Phenomenological implications of an electroweak theory without Higgs

V01ume 248, num6er 1,2

PHY51C5 LE77ER5 8

27 5eptem6er 1990

Phen0men01091ca11mp11cat10n5 0f an e1ectr0weak the0ry w1th0ut H1995 6 . Cvet1~ ~, R. K/59er1er and J. 7rampet16 2,3 Faku1t11tf11rPhy51k, Un1ver51t•1t 81e1efe1d, P05tfach 8640, D-4800 81e1efe1d 1, FR6

Rece1ved 24 May 1990

An a1ternat1ve t0 the 5tandard e1ectr0weak9au9e the0ry 15c0n51dered wh1ch d0e5 n0t 1nv01vethe phy51ca1H1995part1c1e, 6ut c0nta1n5 heavy vect0r 6050n re50nance5 man1fe5t1n9the 5tr0n9 9au9e 6050n 5e1f-1nteract10n5.We 1nve5t19atephen0men01091ca1 1mp11cat10n50f 5uch a the0ry and dem0n5trate that pr0ce55e51nc1ud1n9heavy 4uark5 (1n part1cu1arheavy 4uark pa1r pr0duct10n and a5ymmetry 1n e+e - ann1h11at10n)c0n5t1tute a 900d te5t1n99r0und f0r exper1menta11yd15cr1m1nat1n9th15the0ry fr0m the 5M at ener91e5ava11a61eat pre5ent 0r w1th1nthe near future.

1 . 7 h e 5tandard m0de1 0f e1eCtr0Weak 1nteraCt10n5 ( 5 M ) 15 pre5ent1y 901n9 t0 6e te5ted 1n 5eVera1 h19h ener9y eXper1ment5 w1th 1nCrea51n9 preC1510n. NeVerthe1e55, 0n1y a part 0f th15 the0ry C0U1d 6e 5U6jeCted t0 emp1r1Ca1 1nve5t19at10n5 Up t0 n0W; the 9aU9e 6050n 5e1f-1nteraCt10n5 (a1th0U9h 1nd1rCCt1y 1nd1Cated 6y rad1at1ve c0rrect10n5) and, 1n part1cu1ar, the 5ymmetry 6reak1n9 5ect0r are 5t111 wa1t1n9 f0r the1r exper1menta1 ver1f1cat10n. Meanwh11e, 1t 15 w0rth 100k1n9 f0r a1ternat1ve5 t0 0r var1ant5 0f the 5M wh1ch are a150 1n acc0rdance w1th the a1ready 06ta1ned exper1menta1 f1nd1n95. 1t 151n part1cu1ar the H1995 5ect0r wh1ch pr0v0ke5 cr1t1c15m5 and 5t1mu1ate5 new 1dea5 6ecau5e 0f 1t5 the0ret1ca11y and phen0men01091caUy d155at15fy1n9 feature5. Dur1n9 the 1a5t year5, an a1ternat1ve the0ry t0 the 5M ha5 6een adv0cated and 1nve5t19ated w1th 1ncrea51n9 1nten51ty [ 1 ], wh1ch can 6e c0n51dered a5 the m05t ec0n0m1ca1 var1ant 0f an 5 U ( 2 ) × U ( 1 ) y 9au9e the0ry 0f e1ectr0weak 1nteract10n5, name1y the 0ne w1th n0n11near1y rea112ed 5ymmetry 6reak1n9 5ect0r. Due t0 the 1atter pr0perty, n0 phy51ca1 H1995 part1c1e appear5 at a11, and a11 5ca1ar5 are (n0n11near1y tran5f0rm1n9) w0u1d-6e601d5t0ne 6050n5 wh1ch f1na11y 9et a650r6ed 1n the ma55 creat10n mechan15m. 7he 1a9ran91an 0f th15 m0de1 15 f0rma11y 1dent1ca1 t0 the (9106a1) 5U (2) LX 5U (2) R 5ymmetr1c n0n11near ( N L ) a-m0de19au9e-1nvar1ant1y c0up1ed t0 5 U ( 2 ) L X U ( 1 )y. Fr0m the phen0men01091ca1 p01nt 0f v1ew, th15 the0ry can 6e under5t00d a5 the 5M 1n the 11m1t 0f 1nf1n1te1y heavy H1995 ma55 [ 2 ] "~. 7heref0re 1t 15 r0u9h1y c0mpat161e w1th a1110w-ener9y pred1ct10n5 0f the 5M and 15 1n acc0rdance w1th ava11a61e exper1menta1 re5u1t5. 1n the pre5ent paper we 1dent1fy 50me phen0men01091ca1 c0n5e4uence5 0 f t h e N L m0de1 wh1ch may 6e atta1na61e 6y pre5ent day (LEP 1) 0r near future exper1ment5 (LEP11, 55C, L H C ) and wh1ch may eventua11y ena61e u5 t0 d15t1n9u15h exper1menta11y 6etween the 5M and the N L the0ry. 7heref0re, we c0ncentrate 0n effect5 wh1ch ref1ect the 5pec1f1c feature5 0 f t h e N L m0de1 1n the c1eare5t p055161e way. 1t 15 we11 kn0wn that there are tw0 part1cu1ar1y 5pectacu1ar p01nt51n wh1ch the N L the0ry d1ffer5 fr0m the 5M. 7 h e f1r5t 15 the ex15tence 0f a heavy 9au9e 6050n tr1p1et f/~, (0r 0f even further 5uch tr1p1et5) w1th1n the N L 5upp0rted 6y Deut5che F0r5chun959eme1n5chaft,pr0ject K5 1062/1-1. 2 5upp0rted 6y Deut5che F0r5chun959eme1n5chaft,pr0ject 5ch1 189/4-1. 3 0n 1eave0fa65ence fr0m Rudjer 80~k0v1~1n5t1tute, Yu-41001 2a9re6, Cr0at1a, Yu9051av1a. ~ N0te that the pr061em50f a heavy 6ut f1n1teH1995ma55need n0t emer9e 1nthe NL the0ry 51nee1t 15n0nren0rma112a61eand a5ympt0t1c freed0m ar9ument5 are n0t d1rect1yapp11ca61e. 128

0370-2693/90/$ 03.50 • 1990 - E15ev1er5c1encePu6115her58.V. ( N0rth-H011and )

V01ume248, num6er 1,2

PHY51C5LE77ER5 8

27 5eptem6er 1990

the0ry [ 3 ]. 7he5e 6050n5 are c0nnected t0 an add1t10na1 (••h1dden••) 9au9e 5ymmetry (5U (2) v) 0f the under1y1n9 NL 1a9ran91an [4]. 0 f c0ur5e, they man1fe5t them5e1ve5 m05t c1ear1y at 5uff1c1ent1y h19h ener91e5. Yet, thr0u9h m1x1n9 w1th the 0rd1nary 9au9e 6050n5 (~7Vu,¢9u) they are a61e t0 1nfect even the 10w-ener9y 4uant1t1e5 t0 a certa1n extent [ 1 ]. 7he 0ther 5pec1f1c pr0perty 0f the NL the0ry 6ear1n9 c0n51dera61e phen0men01091ca1 re1evance 15 1t5 n0nren0rma112a6111ty. Fr0m the c0nceptua1 p01nt 0f v1ew th15 1mp11e5 that the the0ry can 0n1y 6e 1nterpreted a5 an effect1ve 0ne, 11m1ted t0 50me f1n1te ener9y ran9e. 7echn1ca11y th15 mean5 that the the0ry 15 we11def1ned 0n1y 1f 50me ener9y cut-0ffA 15 1ntr0duced wh1ch 1nd1cate5 the 0n5et 0f a m0re fundamenta1 5tructure. 1f amp11tude5 are ca1cu1ated t0 h19her 0rder, new 1nteract10n5 n0t ex15t1n9 at the tree 1eve1 appear at each 100p 1eve1.0r f0rmu1ated 0therw15e, at each 100p 0rder, c0unterterm5 have t0 6e 1ntr0duced (f0r cance111n9 the cut-0ff dependence), wh1ch cann0t 6e c0mp1ete1y a650r6ed 1n the ren0rma112at10n 0f tree 1eve1 parameter5 due t0 n0nren0rma112a6111ty 0f the the0ry. 7he n0na650r6ed effect1ve term5 (ne9at1ve5 0f the c0rre5p0nd1n9 c0unterterm5) c0n5t1tute new mea5ura61e 1nteract10n5 [ 5,6 ]. 1t 15 the ma1n purp05e 0f the pre5ent paper t0 ana1y5e 50me phen0men01091ca1 1mp11cat10n5 0f the5e new (0ne-100p) 1nduced 1nteract10n5. 7here6y we c0ncentrate 0n the ferm10n1c 0ne5, 51mp1y 6ecau5e 0n1y pr0ce55e5 1nv01v1n9 ferm10n5 5eem t0 6e fea5161e at pre5ent day ener91e5. 1nduced (5e1f-)1nteract10n5 0f 9au9e 6050n5 w1116e d15cu55ed e15ewhere. 2. Let u5 n0w 5ketch the 0vera11 phy51ca15tructure 0f the NL the0ry [ 1,5,6 ]: 1t c0nta1n5 5even phy51ca1 vect0r me50n5, three neutra1 0ne5, wh1ch we ca11A u (the ma551e55 ph0t0n), 2 ° (the 2-6050n w1th ma55 ~ 91 6 e V ) and V ° (a neutra1 heavy 0ne) and tw0 pa1r5 0f char9ed 0ne5: W~ (the char9ed 1V85 at ~ 81 6 e V ) and V~ (the1r heavy partner5). 7hey are c0nnected w1th the 9au9e 6050n5 0f 5 U ( 2 ) L × U ( 1 ) r × 5 U ( 2 ) v , Wu, Yu, ~2u, v1a m1x1n9 (5eparate 1n the neutra1 and char9ed 5ect0r5). 7he c0rre5p0nd1n9 m1x1n9 an91e5 are den0ted 6y ~0 (f0r the char9ed part1c1e5) and ~, 9 (f0r the neutra1 0ne5). 7he fu11 the0ry (1nc1ud1n9 the ••h1dden•• 9au9e 5ect0r) 15 character12ed 6y f1ve parameter5: 9, 9• [the 9au9e c0up11n95 c0rre5p0nd1n9 t0 5U ( 2 )L × U ( 1 ) r], f 2 (p1ay1n9 the r01e 0f the vacuum expectat10n va1ue 1n the 5M ), 9•• [ the c0up11n9 c0n5tant 0f the ••h1dden•• 9au9e 9r0up 5U (2) v ], and 22 wh1ch character12e5 the 5tren9th 0f the ••h1dden•• 5ymmetry 5ect0r. 1t 15 p055161e t0 expre55 60th the ma55e5 and the m1x1n9 an91e5 1n term5 0f the5e parameter5.7he c0rre5p0nd1n9 f0rmu1ae can 6e f0und 1n ref. [ 1 ]. 51nce the ferm10n and the 9au9e 6050n 5ect0r (except f0r the1r 5e1f-c0up11n9) 0f the 5M 5eem5 t0 6e 50 we11 e5ta6115hed exper1menta11y, 0ne expect5 that 9, 9•, f 2 are very c105e t0 the1r 5M va1ue5. Fr0m prec1510n exper1ment5 0ne can der1ve 60und5 f0r the rema1n1n9 parameter5 [ 7 ]. 7hey re5tr1ct 9• t0 va1ue51ar9er than 109, 1.e.

9/9••<<,0.1,

(1)

6ut d0 n0t re5tr1ct 519n1f1cant1y22(Mv). 0n1y 1fwe 5t1ck t0 the a55umpt10n that Mv 5h0u1d 11e h19her than 800 6 e V and 6e 51tuated 50mewhere 1n the 7eV re910n (0.8-10 7eV), we 06ta1n

1 ~ 2 < 100.

(2)

7he 60und5 ( 1 ) and (2) are 5uff1c1ent t0 a110w f0r an expan510n 0f60th ma55e5 and m1x1n9 an91e5 1n p0wer5 0f

9/9". 0 n e 06ta1n5 [a1way5 up t0 fact0r5 1 + 0 (9/9••)2] M~v=~f292(1~,2,2) M22=~f262( 1 (92~9,2) 2~ ,

¢p=-- ~-,, [1 + ( ~

299• r[1- 1 .

629,,2

]•

M~ = a1f 2 9 •• 2 2 2,

(3)

1~(9~2+ 1 wj .-.j,

(4a)

.l'2gg'~2+ 1 -.-j,

(46)

- $/

129

V01ume248, num6er 1,2

929,2 { ~=

~

PHY51C5 LE77ER5 8 1

k 1d- )2629,,2

\ [64•222929,2• ~.2(92 9,2)2] d-...)

27 5eptem6er 1990 (4c)

where 6 2 =92 d-9, 2. N0te that a11 m1x1n9 an91e5 90 t0 2er0 a5 9/9"--,0. Ferm10n5 are 1ntr0duced 1nt0 the NL the0ry exact1y 11ke 1n the 5M. 1n part1cu1ar, they tran5f0rm a5 51n91et5 under the ••h1dden•• 9au9e 9r0up 5 U ( 2 ) v and d0 n0t c0up1e d1rect1y t0 the c0rre5p0nd1n9 (unm1xed) 9au9e 6050n5 9u. 7hey can 1nteract, h0wever, w1th the phy51ca16050n5 V °, V# at the tree 1eve1v1a m1x1n9 (1.e. thr0u9h the ~¢¢u"and ~u-c0ntent 0f V u). 1n add1t10n, there are the 100p-1nduced 1nteract10n5 ment10ned 6ef0re. 7hu5, 1n 9enera1, the effect1ve 1nteract10n 1a9ran91an 0f ferm10n5 w1th (phy51ca1) 9au9e 6050n5 c0n515t5 0f tw0 part5: the ••d1rect•• c0up11n95 (a1ready repre5ented 6y the tree 1eve1 1a9ran91an 6ut m0d1f1ed 1n f0rm due t0 m1x1n9) and the new 1nduced 0ne5. 1n recent paper5 [ 5,6 ] tw0 0f the auth0r5 have ana1y5ed the 0ne-100p 1nduced 1nteract10n 0f ferm10n5 w1th 9au9e 6050n5. 7here, 1t wa5 e55ent1a1 t0 5eparate the (cut-0ff-dependent) c0ntr16ut10n5 c0m1n9 fr0m 100p d1a9ram51nt0 tw0 part5: 0ne part ha5 t0 6e a650r6ed 1n ren0rma1121n9 the fundamenta11a9ran91an and the rema1n1n9 part 1ead5 t0 mea5ura61e new 1nteract10n5. 0 n e had t0 1dent1fy a11 0ne-100p-1nduced 1nvar1ant 1nteract10n 1a9ran91an5 and ca1cu1ate the1r c0rre5p0nd1n9 c0up11n9 parameter5 (wh1ch depend 109ar1thm1ca11y 0n the cut0ffA ). 7he re5u1t1n9 expre5510n5 can 6e f0und 1n e4. (4.9) 0f ref. [ 5 ] ~2. An unexpected 0utc0me 0f th15 ana1y515 wa5 the 065ervat10n that a11 the5e parameter5 are pr0p0rt10na1 t0 (mf/Mw) 2, where mr den0te5 the ma55 0fthe part1c1pat1n9 ferm10n5 ~3 A5 an 1mmed1ate c0n5e4uence 0f th15 (5uppre5510n) fact0r, the5e 1nduced effect5 are c0mp1ete1y ne9119161e f0r 119ht ferm10n5, part1cu1ar1y 1ept0n5, and they can 0n1y 6e 512ea61e f0r 6- 0r t-4uark5. We w111theref0re c0ncentrate 1n the f0110w1n9 0n react10n51nv01v1n9 heavy 4uark5. Add1n9 the ••d1rect•• and the 1nduced 1nteract1n91a9ran91an5 f0r the 5evera1 (phy51ca1) neutra1 9au9e 6050n5, we 06ta1n the f0110w1n9 NC ferm10n-9au9e-6050n 1nteract10n 1a9ran91an:

£fNc=~ f=v,~,4 E AufYueQff-• 8=2~v0 8uf~u(V~-A~5) f

(5)

f=v,~,4

7he f1r5t term de5cr16e5 the 1nteract10n 0f the phy51ca1 ph0t0n t0 ferm10n5. 1t 15 0f pure vect0r type a5 1t 5h0u1d 6e. 7he c0rre5p0nd1n9 c0up11n9 c0n5tant e 15 w1th1n the pre5ent the0ry 91ven 6y 99¢ e = --6--c05 4/.

(6)

1t 15 1mp0rtant t0 n0te here that the 5tandard m0de1 c0up11n95 9 and 9• are c0nnected w1th e1ectr1c char9e e and ma55 rat10 M2/M 2, 1.e. exper1menta11y kn0wn 4uant1t1e5, 1n the f0110w1n9 way: e=

~

1 + - -

392•9,2

1-

M2 92+9,2]1.

(7)

7he 5ec0nd term c0nta1n5 the ferm10n1c c0up11n95 0f the ma551ve neutra19au9e 6050n5 2 ° and V °u. 7he emer91n9 vect0r and ax1a1 c0up11n9 c0n5tant5 f0r the d1fferent ferm10n type5 have the f0110w1n9 f0rm: f0r neutr1n05:

Vv8--A v8=N8( ~~ 2 ) 2m2K8),

(8)

¢2 7he expre5510n5 (3.101, j ) 1n ref. [ 5 ], c0nta1n1n9 the parameter5 r/2 and r/[, are n0t phy51ca11ymea5ura61e6ecau5e the1r der1vat1ve part5 can 6e a650r6ed 1nt0 the tree 1a9ran91an6y appr0pr1ate redef1n1t10n50f ferrn10n1c f1e1d5 (~,[~w= ~R+ • r/[ 23~R; ¥ ~ = ~L+ •42 U23U+~L). 7h15 re5u1t51n an add1t10na1chan9e: (y2)phy,=Y2-42. #3 Actua11y,the ma55e50f60th 13=+-• mem6er50fthe d0u61et are 1nv01vedw1th1na 91venvertex. 130

V01ume248, num6er 1,2

PHY51C5 L E 7 7 E R 5 8

275eptem6er 1990

f0r (char9ed) 1ept0n5: 2 V~8 = N 8 ( - ~1+ 2R5 + ~1 2 rn~), A~ =N8(-~+~1 2rn~)

(9a,6)

f0r up-type 4uark5 (u, c, t):

V u8 = N 8 [ ~1 - ~4 R 8 -

~2/~2

(m2+m2)K8],

A~=N5{1-~2[m2-22(m2-m2)K8]},

(10a,6)

f0r d0wn-type 4uark5 (d, 5, 6):

V ~ = N n [ - ~ + 2R8+E222(m2 +m~)K5], A~=N8{-~+E2[m2 +22(m2-m2)K8]},

(11a,6)

where N20 = 6 c05 ~, Nv0 = 6 51n ~, R20 = 9,2 ~5 (1 - 99 t9 ~ 51n ~u) 1

K20=~-~-- 1 1

9•51nV/•

(12) 9, 2

Rv0 = ~-~ (1 + ff, ct9,51n

~//)

,

1 ~222( 9 ct9~) Kv0 = ~ 1 + 9• 51n ¢t/

(13) (14)

and

~2=

1 M~ 16~2f~1n •

(15)

N0te that the term5 pr0p0rt10na1 t0 ~2 repre5ent the 0ne-100p 1nduced 1nteract10n5 and that they are pr0p0rt10na1 t0 m2/M2w, 51nce the fact0r 1/f2 c0nta1ned 1n E2 15pr0p0rt10na1 t0 1/M E.

3. N0w we are 91v1n9 the 9enera1 f0rmu1a f0r the d1fferent1a1 cr055 5ect10n f0r e+e - ~ff, wh1ch we 06ta1ned fr0m the LPNc: d-~0(e+e-~ff)= a2f13------f1 ~ 0 0 x°(5)0U(x•5)• d0 45 2 1a=v.2,v

(16)

where

( 5 - M 2) (5-M~) +~F1~M1Mj

X0 =52 [ (5•M2)2 + F2M2 ] [ (5•M2)2 + F2M2 ] ,

(17)

01j = ( V~ V~ +A~A{) [ VjcVJy9~( x, 5) + A ~AJy92( x, 5) ] + ( V~A{ + V~A1e) ( V¢cAJf+ VJfA1y)93 ( x, 5).

(18)

1n e45. ( 16 )- ( 18 ), x/~ 15 the 1nc0m1n9 ener9y, 0t the f1ne-5tructure c0n5tant, f12= 1 - 4m 2/5, V~ = - 1, V~ = Qf and A9 =A~ = 0.7he an9u1ar dependence 15 c0nta1ned 1n the funct10n59~, 92 and 93:

9.(x, f1)=2 + (x2-1)f12,

92(x, f1)=(1+x1)f12,

93(x, f1)=2xf1,

(19a,6,c)

where x = c 0 5 0 and 0= ~ ( e - , f ) . Fr0m (16) we can deduce the f0rward-6ackward a5ymmetry A 1n the e + e - ~ f f p a 1 r pr0duct10n wh1ch 15 def1ned 6y 2~(f0~• f 0 ) (da/d£2) (e + e- ~ f f ) AF8 =

a(e+e-•,ff)

dx (20)

51nce 60th 9~ and 91 are even 1n x, 1t 15 c1ear that the 0n1y c0ntr16ut10n t0 the numerat0r 1n A 15 c0m1n9 fr0m

93 (x, f1), 91v1n9 131

v01ume 248, num6er 1,2

Av5(5) =

PHY51c5 LE77ER5 8

°t2f12(3-f1)

1 j

1 5

45cr(e+e•.•.f f ) ~1,j 20(5)( VeAe+AeVe) ( V}AJf+A~Wf).

27 5eptem6er 1990

(21)

Phen0men01091ca11y, the V-exchan9e c0ntr16ut10n w1110n1y 6e 512ea61e f0r ff pr0duct10n and c0mp1ete1y ne9119161e f0r 66 pr0duct10n. U51n9 e45. ( 16 ) and (21 ), we have ca1cu1ated numer1ca11y pr0duct10n cr055 5ect10n5 and a5ymmetr1e5 f0r 66 and ff. 7here6y, the numer1ca1 va1ue5 0fthe m0de1 parameter5 were d15p05ed a5 f0110w5: we t00k the exper1menta1 va1ue5 f0r Mw (80.43 6 e V ) , M2 (91.15 6 e V ) and 0~=e2/4n= 1/137.036 a5 1nput5. U51n9 (3), (6) and (46), we 06ta1ned the f0110w1n9 re1at10n5 f0r M2/Mw 2 2 and e 1n term5 0f9, 9,9••:

ME

6 2(

(392-9~2)9~2"~

M 2 w - ~ - ~ 1+

99•(1

e = 6 ~, -

6--~

2 929,2~

6---~)~

1•

(22)

(23)

where term5 0fh19her 0rder 1n (9/9••)2 are ne91ected. 7he5e e4uat10n5 can 6e u5ed t0 determ1ne, f0r any ch05en va1ue f0r 9••, the c0rre5p0nd1n9 va1ue5 f0r 9 and 9•. C1ear1y, th15 pr0cedure 15 c0n515tent 0n1y 1f9/9•• and 9•/9•• c0me 0ut t0 6e 5ma11. 7he rema1n1n9 parameter5 0f the m0de1 [9•• and Mv (0r e4u1va1ent1y 22 ) ] were taken t0 vary w1th1n ••rea50na61e•• re910n5, 1.e. 9• wa5 ch05en 5uch that 9/9•• 11e51n the 1nterva1 6etween 0.04 and 0.2 (w1th 0.1 a5 the reference va1ue) and Mv wa5 a55umed t0 take va1ue5 6etween 500 6 e V and 3000 6 e V (w1th M v = 1000 6 e V a5 the reference va1ue). F0r the 4uark ma55e5 we t00k m6= 5 6 e V and mt t0 vary 6etween 90 6 e V and 190 6 e V (w1th the reference ch01ce 135 6 e V ) . F1na11y, the vect0r 6050n w1dth5 had t0 6e 5pec1f1ed. F0r F2 the exper1menta1 va1ue F2 = 2.55 6 e V wa5 ch05en, wherea5 f0r Fv we had t0 u5e the re5u1t5 0f the ca1cu1at10n5 t0 6e pre5ented 1n the 1a5t 5ect10n 0f th151etter. Let u5 n0w d15cu55 0ur re5u1t5: - 7he 66-pr0duct10n cr055 5ect10n wa5 ca1cu1ated f0r V/5 ar0und M2 where 1t 5h0u1d 500n 6e determ1ned at LEP 1 w1th 9reat prec1510n. 1t 5h0w5 the typ1ca1 f0rward-6ackward enhancement and d1ffer5 0n1y weak1y fr0m the 5M pred1ct10n5, the dev1at10n 6e1n9 6% 1n f0rward and 2% 1n 6ackward d1rect10n. 7h15 (5ma11) d1fference t0 the 5M 15 m05t1y due t0 m1x1n9 effect5 (6etween 2 ° and V ° and 7); 1nduced 1nteract10n5 [1.e. the term5 pr0p0rt10na1 t0 e2 1n e45. ( 8 ) - ( 11 ) ] have a1m05t ne9119161e 1nf1uence 51nce they are 5uppre55ed 6y m6/Mw.2 2 7hu5, the 6u1k 0f the effect ha5 a1ready 6een taken 1nt0 acc0unt 1n ref. [ 8 ] where the pure m1x1n9 effect5 0n AF8 have 6een ca1cu1ated. 0 u r re5u1t5 are pract1ca11y 1n5en51t1ve t0 var1at10n5 0f Mv (0r 2 2 ) (51nCe we are far away fr0m the V p01e) and are a150 n0t 5en51t1ve t0 mt. 0 n the 0ther hand, they chan9e dra5t1ca11y 1f9/9" 15 var1ed (e.9. the pred1ct10n 0f the N L m0de1 d1ffer5 fr0m the 5M pred1ct10n5 6y 25% 1n the f0rward d1rect10n 1f 9/ 9•• = 0 . 2 ) #4. 51nce the dev1at10n fr0m the 5M re5u1t5 1ncrea5e5 w1th 1ncrea51n9 c05 0, we expect a m0re 519n1f1cant dev1at10n 1n the 66 f0rward-6ackward a5ymmetry. 7h15 15 c0nf1rmed 6y 0ur re5u1t5 wh1ch are dep1cted 1n f19. 1. We 065erve that a carefu1 mea5urement 0 f 6 6 a5ymmetry near the 2 re50nance may a110w an exper1menta1 d15t1nct10n 6etween the tw0 m0de15, 51nce th15 a5ymmetry w111 6e mea5ured w1th an accuracy 0f 0.5% at LEP. 0 n e remark 15 1n 0rder here. 1t 15 we11 kn0wn fr0m 5M ca1cu1at10n5 that the 6ehav10ur 0f the a5ymmetry (1n part1cu1ar near the 2 re50nance) 15 c0n51dera61y 1nf1uenced 6y h19her-0rder c0rrect10n5. W1th1n the pre5ent paper we have n0t taken the5e (cut-0ff 1ndependent) c0rrect10n51nt0 acc0unt - ne1ther f0r the 5M n0r f0r the NL-m0de1 pred1ct10n5. Neverthe1e55, 0ur re5u1t5 5h0u1d 6e 0f 1nd1cat1ve re1evance 51nce the (f1n1te) rad1at1ve c0rrect10n5 -

~4 N0te that AFaf0r the 66-pr0duct10n cr0555ect10nha5 a1ready6een ca1cu1atedf0r certa1n va1ue50f parameter5 (9/9"= 0.3, Mv= 250 6ev, F2 and Fv ne91ected) 6y 86n15ch [9 ]. 132

V01ume 248, num6er 1,2 %

PHY51C5 LE77ER5 8

27 5eptem6er 1990

AF8(66)

30

n6 20

d0 (t~)

~rx

10 -a

J 10

/1 80

1

-20

1010

~

10~"1

t M2

(6eV)

/

J

1

6)

/.//

1 1

1 0

F19. 1. F0rward-6ackward a5ymmetry AEa 1n 66 pr0duct10n (1n percent) near the 2 re50nance. 7he fu1111ne 15 the pred1ct10n 0f the NL m0de1, the da5hed 11ne den0te5 the 5M re5u1t. 7he f0110w1n9 parameter va1ue5 have 6een u5ed: 9/9" = 0.1, M v = 1000 6eV, mt = 135 6eV.

1 1

1

+1

x

F19. 2. D1fferent1a1 cr0555ect10n da1dx f0r e+e - ~ t t a5 a funct10n 0f x = c 0 5 0 f0r tw0 va1ue5 0f x/~, under the a55umpt10n that M v = 1000 6eV, 9/9•• =0.1 and mt= 135 6eV. (a) NL m0de1 at x/~= 1000 6 e V (1.e. at the V re50nance); (6) 5M at x/~= 1000 6eV; ( c ) NL m0de1 at x/5 = 800 6eV; (d) 5M at x/~ = 800 6eV.

are expected t0 y1e1d a1m05t e4ua1 c0ntr16ut10n5 t0 the c0n51dered 4uant1t1e5 1n 60th m0de15. 7h15 expectat10n 15 r00ted 1n the 065ervat10n that the 0n1y d1fference 1n the rad1at1ve c0rrect10n5 0r191nate5 fr0m d1a9ram5 w1th 1nterna1 V 6050n5, the 512e 0f wh1ch 15 5uppre55ed 6y the 1ar9e ma55 0f Mv. 7hu5, the cut-0ff-1ndependent c0ntr16ut10n5 5h0u1d n0t 1nf1uence the d1fference 6etween the tw0 m0de15 1n a 512ea61e way., - 7he tt-pr0duct10n cr055 5ect10n w111 0n1y 6e mea5ura61e 1f e +e- c0111der5 w1th x/~ > 2mt are ava11a61e. 50, we ca1cu1ate 1t f0r x/~ > 300 6eV. At 10w ener91e5 (1.e. x/~ 0n1y 5119ht1yh19her than 2mt) the d1fference 6etween the tw0 m0de15 15 a9a1n very 5ma11, 6ut 1t 1ncrea5e5 dra5t1ca11y when we appr0ach the ener9y re910n where the new vect0r re50nance V ° emer9e5. 1n f19. 2 we have p10tted ( d a / d x ) (t1) a55um1n9 Mv = 1000 6 e V f0r the tw0 ener9y ch01ce5 x/~ = 800 6 e V and x/~ = 1000 6eV. 7he 1ar9e enhancement 0f the cr055 5ect10n at the re50nance 15 c1ear1y 5een. 7h15 re50nance 519na1 6ec0me51e55 pr0n0unced 1fMv 11e5 at h19her ener91e5, ma1n1y 6ecau5e the w1dth Fv 1ncrea5e5 5teep1y w1th Mv. 1t 15 w0rth ment10n1n9 that th15 enhancement 15 ma1n1y cau5ed (t0 60% ) 6y the new 1nduced 1nteract10n5 0f V t0 ferm10n5 and 0n1y t0 a 5ma11er de9ree 6y m1x1n9 #5. N0te a150 that the typ1ca1 6ackward enhancement 0f da/d,Q 15 105t f0r 10w ener91e5 and 0n1y 510w1y re5t0red f0r 1ncrea51n9 ener9y va1ue5. - 7he new re50nance a150 1nfect5 dra5t1ca11y the f0rward-6ackward a5ymmetry at ener91e5 near the re50nance ma55. F19. 3 5h0w5 AFa(tt) under the a55umpt10n that M v = 1000 6eV. 7he re5u1t1n9 6ehav10ur 15 pract1ca11y #5 7h1515 c1ear fr0m the fact that the 1nduced 1nteract10n5 [the term5 pr0p0rt10na1 t0 e21n ( 8 ) - ( 11 ) ] are ma1n1y pr0p0rt10na1 t0 m 2 f0r tt and 0n1y pr0p0rt10na1 t0 m62 f0r 66.

133

V01ume 248, num6er 1,2

%

PHY51C5 L E 7 7 E R 5 8

27 5eptem6er 1990

AF8( t~ )

60

50

40

30

20

10

1

0

1

400

1

1

600

1

1 000

800

,/5 (6ev)

-10

F19. 3. F0rward-6ackward a5ymmetry AFa 1n t1 pr0duct10n a5 a funct10n 0f x/~. 7 h e fu1111ne de5cr16e5 the pred1ct10n 0f the N L m0de1, the da5hed 11ne the 5M re5u1t. Parameter va1ue5: Mv = 1000 6 e V ( ~ F v = 2.54 6 e V ) , 9/9" = 0.1, mt = 135 6 e V .

AF8( t t )

30

25 300

1

1

1

1

500

1000

1500

2000 ,F5

(6eV)

F19. 4 . 5 a m e a5 f0r f19. 3 6ut w1th Mv = 2000 6 e V ( ~ F v = 150 6 e V ) .

1n5en51t1ve t0 the exact va1ue5 0f mt (at 1ea5t a510n9 a5 2mt << Mv), 6ut 15 a9a1n 5119ht1ywa5hed 0ut w1th 1ncrea51n9 Mv, a5 can 6e 5een fr0m f19. 4. 7he5e ca1cu1at10n5 5h0w that tt-pr0duct10n pr0ce55e5 w111 c0n5t1tute a 900d 1n5trument f0r exper1menta11y d15t1n9u15h1n9 6etween the 5M and the NL m0de1. A11 we have 5a1d a60ut tt pr0duct10n 1n e + e - c0111510n5can 6e tran5ferred w1th 50me care t0 hadr0n1c t-pa1r 134

v01ume 248, num6er 1,2

PHY51c5 LE77ER5 8

27 5eptem6er 1990

pr0duct10n. 1n th15 ca5e, 0f c0ur5e, there are c0mpet1n9 pr0duct10n mechan15m5 (91u0n fu510n 0r 91u0n1c 4uark pa1r ann1h11at10n) 6ut the1r c0ntr16ut10n5 dr0p w1th 1ncrea51n9 t0p ma55. 7hu5, 1f there 15 an add1t10na1 vect0r re50nance V w1th ma55 a60ve the tt thre5h01d, then tt pa1r5 w111d0m1nant1y 6e pr0duced v1a the pr0duct10n and 5ucceed1n9 decay 0f rea1 V 6050n5, and the c0rre5p0nd1n9 pr0duct10n rate5 and a5ymmetr1e5 w1116e 51m11ar t0 th05e 0 f t t pr0duct10n 1n e % - ann1h11at10n. 1t 151mp0rtant t0 keep 1n m1nd that a11 effect5 0fthe new re50nance V ° are cruc1a11ydepend1n9 0n the V w1dth. 7heref0re we are n0w ca1cu1at1n9 th15 1mp0rtant 4uant1ty a5 a funct10n 0f Mv. 4. Ca1cu1at1n9 the decay w1dth 0f V °, we n0te that there are three e55ent1a11yd1fferent m0de5 0f decay: V°~ff (f15 a ferm10n), V ° ~ W + W - and V ° ~ W + W - 7 / 2 ° . 0ther decay5 are n0t p055161e due t0 the a65ence 0f the c0rre5p0nd1n9 c0up11n951n the 1a9ran91an (the 1atter 1nc1ude5 the new 0ne-100p-1nduced effect1ve 1nteract10n5) 0r due t0 ener9y c0n5tra1nt5. 7he c0up11n9 0f V ° t0 ferm10n5 15 c0nta1ned 1n e45. ( 5 ) - ( 11 ). 7he1r va1ue5 are determ1ned 1n the ca5e 0f 1ept0n5 a1m05t ent1re1y 6y the 9au9e 6050n m1x1n9 effect5, wh11e 1n the ca5e 0f heav1er 4uark5 (6 and t 4uark5) the 0ne-100p-1nduced effect5 are apprec1a61e. 7he re1evant part5 0f the 1a9ran91an f0r the 6050n1c decay m0de5 are [ 5,6 ] ~ ( v ° - - , w + w - ) =1c1[ ( w ; . W ~ - w + . w 5 ) v ° ~ + w .~ w ~ v%1, L:(V°--,W+W-2 ° ) = c 2 [ ( W ~+ V 0~ - V ~0W ~+) ( W -/, 2 -0~ 2

(24)

0~w -~ ) + ( w . + 2 ° - 2 ~0W ~+ ) (W-~V°~- V°"W -~) ], (25)

L#(V°--+W + W - 7 ) = Y•(V°-+W+W-2 °, 2 ° ~ 7 , c2 --,c3 ).

( 26 )

7he c0up11n9 parameter5 cj (j= 1, 2, 3) are determ1ned 6y 9au9e 6050n m1x1n95 c1 = ~ 6 c°52~° (9 51n ~-9~ 51n ~/c05 ~) + •9•• 51n2~ c05 ~, c05 ~,

c2=-• -~

(27)

(951n~-9~ 51n ~,c05~)(9c05~+9~ 51n ~51n~) c052~-~9~2c05~51n~c052~u51n2~0

,

(28)

c3 = - • ~-5 (9 51n ~-9~ 51n ~ c05 ~) 9• c05 ~uc052~+ •9,,2 c05 ~ c05 ~ 51n ~u51n2~ .

(29)

A1th0u9h the 0ne-100p-1nduced 9au9e 6050n 5e1f-c0up11n95 may c0ntr16ute a c0n51dera61e fract10n t0 the 5e1fc0up11n95 V°W +W-, V°W +W-7, and V°W + W - 2 °, we w11119n0re them. 1t can 6e 5h0wn [ 10] that the L0rent2 5tructure 0fthe 1nduced cu61c, and pre5uma61y a150 4uart1c 5e1f-c0up11n9515 the 5ame a5 1n ( 2 4 ) - ( 2 6 ) . 7heref0re, the f0110w1n9 f0rmu1ae (31 ) and (32) f0r part1a1 w1dth5 w0u1d rema1n va11d 1n 9enera1, w1th 50mewhat chan9ed va1ue5 0f c0n5tant5 cj ( j = 1, 2, 3). 7he re5u1t1n9 expre5510n5 f0r the decay w1dth5 w1th tw0 60dy f1na1 5tate5 are

F(V0••,ff) = 1

[

1-~nMv0

(2mf~1]~/2r(1 m}~ 1 - k,M~v0,/ J

(<)2~,,v0 (-v0 •Mw ]

F(V°-+W+W - ) = ~

1••

m~

]

- M20] (V)v°)~+A}v°)2)+3 -M2v0(V}v°)2-A}v°)2) ,

( 1-422)3/2( 1-422 + 1222).

(30) (31)

7he 9enera1 expre5510n f0r the decay w1dth 0f a neutra1 vect0r 6050n (w1th ma55 Mv) 1nt0 three 0ther vect0r 6050n5 2 °, W~-, W~- (ma55e5 M1, M2, M3) a5 der1ved fr0m the 1a9ran91an (25) 6ec0me5 a very c0mp11cated 135

V01ume 248, num6er 1,2

PHY51C5 LE77ER58

275eptem6er 1990

f0rmu1a 1n ca5e that a11 ma55e5 are d1fferent. We neverthe1e554u0te 1t here 51nce 1t may 6e 0f c0nven1ence f0r 1ater purp05e5 ~6: 2rn1ut

F(V0.~.201W~-W~-)=

1c212 My ~ d2W[(2--1--21)2--421] 1/2 48(2n) 3 M~M~M~ 2 2m1n

/ X ~46212223 + ~ [ (2-- 1 --21 )2+22/] [ (2--22 --23)2"•[-22223]

2 322 { [23(2~22 --23)2"~22(2~[-23--22)2--222123] [21 ( 2 + 1 --21 )2+ (2-- 1 +21 )2] --2221 (22 +23)W 2} + ~

1

{ 2 ( 2 - 1-21 ) [ 2 2 - ( 1-21 )2] [22 (22 •1•23 ) •22(22 +22 -- 2223) + (22 --23)1(21 +23) ]

"•[•" (2--22 --23) [22"3L2(22J1-23) - 2 ( 2 1 - 2 3 ) 2 ] [21 (2+ 1-21 )2+ ( 2 - 1 +21 )2] + 2W21421( 2-- 22 -- 23 ) -- (22 +23) (2-- 1--21)2]} W4 + ~ [ (2-- 1 --21)2+221 ] 1

- 622 (2-22 - 2 3 ) { 2 ( 2 - 1 - 2 1 ) 1 w 2 + [ ( 2 - 1 )2-22] (2+ 1-21) [22+2(22 +23) - 2 ( 2 2 -23)2] }

+~

1

(2+ 1-2~)2(2-1 +2~) 2

~( [ ( 2~- 22 --23)2(1022+6W2+6222)+ 6222~ -- 102322 -- 152(2+22 --23) ( W2+ 2221) ]

W2 + 6--~323[ (2+22 -23) ( 2 2 - 322 +323) + 2222] { ( 2 - 1 +21 )2+21 (2+ 1 --21 )2+ 2 [ (2-- 1 )2--22] (2•3L 1 --21 )} 5w2 ) 1202 { ( 2 - 1 +21 )2+2~ (2+ 1 - 2 1 ) 2 + [ ( 2 - 1 )2-22] (2+ 1-21)} ,

(32)

~61n the c0ur5e 0f ca1cu1at1n9F( V 0••, W+W- 2 °) we u5ed the pha5e 5pace f0rmu1aefr0m ref. [ 11 ], 6ut had t0 der1vea150the f0110w1n9 pha5e-5pace 1nte9ra1: f d3PLd3p26~'(P -

~-ff~n2f(~(P2 m2 m2) ) ~15P (P3 pup,p.p, [ (P2+m~-rn2)2(w2+p2m2)+p4m4] 2P • 8p1~2 (g```~P~P~+g~P~+guW~P~+g`~P```P'+g~Wu~'~+g'~PuP`')w2[3(~+m2-m~)~-2~2m2]

2 2 2 2 2 1/2 2 2 2 wherew=[(P 2 -m1-rn2) -4m/m2] ,P1=m1,P2=rn~.

136

V01ume 248, num6er 1,2

PHY51C5 LE77ER5 8

27 5eptem6er 1990

7a61e 1 Mv (6eV)

P ~ (6eV)

500 700 1000 1500 2000 3000

0.231 0.54 2.54 24.9 149.6 2133

where 2,= (M1/Mv) 2 (1= 1, 2, 3), 2m1n= ( ~ 2 J f - ~ 3 ) 2 2max= (1 - - X / ~ )2 and W----[ (2--22--2"3)2--422231 ,/2. 7h15 f0rmU1a 51mp11f1e5 C0n51dera61y When MV 15 mUCh 1ar9er than M~2 (1.e. M 2 >> M 2, M~V 1n 0Ur Ca5e), Where 0ne Can ne91ect 2,. 1n th15 11m1t 0ne 9et5 1

1c212

M~

F ( V ° - • 2 ° W + W - ) = 48.1~(27~) 3 M-~--M2

d2 ( 1 - 2 ) 3 ( 2 7 2 2 + 1 6 2 + 2 ) .

(33)

0 F0r the ca5e 0 f W + W - 7 f1na1 5tate we 06ta1n fr0m (26) F ( V ° - - * 7 W + W - ) = 192(22t)3M 4 [ C 3 • 2 M5v 1 d2 ( 1 - 2 ) ( 14- +2 ) 2ma1n

•/2 [23+4(4-22)22+(1-6422)2-422(1-7222)], (34)

where n0w 22= (Mw/Mv) 2 and 2m1,= 422. 7 h e 1mp0rtant p01nt t0 n0te 1n c0nnect10n w1th the5e part1a1 w1dth5 (f0r vect0r 6050n f1na1 5tate5) 15 the1r 5tr0n9 Mv dependence: the part1a1 w1dth5 f0r W + W - - and W + W - 7 - d e c a y m0de5 are pr0p0rt10na1 t0 M5v and f0r the W + W - 2 ° f1na1 5tate 1t even 90e5 w1th the 5eventh p0wer 0f Mv• C0n5e4uent1y, the V re50nance w111 6ec0me very 6r0ad 1f 1t5 ma55 15 5uff1c1ent1y 1ar9e and 1t w111hard1y 6e detecta61e then. We have ca1cu1ated the w1dth 0f V f0r 0ur reference va1ue5 9/9•• = 0.1, Mv = 1 7eV, m t = 13 5 6 e V and M H = 2 7eV (the dependence 0n MH 15 very weak) and 06ta1n Ft0t (V) = 2 . 5 4 6 e V W1th the f0110W1n9 6ranCh1n9 fract10n5: 8R(V°~W+W-)~52.7%,

8R(V°~W+W-2°)=32.5%,

8 R ( V ° ~ 1 e p t 0 n 5 ) = 4.2%,

8R(V°~119ht 4Uark5) = 7.3%,

8R(V°--,W+W-y)~0.8%, 8R(V°~66,

t1)-2.5%.

70 dem0n5trate the 5tr0n9 Mv dependence, we have 4u0ted the t0ta1 w1dth 0f V var10u5 ch01ce5 f0r Mv 1n ta61e 1.7he5e va1ue5 have 6een u5ed 1n the numer1ca1 ca1cu1at10n 0f da/d12 and Av8. 1t 15 a150 1ntere5t1n9 t0 n0te that f0r Mv < 1000 6 e V , the ferm10n1c decay m0de5 c0ntr16ute 519n1f1cant1y, wherea5 f0r Mv > 1000 6 e V they are a1m05t ne9119161e c0mpared t0 the 6050n1c m0de5.

Reference5 [ 1] R. Ca5a16u0n1,5. de Curt15, D. D0m1n1c1and R. 6att0, Phy5. Len. 8 155 (1985) 95; Nud. Phy5. 8 282 (1987) 235. [2] 7. Appe14u15tand C. 8ernard, Phy5. Rev. D 22 (1980) 200. [ 3 ] 7he0r1e5 w1th add1t10na19au9e 6050n tr1p1et5have a1506een 5tud1ed 0n d1fferent 9r0und5; 5ee e.9., v. 8ar9er, w.Y. Keun9 and E. Ma, Phy5. Rev. Lett. 44 (1980) 1169; Phy5. Rev. D 22 (1980) 727; E.H. de 6r00t and D. 5ch11dknecht,2. Phy5. C 10 ( 1981 ) 139; U. 8aur, D. 5ch11dknechtand K.H.6. 5chwar2er, Phy5. Rev. D 35 (1987) 297. 137

V01ume 248, num6er 1,2

PHY51C5 LE77ER5 8

27 5eptem6er 1990

[ 4 ] A.P. 8a1aehandran, A. 5tern and 6.7rahem, Phy5. Rev. D 19 (1979) 2416. [ 5 ] 6. Cvet1~ and R. K69eder, Nuc1. Phy5.8 328 (1989) 342. [6] 6. Cvet1~ and R. Kt19er1er, 81e1efe1dprepr1nt 81-7P 89/16 (June 1989). [ 7] R. Ca5a16u0n1, P, Ch1appetta, D. D0m1n1c1, F. Feru9110 and R. 6att0, CERN prepr1nt CERN-7H.4876/87 (0ct06er 1987); Nuc1. Phy5. 8 310 (1988) 181. [8 ] 6. A1tare111,R. Ca5a16u0n1, D. D0m1n1c1, F. Feru9110 and R. 6att0, CERN prepr1nt CERN-7H.5626/90 (January 1990). [ 9 ] R. 86n15ch, D1p10mar6e1t, Faku1t11t f•dr Phy51k, Un1ver51t11t81e1efe1d (Fe6ruary 1990 ), unpu6115hed. [ 10] 6. Cvet1~ and R. K69er1er, 81e1efe1dprepr1nt 81-7P 90/01 (1990). [ 11 ] H. P1et5chmann, Weak 1nteract10n5; f0rmu1ae, re5u1t5 and der1vat10n5 (5pr1n9er, 8er11n, 1983 ).

138