Duality between Wilson Loops and Scattering Amplitudes in QCD

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1 Duality between Wilson Loops and Scattering Amplitudes in QCD by Yuri Makeenko (ITEP, Moscow) Based on: Y. M., Poul Olesen Phys. Rev. Lett. 12, 7162 (29) [arxiv: [hep-th]] arxiv: [hep-th] further developments

2 39 years of QCD/string correspondence

3 Motivations since 1979 Reformulation of QCD in term of Wilson loops Wilson (1975) (lattice) Y.M., Migdal (1979) (continuum) QCD string is not Nambu Goto but the asymptote at large Wilson loops is universal W (C) large C e KS min(c) = the area law What are the consequences for correlators of composite operators? Regge behavior of scattering amplitudes at high energy and fixed momentum transfer under a few controllable approximations. Why not 3 years ago? Two essential ingredients: Wilson-loop/scattering-amplitude duality in N = 4 SYM Alday, Maldacena (27) Representation of minimal area as quadratic functional of x µ ( ) Douglas (1931)

4 Scattering amplitudes in N = 4 SYM N = 4 super Yang Mills: gluons + 6 scalars + 4 fermions in adjoint representation of SU(N) no asymptotic freedom no confinement but perturbation theory resembles QCD 4-gluon (on-shell) scattering amplitude in multicolor limit N A(s, t) = A IRdiv A tree e f(λ) log2 (s/t)/2 s = (p 1 + p 2 ) 2 and t = (p 1 + p 4 ) 2 are Mandelstam s variables as conjectured by Bern, Dixon, Smirnov (25) based on three loops, verified at four loops and by the others f(λ) (λ = g 2 N the t Hooft coupling) also appears in the anomalous dimensions of cusped Wilson loops and operators of twist two. It has been recently exactly calculated from the spin chains. This reproduces three and four loop SYM results and the λ behavior for large λ originally found using the AdS/CFT correspondence. The next orders in 1/ λ also agree with superstring calculations.

5 Wilson-loop/scattering-amplitude duality Alday, Maldacena (27) Scattering amplitudes = Wilson loops A (p 1,..., p n ) /A tree = W (x 1,..., x n ) for rectangles whose vertices x i are related to the momenta p i = K ( x i x i 1 ), where K = 1/2πα is string tension. Large p s correspond to large x s. p 2 i = = light-like edges of polygon WL/SA duality was originally suggested for large λ but is also verified in perturbation theory (up to two loops) Drummond, Korchemsky, Sokatchev (28); Brandhuber, Heslop, Travaglini (28).

6 QCD amplitudes via Wilson loops Y.M., Migdal (1981) Green s functions of M colorless composite quark operators q(x i )q(x i ) q(x i )γ 5 q(x i ) q(x i )γ µ q(x i ) q(x i )γ µ γ 5 q(x i ) are given by the sum over Wilson loops passing via x i (i = 1,..., M) G T M i=1 dτ M 1 q(x i )q(x i ) M 2 i=1 τi+1 conn dτ i = z()=z(t )=x z(τ i )=x i The weight for the path integration is J[z(τ)] = Dk(τ) sp P e i T dt e mt Dz(τ) J[z(τ)] W [z(τ)]. dτ [ż(τ) k(τ) γ(τ) k(τ)] for spinor quarks of mass m and scalar operators or J[z(τ)] = e 1 2 for scalar quarks. τ is the proper time. T dτ ż2 (τ)

7 QCD amplitudes via Wilson loops (cont.1) The Wilson loop W (C) is in pure Yang Mills at large N (or quenched). For finite N, correlators of several Wilson loops are present. The on-shell scattering amplitudes can be obtained by the Lehman Symanzik Zimmerman reduction. We represent M momenta of the (all incoming) particles by the differences p i = p i 1 p i. Then momentum conservation is automatic while an (infinite) volume V is produced, say, by integration over x = x M : G ( p 1,..., p M ) = 1 V M 1 i=1 d 4 x i e i i p ix i M i=1 q(x i )q(x i ) conn

8 QCD amplitudes via Wilson loops (cont.2) Piecewise constant momentum-space loop Because p(τ) = p i for τ i < τ < τ i+1. ṗ(τ) = i p i δ(τ τ i ) with p i p i 1 p i we write i p i x i = which is manifestly parametric invariant. dτ p(τ) ż(τ) Making the Fourier transformation (for piecewise constant p(τ) ) G ( p 1,..., p M ) = M 2 τi+1 dτ i i=1 z()=z(t )= dt e mt T dτ M 1 Dz(τ) e i T dτ ż(τ) p(τ) J[z(τ)] W [z(τ)], No integration over z() = z(t ) which would produce a volume

9 Minimal area and boundary functional Douglas (1927) To calculate the scattering amplitudes, we substitute the area-law behavior of asymptotically large Wilson loops: and integrate over the paths. W (C) large C e KS min(c), S(C) is highly nonlinear functional = hopeless to calculate Douglas algorithm for solving the Plateau problem (finding the minimal surface) is to minimize the boundary functional A[x(θ)] = 1 8π 2π dφ 2π dφ [ x(θ(φ)) x(θ(φ )) ] 2 1 cos(φ φ ) with respect to the reparametrizations θ(φ) ( θ(φ) ). In general A[x(θ)] A[x(θ )] = S min (C) The minimum is reached at θ(φ) = θ (φ) which is contour-dependent.

10 Minimal area and boundary functional (cont.) By varying A[θ] with respect to θ(φ) at given C, we get the equation 2π dφ ẋ(θ (φ)) [x(θ (φ)) x(θ (φ )) ] 1 cos(φ φ ) = We can rewrite the Douglas functional equivalently as A = 1 4π 2π 2π dθ 1 dθ 2 ẋ(θ 1 ) ẋ(θ 2 ) ln (1 cos [φ(θ 1 ) φ(θ 2 )]) when only φ(θ) is sensitive to reparametrizations. The variation with respect to φ(θ) results in the equation for the inverse function φ (θ) 2π dα ẋ(θ) ẋ(α) cot Simplest example: φ (θ) = θ for a circle. ( φ (θ) φ (α) 2 ) =

11 Example of an ellipse An ellipse x 1 = a cos θ(φ), x 2 = b sin θ(φ) ɛ = a2 b 2 resulits in the equation 2π dα sin [θ (φ + α) θ (φ)] 1 cos α = ɛ Both sides vanish for a circle ɛ = for θ (φ) = φ. The solution for ɛ is θ (φ) = 2π π 1 2K(s) (1 s) 2 + 4s sin 2 φ a 2 + b 2 dα sin [θ (φ + α) + θ (φ)] sin [2θ (φ)] 1 cos α πk ( 1 s 2 ) 2K (s) where K(s) is the complete elliptic integral of the first kind. = log a + b a b Elliptic integrals also emerge for a rectangle (the Schwarz Christoffel mapping).

12 Relation to boundary action in string theory A[x] is well-known as the classical boundary action in string theory Tree-level disk amplitude with Dirichlet b.c. for Polyakov string: integrate over fluctuations inside the disk, X(r, φ) with r < 1 fix the value X(1, φ) x(θ(φ)) at the boundary. Subtlety with fixing conformal gauge: Liouville field decouples only in the interior of the disk. Its boundary value determines θ (φ) at the classical level. The path integral over the boundary values restores the invariance under reparametrizations of the boundary. The path integral over the boundary values decouples in d = 26 for on-shell tachyon = usual Koba Nielsen amplitudes.

13 Relation to boundary action in string theory (cont.) The proof of A[x(θ )] = S min (C) reconstruct the surface X µ (r, φ) from the boundary value X µ (1, φ) = x µ (θ (φ)) by the Poisson formula thus constructed X µ automatically obeys conformal gauge X r X φ =, r2 X r X r = X φ X φ. = the Nambu Goto action = the quadratic action = the boundary action for Polyakov string = the minimal area

14 Integration over reparametrizations Polyakov (1997) Wilson loop in large-n QCD the tree-level string disk amplitude integrated over reparametrizations of the boundary contour. Conformal map of the disk into the upper half-plane: the disk boundary = the real axis t(τ) = tan πτ < t < + T Reparametrization-invariant ansatz + W (C) = Ds(t) exp ( K 2π dt 1dt 2 ẋ(t 1 ) ẋ(t 2 ) ln s(t 1 ) s(t 2 ) ) where the path integral is over reparametrizations s(t) (with s (t) ). This classical boundary action is derivable for: bosonic string in d = 26, superstring in d = 1. Area law for asymptotically large C (or very large K) = a saddle point in the integral over reparametrizations at s(t) = s (t).

15 Large loops and minimal area Gaussian fluctuations around the saddle-point θ (σ) result in a preexponential factor W [x( )] large loops = F [ Kx( ) ] e KS min [x( )] [ 1 + O ( (KS min ) 1)], which is contour dependent F [circle] KR 2 for a circle Asymptotic area law is recovered modulo the pre-exponential which is not essential for large loops. Alternatively the Douglas minimization min t(s) 4π K e + ds + ds [x(t(s)) x(t(s ))] 2 (s s ) 2 = e KS min[x( )] gives an exact area law

16 Functional Fourier transformation Reparametrization-invariant functional Fourier transformation W [p( )] = Dx e i p dx W [x( )] of the disk amplitude for piecewise constant p(t). Performing the Gaussian integration: W [p( )] = Ds(t) exp (α + dt 1 + dt 2 ṗ(t 1 ) ṗ(t 2 ) ln s(t 1 ) s(t 2 ) ) It looks like the disk amplitude with K replaced by 1/K = 2πα. The determinant is a s-independent constant. The principal-value prescription will be important for stepwise p(t). p(t) = p j at the j-th interval for the stepwise discretization = reparametrization changes t j s for s j s keeping their cyclic order discrete reparametrization transformation. Stepwise discretization of x(t) itself would violate the continuity of the string end world line.

17 Derivation of Koba Nielsen amplitudes Stepwise discretization of p(t) naturally results in M-particle (off-shell) Koba Nielsen amplitudes invariant under the P SL(2; R) projective (or Möbius) transformation s as + b cs + d with ad bc = 1 because the projective group is a subgroup of reparametrizations. First note that + + dt = 1 2 dt 2 ṗ(t 1 ) ṗ(t 2 ) ln s(t 1 ) s(t 2 ) ds 1 ds 2 (s 1 s 2 ) 2 [p (t(s 1)) p (t(s 2 ))] 2 Integration over s 1 or s 2 has divergences for adjacent sides k = l ± 1. A regularization by the principal-value prescription is needed

18 Derivation of Koba Nielsen amplitudes (cont.1) Omit the sides with k = l ± 1 the integrals over s 1 and s 2 are finite: 1 2 k l±1 sk s k 1 ds 1 sl s l 1 ds 2 (p k p l ) 2 (s 1 s 2 ) 2 = k l p k p l log s k s l + j p 2 j log (s j s j 1 )(s j+1 s j ) (s j+1 s j 1 ) which is projective invariant. The main formulas: (s i s j ) ds i under the projective transformation. (s i s j ) (cs i + d)(cs j + d) ds i (cs i + d) 2

19 Derivation of Koba Nielsen amplitudes (cont.2) Integrating over reparametrizations at intermediate points (s i 1, s i ) results in the following measure D (M) s = M i=1 ds i s i s i 1 for the integration over s i s. It is invariant under the projective transformation and gives W ( p 1,..., p M ) = s i 1 <s i i ds i s i s i 1 k l ( s k s l α p k p sj s l j 1 s j+1 s j s j+1 s j 1 j ) α p 2 j where the integration over s i emerges from the path integral over reparametrizations. Fixing the P SL(2; R) invariance in the standard way s 1 =, s M 1 = 1, s M = = scalar amplitudes in the Koba Nielsen variables.

20 Path integrals over reparametrization The measure on Diff(R) t(s )=t t(s f )=t f tf 1 D diff t(s) = lim L t (t f t L ) is invariant under reparametrizations L j=1 s t(s), t(s ) = s, t(s f ) = s f, tj+1 t dt j 1 (t j t j 1 ) dt ds The main integral for the integration at the intermediate point t i ti+1 δ dt i t i 1 (t i+1 t i ) 1 δ (t i t i 1 ) 1 δ = 2 (t i+1 t i 1 ) 1 2δ where small δ is introduced to control a logarithmic divergence. This is an analogue of the well-known formula + dt i e (t f t i ) 2 /2ν 1 e (t i t ) 2 /2ν 2 = e (t f t ) 2 /2(ν 1 +ν 2 ) 2π ν1 ν2 (ν 1 + ν 2 ) which is used for calculations with the usual Wiener measure

21 Projective-invariant off-shell amplitudes For 4 scalars this reproduces the Veneziano amplitude A( p 1, p 2, p 3, p 4 ) = 1 dx x α(s) 1 (1 x) α(t) 1, where α(t) = α t linear Regge trajectory and s = ( p 1 + p 2 ) 2, t = ( p 2 + p 3 ) 2 are usual Mandelstam s variables (for Euclidean metric). The tachyonic condition α p 2 j = 1 has not to be imposed. An arbitrary value of the intercept α() can be reached by inserting in the measure (by hands) ( si+1 s i 1 ) α() i s i+1 s i

22 Projective-invariant off-shell amplitudes (cont.) Another choice of the measure (with α() = 1) D (M) s = M i=1 ds i (s i+1 s i 1 ) (s i s i 1 )(s i+1 s i ) also invariant under the projective transformation results in W ( p 1,..., p M ) = ds i s k s l α p k p l s i 1 <s i ( (sj s j 1 )(s j+1 s j ) i k l ) α p 2 j 1 j (s j+1 s j 1 ) where the integration over s j emerges from the path integral over reparametrizations. The Lovelace choice reproduces some projective-invariant off-shell string amplitudes known since late 196 s. The more familiar on-shell tachyon amplitudes can be obtained by setting α p 2 j = 1.

23 Regge Veneziano behavior from the area law To substitute the area-law behavior of W (C) into the path integral and to find out for what momenta the asymptotically large loops dominate. Typical momenta will be large for large loops. Interchanging the order of integration over z(τ) and σ(τ), we obtain G ( p 1,..., p M ) = dt e mt α 2 sp P e T dτ 1 T dτ M 1 M 2 τi+1 dτ i σ(τ i )=τ i Dσ(τ) Dk(τ) i=1 T dτ 2( k(τ 1 )+ṗ(τ 1 )) ( k(τ 2 )+ṗ(τ 2 )) ln( 1 cos[ 2π T (σ(τ 1 ) σ(τ 2 )) e i T dτ γ(τ) k(τ) τ M = T This is rather close to the disk amplitude, except for the additional integration over k. ])

24 Regge Veneziano behavior from the area law (cont.1) The QCD scattering amplitude has the form (fixed s M ) G( p 1,..., p M ) M 1 i=1 s i+1 ds i 1 + s 2 i [ si+1 s i s i s i 1 s i+1 s i 1 s i s j p i p j /2πK K(s 1,..., s M 1 ; p 1,..., p M ) ] p 2 i /2πK which is a convolution of the Koba Nielsen integrand and a kernel. The spectrum is still of the Regge Veneziano type (= linear). For small m (and/or very large M), the integral over T is dominated by large T (M 1)/m because M 1 i dτ i T M 1. Typical values of k 1/T are essential in the path integral over k for large T = K becomes momentum independent: K(s 1,..., s M 1 ; p 1,..., p M ) = M i=1 1 s i+1 s i so the (off-shell) Koba Nielsen amplitudes are reproduced.

25 Regge Veneziano behavior from the area law (cont.2) M = 4 QCD scattering amplitude (with no P SL(2; R)) G 4 = + 1 ds 1 ds 2 ds 3 θ c (s 1, s 2, s 3, s 4 ) (1 + s 2 1 )(1 + s2 2 )(1 + s2 3 ) 1 s 43 s 32 s 21 s 41 ( s21 s 43 s 31 s 42 ) α s ( s 41 s 32 s 31 s 42 ) α t s ij = s i s j. Introducing the variable x = s 21s 43 s 31 s 42 x 1 x = for s 2 = s 1 x = 1 for s 2 = s 3 we get G 4 = 1 dx x α s 1 (1 x) α t 1 + ds 1 ds 3 θ c (s 1, s 3, s 4 ) 1 s 43 s 31 s 41 because of the linear divergence of the integral over s 1 and s 3

26 Dominating C in the sum over paths Loop C dominating in the sum over paths: x = i K G ṗ, x µ (τ (σ)) = i K with arbitrary τ (σ): τ (σ i ) = σ i. j p µ j G ( σ σ j ) It bounds the minimal surface of the area KS min (C) = α t [ln st ] + 1 for s t K. It is large for very large s Actually, the integrand oscillates because of e i p dx that results in the factor of i. But an estimate of the order of magnitude is correct Probably t has to be large but s for the width of C to be K. Then the value of α() (coming from the measure) is not essential.

27 Dominating C in the sum over paths (cont.1) Typical loop C dominating in the sum over paths for t/s =

28 Dominating C in the sum over paths (cont.2) Typical loop C dominating in the sum over paths for t/s =

29 Dominating C in the sum over paths (cont.3) Typical loop C dominating in the sum over paths for t/s =

30 Dominating C in the sum over paths (cont.4) Typical loop C dominating in the sum over paths for t/s =

31 Discussion It was crucial for the success of calculations that all integrals are Gaussian except for the one over reparametrizations which reduces to integration over the Koba Nielsen variables. Derivation is legible for those momenta p i for which asymptotically large loops are essential in the sum over C, estimated to be: KS min (C ) = α t ln s max {t,k}. Spectrum of classical string suggests this should be indeed the case at least for asymptotically large s and large t s. The value of α() is then not essential in this region. But the domain of applicability is broader and extends at least to t K. 4-point scattering amplitude is valid only for asymptotically large s and fixed t associated with small angle or fixed momentum transfer. When t s becomes large, there are no longer reasons to expect the contribution of large loops to dominate over perturbation theory, which comes from integration over small loops.

32 Effective ρ-trajectory and pqcd prediction The figure taken from A. B. Kaidalov, hep-ph/ It is hard to believe that pqcd reggeization is relevant