Karol Kampf, Marc Knecht, Jiri Novotny, Martin Zdrahal
The NA48/2 collaboration has shown clear experimental evidence for a cusp in the data for K -> pi pi^0 pi^0. This effect can be used to extract information on the pi-pi scattering lengths. We address this issue using a two-loop dispersive construction of pi pi -> pi pi and K -> pi pi pi amplitudes in the presence of isospin breaking.
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We report on the calculation of the two-loop electroweak, two-photon mediated short-distance dispersive K_L \to μ^+μ^- decay amplitude. QCD corrections change the sign of this contribution and reduce it by an order of magnitude. The resulting amplitude enables us to provide a constraint on the otherwise uncertain long-distance dispersive amplitude.
We present a full account of the two-loop electroweak, two-photon mediated short-distance dispersive $K_L \toμ^+μ^-$ decay amplitude. QCD corrections change the sign of this amplitude and reduce it by an order of magnitude. Thus, the QCD-corrected two-loop amplitude represents only a small fraction (with the central value of 5 %) of the one-loop weak short-distance contribution, and has the same sign. In combination with a recent measurement, the standard-model prediction of the short-distance amplitude, completed in this paper, provides a constraint on the otherwise uncertain long-distance di…
We reevaluate the two form factors relevant for the $ηπ$ second-class $τ$ decay mode, making systematic use of analyticity, unitarity, combined with updated inputs to the NLO chiral constraints. We focus, in particular, on the shape of the $ρ$ resonance peak which is a background-free signature of a second-class current. Its dispersive construction requires the $ηπ\toππ$ scattering amplitude which we derive from a family of Khuri-Treiman equations solutions constrained with accurate recent results on the $η\to3π$ Dalitz plot.
The dispersive approach based only on very general principles, unitarity, analyticity and crossing symmetry, combined with chiral counting enables to construct fully relativistic model-independent representations of the eta and K three-pion decays valid up to and including two-loop corrections. Here we demonstrate the procedure on the eta->3pi^0 decay. We perform possible matchings of our dispersive approach with the two-loop ChPT result and briefly discuss corresponding predictions of the Dalitz plot slope parameter alpha.
We propose a new method to obtain the K -> pi pi amplitude from K -> pi which allows one to fully account for the effects of final state interactions. The method is based on a set of dispersion relations for the K -> pi pi amplitude in which the weak Hamiltonian carries momentum. The soft pion theorem, which relates this amplitude to the K -> pi amplitude, can be used to determine one of the two subtraction constants - the second constant is at present known only to leading order in chiral perturbation theory.
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…
The relativistic amplitudes of pion photo- and electroproduction are calculated by dispersion relations at constant t. Several sum rules and low-energy theorems for the threshold amplitudes are investigated within this technique. The continuation of the amplitudes to sub-threshold kinematics is shown to provide a unique framework to derive the low-energy constants of chiral perturbation theory by global properties of the excitation spectrum.
It has been recently suggested that it is possible to calculate the effect of final state interactions in K -> pi pi amplitudes by applying dispersive methods to the amplitude with the kaon off-shell. We critically reexamine the procedure, and point out the effects of the arbitrariness in the choice of the off-shell field for the kaon.
The pion mass difference generates a pronounced cusp in K --> 3 pi decays, the strength of which is related to the pi pi S-wave scattering lengths. We apply an effective field theory framework developed earlier to evaluate the amplitudes for K_L --> 3 pi decays in a systematic manner, where the strictures imposed by analyticity and unitarity are respected automatically. The amplitudes for the decay eta --> 3 pi are also given.
A dispersive study of final-state interactions in $K\toπππ$ amplitudes
本文在CP和同位旋守恒假设下，构建了Omnès-Khuri-Treiman-Sawyer-Wali类型的色散表示系统，用于研究$K\ oπππ$弱跃迁振幅中的末态相互作用，给出了耦合积分系统的数值解及减除常数的确定方法。
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, …
Dispersive approaches provide model independent description of meson-meson scattering. We first review here the use of dispersion relations to obtain a model independent unitarization of Chiral Perturbation Theory amplitudes, that establish the existence of light scalar mesons and whose leading 1/N_c behavior suggest they ahve a non q-qbar dominant component. We also review the forward dispersion relation checks on conflicting experimental data and the resulting very precise pion-pion scattering scattering amplitudes.
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…
Dispersive description of the $K \to π\ell^+ \ell^-$ radiative amplitudes
本文通过色散方法确定K→πℓ⁺ℓ⁻辐射幅度的形式因子W_+和W_S的符号及能量依赖性，并利用实验数据推导K_S→π⁺π⁻π⁰的ΔI=1/2部分。
We have evaluated forward dispersion relations for scattering amplitudes that follow from direct fits to several sets of pion-pion scattering experiments, together with the precise K decay results, and high to energy data. We find that some of the most commonly used experimental sets, as well as some recent theoretical analyses based on Roy equations, do not satisfy these constraints by several standard deviations. Finally, we provide a consistent pion-pion amplitude by improving a global fit to data with these dispersion relations.
We express the leading electromagnetic corrections in K-> pi pi amplitudes as integrals over the virtual photon squared-momentum Q^2. The high Q^2 behavior is obtained via the operator product expansion. The low Q^2 behavior is calculated using chiral perturbation theory. We model the intermediate Q^2 region using resonance contributions in order to enforce the matching of these two regimes. The results confirm our previous estimates that the electromagnetic corrections provide a reasonably small shift in the delta I - 3/2 amplitude.
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…
A representation of the amplitude $γ^*(q^2)\toγππ$ is proposed which combines large $N_c$ chiral resonance Lagrangian modelling with general unitarity and analyticity properties. The amplitude is constrained from $γγ$ scattering results and $e^+ e^- \toγπ^0π^0$ measurements by the CMD-2 and SND collaborations. As an application, the contribution of the $ππ+γ$ states in the HVP contribution to the muon $g-2$ are reconsidered, taking into account the effect of the strong $S$-wave $ππ$ rescattering in a model independent way.
Previous representations of pion pair production amplitudes by two real photons at low energy, which combine dispersion theoretical constraints with elastic unitariy, chiral symmetry and soft photon constraints are generalized to the case where one photon is virtual. The constructed amplitudes display explicitly the dependence on the $ππ$ phase-shifts, on pion form factors and on pion polarizabilities. They apply both for space-like and time-like virtualities despite the apparent overlap of the left and right-hand cuts, by implementing a definition of resonance exchange amplitudes complying wi…
A reanalysis of $ππ$ amplitudes for all important partial-waves below about 2 GeV is presented. A set of once subtracted dispersion relations with imposed crossing symmetry condition is used to modify unitary multi-channel amplitudes in the $S$, $P$, $D$, and $F$ waves. So far, these specific amplitudes constructed in our works and many other analyzes have been fitted only to experimental data and therefore do not fulfill the crossing symmetry condition. In the present analysis, the self consistent, i.e. unitary and fulfilling the crossing symmetry, amplitudes for the $S$, $P$, $D$, and $F$ wa…
The behavior of scattering amplitudes in the vicinity of a physical-region Landau singularity is considered. The impact on LHC processes is discussed.
The dispersive representation of the virtual Compton forward scattering amplitude has been recently reexamined in connection with the evaluation of the Cottingham formula for the proton-neutron electromagnetic mass difference. The most difficult part of the analysis is related to one of the invariant amplitudes, denoted as $T_1(ν, Q^2)$, which requires a subtraction in the standard dispersion relation with respect to the energy $ν$ at fixed photon momentum squared $q^2=-Q^2$. We propose an alternative dispersive framework, which implements analyticity and unitarity by combining the Cauchy inte…
In this work, we present different representations of Khuri-Treiman equation, the advantage and disadvantage of each representations are discussed. Using a toy model for scattering amplitude, we study the sensitivity of solution of KT equation to the left hand cut of this toy model and to the different approximate methods. At last, we briefly discuss Watson's theorem when three particles in final states are involved.
The velocity-dependent cusp anomalous dimension is by definition the ultraviolet (UV) anomalous dimension of a Wilson loop with a cusp. It appears in many physically interesting processes. In this talk we present recent progress in the analytic calculation of that quantity.
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 consider the $ππ$ scattering amplitude obtained by Colangelo, Gasser and Leutwyler (CGL), using dispersion relations and chiral perturbation theory to two loops. We point out that the input used by CGL for energies above 1.42 GeV is incompatible with a number of experimental results. Moreover, the error they adscribe to the $δ_0^{(0)}$ phase at 0.8 GeV, an important part of their input, is much underestimated, and its central value is probably displaced. Then, we compare CGL results with direct fits to experimental $ππ$ data; the outcome of this test is mismatch of the CGL amplitude, for se…
We study the factorization of soft and collinear singularities in dimensionally-regularized fixed-angle scattering amplitudes in massless gauge theories. Our factorization is based on replacing the hard massless partons by light-like Wilson lines, and defining gauge-invariant jet and soft functions in dimensional regularization. In this scheme the factorized amplitude admits a powerful symmetry: it is invariant under rescaling of individual Wilson-line velocities. This symmetry is broken by cusp singularities in both the soft and the eikonal jet functions. We show that the cancellation of thes…
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…
A dispersive estimate of the $a_0(980)$ contribution to hadronic light-by-light scattering in $(g-2)_μ$
通过修正的耦合道Muskhelischvili-Omnès形式和数据驱动的$N/D$方法，估计了$a_0(980)$对$(g-2)_μ$中强子光-by-光散射的色散贡献为$-0.46(2)\ imes 10^{-11}$。
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…
Because of their small electromagnetic corrections, the isospin-breaking decays $η\to3π$ seem to be good candidates for extracting isospin-breaking parameters $ (m_d-m_u)$. This task is unfortunately complicated by large chiral corrections and the discrepancy between the experimentally measured values of the Dalitz parameters describing the energy dependence of the amplitudes of these decays and those predicted from chiral perturbation theory. We present two methods based on an analytic dispersive representation that use the information from the NNLO chiral result and the one from the measurem…
We propose a new method to compute amplitudes of electroweak processes in the strong background magnetic field, using $γ\to e^+e^-$ as an example. We show that the {\it moments} of $γ\to e^+e^-$ width are proportional to the derivatives of photon polarization function at the zero energy. Hence, the pair-production width can be easily calculated from the latter by the inverse Mellin transform. The prospects of our approach are commented.
We present an expression for the leading-color (planar) four-loop four-point amplitude of N=4 supersymmetric Yang-Mills theory in 4-2 e dimensions, in terms of eight separate integrals. The expression is based on consistency of unitarity cuts and infrared divergences. We expand the integrals around e=0, and obtain analytic expressions for the poles from 1/e^8 through 1/e^4. We give numerical results for the coefficients of the 1/e^3 and 1/e^2 poles. These results all match the known exponentiated structure of the infrared divergences, at four separate kinematic points. The value of the 1/e^2 c…
Two-photon exchange (TPE) contributions to elastic electron-proton scattering in the forward regime are considered. The imaginary part of TPE amplitude in these kinematics is related to the DIS nucleon structure functions. The real part of the TPE amplitude is obtained from the imaginary part by means of dispersion relations. We demonstrate that the dispersion integrals for the relevant elastic $ep$-scattering amplitude converge and do not need subtraction. This allows us to make clean prediction for the real part of the TPE amplitude at forward angles. We furthermore compare $e^+p$ and $e^-p$…
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…
Dispersive $ππ\rightarrow K\bar K$ amplitude and giant CP violation in B to three light-meson decays at LHCb
本文提出将模型无关的色散 $ππ\ ightarrow K\ar K$ 振幅分析应用于 LHCb 公式体系，更准确地描述了 $B$ 介子三体衰变中的巨大 CP 破坏，并证实了末态相互作用的关键作用。
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…
Analytical dispersive parameterization for elastic scattering of spinless particles
本研究针对自旋为0粒子弹性散射，提出基于倒数散射振幅色散表示的改进参数化方法，用共形变量展开处理左支割线贡献，正确实现Adler零点与阈值因子，为格点数据分析提供优于传统方法的工具。
Isospin-breaking effects on the threshold cusp structures in $ΛN$-$ΣN$ scattering
本文利用K矩阵表示推导多道散射中阈值附近的散射振幅通用表达式，研究同位旋破缺导致的阈值分裂对尖点结构的影响，并应用于ΛN-ΣN耦合系统，发现同位旋破缺可显著改变尖点的相对锐度甚至类型。
A dispersive estimate of the $a_0(980)$ contribution to $(g-2)_μ$
该论文通过修正的耦合道 Muskhelishvili-Omnes 形式和 N/D 方法，利用实验数据拟合得到 $a_0(980)$ 对 $(g-2)_μ$ 的色散贡献为 $-0.43(1)(2)\ imes 10^{-11}$，精度较窄共振近似提升一个数量级。
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…
Application of $K \to 3π$ amplitudes to semileptonic kaon decays
本文通过色散表示法改进中微子衰变的理论描述，利用 $K \to 3π$ 振幅和非局域形式因子关系进行参数化与约束。
Correlation functions of Wilson lines are relevant for describing the infrared structure of scattering amplitudes. We develop a new method for evaluating a wide class of such Wilson line integrals, and apply it to the calculation of the velocity-dependent cusp anomalous dimension in maximally supersymmetric Yang-Mills theory. We compute the four-loop non-planar correction in a recently introduced scaling limit. Moreover, we derive the full planar four-loop result by means of an ansatz which is based on the structure of known analytic results. We determine the coefficients in this ansatz by mak…
We present a new dispersive analysis of the isospin breaking decay eta --> 3 pi. The resulting representation of the decay amplitude allows us to determine the quark mass double ratio Q and we find as a preliminary result Q = 22.3 +/- 0.4. Finally, we discuss a number of improvements that we intend to implement in the future.
We present a dispersive representation of the D-term form factor for hard exclusive reactions, using unsubtracted $t$-channel dispersion relations. The $t$-channel unitarity relation is saturated with the contribution of two-pion intermediate states, using the two-pion distributions amplitude for the $γ^*γ\rightarrow ππ$ subprocess and reconstructing the $ππ\rightarrow N\bar N$ subprocess from available information on pion-nucleon partial-wave helicity amplitudes. Results for the D-term form factor as function of $t$ as well as at $t=0$ are discussed in comparison with available model predicti…
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…
Planar gluon amplitudes in N=4 SYM are remarkably similar to expectation values of Wilson loops made of light-like segments. We argue that the latter can be determined by making use of the conformal symmetry of the gauge theory, broken by cusp anomalies. We derive the corresponding anomalous conformal Ward identities valid to all loops and show that they uniquely fix the form of the finite part of a Wilson loop with n cusps (up to an additive constant) for n=4 and n=5 and reduce the freedom in it to a function of conformal invariants for n>=6. We also present an explicit two-loop calculation f…
We review the use of new and precise dispersive equations, which also implement crossing symmetry, in order to shed further light on the long-standing puzzle in the parameters of the f0(500), as well as the f0(980). This puzzle is finally being settled thanks to several analyses carried out during the last years. In this talk we show how our very recent dispersive data analysis allowed for a precise and model independent determination of the amplitudes for the S,P,D and F waves. In particular, we show how the analytic continuation of once subtracted dispersion relations for the S0 wave to the …
Amplitude structure of $3\to 3$ scattering in a Mandelstam variable representation
该研究构建了S波三全同粒子3→3散射的色散表示，推导单粒子交换驱动的线性积分方程，证明其满足三体幺正性，并给出极点扣除方案以获取三体共振极点的解析结构。
Recent LHCb's amplitude analysis on $B^+\to J/ψϕK^+$ suggests the existence of exotic $X$ and $Z_{cs}$ hadrons, based on an assumption that Breit-Wigner resonances describe all the peak structures. However, all the peaks and also dips in the spectra are located at relevant meson-meson thresholds where threshold kinematical cusps might cause such structures. This points to the importance of an independent amplitude analysis with due consideration of the kinematical effects, and this is what we do in this work. Our model fits well $J/ψϕ$, $J/ψK^+$, and $K^+ϕ$ invariant mass distributions simulta…
We provide a dispersion-theoretical representation of the reaction amplitudes $γK\to K π$ in all charge channels, based on modern pion-kaon $P$-wave phase shift input. Crossed-channel singularities are fixed from phenomenology as far as possible. We demonstrate how the subtraction constants can be matched to a low-energy theorem and radiative couplings of the $K^*(892)$ resonances, thereby providing a model-independent framework for future analyses of high-precision kaon Primakoff data.
Neuro-dispersive extractions of light-meson resonances
该研究提出基于S矩阵引导的神经网络（SINNs）的色散提取方法，确定ππ散射的σ/f₀(500)等共振极点，结果稳健且可适配其他新物理相关反应分析。
We evaluate the two-photon exchange correction to the elastic electron-proton scattering cross section within a dispersive framework. Besides the elastic contribution, we account for all $πN$ intermediate state contributions using the phenomenological MAID fit as an input. We develop a novel method for the analytical continuation of the two-photon exchange amplitudes into the unphysical region and generalize our previous work to the momentum transfer region $ 0.064~\mathrm{GeV}^2 \lesssim Q^2 \lesssim 1~\mathrm{GeV}^2$. We compare our results with recent OLYMPUS, CLAS and VEPP-3 data as well a…
Dispersive analysis of the $\boldsymbol{J/ψ\to γπ^0 π^0}$ process
利用色散振幅分析，结合BESIII数据，通过耦合道Muskhelishvili-Omnès表示描述$π^0π^0$系统的$S$波和$D$波，解决了$0^{++}-2^{++}$相位歧义，为引力形状因子中的双π贡献提供了输入。
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