IL NUOVO CIMENTO VOL. 110 A, N. 2 Febbraio 1997
The coherent nucleus: the low-energy photoabsorption cross-section (*)
R. ALZETTA(1)(**), R. LEPERA(1), G. LIBERTI(1) and G. PREPARATA(2)
(1) Dipartimento di Fisica, Università della Calabria
I-87030 Roges di Rende, Cosenza, Italy INFN, Sezione di Cosenza - Cosenza, Italy
(2) Dipartimento di Fisica, Università di Milano - Milano, Italy
INFN, Sezione di Milano - Milano, Italy
(ricevuto il 2 Gennaio 1997; approvato il 3 Marzo 1997)
Summary. — We apply the theory of the coherent nucleus (CN) to low-energy (EgE
1 GeV) photoabsorption on nuclei. We are able to account in a parameterless way for both the positive mass-shift and the increased width of the photoproduced D( 1232 )-resonances, as recently observed.
PACS 25.20 – Photonuclear reactions.
1. – Introduction
The physics of the atomic nucleus appears as a relatively venerable subject to make one expect surprising revelations from well-established research protocols such as low-energy (EgG 1 GeV) photoabsorption experiments. Instead such expectations do
not look so far-fetched to scientists, like us, who since the beginning of the ’90’s are pursuing a research program based on a completely different picture of the dominant nuclear interaction, mediated by the exchange among nucleons of quasi-real p’s [1, 2]. Thus any new experimental information on the interaction of nuclei with external probes is potentially very interesting, for the ease with which the new picture—the coherent nucleus (CN)—may eventually account for it as opposed to the multiple and unconfortable epicycles required by the conventional, short-range nuclear forces, and it may well become a powerful means to finally shift a paradigm, which we deem completely inadequate.
In this paper we wish to show that the recent data gathered on low-energy photoabsorption [3], with their simple and straightforward interpretation/explanation within the CN framework (compared with the precarious and ambiguous mechanisms invoked in the conventional picture) do constitute a fine confirmation of the remarkable
(*) The authors of this paper have agreed to not receive the proofs for correction. (**) E-mail address: AlzettaHfis.unical.it
Fig 1. – The energy gap D as a function of the nuclear density r.
ability of the new, long-range nuclear interaction to account for apparently subtle aspects of nuclear physics.
Before we deal with the details of the nuclear photoabsorption cross-section, let us very briefly sketch the picture that the long-range coherent p8D-interaction paints of a nucleus. According to the new approach, the atomic nucleus can be looked at as a kind of pionic laser whose “two-level system” is 8-D(1232), and its lasing becomes spontaneous (i.e. it needs neither pump nor cavity) when the density (A/V) exceeds 0.2r0, r0being the nuclear matter density (r0A 0.16 fm23) [4].
As a result of the coherent laser interaction an energy gap D gets generated, depending on the density (see fig. 1), which measures the energy necessary to remove a nucleon 8 from the laser ground state, the so-called coherent ground state (CGS). It is clear that such a gap D is also equal to the additional energy that one needs to spend in order to excite (incoherently) any resonance from the nuclear ground state: in the non-relativistic kinematics, relevant in the type of experiment under consideration, this gap will show up as an apparently increased mass of the produced resonance, as experimentally observed.
Furthermore, in the CGS there condenses a coherent p-field that can easily be shown to modify (increase) the width G of the produced resonances as a function of the
THE COHERENT NUCLEUS:THE LOW-ENERGY PHOTOABSORPTION CROSS-SECTION 181
mass number A. Thus, from the sketchy picture just presented, we are led qualitatively to expect that, comparing with photoabsorption in nucleons, for nuclei:
1) the resonance mass M is higher by the magnitude of the gap D; 2) the larger the resonance width, the larger the nucleus.
All such “trends” appear in the data. We shall now show how to such expectations can be given a successfully quantitative form.
2. – Photoproduction of the D(1232) in the coherent nucleus
In the theory of the coherent nucleus (CN), when a “low-energy” g is absorbed on a nucleus of atomic number A to produce a D(1232)-resonance, it “sees” a coherent state of nucleons, D’s and p’s, whose component of “bare” (free) nucleons is given by cos2u(r) (r is the nuclear density A/V), which at nuclear matter densities r 40.16 fm23is about 0.79, the additional 21 % of the nuclear matter wave field being “bare” D’s. Thus a “bare” D is produced through two distinct mechanisms, according to whether the absorbing baryon is a nucleon or a D. Lacking experimental information on the
Fig. 2. – The D-mass shift as a function of the nuclear density r compared with the experimental data [3].
Fig. 3. – The “incoherent” D-width GDincas a function of the nuclear density r.
transition amplitude gD KD, we cannot at present completely evaluate the photoproduction cross-section for a D, but we may certainly determine its excitation spectrum, i.e. its mass and width.
Let us consider the problem of the D-mass first. Due to the prevalence of the N-components of the nucleus these latter are arranged in shells up to the Fermi momentum pF`
k
3 p2r 2 cos 2u(r)l
1 /3 , (1)their average energy being given by
e(r) ` 3 5 pF2 2 m8 . (2)
According to the discussion in the introduction the mass of the D, mD, in the photoproduction process will just be the given by the “bare” mass mD( 0 )increased by the energy necessary to “free” the nucleon from its nuclear prison. Thus the (positive)
THE COHERENT NUCLEUS:THE LOW-ENERGY PHOTOABSORPTION CROSS-SECTION 183
Fig. 4. – The “renormalized frequency” vRof the coherent p-field.
D-energy shift dE(r) in nuclear photoproduction is given by dE(r) 4D(r)2e(r) , (3)
where D(r) is the binding energy per nucleon due to the coherent 8-D-p interaction (see fig. 1), which is reported in fig. 2. In view of the approximations employed, the agreement between theory and experiment [3] is rather satisfactory.
We turn next to the D-width GD. In the coherent nucleus the photoproduced D can deexcite itself through two mechanisms:
D K8(p)1p(q) , (4)
the “incoherent” decay; and
D 1pcK 8 , (5)
where pcis the p-field that “condenses” in the CN [1, 2]. Thus GDis given by the sum of two terms:
GD4 GDinc1 GDcoh, (6)
Fig. 5. – The “coherent” D-width GDcohas a function of the nuclear density r.
GDinc(GDcoh) referring to the “incoherent” process (2.4)
(
the “coherent” process (2.5))
. The calculation of GDinc is quite straightforward and shall be reported in detail elsewhere [5], it involves the evaluation of a simple “form factor”:FA4 1 VA
VA d3x e2i(p 1 q) Q x, (7)where the integration is extended over the nucleus of volume VA, and the exclusion
from integration over the nucleon momentum p of the forbidden (“Pauli blocked”) region NpNGpF
(
see eq. (2.1))
. The behaviour of GDinc as a function of r 4A/V is given in fig. 3. As expected GDinc(r) markedly decreases with the increasing r and this behaviour is just opposite to what is observed.Regarding the “coherent” transition (2.5) we recall that within a coherence domain [1, 2] the coherent p-field is given by (k 41, 2, 3 is the isospin index)
fk(x , t) 4i
o
r 2 v0
4 p
[
(x× Qak(r))
j1(qr) eivR(r) t1c.c.]
, (8)THE COHERENT NUCLEUS:THE LOW-ENERGY PHOTOABSORPTION CROSS-SECTION 185
Fig. 6. – The broadening of the D-width dGDas a function of the nuclear density r compared with the experimental data [3].
“renormalized frequency”, which is a function of the density r and is reported in fig. 4.
While it is totally straightforward to compute the D-N transition amplitude in the coherent p-field (2.8), the calculation of GDcoh requires a detailed analysis of the coherence domain of a large nucleus as a function of A, which shall be reported elsewhere. Here we only report the behaviour of GDcoh(r), which is displayed in fig. 5. Please note the rapid increase of the coherent D-width with r. In fig. 6 we plot the calculated dGD4 GD2 GDfreeas a function of r. Again the agreement with experiment [3] appears satisfactory.
Finally our CN theory predicts for the peak cross-section on a nucleus of atomic number A the general form
speak
A 4 a 1 c GD GDA (9)
with speakNproton4 a 1 c C 527 mb [6]. In terms of one parameter a 4
c
Fig. 7. – Peak of total cross-section on nuclei speak/A as a function of the nuclear density r compared with the experimental data [3].
value is about 0.55, we obtain the prediction that is compared with the experimental data in fig. 7.
3. – Conclusion
A survey of figs. 2, 6 and 7 allows us to state with good confidence that we have been able to obtain an adequate and simple understanding of D-photoproduction in nuclei. Compared with rival approaches (1), in terms of quite complicated interaction mechanisms involving the D and the nucleons of the nucleus, the simplicity, one might even say the obviousness of the D-8-p dynamics of the CN once more underscores, in our opinion, its ability to yield a description of a seemingly complicated physical object—the atomic nucleus—that is at the same time simple, realistic and faithful.
As stressed in the introduction two are the main dynamical aspect of low-energy (D-production) photoabsorption in nuclei: the positive mass shift of the D-resonance,
THE COHERENT NUCLEUS:THE LOW-ENERGY PHOTOABSORPTION CROSS-SECTION 187
increasing with increasing nuclear density, and the larger D-width the higher the density. According to the CN, the D mass shift is nothing but the energy that must be given to the photoabsorbing nucleon in order to remove it from the coherent ground state (CGS) in which he finds itself inside the nucleus, decreased by the average kinetic energy (Fermi motion), as one sees from eq. (2.3). As for the D-width the effect of Pauli-blocking on the incoherent decay process is overcompensated by the D-8 transition induced by the coherent p-field that condenses in the nucleus CGS: very simple indeed!
If simplicity were still a quality loved and cherished by the scientific community (but, alas, how much the massive use of the computer appears to have changed all this!), the theory of low-energy photoabsorption on nuclei presented in this paper, based on the CN, should no doubt be considered with interest and relief.
* * *
We would like to thank Drs. E. DE SANCTIS and N. BIANCHI for very useful discussions on their experimental results.
R E F E R E N C E S
[1] PREPARATAG., Nuovo Cimento A, 103 (1990) 1213.
[2] PREPARATAG., QED Coherence in Matter (World Scientific) 1995. [3] BIANCHIN. et al., Phys. Rev. C, 54 (1996) 1688.
[4] DELGIUDICEE., GUALDIC., MANGANOG., MELER., MIELEG. and PREPARATAG., Int. J. Mod.
Phys. D, 4 (1995) 531.
[5] ALZETTAR., LEPERAR., LIBERTIG. and PREPARATAG., in preparation. [6] ARMSTRONGT. A. et al., Phys. Rev. D, 5 (1972) 1640.
