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Register a new userDispersion relation for hadronic light-by-light scattering: two-pion contributions
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In this third paper of a series dedicated to a dispersive treatment of the hadronic light-by-light (HLbL) tensor, we derive a partial-wave formulation for two-pion intermediate states in the HLbL contribution to the anomalous magnetic moment of the muon $(g-2)_mu$, including a detailed discussion of the unitarity relation for arbitrary partial waves. We show that obtaining a final expression free from unphysical helicity partial waves is a subtle issue, which we thoroughly clarify. As a by-product, we obtain a set of sum rules that could be used to constrain future calculations of $gamma^*gamma^*topipi$. We validate the formalism extensively using the pion-box contribution, defined by two-pion intermediate states with a pion-pole left-hand cut, and demonstrate how the full known result is reproduced when resumming the partial waves. Using dispersive fits to high-statistics data for the pion vector form factor, we provide an evaluation of the full pion box, $a_mu^{pitext{-box}}=-15.9(2)times 10^{-11}$. As an application of the partial-wave formalism, we present a first calculation of $pipi$-rescattering effects in HLbL scattering, with $gamma^*gamma^*topipi$ helicity partial waves constructed dispersively using $pipi$ phase shifts derived from the inverse-amplitude method. In this way, the isospin-$0$ part of our calculation can be interpreted as the contribution of the $f_0(500)$ to HLbL scattering in $(g-2)_mu$. We argue that the contribution due to charged-pion rescattering implements corrections related to the corresponding pion polarizability and show that these are moderate. Our final result for the sum of pion-box contribution and its $S$-wave rescattering corrections reads $a_mu^{pitext{-box}} + a_{mu,J=0}^{pipi,pitext{-pole LHC}}=-24(1)times 10^{-11}$.
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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 $a_0(980)$: $a_mu^text{HLbL}[text{scalars}]=-9(1)times 10^{-11}$.
The $pi^0$ pole constitutes the lowest-lying singularity of the hadronic light-by-light (HLbL) tensor, and thus provides the leading contribution in a dispersive approach to HLbL scattering in the anomalous magnetic moment of the muon $(g-2)_mu$. It is unambiguously defined in terms of the doubly-virtual pion transition form factor, which in principle can be accessed in its entirety by experiment. We demonstrate that, in the absence of a direct measurement, the full space-like doubly-virtual form factor can be reconstructed very accurately based on existing data for $e^+e^-to 3pi$, $e^+e^-to e^+e^-pi^0$, and the $pi^0togammagamma$ decay width. We derive a representation that incorporates all the low-lying singularities of the form factor, matches correctly onto the asymptotic behavior expected from perturbative QCD, and is suitable for the evaluation of the $(g-2)_mu$ loop integral. The resulting value, $a_mu^{pi^0text{-pole}}=62.6^{+3.0}_{-2.5}times 10^{-11}$, for the first time, represents a complete data-driven determination of the pion-pole contribution with fully controlled uncertainty estimates. In particular, we show that already improved singly-virtual measurements alone would allow one to further reduce the uncertainty in $a_mu^{pi^0text{-pole}}$.
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 present a detailed analysis of $e^+e^-topi^+pi^-$ data up to $sqrt{s}=1,text{GeV}$ in the framework of dispersion relations. Starting from a family of $pipi$ $P$-wave phase shifts, as derived from a previous Roy-equation analysis of $pipi$ scattering, we write down an extended Omn`es representation of the pion vector form factor in terms of a few free parameters and study to which extent the modern high-statistics data sets can be described by the resulting fit function that follows from general principles of QCD. We find that statistically acceptable fits do become possible as soon as potential uncertainties in the energy calibration are taken into account, providing a strong cross check on the internal consistency of the data sets, but preferring a mass of the $omega$ meson significantly lower than the current PDG average. In addition to a complete treatment of statistical and systematic errors propagated from the data, we perform a comprehensive analysis of the systematic errors in the dispersive representation and derive the consequences for the two-pion contribution to hadronic vacuum polarization. In a global fit to both time- and space-like data sets we find $a_mu^{pipi}|_{leq 1,text{GeV}}=495.0(1.5)(2.1)times 10^{-10}$ and $a_mu^{pipi}|_{leq 0.63,text{GeV}}=132.8(0.4)(1.0)times 10^{-10}$. While the constraints are thus most stringent for low energies, we obtain uncertainty estimates throughout the whole energy range that should prove valuable in corroborating the corresponding contribution to the anomalous magnetic moment of the muon. As side products, we obtain improved constraints on the $pipi$ $P$-wave, valuable input for future global analyses of low-energy $pipi$ scattering, as well as a determination of the pion charge radius, $langle r_pi^2 rangle = 0.429(1)(4),text{fm}^2$.
We present a first model-independent calculation of $pipi$ intermediate states in the hadronic-light-by-light (HLbL) contribution to the anomalous magnetic moment of the muon $(g-2)_mu$ that goes beyond the scalar QED pion loop. To this end we combine a recently developed dispersive description of the HLbL tensor with a partial-wave expansion and demonstrate that the known scalar-QED result is recovered after partial-wave resummation. Using dispersive fits to high-statistics data for the pion vector form factor, we provide an evaluation of the full pion box, $a_mu^{pitext{-box}}=-15.9(2)times 10^{-11}$. We then construct suitable input for the $gamma^*gamma^*topipi$ helicity partial waves based on a pion-pole left-hand cut and show that for the dominant charged-pion contribution this representation is consistent with the two-loop chiral prediction and the COMPASS measurement for the pion polarizability. This allows us to reliably estimate $S$-wave rescattering effects to the full pion box and leads to our final estimate for the sum of these two contributions: $a_mu^{pitext{-box}} + a_{mu,J=0}^{pipi,pitext{-pole LHC}}=-24(1)times 10^{-11}$.
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