A Nonperturbative Analogy

By Mark Trodden | July 31, 2006 8:35 pm

In the Fall semester, which is approaching at a far too rapid rate, I am teaching a new advanced graduate course PHY795: Modern Cosmology. Although this will certainly be a huge amount of work, I’m very much looking forward to exposing our graduate students to the topics I spend most of my life thinking about.

As part of my preparation, I have recently turned to explaining the question of baryogenesis – how did the measured excess of matter over antimatter in the universe arise dynamically as the universe evolved? I have discussed this question once before, and also described one of the possible and testable ways in which this may have occurred, through nonperturbative finite temperature dynamics in small extensions of the standard model of particle physics.

In explaining these ideas, I chose to avoid anything more than a passing reference to how baryon number violating transitions take place in such theories, writing

In the standard electroweak theory baryon number is an exact global symmetry. However, baryon number is violated at the quantum level through nonperturbative processes – it is an anomalous symmetry. This feature is closely related to the nontrivial vacuum structure of the electroweak theory.

At zero temperature, baryon number violating events are exponentially suppressed (this is most certainly a good thing, since we would like the protons making up our bodies to remain stable). However, at temperatures above or comparable to the critical temperature of the electroweak phase transition, B-violating vacuum transitions may occur frequently due to thermal activation.

I chose this route because the physics involved – that of nonperturbative physics in chiral quantum field theories – is a difficult one, the technicalities of which I thought would be beyond the level of the post.

However, in preparing for my class, I found myself going over a favorite analogy for some of this physics and thought I’d give it a shot here. I don’t quite recall where I first came across this analogy, although it may have been in a review article by either Emil Mottola or Peter Arnold, which I read when I was a graduate student. What follows will be a little more technical than usual, but I’m hoping that most people with some physics training will get something out of it.

The vacuum of the electroweak theory is degenerate – there are infinitely many vacua, related by large gauge transformations. The field theories constructed around these vacua are entirely equivalent, but transitions between these vacua result in the anomalous production of fermions, which is the method by which the baryon number may change.

Fortunately for us (who wants our protons to spontaneously decay away?), these baryon number violating transitions are forbidden classically and, in fact, even at the perturbative quantum level – baryon number is an exact global symmetry of the theory. Therefore, at zero temperature, the only way baryon number violating processes can occur is through quantum tunneling between the classical vacua of the theory. This in itself is nonperturbative physics, and the relevant calculation yields that if the universe were always close to zero temperature, not one event would have occurred within the present Hubble volume ever in the history of the universe. However, when we include the effects of nonzero temperature, classical transitions between vacua become possible due to thermal activation.

For an analogy to this heady mix of perturbative-nonperturbative physics and finite temperature field theory, it turns out we can lean on a physical system about which most physicists learn in high school or college – the simple pendulum!

This system is a mass m suspended at the end of an arm of length l, and confined to rotate ideally in the plane. The system possesses a periodic vacuum structure labeled by integer n since, measuring angles in radians as scientists do, the system is identical in a minimum energy state whenever the angle θ in the figure is given by 2nπ. It is in this sense analogous to the electroweak theory, and you are to imagine that fermions would be produced should the system make a transition between vacua labeled by different values of n.

If we say that the vacua have zero gravitational potential energy, then the potential energy for any value of the angle θ is simply given by mg[1-cos(θ)], where g is the acceleration due to gravity. Notice that, because there is a cosine in this expression, all the information about the multiple vacua I mentioned is right there in the energy.

If a physicist wanted to understand the classical or low energy quantum mechanics of this system, they might take this potential energy and make it simpler with the approximation that the angle is always small (really what one means by low energy). In that case, if we keep only up to second order in the angle (i.e. do perturbation theory), the potential energy becomes the simplest one we know of – the harmonic oscillator – and is easily solvable. (Actually, the truth is that the full problem is solvable, but anything more complicated probably won’t be, and the approximation is what I’m trying to explain here).

But we do play a price for this approximation – all information about the periodic vacua is lost.

If we think about raising the temperature of the system, we soon run into trouble. Suppose the pendulum is coupled to a thermal bath. Then it will be thermally excited to states of higher and higher energy as the temperature is raised. Obviously, before we start calculating, we can see that as the temperature becomes comparable with the height of the barrier preventing transitions between vacua, it becomes possible for the pendulum to make transitions between vacua, crossing the point θ = π randomly, at an unsuppressed rate. But note that these have to be nonperturbative transitions, since they probe the periodic structure of the vacuum, which is not captured at all by perturbation theory.

This situation is analogous to most familiar calculations in the electroweak theory, in which perturbation theory is usually a safe tool to use. Such an approximation scheme is only valid when the energy of the system is much less than the height of the barrier separating vacua (in the electroweak theory this is known as the sphaleron). In that limit, quantum tunneling between vacua is exponentially suppressed as expected. But in the early universe, when temperatures are much higher than the barrier height, perturbation theory must be abandoned, and these effects, and the associated violation of baryon number, are rampant.

  • Anonymous

    That has to be the only attempt I’ve ever seen for someone to actually explain what a sphaleron is. Most phenomenologists of my acquaintance just shrug their shoulders …

  • http://blogs.discovermagazine.com/cosmicvariance/mark/ Mark

    If it helps any more, a more precise attempt at a definition would be:

    In the infinite dimensional gauge and Higgs field configuration space, adjacent vacua of the electroweak theory are separated by a ridge of configurations with energies larger than that of the vacuum. 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.

  • http://www.valdostamuseum.org/hamsmith/ Tony Smith

    Mark said “… baryogenesis … may have occurred, through nonperturbative finite temperature dynamics in small extensions of the standard model of particle physics …”
    and in a previous orangequark blog entry
    “… CP-violation … appears to be much too small to account for the observed BAU and so it is usual to turn to extensions of the minimal theory. In particular the minimal supersymmetric standard model (MSSM). …”.

    Could the minimal (non-supersymmetric) standard model be used for baryogenesis as suggested in hep-ph/0008142 in which Ayala and Pallares say:
    “… in the presence of strong, large scale primordial hypermagnetic fields, it is possible to generate a large amount of CP violation that combined with a stronger first order EWPT —also produced by the hypermagnetic fields— could account for the observed baryon number to entropy ratio within the SM.
    The fact that fermions couple chirally to background hypermagnetic fields in the symmetric phase makes it possible to build a CP violating asymmetry by considering two fermion interfering processes in an equivalent way to the Bohm-Aharanov effect.
    This asymmetry is converted into baryon number by sphaleron induced processes in the symmetric phase and preserved when these fermions are recaught by the expanding bubble wall. …
    … the background field strength …[is]… compatible with a Higgs mass of order 100 GeV and a phase transition analog to a type I superconductor …”

    If standard model mechanisms such as that suggested by Ayala and Pallares do not work,
    even if the LHC were to see only standard model Higgs type stuff,
    would the fact that we are made up of particles (no antiparticles) be the first clear sign that the standard model needs to be modified ?

    Tony Smith

  • http://www.pieterkok.com/index.html PK

    Very nice analogy!

  • sphaleron

    I prefer to write the key equation, tell that to understand it one needs a dedicated study of anomalies and extended field configurations (sorry, looking at a pendulum is not enough), stating what it means, explaining why a practical man can proceed even without understanding it.

  • http://eskesthai.blogspot.com/2004/11/quantum-harmonic-oscillator.html Plato
  • http://quasar9.blogspot.com/ Quasar9

    Baryogenesis, Baryongenesis

  • Paul Valletta

    a href=””>

  • Paul Valletta

    Quote: But in the early Universe…

    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 😉

  • noname


    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)?

  • http://blogs.discovermagazine.com/cosmicvariance/mark/ Mark

    Hi noname. The instanton is the solution in Euclidean time that describes the path that the field configuration takes between the two states. This helps one to calculate the zero-temperatue tunneling rate. The sphaleron refers to the actual saddle point configuration – the minimax solution – on the ridge.

  • noname

    Thanks Mark!

    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!

  • rillian

    Thanks, I liked the pendulum analogy. Very clear indeed, though I’m at the level of just understanding what a gauge field theory is. However, your more technical version in #2 lost me. Could you could expand on it a bit?

    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?

  • http://blogs.discovermagazine.com/cosmicvariance/mark/ Mark

    Hi rillian. There are a lot of ways to explain this, but let me try to take a more physics-y one. The sphaleron is a particular configuration of gauge and Higgs fields. I refer to it as a saddle point solution with a single negative eigenvalue. What does this mean?

    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.

  • rillian

    Mark, thanks for the additional explanation. Physics-y is Good. :-)

    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?

  • http://blogs.discovermagazine.com/cosmicvariance/mark/ Mark

    Yes, I’d agree with al that rillian. As for the particle-like name, it’s an unstable field configuration that one can think of as decaying away – that’s about as well as I can do. There are much stranger things named like particles!

  • rillian

    Ok. Thanks again for answering my questions. It’s great to have a clear explanations of these things!

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Cosmic Variance

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About Mark Trodden

Mark Trodden holds the Fay R. and Eugene L. Langberg Endowed Chair in Physics and is co-director of the Center for Particle Cosmology at the University of Pennsylvania. He is a theoretical physicist working on particle physics and gravity— in particular on the roles they play in the evolution and structure of the universe. When asked for a short phrase to describe his research area, he says he is a particle cosmologist.


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