Negative anomalous dimensions in N =4 SYM Yusuke Kimura (OIQP) - PowerPoint PPT Presentation

13 Nov., 2015, YITP Workshop - Developments in String Theory and Quantum Field Theory Negative anomalous dimensions in N =4 SYM Yusuke Kimura (OIQP) 1503.0621 [hep-th] with Ryo Suzuki 1

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

Anomalous dimensions of N =4 SYM (Conformal Field Theory) 2 𝑂 Two-point functions 𝜇 = 𝑕 𝑍𝑁 𝑑 𝛽 𝜀 𝛽𝛾 Δ = Δ (0) + 𝜇Δ (1) + 𝜇 2 Δ (2) + ⋯ 𝑃 𝛽 𝑦 𝑃 𝛾 𝑧 = 𝑦 − 𝑧 2Δ 𝛽 Scaling dimension Dilatation generator ⊂ Conformal symmetry, so(4,2) 𝐸 = 𝐸 (0) + 𝜇 𝐸 (1) + 𝜇 2 𝐸 2 + ⋯ 𝐸𝑃 𝛽 (0) = Δ 𝛽 𝑃 𝛽 (0) Via the radial quantisation, 𝑆 4 → 𝑇 3 × 𝑆 𝐼|𝜔〉 = 𝐹|𝜔〉 3

AdS/CFT correspondence 4D 𝑂 = 4 S𝑉(𝑂) SYM (CFT) ⇔ string theory on 𝐵𝑒𝑇 5 × 𝑇 5 4 𝑆 𝐵𝑒𝑇 𝜇 𝜇 = 4𝜌𝑂 = 𝑕 𝑡 Δ 𝜇, 𝑂 = 𝐹(𝑕 𝑡 , 𝑆 𝐵𝑒𝑇 /𝑚 𝑡 ) 𝑚 𝑡 2 𝑂 ) ( 𝜇 = 𝑕 𝑍𝑁 Several ways of looking at the equation  Check the duality  Use to understand something new  String theory at small curvature 𝑆 𝐵𝑒𝑇 ≪ 𝑚 𝑡 is difficult  Gauge theory description is easier at 𝜇 ≪ 1 4

Operator mixing problem 𝐵 𝜈 , Φ 𝑏 𝑏 = 1, ⋯ , 6 , 𝜔 𝛽 For the SO(6) sector, the 1-loop dilatation operator is given by 𝐸 1−𝑚𝑝𝑝𝑞 = 1 𝑂 𝐼 𝐼 = − 1 2 : 𝑢𝑠 Φ 𝑛 , Φ 𝑜 𝜖 𝑛 , 𝜖 𝑜 : − 1 4 : 𝑢𝑠 Φ 𝑛 , 𝜖 𝑜 Φ 𝑛 , 𝜖 𝑜 : 𝜖 𝑛 𝑗𝑘 Φ 𝑜 𝑙𝑚 = 𝜀 𝑛𝑜 𝜀 𝑗𝑚 𝜀 Φ 𝑛 are just matrices 𝑘𝑙 𝐸𝜓 𝛽 = 𝑁 𝛽𝛾 𝜓 𝛾 1) for a general basis Diagonalise 𝑁 𝛽𝛾 to obtain 𝐸𝑃 𝛽 = Δ 𝛽 𝑃 𝛽 2) Eigenvalue problem of the matrix model 5

Will study the spectrum of anomalous dimensions, focusing on the sign of them Δ = Δ (0) + 𝜇Δ 1 (1/𝑂) + 𝑃(𝜇 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

Outline  Brief review of N=4 SYM (CFT)  Anomalous dimensions  Dilatation operator  Operator mixing problem  Planar vs Non-planar  Negative anomalous dimensions [1503.0621, YK-R.Suzuki] 7

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

Operator mixing 𝑢𝑠 Φ 𝑛 , Φ 𝑜 𝜖 𝑛 , 𝜖 𝑜 = 2𝑢𝑠 Φ 𝑛 Φ 𝑜 𝜖 𝑛 𝜖 𝑜 − 2𝑢𝑠 Φ 𝑛 Φ 𝑜 𝜖 𝑜 𝜖 𝑛 𝑢𝑠 Φ 𝑛 , 𝜖 𝑜 Φ 𝑛 , 𝜖 𝑜 = 2𝑢𝑠 Φ 𝑛 𝜖 𝑜 Φ 𝑛 𝜖 𝑜 − 2𝑢𝑠 Φ 𝑛 Φ 𝑛 𝜖 𝑜 𝜖 𝑜 𝑢𝑠 Φ 𝑛 Φ 𝑜 𝜖 𝑛 𝜖 𝑜 𝑢𝑠 𝐵Φ 𝑏 𝑢𝑠 𝐶Φ 𝑐 = 𝑢𝑠 𝐶𝐵Φ 𝑐 Φ 𝑏 + 𝐵𝐶Φ 𝑏 Φ 𝑐 𝑢𝑠 Φ 𝑛 Φ 𝑜 𝜖 𝑛 𝜖 𝑜 𝑢𝑠 𝐵Φ 𝑏 𝐶Φ 𝑐 = 𝑢𝑠 𝐵 𝑢𝑠 𝐶Φ 𝑐 Φ 𝑏 + 𝑢𝑠 𝐶 𝑢𝑠 𝐵Φ 𝑏 Φ 𝑐 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 𝑢𝑠 Φ 𝑛 Φ 𝑜 𝜖 𝑛 𝜖 𝑜 𝑢𝑠 𝐵Φ 𝑏 Φ 𝑐 = 𝑢𝑠 𝐵 𝑢𝑠 Φ 𝑐 Φ 𝑏 + 𝑂𝑢𝑠 𝐵Φ 𝑏 Φ 𝑐 The two derivatives acting on non-nearest neighbor matrices, the trace structure changes 9

𝐼𝑢𝑠 𝐵Φ 𝑏 Φ 𝑐 = 𝑂 𝑢𝑠 𝐵Φ 𝑏 Φ 𝑐 − 𝑢𝑠 𝐵Φ 𝑐 Φ 𝑏 + 1 2 𝜀 𝑏𝑐 𝑢𝑠 𝐵Φ 𝑛 Φ 𝑛 +𝑒𝑝𝑣𝑐𝑚𝑓 𝑢𝑠𝑏𝑑𝑓𝑡 Nearest neighbour transpositions 𝑄 and contractions 𝐷 on flavor indices 𝑄 ⋅ Φ 𝑏 Φ 𝑐 = Φ 𝑐 Φ 𝑏 𝐷 ⋅ Φ 𝑏 Φ 𝑐 = 𝜀 𝑏𝑐 Φ 𝑛 Φ 𝑛 𝐼 𝑄 = 1 − 𝑄 + 1 2 𝐷 𝐼 = 𝑂𝐼 𝑄 + 𝐼 𝑂𝑄 𝐼 𝑄 : not changing the trace structure, but changing the flavour structure 𝐼 𝑂𝑄 : changing the trace structure and the flavour structure 10

Planar limit 𝐷  Dilatation operator is 𝐼 𝑄 = 1 − 𝑄 + 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) 𝑢𝑠(Φ 6 ) Block-diagonal mixing matrix ∼ 𝑢𝑠 Φ 3 𝑢𝑠(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

Remark 𝐼 = 𝑂𝐼 𝑄 + 𝐼 𝑂𝑄 Acting on a small operator 𝐹 = 𝑃(𝑂) + 𝑃(1) 𝐹 = 𝑃(𝑂𝑀) + 𝑃(𝑀 2 ) Acting on a very large operator The planar limit is 𝑂 ≫ 𝑀 𝑕 𝑓𝑔𝑔 = 𝑀/𝑂 D-branes are considered to be described by large operators 𝑀 ∼ 𝑃 𝑂 , 𝑂 ≫ 1 - One can not use the planar limit 12

3. Negative anomalous dimensions [1503.0621, YK-R.Suzuki] 13

Spectral problem in the so(6) singlet sector 𝑢𝑠(Φ 𝑏 Φ 𝑐 )𝑢𝑠(Φ 𝑏 Φ 𝑐 ) , 𝑢𝑠(Φ 𝑏 Φ 𝑐 Φ 𝑏 Φ 𝑏 )  SO(6) singlet operators Singlets are mapped to singlets under dilatation  Consider planar zero modes at one-loop - 𝐼 𝑄 𝜔 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/𝑂 corrections, the planar zero modes will get anomalous 1 1 dimensions of the form: 𝛿 = 0 + 𝑂 𝛿 1 + 𝑂 2 𝛿 2 + ⋯  Sign of 𝛿 1 , 𝛿 2  Operator mixing among the planar zero modes 14

𝑀 = 4 There are 4 singlet operators 𝑢 1 = 𝑢𝑠(Φ 𝑏 Φ 𝑐 )𝑢𝑠(Φ 𝑏 Φ 𝑐 ) , 𝑢 2 = 𝑢𝑠(Φ 𝑏 Φ 𝑏 )𝑢𝑠(Φ 𝑐 Φ 𝑐 ) 𝑢 3 = 𝑢𝑠(Φ 𝑏 Φ 𝑐 Φ 𝑏 Φ 𝑏 ) , 𝑢 4 = 𝑢𝑠(Φ 𝑏 Φ 𝑏 Φ 𝑐 Φ 𝑐 ) 𝐼𝑢 𝑏 = 𝑂𝛿 𝑏𝑐 𝑢 𝑐 0 2 −10/𝑂 10/𝑂 0 2 −12/𝑂 12/𝑂 𝛿 = −12/𝑂 2/𝑂 4 −2 −2/𝑂 7/𝑂 −2 9 block-diagonal if 𝑂 ≫ 1 , where the single-traces are orthogonal to the double-traces. Use Mathematica to compute eigenvalues 15

𝛿 (one-loop anomalous dimension) vs 𝑂 𝑀 = 4 Planar zero mode Negative mode 16

𝑢 1 = 𝑢𝑠(Φ 𝑏 Φ 𝑐 )𝑢𝑠(Φ 𝑏 Φ 𝑐 ) , 𝑢 2 = 𝑢𝑠(Φ 𝑏 Φ 𝑏 )𝑢𝑠(Φ 𝑐 Φ 𝑐 ) 𝑀 = 4 𝑢 3 = 𝑢𝑠(Φ 𝑏 Φ 𝑐 Φ 𝑏 Φ 𝑏 ) , 𝑢 4 = 𝑢𝑠(Φ 𝑏 Φ 𝑏 Φ 𝑐 Φ 𝑐 ) The negative mode looks like 𝑏𝑐 + 3 5 𝜔 = 6𝑈 𝑏𝑐 𝑈 4𝑂 14𝑢 3 − 4𝑢 4 + 168𝑂 2 978𝑢 1 − 107𝑢 2 + ⋯ The leading term is given by the energy-momentum tensor, which is traceless and symmetric 𝑏𝑐 = 𝑢𝑠 Φ 𝑏 Φ 𝑐 − 1 𝑈 6 𝜀 𝑏𝑐 𝑢𝑠(Φ 𝑛 Φ 𝑛 ) 𝑄𝑈 𝑏𝑐 = 𝑈 𝑏𝑐 , 𝐷𝑈 𝑏𝑐 = 0 It is annihilated by the planar dilatation operator, 𝐼 𝑄 = 1 − 𝑄 + 𝐷/2 𝐼 𝑄 𝑈 𝑏𝑐 𝑈 𝑏𝑐 = 𝐼 𝑄 𝑈 𝑏𝑐 𝑈 𝑏𝑐 + 𝑈 𝑏𝑐 𝐼 𝑄 𝑈 𝑏𝑐 = 0 17

Planar zero modes Number of planar zero modes and singlet operators 𝐼 𝑄 = 1 − 𝑄 + 𝐷/2 𝐼 𝑄 𝜔 0 = 0 𝐷 𝑗 1 𝑗 2 ⋯𝑗 𝑚 = 𝑢𝑠 Φ (𝑗 1 Φ 𝑗 2 ⋯ Φ 𝑗 𝑚 ) 1/2 BPS : symmetric and traceless 𝐼 𝑄 𝐷 𝑗 1 𝑗 2 ⋯𝑗 𝑚 = 0 𝑀 = 4 : 𝐷 𝑗𝑘 𝐷 𝑗𝑘 𝑀 = 6 : 𝐷 𝑗𝑘𝑙 𝐷 𝑗𝑘𝑙 , 𝐷 𝑗𝑘 𝐷 𝑘𝑙 𝐷 𝑙𝑗 𝑀 = 8 : 𝐷 𝑗𝑘𝑙𝑚 𝐷 𝑗𝑘𝑙𝑚 , 𝐷 𝑗𝑘𝑙𝑚 𝐷 𝑗𝑘 𝐷 𝑙𝑚 , 𝐷 𝑗𝑘𝑙 𝐷 𝑗𝑘𝑚 𝐷 𝑙𝑚 , 𝐷 𝑗𝑘 𝐷 𝑘𝑗 𝐷 𝑙𝑚 𝐷 𝑚𝑙 , 𝐷 𝑗𝑘 𝐷 𝑘𝑙 𝐷 𝑙𝑚 𝐷 𝑚𝑗  Can not construct single-trace planar zero modes 18

𝑀 = 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

𝑀 = 8 71 singlets 5 planar zero modes 20

On the planar zero modes Mathematica computation at 𝑀 = 4,6,8,10 and analytic computation with some approximation 𝐼 0 𝜔 0 = 0 𝐼𝜔 = 𝑂𝛿𝜔 𝛿 = 𝛿 0 + 𝛿 1 𝑂 + 𝛿 2 𝜔 = 𝜔 0 + 𝜔 1 𝑂 + 𝜔 2 𝑂 2 + ⋯ 𝑂 2 + ⋯ o 𝛿 0 = 0 , 𝛿 1 = 0 , 𝛿 2 ≤ 0 • Planar zero mode → negative mode o 𝜔 0 is a linear combination of the planar zero modes with a fixed number of traces 𝑢𝑠 Φ 4 𝑢𝑠 Φ 4 𝑢𝑠(Φ 2 ) is orthogonal to 𝑢𝑠 Φ 5 𝑢𝑠 Φ 3 𝑢𝑠(Φ 2 ) in the • planar limit, but they mix by the 1/𝑂 effect. • the number of traces might be a good quantity 21

A possible interpretation of negative modes Based on the standard correspondence of AdS/CFT Δ = Δ 0 + 𝜇 𝛿 2 2 1 𝑂 2 + ⋯ 𝛿 2 < 0 𝐹 = 𝐹 0 + 𝛿 2 𝑕 𝑡 𝛽 ′ + ⋯ 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 𝜔 0 22