UCRHEP-T257
May, 1999




Can the Higgs Boson Bootstrap itself ?


Bipin R. Desai and Alexander R. Vaucher


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




Abstract


It is pointed out that Higgs bootstrap is inherent in the top-
condensate models of Nambu and Bardeen et al in that the
ladder sum of Higgs pole diagrams in one channel reproduces
a Higgs pole in the crossed-channel. This result is exact
whenever color flows simultaneously in both channel
directions e.g. for N 1 . Bootstrap solutions for the Higgs
c

boson mass are obtained that are compatible with the
condensate models such as the top-color models which imply
large top-Yukawa couplings at the top mass.





1


I. Introduction


The suggestion [1], [2], [3] that the top-quark can couple to the Higgs


boson more strongly than in the standard model leads to many interesting


questions. One of these is the role the Higgs boson will play by providing a


strong (attractive) force through t-channel exchange that can create bound


states in J P  O channel (s-channel) of the top-antitop (tt ) scattering


amplitude. A natural question to ask then is whether the Higgs boson will


itself be among these bound states; that is, whether the dynamics will permit


the Higgs boson to bootstrap itself.


Interestingly under certain circumstances a Higgs bootstrap is inherent


in the tt condensate approach of Nambu [4] and Bardeen et al [5].


In the condensate model the Higgs boson is generated through a four-fermi


interaction at a high scale by summing the tt bubble diagrams as in Fig. 1.


This figure then represents the lowest order diagram for the Higgs-pole in the


s-channel.


Consider now the t-channel Higgs exchange amplitude as indicated in


fig.2. It is a t-channel sum of fermion bubbles in the same manner as the s-


channel sum of fig.1. In terms of ,  , the top-Yukawa coupling and the
t



Higgs mass, m , it can be expressed as follows
H





2


 2 u uvv
t (1)
t m2
2  H





where u and v are t and t spinors respectively. If we now try to solve the


bound state problem in the ladder approximation by iterating this amplitude


then an interesting thing happens. In fig.3 each term in the ladder sum (the


left column) is itself, according to fig.2, expressed as a sum of fermion


bubbles (the right hand side). We notice that the first term in each sum on the


right hand (i.e. the first column on the right after the equal-sign) when added


up will give the Higgs amplitude in the s-channel as in fig.1, the subsequent


terms on the right being unitarity corrections. That is, a sum of t-channel


Higgs-exchanges produces an s-channel Higgs amplitude (see fig.4).




This implies that the Higgs exchange potential will produce Higgs


boson as a bound state in the s-channel provided the coupling is strong


enough. Therefore a Higgs bootstrap is inherent in the four fermi condensate


model, at least in the ladder approximation. Of course, since color flows





3


either in the s-channel direction or the t-channel but not both simultaneously


in these diagrams, a strict bootstrap occurs only when N  1.
c



We note, however, that in the condensate model once the gap equation


is imposed, the expression for the mass of the Higgs boson itself in terms of


top-mass i.e.


m 2m (2)
H 



is independent of N . The coupling, of course, depends on N [5].
c c



How strong the top-Yukawa coupling should be to create bound states


and therefore to accomplish the bootstrap will be considered below within the


framework of the ladder approximation.


Instead of carrying out a general bootstrap program by varying m and
H



 , we will consider only the question whether the Higgs boson mass (2)
t



reproduces itself.




II. The Bound State Problem


The bound state problem in the ladder approximation has been studied


in certain field theories [6], in non-relativistic potential theories with Yukawa


potentials [7] and in S-matrix theories [8], [9]. The solution is obtained


through dispersion relations by imposing the known analytic and unitarity




4


properties of partial wave scattering amplitudes expressed in a formalism of


the ratio N/D. The zeroes of D, then determine the bound states.




Ghergetta [10] has shown that the dispersion relations methods of the


type indicated above are better suited than the traditional methods in Nambu


Jona-Lasinio type models [11], specifically in the top quark condensation


models of Nambu [4] and Bardeen et al [5].




For our bound state calculation we use the Jacob and Wick [12]


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


scattering amplitude a b  c d 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





5


1 1
where and indicate  and respectively. For our scattering
 
!





2 2


amplitude, therefore,   
0 and












d j  (cos )
Pj




Denoting this amplitude as T , we write






"#$ %& '





1 1






T E  j   T E 
( , ) ( ) P (cos ) (3)
 p  j j
2
j





The Higgs-exchange amplitude T H can be obtained from (1) as







2 2
1 1
 m 
( cos ) ( cos )
 E 
T (H) (E 2

, ) (4)
 2
 t t  mH


( )


where
t 2
 2 1 cos is the momentum transfer and p and the center of
 p 





mass momentum and scattering angle respectively.




The partial-wave projection is given by





6


1 1
T (H) (E ( )
 0
) cos ( , ) (cos ) (5)
d T H E P
 j  j
2 1




If we write m  2m (condensate value), and
H





2
 p
x (6)
m2


then the j  0 value is given by


CDE FG H





123 45 6 789 @A B
 2
( ) 2 2 2
T H  t   
1 ln 1
( ) (7)
 x
0 2
 2 x x x


where we have assumed  2 to be a constant (we will return to this assumption
t



later).


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


the N/D method mentioned earlier. Typically, in this formalism, a partial


wave amplitude




ei sin
T I P Q (8)
l
R





1
I S


R R
cot i
Q





where is the phase-shift and the phase space factor, is expressed as a
T U





ratio



7


N
T (9)
l V D





where D, has a branch cut from the threshold at p2 0 to (right hand cut) as
W





V





it acquires a phase given by (5). But it is real for p2 X 0. N is real for p2 Y 0


but has a (left hand) cut arising from singularities of the partial wave


projection of the t-channel exchange amplitude. This cut extends from to
 W





 m2
v , where v is related to the mass of the exchanged particle (v H in our
0 0 0 
4


case). In terms of x defined in (6) the right hand cut is along (0, ). And the
`





left hand cut along ( , 1) with m 2m, as can be confirmed from (7).
a b a H c





The discontinuity of D across the right-hand cut is easily seen from (8)


and (9) to be N . The left hand discontinuity of N is D ImT , where "Im"
d e l



indicates imaginary part in the interval ( v .
 W , )
0





Specifically, normalizing the amplitude by taking D 1 as x , one
f f g





writes





8


p





q
1 dx'
D( x) h i
1 (10)
e N (x' )
r x' i x
0




u w

1 1 dx'
N (x) s t D(x' ) ImT (x' ) (11)
r
u v x' x l




1 x
with x y
e





r
16 x 1





N is usually taken simply as the partial wave projection of the t-channel


one particle exchange amplitude (in our case, the partial wave projection of


the amplitude (1) given by (7)) without going through the dispersion integrals


in (11). This is what we will also assume for the Higgs exchange [9].


Therefore, we will take


N x T H
( ) ( x).
0



We note from (7) that N ( x) has the proper analytic structure. We need now


to obtain D(x). Before we insert N ( x) given above in the dispersion integral


(10) we note that  , which was assumed constant, actually evolves according
t



to the renormalization group equations as









d 
2 t  3
b (12)
2 t
d





9


where the top-Yukawa coupling  is evaluated at mass . Since  is no
t t



longer restricted to its standard model value ( 1) at the top mass but can be






much larger, according to the top-color models [1],[2],[3], the above equation


implies a rapid increase in  for larger values of 2 [13].
t





In (4) and (12) we note that since   2 2
 ( ), where  t , the t-

t t





dependence of  in the Higgs-exchange amplitude given by (4) must be taken
t



into account in the partial wave projection (5), and, therefore, in (7). The


integration in (5) of
cos from 1 to +1 converts to





1 1

 1 0
d cos
dt
2 4 2
p
1 4 2
p





This implies that it is the region 2 2
0 which is relevent for  ( ). From the

t



renormalization group equations (12) we note that  ( 2 ) does not depend on

t



the sign of 2 and its explicit 2 dependence can be determined from the






appropriate boundary conditions.




Clearly then, depending on the functional form of  ( 2 ), the above

t



integration and the subsequent dispersion relations integral (10) for p2 from 0


to ( i.e. x from 0 to ) from can be very complicated. A reasonable
W W





10


approximation would be to replace  2 by some sort of average ,  2 , over 2 .
t t





Thus we change  
2  2 .
t t





It is also important to note that in addition to the possibility that  can
t



be large at the top mass m [1], [2], [3], its evolution given by (12), can






make  even larger as 2 increases. Bardeen et al [5] have emphasized that
t





if the tt system forms a bound state, then  will blow up as  where
t





is the appropriate composite scale. Therefore,




 2  very large.
t




We will return later to the question of determining  2 .
t




From (7) we note that




 2
T (H) t
 constant ( ) as x 
0 W
 2





The integral in (10) for D, therefore, will not converge making it necessary to


introduce a subtraction to represent the unknown high scale dynamics with D


normalized so that D=1 at the subtraction point. Since we are looking for


possible bound states i.e. zeroes in D in the region
1 x 0 it is essential that






11


the subtraction point not include this region. We will take, in the following,


the subtraction point at x
 , such that x 1, and discuss the bound state
 x1 1



energies as well as any sensitivities of the results to the subtraction point.


Taking account of the above comments we express D given by (10),


using (7) as













2 '  '  '
t 2 1 1
  dx
 ( x ) ln( x )
D( x) 1 2 (13)
 (x x ) (x' ) 
r 2 1 '
32 0 ( x '   1
 x)(x' x ) x' (x' x
)
1





We write the above relation as,




 2
t
D( x)  
1 (14)
 (x x ) I (x, x )
32r 2 1 1





where I (x, x ) is the integral on the right hand side of expression (13). To
1



accomplish a bootstrap vis--vis the condensate model with m 2m (i.e.
H



x 0) we must have






D( )
0  0


Therefore the Yukawa coupling must satisfy





12


 2
t
 2 (15)
2
r
16 x I (0, x )
1 1





A plot of the right side of equation (15) is given in fig. 5 for x 1. We
1



find that it is insensitive to the precise choice of x for x 2 , being roughly a
1 1



constant over a large range of x , decreasing slowly, logarithmically. For a
1



2
typical x associated with an s-channel scale of 1 Tev ( x e f ), for
1 d 1 m2
4


which x g 8, we obtain
1





 2
t
h 1.5 (16)
2
r
16





Thus a Higgs bootstrap is possible as long as the averaged coupling


constant is consistent with (16). Next we consider whether this value is


reasonable based on the renormalization group equations.




III. Estimating  2t


To estimate  2 we note that the solution for (12) with the
t




compositeness boundary condition   as  is given by [5]
t W





13


 1
 2 ( ) ijk lm n , b 0 (17)

t
2

b
2 ln 2







If we try to evaluate  2 by defining it as a simple average,
t





2
 1
 
2 o d 2 2 2
( ) (18)

t
2 p t
2
0





then we find that substituting (17) in (18) will make the integral infinite


because of the singularity at
 in (17).






A more convergent way to estimate  2 may be by extrapolating  2 at
t t




2 2
 2
 2 linearly through by using (12) and then taking the average as
0





given by (18). If we write  
2 2 2
( )  we obtain from (12)

t 0 0



2
  






2  2
( )  b 4
2 lnqrs tu v (19)

t 0 0 2
0





Inserting the above expression in (18) we obtain, for large ,





14


2
  
2  xyz {| }
w 2 b 4
2 ln (20)
t 0 0 2
0





For our Higgs problem, we take  m, the top mass, then  is the top-
0 0



Yukawa coupling at the top-mass ( 
~ 2
(m ) ).
0 t



For the condensate model, Pagles and Stockar [14], and Gherghetta


[10], have derived a relation for the coupling in terms of the composite scale








2
r
16 1
 
2 2 2

(m ) (21)
0 t 2
Nc ln 2
m


Eliminating the logarithm from (20), using (21) (with N 1) we obtain, using
c



the standard model value for b, the following estimate for the top-Yukawa


coupling at the top mass,



4 .87 (27)
0



This is a rather large value but consistent with the top-color model estimates


[1], [2], [3].


We, therefore, conclude that the Higgs bootstrap which is inherent in


the ladder diagram approach of the condensate models is actually feasible for


sensible values of  .
0





15


We thank Professor Jose Wudka for his helpful comments. This work


was supported in part by the U.S. Department of Energy under Contract No:


DE-F603-94ER40837.





16


References


[1] C. T. Hill, Phys. Lett. B 266, 419 (1991)


[2] C. T. Hill, Phys. Lett. B 345, 483 (1995); D. Kominis Phys. Lett. B 358,


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] B. Lee and R. Sawyer, Phys. Rev. 127, 2266 (1962); Also see D.


Amati, S. Fubini and A. Stanghellini, Phys. Lett. 29 (1962).





17


[7] R. Blankenbecler and M.L. Goldberger, Phys. Rev. 126, 766 (1962);


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


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


[8] G.F. Chew, S-matrix Theory of Strong Interactions, Benjamin, New York


(1962).


[9] S. Gasiorowitz, Elementary Particle Physics, John Wiley & Sons Inc,


New York (1966).


[10] T. Gherghetta, Phys. Rev. D50, 5985 (1994).


[11] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345 (1961); ibid Phys.


Rev 124, 246 (1961).


[12] M.Jacob and G.C. Wick, Ann. Phys. (NY) 7, 404 (1959).


2 m
[13] We are taking the conventional definition,  , for the top-Yukawa
t v


coupling so that for the standard model value, v 246 Gev, we have








1 at the top mass. If, on the other hand, v is much smaller than this
t



value, as in the top-color models, then  will be quite large.
t



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





18


Figure Captions:


Fig. 1. Higgs boson as a fermion bubble summation in the s-channel.




Fig. 2. Higgs boson as a fermion bubble summation in the t-channel.




Fig. 3. Individual terms in the ladder summation of t-channel Higgs-exchange


diagrams expressed in terms of fermion bubble diagrams.




Fig. 4. Graphical representation of the bootstrap result.










2
2
Fig. 5. t .
=
(0, ) 16 2
x I x
vs. x1
1 1





19


t





s = + + . .
+

H





fig. 1





20


t





s H = + + . .
+





fig. 2











21


= + + . . .





= + + + . .





= + . . .





F ig. 3.



.





.





.





22


+ + . . . = + . . .





F ig . 4





23


Fig. 5





24



