Regge Recurrences Research Articles

Meson-Baryon Interactions with BrokenSU(3)and the Baryon Spectrum in Relativistic Quantum Mechanics

The multichannel relativistic Schr\"odinger equation is solved for the ${\frac{1}{2}}^{+}$ and ${\frac{3}{2}}^{+}$ partial-wave amplitudes and their Regge recurrences with an energy-dependent potential obtained by computing the baryon-exchange contribution to the pseudoscalar-meson-baryon interaction. As discussed previously, the model yields the usual ${\frac{3}{2}}^{+}$ decuplet and predicts in addition a 27-dimensional representation and a radially excited decuplet in this partial wave. It is shown that in the range of parameters which fit the usual decuplet, there are also decuplet orbital excitations in the ${\frac{7}{2}}^{+}$ and ${\frac{11}{2}}^{+}$ partial-wave amplitudes which correspond to the known experimental resonances. The ${\frac{1}{2}}^{+}$ octet is obtained as deeply bound states, and a second ${\frac{1}{2}}^{+}$ octet or a $\overline{1}\overline{0}$ representation is predicted, depending on whether the $\frac{F}{D}$ ratio is less than or greater than 0.34. In addition, an orbital excitation of the octet occurs in the ${\frac{5}{2}}^{+}$ partial-wave amplitude at very high energies. The $P$-wave phase shifts are in qualitative agreement with experiments, but there are deviations for the ${P}_{33}$ effective range near threshold and for the ${P}_{13}$ phase shifts at higher energies.

Two-Body Unitarity and the Asymptotic Duality Series

The discontinuity of the Regge trajectory obtained recently from a duality series by Kikkawa, Sakita, and Virasoro is investigated. The trajectory is shown to possess all the thresholds corresponding to two-particle intermediate states, i.e., Regge recurrences or daughters. A correction to a linear trajectory is also shown to arise by requiring the Born term (the Veneziano amplitude) to be modified to satisfy two-body partial-wave unitarity near the leading pole in the $J$ plane. This discontinuity is evaluated up to the third two-particle threshold and found to be identical to that obtained from the duality series. Above higher thresholds, such agreement is dependent on the function used to ensure factorization at internal vertices in each term of the duality series.

Baryon Decays and Regge Recurrence Series

Baryon Decays and Regge Recurrence Series

Why do trajectories rise?

Infinite sequences of narrow-width Regge recurrences are shown to arise in scattering amplitudes derived from energy-dependent local potentials, according to two distinct mechanisms.

Strong Coupling of the Multi-Partial-Wave Meson Isotriplet

The algebraic formulation of strong coupling is applied to the strong coupling of the ${\mathrm{SU}}_{2}$-symmetric model in which all the partial waves of the $\ensuremath{\pi}$ mesons are included. The strong-coupling group is a semidirect product of the ${\mathrm{SU}}_{2}\ensuremath{\bigotimes}{\mathrm{SU}}_{2}$ internal symmetry group and an Abelian group which is generated by an infinite number of commuting generators corresponding to the vertices of the $\ensuremath{\pi}$ mesons in different orbital angular momentum states. A physically interesting irreducible representation of the group is obtained which consists of a series of irreducible representations of the $P$-wave strong-coupling group. The Regge recurrences of isobars appear in this series. Each degenerate multiplet of isobars is specified by three quantum numbers---spin $s$, isospin $i$, and an additional quantum number $v$---which satisfy the angular momentum triangular relation. The following form of mass formula is obtained: $M(s,i,v)={M}_{0}+{m}_{0}[x(x\ensuremath{-}1)s(s+1)+(1\ensuremath{-}x)i(i+1)+xv(v+1)]$.

Possible Existence of a Spin-2 Regge Recurrence of theS01 p−pPole (The Unbound Diproton)

A model is developed wherein the $^{1}D_{2}p\ensuremath{-}p$ partial-wave amplitude is coupled through $\frac{N}{D}$ equations to a second, inelastic channel containing a proton and a second particle in a relative $S$ wave. The mass of the second particle is smeared out according to an appropriate Breit-Wigner form around the (3,3) resonance at $1260\ensuremath{-}i120$ MeV. The phenomenological potential is adjusted to fit the elastic $^{1}D_{2}$ phase shift to 400 MeV and the existing imprecise data at 660 MeV. The model, when adjusted to fit experiment, predicts a pole on the second Riemann sheet at $400\ensuremath{-}i300$ MeV. This pole may be interpreted as a bound state in the inelastic channel and has quantum numbers ($I=1$, $B=2$, $J=2$) consistent with the spin-2 Regge recurrence of the pole in the $^{1}S_{0}$ amplitude, the unbound diproton. An alternative coupled-channel model is developed which fits the elastic data but which does not predict the pole. The high-energy predictions for elastic scattering which are derived from the second model are found to differ drastically from those of the first model and from the predictions of various theories for high-energy inelastic processes.

Kinematic Constraints for Infinitely Rising Regge Trajectories

A kinematic constraint on the energy dependence of infinitely rising Regge trajectories is established. Under quite general conditions it is shown that asymptotically $\ensuremath{\alpha}(s)\ensuremath{\le}C{s}^{\frac{1}{2}}$, provided that certain channels couple to high-spin Regge recurrences. The implications of this result for dynamical models of ever-rising trajectories and for the decay rates of resonances are discussed in the light of experimental indications that $\ensuremath{\alpha}(s)\ensuremath{\sim}\mathrm{Cs}$.

Degenerate Regge trajectories and parity doublets of baryon resonances

Abstract The MacDowell reflection symmetry as applied to Regge recurrences of SU(3) multiplets is suggested as a natural explanation of recent experimental observations of parity doublets of baryon resonances (resonance pairs with approximately equal masses, the same Y , I and J values and opposite parities).

High-Spin Baryons. V. Dynamical Model for Regge Recurrences of Inelastic Resonances: Numerical Results

Virtual transitions of the type $\ensuremath{\pi}{N}_{i}\ensuremath{\rightarrow}\ensuremath{\rho}{N}_{j}$, where ${N}_{i}$ and ${N}_{j}$ are arbitrary nucleon isobars, are considered as possible mechanisms for Regge recurrences. It is found that the largest one-pion-exchange forces coupling to ${(\ensuremath{\rho}{N}_{1})}_{L=0},{({\ensuremath{\rho}N}_{2})}_{L=0},\ensuremath{\cdots}$ [where ${N}_{1},{N}_{2},\ensuremath{\cdots}$ have spin-parity ${J}^{P} \mathrm{of} {S}^{P},{(S+2)}^{P},\ensuremath{\cdots}$] occur in the opposite-parity states ${(S+1)}^{\ensuremath{-}P},{(S+3)}^{\ensuremath{-}P},\ensuremath{\cdots}$. The most important couplings seem to be "diagonal," i.e., ${N}_{i}={N}_{j}$. In this case, isospin \textonehalf{} always dominates. If all $\ensuremath{\pi}{N}_{i}{N}_{i}$ couplings are of the same order of magnitude, then there are negative-parity trajectories that follow along with the positive-parity trajectories. It is found that the one-pion-exchange amplitudes lead to substantial violations of unitarity and to spectrum inversion for "off-diagonal" couplings (${N}_{i}\ensuremath{\ne}{N}_{j}$) involving large momentum transfers. To deal with this problem we introduce a one-parameter "form factor." Because of the uncertainties of coupling constants and form factor, predictions of resonance energies and branching ratios are somewhat adjustable. However, resonances occur in states having the correct quantum numbers, and the calculated widths are surprisingly good. Given the nucleon (${{N}_{i}}^{+}$) and ${{\ensuremath{\Delta}}_{i}}^{+}$ trajectories, we are led to the ${{N}_{i}}^{\ensuremath{-}}$ trajectory with $(T,{J}^{P})=(\frac{1}{2},{\frac{3}{2}}^{\ensuremath{-}}),(\frac{1}{2},{\frac{7}{2}}^{\ensuremath{-}}),\ensuremath{\cdots}$ and the ${{N}_{i}}^{\ensuremath{'}\ensuremath{-}}$ trajectory $(\frac{1}{2},{\frac{5}{2}}^{\ensuremath{-}}),(\frac{1}{2},{\frac{9}{2}}^{\ensuremath{-}}),\ensuremath{\cdots}$. In addition there is possibly another $(\frac{1}{2},{\frac{1}{2}}^{\ensuremath{-}}),(\frac{1}{2},{\frac{5}{2}}^{\ensuremath{-}}),\ensuremath{\cdots}$ sequence at rather high mass. The "exothermic" reactions for which $M({N}_{i})>M({N}_{j})$ lead to possible isospin-$\frac{3}{2}$ resonances $(\frac{3}{2},{\frac{1}{2}}^{\ensuremath{-}}),(\frac{3}{2},{\frac{5}{2}}^{\ensuremath{-}}),\ensuremath{\cdots}$, but these calculations are not considered very reliable. The virtual-$f$ configurations $f{N}_{j}$ are also considered. These characteristically involve considerably larger momentum transfers than do the corresponding $\ensuremath{\rho}$-meson configurations. If the $f{N}_{j}$ amplitudes are modified by the same function used to repair the $\ensuremath{\rho}{N}_{j}$ couplings, no resonances result from the $f{N}_{j}$ mechanism. However, present methods of computing the magnitude of this high-momentum-transfer effect are unreliable. Extension of the vector-meson model to $\mathrm{SU}(3)$ leads to 8 and 1 trajectories with ${\frac{3}{2}}^{\ensuremath{-}},{\frac{7}{2}}^{\ensuremath{-}},\ensuremath{\cdots}$ and 8 alone for ${\frac{5}{2}}^{\ensuremath{-}},{\frac{9}{2}}^{\ensuremath{-}},\ensuremath{\cdots}$ and the doubtful companion ${\frac{1}{2}}^{\ensuremath{-}},{\frac{5}{2}}^{\ensuremath{-}},\ensuremath{\cdots}$. Generalization of the $f$ production mechanism to a tensor nonet ${T}_{9}$ leads to hitherto unobserved representations. ${P}_{8}\ensuremath{\rightarrow}{B}_{8}\ensuremath{\rightarrow}{T}_{9}+{B}_{8}$ would give 8, ${10}^{*}$, and 1 representations with spin ${J}^{P}$ of ${\frac{5}{2}}^{+},{\frac{9}{2}}^{+},\ensuremath{\cdots}$, and possibly ${\frac{7}{2}}^{\ensuremath{-}},{\frac{11}{2}}^{\ensuremath{-}},\ensuremath{\cdots}$, while ${P}_{8}+{B}_{10}\ensuremath{\rightarrow}{T}_{9}+{B}_{10}$ would give 10 and 27 representations with ${\frac{7}{2}}^{+},{\frac{11}{2}}^{+},\ensuremath{\cdots}$ if we have underestimated the importance of this effect. Hence present experimental evidence seems to support our conclusions about the relative importance of the vector- and tensor-meson production. The resulting baryon mass spectrum includes most of the prominent observed states. No wrong states are predicted. Some new resonances in the 2-BeV region are suggested.

Backward π±pScattering and theY= 1 Fermion Regge Pole Hypothesis

In this note, we discuss phenomenologically the high energy n±p backward scattering 1 >5 > using the Regge pole hypothesis of Y = 1 Fermion resonances. Following Barger and cline, we assume that a nucleon (N) and rr-N resonances can be assigned to the Regge recurrences, Na, N 1 and Ll. 6 > Chiu and Stack have already analyzed the rr+p backward scattering with the Na trajectory ex­ change/> but its residue function has been determined only so as to fit the experimental data, and the problem as to why the N 1 trajectory can be neglected has not been discussed. In this article, the Regge parameters, in particular the reduced residue functions ri (W)* as well as the pole positions a.z (W) are fitted from resonance parameters 8 > (mass Jilin, decay width r and elasticity x) in the direct channel. For the crossed channel (u-channel) physical region, the func­ tions ai (W) and ri (W) are assumed to be extrapolated. The function ai (W) is parametrized with the usual method of Chew and Frautschi:

Baryonic Regge Recurrences andU(6)W×O(2)WSymmetry with Applications to Decays and Photoproduction

It is shown that if ${{N}_{{\frac{5}{2}}^{+}}}^{*}(1688)$ and ${{\ensuremath{\Delta}}_{{\frac{1}{2}}^{+}}}^{*}(1920)$ are the Regge recurrences of ${N}_{{\frac{1}{2}}^{+}}(940)$ and ${\ensuremath{\Delta}}_{{\frac{3}{2}}^{+}}(1238)$, respectively, they can be classified along with other particles in the (56, 1; $L=2$) representation of the rest-symmetry group $U(6)\ifmmode\times\else\texttimes\fi{}U(6)\ifmmode\times\else\texttimes\fi{}O(3)$. The couplings of these baryons are then fixed by the collinear ${U(6)}_{W}\ifmmode\times\else\texttimes\fi{}{O(2)}_{W}$ group. This leads to the following predictions: ${(R1) {(\frac{D}{F})}_{{{B}^{*}}_{\frac{5}{2}+}\ensuremath{\rightarrow}{B}_{\frac{1}{2}+\mathrm{pseudoscalar}\mathrm{meson}}}=\frac{3}{2};}{(\mathrm{R}2) \frac{\ensuremath{\Gamma}({\ensuremath{\Delta}}^{*}(1920)\ensuremath{\rightarrow}N\ensuremath{\pi})}{\ensuremath{\Gamma}({N}^{*}(1688)\ensuremath{\rightarrow}N\ensuremath{\pi})}=1.84;}{(R3\ensuremath{-}R4) \mathrm{the}\mathrm{photoformation} (\mathrm{i}.\mathrm{e}.,\phantom{\rule{0ex}{0ex}}\mathrm{t}\mathrm{h}\mathrm{e}\phantom{\rule{0ex}{0ex}}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s} \ensuremath{\gamma}+p\ensuremath{\rightarrow}\mathrm{resonance}\ensuremath{\rightarrow}N\ensuremath{\pi}) \mathrm{of} {\ensuremath{\Delta}}^{*}(1920) \mathrm{and} {N}^{*}(1688) \mathrm{should}\mathrm{occur}\mathrm{with}\mathrm{the}\mathrm{multipole}\mathrm{ratios} {{{(\frac{{E}_{3+}}{{M}_{3+}})}_{\ensuremath{\Delta}}}^{*}}_{(1920)}=0 \mathrm{and} {(\frac{{M}_{3\ensuremath{-}}}{{E}_{3\ensuremath{-}}})}_{{N}^{*}(1688)}\ensuremath{\approx}0.1, \mathrm{respectively}.}$These predictions are shown to be in agreement with experiment. Generalizations to higher Regge recurrences are presented.

Possibility of an Infinite Sequence of Regge Recurrences

We show that a Regge trajectory, $\ensuremath{\alpha}(s)$, cannot have the property $\mathrm{Re}\ensuremath{\alpha}(s)\ensuremath{\rightarrow}+\ensuremath{\infty}$ as $s\ensuremath{\rightarrow}+\ensuremath{\infty}$ without leading to inconsistencies with two features of dispersion theory and Regge pole theory: that both $\ensuremath{\alpha}(s)$ and the reduced residue function, $\ensuremath{\gamma}(s)$, are analytic in the cut plane with one cut, and that they and the partial wave amplitude, $a(l, s)$, for $\mathrm{Re}l=\ensuremath{-}\frac{1}{2}$, are bounded for large $|s|$ by $\mathrm{exp}[{|s|}^{\frac{1}{2}}\ensuremath{-}\ensuremath{\epsilon}]$.

Regge recurrences and π-p elastic scattering at high momentum transfers

The influence of the resonance formation on the high momentum transfer elastic π-p scattering is investigated. A new model, in which the known Regge recurrences are extrapolated to higher energies, is proposed. It is shown that the model can account for the present data on the π-p elastic scattering at high momentum transfers. Some implications of such a mechanism are discussed.

Regge recurrences and πp charge-exchange polarization at high energies

Regge recurrences and πp charge-exchange polarization at high energies

Fermion Regge-Pole Model for the Structure of Pion-Nucleon Elastic Scattering in the Backward Hemisphere

The $Y=+1$ fermion resonances that show up strongly in total-cross-section data are classified as Regge recurrences on three straight-line trajectories (namely, ${\ensuremath{\Delta}}_{\ensuremath{\delta}}$, ${N}_{\ensuremath{\gamma}}$, ${N}_{\ensuremath{\alpha}}$) in a Chew-Frautschi plot. From extrapolations of the trajectories, resonance doublets are predicted in the vicinity of 2200 MeV (with ${J}^{P}={\frac{7}{2}}^{\ensuremath{-}} \mathrm{and} {\frac{9}{2}}^{+}$) and 2630 MeV (with ${J}^{P}=\frac{11}{{2}^{\ensuremath{-}}} \mathrm{and} \frac{13}{{2}^{+}}$), due to recurrences of the ${N}_{\ensuremath{\gamma}}$ and ${N}_{\ensuremath{\alpha}}$ trajectories at similar mass values. A model is constructed for ${\ensuremath{\pi}}^{\ensuremath{-}}p$ elastic scattering near the backward direction based on interference of the direct-channel resonance amplitude (${\ensuremath{\Delta}}_{\ensuremath{\delta}}$, ${\ensuremath{\Delta}}_{\ensuremath{\gamma}}$, ${N}_{\ensuremath{\alpha}}$) with the amplitude due to fermion Regge exchange (${\ensuremath{\Delta}}_{\ensuremath{\delta}}$) in the crossed channel. The predictions of the model compare favorably with existing data on the energy dependence of the ${\ensuremath{\pi}}^{\ensuremath{-}}p$ differential cross section at 180\ifmmode^\circ\else\textdegree\fi{} center-of-mass scattering angle and the general shape of the ${\ensuremath{\pi}}^{\ensuremath{-}}p$ angular distributions near 180\ifmmode^\circ\else\textdegree\fi{}. The results confirm the consistency of the Regge-recurrence parity assignments with the scattering data. The resonance elasticities used in the calculations are roughly the same as the elasticities determined from total-cross-section data. The model is extended to ${\ensuremath{\pi}}^{+}p$ elastic scattering at backward angles. In the ${\ensuremath{\pi}}^{+}p$ process, the direct-channel ${\ensuremath{\Delta}}_{\ensuremath{\delta}}$ resonance contribution alone saturates the experimental differential cross section in the backward cone at momenta below 4 $\frac{\mathrm{BeV}}{c}$. Comparison with the ${\ensuremath{\pi}}^{+}p$ backward-scattering data gives additional confirmation for the proposed ${\ensuremath{\Delta}}_{\ensuremath{\delta}}$ Regge-recurrence parity assignments. In addition, the model supports the existence of an $I=\frac{3}{2} s$-wave resonance at 1690 MeV. Finally, the polarization is predicted for ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$ elastic scattering in the backward cone.

High-Spin Baryons. II. Dynamical Mechanisms for Regge Recurrences of Inelastic Resonances; General Theory

Virtual transitions of the type $\ensuremath{\pi}{N}_{i}\ensuremath{\rightarrow}\ensuremath{\rho}{N}_{j}$, where ${N}_{i}$ represents an arbitrary nucleon isobar, are considered as the driving force for Regge recurrences of inelastic resonances. We consider only the $s$-wave $\ensuremath{\rho}{N}_{j}$ configuration. Since most of the low-lying isobars have even parity, we are therefore concerned mainly with odd-parity resonances. Regge trajectories of the latter are a consequence of Regge recurrences of the even-parity states. An important example is given by taking ${N}_{i}$ to be the sequence: nucleon ${\mathrm{\textonehalf{}}}^{+}$ and its recurrence ${\frac{5}{2}}^{+}$ (quantum numbers ${P}_{11}$ and ${F}_{15}$ in conventional notation). Isospin and centrifugal-barrier considerations lead directly to the existence of the ${D}_{13}\ensuremath{-}{G}_{17}$ sequence. In this paper the general formalism is set up. Numerical results are given separately. First, however, a simpler derivation is given of the elastic forces due to isobar exchange in the quasistatic approximation. Previous work has shown the relation of these forces to the existence of even-parity Regge trajectories. Then the unitarity relations are set up for helicity amplitudes describing the reactions ${N}_{a}\ensuremath{\pi}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\pi}$, ${N}_{a}\ensuremath{\pi}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\rho}$, and ${N}_{a}\ensuremath{\rho}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\rho}$. Next a general explicit expression is given for the onepion-exchange (OPE) approximation for ${N}_{a}\ensuremath{\pi}\ensuremath{\rightarrow}{N}_{c}\ensuremath{\rho}$. Expressions are given for the above amplitudes in several models, in which only the OPE coupling is considered. The pole approximation is used to solve the many-channel $\frac{N}{D}$ equations. Techniques for handling the general isospin problem are developed and their relevance to the present problem discussed.

Regge pole and single particle exchange mechanisms in high energy collisions

It is shown that if particles and resonances occur in infinite Regge recurrence series, the well-known contradiction between Regge pole and single particle exchange models for high energy two-body collisions can be avoided, and double particle exchange is depressed with respect to single exchange.

Algebraic Aspects of Regge Recurrences

The algebraic structure of Regge recurrences is examined, and its relation to symmetry schemes underlying various models is established. The relationship between the Spectrum Generating Algebras (SGA) and the noncompact subalgebras of the rotational states are discussed for the Coulomb potential, the three-dimensional oscillator, the signature which appears in exchange potentials, as well as the U(6, 6) and the other suggested SGA of hadron physics.

Interference of ReggeϱExchange with Direct Channel Fermion Resonances

Using a Reggeized $\ensuremath{\rho}$-meson-exchange background amplitude, information on $Y=+1$ fermion resonances is obtained from a calculation of the energy dependence of (i) the difference of the ${\ensuremath{\pi}}^{\ensuremath{-}}p$ and ${\ensuremath{\pi}}^{+}p$ total cross sections and (ii) the ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}n$ differential cross section at zero degrees. Some evidence is found for a new $I=\frac{1}{2}$ resonance at 3350 MeV that might be a further Regge recurrence of the ${N}_{\ensuremath{\gamma}}$ trajectory. Study of the charge-exchange data confirms the validity of the Regge prediction for the phase of the background amplitude.

Possible regge recurrences of the vector and tensor mesons

Possible regge recurrences of the vector and tensor mesons