V01ume 188, num6er 3 PHY51C5 LE77ER5 8 16 Apr11 1987

7HE 6 E N E R A 7 0 R C 0 0 R D 1 N A 7 E 5P1N AND 1505P1N P R 0 J E C 7 1 0 N 1N 7HE CH1RAL 8 A 6 PLU5 5KYRM10N HY8R1D M 0 D E L

A. H05AKA and H. 70K1 Department 0f Phy51c5, 70ky0 Metr0p011tan un1ver51ty, 5eta9aya, 70ky0 158, Japan

Rece1ved 17 N0vem6er 1986; rev15ed manu5cr1pt rece1ved 30 January 1987

We 5tudy the 6ary0n pr0pert1e5 1n the ch1ra1 6a9 p1u5 5kyrm10n hy6r1d m0de1. 70 de5cr16e the phy51ca1 6ary0n 5tate, we app1y the 9enerat0r c00rd1nate pr0ject10n meth0d (6CM) 1n5tead 0f the w1de1y u5ed 5em1c1a551ca1 c011ect1ve c00rd1nate meth0d (CCM). 7he 1ntr1n51c 5tate f0r the 5kyrm10n 15 c0n5tructed a5 the c0herent 5tate and the va1ence 4uark5 are treated exp11c1t1y. 7he nuc1e0n and de1ta ma55e5 and the ax1a1 c0up11n9 c0n5tant 9A are 1nve5t19ated. 7he ma55e5 are 519n1f1cant1y reduced fr0m the 1ar9e hed9eh09 ma55 due t0 the 4uantum effect5 1n the 6CM and a 5ma11 9A 1n the CCM 15 enhanced a5 a re5u1t 0f f1n1te Arc. A9reement w1th exper1menta1 data 0f the5e 4uant1t1e5 15 much 1mpr0ved c0mpared t0 the 5em1c1a551ca1 CCM.

7he ch1ra1 6a9 m0de1 0r191na11y 1ntr0duced 6y Ch0d05 and 7h0rn and 1ndependent1y 6y 1n0ue and Ma5kawa [ 1 ] ha5 recent1y 6een 5tud1ed exten51ve1y 6y a num6er 0f auth0r5 [ 2-4 ] w1th the deve10pment 0f the effect1ve the0ry 0f the 1ar9e Nc QCD [5]. 1t 1nc0rp0rate5 the free 4uark5 w1th1n the a5ympt0t1c re910n, the 6a9, and the p10n 0ut51de 1t. 5pec1f1ca11y the p10n 06ey5 the 5kyrme 1a9ran91an [6] wh1ch adm1t5 the 5011t0n 501ut10n at 2er0 6a9 rad1u5, 7he ch1ra1 1nvar1ance en5ure5 the ax1a1 f1ux c0nt1nu1ty at the 6a9 5urface. 1n the f0110w1n9, th15 m0de115 referred t0 the ch1ra1 6a9 p1u5 5kyrm10n hy6r1d m0de1 (C5H). 7ak1n9 1nt0 acc0unt 5er10u51y the ch1ra1 Ca51m1r effect5 1n the 4uark vacuum [7] due t0 the 5tr0n9 p10n f1e1d, the C5H m0de1 6ec0me5 we11 def1ned at a11 6a9 rad11 R and ha5 a 5m00th c0nnect10n t0 the 5kyrm10n at R--,0, wh11e 1n the 0ther 11m1t R-- ,~ 1t reduce5 t0 the M17 6a9 m0de1 w1th the weak p10n f1e1d [ 8 ].

U5ua11y th15 m0de1 15 501ved under the hed9eh09 an5at2 where the 5p1n and the 1505p1n are c0rre1ated 1n 5uch a way that the 9rand an9u1ar m0mentum K=1+1 van15he5, where J 15 the u5ua1 an9u1ar m0mentum and 1 the 1505p1n. 51nce the hed9eh09 5tate 15 a 5uperp051t10n 0fth05e 5tate5 w1th var10u5 1 and J, 1t d0e5 n0t c0rre5p0nd t0 the phy51ca1 6ary0n 5tate w1th the def1n1te 5p1n and 1505p1n. 70 9et the de51red 5tate5, 0ne 5h0u1d re5t0re the 6r0ken

0370-2693/87/$ 03.50 • E15ev1er 5c1ence Pu6115her5 (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)

8.V.

5ymmetry. W1de1y u5ed 15 the 5em1c1a551ca1 c011ect1ve c00rd1-

nate meth0d (CCM), 0r e4u1va1ent1y, the crank1n9 meth0d deve10ped ma1n1y 1n the nuc1ear phy51c5, where the hed9eh09 1ntr1n51c 5tate 15 treated c1a551- ca11y and the phy51ca1 6ary0n5 are de5cr16ed a5 1t5 r0tat10na1 exc1tat10n5. Adk1n5 et a1. app11ed th15 meth0d f0r the pure 5kyrm10n [9], wh11e recent1y Kahana et a1. and Ku5aka et a1. have perf0rmed 51m- 11ar ana1y5e5 f0r the C5H [ 10]. 7hey have f0und, h0wever, that 1f 0ne u5e5 the rea115t1c parameter5 e5pec1a11y f0r the p10n decay c0n5tant f~ = 93 MeV, the nuc1e0n and de1ta ma55e5 are 0vere5t1mated 6y a 519n1f1cant am0unt and the ax1a1 c0up11n9 9A 6ec0me5 5ma11. 7he5e 4uant1tat1ve d1ff1cu1t1e5 are n0w c1a1med a5 a re5u1t 0f the 5em1c1a551ca1 treatment 0f the pr0- ject10n meth0d. 1t rem1nd5 u5 0fthe 51tuat10n 0f the def0rmed nuc1e1, where the 9enerat0r c00rd1nate meth0d (6CM) 15 5ucce55fu1 e5pec1a11y f0r the weak1y def0rmed re910n and the CCM 15 rec0vered a5 1t5 5tr0n91y def0rmed 11m1t [ 11 ].

F1r5t 1et u5 c0n51der the ener9y 0f the 5y5tem. 1n the 6 C M the hed9eh09 1ntr1n51c 5tate 1h) 15 91ven exp11c1t1y a5 the 5uperp051t10n 0f var10u5 5p1n and 1505p1n 5tate5 11J)

1h >= ~a1j11J> . (1) / j

301

V01ume 188, num6er 3 PHY51C5 LE77ER5 8 16 Apr11 1987

7ak1n9 the matr1x e1ement 0f the ham11t0n1an, we 06ta1n

Mh = (h1H1h) = ~ 1a~J12 <1J1H11J), (2) /J

50 that the hed9eh09 ener9y 15 the we19hted avera9e 0ver the ener91e5 0f the 5tate5 11J). 7heref0re the ener9y 0f the 5tate w1th the 10we5t 5p1n and 1505p1n ( 1 = J = 1/2) 5h0u1d 6e 5ma11er than that 0f the hed- 9eh09 5tate. 0 n the 0ther hand, the ener9y 5pectrum 1n the CCM 15 91ven a5

E(1) =Mh +1(1+ 1 )/2A , (3)

where A 15 the m0ment 0f 1nert1a. 7h15 1mp11e5 E( 1/2) > Mh, wh1ch c0ntrad1ct5 e4. ( 2 ).

An0ther pr061em 15 f0und 1n the ax1a1 c0up11n9 c0n5tant 9A. 7he C5H m0de1 5h0u1d reduce t0 the M17 6a9 p1u5 weak p10n 5y5tem at R~00, 51nce the 1a9ran91an5 c01nc1de except f0r the 5kyrme f0urth 0rder term, the r01e 0f wh1ch d1m1n15he5 a5 the p10n c0ntr16ut10n decrea5e5. 7here 9A 15 exact1y 91ven a5 9A = 1.63= 1.5 9A(M17) 6ecau5e 0f the ax1a1 vect0r c0n5ervat10n [8], wh11e the CCM pr0v1de5 9A=0.98= ] X 1.63 [4]. A fact0r ] =Nf1(N¢+2) ha5 6een C1a1med 6y JaCk50n et a1. a5 the f1n1te N~ c0rrec- t10n [ 12 ]. 1n5tead, 1t wa5 ar9ued 1n ref. [ 13 ] that the 9enerat0r c00rd1nate pr0ject10n may natura11y repr0duce the c0rrect 9A at 1ar9e R. 7h15 ar9ument w111 6e exp11c1t1y pr0v1ded w1th numer1ca1 re5u1t5 1ater.

1n th15 1etter we w0u1d 11ke t0 pr0p05e the 9enera- t0r c00rd1nate meth0d ( 6 C M ) a5 a 5p1n and 1505p1n pr0ject10n meth0d 1n5tead 0f the 5em1c1a551ca1 c01- 1ect1ve c00rd1nate meth0d t0 0verc0me the5e d1ff1- cu1t1e5. 51nce the 6 C M need5 the fu11 4uantum 5tate t0 pr0ject 0ut the de51red 5tate, 1et u5 5tart w1th c0n- 5truct1n9 the hed9eh09 1ntr1n51c 5tate 1h), wh1ch 15 0ne 0f the centra1 155ue51n the f0110w1n9 d15cu5510n5. 1n the C5H m0de1 1 h ) 15 a pr0duct 0f the 4uark part 1 h4) and the p10n part 1 h~ ). 70 c0n5truct 1 h~ ) we 5ha11 4uant12e the 5kyrm10n 5y5tem.

We wr1te the 5kyrme 1a9ran91an a5

+ (114e2f~)[(9a60u(6a0u(1)6) 2

- - (9a60#(/)a0u~)6) 2] , ( 4 )

where the p10n decay c0n5tant f~ and the 5kyrme parameter e are taken f~ =93 MeV and e=4.5 a5 1n ref. [ 4 ]. A c0nven1ent ch01ce 0f the metr1c ten50r 9a6 15

9 a 6 = f ~ a 6 ~ 0 a ~ 6 / ( f ~ ~ 2 ) (5)

wh1ch y1e1d5 the 1a9ran91an wr1tten 1n term5 0f the 4uatern10n (a, ¢) w1th a 6e1n9 a funct10n 0 f ¢ 5uch that a = (f2 ~¢2 ) ~/2. N0te that the f0urth 0rder t1me der1vat1ve d15appear51n the 1a9ran91an (4). 7he can- 0n1ca1 m0mentum 15 def1ned a5 na=0£P5k0~a= Ma6~) 6 w1th Ma6 6e1n9 a funct10n 0f 0 a and 010 a. We then make the 4uant12at10n f0110w1n9 the can0n1ca1 4uant12at10n meth0d

[¢a(X), ff6(y)] =1~a6~(x-y) •

1n the u5ua1 5kyrme m0de1 0n1y the c1a551ca1 f1e1d 0¢ 15 91ven: (h~ 101 h~ ) = ~c. We pr0p05e t0 c0n51der here that the t0p01091ca11y n0ntr1v1a1 06ject (the 5kyrm10n 1n th15 ca5e) 15 a vacuum 5tate 0f a new pha5e, where 4~c 15 re9arded a5 an 0rder parameter 0f a new pha5e and 1h~ ) 15 the new vacuum 5tate. We may then f0110w the 5tandard pr0cedure 1n de5cr16- 1n9 the pha5e tran51t10n; 1.e. the 8C5 the0ry [14], and wr1te 1 h~ ) 1n term5 0f a c0herent 5tate

1h,~)=exp(1f~fd3xrta~a51n0)10), (6)

where the n0rma1 vacuum 10) 15 def1ned a5 (0 [ ~ 10) = (01 n 10) = 0. 7h15 5tate pr0v1de5 the c1a551ca1 hed9eh09 f1e1d; (h~ [ ~a [ h~ ) = ~¢~ = f1a 0(r), w1th 0(r) 6e1n9 the ch1ra1 an91e. N0te here that we are c0n5truct1n9 a var1at10na1 wave funct10n a5 a c0mp11cated 06ject fr0m the n0rma1 vacuum a5 the ca5e 0f 8C5 the0ry. 7he var1at10na1 pr1nc1p1e

(h~ 1H1 h~ ) = 0 re5u1t5 1n the we11-kn0wn d1fferen- t1a1 e4uat10n [9] f0r the ch1ra1 an91e 0(r). We n0te that e4. (6) 15 0ne 0f the many p05516111t1e5 0f var1a- t10na1 funct10n5, a5 u5ua1. 1ndeed, e4. (6) depend5 0n the ch01ce 0fthe metr1c. Furtherm0re, a d1fferent f0rm 0f the c0herent 5tate may 6e dev15ed t0 make an exact e19en5tate 0f the 6ary0n num6er, where the 5U(2) 9r0up current 15 u5ed 1n the exp0nent 1n5tead 0f the m0mentum [ 15 ]. A1th0u9h the5e var10u5 ch01ce5 91ve the 5ame re5u1t at the c1a551ca11eve1, they may pr0v1de d1fferent 4uantum effect5 f0r exc1ta- t10n5 5uch a5 r0tat10n. Here we 5ha11 a55ume that the

302

V01ume 188, num6er 3 PHY51C5 LE77ER5 8 16 Apr11 1987

e55ent1a1 feature 0f the 6CM d0e5 n0t depend much 0n th15 ar61trar1ne55.

Next 1et u5 c0n51der the 4uark part [h4). A5 the 5tr0n9 p10n f1e1d 1ndUCe5 the Ch1ra1 Ca51m1r effect, the hed9eh09 VaCUUm 5h0U1d 6e appr0pr1ate1y treated t0 make the C5H m0de1 We11 def1ned. Unf0rtUnate1y We d1d n0t reach, Up t0 n0W, a fU11 Under5tand1n9 0f 1t5 pr0per treatment Under the r0tat10n; We 5ha11 take 1nt0 aCC0Unt 0n1y the Va1enCe part a5

1h4 ) = [ (1/x/~)(uJ,--d1~)] 3 (7)

1n the ca1cu1at10n 0f the r0tat10na1 effect. 70 e5t1mate the 4uark vacuum c0ntr16ut10n due t0

the r0tat10n, we may rep1ace 1t 6y the ant15011t0n (1n 0ur ca5e the ant15kyrm10n) c0nf19urat10n 1n51de the 6a9, 01.(r) wh1ch 15 def1ned a5 the 501ut10n w1th 60undary c0nd1t10n 0 1 , ( r = 0 ) = 0 and 01,(r=R) = 0(R) [ 15 ]. 7h15 c0nd1t10n 1nd1cate5 that the 6ar- y0n num6er 0f the t0ta1 5y5tem 15 a1way5 kept 0ne c0m1n9 fr0m the va1ence 4uark5, wh11e the fract10na1 6ary0n num6er5 0f 5kyrm10n and ant15kyrm10n can- ce1 0ut. A5 f0r the ener9y, the 51m11ar1ty 6etween the ant15kyrm10n pre5cr1pt10n and the C5H m0de1 wa5 dem0n5trated a60ve a certa1n rad1u5 (R>0.4 fm) [ 16 ]. 51nce th15 pre5cr1pt10n 15 n0t certa1n, h0wever, we 5ha11 pr0v1de numer1ca1 re5u1t5 w1th0ut the r0ta- t10na1 effect 0f the vacuum c0ntr16ut10n and u5e the re5u1t5 0f th15 pre5cr1pt10n 0n1y a5 an 1nd1cat10n 0f the 512e 0f the r0tat10na1 effect 0f the vacuum c0ntr16ut10n.

0nce hav1n9 06ta1ned the fu11 1ntr1n51c 5tate 1 h) = 1 h~ ) [ h4 ), 0ne Can pr0ject 0Ut the de51red e19en 5tate W1th def1n1te 5p1n (Jm) and 1505p1n (/t) 6y the 6CM a5 [ 17 ]

18m) = ~ d [ 9 ] D~m(9)R(9)1h) , (8)

where 9 15 an e1ement 0f 5U (2) D1 1t5 repre5en- , 1, - - r t 1

tat10n and R(9) the r0tat10na1 0perat0r 1n the 150- 5pace. 1n 9enera1 0ne need5 d0u61e pr0ject10n 1n the 5p1n and 1505p1n 5pace, 6ut, due t0 the1r c0rre1at10n 1n the hed9eh09 5tate, 0ne 0f them dr0p5 0ut and the re5u1t1n9 5tate ha5 e4ua1 5p1n and 1505p1n 1= J.

W1th the 5tate (8), var10u5 phy51ca1 4uant1t1e5 are eva1uated u51n9 the techn14ue5 0f the 5U (2) a19e6ra and pr0pert1e5 0fthe c0herent 5tate. F0r 1n5tance the ener9y 0fthe p10n part w1th 1=J15 91ven a5 (f0r 51m- p11c1ty we take t = - m),

E~(1)=(~d[9] D~(9)

× (h1fd3x H~(x)R(9)1h) )

×( f d[9] D~ (h1R(9)1h ) ) -1 ,

where

11. = ~1caMa611~ 6 + • [ (010•) 2 + (0/#) 2 ]

(9)

+ (1/4e2f 4) [((01a) 2 + (01#)2) 2

- ( 0 , a 0 j a + 0 , # 0j#) ~] . (10)

N0w 1et u5 6r1ef1y exp1a1n the ca1cu1at10na1 pr0ce- dure. F1r5t we expand ~a and 7t a 6y the ma551e55 p1ane wave5 t0 rewr1te e4. (6) 1n the m0mentum 5pace. 7hen the p10n part 0f the 0ver1ap funct10n

(h1R(9) 1h) = (h~ 1R(9)[h~ ) (h4 1R(9)[h4 )

15 91ven a5

(h~ 1R(9)1h, ) = e x p ( - ~ f ~ fk3dk u2(k)

\ X [1--c052•p c052 ( ~ ( a + ~ ) ) ] ) ,

where u(k) 15 thej1 tran5f0rm 0f51n 0(r) and a, f1, y 15 the 5U (2) Eu1er an91e. At f1n1te R, 6ecau5e 0f the 5harp 1nCrea5e 0f 51n 0 at the 6a9 5urface, the 1nte9ra1 weak1y d1ver9e5. 7h15 6ehav10r 15 rem0ved 6y, f0r 1n5tance, the art1f1C1a1 5m00th1n9 0f 51n 0(r) (0r 0(r)). 5uch a ren0rma112ed 1nte9ra1 5h0u1d have a 5m00th c0nnect10n t0 the 5kyrm10n at R--*0, where the 1nte9ra1 15 we11 def1ned. A5 a typ1ca1 examp1e we 5h0w 1n f19. 1 the exact 501ut10n 0(r) and the 5m00thed 0ne 0(r) at R = 0.7 fm. R0u9h1y 5peak1n9, 1n the 5ma11 R re910n < 1 fm, the p10n 0ver1ap func- t10n (h= 1R(r) 1h~ ) 15 51m11ar t0 that 0fthe 5kyrm10n (R- ,0 ) and 1t 9radua11y appr0ache5 t0 1 a5 R 1ncrea5e5.

7he 1nte9rat10n 0ver x and 9 wh1ch 15, 1n 9enera1, 51x-f01d can 6e reduced t0 0ne rad1a1 and 0ne (0r tw0)

303

V01ume 188, num6er 3 PHY51C5 LE77ER5 8 16 Apr11 1987

0.) 1.0

C

<£

2 (3.5 (D

••Exact

0.5 1.0 1.5 r [ fm]

F19. 1.7he exact 501ut10n 0fthe ch1ra1 an91e 0(r) and the 5meared 0ne 0(r) at R = 0.7 fm f0r the ca1cu1at10n 0f the 0ver1ap funct10n.

Eu1er an91e 1nte9rat10n f0r the term depend1n9 0n 0n1y the ~ f1e1d. H0wever, f0r the 9 dependent term 0r the f1r5t term ~ ~2M-1~, fu11 51x-f01d 1nte9rat10n 15 nece55ary 51nce they can n0t 6e expre55ed a5 f1n1te p01yn0m1a15 0f ~. 70 av01d th15 c0mp1ex1ty we 51m- p1y dr0p the 9 dependence 1n t7 = (f2 ~ 2 ) 1/2. W1th th15 appr0x1mat10n we can reduce the 1nte9rat10n 0f the tr dependent term up t0 tracta61e d1men510n5. 7he ~2cM-~2 term, h0wever, 5t111 1nv01ve5 the c0up11n9 0f the x and 9 1nte9ra1 1n M - 1. We have e5t1mated 1t 6y 19n0r1n9 1t5 9 dependence and f0und that the c0n- tr16ut10n 15 5ma11, ar0und - 5 0 MeV. 7heref0re we 5ha11 dr0p th15 c0ntr16ut10n 1n the f0110w1n9 d15cu5510n5.

7he t0ta1 ener9y 0f the 5y5tem w1th 1=J c0nta1n5 the p10n part E~ and the 4uark part E4 1nc1ud1n9 the va1ence and the vacuum w1th0ut r0tat10na1 effect [3,4]

Et0t(1) =E4 + E,~(1) + ~ R 3 8 , (11)

where the v01Ume term 151nC1Uded w1th 6a9 C0n5tant 81/4--- 170 MeV. 1n f19. 2 we 5h0w the R dependence 0f the nuc1e0n and de1ta ma55e5 t09ether w1th the hed9eh09 ma55. A5 n0ted 1n the prev10u5 d15cu5510n, the nuc1e0n ener9y appear5 ar0und 1.2 6 e V 6e10w the hed9eh09 ma55 6y a1m05t 200 MeV, wh1ch 15 a 9reat advanta9e 0f the 6 C M . 1ndeed the CCM pre- d1ct5 the nuc1e0n ma55 ar0und 1.6 6 e V [ 10]. 1n the CCM the ener9y 5h1ft t0 the 10wer 51de 15 expected 6y the h19her 0rder 4uantum c0rrect10n5. 7he NA 5p11t- t1n9 15 dep1cted at the 60t t0m 0f f19. 2 6y the 5011d 11ne. A60ut 200 MeV 15 repr0duced at 5ma11 R and 1t

1 1 •

>Q 1.5 c~

R ~ ~NNuc[e0n 1.0

~- DeLta L1J

0.5

0.5 1.0 8a9 Rad1U5 R [ fm]

F19. 2.7he t0ta1 ener91e5 0f the nuc1e0n, the de1ta and the hed- 9eh09 5y5tem a5 a funct10n 0f the 6a9 rad1u5 R. 7he NA ma55 5p11tt1n915 a150 dep1cted at the 60tt0m 0fthe f19ure 6y 5011d 11ne. 70 9et the fee11n9 0f the vacuum c0ntr16ut10n, a 51mp1e e5t1ma- t10n 1n the ant15011t0n pre5cr1pt10n 15 added t0 the 5011d 11ne and dep1cted 6y da5hed 11ne.

m0n0t0n1ca11y decrea5e5 a5 R 1ncrea5e5.1f we add the vacuum c0ntr16ut10n 1n the ant15011t0n appr0x1ma- t10n, we f1nd a f1atter curve up t0 ar0und 0.5 fm a5 dep1cted 6y the da5hed 11ne. 1n 0ur e5t1mat10n, the ma55 5p11tt1n9 15 5ma11er than the exper1menta1 data. 8ut we n0te that the r0tat10na1 effect5 0f a have 6een ne91ected. Appr0pr1ate treatment 0f th15 term 15 expected t0 1ncrea5e 1t and 1ar9e1Y 15 a c0n5e4uence 0f the r0tat10na1 m0t10n 0f the ••def0rmed p10n••. Acc0rd1n91Y the 91u0n1c c0ntr16ut10n and the c010r c0up11n9 c0n5tant ac 6ec0me5 5ma11.

A5 an0ther examp1e we 5tudy the ax1a1 c0up11n9 9A. 7he def1n1t10n 15 [ 9 ]

(N1cr1~2a1N>9A=~(N1~d~xAa(x) 1N> , (12)

where A a 15 the ax1a1 vect0r current and the fact0r 3/2 15 a c0n5e4uence 0f the ax1a1 vect0r c0n5ervat10n. After 50me a19e6ra w1th carefu1 treatment 0f the 4uark 5p1n and 1505p1n 5tructure we f1nd [ 18 ]

304

V01ume 188, num6er 3 PHY51C5 LE77ER5 8 16 Apr11 1987

2 .0

c 1.5 tm

0

0

X

<

0.5

Weak C0upL1n9

1

5tr0n9 C0upL1n9

1 1

0.5 1.0 8 a 9 Rad1u5 R [fm]

F19. 3.7he ax1a1 c0up11n9 c0n5tant 9A a5 a funct10n 0f the 6a9 rad1u5 R. 7he upper and the 10wer da5hed 11ne5 are th05e 0f the weak and the 5tr0n9 c0up11n911m1t5, re5pect1ve1y.

9A=(fd[9](C4+C252)(h~1R(9)1h~5)

×(fd[9]C4(h~1R(9)1h~)) -~

x ~ a , (13)

C=c 05(~ (0~+7)) , 5=51n(~f1) ,

where a determ1ne5 the a5ympt0t1c f0rm 0f 0(r) at 1nf1n1ty a5 0(r-,a/r 2. An 1ntere5t1n9 p01nt 15 that th15 ha5 5m00th c0nnect10n5 t0 60th 9A 1n the 5em1c1a55- 1ca1 CCM [ 4,10 ] and that 0f the M17 p1u5 weak p10n m0de1 [8,13]. 7he f0rmer 15 ach1eved 1f the p10n 15 fu11y def0rmed a5 (h~ 1R(9)1h~ ) ~ ~(9) wh1ch c0r- re5p0nd5 t0 the 5tr0n9 c0up11n911m1t, wh11e the 1atter c0rre5p0nd5 t0 the weak c0up11n9 11m1t (h~ 1R(9)1h~ ) = 1 .7he5e re1at10n5 are we11 kn0wn fr0m the 5tudy 0f the def0rmed nuc1eu5 [ 1 1 ]. 7he d1fference 6etween the5e extreme5 15 a fact0r 5/3, wh1ch can 6e ea511y 06ta1ned fr0m e4. (13). 7he rea1- 1ty w0u1d 11e 1n6etween. We 5h0w 1n f19. 3 9A a5 a funct10n 0f R t09ether w1th th05e 0f the 5tr0n9 and the weak c0up11n9 11m1t5. At 5ma11 R < 0 . 5 fm the

a9reement t0 the exper1menta1 va1ue 9A(exp) = 1.25 15 06ta1ned. A 5ma11 9A 1n the CCM ar0und 0.9 15 enhanced t0 a60ut 1.3.1t 151ntere5t1n9 t0 5ee that the rea1 w0r1d Nc = 3 15 rather c105e t0 the weak c0up11n9 11m1t and the f1n1te Nc c0rrect10n 0r191na11y 1ntr0- duced 6y Jack50n et a1. [ 12] may 9et a 5upp0rt 1n th15 way. M0re 0n th15 p01nt w111 6e deta11ed 1n an0ther paper [ 18 ].

1n 5ummary we have pr0p05ed the 9enerat0r c00r- d1nate 5p1n and 1505p1n pr0ject10n meth0d f0r the ch1ra16a9 p1u5 5kyrm10n hy6r1d m0de11n5tead 0f the 5em1c1a551ca1 c011ect1ve c00rd1nate meth0d. 7h15 w111 6e an0ther p055161e way t0 re5t0re the 6r0ken 5ym- metry and 1nc0rp0rate5 appr0pr1ate 4uan tum effect5 wh1ch 15 n0t 1nc1uded 1n the 5em1c1a551ca1 meth0d. 7hen we 5tud1ed the ener9y 0f the nuc1e0n and de1ta and the ax1a1 c0up11n9 c0n5tant. 80 th are we11 repr0- duced 6y the 6 C M rather than 6y the CCM.

1n 0rder t0 c0n5truct the hed9eh09 p10n wave funct10n f0r the ca5e 0f the 6 C M ca1cu1at10n, we have pr0p05ed t0 c0n51der the t0p01191ca1 06ject (5kyr- m10n) a5 a vacuum 5tate 0f a new pha5e w1th the c1a551ca1 p10n f1e1d a5 the 0rder pa ramete r 0f the new pha5e. 7h15 p01nt 0u9ht t0 6e 1nve5t19ated 1n the future. 0 n c e th15 a55umpt10n 15 accepted, the ca1cu- 1at10na1 pr0cedure 15 5tra19htf0rward and we have pr0v1ded the numer1ca1 re5u1t5 0n the nuc1e0n and de1ta ma55e5 t09ether w1th the ax1a1 c0up11n9 9A. 1n add1t10n, we w0u1d 11ke t0 ment10n that we need exper1menta1 5upp0rt 0n the u5e 0f the hed9eh09 an5at2 f0r the 1ntr1n51c 5tate 0 f t h e 6 C M pr0ject10n, wh1ch 15 a detect10n 0f the 1= J = 5/2 5tate at the ma55 M(1=J= 5/2) ~ 1700 MeV.

We thank Dr. H. M1nakata and Dr. 7. Hat5uda f0r var10u5 d15cu5510n5. 7he numer1ca1 ca1cu1at10n5 were per f0rmed 6y F A C 0 M M380 at 1N5, un1ver51ty 0f 70ky0.

Reference5

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V01ume 188, num6er 3 PHY51C5 LE77ER5 8 16 Apr11 1987

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