1. Super Yang Mills Scattering amplitudes at strong coupling Juan Maldacena Based on L. Alday & JM arXiv:0705.0303 [hep-th] & to appear Strings 2007, Madrid Length of talk: 30 minutes

2. Goal Compute planar, color ordered, gluon scattering amplitudes at strong coupling using strings on AdS 5 xS 5Compute planar, color ordered, gluon scattering amplitudes at strong coupling using strings on AdS 5 xS 5 A = X P T r [ T a 1 ¢¢¢ T a n ] A n ( k 1 ; ¢¢¢ ; k n )

4. 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 IR finite observables  Jet observables.They are building blocks for closely related IR finite observables  Jet observables. There is a conjecture for the all order form of these amplitudes (in the MHV case):There is a conjecture for the all order form of these amplitudes (in the MHV case): Bern, Dixon, Smirnov 2005 Also … Anastasiou, Bern, Dixon, Kosower; Catani A = A T ree ex p f f ( ¸ ) a ( s ; t ; ¹ )g (more precise version later…)

5. Final Answer The scattering process is described by a classical string worldsheet in AdS A » exp f ¡ R 2 2 ¼® ( A rea )g( 1 ) A » exp f ¡ p ¸ S c l ( s ; t ; ¹ )g,

6. Two IR regulators 1 st regulator: Brane in the IR  for motivation1 st regulator: Brane in the IR  for motivation 2 nd regulator: Dimensional regularization  for computations2 nd regulator: Dimensional regularization  for computations

7. 1 st IR regulator D-brane in the bulkD-brane in the bulk Z IR Z=0(boundary of AdS) of AdS) high energy scattering in the bulk  classical string solution Gross Mende ( see also Polchinski Strassler) d s 2 = d x 2 z 2 p proper = zp Scatter with fixed gauge theory energy  proper energy is very large 

8. T-dual AdS space It is convenient to introduce the T-dual coordinates We get again AdS but boundary and IR are exchanged Strings wind in y by an amount proportional to the momentum of the gluon. y represents momentum space. (Kallosh-Tseytlin) d s 2 = d x 2 + d z 2 z 2 d y = ¤ d x z 2 ; r = 1 z d s 2 = d y 2 + d r 2 r 2

9. Final Prescription Gluon momenta, define sequence of light like segments on the boundary. Two consecutive vertices are separated by the momentum k i. Sequence given by color ordering. Formally similar to the computation of light-like Wilson loops This is all in the T-dual coordinates. We have temporarily removed the IR regulator k 1 k 2 k n

10. Leading order answer is independent of the polarization of the gluons.Leading order answer is independent of the polarization of the gluons. The dependence on the polarizations should appear at the next order.The dependence on the polarizations should appear at the next order. Same for MHV and non-MHVSame for MHV and non-MHV

11. Symmetries The classical problem is invariant under SO(2,4) of the “momentum space” or T- dual coordinates.The classical problem is invariant under SO(2,4) of the “momentum space” or T- dual coordinates. Is not a symmetry of the full theory (there is a non-invariant dilaton in the T-dual AdS).Is not a symmetry of the full theory (there is a non-invariant dilaton in the T-dual AdS). Was observed at weak coupling in the planar limit.Was observed at weak coupling in the planar limit. Drummond, Henn, Smirnov, Sokatchev

12. Finding the worldsheet -Start with the cusp in poincare coordinates - Apply 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 segments - (The n gluon solution would require more work or ingenuity) or ingenuity) - Area is divergent…. Kruczenski (for s=t) ¡ Y 2 ¡ 1 ¡ Y 2 0 + Y 2 1 + Y 2 2 + Y 2 3 + Y 2 4 = ¡ 1 Y 0 Y ¡ 1 = Y 1 Y 2 ; Y 3 = Y 4 = 0 s t

13. 2 nd Regulator: Dimensional regularization Amplitudes are usually defined in the gauge theory using dimensional regularization to dimensions 4 + ² Dimensional reduction from 10d SYM to dimensions Dimensional reduction from 10d SYM to dimensions 4 + ² p = 3 + ² d s 2 = f ¡ 1 = 2 d x 2 4 + ² + f 1 = 2 ( d r 2 + d ­ 2 5 ¡ ² ) f = c p g 2 p N 1 r 4 ¡ ² where Use the metric for a D-p-brane with in 10 dim.

14. We then find g 2 p » g 2 ¹ ¡ ² Introducing a scale i S = d i v + p ¸ 4 ¼ ( l og s t ) 2 + cons t an t

15. IR Divergence S j,j+1 We get one per “pizza slice”: We get one per “pizza slice”: They suppress the amplitude  it is very improbably not to emit other gluons Cusp anomalous dimension ¢ ¡ S = f ( ¸ ) l og S + ¢¢¢ ( ) µ ¸ d d¸ ¶ 2 F ( ¸ ) = f ( ¸ ) ; µ ¸ d d¸ ¶ G ( ¸ ) = g ( ¸ ) Sudakov, A. Sen Korchemsky, Catani, Sterman, Magnea, Tejeda Yeomans Bern, Dixon, Smirnov D i v = n X j = 1 i S d i v ( s j ; j + 1 ) i S d i v ( s ) = ¡ 1 2 ² 2 F µ ¸ ( ¡ s ) ² = 2 ¹ ² ¶ + 1 2 ² G µ ¸ ( ¡ s ) ² = 2 ¹ ² ¶ jj+1 They are characterized by two functions of the coupling

16. Why this form? Replace gluons by Wilson lines Configuration invariant under two non compact symmetries: - Boosts - Boosts - Scale transformations - Scale transformations Both are cutoff in the UV direction by the momenta or s. Each gives a factor of log. A » h W i » exp f ¡ f ( ¸ ) ¢ ¿ ¢ Â g » exp f ¡ f ( ¸ )( l og¹ 2 = s ) 2 g

17. The second function, g(λ), characterizes the subleading IR divergencies It was computed to 3 loops at weak coupling It also seems to obey a “maximal transcendentality” principle We can compute it at strong coupling and we find µ ¸ d d¸ ¶ 2 F ( ¸ ) = f ( ¸ ) ; µ ¸ d d¸ ¶ G ( ¸ ) = g ( ¸ ) i S d i v = ¡ 1 2 ² 2 F µ ¸ ( ¡ s ) ² = 2 ¹ ² ¶ + 1 2 ² G µ ¸ ( ¡ s ) ² = 2 ¹ ² ¶ Bern, Dixon, Smirnov g ( ¸ ) » p ¸ 2 ¼ ( 1 ¡ l og 2 )

18. The Bern-Dixon-Smirnov ansatz Agrees with our answer once we use the strong coupling Agrees with our answer once we use the strong coupling form for f(λ). form for f(λ). 4 gluon amplitude is determined by momentum space conformal invariance and the form of the IR divergencies. For n gluons BDS propose an explicit but more complicated function of the kinematic invariants. Gubser, Klebanov, Polyakov ; Kruczenski Beisert, Eden, Staudacher 4 gluons: l og A A T ree = d i v + f ( ¸ )( l og s t ) 2

19. 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.

20. 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. Future