The pion form factor from lattice QCD with two dynamical flavours

We compute the electromagnetic form factor of the pion, using non-perturbatively O(a) improved Wilson fermions. The calculations are done for a wide range of pion masses and lattice spacings. We check for finite size effects by repeating some of the measurements on smaller lattices. The large number of lattice parameters we use allows us to extrapolate to the physical point. For the square of the charge radius we find fm², in good agreement with experiment. PACS 12.38.Gc; 13.40.Gp; 14.40.-n     

Pion radii in the nonlocal chiral quark model

The electromagnetic radius of a charged pion and the transition radius of a neutral pion are calculated in the framework of the nonlocal chiral quark model. It is shown in this model that the vector meson contributions to the pion radii are noticeably suppressed in comparison with a similar contribution in the local Nambu-Jona-Lasinio model. The form factor for the process is calculated for -1 GeV²< q ² < 1.6 GeV². Our results are in satisfactory agreement with experimental data.Received: 28 November 2003, Published online: 13 July 2004PACS: 14.40.-n Mesons - 11.10.Lm Nonlinear or nonlocal theories and models - 12.39.Ki Relativistic quark model     

Weakly model dependent pion form factor with physically interpretable parameters

Conclusion We have constructed the pion form factor model satisfying fundamental principles, providing some freedom for the asymptotic behaviour and depending only on the adjustable parameters with a definite physical interpretation. The explicit incorporation of the pion form factor left-hand cut from the second Riemann sheet into a model was carried out for the first time. As a result we have described (contrary to the previous attempts [18, 19]) the pion form factor data from the range of momenta –9·770 GeV² t 9·579 GeV² and data on theP-wave isovector phase shift in the elastic region simultaneously with the adjustable parameters acquiring very reasonable values. The asymptotic behaviour was enforced by the data to be consistent with the naive quark model prediction. The calculated values of the pion charge radius as well as theP-wave isovector scattering length coincide with the world averaged values. With regard to these positive features of our pion form factor model it can be employed for the reliable calculation (via unitarity) of theP-wave isovector scattering partial amplitude outside the physical region, which could be useful in a saturation of various sum rules in scattering physics.     

Form factors in lattice QCD

Lattice simulations of QCD have produced precise estimates for the masses of the lowest-lying hadrons which show excellent agreement with experiment. By contrast, lattice results for the vector and axial vector form factors of the nucleon show significant deviations from their experimental determination. We present results from our ongoing project to compute a variety of form factors with control over all systematic uncertainties. In the case of the pion electromagnetic form factor we employ partially twisted boundary conditions to extract the pion charge radius directly from the linear slope of the form factor near vanishing momentum transfer. In the nucleon sector we focus specifically on the possible contamination from contributions of higher excited states. We argue that summed correlation functions offer the possibility of eliminating this source of systematic error. As an illustration of the method we discuss our results for the axial charge, g _A , of the nucleon.     

Electromagnetic Properties of Diquarks

Diquark correlations play an important role in hadron physics. The properties of diquarks can be obtained from the corresponding bound-state equation. Using a model for the effective quark-quark interaction that has been proved successfully in the light meson sector, we solve the scalar diquark Bethe-Salpeter equations and use the obtained Bethe-Salpeter amplitudes to compute the diquarks electromagnetic form factors. The scalar ud diquark charge radius is about 8% larger than the pion charge radius, indicating that these diquarks are somewhat larger in size than the corresponding mesons. We also provide analytic fits for the form factor over a moderate range in Q², which may be useful, for example, in building quark-diquark models of nucleons.     

Constraining the low-energy pion electromagnetic form factor with space-like and phase of time-like data

The Taylor coefficients c and d of the EM form factor of the pion are constrained using analyticity, knowledge of the phase of the form factor in the time-like region, 4m _π ²≤t≤t _in and its value at one space-like point, using as input the (g−2) of the muon. This is achieved using the technique of Lagrange multipliers, which gives a transparent expression for the corresponding bounds. We present a detailed study of the sensitivity of the bounds to the choice of time-like phase and errors present in the space-like data, taken from recent experiments. We find that our results constrain c stringently. We compare our results with those in the literature and find agreement with the chiral perturbation-theory results for c. We obtain when c is set to the chiral perturbation-theory values.     

Stranger still: Kaon loops and strange quark matrix elements of the nucleon

Intrinsic strangeness contributions to lowenergy strange quark matrix elements of the nucleon are modelled using kaon loops and meson-nucleon vertex functions taken from nucleon-nucleon and nucleon-hyperon scattering. A comparison with pion loop contributions to the nucleon electromagnetic (EM) form factors indicates the presence of significantSU(3)-breaking in the mean-square charge radii. As a numerical consequence, the kaon loop contribution to the mean square Dirac strangeness radius is significantly smaller than could be observed with parity-violating elastic p experiments planned for CEBAF, while the contribution to the Sachs radius is large enough to be observed with PV electron scattering from (0+, 0) nuclei. Kaon loops generate a strange magnetic moment of the same scale as the isoscalar EM magnetic moment — a scale large enough to be observed at CEBAF — and a strange axial vector form factor having roughly one-third of the magnitude extracted from elastic scattering. In the chiral limit, the loop contribution to the fraction of the nucleon's scalar density arising from strange quarks has roughly the same magnitude as the value extracted from analyses of _N . The importance of satisfying the Ward-Takahashi Identity, not obeyed by previous calculations, is also illustrated, and the sensitivity of results to input parameters is analyzed.This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E) under contracts #DE-AC02-76ER 03069 and #DE-AC05-84ER 40150     

A new determination of the γπ → ππ anomalous amplitude via πe<Superscript>-</Superscript> → π<Superscript>-</Superscript>e<Superscript>-</Superscript>π<Superscript>0</Superscript> data

We discuss the reaction π^-e^- → π^-e^-π⁰ with the purpose of obtaining information on the γπ → ππ anomalous amplitude _3π. We compare a full calculation at (p⁶) in chiral perturbation theory and various phenomenological predictions with the existing data of Amendolia et al. By integrating our theory results using Monte Carlo techniques, we obtain σ = 2.05 nb at (p⁶) and σ = 2.17 nb after including the dominant electromagnetic correction. Both results are in good agreement with the experimental cross-section of σ = (2.11±0.47) nb. On the basis of the ChPT results one would extract from the experimental cross-section as amplitudes _3π^(0)extr = (9.9±1.1) GeV^-3 and _3π^(0)extr = (9.6±1.1) GeV^-3, respectively, which have to be compared with the low-energy theorem _3π = e/(4π²F_π³) = 9.72GeV^-3. We emphasize the need for new data to allow for a comparison of experimental and theoretical distributions and to obtain _3π with smaller uncertainty.     

Electromagnetic vertex function of the pion at T>0

The matrix element of the electromagnetic current between pion states is calculated in quenched lattice QCD at a temperature of T=0.93 T_c. The non-perturbatively improved Sheikholeslami–Wohlert action is used together with the corresponding improved vector current. The electromagnetic vertex function is extracted for pion masses down to 360 MeV and momentum transfers Q²≤2.7 GeV².     

On neutron charge structure

The aim of the present work is to investigate the charge structure of neutron by calculating its root mean square charge radius r _n ^{2 1/2} and the corresponding charge form factor,G _n(q), within the general framework of quark-parton model. The neutron (the nucleon in general) is considered to be disc-like in conformity with the idea of Lorentz invariance. Using the proton charge radius, r _p ² =0·707 fm² as input, our calculations not only reproduce the observed negative sign correctly but also give reasonable value for the magnitude of r _n ² when compared with the experimental limits. –0·16r _n ² –0·11 (fm²). Predictiona are also made for the proton and the neutron charge form factors and the results compared with the data.One of us (D. P.) would like to express his gratitude to Professor A. N. Mitra for his interest and advice in the work, and to Dr. S. M. Mustafa for the warm hospitality at the University of Mosul.     

Radiative corrections to pion-nucleus bremsstrahlung

We calculate the one-photon loop radiative corrections to virtual pion Compton scattering → , that subprocess which determines in the one-photon exchange approximation the pion-nucleus bremsstrahlung reaction Z → Z . Ultraviolet and infrared divergencies of the loop integrals are both treated by dimensional regularization. Analytical expressions for the O() corrections to the virtual Compton scattering amplitudes, A(s, u, Q) and B(s, u, Q) , are derived with their full dependence on the (small) photon virtuality Q from 9 classes of contributing one-loop diagrams. Infrared finiteness of these virtual radiative corrections is achieved (in the standard way) by including soft photon radiation below an energy cut-off . In the region of low center-of-mass energies, where the pion-nucleus bremsstrahlung process is used to extract the pion electric and magnetic polarizabilities, we find radiative corrections up to about -3% for = 5 MeV. Furthermore, we extend our calculation of the radiative corrections to virtual pion Compton scattering → by including the leading pion-structure effect in the form of the polarizability difference - . Our analytical results are particularly relevant for analyzing the data of the COMPASS experiment at CERN which aims at measuring the pion electric and magnetic polarizabilities with high statistics using the Primakoff effect.     

Neutrino charge radius in the minimal supersymmetric standard model

As recently suggested by Degrassi, Marciano, and Sirlin, the electromagnetic form factor of the neutrino extracted from low energy elastic scattering cross sections ofv e and can be used to define the neutrino charge radius in the Standard Model in a gauge-invariant way. The complete one-loop contributions to the form factor consist not only the induced vertex but also the Z vertex corrections and theW andZ propagators arise from the counterterm diagrams, and theWW andZZ box diagrams. We show that the neutrino charge radius is a finite and gauge independent quantity in the Minimal Supersymmetric Standard Model. We consider only the oblique corrections to the neutrino charge radius in our numerical work, because the vertex and box contributions are small. The dependence of the oblique corrections on the supersymmetric parameters in each of the three sectors: Higgs, scalar matter and gaugino sectors are studied by taking into account of the current experimental bounds from LEP.     

decays in Chiral Perturbation Theory

We study the decays K→ππ in one-loop two-flavour Chiral Perturbation Theory. We provide arguments why the calculation of the coefficient of the pionic chiral logarithm ℓ_M=M²logM² is unique and then perform the calculation. As a check we perform the reduction of the known three-flavour result. Our result can be used to perform the extrapolation to the physical pion mass of direct lattice QCD calculations of K→ππ at fixed m_s or . The underlying arguments are expected to be valid for heavier particles and other processes as well.     

Vertex form factors of the rho-meson nucleon interaction at low momentum transfer

We calculated the one pion loop contributions to the form factors of the rho-nucleon interaction within a field theoretical model using effective Lagrangians. The results show that the tensor interaction is likely to be described well within such an approach. We obtain a magnetic coupling in the range of 3.5–7.1 atQ ²=0, depending on the choice of pion-nucleon form factor and a dipole form for theQ ² dependence tensor form factor with a scale of roughly 900 MeV for space-like momentum transfer. The results agree with the findings of the recent analysis of thep-N interaction and support a small scale parameter (0.5 GeV) for the pion-nucleon form factor.This work is supported by BMFT (06BO7027) and COSY-KFA Jülich (41140512)     

Measurement of and the dipion contribution to the muon anomaly with the KLOE detector

We have measured the cross section σ(e⁺e⁻→π⁺π⁻γ(γ)) at DAΦNE, the Frascati -factory, using events with initial state radiation photons emitted at small angle and inclusive of final state radiation. We present the analysis of a new data set corresponding to an integrated luminosity of 240 pb⁻¹. We have achieved a reduced systematic uncertainty with respect to previously published KLOE results. From the cross section we obtain the pion form factor and the contribution to the muon magnetic anomaly from two-pion states in the mass range 0.592<M_ππ<0.975 GeV. For the latter we find Δ^ππa_μ=(387.2±0.5_stat±2.4_exp±2.3_th)×10⁻¹⁰.     

Bose-Einstein correlations in neutrino and antineutrino interactions with nucleons

Four data sets of charged current neutrino and antineutrino interactions with neon, deuterium and hydrogen collected from BEBC and the 15 Foot Bubble Chamber are used to study the Bose-Einstein correlations between like-sign charged pions. Two forms of the parametrization for the effect are used. No substantial differences are found between the data sets obtained with neon, deuterium and hydrogen targets. The Lorentz invariant parametrization of Goldhaber gives for the radius of the pion emission region the valuer _G =0.80 ±0.04±0.16 fm and for the chaoticity parameter the value =0.61±0.04±0.15. Using the Kopylov-Podgoretskii parametrization yieldsr _K =1.27±0.06±0.12 fm, =0.58±0.03±0.12 and for the pion source lifetimec =0.52±0.05±0.12 fm. The Goldhaber parametrization was used to study the effect in further detail. The same emission radius and the same strength of the effect were found for particles produced in interactions on neutrons and protons. The data are compatible with a spherical shape of the pion emission region. No multiplicity or forward-backward dependences are found. No dependence of the effect on the event kinematical variables is seen.     

Boost operators in Coulomb gauge QCD: The pion form factor and Fock expansions in \phi radiative decays

In this article we re-derive the Boost operators in Coulomb Gauge Yang-Mills theory employing the path integral formalism and write down the complete operators for QCD. We immediately apply them to note that what are usually called the pion square, quartic ... charge radii, defined from derivatives of the pion form factor at zero squared momentum transfer, are completely blurred out by relativistic and interaction corrections, so that it is not clear at all how to interpret these quantities in terms of the pion charge distribution. The form factor therefore measures matrix elements of powers of the QCD boost and M?ller operators, weighted by the charge density in the target’s rest frame. In addition we remark that the decomposition of the $ \eta{^\prime}$ wave function in quarkonium, gluonium, ..., components attempted by the KLOE Collaboration combining data from $ \phi$ radiative decays, requires corrections due to the velocity of the final-state meson recoiling against a photon. This will be especially important if such decompositions are to be attempted with data from J/ $ \psi$ decays.     

Extraction of the neutron charge form factor G<Superscript>n</Superscript><Subscript>E</Subscript>(Q<Superscript>2</Superscript>) from the charge form factor of deuteron G<Superscript>d</Superscript><Subscript>C</Subscript>(Q<Superscript>2</Superscript>)

We extract the neutron charge form factor G ⁿ _E(Q ²) from the charge form factor of deuteron G ^d _C(Q ²) obtained from T ₂₀(Q ²) data at 0≤Q ²≤ 1.717 (GeV/c)². The extraction is based on the relativistic impulse approximation in the instant form of the relativistic Hamiltonian dynamics. Our results (12 new points) are compatible with existing values of the neutron charge form factor of other authors. We propose a fit for the whole set (36 points) taking into account the data for the slope of the form factor at Q ² = 0. Received: 26 July 2002 / Accepted: 18 September 2002 / Published online: 4 February 2003 RID="a" ID="a"e-mail: krutov@ssu.samara.ru RID="b" ID="b"e-mail: troitsky@theory.sinp.msu.ru Communicated by V.V. Anisovich     

Calculation of nonlinear modes guided by step-index fibers with finite cladding thickness

A circular optical fiber consisting of a nonlinear core bounded by a nonlinear cladding with a finite thickness is considered here. In this paper, the influence of the cladding radius in comparison with the core radius in the Power – effective index relations P(/k ₀) is examined. Numerical results indicate that the radius ratio between the cladding and the core is an important factor in changing the Power – effective index relations P(/k ₀) when the cladding has a stronger nonlinearity than that of the core.     

New constraints on the pion EM form factor using $ \Pi^{{ \prime}}_{}$(- Q2)

We study the constraints arising on the expansion parameters c and d of the pion electromagnetic form factor from the inclusion of pure spacelike data and the phase of timelike data along with one spacelike datum, using as input the first derivative of the QCD polarization amplitude (- Q ²) . These constraints when combined with other analyses, provide a valuable check on a determination of c due to Guo et al. and on our previous work where pionic contribution to the (g - 2) of the muon was used as the input. This work further illustrates the power of analyticity techniques in form factor analysis.