Comment on Inverse problem from discrete spectrum in the D = 2 dimensional space (2014 Phys. Scr. 989 085201)

We analyse a method for the construction of the potential-energy function from the moments of the ground-state density. The sum rule on which some expressions are based appear to be wrong, as well as the moments and potential-energy functions derived for some illustrative examples.

Energy-momentum tensor in QCD: nucleon mass decomposition and mechanical equilibrium

We review and examine in detail recent developments regarding the question of the nucleon mass decomposition. We discuss in particular the virial theorem in quantum field theory and its implications for the nucleon mass decomposition and mechanical equilibrium. We reconsider the renormalization of the QCD energy-momentum tensor in minimal-subtraction-type schemes and the physical interpretation of its components, as well as the role played by the trace anomaly and Poincaré symmetry. We also study the concept of “quantum anomalous energy” proposed in some works as a new contribution to the nucleon mass. Examining the various arguments, we conclude that the quantum anomalous energy is not a genuine contribution to the mass sum rule, as a consequence of translation symmetry.

Mathematics Refresher

The chapter “Mathematics Refresher” provides a brief reminder of operations with logarithms, matrices, and calculus, for student reference. It starts off by reviewing the differences between regular logarithms and natural logarithms and provides some examples of common operations with logarithms. It then introduces derivatives and integrals (although it is never necessary to compute an integral in this book, it is still useful to know what an integral is) and explains the sum rule, the product rule, the quotient rule, and the chain rule. Next, it provides a brief overview of matrices and matrix operations, including matrix dimensions, and addition and multiplication of matrices. It concludes with a discussion of the identity matrix.

Radiative decays of heavy-light mesons and the $$ {f}_{H,{H}^{\ast },{H}_1}^{(T)} $$ decay constants

The on-shell matrix elements, or couplings $$ {g}_{H{H}^{\ast}\left({H}_1\right)\upgamma} $$ g H H ∗ H 1 γ , describing the $$ B{(D)}_q^{\ast } $$ B D q ∗ → B(D)qγ and B1q → Bqγ (q = u, d, s) radiative decays, are determined from light-cone sum rules at next-to-leading order for the first time. Two different interpolating operators are used for the vector meson, providing additional robustness to our results. For the D*-meson, where some rates are experimentally known, agreement is found. The couplings are of additional interest as they govern the lowest pole residue in the B(D) → γ form factors which in turn are connected to QED-corrections in leptonic decays B(D) → ℓ$$ \overline{\nu} $$ ν ¯ . Since the couplings and residues are related by the decay constants $$ {f}_{H^{\ast}\left({H}_1\right)} $$ f H ∗ H 1 and $$ {f}_{H^{\ast}\left({H}_1\right)}^T $$ f H ∗ H 1 T , we determine them at next-leading order as a by-product. The quantities $$ \left\{{f}_{H^{\ast}}^T,{f}_{H_1}^T\right\} $$ f H ∗ T f H 1 T have not previously been subjected to a QCD sum rule determination. All results are compared with the existing experimental and theoretical literature.

Twist-2 relation and sum rule for tensor-polarized parton distribution functions of spin-1 hadrons

Sum rules for structure functions and their twist-2 relations have important roles in constraining their magnitudes and x dependencies and in studying higher-twist effects. The Wandzura-Wilczek (WW) relation and the Burkhardt-Cottingham (BC) sum rule are such examples for the polarized structure functions g1 and g2. Recently, new twist-3 and twist-4 parton distribution functions were proposed for spin-1 hadrons, so that it became possible to investigate spin-1 structure functions including higher-twist ones. We show in this work that an analogous twist-2 relation and a sum rule exist for the tensor-polarized parton distribution functions f1LL and fLT, where f1LL is a twist-2 function and fLT is a twist-3 one. Namely, the twist-2 part of fLT is expressed by an integral of f1LL (or b1) and the integral of the function f2LT = (2/3)fLT− f1LL over x vanishes. If the parton-model sum rule for f1LL (b1) is applied by assuming vanishing tensor-polarized antiquark distributions, another sum rule also exists for fLT itself. These relations should be valuable for studying tensor-polarized distribution functions of spin-1 hadrons and for separating twist-2 components from higher-twist terms, as the WW relation and BC sum rule have been used for investigating x dependence and higher-twist effects in g2. In deriving these relations, we indicate that four twist-3 multiparton distribution functions FLT, GLT, $$ {H}_{LL}^{\perp } $$ H LL ⊥ , and HTT exist for tensor-polarized spin-1 hadrons. These multiparton distribution functions are also interesting to probe multiparton correlations in spin-1 hadrons. In the near future, we expect that physics of spin-1 hadrons will become a popular topic, since there are experimental projects to investigate spin structure of the spin-1 deuteron at the Jefferson Laboratory, the Fermilab, the nuclotron-based ion collider facility, the electron-ion colliders in US and China in 2020’s and 2030’s.

A novel determination of non-perturbative contributions to Bjorken sum rule

Based on the operator product expansion, the perturbative and nonperturbative contributions to the polarized Bjorken sum rule (BSR) can be separated conveniently, and the nonperturbative one can be fitted via a proper comparison with the experimental data. In the paper, we first give a detailed study on the pQCD corrections to the leading-twist part of BSR. Basing on the accurate pQCD prediction of BSR, we then give a novel fit of the non-perturbative high-twist contributions by comparing with JLab data. Previous pQCD corrections to the leading-twist part derived under conventional scale-setting approach still show strong renormalization scale dependence. The principle of maximum conformality (PMC) provides a systematic and strict way to eliminate conventional renormalization scale-setting ambiguity by determining the accurate $$\alpha _s$$ α s -running behavior of the process with the help of renormalization group equation. Our calculation confirms the PMC prediction satisfies the standard renormalization group invariance, e.g. its fixed-order prediction does scheme-and-scale independent. In low $$Q^2$$ Q 2 -region, the effective momentum of the process is small and in order to derive a reliable prediction, we adopt four low-energy $$\alpha _s$$ α s models to do the analysis, i.e. the model based on the analytic perturbative theory (APT), the Webber model (WEB), the massive pQCD model (MPT) and the model under continuum QCD theory (CON). Our predictions show that even though the high-twist terms are generally power suppressed in high $$Q^2$$ Q 2 -region, they shall have sizable contributions in low and intermediate $$Q^2$$ Q 2 domain. Based on the more accurate scheme-and-scale independent pQCD prediction, our newly fitted results for the high-twist corrections at $$Q^2=1\;\mathrm{GeV}^2$$ Q 2 = 1 GeV 2 are, $$f_2^{p-n}|_{\mathrm{APT}}=-0.120\pm 0.013$$ f 2 p - n | APT = - 0.120 ± 0.013 , $$f_2^{p-n}|_\mathrm{WEB}=-0.081\pm 0.013$$ f 2 p - n | WEB = - 0.081 ± 0.013 , $$f_2^{p-n}|_{\mathrm{MPT}}=-0.128\pm 0.013$$ f 2 p - n | MPT = - 0.128 ± 0.013 and $$f_2^{p-n}|_{\mathrm{CON}}=-0.139\pm 0.013$$ f 2 p - n | CON = - 0.139 ± 0.013 ; $$\mu _6|_\mathrm{APT}=0.003\pm 0.000$$ μ 6 | APT = 0.003 ± 0.000 , $$\mu _6|_{\mathrm{WEB}}=0.001\pm 0.000$$ μ 6 | WEB = 0.001 ± 0.000 , $$\mu _6|_\mathrm{MPT}=0.003\pm 0.000$$ μ 6 | MPT = 0.003 ± 0.000 and $$\mu _6|_{\mathrm{CON}}=0.002\pm 0.000$$ μ 6 | CON = 0.002 ± 0.000 , respectively, where the errors are squared averages of those from the statistical and systematic errors from the measured data.

A closer look at the extraction of |Vub| from B → πℓν

To extract the Cabibbo-Kobayashi-Maskawa (CKM) matrix element |Vub|, we have re-analyzed all the available inputs (data and theory) on the B → πℓν decays including the newly available inputs on the form-factors from light cone sum rule (LCSR) approach. We have reproduced and compared the results with the procedure taken up by the Heavy Flavor Averaging Group (HFLAV), while commenting on the effect of outliers on the fits. After removing the outliers and creating a comparable group of data-sets, we mention a few scenarios in the extraction of |Vub|. In all those scenarios, the extracted values of |Vub| are higher than that obtained by HFLAV. Our best results for |Vub|exc. are (3.94 ± 0.14) × 10−3 and $$ \left({3.93}_{-0.15}^{+0.14}\right) $$ 3.93 − 0.15 + 0.14 × 10−3 in frequentist and Bayesian approaches, respectively, which are consistent with that extracted from inclusive decays |Vub|inc. within 1 σ confidence interval.