LBL-16743 c.do­ Preprint Lawrence Berkeley Laboratory UNIVERSITY OF CALIFORNIA Physics, Computer Science & RECEIVED Mathematics Division LAWRENCE BERKE"lEY LAA('Io,dTOqy DEC 1 3 1983 Submitted for publication LIBRARY AND DOCUMENTS SECTION

THE HIGGS-HIGGS BOUND STATE

R.N. Cahn and M. Suzuki

October 1983 TWO-WEEK LOAN COpy

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The Higgs-Higgs Bound State

R. N. Cahn Lawrence Berkeley Laboratory, Univ. of California Berkeley, California 94720

and

M. Suzuki Physics Department, Univ. of California, Berkeley, and Lawrence Berkeley Laboratory, Univ. of California Berkeley, California 94720

Abstract

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 their masses. Accordingly, so do their bound states.

• 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 U. S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under research grant No. PHY-81-18547.

-1- I. Introduction

The continuing success of the standard model of electroweak interactions increases the importance of understanding spontaneous symmetry breaking mechan­ isms. 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 " in '" ... H'Y or T ... H'Y, or if a it state, a, were found, in a ... H'Y. It the Higgs boson has a mass mH < 35 GeV, e+e- colliders could tlnd it by observing the decayS Z ... Z· H ... (p + p- or e+ e-) H. At a very high energy e+ e- machine, say with Va ~ 2mw, the process e+e- .... ZH could be seen for mH S 60GeV.s It appears that very high energy pporpp colliders (../8 ~ 20-40TeV) could tlnd a Higgs boson with a mass mH ~ 2mw through the signature H ... W+W- or H ... ZZ.7 Ordinary Drell-Yan production of W+W- and ZZ would be formidable backgroUnds since the width of a very heavy Higgs boson is enormous:

r(H ... W+W- + H ... ZZ) ~ 50GeV(50~ev r (1) and thus identifying an excess of events would be extremely di1Ilcult if mH ~ 500GeV.8 (For 2mb < mH < 2mw, the Higgs boson would decay primarily into bb or it, which would make it rather di1Ilcult to identify.)

The problem of identifying a very heavy Higgs boson, mH ~ 500GeV, 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 OJ existence of a very heavy Higgs boson. This has received much theoretical attention, especially from Veltman and co-workers9, and Appelquist and co-workers10 • T'heir perturbative calculations indicate that the effects of very heavy Higgs bosons are small, of order (g2/161J"2)10g(mh/mrv).

-2- A large ffiggs boson mass requires a strong seU-eoupling of the ffiggs boson and for sufficiently large ffiggs 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 ffiggs bosons should bind it fflH ~ 1.3 TeV. The I..' binding energy, however, is not very great even if fflH = 2 TeV. As a result, the bound state will have a width roughly twice the width of the ffiggs boson itself, and thus probably not be any easier to observe than the ffiggs boson although the signature of its decay into four W's might be striking.

n. The Higgs Sector

If the gauge interactions are ignored, the conventional ffiggs doublet is equivalent to the linear O'-model. We write

ifJ = (¢+, ) = (*(¢1 + i¢2») ¢o *(0' + iifJ3) ,

ifJ2 = ¢+¢-+ ¢o¢o = !(¢i + ¢~ + ¢~ +0-2) 2

(2)

.. With p.2 > 0, this produces spontaneous symmetry breaking and at tree level

=~ (3)

-8- When gauge interactions are included; < u > gives rise to the gauge boson masses:

(4)

Thus

< u >=247GeV (5)

In terms at the shifted u field, u = < u > +0',

>. .... t2 Lint --(¢. ¢+u 2u' < u (6) = 4 + - >1- so the 01 has a mass squared m~ = 2>' < u >2= 2p2.

We. shall need several diagrams for IDggs - IDggs elastic scattering, which are shown in Fig. 1. The associated amplitudes arell

-i.MG = (-i6>. < u »2 i (7a) t-m'h

-iMb = (-i6>.) X !. (76) 2

, • II (·6' \2 i 1 -'.Me = -I 1\ < U » X - (7c) , 8- m'h . 2

. (6).)2 A2 ] -,.Md = ----i~ 11 do[ log - 1 (7d) 2 (21r)4 0 m'h - 0(1 - o)t

(2).)2 i1r2 11 [ A2 ] -i.Me = 3---- do log - 1 (7e) 2 (21r)4 0 -0(1 - o)t

-4- The last two have been cutoft' at A. It we render them fillite by making a subtraction at t = -m~ we have

.M (6),)2 i1T2 fl d I 1 a(1 - a) + (7d') -I II = -2-(21T)4 10 a og 1 + a(l- a)(-tlm'h)

. (2),)2 i~ ( 2 I ) -IMe 3----log -m t (7e1) = 2 (21T)4 . H

When the subtraction point is chosen to be t = - m~, it is understood that our 0'4 coupling constant, )', is the one which is renormalized at at a mass scale mHo

m. N ID Calculation

To find the approximate locations of the bound states, we use the N /D method, 12 which we review briefly. The elastic s-wave scattering amplitude,

i6 t? 0 - 1 11 - ---,-- (8) o - 2ik satisfles

1 ' Im-=-k (9) 110 where k is the center of mass momentum. The partial wave amplitude for j = 0 is related to the Lorentz invariant amplitude, M, by

. 1 111 110 = - --. - d cos 0M (k, 0) (10) 81TJS 2 -1

-5- and we use finally rather than ao

Ao =-aoIS (11) 2 so that

1 2k Im- = -- == -p(s) (12) Ao IS

In the N/D method, Ao is written as N(s)/D(s) where N(s) has only lett hand cuts and D( s) has only right hand singularities. Thus on the right hand cut

Im2... = -p(s) = ImD(s) (13) Ao N(a)

A subtracted dispersion relation may be written for D. If at the subtraction point, So, D(ao) = 1

00 D(s) = 1- -11 da'p(s')N(a') [1---, 1], (14) 1r '. a' - s 8 - 80

Ao(a) = 00 N(s) • (15) 1 - .!. ( ds' P(S')N(S,)[_I_ _ 1 ] 1r l.. a' - 8 s' - So

If we approximate N(s) by the "Born amplitude" - the appropriate s-wave projection of the sum of the five amplitudes, Eqs. (7a)-(7c), (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 (f as a bound state of two (f'S, we are calculating a loosely bound state of two elementary (f'S, so the s-channel C! pole is included in the "Born amplitude". Thus for N( a) we take

-6- 0 N (1 ) 1 1 dt[ 18m1£)' . 3), + 9m1£)' (8) = -1611" 4k2 -4A:2 t - m'h + 8 - m'h

9).2 t do 10 1 + 0(1- 0)( -t/m1£) + 3).2 10 (-t/m2 )]- (16) +811"210 g 1+0(1-0) 811"2 g H

Letting

(17)

we have

2 2 ) 9), ) 3), N ( s=4mH+zmH = -log(l+z -- 811"z 1611" 9), 1 9).2 3).2 -1611"3+z - 12811"3g(Z)-128-rr3(1ogz-1) , (18) where

[1 d [1 d I 1 + 0(1 - O)ZX g() Z = 10 0 10 x og -1"":'+--""';0:""'(-1 -_-'0;"")- (19)

Making the subtraction at So = ~m1t and indicating zI = (s' - 4m1t)/m1t,

D(s = 4m)'r + zm)'r) = 1 - !. i' dl J-z-'_[_1 ____ 1 __ ] 11" 10 z' + 4 z, - z z' + 4 - ~

X -9( -). )-log(I+z')- 1 -----3( ).) 9( ).) --1 [2 411" Z, 4 411" 4 411" 3 + z,

--3 ( -). )2 (Iogz' - 1)- --9 ( ). )2 g(z')] (20) 811" 411" 811" 411"

The bound states for the s-wave occur when D(s) = 0 for 0 < 8 < 4m1£.

-7- N. Binding Energies

For sufficiently large values of )'/41r, D does develop a zero. The cor­ responding value of the Higgs boson mass is

(21)

In Fig. 2, the binding energy ,A, of the Higgs-Higgs system is displayed as a function of the mass of the Higgs boson. The binding energy is related to the value, Zs, for w hieh D vanishes:

(22)

We note that the bound state first occurs when mH 11::$ L3 TeV and the binding remains modest even for mH 11::$ 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, So. 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 by 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 80. A reasonable choice is to take 80 = -mho Then the N /D amplitude coincides with the "Born amplitude" at this point. In Fig. 2, we see that the bindng energy is insensitive to the choice of So as long as 80 does not approach the location of the zero itself, near 8 = 4mh. This is understandable since we should not force D to be unity near the location of its zero. We take the curve for 8 ~ -mh to be a reliable choice.

-8- V. Widths

The small binding of the Higgs bosons in the bound state means that the lifetime of the bound state will be essentially half the lifetime of a constituent Higgs boson. Consequently, the bound state has a width at least half its mass. The di1ficulty of recognizing such an object is readily demonstrated by recalling the

problem of finding resonances in the 1r - 1r s-wave system.

It should be noted that the great width which obscures the bound state is an intrinsic problem. The dominant contribution to the width of a very heavy Higgs boson comes from the decay into longitudinal W's, that is, the Higgs bosons of the l1-model. The A¢4 coupling which gives the binding also produces the width.

J

-9- References

1. S. Weinberg, Phys. Rev. Lett. 19 (1967)1264. 2. A. Salam, in Proc. 8''' Nobel Symp., 2nd Ed., ed. N. Svanholm (Almqvist and • Wiksells, Stockholm, 1968), p.361.

3. .See, Cor example, Proceedings of the 1982 DPF Summer Study on Elementary Particle Physics and Future Facilities, 00. R. Don.aldson et cU., Snowmass, Colorado, June 28 - July 16, 1982. of. F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304.

5. B.L. Iofre and V. A. Khoze, Leningrad Preprint 274 (1976); J. D. Bjorken,in Proceedings of the SLAe Summer Institute, 1976.

6. H. Georgi, et cU., Phys. Rev. Lett. 40 (1978).

7. H. A. Gordon, et al., in ReC.3, p. 161.

8. R. Cahn and I. ffinchliffe, unpublished.

9. M. Veltman, Acta Phys. Polan. B8 (1977) 475. J. van der Bij and M. Veltman, Univ. oC Michigan pre-print, UM-TH 83-4.

10. See, Cor example, T. Appelquist and C. Bernard, Phys. Rev. D22 (1980) 200 ..

11. We have ignored the diagrams obtained by crossing t - u and compensated in the overall combinatorial Cactor since we in the end look only at s-wave amplitudes.

12. G. F. Chew and S. Mandelstam, Phys. Rev. 119 (1960) 467.

- 10- Figure Captions

1. The diagrams contributing to the "Born amplitude" for elastic scattering used in the N / D equations.

2. The binding energy, a, of the IDggs-Higgs system as a function of the IDggs boson mass. The different curves represent different choices of the subtraction point, 80: solid curve (so = -3mh), dotted curve (so = ~mh), dashed curve (80 = mh), dot-dash curve (80 = 3m1I ).

-u- + +. + a c

+

d e

Fig. 1

- 12 - c.

140 r------~~------

120 .f I I 100 I I."7 I- -~ 1/ 0 80 I.' II - II I -

Fig. 2

- 13 - This report was done with support from the Department of Energy. Any conclusions or opinions expressed in this report represent solely those of the author(s) and not necessarily those of The Regents of the University of California, the Lawrence Berkeley Laboratory or the Department of Energy. Reference to a company or product name does not imply approval or recommendation of the product by the University of California or the U.S. Department of Energy to the exclusion of others that may be suitable. $.- - ,......

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