Calculation Of Feynman Integrals And Its Applications To Particle Physics Phenomenology

The Standard Model of particle physics,the most profound human knowledge about nature,unifies the three types of interactions other than gravity,i.e.,strong interaction,electromagnetic interaction,and weak interaction,and very accurately describes the laws of operation of the microscopic world.However,the issues like neutrino mass,dark matter,and dark energy also indicate the imperfection of the Standard Model,and the search for new physics is a common topic of interest.The precision test of the Standard Model has become one of the means to discover signals of new physics,which requires both accurate experimental measurements and reliable theoretical predictions.Experimental technology is fast evolving,and with the upcoming upgrade of the Large Hadron Collider and the appearance of new colliders,human exploration of the microscopic world will go one step further,mounting great challenges on theoretical calculations.The Feynman diagram provides a systematic way for perturbatively calculating the cross section and decay width.Accurate theoretical predictions always imply the calculation of higher-order Feynman diagrams,of which an extremely important part is the handling of multi-loop Feynman integrals,the central object of study in this thesis,which is often where the computational bottleneck lies.The tensor Feynman integrals in the scattering amplitude are represented by tensor decomposition as a combination of scalar Feynman integrals.In a typical 2 to 2 scattering process,the number of scalar Feynman integrals is in the hundreds,and it is obviously not desirable to calculate them one by one,not to mention that the calculation of Feynman integrals is not an easy task in the first place.Fortunately,the Feynman integrals in a family defined by a set of propagator topology are not independent of each other and are related by the IBP relations.For a given family,there exists a finite number of master integrals,and all the integrals in this family can be linearly expressed by the master integrals,so the calculation of the master integrals becomes the key to higher-order perturbative studies.The difficulty of calculating master integrals is reflected in different dimensions:multiple loops,multiple legs,multiple scales and complexity of the structure.In the study of higher-order corrections to one process,the relevant master integrals may have all these characteristics at the same time,making the process of obtaining physical results struggling.Therefore,efficient methods for calculating master integrals are particularly important.Direct calculation using only representations of Feynman integral(e.g.,Feynman parametric representation,Mellin-Barnes representation,etc.)is not competent in the vast majority of cases,while the differential equation method based on the IBP relations between Feynman integrals makes the calculation of complex Feynman integrals possible.There are two means for solving the differential equations satisfied by master integrals,numerical and analytic approach.The Sector Decomposition method,although in principle applicable to the calculation of arbitrary Feynman integrals,can not provide sufficiently accurate results for the master integrals in physical region to meet practical needs.However,in the Euclidean region,the numerical results obtained by Sector Decomposition are accurate enough to be used as initial values to obtain more accurate results in the physical region using the evolution of the differential equations.In this thesis,we apply the numerical differential equation method to the phenomenological studies of heavy neutral boson decaying into Higgs plus photon in the littlest Higgs model,and successfully overcome the problems of low accuracy and slow convergence of Sector Decomposition method in the physical region,solving the biggest technical problem in this work.Our study shows that the two-loop QCD correction effect of this process is obvious and should be included in the accurate prediction of the decay width.Whether from the practical point of view of phenomenological study,from the theoretical point of view of analyzing the nature of scattering amplitudes,or even from the understanding of the Feynman integrals themselves,it is very important to obtain the analytic expressions of Feynman integrals.Generally,the dependence of the system of differential equations on kinematic variables and spacetime dimensions is entangled,making it difficult to solve and thus limiting their application.When the kinematic information is completely decoupled from the spacetime dimensions and the differential equations become in canonical form,the system of equations is greatly simplified and their solutions become very easy to implement.When no square root exists in a system of differential equations,or when all square roots introduced by the construction of canonical differential equations can be rationalized simultaneously,the solutions can be expressed as a special class of iterated integrals,i.e.,multiple polylogarithms.It turns out that the vast majority of Feynman integrals can be expressed as this class of special functions which are widely used in the field of precision calculations thanks to their good mathematical properties.Based on the canonical differential equation method,the Feynman integrals involved in the two-loop corrections to two important processes at the LHC are calculated analytically in this thesis.One is the two-loop QCD corrections to the associated production of the top quark and the W boson,and the other is the two-loop mixed QCD-QED corrections to the charged-current Drell-Yan process.Top quark associated production with W boson is one of the mechanisms of single-top quark production at the LHC and is important for studying the electroweak properties of top quarks.One of the important reasons for the absence of its complete two-loop QCD corrections is that the related Feynman integrals are too complicated,rendering it challenging to calculate analytically.We focus on one type of planar Feynman diagrams and obtain two families and construct canonical differential equations for all master integrals,which involve at most three square roots.They are rationalized simultaneously with suitable transformations.And by reduction of the canonical differential equations,we successfully optimize the final result,which reduces the size of the expressions by 74%and greatly improves their application value in the study of phenomenology.Our results provide a strong basis for achieving the goal of complete two-loop QCD corrections to tW production.The Drell-Yan process is the standard candle process at the LHC,and the importance of its exact theoretical predictions cannot be overstated.In previous studies of mixed QCD-EW corrections to Drell-Yan processes,lepton mass is often neglected for computational simplicity,and the mass singularity terms caused by it may have some impacts on the predictions of observables.In order to understand the contribution of these mass singularities,the dependence of lepton mass must be retained.However,the addition of a new scale undoubtedly increases the computational difficulty of the master integrals.With non-zero lepton mass,we successfully calculate the relevant master integrals analytically,and obtain an expansion of the master integrals with respect to the ratio of lepton mass to W boson mass due to the huge gap between them.The singular terms of the lepton mass are clearly included in the final results,and in this way their influence on the observables in phenomenology studies can be under control.Our results are an integral part of the accurate theoretical predictions of the charged-current Drell-Yan process.High-energy physics has entered the era of precision studies,and reliable theoretical predictions are required to meet the requirements of comparison with increasingly precise experimental measurements,in which Feynman integrals play a pivotal role,and a deeper understanding of Feynman integrals will certainly promote another flourishing development of high-energy physics.