I am a total beginner with non-abelian gauges. To write down a process from a neutral pion ($\pi^0 = \frac{1}{\sqrt{2}}(u\overline{u}-d\overline{d})$) I expect to have to write it as this superposition. I would then expect to have to compute the difference of two feynman-diagramms. I mean something like: \begin{align} \langle \pi^0 | iT| \gamma_1 \gamma_2 \rangle = \frac{1}{\sqrt{2}} \left(\langle u\overline{u} | iT| \gamma_1 \gamma_2 \rangle - \langle d\overline{d} | iT| \gamma_1 \gamma_2 \rangle\right) \end{align} Here $T$ is meant in the sence as introduced in Peskin-Schroeder: Introduction to quantum field theory, where the M-Matrix is defined by something like $iT = \delta(\text{input-momentum}-\text{output-momentum})\cdot M$ if I recall it correctly. Now I would proceed to insert the interaction lagrangian and compute the, say, lowest order. I am not particularly interested in the result of the amplitudes of such processes. Also this is just a generic example of a $\pi^0$-Process that came to my mind. I simply want to know if my approach is right.

My supervisor (experimental physicist) told me, that I need only to regard the $\pi^0$-Particle as $u\overline{u}$ OR $d\overline{d}$, since both interchange very easily and the diagrams are not computable anyways. I was not very convinced by this. I understand that, because of the running of QCD coupling constant, we have a problem with perturbation theoretical approaches. My point is not really the actual computation of the scattering but more the fact, that I really dont see how I can link my idea to what my supervisor told me. Especially I can not see in which way the minus in the superposition affects this, if my supervisor was right. What difference would a plus make? Thanks!


1) In general, hadronic decay processes involve strong interactions (they are not perturbative), and the decay rates cannot be computed in terms of Feynman diagrams. The particular reaction you refer to, the decay $\pi^0\to\gamma\gamma$, proceeds via the axial anomaly, and the rate can be expressed in terms of a single experimentally measured parameter, the weak (charged) pion decay constant $f_\pi$. This is discussed in most field theory text books.

2) Because the pion is a non-perturbative bound state you cannot just write $|\pi^0\rangle =(|\bar{u}u\rangle - |\bar{d}d\rangle/\sqrt{2}$. The real pion wave function contains gluons and an arbitrary number of $q\bar{q}$ pairs. However, this combination of quark fields has the right quantum numbers, so you can write $$ \frac{1}{\sqrt{2}}\langle 0|(\bar{u}u-\bar{d}d)|\pi^0\rangle = g_\pi $$ where $g_\pi$ is a constant (that can be related to $f_\pi$). An expression you frequently hear is that $j=(\bar{u}u-\bar{d}d)/\sqrt{2}$ is an interpolating current for the $\pi^0$.

3) Because of isospin symmetry, the coupling of $j_s=(\bar{u}u+\bar{d}d)/\sqrt{2}$ to the $\pi^0$ vanishes. This means that the $\bar{u}u$ and $\bar{d}d$ terms make equal contributions to the matrix element that defines $g_\pi$. In that sense, $\bar{u}u$ all by itself is also an interpolating current for the $\pi^0$.

4) The coupling to $\gamma\gamma$ violates isospin (because of the different quark charges). The $u\bar{u}$ coupling is proportional to $4e^2/9$, and the $\bar{d}d$ coupling is $e^2/9$. So the $\pi^0$ decay into $\gamma\gamma$ proceeds mainly via the $u$ quark component.


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