The SL(2) sector of at strong coupling

The SL(2) sector of at strong coupling Ivan Kostov IPhT-Saclay with

Didina Serban and Dmytro Volin arXiv:hep-th/0703031, arXiv:0801.2542

GGI, Florence, 30 November 2008

integral of L= 2 in [33]) π 4 πthe g density (a representation discovered based on thefor exact integrability, for the first time we are at th # # solution of a gauge theory in four dimensions, 2 coupling results for perturbative, nt. Both the weak coupling and the strong f (g) = (ρ − ρ)du, (3.6) 2 0 a non-trivial be reproduced from the conjectured logBethe M ansatz fequations. (g) = string theory (ρ0in− ρ)du, curved background. the integral equation, written down by Eden and Staudacher log M Spinning Strin

Integral equation for the sl(2) sector The sl(2) sector of PSU(2,2|4)

(2/π) M is the valueisofknown the density K(u, v) log determined in asymptotic [12], this equation as the at infinity (more strictly when caling function BES/FRS Integral equation forstrictly the s Excitations the sl(2) sector: S) scaling function is given by theall3higher ρefficient = in (2/π) log M is the asymptotic value of the density at infinity (more "equation. Mwhere ). AnThe recursive procedure to obtain orders was constructed 0 universal 2

Many examples investigated:

2 for 2 tion discovered L2=reproduced 2 inM [33]) result of [44] was using a linearized formprocedure of the BA equations 1+g " u " ). An efficient recursive to obtaininall[37]. higher orders was const # Lorentz spin 3.1 Universal scaling function Twist on for 2L ∼ log M given is was reproduced resultbyof[49] [44] using afolded linearized form of the BA equations in g) = in [44].(ρ0The − ρ)du, (3.6) Classical ! " log M Theissl(2) sector is spanned on the states Gubserstrings propagating The expression for1L ∼ log M# given by [49] Klebanov! " Mdensity (g) = L at infinity L+ 2 strictly (ρ0 −when ρ)du in!.AdS x #S1 (3.7) ymptotic value of f the (more M L 30 tr"D+ Z +Polyakov’02 ... , D+ = D0 log M + 1 ecursive procedure to+obtain all higher orders was constructed 1 p folded circular f (g) = in [37]. obtained L + 2by acting (ρ0 − ρ)du . oduced using a linearized form of the BA equations with M covariant derivatives D act in al + log M nBethe by [49] is equations scalar fields Z. Here L is a su(4) R-charge and M is a Loren particular ansatz for a spinning strin ! theAnsatz " +loop: [XXX]-½A spin # Bethe equations: At one chain ations in sl(2) sector are picture, L is the length of the chain and M is the magnon num 1 ations have root singularity at u = ±2g. expansion in g L + 2a square (ρ0 − ρ)du . (3.7) The perturbative When M → ∞ withmotion L fixed, the dimension In • uniform in anomalous AdS-time: WS en log! M " $On the contrary, % $ in the large%g2 limit they become essential. these singularities. We M − + + − + L # u − uj 17 this quantity logarithmically scaling) [21,341 − 1/x xj x • scales uniform rotation (Sudakov in 12-plane and

anned on the states ! " tr D Z + . . . ,

D =D +D ,

th M covariant derivatives D act in all possible ways on the L com The R-charge Bethe equations L is3.2 a su(4) and M is a Lorentz spin. In the magnetic spin c k k ,u k Dressing so and that the positions singularities do) at notudepend on(3.10) g:Thephase he rapidity ofThe thevariable chain M theof the magnon number. = e2 i θ(u equations have aissquare root = ±2g. perturbative expansio − BA + − − + singularity xk u − u 1 − 1/x x • stretched in 12/34 plane: profile ψ(σ ∆ = M+ L + f (g, L) ln M + j j k k j"see =k these singularities.  does not On the contrary, in the large g limit they become essentia 1 u old ithsingularity L fixed, anomalous In the limit of(3.8)large Lorentz spi ot at u = the ±2g. %The ≡ perturbative , u expansion = dimension . in g where g is the constant, related thedepend ’t Hooft cos coupl normalize variable so2gthat We the positions ofcoupling the singularities do to not on gψ the contrary, in the the largerapidity g limit4g they become essential.  garithmically (Sudakov scaling) hat the positions of the singularities do not depend on g: [21, 17, 18, 29, 19, 20, 22]  cos ψ k

j

2 & 2 τ )g= N ukowsky” defined by ±variable x(u) is ± ± YM  t(σ, τ ) = ετ, X(σ, 1 u old u = u ± i!, x = x(u ) (3.11) g = . sin ψ ± ± ± 1 uuold = u ± i!, 2  % ≡ , u = . x = x(u ) (3.11) $ & 16 π %4g %≡ , u= (3.8) !. " 2g 4g 2g 1 1 g factor. In the∆ sl(2) sector there are+ no f bound states and M the1rapidities = M + L (g, L) ln + . . . sin ψ u(x) ≡ x + , x(u) = u 1 + . (3.9) 1 − 1 dressing factor. the sl(2) there no bound and the rapidities eal axis. The mode magnons forare theis ground states, is defined by the 2In numbers x forsector u2states Then “Jukowsky” variable x(u) defined by when Large M limit: & n the mode numbers for magnons for the ground states, I/III, when " real axis. $The % AdS/CFT$ Integrability Niklas Beisert & % ! " 1 1 ng constant, related to the ’t Hooft coupling constant λ by ations in the sl(2) sector are Beisert-Eden-Staudacher’06 en by (BES) (L finite ) + 1 , x(u) = u 1 + 1 − 2 . 1 2, ..., ± 12(3.9) 1M . 1 for u k = ±1 ± (3.12) +x2 (L − 3) sgn(k) u(x) x ++ − &Freyhult-Rej-Staudacher’07 ,2 x(u) = u 1 + 1 − . (L~Log M) $ &≡ $ ! +1 "L ' (FRS) 2 M − + 1 2 x u utok n−1 −n 1− ujfor − 3) sgn(k) k21/x = ±1 ..., ±Among M. (3.12) n = numbers k +xk2 (Lcorresponds j ± 2, kx mode holes near n e= 2 i0. σ(u r kare −1 = L−2 k 2,uj ) . = (3.10) g N − which occupy $’universal& $ &u+ −the xholes’ u2−highest1allowed −YM 1/x−mode x+ numbers

Anomalous dimension for large M:

∆ = M + L + f (g, L) ln M + . . .

where g is the coupling constant, related to the ’t Hooft coupling constant λ universal scaling function = cusp anomalous dimension

2 gYM

Korchemsky’89; GKP’02

N g = . 16 π 2 2

1

%"#3# g &0 '"# :,/7.&$4 :,$0'+$'* 3#.+'#) ', '"# J' K,,2' :,/7.&$4 :,$0'+$' λ L9 &0 + 2/$:'&,$ '"# 'JK,,2' :,/7.&$4 ∆ = M + L + f (g, L) ln M + . . . Anomalous dimension for large M:

where g is

2 gYM λ N g = the coupling constant, related2 to=the ’t2 .Hooft 8 πfunction 16 π universal scaling 2

coupling constant ;HI λ

Korchemsky’89;

anomalous 2 N,3 '"# -&$&-+. '%&0' L = =2 cusp &' #8/+.0 '%&:# dimension '"# :/07 ,2 .&4"'O.&=# !&.0 GKP’02 gYM N )&-#$0&,$ 2 +$,-+.,/0

g = . .,,70 @BCGI 2 16 π Provides a critical test of AdS/CFT: N,3 0-+.. L* '"# /$&>#30+. 0:+.&$4 2/$:'&,$ f (g) %+0 :,-7/'#) 7#3'/3L+'&>#.9 &$ '"# 4+/ '"#,39 /7 ', '"# 2,/3'" ,3)#3 &$ g 2 @AC* ADG 1 ! " 73 8 88 Weak coupling π 4 g 6 − 16 π 6 + 4 ζ(3)2 g 8 ± . . . . ;HI f (g) = 8 g 2 − π 2 g 4 + 3 45 630 expansion:

3-loop From perturbative Q$ '"# 0'3&$4 0&)#* '"#guess /$&>#30+. 0:+.&$4 2/$:'&,$ %+0 +.0, :,-7/'#) 2,3 '"# !30' '"3## $,$O'3&> 8 [Moch, Vermasseren, Vogt’04; 4-loop result SYM up to g ,3)#30 @AE* HF* HB* HAG Lipatov at al’04]

[Bern et al’06]

K 1 3 log 2 − + ... , f (g) = 4 g − π 4 π2 g

;HI

%"#3# S= β(2) &0 T+'+.+$J0 :,$0'+$'I U,'" '"# %#+= :,/7.&$4 +$) '"# 0'3,$4 :,/7.&$4 3#0/.'0 '"# /$&>#30+. 0:+.&$4 2/$:'&,$ :+$ L# 3#73,)/:#) 23,- '"# :,$V#:'/3#) U#'"# +$0+'6 #8/+'&, 1$ '"&0 :,$'#('* &' &0 )#'#3-&$#) L9 '"# &$'#43+. #8/+'&,$* %3&''#$ ),%$ L9 W)#$ +$)