# I Magnetic moment of a massless charged Fermion?

1. Dec 5, 2017

### nikkkom

The spin magnetic moment of a charged, spin-1/2 particle is
$$g \frac e {2m} \frac \hbar 2$$
where g is the g-factor (2 for any particle in tree-level approximation, 2.00231930436182 for electron),
e is charge
m is mass
$\frac \hbar 2$ is spin

But with zero mass this expression does not make sense.
What does it mean?
"Massless charged fermions are not allowed in quantum theories"?
Or "massless charged fermions have infinite magnetic moment"??

2. Dec 7, 2017

Staff Emeritus
Or in classical theories.

An electric field has an energy density, and thus an energy, and thus a mass. While actually calculating this mass has its problems, the fact that it is non-zero is manifest.

3. Dec 7, 2017

### nikkkom

A similar argument would apply to color charge of gluons, making them massive as well?

4. Dec 7, 2017

Staff Emeritus
So the reason you want to drag in non-Abelian theories is you feel that my answer isn't complicated enough? This is an I-level thread.

5. Dec 9, 2017

### nikkkom

I'm trying to understand the implications. Are all charged particles necessarily massive?

6. Dec 9, 2017

Staff Emeritus
1. Because of confinement, there are no free gluons
2. Because gluons carry a flux tube with them, on small scales they appear massive.
3. Because of asymptotic freedom, at even smaller scales their effective mass approaches zero.
OK, now we've added all the complications of a non-Abelian theory. I don't understand how this helps, though.

7. Dec 9, 2017

### Staff: Mentor

A massless particle would move at the speed of light. How do you imagine would its electromagnetic field look like? It doesn’t work.

8. Dec 10, 2017

Staff Emeritus
Oops..missed part. "...but so does their effective charge."

9. Dec 10, 2017

### king vitamin

There are certainly valid QFTs with gauge fields coupled to massless matter fields. One example I'm familiar with is (2+1)-dimensional QED coupled to $N$ spin-1/2 Dirac fermions, which is conformally invariant for $N$ large enough (the exact minimum value of $N$ needed is an open question). Condensed matter theorists call this a U(1) spin liquid, and may describe the low-energy physics of certain frustrated magnets. But this is a strongly-interacting theory without a well-defined notion of a "particle," so I'm not sure asking about the magnetic moment still makes sense.

I strongly suspect that a classical theory of massless charged particles doesn't exist. I don't think the above theory has a classical limit. In (3+1) dimensions, QED coupled to spin-1/2 fermions should flow to weak coupling, and without a mass to cut off the RG flow in the IR I suspect it would flow to a trivial theory (the "classical charge" is just zero).