Topcolor Models and Scalar Spectrum

Gustavo Burdman Department of Physics, University of Wisconsin, Madison, WI 53706

ABSTRACT high precision. One way to avoid this problem within Topcolor models is to give up the idea that ESB is fully driven by the

We review the motivation and main aspects of Topcolor mod-

tt > Λ 'O( ) < Å condensate. For instance, a cutoff scale 1TeV els with emphasis on the spectrum of relatively light scalars and gives anon ®ne-tuned coupling (afew percent abovecritical), but

pseudo-scalars.

f ) ' (

a top-pion decay constant of the order of  50 60 GeV,

t Z

whichgivesonlysmallmassestothe W and the .Inthis m

I. DYNAMICAL GENERATION OF t . version of Topcolor [3], a separate mechanism must be invoked Z

to generate most of the W and masses. This is the case of m

The generation of a large fermion mass like t is an extremely

m m Z dif®cult problem in theories of dynamical Electroweak Symme- Topcolor-assisted Technicolor. Most of W and ,aswell

try Breaking (ESB). For instance, in Technicolor theories an as small ( ' 1 GeV) quark masses come from the Technicolor m Extended Technicolor (ETC) interaction is required in order to sector. Thus a small portion of t also comes from ETC terms, obtain fermion masses. The interaction of the ETC gauge bosons but most of the top-quark mass is dynamically generated by the with fermions and technifermions gives rise to fermion masses Topcolor mechanism. Theexplicit ETC quark mass terms for top through terms of the form and bottom turn the top-pions into massive pseudo-Goldstone bosons. Their masses can be estimated in the fermion loop 2

g approximation to be [3]

ET C

= hT T i ( ) m Å

L R

f 2 1

M

ET C

N m m c

2 ETC t 2

' Λ ( )

m 4 

t 2 2 f

8 

Å t

M hT T i

L R where ET C is the ETC gauge boson mass and is the technifermion condensate. Thus, in order to generate the

where mETC is the effective value of the ETC quark masses.

m O ( ) correct value of t , the ETC scale has to be of 1TeV ,which Although these are initially of the order of 1 GeV, the ETC is uncomfortably low. Several modi®cations of the dynamics top-quark mass receives large radiative enhancements from the within technicolor have been proposed in order to accommodate Topcolor interactions [3]. Thus one can have top-pion masses

such a large fermion mass [1]. On the other hand, in top-

m ' ( ) m

 t

in the range  100 300 GeV. If , then the

t t

condensation models, the large top-quark mass is obtained from +

t ! 

top quark would primarily decay as b. The current CDF

t

+

t t i

h Å R

a L arising as a consequence of a new gauge interaction,

Br(t ! W b) m >

measurement of implies  150 GeV at t Topcolor, which couples strongly to the top quark. Topcolor 68% con®dence level [4]. In what follows we will assume that

generates four-fermion interactions of the form the Topcolor enhancement to the ETC top mass is enough to

m >m t make  .

2 t g

 0 

Å ;

t t ( )

The existence of the top-pions ( ) is an essential in-

t Å t

R R L L 2 Λ2 gredient in the Topcolor scenario, regardless of the dynamics

responsible for the ESB sector and the other quark masses. The

(t; b) SU ( ) Λ where is the 2doublet and is the typical scale presence of other low-lying states depends on the details of the of the new interactions. If the coupling g is strong enough the top condensation occurs. The top chiral symmetry is spon- model. A Topcolor model is greatly speci®ed by choosing a

mechanism of isospin breaking, which selects the top-quark di- m

taneously broken and a large t is generated. This implies the presence of Goldstone bosons. Originally [2], it was pro- rection for condensation, leaving the bottom quark unaffected. posed that these were identi®ed with the longitudinal compo- The complete anomaly-free fermion content is necessary in or- nents of the electroweak gauge bosons so that Topcolor would der to know the scalar spectrum of the model. These low lying Λ also be fully responsible for ESB. With the ESB scale de®ned states have, in most cases, masses well below the cutoff scale ,

which points at them as possibly the ®rst signal for Topcolor.  as v 246 GeV, the decay constant of these top-pions is given by the Pagels-Stokar formula II. TOPCOLOR MODELS AND SCALAR

2 N Λ

2 c 2 SPECTRUM

' ( m )

f ln 3

 t

t 2 2 

16 m

t

( ) SU ( )

In all models theTopcolor group contains an SU 31 32

N Λ SU ( ) c

with c the number of colors. From (3) it can be seen that, in whichatanenergyscale breaks down to ordinary 3.The

p

= v= f m SU ( ) t order for  2and to be close to the measured value, 3is assumed to interact strongly with the third generation

t 1 15 the Topcolor scale Λ has to be extremely large (' 10 GeV). quarks. After Topcolor breaking there is, in addition to the

This translates into an acute ®ne-tuning of the coupling g in (1), massless gluons, an octet of massive colored vector particles: which has to be adjusted to the critical value with unnaturally the top-gluons. At this stage, if the Topcolor coupling is above

979

b  critical we would have both t and condensation. To avoid The 1 doublet acquires a Vacuum Expectation Value (VEV)

the latter, the effective Topcolor interaction must be isospin giving mass to the top quark through the coupling in (7). It is of

breaking. We describe two typical scenarios to implement this the form 

1 0 

p

f h + i +

 t

t t

aspect of the theory. r 2

 = ) (9

1



t

( ) A. Models with an additional U 1 As mentioned earlier, the set of three top-pions acquires a An effectively isospin breaking interaction is obtained by em- mass from explicit small quark mass terms (the ETC masses

in Topcolor-assisted Technicolor). There is also a scalar, the

( )

bedding two new U 1 interactions in the weak hypercharge h

t or top-Higgs, which mass is estimated in the Nambu±Jona-

U ( )  U ( ) ! U ( ) U ( ) group such that 1 1 1 2 1 Y , with the 1 1

strongly coupled to the third generation. This leaves an ad- Lasinio (NJL) approximation to be 0

ditional color-singlet massive vector boson, a Z . Both the

m ' m ( )

h t 0 Λ 2 10 top-gluon and the Z have masses of order . After integrating out these heavy particles, the interesting effective four-fermion The second doublet is present as long as the Topcolor interaction

interactions that are induced have the form [8] b

couples to R , as is the case in (5). It is given by

 

 

 

4 2 1 +

Å H

=  + t t

L Å

r L R R L

2

 = ( ) N M 11

9 c 1 0 0

2 p

B

H + iA

   2

1 Å Å

+  b b

R R L L (5)

N These states are deeply bound by the Topcolor interactions and

9 c therefore can be light. Their masses can be estimated once

22 22 0

 =(g= )   =(g= )  g '

Here, 34cot and 1 14cot , with 3 again within the NJL approximation. For instance for 1 1

g QC D U ( )

) Λ =(

and 1 the and 1 Y couplings respectively. The angles and 23TeV one has, for the neutral states [9],

0 

 and characterize the embedding of the Topcolor and the

m ' ( ) ; ( )

H;A

( )  U ( ) SU ( ) U ( )

U 150 330 GeV 12 Y

1 1 1 2 groups in 3c and 1 . The requirement

( ) U ( ) that SU 31and 1 1 couple strongly to the third generation

2 2 0 whereas the mass of the charged states is determined by the

   translates into the conditions cot  1, cot 1. The relation

criticality condition 2 2 2

m = m m ) + : (

 2 13

H;A t

H 

1 21

< <+ ( )

 critical 6The couplings to quarks can be read off equation (7). Calculating

N N c

9 c 9 z

the ®eld renormalization constants i to one loop and using (3),

m =f m

 t the couplings are simply t where is the dynamically

Å t

tti6= hbbi = must be satis®ed in order to obtain h Å 0and 0. generated top quark mass. This is a typical Goldberger-Treiman

Constraints on the top-gluon sector come from t production,

f =v '

factor. Recalling that in Topcolor models we expect  3,

t b as well as t and dijet mass distributions at the Tevatron [5].

0 we see that the coupling of the Topcolor ªHiggsº sector to the

This speci®c model is also constrained by the effects of the Z top quark is considerably larger than that of the SM Higgs boson. on low energy data, both at the Z pole [6] and at low energies Moreover, in models where the second doublet is present this

through FCNC [7, 8]. However, it is possible to accommodate 0

b (h ; )

couples to quarks with the same strength as t couple t Λ >

all these constraints with a Topcolor scale  1 TeV and still 0 0 A to top. This implies that H and decays are dominated by

not have a ®ne tuning problem. For instance, in the most general Å b the b ®nal state. The existence of these relatively light scalar case, a 2 TeV top-gluon gives a Topcolor coupling 4% above its states strongly coupled to third generation quarks implies a very critical value. Therefore, one can imagine a scenario where the rich phenomenology. In what follows we analyze the implica- Topcolor gauge bosons are at the few-TeV scale, making their tions of the scalar spectrum in the model described above. We direct detection dif®cult. In this scenario it is possible that the discuss other alternatives in model building in the next section, effects of Topcolor dynamics will appear at lower energy scales with emphasis on the differences in the scalar spectrum and due to the presence of a relatively light scalar spectrum. phenomenology. The effective interactions of (5) can be written in terms of two

Top-pions: As discussed earlier, these are the pseudo-  auxiliary scalar doublets  and . Their couplings to quarks 1 2 Goldstone bosons of the breaking of the top chiral symmetry. are given by [9] They couple to the third generation quarks as

Å Å

L =   t +   b + : : ( )

: L R L R m

e h c 7

1 1 2 2 t

0 +

p Å

( t t + it b  + ib t  )

i Å Å

L L R

5 R 14 f

2 

2 2 t

   ( +  = N )    (  = N ) c where 1 4 2 1 9 c and 2 4 1 9 .At energies below Λ the auxiliary ®elds acquire kinetic terms, be- Although their masses are lifted by the ETC interactions, they coming physical degrees of freedom. With the properly renor- can still play an important role in low energy observables such

1=2

r

 = z  B

malized ®elds i the criticality conditions (6) are

i i as as rare decay branching fractions and angular distributions

equivalent to [7, 8], as well as in electroweak precision observables at the

r r

h h i = f i = ( ) Z R b  0 8 pole like [10]. Top-pions do not have two-gauge-boson 1 t 2

980 

couplings, and thus single t production must involve a triangle extremely broad. Their production proceeds in similar ways to

h  t diagram. At hadron colliders they are mostly produced through that of t and , with the important difference that the quark

0 p

b s

the gluon-gluon- effective coupling induced by the top loop. inside the triangle loop, the quark, is much lighter than t

For instance, the gluon-gluon fusion has a cross section larger which translates into an effective suppression of the amplitude

m =f 

than that of the s-channel production of the SM Higgs boson by from its ªhardº value of t . t

a factor of [11] In addition to these low lying scalar states, the Topcolor in-   2 teractions in principle lead to the formation of heavier bound

2 v

p

 : ( )

r 15 states. For instance, there will be a color singlet vector meson, f

2 

t

  t

the top-rho t .However, not only couples to top-pions but 

This enhancement is also present in the t production in associa- also couples directly to third generation quarks and with strength

m =f  proportional to t . Moreover, their mass can be estimated

tion with top quarks, or in any production mechanism involving t

0 + Λ

e e

the ttÅ coupling. Production at colliders is discussed in in the NJL approximation to be of the order of the cutoff [10].

t 

[11]. This suggests that the in¯uence of t in low energy observables

<

m m Γ

< as well as in production of Topcolor bound-states (e.g. top-pions

  t

If 2 the top-pion would be narrow (  1GeV).

t t

b through vector meson dominance) is largely suppressed.

In models as the one presented above, where R couples to to the

( )

strong SU 31interaction, instanton effects induce a coupling

 b R of t to , which is not present in (14). This implies that, as B. ªAxialº Topcolor

long as the ttÅ channel is not open, the dominant decay mode of

( )

0 Å In this type of Topcolor models, the strong SU 31group bb the  is to [11]. Also, and independently of the presence of

t Å

b hb b i

L R

0 does not couple to R , barring the possibility of a con-  instanton effects, there is a coupling ofcc Å to given by (14)

densate. An example was presented in [8]. The cancelation

! c times two powers of the t mixing factor arising from the of anomalies requires the introduction of a new set of fermion

rotation of weak to mass quark eigenstates. However, in the a

Q a = ; :::; N

®elds, , with 1 Q. There is a new interaction, Å L;R

present model the bb mode is expected to dominate.

SU (N ) N Q Q . The novelty is that, depending on the choice of ,

Top-Higgs: It corresponds to a loosely bound, CP even, ttÅ

p light quarks might have to ªfeelº the strong Topcolor interac-

m =( f ) 

state, coupling to the top quark with strength t 2 .Inthe

N =

tion. For instance for Q 3 the fermions must transform

NJL approximation its mass is given by (10), and therefore de-

SU ( )  SU ( )  SU ( )

under 3Q 3 3as

tails of the non-perturbative dynamics are crucial to understand 1 2

h

(t; b) (c; s) ' ( ; ; )

the decay modes and width of the t . The top-Higgs produc- L

L 1 3 1 

tion is analogous to the t case. The main difference between

t ' ( ; ; )

R 1 3 1 h

these two states is that t couples to gauge boson pairs. These

Q ' ( ; ; )

R 3 3 1

couplings are suppressed with respect to the case of a SM Higgs

(u; d) ' ( ; ; )

L tt by a factor of 1=r .However,iftheÅchannel is not open this 1 1 3

Γ

(u; d) (c; s) ' ( ; ; )

would be the dominant decay mode. If this is the case, h will

R R t 1 1 3

be considerably smaller than the width of a SM Higgs, whereas

b ' ( ; ; )

R 1 1 3 2

its production cross section will still be r times larger. On the

Q ' ( ; ; ):

L 3 1 3

m > m t

other hand, if h 2 the cross section is still the same but t

2 Γ r

h is larger than the width of the SM Higgs of the same

t b

As advertised above, R is not coupled to the strong Top- tt h Å mass, so that t could only be detected as an excess in the in color interaction, whereas the cancelation of anomalies now

a given channel.

(c; s) SU ( )

requires that L transforms as a triplet under 31.The

0 0

;A ;H

The ªb-pionsº H :If these states are present (i.e.

b b b b

if Topcolor couples to R ) they give potentially dangerous con- tributions to various low energy observables. Their couplings to quarks are analogous to those in (14). The strongest constraint

0 Å 0

B on a model containing these bound-states comes from B mixing [9] and implies the existence of large suppression factors

in the quark mixing matrices [8]. A constraint independent of

= T ∆

these details is the contribution to  , the parameter

measuring deviations from the SM  parameter, due to the split-

ting between the neutral and charged states. As an illustration ∆

of the size of the effect, we plot  in Fig. 1 as a function of the mass of the neutral scalars and making use of the NJL result (13)

for the mass splitting. The horizontal lines represent the 95%

(M )= : Z

c.l. interval obtained using s 0115 [12]. Although

(M ) Z

the bound is tighter as s increases, the b-pion splitting,

( )

always present in models with the additional U 1 's, is not in ∆ contradiction with electroweak precision measurements. Figure 1: The contribution to  due to the mass splitting

These states decay almost exclusively to b pairs and they are among b-pions.

981

U ( ) Q L;R

quarks have standard 1 Y assignments, whereas the case, may be dif®cult to separate from the QCD background,

Y = SU ( )

have 0. Furthermore, they are 2L singlets and there- especially given that these scalars are expected to be very broad.

fore electrically neutral. If leptons are incorporated with their More detailed studies are necessary. Charm-top-pions can also

SU ( ) U ( ) Y

standard 2 1quantum numbers and as singlets un- be pair produced, just like the top-pions and the top-Higgs, via

SU ( )  SU ( )  SU ( ) der 3Q 31 32, all anomalies cancel. The their model independent couplings to gauge bosons. However,

Å

SU ( ) hQQi

3Q forms a condensate which, in turn, breaks Top- as in the case of the previous section, these couplings are also

SU ( ) f =v 

color down to 3c dynamically. suppressed by the ratio .

t

( )

The SU 31is chiral-critical and leads to the formation It is possible to extend this model to the ®rst family by just

tt m

u; d)

Å ( t

of a condensate and a dynamical . This breaks an including the L among the quarks strongly coupled to the

SU ( )  U ( )  U ( )

(c; s)

L L R 4 1 1global chiral symmetry and leads Topcolor interactions, in addition to L . Such Topcolor

to the existence of a composite scalar ®eld F , quadruplet un- models are partly motivated by the possibility of the existence

SU ( )

der 4L. This can be decomposed into two doublets, one of of an excess of events at high energies in the CDF data for the

r

 h which is just 1 of Section II.A, containing t and the top-pions. inclusive jet cross section [13]. In Ref. [14] the consequences

The additional doublet, C , contains a set of three pseudo-scalars, of a universally coupled top-gluon on the inclusive jet produc-

~ h c the ªcharm-top-pionsº or c , as well as a ªcharm-top-Higgsº : tion were studied. It is possible to write down an anomaly

free realization of this proposal. The only other modi®cation 

 1

p

(h + i ) c

2 c needed to insure the cancelation of all anomalies is that now

= ( )

C 16

N = (u; d) ( ; ; )

 L

we need Q 5. Thus, transforms as 1 3 1 under

c

SU ( )  SU ( )  SU ( )

5Q 31 32, and the other fermions trans-

The masses of the charm-top-pions are generated by the same Q

form as before, with the exception of the L;R that transform

( )

terms that break chiral symmetry explicitly and induce the top- SU

as quintuplets under 5Q . The global chiral symmetry bro-

( )  U ( )

pion masses, so they are expected to be of the same order as SU R

ken by the dynamical top quark mass is now 6L 1.

m h  m

c t  . The mass of the in the NJL approximation is . t Therefore, besides the scalar content of the two previous mod-

The couplings to quarks are analogous to those of the top-pions els, there is a new scalar doublet, U , the ªup-top-pionsº. Their m

and have the form [8] h 

masses are very similar to those of t and . Their couplings c

to quarks can be read off (17), with the replacements C!Uand

m

t

c s) t C + : : ( )

( Å

R (c s ) ! (u d)

Å Å L h c 17 L

Å Å L Å . Although these couplings imply an additional

f 

t 0 Å 0

D contribution to D mixing, it is governed by the same Important constraints on this scenario come from low energy product of up-quark rotation matrix elements, and therefore can

0 Å 0 be avoided by the same choice of mass matrices that suppressed

D observables. For instance, D mixing is mediated by

0 the charm-top-pion contributions. There will also be new con-

h 

three-level s-channel exchange of c and .However,the

c

Z h

mixing amplitude is proportional to the product of the various tributions to the hadronic width [10]. The production of u  unknownquark rotationfactors entering thetransformation from and u has the same features as that of the charm-top-pions. In the weak to the mass eigen-basis for quarks [8]. These factors both cases, one expects these resonances to be very broad and depend on the sector of the theory responsible for light quark a careful study is needed in order to establish their detectability masses, e.g. ETC. Thus, charm mixing is a direct constraint in the various channels and in different environments.

on the light-quark mass matrices generated by this sector. On

R R s the other hand, charm-top-pion contributions to c and are III. CONCLUSIONS potentially large [10] and independent of these details. The production of top-pions in this model is completely anal- We have reviewed the main aspects of Topcolor models and ogous to the model in the prevoius section. However, here the the constraints derived from the spectrum of low laying scalars

b and pseudo-scalars. A basic feature of all Topcolor models top-gluons do not couple to R , so there will be no instanton in- 0

Å h

b  ! bb

duced quark mass term and therefore no decay mode. is the existence of the loosely bound state t and a triplet of

t

~ tt Thus, if top-pions have a mass below the Å threshold, the neutral top-pions t . The latter is present in the physical spectrum in states will primarily decay to gluons. Also and as we mentioned models where Topcolor is not solely responsible for ESB (e.g.

in the previous model, thecc Å decay mode is suppressed by two Topcolor-assisted Technicolor).

h

t ! c factors coming from the rotations. Although these fac- Although the production of t shares several features with that tors are not very constrained, naive estimates lead to values of of the SM Higgs, it has a larger cross section and a very different

0 2

r m

 ! cc) Γ (

width (larger or smaller by depending on h ). On the other

Å that indicate that this mode is competitive with the t c 0

hand, the observation of  at hadron colliders is problematic gluon-gluon channel [11]. t

Å cc The production of charm-top-pions does not proceed via given that its decay modes, bb,Å and gluon-gluon, are very gluon-gluon fusion. Production through the anomaly is no only hard to extract from the background. Its production at lepton

suppressed by loop factors and couplings but also by CKM fac- colliders is discussed in [11]. h

tors. The most ef®cient mechanism for single production of c Speci®c realizations of Topcolor tend to have additional, rel-  0 ;

or  is by emission off a top quark line, with this turning atively light, scalars. Given that these tend to be very broad c

into a charm or strange quark for neutral and charged emission objects, their detectability at various facilities requires a careful

tcc respectively. The multijet ®nal state, e.g. t Å Å for the neutral study, particularly at hadron colliders. Considering that Top-

982 color theories are still at relatively early model building stages, the study of low energy signals of and constraints on the scalar spectrum in the various models will play a central role in deter- mining what the dominant Topcolor phenomenology will be in future experiments. ACKNOWLEDGMENTS: Much of this work is based on collaborations and/or conversations with Sekhar Chivukula, Chris Hill, Dimitris Kominis and Elizabeth Simmons. The au- thor thanks Michael Peskin for pointing out the possible effects

∆ 

of b-pions in  . The ®rst part of this work was done while the author was at the Fermilab Theory Group. This research was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-95ER40896 and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation.

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983