

 19 Oct 1995

University of California - Davis UCD-95-36October 1995

PROBING LEPTON-NUMBER-VIOLATING COUPLINGS OF DOUBLY CHARGED HIGGS BOSONS AT AN e\Gamma e\Gamma COLLIDER1

J.F. Gunion Davis Institute for High Energy Physics University of California, Davis, CA 95616

Abstract The doubly charged Higgs bosons \Delta \Gamma \Gamma that are present in exotic Higgs representations can have lepton-number-violating couplings to e\Gamma e\Gamma . We discuss general constraints and phenomenology for the \Delta \Gamma \Gamma and demonstrate that extremely small values for the e\Gamma e\Gamma ! \Delta \Gamma \Gamma coupling (some 8 orders of magnitude smaller than the current limit) would produce observable signals for \Delta \Gamma \Gamma production in direct s-channel production at an e\Gamma e\Gamma collider.

1 Introduction

Doubly-charged scalar particles abound in exotic Higgs representations and appear in many models [1, 2, 3]. For example, a Higgs doublet representation with Y = \Gamma 3 contains a doubly-charged \Delta \Gamma \Gamma and a singly-charged \Delta \Gamma . If part of a multiplet with a neutral member, a \Delta \Gamma \Gamma would immediately signal the presence of a Higgs representation with total isospin T = 1 or higher. Most popular are the complex Y = \Gamma 2 triplet Higgs representations, such as those required in left-right symmetric models, that contain a \Delta \Gamma \Gamma , a \Delta \Gamma and a \Delta 0.

Of course, in assessing the attractiveness of a Higgs sector model containing a \Delta \Gamma \Gamma many constraints need to be considered. For triplet and higher representations containing a neutral member, limits on the latter's vacuum expectation value required to maintain ae = 1 are generally severe. (The first single representation beyond T = 1=2 for which ae = 1 is automatic regardless of the vev is T = 3; Y = \Gamma 4, whose T3 = 0 member is doubly-charged.) Models with T = 1 and T = 2 can `automatically' have ae = 1 at tree-level by combining representations (the most well-known example being a Higgs sector containing both Y = 0 and Y = \Gamma 2 triplets whose neutral members have the same vacuum expectation value). However, such models generally require fine-tuning in order to preserve ae = 1 at one-loop. The simplest way to avoid such problems is to either consider representations that simply do not have a neutral member (for example, a Y = \Gamma 3 doublet or a Y = \Gamma 4 triplet representation), or else models in which the vacuum

1To appear in the Proceedings of the Santa Cruz Workshop on e\Gamma e\Gamma Physics at the NLC, University of California, Santa Cruz, Sept. 4-5, 1995.

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expectation value of the neutral member is precisely zero. We will only consider models of this type in what follows.

Another source of motivation for and constraints on Higgs representations arises if we require unification of the coupling constants without intermediate scale physics. In the Standard Model, it is worth noting that quite precise unification is possible for a relatively simple Higgs sector that includes a triplet Higgs representation -- namely, a single jY j = 2 triplet in combination with either one or two jY j = 1 doublets (the preferred number of doublets depends upon the precise value of ffs(mZ)). The neutral member of this single triplet would need to have zero vacuum expectation value to avoid ae problems. In the case of the minimal supersymmetric extension of the Standard Model, the only Higgs sector that gives precise unification is that comprised of exactly two doublet Higgs representations (plus possible singlet representations); any extra doublet representations (including ones with a doubly-charged boson) or any number of triplet or higher representations (whether or not any doublets are present) would destroy unification. Thus, doubly-charged Higgs bosons would seem to be very unlikely in this context. However, by including appropriate intermediate scale physics, supersymmetric models with triplet and higher representations can often be made consistent with unification. In particular, supersymmetric extensions of the left-right symmetric models, which must contain triplet Higgs representations, typically do have matter at intermediate scales and the exact masses can comfortably be adjusted to achieve coupling unification.

Thus, allowing for the possibility that the two-doublet MSSM might not be nature's choice, experimentalists should be on the look-out for signatures of exotic Higgs representations. Since many of the more attractive higher representations include a doubly-charged Higgs boson, it is important to consider how to search for and study such a particle. The phenomenology of the \Delta \Gamma \Gamma derives from its couplings. Tri-linear couplings of the type W \Gamma W \Gamma ! \Delta \Gamma \Gamma are not present in the absence of an enabling non-zero vacuum expectation value for the neutral member (if present) of the representation, and q0q\Delta \Gamma \Gamma couplings are obviously absent. There are always couplings of the form Z; fl ! \Delta \Gamma \Gamma \Delta ++, and these can be useful for production of the \Delta \Gamma \Gamma , as outlined later. However, an especially interesting possibility is the lepton-number-violating e\Gamma e\Gamma ! \Delta \Gamma \Gamma coupling that is sometimes allowed by symmetry. For Q = T3 + Y2 = \Gamma 2 the allowed cases are:

e\Gamma Re\Gamma R ! \Delta \Gamma \Gamma (T3 = 0; Y = \Gamma 4) ; e\Gamma L e\Gamma R ! \Delta \Gamma \Gamma (T3 = \Gamma 12; Y = \Gamma 3) ; e\Gamma L e\Gamma L ! \Delta \Gamma \Gamma (T3 = \Gamma 1; Y = \Gamma 2) :

(1.1)

Note that the above cases include the T = 3; Y = \Gamma 4 representation that yields ae = 1, the T = 1=2; Y = \Gamma 3 doublet and T = 1; Y = \Gamma 4 triplet representations with no neutral member, and the popular T = 1; Y = \Gamma 2 triplet representation.

The above list of allowed couplings is expanded in the left-right symmetric models where Q = T L3 + T R3 + Y2 . Indeed, in left-right symmetric models there is a

2

jY j = 2 Higgs triplet representation that has lepton-number-violating couplings to right-handed leptons. The right-handed neutrino states acquire a large mass when the neutral member of this `right-handed' triplet acquires a non-zero vev. This large mass in turn leads to a light neutrino mass eigenstate via the popular seesaw mechanism. The phenomenological analysis we present below would have to be extended in the case of this triplet since its neutral member has a vev. However, in the left-right symmetric models there is a second, `left-handed', jY j = 2 triplet that couples to left-handed neutrinos and leptons. The strength of the coupling is the same as that associated with the right-handed sector. As usual, the neutral member of this `left-handed' triplet must have a very small vev in order to preserve ae = 1, and the phenomenology of this triplet's doubly-charged member would be as described below. The strength of the lepton-number-violating coupling (common to the right and left sectors) required to make the see-saw mechanism work properly in the right-handed sector is typically such as to fall into the range that we shall claim can be probed in e\Gamma e\Gamma collisions through production of the doubly-charged member of the left-handed triplet.

In the case of a jY j = 2 triplet representation (to which we now specialize) the lepton-number-violating coupling to (left-handed) leptons is specified by the Lagrangian form: L

Y = ihijTiLCo/2\Delta jL + h:c: ; (1.2)

where i; j = e; _; o/ are generation indices, the 's are the two-component lefthanded lepton fields (`L = ( *`; `\Gamma )L), and \Delta is the 2 \Theta 2 matrix of Higgs fields:

\Delta = ` \Delta

+=p2 \Delta ++

\Delta 0 \Gamma \Delta +=p2 ' : (1.3)

In left-right symmetric models \Delta would have to be subscripted as \Delta L, and there would be a Lagrangian component analogous to that given in Eq. 1.2 with L ! R everywhere.

The strengths of the couplings in Eq. 1.2 are specified by the hij; e\Gamma e\Gamma ! \Delta \Gamma \Gamma will be controlled by hee. Limits on the hij come from many sources. Experiments that directly place limits on the hij by virtue of the \Delta \Gamma \Gamma ! `\Gamma `\Gamma couplings include Bhabbha scattering, (g \Gamma 2)_, muonium-antimuonium conversion, and _\Gamma ! e\Gamma e\Gamma e+. For some details and references see, for example, Refs. [2, 4]. One finds limits of

jh\Delta

\Gamma \Gamma

ee j2 !, 10\Gamma 5m2\Delta \Gamma \Gamma ( GeV)j

h\Delta

\Gamma \Gamma

__ j2 !, 4 \Theta 10\Gamma 5m2\Delta \Gamma \Gamma ( GeV)j

h\Delta

\Gamma \Gamma

ee h\Delta

\Gamma \Gamma __ j !, 6 \Theta 10\Gamma 5m2\Delta \Gamma \Gamma ( GeV)j

h\Delta

\Gamma \Gamma

e_ h\Delta

\Gamma \Gamma ee j !, 5 \Theta 10\Gamma 11m2\Delta \Gamma \Gamma ( GeV)

(1.4)

from the above respective sources. (In our notation, h\Delta

\Gamma \Gamma

ij refers explicitly to theh

ij couplings as they determine \Delta \Gamma \Gamma interactions.) The last of these limits clearly

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suggests small off-diagonal couplings, and in what follows we shall assume that the couplings are entirely diagonal. We shall adopt the conventional form for these couplings of j

h\Delta

\Gamma \Gamma

`` j2 j c``m2\Delta \Gamma \Gamma ( GeV) ; (1.5)

where it will be useful to keep in mind that cee !, 10\Gamma 5 is the strongest of the limits.

Finally, we remark that constraints on the hij through couplings for the \Delta 0 and \Delta \Gamma should also generally be incorporated for the specific model being considered. The only such constraint that is potentially stronger than those outlined above is that associated with Majorana mass terms for the neutrinos coming from their couplings to the \Delta 0. One finds [3] mij = 2hij h\Delta 0i. If h\Delta 0i were of order the SM vev, then, for example, m*e !, 1 eV (as required to prevent neutrinoless double-beta decay from being observable) would imply hee !, 10\Gamma 14. However, this constraint clearly goes away if \Delta 0 does not have a significant vev. In the present study, we assume (as stated earlier) that the vev is, in fact, exactly zero, so as to avoid problems associated with maintaining ae = 1 naturally.

2 General Phenomenology for a \Delta

\Gamma \Gamma

Given adequate machine energy, production of \Delta \Gamma \Gamma \Delta ++ via fl; Z exchange at either an e+e\Gamma or pp collider will yield an observable signal. At an e+e\Gamma collider the cross section for pair production of a boson with weak isospin T3 and charge Q and its conjugate is given at energies s AE m2Z by:

oepair = 43 ssff

2

s !

2fi3 sin4 2`W (`

1 2 T3 + sin

2 `W ^ 1

2 Q \Gamma T3*'

2

+ (Q \Gamma T3)2 sin4 `W ) :

(2.1) For the case of a T3 = \Gamma 1; Y = \Gamma 2 \Delta \Gamma \Gamma we find f: : :g ! 14 + sin4 `W and oepair , 488 fbfi3 at ps = 500 GeV, which yields about 11 fb for m\Delta \Gamma \Gamma , 240 GeV. In other words, we would have at least 50 events for L = 50 fb\Gamma 1 for masses up to within 10 GeV of threshold. We shall discuss signatures in a moment, but this number of events would generally be adequate for \Delta \Gamma \Gamma \Delta ++ detection. However, the e+e\Gamma ! \Delta \Gamma \Gamma \Delta ++ process has a crucial limitation. It allows detection of the \Delta \Gamma \Gamma only up to m\Delta \Gamma \Gamma !, ps=2. This is only half the kinematical reach of schannel production of the \Delta \Gamma \Gamma in the e\Gamma e\Gamma mode of operation at the same ps value. Further, detection of the \Delta \Gamma \Gamma prior to the construction and operation of the e+e\Gamma ; e\Gamma e\Gamma collider NLC complex would be very important in determining the energy range over which good luminosity and good energy resolution for e\Gamma e\Gamma collisions should be a priority. Thus, it is fortunate that observation of \Delta \Gamma \Gamma \Delta ++ pairs with m\Delta \Gamma \Gamma , 500 GeV is straightforward at the LHC. The mass reach for pair production at a pp collider increases rapidly with machine energy. At the LHC, oepair , 1 fb at m\Delta \Gamma \Gamma = 800 GeV, the precise number depending upon T3. With L = 100 fb\Gamma 1, there would clearly be a large number of \Delta \Gamma \Gamma \Delta ++ events for any

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\Delta \Gamma \Gamma with m\Delta \Gamma \Gamma !, 500 GeV. For sufficiently small m\Delta \Gamma \Gamma , observation of \Delta \Gamma \Gamma \Delta ++ pair production might even be possible at the Tevatron.

Another potential production mechanism for the \Delta \Gamma \Gamma at pp colliders is the fusion process, W \Gamma W \Gamma ! \Delta \Gamma \Gamma . However, the required tri-linear coupling is zero given our assumption that the vev of the neutral member (if there is one) of the Higgs representation is zero (thereby avoiding naturalness problems associated with maintaining ae = 1). A general discussion of event rates for W \Gamma W \Gamma ! \Delta \Gamma \Gamma fusion for typical models in which the vev is not zero can be found in Refs. [2, 3, 1].

Decays of a \Delta \Gamma \Gamma are generally quite exotic [2, 3]. If there is an enabling nonzero vev, then \Delta \Gamma \Gamma ! W \Gamma W \Gamma decays can be very important. If this coupling is absent (as we assume), then possible two-body decays include \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma , \Delta \Gamma \Gamma ! \Delta \Gamma \Delta \Gamma and, if the lepton coupling is present, \Delta \Gamma \Gamma ! `\Gamma `\Gamma . Assuming some reasonable amount of degeneracy of the masses of different members of the multiplet, the \Delta \Gamma \Gamma ! \Delta \Gamma \Delta \Gamma decay is likely to be disallowed. Thus, we will focus on the \Delta \Gamma W \Gamma and `\Gamma `\Gamma final states. (Generalization of our discussion if other decays are present will be apparent.) For a T = 1; Y = \Gamma 2 triplet we find (see, for example, Refs. [2, 3])

\Gamma (\Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma ) = g

2

16ss

m3\Delta \Gamma \Gamma fi3\Delta \Gamma W \Gamma

m2W ; \Gamma (\Delta

\Gamma \Gamma ! `\Gamma `\Gamma ) = fififih

\Delta

\Gamma \Gamma

`` fififi

2

8ss m\Delta

\Gamma \Gamma ;

(2.2) where fi\Delta \Gamma W \Gamma is the usual phase space suppression factor. It is convenient to rewrite these widths, using Eq. 1.5, in the forms:

\Gamma (\Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma ) = (1:3 GeV) i m\Delta

\Gamma \Gamma

100 GeV j

3 fi3

\Delta \Gamma W \Gamma ;

\Gamma (\Delta \Gamma \Gamma ! ``) = (0:4 GeV) i c``10\Gamma 5 j i m\Delta

\Gamma \Gamma

100GeV j

3 : (2.3)

In order to gain a rough idea of the relative magnitude of these widths, consider the case [2] m\Delta \Gamma \Gamma = 360 GeV, m\Delta \Gamma = 250 GeV. From Eq. 2.3 we find \Gamma (\Delta \Gamma \Gamma !

\Delta \Gamma W \Gamma ) , 2 GeV and \Gamma (\Delta \Gamma \Gamma ! `\Gamma `\Gamma ) = 19 GeV i c``10\Gamma 5 j. This makes it clear that if any c`` is near 10\Gamma 5 then that `\Gamma `\Gamma mode is very likely to have a partial width larger than the \Delta \Gamma W \Gamma partial width. Since there are currently no limits on co/o/ , the o/ \Gamma o/ \Gamma channel could easily have the largest partial width and be the dominant decay of the \Delta \Gamma \Gamma . On the other hand, when we discuss probing very small cee values, we must keep in mind that if the other c's are of similar size then the \Delta \Gamma W \Gamma mode is quite likely to be dominant if it is kinematically allowed.

The implications for detection of \Delta \Gamma \Gamma \Delta ++ pairs in e+e\Gamma or pp collisions are obvious. If one or more of the c``'s is ?, 10\Gamma 5, the `\Gamma `\Gamma channel with the largest c`` will dominate \Delta \Gamma \Gamma decays. For ` = e or _, we will have spectacular signatures of two like-sign lepton pairs of equal mass. Even a very few events of this type will constitute an unambiguous signal. If it is co/o/ that is largest, the 4o/ final state would have four energetic leptons and/or isolated pions plus missing energy a large

5

fraction of the time and be clearly distinct from possible backgrounds. If all the c``'s are small and the \Delta \Gamma W \Gamma mode is allowed, then since the \Delta \Gamma would most probably decay via \Delta \Gamma ! ZW \Gamma ; \Delta 0W \Gamma , we would have final states containing two W \Gamma 's, two W +'s, and ZZ, Z\Delta 0, or \Delta 0\Delta 0. Only a fraction (2=9)4 = 0:0025 of the time would all the W 's decay to ` = e or _; although this is a very backgroundfree channel, the event rate would not generally be adequate. Reconstruction in hadronic channels of some of the W 's would be necessary. Still, at least one or two leptons could be required, along with pairs of energetic jets having mass mW , and a viable signal is likely to emerge from a sample of 100 or more \Delta \Gamma \Gamma \Delta ++ pair events at the LHC. Thus, as stated earlier, we believe it is entirely reasonable to suppose that a \Delta \Gamma \Gamma in the m\Delta \Gamma \Gamma !, 500 GeV mass range relevant for a ps = 500 GeV e\Gamma e\Gamma collider would already have been observed at the LHC, regardless of the magnitude of the c``'s.

If a \Delta \Gamma \Gamma is found, we shall certainly want to learn all about it. However, only limited information concerning the c``'s will be available from the \Delta \Gamma \Gamma \Delta ++ pair production process. The most optimistic scenario is that in which the \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma decay channel is observed, and yet some 4` final state has a significant branching ratio in \Delta \Gamma \Gamma \Delta ++ pair production. (This would imply that the corresponding c`` is at or above the 10\Gamma 5 level. This is perhaps most probable for the 4o/ final state since co/o/ is intuitively likely to be the largest of the c's and co/o/ has no significant bounds at the moment.) In order to convert a measurement of or (more generally) limit on BR(\Delta \Gamma \Gamma ! `\Gamma `\Gamma ) for a given ` into a determination or bound on the corresponding c``, the total width, \Gamma \Delta \Gamma \Gamma , of the \Delta \Gamma \Gamma must be known. If \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma is observed, the partial width \Gamma (\Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma ) could be computed in any given model and combined with the BR(\Delta \Gamma \Gamma ! `\Gamma `\Gamma ) measurements and limits to determine \Gamma \Delta \Gamma \Gamma . One would then be able to give model-dependent results/limits for the c``'s. If all the c's are small, we would have only limits. The above procedure would fail if the \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma final state has too small a branching ratio to be measured and \Delta \Gamma \Gamma decays are dominated by one or more `\Gamma `\Gamma final states. Without a direct measurement of \Gamma \Delta \Gamma \Gamma , the magnitudes of the c``'s could not be determined (although, when more than one channel is seen, ratios could be obtained). The most that could be said is that the c``'s associated with the important decay channels would have to be large enough to overwhelm the \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma decay channel. However, if this decay channel is not kinematically allowed, the semi-virtual \Delta \Gamma ?W \Gamma and \Delta \Gamma W \Gamma ? alternatives would have very tiny partial widths and this constraint would be satisfied for an enormous range of c`` values. Clearly, a direct and model-independent means for probing the c`` values regardless of size is needed.

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3 Detecting the \Delta

\Gamma \Gamma in e\Gamma e\Gamma Collisions

In this section,2 we shall show that e\Gamma e\Gamma collisions are capable of probing extremely small cee values -- values, for example, that span a very large portion of the parameter space for which the see-saw mechanism for neutrino mass would be natural.

A crucial ingredient in the potential of e\Gamma e\Gamma collisions for producing the \Delta \Gamma \Gamma at an observable rate is the ps spectrum. This is determined by the amount of bremsstrahlung and beamstrahlung of photons from the initial e\Gamma 's. Possible designs for the e\Gamma e\Gamma collider are still being developed, but typically point to a spectrum that can be approximated by a Gaussian in the vicinity of the peak energy, with a 1 sigma rms resolution given by oe , 0:2%ps, accompanied by a tail (coming from the beamstrahlung and bremsstrahlung). Current estimates [6] for a 250 GeV \Theta 250 GeV machine are that roughly 38% of the total luminosity will reside in the narrow Gaussian centered at the nominal machine energy, with the tail being such that the average energy loss from beamstrahlung and bremsstrahlung will be of order 3%. If the e\Gamma e\Gamma collider is run at lower energies, more of the luminosity would remain in the central Gaussian peak. The instantaneous luminosity of the design now being considered is L , 6 \Theta 1033cm\Gamma 2 sec\Gamma 1 for the 250 GeV \Theta 250 GeV case, leading to a total yearly luminosity of order L = 60 fb\Gamma 1, of which roughly L = 25 fb\Gamma 1 would reside in the central Gaussian peak. For the estimates made below, we adopt the working hypothesis that a few years of running will provide a total L = 50 fb\Gamma 1 in the central 0:2% Gaussian peak, for all machine energies below ps , 500 GeV. Further, we shall ignore the extra luminosity that resides outside the Gaussian peak; this luminosity would act to increase the rate for \Delta \Gamma \Gamma events, beyond the estimates to be given, when the \Delta \Gamma \Gamma has a total width \Gamma \Delta \Gamma \Gamma ?, 0:002m\Delta \Gamma \Gamma .

A useful mnemonic for the Gaussian rms resolution, taking ps = m\Delta \Gamma \Gamma , is

oe , 0:2 GeV ` m\Delta

\Gamma \Gamma

100 GeV ' `

R 0:2% ' ; (3.1)

where R is the resolution in percent. The crucial issue is how oe compares to \Gamma \Delta \Gamma \Gamma . For c`` = 10\Gamma 5 and R = 0:2%, Eq. 2.3 predicts that \Gamma (\Delta \Gamma \Gamma ! `\Gamma `\Gamma ) = oe for m\Delta \Gamma \Gamma , 70 GeV. If all the c' are much smaller than 10\Gamma 5, the \Delta \Gamma \Gamma is light, and the \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma decay is either strongly suppressed or disallowed, then the \Delta \Gamma \Gamma will have a width much smaller than oe. Conversely, if m\Delta \Gamma \Gamma , 500 GeV and the \Delta \Gamma W \Gamma decay channel has fi\Delta \Gamma W \Gamma * 0:3, then (even if all the c``'s are extremely small) \Gamma \Delta \Gamma \Gamma * 4:4 GeV, i.e. substantially larger than oe , 1 GeV. We will present approximate results for e\Gamma e\Gamma ! \Delta \Gamma \Gamma in the limits \Gamma \Delta \Gamma \Gamma AE oe and \Gamma \Delta \Gamma \Gamma o/ oe.

Using the Gaussian approximation, the effective cross section for \Delta \Gamma \Gamma production in the s-channel is obtained by convoluting the standard s-channel pole form

2Neutral Higgs detection via direct s-channel production in _+_\Gamma collisions has been considered in Ref. [5]. This section employs some of the ideas developed there.

7

with the Gaussian distribution in ps of rms width oe. The resulting cross section is denoted by oe\Delta \Gamma \Gamma . For \Gamma \Delta \Gamma \Gamma AE oe, \Gamma \Delta \Gamma \Gamma o/ oe, oe\Delta \Gamma \Gamma at ps = m\Delta \Gamma \Gamma is given by:

oe\Delta \Gamma \Gamma = 8??!??:

4ssBR(\Delta

\Gamma \Gamma !e\Gamma e\Gamma )

m2\Delta

\Gamma \Gamma ; \Gamma \Delta

\Gamma \Gamma AE oe;

pss 2p2

4ss \Gamma (\Delta

\Gamma \Gamma

!e

\Gamma e\Gamma )

oe m2\Delta

\Gamma \Gamma ; \Gamma \Delta

\Gamma \Gamma o/ oe . (3.2)

In terms of the integrated luminosity L, total event rates are given by Loe\Delta \Gamma \Gamma . As stated earlier, we will assume that L = 50 fb\Gamma 1 can be accumulated in the Gaussian peak centered at the nominal e\Gamma e\Gamma energy.

Consider first the case where \Gamma \Delta \Gamma \Gamma AE oe. We find an event rate coming from the luminosity of the central Gaussian peak (the rate would actually be augmented in this case by a contribution from the beamstrahlung/bremsstrahlung tail) given by

N (\Delta \Gamma \Gamma ) , 2:5 \Theta 1010 `100 GeVm

\Delta \Gamma \Gamma '

2

BR(\Delta \Gamma \Gamma ! e\Gamma e\Gamma ) : (3.3)

If the \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma decay mode dominates the total width, then BR(\Delta \Gamma \Gamma ! e\Gamma e\Gamma ) , 0:3fi\Gamma 3W \Gamma \Delta \Gamma (cee=10\Gamma 5). For L = 50 fb\Gamma 1 we would then have 3 \Theta 108 \Delta \Gamma \Gamma events (dominated by the \Delta \Gamma W \Gamma final state) if cee , 10\Gamma 5, fi\Delta \Gamma W \Gamma = 1 and m\Delta \Gamma \Gamma = 500 GeV. As fi\Delta \Gamma W \Gamma decreases below 1, the number of events grows rapidly. A total of 100 \Delta \Gamma \Gamma events are produced for cee = 1:3 \Theta 10\Gamma 13(m\Delta \Gamma \Gamma =100 GeV)2fi3\Delta \Gamma W \Gamma , i.e. an observable signal would be present for incredibly small cee values. We emphasize that the scenario of a large \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma width is not so unlikely. If the o/ o/ decay mode dominates \Gamma \Delta \Gamma \Gamma , then BR(\Delta \Gamma \Gamma ! e\Gamma e\Gamma ) , cee=co/o/ . For L = 50 fb\Gamma 1, 108 \Delta \Gamma \Gamma events (almost entirely o/ \Gamma o/ \Gamma ) would be obtained for cee=co/o/ = 0:1 and m\Delta \Gamma \Gamma = 500 GeV. In this case, 100 \Delta \Gamma \Gamma events would correspond to cee=co/o/ = 4 \Theta 10\Gamma 9(m\Delta \Gamma \Gamma =100 GeV)2, again a very respectable sensitivity. Note that the phenomenology of this latter case of o/ \Gamma o/ \Gamma dominance of \Delta \Gamma \Gamma decays is essentially independent of \Gamma (\Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma ).

In this \Gamma \Delta \Gamma \Gamma AE oe case, it is important to note that a measurement or calculation of \Gamma \Delta \Gamma \Gamma is required in order that the value of BR(\Delta \Gamma \Gamma ! e\Gamma e\Gamma ) determined from N (\Delta \Gamma \Gamma ) (see Eq. 3.3) and/or direct observation can be used to compute \Gamma (\Delta \Gamma \Gamma ! e\Gamma e\Gamma ) and, thence, cee. A calculation of \Gamma (\Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma ) is possible given a specific choice of the Higgs representation and the observational knowledge of the masses m\Delta \Gamma \Gamma and m\Delta \Gamma . If \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma is the dominant decay mode, then this yields a fairly accurate value for \Gamma \Delta \Gamma \Gamma . However, if the Higgs representation is not known, or \Delta \Gamma \Gamma ! e\Gamma e\Gamma , _\Gamma _\Gamma , and/or o/ \Gamma o/ \Gamma decays dominate to such an extent that BR(\Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma ) cannot be extracted from the data, then determination of \Gamma \Delta \Gamma \Gamma will require its direct measurement. We return to this issue shortly.

The other useful benchmark scenario is that in which \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma is either highly suppressed or forbidden, and all of the c``'s are relatively small. In this case, \Gamma \Delta \Gamma \Gamma o/ oe is probable, with very narrow widths being predicted if the \Delta \Gamma W \Gamma mode

8

is forbidden. Taking L = 50 fb\Gamma 1, and using Eq. 3.1 for oe and the result in Eq. 2.3 for \Gamma (\Delta \Gamma \Gamma ! e\Gamma e\Gamma ), we find from Eq. 3.2 an event rate of

N (\Delta \Gamma \Gamma ) , 3 \Theta 1010 ` cee10\Gamma 5 ' 0:2%R ! ; (3.4) clearly an enormous event rate results if cee is within a few orders of magnitude of its upper bound. If the \Delta \Gamma W \Gamma decay is two-body allowed but all c``'s are very small, the \Delta \Gamma \Gamma final state would be dominated by the real \Delta \Gamma W \Gamma mode (even though fi\Delta \Gamma W \Gamma o/ 1); if \Delta \Gamma W \Gamma is two-body forbidden, one or several of the `\Gamma `\Gamma modes would dominate unless all the c``'s are extremely small, in which case the \Delta \Gamma \Lambda W \Gamma , \Delta \Gamma W \Gamma \Lambda semi-virtual three-body modes would be dominant. The precise cross-over point between the `\Gamma `\Gamma modes and the semi-virtual modes depends on details and will not be pursued here. (Some sample scenarios illustrating this cross-over were explored in Ref. [3].) Note that if the \Delta \Gamma \Gamma is observed at the LHC or NLC, we will know ahead of time what final state to look in and its detailed characteristics, even if the semi-virtual final state is dominant. Only the latter semi-virtual modes and the e\Gamma e\Gamma final state would have significant backgrounds at an e\Gamma e\Gamma collider.

We emphasize that, in the \Gamma \Delta \Gamma \Gamma o/ oe case, Eq. 3.4 shows that the event rate alone is sufficient to determine cee, unlike in the \Gamma \Delta \Gamma \Gamma AE oe case. Direct measurement of \Gamma \Delta \Gamma \Gamma is not required, but would yield important additional information, as described shortly.

To estimate our ultimate sensitivity to cee when \Gamma \Delta \Gamma \Gamma o/ oe, let us suppose that 100 events are required for observation in the real \Delta \Gamma W \Gamma and `\Gamma `\Gamma final state scenarios, and 1000 events if the semi-virtual final states dominate. From Eq. 3.4, we predict 100 \Delta \Gamma \Gamma events for cee , 3:3 \Theta 10\Gamma 14(R=0:2%); note that this result does not depend upon m\Delta \Gamma \Gamma . Once again, we have dramatic sensitivity. Even in the worst case scenario of requiring 1000 events when the semi-virtual modes dominate the final state, we are able to achieve a nearly 8 orders of magnitude improvement over the current limits on cee. Due to the large direct e\Gamma e\Gamma ! e\Gamma e\Gamma background, ?, 1000 events might also be required if e\Gamma e\Gamma final states dominated the \Delta \Gamma \Gamma decay. However, it seems rather likely that cee ! c__ and co/o/ , in which case this situation would not arise. If the _\Gamma _\Gamma final state were dominant, as few as 10 events would probably constitute a viable signal.

In practice, the LHC determination of m\Delta \Gamma \Gamma in \Delta \Gamma \Gamma \Delta ++ pair production will be imperfect. This is not too important if \Gamma \Delta \Gamma \Gamma is large, but could be a significant factor if \Gamma \Delta \Gamma \Gamma ! oe since then a limited scan would become necessary in order to be certain that at least one energy setting corresponded to ps ' m\Delta \Gamma \Gamma to within a fraction of oe. In \Delta \Gamma \Gamma \Delta ++ production, the smallest error, ffim\Delta \Gamma \Gamma for m\Delta \Gamma \Gamma will be achievable in 4e or 4_ final states. The worst case scenario would be dominance of \Delta \Gamma \Gamma ! \Delta \Gamma W \Gamma decays coupled with a very narrow partial width (due to fi\Delta \Gamma W \Gamma o/ 1). The minimum cee for which we will be able to detect the \Delta \Gamma \Gamma

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increases proportionally to the number of scan points required to span 2ffim\Delta \Gamma \Gamma at intervals of , oe. One could conceivably lose as much as a factor of 10 in cee sensitivity in some cases.

In the very unlikely event that we are unable to exclude the existence of a \Delta \Gamma \Gamma with m\Delta \Gamma \Gamma !, 500 GeV by searching for \Delta \Gamma \Gamma \Delta ++ pairs at the LHC, then directly searching for a \Delta \Gamma \Gamma at the e\Gamma e\Gamma collider could be considered. This would require scanning over a broad energy range. Assuming that we could be confident from the NLC that there is no \Delta \Gamma \Gamma with mass below about 250 GeV, then for R = 0:2% we would need about 350-400 energy settings to cover the 250 \Gamma 500 GeV mass range. This would obviously increase the minimum cee value for which a signal could be detected at each scan point by a similar factor. However, if the \Delta \Gamma \Gamma has a large \Gamma \Delta \Gamma \Gamma , a signal would emerge for smaller cee by combining individual scan points. If the \Delta \Gamma \Gamma is very narrow, the smallest possible R value coupled with a finer scan would maximize the chance of seeing a signal in the \Delta \Gamma \Lambda W \Gamma and e\Gamma e\Gamma channels for which the background is significant.

Smaller R would also allow a direct measurement, by scanning, of smaller \Gamma \Delta \Gamma \Gamma . Measurement of \Gamma \Delta \Gamma \Gamma would provide very important additional information regarding the \Delta \Gamma \Gamma in many of the partial width scenarios we have described, but, of course, might also be simply impossible if \Gamma \Delta \Gamma \Gamma is very tiny. As noted earlier, the case in which a direct \Gamma \Delta \Gamma \Gamma determination by scanning would be most important is that in which \Gamma \Delta \Gamma \Gamma ? oe, since the magnitude of \Gamma \Delta \Gamma \Gamma is needed in order to convert the value of BR(\Delta \Gamma \Gamma ! e\Gamma e\Gamma ) into a determination of cee. Fortunately, when \Gamma \Delta \Gamma \Gamma ? oe measurement of the \Delta \Gamma \Gamma total width by scanning will be straightforward. In contrast, if \Gamma \Delta \Gamma \Gamma o/ oe we find from Eq. 3.4 that N (\Delta \Gamma \Gamma ) provides a direct determination of cee. This is a crucial fact given that a very small \Gamma \Delta \Gamma \Gamma might not be measurable directly. In general, if \Gamma \Delta \Gamma \Gamma is known, values for BR(\Delta \Gamma \Gamma ! `\Gamma `\Gamma ) for any of the ` = e; _; o/ channels will then allow us to determine the corresponding c``. This applies, in particular, for ` = _; o/ in the \Gamma \Delta \Gamma \Gamma o/ oe case where event rate alone is adequate to determine cee. It is important to note that direct measurement of \Gamma \Delta \Gamma \Gamma combined with measurement of the _\Gamma _\Gamma and o/ \Gamma o/ \Gamma branching ratios is the only means for determining c__ and co/o/ in e\Gamma e\Gamma collisions. We will not discuss here the luminosity required for determining \Gamma \Delta \Gamma \Gamma by scanning except to note that it will increase rapidly as \Gamma \Delta \Gamma \Gamma decreases below oe. The simple process of centeringp

s to a value ' m\Delta \Gamma \Gamma will already have provided a first crude measurement of \Gamma \Delta \Gamma \Gamma if \Gamma \Delta \Gamma \Gamma is not too much smaller than oe.

Before concluding this section, we note that the ability to polarize the beams could prove very valuable. Returning to Eq. 1.1, we see that the hypercharge of the \Delta \Gamma \Gamma could be determined directly (up to possible extensions of the charge formula such as in the left-right symmetric models), not to mention the fact that the cross section would be enhanced by a factor of four.

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4 Final Remarks and Conclusions

Although currently out of favor because of the success of the minimal supersymmetric model, there are well-motivated models containing triplet and other Higgs representations which include a \Delta \Gamma \Gamma Higgs boson. If such a boson exists in the mass range !, 500 GeV accessible to the e\Gamma e\Gamma collider option at the NLC, it is very likely to be observed at the LHC, even if too heavy (?, 240 GeV) to be seen in pair production in e+e\Gamma collisions at the NLC. If a \Delta \Gamma \Gamma is detected at either the NLC or LHC, we have demonstrated that the e\Gamma e\Gamma collider could be employed as a \Delta \Gamma \Gamma factory, producing potentially billions of \Delta \Gamma \Gamma 's per year if the e\Gamma e\Gamma ! \Delta \Gamma \Gamma coupling is near its current upper bound. More generally, limits on this lepton-number-violating coupling could be improved by roughly 8 orders of magnitude at the e\Gamma e\Gamma collider, with some dependence on the \Delta \Gamma \Gamma total width and decay pattern. In left-right symmetric models, this implies sensitivity to much of the coupling strength range for which the see-saw mechanism for neutrino mass generation operates most naturally.

Further, if the total width, \Gamma \Delta \Gamma \Gamma , of the \Delta \Gamma \Gamma can be measured by scanning, and if a given `\Gamma `\Gamma final state has measurable branching ratio, then we can combine these quantities to obtain the `\Gamma `\Gamma ! \Delta \Gamma \Gamma coupling. This is the only technique for determining this coupling in the ` = _; o/ cases. It is also the only way to directly determine the e\Gamma e\Gamma coupling when \Gamma \Delta \Gamma \Gamma is larger than the ps resolution.

We emphasize that if m\Delta \Gamma \Gamma ?, ps=2 for the e+e\Gamma collider, then in order to avoid a broad scan search for a \Delta \Gamma \Gamma we must have an approximate determination of m\Delta \Gamma \Gamma via detection of \Delta \Gamma \Gamma \Delta ++ pair production at the LHC. (Such a determination would be especially crucial if \Gamma \Delta \Gamma \Gamma is much smaller than the energy resolution oe of the e\Gamma e\Gamma collider.) This fact provides yet another example of the complementarity of the NLC and the LHC.

Of course, an exactly parallel set of results would apply to a _\Gamma _\Gamma collider.3 Indeed, there are two potential advantages of a _\Gamma _\Gamma collider over an e\Gamma e\Gamma collider. Both advantages derive from the much reduced beamstrahlung and bremsstrahlung associated with muon beams. First, the resolution R for a _\Gamma _\Gamma collider could potentially be much smaller than that for an e\Gamma e\Gamma collider, and, in addition, more of the total luminosity will reside in the Gaussian peak centered at the nominal machine energy. Preliminary studies of _+_\Gamma colliders indicate that R values as small as R , 0:01% might be achievable [8]. In the case that \Gamma \Delta \Gamma \Gamma is very small, a factor of roughly R(__)=R(ee) increase in the \Delta \Gamma \Gamma production rate and corresponding sensitivity to c__ vs. cee would result from the superior resolution alone. In the absence of on-shell \Delta \Gamma W \Gamma decays of the \Delta \Gamma \Gamma , c__ values in the 10\Gamma 15 range would be probed for R(__) , 0:01% assuming that the \Delta \Gamma \Gamma is already discovered at the LHC or NLC so that a broad scan is not necessary. The second advantage of a __ collider might turn out to be larger energy reach. It is anticipated [7] that

3We note that _+_\Gamma colliders are already being actively considered [7].

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ps as high as 4 TeV might eventually prove feasible. Thus, \Delta \Gamma \Gamma \Delta ++ pair production could be detected in the _+_\Gamma mode of operation up to very high m\Delta \Gamma \Gamma , and then the _\Gamma _\Gamma mode of operation would allow a high-sensitivity probe of c__. Of course, if the \Delta \Gamma \Gamma could be detected in both e\Gamma e\Gamma and _\Gamma _\Gamma collisions, then we would measure both cee and c__. These determinations of cee and c__ could then be compared to those independently extracted by measuring \Gamma \Delta \Gamma \Gamma , BR(\Delta \Gamma \Gamma ! e\Gamma e\Gamma ) and BR(\Delta \Gamma \Gamma ! _\Gamma _\Gamma ) at the e\Gamma e\Gamma and/or _\Gamma _\Gamma colliders.

Overall, it is apparent that if a doubly-charged Higgs boson is found at the NLC or LHC, e\Gamma e\Gamma and _\Gamma _\Gamma colliders would separately and in combination provide enormously important information concerning the structure and interactions of the Higgs sector.

5 Acknowledgements This work was supported in part by Department of Energy grant DE-FG03-91ER40674 and the Davis Institute for High Energy Physics.

References

[1] Much of the introductory material given here is extracted from J.F. Gunion,

H.E. Haber, G.L. Kane and S. Dawson, The Higgs Hunters Guide, AddisonWesley Publishing. Detailed references to the original papers can be found therein.

[2] Many of the relevant phenomenological considerations for Higgs triplet representations, especially in the context of left-right symmetric models, are outlined in J.F. Gunion, J. Grifols, B. Kayser, A. Mendez, and F. Olness, Phys. Rev. D40, 1546 (1989).

[3] Phenomenology for Higgs triplet models, focusing on the model with one jY j =

0 and one jY j = 2 triplet, is considered in: J.F. Gunion, R. Vega, and J. Wudka, Phys. Rev. D42, 1673 (1990.)

[4] K. Huitu and J. Maalampi, preprint . [5] V. Barger, M. Berger, J. Gunion and T. Han, Phys. Rev. Lett. 75, 1462 (1995);

UCD-95-27, in preparation.

[6] Jim Spencer, private communication. [7] Proceedings of the Second Workshop on the Physics Potential and Development of _+_\Gamma Colliders, Sausalito, California (1994), ed. by D. Cline, to be published.

[8] G.P. Jackson and D. Neuffer, private communications.

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