A phenomenological profile of the Higgs boson

Nuclear Physics 0 North-Holland

B106 (1976) 292-340 Publishing Company

A PHENOMENOLOGICAL

PROFILE

John ELLIS, Mary K. GAILLARD

OF THE HIGGS BOSON

* and D.V. NANOPOULOS

**

CERN, Geneva Received

7 November

1975

A discussion is given of the production, decay and observability of the scalar Higgs boson H expected in gauge theories of the weak and electromagnetic interactions such as the Weinberg-Salam model. After reviewing previous experimental limits on the mass of the Higgs boson, we give a speculative cosmological argument for a small mass. If its mass is similar to that of the pion, the Higgs boson may be visible in the reactions n-p + Hn or yp --t Hp near threshold. If its mass is < 300 MeV, the Higgs boson may be present in the decays of kaons with a branching ratio 0(10-T), or in the decays of one of the new particles: 3.7 + 3.1 + H with a branching ratio 0(10e4). If its mass is <4 GeV, the Higgs boson may be visible in the reaction pp --f H + X, H --f n+p-. If the Higgs boson has a mass <2m , the decays H -+ e+e- and H + y-r dominate, and the lifetime is 0(6 X 10m4 to 2 X ib-12) seconds. As thresholds for heavier particles (pions, strange particles, new particles) are crossed, decays into them become dominant, and the lifetime decreases rapidly to O(lO-*o) set for a Higgs boson of mass 10 CeV. Decay branching ratios in principle enable the quark masses to be determined.

1. Introduction Many people now believe that weak and electromagnetic interactions may be described by a unified, renormalizable, spontaneously broken gauge theory [l]. This view has not been discouraged by the advent of neutral currents, or the existence of the new narrow resonances [2]. These latter may well be a manifestation of some form of “charm”, a new hadronic degree of freedom [3] favoured by constructors of weak and electromagnetic interaction models. A comprehensive discussion of the phenomenology of conventional charm has been given by Gaillard, Lee and Rosner [4] At the time of writing, the discovery of charm has not been confirmed, but gauge theorists are not yet discouraged. Other particles have been suggested by gauge theorists, including heavy leptons [5], Higgs bosons [6] and intermediate vector bosons. Experimental searches for heavy leptons M+ coupled to muon neutrinos have ruled out [7] masses below 8 GeV. From

l

* And Laboratoire de Physique Theorique et Particules Elementaires, * Address after 1 January 1976: Ecole Normale Superieure, Paris.

292

associe au CNRS, Orsay.

J. Ellis et al. / Riggs boson

293

e+ee collisions, the lower mass limit for charged heavy leptons is about 1% GeV [S], but events are now [9] being seen at storage rings which are consistent with the production and decays of heavy leptons with masses O(2) GeV. The situation with regard to Higgs bosons is unsatisfactory. First it should be stressed that they may well not exist. Higgs bosons are introduced to give intermediate vector bosons masses through spontaneous symmetry breaking. However, this symmetry breaking could be achieved dynamically [lo] without elementary Higgs bosons. Thus the confirmation or exclusion of their existence would be an important constraint on gauge theory model building. Unfortunately, no way is known to calculate the mass of a Higgs boson, at least in the context of the popular Weinberg-Salam [ 111 model, and experimental lower limits [ 1Z-14] on its mass are around 15 MeV, piffling compared with the intermediate vector boson masses expected to be O(50 to 100) GeV. Furthermore, indirect evidence of the Higgs boson’s existence (g - 2 [15], muonic atoms [ 15,161, etc.) is notoriously difficult to find. In this paper we do suggest one speculative argument why the Higgs mass may be small, based on considerations [ 17, 181 about the cosmological term in Einstein’s equations for theories with spontaneous symmetry breaking. Most of this paper is phenomenological, however, and we discuss ways of looking directly for the Higgs boson, pushing the experimental lower mass limit up to a few hundred MeV or a few GeV. We assume couplings of the Higgs boson to leptons and hadrons like those found in gauge theories including the SU(2) X U(1) WeinbergSalam model [ 111. We are motivated to make this assumption by the consistency of the Weinberg-Salam model with present experimental data [ 191. Then the Higgs couplings to fundamental fermions (leptons, quarks) are proportional to their masses, and amplitudes for processes involving the Higgs boson can be shown proportional to matrix elements of the trace 0: of the energy-momentum tensor. This enables phenomenological estimates using the ideas of broken scale invariance [20]. First we consider direct production of the Higgs boson. If its mass is O(m,) then we find that in the reaction n-p + Hn close to threshold,

!?(,-p + Hn)

z’(~‘-~) s

IpR

(1.1)

I2

As a light Higgs boson often decays to e+e- it therefore might be visible in experiments [21] designed to look for n-p -+ e+e-n close to threshold. Another possibility is yp + Hp below or close to the pion production threshold. At higher energies, we find that under favourable circumstances a Higgs boson with mass >2m, might be visible as a small peak above the continuum in the reaction pp + (p’p-)+X. On the other hand, direct production in electron-positron collisions is found very small, a,,,k(e+ee .

+ Higgs boson)

lr\-7 I” =-

op,,k(e+e-

--f 3.1 resonance)

mH

(mH in GeV)

.

(1.2)

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J. Ellis et al. / Higgs boson

Another possible way of producing Higgs bosons is by bremsstrahlung in processes involving massive particles such as an intermediate vector boson. Production by bremsstrahlung along with one of the new narrow resonances (e.g., e+e- --f 3.1 + H) does not seem to be large. However, one of the best ways of looking for the Higgs boson if it has a mass GO0 MeV may be in decays of the 3.7 resonance: then we can estimate r(3.7 j 3.1 + H), r(3.7 --f all)

0(10_4)

(1.3)

The branching ratio is relatively large because competing decay modes are suppressed by the Zweig [22] mechanism. The Higgs boson may also show up in K decays if it has a mass <300 MeV: r(K+ * rr+ + H)

= 0(10-q,

(1.4)

T’(K+ + all)

which is not far bplow limits from present K decay data [23]. On the other hand, the Higgs boson seems unlikely to turn up in the decays of other particles: for example we find

r(q+ 7r"t H) r(7j-+all)

= O(lO-8)

New particle

threshold

Higgs Boson Moss (MeV )

Fig. 1. Branching ratios of the Higgs boson culated from the decay rates of sect. 4.

for different

values ot its mass. The curves are cal-

J. Ellis et al. / Higgs boson

T

295

(set)

)I

1

1

e’ e-

Threshold

Thresholds

K’K-

ld

New particle Threshold

Thresold

Fig. 2. The total lifetime of the Higgs boson culated from the decay rates of sect. 4.

for different

values of its mass. The curves are cal-

for mH <400 MeV, which is way below present experimental limits. We also discuss the decays of Higgs bosons and their experimental signatures. If 2m, < mH < 2m, then the dominant decays are H -+ e+e- and yy. The yy branching ratio is quite large, thanks to contributions to the amplitude from hadrons and from a virtual intermediate vector boson loop. The e+e- and yy rates are generally comparable in order of magnitude in this mass range (for details see fig. 1). For 2m, < mH < 2m,, the j_~+p- mode is dominant, while rrrr takes over for 2m, < mH < 2m,. -1000 MeV decays into strange particles are expected to dominate For mH>2mK until1 the threshold of 4 GeV for new particles (hadrons, heavy leptons?) is reached, whereafter they dominate the decays. This is because decay amplitudes are directly proportional to the masses of fundamental fermion fields. Thus branching ratios to p+p- and different types of hadrons in principle enable the quark masses to be measured. For Higgs boson masses 54 GeV the y+,~ decay mode is expected to be 0(1 to 30)%, and may be the least invisible experimental signal. The main uncertainties in estimating decay rates and the total Higgs boson lifetime arise from our uncertainty as to the quark masses [24,25]. The calculated Higgs lifetime is plotted in fig. 2, and we see that it varies between 6 X 10m4 and 2 X lo-l2 set for 2m, <

J. Ellis et al. / Higgs boson

296

mH < 2m,, so that the Higgs boson could leave a discernible track. For mH > 2m, the lifetime decreases rapidly to 0(10-20)sec for mH -10 GeV. The organization of the paper is as follows. Sect. 2 discusses the role of the H&s boson in gauge theories of weak and electromagnetic interactions, reviews present experimental limits on its mass, and discusses the cosmological connection. Sect. 3 contains calculations of Higgs boson yields in various production and decay processes, while sect. 4 considers its decay modes, rates and branching ratios. Sect. 5 discusses the conclusions and prospects for Higgs boson phenomenology.

2. Couplings and mass of the Higgs boson 2.1. The Weinberg-Salam model Gauge theories of the weak and electromagnetic interactions generally start off with a collection of massless vector fields Wi, and massless spinor fields 4, e, 1-1,etc. for hadrons and leptons, arranged in a symmetric Lagrangian. The physical hadrons, leptons and presumably intermediate vector bosons have masses generated by breaking the gauge symmetry. This may be done either dynamically [lo] or by introducing a new scalar field and breaking the symmetry spontaneously - the Higgs [6] mechanism. Consider for example the simple SU(2) X U(1) gauge model of Weinberg and Salam [ 111. This starts with a triplet (W +, W O, W-) and a singlet Y o of zero-mass gauge fields, to which is added a doublet H= (H+, Ho) o f scalar Higgs fields. In the spontaneous symmetry breakdown Ho = &(u + H + i@ where v = (OlHulO), the vacuum expectation of Ho, arises from the minimum of the potential Ir(H)=~2(H+.H)+x(~+.H)2~2o).

(2.1)

The field H+ and its complex conjugate H- are absorbed to give IV+ and W- their physical masses, gis absorbed to give one linear combination of Wo and Y” a mass, the neutral intermediate vector boson Z”. The other linear combination of W” and Yo stays massless (the photon) and we are left with one neutral scalar field H representing a physical particle, the Higgs [6] boson. The magnitude of u reflects the acquired mass of the W’ bosons:

(2.2)

V2=&,

with rn$ =ig2v2

,

(2.3)

for the W boson mass, g being the non-Abelian weak SU(2) coupling constant, G, = 1.02 X 1O-5/mz, and the W+W- H coupling is specified, gw+-

H

=

2m$lv.

(2.4)

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J. Ellis et al. / Higgs boson

Correspondingly

(2.5) where Bw is the Weinberg angle, and g,o zo H = 2rnglv

(2.6)

for the neutral weak intermediate (2.1) are related to u,

boson Z”. The parameters

of the potential

xv2 = -j.l . The potential down,

V(H)

(2.7)

acquires a non-zero vacuum expectation

(01VI01 = $&J2

value after spontaneous

)

break-

cw

and the Higgs boson mass is given by rni = -2~~

.

After spontaneous breakdown fare g&g (u + H, >

(2.9) the couplings of the Higgs field to fundamental

which gives Higgs-fermion-antifermion masses mf, gffH

=

fermions

(2.10) couplings qfH proportional

to the fermion

mflv .

In all the above equations (2.1) to (2.11) there is one unknown equivalently the Higgs boson mass mH.

(2.11) parameter:

P or

2.2. Ambiguities

The model described above is the simplest version of the Weinberg-Salam model. As soon as we consider more complicated versions of this model, or other models of weak and electromagnetic interactions, then considerable ambiguities arise in the Higgs boson couplings. For example: (i) Even in the context of the Weinberg-Salam [ 1l] model we can choose to have several Higgs fields Hi belonging to several multiplets i with weak isospins Zi. Then if the uncharged member HP of each multiplet has as its third component of isospin Isl

J. Ellis et al. / Higgs boson

298

and acquires a vacuum expectation

C uf(Zf + Ii

rn$ = fg2

mi =-

i

g2 &;I&. i

cos20w

value (OlH,olO)= Ui we find

- I$) ,

(2.12)

(2.13)

After the spontaneous breakdown of the theory in general there remain physical charged Higgs bosons as well as neutral ones Hi. The couplings (2.4) (2.6) and (2.11) are then shared out between the physical neutral Higgs particles:

Ct7.p ZO i

Hi

Vi = 2m2, ,

Es-T,,, ui = mf. i

(2.14)

There is no reason except optimism to believe that the couplings (2.4), (2.6) and (2.11) would be even qualitatively correct for any of the physical neutral Higgs bosons in such a theory. In this paper we are optimistic, and assume either that Nature chooses the simple spontaneous breaking via a Higgs doublet as done by Weinberg and Salam [l l] and discussed in subsect. 2.1, or, if not, that the couplings in Nature are qualitatively similar to this simplest possibility. (ii) If weak and electromagnetic interactions are not unified via the Weinberg-Salam model or some extension thereof, then the Higgs couplings may be very different from those assumed above. A statistical sampling [26] of such models shows for example that in theories with heavy leptons the Higgs coupling to electrons may be governed by a “heavy electron” mass. In this case most of our remarks about the Higgs boson in situations involving electrons would be invalid. Some recent experimental data [9] may be connected with a heavy lepton, but this is as yet unclear, and the WeinbergSalam model can also incorporate sequential heavy leptons without affecting our conclusions. We remain buoyed by the general consistency of the Weinberg-Salam model with experimental data [19]. (iii) It may be that the spontaneous breakdown responsible for the intermediate vector boson masses does not arise from the introduction of a fundamental Higgs field, but from a dynamical mechanism [lo]. Indeed there are theorems [27] that if,

299.

J. Ellis et al. / Higgs boson

for example, the Weinberg-Salam model is embedded in a model with a higher simple group symmetry broken by the Higgs mechanism down to SU(2) X U(l), then this residual symmetry cannot also be broken by the Higgs mechanism. This and other aesthetic reasons including economy lead some people to prefer dynamical symmetry breaking. Unfortunately no calculable model exists, and the couplings of any physical composite Higgs fields in such a situation are unknown. If experiments do not find a Higgs boson of the type discussed here, dynamical symmetry breaking may be more attractive. 2.3. Restrictions

on the Higgs boson mass

If we accept the simplest model discussed in subsect 2.1, what theoretical and phenomenological arguments restrict the Higgs mass m,? Jackiw and Weinberg [ 151 considered the effect of the Higgs on calculations of the muon magnetic moment. They found that for mH sO(mp)

(2.15) comparable with the effects of virtual W’ and Z0 exchanges, and impossible to disentangle from hadronic contributions in standard QED. If mH 9 m,, then (AgJH

electran scattering xcluded

by neutron -nucleus

scattering

r 0+-o’

transitions

essible in n-p-Hn

accessible

In

at low

K-n+H

energies?

decay?

sible in 3.7- 3.1.H decay?

experiment ??

Higgs

Boson

Mass

(MeV)

Fig. 3. Present and possible future limits on the Higgs boson mass.

300

J. Ellis et al. / Higgs boson

is still smaller, so no limit on mH from the muon magnetic moment. Several authors [12,15,16] have wondered whether the Higgs boson could be responsible for apparent discrepancies in X rays from muonic atoms. With the couplings of subsect. 2.1 this could be arranged with mH 5 O(20) MeV. However, the discrepancies now seem to be disappearing [28], and such a Higgs could have difficulties with data on the tine structure of muonic helium [29]. Other restrictions on the existence of a low-mass Higgs boson include the validity of the usual gravitational coupling which excludes [ 121 a long-range scalar force and entails mH > lop4 eV. More stringently, data on neutron-electron scattering require [ 161 mH > 0.7 MeV. Following a suggestion by Sundaresan and Watson [ 121, KohIer et al. [ 131 searched unsuccessfully in Of -+ O+ nuclear transitions for Higgs production and subsequent decay into e+e-. The absence of a signal excluded 2m, < mH < 18 MeV. Another argument from nuclear physics is given by Barbieri and Ericson [ 141, who argue that a low-mass Higgs is inconsistent with angular distribution measurements in low-energy neutron-nucleus scattering. They conclude that almost certainly mH > 5 MeV and very probably >13 MeV. Present experimental constraints on mH are tabulated in fig. 3. Thus phenomenological arguments from macroscopic, atomic and nuclear physics require mH > O(15) MeV. The main purpose of this paper will be to study how this limit could be improved using the mass scales of high energy physics. But first we would like to mention one theoretical reason that may motivate a low value of mH. 2.4. Cosmology and the Higgs boson mass A connection between the Higgs boson and general relativity and cosmology has been made independently by Linde, Dreitlein and Veltman [ 171. They point out that a constant term C in the world’s Lorentz invariant Lagrangian f? gives a term CG in the generally covariant I-agrangian, where g = det(g,,) and gPv is the metric tensor. Such a term in turn introduces a cosmological term into the general relativistic equations of motion R /.lu -‘Rg 2

pv

+‘K2cg 2

w

= K2

(2.16)

%,,*,

where %,, is the energy momentum tensor, R,, is the Riemann tensor, and K z 5.8 X 1O-22 MeV-1 is the gravitational coupling. Cosmological considerations limit u2C< 10es7 cm2, which implies (2.17)

C < 0.23 X. 1O-35 MeV4 . In a spontaneously

broken gauge theory, eqs. (2.8) and (2.9) give (2.18)

and using eq. (2.2) and the value of G,, the cosmological

constraint

(2.17) implies

J. Ellis et al. / Higgs boson

301

mH < 2.35 X 1O-27 MeV if C= C,. There are at least three possible attitudes to this result. (i) Accept it, and take it as an argument against the simple Higgs mechanism of spontaneous symmetry breaking. One might then seek either a more complicated Higgs structure for the theory [ 181, or perhaps a dynamical symmetry breaking mechanism [lo] instead. (ii) Circumvent the result by supposing that the world’s Lagrangian P had a constant term N - tO]v]O> before spontaneous symmetry breakdown, which was then essentially cancelled by the Higgs term C, = (01ll0) to produce a net C E 0. Perhaps the gauge theory’s spontaneous breakdown was cosmologically necessary? (in) Observe that other dynamical terms in 8 can have non-zero vacuum expectation values, and suggest that an interplay between them is necessary to ensure C = (01 L? 10) N 0. In this case the vanishing of the cosmological term would not be the responsibility of weak and electromagnetic interactions alone, or the result of a fiat, but a consistency requirement to be imposed on the ensemble of the world’s interactions. Such extra terms in (OlL?]O) will arise from any other spontaneous symmetry breaking in 2, an example being chiral symmetry breaking in the strong interactions. If the strong interaction part of L? is split into chiral W(N) X SU(N) symmetric and breaking pieces, (2.19)

L?strong = Psym+ P break )

then (01L?bEaklO) f 0 in general [30]. For instance, in an SU(3) X SU(3) theory, assuming pole dominance for the axial current divergences apAtzl+i2

apAti=4+i5,6+i7

’

with 71and K quantum (oI,f.?break~o)z

numbers

gives [20,30]

-$(?!mg +mz)fz ,

(2.20)

where (OIa,A;,,In,

K)

=fp$ fpi .

There is also no reason why (01.@)sym10 > # 0, and this happens in spontaneously broken scale invariance [20] models. This observation suggests another speculative argument. on mH. Let us suppose that, in order to ensure C N 0 we must impose l

‘O]_&,,,,,10) + (O]&b,,,]O> + (01UO) = 0 ;

l

(2.21)

This would not be true if there were yet another spontaneous symmetry breakdown, as for example if there was a higher symmetry broken down to SU(2) X U(1) by the Higgs mechanism.

J. Ellis et al. / Higgs boson

302

We do not know how to calculate (Ol.c’syr,,IO), so we assume it is similar in order of magnitude to the other strong interaction piece (1pbreaklO), (Ol-@sv, IO)= 0wQ,,,10))

.

(2.22)

To estimate ~Ol_@,,,,lO) we naively extend eq. (2.20) to SU(4) [3,4], assuming pole dominance for the axial divergences a,jb and a,/$ coupled to D and F mesons by

and taking fD, fF: = O(f,).

In this case

m;>.

(01J&*&IO) = o(-,Lf;

Substituting (2.22) and (2.23) into the consistency eq. (2.8) for (01VlO) we get

(2.23) condition

(2.21), and using

(2.24) Using (2.2) and (2.9) this yields (2.25) With f, 21 m, and assuming mD e 2 GeV this formula gives mH = O(2) MeV .

(2.26)

Several remarks about this speculative estimate are in order: (a) Personally, we find this argument interesting rather than persuasive. (b) The argument needs (01,@s,_, 10) # 0 in an essential way, for it is evident from (2.8) and (2.20) that (O(IlO) and (OIL?t,,,k (0) are both < 0. This is expected to be a general feature of spontaneous breakdown. Indeed it is a theorem [31] that if a field theory has two possible vacua, it will always choose the lower one, whatever the potential barrier between them. Hence, either this theorem must be circumvented, or the strong and weak spontaneous breakdowns are not independent, and the asymmetric vacuum must have the same level as the symmetric one. We have been unable to construct a theory which has such a property, respects theorems [32,33] ensuring that parity and hadronic symmetry breakdown occur only in O(a), and involves fundamental quark fields. If the vacuum theorem [31] were not circumvented, and the strong and weak spontaneous breakdowns were independent, then the cosmological argument would require both the strong and weak breakdowns separately to leave
J. Ellis et al. / Higgs boson

303

tion in e+e- experiments, they would need masses >8 GeV and the associated “supercharmed” pseudoscalar analogues of the conjectured D and F mesons would probably have masses 25 GeV. The estimate (2.26) is then modified to mtt >0(7) MeV. This is just one example of the sensitivity of (2.26) to the presence of additional structure in the world Lagrangian. Regardless of the above arguments, a low mass for the Higgs boson is no less likely than a high mass, and it is worth while trying systematically to improve the lowermass limits from nuclear physics. Ways of doing this are studied in sect. 3.

3. Production

of the Higgs boson

In this section we discuss how the Higgs boson might be produced in various reactions, mainly involving hadrons. Typical hadronic mass scales >O (few hundred) MeV can be used either to exclude possible mass ranges for the Higgs, or perhaps to see it. We start by proving a theorem about Higgs couplings, relating them to matrix elements of the energy momentum tensor, and then go on to consider production both directly and via decays of other particles. 3.1. The Higgs boson as a dilaton (201 In the discussion of subsect 2.1 it was apparent that in the simplest version of the Weinberg-Salam model the couplings of the Higgs boson to fundamental fermion and boson fields via their masses (2.4), (2.6), (2.11) mean that if the renormalization due to interactions is ignored, the Higgs boson couples to other particles via the trace of the energy momentum tensor

lPumen. =

Ir

Tm,ff+ 2&W+

W- t 2miZo2

.

(3.1)

This observation can be extended in the presence of interactions to give the following theorem: if A and B are states not containing Higgs bosons, then to the lowest relevant order in the semi-weak coupling constant g, the amplitude for A + B t H is given by (3.2) where 0: is the trace of the renormalized improved [35] energy momentum tensor. To prove (3.2), consider first the Weinberg-Salam model, including strong interactions via a quark-gluon coupling, in the unitary gauge [36] where no unphysical Higgs bosons appear. The Lagrangian is of the form _rZ”= L”(Htu)-

V(H),

(3.3)

304

J. Ellis et al. / Higgs boson

where +_!-H4

v(H)=$mi

49

1 ,

(3.4)

and the vacuum expectation value v is the only dimensional parameter appearing in f?. Since v has the dimension of mass, the trace of the improved energy momentum tensor [35] is determined by 84

f (co - 4)_P”

= _v

a-@(H+ v) -(Q-4)1/ &I

=-vE$to(H),

(3.5)

where (D is an operator which effectively multiplies each term in L?” by its dimension, The linear couplings of the Higgs boson are determined by

_e;

=Hg(Htv)IHzO

+ 0(H2)

=+$)to(@).

(3.6)

Comparing (3.5) and (3.6) we see we can write the part of L?” containing fields in the form _@; = -;H+

t O(H2) .

Higgs

(3.7)

Then, for processes in which higher order couplings in H can be neglected, the general amplitude for A -+ B + H is given by (3.2). This result has been obtained by formal manipulations in the unitary gauge. In the general renormalizable gauge [37] Rc, the gauge conditions have the form a,Wp

t i(mw/.$)H=O,

(3.8)

where the H are unphysical Higgs scalars. Condition (3.8) is not scale invariant, and the definition of 0: is not straightforward. However, if we choose the “R gauge” defined by g + *, the gauge condition a, W p = 0 is invariant and we may proceed as before. One finds that eq. (3.8) is replaced by (3.9)

30.5

J. Ellis et al. / Higgs boson

where

and if P R is the complete Lagrangian in the R gauge, +

= (co - 4)PR

(3.11)

.

Now consider a transition amplitude is given by

between quarks 4, -+ qP + ZZ. According to eq. (3.7) the

(3.12) Using the properties (a) that BP” is a symmetric, divergenceless tensor; (b) that the space-integrated operator Jd3x00p E P” is the total four-momentum; (c) that in the limit of vanishing quark mass m, -+0, only the left-handed component of 4, is coupled; (d) CP invariance and crossing symmetry; the matrix element (3.12) can be shown to take the form

(q8 IO:(k) 14,) = rnz 6,@ + k2A(m,R

+ mpL)

+ (m2oL- m,$ [B(m,L - mpR) + m,mpC(m,R

(3.13)

- mpL)] ,

-rs)andR=i(l tY5),andA,BandCarefunctions where k = p, -pp,L=i(l of the momenta, arbitrary except that they contain no singularities for k -+0. The matrix element of 0: is not in itself a physical observable and may be gauge dependent, except in the limit k -+0, where it is determined by condition (b) above. However, the matrix element for 4, + qp + H with all particles on the mass shell must be gauge invariant. Therefore the contribution of 6 _QH (3.10) in eq. (3.9) must be of the form of (3.13) without the zero momentum transfer term 6,pmz. Of the amplitudes we will consider, weak and electromagnetic couplings are significant only for those with IAZI = 1 and [ASI = 1. Then the diagrams which can contribute are those of fig. 4. We have found by explicit calculation of their contribution to the quark amplitudes * (UlMU) - (dlHld)

(dlHb) * Where necessary, and n.

(AZ= I), (AS = 1))

we denote

the usual quarks

by u, d and s, and the proton

and neutron

by p

306

J. Ellis et al. 1 H&s

Fig. 4. Contributions

of 6H (3.10)

boson

to eq. (3.13).

that the contribution of 6 L? H is of the required form to lowest coupling and in m*/m* . We therefore conclude that the formal the unitary gauge ?s va;d. We will later use the properties (a) to theorem (3.2) to estimate various physical amplitudes, both for sequent parts of this section) and for decays (in sect. 4). 3.2. Production

order in the weak result obtained in (d) above, and the production (in sub-

in hadronic collisions

In this section we discuss production of H in various hadronic reactions (rrp, pp collisions) and photoproduction in several kinematic conditions. 3.2.1. Low energies We first consider the possibility that m H = O(m,), in the range 15 MeV
@pH t iinH]

(3.14)

+ 2PZ, - 2mz)n 1 * 7r2H ,

where we have denoted mp N m, f M. The Feynman diagrams relevant to n-p + Hn are shown in fig. 5. The contributions of these graphs to the differential cross section are found to be (3.15)

* The couplings to p and n follow from the dilat,on theorem the form of the Hnn coupling is discussed in subsect. 4.2.

of subsect.

3.1 while motivation

for

J. Ellis et al. / Higgs boson

(a.)

t-channel

lb)

Fig. 5. Diagrams

s - channel

contributing

where G is the usual j%nr+ coupling constant

to the process

307

Cc) u-channel

~‘p

+ Hn.

and

(3.16)

l9w

=.

M2m2n

(3.16b)

(s - lb@)2 M2m2n (u -M2)2

(3.16~) ’

for diagrams Sa, b and c, respectively. We do not believe the precise numerical value of the cross section given by (3.15) and (3.16), but we do believe it is likely to be of the right order of magnitude. In each of the cases (3.16a, b, c) 19;17]2 = O(m;/M2). Substituting

(3.17)

this into (3.15) and using

$=fiG’ and the known values of G and f,, we find

(3.18)

This value may seem very small, but there is a proposal [21,38] to look at the reaction n-p + rr”n near threshold with sufficient sensitivity to see the decay no + e+eif it has a branching ratio O(lO-*). This experiment then has a fair chance of seeing n-p -+ Hn, H + e+e- if mH 5 O(m,).

J. Ellis et al. / Higgs boson

3083

Another possibility

!?(yp’Hp)”

is to study yp + Hp *. In this case

fi _%?i-

(3.19)

7rs’Gm.‘2 ’

which is very small. Nevertheless the availability of high-intensity low-energy photon beams make the reaction accessible, especially if we recall that the reaction could be studied below rr production threshold to reduce background and look for a Higgs boson less massive than the pion. 3.2.2. Intermediate energies We now go to higher energies, say of the order of a few GeV. One candidate reaction is again n-p -+ Hn, where one looks for production via one-pion exchange in the forward direction. If we now compare (3.15) and (3.16a) with the corresponding formula for n-p + pan via one-pion exchange **, we find Hn)

$(r-p

+

$(rr-p

+p”n)

(rni + 2t)2&GFmz N [t - (mp - mn)21 [t - (mp f mT)2]g&n

(3.20) ’

so that if rnh = O(mz) and we consider It1 = O(mz) then (da/dt)

(n-p

-+ Hn) _ “?l

(do/dt) (n- p + PO,)

--x&f;>

if we assume the KSFR relation gPRrr N (doldt) (n-P

+ Hn) =

(do/dt) (n-p

+P”n)

(3.21-

m4p

trzp/fn. Hence we again get

o(G,f;)= o(10-7).

(3.22)

This process therefore seems to have an unappealingly low cross section, but for future reference we point out that (GF fz) seems to be a typical ratio for Higgs boson production relative to the dominant hadronic reactions in similar kinematical conditions. 3.2.3. High energies We now consider H production in higher-energy reactions, typically inclusive reactions at the higher CERN PS/BNL, Fermilab, ISR and CERN SPS energies. First we consider pp + H t X in a case where mH = O(m,). In this case the lowest-order

We thank T. Ericson for suggesting * Absorptive corrections to one-pion

l l

to us this reaction. exchange may well cancel out in the ratio of cross sections.

309

J. Ellis et al. / Higgs boson

contribution

to the matrix element is expected from the theorem (3.2) to be (3.23)

In this mass range matrix elements of 0: may 1201 be dominated by the e(700?) meson. If so, then previous analyses [20] suggest (0 10: Id = fern: with f, = OC&). We would therefore have (3.24) and thence (3.25) This ratio accords with the production ratio (3.22) obtained previously by different methods in a different kinematic region, so we may be prepared to believe it independent of the somewhat questionable derivation. If we guess a(pp + e t X) = 0(1 to 10) mb at ISR energies by analogy with o(pp -+ p + X) then we find o(pp --f H + X) = O(O.l to 1) nb. One way of expressing this result is to observe that since the branching ratio for p + p’+/.- is O(5 X 10e5) an d we have estimated (o(pp + H + X)/ u(pp + p + X)) = 0(1 O- 7, then the Higgs meson is 500 X (I’(H + all)/r(H --f p+/.-)) more difficult to see than the p via its (/.L+P-) decay mode. Let us now consider pp -+ H + X in the case where mH -> O(1) GeV, and we may use Drell-Yan [39] scaling arguments to estimate the production cross section. Assuming that the standard quark parton-antiparton annihilation graphs of fig. 6 dominate the production of H and of the (p+/._-) continuum background, we find

‘?

rnit

NPP + H + X) O(PP + @‘cl-)

p2Cmil H

+ X) =

ufe+eu(e+e-

-+ H)

(3.26)

+ j~+j.-) ’

In deriving (3.26) we have assumed the form (mq/u)@Hfor the couplings of H to quark partons with charges Q,, have grouped partons into “light” (L = u, d, s) and “heavy” (H = c, c’ ...?). and have assumed production at rest in the centre of mass

Fig. 6. Drell-Yan

contributions

to

pp + H + X and pp --L(M+P-) + X.

J. Ellis et al. / Higgs boson

310

from “sea” partons and anti-partons %I =uu =IJ d=U; % =

UC

=

UC,

=uu,=uc

= UC’

=

with distributions j

UC/U,=

uq such that

p

.

(3.27)

. ..

i

If we now use o(e’e-

-f H) = ‘, rr& G, m,26(q2 - mi),

o(e+e-

--f n+1_1-) = 547~ cx2/q2 ,

and integrate numerator we find

NPP

and denominator

--, JN-tcl+~)

+ X>

O(PP j @‘cl-)

in (3.26) over a range AQ in Q z @,

then

=

+ X)

Then if an experimental (n’p-) pair spectrum can be binned in mass ranges AQ, eq. (3.28) will enable one to estimate the Higgs boson signal to background ratio. The numerical value of (3.28) is very sensitive to the value assumed for the parameter p, because mH 3 mL. If we suppose p = 0 (an SU(3) symmetric “sea”] or that there is one new quark c with m, z 2 GeV, and p = 1 (SU(4) symmetric “sea”] then we find O(PP + H(+

I-‘+/.-)

O(PP * @‘cl-

I+

+ X> N IV x>

--f /J+CC-1

10e3 SU(3) case (3.29) lo- l SU(4) case ’

The sophistication of present experiments is clearly insufficient to see H --, c(+p- if formula (3.29) is accepted. However, it is not impossible that if the SU(4) (or still better SU(6) or higher) symmetry estimate is correct, then the Higgs boson may show up * in future pp -f @‘cl-) + X experiments at least as long as mH < 4 GeV so that, as will be discussed in sect. 4, the H --f $1-1~ branching ratio may not be completely negligible.

* We will see in sect. 4 that the branching ratio for H -fe + e - is expected to be minute for mH > %I,, so that a pp --t (e+e-) + X experiment should not be vulnerable to the Higgs boson.

311

J. Ellis et al. / Higgs boson

3.3. Production via decays of other particles Since the Higgs coupling is semi-weak, one might a priori expect branching ratios into decay modes involving the Higgs boson to be O(1O-5). Such rates might not be inaccessible, for example in 17decay where 77+ y+p- has been observed with a branching ratio of lo- 5, and particularly in K decay where branching ratios of 0(10M7) (K+ + n+e’e-) and O(10d8) (KL +/J+/.-) have been observed. Unfortunately the dilaton role (3.2) of the Higgs boson has the effect of suppressing these decay modes more strongly. We shall see that the decay K+ + rr+ t H might be accessible to experiment, but that the most favourable source of low-mass Higgs particles may be the newly discovered [2] heavy vector mesons, where the competing modes are suppressed by Zweig’s rule [22]. 3.3. I. AI = I transitions: 17and C decays There are a priori two ways in which the decays E”+A+H,

q-tnO+H,

(3.30)

might proceed. First, suppose [40] there is a tadpole term e3u3 in the chiral symmetry breaking part pbreak (2.19) of the strong Lagrangian, corresponding in the quark model to a u - d quark mass difference. Then the conventional scheme (2.11) for coupling the Higgs to hadrons is _e” = _Pbre&

(v+H> V

which gives a coupling e3u3H/v. There is then a contribution with Higgs bosons of the form i(finall

_Qbreakl initial) .

(3.31) to 77and 20 decays

(3.32)

The other contributions to 77and Co decays come from higher order weak and electromagnetic effects. However, it is clear that the contributions of (3.32) to TJ-+ no t H and E”+At H in fact vanish. This is because the matrix elements of Pbreak between physical states which are eigenstates of the strong Hamiltonian must vanish. Consider, for example, a phenomenological Lagrangian model for L? break, including a u3 term as in the work of Osborne and Wallace [41]. Introducing an octet Pi of pseudoscalar fields, L?break receives contributions (3 33) where the Pz and Pi terms come from the SU(3) and SU(3) X SU(3) breaking terms L40and cu8 in -@break, and the P3P8 term comes from the conjectured [40,41] tadpole I spin violating term e3u3. The mass terms (3.33) must then be diagonalized to get

J. Ellis et al. / Higgs boson

312

Fig. 7. The effective

AI = 1 quark

operator,

the physical 7~0and n masses, 2 -i m,,0 rr02 -+lf~2.

(3.34)

The Higgs couplings (3.31) being proportional to (3.33) will also be proportional to (3.34) and hence diagonalized away. Thus the e3u3 contribution to n + no t H vanishes, and a similar argument applies to X0 + A + H. For the other, higher-order weak and electromagnetic, contributions to these decays, we will invoke the more general dilaton-like nature (3.2) of the Higgs boson. The effective AI = 1 quark operator which can induce the decays (3.30) is represented diagrammatically in fig. 7. By virtue of eqs. (3.2) and (3.13) this operator may be written in the form .Q,,(AI=

1)=f$j+3qH(m,-

md+amimq),

W

q=

u

0

d

’

(3.35)

The term proportional to m, - md is just the u3 term discussed above. The remaining term represents the radiative corrections and must be O(a), so that we can neglect the (U - d) mass difference, as was done in writing this part of (3.35). The parameter a has dimension [masslP2, and allowing for strong interaction effects a reasonable guess is (3.36)

where m, = O(700) MeV. Then we obtain for TJ+ rru + H, for example, (nO,fln~‘_%%

m2H 2 (7r”Imq4~r3417)) 2mw m,

313

J. Ellis et al. / Higgs boson

(3.37)

yielding a branching _

R

r(v -+

ratio

no+H) = O(lO-8)

r(rj --f all)

II

(3.38)

for rni= m2. However, as photon exchange is not a characteristically short-range phenomenon: the relevance of the quark operator (3.35) is not obvious. More generally, the requirements (a) to (d) of subsect. 3.2 allow a contribution of the form

(3.39)

yielding (3.40)

R, =O(S X 10-7)

for rn&= mz. The somewhat different estimates (3.38) and (3.40) bracket our expectations, indicating a small branching ratio for 1) + rr + H. For Z” + A + H the branching ratio is even lower since the dominant amplitude Co + A t y is only first order in e. Under assumptions analogous to those leading to (3.38) and (3.40) we estimate

Rx0 =

r(ZO+q+H) r(C”

< 10-g )

(3.41)

--f all)

where we have taken [42] rce = (0.6 + 0.3) X lo-l9 sec. The amplitude for Co + A + H + y is of lower order in e, and has a l/k dependence in the H momentum, but the decay is inhibited by three-body phase space. From the diagrams of fig. 8 we find the model-independent result r(cO

--f A +

H + 7)

r(I;lO --fA + y)

“2.6X

IO-‘p(:;y-J,

Fig. 8. Diagrams for X0 + A + H + y.

(3.42)

314

J. Ellis et al. / Higgs boson

Fig. 9. Contributions

to the

one-body I ASI = 1 quark operator.

where + 3x - :x2 +$x3 .

F(x) = -In x -a

For a Higgs boson mass in the range mH = 20 - 60 MeV we find a branching the range (6 X 10dg to 3 X 10-ll).

ratio in

3.3.2. (ASI = 1 transitions: K decay The AQ = 0, laSl= 1 decay K+ + n+e+e- has now [23] been seen with a branching ratio O(lO@ to 10W7), comparable with predictions [43] made using gauge theories and charm [ 1,3] . Reference to fig. 1 shows * that over a range in mH < 2m, H -+ e+e- is a substantial decay mode. Hence it is interesting to use the gauge theory and charm [ 1,3] framework to estimate K+ -+ nf + H, and see whether present or future experiments on K decay can eliminate a range of mH. Unfortunately, we are again hurt by the dilaton property (3.2) of the Higgs boson. Consider first the one-body quark operator derived from the diagrams of fig. 9. Reference to eq. (3.13) tells us that the effective transition operator for s -+ d + H is of the form P.&AS

= 1) = --ig2 m, dRs H[bmz + cm;] 2m$

.

(3.43)

(For simplicity, in this section we will assume ** m, N md N 0: R was defined after eq. (3.13).) Moreover, the Glashow-Iliopoulos-Maiani mechanism [2] requires that b and c --t 0 in the limit of quark mass degeneracy (m, -+ 0). The relevant question is whether this leads to an additional strong suppression b, c = mz/rn& as for K, + p+p-

(3.44)

,

and K+ + n+S v, or whether the dependence

* Decay modes of H are discussed in more detail in sect. 4. l

* The topic of quark

masses is taken

up in sect. 4.

on the exchanged

J. Ellis et al. / Higgs boson

315

quark mass is only logarithmic as in K+ + n+e+e- [43]. To answer this question we calculated explicitly the diagrams of fig. 9 in the ‘t Hooft-Feynman gauge and found in fact that the operator (3.43) vanishes to lowest order in l/mW so that

P&AS

= 1) =

Ok3mi, H>

(3.45)

m:, Then taking the K - 71matrix element of the quark operator

m,dRs = ia,@ypLLs), (R and L were defined after eq. (3.13)) which is the divergence of the weak current, we find a contribution

to the decay amplitude

of order

(3.46) Could strong interactions modify the strong suppression (3.46)? It seems unlikely that corrections to the short-distance behaviour are important: the cancellation holds in the presence of gluon exchange as long as all diagrams are effectively cut off at the same momenta, and the situation here is similar to that for K, + p+~- *. To account for low-energy effects, one might correct the EH vertex by **

m c -+m C (1 tk2/m2)-l

EC

’

where ee is a O++(Z) bound state presumably is destroyed for k2 # 0, and we estimate (n+, HlK+)=4mwSC

of mass -3 GeV. Then the cancellation

m2 F m2,2-_-!!_ s Km2

(3.47)

5

probably with an extra factor of l/n2 from the loop integration. There will also be contributions from two-body operators as illustrated in fig. 10. The effective operator arising from the diagram of fig. 10a is simply related to the usual non-leptonic ]AS] = 1 transition operator:

&&? = -ig3 H@y,Ld) (SyGLu) t (h.c.) + 0 2m$ =&

H W

~~!-+$lkptOniC

(3.48)

* The behaviour of the 3H vertex is similar to that of the Fd$q vertex discussed by Gaillard et al. [44]. However, opinion is not unanimous, see ref. [45]. l * This is consistent only if the contributions involving this vertex are separately gauge invariant. We have not checked this.

J. Ellis et al. / Higgs boson

316

(a)

(b)

Fig. 10. Contributions

to the two-body

IASl

=

1 quark operator.

Diagrams of the type in fig. lob yield non-local operators which induce, among others, the pole diagrams of fig. 11. A simple estimate, consistent with the dilation-like coupling of the Higgs boson, is to sum the contributions of figs. 10a and 11. The result vanishes for mH + 0: defining

h(k2) = (n+(p -

k)lSC;-peptonic IK+(p))

= h 1 - $(

, K

1

where K is a conjectured scalar nK resonance with mK = 0( 1) GeV, and using broken scale invariance [20] (see sect. 4, especially eq. (4.5)) to estimate the k2 dependence of the nnH and I&H vertices, we find

-igm&h bl+,HIK+)=~--. w

1 (3.49)

m:

The parameter h is related through soft-pion theorems to the K, + 2n decay amplitude. Then using either (3.47) or (3.49) we obtain a branching ratio

r(K+ + QK+

n++ H)

~

1o-7

for rni N rnz ,

(3.50)

-+ all)

i.e., not far below the observed level 1231 for K+ -+ n+e+e-.

We should observe that

a ptioti an amplitude like (3.47) or (3.49) but with rni replaced by (rni - mi)2/mi where mh = O(1) GeV IS . a1so possible. Such an amplitude would raise the branching ratio (3.50) above the experimental [23] rate for K+ + ?r+e+e-, but we have been unable to construct a phenomenological model for such a term. A similar estimate to (3.50) would hold for K, + no + H. The process K” + 2H might be thought competitive with K” --, p+/.-, but by CP invariance contributes only to K, decay.

Fig. 11. Pole diagrams from the graphs of fig. lob.

J. Ellis et al. / Higgs boson

317

3.3.3. Heavy particle decay We conclude this section with the more optimistic possibility that the decay of one of the new [2] narrow vector resonances 3.7 + 3.1 + H may be observable. We estimate the branching ratio by comparison with the decays ~‘(1600) + p + rr + rr and 3.7 -+ 3.1 + r-r+ rr, where we assume the (7rn) system is dominated by the e(700?) meson. Zweig’s rule [22] tells us that the hadronic mass scale associated with the amplitude for 3.7 + 3.1 + H is 0(3 GeV), and since we must have mll 5 600 MeV for the process to occur, we write (3.1, H13.7) =

$ W

(3.51)

(3.11e;13.7~1,@=0t oC&)].

Now consider the decay p’ + p + rr + rr. The assumption coupling to 19; determines the p’pe coupling by (PlqyP’)

=gep’Je

of E dominance

of the p’ - p

(3.52)

,

where (010; Id = f,m$ with [20] f, = O(m,). Defining g,rP,=(vIe~I~~I,*=, we obtain r(3.7 -+ 3.1 t H) QP’

+ P + E)

where PH and pE are the momenta of the Higgs and E in the respective decays. The divergence conditions for CPU tell us that the matrix elements (3.52) must be at least quadratic in the momentum transfer k, gey~V=aV[(P+p’)~k]2(~~~‘)+bV(~~k)(~’~k)t...

(3.54)

,

where e and E’ are the polarization vectors of V and V’. Since the polarization vector e will be averaged over, the effective momentum dependence can only be through the variable (p t p’) . k = m$ - m$. So, without loss of generality we retain only the first term on the right-hand side of (3.54) getting r(3.7+3.1 r@‘+P+E)

+H)

“4.6X

,,--7;

(3.55) E

In the SU(4) limit, a3,1 = aP: but aV has dimension [mass] -*, and we are faced with the usual problem of how to parametrize in the broken symmetry case. If PH is a few hundred MeV, we find for example 2 x 10-d r(3.7 + 3.1 t H) = a few x r(3.7-+all) 0.8X 10-S

a3.1 = OP

m2, a3.1~ up (3.7 G:V)*

(3.56) ’

J. Ellis et al. / Higgs boson

318

An alternative and assuming 83.7,3.1,E

estimate is obtained

=Zg3.7,

3.1, Ec

by starting from the decay 3.7 -+ 3.1 + 71t a

(3.57)

’

where Z is a Zweig [22] suppression factor. Next we determine suming e, dominance of the 13; matrix element, getting g0, 3.7, 3.1 =fcc

g3.7,3.1,

where we have introduced

Combining

the E, coupling by as-

(3.58)

Ec ’

f,e by

analogy withf,,

(3.57) and (3.58) we can then write

r(3.7 + 3.1 + H) =-figs q3.7 +3.1 + E) 22

(3.59)

where p, is the effective E momentum, determined by the phase-space integral weighted by the resonance factor to be p, ~60 MeV. Then for pH = O(few hundred MeV) and Z2 z 10e3 we find cc

r(3.7 + 3.1 + H) r(3.7+all)

=f,2

me,fzc=mef,2.

(3.60)

The second assumption for fe, is made in analogy with the observed supression of the photon coupling to the 3.1 vector meson relative to the y - p” coupling. We could, of course, choose mass factors so as to suppress further the estimates (3.56) and (3.60). However, the important point is that the Zweig [22] rule, by suppressing the total hadronic width of the 3.7 state, is helping to overcome the effective Higgs coupling O(G$ f:)= O(10P7) so that a branching ratio for 3.7 + 3.1 + H of 0(10S4) is not unthinkable. 3.4. Production by bremsstrahlung Just as conservation of the electromagnetic current leads to low-energy theorems for photon emission, energy momentum conservation leads to low-energy theorems for Higgs boson emission. Following Low [46], we define the general amplitude * 57P”(p,, ... , pn, k) illustrated in fig. 12a, and expand in powers of k. The momenta pi are taken on their mass shells pl2 - m,2, and B,,(k) is the Fourier transform of the l

The low-energy different form.

theorem

for matrix

elements

of 0: has been given by Mack [47

J in a slightly

J. Ellis et al. / Higgs boson

improved [35] energy momentum two contributions,

m”“(pl, ....pn. k)=C + mt’(pl>

319

tensor. Then 9?i,“‘” may be expressed in terms of

7?Z(p,, ...>pi- k, ....pn)iAi(Pi-

i

k)df”(P,

(3.61)

...>P,,, k) >

where Ai is the propagator for the ith particle, dfV is the vertex function and 311 is the n-point function of fig. 12~. By definition the amplitude%f singularities as k + 0. The vertex function sQf* obeys the Ward identity [48] k,df’(pi,

k)

k) = i[Ai(pi - k)]-l

of fig. 12b, has no

Ef”} ,

{py - ik,,

(3.62)

when pf = mf, where CPU is the covariant spin matrix for the ith particle. The divergence condition for the full amplitude (3.61) then gives k,Q’P’“(P,,

=-F

.... pn, k) = 0

m(P,,

....pi-

k, .,.,Pn) {PY-ik,

~r”)ik,L)np”(P,,..,,~~k::.

For a symmetric amplitude F”(k) with no singularities, k,I’pV(k) = k,,F’p(k) = F’(k) determines P’ to O(k), F’(k)

=; [VI’“(k)

t d”F(k)

- k,NY’I’~(k)]

the divergence condition

+ 0(k2),

aP &. P

Therefore we may use eqs. (3.61) and (3.62) to determine

Ah

3?i s

&4-f’”(P, ,...,P".ct) =

&pJ” and%{“,

(q)

P,

9

(a)

dp’ (P,@= -b-L-P

If W(q) P*q

(b)

Fig. 12. Graphical

(c) definition

of quantities

appearing

in eq. (3.61).

respectively,

J. Ellis et al. /‘H&s

320

boson

to O(k). Since Ai(pi - k) * l/k, the full amplitude Expanding the n point function

+ ...

to O(kO).

(3.61) is thus determined

(

and using conservation mentum,

Cy! 1pi = k and of total angular mo-

of total four-momentum:

where Lr*zpY--

a a ’ aPi, -p’G

is the orbital angular momentum,

the amplitude

(i+1 , .... n, HJl, 2, .... 17=z

%I$P,,

for Higgs boson emission .... pn, k)

W

may be cast in the form (for spins
2 /J mik

+4-n&+

2pi.k-mi

x W(Pl, + O(k)

. ..> P,)

-2$

-(PI>

W

>

.“, P,)

c

Rmi

i,s=i 2p, . k _ mk

pPkp tc i,s=l 2pi.

k - m$

(3.64)

where nB and nf are the total numbers of bosons and fermions, respectively, and the momenta pi are defined to be flowing inwards. As in the case of photon emission, derivatives of ‘Vi? with respect to the external masses cancel out, as do unphysical amplitudes which may appear for spin > 0. The terms which are singular as k -+ 0 are just those expected for the bramsstrahlung emission of a Higgs boson from an external line with coupling proportional to 2mF(mi) for bosons (fermions). The nonpole terms depend on the dimension neff of the effective coupling constant associated

321

J. El‘lliset al. / Higgs boson

with the amplitude %? . It is readily verified that they vanish for PZ,,~~ = 4 as for effective couplings of the type

Now let us consider the leading terms in eq. (3.64) i.e., those of order l/k. If we are interested in the Higgs bremsstrahlung associated with a single particle in an inclusive reaction, p2=,2

A+B+C(p)+H(k)+X, the bremsstrahlung

c ’

cross section is m;

do,H(p, k, 0) = 4

kdkud(cos

0)

G, _

27r2(2p.

kt

da,(p) + W”)

mf$2

2

(3.65)

where do,(p)

= do(A + B --f C(p) + X) ,

and 0 is the angle between p and k. In the rest frame of C, (3.65) becomes

do,H(P, k) = 7 4

m2

G,

e2

da,(p)

(3.66)

.

0

Formula (3.66) yields, for example, bremsstrahlung at the low level of

associated with the 3.1 resonance

kdkO d&(p,

k)“--

(4 X IO@)

(3.67)

d+Jp).

k8

The ideal sources of Higgs bosons are, of course, the heavy intermediate bosons for which the Higgs couplings (2.4) (2.6) become strong. For mvv = O(100) GeV kdk,

dug@,

k) =-

(3.68)

(4 X 10-3)daw(p). ko2

The specific examples of (3.67) and (3.68) are taken up in detail for e+e- annihilation in subsect. 3.5. For the moment we make some general remarks about formulae (3.65) and (3.66). (i) For a relativistic particle with E2 S m 2, the associated Higgs emission is peaked at an energy k, = EmHIm and at an angle 0
&-G,

kdk, F (zmi 0

- F

mi)2 dU(pi cx 0)

(3.69)

322

J. Ellis et al. / Higgs boson

Thus bremsstrahlung associated with elastic scattering off a heavy nucleus for example is suppressed, as is coherent photoproduction y + Z + Z + H. (iii) On the other hand, t h e k-’m d ependent term in (3.64) presents the amusing possibility of enhancing Higgs production in multiparticle reactions. Consider, for example, pj? annihilation at rest: pp + nrr where n is odd. For sufficiently large n the final state will be in an s wave. Then m( pi) is momentum independent and (3.70) if k2max is not so large as to appreciably reduce (nrr) phase space. For k,, and n = 9 we find o(H)/0 - 10e6 : still discouragingly small. 3.5. Production in efe-

- 300 MeV

collisions

There are at least three ways the Higgs boson could be produced in efeone of them is indirectly via the production and decay of the 3.7 resonance in subsect. 3.3.3. Two other natural ways to consider are the following.

collisions: discussed

3.5.1. Direct production e+e- + H + all The situation here is far worse than in pp collisions, because of the small coupling of the Higgs boson to electrons. For e+e- + H + all [49] we have upeak(e+e-

+ H + all) =

f$f r(Hzfee) ,

(3.71)

H where AE is the beam-beam c.m. energy resolution, < AE. Then by comparison with the 3.1 resonance, upeak(e+eupeak(e+e-

+ H -+ all) + 3.1 -+ all)

=J

F(H + e+e-)

3 F(3.1 + e+e-)

and we have assumed f’(H -+ all)

m$.~

(3.72)

rnL .

Taking F(3.1 + e+e-) = 5 keV and presuming f’(H + e’e-) (see sect. 4) we get ‘peak

cH)

‘pd(3.l)

N 1.7 X lo-l3

~ 10-7

mH

(3.73)

mH ’

when mH is expressed in units of 1 GeV. Thus the prospects of seeing the Higgs boson produced directly in efe- collisions seem negligible. 3.5.2. Production by efe- + Z” + H or 3.1 + H These processes would proceed by the bremsstrahlung

mechanism

studied in sub-

J. Mis et al. / Higgs boson

323

sect 3.4, and come from the diagrams of fig. 13. We have calculated these in two ways: from first principles and using the general bremsstrahlung formula (3.66). We find that for e”e - -+ Z” + Z” + H in the Wem . b erg-Salam [l l] model [u~+~- is the pure QED u(e+e- + y + p’p-)]

(3.74)

where mH is expressed in units of GeV, and we have allowed for the effects of a finite Z” decay width F, expected to be O(1) GeV. The cross-section-ratio (3.74) has been evaluated at the optimum c.m. energy, which is &= mZo + 42 drn$ + ir2. In the Weinberg-Salam model with the experimentally favoured value of Bw,mZo ~75 GeV. A similar application of the bremsstrahlung formula (3.66) can be made to e+e- + V + V + H, where V is some narrow hadronic vector meson (e.g., the 3.1 resonance [2]). We find

$

=(0.048)1 /J fi

mC F(V -+ e+e-) mH

.

(3.75)

mP

For the 3.1 resonance with F(3.1 -+ e+e-) 2 5 keV, eq. (3.75) yields o&

_ 1.6 X 1O-3

(3.76) ’

%

“~+I.-

mH is expressed in units of MeV, and the cross section has been evaluated at the optimal energy 4s z 3.1 + &mH. There are two other ways one might imagine looking for the Higgs boson in efecollisions. One is bremsstrahlung from a heavy hadron or lepton pair L’ : e+e- + L+L-H. The basic bremsstrahlung formula (3.66) indicates this would be small for rnLk = 0(2 GeV). Another possibility is the two-photon process yy + H, where the yy pair comes either from colliding e+e- rings, or from the Primakoff effect. Anticipating the H + yy decay rate calculation of sect. 4, we find that characteristically, for mH =3m, to 3 GeV, the ratio of F(H + 7-y) to the yy decay width of a typical hadron h of similar mass, is where

W+77)

r(h+m)-

<

10-S

.

(a)

Fig. 13. The dominant bremsstrahlung

(b)

diagrams (a) for e+e-+

Z” + H, (b) for e+e--+

3.1 + H.

J. Ellis et al. / Higgs boson

324

Thus Primakoff production of a low-mass Higgs boson is barely conceivable: process looks hopeless otherwise.

the y-y

4. Decays of the Higgs boson In the simplest Weinberg-Salam [l l] scheme of sect. 2, the Higgs boson was directly responsible for giving e, 1-1and quarks their masses, and had couplings to fundamental fermions g,ff = l/u mf where v = (&! GF)-1/2. Many decays of the Higgs boson are then easy to calculate. 4.1. Decays to e+e- , p’pThese have been calculated

[ 121 as

(4.1) with T’(H + #p-)

given by replacing m, -+ m, in (4.1). Thus

r(H-+e+e-)=

1.7X lo-l3

r(H+p+~.-)=7.3X

mH

lo-‘mH

,

and the decay to pcl+p- already dominates

the decay to e+e-

(4.2)

for mH just above 2m,.

4.2. Decays to hadrons The inclusive decays to hadrons can be estimated using the quark parton model analogously to e+e- + hadrons. Assuming the usual three coloured quarks, for mH 3 m,, mUYma? T’(H + non-strange

hadrons)

F(H + p+I--)

~ 3(mi + m$) rnz

’

(4.3)

Estimates of m, and md vary between O(10) MeV [24] and O(300) MeV [25], the former corresponding more to the “current” quark picture used in deriving (4.3). For this range of quark masses, the ratio (4.3) varies between about 0.06 and about 50. We do not know at what value of mH the formula (4.3) should start being applicable, but estimates of exclusive contributions to T’(H + non-strange hadrons) favour a large rate in the region mH < 2 GeV.

J. Ellis et al. / Higgs boson

Fig. 14. The decay

H --f nn via an intermediate

Consider H + 7171close to threshold:

325

e(700?)

meson,

to estimate this we use the dilaton theorem

(3.2), (7r+n- IH) = ;

cn+77- lf$lO,

and then use a broken scale invariance (1~+(~)n-(q)lB;(k)I0)

(4.4) [20] estimate of (n4r-

= 2(p2 + q2 - m;) + k2 f

,

l@ElO), (4.5)

as a power series in p2 q2 and k 2 *. Evaluating (4.5) at the physical point p2 =q2 =mi and k2 = rnk, and subs’tituting into (4.4) we get

br+n- If3

N

$

(rnfj t 2m~) ,

and so, close to the 7177threshold

Comparing the decay rates (4.1) and (4.6) it is evident that the decay to 7-mdominates the decay to P+P(- already for “H not far above 7771 threshold. For higher (nr) energies we might expect the H + 7~ decay to proceed via an intermediate ~(700?) meson if it exists, as indicated in fig. 14. The dilaton theorem (3.2) implies a transition (EIH) =;

f, m,2 ,

(4.7)

and we find (4.8) for mH e m,. The formula (4.8) again indicates that H -+ 717~ strongly dominates over H -+ p+p- in this mass range. The upper (nn) branching ratio curve in fig. 1 is obtained from the formulae (4.6) and (4.8) with a reasonable interpolation between

* This estimate

was used for the nnH vertex

in subsect.

3.2.

J. Ellis et al. / Higgs boson

326

them and an extrapolation to mH 21 1 GeV. The lower (nrr) branching ratio curve in ‘rig. 1 was obtained by arbitrarily reducing these calculations by a factor of 10 to 100, so as to connect up more naturally with the light quark value for the branching ratio (4.3) at higher values of mH. We are rather sceptical about a very low value for (4.3) in the region up to m - 2 GeV, as estimates of contributions like H -+ pp also give a large branching ratio.H For m H >, 1 GeV, the coupling to strange quarks presumably becomes dominant: asymptotically l?(H -+ strange hadrons)

W

-+P+P-)

zs

3m2 (4.9)

rn2 ’

with estimates of m, varying between O(150) MeV [24] and O(500) MeV [25] yielding values for (4.9) between 6 and 60. The corresponding branching ratios (4.9) are plotted in fig. 1: an estimate of the KK contribution lies in the same ball-park close to threshold. At mH z 4 GeV, the couplings to new hadronic (charm?) or leptonic (heavy lepton?) degrees of freedom presumably take over. In the simplest SU(4) charm picture we would have T’(H + charmed hadrons)

_ 3mz _ 6oo

r(H + P+P-)

(4.10)

mE

for m, N 1.5 GeV. We conclude that in view of the invisibility of charmed particles to date, the detection of a Higgs boson with mass mH > 4 GeV may be non-trivial. 4.3. Decays to 77 Since the photon is massless, these decays do not proceed directly, but via virtual intermediate states, such as electrons, muons, hadrons and intermediate bosons. The decay rate is [12] r(H+27)=

cY2 8&rr3

G,&1112

(4.11)

,

where I = I, + I,, + Ihadronst Iw + . is calculated from the diagrams of fig. 15. The

_:$

_;$

I Fig. 15. Contributions

__<& WMrans to the quantity

1, I of eq. (4.11).

J. Ellis et al. / Higgs boson

321

contributions I, of leptons (e, I*, heavy lepton?) are known [ 121: the function Z,(m~/m~) is plotted in fig. 16. To estimate the contribution Zhadrons we use the dilaton theorem (3.2) and broken scale invariance arguments [20,50]. In ref. [SO] it was shown that the matrix element (77leK(k)]O) for k2 small was determined by a scale invariance anomaly related to the high-energy behaviour of e+e- + 7 -+ hadrons. In the notation of (4.11) this corresponds to

Zhadrons

‘R,

o(e’e-

R = limit

= Z

u(e+e-

s-+m

+ 7 -+ hadrons)

(4.12)

+ 7 + p+p- )

In the simplest SU(4) charm model we would get Ihadrons = 10/6. This estimate is supposed to work for rni5 a typical hadronic (mass) scale. For large values of rn;,Ihadrons presumably -+ 0 in a way similar to the behaviour of I, shown in fig. 16. Finally we must include the intermediate vector boson loop contribution to the 77 decay mode. This involves a long but straightforward calculation of the Feynman diagrams of fig. 17, where H’ is an unphysical Higgs boson and I$’ a Fadeev-Popov ghost. The relevant Feynman rules have been extracted from appendix C.2 of Fujikawa et al. [37], and are displayed in fig. 18, where the ‘t Hooft-Feynman [.51] gauge has been adopted. There is an extra minus sign for each closed loop of the ghost field @‘. To handle divergent integrals we use dimensional regularization. After Feynman parametrization, the relevant integrals are of the form [52] (our metric is

-

l.O-

;; z T) 5 --

2 Ret,

-c

213 as m:lrn~ -

a” N 0.5 --

I

2 Im I, I

0

1

Fig. 16. A graph

I 2

I 3

of the real and imaginary

I 4

parts of Ia.

b

-0

328

J. Ellis et al. / Higgs boson

+ crossed

graphs

(k,

.p--

k, .v ) + 0 t-$

1 w

Fig. 17. Feynman

diagrams for fw.

gP” = +- - -)

s

*,p

s

*np

a t bup,,

_

cp2 +

dpvp,Pv

(-1)”

id2

(m2 + k2)a-n/2

(p!-2k*p-m2)Q

=

(p2-2k.p-m2)@

x {r(a - ffi> (ck2 + d”‘k,k,)

(-1)“i7d2 (m2 + k2)a-nn/2

r(;;a$)

[a t bp kJ

,

1 r(a)

- I’(c~ - 1 - fn) (m2 t @)

($nc

t ~dWgpy)}

(4.14)

,

J. Ellis et al. / Higgs boson

p-v

w (P) H- (p)

--_-*.--_ . . ...)

F

p’-rni

I

=

K

pz- m$

@’(P) . . . .

‘J W-CP') Y(k) p =

-ig,,

=

329

le [gry (P*P’)P

w-(P) -p,g,,

-p;

g,P

A

=

p’-mi

, H-(p’) h&v

5 -ie(p+p’)v

4 ‘H-(p)

=2

(P-q&

‘q(P) P

H(q)

W-

--f

= 1g In, gvv

W-

< _H!q,_

V

<+;I- =-ig m; w 2mw

.

/ 4-

H(q) -t-

;

v

-*-

Fig. 18. Feynman

where n is the dimension

xezI’(Z)

=

IYz+ z

E

-$g

gp*

rules for the graphs of fig. 17.

of space-time, and l)

e-Zlnx

-

z+o

-!-lnx+O(Z). Z

(4.15)

To extract the finite terms in the limit n + 4, we retain only the lowest relevant order in (/cl X k2)/m$. Then, after integration over the Feynman parameters, each diagram

J. Ellis et al. / Higgs boson

330

of fig. 17 gives a contribution

of the form

AK2 - $>gpv+ Bg,, -

+O(n-4),

(4.16)

where we have dropped terms which vanish on the photon mass shell and set k, . k2 = $rni. In order that the result be finite and gauge invariant the summed coefficients must obey

c Aa(n-4),

Cc=-CB-%&CA.

(4.17)

mH The contributions to A, B and C from each graph of fig. 17 are listed in table 1. Their sum is seen to obey the conditions (4.17). Thus while the individual contributions are neither finite nor gauge invariant, they sum to give a finite gauge invariant result, (4.18) In the notation I,

= -;

of eq. (4.1 l), the result (4.18) corresponds .

to (4.19)

The diagrams of fig. 19 involving a closed H’ loop are separately finite and gauge invariant, as they must be by virtue of the renormalizability of scalar electrodynamics. Their contribution is of order m$/m$ relative to the amplitude (4.18) *. We conclude from (4.19) that the contribution of the W boson loop to H + yy may be very important. We have not calculated corrections to I, as mH approaches the W+W- threshold: we expect the behaviour to be similar to that for I,, fig. 16. We see that in the minimal model assumed here the W loop contribution is large and interferes destructively with the hadronic and leptonic contributions. In fact fermion (as well as scalar meson) loop contributions are directly related to a mass insertion in the corresponding contribution of the vacuum polarization, and since convergent, are determined by the absorptive part which is positive definite. For the W contribution, the vacuum polarization is not defined in the unitary gauge; one must instead consider a W mass insertion in an S matrix element such as e+e- elastic scatter. ing. But there are now several different diagrams which can contribute to the W+Wintermediate state and the direct connection with I is lost. Unfortunately, this sign indeterminacy renders the yy decay rate prediction extremely model dependent. The value of I is very sensitive to the addition of heavy

* We have also repeated the fermion loop calculation of Resnick et al. [12] in order to determine the relative signs of the amplitudes. This serves as a further check on the over-all normalization of (4.19).

J. Ellis et al. / Higgs boson Table 1 Contributions Graph a + crossed b c +

d + crossed

to the H + yy amplitude

from the graphs

c

3(n - 1) -2(n

- 1)

-&I

- 1)

23

$

-2 19

E 2r?@4

e + crossed

0

f + crossed

0

1

1

-B 0

g +

h + crossed

i + crossed 2 X j + crossed

of fig. 17

B

A

-2 -f

Fig. 19. Contributions

331

+f

0” ~-23 12 0 0

1

z 0

1 -13

1 12

to Iw from unphysical

Higgs bosom.

leptons (for mH < 2mL), more hadronic degrees of freedom (for mH 5 new hadronic mass scale), and especially to the choice of the intermediate boson model. To guess a reasonable range of predictions we take

using la from fig. 16, lhadrons from (4.12) and Zw from (4.19). Then reference to fig. 1 indicates that the yy decay mode could be significant for certain ranges of mH*.

5. Discussion Let us now discuss the observability of the Higgs boson in the light of the calculations of its production (sect. 3) and decays (sect. 4). Previous limits on the Higgs boson mass, together with our proposals for improving these limits, are tabulated in fig. 3.

* However, the value of r(H -+ yy) is at most O(lO-‘) parable mass, as was mentioned in subsect. 3.5.1.

X r(h --* yy) for a hadron

h of com-

J. Ellis et al. i Higgs boson

332

If the Higgs boson has a mass 5500 MeV, then we could identify three possible places to look for it: production at threshold in hadronic collisions (subsect. 3.2.1.) in the decay K + rr t H (subsect. 3.3.2.) and in a decay of the 3.7 resonance to 3.1 tH (subsect. 3.3.3.). Higgs bosons with masses -< 2m, decay predominantly into e+eand yy (see fig. l), and have lifetimes 2 2 X IO- l2 set (see fig. 2). Hence one could try looking for narrow bumps in these channels, possibly coming from a particle with a decay path of delectably non-zero length. One possible experiment [38] would be n- capture at rest in a light nucleus such as 9Be or 12C, in which the production of rr” is forbidden. From the cross section (3.18) the ratio of Higgs production to the dominant process n-p -+ yn is * r(n-p

+ Hn)

= O(lO-6)

~(T’-P + 7 n) for Higgs bosons with masses 0(15 to 100) MeV. Detection of a Higgs boson with a mass in the lower part of this range would be aided by its lifetime of O(lO-10)sec which means that it travels several centimeters before decaying. A Higgs boson in the higher part of this mass range would live 0(1 O- 1l) set and not travel very far, but the background y spectrum would be lower. For m H >, 201, thee+ e- and y-y decay modes are disfavoured with respect to H + /.L+/J-, and H + rr+n- should dominate above (rm) threshold. These modes are beset by severe background problems in 3.7 decays from the dominant cascade 3.7 + 3.1 + TT+ n. Also, the lifetime would be too short for the Higgs boson to leave a discernible track. One might have expected [53] K or 7) decays to have already ruled out light Higgs bosons, but this is not so. Reference to subsect. 3.3.2. reveals the estimate r(K+ -+ n+ t H) l?(K+ 4 all) while experiment

x10-7

)

[23] sees

r(K+ + n+e+ee)

= (2.6 + 0.5) X 1O-7 .

r(K+ -+ all)

For the range mn < meee_ < mK - m, to wh‘ic h t h e experiment

was sensitive to e+e- pairs, we find from fig. 1 that (r(H + e’e-))/(I’(H + all)) may vary between 30-100% at mH = m, and 12-95% at mH = 2m,, and is negligible for mH > 2m,. Hence we would estimate r(K+ + rr+ t H, H --f e+ee) --r(K+ 3 all)

-0(1

for 2m, > mH > m,, and the published l

We particularly

thank

to 10) x 10-g [23] data seem unable to rule out a narrow

John Bailey for giving us this number.

J. Ellis et al. / Higgs boson

333

peak in me+ee produced with such a branching ratio. However, future experiments on AS = 1 neutral currents in K decays might well be sensitive to Higgs boson production if its mass is appropriately small. If the Higgs boson has a mass in the range 500 MeV
= O(103 - 105)

(5.1)

o(H) F(H + p+p-) F(H + all) Thus it would have to be a very high statistics experiment indeed which could see a small narrow Higgs bump above the tail of the p meson *. In fact there seems little advantage in looking for a Higgs boson with a mass in the range 500 MeV to 1500 MeV at a very-high-energy accelerator. An experiment at moderate energies, where the dominant production mechanism would be one pion exchange (as discussed in subsect. 3.2.2.) and the backgrounds might be smaller, might be an equally good possibility. For 1.5 GeV < mH < 4 GeV, production at very high energies and detection as a small bump sitting on top of the Drell-Yan [39] (/.~l’y~) continuum seems the only possibility **. Combining the cross section estimate (3.29) with the branching ratios of fig. 1, we find ~(PP + H(-’ IJ+~-)

_ Q -AQ O(3 x 10-T to 10-S)

(5.2)

O(PP + 01%-) + x) which is not encouraging. Even if such a bump were seen, how would one know it was a Higgs boson, and not some random [2] hadronic vector meson? There would be two distinctive features of the Higgs boson: it would not be seen in (e+e-) if it had such a mass, and it would be much narrower than some vector mesons, though not visibly narrower than others [2]. For mH > 4 GeV the Higgs boson production cross section by any mechanism we have been able to think of is minute, and it decays predominantly to as yet conjectural [3-51 massive new particles.

* For this reason and because of its low e+e- branching ratio, there seems little likelihood that the Higgs boson could be the mysterious source of directly produced leptons sought by Lederman and White [54]. l * At least until intermediate vector bosom are discovered, and the bremsstrahlung production of subsect. 3.4 becomes thinkable!

J. Ellis et al. 1 Higgs boson

334

We should perhaps finish with an apology and a caution. We apologize to experimentalists for having no idea what is the mass of the Higgs boson, unlike the case with charm [3,4] and for not being sure of its couplings to other particles, except that they are probably all very small. For these reasons we do not want to encourage big experimental searches for the Higgs boson, but we do feel that people performing experiments vulnerable to the Higgs boson should know how it may turn up. We would like to thank B.W. Lee, J. Prentki, B. and F. Schrempp, G. Segrd and B. Zumino for valuable remarks, comments and suggestions.

Note added in proof Since writing our paper we have learnt of some more considerations [SS-571 about the mass of the Higgs boson. Also, we have been encouraged [58] to calculate its production in neutrino collisions. We also make here some further remarks about the model dependence of our previous results. In two papers [55,56], Sato and Sato have given astrophysical arguments against very light Higgs bosons. They argue that present understanding of the cosmic background radiation excludes 0.1 eV < mH < 100 eV [55], and that stellar evolution would be drastically affected if mH < 0.1 X m, [56]. Most recently, Linde and Weinberg have derived [57] an approximate lower bound on mH from an analysis of Coleman and Weinberg [59]. These authors pointed out that a simple Higgs potential V,(H) = p2@

+ AH4 ,

acquires radiative corrections yield

(2

< 0, x > 0) ,

in perturbation

(A.1)

theory. The one-loop graphs of fig. 20

V,(H) = p2H2 t BH4 In (H2/M2) ,

(A-2)

where M is a mass parameter chosen to absorb all H4 terms, and

&_L 64n2 v4 where v2 = l/fiGF

[3

C v=w,z

rnt - 4 Frn:]

(A.31

as before l. Then by requiring that the value H = v be a global

* The potential is actually gauge dependent, the original calculations of Coleman and Weinberg [59] being performed in the Landau gauge so that no ghost loops appear in fig. 20. However, the conclusions of physical interest are gauge independent to all orders in perturbation theory [60]. There is also a H&s contribution to (A.3) which is negligible for the comparatively light Higgs bosons we are interested in.

J. Ellis et al. / Higgs boson

f l

H,

f

335

/H

__JJ_;__+)y-J

* \H

f

Fig. 20. Diagrams which contribute [59] to the effective Higgs potential in the Landau gauge and in the one-loop approximation. (Diagrams with an internal Higgs loop also contribute, but are irrelevant [ 571 in bounding the Higgs mass.)

minimum

of the potential

(A.2)

Linde and Weinberg [57] showed that if mW z

Srnf 2

mH_

3a2(2 t sec4 Bw)

>3&1: mi TS

(2

+ mi)=

16&iGF sin4 0,

’

(A.4)

which is 0(4.9) GeV if 0 - 3.5” as currently favoured by experiment. Higher order loops introduce H4 [ln(H B/M2)]” t erms into the effective potential, but the limit (A.4) is not greatly affected thereby. Is there any other way of evading the lower limit (A.4)? One might entertain the idea of relaxing the condition that V(u) be a global miirimum, and just demand that it be a local minimum, with the global minimum at the origin H = 0 l . According to standard arguments [31], the minimum at H= u would not then be stable, but the theory with no spontaneous breakdown would be probably so singular as to be undefined [61]. The only way for the local minimum at H = u to be relevant might be if, as suggested in sect. 2.4 for other reasons, the world Lagrangian has another spontaneous breakdown which cannot occur independently of the one relevant to the weak and electromagnetic interactions. The bound (A.4) is significantly altered if there is at least one fundamental fermion field with mass mf 2 mW. There are no theoretical arguments for or against such an object. Experiment has so far revealed only small fermion masses, but our sampling techniques are clearly biased in favour of low masses. It would be an act of bravado to suggest that the discovery of a Higgs boson with a mass appreciably below the limit (A.4) would be an argument for the existence of ultra-high mass fermions. We now turn to Higgs boson production in neutrino collisions. Three relevant Feynman diagrams are shown in fig. 21, The Higgs boson may be emitted either * Note that in deriving the bound (A.4) it is not necessary that fi2 < 0. If p2 > 0 could be excluded, then the bound on nrH would be improved by a factor J2.

.I. Ellis et al. / Higgs boson

336

L&-zdro”s ;!ijgighadr (a)

(b)

hadrons

(cl Fig. 21. Diagrams for Higgs production in neutrino-proton line, (b) the hadronic system, and (c) the virtual W boson.

scattering

via emission

by (a) the muon

from the muon line (fig. 21a), or the hadronic system (fig. 21b) or from the virtual exchanged W boson * (fig. 21~). The first two graphs will give a cross section rising linearly with E, like the total neutrino cross section, and we expect them to contribute a cross section ratio resembling that in purely hadronic processes: u(VtN-‘/C

+H+x)la+b

< N

o(v+N+p-

10-T .

(A.9

tX)

On the other hand, the diagram of fig. 21c can give a contribution rising as Ez. We assume scaling for the deep inelastic structure functions as in the quark parton model with negligible antiquark distributions, we ignore mH, and assume rn$ < mNEv < m$. Introducing the v, N, p-, W and H momenta as in fig. 21 c, and definmg v=2q.p,,

v’-2q’*pN)

x = -q+

y 2 v/2mE V’

y’ = v’/2mE V’

,$ = _q’2/,’

) )

G4.6)

then we have the standard result

&

G;mNEv ‘II

(vtN+p-+X)=

for the total cross section with no H production, l

This diagram

has been mentioned

[ 181, and since submission

(A.71

F~(x)

as a possible

of our paper

source

and of Higgs bosons

it has also been calculated

by Ross and Veltman

by LoSecco

[58].

337

J. Ellis et al. / Riggs boson

d4,(,

x

+ N + /_L- + H + X)1, _= dxdydx’dy’

[

G;M2,E; F+‘) fin3

l_y+y’_x_A+_2y’x x’

.J’

yx’

+xy _xy’ x’ x’

1

(‘4.8)

for the inclusive H production from fig. 21~. The Higgs boson would emerge at an angle OH to the neutrino beam, where BH -m is largest when y y 0 or y - y’. Integrating (4.8) with the kinematical restrictions x’>x,

Y>Yl>

we find

o(vtN+/.- tHtX)I, a(v+N-+p-

+X)

1 =

GFMN E, (r$),

___

(A.9)

6&rr2

where 1

s O

W2WdE

([)

z

1

s 0

= 0.2 experimentally

.

F2(WC

Putting numbers into (A.9), we get

a(vtN-,p- +H+X)I, o(v+N+/.-

+X)

-3

x ,O&

(A.lO) MN

2 Thus the diagram of fig. 21c dominates over the in the range rni < mNE, < mw. other diagrams (AS) if E, x O(100 GeV), but the Higgs boson production rate is not enormous. We should add some remarks about the sensitivity of some results of our paper to specifics of the simplest Weinberg-Salam model we discussed. Generally our production estimates might apply to any semi-weakly coupled particle of similar mass. Possible differences occur when we consider couplings to the electron, which is am, and hence very small in the simplest Weinberg-Salam model. If the He+e- coupling were characteristic of other masses and hence much larger, as in some gauge theories, then our estimates of r(H + e+e-) and T(H + 77) would be altered, decreasing the lifetimes and changing the decay branching ratios of low mass Higgs bosons. Also, direct production in efe- collisions [49] might no longer be negligible. Another sensitive area is the assumption in the model that the Higgs quark couplings Hiiu and

338

J. Ellis et al. / Higgs boson

Hdd are essentially equal. If they were unequal, giving the Higgs couplings a large 13 = 1 component au3 in the (3,3) + (3,3) re p resentation of SU(3) X SU(3), then the estimates in subsect. 3.3.1 on the decays n -+ rr” + H, Co + A + H could be modified, the branching ratios becoming much larger [61]. We should also observe that in a model with physical charged Higgs bosons H’ the production rates in related reactions should be similar to those for the neutral Higgs boson discussed here. On the other hand, the leptonic decay modes of H’ into final states e’y, ~.l*v would be less distinctive than the e+e-, /_i+/_- decays of the neutral Higgs boson. Finally, since our paper was published, the authors of ref. [23] have studied the e+e- invariant mass distribution in K+ + rr+e+e-. With a mass resolution of 4 MeV, they are able to set a limit [62] (at 90% confidence level): r(K+ + rr+ X0(+ e’e-)) .~ T’(K+ + all)

<0.4

x 10-7

for 140 MeV < Mx,, < 340 MeV. This level is still compatible sect. 5, and therefore, unfortunately, inconclusive *.

with the estimate of

We thank D. Cline, B.W. Lee, C. Rubbia, M. Veltman and B. Zumino for useful comments and discussions, and A.M. Diamant-Berger and R. Turlay for providing us with a limit from their K+ decay data.

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to have a significant

branching

ed.

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