Negative anomalous dimensions in N=4 SYM

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1 13 Nov., 2015, YITP Workshop - Developments in String Theory and Quantum Field Theory Negative anomalous dimensions in N=4 SYM Yusuke Kimura (OIQP) [hep-th] with Ryo Suzuki 1

2 1. Introduction brief review of N=4 SYM 2

3 Anomalous dimensions of N=4 SYM (Conformal Field Theory) Two-point functions O α x O β y = c αδ αβ x y 2Δ α λ = g 2 YM N Δ = Δ (0) + λδ (1) + λ 2 Δ (2) + Scaling dimension Dilatation generator Conformal symmetry, so(4,2) DO α (0) = Δ α O α (0) D = D (0) + λ D (1) + λ 2 D 2 + Via the radial quantisation, R 4 S 3 R H ψ = E ψ 3

4 AdS/CFT correspondence 4D N = 4 SU(N) SYM (CFT) string theory on AdS 5 S 5 Δ λ, N = E(g s, R AdS /l s ) λ = R AdS l s 4 λ 4πN = g s (λ = g 2 YM N) Several ways of looking at the equation Check the duality Use to understand something new String theory at small curvature R AdS l s is difficult Gauge theory description is easier at λ 1 4

5 Operator mixing problem A μ, Φ a a = 1,, 6, ψ α For the SO(6) sector, the 1-loop dilatation operator is given by D 1 loop = 1 N H H = 1 2 : tr Φ m, Φ n m, n : 1 4 : tr Φ m, n Φ m, n : m ij Φ n kl = δ mn δ il δ jk Φ m are just matrices 1) Dχ α = M αβ χ β for a general basis 2) Diagonalise M αβ to obtain DO α = Δ α O α Eigenvalue problem of the matrix model 5

6 Will study the spectrum of anomalous dimensions, focusing on the sign of them Δ = Δ (0) + λδ 1 (1/N) + O(λ 2 ) + 1. In the planar limit, anomalous dimensions are all positive 2. But it is not the case when you include non-planar corrections To understand physics of negative anomalous dimensions is to understand nonplanar corrections 6

7 Outline Brief review of N=4 SYM (CFT) Anomalous dimensions Dilatation operator Operator mixing problem Planar vs Non-planar Negative anomalous dimensions [ , YK-R.Suzuki] 7

8 2. Operator mixing problem planar vs non-planar 8

9 Operator mixing tr Φ m, Φ n m, n = 2tr Φ m Φ n m n 2tr Φ m Φ n n m tr Φ m, n Φ m, n = 2tr Φ m n Φ m n 2tr Φ m Φ m n n tr Φ m Φ n m n tr AΦ a tr BΦ b = tr BAΦ b Φ a + ABΦ a Φ b tr Φ m Φ n m n tr AΦ a BΦ b = tr A tr BΦ b Φ a +tr B tr AΦ a Φ b Dilatation operator changes the number of traces by one - Joining and splitting When the two derivatives act on nearest neighbor matrices, we get a term whose trace structure is not changed tr Φ m Φ n m n tr AΦ a Φ b = tr A tr Φ b Φ a +Ntr AΦ a Φ b The two derivatives acting on non-nearest neighbor matrices, the trace structure changes 9

10 Htr AΦ a Φ b = N tr AΦ a Φ b tr AΦ b Φ a δ abtr AΦ m Φ m +double traces Nearest neighbour transpositions P and contractions C on flavor indices P Φ a Φ b = Φ b Φ a H = NH P + H NP C Φ a Φ b = δ ab Φ m Φ m H P = 1 P C H P : not changing the trace structure, but changing the flavour structure H NP : changing the trace structure and the flavour structure 10

11 Planar limit Dilatation operator is H P = 1 P + C 2 Mapped to the Hamiltonian of an integrable spin chain [02 Minahan-Zarembo] the mixing is only among operators with the same trace structure (i.e. trace structure has a meaning) Block-diagonal mixing matrix tr(φ 6 ) tr Φ 3 tr(φ 3 ) Non-planar we do not have the above properties (We can find a nice mixing pattern in non-planar situations in terms of Young diagrams) 11

12 Remark H = NH P + H NP Acting on a small operator Acting on a very large operator E = O(N) + O(1) E = O(NL) + O(L 2 ) The planar limit is N L g eff = L/N D-branes are considered to be described by large operators L O N, N 1 - One can not use the planar limit 12

13 3. Negative anomalous dimensions [ , YK-R.Suzuki] 13

14 Spectral problem in the so(6) singlet sector SO(6) singlet operators tr(φ a Φ b )tr(φ a Φ b ), tr(φ a Φ b Φ a Φ a ) Singlets are mapped to singlets under dilatation Consider planar zero modes at one-loop - H P ψ 0 = 0 o Giving an interesting class of operators o Thanks to integrability, there is a large degeneracy in the planar spectrum. Turning on 1/N corrections, the planar zero modes will get anomalous dimensions of the form: γ = γ N γ N Sign of γ 1, γ 2 Operator mixing among the planar zero modes 14

15 L = 4 There are 4 singlet operators t 1 = tr(φ a Φ b )tr(φ a Φ b ), t 2 = tr(φ a Φ a )tr(φ b Φ b ) t 3 = tr(φ a Φ b Φ a Φ a ), t 4 = tr(φ a Φ a Φ b Φ b ) Ht a = Nγ ab t b γ = /N 10/N /N 12/N 12/N 2/N 4 2 2/N 7/N 2 9 block-diagonal if N 1, where the single-traces are orthogonal to the double-traces. Use Mathematica to compute eigenvalues 15

16 L = 4 γ (one-loop anomalous dimension) vs N Negative mode Planar zero mode 16

17 L = 4 The negative mode looks like ψ = 6T ab T ab + 3 4N 14t 3 4t N 2 978t 1 107t 2 + T ab = tr Φ a Φ b 1 6 δ abtr(φ m Φ m ) PT ab = T ab, CT ab = 0 t 1 = tr(φ a Φ b )tr(φ a Φ b ), t 2 = tr(φ a Φ a )tr(φ b Φ b ) t 3 = tr(φ a Φ b Φ a Φ a ), t 4 = tr(φ a Φ a Φ b Φ b ) The leading term is given by the energy-momentum tensor, which is traceless and symmetric It is annihilated by the planar dilatation operator, H P = 1 P + C/2 H P T ab T ab = H P T ab T ab + T ab H P T ab = 0 17

18 Planar zero modes Number of planar zero modes and singlet operators H P ψ 0 = 0 H P = 1 P + C/2 C i1 i 2 i l = tr Φ (i1 Φ i2 Φ il ) L = 4 : C ij C ij 1/2 BPS : symmetric and traceless H P C i1 i 2 i l = 0 L = 6 : C ijk C ijk, C ij C jk C ki L = 8 : C ijkl C ijkl, C ijkl C ij C kl, C ijk C ijl C kl, C ij C ji C kl C lk, C ij C jk C kl C li Can not construct single-trace planar zero modes 18

19 L = 6 There are 15 singlet operators There are 2 planar zero modes. One stays on the zero, and the other gets a negative anomalous dimension. 19

20 L = 8 71 singlets 5 planar zero modes 20

21 On the planar zero modes Mathematica computation at L = 4,6,8,10 and analytic computation with some approximation Hψ = Nγψ H 0 ψ 0 = 0 ψ = ψ 0 + ψ 1 N + ψ 2 N 2 + γ = γ 0 + γ 1 N + γ 2 N 2 + o γ 0 = 0, γ 1 = 0, γ 2 0 Planar zero mode negative mode o ψ 0 is a linear combination of the planar zero modes with a fixed number of traces tr Φ 4 tr Φ 4 tr(φ 2 ) is orthogonal to tr Φ 5 tr Φ 3 tr(φ 2 ) in the planar limit, but they mix by the 1/N effect. the number of traces might be a good quantity 21

22 A possible interpretation of negative modes Based on the standard correspondence of AdS/CFT Δ = Δ 0 + λ γ 2 N 2 + E = E 0 + γ 2 g s 2 1 α + γ 2 < 0 No interaction (planar zero mode) Negative mode The negative modes would describe multi-particle (multi-string) states with non-zero binding energy. [02 Arutuynov, Penati, Petkou, Santambrogio, Sokatchev] The number of states might be related to the number of traces in ψ 0 22

23 Summary Studied the non-planar mixing so(6) singlet sector Spectrum explicitly up to L = 10 Planar zero modes negative modes γ γ 2 /N 2 is consistent with an analysis of 4-pt functions Anomalous dimensions are always positive in the planar limit Also positive in su(2) sector, so(6) non-singlet sector even with nonplanar corrections Another sector with negative anomalous dimensions: large spin operators like φ S φ, s 1 23