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An SO(10) supersymmetric grand unified theory
Volume 115B, number 1
PHYSICS LETTERS
19 August 1982
AN SO(10) SUPERSYMMETRIC GRAND UNIFIED THEORY T.E. CLARK, T.K. KUO and N. N A K A G A W A
Department of Physics, Purdue University, West Lafayette, IN 47907, USA Received 20 May 1982
A realistic supersymmetric SO(10) grand unified theory is explicitly constructed. The neutrino masses turn out to be 10-5 eV or less. It is also found that the proton life time can be longer than 1031 yr.
Supersymmetry (SUSY) has proven effective in solving the gauge hierarchy problem in grand unified theories (GUTS) [1,2] owing to a class o f supersymmetric "no-renormalization" theorems [3]. SUSY has been applied mainly to SU(5) GUTS, and in particular the " m i n i m a l " SU(5) SUSY GUT has been fully investigated with respect to its low-energy phenomenology
[4]. On the other hand, it is well-known that SO(10) GUTS are also interesting in view of the questions of flavor unification and neutrino masses [ 5 - 7 ] . In addition, anomalies are absent for SO(10). There have been attempts to construct SUSY SO(10) models [8], which eventually involved some unwanted light fields. It is the purpose o f this letter to propose a SO(10) SUSY GUT which is compatible with low energy phenomenology. In regard to p r o t o n decay we will not use the extra symmetries suggested by Sakai and Yanagida and b y Weinberg [9,10]. Instead, we will present an alternative mechanism to control the proton decay b y exploiting the abundance o f heavy Higgs fields * 1 The model contains a number of supermultiplets. First, several 16 dimensional spinor representations [11] are used for matter fields (M) carrying the generation structure. Next, we introduce a 10(xP) and a ~ ( ~ b ) which have Yukawa couplings with M ,2 • 1 In the minimal SU(5) SUSY GUT, the authors of ref. [4] carefully chose masses of SUSY partners of gauge and matter fields to push ~-p up above the experimental lower limit. ,2 Besides the l ~ , we need either a 10 or a 120 in order to have a Cabibbo rotation. 26
The introduction of q~ simultaneously solves two important questions, one about the gauged B - L symmetry and the other concerning right-handed neutrinos [6] , 3 . We must also have a 126(~b') to avoid unwanted light fields and to prevent a SUSY breaking D-term. Finally, in order to break SO(10) to SU(3) × SU(2) × U y ( 1 ) (× U B _ L ( I ) ) while keeping SUSY, we introduce a 210(q~), which turns out to be the minimal choice in order to achieve the symmetry breaking pattern desired , 4 . Now our superpotential reads c)9= 2mldp2 + 4m2~bqS' + 2 m 3 ~ 2 + ~ g l ~3 +g2qbqS¢ ' +g3qbq~xIt + g4qb~b'~ + ~ ~iMiMid? + ~. AijMiM] xp,
i
i,l
(1)
where indices i and j refer to the generation , s All parameters can be chosen to be real except Ai/ which is a complex symmetric matrix. All masses rn a are superheavy. The gauge field will be denoted b y V, which is the 45 of SO(10). For convenience, we use the notation 5~p(3,1".- 1/3), ,3 Though a 16 may also break B-L, it does not induce large vR masses through Witten's mechanism [7] because no such radiative corrections axe available in SUSY. ,4 For this purpose, q~ should at least contain a 24 of SU(5). The candidates 45 and 144 do not have triple serf-couplings and so break SUSY. Also a 54 leads to SUSY breakdown through F-terms ofq~ and 9' since it does not have a SU(5) singlet. ~5 This c'19has a discrete symmetry M ~ -M automatically, while in the SU(5) model it had to be imposed additionally in order to remove the unwanted terms, 5H5 M and g'M~'MI°M [2,10]. 0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland
Volume 115B, number 1
PHYSICS LETTERS
for instance, where 5 refers to its SU(5) content and O, 1: -1/3), that of SU(3) X SU(2) × U(1). We shift 1~ 24qb 75dO 1,~ (1 1:0)' (1,1:0)' (1 1:0)' ~'(1 1:0) a n d q5(1,1:0) by P, Q, R, p and p , respectwely, to break SO(10) directly to SU(3) × SU(2) × Uy(1), and require the vacuum expectation values (vev) of all F components to vanish (only the above five fields have nontrivial contributions) in order to keep SUSY unbroken [12]. For example, both (Fea) = 0 and (F4;) = 0 tell us that P = -(4/V'-~) m2/g 2. The other three vevs provide us with a set of equations for Q, R and (pp'), which can be solved to give non-trivial solutions, Q 0, R :~ 0 and (pp') ~ O. On the other hand, as p and p' break B - L , the D-term of V develops a vev in general [13]. However, ~ and q~' have opposite B - L charge, and therefore (DB_ L) is proportional to (Ipl 2 _ ]p'] 2). Consequently, the requirement (D)= 0 fixesp and p' up to their phases, to be IP] = IP'I = (IPp'I) 1/2 [14]. As SO(10) X SUSY is broken to SU(3) X SU(2) × U(1) × SUSY, aU fields are now superheavy except the remaining gauge fields and the matter fields, 10M = (Q(3,2: 1/6)' U~j,I: -2/3)' E~I,I:I)}' 5M -= {D~,I : 1/3), L(1,2:_1/2) }. In particular, the right-handed neutrinos 1M ~ N c acquire the superheavy Majorana mass [6], m Maj°rana = ~ / 2 p X i •
19 August 1982
.~- [A~)U~Qi H + A'('D)D'C' QH' 1u t,]
+ A!.N)N.CL.H+ A!.E)E.CL.H'] II
11
tl
11
'
(3)
where A~g ) = 12X/3 ~XiSiy + 2X/~ 8 Aiy, A P) = -
+
a'A;j,
A(N) i] = ( l O0 /x'/5) ~ Xi S iy + 2 x/2 8 A/I, Ai(/.E)= 4X/1-O[3'XiSiy + 2X/~ 8' Aii. Here, we have NCLH terms which were absent in the SU(5) model. Another point is that the E-coupling and D-coupling are different ,8 As in the minimal SU(5) SUSY GUT, we break SUSY explicitly but Softly, and then break SU(2) X U(1). The MSUM and sin20w can be also obtained as usual [15]. Just as in an ordinary SO(10) GUT, our SO(10) SUSY GUT is characterized by massive neutrinos. Since we have given large Majorana masses to UR, the Gell-Mann-Ramond-Slansky mechanism works [6], yielding
m ,L = mDirac2 /m Majorana
(2)
While, in order to pick up one pair of light doublet Higgs, denoted by H(1,2:1/2 ) and H~1,2:_1/2) , we must make a fine tuning as in the SU(5) SUSY GUT [2]. A mass matrix can be written down for the following eight doublets; 5 qb(1,2 : 1/2), 5~b(..)_' 45~b~..)' 5 ~ ( . . ) , 5 q b ( 1 , 2 : _ 1 / 2 ) ' 4 5 ~ ( . . ) , 5q}{ ) a n d 5,I,(). We
then adjust one eigenvalue to be m H ,6, which is the mass of the superpartners of the Higgs (shiggs) and is of the order of the SUSY breaking scale Ms.The above eight fields contain either H or H' components with fractional coefficients c~,/3, 3', 8, c~', 13',3'' and 8', respectively * v, By decomposing XMM~ and AMMq~ of (1), we find the following Yukawa couplings of H and H': #6 Namely, we impose one constraint among eight parameters, m l ..... g4 and mH. ,7 These eight fractions a .... , 6 ' can be expressed in terms of seven independent parameters from m l ..... g4 and m H.
(10 GeV)2/1016 GeV = 10 -5 eV. Here mMaj°rana is given in (2), while m Birac is obtained from the Yukawa coupling NCLH ,9 Let us now turn to a discussion of the proton decay In the SU(5)model, the dimension-five operators [9, 10] are induced by the exchange of 5 and 5 Higgs fields, the couplings of which are determined by the fermion masses (and the Kobayashi-Maskawa matrix). As a result, the fermion masses completely specify the structure of the d = 5 operators. In our models, however, we have the two fields, q~and q~. Further, they contribute to the fermion masses and the decay opera*8 The s,I, gives the same couplings, as in SU(5), while our asq~ contributes to differentiate between them. .9 In our model, u L can acquire radiatively induced Majorana masses. However, they are even smaller than those of non-SUSY GUTS [ 10], because they must have an extra factor ( M s / M s u M ) as a consequence of the norenormalization theorem.
27
Volume 115B, number 1
PHYSICS LETTERS
tors in different combinations. Therefore we have the freedom to change the magnitude of the d = 5 operators without changing the fermion masses. Consequently, by choosing appropriate parameters, we can, if necessary, make the proton life time longer than 1031 yr, in contrast to the results obtained in the SU(5) model [4]. Details of this analysis will be given elsewhere. Two comments are in order. One concerns sin20w, which is predicted to be about 0.23 in our model (see also ref. [15] ). As it is slightly too large, the authors of ref. [16] introduced some "semi-light" fields to make it smaller. In this note, however, we do not follow this seemingly arbitrary line of reasoning, although we do have enough extra fields, some of which can be adjusted to be semi-light * lo, in order to implement it. The other is about the flavor unification, which has not been touched upon so far. In the mass matrices, we have enough parameters to adjust the physical masses. But at the same time,we lose predictive power. One way to control those parameters is to extend the group. Along the lines of our SO(10)model, higher SO-groups, particularly SO(14), are being investigated. After completion of this work, we received a paper by Aulakh and Mohapatra who also studied a SO(10) SUSY GUT model [17]. They intend to introduce an intermediate energy scale, and in doing so seem to break SUSY at this scale. While, in our case, we break SO(10) down to SU(3) × SU(2) × U(1) at once, and keep SUSY unbroken until the TeV region. This work is supported in part b y the US Department of Energy.
4:1o Those semi-light fields must not induce the d = 5 operators, and so cannot be (3,1;-1/3).
28
19 August 1982
References [1] M. Veltman, Acta Phys. Pol. B12 (1981) 437; N. Sakai, Z. Phys. Cll (1981) 153; E. Witten, Phys. Lett. 105B (1981) 267. I2] S. Dimopoulos and H. Georgi, Nucl. Phys. B193 (1981) 150. [3] J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310; K. Fujikawa and W. Lang, Nucl. Phys. B88 (1975) 61; M.T. Grisaru, W. Siegel and M. Ro?ek, Nucl. Phys. B159 (1979) 429. [4] J. Ellis, D.V. Nanopoulos and S. Rudaz, GUTS 3: SUSY GUTS 2, CERN TH. 3199 (1981). [5] H. Fritzseh and P. Minkowski, Ann. Phys. (NY) 93 (1975) 193; M.S. Chanowitz, J. Ellis and M.K. Gaillard, Nucl. Phys. B128 (1977)506; H. Georgi and D.V. Nanopoulos, Phys. Lett. 82B (1979) 392; Nucl. Phys. B155 (1979) 52. [6] M. Gell-Mann,P. Ramond and R. Slansky, in: Supergravity, eds. P. van Nieuwenhuizen and D.Z. Freedman (North-Holland, Amsterdam, 1979). [7] E. Witten, Phys. Lett. 91B (1980) 81. [8] Z.-Y. Zhao, Gauge hierarchy in an SO(10) supersymmettic grand unified model, Trieste 9/81/E.P. (1981); R.N. Cahn, I. Hinchliffe and L.J. Hall, Phys. Lett. 109B (1982) 426; T.E. Clark, T.K. Kuo and N. Nakagawa, Towards a SO(10) supersymmetric grand unified theory, Purdue University PURD-TH-82-1 (1982). [9] N. Sakai and T. Yanagida, Nucl. Phys. B197 (1982) 533. [10] S. Weinberg Supersymmetry at ordinary energies I, Harvard University HUTP-81/A047 (1981). [11] See, for example, F. Wilczek and A. Zee, Phys. Rev. D25 (1982) 553. [12] L. O'Raifeartaigh, Nucl. Phys. B96 (1975) 331. [13] F. BucceUa, J.-P. Derendinger, C.A. Savoy and S. Ferrara, Symmetry breaking in supersymmetric GUTS, CERN TH. 3212 (1981), to be published in: Unification of the fundamental interactions II, eds. J. Elles and S. Ferrara. [14] B.A. Ovrut and J. Wess, Phys. Rev. D25 (1982) 409. [15] M.B. Einhorn and D.R.T. Jones, Nucl. Phys. B196 (1982) 475. [16] D.V. Nanopoulos and K. Tamvakis, Phys. Lett. 113B (1982) 151. [17] C.S. Aulakh and R.N. Mohapatra, Implications of supersymmetric SO(10) grand unification, CCNY HEP 82/4 (1982).
PHYSICS LETTERS
19 August 1982
AN SO(10) SUPERSYMMETRIC GRAND UNIFIED THEORY T.E. CLARK, T.K. KUO and N. N A K A G A W A
Department of Physics, Purdue University, West Lafayette, IN 47907, USA Received 20 May 1982
A realistic supersymmetric SO(10) grand unified theory is explicitly constructed. The neutrino masses turn out to be 10-5 eV or less. It is also found that the proton life time can be longer than 1031 yr.
Supersymmetry (SUSY) has proven effective in solving the gauge hierarchy problem in grand unified theories (GUTS) [1,2] owing to a class o f supersymmetric "no-renormalization" theorems [3]. SUSY has been applied mainly to SU(5) GUTS, and in particular the " m i n i m a l " SU(5) SUSY GUT has been fully investigated with respect to its low-energy phenomenology
[4]. On the other hand, it is well-known that SO(10) GUTS are also interesting in view of the questions of flavor unification and neutrino masses [ 5 - 7 ] . In addition, anomalies are absent for SO(10). There have been attempts to construct SUSY SO(10) models [8], which eventually involved some unwanted light fields. It is the purpose o f this letter to propose a SO(10) SUSY GUT which is compatible with low energy phenomenology. In regard to p r o t o n decay we will not use the extra symmetries suggested by Sakai and Yanagida and b y Weinberg [9,10]. Instead, we will present an alternative mechanism to control the proton decay b y exploiting the abundance o f heavy Higgs fields * 1 The model contains a number of supermultiplets. First, several 16 dimensional spinor representations [11] are used for matter fields (M) carrying the generation structure. Next, we introduce a 10(xP) and a ~ ( ~ b ) which have Yukawa couplings with M ,2 • 1 In the minimal SU(5) SUSY GUT, the authors of ref. [4] carefully chose masses of SUSY partners of gauge and matter fields to push ~-p up above the experimental lower limit. ,2 Besides the l ~ , we need either a 10 or a 120 in order to have a Cabibbo rotation. 26
The introduction of q~ simultaneously solves two important questions, one about the gauged B - L symmetry and the other concerning right-handed neutrinos [6] , 3 . We must also have a 126(~b') to avoid unwanted light fields and to prevent a SUSY breaking D-term. Finally, in order to break SO(10) to SU(3) × SU(2) × U y ( 1 ) (× U B _ L ( I ) ) while keeping SUSY, we introduce a 210(q~), which turns out to be the minimal choice in order to achieve the symmetry breaking pattern desired , 4 . Now our superpotential reads c)9= 2mldp2 + 4m2~bqS' + 2 m 3 ~ 2 + ~ g l ~3 +g2qbqS¢ ' +g3qbq~xIt + g4qb~b'~ + ~ ~iMiMid? + ~. AijMiM] xp,
i
i,l
(1)
where indices i and j refer to the generation , s All parameters can be chosen to be real except Ai/ which is a complex symmetric matrix. All masses rn a are superheavy. The gauge field will be denoted b y V, which is the 45 of SO(10). For convenience, we use the notation 5~p(3,1".- 1/3), ,3 Though a 16 may also break B-L, it does not induce large vR masses through Witten's mechanism [7] because no such radiative corrections axe available in SUSY. ,4 For this purpose, q~ should at least contain a 24 of SU(5). The candidates 45 and 144 do not have triple serf-couplings and so break SUSY. Also a 54 leads to SUSY breakdown through F-terms ofq~ and 9' since it does not have a SU(5) singlet. ~5 This c'19has a discrete symmetry M ~ -M automatically, while in the SU(5) model it had to be imposed additionally in order to remove the unwanted terms, 5H5 M and g'M~'MI°M [2,10]. 0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland
Volume 115B, number 1
PHYSICS LETTERS
for instance, where 5 refers to its SU(5) content and O, 1: -1/3), that of SU(3) X SU(2) × U(1). We shift 1~ 24qb 75dO 1,~ (1 1:0)' (1,1:0)' (1 1:0)' ~'(1 1:0) a n d q5(1,1:0) by P, Q, R, p and p , respectwely, to break SO(10) directly to SU(3) × SU(2) × Uy(1), and require the vacuum expectation values (vev) of all F components to vanish (only the above five fields have nontrivial contributions) in order to keep SUSY unbroken [12]. For example, both (Fea) = 0 and (F4;) = 0 tell us that P = -(4/V'-~) m2/g 2. The other three vevs provide us with a set of equations for Q, R and (pp'), which can be solved to give non-trivial solutions, Q 0, R :~ 0 and (pp') ~ O. On the other hand, as p and p' break B - L , the D-term of V develops a vev in general [13]. However, ~ and q~' have opposite B - L charge, and therefore (DB_ L) is proportional to (Ipl 2 _ ]p'] 2). Consequently, the requirement (D)= 0 fixesp and p' up to their phases, to be IP] = IP'I = (IPp'I) 1/2 [14]. As SO(10) X SUSY is broken to SU(3) X SU(2) × U(1) × SUSY, aU fields are now superheavy except the remaining gauge fields and the matter fields, 10M = (Q(3,2: 1/6)' U~j,I: -2/3)' E~I,I:I)}' 5M -= {D~,I : 1/3), L(1,2:_1/2) }. In particular, the right-handed neutrinos 1M ~ N c acquire the superheavy Majorana mass [6], m Maj°rana = ~ / 2 p X i •
19 August 1982
.~- [A~)U~Qi H + A'('D)D'C' QH' 1u t,]
+ A!.N)N.CL.H+ A!.E)E.CL.H'] II
11
tl
11
'
(3)
where A~g ) = 12X/3 ~XiSiy + 2X/~ 8 Aiy, A P) = -
+
a'A;j,
A(N) i] = ( l O0 /x'/5) ~ Xi S iy + 2 x/2 8 A/I, Ai(/.E)= 4X/1-O[3'XiSiy + 2X/~ 8' Aii. Here, we have NCLH terms which were absent in the SU(5) model. Another point is that the E-coupling and D-coupling are different ,8 As in the minimal SU(5) SUSY GUT, we break SUSY explicitly but Softly, and then break SU(2) X U(1). The MSUM and sin20w can be also obtained as usual [15]. Just as in an ordinary SO(10) GUT, our SO(10) SUSY GUT is characterized by massive neutrinos. Since we have given large Majorana masses to UR, the Gell-Mann-Ramond-Slansky mechanism works [6], yielding
m ,L = mDirac2 /m Majorana
(2)
While, in order to pick up one pair of light doublet Higgs, denoted by H(1,2:1/2 ) and H~1,2:_1/2) , we must make a fine tuning as in the SU(5) SUSY GUT [2]. A mass matrix can be written down for the following eight doublets; 5 qb(1,2 : 1/2), 5~b(..)_' 45~b~..)' 5 ~ ( . . ) , 5 q b ( 1 , 2 : _ 1 / 2 ) ' 4 5 ~ ( . . ) , 5q}{ ) a n d 5,I,(). We
then adjust one eigenvalue to be m H ,6, which is the mass of the superpartners of the Higgs (shiggs) and is of the order of the SUSY breaking scale Ms.The above eight fields contain either H or H' components with fractional coefficients c~,/3, 3', 8, c~', 13',3'' and 8', respectively * v, By decomposing XMM~ and AMMq~ of (1), we find the following Yukawa couplings of H and H': #6 Namely, we impose one constraint among eight parameters, m l ..... g4 and mH. ,7 These eight fractions a .... , 6 ' can be expressed in terms of seven independent parameters from m l ..... g4 and m H.
(10 GeV)2/1016 GeV = 10 -5 eV. Here mMaj°rana is given in (2), while m Birac is obtained from the Yukawa coupling NCLH ,9 Let us now turn to a discussion of the proton decay In the SU(5)model, the dimension-five operators [9, 10] are induced by the exchange of 5 and 5 Higgs fields, the couplings of which are determined by the fermion masses (and the Kobayashi-Maskawa matrix). As a result, the fermion masses completely specify the structure of the d = 5 operators. In our models, however, we have the two fields, q~and q~. Further, they contribute to the fermion masses and the decay opera*8 The s,I, gives the same couplings, as in SU(5), while our asq~ contributes to differentiate between them. .9 In our model, u L can acquire radiatively induced Majorana masses. However, they are even smaller than those of non-SUSY GUTS [ 10], because they must have an extra factor ( M s / M s u M ) as a consequence of the norenormalization theorem.
27
Volume 115B, number 1
PHYSICS LETTERS
tors in different combinations. Therefore we have the freedom to change the magnitude of the d = 5 operators without changing the fermion masses. Consequently, by choosing appropriate parameters, we can, if necessary, make the proton life time longer than 1031 yr, in contrast to the results obtained in the SU(5) model [4]. Details of this analysis will be given elsewhere. Two comments are in order. One concerns sin20w, which is predicted to be about 0.23 in our model (see also ref. [15] ). As it is slightly too large, the authors of ref. [16] introduced some "semi-light" fields to make it smaller. In this note, however, we do not follow this seemingly arbitrary line of reasoning, although we do have enough extra fields, some of which can be adjusted to be semi-light * lo, in order to implement it. The other is about the flavor unification, which has not been touched upon so far. In the mass matrices, we have enough parameters to adjust the physical masses. But at the same time,we lose predictive power. One way to control those parameters is to extend the group. Along the lines of our SO(10)model, higher SO-groups, particularly SO(14), are being investigated. After completion of this work, we received a paper by Aulakh and Mohapatra who also studied a SO(10) SUSY GUT model [17]. They intend to introduce an intermediate energy scale, and in doing so seem to break SUSY at this scale. While, in our case, we break SO(10) down to SU(3) × SU(2) × U(1) at once, and keep SUSY unbroken until the TeV region. This work is supported in part b y the US Department of Energy.
4:1o Those semi-light fields must not induce the d = 5 operators, and so cannot be (3,1;-1/3).
28
19 August 1982
References [1] M. Veltman, Acta Phys. Pol. B12 (1981) 437; N. Sakai, Z. Phys. Cll (1981) 153; E. Witten, Phys. Lett. 105B (1981) 267. I2] S. Dimopoulos and H. Georgi, Nucl. Phys. B193 (1981) 150. [3] J. Iliopoulos and B. Zumino, Nucl. Phys. B76 (1974) 310; K. Fujikawa and W. Lang, Nucl. Phys. B88 (1975) 61; M.T. Grisaru, W. Siegel and M. Ro?ek, Nucl. Phys. B159 (1979) 429. [4] J. Ellis, D.V. Nanopoulos and S. Rudaz, GUTS 3: SUSY GUTS 2, CERN TH. 3199 (1981). [5] H. Fritzseh and P. Minkowski, Ann. Phys. (NY) 93 (1975) 193; M.S. Chanowitz, J. Ellis and M.K. Gaillard, Nucl. Phys. B128 (1977)506; H. Georgi and D.V. Nanopoulos, Phys. Lett. 82B (1979) 392; Nucl. Phys. B155 (1979) 52. [6] M. Gell-Mann,P. Ramond and R. Slansky, in: Supergravity, eds. P. van Nieuwenhuizen and D.Z. Freedman (North-Holland, Amsterdam, 1979). [7] E. Witten, Phys. Lett. 91B (1980) 81. [8] Z.-Y. Zhao, Gauge hierarchy in an SO(10) supersymmettic grand unified model, Trieste 9/81/E.P. (1981); R.N. Cahn, I. Hinchliffe and L.J. Hall, Phys. Lett. 109B (1982) 426; T.E. Clark, T.K. Kuo and N. Nakagawa, Towards a SO(10) supersymmetric grand unified theory, Purdue University PURD-TH-82-1 (1982). [9] N. Sakai and T. Yanagida, Nucl. Phys. B197 (1982) 533. [10] S. Weinberg Supersymmetry at ordinary energies I, Harvard University HUTP-81/A047 (1981). [11] See, for example, F. Wilczek and A. Zee, Phys. Rev. D25 (1982) 553. [12] L. O'Raifeartaigh, Nucl. Phys. B96 (1975) 331. [13] F. BucceUa, J.-P. Derendinger, C.A. Savoy and S. Ferrara, Symmetry breaking in supersymmetric GUTS, CERN TH. 3212 (1981), to be published in: Unification of the fundamental interactions II, eds. J. Elles and S. Ferrara. [14] B.A. Ovrut and J. Wess, Phys. Rev. D25 (1982) 409. [15] M.B. Einhorn and D.R.T. Jones, Nucl. Phys. B196 (1982) 475. [16] D.V. Nanopoulos and K. Tamvakis, Phys. Lett. 113B (1982) 151. [17] C.S. Aulakh and R.N. Mohapatra, Implications of supersymmetric SO(10) grand unification, CCNY HEP 82/4 (1982).