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Electron dipole moment: lessons

If you missed the second season of Young Sheldon and 12th season of The Big Bang Theory, it's about 2x 5 episodes to fill the gap. The S02E05 episode of Young Sheldon has a title involving "Czechoslovakian pastries" – Sheldon's dad eats some koláče from a wedding at the beginning (Texas and Nebraska are a top destinations for 19th century Czech emigrants) – and Sheldon and Missy undergo some twins testing under Dr Pilsen. A good greeting to the viewers and Sheldon's inspirers who live in Pilsen, Czechoslovakia. ;-)
A few days ago, you could see reports about the new measurements of the electron electric dipole moment: Google News. The ACME II experiment published its findings in Nature
Improved limit on the electric dipole moment of the electron
and the upper bound is about 10 times smaller (more constraining) than measured a year earlier, namely\[

|d_e|\leq 1.1 \times 10^{-31} e\ {\rm m}.

\] The dipole moment has the units of "charge times distance" and with the natural charge factor of \(e\), the electron charge, the distance is just some 10,000 times the Planck scale! This distance is heuristically the separation between the electron's center-of-mass and its center-of-charge-distribution.

The ACME II collaboration consists of David DeMille (Yale) and my former Harvard colleagues John Doyle and Gerry Gabrielse.

A minute ago, I mentioned a classical way to visualize the dipole moment. I hope you know that such classical pictures are inadequate and you should use the formalism of quantum field theory (or better) to discuss such issues.

Well, in that language, the electron electric dipole moment may be found as one of the terms in the matrix elements of the electromagnetic current operator \(j^\mu\) between two states of an electron with well-defined momenta and spins:\[

\bra{p_f} j^\mu \ket{p_i} = \bar u_f \zzav{ F_1 \gamma^\mu \!+\! F_2 \sigma^{\mu\nu}\gamma_\nu\!+\!F_3 \gamma_5 \sigma^{\mu\nu}\gamma_\nu\! +\!\dots } u_i

\] The "dots" term represents \(F_4\) multiplied by \(q^\mu-q^2 \gamma^\mu / 2m\). Now, the Dirac spinors \(u_f,u_i\) depend on the final and initial momenta \(p_f,p_i\), the form factors \(F_{1,2,3,4}\) depend on \(q^2\), the invariant length of the vector \(q^\mu=p^\mu_f-p^\mu_i\), and: \(F_1\) represents the electric charge, \(F_2\) (plus a part of \(F_1\)) represents the magnetic dipole moment, \(F_3\) is the electric dipole moment. The extra term \(F_4\) would be the so-called anapole moment. Note that \(F_2,F_3\) only differ by \(\gamma_5\) which makes the electromagnetic flip.

In my sketch, factors of two etc. may differ from the most widespread convention.

The magnetic dipole moment is straightforward to measure – it's what bends the electrons' trajectories or what causes the precession of the spins in a magnetic field etc. The electric dipole moment might be impossible to measure, you might think, because the electron primarily reacts to electric fields through its huge charge.

However, the ACME II experimenters cleverly use the other methods, the precession of the electron's spin in strong electric fields – namely those in some molecules.

Any nonzero value of \(F_3\) means that the CP-symmetry (replacement of particles by their antiparticles, and in arrangements that are the mirror of the original one) must be violated. We know that the CP-symmetry is violated in Nature. But so far, the only known source of CP-violation in Nature is the complex phase in the CKM matrix defining the relationship between quarks' mass eigenstates.

This complex phase contributes a hopelessly tiny contribution to the electron electric dipole moment. The \(\theta\)-angle of QCD and new physics could generate greater contributions – that could have been seen in ACME II and similar experiments – but so far those have been seen to be indistinguishable from zero in the experiments.

Phenomenologists have quickly posted a preprint
Interpreting the Electron EDM Constraint
that argues that lots of things in new physics are indeed excluded, even some generic theories that would only contribute to the dipole moment at the two-loop level. So unsurprisingly, we have seen some oversimplified reports in the media that supersymmetry has been almost killed again.

As far as I am concerned, I am not surprised by the experimental result at all. I think that already the experimental limits on the QCD angle shows that the CP-violation is not broken in the most generic way. The complex phase is nonzero in the CKM matrix but it must be a rather special CP-violating parameter. Many others may be zero due to some reason. We don't know the exact reasons but they exist.

Within QCD, the reason why the \(\theta\)-angle may be so tiny is arguably the Peccei-Quinn (half-female) mechanism. Supersymmetry must have something similar and possible explanations exist. I think that the experimental proofs that the CP-violation due to SUSY effects is "far smaller than the generic one" is not a much more serious problem than the strong CP-problem of QCD. So if you said that supersymmetry is excluded by these measurements, you should also say that QCD is excluded! ;-)

OK, so there is some level at which the CP-symmetry is exact and its breaking is rather minor and only significantly affects some parameters, not others. The upper bound for the electron electric dipole moment can keep on shrinking – I expect it will, and that's what I correctly expected before this new ACME II measurement, too.

On the other hand, we don't really want to believe that the CP-violation is too tiny. We need the leptogenesis and baryogenesis to produce a sufficient matter-antimatter asymmetry at the beginning of our Universe's life. So in some regimes, some CP-violating parameters – beyond those in the CKM matrix – are significant, too.

The devil is in the details. But note that my approach to all these unknowns is largely theory-independent and phenomenological. There are very many possible scenarios involving new fields, particles, and symmetries. I don't know what the right one exactly is – nevertheless, the experimental constraints seem to tell us some general if not philosophical facts about the right solution, whatever it is, and I tried to sketch what this "limited CP-violation" looks like in my imagination.

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