On the dynamical symmetry breaking of the electroweak interactions by the top quark

16 March 2000

Ž .Physics Letters B 476 2000 331–338

On the dynamical symmetry breaking of the electroweakinteractions by the top quark

Nguyen Van Hieu a,1, Pham Xuan Yem b,2

a Institut of Physics, NCST, PO Box 429, Bo Ho, Hanoi 10000, Viet Namb 3 er ( )Laboratoire de Physique Theorique et Hautes Energies , Tour 16r1 etage, UniÕersites Paris 6 UniÕersite P. et M. Curie et Paris 7,´ ´ ´ ´

BP 126, 4 place Jussieu, F-75252 Paris cedex 05, France

Received 14 December 1999; received in revised form 26 January 2000; accepted 1 February 2000Editor: R. Gatto

Abstract

We discuss the electroweak gauge symmetry breaking triggered by a new strong attractive interaction to condensatefermion-antifermion, and topcolor is a prototype. To deal with the fermion pairing, a general method based on theHubbard–Stratonovich transformation in the functional integral approach is used.

We derive a formula which relates the W ", Z 0 weak boson masses to that of the condensated fermion, thus generalizingŽ .the Pagels–Stokar formula obtained in QCD. The custodial SU 2 electroweak symmetry turns out to be systematically

2 Ž 2 2 .violated, the deviation of r'M r M cos u from unity is related to the new physics scale L. Some phenomenologicalW Z W

consequences of the top-pair condensation models are discussed. Distinctive signatures of the tt scalar bound state, a Higgs0 ) )boson like denoted by H , are the dominant decay modes H™Fqg , H™FqZ , and H™B qB . q 2000 Elseviert t t t

Science B.V. All rights reserved.

PACS: 11.15.Ex; 11.15.Tk; 12.60.Fr; 14.80.Cp

w xThe well-known Higgs mechanism 1–3 that in-volves an elementary scalar field may not be theunique scenario to spontaneously break the gauge

w xsymmetry of the standard electroweak theory 4 . AŽ .dynamical symmetry breaking DSB due to the

condensation of some fermion-antifermion pair mayalso generate masses to the gauge bosons, a typicalexample borrowed from superconductivity is the

1 E-mail: [email protected] E-mail: [email protected] Unite associee au CNRS, UMR 7589.´ ´

Cooper electron pair. Another example is the nonzerovacuum expectation value of massless quark-anti-quark pair, its condensate breaks the QCD chiralsymmetry. In all cases, nonzero numbers must be

² :assumed to break the symmetries, i.e. 0 F 0 /0,y y² : ² :0 e e 0 /0, 0 qq 0 /0 respectively in the

standard Higgs mechanism, the Cooper electron pairin superconductivity, and quark-antiquark pair inQCD.

With DSB, in order to have large values for theW " and Z0 weak boson masses, there must existbeyond the standard model some new attractive in-teraction with sufficiently massive fermions involved

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00142-8

( )N. Van Hieu, P. Xuan YemrPhysics Letters B 476 2000 331–338332

which replaces the Higgs potential lF 4 qm2F 2

with the wrong sign m2 -0. This idea has motivatedw x w xthe topcolor interaction 5–8 and its extension 9–11

as the dynamical breaking of the electroweak sym-metry due to the condensation of the top-antitop pairsince the top quark is the most massive elementaryparticles. A review of the top-condensation models

w xwith extensive references is recently available 12 .An attempt is made in this note to establish a

relation between the masses of the condensatedfermion and the weak vector bosons W ", Z 0. The

w xHubbard–Stratonovich transformation 13 applied tothe functional integral method, previously devel-

w xopped by one of us 14 for condensed matter physics,turns out to be particularly powerful for treating theproblem of fermion pairing considered here.

We start by introduce a system of left-handed topŽ .and bottom quarks put into an SU 2 doublet:

1yg5c , as1,2 with c x s c x ,Ž . Ž .a 1 b2

1yg5c x s c x , 1Ž . Ž . Ž .2 t2

and a singlet fermion x which is the right-handedtop quark,

1qg5x x s c x . 2Ž . Ž . Ž .t2

We further postulate that the effective four-fermiontopcolor interaction – mediated by topgluons GG –t

may be written in the form

a ,a db gLL sx x c x V c x x x , 3Ž . Ž . Ž . Ž . Ž .int d ag a ,b

b ddb mV sG g g 4Ž . Ž . Ž .gag m a

with some strong coupling constant G having theŽ .y2mass dimension. The quark color index is im-plicitly understood, however it is convenient to ex-plicit the spinor indices a , PPP ,ds1 PPP 4. Startingfrom massless fields, the role of the topcolor interac-

Ž .tion 4 is to dynamically generate masses to boththe top quark as well as to the gauge bosons. Ongeneral ground, one would expect that these masses

are functions of the coupling constant G and of thenew physics energy scale L, as we will see later. Weconsider now the functional integral of the system

w x w x w xZs Dc Dc Dx DxH

b4 a ,a=exp i d x c x Eu c xŽ . Ž . Ž .H a a ,b½baqx x Eu x xŽ . Ž . Ž .a b 5

4 a ,a=exp i d x x x c xŽ . Ž .H d½= db gV c x x x . 5Ž . Ž . Ž .ag a ,b 5

For the free fermion system without V db interaction,ag

the functional integral becomes

w x w x w xZ s Dc Dc Dx DxH0

b4 a ,a=exp i d x c x Eu c xŽ . Ž . Ž .H a a ,b½baqx x Eu x x . 6Ž . Ž . Ž . Ž .a b 5

w x Ž .3Let us introduce 14 the dimensional mass auxil-g a,aŽ . Ž .iary fields denoted by F x and F x whicha,a g

represent the fermion-antifermion system, where F

is defined from F as follows:

ada ,a †a ,bF x s g F x gŽ . Ž . Ž . Ž .g bg 0 d 0

) ad ds g F x g .Ž . Ž . Ž .Ž .g b0 a ,b 0

The associated functional integral for these auxiliaryfields is

F w x w xZ s DF DFH0

4 a ,a db g=exp yi d x F x V F x . 7Ž . Ž . Ž .H d ag a ,b½ 5

( )N. Van Hieu, P. Xuan YemrPhysics Letters B 476 2000 331–338 333

Now we apply the Hubbard–Stratonovich transfor-w x Ž .mation 13 to the interacting part of the action 3 in

the functional integral, and get

4 a ,a db gexp i d x x x c x V c x x xŽ . Ž . Ž . Ž .H d ag a ,b½ 51

w x w xs DF DFHFZ0

4 a ,a db g=exp yi d x F x V F xŽ . Ž .H d ag a ,b½ 5= 4 a ,b gexp yi d x D x c x x xŽ . Ž . Ž .H g a ,b½

a ,a dqx x c x D x , 8Ž . Ž . Ž . Ž .d a ,a 5where

Dd x sV dbF g x ,Ž . Ž .a ,a ag a ,b

a ,b a ,a dbD x sF x V . 9Ž . Ž . Ž .g d ag

d Ž .Physically, the D x defined above represents aa,a

bosonic field which is a bound state of the fermion-antifermion pair due to the strong interaction V db. Itag

is not necessarily a scalar field and it has the canoni-Ž .1cal mass dimension.

Ž .Substituting the expression 8 into the r.h.s. ofŽ .5 and integrating out over all the fermionic func-

ational variables c ,c ,x ,x , we can express Z as aa

functional integral over only the auxiliary fieldsg a,aŽ . Ž .F x and F x , thusa,b g

Z0 w x w x w xZs DF DF exp iS F ,F 10Ž .Ž .H effFZ0

w xwith some effective action 14

4 a ,a db gw xS F ,F sy d x F x V F xŽ . Ž .Heff d ag a ,b

`

Ž2 n.w xq W D,D , 11Ž .Ýns1

Ž2 n.w xand W D,D is a functional of the n-th orderd Ž .with respect to each kind of fields D x anda,a

a,b Ž2 n.Ž . w xD x . In order to write W D,D in a compactg

ˆ Ž .form, we introduce the 4=4 matrices D x andaaˆ Ž .D x with the elements

agg a a ,aˆD̂ x sD x , D x sD x .Ž . Ž . Ž . Ž .a a ,a gga

Then we have

Ž2. 4 4 a Lˆw xW D,D s i d x d y Tr D x S xyyŽ . Ž .H= RD̂ y S yyx , 12Ž . Ž . Ž .a

iŽ4. 4 4 4 4w xW D,D s d x d y d x d yH 1 1 2 22

= a L1̂ ˆTr D x S x yy D yŽ . Ž . Ž .1 1 1 a 11

= R a 2̂S y yx D xŽ . Ž .1 2 2

= L ˆS x yy D yŽ . Ž .2 2 a 22

= RS y yx , 13Ž . Ž .2 1

iŽ2 n. 4 4 4 4w xW D,D s d x d y PPP d x d yH 1 1 n nn

a L1̂ ˆ=Tr D x S x yy D yŽ . Ž . Ž .1 1 1 a 11

= R a n̂S y yx PPP PPP D xŽ . Ž .1 2 n

= L ˆS x yy D yŽ . Ž .n n a nn

= RS y yx , 14Ž . Ž .n 1

LŽ . RŽ .where S xyy and S xyy being respectivelythe propagators of left-handed and right-handedmassless fermions,

1yg5LEuS xyy s d xyy ,Ž . Ž .2

1qg5REuS xyy s d xyy . 15Ž . Ž . Ž .2

Ž .From the expression 11 of the auxiliary fieldseffective action, we derive the field equations

Ž2 n.` w xd W D,Dd dbD x sV . 16Ž . Ž .Ýa ,a ag a ,bd D xŽ .gns1

d a,bŽ . Ž .The bosonic fields D x and D x which de-a,a g

scribe the quark-antiquark systems bound by thestrong interaction V db must be the solutions of theag

Ž .field Eq. 16 .


g Ž .Among the most general bosonic field D x ,a,a

let us consider now a special class of the scalarŽ .D x by making the projectiona

g1qg5gD x s D x ,Ž . Ž .a ,a až /2 a

a1yg5a ,a )D x s D x . 17Ž . Ž . Ž .g až /2 g

In some sense, as we will see, this compositeŽ .scalar D x substitutes the standard elementarya

Higgs field to generate masses to both the top quarkand the gauge bosons. Associated to this particularŽ . Ž . Ž .D x case, the functionals 12 and 13 becomea

Ž2.w xW D,D

s i d4 x d4 y D) x D y l yyx , 18Ž . Ž . Ž . Ž .H a a

Ž4.w xW D,D

i4 4 4 4 )s d x d y d z d w D x D yŽ . Ž .H a a2

=D) z D w P xyy , yyz , zyw ,Ž . Ž . Ž .b b

19Ž .where

L Rl xyy sTr S xyy S yyx , 20Ž . Ž . Ž . Ž .P xyy , yyz , zywŽ .

L R LsTr S xyy S yyz S zywŽ . Ž . Ž .R=S wyx . 21Ž . Ž .

Ž2 n.w xFor the other functionals W D,D , n)2, wehave similar expressions easily generalized. They are

Ž .all nonlocal functionals of the scalar fields D xa) Ž .and D x . Each of them can be expressed in termsa

Ž .of a corresponding local functional of D x anda) Ž . m n l Ž . m nD x and their derivatives E E PPP E D x , E Ea a

l ) Ž .PPP E D x to all orders.a

In the presence of the vector gauge boson fieldsb Ž .W x and B x , the ordinary derivatives EŽ .m m ma

must be replaced by the covariant ones D , thusm

E D x ™D D xŽ . Ž .m a m a

bsE D x q i A x D x , 22Ž . Ž . Ž . Ž .m a m ba

gXb b bA x sg W x q B x d . 23Ž . Ž . Ž . Ž .m m m aa a 2

Up to the second order with respect to the firstŽ . ) Ž .derivatives D D x , D D x and the first orderm a m a

Ž .with respect to the second derivatives D D D x ,m n a) Ž .D D D x , we obtainm n a

Ž .Ž2. 4 0 )w xW D,D s i d x l x D x D xŽ . Ž . Ž .�H a a

1 Ž2. ) mq l x D x D D D x ,Ž . Ž . Ž . 4a m a2

24Ž .Ž4.w xW D,D

iŽ .4 0 ) )s d x P x D x D x D x D xŽ . Ž . Ž . Ž . Ž .�H a a b b2

Ž .2 ) ) mqP x D x D x D x D D D xŽ . Ž . Ž . Ž . Ž .a a b m b

qV x D) x D mD x D) x D D x ,Ž . Ž . Ž . Ž . Ž . 4a a b m b

25Ž .

where

lŽ0. x s d4 y l xyy , 26Ž . Ž . Ž .HmŽ2. 4l x s d y yyx yyx l xyy ,Ž . Ž . Ž . Ž .mH

27Ž .

P Ž0. x s d4 y d4 z d4 w P xyy , yyz , zyw ,Ž . Ž .H28Ž .

mŽ2. 4 4 4P x s d y d z d w yyx yyxŽ . Ž . Ž . mH=P xyy , yyz , zyw , 29Ž . Ž .

m4 4 4V x s d y d z d w yyx wyzŽ . Ž . Ž . mH=P xyy , yyz , zyw . 30Ž . Ž .

Ž2 n.w xFor W D,D with n)2, we have expressionsŽ . Ž .generalizing 24 and 25 .

Ž .Now in both sides of the field Eq. 16 , as well asŽ . Ž .in 24 , 25 and their generalizations for

Ž2 n.w xW D,D ,n)2, we set the scalar composite fieldsŽ .D x to be equal to their vacuum expectation valuesa

DŽ0. which describe the condensate of quark-anti-aŽ . Ž .quark pairs. From 1 and 2 , we note that among


the two components DŽ0. of the doublet, only thea

second component DŽ0. is a neutral field and couldas2

have non-vanishing vacuum expectation value, thus

DŽ0.sd D , 31Ž .a a2

where D is some real constant that we put equalto the top mass DsM . This point can be seent

a , b gŽ . Ž . Ž .from the term D x c x x x q xg a , b da,a dŽ . Ž . Ž . Ž .x c x D x figured on the second line of 8 .a,a

Ž .With this constant value 31 of the scalar fieldsŽ . Ž .D x , the field Eq. 16 reduces toa

1 N 2 y1 yi d4 pcs , 32Ž .H4 2 24G 2 N p yM q ie2pŽ .c t

where the implicit number of quark colors N s3 iscdb Ž . Ž .now taken into account in V of 4 and 16 . Theag

Ž .divergence of the integral in the r.h.s. of 32 may behandled by a momentum cutoff L written in acovariant manner, and we obtain the algebraic equa-tion

N 2 y1 1 x d x2c LH2 22 N 4p xqM0c t

N 2 y1 1cs 22 N 4pc

=

2 2L qM 1t2 2L yM ln s . 33Ž .t 2 GMt

The solution of this equation does exist if and only ifL satisfies the condition

4p 2 2 Nc2L ) . 34Ž .2G N y1c

Ž . Ž .From 33 , let us denote by ys f x the positiveŽ .solution of the equation ln 1qy sxy in the interval

0-x-1, then we have

L2 4p 2 2 Nc2M s , x s1y . 35Ž .t 0 2 2f x GL N y1Ž .0 c

w x 2This equation, reminiscent of 15 , determines M int

terms of L2 and G.In order to derive the gauge boson masses, we

Ž .substitute the constant value 31 of the scalar fieldsŽ2 n.Ž . w xD x into the expressions of W D,D , sum upa

all these expressions and separate out the quadratic

term A A m which contributes to the vector gaugemA Ž .boson mass in the effective Lagrangian LL x .mass

We finally obtain

y1 c 2A 2 mLL x s M II A x A xŽ . Ž . Ž .½mass t m c22

2 24 mqM JJ A x A x , 36Ž . Ž . Ž .5t m 22

where

iIIsy 42pŽ .

=

L R L R˜ ˜ ˜ ˜Tr S p S p S p S pŽ . Ž . Ž . Ž .4d p ,H 2Mt

1y 2p

37Ž .

y1 iJJs

42 Ž .2p

=˜L ˜R ˜L ˜R ˜L ˜Rw xŽ . Ž . Ž . Ž . Ž . Ž .Tr S p S p S p S p S p S p

4d p .H 22Mt1y

2ž /p

38Ž .

˜L ˜RŽ . Ž .S p and S p being the propagators of the freeleft-handed and right-handed massless fermions inmomentum space:

1yg i pu5LS̃ p s ,Ž . 22 p q ie

1qg i pu5RS̃ p s . 39Ž . Ž .22 p q ie

Since II is divergent, we again use the cutoff L in acovariant manner and get

N d x N L2 qM 22c c tL

IIs s ln . 40Ž .H2 2 2 28p xqM 8p M0 t t

The integral JJ is convergent and equals

`N d x N 1c cJJsy sy .H2 2 2 2216p 16p M0 txqMŽ .t

41Ž .


Setting

1 11 2q yW s W , W s W , 42Ž .Ž . Ž .m m m m2 1' '2 21 2 1W sy W s sinu A qcosu Z ,Ž . Ž .m m W m W m21 2

43Ž .B scosu A ysinu Z , 44Ž .m W m W m

and using gsinu ygXcosu s0, we obtain fromW WŽ .36

LL A x syM 2 Wq x W m xŽ . Ž . Ž .mass W m

1 2 my M Z x Z x , 45Ž . Ž . Ž .Z m2

where the gauge boson masses are found to be

g 2 L2 qM 2t2 2M sN M ln , 46Ž .W c t2 232p Mt

X2 2 2 2g qg L qMt 12 2M sN M ln y . 47Ž .Z c t 22 232p Mt

Ž .Eq. 46 is reminiscent of the Pagels–Stokar formulaw x16 which relates the charged pion decay constantf f 131 MeV to the dynamically generated massp

obtained in QCD for the up down quarks. FromQCD to electroweak interactions, the f of the for-p

mer is replaced by the vacuum expectation valuey1r2'w xÕs 2 G s2 M rgf246 GeV of the latter.F W

This may be schematically transcribed as: In QCD,the Pagels–Stokar formula relates f to m . Inp u,d

dynamical symmetry breaking of the electroweakŽ .interactions, Eq. 46 gives Õ in terms of M .t

Furthermore, with g 2 qg X 2 sg 2rcos2u , we getWŽ . Ž .from 46 and 47

M 2 1Wr' s1q . 48Ž .2 2 2M cos u LZ W

2ln q12ž /Mt

The precision electroweak measurements at the 10y3

level force the scale L of top-condensation modelsto be very high about 1015 GeV when we compare

Ž . Ž . Ž . 15experimental data with 46 , 47 and 48 . This 10GeV of L together with the top quark mass M s174t

Ž 2 2 . Ž .GeV implies that the solution y' L rM s f xt 0Ž . Ž .of the Eqs. 33 and 35

ln 1qyŽ .sx0y

must force x to be0

4p 2 2 Ncx s1y0 2 2GL N y1c

M 2 L2t y24s ln 1q f10 .2 2ž /L Mt

This is equivalent to fine-tune the coupling constantG to be Gs3p 2rL2 for N s3. This fine-tuning ofc

G naturally has the following dynamical explanationw x7 :

Ž .The four-fermion interactions as defined in 4could be generated by the exchange of a very mas-sive topgluon GG with mass M f1014 GeV sucht X

that

g 2 3p 2t

Gs s ,2 2M LX

where g is the topcolor coupling constant which istŽ .presumably of order OO 1 for a strong attractiveŽ .interaction. The gap Eq. 33 could require a large

w xhierarchy for stability 7 .It is gratifying to note that using a different

method, we recover some results previously obtainedw xin the literature 7,15,16 . We also remark that in the

standard Higgs mechanism, at the tree-level the pa-<rameter r is equal to unity. The correction D rs r

<y1 can only come from higher order loops towhich the top quark contribution at order g 2 is

w xquadratic in its mass M , thus 17t

3g 2 M 2t

Drs . 49Ž .2 2 264p cos u MW W

Ž .The result 49 is in contrast with the dynamicalsymmetry model discussed here for which the r

parameter is already different from unity to zeroorder of the coupling constants g, gX, as shown by

Ž .Eq. 48 . The unbroken global symmetry of theŽ .Higgs sector translated by rs1 usually called

Ž . w xcustodial SU 2 symmetry 18 of the electroweaktheory is systematically violated by a smooth loga-

Ž .rithm of M . The patterns of custodial SU 2 sym-tŽ . Ž .metry breaking as illustrated by 48 and 49 are

conceptually not the same.We now briefly discuss some phenomenological

consequences of the top-condensation models, in


Ž .Fig. 1. H™F q Z g .t

particular the production and decay modes of theneutral scalar denoted by H which replaces thet

elementary Higgs boson H 0 of the standard model.1. In general, whatever the schema invoked to

dynamically generate the top and the W,Z masses,there must exist a triplet of new Nambu–Goldstonebosons resulting from the breaking of chiral symme-

Ž .try in the top-bottom system postulated in 1 PPPŽ .4 . They are absorbed by the W,Z to acquire masses

Ž . Ž .as shown in 46 , 47 . Furthermore, a neutral CP-even state analogous to the s boson in QCD mustexist, it is a scalar tt bound state denoted by H .t

Ž .Since the force postulated in 4 that ties top-antitoppair is so strong that it can dynamically generatesuch a huge 175 GeV mass, it is likely that thebinding energy is large in H and its mass could bet

w x w xsmaller 10 than twice 5 the top mass. This is takenas a very rough indication of where to find H .t

2. The production cross section of H is governedtw xby gluon-gluon fusion 17 into tt pairs, so that the

Žhadron colliders Tevatron at FermiLab and LHC at.Cern are appropriately the right places for the Ht

searches. On the other hand, the ZqH associatedt

production by lepton colliders, eqey™Z )

™ZqHt

is largely suppressed, since a direct coupling ZZH ist

absent. This is also in sharp contrast with the ele-mentary Higgs boson H 0 production which is domi-

w x q y ) 0nated 17 by e e ™Z ™ZqH , because thedirect coupling ZZH 0 is large.

3. Due to the nature of a strongly bound tt state,the H is leptophobic, and ‘‘almost’’ hadrophobic,t

i.e. it cannot decay into leptons and ‘‘light’’ hadronsmade up by the first two families up, down, charm,

Žstrange quarks at the tree level lowest order of theX.coupling constants g, g . Indeed, the Yukawa direct

couplings of H with leptons and the first twot

families of quarks are absent, contrarily to the stan-

dard elementary Higgs boson H 0 case. Therefore theH although so massive would have a very narrowt

Ž .width only a few GeV in sharp contrast with thestandard elementary Higgs boson H 0 which has a

w xwidth 17 around 18 GeV for a hypothetical 350GeV f2 M mass. The decays of H into ‘‘light’’t t

hadrons can only proceed through gluons emission,similarly to quarkonium Jrc and F decays which

Ž .are suppressed by the Okubo–Zweig–Iizuka OZIrule reflecting QCD asymptotic freedom.

4. The most distinctive signatures of H would bet

its dominant decay modes into the bottomonium Fand an energetic photon or F accompanied by the Zweak boson, as depicted by Fig. 1. This comes fromthe special situation of the third family top-bottomquarks which have the additional topcolor interaction– mediated by topgluons GG – thus destroying thet

universality between the three quark families. Due tothe non-universal character of the top-bottom systemwhich does not possess the Glashow–Iliopoulos–

Ž .Maiani GIM cancellation, flavor changing neutraldecay of H into tcq tc channels is another spectac-t

w xular signature 19 of the top condensation models.Ž . Ž .The ratio G H™FqZ rG H™Fqg ist t

found to be

G H™FqZŽ .t

G H™FqgŽ .t

23s ž /4sinu cosuW W

=

2M2 Z1 4 12y sin u q 1y f0.8 .Ž .W2 3 4 2ž /MHt

50Ž .

Finally we remark that hadronic decays of H wouldt

proceed into bb pair through two topgluons GG ex-t

change in Fig. 2. These decay modes into B) , B

) )Ž . Ž .Fig. 2. The OZI unsuppressed H™B B q B B by topglu-t

ons GG exchange.t


) )Ž . Ž .mesons: H™B B q B B are OZI unsup-t

pressed.

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Documents

On the dynamical symmetry breaking of the electroweak interactions by the top quark