PHYSICAL REVIEW D VOLUME 50, NUMBER 1 1 1 DECEMBER 1994

Radiative corrections to I'(Z + b5) from colored scalars in a model with dynamical symmetry breaking

Anirban Kundu,' Sreerup Raychaudhuri,t Triptesh De, and Binayak Dutta-Roy$ Theory Group, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta-700064, India

(Received 14 January 1994; revised manuscript received 18 July 1994)

Isodoublet color-octet scalar bosons appear in the low-energy limit of a natural extension of the standard model in which the electroweak symmetry is broken by a tf condensate. We show that radiative corrections (involving these scalars) to the branching ratio Rb = r ( Z + b 6 ) / r ( Z + hadrons) are negative and thus place a stringent lower bound on the masses of the colored scalars. This turns out to be N 400 GeV with mt = 150 GeV (and - 700 GeV for mt = 175 GeV) and increases quadratically with mt. It is emphasized that the parameter Rb is well determined experimentally and that theoretical estimates are relatively free from uncertainties emanating from hadronic corrections; thus, it is able to single out the vertex correction sensitive to the nondecoupling effects induced by the heavy top quark.

PACS number(s): 13.38.Dg, 12.15.Lk, 12.60.Fr

Recently, it has been shown [I-41 that the electroweak symmetry of the standard model (SM) may be broken dy- namically by a tf condensate. This is referred to in the literature [5] as the "top-mode standard model." The top quark, being much heavier than the other known fermions (and lying close in the mass spectrum to the electroweak scale v = 246 GeV), may, in this picture, be respon- sible for the breaking of the SU(3), x SU(2)L x U(l )y to SU(3), x U(l),,. It has been shown [2] that in this model, where the presence of a four-fermion interaction of the form G ( ~ L ~ R ) ( ~ R + L ) induces the symmetry break- ing, the bound-state spectrum consists of three mass- less Nambu-Goldstone bosons, which give masses to the massless gauge bosons, and one massive neutral scalar. which may be identified as the Higgs boson.

This model has several attractive features. First, the naturalness problem arising in the elementary scalar sec- tor of the standard model can be isolated in the coupling constant G once for all. Second, no elementary scalar is necessary for the theory, and there is no problem regard- ing the violation of the unitarity bound in WW -+ WW scattering. Third, a definite relationship can be estab- lished between the top mass and the Higgs boson masses reducing thereby (to some extent) the embarrassingly great laxity in the choice of parameters of the other- wise quite successful standard model. Finally, there are physical examples of dynamical symmetry breaking on the eV scale (BCS theory of superconductivity) and the MeV scale (the breaking of chiral symmetry for nucle- ons), and it would be aesthetically satisfying if the mech- anism should recur again at this higher scale of energy.

Although the above model is elegant and economical in

the sense that it does not predict any new particle (even the Higgs scalar is a composite object), unfortunately the top-quark mass mt in this model, as determined from the renormalization-group flow of the coupling constants, appears to be untenable with the present experimental upper bound of 190 GeV. To resolve this difficulty within the same framework, it was proposed [6-81 that one can include an additional SU(3), x s U ( 2 ) ~ x U ( ~ ) Y invariant term in the Lagrangian, which is of the form

Here G' is the coupling constant (of mass dimension-2) for the new interaction, i is the SU(2)L index and I, J, P, Q are the SU(3), indices running bom 1 to 3. The A matrices are the real generators of SU(3) in the manner of Okubo, which we find more convenient for our prob- lem than the usual Gell-Mann matrices [9]. The four- fermionic interaction being nonrenormalizable in 3 + 1 dimensions, a high-energy cutoff A is needed for the reg- ularization of this theory. Effectively, this means that the theory ceases to be valid beyond A. For simplicity, we will use the same cutoff for all four-fermionic operators.

The essential features of this model are threefold. First, a set of color-octet isodoublet scalar effective fields, generically denoted as X, appear in the low-energy limit. Their properties are discussed in detail in Refs. [7,8]. Sec- ond, two types of Yukawa couplings are present, which we denote as gt(4LitR& +H.c.) (the usual SM coupling) and g ~ [ + ~ i ( ~ ~ ) I J t & X ~ Q + H.c.]. Apart &om them, there are six four-boson couplings in the model. We can numeri- cally treat the evolution of these eight couplings through a set of one-loop P functions, which yield the standard Higgs boson mass to be 209 GeV for A = 1015 GeV. Third, mi is a free parameter of the model, as is evident from the infrared quasifixed-point solution [7]

'Electronic address: akundu Osaha.ernet.in 9 2

'Present address: Tata Institute of Fundamental Research, q g t f 2d2 = % ( g t ) & ~ ~ j2) Homi Bhabha Road, Bombay-400085, India. Electronic ad- dress: sreerupOtifrvax.bitnet where BHL denotes Bardeen, Hill, and Lindner [2]. Thus

r~lectronic address: bnyk@saha.ernet.in it is not possible to predict mt in this model, but in

0556-2821/94/50( 1 1)/6872(5)/$06.00 - 50 6872 @ 1994 The American Physical Society

50 - RADIATIVE CORRECTIONS TO ~ ( z - P b&) FROM COLORED . . . 6873

the succeeding analysis, we take some phenomenologi- cally plausible values for mt (130 GeV < mt < 200 GeV). The analysis will also include the recent Collider Detector a t Fermilab (CDF) result [lo], though not yet fully confirmed, about the signal of the top quark with m+ = 174 f 17 GeV.

The mass of x will be an important theme in our dis- cussion. We immediately note that though the field is an effective one arising as a composite of two spinor fields in an SU(3), octet combination, it is not possible to pre- dict the masses as was done for the "Higgs scalar" in Ref. [2] because here the strong interaction plays a non- trivial part and the 1 /N approximation is not valid. Not being determinable from the renormalization-group equa- tions, the mass of x remains a free parameter of the the- ory. When the symmetry is broken, the two partners of the isodoublet get mass splitted.

These new bosons can, through various one-loop ef- fects, have profound consequences, which are experimen- tally observable, and we can place lower bounds on their masses. One notes that the interaction in the minimal condensate scheme is confined only to the third genera- tion of quarks. The other quark generations take part in the one-loop effects through the mixing between the mass eigenstates and the weak eigenstates of the quark wave functions [ll]. This means that the physics of K0- K O , B:-B:, and B:-B: mixing will be affected by, X, and the same is true for the CP-violating E parameter. In an earlier paper 181 we have discussed these effects in de- tail and showed-that we can obtain a lower bound on the mass of the charged scalar x+, which is of the or- der of a few hundreds of GeV. Another bound can be extracted from the observed rate of the radiative B de- cays [12], which is of the same order of magnitude and which is free from a number of undetermined or poorly determined parameters, which entered in Ref. [8]. We have also shown that the maximum mass splitting in the doublet cannot be greater than 47 GeV [13]. This last re- sult is obtained from the present experimental bounds on the oblique electroweak parameters [14]. The oblique pa- rameter S vanishes if x+ and x' are degenerate in mass, and hardly puts any significant bound on their individ- ual masses. For the allowed range of m,+ deduced in this paper, and from the constraint on mass splitting, it can be shown that [13] S gets a negligible contribution.

The chief obstacle to putting phenomenological con- straints on the model from low-energy data such as B:- B: mixing arises, as usual, from uncertainties in the hadronic parameters such as the meson decay constants fK, fB and the bag parameters BK, BB. In this note we investigate a different observable, viz., the ratio Rb, defined as

r (2 + b6) Rb =

r ( Z + hadrons) '

Rb is relatively free from uncertainties in hadronic pa- rameters, which tend to cancel out from numerator and denominator and is also relatively insensitive to QCD corrections. It can be shown [15] that Ra singles out

the nondecoupling part of the vertex correction induced by the heavy top quark flowing in the vertex loop (and that is why one is to introduce, apart from the standard oblique parameters, another parameter [16] to specify the one-loop radiative corrections completely). However, the dependence on mt is rather flat; Rb changes by less than 2% for an mt range of 130-200 GeV. Thus, the ef- fects of new physics can show up in Rb without being masked by hadronic uncertainties. An analysis of the model using Rb is also facilitated by the fact that the experimental error in its determination has come down drastically with the measurements from the CERN e+e- collider LEP and the advent of microvertex detectors, and now stands at [17]

a t 95% C.L., which is remarkably precise. More recent but preliminary result is Rb = 0.2207 f 0.0026 [18]. Re- sults obtained using the latter value of Rb will also be mentioned.

To fix ideas and notations, let us briefly discuss the features of r ( Z + b6) and Rb in the standard model [15,18-201. The tree-level contribution to r ( Z -+ b6) is

where pb = m i l m i , G, is the Fermi coupling constant as obtained from muon decay and Ow is the weak mixing angle.

The electroweak radiative corrections appear in the form of two form factors tcb and pb, respectively, for effec- tive mixing angle and the overall renormalization. Thus, the decay width, calculated to one loop, is given by

G m3, r1(2 -+ bb) = Y - p b J = 87~&

where sin20w is determined from

AT is the electroweak correction to p* decay, given by

[I51

Thus sin28w contains an mt dependence, which has a 1% effect for mt ranging between 130 and 200 GeV. The dependence on Higgs boson mass is completely negligible.

6874 KUNDU, RAYCHAUDHURI, DE, AND DUTTA-ROY

The one-loop correction is dominated by the top quark contribution. The vacuum polarization effect, which is common to all fermionic final states, is denoted by A p t , which is given by

the b mass having been neglected. The Il functions are the standard ones used to denote the vacuum polarization of the gauge bosons. For Z -t bb, the vertex corrections give

Taking both these factors into account, we can write

The corrections introduced by the deviation of these form factors from unity can be factorized:

r l ( z -+ bb) = r O ( z + bb)(i + + 6; ) . (15)

6; contains the effects common to other down-type

quarks ( s and d), and 6f) separates out only the top-

dependent contribution. 6f) is approximately given by

1151

which is correct up to a precision less than 1% in the mt range of 130-200 GeV.

There are four more corrections to the decay ampli- tude, of which the QED and the QCD corrections cancel out when we take ratios. The correction due to nonzero mass of the outgoing fermions is only significant (- 0.5%) for the b quark, and is given by

The last correction is an O(CY?) one, arising from the large t-b mass difference through triangle quark loops [21]:

where

Taking all these corrections into account, Rb comes out to be

In the computation we have taken a, = 0.117, mz = 91.187 GeV, G, = 1.16637 x lo-' GeV-2, mb = 4.7 GeV, and m H = 200 GeV. The top-dependent contri- bution is of the order of lop3, the same as the order of experimental errors, and the variation of Rb with mt over the allowed range of the latter is rather flat. As stated above, this is one of the reasons for which Rb is phenomenologically interesting.

In our model, another term of the form

gets added to the above contribution. We take only the nonoblique part as it is known [13] that the oblique part has negligible contribution. It is noteworthy that the effective Lagrangian only favors the production of left- handed b quarks, but since the same is true for the tree- level case, it will not cause any significant change in the electroweak asymmetries.

In the limit m b -+ 0, we can introduce the effects of the new physics through a change in the vertex factors for the Z -t b6 coupling:

where P is the four-momentum of the Z boson, and the color factor of $ comes from the octet nature of x under SU(3),. The right-handed coupling is not changed as no term of the form ELbRx is allowed in the Lagrangian. The function FL represents the total of all one-loop correction effects, depicted in Fig. 1. It can be written as the sum of three terms:

where F?), F?), and ~ 2 ) denote the contributions from Figs. l ( a ) , 1 (b), and l(c), respectively. The correction works out to be

with

The FL functions are

RADIATIVE CORRECTIONS TO r (Z+b@ FROM COLORED . . . 6875

where AL = gi/g, g being the usual s U ( 2 ) ~ coupling constant, mx is the mass of the charged X, p~ is the mass scale arising in dimensional regularization, and

The two- and three-point functions bl, cg, c2, and c,j in terms of the well-known Passarino-Veltman functions [22] are [19]

~6(ml ,m2,m3) = -~$[C23 + Cll](m2,ml, m3) 1 (37)

where A = 2/(4 - d) - 7 - l n r in d dimensions, and this

FIG. 2. The contribution of the to the parameter Rb

as a function of m,+. The hatched region depicts the CDF bound for the top mass.

divergence cancels out in the final formula for FL. In Fig. 2 we show the plot of Sx(Rb) with m, for the

top mass ranging from 130 to 200 GeV. The correspond- ing gi values can be obtained from Eq. (2). In Fig. 3 we plot the lower bound on m, for the same range of mt. This bound has a negligible dependence on the un- certainties of mz , sin2ew, mb, and mH, but is decreased by 30 GeV if we take a, = 0.124 instead of 0.117. It may be noted that this bound goes as m: and mt = 190 GeV is the maximum allowed limit for Rb = 0.2201 & 0.0031. For mt = 150 GeV, we get m, = 380 GeV as the lower limit. For Rb = 0.2207f 0.0026, the lower bounds on the mass of X+ are 473, 614, and 910 GeV and more than 2 TeV for mt = 130, 140, 150, and 160 GeV, respectively. For all these computations, we have taken the minimum possible value of Rb as Gx(Rb) is negative. However, if the CDF bound [lo] on the top mass is verified, the model will be strongly constrained as the x contribution, being negative, makes the bound very stringent in nature. The new physics decouples in the limit m, + oo; this result is in conformity with those obtained earlier [8,12,13]. f iom Fig. 2 this decoupling is evident; 6,(Rb) goes as m$. Of course, this is just a technical point, since x is a compos- ite object, and it is meaningless to carry m, beyond the compositeness scale. So it can be claimed that within the

FIG. 1. The one-loop diagrams involving the colored FIG. 3. The lower bound m, on the mass of xi as a func- scalara which contribute to r ( Z + a). tion of mt for Rb = 0.2170.

6876 KUNDU, RAYCHAUDHURI, DE, AND DUTTA-ROY - 50

framework of this model, mx has both a n upper a s well as a lower bound, and these come closer and finally coincide as the cutoff A is decreased. This behavior can be eas- ily explained; if we decrease A, (mt)BHL will increase, so there will be a corresponding increase in gi, and 6; func- tions are proportional t o gi2. The coincidence occurs a t about A = 1 TeV.

I n this work, therefore, we have investigated the effects of isodoublet color-octet composite scalars arising in a realistic model with dynamical breaking of electroweak symmetry. The specific process focused on is the decay Z -+ bb, since the ratio Rb is precisely determined and well known t o be free from the QED and the QCD un- certainties and also is rather flat with the variation of mt. We find tha t a stringent lower bound can be placed on the masses of these composite colored scalars, which is around 700 GeV for a top mass of 175 GeV, and in-

creases quadratically with mt. At the present moment, one of the main issues confronting Z decay experiments is t o determine the difference between the standard model prediction of Rb and the experimental number, because this is one more possible gateway t o look into the new physics. The minimal supersymmetric standard model predicts a positive (but small) dRb, and nearly all ex- tensions and modifications in the scalar sector (whether elementary or composite) predict 6Rb to be slightly neg- ative. None of the alternatives can be ruled out a t the present moment, and further precision experiments could help discriminate between models.

We are indebted to Gautam Bhattacharyya for valu- able discussions. The two- and three-point functions were calculated using the code CN developed by Biswarup Mukhopadhyaya and Amitava Raychaudhuri.

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