Consistency checks for two-body finite-volume matrix elements: II. Perturbative systems

Abstract

Using the general formalism presented in Refs. [1,2], we study the finite-volume effects for the $2+\mathcal{J}\to 2$ matrix element of an external current coupled to a two-particle state of identical scalars with perturbative interactions. Working in a finite cubic volume with periodicity $L$, we derive a $1/L$ expansion of the matrix element through $\mathcal{O}(1/L^5)$ and find that it is governed by two universal current-dependent parameters, the scalar charge and the threshold two-particle form factor. We confirm the result through a numerical study of the general formalism and additionally through an independent perturbative calculation. We further demonstrate a consistency with the Feynman-Hellmann theorem, which can be used to relate the $1/L$ expansions of the ground-state energy and matrix element. The latter gives a simple insight into why the leading volume corrections to the matrix element have the same scaling as those in the energy, $1/L^3$, in contradiction to earlier work, which found a $1/L^2$ contribution to the matrix element. We show here that such a term arises at intermediate stages in the perturbative calculation, but cancels in the final result.

arXiv comment17 pages, 4 figures

References in corpus (84)

• Ab-initio Determination of Light Hadron Masses
• Resonance Scattering Phase Shifts on a Non-Rest Frame Lattice
• Ab initio calculation of the neutron-proton mass difference
• Finite-Volume Effects for Two-Hadron States in Moving Frames
• Weak transition matrix elements from finite-volume correlation functions
• Scattering processes and resonances from lattice QCD
• Multiple-channel generalization of Lellouch-Luscher formula
• A relativistic, model-independent, three-particle quantization condition
• Two-particle multichannel systems in a finite volume with arbitrary spin
• Moving Multi-Channel Systems in a Finite Volume with Application to Proton-Proton Fusion
• Expressing the three-particle finite-volume spectrum in terms of the three-to-three scattering amplitude
• Two Nucleons on a Lattice
• Energy dependence of the ρ resonance in ππ elastic scattering from lattice QCD
• Coupled $\pi \pi ,K\overline{K}$ scattering in $P$-wave and the $\rho$ resonance from lattice QCD
• Interactions of Charmed Mesons with Light Pseudoscalar Mesons from Lattice QCD and Implications on the Nature of the D_{s0}^*(2317)
• Isoscalar $\pi \pi$ scattering and the $\sigma$ meson resonance from QCD
• Two Particle States and the $S$-matrix Elements in Multi-channel Scattering
• Relating the finite-volume spectrum and the two-and-three-particle $S$ matrix for relativistic systems of identical scalar particles
• Finite Volume Corrections to the Two-Particle Decay of States with Non-Zero Momentum
• Three particle quantization condition in a finite volume: 2. general formalism and the analysis of data
• Standard-model prediction for direct CP violation in $K\to \pi \pi$ decay
• Resonances in coupled πK,ηK scattering from quantum chromodynamics
• Three-body Unitarity in the Finite Volume
• Light Hadron Masses from Lattice QCD
• Scattering phase shifts for two particles of different mass and non-zero total momentum in lattice QCD
• Resonances in coupled $\pi K,\eta K$ scattering from lattice QCD
• Lattice QCD and Three-particle Decays of Resonances
• An $a_0$ resonance in strongly coupled $\pi \eta$, $K\overline{K}$ scattering from lattice QCD
• Coupled-Channel $D\pi$, $D\eta$ and $D_s\bar{K}$ Scattering from Lattice QCD
• $K\to \pi \pi$ $\Delta I=3/2$ decay amplitude in the continuum limit
• Multichannel one-to-two transition amplitudes in a finite volume
• Two Nucleon Systems at $m_{\pi }\sim 450\mathrm{MeV}$ from Lattice QCD
• Isoscalar $\pi \pi ,K\overline{K},\eta \eta$ scattering and the $\sigma ,f_0,f_2$ mesons from QCD
• Two-Nucleon Higher Partial-Wave Scattering from Lattice QCD
• Multichannel 0-to-2 and 1-to-2 transition amplitudes for arbitrary spin particles in a finite volume
• Improving the Volume Dependence of Two-Body Binding Energies Calculated with Lattice QCD
• $I=1$ and $I=2$ $\pi -\pi$ scattering phase shifts from $N_{\mathrm{f}}=2+1$ lattice QCD
• Finite-volume spectrum of ${\pi }^{+}{\pi }^{+}$ and ${\pi }^{+}{\pi }^{+}{\pi }^{+}$ systems
• Resonance parameters of the rho-meson from asymmetrical lattices
• Nucleon-Nucleon Scattering Parameters in the Limit of SU(3) Flavor Symmetry
• Three-particle systems with resonant subprocesses in a finite volume
• Two- and three-pion finite-volume spectra at maximal isospin from lattice QCD
• $I=3$ three-pion scattering amplitude from lattice QCD
• Predicting positive parity $B_s$ mesons from lattice QCD
• n-Boson Energies at Finite Volume and Three-Boson Interactions
• Masses and decay constants of the $D_{s0}^{\ast }(2317)$ and $D_{s1}(2460)$ from $N_f=2$ lattice QCD close to the physical point
• Timelike pion form factor in lattice QCD
• The $I=1$ pion-pion scattering amplitude and timelike pion form factor from $N_{\mathrm{f}}=2+1$ lattice QCD
• $P$-wave $\pi \pi$ scattering and the $\rho$ resonance from lattice QCD
• Effects of finite volume on the $K_L$-$K_S$ mass difference
• Implementing the three-particle quantization condition including higher partial waves
• The $K\to (\pi \pi {)}_{I=2}$ Decay Amplitude from Lattice QCD
• Baryon-Baryon Interactions and Spin-Flavor Symmetry from Lattice Quantum Chromodynamics
• The I=2 pipi S-wave Scattering Phase Shift from Lattice QCD
• Numerical study of the relativistic three-body quantization condition in the isotropic approximation
• Three-body unitarity versus finite-volume ${\pi }^{+}{\pi }^{+}{\pi }^{+}$ spectrum from lattice QCD
• Two-Nucleon Systems in a Finite Volume: (I) Quantization Conditions
• The $\pi \pi \to \pi {\gamma }^{\star }$ amplitude and the resonant $\rho \to \pi {\gamma }^{\star }$ transition from lattice QCD
• Relativistic, model-independent, multichannel $2\to 2$ transition amplitudes in a finite volume
• The elastic $I=3/2$ $p$-wave nucleon-pion scattering amplitude and the $\Delta (1232)$ resonance from $N_{\mathrm{f}}=2+1$ lattice QCD
• The phase-shift of isospin-2 pi-pi scattering from lattice QCD
• The resonant ${\pi }^{+}\gamma \to {\pi }^{+}{\pi }^0$ amplitude from Quantum Chromodynamics
• Emerging understanding of the ΔI = 1/2 Rule from Lattice QCD
• Two-Nucleon Systems in a Finite Volume: (II) 3S1-3D1 Coupled Channels and the Deuteron
• Matrix elements of unstable states
• Extraction of isoscalar $\pi \pi$ phase-shifts from lattice QCD
• Determination of $s$- and $p$-wave $I=1/2$ $K\pi$ scattering amplitudes in $N_{\mathrm{f}}=2+1$ lattice QCD
• The quark-mass dependence of elastic $\pi K$ scattering from QCD
• The $b_1$ resonance in coupled $\pi \omega$, $\pi \varphi$ scattering from lattice QCD
• The Energy of n Identical Bosons in a Finite Volume at O(L^{-7})
• Perturbative results for two and three particle threshold energies in finite volume
• The $\pi \gamma \to \pi \pi$ transition and the $\rho$ radiative decay width from lattice QCD
• A cross-channel study of pion scattering from lattice QCD
• Dynamically-coupled partial-waves in $\rho \pi$ isospin-2 scattering from lattice QCD
• Form factors of two-hadron states from a covariant finite-volume formalism
• Two-flavor Simulations of the $\rho (770)$ and the Role of the $K\bar{K}$ Channel
• Long-range electroweak amplitudes of single hadrons from Euclidean finite-volume correlation functions
• Hadron-Hadron Interactions from $N_f=2+1+1$ Lattice QCD: The $\rho$-resonance
• Light-Neutrino Exchange and Long-Distance Contributions to $0\nu 2\beta$ Decays: An Exploratory Study on $\pi \pi \to ee$
• The $B\to K^{\ast }$ form factors on the lattice
• Testing the threshold expansion for three-particle energies at fourth order in ${\varphi }^4$ theory
• Consistency checks for two-body finite-volume matrix elements: I. Conserved currents and bound states
• Vector particle scattering on the lattice
• Finite-volume matrix elements in multi-boson states

Cited by in corpus (9)

• Towards grounding nuclear physics in QCD
• Nuclear Forces for Precision Nuclear Physics -- a collection of perspectives
• Decay amplitudes to three hadrons from finite-volume matrix elements
• Few-nucleon matrix elements in pionless effective field theory in a finite volume
• Constraining $1+\mathcal{J}\to 2$ coupled-channel amplitudes in finite-volume
• On-shell representations of two-body transition amplitudes: single external current
• Variations on the Maiani-Testa approach and the inverse problem
• Prospects for ${\gamma }^{\star }{\gamma }^{\star }\to \pi \pi$ via lattice QCD
• Form factors of the $\rho$ meson from effective field theory and the lattice