New sphalerons in the Weinberg-Salam theory

V01ume 216, num6er 3,4

PHY51C5 LE77ER5 8

12 January 1989

NEW 5 P H A L E R 0 N 5 1N 7 H E W E 1 N 8 E R 6 - 5 A L A M 7 H E 0 R Y J. K U N 2 1 A7KHEF-K, P. 0. 80x 41882, NL- 1009 D8 Am5terdam, 7he Nether/and5

and Y. 8R1HAYE Phy514ue 7h00r14ue et Math0mat14ue. Un1ver51t0de/•Etat 71M0n5, 8-7000 M0n5, 8e191um

Rece1ved 14 Ju1y 1988, rev15edmanu5cr1pt rece1ved 19 5eptem6er 1988

We d15c0vered new 5pha1er0n5 1n the We1n6er9-5a1am the0ry, 1n the 11m1t0f van15h1n9 m1x1n9 an91e. Dependent 0n the ma55 0fthe H1995f1e1d,a wh01e 5et 0f 5add1e-p01nt 501ut10n5 ex15t5. 7he Da5hen-Ha551acher-Neveu 5pha1er0n f0rm5 the 6a51c 6ranch 0f501ut10n5 fr0m wh1ch, at cr1t1ca1va1ue5 0fthe H1995ma55, new 6ranche5 0f501ut10n5 5y5temat1ca11yemer9e.

1.1ntr0duct10n 7 h e 5tandard m0de1 5UCCe55fU11y 5Ummar12e5 0Ur pre5ent Under5tand1n9 0 f t h e 5tr0n9 and e1eCtr0Weak 1nteraCt10n5. 1t 15 6a5ed 0n the 9aU9e pr1nC1p1e and the C0nCept 0f 5p0ntane0U5 5ymmetry 6reak1n9. 1n part1CU1ar, the e1eCtr0Weak 1nteraCt10n5 are de5Cr16ed 1n term5 0 f a 5p0ntane0U51y 6r0ken 9aU9e the0ry, the We1n6er9-5a1am the0ry. 8Ut a150 the 9rand Un1f1ed the0r1e5 are f0rmU1ated a5 5p0ntane0U51y 6r0ken 9aU9e the0r1e5. 7 h e phy51Ca1 re1evance 0f 5p0ntane0U51y 6r0ken 9aU9e the0r1e5 ha5 5pUrred 9reat 1ntere5t 1n the 5earCh f0r n0n-pertUr6at1Ve 5trUCtUre5 pre5ent 1n the5e the0r1e5. 5uch n0n-pertur6at1ve 5tructure5 are 6a5ed 0n the 5et 0f n0n-11near c1a551ca1 e4uat10n5 0fm0t10n, t0 wh1ch they f0rm 501ut10n5 w1th f1n1te ener9y 0r f1n1te act10n. 7 h e ma1n cate90r1e5 0f5uch n0n-pertur6at1ve 501ut10n5 are 5011t0n5 and 5pha1er0n5 1n M1nk0w5k1 5pace and 1n5tant0n5 1n euc11dean 5pace. 5011t0n5 c0rre5p0nd t0 m1n1ma 0 f t h e ener9y funct10na1.7hey are character12ed 6y a c0n5erved char9e, wh1ch m a y 6e 0f t0p01091ca1 nature. 1n the 5em1c1a551ca1 expan510n 0 f t h e 4 u a n t u m the0ry they are 1nterpreted a5 91v1n9 r15e t0 part1c1e-11ke 5tate5. 7 h e m05t W0rk 5upp0rted 6y DF6. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 • E15ev1er 5c1ence Pu6115her5 8.V. ( N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n )

pr0m1nent t0p01091ca1 5011t0n 1n a 5p0ntane0u51y 6r0ken 9au9e the0ry 15 the •t H00f1-P01yak0v m0n0p01e [1] 0f an 5 U ( 2 ) 9au9e the0ry w1th a H1995 tr1p1et. 1n c0ntra5t, 5pha1er0n5 c0rre5p0nd t0 5add1e p01nt5 0f the ener9y funct10na1, wh1ch make5 a part1c1e-11ke 1nterpretat10n m05t un11ke1y. 5pha1er0n5 have, h0wever, 6een a550c1ated w1th v a c u u m t0 v a c u u m tran51t10n5 and ferm10n n u m 6 e r v101at10n. 7 h e m05t pr0m1nent 5pha1er0n 15 the Da5hen-Ha551acherNeveu ( D H N ) 5pha1er0n [2] 0f an 5 U ( 2 ) 9au9e the0ry w1th a c0mp1ex H1995 d0u61et. A1th0u9h the •1 H00ft-P01yak0v m0n0p01e and the D H N 5pha1er0n were d15c0vered at the 5ame t1me, 0n1y the part1c1e-11ke 5011t0n5 then rece1ved attent10n. 1t wa5 the 1mp0rtant w0rk 0 f 7 a u 6 e 5 [ 3 ], wh1ch f1na11y 9enerated 1ntere5t 1n 5pha1er0n5. 7au6e5 r190r0u51y pr0ved the ex15tence 0f an un5ta61e 501ut10n 1n the 2er0 m0n0p01e 5ect0r 0 f a n 5 U ( 2 ) 9au9e the0ry w1th a H1995 tr1p1et, 1n the 11m1t 0fvan15h1n9 H1995 ma55 [ 3 ]. 7h15 501ut10n wa5 phy51ca11y 1nterpreted a5 a m0n0p01e-ant1m0n0p01e pa1r. Mant0n [4] then app11ed a 51m11ar rea50n1n9 t0 dem0n5trate the ex15tence 0 f a 5add1e-p01nt 501ut10n, a 5pha1er0n [ 5 ], 1n the We1n6er9-5a1am the0ry. 1n the 11m1t 0f van15h1n9 We1n6er9 an91e th15 501ut10n reduce5 10 the D H N 5pha1er0n. 353

V01ume216, num6er 3,4

PHY51C5 LE77ER5 8

7he phy51ca1 pr0pert1e5 0f the 5pha1er0n 1n the We1n6er9-5a1am the0ry and 1t5 1mp11cat10n5 f0r 6ary0n decay have recent1y rece1ved much attent10n [ 59 ]. 7here rema1n5, h0wever, the 1ntere5t1n9 4ue5t10n 0f whether the We1n6er9-5a1am the0ry p055e55e5 m0re un5ta61e 5tat1c 501ut10n5. 0 n the 0ne hand 7au6e5 recent1y pr0ved the ex15tence 0f a wh01e 5et 0f un5ta61e 501ut10n5 1n the 2er0 m0n0p01e 5ect0r 0f an 5U (2) 9au9e the0ry w1th a H1995 tr1p1et, f0r van15h1n9 H1995 ma55 [10]. 7h15 5u99e5t5 that a150 the We1n6er9-5a1am the0ry may have an ana1090u5 5et 0f 5pha1er0n5.0n the 0ther hand D H N have a1ready 5u99e5ted that there c0u1d 6e add1t10na1 5tat1c 501ut10n5 0f the f1e1d e4uat10n5 f0r a m0re 9enera1 an5at2 [ 2 ]. 7hu5 the We1n6er9-5a1am the0ry may p055e55 a r1ch var1ety 0f n0n-pertur6at1ve 501ut10n5. 1n th15 1etter, we 5h0w that the We1n6er9-5a1am the0ry, 1n the 11m1t0fvan15h1n9 We1n6er9 an91e, d0e5 1ndeed p055e55 add1t10na15add1e-p01nt 501ut10n5.8ut un11ke the 5et 0f 5pha1er0n5 expected 1n the van15h1n9 H1995 ma55 11m1t 1n ana109y w1th 7au6e5• 501ut10n5, the ex15tence 0fthe 501ut10n5 f0und here cruc1a11ydepend5 0n the H1995 ma55 a5 an externa1 parameter 0f the the0ry. We dem0n5trate that the DHN 5pha1er0n pre5ent5 a 11m1t1n9 501ut10n fr0m wh1ch new 10wer ener9y 5add1e-p01nt 501ut10n5 6ranch 0ff, when the H1995 ma55 15 1ncrea5ed. A 51m11ar emer9ence 0f 5add1e-p01nt 501ut10n5 fr0m a 11m1t1n9 - 6ut c0n5tant 501ut10n a5 a funct10n 0f an externa1 parameter wa5 recent1y f0und 6y Mant0n and 5am015 [ 1 1 ] 1n the1r 1ntere5t1n9 5tudy 0f 5pha1er0n5 1n a 04 the0ry 0n a c1rc1e. 7he new 5et 0f 501ut10n5 15 6a5ed 0n the m0re 9enera1 an5at2 6y DHN [ 2 ], 5t1111ead1n9 t0 a 5et 0f n0n11near 0rd1nary d1fferent1a1 e4uat10n5.1n 5ect10n 2 we d15cu55 the an5at2, the e4uat10n5 0f m0t10n and the 60undary c0nd1t10n5.1n 5ect10n 3 we pre5ent the new 5et 0f 501ut10n5 and a 5y5temat1c 1terat1ve meth0d 0f c0n5truct10n. We n0te that the 10we5t new 5pha1er0n appr0ache5, f0r 1ar9e va1ue5 0f the H1995 ma55, the E11am-K1a6ucar-5tern (EK5) 501ut10n [2-14 ], 1n wh1ch the 1en9th 0fthe H1995 f1e1d wa5 fr02en. 1n 5ect10n 4 we pre5ent 0ur c0nc1u510n5. 1n part1cu1ar, we p01nt 0ut the re1evance 0f the new 501ut10n5 1n effect1ve the0r1e5 0f the weak and 5tr0n9 1nteract10n5, where the 5add1e p01nt5 can chan9e t0 10ca1 m1n1ma, a110w1n9 f0r a part1c1e-11ke 1nterpretat10n 0f the 501ut10n5. 354

12 January 1989

2. C1a551ca1 e4uat10n5 We c0n51der the 5 U ( 2 ) L × U ( 1 ) v 10ca11y1nvar1ant We1n6er9-5a1am the0ry [ 15 ] 1n the 11m1t 0f van15h1n9 m1x1n9 an91e 0w. 7hen the U ( 1 ) f1e1d dec0up1e5 fr0m the heavy 5U(2) vect0r 6050n5 and may c0n515tent1y 6e 5et t0 2er0.7he We1n6er9-5a1am the0ry thu5 reduce5 t0 an 5U (2) 9au9e the0ry w1th a c0mp1ex H1995 d0u61et 0. 7he 1a9ran91an read5

1

~ = - -292 - 7r ( F , , F ~ " ) + (D,0)~(D,0) --/~ ( 0 ~ ) - - 1U2) 2

(1)

w1th the c0var1ant der1vat1ve

D,0=(0,,-1V,,)0

(2)

and the f1e1d 5tren9th ten50r

F,,=0, V , - 0 , V,-1[V,, V,1.

(3)

7he 9au9e 5ymmetry 15 5p0ntane0u51y 6r0ken v1a the H1995 p0tent1a1, 1ead1n9 t0 the H1995 f1e1d vacuum expectat10n va1ue

v (0)

<0>=~

.

(4)

7he ma55e5 0f the W 6050n and the H1995 6050n are

Mw = •9v and Mn = , , ~ v, re5pect1ve1y. 70 c0n5truct c1a551ca1 501ut10n5 t0 the f1e1d e4uat10n5 we ch005e the 9enera1 an5at2 5u99e5ted 6y DHN [2 ]. 7he an5at2 f0r the H1995 f1e1d

0= ~vL ( r )

exp[17.1F(r) ]

C)

(5)

c0nta1n5 tw0 rad1a1 funct10n5, a 1en9th L(r) and a pha5e F(r). F0r the 9au9e f1e1d 1 a V,a 1(,,=~92

(6)

the 9enera1 DHN an5at2 c0nta1n5 three rad1a1 funct10n5 1/•" = 6(r) 6,,~,+ H(r) ( ~ , , - ~ , ) +

9r

9r

K(r) ~ 6r,, 9r (•7)

v8 = 0 .

(8)

7he5e An5~1t2e 5at15fy the e4uat10n5 0f m0t10n5 [ 2 ],

V01ume216, num6er 3,4

PHY51C5 LE77ER5 8

7he a60ve An5~1t2e 1ead t0 the 5pher1ca11y5ymmetr1c ener9y funct10na1

2,~M,vf

E=

92

[[(6--1)2+H2--1] 2

dx

x2

+ 2 L 2 ( 6 - 2 51n2F) 2 + 2 L 2 ( H - 5 1 n 2F) 2

+ L 2 ( 2 x F •-K)2+4x2L •2 +2ex2(L 2 - 1)2] .

12 January 1989

7he 4ue5t f0r re9u1ar 501ut10n5 w1th f1n1te ener9y 1mp05e5 60undary c0nd1t10n5 0n the rad1a1 funct10n5 at the 0r191n and at 1nf1n1ty. We f1nd tw0 p055161e 5et5 0f 60undary c0nd:a10n5 wh1ch d1ffer at the 0r191n. 7he rea50n 15 that the vect0r 6050n ma55 term 1ead5 t0 a term 1n the ener9y den51ty pr0p0rt10na1 t0 L 2 6 2 / x 2, wh1ch van15he5 at the 0r191n f0r e1ther L ( 0 ) = 0 0r 6 ( 0 ) =0. 7he f1r5t 5et 0f60undary c0nd1t10n5 6 ( 0 ) =2,

6(00) = 0 ,

H(0) =0.

H(00) = 0 ,

L(0)=0,

L(00)=•

(9) Here we have 1ntr0duced the d1men510n1e55 c00rd1nate x = Mwr and the parameter e = 42/92 = • (MH/ Mw) 2 determ1n1n9 the H1995 ma55 f0r f1xed ma55 0f the vect0r 6050n5. 7he a60ve ener9y den51ty c0nta1n5 a re51dua1 U ( 1 ) 9au9e 5ymmetry. 7h15 9au9e freed0m can 6e u5ed t0 e11m1nate 0ne 0f the rad1a1 funct10n5, except f0r the 1en9th 0fthe H1995 f1e1d L ( x ) , 0fc0ur5e. 1n the f0110w1n9 we ch005e the 9au9e F(x)=0.

(10)

N0te, that the DHN 5pha1er0n wa5 06ta1ned 1n the 9au9e F(x) = ~ w1th H(x) = K ( x ) = 0 [ 2 ] . Fr0m the ener9y funct10na1 we then 06ta1n the f0110w1n9 5et 0f e4uat10n5 0f m0t10n f0r the dynam1ca1 var1a61e5:

6 ( 0 ) =0,

6((x)) = 0 ,

H(0)=0,

H(00)=0,

L• ( 0 ) = 0 ,

L(~)=1

91ve5 r15e t0 the new 501ut10n5.1t c0nta1n5 a5 a 11m1t1n9 ca5e the EK5 60undary c0nd1t10n5 [ 12 ]. 1n the ca1cu1at10n5 we ch005e the vect0r 6050n ma55 Mw= 83 6eV and the c0up11n9 c0n5tant 9 = 0.67.7he H1995 ma55 15 then var1ed v1a the parameter e=

( M n / Mw ) ~.

3. Re5u1t5

x26 • = x2L 26 + 2xH• K + H ( xK• - K) +(6-1)[(6-1)2+H2+K2-1]

15c0n515tent w1th the DHN 5pha1er0n 60undary c0nd1t10n5.7he 5ec0nd 5et 0f60undary c0nd1t10n5

,

(11)

3.1. DHN 5pha1er0n

x2H ~ = x 2 L 2 H - 2 x 6 • K - ( 6 - 1 ) (xK• - K ) +H[(6-1)2+H2+K2-1], x2L••=-2xL•+

(12)

~L( 62 + H2 + •K 2)

+eX2L(L 2- 1 ) .

(13)

7he a65ence 0f K• 1n the ener9y funct10na1 1ead5 t0 c0n5tra1nt e4uat10n5 f0r the funct10n K and 1t5 der1vat1ve K•:

6•H-(6-1)H• K = x x 2 L 2 / 2 + H 2 + ( 6 - 1)2

( 14)

XK~=211-K(1+2XL~-L).

(15)

We 6e91n w1th a 6r1ef rev1ew 0f the DHN 5pha1er0n, 51nce 1t 15 deep1y re1ated t0 the ex15tence 0f the new 501ut10n5.7he DHN 5pha1er0n 15 an un5ta61e 501ut10n t0 the e4uat10n5 0f m0t10n ( 11 ) - ( 15 ) 5u6ject t0 5et 1 60undary c0nd1t10n5, where the funct10n H5(x) and thu5 the c0n5tra1nt K5(x) are tr1v1a1: H 5 ( x ) = 0 , K 5 ( x ) = 0 [2]. When 5tud1ed a5 a funct10n 0fthe H1995 ma55, the ener9y 0fthe DHN 5pha1er0n chan9e5 5m00th1y fr0m E°=7.065 7eV 1n the Pra5ad-50mmerf1e1d 11m1t, e=0, t0 E~-~= 12.576 7eV f0r 1nf1n1te H1995 ma55, e=00 [5]. 7he DHN 5p6a1er0n 15re9u1ar f0r e< 00, 6ut 1n the 1nf1n1te H1995 ma55 11m1tthe 1en9th funct10n L 5(x) 6ec0me5 51n9u1ar: 7he p0tent1a1 then enf0rce5 L 5 ( x # 0 ) = 1 , wh11e the 355

V01ume 216, num6er 3,4

PHY51C5 LE77ER5 8

12 January 1989

60undary c0nd1t10n at the 0r191n re4u1re5 L 5 ( 0 ) = 0 [5]. 1t 15 un11ke1y that there ex15t re9u1ar 501ut10n5 w1th n0n-tr1v1a1 H ( x ) 5u6ject t0 5et 1 60undary c0nd1t10n5 [ 16 ].

1

-

-

1

]

a

3.2. 501ut10n 1 We n0w turn t0 the new 501ut10n5, wh1ch are 5u6ject t0 5et 2 60undary c0nd1t10n5. We 5tart 6y n0t1n9 that the EK5 501ut10n [ 12 ] 06ta1ned 1n the 1nf1n1te H1995 ma55 11m1t, ~=00, 15 c0n515tent w1th 5et 2 60undary c0nd1t10n5. 7h15 11m1t 1mp11e5 a c0n5tant 1en9th 0 f t h e H1995 f1e1d thr0u9h0ut 5pace: L ( x ) = 1. 7he funct10n H ( x ) and the c0n5tra1nt5 K(x) are n0ntr1v1a1 f0r th15 501ut10n. We d15p1ay the numer1ca1 501ut10n f0r e = ~ 1n f19. 1a t09ether w1th the c0rre5p0nd1n9 D H N 5pha1er0n. 1n c0ntra5t t0 th15 5pha1er0n the EK5 501ut10n 15 re9u1ar, and 1t ha5 the 10wer ener9y E ~ = 1 1.782 7eV. C1a551ca1 501ut10n5 0f n0n-11near f1e1d the0r1e5 are we11 kn0wn f0r exh161t1n9 a var1ety 0f cata5tr0phe5 when the parameter5 0f the the0ry are var1ed. 7hu5 the ex15tence 0f the EK5 501ut10n f0r e = 00 d0e5 n0t 9uarantee, that ana1090u5 501ut10n5 ex15t f0r ar61trary va1ue5 0fe. 8 u t a510n9 a5 the dependence 0fthe 501ut10n5 0n the parameter5, here e, 15 5m00th, they can 6e c0n5tructed 6y an 1terat1ve pr0cedure. 7he f1r5t new 6ranch 0f501ut10n5 repre5ent5 the exten510n 0f the EK5 501ut10n t0 f1n1te H1995 ma55e5, e < ~ . We exh161t the 501ut10n5 f0r e = 7 5 and e = 72.38 1n f195. 16 and 1c, re5pect1ve1y. F0r c0mpar150n we a150 5h0w 1n the5e f19ure5 the c0rre5p0nd1n9 D H N 5pha1er0n 501ut10n5. Ev1dent1y, the 5hape 0f the new 501ut10n5 chan9e5 dra5t1ca11y, wh11e ~ 15 decrea5ed. 1t 1n fact appr0ache5 the 5hape 0f the D H N 5pha1er0n 501ut10n and a cata5tr0phe: At the f1n1te cr1t1ca1 va1ue eJ.,-= 72.36 the new 6ranch 0f 501ut10n5 mer9e5 w1th the D H N 5pha1er0n. Let u5 take a c105er 100k at what happen5 c105e t0 the cr1t1ca1 va1ue e~,.. 7 h e key 065ervat10n 15, that the va1ue 0f L ( 0 ) rap1d1y appr0ache5 2er0 c105e t0 e~, thu5 at e~, the 5et 1 60undary c0nd1t10n 0 f t h e D H N 5pha1er0n L ( 0 ) = 0 15 met. We dem0n5trate th15 chan9e 0 f L ( 0 ) a5 a funct10n 0fe 1n f19. 2. Para11e1 t0 the chan9e 1n L ( x ) we 065erve that the 510pe 0f the funct10n 6 ( x ) 1ncrea5e5 5teep1y at the 0r191n, the 10cat10n 0f 1t5 max1mum 5h1ft5 rap1d1y t0ward5 the 0r356

J

L

05

1

1,5

2

x

1

F

7

6

H

1

1 05

1

1.5

x

]

C

H

05

1

1 5

x

F19. 1.7he rad1a1 funct10n5 6 ( x ) , H(x) and L(x) (5011d 11ne5) 0f 501ut10n 1 are 5h0wn f0r three va1ue5 0f the parameter e a5 a funct10n 0fx=Mwr. F0r c0mpar150n, the rad1a1 funct10n5 0fthe DHN 5pha1er0n 6 5 ( x ) , H5(x) and L5(x) (da5hed 11ne5) are 5h0wn f0r the 5ame va1ue5 0f e. 7he5e va1ue5 are e=v0 1n (a), e=75 1n (6), e=72.38 1n (c). N0te, that H 5 ( x ) = 0 , and L5(x) 15 d15c0nt1nu0u5 at the 0r191n 1n (a).

191n, and the va1ue 0f 1t5 max1mum appr0ache5 the va1ue t w 0 : 6 m a x (xm~x 0 ) - - 2 . 7 h u 5 at e~, 1he 5et 1 60undary c0nd1t10n 0f the D H N 5pha1er0n 6 (0) = 2

V01ume 216, num6er 3,4

1.5

•

• • •••••1

PHY51C5 LE77ER5 8

~

1

•

~w71~1

•

3.3. 501ut10n 2

• • •••••

J

0.5

0

1

10

10 2

, 1~,1,1[

E

E

1 ~,,2

10~

10 3

12 January 1989

105

>

F19. 2.7he va1ue0fthe 1en9th funct10n0fthe H1995f1e1dat the 0r191n,L(0), 155h0wn a5 a funct10n0fthe parameter e= ~(M./ Mw) 2 f0r 501ut10n 1. When L(0) reache5 the va1ue 2er0, the 6ranch 0f 501ut10n 1and the 6ranch 0fthe DHN 5pha1er0nmer9e. 15 appr0ached, wh11e 6 ( x ) 6ec0me5 d15c0nt1nu0u5 at the 0r191n. 7 h e mer91n9 0f the new 6ranch 0f 501ut10n5 w1th the D H N 5pha1er0n 6ranch, 0r, v1ewed the 0ther way r0und, the emer91n9 0 f a new 6ranch 0f 501ut10n5 fr0m the D H N 5pha1er0n 6ranch, 15 c1ear1y 5een 1n an ener9y d1a9ram. We theref0re exh161t the ener91e5 0 f t h e tw0 6ranche5 0f501ut10n51n f19. 3. At the cr1t1ca1 p01nt e~, the ener9y 15 E~,- = 11.291 7eV. A60ve e].,- the new 6ranch 15 ener9et1ca11y 10wer than the 6ranch 0f the D H N 5pha1er0n.

7 h e ex15tence 0f the f1r5t new 6ranch 0f 501ut10n5 emer91n9 fr0m the D H N 5pha1er0n at e~,. natura11y 1ead5 t0 the 4ue5t f0r further new 501ut10n5.8ut what w0u1d the5e 501ut10n5 100k 11ke• We expect that the next new 6ranch w111 emer9e fr0m the D H N 5pha1er0n 6ranch at a h19her cr1t1ca1 va1ue e2~ and that 1t w1116e, w1th re5pect t0 ener91e5, 6etween the f1r5t new 6ranch and the 6ranch 0f the 5pha1er0n. 51nce we further expect that the next new 6ranch 0f 501ut10n5 w111 a150 6e 5m00th 1n the e~Q0 11m1t, 1et u5 1n5pect the EK5 501ut10n 1n f19. 1a 0ne m0re t1me: 7 h e funct10n 6 ( x ) ha5 0ne extremum, 6ey0nd th15 max1mum 6 ( x ) 15 remarka61y c105e t0 the funct10n 6 5 ( x ) 0 f t h e D H N 5pha1er0n; the funct10n H(x) ha5 tw0 extrema and 0ne n0de; 6ey0nd the n0de H(x) rema1n5 5ma11, thu5 c105e t0 H 5 ( x ) = 0 . We n0w rea50n that the funct10n5 6 ( x ) and H(x) 0f the 5ec0nd new 501ut10n 5h0u1d 6e even c105er t0 6 5 ( x ) and H 5 ( x ) , and that H(x) 5h0u1d deve10p a 5ec0nd n0de. 1ndeed, the 5ec0nd new 501ut10n can 6e c0n5tructed acc0rd1n91y f0r e = 00. We 5h0w th15 501ut10n 1n f19. 4.1t5 ener9y 15 E ~ = 12.504 7eV. 7h15 501ut10n 15 the end p01nt 0f the 5ec0nd new 6ranch, wh1ch emer9e5 fr0m the D H N 5pha1er0n at the cr1t1ca1 va1ue e~,- ~ 9570. We 1nd1cated the cr1t1ca1 p01nt 1n the ener9y p10t, f19. 3.

15 1

1

1

~--~

x 1

09 C

1()~

1(]2

100

102

104

106 "

F19. 3.7he ener91e50fthe DHN 5pha1er0nand 0f501ut10n 1 are 5h0wn a5 a funct10n0ft6e parameter e= • (MH/Mw)27he DHN 5pha1er0n ex15t5 1n the wh01e ran9e 0fthe parameter: 0~
5jj J

1

]

1

11 ~

L

1 5

:1

X

F19. 4. 7he rad1a1 funct10n56(x), H(x) and L(x) (5011d11ne5) 0f 501ut10n 2 are 5h0wn f0r e=Q0 a5 a funct10n0fx=Mwr. F0r c0mpar150n, the rad1a1 funct10n50fthe DHN 5pha1er0n 65(x), H5(x) and L5(x) (da5hed 11ne5)are a1505h0wn. 357

V01ume 216, num6er 3,4

PHY51C5 LE77ER5 8

0 0

•

05

0 (]2

J

J

1

1 ~]

0

04

2

F19. 5.7he rad1a1funct10n5 6(x), H(x) and L(x) (5011d11ne5) 0f 501u110n 3 are 5h0wn f0r e=0e a5 a funct10n 0f x=Mwr. F0r c0mpar150n, the rad1a1 funct10n5 0f the DHN 5pha1er0n 65(x), H5(x) and L5(x) (da5hed 11ne5) arc a150 5h0wn. 7he area c105e t0 the 0r191n 15ma9n1f1ed 1n the upper r19ht c0rner.

3.4. 501ut10n n We have n0w c0n5tructed tw0 6ranche5 0f 501ut10n5, th0u9h we expect n 6ranche5 0f 501ut10n5 t0 ex15t. 7 h e way t0 pr0ceed t0 c0n5truct the next 6ranche5 appear5 06v10u5:70 c0n5truct the nth 6ranch we 90 t0 the 11m1t e = 0 e ; we 1n5pect 501ut10n n - 1 and we re4u1re that f0r 501ut10n n 60th funct10n5 6 ( x ) and H(x) appr0ach 6 5 ( x ) and H 5 ( x ) c105er and that H(x) ha5 0ne m0re n0de. 7h151terat1ve pr0ce5515 fac111tated 6y the 1ntere5t1n9 065ervat10n that n - 1 0f the n n0de5 0f 501ut10n n are 1dent1ca1 t0 the n0de5 0f 501ut10n n - 1, and that the new n0de 15 c105e5t t0 the 0r191n. A5 an examp1e 0f the a60ve m e t h 0 d 0f c0n5truct10n we 5h0w 1n f19. 5 the th1rd new 501ut10n f0r e = 0c. 1t ha5 an ener9y 0f E ~ = 12.569 7eV. 7 h e c0rre5p0nd1n9 th1rd new 6ranch emer9e5 fr0m the D H N 5pha1er0n at the cr1t1ca1 va1ue e3r ~ 1.1 15 × 106. We have a9a1n m a r k e d th15 cr1t1ca1 p01nt 1n f19. 3. W h e n we c0nt1nue th15 1terat1ve pr0ce55, we f1nd the next 501ut10n w1th 4 n0de5 and E ~ = 12.575 7eV, etc.

4. C0nc1u510n5 We have c0n51dered an 5U (2) 9au9e the0ry w1th a c0mp1ex H1995 d0u61et. 7 h e H1995 p0tent1a1 pr0v1de5 a n0n-tr1v1a1 p a r a m e t e r d e p e n d e n c e 1n the c1a551ca1 the0ry: e = • (M,/Mw)2 enter5 a5 the 0n1y p a r a m e t e r 358

12 January 1989

1n the (5ca1ed) e4uat10n5 0f m0t10n. We have dem0n5trated that d e p e n d e n t 0n th15 p a r a m e t e r the e4uat10n5 0f m0t10n p055e55 a 5et 0ff1n1te ener9y 501ut10n5, a11 0f wh1ch are 5add1e p01nt5.7he we11-kn0wn D H N 5pha1er0n ex15t51n the fu11 p a r a m e t e r 5pace 0 ~<~ ~<~ , where 1t f0rm5 the 6a51c 6ranch 0f 501ut10n5. F r 0 m th15 6ranch 0f the D H N 5pha1er0n new 6ranche5 0f 501ut10n5 emer9e at cr1t1ca1 va1ue5 e~,. 5tart1n9 w1th the f1r5t 6ranch at %~ = 72.36 the5e 6ranche5 a p p e a r rather re9u1ar1y 5paced 1n ~ and can 6e c0n5tructed 1n a 5y5temat1c fa5h10n. We 5h0u1d 11ke t0 draw attent10n t0 51m11ar re5u1t5 f0und 6y M a n t 0 n et a1. [ 11 ] 1n the1r 5tudy 0f 5pha1er0n5 1n a ~ 4 the0ry 0n a c1rc1e. 1n the1r ca5e, the par a m e t e r 15 the 1en9th L 0f the c1rc1e and the 6a51c 6ranch 0f 501ut10n5, ex15t1n9 f0r ar61trary L, 15 f0rmed 6y the c0n5tant 501ut10n 0 = 0 . F r 0 m th15 6ranch new 6ranche5 0f un5ta61e 501ut10n5 emer9e at re9u1ar1y 5paced cr1t1ca1 va1ue5 L,,. We c0nc1ude that the d15c0vered 5tructure 0f the 5pace 0f un5ta61e 501ut10n5 m a y 6e a rather 9enera1 feature 0f 5p0ntane0u51y 6r0ken the0r1e5 w1th externa1 parameter5. Let u5 end w1th a d15cu5510n 0f the phy51ca1 1mp11cat10n5 0 f t h e new 501ut10n5. A11 the new 501ut10n5 are 5add1e p01nt5. 8 u t wh11e the D H N 5pha1er0n repre5ent5 the max1mum ener9y c0nf19urat10n 0n a n0nc0ntracta61e 100p, ana1090u5 100p5 f0r the new 501ut10n5 appear t0 6e a1way5 c0ntracta61e, 51nce the 1en9th 0 f t h e H1995 f1e1d van15he5 n0where: L(x) ¢ 0. 1t 15 theref0re un11ke1y that the new 5pha1er0n5 9enerate 6ary0n decay. What 0ther phy51ca1 re1evance may the new 501ut10n5 have• We reca11 that the EK5 501ut10n wa5 d15c0vered 1n an effect1ve the0ry • t h e weak 1nteract10n5 c0nta1n1n9 a 5kyrme type 5ta611121n9 term [ 12,13 ]. 1n th15 the0ry the p a r a m e t e r ~ 15 f1xed, e = ~ , 6ut the 5ta61112at10n term 1ntr0duce5 a new p a r a m e t e r dependence. A5 a funct10n 0fth15 new p a r a m e t e r e5k the EK5 501ut10n deve10p5 a 5p1ke, and chan9e5 - 0n the 10wer 6ranch - t0 a 10ca1 m1n1mum. 7hu5 the part1c1e-11ke 1nterpretat10n 0f the 501ut10n 6ec0me5 ade4uate, and the p05516111ty ar15e5 that 5uch ••weak 5kyrm10n5•• m19ht 6e f0und at ener91e5 0f 5evera1 7eV [ 12 ]. We n0w app1y a 51m11ar rea50n1n9 t0 the 0ther 501ut10n5.7he5e have h19her ener9y and m0re n0de5 than the EK5 501ut10n, thu5 we 1nterpret them a5 exc1ted 5tate5 0 f t h e ••weak 5kyrm10n5••; a deta11ed 5tudy 15 1n preparat10n [16]. An ana1090u5 part1c1e-11ke

V01ume 216. num6er 3,4

PHY51C5 LE77ER5 8

1 n t e r p r e t a t 1 0 n 0 f t h e n e w 501ut10n5 15 p055161e 1n effect1ve the0r1e5 0 f t h e 5 t r 0 n 9 1nteract10n5, w h e r e t h e D H N 5pha1er0n a n d t h e E K 5 501ut10n h a v e 6 e e n d15cu55ed prev10u51y a n d h a v e 6 e e n 6 a p t 1 2 e d h a d r 0 1 d 5 [17].

Ackn0w1ed9ement W e 9ratefu11y a c k n 0 w 1 e d 9 e d15cu5510n5 w1th P. K051n5k1, A. L a n d e a n d P.J. Mu1der5.

Reference5 [1 ] 6. •t H00ft, Nuc1. Phy5.8 79 (1974) 276; A.M. P01yak0v, 50v. Phy5. JE7P Len. 20 (1974) 194. [ 2] R.F, Da5hen, 8. Ha551acher and A. Neveu, Phy5. Rev. D 10 (1974) 4138. [31C.H. 7au6e5, C0mmun. Math, Phy5.86 (1982) 257, 299. [4] N.5, Mant0n, Phy5. Rev. D 28 (1983) 2019.

12 January 1989

[ 5 ] F.R. K11nkhamer and N.5.Mant0n, Phy5. Rev. D 30 ( 1984 ) 2212. [6] J. 809uta and J. Kun2, Phy5. Lett. 8 154 (1985) 407. [7] P. Arn01d and L. McLerran, Phy5. Rev. D 36 (1987) 581; D37 (1988) 1020. [8] H. A0yama, H. 601d6er9 and 2. Ry2ak, Phy5. Rev. Lett. 60 (1988) 1902. [9] 8. Ratra and E.6. Yaffe, Phy5. Len. 8 205 (1988) 57. [ 10] C.H. 7au6e5, C0mmun. Math. Phy5.97 (1985) 473. [ 11 ] N,5. Mant0n and 7.M. 5am015. Phy5. Lett. 8 207 (1988) 179. [12] 6. E11am, D. K1a6ucar and A. 5tern, Phy5. Rev. Lett. 56 (1986) 1331: 6. E11am and A. 5tern, Nuc1. Phy5. 8 294 (1987) 775. [13] J. Am6j0rn and v.A. Ru6ak0v, Nuc1. Phy5. 8 256 (1985) 434. [ 14] Y. 8r1haye and J. Kun2, 2. Phy5. C, t0 6e pu6115hed. [15] 5. we1n6er9, Phy5. Rev. Len, 19 (1967) 1264; A. 5a1am, 1n: E1ementary pan1c1e the0ry, N06e1 5ymp. N0. 8, ed. N. 5varth01m (w11ey, New Y0rk, 1969 ). [ 16 ] Y. 8r1haye and J. Kun2, 1n preparat10n. [ 17] J. 809uta, Phy5. Rev. Len. 50 (1983) 148; Phy5. Eett. 8 198 (1987) 384J. 809u1a and J. Kun2, Phy5. Lett. 8 166 (1986) 93.

359