Z. Phys. A 344, 317-323 (1993)
""o""' Hadrons and Nuclei fQr OhysikA
9 Springer-Verlag 1993
Scattering length expansion of the coupled channels /l/l and AA AA near the A production threshold A.G. Schneider-Neureither t, H.J. Pirner 1, B. Kerbikov ~' *, J. Mahalanabis 2 Institut f/Jr Theoretische Physik, UniversitfitHeidelberg, Philosophenweg 19, W-6900 Heidelberg, Federal Republic of Germany 2 Saha Institute of Nuclear Physics, Calcutta, India Received: 26 May 1992/Revised version: 20 July 1992
Abstract. We fit the scattering lengths in the triplet s-, pand d-waves for the two channels ~p--~flA and flA--~AA near the A production threshold to the differential cross da section )-~ (fip--,.#A) and to the polarization P. PACS: i3.75.Cs; 14.20.In; ll.80.Gw
1. Introduction The experimental study of the production of hyperons in/~p collisions from LEAR is a particularly attractive channel for investigating the /VN reaction mechanism. The strangeness produced in the final state offers an opportunity to gain information on the dynamics of strangeness creation. Experiment PS185 at CERN has measured the formation of YTpairs from pp reactions. Total and differential cross sections, polarizations and spin correlation parameters for AA production around the threshold region have been reported recently [1]. One expects that close to the AA reaction threshold, a relatively simple creation mechanism can be detected due to the presence of few partial waves. There are mainly two types of theoretical models used to explain the reaction: (i) the "conventional" t-channel meson exchange picture [2 10] (ii) The quark level picture: the gluon exchange model 3S 1 (effective gluon) [5, 11] or 3Po-Etq creation [12-15] or a superposition of both [16]. A low momentum ~s pair can be created with the annihilation of a g/q pair with 3St (effective gluon) and/or 3P0 (vacuum) quantum numbers. Models based on these Supported by the BMFT-grant 06 HD 756 * Permanent address: Institute for Theoretical and Experimental Physics (ITEP), Moscow, Russia
mechanisms describe reasonable well the differential cross sections but sometimes do not fit well the polarization data. Both of these models depend strongly on the initial and final state interaction. Thus, the calculations of transition potentials are embedded in the framework of a coupled channel model. Although these models are supported theoretically, very little experimental information on the final state interaction .#A ~ Z A is available. In general a realistic description of/Sp-*AA requires an appropriate treatment of initial and final state correlations effect. The experimental data on differential cross-section are characterized by strong forward peaking and calculations indicated a substantial p-wave contribution at threshold and the suppression of the s-wave. This has led us to use the well known scattering length expansion of the s-, p- and d-wave amplitudes for the study of the /~p ~ AA and AA ~ AA reactions at threshold. A determination of the low energy parameters is important of establishing relations to the potential model and to the underlying theoretical mechanism. Recently Tabakin et al. [17] approach the ~ p ~ A A reaction with a phenomenological amplitude analysis. In total 14 parameters were included to fit all of the AA data at six momenta for the partial waves aS1, 3D1, 3S1~3D 1 and 3P0,1,2. Our work tries to separately fit the ~ p ~ Z A and AA-~ AA amplitudes by using the knowledge acquired about elastic/~p ~/~p scattering [18]. Finally, information about the off-diagonal channel is the key for understanding strangeness production. In the present work we leave out the tensor coupling 3 D 1 ~ - 3 S 1 . We have checked that the inclusion of this tensor transition for the description of the presently available data set does not improve the quality of the fit and results in strong correlations between parameters. In an alternative work one could include all tensor transitions 3D1~3S~, 3F2 ~3P2, 3G2~-3D3 by making use of a more complete data set anticipated.
318 2. Construction of the S-matrix and the scattering formalism
f ( O ) = ~ [ ( l + 1) R~a+ 1 --~Iel,l_l]
We define the c.m. momenta of the incoming/~p-system and of the AA-system as ks, kI with
g (O) = 2 ~ [RI,I +I -- RLI -1] PII (COSO) .
ky=/k~o+m2-m~.
(1)
S~+_1 h(t)+e(l_h(t))+it~+ dh(t) ll = dt '
We limit our investigation to 1435.15MeV/c
s=(S~'-~P S~v_.~A) \SaA S~a aA/" ~p
(4)
The amplitudes Rz,s are in turn related to the S-matrix elements SSs : R i j = (S{s-1)/2ikl. The grey-disk parametrization for S{s reads (t = l + 89
k~= V / 1 mp (]//PlZb+ m~ -- mp) and
+
P/(CON O )
l
(6) Here R describes the radius, d the diffuseness and s the transparency of the grey disk. The parameters # -+ determine the imaginary part of the S-matrix in the respective spin-orbit channels J = l + 1. By fitting the iOp differential cross section and polarization at P~ab= 1449.0 MeV/c we get the following set of parameters
(2)
+
The S-matrix elements are analyzed in the following way. The elastic scattering process/~p ~/Sp is determined by spin-dependent diffraction scattering on a grey disk (Frahn-Venter model) [191. In the usual parametization of the spin-scattering matrix [-20] the Frahn-Venter/Sp amplitude contains two terms M = f ( O ) + g(O)(a~ + a2)n, k~ x k s n =% x kyl" It can be easily shown that the partial wave decomposition of this amplitude is similar to that of the sum of central and spin-orbit terms commonly used in nuclear physics. Thus one has
(5)
R = l . 1 2 f m , d=0.13fm, /~- = --0.04.
~=0.26,
/~+=0.65, (7)
In Fig. 1 we show a typical differential cross section for /~p scattering at Pl,b = 1449.0 MeV/c. The polarization is only moderately well fitted with this parametrization. As compared to the set of parameters obtained by Kunne [18] for the range of momenta 497 MeV/c _--
p = 1449.0 MeV/c 10:
,
theoretical values experimental data
/
,/
,.Q
0,1 -
b 9
0,01
-1
9
A9
'
'
'
'
-0,6
-0,2
0,2
0,6
COS
0
Fig. L Fit of the differential cross section p p ~ p Plab= 1449.0 MeV/c (lab-momentum)
at
319 triplet amplitude S11 with J = 1 are expressed in terms of SI ~ t resulting in the following parametrization
as
i S,o=h(t)+e(1-h(t))+~771((l+l)#+ i
SIl=h(t)§247
+
The /Sp-+AA scattering amplitudes Fl~ are defined
, _, dh(t) +t# ) ~
(8)
F~ = 2i k l / ~k~ .
_, dh(t) +(/+1)# ~dT-"
(9)
Using the spin dependent scattering formalism of Hoshizaki [20] we obtain the differential cross section
The imaginary parts of the amplitudes (5), (8), (9) are peaked at l=kR. At the AA threshold approximately l< 9 angular momentum states contribute. The final state interaction AA ~ AA is totally unknown and it seems impossible to think about an experiment to measure it. It is interesting to compare the size of the scattering length in this process with the (I = 0) PTN ~ NN-scattering process. We recall that for (I = 0) NN--+ N N scattering we found the following scattering length and volume [21, 22] a~V=o= ( - 1.1 -I- i0.4) fm af=x = (0.5 § i0.2) fm 3 .
(10)
(~)
S(O
(17)
with 5g = llF~ "§
+ 3(IFll 12+ IFlll12+ IF~xl2)
5
2 2
o, 3 1, 5 ~ = ~ 1 Re(fo*l F~, ) + ~ Re(N, f,~ ) + g Re(Fo~l f~*) 3
+ Re (F~ 3
12
7
22
Re (Fil F2'1")§ 1 Re(f21 F2~.) 3
12
3
(11) § Re(tO F 2 . ) + 9 Re(F)1 V2.)
We do not differentiate between singlet and triplet AA scattering. The strangeness production matrix element has near the creation threshold of .dA the following low energy expansion [23]
S(~p ~ AA)Ss = i ~
= dV ( ~ ) { suCPo (cos O ) + ~ Pl (cos O )
+ cgPz(cos O) + ~Pa (cos 6))}
We parametrize the final state matrix in the conventional form with unknown scattering parameters 1 § ia t k 21+ 1 - ia~ Ir2~+ ~ "
(16)
27 2 @ = 1 0 ae(Fil f2**).
(18)
and the polarization
k) +-~ exp(iaSs(pp) + i6Ss(AA)), (12)
where exp (2 i (5[s (PP)) = Sis
+ cg, pal (cos 6))} (13)
with Sfs given by (5), (8), (9), and 6Js(AA) by (11), i.e. exp (2 i ass (AA)) = S (~(AA).
with d ' = 1 Im (F~ Fo11")+3 Im(F~l For*)+ 5 Im (Foil F~I*)
(14)
The real parameters ass have the same dimensions fm21+ 1 as the scattering lengths or volumes at. Using the experimental information that the singlet fraction
+ ~Im(F~ll1 FO.)+ 3 Im(F~x F~*)+ 1 Re(F~l F~*) N,=3Im(F2, ' Fol,)+3im(F,1 Fi,2, ) + ~1 Im(F~, o F 2.) g, = 9 Im(F2** F2*).
Fs = 1(1 --
)
(19)
(15)
is small for low momenta and neglecting the tensor transitions 3S1,-~-3D1 we have five real strangeness production parameters ass for the partial waves 3S,, 3po, 3p,, 3P2 and 3D 1 and three complex final state scattering parameters at for the reaction AA ~.4A with angular mom e n t u m / = 0 , 1, 2.
(19)
To take into account the relativistic kinematics properly we use as normalization factor Y (~.)=
4kfm2
At AA-threshold this factor ~/" (k~-)reduces to kl
(20)
320
3. Numerical analysis
'y
In the numerical analysis eleven parameters (Np = 11) are fitted to the data. These are the three complex AA - , AA scattering lengths and the five transition matrix elements /Sp ~ AA (see Table 1). In the momentum region below Plab=1546.20MeV/c there are 160 data points (ND = 160). Sixteen data are total cross sections, one hundred data points are differential cross sections and forty four are polarizations. Minimization of the z2/d.f, gives zZ/d.f. = 2.4 with the values of the parameters presented in Table 1. The differential cross sections contribute most to be total zZ/d.f, with z2/d.f. = 1.5. The Z2 for the polarization is small because of the larger experimental errors. The covariance matrix indicates strong correlations between the imaginary part of the AA ~ AA scattering parameters and the pp~AA transition matrix elements. The importance of the strangeness production process can be compensated by more or less absorption in the final state. The absolute sizes of the imaginary parts in the AA -~ AA scattering are not very different from the same parameters in NN-scattering. The real p-wave scattering volume of Re aa = 0.63 fm 3 is remarkable. It may indicate a similar p-wave attraction in AA as in NN. In the AA case isospin ( I = 1) meson exchanges cannot contribute. The AA ~ Z A and NN-~NN total cross sections C.f. (10) as functions of P~,b are shown in Fig. 2. Due to the
~' 0
la
50- a 4030-
20 100 1430
2,5-
b
1460
1490
1520
1550
1438
1439
b
1,5
1+ N
1
I
0,5 0 1435
1436
1437
P0b [Mev/o] Fig. 3. The total cross section/~p -+ AA a for the total m o m e n t u m region b for the threshold region
Table 1. Values of the eleven fitted parameters Reaction
Partial waves
Scattering lengths
AA--+AA
l = 0(s-wave) / = 1 (p-wave) l = 2 (d-wave)
ao = (0.05 -+ 0.03 + i(1.48 _+0.06)) fm al = (0.63_+ 0.06 + i(0.91 _+0.07)) fm3 a2 = (0.01 _+0.05 + i(0.28 _+0.08)) fm 5
i0p ~ A A
3S 1 ~ aS 1 3Po ~ aPo
aoll =0.020_+0.001 fm a~ =0.044-0.01 fm a
3P1~ 3P1 3 P2 --+ 3 P2
a~l = -0.086+0.006 fm3 a~1= 0.009 ___0.001 fm3
3D1--+3D 1
a~l = --0.031 _+0.006 fm s
1000 c~
\
800
\,
',,\,
o
-.\
600
+J
",.,,.
~
b
.
-...
XA--> XA
400
200
~ N - - > ~N (1=0)
100
200
300
Plab
[MeV/c]
Fig. 2. The total cross section of AA and N N -+ 2VN for I = 0
larger imaginary AA scattering lengths the AA total cross section is almost twice as big as the ( I = 0 ) ) V N cross section. The transition matrix elements are in general a factor of 10 to 50 smaller than the low energy/~p--+ i0p diagonal elements. This is a consequence of a total strangeness production cross section of O(10 btb) compared to an elastic cross section of O (100 mb). In Fig. 3 the total cross section /~p--+AA is represented. The full momentum region and the threshold region are shown separately in Fig. 3a and b. In the same figures the theoretical fits are shown using the above parameters. The fit to the total production cross section is very good. In the very near threshold region it may be interesting to remeasure carefully the total cross section. In Fig. 4 the contributions of all considered partial waves in the total cross section are represented. One sees that only at very low energies the s-wave dominates. The minimum in the s-wave contribution at Plab 1470 MeV/c originates from the interplay of the phase space and the final state interaction factors in the AA channel. As can be easily seen from (11) the AA production s-wave cross section close to threshold has a minimum at k s = (Ira ao)- 1 ~ 130 MeV/c which corresponds to P~.b~ 1470 MeV/c. This position agrees with the one in Fig. 4.
321 1E2
In Figs. 5-7 we show the differential cross sections and the polarizations at six momenta. The fit to these data is better for higher momenta. The two lower mo-
3Pl
mr. 1El
......5.-.-.-..........ii; ........................................
.,.o
menta show some deviations, especially 'd(~) ' at Plab = 1436.95 MeV/c and the polarization at the lowest momentum Plab= 1435.95 MeV/c. The forward peaking of the cross section seems to develop in the theoretical fit more slowly, than in the data. Concerning the polarization it is nice to see the zero in P move towards more forward angles. One can define a shifted momentum transfer variable t' [18]
lEO
1E-1
1E 8 1430
1460
1490
1580
1550 P
[MeV/e]
lab
t ' = t [ c o s ( O ) ] - t [ c o s ( O ) = 1]
Fig. 4. Absolute theoretical cross sections of the lower partial waves 3S~ ' 3po ' 3p~, 3P2 and 3D 1. For the theoretical errors from the fitted parameters see Table 1
with
t=m~+mA--~+~2s The early onset o f p-wave production is a consequence of the strong central absorptive part in the f p ~ / S p initial state interaction. At Pl,b of 1444 MeV/c the p-waves dominate the cross section. The contribution of the p-waves to the total cross section follows the ordering o - ( 3 p2 ) , . ~ o ( 3 po ) < O- ( 3 p 1 ) . The amount of 0-(3/~ is about six times as big as the individual o-(3P2) or o-(3po) cross sections. The d-waves become important at Plab 1470 MeV/c. The 3pl-wave contribution however still dominates at all higher momenta.
,15 a
1
~
b
~,2-
,06 -
~ +
,03-
0
I
-1
s is the invariant mass of the reaction. The variable t' is constructed to vanish at forward angles and therefore scattering data at different energies can be compared for small [t'l. The data show positive polarizations for 1(1<0.2 (GeV/c) 2. This property is also approximately reproduced in our fit (see Fig. 8). In general such a behavior is seen in the presence of a spin orbit potential which is surface peaked, i.e. proportional to the derivative of the central potential.
t
-
-,6-
J
-,6
-,2
S
I
I
I
-,2
,2
I
r
I
t
-1
t
,6
-1
I
-,6
[
I
I
-,2
I
I
,2
I
,6 COS
COSO
i
C 1436.95 MeV/c
,16 84
~
'
,6-
,09
,2
l l/(s_4m~).(s_4mZ).cos(O)
I435.95 MeV/c
1435.95 MeV/c
,12-
(21)
,12
_a
1436.95 MeV/c
,6-
"" "" "i .....
-~ ,08
.....
2, 2
,04
-,6
Fig. 5 a - d . Differential cross-sections ffp--+ A A and A-polarizations at labm o m e n t a between 1435.95 M e V and
-~-
b 0
I
-1
-,6
I
I
-,2
i
i
I
,2
,6
r
COS
_1 I, -1
. . . . . -,6 -,2
i
,2
I
r
1436.95 M e V
i
,6 COS @
322 1,2 a 1445.35 MeV/c ,6,2-
,6%2
1445.35MeV/c
,3%6
0 -,6
-1
-,2
,2
-1
,6
i
i
t
I
-,6
-1
I
i
-,2
i
I
,2
I
,6 COS t9
COS
3 I,e 1476.50 MeVlc
_d
,6
1476.50 MeV/c
,2 -,2
1Fig. 6 a - d . Differential cross-sections p p ~ AA and _~-polarizations at labm o m e n t a between 1445.35 M e V and 1476.50 M e V
%6
r
-,6
-1
-,2
,2
,6
i
p
-,6
1
i
i
-,2
i
i
,2
F
,6
COS O
6-
a
COS 0
a
t13
1507,50 MeV/c
4-
2
0
t
i
-1
i
I
-,6
i
r
-,2
~
i
,2
i
i
-1
,6
i
i
-,6
i
I
-,2
I
I
,2
I
,6 COS 0
COS 0
1
12
d
1546.20 MeV/c
1546.20 MeV/c
,6 .~
,2
63
Fig. 7 a - d . Differential cross-sections and A-polarizations at labm o m e n t a between 1507.50 M e V and 1546.20 M e V
pp~AA 0
i
-1
i
I
r
-,6
r
-,2
r
I
i
,2
-I
i
p
-1
,6
i
-,6
i
r
i
-,2
i
,2
i
i
COS 0
COS
In our case the initial state /~p-potential has this behaviour. The theoretical final state interaction does not matter since it is independent of p-polarization. It seems that the p-wave transition parameters for AA ~ / ~ p do not change this feature of the initial state interaction. The p-wave parameters a tss - - a 01 1 , a l l , a ~ follow neither a spin-orbit (3P2, 3P1 , 3Po) nor a tensor pattern.
06i I
I<
o2~ ,
~
b
t
,6
c
I -06
108
-04
O2
0
4. C o n c l u s i o n s
[~v/cl ~
Fig. 8. The polarizations at three m o m e n t a as functions of the shifted m o m e n t u m transfer t' = t [cos (O)] - t [cos (O) = 1]. [a P~,b = 1546,20 MeV/c, b Plab = 1507.50 MeV/c, c Plab = 1476.50 MeV/c)]
A theoretical explanation of our fit parameters is not obvious. The restriction of our fit to central spin dependent interactions without tensor transitions does not al-
323 low to exclude meson exchange models [10, 24]. We demonstrate that a model without tensor terms is able to reproduce the existing data. In perturbative Q C D the light quark antiquark annihilation proceeds via a gluon (3S~). The light diquarks or antiquark~ play the role of spectators only supplying angular m o m e n t u m to the reaction. Such a model would explain the triplet behaviour of partial waves. A simple estimate in the quark model gives for the/~p ~ AA matrix element Moce -262q2
(22)
where q2 is the c.m. m o m e n t u m transfer q = (ki--ky) and b the root mean square distance in the harmonic oscillator model between the light or strange quark and the residual diquark e.m. system b ~ 2 . 5 GeV -1 . The fitted partial wave cross sections add up to the total cross sections with different powers of ky. Using (12), (16), (18) without initial and final state interactions we get
(23) and
6t~ (tip --+ AA) = 4realm r .
(24)
Expanding the total cross section with the quark model (22) we get an equivalent expansion in powers of k s . We do not believe that the absolute magnitude in the quark model is calculable, therefore we compare the expansion normalized to the first coefficient c~= 1. (0~:fl:Y)quark = (1:6.0 fm2:9.4 fm 4) (0{:/~:~)fit : (1 : 5.8 fm 2 : 1.5 fro4). The agreement for the/~/e ratio between this theoretical model and the fit is good. The d-wave in the fit is much smaller than in the quark model. It also seems to us difficult to understand the origin of the different sizes of the 3p amplitudes. Indeed a recent calculation with the quark model in [24] could only fit the data by strongly modifying the initial state PPinteraction. The same calculation, by the way, also gives
a dominant 3p~ ~p--+AA partial wave cross section in the quark model. There are forthcoming results on t p ~ X ~ ~ and t p - ~ S + S § + Z - Z - . These reactions can also be described within the framework of the formalism developed in our paper. The final state interaction m a y change depending on isospin. Since both initial and final states are high in energy compared to the inelastic channels we would not expect strong deviations of the absorptive potentials. The real potential of the final hyperon antihyperon pair may, however, vary near threshold in an unpredictable way. C o u l o m b effects are important when final particles are slow and carry opposite charges.
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