In this dissertation, we study the small x resummation in perturbative Quantum Chromodynamics (QCD). In the past decades, the particle accelerators with increasing center of mass energies have opened up the so-called small Bjorken x regime. We are interested in the small x evolution of gluon density, which is described by the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation. Currently, the BFKL kernel eigenvalue is known up to the next-to-leading logarithm (NLL) accuracy. However, it leads to instabilities in calculations and indicates the necessity of resummation of higher order terms. We present the BFKL formalism, parton evolution, and QCD in general in the introductory Chapter 1 of this thesis.

In Chapter 2, we introduce the kinematical constraints and renormalization group improved resummation. We analyze the different types of kinematical constraints in the BFKL equation and show their impact on the BFKL effective eigenfunction and gluon density by numerical calculations. Furthermore, we describe the scale changing transformation in the renormalization group improved resummation and the corresponding expansion technique. We demonstrate the pole structure of the BFKL equation with kinematical constraints in Mellin space in BFKL up to NLL and in N = 4 supersymmetric Yang-Mills theory (sYM) up to next-to-next-to leading logarithmic level (NNLL). We investigate the scale changing transformations and give proof of the vanishing sub-leading poles in N=4 sYM to all orders.

In Chapter 3, we present a more detailed description of the renormalization group improved resummation. We perform the numerical calculation of the structure function F2 in Deep Inelastic Scattering (DIS) and subsequently a fit to the data from HERA collider.  We achieve a very good description of the structure function F2 and its charm component F2c simultaneously. The resulting unintegrated gluon density is consistent with the calculations based on similar approaches available in the literature.

In Chapter 4, we perform the renormalization group improved resummation of the photon-gluon impact factors.  We construct the resummed cross section for virtual photon-photon scattering which incorporates the resumed impact factors and BFKL gluon Green’s function up to the NLL. Conditions on the resummed cross section are constructed by requiring consistency with standard high energy factorization in the collinear limits. Our result is consistent with previous impact factor calculations at the next-to-leading order (NLO), apart from a new term proportional to CF for the longitudinal photon polarization. We compute the resummed cross section and compare it with the LEP data and previous calculations. Our result is lower than the LL approximation but higher than the pure NLL one, being more consistent with the experimental data.

Besides, one of my committee member, Prof. Strikman, needs to attend my defense remotely due to some conditions. I will discuss if it is convenient to set zoom to both my committee member and other people at the same time.