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 Answer 1


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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