INVISIBLE HIGGS DECAYS and NEUTRINO PHYSICS Anjan S. Joshipura1 J. W. F. Valle Y ABSTRACT

Home , Majoron, Neutrino

CERN-TH.6652/92

FTUV/92-35

IFIC/92-34

INVISIBLE HIGGS DECAYS AND NEUTRINO PHYSICS

Anjan S. Joshipura

Theoretical Physics Division, CERN

CH-1211 Geneve 23, Switzerland

and

J. W. F. Valle

Instituto de Fsica Corpuscular - IFIC/CSIC

Dept. de Fsica Teorica, Universitat de Valencia

46100 Burjassot, Valencia, SPAIN

ABSTRACT

A wide class of neutrino physics-motivated mo dels are characterized by the sp onta-

neous violation of a global U (1) lepton numb er symmetry at or b elow the electroweak

scale byan SU (2) U (1) singlet vacuum exp ectation value h i O (1) TeV. In all

these mo dels the main Higgs decaychannel is likely to b e "invisible", e.g. h ! JJ,

where J denotes the asso ciated weakly interacting pseudoscalar Goldstone b oson -

the ma joron. This leads to events with large missing energy that could b e observable

at LEP and a ect the Higgs mass b ounds obtained, as well as lead to novel ways to

search for Higgs b osons at high energy sup ercolliders such as the LHC/SSC.

CERN-TH.6652/92

Septemb er 1992

Permanent address: Theory Group, Physical Research Lab., Ahmedabad, India

Bitnet JOSHIPUR@CERNVM

Bitnet VALLE@EVALUN11 - Decnet 16444::VALLE

1 Intro duction

One of the main puzzles in particle physics to day is the problem of mass generation.

It is b elieved that the masses of the fermions as well as that of gauge b osons arise as

a result of the sp ontaneous breaking of the gauge symmetry. The key ingredient for

this scenario, namely the Higgs b oson [1], has not yet b een found. It is only recently,

with the LEP exp eriments, that one has seriously started constraining the relevant

parameters, including the Higgs b oson mass [2]. The limits on the Higgs mass are,

however, rather mo del dep endent. The present limit on the standard mo del Higgs

coming from the data on e e collisions at LEP is 60 GeV.

An extension of the minimal standard mo del is desirable for many reasons.

One is the question of neutrino masses. Indeed neutrino masses vanish in the mini-

mal standard mo del and almost all attempts to induce them require an enlargement

in the Higgs sector of the theory [3]. Among these, mo dels known as ma joron mo dels

are particularly interesting and have b een extensively studied [3]. The ma joron is

a Goldstone b oson asso ciated with the sp ontaneous breaking of the lepton numb er.

In the mo dels we shall consider it has very tiny couplings to the charged fermions

as well as to the gauge b osons. As a consequence, the ma joron remains invisible.

The ma joron can however have signi cant couplings to Higgs b osons even if its

other couplings are suppressed. This could have imp ortant implications for Higgs

physics. In particular, the normal doublet Higgs b oson could decayinvisibly as

h ! J + J; (1)

where J denotes the Goldstone b oson {the ma joron { asso ciated with sp ontaneously

broken lepton numb er symmetry. The p ossibility of a Higgs b oson decaying invisibly

was raised by Shro ck and Suzuki and reconsidered by Li, Liu and Wolfenstein [4] in

the context of the triplet ma joron mo del [5]. This typ e of mo dels are now excluded

since they lead to an invisible Z width in con ict with LEP observations [6]. Despite

this, the p ossibilityofinvisible Higgs decay still remains op en and exp erimentally

very amusing [7]. A concrete example [8] was recently provided in the context of

sup ersymmetric SU (2) U (1) mo dels where the R parityisspontaneously violated

at (or b elow) the electroweak scale [9]. The lightest Higgs b oson h decays in this

mo del through ma joron emission. Unfortunately, its pro duction rates are likely to

b e small in this case, esp ecially in the low mass region. While this completely avoids 1

the existing LEP1 limits, it is not so useful for the exp erimental detection of the new

e ects at LEP1 (prosp ects of observing such decays are b etter at higher energies).

The ab ovetyp e of suppression in the pro duction of the low mass Higgs b oson

need not o ccur in all mo dels. An example where such suppression can b e absentis

provided [10 ] by the seesaw ma joron mo del [11 ], provided the scale of lepton number

O (1) TeV. This mo del, with suchvacuum exp ectation value violation ob eys h i

(VEV), mayhaveinteresting physical implications including neutrinos with masses

very near their present exp erimental limits [12 ]. However, this is not the most

natural choice for the lepton numb er violation scale if neutrino masses are very tiny

O (1) eV. The masses of the light neutrinos are given by

m ; (2)

where m = h i and M hi. Here hi is the VEV that breaks the SU (2) U (1)

D R

symmetry while h i breaks the global lepton numb er symmetry. Barring unnaturally

small Yukawa couplings , the smallness of neutrino masses follows only if h i

O(1) TeV. Typical mo dels asso ciate h i to a large mass scale at which some higher

symmetry such as left-right, Peccei-Quinn or grand-uni ed symmetries get realized.

As we shall discuss the ma joron-Higgs coupling is suppressed in this case.

In this pap er, we note that there exists a wide class of interesting mo dels

for neutrino masses in which lepton numb er breaking is driven by an isosinglet

VEV (as required by the LEP constraints), but in which the asso ciated scale ob eys

h i O (1) TeV. The distinguishing feature of these new mo dels is that, unlike

seesaw mo dels, where m /hi ,inany of the present mo dels m ! 0as hi!0.

As a result a lowvalue of h i is required in order to obtain a small neutrino mass

either at the tree level or radiatively.

We discuss the invisible decay of the Higgs b osons in this typ e of mo dels. In

contrast with the two situations discussed ab ove, neither the invisible decay nor the

pro duction of the Higgs b osons need to b e suppressed in these mo dels. Moreover, this

feature p ersists even when the lepton numb er symmetry is broken at a scale much

smaller than the weak scale. The latter would lead to the p ossibility of enhanced

ma joron-neutrino couplings. These could, in turn, haveinteresting implications in

neutrinoless double b eta decayaswell as astrophysics [13 ]. In addition, in all cases

these mo dels can lead to interesting physical e ects such as large rates for zen events

at LEP, and avour{violating muon and tau decays with large branching ratios. The 2

former would b e asso ciated to single neutral heavy lepton pro duction and the latter

to neutral heavy lepton exchange in higher order weak pro cesses. The corresp onding

rates can b e large enough to b e exp erimentally measurable. [14].

In the next section, we discuss the main features of various mo dels of neutrino

masses with the lepton numb er broken at a relatively low scale. The third section

contains details of the Higgs p otentials and the ma joron couplings to the Higgs

b osons. The corresp onding decay and pro duction rates are studied in section 4. The

last section contains a discussion of some of the phenomenological implications. The

technical details related to very low-scale breaking of the lepton numb er symmetry

are given in the app endix.

2 Mo dels

Wenow consider several SU (2) U (1) mo dels that have b een suggested in neutrino

physics in order to generate naturally small neutrino masses, either as a result of

radiative corrections or at the tree level . In all these mo dels lepton number is a

symmetry of the Lagrangian. This is sp ontaneously broken by h i O (1) TeV,

thus generating a ma joron given by

J =Im: (3)

In most resp ects, these mo dels all share the existence of a massless isosinglet pseu-

doscalar ma joron, very much the same as the original one in ref. [11 ]. As a result,

all astrophysical constraints [15] related to stellar co oling by ma joron emission can

easily b e ob eyed.

However, there is an imp ortant di erence. In the seesaw ma joron mo del, the

smallness of neutrino masses is linked to the lepton numb er violation at a very high

mass scale, i.e. m ! 0as hib ecomes large. In all the present mo dels m ! 0

as h i!0. The remarkable fact that no mass scale is intro duced ab ove the weak

scale in any of these mo dels is crucial in ensuring the imp ortance of the invisible

Higgs decay (1) relative to the standard mo des suchas h ! ff.Aswe will see,

the invisibly decaying Higgs b oson signature may p ersist even in the limit where

For simplicitywe assume that all of the Yukawa coupling constants are real in all that follows. 3

h iO (1) TeV. This maybeinteresting from the p oint of view of pro cesses such

as neutrinoless double b eta decay with ma joron emission [13].

2.1 Singlet ma joron in "-mo del"

This is an SU (2) U (1) variant [16] of the "seesaw" mo del with the heavy Dirac

lepton suggested in ref. [17]. The relevant terms in the Lagrangian are

T c c T

h ` C H + M CS + fS CS + h:c:; (4)

They involve a bare Dirac mass term M and the Yukawa couplings h ;f . These are

describ ed by arbitrary matrices in generation space. The rst coupling generates

the neutrino Dirac mass term D = h h i, while the third term gives rise to the

Ma jorana mass term for the isosinglet S . This violates lepton number bytwo

units, = f h i. The full mass matrix in two-comp onent basis ; ;S can b e

written as

1 0

C B

0 D 0

C B

: (5)

D 0 M

C B

A @

0 M

For small values of the parameter D M , the heavy leptons here are of quasi-

Dirac typ e and the Ma jorana mass of the light, mostly iso doublet neutrino arises

from the exchange of the heavy leptons, as indicated in Fig. 1(a). The resulting

{

mass is given by

m = ; (6)

Note the di erent relationship b etween m and h i, the lepton numb er breaking

scale. This is a crucial feature of this mo del, which contrasts with the simplest

seesaw mo del [11 ]. This di erence arise from the fact that the the mo del contains a

quasi-Dirac heavy lepton whose mass M is invariant under lepton numb er and is

unrelated to neutrino masses. In contrast, the minimal seesaw mo del has a heavy

Ma jorana lepton whose mass M is inversely related to that of the iso doublet

x c c

A entry would not give a mass to the light neutrinos and can b e forbidden by requiring

sup ersymmetry.

{

For simplicitywe assume here only one generation. The complete form can b e found in ref.

[16]. 4

neutrino. In b oth cases the heavy lepton admixture in the weak charged currentis

determined by the ratio D=M .However, in the "-mo del" this value is restricted

only byweak universality constraints, and not by limits on the neutrino masses [3].

The crucial p oint is that the heavy lepton mass here arises mostly from the entry

M whichisinvariant under lepton numb er, unlike the case of the seesaw mo del. As

a result M can b e relatively low without implying to o large m values. In fact, in

the limit where lepton numb er is exact neutrino masses are strictly forbidden [17].

As a result, there is a rich class of pro cesses that can b e enhanced for values

0:1, well consistent with presentweak universality constraints [14 ]. The of D=M

resulting phenomenology has b een considered in several pap ers. For masses b elow

m the heavy quasi-Dirac leptons may b e singly pro duced at LEP1, giving rise to

striking events characterized by a large amount of missing energy [18]. For higher

masses the existence of such neutral heavy leptons can at present only b e prob ed

through their indirect e ects. For example, if we include mixing b etween the var-

ious generations wehave the interesting p ossibilityof avour and/or CP violation

[19], even in the limit where ! 0, while the iso doublet neutrinos b ecome strictly

massless. As a result, pro cesses suchas !e , ! e , {e conversion in nuclei,

Z ! e , etc. are not only allowed, but their rates are restricted only by the preci-

sion of weak universality tests. As a result, they can all b e within the sensitivityof

present exp eriments as well as of those exp ected at the up coming factory [20].

2.2 Singlet ma joron in Zee-typ e mo dels

Wenow turn to mo dels where neutrino masses are radiatively induced. The simplest

p ossibilitywould b e to consider mo dels with just the three usual neutrinos. The

prototyp e of these mo dels was originally suggested by Zee [21 ]. Here we consider a

variant whichintro duces the sp ontaneous violation of lepton numb er [22 ], so as to

generate the ma joron eq. (3). The relevant terms in the Lagrangian are the Yukawa

couplings

+ T

Ci ` + h:c: ; (7) ` e + h ` e + f `

2 j i Ri ij i Ri ij

where i; j = e; ; . The rst term is the canonical one, resp onsible for generating the

charged lepton masses m when the SU (2) U (1) gauge symmetry is broken by h i.

i Y

The additional couplings involve another Higgs doublet as well as the Zee singlet;

they are sp eci ed by matrices f; g (in generation space), f b eing antisymmetric. In 5

addition we use the following quartic term in the scalar p otential

+ h:c: (8)

instead of the usual cubic term that would explicitly violate lepton numb er.

The quartic coupling xes L( )=2=L() and induces a mixing b etween the

physical singly charged scalars, which plays a crucial role in the radiative generation

of neutrino mass, through the diagram in Fig. 1(b).

One can also consider the singlet ma joron in a coloured version of the Zee

mo del suggested in [13]. This is a variant of the previous mo del which is de ned by

the Lagrangian

T c T

` e + h ` Cb + f ` CQ + + h:c:; (9)

i Ri ij D ij Lj S 2 D S

i Lij i

2=3

where i; j = 1...3, =( ) and are colour triplet lepto quark scalar b osons;

D S

1=3

is an SU (2) U (1) doublet with Y =1/3 and L=1 while is a singlet with

D S

Y =2/3 and also L=1; b are SU (2) singlet charged 1/3 quarks. Again lepton

number is spontaneously broken by h i and this generates the ma joron, as in the

previous cases. Neutrino masses are now induced from the second graph in Fig.

1(b).

In this mo del the ma joron-neutrino couplings are enhanced with resp ect to the

rst case of eq. (7), leading to the observability of neutrinoless double b eta decay

with ma joron emission [13 ].

2.3 Singlet ma joron in mo dels with sterile neutrinos

Recently there has b een a lot of interest in the p ossible existence of light ster-

ile neutrinos [23, 24 ]. These mo dels provide a common framework in terms of

which to explain the solar neutrino data and the existence of a hot dark matter

neutrino comp onent in the universe, as recently suggested by COBE results on the

large-scale structure of the universe [24]. They may also b e relevant in connection

with the atmospheric muon neutrino de cit as well as with the p ossible existence of

anomalies in b eta decays asso ciated with heavy neutrinos .

In fact they have b een originally suggested in relation with the 17 keV decay anomaly. 6

The simplest mo del [23] is again based on the SU (2) U (1) gauge group, but

++ + +

extends the standard mo del by adding four singlet Higgs b osons k ; ;h and

and one SU (2) U (1) singlet neutrino nu . The Yukawainteractions are given by

2 m

T + T ++ T +

` e + f ` Ci ` h + h e Ce k + Ce + h:c:; (10)

i Ri ij 2 j ij Rj i Ri

i Ri S

where ` denotes a lepton doublet and f ; h; are dimensionless Yukawa couplings. In

addition the mo del contains the following crucial scalar self interactions

++ +

h k + h + h:c: ; (11)

where is dimensionless and has dimensions of mass and its magnitude is at

the weak scale. The mo del has a global lepton numb er symmetry U (1) assigned

canonically to the standard mo del states. The quantum numb er assignments and

particle content are summarized in Table 1.

The neutrino mass matrix that follows from electroweak and U (1) violation

has the following form in the basis ( ; ; ; )

e S

0 1

B C

m M

ij i

B C

M = : (12)

B C

@ A

Before U (1) is broken the only non-zero entries are M . In this limit, two of the

G i

neutrinos are massless and the other two form a Dirac state with mass

2 2 2

m M + M + M : (13)

This mass is induced by the diagram in Fig. 2(b). This state is identi ed bytwo

angles and ' de ned as

M M

sin = tan ' = : (14)

m M

Here

M f m

H ia a a

M = sin 2 ln ; (15)

32 M

where is the mixing angle of the scalar b osons and M are their mass eigenvalues.

For suitable choices of Yukawa couplings, the Dirac neutrino mass can b e at the dark

matter scale.

The entries m and only arise at the 2-lo op level from the diagrams in Fig.

2(a) and 2(c). They are more highly suppressed also b ecause they involve additional 7

electroweak violating lepton mass insertions m or U (1) -violating h i insertions .

ij G

These are estimated as

h i h

ab a b

a;b

I sin 2 (16)

128

hi f f h m m

ia jb ab a b

a;b

I ; (17) m

m ij

256 M

where M is a typical Higgs b oson mass, I and I are the relevantFeynman in-

0 m

tegrals, typically of order one (I ! 0, when M ! M ). These terms give

H H

1 2

masses to the lowest-lying neutrinos resp onsible for the explanation of the solar

neutrino data. For suitable values of the parameters, these are in the right range to

give a solution of the solar neutrino de cit via matter enhanced to transitions.

These MSW transitions involve an additional angle needed to diagonalize the re-

sulting light mass matrix. For alternativechoices of parameters, there can also b e

an explanation of the muon de cit in atmospheric neutrinos [24].

3 Scalar Higgs p otential

To complete the sp eci cation of the mo dels, wemust now discuss the asso ciated

scalar p otential. It is clear that all these mo dels are characterized bytwo basic

typ es of scalar p otentials, when one considers only the neutral sector needed to

determine the vacuum. The mo dels in sections 2.1 and 2.3 are characterized bya

scalar p otential with one doublet and one singlet Higgs multiplet, while the scalar

p otential of the remaining mo dels contains an additional Higgs doublet. We discuss

b oth of these in turn.

3.1 One scalar Higgs doublet and one singlet

This case was discussed in [10]. We recall the basic asp ects. The scalar p otential is

given by

2 y 2 y y 2 y 2 y y

V = + + ( ) + ( ) + ( )( ) : (18)

N 1 2

Terms like are omitted ab ove in view of the imp osed U (1) invariance under which

w R +iI

2 2

p p

we require to transform non-trivially and to b e trivial. Let + ,

2 2

v R +iI w v

0 0

1 1

p p p p

+ , where wehave set h i = and h i = . The ab ove p otential

2 2 2 2 8

then leads to a physical massless Goldstone b oson, namely the ma joron J Im

and two massive neutral scalars H (i = 1,2):

H = O R : (19)

i ij j

The mixing O can b e parametrized as

0 1

B C

cos sin

B C

^ B C

O = (20)

B C

@ A

sin cos

The mixing angle as well as the Higgs masses M are related to the parameters of

the p otential in the following way:

2 2

2vw = (M M ) sin 2;

2 1

2 2 2 2 2

2 v = M cos + M sin ;

1 2

2 2 2 2 2

2 w = M cos + M sin :

2 1

tan 2 = : (21)

2 2

v !

1 2

The masses M , the mixing angle , and the ratio of twovacuum exp ectation values

1;2

tan = can b e taken as indep endent parameters in terms of which all couplings

can b e xed. There are no physical charged Higgs b osons in this case.

The p otential in eq. (18) generates the following coupling of H to the ma joron

J :

1=2

2G ) (

2 2 2

tan [M cos H M sin H ]J : (22) L =

2 1 J

2 1

3.2 Two scalar Higgs doublets and one singlet

The part of the scalar p otential containing the neutral Higgs elds is given in this

case by

y y

y 2 y 2 2 2

+ ( ) + ( ) + V =

i i i N 2

i i

y y y y

y y

+ ( )( )+ ( )( )+ ( )( )

12 1 2 13 1 23 2

1 2 1 2

y y y

+ ( )( )+ [( ) + h:c:]; (23)

2 1 2

1 2 1

where a sum over rep eated indices i=1,2 is assumed. Here are the doublet elds

1;2

and corresp onds to the singlet carrying non-zero lepton numb er. 9

In writing down the ab ove equation, wehave imp osed a discrete symmetry

! needed to obtain natural avour conservation in the presence of more

2 2

than one Higgs doublets. For simplicity,we assume all couplings and VEVs to b e

real. Then the conditions for the minimization of the ab ove p otential are easy to

work out and are given by

1 1 1

2 2 2 2 2

+ v + ( + )v + v + v =0; (24)

1 12 13

1 1 2 3 2

2 2 2

1 1 1

2 2 2 2 2

+ v + ( + )v + v + v =0; (25)

2 12 23

2 2 1 3 1

2 2 2

1 1

2 2 2 2

+ v + v + v =0: (26)

3 13 23

3 3 1 2

2 2

These conditions can b e used to work out the mass matrix for the Higgs elds.

To this end we shift the elds as (i=1,2):

v R + iI

i i i

p p

= + ; (27)

2 2

! R + iI

3 3

p p

= + : (28)

2 2

The masses of the CP-even elds R (a=1...3) are obtained from

T 2

R M R; (29) L =

mass

with

0 1

B C

2 v ( + + )v v v v

1 12 1 2 13 1 3

B C

M = : (30)

R B ( + + )v v 2 v v v C

12 1 2 2 23 2 3

B C

@ A

v v v v 2 v

13 1 3 23 2 3 3

The physical mass eigenstates H are related to the corresp onding weak eigenstates

H = O R ; (31)

a ab b

where O isa33 matrix diagonalizing M

2 2 T 2 2

) : (32) ;M ;M O =diag (M OM

3 2 1 R

The ma joron is given in this case by J = I . The coupling of the physical Higgses

to J follows from eq. (23). As in the previous case, it is p ossible to express this 10

coupling entirely in terms of the masses M and the mixing angles characterizing

the matrix O

L = J (2 v R + v R + v v R ) ; (33)

J 3 3 3 13 1 1 23 2 3 2

(M ) R ; (34) =

3a a

1=2 T 2 2

= ( 2G ) tan (O ) M H J : (35)

F 3a a

2 2 1=2

tan ; V =(v +v ) .Wehave made use of eq. (31) and eq. (32) in writing

1 2

the last line.

Unlike in the previous case, there now exists also a massive CP-o dd state A,

related to the doublet elds as follows:

(v I v I ) : (36) A =

2 1 1 2

Its mass is given by

2 2

M = V : (37)

When ! 0 this pseudoscalar b oson b ecomes massless, as the p otential acquires a

new symmetry.

4 Higgs pro duction and decay

The Higgs can b e pro duced at the e e collider through its couplings to Z . Although

the SU (2) U (1) singlet eld do es not couple to Z , all of the CP-even mass

eigenstates H intro duced in the last section have couplings to the Z through mixing.

The couplings relevant for their pro duction through the Bjorken pro cess are given

as follows (a=1...3)

v v

1 2

1=2 2

2G ) Z Z O + O H (38) M L =(

F 1a 2a a HZZ

V V

in the two doublet case and (i=1,2)

1=2 2

L =( 2G ) M Z Z O H (39)

HZZ F i1 i

for the case considered in section 3.1. As long as the mixing app earing in eq. (39)

and eq. (38) are O (1), all Higgs b osons can have signi cant couplings and hence

appreciable pro duction rates through the Bjorken pro cess. 11

In case of the two doublet mo del, the H can also b e pro duced in asso ciation

with the CP-o dd eld A through the coupling

g v v

2 1

@ A: (40) L = Z H O O

HAZ a 1a 2a

cos V V

The width for the invisible H decay can b e parametrized by

3 2

(H ! JJ)= M g : (41)

H HJJ

The one doublet mo del contains two Higgses H whose couplings are given by

g = tan O : (42)

H JJ i2

The analogous couplings in the case of the mo del of section 3.2 are given by(a=

1:::3)

g = tan O : (43)

H JJ a3

The rate for H ! bb also gets diluted in comparison to the standard mo del predic-

tion, b ecause of the mixing e ects. Explicitly one has

3 2G

2 2 2 3=2 2

(H ! b M m (1 4m =M ) g b)= ; (44)

b b H

Hb b

with

= O (one-doublet mo del) g

b H b

(45)

g = O (two-doublet mo del)

H b b

The last coupling dep ends up on how the charged 1/3 quarks transform under the

symmetry whichavoids the avour changing neutral currents in the presence of the

two Higgs doublets. Wehave assumed that this symmetry allows only the to

couple to the d-typ e quarks.

The width of the Higgs decaytotheJJ relative to the conventional bb mo de

dep ends up on the mixing angles. The invisible mo de is exp ected to dominate if the

lepton numb er is broken around or b elow the weak scale. In order to appreciate this

p oint, let us consider the relatively simple situation [10 ] with only one Higgs doublet,

as in the mo del of section 3.1. One could imagine three cases: (i) ! v , (ii) ! v

and (iii) ! v . It follows from eq. (21) that in the rst case, the mixing among the

doublet and singlet eld will b e O (1) if the parameters of the quartic terms in the

Additional contributions due to the decayofH to AA may exist.

a 12

Higgs p otential are similar in magnitude. As a result, the pro duction as well as the

decay of b oth physical Higgs b osons H will b e comparable and could b e observable.

The relative branching ratio in this case is given by

(H ! JJ) 1 M

1 1

2 2 3=2 2

= (1 4m =M ) (tan tan )

b 1

12 m

(H ! bb)

8 (tan tan ) : (46)

50GeV

A similar expression with tan replaced by cot holds in the case of H . It is clear

that a Higgs b oson with M > 50 GeV decays mostly invisibly if tan and tan

are O (1). The pro duction of H (H ) gets diluted compared to the standard mo del

1 2

prediction by cos (sin ). If ! and v are very di erent from each other then

the mixing angle in eq. (21) is very small. Hence in cases (ii) and (iii), only the

predominantly doublet comp onent(H ) will b e pro duced. Use of eq. (21) in the

basic ma joron coupling, eq. (22), reveals that

1=2 2

lim L =( G ) v! H + H O( ) J ; (47)

J F 2 2 1

! v

! 1

1=2 2 2

H +H O( lim L = ( 2 G ) v ) J : (48)

1 2 J F

! v

2 v

It follows that if ! v then only the singlet eld decays to two ma jorons. But

this cannot b e pro duced. In the converse case, ! v , the doublet eld mainly

decays to ma jorons and this also gets pro duced without any substantial suppression

relative to the standard mo del predictions, in view of its small mixing with the singlet

comp onent. This case is then the ideal from the p oint of view of the observability

of the invisible decay and may also lead to the observability of neutrino-ma joron

couplings and to that of neutrinoless double b eta decay with ma joron emission [13 ].

As would b e exp ected, the presence of one more doublet do es not qualitatively

change this conclusion. This is demonstrated in the app endix.

5 Discussion

The Higgs can decayinvisibly in a wide class of SU (2) U (1) singlet ma joron mo dels.

Wehave shown that the pro duction of such Higgs b osons as well as their invisible

decay width could b e sizeable. In fact the Higgs decay width arising from eq. (1)

yy 2

The factor (tan tan ) app earing in eq. (46) go es to a constantvalue in this limit.

1 13

can dominate over that of the standard bb mo de in all such mo dels. This leads to

events with large missing energy carried by the ma joron pair. Since this signature

is very di erent from the conventional Higgs decay, a reanalysis of the present Higgs

search strategies is needed. A comparison of some of the existing LEP data with

the one-doublet mo del of section (3.1) has already b een [10] given, treating sin and

tan as indep endent parameters. Use of more data and a similar comparison of the

two doublet mo del is worth while. While this is not the aim of the presentwork, we

will make some comments on the main phenomenological implications.

The search strategy for the Higgs dep ends up on its mass and on the decay

characteristics relevant for this mass. Accordingly, a Higgs b oson with M 80

GeV can b e lo oked for at LEP1 or LEP2. Heavier Higgs b osons can b e searched at

hadron colliders. If M is less than twice the mass of the W , one has to rely up on its

rare decays suchas , while a heavier Higgs b oson can b e found through its WW

and ZZ decay mo des. All of these searches can b e substantially a ected once the

Higgs decay to ma jorons b ecomes p ossible. Consider, for example, the searches b eing

carried out at LEP for Higgs with masses greater than ab out 10 GeV. This dep ends

up on detecting the Higgs decay to the b b pair. Since the branching ratio for this

mo de gets diluted in the presence of the invisible decay, the Higgs could have escap ed

detection at LEP. This was analysed in ref. [10]. It was shown that a large region

in the parameter space still remains unconstrained in the simple mo del of section

3.1. In particular, anyvalue of M is allowed for a suitable range in and tan .

The mo dels with two Higgs doublets (section 2.2) are even less constrained owing

to the presence of more parameters. But in this case there exists an additional way

to pro duce the Higgs, namely through the asso ciated pro cess Z ! AH , eq. (40). In

the general two Higgs doublet mo dels and in the sup ersymmetric ones, the pro cesses

Z ! Z H and Z ! AH are known to provide complementary information on the

relevant mixing parameters. In the present case, since there exists an additional

mixing involving the singlet eld , the mixings app earing in eq. (38) and eq. (40)

are not related in a simple way, as in the two doublet case. However, if the lepton

numb er breaking do es o ccur at very low scale then the mixing b etween doublet and

singlet is very small (see app endix) and the ab ove pro duction mechanisms can b e

used simultaneously in order to restrict the parameters of the mo del.

Invisible Higgs decay could b e directly observed if the Higgs b oson is pro duced

in asso ciation with a photon, W or Z . The latter can b e used as a tag of the 14

invisible mo de. The pro duction of the Higgs in asso ciation with the photon at LEP

is unfortunately quite suppressed. But the other decays could in principle b e used.

The p ossibilityofinvisible Higgs decays is sp ecially interesting for the case of hadron

colliders [14 ]. In fact, a recent analysis has studied the feasibility of detecting an

invisible Higgs pro duced in asso ciation with W;Z at hadron colliders [25 ]. They

conclude that, with reasonable assumptions, a dominantly decaying Higgs b oson

lighter than 2m can b e detected at b oth LHC and SSC. It is interesting to note

that this intermediate Higgs mass region is one where the traditional tree level f f

mo de is not useful.

The main conclusion of the presentwork, whichwewould like to stress again,

is that the invisible Higgs decays are a generic feature of a wide class of singlet

ma joron mo dels of neutrino mass generation, which are interesting in their own

right. Since the ma joron do es not appreciably couple to fermions nor to gauge

b osons, suchinvisible decaymaybea very go o d way to test the validity of the

ma joron hyp othesis itself. Moreover, since such decay could hide the Higgs, sp ecial

e orts would b e needed to lo ok for the Higgs b oson in this case. But the Higgs b oson,

if discovered through its invisible decay,would tell us that not only SU (2) U (1)

but also the lepton number is a spontaneously broken symmetry.

This work was partially supp orted by CICYT (Spain). We thank Riccardo

Barbieri for useful discussions. A.S.J. thanks S. Rindani for discussions related to

this work. 15

App endix

We discuss here the case of very small violation of lepton numb er symmetry. Sp ecif-

ically,we shall assume v v . The mass matrix M for the CP-even elds can

3 1;2

b e written in the (R ;R ;R ) basis as

1 2 3

0 1

B C

A B

B C

2 2

B C

M = V ; (49)

R B B C C

B C

@ A

1 2

where,

v v v v v

1 3 2 3 3

; ; =2 : (50)

1 13 2 23

2 2 2

V V V

The parameters A; B ; C can b e read from eq. (30) and are O (1), while are

1;2;3

similar in magnitude but are much smaller than A; B ; C . The matrix M can b e

approximately diagonalized by the O given by

O = R ( )R ( )R ( ); (51)

13 3 23 2 12 1

where R denotes a rotation in the ab plane. To leading order, the masses and

mixing angles are given by

; tan 2

C A

2(sin + cos )

1 1 1 2

tan 2 ; (52)

2 cos ( sin + cos )

2 1 2 1 1

tan 2 ;

2 2 2

M A cos + C sin sin 2 B;

1 1 1

2 2 2

M Asin + C cos + sin 2 B: (53)

1 1 1

2 2

The other eigenvalue M is of O ( ). The elements of the mixing matrix relevant

3 3

to determine the Higgs coupling to a ma joron [eq. (33)] can b e read o from eq.

(51) and eq. (52).

2 2

v v cos ( sin + cos )

1 3 1 2 1 2 1 1

; O

2 2

M M

1 1

sin + cos v v

1 1 1 2 2 2 3

O : (54)

2 2

M M

2 2 16

The parameters are O (1). Using eq. (33) and eq. (54), it is seen that in the

1;2

limit 1, the ma joron mainly couples to the predominantly doublet elds as

1;2;3

follows:

L [ v H + v H ] J : (55)

J 1 1 1 2 2 2

The coupling of Z to H and A, eq. (33) and eq. (54), reduces in this case to

1;2

L Z [sin( + )H cos ( + )H ] @ A; (56)

HAZ 1 1 1 2

cos

tan = . Likewise, the HZZ coupling of eq. (38) reduces to

1=2 2

2G ) M Z Z [cos( + )H + sin( + )H ] : (57) L (

F 1 1 1 2 HZZ

This is similar to the conventional couplings in the two doublet case. 17

Figure captions

Figure 1

Diagrams in 1(a) and 1(b) generate non-zero neutrino masses in the mo dels of sec-

tions 2.1 and 2.2, resp ectively.

Figure 2

The diagram in 2(b) generates the Dirac neutrino mass in the mo del of section 2.3,

while those in 2(a) and 2(c) give the small Ma jorana entries m and that can b e

O (10 ) eV. 18

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Table 1: SU (2) U (1) and lepton numb er assignments of the leptons and Higgs

scalars. Quarks are U (1) singlets.

T Y L

0 1

B C

` 1 1

B C

@ A

e 0 2 1

0 0 3

0 1

B C

1 0

B C

@ A

h 0 2 2

0 2 4

k 0 4 2

0 0 2 σ HH

l νc SSνcl Fig. 1a

H σ H σ

φ η φ η

ec' ll' lldc Q

H H

Fig. 1b σ

h+ h+

h++ (a)

ν e e e e ν L L R R L L

h+ η+ (b)

ν e e nc L L R L

h+ h+ σσ

++ η+h η+ (c)

ncce e n L R R L

Fig. 2