Vector Meson Dominance

I’m only now learning about ‘vector meson dominance’—a big idea put forth by Sakurai and others around 1960.

Here’s a family of 9 mesons called the ‘vector nonet’. Each one is made of an up, down or strange quark and an antiup, antidown or antistrange antiquark. That’s 3 × 3 = 9 choices.

In this chart, S is strangeness (the number of strange quarks minus the number of antistrange antiquarks in the particle) and Q is electric charge. I’ll focus on the neutral rho meson, the ρ⁰, which has no strangeness and no charge.

But why are these called ‘vector’ mesons? It’s because the quark and antiquark have spin 1/2, and in this kind of meson their spins are lined up, so together they have spin 1. A spin-1/2 particle is described by a spinor, which is a bit weird, but spin-1 particle is described by something more familiar: a vector!

The most familiar spin-1 particle is a photon. And in fact, the photons around us are slightly contaminated by neutral rho mesons! That in fact is the point of vector meson dominance. But more on that later.

First, if you’ve read a bit about mesons, you may wonder why your friends the pion and kaon weren’t on that last chart. Don’t worry: they’re on this chart! This is the ‘pseudoscalar nonet’.

In these mesons, the spins of the quark and antiquark point in opposite directions, so the overall spin of these mesons is 0. That means they don’t change when you rotate them, like a ‘scalar’. But these mesons do change sign when you reflect them, because then you’re switching the quark and antiquark, and those are fermions so you get a minus sign whenever you switch two of them. So these mesons are ‘pseudoscalars’.

If you don’t get that, don’t worry. I’m going to tell the tale of rho mesons and especially the neutral one, the ρ⁰.

A photon will sometimes momentarily split into a quark-antiquark pair. Since the neutral rho meson is the lightest meson with the same charge, spin and other quantum numbers as a photon, this quark-antiquark pair will usually be a neutral rho! This is basic idea behind ‘vector meson dominance’.

In short, the light you see around you is subtly spiced by a slight mix of neutral rho mesons!

More precisely, real-world photons are a superposition of the ‘bare’ photons we’d have in a world without quarks, and neutral rho mesons.

But you might ask: how do we know this?

When you shoot a low-energy photon at a proton, its wavelength is long, so it sees the proton almost as a point particle.

But a high-energy photon has a short wavelength, so it notices that the proton is made of quarks. And the photon may interact with these as if it were a rho meson—because sometimes it is! This changes how high-energy photons interact with protons, in a noticeable way.

The same thing happens when you slam charged pions at each other. You’d expect them to interact electromagnetically, by exchanging a photon. But if you collide them at high energies you get deviations from purely electromagnetic behavior, since the photon is slightly contaminated by a bit of neutral rho!

In fact this is how the neutral rho was found in the first place. In 1959, William Franzer and Jose Fulco used results of pion collisions to correctly predict the existence and mass of the neutral rho!

They used a lot of cool math, too—complex analysis.

Then in 1960, Sakurai argued that the three rho mesons ρ⁺,ρ⁰,ρ⁻ form an SU(2) gauge field!

So: we can combine these 3 vector mesons into an 𝔰𝔲(2)-valued 1-form… which describes an 𝔰𝔲(2) connection! This is how gauge theory works.

Now, Sakurai’s paper showed up before quantum chromodynamics appeared (1973), or even quarks (1964). But Yang–Mills theory had been known since 1954, so it was natural for him to cook up a Yang–Mills theory with rho mesons as the gauge bosons.

Just one big problem: they’re not massless, as Yang–Mills theory says they should be. Sakurai tried to treat the rho mesons as gauge bosons in a Yang–Mills theory of the nuclear force, and give them a mass ‘by hand’.

Only in 1985, after quantum chromodynamics had solved the fundamental problem of nuclear forces, did people come up with a nice approximate theory in which the rho mesons were gauge bosons for the strong force, with a Higgs serving to give them mass (Hidden Local Symmetry).