Vyacheslav "Slava" Rychkov (Вячеслав Рычков)

Professeur, Faculté de Physique, Université Pierre et Marie Curie (Paris VI), Laboratoire de Physique Théorique de l'Ecole Normale Supérieure (on leave of absence until 2017)

What's new 

22/08/2015

Some months ago, I put a paper out, based on the results obtained under my guidance by Zhong Ming Tan, an outstanding student from the ENS master program, during his two month internship at CERN. I was invited to submit the paper to the newly created Fast Track Communication section of Journal of Physics A, which I did. All three (!) referee reports were positive, so the paper was published. It was also featured in the Publisher's Pick section of the journal's website. In case you don't have access, I copy our featured comments below:

What prompted you to pursue this field of research?

We wanted to bridge a gap between two famous subjects: the epsilon-expansion first used to study the critical points of scalar field theories by Wilson and Fisher in 1972, and conformal symmetry whose presence in these critical points was first pointed out by A M Polyakov in 1970. The importance of conformal symmetry in two-dimensional critical phenomena is long understood, while its utility in dimensions larger than 2 has been recognized only recently. Last year, the world's best determinations of the critical exponents in the 3d Ising model have been obtained using conformal field theory. Much of these recent developments have been numerical. We wanted to find an analytically solvable example, and this naturally led us to study conformal invariance of the epsilon-expansion.

What is this latest paper all about?

One nice feature about the Wilson–Fisher critical points is that they are weakly coupled—their properties allow for a systematic expansion in epsilon, the deviation of the space dimensionality from 4 (hence the name of the method—the epsilon-expansion). In the traditional approach, the coefficients of these series are computed using perturbative field theory, by evaluating Feynman diagrams. In our paper we present an alternative way to compute the leading coefficients, which does not use the Feynman diagrams at all. Instead, our method exploits the consequences of the conformal symmetry of the critical point, and in particular one phenomenon which we call 'multiplet recombination'. It's a conformal field theoretic analogue of the Higgs mechanism familiar from particle physics. We were quite surprised to find out that it leads to such strong constraints on the structure of the epsilon-expansion series.

What do you plan to do next?

It would be very interesting to explore the consequences of 'multiplet recombination' for the conformal field theories other than the Wilson–Fisher fixed point, and to extend our calculations to higher-order terms of the epsilon-expansion series. The first of these tasks should be relatively straightforward, while the second will likely require significant new ideas.


27-28/07/2015 I gave 3 introductory lectures about the conformal bootstrap at a PhD school in DESY: 
  1. Introduction and the current status of the 3d Ising model project. 
  2. The structure of the conformal blocks and the convergence of the OPE. 
  3. Bootstrap in the Minkowski space and the large spin asymptotics of the CFT spectrum. 
The lectures were recorded and can be downloaded here.

 6/02/2015 I'd like to report a new dramatic success of the conformal bootstrap program, appeared in a paper of David Simmons-Duffin, building on his earlier work with Filip Kos and David Poland. He performed a joint analysis of three correlation functions <ssss>, <ssee>, <eeee> where s and e are the two primary scalars of the critical 3d Ising model. Conformal bootstrap proves, rigorously, that the dimensions of these operators must lie in a tiny sliver-like region in the shown plot. The previous best Monte-Carlo determinations of the same dimensions are the dashed rectangle. The best RG determination from the epsilon-expansion is two orders of magnitude less accurate. Notice how much more precise conformal bootstrap has become compared to those old techniques.  





10/12/2014 I'm getting more and more interested in the development and application of Hamiltonian truncation methods in QFT. After last month's paper on TCSA in d>2, today Lorenzo Vitale from EPFL and myself put out a paper where we study the \phi^4 theory in 2 dimensions using the Fock-space Hamiltonian truncation. The method is so simple that it should be taught in introductory QFT classes. The most technical part is a renormalization procedure which allows to improve the accuracy of the method. The python code that we used to carry out the computations was so clean (thanks to Lorenzo) that we included it with the submission. Try it out for yourself!
On the left: the spectrum of the \phi^4 theory as a function of the quartic coupling. Notice the phase transition from the Z2-unbroken to the Z2-broken phase for g~3.




11/11/2014 It's been a busy period of life since my last post. I've been mostly busy working with Matthijs Hogervorst and Balt van Rees on generalizing Truncated Conformal Space Approach from d=2 to d>2 spacetime dimensions. Our first paper has been out in September, and after a short scuffle with the Phys.Rev.D editor and the referees who did not like our intentionally provocative title (were they afraid that we will soon put them out of business? they should! :)) will soon appear in print.

I've also recently attended an interesting meeting on the conformal bootstrap organized by the Princeton theory group. It was nice to see everyone so genuinely interested in what we are doing. 

1/5/2014 Our 3d Ising work has been honored by an appreciating comment by one of the pioneers in the field of critical phenomena - Leo Kadanoff.

19/3/2014 Big day for the 3d Ising collaboration - our paper: "Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents" has appeared on the arxiv. With this work, the bootstrap study of the critical 3d Ising model is entering the precision stage. We beat the precision world records for several previously considered exponents. We also determine precisely several OPE coefficients, about which nothing was known before. We also observe mysterious operator decouplings taking place, which may be a hint of the exact solvability of the theory. 

Feb 2014 I'm attending the KITP program "New methods in non-perturbative quantum field theory", where the bootstrap is featured prominently. Overview talks on the subject were given by David Poland and myselfLeonardo Rastelli and Balt van Rees talked about their recent work on the bootstrap in SUSY theories.  Finally, David Simmons-Duffin and myself gave talks about the ongoing project of solving the 3d Ising model at criticality (the second paper in the series is due soon). All the talks of the program are recorded.

Sep 2013 When you do dimensional regularization, is it just a trick or do fractional dimensions make mathematical sense? Can quantum field theory in fractional spacetime dimensions be non-perturbatively defined? In our recent paper we do exactly that, for conformal field theories. The idea is very simple: four point function in a conformal field theory is fixed up to a function g(u,v) of two conformal cross ratios. It's not necessary to think about what it means to have a vector with a fractional number of components, the number of cross-ratios - two - is always the same. In the paper we combined this idea with the conformal bootstrap techniques to study the famous Wilson-Fisher family of fixed points, which is supposed to interpolate between the free 4D scalar theory and critical 2D Ising model, by way of critical Ising model in D=3. Normally this family is studied by a perturbative method called the epsilon-expansion technique. Our method is in agreement, but is more precise, as the following plot shows. Red bands - predictions for the leading critical exponents using the epsilon-expansion. Black dots - our results. 

6/7/2013 This week I was lecturing about "Phases of Quantum Field Theory" at the Italian Ph.D. school of theoretical physics, newly founded by my friends and collaborators Roberto Contino, Enrico Trincherini and Andrea Wulzer. My last lecture was about the Coleman-Mandula theorem, a surprisingly poorly documented result, given its fundamental status. The only detailed expositions of interest are the original 1967 paper of Coleman and Mandula, and an Appendix in volume 3 of Weinberg's QFT series, which follows the same logic spelling out some missing steps. Both are exceedingly formal and difficult to follow. Is the rigmarole really worth it? To quote from Coleman-Mandula: "Although [our proof] at times attains mathematical levels of obscurity, we make no claims for corresponding standards of rigor". In my opinion, the theorem can be explain physically and simply in a two-hour blackboard lecture, and that's what I did in Padova. I hope to write this up sooner or later.

24/6/2013 A talk about bootstrap at "Strings 2013" in Seoul, South Korea. Here is the link to the video and slides.

18/6/2013 I gave a review talk about the bootstrap program for CFTs at the "String-Math 2013" conference at Stony Brook to a mixed audience of theoretical physicists and mathematicians. Here are the links to the video and slides.

21-31/5/2013 "Back to the Bootstrap 3", the third workshop in the series, has just finished at CERN Theory Division. It was incredible fun and a chance to meet all friends, collaborators, and likeminded researchers working on strongly coupled quantum field theories. 

18/5/2013 This year, new developments abound in the theory of d-dimensional conformal blocks. First Matthijs Hogervorst and I talked the magic rho-coordinate (see the post below). Then there was intriguing work by Liam Fitzpatrick, Jared Kaplan and David Poland about the large spacetime dimension limit. Finally, a few days ago Matthijs, Hugh Osborn  and I put out a paper presenting, among other things, an extremely efficient algorithms for the numerical  evaluation of conformal blocks. 

5/3/2013 Conformal blocks are mysterious special functions. In today's paper with my student Matthijs Hogervorst we are trying to demystify them a bit. Conformal blocks are, first and foremost, sums of contributions of radial quantization states to a matrix element computing a four point function. This leads to power series representations with Gegenbauer polynomial coefficients. Convergence of these series can be optimized by a judicious choice of the radial quantization origin. We argue that the best choice is to insert the operators symmetrically. We analyze in detail the resulting "ρ-series" and show that it converges much more rapidly than for the commonly used variable z. Taking only the first term already approximates the block with a few % accuracy.
We believe the "ρ-series" will be very useful for the conformal bootstrap. Most excitingly, we use them to derive analytically some bootstrap bounds whose existence was previously found numerically.

13/12/2012 I finished a four-lecture PhD course at EPFL about Conformal Field Theory techniques in D>=3 dimensions. The lecture notes (65 pp) can be found here.

26/11/2012 I put out a paper with my student Axel Orgogozo about computing the S parameter in composite Higgs models.

16/11/2012 I gave a review talk in Padova about the current status and outlook of the bootstrap program. The slides can be found here.

13/11/2012 Today, my friends and collaborators Sheer El-Showk and Miguel Paulos put out an exciting paper: Bootstrapping Conformal Field Theories with the Extremal Functional Method. Let's see if I can summarize it here without going into much detail. Most of the work on the conformal bootstrap has so far been going along two directions: 
  • Carving out the space of CFTs, borrowing the name of a paper by David Poland, David Simmons-Duffin and Alessandro Vichi, and consists in deriving all sorts of a priori universal bounds on operator spectrum, OPE coefficients, central charges etc. 
  • Hunting for kinks, i.e. special points in these bounds. So far three such kinks have been identified: one in D=2, one in D=3, and one in the D=4 SUSY case (see Fig. 7 in the "Carving out..." paper). The positions of the D=2 and D=3 kinks correspond to the operator dimensions of the critical Ising model, while the interpretation of the SUSY kink is unclear.
Miguel and Sheer's paper adds a new direction to the list. They show that the theories saturating the bouns are so tightly constrained, that full spectrum can be recovered. In practice "full" means a couple dozen low-lying operators in the OPE of the lowest dimension scalar with itself, at which point the method runs out of steam due to a loss of numerical precision. They have demonstrated how this works for the 2D Ising model case, where it gives results in good agreement with the known exact solution, both for the operator dimensions and for the OPE coefficients. 

17/10/2012 There was an interesting paper today related to my work, so I'd want to mention it here. Pedro Liendo, Leonardo Rastelli and Balt van Rees add an exciting new twist to the bootstrap program. They propose to study bootstrap in a half-space with conformally invariant boundary conditions, rather than in full space. The point if that the two point function in half-space is morally equivalent but somewhat simpler than the four point function in full space that one usually studies. Conformal blocks in half-space are also simpler and explicitly known in any number of spacetime dimensions. This allows the Stony Brook team to push through a number of analytic analyses which are currently too complicated to do in full space. E.g. they solve bootstrap to the leading order in 4-epsilon dimensions. They analyze the conformal block decomposition for the stress tensor two point function. They also derive quite a few bounds under the assumption that the boundary preserves positivity, which they conjecture to be true for the extraordinary (Z2-breaking) and special (Neumann) boundary condition.

3/09/2012 In QFT we are often resigned to dealing with perturbative series which are only asymptotic rather than convergent. The irreducible theoretical error is related to nonperturbative effects, which cannot be captured by summing Feynman diagrams. The situation is better in Conformal Field Theory, as I discuss in my last paper with Duccio Pappadopulo and Riccardo Rattazzi. There, Operator Product Expansion and conformal block decomposition provide expansions for correlation functions which do converge, and even very rapidly (exponentially fast). This technical result can be proven in complete generality. It is important for justifying the conformal bootstrap calculations.

29/03/2012  "Solving 3D Ising Model with Conformal Bootstrap", a paper I've been working on for the last six months with Sheer El-Showk, Miguel Paulos, David Poland, David Simmons-Duffin and Alessandro Vichi, appeared on the Arxiv. 
This is the first paper of a project whose long-term goal is to solve the critical 3D Ising model using conformal field theory techniques. Here's one plot, showing how the 3D Ising lives at a corner point on the boundary of the parameter space allowed by conformal bootstrap:
See the paper for the zoom of the dashed rectangle region, and for the plots showing impact of various assumptions about the gaps in the operator spectrum. While we have not solved the 3D Ising model yet, we have definitely cornered it!

21/02/2012 Today I begin a six-lecture PhD course on "Conformal field theory techniques in D>=3 dimensions". I will talk about the physical foundations of conformal symmetry, relation between scale and conformal invariance, constraints imposed by conformality on the dimensions of local operators and their correlation functions. I will also discuss conformal blocks and their uses in the conformal bootstrap program I am developing since 2008. I will finish by presenting some brand-new results about 3D Ising model obtained via this approach.
 
7/02/2012 Dispersion relations are fun tools in quantum field theory - it's amazing how much one can learn just from analyticity and unitarity of the S-matrix. In my today's paper with Adam Falkowski and Alfredo Urbano, I use this technique to derive a general result about the Higgs boson coupling to the W and Z bosons.

30/11/2011 Komargodski-Schwimmer proof of the a-theorem is definitely a highlight of the year. In October-November I gave a PhD course about it, and today I presented the proof to the interested colleagues in a journal club at the Institut Henri Poincaré.

16/11/2011 
LHC has not found the Higgs boson yet, and perhaps it does not exist. If no Higgs then what? My 
today's paper (the first one with my PhD student Axel Orgogozo) is about strongly coupled, Higgsless, models of Electroweak Symmetry Breaking. 

10/11/2011 In a small research manifesto based on two recent talks, I argue that one can compute 3D critical exponents using conformal field theory techniques. The proposed computation (which will still take some time to perform) is mathematically well defined, unlike standard methods based on the epsilon-expansion which manipulate divergent series.

25/10/2011 My friend and colleague Roberto Contino (Roma La Sapienza) will hold an invited professorhsip of Ecole Normale Supérieure for a month. 

1/10/2011 Alfredo Urbano joins our lab as a postdoc. He has been hired thanks to my project "Radiation problem in Transplanckian scattering" supported by the UPMC program Emergence 2011.  
 
30/09/2011 A new step forward for the bootstrap program in my today's paper "Spinning Conformal Blocks" with Miguel Costa (Porto), Joao Penedones (Perimeter) and David Poland (IAS). For the first time, we are able to compute conformal blocks for correlators of vector and tensor fields. 

23/09/2011 David Poland (IAS), David Simmons-Duffin (Harvard) and my former student Alessandro Vichi (Berkeley) put out an interesting paper with a catchy title "Carving Out the Space of 4D CFTs". In this paper they develop to perfection the art of extracting bounds on operator dimensions in 4D CFTs (with or without extended symmetry, including SUSY) from conformal bootstrap. The field has come a long way from the first paper on this subject which I wrote back in 2008.

19/09/2011 I gave a talk at the "Hierarchies and Symmetries" workshop explaining why I think it should be possible to solve the 3D Ising model at criticality by conformal bootstrap methods which I have been developing over a number of years.

14/09/2011 First day of teaching

06/09/2011 I put on the arXiv the writeup of my plenary talk "EWSB Theory on the Eve of Higgs Boson Exclusion/Discovery" at EPS HEP 2011 conference in Grenoble this summer. What should we conclude from the fact that new physics has not been seen in the first inverse femtobarn of LHC data? Read the proceedings to learn my opinion...

30/08-04/09/2011  In the mountains (Pralognan-la-Vanoise). 
 
26/08/2011 Attended a really interesting seminar by Juan Maldacena at CERN today. He is trying to extend the Coleman-Mandula theorem to interacting CFTs (i.e. to prove that there are no interacting CFTs with conserved higher spin currents). We had exactly the same urge with Riccardo Rattazzi earlier this year but we got only about 30% as far as Juan did.

24/08/2011 "Conformal field theories as building blocks of Nature". Colloquium at the CERN Theory division. Trying to argue, among other things, that we should move on from AdS/CFT to make progress...

18/07/2011 My new paper "Spinning Conformal Correlators" (with Miguel Costa, João Penedones and David Poland) appeared on the arXiv. If you ever worked in Conformal Field Theory, you must know that things are simple for scalars, but quickly become unwieldy for fields of nonzero spin. You must have also heard about the old idea of Dirac to use the auxiliary 6D space to make 4D conformal symmetry manifest. In this paper, we put this idea to good use in bringing the spinning correlators under control.  Basically, our formalism makes computations for nonzero spin as easy as they are for scalars. This is neat, and will also be indispensable for the project of working out conformal blocks for correlators of nonzero spin (work in progress with the same collaborators). 

14/07/2011 Today I gave a talk at KITP, Santa Barbara, about my work with Riccardo Rattazzi and Alessandro Vichi on constraining the Conformal Technicolor scenario, proposed back in 2004 by Markus Luty (who was in the audience) and Takemichi Okui.