Two loop scattering in the near flat sigma-model

J. Minahan, Uppsala Universitet2007 Itzykson Meeting

Based on arXiv:0704.3891

T. Klose, T. McLoughlin, K. Zarembo, J.M.

Outline• Introduction

• The near flat limit of AdS₅xS⁵

• S-matrix in the near flat limit

• σ-model Lagrangian

• Computing mass-shifts to 2 loops

• Computing scattering amplitudes and comparison to BHL results

• Conclusions

Introduction• Exciting results over the last year

BES 0610251 BCDKS 0610248 BHL 0609044

• BHL finds solutions to Janik’s eq. 0603038

consistent with AFS 04060256 and HL 0603204

Consistent at 2 ws loops and higher?

• General AdS₅ x S⁵ world-sheet scattering initiated by KMRZ 0611169 using the light-cone Hamiltonian of FPZ 0603008 AFPZ 0609157

Same structure as the spin-chain 0511082

• One or more loops is computationally hard

• Look for a simplification (and yet is still nontrivial)

• Near flat limit proposed by MS 0612079

• Simplification of crossing eqs. 0603038, 0609044

-- even crossing has rational solution

• Explicitly show that the two loop near flatis consistent with BHL

The near-flat limit (Maldacena & Swanson)

g >> 1 p ∼ g−1/2

(p < 0)ε(p) ≈ −2gp− 14gp

+gp3

6

Rescale: p→ p/2g γ = 1/4gε(p) ≈ −p− 12p

+γ2p3

3

Let: p± =12(ε± p)

⇒ p− = −p p+ =1

4p−−

γ2p3−

12

Spin +2γ++

Single Magnon: ε(p) ≡ E − J =√

1 + 16g2 sin2 p

2g2 =

λ

16π2

Symmetry Algebra

SL(2,R) TransformedBeisert Algebra

Near Flat (after rescaling):

{QI−, QJ

−} = 2δIJp−

{QI−, QJ

+} = γ δIJp−(p− + 2p−l) + SU(2) factors

{QI+, QJ

+} = 2δIJ [p+ + γ2p−(p2−/3 + p−p−l + p2

−l)]

NormalSusy madefrom localcurrents

{QI−, QJ

−} = δIJκ−

{QI−, QJ

+} = δIJκ2 + SU(2) factors

{QI+, QJ

+} = δIJκ+ κ± = ε± κ1

κ1 + iκ2 = −i2geipl(eip − 1)

S-Matrix

x±(p) =1 +

√1 + P 2

Pe±iγp , P =

1γ

sin γp

SU(2|2)xSU(2|2) Structure (Beisert hep-th/0511089)

Zαα, Yaa,Ψαa,ΥaαImpurities:p

p′

p

p′

All terms are linear in γ =π√λ

Near Flat : x± = −1− 1p−± iγ p−

CrossingAntipode (particle antiparticle): x→ 1/x

Near flat: (p−, p+)→ (−p−,−p+)

Janik hep-th/0603038 σ12σ12 = h12, σ12 = eiθ(p,p′)/2

Beisert, Hernandez & Lopez hep-th/0609044

Near flat: heven12 =

√1− 4iγ p′2p2

p′2−p2 + γ2p′2p2

√1 + 4iγ p′2p2

p′2−p2 + γ2p′2p2

Near flat:

χodd(x, y) =

θodd(p, p′) =8γ2

π+O(γ4)

solution to half-flat crossing eq. consistent with even BHL result

eiθeven(p,p′) =1 + iγ p′pp′−p

p′+p

1− iγ p′pp′−pp′+p

e iθodd(p,p′)

1 + γ2 p2−p′2−

(p′−+p−p′−−p−

)2 S ⊗ S

Prefactor only has even powers of γ

e iθodd(p,p′)

1 + γ2 p2−p′2−

(p′−+p−p′−−p−

)2 S ⊗ S

Near flat action Maldacena & Swanson hep-th/0612079

After gauge fixing, imposing constraints and integrating out left-handed fermions :ψ+

• 8 right-handed susy’s for the free action are preserved by interactions

• 8 left-handed susy’s are modified by interactions

ψ ≡ ψ−

• Interactions break 2d Lorentz invariance (Breaking is mild)

• Interactions are quartic, only ∂− derivatives• Fermion propagator is ip−

p2 −m2

1-loop propagator

p >> p = 0

Because of the form of the interactions and fermion prop.only can appear in the numeratork−

∫d2k

(2π)2(k−)n

k2 −m2= 0 by Lorentz inv.if n > 0

Loop integrals with n = 0 all come from interactions with Y 2 − Z2

2-loop propagator

• Integrals are finite due to Lorentz inv. of free action

• Numerator factors appear because of fermion propagatorsor terms in interactions∂−

Useful identity:

✓(m = 1)

I000(p2) p2 = m2

n.b. There are still higher loop corrections

is evaluated at

for the 2-loop mass-shift

1-loop amplitudes

Permutation:

Forward Klose & Zarembo hep-th/0701240 :

Trace:

2-loop amplitudes

DoubleBubble

WineglassInverse

Wineglass

s

t

u

Bubbles

Double log

Absorbtive

2-loop amplitudes

DoubleBubble

WineglassInverse

Wineglass

s

t

u

Wineglasses

Double logs canceled!

First line is precisely what we want

Unwanted piece:

Jacobian FactorWave function ren.

of ext. legs

X X

✓

Conclusions• 2 loop 2→2 agreement with BHL in near flat limit

• Why do the individual 2-loop channels have the same relation to the tree-level amps?

• All loop proof ?

p+ = −γ2p3−/12

1/4p−

• Giant magnons : classical result is 2 loop correction

• 3 →3 scattering and factorization (Giangreco Marotta Puletti, Klose, Ohlsson Sax)

• Extend results to full AdS₅ x S⁵ ?

• Other 2-loop results by Roiban, Tirziu and Tseytlin