On the particle spectrum of E(6) superstrings

Nuc1ear Phy51c5 8298 (1988) 333-356 N0rth-H011and, Am5terdam

0 N 7 H E PAR71CLE 5 P E C 7 R U M 0 F E(6) 5 U P E R 5 7 R 1 N 6 5 M. DREE5 P1~v51c5 Department, Un1ver51tv 0f W15c0n51n, Mad150n, W15c0n51n 53706, U5A

Rece1ved 10 March 1987 (Rev15ed 4 June 1987)

7he a110wed parameter 5pace 0f a 5uper5tr1n9-1n5p1red 5uper9rav1ty m0de1 w1th 0ne add1t10na1 neutra1 9au9e 6050n 2• 15 1nve5t19ated 1n deta11, under the a55umpt10n that n0 1ntermed1ate 5ca1e ex15t5 and that the 5uper5ymmetry 6reak1n9 1n the v15161e 5ect0r can 6e parameter12ed 6y m 0, m1/2 and A a5 1n u5ua1 5uper9rav1ty m0de15. 1t 15 5h0wn that the m0de1 a110w5 f0r 1ar9e Yukawa c0up11n95 and thu5 a re1at1ve1y 5ma11 5uper5ymmetry• 6reak1n9 5ca1e. Add1t10na1 5uperp0tent1a1 c0up11n95 wh1ch d0 n0t 1nv01ve H1995 5uperf1e1d5 a110w t0 reduce 51ept0n and 54uark ma55e5 even further. 7heref0re, 5part1c1e ma55e5 can 11e we11 w1th1n the ran9e 0f the 5LC, 7evatr0n, and LEP even 1f the 2•-6050n 15ve~~heavy. 80und5 0n the ma55e5 0f ex0t1c ferm10n5 and the t0ta1 decay w1dth 0f the 2• are a150 91ven.

1. 1ntr0duct10n 51nCe the f1r5t d15C0Very 0f an an0ma1y-free 5Uper5tr1n9 the0ry [1] a 10t 0f 1ntere5t ha5 6een pa1d t0 the 1nVe5t19at10n 0f the emer91n9 10W-ener9y the0ry a n d 1t5 phen0men01091Ca1 1mp11Cat10n5. E5peC1a11y the heter0t1C 5tr1n9 the0ry 0f ref. [2] ha5 6een 5h0Wn [3] t0 y1e1d an at 1ea5t p0tent1a11y rea115t1C m0de1 after C0mpaCt1f1Cat10n t0 4 d1men510n5. 1t5 m05t pr0m1nent feature5 are a 10Ca1 N = 1 5Uper5ymmetry a n d an effect1Ve E(6) 9rand Un1f1ed 9aU9e 9r0Up; the 1atter 15 6r0ken 6y the H050tan1 mechan15m [4] t0 a 5U69r0Up W1th r a n k > 5 [5]. 7hU5 the part1C1e 5peCtrUm 6e10W the Un1f1Cat10n 5Ca1e M x C0nta1n5 n 0 9enerat10n5 0f C0mp1ete 27-d1men510na1 repre5entat10n5 0f E(6) and at 1ea5t 0ne add1t10na1 U(1) 9aU9e 6050n 1n add1t10n t0 the 5 t a n d a r d m0de1 9aU9e 6050n5, a5 We11 a5 the C0rre5p0nd1n9 5Uperpartner5. F U r t h e r m 0 r e , there m19ht 6e [3, 5] ••5UrV1v0r5•• fr0m add1t10na1 27 + 27 repre5entat10n5, a5 we11 a5 9au9e 51n91et5. 7 h e r e have a150 6een attempt5 [6] t0 f1nd d1fferent c0mpact1f1cat10n 5cheme5 wh1ch 1ead t0 5 0 ( 1 0 ) 0r even 5U(5) a5 effect1ve 6 U 7 9r0up. H0wever, the5e m0de15 face the0ret1ca1 [7] a5 we11 a5 phen0men01091ca1 [8] pr061em5; furtherm0re, the emer91n9 part1c1e 5pectra [9] tend t0 re5em61e th05e 0f N = 1 5uper9rav1ty m0de15 [10] 0f the pre-5tr1n9 era. 1 w111 theref0re n0t d15cu55 th15 p05516111ty any further. 1n5tead, 1 w111 f0cu5 0n the 51mp1e5t E(6) m0de1, w1th 10w ener9y 9au9e 9r0up [11] 5 U ( 3 ) c × 5U(2)L × U ( 1 ) y X U(1)E. 1 w111 a150 d15card the p05516111ty 0f an 1nter0550-3213/88/$03.50 • E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)

334

M. Dree5 / E(6) 5uper5tr1n95

med1ate 5ca1e [12] wh1ch c0u1d 91ve ma55e5 0f 0(101° 6 e V ) t0 a11 ex0t1c part1c1e5, a9a1n 1ead1n9 t0 a 10w-ener9y 5pectrum 0f the pre-5tr1n9 type. Furtherm0re, the 1ntr0duct10n 0f an 1ntermed1ate 5ca1e M 1 dra5t1ca11y reduce5 the pred1ct1ve p0wer 0f the the0ry, 51nce many p05516111t1e5 ex15t [12] f0r the 9au9e 9r0up 6etween M x and M~; the va1ue 0f M~ can vary [13] 0ver 5evera1 0rder5 0f ma9n1tude; and 1n 9enera1 the 5ymmetry 6reak1n9 a t M 1 affect5 the ma55e5 0f a11 5ca1ar part1c1e5 6y D-term c0ntr16ut10n5, wh05e prec15e va1ue 15 5tr0n91y m0de1 dependent [14]. 06v10u51y the part1c1e 5pectrum depend5 0n the way 5uper5ymmetry 15 6r0ken 1n the v15161e 5ect0r. 7he dynam1ca1 0r191n 0f the 6reakd0wn 0f 5uper5ymmetry 15 6e11eved t0 6e [15] the f0rmat10n 0f a 9au91n0-c0nden5ate 1n the ••5had0w•• 5ect0r 0f the m0de1. H0wever, at tree 1eve1 the v15161e 5ect0r rema1n5 5uper5ymmetr1c. Unf0rtunate1y the effect1ve 1-100p 1a9ran91an de5cr161n9 the v15161e 4-d1men510na1 w0r1d 5eem5 t0 depend [16] 0n the deta115 0f the c0mpact1f1cat10n fr0m 10 t0 4 d1men510n5, wh1ch are n0t yet c0mp1ete1y under5t00d. Exp11c1t ca1cu1at10n5 1n d1fferent 5cheme5 re5u1t 1n n0nvan15h1n9 ma55e5 f0r 5ca1ar5 [16] 0r 9au91n05 [17]. 1 have theref0re a55umed that a c0mm0n 5ca1ar ma55 m0, a c0mm0n 9au91n0 ma55 m~/2 and a c0mm0n tr111near 50ft 6reak1n9 parameter A can a11 have n0nvan15h1n9 va1ue5 a1ready at the un1f1cat10n 5ca1e M~. F1na11y, the num6er5 0f 9enerat10n5 and 5urv1v0r5 have t0 6e f1xed. 51nce 1 d0 n0t a110w f0r an 1ntermed1ate 5ca1e, the f0rmer ha5 t0 6e [12] three. 1n ref. [18] 1t ha5 6een 065erved that a 5 ( M w ) and 51n20w(Mw) tend t0 6e rather 5ma11 (4 and 0.205, re5pect1ve1y) 1f there are n0 5urv1v0r5. H0wever, mak1n9 u5e 0f the p05516111ty [18] t0 keep tw0 add1t10na1 5U(2) d0u61et5 0ut 0f 27 + 2-7 119ht, 0ne f1nd5 51n20w(Mw) = 0.23, M x = 2 × 1016 6eV, w1th a 5 ( M w ) = ~. 7hey can ach1eve ma55e5 0f 0 ( M w ) thr0u9h c0up11n95 t0 the 9au9e 51n91et f1e1d5 ment10ned a60ve; h0wever, the1r ex15tence 15 n0t cruc1a1 f0r th15 ana1y515. 1t ha5 6een ar9ued [19] that c0n515tency 0f the c0mpact1f1cat10n 5cheme re4u1re5 M x t0 6e n0t much 5ma11er than the P1anck ma55 Mp; the 91ven va1ue 5eem5 t0 6e 1n c0nf11ct w1th th15 re4u1rement. N0te, h0wever, that the 91ven num6er 15 6a5ed 0n a 0ne-100p ca1cu1at10n; 51nce/~3 = 0 at 0ne-100p, tw0-100p effect5 w111 1ncrea5e M x 6y r0u9h1y an 0rder 0f ma9n1tude. Furtherm0re a11 thre5h01d effect5 have 6een ne91ected. 7heref0re the 5ma11ne55 0f M x d0e5 n0t 5eem t0 6e a 5er10u5 pr061em 0f the m0de1. 1n any ca5e the va1ue5 0f the parameter5 at the weak 5ca1e depend 0n1y 109ar1thm1ca11y 0n the va1ue 0f Mx; theref0re even an uncerta1nty 0f a fact0r 0f 100 1n M x 1ead5 0n1y t0 a 15% var1at10n 0f 065erva61e5 at the weak 5ca1e.

* At th15 p01nt tw0 caveat5 are 1n 0rder. 7here are 1nd1cat10n5 [17,18] that 9au91n0 ma55e5 c0u1d d1ffer fr0m each 0ther a1ready at u1trah19h ener91e5; furtherm0re, the v15161e w0r1d 6ec0me5 n0n5uper5ymmetr1c 0n1y at ener9y 5ca1e5 5ma11er than the c0nden5at10n 5ca1e, wh1ch can 1n pr1nc1p1e 6e 5u65tant1a11y 5ma11er [15] than the c0mpact1f1cat10n 5ca1e. H0wever, 1t w111 6ec0me c1ear that even under the a55umpt10n5 115ted 1n the text the parameter 5pace 15 a1m05t unc0n5tra1ned. Re1ax1n9 the5e a55umpt10n5 w0u1d theref0re n0t 9reat1y 1ncrea5e the a110wed ran9e 0f the var10u5 part1c1e ma55e5.

M. Dree5 / E(6) 5uper5tr1n95

335

Hav1n9 f1xed the 9au9e 9r0up, the num6er and 4uantum num6er5 0f the matter 5uperf1e1d5 and the effect1ve 5uper5ymmetry 6reak1n9 mechan15m, the part1c1e 5pectrum 15 determ1ned 6y the ma55 M 2, 0f the add1t10na1 neutra1 9au9e 6050n, the va1ue5 0f the Yukawa c0up11n95 and the d1men510n1e55 rat105 A / m 0 and m0/m1/2. 1n ref. [20] 1t ha5 6een 065erved that n0w M 2, rather than M 2 5et5 the 5ca1e f0r the ma55 5p11tt1n9 6etween the 5uperpartner5. 7he exact re1at10n 6etween the5e 5ca1e5 depend5, h0wever, 5tr0n91y 0n the va1ue5 0f the var10u5 Yukawa c0up11n95. 1n 5ect. 2 the a110wed ran9e f0r the5e c0up11n95 15 1nve5t19ated 1n deta11. 1t 15 f0und that 6e51de5 the re910n 0f 5ma11 c0up11n95 d15cu55ed 1n the 11terature [9,18] there 15 a re910n 0f 1ar9e c0up11n95, wh1ch 1n turn a110w f0r m0derate 5uper5ymmetry 6reak1n9 5ca1e; due t0 the pre5ence 0f 1ar9e ne9at1ve D-term c0ntr16ut10n5 fr0m the new U(1) the ma55e5 0f 1eft-handed 51ept0n5 can 6e ar61trar11y 5ma11, 1ndependent 0f M 2,• 1n 5ect. 3, 1 d15cu55 the p055161e ma55e5 0f ex0t1c ferm10n5 and the1r 5uperpartner5. 1n 9enera1, 60und5 0n ferm10n ma55e5 can 0n1y 6e der1ved fr0m the re4u1rement 0f a pertur6at1ve un1f1cat10n; thu5 the ferm10n5 can ea511y 6e heav1er than the 119hter 0f the1r 5ca1ar 5uperpartner5. 1n 5ect. 4 the 5pectrum 0f ••0rd1nary•• 5ferm10n5 and 9au91n05 15 5tud1ed. Due t0 the pre5ence 0f 5uperp0tent1a1 c0up11n95 6etween ex0t1c and 0rd1nary 4uark and 1ept0n 5uperf1e1d5 the ma55e5 0f 51ept0n5 0r r19ht-handed d-54uark5 can 6e a1m05t ar61trar11y 5ma11. 7he 119hte5t neutra11n0 and char91n0 can 6e a5 119ht a5 a110wed 6y exper1ment. 1n 5ect. 5, 1 d15cu55 60und5 0n the decay w1dth 0f the 2•. 51nce 50me ex0t1c channe15 are a1way5 0pen, the 6ranch1n9 rat10 1nt0 e+e cann0t 6e 1ar9er than 1.8%; a1th0u9h th15 m19ht c0mp11cate the d15c0very 0f the 2•, 1t 1nd1cate5 that 5evera1 0ther new part1c1e5 c0u1d 6e 065erved at the 5ame t1me. F1na11y, 5ect. 6 c0nta1n5 a 5ummary 0f my re5u1t5. 7he ren0rma112at10n 9r0up e4uat10n5 ( R 6 E ) re1evant f0r the ana1y5e5 0f 5ect5. 2 and 4 are 115ted 1n append1ce5 A and 8, re5pect1ve1y. 2. 7he a110wed ran9e 0f Yukawa c0up11n95 7he term5 1n the 5uperp0tent1a1 wh1ch are re1evant f0r the ana1y515 0f the rad1at1ve 6reak1n9 0f 5U(2) × U(1)y × U(1)E t 0 U ( 1 ) e m read: 3

f , = Y2 (X2.1N3H;H1 + )k3,1N3D1[)1) +

htU;H3Q 3 ,

(1)

1=1

where the n0tat10n 0f ref. [9] ha5 6een u5ed. 1t 15 a55umed that 0n1y N3, H 3 and H 3 have n0nvan15h1n9 vacuum expectat10n va1ue5 (VEV); th15 can a1way5 6e ach1eved 6y a r0tat10n 1n 9enerat10n 5pace. 1n pr1nc1p1e, fn 5h0u1d a150 c0nta1n term5 0f the f0rm N2H3H1, etc. [21]; 1 w111a55ume that they can 6e a650r6ed 1n )~2,1 and )~2,2 a5 far a5 the1r effect 0n the R 6 E 15 c0ncerned. 1t 15, h0wever, n0t 5uff1c1ent t0 take 0n1y )~2,3 1nt0 acc0unt, 51nce th15 c0up11n9 15 60unded ma1n1y 6y 1t5 effect5 1n the H1995 p0tent1a1, and n0t 6y the R 6 E . F1na11y, there 15 n0 rea50n t0 a55ume that 0n1y

M. Dree5 / E(6)5uper5tr1n95

336

0ne pa1r 0f ex0t1c D,D-4uark5 ha5 a 512ea61e Yukawa c0up11n9; e5pec1a11y 1n the re910n 0f 1ar9e X3, 5 three X3•5 cann0t 6e 51mu1ated 6y a 51n91e, 1ar9er c0up11n9. F0r 91ven va1ue5 0f A / m 0 and m0/m1/2 the Yukawa c0up11n95 have t0 6e ch05en 5uch that the ma55 parameter5 m2N~, m2H, and m ~2 , taken at the weak 5ca1e, 06ey the f0110w1n9 e4uat10n5 [9] wh1ch can 6e der1ved 6y m1n1m121n9 the H1995 p0tent1a1 [11]: m 2m

= X2,3A2,3n t a n 1 - ~k22,3 ( ~ 2 + n 2 ) __ 41( 9 1 2 4- 9 2 ) ( U 2 - -

U2) -1•- 1 9 • ( 5 n 2

-- U 2 • 4 0 2 ) ,

(2a)

2 1 2 (5n 2 - U2 -- 402) "4•-41 (91-}92)(U2 -- U2) 4- ~91

(26)

m ~ . = X 2 , 3 A 2 , 3 n C0t8 - )k22 , 3 ( / ) 2 + n 2 )

m 2N~ = ~ . 2 , 3 A 2 . 3 V 0 / n ~ ) k22 , 3 ( U 2 4 - 0 2 )

-- ~369 21•[ 5 n 2 ~ V 2 ~ 4 0 2 )

M2,>>M£

25~2. 2 -- 3691 rt ,

(2c) where ( H ° ) - v, ( ~ 0 ) =- 0, (N3) -- n and 0/v - tan/3. 8y def1n1t10n the f1r5t term 1n the r.h.5. 0f e45. (2) 15 a1way5 p051t1ve. F1na11y, the 9au9e c0up11n95 0f the tw0 U(1) 9r0up5 are a55umed t0 6e e4ua1, wh1ch 15 a very 900d appr0x1mat10n. E4. (2c) can 6e u5ed t0 re1ate m~/2, 1.e. the 5uper5ymmetry 6reak1n9 5ca1e, t0 M2,, wh1ch 15 91ven 6y

2592

M22~ M~,>>-£

--n 18

2= -2m 2 N~

(3)

A55ume that the d1men510n1e55 parameter5 are ch05en 5uch that m 2 ( M w ) < 0; 51nce r - m23(Mw)/m2/2 15 1ndependent 0f m1/2, e4. (3) then f1xe5 m1/2 un14ue1y. N0te that the 0ne-100p 8-funct10n f0r 93 van15he5 1n the 91ven m0de1; theref0re m1/2 e4ua15 the 91u1n0 ma55 m 9 . 0 n the 0ther hand, the a6501ute va1ue 0f r w1116e 1ar9er, and theref0re m9 w1116e 5ma11er, 1f the R 6 5ca11n9 0f m23 15 acce1erated. 1n the 91ven m0de1 th15 can m05t ea511y 6e ach1eved 6y 1ncrea51n9 X 3,1. 7h15 15 dem0n5trated 6y f19. 1, where m 9 / M 2, acc0rd1n9 t0 e4. (3) 15 5h0wn a5 a funct10n 0f X3(Mx), f0r m 0 = A = 0 . 7 h e 5011d curve5 c0rre5p0nd t0 the ••can0n1ca1•• [9,18] ca5e X 2,1 = X 3,1 = 0 f0r 1 = 1, 2, wh11e the da5hed curve ha5 6een 06ta1ned w1th X 3,1 = X 3,2 = X 3,3 = X3.06v10u51y, m J M 2, depend5 0n1y weak1y 0n X 2, ~ un1e55 the X3, ~ are very 5ma11 (•< 0.04). 1t 15 1ntere5t1n9 t0 n0te that f0r 1ar9e va1ue5 0f X 3 an 1ncrea5e 1n X 2 1ncrea5e5 m 2 (Mw), wh1ch mean5 that M 2, 15 decrea5ed. 1t 5h0u1d furtherm0re 6e n0ted that the a6501ute 10wer 60und 0n m J M 2, decrea5e5 fr0m 0.36 t0 0.3 1f 0ne a110w5 f0r de9enerate D-4uark5. A 5ma11 5uper5ymmetry 6reak1n9

M. Dree5 / E(6) 5uper5tr1n95 10L11

1

1

337

1

/~=75 °, ht(Mx):0.03 , m0=A=0

m~1M2,

X2=0.2

0

~

f

1

1

0.1

0.2

0.3

1

0.4

0.5

X3(Mx) F19. 1. 7 h e 91u1n0 ma55 1n un1t5 0f the ma55 0f the 2•-6050n 15 5h0wn a5 a funct10n 0f • 3 ( M x ) . 7 h e 5011d curve5 have 6een 06ta1ned w1th X3,1 = X3, 2 = 0, X3, 3 ~ X 3, wh11e the da5hed curve 15 va11d f0r •3,1 = X3,2 ~ X3.3 ~ X3- 1t 15 a1way5 a55umed that )k2.1 = )k2,2 = 0.

5ca1e 15 fav0red f0r phen0men01091ca1 a5 we11 a5 the0ret1ca1 rea50n5, the 1atter 6e1n9 re1ated t0 the 5ta6111ty 0f the 9au9e h1erarchy. H0wever, up t0 n0w 0n1y the 1a5t 0f e45. (2) ha5 6een 501ved. 1t 15 c0nce1va61e that e45. (2a) and (26) f0rce the 7t3,1 t0 6e 5ma11; 1n fact, ref. [18] 5eem5 t0 1nd1cate that th15 happen5. 0 n e 5trate9y f0r 501v1n9 the5e e4uat10n5 15 t0 ch005e X3, 3 5uch that e4. (2a) 15 fu1f111ed, and then tune 7t2,3 t0 fu1f111e4. (26), where 1n 60th ca5e5 m~ 15 ch05en 1n acc0rdance w1th e4. (2c). 1t 5h0u1d, h0wever, 6e n0ted that the f1r5t 5tep 1n th15 pr0cedure 1n n0t un14ue. 7h15 15 dem0n5trated 6y f19. 2 where R H, the rat10 0f the r.h.5. 0f e4. (2a) and m ~ ( M w ) , 15 5h0wn a5 a funct10n 0f )t3.3(MX) , w1th X~. , = X 3 , = 0. f0r 1 = 1 , 2. and m 0 =. A = 0 . F0r very 5ma11 X33, m~ and thu5 m 2H3 6ec0me5 very 1ar9e, acc0rd1n9 t0 f19. 1; 51nce the r.h.5. 0f e4. (2a) 1ncrea5e5 0n1y 11near1y w1th the 91u1n0 ma55, R H 15 very 5ma11. A 5119ht 1ncrea5e 1n ~k3, 3 make5 m~ decrea5e 5harp1y, cau51n9 R H t0 1ncrea5e and f1na11y t0 reach the de51red va1ue 1 at a 5t111 5ma11 va1ue 0f ~k3, 3. H0wever, an 1ncrea5e 1n 2t3,3 a150 decrea5e5 A2, 3 v1a the R 6 5ca11n9; eventua11y, th15 effect 6ec0me5 d0m1nant and R H 5tart5 t0 decrea5e, a9a1n reach1n9 the va1ue 1 at X3,3(Mx) -- 0.1. F0r 1ar9er va1ue5 0f X3,3 the r.h.5, 0f e4. (2a) 6ec0me5 5ma11er and even ne9at1ve; f1na11yA2,3(Mw) van15he5 and R H reache5 a m1n1mum. 1f X3,3 15 1ncrea5ed further, A2,3(Mw) 6ec0me5 ne9at1ve; h0wever, n a150 chan9e5 519n and thu5 R H r15e5 a9a1n. 7h15 1ncrea5e 15 further acce1erated 6y the

M. Dree5 / E(6) 5uper5tr1n95

338 ]

1

1

2F ~

~

1

1

X2,3(MX)=0.2,ht=0.03, ,8=75*, ~

1/ •

-

/

1

0

/

1

1

1

0.1

0.3

X5,3(Mx F19. 2. 7he rat10 0f m ~ ( M w )

and the r.h.5. 0f e4. (2a) 15 5h0wn a5 a funct10n 0f )k3.3(Mx), w1th )k2,1 = X2,2 ~ 0.

fact that an 1ncrea5e 1n ~k3, 3 CaU5e5 ~k2,3(Mw) t0 decrea5e. EVentUa11y, f0r ~ t 3 , 3 ( M W ) = 0.25, R H reache5 1 f0r the th1rd t1me; n0te that th15 va1ue 0f ~k3, 3 a1ready c0rre5p0nd5 t0 the f1at part 0f the curve5 1n f19. 1. 1t 5h0u1d 6e p01nted 0ut that the 501ut10n w1th 1ar9e ~k3, 3 15 5eparated fr0m the 0ther tW0 501Ut10n5 6y a re910n Where the m1n1m12at10n 0f the H1995 p0tent1a1 re4U1re5 m 2 (MW) t0 6e ne9at1Ve, Wh1Ch Cann0t 6e rea112ed v1a the R 6 5Ca11n9. 0 n the 0ther hand, the f1r5t tW0 501Ut10n5 Can C01nC1de 1f the f1r5t max1mUm 0f R H(X 3.3) 15 at R H = 1.7h15 Can 6e aCh1eVed e.9., 6y decrea51n9/3, Wh1Ch decrea5e5 the r.h.5. 0f e4. (2a). N0te that the tW0 CUrVe5 1n f19. 2 (f0r M 2, = 300 6 e V and 600 6eV) a1m05t C01nC1de. 7h15 15 due t0 the fact [20] that the h1erarchy M22, >> M 2 f0rce5 the parameter5 0f the H1995 p0tent1a1 t0 06ey the e4Uat10n A2 2.3

259~

2 [ m2H~

1

36~2,3 mN31 m~ 3 + 5

362t2 3 ) 25912

2

m~3

4 + 5

362t2 3

2597

(4)

at the weak 5ca1e, 1ndependent 0f the actua1 va1ue 0f M2,; 0r, 5tated d1fferent1y, 5ma11 chan9e5 1n the H1995 p0tent1a1 at 5ca1e M w can chan9e the rat10 M 2 , / M 2 dramat1ca11y. F19. 2 5h0w5 that th15 true a150 f0r 5ma11 chan9e5 0f the Yukawa c0up11n95 at M x, at 1ea5t a5 10n9 a5 they are 5ma11. N0te, h0wever, that th15 f1ne-tun1n9 pr061em 6ec0me5 1e55 5evere a5 X3, 3 1ncrea5e5. 7he va1ue5 0f the

M. Dree5 / E(6) 5uper5tr1n95

339

p a r a m e t e r 5 at the Weak 5Ca1e tend t0 6eC0me 1e55 5en51t1Ve t0 )•3,3(MX); th15 15 d e m 0 n 5 t r a t e d 6y the 1nCrea51n9 5paC1n9 6etWeen the CUrVe5 0f f19. 2, a5 We11 a5 the f1atten1n9 0f the CUrve5 1n f19. 1. A5 ment10ned ear11er, the 1a5t 0f e45. (2) Can f1na11y 6e 501Ved 6y Ch0051n9 )•2,3(MX) appr0pr1ate1y.* 50me re5U1t5 are 5h0Wn 1n f195. 3a,6 and 4a, 6 Where )•2,3(MX) and )•3,3(MX), re5pect1Ve1y, are p10tted Ver5U5 m 0 / m ~ f0r A = 0 (f195. 3) and A = 2 m 0 (f195. 4), f0r Var10U5 Va1Ue5 0f 8. 1n 60th f19Ure5, 1 have a55Umed h t ( M X ) = 0.025 Wh1Ch y1e1d5 m t = 40 6 e V , 1n a9reement w1th the t0p-1nterpretat10n 0f Certa1n U A 1 event5 [22]. 7 h e re5U1t5 0f f195. 1 and 2 have 6een 06ta1ned Under the a55Umpt10n that at M x, 9au91n0 ma55e5 are the 0n1y 50UrCe 0f 5uper5ymmetry 6reak1n9 1n the v15161e 5eCt0r. 7h15 51mp11f1ed 51tUat10n Wa5 C0nven1ent f0r the 4Ua11tat1ve d15Cu5510n 0f the preced1n9 para9raph5 51nCe th15 15 the 0n1y rea115t1C Ca5e Wh1Ch a110W5 t0 de5Cr16e 5uper5ymmetry 6reak1n9 w1th 0n1y 0ne parameter, m1/2 = m~. F195. 3 and 4 5h0w that the 4Ua11tat1ve feature5 d15Cu55ed a60ve, 1.e. the ex15tence 0f three 5eparate 501Ut10n5 f0r )•2,3(Mx) and )~3~3(Mx) f0r 91Ven va1Ue5 0f M2,, 13 and ht, a150 5urV1ve f0r n0nVan15h1n9 va1Ue5 0f m 0 / m ~ and A / m 0. A5 1nd1Cated ear11er, the tw0 501ut10n5 W1th 5ma11 va1Ue5 0f )• 3, 3 ( M x ) C01nC1de f0r Certa1n va1Ue5 0f the parameter5 where the max1mum 0f R H 1n f19. 2 15 at R H = 1; 1f th15 m a x 1 m u m C0rre5p0nd5 t0 an even 5ma11er va1ue 0f R H, the5e 501Ut10n5 d15appear. 1t 15 c1ear fr0m e4. (2a) that the he19ht 0f the max1mum 1n f19. 2 1ncrea5e5 w1th A 2.3( M w ) t a n / 3 . Ch0051n9 A = 0 (f195. 3) theref0re y1e1d5 the 10wer 60und

/~>62 °

(m0=A

=0)

(5)

f0r the5e 501Ut10n5 t0 eX15t. 1nCrea51n9 m 0 W1th A = 0 mean5 t0 decrea5e the a6501Ute Va1Ue 0f A2,3(MW) 51nCe m~ 15 decrea5ed. 7heref0re, the 501Ut10n5 W1th 5ma11 )~3 d15appear f0r m 0 >• 0.6m9. N 0 t e that f195. 3 and 4 C0nta1n 0n1y CUrVe5 W1th/~ < 75 °, 1.e. ~/U < 3.8.7h15 d0e5 n0t m e a n that 1ar9er Va1Ue5 0f/3 are eXC1Uded; h0WeVer, the YUkaWa C0Up11n9 0f the 6 0 t t 0 m 4Uark, Wh1Ch 1 have ne91ected, 15 pr0p0rt10na1 t0 tan/3 and m19ht thU5 6eC0me 1mp0rtant f0r /3 > 75 °. 1n th15 re910n, h0WeVer, 1ar9er Va1Ue5 0f r n 0 / m ~ m19ht 6e a110Wed even f0r the 501Ut10n5 W1th 5ma11 )~3" 0 n the 0ther hand, the 501Ut10n W1th 1ar9e )•3 need5 a n0nVan15h1n9 5Ca1ar ma55 a1ready at M x, 51nCe 0therw15e the 54Uared ma55e5 0f 1eft-handed 51ept0n5, m 2, * 0f c0ur5e, the r.h.5. 0f e4. (2a) a150 depend5 0n )~2.3; theref0re e45. (2a) and (26) have t0 6e 501ved 51mu1tane0u51y 6y varY1n9 •t2.3(Mx) and )~3.3(Mx). F0r the actua1 c0mputat10n5 0f f195. 3 and 4, 1 have a150 taken the term5 -t~2, ~2 1nt0 acc0unt; 1n th15 ca5e a11 e45. (2) have t0 6e 501ved 51mu1tane0u51y. 7he 4ua11tat1ve feature5 0f the var10u5 501ut10n5 can, h0wever, m05t ea511y 6e d15cu55ed u51n9 the 51mp11f1edp1cture de5cr16ed 1n the text.

M. Dree5 / E(6) 5uper5tr1n95

340

1

1

1

1,/11// X2,3(Mx1

,8=70 ° 8= 75°.-.--.~

~=650

0 ~ 1 ~ = 7 5

0

/

M2,= 500 6eV 1A-0 ------ M2,=400 6eVJ -

°

1

0.5

(a)

1

1.0

1.5

2.0

m0/m ~ 1 1

1/~ 1 1 "

1

1 /

1

(6)

--

8 =65

1"•••1"/

8= 70 ° / ,e=750-

X3,3(Mx )

8=75 °

0.1 :

#=70 ° ---

• 65°

0

0.5

1

1.0

M2,=300 6eV ~A-0M2,=400 6eV J

1

1.5

2.0

m0/m ~ F19. 3. (a) X 2.3 ( M x ) 15 5h0wn a5 a funct10n 0f m 0/m9, acc0rd1n9 t0 e45. (2), f0r A = 0, h t ( M x ) = 0.025. 7 h e 501ut10n5 w1th 1ar9er X~,3 are term1nated 6y the re4u1rement5 f0r a p051t1ve 54uared ma55 0f 1eft-handed 51ept0n5 and f0r a f1n1te va1ue 0f ~k3,3 (5ee f19. 36). 1t 15 a55umed that X2, ~ = ~3,, = 0 f0r 1 - 1, 2. (6) X3, 3 ( M x ) a5 a funct10n 0f m0/m ~, acc0rd1n9 t0 e45. (2). Va1ue5 0f parameter5 are a5 1n f19. 3a.

M. Dree5 / E(6)5uper5tr1n95 1

1

1

341

1

A: 2m 0

(a)

/~=750 ~8

,8=65°

P--

=5 0 °

, -

X2,3(Mx) 0.1

-

0.01

8:75°

1

1

0

0.5

1

1

A=2m °

1

1

1.0 m0/m~ 1

1.5

1

2.0

1

(6)

/~:Ts 0 #:75..~

X3,3(Mx) 0.1

~r0 0.01

1 11 0

8=55°

18=50°

1 0.5

1/ 1.0

1 1.5

2.0

m0/m~ F19. 4. (a) A5 1n f19. 3a, 6ut f0r A = 2 m 0. M 2, = 300 6 e V a1way5. (6) A5 1n f19. 36, 6 u t f0r A = 2 m 0. M2, = 300 6 e V a1way5.

342

M. Dree5 / E(6) 5uper5tr1n95

w0u1d 6ec0me ne9at1ve due t0 D-term c0ntr16ut10n5 [18, 23] 0f the new U(1).* N0te the 1ar9e 5pac1n9 (0n a 109ar1thm1c p10t•) 6etween the/3 = 65 ° curve5 f0r M 2, = 300 6 e V (fu11) and 400 6 e V (da5hed), re5pect1ve1y, 1n th15 re910n 0f parameter 5pace; the c0rre5p0nd1n9 curve5 1n the re910n 0f 5ma11 2~3 w0u1d a1m05t 11e 0n each 0ther. 7hu5 the 501ut10n w1th 1ar9e 2~3 15 m0re natura1 1n the 5en5e that 0n1y here the d1men510n 0f parameter 5pace 15 the 5ame at 10w and at h19h ener91e5. 0n1y here 1ead d1fferent va1ue5 0f M 2 , / M 2 (w1th n 2 >> 0 2, ~2 a5 re4U1red 6y exper1ment) t0 e4Ua11y d1fferent Va1Ue5 0f 2•3,3(MX) and 2•2,3(MX). 7 h e effect5 0f r151n9 A fr0m 0 t0 2m 0 are dem0n5trated 1n f19. 4. N0w 50me 501ut10n5 w1th 5ma11 2~3 ex15t 1n the wh01e ran9e fr0m m 0 = 0 t0 m 0 / m ~ ~ 0e. N0te furtherm0re the appearance 0f new 501ut10n5 w1th 5ma11 va1ue5 0f /3 (d0wn t0 13 = 48°), wh1ch nece551tate5 a n0nvan15h1n9 va1ue 0f A2, 3 a1ready at M x and thu5 d0 n0t ex15t 1f m 0 6ec0me5 t00 5ma11. (Remem6er that A - m 0, n0t - m 9 , 51nce 0therw15e the p0tent1a1 w0u1d have char9e- and c010r-6reak1n9 m1n1ma at 1ea5t at 2 0 exc1ude5 a ran9e h19h ener9y 5ca1e5.) F0r 1ar9e va1ue5 0f 8 the re4u1rement m•> 0f va1ue5 0f m 0 / m ~ even f0r the 501ut10n w1th 1ntermed1ate 2, 3. 7he rea50n 15 that an 1ncrea5e 1n A2,3(Mw). tan 8 1ncrea5e5 the r.h.5. 0f e4. (2a) wh1ch 1n th15 re910n 0f parameter 5pace can 6e c0mpen5ated 6y an 1ncrea5e 1n 2•3,3 (5ee f19. 2); th15 decrea5e5 the effect1ve 5uper5ymmetry 6reak1n9 5ca1e. F1na11y, the 501ut10n w1th 1ar9e 2•3 a1m05t d15appear5. Here m 0 / m ~ 15 a9a1n 60unded fr0m 6e10w 6y the re4u1rement f0r p051t1ve m•.2 0 n the 0ther hand, A and thu5 m 0 15 60unded fr0m a60ve, t00, 51nce the R 6 5ca11n9 0f A2, 3 d0e5 n0t 5uff1ce t0 chan9e 1t5 519n 1f A 15 t00 1ar9e a1ready at Mx; a5 d15cu55ed a60ve, th15 chan9e 0f 519n 15 cruc1a1 f0r th15 501ut10n t0 ex15t. Reca111n9 that 501ut10n5 w1th 1ar9e va1ue5 0f 2•3 are m0re natura1 0ne may c0nc1ude that 1ar9e va1ue5 0f A are d15fav0red. 1n th15 c0ntext 1t 15 1ntere5t1n9 t0 n0te that the truncat10n 5cheme 0f ref. [16] y1e1d5 1A1 << m0. 7 h e re5u1t5 0f f195. 3 and 4 have 6een 06ta1ned w1th a 5ma11 va1ue 0f h t. 7h15 d0e5, h0wever, n0t mean that the m0de1 pred1ct5 the t0p-4uark t0 6e 119ht. 7h15 15 0n1y true [18] 1f m 0 = A = 0 51nce 1n th15 re910n the 501ut10n w1th 1ar9e 2•3 15 f0r61dden; the 501ut10n5 w1th 5ma11er va1ue5 0f X 3 then 0n1y a110w f0r ht(Mx)•< 0.045, 1.e. m t •< 75 6 e V , 51nce 1ar9er va1ue5 0f h t 1ead t0 t00 5ma11 va1ue5 0f [A2,3(Mw) 1. H0wever, f0r m 0 4= 0, A = 0, the 501ut10n w1th 1ar9e 2•3 a110w5 f0r t0p-4uark ma55e5 up t0 200 6 e V ; th15 va1ue can a150 6e rea112ed 6y the 501ut10n w1th med1an X 3 1f A = 2m0~0. * 1t 5h0u1d 6e ment10ned that m 0 4:0 f0r 1ar9e X3, 3 15 0n1y nece55ary 1n th15 5pec1f1c m0de1. 7 h e U(1) r• m0de1 0f ref. [9] pred1ct5 p051t1ve D-term5 f0r a11 5tandard 5ferm10n5; ne9at1ve c0ntr16ut10n5 t0 ex0t1c 5ferm10n ma55e5 can 6e c0mpen5ated 6y p051t1ve 5uper5ymmetr1c c0ntr16ut10n5 1f the c0rre5p0nd1n9 ferm10n5 are heavy. 5tart1n9 fr0m an E(6) 6 U 7 9r0up th15 1atter U(1) r• can 0n1y 6e rea112ed 1f there 15 an 1ntermed1ate 5ca1e wh1ch make5 a c0mprehen51ve R 6 ana1y515 very d1ff1cu1t, a5 d15cu55ed 1n the 1ntr0duct10n. 1 6e11eve, h0wever, that the 4ua11tat1ve feature5 0f the 501ut10n5 0f e45. (2) (w1th m0d1f1ed U(1)-char9e5) w0u1d rema1n a5 de5cr16ed 1n the text, 1nc1ud1n9 the ex15tence 0f up t0 three 1ndependent 501ut10n5.

M. Dree5 / E(6) 5uper5tr1n95

343

7he 0n1y c0n5tra1nt5 enter1n9 the re5u1t5 0f f195. 3 and 4 are that e45. (2) are fu1f111ed and that 54uared ma55e5 0f a11 5ca1ar5 are p051t1ve. 1n pr1nc1p1e, further c0n5tra1nt5 0n the parameter 5pace c0u1d emer9e fr0m the re4u1rement that the m1n1mum 0f the H1995 p0tent1a1 wh1ch 15 de5cr16ed 6y e45. (2) 15 the a6501ute m1n1mum 0f the 5ca1ar p0tent1a1. 1n ref. [9] a 5et 0f c0nd1t10n5 ha5 6een der1ved wh1ch 9uarantee the a65ence 0f certa1n c1a55e5 0f unwanted m1n1ma. 1t turn5 0ut that 1n the re910n 0f 1ar9e )~3 0ne 0f the5e c0nd1t10n5 15 0ften v101ated; h0wever the c0rre5p0nd1n9 vacuum w1th n = v = 9 ha5 a 1ar9er va1ue 0f ( V ) than the de51red 0ne w1th n 2 >> u 2, ~2 and 15 thu5 d15fav0red. 1t 5h0u1d f1na11y 6e ment10ned that ch0051n9 ~2,1 and ••2,2 t0 d1ffer fr0m 2er0 0r even t0 6e 0f 0rder 0ne pr0duce5 a 519n1f1cant effect f0r the parameter5 0f the H1995 p0tent1a1 0n1y 1f the ~3,1 are very 5ma11. A5 ment10ned ear11er, a110w1n9 f0r 3 de9enerate ex0t1c D-4uark5 reduce5 the 10wer 60und 0n the effect1ve 5uper5ymmetry 6reak1n9 5ca1e 6y a60ut 20%; theref0re, the c0n5tra1nt rn-~ > 0 then re4u1re5 50mewhat 1ar9er va1ue5 0f m0/m ~ 1f the )k3, 1 a r e 1ar9e. 0 n the 0ther hand, the 5tr0n9er R 6 5ca11n9 0f the parameter5 enter1n9 e45. (2) then a110w5 f0r 1ar9er va1ue5 0f m0/m ~ than 1n the ca5e 0f 0n1y 0ne 1ar9e )t 3 c0up11n9. 7hu5 the net effect 0n the 501ut10n w1th 1ar9e )t 3 15 a 5h1ft 0f the curve5 0f f195. 3 and 4 t0ward5 1ar9er va1ue5 0f m0/m ~. 7he 501ut10n5 w1th 5ma11er X3 rema1n a1m05t unchan9ed 1f 0ne rep1ace5 ~k3, 3 6y ~k3,1/V/3. 7he ma1n re5u1t5 0f th15 5ect10n are that the m0de1 d0e5 n0t 91ve any 5tr0n9 c0n5tra1nt5 0n the va1ue5 0f the unkn0wn Yukawa c0up11n95, and that the 5uper5ymmetry 6reak1n9 5ca1e can 6e 5u65tant1a11y 5ma11er than M2,. 1n the f0110w1n9 5ect10n5 1 w111 1nve5t19ate what th15 mean5 f0r the part1c1e ma55e5.

3. Ex0t1c5 and 5ex0t1c5

1 w1116e91n w1th a d15cu5510n 0f the ex0t1c D-4uark5 and the1r 5uperpartner5. 7he 119hte5t 0f the5e part1c1e5 w0u1d 6e a6501ute1y 5ta61e 1f the fu11 5uperp0tent1a1 were 91ven 6y e4. (1), 1ead1n9 t0 an unaccepta61e a6undance 0f ex0t1c nuc1e1.7heref0re at 1ea5t 0ne 0f the f0110w1n9 term5 ha5 t0 6e added t0 the 5uperp0tent1a1 [9]:

h~Dd~R,

)~LLQLD,

)~e~u~D,

)~8QLQLD,

)~8u~d~7). (6)

A5 5tated 1n ref. [9], ~L and )~L 0r ••8 and )~8 have t0 van15h 51nce 0therw15e the pr0t0n w0u1d 6e much t00 un5ta61e. 1n the prev10u5 5ect10n, 1 have ne91ected a11 the5e c0up11n95. 7h15 15 certa1n1y a 900d appr0x1mat10n, a5 far a5 the parameter5 0f the H1995 p0tent1a1 are c0ncerned, f0r the f1r5t three 0f the5e c0up11n95, wh1ch have t0 6e 5ma11; 0therw15e the R 6 5car1n9 w0u1d dr1ve the 54uared ma55 0f the c0rre5p0nd1n9 51ept0n ne9at1ve wh1ch w0u1d 6e d15a5tr0u5 even f0r the ~R 1n the 91ven m0de1 [24]. 0 n the 0ther hand, 512ea61e va1ue5 0f ~8 and )~8 w0u1d decrea5e 2 2 m 6 and m 6 and thu5 510w d0wn the ev01ut10n 0f m 23, 1f N 3 c0up1e5 t0 the 5ame 5et

M. Dree5 / E(6) 5uper5tr1n95

344

0f D-4uark5; th15 w0u1d 1ncrea5e the re4u1red am0unt 0f 5uper5ymmetry 6reak1n9 1n 0rder t0 keep M 2. f1xed (5ee e4. (3)). 1n any ca5e 1t 15, h0wever, p055161e that a11 the c0up11n95 1n e4. (6) are 5ma11.7he 1mpact 0f the5e c0up11n95 0n 5ferm10n ma55e5 w111 6e d15cu55ed 1n 5ect. 4. At th15 p01nt 1t 15 n0t c1ear whether the D-4uark 0r the D-54uark ha5 even R-par1ty: 1f 0n1y h v v~ 0, 0ne can ch005e R ( D ) = R ( 9 R ) = +1, a1th0u9h R ( / ) ) = R0~R) = + 1 15 a150 p055161e; 1f 0ne 0f the 0ther c0up11n95 1n e4. (6) 15 n0nvan15h1n9 0ne ha5 t0 ch005e R(13)= + 1. 7he ma55e5 0f the D-4uark5 are 91ven 6y mD=;k3.1(MW)•n

,

(7)

1= 1,2,3.

A5 ment10ned ear11er, the X3,1(Mx) are n0t c0n5tra1ned 6y the m0de1. H0wever, the R 6 E f0r X 3,1 ha5 an 1nfrared f1xed p01nt at 3

22~23,1+ 3 E ~23,j = 1692 3 3 4.• ~ 9 ~ -

(8)

j=1

A5 5h0wn 1n f19. 5, th15 f1xed p01nt 15 reached a1ready f0r m0derate va1ue5 0f X3(Mx). 1n th15 and the f0110w1n9 f19ure, d D and d H are the num6er 0f de9enerate

1

1

1

1

1

1

1

1

1

1

1

1

1

5

2

mD/M 2,

1

0 0

[

0.2

0.4 X3(M x)

0.6

F19. 5. 7he ma55 0f the ex0t1c D-4uark 1n un1t5 0f M 2, a5 a funct10n 0f X3(Mx). dr) and d H are the num6er5 0f de9enerate ex0t1c D-4uark5 and 171-1ept0n5, re5pect1ve1y. A11 Yukawa c0up11n95 wh1ch are n0t nece55ary t0 pr0duce the5e ma55e5 have 6een ne91ected.

M. Dree5 / E(6) 5uper5tr1n95

345

ex0t1c D-4uark5 and 171-1ept0n5, re5pect1ve1y, a11 0ther Yukawa c0up11n95 6e1n9 a55umed t0 6e ne9119161e. Fr0m th15 f19ure 0ne f1nd5 f0r the ma55 0f the heav1e5t D-4uark m heavy D ~<

3.1M2,

(9)

F0r a 91ven va1ue 0f the 1.h.5. 0f e4. (8) the ma55 0f the 119hte5t D-4uark 15 max1ma1 1f a11 ex0t1c 4uark5 are de9enerate 1n ma55. Fr0m f19. 5 0ne then f1nd5 m~)9ht 6

2.1M 2,

(10)

f0r the 119hte5t D-4uark, th15 11m1t 6e1n9 reached 1f a11 ;~2,1 0. 1t 5h0U1d 6e ment10ned that the5e 60Und5 are Va11d 1n a11 5Uper5tr1n9 1n5p1red m0de15 W1th0Ut an 1ntermed1ate 5Ca1e. 7he 1ntr0dUCt10n 0f add1t10na1 YUkaWa C0Up11n95 Can 0n1y 5tren9then them. 7h15 15 dem0n5trated 6y the 10We5t 0f the CUrVe5 1n f19. 5 Where ) k 2 , 1 ( M X ) = ~ , 2 , 2 ( M x ) = 1 ha5 6een a55Umed. H0WeVer, even the5e 1ar9e va1Ue5 0f 7t2,1 d0 n0t 9reat1y a1ter the eV01Ut10n 0f X3,1" N0te that a5 500n a5 =

m D >• M2,(d D = 1)

and

m D >~0 . 6 M 2 , ( d D = 3),

(11)

re5pect1ve1y, the decay D ~ 02X 0 6eC0me5 p055161e, Where D 2 15 the 119hter 0f the tW0 m1Xed D--D e19en5tate5 [9,18] and 27 15 the 119hte5t e19en5tate 0f the neutra1 9aU91n0-h19951n0 m1X1n9 (5ee 6e10W). 7he preC15e Va1Ue 0f m D a60Ve Wh1Ch th15 decay 6eC0me5 p055161e 5tr0n91y depend5 0n the Way 5Uper5ymmetry 15 6r0ken; 1f A / m 0 15 1ar9e th15 Can a1ready happen f0r D-4Uark ma55e5 5U65tant1a11y 5ma11er than 1nd1Cated 1n e4. (11). 7he ma55 0f the Char9ed eX0t1C 1ept0n /~1+ 15 91Ven 6y mf47=2t2,1(Mw).n,

1=1,2.

(12)

A5 n0ted 1n ref. [21], the 51tuat10n f0r the neutra1 ex0t1c5 15 m0re c0mp11cated 51nce they have t0 m1x w1th the 141,2 51n91et ferm10n5 1n 0rder t0 91ve ma55e5 t0 them*. 7he upper 11m1t5 f0r char9ed 1ept0n ma55e5 pre5ented 6e10w 5h0u1d, h0wever, a150 6e va11d f0r the neutra1 ex0t1c 51nce 1n the 11m1t 0f 1ar9e m 1~ the ma55 0f the heavy e19en5tate5 15 a150 91ven 6y e4. (12) [21]. 7he5e m1x1n9 term5 f0rce the ex0t1c ferm10n5 t0 6e R-0dd, wh1ch exp1a1n5 my n0tat10n. * 51nce the m0de1 d0e5 a1ready c0nta1n three ma551e55 r19ht-handed neutr1n05 [18] 1n add1t10n t0 the u5ua1 1eft-handed 0ne5 the 1ntr0duct10n 0f tw0 add1t10na1 ma551e55 ferm10n5 w0u1d de5tr0y pred1ct10n5 [25] f0r pr1m0rd1a1 nuc1e05ynthe515 un1e55 the 2•-6050n 15 very heavy. A 5ma11 ma55 f0r the ~4a,2 ferm10n5 can 6e 9enerated rad1at1ve1y [26], 6ut then 0ne ha5 t0 6e carefu1 n0t t0 0verc105e the un1ver5e. 1n any ca5e the N1,2-M1.2 m1x1n9 15 de51red 51nce 1t a110w5 the 119hte5t 0f the ex0t1c d0u61et 1ept0n5 and 51ept0n5 t0 decay: w1th0ut the5e term5 the c0n5ervat10n 0f an ex0t1c 1ept0n num6er w0u1d f0rce 1t t0 6e 5ta61e.

M. Dree5 / E(6) 5uper5tr1n95

346

2.4

1

1

1

1

1

1

1

2.0 mmax1M

- 2• 1.6 d H = 2 ~ d D = 3 ~ 1.2 0

1

1

1

0.2

1

0.4 X3(Mx)

1

1

0.6

1

F19. 6. 7 h e max1ma1 ma55 0f the char9ed ex0t1c 1ept0n5 1n un1t5 0f M 2, a5 a funct10n 0f )t3(Mx); the 6 0 u n d emer9e5 fr0m the re4u1rement that 3, 2 rema1n5 f1n1te up t0 M x. 7he n0tat10n 15 a5 1n f19. 5.

7he R 6 E f0r ••2,1 a150 ha5 an 1nfrared f1xed p01nt; h0wever, 5tart1n9 w1th a 1ar9e va1ue 0f • 2,1(Mx) the R 6 5ca11n9 15 n0t fa5t en0u9h t0 dr1ve th15 c0up11n9 t0 1t5 f1xed p01nt. Fr0m f19. 6 0ne read5 0ff that ma55e5 0f the ex0t1c 1ept0n5 can 6e 5u65tant1a11y 1ar9er than the1r f1xed p01nt va1ue; the 1atter w0u1d e.9., 91ve m1~ •< 1.55M 2, f0r ••3 = 0, d H = 1. Fr0m th15 f19ure 0ne f1nd5 1n5tead f0r the ma55 0f the heav1er ex0t1c 1ept0n 2.2M2, m Hheavy< +

•

(13)

mn+ •< 1.8M2, ,

(14)

wh11e the 119hter 0ne ha5 t0 06ey 119ht

wh1ch a9a1n c0rre5p0nd5 t0 the ca5e 0f de9enerate c0up11n95. 7he5e 60und5 are 10wer 1f 50me 0f the ••3 are 1ar9e, a5 5h0wn 1n the f19ure. L1ke the c0rre5p0nd1n9 60und5 (9) and (10), the 60und5 (13) and (14) are va11d f0r any E(6) 5uper9rav1ty m0de1 w1th0ut 1ntermed1ate 5ca1e5. 7 h e ma55 matr1x f0r the ex0t1c 5ferm10n5 ha5 6een 91ven 1n ref. [21], Unf0rtunate1y 1t5 e19enva1ue5 d0 n0t 0n1y depend 0n ~2,1(Mx) and the rat105 m0/m ~,

M. Dree5 / E(6) 5uper5tr1n95 A/m

347

0, 6ut a150 0n )t3,,(Mx) wh1ch determ1ne the 5uper5ymmetry 6reak1n9 5ca1e

(5ee f19. 1) and effect the R 6 5ca11n9 0f A2,1 and X2,1 (5ee e45. (A.7) and (A.4)). 7hu5 the re1at10n 6etween the ma55e5 0f the ex0t1c ferm10n5 and the1r 5uperpartner5 15 c0mp11cated. 1f, h0wever, 0ne 0f the X3,1 15 1ar9e the ma55 0f the ex0t1c ferm10n5 mu5t n0t 6e ar61trar11y 5ma11 51nce 0therw15e the 54uared ma55 0f 0ne 0f the ex0t1c 5ferm10n5 w0u1d 6e ne9at1ve [18]. 1n the re910n 0f 1ar9e X3, mD, >• M2,, 0ne f1nd5 (f0r A = 0) m~+ >• 0.5M2,.

(15)

1t 15 1ntere5t1n9 t0 n0te that the ex0t1c 5ferm10n ma55 matr1x 15 d0m1nated 6y 1t5 5uper5ymmetr1c entr1e5 1f 60th X 2,1 and X3,1 are 1ar9e; th15 mean5 that ne1ther the decay 171+ -+ H,+ X1 ~0 n0r H + H1 X1 15 p055161e. H0wever, n0ne 0f the5e decay - +•° m0de5 15 exc1uded 1n 9enera1. A11 ma55 60und5 1n th15 5ect10n have 6een 91ven 1n term5 0f M2,; up t0 n0w there ex15t 0n1y 10wer 60und5 0n th15 4uant1ty. 1n ref. [27] 1t ha5 6een 5h0wn that ex15t1n9 1a60rat0ry data 91ve the rather m11d 60und M 2, >• 130 6eV. A 5tr0n9er 60und can 6e 06ta1ned fr0m c05m01091ca1 ar9ument5 [25], a5 ment10ned ear11er. 7he rea50n 15 that the ma551e55 PR ferm10n5 w0u1d 1ncrea5e the pr1m0rd1a1 pr0duct10n 0f 4He 6y an unaccepta61e am0unt un1e55 they dec0up1e 5uff1c1ent1y ear1y, 1.e. the 2•-6050n 15 5uff1c1ent1y heavy. 7ak1n9 a11 c05m01091ca1 mea5urement5 at face va1ue 0ne der1ve5 [25] M 2, >•. 400 6eV; h0wever, the actua1 60und c0u1d 6e 10wer due t0 uncerta1nt1e5 [28] ma1n1y 1n the pr1m0rd1a1 pr0duct10n 0f 3He and D.

4. 5uperpartner5 0f 5tandard ferm10n5 and 9au9e 6050n5 7he char9ed h19951n0-w1n0 ma55 matr1x 0f the c0n51dered 5uper5tr1n9 m0de1 15 c0mp1ete1y ana1090u5 t0 that 0f m1n1ma1 5uper9rav1ty [10], where the 5uper5ymmetr1c h19951n0 ma55/, ha5 t0 6e rep1aced [18, 26] 6y X2,3(Mw)n. 51nce 0ne expect5 th15 4uant1ty t0 6e 4u1te a 61t 1ar9er than M w the m1x1n9 15 1n 9enera1 5ma11. 1n the re910n 0f 1ar9e X3 the w1n0-11ke 5tate 15 a1way5 the 119hter 0ne. N0te that 1n th15 re910n 0f parameter 5pace the tw0 term5 1n the determ1nant 0f the ma55 matr1x tend t0 cance1 each 0ther, wh1ch mean5 that the m0de1 a110w5 f0r ar61trar11y 119ht w1n05. 0 f c0ur5e, 1n a rea115t1c 51tuat10n the exper1menta1 60und [29] rnw >- 35 6 e V ha5 t0 6e 06eyed. 7 h e neutra1 9au91n0-h19951n0 ma55 matr1x 15 n0w a 6 × 6 matr1x [18]; h0wever, 1n the 11m1t M22, >> M 2 the 5uperpartner5 0f the 2•-6050n and the N3-H1995 e55ent1a11y dec0up1e fr0m the 0ther f0ur neutra11n05 and 9et ma55e5 - M 2, [21]. 7he rema1n1n9 4 × 4-matr1x 15 a9a1n that 0f m1n1ma1 5uper9rav1ty w1th/, rep1aced 6y )k2,3(Mw)n; n0 10wer 11m1t can 6e 91ven f0r the ma55 0f the 119hte5t e19en5tate wh1ch 15 d0m1nant1y a 9au91n0.

M. Dree5 / E(6) 5uper5tr1n95

348

1f a11 Yukawa c0up11n95 0f e4. (6) are ne9119161e the ma55e5 0f 54uark5 and 51ept0n5 are 91ven 6y: m~2R= m~ + 0.14m~ - 0.23 c05(2/3)M 2 + 0.0072D E 2 = m~ + 0 . 3 8 m ~ - 0.27c05(2/3)M22

m~L

where D E -

0.0036DE,

(16a) (166)

m~2L = rn~ + 0.38m~ + 0.5 c05(2/3)M22 - 0.0036D 2

(16c)

rn~2L = m 9 + 3.81m~ + 0.35 c05(2/3)M 2 + 0.0072D 2

(16d)

maL2= rn~ + 3.81m~ - 0.42c05(2/3)M 2 + 0.0072D2,

(16e)

m-2uR = m2 + 3.51rn~ + 0.15 c05(2/3)M22 + 0.0072DE,

(16f)

ma,2 = m~ + 3.46m 2 - 0.08 c05(2f1)M 2 - 0.0036D E ,

(169)

5n 2 -- U 2 --

4v2 M22.>>M~28LM22~ and 1 have u5ed 51n20w = 0.23.

A5 ment10ned ear11er the D-term c0ntr16ut10n5 due t0 the new U(1) t0 ma55e5 0f 5U(2)-d0u61et 51ept0n5 are ne9at1ve; 1n the re910n 0f 1ar9e X3 they w0u1d dr1ve the 54uared 51ept0n ma55e5 ne9at1ve 1f m 0 = 0 (5ee 5ect. 2). N0te that 1n the 91ven m0de1 c052/3 15 a1way5 ne9at1ve wh1ch 1mp11e5 m~L < m e . 1f m 0 15 c105e t0 1t5 m1n1ma1 va1ue, the 1eft-handed 5neutr1n0 (m0re 5pec1f1ca11y: 7he 5uperpartner 0f the 1efthanded r-neutr1n0) can even 6y the 119hte5t 5uper5ymmetr1c part1c1e• A5 ment10ned a60ve, e45. (16) are 0n1y va11d 1f the c0rre5p0nd1n9 4uark5 and 1ept0n5 d0 n0t have 519n1f1cant Yukawa c0up11n95. 7hu5 the 119hter 5uperpartner 0f the t0p-4uark can 6e 5u65tant1a11y 119hter than 1nd1cated 6y e45. (16d, e). Furtherm0re, 50me 0f the c0up11n95 0f e4. (6) m19ht 6e n0nne9119161e. 7he 51ept0n ma55e5 can ea511y 6e decrea5ed 6y ch0051n9 2~L, ~L ~ 0; 1n fact, 0ne ha5 a9a1n t0 6e carefu1 n0t t0 9et m~L < 0. A5 d15cu55ed 1n 5ect. 2, /3 ha5 t0 6e 1ar9e 1f 9au91n0 ma55e5 are the d0m1nant 50urce 0f 5uper5ymmetry 6reak1n9; c0m61n1n9 the 60und (5) 0n f1 w1th e45. (166, c) 0ne f1nd5 [23] rn% > 60 6 e V

(m0=A =0).

(17)

A v101at10n 0f th15 60und w0u1d 1nd1cate a m0re 9enera1 5uper5ymmetry 6reak1n9, 51nce 1n th15 ca5e/3 can 6e a5 5ma11 a5 48 °, 1.e. meL 0n1y ha5 t0 6e 1ar9er than 25 6 e V ; th15 60und 15 n0t much 5tr0n9er than the 11m1t that emer9e5 fr0m n0n065ervat10n 0f 5e1ectr0n5 at PE7RA [30]. An0ther p05516111ty 15 t0 ch005e Xn, ~8 v~ 0. 1n 0rder t0 5tudy the 1mpact 0f the5e c0up11n95 0n 54uark ma55e5 1 have c0mputed the re1evant R 6 E f0r the 51mp11f1ed

M. Dree5 / E(6) 5uper5trm95

349

ca5e where a11 0ther Yukawa c0up11n95 are ne9119161e; they are 115ted 1n append1x 8. N 0 t e that the R 6 E f0r 7~8, A8, m24 L and m 2 dec0up1e fr0m th05e f0r X~8, A~, m 2uR• 2 2 m a r and m~. F19. 7a 5h0w5 m 6 (da5hed) and m~L (fu11 curve) 1n un1t5 0f the 91u1n0 ma55 a5 a funct10n 0f X 8 ( M x ) f0r A = 0. 1 have a55umed that X8 15 d1a90na1 1n 9enerat10n 5pace; 1ar9e 0ff-d1a90na1 X8 c0u1d pr0duce 1ar9e f1av0r chan91n9 neutra1 current5 v1a D-exchan9e. 1t may 6e 50mewhat 5urpr151n9 t0 n0te that X8 can dr1ve m 2 t0 ne9at1ve va1ue5. Fr0m the re4u1rement that th15 d0e5 n0t happen 0ne der1ve5 the 60und m~,~ >• 1.4m~, (18) f0r the ma55 0f 5U(2)-d0u61et 4uark5, th15 11m1t 6e1n9 reached f0r m 0 = 0. the 0ther hand, the ma55e5 0f r19ht-handed f1- and c1-54uark5 and ex0t1c ~ 0n D-54uark5 a11 ev01ve 1n the 5ame way 1f the 5ma11 U(1) c0ntr16ut10n5 t0 the1r R 6 E are ne91ected; F19. 76 5h0w5 that they can 6e ar61trar11y 5ma11 even f0r A = 0 1f rn~ >> m 0. H0wever, the D-term c0ntr16ut10n5 have 6een ne91ected 1n th15 f19ure; fr0m e45. (16f, 9) 0ne der1ve5 m~, >~ 0.55M 2, 2

(19)

fr0m the re4u1rement m8, >1 0. 1f the c05m01091ca1 10wer 11m1t [25] 0n M 2, 15 1ndeed va11d th15 1mp11e5 that the 5uperpartner5 0f r19ht-handed up-4uark5 are pr06a61y t00 heavy t0 6e detected at the 7evatr0n c0111der [31]. F19. 76 5eem5 t0 1nd1cate that there 15 a n0ntr1v1a1 10wer 60und a150 0n mar 1f m 0 v~ 0; th15 15, h0wever, 0n1y true 1f A -- 0 a5 1n the truncat10n 5cheme 0f ref. [16]. 0 n the 0ther hand the 1ne4ua11t1e5 (18) and (19) f0110w d1rect1y fr0m the re4u1rement that c010r 15 n0t 6r0ken; un11ke e45. (16), wh1ch 0n1y h01d f0r 5ma11 va1ue5 0f )~L, XL,, X8 and )tw, they are thu5 9enu1ne pred1ct10n5 0f the m0de1. 50 1n pr1nc1p1e the U(1)E D-term c0ntr16ut10n5 a5 we11 a5 the var10u5 c0up11n95 1n e4. (6) c0u1d 1ead t0 very d1fferent ma55e5 f0r the var10u5 54uark5 and 51ept0n5. N0te that very 1ar9e d1fference5 6etween ma55e5 0f 1eft- and r19ht-handed 5ferm10n5 can 91ve 1ar9e c0ntr16ut10n5 t0 0 - 1; h0wever, 1n the 91ven m0de1 th15 4uant1ty a150 9et5 c0ntr16ut10n5 fr0m 2-2• m1x1n9 [20,27] a5 we11 a5 the ex0t1c 5ect0r [32]. 1t 15 theref0re n0t ea5y t0 make 4uant1tat1ve 5tatement5. Furtherm0re, 1n pr1nc1p1e, f195. 7 h01d f0r each 9enerat10n 5eparate1y. 1t 15 we11-kn0wn [33], h0wever, that 1ar9e ma55 d1fference5 6etween 54uark5 0f d1fferent f1av0r can pr0duce 1ar9e f1av0r chan91n9 neutra1 current5. 7heref0re, X 8 and ~t~ have t0 6e e1ther 5ma11 0r r0u9h1y the 5ame f0r a11 9enerat10n5. Neverthe1e55 1t 15 we11 p055161e that there are 512ea61e d1fference5 6etween the var10u5 54uark ma55e5. Unf0rtunate1y 1t 15 hard t0 make any 5tatement5 a60ut the1r re1at1ve 0rder1n9 6ey0nd mar > m~1R. Remem6er, h0wever, that Xn, )t~8 • 0 1mp11e5 X L = X~L ---0 and v1ce ver5a; thu5 e1ther the 51ept0n5 06ey e45. (16a-c) 0r the 54uark5 06ey e45. (16d-9). 7heref0re, the5e e4uat10n5 are u5efu1 even 1n the m05t 9enera1 51tuat10n.

M. Dree5 / E(6) 5uper5tr1n95

350

3

1

1

1

1

(a)

~ 0 = 2 r n ~

-

m0= m

•N

• •

•

• •

•

• •

• •

---4:0

]

• •

•

A:0

k•

0.02•

0.0t

•

••

X•

4:uL1dL

0

•

••

X

••

•

•

•

•

1

0.04

0.03

•

0.05

X8(M x) 1

1

1

1

1

1

1

1 1

1

1

1

1

A:0

1

1

1

(6)

rn~/m 1

1

0

0.01

1

J

1

1

1 1 • 1J

0.1

~

1

1

•

•

1 1 1 1 1

1.0

Ke(Mx) F19. 7. (a) Ma55e5 0f ex0t1c f)-54uark5 and 1eft-handed f1-,d-54uark5 1n un1t5 0f the 91u1n0 ma55 a5 funct10n5 0f X~( M x ) f0r A = 0. A11 0ther Yukawa c0up11n95, a5 we11 a5 D-term c0ntr16ut10n5, have 6een ne91ected. (6) 7 h e ma55 0f the r19ht-handed d-54uark 1n un1t5 0f the 91u1n0 ma55 a5 a funct10n 0f X~8( M x ) f0r A = 0. A11 0ther Yukawa c0up11n95, a5 we11 a5 D-term c0ntr16ut10n5, have 6een ne91ected. 7 h e 5ame curve5 are a150 va11d f0r r19ht-handed f1-54uark5 and ex0t1c D-54uark5 1f the 5ma11 U(1) c0ntr16ut10n5 t0 the re1evant R 6 E are ne91ected.

M. Dree5 / E(6)5uper5tr1n95

351

5. Deeay m0de5 0f the2•-6050n 7 h e ex15tence 0f a new 9au9e 6050n 2• w1th a ma55 0f a few hundred 6 e V 0r a 7eV, a5 pred1cted 6y the m0de1, can ea511y6e ver1f1ed [34] at a 1ar9e hadr0n c0111der 5uch a5 the 55C 6y 1t5 decay 1nt0 char9ed 1ept0n5. 06v10u51y, the pred1ct10n f0r the 512e 0f th15 519na1 d0e5 n0t 0n1y depend 0n the ma55 6ut a150 0n the t0ta1 decay w1dth 0f the 2•. 1t 15 theref0re w0rthwh11e t0 5tudy wh1ch ex0t1c decay channe15 have t0 6e 0pen 1n the 91ven m0de1, and wh1ch channe15 have t0 6e c105ed. 7h15 a150 91ve5 50me 1dea 0f what k1nd 0f new phy51c5 can 6e expected t0 6e d15c0vered m0re 0r 1e55 51mu1tane0u51y w1th the new 9au9e 6050n. 0 f c0ur5e, the 2• can decay 1nt0 a11 5tandard ferm10n5; the1r c0up11n95 are, h0wever, rather 5ma11, and 0ne f1nd5 [27]: F(2• ---, 5tandard ferm10n5) -- 0.0064M2,.

(20)

A5 ment10ned ear11er, 1n the 91ven m0de1 the r19ht-handed neutr1n05 are 5tr1ct1y ma551e55. Furtherm0re the 37-ferm10n5 0f the f1r5t tw0 9enerat10n5 9et the1r ma55e5 0n1y fr0m VEV•5 v, ~; theref0re the NN channe1 15 a150 0pen un1e55 the 2• 15 very f19ht wh1ch w0u1d cau5e c05m01091ca1 pr061em5 [25] (5ee end 0f 5ect. 3). 7he5e part1c1e5 c0ntr16ute [27] (21)

F(2• --* 3 ~ , ~ , + 2NA7)= 0.006M2, ,

1.e., the t0ta1 decay w1dth 15 a1m05t d0u61ed. F1na11y, due t0 the re1at10n 6etween the ~3,1 Yukawa c0up11n95 and the 5uper5ymmetry 6reak1n9 5ca1e dep1cted 1n f19. 1, the 2• can a1way5 decay e1ther 1nt0 a11 ex0t1c D-4uark5 0r 1nt0 the 5uperpartner5 0f the 119ht 1ept0n5. N0te that f0r th15 1atter decay the thre5h01d fact0r ( 1 4m~/M2,) 3/2 :-/8 3 5h0u1d 6e taken 1nt0 acc0unt, where m 5 15 the ma55 0f the 5ca1ar under c0n51derat10n; the f1-fact0r f0r decay5 1nt0 ferm10n5 can 6e ne91ected un1e55 the ferm10n ma55 15 very c105e t0 5M2,. 1 1 F0r the ca5e mD, = ~M2,, 1= 1,2,3, 0ne f1nd5 f1 -- 0.6 a5 an avera9e va1ue f0r the 51ept0n5, and thu5 F(2• --* ••0rd1nary••

51ept0n5, m ~= 0.4M 2,) = 0.0007M2,,

(22a)

wherea5 the c0ntr16ut10n5 0f three ex0t1c D-4uark5 w0u1d 6e much 1ar9er [27]. F(2• ~ 3. DD) = 0.0075M2,.

(23)

1n the 91ven m0de1 the m1n1ma1 decay w1dth 0f the 2• 15 thu5 06ta1ned 6y add1n9 the c0ntr16ut10n5 0f e45. (20)-(22a), wh1ch y1e1d5 the max1ma1 6ranch1n9 fract10n f0r decay5 1nt0/~+/~ : 8 .... (2• --*/~+/2-) -- 1.8%.

(24)

352

M. Dree5 / E(6)5uper5tr1n95

0 f c0ur5e, add1t10na1 decay channe15 f0r the 2• can 6e 0pen. 7he ex0t1c 5U(2)d0u61et 1ept0n5 0f the f1r5t tw0 9enerat10n5 can 6e 119ht, c0ntr16ut1n9 [27] F(2~

2.11ft) --- 0.0033M2,.

(25)

8y ra151n9 the ma55 0f 0ne D-4uark wh11e keep1n9 the 0ther tw0 119ht the avera9e f1-fact0r f0r a11 51ept0n5, 1nc1ud1n9 the ex0t1c5, can a1m05t 6e ra15ed t0 0ne, y1e1d1n9 Fma~ (2• --+ a11 51ept0n5) -- 0.0051M2,.

(226)

F1na11y, a5 d15cu55ed 1n 5ect. 4, r19ht-handed d-4uark5 a5 we11 a5 the f3-54uark5 0f the f1r5t tw0 9enerat10n5 can 6e 119ht, 1n wh1ch ca5e the1r c0ntr16ut10n 15 F(2• --+ 54uark5) = 0.003M2,.

(26)

N0te that the decay 0f the 2• 1nt0 W-pa1r5 and 9au91n0-h19951n0 5tate5 15 e1ther k1nemat1ca11y f0r61dden 0r 5uppre55ed 6y 5ma11 m1x1n9 an91e5. Furtherm0re the 2• cann0t decay 1nt0 pa1r5 0f char9ed H1995 6050n5 0r 1nt0 the p5eud05ca1ar and 0ne 0f the heavy neutra1 H1995 6050n5 [21, 35]. 7he decay 1nt0 the p5eud05ca1ar and the 119hter neutra1 5ca1ar H1995 [36] can 6e p055161e, a5 we11 a5 the 2• --+ 2 H ° decay [37], 6ut the1r c0ntr16ut10n5 t0 the t0ta1 decay w1dth 0f the 2• are 5ma11. 0 n e theref0re f1nd5 8m1n(2• -+/x+/,t - ) -~ 0.8%.

(27)

7hu5 the m0de1 pred1ct5 the t0ta1 decay w1dth 0f the 2•-6050n up t0 a fact0r 0f 2; w1th0ut any 1nf0rmat10n a60ut the p055161e va1ue5 0f Yukawa c0up11n95 and 5part1c1e ma55e5 th15 4uant1ty va1ue can 0n1y 6e pred1cted up t0 a fact0r 0f 6.

6. 5ummary and c0nc1u510n5 1n th15 paper the a110wed ran9e 0f part1c1e ma55e5 1n the 51mp1e5t rank-5 E(6) 5uper5tr1n9 m0de1 ha5 6een 1nve5t19ated 1n deta11. 1n 5ect. 2, 1 have 5h0wn that the rad1at1ve 6reak1n9 0f the under1y1n9 5U(2) × U(1)r x U(1)E-5ymmetry a110w5 f0r 1ar9e Yukawa c0up11n95 and thu5 a re1at1ve1y 5ma11 (c0mpared t0 M2,) 5uper5ymmetry 6reak1n9 5ca1e. 7h15 h1thert0 unexp10red re910n 0f parameter 5pace ha5 tw0 n1ce feature5: the m0de5t va1ue5 0f 5part1c1e ma55e5 d0 n0t endan9er the pertur6at1ve 5ta6111ty 0f the 9au9e h1erarchy even 1f M 2, - 0(1 7eV); and due t0 50me 50rt 0f f1xed p01nt 6ehav10r 0f the re1evant R 6 E the h1erarchy M22, >> M 2 can 6e 06ta1ned w1th0ut f1netun1n9 0f the parameter5 at the 6 U 7 5ca1e. H0wever, 1n the 91ven m0de1 the5e 1ar9e Yukawa c0up11n95 1ead t0 ne9at1ve 54uared ma55e5 f0r 1eft-handed 51ept0n5 un1e55 5ca1ar5 have a n0nvan15h1n9 ma55 a1ready at the 6 U 7 5ca1e. 1n 5ect. 3 1t ha5 6een 5h0wn that the heav1e5t ex0t1c 4uark and 1ept0n ha5 t0 6e 119hter than 3.1M 2, and 2.2M2,, re5pect1ve1y, wh11e the 119hte5t 0f the5e part1c1e5

M. Dree5 / E(6) 5uper5tr1n95

353

cann0t 6e heav1er than 2.1M 2, and 1.8M2,, re5pect1ve1y. F0r 5uff1c1ent1y 1ar9e Yukawa c0up11n95 the ex0t1c ferm10n5 w1116e heav1er than the1r 119hter 5uperpartner and m19ht even 6e a61e t0 decay 1nt0 1t p1u5 a neutra11n0. 1n 5ect. 4 the 5pectrum 0f the rema1n1n9 5uperpartner5 ha5 6een d15cu55ed. 1t ha5 6een dem0n5trated that add1t10na1 Yukawa c0up11n95 can make r19ht-handed c154uark5 0r a11 51ept0n5 a1m05t ar61trar11y 119ht; e5pec1a11y 1n th15 1a5t ca5e the 1eft-handed ~--5neutr1n0 m19ht even 6y the 119hte5t 5uper5ymmetr1c part1c1e. 0 n e char91n0 and tw0 neutra11n05 can a150 6e 119ht. F1na11y, 1n 5ect. 5 1t ha5 6een 5h0wn that the 91ven m0de1 pred1ct5 the t0ta1 decay w1dth 0f the 2•-6050n up t0 a fact0r 2. 1t 15 1ntere5t1n9 that 50me n0n-5tandard decay channe15 0f the 2• are a1way5 0pen; e5pec1a11y e1ther 50me ex0t1c D-4uark5 0r 50me 51ept0n5 have t0 6e preduced 1n 2•-decay5• 1t 5h0u1d 6e 5tre55ed that the5e enc0ura91n9 re5u1t5 have 6een 06ta1ned w1th1n the 51mp1e5t, 1.e. m05t pred1ct1ve E(6) 5uper5tr1n9 m0de1. Re1ax1n9 50me 0f the m0de1 5pec1f1c a55umpt10n5, e.9. a110w1n9 f0r an 1ntermed1ate 5ca1e 1n 6etween the weak and the 6 U 7 5ca1e, can 0n1y 1ncrea5e the ava11a61e parameter 5pace. 1t 15 theref0re 5afe t0 c0nc1ude that 5uper5tr1n9 m0t1vated E(6) m0de15 pred1ct a r1ch ••ex0t1c•• phen0men0109y w1th0ut de5tr0y1n9 the chance5 0f the new acce1erat0r5 5LC, 7evatr0n and LEP t0 pr0duce detecta61e 4uant1t1e5 0f 51ept0n5, 54uark5 and 9au91n05 even 1f the 2•-6050n 15 very heavy. 1 am 9ratefu1 t0 X. 7ata f0r u5efu1 d15cu5510n5 and 5u99e5t10n5. 1 thank D. 2eppenfe1d f0r d15cu5510n5.7h15 re5earch wa5 5upp0rted 1n part 6y the Un1ver51ty 0f W15c0n51n Re5earch C0mm1ttee w1th fund5 9ranted 6y the W15c0n51n A1umn1 Re5earch F0undat10n, and 1n part 6y the U5 Department 0f Ener9y under c0ntract DE-AC02-76ER00881.

Append1xA 1n th15 append1x 1 91ve a11 R 6 E re1evant f0r 9au9e 5ymmetry 6reak1n9 at the weak 5ca1e. A5 pa1r 0f 5U(2) d0u61et5, 5urv1v0r5 fr0m 27 + 27, 1ar9er va1ue5 0f a 5 and 51n20w. N0w 0ne ha5 f0r d9 2

619 4

dt

8rr 2

the 1nve5t19at10n 0f the rad1at1ve d15cu55ed 1n 5ect. 1 an add1t10na1 ha5 6een 1ntr0duced t0 a110w f0r the 9au9e c0up11n95,

(A.1)

w1th 61 = 16, 62 = 4, and 6 3 = 0 (92(MX) = 0.692(MX), and t = 1n Q). 7he R 6 E f0r the 9aU91n0 ma55e5 are a5 U5Ua1 [10] 91Ven 6y dM 1

6192M1

dt

87r 2

(A.2)

M. Dree5 / E(6)5uper5tr1n95

354

7he R 6 E f0r the Yukawa c0up11n95, A-parameter5 and 5ca1ar ma55e5 have 6een der1ved u51n9 the 9enera1 f0rmu1a5 0f ref. [38]; 0ne f1nd5 dh 2 h2 (A.31 dt ~-~t 2"(6ht2 + )k22,3- 1692 3 3 - 3 9 2 - 2959•) , dX~1 dt

X2 ( + ~j ( 2 X 2 j + 3 X 2 3 , 1 ) ~ 3 9 2 2 ~ 9 2 81r2•:•,2 3h2813+2X~,1 •

,

dt

1

3,1 2X21+ E(2)t2,j + 3)t2 j ) • 8rr 2 j •

136922~9 ~

)

)

,

(A.5)

,

8rr1 2 (6ht2At + X2,3A2,3 + 169 3 32M3 + 392M2 + ~92M1) ,

dA dt t

(A.4)

(A.6)

dA2,11 (3h2At~13+ 2~2,1A2,1 + y,(2X 2 J12 J+ 3X2,j13,j) dt 8~r2 • j •

•

] (A.7) + 392M2 + Yt°921 M 1], 6A3,1 dt

1 (22t2,1A3,1+~(2X2,jA2,j+3X2 8~r 2 j

A 3 , j ) + 7 3t692M3+~92M1)~ (1.8)

dm~3 1 ( j~ dt -- 8~ 2 2 X22,j(m~3+ m2j+ m 2-Hj+ [A2,j12)

~92M12),

2 2• + 1A3,3[2) +3~,2t23.j(m2 +m6j+mDj J dm~,

d•

1 (

2

2

8~5 3ht31,3(mH3+

(A.9)

m 2. + 2 ~3 mQ3+At2)

4~k2,1(m23+m2H1 + m ~2, +

1A2,112) -

392M2-~-92M 2) , (A.10)

dm 2 ., - ••( d~8¢r 2 X2•1 ( m23 + m2H~+m~+ 1A2,,1=)- 392Mff • ~92M/2) ,

(1.11)

dm~, d~--

(1.12)

•8~r 1 2 (X2,,(m23 + r n ~ + m~ , + 1A3,112) - 51692M2 3 3 -- ~92M2 ) ,

dm~, -- --8vr 1 2 t(X 23,1~{m 2N3 + m6, 2 + m~3 2 + 1A3,112) • ~92M2 • 9~,11~2~tar2]] --dt v115 dm~3dt dm~3 dt

87r 2-1 (2ht2( 2rnH3+ m23 +rnQ32 + 1 1 t [ 2 ) ~ 9 ~ M 2 ~ 9 ~ M 1 2 ) , 1

2

2

2

+

3 3

3

,

(1.13) (A.14)

1911v11] (A.15)

M. Dree5/ E(6)5uper5tr1n95

355

N 0 t e that the tW0 U(1) 9aU9e C0Up11n95 have 6een a55Umed t0 6e e4Ua1.

Append1x8 H e r e 1115t the R 6 E re1evant f0r the d15cu5510n 0f 5ect. 4. 5tart1n9 fr0m the 5uperp0tent1a1 [9] f 8 = X8QQD + X~8ucdcD and a9a1n u51n9 ref. [38] 0ne 06ta1n5: dX2 dt

X2 (8X~ - 89~ - 39~ • 5 9 2 1), 8~r 2

dA 8 dt

8¢r2 ( 8 X ~ A . +

(8.1)

1

5

2

1 (2X2(2m~+m2+

8eff 2

dm 2 dt

1 (4X2(2m~+m2+1A,[2)~92M2~9~M2) 8~r2

dX~

X~

d----~- -- 8w ---1 ( 6 X ~ - 8 9 ~ -

dm~dt

1A,12)

1692M2-39~M22-5-2~2~ 991.tv11 J 3 3 3

dm~ dt

dA~ • 1 (6X~A~ + dt 8w 2

(8.2)

89~M3 + 39~M2 + ~91M1),

--

5~92~ 3 1]~

892M3 + ~9~M1)

8 ~ (2)Va2(m2+md+m2 +1A~12)-~-923M32-a9~M2)~

,

(8.3)

(8.4)

(8.5) (8.6)

(8.7)

where a = ~ f0r 4 = f1 and a = -~ f0r 4 = c],~. 1n der1v1n9 e45. (8.1)-(8.7), 1 have a55umed that the X 8, X~8 are 5tr1ct1y d1a90na1 1n 9enerat10n 5pace wh1ch need n0t 6e true; h0wever, the ma1n re5u1t5 0f 5ect. 4 d0 n0t depend 0n th15 a55umpt10n.

Reference5 [1] M.8. 6reen and J.H. 5chwar2, Phy5. Lett. 1498 (1984) 117 [2] D.J. 6r055, J. A. Harvey, E. Mart1ne2 and R. R0hm, Phy5. Rev. Lett. 54 (1985) 502; Nuc1. Phy5. 8256 (1985) 253 [3] P. Cande1a5, 6.7. H0r0w1t2, A. 5tr0m1n9er and E. W1tten, Nuc1. Phy5. 8258 (1985) 46 [4] Y. H050tan1, Phy5. Lett. 1268 (1983) 309; 1298 (1983) 193 [5] E. W1tten, Nuc1. Phy5. 8258 (1985) 75 [6] E. W1tten, Nuc1. Phy5. 8268 (1986) 79; L. D1x0n, J.A. Harvey, C. Vafa and E. W1tten, Nuc1. Phy5. 8261 (1985) 678 [7] M. D1ne, N. 5e16er9, X.-6. Wen and E. W1tten, Nuc1. Phy5. 8278 (1986) 769; J. E1115,C. 66me2, D.V. Nan0p0u105 and M. Qu1r65, Phy5. Lett. 1738 (1986) 59

356

M. Dree5 / E(6) 5uper5tr1n95

[8] J. E1115, K. En4v15t, DN. Nan0p0u105, K. 011ve, M. Qu1r65 and F. 2w1rner, Phy5. Lett. 1768 (1986) 403 [9] L. E. 16~t~e2 and J. Ma5, Nuc1. Phy5. 8286 (1987) 107 [10] L.E. 16~e2 and C. L0pe2, Nuc1. Phy5. 8233 (1984) 511; C. K0unna5, A.8. Lahana5, D.V. Nan0p0u105 and M. Qu1r65, Nuc1. Phy5. 8236 (1984) 438; L.E. 16~ae2, D. L0pe2 and C. Muf102, Nuc1. Phy5. 8256 (1985) 218; A. 80u4uet, J. Kap1an and C.A. 5av0y, Nuc1. Phy5. 8262 (1985) 299; H.P. N111e5, Phy5. Rep0rt5 110 (1984) 1 and reference5 there1n [11] E. C0hen, J. E1115,K. En4v15t and D.V. Nan0p0u105, Phy5. Len. 1658 (1985) 76 [12] M. D1ne, V. Kap1un0v5ky, M. Man9an0, C. Napp1 and N. 5e16er9, Nuc1. Phy5. 8259 (1985) 519 [13] 6. C05ta, F. Feru9110, F. 6a661an1 and F. 2w1mer, Pad0va prepr1nt DFPD 18/86 [14] M. Dree5, Phy5. Lett. 1818 (1986) 279 [15] J.-P. Derend1n9er, L. 16~e2 and H.P. N111e5, Phy5. Lett 1558 (1985) 65; M. D1ne, R. R0hm, N. 5e16er9 and E. W1tten, Phy5. Lett 1568 (1985) 5; E. C0hen, J. E1115, C. 60me2 and D.V. Nan0p0u105, Phy5. Lett. 1608 (1985) 62 [16] A. F0nt, F. Queved0 and M. Qu1r65, Phy5. Lett. 1888 (1987) 75 [17] J. E1115, D.V. Nan0p0u105, M. Qu1r65 and F. 2w1mer, Phy5. Lett. 1808, (1986) 83; P. 81n6truy, 5. Daw50n, 1. H1nchc11ffe and M.K. 6a111ard, 8erke1ey prepr1nt L8L-22339 (1986) [18] J. E1115,K. En4v15t, D.V. Nan0p0u105 and F. 2w1rner, Nueh Phy5. 8276 (1986) 14 [19] M. D1ne and N. 5e16er9, Phy5. Rev. Lett. 55 (1985) 366; V.5. Kap1un0v5ky, Phy5. Rev. Lett. 55 (1985) 1036 [20] M. Dree5, N.K. Fa1ck and M. 61~ck, Phy5. Lett. 1678 (1986) 187 [21] J. E1115,D.V. Nan0p0u105, 5.7. Petc0v and F. 2w1rner, Nuc1. Phy5. 8283 (1987) 93 [22] UA1 C011a6., 6. Am150n et a1., Phy5. Lett. 1478 (1984) 493 [23] J. E1115,K. En4v15t, D.V. Nan0p0u105 and F. 2w1rner, M0d. Phy5. Lett. A1 (1986) 57 [24] 8. Cam6e11, J. E1115, M.K. 6a111ard, D.V. Nan0p0u105 and K.A. 011ve, Phy5. Lett. 1808 (1986) 77 [25] 6. 5te19man, K.A. 011ve, D.N. 5chramm and M.5. 7urner, Phy5. Lett. 1768 (1986) 33 [26] J.P. Derend1n9er, L.E. 16a~e2 and H.P. N111e5,Nue1. Phy5. 8267 (1986) 365 [27] V. 8ar9er, N. De5hpande and K. Wh15nant, Phy5. Rev. Lett. 56 (1986) 30 [28] J. E1115, K. En4v15t, D.V. Nan0p0u105 and 5. 5arkar, Phy5. Lett. 1678 (1986) 457 [29] A.H. Cham5edd1ne, P. Nath and R. Arn0w1tt, Phy5. Lett. 1748 (1986) 399; H. 8aer, K. Ha91wara and X. 7ata, Phy5. Rev. Lett 57 (1986) 294; Phy5. Rev. D35 (1987) 1598 [30] W. 8arte1 et a1., Phy5. Lett. 1528 (1985) 385 [31] H. 8aer and E.L. 8er9er, Phy5. Rev. D34 (1986) 1361; erratum, Phy5. Rev. D35 (1987) 406; E. Reya and D.P. R0y, 2. Phy5. C32 (1986) 615 [32] 7.6. R1220, Phy5. Rev. D34 (1986) 892 [33] J. E1115and D.V. Nan0p0u105, Phy5. Lett. 1108 (1982) 44; R. 8ar61er1 and R. 6att0, Phy5. Lett. 1108 (1982) 211; J.F. D0n09hue, H.P. N111e5 and D. Wy1er, Phy5. Lett. 1288 (1983) 55 [34] V. 8ar9er, N. De5hpande, J. R05ner and K. Wh15nant, Phy5. Rev. D35 (1987) 2893 8. Adeva, F. De1 A9u11a, D.V. Nan0p0u105, M. Qu1r65 and F. 2w1rner, CERN prepr1nt 7H.4535/86; M. Duncan and P. Lan9acker, Nuc1. Phy5. 8277 (1986) 285 [35] M. Dree5, Phy5. Rev. D35 (1987) 2910 [36] 7.6. R1220, Ame5 La6 prepr1nt 15-J 2340 (1986) [37] 7.6. R1220, Phy5. Rev. D34 (1986) 1438; 5. Nand1, Phy5. Lett. 1818 (1986) 375 [38] N.K. Fa1ck, 2. Phy5. C30 (1986) 247