→ → The quark-antiquark potential in N=4 SYM from an open ...The quark-antiquark potential in...
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![Page 1: → → The quark-antiquark potential in N=4 SYM from an open ...The quark-antiquark potential in N=4 SYM from an open spin-chain Nadav Drukker Based on arXiv:1105.5144- N.D. and V.](https://reader036.fdocuments.us/reader036/viewer/2022071404/60f8ad07aa5a073c34565578/html5/thumbnails/1.jpg)
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→ →
The quark-antiquark potential in N=4 SYM from anopen spin-chain
Nadav Drukker
Based on arXiv:1105.5144 - N.D. and V. Forini
arXiv:1203.1617 - N.D.
See also arXiv:1203.1913 - D.Correa, J. Maldacena and A. Sever
IGST 2012, ETH Zurich
August 22, 2012
→ →
1
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Introduction and motivation
• One of the most fundamental quantities in a quantum field theory is the potential
between charged particles.
• In gauge theories this is captured by a long rectangular Wilson loop, or a pair of
antiparallel lines.
Nadav Drukker 2 Integrable Wilson loops
![Page 3: → → The quark-antiquark potential in N=4 SYM from an open ...The quark-antiquark potential in N=4 SYM from an open spin-chain Nadav Drukker Based on arXiv:1105.5144- N.D. and V.](https://reader036.fdocuments.us/reader036/viewer/2022071404/60f8ad07aa5a073c34565578/html5/thumbnails/3.jpg)
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Introduction and motivation
• One of the most fundamental quantities in a quantum field theory is the potential
between charged particles.
• In gauge theories this is captured by a long rectangular Wilson loop, or a pair of
antiparallel lines.
• Such an object exists also in N = 4 SYM.
– The Wilson loop calculates the potential between two W-bosons arising from a
Higgs mechanism.
Nadav Drukker 2-a Integrable Wilson loops
![Page 4: → → The quark-antiquark potential in N=4 SYM from an open ...The quark-antiquark potential in N=4 SYM from an open spin-chain Nadav Drukker Based on arXiv:1105.5144- N.D. and V.](https://reader036.fdocuments.us/reader036/viewer/2022071404/60f8ad07aa5a073c34565578/html5/thumbnails/4.jpg)
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Introduction and motivation
• One of the most fundamental quantities in a quantum field theory is the potential
between charged particles.
• In gauge theories this is captured by a long rectangular Wilson loop, or a pair of
antiparallel lines.
• Such an object exists also in N = 4 SYM.
– The Wilson loop calculates the potential between two W-bosons arising from a
Higgs mechanism.
• Explicit calculations at weak and at strong coupling:
V (L, λ) =
− λ
4πL+
λ2
8π2Lln
T
L+ · · · λ ≪ 1
4π2√λ
Γ( 14 )4 L
(
1− 1.3359 . . .√λ
+ · · ·)
λ ≫ 1
Nadav Drukker 2-b Integrable Wilson loops
![Page 5: → → The quark-antiquark potential in N=4 SYM from an open ...The quark-antiquark potential in N=4 SYM from an open spin-chain Nadav Drukker Based on arXiv:1105.5144- N.D. and V.](https://reader036.fdocuments.us/reader036/viewer/2022071404/60f8ad07aa5a073c34565578/html5/thumbnails/5.jpg)
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Introduction and motivation
• One of the most fundamental quantities in a quantum field theory is the potential
between charged particles.
• In gauge theories this is captured by a long rectangular Wilson loop, or a pair of
antiparallel lines.
• Such an object exists also in N = 4 SYM.
– The Wilson loop calculates the potential between two W-bosons arising from a
Higgs mechanism.
• Explicit calculations at weak and at strong coupling:
V (L, λ) =
− λ
4πL+
λ2
8π2Lln
1
λ+ · · · λ ≪ 1
4π2√λ
Γ( 14 )4 L
(
1− 1.3359 . . .√λ
+ · · ·)
λ ≫ 1
Nadav Drukker 3 Integrable Wilson loops
![Page 6: → → The quark-antiquark potential in N=4 SYM from an open ...The quark-antiquark potential in N=4 SYM from an open spin-chain Nadav Drukker Based on arXiv:1105.5144- N.D. and V.](https://reader036.fdocuments.us/reader036/viewer/2022071404/60f8ad07aa5a073c34565578/html5/thumbnails/6.jpg)
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Introduction and motivation
• One of the most fundamental quantities in a quantum field theory is the potential
between charged particles.
• In gauge theories this is captured by a long rectangular Wilson loop, or a pair of
antiparallel lines.
• Such an object exists also in N = 4 SYM.
– The Wilson loop calculates the potential between two W-bosons arising from a
Higgs mechanism.
• Explicit calculations at weak and at strong coupling:
V (L, λ) =
− λ
4πL+
λ2
8π2Lln
1
λ+ · · · λ ≪ 1
4π2√λ
Γ( 14 )4 L
(
1− 1.3359 . . .√λ
+ · · ·)
λ ≫ 1
• Recently O(λ3) was calculated.[
Correa, HennMaldacena, Sever
]
• Hard to guess how to connect these two regimes.
Nadav Drukker 3-a Integrable Wilson loops
![Page 7: → → The quark-antiquark potential in N=4 SYM from an open ...The quark-antiquark potential in N=4 SYM from an open spin-chain Nadav Drukker Based on arXiv:1105.5144- N.D. and V.](https://reader036.fdocuments.us/reader036/viewer/2022071404/60f8ad07aa5a073c34565578/html5/thumbnails/7.jpg)
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Introduction and motivation
• One of the most fundamental quantities in a quantum field theory is the potential
between charged particles.
• In gauge theories this is captured by a long rectangular Wilson loop, or a pair of
antiparallel lines.
• Such an object exists also in N = 4 SYM.
– The Wilson loop calculates the potential between two W-bosons arising from a
Higgs mechanism.
• Explicit calculations at weak and at strong coupling:
V (L, λ) =
− λ
4πL+
λ2
8π2Lln
1
λ+ · · · λ ≪ 1
4π2√λ
Γ( 14 )4 L
(
1− 1.3359 . . .√λ
+ · · ·)
λ ≫ 1
• Recently O(λ3) was calculated.[
Correa, HennMaldacena, Sever
]
• Hard to guess how to connect these two regimes.
• Can we do any better?
Nadav Drukker 3-b Integrable Wilson loops
![Page 8: → → The quark-antiquark potential in N=4 SYM from an open ...The quark-antiquark potential in N=4 SYM from an open spin-chain Nadav Drukker Based on arXiv:1105.5144- N.D. and V.](https://reader036.fdocuments.us/reader036/viewer/2022071404/60f8ad07aa5a073c34565578/html5/thumbnails/8.jpg)
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Introduction and motivation
• One of the most fundamental quantities in a quantum field theory is the potential
between charged particles.
• In gauge theories this is captured by a long rectangular Wilson loop, or a pair of
antiparallel lines.
• Such an object exists also in N = 4 SYM.
– The Wilson loop calculates the potential between two W-bosons arising from a
Higgs mechanism.
• Explicit calculations at weak and at strong coupling:
V (L, λ) =
− λ
4πL+
λ2
8π2Lln
1
λ+ · · · λ ≪ 1
4π2√λ
Γ( 14 )4 L
(
1− 1.3359 . . .√λ
+ · · ·)
λ ≫ 1
• Recently O(λ3) was calculated.[
Correa, HennMaldacena, Sever
]
• Hard to guess how to connect these two regimes.
• Can we do any better?
• Shouldn’t integrability allow us to calculate this for all values of the coupling (in the
planar approximation)?
Nadav Drukker 3-c Integrable Wilson loops
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The end
Nadav Drukker 4 Integrable Wilson loops
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Outline
• Introduction and motivation
• Wilson loops
– Cusp anomalous dimensions and the quark-antiquark potential
– Local operator insertions
• Wilson loops in N = 4 SYM
– Perturbative calculation
– String calculation
– Expansions in small angles
• Wilson loops and integrability
– Operator insertions and open spin–chains
– All loop reflection matrix and a twist
– Wrapping effects and the quark-antiquark potential
Nadav Drukker 5 Integrable Wilson loops
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Wilson loops
• In any gauge theory one can define Wilson loop operators
W = TrP exp
[∮
iAµxµ ds
]
• Can be defined for an arbitrary curve in spacetime.
• This is the holonomy of the gauge field.
Nadav Drukker 6 Integrable Wilson loops
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Wilson loops
• In any gauge theory one can define Wilson loop operators
W = TrP exp
[∮
iAµxµ ds
]
• Can be defined for an arbitrary curve in spacetime.
• This is the holonomy of the gauge field.
• For a pair of antiparallel lines
〈W 〉 ≈ exp[
− T V (L, λ)]
• The potential behaves like
V (L, λ) =
g(λ) screeningf(λ)L conformal
α′L confining
T
L
Nadav Drukker 6-a Integrable Wilson loops
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Cusp anomalous dimensions and quark-antiquark potential
• A regular Wilson loop will suffer from linear UV divergences (uninteresting).
• The antiparallel lines suffer also from a linear IR divergence (subtle).
Nadav Drukker 7 Integrable Wilson loops
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Cusp anomalous dimensions and quark-antiquark potential
• A regular Wilson loop will suffer from linear UV divergences (uninteresting).
• The antiparallel lines suffer also from a linear IR divergence (subtle).
• It is simpler to control logarithmic divergences.
Nadav Drukker 7-a Integrable Wilson loops
![Page 15: → → The quark-antiquark potential in N=4 SYM from an open ...The quark-antiquark potential in N=4 SYM from an open spin-chain Nadav Drukker Based on arXiv:1105.5144- N.D. and V.](https://reader036.fdocuments.us/reader036/viewer/2022071404/60f8ad07aa5a073c34565578/html5/thumbnails/15.jpg)
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Cusp anomalous dimensions and quark-antiquark potential
• A regular Wilson loop will suffer from linear UV divergences (uninteresting).
• The antiparallel lines suffer also from a linear IR divergence (subtle).
• It is simpler to control logarithmic divergences.
• Consider Wilson loops with cusps
• All but the black line will suffer from logarithmic divergences.
• Taking φ = iϕ and ϕ → ∞ gives the Lorenzian null cusp.
Nadav Drukker 7-b Integrable Wilson loops
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Cusp anomalous dimensions and quark-antiquark potential
• A regular Wilson loop will suffer from linear UV divergences (uninteresting).
• The antiparallel lines suffer also from a linear IR divergence (subtle).
• It is simpler to control logarithmic divergences.
• Consider Wilson loops with cusps
• All but the black line will suffer from logarithmic divergences.
• Taking φ = iϕ and ϕ → ∞ gives the Lorenzian null cusp.
My talk will focus on the euclidean cusp, but all that I
say can be immediately extended to Minkowski space.
Nadav Drukker 7-c Integrable Wilson loops
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• One can also consider a compact versions of
cusped loops.
• No gauge-invariance subtleties!
−2
−2
−4
−4
2
2
4
4
Nadav Drukker 8 Integrable Wilson loops
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• One can also consider a compact versions of
cusped loops.
• No gauge-invariance subtleties!
• In a conformal theory they will be equivalent
to the previous picture (up to a finite
anomaly).
• In any case, the log divergences, which are a
UV effect will remain the same.
−2
−2
−4
−4
2
2
4
4
Nadav Drukker 8-a Integrable Wilson loops
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• One can also consider a compact versions of
cusped loops.
• No gauge-invariance subtleties!
• In a conformal theory they will be equivalent
to the previous picture (up to a finite
anomaly).
• In any case, the log divergences, which are a
UV effect will remain the same.
• I label the opening angle π − φ.
• φ = 0 is the circle.
• φ → π gives the antiparallel lines.
−2
−2
−4
−4
2
2
4
4
Nadav Drukker 8-b Integrable Wilson loops
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• One can also consider a compact versions of
cusped loops.
• No gauge-invariance subtleties!
• In a conformal theory they will be equivalent
to the previous picture (up to a finite
anomaly).
• In any case, the log divergences, which are a
UV effect will remain the same.
• I label the opening angle π − φ.
• φ = 0 is the circle.
• φ → π gives the antiparallel lines.
−2
−2
−4
−4
2
2
4
4
• In a conformal theory, by the usual conformal Ward identity
〈W 〉 ∼ 1
d2∆, d = r
cos φ2
1− sin φ2
• ∆ is the coefficient of the log divergence.
Nadav Drukker 8-c Integrable Wilson loops
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• By the inverse exponential map we get the gauge theory on S3 × R
• These are parallel lines on S3 × R.
Nadav Drukker 9 Integrable Wilson loops
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• By the inverse exponential map we get the gauge theory on S3 × R
• These are parallel lines on S3 × R.
• From this last picture we expect
〈W 〉 ≈ exp[
− T V (φ, λ)]
• In a conformal theory T is related to divergence at the cusp by the
exponential map
T = logR
ǫ
• Therefore V (φ, λ) is the same as ∆, the coefficient of the log
divergence.
Nadav Drukker 9-a Integrable Wilson loops
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• By the inverse exponential map we get the gauge theory on S3 × R
• These are parallel lines on S3 × R.
• From this last picture we expect
〈W 〉 ≈ exp[
− T V (φ, λ)]
• In a conformal theory T is related to divergence at the cusp by the
exponential map
T = logR
ǫ
• Therefore V (φ, λ) is the same as ∆, the coefficient of the log
divergence.
• This V (φ, λ) is the generalization of V (L, λ) — the quark-antiquark
potential.
• For a conformal theory it has a pole at φ → π and the residue is
LV (L, λ).
• More generally controls all log divergences of all Wilson loops.
• Needed for a proper renormalization program of Wilson loop operators (and to derive
regularized loop equations).
Nadav Drukker 9-b Integrable Wilson loops
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Local operator insertions
• There is another source of log divergences in Wilson loops:
Adjoint valued operators inserted into the Wilson loop.
Nadav Drukker 10 Integrable Wilson loops
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Local operator insertions
• There is another source of log divergences in Wilson loops:
Adjoint valued operators inserted into the Wilson loop.
• For example, one operator in the straight line
W = TrP[
O(0) exp
(∫
iAµxµ ds
)]
= Tr
[
P exp
(∫ 0
−∞
iAµxµ ds
)
O(0)P exp
(∫
∞
0
iAµxµ ds
)]
• O is any adjoint operator, e.g., F23, D2F14, F12(F43)
2, etc.
• If the theory has adjoint scalars and/or fermions, they can be inserted as well.
Nadav Drukker 10-a Integrable Wilson loops
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Local operator insertions
• There is another source of log divergences in Wilson loops:
Adjoint valued operators inserted into the Wilson loop.
• For example, one operator in the straight line
W = TrP[
O(0) exp
(∫
iAµxµ ds
)]
= Tr
[
P exp
(∫ 0
−∞
iAµxµ ds
)
O(0)P exp
(∫
∞
0
iAµxµ ds
)]
• O is any adjoint operator, e.g., F23, D2F14, F12(F43)
2, etc.
• If the theory has adjoint scalars and/or fermions, they can be inserted as well.
• In a conformal theory, a Wilson loop with two operator insertions at a distance d will
have a VEV
〈W 〉 ∼ 1
d2∆
• ∆ is the coefficient of the log divergences — the conformal dimension of the insertions.
Nadav Drukker 10-b Integrable Wilson loops
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Wilson loops in N = 4 SYM
• In addition to the gauge field, N = 4 SYM has six real scalar fields and four fermions,
all in the adjoint of the gauge group.
• The most natural Wilson loops in N = 4 SYM include a coupling to the scalar fields
W = TrP exp
[∮
(
iAµxµ + |x|nIΦI
)
ds
]
nI do not have to be constant.
• For a smooth loop and continuous |nI | = 1, these are finite observables.
• The Wilson loop with the scalar coupling is natural for calculating the potential
between W-bosons.
Nadav Drukker 11 Integrable Wilson loops
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Wilson loops in N = 4 SYM
• In addition to the gauge field, N = 4 SYM has six real scalar fields and four fermions,
all in the adjoint of the gauge group.
• The most natural Wilson loops in N = 4 SYM include a coupling to the scalar fields
W = TrP exp
[∮
(
iAµxµ + |x|nIΦI
)
ds
]
nI do not have to be constant.
• For a smooth loop and continuous |nI | = 1, these are finite observables.
• The Wilson loop with the scalar coupling is natural for calculating the potential
between W-bosons.
• For the loop with cusp can have each line couple to a different scalar field
Φ1 and Φ1 cos θ +Φ2 sin θ
• Gives another parameter: θ.
Nadav Drukker 11-a Integrable Wilson loops
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Wilson loops in N = 4 SYM
• In addition to the gauge field, N = 4 SYM has six real scalar fields and four fermions,
all in the adjoint of the gauge group.
• The most natural Wilson loops in N = 4 SYM include a coupling to the scalar fields
W = TrP exp
[∮
(
iAµxµ + |x|nIΦI
)
ds
]
nI do not have to be constant.
• For a smooth loop and continuous |nI | = 1, these are finite observables.
• The Wilson loop with the scalar coupling is natural for calculating the potential
between W-bosons.
• For the loop with cusp can have each line couple to a different scalar field
Φ1 and Φ1 cos θ +Φ2 sin θ
• Gives another parameter: θ.
• Crucial point: Calculations of V (φ, θ, λ) are no harder than for the antiparallel case!
Nadav Drukker 11-b Integrable Wilson loops
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Perturbative calculation
• Expanding at weak coupling
V (φ, θ, λ) =∞∑
n=1
(
λ
16π2
)n
V (n)(φ, θ)
• And at strong coupling
V (φ, θ, λ) =
√λ
4π
∞∑
l=0
(
4π√λ
)l
V(l)AdS(φ, θ)
Nadav Drukker 12 Integrable Wilson loops
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1–loop
• Just the exchange of a gluon and scalar field
• This graph is given by the integral
∂λ〈W 〉∣
∣
∣
λ=0=
∫
s<t
ds dt⟨
(iAµxµ(s) + |x|ΦInI(s)) (iAµx
µ(t) + |x|ΦJnJ(t))⟩
=λ
8π2
∫
ds dt−xµ(s)x
µ(t) + nI(s)nI(t)
|x(s)− x(t)|2
=λ
8π2
∫
ds dt− cosφ+ cos θ
s2 + t2 + 2st cosφ= − λ
8π2
cosφ− cos θ
sinφφ log
R
ǫ
• Therefore
V (1)(φ, θ) = 2cosφ− cos θ
sinφφ
Nadav Drukker 13 Integrable Wilson loops
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Higher order graphs
• Ladder graphs are relatively easy.
• They dominate a funny double-scaled limit where
θ → i∞ with λθ fixed.[
Correa, HennMaldacena, Sever
]
• They are given by harmonic polylogs apparently to
all orders.[
Henn, Huber
]
• Results at weak and strong coupling match.
Nadav Drukker 14 Integrable Wilson loops
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Higher order graphs
• Ladder graphs are relatively easy.
• They dominate a funny double-scaled limit where
θ → i∞ with λθ fixed.[
Correa, HennMaldacena, Sever
]
• They are given by harmonic polylogs apparently to
all orders.[
Henn, Huber
]
• Results at weak and strong coupling match.
• Interacting graphs are a bit more complicated.
• At two loops there are bubble graphs and the single
cubic vertex graphs.
• they give
V(2)int (φ, θ) = −2
3(π2 − φ2)V (1)(φ, θ)
• Full 3 loop answer was also calculated.[
Correa, HennMaldacena, Sever
]
Nadav Drukker 14-a Integrable Wilson loops
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String calculation[
Maldacena
][
Rey, Yee
][
DrukkerGross, Ooguri
]
• Within the AdS/CFT correspondence Wilson loops are calculated by an infinite open
string extending to the boundary of AdS.
• At the leading order one should find the minimal area surface.
• One loop requires studying the string fluctuations, and so on.
Nadav Drukker 15 Integrable Wilson loops
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String calculation[
Maldacena
][
Rey, Yee
][
DrukkerGross, Ooguri
]
• Within the AdS/CFT correspondence Wilson loops are calculated by an infinite open
string extending to the boundary of AdS.
• At the leading order one should find the minimal area surface.
• One loop requires studying the string fluctuations, and so on.
• In our case the boundary conditions are lines separated by π − φ
on the boundary of AdS and θ on S5.
• All the string solutions fit inside AdS3 × S1
ds2 =√λ(
− cosh2 ρ dt2 + dρ2 + sinh2 ρ dϕ2 + dϑ2)
Nadav Drukker 15-a Integrable Wilson loops
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String calculation[
Maldacena
][
Rey, Yee
][
DrukkerGross, Ooguri
]
• Within the AdS/CFT correspondence Wilson loops are calculated by an infinite open
string extending to the boundary of AdS.
• At the leading order one should find the minimal area surface.
• One loop requires studying the string fluctuations, and so on.
• In our case the boundary conditions are lines separated by π − φ
on the boundary of AdS and θ on S5.
• All the string solutions fit inside AdS3 × S1
ds2 =√λ(
− cosh2 ρ dt2 + dρ2 + sinh2 ρ dϕ2 + dϑ2)
• The equations of motion can be solved by elliptic integrals.
θ =2b q
√
b4 + p2K , φ = π − 2p2
b√
b4 + p2
(
K−Π(
b4
b4+p2 |k2)
)
where b, k, p and q are related by
b2 =1
2
(
p2 − q2 +√
(p2 − q2)2 + 4p2)
k2 =b2(b2 − p2)
b4 + p2
• These are transcendental equations for p, q in terms of θ, φ
Nadav Drukker 15-b Integrable Wilson loops
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• The induced metric is
ds2ind =√λ1− k2
cn2(σ)
[
−dτ2 + dσ2]
.
• The classical action can also be calculated
Scl =
√λ
2π
∫
dt dϕ p cosh2 ρ sinh2 ρ =T√λ
π
√
b4 + p2
b p
[
(b2 + 1)p2
b4 + p2K− E
]
• This determines V(0)AdS as a function of p, q and implicitly in term of φ, θ.
Nadav Drukker 16 Integrable Wilson loops
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• The induced metric is
ds2ind =√λ1− k2
cn2(σ)
[
−dτ2 + dσ2]
.
• The classical action can also be calculated
Scl =
√λ
2π
∫
dt dϕ p cosh2 ρ sinh2 ρ =T√λ
π
√
b4 + p2
b p
[
(b2 + 1)p2
b4 + p2K− E
]
• This determines V(0)AdS as a function of p, q and implicitly in term of φ, θ.
• We can also expand around φ = θ = 0
V(0)AdS(φ, θ) =
1
π(θ2 − φ2)− 1
8π3(θ2 − φ2)
(
θ2 − 5φ2)
+1
64π5(θ2 − φ2)
(
θ4 − 14θ2φ2 + 37φ4)
− 1
2048π7(θ2 − φ2)
(
θ6 − 27θ4φ2 + 291θ2φ4 − 585φ6)
+O((φ, θ)10)
Nadav Drukker 16-a Integrable Wilson loops
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1–loop determinant
• At one–loop we should consider the 8 transverse bosonic and 8 fermionic fluctuation
modes.
• Such a calculation was done long ago for a confining string by Luscher.
• The “Luscher term” is proportional to the number of transverse dimensions and
always has a Coulomb behavior.
• We have to repeat the calculation in the AdS5 × S5 sigma model.
Nadav Drukker 17 Integrable Wilson loops
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1–loop determinant
• At one–loop we should consider the 8 transverse bosonic and 8 fermionic fluctuation
modes.
• Such a calculation was done long ago for a confining string by Luscher.
• The “Luscher term” is proportional to the number of transverse dimensions and
always has a Coulomb behavior.
• We have to repeat the calculation in the AdS5 × S5 sigma model.
• All the differential operators can be written as Lame operators
−∂2τ − ∂2
σ + 2k2 sn2(σ|k2)
• Requires using many elliptic identities, using different ks and rescaling τ and σ.
Nadav Drukker 17-a Integrable Wilson loops
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• The result of a tedious calculation gives
Γreg = −T2
limǫ→0
∫ +∞
−∞
dω
2πln
ǫ2ω2 det8 OǫF
det5 Oǫ0 det
2 Oǫ1 detOǫ
2
where
detOǫ0∼= sinh(2Kω)
ω
detOǫ1∼= − sinh(2K1 Z(α1))
ǫ2√
(ω2 − k2)(ω2 − k2 + 1)(ω − 2k2 + 1)
detOǫ2∼= sinh(2K2 Z(α2))
ǫ2(1− k2)3/2(k1 + 1)3√
(ω22 + k22)(ω
22 + 1)(ω2
2 + k22 + 1)
detOǫF∼= 8K2
√
ω23 + k22 sinh(K2 Z(αF ))
ǫπ(1− k2)(k1 + 1)2√
(ω23 + 1)(ω2
3 + k22 + 1)
ϑ2(0, q2)ϑ4
(
παF
2K2, q2
)
ϑ′
1(0, q2)ϑ3
(
παF
2K2, q2
)
and ωi, ǫi, ki are algebraic in the usual ω, etc. and αi are solutions to some elliptic
equations. . .
Nadav Drukker 18 Integrable Wilson loops
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• The result of a tedious calculation gives
Γreg = −T2
limǫ→0
∫ +∞
−∞
dω
2πln
ǫ2ω2 det8 OǫF
det5 Oǫ0 det
2 Oǫ1 detOǫ
2
where
detOǫ0∼= sinh(2Kω)
ω
detOǫ1∼= − sinh(2K1 Z(α1))
ǫ2√
(ω2 − k2)(ω2 − k2 + 1)(ω − 2k2 + 1)
detOǫ2∼= sinh(2K2 Z(α2))
ǫ2(1− k2)3/2(k1 + 1)3√
(ω22 + k22)(ω
22 + 1)(ω2
2 + k22 + 1)
detOǫF∼= 8K2
√
ω23 + k22 sinh(K2 Z(αF ))
ǫπ(1− k2)(k1 + 1)2√
(ω23 + 1)(ω2
3 + k22 + 1)
ϑ2(0, q2)ϑ4
(
παF
2K2, q2
)
ϑ′
1(0, q2)ϑ3
(
παF
2K2, q2
)
and ωi, ǫi, ki are algebraic in the usual ω, etc. and αi are solutions to some elliptic
equations. . .
• For small φ we can expand
V(1)AdS(φ, 0) =
3
2
φ2
4π2+
(
53
8− 3 ζ(3)
)
φ4
16π4+
(
223
8− 15
2ζ(3)− 15
2ζ(5)
)
φ6
64π6
+
(
14645
128− 229
8ζ(3)− 55
4ζ(5)− 315
16ζ(7)
)
φ8
256π8+O(φ10)
Nadav Drukker 18-a Integrable Wilson loops
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φ → π limit
• V (1), V (2), V(0)AdS and V
(1)AdS all have poles at φ = π
• In perturbation theory
V (φ, θ) → − λ
8π
1 + cos θ
π − φ+
λ2
32π3
(1 + cos θ)2
π − φlog
e
2(π − φ)+O(λ3)
Nadav Drukker 19 Integrable Wilson loops
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φ → π limit
• V (1), V (2), V(0)AdS and V
(1)AdS all have poles at φ = π
• In perturbation theory
V (φ, θ) → − λ
8π
1 + cos θ
π − φ+
λ2
32π3
(1 + cos θ)2
π − φlog
e
2(π − φ)+O(λ3)
• In the case of θ = 0 we get essentially the same as the antiparallel lines with
L → π − φ
V (L, λ) =
− λ
4πL+
λ2
8π2Lln
T
L+ · · · λ ≪ 1
4π2√λ
Γ( 14 )4 L
(
1− 1.3359 . . .√λ
+ · · ·)
λ ≫ 1
• The strong coupling calculations also agree in the limit.
Nadav Drukker 19-a Integrable Wilson loops
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Expansions in small angles
• Consider the expansion of V (φ, θ, λ) at small φ or θ
1
2
∂2
∂θ2V (φ, θ, λ)
∣
∣
∣
φ=θ=0= −1
2
∂2
∂φ2V (φ, θ, λ)
∣
∣
∣
φ=θ=0=
λ
16π2− λ2
384π2+ · · · λ ≪ 1
√λ
4π2− 3
8π2+ · · · λ ≫ 1
Nadav Drukker 20 Integrable Wilson loops
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Expansions in small angles
• Consider the expansion of V (φ, θ, λ) at small φ or θ
1
2
∂2
∂θ2V (φ, θ, λ)
∣
∣
∣
φ=θ=0= −1
2
∂2
∂φ2V (φ, θ, λ)
∣
∣
∣
φ=θ=0=
λ
16π2− λ2
384π2+ · · · λ ≪ 1
√λ
4π2− 3
8π2+ · · · λ ≫ 1
• This quantity was named the bremsstrahlung function B(λ)[
Correa, HennMaldacena, Sever
]
• Calculates the radiation of an accelerated quark.
• Is related to small deformations of BPS Wilson loops and can be calculated exactly
B =1
2π2λ∂λ〈W◦〉
〈W◦〉 =1
NL1N−1
(
− λ
4N
)
eλ
8N
Nadav Drukker 20-a Integrable Wilson loops
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Expansions in small angles
• Consider the expansion of V (φ, θ, λ) at small φ or θ
1
2
∂2
∂θ2V (φ, θ, λ)
∣
∣
∣
φ=θ=0= −1
2
∂2
∂φ2V (φ, θ, λ)
∣
∣
∣
φ=θ=0=
λ
16π2− λ2
384π2+ · · · λ ≪ 1
√λ
4π2− 3
8π2+ · · · λ ≫ 1
• This quantity was named the bremsstrahlung function B(λ)[
Correa, HennMaldacena, Sever
]
• Calculates the radiation of an accelerated quark.
• Is related to small deformations of BPS Wilson loops and can be calculated exactly
B =1
2π2λ∂λ〈W◦〉
〈W◦〉 =1
NL1N−1
(
− λ
4N
)
eλ
8N
• See also Kolya’s talk tomorrow.
Nadav Drukker 20-b Integrable Wilson loops
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Result so far:
Explicit expressions for these families of Wilson loops at weak and strong coupling.
Nadav Drukker 21 Integrable Wilson loops
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Wilson loops and integrability
• We want to apply the tools of integrability to the case of Wilson loops:
– Find a spin–chain model.
– Find the all loop scattering (and reflection) matrix
– Try to solve it exactly.
• This will allow to derive the gauge theory perturbative results from world-sheet
techniques.
Nadav Drukker 22 Integrable Wilson loops
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Wilson loops and integrability
• We want to apply the tools of integrability to the case of Wilson loops:
– Find a spin–chain model.
– Find the all loop scattering (and reflection) matrix
– Try to solve it exactly.
• This will allow to derive the gauge theory perturbative results from world-sheet
techniques.
• Main trick will be to start with the Wilson loop with an arbitrary insertion in it,
which will simplify the steps above and at the end remove the insertion.
• In the case of the straight line, after removing the insertion, the operator is 1/2 BPS,
so no anomalous dimension. So need to know how to treat the cusp.
Nadav Drukker 22-a Integrable Wilson loops
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string picture
• The string dual of a Wilson loop with an insertion is an excited state of the open
string describing the Wilson loop.
Nadav Drukker 23 Integrable Wilson loops
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string picture
• The string dual of a Wilson loop with an insertion is an excited state of the open
string describing the Wilson loop.
Nadav Drukker 23-a Integrable Wilson loops
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string picture
• The string dual of a Wilson loop with an insertion is an excited state of the open
string describing the Wilson loop.
=⇒
Nadav Drukker 23-b Integrable Wilson loops
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string picture
• The string dual of a Wilson loop with an insertion is an excited state of the open
string describing the Wilson loop.
=⇒
• Study the spectrum of open string states all satisfying the same boundary conditions.
Nadav Drukker 23-c Integrable Wilson loops
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• An insertion of ZJ is described by a string ending along the same curve on the
boundary but in the bulk of space rotating around the equator of S5 with
momentum J .
• An excitation traveling along this string will not know that it’s an open string and not
the usual TrZJ vacuum.
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• An insertion of ZJ is described by a string ending along the same curve on the
boundary but in the bulk of space rotating around the equator of S5 with
momentum J .
• An excitation traveling along this string will not know that it’s an open string and not
the usual TrZJ vacuum.
• Once it gets to the end of the string we should impose boundary conditions.
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• An insertion of ZJ is described by a string ending along the same curve on the
boundary but in the bulk of space rotating around the equator of S5 with
momentum J .
• An excitation traveling along this string will not know that it’s an open string and not
the usual TrZJ vacuum.
• Once it gets to the end of the string we should impose boundary conditions.
Gauge theory picture
We take the cusped Wilson loop with an adjoint
valued operator like ZJ at the cusp.
π − φ
O ∼ ZY Z · · ·ZZ
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• It is clear how to see the appearance of the spin–chain by considering the compact
operator in the gauge theory
• In this case the classical dimension is 5.
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• It is clear how to see the appearance of the spin–chain by considering the compact
operator in the gauge theory
• In this case the classical dimension is 5.
• The bulk hamiltonian is like the usual Minahan-Zarembo spin–chain (Beisert S-matrix
Scdab(p1, p2)⊗ S
cdab(p1, p2) ).
• Boundary interaction has to be studied separately.
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• It is clear how to see the appearance of the spin–chain by considering the compact
operator in the gauge theory
• In this case the classical dimension is 5.
• The bulk hamiltonian is like the usual Minahan-Zarembo spin–chain (Beisert S-matrix
Scdab(p1, p2)⊗ S
cdab(p1, p2) ).
• Boundary interaction has to be studied separately.
• The two boundaries interact through wrapping effects at O(g2(J+1)).
• For J = 0 this is at one-loop.
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All loop reflection matrix and a twist
• The one loop bulk hamiltonian is the same as for closed spin–chains
• The boundary reflection matrix was calculated from Feynman graphs only in the
SU(2) sector.
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All loop reflection matrix and a twist
• The one loop bulk hamiltonian is the same as for closed spin–chains
• The boundary reflection matrix was calculated from Feynman graphs only in the
SU(2) sector.
• To do it to all loops we should use the symmetry:
psu(2, 2|4) −→ZJ vacuum
psu(2|2)L × psu(2|2)R
boundary ↓ ↓
osp(4∗|4) −→ psu(2|2)D
• A single boundary breaks the symmetry to a diagonal psu(2|2).
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All loop reflection matrix and a twist
• The one loop bulk hamiltonian is the same as for closed spin–chains
• The boundary reflection matrix was calculated from Feynman graphs only in the
SU(2) sector.
• To do it to all loops we should use the symmetry:
psu(2, 2|4) −→ZJ vacuum
psu(2|2)L × psu(2|2)R
boundary ↓ ↓
osp(4∗|4) −→ psu(2|2)D
• A single boundary breaks the symmetry to a diagonal psu(2|2).• By the usual argument, the boundary
reflection matrix should have the same
matrix structure as the bulk one
Rbbaa(p) = R0(p)S
bbaa(p,−p)
• It replaces psu(2|2)L ↔ psu(2|2)R labels.
-p
-p -p
-p
p
pp
p
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• Need to determine
R0(p) = σB(p)/σ(p,−p).
• Like the crossing relation in the
bulk, there is a boundary
“crossing-unitarity equation”
R(p) = S(p,−p)Rc(p)
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• Need to determine
R0(p) = σB(p)/σ(p,−p).
• Like the crossing relation in the
bulk, there is a boundary
“crossing-unitarity equation”
R(p) = S(p,−p)Rc(p)
• This translates to the conditions on σB
σB(p)σB(p) =x− + 1/x−
x+ + 1/x+, σB(p)σB(p) = 1 .
where the Joukowsky variables are a solution of
eip =x+
x−, x+ +
1
x+− x− − 1
x−=
1
g.
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• Need to determine
R0(p) = σB(p)/σ(p,−p).
• Like the crossing relation in the
bulk, there is a boundary
“crossing-unitarity equation”
R(p) = S(p,−p)Rc(p)
• This translates to the conditions on σB
σB(p)σB(p) =x− + 1/x−
x+ + 1/x+, σB(p)σB(p) = 1 .
where the Joukowsky variables are a solution of
eip =x+
x−, x+ +
1
x+− x− − 1
x−=
1
g.
• The solution which matches the all consistency requirements is
σB(z) =1 + 1/(x−)2
1 + 1/(x+)2e−iχB(x+)+iχB(x−)
where
χB(x) = −i
∮
dz
2πi
1
x− zlog
sinh 2πg(z + 1/z)
2πg(z + 1/z).
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• So far only right boundary. What about the left?
Nadav Drukker 28 Integrable Wilson loops
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• So far only right boundary. What about the left?
• The left boundary is essentially the same.
• The choice of diagonal subgroup psu(2|2)L × psu(2|2)R → psu(2|2)D′ may be different.
• Conjugate the reflection matrix by a twist matrix G acting on the psu(2|2)L labels
G = diag(eiθ/2, e−iθ/2, eiφ/2, e−iφ/2)
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• So far only right boundary. What about the left?
• The left boundary is essentially the same.
• The choice of diagonal subgroup psu(2|2)L × psu(2|2)R → psu(2|2)D′ may be different.
• Conjugate the reflection matrix by a twist matrix G acting on the psu(2|2)L labels
G = diag(eiθ/2, e−iθ/2, eiφ/2, e−iφ/2)
• This is all the information needed to understand the spectrum of asymptotically large
insertions into the Wilson loop.
p1
p1p1
p1p1
p2 p2
p2p2
−p1
−p1−p1
−p1
−p2
−p2
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• So far only right boundary. What about the left?
• The left boundary is essentially the same.
• The choice of diagonal subgroup psu(2|2)L × psu(2|2)R → psu(2|2)D′ may be different.
• Conjugate the reflection matrix by a twist matrix G acting on the psu(2|2)L labels
G = diag(eiθ/2, e−iθ/2, eiφ/2, e−iφ/2)
• This is all the information needed to understand the spectrum of asymptotically large
insertions into the Wilson loop.
p1
p1p1
p1p1
p2 p2
p2p2
−p1
−p1−p1
−p1
−p2
−p2
• But not the case J = 0 . . .
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Wrapping effects and the quark-antiquark potential
• One can derive a set of boundary thermodynamic Bethe ansatz equations for this open
spin–chain.
• This can be simplified in the small angle limit, where the full answer was reproduced.[
Correa, Maldacen,Sever
][
GromovSever
]
• They are the same as the usual TBA equations with several small modifications:
– The Y functions are related by reflection Ya,s(−u) = Ya,−s(u)
– There are chemical potentials dependent on φ and θ.
– There is a complicated driving term for the massive Ya,0 nodes (aka YQ).
• The Y -system equations are unmodified.
– Analytic properties of the functions are different (determined by the asymptotic
solution).
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• To reproduce the one loop answer it is enough to consider Luscher-like corrections.
• This requires to calculate the eigenvalues of the transfer matrix
p
pp −p
−p−p
p1p1 p2p2 −p1−p2
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• To reproduce the one loop answer it is enough to consider Luscher-like corrections.
• This requires to calculate the eigenvalues of the transfer matrix
p
pp −p
−p−p
p1p1 p2p2 −p1−p2
• by repeated use of the Yang-Baxter equation this simplifies to
ppp
−p−p−p
p1p1 p2p2 −p1−p2
• That is just the product of two twisted psu(2|2) transfer matrices.
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• On the ZJ vacuum this is for the Qs bound state
Tφ,θQ (p) = sTr
[
R(R)(p)R(L)c(p)
]
= sTr[
R(R)(p)GR
(R)c(−p)G]
= σB(p)σB(−p)
(
x−
x+
)2
(sTrG)2
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• On the ZJ vacuum this is for the Qs bound state
Tφ,θQ (p) = sTr
[
R(R)(p)R(L)c(p)
]
= sTr[
R(R)(p)GR
(R)c(−p)G]
= σB(p)σB(−p)
(
x−
x+
)2
(sTrG)2
• Simple group theory gives
(sTrQ G)2= 4(cosφ− cos θ)2
sin2 Qφ
sin2 φ
And the Luscher-Bajnok-Janik formula is
δE ≈ − 1
2π
∞∑
Q=1
∫
∞
0
dp log(
1 + T(φ,θ)Q (p)e−2JEQ
)
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• On the ZJ vacuum this is for the Qs bound state
Tφ,θQ (p) = sTr
[
R(R)(p)R(L)c(p)
]
= sTr[
R(R)(p)GR
(R)c(−p)G]
= σB(p)σB(−p)
(
x−
x+
)2
(sTrG)2
• Simple group theory gives
(sTrQ G)2= 4(cosφ− cos θ)2
sin2 Qφ
sin2 φ
And the Luscher-Bajnok-Janik formula is
δE ≈ − 1
2π
∞∑
Q=1
∫
∞
0
dp log(
1 + T(φ,θ)Q (p)e−2JEQ
)
• Normally for small g (or large J) can expand the logarithm
δE ≈ 1
2π
∞∑
Q=1
∫
∞
0
dp T(φ,θ)Q (p)e−2JEQ
For J = 0 the answer will be proportional tog4(cosφ− cos θ)2
sin2 φ. . .
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• Crucial fact is that the dressing factor has a double pole at p = 0
σB(p)σB(− ¯p) = e2i(χB(x+)+χB(x−)) (2πg)2(x+ + 1/x+)(x− + 1/x−)
sinh(2πg(x+ + 1/x+)) sinh(2πg(x− + 1/x−))
= e2i(χB(x+)+χB(x−)) (2π)2(u2 +Q2/4)
sinh2(2πu)∼ Q2
p2
• Then using∫
∞
0
dp log
(
1 +c
p2
)
= π√c ,
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• Crucial fact is that the dressing factor has a double pole at p = 0
σB(p)σB(− ¯p) = e2i(χB(x+)+χB(x−)) (2πg)2(x+ + 1/x+)(x− + 1/x−)
sinh(2πg(x+ + 1/x+)) sinh(2πg(x− + 1/x−))
= e2i(χB(x+)+χB(x−)) (2π)2(u2 +Q2/4)
sinh2(2πu)∼ Q2
p2
• Then using∫
∞
0
dp log
(
1 +c
p2
)
= π√c ,
• The residue is√
T resQ e−2JEQ = 2
cosφ− cos θ
sinφsinQφ (−1)Q
[
(4g2)J+1
Q2J+1− 2(J + 2)
(4g2)J+2
Q2J+3+ · · ·
]
• so
δE ≈ −(4g2)J+1 cosφ− cos θ
sinφ
∞∑
Q=1
(−1)Q sinQφ
Q2J+1
= − (4g2)J+1
2i
cosφ− cos θ
sinφ
(
Li2J+1(−eiφ)− Li2J+1(−e−iφ))
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For J = 0
δE ≈ −4g2
2i
cosφ− cos θ
sinφ
(
Li1(−eiφ)− Li1(−e−iφ))
= 2g2icosφ− cos θ
sinφ
(
− log(1 + eiφ) + log(1 + e−iφ))
= 2g2cosφ− cos θ
sinφφ+O(g4)
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For J = 0
δE ≈ −4g2
2i
cosφ− cos θ
sinφ
(
Li1(−eiφ)− Li1(−e−iφ))
= 2g2icosφ− cos θ
sinφ
(
− log(1 + eiφ) + log(1 + e−iφ))
= 2g2cosφ− cos θ
sinφφ+O(g4)
• This integrability calculation is in exact agreement with the one loop perturbative
calculation.
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Summary
• Generalization I: A two–parameter family of Wilson loops interpolating between the
line and the antiparallel lines.
• They are no more complicated than the antiparallel lines. Explicit results at 3 loops in
perturbation theory and classical and 1 loop in string theory.
• These observables interesting in their own right: Cusp anomalous dimension,
bremstrahlung function, renormalization of general Wilson loops.
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Summary
• Generalization I: A two–parameter family of Wilson loops interpolating between the
line and the antiparallel lines.
• They are no more complicated than the antiparallel lines. Explicit results at 3 loops in
perturbation theory and classical and 1 loop in string theory.
• These observables interesting in their own right: Cusp anomalous dimension,
bremstrahlung function, renormalization of general Wilson loops.
• Generalization II: Including local operator leads to open spin–chain model.
• Surprisingly simple open spin–chain model, where the boundary reflection can be
diagonalized.
• A set of TBA equations which calculate all these quantities.
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• The answer is not very different from that of the usual spectral problem.
• For Konishi wrapping started at 4 loop order. The cusped Wilson loop is given purely
by wrapping from one loop on.
• Other interesting observables given by similar spin–chains?
Nadav Drukker 35 Integrable Wilson loops
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When I talked about my paper with Valentina a year ago I would end with the question
Will there be a gauge theory derivation of the strong coupling potential:
V (L, λ) =4π2
√λ
Γ( 14 )4 L
Nadav Drukker 36 Integrable Wilson loops
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When I talked about my paper with Valentina a year ago I would end with the question
Will there be a gauge theory derivation of the strong coupling potential:
V (L, λ) =4π2
√λ
Γ( 14 )4 L
We are very close to answering Yes!
Nadav Drukker 36-a Integrable Wilson loops
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The end
Nadav Drukker 37 Integrable Wilson loops