61 research outputs found
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 super-Yang-Mills theory at three loops.
Constraints from the 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 , which is closely related to
the parity-even part of the ratio function , 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 , as well as on a few two-dimensional planes. Finally, we
present an empirical formula for in terms of elements of the coproduct of
the six-gluon MHV remainder function at one higher loop, which works
through three loops for (four loops for ).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
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 super-Yang-Mills theory
We describe the hexagon function bootstrap for solving for six-gluon
scattering amplitudes in the large limit of 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
Conformally-regulated direct integration of the two-loop heptagon remainder
We reproduce the two-loop seven-point remainder function in planar, maximally
supersymmetric Yang-Mills theory by direct integration of conformally-regulated
chiral integrands. The remainder function is obtained as part of the two-loop
logarithm of the MHV amplitude, the regularized form of which we compute
directly in this scheme. We compare the scheme-dependent anomalous dimensions
and related quantities in the conformal regulator with those found for the
Higgs regulator.Comment: 22 pages, 1 figure. Detailed results available in an ancillary fil
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
The four-loop six-gluon NMHV ratio function
We use the hexagon function bootstrap to compute the ratio function which characterizes the next-to-maximally-helicity-violating (NMHV) six-point amplitude in planar N=4 super-Yang-Mills theory at four loops. A powerful constraint comes from dual superconformal invariance, in the form of a Q differential equation, which heavily constrains the first derivatives of the transcendental functions entering the ratio function. At four loops, it leaves only a 34-parameter space of functions. Constraints from the collinear limits, and from the multi-Regge limit at the leading-logarithmic (LL) and next-to-leading-logarithmic (NLL) order, suffice to fix these parameters and obtain a unique result. We test the result against multi-Regge predictions at NNLL and N^3LL, and against predictions from the operator product expansion involving one and two flux-tube excitations; all cross-checks are satisfied. We study the analytical and numerical behavior of the parity-even and parity-odd parts on various lines and surfaces traversing the three-dimensional space of cross ratios. As part of this program, we characterize all irreducible hexagon functions through weight eight in terms of their coproduct. We also provide representations of the ratio function in particular kinematic regions in terms of multiple polylogarithms
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
A Novel Algorithm for Nested Summation and Hypergeometric Expansions
We consider a class of sums over products of Z-sums whose arguments differ by
a symbolic integer. Such sums appear, for instance, in the expansion of Gauss
hypergeometric functions around integer indices that depend on a symbolic
parameter. We present a telescopic algorithm for efficiently converting these
sums into generalized polylogarithms, Z-sums, and cyclotomic harmonic sums for
generic values of this parameter. This algorithm is illustrated by computing
the double pentaladder integrals through ten loops, and a family of massive
self-energy diagrams through in dimensional regularization. We
also outline the general telescopic strategy of this algorithm, which we
anticipate can be applied to other classes of sums.Comment: 36 pages, 2 figures; v2: references added, typos corrected, improved
introduction and comparison with existing methods, matches published versio
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