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We study the perturbative structure of threshold enhanced logarithms in the coefficient functions of deep inelastic scattering (DIS) and semi-inclusive e+e− annihilation (SIA) processes and setup a framework to sum them up to all orders in perturbation theory. Threshold logarithms show up as the distributions ((1−z)−1 logi(1−z))+ from the soft plus virtual (SV) and as logarithms logi(1−z) from next to SV (NSV) contributions. We use the Sudakov differential and the renormalisation group equations along with the factorisation properties of parton level cross sections to obtain the resummed result which predicts SV as well as next to SV contributions to all orders in strong coupling constant. In Mellin N space, we resum the large logarithms of the form logi(N) keeping 1/N corrections. In particular, the towers of logarithms, each of the form $$ {a}_s^n/{N}^{\alpha }{\log}^{2n-\alpha }(N),{a}_s^n/{N}^{\alpha }{\log}^{2n-1-\alpha }(N)\cdots $$ a s n / N α log 2 n − α N , a s n / N α log 2 n − 1 − α N ⋯ etc for α = 0, 1, are summed to all orders in as.

We present the resummed predictions for inclusive cross section for Drell–Yan (DY) production up to next-to-next-to leading logarithmic ($${{\overline{\mathrm{NNLL}}}}$$ NNLL ¯ ) accuracy taking into account both soft virtual (SV) and next-to SV (NSV) threshold logarithms. We restrict ourselves to resummed contributions only from quark anti-quark ($$q {{\bar{q}}}$$ q q ¯ ) initiated channels. The resummation is performed in Mellin-N-space. We derive the N-dependent coefficients and the N-independent constants to desired accuracy for our study. The resummed results are matched through the minimal prescription procedure with the fixed-order results. We find that the resummation, taking into account the NSV terms, appreciably increases the cross section while decreasing the sensitivity to renormalisation scale. We observe that, at 13 TeV LHC energies, the SV + NSV resummation at $${\overline{\mathrm{NLL}}} ({{\overline{\mathrm{NNLL}}}})$$ NLL ¯ ( NNLL ¯ ) gives about 8% (2%) corrections respectively to the NLO (NNLO) results for the considered Q range: 150–3500 GeV. In addition, the absence of quark gluon initiated contributions to NSV part in the resummed terms leaves large factorisation scale dependence indicating their importance at NSV level. We also study the numerical impact of N-independent constants and explore the ambiguity involved in exponentiating them. Finally we present our predictions for the neutral Drell–Yan process at various center of mass of energies.

Melt-spun ErNi crystallizes in orthorhombic FeB-type structure (Space group Pnma, no. 62) similar to the arc-melted ErNi compound. Room temperature X-ray diffraction (XRD) experiments reveal the presence of texture and preferred crystal orientation in the melt-spun ErNi. The XRD data obtained from the free surface of the melt-spun ErNi show large intensity enhancement for (1 0 2) Bragg reflection. The scanning electron microscopy image of the free surface depicts a granular microstructure with grains of ∼1 μm size. The arc-melted and the melt-spun ErNi compounds order ferromagnetically at 11 K and 10 K (TC) respectively. Field dependent magnetization (M-H) at 2 K shows saturation behaviour and the saturation magnetization value is 7.2 μB/f.u. for the arc-melted ErNi and 7.4 μB/f.u. for the melt-spun ErNi. The isothermal magnetic entropy change (ΔSm) close to TC has been calculated from the M-H data. The maximum isothermal magnetic entropy change, -ΔSmmax, is ∼27 Jkg-1K-1 and ∼24 Jkg-1K-1 for the arc-melted and melt-spun ErNi for 50 kOe field change, near TC. The corresponding relative cooling power values are ∼440 J/kg and ∼432 J/kg respectively. Although a part of ΔSm is lost to crystalline electric field (CEF) effects, the magnetocaloric effect is substantially large at 10 K, thus rendering melt-spun ErNi to be useful in low temperature magnetic refrigeration applications such as helium gas liquefaction.