Factorization Formula Research Articles

Factorization for jet radius logarithms in jet mass spectra at the LHC

To predict the jet mass spectrum at a hadron collider it is crucial to account for the resummation of logarithms between the transverse momentum of the jet and its invariant mass $m_J$. For small jet areas there are additional large logarithms of the jet radius $R$, which affect the convergence of the perturbative series. We present an analytic framework for exclusive jet production at the LHC which gives a complete description of the jet mass spectrum including realistic jet algorithms and jet vetoes. It factorizes the scales associated with $m_J$, $R$, and the jet veto, enabling in addition the systematic resummation of jet radius logarithms in the jet mass spectrum beyond leading logarithmic order. We discuss the factorization formulae for the peak and tail region of the jet mass spectrum and for small and large $R$, and the relations between the different regimes and how to combine them. Regions of experimental interest are classified which do not involve large nonglobal logarithms. We also present universal results for nonperturbative effects and discuss various jet vetoes.

Non-abelian factorisation for next-to-leading-power threshold logarithms

Soft and collinear radiation is responsible for large corrections to many hadronic cross sections, near thresholds for the production of heavy final states. There is much interest in extending our understanding of this radiation to next-to-leading power (NLP) in the threshold expansion. In this paper, we generalise a previously proposed all-order NLP factorisation formula to include non-abelian corrections. We define a non-abelian radiative jet function, organising collinear enhancements at NLP, and compute it for quark jets at one loop. We discuss in detail the issue of double counting between soft and collinear regions. Finally, we verify our prescription by reproducing all NLP logarithms in Drell-Yan production up to NNLO, including those associated with double real emission. Our results constitute an important step in the development of a fully general resummation formalism for NLP threshold effects.

Radiation View Factor for Co-Axial and Unequal Rectangles in Parallel Planes

ABSTRACTA closed-form and easy-to-use analytical expression is derived for calculating the radiation view factor between a pair of dissimilar rectangular surfaces facing each other from two different parallel planes. The alignment of the rectangles is such that: (i) a line joining their centers would be normal to both surfaces, and (ii) observed in a plan view in the direction of this shared axis, the rectangles are not turned by an angle relative to each other. Strangely enough, a simple formula for finding view factors for this configuration is not available at present in well-known references on this topic. The usefulness of the proposed expression, for quick calculations of view factors, is demonstrated with examples. It is shown that this generalized expression can be readily specialized to a few well-documented view factor formulae. It can be easily extended for view factor calculations involving pairs of unequal and non-aligned parallel rectangular surfaces, making it a flexible and handy result for heat transfer pedagogy.

The effect of choosing three different C factor formulae derived from NDVI on a fully raster-based erosion modelling

The research was aimed at studying the efect of choosing three different C factor formulae derived from NDVI on a fully raster-based erosion modelling of The USLE using remote sensing data and GIS technique. Methods applied was by analysing all factors affecting erosion such that all data were in the form of raster. Those data were R, K, LS, C and P factors. Monthly R factor was evaluated based on formula developed by Abdurachman. K factor was determined using modified formula used by Ministry of Forestry based on soil samples taken in the field. LS factor was derived from Digital Elevation Model. Three C factors used were all derived from NDVI and developed by Suriyaprasit (non-linear) and by Sulistyo (linear and non-linear). P factor was derived from the combination between slope data and landcover classification interpreted from Landsat 7 ETM+. Another analysis was the creation of map of Bulk Density used to convert erosion unit. To know the model accuracy, model validation was done by applying statistical analysis and by comparing Emodel with Eactual. A threshold value of ≥ 0.80 or ≥ 80% was chosen to justify. The research result showed that all Emodel using three formulae of C factors have coeeficient of correlation value of > 0.8. The results of analysis of variance showed that there was significantly difference between Emodel and Eactual when using C factor formula developed by Suriyaprasit and Sulistyo (non-linear). Among the three formulae, only Emodel using C factor formula developed by Sulistyo (linear) reached the accuracy of 81.13% while the other only 56.02% as developed by Sulistyo (nonlinear) and 4.70% as developed by Suriyaprasit, respectively.

Particle production in high-energy collisions beyond the shockwave limit

We compute next to eikonal (NE) and next to next to eikonal (NNE) corrections to the Lipatov vertex due to a finite target thickness. These arise from electric field insertions into the eikonal Wilson lines. We then derive a $k_T$-factorization formula for single inclusive gluon production at NNE accuracy and find that nuclear effects are absent. We also analyze NNE corrections to two-gluon production where some of the contributions are found to exhibit corrections proportional to $A^{2/3}$.

Eshelby’s tensor of a cubic piezoelectric crystal under plane strain condition and its application to elliptic cavity problems

The general solution of two-dimensional problems in cubic piezoelectric crystals under plane strain condition is derived. The connection between the present solution and that for purely cubic elasticity is established. Accordingly, Eshelby’s tensor of an elliptic inclusion is obtained from the corresponding results in orthotropic elasticity. The tensor is expressed explicitly in terms of material constants and shape ratio of the inclusion. As an application, stress concentration problems of an elliptic hole under two different electric conditions (electrically free and electrically clamped) imposed on the end faces of the material are solved by the equivalent inclusion method. Explicit formulas of stress concentration factors are obtained for uniformly applied tension and shear. The link between the stress concentration factor of a slender cavity and the stress intensity factor of the associated slit crack is also developed and discussed.

Forward di-jet production in dilute-dense collisions

We propose a factorization formula for the cross section for forward di-jet production in dilute-dense collisions. The new formula is applicable for an arbitrary value of the momentum imbalance of the two jets, kt. It unifies the previously derived transverse momentum dependent (TMD) factorization for small kt (of the order of the saturation scale), and the High Energy Factorization (HEF) for large kt (of the order of the momentum of the jets). We extend the previous TMD formula, first to finite Nc, and then to all ranges of kt by including off-shell matrix elements. We present previously unpublished analytical expressions for the TMD gluon distributions in the Golec-Biernat-Wusthoff model, and their perturbative behaviour in the McLerran-Venugopalan model. In addition, we show directly the equivalence of the HEF and the color glass condensate formulae in the dilute target approximation.

Testing the running couplingkT-factorization formula for the inclusive gluon production

The inclusive gluon production at midrapidities is described in the color glass condensate formalism using the ${k}_{T}$-factorization formula, which was derived at fixed coupling constant considering the scattering of a dilute system of partons with a dense one. Recent analysis demonstrated that this approach provides a satisfactory description of the experimental data for the inclusive hadron production in $pp/pA/AA$ collisions. However, these studies are based on the fixed coupling ${k}_{T}$-factorization formula. This formula does not take into account the running coupling corrections, which are important to set the scales present in the cross section. In this paper we consider the running coupling corrected ${k}_{T}$-factorization formula conjectured some years ago and investigate the impact of the running coupling corrections on the observables. In particular, the pseudorapidity distributions and charged hadrons' multiplicity are calculated considering $pp$, $dAu/pPb$, and $AuAu/PbPb$ collisions at RHIC and LHC energies. We compare the corrected running coupling predictions with those obtained using the original ${k}_{T}$-factorization assuming a fixed coupling or a prescription for the inclusion of the running of the coupling. Considering the Kharzeev-Levin-Nardi unintegrated gluon distribution and a simplified model for the nuclear geometry, we demonstrate that the distinct predictions are similar for the pseudorapidity distributions in $pp/pA/AA$ collisions and for the charged hadrons' multiplicity in $pp/pA$ collisions. On the other hand, the running coupling corrected ${k}_{T}$-factorization formula predicts a smoother energy dependence for $dN/d\ensuremath{\eta}$ in $AA$ collisions.

Factorization and dispersion relations for radiative leptonic B decay

Applying the dispersion approach we compute perturbative QCD corrections to the power suppressed soft contribution of $B \to \gamma \ell \nu$ at leading twist. QCD factorization for the $B \to \gamma^{\ast}$ form factors is demonstrated explicitly for the hard-collinear transverse polarized photon at one loop, with the aid of the method of regions. While the one-loop hard function is identical to the matching coefficient of the QCD weak current $\bar u \, \gamma_{\mu \perp} \, (1- \gamma_5) \, b$ in soft-collinear effective theory, the jet function from integrating out the hard-collinear fluctuations differs from the corresponding one entering the factorization formula of $B \to \gamma \ell \nu$. Furthermore, we evaluate the sub-leading power contribution to the $B \to \gamma$ form factors from the three-particle $B$-meson distribution amplitudes (DAs) at tree level. The soft contribution to the $B \to \gamma$ form factors from the three-particle $B$-meson DAs is shown to be of the same power compared with the corresponding hard correction, in contrast to the two-particle counterparts. Numerically the next-to-leading-order QCD correction to the soft two-particle contribution in $B \to \gamma$ form factors will induce an approximately $\left (10 \sim 20 \right) \%$ shift to the tree-level contribution at $\lambda_B(\mu_0)=354 \, {\rm MeV}$. Albeit of power suppression parametrically, the soft two-particle correction can decrease the leading power predictions for the $B \to \gamma$ form factors by an amount of $\left (10 \sim 30 \right) \%$ with the same value of $\lambda_B(\mu_0)$. Employing the phenomenological model of the three-particle $B$-meson DAs inspired by a QCD sum rule analysis, the three-particle contribution to the $B \to \gamma$ form factors is predicted to be of ${\cal O} (1 \%)$, at leading order in $\alpha_s$, with the default theory inputs.

Petersson inner products of weight-one modular forms

Abstract In this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that these Petersson products posses remarkable arithmetic properties. Namely, such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. A similar result was obtained independently using a different method by W. Duke and Y. Li [5]. The main result of this paper is an explicit factorization formula for the algebraic number obtained by exponentiating a Petersson product.

Form factors of the monodromy matrix entries in [formula omitted]-invariant integrable models

We study integrable models solvable by the nested algebraic Bethe ansatz and described by gl(2|1) or gl(1|2) superalgebras. We obtain explicit determinant representations for form factors of the monodromy matrix entries. We show that all form factors are related to each other at special limits of the Bethe parameters. Our results allow one to obtain determinant formulas for form factors of local operators in the supersymmetric t–J model.

Factorization of the transition matrix for the general Jacobi system

The Jacobi system on a full‐line lattice is considered when it contains additional weight factors. A factorization formula is derived expressing the scattering from such a generalized Jacobi system in terms of the scattering from its fragments. This is performed by writing the transition matrix for the generalized Jacobi system as an ordered matrix product of the transition matrices corresponding to its fragments. The resulting factorization formula resembles the factorization formula for the Schrödinger equation on the full line. Copyright © 2016 John Wiley & Sons, Ltd.

A performance-based service life design method for reinforced concrete structures under chloride environment

The chloride-induced reinforcement steel corrosion usually causes premature failure of performance in reinforced concrete structures subjected to chloride environment. However, conventional prescriptive design methods or fully probabilistic methods cannot ensure an explicit performance in design stage. The main objective of this study is to develop a performance-based service life design method, by which concrete structures such as bridges can be designed according to a predefined performance level to satisfy the target service life, reliability and deterioration level. The mechanism of chloride-induced reinforcement steel corrosion initiation was analyzed to deduce a time-varying limit state function, and the corresponding stochastic model was developed considering the effects of random factors. Through first-order reliability method, a partial factor formula was deduced to realize performance verification in performance-based design process. Finally, a case study of a concrete bridge deck exposed to marine environment was presented and the sensitivity analysis of steel corrosion was carried out. The proposed performance-based design method can rationally deal with the uncertainties of random parameters and ensure the predefined target performance to be satisfied during design process instead of predicting performance after design.

Factorization for groomed jet substructure beyond the next-to-leading logarithm

Jet grooming algorithms are widely used in experimental analyses at hadron colliders to remove contaminating radiation from within jets. While the algorithms perform a great service to the experiments, their intricate algorithmic structure and multiple parameters has frustrated precision theoretic understanding. In this paper, we demonstrate that one particular groomer called soft drop actually makes precision jet substructure easier. In particular, we derive a factorization formula for a large class of soft drop jet substructure observables, including jet mass. The essential observation that allows for this factorization is that, without the soft wide-angle radiation groomed by soft drop, all singular contributions are collinear. The simplicity and universality of the collinear limit in QCD allows us to show that to all orders, the normalized differential cross section has no contributions from non-global logarithms. It is also independent of process, up to the relative fraction of quark and gluon jets. In fact, soft drop allows us to define this fraction precisely. The factorization theorem also explains why soft drop observables are less sensitive to hadronization than their ungroomed counterparts. Using the factorization theorem, we resum the soft drop jet mass to next-to-next-to-leading logarithmic accuracy. This requires calculating some clustering effects that are closely related to corresponding effects found in jet veto calculations. We match our resummed calculation to fixed order results for both $e^+e^-\to$ dijets and $pp\to Z+j$ events, producing the first jet substructure predictions (groomed or ungroomed) to this accuracy for the LHC.

Investigations of defect structures and g factors for octahedral (CrO6)7− and (VO6)8− clusters in rutile-type crystals

The electron paramagnetic resonance (EPR) [Formula: see text] factor formulas for Cr[Formula: see text] and V[Formula: see text] ions in Al2O3, TiO2 and VO2 crystals are deduced from Jahn–Teller effect and contributions of the charge transfer (CT) levels. The tetragonal distortions. [Formula: see text] and −0.0124 nm, and [Formula: see text], [Formula: see text] and 0[Formula: see text], for Al2O3:Cr[Formula: see text], TiO2:V[Formula: see text] and VO2, respectively. The calculations of the [Formula: see text] factors agree well with the experimental values. The contributions of the CT levels to [Formula: see text] factors increase with the increasing valence state. It must be taken into account in the precise calculations of [Formula: see text] factors for the high valence state [Formula: see text] ions in crystals.

Associated production of electroweak bosons and heavy mesons at LHCb and the prospects to observe double parton interactions

The production of weak gauge bosons in association with heavy flavored mesons at the LHCb conditions is considered, and a detailed study of the different contributing processes is presented including single (SPS) and double (DPS) parton scattering mechanisms. We find that the usual DPS factorization formula needs to be corrected for the limited partonic phase space, and that including the relevant corrections reduces discrepancies in the associated $ZD$ production. We conclude finally that double parton scattering dominates the production of same-sign $W^\pm D^\pm$ states, as well as the production of $W^-$ bosons associated with $B$-mesons. The latter processes can thus be regarded as new useful DPS indicators.

Effective Field Theory for Jet Processes.

Processes involving narrow jets receive perturbative corrections enhanced by logarithms of the jet opening angle and the ratio of the energies inside and outside the jets. Analyzing cone-jet processes in effective field theory, we find that in addition to soft and collinear fields their description requires degrees of freedom that are simultaneously soft and collinear to the jets. These collinear-soft particles can resolve individual collinear partons, leading to a complicated multi-Wilson-line structure of the associated operators at higher orders. Our effective field theory provides, for the first time, a factorization formula for a cone-jet process, which fully separates the physics at different energy scales. Its renormalization-group equations control all logarithmically enhanced higher-order terms, in particular also the nonglobal logarithms.

Jet shapes in dijet events at the LHC in SCET

We consider the class of jet shapes known as angularities in dijet production at hadron colliders. These angularities are modified from the original definitions in e+e- collisions to be boost invariant along the beam axis. These shapes apply to the constituents of jets defined with respect to either k_T-type (anti-k_T, C/A, and k_T) algorithms and cone-type algorithms. We present an SCET factorization formula and calculate the ingredients needed to achieve next-to-leading-log (NLL) accuracy in kinematic regions where non-global logarithms are not large. The factorization formula involves previously unstudied "unmeasured beam functions," which are present for finite rapidity cuts around the beams. We derive relations between the jet functions and the shape-dependent part of the soft function that appear in the factorized cross section and those previously calculated for e+e- collisions, and present the calculation of the non-trivial, color-connected part of the soft-function to O(\alpha_s). This latter part of the soft function is universal in the sense that it applies to any experimental setup with an out-of-jet p_T veto and rapidity cuts together with two tagged jets and it is independent of the choice of jet (sub-)structure measurement. In addition, we implement the recently introduced soft-collinear refactorization to resum logarithms of the jet size, valid in the region of non-enhanced non-global logarithm effects. While our results are valid for all 2 \to 2 channels, we compute explicitly for the qq' \to qq' channel the color-flow matrices and plot the NLL resummed differential dijet cross section as an explicit example, which shows that the normalization and scale uncertainty is reduced when the soft function is refactorized. For this channel, we also plot the jet size R dependence, the p_T^{\rm cut} dependence, and the dependence on the angularity parameter a.

Inclusive Higgs production at large transverse momentum

We present a factorization formula for the inclusive production of the Higgs boson at large transverse momentum $P_T$ that includes all terms with the leading power of $1/P_T^2$. The cross section is factorized into convolutions of parton distributions, infrared-safe hard-scattering cross sections for producing a parton, and fragmentation functions that give the distribution of the longitudinal momentum fraction of the Higgs relative to the fragmenting parton. The infrared-safe cross sections and the fragmentation functions are perturbatively calculable. The most important fragmentation functions are those for which the fragmenting parton is the top quark, gluon, $W$, $Z$, and the Higgs itself. We calculate the fragmentation functions at leading order in the Standard Model coupling constants. The factorization formula enables the resummation of large logarithms of $P_T/M_H$ due to final-state radiation by integrating evolution equations for the fragmentation functions. By comparing the cross section for the process $q\bar{q}\to H t\bar t$ from the leading-power factorization formula at leading order in the coupling constants with the complete leading-order cross section, we infer that the error in the factorization formula decreases to less than 5\% for $P_T>600$ GeV at a future $100$ TeV collider.

Operator constraints for twist-3 functions and Lorentz invariance properties of twist-3 observables

We investigate the behavior under Lorentz tranformations of perturbative coefficient functions in a collinear twist-3 formalism relevant for high-energy observables including transverse polarization of hadrons. We argue that those perturbative coefficient functions can, {\it a priori}, acquire quite different yet Lorentz-invariant forms in various frames. This somewhat surprising difference can be traced back to a general dependence of the perturbative coefficient functions on lightcone vectors which are introduced by the twist-3 factorization formulae and which are frame-dependent. One can remove this spurious frame dependence by invoking so-called Lorentz invariance relations (LIRs) between twist-3 parton correlation functions. Some of those relations for twist-3 distribution functions were discussed in the literature before. In this paper we derive the corresponding LIRs for twist-3 fragmentation functions. We explicitly demonstrate that these LIRs remove the lightcone vector dependence by considering transverse spin observables in the single-inclusive production of hadrons in lepton-nucleon collisions, $\ell N\to hX$. With the LIRs in hand, we also show that twist-3 observables in general can be written solely in terms of three-parton correlation functions.