Series on Axions, Part 2: Weinberg and the U(1) Problem

Now we are back in 1975. Steven Weinberg, one of the greatest physicists of all time, realizes there is a big problem in the standard model, and he calls this problem “The U(1) Problem”.

When you look at the equation that describes the way quantum chromodynamics acts – its Lagrangian – you find this global symmetry in the limit that all the quark masses go to zero. This global symmetry is composed of  a vector symmetry (for isospin and baryon number) times what is called an “axial” symmetry (which basically just corresponds to rotating something about its axis). That is, if you are setting N flavors of quark masses to zero, this symmetry is U(N)v x U(N)a (where v = vector and a = axial).

In QCD, if we have this global symmetry, we need all of the strong interactions to be invariant under U(2)v x U(2)a. We’ve verified that strong interactions are invariant under U(2)v,  but axial symmetries behave differently. In a nutshell, we see a U(2)a symmetry, but not a U(1)a symmetry! Weinberg pointed this problem out, in his paper “The U(1) Problem”, and said that there simply must be no axial U(1) symmetry in quantum chromodynamics.

Read: Weinberg’s original paper:



  1. Calvin Palmer · · Reply

    Susan, would you mind terribly if I requested that you change the font colour from its current dark grey to black? It’s a bit easier to read for those of us whose eyesight isn’t top-form.

    1. Hi Calvin! Thanks so much for reading! I am trying to figure out if I can make the font color any darker – so far I haven’t figured it out but I’ll keep trying – many apologies!

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