[PPT] - Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory PowerPoint Presentation

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Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory

Till Bargheer

Max–Planck–Institut für Gravitationsphysik Albert–Einstein–Institut Potsdam–Golm Germany

Uppsala University November 3, 2009 arXiv:0905.3738

(with N. Beisert, W. Galleas, F. Loebbert, T. McLoughlin)

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory

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Outline

1 Overview and Motivation 2 Intriguing Results about Amplitudes in N = 4 SYM 3 A Closer Look at Tree-Level Symmetries 4 Outlook: Loops

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 0 / 36

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Why N = 4 Super Yang-Mills Theory

◮ N = 4 super Yang-Mills theory (SYM) is a supersymmetric cousin of QCD. ◮ Many things computable due to large amount of symmetry. ◮ Relation to gravity (strings) via the AdS/CFT correspondence.

N = 4 SYM as a mathematical toy model (of more realistic theories) whose structure we can eventually understand.

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 1 / 36

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Features of N = 4 SYM Theory

Symmetry

◮ Superconformal theory, symmetry group psu(2, 2|4). ◮ Conformal to all orders, β(g) = 0.

Duality

◮ Dual to IIB string theory on AdS5 × S5. ◮ Weak ↔ strong duality. ◮ Very successful comparison of string and gauge spectrum.

Integrability in the Planar Nc → ∞ Limit

◮ Appearance of integrability has led to tremendous progress in the study of

anomalous dimensions of local gauge invariant operators.

◮ Will integrability also help us in the case of scattering amplitudes? ◮ Relation between anomalous dimensions and scattering amplitudes:

Cusp anomalous dimension! Realistic hopes that the planar theory can be solved to all orders!

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 2 / 36

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Scattering Amplitudes in Conformal Theories

Conformal: Scattering amplitudes are not well-defined in conformal theories: There are ∞-ranged interactions (massless gauge bosons)

◮ ⇒ No notion of non-interacting asymptotic states. ◮ ⇒ IR divergences.

Regulate: In order to make sense of scattering processes, need to introduce IR-regulator (D = 4 − 2ε) ⇒ Breaks conformal symmetry.

◮ Regulator drops out of tree-level amplitudes. ◮ Loop amplitudes diverge as ε → 0.

1 2 3 4 5 n . . . . An

Scattering amplitudes only make sense in the regulated theory.

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 3 / 36

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Color Ordering

Color Structure

◮ Device for decomposing amplitudes into smaller gauge-invariant pieces. ◮ Feynman rules factorize into a color part and a kinematical part. ◮ Color factor in amplitudes is a product of traces of color matrices T a.

Decomposition In U(Nc), SU(Nc) gauge theory, product of k traces comes with factor 1/N k

c .

⇒ In the planar (large-Nc) limit, only single-trace terms contribute. ⇒ Amplitudes can be color-decomposed: An({pi, ai}) ∼

σ∈Sn/Zn

Tr(T aσ(1) · · · T aσ(n)) ˆ An(pσ(i)) . Color-stripped partial amplitudes ˆ An

◮ Depend only on the kinematics (momenta, polarization, helicity). ◮ Particles have a definite ordering. ◮ Amplitudes are cyclic in their arguments (particles).

From now on, consider only the color-stripped amplitudes ˆ An!

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 4 / 36

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Spinor-Helicity Variables

Particles One-particle states specified by: Momentum pµ, Polarization vector εµ. Gluons G± Fermions ψA/ ¯ ψA Scalars φAB Helicity h ±1 ±1/2 Momentum Spinors [

Witten hep-th/0312171]

◮ Four-momenta can be written as 2 × 2 hermitean matrices

pa˙

b = (σµ)a˙ bpµ ,

⇒ pµpµ = − det(pa˙

b) . ◮ Massless on-shell condition: det(pa˙ b) = 0.

⇒ Can write pa˙

b as product of two spinors λa (chiral), ˜

λ

˙ b (antichiral).

Since pa˙

b is hermitean, can choose ˜

λ = ±¯ λ: pa˙

b k = ±λa k¯

λ

˙ b k .

Polarization

◮ Gauge freedom:

λ → eiφλ , ¯ λ → e−iφ¯ λ .

◮ For given helicity h, choice of λ uniquely determines polarization vector ε.

Trade (pµ, εµ) for (λa, ¯ λ ˙

a, h).

Efficient variables for amplitudes and symmetries!

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 5 / 36

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The Simplest Tree Amplitudes

Gluon Amplitudes: hk = ±1

◮ Can classify amplitudes by helicities of external particles. ◮ Treat all particles as incoming (changes sign of h for outgoing ones).

ˆ A++++...+

n

= 0 (supersymmetry) [Grisaru, Pendleton,

v. Nieuwenhuizen ][ Grisaru

∼ δ4(P)jk4 122334 · · · n1 , Total momentum: P =

n

k=1

λk¯ λk, Lorentz invariants: jk := λa

j εabλb k.

Other Tree-Level Amplitudes [

Britto Cachazo, Feng][Britto, Cachazo Feng, Witten ]

Tree-level amplitudes can be constructed recursively: BCFW relations.

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 6 / 36

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Loop Amplitudes

Loop Technology

◮ Compute loop amplitudes efficiently: Generalized unitarity.

[

Bern, Dixon Dunbar, Kosower]

Planar four-leg amplitude to four loops, [Bern, Dixon

Smirnov ][Bern, Czakon, Dixon Kosower, Smirnov ]

Six-leg amplitude at two loops. [Bern, Dixon, Kosower, Roiban

Spradlin, Vergu, Volovich ][ Vergu 0903.3526]

◮ Massless scattering amplitudes problematic: Single massless particle can

decay into collinear particles. ⇒ IR divergences when integrating over collinear momenta. ⇒ Amplitudes divergent as ε → 0, typically 1/ε2 at each loop order. Exponentiation

◮ IR divergences exponentiate into divergent prefactor.

[

Sterman Tejeda-Yeomans]

AAll−loop = Adivergent(ε) · Afinite Symmetries (conformal) of remainder might be broken by regulator.

◮ Simplification in the planar limit Nc → ∞? Symmetry enhancement?

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 7 / 36

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Loop Amplitudes: BDS Conjecture

Observation in the planar limit Nc → ∞: [Anastasiou, Bern

Dixon, Kosower ][Bern, Dixon Smirnov ]

Higher loops of 4-gluon amplitude have iterative structure:

◮ Each loop order only depends on previous loop orders. ◮ Surprising! Usually new functional dependence at each loop level.

BDS Conjecture: Finite part of n-gluon amplitude vanishes to all loops: An(p) ∼ Atree

n

(p) exp

2Dcusp(λ)M (1)

n (p) + F(p)

.

Depends only on tree-level Atree

n

, one-loop M (1)

n

and cusp dimension Dcusp(λ). Cusp Dimension

◮ Dcusp(λ) is independent of particle number and kinematics! ◮ Integrability: All-order integral equation for Dcusp(λ).

[

Eden Staudacher]

◮ Expansions for weak and strong coupling.

[Beisert, Eden

Staudacher ][Basso, Korchemsky Kotanski

] Correctness BDS conjecture falsified for n ≥ 6. . . . [

Bern, Dixon, Kosower Roiban, Spradlin, Vergu, Volovich][ Drummond, Henn Korchemsky, Sokatchev] November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 8 / 36

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Amplitudes in the AdS/CFT Dual

AdS/CFT: Equates N = 4 SYM and type IIB strings on AdS5 × S5. [Maldacena

9711200 ]

Gluon Scattering [

Alday Maldacena]

◮ Gluons ↔ Lightlike open strings ending on D3-branes at z = ∞. ◮ IR regularization: One D3-brane xµ away from the boundary, z = zIR. ◮ zIR → ∞: Amplitude dominated by minimal surface over boundary. [ Gross

Mende]

◮ “T-Dualize” xµ → yµ: Boundary of surface becomes polygonal:

− →

◮ Polygon composed of lightlike gluon momenta: yj − yj−1 = pj.

AdS/CFT: Polygon is Wilson loop in N = 4 SYM! String amplitude ≡ VEV of Wilson loop. Computation of the minimal surface agrees with the BDS conjecture, taking the cusp anomalous dimension (computed from integrability) at strong coupling.

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 9 / 36

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Dual Superconformal Symmetry

Dual Superconformal Symmetry [

Drummond, Henn Korchemsky, Sokatchev][ Brandhuber Heslop, Travaglini]

Introduce dual coordinates: xk+1 − xk ≡ pk = λk¯ λk , xn+1 ≡ x1 .

◮ Amplitudes only depend on pk

⇒ Invariance under dual translations.

◮ Find that tree amplitudes are invariant under

conformal transformations in xk!

◮ Can be extended to

dual superconformal symmetry. x1 x2 x3 x4 x5 p1 p2 p3 p4 p5 Loop Integrals BDS: Many loop integrals do not contribute. The ones that do contribute are dual superconformally invariant. [ Drummond, Henn

Smirnov, Sokatchev]

Imposing dual superconformal symmetry, one recovers BDS!

◮ 4 and 5-point amplitudes: Fully constrained to BDS. ◮ n ≥ 6: Dual superconformal symmetry has additional invariants. ◮ Indeed, finite remainder for n = 6. . . . [

Bern, Dixon, Kosower Roiban, Spradlin, Vergu, Volovich][ Drummond, Henn Korchemsky, Sokatchev] November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 10 / 36

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Yangian Symmetry

Standard and dual superconformal symmetry together: Yangian symmetry! Yangians

◮ ∞-dimensional symmetry-algebras. ◮ Distinct feature of integrable systems. ◮ Generators arranged in levels. Level zero: Lie symmetry algebra.

Gauge Theory Picture [ Drummond

Henn, Plefka]

◮ Closure of standard and dual superconformal symmetry generates Yangian! ◮ Yangian symmetry established for tree-level amplitudes.

String Picture

◮ Supersymmetric T-duality maps

[

Alday Maldacena][ Berkovits Maldacena]

string model to itself. Standard and dual superconformal symmetry exchange.

◮ Closure of standard and dual

[ Beisert, Ricci

Tseytlin, Wolf][ Beisert 0903.0606]

superconformal symmetry: Loop algebra, ∞-dimensional, represents classical string integrability.

ℓ = 1 ℓ = 2 ˜ ℓ = 1 ˜ ℓ = 2 November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 11 / 36

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Summary and Motivation

Scattering Amplitudes in N = 4 SYM

◮ MHV, MHV amplitudes extremely simple. ◮ Loops: Divergences exponentiate. ◮ Planar limit: Remainder constrained by dual superconformal symmetry. ◮ Yangian symmetry at strong coupling and for tree amplitudes.

Motivation

◮ N = 4 SYM supposedly is integrable. ◮ Integrability has proven extremely useful for finding the planar spectrum. ◮ Amplitudes perhaps completely determined by symmetry?

Plan

◮ Investigate symmetries more closely! ◮ In fact, there is an important subtlety already at tree-level. . .

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 12 / 36

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Superfield, Superamplitude

Field content (on-shell): 1 gluon G±, helicity ±1 , 4 fermions ψA/ ¯ ψA, helicity ± 1

2 ,

6 scalars φAB, helicity 0 . Superfield [

Mandelstam Nucl.Phys.B213][Brink, Lindgren, Nilsson

] Amplitudes for different external particles are related through supersymmetry. Combine particles into superfield (on-shell supermultiplet) Φ(λ, ¯ λ, η) = G+(λ, ¯ λ) + ηAψA(λ, ¯ λ) + 1

2ηAηBφAB(λ, ¯

λ) + 1

6εABCDηAηBηC ¯

ψD(λ, ¯ λ) +

1 24εABCDηAηBηCηDG−(λ, ¯

λ), where ηA, A = 1, . . . , 4 is a fermionic su(4)-spinor. ⇒ Amplitude for n external particles is superspace function (Superamplitude) An = An(Λ1, . . . , Λn) , Λk = {λk, ¯ λk, ηk} . Helicity measured by helicity charge B = ηC

∂ ∂ηC .

Individual Amplitudes

◮ Superamplitude An(Λk) is a polynomial of degree 4n in ηA k . ◮ Amplitudes of specific external particles can be extracted from An(Λk) as

coefficients of appropriate monomials in ηA

k . ◮ Example: Amplitude with scalar SAB on leg 3 contains coefficient ηA 3 ηB 3 .

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 13 / 36

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Tree of Amplitudes

Helicity expansion: An = An,2 + An,3 + . . . + An,n−2.

# particles n helicity h A4,2 A5,2 A5,3 A6,2 A6,3 A6,4 A7,2 A7,3 A7,4 A7,5 . . . . . . . . . . . . . . .

MHV,tree

n

= δ4(P)jk4 122334 · · · n1 , jk := λa

j εabλb k .

Generalization to MHV superamplitude (degree eight in η’s): [

Nair

Phys. Lett.

B214, 215]

A

MHV,tree

n

= δ4(P)δ8(Q) 122334 · · · n1 , QaA :=

n

k=1

λa

kηA k .

Gluon amplitude can be extracted ∂4 (∂ηj)4 ∂4 (∂ηk)4 δ4(P)δ8(Q) 122334 · · · n1 = δ4(P)jk4 122334 · · · n1 . Taking other η-derivatives yields supersymmetric cousins.

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 15 / 36

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Symmetry Algebra

The symmetry algrebra of N = 4 SYM is psu(2, 2|4): Conformal Super Internal La

b, ¯

L ˙

a ˙ b, Pa˙ b, Ka˙ b, D

QaA, ¯ Q ˙

a A, SaA, ¯

SA

˙ a

RA

B

[Lab, Jc] = −δa

c Jb + 1 2 δa b Jc,

[Lab, Jc] = δc

bJa − 1 2δa b Jc,

[RAB, JC] = −δA

CJB + 1 4 δA BJC,

[RAB, JC] = δC

BJA − 1 4 δA BJC,

[¯ L˙

a ˙ b, J˙ c] = −δ ˙ a ˙ c J˙ b + 1 2 δ ˙ a ˙ b J˙ c,

[¯ L˙

a ˙ b, J˙ c] = δ ˙ c ˙ bJ˙ a − 1 2δ ˙ a ˙ b J˙ c.

{QaA, ¯ Q˙

a B} = δA BPa ˙ a,

{SaA, ¯ SB

˙ a } = δB AKa ˙ a,

[Pa ˙

a, SbA] = δa b ¯

Q˙

a A,

[Ka ˙

a, QbA] = δb a ¯

SA

˙ a ,

[Pa ˙

a, ¯

SA

˙ b ] = δ ˙ a ˙ b QaA,

[Ka ˙

a, ¯

Q

˙ b A] = δ ˙ b ˙ aSaA,

[Ka ˙

a, Pb˙ b] = δ ˙ b ˙ aLba + δb a¯

L

˙ b ˙ a + δb aδ ˙ b ˙ aD,

{QaA, SbB} = δA

BLab − δa b RAB + 1 2 δa b δA BD,

{ ¯ Q˙

a A, ¯

SB

˙ b } = δB A ¯

L˙

a ˙ b + δ ˙ a ˙ b RBA + 1 2 δ ˙ a ˙ b δB AD,

[D, P] = +P, [D, K] = −K, [D, Q] = + 1

2 Q,

[D, ¯ Q] = + 1

2 ¯

Q, [D, S] = − 1

2 S,

[D, ¯ S] = − 1

2 ¯

S.

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 16 / 36

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a := ∂/∂¯

λ ˙

a.

Transformation of the Amplitude Realized through sum over action on the individual legs, J =

J ,

JAn =

n

k=1

JkAn =

n

k=1

n 1 2 3 n − 1 ... . . . . .. . . . k An J

Invariance Are amplitudes invariant, JAn

?

= 0 , J ∈ psu(2, 2|4) ?

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 17 / 36

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Tree-Level Invariance?

MHV Superamplitude: A

MHV,tree

n

= δ4(P)δ8(Q) 122334 · · · n1 , QaA :=

n

k=1

λa

kηA k .

Invariance [

Witten hep-th/0312171]

◮ L, ¯

L, R: Index contractions.

◮ D: Correct weight in λ. ◮ P, Q: Manifest due to δ(P), δ(Q). ◮ K, ¯

Q: Due to K ∼ {S, ¯ S}, ¯ Q ∼ [P, S].

◮ S: Not trivial, but can be shown. ◮ ¯

S: Due to holomorphy. Holomorphy Amplitude mostly depends on λ’s. ¯ λ only in P =

k λk¯

λk and ¯ SB

˙ a δ4(P) =

k

ηB

k

∂ ∂¯ λ ˙

a k

δ4(P) = QbB ∂δ4(P) ∂P b ˙

a

⇒ ¯ SB

˙ a δ4(P)δ8(Q) = 0 .

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 18 / 36

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Conformal Anomaly

Invariance? Just seen: Tree-level MHV-amplitudes seem invariant under free generators. Conformal Anomaly There is an anomalous contribution: [

Cachazo Svrcek, Witten]

∂ ∂¯ z 1 z = πδ2(z) ⇒ ∂ ∂¯ λ ˙

a k

1 λk, λj = πδ2(λk, λj)ε ˙

a˙ b¯

λ

˙ b j .

Contributes when 0 = kj = λa

kεabλb j, i.e. when λk ∼ λj ⇔ pk, pj collinear.

multiple collinear particles with the same total momentum.

◮ Possible miscounting of states in naive Fock-space construction.

A Minor Issue?

◮ Anomaly only occurs at singular momentum configurations. ◮ Can be avoided at tree level. ◮ But at loops, they will show up in integrals.

Better understand anomalous contributions completely!

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 19 / 36

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Anomalous Contribution

Compute the anomalous contribution to the superconformal symmetry generator ¯ S by acting on the MHV amplitude: [ TB, Beisert, Galleas

Loebbert, McLoughlin]

( ¯ S0)B

˙ a AMHV n

=

n

k=1

ηB

k

∂ ∂¯ λ ˙

a k

δ4(P)δ8(Q) 1223 . . . n1 = −π

n

k=1

ε ˙

a˙ b

¯

λ

˙ b kηb k+1 − ¯

λ

˙ b k+1ηb k

δ2(λk, λk+1) δ4(P) δ8(Q)

12 . . . k − 1, kk, k + 1k + 1, k + 2 . . . n1 .

Due to the δ2(λk, λk+1) in the numerator, the gap in the denominator closes: λk−1, λkλk+1, λk+2 ∼ λk−1, λk,k+1λk,k+1, λk+2 . The remainder contains AMHV

n−1 !

Correction terms Can conformal symmetry be made exact by correcting the free generators? If yes, need to relate amplitudes with different numbers of legs. ⇒ Need formalism!

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 20 / 36

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Generating Functional

Generating functional Collect amplitudes in a generating functional A[J] [Arkani-Hamed, Cachazo

Cheung, Kaplan

] with sources J(Λk), Λk = {λk, ¯ λk, ηk} for individual particles k: A[J] =

∞

n=4

gn−2 n

n

k=1

dΛk

An(Λ1, . . . , Λn) Tr

J(Λ1) . . . J(Λn) = g2 4

A4

J J J J

+ g3 5

A5

J J J J J

+ g4 6

A6

J J J J J J

+ g5 7

A7

J J J J J J J

+ . . . . Functional Form of the Generators Generators act on sources. Sign from partial integration. Local form: ( ¯ S0)B

˙ a = ηB ¯

∂ ˙

a

⇒ Functional form: ( ¯ S0)B

˙ a J(Λ) = −ηB ¯

∂ ˙

aJ(Λ)

¯ S0 =

¯ S0

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 21 / 36

SLIDE 27

Correction for ¯ S

Acting with ¯ S0 on the generating functional gives ( ¯ S0)B

˙ a AMHV n

[J] = −2π2

dΛ12

n

k=3

dΛk

d4η′dα ε ˙

a ˙ c¯

λ ˙

c 1ηB 2 ·

· AMHV

n−1 (Λ12, Λ3, . . . , Λn) Tr

[ ˆ J(Λ1), ˆ J(Λ2)]J(Λ3) . . . J(Λn) with the collinear particles Λ1, Λ2 given by λ1 = λ12 sin α, η1 = η12 sin α + η′ cos α, λ2 = λ12 cos α, η2 = η12 cos α − η′ sin α . Correction Can compensate by correction ¯ S+ that turns one source J(Λ12) into two sources J(Λ1), J(Λ2): ¯ S+ =

¯ S+

, ( ¯ S+)B

˙ a J(Λ12) = 2π2

d4η′dα ε ˙

a ˙ c¯

λ ˙

c 1ηB 2 [ ˆ

J(Λ1), ˆ J(Λ2)] .

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 22 / 36

SLIDE 28

Invariance of MHV Amplitudes

Correction for ¯ S: ¯ S =

¯ S0 + g ¯ S+

⇒ 0 =

An

¯ S0

+

An−1

¯ S+

Correction for K: K must also receive correction: K ∼ {S, ¯ S}: K =

K0

+ g

K+

⇒ 0 =

An

K0

+

An−1

K+ November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 23 / 36

SLIDE 29

Invariance of MHV Amplitudes

# particles n helicity h J0 J+ A4,2 A5,2 A5,3 A6,2 A6,3 A6,4 A7,2 A7,3 A7,4 A7,5 . . . . . . . . . . . . . . .

◮ MHV amplitudes are invariant when J = ¯

S, K are corrected: ¯ SAMHV = 0 , KAMHV = 0 .

◮ ¯

S+ and K+ increase helicity by one.

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 24 / 36

SLIDE 30

Correction for S

MHV amplitudes completely invariant. How about MHV? More corrections? [

Drummond, Henn Korchemsky, Sokatchev]

AMHV

n

= δ4(P) [12] . . . [n1]

d8ω

n

i=1

δ4(ηi − ¯ λ ˙

a i ω ˙ a) ,

[jk] := ¯ λ ˙

a j ε ˙ a˙ b¯

λ

˙ b .

This time, anomalous contribution from SaB = ∂ ∂λa ∂ ∂ηB . As for MHV, correction at collinear momenta: (S0)aBJ(Λ) = ∂a∂BJ(Λ) , S0 =

S0 ,

(S−)aBJ(Λ12) = π2

d4η′dαδ(η′)εacλc∂′

B[ ˆ

J(Λ1), ˆ J(Λ2)] , S− =

S−

Induces for K ∼ {S, ¯ S}: K =

K0

+ g

K+

+ g

K−

+ g2

K+−

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 25 / 36

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Invariance of MHV, MHV Amplitudes

# particles n helicity h J0 J+ J− A4,2 A5,2 A5,3 A6,2 A6,3 A6,4 A7,2 A7,3 A7,4 A7,5 . . . . . . . . . . . . . . .

◮ J+, J− increases/deacreases helicity by one. ◮ Invariance enhanced to:

¯ SAMHV = 0 , SAMHV = 0 , KAMHV = 0 , KAMHV = 0 .

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 26 / 36

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Invariance for all Tree-Level Amplitudes

BCF(W) Relations For analyzing all tree-level amplitudes, use BCFW recursion relations which relate collinear limits of generic tree-level amplitudes. [

Britto Cachazo, Feng][Britto, Cachazo Feng, Witten ]

Invariance [ TB, Beisert, Galleas

Loebbert, McLoughlin]

◮ Can show inductively: Corrections obtained from MHV, MHV amplitudes

indeed cancel all conformal anomalies of all tree-level amplitudes.

◮ Singularities removed from all tree-level amplitudes, invariance restored:

# particles n helicity h J0 J+ J− J+− A4,2 A5,2 A5,3 A6,2 A6,3 A6,4 . . . . . . . . . . . .

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 27 / 36

SLIDE 33

All Tree-Level Corrections

Corrections Necessary to correct generators ¯ S, S and K: [ TB, Beisert, Galleas

Loebbert, McLoughlin]

S =

S0 + g S− ,

¯ S =

¯ S0 + g ¯ S+ ,

K = {S, ¯ S} =

K0 + g K−

+ g

K+

Have shown that generating functional Atree is invariant under deformed symmetry generators. Does deformed algebra close? [ TB, Beisert, Galleas

Loebbert, McLoughlin]

◮ [L, . . .], [¯

L, . . .], [R, . . .]: Index contractions.

◮ [D, . . .]: Correct conformal weights. ◮ { ¯

S, Q}, {S, ¯ Q}: Almost trivial.

◮ {S, Q}, { ¯

S, ¯ Q}: Antisymmetric integral.

◮ {S, S}, { ¯

S, ¯ S}: Non-trivial, closes only onto gauge transformations: G[X]J(Λ) = [X, J(Λ)] , {SaA, SbB} = εabG[∂A∂BJ(0)] . Gauge transformations annihilate the amplitude functional.

◮ {SA, ¯

SB} = δA

BK defines K. ◮ [Pab, . . .] and [K ˙ a˙ b, . . .] close onto gauge transformations.

Superconformal algebra is satisfied, but contains gauge transformations.

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 29 / 36

SLIDE 35

Yangian Invariance

Free Case [ Drummond

Henn, Plefka]

◮ Tree level amplitudes are invariant under Yangian up to anomalies

(at singularities).

◮ Level-One Yangian Generators:

J(1)

α An = 1 2f βγ α n

j,k=1

j<k

n − 1 n 1 2 . . . . . . . .. . . . . . . . . . j Jβ k Jγ An

. Corrections

◮ P(1) only depends on undeformed generators

P, L, ¯ L, Q, ¯ Q, D ⇒ No corrections for P(1).

◮ Other Yangian generators: Commutators of P(1) with level-zero

generators.

◮ ⇒ Corrections inherited.

Invariance Invariance of tree amplitudes under P(1) known. ⇒ All tree amplitudes invariant under corrected Yangian!

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 30 / 36

SLIDE 36

How Strong is Symmetry?

Amplitudes as Symmetry Invariants

◮ Amplitudes are linear combinations

[

Drummond, Henn Korchemsky, Sokatchev][ Brandhuber Heslop, Travaglini][ Drummond Henn, Plefka]

f “naive” Yangian invariants (invariants up to singularities).

◮ Correct linear combination determined

[Korchemsky

Sokatchev ]

by prescribed singularities.

◮ Corrections: Collinear residues packaged into generators. ◮ All other “spurious” singularities should be absent!

Symmetry uniquely determines tree-level amplitudes!

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 31 / 36

SLIDE 37

Conclusions

Exact Symmetry for N = 4 Tree Amplitudes

◮ Free superconformal symmetry violated for collinear momenta. ◮ Symmetry can be made exact by corrections. ◮ Correction relates amplitudes with different numbers of legs. ◮ Scattering amplitudes are compatible with conformal symmetry!

Invariants of Exacted Symmetry

◮ Free symmetry constrains amplitudes to combination of invariants. ◮ Coefficients must be fixed by structure of singularities. ◮ Corrected symmetry uniquely determines all tree-level amplitudes!

Loops

◮ Theory supposedly integrable. ◮ Can symmetry determine both divergent and finite part to all orders? ◮ Loop level: Careful treatment of tree-level necessary preliminary!

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 32 / 36

SLIDE 38

Comparison to Trace Operators

Amplitudes Trace Operators Both

A

∼

Φ Ψ Φ Φ Φ Φ Ψ Φ Φ Φ Φ Φ

O

◮ Definite ordering. ◮ Cyclic. ◮ Symmetry acts as

homogeneous sum: δ4(P)δ8(Q) 1223 · · · n1 Tr ΦΦΨΦ · · · ΨΨΦ J =

n

k=1

Match possible generator corrections with appropriate amplitudes. Collect powers of coupling g and number of legs n: 0 =

◮ Analysis uses only corrections described above,

no other corrections necessary.

◮ Action of corrections at loop level problematic. ◮ ⇒ Off-shell formalism: Small mass for each particle. ◮ Compute “Off-shell S-Matrix”.

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 36 / 36

SLIDE 42

End

Thanks for listening!

Tack!

November 3, 2009 Till Bargheer: Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory 36 / 36

Symmetries of Scattering Amplitudes in N = 4 Super-Yang-Mills Theory

Till Bargheer

Uppsala University November 3, 2009 arXiv:0905.3738

Outline

Contents

Why N = 4 Super Yang-Mills Theory

N = 4 SYM as a mathematical toy model (of more realistic theories) whose structure we can eventually understand.

Features of N = 4 SYM Theory

Symmetry

Duality

Integrability in the Planar Nc → ∞ Limit

anomalous dimensions of local gauge invariant operators.

Cusp anomalous dimension! Realistic hopes that the planar theory can be solved to all orders!

Scattering Amplitudes in Conformal Theories

Conformal: Scattering amplitudes are not well-defined in conformal theories: There are ∞-ranged interactions (massless gauge bosons)

Regulate: In order to make sense of scattering processes, need to introduce IR-regulator (D = 4 − 2ε) ⇒ Breaks conformal symmetry.

1 2 3 4 5 n . . . . An

Scattering amplitudes only make sense in the regulated theory.

Contents

Color Ordering

Color Structure

Decomposition In U(Nc), SU(Nc) gauge theory, product of k traces comes with factor 1/N k

⇒ In the planar (large-Nc) limit, only single-trace terms contribute. ⇒ Amplitudes can be color-decomposed: An({pi, ai}) ∼

Tr(T aσ(1) · · · T aσ(n)) ˆ An(pσ(i)) . Color-stripped partial amplitudes ˆ An

From now on, consider only the color-stripped amplitudes ˆ An!

Spinor-Helicity Variables

Particles One-particle states specified by: Momentum pµ, Polarization vector εµ. Gluons G± Fermions ψA/ ¯ ψA Scalars φAB Helicity h ±1 ±1/2 Momentum Spinors [

pa˙

⇒ pµpµ = − det(pa˙

⇒ Can write pa˙

λ

Since pa˙

λ = ±¯ λ: pa˙

λ

Polarization

λ → eiφλ , ¯ λ → e−iφ¯ λ .

Trade (pµ, εµ) for (λa, ¯ λ ˙

Efficient variables for amplitudes and symmetries!

The Simplest Tree Amplitudes

Gluon Amplitudes: hk = ±1

ˆ A++++...+

= 0 (supersymmetry) [Grisaru, Pendleton,

ˆ A+...+−+...+

= 0 (supersymmetry) ˆ A+...+−+...+−+...+

MHV (Maximal Helicity Violating) ˆ A+...+−+...+−+...+−+...+

Next-to-MHV · · · MHV Gluon Amplitudes Extremely simple form at tree level! [Parke

ˆ A

∼ δ4(P)jk4 122334 · · · n1 , Total momentum: P =

λk¯ λk, Lorentz invariants: jk := λa

Other Tree-Level Amplitudes [

Tree-level amplitudes can be constructed recursively: BCFW relations.

Loop Amplitudes

Loop Technology

[

Planar four-leg amplitude to four loops, [Bern, Dixon

Six-leg amplitude at two loops. [Bern, Dixon, Kosower, Roiban

decay into collinear particles. ⇒ IR divergences when integrating over collinear momenta. ⇒ Amplitudes divergent as ε → 0, typically 1/ε2 at each loop order. Exponentiation

[

AAll−loop = Adivergent(ε) · Afinite Symmetries (conformal) of remainder might be broken by regulator.

Loop Amplitudes: BDS Conjecture

Observation in the planar limit Nc → ∞: [Anastasiou, Bern

Higher loops of 4-gluon amplitude have iterative structure:

BDS Conjecture: Finite part of n-gluon amplitude vanishes to all loops: An(p) ∼ Atree

(p) exp

Depends only on tree-level Atree

, one-loop M (1)

and cusp dimension Dcusp(λ). Cusp Dimension

[

[Beisert, Eden

] Correctness BDS conjecture falsified for n ≥ 6. . . . [

Amplitudes in the AdS/CFT Dual

AdS/CFT: Equates N = 4 SYM and type IIB strings on AdS5 × S5. [Maldacena

Gluon Scattering [

− →

AdS/CFT: Polygon is Wilson loop in N = 4 SYM! String amplitude ≡ VEV of Wilson loop. Computation of the minimal surface agrees with the BDS conjecture, taking the cusp anomalous dimension (computed from integrability) at strong coupling.

Dual Superconformal Symmetry

Dual Superconformal Symmetry [

Introduce dual coordinates: xk+1 − xk ≡ pk = λk¯ λk , xn+1 ≡ x1 .

⇒ Invariance under dual translations.

conformal transformations in xk!