SLC and beyond

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

P R O D U C T I O N A N D D E C A Y O F M I N I M A L S U S Y 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 T h e t h e o r y o f s u p e r s y m m e t r y ( S U S Y ) in p a r t i c l e physics has a p r o m i n e n t status due to its a m e l i o r a t i o n of the t h e o r e t i c a l " h i e r a r c h y " and " f i n e t u n i n g " p r o b l e m s o f the S t a n d a r d M o d e l (SM). T h e s e p r o b l e m s arise as a c o n s e q u e n c e of q u a d r a t i c d i v e r g e n c e s for the r e n o r m a l i z e d scalar (Higgs) particle mass, c o m b i n e d with the p r e j u d i c e that the S M Higgs particle s h o u l d be " l i g h t " O ~< (1 T e V ) so as to a c h i e v e c o r r e c t e l e c t r o w e a k b r e a k i n g and to avoid v i o l a t i o n of t r e e - l e v e l p e r t u r b a tive unitarity. S U S Y r e m o v e s the q u a d r a t i c divergences. Since global S U S Y (even a f t e r S U S Y b r e a k i n g ) is n o t p h e n o m e n o l o g i c a l l y satisfactory ( p r e d i c t i n g e.g. light s f e r m i o n s an d an u n t u n e a b l e n o n v a n i s h i n g c o s m o l o g i c a l constant), it is n at u r al to c o n s i d e r a t h e o r y o f local S U S Y which includes gravity t h r o u g h a local s y m m e t r y b e t w e e n f e r m i o n s a n d bosons. Since S U S Y is manifestly n o t a l o w - e n e r g y s y m m e 0550-3213/90/$03.50 © 1990 - Elsevier Science Publishers B.V, (North-Holland)

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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 T e V energy range. If so, it is possible that tree-level processes may copiously occur in future T e V 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 p a r a m e t e r 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 p a r a m e t e r space give a complete description of each process. For each process we choose two mass variables such that the region of p a r a m e t e r space kinematically available is m a p p e d into a " S U S Y 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 " r e a c h " of each reaction channel into SUSY p a r a m e t e r 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 m t / 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

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the phenomenological analysis of minimal SUSY presented in this p a p e r 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 freedom become the longitudinal components of the W + and Z via the Higgs mechanism. The five remaining physical degrees of freedom 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 ~
m+=m 3 +M w,

O ~
(1), (2) (3)

From these constraints it also follows that m2<~rn3<~ml, mlXm+,

m+>~(Mw,m3),

if

mz~MzsinO w

(4a, b) (4c)

where 0w is the standard weak mixing angle. A graphical construction of SUSY masses satisfying these relations is exhibited in the " S U S Y 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 G U T 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].

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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 G e V 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 p a r a m e t e r 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: =

-~

-

( m3 +Mz2) tan2/3. tanZa =

m23--Mz

,

(5)

(6)

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W i t h o u t loss o f generality, the angle a can be taken to lie in the interval - ~ r / 2 ~ a ~< 0. F r o m 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. R e n o r m a l i z a t i o n g r o u p analysis [12] favors the value e = + 1, but the preferred value of /3 d e p e n d s on the u n k n o w n 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 b e e n calculated u n d e r the assumption that the experimentally observed B / B mixing is dominantly due to charged Higgs exchange. A dynamical generation of the relation v 2 / v I ~ m t / m b (i.e. /3 -~ r r / 2 - m b / m t) in a softly broken supersymmetric theory has recently b e e n p r o p o s e d [6]. Such models are attractive in that they give a natural origin to the mass splittings within each generation [7]. If nature indeed exercises the large tan fi =- v 2 / l , l option, then it is clear from eq. (5) that either (i) m I ~ M z , in which case m 3 ~ m 2 and m+<,Mz+M2; 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. T h e first case offers e n c o u r a g e m e n t to the L E P / S L C Higgs search for h 3 and h2; the second case requires L E P 200 for Higgs production. W e can be a bit m o r e quantitative: To lowest order in v J L , 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 - M z l /2m ? 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 r o m eq. (1) we then learn that the . mass d e g e n e r a c y of rn.~"~ and r n 2j , i is I m 2 - m 2 l / M 2 = S 2 < 2 v , / t , 2 T h e 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 ) = S ?" M z4/ ( m f ( ,M 2 - m~)) + 0 ( 8 4) so coupling constants proportional to sin(a - / 3 ) , i.e. g2zz, g2ww, gl3z and gl +w, are suppressed while coupling constants proportional to c o s ( a - / 3 ) , i.e. g~zz, glww, g23z and g2+w, are enhanced. In case (ii) where m,_ ~ M z ' COS2(Og __ /3) = S SI M z4/v( m l ( m 2 2 Mz2)) + O(64) and just the opposite e n h a n c e m e n t s 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 e n h a n c e d but may or may not be kinematically allowed. In the special case where both m~ and m 2 are nearly d e g e n e r a t e with the Z mass, c0s2(/3 - a ) = a ~" / ( 8 , 2 + 8~) + o ( a 2) and sin2(/3 - a ) = 6 2 / ( 6 2 + 622) + 0 ( 8 2 ) . In the processes to follow, we shall have occasion to c o m m e n t on these extra constraints from the dynamically g e n e r a t e d large t,_,/z,t model. T h e production of the neutral Higgses in this model in e + e - reactions has been studied in ref. [7]. A fortuitous feature of all the processes we study in this p a p e r is that the E = _+ 1 choices flip the overall sign of the total amplitude for each process and thus the E p a r a m e t e r is irrelevant. Thus, any two Higgs masses other than _

_

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(m 3, m +) completely parametrize the SUSY Higgs sector! If a particular process u n d e r study has two different external Higgses, the straightforward choice for i n d e p e n d e n t parameters are the two masses of the external Higgses. Since the Higgs masses m l, rn3, m _+ are u n b o u n d e d from above, we choose instead to create a " S U S Y square" with the two axes labeled by b o u n d e d variables which are simple functions of the Higgs masses. These new variables are chosen such that all of the physical region consistent with the S U S Y constraints of eqs. (1) to (4) and with the " r e a c h " in ~/s of a given accelerator is m a p p e d inside the S U S Y square. For most of the decay processes, we choose variables such that the entire S U S Y square is physical. For some of the production cases, a part of the S U S Y square is kinematically closed; we choose variables to minimize these unphysical regions. C o n t o u r plots of rates are calculated and displayed on each S U S Y square. In this way we explore within the b o u n d e d SUSY square the d e p e n d e n c e of each process on all the allowed p a r a m e t e r space. For the interaction lagrangians, F e y n m a n rules, and other Higgs issues, we refer the reader to the literature [10]. Two c o m m e n t s should be m e n t i o n e d regarding Higgs masses: (i) The above mass relations are tree-level relations. Q u a n t u m corrections are expected to alter these relations somewhat [16]. For example, one-loop renormalization 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 m2
3. Higgs production from Z decay We begin with the processes relevant to L E P and the SLC: Z decay. For Z ~ hihj, there is only one available channel, namely h2h 3 [19]**. All other two-Higgs combinations are prohibited by C P invariance, Bose symmetry, a n d / o r mass relations (see table 1). T h e rate for Z - ~ h2h3, normalized to F(Z--* # + / x - ) is shown in fig. 2a, Mass variables are chosen such that the kinematically allowed and S U S Y 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.

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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 Allowed reactions

Coupling

Forbidden reactions

Forbidden by

Z --+ h2Z* Z --+ h2h 3

-sin(a -/3) ~ cos(a - / 3 )

Z ~*~ ~ h3Z* Z ~ h~Z*

CP mass relations

~ 1, 1

Z ~ h,h,

Bose Symmetry and CP mass relations

e+e

--+ y*, Z* --, h + h

e+e

~ Z* --+ h t Z <*1, h e Z ~*~ e + e ~ Z * ~ h2h3, hlh 3

- cos(a - / 3 ) , ~ sin(a - / 3 ) ~ cos(a - / 3 ) , ~ sin(oe - / 3 )

h t -~ h3Z*, h + W * - ,

Z ~ h t h 3, h + h

h3h 3

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

h 3 -~ h2 Z(*I h +--+ hlW *+, h,W+l*),h~W +*

~ cos(a - / 3 ) ~ sin(a - / 3 ) , ~ cos(~ - / 3 ) , ~ 1

h2h 2,

Z --+ h l h 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

hlh 2 h I --+ h3Z h I -~ h2 z(*l, h e h 3 h 2 -+ hi z~*l, h l h 3 h 2 ~ h3 z(*), h l h I, h3h 3 h~ ~ h l Z <*), h + W ~*~ h 3 -~ h2h e h 3 + h , h E, h i h 2 h, ---, h + h h +--+ hlW. h3W

CP mass relations CP mass relations and CP mass relations mass relations mass relations mass relations CP mass relations and CP mass relations 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 sin(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.

most

of the allowed

matching

the

cos(a-/3) increases

region,

kinematic decreases

boundary.

monotonically

with

in fig. 2 a . I t is i n t e r e s t i n g This

of the large tan/3 double-Higgs

production of a single

via Z ~

with

increasing

to show

increasing

that

the

to recall that

the rate. These

m2~m

"reach"

3
nearly

Zh2h 3 coupling

m 3 regardless

m 2 f o r m 3 <~ M Z. F o r

will then maximize

o f m 2, a n d

either

features

rtl 2 o r

m 3

are evident

m I ~ M Z is o n e

of the

models.

production h2Z*

an heh 3 production

It is e a s y

monotonically

fixed, m 2 ~ m 3 degeneracy options

giving LEP/SLC

rate

can be compared

decay, shown

h 3 to the vector

bosons

i n fig. 2 b . ( T h e r e

in a CP-conserving

to the rate

for single h 2

is n o t r e e - l e v e l model.)

With

coupling single h 2

ILL Pois et al. / S U S Y Higgs bosons

468 1.o

0.8

0.6-

0.4-

0.2-

0.0-

0.0

0.z

0.4 0.6 (ms + m.~)

0.8

1.0

0.0

0.2

0.4 0.8 m2/Mz

0.8

1.0

Mz 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 i n c r e m e n t b e t w e e n each c o n t o u r is 0.5. The rates vanish at the k i n e m a t i c b o u n d a r i e s x = 1. The c o m p u t e r g e n e r a t e d c o n t o u r s fail to display the exact d y n a m i c a l v a n i s h i n g of the rate in (b) at M z / m I = 1. The h 2 h ~ ( d i a g o n a l lines) and h 2 (horizontal lines) A L E P H exclusions are also shown.

p r o d u c t i o n the e n t i r e p a r a m e t e r space of m i n i m a l S U S Y is within the k i n e m a t i c a l r e a c h o f L E P / S L C . T h e Z Z h 2 c o u p l i n g is m a x i m i z e d at the SM value as rn 2 ~ 0 o r M z, or m, ~ oo (the SM limit). In fact, it s a t u r a t e s the S M value very quickly with increasing m~:

m~(Mz2 -m~) (g2zz/

SM)" = sinZ( a --/3) = 1 --

+ O(m[6).

(8)

It vanishes as m~ ~ M z (equivalently, as m 3 ~ m2), in a c c o r d with o u r discussion of the large t a n / 3 models. T h e h o r i z o n t a l axis in fig. 2b c o r r e s p o n d s to the limit of a light (m H < M z ) S M Higgs. A b o v e this axis, m o v e m e n t s of the c o n t o u r s away from vertical c o r r e s p o n d to s u p p r e s s i o n of S U S Y Higgs p r o d u c t i o n c o m p a r e d to the S t a n d a r d M o d e l . O n e can see f r o m t h e figure t h a t this s u p p r e s s i o n is m o d e s t i n d e e d in Z d e c a y except for a very light hi. A very light ml ~ M z in t u r n implies (via eq. (1)) a light m 3 ~ m 2. A s we will discuss shortly, r e c e n t L E P d a t a a l r e a d y excludes a significant r a n g e 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. A s can be g l e a n e d from fig. 2b, the allowed r a t e is now less t h a n a t e n t h o f a p e r c e n t for a light h 2 a n d falls off r a p i d l y with i n c r e a s i n g m 2 d u e simply to p h a s e space suppression. T h u s it a p p e a r s unlikely t h a t Z--* h2Z* alone can be u s e d to distinguish b e t w e e n the S M a n d m i n i m a l S U S Y ; a f u r t h e r m e a s u r e m e n t o f a n o n - S M r a t e for / ' ( h 2 ~ ff) or for the l o o p - i n d u c e d [20] F ( Z ~ h 2 y ) is r e q u i r e d .

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 A L E P H 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 A L E P H limit depends on the sector parameter: if t'e/t, 1 > 1 (i.e. • = + 1) as is theoretically preferred, then F ( Z -~ h 2 h 3 ) / 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 A L E P H 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 -

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

470 1.0

1.0

0.8-

0.8-

0.6-

0.8-

0.4-

0.4-

0.2"

0.2 i

'I 0.0

0.0

0.0

6.z

6.4 6.8 (m~ + m~)

6.8

1.0

8.0

8.2

I

L

8.4

0.6

6.8

i

i

1.0

m2/Mz

Mz Fig. 3. Lines of c o n s t a n t c . : / u = m i n ( t a n / 3 , c o t / 3 ) in the (a) Z ~ h2h~ S U S Y s q u a r e and the (b) Z ~ h~Z* S U S Y square. The h2h 3 ( d i a g o n a l lines) and h~ (horizontal lines) A L E P H exclusions are also shown.

p a r a m e t e r s p a c e w h e r e the SM and S U S Y rates w o u l d be distinguishable. This is u n d e r s t o o d from the fact that h~ a p p r o a c h e s the SM higgs as m~ b e c o m e s large. W e r e m i n d the r e a d e r o n e m o r e t i m e that this h2h 3 e x p e r i m e n t a l exclusion holds only if t , 2 / c ~ > 1. Since L E P a n t i c i p a t e s a ten-fold i n c r e a s e in events d u r i n g this year, the e x p e r i m e n t s should be able to e x p l o r e o n e m o r e full unit of c o n t o u r in the Z ~ h2h 3, h2Z* S U S Y squares. To clarify the r e l a t i o n b e t w e e n o u r S U S Y v a r i a b l e s a n d tan/3 ---t,2/c j, we plot fixed values of c 2 / l ' 1 in the S U S Y s q u a r e s r e l e v a n t to Z decay (figs. 3a, b). L a b e l s on the c o n t o u r s apply directly to the case ~,2/~'~ < 1 (i.e. the • = - 1 sector). T h e simple s y m m e t r y ( U 2 / / _ ' 1 ~ t ' l / t " 2, • ~ - - • ) with masses held fixed t h e n shows that the c o n t o u r s a r e l a b e l e d by ( l , 2 / v 1) l in the case c 2 / t , 1 > 1 (i.e. the • = + 1 sector). Both cases are c o v e r e d by i n t e r p r e t i n g the c o n t o u r labels to be t ' < / z , > , w h e r e c < ( t , > ) is the s m a l l e r (larger) of t' 1 and t' 2. A l s o shown are the regions e x c l u d e d by A L E P H data, s u p e r i m p o s e d over the t a n / 3 contours. It is clear from fig. 3a o r 3b t h a t the single h 2 b o u n d excludes 0.76 < t a n / 3 < 1.3. This restriction is the m 2 < 24 G e V expression o f the m i n i m a l S U S Y r e l a t i o n t ' > / t ' < = m a x ( t a n / 3 , c o t / 3 ) > ¢ ( M z + m 2 ) / ( M z - m 2 ) . It is also clear from figs. 3a and 3b that the y = 1 axis ( w h e r e m~ ~ m 2) of the Z ---, h 2 h 3 S U S Y square a n d the y = 1 ( m 3 ~ m 2) a n d x = 1 (m~ ~ M z ) axes of the Z - ~ h 2 Z * S U S Y square are allowed regions of the large t a n / 3 models. T h e A L E P H d a t a a l r e a d y exclude much of t h e r e a c h of t h e s e m o d e l s for the m 3 ~ m 2 large tan/3 o p t i o n r e l e v a n t to L E P / S L C .

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In fact, if m 2 turns out to have a value below the present m 3 b o u n d 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 b o u n d s 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. W e end this section by noting that Giudice [22] has considered the O ( g 4) processes Z --->h z h z h 3 , 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 - colliders

Consider the CP-allowed (table 1) processes e + e - - - . Z* --. h2h3, h l h 3 ,

e + e - - - , 7 * , Z * --. h + h - .

In most of the plots, the S U S Y 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 normalized ratio we label R(hihj). The related reactions forbidden only by CP (i.e. those with h l h 2, or hih , 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 a m o n g themselves. T h e CP-violating sector of the theory is very model dependent, and we do not consider it in this work. T h e Z h 2 h 3 and Zh~h 3 couplings are proportional to c o s ( a - / 3 ) and sin(a - / 3 ) , respectively. Consequently, one or both of the channels Z * ~ h2h3, h l h 3 are g u a r a n t e e d to have a significant coupling. For a light m 3, c o s ( a - / 3 ) is robust and sin(a - 13) is suppressed; Z* ~ h : h 3 is favored over Z* -~ h l h 3. For a heavy m 3, just the opposite is true. Figs. 4a, b show the cross section contours for h l h 3 p r o d u c t i o n at center of mass energies 200 G e V and 1 TeV. Also shown are the L E P exclusions; they are minimal for this process. O n e could choose rn~ and m 3 values as the i n d e p e n d e n t axes, but then the S U S Y constraints would squeeze the allowed region, and the c o n t o u r s 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- b e c o m e s 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

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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 m l / 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, times 10 -3 away from the phase space edge at ~S-= 200 GeV, m a k i n g this process an easy target for L E P 200 if m 1 ~< 1.3 M z. A t v~-= 1 T e V (fig. 4b), R(h~h 3) is typically several p e r c e n t for m] ~< 5 M z, and rises to O(1) for a light m~ ~ M z. T h e r o b u s t n e s s of R ( h ] h 3) is consistent with the claim [23] that the Z*--* h ] h 3 rate d o m i n a t e s all o t h e r heavy hi or h 3 p r o d u c t i o n reactions u p to Ecru ~ 4 TeV. T h e large t a n / 3 m o d e l lies n e a r the y = 0 axis ( m 3 ~ m~) a n d the x = 0 axis (m~ ~ M z) of figs. 4a, b. In fact the ml ~ M z axis is not experimentally accessible in Z* ~ h ] h 3 b e c a u s e the Z h l h 3 coupling, p r o p o r t i o n a l to s i n ( a - / 3 ) , vanishes as ~/m 1 - M z . O n the o t h e r 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 p r o d u c t i o n at Ecm energies of 200 G e V a n d 1 TeV. T h e axis variables for each figure were chosen to maximize the physical region m z + m 3 < fs- within each S U S Y square. In fig. 5a, the p r e f e r r e d variables are x = ( m 2 + m 3 ) / f s , y = m z / m 3. T h e mass relation 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 S U S Y square at x = 1, y = M z / ( V ~ - M z) a n d y = 1, x = 2Mzv~-. In fig. 5b, y = m 2 / m 3, a n d x = m ~ / M z are the p r e f e r r e d variables: the entire p a r a m e t e r space of m i n i m a l S U S Y maps into, a n d fills the u n i t square labeled with these variables. T h e physical c o n s t r a i n t m 2 q- m 3 < vZs- b e c o m e s y ( v ~ - x M z ) > x M z . For x = 1, the y i n t e r c e p t is M z / ( V Y s - M z ) , and excludes only a small p o r t i o n of the square w h e n V~- >> M z . A t either energy, the cross section peaks with increas-

H. Pois et al. / SUSY Higgs bosons

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ing mass d e g e n e r a c y a n d d e c r e a s i n g total mass at R ( h 2 h 3) -~ 10%. T h e p r e s e n t L E P exclusions are seen to cover a significant fraction of the squares, b u 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 L E P 200. T h e Z h 2 h 3 c o u p l i n g c o s ( a - / 3 ) vanishes as m 2 a p p r o a c h e s zero (the y = 0 axis in fig. 5a, a n d the x = 0 = y origin in fig. 5b) or M z (the k i n e m a t i c a l u p p e r right c o r n e r in fig. 5a a n d the x = 1 axis in fig. 5b). As m 3 increases, the coupling s q u a r e d falls as cos2(a /3) = m 22 ( M z2 - m 2 2 ) / m 4 + O ( m 3 6 ) . T h e large t a n / 3 models lie n e a r m 2 ~ M z (just

discussed) or n e a r the y = 1 axis ( m 2 ~ m 3) in figs. 5a, b. Large h2h 3 p r o d u c t i o n rates are p r e d i c t e d for this latter case since c o s 2 ( a - / 3 ) a p p r o a c h e s u n i t y as m 3 a p p r o a c h e s m 2. Fig. 6 shows the cross section for h + h - p r o d u c t i o n at L E P 200 a n d T e V energies. T h e couplings for h + , h - are mass i n d e p e n d e n t , so a S U S Y square is u n n e c e s s a r y . T h e cross-section ratio so n e a r l y scales in m + / ~ that the two curves are indistinguishable. O f course, the k i n e m a t i c b o u n d a r y of the curves in the m i n i m a l S U S Y model, shown in the figure, does not scale. T h e p r e s e n t L E P exclusion a s s u m i n g r 2 > L,1 t r a n s l a t e s via eq. (2), into m + > 89 GeV, a slight i m p r o v e m e n t o n the mass r e l a t i o n m + > M w. This b o u n d shifts the scaling variable cutoffs a bit, as shown in the figure. O n e can observe an increase in the expected cross section for d e c r e a s i n g Higgs mass for a fixed energy as well as for fixed Higgs mass a n d i n c r e a s i n g energy, in accord with phase space a r g u m e n t s . A light h + h

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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 0w) = 0.265 for sin 2 0w = 0.232. For purposes of comparison, we also show rates for single Higgs production via the Bjorken process e+e-~Z*~hi(Z*~ff),

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

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Fig. 7. - l o g ( o - ( e + e -+ h2tz+p, ) / o ' ( e + e - - + / * +tz )) at center-of-mass e n e r g y (a) 200 GeV, with 0.1 i n c r e m e n t b e t w e e n each contour, and (b) o-(e+e - - ) h z ~ + > ) / o - ( e + e - - ) # + p , ) × l03 at center-ofmass e n e r g y 1 TeV, with 0.3 i n c r e m e n t b e t w e e n each contour. A l i n e a r r a t h e r t h a n log scale is favored for (b) by the f l a t n e s s of the c o n t o u r terrain. The rates vanish d y n a m i c a l l y at y = 1. The regions e x c l u d e d by A L E P H d a t a ( h 2 h 3 - d i a g o n a l ; h z - h o r i z o n t a l ) are also shown.

ratio is a few percent for a light hI, but falls off rapidly with increasing h 1 mass: "~ 2 =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 l z z + g2zz = gSM implies a valid "cannot lose" theorem: Nature may suppress at most one of the processes e+e---+ h i Z * , h 2 Z * , 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 T e V e + e collider for Higgs discovery looks very promising. The complementarity 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 T e V machines. Fortunately, s i n 2 ( a - / 3 ) quickly

g 2l z z ~ C0S2( Og -- 3 )

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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-production of Higgses are not realized. Large rates for one or m o r e Higgs production channels are expected at L E P 200. In the large tan/3 models, the m~ ~ M z, m 3 ~ m 2 option is accompanied by an e n h a n c e d c o s e ( a - / 3 ) , so all three of h l , h 2 , h 3 should be p r o d u c e d at L E P 200. T h e m 2 ~ M z , m 3 ~ r n ~ option is accompanied by an e n h a n c e d cose(a - / 3 ) , so h 2 will be copiously p r o d u c e d at L E P 200 (but obscured by the Z peak); h~ and h 3 will also be p r o d u c e d at L E P 200 if rn~ 4 1.3Mz, and at the T e V machine if m I ~< 5 M z. h + production is large for an m + mass nearly up to the kinematical limit ~ - / 2 . It is interesting to realize that the b o u n d s on h 2 and h2h 3 production from L E P also cause exclusions in the S U S Y squares parametrizing Higgs decay to Higgs or gauge bosons. T h e reason of course is that the same two i n d e p e n d e n t parameters which determine Higgs production rates also determine Higgs decay rates. T h e production rates are directly impacted u p o n by the L E P search, and so the decay rates are indirectly impacted. We now turn our attention to Higgs decay channels. A n early study of Higgs decay to Higgses in the two-doublet model has been d o n e in ref. [24].

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

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h 3 ~ WW" is disallowed by CP conservation, h 1 ~ W W 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

4m 2

F(HsM~bb)=a.5MeV(mb/5GeV)

2 1-

m~

)3/2 (rnH/100GeV)

(9)

Furthermore,

/'(h, --, b~)//'(h, --, tt) = . L

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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 t o / " ( h i ----)b-b). Thus, as a benchmark, a 10 -4 G e V 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 G e V 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 n u m b e r 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.

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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 h 3 / 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 g a u g e - b o s o n c h a n n e l s are possible:

h 1 ~ h3Z* ~ h 3 f f ,

h 1 ~ h + W * ~-~ h ± f f ' "

Both couplings, gl3z a n d g l + w , are p r o p o r t i o n a l to s i n ( a - / 3 ) , a n d so rise or fall together. In fact, they rise, quickly a p p r o a c h i n g unity as m I increases above the Z mass. F r o m eq. (8), s i n ( c ~ - / 3 ) exceeds 0.99 already at rnj = 2 M z. T h e rates for these processes are p r e s e n t e d in figs. 9a, b respectively for the particular Z* a n d W* final states d d a n d dfi. T h e two x variables are just translations of each other: x b = x ~ , - coS20w . T h e variables M z / r n I a n d ( m ~ m ~ ) / M z a p p r o p r i a t e for the n e u t r a l c h a n n e l of fig. 9a m a p the entire SUSY p a r a m e t e r space into the unit square. T h e n e u t r a l c h a n n e l is necessarily off-shell if m 2 4:0 since m I - m 3 = ( M z - m 2 ) [ ( M z + m2)/(ml +/n3)] < M z - m2- Accordingly, the rate is small over most of the square. T h e rate for the n e u t r a l c h a n n e l 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 m u m possible values of M z a n d zero, respectively, a n d the Z just barely goes on-shell. Allowing for all Z* final states e n h a n c e s the rate shown by B-~(Z---, d d ) - - 6 . 6 . T h u s the decay h~ --* h3Z* is potentially m e a s u r a b l e if m I a n d m3 are light. T h e behavior of the coupling g~3z a n d the large t a n / 3 regions are as discussed for figs.

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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 c o n t o u r is 0.5. F o r 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 p a r t i c u l a r , t h e m I ~ M z large t a n / 3 case i n c l u d e s the high rate, u p p e r r i g h t - h a n d corner. T h e c h a r g e d c h a n n e l is a l l o w e d by the S U S Y c o n s t r a i n t s a n d k i n e m a t i c s only if "~ my - m +2 = M z2 sin 2 0 w - m 2 > 0, i.e. m 2 < M z sin 0 w ~ 44 G e V . W e l e a r n f r o m fig. 9b t h a t t h e r a t e is e x t r e m e l y small. This is b e c a u s e the S U S Y m a s s sum rules e n f o r c e a n e a r - d e g e n e r a c y c o n d i t i o n : (m I - m + ) / ( m ~ + m + ) < sin 2 0w/(1 + cos 0w) 2 = 0.07, which forces the W far off-shell over the e n t i r e allowed region. H a v i n g a p a r t i c l e off-shell typically s u p p r e s s e s the r a t e by ~ F / ~ M . ( T h r e e - b o d y decays via f o u r - p o i n t couplings a r e typically s u p p r e s s e d even more, by a / ~ . ) W e can see this o n - t o - o f f shell s u p p r e s s i o n explicitly in the next p r o c e s s we consider, h 3 decay. F o r h 3 d e c a y the only o p e n c h a n n e l is

h 3 --~

h e Z (*) ~ h2ff.

T h e rate for this c h a n n e l is p r e s e n t e d in fig. 10 for on-shell a n d off-shell Z. T h e e n t i r e S U S Y p a r a m e t e r s p a c e is k i n e m a t i c a l l y accessible, a n d m a p s into a n d fills t h e unit S U S Y s q u a r e l a b e l e d with the v a r i a b l e s x = m z / m 3 a n d y = m z / M z. O n e can o b s e r v e a p l a t e a u in the r a t e in the on-shell r e g i o n a n d a d e c r e a s e in the rate as t h e eye moves across the on- a n d off-shell b o u n d a r y ( x = y / ( 1 + y ) , which is r o u g h l y t h e c o n t o u r l a b e l e d 5.5). T h e on-shell p l a t e a u is s u p p r e s s e d by the s q u a r e d

480

H. Pois et al. / SUSY Higgs bosons

coupling gZ_3z~ c o s Z ( a - / 3 ) which decreases monotonically with increasing m3, finally as 1 / m 4 for m~ >> M z2. This suppression is nearly offset by the contraction of the derivative coupling in the amplitude with the longitudinal polarization vector of the Z, which grows as m 32 for m 32 >> M£." The net d e p e n d e n c e of the rate 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. F u r t h e r 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 L E P data excludes about 25% of the square. The charged h +- can decay into Wh2, or W* plus any of the other three Higgses:

h+~hzW+(*),hlW-+*,h3 W+*,

w i t h W - + * - - + f f '.

T h e decay width to h 2 is presented in fig. l l a . T h e unit S U S Y square in the variables x = (m 2 + M w2) / m + 2, y = m 2 / M z covers the entire S U S Y p a r a m e t e r space. Again, one can see a significant decrease in the width at the o n - / o f f - s h e l l b o u n d a r y 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 b o u n d a r y basically follows the 4.0 contour down from the top and ends in the lower right corner. The squared coupling g2+2w ~ cos2(c~ - / 3 ) vanishes at m 2 = 0 or M z, and as 1/m4+ for m +2 > > M z2. In addition, it peaks near m2~_= M z / 2 for fixed m+.2 Because this amplitude also involves the contraction of the derivative coupling with the longitudinal polarization vector of the W, it grows as m2+ for m +2 >> M z2. T h e net d e p e n d e n c e 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. R e f e r e n c e to our earlier discussion of h +---, tb, cb then shows that h +---, h2W + may c o m p e t e favorably with the fermionic modes depending on tan/3 and m t. The L E P exclusions clearly impact u p o n this process. T h e process h + ~ hlW +* is shown in fig. l l b . 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 L E P exclusions do not impinge on this S U S Y square. T h e near degeneracy condition arising from the mass sum rules is ( m + - m l ) / ( m + + m 1) < cosZ0w/(1 + ~/1 + coS20w )2 • 0.14. T h e decay width peaks near the corner where m I = M z, m +2 = M 2 + M 2, and the W feels its on-shell pole. The peaking is inhibited however by the vanishing of g + l w ~ s i n ( a - / 3 ) at rnl ~ M z. Because of this suppression and the fact that the W is

14. Pois et al. / S U S Y Higgs bosons lo

1.0

o.

0.8

481

0.6

0.8-

0.4

0.2

0.2-

0.0

0.0

,

~

0.2

0.4

0.0 •

.....

0.6

(m2 • + mw2)/m+

0.8

1.0

o.o

(m+ 2 - m12)/Mz 2

1.25

m+/Mw

ma/M w

~o co,20w

&,,

&

2

1.5

I

0.5

I

2

1

!

34

I~,

I

2 34

I

N

1 O -~

nil, C

10 -~

lO72

E

\

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) - I o g ( F ( h +-~ I]2W+(*~ ~ h2ud)/GeV) with contour increments of 0.2, (b) - l o g ( F ( h + ~ h]W +* ~ h l u J ) / G e V ) with contour increments of 0.5, and (c) F ( h + ~ h3W +* ~ h~ud)/GeV. For the h,/xv f i n a l state, divide by the c o l o r f a c t o r 3. G r a p h (a) h a s d y n a m i c a l z e r o s at y = 0 a n d 1, a n d at x = 0; A L E P H e x c l u s i o n s ( h 2 h 3 - d i a g o n a l ; h 2 - v e r t i c a l ) a r e d i s p l a y e d ; (b) h a s a d y n a m i c a l z e r o at y = 1 a n d a k i n e m a t i c a l z e r o at x = 0; (c) s h o w s via t h e a r r o w t h e m i n i m u m a l l o w e d m 3 / m + r a t i o a c c o r d i n g to t h e 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. T h e 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 b o u n d e d variable x = m 3 / r n + (fig. llc). 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 m3/m+=m3/v/m~+M2w, the new A L E P H b o u n d (for t ' 2 > c ,) m 3 > 3 8 G e V further implies m 3 / m + > 0 . 4 2 or equivalently, m + > 1.1M w. We indicate this b o u n d 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 u n d e r 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 m e n t i o n e d 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 . 2 M 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 competitive with the tb, cb modes when the W is on-shell (roughly the p a r a m e t e r range 0.2M z ~< m 2 ~< 0.7m3), and potentially measurable in the rest of p a r a m e t e r space, where the W is off-shell. Small portions of the S U S Y square for each of these reactions are already excluded by the L E P data.

5.2. PURE HIGGS PROCESSES In the pure Higgs sector, only h~ decay is allowed. At order g2 there are just two o p e n channels: h l~h2h

2,h3h3.

These two-body processes are presented in figs. 12a, b. For each process, the width is large, 10 to 100 MeV, and competitive with the bb m o d e over most of the allowed region. No simple choice of variables will fill the S U S Y squares. For the h2h 2 final state, the variables chosen are x = m 2 / M z, y = M z / m ~ ; the kinematically allowed region ( 2 m 2 < m 1) corresponds to 2xy ~ 1. T h e r e is a robust local maximum of 100 M e V in the rate for m~ = 2 M z, m 2 = 0.7M z. The coupling gl22 vanishes for m 2 = 0 or M z, approaches 3m 2V / ~ - m ?~/ M z <~ 53 as m 1 --+ ~ and approaches m 2 / m 3 as m t - ~ M z, in units of g M z / c O s 0w; it is maximized for m 2

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

483

1.0

0.8-

0.6-

¢~

0.4-

0.2-

0.0 ~ o.o

0.2

0.4

m2/Mz

o.~

o.8

~.o

1.oo

0.95

0.90

o.~

0.80

0.75

(Mz/ml) =

Fig. 12. (a) F ( h l ~ h 2 h 2 l X 10 2 ( G e V ) with c o n t o u r i n c r e m e n t s of 0.5, and (b) F ( h I ~ h 3 h ; ) × 10 2 ( G e V ) with c o n t o u r i n c r e m e n t s of 1.0. G r a p h (a) has d y n a m i c a l zeros at x = 0 and 1, and a k i n e m a t i c a l l y f o r b i d d e n region; (b) has a k i n e m a t i c a l zero at y = 0, and a region f o r b i d d e n by mass sum rules. T h e regions e x c l u d e d by A L E P H d a t a ( h 2 h 3 - d i a g o n a l ; h2-horizontal) are also shown.

n e a r M z / 2 . T h e large t a n / 3 r e g i o n s a r e y = 1 ( w h e r e m~ ~ M z, m 3 ~ rn 2 a n d the c o u p l i n g is m a x i m i z e d ) a n d x = 1 ( w h e r e m ~ ~ M z a n d the c o u p l i n g is s u p p r e s s e d ) . F o r the h3h 3 final state, the c h o s e n v a r i a b l e s are x = ( M z / m ~ ) 2, y = ( 2 m 3 / m l ) 2. W i t h t h e s e variables, the S U S Y Higgs mass sum rules a n d k i n e m a t i c s c o n s t r a i n y > 4-4x a n d 0.75 < x < 1.0. T h e l a t t e r c o n s t r a i n t t r a n s l a t e s into Mz<~m ~ <~ 1.15M z, if h I --* h3h 3 is to occur. T h e c o u p l i n g g133 is m a x i m i z e d at gMz/cOs 0 w as m I --*M z. T h u s the d e c a y width is m a x i m i z e d d y n a m i c a l l y by m 1 = M z, a n d m a x i m i z e d in p h a s e space by t a k i n g m 3 as light as possible. This f a v o r e d region is e x c l u d e d by A L E P H d a t a if t a n / 3 > 1. Still, the rate fall-off away from the m a x i m u m is gentle, a n d t h e r e r e m a i n s an allowed r e g i o n with 10 to 50 M e V width contours. T h e t a n / 3 r e g i o n s are largest at x = 1, a n d at y = 4 which is unphysical for this c h a n n e l . A c c o r d i n g l y , h 1 --* h3h 3 is large ( d i s a l l o w e d ) if N a t u r e c h o o s e s the large t a n / 3 o p t i o n with m~ - M z (rn 2 ~ M z ) . T h e r e are two allowed decays at o r d e r g4: h~ --* 3h 2, h2h3h 3. T h e s e t h r e e - b o d y c h a n n e l s receive c o n t r i b u t i o n s from a d i r e c t O(g 2) q u a r t i c coupling, a n d f r o m two O ( g ) cubic c o u p l i n g s 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. T h e s e g r a p h s a r e down by O ( g ) f r o m the t w o - b o d y g r a p h s (even n e g l e c t i n g the s u p p r e s s i o n d u e to t h e i n t e r m e d i a t e Higgs p r o p a g a t o r ) . W h e n f o l d e d with the t h r e e - b o d y p h a s e space factors, t h e t h r e e - b o d y d e c a y rate is down f r o m the t w o - b o d y r a t e by the usual O ( ~ / T r ) . W e t h e r e f o r e do not c o n s i d e r the t h r e e - b o d y c h a n n e l s any f u r t h e r , e x c e p t to n o t e that a l t h o u g h h ~ - - * 2 h 3 h 2 a p p e a r s to involve t h r e e i n d e p e n d e n t

484

1-!( Pois et al. / S U S Y 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 m z / m 3 ; h +--* h2W+* plotted against m 2 / M z and (m~ + M w. ) /2r n ", + 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 l Z ~*~

485

H. Pois et al. / SUSYHiggs bosons

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 - P R O D U C T I O N 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-antifermion 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 )

Nc,S

Q

---T-

t

-

- M z)

.,

-,

+ M z F~

S2 + ( r e 2 + a ~ ) (s -Mz2) 2 +Mz2Fz2

l'f2

+ ae2fl2

(A.2)

H. Pois et al. / SUSY Higgs bosons

486

where 1 - 4 sin 2 0 w

Ce=

- 1

4cosOwsinOw '

as=

4cosOwsinO w

(A.3)

are the vector and axial-vector couplings of the electron, and - 2( T3L + T3R) + 4 Q sin 2 0 w t~f =

4 cos 0 w sin 0 w

2( T3L -- TtR ) ,

af = 4 cos 0 w sin 0 w

(A.4)

are those of the p r o d u c e d fermion f, whose left- and right-handed c o m p o n e n t s have third c o m p o n e n t s of weak isospin T3t~ and T3R, respectively. ~/s is the center-of-mass energy a n d / 3 = ~/1 - 4m2/s is the center of mass velocity of either p r o d u c e d fermion of mass m. N c is the n u m b e r of final state colors. For e + e --,p.+# Q=-l,G.=G, af=a~,and/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). T h e cross section for charged Higgs pair production is [27] o_(e+e

with /3+= 1//1 -

4m+/s.

~h+h

)

=

, 1 1 71 ,p' ~ 3+ ~ f [t~, ~7, 5, 1,s)

The cross section for neutral Higgs pair production is

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

with Cij = g,jz cos lowed by CP are

Ow/g,

(A.5)

(A.6)

g = e / s i n 0 w. In minimal SUSY, the only couplings al-

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

(A.7)

The cross section for the single hi, 2 production process e+e

~Z*~h,(Z*~ft'),

f 4 : e , u e,

(i=1,2)

is [28] o-(e+e--+hiff)=

2 ) (re' 2 + a e ) ( U 2 q - a ' ( ) " I ( s , rfl/), e 4 gizzNc 2 2 19art 3 (s-M2z)2+Mzrz

(A.8)

487

H. Pois et al. / SUSY Higgs bosons

where V/(x/2- 4~,)(1 + 3txi-x i + ~x2)dxi

= fl+P"'a2~7 ,

I ( s , m 2)

xi

( x,

2E,/~/s,

+ tx z - tx, - 1)" + / X z 7 z

(/xi,/Xz,y z) =

'

(A.9)

(mr,mz,r£)/s,

(g~zz, g2zz) = gMz(c°s(a -/3),sin(c~ -/3))/cos

0w .

(A.10)

A.2. W AND Z-DECAY RATES T h e Z and W widths to lepton pairs are useful for normalization purposes:

[

g2M z 1 +(1-4sin20w

F(Z~+/x

-) =

192~COS 2 0 w g2M w

F(W

~ ~v)

-

)2]

- -

48~-

eeM w

12 sin 2 0 w "

For single Higgs production via Z decay,

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

(A.11)

g2zz is given in (A.10). 1 2 3/2 2 2 F ( Z --+ h2h3) = 48vr Mz[g23z] A (1, m 2 / M z ,

m2/M2"~ 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 ----Igi~kl2A'/z(1,m2/mZ, H ' n i -+ n j n k ) - 64rrm i 1 + ~jk lrl/" t.

m2/m~).

(A.13)

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

H. Pois et aL / SUSY Higgs bosons

488

CP conservation. The relevant couplings are ( g122, g133) = gMz(2 sin 2 a sin(/3 + a ) - cos(/3 + ~ ) cos 2 a, cos 2/3 cos(/3 + a ) ) / c o s 0w

(A.14)

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

V(h,-, hjZ(*) --+ hjff) 2

mie Ig,jzl 16~-2

2

",

12~-

•

,

Nc(t,? +a?)

f2

,~3/2

1 + ~J d x j

(Xj +/~ z

-

/.zj - - 1) 2 +

(A.15) P, z Y z

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

xj=ZEJm,,

2 "~ (/.tj, p.z, y z ) = ( m j , M z , F2z ) / 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

rz/Mz. F(h +--* h~W (*) + --+ h~ff') and F(h, ~ h+W(*)---* h + f f ') are obtained from F(h, --+ h j Z (*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 sin20w=0.232, and the fine structure constant is taken to be a ( M z2)= (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 g22zz are each proportional to s i n 2 ( a - / 3 ) ,

H. Pois et al. / SUSY Higgs bosons

489

which may be written

m~(m~-M2) sin2( a - / 3 ) =

(mT_m2)(m,, 2 2+mZ_M2z)

.

(A.17)

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

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