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33B, 173 (1971). [17] For a detailed discussion of renormalizability of Yang–Mills theories, see E. S. Abers and B. W. Lee, Phys. Rep. ’t Hooft and M. Veltman, Nucl. Phys. B44, 189 (1972); H. Kluberg-Stein and J. B. Zuber, Phys. Rev. D12, 467, 482, 3159 (1975); C. Becchi, A. Rouet, and R. Stora, Commun. Math. Phys. 42, 127 (1975); J. C. Taylor, Nuel. Phys. B33, 436 (1971); J. Zino-Justin, Lecture Notes, Bonn, 1974. [18] S. Adler, Phys. Rev. 177, 2426 (1969); J. Bell and R. Jackiw, Nuovo Cimento.

Wilczek, Phys. Rev. D30, 130 (1984). [5] G. Gelmini and M. Roncadelli, Phys. Lett. 99B, 411 (1981). [6] R. Barbieri, R. N. Mohapatra, D. V. Nanopoulos, and D. Wyler, Phys. Lett. 107B, 80 (1981). [7] N. Ramsey and R. F. Code, Phys. Rev. A4, 1945 (1971). [8] R. Barbieri and R. N. Mohapatra, Z. Phys. 11, 175 (1981); F. Wilczek, Phys. Rev, Lett. 49, 1549 (1982); D. Reiss, Phys. Lett. 115B, 217 (1982). References 37 [9] J. E. Kim, Phys. Rev. Lett. 43, 103 (1979); M. Dine, W. Fischler, and M. Srednicki, Phys.

6. In four dimensions, there can also be mixed anomalies involving gauge and gravitational vertices. Write the general form of this anomaly and derive the constraints it imposes on the representations of the gauge. References [1] For an excellent discussion of and references on symmetries and spontaneously broken symmetries, see S. Weinberg, Brandeis Lectures, 1970; M. A. B. Beg, Lectures Notes in Mexico, 1971; G. Guralnik, C. R. Hagen, and T. W. B. Kibble, Advances in High-Energy Physics (edited by R.

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