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 calculations 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 degeneracy 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 principal 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 propagator, 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 þ; 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 . The explicit components read Dμ ðnÞ¼n δμ − ð1 − λÞ ; ð8Þ n2 ν Δ ν ∇ ∇ν Fμ ¼½ Hμ þ μ : ð15Þ 2 ρ with n ≔ nρn . The parameter λ controls the degeneracy of the principal symbol, as can be seen easily from the Taking the functional derivative of (13) with respect to 0 determinant Aνðx Þ yields the operator equation

2 4 μ ν det DðnÞ¼λðn Þ : ð9Þ ∇ Fμ ¼ 0: ð16Þ

For λ ¼ 0, the determinant vanishes and therefore DðnÞ is This implies that for P ¼ 0 the total fluctuation operator (2) not invertible. The origin of this degeneracy can be traced is degenerate—not only its principle symbol. Therefore, in back to the fact that for λ ¼ 0, the principal symbol has the case of a gauge degeneracy, in addition to the breakdown of structure of a projector on transversal vector fields. This the perturbative expansion (7) associated to the degeneracy motivates the distinction between the two classes λ > 0 and of D, the inverse operator 1=F does not even exist. In order λ ¼ 0.Forλ ¼ 0, the further distinction between the cases to remove the gauge degeneracy we choose a gauge P 0 P 0 ¼ and > is connected with a degeneracy at the condition linear in Aμ, level of the full operator F, discussed in the next subsection. χðAÞ¼0: ð17Þ B. Gauge degeneracy ≔ The degeneracy at the level of the full operator F is a The total gauge-fixed action Stot S þ Sgb is obtained by general feature of any gauge theory. In the context of the adding the gauge breaking action

025009-3 MICHAEL S. RUF and CHRISTIAN F. STEINWACHS PHYS. REV. D 98, 025009 (2018) Z 1 F div F P div 4 1=2χ2 Tr1 ln j ¼ Tr1 ln ð λ þ Þj Sgb ¼ d xg : ð18Þ 2 1 div div ¼ Tr1 ln Fλj þ Tr1 P Fλ This leads to a modification of the associated Hessian, 1 1 1 div such that the resulting gauge-fixed operator F is non- − P P tot 2 Tr1 : ð25Þ degenerate Fλ Fλ Inserting the operator identity (24), the divergent contri- δ2S A ν ∇x δ 0 −1=2 tot½ butions of the individual terms in (25) can be reduced to the ½Ftotμ ð Þ ðx; x Þ¼g gμρ 0 : ð19Þ δAρðxÞδAνðx Þ evaluation of universal functional traces. The details of this calculation are provided in Appendix B. In this way, we find for the one-loop divergences of the nondegenerate The inclusion of the gauge breaking action must be vector field compensated by the corresponding ghost action Z 1 11 7 1 Z Γdiv 4 1=2 G − μν 2 1 ¼ 2 d xg RμνR þ R ω ω 4 1=2ω ω 32π ε 180 30 20 Sgh½ ; ¼ d xg Q ; ð20Þ 1 γ γ − 1 μν þ 6 þ 12 RP þ 6 RμνP ω ω where ðxÞ and ðxÞ are anticommuting scalar ghost 1 γ γ2 γ2 fields. The ghost operator is defined as − μν − 2 2 þ 4 þ 24 PμνP 48 P : ð26Þ δ x 0 Qð∇ Þδðx; x Þ ≔ χðAðxÞþδεAðxÞÞ: ð21Þ Here, we have defined the Gauss-Bonnet term δεðx0Þ μνρσ μν 2 G ≔ RμνρσR − 4RμνR þ R : ð27Þ

The divergent part of the one-loop contributions to the 1 effective action is given by The result (26) is in agreement with [20,24]. Note that for P ¼ 0, (26) is independent of the parameter γ. This 1 calculation, as well as the calculation via the generalized Γdiv F div − Q div Schwinger-DeWitt algorithm in [20], both critically rely on 1 ¼ Tr1 ln totj Tr0 ln j : ð22Þ 2 the nondegeneracy of the principal symbol (9). III. THE NONDEGENERATE VECTOR FIELD IV. THE ABELIAN GAUGE FIELD We first consider the vector field operator (2) with a nondegenerate principal symbol and arbitrary potential The fluctuation operator F for the Abelian gauge field theory (10) is given by ν Δ ν 1 − λ ∇ ∇ν ν λ 0 Fμ ¼½ Hμ þð Þ μ þ Pμ ; > ; ð23Þ ν Δ ν ∇ ∇ν Fμ ¼½ Hμ þ μ : ð28Þ

Since for λ ≠ 0 the principal symbol of (23) is invertible, In view of the general operator (2), this corresponds to the the generalized Schwinger-DeWitt algorithm can be used case λ ¼ 0 and P ¼ 0. As discussed in Sec. II B, the directly [20]. The power of this algorithm lies in its operator (28) is degenerate due to the gauge symmetry of generality, as it is applicable to any type of field. the action (10). We choose a relativistic gauge condition However, instead of using the general algorithm, here with arbitrary gauge parameter η to break the gauge the calculation can be essentially simplified by directly degeneracy of the operator (28), making use of an operator identity for Fλ, 1 μ χðAÞ¼− pffiffiffiffiffiffiffiffiffiffiffi ∇ Aμ: ð29Þ δν δν 1 þ η μ μ 1 ν − γ∇μ ∇ ; ¼ Δ Δ2 ð24Þ Fλ H According to (19) and (21), the components of the gauge- F fixed fluctuation operator tot and the corresponding ghost where we have defined γ ≔ ð1 − λÞ=λ. Since P ¼ OðM2Þ operator Q read 0 and 1=Fλ ¼ OðM Þ, we can make efficient use of (24).For the calculation of the one-loop divergences, it is sufficient 1Apart from an overall minus sign, the transition to Lorentzian to expand the logarithm up to OðM4Þ, signature corresponds to the replacement P → −P.

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η ν Δ ν ∇ ∇ν Taking the trace of the logarithm on both sides of (36) and ½Ftotμ ¼½ Hμ þ μ ; ð30Þ 1 þ η using that the divergent part of the vector trace can be converted into a contribution from a scalar trace 1 Q ¼ pffiffiffiffiffiffiffiffiffiffiffi Δ: ð31Þ 1 η ∇ ∇ν div þ ν μ 2 div Tr1 ln δμ − ¼ Tr0 ln ðΔ þ m Þj ; ð37Þ m2 Thus, the gauge-fixed fluctuation operator (30) falls into P 0 the class of nondegenerate vector fields (23) with ¼ the divergent part of the one-loop effective action is λ 1 1 η and ¼ =ð þ Þ. Therefore, the divergent part of the reduced to the vector and scalar traces of two minimal one-loop effective action can be calculated with the second order operators, methods presented in Sec. III, 1 div 2 div 1 Γ1 ¼ Tr1 ln ðF0 þ m 1Þj div div div 2 Γ1 ¼ Tr1 ln F j − Tr0 ln Qj : ð32Þ 2 tot 1 Δ 21 div ¼ 2 Tr1 ln ð H þ m Þj The first trace follows from (26) for P ¼ 0, while the ghost 1 trace is evaluated directly with the standard result for a 2 div − Tr0 ln ðΔ þ m Þj : ð38Þ minimal second order operator (A16) and (A17), 2 Z 1 13 1 1 The vector and scalar traces in (38) can be calculated Γdiv 4 1=2 G − μν 2 1 ¼ 32π2ε d xg 180 5 RμνR þ 15 R : directly with the closed form algorithm (A16) and (A17). The final result for the one-loop divergences of the Proca ð33Þ model (34) reads Z The result (33) is in agreement with [20]. Since the 1 1 13 Γdiv 4 1=2 G − μν action for the Abelian vector field corresponds to a 1 ¼ 32π2ε d xg 15 60 RμνR free theory, there are no contributions to the renormaliza- 7 1 3 tion of the square of the field strength tensor (11). Note þ R2 − m2R − m4 : ð39Þ also that the result (33) is independent of the gauge 120 2 2 parameter η. The result (39) is in agreement with [20,26]. It has a clear V. THE PROCA FIELD physical interpretation. The effective action of the massive vector field in four dimensions is that of a four component The Proca action for the massive vector field in curved vector field minus one scalar mode, since the Proca field spacetime is given by the action of the Abelian gauge field has 4 − 1 ¼ 3 propagating d.o.f. Therefore, it is clear that (10) supplemented by a mass term [25], (39) does not reproduce the result for the Abelian gauge Z field (33) in the limit m → 0, as the Abelian gauge field has 1 1 4 1=2 μν 2 μ only 4 − 2 ¼ 2 propagating d.o.f. At the level of the S½A¼ d xg F μνF þ m AμA : ð34Þ 4 2 functional traces, this can formally be seen as follows: while the scalar operator for the longitudinal mode of the The mass term breaks the gauge symmetry. The Hessian of Proca field in (38) indeed reduces to the ghost operator of (34) leads to the fluctuation operator the Abelian gauge field in (32) in the limit m → 0, the ghost trace in (32) is subtracted twice compared to the trace of the ν ≔ Δ ν ∇ ∇ν 2δν Fμ ½ Hμ þ μ þ m μ; ð35Þ longitudinal mode in (38). which corresponds to the case λ ¼ 0 and P ¼ m21 of the 2 VI. THE GENERALIZED PROCA FIELD general vector field operator (2), that is F ¼ F0 þ m 1. The mass term in the Proca operator (35) breaks the gauge The generalized Proca model is defined by the action degeneracy of the gauge field operator F0. Nevertheless, the Z 1 1 principal part of the Proca operator (35) is still degenerate. 4 1=2 μν μν S A d xg F μνF M AμAν : 40 This degeneracy cannot be removed by a gauge fixing—in ½ ¼ 4 þ 2 ð Þ contrast to the Abelian gauge field. Similar to (24), there is an operator identity for the Proca field The action is that of the Proca field (34), but with the scalar 2 ν ρ ν mass term m generalized to an arbitrary positive definite δμ ρ ∇μ∇ δρ μν ¼ δμ − : ð36Þ and symmetric background mass tensor M . The back- 2 2 Δ 2 μν F0 þ m m H þ m ground mass tensor M is completely general and might be

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1 constructed from external background fields. Of particular ˜ ρ −1 ρα Γ μν ¼ ðg˜ Þ ð∂μg˜αν þ ∂νg˜μα − ∂αg˜μνÞ: ð47Þ interest are curvature terms 2

μν μν μν M ¼ ζ1R þ ζ2Rg ; ð41Þ A natural structure is the difference tensor

λ ˜ λ ˆ λ which arise in cosmological models [4–11]. The fluctuation δΓ μν ≔ Γ μν − Γ μν operator for the generalized Proca theory (40) reads 1 ˜−1 λα ∇ˆ ˜ ∇ˆ ˜ − ∇ˆ ˜ ¼ 2 ðg Þ ð μgνα þ νgμα αgμνÞ: ð48Þ ν Δ ν ∇ ∇ν ν Fμ ¼½ Hμ þ μ þ Mμ ð42Þ By construction, the difference tensor satisfies The operator (42) corresponds to the general operator (2) ν ρν for λ ¼ 0 and Pμ ¼ gμρM > 0. The standard techniques 1 δΓα ˜−1 αβ∇ˆ ˜ for the calculation of the one-loop divergences are not μα ¼ 2 ðg Þ μgαβ directly applicable to the degenerate operator of the ˜ −1=2∇ˆ ˜ 1=2 0 generalized Proca theory (42). In particular, for the gen- ¼ðdet gρσÞ μðdet gρσÞ ¼ : ð49Þ eralized Proca operator, there is no simple analogue of the ˜ operator identity (36) for the Proca operator. Therefore, we The Ricci curvatures of the new metric gμν are given by adopt a different strategy and first reformulate the gener- ˜ −1 μν ˜ alized Proca theory as a gauge theory by making use of the R ¼ðg˜ Þ Rμν; ð50Þ Stückelberg formalism. In this formulation, the generalized μν ˜ ˆ α β ˆ α background mass tensor M plays a double role as Rμν ¼ Rμν − δΓ βμδΓ αν þ ∇αδΓ μν: ð51Þ potential in the vector sector and as metric in the scalar “ ” Stückelberg sector. For this effective bimetric formu- In the following, when we work with the two metrics gˆμν lation, the standard heat-kernel techniques are applicable and g˜μν, indices are raised and lowered exclusively with the and the one-loop divergences of the generalized Proca metric gˆμν. action (40) are obtained in a closed form.

A. Weyl transformation and bimetric formalism B. Stückelberg formalism The calculations are simplified by performing a Weyl As we have discussed in the context of the Abelian gauge transformation of the background metric field, the gauge symmetry is responsible for the degeneracy of the total fluctuation operator. Therefore, a gauge fixing is 1 μν 1=4 required to remove this degeneracy. At the same time, the gˆμν ≔ det M gνρ gμν: 43 μ2 ½ ð Þ ð Þ gauge fixing can be used to also remove the degeneracy of the principal symbol. A similar mechanism works in the Here, μ is an auxiliary mass parameter, introduced for case of the generalized Proca model, when artificially dimensional reasons. Note that in what follows indices rewritten as a gauge theory, which is realized by the are raised and lowered only with the metric gˆμν. Since Stückelberg formalism. The Stückelberg scalar field φ is the kinetic term is invariant under a Weyl transformation, introduced by the shift we find 1 Z → ∂ φ 1 μ2 Aμ Aμ þ μ μ : ð52Þ 4 ˆ1=2 ˆμα ˆνβF F ˜−1 μν S ¼ d xg 4 g g μν αβ þ 2 ðg Þ AμAν : ð44Þ In terms of the vector field Aμ and the Stückelberg scalar φ, In the second term, we have defined the action (44) is given by Z −1 μν 2 μν −1=2 μν ðg˜ Þ ≔ μ ½det ðM gνρÞ M ; ð45Þ 1 φ 4 ˆ1=2 ˆμα ˆνβF F S½A; ¼ d xg 4 g g μν αβ ˜ which is the inverse of the new metric gρν. In this way, 2 μ −1 μν −1 μν formally the dependency on the original general mass þ ðg˜ Þ AμAν þ μðg˜ Þ Aμ∂νφ μν 2 tensor M has been replaced by a standard mass term. 1 By construction, we have the important relations ˜−1 μν∂ φ∂ φ þ 2 ðg Þ ν ν : ð53Þ ˆ det g˜μν ¼ det gˆμν; ∇ρ det g˜μν ¼ 0: ð46Þ This action has a gauge symmetry as it is invariant under We define the Christoffel connection associated with g˜μν, the simultaneous infinitesimal gauge transformations

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δ ∂ ε δ φ −με ν ≔ ∇ˆ ν μ2 ˜−1 ν εAμ ¼ μ ; ε ¼ : ð54Þ ½D1μ ½ Hμ þ ðg Þμ; ≔−∇ˆ ˜−1 μν∇ˆ We choose a one-parameter family of gauge conditions D0 μðg Þ ν; ν ˆ −1 μν Π ≔−μ∇μðg˜ Þ ; ð63Þ 1 ˆ μ χ½A; φ¼− pffiffiffiffiffiffiffiffiffiffiffi ð∇μA þ ημφÞ: ð55Þ 1 η † −1 ν ˆ ν þ where Πμ ¼ μðg˜ Þμ∇ν, denotes the formal adjoint of Π with respect to the inner product on the space of vectors. In particular, (55) interpolates between the original vector The component D0 can be simplified by using (49) and φ 0 η → ∞ field theory (44) with ¼ ( ) and the Lorentz defines the scalar Laplace operator with respect to the η 0 gauge ( ¼ ). The corresponding gauge breaking action metric g˜μν, reads Z − ˜−1 μν∇˜ ∇˜ ≕ Δ˜ 1 D0 ¼ ðg Þ μ ν : ð64Þ φ 4 ˆ1=2χ2 Sgb½A; ¼2 d xg : ð56Þ Let us briefly discuss what we have achieved by the Stückelberg formalism. The 4 − 1 ¼ 3 propagating d.o.f. The ghost operator is obtained from (55), of the original generalized Proca field have been converted into the 4 þ 1 − 2 ¼ 3 propagating d.o.f., corresponding to ˆ ˆ 2 Qð∇Þ¼Δ þ ημ : ð57Þ those of a vector field, a scalar, and two scalar ghosts fields. In contrast to the principal part of the original generalized For simplicity, we choose the Lorentz gauge, η ¼ 0.2 In Proca operator (42), the additional gauge freedom present terms of the generalized two-component field in the Stückelberg formalism has been used to render the D principal part of the scalar-vector block operator (61) Aμ nondegenerate and, in particular, minimal—the price to pay ϕA ¼ ; ð58Þ φ is the introduction of the second metric g˜μν. the gauge-fixed action acquires the block form C. One-loop effective action Z In order to calculate the one-loop effective action S½A; φþS ½A; φ¼ d4xg1=2ϕAF ϕB; ð59Þ gb AB 1 Γ F − Q 1 ¼ 2 Tr ln Tr0 ln ; ð65Þ where the block matrix fluctuation operator F has compo- A AC nents F B ¼ γ FCB. The components of the (dedensi- we expand F around D. Perturbation theory in Π is efficient tized) inverse configuration space metric γAB are given by as Π ¼ OðM1Þ. Expanding Tr ln F up to OðM4Þ,we obtain gˆμν γAB ¼ : ð60Þ 1 1 F D D − − Tr ln ¼ Tr1 ln 1 þ Tr0 ln 0 T2 2 T4; ð66Þ Splitting the fluctuation operator according to the number where T ¼ OðMiÞ, i ¼ 2, 4 denote the following traces of derivatives, it can be represented as i 2 β 1 1 α δα † 1 F ¼ D þ Π; ð61Þ T2 ≔ Tr Π ¼ Tr0 Π Πβ ; ð67Þ 2 D D1 D0 with the block matrix structure 4 β 2 1 1 α δα † 1 T4 ≔ Tr Π ¼ Tr0 Π Πβ : ð68Þ † 2 D1 Π D D1 D0 D ¼ ; Π ¼ : ð62Þ D Π 0 Note that odd powers in the expansion are zero, because the block matrix Π1=D has only zeros on the diagonal and that The components of the operators in (62) are given by we have used the cyclicity of the trace to convert vector traces into scalar traces, for example: 2 The calculation can be performed for the general η-family of gauges (55). By using (24), it can be seen already at the level of ν β † 1 α δα α δα † 1 the functional traces that all η-dependent terms cancel and the Tr1 Πμ Π ¼ Tr0 Π Πβ : ð69Þ one-loop divergences are independent of the gauge parameter η. D0 D1 D1 D0

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The first two traces in (66), together with the contribution Here, we have introduced the “interpolation metric” of the ghost operator (57), are evaluated directly with (A16) and (A17). Their sum reads GμνðuÞ ≔ gˆμν þ ug˜μν: ð75Þ

Tr1 ln D1 þ Tr0 ln D0 − 2Tr0 ln Q μ Z The function Ψðu; sjσˆ Þ in the integrand of (74) is the result 1 1 1 of the covariant Taylor expansion in σˆ μ 4 ˆ1=2 Gˆ − ˜−1 μν ˜−1 ρσ ˜ ˜ ¼ 16π2ε d xg 15 60 ðg Þ ðg Þ RμρRνσ X∞ 1 1 μ 1 2 ðkÞ μ μ ˜ 2 4 −2 2 −1 μν ˆ Ψ σˆ ≔ ˆ = Ψ σˆ 1 …σˆ k − R − μ trðg˜ Þ − μ ðg˜ Þ Rμν ðu; sj Þ g μ …μ ðu; sÞ ; ð76Þ 120 2 ð 1 kÞ k¼0 1 1 1 2 −1 ˆ ˆ ˆ μν ˆ 2 þ μ trðg˜ ÞR − RμνR þ R ; ð70Þ μ 6 5 15 where σˆ ðx; x0Þ is tangent to the geodesic connecting x with x0 at the point x. Note that the background field dependent where we have introduced the abbreviation coefficients ΨðkÞ ðu; sÞ only involve positive powers of ðμ1…μkÞ the parameters u and s.Ind 2ω 4 dimensions, only −n μν −1 μα1 −1 α2 −1 ν ¼ ¼ ðg˜ Þ ≔ ðg˜ Þ ðg˜ Þα ðg˜ Þα : ð71Þ 1 n−1 terms of the integrand with total s-dependency 1=s contribute to the logarithmically divergent part.3 Therefore, in In (70), we used that the Euler characteristic view of (74), the divergent contributions originate from the Z Z Ψ 1 1 s-independent parts of . Finally, the Gaussian integrals in χ M ≔ 4xgˆ1=2Gˆ 4xg˜1=2G˜ ; (74) are evaluated ð Þ 32π2 d ¼ 32π2 d ð72Þ Z 1 Y4 1 defined in terms of the Gauss-Bonnet term (27),isa μ μ1 μ2 α β σˆ σˆ σˆ k − Gαβσˆ σˆ 4π 2 d exp 4 topological invariant and therefore independent of the ð Þ μ¼1 metric. This allows to combine both contributions from 1 −1 μ …μ gˆμν and g˜μν in (70). The evaluation of the divergent 1 2k ¼ 1=2 ½symkðG Þ : ð77Þ contributions from the remaining traces (67) and (68) G constitutes the most complex part of the calculation. Here, we have introduced the kth totally symmetrized Here, we only sketch the major steps. The details are power of a general rank two tensor Tμν, defined by provided in Appendix C. There are two main complications associated with the evaluation of the divergent parts of the μ …μ ð2kÞ! μ μ μ μ traces (67) and (68). First, the traces (67) and (68) involve ½sym ðTÞ 1 2k ≔ Tð 1 2 T 2k−1 2kÞ: ð78Þ ν k 2k ! propagators δμ=D1 and 1=D0 with different spin. Second, k the propagators are defined with respect to different metrics. Therefore, we have to explicitly perform the After evaluation of the Gaussians (77), the parameter convolution of the corresponding kernels integral over u remains and the final result is expressed in terms of basic elliptic integrals, defined for 0 ≤ l ≤ 2k, Z μ0 δν † 1 Z 2ω 2ω 0 Πν δˆ 0 Π δ˜ 0 ∞ 1=2 T2 ¼ d xd x ðx;x Þ μ0 ðx ;xÞ ; ð73Þ μ …μ gˆ l −1 μ …μ D1 D0 1 2k ≔ 1 2k I 2k;l du 1 2 u ½symkðG Þ : ð79Þ ð Þ 0 G = where we have defined ω ≔ d=2. Inserting the Schwinger- DeWitt representation for the kernels of 1=D1 and 1=D0, Integrals of the form (79) occur unavoidably in the multi- provided in Appendix C, the traces T2 and T4 are ultimately metric case and are characteristic to the problem. They reduced to Gaussian integrals. In case of a single metric this constitute irreducible structures and can in general not be procedure has been outlined in [20]. In the case of two trivially integrated as for the case of a single metric 4 metric structures, the problem becomes more complicated g˜μν ¼ gˆμν. The evaluation of the trace T4 proceeds in an and has been discussed in [27]. The resulting Gaussian analogue way. The individual results for T2 and T4 are integral is provided explicitly in Appendix C. Z Z Z ∞ ∞ 1 ds 3 2ω ˆ1=2 ω−2 The choice of the parametrization in terms of s and u T2 ¼ 2ω d xg duu ω−1 ð4πÞ 0 0 s guarantees that the divergent contribution is isolated in the Z 2ω s-integration, whereas the u-integration is finite. Y 1 4 ˜ — σˆ μ Ψ σˆ μ − σˆ ασˆ β An interesting observation is that even for general gμν in × d ð Þ exp 4 Gαβ : ð74Þ principle—the integrals (79) can be evaluated explicitly in d ¼ 4 μ¼1 dimensions. We comment on this in more detail in Appendix D.

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D. Final result: One-loop divergences for the the results for the divergent part of the traces T2 and T4, generalized Proca model which are explicitly provided in (C35) and (C45). Using the The final result for the divergences of the generalized integral identities presented in Appendix D, the number of Proca model in curved spacetime (40) are obtained by different integrals (79) can be reduced. This allows us to adding the standard minimal second order traces (70) and represent the final result in the compact form

Z 1 Gˆ 1 1 1 1 1 1 div 4 1=2 −1 μρ −1 νσ ˜ ˜ ˜ 2 ˆ ˆ μν ˆ 2 2 ˜ 2 −1 ˆ Γ xgˆ − g˜ g˜ RμνRρσ − R − RμνR R μ R μ g˜ R 1 ¼ 32π2ε d 15 60 ð Þ ð Þ 120 5 þ 15 þ 3 þ 6 trð Þ 1 1 2 −1 μν ˆ α ˆ 4 −2 ð0;0Þ ð2;1Þ μν ð4;1Þ μνρσ ð4;2Þ μνρσ − μ ðg˜ Þ ð∇αδΓ μν þ 7RμνÞ − μ trðg˜ ÞþC I 0 0 þ Cμν I þ CμνρσI þ CμνρσI ; ð80Þ 6 4 ð ; Þ ð2;1Þ ð4;1Þ ð4;2Þ

1 1 ð0;0Þ μ2 ˜−2 μν ˜ − μ2 ˜−1 ˜ C ¼ 6 ðg Þ Rμν 12 trðg ÞR; ð81Þ

1 1 1 1 1 1 1 1 ð2;1Þ μ4 ˜−2 − μ2 ˜−1 μ2 ˜−1 ˆ μ2 ˜−1 αβ − ˆ − ∇ˆ δΓ ∇ˆ δΓ − ∇ˆ δΓ Cμν ¼ 4 ðg Þμν 4 ðg Þμν 2 trðg Þþ3 R þ ðg Þ 2 Rαμβν 6 α βμν þ 3 β ναμ 3 μ ναβ 5 1 1 1 1 1 ∇ˆ δΓ − δΓ δΓλ δΓλ δΓ − δΓ δΓλ δΓ δΓλ − δΓ δΓλ þ 12 μ αβν 4 αλβ μν þ 4 μα λνβ 12 αλμ νβ þ 3 μλα νβ 2 μλν αβ 1 1 − μ2 ˜−1 α 2 ˜ − 5 ˆ − ∇ˆ δΓλ μ2 ˜−1 δΓα δΓβ 6 ðg Þμð Rαν Rαν λ ανÞþ12 trðg Þð μβ ανÞ; ð82Þ

1 limit of a flat spacetime, as the integrals (79), associated ð4;1Þ − μ2 ˜−1 ∇ˆ δΓ ˜ Cμνρσ ¼ 24 trðg Þ μ νρσ; ð83Þ with the presence of the second metric gμν (the background mass tensor Mμν) remain. The result (80) is considerably 1 1 1 more complicated than the one-loop result for the non- ð4;2Þ − μ2∇ˆ δΓ μ2δΓ Γλ − μ2δΓ Γλ Cμνρσ ¼ 6 μ νρσ þ 2 μλν ρσ 4 λμν ρσ: degenerate vector field. In particular, the result cannot simply be obtained in the limit λ → 0 from (26). ð84Þ μν For special choices of M , the general result (80) simplifies and the integrals (79) can be evaluated explicitly. This constitutes our main result. For compactness, we present the one-loop divergences in terms of the two metrics gˆμν and g˜μν. The result in terms of the original VII. CHECKS AND APPLICATIONS OF THE μν metric gμν and the background mass tensor M can easily GENERALIZED PROCA MODEL be recovered by making use of (43) and (45). In this section we reduce our general result (80) to As expected on general grounds, the result (80) is a local several special cases. This provides important cross checks expression, which contains up to four derivatives. The fact and interesting applications. that (80) is not a polynomial of the invariants of Mμν,is related to the role of Mμν as metric in the scalar Stückelberg sector. We emphasize that this result holds for an arbitrary A. Trivial index structure positive definite symmetric background tensor Mμν. The The simplest case for the general background mass original assumption of a strictly positive Mμν is reflected in tensor Mμν, which goes beyond the constant Proca mass the result (80), as there is no smooth limit Mμν → 0. Note term Mμν ¼ m2gμν, is the reduction to a spacetime depen- that the result (80) is independent of the auxiliary mass dent scalar function X2ðxÞ, such that Mμν acquires trivial scale μ, which can be seen by the invariance of (80) under index structure rescaling of μ → αμ, with arbitrary constant α. The result has been derived in curved spacetime without Mμν ¼ X2gμν: ð85Þ considering graviton loops. Nevertheless, the consistency of the renormalization procedure would require to include In particular, this includes the case where X2 is proportional μν the induced kinetic terms for gˆμν an g˜μν (for gμν and M to the curvature scalar R, which is relevant in cosmological respectively) in the bare action. The result (80) shows that vector field models [4–11]. In view of (85), it is easy to the essential complexity is not reduced considerably in the see that

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2 B. Vector field with quartic self interaction ˆ X gμν ¼ 2 gμν; ð86Þ μ The generalized Proca action for a vector field Aμ with quartic self-interaction is considered in [27], ðg˜−1Þμν ¼ gˆμν; ð87Þ Z 1 α 4 1=2 μν μ 2 S½A¼ d xg F μνF þ ðAμA Þ : ð95Þ λ 4 4 δΓ μν ¼ 0; ð88Þ The part quadratic in the quantum fluctuation reads ðG−1Þμν ¼ð1 þ uÞ−1gˆμν; ð89Þ Z 1 ¯ 4 1=2 μν S2½A; δA¼ d xg F μνðδAÞF ðδAÞ 1 4 ˆ 4 detðGμνÞ¼ð þ uÞ detðgμνÞ: ð90Þ α ¯ ¯ ρ μν 2 ¯ μ ¯ ν δ δ In this case, the integrals (79) are trivially evaluated þ 2 ðAρA g þ A A Þ Aμ Aν ; ð96Þ Z ∞ l ¯ μ1μ2 μ …μ u where the vector field Aμ has been split into background Aμ k ˆ 1 2k I 2k;l jg˜¼gˆ ¼½symkðgÞ du k 2 δ ð Þ 0 ð1 þ uÞ þ and perturbation Aμ, − l !l! ðk Þ μ …μ ¯ ¼ ½sym ðgˆÞ 1 2k : ð91Þ Aμ ¼ Aμ þ δAμ; ð97Þ ðk þ 1Þ! k

FðδAÞ¼∇μδAν − ∇νδAμ: ð98Þ Inserting (86)–(90) and the explicit results (91) for the integrals into the general result (80), we obtain the one-loop In order to establish the connection to the generalized Proca result in terms of the original metric gμν, model (40), we identify the background mass tensor Mμν as Z 1 1 13 μν ρ μν μ ν div 4 1=2 μν ¯ α ¯ ¯ 2 ¯ ¯ Γ xg G − RμνR M ðAÞ¼ ðAρA g þ A A Þ: ð99Þ 1 ¼ 32π2ε d 15 60 7 1 3 According to (43) and (45),wehave 2 − 2 − 4 þ 120 R 2 RX 2 X ρσ gˆμν g ξρξσ gμν; 1 ΔX 1 ΔX 2 ¼ð Þ ð100Þ − R − 3XΔX − : ð92Þ 6 X 2 X −1 μν −1 4 μν μ ν ðg˜ Þ ¼ 3 = ðgˆ þ 2ξ ξ Þ; ð101Þ As an additional (trivial) cross check of this result, we set 2 ˜ 31=4 ˆ − ξ ξ X ¼ m in (92) and recover the one-loop divergences for the gμν ¼ gμν 3 μ ν ; ð102Þ Proca model (39). An independent way to derive the result (92) is to insert 2 δΓλ ξ ∇ˆ λξ − ξ ∇ˆ ξλ − 3ξλ∇ˆ ξ (85) directly into the generalized Proca action (40). This μν ¼ 3 ð ðμ νÞ ðμ νÞ ðμ νÞ leads to the ordinary Proca action (34), but with the 2ξλξαξ ∇ˆ ξ constant mass m promoted to a spacetime dependent þ ðμj α jνÞÞ: ð103Þ function X, We have defined the normalized vector field Z 1 X2 4 1=2 F F μν μ ¯ S½A¼ d xg μν þ AμA : ð93Þ Aμ 4 2 ξ ≔ 31=8α1=2 ξμ ≔ ˆμνξ μ μ ; g ν; ð104Þ By performing the Weyl transformation (86), the reduced ξ ξμ 1 action (93) is identical to the Proca action (44), but with such that μ ¼ .For(100) and (101), the integrals (79) gμν → gˆμν and m → μ, reduce to expressions of the form Z 2 Xk 1 μ μ1…μ2 μ μ μ μ μ μ 4 1=2 μα νβ μν k n ˆð 1 2 ˆ 2n−1 2n ξ 2nþ1 ξ 2kÞ S½A¼ d xgˆ gˆ gˆ F μνF αβ þ gˆ AμAν : ð94Þ Ið2k;lÞ ¼ dð2k;lÞg g : ð105Þ 4 2 n¼0

The one-loop divergences for (94) are obtained from (39) We provide a closed form expression for the coefficients ˆ → n by performing the inverse Weyl transformation gμν gμν dð2k;lÞ in Appendix D. We obtain the divergent part of the and agree with those obtained from the reduction (92) of the one-loop effective action for (95) from the general result by general result. inserting (101)–(103) as well as (105) with (D11) into (80),

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Z 1 1 1 1 1 1 1 pffiffiffi div 4 1=2 ˆ ˆ ˆ μν ˆ 2 −1 μα −1 νβ ˜ ˜ ˜ 2 4 Γ xgˆ G − RμνR R − g˜ g˜ RμνRαβ − R − 9 2 3 μ 1 ¼ 32π2ε d 15 5 þ 15 60 ð Þ ð Þ 120 6 ð þ Þ pffiffiffi 2 pffiffiffi 1 pffiffiffi 1 pffiffiffi 4 3μ2 −31 8 3 ˆ ξμξν 3 − 4 3 ˆ −31 8 3 ∇ˆ ξμ 2 þ 45 ð þ ÞRμν þ 18 ð ÞR þ 45 ð þ Þð μ Þ 1 pffiffiffi 1 pffiffiffi −274 136 3 ∇ˆ ξ ∇ˆ ξρ ξμξν 67 − 36 3 ∇ˆ ξ ∇ˆ μξν þ 45 ð þ Þð μ ρÞð ν Þ þ 45 ð Þð μ νÞð Þ : ð106Þ

Instead of ξμ and g˜μν, the authors in [27] use a different the result for the higher orders of the perturbative expan- parametrization. The conversion between our result (106) sion. In d ¼ 4, the result of any total antisymmetrization and their result is easily accomplished by (100) and (104). among five or more indices is necessarily zero We find that the reduction of our general result to the case of the self-interacting vector field (106) is in agreement α β γ δ ω δ μδνδρδσδλ ≡ 0; ð109Þ with the result obtained in [27]. This provides a powerful ½ check of our general result (80). where the total antisymmetrization is performed with unit weight. By contracting (109) with the background tensors C. Perturbative treatment of the ρσ Rμνρσ and ∇μ ∇μ Y , we can systematically construct generalized Proca model 1 n dimensional dependent invariants, which vanish in d ¼ 4 Finally, we test our method by a perturbative calculation, dimensions. At linear order of the expansion in Y, there is which only relies on the well-established generalized one such invariant Schwinger-DeWitt technique [20]. For this purpose, we assume that the background mass tensor has the form α β γ δ ω μ ν ρ σ λ δ δνδρδ δ I1 ¼ ½μ σ λR α βR γ δY ω μν 2 μν μν μν γδα γδ M ¼ m g þ Y ; ð107Þ ¼ Y ðGgμν − 4Rμ Rνγδα þ 8RμγνδR 8 γ − 4 with Yμν ≪ m2gμν and perform an expansion in Yμν. þ Rμ Rνγ RμνRÞ: ð110Þ

1. Expansion of the general result At quadratic order of the expansion there are two independent dimensional dependent invariants The expansion of the general result (80) up to second order reads α β γ δ ω μ ν ρ σ λ δ δνδρδ δ ∇ ∇ I2 ¼ ½μ σ λR α β γY δ Y ω; ð111Þ Γdiv Γdiv Γdiv Γdiv O Y3 1 ¼ 1;ð0Þ þ 1;ð1Þ þ 1;ð2Þ þ ð Þ; ð108Þ α β γ δ ω μ ν ρ σ λ I3 ¼ δ μδνδρδσδλ R α βY γ∇δ∇ Y ω: ð112Þ Γdiv ½ where 1;ðiÞ is the divergent part of the one-loop effective action, which contains terms of ith order in the perturbation Since use of (111) and (112) does not lead to any Y Γdiv . The zeroth order 1;ð0Þ is simply given by the one-loop simplification of our result, we refrain from presenting divergences (39) for the Proca field. Before we proceed, let the explicit expressions. For the first order of the expansion Γdiv Y us discuss the structure of the invariants used to represent 1;ð1Þ, linear in , we obtain

Z 1 1 5 3 1 div 4 1=2 μν 2 μν ρσ μν Γ ¼ d xg RY − R Yμν − m Y þ ½−8RμρνσR Y þ 2RμνR Y 1;ð1Þ 32π2ε 12 6 4 240m2 μν 2 μν μν − 4RRμνY þ R Y − 4ðΔRÞY þ 8ð∇μ∇νRÞY þ 4ðΔRμνÞY ; ð113Þ

≔ μν Γdiv Y where we have defined the trace Y gμνY . For the second order 1;ð2Þ, quadratic in , we make use of the tensor algebra bundle XACT [28–30] and find

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Z 1 1 1 1 div 4 1=2 μν 2 μν μν ρ 2 Γ ¼ d xg − YμνY − Y þ ½−2RYμνY þ 12R Yμ Yνρ þ RY 1;ð2Þ 32π2ε 8 16 48m2 μν μν ρσ μν ρ νρ μν −4R YμνY − 4RμρνσY Y þ 12Y ∇ν∇ρYμ þ 4Y∇ρ∇νY þ 2Y ΔYμν − YΔY

1 μν 2 2 2 μν 2 μν ν αρ μσ μ νρ þ ½−2RμνR Y − R Y þ 8RμνRYY − 2R YμνY − 16Rα RνμρσY Y − 16RμρR νYY 960m4 μ νρ μν ρσ μν ρσ μν ρσ μν ρσ þ8RμνRY ρY þ 32R RμρνσYY þ 24RμρRνσY Y − 8RμνRρσY Y − 8RRμρνσY Y α β μν ρσ μ ν ρσ μν ρσ μν ρσ −16Rμ ν RρασβY Y − 8RμρR νY σY − 4RμνR YρσY þ 16RμρνσY ΔY μν α ρσ μν α ρσ νρ νρ σ σ ρ μν þ32RμσναY ∇ ∇ρY þ 32RνρσαY ∇μ∇ Y − 8RνρY ΔY − 16RνσY ΔY ρ þ 16RμρνσY∇ ∇ Y ρμ ν σ ρμ νσ μ νρ μ μν þ16RσνY ∇μ∇ Y ρ − 32RνσY ∇μ∇ρY þ 2Rð∇μYÞ∇ Y þ 16Rνρð∇μY Þ∇ Y þ 24Rð∇μYÞ∇νY ν σρ μ μν μν ν σρ μ μ σ νρ −16Rσ ð∇μY Þ∇νYρ − 4RY ∇ν∇μY − 12RY∇ν∇μY − 16Rσ Y ∇ν∇μYρ þ 16Rμνρσð∇ YÞ∇ Y νρ μ μν ρ ν μρ μν σ αρ −8RμρY ∇ν∇ Y þ 16RY ∇ν∇ρYμ − 40Rμ Y∇ν∇ρY þ 32Rμρνσð∇αY Þ∇ Y ν ρμ σ μ ν ρ μν σρ μν −80Rσ Y ∇ν∇ Yρμ þ 4Rμνð∇ YÞ∇ Y þ 8Rð∇νYμ Þ∇ρY þ 24Rσμð∇νY Þ∇ρY νρ σμ μν μν σμ νρ μ νρ þ40RνσY ∇ρ∇μY − 24RY ΔYμν − 16RμνYΔY − 16Rνρð∇μY Þ∇σY þ 16Rμνð∇ YÞ∇ρY μν μν 2 μν ρσ μν ρ 2 þ16YΔ∇ν∇μY − 4Y Δ Yμν − 16Y ∇ν∇μ∇σ∇ρY − 8Y ∇νΔ∇ρYμ − 2YΔ Y : ð114Þ

Similar to the calculation for the nondegenerated vector 2. Perturbative calculation via the generalized field, we use the exact operator identity (36) for the Proca Schwinger-DeWitt technique 2 operator F0 þ m 1, which in flat spacetime reads The second order expansion (108) of the general result (80) can be checked by a direct perturbative calculation, which only δν ∂ ∂ν 1 μ δ ν − μ relies on the well-established generalized Schwinger-DeWitt 2 ¼ μ 2 2 2 : ð117Þ F0 þ m m −∂ þ m technique, introduced in [20]. Although conceptually straight- forward, the complexity grows rapidly with growing powers Note that the similarity to the expansion (25) for the of Y and is already quite involved for the expansion up to Y nondegenerate vector field might be misleading here, as second order in . Moreover, we are mainly interested in a (116) is an expansion in Y, not an expansion in background check of the structures involving derivatives of Y not tested by — dimension. This is seen by comparing the counting of the previous checks apart from the three structures involving background dimension for the two cases. While powers of ΔX that remain in (92) after the reduction of M to 2 Y ¼ OðM Þ, expansion in Y in (116) is not efficient as the trivial index structure (85). Therefore, we restrict the direct 2 −2 1=ðF0 þ m Þ¼OðM Þ, such that the combination perturbative calculation to flat spacetime gμν ¼ δμν, ∇μ ¼ ∂μ. 0 Y1=F0 ¼ OðM Þ. Therefore the perturbative series Starting point is the action (40) for the generalized Proca (116) continues up to arbitrary order in Y. The counting field, but with the background mass tensor M treated 2 −2 of background dimension 1= F0 m O M can be perturbatively as in (107). The one-loop divergences up to ð þ Þ¼ ð Þ understood from (117),as second order in Y are obtained by expanding (42) around the Proca operator defined in (35), 1 O M0 2 2 ¼ ð Þ; ð118Þ 2 −∂ þ m F ¼ðF0 þ m 1ÞþY: ð115Þ

1 ν 2 ν −2 ð−∂μ∂ þ m δμÞ¼OðM Þ: ð119Þ Using the split (115) in the expansion of the logarithm up to m2 second order in Y, we find In contrast, in case of the non-degenerate vector field, the 2 2 Tr1 ln F ¼ Tr1 ln ðF0 þ m 1 þ YÞ analogue expansion (25) in P ¼ OðM Þ is efficient in 0 1 background dimension as 1=Fλ ¼ OðM Þ, such that the F 21 Y 2 3 ¼ Tr1 ln ð 0 þ m ÞþTr1 2 combination P1=Fλ ¼ OðM Þ and terms OðP Þ are F0 þ m already finite. The difference between the background 1 1 1 2 − Y Y counting of 1=Fλ and 1=ðF0 þ m Þ can be traced back 2 Tr1 2 2 : ð116Þ F0 þ m F0 þ m to the fact that DetFλ ≠ 0 while DetF0 ¼ 0.

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ðn;pÞ Inserting the identity (117) into the expansion (116),we In d ¼ 4 dimensions, the Uμ …μ ’s are divergent for a O Y2 1 p obtain the following sum of traces up to ð Þ: degree of divergence 2 Tr1 ln F ¼ Tr1 ln ðF0 þ m 1Þ χ − 2 4 ≤ 0 div ¼ p n þ : ð125Þ δν 1 ρ ρ þ Tr0 B þ Tr1 Yμ −∂2 þ m2 −∂2 þ m2 Each commutator reduces the number of derivatives p in (124) and therefore decreases the degree of divergence δσ ν 1 ρ ρ λ δλ − Tr1 Yμ Yσ (125) by one. This shows that the iterative procedure (122) 2 −∂2 þ m2 −∂2 þ m2 is efficient for the calculation of the divergent contribu- δσ ν ρ ρ λ ∂λ∂ 1 tions. In flat spacetime, divergences only arise for þ Tr1 Yμ Yσ χ 0 −∂2 þ m2 m2 −∂2 þ m2 div ¼ k ¼ ,2,4, 1 1 1 −1 n 22−n−k 2k − ðn;2nþ2kÞ ð Þ m Tr0 B 2 2 B 2 2 : ð120Þ Uμ …μ δ 2 1 2nþ2k ¼ 2 ½symnþkð Þμ1…μ2 2 : ð126Þ −∂ þ m −∂ þ m 16π ε k!ðn − 1Þ! nþ k Here, we have defined the scalar operator Following the strategy outlined above, we first calculate the 1 trivial traces which do not involve the evaluation of any ∂ ≔ ∂ μν∂ O M0 Bð Þ 2 μY ν ¼ ð Þ; ð121Þ commutator: m 1 3 and used the cyclicity of the trace to convert two vector 2 div 4 Tr1 ln ðF0 þ m 1Þj ¼ − m ; traces in (120) into scalar traces. Next we evaluate the 16π2ε 2 divergent contributions of the individual traces in (120) div 1 1 2 separately. The first trace is just that of the Proca operator Tr0 B ¼ ð−m YÞ; −∂2 þ m2 16π2ε (35). The remaining traces can be further reduced by 2 2 δν div 1 1 iterative commutation of all powers of 1=ð−∂ þ m Þ to ρ ρ 2 Tr1 Yμ ¼ m Y ; the right, using the identity −∂2 þ m2 16π2ε 4 δσ ν div ρ ρ λ δλ 1 μν 1 1 2 1 ; Y ¼ − ½−∂ ; Y : Tr1 Yμ −∂2 2 Yσ −∂2 2 ¼ 16π2ε YμνY : −∂2 þ m2 −∂2 þ m2 −∂2 þ m2 þ m þ m ð122Þ ð127Þ Each iteration of (122) generates one additional commu- The remaining two traces require the evaluation of nested tator, which increases the number of derivatives acting on commutators. For the first trace we find Y the background tensor by at least one, δσ ν div ρ ρ λ ∂λ∂ 1 2 2 ρ Tr1 Yμ 2 2 Yσ 2 2 2 ½−∂ ; Y¼ð−∂ YÞ − 2ð∂ YÞ∂ρ: ð123Þ −∂ þ m m −∂ þ m ∂ ∂ 1 1 μν 1 μρ ρ ν ν In this way the calculation of the divergent part of the trace ¼ YμνY þ Y Yμ 16π2ε 2 3 m2 (120) is reduced to the evaluation of a few universal 2 functional traces [20], 1 μν ð−∂ Þ þ Y 2 Yμν : ð128Þ 1 div 12 m ðn;pÞ 2 Uμ …μ ðm Þ¼∂μ …∂μ : ð124Þ 1 p 1 p −∂2 2 þ m x0¼x For the second trace we find

div 2 ∂ ∂ 2 ∂ ∂ 1 1 1 1 μν 1 2 1 μν ð−∂ Þ 1 μν ν ρ ρ 1 ð−∂ Þ 1 μ ν μν Tr0 B B ¼ YμνY þ Y þ Y Yμν þ Y Yμ þ Y Y − Y Y −∂2 þm2 −∂2 þm2 16π2ε 4 8 12 m2 6 m2 24 m2 6 m2 1 ∂ ∂ ∂ ∂ 1 ∂ −∂2 ∂ 1 −∂2 ∂ ∂ 1 −∂2 2 μν μ ν ρ σ ρσ μν νð Þ ρ ρ − ð Þ μ ν μν μν ð Þ þ30Y 4 Y þ60Y 4 Yμ 30Y 4 Y þ120Y 4 Yμν m m m m 1 ð−∂2Þ2 þ Y Y : ð129Þ 240 m4

Adding the contributions (127)–(129) according to (120), we obtain the final result for the one-loop divergences on a flat background up to second order in Yμν,

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Z 1 3 3 1 1 1 −∂2 1 ∂ ∂ 1 −∂2 div 4 4 2 μν 2 μν ð Þ μν ν ρ ρ ð Þ Γ ¼ d x − m − m Y − YμνY − Y þ Y Yμν þ Y Yμ − Y Y 1 32π2ε 2 4 8 16 24 m2 4 m2 48 m2 1 ∂ ∂ 1 ∂ ∂ ∂ ∂ 1 ∂ −∂2 ∂ 1 −∂2 ∂ ∂ μ ν μν − μν μ ν ρ σ ρσ − μν νð Þ ρ ρ ð Þ μ ν μν þ 12 Y 2 Y 60 Y 4 Y 120 Y 4 Yμ þ 60 Y 4 Y m m m m 2 2 2 2 1 μν ð−∂ Þ 1 ð−∂ Þ − Y Yμν − Y Y : ð130Þ 240 m4 480 m4

The result (130) is in perfect agreement with the result B. Comparison: Method of nonlocal field redefinition for second order expansion (114) of the general result (80) The authors of [13] have obtained a result for the one- δ reduced to a flat background gμν ¼ μν. This provides a loop divergences of the generalized Proca model (40), powerful check of our general result. In particular, it probes without relying on any perturbative expansion in Y tensorial derivative structures, not captured by the check for (denoted X in [13]). They use the Stückelberg formulation the trivial index structure, discussed in Sec. VII A. It also to rewrite the generalized Proca theory as a gauge theory for provides an important independent check of our method, as the original Proca field and the Stückelberg scalar field. it has been obtained in a complementary way by a direct They derive the corresponding block matrix fluctuation application of the generalized Schwinger-DeWitt method. operator similar to (52). As the authors discuss, with In particular, it neither relies on the Stückelberg formalism respect to the metric gμν, this operator is both, nonminimal nor on the bimetric formulation. as well as not block-diagonal. The authors perform a “shiftlike” transformation of the quantum vector field Aμ VIII. COMPARISON WITH RESULTS [Eq. (22)], IN THE LITERATURE μ μ μρ A ¼ B þ α ∂ρφ: ð133Þ In this section, we compare our results with the one-loop divergences of the generalized Proca model (40) obtained Here, αμν is an a priori undetermined background tensor. previously by different methods and techniques [12,13]. Next, they derive a condition for which the fluctuation operator is diagonalized [Eq. (25)]. However, in contrast to A. Comparison: Local momentum space method the statement of the authors, we believe αμν has to be an In [12], based on the local momentum space method operator instead of a background tensor in order to [31], an expression for the one-loop divergences of the diagonalize the fluctuation operator. This is critical for generalized Proca model (40) has been obtained. The result the algorithm used in [13]. Consequently, our general result 2 2 (80) does not coincide with the one given in [Eq. (45)] of [Eq. (2.24)] in [12], includes terms of order OðR ;RY;Y Þ and, in our conventions, reads [13]. In particular, the result of [13] contains nonlocal Z structures. 1 1 13 7 αμν div 4 1=2 μν 2 Nevertheless, whether is an operator or a background Γ ¼ d xg G − RμνR þ R 1 32π2ε 15 60 120 tensor is not relevant for the contribution to the one-loop 1 3 5 1 3 divergences at linear order in Y, as according to [Eq. (25)], − 2 − 4 − μν − 2 αμν Y 2 m R 2 m 6 RμνY þ 12 RY 4 m Y is first order in and therefore the nontrivial contribution in [Eq. (26)] only affects higher orders in Y, 1 1 μν 2 O Y2 − YμνY − Y : ð131Þ starting at ð Þ. This explains why the authors reproduce 8 16 their result with the generalized Schwinger-DeWitt tech- Y The result (131) agrees with the second order expansion nique [20] at linear order in . The result at linear order in (108) of our general result (80) under the assumption [Eq. (53)] of [13] is in agreement with our result (113) for the approximation linear in Y. Note that this comparison is 2 2 Y ≪ m ;R≪ m ; ∇ ≪ m: ð132Þ nontrivial in the sense that their result involves additional In particular, this means that no terms proportional to terms proportional to the invariant (110), which vanishes in d 4 inverse powers of m appear in the result of [12] (and ¼ dimensions. therefore, apart from total derivatives, no structures involv- IX. CONCLUSIONS ing derivatives of Y). In contrast, our second order expansion (108) was derived only under the assumption We have investigated the renormalization of generalized Y ≪ m2. The coincidence with the terms in (131) therefore vector field models in curved spacetime. We have intro- provides an independent check of several structures linear duced a classification scheme for different vector field and quadratic in Y. models based on the degeneracy structure of their

025009-14 RENORMALIZATION OF GENERALIZED VECTOR FIELD … PHYS. REV. D 98, 025009 (2018) associated fluctuation operator. The distinction between This model has been studied earlier in [27]. We find perfect different degeneracy classes is partially connected to the agreement. Furthermore, we have expanded our general nonminimal structures present in the principal symbol of result (80) up to second order in the deviation from the the fluctuation operator. The simplest theories, where such Proca model and compared it to a direct perturbative nonminimal structures can appear, are vector field theories, calculation. We find perfect agreement. As the direct but these terms are also important for rank two tensor fields, calculation only relies on standard techniques, the agree- as has been recently discussed for the degenerate fluc- ment does not only provide a powerful check of our general tuation operator in the context of fðRÞ gravity [3]. result (80), but also of the method we used to derive it. The nondegenerate vector field and the Abelian gauge Finally, we have compared our results as well as our field are both representatives of two different degeneracy approach to previous work on the generalized Proca model. classes, for which the calculation of the one-loop divergen- In [12], part of our full result for the one-loop divergences ces can be performed with standard methods, based on the (80) has been obtained by a different method. In the generalized Schwinger-DeWitt technique [20].Wehave corresponding limit, we find that our result reduces to briefly reviewed these cases and have discussed the tech- the one derived in [12].In[13], yet another method, based nical details associated with the underlying algorithm for the on a combination of the Stückelberg formalism and a one-loop calculation. The Proca theory of the massive vector nonlocal field redefinition, has been proposed. However, field is the simplest representative in the class of non- the result for the one-loop divergences, obtained in [13], degenerate fluctuation operators with a degenerate principal does not agree with our result (80). In general, it is quite symbol. Only for this special case, standard methods are remarkably that the simple extension from the Proca model directly applicable. In particular, none of these models in the to the generalized Proca model leads to such a drastic different degeneracy classes can be obtained from one increase of complexity—already in flat space. another in a smooth limit—a fact which is related to the Our result for the one-loop divergences of the general- discontinuity in the number of propagating d.o.f. Therefore, ized Proca model (80) with a background mass tensors of μν μν μν the different classes have to be studied separately. the form M ¼ ζ1R þ ζ2Rg is important for cosmo- The generalized Proca model, which results from the Proca logical models, which, at the classical level, have been 2 μν – model by generalizing the constant mass term m g to a local studied extensively [4 11]. It would also be interesting to background mass tensor Mμν, is considerably more compli- apply the method presented in this paper in the context of cated and can no longer be treated directly by standard massive gravity [32,33], more general vector field models – methods. Therefore, we have applied the Stückelberg for- [34 36] and scalar-vector-tensor models [37,38]. malism in order to reformulate the generalized Proca model as a gauge theory, where the background mass tensor plays a ACKNOWLEDGMENTS double role as potential in the vector sector and additional The authors thank I. L. Shapiro for correspondence. The metric in the scalar sector of the Stückelberg field. At the price work of M. S. R. is supported by the Alexander von of dealing simultaneously with two metrics, the standard Humboldt Foundation, in the framework of the Sofja methods are applicable in this case. Kovalevskaja Award, endowed by the German Federal Our main result is the derivation of the one-loop Ministry of Education and Research. divergences for the generalized Proca model (80).A characteristic feature of this new result is the appearance APPENDIX A: GENERAL FORMALISM of the tensorial parameter integrals (79). The vector field loops induce curvature and M-dependent structures. 1. Heat kernel and one-loop divergences Unless these structures are present in the original action, For an action functional S½ϕ of a general field ϕ with the generalized Proca model is not perturbatively components ϕA, the fluctuation operator Fð∇xÞ, obtained renormalizable—not even in flat space. It is interesting from the second functional derivative has components that the main complication of the generalized Proca model A x AC x F Bð∇ Þ¼γ FCBð∇ Þ, where γAB is a symmetric, non- is not connected to the curved background but originates degenerate and ultralocal bilinear form. The Schwinger from the presence of the second metric structure. integral representation of 1=F reads, We have checked our general result (80) by reducing it to Z ∞ simpler models. The one-loop divergences for the trivial 1 − F μν 2 μν ¼ dse s ; ðA1Þ index case M ¼ X g can be obtained in two ways: by F 0 the reduction of the general result (80) and independently from the Proca model by a Weyl transformation. We find where s is the “proper time” parameter and where we perfect agreement. Moreover, to the best of our knowledge, have indicated the bundle structure of inverse operators 1 δA the trivial index case is by itself a genuinely new result. In by the identity matrix , which has components B. The addition, we have performed the reduction of our result (80) Schwinger representation of higher inverse powers 1=Fn μ 2 to the case of a vector field with a ðAμA Þ self-interaction. with n ∈ N and the logarithm of F are found to be

025009-15 MICHAEL S. RUF and CHRISTIAN F. STEINWACHS PHYS. REV. D 98, 025009 (2018) Z μ 0 1 ∞ σ ∇μa0 x; x 0; a0 x; x 1: A13 ds n−1 −sF ð Þ¼ ð Þ¼ ð Þ n ¼ s e ; ðA2Þ F 0 ðn − 1Þ! Therefore, the parallel propagator matrix Z ∞ ds −sF PA 0 ≔ A ln F ¼ − e : ðA3Þ B0 ðx; x Þ ½a0 B0 ðx; x0Þ ðA14Þ 0 s parallel transports a field ϕAðxÞ at x to a field ½PϕA0 ðx0Þ at The integrand of the proper time integral (A1) defines the x0 along the unique geodesic connecting x with x0. It only heat kernel 0 0 μ A agrees with ϕA ðx Þ if σ ∇μϕ ðxÞ¼0. It satisfies Kðsjx; x0Þ ≔ e−sFδðx; x0Þ: ðA4Þ PA PB0 δA B0 C ¼ C: ðA15Þ With the boundary condition Kð0jx; x0Þ¼δðx; x0Þ, it formally satisfies the heat equation Since for a general bitensor, the primed and unprimed indices indicate the corresponding tensorial structure at a ∂ F K 0 0 ð s þ Þ ðsjx; x Þ¼ : ðA5Þ given point, the arguments are omitted whenever there is no possibility for confusion. The coincidence limits x0 → x of For a minimal second order operator the Schwinger-DeWitt coefficients an and their derivatives F ¼ Δ þ P; ðA6Þ can be obtained recursively. Using dimensional regularization, the Schwinger-DeWitt algorithm gives a closed a Schwinger-DeWitt representation for the corresponding result for the divergent part of the one-loop effective action kernel exists of a minimal second order operator (A6) for a generic ϕ 4 1 2 field .Ind ¼ dimensions, the result is given in terms of = 0 σ 0 0 g ðx Þ 1=2 0 − ðx;x Þ 0 the coincidence limit of the second Schwinger-DeWitt Kðsjx; x Þ¼ D ðx; x Þe 2s Ωðsjx; x Þ; ðA7Þ ð4πsÞω coefficient with ω ¼ d=2. The biscalar σðx; x0Þ is Synge’s world 1 Γdiv Δ P div function [39,40], which is defined by 1 ¼ Tr ln ð þ Þj 2 Z μ μ μν 1 σ σ 2σ σ ≔ ∇ σ σ ∇ σ 4 1=2 μ ¼ ; μ μ ; ¼ g ν : ðA8Þ − xg a2 x; x ; ¼ 32π2ε d tr ð Þ ðA16Þ The biscalar Dðx; x0Þ is the dedensitized Van-Vleck 1 determinant αβγδ αβ a2ðx;xÞ¼ ðRαβγδR −RαβR −6ΔRÞ1 180 2 ∂ σ 0 2 0 −1=2 −1=2 0 ðx; x Þ 1 1 1 1 Dðx; x Þ¼g ðxÞg ðx Þ det μ ν ; ðA9Þ P− 1 R Rαβ ΔP ∂x ∂x0 þ2 6R þ12 αβ þ6 : ðA17Þ which is defined by the equation Here, 1=ε is a pole in dimension ε ¼ d=2 − 2 and the −1 μ R A D ∇μðDσ Þ¼2ω: ðA10Þ bundle curvature αβ with components ½Rαβ B is defined by the commutator All nontrivial physical information is encoded in the 0 ∇ ∇ ϕA A ϕB matrix-valued bitensor Ωðsjx; x Þ, ½ μ; ν ¼½Rμν B : ðA18Þ X∞ In particular, for a scalar and vector field, we have 0 n 0 Ωðsjx; x Þ¼ s anðx; x Þ; ðA11Þ 0 σ n¼ ½∇μ; ∇νφ ¼ 0; ½∇μ; ∇νAρ ¼ Rμνρ Aσ: ðA19Þ where the dependence on the proper time parameter s has been explicitly separated by making a power series ansatz ’ with the matrix-valued Schwinger-DeWitt coefficients 2. Covariant Taylor expansion and Synge s rule 0 anðx; x Þ. Inserting the ansatz (A7) together with (A11) The covariant Taylor expansion of a scalar function fðxÞ and the minimal second order operator (A6) into the heat around x0 ¼ x is given by [20], equation (A5), gives a recurrence relation for the Schwinger-DeWitt coefficients X∞ −1 k 0 ð Þ 0 μ1 μ fðx Þ¼ ½∇μ0 ∇μ0 fðx Þ σ σ k : ðA20Þ k! 1 k x0¼x 1 σμ∇ a D−1=2F D1=2a 0 k¼0 ½ðn þ Þþ μ nþ1 þ ð nÞ¼ ; ðA12Þ 0 ϕA where an ≡ 0 for n<0 implies that a0ðx; x Þ satisfies the The generalization of this expansion for fields ðxÞ PA parallel propagator equation requires use of the parallel propagator B0,

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A00 A00 B0 ½Pϕ ¼ P B0 ϕ : ðA21Þ specify the metric with respect to which we perform the covariant Taylor expansion, as the metric enters the world The right-hand side of (A21) transforms as scalar at x0. function σ and the definition of the covariant derivative ∇μ. Therefore, we can consider the right-hand side as a scalar For a metric g˚μν, not necessarily compatible with the 0 function of x and apply the covariant Taylor expansion connection ∇μ, it is natural to introduce the tensor which 0 (A20) around x ¼ x, measures the difference between the connection ∇μ and the ˚ X∞ −1 k Levi-Civita connection associated with gμν, A00 ð Þ A00 B0 P B0 ϕB0 ¼ ½∇μ0 ∇μ0 P B0 ϕ 0 ! 1 k x ¼x 0 k 1 k¼ δ˚Γρ ˚−1 ρα ∇ ˚ ∇ ˚ − ∇ ˚ μ μ μν ¼ ðg Þ ð μgαν þ νgμα αgμνÞ: ðA27Þ × σ 1 σ k . ðA22Þ 2 Using that the coincide limits of the totally symmetrized We provide the coincidence limits for the world function σ˚ 1 2 covariant derivatives acting on the parallel propagator are and Van-Vleck biscalar D˚ = as well as derivatives thereof zero, up to OðM2Þ, A ½∇ μ0 …∇μ0 P 0 ¼ 0;k>0; ðA23Þ ð 1 kÞ B x0¼x σ˚ 0 ½ x0¼x ¼ ; ðA28Þ the parallel propagator can be freely commuted though the 00 0 ∇ σ˚ 0 derivatives in (A22). Further setting x to x and making use ½ μ x0¼x ¼ ; ðA29Þ of (A13), we find ∞ ∇ ∇ σ˚ ˚ X k ½ μ ν x0¼x ¼ gμν; ðA30Þ 0 ð−1Þ 0 A A B μ1 μk ϕ ¼ P 0 ½∇μ0 ∇μ0 ϕ σ σ : B k! 1 k x0¼x k¼0 ∇ ∇ ∇ σ˚ δ˚Γα ˚ δ˚Γα ˚ δ˚Γα ˚ ½ μ ν ρ x0¼x ¼ μνgαρ þ μρgνα þ νρgμα ðA24Þ 3∇ ˚ ¼ ðμgνρÞ; ðA31Þ A Applying this expansion to a general bitensor T B0 around 0 A0 2 α β coincidence x ¼ x , shows that all the information of T B ∇ ∇ ∇ ∇ σ˚ − ˚ 3˚ δ˚Γ δ˚Γ ½ μ ν ρ σ 0 ¼ Rμ ρ ν σ þ gαβ μ σ νρ is contained in the coincidence limits of its derivatives x ¼x 3 ð j j Þ ð Þ 2˚ ∇ δ˚Γα δ˚Γα δ˚Γβ þ gαðρjð μ ν σ þ μβ ν σ Þ X∞ −1 k j Þ j Þ 0 0 ð Þ 0 μ μ A PA ∇ ∇ C σ 1 σ k α α β T B ¼ C ½ μ0 μ0 T Bx0 x : 2˚ ∇ δ˚Γ δ˚Γ δ˚Γ k! 1 k ¼ þ gαðμð νÞ ρσ þ νÞβ ρσÞ; k¼0 A32 ðA25Þ ð Þ ∇ D˚ 1=2 0 3. Coincidence limits ½ μ x0¼x ¼ ; ðA33Þ The coincidence limits for σ, D1=2, a as well as n 1 derivatives thereof can be obtained recursively by repeat- ˚ 1=2 ˚ ½∇μ∇νD 0 ¼ Rμν: ðA34Þ edly taking derivatives of the defining equations (A8), x ¼x 6 (A10), (A12), and (A13). Commutation of covariant derivatives to a canonical order induces curvature terms. The corresponding coincidence limits of the Schwinger- Inserting the coincidence limits from lower orders of the DeWitt coefficients read recursion, higher coincidence limits are obtained system- a˚ 1 atically [14]. In case a bitensor involves derivatives at ½ 0x0¼x ¼ ; ðA35Þ different points, we can recursively reduce the coincidence ∇ a˚ 0 limits of primed derivatives to coincidence limits involving ½ α 0x0¼x ¼ ; ðA36Þ only unprimed derivatives by Synge’s rule [39,40], 1 ∇ ∇ a˚ R ∇ ∇ − ∇ ½ α β 0 0 ¼ αβ; ðA37Þ ½ μ0 Tx0¼x ¼ μ½Tx0¼x ½ μTx0¼x: ðA26Þ x ¼x 2

Here, T represents an arbitrary bitensor. The first 1 a˚ ˚ 1 − P 1=2 ½ 1x0¼x ¼ R : ðA38Þ few coincidence limits of σ, D and an are easily 6 obtained. In this article, apart from the coincidence limit 4 ˚ for a2 ¼ OðM Þ, provided already in (A17), we only need In case the metric gμν is compatible with the connection 2 coincidence limits up to OðM Þ. Note that, when perform- ∇μg˚νρ ¼ 0, the coincidence limits (A28)–(A38) reduce to ing the covariant Taylor expansion (A20), we have to the well-known results [20].

025009-17 MICHAEL S. RUF and CHRISTIAN F. STEINWACHS PHYS. REV. D 98, 025009 (2018) Z 1 11 7 1 APPENDIX B: DETAILS OF THE CALCULATION div 4 1=2 μν 2 1 Fλ xg G− RμνR R ; FOR THE NONDEGENERATE VECTOR FIELD Tr ln j ¼ 16π2ε d 180 30 þ20 The traces in (25) can be systematically reduced to the ðB9Þ evaluation of tabulated universal functional traces Z 1 div 1 1 γ F div Δ div P 4 1=2 Tr1 ln λj ¼ Tr1 ln Hj ; ðB1Þ Tr1 ¼ 2 d xg þ RP Fλ 16π ε 6 12 1 div δν div 1 div γ ν μ μν − 1 μν 1 P Pμ − γP ∇μ∇ν ; þ RμνP ; ðB10Þ Tr ¼ Δ Δ2 ðB2Þ 6 Fλ H x0¼x x0¼x Z 1 1 1 1 div 1 P P 4 1=2 Tr1 P P div Tr1 ¼ 2 d xg Fλ Fλ Fλ Fλ 16π ε 1 div 1 div γ γ2 γ2 P νP ν − 2γPνρP μ∇ ∇ 1 μν 2 ¼ μ μ Δ2 ρ μ ν Δ3 × þ 2 þ 12 PμνP þ 24 P : x0¼x x0¼x 1 div 2 μν ρσ ðB11Þ γ P P ∇μ∇ν∇ρ∇σ : þ Δ4 ðB3Þ x0¼x Adding all traces according to (25), we obtain the final In (B1), we have used the definition of the Hodge operator result (26). (4) along with the identity (24) and the fact that 1 div ν APPENDIX C: MULTIPROPAGATOR Tr1 ln 1 þ λ∇μ ∇ ¼ 0; ðB4Þ Δ BIMETRIC TRACES OF THE GENERALIZED PROCA MODEL which is formally seen by expanding the logarithm, making use of the cyclicity of the trace and resumming the terms. 1. Divergent part of the second order trace 4 The traces in (B3) are already OðM Þ, which allows to The evaluation of the second order trace T2 constitutes freely commute all operators and use the most complex part of the calculation. The functional trace T2 is divergent and needs to be regularized. We use μ δν 1 regularization in the dimension d 2ω. Explicitly, the δμ O M ¼ Δ ¼ ν Δ þ ð Þ: ðB5Þ functional trace of the convoluted integral kernels is H given by The logarithmic trace (B1) is evaluated directly with (A16), β while for the remaining traces we use the following α δα † 1 T2 ¼ Tr0 Π Πβ universal functional traces D1 D0 Z ν 1=2 β0 δμ div g 1 2ω 2ω 0 Σ 0 Σ2 0 ν ν ¼ d xd x ½ 1 ðx; x Þ β0 ðx ;xÞ; ðC1Þ Rδμ − Rμ ; Δ ¼ 16π2ε 6 ðB6Þ H x0¼x β0 1 2 Σ 0 Σ2 0 1 div = 1 1 where the kernels 1 ðx; x Þ and β0 ðx ;xÞ are defined as ∇ ∇ g − μ ν 2 ¼ 2 Rμν Rgμν ; ðB7Þ Δ 0 16π ε 6 12 x ¼x β0 0 δα 1 Σβ 0 ≔ Πα δˆ 0 Σ2 0 ≔ Π† δ˜ 0 1 div 1=2 −1 n 1 ðx; x Þ ðx; x Þ; β0 ðx ;xÞ β0 ðx ;xÞ: g ð Þ D1 D0 ∇μ ∇μ ¼ 1 2n−4 Δn 16π2ε 2n−2 − 1 ! x0¼x ðn Þ ðC2Þ × sym −2 g : B8 ½ n ð Þμ1…μ2n−4 ð Þ First, we insert the integral representations (A1), (A4), and Inserting (B6)–(B8) into (B1)–(B3), we find (A7) for the kernels of the inverse propagators

Z μ0 ∞ δν σˆðx;x0Þ ˆ 0 ds − ˆ 1=2 0 ˆ μ0 0 1=2 0 δðx; x Þ¼ ω e 2s D ðx; x ÞΩν ðsjx; x Þgˆ ðx Þ; ðC3Þ D1 0 ð4πsÞ

025009-18 RENORMALIZATION OF GENERALIZED VECTOR FIELD … PHYS. REV. D 98, 025009 (2018) Z 1 ∞ σ˜ 0 ˜ 0 dt − ðx ;xÞ ˜ 1=2 0 ˜ 0 1=2 δðx ;xÞ¼ ω e 2t D ðx ;xÞΩ0ðtjx ;xÞg˜ ðxÞ; ðC4Þ D0 0 ð4πtÞ Πα Π† together with the explicit expressions (63) for and μ0 into (C2). Then, we expand the derivatives in each factor in (C2) according to the Leibniz rule and collect terms up to OðM3Þ, β0 0 δα β 0 ˆ −1 σα ˆ 0 Σ1 ðx; x Þ¼μ∇σðg˜ Þ δðx; x Þ D1 Z ∞ ds σˆ 1 2 −1 2 1 2 1 0 1=2 −2 ˆ = ˆ −1 σα −1 σα ˆ = ˆ ˆ = −1 σα β ¼ μ ω gˆ e sD ∇σðg˜ Þ þðg˜ Þ D ð∇σD Þ − σˆ σðg˜ Þ Pα 0 ð4πsÞ 2s 1 ˜−1 σα ∇ˆ Pβ0 − σˆ ˜−1 σα β0 O M4 þðg Þ ð σ α Þ 2 σðg Þ ½a1α þ ð Þ; ðC5Þ

1 Σ2 0 −μ ˜−1 σ0 ∇ˆ δ˜ 0 β0 ðx ;xÞ¼ ðg Þβ0 σ0 ðx ;xÞ D0 Z ∞ dt 1 2 σ˜ 0 1 0 0 −1 2 1 2 1=2 ˜ = −2 −1 σ ˆ −1 σ −1 σ ˜ = ˆ ˜ = ¼ −μ ω g˜ D e t ðg˜ Þβ0 ∇σ0 a˜ 0 − ðg˜ Þβ0 σ˜ σ0 a˜ 0 þðg˜ Þβ0 D ∇σ0 D a˜ 0 0 ð4πtÞ 2t −1 σ0 4 −ðg˜ Þβ0 σ˜ σ0 a˜ 1gþOðM Þ: ðC6Þ

Next, we apply the covariant Taylor expansion (A25) 1 ˜ 1=2 ˜ α β separately to the terms in the curly brackets in (C5) and D ¼ 1 þ Rαβσˆ σˆ ; ðC10Þ (C6) up to terms with background dimension OðM2Þ. This 12 requires knowledge of the covariant Taylor expansion of the basic geometrical bitensors up to OðM2Þ, 1 ˜−1 σ0 ∇˜ D˜ 1=2 − P ρ ˜−1 α ˜ σβ ðg Þβ0 ρ0 ¼ 6 β0 ðg Þρ Rαβ ; ðC11Þ 1 σ˜ ˜ σˆ ασˆ β − ˜ δΓλ σˆ ασˆ βσˆ γ ¼ gαβ 2 ðgαλ βγÞ 1 1 ˆ ˆ 1=2 ˆ β 4˜ δΓν δΓλ 3˜ δΓν δΓλ ∇αD ¼ Rαβσˆ ; ðC12Þ þ 24 ð gαν βλ γδ þ gνλ αβ γδ 6 ˆ λ α β γ δ 4g˜αλ∇βδΓ γδ σˆ σˆ σˆ σˆ ; C7 þ Þ ð Þ 1 ˆ β0 β0 ˆ λ γ ð∇σPα Þ¼− Pλ Rσγα σˆ ; ðC13Þ 1 1 2 −1 ρ0 ρ α β λ ðg˜ Þβ0 σ˜ ρ0 ¼ Pβ0 −σˆ ρ − δΓραβσˆ σˆ − ðδΓραλδΓ βγ 2 6 1 β0 ˆ ρ ˆ ρ 2 −1 ρ β0 ˆ α β γ a1 Rα − Rδα μ g˜ Pρ : C14 − 2∇γδΓραβÞσˆ σˆ σˆ ; ðC8Þ ½ α ¼ 6 þ ð Þα ð Þ

1 ˆ 1=2 ˆ α β For the derivation of (C7)–(C14), we made use of (A28)– D ¼ 1 þ Rαβσˆ σˆ ; ðC9Þ 12 (A38). Inserting (C7)–(C14) into (C5) and (C6) yields

Z ∞ 1 1 β0 0 β0 ds 1=2 − σˆ ˆ 1=2 ˆ −1 αρ −1 α ˆ νρ γ −1 αρ ˆ γ Σ1 ðx ;xÞ¼μP ρ ω gˆ e 2sD ∇αðg˜ Þ − ðg˜ Þν Rαγ σˆ þ ðg˜ Þ Rαγσˆ 0 ð4πsÞ 2 6 1 1 1 −1 ρ γ ˆ νρ ˆ νρ 2 −1 νρ −1 γ 4 − ðg˜ Þγ σˆ þ R − Rgˆ þ μ ðg˜ Þ ðg˜ Þνγσˆ þ OðM Þ; ðC15Þ 2s 2 6 Z ∞ D˜ 1=2 σˆ 1 1 2 dt 1 2 1=2 − σ˜ ρ γ δ λ ˆ γ δ ϵ Σ 0 −μ ˜ = Dˆ 2t δΓ σˆ σˆ δΓ δΓ − 2∇ δΓ σˆ σˆ σˆ β0 ðx; x Þ¼ ω g 1 2 e þ ργδ þ ð ργλ δϵ ϵ ργδÞ 0 ð4πtÞ Dˆ = 2t 4t 12t 1 1 − ˜−1 ργ ˜ σˆ δ σˆ ˜ Pρ O M4 6 ðg Þ Rγδ þ 12 ρR β0 þ ð Þ; ðC16Þ

025009-19 MICHAEL S. RUF and CHRISTIAN F. STEINWACHS PHYS. REV. D 98, 025009 (2018) where we have artificially separated a factor of Dˆ 1=2 in (C16). This procedure of dealing with multiple propagators in a functional trace was proposed in [20]. Here, an additional complication is due to the presence of the two metrics gμν and g˜μν, which in addition requires to expand the world function σ˜ in the exponent of the second product as well as the ratio D˜ 1=2=Dˆ 1=2, D˜ 1=2 σ˜ ˜ σˆ μσˆ ν 1 1 gμν λ α β γ ν λ ν λ ˆ λ α β γ δ exp − ¼ exp − 1 þ g˜αλδΓ σˆ σˆ σˆ − ð4g˜ανδΓ δΓ þ 3g˜νλδΓ δΓ þ 4g˜αλ∇βδΓ Þσˆ σˆ σˆ σˆ Dˆ 1=2 2t 2t 4t βγ 48t βλ γδ αβ γδ γδ 1 1 λ ν α β γ δ ϵ η ˜ ˆ μ ν 3 þ g˜αλg˜βνδΓ δΓϵησˆ σˆ σˆ σˆ σˆ σˆ þ ðRμν − RμνÞσˆ σˆ þ OðM Þ: ðC17Þ 32t2 γδ 12

Finally, combining (C15), (C16), and the expansion (C17), we collect all terms up to order OðM2Þ in a function Ψðs; tjσμÞ, which allows us to write the trace (C1) as Z Z ∞ dsdt 1 −μ2 2ω 2ω 0Dˆ ˆ1=2Ψ σˆ μ − σˆ ασˆ β T2 ¼ 2ω ω ω d xd x g ðs; tj Þ exp Gαβðs=tÞ : ðC18Þ 0 ð4πÞ s t 4s

The main complexity related to the presence of the two metrics manifests in the “interpolation metric” Gμν appearing in the exponential of (C18). The interpolation metric relates the two metrics gˆμν and g˜μν via the parameter z,

GμνðzÞ ≔ gˆμν þ zg˜μν: ðC19Þ

By construction, Ψðs; tjσˆ μÞ is a polynomial in σˆ μðx; x0Þ,

X8 μ 1 2 − 2 ðkÞ μ μ Ψ σˆ ≔ ˆ = k= Ψ σˆ 1 σˆ k ðs; tj Þ g s μ1…μk ðs; tÞ : ðC20Þ k¼2 ΨðkÞ The coefficients μ1…μk ðs; tÞ are local tensors parametrically depending on s and t. The nonzero even coefficients are 1 1 Ψð2Þ − ˜−1 − s ˜−1 ˆ λ η s ˜−2 η ˜ s ˜−1 ˆ λ αβ ðs; tÞ¼ 4 ðg Þαβ 4 ðg ÞληR α β þ 12 ðg Þα Rβη þ 3 ðg ÞαλR β t t t t 1 −1 s ˆ ˜ s 2 −2 s η ˆ −1 λ 3 − ðg˜ Þαβ R þ R þ μ ðg˜ Þαβ þ δΓ ∇λðg˜ Þη þ OðM Þ; ðC21Þ 24 t 4t 4t αβ

ð4Þ s η ˆ −1 λ s −1 λ η s −1 ˆ λ s −1 ˜ ˆ 3 Ψ ðs; tÞ¼ g˜βηδΓ ∇λðg˜ Þα − ðg˜ ÞαλδΓ δΓ þ ðg˜ Þαλ∇βδΓ − ðg˜ ÞαβðRγδ − RγδÞþOðM Þ; ðC22Þ αβγδ 8t γδ 24t βη γδ 12t γδ 48t

ð6Þ s −1 λ η λ η ˆ λ s −1 λ η 3 Ψ ðs; tÞ¼ ðg˜ Þαβð4g˜γλδΓ δΓμν þ 3g˜ληδΓ δΓμν þ 4g˜γλ∇δδΓμνÞ − ðg˜ Þαλg˜βηδΓ δΓμν þ OðM Þ; ðC23Þ αβγδμν 192t δη γδ 32t γδ

ð8Þ s −1 λ η 3 Ψ ðs; tÞ¼− ðg˜ Þαβg˜δλg˜γηδΓμνδΓρσ þ OðM Þ: ðC24Þ αβγδμνρσ 128t Changing integration variables from xμ0 → σˆ μðx; x0Þ leads to the Jacobian ∂ μ0 ∂2σˆ 0 −1 x ρν ðx; x Þ 1 2 −1 2 ˆ −1 J ¼ det ¼ det gˆ ¼ gˆ = ðxÞgˆ = ðx0ÞD ðx; x0Þ; ðC25Þ ∂σˆ ν ∂xμ0 ∂xρ

ˆ 1 2 μ which cancels the factor Dðx; x0Þgˆ = ðx0Þ in (C18). Thus, we write (C18) as Gaussian integral over σˆ ,

Z Z Z 2ω μ2 ∞ dsdt Y 1 − 2ω ˆ1=2 σˆ μ Ψ σˆ μ − σˆ ασˆ β T2 ¼ 2ω d xg ω ω d ðs; tj Þ exp Gαβðs=tÞ : ðC26Þ 4π 0 s t 4s ð Þ μ¼1

We further perform a reparametrization ðs; tÞ → ðs; uÞ, which allows to easily extract the divergent structure

025009-20 RENORMALIZATION OF GENERALIZED VECTOR FIELD … PHYS. REV. D 98, 025009 (2018) s s ∂ðs; tÞ s u ¼ ;t¼ ; det ¼ : ðC27Þ t u ∂ðs; uÞ u2

All ultraviolet divergences are captured by the lower bound s → 0 of the s-integral and (C26) acquires the form Z Z Z Z 2ω μ2 ∞ ∞ ds Y 1 − 2ω ˆ1=2 ω−2 σˆ μ Ψ σˆ μ − σˆ ασˆ β T2 ¼ 2ω d xg duu 2ω−1 d ðu; sj Þ exp GαβðuÞ : ðC28Þ 4π 0 0 4s ð Þ s μ¼1 In order to extract the divergent part, we reparametrize the world function by absorbing a factor of s−1=2, Y2ω Y2ω σˆ → σˆs1=2; dσˆ μ → sω dσˆ μ : ðC29Þ μ¼1 μ¼1 Finally, inserting the covariant Taylor expansion for Ψ, (C28) reads Z Z Z Z 2 ∞ X4 ∞ Y2ω μ 2ω 1 2 ω−2 ds ð2kÞ μ μ 1 α β − ˆ = Ψ σˆ σˆ u1 σˆ 2k − σˆ σˆ T2 ¼ 2ω d xg duu ω−1 μ1μ2k ðu; sÞ d exp GαβðuÞ : 4π 0 0 s 4 ð Þ k¼0 μ¼1 ðC30Þ Dimensional regularization annihilates all power law divergences and turns the logarithmically divergent s-integrals for ω → 2 into poles 1=ε in dimension. Thus, we extract the divergent part of (C30) in d ¼ 2ω ¼ 4 dimensions by collecting all terms in the integrand with total s-dependency 1=s.Forω ¼ 2, the prefactor is already of this form. Therefore, only the parts Ψð2kÞ ≔ Ψð2kÞ 0 μ1…μ2k ðuÞ μ1…μ2k ð ;uÞ, which are independent of s, contribute to the divergent part Z Z Z 2 ∞ X4 Y4 μ 4 1 2 ð2kÞ μ μ 1 α β div − ˆ = Ψ σˆ σˆ u1 σˆ 2k − σˆ σˆ T2 ¼ 4 d xg du μ1…μ2kðuÞ d exp GαβðuÞ : ðC31Þ 4π ε 0 4 ð Þ k¼0 μ¼1 Performing the Gaussian integrals yields Z Y4 1 μ μ μ 1 α β 1 −1 μ …μ dσˆ σˆ 1 σˆ 2k exp − Gαβσˆ σˆ ¼ ½sym ðG Þ 1 2k ; ðC32Þ 4π 2 4 1=2 k ð Þ μ¼1 G with the kth totally symmetrized power of the inverse interpolation metric ðG−1Þμν,

−1 μ …μ ð2kÞ! −1 μ μ −1 μ μ ½sym ðG Þ 1 2k ¼ ðG Þð 1 2 ðG Þ 2k−1 2kÞ: ðC33Þ k 2kk! The fact that the Gaussian averages vanish for an odd number of σˆ μ ’s, a posteriori justifies that we have neglected these terms in the covariant Taylor expansion (C20). Note that the resulting expressions are background tensors parametrically depending on u. The divergent part of the trace (67) is given by Z Z 2 ∞ X4 μ 4 1 2 1 −1 μ …μ ð2kÞ div − ˆ = 1 2k Ψμ …μ T2 ¼ 2 d xg du 1 2 ½symkðG Þ 1 2k ðuÞ : ðC34Þ 16π ε 0 = k¼0 G

div Finally, the result for T2 can be expressed as linear combination Z μ2 X 4 1 2 ð2k;lÞ μ1…μ2 Tdiv ¼ − d xgˆ = Cμ …μ I k ; ðC35Þ 2 16π2ε 1 2k ð2k;lÞ k;l with u-integrals

Z ∞ 1=2 μ …μ gˆ l −1 μ …μ 1 2k ≔ 1 2k I 2k;l du 1 2 u ½symkðG Þ ; ðC36Þ ð Þ 0 G =

025009-21 MICHAEL S. RUF and CHRISTIAN F. STEINWACHS PHYS. REV. D 98, 025009 (2018)

ð2k;lÞ and u-independent coefficient tensors Cμ1…μ2k , 1 1 ð2;0Þ − ˜−1 ˜ ˜−2 β ˜ Cμν ¼ 12 ðg ÞμνR þ 6 ðg Þμ Rνβ; ðC37Þ 1 2 1 1 1 ð2;1Þ μ2 ˜−2 ˜−1 α ˆ − ˜−1 ˆ − ˜−1 αβ ˆ δΓα ∇ˆ ˜−1 β Cμν ¼ 2 ðg Þμν þ 3 ðg ÞμRαν 12 ðg ÞμνR 2 ðg Þ Rμανβ þ 2 μν βðg Þα; ðC38Þ 1 1 1 ð4;1Þ − δΓα δΓβ ˜−1 ˜−1 ˆ − ˜ ˜−1 ∇ˆ δΓα Cμνρσ ¼ 6 μν ραðg Þσβ þ 12 ðg ÞμνðRρσ RρσÞþ3 ðg Þμα ρ νσ; ðC39Þ 1 ð4;2Þ δΓα ˜ ∇ˆ ˜−1 β Cμνρσ ¼ 2 μνgασ βðg Þρ; ðC40Þ

1 1 1 1 ð6;2Þ δΓλ δΓη ˜ ˜−1 δΓλ δΓη ˜ ˜−1 − δΓλ δΓη ˜ ˜−1 ˜ ˜−1 ∇ˆ δΓλ Cμνρσαβ ¼ 6 μν σλgρηðg Þαβ þ 8 μν σρgληðg Þαβ 4 μν σρgαλðg Þβη þ 6 gμλðg Þνσ β ρα; ðC41Þ

1 it is possible to derive a sequence of useful integral ð8;3Þ − δΓλ δΓη ˜ ˜ ˜−1 Cμνρσαβγδ ¼ 8 μν ρσgαλgβηðg Þγδ: ðC42Þ identities αβμ …μ μ …μ ∇ˆ ˜ 1 2k −2∇ˆ 1 2k ð λgαβÞIð2kþ2;lÞ ¼ λIð2k;l−1Þ; ðD2Þ μ …μ 2. Divergent part of the fourth order trace 2 − l 1 2k l αβμ …μ ðk ÞI 2 l k> 4 1 2k ð k; Þ O M gˆαβI ; D3 Since the trace (68) is T4 ¼ ð Þ, the operators in the ð2kþ2;lÞ ¼ −1 μ …μ ð Þ 2 g˜ 1 2k k l trace T4 can be freely commuted and we can use ½symkð Þ ¼ αμ …μ μ …μ νμ …μ −1 ν 1 2kþ1 −1 νðμ1 2 2kþ1Þ 1 2kþ1 ðg˜ ÞαI ¼ 2kðg˜ Þ I − I ; δβ 1 ð2kþ2;lÞ ð2k;lÞ ð2kþ2;lþ1Þ α δβ O M ¼ α þ ð Þ: ðC43Þ D4 D1 Δˆ ð Þ αμ …μ μ …μ νμ …μ ν 1 2kþ1 νðμ1 2 2kþ1Þ 1 2kþ1 g˜αI 2 2 l ¼ 2kgˆ I 2 l−1 − I 2 2 l−1 ; ðD5Þ Explicitly, the divergent part of T4 acquires the form ð kþ ; Þ ð k; Þ ð kþ ; Þ αβμ …μ μ …μ μ …μ λ Z δΓ 1 2k ∇ˆ 1 2k 2 δΓ ðμ1 1 2kÞ 1 1 div ανβIð2kþ2;lÞ ¼ λIð2k;lÞ þ k μν Ið2k;lÞ : ðD6Þ div 4 4 −2 αβ −2 γδ ˆ ˆ ˆ ˆ T ¼ μ d xðg˜ Þ ðg˜ Þ ∇α∇β∇γ∇δ : 4 ˆ 2 Δ˜ 2 Δ x0¼x ðC44Þ 2. Evaluation of the integrals for the general case An interesting observation is that in d ¼ 4 dimensions div div The trace T4 is evaluated in the same way as T2 . We only the integrals (79) can be evaluated in terms of invariants of state the final result the metric g˜μν. The evaluation of all tensor integrals (79) Z can be reduced to the evaluation of the fundamental scalar 1 μ4 4 div 4 ˆ1=2 ˜−1 ˜−2 integral Ið0;0Þ.Ind ¼ dimensions, the Cayley-Hamilton T4 ¼ 2 d xg ½trðg Þtrðg ÞIð0;0Þ ν 16π ε 4 theorem guarantees that the eigenvalues λ1; …; λ4 of g˜μ − ˜−1 ˜−2 ρσ − ˜−2 α can be expressed in terms of the invariants e ≔ trðg˜−kÞ trðg Þðg Þ Ið2;0Þρσ trðg ÞIð2;1Þα k with k ¼ 1; …; 4. Therefore, the fundamental integral I 0 0 ˜−2 μν α ð ; Þ þðg Þ Ið4;1Þμνα : ðC45Þ can be expressed in terms of the eigenvalues λkðejÞ, Z Z ∞ ˆ1=2 ∞ g qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidu APPENDIX D: FUNDAMENTAL INTEGRALS Ið0;0Þ ¼ du 1=2 ¼ Q : ðD7Þ 0 G ðuÞ 0 4 1 λ k¼1 ð þ u kÞ 1. Integral identities Starting from the fundamental integral identities The integral (D7) can be evaluated explicitly and expressed in terms of the incomplete elliptic function of the first kind. ∂ The general integrals (79) can then be obtained by differ- αβμ1…μ2k μ1…μ2k Il ¼ −2 Il−1 ˜ ˆ ∂g˜αβ entiating the result with respect to gμν and gμν and by making use of (D1). We refrain from performing these ∂ μ …μ −2 1 2k operations, as the resulting expressions are horrendously ¼ ∂ ˆ Il ; ðD1Þ gαβ complicated, impractical and not very illuminating. Instead,

025009-22 RENORMALIZATION OF GENERALIZED VECTOR FIELD … PHYS. REV. D 98, 025009 (2018) we choose to present the final result in terms of the much Xk μ1…μ2 μ μ μ μ μ μ k n ˆð 1 2 ˆ 2n−1 2n ξ 2nþ1 ξ 2kÞ more compact integrals (79). Ið2k;lÞ ¼ dð2k;lÞg g : ðD8Þ n¼0 3. Evaluation of the integrals for special cases n In the case of the self-interacting vector field considered The general coefficients dð2k;lÞ are given in a closed form in in Sec. VII B, the integrals have the form terms of the hypergeometric function 2F1,

Z 2k ! k ∞ n 3−ðl−1Þ=4 ð Þ lþk−n 3 n−k−1=2 1 −ðkþ3=2Þ d 2k;l ¼ n duu ð þ uÞ ð þ uÞ ð Þ 2 k! n 0 2 ! Γ − l 1 Γ l 1 2 −n ð kÞ ðlþ1Þ=4 ðk þ Þ ð þ = Þ ¼ 2 3 2F1ðk − l þ 1;k− n þ 1=2; −l þ 1=2; 3Þ n!ðk − nÞ! Γðk þ 3=2Þ Γ −l − 1 2 Γ l − 1 lþ1=2 ð = Þ ðk þ n þ Þ þ 3 2F1ðk þ 3=2;kþ l − n þ 1; l þ 3=2; 3Þ : ðD9Þ Γðk − n þ 1=2Þ

The hypergeometric function 2F1 is defined as X∞ ΓðcÞ Γða þ kÞΓðb þ kÞ zk F1ða; b; c; zÞ¼ : ðD10Þ 2 Γ a Γ b Γ c k k! k¼0 ð Þ ð Þ ð þ Þ

For the one-loop divergences, we only need the following coefficients pffiffiffi pffiffiffi 22 pffiffiffi 4 pffiffiffi d0 4 3 −1 3 ;d0 − 4 3;d1 − 3; ð0;0Þ ¼ ð þ Þ ð2;1Þ ¼ 3 ð2;1Þ ¼ 3 þ 4 pffiffiffi 4 pffiffiffi 1 pffiffiffi d0 ¼ ð73 − 42 3Þ;d1 ¼ ð−41 þ 24 3Þ;d2 ¼ ð7 − 3 3Þ; ð4;1Þ 5 ð4;1Þ 5 ð4;1Þ 5 4 pffiffiffi pffiffiffi 8 pffiffiffi pffiffiffi 9 pffiffiffi pffiffiffi d0 ¼ 4 3ð72 − 41 3Þ;d1 ¼ 4 3ð−27 þ 16 3Þ;d2 ¼ 4 3ð2 − 3Þ: ðD11Þ ð4;2Þ 5 ð4;2Þ 5 ð4;2Þ 5

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