Gravitational collapse in quantum mechanics with relativistic kinetic energy

It is proved that the quantum mechanical Hamiltonian $\text{H = Σ}{}_{\mathrm{i=1}}{}^{\mathrm{N}}\text{(p}{}^2\text{+ m}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? κ Σ}{}_{\mathrm{i>j}}\text{|x}{}_{\mathrm{i}}\text{? x}{}_{\mathrm{j}}\text{|}{}^{\mathrm{?1}}$ for bosons (resp, fermions) is bounded from below if N ≤ c_bκ^?1 (resp. $\text{N ≤ c}{}_{\mathrm{f}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). H is unbounded from below if N ≥ c_b^lκ^?1 (resp. $\text{N ≥ c}{}_{\mathrm{f}}{}^{\mathrm{l}}\text{κ}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$). The constants c_b and c_b^l (resp. c_f and c_f^l) differ by about a factor 2 (resp. 4).     

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→∞.     

A note on the momentum distribution function for an electron gas

The one-electron momentum distribution function $\text{〈a}{}_{\mathrm{k\sigma }}{}^2\text{a}{}_{\mathrm{k\sigma }}\text{〉}$ for an electron gas is investigated by a diagrammatic analysis of perturbation theory. It is shown that $\text{〈a}{}_{\mathrm{k\sigma }}{}^2\text{a}{}_{\mathrm{k\sigma }}\text{〉}$ has the following exact asymptotic form for large k (k ? p_F; p_F, the Fermi momentum): $\text{〈a}{}_{\mathrm{k\sigma }}{}^2\text{a}{}_{\mathrm{k\sigma }}\text{〉 =}\text{4}\text{9}\text{(}\text{αr}{}_{\mathrm{s}}\text{π}\text{)}{}^2\text{×(}\text{p}{}_{\mathrm{<mtext>F</mtext>}}{}^8\text{k}{}^8\text{) g?(0) + ?}$, where g?(0) is the zero-distance value of the spin-up-spin-down pair correlation function. The physical implications of the above asymptotic form are discussed.     

Inclusive pion distributions in e+e− annihilation from sum rules and duality

The threshold behaviour of pion production presented in our earlier work is successfully compared with the new SPEAR data. By using duality and sum rules we derive F_T^(π+)(x) ≈ F_L^(π+)(x) ≈ F_T^(π0)(x) ? F_L^(π0)(x) for x near 1. An accompanying results is $\text{σ}{}_{\mathrm{<mtext>\pi </mtext><mtext>A</mtext>}}{}_2\text{(s) ≈ 2σ}{}_{\mathrm{\pi \omega }}\text{(s) ≈ 4σ}{}_{\mathrm{\pi \pi }}\text{(s) ≈ 9(m}{}_{\mathrm{?}}{}^2\text{/s)}{}^3\text{σ}{}_{\mathrm{<mtext>\mu </mtext><mtext>\mu </mtext>}}$ for large s.     

Deep inelastic sum rules at fixed hadron energy

Experimentally, it is much easier to measure structure functions at fixed hadron energy E_h, rather than at fixed Q². We prove the sum rules

,

$\text{h}{}_{\mathrm{NS}}\text{=}\text{4}\text{3}\text{(}\text{1}\text{6}\text{π}{}^2\text{?}\text{5}\text{8}\text{),}$

$\text{h}{}_{\mathrm{S}}\text{=}\text{4}\text{16+3n}{}_{\mathrm{<mtext>f</mtext>}}\text{(2π}{}^2\text{+}\text{1}\text{6}\text{π}{}^2\text{n}{}_{\mathrm{<mtext>f</mtext>}}\text{?}\text{65}\text{12}\text{?}\text{5}\text{72}\text{n}{}_{\mathrm{<mtext>f</mtext>}}\text{).}$

Here s = 2m_pE_h and F_2,3 are the standard functions for scattering of neutrinos on an isoscalar target. K is an unknown constant, and the corrections to the sum rules are O(α_s²), O(α_s^1.75), respectively.     

Observation of proton emission following thermal neutron capture in 34.5 d 84Rb

Using a target prepared by on-line isotope separation, thermal neutron capture in ⁸⁴Rb (I^π = 2^?) has been shown to induce proton emission to the ground state (0⁺) and first excited state (2⁺) of ⁸⁴Kr. The branching ratio was measured as $\text{Γ}{}_{\mathrm{<mtext>p</mtext>}}\text{(0}{}^{\mathrm{+}}\text{)}\text{Γ}{}_{\mathrm{<mtext>p</mtext>}}\text{(2}{}^{\mathrm{+}}\text{)}\text{= 4.7 ± 0.7}$, favouring a $\text{3}\text{2}{}^{\mathrm{?}}$ assignment of the capturing state without excluding $\text{5}\text{2}{}^{\mathrm{?}}$, and the (n_th, p) cross section as 12 ± 2 b. The energy available for the process was determined to be 3.45 ± 0.01 MeV, in agreement with other mass data in the region.     

Anomalous stoner enhancements in amorphous systems

An n-orbital model describing both elastic impurity scattering and exchange interaction is examined near its instability for itinerant ferromagnetism. At the critical point and at zero temperature, long range spin fluctuations cause anomalous enhancements of the density of states near the Fermi energy with $\text{?(E) - ?(E}{}_{\mathrm{F}}\text{) ∝ |E ? E}{}_{\mathrm{F}}\text{|}{}^{\mathrm{<mtext>1</mtext><mtext>4</mtext>}}$ in three-dimensions (3D) and with ln²|E ? E_F| in 2D. An estimation of the conductivity $\text{σ(Ω)}$ in a continuum analog model reveals Ω^{$\text{1}\text{4}$} -and ln²$\text{|Ω}{}_{\mathrm{\tau }}\text{|}$-corrections in 3D and 2D respectively.     

Vector-axial vector interference terms in inelastic neutrino reactions extrapolated from single-pion photoproduction

By use of the generalized scaling sum rules the vector-axial vector interference terms in the scaling limit of the inelastic neutrino reactions are calculated from the absorptive parts of the single-pion photoproduction amplitudes at zero momentum transfer. The photoproduction amplitudes are evaluated by use of the results of multipole analysis. With the optimum values for the two parameters of the generalized scaling as determined in previous works, we have obtained F₃^νN(ω) which leads to good agreement with experiment in the ration of $\text{σ}{}^{\mathrm{<mtext>\nu </mtext><mtext>N</mtext>}}\text{/σ}{}^{\mathrm{<mtext>\nu </mtext><mtext>N</mtext>}}$. However, the zeroth moment in $\text{x( = 1/ω), ∫}{}_0{}^1\text{d}\text{x(F}{}_3{}^{\mathrm{<mtext>\nu </mtext><mtext>p</mtext>}}\text{+ F}{}_3{}^{\mathrm{<mtext>\nu </mtext><mtext>p</mtext>}}\text{)}$ turns out to be smaller than half of the value predicted in the fractionally charged quark model.     

Nuclear matrix elements from L/K electron capture ratios in 59Ni decay

Using a recent theoretical method, the ratio of nuclear matrix elements $\text{R = (}{}^{\mathrm{v}}\text{F}{}^0{}_{220}\text{?√}\text{3}\text{2}{}^{\mathrm{A}}\text{F}{}^0{}_{221}\text{/}{}^{\mathrm{v}}\text{F}{}^0{}_{211}\text{)}$ was determined to be either 20.50^+0.35_?0.55 or 25.22^+0.28_?0.17 in the second-forbidden nonunique decay of 8 × 10⁴ y ⁵⁹Ni. These values of R were obtained from a value of L₃/K = 0.008 ± 0.002 found by subtracting the theoretical ratio $\text{(L}{}_1\text{+ L}{}_2\text{)}\text{K}$ = 0.113 (based on Q_PEC = 1070 ± 8 keV) from the total ratio L/K = 0.121 ± 0.002, which was measured with a reactor-produced, doubly-mass-separated ⁵⁹Ni source introduced as gaseous nickel-ocene, (C₅H₅)₂, into a wall-less, anticoincidence, multiwire proportional counter. The 854–1008 eV L and the 8.33 keV K peaks were measured simultaneously.     

Scaling violation,the Callan Gross relation,and color excitation in electroproduction

The Callan-Gross relation is shown to be consistent with MIT-SLAC data for $\text{σ}{}_{\mathrm{<mtext>L</mtext>}}\text{(Q}{}^2\text{)}\text{σ}{}_{\mathrm{<mtext>T</mtext>}}\text{(Q}{}^2\text{)}$ for x ? 0.33 in deep inelastic eN scattering, despite the fact that these data are taken in the large Q² region where F₁ and F₂ individually exhibit scaling violation. Comparison is made with asymptotic freedom predictions, and color excitation is proposed to explain large values of $\text{σ}{}_{\mathrm{<mtext>L</mtext>}}\text{σ}{}_{\mathrm{<mtext>T</mtext>}}$ at small x.     

Evidence for a narrow peak in Ksoπ±π+π- at 2.6 gev in 12 gev/cpp interactions

We report the evidence for a narrow charged peak (5.5 s.d.), which we suggest calling the I, in the 6-prong-V⁰ topology of $\text{p}$p interactions at 12 GeV/c. The mass, widht and the product of cross section σ_I times the branching ratio BR into the final state (K_s^oτ±τ⁺τ^? are found to be: $\text{M}{}_{\mathrm{I}}\text{=2.60 ±}\text{0.01}\text{GeV}\text{/}\text{c}{}^2$, $\text{Γ}{}_{\mathrm{I}}\text{?}\text{0.018}\text{GeV}\text{/}\text{c}{}^2$, σ_I·BR≈20 μbarn     

Laser magnetic resonance spectrum of BrO (2Π12 ← 2Π32)

Magnetic dipole transitions between the ${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and ${}^2\text{Π}{}_{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$ components of the ground electronic state of BrO have been detected using the technique of laser magnetic resonance on three CO₂ laser lines between 964 and 970 cm^?1. This is the first direct observation of the ${}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ state in BrO. The spin-orbit splitting parameter, A, is determined to be ?967.9831(2) cm^?1 for ⁷⁹Br¹⁶O and ?967.9981(2) cm^?1 for ⁸¹Br¹⁶O. Accurate values for the rotational constant $\text{B}{}^{\mathrm{<mtext>eff</mtext>}}\text{(}{}^2\text{Π}{}_{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, the hyperfine parameters ($\text{b}{}_{\mathrm{F}}\text{+}\text{2c}\text{3}$) and d, the Λ-doubling parameter p, and the Zeeman parameter g| are also determined from an analysis of the measured spectra.     

Production and decay of quark-gluon strings: inclusive distributions of hadrons in p⊥ and x

The dependence of inclusive cross sections of the production of hadrons $\text{p}\text{p}\text{→ hX}$ on p_⊥ (and also on x and √s) is calculated at high energy in the region of small p_⊥ ? 1–2 GeV. The model of production and decay of quark-gluon strings is used under the simplest assumptions about the k_⊥ dependence of the quark distributions in nucleons ～ exp(?γ₁k²_⊥) and about the form of the string fragmentation function $\text{G}\text{?}{}^{\mathrm{<mtext>h</mtext>}}\text{～}\text{exp}\text{[?γ}{}^{\mathrm{<mtext>h</mtext>}}\text{(}\text{p}{}_{\mathrm{\perp }}\text{? z}\text{k}{}_{\mathrm{\perp }}\text{)}{}^2\text{]}$ where γ₁ and γ^h are some constants. The theory reproduces all existing experimental data and yields the “seagull effect” for the dependence of 〈p_⊥〉 on x. Predictions are given for the p_⊥ dependence of the spectra of π^± mesons produced at high energies at SPS and other future colliders.     

On the ratio Σ0/Λ in electroproduction off protons

Recent experimental results on electroproduction off protons show that the ratio σ(ep → eK⁺Σ⁰)/σ(ep → eK⁺Λ) decreases strongly with increasing Q². A simple argument is given in the framework of the quark parton model which could provide a qualitative understanding of this fact. The decrease of the Σ⁰/Λ ratio is related to the decrease of the ratio $\text{F}{}_1{}^{\mathrm{<mtext>\gamma </mtext><mtext>n</mtext>}}\text{/F}{}_1{}^{\mathrm{<mtext>\gamma </mtext><mtext>p</mtext>}}$ as Q² increases, where $\text{F}{}_1{}^{\mathrm{<mtext>\gamma </mtext><mtext>p</mtext>}}\text{and F}{}_1{}^{\mathrm{<mtext>\gamma </mtext><mtext>n</mtext>}}$ are the usual structure functions for deep inelastic electron-nucleon scattering.     

Interference effects in 12C(p, γ)13N and direct capture to unbound states

Excitation functions of the capture reaction ¹²C(p, γ₀)¹³N have been obtained at θ_γ = 0° and 90° and E_p = 150–2500 keV. The results can be explained if a direct radiative capture process, E1(s and d → p), to the ground state in ¹³N is included in the analysis in addition to the two well-known resonances in this beam energy range $\text{[E}{}_{\mathrm{p}}\text{= 457(}\text{1}\text{2}{}^{\mathrm{+}}\text{)}$ and 1699 $\text{(}\text{3}\text{2}{}^{\mathrm{?}}\text{) keV]}$. The direct capture component is enhanced through interference effects with the two resonance amplitudes. From the observed direct capture cross section, a spectroscopic factor of C²S(l = 1) = 0.49 ± 0.15 has been deduced for the $\text{1}\text{2}{}^{\mathrm{?}}$ ground state in ¹³N. Excitation functions for the reaction ${}^{12}\text{C(p,γ}{}_1\text{p}{}^1\text{)}{}^{12}\text{C}$ have been obtained at θ_γ = 0° and 90° and E_p = 610–2700 keV. Away from the 1699 keV resonance the capture γ-ray yield is dominated by the direct capture process E1 (p → s) to the $\text{2366 (}\text{1}\text{2}{}^{\mathrm{+}}\text{) keV}$ unbound state. Above E_p = 1 MeV, the observed excitation functions are well reproduced by the direct capture theory to unbound states (bremsstrahlung theory). Below E_p = 1 MeV, i.e., E_p → 457 keV, the theory diverges in contrast to observation. This discrepancy is well known in bremsstrahlung theory as the “infrared problem”. From the observed direct capture cross sections at $\text{E}{}_{\mathrm{p}}\text{? 1 MeV}$, a spectroscopic factor of C²S(l = 0) = 1.02 ± 0.15 has been found for the $\text{2366 (}\text{1}\text{2}{}^{\mathrm{+}}\text{) keV}$ unbound state. A search for direct capture transitions to the $\text{3512 (}\text{3}\text{2}{}^{\mathrm{?}}\text{)}$and $\text{3547 (}\text{5}\text{2}{}^{\mathrm{+}}\text{) keV}$ unbound states resulted in upper limits of C²S(l = 1) ≦ 0.5 and C²S(l = 2) ? 1.0, respectively. The results are compared with available stripping data as well as shell-model calculations. The astrophysical aspect of the ¹²C(p, γ₀)¹³N reaction also is discussed.     

The closure of cones in the algebra of test functions

We prove that the topological closure of the cone K of positive elements in the algebra of test functions is the same for several topologies (K?τo° = K?τ∞). Further we show that all elements of the closure of K are of the form

$\text{∑}\text{I=1}\text{∞}\text{f}{}^{\mathrm{(i)1}}\text{f}{}^{\mathrm{(i)}}\text{with f}{}^{\mathrm{<mtext>(</mtext><mtext>i</mtext><mtext>)</mtext>}}\text{?}\text{L}{}_{\mathrm{?}}$

and the sum converges. Applying these results, we give a characterization of the elements of K?^τ_o too.     

Dielectric susceptibility and correlation length of one-dimensional CDW with impurties. II. weak disorder

Dependence of static dielectric susceptibility and correlation length of charge density waves (CDW) with weak defects on parameter of incommensurability with lattice is investigated. In almost commensurate phase (h?h_ch_c), $\text{χ ～ (h?h}{}_{\mathrm{c}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{In}{}^{\mathrm{<mtext>-4</mtext><mtext>3</mtext>}}\text{h}{}_{\mathrm{c}}\text{/h?h}{}_{\mathrm{c}}$ and R_c ～ (h ? h_c)^{$\text{2}\text{3}$}. In${}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{h}{}_{\mathrm{c}}\text{/h ? h}{}_{\mathrm{c}}$. Far from commensurability $\text{(h?h}{}_{\mathrm{c}}\text{) χ～ (a+h}{}^2{}_{\mathrm{c}}\text{/h}{}^2\text{)}{}^{\mathrm{<mtext>-2</mtext><mtext>3</mtext>}}$, $\text{R}{}_{\mathrm{c}}\text{～ (a + h}{}^2{}_{\mathrm{c}}\text{/h}{}^2\text{)}{}^{\mathrm{<mtext>-2</mtext><mtext>3</mtext>}}$, where a is the dimensionless ratio of random potential intensities, corresponding to backward and forward scattering impurities.     

A least upper bound on the superconducting transition temperature

It is rigorously shown that the superconducting transition temperature of any material for which the Eliashberg theory is valid must satisfy k_BT_c ? 0.2309 A, where A is the area under its electron-phonon spectral function α²F(ω). This relation is a least upper bound, not just an upper bound, in the sense that there is an optimal situation in which the equality holds. This occurs when the Coulomb pseudopotential parameter $\text{μ}{}^1$ is zero and the spectral function is the Einstein spectrum Aδ(ω ? 1.750 A). These results are generalized in an approximate, but sufficiently accurate, way to the case $\text{μ}{}^1\text{≠ 0}$ to obtain the more useful least upper bound $\text{k}{}_{\mathrm{<mtext>B</mtext>}}\text{T}{}_{\mathrm{<mtext>c</mtext>}}\text{? c(μ}{}^1\text{) A}$ and the corresponding optimal spectrum $\text{Aδ[ω ? d(μ}{}^1\text{)A]}$. Numerical results for the functions $\text{c(μ}{}^1\text{)}$ and d(μ¹) are presented for $\text{0 ? μ}{}^1\text{? 0.20}$. It is shown that the T_c's of many materials (including Nb₃Sn), for which experimental values of A and $\text{μ}{}^1$ are available, do not lie very far below the upper bound.     

Quark masses

In quark gluon theory with very small bare masses, -$\text{ψ}$$\text{M}$ψ, spontaneous breakdown of chiral symmetry generates sizable masses M_u, M_d, M_s, … We find ($\text{M}{}_{\mathrm{<mtext>u</mtext>}}\text{+ M}{}_{\mathrm{<mtext>d</mtext>}}\text{) /2 ≈ m}\text{p}\text{/ √6 ≈ 312}\text{MeV}\text{,}\text{and}\text{M}{}_{\mathrm{<mtext>s</mtext>}}\text{≈ 432}\text{MeV}$. Scalar densities have well determined non-zero vaccum expectations $\text{〈0|u}{}^{\mathrm{a}}\text{|0〉 ≡ 〈0|ψ(x) (λ}{}^{\mathrm{a}}\text{/2)ψ(x)/0〉 ≈ ?}{}_{\mathrm{\pi }}{}^2\text{M}{}^{\mathrm{a}}\text{,}\text{i.e}\text{〈0? u}{}^{\mathrm{o}}\text{/vb0〉 ≈ 8 × 10}{}^{\mathrm{?3}}\text{(}\text{GeV}\text{)}{}^3$ at an SU(3) breaking of the vacuum c′ ≡ 〈0|u⁸|〉/〈0|u^o|0〉 ≈ ? 16%     

Spectroscopy of 42Ca from 41Ca(d,p)

Energy levels in ⁴²Ca up to 7.8 MeV have been studied in the neutron capture reaction ⁴¹Ca(d, p)⁴²Ca with 12 MeV bombarding energy. Ninety-four excited states have been identified and angular distributions have been measured in the interval from 5° to 110° by means of a broad-range magnetic spectrograph. The angular distributions together with DW calculations have been used to determine I_n values and spectroscopic factors. The $\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}$ strength sum agrees with shell-model expectations if the f_{$\text{7}\text{2}$} spectroscopic factors are renormalized by $\text{1}\text{0.75}$, in line with other $\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}$. transfer experiments on ⁴⁰Ca and ⁴¹Ca. A similar renormalization of the l_n = 1 spectroscopic factors brings this strength sum in accordance with the shell-model calculations. The effective ($\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}{}^2$) matrix elements for ⁴²Ca are compared with the corresponding matrix elements of ⁴²Sc and ⁴⁸Sc. The differences between the three sets of matrix elements are of the order of a few hundred keV or less. The monopole centroid energy of the ($\text{f}{}_{\mathrm{<mtext>7</mtext><mtext>2</mtext>}}\text{)}{}^2$ multiplet is shifted downwards in the mass-42 nuclei compared to ⁴⁸Sc, possibly indicating the importance of the monopole pairing force near ⁴⁰Ca.