174 research outputs found

### Evaluating the 6-point Remainder Function Near the Collinear Limit

The simplicity of maximally supersymmetric Yang-Mills theory makes it an
ideal theoretical laboratory for developing computational tools, which
eventually find their way to QCD applications. In this contribution, we
continue the investigation of a recent proposal by Basso, Sever and Vieira, for
the nonperturbative description of its planar scattering amplitudes, as an
expansion around collinear kinematics. The method of arXiv:1310.5735, for
computing the integrals the latter proposal predicts for the leading term in
the expansion of the 6-point remainder function, is extended to one of the
subleading terms. In particular, we focus on the contribution of the 2-gluon
bound state in the dual flux tube picture, proving its general form at any
order in the coupling, and providing explicit expressions up to 6 loops. These
are included in the ancillary file accompanying the version of this article on
the arXiv.Comment: 6 pages, 1 figure, 1 ancillary file; based on talk given at Moriond
QCD 2014. v2: typo corrections, addition of an appendix on the contribution
of two same-helicity gluons; to appear in Int.J.Mod.Phys.

### Exact solutions for N-magnon scattering

Giant magnon solutions play an important role in various aspects of the
AdS/CFT correspondence. We apply the dressing method to construct an explicit
formula for scattering states of an arbitrary number N of magnons on R x S^3.
The solution can be written in Hirota form and in terms of determinants of N x
N matrices. Such a representation may prove useful for the construction of an
effective particle Hamiltonian describing magnon dynamics.Comment: 19 pages, 1 figur

### The Two-Loop Symbol of all Multi-Regge Regions

We study the symbol of the two-loop n-gluon MHV amplitude for all Mandelstam
regions in multi-Regge kinematics in N=4 super Yang-Mills theory. While the
number of distinct Mandelstam regions grows exponentially with n, the increase
of independent symbols turns out to be merely quadratic. We uncover how to
construct the symbols for any number of external gluons from just two building
blocks which are naturally associated with the six- and seven-gluon amplitude,
respectively. The second building block is entirely new, and in addition to its
symbol, we also construct a prototype function that correctly reproduces all
terms of maximal functional transcendentality.Comment: 20 pages, v2: include details on functions f and g, make appendix
consistent with main text, typesetting (published version

### A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon

Seven-particle scattering amplitudes in planar super-Yang-Mills theory are
believed to belong to a special class of generalised polylogarithm functions
called heptagon functions. These are functions with physical branch cuts whose
symbols may be written in terms of the 42 cluster A-coordinates on Gr(4,7).
Motivated by the success of the hexagon bootstrap programme for constructing
six-particle amplitudes we initiate the systematic study of the symbols of
heptagon functions. We find that there is exactly one such symbol of weight six
which satisfies the MHV last-entry condition and is finite in the $7 \parallel
6$ collinear limit. This unique symbol is both dihedral and parity-symmetric,
and remarkably its collinear limit is exactly the symbol of the three-loop
six-particle MHV amplitude, although none of these properties were assumed a
priori. It must therefore be the symbol of the three-loop seven-particle MHV
amplitude. The simplicity of its construction suggests that the n-gon bootstrap
may be surprisingly powerful for n>6.Comment: 30 pages, 3 ancillary files, v3: minor corrections, including a typo
in (33

### Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry

We further exploit the relation between tropical Grassmannians and
$\operatorname{Gr}(4,n)$ cluster algebras in order to make and refine
predictions for the singularities of scattering amplitudes in $\mathcal{N}=4$
planar super Yang-Mills theory at higher multiplicity $n\ge 8$. As a
mathematical foundation that provides access to square-root symbol letters in
principle for any $n$, we analyse infinite mutation sequences in cluster
algebras with general coefficients. First specialising our analysis to the
eight-particle amplitude, and comparing it with a recent, closely related
approach based on scattering diagrams, we find that the only additional letters
the latter provides are the two square roots associated to the four-mass box.
In combination with a tropical rule for selecting a finite subset of variables
of the infinite $\operatorname{Gr}(4,9)$ cluster algebra, we then apply our
results to obtain a collection of $3,078$ rational and $2,349$ square-root
letters expected to appear in the nine-particle amplitude. In particular these
contain the alphabet found in an explicit 2-loop NMHV symbol calculation at
this multiplicity.Comment: v2: corrected minor typos, added references and acknowledgements,
improved conclusion, version to appear in JHE

### Landau Singularities from Whitney Stratifications

We demonstrate that the complete and non-redundant set of Landau
singularities of Feynman integrals may be explicitly obtained from the Whitney
stratification of an algebraic map. As a proof of concept, we leverage recent
theoretical and algorithmic advances in their computation, as well as their
software implementation, in order to determine this set for several nontrivial
examples of two-loop integrals. Interestingly, different strata of the Whitney
stratification describe not only the singularities of a given integral, but
also those of integrals obtained from kinematic limits, e.g.~by setting some of
its masses or momenta to zero.Comment: 7 pages, 6 figure

### 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

### Heptagons from the Steinmann Cluster Bootstrap

We reformulate the heptagon cluster bootstrap to take advantage of the
Steinmann relations, which require certain double discontinuities of any
amplitude to vanish. These constraints vastly reduce the number of functions
needed to bootstrap seven-point amplitudes in planar $\mathcal{N} = 4$
supersymmetric Yang-Mills theory, making higher-loop contributions to these
amplitudes more computationally accessible. In particular, dual superconformal
symmetry and well-defined collinear limits suffice to determine uniquely the
symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes. We
also show that at three loops, relaxing the dual superconformal ($\bar{Q}$)
relations and imposing dihedral symmetry (and for NMHV the absence of spurious
poles) leaves only a single ambiguity in the heptagon amplitudes. These results
point to a strong tension between the collinear properties of the amplitudes
and the Steinmann relations.Comment: 43 pages, 2 figures. v2: typos corrected; version to appear in JHE

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