We construct numerically the minimal surface in AdS spacetime surrounded by the light-like segments, which are dual to the 4, 6 and 8-point gluon scattering amplitudes in N=4 super Yang-Mills theory. We evaluate the area of the minimal surface in the radial cut-off regularization and compare these areas with the formula conjectured by Bern, Dixon and Smirnov (BDS), which is modified by the remainder function of cross-ratios of external momenta for n(\geq 6)-point amplitudes. In our momentum configuration cross-ratios are constant. We calculate the difference of areas with different conformal boost parameters, which is independent of the remainder function, and find that its dependence on the boost parameter is numerically consistent with the BDS formula.
We perform an examination of discontinuities of multiple production amplitudes, which are required for further development of the BFKL approach. It turns out that the discontinuities of 2 $\to$ 2 + n amplitudes obtained in the BFKL approach contradict to the BDS ansatz for amplitudes with maximal helicity violation in N = 4 supersymmetric Yang-Mills theory with large number of colours starting with n = 2. Explicit expressions for the discontinuities of the 2 $\to$ 3 and 2 $\to$ 4 amplitudes in the invariant mass of pairs of produced gluons are obtained in the planar N=4 SYM in the next-to-lead…
We further investigate, in the planar limit of N=4 supersymmetric Yang Mills theories,the high energy Regge behavior of six-point MHV scattering amplitudes. In particular, for the new Regge cut contribution found in our previous paper, we compute in the leading logarithmic approximation (LLA) the energy spectrum of the BFKL equation in the color octet channel, and we calculate explicitly the two loop corrections to the discontinuities of the amplitudes for the transitions 2 to 4 and 3 to 3. We find an explicit solution of the BFKL equation for the octet channel for arbitrary momentum transfers…
We consider the two-loop four-point amplitude in N=2 super QED, and show that there exists an approximate recursive structure similar to that captured by the ABDK/BDS ansatz for MHV amplitudes in N=4 super Yang-Mills. Furthermore, we present a simple relation between the box coefficients of one-loop photon MHV amplitudes in (super) QED, and sums of box coefficients of one-loop MHV amplitudes in (super) Yang-Mills.
A unified theory of the non-Abelian gauge interactions with gravity in the framework of a discretized Kaluaza-Kleine theory is constructed with a modified Dirac operator and wedge product. All the couplings of chiral spinors to the non-Abelian gauge fields emerge naturally as components of the couplings of the chiral spinors to the generalized gravity together with some new interactions. In particular, the currently prevalent gravity-QCD-quark and gravity-electroweak-quark-lepton models are shown to follow as special cases of the general framework.
We define supersymmetric Yang-Mills theory on an arbitrary two-dimensional lattice (polygon decomposition) with preserving one supercharge. When a smooth Riemann surface $Σ_g$ with genus $g$ emerges as an appropriate continuum limit of the generic lattice, the discretized theory becomes topologically twisted $\mathcal{N}=(2,2)$ supersymmetric Yang-Mills theory on $Σ_g$. If we adopt the usual square lattice as a special case of the discretization, our formulation is identical with Sugino's lattice model. Although the tuning of parameters is generally required while taking the continuum limit, t…
We study the effect of loop corrections to conformal correlators on the celestial sphere at null infinity. We first analyze finite one-loop celestial amplitudes in pure Yang-Mills theory and Einstein gravity. We then turn to our main focus: infrared divergent loop amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory. We compute the celestial one-loop amplitude in dimensional regularization and show that it can be recast as an operator acting on the celestial tree-level amplitude. This extends to any loop order and the re-summation of all planar loops enables us to write down an express…
The precise form of the multi-Regge asymptotics of the two-loop six-point MHV amplitude in N = 4 Super-Yang-Mills theory has been a subject of recent controversy. In this paper we utilize the amplitude/Wilson loop correspondence to obtain precise numerical results for the imaginary part of these asymptotics. The region of phase-space that we consider is interesting because it allowed Bartels, Lipatov, and Sabio Vera to determine that the two-loop six-point MHV amplitude is not fixed by the BDS ansatz. They proceeded by working in the framework of a high energy effective action, thus side-stepp…
We study cluster adjacency conjectures for amplitudes in maximally supersymmetric Yang-Mills theory. We show that the n-point one-loop NMHV ratio function satisfies Steinmann cluster adjacency. We also show that the one-loop BDS-like normalized NMHV amplitude satisfies cluster adjacency between Yangian invariants and final symbol entries up to 9-points. We present conjectures for cluster adjacency properties of Plücker coordinates, quadratic cluster variables, and NMHV Yangian invariants that generalize the notion of weak separation.
As a consequence of the AdS/CFT correspondence, planar ${\cal N} =4$ super Yang-Mills SU(N) theory is expected to exhibit stringy behavior and multi-Regge asymptotic. In this paper we extend our recent investigation to consider issues of analyticity, a central feature of Regge asymptotics. We contrast flat-space open string theory in the planar limit with the ${\cal N}=4$ super Yang-Mills theory, as represented by the Bern, Dixon and Smirnov \cite{Bern:2005iz} (BDS) conjecture for n-gluon scattering, believed to be exact for $n=4,5$ and modified only by a function of cross-ratios for $n\geq 6$…
This work is devoted to the six-gluon scattering amplitude in strongly coupled N=4 supersymmetric Yang-Mills theory. At weak coupling, an appropriate high energy limit of the so-called remainder function, i.e. of the deviation from the BDS formula, may be understood in terms of the lowest eigenvalue of the BFKL hamiltonian. According to Alday et al., amplitudes in the strongly coupled theory can be constructed through an auxiliary 1-dimensional quantum system. We argue that certain excitations of this quantum system determine the Regge limit of the remainder function at strong coupling and we …
We revisit the next-to-leading order~(NLO) correction to the eigenvalue of the BFKL equation in the adjoint representation and investigate its properties in analogy with the singlet BFKL in planar $\mathcal{N}=4$ super Yang-Mills Theory~(SYM). We show that the adjoint NLO BFKL eigenvalue is needed to be slightly modified in order to have a property of hermitian separability present for the singlet BFKL. After this modification the adjoint NLO BFKL eigenvalue is expressed through holomorphic and antiholomophic parts of the leading order eigenvalue and their derivatives. The proposed choice of t…
We study, for $SU(2)$ Yang-Mills theories discretized on a lattice, a non-local topological order parameter, the center flux ${z}$. We show that: i) well defined topological sectors classified by $π_1(SO(3))=\mathbb{Z}_2$ can only exist in the ordered phase of ${z}$; ii) depending on the dimension $2 \leq d\leq 4$ and action chosen, the center flux exhibits a critical behaviour sharing striking features with the Kosterlitz-Thouless type of transitions, although belonging to a novel universality class; iii) such critical behaviour does not depend on the temperature $T$. Yang-Mills theories can …
A quantum algorithm of SU(N) Yang-Mills theory is formulated in terms of quantum circuits. It can nonperturbatively calculate the Dyson series and scattering amplitudes with polynomial complexity. The gauge fields in the interaction picture are discretized on the same footing with the lattice fermions in momentum space to avoid the fermion doubling and the gauge symmetry breaking problems. Applying the algorithm to the quantum simulation of quantum chromodynamics, the quark and gluon's wave functions evolved from the initial states by the interactions can be observed and the information from w…
Wilson loops with lightlike polygonal contours have been conjectured to be equivalent to MHV scattering amplitudes in N=4 super Yang-Mills. We compute such Wilson loops for special polygonal contours at two loops in perturbation theory. Specifically, we concentrate on the remainder function R, obtained by subtracting the known ABDK/BDS ansatz from the Wilson loop. First, we consider a particular two-dimensional eight-point kinematics studied at strong coupling by Alday and Maldacena. We find numerical evidence that R is the same at weak and at strong coupling, up to an overall, coupling-depend…
We construct an energy-momentum tensor on the lattice which satisfies the appropriate Ward Identities (WIs) and has the right trace anomaly in the continuum limit. It is defined by imposing suitable WIs associated to the Poincare` invariance of the continuum theory. These relations come forth when the length of the box in the temporal direction is finite, and they take a particularly simple form if the coordinate and the periodicity axes are not aligned. We implement the method for the SU(3) Yang--Mills theory discretized with the standard Wilson action in presence of shifted boundary conditio…
Recently it has been shown how a topologically twisted version of ${\cal N}=4$ super Yang-Mills may be discretized in such a way as to preserve one scalar supersymmetry at nonzero lattice spacing. The remaining fifteen supersymmetries are broken by terms of ${\cal O}(a)$ where $a$ is the lattice spacing. One would like to know whether these remaining supersymmetries are regained in the continuum limit $a \to 0$ and, if not, how much tuning of the couplings in the lattice action is required. In this paper we derive the form of these additional twisted supersymmetries by combining a set of discr…
We examine the possibility of dynamical supersymmetry breaking in two-dimensional $\mathcal{N} = (2, 2)$ supersymmetric Yang-Mills theory. The theory is discretized on a Euclidean spacetime lattice using a supersymmetric lattice action. We compute the vacuum energy of the theory at finite temperature and take the zero temperature limit. Supersymmetry will be spontaneously broken in this theory if the measured ground state energy is non-zero. By performing simulations on a range of lattices up to $96 \times 96$ we are able to perform a careful extrapolation to the continuum limit for a wide ran…
We present the first nonperturbative large-$N$ calculation of the $\mathcal{N}=1$ supersymmetric (SUSY) $\mathrm{SU}(N)$ Yang$-$Mills gluino condensate obtained by means of numerical simulations of the lattice-discretized theory, exploiting large-$N$ twisted volume reduction. We present two different determinations based, respectively, on the Banks$-$Casher formula and on the Gell-Mann$-$Oakes$-$Renner relation, both giving perfectly consistent results. By expressing the lattice results in the Novikov$-$Shifman$-$Vainshtein$-$Zakharov (NSVZ) scheme, we are able for the first time to compare nu…
We consider gauge/gravity correspondence for general Dp-branes with $0\le p\le 4$, namely, the duality between the (p+1)-dimensional maximally supersymmetric Yang-Mills theory and superstring theory on the near-horizon limit of the Dp-brane solution. It has been known that in the large N gauge theory at zero temperature and strong 't Hooft coupling, the two-point functions of operators corresponding to supergravity modes obey power-law, with a power different from the free-field value when $p\neq 3$. In this work, we show that the free-field result of gauge theory can be reproduced from the bu…
From One to Eight: Supersymmetry Restoration in Lattice 3D ${\cal N} = 4$ Super Yang--Mills
National Institute for Theoretical and Computational Sciences, School of Physics, and Mandelstam Institute for Theoretical Physics, University of the Witwatersrand（威特沃特斯兰德大学理论及计算科学国家研究所、物理学院和曼德尔斯坦理论物理研究所）
We compute for the first time the two-loop corrections to arbitrary n-gon lightlike Wilson loops in N=4 supersymmetric Yang-Mills theory, using efficient numerical methods. The calculation is motivated by the remarkable agreement between the finite part of planar six-point MHV amplitudes and hexagon Wilson loops which has been observed at two loops. At n=6 we confirm that the ABDK/BDS ansatz must be corrected by adding a remainder function, which depends only on conformally invariant ratios of kinematic variables. We numerically compute remainder functions for n=7,8 and verify dual conformal i…
