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Register a new userIdentification of shallow two-body bound states in finite volume
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Shoichi Sasaki
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We discuss signatures of bound-state formation in finite volume via the Luscher finite size method. Assuming that the phase-shift formula in this method inherits all aspects of the quantum scattering theory, we may expect that the bound-state formation induces the sign of the scattering length to be changed. If it were true, this fact provides us a distinctive identification of a shallow bound state even in finite volume through determination of whether the second lowest energy state appears just above the threshold. We also consider the bound-state pole condition in finite volume, based on Luschers phase-shift formula and then find that the condition is fulfilled only in the infinite volume limit, but its modification by finite size corrections is exponentially suppressed by the spatial lattice size L. These theoretical considerations are also numerically checked through lattice simulations to calculate the positronium spectrum in compact scalar QED, where the short-range interaction between an electron and a positron is realized in the Higgs phase.
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Recently, a framework has been developed to study form factors of two-hadron states probed by an external current. The method is based on relating finite-volume matrix elements, computed using numerical lattice QCD, to the corresponding infinite-volume observables. As the formalism is complicated, it is important to provide non-trivial checks on the final results and also to explore limiting cases in which more straightforward predications may be extracted. In this work we provide examples on both fronts. First, we show that, in the case of a conserved vector current, the formalism ensures that the finite-volume matrix element of the conserved charge is volume-independent and equal to the total charge of the two-particle state. Second, we study the implications for a two-particle bound state. We demonstrate that the infinite-volume limit reproduces the expected matrix element and derive the leading finite-volume corrections to this result for a scalar current. Finally, we provide numerical estimates for the expected size of volume effects in future lattice QCD calculations of the deuterons scalar charge. We find that these effects completely dominate the infinite-volume result for realistic lattice volumes and that applying the present formalism, to analytically remove an infinite-series of leading volume corrections, is crucial to reliably extract the infinite-volume charge of the state.
In this talk, we present a framework for studying structural information of resonances and bound states coupling to two-hadron scattering states. This makes use of a recently proposed finite-volume formalism to determine a class of observables that are experimentally inaccessible but can be accessed via lattice QCD. In particular, we shown that finite-volume two-body matrix elements with one current insertion can be directly related to scattering amplitudes coupling to the external current. For two-hadron systems with resonances or bound states, one can extract the corresponding form factors of these from the energy-dependence of the amplitudes.
In this work, based on consideration of periodicity and asymptotic forms of wave function, we propose a novel approach to the solution of finite volume three-body problem by mapping a three-body problem into a higher dimensional two-body problem. The idea is demonstrated by an example of two light spinless particles and one heavy particle scattering in one spatial dimension. This 1D three-body problem resembles a two-body problem in two spatial dimensions mathematically, and quantization condition of 1D three-body problem is thus derived accordingly.
Using the general formalism presented in Refs. [1,2], we study the finite-volume effects for the $mathbf{2}+mathcal{J}tomathbf{2}$ 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.
In this work we develop a Lorentz-covariant version of the previously derived formalism for relating finite-volume matrix elements to $textbf 2 + mathcal J to textbf 2$ transition amplitudes. We also give various details relevant for the implementation of this formalism in a realistic numerical lattice QCD calculation. Particular focus is given to the role of single-particle form factors in disentangling finite-volume effects from the triangle diagram that arise when $mathcal J$ couples to one of the two hadrons. This also leads to a new finite-volume function, denoted $G$, the numerical evaluation of which is described in detail. As an example we discuss the determination of the $pi pi + mathcal J to pi pi$ amplitude in the $rho$ channel, for which the single-pion form factor, $F_pi(Q^2)$, as well as the scattering phase, $delta_{pipi}$, are required to remove all power-law finite-volume effects. The formalism presented here holds for local currents with arbitrary Lorentz structure, and we give specific examples of insertions with up to two Lorentz indices.
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