Width of the $\Delta$ resonance in nuclei

In this work we evaluate the imaginary part of the isobar $\Delta$ self-energy $\Sigma_{\Delta}$ from the two-body absorption process $\Delta+N\rightarrow 2N$. This contribution is calculated using a recently developed non-relativistic scheme, which allows for an evaluation of the self-energy with a basis of single-particle states appropriate for both bound hole states and for particle states in the continuum. In order to test the medium dependence of the self-energy, we calculate the two-body absorption term $\Sigma_{\Delta}^{A2}$ for several finite nuclei with $N=Z$, i.e.\ $^{16}$O, $^{40}$Ca and $^{100}$Sn. The resulting self-energy, which is energy dependent and non-local, is compared with a simple parameterization derived from nuclear matter.

In this work we evaluate the imaginary part of the isobar ∆ self-energy Σ∆ from the two-body absorption process ∆ + N → 2N .This contribution is calculated using a recently developed nonrelativistic scheme, which allows for an evaluation of the self-energy with a basis of single-particle states appropriate for both bound hole states and for particle states in the continuum.In order to test the medium dependence of the self-energy, we calculate the two-body absorption term Σ A2 ∆ for several finite nuclei with N = Z, i.e. 16 O, 40 Ca and 100 Sn.The resulting self-energy, which is energy dependent and non-local, is compared with a simple parameterization derived from nuclear matter.14.20.Gk, 21.60.-n,24.10.Cn The self-energy of the ∆ resonance is central in the understanding of nucleus-nucleus collisions, pion-nuclear processes and photonuclear reactions at intermediate energies, see e.g.Ref. [1] for a recent review.The self-energy Σ ∆ is given by a real part and an imaginary part where ω and k, k ′ are the energy and momenta of the isobar, respectively.In a microscopic derivation of the self-energy, it is common to represent various contributions to the self-energy by way of Feynman diagrams.Typical examples are shown in Fig. 1.Experimental data for pion scattering and absorption at various energies covering the isobar resonance region, suggest that the dominant process for absorption of pions at low energies is represented by diagram (a), which couples the isobar to two-nucleon one-hole states (∆ + N → 2N ).Within the terminology of the isobar-hole model [2][3][4], diagram (b) is then supposed to represent the rescattering of e.g. a real pion, so-called reflection contributions to quasi-free scattering.Diagram (c) is an example of a self-energy contribution arising from three-body absorption mechanisms.
Unfortunately, there is no such thing as an experimental measurement of the ∆ self-energy, although indirect information about the self-energy can be derived from e.g.pion-nucleus scattering.In the extensive analyses of Ref. [5], contributions arising from diagrams like (a) and (b) in Fig. 1, are represented by way of a ∆-spreading potential, fitted to provide best results for pion-nucleus elastic scattering.
In Ref. [6], a parameterization for the imaginary part of the ∆ self-energy is obtained in nuclear matter by considering the contributions from the diagrams in Fig. 1, accounting thus for quasielastic corrections, two-body and threebody absorption.The nuclear matter results are then compared to the ∆-spreading potential from the empirical determination in Ref. [5] by allowing for a density dependent self-energy, given by the analytical expression The term Q accounts for the quasielastic part while A2 is the two-body absorption part.A3 represents threebody absorption.A numerical parameterization for these terms was given by Oset and Salcedo [6], while analytical expressions based on the results of Ref. [6] were recently presented by Nieves et al. [7].In the latter work the Σ A2 ∆ term of Eq. ( 2) has been calculated for a "typical" ∆, which is excited in the ∆-hole model, when a pion of a certain kinetic energy is absorbed.This parameterization reads with x = Tπ mπ , T π and m π being the kinetic energy and mass of the pion, respectively.ρ is the density of particles and ρ 0 the corresponding density at nuclear matter saturation.Further, the function Imα ∆ is and ( The parameterization of eq.( 3) can be interpreted as the width of a ∆, which is excited when an energy ω = T π + m π is deposited at a nucleon in nuclear matter of density ρ.
The intention behind this brief report is to study the Σ A2 ∆ term directly for finite nuclei, since this contribution is expected to be the dominant one at energies below the isobar resonance [6].For that purpose we will here employ a recently developed method to calculate the self-energy of the ∆ in a finite nucleus.Microscopic calculations of e.g. the nucleon or ∆ self-energy have preferentially been carried out in nuclear matter, the results of Ref. [6] being one example.One of the advantages of studies in nuclear matter is the possibility to describe the single-particle wave functions by plane waves.For a microscopic calculation in finite nuclei one has to take into account the fact that one needs different representations for bound hole states and particle states in the continuum.A method which allows for this has recently been developed [8,9].The bound hole states are described in terms of harmonic oscillator (h.o.) wave functions while particle states are given by plane waves.The basic ingredients in our microscopic calculation of Σ A2 ∆ are briefly outlined below.The finite nuclei we consider are 16 O, 40 Ca and 100 Sn.
In order to calculate Σ A2 ∆ , we need first to define the transition potential V N N N ∆ for the ∆ + N → 2N process.For a nucleon and an isobar ∆ interacting through the exchange of π plus ρ mesons, the transition potential V N N N ∆ is usually written, in the static non-relativistic limit, as [3] V with S(T ) the transition matrix which creates a spin (isospin) 3/2 object from a spin (isospin) 1/2 one.The meson propagator D π,ρ (k) is defined as in Ref. [9] as The coupling constant for the π meson is given by a relation obtained from the non-relativistic quark model [10] and similarly for the ρ meson we have with In addition we include monopole form factors in order to regularize the potentials at short distances.The cutoff masses are Λ π = 1.2 GeV and Λ ρ = 1.3 GeV, while the coupling constants are g 2 πN N /4π = 14.6 and g 2 ρN N /4π = 0.95, which are equal to the parameters which define the Bonn B nucleon-nucleon potential V N N of table A.2 in Ref. [11].Further, fρNN gρNN = 6.1.The Bonn B potential is used to calculate the G N ∆ -matrix, the next ingredient in our calculations.To calculate the G N ∆ -matrix, we need first to evaluate the nucleon-nucleon G-matrix G N N .This is done by solving the Bethe-Goldstone equation Here V N N is the free nucleon-nucleon interaction.In this work V N N is defined by the parameters of the Bonn B potential in table A.2 of Ref. [11].The term H 0 is the unperturbed hamiltonian.This equation is solved with an angle-average nuclear matter Pauli operator Q with a fixed starting energy Ω = −10 MeV and a Fermi momentum k F = 1.4 fm −1 .From the nucleon-nucleon G N N matrix, we can evaluate the G N ∆ matrix [9] through the relation Having accounted for the short-range correlations through the introduction of the G N ∆ -matrix, we are then able to set up the expression for the imaginary part of The single-hole energy ε h is given by the eigenvalues of the harmonic oscillator minus a constant shift to place the Fermi energy at zero, while the energies of the particle states are represented by the pure kinetic energy.The variables k, K are the relative and center-of-mass momenta of the intermediate particle states p 1 and p 2 in Fig. 1.Further, l and L are the corresponding orbital momenta of the relative and center-of-mass motion.S, J and T are the total spin, total angular momentum and isospin, respectively.Finally, M N is the average proton and neutron masses.A h.o.single-particle state is defined by the quantum numbers n h l h j h , while plane waves are defined by k a l a j a .For further details, see Ref. [9].The energy variable ω refers to the energy of the ∆ relative to the mass of a nucleon.Only positive energies ω contribute, as can be deduced from the δ function in Eq. ( 13).
In the discussion presented here, we only consider ∆ isobar states with orbital angular momentum l b = 0.A Fourier transformation of ImΣ A2 ∆ in Eq. ( 13) leads to an imaginary part, which depends on energy ω and is non-local in the coordinate r, the distance from the center of the nucleus.From the inspection of this function we observe that the non-locality is weak in the sense that it is different from zero only for distances r and r ′ , which are close to each other.Therefore it makes sense to look at the local component of ImΣ A2 ∆ for the various energies as a function of the distance r [12].As an example we present in the left part of Fig. 2 this local approximation obtained for 40 Ca.The shape of these functions is not really identical to a Woods-Saxon shape or a conventional density distribution.In particular at lower energies (ω below 200 MeV) one observes a clear surface contribution to ImΣ A2 ∆ .Similar results are also obtained for the other nuclei (see right side of Fig. 2).
Finally, in Fig. 3, we compare the imaginary part of the ∆ self-energy calculated in the local approximation for a typical radius of r=1.5 fm at various energies ω with the parameterization of Nieves et al. [7].This comparison must be considered with some care.As discussed above, the parameterization of eq.( 3) represents the average imaginary part of a ∆, which is typically excited, when a pion is absorbed in nuclear matter of density ρ, depositing an energy ω = T π + m π .On the other hand, the results for finite nuclei show a non-trivial radial dependence and Fig. 3, just displays results for one "typical" radius.The nuclear matter parameterization is presented for a density ρ= 0.75 ρ 0 which is the average density of nucleons in 100 Sn.The agreement between the microscopic calculation for this nucleus and the parameterization, which is based on studies of nuclear matter is remarkable.This is true for both the absolute value as well as the shape of the energy-dependence.Only for the lightest nucleus which we considered, 16 O, the calculated width lies considerably below the parameterization.
In conclusion we would like to point out that our microscopic evaluation of the ∆-spreading potential in finite nuclei supports the parameterization of [7], which is based on studies of nuclear matter.For low energies, however, it may be important to consider a surface enhancement of the imaginary part in the self-energy of the ∆ isobar.This work has been supported by the NorFA (Nordic Academy for Advanced Study) through grant #93.40.018/00 , the "Bundesministerium für Forschung und Technologie" (06 Tü 736, Germany) and by the contract DGICYT no.PB92-0761 (Spain).
FIG. 1. Example of diagrams which arise in the evaluation of the self-energy of the isobar.(a) is the two-body absorption term evaluated in this work, (b) is an example of a so-called reflection contribution to quasi-free scattering while (c) stands for a three-body contribution.The wavy line represents the GN∆-matrix, the single line is a nucleon while the double stands for the isobar ∆. ∆ at various energies.Results obtained for the local representation at r=1.5 fm, derived from the microscopic calculation of the ∆ self-energy in the finite nuclei 16 O, 40 Ca and 100 Sn are compared to the parameterization of Nieves et al. [7], displayed in eq.( 3) for a density ρ = 0.75ρ0.This parameterization has been extrapolated from ω = mπ to ω=0.

FIG. 2 .
FIG. 2. Local representation of the imaginary part of −Σ A2∆ as function of the distance r from the center of the nucleus.In the left part of the figure results are shown for the nucleus 40 Ca, considering the energies ω = 100, 200 and 400 MeV.In the right part of the figure the results are displayed for the nuclei 16 O, 40 Ca and 100 Sn assuming an energy ω=300 MeV.