Casimir effect and the quarkanti-quark potential Zolt an Bajnok , - - PowerPoint PPT Presentation

Gauge/gravity duality, Munich, July 30, 2013

Casimir effect and the quark–anti-quark potential Zolt´ an Bajnok,

MTA-Lend¨ ulet Holographic QFT Group, Wigner Research Centre for Physics, Hungary with L. Palla, G. Tak´ acs, J. Balog, A. Heged˝ us, G.Zs. T´

• th

1

Gauge/gravity duality, Munich, July 30, 2013

Casimir effect and the quark–anti-quark potential Zolt´ an Bajnok,

MTA-Lend¨ ulet Holographic QFT Group, Wigner Research Centre for Physics, Hungary with L. Palla, G. Tak´ acs, J. Balog, A. Heged˝ us, G.Zs. T´

• th

W = e−TVq¯

q(L,λ)

L q q T

= e−TE0(L,λ)

extra dimension quark space time anti−quark L

Gauge/gravity duality, Munich, July 30, 2013

Casimir effect and the quark–anti-quark potential Zolt´ an Bajnok,

MTA-Lend¨ ulet Holographic QFT Group, Wigner Research Centre for Physics, Hungary with L. Palla, G. Tak´ acs, J. Balog, A. Heged˝ us, G.Zs. T´

• th

F(L) = dE0(L)

dL

=

d

k(L) ω(k(L))

2dL

E0(L) = −

d˜

p 2π log(1 + R(−˜

p)R(˜ p)e−2E(˜

p)L)

Motivation: Casimir-Polder effect

Hendrik Casimir Dirk Polder colloidal solution: neutral atoms force not like Van der Waals

F(L) A

= − cπ2

240L4

not a theoretical curiosity!

2

Motivation: Casimir-Polder effect

Hendrik Casimir Dirk Polder colloidal solution: neutral atoms force not like Van der Waals

F(L) A

= − cπ2

240L4

not a theoretical curiosity! Gecko legs

Motivation: Casimir-Polder effect

Hendrik Casimir Dirk Polder colloidal solution: neutral atoms force not like Van der Waals

F(L) A

= − cπ2

240L4

not a theoretical curiosity! Gecko legs Airbag trigger chip

Motivation: Casimir-Polder effect

Hendrik Casimir Dirk Polder colloidal solution: neutral atoms force not like Van der Waals

F(L) A

= − cπ2

240L4

not a theoretical curiosity! Gecko legs Airbag trigger chip micromechanical device: pieces stick friction, levitation

Motivation: Casimir-Polder effect

Hendrik Casimir Dirk Polder colloidal solution: neutral atoms force not like Van der Waals

F(L) A

= − cπ2

240L4

not a theoretical curiosity! Gecko legs Airbag trigger chip micromechanical device: pieces stick friction, levitation Maritime analogy:

Aim: understand/describe planar Casimir effect

3

Aim: understand/describe planar Casimir effect

Usual explanation: energy of the vacuum: E0(L) = 1

2

• k(L) ω(k(L)) ∝ ∞

Aim: understand/describe planar Casimir effect

Usual explanation: energy of the vacuum: E0(L) = 1

2

• k(L) ω(k(L)) ∝ ∞

E0(L) − E0(∞) − 2Eplate = ∆E0(L) ; ∆E0(L) A

Aim: understand/describe planar Casimir effect

Usual explanation: energy of the vacuum: E0(L) = 1

2

• k(L) ω(k(L)) ∝ ∞

E0(L) − E0(∞) − 2Eplate = ∆E0(L) ; ∆E0(L) A Lifshitz formula: QED, Parallel dielectric slabs (ǫ1, 1, ǫ2)

∆E0(L) A =

∞

d2q 8π2dζ log

• 1 − ǫ1
• ω2 − q2 −
• ǫ1ω2 − q2

ǫ1

• ω2 − q2 +
• ǫ1ω2 − q2

ǫ2

• ω2 − q2 −
• ǫ2ω2 − q2

ǫ2

• ω2 − q2 +
• ǫ2ω2 − q2e−2L√

q2+ζ2

• +

∞

d2q 8π2dζ log

• 1 −
• ω2 − q2 −
• ǫ1ω2 − q2
• ω2 − q2 +
• ǫ1ω2 − q2
• ω2 − q2 −
• ǫ2ω2 − q2
• ω2 − q2 +
• ǫ2ω2 − q2e−2L√

q2+ζ2

Aim: understand/describe planar Casimir effect

Usual explanation: energy of the vacuum: E0(L) = 1

2

• k(L) ω(k(L)) ∝ ∞

E0(L) − E0(∞) − 2Eplate = ∆E0(L) ; ∆E0(L) A Lifshitz formula: QED, Parallel dielectric slabs (ǫ1, 1, ǫ2)

∆E0(L) A =

∞

d2q 8π2dζ log

• 1 − ǫ1
• ω2 − q2 −
• ǫ1ω2 − q2

ǫ1

• ω2 − q2 +
• ǫ1ω2 − q2

ǫ2

• ω2 − q2 −
• ǫ2ω2 − q2

ǫ2

• ω2 − q2 +
• ǫ2ω2 − q2e−2L√

q2+ζ2

• +

∞

d2q 8π2dζ log

• 1 −
• ω2 − q2 −
• ǫ1ω2 − q2
• ω2 − q2 +
• ǫ1ω2 − q2
• ω2 − q2 −
• ǫ2ω2 − q2
• ω2 − q2 +
• ǫ2ω2 − q2e−2L√

q2+ζ2

• Physics can be understood in 1+1 D QFT

L

integrability helps to solve the problem even exactly

A simple calculation

Free bulk + interacting boundaries (QED) E0(L) = 1

2

• k(L) ω(k(L))

eikx eikx

+

L R e

−ikx −ikx −

R e

Q(k) = e2ikLR−(k)R+(−k) − 1 = 0

4

A simple calculation

Free bulk + interacting boundaries (QED) E0(L) = 1

2

• k(L) ω(k(L))

eikx eikx

+

L R e

−ikx −ikx −

R e

Q(k) = e2ikLR−(k)R+(−k) − 1 = 0 E0(L) = 1

2

• j
• j

dk 2πiω(k) d dk log Q

k k i Ci

A simple calculation

Free bulk + interacting boundaries (QED) E0(L) = 1

2

• k(L) ω(k(L))

eikx eikx

+

L R e

−ikx −ikx −

R e

Q(k) = e2ikLR−(k)R+(−k) − 1 = 0 E0(L) = 1

2

• j
• j

dk 2πiω(k) d dk log Q

k k i Ci

E0(L) = 1

2

• +,−

dk

2πi dω(k) dk

log Q

k k i C+ C−

A simple calculation

Free bulk + interacting boundaries (QED) E0(L) = 1

2

• k(L) ω(k(L))

eikx eikx

+

L R e

−ikx −ikx −

R e

Q(k) = e2ikLR−(k)R+(−k) − 1 = 0 E0(L) = 1

2

• j
• j

dk 2πiω(k) d dk log Q

k k i Ci

E0(L) = 1

2

• +,−

dk

2πi dω(k) dk

log Q

k k i C+ C−

E0(L) = ebulkL + ebdry +

dk

2πi dω(k) dk

log Q

k k i C

A simple calculation

Free bulk + interacting boundaries (QED) E0(L) = 1

2

• k(L) ω(k(L))

eikx eikx

+

L R e

−ikx −ikx −

R e

Q(k) = e2ikLR−(k)R+(−k) − 1 = 0 E0(L) = 1

2

• j
• j

dk 2πiω(k) d dk log Q

k k i Ci

E0(L) = 1

2

• +,−

dk

2πi dω(k) dk

log Q

k k i C+ C−

E0(L) = ebulkL + ebdry +

dk

2πi dω(k) dk

log Q

k k i C

Eren (L) =

d˜

k 2π log(1 − R−(˜

k)R+(−˜ k)e−2˜

ǫ(˜ k)L)

k k i C

−i ω

A simple calculation

Free bulk + interacting boundaries (QED) E0(L) = 1

2

• k(L) ω(k(L))

eikx eikx

+

L R e

−ikx −ikx −

R e

Q(k) = e2ikLR−(k)R+(−k) − 1 = 0 E0(L) = 1

2

• j
• j

dk 2πiω(k) d dk log Q

k k i Ci

E0(L) = 1

2

• +,−

dk

2πi dω(k) dk

log Q

k k i C+ C−

E0(L) = ebulkL + ebdry +

dk

2πi dω(k) dk

log Q

k k i C

Eren (L) =

d˜

k 2π log(1 − R−(˜

k)R+(−˜ k)e−2˜

ǫ(˜ k)L)

k k i C

−i ω

interacting but integrable: similar formula

Integrable boundary field theory: Bootstrap

5

Integrable boundary field theory: Bootstrap

Boundary multiparticle state: with n particles

• v1

> v2 > ... > vn >

Integrable boundary field theory: Bootstrap

Boundary one particle state:

• v1

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

times develop

• v1

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

times develop further

• v1

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

Boundary one pt out state: t → ∞

• v1

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

Boundary one pt out state: t → ∞

• v1
• θ

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

Boundary one pt out state: t → ∞

• v1
• θ

Free in particle Free out particle

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

Boundary one pt out state: t → ∞

• v1
• θ

Free in particle ← R-matrix → Free out particle

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

Boundary one pt out state: t → ∞

• v1
• θ

Free in particle ← R-matrix → Free out particle |θB = R(θ)| − θB

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

Boundary one pt out state: t → ∞

• v1
• θ

Free in particle ← R-matrix → Free out particle |θB = R(θ)| − θB Unitarity R∗(θ) = R−1(θ) = R(−θ) Boundary crossing unitarity R(iπ

2 + θ) = S(2θ)R(iπ 2 − θ)

• −θ
• 2θ

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

Boundary one pt out state: t → ∞

• v1
• θ

Free in particle ← R-matrix → Free out particle |θB = R(θ)| − θB Unitarity R∗(θ) = R−1(θ) = R(−θ) Boundary crossing unitarity R(iπ

2 + θ) = S(2θ)R(iπ 2 − θ)

• −θ
• 2θ

sinh-Gordon S(θ) = sinh θ−i sin πp

sinh θ+i sin πp = [−p] = −(−p)(1 + p) ; (p) = sinh(

θ 2+ iπp 2 )

sinh(

θ 2− iπp 2 )

reflection factor R(θ) = 1

2 1+p 2

1 − p

2

3

2 − ηp π 3 2 − Θp π

• Ghoshal-Zamolodchikov ’94

Lagrangian: L = 1

2(∂φ)2−µ(cosh bφ − 1) − δ{µB +e

b 2φ + µB

−e− b

2φ}

• µ

sin b2 cosh b2(η ± Θ) = µB ±

Integrable boundary field theory: Bootstrap

Boundary one particle in state: t → −∞

• v1

Boundary one pt out state: t → ∞

• v1
• θ

Free in particle ← R-matrix → Free out particle |θB = R(θ)| − θB Unitarity R∗(θ) = R−1(θ) = R(−θ) Boundary crossing unitarity R(iπ

2 + θ) = S(2θ)R(iπ 2 − θ)

• −θ
• 2θ

sinh-Gordon S(θ) = sinh θ−i sin πp

sinh θ+i sin πp = [−p] = −(−p)(1 + p) ; (p) = sinh(

θ 2+ iπp 2 )

sinh(

θ 2− iπp 2 )

reflection factor R(θ) =

1

2 1+p 2

1 − p

2

3

2 − ηp π 3 2 − Θp π

• Ghoshal-Zamolodchikov ’94

Lagrangian: L = 1

2(∂φ)2−µ(cosh bφ − 1) − δ{µB +e

b 2φ + µB

−e− b

2φ}

• µ

sin b2 cosh b2(η ± Θ) = µB ±

Integrability →factorizability: |θ1, θ2, . . . , θnB =

i<j S(θi−θj)S(θi+θj) i R(θi)|−θ1, −θ2, . . . , −θnB

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) =

6

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) =

L

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R log(e−E0(L)R)

L R

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R logB|e−H(R)L|B

L R

R L

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R logB|e−H(R)L|B

L R

R L

Boundary state |B = exp

∞

−∞ dθ 4πR(iπ 2 − θ)A+(−θ)A+(θ)

• |0
• +

+ 1

• +

+ 1

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R logB|e−H(R)L|B

L R

R L

Boundary state |B = exp

∞

−∞ dθ 4πR(iπ 2 − θ)A+(−θ)A+(θ)

• |0
• +

+ 1

• +

+ 1

Dominant contribution for large L: two particle term B|e−H(R)L|B = 1 +

k R(iπ 2 − θ)R(iπ 2 + θ)e−2m cosh θkL + . . .

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R logB|e−H(R)L|B

L R

R L

Boundary state |B = exp

∞

−∞ dθ 4πR(iπ 2 − θ)A+(−θ)A+(θ)

• |0
• +

+ 1

• +

+ 1

Dominant contribution for large L: two particle term B|e−H(R)L|B = 1 +

k R(iπ 2 − θ)R(iπ 2 + θ)e−2m cosh θkL + . . .

quantization condition: m sinh θk = 2π

R

• k → Rm

4π

dθ cosh θ

E0(L) = −

m cosh θdθ

4π

R(iπ

2 − θ)R(iπ 2 + θ) e−2mL cosh θ; [Z.B, L. Palla, G. Takacs]

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R logB|e−H(R)L|B

L R

R L

Boundary state |B = exp

∞

−∞ dθ 4πR(iπ 2 − θ)A+(−θ)A+(θ)

• |0
• +

+ 1

• +

+ 1

Dominant contribution for large L: two particle term B|e−H(R)L|B = 1 +

k R(iπ 2 − θ)R(iπ 2 + θ)e−2m cosh θkL + . . .

quantization condition: m sinh θk = 2π

R

• k → Rm

4π

dθ cosh θ

E0(L) = −

m cosh θdθ

4π

R(iπ

2 − θ)R(iπ 2 + θ) e−2mL cosh θ; [Z.B, L. Palla, G. Takacs]

Ground state energy exactly: E0(L) = −m

dθ

4π cosh(θ) log(1 + e−ǫ(θ))

ǫ(θ) = 2mL cosh θ − log(R(iπ

2 − θ)R(iπ 2 + θ))

−

dθ′

2πϕ′(θ − θ′) log(1 + e−ǫ(θ′)) [LeClair, Mussardo, Saleur, Skorik]

L R R

Casimir effect: Boundary finite size effect

7

Casimir effect: Boundary finite size effect

Extension to higher dimensions: k label

k k L

Dispersion ω =

• m2 +

k2

+ k2 ⊥ =

• m2

eff + k2

⊥

rapidity ω = meff(k) cosh θ, k⊥ = meff(k) sinh θ Reflection R(θ, meff(k))

Casimir effect: Boundary finite size effect

Extension to higher dimensions: k label

k k L

Dispersion ω =

• m2 +

k2

+ k2 ⊥ =

• m2

eff + k2

⊥

rapidity ω = meff(k) cosh θ, k⊥ = meff(k) sinh θ Reflection R(θ, meff(k)) Bstate: |B =

• 1 +
• dD−1k

(2π)D−1 dθ 4πR(iπ 2 − θ, meff(k))A+(−θ, −

k)A+(θ, k) + ...

• |0

Casimir effect: Boundary finite size effect

Extension to higher dimensions: k label

k k L

Dispersion ω =

• m2 +

k2

+ k2 ⊥ =

• m2

eff + k2

⊥

rapidity ω = meff(k) cosh θ, k⊥ = meff(k) sinh θ Reflection R(θ, meff(k)) Bstate: |B =

• 1 +
• dD−1k

(2π)D−1 dθ 4πR(iπ 2 − θ, meff(k))A+(−θ, −

k)A+(θ, k) + ...

• |0

Ground state energy (for free bulk): E0(L) =

• dD−1k

(2π)D−1 dθ 4π log(1 + R1(iπ 2 + θ, meff)R2(iπ 2 − θ, meff)e−2meff cosh θL)

Casimir effect: Boundary finite size effect

Extension to higher dimensions: k label

k k L

Dispersion ω =

• m2 +

k2

+ k2 ⊥ =

• m2

eff + k2

⊥

rapidity ω = meff(k) cosh θ, k⊥ = meff(k) sinh θ Reflection R(θ, meff(k)) Bstate: |B =

• 1 +
• dD−1k

(2π)D−1 dθ 4πR(iπ 2 − θ, meff(k))A+(−θ, −

k)A+(θ, k) + ...

• |0

Ground state energy (for free bulk): E0(L) =

• dD−1k

(2π)D−1 dθ 4π log(1 + R1(iπ 2 + θ, meff)R2(iπ 2 − θ, meff)e−2meff cosh θL)

QED: Parallel dielectric slabs (ǫ1, 1, ǫ2) reflections E,⊥, B,⊥ − → R,⊥ look it up in Jackson: R⊥(ω, k = q) = √

ω2−q2−√ ǫω2−q2

√

ω2−q2+√ ǫω2−q2

R(ω, k = q) = ǫ√

ω2−q2−√ ǫω2−q2 ǫ√ ω2−q2+√ ǫω2−q2

Casimir effect: Boundary finite size effect

Extension to higher dimensions: k label

k k L

Dispersion ω =

• m2 +

k2

+ k2 ⊥ =

• m2

eff + k2

⊥

rapidity ω = meff(k) cosh θ, k⊥ = meff(k) sinh θ Reflection R(θ, meff(k)) Bstate: |B =

• 1 +
• dD−1k

(2π)D−1 dθ 4πR(iπ 2 − θ, meff(k))A+(−θ, −

k)A+(θ, k) + ...

• |0

Ground state energy (for free bulk): E0(L) =

• dD−1k

(2π)D−1 dθ 4π log(1 + R1(iπ 2 + θ, meff)R2(iπ 2 − θ, meff)e−2meff cosh θL)

QED: Parallel dielectric slabs (ǫ1, 1, ǫ2) reflections E,⊥, B,⊥ − → R,⊥ look it up in Jackson: R⊥(ω, k = q) = √

ω2−q2−√ ǫω2−q2

√

ω2−q2+√ ǫω2−q2

R(ω, k = q) = ǫ√

ω2−q2−√ ǫω2−q2 ǫ√ ω2−q2+√ ǫω2−q2

Lifshitz formula

Conclusion about Casimir

8

Conclusion about Casimir

Usual derivation: summing up zero freqencies E0(L) = 1

2

• k(L) ω(k(L)) ∝ ∞

Complicated finite volume problem + divergencies

Conclusion about Casimir

Usual derivation: summing up zero freqencies E0(L) = 1

2

• k(L) ω(k(L)) ∝ ∞

Complicated finite volume problem + divergencies as a boundary finite size effect E0(L) = −

d˜

p 2π log(1 + R(−˜

p)R(˜ p)e−2˜

ǫ(˜ p)L)

Reflection factor of the IR degrees of freedom: semi infinite settings, easier to calculate, no divergences

Main problem: q − ¯ q potential in N = 4 SYM

L q q T

W = e−T

LVq¯ q(λ)

extra dimension quark space time anti−quark L

= e−TE0(L,λ)

9

AdS/CFT correspondence (Maldacena 1997) IIB superstring on AdS5 × S5

time space extra dimension anti de Sitter 5D space 5D sphere 4D Minkowski space

6

1 Y 2 i = R2

− + + + +− = −R2

R2 α′

dτdσ

4π

• ∂aXM∂aXM + ∂aY M∂aYM
• + . . .

≡ N = 4 D=4 SU(N) SYM

Ψ1,2,3,4 ր ց Aµ Φ1,2,3,4,5,6 ց ր Ψ1,2,3,4

2 g2

Y M

• d4xTr

−1

4F 2 − 1 2(DΦ)2 + iΨD

/ Ψ + V V (Φ, Ψ) = 1

4[Φ, Φ]2 + Ψ[Φ, Ψ]

β = 0 superconformal

PSU(2,2|4) SO(5)×SO(1,4)

gaugeinvariants:O = Tr(ΦL), det( )

11

AdS/CFT correspondence (Maldacena 1997) IIB superstring on AdS5 × S5

time space extra dimension anti de Sitter 5D space 5D sphere 4D Minkowski space

6

1 Y 2 i = R2

− + + + +− = −R2

R2 α′

dτdσ

4π

• ∂aXM∂aXM + ∂aY M∂aYM
• + . . .

≡ N = 4 D=4 SU(N) SYM

Ψ1,2,3,4 ր ց Aµ Φ1,2,3,4,5,6 ց ր Ψ1,2,3,4

2 g2

Y M

• d4xTr

−1

4F 2 − 1 2(DΦ)2 + iΨD

/ Ψ + V V (Φ, Ψ) = 1

4[Φ, Φ]2 + Ψ[Φ, Ψ]

β = 0 superconformal

PSU(2,2|4) SO(5)×SO(1,4)

gaugeinvariants:O = Tr(ΦL), det( ) Dictionary Couplings: √ λ = R2

α′ , gs = λ N → 0

2D QFT String energy levels: E(λ)

E(λ) = E(∞) + E1

√ λ + E2 λ + . . .

strong↔weak ⇓ λ = g2

Y MN , N → ∞ planar limit

On(x)Om(0) =

δnm |x|2∆n(λ)

Anomalous dim ∆(λ)

∆(λ) = ∆(0) + λ∆1 + λ2∆2 + . . .

AdS/CFT correspondence (Maldacena 1997) IIB superstring on AdS5 × S5

time space extra dimension anti de Sitter 5D space 5D sphere 4D Minkowski space

6

1 Y 2 i = R2

− + + + +− = −R2

R2 α′

dτdσ

4π

• ∂aXM∂aXM + ∂aY M∂aYM
• + . . .

≡ N = 4 D=4 SU(N) SYM

Ψ1,2,3,4 ր ց Aµ Φ1,2,3,4,5,6 ց ր Ψ1,2,3,4

2 g2

Y M

• d4xTr

−1

4F 2 − 1 2(DΦ)2 + iΨD

/ Ψ + V V (Φ, Ψ) = 1

4[Φ, Φ]2 + Ψ[Φ, Ψ]

β = 0 superconformal

PSU(2,2|4) SO(5)×SO(1,4)

gaugeinvariants:O = Tr(ΦL), det( ) Dictionary Couplings: √ λ = R2

α′ , gs = λ N → 0

2D QFT String energy levels: E(λ)

E(λ) = E(∞) + E1

√ λ + E2 λ + . . .

strong↔weak ⇓ λ = g2

Y MN , N → ∞ planar limit

On(x)Om(0) =

δnm |x|2∆n(λ)

Anomalous dim ∆(λ)

∆(λ) = ∆(0) + λ∆1 + λ2∆2 + . . .

2D integrable QFT

AdS/CFT integrability: q − ¯ q potential quark-antiquark potential

extra dimension quark space time anti−quark L

V (L) = −λ

4πL + . . .

≡ Wilson loop: Pe

• C Aµdxµ+

Φ n| ˙ x|ds ∝ e−T

LVq¯ q(λ,θ)

strong coupling V (r) = −4π2√

2λ Γ(1

4)4

1 L(1 − 1.3359 √ λ

+ . . .) minimal surface+fluctuations Integrable system on the strip

quark time space

E0(L) =

d˜

k 2π log(1 − R−(˜

k)R+(−˜ k)e−2˜

ǫ(˜ k)L)

12

R-matrix bootstrap program

Boundary asymptotic states |p1, p2, . . . , pnin/out form a representation of global symmetry:

p1 > pn>0

13

R-matrix bootstrap program

Boundary asymptotic states |p1, p2, . . . , pnin/out form a representation of global symmetry:

p1 > pn>0

Lorentz → E =

i E(pi)

dispersion relation E(p) =

• m2 + p2

R-matrix bootstrap program

Boundary asymptotic states |p1, p2, . . . , pnin/out form a representation of global symmetry:

p1 > pn>0

Lorentz → E =

i E(pi)

dispersion relation E(p) =

• m2 + p2

p1 > pn>0 ∆( ) Q

R-matrix bootstrap program

Boundary asymptotic states |p1, p2, . . . , pnin/out form a representation of global symmetry:

p1 > pn>0

Lorentz → E =

i E(pi)

dispersion relation E(p) =

• m2 + p2

Reflection matrix R: |out → |in commutes with sym. [R, ∆(Q)] = 0

R-matrix bootstrap program

Boundary asymptotic states |p1, p2, . . . , pnin/out form a representation of global symmetry:

p1 > pn>0

Lorentz → E =

i E(pi)

dispersion relation E(p) =

• m2 + p2

Reflection matrix R: |out → |in commutes with sym. [R, ∆(Q)] = 0

1 >

pn>0 R p

1

p pn

R-matrix bootstrap program

Boundary asymptotic states |p1, p2, . . . , pnin/out form a representation of global symmetry:

p1 > pn>0

Lorentz → E =

i E(pi)

dispersion relation E(p) =

• m2 + p2

Reflection matrix R: |out → |in commutes with sym. [R, ∆(Q)] = 0

1 >

pn>0 R p

1

p pn ∆( ) ∆( ) Q Q

R-matrix bootstrap program

Boundary asymptotic states |p1, p2, . . . , pnin/out form a representation of global symmetry:

p1 > pn>0

Lorentz → E =

i E(pi)

dispersion relation E(p) =

• m2 + p2

Reflection matrix R: |out → |in commutes with sym. [R, ∆(Q)] = 0

1 >

pn>0 R p

1

p pn ∆( ) ∆( ) Q Q

Higher spin concerved charge factorization + Bdry Yang-Baxter equation R12 = S12R1 ¯ S21R2 = R2 ¯ S21R1S12

1

p

1

p p2

2

p

1

p

1

p p2 p 2

R-matrix = scalar . Matrix

R-matrix bootstrap program

Boundary asymptotic states |p1, p2, . . . , pnin/out form a representation of global symmetry:

p1 > pn>0

Lorentz → E =

i E(pi)

dispersion relation E(p) =

• m2 + p2

Reflection matrix R: |out → |in commutes with sym. [R, ∆(Q)] = 0

1 >

pn>0 R p

1

p pn ∆( ) ∆( ) Q Q

Higher spin concerved charge factorization + Bdry Yang-Baxter equation R12 = S12R1 ¯ S21R2 = R2 ¯ S21R1S12

1

p

1

p p2

2

p

1

p

1

p p2 p 2

R-matrix = scalar . Matrix Unitarity R(p)R(−p) = Id Boundary crossing symmetry R(p) = S(p, −p)R(¯ p) Maximal analyticity: all poles have physical origin → boundstates, anomalous thresholds

R-matrix bootstrap program: AdS

Nondiagonal scattering: R-matrix = scalar . Matrix

14

R-matrix bootstrap program: AdS

Nondiagonal scattering: R-matrix = scalar . Matrix R-matrix: [Correa-Maldacena-Sever,Drukker] global symmetry PSU(2|2)diag Q = 1 reps

  

b1 b2 f3 f4

   ⊗   

b˙

1

b˙

2

f˙

3

f˙

4

  

[R, ∆(Q)] = 0

1 >

pn>0 R p

1

p pn ∆( ) ∆( ) Q Q

R-matrix bootstrap program: AdS

Nondiagonal scattering: R-matrix = scalar . Matrix R-matrix: [Correa-Maldacena-Sever,Drukker] global symmetry PSU(2|2)diag Q = 1 reps

  

b1 b2 f3 f4

   ⊗   

b˙

1

b˙

2

f˙

3

f˙

4

  

[R, ∆(Q)] = 0

1 >

pn>0 R p

1

p pn ∆( ) ∆( ) Q Q

Rβ ˙

β α ˙ α(p) = Sβ ˙ β α ˙ α(p, −p)R0(p)

R-matrix bootstrap program: AdS

Nondiagonal scattering: R-matrix = scalar . Matrix R-matrix: [Correa-Maldacena-Sever,Drukker] global symmetry PSU(2|2)diag Q = 1 reps

  

b1 b2 f3 f4

   ⊗   

b˙

1

b˙

2

f˙

3

f˙

4

  

[R, ∆(Q)] = 0

1 >

pn>0 R p

1

p pn ∆( ) ∆( ) Q Q

Rβ ˙

β α ˙ α(p) = Sβ ˙ β α ˙ α(p, −p)R0(p)

Unitarity R(z)R(−z) = 1 Crossing symmetry R(z) = S(z, −z)R(ω2 − z) R0(p) =

σB(p) σ(p,−p)

σB = eiχ(x+)−iχ(x−) boundary dressing phase χ(x) =

dz

2π 1 x−z sinh(2πg(z+z−1)) 2πg(z+z−1)

R-matrix bootstrap program: AdS

Nondiagonal scattering: R-matrix = scalar . Matrix R-matrix: [Correa-Maldacena-Sever,Drukker] global symmetry PSU(2|2)diag Q = 1 reps

  

b1 b2 f3 f4

   ⊗   

b˙

1

b˙

2

f˙

3

f˙

4

  

[R, ∆(Q)] = 0

1 >

pn>0 R p

1

p pn ∆( ) ∆( ) Q Q

Rβ ˙

β α ˙ α(p) = Sβ ˙ β α ˙ α(p, −p)R0(p)

Unitarity R(z)R(−z) = 1 Crossing symmetry R(z) = S(z, −z)R(ω2 − z) R0(p) =

σB(p) σ(p,−p)

σB = eiχ(x+)−iχ(x−) boundary dressing phase χ(x) =

dz

2π 1 x−z sinh(2πg(z+z−1)) 2πg(z+z−1)

Maximal analyticity: no boundstates

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) =

15

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) =

L

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R log(e−E0(L)R)

L R

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R logB|e−H(R)L|B

L R

R L

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R logB|e−H(R)L|B

L R

R L

Boundary state |B = exp

∞

−∞ dq 4πRb a(¯

q)CadA+

b (−q)A+ d (q)

• |0
• +

+ 1

• +

+ 1

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R logB|e−H(R)L|B

L R

R L

Boundary state |B = exp

∞

−∞ dq 4πRb a(¯

q)CadA+

b (−q)A+ d (q)

• |0
• +

+ 1

• +

+ 1

Folding trick:

Boundary thermodynamic Bethe Ansatz

Groundstate energy for large L from IR reflection: E0(L) = − limR→∞ 1

R log(Tr(e−HB(L)R)) = − limR→∞ 1 R logB|e−H(R)L|B

L R

R L

Boundary state |B = exp

∞

−∞ dq 4πRb a(¯

q)CadA+

b (−q)A+ d (q)

• |0
• +

+ 1

• +

+ 1

Folding trick:

[Correa,Maldacena,Sever ’12][Drukker ’12]

Ground state energy exactly: E0(L) = −

Q

d˜

p 4π log(1 + e−ǫQ(˜ p))

ǫj(˜ p) = δj

Q(σQ(˜

p) + 2 ˜ EQ(˜ p)L) −

Kj

i (˜

p, ˜ p′) log(1 + e−ǫi(˜

p′))d˜

p′

Regularized q − ¯ q BTBA equations Singular boundary fugacity: σQ(0) = ∞, no-obvious weak coupling expansion shifting countours → regularization (extra source terms, ∼ excited state TBA)

log YQ = −2(f + Ψ)Q − R˜ ǫQ + log σQ + DQ′Q(iuQ′) + log(1 + YQ′) ⋆η KQ′Q +

• 2 log
• 1 + Yv|1
• ⋆ sˆ

⋆KyQ + 2 log(1 + Yv|Q−1) ⋆ s − 2 log 1 − Y− 1 − Y+ ˆ ⋆s ⋆ K1Q

vx

+ log 1 − 1

Y−

1 − 1

Y+

ˆ ⋆KQ + log(1 − 1 Y− )(1 − 1 Y+ )ˆ ⋆KyQ

• log Y−Y+

= 2Dxvs(iuQ) − DQ(iuQ) − log(1 + YQ) ⋆η KQ + 2 log(1 + YQ) ⋆ KQ1

xv ⋆ s + 2 log 1 + Yv|1

1 + Yw|1 ⋆ log Y+ Y− = DQy(iuQ) + log(1 + YQ) ⋆η KQy log Yv|M = −Ds(iuM+1) − log(1 + YM+1) ⋆η s + IMN log(1 + Yv|N) ⋆ s + δM1 log 1 − Y− 1 − Y+ ˆ ⋆s log Yw|M = IMN log(1 + Yw|N) ⋆ s + δM1 log 1 − 1

Y−

1 − 1

Y+

ˆ ⋆s f = i(π − φ) ; Ψ = −i(π − φ) ; R = 2L [ZB, Balog, Hegedus, Toth ’13]

16

q − ¯ q potential: weak coupling expansion quark-antiquark potential Pe

• C Aµdxµ+

Φ nθZJPe

• C Aµdxµ+

Φ nθ

φ

Vq¯

q(g, φ, θ) = Γkg2k

Γ = k

n=1(cos φ−cosh θ sin φ

)nγ(n)

k γ(1)

1

= φ

2

; γ(1)

2

=

φ 12(φ2 − π2)

γ(2)

2 (0) = γ(2)′ 2

(0) = 0 γ(2)′′

2

(φ) = φ

2 cot φ

≡ QMinimal surface Wilson loop strong coupling

V (r) = −4π2√

2λ Γ( 1

4)4

1 L(1 − 1.3359 √ λ

+ . . .)

minimal surface+fluctuations

17