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Register a new userVirtual Hadronic and Leptonic Contributions to Bhabha Scattering
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Stefano Actis
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Using dispersion relations, we derive the complete virtual QED contributions to Bhabha scattering due to vacuum polarization effects in photon propagation. We apply our result to hadronic corrections and to heavy lepton and top quark loop insertions. We give the first complete estimate of their net numerical effects for both small and large angle scattering at typical beam energies of meson factories, LEP, and the ILC. The effects turn out to be smaller, in most cases, than those corresponding to electron loop insertions, but stay, with amounts of typically one per mille, of relevance for precision experiments. Hadronic corrections themselves are typically about 2-3 times larger than those of intermediate muon pairs (the largest heavy leptonic terms).
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Virtual hadronic contributions to the Bhabha process at the NNLO level are discussed. They are substantial for predictions with per mil accuracy. The studies of heavy fermion and hadron corrections complete the calculation of Bhabha virtual effects at the NNLO level.
Effects of vacuum polarization by hadronic and heavy-fermion insertions were the last unknown two-loop QED corrections to high-energy Bhabha scattering and have been first announced in cite{Actis:2007fs}. Here we describe the corrections in detail and explore their numerical influence. The hadronic contributions to the virtual O(alpha^2) QED corrections to the Bhabha-scattering cross-section are evaluated using dispersion relations and computing the convolution of hadronic data with perturbatively calculated kernel functions. The technique of dispersion integrals is also employed to derive the virtual O(alpha^2) corrections generated by muon-, tau- and top-quark loops in the small electron-mass limit for arbitrary values of the internal-fermion masses. At a meson factory with 1 GeV center-of-mass energy the complete effect of hadronic and heavy-fermion corrections amounts to less than 0.5 per mille and reaches, at 10 GeV, up to about 2 per mille. At the Z resonance it amounts to 2.3 per mille at 3 degrees; overall, hadronic corrections are less than 4 per mille. For ILC energies (500 GeV or above), the combined effect of hadrons and heavy-fermions becomes 6 per mille at 3 degrees; hadrons contribute less than 20 per mille in the whole angular region.
We derive the two-loop corrections to Bhabha scattering from heavy fermions using dispersion relations. The double-box contributions are expressed by three kernel functions. Convoluting the perturbative kernels with fermionic threshold functions or with hadronic data allows to determine numerical results for small electron mass m_e, combined with arbitrary values of the fermion mass m_f in the loop, $m_e^2<<s,t,m_f^2$, or with hadronic insertions. We present numerical results for m_f = m_{mu}, m_{tau}, m_{top} at typical small- and large-angle kinematics ranging from 1 GeV to 500 GeV.
Using an effective sigma/f_0(500) resonance, which describes the pipi-->pipi and gammagamma-->pipi scattering data, we evaluate its contribution and the ones of the other scalar mesons to the the hadronic light-by-light (HLbL) scattering component of the anomalous magnetic moment a_mu of the muon. We obtain the conservative range of values: sum_S~a_mu^{lbl}vert_S = -(4.51+- 4.12) 10^{-11}, which is dominated by the sigma/f_0(500) contribution ( 50%~98%), and where the large error is due to the uncertainties on the parametrisation of the form factors. Considering our new result, we update the sum of the different theoretical contributions to a_mu within the standard model, which we then compare to experiment. This comparison gives (a_mu^{rm exp} - a_mu^{SM})= +(312.1+- 64.3) 10^{-11}, where the theoretical errors from HLbL are dominated by the scalar meson contributions.
We consider the contribution of scalar resonances to hadronic light-by-light scattering in the anomalous magnetic moment of the muon. While the $f_0(500)$ has already been addressed in previous work using dispersion relations, heavier scalar resonances have only been estimated in hadronic models so far. Here, we compare an implementation of the $f_0(980)$ resonance in terms of the coupled-channel $S$-waves for $gamma^*gamma^*to pipi/bar K K$ to a narrow-width approximation, which indicates $a_mu^{text{HLbL}}[f_0(980)]=-0.2(2)times 10^{-11}$. With a similar estimate for the $a_0(980)$, the combined effect is thus well below $1times 10^{-11}$ in absolute value. We also estimate the contribution of heavier scalar resonances. In view of the very uncertain situation concerning their two-photon couplings we suggest to treat them together with other resonances of similar mass when imposing the matching to short-distance constraints. Our final result is a refined estimate of the $S$-wave rescattering effects in the $pi pi$ and $bar K K$ channel up to about $1.3$ GeV and including a narrow-width evaluation of the $a_0(980)$: $a_mu^text{HLbL}[text{scalars}]=-9(1)times 10^{-11}$.
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