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Renormalization of Generalized Vector Field Models in Curved Spacetime

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Renormalization of Generalized Vector Field Models in Curved Spacetime PHYSICAL REVIEW D 98, 025009 (2018) Renormalization of generalized vector field models in curved spacetime Michael S. Ruf and Christian F. Steinwachs* Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany (Received 12 June 2018; published 13 July 2018) We calculate the one-loop divergences for different vector field models in curved spacetime. We introduce a classification scheme based on their degeneracy structure, which encompasses the well-known models of the nondegenerate vector field, the Abelian gauge field, and the Proca field. The renormalization of the generalized Proca model, which has important applications in cosmology, is more complicated. By extending standard heat-kernel techniques, we derive a closed form expression for the one-loop divergences of the generalized Proca model. DOI: 10.1103/PhysRevD.98.025009 I. INTRODUCTION nonminimal and higher order operators, a generalization of the Schwinger-DeWitt algorithm, which allows us to reduce Most models of inflation and dynamical dark energy are the calculation to the known case of the minimal second based on scalar-tensor theories and fðRÞ gravity, which order operator, has been introduced in [20]. The direct have an additional propagating scalar degree of freedom application of the generalized Schwinger-DeWitt algorithm (d.o.f.). The one-loop quantum corrections to these models requires the non-degeneracy of the principal part—the on an arbitrary background manifold have been derived for highest-derivative term of the fluctuation operator. a general scalar-tensor theory in [1,2] and recently for fðRÞ However, there are important cases, where the fluc- gravity in [3]. tuation operator has a degenerate principal part—notably Aside from models based on an additional scalar field, fðRÞ gravity [3] and the generalized Proca model consid- vector fields have been studied in cosmology [4–11]. Most ered in this article. Therefore, we make use of the of these models are characterized by a nonminimal cou- Stückelberg formalism [23] to reformulate the generalized pling of the vector field to gravity and are particular cases of Proca model as a gauge theory such that standard heat- the generalized Proca model. kernel techniques become applicable again. The price to The quantum corrections for the generalized Proca pay is the introduction of a second metric tensor. model are difficult to calculate and have been studied This article is organized as follows: In Sec. II, we discuss recently in [12,13] by different approaches. In this article, different vector field models in curved spacetime. In we use another approach, which allows us to derive the particular, we introduce a classification based on their one-loop divergences for the generalized Proca model in a degeneracy structure. In Sec. III, we calculate the one-loop closed form. divergences for the nondegenerate vector field with an We use a combination of the manifest covariant back- arbitrary potential. In Sec. IV, we consider the case of the ground field formalism and the heat kernel technique Abelian gauge field and calculate the one-loop divergences. [14–22]. This general approach can be applied to any type In Sec. V, we derive the one-loop divergences for the Proca of field. The central object in this approach is the differential model of the massive vector field. In Sec. VI, we introduce operator, which propagates the fluctuations of the fields. For the generalized Proca model, calculate the one-loop diver- most physical theories, this fluctuation operator acquires the gences in a closed form and present our main result. In form of a second order minimal (Laplace-type) operator. For Sec. VII, we perform several reductions of our general result this simple class of operators, a closed algorithm for the for the generalized Proca model to specific cases. These calculation of the one-loop divergences exists [14].For reductions provide strong cross checks of our general result and entail applications to cosmological models. In Sec. VIII, *[email protected] we compare our result and our method to previous calcu- lations of the one-loop divergences for the generalized Proca Published by the American Physical Society under the terms of model. Finally, in Sec. IX, we summarize our main results the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to and give a brief outlook on their implications. the author(s) and the published article’s title, journal citation, Technical details are provided in several Appendices. and DOI. Funded by SCOAP3. In Appendix A, we introduce the general formalism and 2470-0010=2018=98(2)=025009(24) 025009-1 Published by the American Physical Society MICHAEL S. RUF and CHRISTIAN F. STEINWACHS PHYS. REV. D 98, 025009 (2018) ν Δ ν 1 − λ ∇ ∇ν provide a collection of tabulated coincidence limits, which ½Fλμ ¼½ Hμ þð Þ μ : ð3Þ arise in the calculation. In Appendix B, we present the Δ details of the calculation for the one-loop divergences of the Here, the Hodge operator on vector fields H is defined in nondegenerate vector field considered in Sec. III.In terms of the positive definite Laplacian Δ by Appendix C, we provide a detailed calculation of the most complex functional traces, which contribute to the one-loop Δ ν ≔ Δδ ν ν Δ ≔− μν∇ ∇ ½ Hμ μ þ Rμ ; g μ ν: ð4Þ divergences of the generalized Proca model considered in Sec. VI. In Appendix D, we collect several important In the terminology of [20], the operator (2) is called integral identities. nonminimal for λ ≠ 1 and minimal for λ ¼ 1. Note that if not indicated otherwise, derivative operators act on II. VECTOR FIELD MODELS IN CURVED everything to their right. The Hodge operator satisfies SPACETIME: GENERAL STRUCTURE ν ½Δ ∇ν ¼ ∇μΔ: ð5Þ In this article we calculate the one-loop divergences for H μ generalized vector field models in a curved spacetime. The An important property of the general second order vector divergent part of the one-loop contribution to the effective field operator (2) is its degeneracy structure, which is action is defined by controlled by the parameter λ and the potential P. The λ ≥ 0 P ≥ 0 F 1 conditions and ensure that the operator is Γdiv F ∇ div positive semidefinite. Therefore, there are three different 1 ¼ 2 Tr ln ð Þj ; ð1Þ degeneracy classes: (1) The nondegenerate vector field: λ > 0, P ≥ 0 where Fð∇Þ is the differential operator that controls the (2) The Abelian gauge field: λ ¼ 0, P ¼ 0 propagation of the fluctuations. For the vector field models (3) The (generalized) Proca field: λ ¼ 0, P > 0 discussed in this work, this fluctuation operator has the The relation among the classes is depicted graphically in particular form Fig. 1. Physically, the different classes correspond to inequivalent theories with a different number of propagat- Fð∇Þ¼Fλð∇ÞþP; ð2Þ ing d.o.f. Mathematically, this is reflected by the degen- eracy structure of the operator (2), which is discussed in the ν where P is a potential with components Pμ and Fλð∇Þ is a following sections in detail. In particular, there is no differential operator with components smooth transition between the classes in the limits λ → 0 FIG. 1. Overview of the different degeneracy classes and reductions of the generalized Proca model. 025009-2 RENORMALIZATION OF GENERALIZED VECTOR FIELD … PHYS. REV. D 98, 025009 (2018) and P → 0. Therefore, the different classes have to be vector field operator (2), the relevant gauge theory is treated separately. defined by the Euclidean action for the Abelian gauge field AμðxÞ, A. Degeneracy of the principal symbol Z 1 An important structure in the theory of differential 4 1=2F F μν S½A¼4 d xg μν : ð10Þ operators Fð∇Þ is the leading derivative term—the princi- pal part Dð∇Þ. Separating the principal part from the lower The Abelian field strength tensor F μν is defined as derivative terms Πð∇Þ, the operator Fð∇Þ takes the form F ≔ ∇ − ∇ Fð∇Þ¼Dð∇ÞþΠð∇Þ: ð6Þ μν μAν νAμ: ð11Þ Physically, the principal part Dð∇Þ contains the informa- The action (10) is invariant under infinitesimal gauge tion about the dominant ultraviolet behavior of the under- transformations lying theory. Therefore, it is the natural starting point for the generalized Schwinger-DeWitt method [14,20], which δεAμ ¼ ∂με; ð12Þ relies on the expansion D ≫ Π of the associated propa- gator, schematically where εðxÞ is the infinitesimal local gauge parameter. Gauge invariance of (10) implies the Noether identity 1 1 1 1 1 ¼ ¼ − Π þÁÁÁ; ð7Þ F D þ Π D D D δS A δS A ∂ ½ ∇ −1=2 ½ 0 μ δ ¼ μ g δ ¼ : ð13Þ where 1=F denotes the inverse of the linear operator F. AμðxÞ AμðxÞ Essential for this perturbative treatment is the notion of F background dimension M, which is understood as the mass The components of the fluctuation operator are obtained dimension of the background tensorial coefficients of the from the Hessian F O Mk differential operator. We write ¼ ð Þ for any oper- 2 k δ S A ator F, which has at least background dimension M . The ν ∇x δ 0 ≔ −1=2 ½ Fμ ð Þ ðx; x Þ g gμρ 0 ; ð14Þ expansion (7) critically relies on the invertibility of D, δAρðxÞδAνðx Þ which can be discussed at the level of the principal symbol DðnÞ, formally obtained by replacing derivatives ∇μ by a where the delta function is defined with zero density weight 0 constant vector field inμ. For the vector field operator (2) at x and unit density weight at x . The fluctuation operator the components of the principal symbol read for the Abelian gauge field is given by F ¼ F0, which λ 0 corresponds to the vector field operator (2) with ¼ and ν P 0 ν 2 ν nμn ¼ .
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