We study the time-momentum representation of the kernel needed to compute the hadronic vacuum polarization contribution to the muon g-2 in the space-like region at next-to-leading order. For small values of the time, we present analytical series expansions; for large values of the time, we present numerical series expansions which overcome the problems showed by naïve asymptotic expansions. These results are to be employed in lattice QCD determination of the hadronic vacuum polarization contribution to the muon g-2 at next-to-leading order.

Italy

Elisa Balzani

Stefano Laporta

Massimo Passera

Time-kernel for lattice determinations of NLO hadronic vacuum polarization contributions to the muon -2

We present simple analytic expressions to compute the hadronic vacuum polarization contribution to the muon <italic>g</italic>-2 in the space-like region up to next-to-next-to-leading order. These results can be employed in lattice QCD calculations of this contribution as well as in space-like determinations based on scattering data, like that expected from the proposed MUonE experiment at CERN.

Hadronic vacuum polarization contributions to the muon <italic>g</italic>-2 in the space-like region

We present a complete reevaluation of the irreducible two-loop vacuum-polarization correction to the photon propagator in quantum electrodynamics, i.e. with an electron-positron pair in the fermion propagators. The integration is carried out by reducing the integrations to a limited set of master integrals, which are calculated using integration-by-parts identities. Dimensional regularization is used in <math><mrow><mi>D</mi><mo>=</mo><mn>4</mn><mo>−</mo><mn>2</mn><mi>ε</mi></mrow></math> dimensions, and on-mass shell renormalization is employed. The one-loop effect is given to order <math><mi>ε</mi></math>, to be combined with the <math><mn>1</mn><mo>/</mo><mi>ε</mi></math> divergence of the two-loop amplitude. Master integrals are given. Final evaluations of two-loop energy shifts for <math><mn>1</mn><mi>S</mi></math>, <math><mn>2</mn><mi>S</mi></math>, and <math><mn>2</mn><mi>P</mi></math> states are done analytically, and results are presented, with an emphasis on muonic hydrogen. For relativistic Dirac-Coulomb reference states, higher-order coefficients are obtained for the <math><mi>Z</mi><mi>α</mi></math>-expansion. We compare the results obtained to the existing literature.

United States

Ulrich D. Jentschura

Dimensional regularization and two-loop vacuum polarization operator: Master integrals, analytic results, and energy shifts

We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

Hjalte Frellesvig

Federico Gasparotto

Manoj Mandal

Pierpaolo Mastrolia

Luca Mattiazzi

Sebastian Mizera

Decomposition of Feynman integrals by multivariate intersection numbers

The 4-loop QED mass and wave function renormalization constants <math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></math> have been evaluated in the on-shell subtraction scheme with 1100 digits of precision. We also worked out the coefficients of the five color structures of the QCD renormalization constants <math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>OS</mi></mrow></msubsup></math> and <math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>OS</mi></mrow></msubsup></math> which can be obtained from QED-like diagrams. The results agree with lower precision results available in the literature. Analytical fits were also obtained for all these quantities.

High-precision four-loop mass and wave function renormalization in QED

We have evaluated with 1100 digits of precision the contribution of all the 891 mass-independent 4-loop Feynman diagrams contributing to the slope of the Dirac form factor in QED. The total 4-loop contribution is<math><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><msubsup><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow><mrow><msup><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mrow><mo>′</mo></mrow></msup></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0.886545673946443145836821730610315359390424032660064745</mn><mo>…</mo><msup><mrow><mo>(</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mi>π</mi></mrow></mfrac><mo>)</mo></mrow><mrow><mn>4</mn></mrow></msup><mspace width="0.25em"></mspace><mo>.</mo></math> We have fit a semi-analytical expression to the numerical value. The expression contains harmonic polylogarithms of argument <math><msup><mrow><mi>e</mi></mrow><mrow><mfrac><mrow><mi>i</mi><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></math>, <math><msup><mrow><mi>e</mi></mrow><mrow><mfrac><mrow><mn>2</mn><mi>i</mi><mi>π</mi></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></msup></math>, <math><msup><mrow><mi>e</mi></mrow><mrow><mfrac><mrow><mi>i</mi><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math>, one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits. We show the correction on the shift of the energy levels of the hydrogen atom due to the slope.

High-precision calculation of the 4-loop QED contribution to the slope of the Dirac form factor

We complete the analytic evaluation of the master integrals for the two-loop non-planar box diagrams contributing to the top-pair production in the quark-initiated channel, at next-to-next-to-leading order in QCD. The integrals are determined from their differential equations, which are cast into a canonical form using the Magnus exponential. The analytic expressions of the Laurent series coefficients of the integrals are expressed as combinations of generalized polylogarithms, which we validate with several numerical checks. We discuss the analytic continuation of the planar and the non-planar master integrals, which contribute to q q → tt in QCD, as well as to the companion QED scattering processes ee → μμ and eμ → eμ.

Stefano Vita

Switzerland

Thomas Gehrmann

Amedeo Primo

Ulrich Schubert

Master integrals for the NNLO virtual corrections to   <math> <mi>q</mi> <mover> <mi>q</mi> <mo>¯</mo> </mover> <mo>→</mo> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ q\overline{q}\to t\overline{t} $$  scattering in QCD: the non-planar graphs

We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss 2 F 1 hypergeometric function, and the Appell F 1 function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to n-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.

Canada

Decomposition of Feynman integrals on the maximal cut by intersection numbers

We evaluate the master integrals for the two-loop non-planar box-diagrams contributing to the elastic scattering of muons and electrons at next-to-next-to-leading order in QED. We adopt the method of differential equations and the Magnus exponential to determine a canonical set of integrals, finally expressed as a Taylor series around four space-time dimensions, with coefficients written as combination of generalised polylogarithms. The electron is treated as massless, while we retain full dependence on the muon mass. The considered integrals are also relevant for crossing-related processes, such as di-muon production at e + e − colliders, as well as for the QCD corrections to top-pair production at hadron colliders. In particular our results, together with the planar master integrals recently computed, represent the complete set of functions needed for the evaluation of the photonic two-loop virtual next-to-next-to-leading order QED corrections to μe → μe and e + e − → μ + μ −.

Master integrals for the NNLO virtual corrections to μe scattering in QED: the non-planar graphs

I have evaluated up to 1100 digits of precision the contribution of the 891 4-loop Feynman diagrams contributing to the electron g -2 in QED. The total mass-independent 4-loop contribution is ae=−1.912245764926445574152647167439830054060873390658725345…(απ)4. I have fit a semi-analytical expression to the numerical value. The expression contains harmonic polylogarithms of argument eiπ3 , e2iπ3 , eiπ2 , one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits.