$P$-wave $\pi \pi$ scattering and the $\rho$ resonance from lattice QCD

We calculate the parameters describing elastic $I=1$, $P$-wave $\pi \pi$ scattering using lattice QCD with $2+1$ flavors of clover fermions. Our calculation is performed with a pion mass of $m_{\pi }\approx 320\mathrm{MeV}$ and a lattice size of $L\approx 3.6\mathrm{fm}$. We construct the two-point correlation matrices with both quark-antiquark and two-hadron interpolating fields using a combination of smeared forward, sequential and stochastic propagators. The spectra in all relevant irreducible representations for total momenta $|\overrightarrow{P}|\le \sqrt{3}\frac{2\pi }{L}$ are extracted with two alternative methods: a variational analysis as well as multiexponential matrix fits. We perform an analysis using Lüscher’s formalism for the energies below the inelastic thresholds, and investigate several phase shift models, including possible nonresonant contributions. We find that our data are well described by the minimal Breit-Wigner form, with no statistically significant nonresonant component. In determining the $\rho$ resonance mass and coupling we compare two different approaches: fitting the individually extracted phase shifts versus fitting the $t$-matrix model directly to the energy spectrum. We find that both methods give consistent results, and at a pion mass of $a{m}_{\pi }=0.18295(36{)}_{\text{stat}}$ obtain $g_{\rho \pi \pi }=5.69(13{)}_{\text{stat}}(16{)}_{\text{sys}}$, $a{m}_{\rho }=0.4609(16{)}_{\text{stat}}(14{)}_{\text{sys}}$, and $a{m}_{\rho }/a{m}_N=0.7476(38{)}_{\text{stat}}(23{)}_{\text{sys}}$, where the first uncertainty is statistical and the second is the systematic uncertainty due to the choice of fit ranges.

Figure 1

Pion dispersion relation. The $\pi$ mass and speed of light determined from the dispersion relation are consistent with a relativistic dispersion relation with the rest frame $\pi$ energy.

Figure 2

The Wick contractions corresponding to the correlation matrix elements of type $C_{\overline{q}q-\overline{q}q}$, $C_{\pi \pi -\overline{q}q}$, $C_{\pi \pi -\pi \pi }^{\text{direct}}$ and $C_{\pi \pi -\pi \pi }^{\mathrm{box}}$.

Figure 3

Sample matrix fit with $N_{\text{states}}=3$ for $|\overrightarrow{P}|=\frac{2\pi }{L},\mathrm{\Lambda }=A_2$ in the range between $t_{\min }/a=8$ and $t_{\min }/a=20$.

Figure 4

Comparison between MFA and GEVP for the momentum frames and irreps $\frac{L}{2\pi }|\overrightarrow{P}|=0,1,\sqrt{2}$ and $\mathrm{\Lambda }=T_1,A_2,E,B_1$, respectively. The green circles on the left panel show the effective energies $E_{\mathrm{eff}}^n$ determined from the principal correlators. In the right panel we present the fitted energies as they depend on the choice of $t_{\min }$. Black diamonds are obtained from MFA, red squares are obtained from the single exponential fits to the principal correlator [see Eq. (40)], and blue circles are from two-exponential fits to the principal correlator [see Eq. (41)]. Note that not all two-exponential fits are shown, as they can become unstable. The red horizontal bands give the $1\sigma$ statistical-uncertainty ranges of the selected one-exponential GEVP fits listed in Table 3.

Figure 5

As in Fig. 4, but for $\frac{L}{2\pi }|\overrightarrow{P}|=\sqrt{2},\sqrt{3}$ and $\mathrm{\Lambda }=B_2,B_3,A_2,E$.

Figure 6

Comparison between MFA and GEVP for the $B_1$ irrep with $|\overrightarrow{P}|=\sqrt{2}\frac{2\pi }{L}$ as in Fig. 4, but with $O_2$ removed from the basis for the GEVP. The reduced basis gives a better extraction of $a{E}_3$ compared to the full basis only in this irrep.

Figure 7

Comparison of fitting Breit-Wigner model BW I versus fitting Breit-Wigner model BW II to the phase shift data. The bands indicate the $1\sigma$ statistical uncertainty.

Figure 8

Contribution of nonresonant background models as described in Sec. 2 to the resonant Breit-Wigner BW I. None of the background phase shift models shows a strong sign of deviation away from 0.

Figure 9

Contribution of nonresonant background models as described in Sec. 2 to the resonant Breit-Wigner model BW II. None of the background phase shift models shows a strong sign of deviation away from 0.

Figure 10

Comparison of $t$-matrix fit and fit to the phase shifts for Breit-Wigner models I and II.

Figure 11

Final result of fitting the resonant model BW I to the spectrum via the $t$-matrix fit. The gray data points are the results of the individual phase shift extractions for each energy level, and are not used in the $t$-matrix fit.

Figure 12

Comparison of our results for the $\rho$ mass and coupling with previous lattice QCD calculations. In the two left panels, we use the dimensionless ratios $a{m}_{\rho }/a{m}_N$ and $a{m}_{\pi }/a{m}_N$, while in the two right panels we use $m_{\pi }$ and $m_{\rho }$ in MeV as reported by each collaboration (with different scale setting methods; the error bars do not include the scale-setting ambiguities). The open red symbols mark calculations with $N_f=2$ gauge ensembles, while the filled blue symbols denote calculations with $N_f=2+1$ sea quarks; the only study so far that explicitly included the $K\overline{K}$ channel, HadSpec ’15, is presented as a purple upward facing triangle. The results of our present work are shown with filled black hexagons. In the left-hand plots, the HadSpec ’15 results are offset horizontally by $-1.8\%$ so that they do not overlap with the result of Bulava et al. ’16. In the right-hand plots, we offset our results by $-8\mathrm{MeV}$ to avoid overlap with Guo et al. ’16. The experimental values [1], where $g_{\rho \pi \pi }$ was calculated from $\mathrm{\Gamma }$ using Eq. (5), are shown with filled green circles.