Bootstrapping an NMHV amplitude through three loops

We extend the hexagon function bootstrap to the next-to-maximally-helicity-violating (NMHV) configuration for six-point scattering in planar ${\cal N}=4$ super-Yang-Mills theory at three loops. Constraints from the $\bar{Q}$ differential equation, from the operator product expansion (OPE) for Wilson loops with operator insertions, and from multi-Regge factorization, lead to a unique answer for the three-loop ratio function. The three-loop result also predicts additional terms in the OPE expansion, as well as the behavior of NMHV amplitudes in the multi-Regge limit at one higher logarithmic accuracy (NNLL) than was used as input. Both predictions are in agreement with recent results from the flux-tube approach. We also study the multi-particle factorization of multi-loop amplitudes for the first time. We find that the function controlling this factorization is purely logarithmic through three loops. We show that a function $U$, which is closely related to the parity-even part of the ratio function $V$, is remarkably simple; only five of the nine possible final entries in its symbol are non-vanishing. We study the analytic and numerical behavior of both the parity-even and parity-odd parts of the ratio function on simple lines traversing the space of cross ratios $(u,v,w)$, as well as on a few two-dimensional planes. Finally, we present an empirical formula for $V$ in terms of elements of the coproduct of the six-gluon MHV remainder function $R_6$ at one higher loop, which works through three loops for $V$ (four loops for $R_6$).Comment: 69 pages, 12 figures, 1 table, 3 ancillary files; v2, minor typo's correcte

A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries

We define the rigidity of a Feynman integral to be the smallest dimension over which it is non-polylogarithmic. We argue that massless Feynman integrals in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show that this bound may be saturated for integrals that we call marginal: those with (L+1)D/2 propagators in (even) D dimensions. We show that marginal Feynman integrals in D dimensions generically involve Calabi-Yau geometries, and we give examples of finite four-dimensional Feynman integrals in massless $\phi^4$ theory that saturate our predicted bound in rigidity at all loop orders.Comment: 5+2 pages, 11 figures, infinite zoo of Calabi-Yau manifolds. v2 reflects minor changes made for publication. This version is authoritativ

Bootstrapping a Five-Loop Amplitude Using Steinmann Relations

The analytic structure of scattering amplitudes is restricted by Steinmann relations, which enforce the vanishing of certain discontinuities of discontinuities. We show that these relations dramatically simplify the function space for the hexagon function bootstrap in planar maximally supersymmetric Yang-Mills theory. Armed with this simplification, along with the constraints of dual conformal symmetry and Regge exponentiation, we obtain the complete five-loop six-particle amplitude.Comment: 5 pages, 2 figures, 1 impressive table, and 2 ancillary files. v2: a few clarifications and references added; version to appear in PR

The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms

We derive an analytic representation of the ten-particle, two-loop double-box integral as an elliptic integral over weight-three polylogarithms. To obtain this form, we first derive a four-fold, rational (Feynman-)parametric representation for the integral, expressed directly in terms of dual-conformally invariant cross-ratios; from this, the desired form is easily obtained. The essential features of this integral are illustrated by means of a simplified toy model, and we attach the relevant expressions for both integrals in ancillary files. We propose a normalization for such integrals that renders all of their polylogarithmic degenerations pure, and we discuss the need for a new 'symbology' of iterated elliptic/polylogarithmic integrals in order to bring them to a more canonical form.Comment: 4+2 pages, 2 figures. Explicit results are included as ancillary files. v2: minor changes made for clarification; references adde

Bootstrapping six-gluon scattering in planar N=4{\cal N}=4 super-Yang-Mills theory

We describe the hexagon function bootstrap for solving for six-gluon scattering amplitudes in the large $N_c$ limit of ${\cal N}=4$ super-Yang-Mills theory. In this method, an ansatz for the finite part of these amplitudes is constrained at the level of amplitudes, not integrands, using boundary information. In the near-collinear limit, the dual picture of the amplitudes as Wilson loops leads to an operator product expansion which has been solved using integrability by Basso, Sever and Vieira. Factorization of the amplitudes in the multi-Regge limit provides additional boundary data. This bootstrap has been applied successfully through four loops for the maximally helicity violating (MHV) configuration of gluon helicities, and through three loops for the non-MHV case.Comment: 15 pages, 3 figures, 2 tables; contribution to the proceedings of Loops and Legs in Quantum Field Theory, 27 April - 2 May 2014, Weimar, Germany; v2, reference adde

The Double Pentaladder Integral to All Orders

We compute dual-conformally invariant ladder integrals that are capped off by pentagons at each end of the ladder. Such integrals appear in six-point amplitudes in planar N=4 super-Yang-Mills theory. We provide exact, finite-coupling formulas for the basic double pentaladder integrals as a single Mellin integral over hypergeometric functions. For particular choices of the dual conformal cross ratios, we can evaluate the integral at weak coupling to high loop orders in terms of multiple polylogarithms. We argue that the integrals are exponentially suppressed at strong coupling. We describe the space of functions that contains all such double pentaladder integrals and their derivatives, or coproducts. This space, a prototype for the space of Steinmann hexagon functions, has a simple algebraic structure, which we elucidate by considering a particular discontinuity of the functions that localizes the Mellin integral and collapses the relevant symbol alphabet. This function space is endowed with a coaction, both perturbatively and at finite coupling, which mixes the independent solutions of the hypergeometric differential equation and constructively realizes a coaction principle of the type believed to hold in the full Steinmann hexagon function space.Comment: 70 pages, 3 figures, 4 tables; v2, minor typo corrections and clarification

Stellar dynamo driven wind braking instead of disc coupling

Star-disc coupling is considered in numerical models where the stellar field is not an imposed perfect dipole, but instead a more irregular self-adjusting dynamo-generated field. Using axisymmetric simulations of the hydromagnetic mean-field equations, it is shown that the resulting stellar field configuration is more complex, but significantly better suited for driving a stellar wind. In agreement with recent findings by a number of people, star-disc coupling is less efficient in braking the star than previously thought. Moreover, stellar wind braking becomes equally important. In contrast to a perfect stellar dipole field, dynamo-generated stellar fields favor field-aligned accretion with considerably higher velocity at low latitudes, where the field is weaker and originating in the disc. Accretion is no longer nearly periodic (as it is in the case of a stellar dipole), but it is more irregular and episodic.Comment: 19 pages, 15 figures, some errors corrected, Astron. Nach. (submitted). For higher quality images, see http://www.nordita.dk/~brandenb/papers/stellardyn

Time course of information processing in visual and haptic object classification

Peer reviewedPublisher PD

D = 5 maximally supersymmetric Yang-Mills theory diverges at six loops

The connection of maximally supersymmetric Yang-Mills theory to the (2,0) theory in six dimensions has raised the possibility that it might be perturbatively ultraviolet finite in five dimensions. We test this hypothesis by computing the coefficient of the first potential ultraviolet divergence of planar (large N_c) maximally supersymmetric Yang-Mills theory in D = 5, which occurs at six loops. We show that the coefficient is nonvanishing. Furthermore, the numerical value of the divergence falls very close to an approximate exponential formula based on the coefficients of the divergences through five loops. This formula predicts the approximate values of the ultraviolet divergence at loop orders L > 6 in the critical dimension D = 4 + 6/L. To obtain the six-loop divergence we first construct the planar six-loop four-point amplitude integrand using generalized unitarity. The ultraviolet divergence follows from a set of vacuum integrals, which are obtained by expanding the integrand in the external momenta. The vacuum integrals are integrated via sector decomposition, using a modified version of the FIESTA program.Comment: 31 pages, revtex, 12 figure