Super Yang Mills Scattering amplitudes at strong coupling

Juan Maldacena

Based on L. Alday & JM Based on L. Alday & JM arXiv:0705.0303arXiv:0705.0303 [hep-th]  [hep-th] & to appear& to appear

Strings 2007, MadridStrings 2007, Madrid

Goal• Compute planar, color ordered, gluon Compute planar, color ordered, gluon

scattering amplitudes at strong coupling scattering amplitudes at strong coupling using strings on AdSusing strings on AdS55xSxS55

A =X

P

Tr[Ta1 ¢¢¢Tan ]An(k1;¢¢¢;kn)

• Yes, they are IR divergent. Yes, they are IR divergent.

• We can introduce an IR regulator.We can introduce an IR regulator.

• They are building blocks for closely related They are building blocks for closely related IR finite observables IR finite observables Jet observables. Jet observables.

• There is a conjecture for the all order form There is a conjecture for the all order form of these amplitudes (in the MHV case): of these amplitudes (in the MHV case):

Bern, Dixon, Smirnov 2005Bern, Dixon, Smirnov 2005Also … Anastasiou, Bern, Dixon, Kosower; CataniAlso … Anastasiou, Bern, Dixon, Kosower; Catani

A =AT reeexpf f (¸)a(s;t;¹ )g(more precise version later…)(more precise version later…)

Final AnswerFinal Answer

The scattering process is described by a The scattering process is described by a classical string worldsheet in AdSclassical string worldsheet in AdS

A » expf ¡R2

2¼®(Area)g

A » expf ¡p¸Scl(s;t;¹ )g

,,

Two IR regulatorsTwo IR regulators

• 11stst regulator: Brane in the IR regulator: Brane in the IR for for motivationmotivation

• 22ndnd regulator: Dimensional regularization regulator: Dimensional regularization for computations for computations

11stst IR regulator IR regulator

• D-brane in the bulkD-brane in the bulk

ZZIRIR Z=0Z=0 (boundary(boundary of AdS)of AdS)

high energy scattering in the high energy scattering in the bulk bulk classical string solution classical string solution

Gross MendeGross Mende( see also Polchinski( see also PolchinskiStrassler) Strassler)

ds2 =dx2

z2pproper = zp

Scatter with fixed gaugeScatter with fixed gaugetheory energytheory energy proper energy is very largeproper energy is very large

T-dual AdS space

It is convenient to introduce the T-dual coordinatesIt is convenient to introduce the T-dual coordinates

We get again AdS but boundary and IR are exchangedWe get again AdS but boundary and IR are exchangedStrings wind in Strings wind in yy by an amount proportional to the momentum by an amount proportional to the momentumof the gluon. y represents momentum space. of the gluon. y represents momentum space.

(Kallosh-Tseytlin)(Kallosh-Tseytlin)

ds2 =dx2+dz2

z2

dy= ¤dxz2; r =

1z

ds2 =dy2+dr2

r2

Final PrescriptionFinal Prescription

Gluon momenta, define sequence ofGluon momenta, define sequence oflight like segments on the boundary. light like segments on the boundary. Two consecutive vertices are separatedTwo consecutive vertices are separated

by the momentum kby the momentum kii . .Sequence given by color ordering. Sequence given by color ordering.

Formally similar to the computation Formally similar to the computation of light-like Wilson loopsof light-like Wilson loops

This is all in the T-dual coordinates. This is all in the T-dual coordinates.

We have temporarily removed the IR We have temporarily removed the IR regulatorregulator

k1

k2

kn

• Leading order answer is independent of Leading order answer is independent of the polarization of the gluons. the polarization of the gluons.

• The dependence on the polarizations The dependence on the polarizations should appear at the next order. should appear at the next order.

• Same for MHV and non-MHVSame for MHV and non-MHV

Symmetries

• The classical problem is invariant under The classical problem is invariant under SO(2,4) of the “momentum space” or T-SO(2,4) of the “momentum space” or T-dual coordinates. dual coordinates.

• Is not a symmetry of the full theory (there Is not a symmetry of the full theory (there is a non-invariant dilaton in the T-dual is a non-invariant dilaton in the T-dual AdS).AdS).

• Was observed at weak coupling in the Was observed at weak coupling in the planar limit. planar limit.

Drummond, Henn, Smirnov, SokatchevDrummond, Henn, Smirnov, Sokatchev

Finding the worldsheetFinding the worldsheet-Start with the cusp in poincare coordinatesStart with the cusp in poincare coordinates

- Apply a general conformal transformationApply a general conformal transformation (not the same as as a conformal transformation in the original coordinates )(not the same as as a conformal transformation in the original coordinates )

- Get the solution for four light-like segmentsGet the solution for four light-like segments

- (The n gluon solution would require more work(The n gluon solution would require more work or ingenuity) or ingenuity)

- Area is divergent….Area is divergent….

KruczenskiKruczenski

(for s=t)(for s=t)

¡ Y 2¡ 1 ¡ Y 2

0 +Y 21 +Y 2

2 +Y 23 +Y 2

4 = ¡ 1

Y0Y¡ 1 =Y1Y2 ; Y3 =Y4 =0

ss

tt

22ndnd Regulator: Dimensional regularization Regulator: Dimensional regularization

Amplitudes are usually defined in the gauge theory usingAmplitudes are usually defined in the gauge theory usingdimensional regularization to dimensionsdimensional regularization to dimensions4+²

Dimensional reduction from 10d SYM to dimensionsDimensional reduction from 10d SYM to dimensions4+²

p= 3+²

ds2 = f ¡ 1=2dx24+² +f 1=2(dr2+d­ 25¡ ²)

f = cpg2pN1

r4¡ ²where where

Use the metric for a D-p-brane with in 10 dim. Use the metric for a D-p-brane with in 10 dim.

We then find We then find

g2p » g2¹ ¡ ²Introducing a scaleIntroducing a scale

iS = div+

p¸

4¼(log

st)2+constant

IR DivergenceIR Divergence

SSj,j+1j,j+1

We get one per “pizza slice”: We get one per “pizza slice”:

They suppress the amplitude They suppress the amplitude it is very improbably it is very improbably notnot to emit other gluons to emit other gluons

Cusp anomalous dimensionCusp anomalous dimension¢ ¡ S = f (¸) logS +¢¢¢( )( )

µ¸dd

¶2

F (¸) = f (¸) ;µ¸dd

¶G(¸) = g(¸)

Sudakov, A. SenSudakov, A. SenKorchemsky, Catani, Korchemsky, Catani, Sterman, Magnea, Tejeda YeomansSterman, Magnea, Tejeda YeomansBern, Dixon, SmirnovBern, Dixon, Smirnov

Div=nX

j =1

iSdiv(sj ;j +1)

iSdiv(s) = ¡12²2

Fµ¸(¡ s)²=2

¹ ²

¶+

12²Gµ¸(¡ s)²=2

¹ ²

¶

jjj+1j+1

They are characterized by two functions of the couplingThey are characterized by two functions of the coupling

Why this form?

Replace gluons by Wilson Replace gluons by Wilson lineslines

Configuration invariant under two Configuration invariant under two non compact symmetries:non compact symmetries: - Boosts- Boosts - Scale transformations- Scale transformations

Both are cutoff in the UV direction by the momenta or s. Both are cutoff in the UV direction by the momenta or s.

Each gives a factor of log . Each gives a factor of log .

A » hWi » expf ¡ f (¸)¢ ¿¢Âg» expf ¡ f (¸)(log¹ 2=s)2g

The second function, The second function, g(g(λλ)), characterizes the subleading IR , characterizes the subleading IR divergenciesdivergencies

It was computed to 3 loops at weak coupling It was computed to 3 loops at weak coupling

It also seems to obey a “maximal transcendentality” principleIt also seems to obey a “maximal transcendentality” principle

We can compute it at strong coupling and we findWe can compute it at strong coupling and we find

Bern, Dixon, SmirnovBern, Dixon, Smirnov

g(¸) »p¸

2¼(1¡ log2)

The Bern-Dixon-Smirnov ansatzThe Bern-Dixon-Smirnov ansatz

Agrees with our answer once we use the strong couplingAgrees with our answer once we use the strong coupling form for f(form for f(λλ) .) .

4 gluon amplitude is determined by momentum space 4 gluon amplitude is determined by momentum space conformal invariance and the form of the IR divergencies. conformal invariance and the form of the IR divergencies.

For n gluons BDS propose an explicit but more For n gluons BDS propose an explicit but more complicated function of the kinematic invariants. complicated function of the kinematic invariants.

Gubser, Klebanov, Polyakov ; KruczenskiGubser, Klebanov, Polyakov ; KruczenskiBeisert, Eden, StaudacherBeisert, Eden, Staudacher

4 gluons:4 gluons:

logA

AT ree=div+ f (¸)(log

st)2

Conclusions

• We gave a prescription for computing n gluon scattering amplitudes in AdS

• We checked the Bern Dixon Smirnov ansatz at strong coupling for four gluons.

• We observed the presence of a “momentum space” conformal symmetry.

• We computed the function g(λ) at strong coupling

• We can also consider processes involving local operators n-gluons. Form factors.

• Compute the solutions for n gluons and check the BDS ansatz for n gluons.

• Can the function g(λ) be computed for all values of the coupling using integrability?

• Explicit solutions for form factors. • Understand the crossover from weak to strong coupling

in jet physics.

FutureFuture