Landscape of the Standard Model

On Thursday, Sergei Dubovsky informed us about some new details of their work with Nima Arkani-Hamed et al. about the landscape of the Standard Model.

In science, we determine the number of solutions to some equations by a careful rational analysis of these equations. It's the laws of mathematics and the laws of physics that matter. A theory at a given level of development makes predictions about some aspects of reality but not others. If the predictions disagree with reality, the theory must be abandoned - or significantly modified which is not always possible. If there is no disagreement, the research continues.

Sometimes you might want the predictions to be as strong as possible - ideally, you dream about a unique theory of everything - but so far, it has never been so in the history of science. If the predictions of a theory at a particular level of development are more or less unique or more or less universal than what you have dreamed about, it is your psychological problem, not a problem of the theory. The only thing that matters in science is whether the theory is right or wrong and whether it's the most predictive theory among those that haven't yet been falsified.

Note that this strategy - a part of the scientific method - strikingly contrasts with the approach of the blue and black sources of intellectual sweepings and their disciples. All these people want the truth about the number of solutions in any context to be determined by semi-religious preconceptions unjustified by any theory whatsoever and by and loud, emotional newspaper articles written by authors with high-school physics education.

Quantum gravity - a theory that we also call string theory - is known to have a large but discrete number of classical solutions and stable or metastable semi-realistic solutions, before any cosmological selection mechanism is taken into account. Nowadays, this set is referred to as the landscape. Every person who is familiar with basic facts about the newest insights of theoretical high-energy physics knows about this fact.

Do field theories share this feature?

To some extent, they surely do. A quantum field theory with a potential equal to a quartic potential often has two minima. More complicated theories can have many more.

But Sergei talked about a new context in which the Standard Model and similar field theories have a landscape of solutions. (According to the crackpots, the Standard Model is therefore not a science either, as we will see.) The context is the compactification on a circle.

If the four-dimensional universe is compactified on a circle, we encounter the well-known Casimir energy. This energy density should be added to the original vacuum energy arising from the cosmological constant in four dimensions. The Casimir energy is a one-loop quantum effect whose important contributions are associated with the lightest particles - gravitons and photons followed by neutrinos. But the effect is real and indistinguishable from classical sources of energy. It is treated as a classical source throughout their work.

The scalar field in 2+1 dimensions that controls the size of the circle - the radion - has a potential energy containing the Casimir contributions. If you realize that the Casimir energy has the opposite sign for bosons and fermions and if you count the number of light fields and the numerical coefficients properly, you will find out that there is a new minimum of the potential. Our universe - the limit where the radius goes to infinity - is degenerate with a similar universe whose one spatial coordinate is compactified on a circle of radius comparable to a few microns!

Periodic viruses abound in the fellow universe.

It is somewhat interesting that the four-dimensional vacuum energy density (the cosmological constant) is approximately equal to the mass of the lightest particles (neutrinos) to the fourth power. In fact, it could be more than just a coincidence although so far, no complete solution of the cosmological constant problem based on this observation exists.

Their (and Glashow-Salam-Weinberg-Gross-Wilczek-Politzer-...) theory compactified on a circle has another scalar field in three dimensions - the electromagnetic Wilson line around the circle. The potential for this field is exponentially tiny and the Wilson line is thus, to an extremely good accuracy, a modulus field. Within this approximation, the degeneracy of the vacua is infinite.

The authors also consider various "AdS3 times S1" geometries solving the equations of the Standard Model coupled to gravity and some solutions that interpolate in between them. These interpolating solutions resemble a cone with a tiny opening angle connected to a cylinder or an extremal black hole, depending on your viewpoint.

Also, their new "AdS3 times S1" background of the Standard Model has a holographically dual two-dimensional conformal field theory, "CFT2". Its central charge is something like 10^{90} (a power of the Hubble length in the Planck units) and it has a strongly hierarchical spectrum of dimensions of operators. There are operators whose dimension equals one - the duals of massless fields - and operators whose dimension differs by one by +exp(-billion), such as the operator dual to the Wilson line scalar field. Then there is a big gap and the following operator, dual to the lightest neutrino, has a dimension comparable to 10^{30}. Quite a hierarchy. You may wonder whether "generic" (whatever the adjective exactly means) conformal field theories with huge central charges have this property.

Note that if you found this CFT, you would know everything about the physics of our Universe at short, sub-micron distances.

Sergei has also drawn some Penrose diagrams arising in their context that look just like the infinitely long Penrose diagram for charged black holes except that they're rotated by 90 degrees: space and time are interchanged.