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I have heard that the EWPT in the Standard Model ($m_h=125$ GeV) is a cross-over transition, i.e. no discontinuity in the order parameter. However what happens to the Higgs mass at the critical temperature $T_{cr}$?

I cannot be precisely 0 since this would be a second order phase transition if I am not mistaken. I have seen results of lattice computations which compute the vev $v(T)$, which is continuosly changing, but nowhere there is any mention of the Higgs mass at $T_{cr}$.

Qmechanic
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krabby patty
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    You can transpose the problem to Landau's phase transition theory in symmetric case with $\eta ^{2} \equiv \phi^{\dagger}\phi;;, a(T)\equiv \mu$ with $C_{0}= 0;;;.$ https://en.wikipedia.org/wiki/Landau_theory – The Tiler Jun 16 '23 at 13:40
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    The potential is flat at the critical point and therefore the Higgs is massless. Explain why you conclude this is impossible. – Cosmas Zachos Jun 19 '23 at 14:03
  • It is probably, Ekstedt & Löfgren, A critical look at the electroweak phase transition, J. High Energ. Phys. 2020 136 (2020), a continuous transition... – Cosmas Zachos Jun 19 '23 at 14:19

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By analogy:

  • Lagrangian Higgs density:$$\mathcal{L}_{H}=(D_{u}\phi_{L})^{\dagger}(D^{u}\phi_{L})-V(\phi_{L})\;\;\;\;\;\;\;\;\;\;\;\;(1)$$ and a potential $$V(\phi_{L})=\mu^{2}\phi_{L}^{\dagger}\phi_{L}+\lambda (\phi_{L}^{\dagger}\phi_{L})^{2}$$ If we take $\mu^{2}<0$, we find a minimum at $$\phi_{3}^{2}=-\frac{\mu^{2}}{\lambda}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(A)$$ $$\phi_{L}=\frac{1}{\sqrt{2}}\left(\begin{matrix} \phi_{1}+i\phi_{2}\\ \phi_{3}+i\phi_{4} \end{matrix}\right)$$

  • Free energy in Ginzburg–Landau theory for a homogeneous superconductor where there is no superconducting current $$\alpha \psi+\beta |\psi|^{2}\psi=0$$

The equation has two solutions :$\psi=0$ or $$|\psi|^{2}=-\frac{\alpha}{\beta}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(B)$$ We compare $(A)$ and $(B)$, the case study (B) can be found in (2). So the mass of the Higgs boson [$m=f(\frac{-\mu^{2}}{\lambda})$] is zero at the critical temperature, because $\alpha(T_{c})=0$ by analogy.

(1) For exemple :Mécanique quantique relativiste :Théorie de jauge, Michael Klasen.

(2) https://en.wikipedia.org/wiki/Ginzburg%E2%80%93Landau_theory

The Tiler
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  • The Higgs theory gives the evolution of $\mu^{2}$ as a function of temperature, but it does not predict the critical temperature and, consequently, the value of $\mu^{2}$ remains arbitrary. As with the Ginzburg-Landau theory, the parameter values must be entered by hand in order to reproduce the experimental data. – The Tiler Jun 21 '23 at 20:53