I~VESTIC¡A{'J()~ REVISTA \IEX[('ANA DE FíSICA'¡'¡ 1.'\) '¡'¡') ';'i~ OCTUBRE 1998

W -hoson c1cddc and wcak-electric dipole IlIOlllcnts al\(l helicity amplitudes in¡di --+ w:l:¡ or ZO

J. (,;lrcia and A. ()ul'ijl'iro/Jc/'tlrtal/ll'l/fo dc Física, L:scuela .)"uperiol" dc Fúica y ;\/UfCl/uílicas, InHillllO Polifécnico Nacional,

EdUicio 9, Unidad I'n~lesional AdoUú Lópe::, ¡\tutcos. 07738 México D,F., ;\Jcxico('-1110 i1: l/l/lid @ e.~/ill. il'" .UlX

Recihido el 23 de ahri I d~ 1l)l)X; acept;h.lll el I I d~ junio de IlJ9X

\Ve sludy lhe pt,s"ibility or l1leasuring [he CI'-ndd e1eetric ;1Ildwl'ak-c1ectric thpoll' rnomcnts kdl1l and wcdrn. respectively) 01'lhe \ l. -hosonIhr(lll~h all anaJysis of [he helicity amplitudes in the process J, JI -... \\.f \. (\' = -..or Z\l). wherc J, and JJ arl' elcmclllary fermioll". \Vesho\\' tha!. fOl"ex;ullple, rOl' the inilial slale /'ii, . slld 11l0l1lcntscan he dC(l'l'lcd in Ihe angular dislrihution 01' the final stalc [Mil'when Ihcirhelicities are (O,::i::) ror F = 'y. or (:f:, O) and (O,:f:) rOl"F = 7,lI.

K¡'PI'¡ml.\: \\'.hoson: weak-eJcctric dipole ITltllllcnt:helici!y amplitudes

Estudiamos la posibilidad de IIIl.xJirtanto el momento dipobr cléclrico dd hOS(l1l1\'. así como .su lllomCIlIOdipolar débil eléctrico ti travésdel an,ílisis de Id" amplitudes de helicidau del proceso J,jJ -t U'::l:\ .(F = ...o ZO). donde J, y J, son fermiolles elementales. Moslramospor ejemplo. que si consideramos inicialmente f'I~, tales momentos pueden .'il'l'detectados cn 1;1distrihuci(ín angular del estado final cuandoSil' hcJicidadc" S\111(O.:f:) para \' = J' () (:f::, O) Y(O,:f:) para \' = 7°

/)l'lcnf'lOrr,C Boson W: momento dipolar déhil eléctrico: amplitudes de hdicidad

P;\CS: 1-l.HO.Er: 12.15.Ji: 12.50.Fk

\Vherl' thc dual tensor ('/<1/ ís dcfincd hy

aClioll J,J; -... trI'!, where Ji and!J hclongs to <InSU(2)¡isououhlet nf L'lclllcnlary fermions, il \vas shown 141 Ihat¡Ill' alllplillldc exhibits. al the Born level, a zer<l. Anoma-IOlls cOllplings spoil tllal zero. indicating a possihle non ele-mentary n:lIure of lhe gallge hosons. Olher related proccsses.!ike [.-'1 1\' ........•.J FJ' also exhihil this zero. Recently Bauret uf. [ni stlldied Ihe proccss JiJi -+ H,:t Z, for CP evenanomalolls cOllplings, showing that Ihe amplituLle exhihilSan :lpproximale /.ero. One can see tha! lhe helici(y ampli-ludes ~,'I (h\\ .. 11\.) c(lming fmm Ihe SM couplings and Iheanomalous CP-c\'ell l.ollplings. ~I\\' = 1 - "\' and ,\\',faclori/.cs \vith factors ( I::¡: cos H) ami (l:!: ('os H), for helic-ilies (::f:, O) and (O. ::f:). respcctively, wherc Illr and IIF areIhe helirities of 111L'¡F ami \' hllsons. and e is Ihc renter ofrnass scallering angle 01"the \ F -hosoll. This rcsult allows uslO Ihink in kinematic regions wherc Ihey vanishcs, indcpen-uenlly 01"Ihe v:tlues 01 paramcters .ó,,.,'\' ami ,\\'. In Ihis pa-pel' we includc conlribulions from t¿\" and wc show that theC(lITcsp<Hlding factors (l::f: cosO) ¡¡nu (1 =Fcos O) appears, re-speclivel)', indiGlling where lo look fOl"Ihis special silllalion10 Illea.,ure the cdm ami wedm parameters.

Thc sl<irting poinl is an cllectivc CP-oJd 1F -hoson dipoleintcr<lction [21

1. lntrnduclinn

AII aspecl 01" Ihe slandard model (SM) 01' c1e<.:Lrowcak 1I1li-licatioll. h;¡sed on the gaugc group SU(2)L x U( Ih', lo 11L'tL's\ed cxperimelllally is Ihe Ihree-vector hoson couplingsIr 1\ " and \1'1 F Z, The explicit tensor nalUre of these ver-tices is a cOllsecuence 01' lhe gauge stru<:lure 01' the model.The ven ices. in turn, are rcspol1sihlc rOl' lhe characteristicl'lL'ctroweak pmperties of Ihe spin olle elclllclltary chargcd 1Fbosoll. thL' (weak) electrk allllmagnetic l11ultipole mOlllenls.In panicular ils electrk dipole !nol11enl (cum) is preuicled lohe quile small, arising al (he three loop leve! [1]; an expeclcdreslJlt sincc lhis I11OJ11entdoes not rcspcct lhe CP sYllllllclry.Using Ihe experimenlal \'alue for the neutroll eJm. lile cor~fcsponding cdm vallle for Ihe Ir hOSOll, d') = ('¡,:')/l..'!\\",is hounded ny I'2J ,~'') :s: IO-:{. It is expeclcd thal Ihe anal.ogulIs el' violating weak eLlm, (l:~ = ('¡,~/'2co:;OIl,'\/II'he also houllded hy 1:11 ¡'x ~ lO-:l. On Ihe cOlllrary, theCP~cven (wcak) magnelíc dipolc Illomcnt 11\' = q'( 1 +,,-\' + ,\\.)/:2.\1 H.. and the (\\fcak) cleclric lJuadrllJlole mo-lIlenl (/, = rd",- - Av)/M/,-, wilh ",- = 1+ radia-¡¡ve COITl'CtiollS and '\l' = 0+ radiative correclions are cx-pL'cled lo dominate in proccsses whefe Ihe complete Vl'rfex11"11'\'(F = J' or Zo) is laken inlo accol1llL This is so whcnlllc reslllting cross-scclion is sllmllleu over all helicity stales.Thc sitllatioIl is L1ifferent when we compute detlnite hl'licityampliludes. as \ve shO\v in this work.

Thc IIIcll10d 01"hclicity ampliludes has heen used lo stllUY¡lIlOlllalous couplings of lhe Ir hoson: ror instance, in the re~

( I )

(2)

J. (¡:\[{nA ;\t\1J A ()UEIJEIRO

;ll1d \;, i::::; iJ,,\~j-D.j\;I' 1\'/; i"thl'lieldorall'- hOSOI1.,II\l' lllt' c!t't'lril' ehar,t.!c, (\' = ~,)or y / (,ll~ HI\" (\' ::::; ZO 1.Sueh

[nl'111,uppl~ 1\\'(';11-,:-)'..'ICt'lríc lilpole 11101llL'Il(lo (he 11' -hosoll

,1/1 1: \'

:2,\111'

1,

li

The l'lllTC,pollding vn(l',\ ís given by 'fll' ¡,.\- c"';IU' /,"',

\\ ill.'l't' 1.- l' Ihe \' -bOSOl1r(lur lllulIlelllUlll, In Hg. I \\'e sho\\'

lh\.' dld~r;lJn \\hel"e Ihe \t'U'l' \'erlex ente!'s in (he proce"s,

;111\1lb\,.,(1lIlLrir.ulion lo 111\.';[lIIplilude i, gi\(,1l b.\

;llld r ::::;('f, ",:! ~iIJ:! HII, ¡-!. ,'!!.; l' :lnd 11are Ihe wavc fUllc-

lillll" nI Illl' illL'oming fcrllliolls ami f ~\~ (/J, l/jl'), r;'," (¡,.. 111)lllt' polaril.alion 4-veclors 01' (he hosons, L;',\,,\/ is 11lt'

KohayasiJi-J\bskaw:llllalrix e!l'lllcnt \\,-'henquarks <lre thc L'I-

1'1l1t'lllillY k:'llliolls in lhe 1"l',IL'linll,

TIlt' IOldl '-Implillldc is gi\'L'Jl by adding lo (4) Ihe (011-

lIi!lutiolh frolll SJ\l alHl anomalous ~I,\ and '\\' lllolllenl.,.

,1l1dlIJe t'olllribulions coming fmlll fcrmion cxchangc in Ihe

u-alHI I-dl<tllllels [61. also shown in Fig. l. Procceding as

lhual. \\L' computc the various angular dillcrential cross sec-

liull I1n (hl\', 11\') / 111'0:-:'&:frolll lhcsc 11(1 (:f:, =f) / rlco:-:.1J

I

1:[(,111<1:1, TI1\.'s-cl1;1llJ1d,l-cll;ll1c1 ;Illd Il-channcl FCYIlI1l.IJItlla-~Ialll~ rOl'lhe pnK'l'ss I, I, --'+ 1'" \', FIIUr-1Il0111CIl1;larc ilHliraledil1 pal'l'l1lhcsis,

(;1)

W

\ \, 1

v

(h)

v

19, )

w-

li

1i

onl) rccei\'cs S~I L'oJlI!'ihulion, d(J-(-~,.:f:)/d('o:-:.H ami

l/a(O,O)/I/cosH arc d{lmillil!cd by the S!\I and ~f";\' or "'"(OlllrihuliollS in ¡ho: cnlire rall,,!-!cnI' (hc angular variable, Tht:olhn four conlribuliollS are lile inlCl"esling (lIlCS wl1ich, aflel

nq:dccling fcrrnioll lllassL's. rcsults in the fol\owing:

¡.JI

(51

X 11({ 1 ' ,,,'1 ) f ~\' (1 J, 1Ill' ) f ~',*(/,'. 11\ ' ),

""Iet. (1)11('0,,0

(6)

",,(11. et)

11 nI", ()

J {(I+h ('o"Hf{[( ll-d(,o:-:'fI). .]'-' -F' ----""--.-- - 21" 1"11 - ,J + ¡J,. ("osR +.J ..\ + 2,1\:32 JT,<; :21'11'. 1 - /'11

(71

In Eqs. ((1) and (7) the various paramclers are defined hy Iand

"1+ rll' - 1'\'

_' (.,1' 'IJ).\ ~ -- - + - ,

,1 11 f

,"~,/'.jJ = 1 + 1'1\' + /'¡.-,

1'\' = Jf{ j. ..... (XIj' .lIt!!'

1,,(1-1"11 )

11'.I\()\()N ELEORIC AND IVEAK.ELECTRIC DIP()I EM()MENTS ,\NI' IIEI.IClT\ AMI'I ITl'DES I~ J, J, ~ 11") ()I< Z" ~51

-210

,10

..oa. 100

'" -,l/l 10OÜ'O-- -3+ 10

O 1[:/I:l - 5'O '0

('r~O) .1KVO0.0010.010.1

-,'0-1 -05 O

cas 11

(a)

0.5

102

-,'" 10l/lO .2Ü 10J2 (.1',.-0) '¡-

0-0001- 0.01... 0.1

-0.5 O

cas 11

10\

05 -05 Ocas 8

lo)

05

rlC;l'RL 2. Pío[ nI' lile :lllgular difft:fI ..'ntiaJ rros ...•-..•('ction for'1', .• 11': -, for lhe helki,y conligllralioll ((J, +). \Ve ;1SSUIllC\' - 2110(¡eV, ;llId (a) f~:,¡ ::::: U, .\-, .-::::() ami sL'\'cr:lI valucs (JI'

.:J., ... ami (h) ~I'- ...•= O .. \-, = Il ami sC\'L'r:ll v,llucs 01"¡',.

Flc;t'IH: J Tlll' ...<l~1Il'~l...in Fígs, 2a-2h t'lut wilh ~ = IIlll0 (le\'.

II in I:q.(7)\vcmakcjI\. :::::Owc()htain.forh...,::::: +1.

...hO\\ing Ihal lh(' ctlt.'CIS fmm ".... are adtkd lo Ihose (JI'S~I

wilh ,,"',f. 11 lile l\'1andclslam variables, and .1/ ¡¡nd yJ are Ihel"llllplings orlhe fL'lIniol1s lo lhe neutral bllSOIl. Ncxl, \Ve ,1Ilal-ill' lhe l\\'() caSi:S \' = -,_Z.

COlllribulioll illld can, in principie, be ohserved rol'('(l~ (} = - 1. lIowe\'er. Ihis can be so only when ¡,¡ takesvaluL's IllUdl hi~~cr than lO-:{ 01' wc are al vcry high energyin tlrdcr lo han~1'", loO s111<111lo lIlake Ihe cocrticiCll1 al IheIdl 01 I~'; I¡¡rgc l'lHlllgh lo he cOlllpclilive lo Ihe SM cOIl!rihu-lioll, \V1lL'1l1'0:; H = -+- 1 lile (O, +) helicilY gi\'es informalionon ~I"; alld '\'" In Fig. 2a we have plotted Eq. (X) rOl' !heprocL'ss tI', -y 11' ~'. wilh Ji; = 2{)O GeV, f~'¡ = 0 . .\, = ()tllld SC\'lT;lI valucs of "::''''~r' showing !hal Ihe cllcc! 01' "::'''_1L'an he ,Cl'l1 in c()sH = + 1. In Fig. lb lhe same has hecn donehu! rOl"~¡;,.. = o. ,\, = (J. and several valllcs 01' I~'¡.Here!he angle 01' inlcrcsl is ('n,"; H = -1; ho\Vcver only rOl' hig val~lICSnI' I~'¡\VL'ean see signifkan! clfce!s, as eomrared lo IheSM eOl1lrihulioll. FOI"grealer valllcs of vr; \Ve !lave a si!ua~lioll "lInl' ,1llal1 \'al11CSnI' tlll:' parallll'ICrS can give importanleonlrihlllilllls.:l, C<l1lhc ,ú'll in Figs, .\a-3h,

~ cosH):!Ji ., [((i~,,", "",~I!I" _')--F- -- + ~--,I,- (1:t2iT.'i :2 e;I'\\')

,/"(11. +)ti ("ll~ (}

¡'/tOfO/l ("({Se' \ • ::::: -,

J. GARCIA ANO A C)UEIJEIRO.t52

'02

10'.oa. 10°

'"(/) lO'OÜ .2"O 10 (j"r'O) ¡¡r-- .3 O1 'O6 0.001., 001'o1:: 0.1"O -5

'O-.'O- I -05 O 0.5

GaS 8

(a)

.oa. '0°Olro~

10"2'"(/)OÜ -,"O 'O--O 166-b"O

108

-1

(,1"r=O) ¡¡rO000100101

-0.5 O

GaS 8

(a)

_._._'-

0.5

(¡¡r,olj""0-0001- 00'0,1

2'O

10'.oa. '0°

'" -,(/) 'OOU -2"O 10-.....

.3, 10Ó .,1:: 'O

"O -5'O

106

-, -05 O

GaS 8

(h)

0.5

'02

.oa. 10°

Ol -'- -roM -2- 10'"(/)OÜ -,"O 10

(¡¡r'ol-- .1K)'0-000'

Ó 106 0,01-1:: O'"O

108

-, - 0.5 O 05

GaS 8

(h)

1:1( iUJ< l': 4. Angular di tlcrcntial cross.section I"or 1,,7, --t 11 'x I forlhe hdil"Íty eontiguratioll (O, -). \Ve as sume ,ff; = 2()O CieV, <lnd(a) "')'fi) = (l, ,\) = (J ami sevcral values of ")' ¡¡lid (h) f.) = O,.\~ = 11amI scveral values 01" ..,),/{"y'

For 11) = -1 wc ohlain

Fr(¡uJ<l: 5. Tlle same as in Figs. 4a-l.h huI with ,¡s = 1000 Gcv.

Figs. 2a-lh. Por greatcr \"illues of .fi Ihe efrects are moreevidenl. as can he seen in Fig. 5a-5h.

. Z-hosoll (,(/Je

I!ere \ve havc Ihe choice ('os 8 = + 1 lo get inl"ol"lnalionon1-","1 ;done 17]. This is SllllWIl in Fig. 4a, for ..¡;. = :200 GcV,~f"') = O, '\1 = O ami values of ji¡; we do lhe sallle for.fi = :100 Gev. ¡".) = 0, A") = O and values 01' ~1l.") inFig. --lh. The resulling curves are quite similar lo those 01"

'/,,(U.-)dn)sH

,i! p',{[I!'I'W ('0('011'(" \ ),]= -- . - -- + ---!J./l +,:3:~Jr"" :1 8/Pr,r ¡ ")

.) co{18w.) .)}x (1 - "osO)" + ---¡;,- (1 + I'()sO)- (10)

81"11' ')

Analogolls results rOl" ,\1,. = :\lx are sho\l,.'n in Figs. 6a-6hrOl" hclicil)' (O.:f::) anLi in Fig. 6c-6d ror (::1:.0), for (he pro-cess f'l~ -----¡. 1\'- ZO [SI. l!ere \Ve have :J.1\Z = O. A/. = O,,¡; = 11111GeV, ami ¡;'z = (1.0111.lI.tIl, 0,1. For ({l,:t) we1l01L'lhal Ihe clTcCIS¡¡re presL'nl al one of Ihe exlremes of Ihecurves }~'í': = (l.O!, 0.1, and in lhe tlloLiiiicalioll oí" Ihe L]U<I-sizcnJ. inL"luding f{;( = o.()() I , For (:f::, O) Ihis elTects are lessllolice¡¡hlc, and onl)' lhe hig value ¡:,~/. = 0.1 have the errectin Ihe dip olllhc curve; howevcr. rol' grcater values of '¡;'lheelfeL:ISare higgcr, as in Ihe prcviolls case,

ui'I'. Me.\". ¡'-ú. 44 (5) (ll)l}X) 44<J---t';¡J

U' -BOSON ELECTRIC AND WEAK-ELECTRIC DlPOLE MOMENTS AND HELlClTY AMPLITUDES 1S'¡JI --..Wj:¡ OR ZO -1SJ

0.5

( ..1"Z :0) ii Z

SM0_0010.01O.,

O

cos O

_._.- _._'-

-0.5

'" ..</l 'OOU"O 106~•ci .,

'Otl"O

."'O -,

----

05

-----_.-

oCOSO

-0.5

-,'O -,

'o'IÓ'

tÓ2

.o .,a. 'O

'" -. (.;','0) ;¡,V> 'OO SMu ., 0.001"O 'O 0.01~

-, O,'oci

:: 101

"O

108

(a) Ih)

0.5ocos 8

-0_5

-,'O

->O'O-,

10° (..l"'z=O).ocz0_001 0.01- S M

.....,-•.-. 0.1

.oa.

(,j"'z:O) iiZSM- 0.001

DO'O,

o 05

COSO

-0,5

'O'.oa. -,

'O

'"</l _"O 'OU"O~ -"'OO

,lJ'::"O -"10-,

(e) (ti)

FI(illRE 6. Plol 01" lhe angular diffcrential cross-scction for ('17, -jo nd ZO for (a).{h) Ihe hclicity conliguration (O, ::1:). and (c)-(J) I"or(:f:, O). \Ve aSSllmc v;. = 200 GeV, ,ó,J-i., = O, ,\., = O;)ml several "alIJes of h,.

In slIlllary. \ve !lave cnmplcmcntcd prcvious rcsuhs, byolhcrs aUlhors. to study lhe possibility 01" ohscrving the ep-udd electrie nI' weak-clectric dipole 1ll0l11cnt01' the 1t' -hosollf~.\. Ihrough lhe angular differential cross scction for hclicilYsl;¡lcs in the processes J¡!} -+ U::f:, 01' Zo.

Acknowldgmcnts

\Ve aCKnowledgc partial supp0r! fmm Comisión de Op-eración y Fomento de las Actividades Académicas dcII.P.N,.México,

1. l.B. Khriplovich ami l\1.E. Pospelov. SOl'. 1. Nucl. 1'11)'_\'. 5J119(1) (13X.

,) W..I. Marciano amI A. Qucijeiro. Phn. Re". [)]J (19~6) J+19

::. t\. de RLíjula. '\1,B. Gavela. O. PellC, ami FJ. Vegas, NI/el. pl/V.\.11J57 (19()J) J 11.

,1. lJ. Bam am.! D. Zeppenfcld, Nuel. Phys. lun,", (19SS) 127.;,. F. 130uJjema el al.. ¡'lEys, Rev. U'I/, .'6 (19X9) 1906.

G. lJ. Baur (" al., Phy.\'. Re\'. Lel/. 72 (1994) 3941. anJ fefcrClll."CSthcrcin

l. In Ihis particular cases lhe ¡ero is localized al cos8 = =Fl fort\'::I:. For quarks in Ihe inirial sIale. rcsults are quile similar.wilh Ihe tero al (.OS f) = ::1:*,

8. For Ihis case Ihe quasilcro is al ('(l~ fI = :f: &.

R('\~ ,\fe.\". Fi.\ . .u (5) (1998) +19--l-5J