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Connection between generation number and anomaly cancellation in supersymmetric models
Volume 222, number 1
PHYSICS LETTERS B
11 May 1989
C O N N E C T I O N BETWEEN G E N E R A T I O N N U M B E R AND ANOMALY CANCELLATION IN S U P E R S Y M M E T R I C M O D E L S
X.-G. HE, G.C. JOSHI, B.H.J. McKELLAR and R.R. VOLKAS Research Centrefor High Energy Physics, School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia Received 14 February 1989
We construct supersyrnmetric theories in which the number of generations of quarks and leptons is related by gauge anomaly cancellation to the spectrum of Higgs fields. Models yielding at least three generations with the minimal Higgs spectrum assumed are discussed in detail. This mechanism requires an extension of the standard model gauge group such that an ordinary quarklepton generation is not anomaly-free. We discuss models o f S U ( 3 ) weak isospin, separate isospin groups for quarks and leptons, and chiral colour.
1. Introduction The puzzle of quark-lepton generations (or families) is one of the outstanding unsolved problems of particle physics. The standard model (SM) gives no indication of how many generations of quarks and leptons should exist in nature. To address this question one therefore has to look beyond standard physics. Indeed many solutions have been proposed, ranging from the topological properties of compactifled spacetimes in string theory to composite models . No preferred scenario has yet emerged. In this letter we propose a new approach to the generation problem within the context of supersymmettic extensions of the SM. We hypothesize that the number of generations can be constrained by cancelling certain gauge anomalies [2] induced by quarks and leptons with those induced by the supersymmetric partners of Higgs fields. In the SM, SU(3)c ® S U ( 2 ) L Q U ( 1 ) r gauge anomalies cancel within a generation [ 3 ]. To implement our idea we therefore have to extend the SM gauge group so that a quarklepton family is now anomalous. What makes our idea highly non-trivial is the demand that the exotic fermions needed to cancel gauge anomalies are precisely the supersymmetric partners of boson fields necessary for gauge symmetry breaking and mass generation.
Thus the anomaly cancellation is not ad hoc but a natural consequence of the symmetry structure of the theory, and the requirement of mass generation. We have constructed several examples of theories utilizing this idea. In the present work we will only display an interesting subset of these models. It should be noted, however, that although our anomaly cancellation criterion does not lead to a unique model, it does greatly restrict the structure of successful theories. There are several ways one can extend the SM gauge group so that a quark-lepton family is rendered anomalous. Our examples will embody two distinct methods for doing this. The first method takes one of the factor groups of the SM (or some standard extension like the left-right symmetric model) and embeds it in a larger simple group in such a way that anomaly freedom is lost (e.g. S U ( 2 ) L c S U ( 3 ) L ) . The second method embedded one of the factor groups in a semi-simple group which is a tensor product of identical units (e.g. S U ( 2 ) L c S U ( 2 ) ~ ® S U ( 2 ) 2 ) . Typically the original SM factor group is the diagonal subgroup of the semi-simple extension. In this way the contributions of various multiplets to gauge anomalies which previously cancelled with each other no longer do so.
~ See ref. [ 1 ] for a review.
86
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (Ngrth-Holland Physics Publishing Division)
Volume 222, number 1
PHYSICS LETTERS B
2. SU(3)¢@ SU(3)L® SU(3)R® U(1)e_L A peculiarity of SU (2) is that it has only real representations, so they are necessarily anomaly-free. If one enlarges the SU(2)L group of the SM to SU (3)L ;~2 the extended quark and lepton representations will no longer be anomaly-free. This simple observation is the basis for the first model we will present which satisfies the central criterion of this paper. One can construct a model based on SU(3)c ® S U ( 3 ) L ® U ( 1 ) r which displays quark-leptonhiggsino anomaly cancellation. However, an extra discrete symmetry must be imposed on the theory in order for it to yield at least three generations of quarks and leptons as the minimal solution. For this reason we will not discuss it in detail here, but instead present an analogous extension of the supersymmetric left-right symmetric model. Our theory has the gauge group S U ( 3 L ® S U ( 3 ) L ® S U ( 3 ) R ® U ( 1 ) B _ L . In usual left-right symmetric theories the left-handed quarks and leptons form doublets under SU (2) L and singlets under SU (2)R. The right-handed projections have the opposite assignment. We will do the obvious thing by introducing another species of quark and lepton so that their left-handed components now form SU (3) L triplets, with the right-handed components forming SU (3) R triplets. The spectrum of a generation is qL= (3,3,1)(~),
~L= (1,3,1) ( - - 1 ) ,
(qc)L = ( 3 , 1 , 3 ) ( - - ½),
(~C)L= (1,1,3) ( + 1).
(2.1)
These are, of course, actually multiplets of chiral superfields describing quarks, leptons and their super partners. Throughout the paper we will name quarklepton superfields after their fermionic components, and Higgs superfields after their bosonic components. Note that right-handed fermions are described by their left-handed antiparticles. Quarks and leptons gain mass from a number of Higgs supermultiplets transforming as HL = (1,3,3) (0),
(2.2)
where the quark-lepton-Higgs superpotential is W----2qqL(qC)LHL - [ - , ~ L (Qc) LHL,
(2.3)
22 For a recenl paper using SU (3) weak isospin see ref. [4 ].
11 May 1989
with an implicit sum over q, ~ and H generations (their number as yet undetermined). The Higgs superfield has the superpotential (2.4)
W=J.H HLHLHL,
which will serve to give mass to Higgs supermultiplets after symmetry breaking. We remark parenthetically that we will implicitly assume supersymmetry breaking occurs through effective soft breaking terms induced by supergravity coupling to a hidden sector. The details of this are unimportant for the present paper. HL contains the usual (2,2) (0) bidoublet under the S U ( 2 ) L ® S U ( 2 ) R ® U ( 1 )U-L subgroup as well as a singlet ( 1,1 ) (0). One can arrange for a relatively large vacuum expectation value (VEV) to develop for this singlet so that SU(3)L,R is broken to SU(2)L,R at a suitably high scale, while simultaneously giving the new charge -~ quarks and charge - ½leptons naturally larger masses. A VEV for the bidoublet part of HL then induces electroweak symmetry breakdown in the usual way. Again, by simple extension of the usual left-right case, VEVs for HL are unable to separate the scales of left and right breaking enough to yield acceptable phenomenology. It is usual to introduce additional Higgs fields which can split the right and left scales as well as giving right-handed neutrinos a relatively large Majorana mass. In the standard left-right model one postulates triplets under SU (2) L,R while our extension demands SU (3) L,R sextets: W=20L~L~L OL+2~(~c)L(~c)L(0C)L,
(2.5)
where 9L= ( 1 , 6 , 1 ) ( + 2 ) ,
(0C)L= (1,1,6) ( - - 2 ) .
(2.6)
In order for the O's to gain mass we will also introduce equal numbers of their mirrors, ~L= ( 1 , 6 , 1 ) ( - - 2 ) ,
(0c)L = (1,1,6) ( + 2 ) .
(2.7)
(An alternative scenario introduces mass terms through the VEVs of (1,6,6)(0) multiplets. We choose not to do this however. ) This is the complete list of the matter fields in the model. In order to constrain generation numbers we now impose the central idea of this paper: quark-lepton-higgsino anomaly cancellation. There is only one gauge anomaly which is not automatically zero: 87
Volume 222, number 1
[SU(3)L.R]3:
PHYSICS LETTERS B
4nG--3nH=0,
(2.8)
where no ( nn ) is the number of quark-lepton (Higgs) generations. (Note that anomalies due to ~)'s cancel with those due to 0's. ) The remarkable consequence of eq. (2.8) is that the simplest meaningful solution is nG =3,
nn =4.
(2.9)
The simplest model of this type necessarily has three quark-lepton generations (as well as four Higgs generations). Note that even though the number of ~, mirror pairs is unconstrained, the minimal number (i.e. one) of them suffices.
l 1 May 1989
W=2qqL(qC)LHIL +/].~J~L(Qc) LH2L +2~LQL~IL
+,t~ (~C)L(~c), ((~C),e.
(3.2)
The bidoublets H~L and H2L are anomaly-free and thus their number is unconstrained. One can similarly introduce any number of (1,2,1,2,1)(0)® (1,1,2,1,2)(0) and (1,2,1,1,2)(0)®(1,1,2,2,1)(0) multiplets. These can be used for symmetry breaking purposes, but not for mass generation. ~)lLand (9c)IL has the usual role of inducing Majorana neutrino masses and also separating the SU (2) ~ scale from the SU (2)~R scale. For phenomenological purposes one also has to add fields playing this latter role for SU(2)q.R. This is most simply done through ~)2L= (1,3,1,1,1)(--2),
3. SU(3)c ® SU(2)g ® SU(2)I~ ® SU(2)[ ® SU(2)~ ® U(1)n-L Our second model utilizes the second method for creating a gauge anomalous generation. One again begins with the supersymmetric left-right model, but this time one enlarges each SU(2) to SU(2) q ®SU(2)~, where quarks transform under the first factor and leptons under the second ~3. This spoils the usual cancellation of [ SU ( 2 ) LR ] 2U ( 1 ) e - L anomalies between quarks and leptons. As for the previous model one can actually construct a left-right asymmetric version which displays the required quark-lepton-higgsino cancellation. However, the minimal solution has nG= 1 and so minimality cannot be used to explain generation number. The quark-lepton-Higgs spectrum is qL= (3,2,1,1,1)(~),
(q~)L = (3,1,2,1,1) (-- ½),
~L= (1,1,1,2,1)(--1),
(~C)L= (1,1,1,1,2) ( + 1 ),
H,L = (1,2,2,1,1) (0),
H2L = (1,1,1,2,2) (0),
0,L= (1,1,1,3,1) ( + 2 ) , (0C)~L = (1,1,1,1,3) ( - - 2 ) ,
(3.1)
where the superpotential is ~3 For a similar treatment of weak hyperchargesee ref. [5 ]. 88
(0c)2L = ( 1,1,3,1,1 ) ( + 2 ) ,
(3.3)
although one could, in general, contemplate a different assignment. Manifest left-right symmetry of course demands equal multiplicities (n,) for OiL and (0C)iL. One also needs the anomaly-free multiplets (1,3,3,1,1) and (1,1,1,3,3) in order to give masses to the OiL. It is easy to see that there are only two non-trivial anomaly-cancellation equations:
[SU(2)[]2U(1)8_L:-no+2q(3)n~ =0,
(3.4a)
nG--2q(3)n2=O,
(3.4b)
[SU(2)~]2U(1)~ L:
where q(3) is an SU(2) group factor given by q(N) = ~ N ( N - 1 ) ( N + 1 ). (eq. (3.4b) justifies the choice made in eq. (3.3). The minimal solution is nG=8,
n~ = n 2 = l .
(3.5)
The result eq. (3.5) is interesting in that it easily accommodates the three known generations and has the minimal multiplicity of one for the anomalous higgsinos. This is in contrast to the previous model (see eq. (2.9)) which required a Higgs multiplicity of four. Eight supersymmetric generations is, however, a lot of particles. One of the consequences of this is lack of SU (3)c asymptotic freedom. This does not necessarily spoil agreement with experiment because it is natural to make the excess generations very heavy. As a result the effective theory obtained by integrating out the heavy states is asymptotically free in its region of validity and thus the correct low-energy running of c~s can be maintained.
Volume 222, number 1
PHYSICS LETTERS B
Perhaps an unsatisfactory feature of this model is that the number of bidoublets is arbitrary. The most attractive number to choose would obviously be one. This, however, implies the tree-level mass relations mJmd
= me~ms = m J m b . . . . .
(3.6)
Not only are these ratios phenomenologically incorrect, but they also imply a diagonal KM matrix. (The analogous relation for leptons is not valid because of the 0~L induced Majorana neutrino masses. ) To correct this situation one can introduce the diquark fields qL= (3,1,1,1,1)(--~),
(3.7)
These particles radiatively generate the off-diagonal KM matrix elements via the diagrams of fig. 1 [6 ]. Since 03L+ (0C)3L are a mirror pair their introduction does not affect the previous anomaly-cancellation analysis. In conclusion then, a minimal choice for the Higgs/ mass-generation sector of this model yields the interesting result of eight quark-lepton families.
4. SU(3)~L® SU(3)¢R® SU(2)L® SU(2)R® U(I)B_L Our last model extends the colour sector of the model rather than the electroweak sector, by going to "chiral colour". Several authors have addressed the problem of anomaly cancellation in such a theory [ 7 ]. Again note that we are utilizing a left-right symmetric electroweak gauge group [ 8 ], because we did not find any simple, interesting solutions in the left-right asymmetric case. The most interesting version of this model has radiatively induced off-diagonal KM matrix elements ~
~'--)~""""~
(~C) L
/
\
/
\
I
I >
I
\ <
q'L
via diquarks as in fig. 1. The quarks, leptons, Higgs fields and diquarks are qL = (3,1,2,1) (~), ~L = (1,1,2,1)(-- 1), H L = (3,3,2,2)(0),
(qc)L = (1,3,1,2) (-- ]), (~")L = (1,1,1,2) ( + 1 ), hE = ( 1,1,2,2 ) (0),
ZL = (3,3,1,1)(0), OL= (1,1,3,1)(2),
(OC)L= (1,1,1,3)(--2),
OL= (1,1,3,1 )(--2),
(O~)L= (1,1,1,3) ( + 2 ) ,
qL= (3,1,1,1 )(_~),
(qc)L=(1,3,1,1) (~), 2
(4.1)
(qC)L = (3,1,1,1,1) (~),
14~"=2qLqLqLqL +2R(qC)L(qC)L(q~)L-
~L
11 May 1989
x
>
(#L
I
<
(¢°L
Fig. 1. Feynman diagram generating radiative corrections to the tree-level quark mass matrix through the agency ofdiquark fields.
and the superpotential is W = , ~ q q L ( q c ) LHL +J.~J~L(J~C)LhL "k-~,qLqLqLl]L +2nR(qC)L(qC)L(qC)L +,~nOqL(qC) LZL + mh hLhL "[-2 HxHLHLZ L +2xXLZLZL
+2oL~LQLOL +20R(~)L(~)L(~c)L + rnoL 0LOL + m,R(QC) L(0~) L.
(4.2)
These interactions lead to masses for the particles in eq. (4.1). We will impose the minimality criterion that there be only one HE right from the outset. This choice implies relations (3.6) at tree-level and necessitates the introduction of radiative mass generation. We also include only one ZL" This multiplet serves two purposes: it separates the weak scale from the chiral colour breaking scale and it generates qL(qC)L mixing via the 2~ term in eq. (4.2). This mixing is necessary for the radiative mass mechanism (see fig. 1 ). The various gauge anomalies in this model are displayed in table 1. It is evident that the [SU(3)cL]2U(1)e L anomaly requires there to be equal numbers of quark-lepton and diquark generations. The [ SU (3) d ] 3 anomaly then sets this number at 5 because of the assumption that there is only one HE and ZL supermultiplet. This model succeeds in a highly non-trivial way. Not only does choosing a minimal number of HL'S and ZL'S imply enough quark-lepton generations to accommodate all the known particles, but the resulting spectrum does not contain any unwanted massless charged particles (see eq. (4.2)). The symmetry 89
Volume 222, number 1
PHYSICS LETTERS B
l 1 May 1989
Table 1 The contribution which each multiplet makes to each gauge anomaly. Anomalies related to those displayed by interchanging left and right are not listed. Every entry must be multiplied by the multiplicity of the relevant multiplet. Anomaly
[SU(3)ct.] 3 [SU(3)cL]2U(1)B_L [SU(2)L]2U(1)8_L [U(1 )6,_L] 3
Multiplet qL
(qc)L
~L
(~C)L
HL
~L
qL
(qC)L
2 2/3 1 6/27
0 0 0 -6/27
0 0 -1 -2
0 0 0 +2
-12 0 0 0
-3 0 0 0
1 --2/3 0 -8/9
0 0 0 8/9
breaking fields are intimately tied together with the quarks, leptons and diquarks in a very constrained system. The colour force is, however, again not asymptotically free.
G U T models obeying our criterion which did not necessarily yield any new low-energy physics.
5. Discussion and concluding remarks
X.-G.H. and R.R.V. would like to thank Robert Foot for useful discussions a n d comments. This work was supported by the Australian Research Council.
We have presented a n u m b e r of supersymmetric extensions of the SM which non-trivially connect generation n u m b e r with the Higgs sector by gauge a n o m a l y cancellation. (Note also that all the models above are free of Witten [ 9 ] a n d mixed gauge-gravitational anomalies [ 10 ]. ) The left-right symmetric model of S U ( 3 ) weak isospin produced three q u a r k - l e p t o n generations as the m i n i m a l sensible solution of the a n o m a l y cancellation equations. It also predicted a four-generation structure for the Higgs fields. The other two models featured a m i n i m a l Higgs sector with eight or five q u a r k - l e p t o n generations respectively. M i n i m a l i t y of the Higgs sector required radiative generation of off-diagonal K M matrix elements via diquark-induced effects. The model of section 3 required only one family of diquarks while the other required five. Both of these theories did not display colour asymptotic freedom, although c~swould fall until the heavy mass thresholds were crossed. These attractive models also yield a few intriguing open problems. In particular it would be very interesting to know if any of these theories could be grand unified, and whether intermediate scales of symmetry breaking were necessary to do this. Alternatively, it would be interesting to construct supersymmetric
90
Acknowledgement
References [ 1] R.D. Peccei, DESYpreprint DESY88-078;Invited talk Zuoz Spring School (Zuoz, Switzerland,April 1988). [2] S.L. Adler, Phys. Rev. 117 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cimento 51A ( 1969) 47; W.A. Bardeen, Phys. Rev. 184 (1969) 1848. [3 ] C. Bouchiat, J. Illiopoulos and Ph. Meyer, Phys. Lett. B 38 (1972) 519; D.J. Gross and R. Jackiw, Phys. Rev. D 6 (1972) 477. [4] R. Barbieri and R.N. Mohapatra, Phys. Lett. B 218 (1989) 225. [5] S. Rajpoot, Mod. Phys. Lett. A 1 (1986) 645. [6] X.-G. He, B.H.J. McKellar and P. Pallaghy, Phys. Lett. B 218 (1989) 216. [7] P.H. Frampton and S.L. Glashow,Phys. Lett. B 190 (1987) 192; Phys. Rev. Len. 58 (1987) 2168; S. Rajpoot, Phys. Rev. Len. 60 (1988) 2003; Phys. Rev. D 38 (1988) 417; R. Foot, H. Lew, R.R. Volkas and G.C. Joshi, Phys. Lett. B 217 (1989) 301; C.Q. Geng, TRIUMF preprint ( 1988). [8 ] S. Rajpoot, Phys. Len. B 208 ( 1988 ) 483. [9] E. Wiuen, Phys. Lett. B 117 (1982) 324. [ 10] R. Delbourgo and A. Salam, Phys. Len. B 40 ( 1972) 381; T. Eguchi and P. Frean, Phys. Rev. Lett. 37 (1976) 1251; L. Alvarez-Gaum6and E. Witten, Nucl. Phys. B 234 (1983) 269.
PHYSICS LETTERS B
11 May 1989
C O N N E C T I O N BETWEEN G E N E R A T I O N N U M B E R AND ANOMALY CANCELLATION IN S U P E R S Y M M E T R I C M O D E L S
X.-G. HE, G.C. JOSHI, B.H.J. McKELLAR and R.R. VOLKAS Research Centrefor High Energy Physics, School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia Received 14 February 1989
We construct supersyrnmetric theories in which the number of generations of quarks and leptons is related by gauge anomaly cancellation to the spectrum of Higgs fields. Models yielding at least three generations with the minimal Higgs spectrum assumed are discussed in detail. This mechanism requires an extension of the standard model gauge group such that an ordinary quarklepton generation is not anomaly-free. We discuss models o f S U ( 3 ) weak isospin, separate isospin groups for quarks and leptons, and chiral colour.
1. Introduction The puzzle of quark-lepton generations (or families) is one of the outstanding unsolved problems of particle physics. The standard model (SM) gives no indication of how many generations of quarks and leptons should exist in nature. To address this question one therefore has to look beyond standard physics. Indeed many solutions have been proposed, ranging from the topological properties of compactifled spacetimes in string theory to composite models . No preferred scenario has yet emerged. In this letter we propose a new approach to the generation problem within the context of supersymmettic extensions of the SM. We hypothesize that the number of generations can be constrained by cancelling certain gauge anomalies [2] induced by quarks and leptons with those induced by the supersymmetric partners of Higgs fields. In the SM, SU(3)c ® S U ( 2 ) L Q U ( 1 ) r gauge anomalies cancel within a generation [ 3 ]. To implement our idea we therefore have to extend the SM gauge group so that a quarklepton family is now anomalous. What makes our idea highly non-trivial is the demand that the exotic fermions needed to cancel gauge anomalies are precisely the supersymmetric partners of boson fields necessary for gauge symmetry breaking and mass generation.
Thus the anomaly cancellation is not ad hoc but a natural consequence of the symmetry structure of the theory, and the requirement of mass generation. We have constructed several examples of theories utilizing this idea. In the present work we will only display an interesting subset of these models. It should be noted, however, that although our anomaly cancellation criterion does not lead to a unique model, it does greatly restrict the structure of successful theories. There are several ways one can extend the SM gauge group so that a quark-lepton family is rendered anomalous. Our examples will embody two distinct methods for doing this. The first method takes one of the factor groups of the SM (or some standard extension like the left-right symmetric model) and embeds it in a larger simple group in such a way that anomaly freedom is lost (e.g. S U ( 2 ) L c S U ( 3 ) L ) . The second method embedded one of the factor groups in a semi-simple group which is a tensor product of identical units (e.g. S U ( 2 ) L c S U ( 2 ) ~ ® S U ( 2 ) 2 ) . Typically the original SM factor group is the diagonal subgroup of the semi-simple extension. In this way the contributions of various multiplets to gauge anomalies which previously cancelled with each other no longer do so.
~ See ref. [ 1 ] for a review.
86
0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (Ngrth-Holland Physics Publishing Division)
Volume 222, number 1
PHYSICS LETTERS B
2. SU(3)¢@ SU(3)L® SU(3)R® U(1)e_L A peculiarity of SU (2) is that it has only real representations, so they are necessarily anomaly-free. If one enlarges the SU(2)L group of the SM to SU (3)L ;~2 the extended quark and lepton representations will no longer be anomaly-free. This simple observation is the basis for the first model we will present which satisfies the central criterion of this paper. One can construct a model based on SU(3)c ® S U ( 3 ) L ® U ( 1 ) r which displays quark-leptonhiggsino anomaly cancellation. However, an extra discrete symmetry must be imposed on the theory in order for it to yield at least three generations of quarks and leptons as the minimal solution. For this reason we will not discuss it in detail here, but instead present an analogous extension of the supersymmetric left-right symmetric model. Our theory has the gauge group S U ( 3 L ® S U ( 3 ) L ® S U ( 3 ) R ® U ( 1 ) B _ L . In usual left-right symmetric theories the left-handed quarks and leptons form doublets under SU (2) L and singlets under SU (2)R. The right-handed projections have the opposite assignment. We will do the obvious thing by introducing another species of quark and lepton so that their left-handed components now form SU (3) L triplets, with the right-handed components forming SU (3) R triplets. The spectrum of a generation is qL= (3,3,1)(~),
~L= (1,3,1) ( - - 1 ) ,
(qc)L = ( 3 , 1 , 3 ) ( - - ½),
(~C)L= (1,1,3) ( + 1).
(2.1)
These are, of course, actually multiplets of chiral superfields describing quarks, leptons and their super partners. Throughout the paper we will name quarklepton superfields after their fermionic components, and Higgs superfields after their bosonic components. Note that right-handed fermions are described by their left-handed antiparticles. Quarks and leptons gain mass from a number of Higgs supermultiplets transforming as HL = (1,3,3) (0),
(2.2)
where the quark-lepton-Higgs superpotential is W----2qqL(qC)LHL - [ - , ~ L (Qc) LHL,
(2.3)
22 For a recenl paper using SU (3) weak isospin see ref. [4 ].
11 May 1989
with an implicit sum over q, ~ and H generations (their number as yet undetermined). The Higgs superfield has the superpotential (2.4)
W=J.H HLHLHL,
which will serve to give mass to Higgs supermultiplets after symmetry breaking. We remark parenthetically that we will implicitly assume supersymmetry breaking occurs through effective soft breaking terms induced by supergravity coupling to a hidden sector. The details of this are unimportant for the present paper. HL contains the usual (2,2) (0) bidoublet under the S U ( 2 ) L ® S U ( 2 ) R ® U ( 1 )U-L subgroup as well as a singlet ( 1,1 ) (0). One can arrange for a relatively large vacuum expectation value (VEV) to develop for this singlet so that SU(3)L,R is broken to SU(2)L,R at a suitably high scale, while simultaneously giving the new charge -~ quarks and charge - ½leptons naturally larger masses. A VEV for the bidoublet part of HL then induces electroweak symmetry breakdown in the usual way. Again, by simple extension of the usual left-right case, VEVs for HL are unable to separate the scales of left and right breaking enough to yield acceptable phenomenology. It is usual to introduce additional Higgs fields which can split the right and left scales as well as giving right-handed neutrinos a relatively large Majorana mass. In the standard left-right model one postulates triplets under SU (2) L,R while our extension demands SU (3) L,R sextets: W=20L~L~L OL+2~(~c)L(~c)L(0C)L,
(2.5)
where 9L= ( 1 , 6 , 1 ) ( + 2 ) ,
(0C)L= (1,1,6) ( - - 2 ) .
(2.6)
In order for the O's to gain mass we will also introduce equal numbers of their mirrors, ~L= ( 1 , 6 , 1 ) ( - - 2 ) ,
(0c)L = (1,1,6) ( + 2 ) .
(2.7)
(An alternative scenario introduces mass terms through the VEVs of (1,6,6)(0) multiplets. We choose not to do this however. ) This is the complete list of the matter fields in the model. In order to constrain generation numbers we now impose the central idea of this paper: quark-lepton-higgsino anomaly cancellation. There is only one gauge anomaly which is not automatically zero: 87
Volume 222, number 1
[SU(3)L.R]3:
PHYSICS LETTERS B
4nG--3nH=0,
(2.8)
where no ( nn ) is the number of quark-lepton (Higgs) generations. (Note that anomalies due to ~)'s cancel with those due to 0's. ) The remarkable consequence of eq. (2.8) is that the simplest meaningful solution is nG =3,
nn =4.
(2.9)
The simplest model of this type necessarily has three quark-lepton generations (as well as four Higgs generations). Note that even though the number of ~, mirror pairs is unconstrained, the minimal number (i.e. one) of them suffices.
l 1 May 1989
W=2qqL(qC)LHIL +/].~J~L(Qc) LH2L +2~LQL~IL
+,t~ (~C)L(~c), ((~C),e.
(3.2)
The bidoublets H~L and H2L are anomaly-free and thus their number is unconstrained. One can similarly introduce any number of (1,2,1,2,1)(0)® (1,1,2,1,2)(0) and (1,2,1,1,2)(0)®(1,1,2,2,1)(0) multiplets. These can be used for symmetry breaking purposes, but not for mass generation. ~)lLand (9c)IL has the usual role of inducing Majorana neutrino masses and also separating the SU (2) ~ scale from the SU (2)~R scale. For phenomenological purposes one also has to add fields playing this latter role for SU(2)q.R. This is most simply done through ~)2L= (1,3,1,1,1)(--2),
3. SU(3)c ® SU(2)g ® SU(2)I~ ® SU(2)[ ® SU(2)~ ® U(1)n-L Our second model utilizes the second method for creating a gauge anomalous generation. One again begins with the supersymmetric left-right model, but this time one enlarges each SU(2) to SU(2) q ®SU(2)~, where quarks transform under the first factor and leptons under the second ~3. This spoils the usual cancellation of [ SU ( 2 ) LR ] 2U ( 1 ) e - L anomalies between quarks and leptons. As for the previous model one can actually construct a left-right asymmetric version which displays the required quark-lepton-higgsino cancellation. However, the minimal solution has nG= 1 and so minimality cannot be used to explain generation number. The quark-lepton-Higgs spectrum is qL= (3,2,1,1,1)(~),
(q~)L = (3,1,2,1,1) (-- ½),
~L= (1,1,1,2,1)(--1),
(~C)L= (1,1,1,1,2) ( + 1 ),
H,L = (1,2,2,1,1) (0),
H2L = (1,1,1,2,2) (0),
0,L= (1,1,1,3,1) ( + 2 ) , (0C)~L = (1,1,1,1,3) ( - - 2 ) ,
(3.1)
where the superpotential is ~3 For a similar treatment of weak hyperchargesee ref. [5 ]. 88
(0c)2L = ( 1,1,3,1,1 ) ( + 2 ) ,
(3.3)
although one could, in general, contemplate a different assignment. Manifest left-right symmetry of course demands equal multiplicities (n,) for OiL and (0C)iL. One also needs the anomaly-free multiplets (1,3,3,1,1) and (1,1,1,3,3) in order to give masses to the OiL. It is easy to see that there are only two non-trivial anomaly-cancellation equations:
[SU(2)[]2U(1)8_L:-no+2q(3)n~ =0,
(3.4a)
nG--2q(3)n2=O,
(3.4b)
[SU(2)~]2U(1)~ L:
where q(3) is an SU(2) group factor given by q(N) = ~ N ( N - 1 ) ( N + 1 ). (eq. (3.4b) justifies the choice made in eq. (3.3). The minimal solution is nG=8,
n~ = n 2 = l .
(3.5)
The result eq. (3.5) is interesting in that it easily accommodates the three known generations and has the minimal multiplicity of one for the anomalous higgsinos. This is in contrast to the previous model (see eq. (2.9)) which required a Higgs multiplicity of four. Eight supersymmetric generations is, however, a lot of particles. One of the consequences of this is lack of SU (3)c asymptotic freedom. This does not necessarily spoil agreement with experiment because it is natural to make the excess generations very heavy. As a result the effective theory obtained by integrating out the heavy states is asymptotically free in its region of validity and thus the correct low-energy running of c~s can be maintained.
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Perhaps an unsatisfactory feature of this model is that the number of bidoublets is arbitrary. The most attractive number to choose would obviously be one. This, however, implies the tree-level mass relations mJmd
= me~ms = m J m b . . . . .
(3.6)
Not only are these ratios phenomenologically incorrect, but they also imply a diagonal KM matrix. (The analogous relation for leptons is not valid because of the 0~L induced Majorana neutrino masses. ) To correct this situation one can introduce the diquark fields qL= (3,1,1,1,1)(--~),
(3.7)
These particles radiatively generate the off-diagonal KM matrix elements via the diagrams of fig. 1 [6 ]. Since 03L+ (0C)3L are a mirror pair their introduction does not affect the previous anomaly-cancellation analysis. In conclusion then, a minimal choice for the Higgs/ mass-generation sector of this model yields the interesting result of eight quark-lepton families.
4. SU(3)~L® SU(3)¢R® SU(2)L® SU(2)R® U(I)B_L Our last model extends the colour sector of the model rather than the electroweak sector, by going to "chiral colour". Several authors have addressed the problem of anomaly cancellation in such a theory [ 7 ]. Again note that we are utilizing a left-right symmetric electroweak gauge group [ 8 ], because we did not find any simple, interesting solutions in the left-right asymmetric case. The most interesting version of this model has radiatively induced off-diagonal KM matrix elements ~
~'--)~""""~
(~C) L
/
\
/
\
I
I >
I
\ <
q'L
via diquarks as in fig. 1. The quarks, leptons, Higgs fields and diquarks are qL = (3,1,2,1) (~), ~L = (1,1,2,1)(-- 1), H L = (3,3,2,2)(0),
(qc)L = (1,3,1,2) (-- ]), (~")L = (1,1,1,2) ( + 1 ), hE = ( 1,1,2,2 ) (0),
ZL = (3,3,1,1)(0), OL= (1,1,3,1)(2),
(OC)L= (1,1,1,3)(--2),
OL= (1,1,3,1 )(--2),
(O~)L= (1,1,1,3) ( + 2 ) ,
qL= (3,1,1,1 )(_~),
(qc)L=(1,3,1,1) (~), 2
(4.1)
(qC)L = (3,1,1,1,1) (~),
14~"=2qLqLqLqL +2R(qC)L(qC)L(q~)L-
~L
11 May 1989
x
>
(#L
I
<
(¢°L
Fig. 1. Feynman diagram generating radiative corrections to the tree-level quark mass matrix through the agency ofdiquark fields.
and the superpotential is W = , ~ q q L ( q c ) LHL +J.~J~L(J~C)LhL "k-~,qLqLqLl]L +2nR(qC)L(qC)L(qC)L +,~nOqL(qC) LZL + mh hLhL "[-2 HxHLHLZ L +2xXLZLZL
+2oL~LQLOL +20R(~)L(~)L(~c)L + rnoL 0LOL + m,R(QC) L(0~) L.
(4.2)
These interactions lead to masses for the particles in eq. (4.1). We will impose the minimality criterion that there be only one HE right from the outset. This choice implies relations (3.6) at tree-level and necessitates the introduction of radiative mass generation. We also include only one ZL" This multiplet serves two purposes: it separates the weak scale from the chiral colour breaking scale and it generates qL(qC)L mixing via the 2~ term in eq. (4.2). This mixing is necessary for the radiative mass mechanism (see fig. 1 ). The various gauge anomalies in this model are displayed in table 1. It is evident that the [SU(3)cL]2U(1)e L anomaly requires there to be equal numbers of quark-lepton and diquark generations. The [ SU (3) d ] 3 anomaly then sets this number at 5 because of the assumption that there is only one HE and ZL supermultiplet. This model succeeds in a highly non-trivial way. Not only does choosing a minimal number of HL'S and ZL'S imply enough quark-lepton generations to accommodate all the known particles, but the resulting spectrum does not contain any unwanted massless charged particles (see eq. (4.2)). The symmetry 89
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l 1 May 1989
Table 1 The contribution which each multiplet makes to each gauge anomaly. Anomalies related to those displayed by interchanging left and right are not listed. Every entry must be multiplied by the multiplicity of the relevant multiplet. Anomaly
[SU(3)ct.] 3 [SU(3)cL]2U(1)B_L [SU(2)L]2U(1)8_L [U(1 )6,_L] 3
Multiplet qL
(qc)L
~L
(~C)L
HL
~L
qL
(qC)L
2 2/3 1 6/27
0 0 0 -6/27
0 0 -1 -2
0 0 0 +2
-12 0 0 0
-3 0 0 0
1 --2/3 0 -8/9
0 0 0 8/9
breaking fields are intimately tied together with the quarks, leptons and diquarks in a very constrained system. The colour force is, however, again not asymptotically free.
G U T models obeying our criterion which did not necessarily yield any new low-energy physics.
5. Discussion and concluding remarks
X.-G.H. and R.R.V. would like to thank Robert Foot for useful discussions a n d comments. This work was supported by the Australian Research Council.
We have presented a n u m b e r of supersymmetric extensions of the SM which non-trivially connect generation n u m b e r with the Higgs sector by gauge a n o m a l y cancellation. (Note also that all the models above are free of Witten [ 9 ] a n d mixed gauge-gravitational anomalies [ 10 ]. ) The left-right symmetric model of S U ( 3 ) weak isospin produced three q u a r k - l e p t o n generations as the m i n i m a l sensible solution of the a n o m a l y cancellation equations. It also predicted a four-generation structure for the Higgs fields. The other two models featured a m i n i m a l Higgs sector with eight or five q u a r k - l e p t o n generations respectively. M i n i m a l i t y of the Higgs sector required radiative generation of off-diagonal K M matrix elements via diquark-induced effects. The model of section 3 required only one family of diquarks while the other required five. Both of these theories did not display colour asymptotic freedom, although c~swould fall until the heavy mass thresholds were crossed. These attractive models also yield a few intriguing open problems. In particular it would be very interesting to know if any of these theories could be grand unified, and whether intermediate scales of symmetry breaking were necessary to do this. Alternatively, it would be interesting to construct supersymmetric
90
Acknowledgement
References [ 1] R.D. Peccei, DESYpreprint DESY88-078;Invited talk Zuoz Spring School (Zuoz, Switzerland,April 1988). [2] S.L. Adler, Phys. Rev. 117 (1969) 2426; J.S. Bell and R. Jackiw, Nuovo Cimento 51A ( 1969) 47; W.A. Bardeen, Phys. Rev. 184 (1969) 1848. [3 ] C. Bouchiat, J. Illiopoulos and Ph. Meyer, Phys. Lett. B 38 (1972) 519; D.J. Gross and R. Jackiw, Phys. Rev. D 6 (1972) 477. [4] R. Barbieri and R.N. Mohapatra, Phys. Lett. B 218 (1989) 225. [5] S. Rajpoot, Mod. Phys. Lett. A 1 (1986) 645. [6] X.-G. He, B.H.J. McKellar and P. Pallaghy, Phys. Lett. B 218 (1989) 216. [7] P.H. Frampton and S.L. Glashow,Phys. Lett. B 190 (1987) 192; Phys. Rev. Len. 58 (1987) 2168; S. Rajpoot, Phys. Rev. Len. 60 (1988) 2003; Phys. Rev. D 38 (1988) 417; R. Foot, H. Lew, R.R. Volkas and G.C. Joshi, Phys. Lett. B 217 (1989) 301; C.Q. Geng, TRIUMF preprint ( 1988). [8 ] S. Rajpoot, Phys. Len. B 208 ( 1988 ) 483. [9] E. Wiuen, Phys. Lett. B 117 (1982) 324. [ 10] R. Delbourgo and A. Salam, Phys. Len. B 40 ( 1972) 381; T. Eguchi and P. Frean, Phys. Rev. Lett. 37 (1976) 1251; L. Alvarez-Gaum6and E. Witten, Nucl. Phys. B 234 (1983) 269.