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.     

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.     

Subject index for volume 88

The emission and excitation spectra of the aromatic thioketone xanthione have been measured in Shpolskii matrices at 15 K. Under these conditions a sharp and rich vibrational structure is observed in the lowest triplet and the first and second excited singlet states. The phosphorescence excitation spectrum places the origin of the T₁ ← S₀ transition at 15 143 cm^?1, while that of the $\text{S}{}_1\text{(n, π}{}^1\text{) ← S}{}_0$ absorption is tentatively assigned to the band at 16 093 cm^?1. The phosphorescence spectrum, which shows only a weak CS stretch vibrational band, is dominated by ring vibrations. In accordance with the previous analysis of ODMR measurements, it is suggested that T₁ and T₂ states are energetically very close, thereby resulting in a lowest triplet state of heavily mixed n, $\text{π}{}^1\text{-π}$, $\text{π}{}^1$ character. No mirror-image relationship is found between the relatively strong S₂ → S₀ fluorescence and the excitation spectrum of the $\text{S}{}_2\text{(π, π}{}^1\text{) ← S}{}_0$ transition. The latter is dominated by a long, pronounced 336-cm^?1 progression.     

Correlations between neutral and charged pions in multiparticle production

We make a theoretical and phenomenological study of correlations between neutral and charged pions in multiparticle production in the framework of the so-called σ, π, ? and ?-? models. Following the method of Drijard and Pokorski, we express the predictions of various models in terms of the negative multiplicity distribution, which is known experimentally. In particular we compute the average number of $\text{π}{}^0\text{,}\text{n}{}_0\text{(n\_)}$, and the integral of π⁰?π⁰ correlations, f₀²(n_), as a function of the number of negative pions; we study also the total multiplicity distribution P(N) and its first two moments $\text{N}\text{and}\text{D}{}_{\mathrm{<mtext>tot</mtext>}}{}^2$. We show that with the present experimental accuracy neither $\text{n}{}_0\text{(n\_)}$ allow us to discriminate between the different models.     

On the super symmetry charges in the theory of scalar and spinor free fields

It is shown that for spinorial charges Q^(L))_α (α = 1, 2, L = 1, …, S) satisfying the commutation relations

$\text{{Q}{}^{\mathrm{(L)}}{}_{\mathrm{\alpha }}\text{, Q}{}^{\mathrm{(M)}}{}_{\mathrm{\beta }}\text{} = ε}{}_{\mathrm{\alpha }}\text{βa}{}^{\mathrm{LM}}\text{Q,}$

$\text{{Q}{}^{\mathrm{(L)}}{}_{\mathrm{\alpha }}\text{, Q}{}^{\mathrm{(M)+}}{}_{\mathrm{\beta }}\text{} = cσ}{}^{\mathrm{\mu }}{}_{\mathrm{\alpha \beta }}\text{P}{}_{\mathrm{\mu }}\text{δ}{}^{\mathrm{LM}}\text{,}$

$\text{[Q}{}^{\mathrm{(L)}}\text{)}{}_{\mathrm{\alpha }}\text{, P}{}^{\mathrm{\mu }}\text{] = 0,}$

where Q is a scalar charge commuting with the spinor charges as well aswith the energy- momentum vector Pμ, there can exist several different multiplets for free massive scalar and spinor fields.     

On the multiplicity of hadrons in QCD jets

A three-loop calculation is presented for the jet multiplicity of produced slightly-off-shell gluons for a pure Yang-Mills theory. Planar and non-planar graphs are found to be equally important in an axial gauge. If the three-loop calculation is indicative of what happens at higher orders $\text{n}\text{(Q}{}^2\text{) ∝}\text{exp}\text{{[(}\text{2C}{}_{\mathrm{<mtext>A</mtext>}}\text{πb}\text{) 1}\text{n}\text{Q}{}^2\text{]}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{}}$ where C_A = 3 and $\text{b =}\text{(33 ? 2n}{}_{\mathrm{<mtext>f</mtext>}}\text{)}\text{12π}$ in QCD.     

Classical integrable finite-dimensional systems related to Lie algebras

During the last few years many dynamical systems have been identified, that are completely integrable or even such to allow an explicit solution of the equations of motion. Some of these systems have the form of classical one-dimensional many-body problems with pair interactions; others are more general. All of them are related to Lie algebras, and in all known cases the property of integrability results from the presence of higher (hidden) symmetries. This review presents from a general and universal viewpoint the results obtained in this field during the last few years. Besides it contains some new results both of physical and mathematical interest.The main focus is on the one-dimensional models of n particles interacting pairwise via potentials V(q) = g²ν(q) of the following 5 types: $\text{ν}{}_{\mathrm{<mtext>I</mtext>}}\text{(q)=q}{}^{\mathrm{?2}}\text{, ν}{}_{\mathrm{<mtext>II</mtext>}}\text{(q)=a}{}^{\mathrm{?2}}\text{sinh}{}^2\text{(aq), ν}{}_{\mathrm{<mtext>III</mtext>}}\text{(q)=a}{}^2\text{/}\text{sin}\text{2(aq), ν}{}_{\mathrm{<mtext>IV</mtext>}}\text{=a}{}^2\text{P}\text{(aq), , ν}{}_{\mathrm{<mtext>V</mtext>}}\text{(q)=q}{}^{\mathrm{?2}}\text{+ω}{}^2\text{q}{}^2$. Here $\text{P}$(q) is the Weierstrass function, so that the first 3 cases are merely subcases of the fourth. The system characterized by the Toda nearest-neighbor potential, g_j²exp[-a(q_j?q_j+1)], is moreover considered. Various generalizations of these models, naturally suggested by their association with Lie algebras, are also treated.     

Lifetime and g-factor results for the 132− 1985 keV level in 91Nb and the 152− 2288 keV level in 91Zr

Lifetime and g-factor measurements have been made with pulsed beam-γ time-differential techniques using the ⁸⁹Y(α, 2n)⁹¹Nb and ⁸⁸Sr(α, n)⁹¹Zr reactions. A mean lifetime τ = 14.4 ± 0.5 nsec and a g-factor of 1.26 ± 0.04 were obtained for the $\text{13}\text{2}{}^{\mathrm{?}}$ 1985 keV level in ⁹¹Nb and τ = 41.9 ± 1.2 nsec and g = 0.70 ± 0.01 were obtained for the $\text{15}\text{2}{}^{\mathrm{?}}$ 2288 keV level in ⁹¹Zr. These results are compared to theoretical calculations for $\text{(π}\text{g}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}\text{)}{}^2\text{(π}\text{p}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ and $\text{(π}\text{g}{}_{\mathrm{<mtext>9</mtext><mtext>2</mtext>}}\text{)(π}\text{p}\text{1}\text{2}\text{)(v}\text{d}{}_{\mathrm{<mtext>5</mtext><mtext>2</mtext>}}\text{)}$ configurations in ⁹¹Nb and ⁹¹Zr, respectively.     

Lorentz contracted geometrical model

The model assumes that when two high energy particles collide each behaves as a geometrical object which has a Gaussian density and is spherically symmetric except for the Lorentz-contraction in the incident direction. Folding the two spatial distribution together we obtain the slope (b) of the elastic diffraction peak in terms of the c.m. velocities (β_i and β_j) and the sizes (A_i and A_j) of the two incident particles. These sizes are assumed to have the experimental s-dependence of σ_tot ∝ πA² for each reaction. The combined s-dependence of the σ_tot's and the β's gives the s-dependence of the elastic slope $\text{b}{}_{\mathrm{ij}}\text{(s) =}\text{1}\text{2}\text{(A}{}_{\mathrm{i}}{}^2\text{β}{}_{\mathrm{i}}{}^2\text{+ A}{}_{\mathrm{j}}{}^2\text{β}{}_{\mathrm{j}}{}^2\text{)}\text{σ}{}^{\mathrm{ij}}{}_{\mathrm{<mtext>tot</mtext>}}\text{(s)}\text{σ}{}^{\mathrm{ij}}{}_{\mathrm{<mtext>tot</mtext>}}\text{(∞)}$. This formula agrees with the experimental slope for p-p, $\text{p}$-p, K⁺-p, K^?-p and π^±-p elastic scattering from 3 to 1500 GeV/c, with only 3 parameters: A_π² = 6.1, A_K² = 3.3 and A_p² = 10.5 (GeV/c)^?2.     

Functional derivative δTc/δα2 (Ω)F(Ω) for weak coupling superconductors

We present approximate analytic calculation of the functional derivative $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{(Ω)F(Ω)}$, where T_c is the superconducting critical temperature and $\text{α}{}^2\text{(Ω)F(Ω)}$ is the electron-phonon spectral function, within the “square-well model” for the phonon mediated electron-electron interaction and weak coupling limit $\text{ω}{}_{\mathrm{D}}\text{(2πT}{}_{\mathrm{c}}\text{)? 1 (ω}{}_{\mathrm{D}}$ is the Debye energy). It is found that $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{(Ω)F(Ω) = (1 + λ)}{}^{-1}\text{G(Ω)}$ where λ is the familiar electron-phonon coupling parameter and $\text{G(Ω)}$ is a universal function of the reduced frequency $\text{Ω}\text{=}\text{Ω}\text{T}{}_{\mathrm{c}}$. We compare this formula with accurate numerical results for several weak coupling superconductors. The overall agreement is good     

Observed and calculated interactions between valence states of the NO molecule

No perturbation between two valence states of NO has ever been identified, although many valence-Rydberg and several Rydberg-Rydberg perturbations have been extensively studied. The first valence-valence crossing to be experimentally documented for NO is reported here and occurs between the ¹⁵N¹⁸O B²Π (v = 18) and B′²Δ (v = 1) levels. No level shifts larger than the detection limit of 0.1 cm^?1 are observed at the crossings near $\text{J = 6.5 [B}{}^2\text{Π}\text{(F}{}_1\text{) ～ B′}{}^2\text{Δ}\text{(F}{}_2\text{)]}$ and $\text{J = 12.5 [B}{}^2\text{Π}\text{(F}{}_1\text{) ～ B′}{}^2\text{Δ}\text{(F}{}_1\text{)]}$; two crossings involving higher rotational levels could not be examined. Semi-empirical calculations of spin-orbit and Coriolis perturbation matrix elements indicate that although the electronic part of the $\text{B}{}^2\text{Π}\text{～ B′}{}^2\text{Δ}$ interaction is large, a small vibrational factor renders the ¹⁵N¹⁸O B (v = 18) ? B′ (v = 1) perturbation unobservable. Semi-empirical estimates are given for all perturbation matrix elements of the operators $\text{Σ}{}_{\mathrm{i}}\text{a}\text{?}{}_{\mathrm{i}}\text{l}{}_{\mathrm{i}}\text{·s}{}_{\mathrm{i}}$ and $\text{B(L}{}_{\mathrm{\pm }}\text{S}{}_{\mathrm{?}}\text{? J}{}_{\mathrm{\pm }}\text{L}{}_{\mathrm{?}}\text{)}$ which connect states belonging to the configurations $\text{(σ2p)}{}^2\text{(π2p)}{}^4\text{(π}{}^1\text{2p)}$, $\text{(σ2p)(π2p)}{}^4\text{(π}{}^1\text{2p)}{}^2$, and $\text{(σ2p)}{}^2\text{(π2p)}{}^3\text{(π}{}^1\text{2p)}{}^2$.     

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 $\text{t=(1+}\text{1}\text{2}\text{(q}{}_1\text{?q}{}_2\text{)}{}^2\text{l}{}^2\text{)}\text{2q}{}_1\text{(q}{}_1\text{+q}{}_2\text{)}$.     

Critical spin relaxation in GdCl3

In the critical region of the insulating, uniaxial ferromagnet GdCl₃ the real part of the uniform susceptibility χ(q = 0, ω; T) has been measured by means of a frequency counting method in the ranges $\text{1.7 MHz ?}\text{ω}\text{2π}\text{? 720 MHz, 0.003 ? (}\text{T}\text{T}{}_{\mathrm{c}}\text{? 1 ≡ ?) ? 0.5}$. Parallel to the easy direction at all temperatures, χ′(ω) has a lorentzian shape, its amplitude being equal to the static susceptibility, χ_T. The half-widths τ^?1 exhibit s critical slowing down: above ?_c = 0.03 they are proportional to χ^?1_T, while at ?_c a change-over to $\text{τ}{}^{\mathrm{?1}}\text{∝ χ}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}{}_{\mathrm{T}}$ occurs, the origin of which is uncertain.     

Asymptotic planarity: An S-matrix basis for the Okubo-Zweig-Iizuka rule

A mechanism is exhibited that monotonically depresses the cylinder component of the topological expansion with increasing t, and it is conjectured that all non-planar S-matrix components diminish as t increases, exchange degeneracy and the Okubo-Zweig-Iizuka rule becoming more accurately satisfied. Such asymptotic planarity is compared to the field-theoretical concept of asymptotic freedom. The characteristics low-t cylinder “quenching interval” is found to be the inverse of the mean value over a two-reggeon loop, of $\text{1}\text{2}\text{π}{}^2\text{(α′)}{}^2\text{(t}{}_1\text{? t}{}_2\text{)}{}^2\text{/(?t),}\text{where}\text{t}{}_1\text{and}\text{t}{}_2$ are the squared masses of the loop reggeons and α′ is the trajectory slope. For leading trajectories the low-t cylinder quenching interval is predicted by this formula to be roughly 0.5 GeV²-consistent with the observed pomeron intercept and slope, with the p-ω and f-A₂ mass differences and with the (φ,ω) deviation from ideal mixing. As t grows negatively over a corresponding interval, it is predicted that the pomeron will become nearly a pure SU(3) singlet. If the pion mass helps to set the scale for reggeon loops coupled to unnatural-parity trajectories, the cylinder quenching interval will be larger, explaining the large (η, η′) deviation from ideal mixing as well as the large π-η mass difference. Even when the small-t cylinder quenching is rapid (“precocious planarity”) the large-t approach to the planar limit turns out to be gentle. A by-product of this study is an explanation of the approximate reality and linearity of trajectories at large t.     

νW2 at small ω′ and resonance form factors in a quark model with broken SU(6)

The small ω′ behaviour of F₂^en/F₂^ep and the apparent difference in the q² dependences of the magnetic form factor of the proton and of the transition to Δ⁺(1236) are quantitatively correlated in a model where nucleon consistes of a quarks and a scalar or vector core. The proton and Δ transition form factors suggest that only the scalar core contributes at large q² and small ω′. As a result the ω′ dependence of $\text{F}{}_2{}^{\mathrm{<mtext>en</mtext>}}\text{F}{}_2{}^{\mathrm{<mtext>ep</mtext>}}$ is obtained for ω′ < 3 and predictions for the weak structure functions and polarisation asymmetries at smallω′ are presented. We predict $\text{F}{}^{\mathrm{<mtext>\nu </mtext><mtext>p</mtext>}}\text{F}{}^{\mathrm{<mtext>\nu </mtext><mtext>n</mtext><mtext>\omega \prime \to 1</mtext><mtext>0</mtext>}}$ asymmetries $\text{ω′→1}\text{1}$ and also expect that $\text{G}{}_{\mathrm{<mtext>m</mtext>}}{}^{\mathrm{<mtext>n</mtext>}}\text{G}{}_{\mathrm{<mtext>m</mtext>}}{}^{\mathrm{<mtext>p</mtext>}}\text{→}\text{?1}\text{2}$ as q²→∞.     

Conductivity of a quasi-one-dimensional metal at finite temperatures

The longitudinal conductivity of a quasi-one-dimensional metal is calculated for the case T?ω_D (ω_D being the limiting phonon frequency) and ω_Dl¹/v?1 where l¹ is the effective mean free path determined by impurity and phonon scattering: $\text{l}{}^1\text{= (l}{}^{\mathrm{?1}}{}_{\mathrm{ph}}\text{+ l}{}^{\mathrm{?1}}{}_{\mathrm{i}}\text{)}{}^{\mathrm{?1}}\text{, l}{}_{\mathrm{ph}}\text{= v/λT, l}{}_{\mathrm{i}}$ is the impurity mean free path. The conductivity is σ = (c₁e²/πS)l³_iv^?2ω_DλT for l_i?l_ph, σ = (c₂e²/πS)vω_D(λT)^?2 for l_i?l_ph, λ being the dimensionless electron-phonon interaction constant, c₁, c₂ ～ 1, S = a_xa_y is the (xy) area per one chain.     

π+-Electroproduction of hydrogen near threshold at four-momentum transfers of 0.2, 0.4 and 0.6 GeV2

The reaction e + p → e′ + π⁺ + n was measured near the one pion threshold, detecting the final electron and neutron in coincidence for values of q² = 0.2, 0.4 and 0.6 GeV². The normalized axial vector form factor to the nucleon $\text{F}{}_{\mathrm{<mtext>A</mtext>}}\text{(q}{}^2\text{)}\text{F}{}_{\mathrm{<mtext>A</mtext>}}\text{(0)}$ was determined from the data within a theoretical framework based on PCAC and current algebra.