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.

United States

Sebastian Dawid

Switzerland

Fernando Romero-López

Stephen Sharpe

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.

United Kingdom

Maxwell Hansen

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.

The isospin-3 three-particle K-matrix at NLO in ChPT

We study the interactions of systems of two and three nondegenerate mesons composed of pions and kaons at maximal isospin using lattice QCD, specifically π + K +, π + π + K + and K + K + π +. Utilizing the stochastic LapH method, we determine the spectrum of these systems on two CLS N  f  = 2 + 1 ensembles with pion masses of 200 MeV and 340 MeV, and include many levels in different momentum frames. We constrain the K matrices describing two- and three-particle interactions by fitting the spectrum to the results predicted by the finite-volume formalism, including up to p waves. This requires also results for the π + π + and K + K + spectrum, which have been obtained previously on the same configurations. We explore different fitting strategies, comparing fits to energy shifts with fits to energies boosted to the rest frame, and also comparing simultaneous global fits to all relevant two- and three-particle channels to those where we first fit two-particle channels and then add in the three-particle information. We provide the first determination of the three-particle K matrix in π + π + K + and K + K + π + systems, finding statistically significant nonzero results in most cases. We include s and p waves in the K matrix for π + K + scattering, finding evidence for an attractive p-wave scattering length. We compare our results to Chiral Perturbation Theory, including an investigation of the impact of discretization errors, for which we provide the leading order predictions obtained using Wilson Chiral Perturbation Theory.

Andrew Hanlon

Germany

Ben Hörz

Colin Morningstar

Interactions of πK, ππK and KKπ systems at maximal isospin from lattice QCD

Recently, the formalism needed to relate the finite-volume spectrum of systems of nondegenerate spinless particles has been derived. In this work we discuss a range of issues that arise when implementing this formalism in practice, provide further theoretical results that can be used to check the implementation, and make available codes for implementing the three-particle quantization condition. Specifically, we discuss the need to modify the upper limit of the cutoff function due to the fact that the left-hand cut in the scattering amplitudes for two nondegenerate particles moves closer to threshold; we describe the decomposition of the three-particle amplitude   <math> <mi>K</mi> </math>  $$ \mathcal{K} $$  df,3 into the matrix basis used in the quantization condition, including both s and p waves, with the latter arising in the amplitude for two nondegenerate particles; we derive the threshold expansion for the lightest three-particle state in the rest frame up to   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (1/L 5); and we calculate the leading-order predictions in chiral perturbation theory for   <math> <mi>K</mi> </math>  $$ \mathcal{K} $$  df,3 in the π + π + K + and π + K + K + systems. We focus mainly on systems with two identical particles plus a third that is different (“2+1” systems). We describe the formalism in full detail, and present numerical explorations in toy models, in particular checking that the results agree with the threshold expansion, and making a prediction for the spectrum of π + π + K + levels using the two- and three-particle interactions predicted by chiral perturbation theory.

Tyler Blanton

Implementing the three-particle quantization condition for π + π + K + and related systems

We study two- and three-meson systems composed either of pions or kaons at maximal isospin using Monte Carlo simulations of lattice QCD. Utilizing the stochastic LapH method, we are able to determine hundreds of two- and three-particle energy levels, in nine different momentum frames, with high precision. We fit these levels using the relativistic finite-volume formalism based on a generic effective field theory in order to determine the parameters of the two- and three-particle K-matrices. We find that the statistical precision of our spectra is sufficient to probe not only the dominant s-wave interactions, but also those in d waves. In particular, we determine for the first time a term in the three-particle K-matrix that contains two-particle d waves. We use three N  f  = 2 + 1 CLS ensembles with pion masses of 200, 280, and 340 MeV. This allows us to study the chiral dependence of the scattering observables, and compare to the expectations of chiral perturbation theory.