Particle Form Factor Research Articles

Asymptotic form factors of hadrons and nuclei and the continuity of particle and nuclear dynamics

The large-${q}^{2}$ behavior of the elastic form factor of a hadron or nucleus is related by dimensional counting to the number of its elementary constituents. Using the framework of a scale-invariant quark model, dimensional-scaling predictions are derived for the $\frac{B({q}^{2})}{A({q}^{2})}$ ratio in the Rosenbluth formula, multiple-photonexchange corrections, and the mass parameters which control the onset of the asymptotic power law in the meson, nucleon, and deuteron form factors. A simple "democratic chain" model predicts that for large ${q}^{2}$, $F({q}^{2})\ensuremath{\propto}{(1\ensuremath{-}\frac{{q}^{2}}{{{m}_{n}}^{2}})}^{1\ensuremath{-}n}$, where ${{m}_{n}}^{2}$ is proportional to the number of constituents $n$. In the case of nuclear targets (or systems with several scales of compositeness), we also define the "reduced" form factor ${f}_{A}({q}^{2})=\frac{{F}_{A}({q}^{2})}{\ensuremath{\Pi}{i=1}^{A}[{F}_{i}({{q}_{i}}^{2})]}$ in order to remove the minimal falloff of ${F}_{A}$ due to the nucleon form factors at ${{q}_{i}}^{2}=(\frac{{{m}_{i}}^{2}}{{{M}_{A}}^{2}}){q}^{2}$. Dimensional counting predicts ${({q}^{2})}^{A\ensuremath{-}1}{f}_{A}({q}^{2})\ensuremath{\rightarrow}\mathrm{const}$. A systematic comparison of the data for $\ensuremath{\pi}$, $p$, $n$, and deuteron form factors with the dimensional-scaling quark-model predictions is given. Predictions are made for the large-spacelike-${q}^{2}$ $^{3}\mathrm{He}$ and $\ensuremath{\alpha}$-particle form factors. We also relate the deuteron form factor to (off-shell) fixed-angle $n\ensuremath{-}p$ scattering, and show that the experimental results for ${t}^{5}{F}_{d}(t)$ are consistent with the magnitude of the $s$-wave wave function ${u}^{\ensuremath{'}}(0)$ obtained from soft-core potentials. The relation of the dynamics of an underlying six-quark state of the deuteron to the nucleon-potential and meson-exchange-current contributions is discussed. The scaling of ${q}^{2}{f}_{d}({q}^{2})$ implies that the nuclear potential (after removing the effects of nucleon structure) displays the scale-invariant behavior of a theory without a fundamental length scale. Predictions are also given for the structure functions, fragmentation, and large-angle scattering of a nucleus.

Relativistic form factors of composite particles

Relativistic form factors of composite particles

E+e–annihilation into hadrons

This review is concerned with the various theoretical approaches to the problem of e+e– annihilation into hadrons in the light of the new experimental data obtained in 1973-4. Discussions are given of the behavior of the total cross section for e+e– annihilation into hadrons, the form of the inclusive spectra, the dependence of the particle form factors in the timelike region on the square of the momentum transfer, and the effect of the process of e+e– annihilation into hadrons on scattering by positrons and on the process e+e–→μ+μ–. Other topics considered include the relation between inclusive annihilation and electroproduction, as well as a number of theoretical models: the parton model, the statistical model, etc.

Some problems in nuclear and elementary particle physics involving gravitons

Attention is drawn to dispersion relation techniques which are useful in calculating binding energies of light nuclei, mechanical form factors of particles and nuclei, and mean mass radii.

The calibration by electronic means of Coulter counter for the determination of absolute volume of particles.

An electronic method is described for the calibration of a particle sizing apparatus based on the principle employed by Coulter counters, without any need for reference particles. A capillary factor, depending upon the dimensions of the orifice, has been obtained by reference measurements. Incorporating this capillary factor and the particle form factor, the absolute particle volumes can be deduced.

Physical Interpretation of Higher-Derivative Field Theories

Polynomial-type field equations are shown to have a realistic physical interpretation in terms of particle form factors, both for classical fields and for ``dipole-regularized'' quantized fields. Form factors arising from such field equations are found to give a reasonable description of the electromagnetic structure of the proton.

Wave Functions and Form Factors for Relativistic Composite Particles. II

We extend our relativistic description of composite particles to the case of clusters of spin-\textonehalf{} particles. From this, explicit expressions for the nucleon vector current form factors are derived. We find that the nucleon's electric form factor has the form ${G}^{E}(t)={\ensuremath{\alpha}}^{\ensuremath{-}1}{S}_{0}(\frac{\ensuremath{-}t}{\ensuremath{\alpha}})\ensuremath{\Sigma}\stackrel{3}{i=1}{q}^{i}G_{\mathrm{ib}}^{1}(t)$ where ${S}_{0}$ is the nonrelativistic form factor, $\ensuremath{\alpha}=\frac{1\ensuremath{-}t}{4{M}^{2}}$, and ${q}^{i}$ is the charge and $G_{\mathrm{ib}}^{1}(t)$ the electric form factor for the $i$th quark. A similar expression is obtained for the magnetic form factor.

Wave Functions and Form Factors for Relativistic Composite Particles. I

We give a relativistic description of composite particles which behave as nonrelativistic clusters in their rest frames. We find that the relativistic form factor can be obtained from the nonrelativistic one by the substitution $S(t)={\ensuremath{\alpha}}^{\frac{(1\ensuremath{-}n)}{2}}{S}_{0}(\frac{|t|}{\ensuremath{\alpha}})$, where $\ensuremath{\alpha}=\frac{1\ensuremath{-}t}{(4{M}_{{A}^{2}})}$, and where $n$ is the number of subparticles in the cluster. An excellent one-parameter fit to the magnetic form factor of the proton is found.

Spectral Function of the Photon Propagator—Mass Spectrum and Timelike Form Factors of Particles

We consider contributions to the spectral function of the photon propagator from the states which contain a particle (charge $e$ and mass ${m}_{J}$) of arbitrary spin $J$, and its antiparticle. We express the contributions in terms of timelike form factors ${\mathcal{F}}_{J}(a)$ (where $a=\ensuremath{-}{P}^{2}g0$, and $P$ is the momentum of the photon) with the normalization ${\mathcal{F}}_{J}(0)=(2J+1){e}^{2}$. The unitarity limit of the spectral function can be transformed into the asymptotically bounded condition ${\mathcal{F}}_{J}(a)\ensuremath{\lesssim}O(a)$. The experimental information about the anomalous magnetic moment of the muon gives a restriction on the sum of all the contributions. Using the restriction, we examine various mass spectra of charged particles and obtain simple results. For example: If there is an infinitely rising mass spectrum ${m}_{J}=f(J)$, then the asymptotic form of the mass formula must be bounded by condition ${m}_{J}gO(J)$ (case I or II) or ${m}_{J}gO({J}^{\frac{1}{(2l+1)}})$ (case III), where $l$ is a parameter in the form factors, assumed to be ${\mathcal{F}}_{J}(a)=(2J+1){e}^{2}(\frac{a}{4{{m}_{J}}^{2}\ensuremath{-}1}){|1\ensuremath{-}\frac{a}{{\ensuremath{\mu}}^{2}}|}^{\ensuremath{-}2l}$ for $J=0,1,2,\ensuremath{\cdots}$, and ${\mathcal{F}}_{J}(a)=(2J+1){e}^{2}{|1\ensuremath{-}\frac{a}{{\ensuremath{\mu}}^{2}}|}^{\ensuremath{-}2l}$ for $J=\frac{1}{2},\frac{3}{2},\ensuremath{\cdots}$. For the purpose of experimental observations of the timelike form factors and the spectral function of the photon, the colliding-beam experiments ${e}^{+}+{e}^{\ensuremath{-}} (or {\ensuremath{\mu}}^{+}+{\ensuremath{\mu}}^{\ensuremath{-}})\ensuremath{\rightarrow}\ensuremath{\lambda}+\overline{\ensuremath{\lambda}}$ (where $\ensuremath{\lambda}$ is a particle of arbitrary spin) are discussed in some detail.

Form factors of elementary particles

How BIG ARE the elementary particles? How is their charge distributed? These questions are tackled with form factors, which are measures of the charge and magnetic‐moment distributions in the particles. Scattering of electrons on nucleons, and recent measurements made with electron–positron colliding beams, give form factors for the proton, neutron and pion. The behavior of the pion form factor can be fairly well understood by the “rho‐dominance model,” but the nucleon form factors are not so easy to understand. Different models are currently being examined in an attempt to solve this basic problem in strong‐interaction physics.

Spectroscopic factors and single particle form factors

Exact equations are obtained which relate single particle overlap functions and spectroscopic factors. Truncation of these equations leads to a theorem relating the sum of the spectroscopic factors to the number of underlying single particle bound states.

On the Asymptotic Behavior of Electromagnetic Form Facors

Electromagnetic form factors for composite particles, are discussed. It is shown that the' form factor of 1l'-meson behaves like (log t) n/t 2 Cn = 1 or 2) for the large positive momentum squared t when the 1l'-meson is assumed to be a bound state of the nucleon and anti-nucleon system. For the spinless nucleon this is verified without any approximation other than the B-S equation with a general kernel and for this case it is shown that n = 1 Some approxi­ mations are taken when we include the spin of the nucleon, and we find n=2for this case, but it is argued that this conclusion may be true in general. Some discussions of the large t behavior of the form factor of nucleon in a bound state are added.

Asymptotic behaviour of form factors of half-integer spin particles

A composi te model of hadrons, which reproduces the t 2 decrease of the magnet ic form factor of the proton, has been recent ly proposed by several authors (1,4): s t rongly in te rac t ing part icles arc considered as bound states and the result is obta ined using the asympto t ic propert ies of the Bethe-Salpc tcr wave funct ion or of the F e y n m a n graph. In par t icular in ref. (2) bounds have been obta ined for any e lec t romagnet ic t rans i t ion between integer-spin part icles and for the J = 8 9 89 case. In the f ramework of the same model, the procedure of ref. (2) can be general ized to include the general halfinteger spin case: the bounds which we obtain, and which are the analogous of cqs. (36) of ref. (2), arc (t -(p q)2)

Asymptotic Behavior of Form Factors of Composite Particles

Form factors for composite nucleons are studied in the large-momentum-transfer limit. It is argued that multiparticle intermediate states in the dispersion relation, containing arbitrarily large numbers of particles, must be included even to begin to estimate the asymptotic behavior. In the simplest approximation, this physical idea can be implemented using the ladder approximation in the nucleon legs of the form factor. The resulting form factors behave essentially like ${t}^{\ensuremath{-}2}$ both for spin-0 and spin-\textonehalf{} nucleons.

Form Factors and Vector-Meson Dynamics

It is suggested that the isovector form factors of various particles will have a common shape arising from a common dynamical determinant. This leads naturally to a product-of-poles shape for the form factors. Two models are considered for the isoscalar form factors. The first, which supposes that the $\ensuremath{\varphi}\ensuremath{-}\ensuremath{\omega}$ mixing angle is energy-independent, agrees with the result of Krohn and Ringo for ${{G}_{E}}^{{n}^{\ensuremath{'}}}(0)$ only if the baryon normalization is very different from that of the quark model. The second involves an energy-dependent mixing angle and permits the quark model to be consistent with experiment. The theory assumes that the mixing angle vanishes at $t=0$.

Electromagnetic Corrections to Weak Interactions. The Beta Decays of the Muon, Neutron, andO14

The momentum-dependent radiative corrections to the beta decays of the muon, the neutron, and ${\mathrm{O}}^{14}$ have been calculated to order $\ensuremath{\alpha}$ using the techniques of dispersion theory. The transition matrix elements can be expressed to this order (neglecting some effects of strong interactions) in terms of sets of vertex functions which satisfy once-subtracted dispersion relations. The absorptive parts of the vertex functions can be expressed to the appropriate order in terms of the vertex functions themselves and the amplitudes for electromagnetic scattering of the charged particles. It is a curious feature of the present calculation that the choice of the subtraction points is not arbitrary, but is determined uniquely by the requirement that such physically significant quantities as decay rates and the momentum spectra of the leptons should contain no infrared divergences when calculated including the contributions of processes in which soft photons are emitted [inner bremsstrahlung]. The subtraction constants play the role of renormalized weak coupling constants. The significance of this electromagnetic renormalization, and the connection between the choice of the subtraction point and the infrared divergence is examined in detail in the case of the muon. Two models for the beta decay of ${\mathrm{O}}^{14}$ have been considered. In one model, the nucleon involved in the transition is treated as a free particle insofar as the calculation of radiative corrections is concerned; in the other, the ${\mathrm{O}}^{14}$ and ${\mathrm{N}}^{14*}$ nuclei are treated as point particles, and the effects of the nuclear structure are ignored. The results obtained from the two models differ only slightly. Because of the appearance in the absorptive parts of the vertex functions of the form factors of the charged particles, evaluated for the particles on the mass shell, we are able to study analytically the effects on the transition amplitude of the finite electromagnetic structure of the nuclei. The effects of the finite spacial distribution of the decaying matter are treated using the usual multipole expansion of the nuclear matrix element. The leading electromagnetic structure correction is of the same form as the familiar $Z\ensuremath{\alpha}\mathrm{RW}$ in the correction for finite nuclear structure (finite deBroglie wavelength effect), but is of a different origin, and leads to a near doubling of the total structure corrections. The known theoretical corrections to the deay rates for the 0+ \ensuremath{\rightarrow} 0+ transitions ${\mathrm{O}}^{14}({\ensuremath{\beta}}^{+}){\mathrm{N}}^{14*}$, ${\mathrm{Al}}^{26*}({\ensuremath{\beta}}^{+}){\mathrm{Mg}}^{26}$, and ${\mathrm{Cl}}^{34}({\ensuremath{\beta}}^{+}){\mathrm{S}}^{34}$ are summarized. Using the recent, very accurate data on the decays of the muon, ${\mathrm{O}}^{14}$ and ${\mathrm{Al}}^{26*}$, we obtain the values ${G}_{\ensuremath{\mu}}=(1.436\ifmmode\pm\else\textpm\fi{}0.001)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$, ${G}_{\ensuremath{\beta}}({\mathrm{O}}^{14})=(1.419\ifmmode\pm\else\textpm\fi{}0.002)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$, and ${G}_{\ensuremath{\beta}}({\mathrm{Al}}^{26*})=(1.430\ifmmode\pm\else\textpm\fi{}0.002)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$ for the renormalized vector coupling constants for these transitions. The less accurate data on the neutron yield ${G}_{V}=(1.356\ifmmode\pm\else\textpm\fi{}0.068)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$. The results for ${G}_{\ensuremath{\mu}}$ and ${G}_{\ensuremath{\beta}}$ are not directly comparable in the present theory, but the different values of ${G}_{\ensuremath{\beta}}$ should be. The discrepancy of (0.8\ifmmode\pm\else\textpm\fi{}0.5)% between the effective coupling constants for the decays of ${\mathrm{O}}^{14}$ and ${\mathrm{Al}}^{26*}$ could, therefore, be significant, and may yield information about the still uncertain Coulomb corrections to the nuclear matrix elements. If the validity of the cutoff-dependent results of perturbation theory is assumed, the renormalization constants can be evaluated, and one obtains ${G}_{\ensuremath{\mu},\mathrm{bare}}=(1.431\ifmmode\pm\else\textpm\fi{}0.001)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$ and ${G}_{\ensuremath{\beta},\mathrm{bare}}({\mathrm{O}}^{14})=(1.404\ifmmode\pm\else\textpm\fi{}0.002\ifmmode\pm\else\textpm\fi{}0.007)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}49}$ erg ${\mathrm{cm}}^{3}$. Those coupling constants differ by 1.9\ifmmode\pm\else\textpm\fi{}0.2%, but because of uncertainties regarding the nuclear matrix element for ${\mathrm{O}}^{14}$, the effects of strong interactions, and the possible existence of an intermediate vector meson which mediates the weak interactions, a direct comparison of these numbers may not be relevant to the possible universality of the Fermi interaction.

Symmetries of strong interactions and electromagnetic form factors

It is shown that if the strong interactions are invariant under the simple Lie groups of rank two G2 orC2, the electromagnetic form factors of neutral particles are zero. This is true if the particles belong to not too high-dimensional representations ofG2 or C2.

Structure singularities of electromagnetic form factors

In local field theories the form factors of particles can have singularities which are characteristic for the structure of these particles as composite systems. These « structure singularities » are studied with the help of examples from perturbation theory. Their mathematical properties are described for real and complex values of the mass variable, and their physical implications are discussed.

Electrical Properties of Some Carbon Black-Oil Suspensions

The dc conductivity and ac (1000 cps) properties of suspensions of R-40 carbon black in transformer, silicone, and linseed oils, and suspensions of Shawinigan black in linseed oil, were studied as functions of time, carbon black concentrations, and rotational speed of test cell. The dc conductivity was also studied as a function of voltage. Immediately after agitating the suspension, both ac and dc conductivities increase rapidly with time, then level off and approach a saturation value. For the dc conductivity, this value increases with increasing voltage in all suspensions studied except the highest concentrations (10 percent by weight), which obey Ohm's law. The rate of approach to saturation is independent of voltage but depends on carbon black concentration. Increasing the concentration increases the conductivity. Increasing the speed of rotation decreases the conductivity. Form factors for the carbon black particles are calculated from the dynamic values of the dielectric constant by Voet's method. Agglomeration factors are then determined. At low speeds the agglomeration factor decreases rapidly with increasing speed. At higher speeds it approaches unity asymptotically.