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If the deuteron were an elementary particle then the triplet n−p effective range would be approximately −ZR(I−Z), where R=4.31 F is the usual deuteron radius and Z is the probability of finding the deuteron in a bare elementary-particle state. This formula is model-independent, but has an error of the order of the range mπ−1=1.41 F of the n−p force, so it becomes exact only in the limit of small deuteron binding energy, i.e., R≫mπ−1. The experimental value of the effective range is not of order R and negative, but rather of order mπ−1 and positive, so Z is small or zero and the deuteron is mostly or wholly composite.

If electromagnetic interactions are wholly responsible for all departures from isotopic spin invariance, then the strict conservation law $\ensuremath{\Delta}T=0$ may be replaced, to order ${e}^{2}$, by the rule $|\ensuremath{\Delta}T|l~2$. Consequences of this weaker restriction are discussed for elementary particle masses, scattering processes, and weak-inter-action decay processes. The apparent absence in nature of particles with isotopic spins greater than one makes it difficult to find very practical experimental tests of this rule.

The current commutation ${\mathrm{relations}}^{1}$ and partially conserved axial-vector current (PCAC) ${\mathrm{assumption}}^{2,3}$ allow the calculation of the matrix elements for emission and absorption of any number of soft ${\mathrm{pions}}^{4}$ and, therefore, in particular, determine the scattering length of a pion on any target particle. In this note we give a simple formula for pion scattering on any particle but a ${\mathrm{pion}}^{5}$, and then extend this result to the more difficult case of pion-pion scattering.

The spectral-function sum rules are given for a general Lie algebra, and then applied to the vector and axial-vector mesons coupled to the currents of SU(3) \ensuremath{\bigotimes} SU(3).

Old-fashioned perturbation theory is applied to a relativistic theory in a reference frame with infinite total momentum. It is found that many undesirable diagrams disappear. The contribution of the remaining diagrams is described by a new set of rules with properties intermediate between those of Feynman diagrams and old-fashioned diagrams, e.g., energy denominators become covariant, and Feynman parameters appear naturally. The new rules are used to derive some integral equations.

It is shown that it is possible to interpret the muon absorption rate in liquid hydrogen in terms of the basic (pn mu nu ) interaction. The main problem is to calculate the rate for the process: (p- mu -p) yields n+ nu +p in terms of muon absorption in atoms. The discussion applies to solid as well as liquid hydrogen, but it is confined to the case of isotopically pure (H/sup 1/)/sub 2/. Many of these assumptions may be tested by measuring the total hydrogen absorption rate over a rarge of hydrogen densities. (B.O.G.)

The dispersion relation is obtained for the ``hose'' instability in a modulated beam of charged particles traveling through a finite ohmic plasma channel. For a relativistic beam, the low‐frequency mode obeys a dispersion relation of the same form as for the unmodulated beam, but it involves a constant which will have to be obtained by machine computation. For large enough modulation frequency the dispersion relation is identical to that for an unmodulated beam.

The observable effects of a neutral intermediate vector boson in semileptonic processes are considered in the context of a proposed renormalizable theory of the weak and electromagnetic interactions. With strange particles neglected, this theory allows the calculation of neutral-current form factors in terms of charged-current form factors and electromagnetic form factors. The results are neither confirmed nor refuted by present data. One possible method of incorporating the strange particles is briefly discussed.

A new set of renormalization-group equations is presented. These equations are based on a renormalization procedure in which counterterms are calculated for zero unrenormalized mass. Unlike the Gell-Mann-Low and Callan-Symanzik equations, they can be solved for arbitrary momenta. The solutions involve a momentum-dependent effective mass as well as a momentum-dependent effective coupling constant. By studying these solutions at large momenta, it can be shown that the nonleading terms discarded by previous authors do, in fact, remain negligible when the perturbation series is summed to all orders if, and only if, the effective mass vanishes at large momentum, which will be the case if a certain anomalous dimension is less than unity, as it is in asymptotically free theories. In this case, the new renormalization-group equations can be used at large momentum to derive not only the leading term, but the first three terms in an asymptotic expansion of any Green's function. These results are also applied to Wilson coefficient functions, and an important cancellation of anomalous dimensions is noted.

The eikonal method is extended to waves with several components propagating in inhomogeneous anisotropic media. Formulas are derived for the motion of wave packets, the change in amplitude along a ray path, and the corrections due to diffraction. The method is applied to pure magnetohydrodynamic disturbances, and the problem of computing ray paths in the ionosphere is discussed in some detail. One result is that the fraction of the energy of an isotropic disturbance that eventually gets as low as 200 km drops from 0.99 at 200 km to 0.2 at 450 km, rises again to 0.99 at 700 km, drops to 0.5 at 3000 km, and continues dropping at greater altitudes. Energy trapped between 200 km and 700 km oscillates around 450 km with period 8.5 sec (at the geomagnetic equator). The eikonal method is also applied to a very general problem involving coupled magnetohydrodynamic, electrodynamic, and acoustic modes.

The Feynman rules are derived for massless particles of arbitrary spin $j$. The rules are the same as those presented in an earlier article for $mg0$, provided that we let $m\ensuremath{\rightarrow}0$ in propagators and wave functions, and provided that we keep to the ($2j+1$)-component formalism [with fields of the ($j$, 0) or (0, $j$) type] or the $2(2j+1)$-component formalism [with $(j, 0)\ensuremath{\bigoplus}(0, j)$ fields]. But there are other field types which cannot be constructed for $m=0$; these include the ($\frac{j}{2}, \frac{j}{2}$) tensor fields, and in particular the vector potential for $j=1$. This restriction arises from the non-semi-simple structure of the little group for $m=0$. Some other subjects discussed include: T, C, and P for massless particles and fields; the extent to which chirality conservation implies zero physical mass; and the Feynman rules for massive particles in the helicity formalism. Our approach is based on the assumption that the $S$ matrix is Lorentz invariant, and makes no use of Lagrangians or the canonical formalism.

Perturbation theory always works in nonrelativistic scattering theory, unless composite particles are present. By "composite particle" is meant a bound state or resonance, or one that would exist for an interaction of opposite sign; in fact, this provides a precise definition of resonances. It follows that if fictitious elementary particles (quasiparticles) are first introduced to take the place of all composite particles, then perturbation theory can always be used. There are several ways of accomplishing this, one of which corresponds to the $\frac{N}{D}$ method. In order to prove these results it is necessary to make a detailed study of the eigenvalues of the scattering kernel, and as a by-product we obtain new proofs of the applicability of the Fredholm theorems to scattering theory, of the convergence of the Born series at high energy, of the Bargmann-Schwinger theorem on the number of bound states, of the Pais-Jost theorem on the identity of the Jost function with the Fredholm determinant, and of Levinson's theorem. We also give explicit formulas for binding energies and phase shifts in potential theory, using first-order perturbation theory after insertion of a single quasiparticle; these formulas work well for the lowest bound state and the $S$-wave scattering length of the Yukawa potential, and give precisely 13.6 eV for the hydrogen atom binding energy.

In theories with spontaneously broken gauge symmetries, various masses, or mass differences may vanish in zeroth order as a consequence of the representation content of the fields appearing in the Lagrangian. These masses or mass differences can then be calculated as finite higher-order effects. The mechanism for cancelation of divergences in second-order fermion masses is described explicitly. The weak interactions play an essential role in canceling infinities in electromagnetic masses.

It is shown for a certain class of interactions which might be expected to induce mu -e transformations that the similarity of the mu and e actually forbids the transformations. (C.J.G.)

ABS>A class of non-Abelian gauge theories of strong interactions is described, for which parity sand strangeness are automatically conserved, and for which the nonconservations of parity and strangeness produced by weak interactions are automatically of order alpha /m/sub w/sup 2/ rather than o f order alpha . When such theories are asymptotically free,'' the order- alpha weak corrections to natural zeroth-order symmetries may be calculated ignoring all effects of strong interactions. Speculations are offered on a possible theory of quarks. (auth)

An investigation is made of states in the Lee model representing a "physical" $V$-particle bound to an $N$-particle. It is found that the energy eigenvalue problem can be reduced to a single transcendental equation involving the parity, energy, renormalized coupling, and $N\ensuremath{-}V$ separation. Only states of odd parity are considered. For such states this equation leads to a single-valued real potential if the renormalized coupling constant is less than the critical value for the appearance of the so-called "ghost" states of the free physical $V$-particle. For stronger couplings the potential becomes many-valued with half its real branches corresponding to the presence of "normal" $V'\mathrm{s}$, and half to "ghost" $V'\mathrm{s}$. For still stronger coupling no real energies exist. It is shown that complex energies appear, at least in the case where the virtual $\ensuremath{\theta}$ particles are assumed nonrelativistic and the $N\ensuremath{-}V$ separation is sufficiently small. A possible reason for the appearance of these difficulties is suggested.

Any nonrelativistic theory may be rewritten by introducing fictitious elementary particles with arbitrary properties. No physical predictions are affected, provided that the interaction part of the Hamiltonian is correspondingly modified. The fictitious elementary particle provides a good representation of a real composite particle if the modified interaction is sufficiently weakened for perturbation theory to work. It corresponds to a truly elementary particle with infinite bare mass, and hence with $Z=0$. We show how the latter condition yields a sum rule for the coupling of a composite particle to its constituents as a function of energy. The sum rule can be used to evaluate such coupling constants as that for the proton-electron-hydrogen vertex. The mathematical method used is that developed by Schmidt for the study of the Fredholm equation, and corresponds to the extraction of a single factor from the full Fredholm determinant.

The existence of the ordinary mu decay, mu yields e + nu + nu (bar), seems to prove that the muon andd electron do not differ in any quantum numbers. It follows that weak electromagnetic transitions between muons and electrons could oecurs if there is a mechanism to produce them. The purpose of this effort is to discuss phenomenologically (without attachment to any specific mechanism other kinds of electromagnetic transitions between muon and electron that may be possible even if mu yields e + gamma is somewhat suppressed. (A.C.)

The absence of an Olbers paradox for neutrinos shows that the average neutrino luminosity must fall off as we observe older and more distant parts of the universe. Unlike the situation for light, the « paradox » could not be avoided by any kind of red shift, and hence puts a strong constraint on cosmological speculation. The ambient neutrino flux turns out to be finite in the modern steady state or evolutionary cosmologies, and is estimated at about 103 neutrinos/cm2 s. Taking absorption of neutrinos into account does not affect these remarks, except that in the oscillating cosmologies there must recur periods when neutrino absorption is faster than neutrino emission. It is estimated that the onset of neutrino absorption for an oscillating universe comes when it has contracted by a (linear) factor of about 105, and absorption might not lower the neutrino population very much until the contraction reaches about 1014.

A detailed analysis is made of the possible conversion of muonium into antimuonium in various environments. An assumed - mu e- mu e weak interaction of the usual form and strength gives a probability of 2.5 x 10/sup -5/ in vacuum, even in the presence of reasonable external electric fields. In a solid the probability is less by at least 10, and probably 20, orders of magnitude. In an inert gas the probability is roughly to be divided by the numbers of collisions during a muon lifetime, and hence is quite small unless the pressure at room temperature is less than about 10/sup -4/ atm. Lowering the temperature does not help. A possible experiment is suggested. (auth)

The methods of current algebra are applied to the problem of calculating corrections to the symmetries of the strong interactions in renormalizable gauge theories of the weak and electromagnetic interactions. The strong interactions are described by a neutral-vector-gluon model, so that their symmetries are just the symmetries of the quark mass matrix, and are determined by the vacuum expectation values of the weakly coupled scalar fields. Corrections to these symmetries are calculated to all orders in the gluon coupling, but only to second order in the gauge coupling $e$. After putting the results in a gauge-invariant form, it is found that all divergences cancel in the corrections to "natural" symmetries of the strong interactions. The weak interactions can produce corrections to strong-interaction symmetries of the same order of magnitude as the electromagnetic corrections, but such "order $\ensuremath{\alpha}$" effects occur only as corrections to the quark mass matrix, and therefore necessarily conserve parity, strangeness, charm, etc., and may produce only isovector corrections to isotopic-spin conservation. It is suggested that these weak-interaction effects of order $\ensuremath{\alpha}$ are responsible for the nonelectromagnetic corrections to isotopic-spin conservation which seem to be needed in calculations of mass differences and $\ensuremath{\eta}$ decay.

Free fields, which transform according to general irreducible representations of the homogeneous Lorentz group, are constructed for any spin. The most general interactions may be built up from these fields, without explicit derivaties. The propagators for these fields are calculated in terms of previously calculated propagators for ($2j+1$)-component fields and Racah coefficients. A general formula is given which describes the decomposition of these propagators into various "daughter" spin exchanges.

An effective Lagrangian for soft-pion interactions is constructed such that lowest order perturbation theory precisely reproduces the results of current algebra.

The invariance of strong interactions under $G$, the product of charge symmetry and charge conjugation, has important consequences for strangeness-conserving lepton interactions. According to the $G$-transformation properties of the strongly interacting "currents," we may divide the primary weak interactions into two classes. The first class includes the conventional nucleon-lepton Fermi interaction, and is the only class that contributes to the $\ensuremath{\beta}$-decay coupling constants. Unambiguous tests for the existence of second-class interactions include: (a) induced scalar term in ${\ensuremath{\mu}}^{\ensuremath{-}}$ absorption, (b) inequality of certain small correction terms in ${\mathrm{B}}^{12}$ and ${\mathrm{N}}^{12}$, or in ${\mathrm{Li}}^{8}$ and ${\mathrm{B}}^{8}$ $\ensuremath{\beta}$ decay, (c) inequality in rates of ${\ensuremath{\Sigma}}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\Lambda}}^{0}+{e}^{\ifmmode\pm\else\textpm\fi{}}+\ensuremath{\nu}$. Absence of second-class interactions would indicate a deep relation between isotopic spin and weak interactions; for example, the recent Feynman-Gell-Mann theory predicts that all vector weak interactions are first class. The presence of second-class interactions would mean that the usual Fermi interaction is insufficient, and must be supplemented by terms involving strange particles. Some general remarks are also made about the relations between (${l}^{\ensuremath{-}}, \overline{\ensuremath{\nu}}$) and (${l}^{+}, \ensuremath{\nu}$) processes, and we prove the following useful theorem: no interference between $V$ and $A$ may occur in any experiment which treats both leptons identically and in which no parity nonconservation effects are measured, providing that we may neglect the mass and charge of the leptons.

A large class of broken supersymmetry theories is shown to imply the existence of fermions ${\ensuremath{\lambda}}^{\ifmmode\pm\else\textpm\fi{}}$ and ${\ensuremath{\lambda}}^{0}$, lighter than or nearly degenerate with the ${W}^{\ifmmode\pm\else\textpm\fi{}}$ and ${Z}^{0}$ gauge bosons, and with vanishing baryon and lepton number. If the ${\ensuremath{\lambda}}^{\ifmmode\pm\else\textpm\fi{}}$ is appreciably lighter than the ${W}^{\ifmmode\pm\else\textpm\fi{}}$ it can be readily produced in ${W}^{\ifmmode\pm\else\textpm\fi{}}$ decay, as well as in ${e}^{+}\ensuremath{-}{e}^{\ensuremath{-}}$ collisions.

An examination is made of supersymmetric theories which have a variety of physically inequivalent solutions in zeroth order. For any theory in which the supersymmetry itself is not spontaneously broken, the potential remains zero in the one-loop approximation at all supersymmetric values of the scalar fields, so the ambiguities present in zeroth order are not resolved by higher-order corrections. In one interesting class of models, these ambiguities include a re-scaling of all fields; there is a corresponding pseudo-Goldstone boson whose mass remains zero to all orders, even though the theory is not scale-invariant.

We like to think of physics as moving toward unity, but at any one moment it is fragmented into specialties, which sometimes seem even to define different scientific personalities. This certainly appeared to be the case when I arrived in Palmer Lab at Princeton University as a new graduate student just 30 years ago. Physicists who worked on strong and weak interactions seemed to me to be very different sorts of animals, differentiated especially by the quantity and nutritional value of the data on which they fed.

The mass difference $m({D}^{+})\ensuremath{-}m({D}^{0})$ of the charmed pseudoscalar meson doublet is estimated as about 6.7 MeV, as compared with the value 15 MeV given by De R\'ujula, Georgi, and Glashow. Mass differences are also estimated for charmed baryons.

The stability of the vacuum sets a lower bound of order $\ensuremath{\alpha}{{G}_{\mathrm{F}}}^{\ensuremath{-}\frac{1}{2}}$ on the Higgs-boson mass. For the simplest SU(2) \ensuremath{\bigotimes} U(1) model, this lower bound is $1.738\ensuremath{\alpha}{{G}_{\mathrm{F}}}^{\ensuremath{-}\frac{1}{2}}$, or 3.72 GeV.

How to express superconvergence relations for pion scattering as statements about the generators ${X}_{a}$ of chiral SU (2) \ensuremath{\bigotimes} SU (2) is shown. One of the superconvergence relations has a remarkable algebraic significance: The matrix $m{J}_{y}$ (where $m$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{J}}$ are the mass and spin matrices) is the sum of a chiral scalar and the fourth component of a chiral four-vector.

The weak contributions to the anomalous magnetic moment of the muon are calculated in a proposed theory of the weak and electromagnetic interactions. The result is finite and of order $\ensuremath{\Delta}{g}_{\ensuremath{\mu}}\ensuremath{\approx}{10}^{\ensuremath{-}8}$, too small to be measured at present. The present agreement between theoretical and experimental values of the muon magnetic moment does not even rule out the possibility that the scalar meson required by this theory has a very small mass. If this mass were sufficiently small, then the scalar-meson field would produce shifts of the order of a hundred parts per million in muonic-atom energy levels.

Two sum rules are derived, relating moments of the spectral functions of the vector and axial-vector currents. If it is assumed that the $\ensuremath{\rho}$ and $A1$ mesons dominate these moments, then their masses must be in the ratio $\frac{{m}_{A1}}{{m}_{\ensuremath{\rho}}}=\sqrt{2}$, in very good agreement with experiment.

A multiplicative selection rule for mu meson-electron transitions is proposed. A "muon parity" = -1 is considered for the muon and its neutnino, while the "muon parity" for all other particles is +1. The selection mile then states that (-1) exp STAno. of initial (-1) parity particles! = (-1) exp STAno. of final (-1) parity particles!. Several reactions that are forbidden by an additive law but allowed by the multiplicative law are suggested; these reactions include mu + -- s+ + nu /sub mu / + nu /sub e/, e/sup -/ + e- -- mu - + mu - , and muonium -- antimu onium ( mu /sup +/e -- mu /sup -/e/sup +/). An intermediate-boson hypothesis is suggested. (T.F.H.)

The early history of the universe is discussed in the context of an exponentially rising density of particle states.

The eikonal approximation is applied to the scattering of a fast particle of arbitrary spin in a slowly varying external electromagnetic field. It is found that“exponentiation“ occurs only for the gyromagnetic ratio g = 2, when transitions to excited states are ignored. However, when transitions to excited states are included, exponentiation always occurs, provided that the anomalous magnetic moment obeys a generalized version of the Drell-Hearn sum rule.

A survey of the structures of the intestine of the Goldfish is given. Several morphological techniques have been used. Special attention has been paid to those characteristics of this organ that may be of importance to physiologists working with the same material, with emphasis on cell-to-cell contact in the mucosal epithelium, which appeared to be quite “tight”.