We describe in detail the implementation of the relativistic three-neutron finite-volume quantization condition derived in ref. [1]. In particular, we show how the complications due to Wigner rotations acting on spins are included, and present concrete formulas for the case when the angular momenta within pairs is restricted to be less than 2. We describe the symmetries of the matrices appearing in the quantization condition, and decompose solutions into irreducible representations of the appropriate doubled finite-volume symmetry groups. We present an implementation of the three-particle K matrix, keeping the two lowest-order terms in the threshold expansion. We provide numerical predictions for the finite-volume spectrum for a setup with nearly physical parameters, including two-particle interactions that are based on experimental results. This exploratory study shows the how lattice QCD calculations of the three-neutron spectrum with sufficient precision can provide detailed information on both two- and three-particle interactions.

United States

Wilder Schaaf

Stephen Sharpe

Implementing the three-neutron quantization condition

We extend the relativistic field theoretic finite-volume formalism to $\textit{Nππ}$ scattering states at maximal isospin, $\textit{I}$ = 5$\textit{/}$2. As in previous work using the relativistic field theory approach, we work to all orders in a generic low-energy effective theory, and determine the quantization condition that relates finite-volume energies to intermediate K matrices, and the integral equations connecting the latter to the physical scattering amplitudes. We discuss the parametrization of the K matrices, and explain in detail the new features that arise in implementing the quantization condition due to the spin of the nucleon in combination with the use of non-degenerate particles. As a concrete example, we provide a sample numerical application including the ∆ resonance in the $\textit{Nπ}$ subchannel. The extension to the $\textit{I}$ = 3$\textit{/}$2 and 1$\textit{/}$2 channels is more involved, due to mixing with $\textit{Nπ}$ states, and we do not provide a complete formalism for these cases. We explain why $\textit{Nπ}$ states cannot be included by treating the nucleon as a pole in $\textit{p}$-wave $\textit{Nπ}$ scattering, an approach that has been successful in studying $\textit{DD}$ scattering using the three-particle $\textit{DDπ}$ formalism. We additionally provide results for all isospins under the assumption of no two-to-three mixing, thereby laying the groundwork for a follow-up paper in which all $\textit{Nππ}$ → $\textit{Nπ}$ systems are fully treated. Finally, we study the singularities in $\textit{Nππ}$ amplitudes arising from $\textit{Nπππ}$ intermediate states, and find that our subthreshold cutoff functions must be modified to avoid such singularities.

United Kingdom

Maxwell Hansen

Switzerland

Fernando Romero-López

Finite-volume formalism for $\textit{Nππ}$ at maximal isospin

We perform a detailed comparison between three formalisms used in recent studies of $\textit{DD}$ scattering at heavier-than-physical pion masses, which aim to understand the properties of the doubly-charmed tetraquark, $$ {T}_{cc}^{+}(3875) $$ . These methods are the three-particle relativistic field theory (RFT) formalism, the two-body Lippmann-Schwinger (LS) equation with chiral effective field theory potentials, and the two-particle relativistic framework proposed by Baião Raposo and Hansen (BRH approach). In a simplified single-channel setting, we derive the conditions under which the infinite-volume integral equations from the RFT and BRH approaches reduce to the LS form. We present numerical examples showing that differences between these methods can be largely removed by adjusting short-range couplings. We also address a number of technical issues in the RFT approach.

Sebastian Dawid

Comparison of integral equations used to study $$ {T}_{cc}^{+} $$ for a stable $\textit{D}$

We develop a comprehensive framework for extracting the pole position and properties of the doubly-charmed tetraquark $$ {T}_{\textrm{cc}}^{+}(3875) $$ from lattice QCD data using the relativistic three-particle formalism. This approach incorporates the effect of the one-pion exchange diagram in and scattering, making it applicable at energies coinciding with the left-hand cut in the partial-wave projected amplitude. We present an example application of this framework to existing lattice QCD data at = 280 MeV. We solve the integral equations describing the reaction, use LSZ reduction to determine the corresponding amplitude, and find the values of the infinite-volume two- and three-body matrices that lead to agreement with lattice phase shifts within their uncertainties. Using these matrices in the three-particle quantization condition, we describe the finite- volume spectrum and find good agreement with the lattice QCD energies. Our results suggest that, at this pion mass, the tetraquark appears as a pair of subthreshold complex poles whose precise location strongly depends on the value of the three-particle matrix.

Finite- and infinite-volume study of

We generalize previous three-particle finite-volume formalisms to allow for multiple three-particle channels. For definiteness, we focus on the two-channel ηππ and   <math> <mi>K</mi> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mi>π</mi> </math>  $$ K\overline{K}\pi $$  system in isosymmetric QCD, considering the positive G parity sector of the latter channel, and neglecting the coupling to modes with four or more particles. The formalism we obtain is thus appropriate to study the b 1(1235) and η(1295) resonances. The derivation is made in the generic relativistic field theory approach using the time-ordered perturbation theory method. We study how the resulting quantization condition reduces to that for a single three-particle channel when one drops below the upper (  <math> <mi>K</mi> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mi>π</mi> </math>  $$ K\overline{K}\pi $$ ) threshold. We also present parametrizations of the three-particle K matrices that enter into the formalism.

Zachary Draper

Three-particle formalism for multiple channels: the ηππ +   <math> <mi>K</mi> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mi>π</mi> </math>  $$ K\overline{K}\pi $$  system in isosymmetric QCD

We generalize the relativistic field-theoretic three-particle finite-volume scattering formalism to describe generic DDπ systems in the charm C = 2 sector. This includes the isospin-0 channel, in which the recently discovered doubly-charmed tetraquark T  cc (3875)+ is expected to manifest as a pole in the DDπ → DDπ scattering amplitude. The formalism presented here can also be applied to lattice QCD settings in which the D * is bound and, in particular, remains valid below the left-hand cut in DD * scattering, thus resolving an issue in previous analyses of lattice-determined finite-volume energies.

Incorporating DDπ effects and left-hand cuts in lattice QCD studies of the Tcc(3875)+

Henry Tye

Remembering Stephen Weinberg

The three-particle K-matrix,   <math> <mi>K</mi> </math>  $$ \mathcal{K} $$  df,3, is a scheme-dependent quantity that parametrizes short-range three-particle interactions in the relativistic-field-theory three-particle finite-volume formalism. In this work, we compute its value for systems of three pions in all isospin channels through next-to-leading order in Chiral Perturbation Theory, generalizing previous work done at maximum isospin. We obtain analytic expressions through quadratic order (or cubic order, in the case of zero isospin) in the expansion about the three-pion threshold.

Spain

Jorge Baeza-Ballesteros

Sweden

Johan Bijnens

Czechia

Tomáš Husek

France

Mattias Sjö

The three-pion K-matrix at NLO in ChPT

We generalize the relativistic field-theoretic (RFT) three-particle finite-volume formalism to systems of three identical, massive, spin-1/2 fermions, such as three neutrons. This allows, in principle, for the determination of the three-neutron interaction from the finite-volume spectrum of three-neutron states, which can be obtained from lattice QCD calculations.

Three relativistic neutrons in a finite volume

The three-particle K-matrix,   <math> <msub> <mi>K</mi> <mrow> <mi>df</mi> <mo>,</mo> <mn>3</mn> </mrow> </msub> </math>  $$ {\mathcal{K}}_{\textrm{df},3} $$ , is a scheme-dependent quantity that parametrizes short-range three-particle interactions in the relativistic-field-theory three-particle finite-volume formalism. In this work, we compute its value for systems of three pions at maximal isospin through next-to-leading order (NLO) in Chiral Perturbation Theory (ChPT). We compare the values to existing lattice QCD results and find that the agreement between lattice QCD data and ChPT in the first two coefficients of the threshold expansion of   <math> <msub> <mi>K</mi> <mrow> <mi>df</mi> <mo>,</mo> <mn>3</mn> </mrow> </msub> </math>  $$ {\mathcal{K}}_{\textrm{df},3} $$  is significantly improved with respect to leading order once NLO effects are incorporated.