# Superposition of charge states in the Neutral Pion

The quarks of neutral pions don't exist in a pure flavour state, and instead are described as a superposition of up-antiup with down-antidown:

$$\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}$$

However up and down quarks have different charges. Considering a single quark in isolation, this appears to indicate a superposition of charge states. Such a charge superposition is forbidden for free particles such as electrons/positrons.

1. How does the standard model reconcile this apparent contradiction? Is this a limitation of the quark model?

2. Are there any circumstances in which free particles can exist in a superposition of charge states?

• Both $u \bar{u}$ and $d \bar{d}$ have charge $0$. So, there is no superposition of states with different charges. 1. There is no contradiction and the SM has nothing to reconcile. 2. Look for "super selection rule". Commented Jan 30, 2023 at 8:38
• Commented Jan 30, 2023 at 9:49
• Commented Jan 30, 2023 at 9:57

You might be confused. Ignoring the sign complication for the moment (having to do with SU(2) conjugate representations), consider the rotation group, in the "addition" of angular momentum constituents/components.

A total spin 0 singlet state composed of two spin 1/2 constituents, $$\frac{1}{\sqrt 2}(|\!\uparrow\downarrow\rangle - |\!\downarrow\uparrow\rangle),$$ is really a superposition of two spin 0 states with eigenvalues S=0, and $$m=0$$. Such states may mix and interconvert.

1. Similarly, you are superposing two middle ($$I_3=0$$) states with isospin 1 (isotriplets), and $$Y=0$$, so Q =0. These two states my interconvert into each other, and so they can mix and superpose.

Raising the isospin gives you a Q=1 state, $$|\bar d u\rangle ,$$ for instance. But you may not superpose this on the neutral pion you started with:

1. As @Hyperon has commented, superselection rules prevent mixed charge superpositions: each charge sector lives in a "different world" so so speak. There are several questions illustrating this on this site. To the extent you are considering representations of conserved charge, you'll never have non-vanishing dot products of states of different charge. E.g., superselection sectors of different fermion number will stay apart, as fermion number cannot change.

2. An illustration: You'll presumably learn later on that S is violated by a bit (2nd order) by the weak interactions (box diagram), leading to, e.g. strangeness oscillations between $$K^0=\bar s d$$ and $$\bar K^0= \bar s d$$, at a frequency ~ 3.5μeV. That is to say, this neutral meson can transition to its antiparticle, so it does not form a leak-proof superselection sector! So you can, and do, form superpositions of indistinct strangeness, e.g., $$K_L^0=\frac{1}{\sqrt 2} (|\bar s d\rangle -| \bar d s \rangle ) ,$$ but distinct charge (still inviolable!). This is a propagating state, whose strangeness flip-flops in time, in collusion with $$K_S^0$$, its orthogonal combination. A formally analogous phenomenon occurs for neutrinos.

• Thanks for your answer. The part explaining forbidden mixed charge superpositions was very helpful. However I'm still not clear on the meaning of superposing two (apparently) different flavour states up-antiup and down-antidown. I'm aware that for both "pure" states the total flavour and charge is zero, allowing superposition, but wouldn't deep inelastic scattering allow us to "collapse" the system into a single state up-antiup or down-antidown? Forgive me if I'm misunderstanding DIS. Commented Jan 31, 2023 at 9:21
• If by DIS you mean S channel photon exchange, it is precisely this channel that allows mixing/superposition of uubar with ddbar, ssbar, etc. have you learned about ω-φ mixing? Commented Jan 31, 2023 at 12:19
• What's the relationship between deep inelastic scattering and S channel photon exchange? I've looked for a source mentioning both but failed to find one. Commented Feb 1, 2023 at 6:47
• Tree level photon exchange diagram. Your question is irrelevant to DIS!! Commented Feb 1, 2023 at 8:09