In Glashow-Weinberg-Salam electroweak theory, the relation $$M_W=M_Z\cos\theta_W\tag{1}$$ is said to be remain true only at the tree-level; it receives corrections from the loop diagrams. See here. But shouldn't the relation $(1)$ be always valid if $M_W$ and $M_Z$ are defined to be physical masses i.e., bare+loop corrections? Please correct me if I have some erroneous impression.
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2$\begingroup$ Why do you think loop corrections should preserve such a relationship? After all, the $W$ and $Z$ couple very differently to all the other particles. $\endgroup$– knzhouCommented May 26, 2019 at 15:34
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1$\begingroup$ Edited! @AccidentalFourierTransform $\endgroup$– SRSCommented May 26, 2019 at 15:39
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$\begingroup$ Doesn't $M_Z$ and $M_W$ receive corrections? If so, don't we redefine $M_Z+\delta_1$ and $M_Z+\delta_2$ to be $M_Z$ and $M_W$? @knzhou $\endgroup$– SRSCommented May 26, 2019 at 15:46
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$\begingroup$ ?? You redefine $M_Z$ and $M_W$ to be the tree value plus corrections, so the 1-loop "physical" masses are different than the tree ones. Veltman and Ross show you the former violate the custodial symmetry present at the tree but not the 1-loop level. What on earth are you asking, given the cottage industry you are invoking? $\endgroup$– Cosmas ZachosCommented May 26, 2019 at 18:31
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$\begingroup$ Could you perhaps articulate your "impression" in PDG mainstream contemporary language? $\endgroup$– Cosmas ZachosCommented May 26, 2019 at 18:50
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1 Answer
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Because after EWSB, the Proca Lagrangians that you obtain for those particles imply that relation, so without loop corrections (amin self-energies and that kind of stuff) your propagator goes like
$$ \frac{-ig_{\mu \nu}}{p^2 - M_Z^2 + i\epsilon}, \quad \frac{-ig_{\mu \nu}}{p^2 - (cos\theta_W M_Z)^2 + i\epsilon} $$
Now, you know that these expressions are without corrections, i.e., at tree level. When you introduce self-energies the mass term in propagator changes and therefore their relation.
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$\begingroup$ So, if I understand you correctly, the quantities $M_Z$ and $M_W$ are defined by the tree-level relations, once and for all. Also see my comment to knzhou. @Vicky $\endgroup$– SRSCommented May 26, 2019 at 15:48
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$\begingroup$ @SRS No, what I'm tealling you is that Proca Lagrangians for $W^{\pm}$ and $Z$ drive you to the relation you are asking for but the masses you see there are 'bare' mass. The real mass of those bosons comes from the complete propagator that is given by corrections such as self-energies. When you compute that corrections, then in the complete propagator you see that to the bare mass is added a term. Now from Källen-Lehmann propagator representation you know that the bare mass plus that addtional term (actually its derivative) (cont.) $\endgroup$– VickyCommented May 26, 2019 at 15:55
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$\begingroup$ gives you the correct the mass. Forget what I said about the energy dependent angle. The masses change, the relation among them too becuse if $M_W$ changes and it is equal to $M_Z cos\theta_W$ and to conserve that relation you would need to assume some relation among $\theta_W $ and the energy: $M_W(p^2) = M_Z(p^2)cos\theta_W(p^2)$ $\endgroup$– VickyCommented May 26, 2019 at 16:02
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$\begingroup$ Are these corrections finite? Do you have a reference which shows the relevant computation? @Vicky $\endgroup$– SRSCommented Jul 5, 2019 at 7:07
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$\begingroup$ @SRS Dou you mean if the corrections as self-energies are finite? Not in general, that's why you need renormalization in QFT $\endgroup$– VickyCommented Jan 18, 2020 at 17:04