Clustering in the N=4 SYM three point function D. Serban w/ Y. - PowerPoint PPT Presentation

Clustering in the N=4 SYM three point function D. Serban w/ Y. Jiang, S. Komatsu, I. Kostov, arXiv:1604.03575

Overview • Introduction: AdS/CFT integrability • The three point function proposal by [Basso, Komatsu, Vieira, 15] • Simple example of clustering: the tree level for one non-BPS operator • Virtual excitations in the bottom channel at strong coupling • Repackaging as a generalised Fredholm determinant • Conclusion and outlook

AdS/CFT integrability Weak-strong coupling duality between N =4 SYM in 4d and string theory in AdS 5 x S 5 background [Maldacena; Witten; Gubser, Klebanov, Polyakov; 98] One loop dilatation operator String sigma model = integrable spin chain classically integrable [Minahan, Zarembo, 02] [Bena, Polchinski, Roiban, 02] � ⌘ � � Z = Φ 1 + i Φ 2 e.g. X = Φ 3 + i Φ 4 string solution Tr ZZZXXZZZXXXZXZZZZ . . . [Kazakov, Marshakov, Minahan, Zarembo, 04] finite-gap solution of the classical sigma solution in terms of Bethe Ansatz equations model [Beisert, Staudacher, 03] Solution for the spectrum in the planar limit at any value of λ through integrability

The three point function in N=4 SYM 1 basic building block; 3 dual to three interacting strings X 2 O C 123 ( � ) h O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) i = | x 12 | ∆ 1 + ∆ 2 − ∆ 3 | x 13 | ∆ 1 + ∆ 3 − ∆ 2 | x 23 | ∆ 2 + ∆ 3 − ∆ 1 11 initial data: three states with definite conformal dimensions and psu(2,2|4) charges | � | | � | O α ( x ) , α = 1 , 2 , 3 } each characterised by a set of rapidities u α X and polarizations (or global rotations with respect to some reference BPS state, e.g Tr Z L ) g α = e ζ A α J A

The three point function at weak coupling At tree level the three point function can be computed using gaussian contraction pure combinatorics 1 h ¯ Z ( x ) Z ( y ) i ⇠ | x � y | 2 1 � h ¯ � � X ( x ) X ( y ) i ⇠ | x � y | 2 Spin chain language : the combinatorics can be expressed in terms of scalar products of states of (pieces of) spin chain [Roiban, Volovich, 04] [Escobedo, Gromov, Sever, Vieira, 10] use Algebraic Bethe Ansatz (ABA) to build and cut the chains into pieces “tailoring” of spin chains

The three point function at weak coupling In some special cases, at weak coupling, the correlators allow determinant representations [Foda, 11] whose semiclassical limit is rather straightforward [ESV, 11; Kostov,12; Kostov, Bettelheim, 14] I du e ip (1) ( u )+ ip (2) ( u ) − ip (3) ( u ) � � log C 123 ( g ) ' 2 π Li 2 C (12 | 3) 3 I Z du dz e ip (3) ( u )+ ip (1) ( u ) − ip (2) ( u ) � e 2 ip ( a ) ( z ) � X � 1 � � + 2 π Li 2 2 π Li 2 2 a =1 C (13 | 2) C ( a ) general structure in agreement with the strong coupling result [Kazama, Komatsu, (Nishimura), 13 & 16] Generically, the three point function can be written as sum-over-partitions expressions coming from tailoring or from the recent proposal by [Basso, Komatsu, Vieira, 15] Resuming the sum over partitions for large numbers of magnons and taking the semiclassical limit is a major open problem for the three point function

BKV hexagon proposal [Basso, Komatsu, Vieira, 15] the asymptotic part of the three point function can be written as a sum over partitions for the three groups of rapidities s, u 1 = α 1 ∪ ¯ α 1 , u 2 = α 2 ∪ ¯ α 2 , u 3 = α 3 ∪ ¯ α 3 1 1 � � 2 3 2 3 � � , u 2 , u 3 s, u 1 3 123 ] asympt = ( − 1) | ↵ 1 | + | ↵ 2 | + | ↵ 3 | w ` 31 ( α 1 , ¯ X Y [ C ••• α 1 ) w ` 12 ( α 2 , ¯ α 2 ) w ` 23 ( α 3 , ¯ α 3 ) ↵ i ∪ ¯ ↵ i = u i i =1 H( α 1 | α 3 | α 2 )H(¯ α 2 | ¯ α 3 | ¯ α 1 ) . X Y × ∪ explicit Ansatz for the hexagon amplitudes obeying form-factor-like axioms H( α 1 | α 3 | α 2 )H(

Mirror particle contribution [Basso, Komatsu, Vieira, 15] the asymptotic contribution is “dressed” by the mirror (virtual) particles (one non-BPS state shown) physical excitations bottom mirror excitations Z 1 [ C • �� ] bottom = d w µ ( w � ) e ip ( w γ ) ` B T ( w � ) h 6 = ( w � , w � ) h ( u , w � 3 � ) , e.g. bottom channel: �1 sum over the rapidities w and polarisations of the mirror particles emitted by one hexagon and absorbed by the other

Weak coupling version of the hexagon generic su(2) three point function: six vertex partition function on a lattice with a conical defect methods based on Sklyanin’s Separation of Variables integral representation [Jiang, Komatsu, Kostov, DS, 15] no straightforward way to take the semiclassical limit

Asymptotic/tree level contribution examples shown for su(2), one non-BPS case 1 123 ] asympt ⌘ A = X ↵ | Y Y [ C • �� ( � 1) | ¯ e ip ( u j ) ` R h ( u k , u j ) ↵ [ ¯ ↵ = u j 2 ↵ j 2 ↵ ,k 2 ¯ ↵ u − v 1 h ( u, v ) su (2) = hexagon 2-body amplitude: u − v + i ✏ s ( u, v ) � ( u, v ) tree level value BES dressing phase symmetric part L 1 Y Y Y Y Y Y L 1 L L l + = - 2 R 3 _ tree level: Izergin determinant Y _ L Y 3 L 1 L L 3 Y l + = - L 2 [Escobedo, Gromov, Sever, Vieira, 10-11] L l B L L + - = 2 1 3 [Kostov, 12] [Kostov, Matsuo, 13] _ _ _ _ _ Y Y Y Y Y [Kostov, Bettelheim, 14] L 2

� ' p λ g = 4 π Integral representation is ✏ ∼ 1 /L 1 N n n 1 1 dz j I X ( � 1) | ↵ | Y Y e � ip ( u j )) ` A = h ( u j , u k ) , X Y Y ] = = 2 ⇡✏ F ( z j ) h ( z j , z k ) h ( z k , z j ) n ! C u ↵ [ ¯ ↵ = u j 2 ↵ j 2 ↵ ,k 2 ¯ ↵ n =0 j =1 j<k sum over partitions coupled contour integrals with: F ( z ) = e � ip ( z ) ` µ ( z ) N (1 � 1 /x + x � ) 2 → Y µ ( z ) = , h ( z, u ) ⌘ h ( z, u j ) (1 � 1 / ( x + ) 2 )(1 � 1 / ( x � ) 2 ) h ( z, u ) q j =1 u 2 − (2 g ✏ ) 2 u + x ( u ) = 2 g ✏ integrals are coupled by the factors ∆ all ( z j , z k ) ≡ h ( z j , z k ) h ( z k , z j ) = 1 − c ( u, v, g ✏ ) 2 ✏ 2 + O ( ✏ 3 ) ( u − v ) 2 ∆ all ( u, v ) = ∆ ( u, v ) � � u ) = ∆ ( u, v ) = ( u − v ) 2 + ✏ 2 tree-level contribution

Clustering is ✏ ∼ 1 /L 1 N n n 1 I dz j X Y Y A = 2 ⇡✏ F ( z j ) ∆ ( z j , z k ) , n ! C u n =0 j =1 j<k manipulate the integrals by separating the contours catch the poles of i ✏ / 2 i ✏ / 2 ∆ ( u, v ) = 1 + u − v + i ✏ . u − v − i ✏ − this gives rise to clustered integrals, e.g. [Moore, Nekrasov, Shatashvili, 98] I I dz 1 dz 2 I I dz 1 dz 2 (2 ⇡✏ ) 2 F ( z 1 ) F ( z 2 ) ∆ ( z 1 , z 2 ) = (2 ⇡✏ ) 2 F ( z 1 ) F ( z 2 ) ∆ ( z 1 , z 2 ) [Borodin, Corwin, 11] C u C u C 1 C 2 dz 2 I [Bourgine, 14] [Menegelli, Yang, 14] + (2 ⇡✏ ) F ( z 2 ) F ( z 2 + i ✏ ) ( C 2 [Basso, Sever, Vieira, 13], … cluster of length 2

Combinatorics of clusters is ✏ ∼ 1 /L 1 n k n F q j ( z j ) I dz j X X Y Y C q 1 , ··· ,q k result of clustering: I n = ∆ q i ,q j ( z i , z j ) . n 2 ⇡✏ q j C j q 1 + ··· q k = n j =1 i<j k =1 F n ( z ) = F ( z ) F ( z + i ✏ ) · · · F ( z + ( n − 1) i ✏ ) ✓ ◆ mn i ✏ i ✏ ∆ mn ( u, v ) = 1 − u − v + im ✏ + m + n u − v − in ✏ 1 n ! n ! C q 1 ,...,q k = = l l d l d l ! . is the number of clusters of lengths n Q d 1 ! d 2 ! · · · q 1 · · · q k d q 1 ≤ · · · ≤ q k ( z ) is the wavefu sfy q 1 + · · · + q k = n . the clusters lengths can be alternatively described by the number of clusters of each type · · · { , · · · } 7! ~ { q 1 , · · · , q k } = { 1 · · · 1 , · · · , l, · · · , l d = { d 1 , d 2 , · · · } . | {z } | {z } d 1 d l

Exact result and semiclassics is ✏ ∼ 1 /L 1 Q k k F q j ( z j ) 1 I dz j exact result: X X Y Y A = ∆ q i ,q j ( z i , z j ) A q 2 k ! 2 ⇡✏ C j j q 1 ,... ,q k j =1 k i<j F n ( z ) = F ( z ) n + ✏ n ( n � 1) semiclassical expansion: F ( z ) n − 1 @ z F ( z ) + O ( ✏ 2 ) ( 2 mn ( z i � z j ) 2 ✏ 2 + O ( ✏ 3 ) . ∆ mn ( z i , z j ) = 1 � the leading term exponentiates: k F ( z j ) q j F ( z ) q 1 dz j dz dz I I I X Y X X A A ' = exp = exp 2 ⇡✏ Li 2 [ F ( z )] q 2 k ! 2 ⇡✏ 2 ⇡✏ q 2 C j C u C u j j =1 q j q k subleading terms easily computable: I 2 ⇡✏ Li 2 [ F ( z )] � 1 I dzdz 0 log [1 � F ( z )] log [1 � F ( z 0 )] dz log A = + . . . . (2 ⇡ ) 2 ( z � z 0 ) 2 2 C × 2 C u u

Mirror excitations mirror contribution given by similar integrals, but now all bound states of a constituents contribute Q B[ ~ n ] [ C • �� ] bottom = X Q a n a ! ~ n Z 1 n a dz a j Y Y Y Y H � n ] = ( − 1) n 2 ⇡✏ µ � a ( z a j ) g � a ( z a j ) T � a ( z a H � aa ( z a i , z a ab ( z a i , z b B[ ~ j ) × j ) j ) �1 a j =1 a a<b 1  i<j  n a 1  i  n a 1  j  n b interaction with measure su(2|2) transfer matrix physical particles mirror (virtual particle) dynamics: u � x [+ a ] ! 1 /x [+ a ] x [ − a ] ! x [ − a ] | | [- a ] x cut physical regime x [ a ] = x ( u + ia ✏ / 2) x ± = x ( u ± i ✏ / 2) -2 g' 2 g' mirror regime [+ a ] x cut