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

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

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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

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13 Nov., 2015, YITP Workshop

  • Developments in String Theory and Quantum Field Theory

Negative anomalous dimensions in N=4 SYM

Yusuke Kimura (OIQP)

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1503.0621 [hep-th] with Ryo Suzuki

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  • 1. Introduction – brief review of N=4 SYM
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Anomalous dimensions of N=4 SYM (Conformal Field Theory)

𝑃𝛽 𝑦 𝑃𝛾 𝑧 = π‘‘π›½πœ€π›½π›Ύ 𝑦 βˆ’ 𝑧 2Δ𝛽 Ξ” = Ξ”(0) + πœ‡Ξ”(1) + πœ‡2Ξ”(2) + β‹― 𝐸𝑃𝛽(0) = Δ𝛽𝑃𝛽(0) 𝐼|πœ”βŒͺ = 𝐹|πœ”βŒͺ πœ‡ = 𝑕𝑍𝑁

2 𝑂

Two-point functions Dilatation generator βŠ‚ Conformal symmetry, so(4,2) 𝐸 = 𝐸(0) + πœ‡ 𝐸(1) + πœ‡2 𝐸 2 + β‹― Via the radial quantisation, 𝑆4 β†’ 𝑇3 Γ— 𝑆

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Scaling dimension

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Ξ” πœ‡, 𝑂 = 𝐹(𝑕𝑑, 𝑆𝐡𝑒𝑇/π‘šπ‘‘) πœ‡ = 𝑆𝐡𝑒𝑇 π‘šπ‘‘

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πœ‡ 4πœŒπ‘‚ = 𝑕𝑑 (πœ‡ = 𝑕𝑍𝑁

2 𝑂)

AdS/CFT correspondence

Several ways of looking at the equation οƒΌ Check the duality οƒΌ Use to understand something new

  • String theory at small curvature 𝑆𝐡𝑒𝑇 β‰ͺ π‘šπ‘‘ is difficult
  • Gauge theory description is easier at πœ‡ β‰ͺ 1

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4D 𝑂 = 4 S𝑉(𝑂) SYM (CFT) ⇔ string theory on 𝐡𝑒𝑇5 Γ— 𝑇5

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Operator mixing problem

For the SO(6) sector, the 1-loop dilatation operator is given by 𝐼 = βˆ’ 1 2 : 𝑒𝑠 Φ𝑛, Ξ¦π‘œ πœ–π‘›, πœ–π‘œ : βˆ’ 1 4 : 𝑒𝑠 Φ𝑛, πœ–π‘œ Φ𝑛, πœ–π‘œ : πœ–π‘› π‘—π‘˜ Ξ¦π‘œ π‘™π‘š = πœ€π‘›π‘œπœ€π‘—π‘šπœ€

π‘˜π‘™

𝐸1βˆ’π‘šπ‘π‘π‘ž = 1 𝑂 𝐼 Φ𝑛 are just matrices

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𝐡𝜈, Φ𝑏 𝑏 = 1, β‹― , 6 , πœ”π›½

1) πΈπœ“π›½ = π‘π›½π›Ύπœ“π›Ύ for a general basis 2) Diagonalise 𝑁𝛽𝛾 to obtain 𝐸𝑃𝛽 = Δ𝛽𝑃𝛽 Eigenvalue problem of the matrix model

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Will study the spectrum of anomalous dimensions, focusing on the sign of them 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 Ξ” = Ξ”(0) + πœ‡Ξ” 1 (1/𝑂) + 𝑃(πœ‡2) + β‹―

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Outline

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

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  • 2. Operator mixing problem – planar vs non-planar
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𝑒𝑠 Φ𝑛, Ξ¦π‘œ πœ–π‘›, πœ–π‘œ = 2𝑒𝑠 Ξ¦π‘›Ξ¦π‘œπœ–π‘›πœ–π‘œ βˆ’ 2𝑒𝑠 Ξ¦π‘›Ξ¦π‘œπœ–π‘œπœ–π‘› 𝑒𝑠 Φ𝑛, πœ–π‘œ Φ𝑛, πœ–π‘œ = 2𝑒𝑠 Ξ¦π‘›πœ–π‘œΞ¦π‘›πœ–π‘œ βˆ’ 2𝑒𝑠 Ξ¦π‘›Ξ¦π‘›πœ–π‘œπœ–π‘œ

𝑒𝑠 Ξ¦π‘›Ξ¦π‘œπœ–π‘›πœ–π‘œ 𝑒𝑠 𝐡Φ𝑏 𝑒𝑠 𝐢Φ𝑐 = 𝑒𝑠 𝐢𝐡Φ𝑐Φ𝑏 + 𝐡𝐢Φ𝑏Φ𝑐 𝑒𝑠 Ξ¦π‘›Ξ¦π‘œπœ–π‘›πœ–π‘œ 𝑒𝑠 𝐡Φ𝑏𝐢Φ𝑐 = 𝑒𝑠 𝐡 𝑒𝑠 𝐢Φ𝑐Φ𝑏 +𝑒𝑠 𝐢 𝑒𝑠 𝐡Φ𝑏Φ𝑐 Dilatation operator changes the number of traces by one

  • Joining and splitting

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When the two derivatives act on nearest neighbor matrices, we get a term whose trace structure is not changed 𝑒𝑠 Ξ¦π‘›Ξ¦π‘œπœ–π‘›πœ–π‘œ 𝑒𝑠 𝐡Φ𝑏Φ𝑐 = 𝑒𝑠 𝐡 𝑒𝑠 Φ𝑐Φ𝑏 +𝑂𝑒𝑠 𝐡Φ𝑏Φ𝑐

Operator mixing

The two derivatives acting on non-nearest neighbor matrices, the trace structure changes

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𝑄 β‹… Φ𝑏Φ𝑐 = Φ𝑐Φ𝑏 𝐷 β‹… Φ𝑏Φ𝑐 = πœ€π‘π‘Ξ¦π‘›Ξ¦π‘› 𝐼 = 𝑂𝐼𝑄 + 𝐼𝑂𝑄 𝐼𝑄 = 1 βˆ’ 𝑄 + 1 2 𝐷 𝐼𝑂𝑄 : changing the trace structure and the flavour structure 𝐼𝑄 : not changing the trace structure, but changing the flavour structure Nearest neighbour transpositions 𝑄 and contractions 𝐷 on flavor indices

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𝐼𝑒𝑠 𝐡Φ𝑏Φ𝑐 = 𝑂 𝑒𝑠 𝐡Φ𝑏Φ𝑐 βˆ’ 𝑒𝑠 𝐡Φ𝑐Φ𝑏 + 1 2 πœ€π‘π‘π‘’π‘  𝐡Φ𝑛Φ𝑛 +π‘’π‘π‘£π‘π‘šπ‘“ 𝑒𝑠𝑏𝑑𝑓𝑑

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β‹±

𝑒𝑠(Ξ¦6) 𝑒𝑠 Ξ¦3 𝑒𝑠(Ξ¦3)

Planar limit

οƒΌ Dilatation operator is 𝐼𝑄 = 1 βˆ’ 𝑄 +

𝐷 2

  • Mapped to the Hamiltonian of an integrable spin chain [02

Minahan-Zarembo]

οƒΌ the mixing is only among operators with the same trace structure (i.e. trace structure has a meaning) Block-diagonal mixing matrix ∼

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(We can find a nice mixing pattern in non-planar situations in terms

  • f Young diagrams)

Non-planar – we do not have the above properties

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The planar limit is 𝑂 ≫ 𝑀

Remark

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𝐼 = 𝑂𝐼𝑄 + 𝐼𝑂𝑄 𝐹 = 𝑃(𝑂𝑀) + 𝑃(𝑀2) Acting on a small operator Acting on a very large operator 𝐹 = 𝑃(𝑂) + 𝑃(1) 𝑕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
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  • 3. Negative anomalous dimensions

[1503.0621, YK-R.Suzuki]

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οƒΌ Sign of 𝛿1, 𝛿2 οƒΌ Operator mixing among the planar zero modes

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  • Consider planar zero modes at one-loop - πΌπ‘„πœ”0 = 0
  • Giving an interesting class of operators
  • Thanks to integrability, there is a large degeneracy in the planar

spectrum.

  • Turning on 1/𝑂 corrections, the planar zero modes will get anomalous

dimensions of the form: 𝛿 = 0 +

1 𝑂 𝛿1 + 1 𝑂2 𝛿2 + β‹―

𝑒𝑠(Φ𝑏Φ𝑐)𝑒𝑠(Φ𝑏Φ𝑐), 𝑒𝑠(Φ𝑏Φ𝑐Φ𝑏Φ𝑏)

Spectral problem in the so(6) singlet sector

οƒΌ SO(6) singlet operators Singlets are mapped to singlets under dilatation

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𝑀 = 4 𝑒1 = 𝑒𝑠(Φ𝑏Φ𝑐)𝑒𝑠(Φ𝑏Φ𝑐), 𝑒2 = 𝑒𝑠(Φ𝑏Φ𝑏)𝑒𝑠(Φ𝑐Φ𝑐) 𝑒3 = 𝑒𝑠(Φ𝑏Φ𝑐Φ𝑏Φ𝑏), 𝑒4 = 𝑒𝑠(Φ𝑏Φ𝑏Φ𝑐Φ𝑐) There are 4 singlet operators 𝛿 = 2 βˆ’10/𝑂 10/𝑂 2 βˆ’12/𝑂 12/𝑂 βˆ’12/𝑂 2/𝑂 4 βˆ’2 βˆ’2/𝑂 7/𝑂 βˆ’2 9 𝐼𝑒𝑏 = 𝑂𝛿𝑏𝑐𝑒𝑐 block-diagonal if 𝑂 ≫ 1 , where the single-traces are orthogonal to the double-traces. Use Mathematica to compute eigenvalues

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𝑀 = 4

𝛿 (one-loop anomalous dimension) vs 𝑂

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Negative mode Planar zero mode

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π‘ˆ

𝑏𝑐 = 𝑒𝑠 Φ𝑏Φ𝑐 βˆ’ 1

6 πœ€π‘π‘π‘’π‘ (Φ𝑛Φ𝑛) πœ” = 6π‘ˆ

π‘π‘π‘ˆ 𝑏𝑐 + 3

4𝑂 14𝑒3 βˆ’ 4𝑒4 + 5 168𝑂2 978𝑒1 βˆ’ 107𝑒2 + β‹― The negative mode looks like It is annihilated by the planar dilatation operator, 𝐼𝑄 = 1 βˆ’ 𝑄 + 𝐷/2

𝑒1 = 𝑒𝑠(Φ𝑏Φ𝑐)𝑒𝑠(Φ𝑏Φ𝑐), 𝑒2 = 𝑒𝑠(Φ𝑏Φ𝑏)𝑒𝑠(Φ𝑐Φ𝑐) 𝑒3 = 𝑒𝑠(Φ𝑏Φ𝑐Φ𝑏Φ𝑏), 𝑒4 = 𝑒𝑠(Φ𝑏Φ𝑏Φ𝑐Φ𝑐)

π‘„π‘ˆ

𝑏𝑐 = π‘ˆπ‘π‘, π·π‘ˆ 𝑏𝑐 = 0

The leading term is given by the energy-momentum tensor, which is traceless and symmetric πΌπ‘„π‘ˆ

π‘π‘π‘ˆ 𝑏𝑐 = πΌπ‘„π‘ˆ 𝑏𝑐 π‘ˆπ‘π‘ + π‘ˆ 𝑏𝑐 πΌπ‘„π‘ˆ 𝑏𝑐 = 0

𝑀 = 4

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Planar zero modes

𝐷𝑗1𝑗2β‹―π‘—π‘š = 𝑒𝑠 Ξ¦(𝑗1Φ𝑗2 β‹― Ξ¦π‘—π‘š) 1/2 BPS : symmetric and traceless 𝑀 = 4 : π·π‘—π‘˜π·π‘—π‘˜ 𝑀 = 6 : π·π‘—π‘˜π‘™π·π‘—π‘˜π‘™, π·π‘—π‘˜π·

π‘˜π‘™π·π‘™π‘—

𝑀 = 8 : π·π‘—π‘˜π‘™π‘šπ·π‘—π‘˜π‘™π‘š, π·π‘—π‘˜π‘™π‘šπ·π‘—π‘˜π·π‘™π‘š, π·π‘—π‘˜π‘™π·π‘—π‘˜π‘šπ·π‘™π‘š, π·π‘—π‘˜π·

π‘˜π‘—π·π‘™π‘šπ·π‘šπ‘™, π·π‘—π‘˜π· π‘˜π‘™π·π‘™π‘šπ·π‘šπ‘—

Number of planar zero modes and singlet operators

𝐼𝑄𝐷𝑗1𝑗2β‹―π‘—π‘š = 0 𝐼𝑄 = 1 βˆ’ 𝑄 + 𝐷/2

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πΌπ‘„πœ”0 = 0

οƒΌ Can not construct single-trace planar zero modes

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

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𝑀 = 8 71 singlets 5 planar zero modes

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𝛿 = 𝛿0 + 𝛿1 𝑂 + 𝛿2 𝑂2 + β‹― πœ” = πœ”0 + πœ”1 𝑂 + πœ”2 𝑂2 + β‹― πΌπœ” = π‘‚π›Ώπœ”

  • 𝛿0 = 0, 𝛿1 = 0, 𝛿2 ≀ 0
  • Planar zero mode β†’ negative mode
  • πœ”0 is a linear combination of the planar zero modes with a fixed number
  • f traces
  • 𝑒𝑠 Ξ¦4 𝑒𝑠 Ξ¦4 𝑒𝑠(Ξ¦2) is orthogonal to 𝑒𝑠 Ξ¦5 𝑒𝑠 Ξ¦3 𝑒𝑠(Ξ¦2) in the

planar limit, but they mix by the 1/𝑂 effect.

  • the number of traces might be a good quantity

On the planar zero modes

𝐼0πœ”0 = 0

Mathematica computation at 𝑀 = 4,6,8,10 and analytic computation with some approximation

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A possible interpretation of negative modes

No interaction (planar zero mode) The negative modes would describe multi-particle (multi-string) states with non-zero binding energy. [02 Arutuynov, Penati, Petkou, Santambrogio, Sokatchev] Based on the standard correspondence of AdS/CFT Negative mode

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Ξ” = Ξ”0 + πœ‡ 𝛿2 𝑂2 + β‹― 𝐹 = 𝐹0 + 𝛿2𝑕𝑑

2 1

𝛽′ + β‹― 𝛿2 < 0 The number of states might be related to the number of traces in πœ”0

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οƒΌ Studied the non-planar mixing οƒΌ so(6) singlet sector οƒΌ Spectrum explicitly up to 𝑀 = 10 οƒΌ Planar zero modes β†’ negative modes οƒΌ 𝛿 ∼ 𝛿2/𝑂2 is consistent with an analysis of 4-pt functions

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Summary

  • Anomalous dimensions are always positive in the planar limit
  • Also positive in 𝑑𝑣(2) sector, 𝑑𝑝(6) non-singlet sector even with non-

planar corrections

  • Another sector with negative anomalous dimensions: large spin operators

like πœšπœ–π‘‡πœš, 𝑑 ≫ 1