Eastern Flavor

Oh, that was really long time no see...I guess both my fans have become uneasy. Time was scarce earlier this month, while the talks I heard were not so attractive as to trigger me. Now that I'm on holiday on California beach I can finally catch up on blogging.

Last Thursday Oleg Lebedev discussed his recent paper about Higgs-dependent Yukawa couplings. In the Standard Model, the fermion masses originate from Yukawa interactions between the Higgs field and two fermions, $Y \psi_1 H \psi_2$. Renormalizability implies that $Y$ has to be a constant. The simple idea from Oleg's paper is to drop that assumption and see what happens when the Yukawa couplings depend on the Higgs field. Quite generally, one can write $Y = Y^{(0)} + Y^{(2)} (H^\dagger H/M^2) + ...$, where $M$ is some new physics scale - TeV, or higher. Such a structure can readily arise from integrating out heavy states with masses of order M, one example being heavy vector-like fermions coupled to the light fermions via standard Yukawa interactions.

In a general situation, the Higgs-dependent Yukawa couplings are everything but spectacular. They just show up as small deviations of the Higgs couplings to the fermions - something rather hard to pinpoint at the LHC. Things become interesting when the first terms in the expansion vanish for some reason. Except that it's fun, it could shed some light on the fermion mass puzzle, in a similar spirit to the Frogatt-Nielsen mechanism. Those fermions whose Yukawa couplings lack the first $n$ terms in the expansion will have their masses suppressed by $(v^2/M^2)^n$. By engineering the integers $n$, one can explain the hierachical fermion masses in the Standard Model without introducing small parameters. For example, the bottom quark can be assigned $n = 1$, so that its Yukawa coupling would be $Y_b^{(2)} (H^\dagger H/M^2)+ ...$. If $Y_b^{(2)} \sim 1$, matching to the observed bottom quark mass fixes the new physics scale $M$ to be 1-2 TeV. For lighter quarks one can assign larger $n$, so as to obtain small masses with order one Yukawa couplings.

One consequence of such a scenario is that the Higgs physics at the LHC changes dramatically (for worse, as usual). The couplings of the Higgs boson to the quarks are enhanced by $(1 + 2n)$ with respect to the Standard Model value. Therefore the Higgs decay rates change by $(1+2n)^2$. In the plot shown here, the branching ratios are compared to the Standard Model ones (dashed). As long as the Higgs boson is heavy enough to decay into W or Z, the modifications are not very much pronounced. If the Higgs is fairly light, however, the swampy decay to the bottom quark is boosted by a factor of 9 so that the golden Higgs-to-two-photons decay has a much smaller branching ratio. That's another sleazy attempt to prevent a quick and easy Nobel prize for the LHC.

The other side of the coin is that the modified Higgs couplings induce flavor changing neutral currents (FCNC). The lack of tree-level FCNCs in the Standard Model relies on the fact that the Higgs couplings are aligned with the Yukawa couplings. Once $n$ is different for different quark flavors, the alignment is gone, and the Higgs boson mediates FCNCs. Therefore, the model like that may easily run into trouble with experiment. Oleg argued that he can find patterns of $n$'s for which FCNCs are suppressed. Nevertheless, CP violation in the kaon mixing is ridiculously well measured and some fine-tuning is necessary to pass that test. If you need a reason not to believe in the modified Yukawa couplings, that's probably the best one.

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.

April Fools'08: Commercial Success

One look at the plot above is enough to conclude that Resonaances is getting more and more popular. The resonaance on the 11th can be well explained by a citation from Cosmic Variance. It is clear, hovever that the background has increased significantly in the second half of the month. A careful analysis shows that the increase can be attributed to people looking for porno on Google and ending up at my post Porno at CERN. I guess they imagine respectable scientist having gay sex on paper-laden desks during office hours and, apparently, find the idea thrilling.


Anyway, popularity is precious, and I decided to profit from that. I proudly announce that today the Resonaances shop opens for the first time. It is conveniently located outside the premises of CERN, just beside Microcosm. The shop sells doublet-triplet magnets, semi-conducting electric wires, ping-pong balls and other spare accelerator parts. The offer also includes fossil wormholes trapped in amber, jester hats, monopole magnetic stickers, handwritten scripts of my posts and other imponderabilia. All at affordable prices. Drop by next time you're around.