In this work we have suggested a solution of the Gribov-Levin-Ryskin,Mueller-Qiu

In this work we have solved the nonlinear GLR-MQ evolution equation upto next-to-leading order (NLO) by considering NLO terms of the gluon-gluon splitting functions and running coupling constant α s (Q 2 ). Here, we have incorporated a Regge-like behaviour of gluon distribution in order to obtain a solution of the GLR-MQ equation in the range of 5GeV 2 ≤ Q 2 ≤ 25GeV 2 . We have studied the Q 2 evolution of the gluon distribution function G(x, Q 2 ) and its nonlinear effects at small-x. It can be observed from our analysis that the nonlinearities increase with decrease in the correlation radius R of two interacting gluons, as expected. We have compared our result of G(x, Q 2 ) as Q 2 increases and x decreases, for two different values of R, viz. R= 2 GeV −1 and 5 GeV −1 . We have also checked the sensitivity of the Regge intercept λ G on our results. We compare our computed results with those obtained by the global analysis to parton distribution functions (PDFs) by various collaborations where LHC data have been included viz. PDF4LHC15, NNPDF3.0, ABM12 and CT14. Besides we have also shown comparison of our results with HERA PDF data viz. HERAPDF15.

We present a study of the contribution of antishadowing effects on the gluon distribution functions G(x, Q 2 ) in light of the Gribov-Levin-Ryskin-Mueller-Qiu, Zhu-Ruan-Shen (GLR-MQ-ZRS) nonlinear equation at small-x, where x is the momentum fraction or Bjorken variable and Q 2 is the four momentum transfer squared or photon virtuality. In this work, we have solved the GLR-MQ-ZRS nonlinear equation using Regge like behaviour of gluons in the kinematic range of 10 −2 ≤ x ≤ 10 −6 and 5 GeV 2 ≤ Q 2 ≤ 100 GeV 2 respectively. We have obtained the solution of G(x, Q 2 ) by considering two particular cases: (a) α s fixed; and (b) the leading order QCD dependency of α s on Q 2 . A comparative analysis is also performed where we compare the gluon distribution function due to inclusion of the antishadowing effect with that of the gluon distribution without including the antishadowing effect. Our obtained results of G(x, Q 2 ) are compared with NNPDF3.0, CT14 and PDF4LHC. We also compare our results with the result obtained from the IMParton C++ package. Using the solutions of G(x, Q 2 ), we have also predicted x and Q 2 evolution of the logarithmic derivative of proton's F 2 structure function i.e. dF 2 (x, Q 2 )/d ln Q 2 .We incorporated both the leading order(LO) and next-to-leading order (NLO) QCD contributions of the gluon-quark splitting kernels, in dF 2 (x, Q 2 )/d ln Q 2 . Our result of dF 2 (x, Q 2 )/d ln Q 2 agrees reasonably well with the experimental data recorded by HERA's H1 detector. * mlalung@tezu.ernet.in † pragyanp@tezu.ernet.in ‡ jks@tezu.ernet.in

We suggest a modified form of a unitarized BFKL equation imposing the so-called kinematic constraint on the gluon evolution in multi-Regge kinematics. The underlying nonlinear effects on the gluon evolution are investigated by solving the unitarized BFKL equation analytically. We obtain an equation of the critical boundary between dilute and dense partonic system, following a new differential geometric approach and sketch a phenomenological insight on geometrical scaling. Later we illustrate the phenomenological implication of our solution for unintegrated gluon distribution f (x, k 2 T ) towards exploring high precision HERA DIS data by theoretical prediction of proton structure functions (F 2 and F L ) as well as double differential reduced cross section (σ r ). The validity of our theory in the low Q 2 transition region is established by studying virtual photon-proton cross section in light of HERA data.

We present a phenomenological study of the small-x behaviour of gluon distribution function G(x, Q 2 ) at next-to-leading order (NLO) and next-to-next-to-leading order(NNLO) in light of the nonlinear Gribov-Ryskin-Levin-Mueller-Qiu (GLR-MQ)evolution equation by keeping the transverse size of the gluons (∼ 1/Q) fixed. We consider the NLO and NNLO corrections, of the gluon-gluon splitting function P gg (z) and strong coupling constant α s (Q 2 ). We have suggested semi-analytical solutions based on Regge like ansatz of gluon density G(x, Q 2 ), which are supposed to be valid in the moderate range of photon virtuality(Q 2 ) and at small Bjorken variable(x). The study of the effects of nonlinearities that arise due to gluon recombination effects at small-x is very interesting, which eventually tames down the unusual growth of gluon densities towards small-x as predicted by the linear DGLAP evolution equation. * mlalung@tezu.ernet.in † pragyanp@tezu.ernet.in ‡ jks@tezu.ernet.in