The Electric Charge Distribution of the Lambda Particle

Transcription

1 427 Progress of Theoretical Physics, Vol. 90, No.2, August 1993 The Electric Charge Distribution of the Lambda Particle Seiichi Y ASUMOTO and Michihiro HIRATA Department of Physics, Hiroshima University, Higashi-Hiroshima 724 (Received January 25, 1993) We get the electric charge distribution of the lambda particle in 5U(3) Skyrme model using the "collective approach". We compare it with the electric charge distribution obtained in the nonrelativistic quark model. It is pointed out that the electric charge distribution of the lambda from the collective approach has. an opposite sign to those obtained from the non-relativistic quark model and the bound state approach in the 5U(3) Skyrme model. 1. Introduction The non-relativistic quark modef) is one of the effective baryon models at low energy_ This model can reproduce very roughly the experimental charge distribution for a proton, but gives too small an electric charge radius. A neutron consists of one u quark and two d quarks. If we adopt the usual non-relativistic wave function for the neutron, it trivially gives a zero electric charge density_ However, the experimental results show that its charge distribution has a positive charge near the center and a negative charge in the outer side. These results indicate that we must consider the virtual processes with the degree of freedom for' the meson, that is, the virtual processes in which the proton decays to a neutron and a 7(+ meson, and in which the neutron decays to a proton and a 7(- meson_ However, it is difficult to estimate such virtual processes in the non-relativistic quark model. To avoid the calculations for the contributions of such virtual processes, we take notice of a lambda particle by the following reasons. The virtual processes in the lambda particle are process (a) in which the A decays to a proton and a K- meson, process (b) to a S- and a K+ meson, process (c) to a X+ and a 7(- meson and processs (d) to a X- and a 7(+ meson. Because the transition amplitudes of process (a) and process (b) are numerically almost equal in the non-relativistic quark model, the baryon currents from both processes almost cancel each other, and the meson currents cancel each other. The baryon currents from process (c) and process (d), needless to say, cancel each other, and the meson currents also cancel each other. Therefore, it is considered that the static properties of the lambda particle are obtained from the three body problem of the u, d and s quark. In this sense, it is expected that the non-relativistic quark model gives the meaningful result for the electric charge distribution of the lambda particle. The Skyrme model has very different structure from the non-relativistic quark model. In this model, the baryons are described as the solitons from the nonlinear (] model, and do not include any quark. The 5U(2) Skyrme model has been rather successful in discribing the static properties of a nucleon and a delta particle_ This model can reproduce qualitatively the experimental charge distributions of the proton and the neutron. 2 ),3) Therefore, it is expected to obtain the qualitatively right charge distribution of the lambda particle by extending the 5U(2) Skyrme model to the

2 428 S. Yasumoto and M. Hirata SU(3) model. 4 )-8) If so, it may be meaningful to compare with the charge distribution in the non-relativistic quark model. To calculate the electric charge distribution of the lambda particle, we use the SU(3) "collective approach" and a classical soliton solution of the hedgehog type which is embedded in the upper SU(2) subgroup.s) One of the parameters, the pion decay constant, which were chosen in Ref. 8) so as to give the best fit for hyperron spectra is about three fourth times smaller than that from the SU(2) Skyrme model. The pion decay constant which is determined so as to fit the nucleon and delta masses is still about two third times smaller than the physical value 93 MeV. The theoretical pion decay constant may increase if the quantum corrections are added to the classical soliton mass. The difference of the pion decay constant between the SU(2) and SU(3) models might be able to be reduced by taking into account the quantum corrections. Anyway the qualitative feature of the charge distribution for the lambda does not depend much on the value of the pion decay constant, so we adopt the best fit parameter set in Ref. 8).. In 2, we get the electric charge distribution of the lambda particle in the non-relativistic quark model. In 3, we review the SU(3) Skyrme model by the SU(3) "collective approach" and present the electric charge distribution of the lambda particle. Finally, we summarize in Non-relativistic quark model To obtain the electric charge distribution of the lambda particle, we use the wave function for the Is state baryons/) (2 1) where (]);(ri) is the orbital wave function of ith quark with the Gaussian type. xl and x/ s are flavor and color-spin wave functions, respectively, and A is the antisymmetrizer. When we use this wave function, the electric charge density of the lambda particle is given as follows: 1 _ 2 1 [( 2a M )3/ 2 { (F"e? r 3 --;r mu+ms exp ( a msm )3/2 { - 7r mu2 exp 2aM 1-2} mu+ms (F"e)2 r (2 2) where the symbol l' stands for the spin-up state, S3=1/2. mu and ms are the masses of u- and s-quark, respectively. Q is the SU(3) quark charge matrix. F" and e are defined in the next section. The charge distribution of the lambda particle in this model is shown in Fig. 1. To obtain this distribution, we used the parameter set in Ref. 1), mu=362 MeV, ms=705 MeV and a=1.9 MeV 2 Values of F" and e are given in the next section. The behavior of charge distribution shown in Fig. 1 can be easily understood

3 The Electric Charge Distribution of the Lambda Particle 429 because the mass of s quark which has a negative charge is about two times heavier than those of u and d quarks. Even if a baryon is relativistically treated, the situation that the heavier quark comes to the center of the baryon will be kept. 3. SU(3) Skyrme model We use the SU(3) Skyrme model lagrangian described by the SU(3) chiral field, U(x ),8) L=L(Sk)+ L(SB) + L(wz), (3'1) NON-RELA. QUARK MODEL ~ lambda 2 ~O.05 o 2 3 (fm) Fig. 1. The radial charge distribution p( r) of the A particle in the non-relativistic quark model. where Lp is called the left-current, L p= Ut(opU). The chiral field is written as (3 2) (3'3) (3'4) (3'5) where the p(x),s are the pseudo-scalar octet meson fields and the I\p'S are the Gell-Mann matrices. The first term in the L(Sk) is the kinetic term of the skyrmion and the second is the term to stabilize the skyrmion. The constants Frc and e are the pion decay constant and the non-dimensionless parameter, respectively. When the symmetry breaking term L(SB) is expanded in the power of p(x), it becomes the mass term of the octet mesons. The Wess-Zumino term, L(wz), was added by Witten, which is related to the baryon number. We use the following hedgehog ansatz for the stationary skyrmion,. ( )=(cosf(r)+ ir' rsinf(r) 0) U a x 1 ' (3'6) where r = r/r. The boundary condition of F(r) for the baryon number = 1 is F(O)=7r

4 Yasurnoto and M. Hirata and F(=)=O. The hedgehog ansatz (3 6) transforms according to Uo(x)----> U(x, t)=a(t)uo(x)at(t), (3 7) where AU) is the 5U(3) rotation matrix as the collective coordinate. By substituting Eq. (3 7) into Eq. (3.1), the lagrangian can be separated to the classical part Lc( Uo, (;0) and the quantum fluctuational part LQ(A, A). The chiral angle F(r) is determined from the variational equation for the classical part, setting F(O)=7r and F(=)=O. Its solution, F(r), guarantees that the hedgehog solution (3 6) is the stationary soliton solution. The hamiltonian obtained by the collective coordinate quantization method is given as follows: H=Ho+Hl, (3 8) 1(1 1) 3 1 Ho=Mcl+Z (7-7 C2(5UR(2))- 8(32 + 2(32 C2(5UR(3)), (3 9) HI = ~ r(l- D~~J), (3 10) (3 11) (3 12) (3 l3) - 47r 1 (2 2) r=d- -2(1 F) r-t FI1:e3 mk -mit)o rr -cos, (3 14) where, r =F"er, F'=dF/dr. C2(5UR(2)) and Cz(5UR(3)) are the 5U(2) and 5U(3) Casimir operators, respectively. D~~J is the function defined by At ilaa = D~~~ilp with ct, (3=1 ~8. According to Ref. 9) we define the baryon states as follows, IlJfs(~»=( -1)s-s3rnD~~l, =( -1)S-s3rnD(~~r,I3),(1,S,-S3), (3 15) (3 16) where the function D defined in Eq. (3 16) is the 5U(3) D function. 1o ),1l) The nota tions (Y, 1, 13) and (1,5, -53) are (hyperchange, isospin, the third component of isospin) and (baryon number, spin, the third component of spin), respectively. The wave function of the lambda particle with the spin-up component is 11 (n» - CD(n) 1Ll1/2 -'II n 8,6-;7, -'II - CnD(n)* n (0,0,0),(1,1/2,-1/2), (3 17) (3 18)

5 The Electric Charge Distribution of the Lambda Particle 431 where D ~d-i7=d 'J'>-iD ~i. The n is taken as n=8 for the octet baryons. We obtain the electromagnetic current from the lagrangian (3 '1) using the Gell-Mann-Levy procedure. IZ ) By this procedure, the electromagnetic current is written as the following form, (3'19) where L p= Ut(opU) and Rp= U(opU t ). is where (3 20) (3 21) For this current, the electric charge density7) p( :r)=f e[kc :r) -6 D(S) R + G( :r)..,; D(S) R +_3- D (SlB( :r)] " K..oJ 3,a a G..oJ 3,b b 2;-;)3 3,8 a=1 b=4 V0 + F"e [K( r) -6 D(S) R + G(:r)..,; D(S) R +-.id(slb( -)] ;-;)3 K..oJ S,a a G..oJ S,b b 2 S,8 r, V0 a=1 b=4 (3 22) (3 23) (3 24) (3 25) and the Ra's in Eq. (3' 22) are the right operators defined in Refs. 6) and 7). Making the wave function of the lambda particle,.we take into consideration the two states of 8 and 27 representation, (3 26) where the coefficients as and az7 are the probability amplitude. The wave function (3 26) obtained by the diagonalization of the hamiltonian (3 8) can reproduce values of Case (3) in Table I, in Ref. 8). As the result, the radial charge distribution of the lambda particle is the following form (see the Appendix): <A I ( -)IA )-( )ZF [ 3 K(r)+ 2 G(:r) 1 2. zf F'] I/Z P r I/Z - as "e g- 2O-G--20nsm.

6 432 S. Yasumoto and M. Hirata To obtain this expression, we used the SU(3) Clebsch-Gordan coefficients. 1O ),13) The charge distribution of the lambda particle, obtained from Eq. (3 27), is shown in Fig. 2. To obtain this charge distribution, we used the best fit parameter set in Ref. 8), namely, F,,-=82.9 MeV, e=4.87 and mk=769 MeV. Then, the probability amplitudes as and a27 are and 0.338, respectively. In the SU(3) collective approach model we use, the classical SU(2) solution is embedded in the SU(3) group and the soliton deformation into strange directions is ignored. So, the charge > Q) ~ ~ o SU(3) SKYRME MODEL lambda 2 4 (fm) density of the lambda consists of the Fig. 2. The radial charge distribution p( r) of the proton charge density and that due to A particle in the Skyrme model using the the K- meson fluctuation. The proton "collective approach". is heavier than the K- meson. Therefore, the lambda has a positive charge distribution near the center of particle and a negative one in the outer side of it. 4. Summary The non-relativistic quark model predicts the situation that the lambda particle has a negative charge distribution near the center of the particle and a positive charge distribution in the outer side. In other words, since the mass of s quark is 2 times heavier than those of u and d quarks, the s quark is distributed mainly near the c.m. of three quarks, and the u and d quarks are distributed outside the s quark. This behavior is intuitively understandable. In the SU(3) Skyrme model using the "collective approach", the form of the charge distribution for the lambda particle is similar to that for the neutron as shown in Fig. 2. In fact, restricting the wave functions of the lambda particle and the neutron to the octet representation, we get the following relation, (4 1) Equation (4,1) is independent of the parameter set, F,,-, e and mk, and is only determined by the relation between the SU(3) D-functions and the SU(3) Clebsch Gordan coefficients. We adopted the "collective approach" in treating the SU(3) Skyrme model. The calculations of the charge distribution for the lambda particle using the "bound state approach" have been reported by Kunz, Mulders and Miller. 14) The "bound state approach" treats the kaon fields as the fluctuations around the classical hedgehog pion

7 The Electric Charge Distribution of the Lambda Particle 433 field. Therefore, the distribution of the kaon fields is seen clearly. According to the result of "bound state approach", the kaon fields come inside a SU(2) Skyrme soliton. The nucleon as the SU(2) Skyrme soliton is an extended object, so the kaon field can penetrate inside the soliton. The charge distribution of the lambda particle becomes negative near the center of the particle and positive in the outer side of it by such an effect of the kaon fields. As the result, the charge distribution of the lambda particle in the "bound state approach" is similar to the result in the non-relativistic quark model. It is interesting that the kaon fields penetrate deeply enough to exceed the charge of the central part of the soliton unlike the case of K mesonic atom. Though the charge distribution for the lambda particle in the "collective approach" is different from that in the "bound state approach", the calculations of magnetic moments due to "both approaches" give the negative values which are in good qualitative agreement with the experimental magnetic moment. Expressed in nuclear magnetons 1/2M N the magnetic moment of the lambda particle in the "bound state approach" is / 5 ) and in the "collective approach", using the wave function (3-26), is Therefore, we cannot judge from the magnetic moment which approach is superior. It seems that the bound state approach is more reasonable than the collective approach from the following points of view. (1) The charge distribution from the bound state approach qualitatively shows the same behavior as that from the non-relativistic quark model, while the charge distribution from the collective approach shows the opposite behavior. (2) The pion decay constant used as the best fit parameter in the bound state approach is almost the same as that used in the SU(2) Skyrme rriodel,16) while the pion decay constant used as the fitting parameter in the collective approach is smaller than that from the SU(2) model. Appendix In this appendix, we present concretely the relations between the right operators in Eq. (3-22) and the wave function (3-26). The wave functions for 8 and 27 representation of the lambda particle are IAfm=j8D<,g:0~0),(I,1/2,-1/2), IA (27»- r;:;;:;27d(27)* 1/2 -" L, I (0,0,0),(1,1/2,-1/2) from Eq. (3-18). The operators R 1±i2 and R3 are connected with the spin operators, Sa(a=l, 2, 3), S±= - R1±i2, S3=-R3, where S± are the raising and lowering operators for spin. Ra(a=l, 2, 3) behave like the spin operators for the wave function. Rs corresponds to the baryon number operator B, Rs=(13/2)B, R (A (n»_i3ia(n» s 1/2-2 1/2

8 434 S. Yasumoto and M. Hirata The remaining relations between these wave functions and the right operators are defined as follows, R4+i5IAi~~> = /8Dt3:0;0),(O,l,-1), R4-i5IAi~~>=O, R IA(8»- 2(D(8)* D(8)* ) 6+i7 l/z - - (0,0,0),(0,1,0) -, (0,0,0),(0,0,0), and R 4+i5 IA (Z7»-6 1n2D(Z7)* l/z - V L, (0,0,0),(0,1,-1), R IA (Z7»-3 1r5D(Z7)* 4- i5 l/z - v;) (O,O,O),(Z,l,O), R IA(Z7»--6(D(Z7)* _D(Z7)* ) 6+i7 l/z - (0,0,0),(0,1,0) (0,0,0),(0,0,0), R 6-i7 IA (Z7»-3 r;r:.lod(z7)* l/z - V 1V (O,O,O),(Z,l,-l), References 1) N. Aizawa and M. Hirata, Prog. Theor. Phys. 86 (1991), ) G. S. Adkins, C. R. Nappi and E. Witten, Nue!. Phys. B228 (1983), ) G. S. Adkins and C. R. Nappi, Nue!. Phys. B233 (1984), ) P. O. Mazur, M. A. Nowak and M. Praszalowiez, Phys. Lett. B147 (1985), ) A. P. Balaehandran, F. Lizzi and V. G. ]. Rodgers, Nue!. Phys. B256 (1985), ) N. Toyota, Prog. Theor. Phys. 77 (1987), ) A. Kanazawa, Prog. Theor. Phys. 77 (1987), ) H. Yabu and K. Ando, Nue!. Phys. B301 (1988), ) E. Guadanini, Nue!. Phys. B236 (1984), ) ].]. De Swart, Rev. Mod. Phys. 35 (1963), ) D. F. Holland, ]. Math. Phys. 10 (1969), ) M. Gell-Mann and M. Levy, Nuovo Cim. 16 (1958), ) P. McNamee and F. Chilton, Rev. Mod. Phys. 36 (1964), ) ]. Kunz, P. ]. Mulders and G. A. Miller, Phys. Lett. B255 (1991), ) ]. Kunz and P. ]. Mulders, Phys. Rev. D41 (1990), ) c. G. Callan, K. Hornbostel and 1. Klebanov, Phys. Lett. B202 (1988), 269.