Production and decay of minimal SUSY Higgs bosons at LEP/SLC and beyond

Nuclear Physics B347 (1990) 461-490 North-Holland

P R O D U C T I O N AND DECAY OF M I N I M A L SUSY H I G G S B O S O N S AT

L E P / SLC AND BEYOND

Heath POIS 1, Thomas J. WEILER 1'2 and Tzu Chiang YUAN ~

1Department of Physics and Astronomy, Vanderbilt Unicersity, Nashcille, TN37235, USA

2Santa Cruz Institute for Particle Physics, Unicersity of California, Santa Cruz, CA 95064, USA

"Department of Phystcs and Astronomy, Northwestern Unieersity, Et'anston, IL 60208, USA

Received 18 June 1990

We present the rates for tree-level production and decay of Higgs bosons in the pure Higgs and gauge-Higgs sectors in the minimal N= 1 supergravity models. Two Higgs masses arc sufficient to determine the SUSY parameters in these processes. With appropriate variables defined in terms of the two Higgs masses and relevant center-of-mass energies, we are able to explore the supersymmetric parameter space available to each process by plotting iso-rate contours on a bounded square. Recent limits from LEP are incorporated into our analysis. These limits exclude regions inside or outside particular contours, and impact on the reach of present and future higher-energy accelerators. A single positive experimental result for any one process would in principle determine the point for the square of every other process, which may or may not be physical.

1. Introduction

The theory of supersymmetry (SUSY) in par t ic le physics has a p rominen t status

due to its amel io ra t ion of the theore t ica l " h i e r a r c h y " and " f ine tun ing" p rob lems

of the S tandard Mode l (SM). These p rob lems arise as a consequence of quadra t ic

d ivergences for the r enormal i zed scalar (Higgs) par t ic le mass, combined with the

pre judice that the SM Higgs part ic le should be " l i gh t " O ~< (1 TeV) so as to

achieve correc t e lec t roweak breaking and to avoid viola t ion of t ree- level per turba-

tive unitarity. S U S Y removes the quadra t ic divergences. Since global S U S Y (even

af ter S U S Y breaking) is not phenomeno log ica l ly satisfactory (predic t ing e.g. light

s fermions and an un tuneab le nonvanishing cosmological constant) , it is natural to

consider a theory o f local S U S Y which includes gravity th rough a local symmetry

be tween fe rmions and bosons. Since S U S Y is manifest ly not a low-energy symme-

0550-3213/90/$03.50 © 1990 - Elsevier Science Publishers B.V, (North-Holland)

462 H. Pois et al. / SUSYHiggs bosons

try, it must be broken. The idea of breaking supergravity at the Planck scale in a "hidden sector" has become very popular in the last few years. Gravity carries the message of broken SUSY to the low-energy observable sector and effects the electroweak breaking. Gravity is no longer divorced from particle physics, Nature realizes her maximal symmetry, and the hierarchy ratio Mz/Mpl is understood.

Minimal N = 1 supergravity models have been reviewed by several authors [1]. The absence of experimental evidence for supersymmetry implies lower bounds on the SUSY partner masses and the SUSY breaking scale [2]. Perturbative unitarity arguments suggest that SUSY partners will appear in the TeV energy range. If so, it is possible that tree-level processes may copiously occur in future TeV colliders. Of more immediate interest is production of the light Higgses of SUSY at LEP and SLC. At least one Higgs particle is predicted to have a mass at or below that of the Z. In sect. 2, we review the Higgs mass relations [3] of the minimal N = 1 supergravity model. In the pure Higgs and gauge-Higgs sector the parameter space can be represented by just two independent variables, which may be taken to be the Higgs masses involved in the particular process. As a consequence, iso-rate contours presented on a two-dimensional parameter space give a complete descrip- tion of each process. For each process we choose two mass variables such that the region of parameter space kinematically available is mapped into a "SUSY square". The boundaries of the squares result from either the SUSY mass relations, or from the kinematic limits. Values of the iso-rate contours reveal the " reach" of each reaction channel into SUSY parameter space. The production of Higgses in Z decay is studied in sect. 3. Some experimental limits on Z decay to one or more Higgses have been newly reported by LEP experimenters [4,5]. We include these limits in our analysis of the various Higgs production and decay channels. In sect. 4, we study the Higgs pair production possibilities at O(g 2) in

future e + e - collisions and compare these with single-higgs production processes. In sect. 5, we study all the tree-level decays of the Higgs at O(g 2) in the gauge-Higgs and pure Higgs sectors with at least two different Higgses involved in the processes. For each production and decay process that we study, we direct some attention to the limit of a large vev ratio. This limiting case has been generated in a dynamical composite Higgs model [6], offers an explanation of the large mt /m b ratio [7], and essentially leaves the minimal SUSY model with just one free parameter. We draw our conclusions in sect. 6. Allowed and forbidden reactions in the gauge-Higgs and pure Higgs sectors are listed in table 1. The analytical results for the cross sections and decay widths are collected in appendix A. Most but not all of these results have been derived previously.

Recently it has been demonstrated [8] that if there is no new physics below a scale (Anp) considerably above the SUSY and electroweak (ew) scales, then in any multi-Higgs model where the minimum number of parameters are fine-tuned to set the ew-scale, the effective gauge-Higgs and pure Higgs sectors below Anp are those of minimal SUSY, up to corrections of order (Aew/Anp) 2. Accordingly, we expect

H. Pois et al. / SUSY Higgs bosons 463

the phenomenological analysis of minimal SUSY presented in this paper to have a

rather general applicability*.

2. The minimal SUSY model

To motivate the completeness of SUSY squares for organizing Higgs phenomenology, we review here in some detail the parametrization of the Higgs and Higgs-gauge sectors. In the general CP-conserving two-Higgs doublet model, three of the eight original scalar degrees of f reedom become the longitudinal compo-

nents of the W + and Z via the Higgs mechanism. The five remaining physical

degrees of f reedom manifest themselves as three neutral Higgses hi, h 2, h 3 and a pair of charged Higgses h-+. We assume the model is CP conserving, in which case h I and h 2 are CP even (scalar), while h 3 is CP odd (pseudoscalar) with respect to coupling to the SM fermions. There are seven independent parameters in the

Higgs sector. These are the two vevs L' l and t' 2, the mixing angle a that results from diagonalization of the h~ - h 2 mass matrix, and the masses ml, m2, m 3 and m + of the Higgs particles hi, h 2, h 3 and h + respectively. The r.m.s, value of the vevs is chosen to generate the correct masses of the W and Z bosons, leaving six undetermined parameters. One of these is conventionally chosen to be the vev

ratio a'2/L'l, and renamed tan/3, with /3 obviously restricted to 0 ~</3 .<< ~-/2. The minimal SUSY model has two Higgs doublets and additional constraints [3, 10]**:

m~ +M~ =m~ + m~, m + = m 3 + M w,

O ~< m2 <~ Mz < m 1

From these constraints it also follows that

(1), (2)

(3)

m2<~rn3<~ml, m + > ~ ( M w , m 3 ) , (4a, b)

m l X m + , if m z ~ M z s i n O w (4c)

where 0 w is the standard weak mixing angle. A graphical construction of SUSY masses satisfying these relations is exhibited in the "SUSY mandala" of fig. 1.

These SUSY mass constraints are derived from the special form of the scalar potential which contains the F-term, D-term, and soft breaking terms (A-terms) induced by supergravity breaking. They are not true in a general two-Higgs doublet

* An example of a popular GUT model which reduces to minimal SUSY when Anp is taken to be large is superstring-inspired E 6 unification. The symmetry breaking of E 6 produces an extra U(1) symmetry which subsequently breaks at Ano. If this latter breaking scale is not allowed to be large, but rather is tuned to a moderate-energy scale, then there results a low-energy Higgs sector differing from that of minimal SUSY. A phenomenological analysis of such a Higgs sector can be found in ref. [9].

** A thorough collection of the Higgs phenomenology (standard and nonstandard models) is available in the third paper of ref. [10].

464 H. Pois et al. / S U S Y Higgs bosons

Pl

P2

Fig. 1. The mass relations are summarized by the chord lengths of the SUSY mandala. Construction proceeds as follows: On a circle with a horizontal line connecting its antipodes (A), (i) pick any point PI in the first quadrant of the circle; label either resulting chord from PE to A with Mz, the other with m 3. (ii) Pick any point P2 in the first quadrant below Pi; label the longer resulting chord from P2 to A by m 1, and the shorter chord by m , . (iii) From A, draw a chord of length M z s i n 0,~. Label the chord at

right angle to it connecting to the opposite antipode by m +.

model. Importantly, these SUSY relations guarantee that the neutral Higgs

particle h 2 exists with a mass less than that of the Z. (Recall that in the SM, there is no upper bound on the Higgs mass. The breakdown of perturbative unitarity at a few TeV suggests a scale for the Higgs or new physics at or below a few TeV, and lattice studies [11] suggest a Higgs mass below 600 GeV if a Landau singularity is to be kept above the Planck scale.) Due to the SUSY mass relations of eqs. (1) and (2) there are two less free parameters in the Higgs sector. In fact, supersymmetry imposes further constraints on the parameters, so that any two of the Higgs masses other than the pair (m3, m +) and a sector parameter E, defined to be - 1 if/3 lies in the first octant (i.e. t, 1 > t, 2) and + 1 if/3 lies in the second octant (i.e. t, 2 > ~,~), are sufficient to parametrize the system. The two angles are fixed in terms of the Higgs masses:

= - ~ - , ( 5 )

( m3 +Mz2) tan2/3. (6) t a n Z a = m23--Mz

H. Pois et al. / SUSY Higgs bosons 4 6 5

Without loss o f generality, the angle a can be taken to lie in the interval

- ~ r / 2 ~ a ~< 0. F rom eq. (6) it is seen that if - a and /3 lie in different octants,

then m 3 exceeds M z, whereas if -o~ and /3 lie in the same octant then h 3 is

lighter than the Z. Renormal iza t ion group analysis [12] favors the value e = + 1,

but the preferred value of /3 depends on the unknown top quark mass; typical

values o f / 3 lie in the range 65 ° ~ 80 ° [13], but tan/3 as large a s m t / m b is allowed

by the R G E analysis [14]. A limit [15] on the ratio t ,2 /u 1 <~ 20 (i.e. /3 ~< 87 °) has been calculated under the assumption that the experimentally observed B / B

mixing is dominant ly due to charged Higgs exchange.

A dynamical genera t ion of the relation v2 /v I ~ m t / m b (i.e. /3 -~ r r / 2 - m b / m t)

in a softly broken supersymmetr ic theory has recently been proposed [6]. Such

models are attractive in that they give a natural origin to the mass splittings within

each genera t ion [7]. I f na ture indeed exercises the large tan fi =- v2 / l , l option, then

it is clear f rom eq. (5) that ei ther (i) m I ~ M z , in which case m 3 ~ m 2 and

m + < , M z + M 2 ; or (ii) m 2 ~ M z , in which case m 3 ~ m 1. Case (i) pertains if

m~ < M z, while case (ii) pertains if m 3 > M z. The first case offers encouragement

to the L E P / S L C Higgs search for h 3 and h2; the second case requires LEP 200 for Higgs product ion. We can be a bit more quantitative: To lowest order in v JL , 2, eq.

(5) gives

( m 2 - M z ) ( M } - m 2) = 4 m 2 m 2 ( U 1 / / . ' 2 ) 2 . (7)

Thus the fractional deviation 6 7 - Im 2 2 - M z l / m ? for m 1 or m 2 or both must be less than 2t ,~/ t , 2, i.e. ]m i - M z l / M z < v~/t, 2. F rom eq. (1) we then learn that the

"~ 2 is I m 2 - m 2 l / M 2 = S 2 < 2 v , / t , 2 . mass degeneracy of rn.~ and r n j , i

The near degeneracy of m I or m 2 with M z impacts on the couplings of the

theory in the following way (see eqs. (A.17) and (A.18) or ref. [10]): In case (i) where m I ~ M z , sin2(a f l ) = " 4 , - S ? M z / ( m f ( M 2 - m~)) + 0 ( 8 4) so coupling con-

stants propor t ional to s in(a - / 3 ) , i.e. g2zz, g2ww, gl3z and gl +w, are suppressed while coupling constants propor t ional to c o s ( a - / 3 ) , i.e. g~zz, glww, g23z and g2+w, are enhanced. In case (ii) where m,_ ~ Mz ' COS2(Og __ /3) = S S M z / ( m l ( m I v 4 2 2 _ _

Mz2)) + O(64) and just the opposite enhancements and suppressions occur. In

particular, if m~ ~ M z, then Z--+ h2/.t/z is kinematically allowed but dynamically

suppressed, while Z---, h2h 3 is dynamically enhanced but may or may not be kinematically allowed. In the special case where both m~ and m 2 are nearly

" 2 degenera te with the Z mass, c0s2(/3 - a ) = a~ / (8 , + 8~) + o ( a 2) and sin2(/3 - a )

= 62 / (62 + 622) + 0(82) . In the processes to follow, we shall have occasion to comment on these extra constraints f rom the dynamically genera ted large t,_,/z,t

model. The product ion of the neutral Higgses in this model in e+e - reactions has been studied in ref. [7].

A fortui tous feature of all the processes we study in this paper is that the E = _+ 1 choices flip the overall sign of the total ampli tude for each process and thus the E pa ramete r is irrelevant. Thus, any two Higgs masses o ther than


(m 3, m +) completely parametr ize the SUSY Higgs sector! If a part icular process under study has two different external Higgses, the straightforward choice for

independent parameters are the two masses of the external Higgses. Since the

Higgs masses m l, rn3, m _+ are unbounded from above, we choose instead to create

a " S U S Y square" with the two axes labeled by bounded variables which are simple

functions of the Higgs masses. These new variables are chosen such that all of the

physical region consistent with the SUSY constraints of eqs. (1) to (4) and with the

" r each" in ~/s of a given accelerator is mapped inside the SUSY square. For most of the decay processes, we choose variables such that the entire SUSY square is

physical. For some of the product ion cases, a part of the SUSY square is

kinematically closed; we choose variables to minimize these unphysical regions.

Contour plots of rates are calculated and displayed on each SUSY square. In this way we explore within the bounded SUSY square the dependence of each process

on all the allowed parameter space. For the interaction lagrangians, Feynman

rules, and other Higgs issues, we refer the reader to the literature [10]. Two comments should be ment ioned regarding Higgs masses:

(i) The above mass relations are tree-level relations. Quan tum corrections are

expected to alter these relations somewhat [16]. For example, one- loop renormal-

ization is expected to increase light Higgs masses by roughly the C o l e m a n -

Weinberg value [17] ~ 10 GeV. We may then infer from the tree-level inequality

m 2 < M z l c o s 2 f l l that mass renormalizat ion is significant if Icos2/3l <½, i.e. 1 / ~ < tan/3 < ~/3. To proceed with definite calculations, we assume the validity

of the tree-level mass relations and tree-level rates.

(ii) For minimal SUSY models, there are n o upper bound unitarity constraints on

the Higgs masses*. Accordingly, perturbative calculations remain reliable even for Higgses of arbitrarily large mass, and our SUSY squares include this possibility.

3. Higgs production from Z decay

We begin with the processes relevant to LEP and the SLC: Z decay. For

Z ~ hihj , there is only one available channel, namely h2h 3 [19]**. All o ther two-Higgs combinat ions are prohibi ted by C P invariance, Bose symmetry, a n d / o r mass relations (see table 1). The rate for Z - ~ h2h3, normalized to F(Z--* # + / x - ) is shown in fig. 2a, Mass variables are chosen such that the kinematically allowed

and SUSY allowed region fills the square. There is a substantial decay rate over

* See ref. [18]. This result is not surprising, for h 2 iS necessarily light and so serves to cancel the bad unitarity behavior of the pure gauge sector. Moreover, couplings in the Higgs-gauge sector of minimal SUSY do not grow with the Higgs masses (unlike the SM) and so remain perturbative as Higgs masses increase.

**Drees and Hikasa have considered this channel below the Z at TRISTAN, ~-= 60 GeV. For recent work on the Z resonance, see Drees et al.

H. Pois et aL / SUSY Higgs bosons

TABLE l Higgs reactions allowed and forbidden by minimal SUSY mass relations, a n d /

or CP conservation a n d / o r Bose symmetry

467

Allowed reactions Coupling Forbidden reactions Forbidden by

Z --+ h2Z* - s i n ( a - / 3 ) Z --+ h2h 3 ~ cos(a - / 3 )

e+e --+ y*, Z* --, h + h ~ 1, 1

e+e ~ Z* --+ h t Z <*1, - cos(a - /3 ) , heZ ~*~ ~ sin(a - / 3 )

e+e ~ Z * ~ h2h3, ~ cos(a - /3 ) , h lh 3 ~ sin(oe - / 3 )

h t -~ h3Z*, h + W * - , ~ sin(a - /3 ) , s in(a - / 3 )

h2h 2, ~ 2 sin 2o~ sin(/3 + a ) - cos(/3 + cocos 2a

h3h 3 ~ cos 2/3 cos(/3 + a )

h 3 -~ h2 Z(*I ~ cos(a - / 3 ) h +--+ hlW *+, ~ sin(a - /3) ,

h ,W+l*),h~W +* ~ cos(~ - /3 ) , ~ 1

Z ~*~ ~ h3Z* CP Z ~ h~Z* mass relations

Z ~ h,h, Bose Symmetry and CP

Z ~ h th 3, h+h mass relations

Z --+ h lh 2 mass relations and CP

W +--+ h+h, mass relations

W +--+ h3W +* CP

W +-~ h * Z absent at tree level

e+e-- ---, Z* -~ h,h,, CP

h lh 2 CP h I --+ h3Z mass relations

h I -~ h2 z(*l, heh 3 CP h 2 -+ hi z~*l, h lh 3 mass relations and CP h 2 ~ h3 z(*), h lh I, mass relations

h3h 3 mass relations h~ ~ h l Z <*), mass relations

h + W ~* ~ mass relations

h 3 -~ h2h e CP h 3 + h ,h E, h ih 2 mass relations and CP

h, ---, h+h mass relations h +--+ hlW. h3W mass relations

*denotes a virtual particle, (*~ denotes a real or virtual particle. The dependence of the couplings of allowed reactions on c o s ( a - / 3 ) or s in(oe-/3) or other is also shown. The Higgs self-couplings are complicated but definite functions of a and /3. Allowed single-Higgs couplings to vector bosons are h2VV ~ sin(a - / 3 ) and hlVV ~ cos(a - /3 ) , with V = Z or W.

m o s t o f t h e a l l o w e d r e g i o n , g i v i n g L E P / S L C a n h e h 3 p r o d u c t i o n " r e a c h " n e a r l y

m a t c h i n g t h e k i n e m a t i c b o u n d a r y . I t is e a s y t o s h o w t h a t t h e Z h 2 h 3 c o u p l i n g

c o s ( a - / 3 ) d e c r e a s e s m o n o t o n i c a l l y w i t h i n c r e a s i n g m 3 r e g a r d l e s s o f m 2, a n d

i n c r e a s e s m o n o t o n i c a l l y w i t h i n c r e a s i n g m 2 f o r m 3 <~ M Z. F o r e i t h e r rtl 2 o r m 3

f i x e d , m 2 ~ m 3 d e g e n e r a c y wi l l t h e n m a x i m i z e t h e r a t e . T h e s e f e a t u r e s a r e e v i d e n t

in f ig . 2a . I t is i n t e r e s t i n g t o r e c a l l t h a t m 2 ~ m 3 < M z , m I ~ M Z is o n e o f t h e

o p t i o n s o f t h e l a r g e t a n / 3 m o d e l s .

T h i s d o u b l e - H i g g s p r o d u c t i o n r a t e c a n b e c o m p a r e d t o t h e r a t e f o r s i n g l e h 2

p r o d u c t i o n v i a Z ~ h 2 Z * d e c a y , s h o w n in f ig . 2b . ( T h e r e is n o t r e e - l e v e l c o u p l i n g

o f a s i n g l e h 3 t o t h e v e c t o r b o s o n s in a C P - c o n s e r v i n g m o d e l . ) W i t h s i n g l e h 2

468

1.o

ILL Pois et al. / SUSY Higgs bosons

0.8

0.6-

0.4-

0.2-

0.0- 0.0 0.z 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.8 0.8 1.0

(ms + m.~) m2/Mz M z

Fig. 2. (a) - l o g ( F ( Z -* h 2 h 3 ) / F ( Z - ~ / z + p , - ) ) and (b) - l o g ( F ( Z ---, h2/~+/,t ) / F ( Z -->/z+/,t )). The inc remen t be tween each con tou r is 0.5. The ra tes vanish at the k inemat i c bounda r i e s x = 1. The

c o m p u t e r g e n e r a t e d contours fail to d isplay the exact dynamica l vanish ing of the rate in (b) at M z / m I = 1. The h2h ~ (d iagona l l ines) and h 2 (hor izonta l l ines) A L E P H exclusions are also shown.

produc t ion the ent i re p a r a m e t e r space of min imal S U S Y is within the k inemat ica l

reach of L E P / S L C . The Z Z h 2 coupl ing is maximized at the SM value as rn 2 ~ 0

or M z, or m, ~ oo ( the SM limit). In fact, it sa tu ra tes the SM value very quickly

with increas ing m~:

( g 2 z z / SM)" = sinZ( a - - /3) = 1 --

2 m~(M z -m~) + O ( m [ 6 ) . (8)

It vanishes as m~ ~ M z (equivalent ly, as m 3 ~ m2), in accord with our discussion

of the large tan /3 models . The hor izonta l axis in fig. 2b co r r e sponds to the l imit of

a light (m H < M z) SM Higgs. Above this axis, movemen t s of the contours away

from ver t ical co r r e spond to suppress ion of SUSY Higgs p roduc t ion c o m p a r e d to

the S t a n d a r d Model . One can see f rom the f igure tha t this suppress ion is modes t

i ndeed in Z decay except for a very light hi . A very light ml ~ M z in turn implies

(via eq. (1)) a light m 3 ~ m 2. As we will discuss shortly, r ecen t L E P da ta a l r eady

excludes a signif icant range of d e g e n e r a t e m 3 ~ m 2 p a r a m e t e r space. As can be

g leaned f rom fig. 2b, the a l lowed ra te is now less than a t en th of a pe rcen t for a

light h 2 and falls off rap id ly with increas ing m 2 due simply to phase space

suppress ion. Thus it appea r s unl ikely tha t Z--* h2Z* alone can be used to

dis t inguish be tween the SM and min imal SUSY; a fur ther m e a s u r e m e n t of a

non -SM ra te for / ' ( h 2 ~ ff) or for the l oop - induced [20] F ( Z ~ h2y) is requi red .

H. Pois et al. / SUSYHiggs bosons 469

The Z ---, hzZ* rate and the Z ~ h2h 3 rate are proportional to sinZ(a - / 3 ) and

c o s 2 ( a - / 3 ) respectively. The two processes are therefore complimentary in that

one, but not both, may be suppressed by Nature's choice of the SUSY parameters [21]. However, until m 3 and M z become negligible compared to ~- , suppressions

from massive particle phase space and from the virtual Z propagator will be considerable. We therefore expect the complimentarity of these processes to

become manifest only at higher energy colliders (to be discussed later).

The ALEPH collaboration at LEP has recently obtained 95% confidence level

rate limits on the two Z decay modes just presented [4]. Their limits translate into

bounds on the SUSY Higgs masses m 2 and m 3. We discuss these limits briefly,

and display them on our figures. Their first limit may be written F (Z ~ hzff) / F (Z --* ff) < 5.8 × 10 -4. As noted in the A L E P H paper, this exclusion differs little

from the SM bound [5] (m n < 24 GeV) because the parameter dependence of the

rate is dominated by the single mass m 2. This is evident in the contours of fig. 2b,

as we have discussed. The second ALEPH limit depends on the sector parameter:

if t 'e / t , 1 > 1 (i.e. • = + 1) as is theoretically preferred, then F(Z -~ h2h3) /F (Z --~ tx+/x ) < 0 . 1 2 ; i f t , 2 / t , l < l ( i . e e = - l ) , t h e n t h e r + ~ - decay mode of the higgses

is suppressed relative to the quark modes, and a much weaker bound ensues - m 3

>/2m D = 3.75 GeV. Since the Z mass sets the scale for the SUSY mass relations,

this t, 2 < t,~ bound excludes only tiny regions of our SUSY squares (y > x M z / 2 m D

- 1 in fig. 2a; and y > [1 + 4 r n ~ / M z --Y2] -1/2 in fig. 2b). We do not show these

tiny exclusions. (Indeed, we expect the lightest Higgs mass after one-loop mass

renormalization to exceed 2mD.) To a good approximation the combined ALEPH bounds (assuming t' 2 > v 1) may

be conservatively summarized as m 2 > 20 GeV, m 3 > 38 GeV. Via the mass sum

rules, we deduce from these bounds the further approximate bounds m + > 89

GeV, and tan/3 ~ [1, 1.3]. Thus, any future detection of m2, m3, m+ with a mass below 20, 38, 90 GeV respectively would rule out minimal SUSY with tan/3 > 1.

The true m2, m 3 bounds (shown in fig. 5b of the second reference in ref. [4]) are correlated, and slightly stronger than the approximate bounds. We employ the true

bounds to determine exclusions in our SUSY squares. Since two mass variables determine rates, the A L E P H bounds on the Higgs production rates can be recast

unambiguously in each of our SUSY squares as exclusions depending only on the

two variables labeling the square. In particular, there is no further dependence on

fS- or other parameters.

The h 2 and h2h 3 (if t , 2 / c 1 > 1) A L E P H exclusions are shown in figs. 2a, b. Because the rate terrain for the pair production process Z -* h2h 3 is large and flat over most of the SUSY square, the h2h 3 exclusion eliminates a significant portion of the Z ---, h2h 3 square. Combined with the h 2 exclusion most of the L E P / S L C reach in h2h 3 production is seen to be already ruled out. On the other hand, the h2h 3 exclusion disallows only a tiny region of the single h 2 production square. However, as seen in fig. 2b, the disallowed region is a significant fraction of the

470

1.0

0.8-

H. Pois et al. / SUSY Higgs bosons

1.0

0.8-

0 .6 -

0 . 4 -

0 . 2 "

0.0 0.0

0.2

0 .8 -

0 . 4 -

i

'I

I L 8.4 0.6

m 2 / M z

0 . 0 i i

6.z 6.4 6.8 6.8 1.0 8.0 8.2 6.8 1.0 (m~ + m~)

M z

Fig. 3. Lines of cons tan t c . : / u = m i n ( t a n / 3 , c o t / 3 ) in the (a) Z ~ h2h~ SUSY square and the (b)

Z ~ h~Z* SUSY square. The h2h 3 (d iagonal l ines) and h~ (hor izonta l l ines) A L E P H exclusions are also shown.

p a r a m e t e r space where the SM and SUSY rates would be dis t inguishable . This is

unde r s tood f rom the fact that h~ app roaches the SM higgs as m~ becomes large.

W e remind the r e a d e r one more t ime that this h2h 3 expe r imen ta l exclusion holds

only if t , 2 / c ~ > 1. Since LEP ant ic ipa tes a ten-fold increase in events dur ing this

year, the exper imen t s should be able to explore one more full unit of con tour in

the Z ~ h2h 3, h2Z* SUSY squares.

To clarify the re la t ion be tween our SUSY var iables and tan/3 --- t ,2/c j , we plot

fixed values of c 2 / l ' 1 in the SUSY squares re levant to Z decay (figs. 3a, b). Labels

on the contours apply di rect ly to the case ~,2/~ '~ < 1 (i.e. the • = - 1 sector). The

s imple symmetry ( U 2 / / _ ' 1 ~ t ' l / t " 2, • ~ - - • ) with masses held fixed then shows that

the contours a re l abe led by ( l ,2 /v 1) l in the case c 2 / t , 1 > 1 (i.e. the • = + 1

sector). Both cases are covered by in t e rp re t ing the con tour labels to be t ' < / z , > ,

where c < ( t , > ) is the smal le r ( larger) of t' 1 and t' 2. Also shown are the regions

excluded by A L E P H data , s u p e r i m p o s e d over the tan /3 contours . It is c lear f rom

fig. 3a or 3b tha t the single h 2 b o u n d excludes 0.76 < tan /3 < 1.3. This res t r ic t ion is

the m 2 < 24 G e V express ion of the min imal SUSY re la t ion t ' > / t ' < =

max(tan/3 , co t /3) > ¢ ( M z + m 2 ) / ( M z - m 2 ) . I t is also c lear f rom figs. 3a and 3b

that the y = 1 axis (where m~ ~ m 2) of the Z ---, h 2 h 3 SUSY square and the y = 1

( m 3 ~ m 2) and x = 1 (m~ ~ M z) axes of the Z - ~ h 2 Z * SUSY square are a l lowed regions of the large tan /3 models . The A L E P H da ta a l ready exclude much of the

reach of these mode l s for the m 3 ~ m 2 large tan/3 op t ion re levant to L E P / S L C .


In fact, if m 2 turns out to have a value below the present m 3 bound of 38 GeV,

then the large tan/3 model would be excluded. Exclusions in tan/3 space bear

directly on the strengths of Higgs couplings to fermions. Since the A L E P H bounds place no direct restrictions on Higgses with masses in

excess of 40 GeV, the A L E P H exclusions will be less restrictive in the higher energy, higher mass processes to which we now turn. We end this section by noting

that Giudice [22] has considered the O ( g 4) processes Z ---> hzhzh3 , h3h3h 3, h2h2 Z*,

and h3h3Z*. Only for very light Higgses (now excluded by the A L E P H data), do these processes have branching ratios in excess of 10 -6 .

4. Higgs production, singly and paired, at future e +e - col l iders

Consider the CP-al lowed (table 1) processes

e + e - - - . Z* --. h2h3, h l h 3 , e + e - - - , 7 * , Z * --. h + h - .

In most of the plots, the SUSY square is completely physical. In the exceptions,

the unavoidable forbidden mass regions are clearly indicated. In our numerical

work we normalize all e+e cross sections to

% . = ~r (e + e - ~ 3,*, Z* --./~ +/-t ) ;

the normal ized ratio we label R(hihj) . The related reactions forbidden only by CP

(i.e. those with h lh 2, or h ih , final states) may occur via one- loop graphs if the loop particles, e.g. charginos, neutralinos, squarks and sleptons, have CP-violating

couplings to the W, Z or Higgses, and via two-loop graphs if the loop particles

have CP-violating couplings among themselves. The CP-violating sector of the

theory is very model dependent , and we do not consider it in this work.

The Zh2h 3 and Zh~h 3 couplings are propor t ional to c o s ( a - / 3 ) and sin(a - / 3 ) , respectively. Consequent ly , one or both of the channels Z * ~ h2h3, h lh 3 are guaran teed to have a significant coupling. For a light m 3, cos(a - / 3 ) is robust and

sin(a - 13) is suppressed; Z* ~ h : h 3 is favored over Z* -~ h lh 3. For a heavy m 3,

just the opposi te is true. Figs. 4a, b show the cross section contours for h lh 3 product ion at center of mass energies 200 Ge V and 1 TeV. Also shown are the

LEP exclusions; they are minimal for this process. One could choose rn~ and m 3

values as the independent axes, but then the SUSY constraints would squeeze the allowed region, and the contours within it, into a strip along m 3 ~-- m 1. To expand the allowed region, we choose the variables x = m j / M z and y = (m 2 - m 3 ) / M z.

In terms of these, the physical region constraint rn I + m:~ < fS- becomes M z y +

2 M z v ~ X < s; the x intercepts are ~ / s / 2 M z at y = 0, and (s + M z ) / ( 2 M z f s ) at

y = 1. As evidenced in fig. 4a, a light rn I = x M z and m 3 = g / ~ - y M z maximize

the cross section at V~- = 200 GeV. R ( h l h 3) ranges from a few times 10-2 to a few


1.0

0.8-

" ~ 0.6- ¢q

I

0.4.

0.2-

1.0

vab~

¢q

I

,-q

0 . 0 - 0 . 0 ' ' J ' I ' --' 1 . . . . i

1.00 1.05 1.10 1.15 1.20 t.2~ 1.30 t.o 2.0 a.o 4.0 5.0

m l / M z m l / M z

Fig. 4. -log(o-(e+e ~hlh3)/o-(e+e----,p~+p~-)) at center-of-mass energy (a) 200 GeV and (b) 1 TeV. The increment between each contour is 0.1. The kinematically forbidden region in (a) is indicated in grey. The computer generated contours fail to display the exact dynamical vanishing of the rate at ml /M z = 1. The regions excluded by ALEPH h2h 3 (diagonal lines) and h 2 (vertical lines) data

are also shown. The h2h 3 exclusion in (b) is insignificant and not shown.

or, t imes 10 -3 away from the phase space edge at ~S-= 200 GeV, making this

process an easy target for LEP 200 if m 1 ~< 1.3 M z. At v~-= 1 Te V (fig. 4b),

R(h~h 3) is typically several percent for m] ~< 5 M z, and rises to O(1) for a light

m~ ~ M z. The robustness of R(h]h 3) is consistent with the claim [23] that the

Z*--* h]h 3 rate dominates all o ther heavy hi or h 3 product ion react ions up to

Ecru ~ 4 TeV. The large tan/3 model lies near the y = 0 axis (m 3 ~ m~) and the

x = 0 axis (m~ ~ M z) of figs. 4a, b. In fact the ml ~ M z axis is not experimental ly

accessible in Z* ~ h]h 3 because the Z h l h 3 coupling, propor t ional to s i n ( a - / 3 ) ,

vanishes as ~/m 1 - M z . On the other hand, s i n ( a - / 3 ) is maximized at 1 for

m 3 = m 1 . Figs. 5a, b show the cross section contours for h2h 3 product ion at Ecm energies

of 200 GeV and 1 TeV. The axis variables for each figure were chosen to maximize

the physical region m z + m 3 < fs- within each SUSY square. In fig. 5a, the

preferred variables are x = ( m 2 + m 3 ) / f s , y = m z / m 3. The mass relat ion m 2 <

M z expressed in these variables is y ( x v ~ - M z) < M z for x > 2 M z / v ~ , and cuts

into the SUSY square at x = 1, y = M z / ( V ~ - M z) and y = 1, x = 2Mzv~-. In fig.

5b, y = m 2 / m 3, and x = m ~ / M z are the preferred variables: the ent i re pa ramete r

space of minimal SUSY maps into, and fills the uni t square labeled with these

variables. The physical const ra int m 2 q- m 3 < vZs- becomes y ( v ~ - x M z ) > x M z .

For x = 1, the y in tercept is Mz/(VYs - M z ) , and excludes only a small por t ion of

the square when V~- >> Mz. At e i ther energy, the cross section peaks with increas-


1 . 0 i ~ j ~ / ~ 1 . 0 ~ B ~-~/~/

0.8 " 0.8 X ~ - ~

0.0 0. 9 0,4 0.6 0.8 1.0 0.0 0.9 0.4 0.6 0.8 1.0 (m2 + ran) m2/Mz

Eern Fig. 5. -log(o-(e +e o h 2h3)/o(e +e o g +Iz-) at center-of-mass energy (a) 200 GeV and (b) 1 TeV. The increment between each contour is 0.2. Regions forbidden by mass relations (a) and kinematics (b) are indicated in grey; the regions excluded by ALEPH data (h2h3-diagonal; h2-horizontal) are also

shown. The rate vanishes dynamically at m 2 = 0 (now ruled out by the LEP data).

ing mass degeneracy and decreasing total mass at R(h2h 3) -~ 10%. The presen t

LEP exclusions are seen to cover a significant fract ion of the squares, bu t still

leave plenty of allowed region. Large rates extend over most of the two squares,

showing this process to be well within the reach of LEP 200. The Zh2h 3 coupl ing

c o s ( a - / 3 ) vanishes as m 2 approaches zero (the y = 0 axis in fig. 5a, and the

x = 0 = y origin in fig. 5b) or M z (the kinemat ical uppe r right corner in fig. 5a and

the x = 1 axis in fig. 5b). As m 3 increases, the coupling squared falls as cos2(a - /3) 2 2 = m 2 ( M z - m22) /m 4 + O(m36) . The large tan/3 models lie nea r m 2 ~ M z (just

discussed) or nea r the y = 1 axis (m 2 ~ m 3) in figs. 5a, b. Large h2h 3 product ion

rates are predic ted for this lat ter case since c o s 2 ( a - / 3 ) approaches uni ty as m 3

approaches m 2.

Fig. 6 shows the cross section for h+h - p roduc t ion at LEP 200 and TeV

energies. The couplings for h + , h - are mass i ndependen t , so a SUSY square is

unnecessary. The cross-section ratio so near ly scales in m + / ~ that the two curves

are indis t inguishable . Of course, the k inemat ic bounda ry of the curves in the

min imal SUSY model, shown in the figure, does not scale. The present LEP

exclusion assuming r 2 > L, 1 t ranslates via eq. (2), into m + > 89 GeV, a slight

improvement on the mass re la t ion m + > M w. This b o u n d shifts the scaling variable

cutoffs a bit, as shown in the figure. One can observe an increase in the expected

cross section for decreasing Higgs mass for a fixed energy as well as for fixed Higgs

mass and increasing energy, in accord with phase space arguments . A light h+ h


TeV Machine

_ S A - - U L

S E LEP 200

u A

~ s L Y E

1 @-~i ~ ~

1 @-2

i0 3~ q

I I I c J

C.030.i25 0.250 C.375 J.50C C,62S 0.7~C 0.87~3 ~.6C3

;~m+/g .....

Fig. 6. R = ~ ( e + e ~ h+h ) / ¢ ( e + e - ~ # + / z ) for LEP 200 and TeV energies. The arrows indicate the kinematic limits at the two energies, as dictated by the SUSY mass relation rn + > Mw, and by the

new ALEPH results (assuming c 2 > et).

pair will be copiously produced at LEP 200, but discovery of a heavier h + will

probably need a TeV machine. The value of R(h+h ) is 10% to 20% at the TeV machine for most of the allowed range of m+ mass. For large Ecm , R(h+h )

asymptotes at (1 + 4sin 4 0w)/(2 + 48sin 4 0 w) = 0.265 for sin 2 0 w = 0.232.

For purposes of comparison, we also show rates for single Higgs production via the Bjorken process

e + e - ~ Z * ~ h i ( Z * ~ f f ) , f g : e , u e, i = 1 , 2 .

The entire parameter space of minimal SUSY maps into, and fills, the unit SUSY

square labeled by m 2 / M z and M z / m 1. Figs. 7a, b show single h 2 production at 200 GeV and 1 TeV. The line on the lower axis corresponds to the SM rate ratio which is O(0.4-1.0%). The slopes of the contours away from vertical are a graphic display of the sin2(a - / 3 ) SUSY reduction in the rate for h 2 production compared to the SM. A comparison of figs. 7a and 2b shows the clear advantage of LEP 200

over L E P / S L C for probing the single h z production SUSY square. At LEP 200, the rate ratio is ~ 1% over nearly the entire square! The behavior of the h2ZZ coupling was discussed in reference to fig. 2b, as were the large tan/3 regions. Figs.

8a, b show h 1 production at 200 GeV and 1 TeV. The experiment cuts off kinematically at Ecru ~ m 1 so that not all of SUSY parameter space may be explored in this process. Almost independent of the h 2 mass, the h 1 production

H. Pois et al. / S U S Y Higgs bosons 475

~q

1.0

0.8-

0.6.

0.4. h

0.2. q

0 . 0 i

1.0 - -

- - - - - - - - - -_¢ ,o o ;--,

0 . 0 0.2 0.4 0.6 0 . 8 1.0 0 .0 0.2 0.4 0.6 0.8 1.0

m 2 / M z m 2 / M z

Fig. 7. - l o g ( o - ( e + e -+ h2tz+p, ) / o ' ( e + e - - + / * +tz )) at center -of -mass energy (a) 200 GeV, wi th 0.1

i nc r emen t be tween each contour , and (b) o-(e+e - - ) h z ~ + > ) / o - ( e + e - - ) #+p , ) × l03 at center-of-

mass energy 1 TeV, with 0.3 inc remen t be tween each contour . A l inear r a the r than log scale is favored for (b) by the f la tness of the con tou r ter ra in . The ra tes vanish dynamica l ly at y = 1. The regions

exc luded by A L E P H da ta (h2h3-d iagonal ; hz-hor izonta l ) are also shown.

ratio is a few percent for a light hI, but falls off rapidly with increasing h 1 mass: g 2 C0S2( Og -- 3 ) "~ 2 l z z ~ =m~(Mz-m2)/m4+ O(m;-6). As has been stressed in ref.

[10], this decoupling of a heavy Higgs from a vector-boson line is required if unitarity is to remain valid perturbatively. Special limiting values of c o s i ( a - / 3 )

are 1 for ml = M z, and zero for m 2 = 0 or M z. The discussion of the large tan/3 regions follows that accorded to figs. 2b and 3b.

At ~/7 = 1 TeV, there is little phase-space suppression over most of the SUSY

square. Accordingly, the unitarity sum rule [10] g lzz + g2zz = gSM implies a valid "cannot lose" theorem: Nature may suppress at most one of the processes e+e---+ h iZ* ,h2Z* , but not both. A comparison of figs. 7b and 8b validates this statement. The reactions e + e - - o hlh3, h2h 3 are related by a similar sum rule. However a comparison of figs. 4b and 5b is not so simple because the preferred

SUSY variables are different in the two graphs. To summarize, the reach of LEP 200 or a TeV e+e collider for Higgs discovery

looks very promising. The complementari ty of couplings virtually guarantees large production rates of one or more Higgses at these machines: Z* -+ h2h 3, h~Z* have rates proportional to c o s 2 ( a - / 3 ) , whereas Z * - o hlh3, h2Z* have rates proportional to s i n e ( a - / 3 ) . Only if h 3 and hi were too heavy to be kinematically produced, and if sin2(a - / 3 ) were suppressed, would the discovery of the Higgses be unexpected at LEP 200 and TeV machines. Fortunately, s i n 2 ( a - / 3 ) quickly

476 H. Pois et al. / SUSY Higgs bosons

, ::/[it/!!i! li 0.5 0.6 0.7 0.8 0.9 1.0

M z / r n j .

).E

1.4

1.2

1,0 0.2 0.4 0.6 0.8 1.0

M z / m l

Fig. 8. -log(o-(e+e -, hl#+p, )/o-(e+e--~tz+y, )) at center-of-mass energy (a) 200 GeV and (b) 1 TeV. The increment between each contour is 0.5. The rates vanish dynamically at y= 0 and 1, and kinematically at x = 0. The regions excluded by ALEPH data (h2h3-diagonal; h2-vertical) are

also shown.

approaches unity as m I increases (eq. (8)), and so the twin conditions for non-pro-

duction of Higgses are not realized. Large rates for one or more Higgs product ion

channels are expected at L E P 200. In the large tan/3 models, the m~ ~ M z,

m 3 ~ m 2 opt ion is accompanied by an enhanced c o s e ( a - / 3 ) , so all three of

h l , h 2 , h 3 should be p roduced at L E P 200. The m 2 ~ M z , m 3 ~ r n ~ option is accompanied by an enhanced cose(a - / 3 ) , so h 2 will be copiously produced at LEP

200 (but obscured by the Z peak); h~ and h 3 will also be produced at LEP 200 if

rn~ 4 1.3Mz, and at the TeV machine if m I ~< 5M z. h + product ion is large for an m+ mass nearly up to the kinematical limit ~ - / 2 .

It is interesting to realize that the bounds on h 2 and h2h 3 product ion from LEP also cause exclusions in the SUSY squares parametr iz ing Higgs decay to Higgs or

gauge bosons. The reason of course is that the same two independent parameters which determine Higgs product ion rates also determine Higgs decay rates. The product ion rates are directly impacted upon by the LEP search, and so the decay rates are indirectly impacted. We now turn our at tent ion to Higgs decay channels. A n early study of Higgs decay to Higgses in the two-doublet model has been done

in ref. [24].

5. Higgs decay processes

To establish a feeling for the size of a part icular Higgs decay rate, it is useful to recall [10] the partial widths for Higgs decay to b-b and WW. In minimal SUSY,


h 3 ~ WW" is disallowed by CP conservation, h 1 ~ WW is allowed, but suppressed relative to the SM by (see eq. (8)) c o s 2 ( / 3 - a ) < M 4 / 4 m 4 + O(m16), which is

(64 cos 4 0w)- 1 already at the two-W threshold. On the other hand, the bb mode is enhanced in the E = +1 sector relative to the SM. F(h i ~ bb) is given by F(HsM -~ bb) × (cos 2 a, sin 2 a, sin 2/3)/cos e/3 for i = 1, 2, 3, where

4 m 2 )3/2 F ( H s M ~ b b ) = a . 5 M e V ( m b / 5 G e V ) 2 1 - m ~ ( r n H / 1 0 0 G e V ) (9)

Furthermore,

2 m b ta_n

/ ' (h , --, b~) / / ' (h , --, tt) = . (cot e a , tan 2 ce, tan 2/3) L m,

x phase space factors, for i = 1,2, 3. (10)

Thus, the b-b channel is the dominant mode for h 2 decay, and for h I and h 3 decay below the tt threshold; it remains a significant and possibly dominant mode for h l and h 3 above ff threshold if tan/3 > 1 (i.e. e = + 1). (For a discussion of h, ~ SUSY partners, see ref. [10] and references therein.) In the processes to follow we wish to compare the neutral higgs' partial widths to / " ( h i ----) b-b). Thus, as a benchmark, a 10 -4 GeV neutral Higgs partial width may be considered as potentially significant.

For the charged higgs, the dominant decay mode by far is h +---, tb above tb threshold. The tb rate is approximately tan2/3 + [mt/(mb tan /3)] 2 X phase space

factors XF(HsM ~ bb). If h+ is below the t-b threshold, then the dominant width is h + ~ cb, given by the above factor but with rn t replaced by m c. So, for m+~< m t, we also expect any 10 -4 GeV charged Higgs' partial width to be potentially significant. Let us now turn to the Higgs decay to final states which include a Higgs particle. As summarized in table 1, Bose symmetry, CP invariance and the SUSY mass relations considerably reduce the number of open decay channels.

5.1. GAUGE-HIGGS PROCESSES

We first dismiss h 2 d e c a y : h 2 cannot decay into gauge bosons a n d / o r Higgses since it is the lightest Higgs and is lighter than the Z. Consider next h I decay. There are no open channels with an on-shell gauge boson in the final state.

478 1t. Pois et al. / SUSY Higgs bosons

1.0-

0.8-

0.6-

tq

0.4-

0.2-

0.0- o.o 0.2 o.4 0.8 0.8 1.o

( r n l z - m 3 : l ) / M z 2

1.0

0 .8 -

0 . 6 -

0 . 4 -

0 . 2 -

0.0 0,00 0.05 0.10 0.1.5 0.20 s l n 2 0 w

( r n z 2 -- m I 2 ) / M z 2

Fig. 9. (a) -Iog(F(h I ---' h3Z* --* h3dd)/GeV) and (b) -log(F(h t ~ h+W * --' h+fid)/GeV). The increment between each contour is 0.5. F o r h3 /x+ /x , h3iS~t' #, h3uu, o r h + # u final states, multiply by 0.23, 0.46, 0.77 and 0.33 respectively. The rates vanish dynamically at y = 1, and kinematically at x = 0.

The regions excluded by ALEPH data (h2h3-diagonal: h2-horizontal) are also shown.

However, the following two off-shell gauge-boson channels are possible:

h 1 ~ h3Z* ~ h 3 f f , h 1 ~ h + W * ~-~ h±f f ' "

Both couplings, gl3z and g l+w, are propor t ional to s i n ( a - / 3 ) , and so rise or

fall together. In fact, they rise, quickly approaching unity as m I increases above

the Z mass. From eq. (8), s in (c~- /3 ) exceeds 0.99 already at rnj = 2 M z.

The rates for these processes are presen ted in figs. 9a, b respectively for the

part icular Z* and W* final states dd and dfi. The two x variables are just

t ranslat ions of each other: x b = x ~ , - coS20w . The variables M z / r n I and ( m ~ - m ~ ) / M z appropr ia te for the neut ra l channel of fig. 9a map the entire SUSY

paramete r space into the uni t square. The neut ra l channel is necessarily off-shell if

m 2 4:0 since m I - m 3 = ( M z - m 2 ) [ ( M z + m2)/(ml +/n3)] < Mz - m2- Accord- ingly, the rate is small over most of the square. The rate for the neut ra l channel

peaks n e a r 1 0 - 4 in the region where the masses of h l , h 3 assume their m i n i mum

possible values of M z and zero, respectively, and the Z just barely goes on-shell.

Allowing for all Z* final states enhances the rate shown by B-~(Z--- , d d ) - - 6 . 6 .

Thus the decay h~ --* h3Z* is potent ial ly measurable if m I and m3 are light. The

behavior of the coupling g~3z and the large tan/3 regions are as discussed for figs.


1.0-

479

0.8-

0,6-

0.4 -

0.2-

0.0- 0.0 0.2 0.4 0.6 0.8 1.0

1TI2/In3

Fig. 10. - l o g ( F ( h 3 - + h 2 Z C * ~ - ~ h 2 d J ) / G e V ) . The increment between each contour is 0.5. For h2k¢+/~ , h2~7~, or h2ufi final states, multiply by 0.23, 0.46 and 0.77 respectively. The rates vanish dynamically at ) , = 0 and 1, and kinematically at x = 1. The regions excluded by A L E P H data

(h2b3-diagonak h2-horizontal) are also shown.

4a, b. In par t i cu la r , the m I ~ M z large tan /3 case inc ludes the high rate , u p p e r

r i gh t -hand corner .

The cha rged channe l is a l lowed by the S U S Y cons t ra in ts and k inemat ics only if "~ 2 my - m+ = M z 2 sin 2 0 w - m 2 > 0, i.e. m 2 < M z sin 0 w ~ 44 GeV. W e lea rn f rom fig.

9b tha t the ra te is ex t remely small . This is because the S U S Y mass sum rules

enforce a n e a r - d e g e n e r a c y condi t ion: (m I - m+) / (m~ + m + ) < sin 2 0w/(1 +

cos 0w) 2 = 0.07, which forces the W far off-shell over the ent i re a l lowed region.

Having a par t i c le off-shell typical ly suppresses the ra te by ~ F / ~ M . ( T h r e e - b o d y

decays via four -po in t coupl ings are typical ly supp re s sed even more , by a / ~ . ) W e

can see this on- to -of f shell suppress ion explicit ly in the next process we consider ,

h 3 decay. F o r h 3 decay the only open channe l is

h 3 --~ h e Z (*) ~ h2ff .

The ra te for this channe l is p r e s e n t e d in fig. 10 for on-shel l and off-shell Z. The

ent i re S U S Y p a r a m e t e r space is k inemat ica l ly accessible, and maps into and fills

the unit S U S Y square l abe led with the var iables x = m z / m 3 and y = m z / M z. One

can observe a p l a t eau in the ra te in the on-shel l reg ion and a dec rease in the ra te

as the eye moves across the on- and off-shell b o u n d a r y (x = y / ( 1 + y ) , which is

roughly the con tour l abe l ed 5.5). The on-shel l p l a t e a u is supp re s se d by the squa red

480 H. Pois et al. / SUSY Higgs bosons

coupling gZ_3z ~ c o s Z ( a - / 3 ) which decreases monotonical ly with increasing m3, finally as 1 / m 4 for m~ >> M z 2. This suppression is nearly offset by the contract ion

of the derivative coupling in the ampli tude with the longitudinal polarization

2 for m 2 >> " The net dependence of the rate vector of the Z, which grows as m 3 3 M£. on large m 3 is m~ ~. Multiplied by B - 1 ( Z --* d d ) ~ 6.6 to sum over all the Z* final

states, the rate is potentially significant when the Z is on-shell, even for m 3 large.

Fur ther discussion of the coupling and the large tan/3 regions (x = 1 and y = 1)

follow that of fig. 5b. As evidenced in the figure, the LEP data excludes about 25% of the square.

The charged h +- can decay into Wh2, or W* plus any of the o ther three Higgses:

h + ~ h z W + ( * ) , h l W - + * , h 3 W + * , w i thW-+*- -+ f f ' .

The decay width to h 2 is presented in fig. l l a . The unit SUSY square in the variables x = (m 2 + 2 2 M w ) / m + , y = m 2 / M z covers the entire SUSY paramete r space. Again, one can see a significant decrease in the width at the on- /o f f - she l l boundary x = (y2 + cos 2 0 w ) / ( y + cos 0w )2, with intercepts x ( y = 0) = 1 and

x ( y = 1) = (1 + cos 2 0w)/(1 + cos 0w) 2 -- 0.50. This boundary basically follows the

4.0 contour down from the top and ends in the lower right corner. The squared 2 > > coupling g2+2w ~ cos2(c~ - / 3 ) vanishes at m 2 = 0 or M z, and as 1/m4+ for m+

Mz 2. In addition, it peaks near m2~_ = M z / 2 for fixed m+.2 Because this ampli tude

also involves the contract ion of the derivative coupling with the longitudinal 2 polarization vector of the W, it grows as m2+ for m+ >> M z 2. The net dependence

of the rate on large m+ is m+~, and we expect and find a sizeable rate even for

large m+ values. The large tan/3 regions are at y = 1 where the rate vanishes, and

at x = 1 (where m 2 ~ m 3) where the rate is suppressed by the off-shell W

propagator . The maximum rate plateaus at m 2 = 0.7M z, and rn+>~ 2 M z , in the on-shell region. W h e n multiplied by B - I ( W - - + u d ) - - 3 . 0 to include all W final states, the width is seen to peak in excess of 10 MeV. Reference to our earlier

discussion of h +---, tb, cb then shows that h +---, h2W + may compete favorably with

the fermionic modes depending on tan/3 and m t. The LEP exclusions clearly impact upon this process.

The process h + ~ hlW +* is shown in fig. l lb . The x = (m 2 + rn~)/M2z variable

may also be written as m z / M z - sin 2 0 w, which shows that h + ~ hlW +* is allowed

only if rn z > M z sin 0 w ~ 44 GeV. Accordingly, the LEP exclusions do not impinge on this SUSY square. The near degeneracy condit ion arising from the mass sum

rules is ( m + - m l ) / ( m + + m 1) < cosZ0w/(1 + ~/1 + coS20w )2 • 0.14. The decay 2 M 2 + M 2, and the W feels its width peaks near the corner where m I = M z, m+ =

on-shell pole. The peaking is inhibited however by the vanishing of g+ lw ~ s i n ( a - / 3 ) at rnl ~ M z. Because of this suppression and the fact that the W is

14. Pois et al. / SUSY Higgs bosons 481

l o

o .

0.8-

0.2-

0 .0 , ~ . . . . .

0.0 0.2 0.4 0.6 0.8 1.0

( m 2 • + m w 2 ) / m + 2

1.0

0.8

0.6

0.4

0.2

0.0 • o.o & &,, ~o co,20w

( m + 2 - m 1 2 ) / M z 2

m + / M w 1.25 1.5 2 3 4 I I I I ~ ,

m a / M w 0.5 1 2 3 4 ! I N ni l ,

1 O -~ C

10 -~

lO- E \

7 2 H

10-

10-9~0.00@/125 0.%5@ 0.%75 0.%00 0.%25 0)750 0.%75 1.%0S Fz13 / l'~l +

Fig. 11. (a) - Iog(F(h +-~ I]2 W+(*~ ~ h2ud)/GeV) with contour increments of 0.2, (b) - l o g ( F ( h + ~ h]W +* ~ h luJ) /GeV) with contour increments of 0.5, and (c) F ( h + ~ h3W +* ~ h~ud)/GeV. For the h, /xv f ina l s ta te , divide by the co lo r f a c t o r 3. G r a p h (a) has d y n a m i c a l ze ros a t y = 0 a n d 1, a n d a t

x = 0; A L E P H exc lus ions ( h 2 h 3 - d i a g o n a l ; h2-ver t ica l ) a r e d i sp layed ; (b) has a d y n a m i c a l z e ro a t y = 1 a n d a k i n e m a t i c a l ze ro a t x = 0; (c) shows via the a r r o w the m i n i m u m a l l owed m 3 / m + r a t io a c c o r d i n g

to the r e c e n t A L E P H d a t a ( a s s u m i n g u 2 > c 1 ).

482 tl. Pois et al. / SUSY Higgs bosons

necessarily off-shell, the rate appears too small to be interesting. The large tan/3 regions are at y = 1 and x = cos 2 0 w.

For the process h+--, h3W* we plot the width versus the single bounded variable

x = m3/rn + (fig. l lc) . The two Higgs masses m 3 and m+, according to eq. (2), are

not independent . In terms of x, they are x M w / ~ / l l - x 2 and M w / f l - x 2

respectively, and are shown at the top of the figure. The rate, including the sum on

W* final states is large ( ~ 10 -4) for x ~< 0.6, or equivalently m + < 1.25M w. Since

m 3 / m + = m 3 / v / m ~ + M 2 w , the new A L E P H bound (for t ' 2 > c ,) m 3 > 3 8 GeV further implies m 3 / m + > 0 . 4 2 or equivalently, m + > 1.1M w. We indicate this bound in fig. 11c.

Higgs decay to four massless fermions via coupling to two W's, viz.

hi--+ WC*~W (*)---~ flf ' |f2f2 ( i = 1 ,2 ) ,

or via two Z 's comes under the subheading of this section, but we will discuss these

processes in detail elsewhere, along with the rate for higgs ~ single top quark +

three light fermions [25]. As ment ioned earlier, h 2 ~ W ' W * is suppressed by the

virtuality of both W's, and h~ ~ W W is suppressed by the large m 1 behavior of the

h 1WW coupling. To summarize the gauge-Higgs decay channels, h~ ~ h3Z*, shown in fig. 9a, is

potentially measurable if m I ~< 1.4M z and m3 ~ 0 .6Mz; h 3 ~ h2 zI*~, shown in fig. 10, is potentially measurable if 0 .2M z ~< m 2 -.< 0.4m3; h + ~ h3W*, shown in fig. l l c , is potentially measurable if m + < 1.25M w. The reaction h + ~ h2W <~, shown in fig.

l l a , is almost certainly competit ive with the tb, cb modes when the W is on-shell

(roughly the parameter range 0.2M z ~< m 2 ~< 0.7m3), and potentially measurable in the rest of parameter space, where the W is off-shell. Small portions of the SUSY

square for each of these reactions are already excluded by the LEP data.

5.2. PURE HIGGS PROCESSES

In the pure Higgs sector, only h~ decay is allowed. At order g2 there are just two

open channels:

h l ~ h 2 h 2 ,h3h3 .

These two-body processes are presented in figs. 12a, b. For each process, the width is large, 10 to 100 MeV, and competi t ive with the bb mode over most of the allowed region. No simple choice of variables will fill the SUSY squares. For the

h2h 2 final state, the variables chosen are x = m 2 / M z, y = M z / m ~ ; the kinematically allowed region (2m 2 < m 1) corresponds to 2xy ~ 1. There is a robust local

maximum of 100 MeV in the rate for m~ = 2 M z, m 2 = 0.7M z. The coupling gl22

V / ~ ~ 3 vanishes for m 2 = 0 or M z, approaches 3m 2 - m ? / M z <~ 5 as m 1 --+ ~ and approaches m 2 / m 3 as m t - ~ M z, in units of gMz /cOs 0w; it is maximized for m 2


1.0

0.8-

0 . 6 - ¢~

0.4-

0.2-

0.0 ~ o.o 0.2 0.4 o.~ o.8 ~.o 1.oo 0.95 0.90 o . ~ 0.80 0.75

m 2 / M z ( M z / m l ) =

Fig. 12. (a) F (h l ~ h 2 h 2 l X 10 2 (GeV) wi th con tou r i nc remen t s of 0.5, and (b) F (h I ~ h 3 h ; ) × 10 2 (GeV) wi th con tour i nc remen t s of 1.0. G r a p h (a) has dynamica l zeros at x = 0 and 1, and a

k inemat ica l ly forb idden region; (b) has a k inemat ica l zero at y = 0, and a region forb idden by mass sum rules. The regions exc luded by A L E P H da ta (h2h3-d iagonal ; h2-hor izonta l ) are also shown.

nea r M z / 2 . The large tan /3 regions are y = 1 (where m~ ~ M z, m 3 ~ rn 2 and the

coupl ing is max imized) and x = 1 (where m ~ ~ M z and the coupl ing is suppressed) .

Fo r the h3h 3 final s tate , the chosen var iables are x = ( M z / m ~ ) 2, y = ( 2 m 3 / m l ) 2. With these var iables , the SUSY Higgs mass sum rules and k inemat ics cons t ra in

y > 4 - 4 x and 0.75 < x < 1.0. The l a t t e r cons t ra in t t rans la tes into Mz<~m ~ <~ 1.15M z, if h I --* h3h 3 is to occur. The coupl ing g133 is maximized at gMz/cOs 0 w as m I --*M z. Thus the decay width is maximized dynamica l ly by m 1 = M z, and

maximized in phase space by tak ing m 3 as light as possible. This favored region is

exc luded by A L E P H da ta if t an /3 > 1. Still, the ra te fal l -off away from the

m ax imum is gent le , and there r emains an a l lowed region with 10 to 50 M e V width

contours . The tan /3 regions are larges t at x = 1, and at y = 4 which is unphysica l

for this channe l . Accord ing ly , h 1 --* h3h 3 is large (d isa l lowed) if Na tu re chooses the

large tan /3 op t ion with m~ - M z (rn 2 ~ Mz) .

T h e r e are two a l lowed decays at o r d e r g4: h~ --* 3h 2, h2h3h 3. These t h r ee -body

channe ls receive con t r ibu t ions f rom a d i rec t O(g 2) quar t ic coupl ing, and f rom two

O ( g ) cubic coupl ings c o n n e c t e d by an off-shell i n t e r m e d i a t e higgs. These graphs

are down by O ( g ) f rom the two-body graphs (even neglec t ing the suppress ion due

to the i n t e r m e d i a t e Higgs p ropaga to r ) . W h e n fo lded with the t h r ee -body phase

space factors , the t h r ee -body decay ra te is down f rom the two-body ra te by the

usual O(~ /Tr ) . W e the re fo re do not cons ide r the t h r ee -body channels any fur ther ,

except to note that a l though h~--*2h3h 2 a p p e a r s to involve th ree i n d e p e n d e n t

484 1-!( Pois et al. / SUSY Higgs bosons

masses, according to the SUSY mass relations it does not; and so h I ~ 2h3h 2 would also fit into a two-dimensional SUSY square.

To summarize the pure Higgs decay channels, if allowed by kinematics and the

SUSY mass relations, h I ~ h2h 2 and h I ~ h3h 3 typically have rates in the tens of

MeV range, h I --> h 3 h 3 is allowed if M z ~ r n I <~ 1.15M z and 2 m 3 < m 1. In this parameter range, the rate for h~ --> h3h 3 exceeds that for h~ ~ h2h 2 by a factor of

1 to 5. h~ ~ h 2 h 2 is allowed whenever 2m 2 < m v Either or both modes appear to exceed the h 1 -~ h3Z* width by one to two orders of magnitude. LEP data excludes

a small region of the h I --> h 2 h 2 square, and a large region of the h 1 --> h 3 h 3

square, since the latter is a light Higgs process by virtue of the SUSY mass

relations.

6. Conclusions

To conclude, we have presented the production and decay of Higgs bosons in the minimal N = 1 supergravity model in a novel way. For processes in the

gauge-Higgs and pure-Higgs sector with at least two different Higgses, the two Higgs masses are sufficient to determine all rates without reference to any further

underlying SUSY parameters. As can be seen from table 1, the constraints of SUSY mass relations, CP invariance, and Bose symmetry considerably reduce the

number of these possible processes in the N = 1 supergravity models. For each

process, we have chosen simple variables such that the intersection of SUSY parameter space and the kinematically available region is mapped into, and often

fills, a bounded SUSY square. Contours of constant rate are displayed on each

square, showing for each channel the "reach" available to validate or invalidate

minimal SUSY. For several processes, the entire SUSY parameter space is

kinematically allowed, and so the SUSY square spans the entire model. Examples of such processes with appreciable rates are single h 2 production plotted against

m 2 / M z and M z / m l ; h2h 3 production plotted against m 2 / M z and mz/m3; h +--* h2 W+* plotted against m 2 / M z and (m~ + M w ) / r n + . 2 ",

Each of the contour plots is related to all the others by the use of SUSY mass relationships. A single physical point in any one SUSY square can be used to

calculate a unique point for each of the other SUSY squares. Thus the plots also reveal how the strengths of the various interactions are interrelated. If the point is outside the square, the channel is kinematically closed. If the point is inside the square and not in a kinematically excluded region, the channel is open and the point fixes the rate.

If low-energy supersymmetry exists in Nature, some of the processes we study here should be detected in the near future. It is possible that LEP will produce h 2

and h3, thereby fixing the unique physical point in the Z --* h2Z* and Z ~ hzh 3 squares. It is very probable that LEP 200 will find the physical point in the

Z* --* Zh 2 square if m 1 is heavy, and the points in the Z* ~ hzh 3 and Z* -~ h lZ ~*~

H. Pois et al. / SUSYHiggs bosons 485

squares if m 1 is light. To reveal SUSY in all her splendor, one has to measure these and the additional processes and observe their relationships to each other. The SUSY squares presented here offer a graphical display of these relationships. Through these relationships, it is possible that SUSY may be first revealed in the Higgs sector, rather than in the superpartner sector of squarks, sleptons and

gauginos.

This work was supported in part by the U.S. Department of Energy Grants DE-FG-05-85ER40226 and DE-AM03-76SF00010. Computing time was provided in part by the College of Arts and Sciences, Vanderbilt University. We also wish to thank H. Haber and M. Sher for discussion, and M. Jordan for inspiration.

Appendix A

In order to make this paper self-contained, we summarize all the formulae of the processes we have studied here. Many of these have been obtained previously by other authors. Some are new. In all processes we neglect the tiny direct Yukawa coupling of the Higgs particles to the light fermions. It is convenient to define the triangle function at the outset:

/~(X, y , Z) = X 2 q_y2 q_ Z2 __ 2(xy + y z + zx). (A.1)

A.1. e+e - PRODUCTION CROSS SECTIONS

Since we normalize our cross sections to e+e- - -+7 *, Z*---,/x+tx -, we first present the cross section for the e+e - production of a charged fermion-antiferm- ion pair via the virtual y and Z s-channel exchange. Define Q as the electric charge of the fermion (opposite to that of the anti-fermion), e.g. Q = - 1 for a

+IX- pair. Then the generic cross section for the ff' final state is [26]

~f( Q, T3L, T3R, 3, s )

t Nc,S Q - - - T - - ., -, - M z ) + M z F~

S2 l'f 2 + ae2fl 2 (A.2)

+ ( r e 2 +a ~ ) (s -Mz2) 2 +Mz2Fz2

486

where


1 - 4 sin 2 0 w - 1 (A.3)

Ce= 4cosOwsinO w ' a s = 4cosOwsinO w

are the vector and axial-vector couplings of the electron, and

- 2( T3L + T3R) + 4Q sin 2 0 w 2( T3L -- TtR ) (A.4)

t~f = 4 cos 0 w sin 0 w , af = 4 cos 0 w sin 0 w

are those of the produced fermion f, whose left- and r ight-handed components

have third components of weak isospin T3t ~ and T3R, respectively. ~/s is the

center-of-mass energy and /3 = ~/1 - 4m2/s is the center of mass velocity of either

p roduced fermion of mass m. N c is the number of final state colors. For

e+e - - ,p .+# Q = - l , G . = G , a f = a ~ , a n d / 3 = l . Cross sections for the reactions

e+e---+y*,Z{*)~h+h_,hihi ( i , j = 1 ,2 ,3 )

can be written in terms of the generic cross section of eq. (A.2). The cross section

for charged Higgs pair product ion is [27]

o _ ( e + e ~ h + h ) = 1 , ' ~ 3 f [ , 1 1 7 p + ~ t~, ~7, 5, 1 , s ) (A.5)

with /3+= 1//1 - 4m+/s. The cross section for neutral Higgs pair product ion is

~ f o-(e+e---+hihj)= ¼A3/2(1,m,/s,m7/s)~r (O,Cij,C,j,l,s), (A.6)

with Cij = g,jz cos Ow/g, g = e / s i n 0 w. In minimal SUSY, the only couplings al-

lowed by CP are

( g 13z, g23z ) = g ( sin( a - / 3 ) , cos( a - / 3 ) ) / 2 cos 0 w . (A.7)

The cross section for the single hi, 2 product ion process

e+e ~ Z * ~ h , ( Z * ~ f t ' ) , f 4 : e , u e, ( i = 1 , 2 )

is [28]

4 2 ) ' 2 + a e ) ( U 2 q - a ' ( ) I ( s , r f l / ) , o - ( e + e - - + h i f f ) = e gizzNc (re " 2 2 19art 3 (s-M2z)2+Mzrz

(A.8)


where

V/(x /2- 4 ~ , ) ( 1 + 3 t x i - x i + ~ x 2 ) d x i

I ( s , m 2) = fl+P"'a2~7 , ( x, + tx z - tx, - 1)" + /Xz7 z ' ( A . 9 )

x i 2 E , / ~ / s , ( / x i , / X z , y z ) = (mr,mz,r£)/s,

(g~zz , g2zz) = g M z ( c ° s ( a - / 3 ) , s i n ( c~ - / 3 ) ) / c o s 0 w . (A.10)

A.2. W AND Z-DECAY RATES

The Z and W widths to lepton pairs are useful for normalizat ion purposes:

F ( Z ~ + / x - ) =

[ )2] g2M z 1 + ( 1 - 4 s i n 2 0 w

192~COS 2 0 w

F ( W ~ ~ v ) - - - g 2 M w eeM w

48~- 12 sin 2 0 w "

For single Higgs product ion via Z decay,

Ig2zz]2 l ( M z rn2) r (Z- -+ / , t+ /x - ) F ( Z - + h 2 / x + p , - ) - 16~.2M 2 , _ (A.11)

g2zz is given in (A.10).

1 F ( Z --+ h2h3) = 48vr

2 3 / 2 2 2 m2/M2"~ Mz[g23z] A (1, m 2 / M z , 3 / z ! (A.12)

g23z is given in (A.7)

A.3. HIGGS DECAY RATES

In the pure Higgs sector, the generic two-body rate is

1 1 lrl/" t.

H ' n i -+ n jnk) - 64rrm i 1 + ~jk - - - - I g i ~ k l 2 A ' / z ( 1 , m 2 / m Z , m 2 / m ~ ) . (A.13)

In minimal SUSY, only h~ ~ h2h2, h3h 3 are allowed by the mass constraints and

4 8 8 H. Pois et aL / SUSY Higgs bosons

CP conservation. The relevant couplings are

( g122, g133) = gMz(2 sin 2a sin(/3 + a )

- cos(/3 + ~ ) cos 2 a , cos 2/3 cos(/3 + a ) ) / c o s 0 w (A.14)

In the Higgs gauge-boson sector, the generic on- and off-shell rate is

V(h , - , hjZ(*) --+ hjff)

2 2 ", , mie Ig,jzl Nc(t,? +a?)

16~ -2 12~- f2 1 + ~J d x j

• , ~ 3 / 2

(Xj +/~ z /.zj 1) 2 - - - + P, z Y z

(A.15)

with /.'f, af defined in eq. (A.4) and

2 2 "~ "~ x j = Z E J m , , (/.tj, p.z, y z ) = ( m j , M z , F z ) / m ? .

Here, E~ is the energy of hj. Since F (Z + ff) = [Nc(v I + a~)Mz]/12rr, the sum over light fermions in eq. (A.15) is effected by replacing the square brackets with

r z / M z . F(h +--* h~W (*) + --+ h~ff') and F(h, ~ h+W(*)---* h+ff ') are obtained

from F(h, --+ h jZ (*1 ~ hjff) via the substitutions: h, --+ h +, m, -+ m +, M z ~ Mw, Fz ~ F w, v2 +aZ ~ g 2 / 4 , e o g and gijz--+g+jw, g,+w, respectively. In minimal SUSY, the only nonzero couplings are gl3z and g32z given in eq. (A.7) and

(g+,w,g+2w,g+3w) = g( sin(or - / 3 ) , cos(c~ - / 3 ) , 1 ) / 2 . (A.16

In the above formulae, we have ignored fermion masses, and all the KM angles m the Wff' coupling.

We have taken the Z mass and width to be 91 and 2.5 GeV, respectively, and the W mass and width to be 81 and 2.1 GeV, respectively. The weak angle is taken to be s in20w=0.232, and the fine structure constant is taken to be a ( M z 2)= (g sin Ow)2/4rr = 1/128.

All the couplings in eqs. (A.7), (A.10), (A.14) and (A.16) may be expressed directly in terms of any two Higgs masses other than the dependent pair (m +, m3). This is accomplished through the use of eqs. (1), (2), (5) and (6) in the main text. Intermediate steps may be facilitated by using formulae appearing in ref. [10]. The squared couplings g~3z, g+lw and g2 2zz are each proportional to s in2(a - /3 ) ,


which may be written

m~(m~-M 2) sin2( a - / 3 ) = (mT_m2)(m,, 2 2+mZ_M2z) . (A.17)

The squared couplings 2 2 2 g23z, g+2w and glzz are each proportional to

cos2(a - / 3 ) = 1 - sin2(a - / 3 ) . (A.18)

The mass sum rules in eqs. (1) and (2) can be used to rewrite (A.17) and (A.18) in

terms of other mass pairs. The couplings g133 and g122 have a slightly more complicated angle (or mass) dependence, as shown in table 1. The couplings g+3w

and g+ z are mass independent.

References

[1] P. Nath, R. Arnowitt and A. Charnseddine, Applied N = 1 supergravity (World Scientific, Singa- pore, 1984); H.P. Nilles, Phys. Rep. 110 (1984) 1; H.E. Haber and G.L. Kane, Phys. Rep. 117 (1985) 76

[2] M. Chen et al., Phys. Rep. 159 (1988) 201 and references therein [3] K. Inoue, A. Kakuto, H. Komatsu and S. Takeshita, Prog. Theor. Phys. 67 (1982) 1889;

R. Flores and M. Sher, Ann. Phys. (N.Y.) 148 (1983) 95: M. Sher, Phys. Rep. 179 (1989) 273

[4] ALEPH Collaboration, D. Decamp et al., [5] ALEPH Collaboration, D. Decamp et al.,

OPAL Collaboration, M.Z. Akrawy et al., [6] T.E. Clark, S.T. Love and W.A. Bardeen,

M. Luty, Phys. Rev. D41 (1990) 2893. [7] J. Kalinowski and S. Pokorski, Phys. Lett.

Phys. Lett. B237 (1990) 291; B241 (1990) 14l Phys. Lett. B236 (1990) 233; Phys. Lett. B236 (1990) 224 Phys. Lett. B237 (1990) 235;

B219 (1989) 116; P. Chiappetta and J. Kalinowski, Z. Phys. C43 (1989) 319; J. Kalinowski, B. Grzadkowski and S. Pokorski, Phys. Lett. B241 (1990) 534

[8] H.E. Haber and Y. Nir, Nucl. Phys. B335 (1990) 363 [9] V. Barger and K. Whisnant, Int. J. Mod. Phys. A3 (1986) 1907: Phys. Rev. D42 (1990) 138

[10] J.F. Gunion and H.E. Haber, Nucl. Phys. B272 (1986) 1; B278 (1986) 449; J.F. Gunion, H.E. Haber, G.L. Kane and S. Dawson, The Higgs hunter's guide (Addison-Wesley, Reading, MA, 1990)

[11] J. Kuti, L. Lin and Y. Shen, Phys. Rev. Lett. 61 (1988) 678 [12] L. AIvarez-Gaume, J. Polchinski and M. Wise, Nucl. Phys. B221 (1983) 495:

J. Ellis, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. BI21 (1983) 123; H.P. Nilles and M. Nusbaumer, Phys. Lett. B145 (1984) 73; P. Majumdar and P. Roy, Phys. Rev. D30 (1984) 2432: M. Drees, M. Gluck and K. Grassie, Phys. Lett. B159 (1985) 118; E. Reya, Phys. Rev. D33 (1986) 773: M. Drees and M. Gluck, Phys. Lett. BI81 (1986) 98 and references therein; J. Lopez and D. Nanopoulos, Mod. Phys. Lett. A5 (1990) 1259

[13] P. Nath, R. Arnowitt and A.H. Chamseddine, Northeastern-Harvard preprint, HUTP-83/A077 NUB # 2588 (1983) unpublished

[14] G.F. Giudice and G. Ridolfi, Z. Phys. 41C (1988) 447: M. Olechowski and S. Pokorski, Phys. Lett. B214 (1988) 393

490 H. Pois et a L / SUSY Higgs bosons

[15] P. Krawczyk and S. Pokorski, Phys. Rev. Lett. 60 (1988) 182; C. Busch, Nucl. Phys. B319 (1989) 15; see third ref. in [10]

[16] M.S. Berger, Phys. Rev. D41 (1990) 225; J. Gunion and A. Turski, Phys. Rev. D40 (1989) 2325:D40 (1989) 2333; Phys. Rev. D39 (1989) 2701: S. Li and M. Sher, Phys. Lett. B140 (1984) 339; H. Pois, M. Sher and T.J. Weiler, one-loop calculations, now in progress

[17] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888; see also third ref. in [10], third ref. in [3]

[18] P. Langacker and H.A. Weldon, Phys. Rev. Lett. 52 (1984) 1377; H.A. Weldon, Phys. Rev. D30 (1984) 1547; Phys. Lett. B146 (1984) 59; R. Casalbuoni, D. Domenici, F. Feruglio and R. Gatto, Nucl. Phys. B299 (1988) 117; L. Durand and J. Lopez, Phys. Rev. D40 (1989) 240: see also the third reference in [10]; J.F. Gunion, H.E. Haber and J. Wudka, UC-Davis preprint UCD-90-18 (1990)

[19] M. Drees and K.-I. Hikasa, Phys. Rev. D40 (1989) 47; M. Drees et al., Proc. Workshop on Z physics at LEP, ed. G. Altarelli, R. Kleiss and C. Verzegnassi (1989l

[20] G. Gamberini, G. Giudice and G. Ridolfi, Nucl. Phys. B292 (1987) 237; T.J. Weiler and T.-C. Yuan, Nucl. Phys. B318 (1989) 337

[21] G.F. Giudice, Phys. Lett. B208 (1988) 315; A. Bartl, W. Majerotto and N. Oshimo, Phys. Lett. B237 (1990) 229

[22] G.F. Giudice, Phys. Rev. D41 (1990l 2174 [23] J. Gunion et al., Phys. Rev. D38 (t988) 3444 [24] G. Pocsik and G. Zsigmond, Phys. Lett. BI12 (1982) 157 [25] H. Pois, T.J. Weiler, T.C. Yuan, in preparation [26] N. Deshpande, X. Tata and D. Dicus, Phys. Rev. D29 (1984) 1527 [27] H. Baer et al., Physics at LEP, Vol. 1, ed. J. Ellis and R. Peccei, CERN 86-02 (1986) [28] J.D. Bjorken, Proc. 1976 SLAC Summer Institute on Particle Physics, ed., M.C. Zipf (SLAG-198,

1977) D.R.T. Jones, S.T. Petcov, Phys. Lett. B84 (1979) 440; F.A. Berends and R. Kleiss, Nucl. Phys. B260 (1985) 32

Production and decay of minimal SUSY Higgs bosons at LEP/SLC and beyond

Documents

Transcript of Production and decay of minimal SUSY Higgs bosons at LEP/SLC and beyond