Notes on EDMs. Matt Reece. October 20, 2013

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1 Notes on EDMs Matt Reece October 20, 2013 EDMs and the mass scale of new physics The electron EDM in QED is the dimension 5 operator L = d e i 2 ψσ µν γ 5 ψf µν, (1) where ψ is the electron field and F µν the electromagnetic field strength. The EDM d has dimensions of length and the current bound is [1]: d e < e cm. (2) If we expand L and decompose ψ into left- and right-handed parts, we find that the EDM operator always mixes left- and right-handed electrons, containing terms like ψ R σµ σ ν ψ L F µν. Because of this, we can assume that d e is always proportional to the electron mass: d e = em e Λ 2, (3) where Λ is proportional to the mass scale of new physics contributing to the EDM. If we tried to construct a model where this ansatz d e m e was not true, we would find that our model predicted large corrections to the electron mass, so the assumption turns out to be reasonable. Now, using GeV fm = c, the bound Eq. 1 translates to: Λ 2 > em e c d e and using m e = 511 kev, we find: ( ) me GeV cm =, (4) cm Λ > 79 TeV (310 TeV). (5) Here the first number is the current bound and the number in parentheses is what the bound would be with a projected next-generation limit d e < e cm. At this level, the bound is very robust: an observation of a nonzero electron EDM would imply new physics 1

2 with a mass near or below 300 TeV. Notice, in particular, that the improvement in reach in the scale of new particle masses goes like the square root of the improvement in the bound on d e itself. Here and elsewhere in these notes, an order-one CP violating phase has been assumed. More generally, for a small CP phase δ, the estimate would be simply d e = δ em e Λ 2, (6) and so the bound on the mass scale probed goes like δ. We have direct experimental information about only two CP phases: the CKM phase, which is order one, and the QCD theta angle, which is < Notably, a simple dynamical mechanism for relaxing the theta angle to zero could exist (in the form of axions), while we know of no such simple mechanism for the CKM phase. Thus, it seems very reasonable to think about order-one CP phases, although it could always turn out that phases are small for either dynamical or accidental reasons. One-loop new physics Our estimate above simply puts a scale Λ in the denominator of the EDM operator. However, in almost every model of interest, EDMs arise through loop diagrams, and as such are quantum effects suppressed by small coupling and a factor of 16π 2 that arises from the phase space integral for the particles running in the loop. In other words, we can think about new physics entering through one-loop diagrams, with an EDM related to the mass scale of new particles by d e = g2 em e, (7) 16π 2 M 2 1 loop where g is a typical coupling constant in the loop (e.g., the weak coupling 0.65, although the estimate here is only valid up to order-one numbers that must be calculated in a given model). In relation to the estimate for Λ above, this means that the true mass scale of new physics entering in the loop would usually be lower, M 1 loop g Λ > 4 TeV (16 TeV), (8) 4π where again the number in parentheses is a projected limit at e cm precision. Here I have plugged in g 0.65, but again, the bound in a given model can differ from this by an order-one number. An example of a one-loop EDM in SUSY theories is shown in Figure 1. This illustrates an important general point: one-loop diagrams for the EDM will generally contain some new particle with lepton quantum numbers, like the electron superpartner appearing in this diagram. If all new particles with lepton quantum numbers are heavy, there may be no important one-loop diagrams contributing to the EDM, and the most important contributions may arise at two loops (as discussed below). 2

3 H u B, W 0 M 2 W 0 H 0 u µ A e H 0 d ẽ L ẽ R H d ẽ L y e γ Figure 1: One-loop EDMs in supersymmetric theories. To unpack the diagrams a bit more: the electron splits into a virtual pair of its superpartner (the selectron) and a neutralino (the superpartner of the photon, Z, or Higgs boson). The diagram at right contains a selectron electron Higgsino interaction, which depends on the electron Yukawa coupling y e = m e /v. So it is proportional to m e, as we assumed. The diagram at left, on the other hand, transforms the left-handed selectron to the right-handed selectron using the A-term trilinear coupling, A e H d ẽ L ẽ R. In a general supersymmetric theory, A e is formally independent of the Yukawa coupling y e, although in many models they are proportional: A e y e m SUSY, where m SUSY is some measure of the SUSY-breaking scale. Again, attempting to break this proportionality would lead to large corrections to m e, so it is reasonable to assume the proportionality. In the diagram at left, the invariant phase that would contribute to CP violation is arg(a em 1,2 ). In many particular models of SUSY breaking, like gauge mediation, this CP phase is zero, and the contribution is absent. In more general models, like gravity mediation, it is unclear whether we should expect this phase to be small. The diagram at right is sensitive to the phase arg(µ M 2 ). Generation of µ, the Higgsino mass parameter, is typically one of the thorniest problems in building a supersymmetric model, and it seems very plausible that it could have a CP phase different from other SUSY-breaking parameters. Two-loop new physics If there are no new particles with lepton quantum numbers contributing a large EDM at one loop, important effects may arise at two loops. In this case there is an estimate similar to the above, but with the loop factor squared: ( ) g 2 2 em e d e = 16π 2, (9) M2 loop 2 where again g is a typical coupling constant in the loop and M 2 loop is now some effective mass scale of particles appearing in the loop. Here the mass scale probed by EDM measurements 3

4 is smaller than the one-loop scale by an additional factor of g/(4π): M 2 loop ( g ) 2 Λ > 210 GeV (850 GeV). (10) 4π Notice that now we ve switched units from TeV to GeV, so these scales are considerably smaller. Although thhc directly probes scales of order hundreds of GeV, the bounds it sets on new particles that don t interact through the strong nuclear force are quite weak, so two-loop EDMs are still a powerful probe of territory that thhc has difficulty covering. A plausible scenario in supersymmetric theories (now sometimes referred to as minisplit or semi-split supersymmetry) is that the new scalar fields are significantly heavier than the new fermionic fields, often by a loop factor (i.e. roughly two orders of magnitude). The one-loop contributions we have considered so far involve the electron s scalar partner. On the other hand, there are two-loop contributions that involve only the fermions. Roughly, we expect that these dominate whenever 1 1 (16π 2 ) 2 Λ 2 fermion > 1 1, (11) 16π 2 Λ 2 scalar i.e. when Λ scalar > 4πΛ fermion. Since the scalars could be a factor of 16π 2 heavier, this is easily satisfied. This motivates us to take a look at two-loop processes with loops of fermionic superpartners, like the one in Figure 2. γ χ + H u Z, γ h Figure 2: Two-loop EDMs in supersymmetric theories. The one-loop diagram in the dashed box is a CP-violating analogue of familiar electroweak precision corrections. The interesting feature of the various two-loop diagrams, of which we show only one, is that the electron is playing an essentially extraneous role. The gray dashed boxed part of Figure 2 illustrates this. The new physics and the CP violation lives in a one-loop subgraph with gauge and Higgs bosons as its external states; we turn this into an electron EDM by stringing an electron line between two of the external legs. Because the electron is added on at this second stage, no new fields with lepton quantum numbers are needed to generate these two-loop EDMs. Thus, a completely generic model of new particles with electroweak 4

5 interactions can usually produce two-loop EDMs, and there is no need to carry the analysis on to three or higher loops. In the supersymmetric context, the phase contributing to the 2-loop EDM is generally a relative phase of the higgsino mass and gaugino masses, and the bound on M 2 loop is effectively a bound on the geometric mean of these two masses. In split supersymmetry models, the higgsinos can either be at around the same mass as the gauginos or significantly heavier, so one outcome of an EDM measurement could be favoring models where the higgsinos are significantly heavier. Summary An EDM measurement at the e cm level effectively probes two important scales. The first is associated with one-loop effects due to particles with lepton quantum numbers, and it probes their masses up to 16 TeV. This is a very interesting scale. The large Higgs boson mass, in supersymmetric theories, is most easily explained through heavy scalar superpartners and mini-split supersymmetry. A scale 10 TeV is roughly the minimum mass for the scalar top quark in such a scenario. In many models, the scalar partner of the electron will be in the same ballpark, so the EDM is a very useful probe of this mass scale. The main idea competing with supersymmetry as an explanation of the hierarchy problem, compositeness, is typically also associated with new particles at a mass scale at or slightly below 10 TeV. Thus, this is a very interesting scale, out of reach of the current generation of colliders but very much a part of many of the most popular theoretical ideas. On the other hand, a generic model might not produce a one-loop EDM but could have the first contribution to the EDM at two loops. In this case, a new measurement would probe masses up to 850 GeV. This constitutes a powerful probe of new physics at the electroweak scale, including possibly some particles that would be very difficult to discover at thhc due to a small production rate. Furthermore, the same mini-split supersymmetry models that favor 10 TeV scalars frequently have the other superpartners at the 100 GeV scale. The EDM measurement, then, is probing two different mass scales that are both of great interest for understanding physics at the weak scale and the possible nature of the Higgs boson. References [1] B. C. Regan, E. D. Commins, C. J. Schmidt and D. DeMille, New limit on the electron electric dipole moment, Phys. Rev. Lett. 88, (2002). 5