Modern Methods of QCD Tomography for Electron-Ion Collder Phenomenology

Hadrons are the bound states of QCD and are a fundamental building block of nature. Through dedicated experimental and theoretical endeavors over the past century, the field of nuclear physics has begun to understand the three-dimensional structure of hadrons in momentum space, which is encoded in Transverse Momentum Dependent PDFs (TMD PDFs). To perform full three dimensional imaging within hadrons, one must note that TMD PDFs contain a mixture of perturbative and non-perturbative contributions. To de-couple these two contributions, theorists rely on factorization theorems, which allow us to write QCD cross sections as a convolution of a perturbative and non-perturbative contributions. Factorization theorems have long since been understood at leading power (LP) for Semi-Inclusive DIS, Drell-Yan, and double inclusive leptonic annihilation. With the use of these formalisms, in the past decade the imaging of the unpolarized quark TMD PDF has moved into an era of precision, where the perturbative contributions are close to N$^4$LL accuracy and non-perturbative extractions have been performed at the accuracy of NNLO+N$^3$LL. Despite this progress, many questions remain to be addressed in the field of TMDs. How does the introduction of spin affect the transverse dynamics of the quarks? What processes are optimal perform imaging of the gluons and how do we establish factorization theorems for these processes? How do we establish factorization theorems beyond LP? The interest in understanding the underlying structure of hadrons has led to the development of the Electron-Ion Collider (EIC), a future facility which will allow us to measure the internal structure of protons at never before seen precision. Due to the high-luminosity, high center of mass energy, and precision beam control, this new facility opens the possibility of measuring spin-dependent TMDs in a wide range of new processes. Furthermore, the high luminosity open the possibility of performing high precision measurements of angular correlations which will allow us to probe the next-to-leading power (NLP) structure of hadrons. To measure the spin-dependent structure of hadrons, experimental measurements are performed for azimuthal spin asymmetries. In this dissertation, I quantify the extent to which we understand the quark Sivers function, a spin-dependent TMD PDF, by performing the first ever global analysis of this function from Semi-Inclusive DIS and Drell-Yan. To allow for a future global analysis of the gluon Sivers asymmetry, I establish a factorization and resummation formalism for heavy-flavor di-jet production at the future EIC using Soft-Collinear Effective Theory (SCET). I then establish a factorization and resummation formalism for the Sivers asymmetry in di-jet production at RHIC and discuss how such a process can be used to test factorization breaking effects and the extent to which theorist understand the universality arguments of the Sivers function. Additionally, while spin asymmetries can be generated by the spin of initial-state particles, they also arise due to hadronization effect. However, probing the spin-dependent final-state hadrons introduces additional experimental complications associated with reconstructing the spin of the hadron. In the past several years, $\Lambda$ baryons, which undergo self-analyzing decay, have been explored as a method of probing spin-dependent fragmentation functions. In this dissertation, I perform one of the first global analyses of the TMD Polarizing Fragmentation Function (TMD PFF) and the first global analysis of the transversity TMD FF. Using SCET, I establish a factorization and resummation formalism for the distribution of hadrons in a jet and use this formalism to make predictions at the EIC. Lastly, I perform direct calculations of the evolution equations of kinematic and intrinsic sub-leading twist distribution functions for the first time