1

Wilson loops at strong coupling

for curved contours with cusps

based on H.D. 1509.00222

• Motivation and introduction

• Contour with two cusps, formed by segments of two circles

• Cusp anomalous dimension in the generic case

• Conclusions

2Introduction

Local supersymmetric Wilson (Maldacena) loop in N = 4 SYM

W [C] = 1N 〈trP exp

∫

C i(Aµ(x(τ))xµ + ΦIθI(x)|x|)dτ〉

C : xµ(τ) closed path in R1,3, θI(x) closed path in S5.

In non-SUSY gauge theories like QCD blue terms absent.

UV properties for smooth contours C:

N = 4 SYM: finite, invariance under conformal maps of C

QCD: No further renormalisation, beyond that necessary for local

correlation functions Polyakov 1979, Dotsenko, Vergeles 1980 ...

Note: Everything disregarding a linear divergence proportional to the length of C.

(It is anyway absent in the SUSY case. Drukker, Gross, Ooguri 1999)

3Introduction

UV properties for contours with cusps: Both in N = 4 SYM and QCD

renormalisation requires cusp anomalous dimension Γ(g, θ), depending on

the coupling and the angle

QCD: one loop Polyakov 1980, two loops Korchemsky, Radyushkin 1987,

three loops Grozin, Henn, Korchemsky, Marquard 2014

N = 4 SYM: ..., four loops (planar limit) Henn, Huber 2013

Strong coupling, input from AdS/CFT:

W [C] in N = 4 SYM ⇐⇒ string partition fct. in AdS5 × S5 with b. c.:

string surface appr. contour C on conformal boundary of AdS5 × S5

Maldacena; Rey/Yee 1998

For λ = g2N → ∞ : logW = −√λ

2πA + O(logλ) , A area

4Introduction

The construction uses Poincare coordinates (conf. bound. at r = 0)

ds2 =1

r2

(

dxµdxµ + dr2)

.

UV divergences ⇐⇒ div. of area due to blow up of metric near r = 0.

Regularised area Aǫ: cut off parts of surface with r < ǫ.

Surfaces for smooth contours near boundary Graham, Witten 1999;

Polyakov, Rychkov 2000

xµ(σ, r) = xµ(σ,0) +1

2

d2

dσ2xµ(σ,0) r2 + O(r3) .

Crucial: no terms linear in r !!

This implies for smooth contours

Aǫ =l

ǫ+ Aren + O(ǫ) .

5Introduction

Then Aren is conformally invariant For infinitesimal trafos see:

Muller, Munkler, Plefka, Pollok, Zarembo 2013

For finite conformal trafo:

Out[56]= Out[58]=

Circle with radius 0.3, centered at (0.5,0) and its image under inversion on unit circle.

Red lines are mapped to each other: r = 0.1 on the left. Blue line: r = 0.1 on the right

6Introduction

With r, σ as coordinates on the string surface one gets for the induced metric

√h =

√

~x0′2

r2(1 +O(r2)) = ∂r

(

−√

~x0′2

r + O(r)

)

Via Stokes the difference of the reg. areas is:

∆A =∫

r′=ǫ

√

~x0′2

rdσ −

∫

r=ǫ

√

~x0′2

rdσ + O(ǫ) = L′ − L +O(ǫ) .

L, L′ AdS length of lines r = ǫ and r′ = ǫ

L = lǫ + O(ǫ) , l length of boundary curve in sense of R

4 .

Line r′ = ǫ (AdS)-isometrically mapped to a line with const. dist. from

boundary, hence L′ = l′ǫ + O(ǫ) , with l′ length of boundary curve

of the mapped surface.

∆A =l′

ǫ− l

ǫ+ O(ǫ)

i.e. limǫ→0(A(ǫ)− lǫ) is conformally invariant.

7Introduction and Motivation

Contours with cusps, one expects

Aǫ =l

ǫ+

n∑

i=1

Γcusp(θi) log ǫ + Aren + O(ǫ) .

Γcusp calculated via cusp between straight halflines

Drukker, Gross, Ooguri 1999

Question: Is this the correct factor for the log divergences also if cusps

have curved legs ?

(in particular with coeff. of logǫ depending on the θj alone)

Positive answer in two steps:

- consider an explicit example:

contour formed out of segments of intersecting circles

- proof for generic curved contours with cusps in an Euclidean plane

8Contour with two cusps, formed by segments of two circles

- special conformal transformations map circles to circles,

straight lines are circles with infinite radius

- special conformal transformations on the boundary continued into

the bulk of AdS act there as isometries

Apply xµ 7→ xµ/x2 to two halflines starting at (q,0), their angles w.r.t.

x1-axis: γ1 < γ2

Q = 1q , Rj =

12q|sinγj|

, D = 12q |cotγ1− cotγ2| , l = 1

q

(

γ1sinγ1

+ γ2sinγ2

)

.

Rj radii of circles, D distance of their centers,

Q distance of cusps, l length of two-cusped contour, θ = γ2 − γ1 cusp angle

9Contour with two cusps, formed by segments of two circles

10Contour with two cusps, formed by segments of two circles

Cusp between straight halflines:

x1 = ρ cos ϕ , x2 = ρ sin ϕ , r = ρf(ϕ)

ϕ = E∫ ∞

f

df√

(f4 + f2)2 − E2(f4 + f2), E = f0

√

1+ f20

f(ϕ) =

(

3ϕ

E

)−1/3+ O(ϕ1/3)

θ = 2E∫ ∞

f0

df√

(f4 + f2)2 − E2(f4 + f2)

Γcusp(θ) = 2f0 − 2

∫ ∞

f0

√

f4 + f2

f4 + f2 − E2− 1

df =π

2

f20

√

1+ f20

2F1

(1

2,3

2,2,

−f20

1+ f20

)

11Contour with two cusps, formed by segments of two circles

Aǫ,L =∫

ρ/f(ϕ)>ǫ, ρ<Ldρ dϕ

√

f4 + f2 + (f ′)2

ρ

=2L

ǫ+ Γcusp(θ) log

ǫ

L+ A0(θ) + . . . .

A0(θ) = 2f0 (logf0 − 1) − 2

∫ ∞

f0

logf

√

f4 + f2

f4 + f2 − E2− 1

df .

Now

x1 = q + ρ cos(ϕ+ γ1) , x2 = ρ sin(ϕ+ γ1) , r =ρ

f(ϕ)

Wanted: regularised area (r > ǫ) of image under

xµ 7→ xµ

x2 + r2, r 7→ r

x2 + r2

12Contour with two cusps, formed by segments of two circles

Since map is an isometry, we can calculate on the original surface. Then

the wanted area is

Aǫ =∫

r′>ǫ

√h dρdϕ

r′ =ρ/f(ϕ)

ρ2 + q2 +2qρ cos(ϕ+ γ1) + (ρ/f(ϕ))2

For each ϕ condition r′ = ǫ has two solutions, hence : ρ−(ϕ) < ρ < ρ+(ϕ)

Complete analysis needs some care, since ϕ enters directly

and via f(ϕ), which is known only implicitly. .... =⇒ ....

13Contour with two cusps, formed by segments of two circles

Aǫ = 2Γcusp(θ) log(µǫ) +l

ǫ+ Aren + O(ǫ)

Aren = −2Γcusp(θ) log(

µQ)

− 2π√

1+ f20

+ 2∫ ∞

0dz

1−√

√

√

√

1+ z2 + f201+ z2 +2f20

log(1 + z2 + f20)

• Most natural RG scheme: minimal subtraction in l/ǫ i.e. µ = 1/l.

Then Aren sum of a term depending only on θ plus a term depending on θ and Q/l.

No conformal invariance, but covariance.

• Limit θ → π i.e. one circle: Aren = −2π as known before.

14Contour with two cusps, formed by segments of two circles

Symmetric case R1 = R2, Aren and Γcusp as functions of θ.

15Cusp anomalous dimension in the generic case

Put cusp at origin of (x1, x2)-plane. Parameterise curved legs of cusp by

x(j)1 = ρ cos

(

φ(j)(ρ))

, x(j)2 = ρ sin

(

φ(j)(ρ))

, j = 1,2 .

Choose small ρ0 (fix at ǫ → 0!!), divide surface in ρ =√

x21 + x2

2 smaller or larger ρ0

Aǫ = Acuspǫ (ρ0) + Asmooth

ǫ (ρ0)

Asmoothǫ (ρ0) =

l − lρ0ǫ

+ O(1)

lρ0 = 2ρ0 +1

6(c21 + c22)ρ

30 + O(ρ40)

2cj = 2dφ(j)

dρ

∣

∣

∣

∣

ρ=0

are the ρ → 0 limits of the curvatures of both legs.

θ = φ2(0) − φ1(0)

16Cusp anomalous dimension in the generic case

Wanted: coordinate system

x1 = ρ u(ρ, ϕ) , x2 = ρ

√

1− u2 , with

u(ρ,0) = cos(

φ(1)(ρ))

, u(ρ, θ) = cos(

φ(2)(ρ))

The additional AdS-coordinate r then parameterised by

r = ρ F (ρ, ϕ) , F (ρ, ϕ) = F1(ϕ) + ρ F2(ϕ) + . . . .

Boundary condition F (ρ,0) = F (ρ, θ) = 0 .

Plan: Insert parameterisation into minimal surface condition (eq. of motion),

expand in ρ, get set of ordinary diff. eq. for the Fn(ϕ).

(expect F1(ϕ) = 1/f(ϕ))

17Cusp anomalous dimension in the generic case

First naive ansatz for u(ρ, ϕ)

u(ρ, ϕ) = cos

(

ϕ

θ

(

φ2(ρ)− φ1(ρ))

+ φ1(ρ)

)

.

But then the boundary conditions for F2(ϕ) cannot be satisfied !!!

More elaborate ansatz for u(ρ, ϕ)

u(ρ, ϕ) = cos

(

ρ s(ϕ) + ϕ

θ

(

φ2(ρ)− φ1(ρ))

+ φ1(ρ)

)

,

where the function s(ϕ) has to be chosen with the behaviour

s(ϕ) = a1 ϕ2/3 + . . . , ϕ → 0 ,

s(ϕ) = a2 (θ − ϕ)2/3 + . . . , ϕ → θ .

Then, after solving the diff. eq. for F2(ϕ), it turns out that its b.c. can be realised if

aj = (3/E)2/3 cj .

18Cusp anomalous dimension in the generic case

Example for θ = 1.2, φ1(ρ) = 5ρ− 15ρ2 , φ2 = 1.2+ ρ+2ρ2 (red curves). Lines of const. ϕ in step size

0.2 are in blue. Green lines show the first steps with size 0.02. On the left we see the situation for s = 0,

on the right for s(ϕ) := a1ϕ2/3(1− ϕ/θ)10 + a2(θ − ϕ)2/3(ϕ/θ)10. aj adapted as described in the text.

19Cusp anomalous dimension in the generic case

Eq. for F1(ϕ) is the same as for 1/f(ϕ) in straight case ⇒ F1(ϕ) = 1/f(ϕ).

F ′′2(ϕ) +G1(ϕ)F

′2(ϕ) +G(ϕ)F2(ϕ) +M(ϕ) = 0 ,

M(ϕ) =−1

θ(F1 + F 31 )

{

θ(F1 + F 31 )F

′1 s′′(ϕ)

+(

c1(θ − ϕ) + c2ϕ+ θs(ϕ))(

2F1(F′1)

3 + F1F′1(7 + 3F 2

1 ))

+(

c1 − c2 − θs′(ϕ))

(1 + F 21 )(

6+ 3F 21 +2(F ′

1)2 + F1F

′′1

)

}

,

G(ϕ) =13F1 +7F 3

1 +2F1(F ′1)

2 + (1+ 5F 21 )F

′′1

F1 + F 31

,

G1(ϕ) =2(2− F 2

1 ) F ′1

F1 + F 31

.

20Cusp anomalous dimension in the generic case

Asymptotic behaviour of F2 for ϕ → 0: (a = (2/E)1/3)

F2(ϕ) =1

3(aa1 − a3c1)−

(4a3a1 − 2a5c1)

15ϕ2/3 +O(ϕ)

+B1

(

ϕ−2/3 +8

5a2 +O(ϕ2/3)

)

+B2

(

ϕ1/3 + a2ϕ+O(ϕ5/3))

.

With a1 = a2c1 (constraining the coordinate system) and B1 = 0 (fixing an integration

constant) we get

F2(ϕ) = B2 ϕ1/3 + . . . i.e. the same power as for F1

21Cusp anomalous dimension in the generic case

Acuspǫ (ρ0) =

∫

ρ<ρ0, ρF (ρ,ϕ)>ǫL(ρ, ϕ)dρdϕ

L(ρ, ϕ) =1

ρL1(ϕ) + L2(ϕ) + . . .

L1(ϕ) =1

3aϕ4/3+ O(ϕ−2/3) , L2(ϕ) = − B2

3a2ϕ4/3+ O(ϕ−2/3)

Subtleties:

- divide ϕ integration into ϕ ∈ (0, θ/2) and ϕ ∈ (θ/2, θ) to control

singularities from both legs of the cusp

- lowest order contribution: integrand as in straight case, but boundaries

depend on F2 =⇒

log div. as in straight case, 1/ǫ div. with ρ0 and B2 dependent factor

22Cusp anomalous dimension in the generic case

- nextleading contribution: only 1/ǫ divergence, its B2 dependent term

cancels with that from leading contrib.

- remaining ρ0 dependence cancels with that from Asmoothǫ

Aǫ =l

ǫ+ Γcusp(θ) logǫ + O(1)

23Conclusions

• First calculation of Wilson loop for strong coupling and a

curved contour with cusps including all divergent and finite terms

• Developed perturbative technique for minimal surfaces near a cusp

• Proof for commonly expected divergence structure of Aǫ for generic

curved contours with cusps in a Euclidean plane

Open:

• Extension to nonplanar cont., Lorentzian, cusps with null tangents

• Dependence of renormalised Wilson loops for segments of intersecting

circles on R1, R2, D in small coupling perturbation theory

• Similar issues for corner contributions to entanglement entropy