Factorization Theorem Research Articles

Small-x factorization from effective field theory

We derive a factorization theorem that allows for resummation of small-x logarithms by exploiting Glauber operators in the soft collinear effective field theory. Our analysis is carried out for the hadronic tensor Wμν in deep inelastic scattering, and leads to the definition of a new gauge invariant soft function Sμν that describes quark and gluon emission in the central region. This soft function provides a new framework for extending resummed calculations for coefficient functions to higher logarithmic orders. Our factorization also defines impact factors by universal collinear functions that are process independent, for instance being identical in small-x DIS and Drell-Yan.

QCD light-cone distribution amplitudes of heavy mesons from boosted HQET

Light-cone distribution amplitudes (LCDAs) frequently arise in factorization theorems involving light and heavy mesons. The QCD LCDA for heavy mesons includes short-distance physics at energy scales of the heavy-quark mass. In this paper we achieve the separation of this perturbative scale from the purely hadronic effects by matching the QCD LCDA to the convolution of a perturbative function with the universal, quark-mass independent LCDA defined in heavy-quark effective theory. This factorization allows to resum potentially large logarithms between ΛQCD and mQ as well as between mQ and the scale Q of the hard process in the production of boosted heavy mesons at colliders. As an application we derive new theoretical predictions for the branching ratio of the decay W± → B±γ. Furthermore, we provide phenomenological models for the QCD LCDAs of both the B¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{B} $$\\end{document} and D mesons expressed as expansions in Gegenbauer polynomials.

Deciphering the Long-Distance Penguin Contribution to B[over ¯]_{d,s}→γγ Decays.

We compute for the first time the long-distance penguin contribution to the double radiative B-meson decays by applying the perturbative factorization theorem. The numerically dominant penguin amplitude arises from the soft-gluon radiation off the light up-quark loop rather than the counterpart charm-loop effect. Importantly, the long-distance up-quark penguin contribution brings about the substantial cancellation of the known factorizable power correction, thus enabling B_{d,s}→γγ to become new benchmark probes of physics beyond the standard model.

Timoshenko & Lekhnitskii's puzzle ‒ Rule of swapping for the torsional rigidity of a rectangular bar

Timoshenko presented the torsional rigidity of an isotropic rectangular bar, and Lekhnitskii presented that of an orthotropic rectangular bar. The solutions of Timoshenko and Lekhnitskii (T&L) are functions of the bar's length, width, thickness and shear modulus or moduli. However, the functions of T&L solutions become different from their original ones when the width and thickness are swapped. Swapping the width and thickness definitions does not alter the bar's physical properties, named the “rule of swapping” by the authors. In the last century, no research has shown the T&L solutions to satisfy the rule of swapping, an observation hereinafter referred to as the “Timoshenko & Lekhnitskii Puzzle”. Roughly 90 years later, Tsai et al. re-solved T&L cases using the TSAI technique. The derived solutions are nearly if not completely identical to T&L's numerically and satisfy the rule of swapping automatically. The rule of swapping is a novel issue and has never been mentioned before. Based on the Weierstrass factorization theorem, this study mathematically proves that they are identical for isotropic and orthotropic bars and satisfy the rule of swapping. The result of a torsional pendulum test is analyzed to support the rule.

On the holomorphic convexity of reductive Galois coverings over compact Kähler surfaces

This article generalizes the result of Katzarkov and Ramachandran from algebraic surfaces to Kähler surfaces. We follow their argument to prove the holomorphic convexity of a reductive Galois covering over a compact Kähler surface which does not have two ends, except that we replace the p-adic factorization theorem by an analysis of the singularities of the continuous subanalytic plurisubharmonic exhaustion function.

Maximally edge‐connected realizations and Kundu's k $k$‐factor theorem

AbstractA simple graph with edge‐connectivity and minimum degree is maximally edge‐connected if . In 1964, given a nonincreasing degree sequence , Jack Edmonds showed that there is a realization of that is ‐edge‐connected if and only if with when . We strengthen Edmonds's result by showing that given a realization of if is a spanning subgraph of with such that when , then there is a maximally edge‐connected realization of with as a subgraph. Our theorem tells us that there is a maximally edge‐connected realization of that differs from by at most edges. For , if has a spanning forest with components, then our theorem says there is a maximally edge‐connected realization that differs from by at most edges. As an application we combine our work with Kundu's ‐factor theorem to show there is a maximally edge‐connected realization with a ‐factor for and present a partial result to a conjecture that strengthens the regular case of Kundu's ‐factor theorem.

Euclidean Structures and Operator Theory in Banach Spaces

We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators Γ \Gamma on a Hilbert space. Our assumption on Γ \Gamma is expressed in terms of α \alpha -boundedness for a Euclidean structure α \alpha on the underlying Banach space X X . This notion is originally motivated by R \mathcal {R} - or γ \gamma -boundedness of sets of operators, but for example any operator ideal from the Euclidean space ℓ n 2 \ell ^2_n to X X defines such a structure. Therefore, our method is quite flexible. Conversely we show that Γ \Gamma has to be α \alpha -bounded for some Euclidean structure α \alpha to be representable on a Hilbert space. By choosing the Euclidean structure α \alpha accordingly, we get a unified and more general approach to the Kwapień–Maurey factorization theorem and the factorization theory of Maurey, Nikišin and Rubio de Francia. This leads to an improved version of the Banach function space-valued extension theorem of Rubio de Francia and a quantitative proof of the boundedness of the lattice Hardy–Littlewood maximal operator. Furthermore, we use these Euclidean structures to build vector-valued function spaces. These enjoy the nice property that any bounded operator on L 2 L^2 extends to a bounded operator on these vector-valued function spaces, which is in stark contrast to the extension problem for Bochner spaces. With these spaces we define an interpolation method, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem, we prove a transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We for example extend and refine the known theory based on R \mathcal {R} -boundedness for the joint and operator-valued H ∞ H^\infty -calculus. Moreover, we extend the classical characterization of the boundedness of the H ∞ H^\infty -calculus on Hilbert spaces in terms of B I P BIP , square functions and dilations to the Banach space setting. Furthermore we establish, via the H ∞ H^\infty -calculus, a version of Littlewood–Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our abstract setup allows us to reduce assumptions on the geometry of X X , such as (co)type and UMD. We conclude with some sophisticated counterexamples for sectorial operators, with as a highlight the construction of a sectorial operator of angle 0 0 on a closed subspace of L p L^p for 1 > p > ∞ 1>p>\infty with a bounded H ∞ H^\infty -calculus with optimal angle ω H ∞ ( A ) > 0 \omega _{H^\infty }(A) >0 .

Discrete representations of finitely generated groups into PSL(2,R)${\rm PSL}(2,\mathbb {R})$

AbstractWe prove a factorization theorem for Fuchsian groups similar to those proved by Agol and Liu for 3‐manifold groups. As an application, we build Makanin–Razborov diagrams, which parameterize the collection of all discrete representations from an arbitrary but fixed finitely generated group to . We define a new class of groups called ‐discrete limit groups and then use the factorization theorem to obtain useful information about this class of groups.

Unfoldings and Coverings of Weighted Graphs

Coverings of undirected graphs are used in distributed computing, and unfoldings of directed graphs in semantics of programs. We study these two notions from a graph theoretical point of view so as to highlight their similarities, as they are both defined in terms of surjective graph homomorphisms. In particular, universal coverings and complete unfoldings are infinite trees that are regular if the initial graphs are finite. Regularity means that a tree has finitely many subtrees up to isomorphism. Two important theorems have been established by Leighton and Norris for coverings of finite graphs. We prove similar results for unfoldings of finite directed graphs. Moreover, we generalize coverings and similarly, unfoldings to graphs and digraphs equipped with finite or infinite weights attached to edges of the covered or unfolded graphs. This generalization yields a canonical “factorization” of the universal covering of any finite graph, that (provably) does not exist without using weights. Introducing ω as an infinite weight provides us with finite descriptions of regular trees having nodes of countably infinite degree. Regular trees (trees having finitely many subtrees up to isomorphism) play an important role in the extension of Formal Language Theory to infinite structures described in finitary ways. Our weighted graphs offer effective descriptions of the above mentioned regular trees and yield decidability results. We also generalize to weighted graphs and their coverings a classical factorization theorem of their characteristic polynomials.

Factorization at next-to-leading power and endpoint divergences in gg → h production

We derive a factorization theorem for the Higgs-boson production amplitude in gluon-gluon fusion induced by a light-quark loop, working at next-to-leading power in soft-collinear effective theory. The factorization is structurally similar to that obtained for the h → γγ decay amplitude induced by a light-quark loop, but additional complications arise because of external color charges. We show how the refactorization-based subtraction scheme developed in previous work leads to a factorization theorem free of endpoint divergences. We use renormalization-group techniques to predict the logarithmically enhanced terms in the three-loop gg → h form factor of order {alpha}_s^3{ln}^kleft(-{M}_h^2/{m}_b^2right) with k = 6, 5, 4, 3. We also resum the first three towers of leading logarithms, {alpha}_s^n{ln}^{2n-k}left(-{M}_h^2/{m}_b^2right) with k = 0, 1, 2, to all orders of perturbation theory.

Semi-inclusive deeply inelastic neutrino and antineutrino nucleus scattering

The (anti)neutrino nucleus scattering plays a very important role in probing the hadronic structure as well as the electroweak phenomenologies. To this end, we calculate the jet production semi-inclusive deeply inelastic (anti)neutrino nucleus scattering process. The initial (anti)neutrino is assumed to be scattered off by a target particle with spin 1. Due to the limitation of the factorization theorem, calculations are carried out in the quantum chromodynamics parton model framework up to tree level twist-3. We consider both the neutral current and the charged current processes and write them into a unified form due to the similar interaction forms. Considering the angular modulations and polarizations of the cross section, we calculate the complete azimuthal asymmetries. We also calculate the intrinsic asymmetries which reveal the imbalance in the distribution of the intrinsic transverse momentum of the quark. We find that these asymmetries can be expressed in terms of the transverse momentum-dependent parton distribution functions (TMD PDFs) and the electroweak couplings. With the determined couplings, these asymmetries can be used to extract the TMD PDFs and further to study the hadronic structures.

Quark and gluon two-loop beam functions for leading-jet pT and slicing at NNLO

We compute the complete set of two-loop beam functions for the transverse momentum distribution of the leading jet produced in association with an arbitrary colour-singlet system. Our results constitute the last missing ingredient for the calculation of the jet-vetoed cross section at small veto scales at the next-to-next-to-leading order, as well as an important ingredient for its resummation to next-to-next-to-next-to-leading logarithmic order. Our calculation is performed in the soft-collinear effective theory framework with a suitable regularisation of the rapidity divergences occurring in the phase-space integrals. We discuss the occurrence of soft-collinear mixing terms that might violate the factorisation theorem, and demonstrate that they are naturally absorbed into the beam functions at two loops in the exponential rapidity regularisation scheme when performing a multipole expansion of the measurement function. As in our recent computation of the two-loop soft function, we present the results as a Laurent expansion in the jet radius R. We provide analytic expressions for all flavour channels in x space with the exception of a set of R-independent non-logarithmic terms that are given as numerical grids. We also perform a fully numerical calculation with exact R dependence, and find that it agrees with our analytic expansion at the permyriad level or better. Our calculation allows us to define a next-to-next-to-leading order slicing method using the leading-jet pT as a slicing variable. As a check of our results, we carry out a calculation of the Higgs and Z boson total production cross sections at the next-to-next-to-leading order in QCD.

Basics of factorization in a scalar Yukawa field theory

The factorization theorems of QCD apply equally well to most simple quantum field theories that require renormalization but where direct calculations are much more straightforward. Working with these simpler theories is convenient for stress testing the limits of the factorization program and for examining general properties of the parton density functions or other correlation functions that might be necessary for a factorized description of a process. With this view in mind, we review the steps of factorization in a real scalar Yukawa field theory for both deep inelastic scattering and semi-inclusive deep inelastic scattering cross sections. In the case of semi-inclusive deep inelastic scattering, we illustrate how to separate the small transverse momentum region, where transverse momentum dependent parton density functions are needed, from a purely collinear large transverse momentum region, and we examine the influence of subleading power corrections. We also review the steps for formulating transverse momentum dependent factorization in transverse coordinate space, and we study the effect of transforming to the well-known ${b}_{*}$ scheme. Within the Yukawa theory, we investigate the consequences of switching to a generalized parton model approach and compare it with a fully factorized approach. Our results highlight the need to address similar or analogous issues in QCD.

Refactorisation in subleading [formula omitted

We establish refactorisation conditions between the subleading O8-O8 contributions to the inclusive B¯→Xsγ decay suffering from endpoint divergences and prove a factorisation theorem for these contributions to all orders in the strong coupling constant. This allows for higher-order calculations of the resolved contributions and consistent summation of large logarithms, consequently reducing the recently found large-scale dependence in these contributions. We implement the concept of refactorisation in a heavy flavour application of SCET, which includes nonperturbative functions as additional subtlety not present in collider applications.

Toeplitz separability, entanglement, and complete positivity using operator system duality

A new proof is presented of a theorem of L. Gurvits [LANL Unclassified Technical Report (2001), LAUR–01–2030], which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system C ( S 1 ) ( n ) C(S^1)^{(n)} of n × n n\times n Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems C ( S 1 ) ( n ) ⊗ min B ( H ) C(S^1)^{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) and C ( S 1 ) ( n ) ⊗ min B ( H ) C(S^1)_{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) , where H \mathcal {H} is an arbitrary Hilbert space and C ( S 1 ) ( n ) C(S^1)_{(n)} is the operator system dual of C ( S 1 ) ( n ) C(S^1)^{(n)} . Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from B ( H ) \mathcal {B}(\mathcal {H}) when H \mathcal {H} has infinite dimension. In particular, we prove that normal positive linear maps ψ \psi on B ( H ) \mathcal {B}(\mathcal {H}) are partially completely positive in the sense that ψ ( n ) ( x ) \psi ^{(n)}(x) is positive whenever x x is a positive n × n n\times n Toeplitz matrix with entries from B ( H ) \mathcal {B}(\mathcal {H}) . We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), showing an equivalence between the Gurvits approach to separation and an earlier approach of T. Ando [Acta Sci. Math. (Szeged) 31 (1970), pp. 319–334] to universality.

Limiting Conditions of Muckenhoupt and Reverse Hölder Classes on Metric Measure Spaces

The natural maximal and minimal functions commute pointwise with the logarithm on A_infty . We use this observation to characterize the spaces A_1 and RH_infty on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt A_p and reverse Hölder classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between A_p and reverse Hölder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of A_1 and RH_infty , including a refined Jones factorization theorem. In addition, we show a boundedness result for the natural maximal function.

Optimal extensions of Lipschitz maps on metric spaces of measurable functions

We prove a factorization theorem for Lipschitz operators acting on certain subsets of metric spaces of measurable functions and with values on general metric spaces. Our results show how a Lipschitz operator can be extended to a subset of other metric space of measurable functions that satisfies the following optimality condition: it is the biggest metric space, formed by measurable functions, to which the operator can be extended preserving the Lipschitz constant. As an application, we show the coarsest metric that can be given for a metric space in which an order bounded lattice-valued-Lipschitz map is defined. Concrete examples involving the relevant space L0(μ) are given.

The noncommutative factor theorem for lattices in product groups

We prove a noncommutative Bader-Shalom factor theorem for lattices with dense projections in product groups. As an application of this result and our previous works, we obtain a noncommutative Margulis factor theorem for all irreducible lattices Γ<G in higher rank semisimple algebraic groups. Namely, we give a complete description of all intermediate von Neumann subalgebras L(Γ)⊂M⊂L(Γ↷G/P) sitting between the group von Neumann algebra and the group measure space von Neumann algebra associated with the action on the Furstenberg-Poisson boundary.

Sufficient Statistics and Split Idempotents in Discrete Probability Theory

A sufficient statistic is a deterministic function that captures an essential property of a probabilistic function (channel, kernel). Being a sufficient statistic can be expressed nicely in terms of string diagrams, as Tobias Fritz showed recently, in adjoint form. This reformulation highlights the role of split idempotents, in the Fisher-Neyman factorisation theorem. Examples of a sufficient statistic occur in the literature, but mostly in continuous probability. This paper demonstrates that there are also several fundamental examples of a sufficient statistic in discrete probability. They emerge after some combinatorial groundwork that reveals the relevant dagger split idempotents and shows that a sufficient statistic is a deterministic dagger epi.

The hadron pair forward-backward asymmetry in the electron positron annihilation process

The forward-backward asymmetry is an important measurable quantity which can enable independent determination of the neutral-current couplings of fermions. In this paper, we extend the definition of the asymmetry from the partonic level to the hadronic level by calculating this asymmetry of the hadron pair in the semi-inclusive electron position annihilation process. Semi-inclusive implies that a back-to-back jet is also measured in addition to the hadron pair. Due to the limitation of the factorization theorem, we calculate this process up to leading order twist-4 level by applying the collinear expansion formalism. After obtaining the differential cross section, we calculate the forward-backward asymmetries and show them in terms of the corresponding di-hadron fragmentation functions. Di-hadron fragmentation functions are introduced to describe the hadron pair productions in the fragmentation process. With available parameterization of the functions, we present a numerical estimation of the forward-backward asymmetry. On the flip side, measurements of the forward-backward asymmetry would give strict restrictions of parameterizations of the di-hadron fragmentation functions. We also present a numerical estimation of the twist-4 di-hadron fragmentation functions in order to illustrate their contributions.