Lee-Wick Standard Model

Apologies for the last two weeks without fresh posts - I was overworked more than usual. I'm coming back with a report on the last Wednesday seminar about the Lee-Wick Standard Model. Ben Grinstein began with a quote from a local philosopher:

....Lee, Wick, Coleman, Gross... not everyone who worked on that was a crackpot...

In such circumstances, since there's no room for a jester to further ridicule, I'll try to be as serious as I can about the subject.

The Lee-Wick extension consists in adding higher-derivative quadratic terms like $ \frac{1}{M^2} \Phi \partial^4 \Phi$ to the lagrangian. It was originally introduced in this early mesozoic paper in an attempt to give a physical meaning to the Pauli-Villars regulator in QED. The new quadratic term in the lagrangian modifies the propagator, so that it look like $i/(p^2 - p^4/M^2)$. The modification seems innocuous, but in fact one has to struggle hard to make sense of it. A field with four-derivative quadratic terms has twice as many degrees as the one with only two-derivative terms. It can be equivalently rewriten as two fields, one healthy, the other a ghost. The ghost has a wrong sign kinetic term in the lagrangian $-1/2 (\partial_\mu \chi)^2$, rather than $+1/2 (\partial_\mu \phi)^2$ for the healthy one.

Ghosts are scary because they violate unitarity of the S-matrix. Due to the negative residue in its propagator $-i/(p^2 - M^2)$, a ghost particle contributes a negative term to the imaginary part of the scattering amplitude. By optical theorem (that follows from unitarity), the latter is related to the total forward scattering cross section, which cannot be negative.

Lee and Wick proposed a solution to the unitarity problem by sweeping it under a series of carpets. First, the ghost is treated as an unstable particle, and its propagator is resummed, $-i/(p^2 - M^2 - i M \Gamma)$. The negative residue of the original propagator is reflected in the unusual sign of the width (a healthy particle would have $i/(p^2 - M^2 + i M \Gamma)$). Unitarity is now saved because the negative sign in the numerator cancels against the negative sign of the width, so that the imaginary part of the scattering amplitudes ends up being positive. A new problem that emerges is that the pole of the resummed propagator is on the physical sheet $p_0 > 0$, so that one has to modify the usual Feynmann prescription for the momentum integration. Lee and Wick proposed deforming the integration contour as in the figure. One can show that this prescription is equivalent to imposing a condition of no exponentially growing outgoing modes. So, at the end of the day, unitarity is saved at the price of sacrificing microcausality. One would expect that the lack of microcausality will sooner or later generate some inconsistencies (for example, violation of causality at the macroscopic scales), but so far nobody could show that it does.

In a recent paper, Ben and company extended the Lee-Wick approach to the entire Standard Model. Their motivation is the hierarchy problem. Because of the higher derivative terms in the lagrangian, the propagators carry more powers of momentum in the denominator, which makes the loop integrals less divergent. In particular, the higgs boson mass receives only logarithmically divergent contributions at one loop. Furthermore, in spite of adding higher derivative terms for the fermions, the flavour changing neutral currents are within the experimental bounds. In fact, the impact of the Lee-Wick extension seems so benign that it is hardly testable. In particular, it is not clear to me if the effects of the modified propagators can be in practice discriminated against the effects of convential higher-order terms.

As a closing remark, I find the subject weird but not crackpotty. Of course, the chances that something like that will show up at the LHC are exponentially slim: $e^{-1/\epsilon}$, compared e.g. to $\epsilon$ for technicolor-like theories, or $\epsilon^2$ for supersymmetry. On the other hand, the subject is certainly amusing because it touches on some fundamental issues in quantum field theory. Anyway, as long as the seminar is entertaining and clear, I don't complain.