Precious Little Moments

Actually, electric dipole moments.... The new limit on the electric dipole moment (EDM) of the electron published by the ACME collaboration was already mentioned on blogs and in popular press. Here I will discuss in slightly more technical terms the significance of this result.

For particle physicists the EDM looks like this:

 So, the EDM of the electron is defined  as a coefficient de of a particular operator that couples the electron field e to the electromagnetic (photon) field Aμ.  By dimensional counting, de is expressed in units of 1/Energy or, equivalently, in units of length. The ACME limit reads |de| ≤ 8.7×10^−29 e*cm  or |de| ≤ e/(2×10^14 GeV), where e is the electric charge. This is 12 times stronger than the previous best limit.

The connection to what ordinary people understand as the EDM  is that, in the limit of non-relativistic quantum mechanics, the EDM operator above reduces to the interaction Hamiltonian

 Thus, at non-relativistic energies the EDM corresponds to a shift of energy levels of the electron in an external electric field E that depends on the direction of electron's spin Se.  This is how the EDM is measured in practice, up to a few unexciting details that those with a passion for craftsmanship can find in the ACME paper ;)   

The EDM operator is non-renormalizable as it has dimension 5 (actually, dimension 6 when it is rewritten in a form that is invariant under the Standard Model gauge symmetry). For that reason it cannot appear in the Standard Model Lagrangian, and  the lowest order prediction is  de=0.  However,  it can be generated by loop effects, much as the closely related operator responsible for  the anomalous magnetic dipole moment of the electron. The important difference between these two is that the EDM, unlike the magnetic moment,  violates CP symmetry. The Standard Model contains only one source of CP violation -  the invariant phase in the CKM matrix. This invariant is proportional to the product of all the CKM mixing angles which are small numbers, therefore CP violating effects in the Standard Model are severely suppressed.  Moreover, it turns out that the electron EDM arises only at the 4th order (4 loops) in perturbation theory. All in all, the contribution from the CKM phase is estimated as  |de| ≤ 10^-38 e*cm, many orders of magnitude below the current experimental sensitivity. The neutrino sector may bring new sources of CP violation - the Dirac and Majorana phases in the PMNS neutrino mixing matrix - and these contributions to the electron EDM may arise already at 2 loops. Since we don't know the phases in the neutrino sector (or even the absolute values of the neutrino masses for that matter) we cannot compute  that contribution exactly. However, we know it should be suppressed by neutrino masses squared and one can estimate it leads to Δde ≲ 10^-43 e*cm, even smaller than the CKM phase contribution. At the end of the day, the Standard Model prediction is de=0 for all practical purpose. That's great, because it means that the electron EDM offers a clean test of the Standard Model: a measurement of  a non-zero value would be an unequivocal proof of new physics beyond the  Standard Model.

So what does the new ACME measurement tell us about new physics?  Generally, new physics contributions to the EDMs must be of the form

 where v=246 GeV is the electroweak scale, M is the mass scale of new physics and c is a numerical coefficient that may be of order 1, or may be smaller depending on a model.  For c∼1, the ACME result translates to the mind blowing limit of M ≳ 10^5 TeV. This is the best case scenario that arises when new physics has a  completely generic flavor and CP violating structure and couples strongly to the electron.  But not every new physics model is constrained so stringently. It is often automatic that new physics contributions to the electron EDM are proportional to the electron mass (if it isn't, an appropriate protection mechanism can be rather easily incorporated into the model). Moreover, the EDM is typically suppressed by a 1-loop factor at least.  Then a  better estimate is

  which leads to the limit M≳10 TeV for the mass scale of CP violating new physics. That's a less impressive but still a non-trivial constraint on models of new physics at the TeV scale. 

It is worth commenting on supersymmetry as a particular example of new physics that may come in a package with new sources of CP violation. Even in the minimal supersymmetric model there are dozens of potential new phases that could show up in the electron EDM.  To avoid a too complicated and too pessimistic analysis one always introduces additional constraints on the parameter space that hopefully can be explained by an underlying model of supersymmetry breaking. As an example, let's see what happens in the very restricted framework where all superpartner masses are the same, and the only new source of CP violation is the relative phase θ between the μ term and Bμ term in the Higgs sector. Then one obtains

where g is the weak coupling in the Standard Model, and tanβ is a free parameter describing the supersymmetric Higgs sector. This is a very severe constraint for large tanβ, but also a non-trivial one for moderate tanβ.  Of course, this kind of arguments does not robustly exclude supersymmetry showing up at the energies achievable at the LHC. The supersymmetric model could have a more elaborate protection mechanism  (e.g. leading to a suppression by additional loop factors), or simply the CP violating phases could vanish because of the way how supersymmetry breaking is transferred to the observable sector (see here for one concrete example).  But the simplest explanation of the current data is that there are no superpartners up to at least ~10 TeV.

The most exciting aspect about the electron EDM is that there's still a lot of room for improving the experimental sensitivity. Therefore we will be able to indirectly probe new physics at even higher scales.  ACME itself boasts that they can improve the limit by another factor of 10 soon.  That would translate to a factor of 3 improvement in the new physics reach; that's by the way more  than the energy jump from 8  to 13  TeV LHC...  Clearly, the EDMs are one of our best chances to pinpoint the scale of new physics in this century.