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Complex rishon-type unification
Volume 123B, number 5
PHYSICS LETTERS
7 April 1983
COMPLEX RISHON-TYPE UNIFICATION Aharon DAVIDSON 1 and Jacob SONNENSCHEIN Department of Nuclear Physics, The WeizmannInstitute of Science, Rehovot, Israel Received 3 January 1983
We present a flavor-chiral version of the rishon model. The troublesome c~C ~ 0 't Hooft limit is satisfied in a non-exotic manner. The scheme is SU(7) grand unified. Its uniqueness stems from the fact that a realistic SU(n)HC scenario is SU(N) embeddable only provided n = 3, with 4 (or 5) preons involved. A total number of three quark/lepton families is dictated by the overall renormalizability. The electric charge serves as the fourth color.
If the quark/lepton compositness [ 1 ] is a reality, let the underlying hypercolor (HC) interaction be unified. With this motivation, the group theoretical problem of consistently embedding SU(n)H C within a grand unified SU(N) theory has been recently analysed [2,3]. The fermionic content of the theory is obviously a crucial issue. The minimal anomaly-free complex representation of left-handed fermions is given by ffL = f~t~ + (N - 4 ) f L c ~
(0t, 13= 1..... N ) ,
(1)
with f ' ~ = _ f ~ a . In fact, for practical N, the above appears to be the only tenable choice [3] which would not endanger [4] color-asymptotic-freedom. After several symmetry breaking stages, SU(n)H C is born and becomes the first interaction to confine. It is thus important, for the sake of decoding the fermionic spectrum, to study the associated 't Hooft equations [5]. Indeed, for n > 3, the SU(N - n) X SU(N - 4) × U(1)a X U(1)b anomaly matching equations exhibit a unique solution [6]. Unfortunately, while starting from fL(preon) + fiL(spectator) = complex,
(2a)
the solution is such that [3] ffL(composite) + fL(spectator) = real,
(2b)
with respect to the color-flavor gauged symmetry. 1 Incumbent of the Maurice M. Boukstein Career Development Chair. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
This conspiracy of parity doublet formation is of course physically unacceptable. The n = 3 case, however, is a completely different story. Its exclusiveness originates from the fact that now only two (rather than three) types of irreducible HC representations are involved. The modified nfpreon maximal flavor symmetry SU(nf) × SU(nf) × U(1) does allow for several 't Hooft-consistent solutions (note that the restrictive Apellquist-Carazzone decoupling condition [7] is empty here since the masslessness of the preons is still protected by color-flavor gauge invariance), out of which the non-exotic solution may be selected. Such a "realistic flavor requirement" leads [3] to the following operative conclusions: (i) SU(3)C must live in one SU(nf) factor. (ii) Integer solutions exist only provided nf = 4 or 5. (iii) q/L(composite) + fit(spectator) = complex and multigenerational. Unified SU(8) is then singled out [3] as the parent symmetry capable of giving birth to the five-preon SU(3)H C scheme, with broken Georgi-Glashow SU(5) [8] emerging as the fundamental color flavor background. In particular, SU(2)W s is fully gauged and W± are therefore elementary. The left over four-preon alternative is of our concern in the present paper. It leads to a complex rishon-type [9] grand unified scheme enjoying proper a C -~ 0 and AHC ~ ~ behaviour. Exactly three ordinary quark/lepton generations are predicted. The electric charge Q, rather than P a t i 299
Volume 123B, number 5
PHYSICS LETTERS
7 April 1983
Salam B - L, plays the role of the fourth color [ 10]. Consider a grand unified theory respecting a local SU(7) gauge invariance, with the chiral fermions furnishing the reducible representation
provided nf = 4 (or 5). To be specific, starting from the given set qJL(preon) = (rq, 1) -- (1,[]) under SU(nf) × SU(nf) × U(1)V [following the conventional notation [5] - ( r l , r2) t = (rl, r2) t ,~, (rl, r2)R] , we end up with
ffL = 21 + 3" 7* .
nf-- 1 n CL(c°mp°site) = ~ [(~, 1) -- (1,~])]
(3)
The apparent global SU(3) which accompanies the fermionic sector does not express a superfluous replication; it is forcefully dictated by the overall vanishing anomaly required by renormalizability. A twofold spontaneous symmetry breaking pattern M
SU(7) -+ SU(3)C X SU(4) × U(1)I M' SU(3)H C × SU(3)C × U(1)I X U(1)II ,
(4)
is invoked to induce a proper HC ~ C asymmetry, namely AHC > A C. Defining the electric (Q) and the weak (W) charges in terms of the normalized U(1)I,I I generators, namely Q = -(~)1/2y I + (3)1/2 Yit ,
W = (~)1/2 YII ,
(5)
we can identify our preons as
Y[ = (3,3)Q=_l/3,W=_l/6 ,
T L = (3,1)Q=I, W=l/3,
V~La = 3(3* ,I)Q=O,W=I/6,
VL = (3* ,1)Q=O,W=_I/3 .
(6) i = 1, 2, 3 is the usual color index, while a = 1, 2, 3 is the initial global SU(3) index. The attached spectatorial set is traced to be ~bL(spectator ) = (1,3)Q=2/3, W=l/2 + (1,3*)a=_2/3,w= 0 +3(1,3*)Q=l/3,W= 0 + 3(1,1)Q=_I,W=_I/2 .
(7)
At this stage, in order to gain some insight into the fermionic spectrum, we idealize the situation. We momentarily neglect aC,Q, w along with possible Yukawa couplings, and also ignore effective terms remnants of grand unification. This is known as the 't Hooft limit. The four-preon QHCD lagrangian then acquires a global SU(4) × SU(4)' × U(1)V symmetry whose anomalies at the preon level must be reproduced by the newly created chiral fermionic bound states. Recall that we have actually arranged for a four-preon scheme just to take advantage of a notable property of 't Hooft equations. Namely, the possibility for a non-exotic composite solution can be realized only 300
_
_
(8)
Note that nf = 4 implies (nf - 1)/(nf - 3 ) = 3, indicating the spontaneous birth of another global SU(3) at the composite level. To proceed, it is necessary to clarify how SU(3)C X U(1)Q × U(1) w lives inside SU(4) × SU(4)' X U(1)V. In particular, observe that SU(4) D SU(3)C × U(1)Q,
(9)
so that the electric charge (rather than Pati-Salam [10] B - L) plays the role of the fourth color. SU(4)', on the other hand, is the nest for the initial global SU(3), with the analogous Q' being such that W = 9 ( Q + Q, _ ~_V). This analysis makes it easier for us to finally uncover the explicit preonic structure and consequently the electro/nuclear content of the fermionic bound states. Taking into account the equally important spectators (7), which automatically absorb the ~bL(composite ) anomalies, the combined list of HC-singlet fermions is specified in table 1. The fermions appearing in table 1 can be rearranged according to ~bL(composite ) + ~bL(spectator ) = ffc(complex) + ~L(real) under SU(3)C × U(1)Q X U(1)W gauge symmetry. Notice that U(1) w is exclusively responsible for the presence of the complex piece. No wonder, W turns out to substitute the third component T 3 of G l a s h o w Weinberg-Salam (GWS) weak isospin. In the light of eq. (5), however, this substitution holds only at the composite level. Consequently, Z is elementary, whereas its would be charged partners W e are expected to be composite objects. A closer look at table 1 reveals that ~bL(real) consists of nine right-handed neutrinos and a set of parity-doublets. The latter are such that each spectator uniquely finds a proper conjugate composite partner. This pairing feature seems to generally accompany unified compositeness. The mass scale m of the
Volume 123B, number 5
Table 1 The content of ¢ L(composite) + ¢ L(spectator). i, j, k are color indices, while a, b, c are global-SU(3) indices. The parity doublets are indicated by the arrows.
uL
fiL
Internal structure
W
#
ei]kTR] TRkVLa
1/ 2
3
eijkTRj TRkV L
0
1
spectator
1/2
1
TRiTRVLa
0
3
TRiTRV L
-1/2
1
0
1
--1/2
3
0
3
j k 3eijkTLTLT L spectatorsa
0 0
3 3
i j g 3eijkTLTLTL
- 1/2
3
spectatorsa
- 1/2
3 ~]
spectator dL
aL
eL
eL
~,b.tre,.,-i eabeVR VR t L a i VRVRT L
b c eabcVRVRTL
7 April 1983
PHYSICS LETTERS
J J
izability, while for the (dL, eL, VL)'families these are the derived 't Hooft multiplicities which actually matter. It is amusing though to watch the way the horizontal structure respects the well-known SU(5) G e o r g i - G l a s h o w classification. A particular attention is to be given to the left-handed leptons e L and vL which, by virtue o f Fermi statistics, cannot have an internal derivative-free structure. We therefore better emphasize that without paying such a price, it is evident from the 't Hooft equations that the chiral symmetry gets necessarily broken. The weak mixing angle is our next target. To calculate 0w, stemming from the W = T 3 effective identification, we still do not need the explicit W+--scenario. At the fully perturbative region AHC < / a < M ' , the relevant running coupling constants evolve via [ 13] all1 (/.0 = Ot~1 + b 4 ln(M/M') + b 0 l n ( M ' / t 0 , a i l (U) = a ~ 1 + b 0 In(M/u),
(1 l )
with b k = - ( 1 / 6 u ) ( 1 l k - 8). Note the relative fertalon-loop contribution o f 5 × 1 + 1 X 3 = 8. Now, from eq. (5) we learn that
0
3
V~VRT L
1/2
3
vL
3eabCVLaVLbVLc
1/2
3
Mtogether, using the definition sin20 w = a/c~w, we obtain
FL
3eabcVLbVLcV L
0
9
sin20w = 1 [ 1 - (77/9~r)a ln(M/M')] .
members o f ffL(real) can be estimated on general grounds, m must first of all vanish on the AHC scale once aC,Q,W are switched off. On the other hand, it is not protected by gauge invariance, and hence, following the so-called "survival hypothesis" [ 11 ], presumably becomes superheavy and decouple from the effective low-energy theory at the AHC -~ ~o limit. These two aspects are consistent with m(vR) --~ a~C,Q,w(AHc)AIac ,
(10)
for some integer k, an order of magnitude which reminds us of the Weinberg mechanism [12]. At any rate, m = m(vR) is believed to be quite heavier than the masses of the U(1) w protected fermions. ~kL(complex) exhibits exactly three ordinary quark/lepton generations. The number three is no magic. For the (u L, dL, ilL, ~L) "families it reflects the primitive preon-replication requested by renormal-
0t-1 = gOq 7 - 1 + ~Otil 3 -1 ,
Ot~1 = ]Ot• 1
(12)
(13)
This result expresses a novel conception. First of all, to be contrasted with the elementary-family value of 3/8, sin20w ~ I/4 at the unification scale. This can be confirmed directly by evaluating tr W2/tr Q2 over all preons and spectators. Second, it so happens that sin20w is (almost)/~-independent! These two features which seem to compensate for each other, i.e. sin20w starts relatively low but would not change too much, will hopefully allow us to make contact with the experimental value sin20w(mw)exp ~ 0.21. With regard to this point, it is understood of course that for/~ < AHC , eqs. (11) are yet to be modified to include non-perturbative HC-corrections. At any rate, the lowenergy physics might be sensitive to the M/M' ratio, which we now try to estimate. Eq. (13) suggests that M/M' must be bound above, namely <~ 103, in order not to endanger the observed value of the weak angle. On the other hand, Otnc C are such that a ~ 1 (/a) = a H l ( ~ ) + (11/67r) ln(~//M') prior to the nC-confine301
Volume 123B, number 5
PHYSICS LETTERS
ment, and hence M/M' must be bound below as otherwise a c ( A H c ) would be too close to unity, The compromise is achieved for
M/M'~-- 10 3 ,
(14)
meaning a characteristic "energy oasis" around M ' "~ 1012 GeV. We note in passing that ~C is expected to temporarily decrease right after crossing/2 "~ AHC. This is because the relative fermionic contribution to the B-function is increased from its preon value of 8 to the effective quark/lepton three generational value of 12. We come now to the charged weak interaction sector. Here we follow as close as possible the spirit of the rishon model [9]. The central point is that the symmetry of the effective low-energy lagrangian is actually larger than SU(3)C × U(1)I X U(1)I I. This has to do with the observation that the preon content of both (ULLdL) as well as (VLLeL), for each generation separately, is precisely the same. It closes upon an SU(3)HC,C-singlet six-preon creature a W - ~ eabcedk VR VRb VRc TLi T~" T kL ,
(15)
which is furthermore a GIM-singlet in the global SU(3) X SU(3) generation space. Thus, taking into account the u L ~ d L and vL ~ e L transitions, U(1)I I is enlarged to SU(2)L. The latter symmetry is special. It induces no SU(2)L" (SU(2)L" U(1)I anomalies at the effective quark/lepton level. Now, following 't Hooft's discussion [5], the effective interactions for/a ~ All C are vanishingly small or alternatively renormalizable. The consequent line of thought is therefore that at present energies it may look as if SU(2)L is fully gauged. A similar conclusion could not have been derived with regard to the broken generation symmetry. It does carry quark/lepton anomalies, and therefore the effective horizontal interactions are highly suppressed. Our SU(7) theory is intrinsically flavor-chiral. It is therefore not a surprise that, unlike in the original rishon model, the quark/lepton internal structure would not provide room for SU(2)R. Nonetheless, passing through a vector-like QHCD stage, we better remark that L ~ R symmetry violation is switched on along with the color (but not necessarily the electro/ weak) forces. After all, ffL(preon) + ffL(spectator) is SU(3)ttC,C real, but SU(3)H C × SU(3)C complex. Now, although (u, d, e, V)L and We happen to essentially 302
7 April 1983
match the rishon description, (u, d, e)R do not. Yet, we formally have ( ~ e L ) = ( d R d L ) = ( ~ U L ) * . These Q = 0, W = - 1 / 2 "mass terms" suggest that U(1)I X U(1)I I -+ U(1)Q should be governed by a VEV o f a scalar singlet having Q = 0 and W = - 1 / 2 if the badly needed mass relation m w = rnZ cos 0 w is to be effectively recovered. Using the SU(7) language, this can be realized via a threefold antisymmetric Higgs 35. The latter automatically co-introduces the desirable charge one HC, C-singlet piece as well. To summarize, we have specified the minimal grand unifying theory whose nested SU(n)H C composite scenario leads to non-exotic f l a v o r - c o l o r assignments in a 't Hooft consistent manner. While focusing on the algebraic successes of the resulting flavor-chiral rishontype scheme and emphasizing its uniqueness, a lot of theoretical and phenomenological problems are still to be settled. At any rate, we do hope we were able to demonstrate how restrictive as well as constructive group theory can be in trying to merge the trails of compositeness and unification. Useful discussions with Professor H. Harari and Professor S. Yankelowicz are very much appreciated. This work has been supported in part by the U S A Israel Binational Science Foundation.
References [1] J.C. Pati, A. Salam and J. Strathdee, Phys. Lett. 59B (1975) 265. [2] Y. Tosa and R.E. Marshak, VPI-HEP-82/9, to be published; I. Bars, Proc. Recontres de Moriond (1982); F. Bordi, R. Casalbuoni, D. Dominici and R. Gatto, Phys. Lett. l17B (1982) 219. [3] A. Davison and J. Sonnenschein, WLS-82/42, to be published. [4] P. Frampton, Phys. Rev. Lett. 43 (1979) 1912. [5] G. 't Hooft, Proc. Carg~se Summer Institute (1979). [6] I. Bars and S. Yankelowicz, Phys. Lett. 101B (1981) 159. [7] T. Appelquist and J. Carazzone, Phys. Rev. 11D (1975) 2865. [8] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [9] H. Harari and N. Seiberg, Phys. Lett. 98B (1981) 269. [10] J.C. Pati and A. Salam, Phys. Rev. 10D (1974) 275. [11] H. Georgi, Nucl. Phys. 156B (1979) 126. [12] S. Weinberg, Phys. Lett. 102B (1981) 401. [ 13] H. Georgi, H.R: Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451.
PHYSICS LETTERS
7 April 1983
COMPLEX RISHON-TYPE UNIFICATION Aharon DAVIDSON 1 and Jacob SONNENSCHEIN Department of Nuclear Physics, The WeizmannInstitute of Science, Rehovot, Israel Received 3 January 1983
We present a flavor-chiral version of the rishon model. The troublesome c~C ~ 0 't Hooft limit is satisfied in a non-exotic manner. The scheme is SU(7) grand unified. Its uniqueness stems from the fact that a realistic SU(n)HC scenario is SU(N) embeddable only provided n = 3, with 4 (or 5) preons involved. A total number of three quark/lepton families is dictated by the overall renormalizability. The electric charge serves as the fourth color.
If the quark/lepton compositness [ 1 ] is a reality, let the underlying hypercolor (HC) interaction be unified. With this motivation, the group theoretical problem of consistently embedding SU(n)H C within a grand unified SU(N) theory has been recently analysed [2,3]. The fermionic content of the theory is obviously a crucial issue. The minimal anomaly-free complex representation of left-handed fermions is given by ffL = f~t~ + (N - 4 ) f L c ~
(0t, 13= 1..... N ) ,
(1)
with f ' ~ = _ f ~ a . In fact, for practical N, the above appears to be the only tenable choice [3] which would not endanger [4] color-asymptotic-freedom. After several symmetry breaking stages, SU(n)H C is born and becomes the first interaction to confine. It is thus important, for the sake of decoding the fermionic spectrum, to study the associated 't Hooft equations [5]. Indeed, for n > 3, the SU(N - n) X SU(N - 4) × U(1)a X U(1)b anomaly matching equations exhibit a unique solution [6]. Unfortunately, while starting from fL(preon) + fiL(spectator) = complex,
(2a)
the solution is such that [3] ffL(composite) + fL(spectator) = real,
(2b)
with respect to the color-flavor gauged symmetry. 1 Incumbent of the Maurice M. Boukstein Career Development Chair. 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
This conspiracy of parity doublet formation is of course physically unacceptable. The n = 3 case, however, is a completely different story. Its exclusiveness originates from the fact that now only two (rather than three) types of irreducible HC representations are involved. The modified nfpreon maximal flavor symmetry SU(nf) × SU(nf) × U(1) does allow for several 't Hooft-consistent solutions (note that the restrictive Apellquist-Carazzone decoupling condition [7] is empty here since the masslessness of the preons is still protected by color-flavor gauge invariance), out of which the non-exotic solution may be selected. Such a "realistic flavor requirement" leads [3] to the following operative conclusions: (i) SU(3)C must live in one SU(nf) factor. (ii) Integer solutions exist only provided nf = 4 or 5. (iii) q/L(composite) + fit(spectator) = complex and multigenerational. Unified SU(8) is then singled out [3] as the parent symmetry capable of giving birth to the five-preon SU(3)H C scheme, with broken Georgi-Glashow SU(5) [8] emerging as the fundamental color flavor background. In particular, SU(2)W s is fully gauged and W± are therefore elementary. The left over four-preon alternative is of our concern in the present paper. It leads to a complex rishon-type [9] grand unified scheme enjoying proper a C -~ 0 and AHC ~ ~ behaviour. Exactly three ordinary quark/lepton generations are predicted. The electric charge Q, rather than P a t i 299
Volume 123B, number 5
PHYSICS LETTERS
7 April 1983
Salam B - L, plays the role of the fourth color [ 10]. Consider a grand unified theory respecting a local SU(7) gauge invariance, with the chiral fermions furnishing the reducible representation
provided nf = 4 (or 5). To be specific, starting from the given set qJL(preon) = (rq, 1) -- (1,[]) under SU(nf) × SU(nf) × U(1)V [following the conventional notation [5] - ( r l , r2) t = (rl, r2) t ,~, (rl, r2)R] , we end up with
ffL = 21 + 3" 7* .
nf-- 1 n CL(c°mp°site) = ~ [(~, 1) -- (1,~])]
(3)
The apparent global SU(3) which accompanies the fermionic sector does not express a superfluous replication; it is forcefully dictated by the overall vanishing anomaly required by renormalizability. A twofold spontaneous symmetry breaking pattern M
SU(7) -+ SU(3)C X SU(4) × U(1)I M' SU(3)H C × SU(3)C × U(1)I X U(1)II ,
(4)
is invoked to induce a proper HC ~ C asymmetry, namely AHC > A C. Defining the electric (Q) and the weak (W) charges in terms of the normalized U(1)I,I I generators, namely Q = -(~)1/2y I + (3)1/2 Yit ,
W = (~)1/2 YII ,
(5)
we can identify our preons as
Y[ = (3,3)Q=_l/3,W=_l/6 ,
T L = (3,1)Q=I, W=l/3,
V~La = 3(3* ,I)Q=O,W=I/6,
VL = (3* ,1)Q=O,W=_I/3 .
(6) i = 1, 2, 3 is the usual color index, while a = 1, 2, 3 is the initial global SU(3) index. The attached spectatorial set is traced to be ~bL(spectator ) = (1,3)Q=2/3, W=l/2 + (1,3*)a=_2/3,w= 0 +3(1,3*)Q=l/3,W= 0 + 3(1,1)Q=_I,W=_I/2 .
(7)
At this stage, in order to gain some insight into the fermionic spectrum, we idealize the situation. We momentarily neglect aC,Q, w along with possible Yukawa couplings, and also ignore effective terms remnants of grand unification. This is known as the 't Hooft limit. The four-preon QHCD lagrangian then acquires a global SU(4) × SU(4)' × U(1)V symmetry whose anomalies at the preon level must be reproduced by the newly created chiral fermionic bound states. Recall that we have actually arranged for a four-preon scheme just to take advantage of a notable property of 't Hooft equations. Namely, the possibility for a non-exotic composite solution can be realized only 300
_
_
(8)
Note that nf = 4 implies (nf - 1)/(nf - 3 ) = 3, indicating the spontaneous birth of another global SU(3) at the composite level. To proceed, it is necessary to clarify how SU(3)C X U(1)Q × U(1) w lives inside SU(4) × SU(4)' X U(1)V. In particular, observe that SU(4) D SU(3)C × U(1)Q,
(9)
so that the electric charge (rather than Pati-Salam [10] B - L) plays the role of the fourth color. SU(4)', on the other hand, is the nest for the initial global SU(3), with the analogous Q' being such that W = 9 ( Q + Q, _ ~_V). This analysis makes it easier for us to finally uncover the explicit preonic structure and consequently the electro/nuclear content of the fermionic bound states. Taking into account the equally important spectators (7), which automatically absorb the ~bL(composite ) anomalies, the combined list of HC-singlet fermions is specified in table 1. The fermions appearing in table 1 can be rearranged according to ~bL(composite ) + ~bL(spectator ) = ffc(complex) + ~L(real) under SU(3)C × U(1)Q X U(1)W gauge symmetry. Notice that U(1) w is exclusively responsible for the presence of the complex piece. No wonder, W turns out to substitute the third component T 3 of G l a s h o w Weinberg-Salam (GWS) weak isospin. In the light of eq. (5), however, this substitution holds only at the composite level. Consequently, Z is elementary, whereas its would be charged partners W e are expected to be composite objects. A closer look at table 1 reveals that ~bL(real) consists of nine right-handed neutrinos and a set of parity-doublets. The latter are such that each spectator uniquely finds a proper conjugate composite partner. This pairing feature seems to generally accompany unified compositeness. The mass scale m of the
Volume 123B, number 5
Table 1 The content of ¢ L(composite) + ¢ L(spectator). i, j, k are color indices, while a, b, c are global-SU(3) indices. The parity doublets are indicated by the arrows.
uL
fiL
Internal structure
W
#
ei]kTR] TRkVLa
1/ 2
3
eijkTRj TRkV L
0
1
spectator
1/2
1
TRiTRVLa
0
3
TRiTRV L
-1/2
1
0
1
--1/2
3
0
3
j k 3eijkTLTLT L spectatorsa
0 0
3 3
i j g 3eijkTLTLTL
- 1/2
3
spectatorsa
- 1/2
3 ~]
spectator dL
aL
eL
eL
~,b.tre,.,-i eabeVR VR t L a i VRVRT L
b c eabcVRVRTL
7 April 1983
PHYSICS LETTERS
J J
izability, while for the (dL, eL, VL)'families these are the derived 't Hooft multiplicities which actually matter. It is amusing though to watch the way the horizontal structure respects the well-known SU(5) G e o r g i - G l a s h o w classification. A particular attention is to be given to the left-handed leptons e L and vL which, by virtue o f Fermi statistics, cannot have an internal derivative-free structure. We therefore better emphasize that without paying such a price, it is evident from the 't Hooft equations that the chiral symmetry gets necessarily broken. The weak mixing angle is our next target. To calculate 0w, stemming from the W = T 3 effective identification, we still do not need the explicit W+--scenario. At the fully perturbative region AHC < / a < M ' , the relevant running coupling constants evolve via [ 13] all1 (/.0 = Ot~1 + b 4 ln(M/M') + b 0 l n ( M ' / t 0 , a i l (U) = a ~ 1 + b 0 In(M/u),
(1 l )
with b k = - ( 1 / 6 u ) ( 1 l k - 8). Note the relative fertalon-loop contribution o f 5 × 1 + 1 X 3 = 8. Now, from eq. (5) we learn that
0
3
V~VRT L
1/2
3
vL
3eabCVLaVLbVLc
1/2
3
Mtogether, using the definition sin20 w = a/c~w, we obtain
FL
3eabcVLbVLcV L
0
9
sin20w = 1 [ 1 - (77/9~r)a ln(M/M')] .
members o f ffL(real) can be estimated on general grounds, m must first of all vanish on the AHC scale once aC,Q,W are switched off. On the other hand, it is not protected by gauge invariance, and hence, following the so-called "survival hypothesis" [ 11 ], presumably becomes superheavy and decouple from the effective low-energy theory at the AHC -~ ~o limit. These two aspects are consistent with m(vR) --~ a~C,Q,w(AHc)AIac ,
(10)
for some integer k, an order of magnitude which reminds us of the Weinberg mechanism [12]. At any rate, m = m(vR) is believed to be quite heavier than the masses of the U(1) w protected fermions. ~kL(complex) exhibits exactly three ordinary quark/lepton generations. The number three is no magic. For the (u L, dL, ilL, ~L) "families it reflects the primitive preon-replication requested by renormal-
0t-1 = gOq 7 - 1 + ~Otil 3 -1 ,
Ot~1 = ]Ot• 1
(12)
(13)
This result expresses a novel conception. First of all, to be contrasted with the elementary-family value of 3/8, sin20w ~ I/4 at the unification scale. This can be confirmed directly by evaluating tr W2/tr Q2 over all preons and spectators. Second, it so happens that sin20w is (almost)/~-independent! These two features which seem to compensate for each other, i.e. sin20w starts relatively low but would not change too much, will hopefully allow us to make contact with the experimental value sin20w(mw)exp ~ 0.21. With regard to this point, it is understood of course that for/~ < AHC , eqs. (11) are yet to be modified to include non-perturbative HC-corrections. At any rate, the lowenergy physics might be sensitive to the M/M' ratio, which we now try to estimate. Eq. (13) suggests that M/M' must be bound above, namely <~ 103, in order not to endanger the observed value of the weak angle. On the other hand, Otnc C are such that a ~ 1 (/a) = a H l ( ~ ) + (11/67r) ln(~//M') prior to the nC-confine301
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ment, and hence M/M' must be bound below as otherwise a c ( A H c ) would be too close to unity, The compromise is achieved for
M/M'~-- 10 3 ,
(14)
meaning a characteristic "energy oasis" around M ' "~ 1012 GeV. We note in passing that ~C is expected to temporarily decrease right after crossing/2 "~ AHC. This is because the relative fermionic contribution to the B-function is increased from its preon value of 8 to the effective quark/lepton three generational value of 12. We come now to the charged weak interaction sector. Here we follow as close as possible the spirit of the rishon model [9]. The central point is that the symmetry of the effective low-energy lagrangian is actually larger than SU(3)C × U(1)I X U(1)I I. This has to do with the observation that the preon content of both (ULLdL) as well as (VLLeL), for each generation separately, is precisely the same. It closes upon an SU(3)HC,C-singlet six-preon creature a W - ~ eabcedk VR VRb VRc TLi T~" T kL ,
(15)
which is furthermore a GIM-singlet in the global SU(3) X SU(3) generation space. Thus, taking into account the u L ~ d L and vL ~ e L transitions, U(1)I I is enlarged to SU(2)L. The latter symmetry is special. It induces no SU(2)L" (SU(2)L" U(1)I anomalies at the effective quark/lepton level. Now, following 't Hooft's discussion [5], the effective interactions for/a ~ All C are vanishingly small or alternatively renormalizable. The consequent line of thought is therefore that at present energies it may look as if SU(2)L is fully gauged. A similar conclusion could not have been derived with regard to the broken generation symmetry. It does carry quark/lepton anomalies, and therefore the effective horizontal interactions are highly suppressed. Our SU(7) theory is intrinsically flavor-chiral. It is therefore not a surprise that, unlike in the original rishon model, the quark/lepton internal structure would not provide room for SU(2)R. Nonetheless, passing through a vector-like QHCD stage, we better remark that L ~ R symmetry violation is switched on along with the color (but not necessarily the electro/ weak) forces. After all, ffL(preon) + ffL(spectator) is SU(3)ttC,C real, but SU(3)H C × SU(3)C complex. Now, although (u, d, e, V)L and We happen to essentially 302
7 April 1983
match the rishon description, (u, d, e)R do not. Yet, we formally have ( ~ e L ) = ( d R d L ) = ( ~ U L ) * . These Q = 0, W = - 1 / 2 "mass terms" suggest that U(1)I X U(1)I I -+ U(1)Q should be governed by a VEV o f a scalar singlet having Q = 0 and W = - 1 / 2 if the badly needed mass relation m w = rnZ cos 0 w is to be effectively recovered. Using the SU(7) language, this can be realized via a threefold antisymmetric Higgs 35. The latter automatically co-introduces the desirable charge one HC, C-singlet piece as well. To summarize, we have specified the minimal grand unifying theory whose nested SU(n)H C composite scenario leads to non-exotic f l a v o r - c o l o r assignments in a 't Hooft consistent manner. While focusing on the algebraic successes of the resulting flavor-chiral rishontype scheme and emphasizing its uniqueness, a lot of theoretical and phenomenological problems are still to be settled. At any rate, we do hope we were able to demonstrate how restrictive as well as constructive group theory can be in trying to merge the trails of compositeness and unification. Useful discussions with Professor H. Harari and Professor S. Yankelowicz are very much appreciated. This work has been supported in part by the U S A Israel Binational Science Foundation.
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