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Study of in-medium ΔΔ elastic scattering cross section

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20June1996 Physics Letters B 378 (1996) 5-11
PHYSICS LElTERS B
Study of in-medium AA elastic scattering cross section
Guangjun Mao a,b, Zhuxia Li ‘, Ylzhong Zhuo ‘, Enguang Zhao a,b a
CCAST (World Laboratory), PO. Box 8730, Beijing 100080, People’s Republic of China b Institute of Theoretical Physics, Academia Sinica, P.0. Box 2735, Bezjing 100080, People’s Republic of China
1 c
Ins ate of Atomic Energy, PO. Box 275(18), Beijing 102413, People’s Republic of China
Received 25 October 1995; revised manuscript received 24 January 1996 Editor: G.F. Bertsch
bstract
Within the framework of self-consistent relativistic Boltzmann-Uehling-Uhlenbeck approach we have derived the explicit expressions for calculating the in-medium AA elastic scattering cross section for the first time. In contrast to the in-medium NN elastic scattering cross section, our numerical results show that the density dependence of in-medium AA elastic scattering cross section is not very pronounced.
PACS:
24.10.Cn; 25.70.-z; 21.65.+f Recently, the investigation of resonance matter, especially A matter, as well as its influence on the particle production [ in heavy ion collisions (WC) has received more and more attention. Both theoretical calculations [2,3] and experimental data [4] indicated that a gradual transition to resonance matter would occur in the collision zone at kinetic energy ranging from SIS up to AGS. The name ~e,s~~ance matter is justified by the fact that at an incident energy of 2 GeV/u more than 30% of the nucleons are excited to resonance state [ 51, which implies the onset of resonance-resonance interaction and resonance-resonance collisions. The A( 1232), as the lowest baryon resonance, plays an important role in intermediate- and high-energy heavy ion collisions. The effects of A resonance on the production of kaons
[
61, antikaons
[
71 as well as antiprotons [ 81 at subthreshold energies have been stressed frequently in the literature. Thus, it is interesting and worthwhile to treat the A more carefully and explicitly within the transport model which is used to simulate the HIC processes. Moreover, since the A-incident scattering cross section is experimentally inaccessible, the theoretical investigation becomes even more important. In view of it, in Ref. [9] we have developed a self-consistent relativistic Boltzmann-Uehling-Uhlenbeck (RBUU) equation for delta distribution function within the framework in which we have done for nucleon’s
[
10-121, the obtained REKJU equation for delta is coupled with that for nucleon through self-energy term and collision term. The in-medium nucleon-A (NA) elastic and inelastic scattering cross section have been presented in Ref. [9] and their medium effects have been studied in a self-consistent way. ’ Mailing address. 0370-2693/96/ 12.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved.
PII SO370-2693(96)00347-4
6 G.
Mao et al. Physics Letters B 378 (1996) 5-l 1
In this work, we shall concentrate on the in-medium AA elastic scattering cross section within the framework of self-consistent RBUU approach presented in Ref. [9]. We start with the interaction Lagrangian density for a system of nucleons and deltas interacting through the CJ, w and QT mesons = &.J (~) (x)~(~) - & (X)r&(X)@%) + &i J(+Y~YS~. tl/(x)@m(x) +&&&)~~(x)c+(x) -g~~~~~(x)Y,~~(x)w”(x) +g;;,~~~(x)y,ysT.~~(x)dc”?r(x) - && (xPm(x)
. s+m> - s&mw*, (xl . @wx) =g&q~wy w>%4(4 +s~*~*v(X>r::~~(x)~A(X) - &&*p (x>+wx) . s+w> - &.&e/hp (xl . dWxl7 (1)
where
gf =fr/mr, gxN=f*/m,; ry = YATA, ri = YATA,
A=(T, o, QT. + is the Rarita-Schwinger spinor of the A-baryon, the definition of symbols and notation can be found in Table 1 of Ref.
[
91. Based on this effective Lagrangian, by means of the closed time-path Green’s function technique [ 13,141 we have derived the self-consistent RBUU equation for delta distribution function, which reads [ 91
(2)
The left hand side of Eq. (2) contains the mean field terms and the right hand side is the collision term. ~rr.~~(s, t) represents the NA scattering cross section and its medium effects have been studied in Ref. [9].
CA& s, t)
is the in-medium AA scattering cross section which includes three parts:
UAA (s, t> = ~AA-NN (s, t> + cAA-+NA (s, f> + gAA,AA (ST t>.
(3) The AA -+ NA scattering cross section can be obtained through detailed balance from NA + AA scattering cross section which has been given in Ref. [ 91. The calculations on the ~NN+AA (s, t) from nucleon’s RBUU equation are in progress and its vice versa cross section can be obtained simultaneously. However, the (+AA+AA s, t) can only be obtained from the A’s RBUU equation. Therefore in this letter we shall calculate ~AA_+AA s, t) within the same approach as given in Ref. [ 91. The lowest-order delta self-energy which contributes to the AA elastic scattering cross section can be written as
X
tr[(T6 1
&Art
1
+“‘“p(5,6)(zi / g:Ar$
1
T~)G~,,(~,~>IDADBA~(~,~)A~(~,~).
(4) The zeroth-order Green’s function for delta and mesons can be found in Refs. [ 9,121;
DA, DB are
given in Table 1 of Ref.
[
91. Here we have ignored the contribution of exchange term, as our previous calculations [ 91 show that it only cause minor influence on the predicted quantities. Following the derivation procedure of Refs. [9,12] we further write down the corresponding transition probability for AA --+ AA scattering (5)
;A(p,p2>p3>p4) = ( &)2(&)2 ~~E~(P>E~\(Pz>E~(P~)~~(P~) (Tama + ~3 -
~4)> Ta = c (T
TA I T3) fi I TE 1 T) 7” I
TB
1 W (T2 I
TA
I
T4)9 TzfiT4
(6)
G.
Mao et
al./Physics Letters B 378 (1996) 5-11 I
x (4 +
m:)D/w(~)D i@;;)
1 1
(p-p3)2-mi(p-p3)2-m2B’
here
a
is the isospin matrix, @a is the spin matrix and
(7)
where rn is the effective mass of A by taking account its decay width. The definition of
rnz can
be found in Ref. [9]. Through computing the Eqs. (5)-(7) and then transforming it into the center of mass of two particle system we have got the analytical expressions for (+AA_.,AA s, t) as
aAA+AA (s, t) =
wg4
~~81m~so2(18m4-~~~2~+~2)2(4m~2-~)?
(T + 8(g;A)4
81m:‘ t -
m2,)2 [4(
169t2 + 450ts + 162s2)mz8 - 16( llt2 + 44~ + 18s2)mz6t + 4( llt2 +
42ts + 26s2)mL;4t2 - 864(2t + 3s)rn;l’ - 8(t+
2s)(t+
s>mz2t3 +2592m$12 +t4(t2+2tS+2S2)] + 32(&)2(&A)2 81mz6(t - m$)(t - m2,) (
1
8mi4 - 6m:2t + t2)2(4m22 - t - 2s) +
1oo(g:A)4 ( 1,jm;4
27m:4(t - m2)2 - 2mz2t + t2)2t2 + (s, t c u)}, T
here
(9)
s=(P+P2)2=[mP)+mP2)12-(p+p2)2> (10)
t=(p-~3)~=i(S-4m;~)(cos8-1), (11) u = 4mz2 -s-t, (12) 6
is the scattering angle in c.m. system. After averaging over the initial state and eliminating the double counting at the final states, the in-medium AA elastic scattering cross section can be expressed by
dA+AA = i j s ~AA-PAA (s, t> d fl.
(13)
In numerical calculation a commonly used form factor
A”A
FNNA(t) = - AS-t
(14)
for nucleon-nucleon-meson vertex is assumed to be used also for the delta-delta-meson vertex and the cut-off mass A, = 1200 MeV, A, = 808 MeV, A, = 510 MeV fixed in Refs. [ 11,121 are used. The effective mass of A is determined by the mean field given in Ref. [9] by taking into account the A degree of freedom explicitly. The relevant parameters are the coupling strengths of nucleons and deltas. In the srcinal (T - w model the coupling strengths of nucleons are obtained by fitting the saturation properties of nuclear matter. Three sets of parameters for ggN, flm and
b, c
and corresponding saturation properties are presented in Table 1 which will be used in the following calculations. For the coupling strengths of deltas, in principle, they should be determined by considering delta-nucleus interaction. Unfortunately, up to now there is no any such information available.
8 ‘Table 1 G.
Mao et al. Physics Letters B 378 (I 996) 5-l 1
Three sets of mean-field parameters and the corresponding nuclear saturation properties
GN
g”
N
b(g”NN)3
&in
K(MeV)
PO
1 9.40 10.95 -0.69 40.44 -15.57 0.70 380 0.145 2 6.90 7.54 -40.49 383.07 -15.76 0.83 380 0.145 3 7.937 6.696 42.35 157.55 -16.00 0.85 210 0.153
Three different choices based on the quark model and mass splitting have been discussed in Refs. [ 15,161. If the SU(6) symmetry is exact for baryons, then one should use the same coupling strengths for nucleons and deltas, that is (15) here we have defined the dimensionless coupling strengths (Y and j?. However, the mass splitting of the multiplets shows that the SU(6) symmetry is not exactly fulfilled. Therefore, one may have two alternative choices
MA
Ly=p= - M 1.31
N
and a= 1, p= 2 M 1.31. (17) Eq. ( 17) is based on the argument that the w meson has a clear quark-antiquark composition while the structure of the hypothetical g is not quite clear. Recently, Jin [ 171 studied the delta coupling strengths by using the finite-density QCD sum rule methods and claimed that (Y N 0.4-0.5. Different models give rather different results and until now the values of (Y and p are still ambiguous. In this work we will adopt mainly the first choice and the influence of different choices of the delta coupling strengths on the predicted quantities will be checked in the following. Since we will perform our numerical calculations at zero temperature for symmetric nuclear matter, the contribution of the pion vanishes in the mean field approximation. In the following (see Fig. 2) one can also find that the pion only plays a very small role in the u.T\~+~~. Therefore, the choice of pion coupling strengths will not affect the results much. Throughout this work, we assume gxA = gGN = f,/m, and take a commonly used value, j /47r = 0.08. Fig. 1 shows the in-medium AA elastic scattering cross section at normal density with different sets of CT, w coupling strengths presented in Table 1 and a = ,0 = 1. One can find a clear dependence of gT\A_AA on the parameters used. The bend around S N 6 GeV2 is caused by the change of centroid delta mass (Md) with S, the total energy of two particle system in free space (see Fig. 4 of Ref. [ 91). The behavior of different curves can be understood by the fact that the main contribution to the effective cross section comes from (T meson at low energy and w meson at high energy. To see this point more clearly, we present in Fig. 2 the contributions from different mesons. Unlike the in-medium NA elastic scattering cross section where the r meson contributes most, from Fig. 2 one can easily find that the effects of pion is nearly negligible. That is not in conflict with the results of Ref.
[
91. In ~~A._+NA he main contribution of pion comes from the NAa coupling, but for the a ,,, the effect of pion enters only through AAn- coupling which, however, is very small. From Fig. 2 it is also noted that the large cancellation effects come from the mixed terms of u + w. Fig. 3 depicts the influence of different choices of delta coupling strengths on the effective cross section. It is found that the effective cross section is very sensitive to the values of cz and p especially at lower energy. As
G. Mao et
al. /Physics
Letters B 378 (1996) 5-I 1
9 0
‘L”’ ““(““I”“(““’
5
6 7 8 9 10
S
(GeV’)
S
(GeV2)
Fig. 1. The in-medium AA elastic scattering cross section at nor- mal density with different sets of nucleon coupling strengths and (Y = /3 = 1. The
dotted, dash-dotted and solid lines correspond to
the first, second and third set of parameters in Table 1, respectively.
t
*
5
6 7 8 9 10
S
(GeV2)
Fig. 3. The dependence of in-medium AA
elastic scattering cross
Fig. 4. The in-medium AA
elastic scattering cross section with section on the values of LY nd p defined in the text; the second different denshies and energies. The calculations are performed set of parameters in Table 1 is used as nucleon coupling strength. with the second set of parameters in Table 1 and (Y = /3 =
1.
/ /
-300 ‘,,‘l....l....l....l..., 5 6 7 8 9 10
Fig. 2. The contributions of different mesons. The calculations are
performed with the second set of parameters in Table 1,
a = p =
1 andp=po.
‘b”
60
0
”
”
”
” ”
”
”
(
” ”
I
” ”
Yl
5
6 7 8 9 10
S
(GeV2) the values of CY nd p vary in a quite large range according to different assumptions, the sensitivity of a&,,, to LY, 3 brings our theoretical predictions with large quantitative uncertainty. From Fig. 1 and Fig. 3 we can find that the quantitative values of u&_*~~ are also sensitive to the nucleon and delta coupling strengths used. Since all three sets of coupling strengths for nucleon in Table 1 can reproduce the empirical saturation properties equally well but have rather different behaviors in effective cross section, one has to consider further testing ground in order to select a reasonable one. In Ref. [ 181 we have probed nuclear equation of state (EOS) and nucleon-nucleon (NN) scattering cross section by analyzing the collective flow at intermediate and high energy in the RBUU approach, the results turned out to favor the second and third sets of parameters. In Ref. [ 1 l] we introduced both momentum and density dependence for nucleon-scalar and nucleon-vector coupling strengths to fit the energy dependence of the empirical optical potential, nucleon mean free path as well as the density-dependent properties of nuclear matter predicted by Dirac-Brueckner-Hartree- Fock (DBHF) theory in addition to the saturation properties of nuclear matter, and found that only with the

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