Schematic Project of PhD Thesis: Two-Loop QCD Corrections with the Differential Equations Method

Transcription

1 Schematic Project of PhD Thesis: Two-Loop QCD Corrections with the Differential Equations Method Matteo Becchetti Supervisor Roberto Bonciani University of Rome La Sapienza 24/01/2017

2 1 The subject of this PhD thesis is the study of the analytical structure of higher-order perturbative corrections in QCD and their application to observables relevant for the Phenomenology at present colliders. The work is divided in two parts: the study of the methods of resolution of the system of differential equations associated to a generic loop correction in QCD; the application of these methods to the computation of the two-loop QCD corrections for the process q q γγ(jets) with massive top loops, and the study of two-loop QCD corrections for the process gg t t, always with massive top loops. The calculations of higher-order perturbative corrections in perturbative Quantum Field Theory can be divided into two steps: i) the Feynman amplitudes can be written as a sum of all tensorial structures compatible with the process and the Lorentz invariance, times combinations of scalar integrals that are finite in number, the so called master integrals; ii) the computation of such master integrals. Our work focuses on this second step. It is known that the master integrals form a minimal finite basis {f 1 ( x),..., f n ( x)} and that all the scalar integrals in the process can be written has a linear combination of them using Integration-by-Parts Identities (IBP) [2]. Differentiating the master integrals with respect to the kinematical invariants of the process { x}, it is possible to construct a linear system of first order coupled PDE [1] d f( x, ɛ) = A( x, ɛ) f( x, ɛ), (1) where d is the total differential operator, A( x, ɛ) is the matrix relative to the partial differential equations. The matrix A( x, ɛ) depends on rational functions of the kinematical invariants { x} and on the regularization parameters ɛ. There are various approaches to the resolution of the system (1), in this work of thesis we choose to focus on the so called canonical basis approach. It was argued in [3] that it is possible to choose a particular basis of master integrals in such a way that the system of PDE for the master integrals has the canonical form d f( x, ɛ) = ɛ dã( x) f( x, ɛ). (2) In this form the system has a manifest solution in terms of iterated Chen integrals [4] [ ] f( x, ɛ) = P exp ɛ dã f 0 (3) γ where P stands for path ordered integration along the contour γ, and f 0 is a vector of initial conditions.

3 2 In (2) and (3) we refer to the matrix relative to the system of PDE as dã; indeed, in canonical basis this matrix has the form of a logarithmic differential one-form, with à = k A k ln α k ( x) (4) where A k are rational numbers and α k ( x) are functions of the kinematical invariants only. As it can be seen from (3)(4), in canonical basis it is clear that the system admits a solution in terms of a special class of functions, the multiple polylogarithms [6, 7] G(a 1,..., a n ; z), which are defined as G(a 1,..., a n ; z) = G(a 1 ; z) = z 0 z 0 dt G(a 2,..., a n ; t), (5) t a 1 dt a 1 0. (6) t a 1 If (a 1,, a n ) = 0 we have G( 0, z) = 1 n! lnn z. The number of a i in the polylogarithm is called the weight of the function. Exploiting this last concept, from (3) it can be seen that every term in the ɛ expansion of the solution has a fixed weight. Therefore, the solution of the system (2) can be found in terms of an expansion in ɛ whose coefficients are expressed in terms of multiple polylogarithms. This is a suitable property of the solution both from a theoretical point of view (having as solution a class of function with useful algebraic properties) and from a pratical point of view (the multiple polylogarithms can be easily evaluated numerically). In this context our work concerns mainly the following three open issues: The transformations that brings a generic system into canonical form (2). Techincal issues connected with the kind of functional space to which the functions α k ( x) belong. Elliptic solutions: the property of some processes to have master integrals that admits a solution in terms of elliptic integrals [11] instead of multiple polylogarithms. Regarding the first point, at the moment do not really exist algorithms or effective criteria to find such a transformations. Therefore a systematic study of how actually getting into the canonical basis is of great interest. Our approach is to impose the largest number of constraints, as possible, on the integral basis choise that can be deduced from generic properties of solution (3). Two of the basic constraints are: adimensionality of the integrals in units of energy, a well defined finite behavior with respect to ɛ expansion. Anyhow, the strongest constraint that can be imposed on the integral basis is on the leading singularities of the master integrals [5]. The master integrals are scalar

4 3 integrals whose integrand is basically a product of propagators in momentum space representation. The leading singularities can be calculated replacing such propagators with appropriate Dirac delta functions and performing the integral [5, 8]. The statement is that the master integrals of the canonical basis have constant leading singularities [3]. Therefore, the computation of the latter quantity gives a strong test for the choise of the integral basis. However, there is no a priori connection between a master integral and its leading singularities that can guide our choice. My work aims at the establishment of this connection. As already mentioned, there are also issues connected to the class of function to which α k ( x) belong. Indeed, if they are rational functions of the kinematical invariants there are no real issues and the solution can be directly written in terms of polylogarithms. However, if α k ( x) are algebraic functions (i.e. functions that involve roots of the kinematical invariants) the integration in terms of multiple polylogarithms cannot be done directly. In this situation there are two possibilities: search for a change of variables that linearizes all the roots[12], otherwise integrate the matrix dã using the concept of symbol [10]. Finally, we mention the issue connected with the elliptic solutions of master integrals. Recently, processes have been found with master integrals that do not admit a solution in terms of multiple polylogarithm but, instead, in terms of elliptic integrals [11]. There are many points of interest connected with this issue: Is it possible to determine a priori which process involves master integrals with elliptic solutions? Which is the better approach to solve these master integrals? Does exist a class of functions that generalizes the multiple polylogarithm to include also the elliptic case? Besides the theoretical aspects just mentioned, we plan to compute the two-loops QCD corrections to the process q q γγ (or into jets), with massive top loop. For this process there are three relevant topologies to be solved, two of them planar and one nonplanar. We choose to solve the topologies with the canonical basis approach, however the non-planar one have master integrals with elliptic solutions, as a consequence the canonical basis cannot be find the whole set of master integrals. As second process under investigation in this work, we study the two-loop non-planar QCD corrections to the gluon fusion production of t t with a complete dependence on the top mass.

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