ππ P-wave resonant scattering from lattice QCD

We present a high-statistics analysis of the ρ resonance in ππ scattering, using 2 + 1 flavors of clover fermions at a pion mass of approximately 320 MeV and a lattice size of approximately 3.6 fm. The computation of the two-point functions are carried out using combinations of forward, sequential, and stochastic propagators. For the extraction of the ρ-resonance parameters, we compare different fit methods and demonstrate their consistency. For the ππ scattering phase shift, we consider different BreitWigner parametrizations and also investigate possible nonresonant contributions. We find that the minimal Breit-Wigner model is sufficient to describe our data, and obtain amρ = 0.4609(16)stat(14)sys and gρππ = 5.69(13)stat(16)sys. In our comparison with other lattice QCD results, we consider the dimensionless ratios amρ/amN and amπ/amN to avoid scale setting ambiguities.


Introduction
The simplest QCD-unstable hadron is the ρ resonance, which decays into two pions with a branching ratio of 99.9%. As such, it is considered a benchmark for hadron spectroscopy on the lattice.
The ρ resonance appears as a pole in the I = 1 elastic P-wave ππ scattering amplitude. Because the scattering is elastic, the energy dependence of the scattering amplitude can be expressed in terms of a single real number, the phase shift. For a simple resonance, such as the ρ, the phase shift starts near 0 at the threshold, then goes through π/2 when the invariant mass is near the mass of the resonance, and continues toward π as the invariant mass leaves the resonance region.
While the Maiani-Testa theorem [1] prohibits the extraction of scattering amplitudes directly from Euclidean correlation functions in infinite volume (except at the threshold), Lüscher's method [2] circumvents the basic assumptions of this theorem and takes advantage of the finite lattice volume.
Speaker, e-mail: s.paul@hpc.leap.eu The Lüscher quantization condition provides a (many-to-one) mapping between the discrete finitevolume multi-hadron spectrum and the elastic scattering amplitude. The relation was initially derived in the rest frame, and was extended to moving frames and coupled channels in Refs. [3][4][5][6][7][8][9].
Here, we present a high-statistics lattice QCD calculation of I = 1 P-wave ππ scattering, using 2+1 flavors of clover fermions at a pion mass of approximately 320 MeV and a lattice size of approximately 3.6 fm [10]. At each stage of the analysis, we compare different fit methods and demonstrate their consistency. We explore different parametrizations for the energy dependence of the scattering phase shift, and investigate whether a nonresonant background contribution is present.

Lattice methods and parameters
We use an ensemble of 1041 gauge-field configurations with N f = 2 + 1 clover-improved Wilson fermions on a large 32 3 × 96 lattice with a ≈ 0.114 fm. The pion mass is m π ≈ 320 MeV, and the strange-quark mass is consistent with its physical value as determined by the "η s " mass.
To extract the energy spectrum in the I = 1, J P = 1 − channel, we use two types of interpolating operators: i) single-hadron operatorsdΓu, where Γ ∈ {γ i , γ 0 γ i }, and ii) two-hadron operators of the form (π + π 0 − π 0 π + )/ √ 2, where π + =dγ 5 u and π 0 = (ūγ 5 u −dγ 5 d)/ √ 2 are individually projected to specific momenta. We build (3 × 3) or (4 × 4) two-point correlation matrices Fig. 1, are implemented using combinations of forward, sequential, and stochastic propagators. Compared to the distillation approach [11], our method has the advantage that the computational cost is independent of the chosen smearing width and scales linearly with the lattice volume. To map out the energy dependence of the ππ scattering amplitude, we use several moving frames with total momenta P and irreducible representations as listed in Table 1.

Spectrum results
To determine the energy levels E n from the two-point correlation matrices C i j , we compare three different fit methods and carefully study the dependence on the fit ranges. In the first two methods, we fit the principal correlators λ n (t, t 0 ) obtained from the generalized eigenvalue problem Table 1. Moving frames and corresponding symmetry groups and irreducible representations.
. In our third method, we directly fit the correlation matrix (or a submatrix thereof) using the multi-exponential form C i j (t) = N states n=1 Z i, n Z j, n e −E n t , with N states equal to the dimension of the matrix. As shown in Fig. 2, the different methods give consistent results. After choosing the nominal fit result, we estimate the remaining systematic uncertainty by varying t min .
The green data points on the left panel correspond to the effective energies of the GEVP principal correlators, aE n e f f (t) = ln λn(t, t 0 ) λn(t+a, t 0 ) . In the right panel we present the fitted energies as a function of t min /a. The extracted energies, indicated with the bands, are taken from the single-exponential principal-correlator fits with t min /a = 8.

Extraction of resonant parameters using the Lüscher method
The discrete multi-hadron spectrum in the finite volume is related to the infinite-volume elastic scattering phase shifts δ via the quantization condition first derived by Lüscher [2] and recently reviewed    Table 2. Fit results for the Breit-Wigner models I and II, obtained either by first extracting individual phase shift points and then fitting the model to the points ("Fit to δ 1 "), or by directly fitting the model to the whole energy spectrum ("t-matrix fit").
in Ref. [12]: Above, t (s) = 1 cot δ (s)−i is the -wave scattering amplitude and M P is a matrix-valued function that depends on the spectrum and volume. The quantization condition can be simplified for each momentum frame according to the symmetries in that frame. The resulting equations for the cases listed in Table 1 are given in Ref. [10].
The numerical analysis can be performed in two different ways: i) mapping each individual energy level to a single phase-shift point, and then performing a secondary fit of a model for δ(s), and ii) directly fitting the parameters of the model to the whole energy spectrum ("t-matrix fit"). We confirmed that both approaches give the same results.
We first considered purely resonant Breit-Wigner-parametrizations for the phase shift: with two different models for Γ: • BW I: This is the simplest form determined by the phase space for a P-wave decay: where g ρππ is the resonance coupling and k is the scattering momentum, 2 m 2 π + k 2 = √ s. • BW II: The above P-wave decay width modified with Blatt-Weisskopf barrier factors [13]: where k R is the scattering momentum at the resonance position and r 0 is the radius of the centrifugal barrier.
The two models are compared in Fig. 3, where we can see that both models describe the data points well. The fitted parameters are presented in Table 2. We find that the centrifugal barrier radius r 0 is not statistically significant. Additionally, the other parameters are consistent between the two models.
The ππ scattering amplitude could in principle also contain a contribution from a non-resonant (NR) background phase. Phenomenologically, this contribution would correspond to the two pions scattering without any resonances being created. We therefore continue the analysis by adding three different background phase models to the resonant phase shift model BW I, writing δ(s) = δ BW (s) + δ NR (s), with • NR I: a constant nonresonant phase A, • NR II: a nonresonant phase depending linearly on s, • NR III: zeroth order nonresonant effective-range expansion (ERE), where a −1 1 is the inverse scattering length and s thres = 4m 2 π is the ππ threshold invariant mass. The fits of δ(s) = δ BW (s) + δ NR (s) are shown in Fig. 4 and give the parameters listed in Table 3. We find that the nonresonant contributions are statistically consistent with zero. We therefore quote the results of the t-matrix fit to the pure BW I model, am ρ = 0.4609(16)(14) and g ρππ = 5.69(13)(16), as our final parameters.

Comparison to other studies
In Fig. 5, we compare our results for g ρππ and m ρ with those from previous lattice studies. We use both the values of m π and m ρ in physical units as reported in each lattice QCD study and dimensionless ratios am π am N and am ρ am N to avoid systematic errors in setting the scale. Our calculation gives am π am N = 0.2968 (13),   Our result for the coupling g ρππ is in good agreement with previous studies, both as a function of m π and am π /am N . For the ratio am ρ /am N , the more recent N f = 2 + 1 results obtained with clover fermions appear to approach the physical point quite linearly as a function of am π /am N . Extrapolations of the lattice data to the physical point using unitarized chiral perturbation theory were discussed in Refs. [14] and [15].

Conclusions
A precision lattice QCD study of isospin-1, P-wave ππ scattering with the Lüscher method is presented using a lattice of size (3.6 fm) 3 × (10.9 fm) at a pion mass of approximately 320 MeV. The large volume, in combination with several moving frames, allows us to thoroughly map out the scattering phase shift near the ρ resonance. We find that the amplitude is well described by the simplest Breit-Wigner model without any modifications or additions of a nonresonant background phase. In comparing the results of different lattice calculations for m ρ , a more consistent picture emerges when the dimensionless ratio am ρ /am N is considered as a function of am π /am N .   [26]. The N f = 2 results are plotted in red, the N f = 2 + 1 results are plotted in blue, and the N f = 2 + 1 calculation that explicitly included the KK channel is plotted in purple. The green circles show the experimental values.