Double Collinear Research Articles

Collinear resummations for the non-linear evolution in QCD at high energy

When computed to next-to-leading order in perturbative QCD, the non-linear Balitsky-Kovchegov (BK) equation for the high-energy evolution of the dipole-hadron scattering appears to be unstable. We show that this instability can be avoided by using the rapidity of the dense hadronic target (instead of that of the dilute dipole projectile) as the evolution time. Using this variable, we construct a collinearly-improved version of the BK equation, where the dominant radiative corrections to the kernel — those enhanced by double collinear logarithms — are resummed to all orders.

Finding the suitable drag-free acceleration noise level for future low-low satellite-to-satellite tracking geodesy missions

This paper evaluates the impact of residual acceleration noise on the estimation of the Earth’s time-varying gravity field for future low-low satellite-to-satellite tracking missions. The goal is to determine the maximum level of residual acceleration noise that does not adversely affect the estimation error. The Gravity Recovery And Climate Experiment (GRACE) has provided monthly average gravity field solutions in spherical harmonic coefficients for more than a decade. It provides information about land and ocean mass variations with a spatial resolution of ∼350 km and with an accuracy within 2 cm throughout the entire Earth. GRACE Follow-on was launched in May 2018 to advance the work of GRACE and to test a new laser ranging interferometer, which measures the range between the two satellites with higher precision than the K-Band ranging system used in GRACE. Moreover, there have been simulation studies that show, an additional pair of satellites in an inclined orbit increases the sampling frequency and reduces temporal aliasing errors. Given the fact that future missions will likely continue to use the low-low satellite-to-satellite tracking formation with laser ranging interferometry, it is expected that the residual acceleration noise will become one of the largest error contributor for the time-variable gravity field solution. We evaluate three different levels of residual acceleration noise based on demonstrated drag-free systems to find a suitable drag-free performance target for upcoming geodesy missions. We analyze both a single collinear polar pair and the optimal double collinear pair of drag-free satellites and assume the use of a laser ranging interferometer. A partitioned best linear unbiased estimator that was developed, incorporating several novel features from the ground up is used to compute the solutions in terms of spherical harmonics. It was found that the suitable residual acceleration noise level is around 2 × 10−12 ms−2 Hz−1/2. Decreasing the acceleration noise below this level did not result in more accurate gravity field solutions for the chosen mission architecture.

Resumming double non-global logarithms in the evolution of a jet

We consider the Banfi-Marchesini-Smye (BMS) equation which resums ‘non-global’ energy logarithms in the QCD evolution of the energy lost by a pair of jets via soft radiation at large angles. We identify a new physical regime where, besides the energy logarithms, one has to also resum (anti)collinear logarithms. Such a regime occurs when the jets are highly collimated (boosted) and the relative angles between successive soft gluon emissions are strongly increasing. These anti-collinear emissions can violate the correct time-ordering for time-like cascades and result in large radiative corrections enhanced by double collinear logs, making the BMS evolution unstable beyond leading order. We isolate the first such a correction in a recent calculation of the BMS equation to next-to-leading order by Caron-Huot. To overcome this difficulty, we construct a ‘collinearly-improved’ version of the leading-order BMS equation which resums the double collinear logarithms to all orders. Our construction is inspired by a recent treatment of the Balitsky-Kovchegov (BK) equation for the high-energy evolution of a space-like wavefunction, where similar time-ordering issues occur. We show that the conformal mapping relating the leading-order BMS and BK equations correctly predicts the physical time-ordering, but it fails to predict the detailed structure of the collinear improvement.

Formation of slantwise orientated nanoscale ripple structures on a single-crystal 4H-SiC surface by time-delayed double femtosecond laser pulses

An effective control of the nanoscale periodic ripple structures on a single-crystal 4H-SiC surface is carried out by temporally delayed double collinear femtosecond laser pulse trains with different linear polarizations. The observed evolution of the ripple slant angle shows a dramatic decrease within an initial time delay range of ~10 ps between the two laser pulses, but keeps almost steady for larger time elapse. Through the detailed analyses, we propose a theoretical model to reveal the underlying physics: regarding the double femtosecond laser pulses, the first laser pulse transfers the semiconductor into nonequilibrium states associated with a transient refractive index grating on the surface, which subsequently tends to change the direction of the surface plasmon excitation of the second laser pulse. These behaviors can eventually result in the slantwise oriented laser intensity fringes and corresponding periodic ablation of grooves. Such investigations provide additional insights into the ripple formation processes, which is very important for control of nanomanufacturing materials by femtosecond lasers.

Approximate solution to the NLL BK equation in the saturation region

We analytically solve the full next-to-leading logarithmic Balitsky-Kovchegov equation in the saturation regime, which includes corrections from quark and gluon loops, and large double transverse logarithms. The analytic result for the $S$-matrix in the saturation regime shows that the linear rapidity decrease with rapidity of the exponent in running coupling case is replaced by rapidity raised to power of 3/2 decreasing. The collinearly-improved Balitsky-Kovchegov equation are also analytically solved in the saturation regime. It shows that the double collinear logarithms do not contribute to the $S$-matrix and the solution is the same as the one obtained from the leading order Balitsky-Kovchegov equation. The numerical solutions to the leading order and full next-to-leading logarithmic Balitsky-Kovchegov equations are performed in order to test the analytic results derived in the saturation regime.

Enhanced laser-induced breakdown spectroscopy on Co sample with double femtosecond laser pulses

This paper presents the study on the enhancement mechanism of laser-induced breakdown spectroscopy on cobalt sample with double femtosecond pulse. The double-pulse experiments were carried out in the collinear beam geometry in air. The influence of the delay between the two collinear laser pulses was investigated by changing the pulses path length. In cases of no delay between the two pulses, the laser-induced breakdown spectroscopy is equivalent to that induced by a single pulse laser with energy equals to the sum of the double collinear pulse energy. The obtained results show that with an optimized delay, the double-pulse laser-induced breakdown spectroscopy has a very clear enhancement comparing with the single-pulse laser induced breakdown spectroscopy. In addition, the dependence of the electron temperature obtained by linear Boltzmann method on the pulse interval was investigated. The experiment shows that the dependence of spectral enhancement factor on the pulse interval is the same as that of the electron temperature, both of them go through two obvious changing processes. Analysing by the Boltzmann equation, we draw the conclusion that the spectral signal enhancement depends on the electron temperature.

Resummation of double collinear logs in BK evolution versus HERA data

We present a global fit to HERA data on the reduced cross section measured in electron–proton collisions in the region of small Bjorken-x: x≤x0=10−2 and moderate to high values of the virtuality Q2<Qmax2=650 GeV2. The main dynamical ingredients in the fits are two recently proposed improved BK equations for the description of the small-x evolution of the dipole scattering amplitude [1,2]. These two new equations provide an all-order resummation of double collinear logarithms that arise beyond leading logarithmic accuracy. We show that a very good description of data is possible in both cases, provided the parent dipole or smallest dipole prescriptions are employed for the running of the coupling.

Collinearly improved JIMWLK evolution in Langevin form

The high-energy evolution of Wilson line operators, which at leading order is described by the Balitsky-JIMWLK equations, receives large radiative corrections enhanced by single and double collinear logarithms at next-to-leading order and beyond. We propose a method for resumming such logarithmic corrections to all orders, at the level of the Langevin formulation of the JIMWLK equation. The ensuing, collinearly-improved Langevin equation features generalized Wilson line operators, which depend not only upon rapidity (the logarithm of the longitudinal momentum), but also upon the transverse size of the color neutral projectile to which the Wilson lines belong. This additional scale dependence is built up during the evolution, via the condition that the successive emissions of soft gluons be ordered in time. The presence of this transverse scale in the Langevin equation furthermore allows for the resummation of the one-loop running coupling corrections.

Crack spacing threshold of double cracks propagation for large-module rack

Large-module rack of the Three Gorges shiplift is manufactured by casting and machining, which is unable to avoid slag inclusions and surface cracks. To ensure its safety in the future service, studying on crack propagation rule and the residual life estimation method of large-module rack is of great significance. The possible crack distribution forms of the rack in the Three Gorges shiplift were studied. By applying moving load on the model in FRANC3D and ANSYS, quantitative analyses of interference effects on double cracks in both collinear and offset conditions were conducted. The variation rule of the stress intensity factor (SIF) influence factor, RK, of double collinear cracks changing with crack spacing ratio, RS, was researched. The horizontal and vertical crack spacing threshold of double cracks within the design life of the shiplift were obtained, which are 24 and 4 times as large as half of initial crack length, c0, respectively. The crack growth rates along the length and depth directions in the process of coalescence on double collinear cracks were also studied.

Resumming double logarithms in the QCD evolution of color dipoles

The higher-order perturbative corrections, beyond leading logarithmic accuracy, to the BFKL evolution in QCD at high energy are well known to suffer from a severe lack-of-convergence problem, due to radiative corrections enhanced by double collinear logarithms. Via an explicit calculation of Feynman graphs in light cone (time-ordered) perturbation theory, we show that the corrections enhanced by double logarithms (either energy-collinear, or double collinear) are associated with soft gluon emissions which are strictly ordered in lifetime. These corrections can be resummed to all orders by solving an evolution equation which is non-local in rapidity. This equation can be equivalently rewritten in local form, but with modified kernel and initial conditions, which resum double collinear logs to all orders. We extend this resummation to the next-to-leading order BFKL and BK equations. The first numerical studies of the collinearly-improved BK equation demonstrate the essential role of the resummation in both stabilizing and slowing down the evolution.

Double collinear splitting amplitudes at next-to-leading order

We compute the next-to-leading order (NLO) QCD corrections to the $1 \to 2$ splitting amplitudes in different dimensional regularization (DREG) schemes. Besides recovering previously known results, we explore new DREG schemes and analyze their consistency by comparing the divergent structure with the expected behavior predicted by Catani's formula. Through the introduction of scalar-gluons, we show the relation among splittings matrices computed using different schemes. Also, we extended this analysis to cover the double collinear limit of scattering amplitudes in the context of QCD+QED.

Characteristics of micro air plasma produced by double femtosecond laser pulses

Dynamic characteristics of air plasma generated by focused double collinear femtosecond laser pulses with a time interval of 10 ns are experimentally investigated. The air plasma emission changes significantly when altering the energy ratio between the two laser pulses. Time-resolved shadowgraphic measurements reveal that a small volume of transient vacuum is formed inside the air shock wave produced by the first laser pulse, which causes the second laser pulse induced ionization zone to present as two separate sections in space. Also recorded is strong scattering of the second laser pulse by the ionized air just behind the ionization front of the first laser pulse produced shock wave. Due to the high intensity of the scattered light, coherent Thomson scattering enhanced by plasma instabilities is believed to be the main scattering mechanism in this case.