/lIve,~tlYllcióll /lCV1.~t(J A!e.rica7la de Física 41, No, 1 (J995) .75-41

Unitarity constraints for the mass of the Higgs m theSU(2)£ 0 U(l) 0 U(l)' gauge model

!l. .\IAIITÍ:"I';Z*Departamento de Fúica, Centro de htve.'itignciún !J de Estudios Avanzados del 1Pl\'

Apartado po"tal 14740. 07000 México. D.F .. México

N. VA:"E<;'\stf'hy"ic., /)e¡H17'I1I1""t. (jIlCC" Mllry ¡¡lid W".,tjidd ColI"!],,

Mil" E"d ROlld. Lo"doll El 4NSHccibido ('1 :!J cI(' 1l{)\'it'llIhn' <1(' 19!J3: an'ptado ('1 ;y d(' octllbl'(' <1('19D.J.

AUSTHACT. The critical ,'alllc:-; [01' liJe llla~s O[ tile Higg:s bosolls. at w!Jich liJe th('ory hC(,OlIlC:;

stwlIgly interactillg, an' cakllla1l'd lI~illg the ec¡ui,'alpllCt., 111('01"('111. .-\t high ('lH'rgies this allowsliS to r('placc tltc IOllgitudillally polariz('d gauge bO!-iolls in the S lIlatrix [01' tli(' COIT(,spoll<!illgGuldstollc bosolls. :\11 appropriat(' allsatz [01' definillg tllC wunld-be Goldstoll(' Imsolls iH the caseof an aclditiouallleutral ClIlTl'ut, 1)(')'0)1(1 the llIilliJllal stalldard JlI(l(lel, is also pres(,llte<!,

HESUMEN. Se ca1culau los ,'alores críticos para I<l!-ilIlasas de los bosollcs dc Higgs ('11el límiteCHel cual la teoría se hace fucrtelllente interactllall(C \lsando el teorema de equivalcncia, A altasl'IH..'rgí¡L" ~(' reemplazan los call1pos de 1I0rma 10Ilgitll<!iuallllt..'lIte polarizados ell la matriz S por losconespoudiclltes UOSOIlt'Sde GoldstOll<'. Se presenta \111Ilnsat: para definir los bOSOJl('Sde Goldstol1t'ell el ca..'m de teller tilla conicllt(, Ilentral adicional llliís all,í. del 1II0delo estiÍndar.

PACS; 12.15.C

The lIlilli'lla! ,talldard !1I0(it'¡ (~IS~I) gives all ad('l[lIale de"Tiplioll of Ihe a\'ailable experi-lIlelltal data 011 the e1ectro\\'eak ill!eraclioll. IJIII, tbeorists belie\'e, tben' is physics beyolldtbe ~lSM alld hope Ihat the i"h'ellt of IIC\\' acceleralors will sbow SOIll(' experilllelltalevi<1ellc(, of thi:;. This He\\' physics should brillg thl' lH'('d [01' !lew lllodels like t}¡e sllpcrsYlll-mctric Illodels, techllicolor 01' silllpl(' l'xtcllsiollS to th(' gauge grollp. Que can t hillk in Illan)'diffl'l'clIt ext.cllsiollS to t.be gauge gl'Ollp. SOllle of t he lI1Ost, relc\'aut are: the llIodificatioll ofthe Iliggs sector (illrludillg Iwo SU(2JL Higgs dOIl!>lels, olle additiollai Iliggs sillglet. olleadditiollal SU(2)J. triplet. or ilion' l'olllplicatt'd Higgs st.rllctul'l'S [1]), alld t.11l~use of rÍl'bergauge groups. left-right sY"""e!.ric lIlode1s SU(2)L0SU((2)u0U(1)JJ_L [2), or additiollalU( 1) groups SU(2JL 0 U( 1) G! U( 1)' [3). !low",'er. 111('origill of the Spolltalleous sYlIlllletrybreakillg, l1ccdcd f.o gCJlcrate t.he gallge bosolJ IIl.L..;ses, relllaills lluclear.

Iu Ihe ~ISM with olle !liggs doublel [4) Ihere are three would-be Goldstolle bosolls'caten' by the gallge fiplds, which bl'come lIIassin' il) thC' procc'ss aud lean' olle neutral

•Also Uni\'crsidad Nacional, O('P(o. de Física, A.A. 1-1490, I3ogol<Í, Colombia,'SIIPI)Orted by Colcieucias ,1I1,iU. (le Alltioqllia, A,A. 1~2G, i\ll'dellíll, ColOlIlIJia.

scalar particle: the Higgs oosons. The current experimental lower bOlllld for the mass(mil) of t his bosous stauds at about GO GeV 151. The actual yalue of '" 11is crucial for theyalidity of the :-'IS~1at high eu"rgies. H'calliug the fael that the uuitarity of th" th"oryis not pn'ser\"ed at Itigh clIergies if mil exceeds a critical valuc of ai>ollt 1 Te\' [U]. Int.his eaSl' the scalar sector of tll(' ~lS~Ibccomcs strongly illtrraclillg alld the pertllrbati\"ccxpallsioll of t he S matrix is no longer valid, t.lms chiral [7J 01' ('Ireet i\"p ¡ti] Lagrangianapproaches may bl' a1l appropriate dcscriptioll for tlle gattge hosons physics sillce t.Itey0111)' use tlle symlllet.ry bn'aking sclteme.

This can he Sl'CII if we consider the high energy limit (comparpd witil lile mass scalc oftIte part.ic1cs illvolved). The polarizatioll vect.or of a vcclor 1>oSOIlis

(k'f k(k'i'))

E" = -:\J' E + .\I(Q + M) .(1)

\\'ith /\" = (Q, k) and /(2 = .\(!. \\"here it is t'as)" to SP!' that. at high ('IH'rgil':-i. the dOlllillantpart of tite gallgc i>OSOIlSis tllat of longitudinal polarizatioll wbich. in tll(' Salllt' limil, cau

be writ t('lI as

(2)

:-'loreo\'l'r, iu th" 't-lIoofl-f'eyull1an gaug" (a s¡H'cial caso of Ih" so call"d 1I( gaug"s \\'hieh\\'(' wiIl disCllSS latcr in a more general form), the gallgf' fixing tertll is gi\'en hy

O,Y/, + i;\f </Ji = 0, (:3)

\\'11('1'(' F¡ and Q¡ are au)' n'ctor b0501l field aud it.s (,OlTf'SIHHldillg Coldstolle \)OSOIl; oIlecall r('pla('(' V¡ witiJ q)¡ iu auy S Illat.l'ix ('akl\lati0l1 at higll ('Il(~rgi('s,

S[V,j = S!1J,J + S[O(¡\}jQ)]' (4)

therdore the seatteriug amplitudes for lougitudiual \V's aud Z's eau he calculated fromthe scatteriug aIuplitudcs of the would-be Goldstouo bosous [!JI. up lo ord"r ¡\}IQ (oulyin tltis gauge); this is k110\\'U as tIle equivalencc theol'clll, proposcd by Lee, Quigg and

TiJaeker iu 1!Jii.TItemaillres11ltofthispapel.istopl.cScnt.allilllsatzforobt.aillillg t1le ('orresponding

would-I,,' Gulds! oue hosous iu t he SU(2)[ 0 U(1JI' 0 U(1)' model. wi l h oue SU(2) dou hletalld two si11glets. \Vc also filld tiJe upper bOlllld fol' thc lIIa:;s of t he lliggs i>osons ill tilelilllit ..¡;; » 1I111, wlterc jS is tlle ClIl. encrgy, hy lookillg at t.ltl' scattcrillg: pro('('ss oftwo neutral GoldstOllC hosolls. Via thc cqui\'aleuce t1lCO¡-CIIIt.ltis is a good apprOXilllatiollfoI' the scatlcring alllplitude of t\\.o llcutral high clIcrgy gallge bosons, \\'ith lOllgit.udiualpolarizatioll, Finall)', wc find a relatioll betwl'l'1l the llIass of tite extra lIiggs bOSOllS, theHe\\' neutral gallgc field and the mixil1g angle betwcen tite lIl'lltral currellts.

The SU(2) 0 V(l) 0 V(l)' model is importaut beeause futnre experimeuts may tiudadditiollillllculral ClIrrellt.s aud, even if it is 1I0t tIte exaet gallge grol1p of tite l'lectroweak

U~ITAIlITY CO~STIlAII'TS FOil TIIE MASS O, TIIE HIGGs... 37

interaction, it providcs thc minimal lllodel with sllch an extra current; thereforc it. miglltbe userul when considering sOllle snperstring theory-Imsed nlOdels, specially those withEG s)'mmetry nr lllodcls \vith left-rigltt s}'mmetry mcntioncd abovc. At low cllcrgics thcscmodels mighl behave like SU(2)1. C9U(l) 0 U(I)' [151.The most general expression fOl"the electrie charge in SU(2JL C9 U(I) (O) U(1)' is

(5)

where T31., }/I and }/2 are the diagonal generalors ofSU(2lL, U(I) aIl(1 U(I)', respectiYl'ly.The second te1"111is aY¡ +bY2 = YCIVS for a givell mn1tiplet, where YCIVS is tlw hyperchargein the \Veinberg-Salam model. The most general Lagrangian for the bosollic sector in thismodel with olle SU(2) doublel and one sillglel is givell by

L = L ((D,,'I>;)t(D"'I>;)+A;['I>!'I>; - en') + A3('I>l'I>¡)(Q>1'I>2), (6)i=1,2

\vhere thc paralllcters Ai_ i = 1,2,3, defilH' tite lIiggs potcntial, thc lIiggs field componentsare

and thc covariallt dcrivat.ivc is [11]

D ° . 7~ -1 .!JI}'B .92}"CJi = Ji - 1.'} .•. JI - /. 2 1 l' - 1 2 2 JI'

TIte Higgs (¡clds have vaCIIlIlII cxpect.atioll vallll's

(11)0 = VI,

(')0 = "2,

(7)

(8)

(9)

and thell the sYllllnetry is spont.alleonsly broken to U(1)Q.The mass matrix obtailled frolll the lIiggs potential for the fields "'3, "'4 (the neutral

would-bc Golstollc bOSOllS) is idclltically zcro, alld it is llllknown which of the fields are'eaten' by tite longitlldinally ¡>olarized gallge fields. 1'0 get tite renol"lnalizaole tlteory, intite 1l( gallge \Ve need to rancl'! tite lllixing terms '/:0 (D"Z"), 'lz' (O"Z,,,) and ",+ (D"IV"-)from tite kinetic Lagrangian of tlH' lIiggs fields, \Vltere '/:0, '1:0' ami ",+ are tite Goldstonebosolls of t}¡c ZIt1 Z;I <llld lV,; n'spedivel.y. 'fhe gauge fixing terlll in tIle lagrangian is f12J

L 1 (O ')' 1 (' Z" )'GF = - 2~" "A' - 2~, 0"" - ~,Mz'IZ

38 R. ~L\HTí,EZ ",D N. V",EGAS

allc1, llsing the ('ovariallt. derivativt' as a fUllctiOlI of tile real ficlds [11]' \Ve gel

q Sillt-,l ~ ( Sill O cot. ¡;: )'lz = -'- -' - ". /-¡ - . ["Y," - 2YGWS sinl ~i <;. JI;¿ cas () . l. I si 11 2~ 'r". •. i - ¡,

, 1::::::1,,1

9 cos I;J ~ ( si/l 8 1a/l ,', , . o )11z' = -- --o - ¡'i (li +. la} LPi - 2} G\VS. Slll- (j 'Pi.MZ' cos 8 s,n 2~

. i=:JA

(11 )

where ..Hz and Jlz' are tlle ZjI alld Z;t lllasscs respectiveI)' alld t3i are th(' T3I. C¡llautulll/luInuers of the Higgs fields. TII(' anglcs 11, ¡!JamI ~ are given uy

(l (1'2tan~=-'---.

b 'JI

esi/lll =~.

!J(12)

4J/",jcoslll sin 11t- ') i _ _ \\ -- ( l' _.' 2 <)

dll_UJ - ,'1. _ '2 ') '1. . ,2 " 2 1I I Slll l.., 1

.\/ X' + .\/X - _.\/11' j lOS 11 Sin ~

where 9'. 92 and e are thc conpli/lg constants aS'ociated with U(I), U(I)' a/ld U(I)Q,respcctivcly, O is tiJe \\'cillhcrg: angle alld tile élllglc¡jJ gives tile lllixillg bCt.WCCll tite '....cakneutral currcnts. Ir 'l/J = O thcrc is 110 lllixing and \\'c get tile SM \vith OIlC addit.iollallllultiplet plus one extra ten/l dm' Z'. According to some recent result, from LEP andlhe L3 collauoralio/l data [131 the poS'i"le valucs of the mixing angle are in thc rangcI¡!JIS .03-0.01.

From Eq. (i) and th" <¡uantlllll ll11mi)('l'sof th" lIiggs uudt.iple!.',

<;-1,= (I/IZ + al//,' (1:l)

ill t.lte lilllit \vhclI t.lw lIlixillg allgl<~ het.\Vt'Cll Z¡! aud Z;t is 4) ::::::01 \Ve have

2 cos 8 cos 1j;MXQ: = -------;::::: -11

yv]

2 cot 8MzfJ = ---- [- sin ~ cos ~ siu 1j;+ sin 8 cos 1/;("Ylf:¡ - 2 sinz O]

ya l'l¡fJ4 1J'2

MXsin¡jJ

M/"

[sill~cos~('os¡!J + SiIlOsill1j;("Y',p) - 2sin20ia=

2cosllsill¡!JMx' Mx' sill¡!J13=- . "'---

!J"l ;\/z

2cot8M",yaYIQ4 V'2

::::::cos -¡j;. ( 14)

UNITAHITY CONSTHAINTS FOR TIIE MASS OF TIIE HIGGs. . . 39

Now let us consider the processes '7z'7z ~ '7z'7z and 'Iz''Iz' ~ 'IZ"/Z', the scatteringamplitudes are

+ -t -_-t-,,-,2-. + 11 _"m 2 ]\ \

(15 )

where 11tH, In\: are the neutral lIiggs OOSOIlS masscs. Rccalling that the llllXIllg i:tllgle

betwccn the neutral gauge OOSOIlS is small, £01' simplicity \\'C asSUlllC tItat thc llllxmg

between JI and ~ is zero, i.e., .\3 = O.The scattering amplitude can be decomposed in partial waves, according to

00

T(8,e) = 1G7r¿([j Pj (cos e),j:::O

at high energy (8 » "'11. m~) the tree-level contributions to the Jamplitudes are given by

[4 2 4 2]Q m,u p 111,\

2 + 2 1

VI V2

. [42 f342]37, a 1nx 111'11([0('7Z"IZ' ~ 'IZ"7Z') = --- -- + -- .

sJ27r "i "1

(lG)

o partial wavc

(1 i)

From unitarity requirement the upper bound for the mass of the Higgs is therefore,

2 /O4íTv¡ 47iv2-- '" -----,3a' 3GFeos" 1/J

4íTV~ 4iTv~

3<7' '" 3 ros' 1/J

. 2 2 \[247r sm ~i .,< h

- 3J2G F sin2 eM1 cos' 1/J

( 18)

whcrc \ve llave made use of thc lIlass rclat.ioll [01' Jfz' obtaincd in flef. [11J, giVCll by

( 19)

40 R. MARTÍNEZ ANO N. VANEGAS

;/' "/ ,

/ \

6 / \I \

/ \

I\

~ 5 /\

~ \- / \"SO 4 / \x / \

~ / \~ 3 I \~~ / \~ / \" /~ ~ \~ I

I - \\/ / - \-I / "- \

/'/

""> / ,<'.\OO 0.2 0.4 0.6 O.S 1.0 1.2 1.4 1.6

~ (tad)

FIGURE l. The regioll betweell the horizontal Hne and the 'parabolic' curves gives the allowedvalues for lhe "', and lhe mixing angle e for differenl values of 1I1z', The dashed, lhe doled and,lhe dOI-dashed curves correspoud to lhe 1I1z' = 150 GeV, 300 GeV and 600 GeV, respeclively.

An interesting case occurs when DIle considers thc l¡mit ITlH ~ 1nXl

(20)

where we find a relation between the "', and the ~ angle for different values of Mz',lu Fig. 1, the regio n belween the horiwntalline and the 'parabolic' curves gives the P05-

sible values for the "'x aIHI the ~ angle. FOl' example, if ~ = 7[/4 then the "'x = 1728 GeV,3458 GeV and 6916 GeV for Ah, = 150 GeV, 300 GeV and 600 GeV, respectively.

In conc1usion, a relation between the masses and the mixing angles of lhe gauge fieldsand the would-be Goldslone bosons, for a theOl'Y with one additional ueutral current, isfound, u5ing the R( gauge fixing, aud we are able to recover the MSM constraiuts on theHiggs mass in the 1/J '" O case. Thc latter relation is valid fOl' any extra ueutral currentand the differences e!epene! on the coupling constants 91,92 and the coefficients (L, b whichdefine the electromaguetic charge. This coefficients depene! on the fermionic content ofthe Illodel and the cancellation of the anomalies.

ACKNO\VLEOGMENTS

R.M. wishes to thank Professor A. Salam for the opportunity of visiting the InternationalCenter fOl'Theoretical Physics and also COLCIENCIAS and The Third \Vorle! Academy

UNITARITY CONSTRAINTS FOR TIIE MASS OF TlIE HIGGs. . . 41

of Sciellces for their partial financial suppor!. N.V. would like to thank the Universidadde Antioquia for thejr support and the Fundación Mazda for their generous scholarship.

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