Holographic Mellin Amplitudes in Various Dimensions Xinan Zhou C. - PowerPoint PPT Presentation

Holographic Mellin Amplitudes in Various Dimensions Xinan Zhou C. N. Yang Institute for Theoretical Physics Stony Brook University Based on PRL 118, (2017) 091602, JHEP 1804 (2018) 014, arXiv:1712. 02788 (with L. Rastelli), and arXiv:1712.02800, arXiv:1804.02397 Workshop on higher-point correlation functions and integrable AdS/CFT Trinity College Dublin April 18, 2018 Xinan Zhou YITP , Stony Brook University

The story of gluon scattering in flat spacetime n-point process 3 4 5 6 7 … 1 3 10 38 154 … # of diagrams (cyclic ordered) Maximally Helicity Violating (MHV) Parke-Taylor Formula: h ij i 4 A n [1 + . . . i − . . . j − . . . n + ] = h 12 ih 23 i . . . h n 1 i Complicated ways to write zeros! Amplitudes can be “ bootstrapped ”: - dimensional analysis - Lorenz invariance - locality - etc. Xinan Zhou YITP , Stony Brook University

Holographic correlators Holographic correlators = on-shell scattering amplitudes in a maximally symmetric spacetime We expect nice properties too: - Flat space limit. - SYM should have some AdS 5 × S 5 ↔ N = 4 hidden simplicity. Scattering in AdS Xinan Zhou YITP , Stony Brook University

Holographic correlators Infinitely many “particles”: Kaluza-Klein modes from the n-sphere. 1/2 BPS operators: ✏ = d O I 1 ...I k 2 − 1 ∆ = ✏ k k k-fold symmetric-traceless representation of SO(n+1). m 2 = ∆ ( ∆ − d ) Dual to scalar fields with . The analogue of S-matrix: h O k 1 ( x 1 ) . . . O k n ( x n ) i Scattering in AdS Xinan Zhou YITP , Stony Brook University

Four-point functions The standard algorithm: Witten diagram expansion. n=2,3 is boring, the form is determined by symmetry. Starting from n=4, the dependence on coordinates becomes non-trivial. At sub-leading order in 1/N, X h O k 1 ( x 1 ) . . . O k 4 ( x 4 ) i = + G ∆ i External legs: bulk-to-boundary propagators B ∂ ( x i , Z ) G ∆ , ` Internal legs: bulk-to-bulk propagators BB ( Z, W ) Vertices: expand the e ff ective Lagrangian. Integrate over the AdS. Xinan Zhou YITP , Stony Brook University

Four-point functions Contact diagrams: D ∆ 1 ∆ 2 ∆ 3 ∆ 4 = Z dZ G ∆ 1 B ∂ ( x 1 , Z ) G ∆ 2 B ∂ ( x 2 , Z ) G ∆ 3 B ∂ ( x 3 , Z ) G ∆ 4 = B ∂ ( x 4 , Z ) AdS d +1 is the scalar one-loop box diagram in four dimensions. D 1111 Exchange diagrams: when the quantum numbers satisfy special relations, an exchange diagram can be expressed as a finite sum of contact Witten diagrams [ D’Hoker Freedman Rastelli ] e.g., condition in the s-channel or ∆ − ` = ∆ 3 + ∆ 4 − 2 m 0 ∆ − ` = ∆ 1 + ∆ 2 − 2 m m 0 for positive integers and . m Xinan Zhou YITP , Stony Brook University

Four-point functions X h O k 1 ( x 1 ) . . . O k 4 ( x 4 ) i = + A total nightmare to implement! - Exchange diagrams may or may not simplify. The diagrams proliferates quickly with increased external weights. - Obtaining the vertices are extremely complicated ( the 15-page results of Arutynov and Florov for AdS5 quartic vertices ). Only a handful of explicit 1/2 BPS correlators over the last 20 years: - AdS5: k=2,3,4 [ Arutyunov Frolov, Arutyunov Dolan Osborn Sokatchev, Arutyunov Sokatchev ] , ( higher k conjecture) [ Dolan Nirchl Osborn ], near-extremal : (n+k, n-k, k+2, k+2) [ Berdichevsky Naaijkens, Uruchurtu ] . - AdS7: k=2 (stress-tensor multiplet) [ Arutynov Sokatchev ] . - AdS4, AdS6: none. What is the organizational principle? Where is the hidden simplicity?? Xinan Zhou YITP , Stony Brook University

Ingredient 1: Mellin representation Mellin representation of conformal correlators [ Mack, Penedones ] Z Y ij ) − δ ij M [ δ ij ] ( x 2 G conn ( x i ) = [ d δ ij ] i<j The integration variables are constrained by n X δ ij = δ ji , δ ii = − ∆ i , δ ij = 0 j =1 M [ δ ij ] is called the reduced Mellin amplitude. Consider OPE ∆ i + ∆ j − ∆ k ⇣ ⌘ X c k ( x 2 O i ( x i ) O j ( x j ) = ij ) − O k ( x k ) + descendants 2 ij k Then should have simple poles at M [ δ ij ] δ ij = ∆ i + ∆ j − ( ∆ k + 2 n ) n ≥ 0 , 2 ⌧ k = ∆ k − ` k Xinan Zhou YITP , Stony Brook University

Mellin representation The constraints are solved by introducing auxiliary “momentum” variables δ ij = p i p j They satisfies “momentum conservation” and “on-shell” condition X p 2 i = − ∆ i p i = 0 , i Let us take n=4, there are two independent variables (“Mandelstam variables”) s = ∆ 1 + ∆ 2 − 2 δ 12 , t = ∆ 1 + ∆ 4 − 2 δ 14 , u = ∆ 1 + ∆ 3 − 2 δ 13 with the constraint . s + t + u = ∆ 1 + ∆ 2 + ∆ 3 + ∆ 4 In the s-channel OPE limit (1,2 come close), a primary operator leads to poles at s = ⌧ O + 2 m , ⌧ O = ∆ O − ` m ≥ 0 , Similar statements for the t and u channels. Xinan Zhou YITP , Stony Brook University

Mellin representation Further define the Mellin amplitude [ Mack ] M Y M ( δ ij ) = M ( δ ij ) Γ ( δ ij ) i<j Two benefits: - The Gamma factors are such that has polynomial residues for M conformal blocks. - At large N, the Gamma functions precisely account for the double-trace operators [ Penedones ] δ 12 = ∆ 1 + ∆ 2 − s For example, has poles that correspond to (recall ) Γ ( δ 12 ) 2 O ∆ 1 ∂ ` ⇤ n O ∆ 2 τ = ∆ 1 + ∆ 2 + 2 n + O (1 /N 2 ) with twist . At large N, the Mellin amplitude is meromorphic, with simple poles corresponding to single-trace operators. Xinan Zhou YITP , Stony Brook University

Witten diagrams in Mellin space Contact diagrams: for a vertex with 2k derivatives, the Mellin amplitude is a degree-k polynomial [ Penedones ] Exchange diagrams (s-channel): simple analytic structure ∞ Q ` ,m ( t ) X M ( s, t ) = s − τ − 2 m + P ` − 1 ( s, t ) m =0 Same pole and same residue as a conformal block with the same quantum number! A remarkable simplification: when , the infinite series truncate τ + 2 m 0 = 2 ∆ into a finite sum. Equivalent to the previous truncation in position space. It has a very natural explanation in Mellin space. Xinan Zhou YITP , Stony Brook University

Witten diagrams in Mellin space Z i ∞ 1 2 − ∆ M ( s, t ) Γ 2 [2 ∆ − s ] Γ 2 [2 ∆ − t ] Γ 2 [2 ∆ − u ds dt s t 2 V G ( x i ) = ] 2 U x 2 ∆ 12 x 2 ∆ 2 2 2 2 − i ∞ 34 U = ( x 12 ) 2 ( x 34 ) 2 V = ( x 14 ) 2 ( x 23 ) 2 ( x 13 ) 2 ( x 24 ) 2 , ( x 13 ) 2 ( x 24 ) 2 The truncation of poles must happen in order to be consistent with the 1/N expansion: - Exchanging an operator contributes ∆ − ` 2 g coll ∆ , ` ( V ) + . . . U If there is a small anomalous dimension U τ / 2 → U τ / 2 + 1 2 γ U τ / 2 log( U ) + 1 8 γ 2 U τ / 2 log 2 ( U ) + . . . ∆ → ∆ + γ - Let’s see how this is reproduced in the inverse Mellin transformation: is produced by a double pole, is produced by a triple pole, log( U ) 2 log( U ) etc. Xinan Zhou YITP , Stony Brook University

Witten diagrams in Mellin space Z i ∞ 1 2 − ∆ M ( s, t ) Γ 2 [2 ∆ − s ] Γ 2 [2 ∆ − t ] Γ 2 [2 ∆ − u ds dt s t 2 V G ( x i ) = ] 2 U x 2 ∆ 12 x 2 ∆ 2 2 2 2 − i ∞ 34 On the other hand, using the counting of 4d N=4 SYM, the tree-level Witten diagrams are of order , and the anomalous dimension is also of O (1 /N 2 ) order . O (1 /N 2 ) Because is multiplied by , only n=1 is allowed at tree level. In γ n log n ( U ) terms of the Mellin representation, this means at most double poles are allowed in the integrand. The truncation must occur: the Gamma functions also contains an infinite series of double poles; if the simple poles in overlaps with these M ( s, t ) double poles, they have to terminate. Xinan Zhou YITP , Stony Brook University