$\chi$PT and the $\pi \pi \rightarrow \gamma \gamma$ Graph and Pion's Electric Polarizability $\alpha _E^{\pi ^\pm }$

Low energy pionic interactions are fundamental tests of chiral symmetry. Chiral perturbation theory ($\chi$PT) predicts a number of basic properties of the pion such as $\pi\pi$ scattering lengths, pion weak and electromagnetic form factors, and pion electric ( $\alpha^+_{\pi}$) and magnetic ($\beta^+_{\pi}$) polarizability. While generally theory and experiment agree, for the charged pion electric polarizability they do not (28,25).

The charged pion's electric and magnetic polarizabilities $\alpha^{\pm}_{\pi}$ and $\beta^{\pm}_{\pi}$ represent pion's response to an external electric or magnetic field and reveal the internal structure of the pion. In the framework of chiral perturbation theory, the electric and magnetic pion polarizability is evaluated as (25),

$$\alpha _E \ = \ - \beta _M \ = \ \frac{\displaystyle 4 \alph... ...(L^r_9 + L^r_{10}) \: = \: 2.8 \times 10^{-4} \quad {\rm fm^3}$$ (2.9)

where $L^r_9 \equiv$ (6.9 $\pm$ 0.2)$~\times$10$^{-3}$ and $L^r_{10} \equiv$ -(5.2 $\pm$ 0.3)$~\times$10$^{-3}$ are known as low energy constants which are empirically determined respectively from the charge pion radius ( $<r^2 _\pi>^{\rm exp}$ = 0.44 $\pm$ 0.03 fm$^2$) and the experimental value of the axial structure constant in radiative pion decay $\pi \rightarrow e^+ \mu_e \gamma$. Chiral perturbation theory predicts $\alpha^{\pm}_{\pi} = \beta^{\pm}_{\pi} = 2.8 \pm 0.4$ fm$^3$ (25).

Pion's polarizability could be observed directly in a pion-Compton scattering experiment, but, in the absence of free pion targets, such experiments are difficult and indirect. Current determinations of the pion polarizability include three approaches:

• $(1)$ $\pi^-A\rightarrow\gamma\pi^-A$ where similar to the well known Primakoff effect, the real pion is scattered off a virtual photon yielding a real pion and real photon. The four momentum transferred squared ${\rm q}^2$ is in the kinematical range q$^2$ $<$ 0 (Figure 2.11a and region I in Figure 2.12).
• $(2)$ $\gamma p \rightarrow \gamma\pi^+ n$ where the real photon is scattered off a virtual pion yielding a real pion and real photon. q$^2$ $<$ 0 (Figure 2.11a and region I in Figure 2.12).
• $(3)$ The cross channel photon-photon reaction $\gamma\gamma\rightarrow \pi^+\pi^-$. q$^2$ $>$ $+4$ m$_{\pi}^2$ (Figure 2.11c and region III in Figure 2.12).

Over recent years a Sepurkov experiment (20) using approach $(1)$ has obtained $\alpha^{\pm}_{\pi} = 6.8 \pm 1.4 \pm 2.3$ fm$^3$, a Lebedev experiment (22) using approach $(2)$ has obtained $\alpha^{\pm}_{\pi} = 20 \pm 12$ fm$^3$, and MARKII (24) and PLUTO (21) experiments using approach $(3)$ have obtained $\alpha^{\pm}_{\pi} = 2.8 \pm 1.6$ fm$^3$ and $\alpha^{\pm}_{\pi} = 19.1 \pm 4.2 \pm 5.8$ fm$^3$ (27) respectively. The measured values of $\alpha^{\pm}_{\pi}$ are generally larger than the CHPT prediction for $\alpha^{\pm}_{\pi}$; however, the different polarizability determinations from the different experimental methods are far from consistent.

Figure 2.11: a) Pion Compton scattering, b) $\pi ^-p\rightarrow \gamma \gamma n$ annihilation graph c) $\gamma \gamma$ $\rightarrow \pi \pi$

Drechsel et al. (10) investigated the contribution of pion polarizability to radiative pion photoproduction via $\gamma p \rightarrow \gamma \pi n$ and explored ways of extrapolating to the $\gamma \pi^+$ scattering cross-section. Wolfe et al. (9) considered radiative pion photoproduction via the reaction mechanism $\gamma N \rightarrow \gamma N \pi$ (Figure 2.11b and region II in Figure 2.12) as a probe of pion's electromagnetic polarizability. They constructed a tree-level amplitude for radiative pion photoproduction using a pseudovector $\pi NN$ Lagrangian for photon energies $E_{\gamma} < 250$ MeV, i.e., for energies below the $\Delta$ resonance. The authors noted that background contributions unrelated to the pion polarizability dominate the photoproduction amplitude. The contribution of the pion polarizability was found to be $\sim$ 0.1%, too small a sensitivity however, to be accessible experimentally.

Figure 2.12: The different kinematical regions for the reactions. region I: $\pi$A $\rightarrow\gamma\pi$A, $\gamma p\rightarrow\gamma\pi p$ region II: $\pi ^-p\rightarrow \gamma \gamma n$ region III: $\gamma\gamma\rightarrow\pi\pi$

The $\pi ^-p\rightarrow \gamma \gamma n$ reaction is potentially sensitive to pion's polarizability. The $\pi \pi \rightarrow \gamma \gamma$ annihilation diagram (Figure 2.11b) which is predicted to dominate the $\pi ^-p\rightarrow \gamma \gamma n$ process in the small opening angle region (7), can be viewed as the annihilation of a real pion with a virtual pion $\pi^-\pi^+\rightarrow\gamma\gamma$ or, via crossing symmetry, as the transition of a real pion to a virtual pion via Compton scattering $\gamma\pi\rightarrow\gamma\pi$ (Figure 2.11a). Therefore even though an indirect tool, this $\pi ^-p\rightarrow \gamma \gamma n$ reaction, with its predicted dominance of the annihilation graph in the small opening angle region may provide with valuable information regarding charged pion's electric polarizability as predicted from the framework of chiral perturbation theory.