Symmetries and scattering amplitudes in N=4 sYM - rbi-t

Transcription

Symmetries and scattering amplitudes
in N=4 sYM
Livia Ferro
Ludwig-Maximilians-Universität München
Theoretical Physics Divn, Ruđer Bošković Institute, Zagreb
Scattering amplitudes
Scattering amplitudes: central objects in GTs
describe interactions between particles
huge number of diagrams
high complexity already at two loops
symmetries and good formalism can help
planar N=4 sYM
N=4 sYM
sYM
massless
massive
(picture of L. Dixon)
L. Ferro (LMU München)
Zagreb, 05.02.2016
Scattering amplitudes in N=4 sYM
Scattering amplitudes: central objects in GTs
describe interactions between particles
N=4 sYM: interacting 4d QFT with
highest degree of symmetry
Features:
scale invariant
hidden symmetries in planar limit - integrable
structure
triality ampls/Wilson loops/correlation fcs
AdS/CFT correspondence
Zagreb, 05.02.2016
On-shell supermultiplet described by a superfield:
= G+ + ⌘ A
A+
p2 = 0 () p↵↵˙ =
1 A B
1
⌘ ⌘ SAB + ⌘ A ⌘ B ⌘ C ✏ABCD
2!
3!
↵ ˜↵
˙
q ↵A =
,
D
+
1 A B C D
⌘ ⌘ ⌘ ⌘ ✏ABCD G
4!
↵ A
⌘
on-shell superspace: ( ↵ , ˜ ↵˙ , ⌘A )
An = h (
1,
˜ 1 , ⌘1 ) (
2,
˜ 2 , ⌘2 )... (
n,
˜ n , ⌘n )i
Amplitudes labeled by two numbers:
number of particles - n
4k
⌘
helicity L. Ferro (LMU München)
Zagreb, 05.02.2016
On-shell supermultiplet described by a superfield:
= G+ + ⌘ A
A+
p2 = 0 () p↵↵˙ =
1 A B
1
⌘ ⌘ SAB + ⌘ A ⌘ B ⌘ C ✏ABCD
2!
3!
↵ ˜↵
˙
,
q ↵A =
D
+
1 A B C D
⌘ ⌘ ⌘ ⌘ ✏ABCD G
4!
↵ A
⌘
on-shell superspace: ( ↵ , ˜ ↵˙ , ⌘A )
Amplitudes labeled by two numbers:
number of particles - n
4k
helicity - ⌘
MHV tree level:
[Parke-Taylor]
Zagreb, 05.02.2016
Symmetries and ampls in N=4 sYM
Important in discovering the characteristic of ampls
Superconformal symm.
expected
Dual superconformal symm.
hidden
Yangian symmetry
level-zero generators
level—one generators
superconformal
dual superconformal
+ Serre relations for higher commutators
The Yangian Y(g) of the Lie algebra g is generated by
and
Tree-level holomorphic anomalies, broken at loop level
Zagreb, 05.02.2016
Symmetries and ampls in N=4 sYM
Important in discovering the characteristic of ampls
Superconformal symm.
expected
Dual superconformal symm.
(planar)
hidden
Yangian symmetry
!
Hallmark of integrability
}
Integrable deformation
of amplitudes
}
Best formalism:
Grassmannian
(J. Drummond, LF)
Infinite-dimensional algebra
Present also in spin chains
„Solvability“ of the theory
(LF, T. Lukowski, C. Meneghelli, J. Plefka, M. Staudacher)
Zagreb, 05.02.2016
Integrability
Exact solvability of MSYM
Yang-Baxter equation
zi complex
parameters
R-matrix describes the „scattering“ of particles
It is the essential ingredient for all integrable models
Amplitudes and integrability: spectral parameter?
Zagreb, 05.02.2016
Integrability
Exact solvability of MSYM
Yang-Baxter equation
Solution for the (integral kernel) of R12 of a (specific) YBe:
Spectral parameter deformation of scattering amplitude A4,2!
Scattering in 4d <-> scattering in 2d!
It can be used for loop amplitudes: new regularization method?
Zagreb, 05.02.2016
Grassmannian
(Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Cheung, Goncharov, Hodges, Kaplan,Postnikov,Trnka)(Mason,Skinner)
In twistor space:
Grassmannian: space of k-planes in n dim.s
Integration over complex parameters
It computes all tree-level amplitudes
Yangian symmetry nice geometric interpretation
Zagreb, 05.02.2016
Grassmannian
(Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Cheung, Goncharov, Hodges, Kaplan,Postnikov,Trnka)(Mason,Skinner)
In twistor space:
Example: MHV tree-level four points
=
I
dc13 dc14 dc23 dc24
c13 c24 (c13 c24 c14 c23 )
⇥
2
Y
k=1
2
2
(
⇣
3
c13
1
c23
2)
˜ k + ck3 ˜ 3 + ck4 ˜ 4
⌘
2
(
4
(⌘k + ck3 ⌘3 + ck4 ⌘4 )
4
c14
1
c24
2)
Zagreb, 05.02.2016
Grassmannian
In twistor space:
Example: MHV tree-level four points
Zagreb, 05.02.2016
Grassmannian
In momentum twistor space:
MHV part factored out
space associated to Wilson loop
Zagreb, 05.02.2016
Amplituhedron
(picture of A. Gilmore)
Conjecture
the volume of the amplituhedron
=
amplitude in planar N=4 sYM
(N. Arkani-Hamed, J. Trnka)
Geometric interpretation of amplitudes!
Idea: NMHV tree-level amplitudes = volume of
polytope in dual momentum twistor space (Hodges)
Related to (positive) Grassmannian
Zagreb, 05.02.2016
Amplituhedron
In a nutshell
~x3
center of mass of
3 points in 2d
·~x
~x2
~x1
c1 ~x1 + c2 ~x2 + c3 ~x3
~x =
c1 + c2 + c3
Interior: ranging over all positive c1, c2, c3
Zagreb, 05.02.2016
Amplituhedron
In a nutshell
Z3
triangle in
Zi =
✓
1
~xi
·Y
projective space
Z2
Z1
I
Y =
I
c1 Z1
+
I
c2 Z2
+
◆
I
c3 Z3
I = 1, 2, 3
Interior: ranging over all positive c1, c2, c3 (GL(1))
Zagreb, 05.02.2016
Amplituhedron
In a nutshell
Generalizations:
triangles in 2d -> polygons + higher dimensions
most general at tree level:
I
Y↵
=
X
a
Geometric
requirements
{
I
c↵a Za
I = 1, 2, ..., k + m
a = 1, ...n
„Internal" positivity (c) = interior
„External" positivity (Z) = convexity
Volume = amplitude in planar N=4 sYM
Zagreb, 05.02.2016
Amplituhedron
In a nutshell
Generalizations:
Positive grassmannian -> tree amplituhedron Ωn,k,L
I
Y↵
=
X
I
c↵a Za
a
G(k, k + m)
G+ (k + m, n)
G+ (k, n)
physics: m=4
tree: k=1 polytope, k>1 more complicated object
loops: similar, more complicated formulae
Zagreb, 05.02.2016
Amplituhedron
In a nutshell
Generalizations:
Positive grassmannian -> tree amplituhedron Ωn,k,L
I
Y↵
=
X
I
c↵a Za
a
G(k, k + m)
„Amplitude
coords"
0
B
B
B
Z=B
B
B
@
za
'A
1 ⌘aA
.
.
.
'A
k ⌘aA
1
C
C
C
C
C
C
A
G+ (k + m, n)
G+ (k, n)
Bosonized momentum twistors
Zagreb, 05.02.2016
Amplituhedron
(
Integral representation:
(Some) Open questions
⌦tree
n,k
which symmetries are preserved?
how to evaluate volume for k>1 and for loops?
how to perform the integral (prescription)?
(LF, T. Lukowski, A. Orta, M. Parisi)
Zagreb, 05.02.2016
Conclusions
N=4 super Yang-Mills has a lot of good properties
„harmonic oscillator" of 21st century
Driving force: symmetries!
A lot of work still has to be done!
Zagreb, 05.02.2016

Symmetries and scattering amplitudes in N=4 sYM - rbi-t

Transcription

Similar documents