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Conformal Field Theories As Building Blocks of Nature Slava Rychkov

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Conformal Field Theories As Building Blocks of Nature Slava Rychkov Conformal Field Theories as Building Blocks of Nature Slava Rychkov Université Pierre et Marie Curie and École Normale Supérieure, Paris 1 November 18, 2010 Re: Recommendation Letter for Paolo Lodone (SNS Pisa) Dear Colleagues, Paolo Lodone is about to graduate from the Scuola Normale of Pisa. His expertise is Beyond the Standard Model phenomenology. He has worked on composite Higgs, low- energy supersymmetry, and TeV scale gravity. I know him quite well since I was an assistant professor at the Scuola Normale during his Master’s thesis and the beginning of his PhD, and because we have a history of collaboration (see below). I think he is a very talented student, with strong technical skills. He is not afraid of hard problems which involve conceptual issues and which do not have ready-made solutions. Actually, I believe he prefers this kind of problems. I would classify him a “thinker”, i.e. someone who may not be incredibly fast at the onset of a project, but is eventually able to see farther, endure longer, and make nontrivial contribution rather than just follow advisor’s directions. He’s the strongest among several phenomenology students who are finishing in Pisa this year, including as well the students of the previous years going back perhaps as far as the stellar generation of Contino, Papucci etc. Here’s how I got to know Paolo. For his first PhD project I proposed him the problem of QCD radiation in trans-Planckian scattering. As explained in a 2001 paper of Giudice, Rattazzi and Wells, small-angle trans-Planckian scattering is a complementary signal of TeV-scale gravity scenarios. It is not as widely acclaimed as the black hole production, but it has an advantage of being under theoretical control. However, QCD effects were never properly included in the calculation of the scattering amplitude. I knew about this problem from Riccardo Rattazzi, and I was also interested in it as a warmup for the more difficult problem of gravitational radiation. I had an idea how this could be approached so I suggested that Paolo look into this. Now, this was not the simplest thing to start your PhD with, but it proved an impressive test of Paolo’s abilities. (Paolo’s Master’s thesis - his first paper - was on the electroweak precision tests in the composite Higgs boson scenario. This was an interesting contribution to the subject and is recognized as such in the later literature. But he had no prior experience with Extra-dimensional theories.) (see next page) Laboratoire de Physique Théorique de l'École Normale Supérieure 24 rue Lhomond, 75005 Paris, France Suppose LHC fails... to find physics beyond the Standard Model Higgs... We will need a good exit strategy 2 /33 Exit strategy 1 go anthropic However, likely to become crowded... Exit strategy 2 (this talk): become a condensed matter physicist 3 /33 Particle physics = condensed matter physics of the vacuum R EFERENCE F RAME BrainwashedBut beware the byopinion Feynman? of condensed matter Philip W. Andersonphysicists: very morning I walk along the approximation to it. I found a very Ecorridor toward my office, past a useful rough approximation using my wall chart advertising the Standard “chemical pseudopotential” method, Model of the elementary particles, perturbationwhich is based on findingtheory a “most the model that is a candidate for one localized” approximation to the atomic of the great scientific achievements wavefunctionsis boring in the presence of the of the 20th century. The chart was other atom or atoms. But there seems put out by the Department of Energy to be no formalism to generalize this. and has the endorsement of the Yet another case of a force that American Association of Physics cannot be calculated by some kind of Teachers. It is very clear and concise, renormalized diagram is the attrac- explaining everything beautifully. tive potential of a covalent bond Unfortunately, there is one (but only necting by means of the emission and between atoms. The best way to cal- one) comment on it that is not true, absorption of photons. culate this potential is to abandon and accepted by many not to be true, But the nucleon is not an elemen- the separate atoms and let the elec- although it has also become a piece of tary object. It is a bound state of trons completely change their wave- folklore: It says that the interactions three elementary quarks, and bound functions to “bonding orbitals.” This between nucleons “may be modeled states cannot be described using dia- is the Hund–Mulliken scheme of the 4 /33 by the exchange of mesons.” grammatic perturbation theory. The 1920s, which lies behind modern It is indeed true that the domi- essential step in a derivation of energy band theory. But Walter nant long-range attraction of nucle- Feynman diagrams, or in fact of any Heitler and London found a useful ons is well modeled by the classic other form of diagrammatic pertur- model by leaving the atoms alone, OPEP—one-pion exchange poten- bation theory, is to imagine turning and calculating a spin-dependent tial—but after four decades of on the interactions gradually and to “exchange” interaction between the attempts to fit more and more com- assume that nothing discontinuous spins of atomic electrons. This model plicated meson exchange models to happens as we do. But bound states was extensively exploited by Linus the observations on shorter-range don’t form continuously. Therefore, Pauling in his valence bond methods, interactions, nuclear physicists now there are usually serious difficulties which are much valued by chemists accept that short-range repulsion in perturbative treatments of scat- for their qualitative insights. needs to be understood through the tering when bound states are pres- The approach using localized elec- messy business of overlap of the ent. When bags of quarks come close tronic orbits with spins is necessarily nucleons, which are bags that con- together, quarks from one nucleon the correct way to describe insulat- tain three quarks apiece. In fact, overlap into the bound-state wavefunc- ing magnetic materials, such as Mott there is as yet no unique microscopic tion of the other. This gives a repulsive insulators like La CuO . But the theory of nucleon–nucleon interac- 2 4 “overlap” force that gets its strength interaction between the localized tions; there are only empirical model from the strong binding by means of spins has no diagrammatic perturba- potentials, which, try as they may, tive description of the normal kind, never quite fit all the facts (see Hans gluons that holds the nucleon together, since quarks are fermions and can’t again because the electrons are in Bethe’s article in the American Phys- bound states. I calculated the inter- ical Society’s centennial issue of occupy the same phase space. action in 1958 (and called it “super- Reviews of Modern Physics.1) Just as simple and straightfor- exchange”), but my method produces The reason this meson theory of ward a case where Feynman dia- a power series that is upside down interactions so quickly became folk- grams don’t work is presented by the relative to the diagram series: T/U is lore—aside from the fact that Hideki well-known van der Waals potential the parameter, where T is kinetic Yukawa did earn a Nobel Prize for between filled shell atoms. The energy and U is the interaction predicting the pion from it—is that it attractive part, which falls off like –6 potential, while Feynman diagrams fits so well with Richard Feynman’s R , is the London dispersion force, diagrammatic, perturbative picture indeed coming from photon exchange lead to series in U/T. This inversion of physics as exemplified by quantum and calculated from perturbation occurs because I started by assuming electrodynamics, where everything is theory by Fritz London. But there is the localizing interaction potential calculated using diagrams with elec- no diagrammatic way to treat the was infinitely strong and reintro- trons represented by lines moving other half, the short-range overlap duced the kinetic energy as a pertur- backward and forward in time, con- repulsion, because that is caused by bation. This means that I assume the overlapping of electrons from one continuity with a state in which part PHILIP ANDERSON, a condensed matter theo- atom into bound orbitals on the other. of phase space is separated out: a so- rist, is the Joseph Henry Professor of Physics The repulsion can be calculated with called projective transformation. Emeritus at Princeton University in Prince- reasonable success, but laboriously, Physicists for a couple of genera- ton, New Jersey. and there is no good perturbative tions—starting with John Slater and FEBRUARY 2000 PHYSICS TODAY © 2000 American Institute of Physics, S-0031-9228-0002-210-9 11 This is a talk about strongly coupled, conformally invariant dynamics And how it can be relevant for - condensed matter - statistical mechanics - quantum gravity - Beyond the Standard Model (jokes aside) 5 /33 Scale Invariance Ferromagnet (Ising Model) Magnetization High temperature T>Tc Critical point T→Tc Scale invariance 6 /33 Conformal Invariance - emergent at the critical point Conformal transformation (preserves orthogonality of coordinate grid; locally looks like dilation) Why this extra symmetry? - not yet fully understood Generically but not always true Polchinski 1988, Dorigoni, S.R. 2009, El-Showk, Nakayama, S.R. 2011, Antoniadis, Buican 2011 Fortin, Grinstein, Stergiou 2011 7 /33 Power of conformal symmetry e.g. constrains 3-point correlation functions ε(x) energy density field in
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