Angularity in DIS at next-to-next-to-leading log accuracy

Angularity is a class of event-shape observables that can be measured in deep-inelastic scattering. With its continuous parameter a one can interpolate angularity between thrust and broadening and further access beyond the region. Providing such systematic way to access various observables makes angularity attractive in analysis with event shapes. We give the definition of angularity for DIS and factorize the cross section by using soft-collinear effective theory. The factorization is valid in a wide range of a below and above thrust region but invalid in broadening limit. It contains an angularity beam function, which is new result and we give the expression at $$ \mathcal{O} $$ O (αs). We also perform large log resummation of angularity and make predictions at various values of a at next-to-next-to-leading log accuracy.

BENDING OF ORTHOTROPIC RECTANGULAR CANTILEVER PLATE

An iterative method of superposition of correcting functions is proposed. The partial solution of the main differential bending equation is represented by a fourth-degree polynomial (the beam function), which gives a residual only with respect to the bending moment on parallel free faces. This discrepancy and the subsequent ones are mutually compensated by two types of correcting functions-hyperbolic-trigonometric series with indeterminate coefficients. Each function satisfies only a part of the boundary conditions. The solution of the problem is achieved by an infinite superposition of correcting functions. For the process to converge, all residuals must tend to zero. When the specified accuracy is reached, the process stops. Numerical results of the calculation of a square ribbed plate are presented.

A toolbox for $$q_{T}$$ and 0-jettiness subtractions at $$\hbox {N}^3\hbox {LO}$$

We derive the leading-power singular terms at three loops for both $$q_T$$ q T and 0-jettiness, $$\mathcal {T}_0$$ T 0 , for generic color-singlet processes. Our results provide the complete set of differential subtraction terms for $$q_T$$ q T and $$\mathcal {T}_0$$ T 0 subtractions at $$\hbox {N}^3\hbox {LO}$$ N 3 LO , which are an important ingredient for matching $$\hbox {N}^3\hbox {LO}$$ N 3 LO calculations with parton showers. We obtain the full three-loop structure of the relevant beam and soft functions, which are necessary ingredients for the resummation of $$q_T$$ q T and $$\mathcal {T}_0$$ T 0 at $$\hbox {N}^3\hbox {LL}'$$ N 3 LL ′ and $$\hbox {N}^4\hbox {LL}$$ N 4 LL order, and which constitute important building blocks in other contexts as well. The nonlogarithmic boundary coefficients of the beam functions, which contribute to the integrated subtraction terms, are not yet fully known at three loops. By exploiting consistency relations between different factorization limits, we derive results for the $$q_T$$ q T and $$\mathcal {T}_0$$ T 0 beam function coefficients at $$\hbox {N}^3\hbox {LO}$$ N 3 LO in the $$z\rightarrow 1$$ z → 1 threshold limit, and we also estimate the size of the unknown terms beyond threshold.

Transverse-energy-energy correlations in deep inelastic scattering

Event shape observables have been widely used for precision QCD studies at various lepton and hadron colliders. We present the most accurate calculation of the transverse-energy-energy correlation event shape variable in deep-inelastic scattering. In the framework of soft-collinear effective theory the cross section is factorized as the convolution of the hard function, beam function, jet function and soft function in the back-to-back limit. A close connection to TMD factorization is established, as the beam function when combined with part of the soft function is identical to the conventional TMD parton distribution function, and the jet function is the second moment of the TMD fragmentation function matching coefficient. We validate our framework by comparing the obtained LO and NLO leading singular distributions to the full QCD calculations in the back-to-back limit. We report the resummed transverse-energy-energy correlation distributions up to N3LL accuracy matched with the NLO cross section for the production of a lepton and two jets. Our work provides a new way to precisely study TMD physics at the future Electron-Ion Collider.

Collinear expansion for color singlet cross sections

We demonstrate how to efficiently expand cross sections for color-singlet production at hadron colliders around the kinematic limit of all final state radiation being collinear to one of the incoming hadrons. This expansion is systematically improvable and applicable to a large class of physical observables. We demonstrate the viability of this technique by obtaining the first two terms in the collinear expansion of the rapidity distribution of the gluon fusion Higgs boson production cross section at next-to-next-to leading order (NNLO) in QCD perturbation theory. Furthermore, we illustrate how this technique is used to extract universal building blocks of scattering cross section like the N-jettiness and transverse momentum beam function at NNLO.

Backward Mechanical-Electric Coupling Effect of a Frequency-Up-Conversion Piezoelectric Energy Harvester

This paper developed an analytical model for a frequency-up-conversion piezoelectric energy harvester (PEH) composed of a piezoelectric bimorph and a stopper as shown in Fig.1. The whole system was subjected to a harmonic excitation. A bimodal approach was adopted to animate the beam stopper reaction. When the tip of the bimorph vibrates in free space or impacts with the stopper, a cantilever beam function was adopted. On the other hand, if the tip of the bimorph sticks with the stopper, a clamped-pinned beam function was applied to model the piezoelectric bimorph. To exam the effect of backward mechanical-electric coupling on power output, the dynamics and output energies are compared for two cases: 1) neglecting the backward mechanical-electric coupling effect in the model; 2) including the backward mechanical-electric coupling effect in the model. To obtain maximum output energy, the steady-state analytical solutions were studied to obtain the optimum gap between the piezoelectric beam and the stopper. From the results, we found that if the beam impacts and/or sticks with the stopper, the PEH model without the backward mechanical-electric coupling will exaggerate the output energy.

Bimodal Approach of a Frequency-Up-Conversion Piezoelectric Energy Harvester

This paper developed an analytical model for a piezoelectric energy harvester (PEH) composed of a piezoelectric bimorph and a stopper as shown in Fig. 1, which was subjected to a harmonic excitation. Frequency-up-conversion, which has proved to improve the energy harvesting efficiency, was achieved due to the mechanical impact between the piezoelectric bimorph and the stopper. The piezoelectric bimorph was modeled as Euler–Bernoulli beam. A bi-modal approach was adopted to animate the beam stopper reaction. When the tip of the bimorph is free for motion, a cantilever beam function is adopted, while the tip encounters a stop, a clamped-pinned beam function is used to model the bimorph. The periodic solutions and their corresponding output voltage and power were obtained. With the same initial conditions and base excitations, the output energies of transient vibrations are compared for two cases: (1) without impact between the piezoelectric beam and the stopper; (2) with impact by reducing the gap distance between the piezoelectric beam and the stopper. With the purpose of maximizing the output power, from the steady-state analytical solutions, we studied the optimum gap between the piezoelectric beam and the stopper when the base excitations are fixed and initial conditions are set to zero.