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In Matthew Schwarz QFT textbook on page 588-9, $W^+_L Z_L \rightarrow W^+_L Z_L$ was illustrated but the kinds of weak bosons are absent. The diagram of the problem is as follows:
enter image description here

This is s-channel, so the time is flowing from left to the right.

Question

1) Charge conservation

If I assign the $W^+$ as coming from the top left to the first vertex and $W^-$ as intermediate particle flowing to the left and again $W^+$ created at the second vertex going out to top right, then I think the charge conservation is violated at each vertex because $W^+$ carries positive electric charge however, $W^-$ boson carries the negative charge.

Can anybody explain this diagram?

2) Particle flow arrow

Also, I am curious about the particle-flow-arrow drawn on this diagram. Each boson can be seen as a particle, but each is anti-particle to each other, so there is ambiguity in specifying the direction of particle flow.

Liberty
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1 Answers1

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If $W^+$ and $Z$ are incoming on the right, then the only single particle intermediate possible is $W^+$. No other single particle would conserve all of the relevant quantum numbers (electric charge, baryon number, and lepton number, especially). You could have pairs of particles whose net charge is 1 (for example, a positron-neutrino pair), but those are higher order diagrams.

With the particle flow arrow, I think that's just indicating charge flow. $Z$s are their own anti-particle, and $W^-$ is the antiparticle for $W^+$, but none of them have conserved particle number.

To understand this diagram better, think of the $Z$ as a massive photon analogue, and the $W^+$ playing a positron like role. That is, the lowest order tree diagrams for all of these are analogous, even if they differ in details:

  • $e^- \gamma \rightarrow e^- \gamma$
  • $W^+ \gamma \rightarrow W^+ \gamma$
  • $p^+ Z \rightarrow p^+ Z$.
Sean E. Lake
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  • Thanks. But I don't see any $W^+ W^+ Z$ interaction in the Electroweak symmetry broken Lagrangian. And the calculation in textbook uses coupling term such as $\partial_\mu Z (W^+\mu W^-\nu)$ . So is it possible to have the same two $W^+$ interacting with Z? – Liberty Jan 29 '17 at 10:55
  • See the $\mathcal{L}_{WWV}$ term. Recall that $W^+$ and $W^-$ are antiparticles of each other, and the $Z$ is it's own anti-particle (else it couldn't mix with the photon in the symmetry breaking), so a term that allows $W^+ + W^- \rightarrow Z$ also allows $W^+ + Z \rightarrow W^+$. – Sean E. Lake Jan 29 '17 at 22:13