system
Given the following post, respond as if you're commenting on Reddit:
human
What does it mean for the Higgs field to be "tachyonic"?
gpt
To answer this question, one has to first understand something about how masses are determined in quantum field theory.

In quantum field theory, a scalar field like the Higgs field has a potential energy function.  The vacuum is the state that minimizes this function, and so the field will settle throughout space to the value that minimizes the potential.  Particles can arise as excitations around this vacuum value.  The mass of these particles is determined by the curvature of the potential function at this point; in particle, the square of the mass is equal to the second derivative of the potential function at this minimizing value.

Now in the early universe, the potential for the Higgs field looks like a "U" (well, actually, a bowl), having its minimum at the origin, and so the Higgs field has a vacuum value of zero.  There are also particle excitations around this value, and since the curvature is positive at this point, the mass of these excitations satisfies the relation m^2 > 0.

As the universe expands and cools, there is a critical temperature at which the shape of the potential energy function changes from a "U" to a "W" shape (well, actually, from a bowl shape to a Mexican hate shape).  Now the Higgs field has settled to a value of zero throughout space, so we could ask ourselves what the excitations about this value look like now.  Because the shape of the potential has changed, we are now at a local maximum (the central tip of the "W") rather than a minimum, and so the curvature of the potential has the opposite sign.  Thus, if we calculate the mass of excitations around this point, we get the relation m^2 < 0, and particle with m^2 < 0 would be tachyonic.

But we don't actually get tachyons.  As you see from the above, finding that the particle excitations would be tachyonic is an indication that we are looking for excitations around a local maximum rather than a minimum.  What this actually means is that the vacuum value of the Higgs field is no longer zero, but is a value that takes takes you a minimum of the "W" (actually, Mexican hat).  So the vacuum value of the Higgs field changes from zero to something non-zero.  We are once again at a local minimum, and all the particle excitations have non-negative mass squared.
 
Edit: Fixed formatting.