An Analytic Result for the Two-Loop Hexagon Wilson Loop in N = 4 SYM

In the planar N=4 supersymmetric Yang-Mills theory, the conformal symmetry constrains multi-loop n-edged Wilson loops to be basically given in terms of the one-loop n-edged Wilson loop, augmented, for n greater than 6, by a function of conformally invariant cross ratios. We identify a class of kinematics for which the Wilson loop exhibits exact Regge factorisation and which leave invariant the analytic form of the multi-loop n-edged Wilson loop. In those kinematics, the analytic result for the Wilson loop is the same as in general kinematics, although the computation is remarkably simplified with respect to general kinematics. Using the simplest of those kinematics, we have performed the first analytic computation of the two-loop six-edged Wilson loop in general kinematics.Comment: 17 pages. Extended discussion on how the QMRK limit is taken. Version accepted by JHEP. A text file containing the Mathematica code with the analytic expression for the 6-point remainder function is include

Effective action for the Regge processes in gravity

It is shown, that the effective action for the reggeized graviton interactions can be formulated in terms of the reggeon fields $A^{++}$ and $A^{--}$ and the metric tensor $g_{\mu \nu}$ in such a way, that it is local in the rapidity space and has the property of general covariance. The corresponding effective currents $j^{-}$ and $j^{+}$ satisfy the Hamilton-Jacobi equation for a massless particle moving in the gravitational field. These currents are calculated explicitly for the shock wave-like fields and a variation principle for them is formulated. As an application, we reproduce the effective lagrangian for the multi-regge processes in gravity together with the graviton Regge trajectory in the leading logarithmic approximation with taking into account supersymmetric contributions.Comment: 39 page

Analytic properties of high energy production amplitudes in N=4 SUSY

We investigate analytic properties of the six point planar amplitude in N=4 SUSY at the multi-Regge kinematics for final state particles. For inelastic processes the Steinmann relations play an important role because they give a possibility to fix the phase structure of the Regge pole and Mandelstam cut contributions. These contributions have the Moebius invariant form in the transverse momentum subspace. The analyticity and factorization constraints allow us to reproduce the two-loop correction to the 6-point BDS amplitude in N=4 SUSY obtained earlier in the leading logarithmic approximation with the use of the s-channel unitarity. The exponentiation hypothesis for the remainder function in the multi-Regge kinematics is also investigated. The 6-point amplitude in LLA can be completely reproduced from the BDS ansatz with the use of the analyticity and Regge factorization.Comment: To appear in the proceedings of 16th International Seminar on High Energy Physics, QUARKS-2010, Kolomna, Russia, 6-12 June, 2010. 15 page

Multi-Regge kinematics and the moduli space of Riemann spheres with marked points

We show that scattering amplitudes in planar N = 4 Super Yang-Mills in multi-Regge kinematics can naturally be expressed in terms of single-valued iterated integrals on the moduli space of Riemann spheres with marked points. As a consequence, scattering amplitudes in this limit can be expressed as convolutions that can easily be computed using Stokes' theorem. We apply this framework to MHV amplitudes to leading-logarithmic accuracy (LLA), and we prove that at L loops all MHV amplitudes are determined by amplitudes with up to L + 4 external legs. We also investigate non-MHV amplitudes, and we show that they can be obtained by convoluting the MHV results with a certain helicity flip kernel. We classify all leading singularities that appear at LLA in the Regge limit for arbitrary helicity configurations and any number of external legs. Finally, we use our new framework to obtain explicit analytic results at LLA for all MHV amplitudes up to five loops and all non-MHV amplitudes with up to eight external legs and four loops.Comment: 104 pages, six awesome figures and ancillary files containing the results in Mathematica forma

Thermodynamic Bethe Ansatz Equations for Minimal Surfaces in AdS_3

We study classical open string solutions with a null polygonal boundary in AdS_3 in relation to gluon scattering amplitudes in N=4 super Yang-Mills at strong coupling. We derive in full detail the set of integral equations governing the decagonal and the dodecagonal solutions and identify them with the thermodynamic Bethe ansatz equations of the homogeneous sine-Gordon models. By evaluating the free energy in the conformal limit we compute the central charges, from which we observe general correspondence between the polygonal solutions in AdS_n and generalized parafermions.Comment: 25 pages, 4 figures, v2: a figure and references added, minor corrections, v3: references added, minor corrections, to appear in JHE

Bootstrapping the three-loop hexagon

We consider the hexagonal Wilson loop dual to the six-point MHV amplitude in planar N=4 super Yang-Mills theory. We apply constraints from the operator product expansion in the near-collinear limit to the symbol of the remainder function at three loops. Using these constraints, and assuming a natural ansatz for the symbol's entries, we determine the symbol up to just two undetermined constants. In the multi-Regge limit, both constants drop out from the symbol, enabling us to make a non-trivial confirmation of the BFKL prediction for the leading-log approximation. This result provides a strong consistency check of both our ansatz for the symbol and the duality between Wilson loops and MHV amplitudes. Furthermore, we predict the form of the full three-loop remainder function in the multi-Regge limit, beyond the leading-log approximation, up to a few constants representing terms not detected by the symbol. Our results confirm an all-loop prediction for the real part of the remainder function in multi-Regge 3-->3 scattering. In the multi-Regge limit, our result for the remainder function can be expressed entirely in terms of classical polylogarithms. For generic six-point kinematics other functions are required.Comment: 36 pages, 1 figure, plus 8 ancillary files containing symbols of functions; v2 minor typo correction

The Multi-Regge limit of NMHV Amplitudes in N=4 SYM Theory

We consider the multi-Regge limit for N=4 SYM NMHV leading color amplitudes in two different formulations: the BFKL formalism for multi-Regge amplitudes in leading logarithm approximation, and superconformal N=4 SYM amplitudes. It is shown that the two approaches agree to two-loops for the 2->4 and 3->3 six-point amplitudes. Predictions are made for the multi-Regge limit of three loop 2->4 and 3->3 NMHV amplitudes, as well as a particular sub-set of two loop 2 ->2 +n N^kMHV amplitudes in the multi-Regge limit in the leading logarithm approximation from the BFKL point of view.Comment: 28 pages, 3 figure

Single hole dynamics in the t-J model on a square lattice

We present quantum Monte Carlo (QMC) simulations for a single hole in a t-J model from J=0.4t to J=4t on square lattices with up to 24 x 24 sites. The lower edge of the spectrum is directly extracted from the imaginary time Green's function. In agreement with earlier calculations, we find flat bands around $(0,\pm\pi)$, $(\pm\pi,0)$ and the minimum of the dispersion at $(\pm\pi/2,\pm\pi/2)$. For small J both self-consistent Born approximation and series expansions give a bandwidth for the lower edge of the spectrum in agreement with the simulations, whereas for J/t > 1, only series expansions agree quantitatively with our QMC results. This band corresponds to a coherent quasiparticle. This is shown by a finite size scaling of the quasiparticle weight $Z(\vec k)$ that leads to a finite result in the thermodynamic limit for the considered values of $J/t$. The spectral function $A(\vec k, \omega)$ is obtained from the imaginary time Green's function via the maximum entropy method. Resonances above the lowest edge of the spectrum are identified, whose J-dependence is quantitatively described by string excitations up to J/t=2

Quantum Spectral Curve at Work: From Small Spin to Strong Coupling in N=4 SYM

We apply the recently proposed quantum spectral curve technique to the study of twist operators in planar N=4 SYM theory. We focus on the small spin expansion of anomalous dimensions in the sl(2) sector and compute its first two orders exactly for any value of the 't Hooft coupling. At leading order in the spin S we reproduced Basso's slope function. The next term of order S^2 structurally resembles the Beisert-Eden-Staudacher dressing phase and takes into account wrapping contributions. This expansion contains rich information about the spectrum of local operators at strong coupling. In particular, we found a new coefficient in the strong coupling expansion of the Konishi operator dimension and confirmed several previously known terms. We also obtained several new orders of the strong coupling expansion of the BFKL pomeron intercept. As a by-product we formulated a prescription for the correct analytical continuation in S which opens a way for deriving the BFKL regime of twist two anomalous dimensions from AdS/CFT integrability.Comment: 53 pages, references added; v3: due to a typo in the coefficients C_2 and D_2 on page 29 we corrected the rational part of the strong coupling predictions in equations (1.5-6), (6.22-24), (6.27-30) and in Table