Talk presented at the "Beyond the " Conf., TRI-PP-S8-104 Iowa State University, Ames, Iowa, Nov. 18-20 Doc 1988

FROM THE UNIQUENESS OF THE STANDARD MODEL AND GOING BEYOND

C.Q. Geng TRIUMF, Theory Group, 4004 Wcsbrook Mall Vancouver, B.C., Canada V6T 2A3

Abstract We argue that the resolution of the question of the uniqueness of quark and lepton representations and their charge quantizations in the standard model based on the cancellations of the three known chiral anomalies in four dimensions may hint at new physics beyond the standard model. However, it does not help with the generation problem.

Although the standard model1 of SU(3)c x SU (2)i x U(1)Y has been remark- ably successful experimentally, there are many reasons for going beyond it and many models are being pursued. In this report we will present a new theoretical argument for new physics on the basis of freedom from the chiral gauge anomalies. It is well-known that the -free conditions arising from the theoretical requirements of renormalizability and self-consistency are the most elegant tool to test the . Three anomalies thus far have been identified for chiral gauge theories in four dimensions: (1) The triangular (perturbative) cliiral ,2 which must be cancelled to avoid the breakdown of gauge invariance and renormalizability of the theory, we call this the triangular anomaly. (2) The global (non-perturbative) SU(2) chiral gauge anomaly,3 which must be absent in order to define the integral in a gauge invariant way; we call this the global anomaly. This anomaly was first pointed out by Witten,3 and is known as the Witten SU(2) anomaly. He showed in 1982 that any SU(2) gauge theory with an odd number of left-handed fermion (Weyl) doublets is mathematically inconsistent. (3) The mixed (perturbative) chiral gauge-,4'5 which must be cancelled in order to ensure general covariance of the theory; we call this the mixed anomaly. This anomaly was first discussed by Delburgo and Salam4 in 1972 and its consequences studied by Alvarez-Gaume and Witten5 in 1983, who concluded that a necessary condition for consistency of the theory coupled to gravity is that the sum of the U(l) charges of the left-handed vanishes, i.e., TrQ — 0. The triangular anomaly-free of the standard model was first noted6 in 1972 for each quark-lepton family. It was clear that with only the triangular anomaly- free condition7 one could not explain the empirically determined quark-lepton representations and tiieir quantized hypercharges. We now summarize the recent study8 on the question of the uniqueness of quarks and leptons in the standard model by insisting on all three anomaly-free conditions. We begin by allowing an arbitrary number of (left-handed) Weyl representations under the standard group, i.e., SU(3)xSU(2)xU(l) 3 2 3 1 3 1

3 2 Qit i = 1 2 1 1 where the integers j, k,l, m,n and p and the 1^(1) charges are all arbitrary. The triangular anomaly-free condition then leads to the following equations:

i k I m :0, (2a) 1=1

\SU(2)]2U(1) : (2b) i=l i=l 1=1 .= 1 (2c) 1=1

1=1 1=1 1=1 (2d) The global SU(2) anomaly-free condition is 3j + 3m + n = N , (3) where N is an even integer. Finally the mixed anomaly-free condition is

£ £$ 2£ft + f:5i=0. (4)

The requirements of minimality and the three anomaly-free conditions [Eqs. (2)-(4)] lead to the values: j = 1, Jfc = 0,/ = 2,m = 0,n = l,p = 1, and, in an obvious notation, to the four relations (note that there must be lepton as well as quark representations): 2 Q, = -iL.ft = *f,Q2 = - f,-qi = -2qi . (5) It is seen from Eq. (5) that all the U(l) charges are uniquely determined in terms of a single 17(1) charge, qi; choosing the normalization qi — —1, the resulting Weyl representations of SU(3) and SU(2) and their 17(1) charges are those in the standard model (cf. Table 1) and the electric chaiges of quarks and leptons are given in Table 1 after the electroweak symmetry spontaneously breaks down to U(1)EM by the Higgs mechanism. Table 1. The quantum numbers of quark and lepton representations under

x SU(2)L x U(l)Y and SU(3)C x

Particles SU(3)C x SU(2)L x V(\)Y -> SU(3)c x U(1)EM

(i = 1,2,3) 1 2/3\ 3 2 3 c -1/3) o: V3 3 1 4 3 -2/3 ~3 2 3 1 3 1/3 3 o; 1 2 -1 (1 -?) 1 1 2 1 1 We thus find that the requirements of minimality and freedom from all three cliiral gauge anomalies lead to a unique set of Weyl representations (and their U(l)y charges) of the standard group that correspond to the observed quarks and leptons of one family. Furthermore, the U(1)Y charges of these quarks and leptons are quantized and correctly determined by adding the mixed anomaly-free condition and thus a long-standing puzzle of the electric charge quantization of quark and lepton can be solved within the content of the standard model. In spite of the success of the standard model, it is still a mystery why the three anomaly cancellations, especially the global and the mixed ones, should be satisfied. Naturally one hopes that new physics beyond the standard model can provide us an explanation to this question. For this purpose, we now review the three chiral anomalies for the simple Lie groups. 1. The Triangular Anomaly. It has been shown9 that the simple Lie groups:

SU(2), SO(2k + l)(Jk > 2), SO(4k)(k > 2), SO{ik + 2)(fc > 2), Sp{2k), G2, Ft, E6, ET, and E& are safe groupJ. The only simple groups with possible triangular anomaly are the unitary groups SU(n)(n > 3). Therefore, if we start with the groups which do not contain SU(n)(n > 3) group, the theory will be free of triangular anomaly. 2. The Global Anomaly. We classify the simple Lie groups G into the following two classes. (I) Sp(2Jfc)(Sp(2) ~ SU(2)). These groups10 have the property of

n<(Sp(2k)) = Z2 , (6) where II4 is the fourth homotopy group and Z2 is the two-valued discrete group (like parity). According to Witten,3 the group G^ = Sp(2k) has global anomaly if the number of fermion zero modes (for SU(9.) group, it is equal to the number of fermion doublets) is odd. (II) SU(n)(n > 3),SO(2Jfc +l)(Jfc > 2),5O(4Jfc)(Jb > 2),SO(4Jfc + 2)(fc > (//) 2),G2,F4,EB,E7, and E6. These groups (G ) have no global anomaly since their fourth homotopy groups are trivial,3-10 i.e.,

( n4(G ")) = 0. (7) However, the interesting question11 arises as to how one can know at the level of G(/;) whether such a theory is global anomaly-free when G(//) breaks down to groups which contain G'''. Recently, we present a sufficient condition11 that for any simple group G, containing 5p(2fc) as a subgroup, and for which n4(G) = 0, the vanisliing of the triangular perturbative anomaly for Weyl representations of G will guarantee the absence of the global non-perturbative Sp{2k) anomaly. 3. The Mixed Anomaly. This anomaly is non-trivial only for the theory in which there is U(l) symmetry with non-zero total charges.5 Obviously, all the simple Lie groups (G(/)l(//)) are safe groups. Furthermore, when these groups break down to groups which contain //(I), e.g.,

(8) unlike the previous case, there is no mixed anomaly since the Z7(l) operators are the generators of G and must be traceless. From the above studies we see that the three anomaly-free conditions in the standard model may be automatically satisfied if it comes from a large group, especially, a grand unification group. For example, with the E6 grand unification theory, the triangular, the global, and the mixed anomalies are trivial at the level of EG which guarantees their freedom at the standard group level. We thus con- clude that the resolution of the question of the uniqueness of the massless fermion representations and J7(l)v charges for the standard group - when viewed from the standpoint of the perturbative triangular and mixed chiral gauge-gravitational anomalies and the absence of the non-perturbative global 51/(2) chiral gauge anomaly in four dimensions - argues strongly for some new physics beyond the standard model. However, the result may receive criticism, as mentioned by Zee,12 that it is an argument that the world is as it is because it is the way it is. Finally we remark that the imposition of all three anomaly-free conditions for the standard model does not shed any immediate light on the "generation problem". Moreover, if one enlarges the standard group to include an 51^(2) or 5(7(3) group, one can show that the theories are precisely the one family fermion 8 x structure of the left-right symmetric model SU(3)C SU(2)L x SU{2)R x U(l) 13 and the chiral-color model, SU{Z)CL X SU(Z)CR X SU{2)L x U{l)Y, respectively, instead of having a family group. Clearly, some new ideas14 (possibly some as-yet- unidentified anomalies in four dimensions) are needed to constrain on the number of families which would be a key to the new physics. Acknowledgements

I would like to thank Professors R.E. Marshak and J.N. Ng for encouragement and discussions and Professor B.L. Young for inviting me to attend the confer- ence. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.

References 1 Glashow, S. L., Nucl. Phys. 22, 579 (1961); Weinberg, S., Phys. Rev. Lett. 19, 1264 (1964); Salam, A., " Elementary Particle Theory", ed. Svarthom, N. (Almquist and Wilsell, Stockholm, 1968). 2 Adler, S. Phys. Rev. 117, 2426 (1969); Bell, J and Jackiw, R., Nuovo Ci- mento 51 A, 47 (1969); Bardeen, W., Phys. Rev. 184,1848 (1969). 3 Witten, E., Phys. Lett. BUT, 324 (1982). 4 Delbourgo, R. and Salam, A., Phys. Lett. B40, 381 (1972); see also Eguchi, T. and Frend, T., Phys. Rev. Lett. 37, 1251 (1976). 5 Alvarez-Guame, L. and Witten, E., Nucl. Phys. B234, 269 (1983). 6 Bouchiat, C, Illiopoulos, J. and Meyer, Ph., Phys. Lett. B73, 519 (1972); Gross, D. J. and Jackiw, R., Phys. Rev. D6, 477 (1972). 7 For reviews, see Glashow, S. L., "Quarks and Leptons", ed. Levy, M., (Car- gese, 1979); "Fundamental Interactions", ed. Levy, M., (Cargese, 1981). 8 Geng, C. Q. and Marshak, R. E., Phys. Rev. D39 (1989), 9 Gcorgi, H. and Glashow, S. L. Phys. Rev. D6,1159 (1972); Okubo, S., Phys. Rev. D16, 3528 (1977). 10 Atiyah, M. T., Proc. of the XVIJIth Solvay Conference on Physics at Uni- versity of Texas, Austin, Texas, 1982 ( Phys. Rep. 104, 203 (1984) ). 11 Geng, C. Q., Marshak, R. E., Zhao, Z. Y., and Okubo, S., Phys. Rev. D36, 1953 (1987). 12 Zee, A., Phys. Lett. B99, 110 (1981). 13 Geng, C. Q., TRIUMF Report No. TRI-PP-88-103TR , to be published in Phys. Rev. D. 14 For a review, see Peccei, R. D., DESY Report No. DESY-88-078.