What is the physical meaning of Vacuum expectation value. If we have a 2D plot for potential V(\phi) for the scalar field for negative mass square, then which distance is vev in that plot?
The vev is the ``distance'' of the minimum of the Higgs potential from the origin. So in the 2D plot it would be the radius of the circle, where the potential attained its minimum value.
The physical meaning is a bit more subtle.
It is interesting to note that it has a constant value. That means (apparently) the average expected value of this scalar field does not depend on space-time coordinates. However, this may not be the case as, the shape of the Higgs potential in early universe was symmetric about the origin which means the VEV was zero.
The vacuum expectation value of the Higgs field is constant, by Lorentz invariance. But it does receive corrections and, even if it is zero, in the classical action, can become non-zero due to quantum effects. What's interesting is that its measured value is such that perturbation theory about free fields is consistent at that value.
The discovery of Higgs Boson in 2012 completes particle spectrum of Standard model and provides information on electroweak symmetry breaking as well as Fermion mass generation.
Quartic (dimensionless) and trilinear Higgs couplings (proportional to Higgs VEV) of the Higgs potential are still quite unconstrained. In LHC, which is a proton-proton collider, triliear self coupling provides a leading channel for Higgs pair production. This channel then provides a way to probe into the strength of trilinear Higgs self coupling. See attached Feynman graph.
Yukawa couplings of the form fermion-fermion-Higgs are quite small and remains perturbative upto Planck scale except the Top Yukawa which is rather Large. It may become quite large at the Planck scale due to remormalization group evolution I guess.
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