Background Minimal surfaces Conclusions and outlook Scattering amplitudes via AdS / CFT Luis Fernando Alday IAS Annual Theory Meeting - Durham - December 2009 arXiv:0705.0303,..., arXiv:0904.0663, L.F.A & J. Maldacena; arXiv:0911.4708, L.F.A, D. Gaiotto & J. Maldacena Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Minimal surfaces Conclusions and outlook Motivations We will be interested in gluon scattering amplitudes of planar N = 4 super Yang-Mills. Motivation: It can give non trivial information about more realistic theories but is more tractable. Weak coupling: Perturbative computations are easier than in QCD. In the last years a huge technology was developed. The strong coupling regime can be studied, by means of the gauge/string duality, through a weakly coupled string sigma model. Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Minimal surfaces Conclusions and outlook Aim of this project Learn about scattering amplitudes of planar N = 4 super Yang-Mills by means of the AdS / CFT correspondence. Background 1 Gauge theory results String theory set up Explicit example Minimal surfaces 2 Minimal surfaces in AdS 3 Minimal surfaces in AdS 5 Conclusions and outlook 3 Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example Gauge theory amplitudes ( Bern, Dixon and Smirnov, also Anastasiou, Kosower) Focus in gluon scattering amplitudes of N = 4 SYM, with SU ( N ) gauge group with N large, in the color decomposed form ρ Tr ( T a ρ (1) ... T a ρ ( n ) ) A ( L ) A L , Full ∼ � n ( ρ (1) , ..., ρ (2)) n Leading N color ordered n − points amplitude at L loops: A ( L ) n The amplitudes are IR divergent. Dimensional regularization D = 4 − 2 ǫ → A ( L ) n ( ǫ ) = 1 /ǫ 2 L + ... Focus on MHV amplitudes and scale out the tree amplitude n ( ǫ ) = A ( L ) n ( ǫ ) M ( L ) λ L M ( L ) � → M n = n A (0) n L Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example Based on explicit perturbative computations: BDS proposal for all loops MHV amplitudes n � � � � �� λµ 2 ǫ λµ 2 ǫ − 1 − 1 � 8 ǫ 2 f ( − 2) ǫ g ( − 1) + f ( λ ) Fin (1) log M n = n ( k ) s ǫ s ǫ i , i +1 i , i +1 i =1 f ( λ ) , g ( λ ) → cusp/collinear anomalous dimension. Fine for n = 4 , 5, not fine for n > 5. Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example AdS / CFT duality (Maldacena) Four dimensional Type IIB string theory on AdS 5 × S 5 . ⇔ maximally SUSY Yang-Mills √ YM N = R 2 1 � g 2 λ ≡ N ≈ g s α ′ The AdS / CFT duality allows to compute quantities of N = 4 SYM at strong coupling by doing geometrical computations on AdS . Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example Remember a similar problem: Expectation value of Wilson loops at strong coupling (Maldacena, Rey) ds 2 = R 2 dx 2 3+1 + dz 2 z 2 We need to consider the minimal area ending (at z = 0 ) on the Wilson loop. z=0 √ λ 2 π A min � W � ∼ e − Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example Scattering amplitudes can be computed at strong coupling by considering strings on AdS 5 (L.F.A., Maldacena) As in the gauge theory, we need to introduce a regulator. ds 2 = R 2 dx 2 3+1 + dz 2 z 2 Place a D-brane at z = z IR ≫ R . The asymptotic states are open strings ending on the D-brane. Consider the scattering of these open strings (representing the gluons) Need to find the world-sheet representing this process... Z = Z IR Z = 0 Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example The world-sheet is easier to find if we go to a dual space: ˜ AdS (four T − dualities plus z → r = R 2 / z ). AdS → ds 2 = dx 2 3+1 + dz 2 s 2 = dy 2 3+1 + dr 2 → d ˜ z 2 r 2 The problem reduces to a minimal area problem! Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example What is now the boundary of our world-sheet? For each particle with momentum k µ draw a segment ∆ y µ = 2 π k µ Concatenate the segments 2 1 k 4 k 3 according to the particular color 3 4 ordering. k k 2 1 Polygon of light-like edges. Look for the minimal surface 2 r= R / Z IR ending in such polygon. As we have introduced the regulator, the minimal surface ends at r = R 2 / z IR > 0. As z IR → ∞ the boundary of the world-sheet moves to r = 0. Vev of a Wilson-Loop given by a sequence of light-like segments! Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example Prescription √ λ 2 π A min A n ∼ e − A n : Leading exponential behavior of the n − point scattering amplitude. A min ( k µ 1 , k µ 2 , ..., k µ n ): Area of a minimal surface that ends on a sequence of light-like segments on the boundary. Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example Four point amplitude at strong coupling Consider k 1 + k 3 → k 2 + k 4 The simplest case s = t . Y 2 Y 2 Need to find the minimal surface ending on such sequence of light-like segments � Y 0 (1 − y 2 1 )(1 − y 2 r ( y 1 , y 2 ) = 2 ) = y 0 y 1 y 2 Y 1 Y 1 In embedding coordinates ( − Y 2 − 1 − Y 2 0 + Y 2 1 + ... + Y 2 4 = − 1) Y 0 Y − 1 = Y 1 Y 2 , Y 3 = Y 4 = 0 ”Dual” SO (2 , 4) isometries → most general solution ( s � = t ) Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example Let’s compute the area... In order for the area to converge we need to introduce a regulator. Supergravity version of dimensional regularization: consider the near horizon limit of a D (3 − 2 ǫ ) − brane! Regularized supergravity background √ λ D c D � L ǫ =0 � dy 2 D + dr 2 � ds 2 = � → S ǫ = λ D c D r 2+ ǫ r ǫ 2 π The regularized area can be computed and it agrees precisely with the BDS ansatz! Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Gauge theory results Minimal surfaces String theory set up Conclusions and outlook Explicit example What about other cases with n > 4? Dual SO (2 , 4) symmetry constraints the form of the answer (Drummond et. al.) for all n SO (2 , 4) → A strong = A BDS + R ( x ij x kl x ik x jl ) We can construct such cross-ratios for n ≥ 6 so for this case the answer will ( in principle ) differ from BDS. How do we compute the area of minimal surfaces for n ≥ 6? Reduced/baby model: Strings on AdS 3 . Full problem: Strings on AdS 5 Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Minimal surfaces in AdS 3 Minimal surfaces Minimal surfaces in AdS 5 Conclusions and outlook Strings on AdS 3 : The external states live in 2 D , e.g. the cylinder. Consider a zig-zagged Wilson loop of 2 n sides Parametrized by n X + coordinates and n i X − coordinates. i We can build 2 n − 6 invariant cross ratios. ! + X X 2 3 ! + X X 1 4 + ! X X 2 3 1 ! 1 Consider classical strings on AdS 3 . Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Minimal surfaces in AdS 3 Minimal surfaces Minimal surfaces in AdS 5 Conclusions and outlook Strings on AdS 3 Strings on AdS 3 : � Y .� Y = − Y 2 − 1 − Y 2 0 + Y 2 1 + Y 2 2 = − 1 Eoms : ∂ ¯ ∂� Y − ( ∂� Y . ¯ ∂� Y ) � Virasoro : ∂� Y .∂� Y = ¯ ∂� Y . ¯ ∂� Y = 0 , Y = 0 Pohlmeyer kind of reduction → generalized Sinh-Gordon p 2 = ∂ 2 � z ) = log( ∂� ∂� Y .∂ 2 � Y . ¯ α ( z , ¯ Y ) , Y ↓ ∂α − e 2 α + | p ( z ) | 2 e − 2 α = 0 ∂ ¯ p = p ( z ) , α ( z , ¯ z ) and p ( z ) invariant under conformal transformations. e 2 α d 2 z Area of the world sheet: A = � Luis Fernando Alday Scattering amplitudes via AdS / CFT
Background Minimal surfaces in AdS 3 Minimal surfaces Minimal surfaces in AdS 5 Conclusions and outlook Generalized Sinh-Gordon → Strings on AdS 3 ? From α, p construct flat connections B L , R and solve two linear auxiliary problems. ( ∂ + B L ) ψ L a = 0 � e α � ∂α B L z = e − α p ( z ) − ∂α ( ∂ + B R ) ψ R a = 0 ˙ Space-time coordinates � Y − 1 + Y 2 Y 1 − Y 0 � = ψ L a M ψ R a = Y a , ˙ ˙ a Y 1 + Y 0 Y − 1 − Y 2 One can check that Y constructed that way has all the correct properties. Luis Fernando Alday Scattering amplitudes via AdS / CFT
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