Large spin systematics in CFT

Transcription

1 University of Oxford Holography, Strings and Higher Spins at Swansea with A. Bissi and T. Lukowski

2 Introduction The problem In this talk we will consider operators with higher spin: T rϕ µ1 µl ϕ, T rϕϕ µ1 µl T rϕϕ And study their dimension for large values of the spin l. The motivation They form the leading twist sector of many theories. Important role in QCD analysis of deep inelastic scattering. Related to Wilson loops with cusps and UV divergences of scattering amplitudes. Fundamental role in applying integrability to AdS/CF T.

3 Introduction The method The method relies only on symmetries and basic properties of CFT s (conformal symmetry, analiticity, unitarity, structure of the OPE,...) The results will hold for a vast family of theories! Similar in spirit to the conformal bootstrap program.

4 Leading twist operators Generic four dimensional gauge theory: Leading twist operators built from scalars and derivatives O l = T rϕ µ1 µl ϕ + At large values of the spin, they acquire a logarithmic anomalous dimension: l l = f(g) log l + This behavior is valid to all orders in perturbation theory! f(g) appears in many computations! What can we say about sub-leading corrections? l l = f(g) log l + f (0) (g) + f (1) (g, log l) l + f (2) (g, log l) l 2 +

5 Reciprocity Reciprocity principle Odd powers of 1/l are fixed in terms of the even ones! where J 2 0 l l 2 = γ l γ l f(l γ l) f(l) = a 0 (log J 0 ) + a 2(log J 0 ) J 2 0 = l(l + 1). + a 4(log J 0 ) J First observed in QCD [Moch, Vermaseren, Mogt, Gribov, Lipatov, Drell, Levy, Yan, Dokshitzer, Marchesini, Salam...] Then checked for other theories, including MSYM [Basso, Korchemski, Beccaria, Forini, Tirziu, Tseytlin,...] Led to great activity but a proof was still missing!

6 Reciprocity Equivalent formulation Define the full Casimir: Reciprocity principle J 2 = (l + γ l /2)(l + γ l /2 + 1) γ l = α 0 (log J) + 0 J + α 2(log J) J J 3 + α 4(log J) J 4 + Let us prove this for a generic four dimensional CFT!

7 CFT - Symmetries Conformal algebra: Scale transformations dilatation D Poincare Algebra: P µ and M µν Special conformal transformations: K µ [D, P µ ] = P µ, [D, K µ ] = K µ, [K µ, P ν ] = η µν D im µν We may also have flavor symmetries.

8 CFT - Ingredients Main ingredient: Local conformal primary operators: O,l (x), [K µ, O,l ] = 0 Dimension In addition we have descendants P µk...p µi O,l. Operators form an algebra (OPE) Lorentz spin O i (x)o j (0) = k C ijk x k i j O k (0) The set i and C ijk (for all operators) characterizes the CFT and is called the CFT data.

9 CFT - Observables Main observable: Correlation functions of local operators O 1 (x 1 )...O n (x n ) Conformal symmetry + OPE: CFT data all correlation functions! 2 and 3pt-function O i (x 1 )O j (x 2 ) = O i (x 1 )O j (x 2 )O k (x 3 ) = δ ij x 12 2 i C ijk x 12 ij k x13 ik j x 23 jk i

10 OPE factorization φ: Scalar operator of dimension d on a generic CFT, e.g T rϕ 2. Conformal symmetry φ(x 1 )φ(x 2 )φ(x 3 )φ(x 4 ) = G(u, v) x 2d 12 x2d 34, u = x2 12 x2 34 x 2, v = x2 14 x x2 24 x 2 13 x2 24 Crossing relation: v d G(u, v) = u d G(v, u) OPE ( φ(x 1 )φ(x 2 ) = x 2d 1 + ) C φφk x 12 k ij O k (x 1 ) k Identity operator Conformal primaries [K µ, O] = 0 Descendants P µ1...p µn O

11 OPE factorization OPE 4-pt function factorizes! G(u, v) = l 1 3 O l C 12,l C 34,l 2 4 G(u, v) = 1 + C,l 2 G,l(u, v),l 6 Identity operator Conformal primaries Conformal blocks The conformal blocks are explicitly known.

12 Conformal blocks Conformal blocks properties Small u behavior: G,l (u, v) u 1 2 ( l) k,l (1 v) k,l (1 v) = (1 v) l 2F 1 ( 1 2 ( + l), 1 2 ( + l), + l; 1 v) The power of u is controlled by the twist τ = l. There exist a second order operator D such that ( ) D u 1 2 ( l) k,l (v) = J 2 u 1 2 ( l) k,l (v) with J 2 = 1 4 ( + l 2)( + l).

13 Leading twist contribution We can focus in the contribution of leading twist operators by considering the small u limit: G(u, v) = 1 + u τ lead/2 h(v) +... In perturbative CFT they are given by the tower of operators mentioned above! τ l = l l = 2 + γ l G(u, v) = 1 + u h(log u, v) +... a l u 1+γl/2 k l+2+γl,l(v) = u h(log u, v) l

14 Tree level First consider the problem at tree level: γ l = 0 ( ) a (0) 1 l uk l+2,l (1 v) = u v + 1 l=0,2,... We can solve: Γ(l + 1)2 l = 2 Γ(2l + 1) a (0) Each term in the sum diverges logarithmically as v 0. We need an infinite number of terms to reproduce the divergence on the right hand side! The divergence comes from the region l 1. Large l behavior of a (0) l is fixed by the divergence of the r.h.s.

15 Loop level Let us turn on the coupling! τ l = 2 + γ l (g), a l = a (0) l â l (g) G(u, v) = G (0) (u, v) + gg (1) (u, v) +... To any order in perturbation theory: 1 v +1 1 v h(0) (log u, log v)+h (1) (log u, log v)+vh (2) (log u, log v)+ Where only integer powers of v can appear due to analiticity [Dolan-Osborn] a l u γl/2 k l+2+γl,l(v) = 1 v h(0) (log u, log v)+h (1) (log u, log v)+ l

16 Method Consider a l u γl/2 k l+2+γl,l(v) Assume a general large l behavior: l ( ) γ l = p 0 (log J) + p 1(log J) J â l = q 0 (log J) + q 1(log J) J + p 2(log J) J q 2(log J) J 2 + where J 2 = (l + γ l /2)(l + γ l /2 + 1). Compute the divergent terms in ( ) as v 0. Since they come from the large l region, they should be controlled by the above expansion!

17 Method Method to compute the divergent contributions [ L.F.A, Maldacena, Fitzpatrick et. al., Komargodski at. al.] Focus in the small v/large l region: v = ɛ, l = x ɛ 1/2, Perform a change of variables x j: 1 2 l j 2 ɛ = ( x ɛ 1/2 + γ l)( x ɛ 1/2 + γ l + 1) Use integral representation for the hypergeometric function and use saddle point. 0 dx

18 Method ( ) = 1 ɛ ɛ 1/2 +finite q 0 u p 0/2 jk 0 (2j)dj + 0 u p 0/2 (q 0 p 1 log u + 2q 1 q 0 p 0)K 0 (2j)dj Claim: all divergent terms (not only the leading one) are captured! But ( ) = 1 ɛ h(0) (log u, log ɛ) + h (1) (log u, log ɛ) + 0 u p 0/2 (q 0 p 1 log u + 2q 1 q 0 p 0)K 0 (2j)dj = 0

19 Results At any order in perturbation theory p i, q i are polynomials in log j 2 /ɛ. 0 P (log j 2 /ɛ)k 0 (2j)dj = 0 P (log j 2 /ɛ) = 0 p 1 = 0, q 1 = 1 2 q 0p 0 Acting with D multiplies the integrand by j 2 /ɛ and increases the degree of divergence by one (preserving analiticity). We obtain a constraints, involving the higher order terms in the expansions!

20 Results Results 1 γ(l) expanded for large l contains only even powers of 1/J! 2 â(l) 2+γ (l) expanded for large l contains only even powers of 1/J! The results extend to arbitrary dimensions and non-scalar operators! Example: N = 4 SYM γ(l) = 2h 1 (l)λ +... â(l) = h 2 (l)λ +... Both statements can be explicitly checked up to three loops!

21 Generalizations Non-perturbative results Given O of dimension O there are double trace operators O l O of dimension [L.F.A., Maldacena, Fitzpatrick et. al; Komargodski et.al.] l l = 2 O + γ l, γ l = c l τ min + τ min : Twist of the minimal twist operator in O O. Our method applies also to this case! γ l = c ( J τ 1 + p ) 2 min J Extensively checked for the critical O(N) model and theories with holographic dual.

22 Generalizations Non-conformal theories We can consider a theory with non-vanishing beta function. The anomalous dimensions will depend on the scheme. Using DREG with D = 4 2ɛ β ɛ (g) = 2ɛ + β(g) The beta function vanishes at ɛ cr = β(g)/2. Even expansion in terms of the corrected Casimir J 2 β = (l + γ l/2 β/2)(l + γ l /2 β/2 + 1) We have checked this is the case for QCD and N = 0, 1, 2 SYM theories!

23 Conclusions We have derived an infinite number of constraints for the large spin expansion of the anomalous dimension and structure constants of higher spin operators. The derivation relied solely in CFT arguments and symmetries, and applies to a large class of theories. Reciprocity was assumed to make progress in the computation of the dimension of twist-two operators in MSYM. Can use this to make progress in the computation of structure constants? Unexplored prediction for OPE coefficients, can we check it for QCD? It would be interesting to use the full power of crossing-symmetry.