54 research outputs found
Three-point functions in SYM: the hexagon proposal at three loops
Basso, Komatsu and Vieira recently proposed an all-loop framework for the
computation of three-point functions of single-trace operators of
super-Yang-Mills, the "hexagon program". This proposal results in several
remarkable predictions, including the three-point function of two protected
operators with an unprotected one in the and sectors. Such
predictions consist of an "asymptotic" part---similar in spirit to the
asymptotic Bethe Ansatz of Beisert and Staudacher for two-point functions---as
well as additional finite-size "wrapping" L\"uscher-like corrections. The focus
of this paper is on such wrapping corrections, which we compute at three-loops
in the sector. The resulting structure constants perfectly match the
ones obtained in the literature from four-point correlators of protected
operators.Comment: 18 pages, 3 tables; v2: note added, ref. added, (some) misprints
corrected; v3: more ref. added, more misprints correcte
Tessellating cushions: four-point functions in N=4 SYM
We consider a class of planar tree-level four-point functions in N=4 SYM in a
special kinematic regime: one BMN operator with two scalar excitations and
three half-BPS operators are put onto a line in configuration space;
additionally, for the half-BPS operators a co-moving frame is chosen in flavour
space. In configuration space, the four-punctured sphere is naturally
triangulated by tree-level planar diagrams. We demonstrate on a number of
examples that each tile can be associated with a modified hexagon form-factor
in such a way as to efficiently reproduce the tree-level four-point function.
Our tessellation is not of the OPE type, fostering the hope of finding an
independent, integrability-based approach to the computation of planar
four-point functions.Comment: 10 pages, 2 figure
Colour-dressed hexagon tessellations for correlation functions and non-planar corrections
We continue the study of four-point correlation functions by the hexagon
tessellation approach initiated in 1611.05436 and 1611.05577. We consider
planar tree-level correlation functions in supersymmetric
Yang-Mills theory involving two non-protected operators. We find that, in order
to reproduce the field theory result, it is necessary to include colour
factors in the hexagon formalism; moreover, we find that the hexagon approach
as it stands is naturally tailored to the single-trace part of correlation
functions, and does not account for multi-trace admixtures. We discuss how to
compute correlators involving double-trace operators, as well as more general
effects; in particular we compute the whole next-to-leading order in the
large- expansion of tree-level BMN two-point functions by tessellating a
torus with punctures. Finally, we turn to the issue of "wrapping",
L\"uscher-like corrections. We show that colour-dressing reproduces an
earlier empirical rule for incorporating single-magnon wrapping, and we provide
a direct interpretation of such wrapping processes in terms of
supersymmetric Feynman diagrams.Comment: 42 pages, typos correcte
Local integrands for the five-point amplitude in planar N=4 SYM up to five loops
Integrands for colour ordered scattering amplitudes in planar N=4 SYM are
dual to those of correlation functions of the energy-momentum multiplet of the
theory. The construction can relate amplitudes with different numbers of legs.
By graph theory methods the integrand of the four-point function of
energy-momentum multiplets has been constructed up to six loops in previous
work. In this article we extend this analysis to seven loops and use it to
construct the full integrand of the five-point amplitude up to five loops, and
in the parity even sector to six loops.
All results, both parity even and parity odd, are obtained in a concise local
form in dual momentum space and can be displayed efficiently through graphs. We
have verified agreement with other local formulae both in terms of
supertwistors and scalar momentum integrals as well as BCJ forms where those
exist in the literature, i.e. up to three loops.
Finally we note that the four-point correlation function can be extracted
directly from the four-point amplitude and so this uncovers a direct link from
four- to five-point amplitudes.Comment: 29 pages LaTeX, 8 figure
The Correlahedron
We introduce a new geometric object, the correlahedron, which we conjecture to be equivalent to stress-energy correlators in planar N=4N=4 super Yang-Mills. Re-expressing the Grassmann dependence of correlation functions of n chiral stress-energy multiplets with Grassmann degree 4k in terms of 4(n + k)-linear bosonic variables, the resulting expressions have an interpretation as volume forms on a Gr(n+k, 4+n+k) Grassmannian, analogous to the expressions for planar amplitudes via the amplituhedron. The resulting volume forms are to be naturally associated with the correlahedron geometry. We construct such expressions in this bosonised space both directly, in general, from Feynman diagrams in twistor space, and then more invariantly from specific known correlator expressions in analytic superspace. We give a geometric interpretation of the action of the consecutive lightlike limit and show that under this the correlahedron reduces to the squared amplituhedron both as a geometric object as well as directly on the corresponding volume forms. We give an explicit easily implementable algorithm via cylindrical decompositions for extracting the squared amplituhedron volume form from the squared amplituhedron geometry with explicit examples and discuss the analogous procedure for the correlators
Bound State Scattering Simplified
In the description of the AdS5/CFT4 duality by an integrable system the
scattering matrix for bound states plays a crucial role: it was initially
constructed for the evaluation of finite size corrections to the planar
spectrum of energy levels/anomalous dimensions by the thermodynamic Bethe
ansatz, and more recently it re-appeared in the context of the glueing
prescription of the hexagon approach to higher-point functions. In this work we
present a simplified form of this scattering matrix and we make its pole
structure manifest. We find some new relations between its matrix elements and
also present an explicit form for its inverse. We finally discuss some of its
properties including crossing symmetry. Our results will hopefully be useful
for computing finite-size effects such as the ones given by the complicated
sum-integrals arising from the glueing of hexagons, as well as help towards
understanding universal features of the AdS5/CFT4 scattering matrix.Comment: 17 pages; Mathematica notebook attached to the submission; v2:
misprints and references correcte
Positivity of hexagon perturbation theory
The hexagon-form-factor program was proposed as a way to compute three- and
higher-point correlation functions in super-symmetric
Yang-Mills theory and in the dual AdSS superstring theory, by
exploiting the integrability of the theory in the 't Hooft limit. This approach
is reminiscent of the asymptotic Bethe ansatz in that it applies to a
large-volume expansion. Finite-volume corrections can be incorporated through
L\"uscher-like formulae, though the systematics of this expansion is largely
unexplored so far. Strikingly, finite-volume corrections may feature negative
powers of the 't Hooft coupling in the small- expansion, potentially
leading to a breakdown of the formalism. In this work we show that the
finite-volume perturbation theory for the hexagon is positive and thereby
compatible with the weak-coupling expansion for arbitrary -point functions.Comment: v2: misprints corrected, further details on physical magnons adde
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