CDD Pole Research Articles

Resonance on top of thresholds: TheΛc(2595)+as an extremely fine-tuned state

A dedicated study of the $\pi\Sigma_c$ scattering around its threshold is carried out in this work to probe the nature of $\Lambda_c(2595)^+$. We first demonstrate that the effective range expansion approach fails to work near the $\Lambda_c(2595)^+$ pole position, due to the presence of a nearby CDD pole around the $\pi\Sigma_c$ thresholds. We then develop a general framework to properly handle the situation with a CDD pole accompanied by nearby thresholds, which is first elaborated for the single-channel case and then generalized to the coupled-channel study. The isospin breaking effects of the three $\pi\Sigma_c$ channels with different thresholds are specially taken into account in our study. The finite-width effects from the $\Sigma_c$ baryons are considered and found to be relevant to give the $\Lambda_c(2595)^+$ width fully compatible to its experimental value. Through the compositeness analysis, our robust conclusion is that the $\pi^0\Sigma_c^+$ component is subdominant inside the $\Lambda_c(2595)^+$.

CDD pole effects on the P33 channel of the Chew–Low model without crossing

To better understand the role of CDD poles in modern field theories, we investigate the effects of CDD poles on the [Formula: see text] channel in [Formula: see text] scattering. We use the non-crossing Chew–Low model for this investigation. The Chew–Low model is the simplest phenomenological model of [Formula: see text] scattering. In the no crossing limit, the [Formula: see text] channel can be easily solved and displays the predominate [Formula: see text] resonance. It has been shown that the denominator function, used in the [Formula: see text] method of solution, is a generalized R-function. Our research centers on two investigations. The first is whether one can readily associate a CDD pole with the [Formula: see text] resonance. We find that one cannot, since the CDD pole forces the phases to disagree with the experimental values. The second investigation is into the nature of the redundant solutions one can generate by adding CDD pole to the denominator function. We find that the solutions with the added CDD pole cannot reproduce the phases of the original Chew–Low Model at low energies. Thus, there are no redundant solutions to this model. These results are contrary to what is generally assumed. We question the use of adding a CDD pole to produce resonances and the ideal that adding such poles do not effect the low energy phases.

Castillejo, Dalitz, and Dyson ambiguity in field theoretic models of scattering

We examine the numerical solutions of a field theoretic model to better understand the role played by the poles associated with the ambiguity identified by Castillejo, Dalitz, and Dyson (CDD). By analytically analyzing the numerical solutions to the crossing-symmetric Chew-Low model previously found, we show that the solutions are unique. This is done by requiring a second solution that would reproduce the original threshold and resonance. This uniqueness applies to the family of solutions, found by varying the parameters, that produces a resonance. The $T$ matrix for this model is not a generalized $R$ function. and no analogy can be made to the CDD analysis. Adding a CDD pole to the solution would constitute a new model, so there is no CDD ambiguity. None of the family of solutions from this new model will reproduce the original threshold and resonance. We also find that one cannot treat an individual channel as an $R$ function, nor can one add a CDD pole to an individual channel alone. Both a pole and a crossed pole must be added to all channels to maintain crossing symmetry. Crossing symmetry analytically connects all of the channels. The Chew-Low model being phenomenological is more pertinent to modern field theoretic models.

Implications of the present bound on the width of theΘ+(1540)

The recently reported Theta(1540) exotic baryon seems to be very narrow: Gamma(Theta)< 1 MeV according to some analyses. Using methods of low energy scattering theory, we develop expectations for the width of the Theta, an elastic resonance in KN scattering in a theory where the characteristic range of interactions is ~1 Fermi. If the Theta is a potential scattering resonance, generated by the forces in the KN channel, its width is hard to account for unless the KN-channel orbital angular momentum, l, is two or greater. If the Theta is a CDD pole, its coupling to the scattering channel is at least an order of magnitude less than the coupling of the Lambda(1520) unless l is two or greater. Either way, if the Theta proves to be in the KN s- or p-wave, new physics must be responsible for its narrow width.

Do K-Matrix Fits Exclude a CDD Type of Λ (1405) Resonance?

We compare s-wave N/D amplitudes with 1 left-hand pole with a recent K-matrix fit to the low energy K̄N data. The isolated solution does not agree very well with the fitted K-matrix but much better agreement is obtained if a CDD pole is added. In this solution the Λ (1405) disappears if the CDD pole is removed. We also point out the effect of the CDD degrees of freedom on the K-matrix and show that there has been some misunderstanding of the situation.

The occurence of a CDD singularity in the neutron-deuteron doublet scattering amplitude

It is shown that for the Amado model of elastic neutron-deuteron scattering in the s-wave doublet channel the single channel N D solution is not equivalent to the exact amplitude. Equivalence of the N/ D solution is demonstrated under the insertion of a CDD pole which simultaneously generates the triton pole and the effective range anomaly, thus establishing a link between these apparently unrelated features of the doublet amplitude.

An improvedN/D method for inelastic partial-wave amplitudes

The problem of constructing the partial-wave amplitudes exhibiting a given left-hand cut and a given inelasticity is reformulated in terms of an improvedN/D method. This method presents the following advantages. The form of the basic equations is independent of the asymptotic behaviour of the left-hand discontinuity. The effects of inelasticity are evaluated explicitly and disconnected from the effects of the left-hand cut. The inherent ambiguities of anN/D decomposition do not affect the final expression of the scattering amplitude. The CDD pole ambiguity is reduced to the arbitrariness in the choice of a meromorphic function in this final expression. The description of elastic or inelastic resonances serves as illustration of the method.

Models of Y ∗0 (1405) and low-energy [formula omitted]N interactions

Two models have been proposed for the Y ∗ 0 (1405) resonance: 1. (a) as an s-wave virtual bound state of the K N-πΣ system, and 2. (b)as a CDD pole arising from high-lying channels. The compatibility of these interpretations with low-energy K N scattering data is analyzed, and it is found that the virtual bound state interpretation is strongly favoured.

Generalized potential and uniqueness in the calculation of the pion-nucleon P 33 state

Two different iteration procedures are used to calculate the P 33 phase shift of pion-nucleon scattering: a modified N/ D method and the Newton-Kantorovich method. It is possible to find the correct generalized potential for the P 33 partial wave amplitude from threshold up to T π = 870 MeV without using the P 33 phase shift. With this input the iterations converge exactly to the phase-shift-analysis values [1] only if we introduce two parameters. It turns out that the output phase shift is very sensitive to small variations in the potential. Since we have to use parameters we conclude that in the usual N/ D method a solution is only possible if one includes at least one CDD pole.

Argand loops in the Lee model

We study with the help of Argand diagrams the elastic amplitudes in the scalar Lee model in order to determine the relative importance of inelastic thresholds and the Peierls (P) singularity. An input Nθ resonance, constructed separately by either the choice of the cut-off function or by a CDD pole, leads to non-resonant loops in the Argand diagram of the elastic Vθ amplitude which are associated with the N ∗θ threshold. The P-mechanism works neither in the non-CDD case because the Nθ resonance is necessarily too large, nor in the CDD case because of an explicit cancellation. The low-energy part of the Vθ spectrum, in the absence of input resonances, is characterized by simultaneous Vθ boundstates and anti-resonances. The absence of low-energy Vθ resonances, in the non-CDD case, is connected to the cusp in the elastic Nθ denominator at the Nθ threshold.

Lagrangian formulation of the Zachariasen model with a CDD pole

The generalization of Thirring's formulation of the Zachariasen model in the presence of a CDD pole is given. The local field operator, describing an interacting CDD particle, is introduced. It is shown that the CDD pole can be interpreted as a particle only if the mass renormalization in the Zachariasen model is finite.

Der CDD-Pol in der Zweikanalstreuung

Comparing the inelastic one channel N/D method of Frye and Warnock, and the many channel matrix N/D method of Bjorken it is shown, that for a large class of many channel models the solution of the Bjorken method is a solution of the Frye Warnock method. We demonstrate at a simple model of the two channel scattering, that one dynamical CDD pole exists in the Frye Warnock method.

Rho-meson shape

A detailed description is developed for the invariant dipion mass distribution in the reaction πp → ππN. Based on OPEM and modified phenomenologically this “generalized Breit-Wigner” formula accounts for peripheral production, t-dependence of the ϱ parameters, skewness of the ϱ peak, energy dependence of the ϱ width, energ-dependent departure from OPEM, background, and background interference. Contributions from other ππ partial waves are included and the possibility of ϱ being a CDD pole considered. The formula is used on a data compilation of large statistics and conclusions concerning the presence of the various terms and the ϱ parameters are reached. Finally, we compare the width of the pion-produced ϱ meson with the width of the e +e − produced ϱ meson.

Low-energy π-π scattering

We consider the forward π-π dispersion relations for T = 0 and T = 2 in a soluble model. The model is based on two assumptions. (i) The π-π cross section has the same constant asymptotic value for all isobaric states (Pomeranchuk theorem). (ii) The forward amplitude becomes pure imaginary at infinite energies. For simplicity we only consider cases with one or no CDD poles in each isobaric state. We find two solutions which are consistent with the constraints of crossing: Solution A has a T = 0, CDD pole, the T = 0 scattering length ( a o) = −0.8. the T = 2 scattering length ( a 2) = −0.4. Solution B has a T = 0 and a T = 2. CDD pole, a o ≈ +1 and a 2 ⪅ 0. There is an S-wave resonance in the region 3 m π to 8 m π for T = 0. Solution B is in qualitative agreement with the experimental evidence.

N/D approach to πN scattering in the P 11 state with short-range forces

The πN scattering in the P 11 state is treated in an one-channel N/ D approach in elastic approximation. It is shown that the compositeness condition δ 11(∞) = − π for the nucleon and the zero of δ 11( ω) in the low-energy region can be made consistent with each other by taking into account short-range forces appropriately. We find (without introducing a CDD pole or a cut-off) for a given nucleon-pole position and given contributions from N, N ∗ and ϱ exchange, the required value for the πN coupling constant and two zeros of δ(ω) for physical ω, one of them in the low-energy region at the right position. The parameters of the short-range forces approximated by a pole term are fixed by asymptotic conditions on the behaviour of the scattering amplitude at large energies.

Effect of a Castillejo-Dalitz-Dyson Pole in theπNI=12,J=12+Amplitude

Recent phase-shift analyses of the $I=\frac{1}{2}$, $J={\frac{1}{2}}^{+}$ $\ensuremath{\pi}N$ amplitude imply that the real part of the phase shift changes sign at about 150-MeV pion lab kinetic energy and either passes through 90\ifmmode^\circ\else\textdegree\fi{} or reaches a maximum slightly below 90\ifmmode^\circ\else\textdegree\fi{} at about 600 MeV. These features of the phase shift strongly contradict the predictions of dispersion-theory calculations of the nucleon mass, which generally yield a negative phase shift decreasing with increasing energy. We assume that the observed phase-shift behavior is due to multichannel-coupling effects which can be accounted for in a single-channel calculation only by including a Castillejo-Dalitz-Dyson pole. A previous determination of the nucleon mass, $m$, and coupling with pions, $\frac{{g}^{2}}{4\ensuremath{\pi}}$, based on the assumed knowledge of the $\ensuremath{\pi}N$ $I=\frac{3}{2}$, $J={\frac{3}{2}}^{+}$ (${N}^{*}$) resonance parameters, is repeated with the inclusion of a CDD pole. A variant of the Bal\'azs method is used. The CDD pole parameters are determined so as to give the phase-shift behavior mentioned above. The Bal\'azs-pole residues are determined by requiring crossing symmetry in a form relating the $\ensuremath{\pi}N$ amplitude on the nearby left-hand cut to physical $\ensuremath{\pi}N$ scattering. We find $m=0.97$ in units of the physical nucleon mass; $\frac{{g}^{2}}{4\ensuremath{\pi}}=10.5$; and a $\ensuremath{\pi}N$ $I=\frac{1}{2}$, $J={\frac{1}{2}}^{+}$ scattering length between about -0.038 and -0.048 (in units of the cube of the pion Compton wavelength), depending on the assumed energy at which the phase shift changes sign. These parameters appear to be very insensitive to the choice of Bal\'azs-pole positions.

I = J = 0 dipion state with a CDD pole

I = J = 0 dipion state with a CDD pole