Experimental detection and investigation of muon catalyzed fusion of deuterium and tritium

Measurement of the neutron yield of the reaction $\text{dt}\text{μ →}{}^4\text{He}\text{+ μ}{}^{\mathrm{?}}\text{+ 17.6}\text{MeV}$, induced by negative muons in a mixture of gaseous D₂ and T₂, has shown that the rate of muon transfer from deuterium to tritium is λ_dt = (2.7 ± 0.9) × 10⁸s^?1 and that the lower limit of the formation rate of dt_μ molecules is λ_dtμ > 10⁸s^?1.     

Measurement of the (K → πμν)/(K → πeν) branching ratio

Partial branching ratios $\text{(}\text{K}{}^{\mathrm{+}}\text{→ π}{}^0\text{μ}{}^{\mathrm{+}}\text{ν)}\text{(}\text{K}{}^{\mathrm{+}}\text{→ π}{}^0\text{e}{}^{\mathrm{+}}\text{ν)}$ have been measured in six lepton momentum bins between 120 and 180 MeV/c. From these, six relations between the form factor slopes λ₊ and λ⁰ have been derived. Using the world average value λ₊ = 0.029 ± 0.003 and combining the six partial results on λ₀, we obtain λ₀ = 0.019 ± 0.010. This value is in agreement with the Callan-Treiman relation and other predictions. Together with recent K_$\text{l}$3⁰ results, it supports the $\text{ΔI =}\text{1}\text{2}$ rule. The agreement of our result with the value of λ₀ obtained from K_μ3 Dalitz-plot studies supports muon-electron universality.     

On some recent results concerning the superconducting transition temperature

The Eliashberg gap equations relate the transition temperature T_c of an isotropic superconductor to its electron-phonon spectral function α²F(ω) and Coulomb pseudopotential parameter $\text{μ}{}^1$. Recently the Eliashberg theory has been used to derive some supposedly rigorous results bearing on the problem of attaining higher superconducting transition temperatures: Bergmann and Rainer derived an expression for the functional derivative $\text{δT}{}_{\mathrm{c}}\text{δα}{}^2\text{F(ω)}$; Allen and Dynes showed that in the asymptotic limit of very large $\text{λ(λ?10)k}{}_{\mathrm{B}}\text{T}{}_{\mathrm{c}}\text{=f(μ}{}^1\text{)(λ〈ω}{}^2\text{〉)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and Leavens proved that for any isotropic superconductor k_BT_c ?0.2309A, where A is the area under its electron-phonon spectral function. In this letter we show that the result of Allen and Dynes is not compatible with the other results and is, in fact, incorrect.     

Relativistically invariant equations (βμ∂μ+ϱ)Ψ(x)=0 with singular ϱ, and detβ0 ≠ 0

We study relativistically invariant equations (β^μ?_μ+?)Ψ(x)=0 with a singular ?. The absence of a simple relation between these equations and their “conjugate” $\text{((β}{}^{\mathrm{\mu }}\text{)}{}^1\text{?}{}_{\mathrm{\mu }}\text{? ?}{}^1\text{)?(x) = 0}$ is pointed out. Assuming that Ψ(x) transforms under an arbitrary finite-dimensional representation of ISL(2,C), the problem of existence is investigated. It is shown that the assumption detβ⁰ ≠ 0 implies: all β^μ are invertible and the representation of ISL(2,C) is even- dimensional. The difficulties in the general treatment of the problem are discussed. A particular class of equations is illustrated by several examples.     

Experimental limits on the mass and decay lifetime of νμ

The nonobservation of photon-induced e⁺e^? pairs pointing in the neutrino beam direction in large bubble chambers would allow one to set an upper limit on the product m_νμλ^CM for muon-type neutrinos, where m_νμ is the mass and λ^CM is the intrinsic partial decay width in sec^?1 for ν_μ→γ+X. In the theory of Eliezer and Ross where , this implies an upper limit on m_νμ considerably smaller than the present upper limit, and a large lower limit on the lifetime τ_νμ.     

Phase transition and phonon-drag conductivity in quasi-unidimensional systems

We investigate the properties of a quasi-unidimensional system which exhibits Peierls instability at T_c(μ), where μ is the Fermi energy as measured from the middle of the conduction band. T_c(μ) decreases as 6μ6 increases. The phonon-drag diagrams contribute to fluctuation conductivity, which is proportional to $\text{(T ? T}{}_{\mathrm{c}}\text{)}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$, if |μ| > 1.056 T_c(0). The Kohn Anomalies do not occur at 2k_F unless μ = 0 or |μ| ? T.     

Two theorems on the level order in potential models

We study the potentials of the form U(r)=?r^?1+λV(r), $\text{(}\text{d}\text{d}\text{r}\text{)(}\text{r}{}^2\text{d}\text{V}\text{d}\text{r}\text{)?0}$, and show that the energy levels satisfy the inequalities $\text{E(N}{}_{\mathrm{c}}\text{, l)?E(N}{}_{\mathrm{c}}\text{, l+1)}$ to first order in λ, where N_c denotes the coulombic principal quantum number and l the angular momentum. Similarly for potentials $\text{U(r)=r}{}^2\text{+λV(r), (}\text{d}\text{d}\text{r}{}^2\text{)}{}^2\text{V(r)?0}$, we prove to first order in λ that $\text{E}\text{?}\text{(N}{}_{\mathrm{H}}\text{,}\text{l}\text{)?}\text{E}\text{?}\text{(N}{}_{\mathrm{H}}\text{,}\text{l}\text{+2)}$, where NH denotes the harmonic oscillator quantum number. In the latter case, we give also quantitative restrictions on the relative positions at the lth and (l+1)th states.     

Test of a simple relation between gap anisotropy and mass enhancement anisotropy

A simple formula due to Butler and Allen relates anisotropy of the superconducting energy gap Δ_k to the anisotropy of the mass enhancement λ_k and Coulomb pseudopotential μ_k¹. This formula is tested against the numerical calculations of Peter, Ashkenazi and Dacorogna, and agrees quite well. The discrepancies are traced to two sources. (1) In regions of the Fermi surface where | λ_k-λ | or $\text{| μ}{}_{\mathrm{k}}{}^1\text{-μ}{}^1\text{|}$ are particularly large, the linear expansion (weak-anisotropy approximation) fails. (2) In regions of the Fermi surface where the important phonon frequencies happen to be especially small, the two-square-well model becomes less valid.     

Inclusive K10(890) production in π−p interactions at 16 GeV/c

Inclusive $\text{K}{}^{10}$ production is studied in π^?p interactions at 16 GeV/c with x_? > 0.2. The $\text{K}{}^{10}$ is found to be pre-dominantly centrally produced with cross section $\text{σ(}\text{K}{}^{10}\text{) = (72 ± 12) μb}$ for x_? > 0.2 and compares closely to data on $\text{K}{}^{10}$ production in π⁺p interactions at the same energy.     

The ′T Hooft-Polyakov monopole near the Prasad-Sommerfield limit

A perturbative classical monopole solution for the SO(3) gauge theory is constructed in the limit of small but non-vanishing Higgs potential. This corresponds to the limit $\text{μ}{}^2\text{2M}{}_{\mathrm{W}}{}^2\text{=}\text{λ}\text{? 1}$, where μ equals the mass of the scalar particle and M_W equals the mass of the intermediate vector particles. The monopole solution and mass are found to involve non-analytic functions of λ: $\text{γ}$ and λ ln λ. The monopole mass M_m is calculated to order $\text{μ}{}^2\text{M}{}_{\mathrm{W}}$ as

$\text{M}{}_{\mathrm{m}}\text{=}\text{4π}\text{e}{}^2\text{M}{}_{\mathrm{w}}\text{1+}\text{1}\text{2}\text{μ}\text{M}{}_{\mathrm{w}}\text{+}\text{1}\text{2}\text{μ}{}^2\text{M}{}^2{}_{\mathrm{w}}\text{ln}\text{μ}\text{M}{}_{\mathrm{w}}\text{+0.7071}\text{μ}{}^2\text{M}{}^2{}_{\mathrm{w}}$

.     

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.     

Complexified Pöschl–Teller II potential model

With the help of complexifying a five-parameter exponential-type potential model, we obtain a general complex version of the Pöschl–Teller II potential, , where q_c=q₀e^2iαε, real V₁>0, q₀>0 and $\text{0<λε<}\text{π}\text{2}$. It has been shown that this complex potential is P-pseudo-Hermitian and PT-symmetric, where the parity operator P acts on the position operator as $\text{PxP}{}^{\mathrm{-1}}\text{=}\text{ln}\text{q}{}_0\text{λ}\text{−x}$. The discrete energy eigenvalues are shown to be real when $\text{V}{}_2\text{⩾−}\text{q}{}_0\text{λ}{}^2\text{4}$ while they are complex conjugate pairs if $\text{V}{}_2\text{<−}\text{q}{}_0\text{λ}{}^2\text{4}$.     

Remarks on an s-wave Kπ resonance at 1360 MeV

The $\text{K}{}^{\mathrm{O}}\text{π}{}^{\mathrm{?}}\text{and K}{}^{\mathrm{?}}\text{π}{}^{\mathrm{+}}$ mass distributions obtained from a K^?p experiment at 5.5 GeV/c are used to study the recent suggestion of an s-wave Kπ resonance at 1360 MeV. Using a model including $\text{I =}\text{3}\text{2}$ s- and p- wave and $\text{I =}\text{1}\text{2}$, s-, p- and d-wave amplitudes, we analyze the spherical harmonic moments 〈Y_l⁰〉 of the Kπ decay angular distribution in the K¹ (1420) mass region and obtain the result that the reported mass of 1360 MeV does not necessarily represent the resonance position and is influenced by cuts made in the Kπ decay angle.     

Necessary and sufficient conditions for the existence of a lagrangian. The case of quasi-linear field equations

Necessary and sufficient conditions for the existence of the Lagrangian associated with given field equations of motion are investigated. For the quasi-linear equations A_ab^μν(x^λ, φ^c, φ_?^c)φ_μν^b + B_a(x^μ, φ^b, φ_ν^b) = 0, the complete necessary and sufficient conditions are obtained, resorting to the formalism of an exterior derivative. It is emphasized that, to find expressions of these conditions, the anti-symmetric parts of the second derivatives of a Lagrangian, $\text{R}{}_{\mathrm{\mu \nu }}{}^{\mathrm{ab}}\text{= (}\text{?}{}^2\text{L}\text{?φ}{}_{\mathrm{\mu }}{}^{\mathrm{a}}\text{?φ}{}_{\mathrm{\nu }}{}^{\mathrm{b}}\text{?}\text{?}{}^2\text{L}\text{?φ}{}_{\mathrm{\nu }}{}^{\mathrm{a}}\text{?φ}{}_{\mathrm{\mu }}{}^{\mathrm{b}}\text{)/2}$, which disappear in the field equations, take an important role. The procedure to construct the Lagrandian associated with the field equations is also presented.     

The bound state of weakly coupled Schrödinger operators in one and two dimensions

We study the unique bound state which $\text{(}\text{?d}{}^2\text{dx}{}^2\text{) + λV}$ and ?Δ + λV (in two dimensions) have when λ is small and V is suitable. Our main results give necessary and sufficient conditions for there to be a bound state when λ is small and we prove analyticity (resp. nonanalyticity) of the energy eigenvalue at λ = 0 in one (resp. two) dimensions.     

Generalized renormalization group equation for period-doubling bifurcations

Given a mapping $\text{x → ?(λ,x)}$, whose bifurcation points accumulate at λ = Λ_∞, it is shown that the iterates $\text{(?α)}{}^{\mathrm{p\times }}\text{[?}{}^{\mathrm{2p}}\text{(Λ}{}_{\mathrm{\infty }}\text{?μ/δ}{}^{\mathrm{p}}\text{, X}{}_{\mathrm{\infty }}\text{+y/(?α)}{}^{\mathrm{p}}\text{)?X}{}_{\mathrm{\infty }}\text{]}$ converge to a function ψ(μ,y)asp→∞, where X_∞ maximizes $\text{?(Λ}{}_{\mathrm{\infty ,x}}\text{)}$. The function ψ is universal up to scaling in μ and y, and satisfies ψ(μ/δ,ψ(μ/δ,?y/α)) = ?α^?1ψ(μ,y). This result generalizes the well-known Feigenbaum universal function g(y) with g(g(?y/α)) = ?g(y/α, which is the special case for μ = 0.     

Transverse spin correlation function of the one-dimensional spin-12 XY model

The transverse spin pair correlation function p^x_n=<S^x_mS^x_m+n>=<S^x_mS^x_m+n> is calculated exactly in the thermodynamic limit of the system described by the one-dimensional, isotropic, spin-$\text{1}\text{2}$, XY Hamiltonian

$\text{H=?2J}\text{∑}\text{l=1}\text{N}\text{(S}{}^{\mathrm{x}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{x}}{}_{\mathrm{l+1}}\text{+S}{}^{\mathrm{y}}{}_{\mathrm{l}}\text{S}{}^{\mathrm{y}}{}_{\mathrm{l+1}}\text{)}$

. It is found that at absolute zero temperature (T = 0), the correlation function ρ^x_n for n ≥ 0 is given by

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p}}\text{=}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p}}\text{Π}\text{j=1}\text{p?1}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p?2j}}\text{if}\text{n=2p}$

,

$\text{ρ}{}^{\mathrm{x}}{}_{\mathrm{2p+1}}\text{=±}\text{1}\text{4}\text{2}\text{π}{}^{\mathrm{2p+1}}\text{Π}\text{j=1}\text{p}\text{4j}{}^2\text{4j}{}^2\text{?1}{}^{\mathrm{2p+2j}}\text{if}\text{n=2p+1}$

, where the plus sign applies when J is positive and the minus sign applies when J is negative. From these the asymptotic behavior as n → ∞ of |?^x_n| at T = 0 is derived to be $\text{|ρ}{}^{\mathrm{x}}{}_{\mathrm{n}}\text{| ～}\text{a}\text{n}$ with a = 0.147088?. For finite temperatures, ρ^x_n is calculated numerically. By using the results for ?^x_n, the transverse inverse correlation length and the wavenumber dependent transverse spin pair correlation function are also calculated exactly.     

Elementary-particle treatment of the radiative muon capture in 12C

We consider radiative muon capture to a definite nuclear final state. The example chosen is the ${}^{12}\text{C}\text{(0}{}^{\mathrm{+}}\text{T = 0)(μ}{}^{\mathrm{?}}\text{, γν)}{}^{12}\text{B}\text{(1}{}^{\mathrm{+}}\text{T = 1)}$ transition. The elementary particle treatment adopted in this work discloses several aspects of the reaction mechanism, which remain hidden in the usual impulse approximation calculations. In particular, one sees a considerable enhancement of the capture rate when the q² dependence of the weak form factors F_A(q²) and F_P(q²) is taken into account. The branching ratio of the radiative capture in the region sensitive to the possibly non-zero mass m_νμ of the muon-neutrino is estimated to be ≈1.36 × 10^?10.