In my last post, I discussed the puzzle posed for cosmologists and particle physicists by the observation of the baryon asymmetry of the universe (BAU) – the fact that the universe is composed almost entirely of matter, with a negligible amount of anti…

So the single negative eigenvalue refers to the spectrum of the “matrix” of partial second derivatives (Hessian) of the potential? Extended to whatever infinity of dimensions the system has in this case?

Then having a single negative eigenvalue is just the mathematical handle on your statement that there’s only one path along which sphaleron is unstable. It is, to switch analogies, the “mountain pass” through which the system will tunnel between vacua?

If it’s just a saddle point in the potential, why is it named like a particle?

Well, suppose you have some system and you’d like to know whether it is locally stable. To figure this out, you could imagine giving the system little pushes in every conceivable direction. If it always comes back, then it is stable (think of a ball at the bottom of a bowl), If it evolves away from the starting configuration in every direction that you push it, then it is completely unstable (think of a ball on top of a hill). However, if, when you do your pushing, the system evolves back to the original in every direction but a single one, in which it evolves away, then we say that there is a single negative eigenvalue – there is a single direction in which the system in unstable. (The word eigenvalue just refers to the mathematical quantity that one solves for to find out which directions behave in which way.)

In the standard model, the system has infinitely many degrees of freedom, and so there are infinitely many ways one can disturb the sphaleron. Nevertheless, only one of them leads to an instability.

In my pendulum example, which is 1-dimensional, the relevant configuration is just completely unstable (the pendulum standing straight up), because there’s only one eigenvalue in 1-d. For an analogy with the sphaleron, let’s go back to a 2-D example – the ball. Think of two valleys, separated by a smooth ridge of varying height. At a random point on the ridge, if you move the ball in a direction away from the ridge you’ll fall off into a valley (unstable), and if you move a little along the ridge, you’ll also continue to roll, because the ridge itself is sloped. However, there will be a point that is the lowest point on the ridge. Here, if you move along the ridge a little, you’ll fall back to the lowest point (stable in that direction ), and if you move off the ridge you’ll fall away, towards a valley (unstable in that direction). That lowest point on the ridge is the saddle point, and an analogy for the sphaleron.

The lowest energy point on this ridge is a saddle point solution to the equations of motion with a single negative eigenvalue, and is referred to as the sphaleron.

Eigenvalue of what? And what does it mean to have a single negative eigenvalue?

The sphaleron refers to the actual saddle point configuration – the minimax solution – on the ridge.

You have cleared up something I wanted to know for a long time, but was afraid to ask!

Thanks for the explanation in the article AND in comment #2.

I had a follow-up question:

What then is the difference between a sphaleron and an instanton(that are also obtained by extremizing the action and solving for the equation of motion with two different vacua as boundary conditions)?

So if there is a imminent transition, say at “late universe times”, then the information the temperature of ,say the very last radiative/heat product?..would be able to violate the barrier, in that the “heat_death” scenario of the late time Universe, has the effect of Thermal Entropy tends from COLD to Hot?

The fact remains close to the Big Bang, the transition of heat is from Hot to Cold, but this cannot be the case for the Universe’s ‘Endgame’ ?..it’s surely the reverse Cold to Hot, for cyclic comminication of Heat Bath potentials?

Obviously any “future” potential has MORE information available to transpose than a “past” sink ?

There is another analogy I have come across before, it involves Two Tennis players hitting a ball over a deviding net, each has the potential to “slam” the ball, but do not commit to this act due to the desire/compulsion of maintaining the Rally, for if either of them slams, it will be “break_point”, and thus a new game is set in motion