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Axions, R invariance and Witten-O'Raifeartaigh models
Volume 132B, number 1,2,3
PHYSICS LETTERS
24 November 1983
AXIONS, R INVAR1ANCE AND WITTEN-O'RAIFEARTAIGH MODELS R. HOLMAN 1 Department o f Physics, Johns Hopkins University, Baltimore, MD 21218, USA Received 29 July 1983
We show that the Witten-O'Raifeartaigh models of supersymmetry breaking with a spontaneously broken R symmetry, the R current QCD anomaly can only take on a certain values if one is to solve the strong CP problem via the Peccei-Quinn mechanism with a cosmologically acceptable axion. We then apply these criteria to specific models.
Witten-O'Raifeartaigh ( W - O ' R ) models [ 1], have of late become the prime candidates for constructing viable globally supersymmetric models. The most important reason for this is the ability of these models to explain the existence of such widely differing mass scales as the grand unification scale M G ~> 1015 GeV and the weak scale M w ~ 102 GeV. A variant of these models, known as the,geometric hierarchy model (GH) [2] generates both M w and M G from an intermediate scaleM I ~ 1012 GeV. A characteristic feature of these models is that they have an R symmetry [3]. This is a chiral U(1) symmetry which acts as follows on chiral superfields (~i} and on vector superfields V:
R~i(x, O) = exp( 2inic~)dPi(x, e - i ~ o ) ,
(1)
R V(x, O, O) = V(x, e - ia 0, e ia 0 ) .
(2)
In terms of the components (9i, ~i, Fi of ~i and An, k. D of V, we have:
Rfbi = exp(2inioO(9 i ,
Rff i = exp[i(2n i - 1)or] ~i, (3, 4)
RF i = exp [i(2n i - 2)t~] F i , RA n = A n ,
Rk =exp(ia)k,
(5) RD=,D.
(6, 7, 8)
The n t's are known as the R character of ~i. A term 1 Address after June 1983: Department of Physics, University of Florida, GainesviUe, FL 32611, USA.
in the superpotential is R invariant if and only if the sum of the R characters of the fields in that term is one.
If U(1)R has a non-zero gluon anomaly, NR, then it might be possible to use it, in conjunction with the Peccei-Quinn (PQ) mechanism [4] to solve the strong CP problem. What makes this idea even more intriguing is the fact that various groups [5] have found that the PQ symmetry must be spontaneously broken at a scale)Ca which cannot be greater than 1012 GeV, in order to keep the axion energy density from overclosing the universe. I f f a is equal to 1012 GeV, then axions could form the dark matter of the universe and could possibly be used for galaxy formation purposes [6]. But this critical value o f f a is]ust that value of M 1 necessary in GH models to generate M G and M w ! Thus we have the possibility of being able to use GH models and their R symmetries to solve the gauge hierarchy and strong CP problems and explain the existence of galaxies!. Unfortunately, this program cannot be realized as states. We shall show below that in W O'R and GH models with a spontaneously broken R symmetry and no other PQ symmetry, the R current must be anomaly free, which of course precludes its use as a PQ symmetry. However, we shall see that this condition can be of great use in model building just as the condition that the gauge symmetry currents be anomaly-free is of great use in unified model building. If the model has another PQ symmetry, then we shall show that the R anomaly must either be zero or + 1 and if it takes the values -+1, the PQ anomaly must be 141
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PHYSICS LETTERS
+-1 also (with no sign correlations between the two anomalies). We first consider the case of a pure W - O ' R or GH model with no other PQ symmetries imposed on it. The characteristic of W-O'R/GH models which allows them to generate the unification mass scale from a smaller mass scale is the fact that the superpotential contains a part of the generic form:
W(A, X, Y) : XX(A 2 - M 2) + 2g {A2}Y,
(9 3
where all superfield products are singlets under the gauge group G as is the superfield X. In order for the model to be R-invariant, X, Y must have R character one, while that of A must be zero. However, the scalar part of X, is the field whose VEV is driven to M G by quantum corrections to the effective potential. Hence R is always spontaneously broken at M G. This will then give rise to axions whose energy density will be larger than the critical density for a closed universe by several orders of magnitude unless the R current is anomaly-free. Of course in this case there is no axion or PQ mechanism so that the strong CP problem will remain unsolved. Next, we consider the case where, in addition to the R symmetry, we have another PQ symmetry with current JpQ and gluon anomaly N p Q ~: 1. Then the linear combination - N p Q J R + NRJpQ, denoted by J1, is anomaly free while the orthogonal combination NRJ R + NpQ JpQ, denoted by J2, has anomaly N 2 + NI~Q * z Let qi be t h e U ( 1 ) p Q charge of the chiral superfield dPi (the vector superfield has PQ charge 0). Then, under J2, the components of q~i transform as:
J2dPi= exp(2in~&)~bi, J2~i = exp[i(2n~ - N R ) & ] ffi, (lO, 1 1) J2Fi = expii(2n~ - 2NR)a ] Fi, (12) where we have defined (ni is the R character of ~i): 1 n i,' = N R n i + ~NpQq i .
(13)
We may identify J2 with a generalized R symmetry, denoted RNR which acts o n dPi and V as follows:
RN Rrbi(X, O) = exp( 2in ~oOebi(x, exp(-iNR&)0 ) , (14) ,1 Where the PQ and R charges are chosen so t h a t N R and NpQ are integers. , 2 If (N~ + N~Q) 1/2 is an integer, we may normalize J l and J2 by this factor and J2 will have anomaly (N~ + N~Q) I/2.
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RNR V(X, O, if)= V(x, e x p ( i N R a ) 0 , exp(iNRO0ff) , (15)
RNRA u=A u,
RNRk = e x p ( i a N R ) k ,
RNRD=D. (16, 17, 18)
Now, the RNR character of the field X in eq. (9) is
N R +1~NpQqx, where qx is the PQ charge of X, If
this is non-zero and N 2 + N2Q is also non-zero, then RNR will be spontaneously broken at M G and the resulting axions will again overclose the universe. Hence either N 2 + N2Q is zero, in which case we are again left with a strong CP problem since both N R and NpQ must vanish or N R + 5NpQqx must vanish. In order to be able to solve the strong CP problem, we try to see if the latter choice is consistent with other constraints on axion physics, such as the elimination of axion domain walls [7]. These come about because a discrete subgroup of U(I)pQ is left unbroken by the QCD gluon anomaly, but is spontaneously broken. The resulting domain walls will tend to seriously disrupt standard cosmology [8] so that they must be eliminated somehow. Inflationary universe scenarios [9] have been suggested as a way to eliminate the axion the domain wall problem but we do not consider these methods viable for the reason that it is not clear that one may reheat the universe to a high enough temperature to produce baryon asymmetry without restoring the U(1)pQ symmetry. Should this symmetry be restored, of course, we shall have the axion domain wall problem all over again. The most natural of ridding the universe of axion domain walls seems to be that of Lazarides and Shafti (LS) [ 10] and its variants [11] and that of Georgi and Wise [12]. The LS mechanism consists of identifying the residual discrete subgroup of U(1)pQ with a subgroup of the center ,3 of a continuous group (which may be the gauge group G or it may be a continuous global flavor group, say). The various domains then become continuously equivalent. When U(1)pQ is spontaneously broken, a system of walls bounded by strings forms which then decays away [13]. Implementation of the LS mechanism or any of its variants tells that N R, and hence qx must be zero. We show this for the original LS mechanism ,4. Since, for any superfield qs, its fermionic and bosonic components must be in the same , 3 The center of a group is the subgroup consisting of all elements which commute with the entire group.
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PHYSICS LETTERS
G representation, we must have (taking the vector superfield, say) N R ~ 0 (modN2R + N 2 Q ) .
(19)
However, since N R and NpQ are integers, this necessarily implies N R = 0. Since we also choose N R + ~ N p Q q x to be zero, we choose q x = 0 or else NpQ will be zero and the PQ mechanism will fail. In the method of Georgi and Wise, the residual discrete subgroup of U(1)pQ is constructed so as to be identical to Z f = ( 1)Y, where F is the fermion number operator, in its action on the fields of the theory. This Z f can never be spontaneously broken in a Lorentz invariant field theory so that no domain walls appear. In order to implement this m e t h o d with RNR as the PQ symmetry, we must demand the following: (a) All RNR characters must be integers (14). (b) N 2 + N~pQ = 2, which implies JNR[ = INpQ = l . 4 , since N R, NpQ are integers. This implies that [qev [ is two, since we have demanded that N R + 2NPQ q x vanish. We now examine the implications of this constraint on two models, the original GH model of Dimopoulos and Raby [2] and the SO(10) version of it devised by Kalara and Mohapatra [14]. We have previously [ 15] calculated the R-anomaly o f the D i m o p o u l o s - R a b y model under the assumption that all matter superfields had the same R character, and found: N R = 2(5
2ng).
(20)
24 November 1983
clearly unphysical. Hence the model must be modified if it is to remain viable. The superfields of the K a l a r a - M o h a p a t r a (KM) model are given in table 1 below together with their SO(10) representations, multiplicities, R characters and U(1)pQ charges (suitably generalized). The values of these quantities were found by examining the superpotential of the model (for more details see ref. [14] ). N R and NpQ may be c o m p u t e d to be N R = 2[3 + 2 n G ( n g - m + 1 ) - 2rig] ,
(21)
NpQ=2[qG(ng
(22)
m+l)+qx(12+m)],
where q x is the U(I)pQ charge of X, m is the number of 10's of Higgs and n G and q G are the R character and U(1)pQ charge of the superfield G. There are two cases to be considered. In the first case N R is zero as is q x " This then tells us that 2(ng - m + I) = (2ng - 3 ) / n G ,
(23)
which tells us that n G c a n n o t be an integer. In order to use the LS mechanism, we must choose INpQ I equal to four [center of SO(10) is Z4] and IqG I equal to two. Using eq. (22), we find ng-m+l
=+1,
(24)
or ng=
(25) m-1
Here, ng represents the number o f generations o f light fermions. Setting N R = 0 gives us n g = 5 / 2 which is
This then tells us that n G must be given by
#4 If (N~ +N~Q) 1/~ is an integer and we normalize the RNR current appropriately, we find that for the Georgi-Wise method to work, either INpQI = 2, N R = 0, or INRI = 2, NP0 = 0. However, from the previous case treated here, the latter condition cannot be true.
Hence we may determine all R characters in terms of ng, and we have correlated the number of 10's o f Higgses with the number o f generations. At the electroweak level, this gives us a correlation between the
In G I = 1 (2ng - 3 ) .
(26)
Table 1 Fields in the Kalara-Mohapatra model. Field
SO (10)
Multiplicity
R character
PQ charge
Y, A
54 16 16 10
1 1, ng 1 m, 1
1 (1 - nG)/2, nG/2 (1 - nG)/2 i -nG, nG
1 1
1 O, 1 - 2n G, 2n G - I
qx, 0 ( q g - qG)/2, qG/2 (qG - q x ) / 2 q x - qG, qG qx O, q x - 2qG, q x + 2qG
qJ, Xa ff Hi, G ~ X,Z,Z B, C, ~
1 1
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numbers of light Higgs doublets and the number of generations. We now consider the cases where Lqxi = 2 and INR L= [NpQ I = 1, subject to the constraint that N R + 1NpQqx vanish. In this case, we must also demand that the RNR characters of all fields be integers. This can be satisfied in this case by demanding that n G 1 + 5qG be an even integer. We find that neither n G nor qG can be integers and that ng + m must be an even integer. Hence, we see in this case that although we still get a correlation between the number of generations and the number of Higgs 10's, it is of a much weaker sort. In conclusions, we have found the conditions on W - O ' R / G H models under which they may be used to solve the strong CP problem in a cosmologically acceptable manner. We saw that if the natural R symmetry of these models is used as a PQ symmetry, then axions appear whose energy density will exceed the critical density by many orders of magnitude. If a second PQ symmetry is imposed, we saw that the elimination of axion walls demands that the QCD anomaly either vanish or be equal to +1, with the anomaly of the second PQ current also equal to -+1 in the latter case. We then applied these criteria to specific models and found some interesting constraints on them. A final point that should be dealt with is the relation of this work with that of ref. [15]. In that work, constraints on the R anomaly were found that allowed the R symmetry to eliminate axion domain walls via the LS and Georgi-Wise mechanisms. Those constraints are clearly not applicable to W - O ' R / G H models with N R = 0. However, there are other globally supersymmetric models where the constraints of ref. [15] can still be enforced such as the N g - O v r u t [16] supersymmetric subconstituent model.
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We would like to thank Professor C.W. Kim and Dr. D. Dominici for helpful conversations on this topic. This work was supported in part by the National Science Foundation.
References [1] E. Witten, Phys. Lett. 105B (1981) 267; O'Raifeartaigh, Nucl. Phys. B96 (1975) 331 [2] S. Dimopoulos and S. Raby, Los Alamos preprint AL-UR82-1282. [3] See e.g.: J.B. Wess and J. Bagger, Supersymmetry (Princeton, U.P., Princeton, 1983). [4] R.D. Peccei and H. Quinn, Phys. Rev. Lett. 38 (1977) 1440, S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. [5] J. Presskill, M.B. Wise and F. Wilczek, Phys. Lett. 120B (1983) 127; L.F. Abbott and P. Sikivie, Phys. Lett. 120B (1983) 133; M. Dine and W. Fischler, Phys. Lett. 120B (1983) 137. [6] J. Ipser and P. Sikivie, Phys. Rev. Lett. 50 (1983) 925; F.W. Stecker and Q. Shaft, Phys. Rev. Lett. 50 (1983) 928; M.S. Turner, F. Wilczek and A. Zee, Phys. Lett. 125B (1983) 35. [7] P. Sikivie, Phys. Rev. Lett. 48 (1982) 1156. [8] T.W.B. Kibble, J. Phys. A9 (1976) 1387; Ya.B. Zeldovich, I. Yu. Kobzarev and L.B. Okun, Soy. Phys. JETP 40 (1975) 1. [9] A.H. Guth, Phys. Rev. D23 (1981) 347; A. Linde, Phys. Lett. 108B (1982) 389; A. Albrecht and P. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. [ 10] G. Lazarides and Q. Shaft, Phys. Lett. 115B (1982) 21. [ 11] S. Dimopoulos, P.H. Frampton, H. Georgi and M.B. Wise, Phys. Lett. l17B (1982) 185. [12] H. Georgi and M.B. Wise, Phys. Lett. l16B (1982) 123. [13] T.W.B. Kibble, G. Lazarides and W. Shaffi, Phys. Rev. D26 (1982) 435; A. Vilenkin and A. Everett, Tufts University preprint TUTP-82-3 (1982). [ 14] S. Kalara and R.N. Mohapatra, University of Maryland preprint No. 128 (1983). [15] D. Dominici, R. Holman and C.W. Kim, Johns Hopkins preprint JHU-HET 8303 (1982). Y.J. Ng and B.A. Ovrut, Rockefeller University preprint RU 83/B/48 (1983).
PHYSICS LETTERS
24 November 1983
AXIONS, R INVAR1ANCE AND WITTEN-O'RAIFEARTAIGH MODELS R. HOLMAN 1 Department o f Physics, Johns Hopkins University, Baltimore, MD 21218, USA Received 29 July 1983
We show that the Witten-O'Raifeartaigh models of supersymmetry breaking with a spontaneously broken R symmetry, the R current QCD anomaly can only take on a certain values if one is to solve the strong CP problem via the Peccei-Quinn mechanism with a cosmologically acceptable axion. We then apply these criteria to specific models.
Witten-O'Raifeartaigh ( W - O ' R ) models [ 1], have of late become the prime candidates for constructing viable globally supersymmetric models. The most important reason for this is the ability of these models to explain the existence of such widely differing mass scales as the grand unification scale M G ~> 1015 GeV and the weak scale M w ~ 102 GeV. A variant of these models, known as the,geometric hierarchy model (GH) [2] generates both M w and M G from an intermediate scaleM I ~ 1012 GeV. A characteristic feature of these models is that they have an R symmetry [3]. This is a chiral U(1) symmetry which acts as follows on chiral superfields (~i} and on vector superfields V:
R~i(x, O) = exp( 2inic~)dPi(x, e - i ~ o ) ,
(1)
R V(x, O, O) = V(x, e - ia 0, e ia 0 ) .
(2)
In terms of the components (9i, ~i, Fi of ~i and An, k. D of V, we have:
Rfbi = exp(2inioO(9 i ,
Rff i = exp[i(2n i - 1)or] ~i, (3, 4)
RF i = exp [i(2n i - 2)t~] F i , RA n = A n ,
Rk =exp(ia)k,
(5) RD=,D.
(6, 7, 8)
The n t's are known as the R character of ~i. A term 1 Address after June 1983: Department of Physics, University of Florida, GainesviUe, FL 32611, USA.
in the superpotential is R invariant if and only if the sum of the R characters of the fields in that term is one.
If U(1)R has a non-zero gluon anomaly, NR, then it might be possible to use it, in conjunction with the Peccei-Quinn (PQ) mechanism [4] to solve the strong CP problem. What makes this idea even more intriguing is the fact that various groups [5] have found that the PQ symmetry must be spontaneously broken at a scale)Ca which cannot be greater than 1012 GeV, in order to keep the axion energy density from overclosing the universe. I f f a is equal to 1012 GeV, then axions could form the dark matter of the universe and could possibly be used for galaxy formation purposes [6]. But this critical value o f f a is]ust that value of M 1 necessary in GH models to generate M G and M w ! Thus we have the possibility of being able to use GH models and their R symmetries to solve the gauge hierarchy and strong CP problems and explain the existence of galaxies!. Unfortunately, this program cannot be realized as states. We shall show below that in W O'R and GH models with a spontaneously broken R symmetry and no other PQ symmetry, the R current must be anomaly free, which of course precludes its use as a PQ symmetry. However, we shall see that this condition can be of great use in model building just as the condition that the gauge symmetry currents be anomaly-free is of great use in unified model building. If the model has another PQ symmetry, then we shall show that the R anomaly must either be zero or + 1 and if it takes the values -+1, the PQ anomaly must be 141
Volume 132B, number 1,2,3
PHYSICS LETTERS
+-1 also (with no sign correlations between the two anomalies). We first consider the case of a pure W - O ' R or GH model with no other PQ symmetries imposed on it. The characteristic of W-O'R/GH models which allows them to generate the unification mass scale from a smaller mass scale is the fact that the superpotential contains a part of the generic form:
W(A, X, Y) : XX(A 2 - M 2) + 2g {A2}Y,
(9 3
where all superfield products are singlets under the gauge group G as is the superfield X. In order for the model to be R-invariant, X, Y must have R character one, while that of A must be zero. However, the scalar part of X, is the field whose VEV is driven to M G by quantum corrections to the effective potential. Hence R is always spontaneously broken at M G. This will then give rise to axions whose energy density will be larger than the critical density for a closed universe by several orders of magnitude unless the R current is anomaly-free. Of course in this case there is no axion or PQ mechanism so that the strong CP problem will remain unsolved. Next, we consider the case where, in addition to the R symmetry, we have another PQ symmetry with current JpQ and gluon anomaly N p Q ~: 1. Then the linear combination - N p Q J R + NRJpQ, denoted by J1, is anomaly free while the orthogonal combination NRJ R + NpQ JpQ, denoted by J2, has anomaly N 2 + NI~Q * z Let qi be t h e U ( 1 ) p Q charge of the chiral superfield dPi (the vector superfield has PQ charge 0). Then, under J2, the components of q~i transform as:
J2dPi= exp(2in~&)~bi, J2~i = exp[i(2n~ - N R ) & ] ffi, (lO, 1 1) J2Fi = expii(2n~ - 2NR)a ] Fi, (12) where we have defined (ni is the R character of ~i): 1 n i,' = N R n i + ~NpQq i .
(13)
We may identify J2 with a generalized R symmetry, denoted RNR which acts o n dPi and V as follows:
RN Rrbi(X, O) = exp( 2in ~oOebi(x, exp(-iNR&)0 ) , (14) ,1 Where the PQ and R charges are chosen so t h a t N R and NpQ are integers. , 2 If (N~ + N~Q) 1/2 is an integer, we may normalize J l and J2 by this factor and J2 will have anomaly (N~ + N~Q) I/2.
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RNR V(X, O, if)= V(x, e x p ( i N R a ) 0 , exp(iNRO0ff) , (15)
RNRA u=A u,
RNRk = e x p ( i a N R ) k ,
RNRD=D. (16, 17, 18)
Now, the RNR character of the field X in eq. (9) is
N R +1~NpQqx, where qx is the PQ charge of X, If
this is non-zero and N 2 + N2Q is also non-zero, then RNR will be spontaneously broken at M G and the resulting axions will again overclose the universe. Hence either N 2 + N2Q is zero, in which case we are again left with a strong CP problem since both N R and NpQ must vanish or N R + 5NpQqx must vanish. In order to be able to solve the strong CP problem, we try to see if the latter choice is consistent with other constraints on axion physics, such as the elimination of axion domain walls [7]. These come about because a discrete subgroup of U(I)pQ is left unbroken by the QCD gluon anomaly, but is spontaneously broken. The resulting domain walls will tend to seriously disrupt standard cosmology [8] so that they must be eliminated somehow. Inflationary universe scenarios [9] have been suggested as a way to eliminate the axion the domain wall problem but we do not consider these methods viable for the reason that it is not clear that one may reheat the universe to a high enough temperature to produce baryon asymmetry without restoring the U(1)pQ symmetry. Should this symmetry be restored, of course, we shall have the axion domain wall problem all over again. The most natural of ridding the universe of axion domain walls seems to be that of Lazarides and Shafti (LS) [ 10] and its variants [11] and that of Georgi and Wise [12]. The LS mechanism consists of identifying the residual discrete subgroup of U(1)pQ with a subgroup of the center ,3 of a continuous group (which may be the gauge group G or it may be a continuous global flavor group, say). The various domains then become continuously equivalent. When U(1)pQ is spontaneously broken, a system of walls bounded by strings forms which then decays away [13]. Implementation of the LS mechanism or any of its variants tells that N R, and hence qx must be zero. We show this for the original LS mechanism ,4. Since, for any superfield qs, its fermionic and bosonic components must be in the same , 3 The center of a group is the subgroup consisting of all elements which commute with the entire group.
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PHYSICS LETTERS
G representation, we must have (taking the vector superfield, say) N R ~ 0 (modN2R + N 2 Q ) .
(19)
However, since N R and NpQ are integers, this necessarily implies N R = 0. Since we also choose N R + ~ N p Q q x to be zero, we choose q x = 0 or else NpQ will be zero and the PQ mechanism will fail. In the method of Georgi and Wise, the residual discrete subgroup of U(1)pQ is constructed so as to be identical to Z f = ( 1)Y, where F is the fermion number operator, in its action on the fields of the theory. This Z f can never be spontaneously broken in a Lorentz invariant field theory so that no domain walls appear. In order to implement this m e t h o d with RNR as the PQ symmetry, we must demand the following: (a) All RNR characters must be integers (14). (b) N 2 + N~pQ = 2, which implies JNR[ = INpQ = l . 4 , since N R, NpQ are integers. This implies that [qev [ is two, since we have demanded that N R + 2NPQ q x vanish. We now examine the implications of this constraint on two models, the original GH model of Dimopoulos and Raby [2] and the SO(10) version of it devised by Kalara and Mohapatra [14]. We have previously [ 15] calculated the R-anomaly o f the D i m o p o u l o s - R a b y model under the assumption that all matter superfields had the same R character, and found: N R = 2(5
2ng).
(20)
24 November 1983
clearly unphysical. Hence the model must be modified if it is to remain viable. The superfields of the K a l a r a - M o h a p a t r a (KM) model are given in table 1 below together with their SO(10) representations, multiplicities, R characters and U(1)pQ charges (suitably generalized). The values of these quantities were found by examining the superpotential of the model (for more details see ref. [14] ). N R and NpQ may be c o m p u t e d to be N R = 2[3 + 2 n G ( n g - m + 1 ) - 2rig] ,
(21)
NpQ=2[qG(ng
(22)
m+l)+qx(12+m)],
where q x is the U(I)pQ charge of X, m is the number of 10's of Higgs and n G and q G are the R character and U(1)pQ charge of the superfield G. There are two cases to be considered. In the first case N R is zero as is q x " This then tells us that 2(ng - m + I) = (2ng - 3 ) / n G ,
(23)
which tells us that n G c a n n o t be an integer. In order to use the LS mechanism, we must choose INpQ I equal to four [center of SO(10) is Z4] and IqG I equal to two. Using eq. (22), we find ng-m+l
=+1,
(24)
or ng=
(25) m-1
Here, ng represents the number o f generations o f light fermions. Setting N R = 0 gives us n g = 5 / 2 which is
This then tells us that n G must be given by
#4 If (N~ +N~Q) 1/~ is an integer and we normalize the RNR current appropriately, we find that for the Georgi-Wise method to work, either INpQI = 2, N R = 0, or INRI = 2, NP0 = 0. However, from the previous case treated here, the latter condition cannot be true.
Hence we may determine all R characters in terms of ng, and we have correlated the number of 10's o f Higgses with the number o f generations. At the electroweak level, this gives us a correlation between the
In G I = 1 (2ng - 3 ) .
(26)
Table 1 Fields in the Kalara-Mohapatra model. Field
SO (10)
Multiplicity
R character
PQ charge
Y, A
54 16 16 10
1 1, ng 1 m, 1
1 (1 - nG)/2, nG/2 (1 - nG)/2 i -nG, nG
1 1
1 O, 1 - 2n G, 2n G - I
qx, 0 ( q g - qG)/2, qG/2 (qG - q x ) / 2 q x - qG, qG qx O, q x - 2qG, q x + 2qG
qJ, Xa ff Hi, G ~ X,Z,Z B, C, ~
1 1
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Volume 132B, number 1,2,3
PHYSICS LETTERS
numbers of light Higgs doublets and the number of generations. We now consider the cases where Lqxi = 2 and INR L= [NpQ I = 1, subject to the constraint that N R + 1NpQqx vanish. In this case, we must also demand that the RNR characters of all fields be integers. This can be satisfied in this case by demanding that n G 1 + 5qG be an even integer. We find that neither n G nor qG can be integers and that ng + m must be an even integer. Hence, we see in this case that although we still get a correlation between the number of generations and the number of Higgs 10's, it is of a much weaker sort. In conclusions, we have found the conditions on W - O ' R / G H models under which they may be used to solve the strong CP problem in a cosmologically acceptable manner. We saw that if the natural R symmetry of these models is used as a PQ symmetry, then axions appear whose energy density will exceed the critical density by many orders of magnitude. If a second PQ symmetry is imposed, we saw that the elimination of axion walls demands that the QCD anomaly either vanish or be equal to +1, with the anomaly of the second PQ current also equal to -+1 in the latter case. We then applied these criteria to specific models and found some interesting constraints on them. A final point that should be dealt with is the relation of this work with that of ref. [15]. In that work, constraints on the R anomaly were found that allowed the R symmetry to eliminate axion domain walls via the LS and Georgi-Wise mechanisms. Those constraints are clearly not applicable to W - O ' R / G H models with N R = 0. However, there are other globally supersymmetric models where the constraints of ref. [15] can still be enforced such as the N g - O v r u t [16] supersymmetric subconstituent model.
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We would like to thank Professor C.W. Kim and Dr. D. Dominici for helpful conversations on this topic. This work was supported in part by the National Science Foundation.
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