Tuesday, November 8, 2011

The Millenium Problem In Yang-Mills Theory

Marco Frasca and some of his colleagues are making good progress in solving the Millenium Problem in Yang Mills Theory:

Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap:" the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics.

Basically, this means that quantum chromodynamics implies the existence of at least one kind of massive glueball, which is a composite particles made of massless gluons, that the numerical solutions to the theory imply, but that have not yet been definitively observed and have not been proven to exist analytically, although there have been strongly suggestive experimental observations of some kinds of glueballs and many steps in the analytical proof have been completed.

Frasca's primary contribution relates to the relationship between the strength of the strong force and the energy scale of the transaction, called that "beta function" that describes the "running coupling constant" of the strong force. Essentially the strong force is very strong at a scale associated with quark confinement, but quarks are not bound by the strong force at low energies and short distances, or beyond a certain long distance range.

When he says that a function is trivial he means that under certain conditions (one in the low energy "infrared" regime in which quarks move more or less freely within hadrons, and one in the high energy "ultraviolet" regime in which quarks are too distant from each other to be bound by the strong force), the strength of the strong force is zero and it does not interact with the particles in the system. Questions of quantum trivality are central to both the Higgs mechanism of the Standard Model, and the high energy behavior of the electroweak forces.

This beta function is essentially the same as function that describes a harmonic oscillator, a function that also has an important role to play in classical electromagnetism (i.e. Maxwell's equations) and Newtonian gravitational fields. Kepler's law (which says that two body systems orbit in perfect ellipses) is a special case of the more general relationship.

The weak force and electromagnetic force coupling constants also run, but according to different formulas.

Particle Width and Decay Rates

Footnote: Tommaso Dorigo again kindly answers one of my questions, about the unhadronized nature of the top quark, in the comments to this post. I've heard it explained before, but never as clearly with the relevant values of the physical constants set out.

He writes:

[T]he top quark is the only quark that lives its life free of QCD infrared slavery. The price to pay for this luxury is that it lives a very short life!

It is the width of the top quark (the inverse of its lifetime) what guarantees that the top quark does not hadronize before decaying: G_t is 1.5 GeV, while Lambda_QCD is 0.2 GeV. So the scale of its lifetime is one order of magnitude shorter than the scale of time needed for QCD interactions.

But saying that the top quark is free does not lead us very far. QCD still applies. Yes, we cannot observe them directly, because we do not have cameras with a 1/10000000000000000000000000 sec shutter...


A post on particle width, decay rates and some of the general patterns they show is on my to do list. Generally speaking, the heavier something is the more rapidly it decays, and the more spin a composite particle has, the more rapidly it decays, even if its component parts have the same masses. Higher generation fundamental particles are the most empheral of all, with the top quark being the most short lived of any of the massive fundamental particles (photons are eternal, and theoretically massless gluons don't seem to have much meaning separate and apart from their role in the strong force which has a time scale slower than the top quark decay rate).

An important corollary of the top quark's rapid decay rate is that composite particles made of quarks (hadrons) are a combination of five rather than six kinds of quarks (in three colors and three antiparticle versions each).

The half life of the W and Z bosons is about 3×10^−25 seconds. The mean lifetime of a top quark is 5×10^−25 seconds, which translates to a half life of about 7x10^-25 seconds. The time scale comparable to the half life time scale in which the strong force operates to form hadrons is about 53x10^-25 seconds.

As I summed it up six years ago:

Why is the world almost entirely composed of up quarks, down quarks, electrons and neutrinos? Because everything else is highly unstable and rapidly undergoes decay.

There are at least 165 different kinds of unstable particles whose lifetimes have been experimentally measured. Protons, electrons and first generation neutrinos appear not to decay. A neutron is the longest lived unstable particle and lasts on average about 886 seconds.

The muon (a second generation electron) lasts on average for only 10^-6 seconds (a hundred million times shorter than a neutron). The next three longest lived particles that decay, kaons and pions (common two quark particles called mesons), last 1% of the life of a muon (one ten billionth as long as a neutron), which is 10^-8 seconds, and no other particle lasts more than about 1% of the life of a kaon or pion (a trillionth as long as a neutron), which is 10^-10 seconds. Only 36 possible particles last on average longer than 10^-22 seconds. The W and Z particles discussed above last on average 10^-25 seconds.

The data is reviewed at length in this very long paper. In short, anything with a second generation particle in it is unstable, anything with a third generation particle in it is even more unstable, and the more unstable particles there are in something, the more unstable it will be.

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