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Ft». 421* Clffis-Lh'- Fjf.F.Ol ClffiS-lH'- fjf.f.Ol THE ABELIAN HIGGS KIEBLE MODEL. UNITARITY OF THE S- OPERATOR. C. BECCHI* by A. ROUET '•"• R. STORA Centre de Physique Théorique CNRS Marseille. : Results concerning the renormali nation of the abelian Higgs Kibble model in the 't Hooft gauges are presented. A direct combinatorial proof of the unitarity of the physical S -operator is described. ft». 421* x On leave of absence from the University of Genova xx. Boursier de thèse CEA Postal Address i Centre de Physique Théorique 31. chemin J. Aiguier 13274 MARSEILLE CEDEX 2 The Higgs-Kibble model [IJ provides the best known exception to the Goldstone theorem I 2 j concerning the phenomenon of spontaneous symmetry breaking in local field theory. If zero mass vector particles are present in the unbroken theory the degrees of freedom corresponding to the Goldstone boson may be lost, the vector particles becoming massive. In order to discuss this phenomenon in the framework of renormalized quantum field theory [3,4,5j it is necessary to introduce a certain number of non physical degrees of freedom, some of which are associated with negative norm one-particle states. The number of non physical fields and their properties depend on the choice of the gauge [3,4,5,6] , It is however convenient to exclude gauges such as the Stueckelberg gauge which lead [sj to massless particles, in order to avoid unnecessary infrared problems. A choice of gauge satisfying this requirement has been proposed by G. t'Hoofc 14 . In this gauge, according to the Faddeev-Popov fTJ prescription, a system of anticommuting scalar ghost fields must be introduced, which are coupled to the remaining fields. Thus we pay for the absence of masslese particles by increasing the number of non physical degrees of freedom, which are now the Goldstone boson, the scalar photon and. the Faddeev-Popov ghosts. Another difficulty which is connected with this choice of t.-e gauge is that the gauge in­ variance of the lagrangian is now broken nor only by the gauge fixing terras, but also by the Faddeev-Popov compensating r^nns. As a consequence the original Ward identity is char.j..:d in a way which is more difficult to handle. However, modifying the Ward identity in the way proposed by P'.avnov [_8J , it can be seen flOj that the complete lagrangian is invariant under some infinitesimal transformations of the fields, of the super gauge type |9J (Slavnov transformations), in which one of the Faddeev-Popov fields plays the role of the gauge parameter, the other one transforming in a way specified by the choice of the.gauge function. It is very tempting to use this symmetry property as the essential feature of the Higgs-Kibble model. The renormalizability of the model will then be proved by showing that the invariance of the lagrangjan under Slavnov transformations can be extended beyond the tree approximations. It turns 74/P.625 2 out however that this invariance property does not completely charac- teri2e the model, at least in the abelian case. The other essential feature of the. model, which expresses the condition for spontaneous symmetry breakdown is the degeneracy of the masses of the non physi­ cal particles. The authors f]0} have just carried out a complete treatment of the renormalization of the abelian Higgs-Kibble model in the Bogoliubov-Parasiuk - Hepp-Zimmermann scheme £lll , They have proved tha* : i) There exists to all orders of the perturbation expansion an effective lagrangian which is invariant under Slavnov transformations, ii) The free parameters can be fixed by a set of normalization conditions, which includes a complete mass renormalization procedure for both physical and nor. physical particles» the non physical masses being degenerate. iii) The matrix elements of the scattering operator between physical particles states £12} do not depend upon the parameters which specify the choice of the gauge. ivjAll the previous results are extended to the ir.ost general gauge com- pati-olc vit:'.: rcncmalizitility and charge conjugation invariance [I3j . In this letter we show how the unitarity of the phy­ sical S-operator can be proved by a direct combinatorial method. The complete proof will be given in the paper previously referred to in C.10J. First we give some definitions and symbols- Let us denote by s tlie î£ (% V fA« Ct"c ) fields; (£, is the physical scalar field, (0 is the Goldstone boson, /L is the vector meson, C and C are the Faddeev- Popov boosts; let J •= MjrJ J ; £, 3) be the correcponding sources. The physical sector of Fock space is generated up to zero norm states by <£, and transverse vector quanta A» , the non physical sector is generated by the (p. , "oVA , C , O quanta. The Slavnov transformation is the usual gauge transformation i"or the fields <p4/ <pz and A. where G plays the role of the gauge parameter, the C-field variation being proportional to the gauge function ^£= Q-^-A + b(pz 74/P.625 Lee be the generating functional of the disconnected Green functions. The definition of the time ordering symbol "T" flmi of the interaction lagrangien *t|nt summarize the renormalization procedure. The perturbation parameter ia chosen to be ri.. The LSZ reduction formulae allow to write down the non physical contribution to the physical unitarity relation. Let us first define the operator valued functional where r\j and Kuy are the differential operators ôccuring in the asymptotic equations of the ^P. and the f\u fields; Cp and f\ are the corresponding asymptotic field operators. The physical S-operator iB The contribution of non physical particles to physical unitarity is explicitly given in the expression : Here I— and O. are respectively the differential operators occuring in the asymptotic field^equations and the positive frequency pirt of the asymptotic field commutators. The idea of the proof is to consider : and Co evaluate Ve choose as independent non physical fields the gauge function 0 , and tne G— -CL^A + D ^f2 Faddeev-Popov fields with C . normalized in i such a way that Che paropagatora have the saw-.: resiiiue^ at their common pole [lOj . This condition can be fulfilled âince<*^^X. has a simple pole as a consequence of the Slavnov identity. Using the symmetry of the model under the substitution C-*> C anc. C-»-*G , ve can rewrite the operator in the form : <*- /«ft l\ CL $>*% ^l^^lfi,., (7) Ltttyn a ( t s j% = (t stt)cd jTX*^ Now making extensive uae of the Slavnov identity which essentially correspond» to the transformations oC r Q and oQ - C Eq. (5) uimplifies to : V ^ (B) Integrating with respect to *A yields Further use of die Slavnov identity, according to vhich the expectation value of the time ordered product of an arbitrary number of gauge operators between physical states vanishes, yieldfi vhich complete» the proof. 5 FOOTNOTES and REFERENCES tO P. HIGGS Phys. Letters _12.. 132 (1964) Phys. Rev. 145, 1156 (1966) T.W.B. KIBBLE Phys. Rev. _I5J5, 1554 (1967) [2] J. GOLDSTONE Nuovo Cimento 19, Î54 (195!) J. GOLDSTONE, A. SALAM, S. WEINBERG Phys. Rev. 127, 965 (1962) B.W. LEE Phys. Rev. D5, 823 (1972) J.H. LOWENSTEIN, B. SCHROER, M. WEINSTEIN, W. ZIMMERMANN Vorking in the massive Stuekelberg gauge have been able to study the zero ghost mass limit: (private communication) See also : 0. PIGUET, CERN/TH 1798 (January 1974) G't HOOFT Nuclear Phys. B35, 167 (1971) 1.5} B.W. LEE, J. ZINN JUSTIN Phys. Rev. D5, 3137 (1972) Phys. Rev. D5, 3155 (1972) Phys. Rev. D7, 1049 (1973) B.K. LEE Phys. Rev. D9, 933 (1974) Co] L. GERVAIS, J. NEVEU Nuclear Physics £46, 381 (1972) [7] L.B. FADDEEV, V.N. POPOV Phys. Letters 25B, 29 (1967) [8"] A.A. SLAVNOV Kiev Preprint IIP 71-83 E (1971) TMO VO, 153 (1972) £9] J. KESS, B. ZUMINO CERN TH 1753 (September 1973) Co be published in Nuclear Phys. B 74/P.62S 6 [10] C. BECCHI, A. ROUET, R. SIORA (in preparation) [il] W. ZIMMERMANN Ann. Phys. 77., 536 (1973) Ann. Phys. 77, 570 (1973) Ï-M. 1. LAM Phys. Rev. D6, 2145 D7, 2943 (1973) [l2] G't H00FT, M. VELTMAN Nuclear Phys. B50, 318 (1972) []3~\ This excludes in particular the interesting family of gauges introduced in ref. (6) [l4J Due to the occurence of double poles in the ^ ^ propagator» a regularization which will be removed at the end, is necessary to give a meaning separately to the first three terms of Eq. (6). .
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