Novel insights into the γγ?→π0 transition form factor

BaBar’s observation of significant deviations of the pion transition form factor (TFF) from the asymptotic expectation with Q ²>9 GeV² has brought about a serious crisis to the fundamental picture established for such a simple q[`(q)]q\bar{q} system by perturbative QCD, i.e. the dominance of collinear factorization at high momentum transfers for the pion TFF. We show that non-factorizable contributions due to open flavors in γγ ^∗→π ⁰ could be an important source that contaminates the pQCD asymptotic limit and causes such deviations with Q ²>9 GeV². Within an effective Lagrangian approach, the non-factorizable amplitudes can be related to intermediate hadron loops, i.e. K ^(∗) and D ^(∗) etc., and their corrections to the π ⁰ and η TFFs can be estimated.     

Photon-pion transition form factor at high photon virtualities within the nonlocal chiral quark model

Recently, the BABAR collaboration reported the measurements of the photon-pion transition form factor F _πγγ*(Q ²), which are in strong contradiction to the predictions of the standard factorization approach to perturbative QCD. In the present work, based on a nonperturbative approach to the QCD vacuum and on rather universal assumptions, we show that there exist two asymptotic regimes for the pion transition form factor. One regime with the asymptotic behavior F _πγ*γ(Q ²) ∼ 1/Q ² corresponds to the result of the standard QCD factorization approach, while other violates the standard factorization and leads to asymptotic behavior as F _πγ*γ(Q ²) ∼ ln(Q ²)/Q ². Furthermore, considering specific nonlocal chiral quark models, we find the region of parameters, where the existing CELLO, CLEO and BABAR data for the pion transition form factor are successfully described.     

Relativistic constituent quark model and experiments at JLab

Within the relativistic-quantum-mechanics version developed by the present authors in their previous studies, the pion electromagnetic form factor is calculated in the region of high momentum transfers reached and planned to be reached in experiments at the Thomas Jefferson Laboratory (JLab). An asymptotic expansion was obtained for the pion form factor for Q ² → ∞. This expansion describes well existing experimental values and the results of a QCD simulation of future experiments at JLab. It is shown that the region of experiments at JLab is an asymptotic region for the relativistic constituent quark model and that the behavior of the pion form factor, F _π (Q ²)Q ² = const, can be obtained within this model.     

On the possibility to measure the π0→γγ decay width and the γ?γ→π0 transition form factor with the KLOE-2 experiment

A possibility of KLOE-2 experiment to measure the width \varGamma_{p⁰ ?gg}\varGamma_{\pi^{0} \to\gamma\gamma} and the π ⁰ γγ ^∗ form factor F(Q ²) at low invariant masses of the virtual photon in the space-like region is considered. This measurement is an important test of the strong interaction dynamics at low energies. The feasibility is estimated on the basis of a Monte-Carlo simulation. The expected accuracy for \varGamma_{p⁰ ?gg}\varGamma_{\pi^{0} \to\gamma\gamma} is at a per cent level, which is better than the current experimental world average and theory. The form factor will be measured for the first time at Q ²≤0.1 GeV² in the space-like region. The impact of these measurements on the accuracy of the pion-exchange contribution to the hadronic light-by-light scattering part of the anomalous magnetic moment of the muon is also discussed.     

On the pion distribution amplitude shape

We argue that the recent BaBar data on γ → π e.m. transition form factor at large photon virtuality supports the idea that pion distribution amplitude (DA) is close to unity with ϕ_π^′(0)/6 ≫ 1 at a normalization point of μ = 0.6–0.8 GeV. Such pion DA can be obtained in the effective chiral quark model. The possible flat shape of the pion DA implies that the standard expansion of the DA in Gegenbauer polynomials can be divergent. On basis of chiral models we predict that the two-pion DA should exhibit anomalous endpoint behaviour for pions in the S-wave and that such feature is absent for higher partial waves. The latter implies that the ρ, f ₂, etc. meson DAs have no anomalous endpoint behaviour. Possible implications of such pion DA for other hard exclusive processes are shortly discussed.     

Pion form factor in QCD: How to calculate?

The calculation of the pion form factor F _π(Q ²) in QCD is discussed. The main points of the nonlocal condensate QDC sum rule approach are considered and its results for the pion form factor are shown compared with the predictions of the perturbative and lattice QCD. The local duality (LD) approach for the pion FF in QCD is studied. It is shown that the main parameter of the approach for Q ² ≥ 2 GeV², namely, s ₀^LD(Q ²) should grow with an increase in Q ², rather than remain constant.     

Radiative corrections in Ke4 decay

The final state interaction of pions in the decay K ^±→π ⁺ π ⁻ e ^± ν allows one to obtain the value of the isospin and angular momentum zero pion–pion scattering length a ₀⁰. To extract this quantity from experimental data the radiative corrections (RC) have to be taken into account. Based on the lowest order results and the factorization hypothesis, we get the expressions for RC in the leading and next-to leading logarithmical approximation. It is shown that the decay width dependence on the lepton mass m _e through the parameter s = \fraca2p(ln\fracM²m_e²-1)\sigma=\frac{\alpha}{2\pi}(\ln\frac{M^{2}}{m_{e}^{2}}-1) has the standard form of the Drell–Yan process and is proportional to the Sommerfeld–Sakharov factor. The numerical estimations are presented.     

Roy–Steiner equations for <Emphasis Type="Italic">γγ</Emphasis>→<Emphasis Type="Italic">ππ</Emphasis>

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 equations are valid. As an application, we consider the resolution of the γγ→ππ partial waves by a Muskhelishvili–Omnès representation with finite matching point. We find a sum rule for the isospin-two S-wave, which, together with chiral constraints, produces an improved prediction for the charged-pion quadrupole polarizability (a₂-b₂)^p±=(15.3±3.7)×10^-4(\alpha_{2}-\beta_{2})^{\pi^{\pm}}=(15.3\pm3.7)\times 10^{-4} fm⁵. We investigate the prediction of our dispersion relations for the two-photon coupling of the σ-resonance Γ _σγγ . The twice-subtracted version predicts a correlation between this width and the isospin-zero pion polarizabilities, which is largely independent of the high-energy input used in the equations. Using this correlation, the chiral perturbation theory results for pion polarizabilities, and our new sum rule, we find Γ _σγγ =(1.7±0.4) keV.     

Five Dimensional Cosmological Models in General Relativity

A five dimensional Kaluza-Klein space-time is considered in the presence of a perfect fluid source with variable G and Λ. An expanding universe is found by using a relation between the metric potential and an equation of state. The gravitational constant is found to decrease with time as G∼t ^−(1−ω) whereas the variation for the cosmological constant follows as Λ∼t ⁻², L ~ ([(R)\dot]/R)²\Lambda \sim (\dot{R}/R)^{2} and L ~ [(R)\ddot]/R\Lambda \sim \ddot{R}/R where ω is the equation of state parameter and R is the scale factor.     

Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

We study random spatial permutations on ℤ³ where each jump x↦π(x) is penalized by a factor e^{-T|| x-p(x)||2}\mathrm{e}^{-T\| x-\pi (x)\|^{2}}. The system is known to exhibit a phase transition for low enough T where macroscopic cycles appear. We observe that the lengths of such cycles are distributed according to Poisson-Dirichlet. This can be explained heuristically using a stochastic coagulation-fragmentation process for long cycles, which is supported by numerical data.     

D?→Dπ and D?→Dγ decays: axial coupling and magnetic moment of D? meson

The axial coupling and the magnetic moment of D ^∗-meson or, more specifically, the couplings g_D^*Dpg_{D^{\ast}D\pi} and g_D^*Dgg_{D^{\ast}D\gamma }, encode the non-perturbative QCD effects describing the decays D ^∗→Dπ and D ^∗→Dγ. We compute these quantities by means of lattice QCD with N _f=2 dynamical quarks, by employing the Wilson (“clover”) action. On our finer lattice (a≈0.065 fm) we obtain g_D^*Dp⁺=20±2g_{D^{\ast}D\pi^{+}}=20\pm2, and g_{D^*0 D⁰g}=2.0±0.6 GeV^-1g_{D^{\ast0} D^{0}\gamma}=2.0\pm 0.6~{\rm GeV}^{-1}. This is the first determination of g_{D^*0 D⁰g}g_{D^{\ast0} D^{0}\gamma} on the lattice. We also provide a short phenomenological discussion and the comparison of our result with experiment and with the results quoted in the literature.     

Search for the f1(1285) → π+π-π0 decay with the VES detector

The isospin-violating decay f ₁(1285) → π⁺π^-π⁰ has been studied at the VES facility. This study is based on the statistics acquired in π^- Be interactions at 27, 36.6 and 41 GeV/c in the diffractive reaction π^- N → (f ₁π^-)N . The f ₁(1285) → π⁺π^-π⁰ decay is observed. The preliminary ratio of decay probabilities BR(f ₁(1285) → π⁺π^-π⁰) to BR(f ₁(1285) → ηπ⁺π^-)^. BR(η → γγ) is ∼ 2% . Original article based on material presented at HADRON 2007.     

Anomalous dipion invariant mass distribution of the ?(4S) decays into ?(1S)π+π? and ?(2S)π+π?

To solve the discrepancy between the experimental data on the partial widths and lineshapes of the dipion emission of ϒ(4S) and the theoretical predictions, we suggest that there is an additional contribution, which had not been taken into account in previous calculations. Noticing that the mass of ϒ(4S) is above the production threshold of B[`(B)]B\bar{B}, the contribution of the sequential process \varUpsilon(4S)? B[`(B)]? \varUpsilon(nS)+S?\varUpsilon(nS)+p⁺p^-\varUpsilon(4S)\to B\bar{B}\to \varUpsilon(nS)+S\to\varUpsilon(nS)+\pi^{+}\pi^{-} (n=1,2) may be sizable, and its interference with that from the direct production would be important. The goal of this work is to investigate if a sum of the two contributions with a relative phase indeed reproduces the data. Our numerical results on the partial widths and the lineshapes d\varGamma(\varUpsilon(4S)?\varUpsilon(2S,1S)p⁺p^-)/d(m_p⁺p^-)d\varGamma(\varUpsilon(4S)\to\varUpsilon(2S,1S)\pi^{+}\pi^{-})/d(m_{\pi ^{+}\pi^{-}}) are satisfactorily consistent with the measurements; thus the role of this mechanism is confirmed. Moreover, with the parameters obtained by fitting the data of the Belle and BaBar collaborations, we predict the distributions dΓ(ϒ(4S)→ϒ(2S,1S)π ⁺ π ⁻)/dcosθ, which have not been measured yet.     

Dimer state in the two-dimensional anisotropic alternated-exchange Heisenberg model

An analysis is made of the two-dimensional Heisenberg model with S=1/2, anisotropic exchange interaction between nearest neighbors, and alternating exchange in two directions, [100] and [010] (corresponding to condensation of the (π, π) mode) and in one direction [100] (corresponding to condensation of the (π, 0) mode). The quantum Monte Carlo method is used to calculate the thermodynamic characteristics and the spin correlation functions which are used as the basis to determine the boundary of stability of an anisotropic antiferromagnetic with respect to alternation of exchange δ=(1−J ^x,y /J ^z )^0.4 in the (π,π) model and δ=(1−J ^x,y /J ^z )^0.31 in the (π,0) model. In the (π,0) model a disordered quantum state exists in the range (1−J ^x,y /J ^z )^0.31<δ<(0.3–0.35). The energy (E−0.68)=0.36δ ^1.80(6) and 0.21 δ ^2.0(5), the energy gap between the ground and excited states H _c (δ)=1.96δ ^2.(1), 1.8(1) (δ−0.35(3))^0.67(2) were determined as a function of the alternation of exchange in the (π,π)-and (π,0) models, respectively. Fiz. Tverd. Tela (St. Petersburg) 40, 1080–1085 (June 1998)     

Chiral phenomenological relations between rates of rare radiative decays of kaon to pion and leptons and the meson form factors

In the framework of the chiral perturbation theory we obtain the phenomenological relations between decay branches of rare radiative kaon to pion and leptons K ⁺ → π⁺ l ⁺ l ⁻ and K _S ⁰ → π⁰ l ⁺ l ⁻ and meson form factors. The comparison of these results with the present-day experimental data shows us that the ChPT relations for a charge kaon can determine meson form factors from already measured decay rates at high precision level. However, in the case of the neutral kaon decays K ⁰ → π⁰ e ⁺ e ⁻(μ⁺μ⁻) the form factor data are known to a higher precision than data on the differential rates of radiative kaon decay K ⁰ π⁰ e ⁺ e ⁻(μ⁺μ⁻).     

Di-pion emission in heavy quarkonia decays

The dipion spectrum for the ϒ(nS) → ϒ(n′S) transition with n < 4 has the form dw/dq ∼ (phase space) |η − x|², with x = q ² − 4m _π² / (ΔM)² − 4m _π² < q ² ≡ M _ππ², and ΔM = M(nS) − M(n′S). The parameter η is calculated and the spectrum is shown to reproduce the experimental data for all three types of decays: 3 → 1, 2 → 1, and 3 → 2 with η ≈ 0.5, 0, and −3, respectively. The text was submitted by the author in English.     

Calculation of the structure and vibrational states for anionic forms of Co-, Ni-, and Cu-porphines

The structure and vibrations of neutral porphine metal complexes (Me-P, Me = Co, Ni, Cu) and their d-anionic forms with an additional electron localized in vacant d_x² -_y² - d_x^2 -_y^2 - and d_z² d_z^2 -orbitals are compared based on calculations by a DFT method. It is shown that such electron population causes a significant increase of the electronic charge on the macrocycle rather than on the Me atom and is accompanied by a considerable redistribution of π- and σ-electron densities (ρ_π ρ_σ). A predominant gain of ρ_π (0.49e) is found for the monoanion of Co-P (Co-P^–, d_z² d_z^2 -monoanion); of ρ_σ (0.6e), for Ni-P^–( d_x² -_y² - d_x^2 -_y^2 - monoanion). These features are reflected in both the structure of the anions and the behavior of their vibrational frequencies. The greatest frequency shifts among IR active modes when populating the d_z² d_z^2 - and d_x² -_y² - d_x^2 -_y^2 - orbitals occur for out-of-plane vibrations (>30 cm^–1) and in-plane modes (34–46 cm^–1) involving MeN- and C_αC_m-bonds, respectively. Abnormally large frequency lowering is found for B_1g-type modes (active in the resonance Raman spectrum) involving mainly C_αC_m-, C_βC_β-, C_αC_β-, and MeN-bonds. This is related to a change in the d_π-e_g interaction strength during such vibrations that contributes to a decrease in the corresponding force constants.     

Pion and kaon electromagnetic form factors in a SUL(3) ⊗ SUR(3) effective Lagrangian

A SU(2) effective Lagrangian is extended to a SU _L(3) ⊗SU _R(3) by including the vector and axial vector meson. With this effective Lagrangian, electromagnetic form factors of charged pion and kaon are calculated in both time- and space-like regions. The pseudoscalar meson loops are taken into account. Good agreement with experimental data is obtained for those form factors and charged pseudoscalar meson radii. Decay widths of ρ→ππ and φ→K ⁺ K ^- are also calculated and shown to agree with experimental data very well. Received: 20 December 1999 / Accepted: 12 October 2000