Internal Quantum Numbers Research Articles

Can Quarks Be Kept Inside Hadrons?

A self-consistent dynamical mechanism is suggested which appears to be compatible with conventional ideas about relativistic field theory, and which prevents the appearance of the quanta of the field in outgoing or incoming scattering states. A model calculation is provided which illustrates the mechanism. The model produces a spectrum of conventional hadron states which corresponds to an infinitely rising Regge trajectory. All states are physical. When SU(3) internal quantum numbers are included, it is argued that the mechanism is stable only if the physical states have zero triality.

Tests for the Internal Quantum Numbers of Partons

The restrictions for the structure functions of deep-inelastic electroproduction and neutrino-induced production following from assigning partons to any representation of $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}Y$ are derived. Apart from the general parton-model assumptions only isospin invariance and positivity are used. Comparison with experiment allows us to exclude all models which have only partons of spin \textonehalf{}, isospin \ensuremath{\le}\textonehalf{}, and integral charge.

A duality formulation of the parity-change rule for baryons

Diffraction dissociation reactions can be expected to gain particular importance at high energies because of the asymptotic constancy of their cross-sections. In Regge theory such reactions go by Pomeranchuk exchange, for they involve no change in the internal quantum numbers (B, S, 1, G, etc.) of the dissociating particle. Whether there exist selection rules governing changes of spin and pari ty has been a controversial question, various rules having been proposed (1), in particular the empirical rule AP = (-1) AJ proposed by GRIBOV and ~r (2). This rule can be derived from Regge theory (3) for a spinless particle dissociating at 0 ~ and it follows (at all angles) from simple angular-momentum considerations on making a j e = 0 + assignment for a pomeron. When applied to baryons the rule A P = (--1) A: requires that diffractively produced I = 89 nucleon resonances must belong to the (~ natural ~> pari ty sequence 1+ a~ , 2 , 5+ 7-2 , 2 . . . . , i.e. P = (-1) J-~. Experiments (4) show peaks at about 1400, 1520, 1690 and 2190 MeV suggestive of known resonances in the vicinity of these masses having this sequence of quantum numbers, but in the absence of proper spin-parity analyses the evidence must be taken as inconclusive. Furthermore, simple angular-momentum arguments fail to exclude resonances of opposite (~ unnatura l ~>) parity, and as yet the rule has no theroetical foundation. The aim of this paper is to consider the pomeron (~ : I e = 0 +, J P = 0 +) exchange dissociation of a nucleon into a r:J~ final state

QUARKS, LEPTONS AND WEAK INTERACTIONS

Using a fourth neutral quark, we classify weakly interacting particles, without changing the strong interactions content of the quark model. Leptons are in the octet 3 x [MATH] x 1, which includes heavy leptons as well. Weak interactions are represented graphically by quark diagrams in an analogous way to the strong ones. They are practically of current x current nature. The octet of bare weak hadronic currents has exactly the same quark structure as the leptonic ones (generalized Universality ). The usual selection rules of weak interactions follow without any additional assumption. ΔQ = - ΔS is possible, but only for meson-lepton interactions, in accord with the only experimental possibility open, also K02[MATH]. The representation deals with internal quantum numbers only and may be used as a natural frame-work for different theories of weak interactions, using different assumptions about the external dynamical nature of the theory.

Determination of Asymptotic Parameters in the Statistical Bootstrap Model

The statistical bootstrap model predicts that the density of hadron states $\ensuremath{\rho}(m)$ approaches $c{m}^{a}{e}^{\mathrm{bm}}$ asymptotically. We consider the consequences of extending the bootstrap condition in the model from asymptotic down to finite masses. This allows us to determine the parameters $a$, $b$, and $c$ for various assignments of the hadron volume and low-mass input spectrum, and for the extreme cases of excluding all exotic particles or including all of them. In all cases, $a=\ensuremath{-}3$ for ${\ensuremath{\rho}}_{\mathrm{tot}}$ (summed over all internal quantum numbers). The parameter $b$ varies somewhat from case to case but is always of order $m_{\ensuremath{\pi}}^{}{}_{}{}^{\ensuremath{-}1}$; thus we predict the maximum temperature ${T}_{0}={b}^{\ensuremath{-}1}\ensuremath{\approx}{m}_{\ensuremath{\pi}}$ in rough agreement with Hagedorn's empirical determination. The inclusion of exotic states has little effect on ${\ensuremath{\rho}}_{\mathrm{tot}}$ but does redistribute the partial level densities according to a simple rule. The predicted level densities (excluding exotic states) are compared with present data.

Formulation of the Many-Body Problem for Composite Particles. III. The Projected Hamiltonian

A previously developed representation for many-atom problems is extended by incorporation of all effects of interatomic exchange of electrons and nuclei into a ``projected Hamiltonian.'' An expression for the projected Hamiltonian is derived which is exact on the two-atom subspace, and hence incorporates those exchange effects which are important at low densities. By the use of the center-of-mass separation of atomic wavefunctions, this expression for the projected Hamiltonian is transformed into a representation in terms of atomic field operators ψν(r) and ψν†(r), where r is the center-of-mass position of an atom and v stands for all of its internal quantum numbers.

The analytic properties of the scalar vertex function

A single spectral representation of the vertex function F, for the conversion of a scalar particle binto a scalar particle d via interaction with a third scalar particle s, is obtained heuristically by using unitarity to evaluate the contribution to Im F from a state ( a c ) with the same internal quantum numbers as ( d b ) and then making a one-pole approximation to the amplitude for d b → a c and a constant approximation to the vertex function for the vertex (cas). For the case where the masses satisfy the inequality m a + m c ⩾ m b + m d, the above approximations are shown to give the same result as that obtained from the Feynman parametrisation of the vertex function arising from the triangle diagram. Two methods are then used to obtain a spectral representation for the triangle diagram vertex function for all allowed mass configurations, including cases for which the threshold is anomalous. The first is to evaluate the triangle diagram vertex function directly; the second is to continue in m b 2 and m d 2 the result for the case m a + m c ⩾ m b + m d.

Charges and Generators of Symmetry Transformations in Quantum Field Theory

Within the Wightman approach to quantum field theory, we review and clarify the properties of formal charges, defined as space integrals for the fourth component of a local current. The conditions for a formal charge to determine an operator (generator) are discussed, in connection with the well-known theorems of Goldstone and of Coleman. The symmetry transformations generated by this operator---given its existence---are also studied in some detail. For generators in a scattering theory, we prove their additivity and thus provide a simple way to characterize them from their matrix elements between one-particle states. This characterization allows an immediate construction of the unitary operators implementing the symmetry transformations, and implies that all internal symmetry groups are necessarily compact. We also indicate how to construct interacting fields having definite internal quantum numbers. The present status of the proof of Noether's theorem and of its converse is discussed in the light of the rather delicate properties of formal charges.

Comment on "Tests for Eightfold-Way Octets in the Baryon Spectrum"

It is pointed out that the phenomenon of multiplet mixing, when there are a number of relatively closelying resonances with the same values of the internal quantum numbers, renders inappropriate the use of the Gell-Mann-Okubo mass formula in attempting to judge the validity of the $\mathrm{SU}(3)$ symmetry scheme.

The ( 3He, 7Be) reaction on light nuclei

An experimental study of the ( 3He, 7Be) reaction has been performed at 30 MeV with 10B, 12C, 16O, 19F, 20Ne, 28Si and 40Ca target nuclei.The doublet arising from the occurrence of two bound states of the 7Be nucleus, the 7Be (0) 3 2 − ground state, and the 7Be (1) 1 2 − first excited state, was resolved in all the spectra. Angular distributions up to about 90° or more in the c.m. system are presented for the strongest transitions. A mechanism of one-step pick-up of four correlated nucleons, with the internal quantum numbers of a free α-particle, in short α-pick-up, is consistent with the main characteristics of the data. Unambiguous experimental evidence, though, calls for at least a partial correction to this simplified picture: the angular distributions of 7Be (0) and 7Be (1) are found to be somewhat different, and the shapes of the angular distributions vary rather quickly with the incident energy. The nature and validity of the assumptions of an α-pick-up mechanism are discussed, and tentative DWBA fits to the data are obtained with a “fixed range” interaction approximation. The main spectroscopic information is that the α-transfer spectroscopic factor does not decrease between 12C and 40Ca. nearly as strongly as shell-model wave functions would suggest.

Crossing Symmetry in theO(4, 2)Formulation of the Dirac Theory

The formulation of the Dirac theory based on the group $O(4, 2)$, in which an internal additive quantum number $n$ appears, distinguishing particles and antiparticles, is discussed in detail. The substitution law then becomes an analytic continuation in $n$.

Selection Rules for Diffraction Dissociation

The concept that internal quantum numbers do not change in diffractive production is extended to the quantum numbers of the quark model. The resulting approximate selection rules are discussed and compared with the predictions of other models.

Spinor space and strong interactions

In order to explain the existence of internal quantum numbers of elementary particles, the ordinary space–time position vector is assumed to be a bilinear combination of two basic spinors. It is shown that this assumption leads to the replacement of the inhomogeneous Lorentz group by the (4 + 1) de Sitter group. Using the idea of external and internal space for the description of strongly interacting particles, the assumption provides a simple way of introducing the internal symmetries. The internal symmetry group SU(3) appears naturally connected with the external space, yielding the Gell-Mann -Okubo mass formula.

Reggeization of External Particles

The possibility of determining the Regge trajectory of an external particle is raised by considering the constraints imposed by bootstrap conditions on the two-body scattering of particles of arbitrary spins and masses. Assuming that the trajectory is the solution of a certain system of equations, it is made plausible that a reduced system corresponding to the continuation in spin and mass of only one external particle can equivalently generate the trajectory if one point on the trajectory is known. The particular system of $\ensuremath{\pi}\ensuremath{\pi}$ scattering is taken up as a test case where one external pion has arbitrary mass and spin but otherwise carries the pion internal quantum numbers, and where the bootstrap conditions are based on the self-supporting mechanism of the $\ensuremath{\rho}$. Linear combinations of the two independent helicity amplitudes, having kinematic singularities that are uniformly factorizable for arbitrary $J$, are constructed and unitarized by a phase $N{D}^{\ensuremath{-}1}$ method. An implicit solution of the bootstrap matching equations is found, yielding a trajectory which identically passes through the physical pion when $J\ensuremath{\rightarrow}0$. Only the ratio of the Regge-pole energy-plane residues of the two helicity states is determinable. Finally, a test for uniqueness, largely based on the asymptotic behavior of this ratio as $|J|\ensuremath{\rightarrow}\ensuremath{\infty}$, is established.

Classification of Particle Multiplets

In the present paper we observe that in considering how particle states transform under $\ensuremath{\Theta}=CPT$ one cannot neglect internal symmetries, since $\ensuremath{\Theta}$ changes the signs of all conserved internal additive quantum numbers; and that, furthermore, if a physical theory admits a parity symmetry $\mathcal{P}$, this symmetry may also, like $\ensuremath{\Theta}$, effect an automorphism on the group of internal symmetries, as does, for example, $\mathrm{CP}$, the correct parity operator in the presence of weak interactions. We find all types of particle multiplets for massive particles that are compatible with local field theory, particle multiplets being defined as the irreducible corepresentations of the group obtained by extending the internal symmetry group by $\mathcal{P}$ and $\ensuremath{\Theta}$. Of the thirteen types that exist, only three occur in the various approximation schemes that are at present used to describe nature. The reason for the nonappearance of the other types is not to be found in the usual postulates of local field theory, because Lagrangian field-theoretical examples of all thirteen types do exist and are given.

Speculations on Particles, Superselection Rules, and Composites

The occurrence of baryon number, electric charge, lepton number, and their association with mass, spin, and internal quantum numbers is investigated from the viewpoint of measurability in quantum field theory and on the basis of a very general composite-particle hypothesis. From this scheme, a very satisfactory classification of all permissible particle states is obtained. The special role played by $s>~1$ massless, neutral, structureless bosons is pointed out.

Structure and Excitation Mechanism of theN*in Photoproduction γ +p→N*

The photoproduction of nucleon resonances r+p-'7N* is theoretically examined in the framework of the fundamental triplet model of hadrons. The partial width of the decays N*-'7N+r and of the Breit-Wigner formula for the processes r+p-'7N*-'7N+rc are obtained in terms of the internal quantum numbers and the structure parameters of the Nand N* 's. The basic assumption is as follows: The nucleon and its resonance levels are well de­ scribed by the non-relativistic bound state wave functions of three constituent particles. Both the nucleon and the (3,3) -resonance, Na3* belong to the 56-dimensional representation of the SU6, and are further characterized by the S-wave internal space wave function. The other resonances are regarded as the single particle excitation mode in which one of constituents is in· a higher angular momentum state. The electromagnetic interaction is U-s pin invariant and its exchange current effect is neglected. First it is shown that the selection rule for electric and magnetic multipoles to induce the process r + N--'7 N* is completely fixed by the above assumption and an additional hypo­ thetical relation L+S=J which may be called the Kycia-Riley relation, where L, S and J are the internal orbital, spin and total angular momentum, respectively. Both the electric and magnetic multipoles (G'J-1/2 and 3JiJ+1/2) contribute to the excitation of the N 1,2J* 's with I= (1/2), while only the magnetic muhipole transition (3/iJ -1/2) is necessary for the excitation of the N 3,2J* 's with I= (3/2), I being the isospin of the resonance levels. Consequently the N 11* (1400) excitation would be much suppressed in agreement with experim~nt. The partial widths for various N* excitations are numerically obtained. It is shown that the existing experimental data on the Naa* and N1a* are consistent with them if the spatial extension of the bound state is put nearly equal to the nucleon Compton wave length. It is also noted that only the partial width of the N 33* photoexcitation is not sensitive to the core radius in the long photon wave-length approximation and in the su6 symmetric limit of the internal space wave function.

On the Connection between External and Internal Symmetries of Strongly Interacting Particles

The relation between the mass and spin-parity Jp, and internal quantum numbers of elementary particles, hints at a nontrivial connection between the external symmetry, namely the Poincaré group or its Lie algebra, and a so-called internal symmetry. Studying this connection mathematically, we find that any extension of (resp. by) the Poincaré Lie algebra 𝒫 by (resp. of) a semisimple Lie algebra 𝒳 is equivalent to the trivial one, 𝒫 ⊕ 𝒳. Moreover, if we are looking for a Lie algebra containing 𝒫 and 𝒳 in an economical and nontrivial way, namely what we call a (nontrivial) unification of 𝒫 and 𝒳, we find restrictions on the possibilities of choice of 𝒳, which exclude compact internal symmetries. An explicit treatment of SL(3, C) as internal symmetry, related to the external symmetry by the unification process of Lie algebras, gives a mass formula which is in very good accordance with the experimental data, and can be theoretically interpreted by means of a so-called ``classification principle''.

Exchange of Even- and Odd-Parity Baryon-Meson Resonances and Backward Elastic Scattering

If there are two meson-baryon resonances or bound states with opposite parities but with the same remaining internal quantum numbers including signature, one at c.m. energy ${W}_{1}$ with $l=J\ensuremath{-}\frac{1}{2}$ and the other at ${W}_{2}$ with $l=J+\frac{1}{2}$, then the residue of the Regge trajectory on which they lie must change sign between ${W}_{1}$ and $\ensuremath{-}{W}_{2}$. Consequences of the exchange of such a trajectory to backward elastic scattering are discussed.

On extracting the infinite-dimensional unitary representations ofSU 2,2 in a free-particle basis. — I

The infinite-dimensional unitary irreducible representations ofSU2,2, which includes the Poincare group properly, are sought in a basis in which the free-particle quantum numbers momentum, spin, and helicity are sharp. None of the standard representation formalisms apply, such as the Wigner-Mackey semidirect product theory or the Weyl-Cartan theory for semisimple groups, so a general method based on Schur’s lemma is used. This paper discusses new insights into the problem of synthesizing the Poincare group and internal symmetries gained by working out this specific problem. These are, very briefly: 1) the explanation of continuous mass spectra within supermultiplets; 2) new sources and possibilities for internal quantum numbers; 3) the existence of massiveSU2,2-invariant fields in spite of the general belief to the contrary (SU2,2=C ≡ conformal group of space-time); 4) the convenience of space-time-dependent similarity transformations to change to familiar bases of irreducible states.