618 research outputs found
Web matrices : structural properties and generating combinatorial identities
In this paper we present new results for the combinatorics of web diagrams and web worlds. These are discrete objects that arise in the physics of calculating scattering amplitudes in non-abelian gauge theories. Web-colouring and web-mixing matrices (collectively known as web matrices) are indexed by ordered pairs of web-diagrams and contain information relating the number of colourings of the first web diagram that will produce the second diagram. We introduce the black diamond product on power series and show how it determines the web-colouring matrix of disjoint web worlds. Furthermore, we show that combining known physical results with the black diamond product gives a new technique for generating combinatorial identities. Due to the complicated action of the product on power series, the resulting identities appear highly non-trivial. We present two results to explain repeated entries that appear in the web matrices. The first of these shows how diagonal web matrix entries will be the same if the comparability graphs of their associated decomposition posets are the same. The second result concerns general repeated entries in conjunction with a flipping operation on web diagrams. We present a combinatorial proof of idempotency of the web-mixing matrices, previously established using physical arguments only. We also show how the entries of the square of the web-colouring matrix can be achieved by a linear transformation that maps the standard basis for formal power series in one variable to a sequence of polynomials. We look at one parameterized web world that is related to indecomposable permutations and show how determining the web-colouring matrix entries in this case is equivalent to a combinatorics on words problem
Infrared singularities in multi-leg scattering amplitudes
I discuss the state-of-the-art knowledge of long-distance singularities in
multi-leg gauge-theory scattering amplitudes and report on an on-going
calculation of the three-loop soft anomalous dimension through the
renormalization of correlators of semi-infinite Wilson lines. I also discuss
the non-Abelian exponentiation theorem that has been recently generalised to
multiple Wilson lines and demonstrate its application in computing the soft
anomalous dimension. Finally, I present recent results for
multiple-gluon-exchange webs and discuss their analytic structure.Comment: 16 page, 5 figure
Progress on soft gluon exponentiation and long-distance singularities
I review the recent progress in studying long-distance singularities in
gauge-theory scattering amplitudes in terms of Wilson lines. The non-Abelian
exponentiation theorem, which has been recently generalised to the case of
multi-leg amplitudes, states that diagrams exponentiate such that the colour
factors in the exponent are fully connected. After a brief review of the
diagrammatic approach to soft gluon exponentiation, I sketch the method we used
to prove the theorem and illustrate how connected colour factors emerge in the
exponent in webs that are formed by sets of multiple-gluon-exchange diagrams.
In the second part of the talk I report on recent progress in evaluating the
corresponding integrals, where a major simplification is achieved upon
formulating the calculation in terms of subtracted webs. I argue that the
contributions of all multiple-gluon-exchange diagrams to the soft anomalous
dimension take the form of products of specific polylogarithmic functions, each
depending on a single cusp angle.Comment: 22 pages, 13 figures; presented at the 11th International Symposium
on Radiative Corrections, RADCOR 2013, 22-27 September 2013, Lumley Castle,
U
Multiparton webs beyond three loops
Correlators of Wilson-line operators are fundamental ingredients for the
study of the infrared properties of non-abelian gauge theories. In perturbation
theory, they are known to exponentiate, and their logarithm can be organised in
terms of collections of Feynman diagrams called webs. We study the
classification of webs to high perturbative orders, proposing a set of tools to
generate them recursively: in particular, we introduce the concept of Cweb, or
correlator web, which is a set of skeleton diagrams built with connected gluon
correlators, instead of individual Feynman diagrams. As an application, we
enumerate all Cwebs entering the soft anomalous dimension matrix for
multi-parton scattering amplitudes at four loops, and we compute the mixing
matrices for all Cwebs connecting four or five Wilson lines at that loop order,
verifying that they obey sum rules that were derived or conjectured in the
literature. Our results provide the colour building blocks for the calculation
of the soft anomalous dimension matrix at four-loop order.Comment: Published version; Published in JHEP; 51 pages, 65 figure
Cwebs beyond three loops in multiparton amplitudes
Correlators of Wilson-line operators in non-abelian gauge theories are known
to exponentiate, and their logarithms can be organised in terms of collections
of Feynman diagrams called webs. In [1] we introduced the concept of Cweb, or
correlator web, which is a set of skeleton diagrams built with connected gluon
correlators, and we computed the mixing matrices for all Cwebs connecting four
or five Wilson lines at four loops. Here we complete the evaluation of
four-loop mixing matrices, presenting the results for all Cwebs connecting two
and three Wilson lines. We observe that the conjuctured column sum rule is
obeyed by all the mixing matrices that appear at four-loops. We also show how
low-dimensional mixing matrices can be uniquely determined from their known
combinatorial properties, and provide some all-order results for selected
classes of mixing matrices. Our results complete the required colour building
blocks for the calculation of the soft anomalous dimension matrix at four-loop
order.Comment: 51 pages, 37 figures, Published version, published in JHE
Position-space cuts for Wilson line correlators
We further develop the formalism for taking position-space cuts of eikonal
diagrams introduced in [Phys.Rev.Lett. 114 (2015), no. 18 181602,
arXiv:1410.5681]. These cuts are applied directly to the position-space
representation of any such diagram and compute its discontinuity to the leading
order in the dimensional regulator. We provide algorithms for computing the
position-space cuts and apply them to several two- and three-loop eikonal
diagrams, finding agreement with results previously obtained in the literature.
We discuss a non-trivial interplay between the cutting prescription and
non-Abelian exponentiation. We furthermore discuss the relation of the
imaginary part of the cusp anomalous dimension to the static interquark
potential.Comment: 39+18 pages, 16 figures; elaborated the discussion of the comparison
of numerical and analytic results for the three-gluon vertex diagram in the
caption of fig. 16; version to be published in JHE
Multiparton Cwebs at five loops
Scattering amplitudes involving multiple partons are plagued with infrared
singularities. The soft singularities of the amplitude are captured by the soft
function which is defined as the vacuum expectation value of Wilson line
correlators. Renormalization properties of soft function allows us to write it
as an exponential of the finite soft anomalous dimension. An efficient way to
study the soft function is through a set of Feynman diagrams known as Cwebs
(webs). We obtain the mixing matrices and exponentiated colour factors for all
the Cwebs at five loops that connect six massless Wilson lines. Our results are
the first key ingredient for the calculation of the soft anomalous dimension at
five loops.Comment: 46 pages, 29 figures, 27 tables and 1 ancillary fil
Deciphering colour building blocks of massive multiparton amplitudes at 4-loops and beyond
The soft function in non-abelian gauge theories exponentiate, and their logarithms can be organised in terms of the collections of Feynman diagrams called Cwebs. The colour factors that appear in the logarithm are controlled by the web mixing matrices. Direct construction of the diagonal blocks of Cwebs using the new concepts of Normal ordering, basis Cweb and Fused-Web was recently carried out in [1]. In this article we establish correspondence between the boomerang webs introduced in [2] and non-boomerang Cwebs. We use this correspondence together with Uniqueness theorem and Fused web formalism introduced in [1] to obtain the diagonal blocks of four general classes of Cwebs to all orders in perturbation theory which also cover all the four loop Boomerang Cwebs connecting four Wilson lines. We also fully construct the mixing matrix of a special Cweb to all orders in perturbation theory
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