•-> ^ IC/94/202

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

MASSES OF HEAVY FERMIONS AND IN FOUR-GENERATION FERMION CONDENSATE SCHEME

INTERNATIONAL Bang—Rong Zhou ATOMIC ENERGY AGENCY

UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE

IC/94/202 The top- condensate scheme [1,2] of electroweak symmetry breaking and its many

International Atomic Energy Agency fermion-generation extention [3-7] provide a dynamical explanation of the origin of the and Higgs boson and the large top-quark mass. In the model with heavier fermions than United Nations Educational Scientific and Cultural Organization the top-quark, the momentum cut-off A could be lower and the fine-tuning problem of INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS the coupling constants will no longer be a serious one [5]. A model of this kind with a heavier degenerate fourth generation of f/-fermions has been expounded in the bubble approximation [5,7], It has been obtained that the acceptable momentum cut-off A could MASSES OF HEAVY FERMIONS AND HIGGS BOSON 6 3 IN FOUR-GENERATION FERMION CONDENSATE SCHEME be in the region 10 -5 x 10 GeV and the f-fermion mass mv in the range 163-353 GEV

if the top-quark mass mt is taken to be 160 GeV [5], The mass mAo of the composite Higgs boson h° turns out to be in the range 324 — 689 GeV and obeys the constraints 1 Bang-Rong Zhou m<7 + m, < niv> < 2mu [7]. Althrough these results in the bubble approximation represent International Centre for Theoretical Physics, Trieste, Italy and the main feature of the model, they can not be considered as precise predictions of the China Centre of Advanced Science and Technology, (World Laboratory), masses of relavant particles since the effects of the color and electroweak gauge interactions P.O.Box 8730, Beijing 100080, People's Republic of China. and the dynamics of the composite Higgs have been omitted from the discussions.

In order to take these effects into account, we will follow the approach of the low-energy ABSTRACT effective Lagrangian proposed in Ref.[2]. When a heavy fourth generation of The analyses based on low-energy effective Lagrangian indicate that a model of electroweak symmetry breaking of Nambu-Jona-Lasinio (NJL) type with and leptons are included, the Lagrangian of Yukawa-form, which is equivalent to the four generations of quarks and leptons could accomodate itself to the top-quark mass effective four-fermion Lagrangian responsible for spontaneous breaking of the electroweak ~ 174 GeV for the acceptable momentum cut-off A ~ 10e - 5 x 103 GeV. The fourth generation of quarks will always be heavier than the top-quark but the corresponding SUL(2) x Uy(\) gauge group at the momentum cut-off A, can be expressed by leptons, though being heavier than the Z boson , can be either lighter or heavier than the top-quark depending on A being cither greater or less than ~ 2.5 x 10* GeV. The mass (I) of the Higgs boson is predicted to be in the region 287-481 GeV which could provide an aF)} + h.c] - important experimental test of the model. J where

(2)

is the static Higgs doublet to be regarded as a manifestation of the composite Higgs field. MIRAMARE - TRIESTE July 1994 The fermion doublets

U Q*L = ( QB ) , QaR = UoR, DaR (a = 3,4) (3)

are assigned in the SU (3) representation R with dimension dQ (R). They are all assumed 'Permanent address: Department of Physics, Graduate School, Academia Sinica, Bei- C a jing 100039, People's Republic of China. to be in their mass eigenstates [6] and limited to only those generations containing heavy

2 fermions, i.e. a = 3 corresponds to the third generation of quarks ((, b) and a = 4 to the By rescaling the Higgs fields by H —* H/\/Zn and taking the unitary gauge H = fourth generation of quarks and leptons (U, D) and (N, E). The denotations g%(Q = Ua, (0, ft°)/\/2 we may find out from the Lagrangian (4) and the expressions (5) the low-

Da) represent the Higgs-Yukawa coupling constants and mB is the bare mass of the Higgs energy physical tree spectrum as follows: fields. mq = gQ < h° > ' (?) From the two-point and four-point functions of the Higgs fields calculated based on 4o = -2m2 /Z = A < h° 2 Cy in the bubble approximation, it is found that the static Higgs fields H will acquire H H o ?« (8) an induced kinetic and quarti< self-interaction term. When all of gauge interactions with are openned, the induced effective Lagrangian at some low-energy scale fi below the 'I AI momentum cut-off A cars be expressed by Eq.(8) can be further reduced to that

f 2 C = £kinrtk! - (H*C + C //) + AL^,ft + ZH\DJI\ - m^tfH - y(&H?, h (io)

C = £ I&.^'MQZLU.R) + S°nJDaLQaR)] (4) Therefore, the tree mass formula (10) of the composite Higgs boson is precisely coincident o=3,4 where D^ is the gauge covariant derivative and all loops are now defined with respect to with the approximate mass formula (7) in Ref.[7]. The complete dynamics contained in the low-energy effective Lagrangian (4) is em- the low-energy scale p. £ki™tk represents the SUC(^)'XSU[J[2)XUY(\) gauge- invariant bodied in the renormalization group evolutions of the gauge coupling constants and the kinetic terms of the gauge and (J-fermion fields. ALfKip, represents the fermion loop contributions after deducting the low-energy loop terms to the gauge coupling constants Higgs-Yukawa and the Higgs-quartic coupling constants. After rescaling H —» Hj\fZ~H, in the bubble approximation of the effective four-fermion interactions [6], The constants Eq.(4) will have the same form as the Lagrangian of the of strong and electroweak interactions. Therefore, we can use the renormalization group evolutions of Zn,mjf and Ao turn out to be as follows these coupling constants in the standard model but necessary compositeness boundary 1 Q I conditions consistent with Eq.(6) must be imposed. Under the conventional normaliza- tion,

l*)-2ln^ (5) where Z#,ZQ and ZQ are respectively the divergent wave function normalization con- where the Q in the sums run over all the heavy fermions in thethird and the fourth L R stants of the Higgs fields, the left-handed and the right-handed Q fermion fields, the generation except the light bottom quark (this is equivalent to assuming g° = 0 ). Eq.(5) kinetic terms in Eq.(4) will have the form of the free field theory. The physical couplings is just the generalization of the formula (3.3) in Ref.[2]. We notice that g'q and X are defined by 2 Zff —> 0, m H —t ml, \0 —• 0, gg —» constant, when /i->A (6) and the Higgs fields H will return to the static one in this case. Hence Eq.(6) simply A- (12) represents the compositeness boundary conditions in the bubble approximation.

3 where ZHQ and ZtH are the proper vertex renomalization constants for the Higgs-Yukawa the Higgs fields will disappear and the Higgs boson will cease to propagate. To one-loop

and the Higgs-quartic interactions separately. order we have the following expressions for these constants

In order to describe the compositeness boundary conditions consistent with Eq.(6),

following the method used in Ref.[2],we may further take the "unconventional" normal- 2 ization convention 2 in H - H/&'(?) (13) (20) where t represents the top-quark. Thus we will obtain the wave function normalization where eg is the value of the electric charge of the Q-fermions defined by constant ZH in the Hiogs kinetic term and the Higgs quartic self-coupling constant A in this convention as follows: (21)

with the sign function

(15) for Q = Ua (22) In addition, we also define that YQL is the Uy(i.) charge of the left-handed Q-fermions and Cj(flg) is the eigenvalue of

the SUC(Z) quadratic Casimir operator in the representation R of the Q-fermions. The

2 2 running coupling constants g~i (n), gi (/i) and gi'(ti) of the fy(l), SUL{2) and SUC(3) It is easy to verify that owing to Eq.(6) we will have gauge group will be determined by the renormalization group evolution formulas in the

standard model [8] . 0, 'X(fi)-> 0 and ZQ(fi) ^ 0 [Q = U,D,N,E), when/J-> A (17)

However, the ratio between the Yukawa coupling constants g'q1 and g2 will be ultraviolet i - 1, A J (li) finite, i.e. with

• constant, when ft —' A (18) - --jg-, A = T and (24)

for the model with four generations of quarks and leptons. Here we have taken the where the parameter 7jQt is defined by iT-boson mass Mz as the low-energy scale parameter and this amounts to specify a sub-

= (9°Q/9?f (19) traction scheme of the renormaiization. Changing the scale parameter will only lead to an additional finite renormalization and not bring about essential influence on the results. In Eq.(18) we have kept only the contributions Z%£F, ZQ+F and ZQ^F coming from In this paper we will also ignore the effect of the large mass difference m, - mb on the the fermion and gauge-boson loops in the corresponding wave function normalization conventional running of the gauge couplings gi2(/i) and g~22[n) which was indicated in constants ZHQ, ZQL and ZQR separately, this is because when fi -+ A, the kinetic term of Ref.[7], " ".>•$.< *: r^" •• "i™"

In the conventional normalization (11), Eq.(4) has the same form as the Lagrangian and the denotations of the standard model, hence by means of standard calculations we can derive out the 1 , 3 , 4 . E=ZH\- + renormalization group evolution equations of the Higgs-Yukawa couplings g~Q(ft) and the Qi-t ZQ "72* "8* 3*'

Higgs quartic coupling A(^). The results are that dQ(R) D = -B (31)

2 «">•'£-{[§• dQ(R)\ gQ We may assume that E > 0 and D > 0 because of weakness of the gauge couplings jfjJ(z" = 1,2). Then from the zero points of the right-handed side of Eq.(29) = t, U,D, N, E (25) = \ {-(1 ± [(1 64(1 (32) and = 12 I A2 - A it is easy to find that i_(£ -+ oo) and x+(t —+ 0) will respectively be the ultraviolet (UV) and infrared (IR) stable fixed point [9], Considering that in the model with heavier ~g \ B = ^( 25,V,2 (26) fermions than the top-quark,the momentum cut-off A may be lower, it is more reliable to where we have used the definitions use the IR stable fixed point z+(t —> 0) to specify the boundary condition of A, i.e.

(27) A(0) = ZH(0) x+(0) (33) and omitted the coupling constant gt, for the light bottom-quark. By means of Eqs.(14)- where x+(0) may be calculated by Eqs.(31) and (32) when Z'H and ZQ(Q = U,D,N,E) (16), Eqs.(25) and (26) can be transformed into coupled equations about ZH and ZQ(Q = have been solved in advance. In this case, as long as i. < i < x+, then x will be driven U, D, N, E) and an equation about A. They must be solved together with the composite- toward the constant r_(A) when |i->A and A(A) = Zff(A)i_(A) will tend to zero, i.e. ness boundary conditions given by Eqs.{17) and (18). In addition, in the equation of A the UV boundary condition (17) may be automatically satisfied. The boundary values of there are the terms like X/Zn and (A/ZH) . Since A(A)/Zj/(A) is of the indefinite form the gauge coupling constants will be taken to be that of 0/0 by Eq.(17), we must also determine the boundary condition satisfied by A/Z// if 9i2{M ) = 0.127, gi\M ) = 0.446, = 1-44 (34) we intend to solve the equation numerically. In fact, the function z 2

and Mz - 91.17 GeV, following Ref.[2]. = \/ZH (28) Without loss of essentiality, we solve the equations of ZQ(Q = H, U, D, N, E) with the obeys the equation assumption that g° = constant, for Q = U,D,N,E (35) (29) Q i.e. the masses of the fourth generation of fermions are degenerate in the bubble ap- where the variant t is defined by proximation. In this case the boundary conditions of ZQ will contain only a single free (30) parameter rjQt expressed by Eq.(19). Once the value of rjq, is specified, all the boundary conditions of ZfijZq will be completely determined by Eq.(18), where the wave function Table I. The masses of the fourth generation of fermions U,D,N,E and the Higgs a normalization constants can be calculated by Eq.(20). Furthermore, after all the Zq are boson h for different values of the momentum cut-off A. The values of parameter r/qt

solved explicitly, the boundary condition of A will also be determined by Eq.(33). There- have been chosen so as to fit the top-quark mass m, = 174 GeV.

fore, rjQt in fact becomes the only free parameter for solving of the total equations. The A(GeV) 107 106 105 5 x lO4 2.5 x 104 104 5 x 103 masses m0 of the heavy fermions and the mass mho of the Higgs boson will repectively

be given by the roots of the equations

T}Qt 0.083 0.149 0.33 0.44 0.60 0.942 1.376 = t,U,D,N,E (36) mu{GeV) 209 229 261 275 293 326 366

and mD(GeV) 207 228 260 274 292 325 365 (37) mjv(GeU) 90 110 142 156 174 207 244

where i> = (\/2GF) = 246 GeV is the standard -model Higgs vacuum expectation mE{GeV) 93 113 144 159 176 209 246 value and GV is the Fermi constant [10]. mhc{GeV) 260 287 333 353 379 427 481 We list in Table 1 the results of masses of the heavy fermions and the Higgs boson

when the top-quark mass is fitted to the present experimental value rWj = 174 GeV [11]

by adopting appropriate parameter IJQI for different momentum cut-off A.

10 It can be seen from Table I that, for the model of minimal dynamical electroweak symmetry breaking of NJL-type with four generations of quarks and leptons, when all the gauge interactions and the dynamics of the composite Higgs boson are included Acknowledgments it is entirely possible to accomodate itself to the present experimental top-quark mass The author would like to thank Professor Abdus Salam, the International Atomic m, ~ 174 GeV for different acceptable momentum cut-off A ~ 106 - 5 x 103 GeV. Under Energy Agency and UNESCO for hospitality at the International Centre for Theoretical the assumption that the fourth generation of fermions are mass-degenerate in the bubble Physics, Trieste. This work was partially supported b\ the National Science Foundation approximation, once mi is specified then the masse of all the other heavy particles in of China. the model will be completely determined by the renormalization group equations. The leptons in the fourth generation without the color interactions are obviously the lightest References particle among them. Their masses, mN and trig, could become smaller than the top- [I] Y.Nambu, in New Theories in Physics, Proceedings of the XI International Symposium quark mass m,, For instance, they could be even down to near or below the Z-boson mass on Elementary , Kazimierz, Poland, 1988, edited by Z.Ajduk, S.Porkorski mz when the momentum cut-off A > 107 GeV. If one expects the masses of all the fourth and A.Trautman (World Scientific, Singapore, 1989); V.A.Miransky, M.Tanabashi and generation of fermions including leptons to be close to or bigger than the top-quark mass, K.Yamawaki.Mod.Phys.Lett. A4,1043(1989); Phys.Lett.B221,177(1989);W.J.Marciano, then the momentum cut-off A must be confined in the region 2.5 x 10* — 5 x 103 GeV and Phys.Rev. Lett.62,2793(1989). this means quite a low compositeness scale. Table I also shows that for all the the values [2] W.A.Bardeen, C.T.Hill and M.Lindler, Phys.Rev.D41, 1647(1990). of A we will have rn^ SB mo and ro.v as nj, This result merely reflects the fact that (3) H.Terazawa, Y.Chikashige and K.Akama, Phys.Rev.Dl5, 480(1977); W.J.Marciano, 3 J i 2 when the gauge interactions are neglected the ratios <7w (/i)/i?b (/i) and g~N (ii)/g~E (p) Phys.Rev.D 41 ,219(1990); M.Suzuki, Phys.Rev.D 41, 3457(1990); M.Luty, Phys.Rev.D have the unities as their infrared stable fixed points and the masses of particles actually 41, 2893(1990); R.F.Lebed and M.Suzuki, Phys.Rev.D45,1744(1992). depend on only the low-energy evolutions of corresponding running coupling constants. [4] B.R.Zhou, Commun.Theor.Phys. 16,337(1993). The Higgs boson mass my> will increase rapidly as the momentum cut-off A decreases. [5] B.R.Zhou, Phys.Rev.D47, 2656(1993); ibid,D4&, 4484(1993)(E).

e 3 When A = 10 - 5 x 10 GeV, mho will vary in the region 287 - 481 GeV. In the mean- [6] B.R.Zhou, Phys.Rev.D47, 5038(1993). time mu will vary in the region 229 — 336 GeV. These results coincide with the general [7] B.R.Zhou, Phys.Rev.D50, 578(1994). pattern of that mkt varies as \(mu) in the bubble approximation [llj. Fn addition, the [8] H.D.Politzer,Phys.Rev.Lett.30,!346(1973); DJ.Gross and F.Wilczek, Phys.Rev.Lett. mass inequalities 2(mg)m|.n < m^ < 2(mQ)maT{Q = t,U,D,N,E) valid in the bubble 30, 1343(1973). approximation are also satisfied in present case. [9} D.J.Gross, Application of the renormalization group, in Methods in Field Theory, In conclusion, if experimentally one can find the Higgs boson with the mass in the p.140, edited by R.Balian and J.Zinn-Justin (North-Holland, 1976). region 287 — 481 GeV, then the theoretical model of dynamical electroweak symmetry [10] Particle Data Group, J.J.Hernandez et a/., Phys. Lett. B239, 1(1990). breaking of NJL-type with the fourth generation of heavy quarks and leptons would get [II] F.Abe et al. (CDF Collaboration), Phys.Rev.Lett. 73, 225(1994). a strong support and deserve to be investigated further.

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