The branching ratio of the η->3πdecay is an important source of information on the value of the quark mass ratio 1/R=(m_d-m_u)/(m_s-\hat m). Furthermore, isospin breaking effects in the decays K->3πprovide information on the pion scattering lengths. The cusp effect in the K->3πdecays is presently being analyzed by the NA48 and KTeV experiments. From the theoretical point of view, these processes have been studied by different methods. We propose a unified and relativistic treatment relying on very general principles, unitarity, analyticity and crossing symmetry, combined with chiral counting, in order to construct model-independent representations of the corresponding amplitudes that are valid at two loops. A general description of the procedure is given and is illustrated in the case of the ηdecay amplitude in the leading order in the isospin breaking.
We present and develop a general dispersive framework allowing us to construct representations of the amplitudes for the processes $Pπ\toππ$, $P=K,η$, valid at the two-loop level in the low-energy expansion. The construction proceeds through a two-step iteration, starting from the tree-level amplitudes and their S and P partial-wave projections. The one-loop amplitudes are obtained for all possible configurations of pion masses. The second iteration is presented in detail in the cases where either all masses of charged and neutral pions are equal, or for the decay into three neutral pions. Iss…
We extend existing dispersive approach in subloop insertion to the case of crossed two-loop box type topologies. Based on the ideas of the Feynman trick, mass shift approach and dispersive representation of two-point Passarino-Veltman function we expressed two-loop scalar diagrams in the compact analytical form suitable for the automatization of the calculations. The results are expressed in a way that the numerical integration over Feynman and dispersive parameters and differentiation with respect to mass shift parameters are required in the final stage only.
The two-loop contributions are now often required by the precision experiments, yet are hard to express analytically while keeping precision. One way to approach this challenging task is via the dispersive approach, allowing to replace sub-loop diagram by effective propagator. This paper builds on our previous work, where we developed a general approach based on representation of many-point Passarino-Veltman functions in two-point function basis. In this work, we have extracted the UV-divergent poles of the Passarino-Veltman functions analytically and presented them as the dimensionally-regula…
As the new-generation precision experiments such as MOLLER and P2 look for physics beyond Standard Model, it is becoming increasingly important to evaluate the higher-order electroweak radiative corrections to a sub-percent level of uncertainty. However, due to propagators with different masses and higher-order tensor Feynman integrals, the two-loop calculations involving thousands of Feynman graphs become a demanding task requiring novel computational approaches. In this paper, we describe our dispersive sub-loop insertion approach and develop two-loop integrals using two-point functions basi…
A dispersive representation based on unitarity and analyticity is used to study the low energy $γN \to πN\ \text{and}\ γ^* N \to πN$ partial wave amplitudes. Final state interactions of the $πN$ system are critical to this analysis. The left-hand cut contribution is estimated by invoking baryon chiral perturbation theory results, while the right-hand cut contribution responsible for final state interaction effects is taken into account via an Omnès formalism with elastic phase shifts as input. It is found that a good numerical fit can be achieved with only one subtraction parameter, and the ex…
We present a dispersive representation of the $γN\rightarrow πN$ partial-wave amplitude based on unitarity and analyticity. In this representation, the right-hand-cut contribution responsible for $πN$ final-state-interaction effect are taken into account via an Omnés formalism with elastic $πN$ phase shifts as inputs, while the left-hand-cut contribution is estimated by invoking chiral perturbation theory. Numerical fits are performed in order to pin down the involved subtraction constants. It is found that good fit quality can be achieved with only one free parameter and the experimental data…
We present and numerically implement a computational method to construct relativistic scattering amplitudes that obey analyticity, crossing, elastic and inelastic unitarity in three and four spacetime dimensions. The algorithm is based on the Mandelstam representation of the amplitude and iterations of unitarity. The input for the iterative procedure is given by the multi-particle double spectral density, the S-wave inelasticity, and the value of the amplitude at the crossing-symmetric point. The output, obtained at the fixed point of the iteration of unitarity, is a nonperturbative scattering…
We derive the dependence of the leading-twist pion light-cone distribution amplitude (LCDA) on a parton momentum fraction $x$ by directly solving the dispersion relations for the moments with inputs from the operator product expansion (OPE) of the corresponding correlation function. It is noticed that these dispersion relations must be organized into those for the Gegenbauer coefficients first in order to avoid the ill-posed problem appearing in the conversion from the moments to the Gegenbauer coefficients. Given the values of various condensates in the OPE, we find that a solution for the pi…
In this proceeding, we illustrate the applicability of a new parameterization of the S-wave amplitude on the example of the $ππ\to ππ$ and $πK \to πK $ lattice data ($m_π\sim 240$ MeV) from the HadSpec collaboration. The applied parameterization follows from the dispersive representation for the inverse scattering amplitude. The left-hand cut contribution is parametrized by the series in a suitably constructed conformal variable. The crucial input in the analysis is the Adler zero, whose position we extracted from the chiral perturbation theory at next-to-leading order with the uncertainties p…
The hadronic contribution to the eight forward amplitudes of light-by-light scattering ($γ^*γ^*\to γ^*γ^*$) is computed in lattice QCD. Via dispersive sum rules, the amplitudes are compared to a model of the $γ^*γ^*\to {\rm hadrons}$ cross sections in which the fusion process is described by hadronic resonances. Our results thus provide an important test for the model estimates of hadronic light-by-light scattering in the anomalous magnetic moment of the muon, $a_μ^{\rm HLbL}$. Using simple parametrizations of the resonance $M\to γ^*γ^*$ transition form factors, we determine the corresponding …
A dispersive analysis of $η\to 3π$ decays has been performed in the past by many authors. The numerical analysis of the pertinent integral equations is hampered by two technical difficulties: i) The angular averages of the amplitudes need to be performed along a complicated path in the complex plane. ii) The averaged amplitudes develop singularities along the path of integration in the dispersive representation of the full amplitudes. It is a delicate affair to handle these singularities properly, and independent checks of the obtained solutions are demanding and time consuming. In the present…
We discuss the simultaneous dispersive analyses of $πK\to πK$ and $ππ\to K \bar{K}$ scattering data and the $κ/K^*_0(700)$ resonance. The unprecedented statistics of present and future Hadron Experiments, modern Lattice QCD calculations, and the wealth of new states and decays, require such precise and model-independent analyses to describe final state interactions. We review the existing and often conflicting data and explain in detail the derivation of the relevant dispersion relations, maximizing their applicability range. Next, we review and extend the caveats on some data, showing their i…
Scattering off the edge of a composite particle or finite-range interaction can precede that off its center. An effective theory treatment with pointlike particles and contact interactions must find that the scattered experimental wave is slightly advanced, in violation of causality (the fundamental underlying theory being causal). In practice, partial-wave or other projections of multivariate amplitudes exponentially grow with imaginary E, so that upper complex-plane analyticity is not sufficient to obtain a dispersion relation for them, but only for a slightly modified function (the modified…
We use a dispersion representation based on unitarity and analyticity to study the low energy $γ^* N\rightarrow πN$ process in the $S_{11}$ channel. Final state interactions among the $πN$ system are critical to this analysis. The left-hand part of the partial wave amplitude is imported from $\mathcal{O}(p^2)$ chiral perturbation theory result. On the right-hand part, the final state interaction is calculated through Omnès formula in $S$ wave. It is found that a good numerical fit can be achieved with only one subtraction parameter, and the eletroproduction experimental data of multipole ampli…
We study the long-distance contribution to $B_s\toμ^+μ^-$ decay, which is generated by the two-photon intermediate state via $B_s\toγ^*γ^*\toμ^+μ^-$ transition. It is found that the dispersive two-photon amplitude can interfere with the dominant short-distance amplitude, which gives rise to new theoretical uncertainty in the branching ratio of $B_s\toμ^+μ^-$. Our analysis shows that, by taking into account present experimental constraints, this uncertainty could be up to the same order of magnitude as some theoretical uncertainties of ${\cal B}(B_s\toμ^+μ^-)$ given in the past literature. Futu…
