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QCD corrections for Higgs-boson production
Volume 91B, number 3,4
PHYSICS LETTERS
21 April 1980
QCD C O R R E C T I O N S F O R HIGGS-BOSON PRODUCTION A. NICOLAIDIS
Laboratoire de Physique Corpuseulaire, CollOgede France, Paris, France Received 20 November 1979
We consider, within QCD, the spectrum of a Higgs boson radiated off a heavy quark. Like in the case of the structure functions of the photon, QCD corrections renormalize the free quark result, supressing it at large x and enhancing it at small x. Also a Higgs boson distribution within a gluon is developed, which is not negligible at small x.
All present models of spontaneously broken gauge theories of weak and electromagnetic interactions contain as an important ingredient Higgs scalars. In the conventional model [ 1] the coupling of the Higgs to fermions (leptons, quarks) is proportional to the fermion mass, while for the intermediate gauge bosons (Z, We ) it is proportional to their mass squared. Evidently the Higgs particle will be produced most abundantly in processes involving heavy particles. Gaemers and Gounaris [2] have considered Higgs boson (mass 34) production in e+e - annihilation through bremsstrahlung off a heavy quark (mass m). However, at high energies, X/7>> M ( P E P - L E P energies), the heavy quark will have enough " t i m e " r -= (1//3) log [a s (M2)/as (02)1 2 (fl = 11 - gNf) to develop its own cloud composed o f QCD quanta and therefore QCD corrections to the Higgs boson spectrum will be quite important. We examine these corrections in this paper. Consider a hard scattering process U where a heavy quark q is produced, which subsequently radiates a Higgs boson H (fig. la). In the leading log approximation (LLA), where all masses are ignored compared to the high energies involved, this process factorizes in two pieces: the cross section for the production o f the heavy quark times the probability density of finding a Higgs boson inside the quark. To extract this probability density (zeroth order in c~s at first) we calculate in the LLA the cross section for the process shown in fig. 1a. The amplitude is given b y
H(k,M..) q(p,m)
U
q(r,m)
(a)
@/.---"@ /-"" .........
(c)
(b)
Fig. l. Higgs boson radiation off a heavy quark. (a) Born term. (b) Gluonic QCD correction. (c) The development of a Higgs distribution within a gluon.
M=gH~(r )
4(+ r+ m
U,
(1)
(k + r) 2 - m 2 where gH = m(GFX/~) 1/2" See fig. la for the rest o f the notation. The cross section is do = IMI 2 d V (2) ,
(2)
and d V (2) is the two-particle phase space. In the limit where all mass terms are ignored and considering collinear emission we have [31 dV (2) = d g ( 1 ) [x (1 - x)/8rr 2 ] p2 d x d z ,
(3)
IMI 2 ~ g 2 ( 2 k . r) -1 Tr (XUU l ,
(4)
where x is the momentum fraction carried b y the Higgs boson, d V (1) is the one-particle phase space and z = cos (k, i). Using the above relations we find 447
Volume 91B, number 3,4
do dx
21 April 1980
PHYSICS LETTERS
1+.
x log~_-uTr(pUU)dV(1),
where v = (1 - M 2 / x 2 Q 2 ) 1/2 and we have assumed throughout that Q2 >>M2, m 2. Therefore we obtain
It is relatively simple to solve the algebraic eqs. (10)-(12) and to find Hqv(n), Hqs(n) and Hg(n). Finally for the Higgs distribution within a quark or gluon we obtain
Ptt/q(X) = (oM/47r)m2xt ,
Hqi(n) = c(n)t
16n2
(5)
(6)
with a w = GFX/~w/4zr and t ~ log Q2/M2. It is remarkable that eq. (6) is the exact result [2] when M/Q, m/Q terms are negligible. Gluon emission by the heavy quark before radiating the Higgs boson will modify the Higgs boson spectrum (fig. lb). Also since the transition g ~ q -* H is possible, a Higg boson distribution within a gluon will be developed (fig. lc). The Higgs distribution within a quark (gluon) Hq (/-/g), will be given by an Altarelli-Parisitype equation, which is summing up to all orders in as, in the LLA, the strong-interaction renormalization effects: dHqi
dt
as(t) -mZe(x) + ~ [Hqi @ Pqq + H g @ Pgql , (7)
dt - 2n
Hqi ® Pqg + Hg ® Pgg .
(8)
The symbol A ® B stands for the convolution 1
A(x) ® B(x) = f ~ A ( y ) B ( x / y )
,
(9)
X
and c(x) = (oM/410 3x (the factor 3 takes the color into account). The above equations are simplified when we take the moments. Defining also the valence contribution H qv = H qi - Hq/and the singlet contribution H qs = ]~iHqi we have dHqVdt- (m2 - m2)c(n)
+ as(t) - ~ - Hqv(n)Aqq(n) ,
(10)
(13)
tm _ m2
X ( i + dqq(~) + (m2) Hg(n) = c(n)t(m 2 )
1 + dgg(n)
I,
[1 + d+(n)] [1 + d - (n)l
J
-dqg(n) [1 + d+(n)] [1 + d - ( n ) ]
(14)
The d(n), A(n) are the well-known anomalous dimensions which can be found for instance in ref. [4]. The similarity of these equations with the corresponding structure functions of the photon (the bestknown colorless particle) in QCD is evident [5]. The free quark model transition q ~ 7 has been replaced by q -~ H and the electric charge of the quark by its mass. To recover the distributions Hq(x, t), Hg(x, t) we have to undo the moment integral. We use a simple method which has been applied already in parametrizing the structure functions of the photon [4]. After extracting the singular behavior at x ~ 0, which is the same for both colorless particles, we expand the distributions in series of Jacobi polynomials. We take also five flavors (u, d, s, c, b) and for the quark masses ,1 we assume m u = 5 MeV, m d = 9 MeV, m s = 180 MeV, m e = 1.5 GeV, m b = 4.7 GeV with (m 2 ) = 4.874 GeV 2 . Writing
xHi(x, t) = (aw/41r)tfH/i(x) ,
(15)
we find for the bottom quark, keeping five terms in the expansion fHPo(X) = 3x-0-6(0.046 + 3.93x (16) -- 4.98x 2 + 40.18x 3 -- 33x 4) .
dHqs dt - (m2)2Nfc(n)
%(0 +~ [Hqs(n)Aqq(n)+2NfHg(n)Agq(n)] , (11)
In fig. 2 the two distributions, with and without strong interactions, are compared. QCD effects suppress the spectrum at large x and enhance it at small X.
The charmed quark fragmentation function into
dHg as(t) [Hqs(n)Aqg(n) + Hg(n)Pgg(n)] . dt 2n
448
(12)
,1 The quark masses are those which appear in the lagrangian. The light quark masses can be determined using PCAC [6]. We use the values quoted in the second paper of ref. [6 ].
Volume 91B, number 3,4
21 April 1980
PHYSICS LETTERS
22.0
2.0
// 19.8
/
• i I
1.8
17.6 1.6
15.4
I
I I
1.4 13.2
/ /, s•
11.0
1.2
•.// 8.8
1.0
/
\
4,4
.8
.6 .4
"%
. . . . ..1
" ' .2 ;
. 3'
' .~
.~
. '=
.;
.7
1.
.;
Fig. 2. The momentum fraction carried by the Higgs boson, radiated by a b quark. Dashed line is the free-quark result rn~x 2 , while the solid line is the QCD result (in units (~w/4~)t, color is not included). the Higgs boson, together with the zeroth order result in as, is shown in fig. 3. The five-term a p p r o x i m a t i o n is
fH/c(X) = 3 x - 0 - 6 ( 0 . 0 9 1 + 0 . 0 4 1 7 x (17) + 0.2x 2 + 3.38x 3 - 3x 4) . Given the smallness o f the light quark masses the fragmentation u (d) ~ H is d o m i n a t e d by the singlet piece. The Higgs distribution within a gluon, eq. (14), is not negligible. It is bigger than the light quark contribution, over almost the entire region o f x (fig. 4). Three terms only are sufficient to reproduce accurately the light quark and gluon fragmentation into the Higgs: 2.5 i/ iI /I
I
111
1.5
11 /##
1.0
.0 ,0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.
Fig. 4. The same as in fig. 2, for the u (or d) quark (a) and the gluon (b). fH/u(d) = 3 x - 0 " 6 ( 0"09 -- 0.25x + 0 . 1 7 5 x 2 ) ,
(18)
fH/g = 3 x - 0 " 6 ( 0"15 + 0.243x -- 0 . 4 1 7 x 2) .
(19)
In table 1 we present the ratios Pi = o ( e + e ~ q i q i H ) / o ( e + e - -+ %.C~.) ( q i = s, c, b quark) for a beam energy o f 100 GeV and t w o different values o f M. The first row corresponds to the exact result, to zeroth order in a s, and the numbers q u o t e d are taken from ref. [2]. The second row gives the same ratios in the L L A (still z e r o t h order in C~s). For the lower mass (M = 3 GeV), L L A is a very reasonable a p p r o x i m a t i o n scheme. For the higher mass (M = 10 GeV) the L L A overestimates the exact result b y a factor 1 . 3 - 1 . 5 . Clearly, it is desirable to find a consistent way to inTable 1 The ratios Pi = o( e+e- --" qiEtiH)/a(e +e- ~ qiEti) (qi = s, c, b) at x/)-= 200 GeV. The first row corresponds to the exact result (zeroth order in C~s), second row is the leading log approximation to the exaet result (zeroth order in a s and the third row is the QCD-corrected prediction (to all orders in a s, in the LLA).
/ 2.0
.2
J
4//
M
s
c
b
10
0 . 1 0 X 10 - 6 0 . 1 5 X 10 - 6 0 . 3 4 × 10 - 6
0 . 9 5 × 10 - 6 1.3 X 10 - 6 1.4 × 10 - 6
0 . 9 2 × 10 -5 1.2 × 10 -5 1.18 × 10 -5
0.17 X 10 -6 0.2 x 10 -6 1.4 X 10 -6
0.16 X 10 -s 0.19 X 10 -5 0.31 X 10 -5
0.17 X 10 -a 0.17 x 10 -4 0.19 X 10 -4
f" °5
0.0
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
Fig. 3. The same as in fig. 2, for the charm quark.
1.
449
Volume 91B, number 3,4
PHYSICS LETTERS
corporate the Higgs mass effects, when the ratio M/Q is not too small. The third row gives the QCD-corrected predictions (in the LLA). For the relatively light quarks (s, c quarks) where the singlet component (small x ) is important, the QCD result is larger than the free-quark model result. Also for the lower mass (M = 3 GeV) the enhancement is larger since a larger protion of the small-x region is available. On the other hand, the heaviest quark (b quark) is dominated by the valence contribution, which is supressed by the QCD corrections and therefore the ratio Pb is not enhanced. (Notice also that the first two rows have been calculated with m b = 4.5 GeV, .while the third with m b = 4.7 GeV.) In any case the QCD corrections are quite significant and they have to be taken into account. Armed with the above distributions we can calculate the Higgs boson production via bremsstrahlung off a heavy quark produced in any hard process [7] (largePT collisions, or 77 collisions). The observation of the Higgs boson in such a process will be not only a test of the gauge theories o f weak and electromagnetic interactions but also a test o f the gauge theory of strong interactions, namely quantum chromodynamics.
450
21 April 1980
Discussions with N. Arteaga-Romero are gratefully acknowledged. This work was supported by a F r a n c e Qu6bec scholarship.
References [1] S. Weinberg, Phys. Rev. Lett 19 (1967) 1264; A. Salam, in: Proc. 8th Nobel Symp. (Stockholm, 1968), ed. N. Svartholm (Almqvist and Wiksell, Stockholm, 1968) o. 367. [2] K.J.F. Gaemers and G.J. Gounaris, Phys. Lett. 77B (1978) 379; for the general properties of the Higgs see: J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B106 (1976) 292; G. BarbieUini et al., DESY report 79/27. [3] A.V. Smilga and M.I. Vysotsky, Nucl. Phys. B150 (1979) 173. [4] A. Nicolaidis, LPC 79-16 preprint, Nucl. Phys. B163 (1980) 156. [5] E. Witten, Nucl. Phys. B120 (1977) 189; C.H. Llewellyn Smith, Phys. Lett. 79B (1978) 83. [6] S. Weinberg, Trans. NY Acad. Sci. Set. II 38 (1977) 185; H. Fritzsch, lectures given at the 10th GIFT Seminar (June 1979), CERN TH 2699 preprint. [7 ] N. Arteaga-Romero and A. Nicolaidis, in preparation.
PHYSICS LETTERS
21 April 1980
QCD C O R R E C T I O N S F O R HIGGS-BOSON PRODUCTION A. NICOLAIDIS
Laboratoire de Physique Corpuseulaire, CollOgede France, Paris, France Received 20 November 1979
We consider, within QCD, the spectrum of a Higgs boson radiated off a heavy quark. Like in the case of the structure functions of the photon, QCD corrections renormalize the free quark result, supressing it at large x and enhancing it at small x. Also a Higgs boson distribution within a gluon is developed, which is not negligible at small x.
All present models of spontaneously broken gauge theories of weak and electromagnetic interactions contain as an important ingredient Higgs scalars. In the conventional model [ 1] the coupling of the Higgs to fermions (leptons, quarks) is proportional to the fermion mass, while for the intermediate gauge bosons (Z, We ) it is proportional to their mass squared. Evidently the Higgs particle will be produced most abundantly in processes involving heavy particles. Gaemers and Gounaris [2] have considered Higgs boson (mass 34) production in e+e - annihilation through bremsstrahlung off a heavy quark (mass m). However, at high energies, X/7>> M ( P E P - L E P energies), the heavy quark will have enough " t i m e " r -= (1//3) log [a s (M2)/as (02)1 2 (fl = 11 - gNf) to develop its own cloud composed o f QCD quanta and therefore QCD corrections to the Higgs boson spectrum will be quite important. We examine these corrections in this paper. Consider a hard scattering process U where a heavy quark q is produced, which subsequently radiates a Higgs boson H (fig. la). In the leading log approximation (LLA), where all masses are ignored compared to the high energies involved, this process factorizes in two pieces: the cross section for the production o f the heavy quark times the probability density of finding a Higgs boson inside the quark. To extract this probability density (zeroth order in c~s at first) we calculate in the LLA the cross section for the process shown in fig. 1a. The amplitude is given b y
H(k,M..) q(p,m)
U
q(r,m)
(a)
@/.---"@ /-"" .........
(c)
(b)
Fig. l. Higgs boson radiation off a heavy quark. (a) Born term. (b) Gluonic QCD correction. (c) The development of a Higgs distribution within a gluon.
M=gH~(r )
4(+ r+ m
U,
(1)
(k + r) 2 - m 2 where gH = m(GFX/~) 1/2" See fig. la for the rest o f the notation. The cross section is do = IMI 2 d V (2) ,
(2)
and d V (2) is the two-particle phase space. In the limit where all mass terms are ignored and considering collinear emission we have [31 dV (2) = d g ( 1 ) [x (1 - x)/8rr 2 ] p2 d x d z ,
(3)
IMI 2 ~ g 2 ( 2 k . r) -1 Tr (XUU l ,
(4)
where x is the momentum fraction carried b y the Higgs boson, d V (1) is the one-particle phase space and z = cos (k, i). Using the above relations we find 447
Volume 91B, number 3,4
do dx
21 April 1980
PHYSICS LETTERS
1+.
x log~_-uTr(pUU)dV(1),
where v = (1 - M 2 / x 2 Q 2 ) 1/2 and we have assumed throughout that Q2 >>M2, m 2. Therefore we obtain
It is relatively simple to solve the algebraic eqs. (10)-(12) and to find Hqv(n), Hqs(n) and Hg(n). Finally for the Higgs distribution within a quark or gluon we obtain
Ptt/q(X) = (oM/47r)m2xt ,
Hqi(n) = c(n)t
16n2
(5)
(6)
with a w = GFX/~w/4zr and t ~ log Q2/M2. It is remarkable that eq. (6) is the exact result [2] when M/Q, m/Q terms are negligible. Gluon emission by the heavy quark before radiating the Higgs boson will modify the Higgs boson spectrum (fig. lb). Also since the transition g ~ q -* H is possible, a Higg boson distribution within a gluon will be developed (fig. lc). The Higgs distribution within a quark (gluon) Hq (/-/g), will be given by an Altarelli-Parisitype equation, which is summing up to all orders in as, in the LLA, the strong-interaction renormalization effects: dHqi
dt
as(t) -mZe(x) + ~ [Hqi @ Pqq + H g @ Pgql , (7)
dt - 2n
Hqi ® Pqg + Hg ® Pgg .
(8)
The symbol A ® B stands for the convolution 1
A(x) ® B(x) = f ~ A ( y ) B ( x / y )
,
(9)
X
and c(x) = (oM/410 3x (the factor 3 takes the color into account). The above equations are simplified when we take the moments. Defining also the valence contribution H qv = H qi - Hq/and the singlet contribution H qs = ]~iHqi we have dHqVdt- (m2 - m2)c(n)
+ as(t) - ~ - Hqv(n)Aqq(n) ,
(10)
(13)
tm _ m2
X ( i + dqq(~) + (m2) Hg(n) = c(n)t(m 2 )
1 + dgg(n)
I,
[1 + d+(n)] [1 + d - (n)l
J
-dqg(n) [1 + d+(n)] [1 + d - ( n ) ]
(14)
The d(n), A(n) are the well-known anomalous dimensions which can be found for instance in ref. [4]. The similarity of these equations with the corresponding structure functions of the photon (the bestknown colorless particle) in QCD is evident [5]. The free quark model transition q ~ 7 has been replaced by q -~ H and the electric charge of the quark by its mass. To recover the distributions Hq(x, t), Hg(x, t) we have to undo the moment integral. We use a simple method which has been applied already in parametrizing the structure functions of the photon [4]. After extracting the singular behavior at x ~ 0, which is the same for both colorless particles, we expand the distributions in series of Jacobi polynomials. We take also five flavors (u, d, s, c, b) and for the quark masses ,1 we assume m u = 5 MeV, m d = 9 MeV, m s = 180 MeV, m e = 1.5 GeV, m b = 4.7 GeV with (m 2 ) = 4.874 GeV 2 . Writing
xHi(x, t) = (aw/41r)tfH/i(x) ,
(15)
we find for the bottom quark, keeping five terms in the expansion fHPo(X) = 3x-0-6(0.046 + 3.93x (16) -- 4.98x 2 + 40.18x 3 -- 33x 4) .
dHqs dt - (m2)2Nfc(n)
%(0 +~ [Hqs(n)Aqq(n)+2NfHg(n)Agq(n)] , (11)
In fig. 2 the two distributions, with and without strong interactions, are compared. QCD effects suppress the spectrum at large x and enhance it at small X.
The charmed quark fragmentation function into
dHg as(t) [Hqs(n)Aqg(n) + Hg(n)Pgg(n)] . dt 2n
448
(12)
,1 The quark masses are those which appear in the lagrangian. The light quark masses can be determined using PCAC [6]. We use the values quoted in the second paper of ref. [6 ].
Volume 91B, number 3,4
21 April 1980
PHYSICS LETTERS
22.0
2.0
// 19.8
/
• i I
1.8
17.6 1.6
15.4
I
I I
1.4 13.2
/ /, s•
11.0
1.2
•.// 8.8
1.0
/
\
4,4
.8
.6 .4
"%
. . . . ..1
" ' .2 ;
. 3'
' .~
.~
. '=
.;
.7
1.
.;
Fig. 2. The momentum fraction carried by the Higgs boson, radiated by a b quark. Dashed line is the free-quark result rn~x 2 , while the solid line is the QCD result (in units (~w/4~)t, color is not included). the Higgs boson, together with the zeroth order result in as, is shown in fig. 3. The five-term a p p r o x i m a t i o n is
fH/c(X) = 3 x - 0 - 6 ( 0 . 0 9 1 + 0 . 0 4 1 7 x (17) + 0.2x 2 + 3.38x 3 - 3x 4) . Given the smallness o f the light quark masses the fragmentation u (d) ~ H is d o m i n a t e d by the singlet piece. The Higgs distribution within a gluon, eq. (14), is not negligible. It is bigger than the light quark contribution, over almost the entire region o f x (fig. 4). Three terms only are sufficient to reproduce accurately the light quark and gluon fragmentation into the Higgs: 2.5 i/ iI /I
I
111
1.5
11 /##
1.0
.0 ,0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.
Fig. 4. The same as in fig. 2, for the u (or d) quark (a) and the gluon (b). fH/u(d) = 3 x - 0 " 6 ( 0"09 -- 0.25x + 0 . 1 7 5 x 2 ) ,
(18)
fH/g = 3 x - 0 " 6 ( 0"15 + 0.243x -- 0 . 4 1 7 x 2) .
(19)
In table 1 we present the ratios Pi = o ( e + e ~ q i q i H ) / o ( e + e - -+ %.C~.) ( q i = s, c, b quark) for a beam energy o f 100 GeV and t w o different values o f M. The first row corresponds to the exact result, to zeroth order in a s, and the numbers q u o t e d are taken from ref. [2]. The second row gives the same ratios in the L L A (still z e r o t h order in C~s). For the lower mass (M = 3 GeV), L L A is a very reasonable a p p r o x i m a t i o n scheme. For the higher mass (M = 10 GeV) the L L A overestimates the exact result b y a factor 1 . 3 - 1 . 5 . Clearly, it is desirable to find a consistent way to inTable 1 The ratios Pi = o( e+e- --" qiEtiH)/a(e +e- ~ qiEti) (qi = s, c, b) at x/)-= 200 GeV. The first row corresponds to the exact result (zeroth order in C~s), second row is the leading log approximation to the exaet result (zeroth order in a s and the third row is the QCD-corrected prediction (to all orders in a s, in the LLA).
/ 2.0
.2
J
4//
M
s
c
b
10
0 . 1 0 X 10 - 6 0 . 1 5 X 10 - 6 0 . 3 4 × 10 - 6
0 . 9 5 × 10 - 6 1.3 X 10 - 6 1.4 × 10 - 6
0 . 9 2 × 10 -5 1.2 × 10 -5 1.18 × 10 -5
0.17 X 10 -6 0.2 x 10 -6 1.4 X 10 -6
0.16 X 10 -s 0.19 X 10 -5 0.31 X 10 -5
0.17 X 10 -a 0.17 x 10 -4 0.19 X 10 -4
f" °5
0.0
.0
.1
.2
.3
.4
.5
.6
.7
.8
.9
Fig. 3. The same as in fig. 2, for the charm quark.
1.
449
Volume 91B, number 3,4
PHYSICS LETTERS
corporate the Higgs mass effects, when the ratio M/Q is not too small. The third row gives the QCD-corrected predictions (in the LLA). For the relatively light quarks (s, c quarks) where the singlet component (small x ) is important, the QCD result is larger than the free-quark model result. Also for the lower mass (M = 3 GeV) the enhancement is larger since a larger protion of the small-x region is available. On the other hand, the heaviest quark (b quark) is dominated by the valence contribution, which is supressed by the QCD corrections and therefore the ratio Pb is not enhanced. (Notice also that the first two rows have been calculated with m b = 4.5 GeV, .while the third with m b = 4.7 GeV.) In any case the QCD corrections are quite significant and they have to be taken into account. Armed with the above distributions we can calculate the Higgs boson production via bremsstrahlung off a heavy quark produced in any hard process [7] (largePT collisions, or 77 collisions). The observation of the Higgs boson in such a process will be not only a test of the gauge theories o f weak and electromagnetic interactions but also a test o f the gauge theory of strong interactions, namely quantum chromodynamics.
450
21 April 1980
Discussions with N. Arteaga-Romero are gratefully acknowledged. This work was supported by a F r a n c e Qu6bec scholarship.
References [1] S. Weinberg, Phys. Rev. Lett 19 (1967) 1264; A. Salam, in: Proc. 8th Nobel Symp. (Stockholm, 1968), ed. N. Svartholm (Almqvist and Wiksell, Stockholm, 1968) o. 367. [2] K.J.F. Gaemers and G.J. Gounaris, Phys. Lett. 77B (1978) 379; for the general properties of the Higgs see: J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B106 (1976) 292; G. BarbieUini et al., DESY report 79/27. [3] A.V. Smilga and M.I. Vysotsky, Nucl. Phys. B150 (1979) 173. [4] A. Nicolaidis, LPC 79-16 preprint, Nucl. Phys. B163 (1980) 156. [5] E. Witten, Nucl. Phys. B120 (1977) 189; C.H. Llewellyn Smith, Phys. Lett. 79B (1978) 83. [6] S. Weinberg, Trans. NY Acad. Sci. Set. II 38 (1977) 185; H. Fritzsch, lectures given at the 10th GIFT Seminar (June 1979), CERN TH 2699 preprint. [7 ] N. Arteaga-Romero and A. Nicolaidis, in preparation.