UCRHEP-T256
April, 1999



"Low Mass Higgs Boson Consistent with Precision Experiments: A
Consequence of Large Top-Yukawa Coupling in Condensate Models"


Bipin R. Desai and Alexander R. Vaucher


Department of Physics, University of California
Riverside, California 92521, USA




Abstract


It is shown, using dispersion relations techniques for
bound states, that the presence of a large top-Yukawa
coupling lowers the Higgs mass from the condensate-model
value of twice the top mass ( 350 Gev) to 100  200 Gev
consistent with the Z0 width precision measurements. The
coupling is found to be  3.7 at the top-mass, much larger than
the Standard Model value  1. It corresponds to a
compositeness scale  1.4 Tev, which is consistent with top-
color models, and implies quite different scales for fermion
mass generation and electroweak symmetry breaking. A
second scalar state around 1 Tev also emerges as a solution in
combination with the low mass Higgs.





1


Introduction


It has been suggested recently, particularly in the framework of the top


color models, that the fermion mass generation and electroweak symmetry


breaking, which have the same origin in the standard model with a single


Higgs doublet, may actually have different origins [1], [2], [3]. In a


composite model such as the top-condensation model of Nambu, [4] and


Bardeen et al [5], this allows the composite scale to be in the Tev range


rather than at the GUT scale [5], and, therefore, allows the top Yukawa


2
 m
coupling  to be large ( , v   246 Gev ). The origin of the
t t v


electroweak symmetry breaking is then assumed to lie elsewhere.


The top Yukawa coupling evolves according to the


renormalization group equations as:










d 
2 t  3
 b (1)
2 t

d





where is the energy at which  is measured , and b is a positive constant,
 t



its magnitude depending on the assumed higher symmetry. The Yukawa




2


coupling of the other quarks is ignored as well as the gauge couplings since  t

is expected to dominate. We note from (1) and from ref. [5] that, with the


standard model restriction  1 lifted, a large coupling at the top mass could
t 



lead to a rapidly diverging  for large , a fact that will have an important
t 





bearing as we will see below.




In top condensation models the Higgs boson is generated from a four-


fermi interaction at a high scale by summing the fermion bubbles in the large


N limit. The tt amplitude in J P  O channel (the s-channel) develops, as a
c



result of the summation, a pole at s 2
 4m , where m is the mass of the top.


The Higgs boson, therefore, will have a mass m  2m , which is about 350
H




Gev. However, precision measurements on the Z0 -width appear to indicate


the Higgs mass to be lower, between 100 and 200 Gev [6], [7].




It is interesting to examine whether , because of the large magnitude of


the top-Yukawa coupling, the condensate value 2m could change, in


particular, whether it would go down. At the same time it is also interesting


to examine whether a strong top-Yukawa coupling could dynamically


generate further scalar states which have been speculated in several models.




3


The indirect measurements on the Higgs mass referred to earlier


assume the standard model to be correct [6], [7]. Our use of the condensate


models and top color models to explain those results will not be incompatible


with those assumptions because they reproduce the standard model at low


energies.




We will try to solve the bound state problem using the ladder


approximation of iterating the lowest order Higgs pole diagram. As for the


strength of the top-Yukawa coupling,  , even though estimates for its
t



magnitude are obtained in terms of the cut-off partameter [8] we will keep it


as a free parameter to be varied in order to produce the bound states at


appropriate energies.




Bound State Calculation


Before we embark on our calculation we will first construct the tt


amplitude which has the same quantum numbers in the s-channel as the Higgs


boson (i.e. J P  O ). For this we use the Jacob and Wick [9] helicity





4


1 1
formalism for states,  
, , with helicities   ,   , with the
1 2 1 2 2 2


scattering amplitude ab cd given by










1 1
T E = j
+ T E ei(-)
( ) ( ) d j (
 )
cd ,ab p c d j a b
2
j




where      P
 ,  . For J  O in the s-channel, the appropriate
a b  c d



linear combination of the states is




























2





1 1
where and indicate and respectively. For our scattering






2 2


amplitude, therefore,   
0 and






d j  (cos )
! " P #
j





Denoting this amplitude as T , we write






$%& '( )





1 1
T ( E, ) = j
+ +T (E)+ P
+ + (cos )
p j j
2
j





5


The j  0 projection T is what interests us:
0 0



1
1
T 3 4
( ) cos (2)
# ( , # )
1 E d T1 E
,0 2 2 1


The bound state problem in the ladder approximation has been studied


by Lee and Sawyer in  5 3 field theory [10]. They showed that it can be cast


in the so called N/D formalism which has a rigorous foundation in potential


theory for well-behaved potentials like the Yukawa potential [11] and which


has been commonly used in S-matrix theories [12], [13]. The solution is


obtained through dispersion relations by imposing the known analytic and


unitarity properties of partial wave scattering amplitudes. The zeroes of D,


then determine the bound state (or resonance) energies.




Ghergetta [14] has shown that the dispersion relations method of the


type indicated above is better suited than the traditional methods to regulate


the Nambu Jona-Lasinio model [15], particularly the top quark condensation


model of Nambu [4] and Bardeen et al [5] by maintaining gauge invariance


without depending on arbitrary shifts in loop momenta involved in the fermion


bubble summation.





6


In the following we will attempt to solve our Higgs problem through


this method. Typically, in this formalism, a partial wave amplitude is


expressed as a ratio




N
T (3)
l 6 D


In order to discuss the dispersion relations we first define a variable v p2 ( p
6





being the c.m. momentum). The partial wave amplitude, T , generally has two
l



types of branch cuts in the complex v plane. One related to v 0 where T
7 l



becomes complex, acquiring a phase,


9 @





ei sin
T 8 , v 0 (4)
l 7
A





@





where is the phase-shift, and A the phase space factor. The associated


branch-cut (the right hand cut) is taken along (0, ) on the real v-axis. The
B





second type of cut exists typically whenever t-channel exchange is involved.


For a particle of mass M , exchanged in the t-channel, the (total) scattering


amplitude has the form


1
2
t C M


The partial wave projection, T , of the above amplitude then has a cut (the left
l



1
hand cut) along ( E 2
C D , v ) on the negative v axis where v C M . Apart from
0 0 4



7


the two types of cuts, T is real along the real v axis but may have poles in
l



(v 0
, ) which correspond to bound states.
0



In the N/D formalism, D inherits the right hand cut, N the left, other


wise they are real along the real v axis. The zeroes of D then correspond to


bound states.


For v 0, one can write T given by (4) as
F l



1
Tl G
H
I P iI
cot


Therefore comparing with (3) the discontinuity of D along the right hand cut


is given by


Im D H N Q v Q R
S 0
G





similarly from (3) the N-discontinuity is given by


Im N R Q Q
T D Im T U v v0




The dispersion relations for N and D can now be written. Normally


one would normalize the amplitude by taking D 1 as
l v , i.e. write the


dispersion relations as




V
1 dv'
D (v) 1 N (v'
= - ) (5)
l v' - v l
0





8


v
W
1 0 dv'
N (v ) D (v' ) Im T (v'
= ) (6)
l v' - v l l
-





However in our case it is essential to take account of the presence of the


Higgs pole at s 2
 m (see expression (7) below). We can accomplish this by
H



making appropriate subtractions.




The Higgs pole for X X
J Y O amplitude, T ( ) defined in (2) in the s-channel is
` E
,0



given by [5]




 2
2 p2
T t (7)
a b c
0 s 2
c mH




In order to properly incorporate the subtraction procedure for D we note that


the ladder approximation (i.e. the underlying N/D formalism) involves a sum


over the powers of the coupling strength,  . It should therefore reproduce
t



(7) for small  . Therefore,
t





D 2
d s e m (8)
H



for small  .
t





9


If we take the condensate value m  2m, then the problem simplifies
H



considerably because the above term is proportional to




s 2 2
m 
4 4 p (9)




where as indicated previously p is the c.m. momentum in tt scattering. If we


define


p2
x  (10)
m2


and, therefore, s 2
f m g
4 1
( x)





then from (7), (8) and (9), for small  ,
t



 2 x
N h i t (11)
2


D p x


Even though the N/D ratio above is a constant we want to retain the


individual x-dependence to examine the possible shifts in x.




We choose the subtraction point for D in the region where D is real,


along the negative x-axis. Let this point be x q r x , then making two
0



subtractions, expression (5) is modified to


10


w
2
u x ' '
1
0
s t u u ' (x x ) dx x
D( x) D( x ) (x x )D (v x ) v N (x' )(12)
0 0 0
y (x' 2 y
v x)(x' u x ) 16 x' u 1
0 0



Similarly for N(x).


To accomplish (11) we write the first two terms in D as x, and write D


(and similarly) N as,








(x x )2 dx' x'
D( x) 0
x N (x' ) (13)
y 2
16 (x' x)(x' x )2 x' 1
0 0






 x
2 x (x 2 0
x )
dx'
N (x 0
t
) D(x' ) ImT(x' ) (14)
2 2
y
2 16 (x' x)(x' 0
x )


We note that for small  we recover (11) and (7).
t





We will examine the zeroes of D in the first iteration i.e. we examine


the zeroes of (13) by substituting for N the first term on the right hand side of


(14). The D-function is then







(x 2 2
x ) dx' x' x'
D x 0
x t
( ) (15)
y 2
32 (x' x)(x' x )2 x' 1
0 0



Notice that when  2 is very small, then the zero of D(x) is given by x=0
t



which gives the condensate mass (m 2m) for the Higgs boson. However, if
H



 is sufficiently large then the zeroes of D are likely to shift.
t



As for the choice of the subtraction point x x , where D(x) equals the
0
d





condensate value x, we take advantage of the fact that the top color model



11


and condensate models are characterized by a scale. It is, therefore,


reasonable to expect that the subtraction point be related to that scale. If we


take to be the scale parameter in the s-variable i.e. s then from (10)
e 0 f g





2
x h i (16)
0 m2
4


We have left  2 inside the integral because in actual fact  depends on
t t



2 as given by (1). That is, it depends on p2 and , therefore, on x. Clearly






then, depending on the functional form of  2 ( ) , the integration in (15) can be

t



very complicated, particularly since (1) predicts a divergent behavior. In fact


if we, as emphasized by Bardeen et al [5], impose the composite condition


 as , where is the compositeness scale then the solution of (1)
t  l m m
j k





turns out to be


1
 2 ( ) n opq rs t , b u (17)
v 0

t 2

b
2 ln 2







If we insert this expression in (15) then the integral becomes infinite because


of the singularity at v .
 w





A more convergent way may be to extrapolate  2 at 2 x 2 linearly
t   0



through 2 
v 2 2 2
x 2 by using (1). If we write  ( ) x we obtain from (1)
 
t 0 0



2
  






2
x 2
( ) y b 4
2 lnz{| }~ (18)

t 0 0 2
 0





12


where


 =top-Yukawa coupling at the top mass
0



= top mass = m
 0



s






Therefore, for  2 inside the integral in (15) we can write
t



  
2 2
( ) (19)
b 4
2 ln 4 1
( x)

t 0 0



The integral in (15) is then cut-off at 2 2 , or equivalently at x .
x
 0



The equation (15) can then be written as



x ( , , ) (20)
F x 0



where


x
2 0
 (x x ) dx' (x' ) x' x'
F (x 0 t
, , (21)
)
0
2 (x' x)(x' x )2 x' 1
0 0



where


2 2
y
(x) 32 4 1 (22)
( b) ln ( x)

t 0 0



 2
0 (23)
0 2
y
16


3
In the following we take b from the MSSM model where there are two






8y 2


Higgs doublets.


From (20) and (21) it is clear that for 1 we recover the
0



condensate result x=0 (i.e. m 2m). As is increased, however, the
H 0






13


solution of (20), that is the intersection of the straight line given by the left


hand side of (20) with 
F (x, , ) , will give a non-zero value of x. The
0



corresponding Higgs mass is then given by


m 2m 1 x . (24)
H





There are two possible solutions of (20): one corresponds to the region


x<0 when both sides of (20) are negative. It involves the low mass region of


the Higgs boson. There is also a possible solution for x>0 in a region where








F(x, , ) , because it is a principal part integral, becomes positive. This is
0



the second Higgs state. The situation is schematically described in Fig.1.


In Figure 2 we have plotted curves for m = 100 and 200 Gev in terms
H



of the two free parameters  and . The dotted line in the figure is the
0





Pagles-Stockar expression [8] which relates  to the scale parameter
0





 2 1
0
2 2
y
16 N ln
c 2
m (25)




with N =3. The intersection of the dotted line with the two curves
c



correspond to the allowed values of  , and , for each value of m .
0 H






Only the following narrow corridor in the ( , ) is found to
0



participate in generating Higgs masses between 100 and 200 Gev:




14



3.6 3 8
. (26)
0



1.2 Gev
1.6 Gev


This parameter range is consistent with the top-color models [1], [2], [3].


There will also be another solution of (20) for positive values of x, as


explained earlier. This corresponds to the second Higgs particle, m*H .

Because the integrand in (21) has an (artificial) infinity at a finite point x x0

along the integration path, we move the upper limit to as the integral is then






convergent. The solution to (20) is found to be


m*
1 .
1 5 Tev
H



for the range given by (26), which is, of course, close to the scale parameter.





Conclusion


We have shown that low mass values for the Higgs boson consistent


with the Z0 -width precision measurements can be obtained in the condensate


models provided that the top-Yukawa coupling is large. The magnitude of


the coupling constant at the top mass ( 3.7 ) is consistent with the
0



estimates of top-color models and corresponds to a scale 1.4 Tev implying


that the fermion mass generation has a different scale from the scale for



15


electroweak symmetry breaking. Because the top-Yukawa coupling is large,


its evolution at high energies through renormalization group equation plays an


important role. Furthermore, in combination with the low mass state a second


Higgs around 1 Tev is also generated.


We are grateful to Professor Jose Wudka for several helpful


discussions. This work was supported in part by the U.S. Department of


Energy under Contract No: DE-F603-94ER40837.





16


References


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312 (1995); J.D. Wells, ; M. Spira and J.D. Wells, hep-


.


[3] R.S. Chivkula, B.A. Dobrescu, H. Georgi and C.T. Hill, .


[4] Y. Nambu, report EFI 88-39 (July1988), published in the proceedings of


the Kazimierz 1988 Conference on New Theories in physics, ed. T.


Eguchi and K. Nishijima; in the proceedings of the 1988 International


workshop on New Trends in Strong Coupling Gauge Theories, Nagoya,


Japan, ed. Bando, Muta, and Yamawaki (World Scientific, 1989); report


EFI-89-08 (1989); Also see V.A. Miransky, M. Tanabashi and K.


Yamawaki, Mod. Phys. Lett. A4 (1989) 1043; Phys. Lett. B 221 (1989)


177; W.J. Marciano, Phys. Lett. 62 (1989) 2793.


[5] W.A. Bardeen, C.T. Hill and M. Lindner, Phys. Rev. D41, 1647 (1990).


[6] G. Giacomelli and R. Giacomelli Results from High energy
e e


Collisions, Lecture Notes, DFUB98/23; LEP Electroweak working


group. A combination of preliminary LEP electroweak measurements


and constraints on the Standard Model, LEPEWWG/98-01 (1998).





17


[7] ALEPH Collaboration, P. Teixeira-Dias,


Search for the SM Higgs Boson at the LEP2 collider at s 189 Gev ,


CERN-EP/98-144 (1998); DELPHI Collaboration, P. Aberu et al. A


search for neutral Higgs Bosons in the MSSM and in models with two


scalar field doublets, Euro. Phys. Journal C5 (1998) 19; L3


Collaboration, M. Acciarri et al. Search for the SM Higgs boson in
e e


interactions at s 189 Gev , CERN-EP98/52 (1998); OPAL


Collaboration, G. Abbiendi et al. Search for Higgs bosons in
e e


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[8] H. Pagels and S. Stockar, Phys. Rev. D20, 2947 ( 1979), also see


reference [14].


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M.L. Godberger, M.T. Grisaru, S.W. McDowell and D. Wong, Phys.




18


Rev. 120, 2250 (1960). Also see R.G. Newton, Scattering Theory of


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19


Figure Captions:


Fig. 1. A schematic plot of 
x and F(x, , ) as a function of x . The
0



intersection points indicate the (bound) state solutions of (20) for a


fixed value of ( , ).
0





Fig. 2. The solutions of (20) corresponding to the Higgs boson mass m 100
H



and 200 Gev plotted as functions of  and . The dotted curve is
0





the Pagles-Stockar [8] expression (25).





20


x





F





Figure 1.





21


3  

100 Gev




2.5



200 Gev


2





1.5 P S






1





0.5





0

2 3 4 5 6





Figure 2.





22



