Exploring scattering amplitudes in gauge theories using soap films Gregory Korchemsky IPhT, Saclay Forum de la Théorie au CEA, Apr 4, 2013 - p. 1/20
The Standard Model ✔ All matter is composed of spin − 1 / 2 fermions ✔ All forces (except gravity) is carried by spin − 1 vector bosons: ν γ electromagnetism weak strong (QCD) ✔ Gauge theory with the symmetry group SU (3) × SU L (2) × U Y (1) ✔ The only missing piece was the Higgs boson ... since July 4th, 2012 we are 99 . 99% certain it is there Forum de la Théorie au CEA, Apr 4, 2013 - p. 2/20
Higgs at LHC One of the decay modes of the Higgs boson: gluon + gluon → Higgs → e + + e − + µ + + µ − P µ + µ − Z g H g e + Z P e − Theory Experiment ✔ Specific feature of proton colliders – lots of produced quarks and gluons in the final state leading to large background ✔ Identification of Higgs boson requires detailed understanding of scattering amplitudes for many scattered high-energy particles – especially quarks and gluons of QCD ✔ Theory should provide solid basis for a successful physics program at the LHC Forum de la Théorie au CEA, Apr 4, 2013 - p. 3/20
Theory toolkit Feynman diagrams: ✔ intuitive graphical representation of the scattering amplitudes ✔ bookkeeping device for simplifying lengthy calculations in perturbation theory (in coupling constants) For LHC physics we need scattering amplitudes with many particles involved! We know how to do this in principle: (1) draw all Feynman diagrams (2) compute them! + . . . Reality is more complicated however... Most often the computation of multi-leg/loop Feynman diagrams is too hard = ⇒ ☞ Feynman diagrams are not optimized for the processes with many particles involved ☞ Important to find more efficient methods making use of hid- den symmetries ☞ Try to consider first the simplest gauge theory Forum de la Théorie au CEA, Apr 4, 2013 - p. 4/20
The simplest gauge theory Maximally supersymmetric Yang-Mills theory ✔ Most (super)symmetric theory possible (without gravity) ✔ Uniquely specified by local internal symmetry group - e.g. number of colors N c for SU ( N c ) ✔ Exactly scale-invariant field theory for any coupling (Green functions are powers of distances) ✔ Weak/strong coupling duality (AdS/CFT correspondence, gauge/string duality) Particle content: massless spin-1 gluon ( = the same as in QCD) 4 massless spin-1/2 gluinos ( = cousin of the quarks) 6 massless spin-0 scalars Interaction between particles: All proportional to same dimensionless coupling g YM and related to each other by supersymmetry Forum de la Théorie au CEA, Apr 4, 2013 - p. 5/20
Why Maximally supesymmetric Yang-Mills theory is interesting? ✔ Four-dimensional gauge theory with extended spectrum of physical states/symmetries ✔ An excellent testing ground for QCD in the perturbative regime relevant for collider physics ✔ Is equivalent to QCD at tree level and serves as one (most complicated) piece of QCD all-loop computation ✔ Why N = 4 SYM theory is fascinating? ✗ At weak coupling , ■ the number of contributing Feynman diagrams is MUCH bigger compared to QCD ■ ... but the final answer is MUCH simpler ✗ At strong coupling , the conjectured gauge/string duality (AdS/CFT correspondence) Weakly coupled ‘dual’ string theory on AdS 5 × S 5 Strongly coupled planar N = 4 SYM ⇐ ⇒ ✔ Final goal (dream): Maximally supesymmetric Yang-Mills theory is an example of the four-dimensional gauge theory that can be/ should be/ would be solved exactly for arbitrary value of the coupling constant!!! Forum de la Théorie au CEA, Apr 4, 2013 - p. 6/20
Conventional approach Simplest example: Gluon scattering amplitudes 4 3 5 + . . . 2 S = 6 1 7 Number of external gluons 4 5 6 7 8 9 10 Number of ‘tree’ diagrams 4 25 220 2485 34300 559405 10525900 ✔ Number of diagrams grows factorially for large number of external gluons/number of loops ✔ If one spent 1 second for each diagram, computation of 10 gluon amplitude would take 121 days! ✔ ... but the final expression for tree amplitudes looks remarkably simple � 12 � 4 (1 + 2 + 3 − . . . n − ) = ˆ ˜ A tree spinor notations: � ij � = λ α ( p i ) λ α ( p j ) � 12 �� 23 � . . . � n 1 � , n Forum de la Théorie au CEA, Apr 4, 2013 - p. 7/20
Four-gluon planar amplitude in N = 4 SYM at weak coupling ‘Mirracle’ at weak coupling: number of Feynman diagrams increases with loop level but their sum can be expressed in terms of a few ‘special’ scalar box-like integrals Example: four-gluon amplitude in N = 4 SYM: ✔ One loop: 2 3 1 4 ✔ Two loops: 2 3 1 4 all-loop iteration structure conjectured ✔ Three loops: 2 3 1 4 Little hope to get an exact all-loop analytical solution... Forum de la Théorie au CEA, Apr 4, 2013 - p. 8/20
How to solve a complicated problem The Feynman Problem-Solving Algorithm (attributed to Gell-Mann) (1) Write down the problem; (2) Think very hard; (3) Write down the answer. Non-Feynman Problem-Solving Algorithm (1) Write down the problem; (2) Ask Feynman; (3) Copy down his solution. Another “Feynman method” due to the mathematician Gian-Carlo Rota: Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say: ’How did he do it? He must be a genius!’ List of relevant ‘favorite problems’: ✔ Harmonic oscillator ✔ Two-dimensional Ising model Forum de la Théorie au CEA, Apr 4, 2013 - p. 9/20
Harmonic oscillator ✔ One of the few quantum mechanical systems for which a simple exact solution is known ✔ The quantum mechanical analogue of the classical harmonic oscillator H = p 2 2 m + 1 2 mω 2 x 2 ✔ Surprising duality between coordinates and momenta x → − ( mω ) − 1 p p → ( mω ) x , ✔ The wave function looks alike in the coordinate and momentum representations „ « „ « − x 2 − p 2 ( mω ) − 1 mω ˜ Ψ 0 ( x ) ∼ exp ⇔ Ψ 0 ( p ) ∼ exp 2 2 � � ✔ The dual symmetry is sufficient to find the eigenspectrum ˛ ˛ Ψ ℓ ( x ) = ˜ Ψ ℓ ( p ) ˛ ˛ p = xmω Forum de la Théorie au CEA, Apr 4, 2013 - p. 10/20
Two-dimensional Ising model Mathematical model in statistical mechanics with a nontrivial critical behaviour Is defined on a discrete collection of “spins” with nearest neighbor interaction X X e − βE [ S i ] , E = − Z = S i S i + � e { S i = ± 1 } i,� e √ Undergoes a phase transition between ordered/disordered phases at β c = ln(1 + 2) / 2 Has a strong/weak duality (between high/low temperatures ), the Kramers-Wannier duality ⇒ = S i + � S i e Dual lattice Forum de la Théorie au CEA, Apr 4, 2013 - p. 11/20
Duality in the Ising model Sum over spin configurations is dual to the random walk of a particle on the square lattice = ⇒ + − The free energy of the Ising model is determined by a free lattice fermion Z d 2 p p + im ∼ m ln 1 1 � E � = − 2 � S i S i +1 � = Tr (2 π ) 2 ˆ m | p |≪ 1 Free propagator of fermion with the mass m ∼ β − β c , sinh(2 β c ) = 1 β c Conformal symmetry at β = β c Forum de la Théorie au CEA, Apr 4, 2013 - p. 12/20
Back to scattering amplitudes: dual conformal symmetry Examine one-loop ‘scalar box’ diagram ✔ Change variables to go to a dual ‘coordinate space’ picture (remember 2D Ising model!): p 1 = x 1 − x 2 ≡ x 12 , p 2 = x 23 , p 3 = x 34 , p 4 = x 41 , k = x 15 x 3 p 2 p 3 Z Z d D k ( p 1 + p 2 ) 2 ( p 2 + p 3 ) 2 d D x 5 x 2 13 x 2 24 = k 2 ( k − p 1 ) 2 ( k − p 1 − p 2 ) 2 ( k + p 4 ) 2 = x 2 15 x 2 25 x 2 35 x 2 x 2 x 5 x 4 45 Unexpected conformal invariance under inversion (for D = 4 ) 5 ) 4 , x µ i → x µ i /x 2 d 4 x 5 → d 4 x 5 / ( x 2 x 2 ij → x 2 ij / ( x 2 i x 2 i , j ) p 1 p 4 x 1 ✔ The integral is invariant under conformal SO (2 , 4) transformations in the dual space! ✔ The symmetry is not related to conformal SO (2 , 4) symmetry of N = 4 SYM ✗ Conventional conformal transformations act locally on the coordinates (preserve angles) ✗ After Fourier transform, conformal transformations act nonlocally on the momenta ✗ Dual conformal transformations act locally on the momenta ✔ The dual conformal symmetry is powerful enough to determine four- and five-gluon planar scattering amplitudes to all loops! Forum de la Théorie au CEA, Apr 4, 2013 - p. 13/20
Dual description of scattering amplitudes Let us introduce dual variables (remember 2D Ising model): x 4 x 4 x 5 x 5 x 3 x 3 ⇒ S = x 6 x 6 x 2 x 2 x 7 x 7 x 1 x 1 Closed contour in Minkowski space-time ( n − gon) C n = [ x 1 , x 2 ] ∪ . . . ∪ [ x n , x 1 ] Built from light-like segments defined by gluon momenta p 2 p i = x i − x i +1 , i = 0 Who “lives” on polygon light-like contour in the dual description? Forum de la Théorie au CEA, Apr 4, 2013 - p. 14/20
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