ЯДЕРНАЯ ФИЗИКА, 2011, том 74, № 1, с. 42-50
ЯДРА
pN-RESONANCE DYNAMICS IN THE PROTON NUCLEUS REACTION
©2011 Swapan Das*
Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai, India
Received March 2, 2010
The strong coupling of rho meson to the nucleon produces s- and p-wave rho-meson—nucleon (pN) resonances. In a nucleus, the pN-resonance—hole polarization generates the optical potential or self-energy for the p meson. The scattering of p meson due to this potential provides valuable information about the pN-resonance dynamics in a nucleus. To investigate it, we use this potential to calculate the mass distribution spectrum for the p meson produced coherently in the proton—nucleus reaction. The cross sections arising due to s- and p-wave pN resonances have been presented. The coherent and incoherent contributions to the cross sections due to these resonances are compared. In addition, the calculated results due to nonrelativistic and relativistic p-meson self-energy are illustrated.
1. INTRODUCTION
The p-meson—nucleon resonances arise because of the strong coupling of this meson to the nucleon. The importance of the hadronic resonances in the particle production phenomena is undoubtedly understood from long ago. For example, the pion production data in the nuclear reaction below 1 GeV had been interpreted successfully by the formation of the A(1232) resonance in the intermediate state [1]. Therefore, the dynamics of the pN resonances can have intimate relationship with the p-meson production in the nuclear reaction. The sub-threshold p-meson emission is well described by the N(1520)-resonance production in the intermediate state [2]. At higher energy, the importance of N(1720) resonance has been discussed [3] in context to the p-meson production in the proton—nucleus reaction. Therefore, the study of the p-meson production in the nuclear reaction can unveil many interesting physics of the pN-resonance dynamics in a nucleus.
The pN resonances were also found significant to investigate the modification of the p-meson in the nuclear medium. Some time back, Kondratyuk et al. [4] estimated the p-meson—nucleon scattering amplitude, i.e., fpN, due to the formation of pN resonances in the intermediate state. Using this amplitude, they have shown the p-meson mass in a nucleus is below 770 MeV in the static limit. This agrees with the result of scaling hypothesis formulated by Hatsuda and Lee [5]. Friman et al. [6] calculated the p-meson self-energy arising due to p-wave pN scattering, via N(1720) and A(1905)
E-mail: swapand@barc.gov.in
resonances. According to their calculation, this self-energy (which appears in the p-meson spectral function) reduces the mass of the p meson at high baryon density. Peters et al. [7] extended this calculation by incorporating contributions from all four-starred s- and p-wave pN resonances. As shown by them, the s-wave pN resonances (specifically, N(1520)) have significant influence on the p-meson spectral function. The p-meson self-energy due to s-wave pN-resonance—hole polarization shows an important feature, as it depends upon both energy and momentum of the p meson. Therefore, the self-energy due to s-wave pN resonance can contribute to the p-meson spectral function in the static limit. This is unlike that occurring due to p-wave pN scattering, where the p-meson self-energy depends only on its momentum.
The self-energy or optical potential is inevitable to describe the elastic scattering of a particle by the nucleus. Using it, we studied various aspects for the coherent (elastic) scattering of p0 meson in the proton—nucleus reaction [8]. In one part of this study, we investigated the sensitivity of the coherent p0-meson mass distribution on its optical potential formulated by various authors [4—6, 9] including Peters et al. [7]. The later authors, as mentioned above, have shown nonrelativistically that various pN resonances can generate the self-energy for the p-meson in a nucleus. Due to lack of scope, we could not show in our earlier work [8] the distinct contribution arising from each pN resonance to the p0-meson production cross section. Therefore, those are illustrated in the present work. In addition, we extend our calculation to accommodate the relativistic evaluation of the p-meson self-energy due to these resonances and discuss how the pN resonance behave relativistically in the above reaction. We compare the nonrelativistic
and relativistic optical potentials for the p0 meson arising due to each pN-resonance—hole polarization in the nucleus. We also present the results for the coherent and incoherent contributions of these resonances to the overall cross sections.
2. FORMALISM
The coherent p0-meson production in the proton— nucleus reaction describes the elastic scattering of the virtual p meson (emitted by the beam proton) to its real state. Symbolically, this reaction is used to express as p + A(g.s.) ^ p' + p0 + A(g.s.). As we worked out earlier [8], the double differential cross section for the coherent p-meson mass distribution in the (p, p') reaction on a nucleus is given by
d(T f f dQpdEp,[KF]S(m2)(\Tfl\2), (1)
dmdQ.
p
where [KF] is the kinematical factor associated with this reaction. The expression for it is
[KF ] =
УПрЕл'
(2)
kpf kp mm
kp\kp(Ei - Ep) - (kp - kp/) • kpEp|
The factor S(m2) in Eq. (1) denotes the mass distribution function for the p-meson of mass m in the free space. It is well addressed by the Breit—Wigner form [10]:
mprp(m2)
S(m2) =
1
(3)
n [(m2 - mp)2 + m2rp(m2)]'
with mp(~770 MeV) being the resonance mass of the p meson in the free-state. rp(m2) represents the total free-space decay width for the p meson of mass m, arising dominantly (~100%) due to p Therefore, rp(m2) can be written as
Гр(т2) w Г(шр)р-жж ~k(m2)
mp
m
(4)
6(m2 - 4m2 )
with r(m?)
p)p-
k(m2) j
■ w 150 MeV; k(m2) is the pion momentum in the rest frame of decaying p0 meson of mass m. This expression intrinsically implies the detection of p meson through its decay products n- in the final state. We have not considered here the distortion due to pion—nucleus scattering since the effect of it (in this kind of reaction) has been found insignificant [11].
Tfi in Eq. (1) is the T matrix for the coherent p-meson production in the proton—nucleus reaction. The annular bracket around \Tfi\2 represents the average and summation over spins and polarization of the initial and final states, respectively. Tfi is given by
Tfl = J J dr'drx(-)* (kp, г')Пр(г') x x Gp(r' - г)фр(я, г).
(5)
In this equation, Gp(r' — r) describes the propagation of the virtual p0 meson from r to r', where the elastic scattering of this meson (to its real state) takes place due to the p-meson—nucleus optical potential VoP(r'). This potential appears in the above equation through the p-meson self-energy, i.e., np(r') = = 2q0Vop(r'). Here, <&(= Ep - Ep/) is the energy of the virtual p-meson. np(r') will be elaborated in the next section.
The symbol ^p(q, r) in Eq. (5) stands for the virtual p-meson production amplitude in the (p,p') reaction. The expression for it is
r)rpNNX(+) (kp, r), (6)
Фр(кр<
= x(-)*
(kp
where rpNN denotes the vertex function at the p°pp' vertex. It is governed by the pNN Lagrangian as described in [8]. Xs represent the distorted wave functions for protons in the continuum. In the energy region of p-meson production, the distortion arising due to proton—nucleus scattering is purely absorptive. Therefore, the distorted wave functions for protons can be approximated by their plane waves multiplied by the attenuation factor Af for them [12], i.e.,
^-)*(kp/, r)x(+)(kp, r) wAfeiq'r;
x
(7)
q
kp kp'
This equation shows q is the momentum of the virtual p meson emitted at the pNN vertex.
The p0 meson in the final state can be emitted in all directions. Therefore, the partial wave expansion is preferred to express its wave function %(-)(kp, r') appearing in Eq. (5), i.e.,
X(-)*(kp,r) =
(8)
4^
p Im
The radial part of this wave function, i.e., ui(kpr), is generated by solving the relativistic wave (Schrodin-ger) equation using the p-meson optical potential VOp(r), described later.
x
x
Table 1. Values for pN resonances' parameters taken from [7]. (L2i,2j is the spectroscopic symbol for the pion-nucleon resonance state. Ss denotes the spin-isospin
transition factor. All other symbols carry their usual meanings)
R(mR)L2i,2J
N (1520)D13 A(1620)S'3i N (1650)Sn Д(1700)^зз N(940) Д(1232)Рзз N (1680)F15 N (1720) Pi3 Д(1905)Рз5
I (J P
1(3-2 v 2
3/12 v 2
if1" 2 v 2
3/32 v 2
1(1 +
2 v 2
3/3 +
2 v 2
1/5 + 2^2 1(3 +
3/5 + 2^2
95 130 135 180 0 120 118 50 140
Гп
25 20 15 120 0 0 12 100 210
fRNp
7.0 2.5 0.9 5.0
7.7 15.3
6.3
7.8 12.2
SE
8/3 8/3 4
16/9 4
16/9 6/5 8/3 4/5
3. RESULTS AND DISCUSSION The attenuation factor AF in Eq. (7), which accounts the distortion for protons, can be estimated by using the following equation [12]:
At
1 A
dbT (b)exp[-Im5(b)j,
(9)
where A is the mass number of the nucleus. T(b)[= = /+» Q(b,z)dz] represents the thickness function for the nucleus at the impact parameter b. Q(b, z) describes the density distribution for the nucleus, given in Eq. (12) for 12C nucleus. ¿(b) denotes the total phase-shift function for the proton—nucleus scattering, i.e., ¿(b) = ¿p(b) + 5p>(b). The form for it is given by
Sp(b) = --
Vop(b,z)dz,
where vp is the velocity of the proton. VOp(b,z) denotes the optical potential for the proton—nucleus scattering.
The imaginary part of the proton—nucleus optical potential VOp(b, z) is required to evaluate the attenuation factor Af for protons (see Eqs. (9) and (10)). Using the high-energy ansatz, i.e., "tQ" approximation, the imaginary part of VOp(b,z) can be written as
ImVpp(r) = -\vaf6( r).
(11
it are taken from the measurements [13, 14]. Q(r) describes the spatial shape of the optical potential VOp (r). It is usually approximated by the density distribution of th
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