Conformal Standard Model and the second \(325\GeV\) Higgs boson

Does Peter Higgs (or God) have a secretive brother?

Křištof Meissner and Hermann Nicolai released a short preprint

\(325\GeV\) scalar resonance seen at CDF?

in which they use a strange accumulation of four events of the type\[

p\bar p \to \ell^+ \ell^- \ell^+ \ell^-

\] observed by CDF, a detector at the defunct Tevatron, that happen to have the invariant energy \(E=325\GeV\) within the detector resolution, to defend some interesting models in particle physics. The probability that four events of this kind are clumped this accurately is (according to the Standard Model and some simple statistical considerations) smaller than 1 in 10,000. I would still bet it's a fluctuation. But it is unlikely enough for us not to consider the authors of papers about this bump to be leaves blown around by a gentle wind.

Three previous TRF articles have discussed possible signals near \(325\GeV\): this very four-lepton signal, a different signal at D0 indicating a different particle, a new top-like quark, and some deficits near that mass at the LHC: yes, a convincing confirmation of the \(325\GeV\) by the European collider doesn't seem to exist.




At the end of their new article, Meissner and Nicolai mention that this bump, if real, could be the heavier new Higgs boson in the Minimal Supersymmetric Standard Model in which, I remind you, the God particle has five faces.

This is the possibility that every particle phenomenologist is aware of but most of the new Polish-German article is actually dedicated to a different, non-supersymmetric explanation. They exploited the excess to promote their old, 2006 idea about the Conformal Standard Model:

Conformal Symmetry and the Standard Model

In this interesting model, one doubles the number of the Higgs boson but the justification is a different one than the supersymmetric justification. And they conclude that this Conformal Standard Model stabilizes the hierarchy because it is conformally invariant classically; and it may remain consistent all the way up to the Planck scale, too.

That paper is designed to explain the unbearable lightness of the Higgs' being in an unusual, yet seemingly very natural way: things are light because in an approximation, they're massless. Their being massless is a consequence of the conformal symmetry and this symmetry should be imposed at the tree level. What does it mean? Which terms violated the conformal invariance at the classical level?

Well, it's easy to answer this question. The only conformally non-invariant terms are those that have dimensionful (i.e. not dimensionless) coefficients and in the Standard Model, the only such classical terms are the \(-\mu^2 h^2\) quadratic terms for the Higgs field. In the Standard Model, this term is the source of the low-energy, electroweak scale and all other masses such as the Higgs mass, Z-boson mass, W-boson mass, and top quark mass (and, with some suppression, other fermion masses) are controlled by this quadratic term.

This quadratic term is also what makes the Standard Model unnatural.

Obviously, particle physics wouldn't work if you just erased this term: the electroweak symmetry couldn't be broken at all. So they have to emulate its functions – well, more precisely, they have to prove that Nature emulates its functions – differently. "Differently" means that the electroweak symmetry is broken by quantum (i.e. virtual loop) effects: they need a "radiative" (="by quantum loops") electroweak symmetry breaking.

This idea has been around for a long time because of the work of two rather well-known men, Coleman and Weinberg. However, in the context of the Standard Model, it's been a failing idea. The most obvious bug is that the radiatively generated quadratic term still had to be rather small compared to the quartic one – because it's just a "quantum correction" – which means that the Coleman-Weinberg model predicted a Higgs boson much lighter than the Z-boson, about \(10\GeV\). That's too bad because we have known that the Higgs mass is \(126\GeV\) since the Independence Day and we have realized that the mass exceeds \(100\GeV\) for more than a decade. If you tried to achieve this heavy Higgs boson in the Coleman-Weinberg framework, you would need such a strong quartic self-interaction for the Higgs that it would die of the Landau pole disease right behind the corner, within the energies that the LHC is already probing.

Meissner and Nicolai chose to incorporate right-handed neutrinos with both Dirac (shared with left-handed neutrinos) and Majorana mass terms and the seesaw mechanism; and the extra scalar field that is helpful for a particular realization of the seesaw mechanism. They have also changed the detailed logic of how the conformal symmetry is allowed to be violated (to arguments centered around the dimensional regularization). I don't quite understand the change yet and I don't see whether this change is quite independent from their other "update", the addition of the new neutrino and scalar fields. However, what I understand is that their model treats the two Higgs doublets "democratically" when it comes to the Higgs potential terms (and there is a quartic term mixing them). However, the Yukawa couplings are different for the two Higgs doublets; the normal one is responsible for the quarks and charged leptons while the new one is responsible for the neutrinos.

At any rate, they compute the one-loop effective potential for the old light Higgs field \(h\) and their new, now arguably \(325\GeV\)-weighing scalar field \(\phi\). These one-loop terms in the potential contain some logarithms and for dimensional reasons, the arguments of the logarithms have to be dimensionless. This forces them to introduce a new scale \(v\). What's different about this \(v\) relatively to "generic" scales that appear in similar quantum field theories is that its powers never enter the effective Lagrangian; it only appears through its logarithm.

The renormalization group flows are modified in the presence of the two scalar doublets. I don't understand the reason "conceptually" but they claim that because of the extra scalar doublet, the Landau pole is delayed to super-high energies above the Planck scale so the theory may be OK up to the scale of quantum gravity.

In the new 2012 paper, the authors also offer some reasons not to worry that the hypothetical new particle at \(325\GeV\) is not showing up in events with missing energy or events with two jets instead of two of the four leptons. If you offered me 1-to-1 odds, I would bet that their model isn't realized in Nature but it is neither impossible nor "insanely implausible", I think.