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Dynamical SU(3) symmetry breaking in the linear baryonic string model
Volume 80B, number 3
PHYSICS LETTERS
1 January 1979
DYNAMICAL SU(3) SYMMETRY BREAKING IN THE LINEAR BARYONIC STRING MODEL
P. ZENCZYKOWSKI Institute of Nuclear Physics, 31-342 Cracow, Poland
Received 29 September 1978
A dynamical mechanism responsible for breaking of the SU(3) symmetry among various states from the s - u - d sector is proposed. In the case when SU(3) is dynamically maximally broken we obtain Ac~-- A,y degeneracy and reproduce also other prominent features of the leading strange baryon trajectories.
The construction of dual models for baryons has been a difficulty since the beginning of the dual models. Recently, following the success of the dual topological unitarization approach [ 1] in describing mesons, interest in finding a correct "planar" theory of baryons has been renewed. Various mechanisms responsible for the deviation from planarity have been proposed [ 2 - 4 ] . Inami, Kawarabayashi and Kitakado (IKK) [3] have suggested a linear string model for baryons. Such linear baryons are consistent with the planar bootstrap condition [5]. In ref. [3] a SU(3)-symmetric interaction kernel has been assumed and the resulting shifts of baryon trajectories have been discussed. In what follows we will relax this assumption and allow for explicit SU(3) symmetry breaking in the kernel. It is the purpose of this paper to show that the linear baryonic string model (LBSM) is capable of explaining the observed pattern of the EXD breaking between leading strange natural (and unnatural) baryon trajectories. In the LBSM one has two active quarks in a spin-1 state at the ends of the string, and a spectator quark which sits in the middle with l = 0 relative to the pair of active quarks. Only the active quarks couple to the mesons [3, 5, 6]. The exchange degeneracy pattern of the model is (1 + 8).y ~ (8 + 10)c~,
(1 + 8)t3 *+ (8 + 10)~ .
(1)
In ref. [3] the EXD breaking has been explained as a result of the interchange between the spectator quark and one of the active quarks (fig. 1). The LBSM has obvious shortcomings, namely it does not describe 298
correctly the L = 0 states since SU(6) symmetry is broken from the beginning by choosing one quark as being a spectator. However, we will not be interested in the L = 0 states and we will consider only the states with angular momentum different from zero. The mechanism proposed by IKK [3] as being responsible for baryon trajectorie splitting is analogous to f-renormalization. To see that, assume first exact SU(3) symmetry at the planar level. The resulting spectrum consists of a degenerate singlet and octet for mesons and a singlet, two octets and a decuplet for baryons. When the SU(3)-symmetric kernel is turned on the mesons split. The SU(3) singlet goes up (down) with respect to the SU(3) octet. As shown in ref. [3], similarly one gets splitting of SU(3) multiplets within the baryon sector. Now, imagine SU(3) breaking of the kernel only. For mesons the singlet and the I = 0 component of the octet mix. In much the same way one gets mixing in the baryon sector: the singlet with the I = 0 component of the antisymmetric octet as well as the I = 1 components of the symmetric octet and de-
8
ta
s
Fig. 1. The interaction kernel in the diagrammatic representation.
Volume 80B, number 3
PHYSICS LETTERS
cuplet. The size of this mixing is controlled by the magnitude of the SU(3) breaking and as a result deviations both from the ideal SU(3) multiplet structure and from the IKK predictions for trajectories will emerge. To break SU(3) dynamically we assume that the kernel corresponding to figs. la, lb takes on the values V and eV respectively, e being the measure of SU(3) breaking (e = 0 corresponds in a sense to "maximal" SU(3) breaking). We do not intend to explain why e < 1 but assume that, for some reasons *I , the kernel is smaller when the strange quark is involved in the interchange active ~+ passive quark. There is no need for e and V t o be the same for natural (eN, VN) and unnatural (eU, VU) parity trajectories, however in both cases we expect eN, U ,~ 1. In fact it is known that VN and VU differ in sign. The mixing must occur separately in symmetric and antisymmetric sectors since they have different signa. tures. Calculating the interaction kernel in the SU(3)symmetric basis one obtains (Asl (11As) = - ( E a [ l l [ Za) = 1 V, (Y.D [ 1)[ ED ) = -(A1 [I)[ A 1) = ½(1 + 2 e ) g ,
(2) (Zs] 1)1 ~s> = -(Aal I)]Aa) = 1(1 -
4e)V,
(~sl I;'l Y.D> -- -- (1/3 x/2) (1 -
e)V.
Here ZD, A1 stands for the decuplet and singlet, respectively; (a) s denotes the (anti)symmetry of the pair of active quarks for octet trajectories. Diagonalizing the 2 X 2 matrices one gets for the symmetric sector Aj+(s) = ~(1 +-(1 + 8 e 2 ) l / 2 ) V , (3a) ~+ = cos ~¢+(s) ~D + sin ~o+(s)Z s ,
Z_ orthogonal,
cot ~o+(s) = [2X/if(1 - e ) ] - I (1 + 8e + 3(1 + 8e2)1/2), and for the antisymmetric sector Aj_+(a) = --Aj+_(s), A+ = cos tp+(a) A 1 + sin ¢+(a)A a , cot ¢+(a) = - c o t ~o+(s).
A_ orthogonal, (3b)
,1 This is in agreement with the picture of strange quarks hardly propagating in rapidity space.
1 January 1979
The shift of A s (Na) is independent of e, A/As, Xa = T ½ V.
(3C)
Let us now take the limit e -+ 0. The comparison of the obtained spectrum (e = 0), the IKK spectrum (e = 1) and the experimental one is shown in figs. 2a, 2b for natural and unnatural parity trajectories, respectively. We have gathered the average experimental data for all well-confirmed resonances on leading trajectories except the L = 0 ones. The same slope a' = 0.95 GeV 2 for all trajectories has been assumed. We note that in the eN,U = 0 case one should observe clustering of the trajectories, i.e. Aa should coincide with A.r, E~ with E~ and so on. It is well seen that in reality e N (eU) is not far from zero. We prefer to begin with the discussion of the case eN, U = 0 which may be considered as a first, crude approximation to nature. It is clear that for eN, U = 0 the planar trajectories corresponding to s sitting in the middle of the string will not acquire any shift. Thus these trajectories (A_t3, E_8 , A . r , Z a ) are "pure" quark states in the sense that for these four types of states the strange quark is always in the middle. This has ~interesting consequences. In the decay of such a state into a ground state baryon plus meson the second cannot be formed from the spectator quark and the newly created antiquark, since that would not change to zero the relative angular m o m e n t u m residing in the pair of active quarks. Therefore decays of the type A_~, E_ z , A_. r, ~-c~ ~ K.N are forbidden. For e :~ 0 there are small admixtures of wavefunctions with the strange quark being active and, consequently, small YKN coupling is allowed. Experimentally we know this is indeed the case [7], namely A.r(3/2- , 1690), Z a (5/2 +, 1915), A~(5/2-, 1830) couple only weakly to KiN. The dominantly octet Z 8 (7/2 +, ?) and A v ( 7 / 2 - , ?) have not yet been observed [7], probably due to their inherent reluctance to decay into KN. In fact from the decay width rates P [Y.~(5/2 +, 1915) ~ KN] / F[_A~(5/2 +, 1815) ~ g~N] ~ 1/5 and F[AI~(5/2- , 1830) K N ] / P [ ~ ( 5 / 2 - , 1 7 6 5 ) + ~N] ~ 1/5 [7] one can infer that eN, U ~ 0.3 -- 0.4. This corresponds to the mixing angle ~0+ ~ 10 ° - 15 °. Even for such a value of e the resulting spectrum should be similar to the e = 0 case since for small e the splitting of trajectories (A a - A m, E 6 - Zt~) is proportional to e 2. It is worth noting that in the limit e ~ 0 the flavour wavefunctions are similar to the ones chosen in the QCD inspired calculations of the p-wave baryonic spectrum in the naive 299
Volume 80B, n u m b e r 3
aZ
PHYSICS LETTERS
/LIc~-A4¢
.Z
A_g Z.,,(,)
O
A_s
E=O
EX£
;b) A
1690)). These states may be considered to be to a large extent planar - the smallness of eN, U being the origin of this approximate planarity. In reality the parameters eu, eN, VU, VN may depend on t. Unfortunately, we do not know of any reliable way of calculating them at present. Even then, however, we might estimate VU, VN from the splitting of nonstrange baryon trajectories, and assuming e N = e U = 0.3 calculate the EXD breaking in the s - u - d sector. There are not enough accurate data to carry out an exact t-dependent analysis of the VN(t), Vu(t ) functions, but from N(5/2 +, 1688), N ( 3 / 2 - , 1520) and A(5/2 +, 1890)we get 2VN(t ) ~ - 0 . 5 in the natural parity sector for t ~ 2 - 4 GeV 2 and similarly from A(7/2 +, 1950) and N(5/2-, 1670)we obtain 2 V u ( t ) 0.3. (We have tacitly assumed linear parallel trajectories.) Then A_~ - A s = 0.01 = E+5 - Et3 and A s E-c~ ~ 0.11 > ~# - A # ~ 0.06 in agreement with the tendency exhibited by the data. There is a missing factor of about 2 when we compare this prediction with experiment. It might be due to the fact that our starting values of the V'.s calculated from the nonstrange baryonic spectrum could have some inherent errors. We recall also that for L = 0 strange baryons the E - A splitting has been explained with the help of coulour magnetic interactions [10]. These, and other similar interactions could change to some extent the firstorder correction to planar baryons. Therefore our results for (A - E)a,~ splitting may be considered encouraging. For charmed baryons with a charmed quark replacing the strange one we expect even more dramatic clustering of the trajectories, the almost exact vanishhag of the e(charm) being the reason of this effect. In this letter we have looked for a rough explanation of the existing pattern of EXD breaking among baryons from the s - u - d sector and we have found that strong dynamical SU(3) breaking, i.e. e2,U ,~ 1 has a close resemblance to the data. We have recognized that the trajectories A t3E_ ~A _ v E _ , ~ are approximately planar. We believe we have identified the origin of the rough structure of EXD breaking among strange baryon trajectories. -
.Z o
~=I
OKK)
£= 0
EXP.
Fig. 2. Splitting of (a) natural and (b) unnatural parity strangeness S = - 1 baryon trajectories.
quark model [8]. Our considerations could also shed some light on the concept of the ideal mixing of baryons [9]. We believe that the arguments given in this paper apply especially well to the highest leading trajectories. Lower lying trajectories may receive some admixtures from the nonleading ones [3] breaking the expected pattern. Therefore we consider the agreement in the A v - A s - E~ and E~ - A n - E~ cases as being more meaningful than discrepancies observed for lower lying trajectories. Besides the kernel interchanging the quarks one could also add a kernel in which the quarks do not change their positions. This could further shift the trajectories without any change in their wave functions. Therefore we regard the strong selection rules A_~E_~A v~ ~ ~- KN as even more important in establishing thevalidity of the whole scheme than the relative positions of their trajectories (c.f. A v ( 3 / 2 - ,
300
1 January 1979
I would like to thank Dr. J. Kwieciflski for numerous valuable discussions.
Volume 80B, number 3
PHYSICS LETTERS
References [1] G.F. Chew and C. Rosenzweig, Phys. Rep. 41C (1978) 263; Chan H.M. and Tsou S.T., RL-76-080, Proc. Bielefeld Summer Institute (1976). [2] G.C. Rossi and G. Veneziano, Nucl. Phys. B123 (1977) 507; M. Imachi, S. Otsuki and F. Toyoda, Prog. Theor. Phys. 57 (1977) 517; K. Konishi, Nucl. Phys. B131 (1977) 143. [3] T. Inami, K, Kawarabayashi and S. Kitakado, Phys. Lett. 72B (1977) 127. [4] C. Rosenzweig, Phys. Lett. 71B (1977) 203.
1 January 1979
[5] A. Nakamura, Nagoya Univ. preprint, DPNU-11-78 (1978). [6] J. Mandula, J. Weyers and G. Zweig, Ann. Rev. Nucl. Sei. 20 (1970) 289; S. Mandelstam, Phys. Rev. D1 (1970) 1734. [7] M. Fukugita and K. Igi, Phys. Rep. 31C (1977) 237; J.L. Rosner, Phys. Rep. 11C (1974) 189. [8] N. Isgur and G. Karl, Phys. Lett. 74B (1978) 353; RL78-045, Rutherford Lab. preprint (1978). [9] W.P. Petersen and J.L. Rosner, Phys. Rev. D6 (1972) 820; D. Faiman, Phys. Rev. D15 (1977) 854. [10] A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147.
301
PHYSICS LETTERS
1 January 1979
DYNAMICAL SU(3) SYMMETRY BREAKING IN THE LINEAR BARYONIC STRING MODEL
P. ZENCZYKOWSKI Institute of Nuclear Physics, 31-342 Cracow, Poland
Received 29 September 1978
A dynamical mechanism responsible for breaking of the SU(3) symmetry among various states from the s - u - d sector is proposed. In the case when SU(3) is dynamically maximally broken we obtain Ac~-- A,y degeneracy and reproduce also other prominent features of the leading strange baryon trajectories.
The construction of dual models for baryons has been a difficulty since the beginning of the dual models. Recently, following the success of the dual topological unitarization approach [ 1] in describing mesons, interest in finding a correct "planar" theory of baryons has been renewed. Various mechanisms responsible for the deviation from planarity have been proposed [ 2 - 4 ] . Inami, Kawarabayashi and Kitakado (IKK) [3] have suggested a linear string model for baryons. Such linear baryons are consistent with the planar bootstrap condition [5]. In ref. [3] a SU(3)-symmetric interaction kernel has been assumed and the resulting shifts of baryon trajectories have been discussed. In what follows we will relax this assumption and allow for explicit SU(3) symmetry breaking in the kernel. It is the purpose of this paper to show that the linear baryonic string model (LBSM) is capable of explaining the observed pattern of the EXD breaking between leading strange natural (and unnatural) baryon trajectories. In the LBSM one has two active quarks in a spin-1 state at the ends of the string, and a spectator quark which sits in the middle with l = 0 relative to the pair of active quarks. Only the active quarks couple to the mesons [3, 5, 6]. The exchange degeneracy pattern of the model is (1 + 8).y ~ (8 + 10)c~,
(1 + 8)t3 *+ (8 + 10)~ .
(1)
In ref. [3] the EXD breaking has been explained as a result of the interchange between the spectator quark and one of the active quarks (fig. 1). The LBSM has obvious shortcomings, namely it does not describe 298
correctly the L = 0 states since SU(6) symmetry is broken from the beginning by choosing one quark as being a spectator. However, we will not be interested in the L = 0 states and we will consider only the states with angular momentum different from zero. The mechanism proposed by IKK [3] as being responsible for baryon trajectorie splitting is analogous to f-renormalization. To see that, assume first exact SU(3) symmetry at the planar level. The resulting spectrum consists of a degenerate singlet and octet for mesons and a singlet, two octets and a decuplet for baryons. When the SU(3)-symmetric kernel is turned on the mesons split. The SU(3) singlet goes up (down) with respect to the SU(3) octet. As shown in ref. [3], similarly one gets splitting of SU(3) multiplets within the baryon sector. Now, imagine SU(3) breaking of the kernel only. For mesons the singlet and the I = 0 component of the octet mix. In much the same way one gets mixing in the baryon sector: the singlet with the I = 0 component of the antisymmetric octet as well as the I = 1 components of the symmetric octet and de-
8
ta
s
Fig. 1. The interaction kernel in the diagrammatic representation.
Volume 80B, number 3
PHYSICS LETTERS
cuplet. The size of this mixing is controlled by the magnitude of the SU(3) breaking and as a result deviations both from the ideal SU(3) multiplet structure and from the IKK predictions for trajectories will emerge. To break SU(3) dynamically we assume that the kernel corresponding to figs. la, lb takes on the values V and eV respectively, e being the measure of SU(3) breaking (e = 0 corresponds in a sense to "maximal" SU(3) breaking). We do not intend to explain why e < 1 but assume that, for some reasons *I , the kernel is smaller when the strange quark is involved in the interchange active ~+ passive quark. There is no need for e and V t o be the same for natural (eN, VN) and unnatural (eU, VU) parity trajectories, however in both cases we expect eN, U ,~ 1. In fact it is known that VN and VU differ in sign. The mixing must occur separately in symmetric and antisymmetric sectors since they have different signa. tures. Calculating the interaction kernel in the SU(3)symmetric basis one obtains (Asl (11As) = - ( E a [ l l [ Za) = 1 V, (Y.D [ 1)[ ED ) = -(A1 [I)[ A 1) = ½(1 + 2 e ) g ,
(2) (Zs] 1)1 ~s> = -(Aal I)]Aa) = 1(1 -
4e)V,
(~sl I;'l Y.D> --
e)V.
Here ZD, A1 stands for the decuplet and singlet, respectively; (a) s denotes the (anti)symmetry of the pair of active quarks for octet trajectories. Diagonalizing the 2 X 2 matrices one gets for the symmetric sector Aj+(s) = ~(1 +-(1 + 8 e 2 ) l / 2 ) V , (3a) ~+ = cos ~¢+(s) ~D + sin ~o+(s)Z s ,
Z_ orthogonal,
cot ~o+(s) = [2X/if(1 - e ) ] - I (1 + 8e + 3(1 + 8e2)1/2), and for the antisymmetric sector Aj_+(a) = --Aj+_(s), A+ = cos tp+(a) A 1 + sin ¢+(a)A a , cot ¢+(a) = - c o t ~o+(s).
A_ orthogonal, (3b)
,1 This is in agreement with the picture of strange quarks hardly propagating in rapidity space.
1 January 1979
The shift of A s (Na) is independent of e, A/As, Xa = T ½ V.
(3C)
Let us now take the limit e -+ 0. The comparison of the obtained spectrum (e = 0), the IKK spectrum (e = 1) and the experimental one is shown in figs. 2a, 2b for natural and unnatural parity trajectories, respectively. We have gathered the average experimental data for all well-confirmed resonances on leading trajectories except the L = 0 ones. The same slope a' = 0.95 GeV 2 for all trajectories has been assumed. We note that in the eN,U = 0 case one should observe clustering of the trajectories, i.e. Aa should coincide with A.r, E~ with E~ and so on. It is well seen that in reality e N (eU) is not far from zero. We prefer to begin with the discussion of the case eN, U = 0 which may be considered as a first, crude approximation to nature. It is clear that for eN, U = 0 the planar trajectories corresponding to s sitting in the middle of the string will not acquire any shift. Thus these trajectories (A_t3, E_8 , A . r , Z a ) are "pure" quark states in the sense that for these four types of states the strange quark is always in the middle. This has ~interesting consequences. In the decay of such a state into a ground state baryon plus meson the second cannot be formed from the spectator quark and the newly created antiquark, since that would not change to zero the relative angular m o m e n t u m residing in the pair of active quarks. Therefore decays of the type A_~, E_ z , A_. r, ~-c~ ~ K.N are forbidden. For e :~ 0 there are small admixtures of wavefunctions with the strange quark being active and, consequently, small YKN coupling is allowed. Experimentally we know this is indeed the case [7], namely A.r(3/2- , 1690), Z a (5/2 +, 1915), A~(5/2-, 1830) couple only weakly to KiN. The dominantly octet Z 8 (7/2 +, ?) and A v ( 7 / 2 - , ?) have not yet been observed [7], probably due to their inherent reluctance to decay into KN. In fact from the decay width rates P [Y.~(5/2 +, 1915) ~ KN] / F[_A~(5/2 +, 1815) ~ g~N] ~ 1/5 and F[AI~(5/2- , 1830) K N ] / P [ ~ ( 5 / 2 - , 1 7 6 5 ) + ~N] ~ 1/5 [7] one can infer that eN, U ~ 0.3 -- 0.4. This corresponds to the mixing angle ~0+ ~ 10 ° - 15 °. Even for such a value of e the resulting spectrum should be similar to the e = 0 case since for small e the splitting of trajectories (A a - A m, E 6 - Zt~) is proportional to e 2. It is worth noting that in the limit e ~ 0 the flavour wavefunctions are similar to the ones chosen in the QCD inspired calculations of the p-wave baryonic spectrum in the naive 299
Volume 80B, n u m b e r 3
aZ
PHYSICS LETTERS
/LIc~-A4¢
.Z
A_g Z.,,(,)
O
A_s
E=O
EX£
;b) A
1690)). These states may be considered to be to a large extent planar - the smallness of eN, U being the origin of this approximate planarity. In reality the parameters eu, eN, VU, VN may depend on t. Unfortunately, we do not know of any reliable way of calculating them at present. Even then, however, we might estimate VU, VN from the splitting of nonstrange baryon trajectories, and assuming e N = e U = 0.3 calculate the EXD breaking in the s - u - d sector. There are not enough accurate data to carry out an exact t-dependent analysis of the VN(t), Vu(t ) functions, but from N(5/2 +, 1688), N ( 3 / 2 - , 1520) and A(5/2 +, 1890)we get 2VN(t ) ~ - 0 . 5 in the natural parity sector for t ~ 2 - 4 GeV 2 and similarly from A(7/2 +, 1950) and N(5/2-, 1670)we obtain 2 V u ( t ) 0.3. (We have tacitly assumed linear parallel trajectories.) Then A_~ - A s = 0.01 = E+5 - Et3 and A s E-c~ ~ 0.11 > ~# - A # ~ 0.06 in agreement with the tendency exhibited by the data. There is a missing factor of about 2 when we compare this prediction with experiment. It might be due to the fact that our starting values of the V'.s calculated from the nonstrange baryonic spectrum could have some inherent errors. We recall also that for L = 0 strange baryons the E - A splitting has been explained with the help of coulour magnetic interactions [10]. These, and other similar interactions could change to some extent the firstorder correction to planar baryons. Therefore our results for (A - E)a,~ splitting may be considered encouraging. For charmed baryons with a charmed quark replacing the strange one we expect even more dramatic clustering of the trajectories, the almost exact vanishhag of the e(charm) being the reason of this effect. In this letter we have looked for a rough explanation of the existing pattern of EXD breaking among baryons from the s - u - d sector and we have found that strong dynamical SU(3) breaking, i.e. e2,U ,~ 1 has a close resemblance to the data. We have recognized that the trajectories A t3E_ ~A _ v E _ , ~ are approximately planar. We believe we have identified the origin of the rough structure of EXD breaking among strange baryon trajectories. -
.Z o
~=I
OKK)
£= 0
EXP.
Fig. 2. Splitting of (a) natural and (b) unnatural parity strangeness S = - 1 baryon trajectories.
quark model [8]. Our considerations could also shed some light on the concept of the ideal mixing of baryons [9]. We believe that the arguments given in this paper apply especially well to the highest leading trajectories. Lower lying trajectories may receive some admixtures from the nonleading ones [3] breaking the expected pattern. Therefore we consider the agreement in the A v - A s - E~ and E~ - A n - E~ cases as being more meaningful than discrepancies observed for lower lying trajectories. Besides the kernel interchanging the quarks one could also add a kernel in which the quarks do not change their positions. This could further shift the trajectories without any change in their wave functions. Therefore we regard the strong selection rules A_~E_~A v~ ~ ~- KN as even more important in establishing thevalidity of the whole scheme than the relative positions of their trajectories (c.f. A v ( 3 / 2 - ,
300
1 January 1979
I would like to thank Dr. J. Kwieciflski for numerous valuable discussions.
Volume 80B, number 3
PHYSICS LETTERS
References [1] G.F. Chew and C. Rosenzweig, Phys. Rep. 41C (1978) 263; Chan H.M. and Tsou S.T., RL-76-080, Proc. Bielefeld Summer Institute (1976). [2] G.C. Rossi and G. Veneziano, Nucl. Phys. B123 (1977) 507; M. Imachi, S. Otsuki and F. Toyoda, Prog. Theor. Phys. 57 (1977) 517; K. Konishi, Nucl. Phys. B131 (1977) 143. [3] T. Inami, K, Kawarabayashi and S. Kitakado, Phys. Lett. 72B (1977) 127. [4] C. Rosenzweig, Phys. Lett. 71B (1977) 203.
1 January 1979
[5] A. Nakamura, Nagoya Univ. preprint, DPNU-11-78 (1978). [6] J. Mandula, J. Weyers and G. Zweig, Ann. Rev. Nucl. Sei. 20 (1970) 289; S. Mandelstam, Phys. Rev. D1 (1970) 1734. [7] M. Fukugita and K. Igi, Phys. Rep. 31C (1977) 237; J.L. Rosner, Phys. Rep. 11C (1974) 189. [8] N. Isgur and G. Karl, Phys. Lett. 74B (1978) 353; RL78-045, Rutherford Lab. preprint (1978). [9] W.P. Petersen and J.L. Rosner, Phys. Rev. D6 (1972) 820; D. Faiman, Phys. Rev. D15 (1977) 854. [10] A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147.
301