I was thinking that one could write down a loop diagram with a top loop and 3 external neutral Higgs particles. Since the Yukawa interactions are also present when the electro-weak symmetry is restored at high temperatures this would give us an effective $hhh$ vertex and break the electroweak symmetry. Why can this not happen? In particular the top Yukawa coupling is also order 1.
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You can't close the top loop. Every Higgs vertex has a left-handed and a right-handed top coming out of it. So any top "triangle" would have two top lines with the same chirality touching.
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1$\begingroup$ Neat. But the OP might retort with a 4-h "square"... $\endgroup$ Commented May 26, 2021 at 15:26
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1$\begingroup$ @CosmasZachos in the Standard Model that would just renormalize the Higgs quartic leaving the SU(2) gauge invariance unbroken. From what I understand OP wanted to know why his triangle diagram doesn't break the gauge SU(2) when working in the unbroken theory. Then we can go into as many details as you want about how the electroweak symmetry breaking can arise at loop level when certain conditions are met $\endgroup$– FrodCubeCommented May 26, 2021 at 16:02
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1$\begingroup$ Fair enough... I thought he was trying to reinvent Coleman-Weinberg, but, you are so right!, what seems to be in mind is symmetry restoration. Neat answer. $\endgroup$ Commented May 26, 2021 at 16:14