In this talk we present the result for the $n_f$ dependent piece of the three-loop cusp anomalous dimension in QCD. Remarkably, it is parametrized by the same simple functions appearing in analogous anomalous dimensions in ${\mathcal N}=4$ SYM at one and two loops. We also compute all required master integrals using a recently proposed refinement of the differential equation method. The analytic results are expressed in terms of harmonic polylogarithms of uniform weight.
I present analytical expressions for the cusp anomalous dimension in QCD through three loops in terms of elementary functions and ordinary polylogarithms. I observe interesting relations between the results at different loops and provide a conjecture for the $n$-loop cusp anomalous dimension in terms of the lower-loop results. I also present numerical results and simple approximate formulas for the cusp anomalous dimension relevant to top-quark production.
We present an analytic derivation of the full four-loop cusp anomalous dimension of $\mathcal{N}=4$ supersymmetric Yang-Mills theory from the Sudakov form factor. To extract the cusp anomalous dimension, we calculate the $ε^{-2}$ pole of the form factor using parametric integrations of finite integrals. We provide uniformly transcendental results for the master integrals through to weight six and confirm a very recent independent analytic result for the full four-loop cusp anomalous dimension of the $\mathcal{N}=4$ model.
We present the full analytic result for the three-loop angle-dependent cusp anomalous dimension in QCD. With this result, infrared divergences of planar scattering processes with massive particles can be predicted to that order. Moreover, we define a closely related quantity in terms of an effective coupling defined by the light-like cusp anomalous dimension. We find evidence that this quantity is universal for any gauge theory, and use this observation to predict the non-planar $n_{f}$-dependent terms of the four-loop cusp anomalous dimension.
We construct an exact analytical solution to the integral equation which is believed to describe logarithmic growth of the anomalous dimensions of high spin operators in planar N=4 super Yang-Mills theory and use it to determine the strong coupling expansion of the cusp anomalous dimension.
We derive an analytic formula at three loops for the cusp anomalous dimension Gamma_cusp(phi) in N=4 super Yang-Mills. This is done by exploiting the relation of the latter to the Regge limit of massive amplitudes. We comment on the corresponding three loops quark anti-quark potential. Our result also determines a considerable part of the three-loop cusp anomalous dimension in QCD. Finally, we consider a limit in which only ladder diagrams contribute to physical observables. In that limit, a precise agreement with strong coupling is observed.
We compute the four-loop $n_f$ contribution proportional to the quartic Casimir of the QCD cusp anomalous dimension as an expansion for small cusp angle $ϕ$. This piece is gauge invariant, violates Casimir scaling, and first appears at four loops. It requires the evaluation of genuine non-planar four-loop Feynman integrals. We present results up to ${\mathcal O}(ϕ^4)$. One motivation for our calculation is to probe a recent conjecture on the all-order structure of the cusp anomalous dimension. As a byproduct we obtain the four-loop HQET wave function anomalous dimension for this color structur…
We compute the nonplanar contribution to the universal anomalous dimension of the SU(4)-singlet twist-two operators in N=4 supersymmetric Yang-Mills theory at four loops through Lorentz spin 18. From this, we numerically evaluate the nonplanar contribution to the four-loop lightlike cusp anomalous dimension and derive the transcendental $ζ_3$ and $ζ_5$ parts of the universal anomalous dimension for arbitrary Lorentz spin in analytic form. As for the lightlike cusp anomalous dimension and the $ζ_5$ part of the universal anomalous dimension, we confirm previous results.
We compute the fermionic contributions to the cusp anomalous dimension in QCD at four loops as an expansion for small cusp angle. As a byproduct we also obtain the respective terms of the four-loop HQET wave function anomalous dimension. Our new results at small angles provide stringent tests of a recent conjecture for the exact angle dependence of the matter terms in the four-loop cusp anomalous dimension. We find that the conjecture does not hold for two of the seven fermionic color structures, but passes all tests for the remaining terms. This provides strong support for the validity of the…
We present the details of the analytic calculation of the three-loop angle-dependent cusp anomalous dimension in QCD and its supersymmetric extensions, including the maximally supersymmetric $\mathcal{N}=4$ super Yang-Mills theory. The three-loop result in the latter theory is new and confirms a conjecture made in our previous paper. We study various physical limits of the cusp anomalous dimension and discuss its relation to the quark-antiquark potential including the effects of broken conformal symmetry in QCD. We find that the cusp anomalous dimension viewed as a function of the cusp angle a…
We calculate the small angle expansion of the four-loop QCD cusp anomalous dimension. As a byproduct of our calculation, we also obtain the four-loop anomalous dimension of the heavy-quark field in HQET. The validity of the calculational setup is cross-checked by the independent calculation of the four-loop QCD beta-function from heavy-quark-gluon vertex renormalization in HQET. We check the obtained results for the cusp anomalous dimension and heavy-quark field anomalous dimension against available analytical and numerical results. Finally, we find that the maximal transcendentality contribut…
I discuss 3 related quantities: the cusp anomalous dimension, the HQET heavy-quark field anomalous dimension, and the quark-antiquark potential. Leading large $n_f$ terms can be calculated to all orders in $α_s$. Next to leading terms with the abelian color structure $C_F^2$ also can be found to all orders (but not non-abelian $C_F C_A$ terms). This talk is based on Appendices C and D in [arXiv:1510.07803].
I present a general method for determining the massive cusp anomalous dimension in QCD to a very high degree of accuracy using its asymptotic behavior at small and large quark velocities. I show that the method works exceedingly well at two and three loops where exact results are already known. I then present a calculation of the massive cusp anomalous dimension using its asymptotics at four loops, and I provide a detailed study of the results for different values of the number of flavors and for separate color structures. The method can be further extended and applied to higher numbers of loo…
We compute the fermionic contribution to the photon-quark form factor to four-loop order in QCD in the planar limit in analytic form. From the divergent part of the latter the cusp and collinear anomalous dimensions are extracted. Results are also presented for the finite contribution. We briefly describe our method to compute all planar master integrals at four-loop order.
We calculate the complete quark and gluon cusp anomalous dimensions in four-loop massless QCD analytically from first principles. In addition, we determine the complete matter dependence of the quark and gluon collinear anomalous dimensions. Our approach is to Laurent expand four-loop quark and gluon form factors in the parameter of dimensional regularization. We employ finite field and syzygy techniques to reduce the relevant Feynman integrals to a basis of finite integrals, and subsequently evaluate the basis integrals directly from their standard parametric representations.
Nonplanar Four-Loop Anomalous Dimensions of Twist-Two Operators in N=4 Super Yang-Mills Theory: Higher Moment, General Result, and Cusp Anomalous Dimension
We compute analytically the matter-dependent contributions to the quartic Casimir term of the four-loop light-like cusp anomalous dimension in QCD, with $n_f$ fermion and $n_s$ scalar flavours. The result is extracted from the double pole of a scalar form factor. We adopt a new strategy for the choice of master integrals with simple analytic and infrared properties, which significantly simplifies our calculation. To this end we first identify a set of integrals whose integrands have a dlog form, and are hence expected to have uniform transcendental weight. We then perform a systematic analysis…
We present numerical results for the nonplanar lightlike cusp and collinear anomalous dimension at four loops in ${\mathcal N} = 4$ SYM theory, which we infer from a calculation of the Sudakov form factor. The latter is expressed as a rational linear combination of uniformly transcendental integrals for arbitrary colour factor. Numerical integration in the nonplanar sector reveals explicitly the breakdown of quadratic Casimir scaling at the four-loop order. A thorough analysis of the reported numerical uncertainties is carried out.
The light-like cusp anomalous dimension is a universal function that controls infrared divergences in quite general quantum field theories. In the maximally supersymmetric Yang-Mills theory this function is fixed fully by integrability to the three-loop order. At four loops a non-planar correction appears which we obtain for the first time from a numerical computation of the Sudakov form factor. Key ingredients are widely applicable methods to control the number-theoretic aspects of the appearing integrals. Our result shows explicitly that quadratic Casimir scaling breaks down at four loops.
We present the complete formula for the cusp anomalous dimension at four loops in QCD and in maximally supersymmetric Yang-Mills. In the latter theory it is given by \begin{equation} Γ^{\rm}_{\rm cusp}\Big|_{α_s^4} = -\left( \frac{α_s N}π\right)^4 \left[ \frac{73 π^6}{20160} + \frac{ ζ_{3}^2}{8} + \frac{1}{N^2} \left( \frac{31π^6}{5040} + \frac{9 ζ_3^2}{4} \right) \right] \,.\nonumber \end{equation} Our approach is based on computing the correlation function of a rectangular light-like Wilson loop with a Lagrangian insertion, normalized by the expectation value of the Wilson loop. In maximally…
We consider the extension of the CMW soft-gluon effective coupling in the context of soft-gluon resummation for QCD hard-scattering observables beyond the next-to-leading logarithmic accuracy. We present two proposals of a soft-gluon effective coupling that extend the CMW coupling to all perturbative orders in the \ms\ coupling $\as$. Although both effective couplings are well-defined in the physical four-dimensional space time, we examine their behaviour in $d=4 -2\ep$ space time dimensions. We uncover an all-order perturbative relation with the cusp anomalous dimension: the (four dimensional…
The main features of the cusp anomalous dimension in N=4 supersymmetri c Yang-Mills theory are reviewed. Moreover, the strong coupling expansion of the cusp derived is presented.
Correlation functions of Wilson lines are relevant for describing the infrared structure of scattering amplitudes. We develop a new method for evaluating a wide class of such Wilson line integrals, and apply it to the calculation of the velocity-dependent cusp anomalous dimension in maximally supersymmetric Yang-Mills theory. We compute the four-loop non-planar correction in a recently introduced scaling limit. Moreover, we derive the full planar four-loop result by means of an ansatz which is based on the structure of known analytic results. We determine the coefficients in this ansatz by mak…
In this work we study the cusp anomalous dimension of the marginally deformed N=4 sYM theory. We find the expression of the cusp anomalous dimension both at the weak and strong coupling limits. On the gravity side we partially map the system of equations to that of undeformed N=4 sYM, by defining in the internal space an effective angle Δθ_{eff} which depends on the deformation parameter. The cusp anomalous dimension then can be read from that of the N=4 sYM theory, by using this effective angle instead of the usual angle of the undeformed 5-sphere. In the weak coupling regime we find no beta …
Recently a new picture has been developed for examining Wilson lines, and the corresponding anomalous dimensions which govern their renormalization properties. By making a particular coordinate transform, the calculation of the cusp anomalous dimension in QED or QCD can be related to the energy of a pair of static charges in Euclidean Anti-de-Sitter (AdS) space. This paper shows how the same picture can be used to describe Wilson lines in quantum gravity. We show how the relevant cusp anomalous dimension (which has recently been shown to be one loop exact) can be obtained using the Newtonian l…
The angle-dependent cusp anomalous dimension governs divergences coming from soft gluon exchanges between heavy particles, such as top quarks. We focus on the matter-dependent contributions and compute the first truly non-planar terms. They appear at four loops and are proportional to a quartic Casimir operator in color space. Specializing our general gauge theory result to U(1), we obtain the full QED four-loop angle-dependent cusp anomalous dimension. While more complicated functions appear at intermediate steps, the analytic answer depends only on multiple polylogarithms with singularities …
University of Oxford（牛津大学）；CERN（欧洲核子研究组织）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Arizona State University（亚利桑那州立大学）；University of Amsterdam（阿姆斯特丹大学）
We study the first sub-leading correction $O((\ln s)^0)$ to the cusp anomalous dimension in the high spin expansion of finite twist operators in ${\cal N}=4$ SYM theory. This term is still governed by a linear integral equation which we study in the weak and strong coupling regimes. In the strong coupling regime we find agreement with the string theory computations
We study the factorization of soft and collinear singularities in dimensionally-regularized fixed-angle scattering amplitudes in massless gauge theories. Our factorization is based on replacing the hard massless partons by light-like Wilson lines, and defining gauge-invariant jet and soft functions in dimensional regularization. In this scheme the factorized amplitude admits a powerful symmetry: it is invariant under rescaling of individual Wilson-line velocities. This symmetry is broken by cusp singularities in both the soft and the eikonal jet functions. We show that the cancellation of thes…
We study the cusp anomalous dimension in N=6 ABJ(M) theory, identifying a scaling limit in which the ladder diagrams dominate. The resummation is encoded into a Bethe-Salpeter equation that is mapped to a Schroedinger problem, exactly solvable due to the surprising supersymmetry of the effective Hamiltonian. In the ABJ case the solution implies the diagonalization of the U(N) and U(M) building blocks, suggesting the existence of two independent cusp anomalous dimensions and an unexpected exponentiation structure for the related Wilson loops. While consistent with previous perturbative analysis…
In the context of AdS/CFT we provide analytical support for the proposed duality between a Wilson loop with a cusp, the cusp anomalous dimension, and the meson model constructed from a rotating open string with high angular momentum. This duality was previously studied using numerical tools in [1]. Our result implies that the minimum of the profile function of the minimal area surface dual to the Wilson loop, is related to the inverse of the bulk penetration of the dual string that hangs from the quark--anti-quark pair (meson) in the gauge theory.
We present an expression for the leading-color (planar) four-loop four-point amplitude of N=4 supersymmetric Yang-Mills theory in 4-2 e dimensions, in terms of eight separate integrals. The expression is based on consistency of unitarity cuts and infrared divergences. We expand the integrals around e=0, and obtain analytic expressions for the poles from 1/e^8 through 1/e^4. We give numerical results for the coefficients of the 1/e^3 and 1/e^2 poles. These results all match the known exponentiated structure of the infrared divergences, at four separate kinematic points. The value of the 1/e^2 c…
This work addresses the resurgent properties of the cusp anomalous dimension's strong coupling expansion, obtained from the integral Beisert-Eden-Staudacher (BES) equation. This expansion is factorially divergent, and its first nonperturbative corrections are related to the mass gap of the $O(6)$ $σ$-model. The factorial divergence can also be analysed from a resurgence perspective. Building on the work of Basso and Korchemsky, a transseries ansatz for the cusp anomalous dimension is proposed and the corresponding expected large-order behaviour studied. One finds non-perturbative phenomena in …
We calculate the anomalous dimension of the cusped Wilson loop in ${\cal N}=4$ supersymmetric Yang-Mills theory to order $λ^2$ ($λ=g^2_{YM}N$). We show that the cancellation between the diagrams with the three-point vertex and the self-energy insertion to the propagator which occurs for smooth Wilson loops is not complete for cusped loops, so that an anomaly term remains. This term contributes to the cusp anomalous dimension. The result agrees with the anomalous dimensions of twist-two conformal operators with large spin. We verify the loop equation for cusped loops to order $λ^2$, reproducing…
We study the velocity-dependent cusp anomalous dimension in supersymmetric Yang-Mills theory. In a paper by Correa, Maldacena, Sever, and one of the present authors, a scaling limit was identified in which the ladder diagrams are dominant and are mapped onto a Schrodinger problem. We show how to solve the latter in perturbation theory and provide an algorithm to compute the solution at any loop order. The answer is written in terms of harmonic polylogarithms. Moreover, we give evidence for two curious properties of the result. Firstly, we observe that the result can be written using a subset o…
We study the first sub-leading correction $O((\ln s)^0)$ to the cusp anomalous dimension in the high spin expansion of finite twist operators in ${\cal N}=4$ SYM theory. Since this approximation is still governed by a linear integral equation (derived already from the Bethe Ansatz equations in a previous paper), we finalise it better in order to study the weak and strong coupling regimes. In fact, we emphasise how easily the weak coupling expansion can be obtained, confirms the known four loop result and predicts the higher orders. Eventually, we pay particular attention to the strong coupling…
We review the computation of the anomalous dimension of twist operators in the planar limit of N=4 SYM using the asymptotic Bethe ansatz and demonstrate how this quantity is obtained at weak, strong and intermediate values of the coupling constant. The anomalous dimension of twist operators in the limit of large Lorentz spin played a major role in the construction as well as in many tests of the asymptotic Bethe equations, this aspect of the story is emphasised.
We derive a set of integral equations of the TBA type for the generalized cusp anomalous dimension, or the quark antiquark potential on the three sphere, as a function of the angles. We do this by considering a family of local operators on a Wilson loop with charge L. In the large L limit the problem can be solved in terms of a certain boundary reflection matrix. We determine this reflection matrix by using the symmetries and the boundary crossing equation. The cusp is introduced through a relative rotation between the two boundaries. Then the TBA trick of exchanging space and time leads to an…
We revisit the strong coupling limit of the cusp anomalous dimension in planar N=4 super Yang-Mills theory. It is known that the strong coupling expansion is asymptotic and non-Borel summable. As a consequence, the cusp anomalous dimension receives non-perturbative corrections, and the complete strong coupling expansion should be a resurgent transseries. We reveal that the perturbative and non-perturbative parts in the transseries are closely interrelated. Solving the Beisert-Eden-Staudacher equation systematically, we analyze in detail the large order behavior in the strong coupling perturbat…
The velocity-dependent cusp anomalous dimension is by definition the ultraviolet (UV) anomalous dimension of a Wilson loop with a cusp. It appears in many physically interesting processes. In this talk we present recent progress in the analytic calculation of that quantity.
We consider the classical limit of the recently obtained exact result for the anomalous dimension of a cusped Wilson line with the insertion of an operator with L units of R-charge at the cusp in planar N=4 SYM. The classical limit requires taking both the 't Hooft coupling and L to infinity. Since the formula for the cusp anomalous dimension involves determinants of size proportional to L, the classical limit requires a matrix model reformulation of the result. We construct such matrix model-like representation and find corresponding classical algebraic curve. Using this we derive the classic…
We introduce a new purely bosonic, $\frac{1}{6}$ BPS Wilson loop for ABJM theory on $S^3$ that couples scalar fields to a latitude at an angle $θ$ on $S^2\in\mathbb{C}P^3$. Through localization of this operator, we relate the expansion of the cusp anomalous dimension at small cusp angles to the logarithmic derivative of the ABJM Wilson loop. This defines, non-perturbatively in the 't Hooft coupling, the bremsstrahlung function $B(λ)$ describing in three dimensions the soft radiation of a $W$-boson undergoing a sudden change in trajectory. We compare our results for $B(λ)$ to the known weak/str…
The full strong coupling expansion of the cusp anomalous dimension and the O(6) mass gap
HUN-REN Wigner Research Centre for Physics（匈牙利科学院维格纳物理研究中心）；Institute of Theoretical Physics and Mark Kac Center for Complex, Systems Research, Jagiellonian University（雅盖隆大学理论物理研究所与马克·卡克复杂系统研究中心）；Department of Theoretical Physics, Budapest University of Technology and Economics（布达佩斯技术与经济大学理论物理系）
We construct the minimal surface in AdS, relevant for the strong coupling behaviour of local supersymmetric Wilson loops in N=4 SYM for a closed contour formed out of segments of two intersecting circles. Its regularised area is calculated including all divergent parts and the finite renormalised term. Furthermore we prove, that for generic planar curved contours with cusps the cusp anomalous dimensions are functions of the respective cusp angles alone. They do not depend on other local data of the cusps.
We introduce a method for extracting the cusp anomalous dimension at L loops from four-gluon amplitudes in N=4 Yang-Mills without evaluating any integrals that depend on the kinematical invariants. We show that the anomalous dimension only receives contributions from the obstructions introduced in hep-th/0601031. We illustrate this method by extracting the two- and three-loop anomalous dimensions analytically and the four-loop one numerically. The four-loop result was recently guessed to be f^4 = - (4ζ^3_2+24ζ_2ζ_4+50ζ_6- 4(1+r)ζ_3^2) with r=-2 using integrability and string theory arguments i…
I present results for soft anomalous dimensions through three loops for many QCD processes. In particular, I give detailed expressions for soft anomalous dimensions in various processes with electroweak and Higgs bosons as well as single top quarks and top-antitop pairs.
We analyze the cusp anomalous dimension in the (leading) ladder limit of $\mathcal N=4$ SYM and present new results for its higher-order perturbative expansion. We study two different limits with respect to the cusp angle $ϕ$. The first is the light-like regime where $x = e^{i\,ϕ}\to 0$. This limit is characterised by a non-trivial expansion of the cusp anomaly as a sum of powers of $\log x$, where the maximum exponent increases with the loop order. The coefficients of this expansion have remarkable transcendentality features and can be expressed by products of single zeta values. We show that…
We discuss an AdS / QCD model that consider anomalous dimensions. The effect of this kind of dimensions is considered as a mass term that depend on holographical coordenate for duals modes in bulk.
We emphasize that non-conformal theories provide a natural playground for the ideas of the Maldacena conjecture, opening the possibility of exploring properties that could potentially be in the same universality class as QCD. In particular, we discuss in detail how light-like Wilson loops, an important ingredient in the prescription for scattering amplitudes, can be described in a number of gravity duals of non-conformal gauge theories. We point out to a few universal properties and the prominent role of the strong scale.
We calculate the collinear anomalous dimensions in massless four-loop QCD and $\mathcal{N} = 4$ supersymmetric Yang-Mills theory from the infrared poles of vertex form factors. We give very precise numerical approximations and a conjecture for the complete analytic results in both models we consider.
Kavli Institute for Theoretical Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
Inequalities are derived for Wilson loops generalizing the well-known Bachas inequality for rectangular contours. The inequalities are compatible with the area law for large contours. The Polyakov cusp anomalous dimension of Wilson lines (playing an important role in QCD applications to hard processes) has a convex angular dependence. This convexity is crucial for the consistency of the inequalities with renormalization. Some parallel properties can be found in the string theory. The Kac-Ray cusp term from the "shape of a drum" problem has the same angular convexity property and plays the role…
Wilson Loops with Lagrangians: Large Spin OPE and Cusp Anomalous Dimension Dictionary
Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器）；Centro de Fisica do Porto e Departamento de Fisica e Astronomia, Faculdade de Ciencias da Universidade do Porto（波尔图大学理学院物理与天文学系及波尔图物理中心）；Department of Mathematics, King’s College London（伦敦国王学院数学系）
We present the result of a calculation for the third and fourth moments of the non-singlet four-loop anomalous dimension of Wilson twist-2 operators in QCD with full color and flavour structures. We discuss also a general expressions for some contributions to the full four-loop anomalous dimension obtained with the help of the method, based on LLL-algorithm, which was proposed earlier by us for the reconstruction of a general form of the anomalous dimension from a fixed values.
