SCATTERING AMPLITUDES FOR ZZ PRODUCTION AT THE LHC AND TOP-QUARK MASS EFFECTS

With the Large Hadron Collider providing experimental data with unprecedented precision, theoretical predictions must improve similarly to keep up.Among a plethora of processes being studied at the LHC, the production of a pair of vector bosons is of particular importance. Consequently, precise theoretical predictions for these processes are necessary. This thesis discusses primarily the calculation of ZZ production through gluon fusion at 2-loops with full top-quark mass dependence as well as the technological improvements required to successfully perform the calculation. Also discussed briefly is the quark initiated production of $\gamma\gamma + \text{jet}$ at 2-loops where some of these technologies allowed to overcome prior bottlenecks in the calculation of the helicity amplitudes.The 2-loop corrections for ZZ production through massless quarks had been known; in this work, the 2-loop corrections through the massive top quark are calculated .To achieve this, a new algorithm to systematically construct linear combinations of integrals with a convergent parametric integral representation is developed. This algorithm finds linear combinations of general integrals with numerators, dots, and dimension shifts as well as integrals from subsectors.To express the amplitudes in terms of these integrals, Integration-By-Parts (IBP) reduction is performed making use of syzygies and finite field based methods.A new algorithm is employed to construct these syzygies using linear algebra. The IBP reductions for $gg\rightarrow ZZ$ are successfully performed using these techniques. Further improvements, including predetermining the structure of the coefficients in IBP reductions, are used to successfully perform the reductions for $\gamma\gamma + jet$. Multivariate partial fractioning is used to simplify the final expressions to more manageable forms and render them suitable for fast numerical evaluation.%\thispagestyle{empty}In the case of $gg\rightarrow ZZ$, due to the presence of structures beyond polylogarithms, sector decomposition is employed to numerically evaluate the finite master integrals.Evaluating the amplitudes, agreement is found with previously calculated expansions specifically in the limit of large and small top mass. Improved results are presented for scattering at intermediate energies and/or for non-central scattering angles. With this calculation, the last building block required for the calculation of the full NLO cross-section for $gg\rightarrow ZZ$ is known.