Renormalization of Generalized Vector Field Models in Curved

Renormalization of generalized vector ﬁeld models in curved spacetime

Michael S. Ruf and Christian F. Steinwachs∗ Physikalisches Institut, Albert-Ludwigs-Universit¨atFreiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany

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.

PACS numbers: 04.60.-m; 04.62.+v; 11.10.Gh; 11.15.-q; 98.80.Qc

I. INTRODUCTION f(R) gravity [3] and the generalized Proca model considered in this article. Therefore, we make use of the Most models of inflation and dynamical dark energy Stückelberg formalism [23] to reformulate the generalized are based on scalar-tensor theories and f(R) gravity, Proca model as a gauge theory such that standard heat- which have an additional propagating scalar degree of kernel techniques become applicable again. The price to freedom. The one-loop quantum corrections to these pay is the introduction of a second metric tensor. models on an arbitrary background manifold have been This article is organized as follows: In Sec. II, we dis- derived for a general scalar-tensor theory in [1, 2] and cuss different vector field models in curved spacetime. In recently for f(R) gravity in [3]. particular, we introduce a classification based on their Aside from models based on an additional scalar field, degeneracy structure. In Sec. III, we calculate the one- vector fields have been studied in cosmology [4–11]. Most loop divergences for the nondegenerate vector field with of these models are characterized by a nonminimal cou- an arbitrary potential. In Sec. IV, we consider the case pling of the vector field to gravity and are particular cases of the Abelian gauge field and calculate the one-loop di- of the generalized Proca model. vergences. In Sec. V, we derive the one-loop divergences The quantum corrections for the generalized Proca for the Proca model of the massive vector field. In Sec. model are difficult to calculate and have been studied VI, we introduce the generalized Proca model, calculate recently in [12, 13] by different approaches. In this arti- the one-loop divergences in a closed form and present our cle, we use another approach, which allows us to derive main result. In Sec. VII, we perform several reductions of the one-loop divergences for the generalized Proca model our general result for the generalized Proca model to spe- in a closed form. cific cases. These reductions provide strong cross checks We use a combination of the manifest covariant back- of our general result and entail applications to cosmolog- ground field formalism and the heat kernel technique ical models. In Sec. VIII, we compare our result and [14–22]. This general approach can be applied to any our method to previous calculations of the one-loop di- type of field. The central object in this approach is the vergences for the generalized Proca model. Finally, in differential operator, which propagates the fluctuations Sec. IX, we summarize our main results and give a brief of the fields. For most physical theories, this fluctua- outlook on their implications. tion operator acquires the form of a second order min- Technical details are provided in several appendices. imal (Laplace-type) operator. For this simple class of In Appendix A, we introduce the general formalism and operators, a closed algorithm for the calculation of the provide a collection of tabulated coincidence limits, which arXiv:1806.00485v2 [hep-th] 11 Jul 2018 one-loop divergences exists [14]. For nonminimal and arise in the calculation. In Appendix B, we present the higher order operators, a generalization of the Schwinger- details of the calculation for the one-loop divergences of DeWitt algorithm, which allows to reduce the calculation the nondegenerate vector field considered in Sec. III. to the known case of the minimal second order opera- In Appendix C, we provide a detailed calculation of the tor, has been introduced in [20]. The direct application most complex functional traces, which contribute to the of the generalized Schwinger-DeWitt algorithm requires one-loop divergences of the generalized Proca model con- the nondegeneracy of the principal part—the highest- sidered in Sec. VI. In Appendix D, we collect several derivative term of the fluctuation operator. important integral identities. However, there are important cases, where the fluctuation operator has a degenerate principal part—notably

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General vector ﬁeld F( ) = F ( ) + P ∇ λ ∇

Non-degenerate vector ﬁeld Degenerate vector ﬁeld λ > 0 λ = 0

Abelian gauge field Generalized Proca field gauge fixing P = 0 P > 0

Trivial index structure Self-interacting vector ﬁeld Perturbative expansion (Y m2) ν 2 ν ν ¯ ¯ρ ν ¯ ¯ν ν 2 ν ν Pµ = X (x)δµ Pµ = α AρA δµ + 2AµA Pµ = m δµ + Yµ

Proca ﬁeld ν 2 ν Pµ = m δµ

FIG. 1. Overview of the diﬀerent degeneracy classes and reductions of the generalized Proca model.

II. VECTOR FIELD MODELS IN CURVED not indicated otherwise, derivative operators act on ev- SPACETIME: GENERAL STRUCTURE erything to their right. The Hodge operator satisﬁes

ν In this article we calculate the one-loop divergences [∆H]µ ν = µ∆ . (5) for generalized vector field models in a curved spacetime. ∇ ∇ The divergent part of the one-loop contribution to the An important property of the general second order vector effective action is defined by field operator (2) is its degeneracy structure, which is controlled by the parameter λ and the potential P. The 1 div Γdiv = Tr ln F( ) , (1) conditions λ 0 and P 0 ensure that the operator F is 1 2 ∇ | positive semidefinite.≥ Therefore,≥ there are three different where F( ) is the differential operator that controls the degeneracy classes: propagation∇ of the fluctuations. For the vector field models discussed in this work, this fluctuation operator has 1. The nondegenerate vector field: λ > 0, P 0 the particular form ≥

F( ) = F ( ) + P , (2) 2. The Abelian gauge field: λ = 0, P = 0 ∇ λ ∇ ν where P is a potential with components Pµ and Fλ( ) 3. The (generalized) Proca field: λ = 0, P > 0 is a differential operator with components ∇

ν ν ν The relation among the classes is depicted graphically in [Fλ] = [∆H] + (1 λ) µ . (3) µ µ − ∇ ∇ Fig. 1. Physically, the different classes correspond to in- Here, the Hodge operator on vector fields ∆H is defined equivalent theories with a different number of propagat- in terms of the positive definite Laplacian ∆ by ing degrees of freedom. Mathematically, this is reflected by the degeneracy structure of the operator (2), which is ν ν ν µν [∆H]µ := ∆δµ + Rµ , ∆ := g µ ν . (4) discussed in the following sections in detail. In particu- − ∇ ∇ lar, there is no smooth transition between the classes in In the terminology of [20], the operator (2) is called non- the limits λ 0 and P 0. Therefore, the different minimal for λ = 1 and minimal for λ = 1. Note that if classes have to→ be treated→ separately. 6 3

A. Degeneracy of the principal symbol is defined by the Euclidean action for the Abelian gauge field Aµ(x), An important structure in the theory of differential 1 operators F( ) is the leading derivative term—the prin- S[A] = d4xg1/2 µν . (10) ∇ 4 Fµν F cipal part D( ). Separating the principal part from the Z lower derivative∇ terms Π( ), the operator F( ) takes The Abelian field strength tensor µν is defined as the form ∇ ∇ F µν := µAν ν Aµ . (11) F( ) = D( ) + Π( ). (6) F ∇ − ∇ ∇ ∇ ∇ The action (10) is invariant under infinitesimal gauge Physically, the principal part D( ) contains the informa- transformations tion about the dominant ultraviolet∇ behavior of the underlying theory. Therefore, it is the natural starting point δεAµ = ∂µε , (12) for the generalized Schwinger-DeWitt method [14, 20], where ε(x) is the infinitesimal local gauge parameter. which relies on the expansion D Π of the associated Gauge invariance of (10) implies the Noether identity propagator, schematically δS[A] δS[A] 1 1 1 1 1 ∂ = g−1/2 = 0 . (13) = = Π + , (7) µ δA (x) ∇µ δA (x) F D + Π D − D D ··· µ µ The components of the fluctuation operator F are ob- where 1/F denotes the inverse of the linear operator F. tained from the Hessian Essential for this perturbative treatment is the notion of background dimension M, which is understood as the δ2S[A] ν x 0 : −1/2 mass dimension of the background tensorial coefficients Fµ ( )δ(x, x ) = g gµρ 0 , (14) ∇ δAρ(x)δAν (x ) of the differential operator. We write F = (Mk) for any operator F, which has at least backgroundO dimen- where the delta function is defined with zero density sion Mk. The expansion (7) critically relies on the in- weight at x and unit density weight at x0. The fluc- vertibility of D, which can be discussed at the level of tuation operator for the Abelian gauge field is given by the principal symbol D(n), formally obtained by replac- F = F0, which corresponds to the vector field operator ing derivatives µ by a constant vector field inµ. For the (2) with λ = 0 and P = 0. The explicit components read vector field operator∇ (2) the components of the principal F ν = [∆ ] ν + ν . (15) symbol read µ H µ ∇µ∇ ν nµn Taking the functional derivative of (13) with respect to D ν (n) = n2 δν (1 λ) , (8) 0 µ µ − − n2 Aν (x ) yields the operator equation µ ν 2 ρ Fµ = 0 . (16) with n := nρn . The parameter λ controls the degener- ∇ acy of the principal symbol, as can be seen easily from This implies that for P = 0 the total fluctuation operator the determinant (2) is degenerate – not only its principle symbol. There- fore, in case of a gauge degeneracy, in addition to the 2 4 det D(n) = λ n . (9) breakdown of the perturbative expansion (7) associated to the degeneracy of D, the inverse operator 1/F does For λ = 0, the determinant vanishes and therefore D(n) not even exist. In order to remove the gauge degeneracy is not invertible. The origin of this degeneracy can be we choose a gauge condition linear in Aµ, traced back to the fact that for λ = 0, the principal symbol has the structure of a projector on transversal vector χ(A) = 0 . (17) fields. This motivates the distinction between the two classes λ > 0 and λ = 0. For λ = 0, the further distinc- The total gauge-fixed action Stot := S + Sgb is obtained tion between the cases P = 0 and P > 0 is connected by adding the gauge breaking action with a degeneracy at the level of the full operator F, 1 discussed in the next subsection. S = d4xg1/2χ2 . (18) gb 2 Z This leads to a modification of the associated Hessian, B. Gauge degeneracy such that the resulting gauge-fixed operator Ftot is nondegenerate

The degeneracy at the level of the full operator F is 2 a general feature of any gauge theory. In the context of ν x 0 −1/2 δ Stot[A] [Ftot]µ ( )δ(x, x ) = g gµρ 0 . (19) the vector ﬁeld operator (2), the relevant gauge theory ∇ δAρ(x)δAν (x ) 4

The inclusion of the gauge breaking action must be com- Here, we have defined the Gauss-Bonnet term pensated by the corresponding ghost action := R Rµνρσ 4 R Rµν + R2 . (27) G µνρσ − µν ∗ 4 1/2 ∗ Sgh[ω, ω ] = d xg ω Q ω , (20) The result (26) is in agreement with [20, 24].1 Note that Z for P = 0, (26) is independent of the parameter γ. This ∗ where ω (x) and ω(x) are anticommuting scalar ghost calculation, as well as the calculation via the generalized fields. The ghost operator is defined as Schwinger-DeWitt algorithm in [20], both critically rely δ on the nondegeneracy of the principal symbol (9). Q( x)δ(x, x0) := χ (A(x) + δ A(x)) . (21) ∇ δε(x0) ε The divergent part of the one-loop contributions to the IV. THE ABELIAN GAUGE FIELD effective action is given by 1 The fluctuation operator F for the Abelian gauge field Γdiv = Tr ln F div Tr ln Q div . (22) 1 2 1 tot| − 0 | theory (10) is given by F ν = [∆ ] ν + ν . (28) µ H µ ∇µ∇ III. THE NONDEGENERATE VECTOR FIELD In view of the general operator (2), this corresponds to the case λ = 0 and P = 0. As discussed in Sec. II B, the We first consider the vector field operator (2) with a operator (28) is degenerate due to the gauge symmetry nondegenerate principal symbol and arbitrary potential of the action (10). We choose a relativistic gauge condi- F ν = [∆ ] ν + (1 λ) ν + P ν , λ > 0 , (23) tion with arbitrary gauge parameter η to break the gauge µ H µ − ∇µ∇ µ degeneracy of the operator (28), Since for λ = 0 the principal symbol of (23) is invertible, 1 the generalized6 Schwinger-DeWitt algorithm can be used χ(A) = µA . (29) −√1 + η ∇ µ directly [20]. The power of this algorithm lies in its gen- erality, as it is applicable to any type of field. However, According to (19) and (21), the components of the gauge- instead of using the general algorithm, here the calcula- fixed fluctuation operator Ftot and the corresponding tion can be essentially simplified by directly making use ghost operator Q read of an operator identity for Fλ, ν ν η ν [Ftot] = [∆H] + µ , (30) δν δν 1 µ µ 1 + η ∇ ∇ µ = µ γ ν , (24) µ 2 1 Fλ ∆H − ∇ ∆ ∇ Q = ∆ . (31) √1 + η where we have defined γ := (1 λ)/λ. Since P = M2 − O and 1/F = M0 , we can make efficient use of (24). Thus, the gauge-fixed fluctuation operator (30) falls into λ For the calculationO of the one-loop divergences, it is suf- the class of nondegenerate vector fields (23) with P = 0 ficient to expand the logarithm up to M4 , and λ = 1/(1 + η). Therefore, the divergent part of O the one-loop effective action can be calculated with the div div Tr1 ln F = Tr1 ln (Fλ + P) methods presented in Sec. III, | | div div 1 div 1 div div = Tr ln F + Tr P Γ = Tr1 ln Ftot Tr0 ln Q . (32) 1 λ| 1 F 1 2 | − | λ div 1 1 1 The first trace follows from (26) for P = 0, while the Tr P P . (25) − 2 1 F F ghost trace is evaluated directly with the standard result λ λ for a minimal second order operator (A16) and (A17), Inserting the operator identity (24), the divergent contri- 1 13 1 1 butions of the individual terms in (25) can be reduced to Γdiv = d4x g1/2 R Rµν + R2 . 1 32π2ε 180G − 5 µν 15 the evaluation of universal functional traces. The details Z of this calculation are provided in Appendix B. In this (33) way, we find for the one-loop divergences of the nonde- The result (33) is in agreement with [20]. Since the action generate vector field for the Abelian vector field corresponds to a free theory, 1 11 7 1 there are no contributions to the renormalization of the Γdiv = d4xg1/2 R Rµν + R2 1 32π2ε 180G − 30 µν 20 square of the field strength tensor (11). Note also that Z the result (33) is independent of the gauge parameter η. 1 γ γ + + RP 1 + R P µν 6 12 − 6 µν 2 2 1 γ γ µν γ 2 1 Apart from an overall minus sign, the transition to Lorentzian + + Pµν P P . (26) − 2 4 24 − 48 signature corresponds to the replacement P → −P. 5

V. THE PROCA FIELD Therefore, it is clear that (39) does not reproduce the result for the Abelian gauge field (33) in the limit m 0, as the Abelian gauge field has only 4 2 = 2 propagating→ The Proca action for the massive vector field in curved − spacetime is given by the action of the Abelian gauge degrees of freedom. At the level of the functional traces, field (10) supplemented by a mass term [25], this can formally be seen as follows: while the scalar operator for the longitudinal mode of the Proca field in 1 1 (38) indeed reduces to the ghost operator of the Abelian S[A] = d4x g1/2 µν + m2A Aµ . (34) 4 Fµν F 2 µ gauge field in (32) in the limit m 0, the ghost trace Z in (32) is subtracted twice compared→ to the trace of the The mass term breaks the gauge symmetry. The Hessian longitudinal mode in (38). of (34) leads to the fluctuation operator

F ν := [∆ ] ν + ν + m2δν , (35) µ H µ ∇µ∇ µ VI. THE GENERALIZED PROCA FIELD which corresponds to the case λ = 0 and P = m21 of the 2 The generalized Proca model is defined by the action general vector field operator (2), that is F = F0 + m 1. The mass term in the Proca operator (35) breaks the 4 1/2 1 µν 1 µν gauge degeneracy of the gauge field operator F0. Nev- S[A] = d x g + M A A . (40) 4Fµν F 2 µ ν ertheless, the principal part of the Proca operator (35) Z is still degenerate. This degeneracy cannot be removed by a gauge fixing—in contrast to the Abelian gauge field. The action is that of the Proca field (34), but with the scalar mass term m2 generalized to an arbitrary positive Similar to (24), there is an operator identity for the Proca µν field definite and symmetric background mass tensor M . The background mass tensor M µν is completely gen- δν ρ δν eral and might be constructed from external background µ = δρ ∇µ∇ ρ . (36) F + m2 µ − m2 ∆ + m2 fields. Of particular interest are curvature terms 0 H µν µν µν Taking the trace of the logarithm on both sides of (36) M = ζ1R + ζ2Rg , (41) and using that the divergent part of the vector trace can be converted into a contribution from a scalar trace which arise in cosmological models [4–11]. The fluctuation operator for the generalized Proca theory (40) reads ν div ν µ 2 div Tr1 ln δµ ∇ ∇ = Tr0 ln ∆ + m , (37) ν ν ν ν − m2 F = [∆H] + µ + M . (42) µ µ ∇ ∇ µ the divergent part of the one-loop effective action is re- The operator (42) corresponds to the general operator duced to the vector and scalar traces of two minimal ν ρν (2) for λ = 0 and Pµ = gµρM > 0. The standard second order operators, techniques for the calculation of the one-loop divergences are not directly applicable to the degenerate operator of 1 div Γdiv = Tr ln F + m21 the generalized Proca theory (42). In particular, for the 1 2 1 0 generalized Proca operator, there is no simple analogue of 1 2 div = Tr1 ln ∆H + m 1 the operator identity (36) for the Proca operator. There- 2 fore, we adopt a different strategy and first reformulate 1 2 div Tr0 ln ∆ + m . (38) the generalized Proca theory as a gauge theory by making − 2 use of the Stückelberg formalism. In this formulation, the µν The vector and scalar traces in (38) can be calculated generalized background mass tensor M plays a double directly with the closed form algorithm (A16)–(A17). role as potential in the vector sector and as metric in the The final result for the one-loop divergences of the Proca scalar Stückelberg sector. For this effective “bimetric” model (34) reads formulation, the standard heat-kernel techniques are applicable and the one-loop divergences of the generalized 1 1 13 Proca action (40) are obtained in a closed form. Γdiv = d4x g1/2 R Rµν 1 32π2ε 15G − 60 µν Z 7 1 3 + R2 m2R m4 . (39) A. Weyl transformation and bimetric formalism 120 − 2 − 2 The result (39) is in agreement with [20, 26]. It has a The calculations are simplified by performing a Weyl clear physical interpretation. The effective action of the transformation of the background metric massive vector field in four dimensions is that of a four component vector field minus one scalar mode, since the 1 µν 1/4 gˆµν := det M gνρ gµν . (43) Proca field has 4 1 = 3 propagating degrees of freedom. µ2 − 6

Here, µ is an auxiliary mass parameter, introduced for realized by the Stückelberg formalism. The Stückelberg dimensional reasons. Note that in what follows indices scalar field ϕ is introduced by the shift are raised and lowered only with the metricg ˆ . Since the µν 1 kinetic term is invariant under a Weyl transformation, we Aµ Aµ + ∂µϕ . (52) find → µ

2 In terms of the vector field Aµ and the Stückelberg scalar 1 µ µν S = d4x gˆ1/2 gˆµαgˆνβ + g˜ −1 A A . ϕ, the action (44) is given by 4 Fµν Fαβ 2 µ ν Z (44) 1 S[A, ϕ] = d4x gˆ1/2 gˆµαgˆνβ 4 Fµν Fαβ Z In the second term, we have defined 2 µ −1 µν −1 µν + g˜ AµAν + µ g˜ Aµ∂ν ϕ −1 µν 2 µν −1/2 µν 2 g˜ := µ det M gνρ M , (45) 1 −1 µν + g˜ ∂ν ϕ∂ν ϕ . (53) which is the inverse of the new metric g ˜ . In this way, 2 ρν formally the dependency on the original general mass tensor M µν has been replaced by a standard mass term. By This action has a gauge symmetry as it is invariant under construction, we have the important relations the simultaneous infinitesimal gauge transformations δ A = ∂ ε, δ ϕ = µε . (54) detg ˜ = detg ˆ , ˆ detg ˜ = 0 . (46) ε µ µ ε − µν µν ∇ρ µν We choose a one-parameter family of gauge conditions We define the Christoffel connection associated withg ˜µν , 1 µ χ[A, ϕ] = ˆ µA + ηµϕ . (55) 1 ρα −√1 + η ∇ Γ˜ρ = g˜−1 ∂ g˜ + ∂ g˜ ∂ g˜ . (47) µν 2 µ αν ν µα − α µν In particular, (55) interpolates between the original vec- A natural structure is the difference tensor tor field theory (44) with ϕ = 0 (η ) and the Lorentz gauge (η = 0). The corresponding→ gauge ∞ breaking action δΓλ := Γ˜λ Γˆλ reads µν µν − µν 1 λα −1 ˆ ˆ ˆ 1 4 1/2 2 = g˜ µg˜να + ν g˜µα αg˜µν . (48) S [A, ϕ] = d x gˆ χ . (56) 2 ∇ ∇ − ∇ gb 2 Z By construction, the difference tensor satisfies The ghost operator is obtained from (55),

1 αβ ˆ ˆ 2 δΓα = g˜−1 ˆ g˜ Q( ) = ∆ + ηµ . (57) µα 2 ∇µ αβ ∇ −1/2 1/2 2 = det g ˜ ˆ detg ˜ = 0 . (49) For simplicity, we choose the Lorentz gauge, η = 0. In ρσ ∇µ ρσ terms of the generalized two-component field The Ricci curvatures of the new metricg ˜µν are given by A φA = µ , (58) ϕ ˜ −1 µν ˜ R = g˜ Rµν , (50) the gauge-fixed action acquires the block form R˜ = Rˆ δΓα δΓβ + ˆ δΓα . (51) µν µν − βµ αν ∇α µν 4 1/2 A B S[A, ϕ] + Sgb[A, ϕ] = d x g φ F φ , (59) In the following, when we work with the two metricsg ˆµν AB andg ˜µν , indices are raised and lowered exclusively with Z the metricg ˆµν . where the block matrix fluctuation operator F has com- A AC ponents F B = γ FCB. The components of the (dedensitized) inverse configuration space metric γAB are B. Stückelberg formalism given by gˆ γAB = µν . (60) As we have discussed in the context of the Abelian 1 gauge field, the gauge symmetry is responsible for the degeneracy of the total fluctuation operator. Therefore, a gauge fixing is required to remove this degeneracy. At the same time, the gauge fixing can be used to also remove 2 The calculation can be performed for the general η-family of the degeneracy of the principal symbol. A similar mech- gauges (55). By using (24), it can be seen already at the level of anism works in the case of the generalized Proca model, the functional traces that all η-dependent terms cancel and the when artificially rewritten as a gauge theory, which is one-loop divergences are independent of the gauge parameter η. 7

Splitting the fluctuation operator according to the num- Note that odd powers in the expansion are zero, because ber of derivatives, it can be represented as the block matrix Π 1/D has only zeros on the diagonal and that we have used the cyclicity of the trace to convert F = D + Π , (61) vector traces into scalar traces, for example: with the block matrix structure 1 δν δβ 1 † † α α α α † D1 Π Tr1 Πµ Π = Tr0 Π Πβ . (69) D = , Π = . (62) D0 D1 D1 D0 D0 Π The components of the operators in (62) are given by The first two traces in (66), together with the contribution of the ghost operator (57), are evaluated directly ν ν ˆ 2 −1 ν with (A16)–(A17). Their sum reads [D1]µ := ∆H + µ g˜ , µ µ h i −1 µν D := ˆ µ g˜ ˆ ν , Tr ln D + Tr ln D 2 Tr ln Q 0 − ∇ ∇ 1 1 0 0 − 0 ν ˆ −1 µν Π := µ µ g˜ , (63) 1 4 1/2 1 ˆ 1 −1 µν −1 ρσ ˜ ˜ − ∇ = d xgˆ g˜ g˜ RµρRνσ 16π2ε 15G − 60 † −1 ν ˆ Z where Πµ = µ g˜ ν , denotes the formal adjoint of µ ∇ 1 ˜2 1 4 −2 2 −1 µν ˆ Πν with respect to the inner product on the space of R µ tr g˜ µ g˜ Rµν −120 − 2 − vectors. The component D0 can be simplified by using 1 2 −1 1 µν 1 2 (49) and defines the scalar Laplace operator with respect + µ tr g˜ Rˆ Rˆµν Rˆ + Rˆ , (70) 6 − 5 15 to the metricg ˜ , µν µν D = g˜ −1 ˜ ˜ =: ∆˜ . (64) where we have introduced the abbreviation 0 − ∇µ∇ν µν µα α ν Let us briefly discuss what we have achieved by the g˜−n := g˜−1 1 g˜−1 2 g˜−1 . (71) Stückelberg formalism. The 4 1 = 3 propagating de- α1 ··· αn−1 grees of freedom of the original− generalized Proca field have been converted into the 4 + 1 2 = 3 propagating In (70), we used that the Euler characteristic degrees of freedom, corresponding− to those of a vector 1 1 field, a scalar, and two scalar ghosts fields. In contrast χ( ) := d4x gˆ1/2 = d4x g˜1/2 ˜ , (72) M 32π2 G 32π2 G to the principal part of the original generalized Proca Z Z operator (42), the additional gauge freedom present in b the Stückelberg formalism has been used to render the defined in terms of the Gauss-Bonnet term (27), is a topo- principal part D of the scalar-vector block operator (61) logical invariant and therefore independent of the metric. nondegenerate and, in particular, minimal—the price to This allows to combine both contributions fromg ˆµν and pay is the introduction of the second metricg ˜µν . g˜µν in (70). The evaluation of the divergent contributions from the remaining traces (67) and (68) constitutes the most complex part of the calculation. Here, we only C. One-loop effective action sketch the major steps. The details are provided in Ap- pendix C. There are two main complications associated In order to calculate the one-loop effective action with the evaluation of the divergent parts of the traces (67) and (68). First, the traces (67) and (68) involve 1 ν propagators δ /D1 and 1/D0 with different spin. Sec- Γ1 = Tr ln F Tr0 ln Q , (65) µ 2 − ond, the propagators are defined with respect to differ- we expand F around D. Perturbation theory in Π is ef- ent metrics. Therefore, we have to explicitly perform the ficient as Π = (M1). Expanding Tr ln F up to (M4), convolution of the corresponding kernels we obtain O O µ0 1 2ω 2ω 0 ν δν 0 † 1 0 T = d x d x Π δˆ(x, x ) Π 0 δ˜(x , x) , Tr ln F = Tr1 ln D1 + Tr0 ln D0 T2 T4 , (66) 2 D µ D − − 2 Z " 1 #" 0 # (73) where T = Mi , i = 2, 4 denote the following traces i O 2 β where we have defined ω := d/2. Inserting the Schwinger- 1 1 α δα † 1 T2 := Tr Π = Tr0 Π Πβ , (67) DeWitt representation for the kernels of 1/D1 and 1/D0, 2 " D # D1 D0 provided in Appendix C, the traces T2 and T4 are ulti- 2 1 1 4 δβ 1 mately reduced to Gaussian integrals. In case of a single T := Tr Π = Tr Πα α Π† . 4 2 D 0 D β D metric this procedure has been outlined in [20]. In the " # " 1 0 # case of two metric structures, the problem becomes more (68) complicated and has been discussed in [27]. The resulting 8

Gaussian integral is Finally, the Gaussian integrals in (74) are evaluated

4 1 µ µ1 µ2k 1 α β dˆσ σˆ σˆ exp Gαβσˆ σˆ ∞ ∞ (4π)2 ··· −4 1 2ω 1/2 ω−2 ds Z µ=1 ! T2 = 2ω d x gˆ du u ω−1 Y (4π) 0 0 s 1 µ1...µ2k Z Z Z −1 = 1/2 symk G . (77) 2ω 1 G dˆσµ Ψ(ˆσµ) exp G σˆασˆβ . × −4 αβ Here, we have introduced the kth totally symmetrized µ=1 ! Z Y µν (74) power of a general rank two tensor T , defined by (2k)! [sym (T )]µ1...µ2k := T (µ1µ2 T µ2k−1µ2k) . (78) k 2kk! ··· Here, we have introduced the “interpolation metric” After evaluation of the Gaussians (77), the parameter integral over u remains and the final result is expressed in terms of basic elliptic integrals, defined for 0 ` 2k, ≤ ≤ G (u) :=g ˆ + ug˜ . (75) µν µν µν ∞ 1/2 gˆ µ ...µ Iµ1...µ2k := du u` sym G−1 1 2k . (79) (2k,`) G1/2 k Z0 The function Ψ(u, s σˆµ) in the integrand of (74) is the Integrals of the form (79) occur unavoidably in the mul- | result of the covariant Taylor expansion inσ ˆµ timetric case and are characteristic to the problem. They constitute irreducible structures and can in general not be trivially integrated as for the case of a single metric 4 ∞ g˜µν =g ˆµν . The evaluation of the trace T4 proceeds in Ψ(u, s σˆµ) :=g ˆ1/2 Ψ(k) (u, s)ˆσµ1 σˆµk , (76) an analogue way. The individual results for T2 and T4 | (µ1...µk) ··· k=0 are provided explicitly in Appendix C. X D. Final result: One-loop divergences for the generalized Proca model whereσ ˆµ(x, x0) is tangent to the geodesic connecting x 0 The final result for the divergences of the generalized with x at the point x. Note that the background field Proca model in curved spacetime (40) are obtained by dependent coefficients Ψ(k) (u, s) only involve posi- (µ1...µk) adding the standard minimal second order traces (70) tive powers of the parameters u and s. In d = 2ω = 4 and the results for the divergent part of the traces T2 dimensions, only terms of the integrand with total s- and T4, which are explicitly provided in (C35) and (C45). dependency 1/s contribute to the logarithmically diver- Using the integral identities presented in Appendix D, the gent part.3 Therefore, in view of (74), the divergent con- number of different integrals (79) can be reduced. This tributions originate from the s-independent parts of Ψ. allows us to represent the final result in the compact form

1 ˆ 1 µρ νσ 1 1 1 Γdiv = d4xgˆ1/2 G g˜−1 g˜−1 R˜ R˜ R˜2 Rˆ Rˆµν + Rˆ2 1 32π2ε 15 − 60 µν ρσ − 120 − 5 µν 15 Z " 1 1 1 µν 1 + µ2R˜ + µ2 tr g˜−1 Rˆ µ2 g˜−1 ˆ δΓα + 7Rˆ µ4 tr g˜−2 3 6 − 6 ∇α µν µν − 4 (0,0) (2,1) µν (4,1) µνρσ (4,2) µνρσ + C I(0,0) + Cµν I(2,1) + CµνρσI(4,1) + CµνρσI(4,2) , (80) #

1 µν 1 C(0,0) = µ2 g˜−2 R˜ µ2 tr g˜−1 R,˜ (81) 6 µν − 12

3 The choice of the parametrization in terms of s and u guarantees dimensions. We comment on this in more detail in Appendix D. that the divergent contribution is isolated in the s-integration, whereas the u-integration is ﬁnite. 4 An interesting observation is that even for generalg ˜µν –in principle–the integrals (79) can be evaluated explicitly in d = 4 9

1 1 1 1 C(2,1) = µ4 g˜−2 µ2 g˜−1 µ2 tr g˜−1 + Rˆ µν 4 µν − 4 µν 2 3 αβ 1 1 1 1 5 1 + µ2 g˜−1 Rˆ ˆ δΓ + ˆ δΓ ˆ δΓ + ˆ δΓ δΓ δΓλ −2 αµβν − 6∇α βµν 3∇β ναµ − 3∇µ ναβ 12∇µ αβν − 4 αλβ µν 1 1 1 1 + δΓλ δΓ δΓ δΓλ + δΓ δΓλ δΓ δΓλ 4 µα λνβ − 12 αλµ νβ 3 µλα νβ − 2 µλν αβ 1 α 1 µ2 g˜−1 2R˜ 5Rˆ ˆ δΓλ + µ2 tr g˜−1 δΓα δΓβ , (82) − 6 µ αν − αν − ∇λ αν 12 µβ αν 1 − C(4,1) = µ2 tr g˜ 1 ˆ δΓ , (83) µνρσ −24 ∇µ νρσ 1 1 1 C(4,2) = µ2 ˆ δΓ + µ2δΓ δΓλ µ2δΓ δΓλ . (84) µνρσ −6 ∇µ νρσ 2 µλν ρσ − 4 λµν ρσ

This constitutes our main result. For compactness, we A. Trivial index structure present the one-loop divergences in terms of the two metricsg ˆµν andg ˜µν . The result in terms of the original met- The simplest case for the general background mass ten- µν ric gµν and the background mass tensor M can easily sor M µν , which goes beyond the constant Proca mass be recovered by making use of (43) and (45). term M µν = m2gµν , is the reduction to a spacetime de- As expected on general grounds, the result (80) is a pendent scalar function X2(x), such that M µν acquires local expression, which contains up to four derivatives. trivial index structure The fact that (80) is not a polynomial of the invariants of M µν , is related to the role of M µν as metric in the M µν = X2gµν . (85) scalar Stückelberg sector. We emphasize that this result holds for an arbitrary positive definite symmetric back- In particular, this includes the case where X2 is pro- ground tensor M µν . The original assumption of a strictly portional to the curvature scalar R, which is relevant in positive M µν is reflected in the result (80), as there is no cosmological vector field models [4–11]. In view of (85), smooth limit M µν 0. Note that the result (80) is inde- it is easy to see that pendent of the auxiliary→ mass scale µ, which can be seen X2 by the invariance of (80) under rescaling of µ αµ, with gˆ = g , (86) arbitrary constant α. → µν µ2 µν µν The result has been derived in curved spacetime with- g˜−1 =g ˆµν , (87) out considering graviton loops. Nevertheless, the con- δΓλ = 0 , (88) sistency of the renormalization procedure would require µν −1 µν −1 µν to include the induced kinetic terms forg ˆµν ang ˜µν (for G = (1 + u) gˆ , (89) µν gµν and M respectively) in the bare action. The result 4 det( G ) = (1 + u) det (ˆgµν ) . (90) (80) shows that the essential complexity is not reduced µν considerably in the limit of a flat spacetime, as the in- In this case, the integrals (79) are trivially evaluated tegrals (79), associated with the presence of the second µν metricg ˜µν (the background mass tensor M ) remain. ∞ ` µ1···µ2k µ1...µ2k u The result (80) is considerably more complicated than I(2k,`) = [symk (ˆg)] du k+2 g˜=ˆg 0 (1 + u) the one-loop result for the nondegenerate vector field. In Z (k `)!`! particular, the result cannot simply be obtained in the µ1...µ2k = − [symk (ˆg)] . (91) limit λ 0 from (26). (k + 1)! → µν For special choices of M , the general result (80) sim- Inserting (86)–(90) and the explicit results (91) for the plifies and the integrals (79) can be evaluated explicitly. integrals into the general result (80), we obtain the one- loop result in terms of the original metric gµν ,

1 1 13 Γdiv = d4xg1/2 R Rµν VII. CHECKS AND APPLICATIONS OF THE 1 32π2ε 15G − 60 µν Z GENERALIZED PROCA MODEL 7 1 3 + R2 RX2 X4 120 − 2 − 2 In this section we reduce our general result (80) to sev- 1 ∆X 1 ∆X 2 eral special cases. This provides important cross checks R 3X∆X . (92) −6 X − − 2 X and interesting applications. # 10

As an additional (trivial) cross check of this result, we In order to establish the connection to the generalized set X = m in (92) and recover the one-loop divergences Proca model (40), we identify the background mass ten- for the Proca model (39). sor M µν as An independent way to derive the result (92) is to insert (85) directly into the generalized Proca action (40). This leads to the ordinary Proca action (34), but with µν ¯ ¯ ¯ρ µν ¯µ ¯ν the constant mass m promoted to a spacetime dependent M (A) = α AρA g + 2A A . (99) function X, 1 X2 S[A] = d4x g1/2 µν + A Aµ . (93) According to (43) and (45), we have 4Fµν F 2 µ Z By performing the Weyl transformation (86), the reduced ρσ action (93) is identical to the Proca action (44), but with gˆµν = (g ξρξσ) gµν , (100) g gˆ and m µ, −1 µν −1/4 µν µ ν µν → µν → g˜ = 3 (ˆg + 2ξ ξ ) , (101) 2 2 4 1/2 1 µα νβ µ µν 1/4 S[A] = d x gˆ gˆ gˆ + gˆ A A . g˜µν = 3 gˆµν ξµ ξν , (102) 4 Fµν Fαβ 2 µ ν − 3 Z (94) 2 δΓλ = ξ ˆ λξ ξ ˆ ξλ 3ξλ ˆ ξ µν 3 (µ ∇ ν) − (µ ∇ν) − ∇(µ ν) The one-loop divergences for (94) are obtained from (39) λ α ˆ +2ξ ξ ξ(µ| αξ|ν) . (103) by performing the inverse Weyl transformationg ˆµν ∇ → gµν and agree with those obtained from the reduction We have deﬁned the normalized vector ﬁeld (92) of the general result.

A¯ B. Vector ﬁeld with quartic self interaction ξ := 31/8α1/2 µ , ξµ :=g ˆµν ξ , (104) µ µ ν

The generalized Proca action for a vector field Aµ with quartic self-interaction is considered in [27], µ such that ξµξ = 1. For (100) and (101), the integrals 1 α 2 (79) reduce to expressions of the form S[A] = d4x g1/2 µν + A Aµ . (95) 4Fµν F 4 µ Z The part quadratic in the quantum fluctuation reads k µ1...µ2k n (µ1µ2 µ2n−1µ2n µ2n+1 µ2k) 1 I = d(2k,`)gˆ gˆ ξ ξ . S [A,¯ δA] = d4x g1/2 (δA) µν (δA) (2k,`) ··· ··· 2 4Fµν F n=0 X (105) Z α + A¯ A¯ρgµν + 2A¯µA¯ν δA δA , (96) 2 ρ µ ν i where the vector field Aµ has been split into background We provide a closed form expression for the coefficients A¯µ and perturbation δAµ, n d(2k,`) in Appendix D. We obtain the divergent part of the A = A¯ + δA , (97) one-loop effective action for (95) from the general result µ µ µ by inserting (101)–(103) as well as (105) with (D11) into (δA) = δA δA . (98) F ∇µ ν − ∇ν µ (80),

1 1 1 1 1 µα νβ 1 1 Γdiv = d4xgˆ1/2 ˆ Rˆ Rˆµν + Rˆ2 g˜−1 g˜−1 R˜ R˜ R˜2 9 + 2√3 µ4 1 32π2ε 15G − 5 µν 15 − 60 µν αβ − 120 − 6 Z 2 1 1 2 +√4 3µ2 31 + 8√3 Rˆ ξµξν + 3 4√3 Rˆ + 31 + 8√3 ˆ ξµ 45 − µν 18 − 45 − ∇µ 1 1 + 274 + 136√3 ˆ ξ ˆ ξρ ξµξν + 67 36√3 ˆ ξ ˆ µξν . (106) 45 − ∇µ ρ ∇ν 45 − ∇µ ν ∇

Instead of ξµ andg ˜µν , the authors in [27] use a diﬀer- ent parametrization. The conversion between our result 11

(106) and their result is easily accomplished by (100) and antisymmetrization among five or more indices is neces- (104). We find that the reduction of our general result sarily zero to the case of the self-interacting vector field (106) is in agreement with the result obtained in [27]. This provides δα δβδγ δδ δω 0 , (109) [µ ν ρ σ λ] ≡ a powerful check of our general result (80). where the total antisymmetrization is performed with unit weight. By contracting (109) with the background ρσ tensors Rµνρσ and µ1 µn Y , we can systematically construct dimensional∇ · · · ∇ dependent invariants, which C. Perturbative treatment of the generalized vanish in d = 4 dimensions. At linear order of the ex- Proca model pansion in Y, there is one such invariant

α β γ δ ω µ ν ρ σ λ Finally, we test our method by a perturbative calcula- I1 = δ[µδν δρ δσδλ]R α β R γ δY ω tion, which only relies on the well-established generalized µν γδα γδ = Y gµν 4Rµ Rνγδα + 8RµγνδR Schwinger-DeWitt technique [20]. For this purpose, we G − +8R γ R 4R R) . (110) assume that the background mass tensor has the form µ νγ − µν At quadratic order of the expansion there are two inde- µν 2 µν µν M = m g + Y , (107) pendent dimensional dependent invariants

I = δα δβδγ δδ δω Rµ ν Y ρ σY λ , (111) with Y µν m2gµν and perform an expansion in Y µν . 2 [µ ν ρ σ λ] α β ∇γ δ ∇ ω µ ν I = δα δβδγ δδ δω R Y ρ σY λ . (112) 3 [µ ν ρ σ λ] α β γ ∇δ∇ ω Since use of (111) and (112) does not lead to any sim- 1. Expansion of the general result plification of our result, we refrain from presenting the explicit expressions. For the first order of the expansion div Γ1,(1), linear in Y, we obtain The expansion of the general result (80) up to second order reads 1 1 5 3 Γdiv = d4xg1/2 RY Rµν Y m2Y 1,(1) 32π2ε 12 − 6 µν − 4 div div div div 3 Z Γ1 = Γ1,(0) + Γ1,(1) + Γ1,(2) + Y , (108) 1 O µν ρσ µν + 2 8RµρνσR Y + 2Rµν R Y 240m − hµν 2 where Γdiv is the divergent part of the one-loop effective 4RRµν Y + R Y 4 (∆R) Y 1,(i) − − action, which contains terms of ith order in the pertur- µν µν div +8 ( µ ν R) Y + 4 (∆Rµν ) Y , (113) bation Y. The zeroth order Γ1,(0) is simply given by ∇ ∇ the one-loop divergences (39) for the Proca field. Before i µν we proceed, let us discuss the structure of the invariants where we have defined the trace Y := gµν Y . For the div used to represent the result for the higher orders of the second order Γ1,(2), quadratic in Y, we make use of the perturbative expansion. In d = 4, the result of any total tensor algebra bundle xAct [28–30] and find

1 1 1 1 Γdiv = d4xg1/2 Y Y µν Y 2 + 2RY Y µν + 12Rµν Y ρY + RY 2 1,(2) 32π2ε −8 µν − 16 48m2 − µν µ νρ Z 4Rµν Y Y 4R Y µν Y ρσ + 12Y µν Y ρ + 4Y Y νρ + 2Y µν ∆Y Y ∆Y − µν − µρνσ ∇ν ∇ρ µ ∇ρ∇ν µν − 1 + 2R Rµν Y 2 R2Y 2 + 8R RYY µν 2R2Y Y µν 16R ν R Y αρY µσ 16R Rµ YY νρ 960m4 − µν − µν − µν − α νµρσ − µρ ν +8R RY µ Y νρ + 32Rµν R YY ρσ + 24R R Y µν Y ρσ 8R R Y µν Y ρσ 8RR Y µν Y ρσ µν ρ µρνσ µρ νσ − µν ρσ − µρνσ 16R α βR Y µν Y ρσ 8R Rµ Y ν Y ρσ 4R Rµν Y Y ρσ + 16R Y µν ∆Y ρσ − µ ν ρασβ − µρ ν σ − µν ρσ µρνσ +32R Y µν α Y ρσ + 32R Y µν αY ρσ 8R Y νρ∆Y 16R Y νρ∆Y σ + 16R Y σ ρY µν µσνα ∇ ∇ρ νρσα ∇µ∇ − νρ − νσ ρ µρνσ ∇ ∇ +16R Y ρµ ν Y σ 32R Y ρµ Y νσ + 2R ( Y ) µY + 16R ( Y νρ) µY + 24R ( Y ) Y µν σν ∇µ∇ ρ − νσ ∇µ∇ρ ∇µ ∇ νρ ∇µ ∇ ∇µ ∇ν 16R ν ( Y σρ) Y µ 4RY µν Y 12RY Y µν 16R ν Y σρ Y µ + 16R ( µY ) σY νρ − σ ∇µ ∇ν ρ − ∇ν ∇µ − ∇ν ∇µ − σ ∇ν ∇µ ρ µνρσ ∇ ∇ 8R Y νρ µY + 16RY µν Y ρ 40R ν Y Y µρ + 32R ( Y µν ) σY αρ − µρ ∇ν ∇ ∇ν ∇ρ µ − µ ∇ν ∇ρ µρνσ ∇α ∇ 12

80R ν Y ρµ σY + 4R ( µY ) ν Y + 8R ( Y ρ) Y µν + 24R ( Y σρ) Y µν − σ ∇ν ∇ ρµ µν ∇ ∇ ∇ν µ ∇ρ σµ ∇ν ∇ρ +40R Y νρ Y σµ 24RY µν ∆Y 16R Y ∆Y µν 16R ( Y σµ) Y νρ + 16R ( µY ) Y νρ νσ ∇ρ∇µ − µν − µν − νρ ∇µ ∇σ µν ∇ ∇ρ +16Y ∆ Y µν 4Y µν ∆2Y 16Y µν Y ρσ 8Y µν ∆ Y ρ 2Y ∆2Y . (114) ∇ν ∇µ − µν − ∇ν ∇µ∇σ∇ρ − ∇ν ∇ρ µ −

2. Perturbative calculation via the generalized can be understood from (117), as Schwinger-DeWitt technique 1 = M0 , (118) The second order expansion (108) of the general re- ∂2 + m2 O − sult (80) can be checked by a direct perturbative calcu- 1 − ∂ ∂ν + m2δν = M 2 . (119) lation, which only relies on the well-established gener- m2 − µ µ O alized Schwinger-DeWitt technique, introduced in [20]. Although conceptually straightforward, the complexity In contrast, in case of the nondegenerate vector field, the 2 grows rapidly with growing powers of Y and is already analogue expansion (25) in P = M is efficient in O 0 quite involved for the expansion up to second order in background dimension as 1/Fλ = M , such that the O Y. Moreover, we are mainly interested in a check of the combination P 1/F = M2 and terms P3 are λ structures involving derivatives of Y not tested by the already finite. The differenceO between the backgroundO 2 previous checks—apart from the three structures involv- counting of 1/Fλ and 1/(F0 + m ) can be traced back to ing powers of ∆X that remain in (92) after the reduction the fact that Det Fλ = 0 while Det F0 = 0. of M to the trivial index structure (85). Therefore, we Inserting the identity6 (117) into the expansion (116), restrict the direct perturbative calculation to flat space- we obtain the following sum of traces up to Y2 : O time gµν = δµν , µ = ∂µ. ∇ 2 Starting point is the action (40) for the generalized Tr1 ln F = Tr1 ln F0 + m 1 Proca field, but with the background mass tensor M ν 1 δ treated perturbatively as in (107). The one-loop diver- + Tr B + Tr Y ρ ρ 0 ∂2 + m2 1 µ ∂2 + m2 gences up to second order in Y are obtained by expanding − σ −ν (42) around the Proca operator defined in (35), 1 δρ δ Tr Y ρ Y λ λ − 2 1 µ ∂2 + m2 σ ∂2 + m2 F = F + m21 + Y . (115) − − 0 δσ ∂ ∂ν 1 + Tr Y ρ ρ Y λ λ Using the split (115) in the expansion of the logarithm 1 µ ∂2 + m2 σ m2 ∂2 + m2 − − up to second order in Y, we find 1 1 1 Tr0 B B . (120) 2 − 2 ∂2 + m2 ∂2 + m2 Tr1 ln F = Tr1 ln F0 + m 1 + Y − − 1 = Tr ln F + m21 + Tr Y Here, we have defined the scalar operator 1 0 1 F + m2 0 1 µν 0 1 1 1 B(∂) := ∂µY ∂ν = M , (121) Tr1 Y Y . (116) m2 O − 2 F + m2 F + m2 0 0 and used the cyclicity of the trace to convert two vector Similar to the calculation for the nondegenerate vector traces in (120) into scalar traces. Next we evaluate the field, we use the exact operator identity (36) for the Proca divergent contributions of the individual traces in (120) operator F + m21, which in flat spacetime reads 0 separately. The first trace is just that of the Proca oper- ν ν δµ ∂µ∂ 1 ator (35). The remaining traces can be further reduced = δν . (117) 2 2 F + m2 µ − m2 ∂2 + m2 by iterative commutation of all powers of 1/( ∂ + m ) 0 − to the right, using the identity − Note that the similarity to the expansion (25) for the 1 1 1 nondegenerate vector field might be misleading here, as , Y = ∂2, Y . (116) is an expansion in Y, not an expansion in back- ∂2 + m2 − ∂2 + m2 − ∂2 + m2 − − − ground dimension. This is seen by comparing the count- (122) ing of background dimension for the two cases. While Y = M2 , expansion in Y in (116) is not efficient Each iteration of (122) generates one additional commu- O as 1/(F + m2) = M−2 , such that the combination tator, which increases the number of derivatives acting 0 O Y 1/F = M0 . Therefore the perturbative series on the background tensor Y by at least one, 0 O (116) continues up to arbitrary order in Y. The count- 2 2 ρ ing of background dimension 1/(F + m2) = M−2 ∂ , Y = ∂ Y 2 (∂ Y) ∂ρ . (123) 0 O − − − 13

In this way the calculation of the divergent part of the δσ δν div 1 Tr Y ρ ρ Y λ λ = Y Y µν . trace (120) is reduced to the evaluation of a few universal 1 µ ∂2 + m2 σ ∂2 + m2 16π2ε µν functional traces [20], − − (127)

1 div The remaining two traces require the evaluation of nested (n,p) (m2) = ∂ ∂ . (124) µ1...µp µ1 µp 2 2 commutators. For the first trace we find U ··· ∂ + m 0 − x =x div δσ ∂ ∂ν 1 (n,p) ρ ρ λ λ Tr1 Yµ Yσ In d = 4 dimensions, the µ1...µp ’s are divergent for a ∂2 + m2 m2 ∂2 + m2 degree of divergence U − − 1 1 1 ∂ ∂ = Y Y µν + Y µρ ρ ν Y ν 16π2ε 2 µν 3 m2 µ χdiv = p 2n + 4 0. (125) − ≤ 1 ( ∂2) + Y µν − Y . (128) Each commutator reduces the number of derivatives p 12 m2 µν in (124) and therefore decreases the degree of divergence (125) by one. This shows that the iterative procedure For the second trace we find (122) is efficient for the calculation of the divergent con- div tributions. In flat spacetime, divergences only arise for 1 1 Tr0 B B χ = k = 0, 2, 4, ∂2 + m2 ∂2 + m2 div − − 1 1 1 1 ∂2 n 2−n−k 2k µν 2 µν − (n,2n+2k) ( 1) 2 m = 2 Yµν Y + Y + Y 2 Yµν µ ...µ = − symn+k (δ) . 16π ε "4 8 12 m U 1 2n+2k 16π2ε k!(n 1)! µ1...µ2n+2k − 2 (126) 1 ∂ν ∂ρ 1 ∂ 1 ∂µ∂ν + Y µν Y ρ + Y − Y Y Y µν 6 m2 µ 24 m2 − 6 m2 1 ∂ ∂ ∂ ∂ 1 ∂ ∂2 ∂ Following the strategy outlined above, we first calculate µν µ ν ρ σ ρσ µν ν − ρ ρ + Y 4 Y + Y 4 Yµ the trivial traces which do not involve the evaluation of 30 m 60 m 2 2 2 any commutator: 1 ∂ ∂µ∂ν 1 ∂ Y − Y µν + Y µν − Y − 30 m4 120 m4 µν div 1 3 2 Tr ln F + m21 = m4 , 1 ∂2 1 0 16π2ε −2 + Y − Y . (129) 240 m4 div # 1 1 2 Tr0 B = m Y , ∂2 + m2 16π2ε − Adding the contributions (127)–(129) according to (120), − δν div 1 1 we obtain the final result for the one-loop divergences on Tr Y ρ ρ = m2Y , a flat background up to second order in Y µν , 1 µ ∂2 + m2 16π2ε 4 −

1 3 3 1 1 1 ∂2 1 ∂ ∂ 1 ∂2 Γdiv = d4x m4 m2Y Y Y µν Y 2 + Y µν − Y + Y µν ν ρ Y ρ Y − Y 1 32π2ε −2 − 4 − 8 µν − 16 24 m2 µν 4 m2 µ − 48 m2 Z " 2 2 1 ∂ ∂ 1 ∂ ∂ ∂ ∂ 1 ∂ν ∂ ∂ρ 1 ∂ ∂µ∂ν + Y µ ν Y µν Y µν µ ν ρ σ Y ρσ Y µν − Y ρ + Y − Y µν 12 m2 − 60 m4 − 120 m4 µ 60 m4 2 2 1 ∂2 1 ∂2 µν − − Y 4 Yµν Y 4 Y . (130) −240 m − 480 m #

The result (130) is in perfect agreement with the result DeWitt method. In particular, it neither relies on the for second order expansion (114) of the general result (80) Stückelberg formalism nor on the bimetric formulation. reduced to a flat background gµν = δµν . This provides a powerful check of our general result. In particular, it probes tensorial derivative structures, not captured by VIII. COMPARISON WITH RESULTS IN THE the check for the trivial index structure, discussed in Sec. LITERATURE VII A. It also provides an important independent check of our method, as it has been obtained in a complementary In this section, we compare our results with the one- way by a direct application of the generalized Schwinger- loop divergences of the generalized Proca model (40) ob- 14 tained previously by different methods and techniques result (80) does not coincide with the one given in [Eq. [12, 13]. (45)] in [13]. In particular, the result of [13] contains nonlocal structures. Nevertheless, whether αµν is an operator or a back- A. Comparison: Local momentum space method ground tensor is not relevant for the contribution to the one-loop divergences at linear order in Y, as according In [12], based on the local momentum space method to [Eq. (25)], αµν is first order in Y and therefore the [31], an expression for the one-loop divergences of the nontrivial contribution in [Eq. (26)] only affects higher generalized Proca model (40) has been obtained. The orders in Y, starting at Y2 . This explains why O result [Eq. (2.24)] in [12], includes terms of order the authors reproduce their result with the generalized R2,RY,Y 2 and, in our conventions, reads Schwinger-DeWitt technique [20] at linear order in Y. O The result at linear order in [Eq. (53)] of [13] is in agree- 1 1 13 7 ment with our result (113) for the approximation linear Γdiv = d4x g1/2 R Rµν + R2 1 32π2ε 15G − 60 µν 120 in Y. Note that this comparison is nontrivial in the sense Z that their result involves additional terms proportional to 1 2 3 4 5 µν 1 3 2 m R m Rµν Y + RY m Y the invariant (110), which vanishes in d = 4 dimensions. −2 − 2 − 6 12 − 4 1 1 Y Y µν Y 2 . (131) −8 µν − 16 IX. CONCLUSIONS The result (131) agrees with the second order expansion (108) of our general result (80) under the assumption We have investigated the renormalization of generalized vector field models in curved spacetime. We have in- Y m2,R m2, m. (132) ∇ troduced a classification scheme for different vector field models based on the degeneracy structure of their as- In particular, this means that no terms proportional to sociated fluctuation operator. The distinction between inverse powers of m appear in the result of [12] (and different degeneracy classes is partially connected to the therefore, apart from total derivatives, no structures in- nonminimal structures present in the principal symbol of volving derivatives of Y ). In contrast, our second order the fluctuation operator. The simplest theories, where expansion (108) was derived only under the assumption such nonminimal structures can appear, are vector field Y m2. The coincidence with the terms in (131) there- theories, but these terms are also important for rank two fore provides an independent check of several structures tensor fields, as has been recently discussed for the degen- linear and quadratic in Y . erate fluctuation operator in the context of f(R) gravity [3]. B. Comparison: Method of nonlocal field The nondegenerate vector field and the Abelian gauge redefinition field are both representatives of two different degeneracy classes, for which the calculation of the one-loop divergences can be performed with standard methods, based The authors of [13] have obtained a result for the one- on the generalized Schwinger-DeWitt technique [20]. We loop divergences of the generalized Proca model (40), have briefly reviewed these cases and have discussed the without relying on any perturbative expansion in Y (de- technical details associated with the underlying algo- noted X in [13]). They use the Stückelberg formulation rithm for the one-loop calculation. The Proca theory to rewrite the generalized Proca theory as a gauge the- of the massive vector field is the simplest representative ory for the original Proca field and the Stückelberg scalar in the class of nondegenerate fluctuation operators with field. They derive the corresponding block matrix fluc- a degenerate principal symbol. Only for this special case, tuation operator similar to (52). As the authors discuss, standard methods are directly applicable. In particular, with respect to the metric g , this operator is both, µν none of these models in the different degeneracy classes nonminimal as well as not block-diagonal. The authors can be obtained from one another in a smooth limit—a perform a “shiftlike” transformation of the quantum vec- fact which is related to the discontinuity in the number of tor field Aµ [Eq. (22)], propagating degrees of freedom. Therefore, the different µ µ µρ classes have to be studied separately. A = B + α ∂ρϕ . (133) The generalized Proca model, which results from the Here, αµν is an a priori undetermined background tensor. Proca model by generalizing the constant mass term Next, they derive a condition for which the fluctuation m2gµν to a local background mass tensor M µν , is con- operator is diagonalized [Eq. (25)]. However, in contrast siderably more complicated and can no longer be treated to the statement of the authors, we believe αµν has to directly by standard methods. Therefore, we have ap- be an operator instead of a background tensor in order plied the Stückelberg formalism in order to reformulate to diagonalize the fluctuation operator. This is critical the generalized Proca model as a gauge theory, where the for the algorithm used in [13]. Consequently, our general background mass tensor plays a double role as potential 15 in the vector sector and additional metric in the scalar ACKNOWLEDGEMENTS sector of the Stückelberg field. At the price of dealing simultaneously with two metrics, the standard methods The authors thank I. L. Shapiro for correspondence. are applicable in this case. The work of M. S. R. is supported by the Alexander von Humboldt Foundation, in the framework of the Sofja Ko- Our main result is the derivation of the one-loop di- valevskaja Award 2014, endowed by the German Federal vergences for the generalized Proca model (80). A char- Ministry of Education and Research. acteristic feature of this new result is the appearance of the tensorial parameter integrals (79). The vector field loops induce curvature and M-dependent structures. Appendix A: GENERAL FORMALISM Unless these structures are present in the original action, the generalized Proca model is not perturbatively 1. Heat kernel and one-loop divergences renormalizable—not even in flat space. It is interesting that the main complication of the generalized Proca For an action functional S[φ] of a general field φ model is not connected to the curved background but with components φA, the fluctuation operator F( x), originates from the presence of the second metric struc- obtained from the second functional derivative has com-∇ ture. A x AC x ponents F B ( ) = γ FCB ( ), where γAB is a symmetric, nondegenerate∇ and ultralocal∇ bilinear form. The We have checked our general result (80) by reducing Schwinger integral representation of 1/F reads, it to simpler models. The one-loop divergences for the trivial index case M µν = X2gµν can be obtained in two 1 ∞ = ds e−sF , (A1) ways: by the reduction of the general result (80) and F 0 independently from the Proca model by a Weyl trans- Z formation. We find perfect agreement. Moreover, to the where s is the “proper time” parameter and where we best of our knowledge, the trivial index case is by itself a have indicated the bundle structure of inverse operators A genuinely new result. In addition, we have performed the by the identity matrix 1, which has components δB. The reduction of our result (80) to the case of a vector field Schwinger representation of higher inverse powers 1/F n with a (A Aµ)2 self-interaction. This model has been with n N and the logarithm of F are found to be µ ∈ studied earlier in [27]. We find perfect agreement. Fur- ∞ thermore, we have expanded our general result (80) up 1 ds n−1 −sF n = s e , (A2) to second order in the deviation from the Proca model F 0 (n 1)! Z ∞ − and compared it to a direct perturbative calculation. We ds −sF find perfect agreement. As the direct calculation only re- ln F = e . (A3) − 0 s lies on standard techniques, the agreement does not only Z provide a powerful check of our general result (80), but The integrand of the proper time integral (A1) defines also of the method we used to derive it. Finally, we have the heat kernel compared our results as well as our approach to previous 0 − F 0 work on the generalized Proca model. In [12], part of K(s x, x ) := e s δ(x, x ) . (A4) | our full result for the one-loop divergences (80) has been With the boundary condition K(0 x, x0) = δ(x, x0), it obtained by a different method. In the corresponding | limit, we find that our result reduces to the one derived formally satisfies the heat equation in [12]. In [13], yet another method, based on a combi- (∂ + F) K(s x, x0) = 0 . (A5) nation of the Stückelberg formalism and a nonlocal field s | redefinition, has been proposed. However, the result for For a minimal second order operator the one-loop divergences, obtained in [13], does not agree with our result (80). In general, it is quite remarkably F = ∆ + P , (A6) that the simple extension from the Proca model to the generalized Proca model leads to such a drastic increase a Schwinger-DeWitt representation for the corresponding of complexity—already in flat space. kernel exists

1/2 0 0 0 g (x ) 1/2 0 − σ(x,x ) 0 Our result for the one-loop divergences of the general- K(s x, x ) = D (x, x ) e 2 s Ω(s x, x ) , ized Proca model (80) with a background mass tensors | (4π s)ω | µν µν µν of the form M = ζ1R + ζ2Rg is important for cos- (A7) mological models, which, at the classical level, have been 0 studied extensively [4–11]. It would also be interesting with ω = d/2. The biscalar σ(x, x ) is Synge’s world to apply the method presented in this paper in the con- function [39, 40], which is deﬁned by text of massive gravity [32, 33], more general vector ﬁeld σµσ = 2σ , σ := σ, σµ = gµν σ . (A8) models [34–36] and scalar-vector-tensor models [37, 38]. µ µ ∇µ ∇ν 16

The biscalar D(x, x0) is the dedensitized Van-Vleck de- Here, 1/ε is a pole in dimension ε = d/2 2 and the bun- terminant − A dle curvature Rαβ with components [Rαβ] B is defined ∂2σ(x, x0) by the commutator D 0 −1/2 −1/2 0 (x, x ) = g (x) g (x ) det µ 0ν , (A9) ∂x ∂x A A B [ µ, ν ] φ = [Rµν ] φ . (A18) which is defined by the equation ∇ ∇ B D−1 (Dσµ) = 2ω . (A10) In particular, for a scalar and vector field, we have ∇µ σ All nontrivial physical information is encoded in the [ µ, ν ] ϕ = 0, [ µ, ν ] Aρ = Rµνρ Aσ . (A19) matrix-valued bitensor Ω(s x, x0), ∇ ∇ ∇ ∇ | ∞ Ω(s x, x0) = sn a (x, x0) , (A11) 2. Covariant Taylor expansion and Synge’s rule | n n=0 X where the dependence on the proper time parameter s The covariant Taylor expansion of a scalar function 0 has been explicitly separated by making a power series f(x) around x = x is given by [20], ansatz with the matrix-valued Schwinger-DeWitt coeffi- ∞ 0 k cients an(x, x ). Inserting the ansatz (A7) together with 0 ( 1) 0 µ1 µk f(x ) = − µ0 µ0 f(x ) σ σ . (A11) and the minimal second order operator (A6) into k! ∇ 1 · · · ∇ k x0=x ··· k=0 h i the heat equation (A5), gives a recurrence relation for X (A20) the Schwinger-DeWitt coefficients A µ −1/2 1/2 The generalization of this expansion for fields φ (x) re- [(n + 1) + σ µ] an+1 + D F D an = 0 , A ∇ quires use of the parallel propagator B0 , (A12) P 00 00 0 0 A A B where a 0 for n < 0 implies that a (x, x ) satisfies the [ φ] = 0 φ . (A21) n 0 P P B parallel propagator≡ equation 0 µ 0 The right-hand side of (A21) transforms as scalar at x . σ µa0(x, x ) = 0, a0(x, x) = 1 . (A13) Therefore, we can consider the right-hand side as a scalar ∇ 0 Therefore, the parallel propagator matrix function of x and apply the covariant Taylor expansion 0 A 0 A 0 (A20) around x = x, B0 (x, x ) := [a0] B0 (x, x ) (A14) P ∞ 0 k A A 0 A00 B0 ( 1) A00 B0 parallel transports a field φ (x) at x to a field [ φ] (x ) B0 φ = − µ0 µ0 B0 φ 0 P 0 P k! ∇ 1 · · · ∇ k P x0=x at x along the unique geodesic connecting x with x . It k=0 A0 0 µ A X h i only agrees with φ (x ) if σ µφ (x) = 0. It satisfies σµ1 σµk . (A22) ∇ × ··· A B0 A 0 = δ . (A15) P B P C C Using that the coincide limits of the totally symmetrized Since for a general bitensor, the primed and unprimed covariant derivatives acting on the parallel propagator indices indicate the corresponding tensorial structure at are zero, a given point, the arguments are omitted whenever there A is no possibility for confusion. The coincidence limits (µ0 ... µ0 ) B0 = 0 , k > 0 , (A23) x0 x of the Schwinger-DeWitt coefficients a and their ∇ 1 ∇ k P x0=x → n h i derivatives can be obtained recursively. Using dimen- the parallel propagator can be freely commuted though sional regularization, the Schwinger-DeWitt algorithm the derivatives in (A22). Further setting x00 to x0 and gives a closed result for the divergent part of the one- making use of (A13), we find loop effective action of a minimal second order operator (A6) for a generic field φ. In d = 4 dimensions, the result ∞ k A0 A0 ( 1) B0 is given in terms of the coincidence limit of the second φ = B − µ0 µ0 φ P k! ∇ 1 · · · ∇ k x0=x Schwinger-DeWitt coefficient kX=0 h i µ1 µk div σ σ . (A24) div 1 × ··· Γ1 = Tr ln (∆ + P) 2 A 1 Applying this expansion to a general bitensor T B0 = d4x g1/2 tr a (x, x) , (A16) around coincidence x = x0, shows that all the informa- 2 2 0 − 32π ε A Z tion of T B is contained in the coincidence limits of its 1 a (x, x) = R Rαβγδ R Rαβ 6 ∆R 1 derivatives 2 180 αβγδ − αβ − ∞ 2 k 1 1 1 1 A0 A0 ( 1) C0 αβ T = − 0 0 T + P R1 + R R + ∆P . B C µ1 µk B 2 − 6 12 αβ 6 P k! ∇ · · · ∇ x0=x kX=0 h i (A17) σµ1 σµk . (A25) × ··· 17

3. Coincidence limits We provide the coincidence limits for the world function ˚σ and Van-Vleck biscalar D˚1/2 as well as derivatives thereof 2 1/2 up to M , The coincidence limits for σ, D , an as well as deriva- O tives thereof can be obtained recursively by repeatedly [˚σ] 0 = 0 , (A28) taking derivatives of the defining equations (A8), (A10), x =x [ ˚σ] 0 = 0 , (A29) (A12) and (A13). Commutation of covariant derivatives ∇µ x =x to a canonical order induces curvature terms. Inserting [ ˚σ] 0 = ˚g , (A30) ∇µ∇ν x =x µν the coincidence limits from lower orders of the recursion, ˚ α ˚ α ˚ α [ ˚σ] 0 = δΓ ˚g + δΓ ˚g + δΓ ˚g higher coincidence limits are obtained systematically [14]. ∇µ∇ν ∇ρ x =x µν αρ µρ να νρ µα In case a bitensor involves derivatives at different points, = 3 (µ˚gνρ) , (A31) we can recursively reduce the coincidence limits of primed ∇ 2 ˚ α ˚ β [ ˚σ] 0 = R˚ + 3˚g δΓ δΓ derivatives to coincidence limits involving only unprimed ∇µ∇ν ∇ρ∇σ x =x − 3 µ(ρ|ν|σ) αβ µ(σ νρ) derivatives by Synge’s rule [39, 40], α α β + 2˚g δ˚Γ + δ˚Γ δ˚Γ α(ρ| ∇µ ν|σ) µβ ν|σ) α α β [ µ0 T ]x0=x = µ[T ]x0=x [ µT ]x0=x . (A26) + 2˚g δ˚Γ + δ˚Γ δ˚Γ , ∇ ∇ − ∇ α(µ ∇ν) ρσ ν)β ρσ (A32) Here, T represents an arbitrary bitensor. The first few 1/2 1/2 coincidence limits of σ, D and an are easily obtained. µD˚ = 0 , (A33) ∇ x0=x In this article, apart from the coincidence limit for a2 = h i M4 ˚1/2 1 ˚ , provided already in (A17), we only need coinci- µ ν D = Rµν . (A34) O 0 dence limits up to M2 . Note that, when performing ∇ ∇ x =x 6 O Theh correspondingi coincidence limits of the Schwinger- the covariant Taylor expansion (A20), we have to specify DeWitt coefficients read the metric with respect to which we perform the covariant Taylor expansion, as the metric enters the world function [˚a0]x0=x = 1 , (A35) σ and the definition of the covariant derivative µ. For ∇ [ α˚a0]x0=x = 0 , (A36) a metric ˚gµν , not necessarily compatible with the con- ∇ nection , it is natural to introduce the tensor which 1 µ ˚ 0 ∇ [ α βa0]x =x = Rαβ , (A37) measures the difference between the connection µ and ∇ ∇ 2 ∇ 1 the Levi-Civita connection associated with ˚gµν , [˚a ] 0 = R˚1 P . (A38) 1 x =x 6 −

ρ 1 ρα ˚ −1 In case the metric ˚gµν is compatible with the connection δΓ µν = ˚g µ˚gαν + ν˚gµα α˚gµν . 2 ∇ ∇ − ∇ µ˚gνρ = 0, the coincidence limits (A28)–(A38) reduce to (A27) the∇ well-known results [20].

Appendix B: DETAILS OF THE CALCULATION FOR THE NONDEGENERATE VECTOR FIELD

The traces in (25) can be systematically reduced to the evaluation of tabulated universal functional traces

Tr ln F div = Tr ln ∆ div , (B1) 1 λ| 1 H| div ν div div 1 ν δµ µν 1 Tr1 P = Pµ γ P µ ν 2 , (B2) F ∆ 0 − ∇ ∇ ∆ 0 λ H x =x x =x div div div div 1 1 1 1 1 ν µ νρ µ 2 µν ρσ Tr1 P P = Pµ Pν 2 2γ P Pρ µ ν 3 + γ P P µ ν ρ σ 4 . (B3) F F ∆ 0 − ∇ ∇ ∆ 0 ∇ ∇ ∇ ∇ ∆ 0 λ λ x =x x =x x =x

In (B1), we have used the deﬁnition of the Hodge operator (4) along with the identity (24) and the fact that

1 div Tr ln 1 + λ ν = 0, (B4) 1 ∇µ ∆∇ which is formally seen by expanding the logarithm, making use of the cyclicity of the trace and resumming the terms. The traces in (B3) are already M4 , which allows to freely commute all operators and use O µ δν µ 1 = δν + (M) . (B5) ∆H ∆ O 18

The logarithmic trace (B1) is evaluated directly with (A16), while for the remaining traces we use the following universal functional traces

ν div 1/2 δµ g 1 ν ν = 2 R δµ Rµ , (B6) ∆ 0 16π ε 6 − H x =x div 1 g1/2 1 1

µ ν 2 = 2 Rµν Rgµν , (B7) ∇ ∇ ∆ 0 16π ε 6 − 12 x =x div 1 g1/2 ( 1)n µ1 µ2n−4 = − sym (g) . (B8) n 2 n−2 n−2 µ1...µ2n−4 ∇ · · · ∇ ∆ 0 16π ε 2 (n 1)! x =x −

Inserting (B6)–(B8) into (B1)–(B3), we ﬁnd 1 11 7 1 Tr ln F div = d4xg1/2 R Rµν + R2 , (B9) 1 λ| 16π2ε 180G − 30 µν 20 Z 1 div 1 1 γ γ Tr P = d4xg1/2 + RP 1 + R P µν , (B10) 1 F 16π2ε 6 12 − 6 µν λ Z div 1 1 1 γ γ2 γ2 Tr P P = d4xg1/2 1 + + P P µν + P 2 . (B11) 1 F F 16π2ε 2 12 µν 24 λ λ Z

Adding all traces according to (25), we obtain the ﬁnal result (26).

Appendix C: MULTIPROPAGATOR BIMETRIC TRACES OF THE GENERALIZED PROCA MODEL

1. Divergent part of the second order trace

The evaluation of the second order trace T2 constitutes the most complex part of the calculation. The functional trace T2 is divergent and needs to be regularized. We use regularization in the dimension d = 2ω. Explicitly, the functional trace of the convoluted integral kernels is given by

β α δα † 1 2ω 2ω 0 β0 0 2 0 T2 = Tr0 Π Πβ = d x d x Σ1 (x, x )Σβ0 (x , x) , (C1) D1 D0 Z h i 0 β 0 2 0 where the kernels Σ1 (x, x ) and Σβ0 (x , x) are deﬁned as

β0 β0 0 α δα ˆ 0 2 0 † 1 ˜ 0 Σ1 (x, x ) := Π δ(x, x ) , Σβ0 (x , x) := Πβ0 δ(x , x) . (C2) D1 D0 First, we insert the integral representations (A1), (A4) and (A7) for the kernels of the inverse propagators

0 µ ∞ 0 δν 0 ds − σˆ(x,x ) 1/2 0 µ0 0 1/2 0 δˆ(x, x ) = e 2s Dˆ (x, x ) Ωˆ (s x, x )ˆg (x ) , (C3) D (4πs)ω ν | 1 Z0 ∞ 0 1 0 dt − σ˜(x ,x) 1/2 0 0 1/2 δ˜(x , x) = e 2t D˜ (x , x) Ω˜ (t x , x)˜g (x) , (C4) D (4πt)ω 0 | 0 Z0 α † together with the explicit expressions (63) for Π and Πµ0 into (C2). Then, we expand the derivatives in each factor in (C2) according to the Leibniz rule and collect terms up to M3 , O β0 β0 0 ˆ −1 σα δα ˆ 0 Σ1 (x, x ) = µ σ g˜ δ(x, x ) ∇ D1 ∞ ds 1/2 − σˆ 1/2 −1 σα −1 σα −1/2 1/2 1 −1 σα β0 2s Dˆ ˆ Dˆ ˆ Dˆ = µ ω gˆ e σ g˜ + g˜ σ σˆσ g˜ α 0 (4πs) ∇ ∇ − 2s P Z σα 0 1 σα 0 + g˜ −1 ˆ β σˆ g˜ −1 [a ] β + M4 , (C5) ∇σPα − 2 σ 1 α O 19

0 2 0 −1 σ 1 0 ˆ 0 ˜ Σβ0 (x , x) = µ g˜ β0 σ δ(x , x) − ∇ D0 ∞ 0 0 0 dt 1/2 1/2 − σ˜ −1 σ 1 −1 σ −1 σ −1/2 1/2 D˜ 2t ˆ 0 0 D˜ ˆ 0 D˜ = µ ω g˜ e g˜ β0 σ a˜0 g˜ β0 σ˜σ a˜0 + g˜ β0 σ a˜0 − 0 (4πt) ∇ − 2t ∇ Z 0 −1 σ 4 g˜ σ˜ 0 a˜ + M . (C6) − β0 σ 1 O o Next, we apply the covariant Taylor expansion (A25) separately to the terms in the curly brackets in (C5) and (C6) up to terms with background dimension M2 . This requires knowledge of the covariant Taylor expansion of the O basic geometrical bitensors up to M2 , O 1 σ˜ =g ˜ σˆασˆβ g˜ δΓλ σˆασˆβσˆγ αβ − 2 αλ βγ 1 + 4˜g δΓν δΓλ + 3˜g δΓν δΓλ + 4˜g ˆ δΓλ σˆασˆβσˆγ σˆδ , (C7) 24 αν βλ γδ νλ αβ γδ αλ∇β γδ 0 −1 ρ ρ 1 α β 1 λ α β γ g˜ σ˜ 0 = 0 σˆ δΓ σˆ σˆ δΓ δΓ 2 ˆ δΓ σˆ σˆ σˆ , (C8) β0 ρ Pβ − ρ − 2 ραβ − 6 ραλ βγ − ∇γ ραβ 1 Dˆ 1/2 = 1 + Rˆ σˆασˆβ , (C9) 12 αβ 1 D˜ 1/2 = 1 + R˜ σˆασˆβ , (C10) 12 αβ 0 −1 σ 1/2 1 ρ −1 α β g˜ ˜ 0 D˜ = 0 g˜ R˜ σ , (C11) β0 ∇ρ − 6Pβ ρ αβ 1 ˆ Dˆ 1/2 = Rˆ σˆβ , (C12) ∇α 6 αβ 0 1 0 ˆ β = β Rˆ λσˆγ , (C13) ∇σPα − 2Pλ σγα 0 1 ρ 0 [a ] β = Rˆ ρ Rδˆ ρ + µ2 g˜−1 β . (C14) 1 α α − 6 α α Pρ For the derivation of (C7)–(C14), we made use of (A28)–(A38). Inserting (C7)–(C14) into (C5) and (C6) yields

∞ β0 0 β0 ds 1/2 − σˆ 1/2 −1 αρ 1 −1 α νρ γ 1 −1 αρ γ Σ (x , x) = µ gˆ e 2s Dˆ ˆ g˜ g˜ Rˆ σˆ + g˜ Rˆ σˆ 1 P ρ (4πs)ω ∇α − 2 ν αγ 6 αγ Z0 1 ρ 1 1 νρ g˜−1 σˆγ + Rˆνρ Rˆgˆνρ + µ2 g˜−1 g˜−1 σˆγ + M4 , (C15) −2s γ 2 − 6 νγ O

∞ ˜ 1/2 2 0 dt 1/2 1/2 D − σ˜ σˆρ 1 γ δ 1 λ γ δ Σ 0 (x, x ) = µ g˜ Dˆ e 2t + δΓ σˆ σˆ + δΓ δΓ 2 ˆ δΓ σˆ σˆ σˆ β ω 1/2 ργδ ργλ δ ργδ − 0 (4πt) Dˆ ! 2t 4t 12t − ∇ Z 1 −1 ργ δ 1 ρ 4 g˜ R˜ σˆ + σˆ R˜ 0 + M , (C16) −6 γδ 12 ρ P β O 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 andg ˜µν , which in addition requires to expand the world functionσ ˜ in the exponent of the second product as well as the ratio D˜ 1/2/Dˆ 1/2,

˜ 1/2 µ ν D σ˜ g˜µν σˆ σˆ 1 λ α β γ 1 ν λ ν λ exp = exp 1 + g˜αλδΓ σˆ σˆ σˆ 4˜gαν δΓ δΓ + 3˜gνλδΓ δΓ Dˆ 1/2 −2t − 2t 4t βγ − 48t βλ γδ αβ γδ 1 +4˜g ˆ δΓλ σˆασˆβσˆγ σˆδ + g˜ g˜ δΓλ δΓν σˆασˆβσˆγ σˆδσˆσˆη αλ∇β γδ 32t2 αλ βν γδ η 1 + R˜ Rˆ σˆµσˆν + M3 . (C17) 12 µν − µν O 20

Finally, combining (C15), (C16), and the expansion (C17), we collect all terms up to order M2 in a function Ψ(s, t σµ), which allows us to write the trace (C1) as O | ∞ 2 dsdt 2ω 2ω 0 1/2 µ 1 α β T2 = µ d x d x Dˆ gˆ Ψ(s, t σˆ ) exp G (s/t)ˆσ σˆ . (C18) 2ω ω ω αβ − 0 (4π) s t | −4s Z Z

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 metricsg ˆµν andg ˜µν via the parameter z,

Gµν (z) :=g ˆµν + zg˜µν . (C19) By construction, Ψ(s, t σˆµ) is a polynomial inσ ˆµ(x, x0), | 8 Ψ(s, t σˆµ) :=g ˆ1/2 s−k/2Ψ(k) (s, t)ˆσµ1 σˆµk . (C20) | µ1...µk ··· kX=2 (k) The coeﬃcients Ψµ1...µk (s, t) are local tensors parametrically depending on s and t. The nonzero even coeﬃcients are

1 s s η 1s Ψ(2)(s, t) = g˜−1 g˜−1 Rˆλ η + g˜−2 R˜ + g˜−1 Rˆλ αβ − 4t αβ − 4t λη α β 12t α βη 3t αλ β 1 − s s − s η − λ g˜ 1 Rˆ + R˜ + µ2 g˜ 2 + δΓ ˆ g˜ 1 + M3 , (C21) − 24 αβ t 4t αβ 4t αβ∇λ η O (4) s λ s s Ψ (s, t) = g˜ δΓη ˆ g˜−1 g˜−1 δΓλ δΓη + g˜−1 ˆ δΓλ αβγδ 8t βη γδ∇λ α − 24t αλ βη γδ 12t αλ ∇β γδ s g˜−1 R˜ Rˆ + M3 , (C22) − 48 t αβ γδ − γδ O s Ψ(6) (s, t) = g˜−1 4˜g δΓλ δΓη + 3˜g δΓλ δΓη + 4˜g ˆ δΓλ αβγδµν 192t αβ γλ δη µν λη γδ µν γλ∇δ µν s g˜−1 g˜ δΓλ δΓη + M3 , (C23) − 32t αλ βη γδ µν O s Ψ(8) (s, t) = g˜−1 g˜ g˜ δΓλ δΓη + M3 . (C24) αβγδµνρσ − 128t αβ δλ γη µν ρσ O

0 Changing integration variables from xµ σˆµ(x, x0) leads to the Jacobian → 0 − ∂xµ ∂2σˆ(x, x0) 1 J = det = det gˆρν =g ˆ1/2(x)ˆg−1/2(x0)Dˆ −1(x, x0) , (C25) ∂σˆν ∂xµ0 ∂xρ ! which cancels the factor Dˆ (x, x0)ˆg1/2(x0) in (C18). Thus, we write (C18) as Gaussian integral overσ ˆµ,

µ2 ∞ dsdt 2ω 1 T = d2ωxgˆ1/2 dˆσµ Ψ(s, t σˆµ) exp G (s/t)ˆσασˆβ . (C26) 2 − 2ω sωtω | −4s αβ (4π) 0 µ=1 ! Z Z Z Y We further perform a reparametrization (s, t) (s, u), which allows to easily extract the divergent structure → s s ∂(s, t) s u = , t = , det = . (C27) t u ∂(s, u) u2 All ultraviolet divergences are captured by the lower bound s 0 of the s-integral and (C26) acquires the form → µ2 ∞ ∞ ds 2ω 1 T = d2ωxgˆ1/2 du uω−2 dˆσµ Ψ(u, s σˆµ) exp G (u)ˆσασˆβ . (C28) 2 − 2ω s2ω−1 | −4s αβ (4π) 0 0 µ=1 ! Z Z Z Z Y In order to extract the divergent part, we reparametrize the world function by absorbing a factor of s−1/2,

2ω 2ω σˆ σsˆ 1/2, dˆσµ sω dˆσµ . (C29) → → µ=1 ! µ=1 ! Y Y 21

Finally, inserting the covariant Taylor expansion for Ψ, (C28) reads

2 ∞ µ 2ω 1/2 ω−2 T2 = 2ω d xgˆ du u − (4π) 0 Z Z 4 ∞ 2ω ds (2k) µ u1 µ2k 1 α β Ψ ··· (u, s) dˆσ σˆ σˆ exp G (u)ˆσ σˆ . (C30) × sω−1 µ1 µ2k ··· −4 αβ " 0 µ=1 ! # kX=0 Z Z Y 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 = 2ω = 4 dimensions by collecting→ all terms in the integrand with total s-dependency 1/s. For ω = 2, the prefactor is already of this form. (2k) (2k) Therefore, only the parts Ψµ1...µ2k (u) := Ψµ1...µ2k (0, u), which are independent of s, contribute to the divergent part

µ2 ∞ 4 4 1 T div = d4x gˆ1/2 du Ψ(2k) (u) dˆσµ σˆu1 σˆµ2k exp G (u)ˆσασˆβ . (C31) 2 − 4 µ1...µ2k ··· −4 αβ (4π) ε 0 " µ=1 ! # Z Z kX=0 Z Y Performing the Gaussian integrals yields

4 1 µ µ1 µ2k 1 α β 1 −1 µ1...µ2k dˆσ σˆ σˆ exp Gαβσˆ σˆ = symk G , (C32) (4π)2 ··· −4 G1/2 Z µ=1 ! Y µν with the kth totally symmetrized power of the inverse interpolation metric G−1 ,

µ ...µ (2k)! (µ µ µ µ ) sym G−1 1 2k = G−1 1 2 G−1 2k−1 2k . (C33) k 2kk! ··· The fact that the Gaussian averages vanish for an odd number ofσ ˆµ’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

2 ∞ 4 µ 1 µ ...µ T div = d4x gˆ1/2 du sym G−1 1 2k Ψ(2k) (u) . (C34) 2 − 16π2ε G1/2 k µ1...µ2k Z Z0 k=0 X div Finally, the result for T2 can be expressed as linear combination µ2 T div = d4xgˆ1/2 C(2k,`) Iµ1...µ2k , (C35) 2 − 16π2ε µ1...µ2k (2k,`) Z Xk,` with u-integrals

∞ 1/2 gˆ µ ...µ Iµ1...µ2k := du u` sym G−1 1 2k , (C36) (2k,`) G1/2 k Z0 (2k,`) and u-independent coeﬃcient tensors Cµ1...µ2k ,

1 1 β C(2,0) = g˜−1 R˜ + g˜−2 R˜ , (C37) µν − 12 µν 6 µ νβ 1 − 2 − α 1 − 1 − αβ 1 − β C(2,1) = µ2 g˜ 2 + g˜ 1 Rˆ g˜ 1 Rˆ g˜ 1 Rˆ + δΓα ˆ g˜ 1 , (C38) µν 2 µν 3 µ αν − 12 µν − 2 µανβ 2 µν ∇β α 1 − 1 − 1 − C(4,1) = δΓα δΓβ g˜ 1 + g˜ 1 Rˆ R˜ + g˜ 1 ˆ δΓα , (C39) µνρσ − 6 µν ρα σβ 12 µν ρσ − ρσ 3 µα ∇ρ νσ 1 − β C(4,2) = δΓα g˜ ˆ g˜ 1 , (C40) µνρσ 2 µν ασ∇β ρ (6,2) 1 η − 1 − 1 − C = δΓλ δΓ g˜ g˜ 1 + δΓλ δΓη g˜ g˜ 1 δΓλ δΓη g˜ g˜ 1 µνρσαβ 6 µν σλ ρη αβ 8 µν σρ λη αβ − 4 µν σρ αλ βη 1 − + g˜ g˜ 1 ˆ δΓλ , (C41) 6 µλ νσ ∇β ρα (8,3) 1 − C = δΓλ δΓη g˜ g˜ g˜ 1 . (C42) µνρσαβγδ − 8 µν ρσ αλ βη γδ 22

2. Divergent part of the fourth order trace

Since the trace (68) is T = (M4), the operators in the trace T can be freely commuted and we can use 4 O 4 β δα β 1 = δα + (M) . (C43) D1 ∆ˆ O

Explicitly, the divergent part of T4 acquires the form div div 4 4 −2 αβ −2 γδ ˆ ˆ ˆ ˆ 1 1 T4 = µ d x g˜ g˜ α β γ δ . (C44) ∇ ∇ ∇ ∇ ∆ˆ 2 ∆˜ 2 x0=x Z div div The trace T4 is evaluated in the same way as T2 . We only state the ﬁnal result 4 1 µ ρσ T div = d4x gˆ1/2 tr g˜−1 tr g˜−2 I tr g˜−1 g˜−2 I 4 16π2ε 4 (0,0) − (2,0)ρσ Z h − − µν tr g˜ 2 I α + g˜ 2 I α . (C45) − (2,1)α (4,1)µνα i Appendix D: FUNDAMENTAL INTEGRALS

1. Integral identities

Starting from the fundamental integral identities

αβµ1···µ2k ∂ µ1···µ2k I` = 2 I`−1 − ∂g˜αβ

∂ µ1···µ2k = 2 I` , (D1) − ∂gˆαβ it is possible to derive a sequence of useful integral identities

ˆ g˜ Iαβµ1···µ2k = 2 ˆ Iµ1···µ2k , (D2) ∇λ αβ (2k+2,`) − ∇λ (2k,`−1) 2(k `) Iµ1···µ2k k > ` gˆ Iαβµ1···µ2k = − (2k,`) , (D3) αβ (2k+2,`) −1 µ1...µ2k (2 symk g˜ k = ` ν ν(µ g˜−1 Iαµ1···µ2k+1 = 2k g˜−1 1 Iµ2···µ2k+1) Iνµ1···µ2k+1 , (D4) α (2k+2,`) (2k,`) − (2k+2,`+1) ··· ··· ··· g˜ν Iαµ1 µ2k+1 = 2kgˆν(µ1 Iµ2 µ2k+1) Iνµ1 µ2k+1 , (D5) α (2k+2,`) (2k,`−1) − (2k+2,`−1) δΓ Iαβµ1···µ2k = ˆ Iµ1···µ2k + 2kδΓ (µ1 Iµ2···µ2k)λ . (D6) ανβ (2k+2,`) ∇ν (2k,`) λν (2k,`)

2. Evaluation of the integrals for the general case

An interesting observation is that in d = 4 dimensions the integrals (79) can be evaluated in terms of invariants of the metricg ˜µν . The evaluation of all tensor integrals (79) can be reduced to the evaluation of the fundamental scalar ν integral I(0,0). In d = 4 dimensions, the Cayley-Hamilton theorem guarantees that the eigenvalues λ1, . . . , λ4 ofg ˜µ −k can be expressed in terms of the invariants ek := tr g˜ with k = 1,..., 4. Therefore, the fundamental integral I(0,0) can be expressed in terms of the eigenvalues λk(ej), ∞ gˆ1/2 ∞ du I(0,0) = du 1/2 = . (D7) 0 G (u) 0 4 Z Z k=1 (1 + uλk) q The integral (D7) can be evaluated explicitly and expressed in termsQ of the incomplete elliptic function of the first kind. The general integrals (79) can then be obtained by differentiating the result with respect tog ˜µν andg ˆµν and by making use of (D1). We refrain from performing these operations, as the resulting expressions are horrendously complicated, impractical and not very illuminating. Instead, we choose to present the final result in terms of the much more compact integrals (79). 23

3. Evaluation of the integrals for special cases

In the case of the self-interacting vector ﬁeld considered in Sec. VII B, the integrals have the form

k Iµ1...µ2k = dn gˆ(µ1µ2 gˆµ2n−1µ2n ξµ2n+1 ξµ2k) . (D8) (2k,`) (2k,`) ··· ··· n=0 X n The general coeﬃcients d(2k,`) are given in a closed form in terms of the hypergeometric function 2F1,

(2k)! k ∞ dn = 3−(`−1)/4 du u`+k−n(3 + u)n−k−1/2(1 + u)−(k+3/2) (2k,`) 2n k! n Z0 (2k)! Γ(k ` + 1)Γ(` + 1/2) = 2−n 3(`+1)/4 − F (k ` + 1, k n + 1/2, ` + 1/2, 3)+ n!(k n)! Γ(k + 3/2) 2 1 − − − − Γ( ` 1/2)Γ(k + ` n + 1) 3`+1/2 − − − F (k + 3/2, k + ` n + 1, ` + 3/2, 3) . (D9) Γ(k n + 1/2) 2 1 − −

The hypergeometric function 2F1 is deﬁned as

∞ Γ(c) Γ(a + k) Γ(b + k) zk F (a, b, c, z) = . (D10) 2 1 Γ(a) Γ(b) Γ(c + k) k! kX=0 For the one-loop divergences, we only need the following coeﬃcients 22 4 d0 = √4 3 1 + √3 , d0 = 4√3 , d1 = + √3 , (0,0) − (2,1) 3 − (2,1) − 3 4 4 1 d0 = 73 42√3 , d1 = 41 + 24√3 , d2 = 7 3√3 , (4,1) 5 − (4,1) 5 − (4,1) 5 − 4 8 9 d0 = √4 3 72 41√3 , d1 = √4 3 27 + 16√3 , d2 = √4 3 2 √3 . (D11) (4,2) 5 − (4,2) 5 − (4,2) 5 −

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