Axial anomaly and the interplay of quark loops with pseudoscalar and vector mesons in the ${\gamma }^{*}\to {\pi }^{+}{\pi }^0{\pi }^{-}$ process

Motivated by the ongoing measurements of the Primakoff process ${\pi }^{-}{\gamma }^{*}\to {\pi }^{-}{\pi }^0$ by the COMPASS Collaboration at CERN, the transition form factor for the canonical anomalous process ${\gamma }^{*}\to {\pi }^{+}{\pi }^0{\pi }^{-}$ is calculated in a constituent quark-loop model. The simplest contribution to this process is the quark “box” amplitude. In the present paper we also explicitly include the vector meson degrees of freedom, i.e., the $\rho$ and the $\omega$, thus giving rise to additional, resonant contributions. We find that in order to satisfy the axial anomaly result, a further subtraction in the resonant part is needed. The results are then compared with the vector meson dominance model as well as the Dyson-Schwinger calculations, the chiral perturbation theory result, and the available data.

Figure 1

One of the box diagrams for the process ${\gamma }^{*}\to {\pi }^{+}{\pi }^0{\pi }^{-}$. There are six different contributing graphs, obtained from Fig. 1 by the permutations of the vertices of the three different pions. The position of the $u$ and $d$ quark flavors on the internal lines, as well as $Q_u$ or $Q_d$ quark charges in the quark-photon vertex, varies from graph to graph, depending on the position of the quark-pion vertices. The physical pion fields are ${\pi }^{\pm }=({\pi }_1\mp i{\pi }_2)/\sqrt{2}$ and ${\pi }^0\equiv {\pi }_3$. Thus, in Eq. (3) one has $\pi =\sqrt{2}({\pi }^{+}{t}_{+}+{\pi }^{-}{t}_{-})+{\pi }^0{t}_3$ where $t_{\pm }=t_1\pm i{t}_2$ (see text).

Figure 2

The form factor $G_{\rho \pi \pi }$ seen as a quark triangle diagram in the CQL model.

Figure 3

One of the “resonant” triangle diagrams for the process $\gamma (q)\to {\pi }^{+}(p_1){\pi }^0(p_2){\pi }^{-}(p_3)$, where the two pions with momenta $p_1$ and $p_2$ are obtained by the decay of the intermediate $\rho$ meson transferring the momentum $Q=p_1+p_2$. Two more analogous graphs are obtained by replacing ${\rho }^{+}\to {\pi }^{+}{\pi }^0$ and ${\pi }^{-}$ by, respectively, ${\rho }^0\to {\pi }^{+}{\pi }^{-}$ and ${\pi }^0$, or by ${\rho }^{-}\to {\pi }^{-}{\pi }^0$ and ${\pi }^{+}$. Also, each of the three graphs has its crossed graph.

Figure 4

One of the graphs for the ${\gamma }^{*}\to 3\pi$ nonresonant triangle contribution depicting a ${\pi }^{+}{\pi }^0$ pair coming out from the $\overline{q}q\pi \pi$ Weinberg-Tomozawa interaction that is needed for the appropriate subtraction. The two analogous graphs are obtained by exchanging the pion pair by ${\pi }^{+}{\pi }^{-}$ and ${\pi }^0{\pi }^{-}$. There are two of each of these graphs.

Figure 5

For various values of the mass parameter $M_q$, our numerically calculated form factors ${\tilde{F}}_{\gamma }^{3\pi }(s,t,u)$ are shown as functions of $t=-(p_2+p_3{)}^2$ $=-(p_{{\pi }^0}+p_{{\pi }^{-}}{)}^2$, i.e., of the invariant mass of the outgoing pion pair ${\pi }^0{\pi }^{-}$ in the Serpukhov and COMPASS experiments. There, all pions are on shell, and for definiteness we fix $u=m_{\pi }^2$. The curves belonging to the lower stripe (blue online) are only the quark box contribution (i.e., the lower stripe represent predictions of the simple CQL approach for various $M_q$). The upper stripe (with black curves) represents the corresponding predictions from the presently pertinent CQL-VMD approach; that is, the black curves represent predictions which include the $\rho$-resonant triangle loops with the subtracted Weinberg-Tomozawa interaction. In the both stripes, the dotted curves correspond to $M_q=300\mathrm{MeV}$, the dashed ones to $M_q=330\mathrm{MeV}$ and the solid ones to $M_q=360\mathrm{MeV}$. The exhibited data point [7] really corresponds to the average value of the form factor over the momentum range covered by the experiment (between the two-pion threshold and 10 $m_{\pi }^2$), which measured the total cross section—see the discussion in Sec. 4b.

Figure 6

The form factors ${\tilde{F}}_{\gamma }^{3\pi }(s,t,u)$ from various approaches are depicted as functions of $t$, the invariant mass of the outgoing pion pair ${\pi }^0(p_2)$, ${\pi }^{-}(p_3)$, as in the Serpukhov and COMPASS experiments, where all pions are on shell. We fix $u=m_{\pi }^2$ for definiteness. The upper shaded stripe covers the results of our CQL-VMD approach for constituent masses between $M_q=360\mathrm{MeV}$ (corresponding to the solid black curve marking the upper edge of that stripe) and $M_q$ equal to the DS scale $\Lambda =565.69\mathrm{MeV}$ of Ref. 29 (corresponding to the lower edge of that stripe). The lower shaded stripe (blue online) covers the results of the pure CQL model [30] for the same $M_q$ interval. That is, the (blue) dashed curve depicts the CQL model form factor for $M_q=360\mathrm{MeV}$, while the lower edge of that stripe is the very slowly varying CQL form factor for the high $M_q=\Lambda =565.69\mathrm{MeV}$, the DS scale of Ref. 29. A comparison is made with results of the modified VMD [32, 39] (green dotted curve) and of DS (in GIA) [29] (red dash-dotted curve). Again, the exhibited Serpukhov point [7] is actually the average value extracted from the total cross section—see Sec. 4b for the comparison with experiment.

Figure 7

Same as previous Fig. 6, but for the CEBAF kinematics, where the two outgoing and on-shell pions are ${\pi }^{+}$ and ${\pi }^0$. The form factor is thus given as the function of their invariant mass squared, the $s$ variable. For the off-shell pion, ${\pi }^{-}$, we use $p_3^2=m_{\pi }^2$. We also fix $t=-m_{\pi }^2$.

Figure 8

Total cross sections for various $M_q$ predicted by CQM is depicted by the lower solid curve, while CQM-VMD yielded the upper solid curve. The (constant) value predicted by VMD alone is depicted by the dotted line, and the ChPTh prediction (electromagnetically and chiral-loop corrected) of $1.33\pm 0.03\mathrm{nb}$ [42], is given by the dashed line. The experimental cross section is denoted by the grey area.