# Where do pions get angular momentum from?

If a neutral meson carrying intrinsic spin equal to 1 decays into a pair of charged pions, how can they possibly conserve angular momentum? Pions have no spin and if they originate from the same point then their relative motion can't generate angular momentum either.

Won't they just go their separate ways? How can there be orbital angular momentum between them?

• They'll go their separate ways, in a p-wave: non-uniformly, mindful of the spherical harmonic $Y_1^m$ angular distribution imposed by the spin of the parent particle. What is the question? Commented Jan 16, 2022 at 14:42
• @CosmasZachos Isn't p related to orbital angular momentum? Commented Jan 16, 2022 at 14:52
• Yes: the unit spin of the parent particle converts to orbital angular momentum in the daughters. Total spin + orbital is conserved, as it should; you recall the separate components interconvert, no? Commented Jan 16, 2022 at 14:54

Let’s see a concrete example:

$$ρ_0 \rightarrow π^+π^-$$

This is an indicative first-order Feynman diagram for this decay .

Pions have no spin and if they originate from the same point then their relative motion can't generate angular momentum either.

It is a quantum mechanical reaction. As you see, the two pions do not originate from the same point. In general, in Feynman diagrams, the vertices are only out of three lines and one of them is the outgoing one.

This leaves leeway for orbital angular momentum to exist between them, when the final two pions are described in terms of a wave function, which will carry the spin 1 of the rho by conservation of angular momentum.

• Note that $\rho^0\to\pi^+\pi^-$ is allowed, but $\rho^0\to\pi^0\pi^0$ is forbidden, probably because exchange symmetry forbids two identical $\pi^0$ to occupy a negative-parity $L=1$ final state.
– rob
Commented Jan 16, 2022 at 19:30
• presumably this just a first-order feynman diagram for the decay? I assume the gluon could also interact with the anti-up quark instead of the up quark Commented Jan 16, 2022 at 21:53
• @user253751 sure, all the quantum number conserving replacements can be drawn, it is indicative to show the quantum complexity. Commented Jan 17, 2022 at 6:02
• @annav Can this be regarded as evidence that virtual particles actually exist, rather than merely being mathematical artefacts? Commented Jan 19, 2022 at 8:59
• @Struggling_Student I do not think so. Incoming and outgoing particles exist, but the quantum number conservation is just algebra, it does not require to name as particle the carrier of thedp/dt with the quantum numbers. Commented Jan 19, 2022 at 9:28