Comment on critical instability

Physics Letters B 277 (1992) 153-157 North-Holland

PHYSICS LETTERS B

C o m m e n t on critical instability S t e p h e n F. g a n g

Department of Phystcs, Untversttyof Southampton, SouthamptonS09 5NH, UK and Mahlko Suzuki

Department of Phystcsand LawrenceBerkeley Laboratory, Umversttyof Cahfornta. Berkeley, CA 94720, USA Received 22 November 1991, revised manuscript received 12 December 1991

We d~scussthe problem of the mass sphttmg between top and bottom quarks, w~thmthe context of Nambu-Jona-Lasmm type models revolving top and bottom quark condensates We interpret the phenomenon of"cntmal mstabdlty" recently proposed to account for such a mass sphttmg as the fine-tuning of two vacuum expectatmn values in a composite two-H~ggsdoublet model

The minimal top quark condensate model o f composite Hlggs bosons [ 1,2 ] provides a prediction for the top quark mass mt, and the Higgs boson mass mn, as a function o f an ultraviolet cut-off mass scale A [ 3 ] The minimal model introduces a single N a m b u Jona-Laslnm (NJL) operator of the form Gt(~_~LtR)(FRQL), where QL = (~LL) and (t'RQL) lS a colour singlet blhnear with the q u a n t u m numbers of a Hlggs doublet ¢t If Gt ts fine-tuned then this results in the /'t scalar component of Ot and the top quark having hght masses compared to A [ 1-3 ] A low energy effective standard model lagrangmn results [ 3 ] However, in the minimal model the only fermlon to receive mass ~s the top quark In order to account for the other fermion masses, one must introduce add~tmnal N J L operators The most general extensmn of the m i m m a l model to include both top and bottom quarks is 5aNJL= G,(QLtR) (i-RQL) + Gb(O_~.b~) (6~Q~)

+G,b[ (QLtR) (6~Q~_) + h c l , [

( 1)

b~

where Q~ = ~-t~ J and c denotes charge conjugation so that E~t~ = TLbR It has been shown [4,5 ] that in general when the coefficients Gt Gb, Gtb are finetuned, one is led to a low energy effective standard model involving two H I S S doublets, ~t~i-RQL, Ob~tQ~. Only m the case GtGb= Gtb wlll a single 0370-2693/92/$0500© 1992ElsevlerScmncePubhshersBV

Hlggs doublet result [6 ] The question o f the top and bottom quark mass sphttlng m such a model is comphcated by presence o f the cross term m eq ( 1 ) proportlonal to G,b, which allows Ot and eb to mix One may define linear combinations of Ot and eb which couple purely to top and b o t t o m quarks by introducing a mixing angle 0 [4,5] where tan20=2Gtb/ ( Gt2 - Gb2 ) In order to clarify the question of the t o p bottom mass sphttmg m the two Higgs doublet model, It is convement to take Gtb=O Th~s corresponds to the choice 0 = 0 If the contact operators arise from heavy gauge boson exchange [ 7 ] then Gtb= 0 follows automatically, since the only gauge boson which can mediate bL--~tL transmons ~s the W In reahstlc models, some amount o f mixing is needed to break the Peccel-Quinn symmetry and avoid an axlon However, our main conclusions are not altered by the presence of Gtb Assuming Gtb=O, then the problem of the top and b o t t o m quark masses appears to decouple into two independent problems There is a "top sector" mvolvlng a composite Higgs doublet 0t ~ I'RQL which couples only to top quarks, and there is a " b o t t o m sector" revolving a second composite Hlggs doublet ¢b ~ b-~tQ~. whmh couples only to bottom quarks If the two sectors are controlled by two independent couphngs Gt and Gb, respectively, then it is clear that All nghts reserved

153

Volume 277, number 1,2

PHYSICS LETTERS B

the top-bottom mass difference is not determined by the theory Recently, however, it has been pointed out [8,9] that for certain gauge models Gt=Gb (and G/b=0) In this case the top-bottom quark mass sphttmg arises solely from the small gauged hypercharge electroweak corrections [8,9] Because of the fine-tuning involved, such small corrections can lead to large mass sphttlngs This effect has been called "crlUcal instability" [ 8,9 ] The effect of critical instabfllty was originally studied using the gap equation [ 8,9 ] But the gap equation approach does not take account of the dynamically generated composite Hlggs exchange contributions A better approach is to study the low energy effective theory (1 e , the standard model with two Hlggs doublets), then use renormallzatlon group ( R G ) methods to make quantitative pre&ctlons [ 3,4 ] Recently the phenomenon of critical lnstablhty has been interpreted in the R G approach [ 10] It is claimed that the mass independent renormahzat~on scheme usually adopted [3,4] is not statable for a proper treatment of critical instability [ 10 ] The argument is as follows [ 10 ] Critical mstablhty is related to the quadratic divergence of the Hlggs mass term But the mass independent renormallzaUon scheme [ 3,4 ] picks up only the logarithmic divergences and loses reformation of the quadratic ones [ 10 ] Therefore one cannot treat the critical instability m the mass independent R G equations [ 10 ] The authors of ref [ 10 ] then propose a mass dependent renormahzatlon scheme, such as the Georgl-Pohtzer scheme [ 1 1 ] In such mass dependent renormahzatlon schemes, the information about cnUcal instability is consistently preserved in the R G equations

rio] In this paper we wish to examine the phenomenon of critical instability within the much simpler framework of the mass independent renormahzatlon scheme which is usually adopted for these models [ 3 ] In the conventional approach [ 3 ], it is true that the R G equations contain no information about finetuning The usual procedure is to use the gap equation in order to derive the composite scalar wavefuncUon renormahzatlon Z¢, from which the "composlteness" boundary condmons of the running couphngs m the effective low energy theory are obtained [3] All the reformation about fine-tuning is contamed in the gap equation We shall argue that this 154

27 February 1992

procedure is perfectly adequate for discussing the specific example of fine-tuning known as cnUcal lnstablhty To be precise, we shall consider an NJL-type model with two gap equations and hence two Hlggs doublets, when they are rewritten in the auxlhary field theory We shall show that the mstablhty can be absorbed into two vacuum expectation values (VEVs), with rob~rot = Vb/Vt, and identical Yukawa couphngs for both t and b In other words, the phenomenon of critical mstabxhty is nothing more than the fine-tuning of two VEVs vb and vt Let us consider the examples of eq ( l ) but with Gtb=O The eqmvalent lagranglan at the scale#=A is then ~Oequlv =

~L tR(~t q_(~_b[t(~b + h c

2 ~" 2 -MtoO,•t --Mboq)~,Ob ,

(2)

where we have normahzed q~ and ~b so that the Yukawa couplings are equal to unity The EulerLagrange equations imply

Gt = 1 / M 2,

ab = 1/M~o

(3)

Alternatively, Gt, Gb may be expressed as

Gt=sI~2).t/A 2, Gb =81t2).b/A 2 ,

(4)

where 2t, 2 b are dimensionless parameters and 2~,b= 1 lS the crlUcal value m each case In writing eq (4) we are implicitly assuming that the ultraviolet cut-offA Is common to both terms, which would be the case if both NJL operators originated from the exchange of a heavy gauge boson of mass ~A For the case of critlcal instability, such a heavy gauge boson would have equal couplings to top and bottom quarks [8,9] However, the effective values of2t, 2b which take account of other corrections due to gluon exchange, and weak hypercharge boson exchange will be different

[8,9], 2,=).+ ~ ).b=).+ ~

1 1

[8g~(A)+]g~(A)], [8g~(A)-½g21(A)]

(5)

Since the chlral symmetry breaking is being driven from physics at the scale A, the QCD couphng g3 and the hypercharge couphng g~ are both evaluated at A Gluon exchange provides an addmonal attractive force, common to both top and bottom quarks Hy-

Volume 277, number 1,2

PHYSICS LETTERSB

percharge boson exchange provides an attractive perturbation for/-t and a repulsive perturbation for b-b [8,9] A common value of 2 = l - 2 o t 3 ( A ) / n may therefore lead to top and bottom quark condensates which are split by

g21(A) J,t--2b = 16/t 2

27 February 1992

where 1

gt=gb = X7=~ '

z~'

(6)

~,

O')t= Z-~'

&' ~b

O)b-- Z~

(11)

Combining (4) and (6),

Gt-Gb=g~(a)/6a

2

(7)

Eq (7) shows how the effect of the weak hypercharge boson exchange perturbation gwes rise to a splitting between the effective couphngs Gt and Gb, as defined above Using eq (3) it is clear that, for large values of A, this effect can lead to rather large mass splittings M 2 >> M~o in the equivalent lagrangian in eq (2) With M 2 >> M 2 , the equivalent lagrangmn in eq (2) gives rise to an effective low energy two-Higgs doublet model at the scale # < A [ 4,5 ] ,~eff= OL tROt + QL -c b eR 0 b + h C

1" --2 + Zo, I DuCt 12+Zcb IOu0b I2 - M--2 , OtOt--MbO~Oo

- ½~t(OD,)~- ½~b(0~00) ~ ,

(8)

where A2

(T•)2

z0,=z~b=z~=

In 7

6

,

, A2

~, =~o= (-a-~)~'n 7 ,

Eq (10) is a simple two-Hxggs doublet extension of the known results in ref [ 3 ] Eq ( 11 ) gaves the parameters of the effective lagrangmn in a region # ~
mb=gbVb, 3

~r,==Mt=o_ ~ 3~t2 = M 2 -

(A~_U~),

3 ~5n2 ( A 2 - u 2)

(9)

After rescahng,

¢,+~,

O,

., .._,

v,o

~¢o,

eq (8) becomes

~e~=&O.LtRO,+goO.tb~¢o+ h

+ l O u O / l z + l O u 0 b I2 -MtOtOt-MoO~Oe 2 t 2 -- Itot(O~Ot)2--

ltob(0~0b)2 ,

(10)

vt,

m~,b=

2x/~bVo,

(12)

where the couplings and VEVs are evaluated on-massshell Note that as t~--,A, gt(a) =go(g), and the low energy values of these two Yukawa couplings will only differ because of hypercharge boson exchange corrections, leading to on-shell differences between gt and gb of a few percent [4 ] Clearly the large t o p - b o t t o m quark mass splitting must originate from the difference between the two VEVs vt and Vo The VEVs vt and Vbare obtained from eq (10) as

vt=

c

met = ~

-

tot

,

Vb=

~/

tob

(13)

The low energy couplings tot, too are easily determined from the appropnate R G equations using the 155

Volume 277, number 1,2

PHYSICS LETTERS B

boundary conditions in eqs (9), (11) [3,4] However, as pointed out in ref [ 10 ], the low energy mass parameters Mr, Mb cannot be obtained from R G equations in the mass independent scheme, since this scheme involves only dimensionless couplings Nevertheless, the results in eqs (9), (11 ) may he used to obtain an estimate for the low energy values of Mr, Mb O f course any such estimate will be very inaccurate since no account is taken of strong, electroweak or nlggs exchange radiative corrections in eqs (9), ( 11 ) However, such an estimate does contam the essential information about critical instability since M E >> M~o in eq (9) Although the values of vt, Vb obtained from eqs (9), ( 1 1 ), (13) are untrustworthy, this is not so crucial since we know from phenomenology that v 2 +v~ = ( 175 GeV) 2

(14)

We are only interested in the difference v 2 - v ~ , or M~ - M 2 (since 09t~ COb) Many of the quadratic mass corrections which we cannot calculate will cancel since they are c o m m o n to both M 2 and M 2 The only corrections which will not cancel are due to the weak hypercharge exchange corrections Since the hypercharge coupling is not asymptotically free, the largest hypercharge corrections are in the high energy region But our procedure has correctly accounted for the corrections in this region according to eq (7) which implies M 2 >> M2o in eq (9) Therefore our procedure would be expected to give a fairly reliable estimate for v 2 - v~, and hence rot2 --mb2 from eq (12) Using eqs (9), ( 11 ) we find 1

m2_m 2= ~

1

(3~t2 _k~t2) = ~

(M~-M~o)

Using COb=Ogt=2/Z,, eq (13) then gives v 2 - v 2 = ½( M 2 - M 2 o )

( 15 )

In this approximation the difference in the squared VEVs Is clearly independent of the renormahzatlon scale# Uslngeqs (3), (4), (15), i)2 _ _ V2b2=

I(GF1_Gvt )

1 ( 3__.3_'~AZ( 1

1)

l(3"~A2(~,t--,,q.b ~

Using eq (6), and the fact that the effective crmcal couplings are 2b..~)tt~ 1, we obtain 3

V2--V~,~ (16~z2)2gZ(A)A 2

(16)

Eq (16) contains the essence of critical instability a small dimensionless coupling gl2 (A) multiplying a large dimensional scale A 2, giving a sizeable sphttmg between the two VEVs The t o p - b o t t o m quark mass splitting is then simply given from eqs ( 12 ), (16),

m t2- - m b 2~~g

2 (Vt--V2), 2

(17)

where gZ.~g2 .~g~, (accurate to a few percent) is the low energy Yukawa coupling determined from the R G equations with the boundary conditions in eqs (9), ( 11 ), namely g--, oo a s / t ~ A [ 4 ] Note that eq (16) should be used with a little care, due to the fine-tuning nature of the problem For example, if we substitute A = 1015 GeV, g2 (A) = 0 2 we obtain a value o f v2 + v~ which greatly exceeds the physical requirement in eq (14) Eq (16) clearly only makes sense in the physical region v2 - v2 < ( 175 GeV) 2 Since the difference in the squared VEVs grows quadratically with the cut-off, rather low values of A are required But for low values of A the top mass itself is too large [ 3 ], so three generation models do not work I f a fourth generation is introduced, this difficulty can be circumvented With the fourth generation quarks (t', b' ), four gap equations are introduced for masses of the third and fourth generation quarks Bando et al [ 9 ] pointed out that such a model predicts a large t-b mass splitting but a moderate t ' b' mass sphttlng if the binding force of the third generation is tuned closely to the critical value This model amounts to introducing four Hxggs doublets, one for each quark mass Since the low energy Yukawa couplings satisfy & =gt and gb =gb as well as gt..~gb (to a few percent accuracy), it holds in the approximation of ignoring generation mixing as well as u p down mixing that

m, /vt = m,/vt ,~ mb /Vb = mb/Vb ,

( 18 )

where the four VEVs are constrained to v2 + v~ + vt2 +Vb2 = ( 175 GeV) 2 The two VEVs, vt and Vb obey the same constraint as eq (16), V2 --V~ ~

156

27 February 1992

3

. 2

(16R.2) 2 gl (

A)A 2

(19)

Volume 277, number 1,2

PHYSICS LETTERS B

and therefore

v~ -v~ ~v~-v~,

(20)

or

mE--rn 2 ,~mZ--m~,

(21)

F r o m the v i e w p o i n t o f gap equaUons, this m o d e l involves fine-tuning o f the three parameters Gt ( = Go ), Gt ( = Gb), a n d A near the cnUcal p o i n t In o u r mterpretaUon, ~t m e a n s the line-tuning o f three VEVs the fourth VEV is o b t a i n e d from eq ( 2 0 ) , which is a real predxctlon o f this m o d e l In conclusion, we have discussed the p h e n o m e n o n o f c r m c a l instability w~thm the f r a m e w o r k o f the conventional mass i n d e p e n d e n t renormahzatxon scheme [ 3 ] We have considered the corresponding two-Hlggs doublet m o d e l with zero mixing angle [4,5 ] a n d shown that the effect o f critical m s t a b d l t y is to reduce a s~zeable sphttlng between the two VEVs given b y eq ( 1 6 ) , with the two Yukawa couphngs r e m a i n i n g a p p r o x i m a t e l y equal [ 4 ] Eq ( 1 6 ) should be fmrly rehable, since the radmtlve correcUons to the Hlggs mass p a r a m e t e r s which cannot be rehably calculated m the mass i n d e p e n d e n t scheme will largely cancel m the difference Mr2 _ M b2 One o f us (S F K ) w o u l d hke to acknowledge discussion with D Ross a n d the s u p p o r t a n d h o s p l t a h t y o f the U m v e r s l t y o f Callforma, Lawrence Berkeley

27 February 1992

Laboratory, where this work was started S F K would also hke to acknowledge the support o f an SERC Adv a n c e d Fellowship This work was s u p p o r t e d in part by the Director, Office o f Energy Research, Office o f High Energy a n d N u c l e a r Physics, D i v i s i o n o f High Energy Physics o f the US D e p a r t m e n t o f Energy under Contract DE-AC03-76SF00098 and m part b y the N a t i o n a l Science F o u n d a t i o n u n d e r G r a n t PHY-9021139

References

[ 1] Y Nambu, m Proc XI Warsaw Symp on Elementary particle physics, eds Z A Ajduk et al (World Scientific, Singapore, 1989) p 1 [2 ] V A Mlransky, M Tanabashl and K Yamawakl, Mod Phys Lett A4 (1989) 1043, Phys Lett B221 (1989) 177 [3] WA Bardeen, C T Halland M Lmdner, Phys Rev D 41 (1990) 1647 [4] M Luty, Phys Rev D 41 (1990)2893 [5] M Suzuki, Phys Rev D 41 (1990) 3457 [ 6 ] J Frohhch and L Lavoura, Phys Lett B 253 ( 1991 ) 218 [ 7 ] S F gang and S H Mannan, Phys Lett B 241 (1990) 249 [ 8 ] Y Nagoshl, K. Nakamshl and S Tanaka, Prog Theor Phys 85 (1991) 131, M Bando, T Kugo, N Maekawa, N Sasakura, Y Watabdo and K Suehlro, Phys Lett B 246 (1990) 466 [9] M Bando, T Kugo and K Suehlro, Prog Theor Phys 85 (1991) 1299 [ 10] M Bando, T Kugo, N Maekawa and H Nakano, Kyoto prepnnt ( 1991 ) [ 11 ] H Georgl and S Pohtzer, Phys Rev D 14 (1976) 1829

157