Analytical Mellin-Barnes techniques with applications to two-loop SU(3) chiral perturbation theory and QED at higher loops
Abstract
The present era is one of precision in particle physics. To account for the lacunae in the otherwise
successful Standard Model, observables are calculated to high precision in various theoretical models,
which are then tested against experimental data to determine whether a given model is realised in
nature. In perturbative quantum eld theoretical models, higher order calculations require the
evaluation of multi-loop diagrams with multiple mass scales. Although an advanced technology has
been developed to evaluate these loop integrals, the majority of techniques are still numerical in
nature. In this thesis, we advance one technology that allows for the analytic evaluation of multi-loop
diagrams with several mass scales, the Mellin-Barnes (MB) technique, by studying and applying it
primarily in the context of three- avoured chiral perturbation theory (SU(3) ChPT). At two loop
order, the expressions for the pion, kaon and eta masses and decay constants depend on 'sunset'
diagrams, which appear with up to three independent masses, and the analytic evaluation of which
provides us the backdrop on which we develop our techniques.
The rst part of this work concerns itself with the development of the MB technology and its
application to the mathematics of sunset diagrams. We begin by developing an approach that allows
one to derive a minimal MB representation of a multi-loop multi-scale integral while retaining straight
line contours throughout the derivation process. After reducing the variety of vector and tensor
sunsets to a set of four scalar master integrals, this is then applied to evaluate all two mass scale
con gurations of the sunset, including (for completeness) those not arising in the ChPT context. The
same approach is used thereafter, with appropriate modi cations, to derive various MB representations
of the three mass scale integrals appearing in SU(3) ChPT. Each of these integrals is evaluated for all
accessible regions of convergence retaining their full dependence arising from dimensional
regularization, and in the ! 0 limit for the expressions that converge with physical meson mass
values. Formulae are also derived that allow one to expand these integrals to arbitrary order in .
The second part of this work focusses on physical applications of the aforementioned results in ChPT.
The sunset results are applied to obtain fully analytic expressions for m2
, m2
K, m2
, F , FK and F ,
which are subsequently truncated appropriately to obtain simpli ed representations that are
particularly suitable for tting with lattice QCD data. Such a preliminary lattice t is performed for the expression FK=F to extract values of the low energy constants (LEC) Lr
5, Cr
14 + Cr
15 and
Cr
15 + 2Cr
17. We also perform a numerical study of the meson masses and decay constants to examine
the relative contributions of their various components, and to investigate their dependence on the
values of the LEC. As another application of these analytic expressions, we nd an expansion of F
and m2
in the strange quark mass in the isospin limit, and perform the matching of the chiral SU(2)
and SU(3) low energy constants. A numerical study on this demonstrates the strong dependence of F
on the LEC in the chiral limit.
In the nal part of the thesis, we develop and demonstrate two methods of analytic continuation that
may be used to obtain results when values of the mass parameters do not allow for convergence of
Feynman integrals calculated using MB techniques. We apply the rst technique to the three mass
scale sunsets, and therefore obtain the full set of results for these integrals, i.e. we get solutions for the
sunsets for all possible values of the mass parameters. The same technique is then applied to
analytically continue the results of the most general four mass scale sunset integral to obtain results
which converge for physical values of the meson masses. We apply the second method of analytic
continuation in a non-ChPT context to demonstrate the general applicability of the methods
developed in this work. We rst calculate the complete result of a class of three-loop QED vacuum
polarisation contributions arising from non-diagonal avour charged leptons to the g 2 of each
charged lepton, and then show how one may obtain the expression for the case with an external muon
or tau leg from the results of the case of external electron leg by means of analytic continuation.