# Is the analytical dependence of the theoretical Higgs potential with temperature known?

At the very beginning of the unvierse, the Higgs potential had a paraboloid shape. After the electroweak Symmetry Breaking, it took the mexican hat shape.

Is the analytical dependence of Higgs potential with temperature known ?

• – G. Smith May 28 at 21:20
• Thank you, but the vev is a consequence of the potential, so my question is different. And also, the answer to the other link needs to pay for accessing the document. – Mathieu Krisztian May 29 at 7:56
• That paper behind the paywall is the original 1974 Weinberg paper that Simon’s answer mentions. It calculates the temperature corrections to the potential, not just the vev. – G. Smith May 29 at 16:19
• @G. Smith : thank you – Mathieu Krisztian May 29 at 19:49

The temperature contribution to the effective potential was calculated by Weinberg and by Jackiw & Dolan independently in 1974. The FTEP at one-loop order and at high temperature ($$T\gg T_c$$) looks like $$V_{\mathrm{T}}(\phi_{\mathrm{cl}})= V_{\mathrm{eff}}(\phi_{\mathrm{cl}}) + \frac{\lambda}{8}T^2\phi_{\mathrm{cl}}^2 - \frac{\pi^2}{90}T^4 + \cdots$$ where $$V_{\mathrm{eff}}(\phi_{\mathrm{cl}})$$ is the zero-temperature one-loop effective potential (which will have the Mexican hat shape). This means that the additional $$\phi_{\mathrm{cl}}^2$$ term due to the temperature contribution will dominate over the original $$\phi_{\mathrm{cl}}^2$$ term due to the "mass term" in the Higgs potential. Here is a plot of the FTEP temperature dependence: Note that the expression above is only valid for high temperatures, so plotting out that potential will not produce the same figure.