Hexagons and 3pt functions

Transcription

1 Hexagons and 3pt functions Benjamin Basso ENS Paris Current Themes in Holography NBI Copenhagen 2016 based on work with Vasco Goncalves, Shota Komatsu and Pedro Vieira

2 The spectral problem is solved O(x) colour bubble O(0) O(x) O(0) = 1 x 2 All N=4 SYM planar 2pt functions are any value of t Hooft coupling Fundamental working assumption : Mixing problem for local (single trace) operator is equivalent to an integrable spin chain problem O =tr L = Scaling dimension String energy Spin chain energy

3 O(x) 2pt function history 2009 O(0) 2005 Thermodynamic Bethe Ansatz Symmetries Beisert S-matrix Beisert-Staudacher Asymptotic Bethe Ansatz Quantum Spectral Curve BMN Vacuum Spin Chain Picture QCD Story Perturbative Integrability Full set of equations : see Final word (?) Two Years Ago [Gromov,Kazakov,Leurent,Volin 14] It leads to a wealth of amazing results / predictions for the gauge / string theory

4 Perturbative predictions Example : Scaling dimension of shortest unprotected operator (so-called Konishi) = lightest massive string state O tr DZDZ Comments : - Z.. stand for single valued multiple zeta values - Could get more loops if needed [Marboe,Volin 14]

5 Exploring non-perturbative territories Example : Scaling dimension of twist two operator for complex spin = leading Regge trajectory O tr ZD S Z BFKL Pomeron branch (high energy forward scattering) DGLAP branch (DIS like set up) Plot of real part of the spin S as a function of the scaling dimension for t Hooft coupling = 6.3 [Gromov,Levkovich-Maslyuk,Sizov 15]

6 Precision test of the gauge / gravity interpolation Example : Pomeron (= Reggeized graviton) intercept strong coupling string prediction graviton has j = 2 weak coupling gauge prediction Pomeron has j = 1 [Gromov,Levkovich-Maslyuk,Sizov 15]

7 Towards solving planar N=4 SYM theory O(x) 2pt functions Solved [Many people here] O(0) 3pt functions Wanted O 1 (x 1 ) [Here also] O 2 (x 2 ) O 3 (x 3 ) Looking forward... Two cross ratios Use OPE / conformal bootstrap or better?... and farther Can we also understand handles = string loops?

8 Plan / Goal / Question Can we find structure constants of single trace operators at finite coupling in planar N=4 SYM theory? ho 1 (x 1 )O 2 (x 2 )O 3 (x 3 )i = C 123 x12 12x 23 23x O 1 (x 1 ) S-matrix TBA O 2 (x 2 ) O 3 (x 3 ) Spin Chain QCD Story Bethe Ansatz QSC Can we follow a similar path for structure constants?

9 Plan / Goal / Question Can we find structure constants of single trace operators at finite coupling in planar N=4 SYM theory? ho 1 (x 1 )O 2 (x 2 )O 3 (x 3 )i = C 123 x12 12x 23 23x O 1 (x 1 ) More recently SFT/Spin vertex O 2 (x 2 ) O 3 (x 3 ) Hexagons Spin chain tayloring 2010 Yes we can!... but it will take time... In this talk I will show you how one can start off using the hexagon bootstrap program

10 2-pt functions S-matrix Spin Chain Bethe Ansatz Focus on long spin chain states

11 Bethe States 0) Pick BMN (= ferromagnetic) vacuum with very big length L 0) Add magnons O tr...zy Z...ZY Z... 1) Write Bethe wave function p 3 p 3 p 2 p 1 p 1 +S(p 1,p 2 ) +... p 2 2) Imposing periodicity conditions gives the Bethe ansatz equations : (i.e. quantization conditions for the magnon momenta) e ip il Y j6=i S(p i,p j )=1 3) Get the energies : E = X i E(p i )

12 Asymptotic solution I It s a cutting procedure of sort : cut open here (off shell edge very far away) Complicated mess S Sort of dilute gas approximation : Zoo of interactions reduces to 2-by-2 elastic scattering events Geodesics to asymptotic solution : Magnon S-matrix Thanks to integrability : This description is correct up to exponentially small in system length (so called wrapping) corrections [Ambjorn,Janik,Kristjansen 05][Bajnok,Janik 08] e L E O(g 2L )

13 Power of symmetry Way to go? Let the symmetries do the job Residual symmetry group of BMN (ferro) vacuum : PSU(2 2) PSU(2 2) n R 3 Left Right [Beisert 05] Central extensions : contain energy (and coupling constant) Each magnon transforms in bi-fundamental irrep p 1 p Left Right Dispersion relation r E = g 2 sin 2 p 2 (Dimension = 16 = 8 bosons + 8 fermions)

14 Power of symmetry Way to go? Let the symmetries do the job Residual symmetry group of BMN (ferro) vacuum : [Beisert 05] Symmetry fixes S-matrix (up to overall scalar factor) S 12 S 0 12 S 12 S 12 p 1 p 2 X Fulfills Yang-Baxter equation = X Scalar factor constrained by crossing symmetry [Janik 05]

15 More recently O 1 (x 1 ) SFT/Spin vertex Spin chain tayloring 2010 Hexagons O 2 (x 2 ) O 3 (x 3 ) From 3-pt functions to hexagons ho 1 (x 1 )O 2 (x 2 )O 3 (x 3 )i = C 123 x12 12x 23 23x 13 13

16 Gauge / String definition O 1 (x 1 ) C 123 O 2 (x 2 ) O 3 (x 3 ) 3-punctured sphere pair of pants ending on 3 spin chains at the boundary

17 Topology : - 3 operators - 3 bridges l 12 Spin chain tayloring O 1 L 1 l 13 = L 1 + L 3 L 2 2 [Many people, see e.g. Escobedo,Gromov,Sever,Vieira, Foda,Fleury,Caetano, Kazama,Komatsu,Nishimura, Jiang,Kostov,Petrovskii,Serban, etc.] L 2 L 3 O 2 O 3 l 23 Example : 3 BPS states (= 3 spin chain vacua) - Contract scalar fields as indicated above - Count number of inequivalent Wick contractions - Normalize by norms C 123 = p L1 L 2 L 3 N

18 Spin chain tayloring O 1 L 1 [Many people, see e.g. Escobedo,Gromov,Sever,Vieira, Foda,Fleury,Caetano, Kazama,Komatsu,Nishimura, Jiang,Kostov,Petrovskii,Serban, etc.] l 12 l 13 = L 1 + L 3 L 2 2 L 2 L 3 O 2 O 3 l 23 More complicated : 3 non-bps states - Same as before - But split each spin chain C tree 123 = H i = H (a) i p L1 L 2 L 3 P p H (b) i (a) 3 (b) 1 Nq 1 1 (a) 1 (b) (a) 2 (b) 3

19 Spin chain tayloring O 1 L 1 [Many people, see e.g. Escobedo,Gromov,Sever,Vieira, Foda,Fleury,Caetano, Kazama,Komatsu,Nishimura, Jiang,Kostov,Petrovskii,Serban, etc.] l 12 l 13 = L 1 + L 3 L 2 2 L 2 L 3 O 2 O 3 l 23 Recipe : Cut spin chain states and compute their overlap following the Wick contractions Use integrability to evaluate partial wave function overlaps How to go to higher loops? Hard... spin chain wave functions are unknown, as well as corrections to splitting vertex

20 Cutting /asymptotic procedure - Start with pair of pants - Cut open 3 times - Get 2 hexagons Cut Open Here 1 pair of pants = 2 hexagons Here Bottom line : For large operators the 3pt function factorizes into 2 disjoint hexagons And Here

21 Cutting /asymptotic procedure Hexagon factorization with magnons u 2 u 1 u 2 + e ip 2` u 1 u 1 u 2 u 1 + u 2 S(u 1,u 2 )e ip 1` + e i(p 1+p 2 )`

22 Hexagon factorization [BB,Komatsu,Vieira 15] - 3pt function = sum of products of 2 hexagons - Leftover information about spin chain state is in the sum over bipartite partition of Bethe roots - Elementary block = hexagon form factor u 1 u 2...u N Amplitude for creating magnons on the edges of an hexagon h A 1Ȧ1,...,A N Ȧ N (u 1,...,u N )=hh A 1 Ȧ 1 1 A N Ȧ N N i 1 0i 2 0i 3 Apply integrable bootstrap to determine it at finite coupling

23 Use super-symmetry 3pt function = (BMN)^3 O 2 =tr Z(1) L 2 2 BMN vacua + 1 twisted BMN vacuum Z = Z + Z + Y (for overlap ops 1 and 2) Residual symmetry : (for BPS condition) Ȳ O(3) O(3) fix a line in spacetime fix three (real) scalars out of six + 8 Supercharges : Q a + ab S b O 1 =trz(0) L 1 O 3 =tr Z(1) L 3 part of family of twisted correlators (and dotted version) see [Drukker,Plefka 09] Total symmetry of hexagon: PSU(2 2) = diagonal subgroup of PSU(2 2) PSU(2 2) Left Right

24 Power of symmetry Two-magnon hexagon form factor fixed matrix part L R SU(2 2) 2 excitation = h 12 S scalar part SU(2 2) Beisert S-matrix up to a scalar factor h A 1Ȧ1,A 2 Ȧ 2 =( 1) f 1 f 2 h 12 Ȧ 2 2 Ȧ 1 1 S 12 A 1 1 A 2 2

25 N-magnon form factor Conjecture for N-magnon form factor : L R = NY h(u i,u j ) i<j SU(2 2) Beisert S-matrix S h A 1Ȧ1 A N Ȧ N =( 1) f N Y i<j h ij ȦN N... Ȧ 1 1 S A A N N

26 Bootstrap for scalar factor Hexagon as a branch point twist field (with conical excess) Two main axioms : [Cardy,Castro-Alvaredo,Doyon 07] I. Watson equation : one can permute magnons using S-matrix h 12 /h 21 = S 0 12 = x+ 1 x 2 x 1 x /x 1 x /x + 1 x 2 II. Decoupling/crossing equation : a pair of a magnon and antimagnon with zero net charges and energy must decouple (also known as kinematical pole condition) One main solution (not unique) : [BB,Komatsu,Vieira 15] h 12 = x 1 x 2 x 1 x + 2 h(u 2 1,u 2)h(u 1,u 2 )= x 1 x 2 x 1 x /x 1 x /x + 1 x /x + 1 x 2 1 1/x + 1 x+ 2 (same as Janik s crossing equation)

27 Gluing hexagons into 3-pt functions

28 Asymptotic formula Consider 2 BPS operators and 1 non-bps operator, e.g. Non-BPS i.e. BPS BPS ho 1 O 2 O 3 i = C 123 tensor x12 12x 23 23x e.g. O 1 =trd S Z L 1 with the rest BPS

29 Asymptotic formula Hexagon prediction : first factor has to do with normalization of spin chain state (i.e. conversion factor from infinite to finite volume normalization) see [Pozsgay,Takacs 08] C = C 123 ui j Q S k=1 µ(u k) Qi<j S(u i,u j ) A2 Hexagon part A = Y i<j h(u i,u j ) sum over partitions of Bethe Roots X [ =u( 1) Y j2 e ip j` Y i2,j2 1 h(u i,u j ) Valid to all loops asymptotically (large enough operators)

30 C 123 C Comparison with data from [Eden,Heslop,Korchemsky,Sokatchev 11] [Eden 12] [Chicherin,Drummond,Heslop,Sokatchev 14] Non-BPS BPS BPS Non-BPS 2-loop mismatch BPS BPS Perfect agreement between asymptotic hexagon description and data up to zeta s

31 Full solution? 20?? Recently Thermodynamical hexagons? SFT/Spin vertex Asymptotic hexagons Quantum Hexagonal Curve? Tayloring 20?? 2010 A complete solution must include finite size corrections (wrapping effect) because spin chains have finite lengths

32 Beyond asymptotic description Include finite size effects = so-called wrapping effects [Ambjorn,Janik,Kristjansen 05] [Bajnok,Janik 08] asymptotic + wrapping +... more mirror particles exchanged vacuum (in mirror = double Wick rotated theory) virtual effect : exchange of 1 particle in mirror channel wrapping corrections : e L E O(g 2L ) (Resummation of all finite size corrections leads to TBA eqs and Quantum Spectral Curve)

33 Finite size effects for 3pt functions O 1 New virtual effects : Exchange of mirror particles between the two hexagons O 2 O 3 These new virtual effects come from the 3 mirror channels (= where we cut)

34 Finite size effects at weak coupling O 1 L 1 New virtual effects : Exchange of mirror particles between the two hexagons l 12 l 13 = L 1 + L 3 L 2 2 L 2 L 3 O 2 O 3 O 2 l 23 O 1 O 1 Here a gluon is passing through the bridge O 3 Virtual effects suppressed with bridge size O(g 2`ij )

35 Mirror effects using hexagons First finite size effect Corrections coming from exchange of a single particle in the three mirror channels Asymptotic = vacuum contribution A!A+ A 12 + A 23 + A 31 Integral over momentum of exchanged particle A = X a>1 Z du 2 µ a(u) 1 x [+a] x [ a] ` u bridge length int a (u {u i }) int includes hexagon interaction between exchanged mirror particle and magnons on spin chain

36 Examples Short (length two or three) operators O 1 =Tr(D S Z 2 )+... O 1 =Tr(D S Z 2 )+... O(g 6 ) O(g 8 ) O(g 6 ) O(g 6 ) O(g 8 ) O(g 6 ) O 2 =Tr( ZY ) O(g 4 ) O 3 =Tr(Ȳ Z) O 2 =Tr( ZY Y ) O(g 6 ) O 3 =Tr(Ȳ Ȳ Z) New 2-loop virtual effect Similar configuration with a slightly bigger bridge delays the new virtual effect to 3-loops Conclusion : the asymptotic result is the same for both - but it is valid up to 1-loop on the left - and up to 2-loop on the right

37 3 loop match At tree level and one loop, the asymptotic result is not corrected At two loops the first single particle virtual corrections in the opposing channel kicks in with finite size effect included we get a perfect match with known results up to 3 loops! At three loops the first single particle virtual corrections in neighboring channels kicks in [Eden,Sfondrini 15], [BB,Goncalves,Komatsu,Vieira 15] [Eden 12], [Chicherin,Drummond,Heslop,Sokatchev 15]

38 Conclusions Integrability comes with powerful new strategies for computing quantities at any value of the coupling in planar N=4 SYM theory It allows us to attack increasingly complicated objects and find allloop expressions (conjectures) for them, like for amplitudes, structure constants, etc. Here we presented a strategy for structure constants : - cut open pair of pants into hexagons - glue hexagons back together in the end

39 Summary hexagon picture 3-pt function = finite volume correlator of two hexagons identify C 123 = Z X H H H H (momentum of) mirror particles where we glue partitions of physical rapidities identify Elementary patch = hexagon form factor (can be found using an integrable bootstrap) magnons in mirror channels H magnons on spin chain 1 and 3

40 THANK YOU!