Conformal Symmetry and Integrability inPerturbative N = 4 Scattering Amplitudes∗,∗∗
∗
‡Max-Planck-Institutfur Gravitationsphysik,Albert-Einstein-Institut,Potsdam, Germany
TillBargheer
NiklasBeisert
WellingtonGalleasFlorian
Loebbert
TristanMcLoughlin
WorkshopAmplitudes
2010
May 72010
∗ arxiv:0905.3738
NiklasBeisert
TristanMcLoughlin
JohannesHenn
JanPlefka
Queen MaryUniversity of London
‡AEI
HU-B
∗∗
∗∗ arxiv:1002.1733
Amp10, Niklas Beisert 1
Scattering Amplitudes in N = 4 SYM
Remarkable N = 4 Super Yang–Mills:
• Exact superconformal symmetry psu(2, 2|4).
• And some mysterious features: AdS/CFT, integrability, T-duality, . . .
• Integrability in the planar limit: psu(2, 2|4) Yangian.
• Simplifications for planar scattering.
• Duality of (some) amplitudes and Wilson loops.
Questions
• Same cusp dimension from amplitudes & integrable system:
How to apply integrability to scattering amplitudes?
• Can one compute remainder function F (p, λ) (like Dcusp(λ))?
• Relation between (dual) superconformal symmetry and integrability?
• What about non-MHV amplitudes?
• What about non-planar corrections?
Amp10, Niklas Beisert 2
Outline
Symmetries of Scattering Amplitudes (S-matrix):
• Understand symmetries of S-matrix.
• Apply symmetries to (fully?) constrain S-matrix.
How to treat (super)conformal symmetry of the S-matrix in N = 4 SYM?
Concretely
• Structure of symmetries: superconformal and Yangian
• Free symmetries
• Symmetries at tree level
• Symmetries at one loop
Amp10, Niklas Beisert 3
Scattering Amplitudes
Colour-ordered scattering amplitudes (1-trace, 2-trace, genus-1):
A
Legs: Field Ω combines on-shell gluons Γ , fermions Ψ & scalars Φ:
Ω(λ, λ, η) = Γ (λ, λ) + ηaΨa(λ, λ) + 12ηaηbΦab(λ, λ) + . . . .
Amplitude A(Λ1, . . . , Λn) on spinor helicity superspace Λ = (λ, λ, η)
pβα = λβλα, qαb = λβηa.
Amp10, Niklas Beisert 5
Free Superconformal Symmetry
Free representation J of superconformal algebra psu(2, 2|4):
J =
n∑k=1 A
JAk
k
• Single-leg representation Jk: multiplet of free on-shell fields Ω.
• Symmetry: Amplitudes are invariant under J.
• Representation independent of colour structure.
Algebra Generators: (Einfuehrung in Fraktur)
• super-Poincare: P, L, Q, Q, R
• superconformal: K, D, S, S
Invariance of tree (N∗MHV) amplitudes:
• P, Q: manifest through delta functions,
• L, R, D: weight and index contractions,
• Q, S: medium difficulty, • K, S: hardest.
PQ Q
KSS
DRL
Amp10, Niklas Beisert 6
Dual Superconformal Symmetry
Remarkable features of disk amplitudes (single-trace, large-Nc):
A
• Simplifications: BDS formula.
• Only particular integrals appear.
Dual conformal symmetry? [DrummondKorchemskySokatchev
]
• Dual superconformal symmetry! [Drummond, Henn
KorchemskySokatchev
]
• Another psu(2, 2|4):
L, R, D, Q, S: shared with conventional,
P, Q: trivial new generators,
K, S: non-trivial new generators.
P
K
SS
DRL
K
S
P
Q
Amp10, Niklas Beisert 7
T-Self-Duality in String Theory
Detour: Consider string theory picture at strong coupling.
• Strings propagate on AdS5 × S5 superspace.
• Background is coset space PSU(2, 2|4)/Sp(2)× Sp(1, 1).
• Isometries of background are Noether symmetries: psu(2, 2|4).
T-duality transformation: [ AldayMaldacena][ Berkovits
Maldacena]
• 4 bosonic + 8 fermionic T-dualities.
• terms at worldsheet boundaries: planar!
• maps AdS5 × S5 string model to itself:
self-duality!
• maps isometries to dual isometries:
dual superconformal symmetryP
Q
K
S K
S
P
S
DRL
Amp10, Niklas Beisert 8
String Theory IntegrabilityIntegrability enhances conserved charges:
Q(z) = zQ0 + z2Q1 + z3Q2 + . . .
• Q0 are Noether charges.
• Qk form (half) loop algebra.
• ∞-dimensional hidden symmetries.
T-self-duality [ NB, RicciTseytlin, Wolf][ Berkovits
Maldacena][ NB0903.0609]
• maps loop algebra to itself.
• It shows that conventional & dual
superconformal symmetry
close into loop algebra.
• Quantum algebra called Yangian.
P0
Q0
Q0
K0
S0
S0
LRD0
P1
Q1
Q1
K1
S1
S1
LRD1
P2
Q2
Q2
K2
S2
S2
LRD2
P3
Q3
Q3
K3
S3
S3
LRD3
P
Q
Q
K
S
SLRD
K
S
P
Q
Amp10, Niklas Beisert 9
Yangian Symmetry
Back to scattering amplitudes in N = 4 SYM:
Free representation J = J0, J = J1 of psu(2, 2|4) Yangian: [Drummond
HennPlefka
]
JA =
n∑k=1 A
JAk
kJA = FABC
n∑k<`=1 A
JBk JC`
k `
· · ·
• Yangian Symmetry:
Amplitudes are invariant under J, J. [Drummond, Henn
KorchemskySokatchev
]
• compatible with cyclic structure (exceptional)!
• J & J all one needs to know.
In fact, only J and P = K sufficient.
• Representation requires planar limit & ordering;
depends on colour structure.
P
Q
Q
K
S
SLRD
P
QQK
S
SLRD
Amp10, Niklas Beisert 10
Invariant(s) & Integrability
Integrability means (pragmatic definition):
• yes we can calculate what we are interested in,
• and we can do it efficiently:
BDS, Graßmannian, TBA, Y-system, . . . ,
• there is a hidden symmetry to constrain observables uniquely.
Tuesday/Wednesday:learned a lot about superconformal/dual/Yangian invariants.
• Truly nice & useful results.
Amp10, Niklas Beisert 12
Nitpicking
Invariants?
• “Invariants” almost (everywhere) invariant.
Symmetries have distributional anomalies.
• Ignore at tree level ⇒ hits you hard at loops.
Smearing anomaly by loop integration.
• Representation requires (potentially ugly) deformations.
Invariant!
• There can be only one invariant: the S-matrix.
• S-matrix assembled from almost-invariants.
Amp10, Niklas Beisert 14
Superconformal Boost Acting on MHV Amplitude
Apply the free superconformal generator (S0)bα =
∑nk=1 η
bk∂k,α on
AMHVn =
δ4(P ) δ8(Q)
〈12〉〈23〉. . . 〈n1〉, P βα =
n∑k=1
λβkλαk , Qβa =
n∑k=1
λβk ηak.
The anti-holomorphic derivative ∂α = ∂/∂λα acts only on λβ.
AMHVn is holomorphic in λ’s except for δ4(P )
(S0)bαδ
4(P ) =n∑k=1
ηbk∂k,αδ4(P ) =
n∑k=1
ηbkλγk
∂δ4(P )
∂P γα= Qγb
∂δ4(P )
∂P γα.
This term vanishes due to δ8(Q). Thus S0AMHV = 0. Well. . .
Amp10, Niklas Beisert 16
Holomorphic Anomaly
Watch out! The holomorphic anomaly yields extra contributions [CachazoSvrcekWitten
]
∂
∂z
1
z= πδ2(z),
∂
∂λα1
〈λ, µ〉= πδ2(〈λ, µ〉) εαβµ
β.
Taking the anomaly into account one obtains [Bargheer, NB, GalleasLoebbert, McLoughlin]
(S0)bαA
MHVn = −π
n∑k=1
εαγ(λγkη
bk+1−λ
γk+1η
bk
)δ2(〈k, k + 1〉) δ4(P ) δ8(Q)
〈12〉. . .〈k, k + 1〉. . . 〈n1〉
.
Free conformal symmetry breaks down when λk ∼ λk+1: Collinear!
Note that gap〈k, k + 1〉 in denominator chain of 〈j, j + 1〉 closes
〈λk−1, λk〉
〈λk, λk+1〉〈λk+1, λk+2〉 ∼ 〈λk−1, λk,k+1〉〈λk,k+1, λk+2〉.
Remains AMHVn−1 (Λ1, . . . , Λk,k+1, . . . , Λn). Calls for An−1 7→ An!
Amp10, Niklas Beisert 17
The S-Matrix
Package all amplitudes into generating functional with sources J(p)
A[J ] =g2
4A4J
J
J
J
+g3
5A5
J
J J
J
J
+g4
6A6
J
J
J
J
J
J
+g5
7A7
J
J
J J
J
J
J
+g6
8A8
J
J
J
J
J
J
J
J
+. . . .
Expected structure of symmetries at tree level
An
J0
+An−1
J1
+An−2
J2
+ . . . = 0.
Type of action also known in classical theory as non-linear (in fields).
Example: Supersymmetry in gauge theories: δψ ∼ ε(Dϕ+ F + [ϕ,ϕ]).
Amp10, Niklas Beisert 18
Symmetry Correction
Formulate by acting on generating functional (S0)bαAMHV
n [J ]
. . . = −π∫d4|4Λ12
n∏k=3
d4|4Λk d4η′ dα dϕ e3iϕεαγλ
γ1 ηb2
× Tr([J(Λ1), J(Λ2)] . . . J(Λn)
)AMHVn−1 (Λ12, Λ3, . . . , Λn)
with the replacements
λ1 = λ12eiϕ sinα, η1 = (η12 sinα+ η′ cosα)e−iϕ,
λ2 = λ12 cosα, η2 = η12 cosα− η′ sinα.
Compensate anomaly by (S+)bαAMHVn−1 [J ] with the action on sources
(S+)bαJ(Λ12) = π
∫d4η′ dα dϕ e3iϕεαγλ
γ1 ηb2 [J(Λ1), J(Λ2)].
Amp10, Niklas Beisert 19
Classical Representation
We find the following corrections for representation of S, S and K
S = S0 + S+ , S = S0 + S− , K = K0 + K+ + K− + K+− .
To be done:
• Does the deformation annihilate all tree amplitudes?
• Is it a consistent representation of superconformal symmetry?
• What does it mean?
Similar deformations expected for classical Yangian representation, e.g.
Q = . . . +
n∑k<`=1
A
Pk S`
k `
· · ·+ . . . .
Amp10, Niklas Beisert 21
Invariance of Tree Amplitudes
Collinear limit is universal for all (tree) amplitudes: splitting function
2
1
An ∼ 12
2
1
An−1split .
Follows e.g. from recursion relation by inheritance.
Exact invariance of all tree amplitudes: [Bargheer, NB, GalleasLoebbert, McLoughlin]
• Collinear singularities
universal, same as for MHV.
• No anomalies from
multi-particle singularities.
• Structure of cancellations −→An,k: n-leg Nk−2MHV.
J0
J+ J−
J+−
A4,2
A5,2 A5,3
A6,2 A6,3 A6,4
. . .. . .. . .. . .
Amp10, Niklas Beisert 22
Symmetry Algebra
Does the deformed representation close? [Bargheer, NB, GalleasLoebbert, McLoughlin]
• [L, . . . ], [R, . . . ]: proper index contractions.
• [D, . . . ]: proper conformal weights.
• Q,S, Q, S, Q,S, Q, S: straight-forward.
• S,S, S, S: rather non-trivial, requires vanishing central charge.
Closes only onto gauge transformation
Sαb,Sγd = εαγGbd,
Sbα, S
dγ = 1
2εαγεbdefGef ,
GabJ(Λ) ∼ [∂a∂bJ(0), J(Λ)].
On gauge invariant objects effectively S,S = 0 = S, S.• Sαb, S
dγ = δdbKαγ: defines conformal boost K.
• [K, . . . ], [P, . . . ]: through Jacobi identities.
Superconformal algebra satisfied, but contains gauge transformations.
Amp10, Niklas Beisert 23
Massless Asymptotic States
Reconsider conceptual problems of massless scattering:
• No mass gap: Massless particles can “decay” into particle showers
• Particle number not well-defined in asymptotic region.
• But: Shower particles are strictly collinear.
Single massless particle physically indistinguishable from shower.
• Overcounting of collinear states leads to IR divergencies at loop level.
• Can cancel IR divergencies in cross sections:
. . .
. . .. . . +. . .
. . .. . . +. . .
. . .. . .
Finite, but scattering amplitudes are more convenient.
Amp10, Niklas Beisert 25
Symmetry for Massless Asymptotic States
Asymptotic space:
• Fock space is too large; overcounts collinear states.
• Should project out collinear states: Conceptually hard!
• Rather embed asymptotic space in larger Fock space.
Keep collinear issues in mind.
Our results are in line with the above:
• Conformal anomaly precisely where asymptotic particles overcount.
• Deformation makes superconformal representation
compatible with embedding of asymptotic space into Fock space. (?)
• Exact conformal invariance incompatible with fixed particle number.
Have to consider generating functional A instead of amplitude(s) A.
Scattering operator for asymptotic space instead of scattering matrix.
Amp10, Niklas Beisert 26
Uniqueness
All tree amplitudes have been constructed by recursion relations [DrummondHenn ]
An,k(p) = AMHVn (p)
∑αcαRα(p).
Each R is (almost) invariant under the free Yangian representation.
How to obtain the correct physical linear combination cα = 1?
• Demand absense of spurious singularities or equivalently [ Hodges0905.1473]
• demand correct collinear behaviour. [KorchemskySokatchev ]
Deformed representation ensures correct collinear behaviour. [Bargheer, NB, GalleasLoebbert, McLoughlin]
Therefore symmetry alone fixes correct linear combination cα = 1!
Very important for construction by symmetry at higher loops:
• Adding any invariant respects symmetry: ambiguity!
• Adding tree level amounts to an overall factor; okay.
Can symmetry fix planar amplitude completely (non)perturbatively?
Amp10, Niklas Beisert 27
Symmetry of Unitarity Cuts
What about conformal symmetry for loop amplitudes?
• Particles in loop are off-shell.
• Conformal symmetry applies to on-shell particles only.
• How to determine loop anomaly?
A(1)n
Idea: Consider unitarity cut of loop amplitude [KorchemskySokatchev ][
BrandhuberHeslop
Travaglini][Sever
Vieira]
disc A(1)n =
n−4∑k=0
A(0)k+4 A
(0)n−k
• External and loop particles on shell!
• Only tree amplitudes used.
• Can act with symmetry to determine anomaly of cut.
• Use dispersion integral to reconstruct anomaly of loop later.
Amp10, Niklas Beisert 29
Anomaly of Unitarity Cuts
Act with superconformal boost on unitarity cut using invariance of trees S0 + S+
n−4∑k=0
A(0)k+4 A
(0)n−k
Three remaining contributions: [ NB, HennMcLoughlin, Plefka]
A(0)4 A
(0)n
S02
• from divergent measure
• no momentum transfer
n−4∑k=1 A
(0)k+3 A
(0)n−k
S+3
• integral localised
• finite, rational
A(0)4 A
(0)n−1S+
3
• one-loop splitting
• ill-defined→zero
Amp10, Niklas Beisert 30
One-Loop Anomaly
The anomaly of unitarity cuts reads
disc
S0 + S+
A(1)n = − A
(0)4 A
(0)n
S02 −
n−4∑k=1 A
(0)k+3 A
(0)n−k
S+3
How to promote it to the loop anomaly? [ NB, HennMcLoughlin, Plefka]
• All integrals are fully localised, cut anomaly is rational.
• Multiply by logs of cut invariants to reproduce cut discontinuity. S0 + S+
A(1)n = − A
(0)4 A
(0)n
S02
log(−s) −n−4∑k=1 A
(0)k+3 A
(0)n−k
S+3
logs
t
Confirmed explicitly for MHV and 6pt NMHV amplitudes.
Amp10, Niklas Beisert 31
One-Loop Invariance
Can rewrite loop anomaly as invariance statement: [ NB, HennMcLoughlin, Plefka]
S0 + S+
A(1)n + A
(0)4 A
(0)n
S02
log +n−4∑k=1 A
(0)k+3 A
(0)n−k
S+3
log = 0.
One-loop superconformal invariance of scattering amplitudes
S(0)A(1) + S(1)A(0) = 0.
• Different from earlier proposal: [SeverVieira]
Uses only on-shell amplitudes, compatible with regulator.
• Tree anomalies: collinearities (1→ 2).
• Loop anomalies: measure (2→ 2), loop collinearities (2→ k).
Amp10, Niklas Beisert 32
Superconformal Algebra
Do the loop corrections of the algebra close?
• Measure contributions take a simple universal form ⇒ closure
J(1)2→2 ∼ [X2→2, J
(0)], X2→2 =
n∑k=1
1
ε2
(sk,k+1
−µ2
)−ε.
• Corrections S(1)2→k are invariant under super-Poincare.
Rest of superconformal algebra is not yet clear:
• Some indications that algebra may not close properly.
• Deformations for S(1)2→k not uniquely determined by amplitudes.
• Closure on which class of functions?
Only amplitude itself: only singlet representation consistent?
• Can superconformal algebra be defined at loop level?
In any case, we understand the one-loop superconformal anomaly.
Amp10, Niklas Beisert 33
Yangian Anomaly
What about integrability at one loop?
One-loop (exact) anomaly of dual conformal generator K
[Drummond, Henn
KorchemskySokatchev
][Drummond, Henn
KorchemskySokatchev
][Brandhuber
HeslopTravaglini
][Elvang
FreedmanKiermaier
][KorchemskySokatchev ][
BrandhuberHeslop
Travaglini]
KβαA(1) ∼n∑k=1
1
ε
(sk,k+1
−µ2
)−ε k∑j=1
pβαj A(0).
Dual conformal generator is bi-local momentum generator K ' P
P ' P ∧D + P ∧ L + Q ∧ Q.
One-loop correction compensates the above anomaly. [ NB, HennMcLoughlin, Plefka]
P(1) ' P ∧D(1)2→2 + P
(1)local
'∑nj<k=1
A
P D(1)
j k k + 1
· · ·+∑nk=1
A
P(1)loc
k k + 1
Amp10, Niklas Beisert 35
Dual Superconformal Anomaly
Loop anomaly of dual superconformal symmetry S complicated. [KorchemskySokatchev ]
Generic form of the Yangian generator S = Q
Q ' P ∧ S + Q ∧ L + Q ∧R + Q ∧D
Follows from commutator Q = [S, P] with anomalous S [ NB, HennMcLoughlin, Plefka]
Q(1) ' P ∧ S(1)2→k + Q ∧D
(1)2→2 + Q
(1)local
'∑j<k
A
Q D(1)
j k k + 1
· · ·
+∑j<k
A
P S(1)
j k k + 1
· · ·+∑k
A
Q(1)loc
k k + 1
Amp10, Niklas Beisert 36
Conclusions
? Superconformal Symmetry at Tree Level• Tree amplitudes almost invariant under free superconformal symmetry.
• Invariance violated for singular configurations: Collinear momenta.
• Transformations can be corrected to make trees fully invariant.
• Dynamic corrections requires, changes number of legs.
• Superconformal algebra closes onto gauge transformations.
• Yangian appears to lead to a unique invariant ⇒ the S-matrix.
? Superconformal Symmetry at One Loop• Transformations can be corrected to make loops fully invariant.
? Open Problems• How does the algebra at loop level close?
• What about conformal inversions?
Amp10, Niklas Beisert 38
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