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The Higgs-Higgs bound state
Volume 134B, number 1,2
PHYSICS LETTERS
5 January 1984
THE HIGGS-HIGGS BOUND STATE ~ R.N. CAHN
Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94 720, USA and M. SUZUKI
Physics Department, University of California, Berkeley, CA 94720, USA and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA Received 17 October 1983
The binding of two Higgs bosons is calculated using the N/D method. Bound states occur for mH > 1.3 TeV. The binding is weak and even when mH = 2 TeV the binding energy is only about 150 GeV. Higgs bosons with masses in the TeV range have widths comparable to their masses. Accordingly, so do their bound states.
1. Introduction. The continuing success of the standard model of electroweak interactions increases the importance of understanding spontaneous symmetry breaking mechanisms. While alternatives are available, the initial proposal of Weinberg [1 ] and Salam [2] that fundamental scalars are responsible remains a viable approach. Indeed, the search for the Higgs boson, the surviving scalar, is one of the most prominent goals at present accelerators and a major motivation for the construction of new ones [3]. Present opinion is that a light Higgs boson could be best found [4] in ~ -~ H7 or T ~ HT, or if a it state, (9, were found, in (9 ~ HT. If the Higgs boson has a mass m H < 35 GeV, e+e - colliders could find it by observing the decay [5] Z -~ Z*H ~ (p+p- or e + e - ) H . At a very high energy e+e- machine, say with x / s ' ~ 2m w, the process e+e - ~ ZH could be seen for m H ~< 60 GeV [6]. It appears that very high energy pp or ~p colliders (x/s'-~ 2 0 40 TeV) could find a Higgs boson with a mass m H I> 2m w through the signature H ~ W+W- or H ~ ZZ [7]. Ordinary D r e l l - Y a n production of W+W- and ZZ would be formidable backgrounds since the width of a very heavy Higgs boson is enormous: r ( U -+ W+W- + H ~ ZZ) ~ 50 GeV(mH/500 GeV) 3
(1)
and thus identifying an excess of events would be extremely difficult if m H i> 500 GeV [8]. (For 2m b < m H < 2row, the Higgs boson would decay primarily into 13b or t-t, which would make it rather difficult to identify.) The problem of identifying a very heavy Higgs boson, m H ~> 500 GeV, is thus a serious one and it is worthwhile to seek indirect tests of its presence. One possibility is to identify low energy parameters whose values might reveal the existence of a very heavy Higgs boson. This has received much theoretical attention, especially from Veltman and co-workers [9], and Appelquist and co-workers [10]. Their2Perturbative calculations indicate that the effects of very heavy Higgs bosons are small, of order (g2/16rr2) l o g ( m u / m 2 ) . A large Higgs boson mass requires a strong self-coupling of the Higgs boson and for sufficiently large Higgs This work supported in part by the Director, Office of Energy Research, Office of High Energy Physics and Nuclear Physics, Division of High Energy Physics of the US Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under research grant No. PHY-82-03424. 0.031-9163/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
115
Volume 134B, number 1,2
PHYSICS LETTERS
5 January 1984
boson mass, perturbation theory is no longer reliable. We have investigated a simple non-perturbative effect: the existence of bound states. Our calculations show that two Higgs bosons should bind if m H/> 1.3 TeV. The binding energy, however, is not very great even if m H -- 2 TeV. As a result, the bound state will have a width roughly twice the width of the Higgs boson itself, and thus probably not be any easier to observe than the Higgs boson although the signature o f its decay into four W's might be striking. 2. TheHiggs sector. If the gauge interactions are ignored, the conventional Higgs doublet is equivalent to the linear o-model. We write (4)+) = ( 2
4)= 4)0
1/2(4)1 +i4)2) )
\ 2 - 1 / 2 ( o + i 4 ) 3)
'
1 2 4)2 = 4)+4)- + ¢ %-0- = ~(4)1 + 4)2 + 4)2 + o2) = } ( ~ . ~ +
o2),
22 = (~/a4))2 + U24)2 -- ~,(4)2)2 =}ata @ .~)/~ ~+ g~taaOlao 1 _ ~-X(d~.@+o 2 -/a2/X) 2 + ~4/4X
(2)
With/22 > 0, this produces spontaneous symmetry breaking and at tree level (o) = (uz/x) 1/2 ,
(3)
When gauge interactions are included, (o) gives rise to the gauge boson masses: m 2 = ¼g2(ay2 = ~ / 2 g 2 / S O F .
(4)
Thus (o) = 247 GeV.
(5)
In terms of the shifted o field, a = (o) + a', 22int = -¼ X(III"d~+ o '2 + 2o'(4))) 2 '
(6)
so the a' has a mass squared m~t = 23,(a) 2 = 2/12. We shall need several diagrams for Higgs-Higgs elastic scattering, which are shown in fig. 1. The associated amplitudes are +1 - iC/~a = (-i6X(o)) 2 i/(t - m 2 ) ,
--iC~b = (--i6X) X ~1 ,
(7a,b)
1 - i c~ c = (-i6X(a)) 2 [i/(s - m 2 ) ] X ~-,
(7c)
,1 We have ignored the diagrams obtained by crossing t --* u and compensated in the overall combinatorial factor since we in the end look only at s-wave amplitudes.
+
a
b
c
d
Fig. 1. The diagrams contributing to the "Born amplitude" for elastic scattering used in t h e
116
•
N/D equations.
Volume 134B, number 1,2
-- iQ?~d
PHYSICS LETTERS
(6X)2 in2 ) ( A2 - - - - d a log ~ 2 (27r)4 0 m~i -- a(1 - a ) t
j(
(2X)2 i7r2 - i9?( e = 3 T (27r)4 0
5 January 1984
) -
1
,
(7d)
)
da log a(~-_a)t
1 .
(7e)
The last two have been cut o f f a t A. If we render them finite b y making a subtraction at t = --m2u we have l
fdalog
- i Q ~ d _ (6X)~2 i7r2 2 (2zr)4 0
l+a(1-o0 1 + a(1 - a)(-t/m2)
_iC~e=3.
'
(2X) 2 in 2 2 (2704
log(_m2H/t)
(7d',e')
When the subtraction point is chosen to be t = - m 2 , it is u n d e r s t o o d that our a 4 coupling constant, X, is the one which is renormalized at at a mass scale m H.
3. N/D Calculation. To
find the a p p r o x i m a t e location of the b o u n d states, we use the N/D m e t h o d , [1 1], which we review briefly. The elastic s-wave scattering amplitude,
(8)
a 0 = [exp(2i60) - 1 ] / 2 i k , satisfies Im a21 = -k Iy
(9)
where k is the center of mass m o m e n t u m . The partial wave amplitude f o r j = 0 is related to the Lorentz invariant amplitude, 9 R , b y 1
ao -
1
1
2 f d cos 0 ctg (k, O)
(10)
-1 and we use finally rather than a 0 _1
A 0 - ~x/~a 0 ,
(11)
so that Im A 61 =
-2kA/s-
-p (s).
(12)
In the N/D m e t h o d , A 0 is written asN(s)/D(s) where singularities. Thus on the right hand cut Im A0q =
-p(s)
= [ N ( s ) ] - I Im
N(s) has
only left hand cuts and
only right hand
O(s).
(13)
A subtracted dispersion relation m a y be written for D. If at the subtraction point,
D(s)=l l__f ds,p(s,)N(s,)( St 1 S lr S#
D(s) has
1 St
St
) $0
S
So, D(so) = 1 (14)
'
St
S0
stx
If we a p p r o x i m a t e
N(s) b y
the "Born a m p l i t u d e " - the appropriate s-wave projection of the sum o f the five 117
Volume 134B, number 1,2
PHYSICS LETTERS
5 January 1984
amplitudes, eqs. ( 7 a ) - ( 7 c ) , (7d'), (7e') - we obtain an amplitude Ao(s ) which satisfies elastic unitarity and which reduces to the "Born amplitude" in the weak coupling limit. Unlike the bootstrap calculation to generate the o as a bound state of two a's, we are calculating a loosely bound state of two elementary o's, so the s-channel a pole is included in the "Born amplitude". Thus for N(s) we take 0 1 1 a( 161r 4k2 _4k2
N(s)
[ 18m2X
dt\t_m+,
9m2X
3X+--+ S -- m 2
9X2
f
1 do, log
87r2 0
1 + a(1 - a ) ( - t / m 2) 1 +a(1 a) --
+
3X2 87r2
,
log(-t/m 2)].
(16)
Letting
(I 7)
z = 4k2/m 2 , /
I 1_
we have
N(s =4m2+zm2H)=8~zlOg(l + z )
3X
9X
167r
16n 3 + z
1
3X2 - (log z - 1)
9X 2
(18)
-1287r - 3 g(z) - 128n 3
where 1
g(z) =
f 0
1
doe
f dxlog 1 1++ o~(I - oOzx (~(1-~) "
(19)
0
Making the subtraction at s o = ~'m2 and indicating z' = (s' - 4 m 2 ) / m 2,
D(s=4m2+zm2)=l-lf dz'[z--It~2( 1-__ 1) o0
0 X 9
X 7711°g(l+z' ) _ 4 3
\z' +4] --4
z'
~
Z
z'+4-
3( )2
, 3 - - ~ z ' - - - 8~
~"
(log z' - 1) -
g(z')l "
(20)
The bound states for the s-wave occur when D(s) = 0 < s < 4m 2.
4. Binding energies. For sufficiently large values o f X/4n, D does develop a zero. The corresponding value o f the Higgs boson mass is m H =
(21)
In fig. 2, the binding energy, A, o f the Higgs-Higgs system is displayed as a function of the mass o f the Higgs boson. The binding energy is related to the value, ZB, for which D vanishes: A = 2m H -- 2mH(1 + lzB)l/2 .
(22)
We note that the b o u n d state first occurs when m H ~ 1.3 TeV and the binding remains modest even for m H ~ 2 TeV. The curve for the binding energy as a function of the Higgs mass has been computed for several values of the subtraction point, s 0. The location of a zero in D should, in principle, not depend on the choice of the subtraction point, if the N / D equations are solved consistently. Setting D(so) = 1 is just a normalization which is compensated b y a change in N. In our calculation, however, since the N function is being approximated by the "Born amplitude", without being iterated, we expect that the numerical value of the binding energy will have some dependence on §0" A reasonable choice is to take s o = - m 2. Then the N / D amplitude coincides with the "Born amplitude" at this point. In fig. 2, we see that the binding energy is insensitive to the choice o f s 0 as long as s o does not approach the location of the zero itself, near s = 4m 2. This is understandable since we should not force D to be unity near the location of its zero. We take the curve for s = - m 2 to be a reliable choice.
118
Volume 134B, number 1,2
PHYSICS LETTERS
5 January 1984
80
<1 60
40
20
t
1.
12
1,4
MH
1 6
(TeV)
1.8
Fig. 2. The binding energy, ,% of the Higgs-Higgs system as a function of the Higgs boson mass. The different curves represent different choices of the subtraction point, so: solid curve (s o = -3m~t), dotted curve (so = - m ~ ) , dashed curve (so = 3m~t).
5. Widths. The small binding o f the Higgs bosons in the b o u n d state means that the lifetime o f the b o u n d state will be essentially half the lifetime o f a c o n s t i t u e n t Higgs boson. C o n s e q u e n t l y , the b o u n d state has a width at least half its mass. The difficulty o f recognizing such an object is readily d e m o n s t r a t e d b y recalling the p r o b l e m o f finding resonances in the 7r-n s-wave system. It should be n o t e d that the great width which obscures the b o u n d state is an intrinsic problem. The d o m i n a n t c o n t r i b u t i o n to the width o f a very heavy Higgs boson comes from the decay into longitudinal W's, that is, the Higgs bosons o f the a-model. The )re 4 coupling which gives the binding also produces the width. References [1 ] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264. [2] A. Salam, in: Proc. 8th Nobel Symp., 2nd Ed., ed. N. Svartholm (Almqvist and Wiksells, Stockholm, 1968) p. 367. [3 ] See, for example, R. Donaldson et al., eds., Proc. 1982 DPF Summer Study of Elementary particle physics and future facilities, (Snowmass, CO, June 28-July 16, 1982). [4] F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304. [5] B.L. Ioffe and V.A. Khoze, Leningrad preprint 274 (1976); J.D. Bjorken, in: Proc. SLAC Summer Institute (1976). [6] H. Georgi et al., Phys. Rev. Lett. 40 (1978) 1. [7 ] H.A. Gordon et al., in: Proc. 1982 DPF Summer study on Elementary particle physics and future facilities, eds. R. Donald et al. (Snowmass, CO, June 28-July 16, 1982) p. 161. [8] R. Cahn and I. I-Iinchliffe, unpublished. [9] M. Veltman, Acta Phys. Polon. B8 (1977) 475; J. van der Bij and M. Veltman, Univ. of Michigan preprint, UM-TH 83-4. [10] See, for example. T. Appelquist and C. Bernard, Phys. Rev. D22 (1980) 200. [11 ] G.F. Chew and S. Mandelstam, Phys. Rev. 199 (1960) 467.
119
PHYSICS LETTERS
5 January 1984
THE HIGGS-HIGGS BOUND STATE ~ R.N. CAHN
Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94 720, USA and M. SUZUKI
Physics Department, University of California, Berkeley, CA 94720, USA and Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA Received 17 October 1983
The binding of two Higgs bosons is calculated using the N/D method. Bound states occur for mH > 1.3 TeV. The binding is weak and even when mH = 2 TeV the binding energy is only about 150 GeV. Higgs bosons with masses in the TeV range have widths comparable to their masses. Accordingly, so do their bound states.
1. Introduction. The continuing success of the standard model of electroweak interactions increases the importance of understanding spontaneous symmetry breaking mechanisms. While alternatives are available, the initial proposal of Weinberg [1 ] and Salam [2] that fundamental scalars are responsible remains a viable approach. Indeed, the search for the Higgs boson, the surviving scalar, is one of the most prominent goals at present accelerators and a major motivation for the construction of new ones [3]. Present opinion is that a light Higgs boson could be best found [4] in ~ -~ H7 or T ~ HT, or if a it state, (9, were found, in (9 ~ HT. If the Higgs boson has a mass m H < 35 GeV, e+e - colliders could find it by observing the decay [5] Z -~ Z*H ~ (p+p- or e + e - ) H . At a very high energy e+e- machine, say with x / s ' ~ 2m w, the process e+e - ~ ZH could be seen for m H ~< 60 GeV [6]. It appears that very high energy pp or ~p colliders (x/s'-~ 2 0 40 TeV) could find a Higgs boson with a mass m H I> 2m w through the signature H ~ W+W- or H ~ ZZ [7]. Ordinary D r e l l - Y a n production of W+W- and ZZ would be formidable backgrounds since the width of a very heavy Higgs boson is enormous: r ( U -+ W+W- + H ~ ZZ) ~ 50 GeV(mH/500 GeV) 3
(1)
and thus identifying an excess of events would be extremely difficult if m H i> 500 GeV [8]. (For 2m b < m H < 2row, the Higgs boson would decay primarily into 13b or t-t, which would make it rather difficult to identify.) The problem of identifying a very heavy Higgs boson, m H ~> 500 GeV, is thus a serious one and it is worthwhile to seek indirect tests of its presence. One possibility is to identify low energy parameters whose values might reveal the existence of a very heavy Higgs boson. This has received much theoretical attention, especially from Veltman and co-workers [9], and Appelquist and co-workers [10]. Their2Perturbative calculations indicate that the effects of very heavy Higgs bosons are small, of order (g2/16rr2) l o g ( m u / m 2 ) . A large Higgs boson mass requires a strong self-coupling of the Higgs boson and for sufficiently large Higgs This work supported in part by the Director, Office of Energy Research, Office of High Energy Physics and Nuclear Physics, Division of High Energy Physics of the US Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under research grant No. PHY-82-03424. 0.031-9163/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
115
Volume 134B, number 1,2
PHYSICS LETTERS
5 January 1984
boson mass, perturbation theory is no longer reliable. We have investigated a simple non-perturbative effect: the existence of bound states. Our calculations show that two Higgs bosons should bind if m H/> 1.3 TeV. The binding energy, however, is not very great even if m H -- 2 TeV. As a result, the bound state will have a width roughly twice the width of the Higgs boson itself, and thus probably not be any easier to observe than the Higgs boson although the signature o f its decay into four W's might be striking. 2. TheHiggs sector. If the gauge interactions are ignored, the conventional Higgs doublet is equivalent to the linear o-model. We write (4)+) = ( 2
4)= 4)0
1/2(4)1 +i4)2) )
\ 2 - 1 / 2 ( o + i 4 ) 3)
'
1 2 4)2 = 4)+4)- + ¢ %-0- = ~(4)1 + 4)2 + 4)2 + o2) = } ( ~ . ~ +
o2),
22 = (~/a4))2 + U24)2 -- ~,(4)2)2 =}ata @ .~)/~ ~+ g~taaOlao 1 _ ~-X(d~.@+o 2 -/a2/X) 2 + ~4/4X
(2)
With/22 > 0, this produces spontaneous symmetry breaking and at tree level (o) = (uz/x) 1/2 ,
(3)
When gauge interactions are included, (o) gives rise to the gauge boson masses: m 2 = ¼g2(ay2 = ~ / 2 g 2 / S O F .
(4)
Thus (o) = 247 GeV.
(5)
In terms of the shifted o field, a = (o) + a', 22int = -¼ X(III"d~+ o '2 + 2o'(4))) 2 '
(6)
so the a' has a mass squared m~t = 23,(a) 2 = 2/12. We shall need several diagrams for Higgs-Higgs elastic scattering, which are shown in fig. 1. The associated amplitudes are +1 - iC/~a = (-i6X(o)) 2 i/(t - m 2 ) ,
--iC~b = (--i6X) X ~1 ,
(7a,b)
1 - i c~ c = (-i6X(a)) 2 [i/(s - m 2 ) ] X ~-,
(7c)
,1 We have ignored the diagrams obtained by crossing t --* u and compensated in the overall combinatorial factor since we in the end look only at s-wave amplitudes.
+
a
b
c
d
Fig. 1. The diagrams contributing to the "Born amplitude" for elastic scattering used in t h e
116
•
N/D equations.
Volume 134B, number 1,2
-- iQ?~d
PHYSICS LETTERS
(6X)2 in2 ) ( A2 - - - - d a log ~ 2 (27r)4 0 m~i -- a(1 - a ) t
j(
(2X)2 i7r2 - i9?( e = 3 T (27r)4 0
5 January 1984
) -
1
,
(7d)
)
da log a(~-_a)t
1 .
(7e)
The last two have been cut o f f a t A. If we render them finite b y making a subtraction at t = --m2u we have l
fdalog
- i Q ~ d _ (6X)~2 i7r2 2 (2zr)4 0
l+a(1-o0 1 + a(1 - a)(-t/m2)
_iC~e=3.
'
(2X) 2 in 2 2 (2704
log(_m2H/t)
(7d',e')
When the subtraction point is chosen to be t = - m 2 , it is u n d e r s t o o d that our a 4 coupling constant, X, is the one which is renormalized at at a mass scale m H.
3. N/D Calculation. To
find the a p p r o x i m a t e location of the b o u n d states, we use the N/D m e t h o d , [1 1], which we review briefly. The elastic s-wave scattering amplitude,
(8)
a 0 = [exp(2i60) - 1 ] / 2 i k , satisfies Im a21 = -k Iy
(9)
where k is the center of mass m o m e n t u m . The partial wave amplitude f o r j = 0 is related to the Lorentz invariant amplitude, 9 R , b y 1
ao -
1
1
2 f d cos 0 ctg (k, O)
(10)
-1 and we use finally rather than a 0 _1
A 0 - ~x/~a 0 ,
(11)
so that Im A 61 =
-2kA/s-
-p (s).
(12)
In the N/D m e t h o d , A 0 is written asN(s)/D(s) where singularities. Thus on the right hand cut Im A0q =
-p(s)
= [ N ( s ) ] - I Im
N(s) has
only left hand cuts and
only right hand
O(s).
(13)
A subtracted dispersion relation m a y be written for D. If at the subtraction point,
D(s)=l l__f ds,p(s,)N(s,)( St 1 S lr S#
D(s) has
1 St
St
) $0
S
So, D(so) = 1 (14)
'
St
S0
stx
If we a p p r o x i m a t e
N(s) b y
the "Born a m p l i t u d e " - the appropriate s-wave projection of the sum o f the five 117
Volume 134B, number 1,2
PHYSICS LETTERS
5 January 1984
amplitudes, eqs. ( 7 a ) - ( 7 c ) , (7d'), (7e') - we obtain an amplitude Ao(s ) which satisfies elastic unitarity and which reduces to the "Born amplitude" in the weak coupling limit. Unlike the bootstrap calculation to generate the o as a bound state of two a's, we are calculating a loosely bound state of two elementary o's, so the s-channel a pole is included in the "Born amplitude". Thus for N(s) we take 0 1 1 a( 161r 4k2 _4k2
N(s)
[ 18m2X
dt\t_m+,
9m2X
3X+--+ S -- m 2
9X2
f
1 do, log
87r2 0
1 + a(1 - a ) ( - t / m 2) 1 +a(1 a) --
+
3X2 87r2
,
log(-t/m 2)].
(16)
Letting
(I 7)
z = 4k2/m 2 , /
I 1_
we have
N(s =4m2+zm2H)=8~zlOg(l + z )
3X
9X
167r
16n 3 + z
1
3X2 - (log z - 1)
9X 2
(18)
-1287r - 3 g(z) - 128n 3
where 1
g(z) =
f 0
1
doe
f dxlog 1 1++ o~(I - oOzx (~(1-~) "
(19)
0
Making the subtraction at s o = ~'m2 and indicating z' = (s' - 4 m 2 ) / m 2,
D(s=4m2+zm2)=l-lf dz'[z--It~2( 1-__ 1) o0
0 X 9
X 7711°g(l+z' ) _ 4 3
\z' +4] --4
z'
~
Z
z'+4-
3( )2
, 3 - - ~ z ' - - - 8~
~"
(log z' - 1) -
g(z')l "
(20)
The bound states for the s-wave occur when D(s) = 0 < s < 4m 2.
4. Binding energies. For sufficiently large values o f X/4n, D does develop a zero. The corresponding value o f the Higgs boson mass is m H =
(21)
In fig. 2, the binding energy, A, o f the Higgs-Higgs system is displayed as a function of the mass o f the Higgs boson. The binding energy is related to the value, ZB, for which D vanishes: A = 2m H -- 2mH(1 + lzB)l/2 .
(22)
We note that the b o u n d state first occurs when m H ~ 1.3 TeV and the binding remains modest even for m H ~ 2 TeV. The curve for the binding energy as a function of the Higgs mass has been computed for several values of the subtraction point, s 0. The location of a zero in D should, in principle, not depend on the choice of the subtraction point, if the N / D equations are solved consistently. Setting D(so) = 1 is just a normalization which is compensated b y a change in N. In our calculation, however, since the N function is being approximated by the "Born amplitude", without being iterated, we expect that the numerical value of the binding energy will have some dependence on §0" A reasonable choice is to take s o = - m 2. Then the N / D amplitude coincides with the "Born amplitude" at this point. In fig. 2, we see that the binding energy is insensitive to the choice o f s 0 as long as s o does not approach the location of the zero itself, near s = 4m 2. This is understandable since we should not force D to be unity near the location of its zero. We take the curve for s = - m 2 to be a reliable choice.
118
Volume 134B, number 1,2
PHYSICS LETTERS
5 January 1984
80
<1 60
40
20
t
1.
12
1,4
MH
1 6
(TeV)
1.8
Fig. 2. The binding energy, ,% of the Higgs-Higgs system as a function of the Higgs boson mass. The different curves represent different choices of the subtraction point, so: solid curve (s o = -3m~t), dotted curve (so = - m ~ ) , dashed curve (so = 3m~t).
5. Widths. The small binding o f the Higgs bosons in the b o u n d state means that the lifetime o f the b o u n d state will be essentially half the lifetime o f a c o n s t i t u e n t Higgs boson. C o n s e q u e n t l y , the b o u n d state has a width at least half its mass. The difficulty o f recognizing such an object is readily d e m o n s t r a t e d b y recalling the p r o b l e m o f finding resonances in the 7r-n s-wave system. It should be n o t e d that the great width which obscures the b o u n d state is an intrinsic problem. The d o m i n a n t c o n t r i b u t i o n to the width o f a very heavy Higgs boson comes from the decay into longitudinal W's, that is, the Higgs bosons o f the a-model. The )re 4 coupling which gives the binding also produces the width. References [1 ] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264. [2] A. Salam, in: Proc. 8th Nobel Symp., 2nd Ed., ed. N. Svartholm (Almqvist and Wiksells, Stockholm, 1968) p. 367. [3 ] See, for example, R. Donaldson et al., eds., Proc. 1982 DPF Summer Study of Elementary particle physics and future facilities, (Snowmass, CO, June 28-July 16, 1982). [4] F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304. [5] B.L. Ioffe and V.A. Khoze, Leningrad preprint 274 (1976); J.D. Bjorken, in: Proc. SLAC Summer Institute (1976). [6] H. Georgi et al., Phys. Rev. Lett. 40 (1978) 1. [7 ] H.A. Gordon et al., in: Proc. 1982 DPF Summer study on Elementary particle physics and future facilities, eds. R. Donald et al. (Snowmass, CO, June 28-July 16, 1982) p. 161. [8] R. Cahn and I. I-Iinchliffe, unpublished. [9] M. Veltman, Acta Phys. Polon. B8 (1977) 475; J. van der Bij and M. Veltman, Univ. of Michigan preprint, UM-TH 83-4. [10] See, for example. T. Appelquist and C. Bernard, Phys. Rev. D22 (1980) 200. [11 ] G.F. Chew and S. Mandelstam, Phys. Rev. 199 (1960) 467.
119