Energy-Dependent π+π+π+ Scattering Amplitude From QCD

Focusing on three-pion states with maximal isospin (π^{+}π^{+}π^{+}), we present the first nonperturbative determination of an energy-dependent three-hadron scattering amplitude from first-principles QCD. The calculation combines finite-volume three-hadron energies, extracted using numerical lattice QCD, with a relativistic finite-volume formalism, required to interpret the results. To fully implement the latter, we also solve integral equations that relate an intermediate three-body K matrix to the physical three-hadron scattering amplitude. The resulting amplitude shows rich analytic structure and a complicated dependence on the two-pion invariant masses, represented here via Dalitz-like plots of the scattering rate.

Introduction.-The three-body problem lies at the core of a broad range of outstanding questions in quantum chromodynamics (QCD). The largest uncertainty in QCD-based structure calculations of light nuclei, for example, is the estimate of the three-nucleon force (see Ref. [1]). In addition, many QCD resonances have significant branching fraction to channels with three or more hadrons. The Roper resonance, for example, has defied simple quark-model descriptions, due in part to its nature as a broad resonance with a ∼30% branching fraction to Nππ. A rigorous QCD calculation would elucidate the role of nonperturbative dynamics in the Roper's peculiar properties, e.g., the fact that it has a lower mass than the negative-parity ground state, which seems unnatural from the perspective of the quark model [2,3].
As a necessary step towards studying a broad class of three-hadron systems, in this work we present the first study of an energy-dependent three-body scattering amplitude from QCD. This nonperturbative result is achieved by the coalescence of three novel techniques: a calculation of finite-volume three-hadron energies based in numerical lattice QCD, a relativistic finite-volume formalism to relate the energies to K matrices, and a numerical evaluation of corresponding integral equations to convert the latter into the three-hadron scattering amplitude. The theoretical basis required to achieve these final two steps was derived in Refs. [4,5]. (A large body of work has investigated general methods for relating finite-volume energies to scattering amplitudes for both two-and three-body states. See Refs. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and Refs. , respectively.) This work considers the scattering of three-pion states with maximal isospin (I ¼ 3) in QCD with three dynamical quarks (N f ¼ 2 þ 1): two degenerate light quarks, with heavier-than-physical mass corresponding to a pion mass m π ≈ 391 MeV, and a strange quark. This channel offers an optimal benchmark case, since both the maximal-isospin three-pion system and its two-pion subsystem are expected to be weakly interacting and nonresonant.
Many numerical studies of three-hadron states have been published over the last decade, ranging from early work deriving and fitting large-volume expansions of the threepion ground state [46][47][48] to more recent results using quantization conditions to study ground [49] and excited states [50][51][52], with the latter set each analyzing the lattice QCD spectrum published in Ref. [53]. Independent sets of finite-volume energies have also been calculated and analyzed in Refs. [54] and [55]. The present investigation goes beyond this previous work, by providing the first complete numerical determination of physical scattering amplitudes for three-body systems.
In the following, we first discuss our determination of two-and three-pion finite-volume energies, before describing the fits used to relate these to infinite-volume K matrices. The latter then serve as inputs to known integral equations, which we solve numerically to extract the 3π þ → 3π þ scattering amplitude. Additional details of the analysis are discussed in the Supplemental Material [56].
Spectral determination.- Figure 1 summarizes the twoand three-pion finite-volume spectra calculated in this work. Two-pion energies on the larger volume have already appeared in Ref. [57].
Computations were performed on anisotropic lattices which have a temporal lattice spacing, a t , finer than the spatial lattice spacing, a s [a t ¼ a s =ξ with ξ ¼ 3.444ð6Þ [57] ]. Two lattice ensembles were used, differing only in the volume: ðL=a s Þ 3 × ðT=a t Þ ¼ 20 3 × 256 (with 256 gauge-field configurations) and 24 3 × 128 (with 512 configurations). We use 2 þ 1 flavors of dynamical clover fermions, with three-dimensional stout-link smearing in the fermion action, and a tree-level Symanzik-improved gauge action. The bare parameters and basic lattice properties are detailed in Refs. [58,59]. Setting the scale via [where a t m latt Ω ¼ 0.2951ð22Þ was measured in Ref. [60] and m exp Ω is the experimentally determined Ω baryon mass from Ref. [61] ] and combining with a t m π ¼ 0.06906ð13Þ [57] and a t m K ¼ 0.09698ð9Þ [62], yields m π ≈ 391 MeV and m K ≈ 550 MeV. The values of a t m π and ξ translate into spatial extents of m π L ¼ 4.76 and m π L ¼ 5.71 for the two ensembles.
The spectrum of energies in a finite volume is discrete and each energy level provides a constraint on the scattering amplitudes at the corresponding center-of-momentum energy.
To obtain more constraints, we compute spectra for systems with overall zero and nonzero momentum, P. Momenta are quantized by the cubic spatial boundary conditions, P ¼ ð2π=LÞðn 1 ; n 2 ; n 3 Þ, where fn i g are integers, and we write this using a shorthand notation as ½n 1 n 2 n 3 .
In this work we restrict attention to S-wave scattering. The reduced symmetry of a cubic lattice means that total angular momentum J is not a good quantum number and instead channels are labeled by the irreducible representation (irrep, Λ) of the octahedral group with parity for P ¼ 0 or the relevant subgroup that leaves P invariant for P ≠ 0 [63,64]. We consider the relevant irreps which contain J ¼ 0: A − 1 ðA þ 1 Þ for πππ (ππ) at rest and A 2 ðA 1 Þ for πππ (ππ) with nonzero P. Isospin I and G parity G are good quantum numbers in our lattice formulation; these distinguish the two-pion (I G ¼ 2 þ ) and three-pion (I G ¼ 3 − ) channels. We neglect higher partial waves here, in particular the twoparticle D wave that mixes with the S wave in the finitevolume energies. As described in Ref. [57], a nonzero D-wave interaction can be extracted, in particular, if aided by the consideration of other, nontrivial finite-volume irreps, but has a small influence on the two-pion energies considered here. There is, in principle, a systematic uncertainty associated with neglecting the D-wave contribution. Given the consistency of our results with Ref. [57], this appears to be below the statistical uncertainty in the present fits. See also Secs. VIII A and B of that work for more discussion.
To reliably extract the finite-volume energies we have computed two-point correlation functions using a large basis of appropriate interpolating operators. From these, the spectra are determined using the variational method [65][66][67], with our implementation described in Refs. [68,69]. This amounts to calculating a matrix of correlation functions, and diagonalizing Mðt; t 0 Þ ¼ Gðt 0 Þ −1=2 GðtÞGðt 0 Þ −1=2 for a fixed t 0 . One can show that the corresponding eigenvalues FIG. 1. The π þ π þ and π þ π þ π þ finite-volume spectra in the center-of-momentum frame for the relevant finite-volume irreps with various overall momenta, as explained in the text. Points are computed energy levels on the two volumes with error bars showing statistical uncertainties. Each rectangular inset shows a vertical zoom of the region indicated by the small neighboring rectangle. Gray curves are the "noninteracting" finite-volume energies, i.e., the energies in the absence of any interactions between pions. Orange curves are predictions from the finitevolume formalism based only on the two-particle scattering length, given in Eq. L/as satisfy λ n ðt; t 0 Þ → e −E n ðLÞðt−t 0 Þ , where E n ðLÞ is the nth energy level with overlap to some of the operators in the basis. This basic methodology has been applied to a wide range of two-hadron scattering observables for several phenomenologically interesting channels [57,[70][71][72][73][74][75][76][77][78][79][80][81]. See Sec. I of the Supplemental Material [56] for some example plots of λ n ðt; t 0 Þ. In order to robustly interpolate the two-and three-pion energy eigenstates we use operators with two-and threemeson-like structures in the appropriate irrep, constructed from products of single-meson-like operators projected to definite spatial momentum. The latter are built from linear combinations, chosen to optimize overlap to the single-pion states, of fermion bilinears of the form,ψΓD…Dψ, where ψ is a quark field and D is a discretized covariant derivative. Details of these operator constructions are given in Sec. V of the Supplemental Material [56] with further details relevant to the three-meson-like operators presented in Ref. [82]. Using such a wide variety of optimized operators, and especially multihadron operators with momentumprojected single-hadron components, allows one to minimize excited state contamination and extract the energies reliably and precisely from small values of t. This approach is made feasible due to the distillation method [83] which we employ to efficiently compute the numerous quark-field Wick contractions that are required. We use 128 distillation vectors for the 20 3 ensemble and 162 for the 24 3 .
Returning to the two-and three-pion spectra summarized in Fig. 1, we observe a one-to-one correspondence between the computed energy levels and the noninteracting energies in all panels, with the computed values slightly higher in energy than the noninteracting levels. This suggests that the system is weakly interacting and repulsive in both the twoand three-hadron sectors.
Analyzing the finite-volume spectra.-We now describe our method for determining two-and three-body K matrices from the extracted finite-volume energies, beginning with an overview of scattering observables: The two-pion scattering amplitude is defined as the connected part of the overlap between an incoming π þ π þ asymptotic state (with momenta p; −p) to an outgoing π þ π þ state (with p 0 ; −p 0 ). Without loss of generality, here we have assumed the center-of-momentum frame. We also define p ¼ jpj ¼ jp 0 j, where we have used that the magnitudes must be equal to satisfy energy conservation. In addition, s 2 ≡ E ⋆2 2 ≡ 4ðp 2 þ m 2 π Þ defines the squared center-of-momentum frame energy. The only remaining degree of freedom is the scattering angle between p and p 0 . In this work we focus on the S-wave scattering amplitude, denoted M 2 , in which this angle is integrated to project onto zero-angular-momentum states. Finally we recall the simple relation between M 2 and the K matrix in the elastic region, K −1 2 ¼ ReM −1 2 . The imaginary part of M −1 2 is completely fixed by unitarity so that K 2 is the only part free to depend on the dynamics of the system. We work with the simple phase space factor, proportional to the momentum magnitude. See, e.g., Ref. [23] for more details. In contrast to M 2 , K 2 is real for real s 2 and is meremorphic in a region of the complex s 2 plane around s 2 ¼ 4m 2 π . In this work we also consider an analogous, three-body K matrix, introduced in Ref. [4] and denoted by K df;3 .
In the two-pion sector, in the case that the S-wave interactions are dominant, the scalar-irrep finite-volume energies satisfy the quantization condition [6,7,9], is the center-of-momentum energy and FðE 2 ; P; LÞ is a known geometric function. For the three-body sector, we use the isotropic approximation of the general formalism derived in Ref. [4], which takes an analogous form, now for pseudoscalar-irrep energies where the notation is meant to stress that F 3;iso ½K 2 ðE 3 ; P; LÞ is a functional of K 2 ðE ⋆ 2 Þ. F 3;iso is defined in Eq. (39) of Ref. [4]. Here K 3;iso is the component of K df;3 that only depends on the total three-hadron energy, i.e., is "isotropic." Equation (3) holds only when K df;3 is well approximated to be isotropic and our fits give evidence that this is a good approximation for this system.
Combining these two conditions with the energies plotted in Fig. 1 allows one to constrain both the two-and threehadron K matrices. One strategy is to fit a parametrization of K 2 and use this to determine the energy dependence of K 3;iso as summarized in Fig. 2. An alternative approach is to parametrize both K matrices and fit these simultaneously to the entire set of finite-volume energies. A detailed discussion with a wide range of fits is given in Sec. II of the Supplemental Material [56]. Both strategies give consistent results and the key message is that the full dataset is well described by a constant K 3;iso that is consistent with zero, together with the leading-order effective range expansion: Here the second equation defines the S-wave scattering phase shift δðpÞ, and the first defines the scattering length a 0 . Our best fit, performed simultaneously to all spectra shown in Fig. 1 but with a cutoff in the center-of-momentum frame energies included, yields m π a 0 ¼ 0.296 AE 0.008 with a χ 2 per degree of freedom of 64.5=ð37 − 2Þ ¼ 1.84. This fit is denoted by B 2þ3 in Sec. II of the Supplemental Material [56]. As explained there, the fitted data include all two-pion energies below E ⋆ 2;cut ¼ 3.4m π and all three-pion energies below E ⋆ 3;cut ¼ 4.4m π , with both cutoffs PHYSICAL REVIEW LETTERS 126, 012001 (2021) applied to energies in the center-of-momentum frame. The square-bracketed matrix gives the correlation between the two fit parameters. This fit is consistent with the previous determination of the scattering length at this pion mass, presented in Ref. [57], and is also the value used to generate the orange curves in Fig. 1 (together with K 3;iso ¼ 0). In Fig. 2 we illustrate the same fit using the darker cyan curves. In addition, we include the lighter bands as a systematic uncertainty, estimated from the spread of various constant and linear fits, as detailed in Sec. II of the Supplemental Material [56]. 3π þ scattering amplitude.-Following the relativistic integral equations presented in Ref. [5], we can write the J ¼ 0 and K 3;iso ¼ 0 amplitude as follows: M 2 is the S-wave two-particle scattering amplitude, introduced above, which depends on the invariant The ðu; uÞ superscript emphasizes that specific spectator momenta, k and p, are singled out in the initial and final states, respectively. The function G s encodes the spectator exchange, projected to the S wave. It inherits a scheme dependence through the smooth cutoff function H, defined in Eqs. (28) and (29) of Ref. [4]. This scheme dependence is matched by that inside of K 3;iso such that the resulting scattering amplitude is universal.
To use Eq. (5) in practice, one requires a parameterization for M 2 . As described in the previous section, the π þ π þ system is well described using the leading order effective range expansion for M 2 , Following the derivation of Ref. [5], the final step is to symmetrize with respect to the spectators, to reach where P p ¼ fp; a 0 ; −p − a 0 g and P k ¼ fk; a; −k − ag. We have presented the left-hand side as a function of the five Lorentz invariants that survive after truncating to J ¼ 0 in both the two and three particle sector: the squared threehadron center-of-momentum frame energy, s 3 , as well as two pion-pair invariant masses for each of the initial and final states. These are defined by introducing the notation fk; a; −k − ag ¼ fp 1 ; p 2 ; p 3 g, then, for example, where the middle expression depends on on-shell fourvectors with p 2 1 ¼ m 2 π . In the top panel of Fig. 3 we show a Dalitz-like plot of jM 3 j 2 as a function of ðm 12 ; m 13 Þ, with all other kinematics fixed as indicated in the caption. In a usual Dalitz description, the incoming energy is fixed by the decaying particle so that only the outgoing kinematics can vary, whereas here we simply fix the other kinematics. The inputs to this plot are the best-fit scattering length, given in Eq. (4), together with K 3;iso ¼ 0. The bottom panel of Fig. 3 shows the same ffiffiffiffi ffi Example of data and fits for K 2 and K 3;iso , as described in the text. The red points are given by substituting finite-volume energies into −1=FðE 2 ; P; LÞ and −1=F 3;iso ðE 3 ; P; LÞ for the two-and three-particle energies, respectively, with the volume and P indicated in the legend. A symbol appearing at the very top or bottom represents a case where the central value falls outside the plotted region. The dark cyan bands represent the fit shown in Eq. (4) while the lighter bands show the spread covered by the various fits described in the Supplemental Material [56]. For the bottom panel we normalize to m 2 π K LO 3;iso ¼ 4608π 2 ðm π a 0 Þ 2 , with m π a 0 taken from Eq. (4). This simple relation between K 3;iso and the two-particle scattering length holds at leading order in chiral perturbation theory at threshold, as was first derived in Ref. [50]. The gray curve gives the full leading-order prediction, which is linear in E

B;/m1r
Additional details concerning the S-wave integral equations are presented in Secs. III and IV of the Supplemental Material [56], where we also describe the propagation of the uncertainties of m π a 0 and K 3;iso into the predicted amplitude. (See also Ref. [84] for more details on expressing the three-particle amplitude via a truncated partial wave series and Ref. [85] for a discussion of integral equations and their solutions in a resonant three-hadron channel.) Summary.-In this work we have presented the first lattice QCD determination of the energy-dependent threeto-three scattering amplitude for three pions with maximal isospin. The calculation proceeded in three steps: (i) determining finite-volume energies with π þ π þ π þ quantum numbers, (ii) using the framework of Ref. [4] to extract two-and three-body K matrices from these, and (iii) applying the results of Ref. [5] to convert these to the threehadron scattering amplitude, by solving known integral equations. The three steps are summarized, respectively, by Figs. 1, 2, and 3.
Having established this general workflow, it is now well within reach to rigorously extract three-hadron resonance properties from lattice QCD calculations. In particular the formalism has recently been extended to three-pion states with any value of isospin in Ref. [42]. This should enable studies, for example, of the ω, h 1 , and a 1 resonances. The main outstanding challenges here include rigorous resonant parametrizations of the intermediate three-body K matrix, as well as a better understanding of the analytic continuation required to identify the resonance pole position. with final kinematics fixed to fp 02 1 ; p 02 2 g ¼ f0.01m 2 π ; 0.7m 2 π g ⇒ fm 0 12 ; m 0 13 g ¼ f2.1m π ; 2.25m π g. Bottom: Same total energy, now with incoming and outgoing kinematics set equal, as discussed in the text.