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The Primakoff effect at high energy
Volume 211, number 4
PHYSICS LETTERS B
8 September 1988
T H E P R I M A K O F F E F F E C T AT H I G H E N E R G Y Mourad TAMAZOUZT Laboratoire de Physique Corpusculaire, Collbge de France, 11, place Marcelin-Berthelot, F- 75231 Paris Cedex 05, France Received 9 April 1988
We calculate the processes n±7--,n~7* and n-+)'-~p-+y*, using a model based on QCD. From the first one, we derive the cross section of the Compton effect n+-7--,n±'t.Considering an experiment where the above processes can be measured via the Primakoff effect, we notice that an appreciable yield may be obtained, at least for the reaction n~+N-,n±TN. within the f r a m e w o r k o f the B r o d s k y - L e p a g e (BL) m o d e l [3], the processes n+-7on+-7 and n-+7 --, n-+ (P-+)7*, where 7" is a virtual time-like p h o t o n identified by its decay into a lepton pair ~ (fig. l a ) . The envisaged experiment would consist o f b o m b a r d i n g a nuclear target with a charged pion b e a m o f a few h u n d r e d GeV, which would interact coherently with the electromagnetic field o f that target. The interest o f such an e x p e r i m e n t a l study would be to check the validity o f a m o d e l based on Q C D in
1. Introduction
Experimentally, the study o f the pion C o m p t o n scattering, via a P r i m a k o f f - t y p e process, has shown that at low energy ( ~ < 1 G e V ) , the reaction n -+7--' n -+7 confirms Q E D predictions. In other words, at that energy the charged pion behaves essentially as a point-like particle [ 1,2 ]. F o r higher energies ( x / ~ v / > 2 G e V ) , we calculate,
"
1.
~ ~ 1 ~ 9 ~
2a.
2b. b •
3.
4. .
.
+
+
+
Fig. 1. (a) Diagram for n ±N -, n ± ( p ± ) 7*N. (b) Different Feynman graphs contributing to n - y-, n - ( p- )y . 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
477
Volume 211, number 4
PHYSICS LETTERS B
a type of process where it has not been investigated up to now.
8 September 1988
T+O = T~-O = 0 , T ~ °=Kx//2e~
(11)
sinO x~b 1 - y 1--cosOI--Rb A
(12)
2. Analytical amplitudes
T~-O=T+O=O,
Here we calculate the process ~-+7~n-+7 *, treating the Compton effect n+-y~n-+y as a limit case. At the same time, we calculate the process n-+y~p-+y*, noting that, according to C-invariance and to helicity conservation, only p~ can be produced. Hereafter, we write down, for the ~ -+7--. n -+y* process, the helicity amplitudes T~' corresponding to the graphs of fig. lb (2, 2' represent the incoming and outgoing photon helicities, respectively). The contributing amplitudes of n+-7-,p+-y* are identical to those of 7t-+y~n-+y*, except for T~-a- which has the opposite sign; the amplitudes T3~' and T4~' have no contribution. Setting K=27rt2acq/3s, where s is the n-+y invariant mass squared, and neglecting the quark masses, one gets
where 0 is the pion emission angle in the ny CM frame; el, e2 are the quark charges, Rb = Q~,/s, with Q2 = +q2 (Qb is the 7* mass),
T~-+ = - K e ~ l + c o s 0 1 1 - y 1 - c o s 0 A Dr2 ' 1 +cos 0 1 x ( 1 - y ) T~-a+ =Key 1-cos-----OA D,------~' T~-b+ =Key 1 +cos 0 1 1 - y -cos-------0 1 1-R~ 7 '
(13)
A=x(1-x)y(1-y), DI2 = 1 - y ( 1 --Rb) ,
D34 = [y+ ( 1 --y)Rb]
×{xy+x+y-l + (-xy+x+y-1) + (1-y) [ l +x+ (l-x)
cos0
cos0]Rb}.
By parity invariance, we derive the remaining helicity amplitudes, namely:
T - - ( x , y , s , O , Rb)=T++(x,y,s,O, Rb) ,
(14)
T-+(x,y,s,O, R b ) = T + - ( x , y , s , O , Rb) ,
(15)
(1)
T-°(x,y,s,O, R b ) = - T + ° ( x , y , s , O , Rb) .
(16)
(2)
The helicity amplitudes at the hadron level are obtained by introducing the only free parameter of the Brodsky-Lepage model, i.e. the meson (n, p) wavefunction; one then gets
(3)
I 1
T~ ÷ =Kel e2 ×x(1-x)y2-(l--y)(1--COS0)(1--Rb) A
D34
(4)
0
0
'
×T~a'(x, y, s, O, Rb) T ++ = 0 ,
(5)
T +- =0,
(6)
T~a-=Ke21 1 x ( 1 - y ) A Dl2 '
(7)
T+b_ =Ke2 1-y .4
(the sum is taken over all graphs of fig. lb after symmetrization ). The q~wavefunctions, derived from non-perturbative QCD, are the following [ 4 ]: 0~ (x) ,~x( 1 - x ) ( 2 x - 1 )2,
'
(8)
1
T ~ - =Kele2 ( 1 - c o s 0) ( 1 - - R b ) D3 4
(9)
T~ - = --Kel e2 × (1--cos0)(1--Rb) 1 (1--x)y(1--y) A
478
(17)
034
,
(10)
(gpL(x)~x(1-x)[O.3(2x-1)2+O.14] .
(18)
The normalisation prescriptions of q~are given in ref. [51. Integrating over the wavefunctions apparently may cause some of the propagators of internal lines in the
Volume 211, number 4
PHYSICS LETTERS B
diagrams to go on-mass-shell for particular non-zero values of the quark m o m e n t u m fractions [6]. Fortunately this is not the case here, since those apparent singularities disappear when analytic integration over one o f the two variables, say x, is carried out.
3. Theoretical predictions
8 September 1988 ' 1 ' 1 ' 1 ' 1
I'
I
I ' l l l
103 102
100
,g i
3.1. n+-7~n+-(p +-)7* Defining N ~' (s, 0, Rb) =S 2 [ M ~' (s, 0, Rb) ]2,
(19)
10-4
the differential cross section reads
da P (N+++N+-+N d cos 0 - 32ns 3
+°)
(20)
2 --4Rbm=(p)/S] 2 1/2
.
(21)
At the outgoing real photon limit (Rb = 0); the angular distributions of the Compton effect n-+7~n +7 ~, for both helicity contributions ( + + ) and ( + - ), are separately given in fig. 2. ~' Accordingto crossing symmetry and to our result for yT-*n+pV [5], we expect the n-+7~p+7 cross section to be zero. This is confirmed by our present calculation. I I l ' l ' l t l
-1.0
, -0.8I , -0.6I , -0.4-I , -0.2I i 0.0I 0.2I -)-I'-i-'l~'l'~ l 0./* 0.6 0.8 1. CosO
with p = [ ( 1 --Rb--m=(p)/S) 2
10-3
[ ' l ' l
I'
106 I0 s
Fig. 3. Angular distributions of the outgoing meson, predicted for the processes n ±7-° n ±~ (solid line ), and n ±7 -~P±~ (dash-dotted line) at v/~=2 GeV. When the outgoing photon is off-shell (R b ~ 0) and decays into a lepton pair I~, the differential cross section of x-+y-on + (p-+) ~ becomes d2a ' oe da dRb d cos/9 = 3nRb d cos 0"
(22)
Integrating the above expression over Rb, between the kinematical limits Rb,.i. = 4 m ~2/ s ( ~ = e ) , and R~ax= w e obtain the corresponding angular distributions for v / s = 2 GeV (see fig. 3).
(w/s-mn(p))2/s,
3.2. n +-N-* n ± (p +-)7*N
10t,
The production cross section o f the n ± (p-+))'* system by incident pions, in the Coulomb field o f the target N, can be written [ 1 ]
~> IO3 I
"~
102
2i
101
--O1,-O
dac dtds-
100 -%. %. 10-1
%.% % %.%
-1.0
-0.8
-0.6
-0.t,
-0.2
0.0
0.2
0.t~
0.6
0.8
Eos 0
Fig. 2. Angular distributions predicted for the ( + + ) (solid line ) and ( + - ) (dash-dotted line) helicities of the Compton effect ~±y--*~±y.
Z 2oe 6'~(p)v*(s) t-train n s t2 IFem(t) 12
(23)
where Z i s the nucleus charge; ~(o)r. is the above calculated cross section integrated over Rb and 0 we take Icos 01 ~<0.8; t is the absolute value o f the n - N fourm o m e n t u m transfer squared, with t
Volume 211, number 4
PHYSICS LETTERS B
8 September 1988
Table 1 Total cross sections, in nb, of Primakoff processes n+-N~n+TN and n + N - ~ n +-( p + ) ~ N , for 500 GeV pion beams bombarding copper and lead targets N ( s , ~ = 2 GeV 2, tin,x= 10 -3 GeV2). The results of the first column are obtained by summing over the two contributions in fig. 2.
Cu Pb
n+yN
n+e+e-N
n±g+g-N
p-+e+e-N
p+~i+g-N
3.06 24.48
0.89× 10 2 0.72× 10 -~
0.53× 10 -2 0.43× 10 -l
0.84X 10 -3 0.67× 10 -2
0.79X 10 -3 0.63× 10 -2
For copper ( Z = 29) and lead ( Z = 82) targets, one takes respectively [ 1 ] F eca m ( t ) -~ e -2°°' and Fem(/)Pb- e-400, (t in GeV2). Within the considered t-range, accounting for the predominant contribution of extremely small values of t, one may take Fe~ = FeF~ 1. Integrating the above relation over t, between tmin and t . . . . and then over s between Stain= 4 GeV 2 and its upper limit given by train= t . . . . i.e. Smax= x[i-0-5 GeV 2, we get a good enough approximation: 2120/(
ac- 3ns3
In
~
-- 5 ) iff(Smin )
,
(24)
where +0.8
~=s 3
f•/
-0.8
d COS 0 ( do' ) \ d cos O/"
(25)
Numerical values of a~ for a 500 GeV pion beam, are displayed in table 1.
- the total cross section is much larger in n+-7~n+-7 than in n+-y~n +-( p + ) ~ , which was to be expected, since the 7" decay introduces an extra a factor; - in n+-y--,n+-~, the total cross section is larger when the massive photon decays into e+e - than into g-+Ix-, whereas it remains practically the same for n -+y ~ p -+~ because if its very small contribution for small values of Rb; - the total cross section for the Compton effect n -+y ~ n -+y appears very small compared to the values given in ref. [ 1 ] ( ~ 0.1 mb); this is first due to the higher energy (x/s) range, and correlatively to the model used (aBL~ 1/S 3, O ' B o r n ~ l/s). (iv) The specific problem of kinematic singularities that may appear in Compton scattering (contrary to 7y reactions) is eliminated by the analytic integration. (v) Finally, similar predictions can be given for an incident kaon beam.
Acknowledgement 4. Concluding remarks On different results, we make the following comments. (i) First, we notice that the shape of the angular distribution for n+-y-,n+-7 (fig. 2) is rather similar to that of the QED Compton effect, favouring backward angles. (ii) In fig. 3, the meson angular distribution in the processes n+-T~n+-£~ and n+-T--,p-+~ favour backward angles as well as forward ones. The first dominates over the second by about one order of magnitude, due to the fact that (i) its range of integration over Rb is much larger (Rb(n)<~0.86, Rb(p) ~<0.36 at x / s = 2 GeV), and (ii) it remains finite near Rb = 0. (iii) Table 1 suggests three remarks: 480
I am very grateful indeed to Professor P. Kessler for his encouraging and generous support, and Professor M. Froissart for having read the manuscript. I am also indebted to Professor E. Maina for having drawn my attention to the problem of possible existence of kinematic singularities. Finally I would like to thank J.C. Couillard for the drawings.
References [ 1 ] R.V. Kowalewski et al., Phys. Rev. D 29 (1984) 1000; M. Zielinski et al., Phys. Rev. D 29 (1984) 2633. [2] Yu.M. Antipov et al., Phys. Lett. B 121 (1983) 445. [3] S.J. Brodsky and P. Lepage, Phys. Rev. D 24 ( 1981 ) 1808. [ 4 ] V.L. Chernyak and A.R, Zhitnitsky, Nucl. Phys. B 222 ( 1983 ) 382. [ 5 ] M. Tamazouzt, preprint LPC/87-30. [6] E. Maina and G.R. Farrar, Phys. Lett. B 206 (1988) 120.
PHYSICS LETTERS B
8 September 1988
T H E P R I M A K O F F E F F E C T AT H I G H E N E R G Y Mourad TAMAZOUZT Laboratoire de Physique Corpusculaire, Collbge de France, 11, place Marcelin-Berthelot, F- 75231 Paris Cedex 05, France Received 9 April 1988
We calculate the processes n±7--,n~7* and n-+)'-~p-+y*, using a model based on QCD. From the first one, we derive the cross section of the Compton effect n+-7--,n±'t.Considering an experiment where the above processes can be measured via the Primakoff effect, we notice that an appreciable yield may be obtained, at least for the reaction n~+N-,n±TN. within the f r a m e w o r k o f the B r o d s k y - L e p a g e (BL) m o d e l [3], the processes n+-7on+-7 and n-+7 --, n-+ (P-+)7*, where 7" is a virtual time-like p h o t o n identified by its decay into a lepton pair ~ (fig. l a ) . The envisaged experiment would consist o f b o m b a r d i n g a nuclear target with a charged pion b e a m o f a few h u n d r e d GeV, which would interact coherently with the electromagnetic field o f that target. The interest o f such an e x p e r i m e n t a l study would be to check the validity o f a m o d e l based on Q C D in
1. Introduction
Experimentally, the study o f the pion C o m p t o n scattering, via a P r i m a k o f f - t y p e process, has shown that at low energy ( ~ < 1 G e V ) , the reaction n -+7--' n -+7 confirms Q E D predictions. In other words, at that energy the charged pion behaves essentially as a point-like particle [ 1,2 ]. F o r higher energies ( x / ~ v / > 2 G e V ) , we calculate,
"
1.
~ ~ 1 ~ 9 ~
2a.
2b. b •
3.
4. .
.
+
+
+
Fig. 1. (a) Diagram for n ±N -, n ± ( p ± ) 7*N. (b) Different Feynman graphs contributing to n - y-, n - ( p- )y . 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing D i v i s i o n )
477
Volume 211, number 4
PHYSICS LETTERS B
a type of process where it has not been investigated up to now.
8 September 1988
T+O = T~-O = 0 , T ~ °=Kx//2e~
(11)
sinO x~b 1 - y 1--cosOI--Rb A
(12)
2. Analytical amplitudes
T~-O=T+O=O,
Here we calculate the process ~-+7~n-+7 *, treating the Compton effect n+-y~n-+y as a limit case. At the same time, we calculate the process n-+y~p-+y*, noting that, according to C-invariance and to helicity conservation, only p~ can be produced. Hereafter, we write down, for the ~ -+7--. n -+y* process, the helicity amplitudes T~' corresponding to the graphs of fig. lb (2, 2' represent the incoming and outgoing photon helicities, respectively). The contributing amplitudes of n+-7-,p+-y* are identical to those of 7t-+y~n-+y*, except for T~-a- which has the opposite sign; the amplitudes T3~' and T4~' have no contribution. Setting K=27rt2acq/3s, where s is the n-+y invariant mass squared, and neglecting the quark masses, one gets
where 0 is the pion emission angle in the ny CM frame; el, e2 are the quark charges, Rb = Q~,/s, with Q2 = +q2 (Qb is the 7* mass),
T~-+ = - K e ~ l + c o s 0 1 1 - y 1 - c o s 0 A Dr2 ' 1 +cos 0 1 x ( 1 - y ) T~-a+ =Key 1-cos-----OA D,------~' T~-b+ =Key 1 +cos 0 1 1 - y -cos-------0 1 1-R~ 7 '
(13)
A=x(1-x)y(1-y), DI2 = 1 - y ( 1 --Rb) ,
D34 = [y+ ( 1 --y)Rb]
×{xy+x+y-l + (-xy+x+y-1) + (1-y) [ l +x+ (l-x)
cos0
cos0]Rb}.
By parity invariance, we derive the remaining helicity amplitudes, namely:
T - - ( x , y , s , O , Rb)=T++(x,y,s,O, Rb) ,
(14)
T-+(x,y,s,O, R b ) = T + - ( x , y , s , O , Rb) ,
(15)
(1)
T-°(x,y,s,O, R b ) = - T + ° ( x , y , s , O , Rb) .
(16)
(2)
The helicity amplitudes at the hadron level are obtained by introducing the only free parameter of the Brodsky-Lepage model, i.e. the meson (n, p) wavefunction; one then gets
(3)
I 1
T~ ÷ =Kel e2 ×x(1-x)y2-(l--y)(1--COS0)(1--Rb) A
D34
(4)
0
0
'
×T~a'(x, y, s, O, Rb) T ++ = 0 ,
(5)
T +- =0,
(6)
T~a-=Ke21 1 x ( 1 - y ) A Dl2 '
(7)
T+b_ =Ke2 1-y .4
(the sum is taken over all graphs of fig. lb after symmetrization ). The q~wavefunctions, derived from non-perturbative QCD, are the following [ 4 ]: 0~ (x) ,~x( 1 - x ) ( 2 x - 1 )2,
'
(8)
1
T ~ - =Kele2 ( 1 - c o s 0) ( 1 - - R b ) D3 4
(9)
T~ - = --Kel e2 × (1--cos0)(1--Rb) 1 (1--x)y(1--y) A
478
(17)
034
,
(10)
(gpL(x)~x(1-x)[O.3(2x-1)2+O.14] .
(18)
The normalisation prescriptions of q~are given in ref. [51. Integrating over the wavefunctions apparently may cause some of the propagators of internal lines in the
Volume 211, number 4
PHYSICS LETTERS B
diagrams to go on-mass-shell for particular non-zero values of the quark m o m e n t u m fractions [6]. Fortunately this is not the case here, since those apparent singularities disappear when analytic integration over one o f the two variables, say x, is carried out.
3. Theoretical predictions
8 September 1988 ' 1 ' 1 ' 1 ' 1
I'
I
I ' l l l
103 102
100
,g i
3.1. n+-7~n+-(p +-)7* Defining N ~' (s, 0, Rb) =S 2 [ M ~' (s, 0, Rb) ]2,
(19)
10-4
the differential cross section reads
da P (N+++N+-+N d cos 0 - 32ns 3
+°)
(20)
2 --4Rbm=(p)/S] 2 1/2
.
(21)
At the outgoing real photon limit (Rb = 0); the angular distributions of the Compton effect n-+7~n +7 ~, for both helicity contributions ( + + ) and ( + - ), are separately given in fig. 2. ~' Accordingto crossing symmetry and to our result for yT-*n+pV [5], we expect the n-+7~p+7 cross section to be zero. This is confirmed by our present calculation. I I l ' l ' l t l
-1.0
, -0.8I , -0.6I , -0.4-I , -0.2I i 0.0I 0.2I -)-I'-i-'l~'l'~ l 0./* 0.6 0.8 1. CosO
with p = [ ( 1 --Rb--m=(p)/S) 2
10-3
[ ' l ' l
I'
106 I0 s
Fig. 3. Angular distributions of the outgoing meson, predicted for the processes n ±7-° n ±~ (solid line ), and n ±7 -~P±~ (dash-dotted line) at v/~=2 GeV. When the outgoing photon is off-shell (R b ~ 0) and decays into a lepton pair I~, the differential cross section of x-+y-on + (p-+) ~ becomes d2a ' oe da dRb d cos/9 = 3nRb d cos 0"
(22)
Integrating the above expression over Rb, between the kinematical limits Rb,.i. = 4 m ~2/ s ( ~ = e ) , and R~ax= w e obtain the corresponding angular distributions for v / s = 2 GeV (see fig. 3).
(w/s-mn(p))2/s,
3.2. n +-N-* n ± (p +-)7*N
10t,
The production cross section o f the n ± (p-+))'* system by incident pions, in the Coulomb field o f the target N, can be written [ 1 ]
~> IO3 I
"~
102
2i
101
--O1,-O
dac dtds-
100 -%. %. 10-1
%.% % %.%
-1.0
-0.8
-0.6
-0.t,
-0.2
0.0
0.2
0.t~
0.6
0.8
Eos 0
Fig. 2. Angular distributions predicted for the ( + + ) (solid line ) and ( + - ) (dash-dotted line) helicities of the Compton effect ~±y--*~±y.
Z 2oe 6'~(p)v*(s) t-train n s t2 IFem(t) 12
(23)
where Z i s the nucleus charge; ~(o)r. is the above calculated cross section integrated over Rb and 0 we take Icos 01 ~<0.8; t is the absolute value o f the n - N fourm o m e n t u m transfer squared, with t
Volume 211, number 4
PHYSICS LETTERS B
8 September 1988
Table 1 Total cross sections, in nb, of Primakoff processes n+-N~n+TN and n + N - ~ n +-( p + ) ~ N , for 500 GeV pion beams bombarding copper and lead targets N ( s , ~ = 2 GeV 2, tin,x= 10 -3 GeV2). The results of the first column are obtained by summing over the two contributions in fig. 2.
Cu Pb
n+yN
n+e+e-N
n±g+g-N
p-+e+e-N
p+~i+g-N
3.06 24.48
0.89× 10 2 0.72× 10 -~
0.53× 10 -2 0.43× 10 -l
0.84X 10 -3 0.67× 10 -2
0.79X 10 -3 0.63× 10 -2
For copper ( Z = 29) and lead ( Z = 82) targets, one takes respectively [ 1 ] F eca m ( t ) -~ e -2°°' and Fem(/)Pb- e-400, (t in GeV2). Within the considered t-range, accounting for the predominant contribution of extremely small values of t, one may take Fe~ = FeF~ 1. Integrating the above relation over t, between tmin and t . . . . and then over s between Stain= 4 GeV 2 and its upper limit given by train= t . . . . i.e. Smax= x[i-0-5 GeV 2, we get a good enough approximation: 2120/(
ac- 3ns3
In
~
-- 5 ) iff(Smin )
,
(24)
where +0.8
~=s 3
f•/
-0.8
d COS 0 ( do' ) \ d cos O/"
(25)
Numerical values of a~ for a 500 GeV pion beam, are displayed in table 1.
- the total cross section is much larger in n+-7~n+-7 than in n+-y~n +-( p + ) ~ , which was to be expected, since the 7" decay introduces an extra a factor; - in n+-y--,n+-~, the total cross section is larger when the massive photon decays into e+e - than into g-+Ix-, whereas it remains practically the same for n -+y ~ p -+~ because if its very small contribution for small values of Rb; - the total cross section for the Compton effect n -+y ~ n -+y appears very small compared to the values given in ref. [ 1 ] ( ~ 0.1 mb); this is first due to the higher energy (x/s) range, and correlatively to the model used (aBL~ 1/S 3, O ' B o r n ~ l/s). (iv) The specific problem of kinematic singularities that may appear in Compton scattering (contrary to 7y reactions) is eliminated by the analytic integration. (v) Finally, similar predictions can be given for an incident kaon beam.
Acknowledgement 4. Concluding remarks On different results, we make the following comments. (i) First, we notice that the shape of the angular distribution for n+-y-,n+-7 (fig. 2) is rather similar to that of the QED Compton effect, favouring backward angles. (ii) In fig. 3, the meson angular distribution in the processes n+-T~n+-£~ and n+-T--,p-+~ favour backward angles as well as forward ones. The first dominates over the second by about one order of magnitude, due to the fact that (i) its range of integration over Rb is much larger (Rb(n)<~0.86, Rb(p) ~<0.36 at x / s = 2 GeV), and (ii) it remains finite near Rb = 0. (iii) Table 1 suggests three remarks: 480
I am very grateful indeed to Professor P. Kessler for his encouraging and generous support, and Professor M. Froissart for having read the manuscript. I am also indebted to Professor E. Maina for having drawn my attention to the problem of possible existence of kinematic singularities. Finally I would like to thank J.C. Couillard for the drawings.
References [ 1 ] R.V. Kowalewski et al., Phys. Rev. D 29 (1984) 1000; M. Zielinski et al., Phys. Rev. D 29 (1984) 2633. [2] Yu.M. Antipov et al., Phys. Lett. B 121 (1983) 445. [3] S.J. Brodsky and P. Lepage, Phys. Rev. D 24 ( 1981 ) 1808. [ 4 ] V.L. Chernyak and A.R, Zhitnitsky, Nucl. Phys. B 222 ( 1983 ) 382. [ 5 ] M. Tamazouzt, preprint LPC/87-30. [6] E. Maina and G.R. Farrar, Phys. Lett. B 206 (1988) 120.