Nonminimal non-Abelian quantum vector fields in curved spacetime

Nonminimal non-Abelian quantum vector fields in curved spacetime L. L. Salcedo * Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada, E-18071 Granada, Spain (Received 27 July 2022; accepted 27 October 2022; published 22 November 2022) The quantum effective action of nonminimal vector fields with Abelian or non-Abelian gauge degrees of freedom in curved spacetime is studied. The Proca or Yang-Mills fields are coupled to a local masslike term acting in both coordinate and gauge spaces. Pathologies due to gauge invariance in the ultraviolet are avoided through the introduction of a non-Abelian version of the Stueckelberg field. It is found that the breaking of gauge invariance induced by the mass term affects only the tree-level part of the effective action. The ultraviolet divergent part of the effective action to one loop is obtained using the method of covariant symbols and dimensional regularization. Formulas are given valid for any spacetime dimension and explicit results are shown for the two-dimensional case. As already happened for a single vector field, the ultraviolet divergences are local but not of polynomial type. DOI: 10.1103/PhysRevD.106.105019 I. INTRODUCTION Vector fields play a prominent role in the Standard Model of particles, as mediators of gauge interactions. In turn there is currently a growing interest in the role that various types of vector fields could play in relativistic gravity and cosmology [1–9].Asnotedin[9],“Imposing the conditions of Lorentz symmetry, unitarity, locality and a (pseudo-)Riemannian spacetime, any attempt of modifying gravity inevitably introduces new dynamical degrees of freedom. They could be additional scalar, vector or tensor fields.”The subject has received a further boost with the discovery of ghost-free consistent nonlinear theories of Proca interactions [10–15]. The crucial issue of the quantum stability of these theories has been analyzed in [16,17]. In this work, we consider a set of Nvector fields in curved spacetime endowed with Abelian (Proca) and/or non-Abelian (Yang-Mills) internal degrees of freedom. No self-interactions are included beyond those implied by the Yang-Mills structure, but the vector fields are coupled to an external masslike x-dependent tensor field which is allowed to arbitrarily mix them [see Eq. (2.1)]. Our focus is on the proper quantization of such a theory and on the structure of the quantum fluctuations. Early work studying the subject of quantum fluctuations for vectors fields was carried out in [18–21] (see [22–29] for recent related work). Particular nonminimal couplings (the minimal case being a standard mass term) were considered in [30] at the classical level and in [31] at the quantum level. The quantized theory for general nonminimal couplings was first studied in [32] for particular spacetime backgrounds. There it was found that pathologies arise in the quantization of the theory since the mass term couples effectively as a metric field. Technically the problem is that the masslike field breaks gauge invariance but does not suppress the fluctuations in the longitudinal polarization at large wave numbers. In other words, the principal symbol of the fluctuation operator is singular. General backgrounds were considered in [33] solving the above-mentioned pathology by means of a Stueckelberg field. In this way, a proper gauge symmetry is present in the theory and one can proceed through a standard gauge fixing procedure. However, approximations were introduced in the analysis of [33] giving rise to a nonlocal result. A full solution to the problem of computing the ultraviolet (UV) divergent part of the effective action, within dimensional regularization, was obtained in [34] using the SchwingerDeWitt technique and later in [35] using the method of covariant symbols, finding perfect concordance in both calculations. The pathologies identified in [32] translate to the fact that the UV divergences are local but not polynomial in the masslike external field. The results just noted refer to a single vector field. Here we address the case of several vector fields. This allows us to consider the non-Abelian scenario. In fact, we consider sets of vector fields organized in Abelian and non-Abelian *[email protected] Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 106, 105019 (2022) 2470-0010=2022=106(10)=105019(28) 105019-1 Published by the American Physical Society multiplets. We treat the Abelian and non-Abelian versions simultaneously since the formalism is identical in both cases. In the absence of a mass term, there would be a gauge symmetry present in the Lagrangian. This symmetry is explicitly broken by the mass term. Nevertheless we obtain the remarkable result that the breaking only affects the effective action at tree level, while the contributions from one or more loops are fully gauge invariant. This results in an important simplification of the calculation. Another insight comes from the introduction of the Stueckelberg field in our present non-Abelian setting (see [36] for a review on this subject). In the Abelian case, the Stueckelberg field appears through Aμ¼Bμþ∂μϕ. In this way, a Uð1Þgauge symmetry arises from Bμ→Bμþ∂μΛ,ϕ→ϕ−Λ. In the non-Abelian case, a literal translation of this prescription would take the form Aa μ¼ðBΩÞa μwhere Ωrefers to a non-Abelian gauge transformation with parameters ϕa(in Lie algebra of the gauge group). The resulting theory enjoys a non-Abelian gauge invariance and one can then proceed to fix the gauge through the Fadeev-Popov method. Such approach is, in principle, correct but exceedingly complicated, as the dependence on ϕais nonlinear. In particular, this would imply a reorganization of the loop expansion from the original theory (field Aa μ) to that with Ba μand ϕa.We develop a completely different approach where the Stueckelberg field is introduced linearly also in the nonAbelian setting. Once the Stueckelberg field is introduced, the (UV regulated) quantum theory is no longer pathological and it is possible to proceed to a systematic computation of its effective action. Our focus is on the UV part of the effective action to one-loop order. As already known from the study of the Abelian case [34,35], the mass term acts as an effective second metric tensor. In the present case, this is in fact a non-Abelian effective metric. Presumably, the generalized Schwinger-DeWitt technique [18] can be adapted to this situation, but such an approach is not presently available. Instead here we apply the method of covariant symbols [37]. This method is simple to use and allows one to formulate the loop momentum integration while preserving manifest covariance under diffeomorphism and gauge transformations. Details of the method are provided below. The problem studied here is an extension of that already solved for N¼1(just one vector field), so some specific features of that case are inherited in the more general setting analyzed in this work. In particular, the loop momentum integrals cannot be written in closed form. For N>1this problem is even worse, as the propagators are now matrices with respect to the gauge indices. Also for this reason some contributions to the effective action (see ΓL;0below) cannot be expressed in a standard form involving just integration over x,p, and traces in internal space, and it is necessary to resort to a parametric form, with integration over one more parameter. Unfortunately, while the problem is well posed and the methodfullyappropriate to solve it, wehave found an unexpected impediment, namely, the number of terms obtained for the physically relevant case of four spacetime dimensions is prohibitively large (at least hundreds of terms are generated). In view of this, we develop the formulas for the general case but only present detailed results for two spacetime dimensions (note that there are no UV divergences for odd dimensions within dimensional regularization). In Sec. II we expose the theory to be analyzed. In Sec. III the background field approach is introduced for the effective action. It is shown that its quantum part admits a gauge-covariant treatment. The effective action to one loop is constructed, showing its limitations in the UV sector. Those obstacles are overcome in Sec. IV by introducing the Stueckelberg field. The nonpathological one-loop effective action is constructed and then decomposed into various contributions to be computed subsequently. Section V introduces some notational conventions. Section VI introduces general considerations to undertake the calculation and presents explicit results for d¼2. Some nontrivial symmetries related to metric deformations are also verified. The actual calculations are worked out in Sec. VII. To this end, the method of covariant symbols is reviewed first, and its application to the various contributions is discussed, including the extraction of the coefficients of the UV divergence. The conclusions are summarized in Sec. VIII. The proof of some formulas is provided in Appendix A. Properties of the operator Zμν and its relation to ˆ Rμν of [18] are discussed in Appendix B. The canonical form of ΓSusing a basis of standard operators is displayed in Appendix C. Explicit results for perturbative mass expansions are presented in Appendix D. Details of the method of noncovariant symbols, used as a check of the calculations, are given in Appendix E. Finally, the method of covariant symbols is illustrated through a sample computation in Appendix F. II. FORMULATION OF THE PROBLEM We consider Nreal vector fields ˆ Aa μðxÞ,a¼1;…;N in an Euclidean d-dimensional spacetime with metric gμνðxÞ and action1 S½ ˆ A;M;g¼Zddxffiffiffi g p1 4 ˆ Fμν a ˆ Fa μν þ1 2Mμν ab ˆ Aa μ ˆ Ab ν;ð2:1Þ where Mμν abðxÞis a positive definite2local mass term fulfilling the symmetry condition Mμν abðxÞ¼Mνμ baðxÞ:ð2:2Þ 1The ugly notation ˆ Aa μand ˆ Fμν awill soon be traded by Aa μ and Fμν. 2As a matrix with indices ðμaÞand ðνbÞ. L. L. SALCEDO PHYS. REV. D 106, 105019 (2022) 105019-2 Coordinate indices are raised, lowered, and contracted with the metric gμν. The Nfields are organized in ngauge sectors. Each sector has gauge symmetry of either SUðniÞor U(1) type and the full gauge group is the direct product of these.3The fields fall in the Lie algebra of the group, i.e., the adjoint representation, with gauge coupling giin the gauge sector i. Without loss of generality, one can choose gi¼1for the Abelian factors. Hence, N¼Pn i¼1Niwhere Ni¼n2 i−1 for an SUðniÞsector and Ni¼1for a U(1) one. With a standard normalization of the fields, the field strength tensor is ˆ Fa μν ¼∂μ ˆ Aa ν−∂ν ˆ Aa μþgafabc ˆ Ab μ ˆ Ac ν;ð2:3Þ where fabc are the structure constants of the gauge group. Since the gauge group is a direct product, the structure constants are block diagonal, one block for each gauge sector, and ga¼giis the coupling of the ith sector (the ga take a common value within each block). Of course, in a U(1) sector the structure constant vanishes. The field strength tensor ˆ Fa μνðxÞis covariant under gauge and coordinate transformations. The kinetic part of the action is block diagonal, while the mass term may mix different gauge sectors. When all the gauge sectors are of the U(1) type, the theory is Abelian and reduces to a generalized Proca field with Nflavors. Nevertheless, since all the cases can be treated within the same scheme, we will refer to the internal space as gauge space. Regarding the symmetries, the kinetic term is fully local gauge invariant but such symmetry is reduced to a global one by the mass term: the action is invariant under M→MΩ¼ ΩMΩ−1for a global transformation Ωin the gauge group. If some gauge sectors are equivalent, namely, with equal gauge group SUðniÞor U(1) and same gi, there is an additional global symmetry under rotations among those equivalent sectors, with a corresponding rotation of M. A further symmetry, special for the case of d¼4spacetime dimensions, is that of local Weyl-like transformations, namely, the action is unchanged under the simultaneous replacements gμνðxÞ→ξðxÞgμνðxÞand Mμν abðxÞ→ξ−2ðxÞMμν abðxÞ. III. THE EFFECTIVE ACTION A. The background gauge field Within the background field approach [38], the field is split as a background plus a fluctuation, ˆ Aa μðxÞ¼ Aa μðxÞþAa μðxÞ. In this approach, the effective action Γ½A;M;gfollows from Z¼e−Γ½A;M;g¼ZDAe−S½AþAþRddxffiffig pJA;ð3:1Þ where Aa μðxÞis the background field and the current Jμ aðxÞ is adjusted so that hAa μðxÞi ¼ 0. As usual, Jμ aðxÞ¼ 1 ffiffiffiffiffiffiffiffiffi gðxÞ p δΓ½A;M;g δAa μðxÞ:ð3:2Þ In the background gauge field approach, the field Aa μðxÞ transforms homogeneously under local gauge transformations, the inhomogeneity being saturated by the transformation of Aa μðxÞ. Correspondingly, the gauge-covariant derivative relies on Aa μðxÞas a gauge connection. We will use a single covariant derivative ∇μcontaining coordinate and gauge connections [39]. The coordinate connection is that of Levi-Civita for the metric and the gauge connection is that of the background field Aa μ. So, for instance, for coordinate-scalar and coordinate-vector fields ϕaand Ba μ, respectively, both in the adjoint gauge representation, ∇μϕa¼∂μϕaþgafabcAb μϕc; ∇μBa ν¼∂μBa ν−Γλ μνBa λþgafabcAb μBc ν:ð3:3Þ Throughout, coordinate indices are contracted with the metric gμν and gauge-vector indices with δab. The effective action can be split into the classical or treelevel component S½Aand the quantum correction ΓQ½A which contains graphs with one or more loops, Γ½A¼S½AþΓQ½A:ð3:4Þ Here we find a fundamental result given by the following Theorem.—ΓQ½Ais invariant under local gauge transformations and all the gauge breaking in the effective action is saturated by the mass term at tree level. That is, ΓQ½A;M;g¼ΓQ½AΩ;MΩ;g;ð3:5Þ where ΩðxÞis any local gauge transformation, and AΩand MΩare the gauge-transformed fields. Proof.—The reason is fairly simple. The semiclassical expansion follows from a Taylor expansion of the action S½AþA−Rddxffiffiffi g pJAin powers of the fluctuation A. The zeroth order gives the classical action, and the first order in Acancels due to the equations of motion, i.e., the choice of Jμ. The quantum component ΓQdepends only on terms that are quadratic or higher order in A. The breaking of gauge invariance would come solely from the mass term 1 2Mμν abAa μAb νbut this is covariant since the field Aa μtransforms homogeneously under gauge transformations. The property (3.5) is important because it allows us to use a gauge-covariant formalism for ΓQ. 3More generally, one could take a Lie subgroup of SOðNÞand the results and formulas derived in this work hold equally well in that case. NONMINIMAL NON-ABELIAN QUANTUM VECTOR FIELDS IN …PHYS. REV. D 106, 105019 (2022) 105019-3 B. One-loop effective action The one-loop effective action follows from the quadratic part of the action: Γ1½A¼1 2log Det1δ2S½AþA δA2A¼0 :ð3:6Þ The subindex 1 in Det1indicates that the determinant is to be evaluated in the space AμðxÞ, i.e., of coordinate vectors. Also, for the gauge degrees of freedom, the determinant is taken in the adjoint gauge representation space. This is not explicitly indicated but will be implicit in all formulas as no other gauge representations will be present. Therefore, we need to isolate the terms quadratic in A from the action S½AþA. After the shift ˆ A¼AþA, the field strength tensor in Eq. (2.3) becomes ˆ Fa μν ¼Fa μν þ∇μAa ν−∇νAa μþgafabcAb μAc νð3:7Þ with Fa μν ¼∂μAa ν−∂νAa μþgafabcAb μAc ν:ð3:8Þ In the shifted variables, the quadratic part of the kinetic term of the action takes the form4 Sð2Þ kin½A¼Zddxffiffiffi g p1 4ð∇μAa ν−∇νAa μÞ2−1 2Fab μν Aa μAb ν; ð3:9Þ where we have the introduced field strength tensor Fab μν as an antisymmetric matrix in gauge space (as well as in coordinate space) Fab μν ≡gcfacbFc μν:ð3:10Þ Note that Fab μν vanishes in the Abelian sectors. In what follows, we adopt the convention that covariant derivatives are indicated by adding new indices to the left, hence ϕa μ≡∇ μϕa,Ba μν ≡∇ μBa ν, etc. The only exceptions to this rule are the operators Zμ1μ2…and ZR μ1μ2….5With this convention, Sð2Þ kin½A¼Zddxffiffiffi g p1 4ðAa μν −Aa νμÞ2−1 2Fab μν Aa μAb ν: ð3:11Þ Using integration by parts and Bianchi identities, the quadratic part of the kinetic term can be written as (see Appendix A) Sð2Þ kin½A¼Zddxffiffiffi g p1 2ðAa μνÞ2−1 2ðAa μμÞ2 −Fab μν Aa μAb νþ1 2RμνAa μAa ν;ð3:12Þ where Rμν is the Ricci tensor. Thus, adding the mass term, Sð2Þ mass½A¼Zddxffiffiffi g p1 2Mμν abAa μAb ν;ð3:13Þ the full quadratic Lagrangian controlling the one-loop fluctuations is Lð2ÞðxÞ¼1 2AμKμν 0Aν;ð3:14Þ where Kμν 0¼−gμν∇2þ∇μ∇ν−2Fμν þRμν þMμν:ð3:15Þ Here, and also in what follows, we use a matrix notation for the gauge indices, which will be implicit. As advertised, the Lagrangian Lð2ÞðxÞis manifestly gauge invariant. The term þ∇μ∇νin Kμν 0is a direct consequence of gauge invariance of the kinetic energy part of the action (2.1) and is needed to retain just three polarizations in the Proca field. While the differential operator K0needs not be singular in the presence of a positive definite mass term Mμν, its principal symbol, i.e., the Oð∇2Þleading UV divergent component is singular, since the longitudinal polarizations are not penalized at large wave numbers. The fact that the principal symbol is singular introduces pathologies in the effective action which prevent one from carrying out an extraction of the UV divergent terms. In the special case of a standard Proca field, with a constant scalar mass, the UV divergent part of the effective action is a polynomial in the mass [18], but this is no longer so for a nonconstant mass term even in the Abelian case [34]. This confirms that K0cannot be directly used as the fluctuation operator. IV. THE STUECKELBERG FIELD A. The non-Abelian Stueckelberg field In order to bypass the above-mentioned pathologies in the UV, we will adapt the Stueckelberg approach introduced in [33] (and also applied in [34,35]) for the nonminimal Proca field to the non-Abelian case. To this end, we rewrite the partition function as 4We will occasionally place all the coordinate indices as lower indices when no ambiguity arises. Repeated coordinate indices are always contracted with the metric gμν. 5Some conventions used in this work are summarized in Sec. V. L. L. SALCEDO PHYS. REV. D 106, 105019 (2022) 105019-4 Z¼ZDAe−S½AþAþRddxffiffig pJAZDχe−Sgf ½χ;ð4:1Þ where Sgf½χcan be any action. The partition function is unchanged as the new factor is just a constant.6We take a standard choice Sgf½χ¼Zddxffiffiffi g p1 2χaχa;ð4:2Þ where χaðxÞis a coordinate scalar and a gauge vector (i.e., in the adjoint representation). Subsequently, a change of variables ðA;χÞ→ðB;ϕÞis applied in (4.1), where Ba μðxÞis a real coordinate-vector and gauge-vector field and ϕaðxÞis real coordinate-scalar and gauge-vector field, Z¼ZDBDϕJ½B;ϕe−S½AþA−Sgf ½χþRddxffiffig pJA; J½B;ϕ¼Det∂ðA;χÞ ∂ðB;ϕÞ:ð4:3Þ By construction, the effective action does not depend on the detailed choice of gauge-fixing function(al) χ½B;ϕ, moreover, the expectation value of any functional written in the form F½A;χis independent of this choice (unless the very functional Fdepends on it). This property provides identities for the gauge-fixing dependence of the expectation values [40]. We choose a linear change of variables. Besides simplicity, the virtue of such a choice is that the loop expansion in the new variables coincides with that in the old ones. Specifically, we take Aa μ¼Ba μþ∇μϕa;χa¼∇μBa μ;ð4:4Þ or, using our notational convention for the covariant derivatives, Aa μ¼Ba μþϕa μ;χa¼Ba μμ:ð4:5Þ The corresponding Fadeev-Popov determinant is easily obtained as (see Appendix A) J½B;ϕ¼Det0ðδab∇α∇αÞ:ð4:6Þ The subindex 0 in Det0indicates that the determinant is to be evaluated in the ϕaspace, i.e., the coordinate-scalar space. As already noted, the reference to the adjoint gauge representation is not explicitly displayed, as its presence is ubiquitous and no other gauge representation will be needed. With our choice of a linear change of variables, the determinant Jdoes not depend on the quantum fields B, ϕ. It depends on Aa μand gμν. As usual, the determinant can be implemented through a complex ghost field with quadratic action. It is worth noticing that one could have introduced the Stueckelberg field in a different manner, to wit, through the change of variable A¼BΩin Eq. (3.1), where ΩðxÞis an arbitrary gauge transformation, and ϕaðxÞenters through Ω¼eiϕ. In addition, the measure DA is replaced by DBDΩ.7In this way, the full theory S½BΩbecomes gauge invariant even in the presence of the mass term. Then one fixes the gauge as usual with the Fadeev-Popov method. A more involved question is how to introduce the background gauge machinery. In such alternative approach, the change of variables from ðA;χÞto ðB;ϕÞis not linear and so it should be considerably more complicated than the method adopted above. In the Abelian case, the two approaches are equivalent. B. The one-loop effective action revisited The introduction of the gauge-fixing action Sgf ½χadds an irrelevant constant to the effective action. Hence, in variables ðB;ϕÞthe effective action is just Γ1½A¼1 2log Det1þ0δ2ðS½AþAþSgf½χÞ δðB;ϕÞ2B¼0 ϕ¼0 −log Det0ðδab∇2Þ;ð4:7Þ where the last term comes from the Fadeev-Popov determinant. The subindex 1þ0indicates the direct sum of coordinate-vector and -scalar spaces. The kinetic energy term (3.12) in variables ðB;ϕÞ becomes (see Appendix A) Sð2Þ kin½A¼Zddxffiffiffi g p1 2ðBa μνÞ2−1 2ðBa μμÞ2−Fab μν Ba μBb ν þ1 2RμνBa μBa νþFab μμνϕaBb νþ1 2Fab μμνϕaϕb ν;ð4:8Þ while Sgf½χis already quadratic, namely, Sgf½χ¼Zddxffiffiffi g p1 2ðBa μμÞ2:ð4:9Þ This contribution removes the problematic longitudinal term in the kinetic energy. The price to pay is the introduction of a kinetic term for ϕwhich has a metriclike coupling to the mass tensor, namely, the last term in 6It does not depend on Aμ aðxÞ,Jμ aðxÞ, nor Mμν abðxÞ. It is also independent of the metric if no derivatives are involved. 7Or just DBDϕ. The two measures DΩand Dϕare equivalent. As is well known, the Jacobian of an ultralocal change of variables such as ∂Ω=∂ϕhas no effect in dimensional regularization. NONMINIMAL NON-ABELIAN QUANTUM VECTOR FIELDS IN …PHYS. REV. D 106, 105019 (2022) 105019-5 Sð2Þ mass½A¼Zddxffiffiffi g p1 2Mμν abBa μBb νþMμν abϕa μBb ν þ1 2Mμν abϕa μϕb ν:ð4:10Þ In summary, the new full quadratic Lagrangian controlling the one-loop fluctuations is Lð2Þ¼1 2ðB;ϕÞKðB;ϕÞTþω a∇2ωa:ð4:11Þ The ghost field ωais a complex fermionic coordinate-scalar and gauge-vector field. On the other hand Kis a second order differential operator acting on the space ðB;ϕÞ, K¼−∇2gμν −2Fμν þRμν þMμν −FααμþMμα∇α Fααν−∇αMαν 1 2fFααβ;∇βg−∇αMαβ∇β:ð4:12Þ The matrix gauge indices are implicit. f;gdenotes the anticommutator. Note that several differential operators can be read from the last term in Eq. (4.8), namely, Fααβ∇β,∇βFααβor 1 2fFααβ;∇βg. All of them are equivalent in the ϕ-ϕsector, since the matrix Fμν is antisymmetric and the Hilbert space spanned by ϕais real. However, the functional integral over Band ϕis only related to the determinant of the symmetric version of K, the one presented in Eq. (4.12). As expected, after the introduction of the Stueckelberg field, the principal symbol of the operator Kis no longer singular. Nevertheless, even though the technical problems have been sorted, the pathologies still will reflect on the effective action; in particular, one finds that the UV divergences do not depend polynomially on M, as already happened in the case N¼1studied in [34,35]. The leading UV divergent terms, with two derivatives, are at the diagonal of the matrix K. Particularly problematic will be the term −∇αMαβ∇βin the ϕ-ϕsector. Because the leading divergence in the covariant derivatives is Abelian, only the symmetric component of Mμν is truly of second order. Hence, we will introduce the separation of the mass tensor into symmetric and antisymmetric components, Mμν ab ¼Mμν ab þQμν ab;M μν ab ¼Mνμ ab ¼Mμν ba; Qμν ab ¼−Qνμ ab ¼−Qμν ba:ð4:13Þ It can be noted that Mμν must be positive definite and should dominate Qμν, which is not. The mass term from Q is subdivergent since it is of first order in the derivatives. Indeed, after integration by parts, Zddxffiffiffi g p1 2Qμν abϕa μϕb ν ¼Zddxffiffiffi g p−1 2Qμμν ab ϕaϕb ν−1 4Qμν acFcb μνϕaϕb:ð4:14Þ Then the (symmetric) fluctuation operator Ktakes the final form K¼−∇2gμν þYμν −ΦμþMμα∇α Φν−∇αMαν −∇2 Mþ1 2fPβ;∇βgþW;ð4:15Þ where we have introduced the following shorthand notation: Yμν ¼Mμν −2Fμν þRμν;Φμ¼Fααμ; Pμ¼Fααμ −Qααμ;W¼−1 4fQμν;F μνg;ð4:16Þ as well as ∇2 M≡∇ αMαβ∇β:ð4:17Þ As already noted, the field MμνðxÞ, which was seemingly UV subdominant in the original action, is in fact UV dominant in the sector of the field ϕin Kand acts effectively as a second (inverse) metric. In the Abelian case (N¼1) such metric is an ordinary one, and even so it introduced a considerable amount of complication in the calculation of the effective action in [34,35]. In the setting discussed in this work, the “effective metric”MμνðxÞis a non-Abelian one in gauge space, so we can certainly expect a higher degree of difficulty in the resources needed to attack this problem. From the Lagrangian in Eq. (4.11), the effective action to one loop is thus Γ1½A;M;g¼ΓK½A;M;gþΓgh½A;g;ð4:18Þ with ΓK½A;M;g¼1 2Tr1þ0logðKÞ; Γgh½A;g¼−Tr0logð∇2Þ:ð4:19Þ L. L. SALCEDO PHYS. REV. D 106, 105019 (2022) 105019-6 C. Contributions to the effective action As just said, the effective action can be split into Γ1¼ΓKþΓgh:ð4:20Þ The operator Kcan be split into UV leading Oð∇2Þand subdivergent Oð∇Þcomponents K¼KLþKS; KL≡diagð−∇2gμν;−∇2 MÞ:ð4:21Þ Correspondingly, we also separate the effective action as ΓK¼1 2Tr1þ0ðKLð1þK−1 LKSÞÞ ¼ ΓLþΓS;ð4:22Þ with ΓL¼1 2Tr1þ0logðKLÞ; ΓS¼1 2Tr1þ0logð1þK−1 LKSÞ:ð4:23Þ The relation TrðlogðABÞÞ ¼ TrðlogðAÞÞþTrðlogðBÞÞ is only guaranteed for sufficiently convergent pseudodifferential operators Aand B. Nevertheless, it is expected to correctly reproduce the UV divergent terms within dimensional regularization when Aand Bare both coordinatescalar operators. ΓLcan be split further as ΓL¼ΓL;1þΓL;0;ð4:24Þ with ΓL;1¼1 2Tr1logð−∇2gμνÞ; ΓL;0¼1 2Tr0logð−∇2 MÞ:ð4:25Þ For the purpose of obtaining the UV divergences of the effective action, the subdivergent terms can be treated perturbatively ΓS¼1 2Tr1þ0logð1þK−1 LKSÞ¼X ∞ n¼1 ΓS;n;ð4:26Þ where ΓS;n ¼−1 2nTr1þ0ðð−K−1 LKSÞnÞ:ð4:27Þ Since K−1 LKS¼Oð∇−1Þ, terms with n>dare UV finite in dspacetime dimensions. Thus, collecting the various contributions, Γdiv 1¼Γdiv gh þΓdiv L;0þΓdiv L;1þX d n¼1 Γdiv S;n:ð4:28Þ The contributions to Γdiv 1½A;M;gare analyzed in the following sections, after introducing some notation. V. SOME NOTATIONAL CONVENTIONS A. Covariant derivatives Let us first recall that the covariant derivative operator ∇μcontains all connections (and not only the Christoffel symbols), and also our convention that covariant derivatives are indicated by adding coordinate indices to the left, e.g., Rρμναβ ≡½∇ρ;R μναβ;Rλλμ ¼1 2Rμ:ð5:1Þ Here Rμναβ,Rμν, and Rdenote the Riemann tensor, the Ricci tensor, and the scalar curvature, respectively. All quantities in the fluctuation operator Kare to be regarded as operators acting on the vector space spanned by the fields Bμand ϕ. Hence ∇μacts on such quantities through the commutator. In particular gμν,Mμν, and Fμν are purely multiplicative operators, which means that they are ordinary functions (possibly matrices in gauge space).8 B. Operators Zμ1μn We will make use of the operator Zμν, which is defined as Zμν ≡½∇μ;∇ν:ð5:2Þ This operator is multiplicative because its action on a quantity does not involve derivatives of that quantity; Zμν is diagonal in xspace. However, it is not purely multiplicative because it acts (is not diagonal) on coordinate indices. For instance, for a purely multiplicative tensor field Vμν, ½Zμν;Vαβ¼RμναλVλβ þRμνβλVαλ þ½Fμν;Vαβ:ð5:3Þ The operator Zμν admits a natural separation between coordinate and gauge actions Zμν ¼ZR μν þFμν;ð5:4Þ where ZR μν acts only on coordinate indices. As illustrated in (5.3), the operator ZR μν acts on every coordinate index in turn. Higher order operators Zμ1μn, with ncovariant derivatives, are defined recursively (see Appendix B) so that they are also multiplicative. Letting I¼μ1μndenote a 8Of class Cð∇; ZÞin the notation of [37]. NONMINIMAL NON-ABELIAN QUANTUM VECTOR FIELDS IN …PHYS. REV. D 106, 105019 (2022) 105019-7 string of coordinate indices, the operators ZIhave again a clean separation between coordinate and gauge ZI¼ZR IþFI;ð5:5Þ and also fulfill ½ZR I;Vμ1μ2¼RIμ1λVλμ2þRIμ2λVμ1λ:ð5:6Þ In fact, these operators are an anti-Hermitian version of the derivatives of the operator ˆ Rμν of [18]. Specifically, ZR α1αnμν ¼∇α1∇αn ˆ Rμν þCα1αnμν;ð5:7Þ where the CIare purely multiplicative operators constructed with the Riemann tensor, Cμν ¼0; Cα1αnμν ¼1 2Rλα2α3αnμνα1λþ1 2Rα1λα3αnμνα2λþ þ1 2Rα1αn−1λμναnλ:ð5:8Þ Eventually we will need to take traces of operators with a factor ZR Ion the left. The formulas for the coordinate-scalar and -vector spaces are, respectively, tr0ðZR IOÞ¼−tr0ðCIOÞ; tr1ðZR IOμνÞ¼tr1ððRIμλ−gμλCIÞOλνÞ:ð5:9Þ Further details and proofs are given in Appendix B. C. Integrals and traces We will use the shorthand notation hXix≡Zddxffiffiffi g pX; ð5:10Þ as well as hXix;g ≡Zddxffiffiffi g ptrgðXÞ¼htrgðXÞix;ð5:11Þ where trgðÞ refers to trace over gauge space. In particular trgð1Þ¼N, the dimension of the gauge space is the number of dynamical real vector fields in the theory. In addition, for integrals over a momentum variable in Sec. VII, hXip≡1 ffiffiffi g pZddþ2εp ð2πÞdX; ð5:12Þ where dþ2εrefers to dimensional regularization. We also use combinations such as hXix;p for hhXipix, etc. VI. RESULTS FOR ΓDIV 1 A. Results for Γgh and ΓL;1 The contributions Γgh and ΓL;1to the one-loop effective action are given by Eqs. (4.19) and (4.25), respectively. The computation of their UV divergent part is straightforward in dimension regularization, in dþ2εdimensions, using the identity Tr logð−∇2Þjdiv ¼1 ð4πÞd=2 1 εZddxffiffiffi g ptrðbd=2ðxÞÞ;ð6:1Þ where the trace is taken in the corresponding space and bn is the nth heat-kernel coefficient of the Laplacian [39].For d¼2and d¼4, the required coefficients are b1¼1 6R; b2¼1 12 Z2 μν þ1 180 R2 μναβ −1 180 R2 μν þ1 72 R2:ð6:2Þ As they stand, these formulas hold for an arbitrary space since Zμν takes care of all required curvatures (coordinate, gauge, or other in more general cases). For the space of coordinate tensors of rank rin ddimensions (and adjoint gauge representation), one easily finds trrðZ2 μνÞ¼trrðF2 μνÞþtrrððZR μνÞ2Þ ¼drtrgðF2 μνÞ−rdr−1NR2 μναβ;ð6:3Þ where as already said trgðÞ denotes the trace over gauge space. Of course, this result is fully consistent with Eq. (5.9). Therefore, for d¼2, Γdiv gh ¼−1 4πε1 6Rx;g ; Γdiv L;1¼1 4πε1 6Rx;g :ð6:4Þ These two contributions cancel each other, as they should in d¼2. For d¼4, Γdiv gh ¼−1 ð4πÞ2ε1 12F2 μν þ1 180R2 μναβ −1 180R2 μν þ1 72R2x;g ; Γdiv L;1¼1 ð4πÞ2ε1 6F2 μν −11 360R2 μναβ −1 90R2 μν þ1 36R2x;g : ð6:5Þ L. L. SALCEDO PHYS. REV. D 106, 105019 (2022) 105019-8 B. Results for ΓS 1. Contributions to ΓS The UV divergent part of ΓSin ddimensions is contained in Pd n¼1ΓS;n, where ΓS;n is given in Eq. (4.27). The quantity KL, defined in Eq. (4.21), is homogeneous in ∇ of degree þ2, while KS≡K−KLcontains terms of degrees 0 and 1. We will expand ΓS;n in powers of ∇ keeping terms up to Oð∇−4Þ, which is sufficient for Γdiv 1in spacetime dimensions d≤4. The trace cyclic property can be used to collect equivalent terms and also we choose whenever possible to bring the trace to the scalarcoordinate space. Introducing the notation Δ≡1 ∇2;ΔM≡1 ∇2 M ;ð6:6Þ this procedure yields the following expressions: ΓS;1¼Tr1−1 2ΔYμνþTr0−1 2ΔMW−1 4ΔMfPμ;∇μg; ΓS;2¼Tr1−1 4ΔYμαΔYανþTr01 2ΔMΦμΔΦμþ1 2ΔM∇μMμνΔMνα∇α −1 2ΔMΦμΔMμν∇ν−1 2ΔM∇μMμνΔΦν −1 4ΔMWΔMW−1 16 ΔMfPμ;∇μgΔMfPν;∇νg−1 4ΔMfPμ;∇μgΔMW; ΓS;3¼Tr01 2ΔM∇μMμνΔYναΔMαβ∇βþ1 2ΔMWΔM∇μMμνΔMνα∇αþ1 4ΔMfPμ;∇μgΔM∇νMναΔMαβ∇β −1 4ΔMfPμ;∇μgΔM∇νMναΔΦα−1 4ΔMfPμ;∇μgΔMΦνΔMνα∇α −1 8ΔMWΔMfPμ;∇μgΔMfPν;∇νg−1 48 ΔMfPμ;∇μgΔMfPν;∇νgΔMfPα;∇αgþOð∇−5Þ; ΓS;4¼Tr0−1 4ΔM∇μMμνΔMνα∇αΔM∇βMβρΔMρσ∇σþ1 8ΔMfPμ;∇μgΔMfPν;∇νgΔM∇αMαβΔMβρ∇ρ −1 128 ΔMfPμ;∇μgΔMfPν;∇νgΔMfPα;∇αgΔMfPβ;∇βgþOð∇−5Þ:ð6:7Þ The functional traces are of the form TrrðAÞ¼Zddxffiffiffi g ptrrðhxjAjxiÞ;r¼0;1;ð6:8Þ where dis the spacetime dimension and tr0or tr1refer to the trace over coordinate labels, scalar or vector, respectively, and also include trace over gauge labels. In detail, the diagonal (in xspace) matrix element hxjAjxiis of the form hx; I0;ajAjx; I; bi, where a,bare gauge labels in the adjoint representation and I,I0are coordinate labels. For Tr0ðÞ these coordinate labels are absent, while for Tr1ðÞ they are of the type I¼μ,I0¼ν. At this point, one could already attempt the computation of the functional traces to obtain Γdiv S. Nevertheless, it is convenient to first simplify the expressions by bringing the operators to a canonical form. This is in the same spirit as the universal functional traces of [18]. The goal is to put together terms involving powers of ∇(to wit, ∇μ,Δ, and ΔM) on one side and the purely multiplicative terms on another. That is, bring the various operators in (6.7) to the form A¼X n OnAn;ð6:9Þ where Onform a basis of pseudodifferential operators and Anare purely multiplicative operators. We have chosen to put the latter on the right-hand side. Thus, for the diagonal matrix elements, hx;I0;ajAjx;I;bi¼X n;c hx;I0;ajOnjx;I;ciðAnðxÞÞcb;ð6:10Þ or just hxjAjxi¼PnhxjOnjxiAnðxÞ. The coefficients An do not modify the UV degree of divergence of the term, which is controlled by On, so the hardest work is computing hxjOnjxi. NONMINIMAL NON-ABELIAN QUANTUM VECTOR FIELDS IN …PHYS. REV. D 106, 105019 (2022) 105019-9 If there are two metrics gμν and Mμν (and even more so when the latter is non-Abelian), the integrals cannot be evaluated in closed form in general. In this case, one should be aware of ambiguities in the final expression due to integration by parts in momentum space. Complicated expressions can occasionally reach a simpler form through integration by parts in pμ. For this reason, the numbers quoted in Table Iare upper bounds. An explicit application of the method of the covariant symbols to illustrate the procedure just described is displayed in Appendix F, by computing one of the universal functional traces of [18]. In what follows, we proceed to give details of the calculation of Γdiv Sand Γdiv L;0. B. Calculation of Γdiv Sin d=2 The starting point is Eq. (6.15). Applying the covariant symbols formula (7.5), hxj ˆ Ojxi¼h¯ Oipð7:7Þ [with hipdefined in Eq. (5.12) and the 1is implicit)], one has14 ΓS;1¼tr1−1 2 ¯ O19Yμν þtr0−1 2 ¯ O2W−1 2ð¯ O1ÞμPμ þ1 4 ¯ O2Pμμ þOðp−3Þx;p ; ΓS;2¼tr01 2ð¯ O3ÞμνMμαMαν −1 4ðO4½PμÞμνPνþOðp−3Þx;p :ð7:8Þ Since ¯ ∇μ¼OðpμÞ, while for multiplicative operators ¯ X¼ Oð1Þfor large pμ, the terms Oð∇−nÞin ˆ Obecome Oðp−nÞ in ¯ O. Within dimensional regularization hp−nipvanishes for all nwith the exception n¼d. Hence, we need to isolate terms 1=p2in d¼2, and in particular more UV convergent terms can be neglected. Of the basic operators present in the formula, O2;3;4;19, are of Oð∇−2Þ. Therefore, the covariant symbols of the latter only require the leading terms of the building blocks, ¯ ∇μ¼pμþOðp−1Þ;¯ Δ¼−NgþOðp−3Þ; ¯ ΔM¼−NMþOðp−3Þ;¯ X¼XþOðp−1Þ;ð7:9Þ where Xis any purely multiplicative operator. Here we have introduced the definitions Ng≡ð−gμνpμpνÞ−1;N M≡ð−MμνpμpνÞ−1:ð7:10Þ Note that NMis a matrix in gauge space. This produces ¯ O2¼−NMþOðp−3Þ; ¯ O3¼NMNgpμpνþOðp−3Þ; O4½Xμν ¼NMXNMpμpνþOðp−3Þ; O19 ¼−NgþOðp−3Þ:ð7:11Þ The terms shown explicitly are homogeneous of degree p−2. The remaining operator O1is Oð∇−1Þand its covariant symbol Oðp−1Þ. To isolate the 1=p2term we have to take one more term in the expansion. The expansion in (7.2) is effectively in powers of ∇=p, so terms with one more covariant derivative are needed. Note that in this counting Zμ1;…;μncounts as Oð∇nÞ. From its definition ðO1Þλ≡ΔM∇λ, one obtains ð¯ O1Þλ¼¯ ΔM¯ ∇λ;ð7:12Þ with ¯ ΔM¼ð¯ ∇μ¯ Mμν ¯ ∇νÞ−1:ð7:13Þ The expansion of ¯ ∇λin Eq. (7.9) is already sufficient, but ¯ ΔMneeds to be expanded to order 1=p3which in turn requires ¯ Mμν to order 1=p, ¯ ΔM¼ðpμðMμν −Mαμν∂αÞpνþOð1ÞÞ−1 ¼−NMþNMpμMνμα∂νpαNMþOðp−4Þ:ð7:14Þ Therefore, ð¯ O1Þλ¼−NMpλþNMpμMνμα∂νpαNMpλþOðp−3Þ: ð7:15Þ To the order needed, the operators ¯ O2;3;4;19 do not have any ∂μ, hence they already coincide with ¯ On1.For ¯ O1the momentum derivatives can be moved to the right using ½∂μ;p ν¼δμ ν;½∂μ;NM¼2pνNMMμνNM:ð7:16Þ 14For a multiplicative X,hxjOXjxi¼hxjOjxiXðxÞ, consistently h¯ O¯ Xip¼h¯ OipXðxÞsince ¯ X−Xcontains ∂μbut not pμ [see Eq. (7.2)], and such terms vanish inside hip. L. L. SALCEDO PHYS. REV. D 106, 105019 (2022) 105019-16 In this way,15 ð¯ O1Þλ1¼−NMpλ−NMMμμνNMpλpν −2NMMμνNMMμαβNMpλpνpαpβ þOðp−3Þ:ð7:17Þ The first term is homogeneous of degree 1=p and the two other explicit terms are homogeneous of degree 1=p2. The first term vanishes within momentum integration due to parity. For the same reason, all odd order terms have been omitted in Table I. In any case, only the contributions 1=pd are relevant in ddimensions for Γdiv 1. No operators ZIappear in the expansions of ¯ O1;2;3;4;19 to the order required in Eq. (7.8). The most UV divergent operator is O1¼Oð∇−1Þ, so its leading term is 1=p and the term relevant in d¼2,1=p2, comes from contributions of the type ∇=p2, while ZIneeds at least two covariant derivatives. Operators ZIdo appear in d¼4. Just as ∂μ, the operators ZR Ihave to be resolved before having a useful expression for hxjOnjxi. To this end, the operators ZR Ican be moved to the left or to the right, using their commutation relations noted in (B4), including (7.4). In accordance with the choice in Eq. (6.9), where the multiplicative operators Anhave been placed at the right, ZR Ishould be moved to the left. This allows one to apply the rules in Eq. (5.9) (see Appendix B). It remains to insert in Eq. (7.8) the various expressions for ¯ On1just obtained. In the resulting expression, the terms involving Pμin ΓS;1are not manifestly Hermitian. This can be fixed by applying integration by parts in pand x spaces16 as well as the trace cyclic property. This gives, Γdiv S;1¼1 2NgYμμ þ1 2NMW−1 4NMPμμ þ1 2NMMμνNMMμαβNMPλpνpαpβpλ −1 2NMMμαβNMMμνNMPλpνpαpβpλx;p;g ; Γdiv S;2¼1 2NgNMMμαMανpμpν−1 4NMPμNMPνpμpνx;p;g : ð7:18Þ The integrand is a homogeneous function of pof degree −2. It only remains to extract the 1=εcoefficient to isolate the UV divergent contributions. The details are given in Sec. VII D. An application of the rules provided there immediately produces the result quoted in Eq. (6.16). In Eq. (7.17) there is one term of degree p−1and two terms of degree p−2. Table Ishows the number of terms of each degree for the diagonal matrix elements of the operators On. Only even orders are displayed since odd orders vanish upon integration over pμin any parity preserving regularization, such as dimensional regularization. For a given operator, the number of terms increases rapidly with (minus) the degree. Nevertheless, the number of terms displayed in the table is an upper bound; this number is subject to variations due to various identities which allow one to write a given expression in different forms. Such identities include integration by parts in momentum space and reordering of the covariant derivatives acting on a tensor due to the Jacobi identity, ½∇μ;½∇ν;X ¼ ½∇ν;½∇μ;Xþ½Zμν;X:ð7:19Þ Furthermore, integration by parts in xspace and trace cyclic property is allowed within the functional trace operations Tr0;1in Eq. (6.13). We have not attempted a systematic minimization of the number of terms as there is no practical procedure to do this, and in any case we do not expect a significant reduction in the length of the expressions. An exception is the operator O2at p−4which is used below, Eq. (7.21), in the computation of ΓL;0, in Sec. VI C 4. Nevertheless, it should be noted that, recently, important progress has been achieved in the counting and classification of allowed independent terms in effective field theories, through the construction of Hilbert series of the operator basis [50]. In this technique, the basic blocks (fields or composite operators) plus their symmetrized derivatives are identified with representations of the d-dimensional conformal group.17 Computation of the Clebsch-Gordan series then allows one to obtain generating functions for basis operators and count them. The key point is that both equation of motion as well as integration by part identities are automatically accounted for, in addition to spacetime and internal group symmetries. The method has been successfully applied to pure Einstein relativity and also to general relativity combined with the Standard Model of particle physics [51]. The adaptation of the Hilbert series technique to obtain basis of operators in diagonal matrix elements and the effective action contributions as those displayed in Eqs. (7.18) or (7.22) would be extremely interesting, and more so in d¼4where the number of terms becomes huge. Serious complications arise due to the presence of an additional momentum variable, with its own integration by parts identities, and also the existence of constraints relating some of the building blocks, such as Mμν and NM. No attempt will be made here to adapt the promising technique of Hilbert 15Actually, for convenience here ∂μhas been moved to the left (and then removed) so the rhs differs from ¯ O1by terms which vanish upon momentum integration. 16As shown in Appendix C of [35],pμcan be treated as a constant when integrating by parts in xspace, when ∇μand ZR I are no longer present. 17Alternatively, cohomological techniques can be applied [50]. NONMINIMAL NON-ABELIAN QUANTUM VECTOR FIELDS IN …PHYS. REV. D 106, 105019 (2022) 105019-17 series to working with covariant symbols; we defer such a study to future work. C. Calculation of Γdiv L;0in d=2 Because of the similarity between the operators O0 2and O2, the expression of their diagonal matrix elements are identical when written in terms of NM, with the only proviso of using the new definition NM¼ð−z−MμνpμpνÞ−1ð7:20Þ for O0 2, instead of that in (7.10). The UV divergent terms in ΓL;0must have exactly d covariant derivatives. The operator O0 2is of UV degree Oð∇−2Þ, so the leading term in O02is Oðp−2Þand no derivatives. Since one still needs to expand its covariant symbol to dcovariant derivatives, the order p−d−2∇dis needed. The proper divergence Oðp−dÞis only recovered after integration over z, a parameter of dimension squared mass. We present results only for the case d¼2. For the terms of order exactly p−4, the calculation gives the following result: hxjO0 2jxið−4Þ¼4NMMμνNMMμαβNMMρσλNMMρηNMpνpαpβpσpλpηþ2NMMμνNMMμαβNMMρρσNMpνpαpβpσ þ2NMMμμνNMMαβρNMMασNMpνpβpρpσþNMMμνNMMμαβρNMMασNMpνpβpρpσ þNMMμνNMMαμβρNMMασNMpνpβpρpσþ2 3NMMμνNMMαβNMMρσNMRμαρλpνpβpσpλ −2 3NMMμνNMMαβNMMρσNMRμλαρpνpβpσpλþNMMμμνNMMααβNMpνpβ −NMMμνFμαNMMαβNMpνpβ−NMMμνNMFμαMαβNMpνpβ−1 6NMMμνNMRμνp :ð7:21Þ The expression obtained after applying the method of covariant symbols has been simplified by using integration by parts in momentum space, also achieving a manifest Hermitian form (since O2is Hermitian). It can be noted that the expression in (7.21) holds for arbitrary d.Ford¼2, one can use the identities in (6.33). Alsowe integrate by parts with respect to xto have at mostone covariant derivative on Mμν. This gives for the effective action Γdiv L;0¼1 2−2NMMμνNMMμαβNMMρσNMMρληNMpνpαpβpσpλpη−2NMMμναNMMμβNMMρσλNMMρηNMpνpαpβpσpλpη −NMMμνNMMμαβNMMαρσNMpνpβpρpσþNMMμνNMMααβNMMμρσNMpνpβpρpσ −NMMμναNMMβμρNMMβσNMpνpαpρpσþNMMμναNMMββρNMMμσNMpνpαpρpσ þNMMμνNMMαβNMMμρNMRpνpαpβpρþNMMμμνNMMααβNMpνpβ−NMMμνFμαNMMαβNMpνpβ −NMMμνNMFμαMαβNMpνpβþ2 3NMMμνNMMμαNMRpνpα−1 12 NMMμμNMRx;p;g;z :ð7:22Þ One could apply further the trace cyclic property and also the identity NMMμνNMpμpν¼−NM−zN2 Min one of the terms, but no simplification would be achieved. The final step is to extract the 1=εcoefficient from the radial part of the momentum integral, as described in the next subsection. This procedure yields the expression quoted in Eq. (6.32). D. Extraction of the UV divergent component Let us consider first the case when the parameter zis not present, as in Eq. (7.18). Once the operators ∂μand ZR Ihave been removed, the structure of a general term ¯ Onto be integrated over pμis a sum of products with factors NM, Ng, and pμ, as well as pμ-independent multiplicative operators MI,RI,FI, etc. Hence, the momentum integral affects only terms of the form Nn gðNMÞa1b1ðNMÞambmpμ1pμ2j≡Nn gN⊗m Mp⊗2j:ð7:23Þ This monomial is homogeneous in pμwith degree 2ðj−n−mÞ. L. L. SALCEDO PHYS. REV. D 106, 105019 (2022) 105019-18 We are only interested in the UV divergent part and to extract it we will use dimensional regularization, namely, in dþ2εdimensions with ε→0−. For the UV divergent part, dcan be used instead of dþ2εin the UV-finite contributions. Within dimensional regularization, the integral Im;j d¼1 ffiffiffi g pZddþ2εp ð2πÞdNn gN⊗m Mp⊗2jð7:24Þ vanishes unless d¼2ðnþm−jÞ, which is assumed in what follows. Recalling that pμ¼ikμand ddpμ≡ddkμ, Im;j d¼ð−1Þj1 ffiffiffi g pZddþ2εkμ ð2πÞdNn gN⊗m Mk⊗2j:ð7:25Þ Let us introduce a standard tetrad field eA μðxÞ, gμν ¼eA μeB νδAB;e A μeμ B¼δA B;ð7:26Þ in such a way that kμ¼eA μkA;d dkμ¼ffiffiffi g pddkA; Ng¼1 k2 A ;N M¼1 kAkBMAB ;ð7:27Þ thus ðIm;j dÞμ1μ2j¼ðIm;j dÞA1A2jeA1 μ1eA2j μ2j; ðIm;j dÞA1A2j¼ð−1ÞjZddþ2εkA ð2πÞdNn gN⊗m Mk⊗2j A:ð7:28Þ The UV divergence comes from the radial part of the integral, hence we introduce spherical coordinates, kA¼k ˆ kA;k¼ffiffiffiffiffi k2 A q¼N−1=2 g; ˆ k2 A¼1:ð7:29Þ This allows one to separate the integral into radial and angular average factors, Im;j d¼ð−1Þj ð4πÞd=2 2 Γðd=2ÞIε ˆ Im;j d;ð7:30Þ where the angular average is ˆ Im;j d¼Γðd=2Þ 2πd=2Zdd−1Ωˆ k ˆ N⊗m M ˆ k⊗2j A≡hˆ N⊗m M ˆ k⊗2j Aiang; ˆ NM≡1 ˆ kA ˆ kBMAB ¼NM=Ng;ð7:31Þ and Iεis the radial part (introducing a cutoff mass m0to avoid a trivial infrared divergence for negative ε), and using the condition 2ðnþm−jÞ¼d, Iε¼Z∞ m0 dkk2ε−1¼−m2ε 0 2ε¼−1 2εþOð1Þ:ð7:32Þ In summary, for the UV divergent part, one obtains Im;j;div d¼1 ð4πÞd=2Γðd=2Þ 1 εð−1Þjþ1ˆ Im;j d:ð7:33Þ As noted, the angular averages ˆ Im;j dare perfectly UV convergent and well defined, but they cannot be written in closed form in general. The analysis is similar for ΓL;0, which involves an additional integration over z. As mentioned, in this case and for d¼2, the relevant terms are of order p−4, and to extract the UV divergent part one must integrate over zand pμ. The point can be elucidated following the steps shown above (for d¼2), noting that now NMcontains z: hp⊗2jN⊗ð2þjÞ Miz;p ¼1 ffiffiffi g pZd2þ2εp ð2πÞ2Zγ dz 2πi × logðzÞp⊗2j1 −z−Mμνpμpν⊗ð2þjÞ ¼ð−1ÞjZ∞ m0 dkk2jþ1þ2εZdΩ ð2πÞ2Zγ dz 2πi × logðzÞ ˆ k⊗2j1 −zþk2Mμν ˆ kμ ˆ kν⊗ð2þjÞ: ð7:34Þ Applying the rescaling z→zk2and noting that the induced term with logðk2Þvanishes since no singularities are enclosed by the path γin the zcomplex plane, hp⊗2jN⊗ð2þjÞ Miz;p ¼ð−1Þjþ1m2ε 0 2εZdΩ ð2πÞ2Zγ dz 2πi × logðzÞˆ k⊗2j1 −zþMμν ˆ kμ ˆ kν⊗ð2þjÞ ¼ð−1Þjþ1 4πε hˆ k⊗2jˆ N⊗ð2þjÞ Miang;z þOð1Þ; ð7:35Þ where ˆ NM¼ð−zþMμν ˆ kμ ˆ kνÞ−1. Applying this angular average in Eq. (7.22) yields Eq. (6.32). VIII. SUMMARY AND CONCLUSIONS In this work we have addressed the problem of quantizing a system of, in general, non-Abelian vector fields with a completely general local nonminimal mass term coupling all of them. The case of NAbelian fields is a particular instance in our formulation. We make use of a background field approach. A remarkable result is that, although the NONMINIMAL NON-ABELIAN QUANTUM VECTOR FIELDS IN …PHYS. REV. D 106, 105019 (2022) 105019-19 mass term breaks gauge invariance, the effective action is fully gauge (as well as coordinate) invariant beyond tree level, Eq. (3.5). The technical problems present in the original theory (namely, the UV region is blind to the mass term and so requires some type of gauge fixing) are satisfactorily removed by introducing a non-Abelian Stueckelberg field, Eqs. (4.1) and (4.4). This is done after background and fluctuation fields have been separated and the Stueckelberg field only affects the latter. This auxiliary field is introduced linearly so that the loop structure of the original theory is preserved. As a consequence, the computation of the UV divergent part of the effective action to one loop can be carried out systematically preserving coordinate and gauge symmetries during the calculation. To this end, we apply dimensional regularization and the method of covariant symbols. This produces terms which are local, i.e., they contain a finite number derivatives of the external fields (no more than din dspacetime dimensions), however, they are not polynomial with respect to the mass term, a fact already observed in the simpler case of a single vector field. The one-loop effective action is expressed in Eq. (4.28) as a sum of four terms, Γgh defined in (4.19),ΓL;1 and ΓL;0in (4.25), and ΓSin (4.26) [and expanded in Eq. (6.7)]. The formalism is developed for arbitrary spacetime dimensions and we present explicit results for the two-dimensional case. Regrettably in the four-dimensional case too many terms are produced (even selecting particular settings, such as purely Abelian, flat spacetime, or perturbative expansions) so their explicit expression would be of little practical use. The explicit two-dimensional results for the UV divergent component are displayed in (6.4) for Γgh and ΓL;1,in(6.16) for ΓS, and in (6.32) for ΓL;0. Checks have been applied to the results of the calculation. Particularly stringent are the tests related to invariance with respect to metric transformations in Sec. VI C 5. Perturbative results are also presented in Appendix D. Although not explicitly discussed in the text, we have repeated most of the calculations of diagonal matrix elements using the method of noncovariant symbols [37,46] to find an identical result, modulo integration by parts in momentum space. In some cases, the latter method has produced shorter expressions. Details of the noncovariant method are discussed in Appendix E. As already pointed out, the nonminimal masslike coupling discussed in this work (or also its Abelian version) respects locality but introduces terms in the effective action which are nonpolynomial in the field MμνðxÞ. All current efforts for an effective field theory description of general relativity or the Standard Model of particles, or both (e.g., [51]), naturally assume an expansion in local and polynomial operators over some power of the cutoff (a newphysics scale), OαðxÞ=Λn, consistent with the renormalization group analysis of Wilson [52]. In this light, the analyses presented in [34,35] and in this work should indicate that a nonminimal coupling of the type Eq. (2.1) can be ruled out in vector field theories, also in the nonAbelian setting. If the presence of such nonminimal coupling could not be prevented through some mechanism (such as the requirement of strict gauge invariance), one would be impelled to assume a much larger class of effective field theories, including local but nonpolynomial operators. On the other hand, even in that case, reexpansions as that in Appendix Dwould bring the expression again to the standard form, requiring only local and polynomial composite operators, provided that the scale mcan be interpreted as a proper cutoff of the theory and a separation of the type Mμν ¼m2gμν þHμν is somehow natural. The fluctuation operator in Eq. (4.15) is a rather involved one, due to the presence of a non-Abelian field Mμν coupling like a metric in the ϕsector. Chan’s method or even the Schwinger-DeWitt technique are not readily available to deal with such term. Yet the formalism of covariant symbols could be applied also to ΓL;0, upon introduction of a parametric form to remove the logarithm. In fact, the method of covariant symbols is a practical and easy-to-use tool to obtain diagonal matrix elements of local operators fð∇;XÞ, provided the dependence on ∇is of rational type. This latter requirement follows from the fact that, in practice, the covariant symbols are only obtained as an expansion in powers of ∇=p. It can also be noted that the covariant symbols depend only on the connection; the presence of a metric is not required, as the Riemann coordinates can be defined directly from the connection [53]. The method can be used to obtain not only the counterterms but also covariant derivative expansions of the effective action itself [45,54,55]. In particular, the treatment of fermionic modes in curved spacetime poses no special problems once the spin connection is included in the covariant derivative [56]. The extension of the method for finite temperature also exists [46,49] (but not yet for temperature and curvature at the same time). The method of covariant symbols should apply whenever the generalized Schwinger-DeWitt technique applies, so it can be used as an alternative approach in the analysis of quantum field theories in curved spacetime, effective field theories involving gravity, or in the study of the newly developed Proca theories noted in the Introduction. ACKNOWLEDGMENTS I thank C. Garcia-Recio for suggestions on the manuscript and A. O. Barvinsky for critical remarks. This work has been partially supported by MCIN/AEI/10.13039/ 501100011033 under Grant No. PID2020–114767GBI00, by the Junta de Andalucía (Grant No. FQM-225), by the FEDER/Junta de Andalucía-Consejería de Economía y Conocimiento 2014-2020 Operational Program under Grant No. A-FQM-178-UGR18, and by the Consejería de Conocimiento, Investigación y Universidad, L. L. SALCEDO PHYS. REV. D 106, 105019 (2022) 105019-20 Junta de Andalucía and European Regional Development Fund (ERDF), Ref. SOMM17/6105/UGR. APPENDIX A: DERIVATION OF SOME FORMULAS 1. Derivation of Eq. (3.12) Applying integration by parts in the first term in (3.11) [using the notation of (5.10)], 1 4hðAa μν −Aa νμÞ2ix ¼1 2hðAa μνÞ2−Aa μνAa νμix¼1 2hðAa μνÞ2þAa νAa μνμix ¼1 2hðAa μνÞ2þAa νAa νμμ þAa νZab μν Ab μix ¼1 2hðAa μνÞ2−Aa ννAa μμ þAa νðFab μν Ab μþRμνμαAa αÞix ¼1 2hðAa μνÞ2−ðAa μμÞ2−Fab μν Aa μAb νþRμαAa αÞix:ðA1Þ Added to the other term in (3.11),h−1 2Fab μν Aa μAb νi, produces (3.12). 2. Derivation of Eq. (4.6) From the change of variables Aa μ¼Ba μþϕa μ;χa¼Ba μμ;ðA2Þ one obtains ∂ðAa μ;χaÞ ∂ðBb ν;ϕbÞ¼δabgμνδab∇μ δab∇ν0ðA3Þ (with rows for the numerator and columns for the denominator). More conveniently, doing the change of variables in two steps ðB;ϕÞ→ðA;ϕÞ→ðA;χÞ, ∂ðAa μ;χaÞ ∂ðBb ν;ϕbÞ¼∂ðAa μ;χaÞ ∂ðAc λ;ϕcÞ ∂ðAc λ;ϕcÞ ∂ðBb ν;ϕbÞ;ðA4Þ using χa¼Aa μμ −ϕa μμ in the first factor, produces δabgμνδab∇μ δab∇ν0 ¼δacgμλ0 δac∇λ−δac∇2δcbgλνδcb∇λ 0δcb :ðA5Þ The second matrix has a unit determinant and likewise for the upper-left block in the first matrix. This produces Eq. (4.6). 3. Derivation of Eq. (4.8) Starting from (3.11), and using Aa μ¼Ba μþϕa μ, one obtains terms of the types BB,Bϕ, and ϕϕ. The terms BB are just those in (3.12) with Aa μ→Ba μ. The terms Bϕare given by twice (3.11) replacing one of the Awith Aa μ→Ba μand the other one with Aa μ→ϕa μ. This produces ðSð2Þ kinÞBϕ¼1 2ðϕa μν −ϕa νμÞðBa μν −Ba νμÞ−Fab μν ϕa μBb νx :ðA6Þ Using ϕa μν −ϕa νμ ¼Fab μν ϕband integration by parts in the other term, ðSð2Þ kinÞBϕ¼h−ϕaFab μν Bb μν þFab μμνϕaBb νþFab μν ϕaBb μνix ¼hFab μμνϕaBb νix:ðA7Þ Finally, the term ϕϕ is half the previous one after the replacement Ba μ→ϕa μ. This produces (4.8). APPENDIX B: THE OPERATORS Zμ1μn 1. Definition and properties of ZR μ1μn The operator Zμν is defined as Zμν ¼½∇μ;∇ν¼ZR μν þFμν:ðB1Þ ZR μν acts on coordinate indices and Fμν on gauge indices. ZR μν is multiplicative but not “purely multiplicative”(by definition) as it is not diagonal in the coordinate indices. The higher rank tensors are defined recursively, namely (recall that I¼μ1μnstands for a string of coordinate indices), ZαI¼½∇α;Z Iþ1 2f∇λ;R Iαλg:ðB2Þ The extra term ensures that ZIis a multiplicative operator. The same formula applies to ZR I. The clean separation between coordinate and gauge sectors, ZI¼ZR IþFI;ðB3Þ holds too for higher rank tensors. The operators ZI,ZR I, and FIare all anti-Hermitian. From its definition, ZR Ihas the property ½ZR I;Vμ1μ2¼RIμ1λVλμ2 þRIμ2λVμ1λ þ;ðB4Þ where Vis a coordinate tensor. In particular, for a scalar field ϕðxÞ, NONMINIMAL NON-ABELIAN QUANTUM VECTOR FIELDS IN …PHYS. REV. D 106, 105019 (2022) 105019-21 ½ZR I;ϕ¼0:ðB5Þ Now, let j0ibe the scalar function that takes the value 1 for all x. This is a coordinate scalar. The relation ∇μj0i¼0 implies ZR μνj0i¼0¼h0jZR μν:ðB6Þ More generally [using Eq. (B2)], ZR Ij0i¼CIj0i;h0jZR I¼−h0jCI;ðB7Þ with Cμν ¼0; Cαμν ¼1 2Rλμναλ; Cα1αnμν ¼1 2Rλα2α3αnμνα1λþ1 2Rα1λα3αnμνα2λþ þ1 2Rα1αn−1λμναnλ:ðB8Þ 2. Relation with the operator ˆ Rμν of [18] The operator ZR μν coincides with ˆ Rμν in [18]. The higher rank operators ð∇α1∇αn ˆ RμνÞρσare introduced in [18], with the convention that the covariant derivative connections do not act on the matrix indices ρ,σ. Using the notation ˆ Rα1αnμν ≡∇ α1∇αn ˆ Rμν;ðB9Þ these operators fulfill the recursion ˆ RαI¼½∇α; ˆ RIþRIαλ∇λ;ðB10Þ as well as ½ˆ RI;Vμ1μ2¼½ZR I;Vμ1μ2 ¼RIμ1λVλμ2 þRIμ2λVμ1λ þ:ðB11Þ The operators ZR Iand ˆ RIare related through ZR I¼ˆ RIþCI:ðB12Þ Therefore, ˆ RIj0i¼0;h0j ˆ RI¼−2h0jCI:ðB13Þ The operators ˆ RIand ZR Ihave identical commutation properties on purely multiplicative fields. The ˆ RIare simpler than ZR Iwhen acting on states on the right (since they vanish on coordinate scalars), while the ZR Ihave the virtue of being anti-Hermitian. 3. Derivation of Eqs. (5.9) The first relation in (5.9), the trace in the coordinatescalar space, follows from the fact that ZR Icoincides with −CIwhen acting on scalars on the left, hϕjZR I¼h0jϕZR I¼h0jðZR Iϕ−½ZR I;ϕÞ ¼ −hϕjCI:ðB14Þ For the second relation, the trace in the coordinate-vector space, consider an operator Oμνacting on the coordinatevector space, ðOVÞμ¼OμνVν. Disregarding the gaugespace sector for simplicity, the trace can be written as tr1ðOμνÞ¼X A uA μOμνuν A;ðB15Þ where uμ AðxÞis any local basis of vectors at xand uA μðxÞis its dual basis, uA μuμ B¼δA B. When Oμνis purely multiplicative (i.e., it does not contain ∇μnor ZR I), the trace is simply tr1ðOμνÞ¼OμνuA μuν A¼Oμνgνμ¼Oμμ:ðB16Þ If the operator has a factor ZR Ion the left, where Iis any string of coordinate indices, without loss of generality we can assume that it has the form ZR IHIμν. That is, the rowcolumn indices μν are not in ZR I, and all the indices in Iare different. For instance, ZR ααμXνcan be rewritten as ZR αβλgαβgμλXν. Using now huA μjZR I¼h0juA μZR I¼h0jðZR IuA μ−½ZR I;u A μÞ ¼h0jð−CIuA μþRIλμuA λÞ¼huA λjðRIλμ−gλμCIÞ; ðB17Þ it follows that tr1ðZR IOμνÞ¼X A uA μZR IOμνuν A ¼X A uA μðRIμλ−gμλCIÞOλνuν A ¼tr1ððRIμλ−gμλCIÞOλνÞ:ðB18Þ This proves Eqs. (5.9). If the operator contains more than one factor ZRon the left, the procedure is applied recursively, tr1ðZR IZR JOμνÞ¼tr1ððRIμλ−gμλCIÞZR JOλνÞ ¼tr1ðZR JðRIμλ−gμλCIÞOλν −½ZR J;R Iμλ−gμλCIOλνÞ:ðB19Þ L. L. SALCEDO PHYS. REV. D 106, 105019 (2022) 105019-22 APPENDIX C: CANONICAL FORM OF ΓS The following formulas display the expression of the operators in ΓS;n,n¼1, 2, 3, 4 in Eq. (6.7) using the basis of operators in (6.12) to bring them to the canonical form in (6.9), by means of the commutation identities in (6.11). These expressions hold in any spacetime dimension. Equations (C1)–(C4) correspond to ΓL;1,ΓL;2,ΓL;3, and ΓL;4, respectively, ΔYμν ¼O19Yμν; ΔMW¼O2W; 1 2ΔMfPμ;∇μg¼ðO1ÞμPμ−1 2O2Pμμ;ðC1Þ ΔYαμΔYμβ ¼O20YαμYμβ þOð∇−5Þ; ΔMΦμΔΦμ¼O10Φ2 μþOð∇−5Þ; ΔM∇μMμνΔMνα∇α¼ðO3ÞμνMμαMαν þ2ðO6ÞμναMνμβMβα þ4ðO12ÞμναβMανμρMρβ −ðO11Þμνð2MνμαMβαβ þMααμβMβν þ2MανμβMβα þMαβMβνRμα −2FμαMαβMβνÞ−ðO5ÞμðMμνMανα þMνμαMανÞþOð∇−5Þ; ΔMΦμΔMμν∇ν¼ðO5ÞμΦνMνμ þO10ð−ΦμMνμν −ΦμνMνμÞþ2ðO11ÞμνΦμαMαν þOð∇−5Þ; ΔM∇μMμνΔΦν¼ðO5ÞμMμνΦνþ2ðO11ÞμνMνμαΦαþOð∇−5Þ; ΔMWΔMW¼ðO13½WÞμνW; 1 4ΔMfPμ;∇μgΔMfPν;∇νg¼ðO4½PμÞμνPνþ1 2ðO7½PμμÞνPν−ðO16½Pμ;M μνναÞαβPβ−ðO9½Pμ;MμναÞναβPβ −ðO16½Pμ;M ναRμναβÞβρPρ þðO18½Pμ;Mμνα;MνβρÞβρασPσþðO18½Pμ;M μνα;MνβρÞαβρσPσ −1 4O13½PμμPνν −1 2ðO7½PμÞμPνν þ1 2ðO16½Pμ;M μναÞναPββ þOð∇−5Þ; 1 2 ΔMfPμ;∇μgΔMW¼ðO7½PμÞμWþ1 2O13½PμμW−ðO16½Pμ;MμναÞναWþOð∇−5Þ;ðC2Þ ΔM∇μMμνΔYναΔMαβ∇β¼ðO11ÞμνMμαYαβMβν þOð∇−5Þ; ΔMWΔM∇μMμνΔMνα∇α¼ðO14½WÞμνMμαMαν þOð∇−5Þ; 1 2ΔMfPμ;∇μgΔM∇νMναΔMαβ∇β¼1 2ðO14½PμμÞναMνβMβα þðO8½PμÞμναMνβMβα þ2ðO15½PμÞμναβMανρMρβ −ðO17½Pμ;M μναÞναβρMβσMσρ −ðO14½PμÞμνðMναMβαβ þMανβMβαÞþOð∇−5Þ; 1 2ΔMfPμ;∇μgΔM∇νMναΔΦα¼ðO14½PμÞμνMναΦαþOð∇−5Þ; 1 2ΔMfPμ;∇μgΔMΦνΔMνα∇α¼ðO14½PμÞμνΦαMαν þOð∇−5Þ; 1 4ΔMWΔMfPμ;∇μgΔMfPν;∇νg¼ðO16½W;PμÞμνPνþOð∇−5Þ; 1 8ΔMfPμ;∇μgΔMfPν;∇νgΔMfPα;∇αg¼ðO16½Pμ;P μνÞναPαþ1 2ðO16½Pμ;P ννÞμαPα þ1 2ðO16½Pμμ;P νÞναPαþðO9½Pμ;P νÞμναPα−ðO18½Pμ;Mμνα;P βÞναβρPρ −ðO18½Pμ;P ν;M ναβÞαβνρPρ−ðO18½Pμ;P ν;M ναβÞμαβρPρ −1 2ðO16½Pμ;P νÞμνPαα þOð∇−5Þ;ðC3Þ NONMINIMAL NON-ABELIAN QUANTUM VECTOR FIELDS IN …PHYS. REV. D 106, 105019 (2022) 105019-23 ΔM∇μMμνΔMνα∇αΔM∇βMβρΔMρσ∇σ¼ðO15½MμνMναÞμαβρMβσMσρ þOð∇−5Þ; 1 4ΔMfPμ;∇μgΔMfPν;∇νgΔM∇αMαβΔMβρ∇ρ¼ðO17½Pμ;P νÞμναβMαρMρβ þOð∇−5Þ; 1 16 ΔMfPμ;∇μgΔMfPν;∇νgΔMfPα;∇αgΔMfPβ;∇βg¼ðO18½Pμ;P ν;P αÞμναβPβþOð∇−5Þ:ðC4Þ APPENDIX D: PERTURBATIVE EXPANSIONS For completeness, we give here the expressions for Γdiv L;0and Γdiv Sin d¼2[Eqs. (6.32) and (6.16), respectively] to second order in an expansion in powers of Hμν, where Mμν ¼m2gμν þHμν ðD1Þ and to all orders in the other fields. m2is a constant c number. In d¼2, the combination Γdiv L;1þΓdiv gh cancels, so there are no further contributions to Γdiv 1. The result is obtained by expanding Eqs. (6.32) and (6.16) and carrying out the angular average, and integration over zin the case of Γdiv L;0, as such evaluations can be done explicitly when Mμν ¼m2gμν, Γdiv L;0¼1 4πε1 12 Rþ1 m4−1 12 FμνHμαHνα −1 96 HμμHννRþ1 48 HμνHμνR −1 24 HμμνHναα −1 96 HμννHμαα −1 48 HμναHμνα þ1 24 HμναHνμαþOðH3Þx;g :ðD2Þ Γdiv S¼1 4πε−1 2m2−1 2R−1 4Hμμ þ1 m2−1 2 ˜ W−1 4QμνQμν −1 16 HμμHνν þ1 8HμνHμν þ1 m4−1 8PμPμþ1 4 ˜ WHμμ þ1 16 QμνQμνHαα þ1 8QμνQμαHνα þ1 4QμνHμαHνα þ1 m61 16 PμPμHνν þ1 8PμPνHμν −1 24 PμHμνHναα þ1 24 PμHμνHανα þ1 96 PμHννHμαα þ1 48 PμHννHαμα þ1 48 PμHναHμνα −1 12 PμHναHνμα −1 96 PμHμννHαα −1 48 PμHμναHνα −1 48 PμHνμνHαα þ1 12 PμHνμαHνα −1 24 PμHνναHμα þ1 24 PμHνααHμν −1 16 ˜ WHμμHνν −1 8 ˜ WHμνHμν −1 96 QμνQμνHααHββ −1 48 QμνQμνHαβHαβ −1 48 QμνQμαHναHββ −1 24 QμνQμαHνβHαβ −1 24 QμνQμαHαβHνβ −1 48 QμνQμαHββHνα þ1 m8−1 96 PμPμHννHαα −1 48 PμPμHναHνα −1 48 PμPνHμνHαα −1 24 PμPνHμαHνα −1 24 PμPνHναHμα −1 48 PμPνHααHμν −1 48 PμHμνPνHαα −1 48 PμHμνPαHνα −1 192 PμHννPμHαα −1 96 PμHναPμHνα −1 48 PμHναPνHμαþOðH3Þx;g :ðD3Þ In this formula ˜ W≡W−1 2Pμμ. The term Rin Γdiv L;0is the Gauss-Bonnet invariant in two dimensions, complying by itself with the transverse and longitudinal symmetry invariance discussed in Sec. VI C 5. In the two-dimensional case, these symmetries do not allow terms with one Hμν nor of order 1=m2in Γdiv L;0. L. L. SALCEDO PHYS. REV. D 106, 105019 (2022) 105019-24 APPENDIX E: METHOD OF NONCOVARIANT SYMBOLS 1. Covariant vs noncovariant method of symbols The method of noncovariant symbols [57,58] allows one to obtain diagonal matrix elements of pseudodifferential operators and it can be extended to curved spacetime [37]. The difference between the covariant and noncovariant versions can be elucidated already in the case of flat spacetime. Let ˆ f¼fðD; XÞbe a pseudodifferential operator constructed out of the gauge-covariant derivative Dμ and one (or more) non-Abelian fields XðxÞ(a purely multiplicative operator). Then, hxj ˆ fjxi¼Zddp ð2πÞdhxj ˆ fjpihpjxi ¼Zddp ð2πÞde−xphxjˆ fjpi ¼Zddp ð2πÞdhxje−xp ˆ fexpj0i ¼Zddp ð2πÞdhxjfðDþp; XÞj0i;ðE1Þ where pμis imaginary and j0iis the state hxj0i¼1. The quantity hxjfðDþp; XÞj0iis the (noncovariant) symbol of ˆ f. After momentum integration, the result is gauge covariant in the sense that Dμwill only appear in the form of a commutator ½Dμ;. This follows from the fact that under the shift Dμ→Dμþaμ, where aμis an arbitrary constant imaginary c number, the dependence on aμcancels upon momentum integration (since aμcan be compensated by a corresponding shift in pμ). The virtue of the covariant symbol, ¯ f¼e−D∂fðDþp; XÞeD∂;ðE2Þ is that it is multiplicative (with respect to x) and is already covariant without momentum integration [37]. To illustrate the point, let us apply the method of noncovariant symbols to ˆ f¼ðD2 μ−XÞ−1, hxjˆ fjxi¼1 ðpμþDμÞ2−Xp :ðE3Þ Defining N¼1=ðp2 μ−XÞand expanding in powers of Dμ, hxjˆ fjxi¼hN−Nð2pμDμþD2 μÞN þNð2pμDμÞNðpνDνÞNþOðD3Þip:ðE4Þ Instead of doing the momentum integration, one can add terms which are identically zero by integration by parts in momentum space to bring the expression to a covariant form. For instance, the first order term18 hxjˆ fjxið1Þ¼h−2pμNDμNipðE5Þ can be supplemented with 0¼h∂μð−DμNÞip¼h2pμDμN2ip;ðE6Þ yielding a manifestly gauge-covariant result hxj ˆ fjxið1Þ¼h2pμ½Dμ;NNip:ðE7Þ This procedure can be carried out systematically. Rewriting Eq. (E4) as hxjˆ fjxi¼hT0þT1þT2þip; T0¼N; T1¼−2NpDN; T2¼−NDDN þ4NpDNpDN; ðE8Þ the systematic integration by parts suggested by the method of covariant symbols can be implemented as19 e−D∂X n Tnj0i¼X n ˜ Tnj0i; ˜ Tn¼X n j¼0 ð−1Þj j!ðD∂ÞjTn−j;ðE9Þ and now hxj ˆ fjxi¼h˜ T0þ˜ T1þ˜ T2þip:ðE10Þ In this way, ˜ T0¼T0¼NðE11Þ is already covariant, and ˜ T1¼T1−D∂T0¼−2NpDN−D∂N¼2½pD;NNðE12Þ is the term obtained previously in Eq. (E7). For the second order, 18Actually this term vanishes by parity; nevertheless, it serves to illustrate the point. 19Equivalently, one can put instead a factor epD at the right and move ∂μto left. NONMINIMAL NON-ABELIAN QUANTUM VECTOR FIELDS IN …PHYS. REV. D 106, 105019 (2022) 105019-25