Lev Fil'kov, Viktor Kashevarov
An analysis of different dispersion sum rules (DSRs) for the dipole and quadrupole pion polarizabilities is carried out. We prove the absence of additional spurious singularities in these approaches. The results of the calculations of the polarizabilities in the framework of DSRs are compared with ChPT predictions and with data obtained in different experiments.
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Data on pion polarizabilities obtained in different experiments are reviewed. The values of the dipole and quadrupole polarizabilities of the pi^0 and pi^{+-} mesons found are compared with predictions of dispersion sum rules (DSRs) and two-loop calculations in the framework of chiral perturbation theory (ChPT). Possible reasons of a difference between the predictions of DSRs and ChPT are discussed.
We discuss main experimental works, where dipole polarizabilities of charged pions have been determined. Possible reasons for the differences between the experimental data are discussed. In particular, it is shown that the account of the $σ$-meson gives a significant correction to the value of the polarizability obtained in the latest experiment of the COMPASS collaboration.
We present a calculation of the virtual Compton scattering amplitude for $γ^\ast+π\to γ+π$ in the framework of chiral perturbation theory at ${\cal O}(p^4)$. We explicitly derive expressions for generalized electromagnetic polarizabilities and discuss alternative definitions of these quantities.
The electric ($α_π$) and magnetic ($β_π$) Compton polarizabilities of both the charged and the neutral pion are of fundamental interest in the low-energy sector of quantum chromodynamics (QCD). Pion polarizabilities affect the shape of the $γπ\toγπ$ Compton scattering angular distribution at back scattering angles and $γγ\toππ$ absolute cross sections. Theory derivations are given of the $γπ\toγπ$ Compton scattering differential cross section, dispersion relations, and sum rules in terms of the polarizabilities. We review experimental charged and neutral polarizability studies and theoretical …
We present a calculation of the cross section and the event generator of the reaction $πe\to πe γ$. This reaction is sensitive to the pion generalized dipole polarizabilities, namely, the longitudinal electric $α_L(q^2)$, the transverse electric $α_T(q^2)$, and the magnetic $β(q^2)$ which, in the real-photon limit, reduce to the ordinary electric and magnetic polarizabilities $\barα$ and $\barβ$, respectively. The calculation of the cross section is done in the framework of chiral perturbation theory at ${\cal O}(p^4)$. A pion VCS event generator has been written which is ready for implementat…
Recent experimental pion polarizability data are reviewed and analyzed. Data on the radiative π^+ meson photoproduction on the proton at MAMI has been used to determine the difference of the electric and magnetic dipole polarizabilities of the charged pion. Fits of the experimental data to the total cross sections of the processes γγ->π^0π^0 and γγ->π^\pmπ^\pm in the energy region from threshold to 2250 MeV and 2500 MeV, respectively, have been carried out using dispersion relations with subtractions for the invariant helicity amplitudes, where the dipole and the quadrupole polarizabilities of…
I discuss recent results regarding the influence of pion-nucleon and and nucleon-resonance physics on the (spin-) polarizabilities of the nucleon as measured in (polarized) Compton scattering.
Nucleon electric polarizabilities and nucleon-pion scattering at physical pion mass
本研究在物理π介子质量下通过格点QCD计算核子电极化率，发现并计算了$Nπ$态的实质性贡献，证实其动态贡献对可靠确定极化率至关重要，为未来相关研究奠定基础。
Virtual Compton scattering off nucleons and pions at low energies is discussed. Predictions for the generalized polarizabilities of the nucleon are presented within the framework of heavy-baryon chiral perturbation theory and the linear sigma model. First results for the generalized polarizabilities of the charged pion in chiral perturbation theory at ${\cal O}(p^4)$ are shown.
Using the fact that the contribution of the states with isospin I=0 in the difference of the amplitudes of the processes γγ->π^+π^- and γγ->π^0π^0 is very small, we have analyzed the dispersion sum rules for the difference between the dipole polarizabilities of the charged and neutral pions as a function of the σ-meson parameters. Then taken into account the current perturbation value of (α-β)_π^0=-1.9, we have found (α-β)_π^(+-)=9.4--8.2 for values of the σ-meson parameter within the region: M_s=400--550 MeV, Γ_s=400--600 MeV, Γ_{\s->γγ}= 0--3 keV. It has been shown that the value of the deca…
The Compton scattering on a charged pion is studied using the dispersion relations. Unknown mass, full width and decay width into γγof the sigma meson are found from a fit to the experimental data for the process γ+γ\to π^0+π^0. These parameters are used for the calculation of the cross section of the Compton scattering on the charged pion. The analysis of the obtained results shows that the data on the elastic γ\pion scattering in the energy region up to \sqrt{s}=1 GeV (together with the data on process γγ\to π^0π^0) could be used both for a determination of the pion polarizability values wit…
The values of charged pion polarizabilities obtained in the framework of chiral perturbation theory at the level of two-loop accuracy are compared with the experimental result recently reported by COMPASS Collaboration. It is found that the calculated value for the dipole polarizabilities $(α-β)_{π^\pm}= (5.7\pm 1.0)\times 10^{-4}\,{\rm fm}^3$ fits quite well the experimental result $(α-β)_{π^\pm} = (4.0 \pm 1.2_{\rm stat} \pm 1.4_{\rm syst}) \times 10^{-4}\,{\rm fm}^3$.
We report the results of the working group on ``Pion (Kaon) and Sigma Polarizabilities''. Interesting possibilities to measure these polarizabilities include the radiative pion photoproduction in the MAMI experiment at Mainz, as well as at the GRAAL facility (actually the latter is being considered for an experimental determination of the pion polarizabilities here for the first time), the experimental plans on Primakoff effect at FNAL, and the measurements at the Frascati $Φ$-factory DA$Φ$NE.
We discuss the virtual Compton scattering amplitude $γ^\ast+π\to γ+π$ which enters the reaction $π^-+e^-\toπ^-+e^-+γ$. The low-energy scattering amplitude is divided into a model-independent part, involving only the electromagnetic form factor, and a residual part which contains structure information related specifically to (virtual) Compton scattering. In the limit of vanishing final-photon energy, the residual amplitude can be expressed in terms of three generalized dipole polarizabilities which are functions of the squared virtual-photon four-momentum. We study the VCS amplitude in the fram…
The polarizabilities of the pion have been predicted in several different theoretical frameworks. The status of these is reviewed.
We evaluate the amplitude for gamma gamma --> pi^+ pi^- to two loops in the framework of chiral perturbation theory.The three new coupling constants that enter the result at this order in the low-energy expansion are estimated via resonance saturation. We discuss in addition the crossed channel processes gamma pi^+ --> gamma pi^+ - in particular the charged pion polarizabilities - to the same accuracy. The predictions are compared with available experimental information.
Recent attempts to determine the pion polarizability by dispersion relations yield values that disagree with the predictions of chiral perturbation theory. These dispersion relations are based on specific forms for the absorptive part of the Compton amplitudes. The analytic properties of these forms are examined, and the strong enhancement of intermediate-meson contributions is shown to be connected to non-analytic structures
We use chiral perturbation theory to study the extraction of pion electromagnetic polarizabilities from lattice QCD. Chiral extrapolation formulae are derived for partially quenched QCD, and quenched QCD simulations. On a torus, volume dependence of electromagnetic observables is complicated by SO(4) breaking, as well as photon zero-mode interactions. We determine finite volume corrections to the Compton scattering tensor of pions. We argue, however, that such results cannot be used to ascertain volume corrections to polarizabilities determined in lattice QCD with background field methods. Con…
The electromagnetic dipole and quadrupole polarizabilities of the neutral and charged pions are calculated in the Nambu--Jona-Lasinio model. Our results agree with the recent experimental analysis of these quantities based on Dispersion Sum Rules. Comparison is made with the results from the Chiral Perturbation Theory.
We present the results for the pion electromagnetic quadrupole polarizabilities, calculated within the Nambu--Jona-Lasinio model. We obtain the sign and magnitude in agreement with the respective experimental analysis based on the Dispersion Sum Rules. At the same time the dipole polarizabilities are well reproduced. Comparison is also made with the results from the Chiral Perturbation Theory.
We analyze the amplitude of radiative charged pion photoproduction within the framework of heavy baryon chiral perturbation theory (HBChPT) and discuss the best experimental setup for the extraction of the charged pion polarizabilities from the differential cross section. We find that the contributions from two unknown low energy constants(LECs) in the $πN$ chiral Lagrangian at order $p^{3}$ are comparable with the contributions of the charged pion polarizabilities. As a result, it is necessary to take these two LECs' effects into account. Furthermore, we discuss the applicability of the extra…
We evaluate the electric and magnetic polarizabilities of charged pions in the framework of chiral perturbation theory at next-to-leading order. This requires a two-loop evaluation of the Compton amplitude near threshold. We estimate the two new low-energy constants which enter the chiral expansion at this order with resonance saturation. The numerical results are compared with presently available experimental information.
Fermilab E781 plans measurements of gamma-Sigma and $γ$-pion interactions using a 600 GeV beam of Sigmas and pions, and a virtual photon target. Pion polarizabilities and radiative transitions will be measured in this experiment. The former can test a precise prediction of chiral symmetry; the latter for a_1(1260) ----> pi + gamma is important for understanding the polarizability. The experiment also measures polarizabilities and radiative transitions for Sigma hyperons. The polarizabilities can test predictions of baryon chiral perturbation theory. The radiative transitions to the Sigma*(1385…
We construct sum rules for the forward vector and tensor polarizabilities for any spin-$S$ target and apply them to the spin-1 deuteron. We calculate these polarizabilities of the deuteron to the next-to-leading order in the pionless effective field theory.
We calculate the spin-polarizabilities of the nucleon by means of dispersion relation, where the pion photoproduction amplitude predicted by chiral soliton model is utilized. We consider the $Nπ$ and $Δπ$ channels in the pion-photoproduction amplitude, and also evaluate the contribution of the anomalous term of the $π^0γγ$ process. In a narrow decay-width limit of the $Δ$ particle the result coincides with that in heavy baryon chiral perturbation theory (HBChPT) except for the interference part of the electric and magnetic amplitudes. A comparison with the multipole analysis is given.
A fit of the experimental data to the total cross section of the process $γγ->π^+π^-$ in the energy region from threshold to 2500 MeV has been carried out using dispersion relations with subtractions for the invariant amplitudes, where the dipole and the quadrupole polarizabilities of the charged pion are free parameters. As a result, the sum and the difference of the electric and magnetic dipole and quadrupole polarizabilities of the charged pion have been found: $(alpha_1+β_1)=(0.18+0.11-0.02)10^(-4)fm^3$, $(alpha_1-β_1)=(13.0+3.6-1.9)10^(-4)fm^3$, $(alpha_2+β_2)=(0.133+- 0.015)10^(-4)fm^5$,…
Neutral pion polarizabilities from four-point functions in lattice QCD
本研究采用四点函数法开展中性π介子极化率的晶格QCD原理验证模拟，发现电极化率与现有方法结果一致，磁极化率存在显著差异，提出夸克分解方法以探究非连通图的作用。
We report on a dispersion relation formalism for the virtual Compton scattering (VCS) reaction on the proton, which for the first time allows a dispersive evaluation of 4 generalized polarizabilities. The dispersion formalism provides a new tool to analyze VCS experiments above pion threshold, thus increasing the sensitivity to the generalized polarizabilities of the nucleon.
Based on Compton scattering and meson photoproduction data the polarizabilities of the nucleon are precisely studied and well understood due to recent experimental and theoretical work based on nonsubtracted dispersion relations. The {\it recommended} experimental values are $α_p=12.0\pm 0.6$, $(12.0)$, $β_p=1.9\mp 0.6$, $(1.9)$, $α_n=12.5\pm 1.7$, $(12.7\pm 0.9)$, $β_n=2.7\mp 1.8$, $(2.5\mp 0.9)$ in units of $10^{-4}$fm$^3$ and $γ^{(p)}_π=-36.4\pm 1.5$, $(-36.6)$, $γ^{(n)}_π=+58.6\pm 4.0$, $(58.3)$, $(γ^{(p)}_0=-0.58\pm 0.20)$, $γ^{(n)}_0=0.38\pm 0.22)$ in units of $10^{-4}$fm$^4$ [1]. The nu…
We present results for pion polarizabilities predicted using dispersion relations from our earlier Amplitude Analysis of world data on two photon production of meson pairs. The helicity-zero polarizabilities are rather stable and insensitive to uncertainties in cross-channel exchanges. The need is first to confirm the recent result on $(α_1-β_1)$ for the charged pion by COMPASS at CERN to an accuracy of 10\% by measuring the $γγ\toπ^+π^-$ cross-section to an uncertainty of ~1\%. Then the same polarizability, but for the $π^0$, is fixed to be $(α_1-β_1)_{π^0}=(-0.9\pm0.2)\times 10^{-4}$ fm$^{3}…
A dispersion relation formalism for the virtual Compton scattering (VCS) reaction on the proton is presented, which for the first time allows a dispersive evaluation of 4 generalized polarizabilities at a four-momentum transfer $Q^2 \leq$ 0.5 GeV$^2$. The dispersive integrals are calculated using a state-of-the-art pion photo- and electroproduction analysis. The dispersion formalism provides a new tool to analyze VCS experiments above pion threshold, thus increasing the sensitivity to the generalized polarizabilities of the nucleon.
Pion polarizabilities from a dispersive analysis of $γγ\to ππ$
本文通过色散分析末态含双π介子的光子聚变反应计算了π介子极化率，利用共形变量展开参数化更重的左手割，并结合拉格朗日方法、Adler零约束和截面数据拟合确定系数，为COMPASS、JLab和BESIII的实验测量提供理论预测。
We perform a fit of the real Compton scattering (RCS) data below pion-production threshold to extract the electric ($α_{E1}$) and magnetic ($β_{M1}$) static scalar dipole polarizabilities of the proton, using fixed-$t$ subtracted dispersion relations and a bootstrap-based fitting technique. The bootstrap method provides a convenient tool to include the effects of the systematic errors on the best values of $α_{E1}$ and $β_{M1}$ and to propagate the statistical errors of the model parameters fixed by other measurements. We also implement various statistical tests to investigate the consistency …
Polarizabilities from kaon Compton scattering
本研究针对K介子康普顿散射极化率测量受$K^*(892)$共振影响的问题，提出利用色散理论从$Kπ$中间态模型无关地重构共振贡献以扩展可用能区的方法，并分析了电、磁偶极极化率的层级特性及角覆盖度对区分二者的灵敏度。
We review various applications of dispersion relations (DRs) to the electromagnetic structure of hadrons. We discuss the way DRs allow one to extract information on hadron structure constants by connecting information from complementary scattering processes. We consider the real and virtual Compton scattering processes off the proton, and summarize recent advances in the DR analysis of experimental data to extract the proton polarizabilities, in comparison with alternative studies based on chiral effective field theories. We discuss a multipole analysis of real Compton scattering data, along w…
A fit of the experimental data to the total cross section of the process γγ->π^0π^0 in the energy region from 270 to 2250 MeV has been carried out using dispersion relations for the invariant amplitudes where the quadrupole polarizabilities are free parameters. As a result the sum and difference of the electric and magnetic quadrupole polarizabilities of the π^0 meson have been found for the first time: (\alpha2+\beta2)_π^0=(-0.181+-0.004)*10^{-4}fm^5, (\alpha2-\beta2)_pi^0=(39.70+-0.02)*10^{-4}fm^5. In addition, dispersion sum rules have been constructed for this sum and difference, respectiv…
The unknown constants in Chiral Perturbation Theory needed for an all orders analysis of the polarizabilities and electromagnetic corrections to the masses of the pseudo-Goldstone bosons are estimated at leading order in $1/N_c$. We organize the calculation in an $1/N_c$-expansion and separate long- and short-distance physics contributions by introducing an Euclidean cut-off. The long-distance part is evaluated using the ENJL model and the short-distance part using perturbative QCD and factorization. We obtain very good matching between both. We then include these estimates in a full Chiral Pe…
Low-energy Compton scattering off the proton is used for determination of the proton polarizabilities. However, the present empirical determinations rely heavily on the theoretical description(s) of the experimental cross sections in terms of polarizabilities. The most recent determinations are based on either the fixed-$t$ dispersion relations (DR) or chiral perturbation theory in the single-baryon sector ($χ$PT). The two approaches obtain rather different results for proton polarizabilities, most notably for $β_{M1}$ (magnetic dipole polarizability). We attempt to resolve this discrepancy by…
We present a phenomenological discussion of spin-independent, generalized dipole polarizabilities of hadrons entering the virtual Compton scattering process gamma* h -> gamma h. We introduce a new method of obtaining a tensor basis with appropriate Lorentz-invariant amplitudes which are free from kinematical singularities and constraints. We then motivate a gauge-invariant separation into a generalized Born term containing ground-state properties only, and a residual contribution describing the model-dependent internal structure. The generalized dipole polarizabilities are defined in terms of …
Starting from hyperbolic dispersion relations, we present a system of Roy--Steiner equations for pion Compton scattering that respects analyticity and unitarity requirements, gauge invariance, as well as crossing symmetry, and thus all symmetries of the underlying quantum field theory. To suppress the dependence on the high-energy region, we also consider once- and twice-subtracted versions of the equations, where the subtraction constants are identified with dipole and quadrupole pion polarizabilities. We consider the resolution of the $γγ\toππ$ partial waves by a Muskhelishvili-Omnès represe…
Starting from hyperbolic dispersion relations, we derive a system of Roy--Steiner equations for pion Compton scattering that respects analyticity, unitarity, gauge invariance, and crossing symmetry. It thus maintains all symmetries of the underlying quantum field theory. To suppress the dependence of observables on high-energy input, we also consider once- and twice-subtracted versions of the equations, and identify the subtraction constants with dipole and quadrupole pion polarizabilities. Based on the assumption of Mandelstam analyticity, we determine the kinematic range in which the equatio…
The main experimental works, where dipole polarizabilities of charged pions have been determined, are considered. Possible reasons for the differences between the experimental data are discussed. In particular, it is shown that the account of the $σ$-meson gives a significant correction to the value of the polarizability obtained in the latest experiment of the COMPASS collaboration. We present also new fit results for the $(γγ\toππ)$ reaction.
Virtual Compton Scattering is studied at the Thomas Jefferson National Accelerator Facility in the energy domain below pion threshold and in the Delta(1232) resonance region. The data analysis is based on the Dispersion Relation (DR) approach. The electric and magnetic Generalized Polarizabilities (GPs) of the proton and the structure functions Pll-Ptt/epsilon and Plt are determined at four-momentum transfer squared Q2=0.92 and 1.76 GeV2. The DR analysis is consistent with the low-energy expansion analysis. The world data set indicates that neither the electric nor magnetic GP follows a simple…
As a guideline for future experiments to extract the four (leading) spin polarizabilities of the nucleon, we have constructed the forward amplitude for polarized Compton scattering by dispersion integrals. These integrals have been saturated by recently measured helicity-dependent photoabsorption cross sections as well as predictions for pion photoproduction multipoles from several phenomenological descriptions and chiral perturbation theory. The comparison of these results corroborates the strategy to extract the spin polarizabilities by fitting them to polarized Compton data and fixing all h…
Charged and neutral, pion and kaon electric polarizabilities are extracted from lattice QCD using an ensemble of anisotropic gauge configurations with dynamical clover fermions. We utilize classical background fields to access the polarizabilities from two-point correlation functions. Uniform background fields are achieved by quantizing the electric field strength with the proper treatment of boundary flux. These external fields, however, are implemented only in the valence quark sector. A novel method to extract charge particle polarizabilities is successfully demonstrated for the first time.
Recent experimental results on the proton and neutron polarizabilities are examined from the point of view of backward dispersion relations. Results are found to be in reasonable agreement with the measured values. A rigorous relationship between the nucleon and pion polarizabilities is derived and shown to be in excellent agreement with several models.
A unified presentation is given on the use of dispersion relations in the real and virtual Compton scattering processes off the nucleon. The way in which dispersion relations for Compton scattering amplitudes establish connections between low energy nucleon structure quantities, such as polarizabilities or anomalous magnetic moments, and the nucleon excitation spectrum is reviewed. We discuss various sum rules for forward real and virtual Compton scattering, such as the Gerasimov-Drell-Hearn sum rule and its generalizations, the Burkhardt-Cottingham sum rule, as well as sum rules for forward n…
A status report on the topic Compton scattering and polarizabilities is presented with emphasis on the scalar t-channel as entering into dispersion theory. Precise values for the polarizabilities are obtained leading to $α_p = 12.0\pm 0.6$ $(12.0)$, $β_p=1.9\mp 0.6$ $(1.9)$, $α_n= 12.5\pm 1.7$ $(13.4)$, $β_n= 2.7 \mp 1.8$ $(1.8)$ in units of $10^{-4}$ fm$^3$ and $γ^{(p)}_π= -36.4 \pm 1.5$ $(-36.6)$, $γ^{(n)}_π= 58.6 \pm 4.0$ $(58.3)$, $(γ^{(p)}_0= -0.58\pm 0.20)$, $(γ^{(n)}_0 = +0.38\pm 0.22)$ in units of $10^{-4}$ fm$^4$, for the proton (p) and neutron (n), respectively. The data given with a…
Dispersion theory of nucleon (nucleus) Compton scattering spin polarizabilities and quasi-optical $γ$-ray polarization plane rotation and birefringence effect in a matter with polarized protons (nuclei)
该研究提出在含极化质子或原子核的物质中观测γ射线偏振面旋转及双折射效应，以探测核子康普顿散射自旋极化率随能量的变化。
We elaborate on a recently proposed extension of the Gerasimov-Drell-Hearn (GDH) sum rule which is achieved by taking derivatives with respect to the anomalous magnetic moment. The new sum rule features a {\it linear} relation between the anomalous magnetic moment and the dispersion integral over a cross-section quantity. We find some analogy of the linearized form of the GDH sum rule with the `sideways dispersion relations'. As an example, we apply the linear sum rule to reproduce the famous Schwinger's correction to the magnetic moment in QED from a tree-level cross-section calculation and o…
Fixed-$t$ subtracted dispersion relations are presented for Compton scattering off the nucleon at energies $E_γ\leq 500$ MeV, as a formalism to extract the nucleon polarizabilities from the data with a minimum of model dependence. The scalar polarizabilities difference $α- β$ and the backward spin polarizability $γ_π$ enter directly as fit parameters in this formalism.
By looking at the complex plane of the pion-mass squared we establish a dispersion relation which the static quantities, such as baryon masses, magnetic moments, polarizabilities, should obey. This dispersion relation yields insight into the differences between the heavy-baryon and relativistic calculations in the baryon sector of chiral perturbation theory.
Below pion threshold virtual Compton scattering off the nucleon gives access to the generalized electromagnetic polarizabilities. Different theoretical results for the generalized polarizabilities have been compared. In particular, the influence of the generalized polarizabilities on the unpolarized cross section and on double polarization observables has been investigated. Predictions for these observables have been obtained in the linear sigma model and chiral perturbation theory.
The statement of the authors of the article Phys. Rev. C 77, 065211 (2008) that spurious singularities occur in the dispersion relation approach, where imaginary parts of the amplitudes of the process γγ->ππare saturated by the contributions of meson resonances by using Breit-Wigner expressions, is analyzed. It is shown that there are no any additional singularities in this approach and the disagreement between the predictions of the dispersion relations and the chiral perturbation theory for (α_1-β_1)_π^{\pm} remains.
Dipole polarizabilities of light pseudoscalar mesons within the Domain Model of QCD vacuum
本文在QCD真空畴模型框架下计算轻赝标介子偶极极化率，重点分析中间标量介子场的贡献，结果与COMPASS实验及手征微扰理论相符。
Recent attempts to determine the pion polarizability by dispersion relations yield values that disagree with the predictions of chiral perturbation theory. These dispersion relations are based on specific forms for the absorptive part of the Compton amplitudes. The analytic properties of these forms are examined, and the strong enhancement of intermediate-meson contributions is shown to be connected with spurious singularities. If the basic requirements of dispersion relations are taken into account, the results of dispersion theory and effective field theory are not inconsistent.
In view of ongoing experimental activities to determine the pion polarizabilities, we have started to recalculate the available two-loop expressions in the framework of chiral perturbation theory, because they have never been checked before. We make use of the chiral Lagrangian at order p^6 now available, and of improved techniques to evaluate the two-loop diagrams. Here, we present the result for the neutral pions. The cross section for the reaction gamma+gamma->pi0+pi0 agrees with the earlier calculation within a fraction of a percent. We present analytic results for the dipole and quadrupol…
We reply to the Comment by Fil'kov and Kashevarov, arXiv:0805.4486 [hep-ph]. We show that the discrepancies between ChPT and dispersion theory, reported for the polarizability of the pion, result from applying dispersion theory to non-analytic functions.
The multichannel S- and P-wave amplitudes for the pion-pion scattering, constructed requiring analyticity and unitarity of the S-matrix and using the uniformization procedure, are elaborated using the dispersion relations with imposed crossing symmetry condition. The amplitudes are modified in the low-energy region to improve their consistency with experimental data and the dispersion relations. Agreement with data is achieved for both amplitudes from the threshold up to 1.8 GeV and with dispersion relations up to 1.1 GeV. Consequences of the applied modifications, e.g. changes of the S-wave l…
The possibility of pion--pair formation in a hot pion gas, based on the bosonic gap equation, is pointed out and discussed in detail. The critical temperature for condensation of pion pairs (Evans--Rashid transition) is determined as a function of the pion density. As for fermions, this phase transition is signaled by the appearance of a pole in the two--particle propagator. In bose systems there exists a second, lower critical temperature, associated with the appearance of the single--particle condensate. Between the two critical temperatures the pion dispersion relation changes from the usua…
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