The Abelian Higgs model under a gauge invariant looking glass: exploiting new Ward identities for gauge invariant operators and the Equivalence Theorem

Abstract

The content of two additional Ward identities exhibited by the U(1) Higgs model is exploited. These novel Ward identities can be derived only when a pair of local composite operators providing a gauge invariant setup for the Higgs particle and the massive vector boson is introduced in the theory from the beginning. Among the results obtained from the above mentioned Ward identities, we underline a new exact relationship between the stationary condition for the vacuum energy, the vanishing of the tadpoles and the vacuum expectation value of the gauge invariant scalar operator. We also present a characterization of the two-point correlation function of the composite operator corresponding to the vector boson in terms of the two-point function of the elementary gauge fields. Finally, a discussion on the connection between the cartesian and the polar parametrization of the complex scalar field is presented in the light of the Equivalence Theorem. The latter can in the current case be understood in the language of a constrained cohomology, which also allows to rewrite the action in terms of the aforementioned gauge invariant operators. We also comment on the diminished role of the global U(1) symmetry and its breaking.

Remarks on the Clauser-Horne-Shimony-Holt inequality in relativistic quantum field theory

“…However, due to the translation invariance of the Wightman function and, consequently, of the scalar product, a pair of test functions can still satisfy eqs. (40) if they are odd and even with respect to a certain point of the space-time which, in the present case, is the location of Bob's lab, see Fig. Due to eqs.…”

“…3 We notice that, since at least one of the labs is not located at the origin of the coordinate system, one pair of test functions will not have the odd and even symmetries with respect to x = 0, as assumed in (40). However, due to the translation invariance of the Wightman function and, consequently, of the scalar product, a pair of test functions can still satisfy eqs.…”

“…Concerning now the scalar products f |f and g|g , we have followed the same prescription adopted in the original work [22] and have taken ( f |f , g|g ) purely imaginary 3 , as described in eqs. (40), namely…”

“…The interest in the U (1) Higgs model is due to a set of articles [33,34,35,41,40,39,38,37,36,42,43,44] where a fully BRST invariant description of the massive gauge boson has been worked out.…”

“…We mention here that a BRST invariant operator O can be introduced also for the Higgs field h [41,40,39,38,37,36]…”