Three-loop contribution of the Faddeev-Popov ghosts to the β\beta-function of N=1{\cal N}=1 supersymmetric gauge theories and the NSVZ relation

We find the three-loop contribution to the $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives produced by the supergraphs containing loops of the Faddeev--Popov ghosts. This is done using a recently proposed algorithm, which essentially simplifies such multiloop calculations. The result is presented in the form of an integral of double total derivatives in the momentum space. The considered contribution to the $\beta$-function is compared with the two-loop anomalous dimension of the Faddeev--Popov ghosts. This allows verifying the validity of the NSVZ equation written as a relation between the $\beta$-function and the anomalous dimensions of the quantum superfields. It is demonstrated that in the considered approximation the NSVZ equation is satisfied for the renormalization group functions defined in terms of the bare couplings. The necessity of the nonlinear renormalization for the quantum gauge superfield is also confirmed.Comment: 20 pages, 4 figures, minor corrections, the final version to appear in Eur.Phys.J.

Results of investigation of muon fluxes of superhigh energy cosmic rays with X-ray emulsion chambers

The overall data from the investigation of the cosmic ray muon flux in the range of zenith angles (0-90) deg within the energy range (3.5 to 5.0) TeV is presented. The exposure of large X-ray emulsion chambers underground was 1200 tons. year. The data were processe using the method which was applied in the experiment Pamir and differred from the earlier applied one. The obtained value of a slope power index of the differential energy spectrum of the global muon flux is =3.7 that corresponds to the slope of the pion generation differential spectrum, gamma sub PI = 2.75 + or - .04. The analysis of the muon zenith-angular distribution showed that the contribution of rapid generation muons in the total muon flux agree the best with the value .2% and less with .7% at a 90% reliability level

Experimental study of intrinsic multiple Andreev reflections effect in GdO(F)FeAs superconductor array junctions

We report the first observation of the intrinsic multiple Andreev reflections effect (IMARE) in S-n-S-...-S-arrays (S = superconductor, n = normal metal) formed by "break-junction" technique in GdO(F)FeAs superconductor (Tc = 48 - 53 K). We show that superconducting gap peculiarities at dI/dV-spectra sharpen dramatically in the arrays as compared with that in the single-contact spectra; this enables to improve significantly accuracy of the bulk superconducting parameters determination. Using IMARE, we determined the large and the small gap values \Delta_L = 11 +- 1.1 meV and \Delta_S = 2.6 +- 0.4 meV. The BCS-ratio 2\Delta_L/kTc^{local} = 5.0 - 5.9 > 3.52 (Tc^{local} is the contact area critical temperature) evidences for a strong electron-boson coupling. The results obtained agree well with our previous data by Andreev spectroscopy for single SnS-contacts.Comment: 6 pages, 6 figure

The Tunka Experiment: Towards a 1-km^2 Cherenkov EAS Array in the Tunka Valley

The project of an EAS Cherenkov array in the Tunka valley/Siberia with an area of about 1 km^2 is presented. The new array will have a ten times bigger area than the existing Tunka-25 array and will permit a detailed study of the cosmic ray energy spectrum and the mass composition in the energy range from 10^15 to 10^18 eV.Comment: 3 pages, 2 figures, to be published in IJMP

Latitudinal decline in stand biomass and productivity at the elevational treeline in the Ural mountains despite a common thermal growth limit

Aim: To quantify tree biomass and stand productivity of treeline ecotones and identify driving factors. Location: treeline ecotones of seven regions from the South to Polar Urals, spanning a latitudinal gradient of 1,500 km. Taxa: Picea obovata, Betula pubescens, Larix sibirica. Methods: Stand biomass and productivity were estimated across 18 elevational transects from the tree species line to the closed forest line based on allometric measurements of 326 trees (including roots for 53 trees), stand structure assessments and demographic patterns of 20,600 trees. Stand growth data were linked to (a) temperatures monitored in situ for five years in the South and Polar Urals, (b) climate variables extrapolated from nearby climate stations and (c) measures of nutrient availability in soils and tree foliage. Results: treeline position along the latitudinal gradient occurred at a similar mean growing season temperature. Despite the common cold limitation of tree distribution along the Ural mountain range, stand biomass and productivity within the treeline ecotone decreased by a factor of three and five from the South to the Polar Urals, mainly due to a declining stand density. Among climatic variables, growing season length decreased by 20% and winter temperatures declined by 4°C towards the Polar Urals, whereas growing degree days > 5°C remained similar, averaging 554 ± 9°C. Soil development was poorer in the Polar than in the South Urals, and plant-available N and P in the soil were 20 and 30 times lower, respectively, probably due to lower winter temperatures. Main conclusions: Our results suggest that once the thermal limitation for tree growth is relieved, soil fertility—restricted by permafrost and low soil temperatures during winter—plays a key and yet underexplored role for stand productivity in treeline ecotones. The observed latitudinal decline in stand productivity is important for above- and belowground diversity and functioning. © 2020 The Authors. Journal of Biogeography published by John Wiley & Sons Lt

Three-loop verification of a new algorithm for the calculation of a β\beta-function in supersymmetric theories regularized by higher derivatives for the case of N=1{\cal N}=1 SQED

We verify a recently proposed method for obtaining a $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories regularized by higher derivatives by an explicit calculation. According to this method, a $\beta$-function can be found by calculating specially modified vacuum supergraphs instead of a much larger number of the two-point superdiagrams. The result is produced in the form of a certain integral of double total derivatives with respect to the loop momenta. Here we compare the results obtained for the three-loop $\beta$-function of ${\cal N}=1$ SQED in the general $\xi$-gauge with the help of this method and with the help of the standard calculation. Their coincidence confirms the correctness of the new method and the general argumentation used for its derivation. Also we verify that in the considered approximation the NSVZ relation is valid for the renormalization group functions defined in terms of the bare coupling constant and for the ones defined in terms of the renormalized coupling constant in the HD+MSL scheme, both its sides being gauge-independent.Comment: 19 pages, 8 figures; title changed, minor corrections; the final version to appear in Nuclear Physics