The composite Higgs boson and the gauge hierarchy problem

Volume 258, number 3.4

PHYSICS LETTERS B

I l April 1991

The composite Higgs boson and the gauge hierarchy problem N.V. Krasnikov Centre de Phvstque Thkonque 2, Ecole Polytechmque, F-91128 Palatseau Cedex. France

Received 21 December 1990

We show that in the Wemberg-Salam model w~th dynamical symmetry breaking due to the t-quark condensate for the specml case when all Yukawa couphngs except the t-quark couphng vamsh, there Is hidden UL( l )®UR( 1) gauge symmetry which protects the smallness of the dynamical Hlggs mass against radmtlve correctmns We propose a toy model with hidden UL( 1)®UR( 1 ) gauge symmetry which solves the gauge h~erarchy problem

O n e o f the m a j o r p r o b l e m s c o n f r o n t i n g the W e l n b e r g - S a l a m m o d e l ~s the gauge h i e r a r c h y problem (see, e.g. refs [ 1 , 2 ] ) . In the W e l n b e r g - S a l a m m o d e l there is one key sector m the theory, the H~ggs sector [ 3 ] This sector is very i m p o r t a n t for the m o d e l since the mass scale o f the W and Z ts g i v e n by the p a r a m e t e r in this sector. T h e Higgs fields i n d u c e a s p o n t a n e o u s b r e a k i n g o f the S U (2) × U ( 1 ) electroweak gauge s y m m e t r y a n d also allow Y u k a w a couphngs to the f e r m l o n s w h i c h lead to mass g e n e r a t i o n after the b r e a k i n g o f the S U ( 2 ) × U ( 1 ) s y m m e t r y . T h e q u a n t u m c o r r e c t i o n s to the Hlggs mass are q u a d r a n c a l l y d i v e r g e n t and at o n e l o o p level we h a v e 8m2=O(o~/n,

h2/470A 2 .

(1)

Here A is s o m e ultraviolet cutt-off T h e c o r r e c n o n ( 1 ) is t e c h m c a l l y natural if 8 m ~ ~
c o r r e c n o n s are included. We can also r e f o r m u l a t e the gauge h i e r a r c h y p r o b l e m in the following way. W h e n we put the f e r m l o n mass equal to zero m Q E D or Q C D we find that the lagranglan in this l i m i t is mv a r i a n t u n d e r the global choral s y m m e t r y ~u-. exp (ieys)~,. T h e chlral s y m m e t r y is b r o k e n by mass t e r m s m a soft way and as a result we find that for the f e r m l o n masses we have only logarithmic dxvergences 5mf~O( o~lzt)mrln(A/mf)

.

F o r the scalar fields we do not o b t a i n a new s y m m e try w h e n we put the scalar mass equal to zero and n o t h i n g protects the smallness o f the scalar mass against r a d i a t i v e corrections. At present the m o s t p o p u l a r e x p l a n a t i o n o f the gauge h i e r a r c h y p r o b l e m is s u p e r s y m m e t r y [ 1,4-6 ] In the s u p e r s y m m e t n c m o d e l s the ( m a s s ) 2 o f the scalars h a v e no longer q u a d r a n c divergence. T h e masses o f the scalars are not r e n o r m a h z e d at all as a result o f the so-called n o n r e n o r m a l i z a n o n t h e o r e m [7]. T h e s u p e r s y m m e t r y has to be b r o k e n m a soft way and the e f f e c n v e ultraviolet c u t - o f f A 2 = I m ~ -m 2 I, where mn and m y are the b o s o n and f e r m i o n masses o f the s a m e s u p e r m u l t l p l e t N o t e that there is also a " t e c h n i c o l o r " e x p l a n a t i o n [ 8 ] o f the gauge hlerarchy p r o b l e m or we can s i m p l y postulate [ 9 ] that the scale A = 1 - 1 0 TeV is a f u n d a m e n t a l scale o f the nature. It has b e e n p r o p o s e d recently [ 1 0 - 1 2 ] that the s y m m e t r y breaking m the W e l n b e r g - S a l a m m o d e l has 399

V o l u m e 258. n u m b e r 3,4

PHYSICS LETTERS B

a dynamical nature and it arises due to the top quark condensate # 0 . The Hlggs boson m such a scenario has a composite nature Bardeen and coauthors [11] have considered the W e l n b e r g - S a l a m model w~thout kinetic and self-interaction terms for the Higgs field and have found that account of radiative correcuons leads to a nonzero ( h > condensate They have found also that the effective Hlggs boson mass acqmres an a d d m v e quadratic dependence upon the ultraviolet cut-off, hence it is necessary to use fine tuning to obtain the small value of the condensate In this paper we show that in the W e l n b e r g - S a l a m model without kinetic term for the Hlggs field for the specml case when all Yukawa coupling constants except the t-quark constant vanish, there is a hidden UL( 1 ) ® U R ( 1 ) gauge symmetry which protects the smallness o f the effective Hlggs boson mass against r a d l a t w e corrections. When we take into account other nonzero Yukawa couplings the hidden gauge symmetry is lost, however due to their smallness we can increase the value of the effective ultraviolet cutoffA m formula ( 1 ) by a factor m t / m b ~ 50 Consider at first the case when all Yukawa couphngs except the t-quark couphng are equal to zero The lagranglan of the W e l n b e r g - S a l a m model is f = Y'~,n~t,~+ZHID,HI2 r n ~ H , H _ ) t o ( H , H ) 2

+ g~( Ct[ tRH, + h.c. ) , ~'L = (tL, bL),

H,= (H,, H2)

(2)

Following Bardeen and coauthors we put Z H = 2 0 = 0 , that corresponds to the composite Hlggs scenario. For ZH = 0 the model is n o n r e n o r m a h z a b l e and we have to introduce some ultraviolet cut-off A At lower energies we obtain the nonzero induced parameters ZH, 20 as a result of the account of loop corrections. The field H, is an auxiliary one and after its ehminatlon we find

f = Gme.c

(3)

--, 2raft

By the introduction of the auxiliary vector fields A , and B,, we can rewrite (3) as

~:= ~ ....... + gt ( FLT, tL + bL f , bL ) A , + gtt-R 7, tR + 2m 2 A , B ,

(4)

In the limit m ~ - - . 0 the lagrangian ( 4 ) is lnvarlant under the additional (besides the standard SU (2) ® U ( I ) ) U L ( 1 ) ® U R ( 1 ) gauge symmetry (tL, bL)--*exp [lgtaL(X) ] (tL, bL), tR--'exp [ Ig~aR (X) ] tR,

A,-*A,+O,OZL,

B/,~B,+O,oL R

(5)

For m n 4- 0 by introducing the two scalar fields ~0Land ~R we can rewrite the term 2m2HA,B, m a gauge invariant form'

2mH (Au--0u~L) ( B , - O,~R) . In the unltare gauge ~ 0 L = ~ 0 R = 0 w e reproduce the original 2 m ~ A , B , term. The S-matrix, being gauge xnvanant, does not depend on the gauge, however the off-shell Green's functions will depend on the gauge. Due to the Ward ldentmes the correlators (Jim, JuL>, (J,R, J,R>, and (J,L, JuR> are transverse (here J,L =/-LY,/L + b-L7,bL, J~R =/-RT, tR ) and hence the term 2m 2 (A,--OuCL)(B,--Ou~0R) IS lnvariant under the renormahzatlon. In other words, the quantum corrections to m h vanish in each order of the perturbation theory Therefore the hidden UL(1 ) ® U R ( 1 ) gauge symmetry of the lagranglan (3) protects the smallness of the Hlggs boson mass and we have at least a technical solution o f the gauge hierarchy problem. In the general case when we take into account all Yukawa coupling constants the lagrangian has the form ..~

f = ~,ne,,~+ mg ~2 ({LIRFRtLq-~LIR{RbL) "

11 A p r i l 1991

Go A _ , 2 / 3 [ , , t t ,~,Z(+ 2 / 3 ) / 4 + h . c . ) = ~ klneHc ~ ~5U k y/L~/R aJ

+g,Tl/3(~kq~-l/3~HC+h c ) + gb/) (q)L~t)q~ t) H ~+ h.c. ) - m 2HH t H ,

Here for slmphcity we neglect color Indices for the tquark. Our key observation is that due to the Flerz mdentitles the lagrangian (3) can be rewritten In the form 400

H~ = e~jH~ ,

(6 )

After the elimination of the H-fields and performing the Flerz transformations we find terms like (for the

Volume 258, number 3,4

PHYSICS LETTERS B

case of the Yukawa coupling for the c-quark) hb(

t-L ~)/aCL -Jr ~L YI, SL ) A ~ -~- hbt-R ~)I~CR B'I,

+ 2 m 2 A~B~.

(7)

The nondlagonal current is not conserved and as a consequence we have a quadratic divergence for the m 2. Besides the t-coupling the most essential is the b-coupling constant hb ~ 0.02. For h b = 0.02 the estimate & n 2 = O ( h 2 / 4 n ) A z leads to a value of the effective ultravtolet cut-off A ~<50-250 TeV. In other words account of the hidden Ue( 1 ) ® U R ( 1 ) gauge group allows to increase the value of the ultravtolet cut-offparameter A by a factor h,/hb ~ 50 To explam the gauge hierarchy problem we can postulate the extstence of the fundamental cut-off A ' = 50-250 TeV which makes in some unkown way the F e y n m a n integrals ultraviolet finite or we can postulate that the supersymmetry is broken at that scale. However there is another posstbthty: we can speculate that beside a nonzero ( i t ) condensate other condensates (l~b), ( f l u ) , ( ~ ) . . . are also different from zero. Namely we can postulate the existence of the four-fermion interaction with the lagranglan

JueA,, + J,,RBu + ~M 2 (A u -- O,u~L ) ( B u - - Ou~R ) , J~,e = tiL ~)~,ue + {eTa, tL + ce h, ce + de 1.. dL+ gI_7~,SL + be 7~,bL + fe 71,re + e-LIn eL + l~e ?I,I~L + Ok }'~ VL -t

~

~

-it~

tt

I 1 April 1991

(3) To obtain nontrivtal quark mlxlngs we must have a nondlagonal mass matrix. To make the toy model (8) renormahzable we can introduce standard kinetic terms for the fields Aa and Ba. There are indications [ 13 ] that the fermlon condensates are p r o p o m o n a l to the corresponding gauge couphng constants

g t ( i t ) ~ g t M 3, g b ( b b ) ~ g b M3 ..... The most nontrlvlal problem for such a scenario ts the explanation of the nontrlvlal mixing between different generations. At present I do not know how to obtain a nondiagonal mass matrix for quarks. To conclude, we have shown that m the WeinbergSalam model with dynamical symmetry breaking for the special case when all Yukawa couplings except the t-quark couphng vanish there is a hidden UL( 1 ) ® UR ( 1 ) gauge symmetry which protects the smallness of the dynamical H~ggs mass against radiative corrections. We have proposed a toy model with hidden UL( 1 ) ®UR( 1 ) symmetry which solves the gauge hierarchy problem. I am mdebted to the collaborators of the Centre de Physique Thdorlque of the Ecole Polytechnlque and especially to Professor G Grunberg for the kind hospttahty during my stay at the Ecole Polytechnlque where this paper has been finished. I thank G Grunberg, A. Kyatkln, P. Sorba and V. Matveev for useful discussions and comments.

-~- P L 7it P L "1"/J L ~ U / / L ,

J~,R = gt t-R7utR + gb6R 7ubR + g~ dR 7u CR + g~ZR YUrR

References

+.

[1]HP Nllles, Phys Rep 110(1983) 1 [2] P Langacker, Phys Rep 72 ( 1981 ) 185 [3] P W Hlggs,Plays Rev Lett 12 (1964) 132 [4] Y A Goldfand and E P Llkhtman, JETP Len 13 (1971) 329 [ 5] D.V Volkovand V P Akulov,JETP Lett 16 ( 1972 ) 438 [6] J Wessand B Zummo, Phys Len B 49 (1974) 52 [7 ] J Wessand J Bagger,Princeton Seriesin Physics (Princeton U P, Princeton, NJ, 1983) [8] L Susskmd, Phys Rev D20 (1979) 2619, S Wemberg,Phys Rev D 19 (1978)1277 [9] N.V Krasmkov, Phys Len B 214 (1988) 363 [ 10] Y Nambu, Bootstrap symmetry breaking m electroweak umficatlon, EF1 preprmt 89-08 (1988) [ I I ] W A Bardeen, CT Hill andM Lmdner, Phys Rev D41 (1990) 1647 [12]V Mlransky, M Tanabashl and K Yamawakl, Phys Lett B221 (1989) 177 [ 13] N V Krasnlkov, m preparation

(8)

The interaction (8) has UL( 1 ) ®UR ( 1 ) gauge symmetry whtch protects the smallness of the M 2 mass term against radiative corrections. Account of radiative corrections for such a model leads to nonzero ( t t ) , ( b b ) , ( e c ) . . . condensates [ 13 ]. Due to the Flerz transformation nonzero condensates induce nonzero quark and lepton masses g' ,

mb ~ gb ...

There are three problems in our scenario: ( 1 ) The toy model (8) is nonrenormallzable. (2) To obtain the correct fermlon masses we have to obtain splitting between different fermlon condensates.

401