Webs and posets Dukes, M.; Gardi, E.; McAslan, H. ...
The journal of high energy physics,
2014, Letnik:
2014, Številka:
1
Journal Article
Recenzirano
Odprti dostop
A
bstract
The non-Abelian exponentiation theorem has recently been generalised to correlators of multiple Wilson line operators. The perturbative expansions of these correlators exponentiate in terms ...of sets of diagrams called webs, which together give rise to colour factors corresponding to connected graphs. The colour and kinematic degrees of freedom of individual diagrams in a web are entangled by mixing matrices of purely combinatorial origin. In this paper we relate the combinatorial study of these matrices to properties of partially ordered sets (posets), and hence obtain explicit solutions for certain families of web-mixing matrix, at arbitrary order in perturbation theory. We also provide a general expression for the rank of a general class of mixing matrices, which governs the number of independent colour factors arising from such webs. Finally, we use the poset language to examine a previously conjectured sum rule for the columns of web-mixing matrices which governs the cancellation of the leading subdivergences between diagrams in the web. Our results, when combined with parallel developments in the evaluation of kinematic integrals, offer new insights into the all-order structure of infrared singularities in non-Abelian gauge theories.
Quantification of tissue properties is improved using the general proportionator sampling and estimation procedure: automatic image analysis and non-uniform sampling with probability proportional to ...size (PPS). The complete region of interest is partitioned into fields of view, and every field of view is given a weight (the size) proportional to the total amount of requested image analysis features in it. The fields of view sampled with known probabilities proportional to individual weight are the only ones seen by the observer who provides the correct count. Even though the image analysis and feature detection is clearly biased, the estimator is strictly unbiased. The proportionator is compared to the commonly applied sampling technique (systematic uniform random sampling in 2D space or so-called meander sampling) using three biological examples: estimating total number of granule cells in rat cerebellum, total number of orexin positive neurons in transgenic mice brain, and estimating the absolute area and the areal fraction of β islet cells in dog pancreas. The proportionator was at least eight times more efficient (precision and time combined) than traditional computer controlled sampling.
We claim that factorization implies that the evolution kernel, defined by the logarithmic derivative of the
Nth moment of the structure function
dln
F
2
N
/
dln
Q
2, receives logarithmically enhanced ...contributions (Sudakov logs) from a single source, namely, the constrained invariant mass of the jet. Available results from fixed-order calculations facilitate Sudakov resummation up to the next-to-next-to-leading logarithmic accuracy. We use additional all-order information on the physical kernel from the large-
β
0 limit to model the behaviour of further subleading logs and explore the uncertainty in extracting
α
s
and in determining the magnitude of higher-twist contributions from a comparison with data on high moments.
Abstract The proportionator is a novel and radically different approach to sampling with microscopes based on the well-known statistical theory (probability proportional to size—PPS sampling). It ...uses automatic image analysis, with a large range of options, to assign to every field of view in the section a weight proportional to some characteristic of the structure under study. A typical and very simple example, examined here, is the amount of color characteristic for the structure, marked with a stain with known properties. The color may be specific or not. In the recorded list of weights in all fields, the desired number of fields is sampled automatically with probability proportional to the weight and presented to the expert observer. Using any known stereological probe and estimator, the correct count in these fields leads to a simple, unbiased estimate of the total amount of structure in the sections examined, which in turn leads to any of the known stereological estimates including size distributions and spatial distributions. The unbiasedness is not a function of the assumed relation between the weight and the structure, which is in practice always a biased relation from a stereological (integral geometric) point of view. The efficiency of the proportionator depends, however, directly on this relation to be positive. The sampling and estimation procedure is simulated in sections with characteristics and various kinds of noises in possibly realistic ranges. In all cases examined, the proportionator is 2–15-fold more efficient than the common systematic, uniformly random sampling. The simulations also indicate that the lack of a simple predictor of the coefficient of error (CE) due to field-to-field variation is a more severe problem for uniform sampling strategies than anticipated. Because of its entirely different sampling strategy, based on known but non-uniform sampling probabilities, the proportionator for the first time allows the real CE at the section level to be automatically estimated (not just predicted), unbiased—for all estimators and at no extra cost to the user.
Multiple gluon exchange webs Falcioni, Giulio; Gardi, Einan; Harley, Mark ...
The journal of high energy physics,
10/2014, Letnik:
2014, Številka:
10
Journal Article
Recenzirano
Odprti dostop
A
bstract
Webs are weighted sets of Feynman diagrams which build up the logarithms of correlators of Wilson lines, and provide the ingredients for the computation of the soft anomalous dimension. We ...present a general analysis of multiple gluon exchange webs (MGEWs) in correlators of semi-infinite non-lightlike Wilson lines, as functions of the exponentials of the Minkowski cusp angles,
α
ij
, formed between lines
i
and
j
. We compute a range of webs in this class, connecting up to five Wilson lines through four loops, we give an all-loop result for a special class of diagrams, and we discover a new kind of relation between webs connecting different numbers of Wilson lines, based on taking collinear limits. Our results support recent conjectures, stating that the contribution of any MGEW to the soft anomalous dimension is a sum of products of polylogarithms, each depending on a single cusp angle, and such that their symbol alphabet is restricted to
α
ij
and 1 −
α
ij
2
. Finally, we construct a simple basis of functions, defined through a one-dimensional integral representation in terms of powers of logarithms, which has all the expected analytic properties. This basis allows us to compactly express the results of all MGEWs computed so far, and we conjecture that it is sufficient for expressing all MGEWs at any loop order.
The large-order behaviour of QCD is dominated by renormalons. On the other hand renormalons do not occur in conformal theories, such as the one describing the infrared fixed-point of QCD at small β0 ...(the Banks–Zaks limit). Since the fixed-point has a perturbative realization, all-order perturbative relations exist between the conformal coefficients, which are renormalon-free, and the standard perturbative coefficients, which contain renormalons. Therefore, an explicit cancellation of renormalons should occur in these relations. The absence of renormalons in the conformal limit can thus be seen as a constraint on the structure of the QCD perturbative expansion. We show that the conformal constraint is non-trivial: a generic model for the large-order behaviour violates it. We also analyse a specific example, based on a renormalon-type integral over the two-loop running-coupling, where the required cancellation does occur.
Perturbative and non-perturbative aspects of differential cross-sections close to a kinematic threshold are studied applying “dressed gluon exponentiation” (DGE). The factorization property of soft ...and collinear gluon radiation is demonstrated using the light-cone axial gauge: it is shown that the singular part of the squared matrix element for the emission of an
off-shell gluon off a nearly on-shell quark is universal. We derive a generalized splitting function that describes the emission probability and show how Sudakov logs emerge from the phase-space boundary where the gluon transverse momentum vanishes. Both soft and collinear logs associated with a single dressed gluon are computed through a single integral over the running-coupling to any logarithmic accuracy. The result then serves as the kernel for exponentiation. The divergence of the perturbative series in the exponent indicates specific non-perturbative corrections. We identify two classes of observables according to whether the radiation is from an initial-state quark, as in the Drell–Yan process, or a final-state quark, forming a jet with a constrained invariant mass, as in fragmentation functions, event-shape variables and deep inelastic structure functions.
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