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

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

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

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

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.

Mathematically, we describe these vertex interactions by

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

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

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

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

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

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)

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

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].

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

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

The corresponding S-matrix elements are obtained from

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

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

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 $T_{\alpha\beta}(k,k^{\prime};E_{\text{pole}})^{-1}=0$.

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

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

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

where

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

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 $l_{\alpha}=1$

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

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

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

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

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

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.

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

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.

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

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.