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I am quite confused about energy scale and vacuum expectation value (VEV). For example, this wiki page says 246Gev is VEV of Higgs field. Does it mean if we go to a higher energy scale (like 24600Gev) for a lower energy scale (like 0.0246Gev), we will not observe Higgs field as well as other electroweak process?

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    $\begingroup$ "Electroweak scale" is a bit vague, as there are several important energy scales all in proportion to the Higgs VEV. At low energies, the weak interaction is short-ranged due to W/Z masses proportional to the VEV. $\endgroup$
    – J.G.
    Commented Nov 12, 2022 at 11:09

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You have $v\equiv 246$GeV, whence the Fermi constant $G_F=v^2/\sqrt 2$ controlling weak interactions with energies much below your conjectural v/100 is all these interactions "know" about the Standard Model. The masses of the W & Z vector bosons are about a third of v and their effect on the relevant interactions is indirect at these energies.

The mass of the Higgs boson happens to be about half of v, by "accident", and it is hard to see any effects of it in your conjectural v/100 energies, beyond the theoretical explanation of all fermions masses; so it is a "pie in the sky" issue at these energies.

By contrast, at your higher conjectural energies of 100v, you are well above threshold for production of W, Z, and the higgs, and, in fact, you would expect to start seeing "equivalence theorem" features of EW unification, as v/E will then be a "small" number! A small perturbative correction to expected SM amplitudes "before breaking" and the inner workings of the SM.

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    $\begingroup$ Just an elaboration: at energies much below $v$ (e.g. $\sim v/100$), the heavy modes (higgs, W/Z, even the top, ..) are much suppressed. The way people talk about this is by integrating out these heavy modes to get an effective theory of the low-energy modes, for example it introduces four-fermi interactions (en.wikipedia.org/wiki/Fermi%27s_interaction) by integrating out W. Historically, this is what people saw first, (this effective interaction), before being able to probe higher energies and "seeing" the W-boson. $\endgroup$
    – QCD_IS_GOOD
    Commented Nov 14, 2022 at 16:39

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