Evolution Kernel Research Articles

Lattice QCD Study of Transverse-Momentum Dependent Soft Function.

In this work, we perform a lattice QCD study of the intrinsic, rapidity-independent soft function within the framework of large momentum effective theory. The computation is carried out using a gauge ensemble of N_{f}=2+1+1 clover-improved twisted mass fermion. After applying an appropriate renormalization procedure and the removal of significant higher-twist contamination, we obtain the intrinsic soft function that is comparable to the one-loop perturbative result at large external momentum. The determination of the nonperturbative soft function from first principles is crucial to sharpen our understanding of the processes with small transverse momentum such as the Drell-Yan production and the semi-inclusive deep inelastic scattering. Additionally, we calculate the Collins-Soper evolution kernel using the quasi-transverse-momentum-dependent wave function as input.

Parton distribution function for topological charge at one loop

We present results for the gg-part of the one-loop corrections to the recently introduced “topological charge” GPD overset{sim }{F} (x, q2). In particular, we give expression for its evolution kernel. To enforce strict compliance with the gauge invariance requirements, we have used on-shell states for external gluons, and have obtained identical results both in Feynman and light-cone gauges. No “zero mode” δ(x) terms were found for the twist-4 gluon GPD overset{sim }{F} (x, q2).

From the Evolution Kernel in (2+1)-Dimensional Spacetime to a Thermal Theory of Two-Dimensional Matter

From the Evolution Kernel in (2+1)-Dimensional Spacetime to a Thermal Theory of Two-Dimensional Matter

Quark Mass Effects in JIMWLK Evolution

Quark mass effects in the Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) evolution appear at the next to leading order (NLO) of QCD, i.e., at 𝒪(α2s) of the QCD coupling for the JIMWLK evolution kernels. Based on the light-cone hadronic wave function approach, we obtain the analytic expressions for the JIMWLK evolution kernels, which are expressed in terms of one dimensional integrals that in principle can be numerically useful for future phenomenological applications. In comparison to the situation with massless quarks, a new divergence arises. We illustrate the origin of this divergence and how the divergence is canceled within the renormalization procedures of light-cone quantization.

Renormalization of the radiative jet function

We show how to compute directly the renormalization/evolution of the radiative jet function that appears in the factorization theorems for $B\to \gamma\ell\nu$ and $H\to \gamma\gamma$ through a $b$-quark loop. We point out that, in order to avoid double counting of soft contributions, one should use in the factorization theorems a subtracted radiative jet function, from which soft contributions have been removed. The soft-contribution subtractions are zero-bin subtractions in the terminology of soft-collinear effective theory. We show that they can be factored from the radiative jet function and that the resulting soft-subtraction function gives rise to a nonlocal renormalization of the subtracted radiative jet function. This is a novel instance in which zero-bin subtractions lead to a nonlocality in the renormalization of a subtracted quantity that is not present in the renormalization of the unsubtracted quantity. We demonstrate the use of our formalism by computing the order-$\alpha_s$ evolution kernel for the subtracted radiative jet function. Our result is in agreement with the result that had been inferred previously by making use of the factorization theorem for $B\to \gamma\ell\nu$, but that had been ascribed to the unsubtracted radiative jet function.

One-loop structure of parton distribution for the gluon condensate and \u201czero modes\u201d

We present results for one-loop corrections to the recently introduced “gluon condensate” PDF F(x). In particular, we give expression for the gg-part of its evolution kernel. To enforce strict compliance with the gauge invariance requirements, we have used on-shell states for external gluons, and have obtained identical results both in Feynman and light-cone gauges. No “zero mode” δ(x) terms were found for the twist-4 gluon PDF F(x). However a q2δ(x) term was found for the ξ = 0 GPD F(x, q2) at nonzero momentum transfer q. Overall, our results do not agree with the original attempt of one-loop calculations of F(x) for gluon states, which sets alarm warning for calculations that use matrix elements with virtual external gluons and for lattice renormalization procedures based on their results.

Extended collinearly-improved Balitsky-Kovchegov evolution equation in target rapidity

An extended collinearly improved Balitsky-Kovchegov evolution equation in the target rapidity representation is derived by including the running coupling corrections during the expansion of the ``real'' $S$-matrix. We find that the running coupling brings important modifications to the evolution equation, as one can see that there are extra contributions to the evolution kernel once the running coupling is included. To identify the significance of the modifications, we numerically solve the evolution equation with and without the running coupling contributions during the $S$-matrix expansion. The numerical results show that the scattering amplitude is largely suppressed by the running coupling corrections, which indicate that one needs to consider the running coupling contributions during the derivation of the nonlinear evolution equation in the target rapidity representation.

Three-loop off-forward evolution kernel for axial-vector operators in Larin’s scheme

Evolution equations for leading-twist operators in high orders of perturbation theory can be restored from the spectrum of anomalous dimensions and the calculation of the special conformal anomaly at one order less using conformal symmetry of QCD at the Wilson-Fisher critical point at noninteger $d=4\ensuremath{-}2ϵ$ space-time dimensions. In this work, we generalize this technique to axial-vector operators. We calculate the corresponding three-loop evolution kernels in Larin's scheme and derive explicit expressions for the finite renormalization kernel that describes the difference to the vector case to restore the conventional modified minimal subtraction scheme. The results are directly applicable to deeply virtual Compton scattering and the transition form factor ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\pi}$.

Analytic continuation and reciprocity relation for collinear splitting in QCD * *Supported in part by National Natural Science Foundation of China (11975200) and the Zhejiang University Fundamental Research Funds for the Central Universities (2019QNA3005)

It is well-known that direct analytic continuation of the DGLAP evolution kernel (splitting functions) from space-like to time-like kinematics breaks down at three loops. We identify the origin of this breakdown as due to splitting functions not being analytic functions of external momenta. However, splitting functions can be constructed from the squares of (generalized) splitting amplitudes. We establish the rules of analytic continuation for splitting amplitudes, and use them to determine the analytic continuation of certain holomorphic and anti-holomorphic part of splitting functions and transverse-momentum dependent distributions. In this way we derive the time-like splitting functions at three loops without ambiguity. We also propose a reciprocity relation for singlet splitting functions, and provide non-trivial evidence that it holds in QCD at least through three loops.

Lattice QCD Calculations of Transverse-Momentum-Dependent Soft Function through Large-Momentum Effective Theory.

The transverse-momentum-dependent (TMD) soft function is a key ingredient in QCD factorization of Drell-Yan and other processes with relatively small transverse momentum. We present a lattice QCD study of this function at moderately large rapidity on a 2+1 flavor CLS dynamic ensemble with a=0.098 fm. We extract the rapidity-independent (or intrinsic) part of the soft function through a large-momentum-transfer pseudoscalar meson form factor and its quasi-TMD wave function using leading-order factorization in large-momentum effective theory. We also investigate the rapidity-dependent part of the soft function-the Collins-Soper evolution kernel-based on the large-momentum evolution of the quasi-TMD wave function.

On Existence and Uniqueness of Fractional Linear Integro Partial Differential Equation with Evolution Kernel Using Modified Bielecki Method and its Numerical Solution

On Existence and Uniqueness of Fractional Linear Integro Partial Differential Equation with Evolution Kernel Using Modified Bielecki Method and its Numerical Solution

Evolutionary Multivariate Kernal Svm Prediction Method for Classification

Thyroid disorders are common among the world wide population. This disorders posses’ significant problems among Indians. Research studies shows that nearly 32% of Indian population suffers from various thyroid disorders. This paper deals with the thyroid data set which in turn classify into three groups as hyper thyroidisim, hypothyroidism and normal. The American Thyroid Association reported twelve percent of their citizens suffer from thyroidism in which 60% population are unaware of their condtions.. Above statistics implies the classification of thyroid disorder is crucial in global perspective too. The thyroid data set are collected from UCI repository and it is multivariate type with 21 attributes. With the 21 attributes only 10 attributes are selected based on their rank. Hybrid Differential Evolution Kernel Based SVM algorithm is used to classify the data set. It takes around 30 epochs to stabilize the errors. The classification accurancy is observed to be 67.97%.

Non-perturbative structure of semi-inclusive deep-inelastic and Drell-Yan scattering at small transverse momentum

We consider semi-inclusive deep inelastic scattering (SIDIS) and Drell-Yan events within transverse momentum dependent (TMD) factorization. Based on the simultaneous fit of multiple data points, we extract the unpolarized TMD distributions and the non-perturbative evolution kernel. The high quality of the fit confirms a complete universality of TMD non-perturbative distributions. The extraction is supplemented by phenomenological analyses of various parts of the TMD factorization, such as sensitivity to non-perturbative parameterizations, perturbative orders, collinear distributions, correlations between parameters, and others.

Transverse-momentum-dependent factorization for lattice observables

Using soft collinear effective field theory, we derive the factorization theorem for the quasi-transverse-momentum-dependent (quasi-TMD) operator. We check the factorization theorem at one-loop level and compute the corresponding coefficient function and anomalous dimensions. The factorized expression is built from the physical TMD distribution, and a nonperturbative lattice related factor. We demonstrate that lattice related functions cancel in appropriately constructed ratios. These ratios could be used to explore various properties of TMD distributions, for instance, the nonperturbative evolution kernel. A discussion of such ratios and the related continuum properties of TMDs is presented.

Higher-order Sudakov resummation in coupled gauge theories

We consider the higher-order resummation of Sudakov double logarithms in the presence of multiple coupled gauge in. The associated evolution equations depend on the coupled β functions of two (or more) coupling constants αa and αb, as well as anomalous dimensions that have joint perturbative series in αa and αb. We discuss possible strategies for solving the system of evolution equations that arises. As an example, we obtain the complete three-loop (NNLL) QCD⊗QED Sudakov evolution factor. Our results also readily apply to the joint higher-order resummation of electroweak and QCD Sudakov logarithms.As part of our analysis we also revisit the case of a single gauge interaction (pure QCD), and study the numerical differences and reliability of various methods for evaluating the Sudakov evolution factor at higher orders. We find that the approximations involved in deriving commonly used analytic expressions for the evolution kernel can induce noticeable numerical differences of several percent or more at low scales, exceeding the perturbative precision at N3LL and in some cases even NNLL. Therefore, one should be cautious when using approximate analytic evolution kernels for high-precision analyses.

Experimental and numerical investigations of double pulse laser energy deposition in air

This study focuses on the laser energy deposition characteristics through single and double pulse at short and long pulse intervals in quiescent air, both experimentally and numerically. A frequency-doubled and Q-switched Nd: YAG laser is used to create the spark in the air. High-speed schlieren imaging technique is used to visualize the shock-wave evolution and plasma kernel expansion. To estimate the energy available for ignition, the energy absorbed is measured and the energy loss through the shock-wave is estimated by using Jones blast wave model. Two-dimensional numerical simulations are carried out by solving the Navier-Stokes equations along with the species conservation equations in finite volume framework. After cessation of the laser pulse, the breakdown zone assumed to consists of seven species (O2, O, N2, N, NO, NO+ and e−). For double pulse at short pulse interval, the second pulse absorbed more energy than the first pulse, and the shock loss is relatively higher. The lifetime of plasma is found to have increased by 13.56 % due to the jump in temperature and pressure after the introduction of the second pulse. Same total energy two successive laser pulses at short time interval are found to be a promising alternative of single pulse for lean limit of combustion. The amount of energy left in the kernel for ignition is 16.02 % and 15.78 % of the absorbed energy in 50 + 50 mJ and 100 + 100 mJ double pulse respectively. In case of double laser pulse deposition at long pulse interval, along with multiple layers of shock-wave, a third lobe and a fourth lobe are observed on either side (upstream and downstream) of the plasma kernel enhancing the kernel surface area and mixing process.

Two-loop evolution equation for the B -meson distribution amplitude

We derive the two-loop evolution equation of the B-meson light-cone distribution amplitude which is the last missing element for the next-to-next-to-leading logarithmic resummation of QCD corrections to B decays in QCD factorization. We argue that the evolution kernel to all orders in perturbation theory can be written as a logarithm of the generator of special conformal transformations times the cusp anomalous dimension, up to a scheme-dependent overall constant. Up to this constant term, the evolution kernel to a given order in perturbation theory can be obtained from the calculation of special conformal anomaly at one order less.

Extraction of unpolarized quark transverse momentum dependent parton distributions from Drell-Yan/Z-boson production

We present the extraction of unpolarized quark transverse momentum dependent parton distribution functions (TMDPDFs) and the non-perturbative part of TMD evolution kernel from the global analysis of Drell-Yan and Z-boson production data. The analysis is performed at the next-to-next-to-leading order (NNLO) in perturbative QCD, using the ζ-prescription. The estimation of the error-propagation from the experimental uncertainties to non-perturbative function is made by the replica method. The importance of the inclusion of the precise LHC data and its influence on the determination of non-perturbative functions is discussed.

Spin budget of the proton at NNLO and beyond

We revisit the scale evolution of the quark and gluon spin contributions to the proton spin, $\frac{1}{2}\Delta \Sigma$ and $\Delta G$, using the three-loop results for the spin-dependent evolution kernels available in the literature. We argue that the evolution of the quark spin contribution may actually be extended to four-loop order, and that to all orders a single anomalous dimension governs the evolution of both $\Delta \Sigma$ and $\Delta G$. We present analytical solutions of the evolution equations for $\Delta \Sigma$ and $\Delta G$ and investigate their scale dependence both to large and down to lower "hadronic" scales. We find that the solutions remain perturbatively stable even to low scales, where they come closer to simple quark model expectations. We discuss a curious scenario for the proton spin, in which even the gluon spin contribution is essentially scale independent and has a finite asymptotic value as the scale becomes large. We finally also show that perturbative three-loop evolution leads to a larger spin contribution of strange anti-quarks than of strange quarks.

Leading-logarithmic threshold resummation of the Drell-Yan process at next-to-leading power

We resum the leading logarithms αsn ln2n − 1(1 − z), n = 1, 2, . . . near the kine-matic threshold z = Q2/ŝ → 1 of the Drell-Yan process at next-to-leading power in the expansion in (1 − z). The derivation of this result employs soft-collinear effective theory in position space and the anomalous dimensions of subleading-power soft functions, which are computed. Expansion of the resummed result leads to the leading logarithms at fixed loop order, in agreement with exact results at NLO and NNLO and predictions from the physical evolution kernel at N3LO and N4LO, and to new results at the five-loop order and beyond.