Finite-range Regularization Research Articles

Octet and decuplet baryon σ terms and mass decompositions

We present a comprehensive analysis of the SU(3) octet and decuplet baryon masses and $\sigma$-terms using high-precision lattice QCD data and chiral SU(3) effective theory with finite range regularization. The effects of various systematic uncertainties, including from the scale setting of the lattice data and the regularization prescriptions, are quantified. We find the pion-nucleon and strange nucleon $\sigma$-terms to be $\sigma_{\pi N} = 44(3)(3)$ MeV and $\sigma_{N s} = 50(6)(1)$ MeV, respectively. The results provide constraints on the energy-momentum tensor mass decompositions of the SU(3) octet and decuplet baryons, where we find the trace anomaly and quark/gluon energies decrease for strange baryons due to their larger strange $\sigma$-terms.

Regularization in nonperturbative extensions of effective field theory

The process of renormalization in nonperturbative Hamiltonian effective field theory (HEFT) is examined in the $\mathrm{\ensuremath{\Delta}}$-resonance scattering channel. As an extension of effective field theory incorporating the L\"uscher 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 $\ensuremath{\pi}N$ analysis, fits to experimental phase shifts withstand large variations in the regularization parameter $\mathrm{\ensuremath{\Lambda}}$, providing an opportunity to explore the sensitivity of the finite-volume spectrum and state composition on the regulator. While the L\"uscher formalism ensures the eigenvalues are insensitive to $\mathrm{\ensuremath{\Lambda}}$ variation in the single-channel case, the eigenstate composition varies with $\mathrm{\ensuremath{\Lambda}}$; the admission of short-distance interactions diminishes single-particle contributions to the states. In the two-channel $\ensuremath{\pi}N$, $\ensuremath{\pi}\mathrm{\ensuremath{\Delta}}$ analysis, $\mathrm{\ensuremath{\Lambda}}$ is restricted to a small range by the experimental data. Here the inelasticity is particularly sensitive to variations in $\mathrm{\ensuremath{\Lambda}}$ and its associated parameter set. This sensitivity is also manifest in the finite-volume spectrum for states near the opening of the $\ensuremath{\pi}\mathrm{\ensuremath{\Delta}}$ scattering channel. Future high-quality lattice QCD results will be able to discriminate $\mathrm{\ensuremath{\Lambda}}$, 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.

Octet and decuplet baryon self-energies in relativistic SU(3) chiral effective theory

The self-energies of the full set of flavor SU(3) octet and decuplet baryons are computed within a relativistic chiral effective theory framework. The leading nonanalytic chiral behavior is derived for the octet and decuplet masses, and a finite-range regularization consistent with Lorentz and gauge invariance is applied to account for the finite size of the baryons. Using a four-dimensional dipole form factor, the relative importance of various meson-baryon loop contributions to the self-energies is studied numerically as a function of the dipole range parameter and meson mass, and comparison is made between the relativistic results and earlier approximations within the heavy baryon limit.

Nucleon electromagnetic form factors with a nonlocal chiral effective Lagrangian

The relativistic version of finite-range regularization is proposed. The covariant regulator is generated from the nonlocal Lagrangian. This nonlocal interaction is gauge invariant and is applied to study the nucleon electromagnetic form factors at momentum transfer up to $2\text{ }\text{ }{\mathrm{GeV}}^{2}$. Both octet and decuplet intermediate states are included in the one-loop calculation. Using a dipole regulator with $\mathrm{\ensuremath{\Lambda}}$ around 0.85 GeV, the obtained form factors, electromagnetic radii, as well as the ratios of the form factors are all comparable with the experimental data. This successful application of chiral effective Lagrangian to relatively large momentum transfer makes it possible to further investigation of hadron quantities at high ${Q}^{2}$.

Chiral extrapolation of nucleon axial charge gA in effective field theory**Supported by National Natural Science Foundation of China (11475186) and Sino-German CRC 110 (NSFC 11621131001)

The extrapolation of nucleon axial charge gA is investigated within the framework of heavy baryon chiral effective field theory. The intermediate octet and decuplet baryons are included in the one loop calculation. Finite range regularization is applied to improve the convergence in the quark-mass expansion. The lattice data from three different groups are used for the extrapolation. At physical pion mass, the extrapolated gA are all smaller than the experimental value.

Spin of the proton in chiral effective field theory

Proton spin is investigated in chiral effective field theory through an examination of the singlet axial charge, $a_0$, and the two non-singlet axial charges, $a_3$ and $a_8$. Finite-range regularization is considered as it provides an effective model for estimating the role of disconnected sea-quark loop contributions to baryon observables. Baryon octet and decuplet intermediate states are included to enrich the spin and flavour structure of the nucleon, redistributing spin under the constraints of chiral symmetry. In this context, the proton spin puzzle is well understood with the calculation describing all three of the axial charges reasonably well. The strange quark contribution to the proton spin is negative with magnitude 0.01. With appropriate $Q^2$ evolution, we find the singlet axial charge at the experimental scale to be ${\hat a}_0 = 0.31^{+0.04}_{-0.05}$, consistent with the range of current experimental values.

Pure sea-quark contributions to the magnetic form factors ofΣbaryons

We propose the pure sea-quark contributions to the magnetic form factors of $\Sigma$ baryons, $G_{\Sigma^-}^u$ and $G_{\Sigma^+}^d$, as priority observables for the examination of sea-quark contributions to baryon structure, both in present lattice QCD simulations and possible future experimental measurement. $G_{\Sigma^-}^u$, the $u$-quark contribution to the magnetic form factor of $\Sigma^-$, and $G_{\Sigma^+}^d$, the $d$-quark contribution to the magnetic form factor of $\Sigma^+$, are similar to the strange quark contribution to the magnetic form factor of the nucleon, but promise to be larger by an order of magnitude. We explore the size of this quantity within chiral effective field theory, including both octet and decuplet intermediate states. The finite range regularization approach is applied to deal with ultraviolet divergences. Drawing on an established connection between quenched and full QCD, this approach makes it possible to predict the sea quark contribution to the magnetic form factor purely from the meson loop. In the familiar convention where the quark charge is set to unity $G_{\Sigma^-}^u = G_{\Sigma^+}^d$. We find a value of $-0.38^{+0.16}_{-0.17}\ \mu_N$, which is about seven times larger than the strange magnetic moment of the nucleon found in the same approach. Including quark charge factors, the $u$-quark contribution to the $\Sigma^-$ magnetic moment exceeds the strange quark contribution to the nucleon magnetic moment by a factor of 14.

Electric form factors of the octet baryons from lattice QCD and chiral extrapolation

We apply a formalism inspired by heavy baryon chiral perturbation theory with finite-range regularization to dynamical $2+1-$flavor CSSM/QCDSF/UKQCD Collaboration lattice QCD simulation results for the electric form factors of the octet baryons. The electric form factor of each octet baryon is extrapolated to the physical pseudoscalar masses, after finite-volume corrections have been applied, at six fixed values of $Q^2$ in the range 0.2-1.3 GeV$^2$. The extrapolated lattice results accurately reproduce the experimental form factors of the nucleon at the physical point, indicating that omitted disconnected quark loop contributions are small. Furthermore, using the results of a recent lattice study of the magnetic form factors, we determine the ratio $\mu_p G^p_E/G^p_M$. This quantity decreases with $Q^2$ in a way qualitatively consistent with recent experimental results.

Magnetic form factors of the octet baryons from lattice QCD and chiral extrapolation

We present a $2+1$-flavor lattice QCD calculation of the electromagnetic Dirac and Pauli form factors of the octet baryons. The magnetic Sachs form factor is extrapolated at six fixed values of ${Q}^{2}$ to the physical pseudoscalar masses and infinite volume using a formulation based on heavy-baryon chiral perturbation theory with finite-range regularization. We properly account for omitted disconnected quark contractions using a partially quenched effective field theory formalism. The results compare well with the experimental form factors of the nucleon and the magnetic moments of the octet baryons.

Strange magnetic form factor of the nucleon in a chiral effective model at next to leading order

The strange magnetic form factor of the nucleon is studied in a chiral effective model motivated by the heavy baryon formalism at next-to-leading order. The one-loop contributions from kaon and intermediate octet and decuplet hyperons are included, using finite-range regularization to deal with the ultraviolet divergences. Drawing on an established connection between quenched and full QCD, this model makes it possible to predict the strange magnetic form factor under the hypothesis that for a dipole regulator mass $\mathrm{\ensuremath{\Lambda}}$ around 0.8 GeV, strangeness in the core is negligible. The strange magnetic form factor is found to be small and negative over a range of momentum transfer, while the strange magnetic moment is consistent with the best lattice QCD determinations.

SU(3) chiral perturbation theory expansion of moments of quark distributions

We present formulas for the chiral extrapolation of spin-dependent and spin-independent moments of quark distributions of octet baryons, including loop corrections and counterterms, to leading nonanalytic order. This analysis allows for isospin breaking, and may be used for the chiral extrapolation of both ($2+1$)- and ($1+1+1$)-flavor lattice QCD results. An example of such an application is given, with the extrapolation formulas applied, using the finite-range regularization scheme, to recent ($2+1$)-flavor QCDSF/UKQCD Collaboration lattice results for the first spin-independent and first two spin-dependent Mellin moments.

First moments of nucleon generalized parton distributions

We extrapolate the first moments of the generalized parton distributions using heavy baryon chiral perturbation theory. The calculation is based on the one loop level with the finite range regularization. The description of the lattice data is satisfactory and the extrapolated moments at physical pion mass are consistent with the results obtained with dimensional regularization, although the extrapolation in the momentum transfer to $t=0$ does show sensitivity to form factor effects which lie outside the realm of chiral perturbation theory. We discuss the significance of the results in the light of modern experiments as well as QCD inspired models.

Strange magnetic form factor of the proton atQ2=0.23 GeV2

We determine the $u$ and $d$ quark contributions to the proton magnetic form factor at finite momentum transfer by applying chiral corrections to quenched lattice data. Heavy baryon chiral perturbation theory is applied at next-to-leading order in the quenched and full QCD cases for the valence sector using finite range regularization. Under the assumption of charge symmetry these values can be combined with the experimental values of the proton and neutron magnetic form factors to deduce a relatively accurate value for the strange magnetic form factor at ${Q}^{2}=0.23 {\mathrm{GeV}}^{2}$, namely, ${G}_{M}^{s}=\ensuremath{-}0.034\ifmmode\pm\else\textpm\fi{}0.021 {\ensuremath{\mu}}_{N}$.

Chiral extrapolation of octet-baryon charge radii

The charge radii of octet-baryons obtained in quenched lattice-QCD calculations are extrapolated within heavy-baryon chiral perturbation theory. Finite-range regularization is applied to improve the convergence of the chiral expansion and to provide estimates of quenching artifacts. Lattice values of quark distribution radii and baryon charge radii for ${m}_{\ensuremath{\pi}}^{2}$ in the range $(0.1,0.7)\text{ }\text{ }{\mathrm{GeV}}^{2}$ are described very well with finite-range regularization. Upon estimating corrections for both finite-volume and quenching effects, the obtained charge radii of the proton, neutron and ${\ensuremath{\Sigma}}^{\ensuremath{-}}$ are in good agreement with experimental measurements. The predicted charge radii of the remaining octet-baryons have not yet been measured and present a challenge to future experiments.

Chiral extrapolation of nucleon magnetic form factors

The extrapolation of nucleon magnetic form factors calculated within lattice QCD is investigated within a framework based upon heavy baryon chiral effective-field theory. All one-loop graphs are considered at arbitrary momentum transfer and all octet and decuplet baryons are included in the intermediate states. Finite range regularization is applied to improve the convergence in the quark-mass expansion. At each value of the momentum transfer (${Q}^{2}$), a separate extrapolation to the physical pion mass is carried out as a function of ${m}_{\ensuremath{\pi}}$ alone. Because of the large values of ${Q}^{2}$ involved, the role of the pion form factor in the standard pion-loop integrals is also investigated. The resulting values of the form factors at the physical pion mass are compared with experimental data as a function of ${Q}^{2}$ and demonstrate the utility and accuracy of the chiral extrapolation methods presented herein.

Chiral extrapolation of lattice data for B-meson decay constant

The B-meson decay constant fB has been calculated from unquenched lattice QCD in the unphysical region. For extrapolating the lattice data to the physical region, we propose a phenomenological functional form based on the effective chiral perturbation theory for heavy mesons, which respects both the heavy quark symmetry and the chiral symmetry, and the non-relativistic constituent quark model which is valid at large pion masses. The inclusion of pion loop corrections leads to non-analytic contributions to fB when the pion mass is small. The finite-range regularization technique is employed for the re-summation of higher-order terms of the chiral expansion. We also take into account the finite volume effects in lattice simulations. The dependence on the parameters and other uncertainties in our model are discussed.

Extrapolation of lattice QCD results beyond the power-counting regime

Resummation of the chiral expansion is necessary to make accurate contact with current lattice simulation results of full QCD. Resummation techniques including relativistic formulations of chiral effective field theory and finite-range regularization (FRR) techniques are reviewed, with an emphasis on using lattice simulation results to constrain the parameters of the chiral expansion. We illustrate how the chiral extrapolation problem has been solved and use FRR techniques to identify the power-counting regime (PCR) of chiral perturbation theory. To fourth-order in the expansion at the 1% tolerance level, we find 0 \le m_pi \le 0.18 GeV for the PCR, extending only a small distance beyond the physical pion mass.