Related to \(\pi^0\to \gamma\gamma\)

Based on Euclidean notation (not the same as the \(\gamma_5\) notation used in the code).

The axial current that couple to \(\pi^0\) is:

\[ \begin{align} J_{\mu}^{5,\pi^0}(x) = \frac{1}{\sqrt 2} \big( \bar u(x) \gamma_\mu \gamma_5 u(x) - \bar d(x) \gamma_\mu \gamma_5 d(x) \big) \end{align} \]

The corresponding pseudo-scalar density

\[ \begin{align} P^{\pi^0}(x) = \frac{1}{\sqrt 2} \big( \bar u(x) \gamma_5 u(x) - \bar d(x) \gamma_5 d(x) \big) \end{align} \]

We have (\(p_t = i \sqrt{m_\pi^2 + \vec p^2 }\))

\[ \begin{align} \langle 0 | i J_\mu^{5,\pi^0}(0) | \pi^0(\vec p) \rangle = i f_\pi p_\mu \end{align} \]

\[ \begin{align} \langle 0 | i P^{\pi^0}(0) | \pi^0(\vec p) \rangle = f_\pi \frac{m_\pi^2}{2 m_l} \end{align} \]

\[ \begin{align} f_\pi = \sqrt 2 F_\pi \end{align} \]

Here \(F_\pi \approx 92~\mathrm{MeV}\).

\[\begin{split} \begin{align} \mathcal H_{\mu,\nu} (x, p) =& \langle 0 | T J_\mu(x/2) J_\nu(-x/2) | \pi(\vec p) \rangle \\ =& -\epsilon_{\mu,\nu,\rho,\sigma} x_\rho p_\sigma H(x^2,p\cdot x) \end{align} \end{split}\]

\[\begin{split} \begin{align} \mathcal{F}_ {\mu,\nu}(q_1,q_2)=& \int d^4x \, e^{-i\left(q_1-q_2\right)\cdot x/2}\mathcal{H}_ {\mu,\nu}(x,q_1 + q_2) \\ \mathcal{F}_ {\mu,\nu}(q_1,q_2)=& i \epsilon_{\mu,\nu,\alpha,\beta}{q_1}_ {\alpha}{q_2}_ {\beta}\mathcal{F}_ {\pi^0\gamma^* \gamma^* }(-q_1^2,-q_2^2) \end{align} \end{split}\]

OPE relation

At small \(x\):

\[\begin{split} \begin{align} S^q(x) =& \langle T\{q(x) \bar q(0) \} \rangle \\ \approx& (-\gamma_\rho \partial_\rho + m_q) G(x) \\ \approx& \frac{2 x_\rho \gamma_\rho + m_q x^2}{4\pi^2 (x^2)^2} \\ \end{align} \end{split}\]

\[ \begin{align} \gamma_\mu \gamma_\rho\gamma_\nu = -\epsilon_{\mu,\nu,\rho,\sigma} \gamma_\sigma \gamma_5 + ( \delta_{\mu,\rho}\gamma_\nu +\delta_{\rho,\nu}\gamma_\mu -\delta_{\mu,\nu}\gamma_\rho ) \end{align} \]

\[\begin{split} \begin{align} T\{J_\mu(x) J_\nu(0)\} =& \sum_q e_q^2 T( \bar q(x) \gamma_\mu q(x) \bar q(0) \gamma_\nu q(0) ) \\ \simeq& -\frac{1}{\pi^2(x^2)^2} \epsilon_{\mu,\nu,\rho,\sigma} x_\rho \sum_q e_q^2 \bar q(\frac{x}{2}) \gamma_\sigma \gamma_5 q(\frac{x}{2}) \end{align} \end{split}\]

\[\begin{split} \begin{align} \mathcal H_{\mu,\nu} (x, p) \approx& -\frac{1}{\pi^2(x^2)^2} \epsilon_{\mu,\nu,\rho,\sigma} x_\rho \sum_q e_q^2 \langle 0 | \bar q(0) \gamma_\sigma \gamma_5 q(0) | \pi(\vec p) \rangle \\ \approx& - \frac{e_u^2 - e_d^2}{\pi^2} \epsilon_{\mu,\nu,\rho,\sigma} \frac{x_\rho}{(x^2)^2} \frac{f_\pi}{\sqrt 2} p_\sigma \end{align} \end{split}\]

Therefore:

\[ \begin{align} H(x^2, p\cdot x) \approx& \frac{e_u^2 - e_d^2}{\pi^2} \frac{1}{(x^2)^2} F_\pi \end{align} \]