Three-body unitarity versus finite-volume π + π + π + spectrum from lattice QCD

2020

J. High Energ. Phys.

We present a generalization of the relativistic, finite-volume, three-particle quantization condition for non-identical pions in isosymmetric QCD. The resulting formalism allows one to use discrete finite-volume energies, determined using lattice QCD, to constrain scattering amplitudes for all possible values of two-and three-pion isospin. As for the case of identical pions considered previously, the result splits into two steps: the first defines a non-perturbative function with roots equal to the allowed energies, E n (L), in a given cubic volume with side-length L. This function depends on an intermediate three-body quantity, denoted K df,3 , which can thus be constrained from lattice QCD input. The second step is a set of integral equations relating K df,3 to the physical scattering amplitude, M 3. Both of the key relations, E n (L) ↔ K df,3 and K df,3 ↔ M 3 , are shown to be block-diagonal in the basis of definite three-pion isospin, I πππ , so that one in fact recovers four independent relations, corresponding to I πππ = 0, 1, 2, 3. We also provide the generalized threshold expansion of K df,3 for all channels, as well as parameterizations for all three-pion resonances present for I πππ = 0 and I πππ = 1. As an example of the utility of the generalized formalism, we present a toy implementation of the quantization condition for I πππ = 0, focusing on the quantum numbers of the ω and h 1 resonances.

Section: Jhep07(2020)047mentioning

confidence: 99%

Generalizing the relativistic quantization condition to include all three-pion isospin channels

Hansen

Romero-López

2020

J. High Energ. Phys.

“…We label theseB 2;L and B 3 respectively, while the left (right) endcaps are denotedÂ 0 (Â). 9 Before presenting the general expression for C 3;L , we first give the definitions of these four segments. The "hat" on these quantities is used to distinguish them from similar, but different, endcaps denoted A 0 and A in HS1.…”

Section: B Expansion Of C 3;l ðE; Pþ In Relevant Cutsmentioning

confidence: 99%

“…Examples of the successful combination of these methods are in Refs. [1][2][3][4][5][6][7][8][9][10][11]. 1 A key output of the theoretical formalism is a quantization condition, an equation whose solutions give the finite-volume three-particle energy levels in terms of infinite-volume scattering quantities.…”

Section: Introductionmentioning

confidence: 99%

“…(3) The "finite-volume unitarity" (FVU) approach of Refs. [6,9,10,49]. This is a relativistic formalism that is based on general forms for the three-particle scattering amplitude developed in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…It has been applied to the 3π þ spectrum from LQCD in Refs. [9,10]. Our aim, as already noted, is to simplify the RFT derivation given in HS1 and HS2.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Alternative derivation of the relativistic three-particle quantization condition

Blanton

2020

Phys. Rev. D

We present a simplified derivation of the relativistic three-particle quantization condition for identical, spinless particles described by a generic relativistic field theory satisfying a Z 2 symmetry. The simplification is afforded by using a three-particle quasilocal K matrix that is not fully symmetrized, K ðu;uÞ df;3 , and makes extensive use of time-ordered perturbation theory (TOPT). We obtain a new form of the quantization condition. This new form can then be related algebraically to the standard quantization condition, which depends on a fully symmetric three-particle K matrix, K df;3. The new derivation is fully explicit, allowing, for example, a closed-form expression for K df;3 to be given in terms of TOPT amplitudes. The new form of the quantization condition is similar in structure to that obtained in the "finite-volume unitarity" approach, and in a companion paper we make this connection concrete. Our simplified approach should also allow a more straightforward generalization of the quantization condition to nondegenerate particles, and perhaps also to more than three particles.

Erratum to: Generalizing the relativistic quantization condition to include all three-pion isospin channels

Hansen

Romero-López

2021

J. High Energ. Phys.

We have found an error in a statement following eq. (2.5) of our paper, concerning the function f (a, b, k) that first appears in that equation. The issue arises in the statement that it is convenient to take the function f (a, b, k) to be exchange symmetric with respect to its three arguments. This has the unwanted consequence of making six of the seven operators in the column vector of eq. (2.4) identically equal. This, in turn, implies that many operators are identically zero in the definite isospin basis, considered in section 2.4. To repair this, the last sentence of the paragraph containing eq. (2.4), starting “It is convenient for the subsequent…”, should be removed, as should footnote 3 and the next paragraph, beginning with “At this point, the reader may wonder why…”.