It has been shown that by using a Lagrange multiplier field to ensure that the classical equations of motion are satisfied, radiative effects beyond one-loop order are eliminated. It has also been ...shown that through the contribution of some additional ghost fields, the effective action becomes form invariant under a redefinition of field variables, and furthermore, the usual one-loop results coincide with the quantum corrections obtained from this effective action. In this paper, we consider the consequences of a gauge invariance being present in the classical action. The resulting gauge transformations for the Lagrange multiplier field as well as for the additional ghost fields are found. These gauge transformations result in a set of Faddeev–Popov ghost fields arising in the effective action. If the gauge algebra is closed, we find the Becci–Rouet–Stora–Tyutin (BRST) transformations that leave the effective action invariant.
Using the background field method, we study in a general covariant gauge the renormalization of the six-dimensional Yang-Mills theory. This requires background gauge invariant counterterms, some of ...which do not vanish on shell. Such counterterms occur, even off shell, with gauge-independent coefficients. The analysis is done at one-loop order and the extension to higher orders is discussed by means of the Becchi-Rouet-Stora-Tyutin identities. We examine the behavior of the beta function, which implies that this theory is not asymptotically free.
The effective potential V is a massless self-coupled scalar theory, and massless scalar electrodynamics is considered. Both the MS¯ and Coleman-Weinberg renormalization schemes are examined. The ...renormalization scheme dependence of V is determined. Upon summing all of the logarithmic contributions to V, it is shown that the implicit and explicit dependence on the renormalization scale μ cancels. In addition, if there is spontaneous symmetry breaking, then the dependence on the background field Φ cancels, leaving V flat but with nonperturbative contributions. The quartic scalar coupling in the Coleman-Weinberg renormalization scheme consequently vanishes.
The zero to four loop contribution to the cross section Re+e− for e+e−→ hadrons, when combined with the renormalization group equation, allows for summation of all leading-log, next-to-leading-log, ...…, next-to-next-to-next-to-leading-log perturbative contributions. It is shown how all logarithmic contributions to Re+e− can be summed and that Re+e− can be expressed in terms of the log-independent contributions, and once this is done, the running coupling a is evaluated at a point independent of the renormalization scale μ. All explicit dependence of Re+e− on μ cancels against its implicit dependence on μ through the running coupling a so that the ambiguity associated with the value of μ is shown to disappear. The renormalization scheme dependency of the “summed” cross section Re+e− is examined in three distinct renormalization schemes. In the first two schemes, Re+e− is expressible in terms of renormalization scheme-independent parameters τi and is explicitly and implicitly independent of the renormalization scale μ. Two of the forms are then compared graphically both with each other and with the purely perturbative results and the renormalization group-summed next-to-next-to-next-to-leading-log results.
We examine the self-consistency of the first-order formulation of the Yang-Mills theory. By comparing the generating functional Z before and after integrating out the additional field Fμνa, we derive ...a set of structural identities that must be satisfied by the Green's functions at all orders. These identities, which hold in any dimension, are distinct from the usual Ward identities and are necessary for the internal consistency of the first-order formalism. They relate the Green's functions involving the fields Fμνa, to Green's functions in the second-order formulation which contain the gluon strength tensor fμνa. In particular, such identities may provide a simple physical interpretation of the additional field Fμνa.
We study the Becchi-Rouet-Stora-Tyutin (BRST) renormalization of an alternative formulation of the Yang-Mills theory, where the matrix-propagator of the gluon and the complementary fields is ...diagonal. This procedure involves scalings as well as nonlinear mixings of the fields and sources. We show, in the Landau gauge, that the BRST identities implement a recursive proof of renormalizability to all orders.
We study the self-consistency of the first-order formulation of quantum gravity, which may be attained by introducing, apart from the graviton field, another auxiliary quantum field. By comparing the ...forms of the generating functional Z before and after integrating out the additional field, we derive a set of structural identities, which must be satisfied by the Green's functions at all orders. These are distinct from the usual Ward identities, being necessary for the self-consistency of the first-order formalism. They relate the Green's functions involving the additional quantum field to those containing a certain composite graviton field, which corresponds to its classical value. Thereby, the structural identities lead to a simple interpretation of the auxiliary field.
The renormalization that relates a coupling “a” associated with a distinct renormalization group beta function in a given theory is considered. Dimensional regularization and mass independent ...renormalization schemes are used in this discussion. It is shown how the renormalization a*=a+x2a2 is related to a change in the mass scale μ that is induced by renormalization. It is argued that the infrared fixed point is to be a determined in a renormalization scheme in which the series expansion for a physical quantity R terminates.
By using the Faddeev-Popov quantization procedure, we demonstrate that the radiative effects computed using the first-order and second-order Einstein-Hilbert action for general relativity are the ...same, provided one can discard tadpoles. In addition, we show that the first-order form of this action can be used to obtain a set of Feynman rules that involves just two propagating fields and three three-point vertices; using these rules is considerably simpler than employing the infinite number of vertices that occur in the second-order form. We demonstrate this by computing the one-loop, two-point function.
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