Lattice calculation of the pion transition form factor with $N_f=2+1$ Wilson quarks

We present a lattice QCD calculation of the double-virtual neutral pion transition form factor, with the goal to cover the kinematic range relevant to hadronic light-by-light scattering in the muon $g-2$. Several improvements have been made compared to our previous work. First, we take into account the effects of the strange quark by using the $N_f=2+1$ coordinated lattice simulation gauge ensembles. Second, we have implemented the on-shell $\mathcal{O}(a)$ improvement of the vector current to reduce the discretization effects associated with Wilson quarks. Finally, in order to have access to a wider range of photon virtualities, we have computed the transition form factor in a moving frame as well as in the pion rest frame. After extrapolating the form factor to the continuum and to physical quark masses, we compare our results with phenomenology. We extract the normalization of the form factor with a precision of 3.5% and confirm within our uncertainty previous somewhat conflicting estimates for a low-energy constant that appears in chiral perturbation theory for the decay ${\pi }^0\to \gamma \gamma$ at the next-to-leading order. With additional input from experiment and theory, we reproduce recent estimates for the decay width $\mathrm{\Gamma }({\pi }^0\to \gamma \gamma )$. We also study the asymptotic large-$Q^2$ behavior of the transition form factor in the double-virtual case. Finally, we provide as our main result a more precise model-independent lattice estimate of the pion-pole contribution to hadronic light-by-light scattering in the muon $g-2$: $a_{\mu }^{\text{HLbL};{\pi }^0}=(59.7\pm 3.6)\times {10}^{-11}$. Using in addition the normalization of the form factor obtained by the PrimEx experiment, we get the lattice and data-driven estimate $a_{\mu }^{\text{HLbL};{\pi }^0}=(62.3\pm 2.3)\times {10}^{-11}$.

Figure 1

Kinematic reach in the photon virtualities ($q_1^2,q_2^2$) for the ensemble N200; see Table 1 for its parameters. Left: Pion rest frame. Right: Moving frame where the pion has one unit of momentum in the $z$-direction $\overrightarrow{p}=(2\pi /L)\widehat{z}$.

Figure 2

The functions ${\tilde{A}}^{(1)}(\tau )$ and ${\tilde{A}}^{(2)}(\tau )$ for two different orbits with $(|{\overrightarrow{q}}_1{|}^2,|{\overrightarrow{q}}_2{|}^2)={(\frac{2\pi }{L})}^2(3,2)$ and $|\overrightarrow{p}|=2\pi /L$ for ensemble D200, whose parameters are given in Table 1. Lattice data are in black. The blue and red lines correspond to a fit of the tail using the VMD and LMD models, respectively, as explained in Sec. 2d.

Figure 3

Numerical integration for data generated using a LMD model with lattice parameters close to N200. Blue (red) points correspond to the transition form factor ${\mathcal{F}}_{{\pi }^0{\gamma }^{*}{\gamma }^{*}}(-Q_1^2,-Q_2^2)$ calculated using ${\tilde{A}}^{(1)}(\tau )$ (${\tilde{A}}^{(2)}(\tau )$), respectively. The exact result is represented by the black line. Top: Using a trapezoidal integration rule. Bottom: Using a Simpson integration rule.

Figure 4

Left: Influence of the discretization of the lattice derivative used to implement $\mathcal{O}(a)$-improvement on the function $A^{(1)}(\tau )$. Black points correspond to $\mathrm{\Delta }{\tilde{A}}^{(1)}(\tau )$ before improvement. Red, blue, and green points correspond to $\mathrm{\Delta }{\tilde{A}}^{(1)}(\tau )$ after improvement using, respectively, the symmetric, forward, or backward derivative at the sink. Right: The function $A^{(1)}(\tau )$ for different discretizations of the vector current. Black and blue (red and orange) points correspond to local-local and local-conserved correlation functions without (with) improvement of the vector currents using the symmetric lattice derivative at the sink. Results correspond to the ensemble H102.

Figure 5

Effective masses given by Eq. (27) for the ensembles C101 (left panel) and D200 (right panel) with $t=x_0-y_0$. Black and blue points correspond, respectively, to $|\overrightarrow{p}|=0$ and $|\overrightarrow{p}|=2\pi /L$. The red lines correspond to the effective masses obtained from the fit using Eq. (28). The vertical lines indicate the plateau region and the dashed horizontal lines the extracted pion energy and its error.

Figure 6

Lattice results for the ensemble D200, with a pion mass of 200 MeV, using two local vector currents at the source and at the sink. Left panel: single virtual; right panel: double virtual. Black points correspond to the results obtained in the pion rest frame while the blue points are obtained in the pion moving frame. Error bands correspond to the global $z$-expansion fitting procedure described in Sec. 3c. Some noisy points, with a small contribution to the fit, are not displayed for clarity.

Figure 7

Extrapolated TFF at the physical point using the modified $z$-expansion (43) with $N=3$. The horizontal black lines correspond to the Brodsky-Lepage and OPE predictions. The red line corresponds to the asymptotic prediction given by Eq. (69) including higher twists and NLO corrections and assuming an asymptotic DA (see Sec. 4c). The result of Ref. [51], obtained in the dispersive framework, is shown for comparison. The dashed black line in the double-virtual case corresponds to the prediction given by Eq. (83) in Ref. [52]. Experimental data from CELLO and CLEO are displayed in the single-virtual case [26].

Figure 8

Study of hypercubic artifacts for different values of the photon virtualities $Q_1^2$ and $Q_2^2$. Within our statistical accuracy, we do not observe hypercubic effects, even at large virtualities. The results correspond to the ensemble N200. The blue and red points correspond to the pion rest frame and moving frame, respectively.

Figure 9

Transition form factor obtained using two different volumes ($L=2.1\mathrm{fm}$ and $L=3.1\mathrm{fm}$ for H200 and N202, respectively) at the same bare lattice parameters $\kappa$ and $\beta$.

Figure 10

Left panel: $\mathrm{\Delta }F(-Q_1^2,-Q_2^2)$ defined through Eq. (47) for the ensembles H105, N203, N200, D200, and N302. Ensembles at the same lattice spacing $a=0.064\mathrm{fm}$ are plotted using plain lines, whereas ensembles at different lattice spacings are plotted using dashed lines. Right panel: TFF with and without including the disconnected contribution for our lightest pion mass ensemble D200. Red points have been shifted slightly to the right for clarity.

Figure 11

Results of the global LMD and $\mathrm{LMD}+\mathrm{V}$ fits in the single-virtual case and for the ensembles N203 (left panel) and N200 (right panel). The data strongly favor the $\mathrm{LMD}+\mathrm{V}$ model, which has the correct asymptotic behavior at these kinematics, over the LMD model.

Figure 12

Transition form factor extrapolated to the physical point using the three phenomenological models described in the main text. In the single-virtual case, we also compare the results with data from the CELLO and CLEO experiments [26].

Figure 13

Left panel: Difference in $a_{\mu }^{\text{HLbL};{\pi }^0}$ with or without including the disconnected diagrams. The red line corresponds to a linear fit and the vertical dashed line to the physical point. Right panel: Continuum and chiral extrapolation of $a_{\mu }^{\text{HLbL};{\pi }^0}$ using the local $z$-expansion performed on each ensemble.

Figure 14

Fits of the $\mathrm{LMD}+\mathrm{V}$ TFF using the modified $z$-expansion in the range $0–4{\mathrm{GeV}}^2$ and for different values of $N$. The $\mathrm{LMD}+\mathrm{V}$ TFF is defined using Eq. (48c) and the fit parameters in Eq. (53).