is well described by a quark-model-like core dressed by $\pi
N$ and $\pi\Delta$ channels. The crucial finding from our analysis is that the
corresponding $2s$ excitation is _not_ the $\Delta(1600)$. Using the lattice
QCD calculation of the first excited state in the $\Delta(3/2^{+})$ spectrum
in Ref. [33], we saw that the $2s$ state sits at $\sim 2.15$ GeV, and can be
explained in HEFT through the inclusion of a second bare state. This bare
state couples to many states in the spectrum, but only weakly to one of the
states close to the $\Delta(1600)$ on the volumes used in Refs. [33] and [37].
This means that the states which are associated with the $\Delta(1600)$ rely
on highly non-trivial mixing in the two-particle basis states. We conclude
then that, for a complete description of the $\Delta(1600)$, it is
insufficient to consider it as a simple $2s$ excitation of the $\Delta(1232)$.
One needs to include the rich resonance properties obtained through coupling
to $\pi N$ and $\pi\Delta$ channels.

This work, along with previous studies of the Roper resonance [19, 20, 26, 27]
and the $\Delta$-spectrum [33] show how LQCD and HEFT suggest new
interpretations of the first excitations observed in the
$J^{P}=1/2^{+}\,(3/2^{+})$ spectrum for nucleons ($\Delta$-baryons). Rather
than seeing the first excitations as $2s$ radial excitations of the
corresponding ground states, one finds that this role is filled by states at
significantly higher energies. The Roper and the $\Delta(1600)$ are then
realised as dynamically generated resonances.

###### Acknowledgements.

LH would like to thank Zhan-Wei Liu for discussions during his visit to
Adelaide. This work was supported by an Australian Government Research
Training Program Scholarship and with supercomputing resources provided by the
Phoenix High Performance Computing service at the University of Adelaide. This
research was undertaken with the assistance of resources from the National
Computational Infrastructure (NCI), provided through the National
Computational Merit Allocation Scheme, and supported by the Australian
Government through Grant No. LE190100021 via the University of Adelaide
Partner Share. This research was supported by the University of Adelaide and
the Australian Research Council through Grants No. DP190102215 and DP210103706
(DBL), and DP230101791 (AWT).

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