Volume 158B, number 5 PHYSICS LETTERS 29 August 1985

L I G H T H I G G S B O S O N S AND SUPERSYMMETRY

John ELLIS, K. ENQVIST, D.V. N A N O P O U L O S

CERN, CH 1211 Geneva 23, Switzerland

and

S. R I T Z 1

University of Wisconsin, Madison, WI 53706, USA

Received 3 May 1985

QCD radiative corrections cause a large reduction in the T-~ H°+ y decay rate which clouds the interpretation of experimental upper limits on this decay. Electroweak radiative corrections make an important contribution to the lightest Higgs mass in models with supersymmetry broken at low energies. Calculating the Higgs mass and the T decay branching ratio incorporating these two effects, we find that a light supersymmetric Higgs may be lurking just beyond the sensitivity of present experiments on T radiative decays. This Higgs boson may be produced in association with the next-to-lightest Higgs in Z ° decays at a large rate and with a distinctive signature.

Many particle physicists now place the mechanism of spontaneous gauge symmetry breaking at the top of their agenda of problems to be solved. In particular, searches for the putative Higgs boson or bosons H are now becoming [1 ] very topical. Experimentally, searches [2] for monochromatic photons in quarkon- ium decay are now approaching the sensitivity re- quired to detect T -~ H + 3' [3 ] in the standard model, and the opportunities for extensive Higgs searches at the SLC and LEP are fast approaching. Theoretically, ideas which make elementary higgses respectable by stabilizing their masses using supersymmetry ,1 enable one to make increasingly specific predictions for their masses, couplings and production rates. In this paper we re-examine the phenomenology of light neutral Higgs bosons in general, and of those expected in supersymmetric theories in particular.

First we reconsider the radiative decay T -~ H + 3'. Two approaches suggest large reductions in the branch- ing ratio for this decay mode B(T ~ H + 3'), namely a calculation [5] of perturbative QCD radiative cor-

i Supported by United States Department of Energy contract number DE/AC02/76ER00881.

*1 For recent reviews, see ref. [4 ].

rections and estimates of effects due to mixing [6] with P-wave bot tomonium states. We show how these two effects can be combined phenomenologlcally without double counting, and present numerical es- timates of the total reduction in B(T ~ H + 3'). Typic- ally we find that B(T ~ H + 3') is reduced by a factor O(2), indicating that the ongoing searches for mono- chromatic photons in T decay will have to become somewhat more sensitive before m H < m T can be ex- cluded.

Next we reconsider the mass to be expected for the lightest neutral Higgs in supersymmetric models which contain squarks and/or gluinos light enough to be detected [7,8] at the CERN pp-collider. Radiative corrections [9] to the light Higgs mass are often larger than its tree-level value, reflecting the closet Cole- man-Weinberg nature of this neutral Higgs scalar H. Its mass is generally < 1 0 GeV, making it vulnerable to early exposure in T decays.

Finally, we point out that in many supersymmetric models [9,10] of the type introduced above, the second lightest neutral Higgs is a pseudoscalar closet axion which is light enough to be produced in Z 0 decay, and the decay Z 0 ~ a + H (closet axion + closet Coleman-Weinberg) is likely to have a large branch-

417

Volume 158B, number 5 PHYSICS LETTERS 29 August 1985

ing ratio. Such decays would have a clear event signa- ture: a -~ bbjets accompanied by H ~ c~-or more dis- tinctively H -~ r+¢ - .

We start with our re-evaluation o f the branching ratio for T ~ H + 3'. Vysotsky [5] has calculated the perturbative QCD radiative corrections to F(T ~ H + 3,) and found them to be large:

F(T ~ H + 3') _ 4as F 0(T ~ H + 3') 1 - ~ a(K),

(1)

with

ro(T-~ H + 3') G F m ~ R 0 ( T ~ n + 7) = F0(T -~ e+e _) - ,V~Trot (1 - K) (2)

2 2 and K = mH/m T, with a(K) a mathematically non- trivial function which takes values between 10 (for K = 0), 15 (for K = 0.87) and 20 (for K = 0.94). The function a(K) exhibits a singularity as K ~ 1 which corresponds to k~ ~ 0 in T -~ H + ~, decay:

a(K) f__,l as(K) - 4rr/3(1 - K) 1/2. (3)

We interpret this singularity as a first order manifesta- tion of binding effects. Note that as(K ) is not negli- gible even at K = 0: as(K) = 4.19. There is also a known correction [11 ] to F(T -* e+e- ) :

16a s F(T ~ e+e- ) _ 1 (4)

F0(T _,. e+e - ) 37r '

so that the overall correction to R ( T ~ H + 3') = F(T H + 3')/F(T -~ e+e- ) can be written in the forms

Ors R ( T ~ H + 3 ' ) = 1 - (-~a(K) - ~ - ) ,

R 0 ( T ~ H + 3") ~- (Sa)

o r

1 - (4as~3 It) a(K)

1 - 16as/31r ' (5b)

which are in principle equivalent at leading order in as. In practice, formulae (Sa) and (Sb) exhibit a sig- nificant numerical difference which reflects higher order effects.

Several authors have recently evaluated the effects

of mixing with P-wave bottomonia on P(T ~ H + 3'). They find a wave-function model-dependent correc- t ion

r ( T ~ H + 3') _ 1 + 0(1/k~¢), (6) FO (T ~ H + 7)

for large k. r. There is no singularity in F(T -~ H + 7) as k 7 -~ 0, since a full treatment of binding effects smooths out the rate as K -~ 1.

The perturbative and mixing corrections to F(T H + 7) share the common features that they are

largest when K ~ 0(1)(k. r ~ 0), and both have a piece o: 1/k. r (at small k.. for perturbative QCD, at large k~ for mixing). Th~se 1 ]k, r pieces have consis- tent normalizations within the theoretical uncertain- ties. We wish to construct a phenomenological form including all these corrections to R ( T ~ H + 3') which does not double-count. We do this in two alternative ways. One is (A) to use (5a) for m H < ~ , and for m H > ~ use a phenomenological parametrization of the functional form of the mixing corrections [6], normalized so that R ( T -+ H + 7) is continuous at m s = t~. The other method (B) is to use (5b) for m H < r~ and to use for m H > r~ the same phenom- enological parametrization of the functional form of the mixing corrections, but now normalized so that F(T-+ H + 3') [1,6] is continuous at m H = ~ , and then divided by the T -'- e+e- correction fac- tor (4). These are two alternative patch-ups which seem to avoid double-counting and are equivalent to leading order in a s . While they both give continuous curves for R ( T -+ H + 7), their derivatives exhibit an unsightly jump at m 0 which we remove by gaussiani smoothing. Our final phenomenological corrections to R ( T ~ H + 3') are insensitive to ~ in the range o f 5 to 7 GeV. Fig. 1 shows the leading order and the corrected ratios R ( T ~ H + 7) = F(T-+H + 7) /F(T -~ e+e- ) calculated using prescriptions (A) and (B) described above. We see that they give similar results, and that both reduce the expected R ( T ~ H + 3') by a factor 0(2). Under these circumstances, we cannot exclude the possibility that higher order corrections might also be important and suppress T ~ H + 3' de- cay even further. However, fig. 1 means that mono- chromatic photon searches [2] must be at least twice as sensitive as was suggested by the zeroth order for- mula (2) before light higgses can be observed or ex- cluded in T -+ H + 3" decay.

418

Volume 158B, number 5 PHYSICS LETTERS 29 August 1985

9 i i i t i i i I i

7 -

C 6 ~ s

2

t I I I I ' t

0 1 2 3 z, 5 6 7 8 9 r~ (GeV)

Fig. 1. R ( T --* H + 3') -= I ' (T ~ H + 7 ) / I ' (T ~ e+e -) p lot ted as a function of m H, calculated using (L) the leading order for- mula (2), and in corporation first order radiative and binding corrections according to the prescriptions (A) and (B) describ- ed in the text. The difference between these two latter curves is a measure of higher order certainties. Also shown is the pre- ferred range of light supersymmetric Higgs masses.

After the above model-independent analysis, we now turn to the model-dependent predictions for the light Higgs masses in supersymmetric theories. The ef- fective scalar potential for neutral higgses in such models is

V = ml 2 IH 112 + m 2 IH 2 [2 _ m2(H1 H2 + b.c.)

1 ~ + ~ ( 2 +g'2XInll2 IH212) +/iV, (7)

where ~ V contains the radiative corrections [9]. The physical spectrum contains [12] two neutrals Ha, K ,2 with masses

m2Hp, K = ½{(m 2 + m 2 + m 2) + [(ml 2 + m 2 + m2) 2

_ 4(ml 2 + m2)m2zcos220 ] 1/2}, (8)

at the tree level and one with mass

m2a = 2m2/sin 20, (9)

where tan 0 = (0 IH 210)/(01H 110) and

co5220 = 1 - 4m4/(m 2 + m2) 2. (10)

We work in the context of minimal supergravity models in which radiative corrections due to a heavy t quark drive weak gauge symmetry breaking [9], for

. 2 The subscript P s tands for t~dnava (heavy in Finnish), while K stands for kevyt (light in Finnish).

which (0IH2I) ~ (01Hll0) and cos 20 < 1, so that m 2 K (8) is small and corrections to m2 K become im- portant. In this framework

2 + m 2 A 2 ) ' (11a) - 3 r 2 (m 2 + m 2 + m ~ L m 2 - m 2 l + 6 r " -

m r + m2 2 = 2m4 2 + 2rn~2~L 3r2 : 2 + m 2 l + 6 r ~[mQ

2 + rrr'~L + m2A2), (1 lb)

m 2 = Bm4m O, (1 lc)

where

1 Bmo 1[ B2m2 4 2 m4 - 2 sin 2~0 + - - - 7-~ ~Im~L 2~sin220 1 * o r L

3r2" 2 + ~ ( m Q m 2 + m 2 A 2-3m~2~L)]) 1/2. ( l l d )

The parameter r 2 is related to the top quark mass by

r 2 = (mt/mw) 2/[N - 6(mt/mw) 2 ], (11 e)

where N is a parameter which is calculable from the renormalization group (see table 1), and the spin-zero sparticle masses

2 =m2(1 + ( l l f ) mQ,U'~L CQ,U,L ~J 2 )

Table 1 Values of renormalization parameters. The values for various parameters renormalized at m w entering into calculation of the low-energy mass spectrum ((for defini- tions, see ref. [9]). These values have been obtained by taking ~em(M w) = 1/127.5, ~a (Mw) = 0.122 and sin 20w(Mw) = 0.22 with m Z = 93.8 GeV.

CQ 6.65 CU 6.23 CD 6.18 C L 0.52 CE 0.15 C A 3.84 Ca 0.59 CH 2.55 c4 1.40 N 23.49

419

Volume 158B, number 5 PHYSICS LETTERS 29 August 1985

with the coefficients C O U,L given in table 1. The parameter m 0 is a model-dependent scalar mass which may (or may not) be identified with the gravitino mass m312, A = A Oa) is a model-dependent parameter expected to be O(1) w i t h A ( m x ) = A x = 3 a theo- retically favoured value. We take B = A - 1 before renormalization, and after renormalization

A = (A x + CA~2)/(1 + 6r2),

B = (A x - 1 + CB~ ) -- 3r2A, (1 lg)

where C A and C B are given in table 1, and ~ is the pri- mordial gaugino/scalar mass ratio ml/2/m 0 which de- termines the gluino mass

m~" ~- [~3(#)/o~3(Mx)] ~m 0 ~ 2.8 ~m 0 . (1 lh)

Our general strategy for solving the eqs. (11) and hence evaluating the I-Iiggs boson masses (8), (9) is (i) assume values of m t and m y motivated by experiment,

and supplement these assumptions with a suitable (like- ly) choice of the dimensionless parameters A x , ~. Now r, M 0, B, mEL and hence m 2, m 2 are determined and we can (ii) solve (10), (11 c) and ( 11 d) iteratively to extract m3, m 4 and 0. Finally we can (iii) calculate m2Hp,K (8) and m a (9). We know from previous ex- perience [10] that consistent solutions for m t = 0(40) GeV and m~" = 0(60) GeV exist for (~ ,Ax) ~ (0.7, - 2 .5 ) and (1, 3). Fig. 2 shows the dependence of milK on m t for fixed m y , ~,A x and on m~t for fixed m t , ~ , A x : typically milK is a few GeV [12]. Fig. 3 shows the dependence o f m a on ~ and A X for fixed m t = 40 GeV and m y = 60 GeV: often m a ~ O(mz). We have found that in the region of interest, m a is in- sensitive to the value of m t but increases approximate- ly linearly with m y at fixed ~, A X-

So far we have only discussed the three level con- tributions to Higgs scalar masses, but the smallness of milK (8) in fig. 2 suggests that radiative corrections

5

3

E 2

0

10

~ 6 L ~

E ~

~=0.7 ~= 60 GeV x=-2.5

t I I I I I I

~=1.0 ~ = 60 GeV Ax=3.0

nlH ~,

~=0.7 mt = 40 G e V ~

AX=-2"5 m ~ ' " ""

I t I I

- ~=1.0 mt=t~O G e V J Ax=3-0

,a. , ~ , "a" .a~ s . ~

s s ~ . j , .

I I I I I | I I I I I I

0 32 36 40 44 48 52 56 60¢0 S0 60 70 80 90 100 m r (GeV) m, 4 (GeV)

Fig. 2. Tree level (milK) and radiative correction (~mH) contributions to the total light Higgs mass m H. Values are shown for typical [10] values of(~,Ax) = (0.7, -2.5) and (1.0, +3.0), as functions ofm t (for my fixed at 60 GeV) and of my (for m t fixed at 40 GeV).

420

Volume 158B, number 5 PHYSICS LETTERS 29 August 1985

e"

A >

120

100

80

60

t~O

7

6

S

t,

3

~--1.0

0.7

0./* I I I I

120

100

80

~ ~ =1.0

0.7

i t 0 . t ' i I t

i 60

~0 I

O.L, _ . _ . ~ - ~ ' P "

I I I I

_ ~ . 0

i I I

-34 -3.2 -3.0 -2.8 -2.6 2.9 3.0 3.1 3.2

Ax A x

4

3 i 2.7 2.8

Fig. 3. Values of the closet axion and light Higgs masses for typical values of m t = 40 GeV, m~ = 60 GeV as functions of ~ and A X in the preferred domains offer. [10].

[9 ] ' f rom 8 Vin eq. (7) may be important , although they are not important for mHp or m a. They con- tribute [9]

8m 2 = -(6ct2/aTrXM22 + m 2 _ M2m 4 - ~ B m o m 4)

- (6t~ 1/20nX M2 + m 2 _ M 1 m4 - ½Bmom 4)

+ [3h2/(nTr) 2] [(CQ + C U - 2 C L ) ~ 2 m 2

- 2(m 2 - Bmom4) + (Am 0 - m4)2], (12)

where M 2 (M 1) is the mass of the SU(2) [U(1)y] gaugino: M i = o q ( # ) / ~ ( m x ) ~ m 0 and h t = g 2 ( m t / m w ) is the top quark Yukawa coupling. This expression is clearly more important for larger m ~ as seen in fig. 2, which also exhibits the dependence of 6m 2 on m t. The total light Higgs mass is

m 2 =m2HK + 6m2H, (13)

which is also plotted in fig. 2. Clearly the radiative

corrections 8m2H (12) make the dominant contribu- tion to m 2 in most of the domain of interest, making it a closet Coleman-Weinberg Higgs boson.

It is this total light Higgs mass m n (13) which is plotted below m a (9) in fig. 3, as a function of ~ and A X for fixed m t = 40 GeV and m~ = 60 GeV. It is clear from figs. 2 and 3 that in general m H is a few GeV, so that the closet Coleman-Weinberg Higgs bo- son is accessible in T ~ H + 7 decay. Because 0 given by eq. (10) is close to ~r]4, it has similar couplings to quarks and leptons as does a conventional minimal Weinberg-Salam Higgs boson [13]:

gHq~/aq2/3 = gH-42/aq2/3 IWS cos or/sin 0,

gH-q-l/3 q- l/a' g H ~

=gH-4_l/aq_,/a,gn~gr-lWS sin or/cos 0, (14)

where a is fixed by [13]

421

Volume 158B, number 5 PHYSICS LETTERS 29 August 1985

and ~ ~ 0 "" 45 ° + 0(2*) in the domain of parameter space of interest in this paper. Thus the rate for 'I ' -* H + 1, and the H decay modes are expected to be al- most indistinguishable from those of a conventional light Weinberg-Salam I-Iiggs boson. The model-inde- pendent analysis of 3" ~ H + ~ decay earlier in this paper therefore suggests that the experimental noose may be tightening around the neck of this closet Coleman-Weinberg ttiggs: see fig. 1 where the range of m H from fig. 3 is indicated. Note also that topon- ium 0 ~ H + 3, and 0 ~ a + ~, could both be observable if m o "" 80 GeV.

Finally we turn to Z 0 -* a + H decay. The relevant decay vertex is [131

cos(~-0) gz°aH = g2 2 cos O w (Pa + PH) ~ (16)

which has no unpleasant suppression factor if e ~ O ~ 1r/4 as we expect. The vertex (16) gives a cross sec- tion

o(s) = (Ir~2/3s)X3/2(1, m2HK/S, m2/s)

[(1 -- 4 sin20w) 2 + 1 ] cos2(e -- 0) X

64 sin40w cos40w

s 2

where X is the conventional phase space function. Fig. 4 exhibits the cross section (17) at its peak, modified by initial state bremsstrahlung corrections [14]. Rather than use the input theoretical parameters m 0 and roll 2 = ~m O, we have labelled the axes in fig. 4 by the physical observables m~ (1 l f ) and m~" (1 lh). The contours correspond (from right to left) to o = 10 -3 , 5 X 10 -3 , 10 -2 , 5 × 1 0 - 2 1 0 -1 and 5 X 10 -1 nb: for comparison, o(e+e - -~ # + # - ) -- 1.0 nb at the Z 0 peak. We see that if the squarks and gluinos have similar masses and are light enough to be produced at the CERN p~- collider, the Z 0 may well have an ob- servably large branching ratio into a + It. For the larger values of m a which occur when m'~ ~" m'~, the maxi- mum of e(e÷e - ~ Z 0 ~ a + H) does not occur on the

100

90

80

60

50

t,O

I I I I I I Ax_--2.s

/ /,

I / / / / m~' < 2 0 G e V / / 7 / 1

,/,-/,// I/ / I !/I / / / / /

I i / I I i i / , ' / / / / I / / I /;'I /,' ,'/,:;,/

50 60 70 80 90 m~' (GeV)

100

70

65

60

50

/,5

L,O

I I " 1 2 p

A/ m~'R<20 GeV ////¢,!t/

t ¢ / l l

I I' //I / / / , f / /?//"/ ,

50 60 m~ (GeV)

70

Fig. 4. Contours of the radiatively corrected [ 14] cross see- lion (17) for e+e - --* a + H at its peak for ranges of values of ra~. and my and two selected values of Ax: see also table 2. We use mZo = 93.8 GeV and FzO = 3.0 GeV. The contours from right to left are for apeak = 10 -3, 5 X 10 -3, 10 --2 , 5 X 10 -2, 10 -1 and5 X 10 -1 nb, respeetively.

Z 0 peak itself, but at some higher centre-of-mass ener. gy. Table 2 shows for representative choices of m~" and m~ the corresponding values o f m H and ma,

422

Volume 158B, number 5 PHYSCIS LETTERS 29 August 1985

Table 2 Predictions for different m~', rn~ values. All values are calculated withA X = 3, m t = 40 GeV. In cases where Opeak and x/sat the peak are not tabulated, the peak cross section occurs above x/s -= 150 GeV and apeak < 10 -3 nb. The angle factors (14), (15) which increase R(T ~ H + 3,) by 2 to 7% are not included.

rn~ m~ m H m a R(T--+H +3,) Opeak x/~ (X 10 -3) (nb) at peak

(GeV) (L) (B)

40 40 3.5 49 6.9 3.4 0.35 94.1 50 4.7 90 6.1 2.9 - 60 5.5 121 5.4 2.5 - 70 6.0 148 4.8 2.2 -

50 50 4.7 61 6.1 2.9 0.17 94.1 60 5.8 105 5.0 2.3 - 70 6.5 137 4.2 1.8 -

60 60 5.9 74 4.9 2.2 0.046 94.2 70 7.0 118 3.6 1.5 -

70 70 7.2 86 3.4 1.3 8.3 x10 --4 96.6

R ( T ~ H + 7) calculated in leading order (L) and in higher order using method (B), and the radiatively cor- rected o and x/s-at peak.

Associated production of a + H in Z 0 decay or else- where should have a distinctive experimental signature. The closet axion a is heavy enough to decay into b b (of interest to precision vertex detectors) but not heavy enough to decay into tt, while the light closet Coleman-Weinberg Higgs mostly decays into c~-or r+r - (vertex detectors again, also with a distinct topology). A final state containing a larger mass bb- jet pair and a lower mass r+r - pair should be easy to fmd.

We have seen in this paper that a light neutral Higgs may lurk just beyond the sensitivity of present T -+ H + 7 searches, that a light Higgs is expected in super- symmetric models with a light sparticle spectrum ac- cessible to the CERN p~ collider, and that such models suggest a large Z 0 -+ a + H decay branching ratio with a distinctive signature. Perhaps the mechanism of spon- taneous gauge symmetry breaking will soon be eluci- dated.

We would like to thank Haimo Zobernig for check- ing some of the calculations in this paper. One of us (K.E.) gratefully acknowledges the financial support

of the Academy of Finland, and one of us (S.R.)

thanks the CERN Theoretical Physics Division for its hospitality.

[1 ] J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B106 (1976) 292; L. Okun, Proc. 1981 Intern. Symp. on Lepton and pho- ton interactions at high energies, ed. W. Pfeil (Universi- t~it Bonn, 1981) p. 1118.

[2] CUSB Collab., S. Youssef et al., Phys. Lett. 139B (1984) 332; P. Franzini, Talk Fifth Moriond Workshop on Heavy quarks, flavour mixing hnd CP-violation (January 1985); Argus Collab., M. Albrecht et al., DESY preprint 85- 014 (1985).

[3] F.A. Wilczek, Phys. Rev. Lett. 39 (1977) 1304. [4] J. Ellis, CERN preprint TH.3802 (1984);

D.V. Nanopoulos, CERN preprint TH.3995 (1984); H.P. Nilles, Phys. Rev. Cll0 (1984) 2.

[5] M.I. Vysotsky, Phys. Lett. 97B (1980) 159. [6] H.E. Haber, G.L. Kane and T. Sterling, Nucl. Phys. B16

(1979) 493; J. Ellis, M.K. Galliard, DN. Nanopoulos and C.T. Sachrajda, Phys. Lett. 83B (1979) 339; J. Pantaleone, M. Peskin and S.-H. Tye, SLAC preprint PUB-3439 (1984); J. Polchinski, S.R. Sharpe and T. Barnes, Harvard pre- print HUTP-84/A064 (1984); J.P. Jackson and J.L. Rosner, CERN preprint TH.3992 (1984); H.J. Lipkin, Phys. Lett. 151B (1985) 155; W. Bernreuther and W. Wetzel, Univ. Heidelberg pre- print HD-THEP-85-2 (1985).

[7] J. Ellis and H. Kowalski, Phys. Lett. 142B (1984) 441; Nucl. Phys. B246 (1984) 189; CERN preprint TH-4072 (1984).

[8] E. Reya and D.P. Roy, Phys. Lett. 141B (1944) 442; Phys. Rev. Lett. 52 (1984) 881 ; Dortmund preprint (1984).

[9] J. Ellis, J.S. Hagelin, D.V. Nanopoulos and K.A. Tamvakis, Phys. Lett. l15B (1983) 275; C. Kounnas, A.B. Lahanas, D.V. Nanopoulos and M. Quiros, Nucl. Phys. B226 (1984) 438.

[10] J. Ellis and M. Sher, Phys. Lett. 148B (1984) 309; L.E. Iba~ez, C. Lopez and C. Mu~oz, CERN preprint TH.4071 (1984); K. Enqvist, A.B. Lahanas and D.V. Nanopoulos, CERN preprint TH.4095 (1984).

[11 ] R. Barbieri, R. Gatto, R. K6gerler and Z. Kunszt, Phys. Lett. 97B (1979) 499.

[12] K. Inoue et al., Prog. Theor. Phys. 68 (1982) 927; 71 (1984) 413; M. Drees, M. Gltick and K. Grassie, Univ. Dortmund pre- print DO-TH85/4. See also: N. Deshpande, X. Tata and D. Dicus, Phys. Rev. D29 (1984) 1527; H.E. Haber and J.F. Gunion, SLAC preprint SLAC-PUB- 3404 (1984). F.A. Berends and R. Keiss, Leiden Univ. preprint (1984).

[131

[141

423