Using the framework of non-relativistic effective field theory, the finite-volume ground-state energy shift is calculated up-to-and-including $O(L^{-6})$ for the system of three pions in the channel with the total isospin $I=1$. The relativistic corrections are included perturbatively, up to the same order in the inverse of the box size $L$. The obtained explicit expression, together with the known result for the system with maximal isospin $I=3$, can be used for the extraction of two independent effective three-body couplings from the measured ground-state spectrum of three pions.
We solve the coupled system of Dyson-Schwinger and Bethe-Salpeter equations for the quark propagator and the pion Bethe-Salpeter amplitude on a finite volume. To this end we use a truncation scheme that includes pion cloud effects in the quark propagator and light mesons. We study volume effects in the quark condensate, the pion mass and the pion decay constant and compare to corresponding results in other approaches. In general we find large effects for volumes below V=(1.8 fm)^4.
We derive an asymptotic formula a la Luscher for the finite volume correction to the pion decay constant: this is expressed as an integral over the < 3 π| A_μ|0 > amplitude after proper subtraction of the pion pole contribution. We analyze the formula numerically at leading and next-to-leading order in the chiral expansion.
We evaluate the pion mass in finite volume to two loops within Chiral Perturbation Theory. The results are compared with a recently proposed extension of the asymptotic formula of Luscher. We find that contributions, which were neglected in the latter, are numerically very small at the two-loop level and conclude that for Mpi*L>2, L>2fm the finite volume effects in the meson sector are analytically well under control.
We calculate the pion and kaon electromagnetic form factors in finite volume for a generic momentum transfer in space-like region, using SU(3) chiral perturbation theory. To this end, we first find the hadronic matrix element in a new form which is suitable for our calculation in finite volume. We present our numerical findings for the chiral behavior and finite volume corrections and compare these for pion and kaon form factors. As a result, for the pion electromagnetic form factor we find the finite volume correction $ΔF^π_{em} = 0.0041$ at momentum transfer $q^2 = -0.1358$ and for the kaon …
We investigate how Lüscher's relation between the finite-volume energy of two pions at rest and pion scattering lengths has to be modified in quenched QCD. We find that this relation changes drastically, and in particular, that ``enhanced finite-volume corrections" of order $L^0=1$ and $L^{-2}$ occur at one loop ($L$ is the linear size of the box), due to the special properties of the $η'$ in the quenched approximation. We define quenched pion scattering lengths, and show that they are linearly divergent in the chiral limit. We estimate the size of these various effects in some numerical examp…
I present higher loop order results for several calculations in Chiral perturbation Theory. 1) Two-loop results at finite volume for hadronic vacuum polarization. 2) A three-loop calculation of the pion mass and decay constant in two-flavour ChPT. For the pion mass all needed auxiliary parameters can be determined from lattice calculations of $ππ$-scattering. 3) Chiral corrections to neutron-anti-neutron oscillations.
We determine the relative pion mass shift $M_π(L)/M_π-1$ due to the finite spatial extent $L$ of the box by means of two-flavor chiral perturbation theory and the one-particle Lüscher formula. We use as input the expression for the infinite volume $ππ$ forward scattering amplitude up to next-to-next-to-leading order and can therefore control the convergence of the chiral series. A comparison to the full leading order chiral expression for the pion mass in finite volume allows us to check the size of subleading terms in the large-$L$ expansion.
We calculate the scalar semileptonic kaon decay in finite volume at the momentum transfer $t_{m} = (m_{K} - m_π)^2$, using chiral perturbation theory. At first we obtain the hadronic matrix element to be calculated in finite volume. We then evaluate the finite size effects for two volumes with $L = 1.83 fm$ and $L= 2.73 fm$ and find that the difference between the finite volume corrections of the two volumes are larger than the difference as quoted in \cite{Boyle2007a}. It appears then that the pion masses used for the scalar form factor in ChPT are large which result in large finite volume co…
The aim of this paper is to investigate an efficient strategy that allows to obtain pi-pi phase shifts and rho meson properties from QCD lattice data with high precision. For this purpose we evaluate the levels of the pi-pi system in the rho channel in finite volume using chiral unitary theory. We investigate the dependence on the pi mass and compare with other approaches which use QCD lattice calculations and effective theories. We also illustrate the errors induced by using the conventional Luscher approach instead of a more accurate one recently developed that takes into account exactly the…
We use chiral perturbation theory to study the extraction of pion electromagnetic polarizabilities from lattice QCD. Chiral extrapolation formulae are derived for partially quenched QCD, and quenched QCD simulations. On a torus, volume dependence of electromagnetic observables is complicated by SO(4) breaking, as well as photon zero-mode interactions. We determine finite volume corrections to the Compton scattering tensor of pions. We argue, however, that such results cannot be used to ascertain volume corrections to polarizabilities determined in lattice QCD with background field methods. Con…
We perform a calculation in one-loop chiral perturbation theory of the two-pion matrix elements and correlation functions of an I=0 scalar operator, in finite and infinite volumes for both full and quenched QCD. We show that major difficulties arise in the quenched theory due to the lack of unitarity. Similar problems are expected for quenched lattice calculations of $K \to ππ$ amplitudes with $ΔI=1/2$. Our results raise the important question of whether it is consistent to study $K\toππ$ amplitudes beyond leading order in chiral perturbation theory in quenched or partially quenched QCD.
We compute finite volume effects relevant for lattice QCD simulations using background fields. Focusing on constant electric and magnetic fields on a periodic lattice, we determine volume corrections to pion two-point functions using chiral perturbation theory. Among such corrections are the finite volume shifts to the electric and magnetic polarizabilities, which are numerically shown to be non-negligible. We additionally find that all terms in the single-particle effective action can be renormalized by infrared effects. This includes Born couplings to the pion current and total charge-square…
The ground state energy of a collection of n pions and m kaons with short range interactions is calculated for a volume with finite spatial extent L and periodic boundary conditions. This calculation is accomplished to order L^-6 in the large volume expansion. With this result one can extract the various two- and three-body interactions between pions and kaons from lattice QCD data.
Pionless effective field theory in a finite volume (FVEFT$_{π\!/}$) is investigated as a framework for the analysis of multi-nucleon spectra and matrix elements calculated in lattice QCD (LQCD). By combining FVEFT$_{π\!/}$ with the stochastic variational method, the spectra of nuclei with atomic number $A\in\{2,3\}$ are matched to existing finite-volume LQCD calculations at heavier-than-physical quark masses corresponding to a pion mass $m_π=806$ MeV, thereby enabling infinite-volume binding energies to be determined using infinite-volume variational calculations. Based on the variational wave…
We evaluate the pion mass in finite volume to two loops within Chiral Perturbation Theory. The results are compared with a recently proposed extension of the asymptotic formula of Luscher. We find that contributions, which were neglected in the latter, are numerically very small at the two-loop level.
In a recent paper by Xiu-Li Gao, Zhi-Hui Guo, Zhiguang Xiao, and Zhi-Yong Zhou, Phys. Rev. D 105, 094002 (2022), here referred to as I, $S$-wave $ππ$ scattering phase shifts obtained in a lattice-QCD calculation are analyzed using dispersive $S$-matrix methods. We question the reliability of the conclusion from this analysis that, for a pion mass of 391 MeV, the lattice phases favor the presence of both a $σ$-meson bound state and a nearby virtual state. Our main criticism concerns the neglect of the $S$-wave $K\bar{K}$ channel, which was considered alongside additional $s\bar{s}$ interpolatin…
Progress in calculating scattering phase shifts on $N_f=2+1$ anisotropic clover Wilson lattices is described. The stochastic LapH method facilitates computations in large volumes and for light pion masses. Results for pion masses down to 240 MeV, keeping $m_πL > 4$, are presented.
We report on the MILC Collaboration calculation of electromagnetic effects on light pseudoscalar mesons. The simulations employ asqtad staggered dynamical quarks in QCD plus quenched photons, with lattice spacings varying from 0.12 to 0.06 fm. Finite volume corrections for the MILC realization of lattice electrodynamics have been calculated in chiral perturbation theory and applied to the lattice data. These corrections differ from those calculated by Hayakawa and Uno because our treatment of zero modes differs from theirs. Updated results for the corrections to "Dashen's theorem" are presente…
In the chiral limit of QCD, the pion decay constant F can be extracted from lattice gauge theory by means of a coupling to isospin chemical potential. Here we compute the leading correction due to finite volume in the epsilon-expansion of chiral perturbation theory. A comparison is made to recent Monte Carlo data.
We present a calculation of the scattering phase shift for the I=2 S-wave pion-pion system in the continuum limit with two-flavor full QCD. Calculations are made at three lattice spacings, using the finite volume method of Lüscher in the center of mass frame, and its extension to the laboratory frame.
We present a study of the finite-volume two-pion matrix elements and correlation functions of the I=0 scalar operator, in full and partially quenched QCD, at one-loop order in chiral perturbation theory. In partially quenched QCD, when the sea and valence light quark masses are not equal, the lack of unitarity leads to the same inconsistencies as in quenched QCD and the matrix elements cannot be determined. It is possible, however, to overcome this problem by requiring the masses of the valence and sea quarks to be equal for the u and d quarks while keeping the strange quark (s) quenched (or p…
We investigate the utility of partially twisted boundary conditions in lattice calculations of meson observables. For dynamical simulations, we show that the pion dispersion relation is modified by volume effects. In the isospin limit, we demonstrate that the pion electromagnetic form factor can be computed on the lattice at continuous values of the momentum transfer. Furthermore, the finite volume effects are under theoretical control for extraction of the pion charge radius.
We present the three-pion spectrum with maximal isospin in a finite volume determined from lattice QCD, including excited states in addition to the ground states across various irreducible representations at zero and nonzero total momentum. The required correlation functions, from which the spectrum is extracted, are computed using a newly implemented algorithm which speeds up the computation by more than an order of magnitude. On a subset of the data we extract a nonzero value of the three-pion threshold scattering amplitude using the $1/L$ expansion of the three-particle quantization conditi…
Finite-volume pionless effective field theory provides an efficient framework for the extrapolation of nuclear spectra and matrix elements calculated at finite volume in lattice QCD to infinite volume, and to nuclei with larger atomic number. In this work, it is demonstrated how this framework may be implemented via a set of correlated Gaussian wavefunctions optimised using differentiable programming and via solution of a generalised eigenvalue problem. This approach is shown to be significantly more efficient than a stochastic implementation of the variational method based on the same form of…
We calculate the finite volume effects for the pion and nucleon mass. For the pion mass we present the results of a full two-loop calculation in chiral perturbation theory. The outcome shows that the resummed version of the Luscher formula we presented in an earlier work does indeed give an excellent approximation to the two-loop result. In view of this result we apply the same resummed formula to the nucleon mass. In the nucleon sector the extension of the chiral expansion to higher quark masses appears to be more problematic and it is therefore more difficult to make reliable predictions for…
We consider finite volume effects on the electro-magnetic pion form factor near the chiral limit, in the so-called $ε$ regime. The pseudoscalar-vector-pseudoscalar three-point function is calculated in the $ε$ expansion of chiral perturbation theory to the next-to-leading order. In the $ε$ regime, finite volume effects are non-perturbatively large in general. However, we find a way to remove its dominant part, by inserting momenta to the correlators, and taking an appropriate ratio of them. The subleading contribution is, then, shown to be perturbatively small, and one can extract the form fac…
We consider finite volume effects on the electromagnetic form factor of the pion. We compute the peudoscalar-vector-pseudoscalar correlator in the $ε$ expansion of chiral perturbation theory up to the next-to-leading order and find a way to remove the dominant part, which comes from a contribution of the pion zero-mode. Inserting non-zero momentum to relevant operators (or taking a subtraction of the correlators at different time-slices), and taking an appropriate ratio of them, one can automatically cancel the zero-mode's contribution, which becomes non-perturbatively large $\sim \mathcal{O}(…
We calculate finite volume effects on the pion masses and decay constant in twisted mass lattice QCD (tmLQCD) at finite lattice spacing. We show that the lighter neutral pion in tmLQCD gives rise to finite volume effects that are exponentially enhanced when compared to those arising from the heavier charged pions. We demonstrate that the recent two flavour twisted mass lattice data can be better fitted when twisted mass effects in finite volume corrections are taken into account.
We present a lattice QCD calculation of the scattering phase shift for the I=2 $S$-wave two-pion system using the finite size method proposed by Lüscher. We work in the quenched approximation employing the standard plaquette action at $β=5.9$ for gluons and the Wilson fermion action for quarks. The phase shift is extracted from the energy eigenvalues of the two-pion system, which are obtained by a diagonalization of the pion 4-point function evaluated for a set of relative spatial momenta. In order to change momentum of the two-pion system, calculations are carried out on $24^3\times 60$, $32^…
We study the finite-volume correction on the hadronic vacuum polarization contribution to the muon g-2 ($a_μ^{\rm hvp}$) in lattice QCD at (near) physical pion mass using two different volumes: $(5.4~{\rm fm})^4$ and $(8.1~{\rm fm})^4$. We use an optimized AMA technique for noise reduction on $N_f=2+1$ PACS gauge configurations with stout-smeared clover-Wilson fermion action and Iwasaki gauge action at a single lattice cut-off $a^{-1}=2.33$ GeV. The calculation is performed for the quark-connected light-quark contribution in the isospin symmetric limit. We take into account the effects of back…
We propose an alternative approach to Lüscher's formula for extracting two-body scattering phase shifts from finite volume spectra with no reliance on the partial wave expansion. We use an effective-field-theory-based Hamiltonian method in the plane wave basis and decompose the corresponding matrix elements of operators into irreducible representations of the relevant point groups. The proposed approach allows one to benefit from the knowledge of the long-range interaction and avoids complications from partial wave mixing in a finite volume. We consider spin-singlet channels in the two-nucleon…
The physics of pions within a finite volume is explored using lattice regularized chiral perturbation theory. This regularization scheme permits a straightforward computational approach to be used in place of analytical continuum techniques. Using the pion mass, decay constant, form factor and charge radius as examples, it is shown how numerical results for volume dependences are obtained at the one-loop level from simple summations.
Using the tadpole improved clover Wilson quark action on coarse anisotropic lattices, the $ππ$ scattering length in the I=2 channel is calculated within quenched approximation. We show that such a calculation is feasible using small lattices on small computers provided that the finite volume and finite lattice spacing errors are under control. Our results are extrapolated towards the chiral, infinite volume and continuum limit. Comparisons of our results with previous lattice results from JLQCD collaboration, the new results from E865 experiment, and the results from Chiral Perturbation Theory…
Knowledge of the electric polarizability is crucial to understanding the interactions of hadrons with electromagnetic fields. The neutron polarizability is very sensitive to the quark mass and is expected to diverge in the chiral limit. Here we present results for the electric polarizability of the neutron, neutral pion, and neutral kaon on eight ensembles with nHYP-smeared clover dynamical fermions with two different pion masses (227 and 306 MeV). These are currently the lightest pion masses used in polarizability studies. For each pion mass we compute the polarizability at four different vol…
We study the scaling behavior of the two-flavor chiral phase transition using an effective quark-meson model. We investigate the transition between infinite-volume and finite-volume scaling behavior when the system is placed in a finite box. We can estimate effects that the finite volume and the explicit symmetry breaking by the current quark masses have on the scaling behavior which is observed in full QCD lattice simulations. The model allows us to explore large quark masses as well as the chiral limit in a wide range of volumes, and extract information about the scaling regimes.In particula…
The question of the exact nature of the phase transition in two-flavor QCD is still under discussion. Recent results for small quark masses in simulations with 2+1 flavors show scaling behavior consistent with the O(4) or O(2) universality class. For a precise determination, an assessment of deviations from the ideal scaling behavior due to finite quark masses and finite simulation volumes is necessary. We study the scaling behavior at the chiral phase transition with an effective quark-meson model. In our Renormalization Group approach, the quark masses in the model can be varied from the chi…
We calculate the volume dependence of the ground-state energy of n identical bosons with short-range repulsive interactions in a periodic spatial volume of side L, up to and including terms of order 1/L^6. With this result, Lattice QCD calculations of the ground-state energies of three or more pions will allow for a systematic extraction of the three-pion interaction at this order in the volume expansion.
We study the finite volume effects on the electric polarizability for the neutron, neutral pion, and neutral kaon using eight dynamically generated two-flavor nHYP-clover ensembles at two different pion masses: 306(1) and 227(2) MeV. An infinite volume extrapolation is performed for each hadron at both pion masses. For the neutral kaon, finite volume effects are relatively mild. The dependence on the quark mass is also mild and a reliable chiral extrapolation can be performed along with the infinite volume extrapolation. Our result is $α_{K^0}^\mbox{phys}=0.356(74) \times 10^{-4} \mbox{fm}^3$.…
We calculate a two-pion wave function for the I=2 $S$-wave two-pion system with a finite scattering momentum and estimate the interaction range between two pions, which allows us to examine the validity of a necessary condition for the finite size formula presented by Rummukainen and Gottlieb. We work in the quenched approximation employing the plaquette gauge action for gluons and the improved Wilson action for quarks at $1/a=1.63 {\rm GeV}$ on $32^3\times 120$ lattice. The quark masses are chosen to give $m_π= 0.420$, 0.488 and $0.587 {\rm GeV}$. We find that the energy dependence of the int…
We propose a new improved model-independent method for calculating the pion charge radius. In a recently-proposed model-independent method for the pion charge radius, we find it difficult to compute the pion charge radius for small pole mass $M_{\rm{pole}}^2$ and volume due to systematic errors coming from finite volume effect and higher-order contamination of the Taylor expansion of the form factor. We circumvent this difficulty by introducing a new appropriate function and propose a modified method that can calculate the pion charge radius with less systematic errors in the small $M_{\rm{pol…
We present preliminary results of scattering length and phase shift for I=2 S-wave $ππ$ system with the Wilson fermions in the quenched approximation. The finite size method presented by Lüscher is employed, and calculations are carried out at $β=5.9$ on a $24^3\times 60$ and $32^3\times 60$ lattice.
We present results of phase shift for I=2 $S$-wave $ππ$ system with the Wilson fermions in the quenched approximation. The finite size method proposed by Lüscher is employed, and calculations are carried out at $β=5.9$ ($a^{-1}=1.934(16)$ GeV from $m_ρ$) on $24^3 \times 60$, $32^3 \times 60$, and $48^3 \times 60$ lattices.
The ab-initio understanding of hadronic three-body systems above threshold, such as exotic resonances or the baryon spectrum, requires the mapping of the finite-volume eigenvalue spectrum, produced in lattice QCD calculations, to the infinite volume. We present the first application of such a formalism to a physical system in form of three interacting positively charged pions. The results for the ground state energies agree with the available lattice QCD results by the NPLQCD collaboration at unphysical pion masses. Extrapolations to physical pion masses are performed using input from effectiv…
Due to a pion-pole contribution, the nucleon axial current matrix elements can be visualized in position space as becoming delocalized as the chiral limit is approached, with one-third of the current at distance of order the inverse pion mass. However, this delocalization effect will not cause calculations of the axial coupling in a finite system with standard boundary conditions (periodic for boson fields, antiperiodic for fermions) to have large finite volume effects. This is seen by calculating axial coupling using a nonstandard current with axial quantum numbers. The matrix elements of thi…
Electromagnetic pion mass splitting using a Pauli-Villars-regulated photon propagator
Helmholtz Institut Mainz（亥姆霍兹美因茨研究所）；Johannes Gutenberg-Universität Mainz（约翰内斯·古腾堡美因茨大学）；GSI Helmholtz Centre for Heavy Ion Research（GSI重离子研究中心）；PRISMA + Cluster of Excellence & Institut für Kernphysik, Johannes Gutenberg-Universität Mainz（PRISMA卓越集群与约翰内斯·古腾堡美因茨大学核物理研究所）；Theoretical Physics Department, CERN（欧洲核子研究中心理论物理部）
Phase shifts for $s$-wave $ππ$ scattering in both the $I=0$ and $I=2$ channels are determined from a lattice QCD calculation performed on 741 gauge configurations obeying G-parity boundary conditions with a physical pion mass and lattice size of $32^3\times 64$. These results support our recent study of direct CP violation in $K\toππ$ decay \cite{Abbott:2020hxn}, improving our earlier 2015 calculation \cite{Bai:2015nea}. The phase shifts are determined for both stationary and moving $ππ$ systems, at three ($I=0$) and four ($I=2$) different total momenta. We implement several $ππ$ interpolating…
The quark mass dependence of hadrons is an important input for lattice calculations. We investigate the light quark mass dependence of the binding energy of the X(3872) in a finite box to next-to-leading order in an effective field theory for the X(3872) with perturbative pions (XEFT). At this order, the quark mass dependence is determined by a quark mass-dependent contact interaction in addition to the one-pion exchange. While there is only a moderate sensitivity to the light quark masses in the region up to twice their physical value, the finite volume effects are significant already at box …
We conduct a two-flavor ($N_f=2$) lattice QCD calculation of the elastic phase-shifts for pion-pion scattering in the scalar, isoscalar channel (the $σ$-meson). The calculation is performed for two quark masses corresponding to a pion mass of $315\text{ MeV}$ and $227\text{ MeV}$. The $σ$-meson parameters are extracted using various parametrizations of the scattering amplitude. The results obtained from a chiral unitary parametrization are extrapolated to the physical point and read $M_σ= (440^{+10}_{-16}(50) - i\,240(20)(25))\text{ MeV}$, where the uncertainties in the parentheses denote the …
It is shown that the strong volume-dependence of the axial charge of the nucleon seen in lattice QCD calculations can be understood quantitatively in terms of the pion-induced interactions between neighbouring nucleons. The associated wave function renormalization leads to an increased suppression of the axial charge as the strength of the interaction increases, either because of a decrease in lattice size or in pion mass.
In an exploratory study, using the tadpole improved clover Wilson quark action on small, coarse and anisotropic lattices, the $ππ$ scattering length in the I=2 channel is calculated within quenched approximation. A new method is proposed which enables us to make chiral extrapolation of our lattice results without calculating the decay constant on the lattice. Finite volume and finite lattice spacing errors are analyzed and the results are extrapolated towards the infinite volume and continuum limit. Comparisons of our lattice results with the new experiment and the results from Chiral Perturba…
We demonstrate the efficacy of the stochastic LapH method to treat all-to-all quark propagation on a $N_f = 2+1$ CLS ensemble with large linear spatial extent $L = 5.5$ fm, allowing us to obtain the benchmark elastic isovector p-wave pion-pion scattering amplitude to good precision already on a relatively small number of gauge configurations. These results hold promise for multi-hadron spectroscopy at close-to-physical pion mass with exponential finite-volume effects under control.
We perform a fit of the finite-volume QCD spectrum of three pions at maximal isospin to constrain the three-body force. We use the unitarity-based relativistic three-particle quantization condition, with the GWUQCD spectrum obtained at 315 MeV and 220 MeV pion mass in two-flavor QCD. For the heavier pion mass we find that the data is consistent with a constant contact term close to zero, whereas for the lighter mass we see a statistically significant energy dependence in tension with the prediction of leading order ChPT. Our results also suggest that with enough three-body energy levels, the t…
