Dilatation Operator. October 5, 2007

Transcription

1 Dilatation Operator Corneliu Sochichiu INFN LNF, Frascati & IAP, Chişinău, Moldova 3d RTN Workshop ForcesUniverse Valencia October 5, , / 16

2 Outline 1 Introduction Main motivation Setup Renormalization & Operator mixing 2 General construction & Tools Operator Product Expansion Differential regularization Results 3 Example 4 Conclusion 5, / 16

3 Related Publications based on: JHEP09(2007)025 BG: hep-th/ (with S. Bellucci), hep-th/ , hep-th/ , hep-th/ , hep-th/ , hep-th/ , / 16

4 AdS/CFT correspondence AdS/CFT correspondence (planar limit: N ): (N = 4 SYM) M1,3 (string theory) AdS5 S 5 Recorded a considerable progress since 97. Can be mapped to a spin chain. Integrability... [Minahan-Zarembo, Staudacher-Beisert,...] Can be extended to include 1/N corrections, which correspond to string interactions. Still can be mapped to a spin system: dynamical model, but no integrability! Available tools: Statistical physics/thermodynamics Object of study: Dilatation operator/mixing matrix [Beisert-Kristjansen-Plefka-Semenoff-Staudacher] { } Spin Two approaches: parametrization. Matrix 5, / 16

5 AdS/CFT correspondence AdS/CFT correspondence (planar limit: N ): (N = 4 SYM) M1,3 (string theory) AdS5 S 5 Recorded a considerable progress since 97. Can be mapped to a spin chain. Integrability... [Minahan-Zarembo, Staudacher-Beisert,...] Can be extended to include 1/N corrections, which correspond to string interactions. Still can be mapped to a spin system: dynamical model, but no integrability! Available tools: Statistical physics/thermodynamics Object of study: Dilatation operator/mixing matrix [Beisert-Kristjansen-Plefka-Semenoff-Staudacher] { } Spin Two approaches: parametrization. Matrix 5, / 16

6 AdS/CFT correspondence AdS/CFT correspondence (planar limit: N ): (N = 4 SYM) M1,3 (string theory) AdS5 S 5 Recorded a considerable progress since 97. Can be mapped to a spin chain. Integrability... [Minahan-Zarembo, Staudacher-Beisert,...] Can be extended to include 1/N corrections, which correspond to string interactions. Still can be mapped to a spin system: dynamical model, but no integrability! Available tools: Statistical physics/thermodynamics Object of study: Dilatation operator/mixing matrix [Beisert-Kristjansen-Plefka-Semenoff-Staudacher] { } Spin Two approaches: parametrization. Matrix 5, / 16

7 Spin vs. Matrix approach Spin description: " The planar limit is easy " (Planar) Integrability is manifest % Non-planar corrections are difficult % Semi-classical limit is difficult Matrix description: % Planar limit requires some effort % No integrability for any finite matrix size " Non-planarity is natural " Semi-classical limit is easy Taking into account the non-planar contribution to AdS/CFT requires matrix model approach. 5, / 16

8 Spin vs. Matrix approach Spin description: " The planar limit is easy " (Planar) Integrability is manifest % Non-planar corrections are difficult % Semi-classical limit is difficult Matrix description: % Planar limit requires some effort % No integrability for any finite matrix size " Non-planarity is natural " Semi-classical limit is easy Taking into account the non-planar contribution to AdS/CFT requires matrix model approach. 5, / 16

9 Spin vs. Matrix approach Spin description: " The planar limit is easy " (Planar) Integrability is manifest % Non-planar corrections are difficult % Semi-classical limit is difficult Matrix description: % Planar limit requires some effort % No integrability for any finite matrix size " Non-planarity is natural " Semi-classical limit is easy Taking into account the non-planar contribution to AdS/CFT requires matrix model approach. 5, / 16

10 Spin vs. Matrix approach Spin description: " The planar limit is easy " (Planar) Integrability is manifest % Non-planar corrections are difficult % Semi-classical limit is difficult Matrix description: % Planar limit requires some effort % No integrability for any finite matrix size " Non-planarity is natural " Semi-classical limit is easy Taking into account the non-planar contribution to AdS/CFT requires matrix model approach. 5, / 16

11 Dilatation Operator corresponds to a dynamical system through the identification RG-flow Dynamics... in terms of a differential operator can be directly interpreted as a Hamiltonian of QM system w Matrix Model description.... is well studied in (mostly compact sectors of) N = 4 SYM.... plays key role in theories with conformal invariance: N = 4 SYM, β-deformed versions, Quiver theories as well as in ordinary gauge theories e.g. QCD.... we need to know it for a generic gauge theory 5, / 16

12 Dilatation Operator corresponds to a dynamical system through the identification RG-flow Dynamics... in terms of a differential operator can be directly interpreted as a Hamiltonian of QM system w Matrix Model description.... is well studied in (mostly compact sectors of) N = 4 SYM.... plays key role in theories with conformal invariance: N = 4 SYM, β-deformed versions, Quiver theories as well as in ordinary gauge theories e.g. QCD.... we need to know it for a generic gauge theory 5, / 16

13 Dilatation Operator corresponds to a dynamical system through the identification RG-flow Dynamics... in terms of a differential operator can be directly interpreted as a Hamiltonian of QM system w Matrix Model description.... is well studied in (mostly compact sectors of) N = 4 SYM.... plays key role in theories with conformal invariance: N = 4 SYM, β-deformed versions, Quiver theories as well as in ordinary gauge theories e.g. QCD.... we need to know it for a generic gauge theory 5, / 16

14 Dilatation Operator corresponds to a dynamical system through the identification RG-flow Dynamics... in terms of a differential operator can be directly interpreted as a Hamiltonian of QM system w Matrix Model description.... is well studied in (mostly compact sectors of) N = 4 SYM.... plays key role in theories with conformal invariance: N = 4 SYM, β-deformed versions, Quiver theories as well as in ordinary gauge theories e.g. QCD.... we need to know it for a generic gauge theory 5, / 16

15 Dilatation Operator corresponds to a dynamical system through the identification RG-flow Dynamics... in terms of a differential operator can be directly interpreted as a Hamiltonian of QM system w Matrix Model description.... is well studied in (mostly compact sectors of) N = 4 SYM.... plays key role in theories with conformal invariance: N = 4 SYM, β-deformed versions, Quiver theories as well as in ordinary gauge theories e.g. QCD.... we need to know it for a generic gauge theory orneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse OctoberValencia) 5, / 16

16 Dilatation Operator corresponds to a dynamical system through the identification RG-flow Dynamics... in terms of a differential operator can be directly interpreted as a Hamiltonian of QM system w Matrix Model description.... is well studied in (mostly compact sectors of) N = 4 SYM.... plays key role in theories with conformal invariance: N = 4 SYM, β-deformed versions, Quiver theories as well as in ordinary gauge theories e.g. QCD.... we need to know it for a generic gauge theory orneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse OctoberValencia) 5, / 16

17 Objective To propose a recipe to construct the dilatation operator as a differential operator acting on the space of normal symbols of local composite operators. The procedure should apply to renormalizable models with massless fields... be as Geometric as possible 5, / 16

18 Objective To propose a recipe to construct the dilatation operator as a differential operator acting on the space of normal symbols of local composite operators. The procedure should apply to renormalizable models with massless fields... be as Geometric as possible 5, / 16

19 Objective To propose a recipe to construct the dilatation operator as a differential operator acting on the space of normal symbols of local composite operators. The procedure should apply to renormalizable models with massless fields... be as Geometric as possible 5, / 16

20 The Setup [Polyakov] Alphabet : {φ A } localized fields and their derivatives in SYM: {W A } = {F µν, φ, ψ, F, φ, ψ... } Language : (gauge invariant) combinations of letters Words : simplest gauge invariants, one-trace composite operators, O A1 A 2...A L = tr W A1 W A2... W AL Phrases : O A1 A 2...A L1 O B1 B 2...B L2... O C1 C 2...C Lr Dual states : Ǒ = O φ ˇφ ˇφ(x) φ(x) 5, / 16

21 The Setup [Polyakov] Alphabet : {φ A } localized fields and their derivatives in SYM: {W A } = {F µν, φ, ψ, F, φ, ψ... } Language : (gauge invariant) combinations of letters Words : simplest gauge invariants, one-trace composite operators, O A1 A 2...A L = tr W A1 W A2... W AL Phrases : O A1 A 2...A L1 O B1 B 2...B L2... O C1 C 2...C Lr Dual states : Ǒ = O φ ˇφ ˇφ(x) φ(x) orneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse OctoberValencia) 5, / 16

22 The Setup [Polyakov] Alphabet : {φ A } localized fields and their derivatives in SYM: {W A } = {F µν, φ, ψ, F, φ, ψ... } Language : (gauge invariant) combinations of letters Words : simplest gauge invariants, one-trace composite operators, O A1 A 2...A L = tr W A1 W A2... W AL Phrases : O A1 A 2...A L1 O B1 B 2...B L2... O C1 C 2...C Lr Dual states : Ǒ = O φ ˇφ ˇφ(x) φ(x) 5, / 16

23 The Setup [Polyakov] Alphabet : {φ A } localized fields and their derivatives in SYM: {W A } = {F µν, φ, ψ, F, φ, ψ... } Language : (gauge invariant) combinations of letters Words : simplest gauge invariants, one-trace composite operators, O A1 A 2...A L = tr W A1 W A2... W AL Phrases : O A1 A 2...A L1 O B1 B 2...B L2... O C1 C 2...C Lr Dual states : Ǒ = O φ ˇφ ˇφ(x) φ(x) 5, / 16

24 The Setup [Polyakov] Alphabet : {φ A } localized fields and their derivatives in SYM: {W A } = {F µν, φ, ψ, F, φ, ψ... } Language : (gauge invariant) combinations of letters Words : simplest gauge invariants, one-trace composite operators, O A1 A 2...A L = tr W A1 W A2... W AL Phrases : O A1 A 2...A L1 O B1 B 2...B L2... O C1 C 2...C Lr Dual states : Ǒ = O φ ˇφ ˇφ(x) φ(x) 5, / 16

25 The Setup [Polyakov] Alphabet : {φ A } localized fields and their derivatives in SYM: {W A } = {F µν, φ, ψ, F, φ, ψ... } Language : (gauge invariant) combinations of letters Words : simplest gauge invariants, one-trace composite operators, O A1 A 2...A L = tr W A1 W A2... W AL Phrases : O A1 A 2...A L1 O B1 B 2...B L2... O C1 C 2...C Lr Dual states : Ǒ = O φ ˇφ ˇφ(x) φ(x) 5, / 16

26 Renormalization & Operator mixing Dimension is given by the renormalization scale. Consider a set of composite local operators O J closed under renormalization (mixing) O Ren J Anomalous part of Dilatation Operator = Z(Λ) J I O I = Z 1 Z(Λ) log Λ Anomalous dimensions O λ = λo λ 5, / 16

27 General Construction We have to analyze the RG-transformation of a composite operator O in perturbation theory. Mixing matrix Z(Λ) can be found considering divergent terms in correlators of O with another probe operator O, : O y (φ) :: O 0 : = : O y : e :V (φ): : O 0 : 0 The source of relevant divergences is the Wick expansion of products e ( :V (φ): : O 0 : = 1 : V (φ) : + 1 ) : V (φ) :: V (φ) : +... : O 0 : 2! So, we should modify O 0 in such a way to cancel divergences and find the scale dependence after the cancelation. Corneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

28 Tools Wick expansion in functional form can be cast into [see Kleinert] : O y :: O x := e ˇφ Ay D AB (y x) ˇφ Bx O y O x O O(x, y) star product resembles one in noncommutative theories! ˇφ Ax = φ A (x) Not a functional derivative! e.g. Euclidean massless propagator for a scalar field, D ab (x y) = 1 δ ab 4π 2 (x y) 2 Functional Wick expansion can be generalized to the product of 3, 4,... factors Corneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

29 Tools Differential regularization/renormalization scheme in real space allows to regularize singular expressions like [Freedman-Johnson-Latorre], 1 x 2k = 1 ln 4 k 1 (k 1)!(k 2)! k 1 µ2 x 2 x 2, k 2 introduces a scale dependence: µ [ ] [ ] 1 1 8π 2 µ x 2k x 2k = 4 k 1 (k 1)!(k 2)! k 2 δ(x) reg where we used the property 1 x 2 = 4π2 δ(x) Corneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

30 One vertex level Regularizing the terms in the Wick Expansion we get for the first order in interaction potential dy [V int (y) ] = = [ e ˇφ y D y ˇφ ] V y dy ( dy ˇφ y [D y ] ˇφ ( ˇφ y ˇφ y ) [D y D y ] ( ˇφ ˇφ) + 1 3! ( ˇφ 3 ) [Dy 3 ] ( ˇφ 3 ) + 1 ) 4! ( ˇφ 4 ) [Dy 4 ] ( ˇφ 4 ) +... V y, Corneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

31 One vertex level Regularizing the terms in the Wick Expansion we get for the first order in interaction potential dy [V int (y) ] = ( = dy dy [ e ˇφ y D y ˇφ ] V y 1 2 ( ˇφ y ˇφ y ) [D y D y ] ( ˇφ ˇφ) + 1 3! ( ˇφ 3 ) [Dy 3 ] ( ˇφ 3 ) + 1 4! ( ˇφ 4 ) [Dy 4 ] ( ˇφ 4 ) ) V y, Corneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

32 Two vertex level Second level yields 1 2! ( ˇφ 3 y 1 dy 1 dy 2 [V int (y 1 ) V int (y 2 ) ] = 1 dy 1 dy 2 2 { ( ˇφ y1 ˇφ y1 ˇφ y2 ) [D y1 D y1 y 2 D y2 ] ( ˇφ ˇφ y2 ˇφ)+ } ˇφ y2 ) [Dy 2 1 D y1 y 2 D y2 ] ( ˇφ 2 ˇφ y2 ˇφ) +... V y1 V y2. Corneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

33 Example: Compact sector of N = 4 SYM at one loop The compact sector is generated by scalar fields only and no derivatives, O a1...a L1 ;b 1...b L2 ;... = tr φ a1... φ al1 tr φ b1... φ bl2... The one-loop part comes from the first term in of the expansion: dy [V int (y) ] = 1 8π ˇδ 2 2 V int (φ) where ˇδf (φ) = ˇφ a (f (φ)) ˇφ a, V (φ) = g 2 1st level Wick 4 tr[φ a, φ b ] 2 Corneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

34 Conclusions Non-planar gauge string duality can be described using matrix models if we know the mixing matrix as a differential operator The problem can be solved by (almost) no Feynman diagram computation of at each order Simple examples give perfect agreement with known results more work to be done: two loops, application to more explicit examples etc Corneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

35 Conclusions Non-planar gauge string duality can be described using matrix models if we know the mixing matrix as a differential operator The problem can be solved by (almost) no Feynman diagram computation of at each order Simple examples give perfect agreement with known results more work to be done: two loops, application to more explicit examples etc Corneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

36 Conclusions Non-planar gauge string duality can be described using matrix models if we know the mixing matrix as a differential operator The problem can be solved by (almost) no Feynman diagram computation of at each order Simple examples give perfect agreement with known results more work to be done: two loops, application to more explicit examples etc orneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

37 Conclusions Non-planar gauge string duality can be described using matrix models if we know the mixing matrix as a differential operator The problem can be solved by (almost) no Feynman diagram computation of at each order Simple examples give perfect agreement with known results more work to be done: two loops, application to more explicit examples etc orneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

38 Conclusions Non-planar gauge string duality can be described using matrix models if we know the mixing matrix as a differential operator The problem can be solved by (almost) no Feynman diagram computation of at each order Simple examples give perfect agreement with known results more work to be done: two loops, application to more explicit examples etc orneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16

39 Conclusions Non-planar gauge string duality can be described using matrix models if we know the mixing matrix as a differential operator The problem can be solved by (almost) no Feynman diagram computation of at each order Simple examples give perfect agreement with known results more work to be done: two loops, application to more explicit examples etc orneliu Sochichiu (INFN LNF, Frascati & IAP, Chişinău, Dilatation Moldova Operator [3cm] 3d RTN Workshop ForcesUniverse October Valencia) 5, / 16