Radiative corrections to τ → ππντ

A bstract Hadronic τ decays present an opportunity to determine the isovector part of the hadronic-vacuum-polarization contribution to the anomalous magnetic moment of the muon in a way complementary to e + e − → hadrons cross sections. However, the required isospin rotation is only exact in the isospin limit, and corrections need to be under control to draw robust conclusions, most notably for τ → ππν τ decays to determine the two-pion contribution, $$ {a}_{\mu}^{\textrm{HVP},\textrm{LO}}\left[\pi \pi, \tau \right] $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mi>HVP</mml:mi> <mml:mo>,</mml:mo> <mml:mi>LO</mml:mi> </mml:mrow> </mml:msubsup> <mml:mfenced> <mml:mi>ππ</mml:mi> <mml:mi>τ</mml:mi> </mml:mfenced> </mml:math> . In this work, we present a novel analysis of the required radiative corrections using dispersion relations, thereby extending in a model-independent way the previous analysis in chiral perturbation theory (ChPT) beyond the threshold region. In particular, we include the dominant structure-dependent virtual corrections from pion-pole diagrams, leading to sizable changes in the vicinity of the ρ (770) resonance. Moreover, we work out the matching to ChPT and devise a strategy for a stable numerical evaluation of real-emission contributions near the two-pion threshold, which proves important to capture isospin-breaking corrections enhanced by the threshold singularity. For the numerical analysis, we use a dispersive representation of the pion form factor including the ρ ′ , ρ ′′ resonances, perform fits to the available data sets for the τ → ππν τ spectral function, and calculate the corresponding radiative correction factor G EM ( s ) in a self-consistent manner. Based on these results, we evaluate the τ -specific isospin-breaking corrections to $$ {a}_{\mu}^{\textrm{HVP},\textrm{LO}}\left[\pi \pi, \tau \right] $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mi>HVP</mml:mi> <mml:mo>,</mml:mo> <mml:mi>LO</mml:mi> </mml:mrow> </mml:msubsup> <mml:mfenced> <mml:mi>ππ</mml:mi> <mml:mi>τ</mml:mi> </mml:mfenced> </mml:math> .