Factorization Formula Research Articles

The universal R-matrix and factorization of the L-operators related to the Baxter Q-operators

We consider the ‘universal monodromy operators’ for the Baxter Q-operators. They are given as images of the universal R-matrix in oscillator representation. We find related universal factorization formulas in the case.

QCD radiative corrections to the soft spectator contribution in the wide angle Compton scattering

We derive the complete factorization formula for the leading power contribution in wide angle Compton scattering. It consists of the soft- and hard-spectator contributions. The hard-spectator contribution is well known and defined in the form of the convolution of a hard kernel with the nucleon distribution amplitudes. The soft-spectator contribution describes the scattering which involves the soft modes. We use the soft collinear effective theory in order to define this term in a field theoretical approach. Using the SCET framework we provide the proof of the factorization formula. We also compute the next-to-leading QCD corrections to the hard coefficient function of the soft spectator contribution and perform a phenomenological analysis of existing experimental data within the developed formalism.

Fluctuations of cell population in a colonic crypt.

The number of stem cells in a colonic crypt is often very small, which leads to large intrinsic fluctuations in the cell population. Based on the model of cell population dynamics with linear feedback in a colonic crypt, we present a stochastic dynamics of the cell population [including stem cells (SCs), transit amplifying cells (TACs), and fully differentiated cells (FDCs)]. The Fano factor, covariance, and susceptibility formulas of the cell population around the steady state are derived by using the Langevin theory. In the range of physiologically reasonable parameter values, it is found that the stationary populations of TACs and FDCs exhibit an approximately threshold behavior as a function of the net growth rate of TACs, and the reproductions of TACs and FDCs can be classified into three regimens: controlled, crossover, and uncontrolled. With the increasing of the net growth rate of TACs, there is a maximum of the relative intrinsic fluctuations (i.e., the Fano factors) of TACs and FDCs in the crossover region. For a fixed differentiation rate and the net growth rate of SCs, the covariance of fluctuations between SCs and TACs has a maximum in the crossover region. However, the susceptibilities of both TACs and FDCs to the net growth rate of TACs have a minimum in the crossover region.

Universality of transverse-momentum resummation and hard factors at the NNLO

We consider QCD radiative corrections to the production of colorless high-mass systems in hadron collisions. The logarithmically-enhanced contributions at small transverse momentum are treated to all perturbative orders by a universal resummation formula that depends on a single process-dependent hard factor. We show that the hard factor is directly related to the all-order virtual amplitude of the corresponding partonic process. The direct relation is universal (process-independent), and it is expressed by an all-order factorization formula that we explicitly evaluate up to the next-to-next-to-leading order (NNLO) in QCD perturbation theory. Once the NNLO scattering amplitude is available, the corresponding hard factor is directly determined: it controls NNLO contributions in resummed calculations at full next-to-next-to-leading logarithmic accuracy, and it can be used in applications of the qT subtraction formalism to perform fully-exclusive perturbative calculations up to NNLO. The universality structure of the hard factor and its explicit NNLO form are also extended to the related formalism of threshold resummation.

FGM/homogeneous bimaterials with systems of cracks under thermo-mechanical loading: Analysis by fracture criteria

Fracture criteria for prediction of extension of the interface crack and of the crack growth direction in a bimaterial consisting of a homogeneous and a functionally graded material (FGM) with systems of internal defects are studied. The bimaterial is subjected to a heat flux and a tensile load applied at infinity. It is assumed that the thermal properties of the FGM have exponential form. The Young’s modulus and Poisson’s ratio are assumed to be constant. In the previous papers (Petrova and Schmauder, 2011a,b, 2012a,b) asymptotic analytical formulas for the stress intensity factors (SIFs) at the interface crack tips were obtained as a series of a small parameter (the ratio between sizes of the internal and interface cracks). These SIFs are used in fracture criteria to obtain the possible direction of crack propagation and critical loads. The maximum circumferential stress criterion is used and some results for the fracture angles is obtained by the minimum strain energy density in order to compare the predictions for the fracture angles by two fracture criteria. The influence of geometry of the problem (location and orientation of cracks) and the parameters of non-homogeneity of FGMs on the main fracture characteristics is investigated.

Thermal-hydro-mechanical coupling stress intensity factor of brittle rock

A new calculation formula of THM coupling stress intensity factor was derived by the boundary collocation method, in which an additional constant stress function was successfully introduced for the cracked specimen with hydraulic pressure applied on its crack surface. Based on the newly derived formula, THM coupling fracture modes (including tensile, shear and mixed fracture mode) can be predicted by a new fracture criterion of stress intensity factor ratio, where the maximum axial load was measured by self-designed THM coupling fracture test. SEM analyses of THM coupling fractured surface indicate that the higher the temperature and hydraulic pressure are and the lower the confining pressure is, the more easily the intergranular (tension) fracture occurs. The transgranular (shear) fracture occurs in the opposite case while the mixed-mode fracture occurs in the middle case. The tested THM coupling fracture mechanisms are in good agreement with the predicted THM coupling fracture modes, which can verify correction of the newly-derived THM coupling stress intensity factor formula.

THE HEART OF FACTORIZATION

Factorization is at the heart of nearly any calculation in pertubative QCD. It follows from the universal behavior of gauge theories in soft and collinear limits. This talk gives a summary of recent progress on producing a more transparent understanding of factorization and connecting traditional approaches to those of Soft-Collinear Effective Theory. The main result is the formulation and proof, at tree-level, of a factorization formula in QCD. The proof exploits the many advantages of spinor helicity methods, but does not use any effective field theory tricks. Once the factorization formula is proven, the transition to an effective theory description is effortless.

Joint resummation for pion wave function and pion transition form factor

We construct an evolution equation for the pion wave function in the $k_T$ factorization theorem, whose solution sums the mixed logarithm $\ln x\ln k_T$ to all orders, with $x$ ($k_T$) being a parton momentum fraction (transverse momentum). This joint resummation induces strong suppression of the pion wave function in the small $x$ and large $b$ regions, $b$ being the impact parameter conjugate to $k_T$, and improves the applicability of perturbative QCD to hard exclusive processes. The above effect is similar to those from the conventional threshold resummation for the double logarithm $\ln^2 x$ and the conventional $k_T$ resummation for $\ln^2 k_T$. Combining the evolution equation for the hard kernel, we are able to organize all large logarithms in the $\gamma^{\ast} \pi^{0} \to \gamma$ scattering, and to establish a scheme-independent $k_T$ factorization formula. It will be shown that the significance of next-to-leading-order contributions and saturation behaviors of this process at high energy differ from those under the conventional resummations. It implies that QCD logarithmic corrections to a process must be handled appropriately, before its data are used to extract a hadron wave function. Our predictions for the involved pion transition form factor, derived under the joint resummation and the input of a non-asymptotic pion wave function with the second Gegenbauer moment $a_2=0.05$, match reasonably well the CLEO, BaBar, and Belle data.

A factorization formula for power series

Given an odd prime p, we give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. Our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.

Boosted top production: factorization and resummation for single-particle inclusive distributions

We study single-particle inclusive (1PI) distributions in top-quark pair production at hadron colliders, working in the highly boosted regime where the top-quark p_T is much larger than its mass. In particular, we derive a novel factorization formula valid in the small-mass and soft limits of the differential partonic cross section. This provides a framework for the simultaneous resummation of soft gluon corrections and small-mass logarithms, and also an efficient means of obtaining higher-order corrections to the differential cross section in this limit. The result involves five distinct one-scale functions, three of which arise through the subfactorization of soft real radiation in the small-mass limit. We list the NNLO corrections to each of these functions, building on results in the literature by performing a new calculation of a soft function involving four light-like Wilson lines to this order. We thus obtain a nearly complete description of the small-mass limit of the differential partonic cross section at NNLO near threshold, missing only terms involving closed top-quark loops in the virtual corrections.

THE 1-JETTINESS EVENT-SHAPE FOR DIS WITH NNLL RESUMMATION

We propose the use of 1-jettiness, a global event shape, for exclusive single jet production in lepton-nucleus deep inelastic scattering (DIS). We derive a factorization formula, using the Soft-Collinear Effective Theory, differential in the transverse momentum and rapidity of the jet and the 1-jettiness event shape. It provides a quantitative measure of the shape of the final-state QCD radiation in the presence of the hard jet, providing a useful powerful probe of QCD and nuclear physics. For example, one expects differences in the observed pattern of QCD radiation between large and small nuclei and these can be quantified by the 1-jettiness event shape. Numerical results are given for this new DIS event shape at leading twist with resummation at the next-to-next-to-leading logarithmic (NNLL) level of accuracy, for a variety of nuclear targets. Such studies would be ideal at a future EIC or LHeC electron-ion collider, where a range of nuclear targets are planned.

Flow characteristics of the raw sewage for the design of sewage-source heat pump systems.

The flow characteristics of raw sewage directly affect the technical and economic performance of sewage-source heat pump systems. The purpose of this research is to characterize the flow characteristics of sewage by experimental means. A sophisticated and flexible experimental apparatus was designed and constructed. Then the flow characteristics of the raw sewage were studied through laboratorial testing and theoretical analyses. Results indicated that raw sewage could be characterized as a power-law fluid with the rheological exponent n being 0.891 and the rheological coefficient k being 0.00175. In addition, the frictional loss factor formula in laminar flow for raw sewage was deduced by theoretical analysis of the power-law fluid. Furthermore, an explicit empirical formula for the frictional loss factor in turbulent flow was obtained through curve fitting of the experimental data. Finally, the equivalent viscosity of the raw sewage is defined in order to calculate the Reynolds number in turbulent flow regions; it was found that sewage had two to three times the viscosity of water at the same temperature. These results contributed to appropriate parameters of fluid properties when designing and operating sewage-source heat pump systems.

Rapidity resummation forB-meson wave functions

Transverse-momentum dependent (TMD) hadronic wave functions develop light-cone divergences under QCD corrections, which are commonly regularized by the rapidity $\zeta$ of gauge vector defining the non-light-like Wilson lines. The yielding rapidity logarithms from infrared enhancement need to be resummed for both hadronic wave functions and short-distance functions, to achieve scheme-independent calculations of physical quantities. We briefly review the recent progress on the rapidity resummation for $B$-meson wave functions which are the key ingredients of TMD factorization formulae for radiative-leptonic, semi-leptonic and non-leptonic $B$-meson decays. The crucial observation is that rapidity resummation induces a strong suppression of $B$-meson wave functions at small light-quark momentum, strengthening the applicability of TMD factorization in exclusive $B$-meson decays. The phenomenological consequence of rapidity-resummation improved $B$-meson wave functions is further discussed in the context of $B \to \pi$ transition form factors at large hadronic recoil.

Rescattering photoelectron spectroscopy of small molecules

We have measured angle-resolved rescattering photoelectron spectra of three small molecules (O2, CO2, and C2H4) using intense near-infrared laser pulses at several laser intensities. Based on the factorization formula for rescattering processes, we have extracted, from the electron spectra, the field-free differential cross sections of elastic electron scattering by the molecular ions. The detail of the extraction procedure is described. The experimentally extracted differential cross sections are compared with theoretical calculations of the field-free differential cross sections. Fairly good agreement between the experimentally extracted and theoretically calculated DCSs for wide range of the collision momentum indicates the validity of the present extraction procedure for the molecules.

A general method for developing friction factor formulas under supercritical conditions and in different geometries

A general method for developing friction factor formulas in different geometries and under supercritical conditions was set up in this paper. The method was based on the two-layer wall function theory which could account for both the laminar viscous layer and turbulent layer. This characteristic led to its good agreement with empirical correlations in a wide range of Reynolds numbers, especially the prediction accuracy was improved at low Reynolds numbers in comparison with the previous work. This method could be extended to the flows over rough surfaces or with axial pressure gradient as long as the appropriate wall function is provided. Moreover, the wall function is based on Van Direst transformation which could take account of the fluid property variation across the boundary layer and thus the friction formula could be applied to supercritical flow. This formula could explain the special features of friction coefficient and showed good agreement with the experimental data performed by Nuclear Power Institute of China.

Modification of existing shape factor models for self-compacting concrete strength by means of Bayesian updating techniques

In this paper, the existing formulas for the shape factors for uniaxial compressive strength of vibrated concrete, in which the influence of the shape, size and damage due to drilling has been included, are adapted for powder-type self-compacting concrete. An extensive test program was executed and Bayesian updating techniques were applied in order to calculate posterior distributions of the parameters involved in the conversion formulas for shape factors. Using this method, differences between the shape factor formulas of vibrated and self-compacting concrete could be analysed despite the large variability in the experimentally obtained test results. The methodology developed in this paper for the Bayesian updating of shape factor formulas can also be applied to other material properties of self-compacting concrete and even other materials, especially in cases where a limited number of samples are available or in cases where a considerable variability in the test data is observed.

Hexagon functions and the three-loop remainder function

Abstract We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar $ \mathcal{N} $ = 4 super-Yang-Mills theory, as a function of the three dual conformal cross ratios. The result can be expressed in terms of multiple Goncharov polylogarithms. We also employ a more restricted class of hexagon functions which have the correct branch cuts and certain other restrictions on their symbols. We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. The three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematic limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multi-Regge factorization directly at the function level, and thereby to fix uniquely a set of Riemann ζ valued constants that could not be fixed at the level of the symbol. The near-collinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with the factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to −7.

An Ideal Model to Measure Humorousness: 1+1 Two Factor Formulae

At present, humor is mainly studied on its internal mechanism and communication function. Humorousness is an indicator of humor quality, usually equivalent to the comic effect generated by the verbal humor in a special context. It is not a scattered random sequence that can be qualitatively defined only, but measurable and adjustable through a set of pragmatic devices and strategies. Based on Relevance Theory, this paper is intended to establish an ideal model “1+1 Two Factor Formulae” to quantify humorousness of verbal humor by deconstructing traditional Incongruity-Resolution Theory. It is expected to shed some light on humor research and give some inspiration to stylists and humorous writers.

Exact form factors of the O(N) σ-model

A general form factor formula for the $O(N)\sigma$-model is constructed and applied to several operators. The large N limits of these form factors are computed and compared with the 1/N expansion of the $O(N)\sigma$-model in terms of Feynman graphs and full agreement is found. In particular, O(3) and O(4) form factors are discussed. For the $O(3)\sigma$-model several low particle form factors are calculated explicitly.

Top quark pair production at small transverse momentum in hadronic collisions

We investigate the transverse momentum resummation for top quark pair production at hadron colliders using the soft-collinear effective theory and the heavy-quark effective theory. We derive the factorization formula for $t\overline{t}$ production at the small pair transverse momentum, and show in detail the procedure for calculating the key ingredient of the factorization formula: the next-to-leading order soft functions. We compare our numerical results with experimental data and find that they are consistent within theoretical and experimental uncertainties. To verify the correctness of our resummation formula, we expand it to the next-to-leading order and the next-to-next-to-leading order, and compare those expressions with the exact fixed-order results numerically. Finally, using the results of transverse momentum resummation, we discuss the transverse-momentum-dependent forward-backward asymmetry at the Tevatron.