Exploring scattering amplitudes in gauge theories using soap films - - PowerPoint PPT Presentation

Forum de la Théorie au CEA, Apr 4, 2013 - p. 1/20

Exploring scattering amplitudes in gauge theories using soap films

Gregory Korchemsky

IPhT, Saclay

Forum de la Théorie au CEA, Apr 4, 2013 - p. 2/20

The Standard Model

✔ All matter is composed of spin−1/2 fermions ✔ All forces (except gravity) is carried by spin−1 vector bosons:

electromagnetism weak strong (QCD) ν γ

✔ Gauge theory with the symmetry group SU(3) × SUL(2) × UY(1) ✔ The only missing piece was the Higgs boson ... since July 4th, 2012 we are 99.99% certain it is

there

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Higgs at LHC

One of the decay modes of the Higgs boson: gluon + gluon → Higgs → e+ + e− + µ+ + µ− g g P P H Z Z e− e+ µ− µ+ 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

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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

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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 Nc for SU(Nc) ✔ 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 gYM and related to each other by supersymmetry

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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)

Strongly coupled planar N = 4 SYM ⇐ ⇒ Weakly coupled ‘dual’ string theory on AdS5 × S5

✔ 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!!!

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Conventional approach

Simplest example: Gluon scattering amplitudes

1 2 3 4 5 6 7

+ . . . S

=

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

Atree

n

(1+2+3− . . . n−) = 124 1223 . . . n1 , ˆ spinor notations: ij = λα(pi)λα(pj) ˜

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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:

1 2 3 4

✔ Two loops:

1 2 3 4

all-loop iteration structure conjectured

✔ Three loops:

1 2 3 4

Little hope to get an exact all-loop analytical solution...

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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

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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

• scillator

H = p2 2m + 1 2 mω2x2

✔ Surprising duality between coordinates and momenta

p → (mω)x , x → −(mω)−1p

✔ The wave function looks alike in the coordinate and momentum representations

Ψ0(x) ∼ exp „ − x2 2 mω

• «

⇔ ˜ Ψ0(p) ∼ exp „ − p2 2 (mω)−1

• «

✔ The dual symmetry is sufficient to find the eigenspectrum

Ψℓ(x) = ˜ Ψℓ(p) ˛ ˛ ˛ ˛

p=xmω

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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 Z = X

{Si=±1}

e−βE[Si] , E = − X

i, e

SiSi+

e

Undergoes a phase transition between ordered/disordered phases at βc = ln(1 + √ 2)/2 Has a strong/weak duality (between high/low temperatures ), the Kramers-Wannier duality Si Si+

e

Dual lattice = ⇒

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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 E = −2SiSi+1 = Tr Z

|p|≪1

d2p (2π)2 1 ˆ p + im ∼ m ln 1 m Free propagator of fermion with the mass m ∼ β − βc βc , sinh(2βc) = 1 Conformal symmetry at β = βc

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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!):

p1 = x1 − x2 ≡ x12 , p2 = x23 , p3 = x34 , p4 = x41 , k = x15

p1 p2 p3 p4 x1 x2 x3 x4 x5

= Z dDk (p1 + p2)2(p2 + p3)2 k2(k − p1)2(k − p1 − p2)2(k + p4)2 = Z dDx5 x2

13x2 24

x2

15x2 25x2 35x2 45

Unexpected conformal invariance under inversion (for D = 4) xµ

i → xµ i /x2 i ,

d4x5 → d4x5/(x2

5)4 ,

x2

ij → x2 ij/(x2 i x2 j)

✔ 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!

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Dual description of scattering amplitudes

Let us introduce dual variables (remember 2D Ising model):

S

= ⇒

x1 x1 x2 x2 x3 x3 x4 x4 x5 x5 x6 x6 x7 x7 Closed contour in Minkowski space-time (n−gon) Cn = [x1, x2] ∪ . . . ∪ [xn, x1] Built from light-like segments defined by gluon momenta pi = xi − xi+1 , p2

i = 0

Who “lives” on polygon light-like contour in the dual description?

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Scattering amplitudes/Wilson loops duality

Gluon (MHV) scattering amplitude = Wilson loop in Minkowski space-time

S S

⇔

p1 p2 p3 p4 p5 p6 p7 x1 x2 x3 x4 x5 x6 x7

✔ What is the Wilson loop?

Wn = 0| exp „ ig I

Cn

dxµAµ(x) « |0

✗ Circulation of (nonabelian) gauge field along polygon like contour ✗ Interaction of a test particle moving along the closed contour Cn with its own radiation ✗ Gauge-invariant scalar function of dual distances = Mandelstam invariants

(x1 − x2)2 = 0 , (x1 − x3)2 = (p1 + p2)2 , (x1 − x4)2 = (p1 + p2 + p3)2 , . . .

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Weak coupling

Weak coupling expansion (back to Feynman diagrams): Wn = 1 + 1 2 (ig)2 I

Cn

dxµ I

Cn

dyνAµ(x)Aν(y) + O(g4)

✔ Simplest example n = 4: four-particle amplitude

ln W(C4) =

x3 x3 x3 x2 x2 x2 x1 x1 x1 x4 x4 x4

(s = x2

13 , t = x2 24)

=g2Nc  − 1 ǫ2 h` −s/µ2´−ǫ + ` −t/µ2´−ǫi + 1 2 ln2 “s t ” + const ff + O(g4) ☞ Identity the light-like segments with the on-shell gluon momenta xi,i+1 = pi ☞ coincide with one-loop four gluon amplitude A4(s, t)

✔ Conformal symmetry of N = 4 SYM =

⇒ All-loop result for light-like Wilson loop W4 ln W4 = ln A4(s, t) = Div(1/ǫ) + 1 4 Γcusp(g2Nc) ln2 “s t ” (BDS ansatz) How to compute the Wilson loop at strong coupling? Gauge/string duality!

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Gauge/string duality

Surprising correspondence of strongly interacting quantum field theories with gravitational theories: Strongly coupled maximally supersymmetric gauge theory ⇔ Weakly coupled ‘dual’ string theory on anti-de Sitter space Dual space is 5 dimensional hyperbolic space with constant negative curvature ds2 = R2 2 6 4e2τ ` −dt2 + dx2 + dy2 + dz2´ | {z }

Minkowski

+ dτ 2 |{z}

5th dim

3 7 5 Hyperbolic space depicted by M.C.Escher:

✔ Each fish has the same coordinate-invariant “proper”

size

✔ Appear to get smaller near the boundary because of

their projection onto the flat surface of this page.

✔ We “live” at the circular boundary, infinitely far from

the center of the disk (τ → ∞).

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From Wilson loops to minimal surfaces

x1 x2 x3 x4 x5 x6 x7 5th dim.

λ→∞ = ⇒

Minimal surface in AdS The Wilson loop at strong coupling from gauge/string duality Amplitude = Wilson loop ∼ exp “ − p g2Nc × Area ”

✔ Defined by the area of minimal surface in anti-de Sitter space ✔ The surface ends at the AdS boundary on a polygon given by a sequence of gluon momenta

How to compute the area of minimal surface (Plateau’s problem)?

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Soap Bubbles

Plateau’s problem in mathematics: show the existence of a minimal surface with a given boundary Physicists solution: soap bubbles always find the smallest surface area with a given boundary

✔ A spherical soap bubble is the least-area for a given volume of air (H.A.Schwarz, 1884). ✔ More complex examples: ✔ Even more complex examples: ✔ Generalization to anti–de Sitter space is straightforward

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Computing scattering amplitudes using soap films

S

= ⇒

p1 p1 p2 p2 p3 p3 p4 p5 p6 p7 p7

✔ Explicit expressions for 4- and 5-particle scattering amplitudes for arbitrary coupling ! ✔ First nontrivial results for n−particle scattering amplitudes at strong coupling

Maximally supersymmetric Yang-Mills theory is the simplest gauge theory – “harmonic

• scillator of 21st century”:

✔ An excellent testing ground for computing QCD scattering amplitudes needed for precise

theoretical predictions at hadron colliders

✔ Unexpected interconnections and hidden symmetries

We are close to finding an exact solution to Maximally supersymmetric Yang-Mills theory Stay tuned!