Limits on interaction strength

Advancements in our understanding of quantum field theory (QFT) are rare these days. Papa Weinberg and others worked out all the easy stuff back in the 70s, and since that time we had only isolated flashes of genius such as Seiberg-Witten or AdS/CFT. Yesterday at ICHEP I heard the talk of Slava Rychkov about one of the two most interesting advancements that happened last year.

In the real world there seems to be some constraint on the interaction strength. In QFT, the interaction strength is represented by a coupling constant which is related to the probability of a particle splitting into 2 or more particles. In the first approximation, the larger the coupling constant, the larger the interaction strength. However in all examples we are aware of the interaction strength cannot be arbitrarily increased. When the coupling constant reaches the value of order 4 $\pi$, rather than interactions becoming stronger and stronger, the theory undergoes a phase transition. A new theoretical theoretical description with new effective particles emerges, and these new particles interact with a finite strength. The well know example is QCD: at low energies when the strong coupling constant grows very large the theory of quarks and gluons rearranges into the theory of mesons and baryons who interact with the final strength.

So, is there some hard-wired limit on the interaction strength in QFT? For the moment, such limits can be derived only in the context of conformal QFTs (abbreviated as CFTs). CFTs do not describe the real worlds particle physics, as conformal invariance is not compatible with massive particles. But CFT may be an effective description of some subsystems of the real world. Condensed matter physicists in their laboratories routinely produce materials that can be well described by CFT. In our field, there are ideas that the Higgs boson could emerge from another sector that is approximately conformal sector over a large range of energy scales. More recently, particle phenomenologists entertained the idea of unparticles, that is a conformal sector that weakly couples to standard model particles so that this weird stuff can be produced in colliders.

It turns that in CFT there are concrete limits on the interaction strength, more precisely on the 3-point function of primary operators. These limits can be derived using operator product expansion (in this context, conformal block decomposition) and requiring that it satisfies crossing symmetry. The rest is some algebra and a clever rearrangement of terms in the expansion. The result as a function of the operator dimension is shown on the plot below.
So, indeed, the interaction strength cannot be arbitrarily large. This result limits the options of what can be observed at the LHC in certain unparticle scenarios. Similar methods can be used to derive bounds on possible operator dimensions in CFT.

See the slides or the original paper.