QCD predictions for four-jet final states in e+e− annihilation

We have calculated the four-jet production processes $\text{e}{}^{\mathrm{+}}\text{e}{}^{\mathrm{?}}\text{→q}\text{q}\text{gg}$ and $\text{e}{}^{\mathrm{+}}\text{e}{}^{\mathrm{?}}\text{→q}\text{q}\text{q}\text{q}$ to lowest-order QCD perturbation theory. We find that (q$\text{q}$q$\text{q}$) production is small compared to the dominant process $\text{e}{}^{\mathrm{+}}\text{e}{}^{\mathrm{?}}\text{→q}\text{q}\text{gg}$ which can in part be traced to the fact that the latter process is more singular as the 2- and 3-jet phase-space limits are approached. We present differential 4-jet acoplanarity distributions and compare them with non-perturbative acoplanarity distributions at maximum PETRA and PEP energies. Leading log cross-section formulae are derived for various cut-off procedures and are compared to the results of our numerical integrations. We also present results on associated heavy quark production in e⁺e^? annihilation.     

The average transverse distance between partons as measured by nucleon form factors

The values of $\text{d}\text{F}{}_1\text{(q}{}^2\text{)}\text{d}\text{q}{}^2$ at q²=0 for the neutron and the proton provide a measure of the average transverse separations squared, 〈$\text{y}{}^2$〉, between a u or d quark and the rest of the partons in a nucleon. Using the measured values of the form factors (together with parton x-distributions), we find that 〈$\text{y}{}^2\text{= 17.4}\text{GeV}{}^{\mathrm{?2}}$ for u quarks and 16.4 GeV^?2 for d quarks in a proton. We speculate that the small difference between u and d quarks is caused by “quark pairing” and discuss other possible experimental signatures of quark pairing.     

The asymptotics of the gap in the Mathieu equation

We provide a simple proof that the kth gap, Δ_k, for the Mathieu operator $\text{?d}{}^2\text{dx}{}^2\text{+ 2κ}\text{cos}\text{(2x)}$ is $\text{Δ}{}_{\mathrm{k}}\text{= 8(}\text{κ}\text{4}\text{)}{}^{\mathrm{k}}\text{[(k ? 1)!]}{}^{\mathrm{?2}}\text{(1 + o(k}{}^{\mathrm{?2}}\text{))}$, a result obtained (up to the value of an integral) by Harrell. The key observation is that what is involved is tunneling in momentum space.     

Reflection of long waves by interfaces

We derive comparison identities for waves satisfying the equation d²Ψ/dz²+q²(z)Ψ=0. One of these identities is used to show that to second order in the product (wavenumber component normal to interface) × (interface thickness), the reflection amplitude is given by $\text{r=(1?2q}{}_1\text{q}{}_2\text{l}{}^2\text{)}\text{(q}{}_1\text{?q}{}_2\text{)}\text{(q}{}_1\text{+q}{}_2\text{)}$, where l is a legnth determined by the deviation of the interface profile from a step, and q₁, q₂ are the normal components of the wave numbers in media 1 and 2 on either side of the interface. For the continuous interfaces discussed, l is about two-fifths of the 10–90 interface thickness. The corresponding formula for the transmission amplitude is $\text{t=(1+}\text{1}\text{2}\text{(q}{}_1\text{?q}{}_2\text{)}{}^2\text{l}{}^2\text{)}\text{2q}{}_1\text{(q}{}_1\text{+q}{}_2\text{)}$.     

Perturbation theory with instantons

We develop “perturbation theory” rules for calculating the effect of instantons in a pure Yang-Mills theory with no fermions, in the “dilute gas” approximation in which the N-instanton solution is assumed to be the sum of N widely separated one-instanton solutions. These rules are then used to compute the gluon propagator and proper vertex function including all orders of the instanton interaction but only to lowest order in the gluon coupling. It is to be expected that such an approximation is valid only for momenta q larger than the physical mass μ. The result is that in this regime instantons cause variations in the propagator and vertex of the form where b is the coefficient in the expansion of the β function: β = bg³ + 3. .     

Hadronic lepton-pair production beyond the leading order in perturbative QCD

Hadronic lepton-pair production in perturbative QCD is calculated to order $\text{g}{}^{\mathrm{<mtext>2</mtext>}}$ to be usable in combination with the existing calculation of the anomalous dimension of order $\text{g}{}^{\mathrm{<mtext>4</mtext>}}$. The structure function of the subprocess $\text{q}\text{+}\text{q}\text{→ γ +}$ anything in an arbitrary covariant gauge for gluons is found to have gauge-dependent contributions which are exactly cancelled by contributions from operator matrix elements to give the gauge-independent “coefficient function” for lepton-pair production in the minimal subtraction scheme. This provides a consistency check of the perturbative QCD recipe. A comparison is made with other existing calculations.     

Kerr-effect and dielectric tensor elements of magnetite (Fe3O4) between 0.5 and 4.3 eV

The complex polar Kerr effect (rotation and ellipticity) of magnetite single crystals has been measured at room temperature between 0.5 and 4.3 eV. From the magneto-optical data and the optical constants, the off-diagonal elements (?_xy) of the dielectric tensor has been derived. Three well separated magneto-optical transitions have been indentified. At about 0.75 eV one strong magneto-optical structure with a diamagnetic line shape is assigned to a $\text{3d}{}^6\text{→3d}{}^5\text{(}{}^6\text{A}{}_{\mathrm{1g}}\text{) 4s}$ transition from Fe²⁺ in octahedral sites. Two other structures with paramagnetic line shapes near 1.85 and 2.90 eV are assigned to the orbital promotion processes $\text{3d}{}^6\text{(Fe}{}^{\mathrm{2+}}{}_{\mathrm{oct}}\text{)→3d}{}^5\text{(}{}^4\text{T}{}_{\mathrm{1g}}\text{) 4s}$ and $\text{3d}{}^5\text{(Fe}{}^{\mathrm{3+}}{}_{\mathrm{tet}}\text{)→3d}{}^4\text{(}{}^5\text{T}{}_2\text{) 4s}$, respectively, which take into account Fe 3d^n?1 final state effects.     

Inner Coulomb corrections to pion-nucleus scattering

A semiclassical model for inner Coulomb corrections to pion-nucleus scattering in the Δ-resonance region is suggested. Its main consequence is a change of the nuclear radius R, giving $\text{R(π}{}^{\mathrm{?}}\text{)}\text{R(π}{}^{\mathrm{+}}\text{)}\text{= 1+2Z}\text{αE}\text{k}{}^2\text{R}$. The Glauber model can be extended to include these effects.     

The evolution of a cluster of particles and the invariants of a time-dependent oscillator

A simple physical realization is found for the time-dependent harmonic oscillator equation $\text{q}\text{?}\text{+ φ(t)q = 0}$, for its auxiliary equation $\text{g}\text{?}\text{9 + φ(t)? = h}{}^2\text{?}{}^{\mathrm{?3}}$, and for the Ermakov invariant. Through generalization of the model, a set of invariants is obtained for a system of equations of the form .     

Relativistic correction to the deuteron magnetic moment and angular condition

The relativistic correction (RC) to the deuteron magnetic moment is calculated using the light-cone dynamics. The restrictions imposed by the angular condition on the electromagnetic current operator of the deuteron are discussed in detail. It is shown that the additive model for the current operator of interacting constituents is consistent with the angular condition only for the two first terms of the expansion of the “good” current component $\text{j}{}_{\mathrm{+}}\text{=}\text{1}\text{2}\text{(j}{}_0\text{+ j}{}_{\mathrm{<mtext>z</mtext>}}\text{)}$ in powers of the momentum transfer q. The RC to μ_d is expressed through the matrix element of the “good” component j₊ and is found to be equal to $\text{(0.6–0.8) × 10}{}^{\mathrm{?2}}\text{e}\text{h}\text{?}\text{/2m}{}_{\mathrm{<mtext>p</mtext>}}\text{c}$ for realistic NN potentials. Taking account of RC decreases essentially the discrepancy between the theoretical and experimental values of μ_d. Possible solutions of the angular condition for squared q-terms of the j₊ current component are also discussed.     

Consistent MFA correlation functions of Ising models for all temperatures

A correct calculation of the Ising model correlation function C(q) = 〈(S(q) ? 〈S(q)〉) (S(-q) ? 〈S(-q)〉)〉 in the MFA results in

$\text{C(q)=}\text{〈S}{}^2\text{〉?〈〉}{}^2\text{1?(〈S}{}^2\text{〉?〈S〉}{}^2\text{βJ(q}\text{1}\text{N}\text{∑}\text{q}\text{1}\text{1?〈S〉}{}^2\text{βJ(q)}{}^{\mathrm{?1}}\text{C(q) fulfills the exact sum rule N}{}^{-1}\text{Σ}{}_{\mathrm{q}}\text{C(q) = 〈S}{}^2\text{〉 ? 〈S〉}{}^2$

. Previous literature supposed a violation of this sum rule to be a characteristic disadvantage of this approximation.     

Stress-optical dispersion in rubidium chloride and rubidium bromide crystals

The stress-optical coefficients $\text{C = (}\text{n}{}^3\text{2}\text{) (q}{}_{11}\text{?q}{}_{12}\text{)}$ and $\text{C′ = (}\text{n}{}^3\text{2}\text{)q}{}_{44}$ of RbCl and RbBr crystals have been measured at room temperature (26°C) over the wave length range 5750-2500 Å.     

Search for 4n IN π− double charge exchange reactions with heavy nuclei

A search for the existence of the tetraneutron has been made using the double charge exchange reaction $\text{π}{}^{\mathrm{?}}\text{+}{}^{208}\text{Pb}\text{→}{}^4\text{n + π}{}^{\mathrm{+}}\text{+}$ residuals for ⁴n production and the capture process in the same target, ${}^{208}\text{Pb}\text{+}{}^4\text{n}\text{→}{}^{212}\text{Pb}\text{+ γ}$, for the ⁴n detection. No event has been found, giving an upper limit for the product of the production cross section σ_p, the detection cross section σ_d and the ⁴n lifetime τ. Assuming 10^?18 ≦ τ ≦ 10^?9 sec it follows that σ_pσ_dτ ≦ 2.5 × 10^?65cm⁴ sec with 90 % confidence, and for $\text{τ ≧ 10}{}^{\mathrm{?9}}\text{sec}\text{, σ}{}_{\mathrm{<mtext>p</mtext>}}\text{σ}{}_{\mathrm{<mtext>d</mtext>}}\text{≦ 2.5 × 10}{}^{\mathrm{?56}}\text{cm}{}^4$ with 90 % confidence. The magnitude of this value is comparable to the experimental limit of the ⁴He(π^?, π⁺)⁴n cross section.     

An experimental study of the form factor in the decay K+ → πoe+ν

The q² variation of the factor $\text{?}{}_{\mathrm{+}}\text{(q}{}^2\text{)}$ in the decay K⁺→π⁰e⁺ν has been studied using a sample of even detected in the CERN 1.1 m³ heavy-liquid bubble chamber. The data are consistent with a linear development $\text{?}{}_{\mathrm{+}}\text{(q}{}^2\text{)=?}{}_{\mathrm{+}}\text{(0) (1+λ}{}_{\mathrm{+}}\text{q/m}{}^2{}_{\mathrm{\pi }}\text{)}$ with λ₊=0.027±0.008.     

Singular perturbation potentials

This is a perturbative analysis of the eigenvalues and eigenfunctions of Schrödinger operators of the form ?Δ + A + λV, defined on the Hilbert space L²(Rⁿ), where $\text{Δ = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{?}{}^2\text{?X}{}_{\mathrm{i}}{}^2$, A is a potential function and V is a positive perturbative potential function which diverges at some finite point, conventionally the origin. λ is a small real or complex parameter. The emphasis is on one-dimensional or separable problems, and in particular the typical example is the “spiked harmonic oscillator” Hamiltonian, $\text{?d}{}^2\text{dx}{}^2\text{+ x}{}^2\text{+}\text{l(l + 1)}\text{x}{}^2\text{+ λ|x|}{}^{\mathrm{?\alpha }}$, where α is a positive constant. When this kind of perturbation is very singular, the first-order Rayleigh-Schrödinger perturbative correction, (u₀, Vu₀), where u₀ is the unperturbed eigenfunction, diverges. This analysis constructs explicitly calculable terms in a modified perturbation series to a finite order, by using linear operator theory in concert with approximation methods for differential equations. Along the way a connection between a W-K-B type approximation and Bessel functions is exploited.     

Activation energy of field emission flicker noise power due to adsorbed potassium on tungsten

The temperature dependence of the field emission flicker noise spectral density functions has been investigated for potassium adsorbed on tungsten (112) planes by a probe hole technique. By integration of the spectral density functions $\text{W(?) = B}{}_{\mathrm{i}}\text{?}{}^{\mathrm{?ge}}\text{i}$ the noise power $\text{(δn}{}^2{}_{\mathrm{\Delta ?}}$ for different frequency intervals Δ? is obtained. From the exponential temperature dependence of $\text{(δn}{}^2{}_{\mathrm{\Delta ?}}$ noise power “activation energies” q_Δ? are determined. Plots of these energies versus coverage show a similar “oscillating” behaviour as recently found for $\text{W(?}{}_{\mathrm{j}}\text{)}$ or which indicates phase transitions of the adsorbed potassium submonolayers. The noise activation energies are discussed in terms of existing models and a comparison is made between the experimental q values and surface diffusion energies E_d as determined by conventional methods.     

Excited vibrational state microwave spectra,structure, and force-field of trifluorosilane

The J = 2?1 microwave spectrum of six isotopic species of HSiF₃ has been observed and assigned in excited states of five of the six fundamental vibrations. The assignment is based on relative intensities, double resonance experiments, and trial anharmonic force constant calculations. Analysis of the spectra leads to experimental values for five of the $\text{α}{}_{\mathrm{r}}{}^{\mathrm{B}}$ constants, all three l-doubling constants q_t, one Fermi resonance constant φ₂₃₃, and one zeta constant $\text{ζ}{}_{\mathrm{6,\; 6}}{}^{\mathrm{(z)}}$.The harmonic force field has been refined to all the available data on vibration wavenumbers, centrifugal distortion constants, and zeta constants. The cubic anharmonic force field has been refined to the data on $\text{α}{}_{\mathrm{r}}{}^{\mathrm{B}}$ and q_t constants, using two models: a valence force model with two cubic force constants for SiH and SiF stretching, and a more sophisticated model. With the help of these calculations, the following equilibrium structure has been determined: $\text{r}{}_{\mathrm{e}}\text{(}\text{SiH}\text{) = 1.4468(±5)}\text{A}\text{?}$, $\text{r}{}_{\mathrm{e}}\text{(}\text{SiF}\text{) = 1.5624(±1)}\text{A}\text{?}$, ∠HSiF = 110.64(±3)°,     

Orbital classical solutions,non-perturbative phenomena and singularity at the zero coupling constant point

We try to extend previous arguments on orbital classical solutions in non-relativistic quantum mechanics to the $\text{1}\text{4}\text{λ|φ|}{}^4$ complex relativistic field theory. The single valuedness of the Green function in the semiclassical $\text{(}\text{h}\text{?}\text{→ 0)}$ limit leads to a Bohr-Sommerfeld quantization. A path integral formalism for the Green functions analogous to that in non-relativistic quantum mechanics is employed and a semiclassical approach which uses our classical solutions indicates non-perturbative effects. They reflect and $\text{e}{}^{\mathrm{<mtext>1</mtext><mtext>\lambda </mtext>}}$ singularity at the zero coupling constant point.