Electromagnetic decays of the neutral pion

We complement studies of the neutral pion transition form factor ${\pi }^0\to {\gamma }^{(*)}{\gamma }^{(*)}$ with calculations for the electromagnetic decay widths of the processes ${\pi }^0\to e^{+}{e}^{-}$, ${\pi }^0\to e^{+}{e}^{-}\gamma$ and ${\pi }^0\to e^{+}{e}^{-}{e}^{+}{e}^{-}$. Their common feature is that the singly or doubly virtual transition form factor serves as a vital input that is tested in the nonperturbative low-momentum region of QCD. We determine this form factor from a well-established and symmetry-preserving truncation of the Dyson-Schwinger equations. Our results for the three- and four-body decays match results of previous theoretical calculations and experimental measurements. For the rare decay we employ a numerical method to calculate the process directly by deforming integration contours, which in principle can be generalized to arbitrary integrals as long as the analytic structure of the integrands are known. Our result for the rare decay is in agreement with dispersive calculations but still leaves a $2\sigma$ discrepancy between theory and experiment.

Figure 1

Kinematic domains in $Q^2$ and $Q^{\prime 2}$ including the symmetric (red line) and asymmetric limits ($Q^2$, $Q^{\prime 2}$ axes). The area with $|\omega |<{\eta }_{+}$ corresponds to spacelike momentum transfer. In the timelike region we show the relevant domains for the Dalitz and double Dalitz decays and the dotted lines indicate the vector-meson pole locations (not to scale).

Figure 2

The transition form factor given by Eq. (7). The nonperturbative input is the Bethe-Salpeter amplitude ${\mathrm{\Gamma }}_{\pi }$ of the pion (grey circle), the dressed quark propagators (straight lines) and the dressed quark-photon vertices ${\mathrm{\Gamma }}_{\nu }$ (blue circles).

Figure 3

On-shell transition form factor in the symmetric limit (${\eta }_{+}=Q^2=Q^{\prime 2}$) and asymmetric limit (${\eta }_{+}=Q^2/2$, $Q^{\prime 2}=0$) for moderate spacelike values ${\eta }_{+}>0$.

Figure 4

The tree-level contributions to the (a) single Dalitz decay, (b) double Dalitz decay and (c) rare decay of the neutral pion. The yellow circle denotes the pion transition form factor $F(Q^2,Q^{\prime 2})$, see Eq. (1).

Figure 5

Transition form factor shown in the region relevant for the three- and four-body decays. The singly virtual form factor for the three-body decay, for which either $Q^2=0$ or $Q^{\prime 2}=0$, is indicated by the heavy blue lines.

Figure 6

Relevant kinematic domain of the transition form factor in the ${\pi }^0\to e^{+}{e}^{-}$ decay. The parabola starting at ${\eta }_{+}=-m_{\pi }^2/4$ is the region that is sampled in the integral.

Figure 7

Sketch of the overlapping branch cuts in the integrand of $\mathcal{A}(t)$, i.e., the complex $\sigma$ plane, for $t=(-1+i)m_{\pi }^2/4$ and $m=40\mathrm{MeV}$. The cut ${\sigma }_l$ (solid, red) is generated by the lepton pole and the cut ${\sigma }_{\gamma }$ (dashed, blue) by the photon poles; the latter opens at $\sigma =t$ but the former does not. The dotted line shows a possible integration path avoiding all singularities. The units are in ${\mathrm{GeV}}^2$.

Figure 8

Result for $\mathcal{A}(t)$ in the complex $t$ plane. The units for $t$ are ${\mathrm{GeV}}^2$ and $\mathcal{A}(t)$ is dimensionless. The on-shell pion point is $t_0=-m_{\pi }^2/4\approx -0.005{\mathrm{GeV}}^2$.

Figure 9

Kinematically accessible regions in the $(\omega ,{\eta }_{+})$ plane for different frames; the units are in ${\mathrm{GeV}}^2$. From left to right: $\alpha =0$, $½$, 1, ${}^3⁄{}_2$, 8. The (blue) triangles are merely drawn for guidance and show the spacelike domain ($Q^2>0$, $Q^{\prime 2}>0$).