Academic literature on the topic 'Three-loop QED vacuum polarisation'

It is important to correct for finite-volume (FV) effects in the presence of QED, since these effects are typically large due to the long range of the electromagnetic interaction. We recently made the first lattice calculation of electromagnetic corrections to the hadronic vacuum polarisation (HVP). For the HVP, an analytical derivation of FV corrections involves a two-loop calculation which has not yet been carried out. We instead calculate the universal FV corrections numerically, using lattice scalar QED as an effective theory. We show that this method gives agreement with known analytical results for scalar mass FV effects, before applying it to calculate FV corrections for the HVP. This method for numerical calculation of FV effects is also widely applicable to quantities beyond the HVP.

We evaluate the one-loop vacuum polarization tensor for three-dimensional quantum electrodynamics (QED), using an analytic regularization technique, implemented in a gauge-invariant way. We show thus that a gauge boson mass is generated at this level of radiative correction to the photon propagator. We also point out in our conclusions that the generalization for the non Abelian case is straightforward.

In this paper we further develop our matrix element and complex angular momentum summation techniques, in order to calculate both the one-loop free field and two-loop interacting vacuum diagrams in field theory on a four-sphere. In the case of the free field diagrams, we show how the sums may be evaluated by integrating over an analytic function with both poles and branch cuts where the discontinuity across the cuts determines the result. We then extend the matrix element formalism to multiple angular momenta involving the addition of angular momenta and the associated Clebsch-Gordon type selection rules. This then allows us to evaluate the matrix elements of two-loop diagrams in spherical QED as a function of the three angular momenta in the diagram. The selection rules allow us to cast the triple angular momentum sum into a form which enables evaluation again by contour integration. The result is obtained in analytic form using dimensional regularization for the previously obtained spinor case, and also for scalar QED, which we believe is a new result. Finally, we discuss the applicability of this method for calculations in non-Abelian field theory which we believe cannot be performed using earlier methods.

We review in a pedagogical way the method of differential equations for the evaluation of D-dimensionally regulated Feynman integrals. After dealing with the general features of the technique, we discuss its application in the context of one- and two-loop corrections to the photon propagator in QED, by computing the Vacuum Polarization tensor exactly in D. Finally, we treat two cases of less trivial differential equations, respectively associated to a two-loop three-point, and a four-loop two-point integral. These two examples are the playgrounds for showing more technical aspects about: Laurent expansion of the differential equations in D (around D = 4); the choice of the boundary conditions; and the link among differential and difference equations for Feynman integrals.

Abstract This addendum provides results complementary to those obtained in [J. Phys. G 49, 055001 (2022)]. Specifically, an equivalent form of the relation, which binds together the ‘spacelike’ kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment a μ HVP , is obtained. It is shown that the infrared limiting value of the ‘spacelike’ and ‘timelike’ kernel functions, which enter the representations for a μ HVP involving the Adler function and the R-ratio, is identical to the corresponding QED contribution to the muon anomalous magnetic moment of the preceding order in the electromagnetic coupling. The next-to-leading order contributions a μ HVP ( 3 b ) (which includes the leptonic and hadronic insertions) and a μ HVP ( 3 c ) (which includes the double hadronic insertion), are studied. The three kernel functions appearing in the representations for a μ HVP ( 3 b ) , which involve the hadronic vacuum polarization function, Adler function, and the R-ratio, are presented for the cases of the electron and τ-lepton loop insertions.

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.