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Dispersion relations and KΛ photoproduction on a proton
Nuclear Physics 82 (1966) 680--682; ( ~ North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher
DISPERSION RELATIONS AND KA P H O T O P R O D U C T I O N ON A PROTON N. F. NELIPA
P. N. Lebedev Physical Institute of the USSR Academy of Sciences, Moscow, USSR Received 19 January 1966 Abstract: Experimental data o n K.A_photoproduction on a proton are analysed with the aid o f t h e approximate set of integral equations for multipoles. The set of parameters is determined (including the KNA coupling constant).
1. Introduction The purpose of this paper is to analyse the available experimental data on the 3'+P --' K + + A ° reaction with the aid of the set of integral equations for multipoles obtained earlier 1) (preliminary results were obtained in ref. 2)). To obtain the exact solution of the above set of equations is a highly complicated problem. Therefore we confine ourselves to the consideration of an approximate case. 2. Approximations and Method of Analysis We take into account the contributions from (i) the pole term for all three channels, (ii) the Krr system in the intermediate state for channel III (with the aid of the resonant state K*) and (iii) the integral terms. The resonant state of the nN-system with I = J = ½ does not appear in the reaction under study. The resonant rcN state with I = ½ makes a contribution only to the higher d, f , . . . multipoles; we neglect it because the experimental data on angular distributions can, as will be clear from the following, be well explained by the s-, p-approximation. Besides we neglect the multi-meson intermediate states as well as the Y* resonance. The available experimental data indicate that the p~; state is resonant z). Therefore, in the integrands of the equations for multipoles we keep the imaginary parts of only the resonant multipoles E1 + and M1 +. Then the set of integral equations for multipoles becomes simpler and its solution can be found (in a different way than in ref. 2)). For the KA scattering phase we choose several expressions in the form of a BreitWigner resonance (arc sin
6P~r~( W)
F
[W-M*I+F
| ½ , exp [ - I W - M * I F -1 ] / ~arc ctg(M* - W)F- 1, 680
(1) (2) (3)
KA-PHOTOPRODUC~ON
681
where M * and F are the position and width of the KA resonance and W the total energy of the system. We have assumed that the resonance state decays primarily into a KA system 3) (i.e., F~:a ~ Ftota0. The expressions for the KA production differential cross sections and A ° particle polarization incorporate nine parameters PA,
MK*,
F,
M*,
gNKA'
5,
21,
23,
24 .
Here #a is the anomalous magnetic moment of the A ° particle, MK. the mass of the K* particle, e the relative parity of the A, 2~particle, 24 __ gNK*A//TKK*
x/4-~= 2Ma
MK.
gNI(APa '
/~K*,/~A~ are the magnetic transition moments, 2 3 __ gNK*Ag~KK* ,
MK* and g the corresponding coupling constants. For the first three parameters we have used the available experimental values (MK. = 1.9 M~, /~a = - 0 . 7 5 "e/2Mp and F = 0.1 MK); the other parameters are sought by using the cross sections for the production of K + mesons 4) and the polarization of a A ° particles 5). The analysis is made with the aid of the method proposed by Sokolov and Silin 6). This method permits the determination of the minimum of X2 defined as follows 2
n~=l L A~'~ 1 where ~-~. and ~ - are experimental and theoretical values of the KA production differential cross sections and A ° particle polarization at the available point of energy and angle, A~'~ the corresponding experimental error and n the total number of experimental points. The calculations were performed with an electronic computer.
3. Results of Analysis The values of the desired parameters for the cases e = + 1 and ~ = - 1 are given in table 1. It is clear that the parameter values slightly depend upon the form of the phase. The differential cross sections and polarizations of the A ° particle obtained with the aid of the parameters coincide in the case e = + 1 with the corresponding experimental quantities within the errors (except in some cases). We considered several values of F and have selected the minimum one, which gives good agreement with the
682
N.F.
NELIPA
e x p e r i m e n t a l results on the A ° particle p o l a r i z a t i o n (because e x p e r i m e n t a l l y F ~ 0.1 M ~ ) . T h e c a l c u l a t e d values o f A ° p o l a r i z a t i o n are one o r d e r less t h a n the e x p e r i m e n t a l ones in the case e = - 1. O n e c a n o b t a i n the values o f P n e a r the e x p e r i m e n t a l ones, b y a s s u m i n g t h a t F > O.1 M K, b u t this is in c o n t r a d i c t i o n with the experiment. T h e r e f o r e the m o d e l u n d e r c o n s i d e r a t i o n predicts results which are in g o o d agreem e n t with the available e x p e r i m e n t a l data, if e = + 1; this is n o t in c o n t r a d i c t i o n with the recent e x p e r i m e n t 7). TABLE 1
Values of parameters obtained ~(IV) e
Z~
ffNKA 4zr
M*
21
2a
2~
1
+1 --1
2 2.04
9.614-0.26 3.424-0.38
3.5424-0.011 3.4164-0.012
0.544-0.19 --0.484-0.01
0.55±0.04 0.124-0.08
0.694-0.13 --0.31-4-0.07
2
+1 --1
2.02 2.04
9.594-0.22 3.654-0.44
3.5414-0.016 3.4304-0.024
0.55-4-0.17 --0.48-4-0.01
0.494-0.05 0.0854-0.09
0.724-0.13 --0.32-4-0.08
3
q-1 --1
2.05 2.1
9.214-0.34 4.294-0.61
3.6954-0.132 3.9214-0.55
0.40 :j:0.23 --0.914-0.07
0.544-0.06 --0.45±0.12
0.604-0.16 --0.584-0.37
T h e o b t a i n e d value g~KA/4rr for e = + 1 is larger t h a n the one u s u a l l y given ( b u t g~Ka/4u ~ 6 if 24 = 0) a n d is n e a r to the p r e d i c t i o n o f u n i t a r y s y m m e t r y . T h e s-wave p r e d o m i n a t e s in the energy range u n d e r consideration. W e w o u l d like to stress t h a t m o r e a c c u r a t e e x p e r i m e n t a l d a t a for P are desirable in o r d e r to o b t a i n final conclusions. W e have u s e d several initial sets o f p a r a m e t e r s for the i t e r a t i o n p r o c e d u r e a n d have n o t o b t a i n e d results different f r o m those given in table 1. I w o u l d like to t h a n k M. A. M a r k o v for stimulating discussions a n d R. M. Borodsi5 for assistance in the n u m e r i c a l calculations.
References 1) 2) 3) 4)
N. F. Nelipa and V. A. Tsarev, Nuclear Physics 45 (1963) 665 N. F. Nelipa and V. A. Tsarev, Nuclear Physics 55 (1964) 155 A. I. Baz, V. G. Vaks and A. I. Larkin, ZhETF 43 0962) 166 R. L. Anderson et aL, Phys. Rev. Lett. 9 (1962) 131 and private communication; C. W. Peck, Phys. Rev. 135 (1964) 830 5) H. Tohm et al., Phys. Rev. Lett. 11 (1963) 433; B. Borgia et aL, Nuovo Cim. 32 (1964) 218 6) S. N. Sokolov and I. N. Silin, preprint Joint Institute for Nuclear Research D-810, Dubna (1961) 7) Cynthia Aeff et aL, Phys. Rev. 137 (1965) Bl105
DISPERSION RELATIONS AND KA P H O T O P R O D U C T I O N ON A PROTON N. F. NELIPA
P. N. Lebedev Physical Institute of the USSR Academy of Sciences, Moscow, USSR Received 19 January 1966 Abstract: Experimental data o n K.A_photoproduction on a proton are analysed with the aid o f t h e approximate set of integral equations for multipoles. The set of parameters is determined (including the KNA coupling constant).
1. Introduction The purpose of this paper is to analyse the available experimental data on the 3'+P --' K + + A ° reaction with the aid of the set of integral equations for multipoles obtained earlier 1) (preliminary results were obtained in ref. 2)). To obtain the exact solution of the above set of equations is a highly complicated problem. Therefore we confine ourselves to the consideration of an approximate case. 2. Approximations and Method of Analysis We take into account the contributions from (i) the pole term for all three channels, (ii) the Krr system in the intermediate state for channel III (with the aid of the resonant state K*) and (iii) the integral terms. The resonant state of the nN-system with I = J = ½ does not appear in the reaction under study. The resonant rcN state with I = ½ makes a contribution only to the higher d, f , . . . multipoles; we neglect it because the experimental data on angular distributions can, as will be clear from the following, be well explained by the s-, p-approximation. Besides we neglect the multi-meson intermediate states as well as the Y* resonance. The available experimental data indicate that the p~; state is resonant z). Therefore, in the integrands of the equations for multipoles we keep the imaginary parts of only the resonant multipoles E1 + and M1 +. Then the set of integral equations for multipoles becomes simpler and its solution can be found (in a different way than in ref. 2)). For the KA scattering phase we choose several expressions in the form of a BreitWigner resonance (arc sin
6P~r~( W)
F
[W-M*I+F
| ½ , exp [ - I W - M * I F -1 ] / ~arc ctg(M* - W)F- 1, 680
(1) (2) (3)
KA-PHOTOPRODUC~ON
681
where M * and F are the position and width of the KA resonance and W the total energy of the system. We have assumed that the resonance state decays primarily into a KA system 3) (i.e., F~:a ~ Ftota0. The expressions for the KA production differential cross sections and A ° particle polarization incorporate nine parameters PA,
MK*,
F,
M*,
gNKA'
5,
21,
23,
24 .
Here #a is the anomalous magnetic moment of the A ° particle, MK. the mass of the K* particle, e the relative parity of the A, 2~particle, 24 __ gNK*A//TKK*
x/4-~= 2Ma
MK.
gNI(APa '
/~K*,/~A~ are the magnetic transition moments, 2 3 __ gNK*Ag~KK* ,
MK* and g the corresponding coupling constants. For the first three parameters we have used the available experimental values (MK. = 1.9 M~, /~a = - 0 . 7 5 "e/2Mp and F = 0.1 MK); the other parameters are sought by using the cross sections for the production of K + mesons 4) and the polarization of a A ° particles 5). The analysis is made with the aid of the method proposed by Sokolov and Silin 6). This method permits the determination of the minimum of X2 defined as follows 2
n~=l L A~'~ 1 where ~-~. and ~ - are experimental and theoretical values of the KA production differential cross sections and A ° particle polarization at the available point of energy and angle, A~'~ the corresponding experimental error and n the total number of experimental points. The calculations were performed with an electronic computer.
3. Results of Analysis The values of the desired parameters for the cases e = + 1 and ~ = - 1 are given in table 1. It is clear that the parameter values slightly depend upon the form of the phase. The differential cross sections and polarizations of the A ° particle obtained with the aid of the parameters coincide in the case e = + 1 with the corresponding experimental quantities within the errors (except in some cases). We considered several values of F and have selected the minimum one, which gives good agreement with the
682
N.F.
NELIPA
e x p e r i m e n t a l results on the A ° particle p o l a r i z a t i o n (because e x p e r i m e n t a l l y F ~ 0.1 M ~ ) . T h e c a l c u l a t e d values o f A ° p o l a r i z a t i o n are one o r d e r less t h a n the e x p e r i m e n t a l ones in the case e = - 1. O n e c a n o b t a i n the values o f P n e a r the e x p e r i m e n t a l ones, b y a s s u m i n g t h a t F > O.1 M K, b u t this is in c o n t r a d i c t i o n with the experiment. T h e r e f o r e the m o d e l u n d e r c o n s i d e r a t i o n predicts results which are in g o o d agreem e n t with the available e x p e r i m e n t a l data, if e = + 1; this is n o t in c o n t r a d i c t i o n with the recent e x p e r i m e n t 7). TABLE 1
Values of parameters obtained ~(IV) e
Z~
ffNKA 4zr
M*
21
2a
2~
1
+1 --1
2 2.04
9.614-0.26 3.424-0.38
3.5424-0.011 3.4164-0.012
0.544-0.19 --0.484-0.01
0.55±0.04 0.124-0.08
0.694-0.13 --0.31-4-0.07
2
+1 --1
2.02 2.04
9.594-0.22 3.654-0.44
3.5414-0.016 3.4304-0.024
0.55-4-0.17 --0.48-4-0.01
0.494-0.05 0.0854-0.09
0.724-0.13 --0.32-4-0.08
3
q-1 --1
2.05 2.1
9.214-0.34 4.294-0.61
3.6954-0.132 3.9214-0.55
0.40 :j:0.23 --0.914-0.07
0.544-0.06 --0.45±0.12
0.604-0.16 --0.584-0.37
T h e o b t a i n e d value g~KA/4rr for e = + 1 is larger t h a n the one u s u a l l y given ( b u t g~Ka/4u ~ 6 if 24 = 0) a n d is n e a r to the p r e d i c t i o n o f u n i t a r y s y m m e t r y . T h e s-wave p r e d o m i n a t e s in the energy range u n d e r consideration. W e w o u l d like to stress t h a t m o r e a c c u r a t e e x p e r i m e n t a l d a t a for P are desirable in o r d e r to o b t a i n final conclusions. W e have u s e d several initial sets o f p a r a m e t e r s for the i t e r a t i o n p r o c e d u r e a n d have n o t o b t a i n e d results different f r o m those given in table 1. I w o u l d like to t h a n k M. A. M a r k o v for stimulating discussions a n d R. M. Borodsi5 for assistance in the n u m e r i c a l calculations.
References 1) 2) 3) 4)
N. F. Nelipa and V. A. Tsarev, Nuclear Physics 45 (1963) 665 N. F. Nelipa and V. A. Tsarev, Nuclear Physics 55 (1964) 155 A. I. Baz, V. G. Vaks and A. I. Larkin, ZhETF 43 0962) 166 R. L. Anderson et aL, Phys. Rev. Lett. 9 (1962) 131 and private communication; C. W. Peck, Phys. Rev. 135 (1964) 830 5) H. Tohm et al., Phys. Rev. Lett. 11 (1963) 433; B. Borgia et aL, Nuovo Cim. 32 (1964) 218 6) S. N. Sokolov and I. N. Silin, preprint Joint Institute for Nuclear Research D-810, Dubna (1961) 7) Cynthia Aeff et aL, Phys. Rev. 137 (1965) Bl105