Higher Order Calculations for Low Energy Precision Physics

This thesis concerns higher order calculations needed for precision physics in the low energy region of particle
physics. Of the four papers it contains, the first two introduce calculations at order p8 in the power counting
of chiral perturbation theory, which is an effective field theory of QCD at low energies. The remaining two
papers concern the hadronic contributions to the muon anomalous magnetic moment, or muon g − 2, which are
responsible for the main uncertainty in the theoretical prediction of the quantity.

Paper I. The pion mass and decay constant are calculated at order p8 within two-flavour chiral perturbation
theory. A small numerical study of the quark mass dependence is performed, and there is good agreement with
lower order results at the physical point.

Paper II. The order p8 mesonic chiral Lagrangian is derived for two, three as well as a general number of flavours.
This is done by explicitly creating all operators allowed by the relevant symmetries, and finding a minimal basis of
operators. Special cases where some of the external fields are set to zero are also considered.

Paper III. The finite volume effects from the next-to-leading order electromagnetic corrections to the hadronic
vacuum polarisation are here calculated in QEDL. This is needed for precision calculations of the muon g − 2
on the lattice. The analytic results are compared to lattice simulations as well as numerical lattice perturbation
theory. There is good agreement between the methods, and it is found that the electromagnetic corrections are
suppressed to such an extent that they for moderately sized lattices and pion masses in principle can be neglected
for the currently sought precision on the hadronic vacuum polarisation.

Paper IV. Short-distance constraints on the hadronic light-by-light contribution to the muon g − 2 are here
derived. Such constraints are useful for the matching of hadronic models valid at low energies to the high energy
region. In particular, the 4-point function entering into the hadronic light-by-light piece is calculated as a 3-point
function in the presence of an external electromagnetic field. We show that the quark loop is the first term in an
operator product expansion, and also consider the next term containing the condensate ⟨q σαβ q⟩ which is related
to the magnetic susceptibility of the QCD vacuum. This latter contribution is found to be negligible due to the
suppression in quark masses and sizes of the condensates.