The Q2 dependence of the flavour singlet pseudoscalar meson wave function in QCD

We calculate the Q² dependence of the flavour singlet pseudoscalar meson wave function in QCD by summing up the leading large logarithms with the aid of the ladder approximation of its Bethe-Salpeter equation. The calculation is performed in an axial gauge. We find that the Q² dependence is diagonalized by expanding the wave function in the space of longitudinal momentum fractions in terms of Gegenbauer polynomials. The anomalous dimensions agree with those of one of the two operators to be considered for polarized deep inelastic lepton-hadron scattering in addition to the operators already contributing to the spin-averaged case.     

On the Q2 dependence of parton fragmentation functions

The Q² dependences of parton fragmentation functions are calculated using lowest-order quantum chromodynamics (QCD). The resulting scaling deviations have a simple intuitive form when a suitable valence-sea decomposition is employed for the quark fragmentation functions.     

QCD predictions for the Q2 dependence of quark and gluon fragmentation functions using Altarelli-Parisi type equations

In the leading logarithmic approximation, the fragmentation functions of quarks and gluons are investigated using Altarelli-Parisi type equations. Using a new method to make the Mellin transformation, the equation is solved. Analytic expressions for the fragmentation functions near z = 0 and z = 1 are also given. Finally, numerical results for the fragmentation functions D_q^π, D_q^K are presented for different Q².     

How can we properly define the Q2 dependent parton distribution function in QCD?

We discuss how we can properly define the Q² dependent parton distribution functions in quantum chromodynamics within the framework of the operator product expansion and renormalization group techniques. It is proposed that the moments of the parton distribution functions at Q² should be defined as the hadronic expectation values of the twist-2 operators renormalized at Q². The integro-differential equations for the parton densities obtained by Altarelli and Parisi are reproduced in the leading logarithmic approximation. An application of our present formalism will be given in the case of a longitudinal structure function.     

Electroproduction of ϱ0 mesons at 0.3 < Q2 < 1.5 GeV2 and 1.7 < W < 2.8 GeV

Production and decay characteristics of electroproduced rho mesons were studied in the final state epπ⁺π^?.     

Δ++π−-electroproduction near threshold and the Q2 dependence of the contact interaction

The Q² dependence of σ (γvp→π^?Δ⁺⁺) was measured. Close to threshold σ (γvp→π^?Δ⁺⁺) is roughly proportional to 1/(m²_? + Q²)². The data are consistent with dominance of the contact term and imply that the contact term has the Q² dependence of the ? propagator.     

Electroproduction of single charged pions from deuterons at Q2 ∼0.5 (GeV/c)2

The ratio of π^? to π⁺ electroproduction cross sections from deuterons has been measured in the resonance region at an average four-momentum transfer squared of 0.5 (GeV/c)². Results are presented over a wide range of pion production angles and comparisons are made with theoretical predictions based on SU(6)_w symmetry and the Melosh transformation.     

Non-perturbative results for the ω2x2 + λx4 oscillator

For the ω²x² + λx⁴, λ >0, oscillator we prove the existence of a basic set of solutions, the set consisting of a dominant and a subdominant solution. This result does not seem to be obtainable by means of some recently developed perturbation methods.     

Intensity perturbation in the B2Σ-X2Σ band system of SrF

The intensity anomaly of the B²Σ-X²Σ system of SrF was analyzed based on the interaction between the A²Π and B²Σ states. The theoretical relative intensity was calculated from overlap integrals and spectroscopic constants previously obtained. Comparison between theoretical and observed intensities showed that the electronic transition moment of the B-X system is the same order of magnitude as that of the A-X system.     

Vierbein singularities in CP2

The compact open four-dimensional manifold $\text{C}$P², with Euclidean metric, has recently attracted attention as an example of a spacetime in which fields of half-integral spin cannot be defined in the absence of additional structure (such as an electromagnetic background). In this note we identify the specific topological anomaly responsible for this phenomenon: $\text{C}$P² contains a class of two-dimensional spheres—“complex lines”— in small neighborhoods of each of which the two transverse degrees of freedom are forced to “twist” in a characteristic way. It is shown in detail how the twists force vierbeins—tetrads of orthonormal vector fields that play a central role in the theory of spinors—to have singularities on every complex line. As an aid to visualization we construct an example of a vierbein with, loosely speaking, the smallest possible set of singularities: It is ill-defined at every point of some one complex line and smooth everywhere else. The behavior of such a vierbein near any one of its singular points is characterized explicitly. The structure of the minimally singular vierbein is used to illuminate the observation of Hawking and Pope that in the presence of an appropriate electromagnetic background, fields of any spin can exist on $\text{C}$P² as long as their electric charges are correctly quantized, but that the charge values available to half-integral-spin fields differ from those available to integral-spin fields.     

Rotational analyses of the A1Π-X1Σ+ and the a3Π-X1Σ+ systems of AlBr

The absorption spectrum of the A¹Π-X¹Σ⁺ system and the emission spectrum of the a³Π-X¹Σ⁺ system of AlBr have been examined at higher resolution than previously used. The present study has produced a significantly improved set of molecular constants for these excited states of AlBr. All data have been fitted in a self-consistent manner for an optimum set of constants. The broadening of features in the A-X system has been attributed to type c⁺ predissociation in the A¹Π state. Also, a comparison is made between the emission and absorption spectra of the A-X system.     

The electronic isotope shift in the A2Π-X2Σ+ bands of BeH,BeD and BeT

The 0-0, 1-1, 2-2, and 3-3 bands of the A²Π-X²Σ⁺ transition of the tritiated beryllium monohydride molecule have been observed at 5000 Å in emission using a beryllium hollow-cathode discharge in a He + T₂ mixture. The rotational analysis of these bands yields the following principal molecular constants.

$\text{A}{}^2\text{Π:}\text{B}{}_{\mathrm{e}}\text{= 4.192}\text{cm}{}^{\mathrm{?1}}\text{; r}{}_{\mathrm{e}}\text{= 1.333}\text{A}\text{?}$

$\text{X}{}^2\text{Σ:}\text{B}{}_{\mathrm{e}}\text{= 4.142}\text{cm}{}^{\mathrm{?1}}\text{; r}{}_{\mathrm{e}}\text{= 1.341}\text{A}\text{?}$

$\text{ω}{}_{\mathrm{e}}\text{′ ? ω}{}_{\mathrm{e}}\text{″ = 16.36}\text{cm}{}^{\mathrm{?1}}\text{; ω}{}_{\mathrm{e}}\text{′X}{}_{\mathrm{e}}\text{′ ? ω}{}_{\mathrm{e}}\text{″X}{}_{\mathrm{e}}\text{″ = 0.84}\text{cm}{}^{\mathrm{?1}}$

From the pure electronic energy difference (E_Π - E_Σ)_BeT = 20 037.91 ± 1.5 cm^?1 and the corresponding previously known values for BeH and BeD, the following electronic isotope shifts are derived

$\text{ΔE}{}_{\mathrm{e}}{}^{\mathrm{i}}\text{(}\text{BeH}\text{?}\text{BeT}\text{) = ?4.7 ≠ 1.5}\text{cm}{}^1\text{, ΔE}{}_{\mathrm{e}}{}^{\mathrm{i}}\text{(}\text{BeH}\text{?}\text{BeT}\text{) = ?1.8 ≠ 1.5}\text{cm}{}^1$

and related to the theoretical approach given by Bunker to the problem of the breakdown of the Born-Oppenheimer approximation.     

Photoluminescence of the A2Π-X2Σ+ system of the yttrium oxide molecule

A cw dye laser, tunable in the region 570–620 nm, has been used to excite photoluminescence in the A²Π-X²Σ⁺ system of the YO molecule. Two methods have been used to obtain spectra. They are (i) photoluminescence and (ii) spectrally selective excitation spectroscopy. The latter maintains the simplicity of photoluminescence, but has higher resolution. The spectra obtained at medium resolution (～0.05 nm) have been surveyed and the identification of transitions has been outlined. A method was devised in which known lower state constants were used in calculating approximate upper state constants whenever the laser simultaneously excited coincident rotational transitions in the same band. From these calculations, an approximate value of α′_e was calculated which was found to be close to, but slightly larger than values previously computed from the Pekeris formula. Many examples of collisional transfer were observed. In particular, maximum transfer from ${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ to ${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ was always observed to involve an increase of one unit in the vibrational quantum number. No transfer was observed between opposite parity components of Λ doublets, in spite of their closeness in energy, indicating that the symmetry of the electronic eigenfunctions was unaffected by collisions. Reanalysis of previous high resolution data has shown that, when the contribution of Λ-doubling parameters to the separation of the ${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and ${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ states is taken into account, the value of the spin-orbit coupling constant, A, changes significantly. It was also found necessary to include the constant, A_D, to account for the effect of centrifugal distortion on the spin-orbit interaction.     

Determination of the B1Π-X1Σ and C1Σ-X1Σ electronic transitions strengths of CaO

The strengths S_e of the B¹Π-X¹∑ and C¹∑-X¹∑ electronic transitions of the CaO molecule have been determined by absorption measurements in a shock tube. Powdered calcium oxide and a gaseous mixture of oxygen and argon in the ratio of 1:5 or 1:9 were introduced into the shock channel. The CaO gas-phase concentration was determined by using the equilibrium constant for CaO?Ca+O; atomic oxygen concentration was calculated and that of calcium was measured spectroscopically. The values of S_e (in atomic units) are for B¹Π-X¹∑, 2.32±0.49; for C¹∑-X¹∑, 2.88±0.67.     

Spectroscopic analysis of the d1Σ+→a1Σ+ and d1Σ+→b1Π green band systems of XeO

A comprehensive high resolution spectroscopic analysis has been made on the XeO green bands photographed in emission from an RF discharge source. Rotation-vibration constants derived from the analysis of the spectrum of the isotopically enriched species ¹²⁹Xe¹⁶O and ¹²⁹Xe¹⁸O were used to give RKR potential curves for the d¹Σ⁺ and b¹Π states. The bond distances and dissociation energies of the d¹Σ⁺ and b¹Π states were respectively found to be $\text{r}{}_{\mathrm{e}}\text{= 2.852 ± 0.002}\text{A}\text{?}$, D_e = 693 ± 10 cm^?1 and $\text{r}{}_{\mathrm{e}}\text{= 2.548 ± 0.002}\text{A}\text{?}$, D_e = 461 ± 10 cm^?1. For the a¹Σ⁺ state it was not possible to establish a unique vibrational numbering or to construct an RKR potential curve, since observed bands of the d¹Σ⁺→a¹Σ⁺ system involve only high vibrational levels of the a¹Σ⁺ state, which are severely predissociated. The observations are consistent with a fairly deep well, in agreement with the latest ab initio calculations which give a well depth of 0.7 eV.     

Theoretical study of the AlO blue-green (B2∑+−X2∑+ band system

Two independent, extensive theoretical calculations are reported for the relative band strengths of the AlO (B²∑⁺?X²∑⁺) blue-green system and for the radiative lifetimes of the lowest few vibrational levels of the B²∑⁺ state. The theoretical lifetimes, which include a small (<0.5%) contribution from bound-bound transitions into the A²Π state, are in excellent agreement with laser fluorescence studies. The theoretical lifetimes increase monotonically and very slowly with increasing vibrational quantum number. The relative band strengths for the blue-green system derived from the two theoretical calculations are in excellent agreement, but differ systematically from the relative band strengths of Linton and Nicholls. Our results suggest that their self-absorption corrections are not large enough, resulting in relative intensities that are too large, especially for the weak bands with r centroids less than 1.5Å.     

Analysis of perturbations in the A2Π-X2Σ+ “Red” system of CN

Deperturbed vibration-rotation constants of the A²Π(v′ = 0 to 12) and X²Σ⁺(v″ = 0 to 8) states of CN are obtained. Specroscopic data from several sources are combined using a weighted, nonlinear least-squares fitting routine. The diagonalized effective Hamiltonian matrix contains as many as two ²Π and four ²Σ⁺ mutually interacting vibronic levels. Perturbations of A²Π by both X²Σ⁺ and B²Σ⁺ are treated simultaneously. The deperturbed constants and interaction matrix elements obtained provide a significantly more accurate representation of all perturbed and unperturbed observed lines than the previously reported values. The electronic factors of the spin-orbit and rotation-electronic perturbation matrix elements for the A ～ X and A ～ B interactions are determined and several previously unreported perturbations are detected and analyzed. Merged constants and Dunham coefficients are calculated; a detailed statistical treatment of the parameters and error estimates has also been carried out.     

Vibrational analysis of the a3Σ+-X1Σ+ and b3Π-X1Σ+ band systems of GeS

Spectra of GeS have been obtained in a chemiluminescent flame produced by the reaction Ge + OCS → GeS + CO. Neither of the known band systems, D¹Π-X¹Σ⁺ and E¹Σ⁺-X¹Σ⁺, was observed, but two new band systems in the regions 350–400 and 420–650 nm were obtained. By comparison with similar systems in isovalent molecules, these were assigned as b³Π₁-X¹Σ⁺ and a³Σ⁺-X¹Σ⁺, respectively. Vibrational assignments were made with the help of the germanium isotope effect and vibrational constants were obtained for the states involved. Approximate Morse potential Franck-Condon factors were computed and were shown to fit the general trend of the intensity distribution for both systems. Addition of active nitrogen to the flame was shown to increase the intensity of the b-X system by an order of magnitude while hardly affecting the a-X system. Constants (in cm^?1) obtained for the two new states are: a³Σ⁺: T_e = 21986.3 ± 2.3, ω_e = 388.9 ± 1.0, ω_ex_e = 1.35 ± 0.11; b³Π₁: T_e = 27192.0 ± 1.8, ω_e = 435.4 ± 1.1, ω_ex_e = 1.68 ± 0.20.     

The phase of the forward π+π− → π+π− scattering amplitude

Using methods developed by Pietarinen, we evaluate the phase of the forward scattering amplitude for the process π⁺π^? → π⁺π^? in the energy range 1.0 GeV ? √ s ? 1.8 GeV. Our results are in disagreement with previous work, unless early onset of the asymptotic behaviour of the phase of this amplitude is imposed. They tend to rule out some solutions for the partial-wave amplitudes in this energy range.