Casimir effect and the quark anti-quark potential. Zoltán Bajnok,

Transcription

1 Gauge/gravity duality, Munich, July 3, 23 Casimir effect and the quark anti-quark potential Zoltán Bajnok, MTA-Lendület Holographic QFT Group, Wigner esearch Centre for Physics, Hungary with L. Palla, G. Takács, J. Balog, A. Hegedűs, G.Zs. Tóth

2 Gauge/gravity duality, Munich, July 3, 23 Casimir effect and the quark anti-quark potential Zoltán Bajnok, MTA-Lendület Holographic QFT Group, Wigner esearch Centre for Physics, Hungary with L. Palla, G. Takács, J. Balog, A. Hegedűs, G.Zs. Tóth T L anti quark W = e T V q q q(l,λ) L q time quark space = e T E (L,λ) extra dimension

3 Gauge/gravity duality, Munich, July 3, 23 Casimir effect and the quark anti-quark potential Zoltán Bajnok, MTA-Lendület Holographic QFT Group, Wigner esearch Centre for Physics, Hungary with L. Palla, G. Takács, J. Balog, A. Hegedűs, G.Zs. Tóth F (L) = de (L) dl = d k(l) ω(k(l)) 2dL E (L) = d p 2π log( + ( p)( p)e 2E( p)l )

4 2 Motivation: Casimir-Polder effect Hendrik Casimir Dirk Polder colloidal solution: neutral atoms force not like Van der Waals F (L) A = cπ2 24L 4 not a theoretical curiosity!

5 Motivation: Casimir-Polder effect Hendrik Casimir Dirk Polder colloidal solution: neutral atoms force not like Van der Waals F (L) A = cπ2 24L 4 not a theoretical curiosity! Gecko legs

6 Motivation: Casimir-Polder effect Hendrik Casimir Dirk Polder colloidal solution: neutral atoms force not like Van der Waals F (L) A = cπ2 24L 4 not a theoretical curiosity! Gecko legs Airbag trigger chip

7 Motivation: Casimir-Polder effect Hendrik Casimir Dirk Polder colloidal solution: neutral atoms force not like Van der Waals F (L) A = cπ2 24L 4 not a theoretical curiosity! Gecko legs Airbag trigger chip micromechanical device: pieces stick friction, levitation

8 Motivation: Casimir-Polder effect Hendrik Casimir Dirk Polder colloidal solution: neutral atoms force not like Van der Waals F (L) A = cπ2 24L 4 not a theoretical curiosity! Gecko legs Airbag trigger chip micromechanical device: pieces stick friction, levitation Maritime analogy:

9 Aim: understand/describe planar Casimir effect 3

10 Aim: understand/describe planar Casimir effect Usual explanation: energy of the vacuum: E (L) = 2 k(l) ω(k(l))

11 Aim: understand/describe planar Casimir effect Usual explanation: energy of the vacuum: E (L) = 2 k(l) ω(k(l)) E (L) E ( ) 2E plate = E (L) ; E (L) A

12 Aim: understand/describe planar Casimir effect Usual explanation: energy of the vacuum: E (L) = 2 E (L) E ( ) 2E plate = E (L) ; Lifshitz formula: QED, Parallel dielectric slabs (ɛ,, ɛ 2 ) E (L) A = + k(l) ω(k(l)) E (L) [ d 2 q 8π2dζ log ɛ ω 2 q 2 ɛ ω 2 q 2 ɛ ω 2 q 2 + ɛ 2 ω 2 q 2 ] ɛ 2 ω 2 q 2 ɛ ω 2 q 2 ɛ 2 ω 2 q 2 + q2 +ζ 2 ɛ 2 ω 2 q 2e 2L [ d 2 q ω 2 q 2 ɛ ω 2 q 2 ω 8π2dζ log ω 2 q q 2 ] ɛ 2 ω 2 q 2 ɛ ω 2 q 2 ω 2 q 2 + ɛ 2 ω 2 q 2e 2L q 2 +ζ 2 A

13 Aim: understand/describe planar Casimir effect Usual explanation: energy of the vacuum: E (L) = 2 E (L) E ( ) 2E plate = E (L) ; Lifshitz formula: QED, Parallel dielectric slabs (ɛ,, ɛ 2 ) E (L) A = + k(l) ω(k(l)) E (L) [ d 2 q 8π2dζ log ɛ ω 2 q 2 ɛ ω 2 q 2 ɛ ω 2 q 2 + ɛ 2 ω 2 q 2 ] ɛ 2 ω 2 q 2 ɛ ω 2 q 2 ɛ 2 ω 2 q 2 + q2 +ζ 2 ɛ 2 ω 2 q 2e 2L [ d 2 q ω 2 q 2 ɛ ω 2 q 2 ω 8π2dζ log ω 2 q q 2 ] ɛ 2 ω 2 q 2 ɛ ω 2 q 2 ω 2 q 2 + ɛ 2 ω 2 q 2e 2L q 2 +ζ 2 Physics can be understood in + D QFT integrability helps to solve the problem even exactly L A

14 4 A simple calculation Free bulk + interacting boundaries (QED) E (L) = 2 k(l) ω(k(l)) e ikx e ikx + L e ikx ikx e Q(k) = e 2ikL (k) + ( k) =

15 A simple calculation Free bulk + interacting boundaries (QED) E (L) = 2 k(l) ω(k(l)) e ikx e ikx + L e ikx ikx e Q(k) = e 2ikL (k) + ( k) = E (L) = 2 j C j dk 2πi ω(k) d dk log Q C i k i k

16 A simple calculation Free bulk + interacting boundaries (QED) E (L) = 2 k(l) ω(k(l)) e ikx e ikx + L e ikx ikx e Q(k) = e 2ikL (k) + ( k) = E (L) = 2 j C j dk 2πi ω(k) d dk log Q C i k i k E (L) = 2 +, C + dk dω(k) 2πi dk log Q k i k C

17 A simple calculation Free bulk + interacting boundaries (QED) E (L) = 2 k(l) ω(k(l)) e ikx e ikx + L e ikx ikx e Q(k) = e 2ikL (k) + ( k) = E (L) = 2 j C j dk 2πi ω(k) d dk log Q C i k i k E (L) = 2 +, C + dk dω(k) 2πi dk log Q k i k C E (L) = e bulk L + e bdry + dk dω(k) 2πi dk log Q k i C k

18 A simple calculation Free bulk + interacting boundaries (QED) E (L) = 2 k(l) ω(k(l)) e ikx e ikx + L e ikx ikx e Q(k) = e 2ikL (k) + ( k) = E (L) = 2 j C j dk 2πi ω(k) d dk log Q C i k i k E (L) = 2 +, C + dk dω(k) 2πi dk log Q k i k C E (L) = e bulk L + e bdry + dk dω(k) 2πi dk log Q k i C k E ren (L) = d k 2π log( ( k) + ( k)e 2 ɛ( k)l ) iω C k i k

19 Free bulk + interacting boundaries (QED) E (L) = 2 k(l) ω(k(l)) A simple calculation e ikx e ikx + L e ikx ikx e Q(k) = e 2ikL (k) + ( k) = E (L) = 2 j C j dk 2πi ω(k) d dk log Q C i k i k E (L) = 2 +, C + dk dω(k) 2πi dk log Q k i k C E (L) = e bulk L + e bdry + dk dω(k) 2πi dk log Q k i C k E ren (L) = d k 2π log( ( k) + ( k)e 2 ɛ( k)l ) iω C interacting but integrable: similar formula k i k

20 Integrable boundary field theory: Bootstrap 5

21 Integrable boundary field theory: Bootstrap Boundary multiparticle state: with n particles v > v 2 >... > v n >

22 Boundary one particle state: Integrable boundary field theory: Bootstrap v

23 Integrable boundary field theory: Bootstrap Boundary one particle in state: t v

24 Integrable boundary field theory: Bootstrap Boundary one particle in state: t times develop v v

25 Integrable boundary field theory: Bootstrap Boundary one particle in state: t times develop further v v

26 Integrable boundary field theory: Bootstrap Boundary one particle in state: t v Boundary one pt out state: t v

27 Integrable boundary field theory: Bootstrap Boundary one particle in state: t v Boundary one pt out state: t v θ

28 Integrable boundary field theory: Bootstrap Boundary one particle in state: t v Boundary one pt out state: t v θ Free in particle Free out particle

29 Integrable boundary field theory: Bootstrap Boundary one particle in state: t v Boundary one pt out state: t v θ Free in particle -matrix Free out particle

30 Integrable boundary field theory: Bootstrap Boundary one particle in state: t v Boundary one pt out state: t v θ Free in particle -matrix Free out particle θ B = (θ) θ B

31 Integrable boundary field theory: Bootstrap Boundary one particle in state: t v Boundary one pt out state: t v θ Free in particle -matrix Free out particle θ B = (θ) θ B Unitarity (θ) = (θ) = ( θ) Boundary crossing unitarity ( iπ 2 + θ) = S(2θ)(iπ 2 θ) θ 2θ

32 Integrable boundary field theory: Bootstrap Boundary one particle in state: t Boundary one pt out state: t v θ Free in particle -matrix Free out particle v θ B = (θ) θ B 2θ Unitarity (θ) = (θ) = ( θ) θ Boundary crossing unitarity ( iπ 2 + θ) = S(2θ)(iπ 2 θ) sinh-gordon S(θ) = reflection factor (θ) = ( 2 sinh θ i sin πp sinh θ+i sin πp = [ p] = ( p)( + p) ; (p) = sinh ( θ + iπp 2 ) ( +p 2 ) ( p 2 ) [ 3 2 ηp ] [ ] 3 π 2 Θp π Lagrangian: L = 2 ( φ)2 µ(cosh bφ ) δ{µ B + e b 2 φ + µ B e b 2 φ } sinh( θ 2 iπp 2 ) 2 ) Ghoshal-Zamolodchikov 94 µ cosh b 2 (η ± Θ) = µ B sin b 2 ±

33 Integrable boundary field theory: Bootstrap Boundary one particle in state: t Boundary one pt out state: t v θ Free in particle -matrix Free out particle θ B = (θ) θ B v 2θ Unitarity (θ) = (θ) = ( θ) θ Boundary crossing unitarity ( iπ 2 + θ) = S(2θ)(iπ 2 θ) sinh-gordon S(θ) = sinh θ i sin πp sinh θ+i sin πp = [ p] = ( p)( + p) ; (p) = sinh ( θ + iπp 2 ) ( +p 2 ) ( p 2 ) [ 3 2 ηp ] [ ] 3 π 2 Θp π reflection factor (θ) = ( 2 Lagrangian: L = 2 ( φ)2 µ(cosh bφ ) δ{µ B + e b φ 2 + µ B e b φ 2 } sinh( θ 2 iπp 2 ) 2 ) Ghoshal-Zamolodchikov 94 µ cosh b 2 (η ± Θ) = µ B sin b 2 ± Integrability factorizability: θ, θ 2,..., θ n B = i<j S(θ i θ j )S(θ i +θ j ) i (θ i) θ, θ 2,..., θ n B

34 6 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) =

35 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L

36 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = lim log(tr(e HB (L) )) = lim log(e E (L) ) L

37 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L lim log(tr(e HB (L) )) = lim log B e H()L B L

38 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L lim log(tr(e HB (L) )) = lim log B e H()L B Boundary state B = exp { dθ 4π (iπ 2 θ)a+ ( θ)a + (θ) } + + L + +

39 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L lim log(tr(e HB (L) )) = lim log B e H()L B Boundary state B = exp { dθ 4π (iπ 2 θ)a+ ( θ)a + (θ) } Dominant contribution for large L: two particle term + + B e H()L B = + k ( iπ 2 θ)(iπ 2 + θ)e 2m cosh θ kl +... L + +

40 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L lim log(tr(e HB (L) )) = lim log B e H()L B Boundary state B = exp { dθ 4π (iπ 2 θ)a+ ( θ)a + (θ) } Dominant contribution for large L: two particle term + + B e H()L B = + k ( iπ 2 θ)(iπ 2 + θ)e 2m cosh θ kl +... quantization condition: m sinh θ k = 2π k m 4π dθ cosh θ L + + E (L) = m cosh θdθ 4π ( iπ 2 θ)(iπ 2 + θ) e 2mL cosh θ ; [Z.B, L. Palla, G. Takacs]

41 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L lim log(tr(e HB (L) )) = lim log B e H()L B Boundary state B = exp { dθ 4π (iπ 2 θ)a+ ( θ)a + (θ) } Dominant contribution for large L: two particle term + + B e H()L B = + k ( iπ 2 θ)(iπ 2 + θ)e 2m cosh θ kl +... quantization condition: m sinh θ k = 2π k m 4π dθ cosh θ L + + E (L) = m cosh θdθ 4π ( iπ 2 θ)(iπ 2 + θ) e 2mL cosh θ ; [Z.B, L. Palla, G. Takacs] Ground state energy exactly: E (L) = m dθ 4π cosh(θ) log( + e ɛ(θ) ) ɛ(θ) = 2mL cosh θ log(( iπ 2 θ)(iπ 2 + θ)) dθ 2π ϕ (θ θ ) log( + e ɛ(θ ) ) [LeClair, Mussardo, Saleur, Skorik] L

42 Casimir effect: Boundary finite size effect 7

43 Casimir effect: Boundary finite size effect Extension to higher dimensions: k label Dispersion ω = m 2 + k 2 + k2 = m 2 eff + k2 k rapidity ω = m eff (k ) cosh θ, k = m eff (k ) sinh θ k eflection (θ, m eff (k )) L

44 Casimir effect: Boundary finite size effect Extension to higher dimensions: k label Dispersion ω = m 2 + k 2 + k2 = m 2 eff + k2 k rapidity ω = m eff (k ) cosh θ, k = m eff (k ) sinh θ k eflection (θ, m eff (k )) Bstate: B = { + dd k dθ (2π) D 4π (iπ 2 θ, m eff (k ))A+ ( θ, k )A + (θ, k ) +... L }

45 Casimir effect: Boundary finite size effect Extension to higher dimensions: k label Dispersion ω = m 2 + k 2 + k2 = m 2 eff + k2 k rapidity ω = m eff (k ) cosh θ, k = m eff (k ) sinh θ k eflection (θ, m eff (k )) Bstate: B = { + dd k dθ (2π) D 4π (iπ 2 θ, m eff (k ))A+ ( θ, k )A + (θ, k ) +... Ground state energy (for free bulk): L } E (L) = dd k dθ (2π) D 4π log( + ( iπ 2 + θ, m eff )2 ( iπ 2 θ, m eff )e 2m eff cosh θl )

46 Casimir effect: Boundary finite size effect Extension to higher dimensions: k label Dispersion ω = m 2 + k 2 + k2 = m 2 eff + k2 k rapidity ω = m eff (k ) cosh θ, k = m eff (k ) sinh θ k eflection (θ, m eff (k )) Bstate: B = { + dd k dθ (2π) D 4π (iπ 2 θ, m eff (k ))A+ ( θ, k )A + (θ, k ) +... Ground state energy (for free bulk): L } E (L) = dd k dθ (2π) D 4π log( + ( iπ 2 + θ, m eff )2 ( iπ 2 θ, m eff )e 2m eff cosh θl ) QED: Parallel dielectric slabs (ɛ,, ɛ 2 ) reflections E,, B,, look it up in Jackson: (ω, k = q) = ω 2 q 2 ɛω 2 q 2 ω 2 q 2 + ɛω 2 q 2 (ω, k = q) = ɛ ω 2 q 2 ɛω 2 q 2 ɛ ω 2 q 2 + ɛω 2 q 2

47 Casimir effect: Boundary finite size effect Extension to higher dimensions: k label Dispersion ω = m 2 + k 2 + k2 = m 2 eff + k2 k rapidity ω = m eff (k ) cosh θ, k = m eff (k ) sinh θ k eflection (θ, m eff (k )) Bstate: B = { + dd k dθ (2π) D 4π (iπ 2 θ, m eff (k ))A+ ( θ, k )A + (θ, k ) +... Ground state energy (for free bulk): L } E (L) = dd k dθ (2π) D 4π log( + ( iπ 2 + θ, m eff )2 ( iπ 2 θ, m eff )e 2m eff cosh θl ) QED: Parallel dielectric slabs (ɛ,, ɛ 2 ) reflections E,, B,, look it up in Jackson: Lifshitz formula (ω, k = q) = ω 2 q 2 ɛω 2 q 2 ω 2 q 2 + ɛω 2 q 2 (ω, k = q) = ɛ ω 2 q 2 ɛω 2 q 2 ɛ ω 2 q 2 + ɛω 2 q 2

48 Conclusion about Casimir 8

49 Conclusion about Casimir Usual derivation: summing up zero freqencies E (L) = 2 k(l) ω(k(l)) Complicated finite volume problem + divergencies

50 Conclusion about Casimir Usual derivation: summing up zero freqencies E (L) = 2 k(l) ω(k(l)) Complicated finite volume problem + divergencies as a boundary finite size effect E (L) = d p 2π log( + ( p)( p)e 2 ɛ( p)l ) eflection factor of the I degrees of freedom: semi infinite settings, easier to calculate, no divergences

51 Main problem: q q potential in N = 4 SYM T anti quark L q q L W = e T L V q q(λ) time quark space extra dimension = e T E (L,λ) 9

52 AdS/CFT correspondence (Maldacena 997) The Illusion of Gravity - Juan Maldacena, Scientific American (25)

53 AdS/CFT correspondence (Maldacena 997) 2 α II B superstring on AdS 5 S 5 4D Minkowski space time space extra dimension 5D anti de Sitter space 5D sphere 6 Y i 2 = = ( ) 2 a X M a X M + a Y M a Y M +... dτdσ 4π 2 g 2 Y M N = 4 D=4 SU(N) SYM Ψ,2,3,4 A µ Φ,2,3,4,5,6 Ψ,2,3,4 d 4 xtr [ 4 F 2 2 (DΦ)2 + iψd/ Ψ + V ] V (Φ, Ψ) = 4 [Φ, Φ]2 + Ψ[Φ, Ψ] P SU(2,2 4) β = superconformal SO(5) SO(,4) gaugeinvariants:o = Tr(Φ L ), det( )

54 AdS/CFT correspondence (Maldacena 997) 2 α II B superstring on AdS 5 S 5 4D Minkowski space time space extra dimension 5D anti de Sitter space 5D sphere 6 Y i 2 = = ( ) 2 a X M a X M + a Y M a Y M +... dτdσ 4π 2 g 2 Y M N = 4 D=4 SU(N) SYM Ψ,2,3,4 A µ Φ,2,3,4,5,6 Ψ,2,3,4 d 4 xtr [ 4 F 2 2 (DΦ)2 + iψd/ Ψ + V ] V (Φ, Ψ) = 4 [Φ, Φ]2 + Ψ[Φ, Ψ] P SU(2,2 4) β = superconformal SO(5) SO(,4) gaugeinvariants:o = Tr(Φ L ), det( ) Couplings: λ = 2 α, g s = λ N 2D QFT String energy levels: E(λ) E(λ) = E( ) + E + E 2 λ λ +... Dictionary strong weak λ = g 2 Y M N, N planar limit O n (x)o m () = δ nm x 2 n(λ) Anomalous dim (λ) (λ) = () + λ + λ

55 AdS/CFT correspondence (Maldacena 997) 2 α II B superstring on AdS 5 S 5 4D Minkowski space time space extra dimension 5D anti de Sitter space 5D sphere 6 Y i 2 = = ( ) 2 a X M a X M + a Y M a Y M +... dτdσ 4π 2 g 2 Y M N = 4 D=4 SU(N) SYM Ψ,2,3,4 A µ Φ,2,3,4,5,6 Ψ,2,3,4 d 4 xtr [ 4 F 2 2 (DΦ)2 + iψd/ Ψ + V ] V (Φ, Ψ) = 4 [Φ, Φ]2 + Ψ[Φ, Ψ] P SU(2,2 4) β = superconformal SO(5) SO(,4) gaugeinvariants:o = Tr(Φ L ), det( ) Couplings: λ = 2 α, g s = λ N 2D QFT String energy levels: E(λ) E(λ) = E( ) + E + E 2 λ λ +... Dictionary strong weak λ = gy 2 MN, N planar limit O n (x)o m () = δ nm x 2 n(λ) Anomalous dim (λ) (λ) = () + λ + λ D integrable QFT

56 quark-antiquark potential AdS/CFT integrability: q q potential time L quark anti quark space extra dimension V (L) = λ 4πL +... Wilson loop: Pe C A µdx µ + Φ n ẋ ds e T L V q q(λ,θ) strong coupling V (r) = 4π2 2λ Γ( 4 )4 L ( ) λ minimal surface+fluctuations Integrable system on the strip quark time space E (L) = d k 2π log( ( k) + ( k)e ɛ( k) ) 2

57 3 -matrix bootstrap program Boundary asymptotic states p, p 2,..., p n in/out form a representation of global symmetry: p > p n >

58 -matrix bootstrap program Boundary asymptotic states p, p 2,..., p n in/out form a representation of global symmetry: p > p n > Lorentz E = i E(p i ) dispersion relation E(p) = m 2 + p 2

59 -matrix bootstrap program Boundary asymptotic states p, p 2,..., p n in/out form a representation of global symmetry: p > p n > Lorentz E = i E(p i ) dispersion relation E(p) = ( Q) m 2 + p 2 p > p n >

60 -matrix bootstrap program Boundary asymptotic states p, p 2,..., p n in/out form a representation of global symmetry: p > p n > Lorentz E = i E(p i ) dispersion relation E(p) = m 2 + p 2 eflection matrix : out in commutes with sym. [, (Q)] =

61 -matrix bootstrap program Boundary asymptotic states p, p 2,..., p n in/out form a representation of global symmetry: p > p n > Lorentz E = i E(p i ) dispersion relation E(p) = m 2 + p 2 p p n eflection matrix : out in commutes with sym. [, (Q)] = p > p n >

62 -matrix bootstrap program Boundary asymptotic states p, p 2,..., p n in/out form a representation of global symmetry: p > p n > Lorentz E = i E(p i ) dispersion relation E(p) = m 2 + p 2 ( Q) p p n eflection matrix : out in commutes with sym. [, (Q)] = ( Q) p > p n >

63 -matrix bootstrap program Boundary asymptotic states p, p 2,..., p n in/out form a representation of global symmetry: p > p n > Lorentz E = i E(p i ) dispersion relation E(p) = m 2 + p 2 ( Q) p p n eflection matrix : out in commutes with sym. [, (Q)] = Higher spin concerved charge factorization + Bdry Yang-Baxter equation 2 = S 2 S 2 2 = 2 S 2 S 2 p p 2 ( Q) p > p n > p p 2 -matrix = scalar. Matrix p p 2 p p 2

64 -matrix bootstrap program Boundary asymptotic states p, p 2,..., p n in/out form a representation of global symmetry: p > p n > Lorentz E = i E(p i ) dispersion relation E(p) = m 2 + p 2 ( Q) p p n eflection matrix : out in commutes with sym. [, (Q)] = Higher spin concerved charge factorization + Bdry Yang-Baxter equation 2 = S 2 S 2 2 = 2 S 2 S 2 p p 2 ( Q) p > p n > p p 2 -matrix = scalar. Matrix p p 2 p p 2 Unitarity (p)( p) = Id Boundary crossing symmetry (p) = S(p, p)( p) Maximal analyticity: all poles have physical origin boundstates, anomalous thresholds

65 4 -matrix bootstrap program: AdS Nondiagonal scattering: -matrix = scalar. Matrix

66 -matrix bootstrap program: AdS Nondiagonal scattering: -matrix = scalar. Matrix -matrix: [Correa-Maldacena-Sever,Drukker] global symmetry P SU(2 2) diag Q = reps b b 2 f 3 f 4 b b 2 f 3 f 4 [, (Q)] = ( Q) ( Q) p p n p > p n >

67 -matrix bootstrap program: AdS Nondiagonal scattering: -matrix = scalar. Matrix -matrix: [Correa-Maldacena-Sever,Drukker] global symmetry P SU(2 2) diag Q = reps b b 2 f 3 f 4 b b 2 f 3 f 4 p p n [, (Q)] = ( Q) ( Q) β β α α (p) = Sβ β α α (p, p) (p) p > p n >

68 -matrix bootstrap program: AdS Nondiagonal scattering: -matrix = scalar. Matrix -matrix: [Correa-Maldacena-Sever,Drukker] global symmetry P SU(2 2) diag Q = reps b b 2 f 3 f 4 b b 2 f 3 f 4 p p n [, (Q)] = ( Q) β β α α (p) = Sβ β α α (p, p) (p) ( Q) p > p n > Unitarity (z)( z) = Crossing symmetry (z) = S(z, z)(ω 2 z) (p) = σ B(p) σ(p, p) σ B = e iχ(x+ ) iχ(x ) boundary dressing phase χ(x) = dz 2π x z sinh(2πg(z+z )) 2πg(z+z )

69 Nondiagonal scattering: -matrix bootstrap program: AdS -matrix = scalar. Matrix -matrix: [Correa-Maldacena-Sever,Drukker] global symmetry P SU(2 2) diag Q = reps b b 2 f 3 f 4 b b 2 f 3 f 4 p p n [, (Q)] = ( Q) β β α α (p) = Sβ β α α (p, p) (p) ( Q) p > p n > Unitarity (z)( z) = Crossing symmetry (z) = S(z, z)(ω 2 z) (p) = σ B(p) σ(p, p) σ B = e iχ(x+ ) iχ(x ) boundary dressing phase χ(x) = dz 2π x z sinh(2πg(z+z )) 2πg(z+z ) Maximal analyticity: no boundstates

70 5 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) =

71 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L

72 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = lim log(tr(e HB (L) )) = lim log(e E (L) ) L

73 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L lim log(tr(e HB (L) )) = lim log B e H()L B L

74 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L lim log(tr(e HB (L) )) = lim log B e H()L B Boundary state B = exp { dq 4π b a ( q)cad A + b ( q)a+ d (q)} + + L + +

75 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L lim log(tr(e HB (L) )) = lim log B e H()L B Boundary state B = exp { dq 4π b a ( q)cad A + b ( q)a+ d (q)} + + L + + Folding trick:

76 Boundary thermodynamic Bethe Ansatz Groundstate energy for large L from I reflection: E (L) = L lim log(tr(e HB (L) )) = lim log B e H()L B Boundary state B = exp { dq 4π b a( q)c ad A + b ( q)a+ d (q)} + + L + + Folding trick: [Correa,Maldacena,Sever 2][Drukker 2] Ground state energy exactly: E (L) = Q d p 4π log( + e ɛ Q( p) ) ɛ j ( p) = δ j Q (σ Q( p) + 2Ẽ Q ( p)l) K j i ( p, p ) log( + e ɛi ( p ) )d p

77 6 egularized q q BTBA equations Singular boundary fugacity: σ Q () =, no-obvious weak coupling expansion shifting countours regularization (extra source terms, excited state TBA) log Y Q = 2(f + Ψ)Q ɛ Q + log σ Q + D Q Q (iu Q ) + log( + Y Q ) η K Q Q + [ 2 log ( ) + Y v s ˆ KyQ + 2 log( + Y v Q ) s 2 log Y Kvx Y +ˆ s Q + log Y Y + ˆ K Q + log( Y )( Y + )ˆ K yq ] log Y Y + = 2D xvs (iu Q ) D Q (iu Q ) log( + Y Q ) η K Q + 2 log( + Y Q ) K Q xv s + 2 log + Y v + Y w log Y + Y = D Qy (iu Q ) + log( + Y Q ) η K Qy log Y v M = D s (iu M+ ) log( + Y M+ ) η s + I MN log( + Y v N ) s + δ M log Y Y +ˆ s log Y w M = I MN log( + Y w N ) s + δ M log Y Y + ˆ s [ZB, Balog, Hegedus, Toth 3] f = i(π φ) ; Ψ = i(π φ) ; = 2L

78 7 quark-antiquark potential Pe C A µdx µ + Φ n Z J Pe C A µdx µ + Φ n θ q q potential: weak coupling expansion V q q (g, φ, θ) = Γ k g 2k Γ = k n= ( φ cos φ cosh θ sin φ ) n γ (n) k γ () = φ 2 ; γ () 2 = φ 2 (φ2 π 2 ) V (r) = 4π2 2λ QMinimal surface Wilson loop strong coupling Γ( 4 )4 L (.3359 λ +...) minimal surface+fluctuations γ (2) 2 () = γ(2) 2 () = γ (2) 2 (φ) = φ 2 cot φ