Wilson loops at strong coupling for curved contours with cusps. Contour with two cusps, formed by segments of two circles

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1 1 Wilson loops at strong coupling for curved contours with cusps based on H.D Motivation and introduction Contour with two cusps, formed by segments of two circles Cusp anomalous dimension in the generic case Conclusions

2 Introduction 2 Local supersymmetric Wilson (Maldacena) loop in N = 4 SYM W[C] = N 1 trp exp C i(a µ(x(τ))ẋ µ + Φ I θ I (x) ẋ )dτ C : x µ (τ) closed path in R 1,3, θ I (x) closed path in S 5. 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 Note: Everything disregarding a linear divergence proportional to the length of C. (It is anyway absent in the SUSY case. Drukker, Gross, Ooguri 1999)

3 Introduction 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 3 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 AdS 5 S 5 with b. c.: string surface appr. contour C on conformal boundary of AdS 5 S 5 For λ = g 2 N : logw = λ 2π A + O(logλ), A area Maldacena; Rey/Yee 1998

4 Introduction The construction uses Poincaré coordinates (conf. bound. at r = 0) 4 ds 2 = 1 r 2 ( dx µ dx µ + dr 2). UV divergences div. of area due to blow up of metric near r = 0. Regularised area A ǫ : cut off parts of surface with r < ǫ. Surfaces for smooth contours near boundary Graham, Witten 1999; Polyakov, Rychkov 2000 x µ (σ,r) = x µ (σ,0) + 1 2dσ 2xµ (σ,0) r 2 + O(r 3 ). Crucial: no terms linear in r!! This implies for smooth contours A ǫ = l ǫ + A ren + O(ǫ). d 2

5 Introduction 5 Then A ren is conformally invariant For infinitesimal trafos see: Müller, Münkler, 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

6 Introduction With r,σ as coordinates on the string surface one gets for the induced metric ( x0 2 h = x0 2 r 2 (1+O(r 2 )) = r Via Stokes the difference of the reg. areas is: x 0 2 x 0 2 r ) + O(r) A = dσ dσ + O(ǫ) = L L +O(ǫ). r =ǫ r r=ǫ r L, L AdS length of lines r = ǫ and r = ǫ 6 L = l ǫ + O(ǫ), l length of boundary curve in sense of R4. Line r = ǫ (AdS)-isometrically mapped to a line with const. dist. from boundary, hence L = l ǫ + O(ǫ), of the mapped surface. A = l ǫ l ǫ + O(ǫ) i.e. lim ǫ 0 (A(ǫ) ǫ l ) is conformally invariant. with l length of boundary curve

7 Introduction and Motivation 7 Contours with cusps, one expects A ǫ = l n ǫ + Γ cusp (θ i ) log ǫ + A ren + O(ǫ). i=1 Γ cusp 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 θ j 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

8 Contour with two cusps, formed by segments of two circles 8 - 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 x µ x µ /x 2 to two halflines starting at (q,0), their angles w.r.t. x 1 -axis: γ 1 < γ 2 Q = 1 q, R j = 1 2q sinγ j, D = 1 2q cotγ 1 cotγ 2, l = 1 q ( γ1 sinγ 1 + γ 2 sinγ 2 ). R j radii of circles, D distance of their centers, Q distance of cusps, l length of two-cusped contour, θ = γ 2 γ 1 cusp angle

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

10 Contour with two cusps, formed by segments of two circles Cusp between straight halflines: 10 x 1 = ρ cos ϕ, x 2 = ρ sin ϕ, r = ρ f(ϕ) ϕ = E f df (f 4 +f 2 ) 2 E 2 (f 4 +f 2 ), E = f 0 1+f 2 0 f(ϕ) = ( ) 3ϕ 1/3 + O(ϕ 1/3 ) E θ = 2E f 0 df (f 4 +f 2 ) 2 E 2 (f 4 +f 2 ) Γ cusp (θ) = 2f 0 2 f 0 f 4 +f 2 f 4 +f 2 E 1 π f 2 ( df = 0 1 2F f 2 0 2,3 2,2, f0 2 ) 1+f0 2

11 Contour with two cusps, formed by segments of two circles 11 A ǫ,l = ρ/f(ϕ)>ǫ, ρ<l dρ dϕ f 4 +f 2 +(f ) 2 ρ = 2L ǫ A 0 (θ) = 2f 0 (logf 0 1) 2 + Γ cusp (θ) log ǫ L + A 0(θ) f 0 logf f 4 +f 2 f 4 +f 2 E 1 df. 2 Now x 1 = q + ρ cos(ϕ+γ 1 ), x 2 = ρ sin(ϕ+γ 1 ), r = ρ f(ϕ) Wanted: regularised area (r > ǫ) of image under x µ x µ x 2 +r 2, r r x 2 +r 2

12 Contour with two cusps, formed by segments of two circles 12 Since map is an isometry, we can calculate on the original surface. Then the wanted area is A ǫ = r >ǫ h dρdϕ r = ρ/f(ϕ) ρ 2 +q 2 +2qρ cos(ϕ+γ 1 )+(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.... =...

13 Contour with two cusps, formed by segments of two circles 13 A ǫ = 2Γ cusp (θ) log(µǫ) + l ǫ + A ren + O(ǫ) A ren = 2Γ cusp (θ) log ( µq ) 2π 1+f dz 1 1+z2 +f z 2 +2f 2 0 log(1+z 2 +f 2 0 ) Most natural RG scheme: minimal subtraction in l/ǫ i.e. µ = 1/l. Then A ren 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: A ren = 2π as known before.

14 Contour with two cusps, formed by segments of two circles 14 Symmetric case R 1 = R 2, A ren and Γ cusp as functions of θ.

15 Cusp anomalous dimension in the generic case Put cusp at origin of (x 1,x 2 )-plane. Parameterise curved legs of cusp by 15 x (j) 1 = ρ cos ( φ (j) (ρ) ), x (j) 2 = ρ sin ( φ (j) (ρ) ), j = 1,2. Choose small ρ 0 (fix at ǫ 0!!), divide surface in ρ = x 2 1 +x2 2 smaller or larger ρ 0 A ǫ = A cusp ǫ (ρ 0 ) + A smooth ǫ (ρ 0 ) A smooth ǫ (ρ 0 ) = l l ρ 0 ǫ + O(1) l ρ0 = 2ρ (c2 1 +c2 2 )ρ3 0 + O(ρ4 0 ) 2c j = 2 dφ(j) dρ ρ=0 are the ρ 0 limits of the curvatures of both legs. θ = φ 2 (0) φ 1 (0)

16 Cusp anomalous dimension in the generic case 16 Wanted: coordinate system x 1 = ρ u(ρ,ϕ), x 2 = ρ 1 u 2, 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) = F 1 (ϕ) + ρ F 2 (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 F n (ϕ). (expect F 1 (ϕ) = 1/f(ϕ))

17 Cusp anomalous dimension in the generic case 17 First naive ansatz for u(ρ,ϕ) u(ρ,ϕ) = cos ( ϕ θ ( φ2 (ρ) φ 1 (ρ) ) + φ 1 (ρ)). But then the boundary conditions for F 2 (ϕ) cannot be satisfied!!! More elaborate ansatz for u(ρ, ϕ) ( ρ s(ϕ)+ϕ u(ρ,ϕ) = cos θ ( φ2 (ρ) φ 1 (ρ) ) + φ 1 (ρ) where the function s(ϕ) has to be chosen with the behaviour ), s(ϕ) = a 1 ϕ 2/3 +..., ϕ 0, s(ϕ) = a 2 (θ ϕ) 2/3 +..., ϕ θ. Then, after solving the diff. eq. for F 2 (ϕ), it turns out that its b.c. can be realised if a j = (3/E) 2/3 c j.

18 Cusp anomalous dimension in the generic case 18 Example for θ = 1.2, φ 1 (ρ) = 5ρ 15ρ 2, φ 2 = 1.2+ρ+2ρ 2 (red curves). Lines of const. ϕ in step size 0.2 are in blue. Green lines show the first steps with size On the left we see the situation for s = 0, on the right for s(ϕ) := a 1 ϕ 2/3 (1 ϕ/θ) 10 +a 2 (θ ϕ) 2/3 (ϕ/θ) 10. a j adapted as described in the text.

19 Cusp anomalous dimension in the generic case Eq. for F 1 (ϕ) is the same as for 1/f(ϕ) in straight case F 1 (ϕ) = 1/f(ϕ). 19 F 2 (ϕ)+g 1(ϕ)F 2 (ϕ)+g(ϕ)f 2(ϕ)+M(ϕ) = 0, M(ϕ) = 1 { θ(f 1 +F1 3) θ(f 1 +F 3 1)F 1 s (ϕ) + ( c 1 (θ ϕ)+c 2 ϕ+θs(ϕ) )( 2F 1 (F 1) 3 +F 1 F 1(7+3F 2 1) ) + ( c 1 c 2 θs (ϕ) ) (1+F 2 1) ( 6+3F (F 1) 2 +F 1 F 1) }, G(ϕ) = 13F 1 +7F F 1(F 1 )2 +(1+5F 2 1 )F 1 F 1 +F 3 1, G 1 (ϕ) = 2(2 F2 1 ) F 1 F 1 +F 3 1.

20 Cusp anomalous dimension in the generic case 20 Asymptotic behaviour of F 2 for ϕ 0: (a = (2/E) 1/3 ) F 2 (ϕ) = 1 3 (aa 1 a 3 c 1 ) (4a3 a 1 2a 5 c 1 ) 15 ϕ 2/3 +O(ϕ) +B 1 ( ϕ 2/ a2 +O(ϕ 2/3 ) ) +B 2 ( ϕ 1/3 +a 2 ϕ+o(ϕ 5/3 ) ). With a 1 = a 2 c 1 (constraining the coordinate system) and B 1 = 0 (fixing an integration constant) we get F 2 (ϕ) = B 2 ϕ 1/ i.e. the same power as for F 1

21 Cusp anomalous dimension in the generic case 21 A cusp ǫ (ρ 0 ) = ρ<ρ 0, ρf(ρ,ϕ)>ǫ L(ρ,ϕ)dρdϕ L(ρ,ϕ) = 1 ρ L 1(ϕ) + L 2 (ϕ) +... L 1 (ϕ) = 1 3aϕ 4/3 + O(ϕ 2/3 ), L 2 (ϕ) = B 2 3a 2 ϕ 4/3 + 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 F 2 = log div. as in straight case, 1/ǫ div. with ρ 0 and B 2 dependent factor

22 Cusp anomalous dimension in the generic case 22 - nextleading contribution: only 1/ǫ divergence, its B 2 dependent term cancels with that from leading contrib. - remaining ρ 0 dependence cancels with that from A smooth ǫ A ǫ = l ǫ + Γ cusp(θ) logǫ + O(1)

23 Conclusions 23 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 A ǫ 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 R 1, R 2, D in small coupling perturbation theory Similar issues for corner contributions to entanglement entropy