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How do we know that the quanta of a quantized Dirac field describe elementary spin-1/2 fermions (or point particles of spin-1/2) and not composite spin-1/2 fermions (or extended structure of spin-1/2) such as a proton or a neutron?

Response to comment Surely Dirac equation doesn't describe the composite spin-1/2 particles. This is because, if it were so, the QED Feynman rule (derived assuming that the interacting charged fermions are described by Dirac theory) for the proton-proton-photon vertex in the case electron-proton scattering would be $-ie\gamma_\mu$. But it is not the case.

Response to the answers I was reading Halzen and Martin. There they said that the vertex factor $ie\gamma^\mu$ cannot be used for protons because unlike electrons it is an extended structure. It is an effective vertex proton-photon vertex $ie \Gamma^\mu$ which contains the information that the proton is non-elementary. You may find the same thing written in the paragraph above equation (345), in the notes here.

But if I understand the existing answers correctly, they are suggesting that $ie\gamma^\mu$ is replaced by the effective vertex $ie\Gamma^\mu$ not because the proton is an extended object but because we are taking loop corrections into account.

I am confused now. What is the correct reason?

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    $\begingroup$ What makes you think it doesn't? $\endgroup$ – user4552 Jan 23 at 12:58
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    $\begingroup$ Related: physics.stackexchange.com/q/506225 $\endgroup$ – tparker Jan 23 at 13:47
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    $\begingroup$ There is a whole branch of nuclear physics (relativistic mean field theory) predicated on using Dirac's equation to describe neutron and proton motions in nuclei and neutron stars. $\endgroup$ – Lewis Miller Jan 23 at 16:47
  • $\begingroup$ I think that by "the information that the proton is non-elementary", they really mean "the information about the loop-level strong interactions that confine the baryon." So it amounts to the same thing. $\endgroup$ – tparker Jan 23 at 16:53
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    $\begingroup$ It would be too good to be true if we could tell from experiments at arbitrarily low energies whether a certain particle was composite, just based on some kind of qualitative difference in behavior. Then nobody would build particle accelerators. What we should expect is that there should be an energy scale set by the excitation energies of the nucleon, and also a length scale set by the size of the nucleon. Experiments far below these scales should reveal this structure only through small corrections. $\endgroup$ – user4552 Jan 23 at 17:32
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One succes of the Dirac equation is that it correctly implies the particle g-factor to be g=2, explaining the lepton g-factor. For protons and neutrons g is very different from 2, so the Dirac equation itself cannot be applied to these.

The squared Dirac equation exhibits a spin dependent term, the relativistic generalization of the Pauli interaction. In this equation the g-factor of 2 can be substituted for by the g-factor of the proton or the neutron. In this case the modification takes account of the fact that these are composite particles. There are also loop corrections to the g-factor. These can also be taken into account in this way - but avoid double counting in perturbation theory. So the answer is: both.

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    $\begingroup$ The point of the OP's question is why the Dirac equation's prediction $g=2$ fails for baryons. This seems more like a consequence than an explanation for that putative failure. $\endgroup$ – tparker Jan 23 at 21:30
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    $\begingroup$ I think the question is: is the vertex altered to account for loop corrections or for composite particle nature. Your comment induced me to extend (and hopefully improve) my answer in this sense. $\endgroup$ – my2cts Jan 23 at 22:03
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The Dirac equation does describe composite spin-1/2 fermions - namely, baryons like the proton and the neutron. Conversely, future experiments might reveal the electron to be composite even though it's described by the Dirac equation (plus perturbative corrections).

The vertex term you describe does appear in the scattering cross-section for proton-photon scattering, but it's corrected by loop-level renormalization terms that stem from interactions, which are tiny (but measurable) for the electron but large for the proton.

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  • $\begingroup$ I was reading Halzen and Martin, where they said that the vertex factor $ie\gamma^\mu$ cannot be used for protons because unlike electrons it is an extended structure. But you are suggesting that $ie\gamma^\mu$ is replaced by the effective vertex $ie\Gamma^\mu$ not because the proton is an extended object but because we are taking loop diagrams into account. Is that correct? $\endgroup$ – mithusengupta123 Jan 23 at 14:30
  • $\begingroup$ @mithsengupta123 That's largely a matter of semantics. The strong interaction leads to both quark confinement into protons and loop-level effects that renormalize the vertex factor. How you want to draw the arrows of causal implication is basically a philosophical matter. $\endgroup$ – tparker Jan 23 at 16:48
  • $\begingroup$ @mithsengupta123 Also, QED also has a renormalized vertex factor at loop level. It's just not as big of a correction as for the baryon case. The two cases are really only quantitatively different in this respect. $\endgroup$ – tparker Jan 23 at 16:50
  • $\begingroup$ Thanks, @tparker ... but for someone doing this electron-proton scattering calculation for the first time, how would he/she convince himself/herself a priori that corrections will be large for proton-electron-photon vertex than electron-electron-photon vertex? Aren't you already using the knowledge that protons are made of smaller constituents to argue this? $\endgroup$ – mithusengupta123 Jan 30 at 7:55
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Just as a complement to tparkers answer, people have been using the Dirac equation for composite particles since very long ago. Just remind yourself of the Yukawa model for hadron-hadron interactions

$$\mathcal L=\bar\psi(i\gamma^\mu\partial_\mu-M)\psi+\frac{1}{2}(\partial^\mu\phi)(\partial_\mu\phi)-\frac{1}{2}m^2\phi^2-ig\phi\bar\psi\gamma^5\psi$$

Note the first term, which is the Dirac equation for the nucleon in question. This theory is what gives us the attractive Yukawa potential

$$V(r)=-\frac{g^2}{4\pi}\frac{e^{-mr}}{r}$$

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