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Vertical-horizontal symmetric minimal grand-unification
Volume 90B, number 1,2
PHYSICS LETTERS
11 February 1980
VERTICAL-HORIZONTAL SYMMETRIC MINIMAL GRAND-UNIFICATION Aharon DAVIDSON Physics Department, Syracuse University, Syracuse, N Y 13210, USA Received 1 October 1979 Revised manucript received 9 November 1979
With the motivation that the traditional generation structure is a fundamental property of nature, namely that it perfectly survivesat ultra-high energiesbeyond the unification mass, we are led to consider semi-simplegrand unified theories. A discrete vertical-horizontal symmetry must then be invoked to ensure a single gauge coupling constant. This in turn is shown to provide a natural origin for fixing the fermion representations and especially the total number of flavors. Within a multigeneration generalization of the Georgi-Glashow SU(5) theory, we predict five ordinary mass fermion families.
The minimal anomaly-free Georgi-Glashow SU(5) model [ 1] is at present the most elegant prototype of a grand unified theory. To list a few of its attractive advantages [2], they are (i) physically correct SU(3)c × [SU(2)L × U(1)]WS decomposition with the right electric-charge assignments, (ii) prediction [3] for the renormalized sin20w in agreement with experimental data, (iii) natural explanation for the extreme smallness of t~QED/t~QCD at ordinary momenta, (iv) exciting prediction on the proton lifetime with observable decay modes [4], (v) existence of a mechanism [5] for generating the observed baryon-to-photon ratio of the universe, and (vi) some renormalization group invariant quark-lepton mass relations [3]. However, the SU(5) model falls by construction into the category of single-generation theories. Consequently, this model is not capable of fixing the total number of the repeated fermionic families, nor even to provide an explanation for the experimental fact that more than just one such family exists. Moreover, inter-generational fundamental quantities like the familiar Cabibbo angle do not have a solid origin to emerge from within the framework of such a model. We are tempted to believe that the various SU(5)generations are in fact completely distinguished by means of an additional so-called horizontal symmetry [6]. The ultimate success of QED and QCD gauge theories naturally suggests that this extra group com-
ponent, which is expected to remove the SU(5) left over degeneracy in the QN's, is a full local gauge group. This is why most flavor-unifying attempts have been primarily aimed towards the discovery of a universal simple Lie-group G, such that G D SU(5) × G h with G h being some unknown proper horizontal symmetry. As far as special unitary groups are concerned, the problem then is to find an optimal physically consistent set of SU(N > 5) representations subject to the generalized SU(5) generation structure, with the hope that some first principle will be available for determining the definite number of the ordinary mass fermion families. Some recent suggestions include (i) an SU(11) symmetry [7] for three ordinary mass generations, and (ii) an SU(9) symmetry [8] also for three generations. All such SU(N > 5) grand unified theories share several fundamental difficulties in common: (1) There is no compelling reason within the SU(N > 5) theories why the SU(5)-generation structure gets selected out among the many alternatives a priori permissible. (2) They do not provide any universal principle capable of fixing the particular N, the physical set of representations, and especially the total number of flavors. (3) They involve a huge number of gauge bosons, some of them may be heavier than the unification mass scale of the standard SU(5), i.e. too close to the 87
Volume 90B, number 1,2
PHYSICS LETTERS
Planck mass, if G is expected to be primarily broken into SU(5) X G h. (4) The conventional generation structure can be realized only after the primary stage of the spontaneous symmetry breaking. With the motivation that the generation structure is in fact a fundamental property o f nature, namely that the conservative generation structure survives at ultra-high energies beyond the unification mass, we are led to the conclusion that a single-generation symmetry is an explicit factor of the grand unifying symmetry. Consequently, the minimal way to produce the desired multi-generation SU(5)-based scheme is to consider the following semi-simple group G = SU(5) I × SU(5)II.
(1)
Such a semi-simple group can serve the idea of unification if and only if a discrete permutation symmetry (I ~ II) is imposed on the theory in order to ensure a single gauge coupling constant. This so-called verticalhorizontal symmetry must be first of all consistent with the choice of the minimal set of fermions, thus establishing a natural non-trivial origin for fixing the total number of flavors. In other words, it allows us to deduce pure horizontal conclusions from information based on the vertical single-generation scheme. The major color constraint [9], namely only singlets and triplets under SU(3)c are allowed, restricts the relevant representations of the vertical SU(5) to be fundamental totally antisymmetric ones. Thus, the most general set f o f fermionic representations is necessarily of the form,
f=.~.nij(i;f) ,
i,j = 1 , 5 , 1 0 , 1 0 , 5 ,
(2)
l,]
where integers nij count the number of (i ;j) G-representations in f. By virtue of the vertical-horizontal symmetry, these integers have the property
11 February 1980
- N/D(j)ni/representations i associated with each SU(5) component of G. Following the Georgi analysis [7], the ordinary mass fermion sector looks like Nf ordinary SU(5)I families, such that Nf -= N(10) - N(]-O)= N ( 5 ) - N ( 5 ) .
(5)
The argument for this important observation is the following: The N(10) 10's will combine with N(]-O) linear combinations of the 10's to form four component super-heavy fermions. The remainder Nf orthogonal linear combinations carry the uncompensated chiral SU(2) × U(1) vertical symmetry, so they are left light and correspond therefore to ordinary fermions. Similarly, N(5) 5's pair with N(5) 5's leaving Nf 5's corresponding to ordinary fermions. Thus, taking into account the explicit dimensions of the relevant SU(5) representations and the fact that A ( 5 ) = -A(10) =A(10) = - A ( 5 ) while A(1) = 0, the total number Nf of ordinary mass fermion families becomes Nf = (nl0,1 - n]-0,1) + 5(n10,5 - n ~ , 5 + nl0, ~ _ ni~,~ )
(6)
+ 10(nl0,10 - n ~ , ~ ) • At this stage we are led to assume the absence of trivial or even semi-trivial fermionic representations which do not carry any vertical and/or horizontal substructure. In the SU(5)I × SU(5)II language it simply means that no fermionic representations of the form (1 ;n) + (n ; 1) are allowed. Such an assumption is actually suggested by the well-known fact that the Georgi-Glashow SU( 5 ) scheme does not involve fermion singlets. Consequently, the optimal number Nf > 1 turns out to be Nf = 5 .
(7)
(4)
This is where the strength of the discrete verticalhorizontal symmetry actually shows off. It forcefully relates the number o f generations with the underlying single-generation theory. Furthermore, and what is even more important, the minimization of eq. (6) consistent with the anomaly constraint (4), the symmetry property (3) and the above no-singlets assumption ,1 fix the representation contents o f f , namely
is fully satisfied. Here, A(i) denotes the anomaly associated with a given SU(5)-representation i whose dimension is D(i). Using these notations, there are N(i)
,1 By giving up this assumption, i.e. by allowing the SU(5) generation to contain singlets, we can arrive at any number of generations by adding to eq. (8) an arbitrary number of [(10 ; 1) + (1 ; 10) + (5 ; 1) + (1 ;5)].
nij = nji .
(3)
Now, it is easy to verify that the theory becomes free of triangular anomalies [10] only if the following condition
.~.A(i)D(j)ni/ = O, t,1
88
(8)
f = (10 ;5-) + (5 ;10) + (5 ;5).
To see this we first notice that the desirable excess of 10's over 10 's can be realized minimally provided we introduce the (10 ; 5) representation. This necessitates the introduction of (5 ; 10) as well. Had we introduced the (10 ; 5) + (5 ; 10) combination, minimality would have been then spoiled as a result from having too many 5's. Next, we must add the (5 ; 5) representation, so that N(10) - N(]-0) = N ( 5 ) - N(5) = 5. This completes the derivation of eq. (8). It is worth noting that the predicted total number of five ordinary mass-fermion families absolutely agrees with some general analysis [11] of the ms,b masses in the standard SU(5). According to this analysis, the conventional Higgs system seems to favor 3 to 4 generations, while a more complicated Higgs system (involving a 45 representation) prefers 5 - 6 fermionic families. It should be also noted that the QCD asymptotic-freedom requires [12] 2Nf ,( ~ which holds trivially for Nf = 5, and that the cosmological limits [13] on the number of neutrino flavors are not violated since N v = 5. At this stage it is relevant to discuss qualitatively the spontaneous breakdown of the grand unified SU(5)I × SU(5)II symmetry into the effective [SU(3) × SU(2) × U(1)] × G h symmetry, where G h is the residual horizontal group factor. The first thing to notice is that the discrete vertical-horizontal symmetry can in fact be spontaneously broken in a complete analogy to the breakdown of left-right symmetry in various SU(2)L × SU(2)R × U(1) electro/weak models [14] and the similar breakdown of flavor-color symmetry within the Pati-Salam [SU(n)] 4 theory [15]. The only question is then the following: Does it occur at the primary stage of the spontaneous symmetry breaking with a typical mass-scale of 1015 GeV, or perhaps at lower energy regime somewhere in the gauge energy desert between 103 GeV and m(GUM). This question is of course still open for a deeper discussion. One attractive possibility is that the SU(5)I × SU(5)II group is primarily broken in an anti-symmetrical fashion using the Higgs system (24 ; 1) + (1 ;24) + (1 ; 10) + (10 ; 1). The introduction of the (1 ; 10) + (10 ; 1) combination comes to assure the Georgi's mechanism [7] discussed earlier. At any rate, we must consider a new intermediate stage of the spontaneous symmetry breaking where GI~ is completely broken while the vertical SU(3) × SU(2)× U(1) survives. This is supt
11 February 1980
PHYSICS LETTERS
Volume 90B, number 1,2
t
.
t
posed to be characterized by a mass-scale m h which can be estimated as follows. The present knowledge concerning horizontal gauge interactions, namely the existence of a low-energy GIM-mechanism and the corresponding suppression of the flavor-changing neutralcurrent interactions, allows the lightest horizontal gauge bosons to be as light as [6] a - l m (W-+) far below the unification mass scale. Consequently, our conjecture is that mh is only 2 - 3 orders of magnitude heavier than the characteristic mass-scale - 3 0 0 GeV associated with SU(3) × SU(2) X U(1) -> SU(3)c X U(1)E M. The door is thus open for a new type of interaction to emerge far below 1015 GeV, serving towards a desirable deviation from the grand gauge plateau approach. It is important to notice that the Higgs multiplet which is expected to give ordinary masses to the five fermionic generations is necessarily of the form ¢=(5;10)+(10;5)+(5;5).
(9)
This is simply because the 10 and 5 SU(5)I fermionic representations have different horizontal substructures. Thus, the scalar multiplet ~ is automatically subjected to the generalized generation structure (ref. [16], Davidson et al. [6]), allowing each fermionic generation to be characterized by a different mass-scale. Finally, several remarks are in order: (1) No additional lepto-quark bosons beyond the ones which accompany the standard SU(5) appear in its semi-simple generalization. This means that B - L remains as a conserved global quantity, while the fermion number B + L is broken with a mass-scale of 1015 GeV. (2) In the presence of a horizontal gauge symmetry we expect a violation of both the GIM-mechanism as well as the e - g - r - . . , universality. Following the above discussion, these are supposed to be characterized by a mass-scale of order m h. (3) At relatively low energies, the lepton number itself is not conserved for each leptonic family separately. We can still have t
F(/l -~ eT)/F@ -~ ev~) < ot[m'h/mw] 4 ,
(10)
in spite of the fact that the neutrinos stay massless.
References [1] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438.
89
Volume 90B, number 1,2
PHYSICS LETTERS
[2] See for example, J. Ellis, CERN preprint TH-2723. 13] H. Georgi, H.R. Quin and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451; A.J. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66. [4] T.J. Goldman and D.A. Ross, Phys. Lett. 84B (1979) 208. [5] J. Ellis, M.K. Galliard and D.V. Nanopoulos, Phys. Lett. 80B (1979) 360. [6] For example, S. Barr and A. Zee, Phys. Rev. D17 (1978) 1854; F. Wilczek and A. Zee, Phys. Rev. Lett. 42 (1979) 421 ; T. Machara and T. Yanagida, Prog. Theor. Phys. 61 (1979) 1434; A. Davidson, M. Koca and K.C. Wali, Phys. Rev. Lett. 43 (1979) 92. [7] H. Georgi, Nucl. Phys. B156 (1979) 126. [8] P.H. Frampton, Ohio State preprint C00-1545-262 (1979). [9] See for example, M. GeU-Mann, P. Ramond and R. Slansky, Rev. Mod. Phys. 50 (1978) 721.
90
11 February 1980
[10] S.L. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cimento 51 (1969) 47. [11] H. Georgi and C. Jarlskog, Harvard preprint HUTP-79/ A026 (1979); P.H. Frampton, S. Nandi and J.J.G. Scanio, Ohio State preprint C00-1545-253 (1979). [12] D.T. Gross and F.A. Wilczek, Phys. Rev. Lett. 30 (1973) 1343; H. Politzer, Phys. Rev. Lett. 30 (1973) 1346. [13] V.F. Shoartsman, JETP Lett. 9 (1969) 184; G. Steigman, D.N. Schraman and J.E. Gunn, Phys. Lett. 66B (1977) 202. [14] J.C. Pati and A. Salam, Phys. Rev. Lett 31 (1973) 661; R.N. Mohapatra and J.C. Pati, Phys. Rev. D l l (1975) 566; G. Senjanovic and R.W. Mohapatra, Phys. Rev. D12 (1975) 1502. [15] J.C. Pati and A. Salam, Phys. Lett. 58B (1975) 333. [16] E. Derman, Phys. Rev. D19 (1979) 317; A. Davidson, M. Koca and K.C. Wali, to be published.
PHYSICS LETTERS
11 February 1980
VERTICAL-HORIZONTAL SYMMETRIC MINIMAL GRAND-UNIFICATION Aharon DAVIDSON Physics Department, Syracuse University, Syracuse, N Y 13210, USA Received 1 October 1979 Revised manucript received 9 November 1979
With the motivation that the traditional generation structure is a fundamental property of nature, namely that it perfectly survivesat ultra-high energiesbeyond the unification mass, we are led to consider semi-simplegrand unified theories. A discrete vertical-horizontal symmetry must then be invoked to ensure a single gauge coupling constant. This in turn is shown to provide a natural origin for fixing the fermion representations and especially the total number of flavors. Within a multigeneration generalization of the Georgi-Glashow SU(5) theory, we predict five ordinary mass fermion families.
The minimal anomaly-free Georgi-Glashow SU(5) model [ 1] is at present the most elegant prototype of a grand unified theory. To list a few of its attractive advantages [2], they are (i) physically correct SU(3)c × [SU(2)L × U(1)]WS decomposition with the right electric-charge assignments, (ii) prediction [3] for the renormalized sin20w in agreement with experimental data, (iii) natural explanation for the extreme smallness of t~QED/t~QCD at ordinary momenta, (iv) exciting prediction on the proton lifetime with observable decay modes [4], (v) existence of a mechanism [5] for generating the observed baryon-to-photon ratio of the universe, and (vi) some renormalization group invariant quark-lepton mass relations [3]. However, the SU(5) model falls by construction into the category of single-generation theories. Consequently, this model is not capable of fixing the total number of the repeated fermionic families, nor even to provide an explanation for the experimental fact that more than just one such family exists. Moreover, inter-generational fundamental quantities like the familiar Cabibbo angle do not have a solid origin to emerge from within the framework of such a model. We are tempted to believe that the various SU(5)generations are in fact completely distinguished by means of an additional so-called horizontal symmetry [6]. The ultimate success of QED and QCD gauge theories naturally suggests that this extra group com-
ponent, which is expected to remove the SU(5) left over degeneracy in the QN's, is a full local gauge group. This is why most flavor-unifying attempts have been primarily aimed towards the discovery of a universal simple Lie-group G, such that G D SU(5) × G h with G h being some unknown proper horizontal symmetry. As far as special unitary groups are concerned, the problem then is to find an optimal physically consistent set of SU(N > 5) representations subject to the generalized SU(5) generation structure, with the hope that some first principle will be available for determining the definite number of the ordinary mass fermion families. Some recent suggestions include (i) an SU(11) symmetry [7] for three ordinary mass generations, and (ii) an SU(9) symmetry [8] also for three generations. All such SU(N > 5) grand unified theories share several fundamental difficulties in common: (1) There is no compelling reason within the SU(N > 5) theories why the SU(5)-generation structure gets selected out among the many alternatives a priori permissible. (2) They do not provide any universal principle capable of fixing the particular N, the physical set of representations, and especially the total number of flavors. (3) They involve a huge number of gauge bosons, some of them may be heavier than the unification mass scale of the standard SU(5), i.e. too close to the 87
Volume 90B, number 1,2
PHYSICS LETTERS
Planck mass, if G is expected to be primarily broken into SU(5) X G h. (4) The conventional generation structure can be realized only after the primary stage of the spontaneous symmetry breaking. With the motivation that the generation structure is in fact a fundamental property o f nature, namely that the conservative generation structure survives at ultra-high energies beyond the unification mass, we are led to the conclusion that a single-generation symmetry is an explicit factor of the grand unifying symmetry. Consequently, the minimal way to produce the desired multi-generation SU(5)-based scheme is to consider the following semi-simple group G = SU(5) I × SU(5)II.
(1)
Such a semi-simple group can serve the idea of unification if and only if a discrete permutation symmetry (I ~ II) is imposed on the theory in order to ensure a single gauge coupling constant. This so-called verticalhorizontal symmetry must be first of all consistent with the choice of the minimal set of fermions, thus establishing a natural non-trivial origin for fixing the total number of flavors. In other words, it allows us to deduce pure horizontal conclusions from information based on the vertical single-generation scheme. The major color constraint [9], namely only singlets and triplets under SU(3)c are allowed, restricts the relevant representations of the vertical SU(5) to be fundamental totally antisymmetric ones. Thus, the most general set f o f fermionic representations is necessarily of the form,
f=.~.nij(i;f) ,
i,j = 1 , 5 , 1 0 , 1 0 , 5 ,
(2)
l,]
where integers nij count the number of (i ;j) G-representations in f. By virtue of the vertical-horizontal symmetry, these integers have the property
11 February 1980
- N/D(j)ni/representations i associated with each SU(5) component of G. Following the Georgi analysis [7], the ordinary mass fermion sector looks like Nf ordinary SU(5)I families, such that Nf -= N(10) - N(]-O)= N ( 5 ) - N ( 5 ) .
(5)
The argument for this important observation is the following: The N(10) 10's will combine with N(]-O) linear combinations of the 10's to form four component super-heavy fermions. The remainder Nf orthogonal linear combinations carry the uncompensated chiral SU(2) × U(1) vertical symmetry, so they are left light and correspond therefore to ordinary fermions. Similarly, N(5) 5's pair with N(5) 5's leaving Nf 5's corresponding to ordinary fermions. Thus, taking into account the explicit dimensions of the relevant SU(5) representations and the fact that A ( 5 ) = -A(10) =A(10) = - A ( 5 ) while A(1) = 0, the total number Nf of ordinary mass fermion families becomes Nf = (nl0,1 - n]-0,1) + 5(n10,5 - n ~ , 5 + nl0, ~ _ ni~,~ )
(6)
+ 10(nl0,10 - n ~ , ~ ) • At this stage we are led to assume the absence of trivial or even semi-trivial fermionic representations which do not carry any vertical and/or horizontal substructure. In the SU(5)I × SU(5)II language it simply means that no fermionic representations of the form (1 ;n) + (n ; 1) are allowed. Such an assumption is actually suggested by the well-known fact that the Georgi-Glashow SU( 5 ) scheme does not involve fermion singlets. Consequently, the optimal number Nf > 1 turns out to be Nf = 5 .
(7)
(4)
This is where the strength of the discrete verticalhorizontal symmetry actually shows off. It forcefully relates the number o f generations with the underlying single-generation theory. Furthermore, and what is even more important, the minimization of eq. (6) consistent with the anomaly constraint (4), the symmetry property (3) and the above no-singlets assumption ,1 fix the representation contents o f f , namely
is fully satisfied. Here, A(i) denotes the anomaly associated with a given SU(5)-representation i whose dimension is D(i). Using these notations, there are N(i)
,1 By giving up this assumption, i.e. by allowing the SU(5) generation to contain singlets, we can arrive at any number of generations by adding to eq. (8) an arbitrary number of [(10 ; 1) + (1 ; 10) + (5 ; 1) + (1 ;5)].
nij = nji .
(3)
Now, it is easy to verify that the theory becomes free of triangular anomalies [10] only if the following condition
.~.A(i)D(j)ni/ = O, t,1
88
(8)
f = (10 ;5-) + (5 ;10) + (5 ;5).
To see this we first notice that the desirable excess of 10's over 10 's can be realized minimally provided we introduce the (10 ; 5) representation. This necessitates the introduction of (5 ; 10) as well. Had we introduced the (10 ; 5) + (5 ; 10) combination, minimality would have been then spoiled as a result from having too many 5's. Next, we must add the (5 ; 5) representation, so that N(10) - N(]-0) = N ( 5 ) - N(5) = 5. This completes the derivation of eq. (8). It is worth noting that the predicted total number of five ordinary mass-fermion families absolutely agrees with some general analysis [11] of the ms,b masses in the standard SU(5). According to this analysis, the conventional Higgs system seems to favor 3 to 4 generations, while a more complicated Higgs system (involving a 45 representation) prefers 5 - 6 fermionic families. It should be also noted that the QCD asymptotic-freedom requires [12] 2Nf ,( ~ which holds trivially for Nf = 5, and that the cosmological limits [13] on the number of neutrino flavors are not violated since N v = 5. At this stage it is relevant to discuss qualitatively the spontaneous breakdown of the grand unified SU(5)I × SU(5)II symmetry into the effective [SU(3) × SU(2) × U(1)] × G h symmetry, where G h is the residual horizontal group factor. The first thing to notice is that the discrete vertical-horizontal symmetry can in fact be spontaneously broken in a complete analogy to the breakdown of left-right symmetry in various SU(2)L × SU(2)R × U(1) electro/weak models [14] and the similar breakdown of flavor-color symmetry within the Pati-Salam [SU(n)] 4 theory [15]. The only question is then the following: Does it occur at the primary stage of the spontaneous symmetry breaking with a typical mass-scale of 1015 GeV, or perhaps at lower energy regime somewhere in the gauge energy desert between 103 GeV and m(GUM). This question is of course still open for a deeper discussion. One attractive possibility is that the SU(5)I × SU(5)II group is primarily broken in an anti-symmetrical fashion using the Higgs system (24 ; 1) + (1 ;24) + (1 ; 10) + (10 ; 1). The introduction of the (1 ; 10) + (10 ; 1) combination comes to assure the Georgi's mechanism [7] discussed earlier. At any rate, we must consider a new intermediate stage of the spontaneous symmetry breaking where GI~ is completely broken while the vertical SU(3) × SU(2)× U(1) survives. This is supt
11 February 1980
PHYSICS LETTERS
Volume 90B, number 1,2
t
.
t
posed to be characterized by a mass-scale m h which can be estimated as follows. The present knowledge concerning horizontal gauge interactions, namely the existence of a low-energy GIM-mechanism and the corresponding suppression of the flavor-changing neutralcurrent interactions, allows the lightest horizontal gauge bosons to be as light as [6] a - l m (W-+) far below the unification mass scale. Consequently, our conjecture is that mh is only 2 - 3 orders of magnitude heavier than the characteristic mass-scale - 3 0 0 GeV associated with SU(3) × SU(2) X U(1) -> SU(3)c X U(1)E M. The door is thus open for a new type of interaction to emerge far below 1015 GeV, serving towards a desirable deviation from the grand gauge plateau approach. It is important to notice that the Higgs multiplet which is expected to give ordinary masses to the five fermionic generations is necessarily of the form ¢=(5;10)+(10;5)+(5;5).
(9)
This is simply because the 10 and 5 SU(5)I fermionic representations have different horizontal substructures. Thus, the scalar multiplet ~ is automatically subjected to the generalized generation structure (ref. [16], Davidson et al. [6]), allowing each fermionic generation to be characterized by a different mass-scale. Finally, several remarks are in order: (1) No additional lepto-quark bosons beyond the ones which accompany the standard SU(5) appear in its semi-simple generalization. This means that B - L remains as a conserved global quantity, while the fermion number B + L is broken with a mass-scale of 1015 GeV. (2) In the presence of a horizontal gauge symmetry we expect a violation of both the GIM-mechanism as well as the e - g - r - . . , universality. Following the above discussion, these are supposed to be characterized by a mass-scale of order m h. (3) At relatively low energies, the lepton number itself is not conserved for each leptonic family separately. We can still have t
F(/l -~ eT)/F@ -~ ev~) < ot[m'h/mw] 4 ,
(10)
in spite of the fact that the neutrinos stay massless.
References [1] H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438.
89
Volume 90B, number 1,2
PHYSICS LETTERS
[2] See for example, J. Ellis, CERN preprint TH-2723. 13] H. Georgi, H.R. Quin and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451; A.J. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B135 (1978) 66. [4] T.J. Goldman and D.A. Ross, Phys. Lett. 84B (1979) 208. [5] J. Ellis, M.K. Galliard and D.V. Nanopoulos, Phys. Lett. 80B (1979) 360. [6] For example, S. Barr and A. Zee, Phys. Rev. D17 (1978) 1854; F. Wilczek and A. Zee, Phys. Rev. Lett. 42 (1979) 421 ; T. Machara and T. Yanagida, Prog. Theor. Phys. 61 (1979) 1434; A. Davidson, M. Koca and K.C. Wali, Phys. Rev. Lett. 43 (1979) 92. [7] H. Georgi, Nucl. Phys. B156 (1979) 126. [8] P.H. Frampton, Ohio State preprint C00-1545-262 (1979). [9] See for example, M. GeU-Mann, P. Ramond and R. Slansky, Rev. Mod. Phys. 50 (1978) 721.
90
11 February 1980
[10] S.L. Adler, Phys. Rev. 177 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cimento 51 (1969) 47. [11] H. Georgi and C. Jarlskog, Harvard preprint HUTP-79/ A026 (1979); P.H. Frampton, S. Nandi and J.J.G. Scanio, Ohio State preprint C00-1545-253 (1979). [12] D.T. Gross and F.A. Wilczek, Phys. Rev. Lett. 30 (1973) 1343; H. Politzer, Phys. Rev. Lett. 30 (1973) 1346. [13] V.F. Shoartsman, JETP Lett. 9 (1969) 184; G. Steigman, D.N. Schraman and J.E. Gunn, Phys. Lett. 66B (1977) 202. [14] J.C. Pati and A. Salam, Phys. Rev. Lett 31 (1973) 661; R.N. Mohapatra and J.C. Pati, Phys. Rev. D l l (1975) 566; G. Senjanovic and R.W. Mohapatra, Phys. Rev. D12 (1975) 1502. [15] J.C. Pati and A. Salam, Phys. Lett. 58B (1975) 333. [16] E. Derman, Phys. Rev. D19 (1979) 317; A. Davidson, M. Koca and K.C. Wali, to be published.