Regularization in nonperturbative extensions of effective field theory

The process of renormalization in nonperturbative Hamiltonian effective field theory (HEFT) is examined in the $\mathrm{\Delta }$-resonance scattering channel. As an extension of effective field theory incorporating the Lüscher formalism, HEFT provides a bridge between the infinite-volume scattering data of experiment and the finite-volume spectrum of energy eigenstates in lattice QCD. HEFT also provides phenomenological insight into the basis-state composition of the finite-volume eigenstates via the state eigenvectors. The Hamiltonian matrix is made finite through the introduction of finite-range regularization. The extent to which the established features of this regularization scheme survive in HEFT is examined. In a single-channel $\pi N$ analysis, fits to experimental phase shifts withstand large variations in the regularization parameter $\mathrm{\Lambda }$, providing an opportunity to explore the sensitivity of the finite-volume spectrum and state composition on the regulator. While the Lüscher formalism ensures the eigenvalues are insensitive to $\mathrm{\Lambda }$ variation in the single-channel case, the eigenstate composition varies with $\mathrm{\Lambda }$; the admission of short-distance interactions diminishes single-particle contributions to the states. In the two-channel $\pi N$, $\pi \mathrm{\Delta }$ analysis, $\mathrm{\Lambda }$ is restricted to a small range by the experimental data. Here the inelasticity is particularly sensitive to variations in $\mathrm{\Lambda }$ and its associated parameter set. This sensitivity is also manifest in the finite-volume spectrum for states near the opening of the $\pi \mathrm{\Delta }$ scattering channel. Future high-quality lattice QCD results will be able to discriminate $\mathrm{\Lambda }$, describe the inelasticity, and constrain a description of the basis-state composition of the energy eigenstates. Finally, HEFT has the unique ability to describe the quark-mass dependence of the finite-volume eigenstates. The robust nature of this capability is presented and used to confront current state-of-the-art lattice QCD calculations.

Figure 1

Self-energy (left) and two-particle interaction (right) contributions to the $\mathrm{\Delta }$ mass.

Figure 2

Dependence of Hamiltonian energy eigenvalues for the one-channel analysis of Sec. 4 on $u_{\min }$, governing the maximum momentum to be considered in constructing the finite-volume Hamiltonian. Our selection of $u_{\min }={10}^{-2}$ ensures a robust consideration of high-momentum basis states regulated by $u(k^2)$.

Figure 3

$P$-wave $\pi N$ phase shifts for a system with a bare state, where the solid points are experimental data obtained from Refs. [37, 38], the solid line is the fit using HEFT to the data, and the dashed line represents a phase shift of 90°. The parameter set producing this curve is given by fit I of Table 1 and gives a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ of 18.22.

Figure 4

Lattice volume dependence of the energy eigenvalues of the Hamiltonian. The solid lines represent the energy eigenvalues following from fit I of Table 1. The horizontal dot-dashed line is the bare mass and the curved dashed lines are the noninteracting $\pi N$ basis states at $k=2\pi /L,2\sqrt{2}\pi /L,\dots$.

Figure 5

Lattice volume dependence of the energy eigenvalues of the Hamiltonian from fit I of Table 1. The solid (red), short-dashed (blue), and long-dashed (green) highlights on the eigenvalues correspond to the states with the largest, second-largest, and third-largest contribution from the bare basis state $|{\mathrm{\Delta }}_0⟩$, respectively.

Figure 6

$P$-wave $\pi N$ phase shifts for a system with no single-particle state, where the solid points are experimental data obtained from Refs. [37, 38], the solid line is the fit using HEFT to the data, and the dashed line represents a phase shift of 90°. The parameter set producing this curve is given by fit IV of Table 1.

Figure 7

Dependence of the lowest-lying eigenvalues of the finite-volume Hamiltonian on the regulator parameter $\mathrm{\Lambda }$ for two different lattice sizes, where $\mathrm{\Lambda }$ is varying from 0.6 to 8.0 GeV. The solid (black) lines are the eigenvalues, the horizontal dashed (blue) lines are $\pi N$ basis states, and the curved dot-dashed (blue) line is the mass of the bare $\mathrm{\Delta }$. The Hamiltonian was constrained to experimental data with $E\lesssim 1350\mathrm{MeV}$.

Figure 8

Dependence of the energy-eigenstate basis-state structure on the regulator parameter $\mathrm{\Lambda }$, where $\mathrm{\Lambda }$ is varying from 0.6 to 8.0 GeV, for a lattice size of $L=2.99\mathrm{fm}$. Only the ground state is shown, as all higher eigenstates lie above the fitting threshold of $E=1350\mathrm{MeV}$ and thus are not physically constrained.

Figure 9

Dependence of the energy-eigenstate basis-state structure on the regulator parameter $\mathrm{\Lambda }$, where $\mathrm{\Lambda }$ is varying from 0.6 to 8.0 GeV, for a lattice size of $L=5\mathrm{fm}$. The two lowest-lying energy eigenstates of the finite-volume Hamiltonian are investigated. The upper plot shows the eigenvectors for the ground state, while the lower plot shows the first excited state.

Figure 10

Pion mass dependence of the finite-volume HEFT eigenvalues at $L=2.99\mathrm{fm}$ for increasing values of the regulator parameter $\mathrm{\Lambda }$. No bare basis state is present for $\mathrm{\Lambda }=8.0\mathrm{GeV}$. The parameters for these fits are given by their corresponding entries in Table 1. The solid black curves illustrate the finite-volume energy levels predicted by HEFT from fits to experimental phase shifts. These lines are dressed by solid (red), short-dashed (blue), and long-dashed (green) highlights indicating states with the largest, second-largest, and third-largest contribution from the bare basis state $|{\mathrm{\Delta }}_0⟩$, respectively. Lattice QCD results for lowest-lying $\mathrm{\Delta }$ masses, denoted by the (black) points, are from the PACS-CS Collaboration [14]. The quoted ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ for each plot are for the lowest-lying energy eigenvalue with respect to the PACS-CS data points. As these lattice results follow from local three-quark operators, they are expected to lie on a solid (red) energy eigenstate in the first case and perhaps on a short-dashed (blue) energy eigenstate when it is the lowest-lying state of the spectrum. The vertical dashed (black) line illustrates the physical pion mass.

Figure 11

Pion mass dependence of the lowest-lying finite-volume HEFT eigenvalue at $L=2.99\mathrm{fm}$ using a dipole regulator, where the data points are the PACS-CS data. Several parameter sets corresponding with each value of $\mathrm{\Lambda }$ are overlapped, each with a corresponding bare mass expansion fit to the PACS-CS data.

Figure 12

$P$-wave $\pi N$ phase shifts for a Gaussian regulator, where the solid points are experimental data obtained from Refs. [37, 38], the solid line is the fit using HEFT to the data, and the dashed line represents a phase shift of 90°. A Gaussian regulator with $\mathrm{\Lambda }=0.8\mathrm{GeV}$ gives a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ of 18.8.

Figure 13

Dependence of the lowest-lying eigenvalues of the finite-volume Hamiltonian on $\mathrm{\Lambda }$ for a Gaussian regulator at $L=2.99\mathrm{fm}$. The solid (black) lines are the eigenvalues, the horizontal dashed (blue) line is the $\pi N(k=1)$ basis state, and the curved dot-dashed (blue) line is the mass of the bare $\mathrm{\Delta }$. The Hamiltonian was constrained to experimental data with $E\lesssim 1350\mathrm{MeV}$.

Figure 14

Pion mass dependence of the lowest-lying finite-volume HEFT eigenvalue at $L=2.99\mathrm{fm}$ using a Gaussian regulator, where the data points are the PACS-CS data. Four parameter sets corresponding to each value of $\mathrm{\Lambda }$ are overlapped, each with a corresponding bare mass expansion fit to the PACS-CS data.

Figure 15

$P$-wave $\pi N$ phase shifts and inelasticities. The solid points are experimental data obtained from Refs. [37, 38]. The solid (red) curve is the fit of HEFT to the scattering data. The horizontal dashed line highlights a phase shift of 90°, while the vertical dashed line illustrates the position of the $\pi \mathrm{\Delta }$ threshold. The upper plots illustrate the best fit when the regulator parameter $\mathrm{\Lambda }=0.8\mathrm{GeV}$ (fit V), while the lower plots illustrate the best fit when the regulator parameter $\mathrm{\Lambda }=1.2\mathrm{GeV}$ (fit VI). Only phenomenologically motivated values associated with the induced pseudoscalar form factor of a baryon are able to give a good description of the inelasticity.

Figure 16

Two-channel lattice volume dependence of the energy eigenvalues of the Hamiltonian for the fit to experimental data with $\mathrm{\Lambda }=0.8\mathrm{GeV}$. The solid lines represent the energy eigenvalues. The horizontal dot-dashed line is the bare mass and the curved dashed lines are the $\pi N$ and $\pi \mathrm{\Delta }$ scattering states at $k=(n_x^2+n_y^2+n_z^2{)}^{1/2}2\pi /L$.

Figure 17

Two-channel lattice volume dependence of the energy eigenvalues of the Hamiltonian for the fit to experimental data with $\mathrm{\Lambda }=0.8\mathrm{GeV}$. The solid (red), short-dashed (blue), and long-dashed (green) highlights on the energy eigenvalues correspond to the states with the largest, second-largest, and third-largest contribution from the bare $\mathrm{\Delta }$ basis state, respectively.

Figure 18

Regulator parameter $\mathrm{\Lambda }$ dependence of the energy eigenvalues of the two scattering-channel HEFT finite-volume energy levels for the 2.99 fm lattice (top) and the 5.0 fm lattice (bottom). The solid (black) lines illustrate the energy eigenvalues, the thin horizontal dashed (blue) lines are the noninteracting energy levels of the $\pi N$ and $\pi \mathrm{\Delta }$ basis states, and the thin dash-dotted (blue) curve is the mass of the bare $\mathrm{\Delta }$ basis state.

Figure 19

Regulator parameter $\mathrm{\Lambda }$ dependence of the eigenvectors describing the composition of the two lowest-lying energy eigenstates in the two-channel case with $\pi N$ and $\pi \mathrm{\Delta }$ scattering channels. The top two plots present results for a lattice volume of 2.99 fm, whereas the bottom two are for $L=5.0\mathrm{fm}$. The left plot shows the eigenvectors for the lowest-lying state, while the right plot shows the next energy eigenstate. The bare basis states contribute most strongly to the lowest energy eigenstate at 2.99 fm. At 5 fm, the lowest state is only just dominated by the bare basis state, with a strong contribution from many scattering states, while the first excited eigenstate is an approximately equal mixing of the bare state and the $\pi N(k=1)$ scattering state.

Figure 20

Pion mass dependence of the finite-volume HEFT energy eigenvalues for $\mathrm{\Lambda }=0.8\mathrm{GeV}$ (top) and $\mathrm{\Lambda }=1.2\mathrm{GeV}$ (bottom) for the PACS-CS lattice length $L=2.99\mathrm{fm}$. The vertical dashed (black) line denotes the physical pion mass. The thin diagonal (blue) dashed lines show the two-particle noninteracting energies and the thin more-horizontal (blue) dot-dashed line illustrates the bare basis-state mass. The solid (black) points are the lowest-lying $\mathrm{\Delta }$ energies from lattice QCD [14]. The solid black curves illustrate the finite-volume energy levels predicted by HEFT from fits to experimental phase shifts and inelasticities. These lines are dressed by solid (red), short-dashed (blue), and long-dashed (green) highlights indicating states with the largest, second-largest, and third-largest contribution from the bare basis state $|{\mathrm{\Delta }}_0⟩$, respectively. As the PACS-CS results follow from local three-quark operators, they are expected to lie on a solid (red) energy eigenstate. The quoted ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ are for the lowest-lying energy eigenvalue with respect to the PACS-CS data points.

Figure 21

Two-channel pion mass dependence of the three lowest-lying finite-volume HEFT eigenvalues at $L=2.99\mathrm{fm}$ using a dipole regulator, where the data points are the PACS-CS data. Four parameter sets corresponding to each value of $\mathrm{\Lambda }$ are displayed, each with a corresponding bare mass expansion fit to the PACS-CS data.

Figure 22

Comparison between the energy eigenvalues calculated in HEFT constrained by scattering data and the PACS-CS results for the quark-mass dependence (solid black lines) and lattice QCD data from the CLS consortium (data points) for ensembles D200 (left) [16] and N401 (right) [15]. The dashed blue lines denote noninteracting basis states, while the dot-dashed blue line is the mass of the bare basis state.