Why the collision of two deuterons cannot produce an alpha particle and a neutral pion? I am not able to understand why the following reaction cannot occur or rather is not observed in nature:
\begin{equation}
d + d \rightarrow \alpha + \pi^0
\end{equation}
I can see that all the quantum numbers are satisfied, but I am not able to understand why this reaction does not take place?
 A: I think you have two issues that are going to make this an insignificant part of the total cross-section.
The main issue is that the deuterium nucleus is very fragile. Its binding energy is only a couple of MeV. But to make this reaction go, you're going to need center of mass energies on the order of 100 MeV. Therefore it's going to be overwhelmingly likely that you will simply break up the deuterons into two neutrons and two protons.
The other issue is that at a center of mass energy that is near the threshold, you're typically bringing in quite a bit of angular momentum. The probability distribution is a triangle that increases as you go up to the maximum geometrically possible angular momentum, which is something like $5\hbar-10 \hbar$. So you have a lot of probability of getting $5\hbar$ or $10\hbar$, and very little probability of $0 \hbar$. But for a pion exiting with a fairly low energy (as needed for a gentle reaction in which the alpha stays stuck together), you only get $0 \hbar$. Therefore none of the angular momenta will work except for the $0 \hbar$ channel, which is a small part of the total reaction cross-section.
So in simple terms, you're hitting it too hard and it's spinning too fast, so it's going to break up.
A: Phys. Rev. C v. 50, no. 2, p. R537 (1994):

The first observation of a positive signal in the
$d+d\rightarrow\alpha+\pi^0$ reaction was made a few years ago [1] at
the French National Laboratory Saturne (LNS). The measured cross
section close to the $\eta$ threshold at a deuteron kinetic energy
($T_d$) of 1100 MeV and a c.m. angle of 107° is
(0.97$\pm$0.20$\pm$0.15) pb/sr. This value seems high and well above
the upper limit of 0.003 pb/sr at 600 MeV expected from a purely
electromagnetic process emphasized by a $\Delta$ excitation in an
intermediate state [2]. Charge symmetry breaking (CSB) is necessarily
concerned with the small $u$-$d$ quark mass difference which is
responsible for the well-known $\rho$-$\omega$ and $\pi^0$-$\eta$
mixing. Charge symmetry (CS) holds in strong interaction in the limit
where the $u$-$d$ quark mass difference is neglected. In this case CS
forbidden transitions like $d+d\rightarrow\alpha+\pi^0$ are strongly
suppressed [3].

