We describe a method to remove non-decoupling heavy fields from a quantized field theory and to construct a low-energy one-loop effective Lagrangian by integrating out the heavy degrees of freedom in the path integral. We apply this method to the Higgs boson in a spontaneously broken SU(2) gauge theory (gauged linear σ-model). In this context, the background-field method is generalized to the non-linear representation of the Higgs sector by applying (a generalization of) the Stueckelberg formalism. The (background) gauge-invariant renormalization is discussed. At one loop the log M H -terms of the heavy-Higgs limit of this model coincide with the UV-divergent terms of the corresponding gauged non-linear σ-model, but vertex functions differ in addition by finite (constant) terms in both models. These terms are also derived by our method. Diagrammatic calculations of some vertex functions are presented as consistency check. *

We integrate out the Higgs boson in the electroweak standard model at one loop and construct a low-energy effective Lagrangian assuming that the Higgs mass is much larger than the gauge-boson masses. Instead of applying diagrammatical techniques, we integrate out the Higgs boson directly in the path integral, which turns out to be much simpler. By using the background-field method and the Stueckelberg formalism, we directly find a manifestly gauge-invariant result. The heavy-Higgs effects on fermionic couplings are derived, too. At one loop the log M H -terms of the heavy-Higgs limit of the electroweak standard model coincide with the UV-divergent terms in the gauged non-linear σ-model, but vertex functions differ in addition by finite constant terms. Finally, the leading Higgs effects to some physical processes are calculated from the effective Lagrangian. *

Within the framework of the path-integral formalism we reinvestigate the different methods of removing the unphysical degrees of freedom from spontanously broken gauge theories. These are: construction of the unitary gauge by gauge fixing; R ξ -limiting procedure; decoupling of the unphysical fields by point transformations. In the unitary gauge there exists an extra quartic divergent Higgs self-interaction term, which cannot be neglected if perturbative calculations are performed in this gauge. Using the Stückelberg formalism this procedure can be reversed, i. e., a gauge theory can be reconstructed from its unitary gauge. We also discuss the equivalence of effective-Lagrangian theories, containing arbitrary interactions, to (nonlinearly realized) spontanously broken gauge theories and we show how they can be extended to Higgs models. * Supported in part by Deutsche Forschungsgemeinschaft, Project No.: Ko 1062/1-2 † E-Mail: knetter@physf.uni-bielefeld.de 0 and then to perform the limit ξ → ∞ [8,9]. In this limit, the unphysical fields get infinite masses and decouple. However, the ghost-ghost-scalar couplings get infinite, too, with the consequence, that the ghost term does not completely vanish: there remains the abovementioned Higgs-self-coupling term.The third way is most similar to the classical treatment: the unphysical fields are decoupled from the physical ones by point transformations [5]. This procedure consists of two subsequent transformations; first the unphysical scalars are paramatrized nonlinearly and then they are decoupled and can be integrated out. Since in this formalism transformations of the functional integrand are performed, the Jacobian determinant due to the change of the integral measure has to be considered [5,10], it yields again the new Higgs-self-interaction term.Thus, all three methods lead to a quantum level Lagrangian (in the U-gauge) which contains, in addition to the classical U-gauge Lagrangian, the extra nonpolynomial quartic divergent Higgs-self-interaction term. The same term was derived by quantizing the classical U-gauge Lagrangian canonically [9,11] where it emerges as a remnant of covariantrization. It has been shown on the one-loop level for three-and four-Higgs-interaction amplitudes that the quartic divergences cancel against quartic divergent N-Higgs-vertices which are quantum induced by gauge-boson loops in the U-gauge [12]; this ensures renormalizability. In fact, a linear SBGT is renormalizable even in its U-gauge because of the equivalence of all gauges [5] (although this is not expected by naive power-counting due to the bad high-energy behaviour of the gauge boson propagator in this gauge). So loop calculations can be performed consistently in the unitary gauge if the extra term (which does not contribute to the presently phenomenological most interesting processes at one-loop level) is taken into account. Loop calculations may be simpler in the U-gauge than in the R ξ -gauge because there are less Feynman diagrams to be considered, on the other hand the resul...

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