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) × SUL (2) × UY (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+
g
Z
µ−
Z
e+
H
g
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 hidden 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 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
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)
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!!!
Forum de la Théorie au CEA, Apr 4, 2013 - p. 6/20
Conventional approach
Simplest example: Gluon scattering amplitudes
4
3
5
S
2
+ ...
=
6
1
7
Number of external gluons
Number of ‘tree’ diagrams
4
4
5
25
6
220
7
2485
8
34300
9
559405
10
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
h12i4
...n ) =
,
h12ih23i . . . hn1i
−
ˆ
˜
spinor notations: hiji = λα (pi )λα (pj )
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
1
p2
+ mω 2 x2
H=
2m
2
✔ Surprising duality between coordinates and momenta
p → (mω)x ,
x → −(mω)−1 p
✔ The wave function looks alike in the coordinate and momentum representations
„
«
x2 mω
Ψ0 (x) ∼ exp −
2 ~
⇔
„
p2 (mω)−1
Ψ̃0 (p) ∼ exp −
2
~
«
✔ The dual symmetry is sufficient to find the eigenspectrum
˛
˛
Ψℓ (x) = Ψ̃ℓ (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
−βE[Si ]
Z=
e
,
E=−
Si Si+~e
{Si =±1}
i,~
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+~e Si
=⇒
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
hEi = −2hSi Si+1 i = Tr
Z
|p|≪1
−
d2 p
1
1
∼
m
ln
(2π)2 p̂ + im
m
Free propagator of fermion with the mass
m∼
Conformal symmetry at β = βc
β − βc
,
βc
sinh(2βc ) = 1
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!):
p1 = x1 − x2 ≡ x12 ,
p2
x2
x3
x5
p2 = x23 ,
p3 = x34 ,
p4 = x41 ,
k = x15
p3
x4
=
Z
dD k (p1 + p2 )2 (p2 + p3 )2
=
k2 (k − p1 )2 (k − p1 − p2 )2 (k + p4 )2
Z
dD x5 x213 x224
x215 x225 x235 x245
Unexpected conformal invariance under inversion (for D = 4)
p4
p1
x1
µ
2
xµ
i → xi /xi ,
d4 x5 → d4 x5 /(x25 )4 ,
x2ij → x2ij /(x2i x2j )
✔ 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):
x4
x4
x5
x3
S
x6
x5
x3
=⇒
x2
x6
x2
x7
x7
x1
x1
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 ,
p2i = 0
Who “lives” on polygon light-like contour in the dual description?
Forum de la Théorie au CEA, Apr 4, 2013 - p. 14/20
Scattering amplitudes/Wilson loops duality
Gluon (MHV) scattering amplitude = Wilson loop in Minkowski space-time
x4
p4
p3
x5
x3
p5
S
p2
⇔
S
x6
x2
p6
p1
x7
p7
x1
✔ What is the Wilson loop?
„
Wn = h0| exp ig
I
Cn
«
dxµ Aµ (x) |0i
✗ 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 ,
...
Forum de la Théorie au CEA, Apr 4, 2013 - p. 15/20
Weak coupling
Weak coupling expansion (back to Feynman diagrams):
I
I
1
Wn = 1 + (ig)2
dxµ
dy ν hAµ (x)Aν (y)i + O(g 4 )
2
Cn
Cn
✔ Simplest example n = 4: four-particle amplitude
x3
x2
x3
x2
x3
x2
(s = x213 , t = x224 )
ln W (C4 ) =
x1
=g 2 Nc
x4

x1
x4
x1
x4
ff
i 1
“s”
´
´
`
1 h`
−ǫ
−ǫ
+ ln2
+ −t/µ2
+ const + O(g 4 )
− 2 −s/µ2
ǫ
2
t
☞ 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
“ ”
1
2
2 s
ln W4 = ln A4 (s, t) = Div(1/ǫ) + Γcusp (g Nc ) ln
4
t
(BDS ansatz)
How to compute the Wilson loop at strong coupling? Gauge/string duality!
Forum de la Théorie au CEA, Apr 4, 2013 - p. 16/20
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
3
2
´
7
6 2τ `
2
2
2
2
−dt + dx + dy + dz + dτ 2 5
ds = R 4e
|{z}
|
{z
}
2
2
Minkowski
5th dim
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 → ∞).
Forum de la Théorie au CEA, Apr 4, 2013 - p. 17/20
From Wilson loops to minimal surfaces
Minimal surface
in AdS
x4
x5
x3
x6
λ→∞
=⇒
x2
x7
x1
5th dim.
The Wilson loop at strong coupling from gauge/string duality
“ p
”
2
Amplitude = Wilson loop ∼ exp − g Nc × 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)?
Forum de la Théorie au CEA, Apr 4, 2013 - p. 18/20
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
Forum de la Théorie au CEA, Apr 4, 2013 - p. 19/20
Computing scattering amplitudes using soap films
p4
p3
p3
p2
p5
S
=⇒
p6
p1
p2
p1
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
oscillator 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!
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