$S$- and $D$-wave phase shifts in isospin-2 $\ensuremath{\pi}\ensuremath{\pi}$ scattering from lattice QCD

$S$ - and $D$ -wave phase shifts in isospin-2 $π π$ scattering from lattice QCD

Jozef J. Dudek, Robert G. Edwards, and Christopher E. Thomas (for the Hadron Spectrum Collaboration)

Phys. Rev. D 86, 034031 – Published 30 August 2012

Abstract

The isospin-2 $π π$ system provides a useful testing ground for determining elastic hadron scattering parameters from finite-volume spectra obtained using lattice QCD computations. A reliable determination of the excited state spectrum of two pions in a cubic box follows from variational analysis of correlator matrices constructed using a large basis of operators. A general operator construction is presented which respects the symmetries of a multihadron system in flight. This is applied to the case of $π π$ and allows for the determination of the scattering phase shifts at a large number of kinematic points, in both $S$ -wave and $D$ -wave, within the elastic region. The technique is demonstrated with a calculation at a pion mass of 396 MeV, where the elastic scattering is found to be well described by a scattering length parametrization.

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Received 2 April 2012

DOI:https://doi.org/10.1103/PhysRevD.86.034031

Authors & Affiliations

Jozef J. Dudek^1,2,*, Robert G. Edwards^1,†, and Christopher E. Thomas^3,‡ (for the Hadron Spectrum Collaboration)

¹Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, Virginia 23606, USA
²Department of Physics, Old Dominion University, Norfolk, Virginia 23529, USA
³School of Mathematics, Trinity College, Dublin 2, Ireland

^*dudek@jlab.org
^†edwards@jlab.org
^‡thomasc@maths.tcd.ie

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Vol. 86, Iss. 3 — 1 August 2012

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Figure 1
Contributions of ground state ( $n = 0$ ) pion (horizontal red solid line) and excited pion states (other solid lines) to the single pion correlator at zero momentum, $C (t) = ⟨ (\bar{ψ} □ γ_{5} □ ψ) (t) \cdot (\bar{ψ} □ γ_{5} □ ψ) (0) ⟩$ and $N_{vecs} = 162$ on the $24^{3}$ lattice. Summed contribution of all states indicated by the top (gray) curve. Excited state pions are observed to contribute significantly until $t ≳ 20 a_{t}$ . (Excited state contributions determined from the results of variational analysis using a large operator basis, see the text.)
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Figure 2
Effective masses [throughout this paper we define the effective mass of a correlator $C (t)$ to be $m_{eff} = \frac{1}{3 a_{t}} \log [\frac{C (t)}{C (t + 3 a_{t})}]$ ] of single-pion correlators computed using $\bar{ψ} □ γ_{5} □ ψ$ (darker shades, squares) and “optimized” operators, $Ω_{n = 0}$ (lighter shades, circles). Shown for a range of momenta on the $L / a_{s} = 24$ lattice.
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Figure 3
Pion mass as a function of lattice spatial volume. Volume dependence fitted with Eq. (10).
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Figure 4
Pion dispersion relation. Fit as described in the text. Lower plot shows $\hat{ξ} (n^{2}, L) \equiv \frac{\frac{2 π}{L / a_{s}} \sqrt{n^{2}}}{\sqrt{(a_{t} E_{n^{2}})^{2} - (a_{t} m)^{2}}}$ and the fitted value of $ξ = 3.444 (6)$ .
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Figure 5
Effective masses of a “ $π π$ ” correlator as described in the text. Raw correlator (red squares) and shifted correlator (green diamonds).
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Figure 6
Fits to diagonal $π π$ correlators with $\vec{P} = [0, 0, 0]$ using the lowest allowed $| \vec{k} |$ that gives rise to irrep $Λ^{P}$ . Correlator is plotted via $e^{2 E_{π} (\vec{k}) t} C (t)$ such that in the limit of noninteracting pions and $T \to \infty$ we would have a horizontal line. The solid red line shows the result of the fit using Eq. (12) while the orange dashed line shows the result of excluding the constant contribution, which should correspond to the $T \to \infty$ behavior. Fit parameters given in Table 4.
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Figure 7
Simulated contributions to a correlator ( ${\vec{k}}_{1} = [0, 0, 0]$ , ${\vec{k}}_{2} = [1, 0, 0]$ ) of the desired ( $T \to \infty$ , red) term, Eq. (19), and two “polluting” (finite- $T$ ) terms from Eq. (20)—the first term (leading, green dashed) and the sum of the two terms (leading plus subleading, blue, dash-dotted). Observe that in the time region we will consider, the leading term dominates over the subleading term.
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Figure 8
Fits to diagonal $π π$ correlators with $\vec{P} = [0, 0, 1]$ using the lowest allowed $| {\vec{k}}_{1} |$ , $| {\vec{k}}_{2} |$ that gives rise to irrep $Λ$ . Correlator is plotted via $e^{(E_{π} ({\vec{k}}_{1}) + E_{π} ({\vec{k}}_{2})) t} C (t)$ such that in the limit of noninteracting pions and $T \to \infty$ we would have a horizontal line. The solid red line shows the result of the fit using Eq. (21) while the orange dashed line shows the result of excluding the contribution proportional to $c$ , which should correspond to the $T \to \infty$ behavior.
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Figure 9
Principal correlators from solution of Eq. (9) applied to the weighted-shifted correlator matrix $\hat{\tilde{C}} (t)$ for $\vec{P} = [0, 0, 1]$ , $Λ = A_{1}$ with $t_{0} = 14 a_{t}$ . Plotted is $e^{E (t - t_{0})} λ (t)$ against $t / a_{t}$ along with fits to the time dependence according to Eq. (25). Also plotted in the bottom-right are the effective masses of the principal correlators (with the energy weighting $Δ E_{\min }$ corrected) and the fit values $E$ superimposed as horizontal bands. All energies are those in the frame in which the lattice is at rest.
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Figure 10
Diagonal elements of the weighted-shifted correlation matrix for $\vec{P} = [0, 0, 1]$ , $Λ = A_{1}$ : ${\hat{\tilde{C}}}_{[{\vec{k}}_{1}, {\vec{k}}_{2}]}^{[{\vec{k}}_{1}, {\vec{k}}_{2}]} (t)$ and their reconstruction using terms in the sum over states in Eq. (14). Plotted is $e^{{\tilde{E}}_{π π}^{(0)} t} \hat{\tilde{C}} (t)$ .
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Figure 11
“Matrix” plot of values of $Z_{[{\vec{k}}_{1}, {\vec{k}}_{2}]}^{(n)}$ normalized according to $\frac{Z_{[{\vec{k}}_{1}, {\vec{k}}_{2}]}^{(n)}}{\underset{n}{max } [Z_{[{\vec{k}}_{1}, {\vec{k}}_{2}]}^{(n)}]}$ so that the largest overlap across all states for a given operator $[{\vec{k}}_{1}, {\vec{k}}_{2}]$ is unity.
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Figure 12
Extracted center-of-momentum frame energy spectra for $\vec{P} = [0, 0, 0]$ irreps $A_{1}^{+}$ , $E^{+}$ , $T_{2}^{+}$ . The curves represent noninteracting pion pair energies, $\sqrt{m_{π}^{2} + | {\vec{k}}_{1} |^{2}} + \sqrt{m_{π}^{2} + | {\vec{k}}_{2} |^{2}}$ whose uncertainty is determined by the uncertainty on $a_{t} m_{π}$ and $ξ$ determined in Sec. 5. The shaded area represents opening of inelastic ( $4 π$ ) threshold.
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Figure 13
As Fig. 12 for $\vec{P} = [0, 0, 1]$ .
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Figure 14
As Fig. 12 for $\vec{P} = [0, 1, 1]$ .
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Figure 15
As Fig. 12 for $\vec{P} = [1, 1, 1]$ .
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Figure 16
Finite-volume spectrum for the toy model of effective-range parametrizations in the irrep $\vec{P} = [0, 0, 1]$ , $A_{1}$ . Green squares indicate the spectrum including only $ℓ = 0$ scattering, blue circles include $ℓ = 0$ , 2, and black diamonds include $ℓ = 0$ , 2, 4. Note that for many of the energy levels the squares, circles, and diamonds lie on top of each other. Red curves show noninteracting energies of pion pairs with momenta ${\vec{k}}_{1}$ , ${\vec{k}}_{2}$ .
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Figure 17
Lowest four energy levels ( $n = 0$ , 1, 2, 3) in the toy model with volume $L = 3.5 fm$ in irrep $\vec{P} = [0, 0, 1]$ , $A_{1}$ . (a) Sensitivity of $δ_{0}$ extracted from Eq. (27) as a function of assumed values of $δ_{2}$ in range $\pm 2 | δ_{2}^{exact} |$ with $δ_{4} = 0$ . (b) Sensitivity of $δ_{0}$ extracted from Eq (27) as a function of assumed values of $δ_{4}$ in range $\pm 2 | δ_{4}^{exact} |$ with $δ_{2} = δ_{2}^{exact}$ . Boxes on far left indicate exact values of $δ_{0}$ at the corresponding scattering momenta. Arrows on $x$ axis indicate exact values of $δ_{2, 4}$ .
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Figure 18
Simultaneous solution of two equations (27) for $δ_{2, 4}$ . Open black square shows exact values with uncertainty indicating the variation in $δ_{2, 4}^{exact}$ over the momentum region between the two determined $p_{cm}$ . (a) $\vec{P} = [0, 0, 1]$ , $n = 0$ in $E_{2}$ and $n = 0$ in $B_{1}$ . (b) $\vec{P} = [0, 0, 1]$ , $n = 1$ in $E_{2}$ and $n = 0$ in $B_{2}$ .
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Figure 19
Phase shifts, $δ_{0, 2, 4} (p_{cm})$ extracted from $L = 3.5 fm$ spectrum using the method described in the text. Uncertainty in $δ_{0}$ indicates the effect of a conservative assumed uncertainty on $δ_{2, 4}$ . Some points with very large uncertainty not shown. Toy model input phase shifts shown by the curves.
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Figure 20
$δ_{2}$ values in elastic scattering region determined from finite-volume spectra. Filled points determined by simultaneous solution of equations (27) (innermost error bar statistical uncertainty, outermost error bar reflects combined statistical uncertainty and uncertainty in $a_{t} m_{π}$ , $ξ$ with all errors added in quadrature). Open points determined from single levels, with effect of $δ_{4}$ estimated (innermost error bar statistical uncertainty, outermost error bar reflects combined statistical uncertainty and uncertainty in $a_{t} m_{π}$ , $ξ$ and $δ_{4}$ ).
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Figure 21
$δ_{4}$ values in elastic scattering region determined from finite-volume spectra. Filled points determined by simultaneous solution of equations (27) (innermost error bar statistical uncertainty, outermost error bar reflects combined statistical uncertainty and uncertainty in $a_{t} m_{π}$ , $ξ$ with errors added in quadrature).
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Figure 22
$δ_{0}$ values in elastic scattering region determined from finite-volume spectra. Innermost error bar is the statistical uncertainty, middle error bar combined statistical uncertainty and uncertainty in ( $a_{t} m_{π}$ , $ξ$ ), outermost error bar reflects total uncertainty including imperfect knowledge of $δ_{2, 4}$ (all errors added in quadrature). Some points with very large uncertainty not shown.
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Figure 23
Upper: $δ_{0} (p_{cm})$ obtained through “global fits” to finite-volume spectra using effective range and scattering length parametrizations. Lower: $δ_{2} (p_{cm})$ obtained through “global fits” to finite-volume spectra using a scattering length parametrizations. Also shown for comparison, the “level-by-level” analysis previously presented in Figs. 20 and 22.
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Figure 24
Extracted $I = 2$ $π π$ elastic scattering phase shift in $S$ -wave, $δ_{0} (p_{cm})$ , as obtained from analysis of finite-volume spectra with $m_{π} = 396 MeV$ . Center-of-momentum frame scattering momentum expressed in units of the temporal lattice spacing. The momentum region plotted is entirely elastic, with the $4 π$ threshold opening at $(a_{t} p_{cm})^{2} = 0.014$ . Points correspond to an analysis treating each energy level independently. The innermost error bar is the statistical uncertainty, the middle error bar reflects combined statistical uncertainty and uncertainty in $(a_{t} m_{π}, ξ)$ , and the outermost error bar shows the total uncertainty including imperfect knowledge of $δ_{2, 4}$ (all errors added in quadrature). Curves indicate a global analysis of all energy levels describing the phase shift by a scattering length or an effective range parametrization.
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Figure 25
Extracted $I = 2$ $π π$ elastic scattering phase shift in $D$ -wave, $δ_{2} (p_{cm})$ , as obtained from analysis of finite-volume spectra with $m_{π} = 396 MeV$ . Center-of-momentum frame scattering momentum expressed in units of the temporal lattice spacing. Momentum region plotted is entirely elastic, with the $4 π$ threshold opening at $(a_{t} p_{cm})^{2} = 0.014$ . Points correspond to an analysis treating energy regions locally as described earlier in the manuscript. The inner error bar is the statistical uncertainty, and the outer error bar reflects the combined statistical uncertainty and uncertainty in $a_{t} m_{π}$ , $ξ$ and the value of $δ_{4}$ (errors added in quadrature). Curves indicate a global analysis of all energy levels describing the phase shift by a scattering length.
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Figure 26
Our $δ_{0}$ extraction (colored points) compared with those of NPLQCD (gray points) over the elastic region (left) and zoomed in to small scattering momentum (right).
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Figure 27
Extracted $I = 2$ $π π$ elastic $S$ -wave(red), $D$ -wave(blue) scattering phase shift (for $m_{π} = 396 MeV$ , all errors combined). Shown in gray the experimental data from [3, 4, 5, 6] and the constrained analysis using Roy equations [34, 35] (black line, gray band). For the heavy pion mass the entire region plotted is elastic while for the experimental data only $p_{cm}^{2} < 0.058 GeV$ is elastic.
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Physical Review D

covering particles, fields, gravitation, and cosmology

$S$ - and $D$ -wave phase shifts in isospin-2 $π π$ scattering from lattice QCD

Jozef J. Dudek, Robert G. Edwards, and Christopher E. Thomas (for the Hadron Spectrum Collaboration)

Phys. Rev. D 86, 034031 – Published 30 August 2012

Abstract

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Physical Review D

covering particles, fields, gravitation, and cosmology

S- and D-wave phase shifts in isospin-2 ππ scattering from lattice QCD

Jozef J. Dudek, Robert G. Edwards, and Christopher E. Thomas (for the Hadron Spectrum Collaboration)

Phys. Rev. D 86, 034031 – Published 30 August 2012

Abstract

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$S$ - and $D$ -wave phase shifts in isospin-2 $π π$ scattering from lattice QCD