No abstract

No abstract

We present a determination of the pion-nucleon (πN ) σ-term σπN based on the Cheng-Dashen low-energy theorem (LET), taking advantage of the recent high-precision data from pionic atoms to pin down the πN scattering lengths as well as of constraints from analyticity, unitarity, and crossing symmetry in the form of Roy-Steiner equations to perform the extrapolation to the Cheng-Dashen point in a reliable manner. With isospin-violating corrections included both in the scattering lengths and the LET, we obtain σπN = (59.1 ± 1.9 ± 3.0) MeV = (59.1 ± 3.5) MeV, where the first error refers to uncertainties in the πN amplitude and the second to the LET. Consequences for the scalar nucleon couplings relevant for the direct detection of dark matter are discussed.

We present a detailed analysis of e + e − → π + π − data up to √ s = 1 GeV in the framework of dispersion relations. Starting from a family of ππ P -wave phase shifts, as derived from a previous Roy-equation analysis of ππ scattering, we write down an extended Omnès 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 ω 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 ππ µ | ≤1 GeV = 495.0(1.5)(2.1) × 10 −10 and a ππ µ | ≤0.63 GeV = 132.8(0.4)(1.0) × 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 ππ P -wave, valuable input for future global analyses of low-energy ππ scattering, as well as a determination of the pion charge radius, r 2 π = 0.429(1)(4) fm 2 .

Abstract. Low-energy pion-nucleon scattering is relevant for many areas in nuclear and hadronic physics, ranging from the scalar couplings of the nucleon to the long-range part of two-pion-exchange potentials and three-nucleon forces in Chiral Effective Field Theory. In this talk, we show how the fruitful combination of dispersion-theoretical methods, in particular in the form of Roy-Steiner equations, with modern high-precision data on hadronic atoms allows one to determine the pion-nucleon scattering amplitudes at low energies with unprecedented accuracy. Special attention will be paid to the extraction of the pion-nucleon σ-term, and we discuss in detail the current tension with recent lattice results, as well as the determination of the low-energy constants of chiral perturbation theory. c

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) µ , 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 γ * γ * → ππ. 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 π-box µ = −15.9(2)×10 −11 . As an application of the partial-wave formalism, we present a first calculation of ππ-rescattering effects in HLbL scattering, with γ * γ * → ππ helicity partial waves constructed dispersively using ππ 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) µ . 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 π-box µ + a ππ,π-pole LHC µ,J=0 = −24(1) × 10 −11 .

The pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon (g − 2) µ is fully determined by the doubly-virtual pion transition form factor. Although this crucial input quantity is, in principle, directly accessible in experiment, a complete measurement covering all kinematic regions relevant for (g − 2) µ is not realistic in the foreseeable future. Here, we report in detail on a reconstruction from available data, both space-and time-like, using a dispersive representation that accounts for all the low-lying singularities, reproduces the correct high-and low-energy limits, and proves convenient for the evaluation of the (g − 2) µ loop integral. We concentrate on the systematics of the fit to e + e − → 3π data, which are key in constraining the isoscalar dependence, as well as the matching to the asymptotic limits. In particular, we provide a detailed account of the pion transition form factor at low energies in the time-and space-like region, including the error estimates underlying our final result for the pion-pole contribution, a π 0 -pole µ = 62.6 +3.0 −2.5 × 10 −11 , and demonstrate how forthcoming singly-virtual measurements will further reduce its uncertainty.

We address the contribution of the 3π channel to hadronic vacuum polarization (HVP) using a dispersive representation of the e + e − → 3π amplitude. This channel gives the second-largest individual contribution to the total HVP integral in the anomalous magnetic moment of the muon (g − 2) µ , both to its absolute value and uncertainty. It is largely dominated by the narrow resonances ω and φ, but not to the extent that the off-peak regions were negligible, so that at the level of accuracy relevant for (g − 2) µ an analysis of the available data as model independent as possible becomes critical. Here, we provide such an analysis based on a global fit function using analyticity and unitarity of the underlying γ * → 3π amplitude and its normalization from a chiral low-energy theorem, which, in particular, allows us to check the internal consistency of the various e + e − → 3π data sets. Overall, we obtain a 3π µ | ≤1.8 GeV = 46.2(6)(6) × 10 −10 as our best estimate for the total 3π contribution consistent with all (low-energy) constraints from QCD. In combination with a recent dispersive analysis imposing the same constraints on the 2π channel below 1 GeV, this covers nearly 80% of the total HVP contribution, leading to a HVP µ = 692.3(3.3) × 10 −10 when the remainder is taken from the literature, and thus reaffirming the (g−2) µ anomaly at the level of at least 3.4σ. As side products, we find for the vacuum-polarization-subtracted masses M ω = 782.63(3)(1) MeV and M φ = 1019.20(2)(1) MeV, confirming the tension to the ω mass as extracted from the 2π channel.