Renormalizable Models with Broken Symmetries

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1. [1] K. Hepp, Theorie de la Renormalisation. Lecture Notes in Physics Vol. 2. Springer VerlagNew York1969. MR277208
2. K. Hepp, in "Statistical Mechanics and Quantum Field Theory", Les Houches1970 
3. K. Hepp, in "Statistical Mechanics and Quantum Field Theory", Gordon BreachNew York (1971). 
4. H. Epstein, V. Glaser, same volume. 
5. W. Zimmermann, in "Lectures on Elementary Particles and Quantum Field Theory", Brandeis (1970) Vol. I. 
6. W. Zimmermann, in "Lectures on Elementary Particles and Quantum Field Theory", MIT Press, Cambridge Mass, (1970) 
7. E. R. Speer : Feynman Amplitudes, Princeton University Press (1965). Zbl0172.27301
8. [2] N. N. Bogoliubov, D. V. Shikov, "Introduction to the theory of quantized Fields", Interscience Pub.New York (1960). Zbl0088.21701
9. [3] H. Epstein, V. Glaser : The role of locality in Perturbation Theory, CERN TH 1400, 16 September 1971, to appear in Ann. Institut Poincaré. Zbl1216.81075MR342091
10. H. Epstein, V. Glaser, "Adiabatic Limit in Perturbation Theory" CERN TH 1344, 10 June 1971, in Meeting on Renormalization Theory, CNRSMarseille (1971). 
11. [4] For a description of the n-dimensional regularization see, for instance, the forthcoming CERN report by G.'t Hooft, M. Veltman, also E. R. Speer, to appear. 
12. [1] W. Zimmermann, Commun. math. Phys.15, 208 (1969), Zbl0192.61203MR255162
13. W. Zimmermann in "Lectures on Elementary Particles and Quantum Field Theory", Brandeis University, Vol. I, MIT Press., Cambridge Mass. (1970), Annals of Physics, to be published. 
14. [2] J. H. Lowenstein, Phys. Rev. D4, 2281 (1971), 
15. J. H. Lowenstein in "Seminars on Renormalization Theory", Vol. II, Maryland, Technical Report 73-068 (1972). 
16. [3] K. Hepp, Commun, Math. Phys.6, 161 (1967). 
17. [4] J. H. Lowenstein, M. Weinstein and W. Zimmermann, to be publ. 
18. [5] The convergence of the finite part (II.22) will be proved in a paper by M. Gomes, J. H. Lowenstein and W. Zimmermann, in preparation. 
19. [6] It is assumed there that the number of mass parameters is not larger than the number of free parameters of the Lagrangian. For a discussion of this point see Chapter III, page 26 
20. [7] In the models studied in Chapter IV one of the parameters ${\lambda }_j$ is not determined by renormalization conditions, but directly given as constant or power series in $g_j$. It can be shown for the cases considered that the Green’s functions do not depend on the value of this parameter 
21. [8] In unrenormalized form the action principle of quantum field theory is due to J. Schwinger, Phys. Rev.91, 713 (1953). 
22. Using Caianiello's renormalization method related formulae were derived in E. Caianiello, M. Marinaro, Nuovo Cimento27, 1185 (1963) Zbl0112.45605
23. F. Guerra, M. Marinaro, Nuovo Cimento42A, 306 (1966). The form (IV.24) is proved in ref. [2], it is also valid in the presence of anomalies. 
24. [9] Y. Lam, Phys. Rev. DM. Gomes and J. H Lovenstein, to be published. 
25. [10] A. Rouet, to be published J. H. Lowenstein, M. Weinstein and W. Zimmermann, to be publ. 
26. [1] This is best explained in K. Symanzik, "Lectures on Symmetry Breaking" in Cargese (1970). 
27. K. Symanzik, "Lectures on Symmetry Breaking" Gordon and Breah, New York (1972). MR464995
28. [2] J. H. Lowenstein, P.R. D4, 2281 (1971). 
29. A. Rouet, R. Stora, Nuovo Cim. Lett.4, 136, 139 (1972). 
30. [3] R. Jost, The General Theory of Quantized Fields. A.M.S.Providence (1965). Zbl0127.19105MR177667
31. [4] The formal "improved" energy momentum tensor was defined in : CG. Callan, S. Coleman, R. Jackiv, Ann. Phys.59, 42 (1970). Zbl1092.83502
32. The present finite version is due to M. Bergere (private communication). The asymmetry identity was first found by A. Rouet, unpublished, and A. Rouet, R. Stora [2]. 
33. The general form of ${\Theta }_{\mu \nu }$ compatible with Ward identities was first given by K. Symanzik and K. Wilson, private communications (1970) and is now best described in terms of normal products, as done here. 
34. [5] That the trace of the energy momentum tensor becomes soft at the GellMann Low value of the coupling constant, is shown in : B. Schroer, Lett. Nuov. Cim.2, 867 (1971). 
35. [6] K. Symanzik, unpublished and J. Lowenstein, Seminars on Renormalization Theory, Vol. II, Maryland Technical Report 73 - 068 (1972). 
36. [7] Y. P. Lam, B. Schroer, to be published. 
37. A. Rouet, Equivalence theorems for Effective Lagrangians, MarseilleCNRS preprint 73/P. 528. March 1973. 
38. [8] Arguments of this type may be found in : A. Becchi : "Absence of strong interactions to the axial anomaly in the $\sigma$ model", CERN TH 1611. January (1973). Comm. Math. Phys. to appear. 
39. [1] The material of this section will be published in a series of papers by J. H. Lowenstein, M. Weinstein, W. Zimmermann (part I and II) 
40. The material of this section will be published in a series of papers J. H. Lowenstein, B. Schroer (part III). 
41. [2] B. Lee, Nucl. Phys. B9, 649 (1969). 
42. [3] K. Symanzik, Lett. Nuovo Cimento2, 10 (1969) 
43. K. SymanzikCommun. Math. Phys.16, 48 (1970). MR266541
44. [4] For the proof see part II of ref. [1]. 
45. [5] For problems concerning unstable particles in perturbation theory we refer to M. Veltman, Physica29, 122 (1969) and part III of ref. [1]. 
46. [6] For the formulation of the renormalization conditions see ref. [1]. 
47. [7] K. Symanzik, Lett. Nuovo Cimento2, 10 (1969) 
48. K. SymanzikCommun. Math. Phys.16, 48 (1970). MR266541
49. F. Jegerlehner and B. Schroer, to be published. 
50. [8] By appropriate choice of $𝒲$ one of the coefficients can be made to vanish in zero order, but not both. 
51. [9] B. Lee, Phys. Rev. D5, 823 (1972). 
52. [10] B. Lee and J. Zinn-Justin, to be published. 
53. [11] The values given are the masses in zero order. Only for stable particles may the zero order values be identified with the exact masses by suitable normalization conditions. 
54. [12] See ref. [1] , part III. 
55. [1] This is the strategy advocated by J. Schwinger in : J. Schwinger "Particles Sources and Fields", Addison Wesley Pub. Co., Reading, Mass. (1970). Zbl0155.32302
56. J. Schwinger, "Particles and Sources", Gordon & BreachNew York (1967) (Brandeis 1967). See also : Zbl0155.32302
57. A. Rouet, R. Stora, Lectures given at the Universities of Geneva and Lausanne (1973). 
58. [2] Ch. III, Refs. [1] and [8]. The determination of an $L_4$ Lagrangian such that Ward identities hold, performed in Ref. [7] of Ch. III requires one more constraint than is allowed by the number of parameters at disposal. That constraint, a relation between some $r$ coefficients, can be shown to be automatically fulfilled by an argument concerning the high momentum behaviour of vertex functions. This argument is due to O. Piguet and similar to one used by him in the construction of an $L_4$ Lagrangian for the Higgs Kibble model with massive photons of Chapter IV. (O. Piguet, private communication). 
59. [3] More details can be found in A. Rouet, R. Stora, Ref. [1] 
60. A. Rouet, article in preparation. 
61. [4] We wish to thank B.W. Lee for a discussion on this point. 
62. [5] This way of obtaining the Slavnov identities can be found in : L. Quaranta, A. Rouet, R. Stora, E. Tirapegui "Spontaneously broken gauge invariance : Ward identities, Slavnov identities, gauge invariance", in "Renormalization of Yang Mills Fields and Applications to Particle Physics", CNRSMarseille, June 19-23 (1972), where however the treatment of renormalization formal due to the a priori possible occurence of infrared difficulties which does not happen in the treatment given in this chapter. Concerning the renormalization of gauge field theories see G.'t Hooft and B.W. Lee's Lectures, in this volume, where the original work by G.'t Hooft, B.W. Lee, M. Veltman, J. Zinn-Justin, is reported. 
63. [6] Such a normalization condition was used in a version of the renormalization of the Higgs Kibble model in Stueckelberg's gauge, by J. H. Löwenstein, M. Weinstein, W. Zimmermann. 
64. [7] J. H. Lowenstein, B. Schröer, PR. D 6, 1553 (1972).