Three-loop vertex integrals at symmetric point

This paper provides details of the massless three-loop three-point integrals calculation at the symmetric point. Our work aimed to extend known two-loop results for such integrals to the three-loop level. Obtained results can find their application in regularization-invariant symmetric point momentum-subtraction (RI/SMOM) scheme QCD calculations of renormalization group functions and various composite operator matrix elements. To calculate integrals, we solve differential equations for auxiliary integrals by transforming the system to the ε-form. Calculated integrals are expressed through the basis of functions with uniform transcendental weight. We provide expansion up to the transcendental weight six for the basis functions in terms of harmonic polylogarithms with six-root of unity argument.

Iterative Methods for Computing the Resolvent of the Sum of a Maximal Monotone Operator and Composite Operator with Applications

Total variation image denoising models have received considerable attention in the last two decades. To solve constrained total variation image denoising problems, we utilize the computation of a resolvent operator, which consists of a maximal monotone operator and a composite operator. More precisely, the composite operator consists of a maximal monotone operator and a bounded linear operator. Based on recent work, in this paper we propose a fixed-point approach for computing this resolvent operator. Under mild conditions on the iterative parameters, we prove strong convergence of the iterative sequence, which is based on the classical Krasnoselskii–Mann algorithm in general Hilbert spaces. As a direct application, we obtain an effective iterative algorithm for solving the proximity operator of the sum of two convex functions, one of which is the composition of a convex function with a linear transformation. Numerical experiments on image denoising are presented to illustrate the efficiency and effectiveness of the proposed iterative algorithm. In particular, we report the numerical results for the proposed algorithm with different step sizes and relaxation parameters.

WEIGHTED WEAK TYPE ENDPOINT ESTIMATES FOR THE COMPOSITIONS OF CALDERÓN–ZYGMUND OPERATORS

Let $T_{1}$, $T_{2}$ be two Calderón–Zygmund operators and $T_{1,b}$ be the commutator of $T_{1}$ with symbol $b\in \text{BMO}(\mathbb{R}^{n})$. In this paper, by establishing new bilinear sparse dominations and a new weighted estimate for bilinear sparse operators, we prove that the composite operator $T_{1}T_{2}$ satisfies the following estimate: for $\unicode[STIX]{x1D706}>0$ and weight $w\in A_{1}(\mathbb{R}^{n})$, $$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log \bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx,\nonumber\end{eqnarray}$$ while the composite operator $T_{1,b}T_{2}$ satisfies $$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1,b}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}^{2}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log ^{2}\bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx.\nonumber\end{eqnarray}$$

Renormalizability of pure 𝒩 = 1 super-Yang–Mills in the Wess–Zumino gauge in the presence of the local composite operators A2 and λ̄λ

The [Formula: see text] super-Yang–Mills theory in the presence of the local composite operator [Formula: see text] is analyzed in the Wess–Zumino gauge by employing the Landau gauge fixing condition. Due to the supersymmetric structure of the theory, two more composite operators, [Formula: see text] and [Formula: see text], related to the SUSY variations of [Formula: see text] are also introduced. A BRST invariant action containing all these operators is obtained. An all-order proof of the multiplicative renormalizability of the resulting theory is then provided by means of the algebraic renormalization setup. Though, due to the nonlinear realization of the supersymmetry in the Wess–Zumino gauge, the renormalization factor of the gauge field turns out to be different from that of the gluino.