# What crossed over at the electroweak crossover?

It's been known for quite some time now that the electroweak "transition" in the early universe is first-order for a Higgs mass of less than about 75 GeV, but for a larger Higgs mass (including the 125 GeV mass that appears to describe our universe), the transition goes away entirely and becomes a crossover at which no physical quantities change non-analytically. (E.g. see here and here.) If understand the first reference correctly, in the $m_H < 75$ GeV regime, the simplest quantity that jumps discontinuously across the transition is the magnetic screening length, or equivalent its inverse, the "magnetic screening mass" (although the mass remains strictly positive both above and below the transition).

But I've never heard anyone actually identify which two physical quantities cross over at the electroweak "crossover" for $m_h > 75$ GeV. What are they? Put another way, if everything changes analytically as a function of temperature in the heavy-Higgs regime, then on what physical basis can we identifying one particular temperature as the "crossover temperature"?

The most natural order parameter for the EW phase transition is the square of the Higgs field, $\langle H^aH^a\rangle$, because in the mean field (weak coupling) limit this is just the VEV squared. To distinguish the order of the phase transition we can study fluctuations (order parameter susceptibilities), for example $$\langle [H^aH^a-\langle H^aH^a\rangle]^2 \rangle$$ In a cross over transitions fluctuations peak at some pseudo-critical temperature, but they do not diverge as the volume $V\to\infty$.
Note that for a sharp phase transition all possible order parameters are non-analytic at the same $T_c$, but in a crossover transition different order parameters may give different pseudo-critical temperatures. This will obviously get worse if the crossover is broad, which is the case as one goes away from the critical endpoint.
Also note that the electroweak transition does not involve a change of symmetry, and $\langle H^aH^a\rangle$ is not a sharp order parameter in the sense that it is not zero in both phases. This is, of course, the reason why the transition can have an endpoint in the first place.