Magnetic monopole vs (g-2)

[Roberto Pavani]

TL;DR

If QED correctly explains $(g-2)\neq 0$, does this constitute indirect evidence that $dF=0$ everywhere; i.e. no magnetic monopole? Or does it only constrain the local region of the experiment?

While working on a preprint, I noticed what seems to be a straightforward logical chain that I haven't seen stated explicitly in the literature. I'd like to understand whether (a) it's well-known but considered too trivial to state, (b) it has been discussed somewhere, or (c) there's a flaw I'm missing.

The chain is:

1. The anomalous magnetic moment $(g-2)_e/2$ is measured to $\sim 10^{-13}$ precision.
2. Its value is reproduced by QED loop corrections involving the photon field $A_\mu$.
3. The standard QED calculation assumes $A_\mu$ is a globally defined potential, so that $F = dA$.
4. $F = dA$ implies $dF = d^2A = 0$ identically; i.e. $\nabla\cdot\mathbf{B} = 0$: no magnetic monopole.

Now, the obvious objection: in Dirac's formulation (1931), monopoles and $A_\mu$ coexist
$A_\mu$ is defined everywhere except on the Dirac string, and the Schwinger loop diagram presumably doesn't "see" a distant
monopole. So the chain might only say: "in the region where the electron interacts with the field, $dF=0$ holds".

This raises the question I'd like to understand:

- Has anyone computed what happens to $(g-2)$ in a theory that admits monopoles (Dirac quantisation, 't Hooft–Polyakov)? Would the agreement with experiment be spoiled, or would the correction be negligible for a distant monopole?
- If a monopole existed at distance $R$ from the experiment, at what $R$ would it start affecting the $(g-2)$ calculation?
- Is the $10^{-13}$ precision of $(g-2)_e$ actually constraining the monopole flux in some indirect way that direct searches (MoEDAL, MACRO) do not?

I'm not claiming anything new, just trying to understand whether this connection has been made explicit somewhere, or whether it's trivial / flawed.

References I've checked without finding an explicit statement:

- Schwinger (1948), original $(g-2)$ calculation
- Jackson, Classical Electrodynamics, Ch. 6 (monopoles)
- Preskill, "Magnetic Monopoles" (1984 review)
- Rajantie, "Introduction to Magnetic Monopoles" (2012)

Any pointers appreciated.

 

[Dale]

I don't see why a dipole moment anomaly would have any bearing on the monopole. Mathematically the various multipole terms are all orthogonal, i.e. they form an orthogonal basis.

 

[renormalize]

I don't see why a dipole moment anomaly would have any bearing on the monopole. Mathematically the various multipole terms are all orthogonal, i.e. they form an orthogonal basis.

Closed loops of virtual low-mass magnetic monopoles would modify the QED prediction of $g-2$ for the electron and muon. @Roberto Pavani should look at:
Mass limit for Dirac-type magnetic monopoles
Abstract
We calculate the contribution of virtual monopole-antimonopole pairs to the anomalous magnetic moment of the muon $(g−2)_μ\,$. If strong coupling QED generates confinement of monopole-antimonopole pairs, as suggested by recent lattice calculations, most of the monopole search experiments would not have been capable of detecting monopoles. However, such confined systems contribute to radiative corrections. We conclude that the effective monopole mass has to be larger than 120 GeV.
as well as the later references in Google Scholar that cite this paper.

 

[Roberto Pavani]

Thank you @renormalize, that's exactly the reference I was looking for.
The Vento & Lacroix paper confirms that virtual monopole-antimonopole loops would contribute to $(g-2)$, and the current precision sets $m_{\rm monopole} > 120$ GeV.

This also means that future improvements in $(g-2)$ precision (e.g. from Muon g-2 at Fermilab, or future electron $g-2$ experiments) would progressively either:
- detect a deviation attributable to virtual monopole loops (evidence for their existence at higher mass), or
- push the lower bound on $m_{\rm monopole}$ ever higher, asymptotically approaching a complete exclusion.

In either case, $(g-2)$ measurements act as an indirect monopole search complementary to direct detection experiments (MoEDAL, MACRO), with the advantage of being sensitive to confined monopole-antimonopole pairs that would escape direct searches; as the paper you cited explicitly notes.