Conformal Symmetry and Integrability in

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?


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


Free

Symmetries


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.


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


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

QQ

K

SS

DRL

K

S

P

Q


http://arxiv.org/abs/0707.0243


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

QQ

S

DRL




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





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




Dual Conformal, Yangian, . . .


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.


Not Correct!


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.


Superconformal Anomaly

at Tree Level


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. . .


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!


http://arxiv.org/abs/hep-th/0409245


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]).


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)].


Classical Deformation for

Superconformal Symmetry


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 `

· · ·+ . . . .


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

. . .. . .. . .. . .



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.



Implications


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.


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.


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?






Superconformal Symmetry

of Loop Amplitudes


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.





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



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.



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).




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.


Integrability

of Loop Amplitudes


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









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




Conclusions


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?


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