1 Introduction↩︎

The nonminimal vector operators emerge in the gauge theories of vector fields, either Abelian or non-Abelian, under the use of the DeWitt-Faddeev-Popov method with the “nonminimal” gauge fixing conditions. The same operators gain notorious importance for the one-loop calculations in higher derivative quantum gravity, emerging in the action of Faddeev-Popov ghosts and in the weight operator [1]. In this case, the heat-kernel representations of vector operators contribute to the one-loop divergences, which may be of the types \(C^2\) (square of the Weyl tensor), \(E_4\) (Gauss-Bonnet topological term), \(R^2\) and \(\mathop{\mathrm{\square}}R\). The last term is a total derivative and is apparently irrelevant. However, there is a big difference between the status of this term in the general theory of fourth-derivative gravity and in its reduced conformal sibling [2]. In the general model, this term is irrelevant. On the other hand, the situation may be quite different in the conformal \(C^2\)-based quantum gravity, which is free from the unphysical ghosts at the tree level. If being multiplicatively renormalizable, conformal quantum gravity would be also ghost-free at the quantum level, which would be an interesting feature. The main obstacle is that the integration of the \(\mathop{\mathrm{\square}}R\)-term in the anomaly produces an \(R^2\)-term in the finite one-loop contribution to the effective action, violating conformal symmetry. If such a term appears, this means the number of degrees of freedom of the theory changes under the first quantum correction, the Källén-Lehmann representation gets broken, one can expect the \(R^2\)-type divergences in higher loops, such that the theory becomes non-unitary. Thus, it is important to understand the status of \(\mathop{\mathrm{\square}}R\) in the one-loop divergences and whether this term is ambiguous or not.

The functional trace of the Schwinger-DeWitt coefficient \(\hat{a}_2\) for a nonminimal vector field operator was first obtained in [1] using Feynman diagrams. The result has been confirmed as an important application of the generalized Schwinger-DeWitt technique [3]. However, in both works [1] and [3] there was no evaluation of the total derivative term \(\mathop{\mathrm{\square}}R\). The first derivation of this term for the Maxwell theory in a one-parameter gauge was given in [4]. This work concluded that the trace anomaly for the Maxwell field depends on the gauge fixing, which led to an alternative consideration with an opposite conclusion in [5]. After this, for a general vector-field operator, the trace of \(\hat{a}_2\) was independently obtained in [6] and subsequently verified in [7], [8]. Recently, the coefficients of the surface terms appearing in the trace of \(\hat{a}_{2}\) were independently derived in [2] using a modification of the generalized Schwinger-DeWitt technique [3], as a part of the first calculation of the \(\mathop{\mathrm{\square}}R\)-divergence in conformal quantum gravity.

All these works confirmed the original result [4]. However, regardless of the methods of [4], [6], [7], [8] and [2] were qualitatively different, we decided to perform one more independent derivation of the same quantity following the approach of Refs. [9] and [10], i.e., using the local momentum representation and normal coordinates.

The paper is organized as follows. The content of Sec. 2 serves as a motivation for the rest of the paper. We show the simple derivation of the desired \(\mathop{\mathrm{\square}}R\)-divergence that is based on the two postulates, namely, the gauge-fixing independence of the trace anomaly for the Maxwell field, assuming the irrelevance of the Jacobian of the linear constant change of variables, and on the absence of multiplicative anomaly (MA) in the divergences of the vector operators. Both postulates may be incorrect, but in our opinion, it is worth seeing how they can be applied. Sec. 3 describes the local momentum representation for the propagator of the nonminimal vector operator. Sec. 4 describes the coincidence limit in the propagator, the derivation of one-loop divergences, and the potentially important gauge-fixing dependence in the IR regulator. Finally, in Sec. 5 we present the concluding discussions, mainly concerning MA.

2 Simple logic for the nonminimal vector operator↩︎

In this section, we discuss a simple indirect derivation of the divergences of the \(\,Tr\,\log \hat{H}\) of the nonminimal second-order vector operator \[\begin{align} \hat{H} \,=\,H^{\mu}_{\nu} = \delta^{\mu}_{\nu} \mathop{\mathrm{\square}} - \lambda \nabla^{\mu} \nabla_{\nu} + P^{\mu}_{\nu}. \label{Hbase} \end{align}\tag{1}\] Here \(P_{\mu\nu}\) is an arbitrary tensor and \(\lambda\) is an arbitrary parameter. In the case \(\lambda=0\), we have a minimal operator, but our interest is to elaborate the general nonminimal case with \(\lambda\neq 0\), and establish the \(\lambda\)-dependence in the one-loop contribution \(\,\,Tr\,\log \hat{H}\,\) of the operator (1 ).

As we have already mentioned in the Introduction, this section is based on the assumptions that i) There is no MA for the one-loop divergences; ii) There is no effect on the divergences from the linear constant change of variables, opposite to the statement of [5]. Both statements may be incorrect, but let us use these assumptions, just to see how it works. In this section, we first formulate the test for the correctness of \(\,Tr\,\log \hat{H}\) for the operator (1 ), and then show how this test can be used in practice.

2.1 Gauge fixing test↩︎

According to the general QFT theorems, the one-loop divergences of effective action do not depend on the gauge fixing or parametrization of quantum fields on the classical mass shell. This theorem was proved in many works; let us mention, in particular, [11]–[14] for the proof of gauge-independence on shell in the Yang-Mills theories and [15] for the Maxwell theory, which is our present case. We can also mention the proof in [16] for the same statement in the background field formalism, [17] for the extension to a curved spacetime, and the proof for generic models of quantum gravity [18]. In quantum (including conformal) gravity, this statement was applied to the separation of essential couplings [1], [19], [20]. A relevant detail is that the proof of the on-shell gauge independence in curved spacetime [17] covers the surface terms in the divergences.⁴

Let us consider a particular example with an Abelian vector model. The vacuum effective action of the free electromagnetic field is related to the path integral over \(A_\mu\) with the action \[\begin{align} S_1\,=\,-\,\frac{1}{4}\int d^4x\sqrt{-g} \,F^2_{\mu\nu} \,\,+\,\, \frac{\lambda-1}{2}\int d^4x\sqrt{-g}\, \chi^2, \label{em} \end{align}\tag{2}\] where \(F_{\mu\nu}=\nabla_\mu A_\nu - \nabla_\nu A_\mu\), \(\chi=\nabla_\mu A^\mu\) is the covariant gauge-fixing condition and \(\lambda\) is an arbitrary gauge-fixing parameter. The one-loop effective action of the theory is given by \[\begin{align} \frac{i}{2}\,\,Tr\,\log \hat{H} \,-\,i \,Tr\,\log \hat{H}_{gh}, \label{1-loop} \end{align}\tag{3}\] where \(\hat{H}\) is the bilinear form of the action (2 ) and the second term is the ghost part, \[\begin{align} \,Tr\,\log \hat{H}_{gh} \,=\,\,Tr\,\log \Big( \frac{\delta\chi}{\delta A_\mu}\,\nabla_\mu \Big) \,=\,\,Tr\,\log \mathop{\mathrm{\square}}, \label{1-loop-ghost} \end{align}\tag{4}\] which does not depend on \(\lambda\). Therefore, our interest will be in the first term in (3 ). The classical equations of motion of the theory are \(\mathop{\mathrm{\square}}A^\mu - \nabla_\nu\nabla_\mu A^\nu= 0\). On the other hand, the divergences that emerge in the free theory (3 ) do not depend on \(A^\mu\), but only on the external metric. Therefore, according to the aforementioned QFT theorem, the divergent part of the effective action \(\Gamma(g)\) does not depend on the parameter \(\lambda\).

The object of our interest is the bilinear form of the action (2 ), \[\begin{align} && H_{\mu\nu}(\lambda) \,=\, g_{\mu\nu} \mathop{\mathrm{\square}}- \lambda\nabla_\nu\nabla_\mu + (\lambda-1) R_{\mu\nu} \,=\, g_{\mu\nu} \mathop{\mathrm{\square}}- \lambda\nabla_\mu\nabla_\nu - R_{\mu\nu}. \label{biem} \end{align}\tag{5}\] Compared to the general operator (1 ), we note that (5 ) corresponds to the particular choice \[\begin{align} P_{\mu\nu} \,=\, -\, R_{\mu\nu}. \label{crit} \end{align}\tag{6}\] As the tensor \(P_{\mu\nu}\) satisfies the condition (6 ), the dependence on the parameter \(\lambda\) has to disappear. This condition is satisfied for Eq. (5.30) of [3], and it was used in this paper as a test of correctness of all the terms except \(\mathop{\mathrm{\square}}R\).

2.2 Doubling of the nonminimal operator↩︎

The divergences of (3 ) with \(\lambda=0\), have the form \[\begin{align} \Gamma^{(1)}_{div,\,min\,vec} \,=\,-\,\frac{1}{\epsilon} \int d^4x\sqrt{-g}\,\, \Big\{ \frac{7}{60}C^2 - \frac{8}{45}E_4 + \frac{1}{36}\,R^2 - \frac{1}{30}\mathop{\mathrm{\square}}R \Big\} \label{divminvec} \end{align}\tag{7}\] for the vector part and \[\begin{align} \Gamma^{(1)}_{div,\,ghost} \,=\,-\frac{1}{\epsilon} \int d^4x\sqrt{-g}\, \Big\{-\frac{1}{60}C^2 + \frac{1}{180}E_4 - \frac{1}{36}\,R^2 - \frac{1}{15}\mathop{\mathrm{\square}}R \Big\} \label{divghost} \end{align}\tag{8}\] for the ghost contribution. Summing up, we arrive at the well-known result \[\begin{align} \Gamma^{(1)}_{div,\,min} \,=\,-\,\frac{1}{\epsilon} \int d^4x\sqrt{-g}\,\, \Big\{ \frac{1}{10}\,C^2 - \frac{31}{180} E_4 - \frac{1}{10} \mathop{\mathrm{\square}}R \Big\}, \label{divmin} \end{align}\tag{9}\] where \(\epsilon=(4\pi)^2(n-4)\) is the parameter of dimensional regularization.

In what follows, we ignore \(C^2\) and \(E_4\) and other non-surface terms since they were evaluated in a consistent way before, and concern only with the total derivative terms \[\begin{align} \mathop{\mathrm{\square}}R, \qquad \nabla_\rho\nabla_\sigma P^{\rho\sigma}, \qquad \mathop{\mathrm{\square}}P, \qquad (here \,\,\, P = P^{\rho\sigma} g_{\rho\sigma} ), \label{surfstruct} \end{align}\tag{10}\] for the general nonminimal operator (1 ). Let us note that \(2\nabla_\rho\nabla_\sigma R^{\rho\sigma} = \mathop{\mathrm{\square}}R\) owing to the Bianchi identity, but there is no such reduction for the term \(\nabla_\rho\nabla_\sigma P^{\rho\sigma}\).

Consider the product \(\hat{F} = \hat{H}(\lambda)\hat{H}^* (\beta)\) of the general operator (1 ) with the parameter \(\lambda\) and a conjugate operator \(\hat{H}^*(\beta)\) of the special form (5 ) and an arbitrary parameter \(\beta\), \[\begin{align} \hat{H}^* (\beta) \,=\, {H}_{\nu}^{*\,\mu}(\beta) = \delta^{\mu}_{\nu} \mathop{\mathrm{\square}} - \beta\nabla^{\mu} \nabla_{\nu} - R^{\mu}_{\nu}. \label{Hbase2} \end{align}\tag{11}\] Assuming the absence of the MA, we get \[\begin{align} \,Tr\,\log \hat{F} &=& \,Tr\,\log \big[ \hat{H}(\lambda)\,\hat{H}^*(\beta)\big] \,=\, \,Tr\,\log \big[H_\alpha^{\,\,\mu}(\lambda)\,{H}_{\mu}^{*\,\nu}(\beta)\big] \nonumber \\ &=& \,Tr\,\,ln\,\hat{H}(\lambda) \,+\, \,Tr\,\,ln\,\hat{H}^*(\beta). \label{prodops} \end{align}\tag{12}\] The l.h.s.of this expression can be easily evaluated if the fourth-derivative operator \(\hat{F}\) is minimal, i.e., the higher derivatives form \(\mathop{\mathrm{\square}}^2\). After a small algebra, the minimality condition for the product (12 ) can be found in the form \[\begin{align} \beta\,=\, \frac{\lambda}{\lambda-1} \quad \Longleftrightarrow \quad \lambda\,=\, \frac{\beta}{\beta-1}. \label{minim} \end{align}\tag{13}\] Taken that this is satisfied, we arrive at the minimal fourth-order operator \[\begin{align} \hat{F} \,=\, \hat{H}(\lambda)\,\hat{H}^*(\beta) &=& \hat{1}\mathop{\mathrm{\square}}^2 \,+\, \hat{V}^{\rho\sigma}\nabla_\rho\nabla_\sigma \,+\, \hat{N}^{\rho}\nabla_\rho \,+\, \hat{U}, \label{4op} \end{align}\tag{14}\] with the following elements \[\begin{align} && \hat{V}^{\rho\sigma} \,=\, [V^{\rho\sigma}]_\alpha^{\,\,\nu} \,=\, g^{\rho\sigma}\big(P_\alpha^{\,\,\nu} - R_\alpha^{\,\,\nu}\big) \,-\, \beta\big( P_\alpha^{\,\,\rho} + R_\alpha^{\,\,\rho}\big)\,g^{\nu\sigma}, \nonumber \\ && \hat{N}^\rho \,=\, {[\hat{N}^\rho]}_\alpha^{\,\,\nu} \,=\, -\, 2 \nabla^\rho R_\alpha^{\,\,\nu}, \nonumber \\ && \hat{U} \,=\, [\hat{U}]_\alpha^{\,\,\nu}\,=\, - \mathop{\mathrm{\square}}R_\alpha^{\,\,\nu}. \label{elemop} \end{align}\tag{15}\] In the first of these formulas, we assume automatic symmetrization over the pair of indices \(\rho\sigma\), i.e., \(A^\rho B^\sigma\) should be replaced by \(\frac{1}{2}\big( A^{\rho} B^{\sigma} + A^{\sigma} B^{\rho}\big)\).

An important detail is that, according to the general QFT theorem discussed before, the contribution of the second term \(\,Tr\,\log {H}_{\mu}^{\,\,\nu}(\beta)\) does not depend on \(\beta\) and can be derived, e.g., for \(\beta=0\), using Eq. (7 ). Thus, all the \(\lambda\)-dependence of \(\,Tr\,\log \hat{F}\) is concentrated in \(\,Tr\,\log \hat{H}(\lambda)\). Then, for the surface terms (10 ) in the divergent part of \(\frac{i}{2}\,Tr\,\,ln\,\hat{F}\), we shall use the result of [21], that gives for the total derivative terms \[\begin{align} \frac{i}{2}\,Tr\,\,ln\,\hat{F}\Big|_{div, \,tot-der} &=& -\,\frac{1}{\epsilon} \int d^4x\sqrt{-g}\,\,tr\,\bigg\{ \frac{\hat{1}}{15}\mathop{\mathrm{\square}}R \,+\, \frac{1}{9}\mathop{\mathrm{\square}}\hat{V} \nonumber \\ && \qquad -\,\, \frac{5}{18}\nabla_\rho\nabla_\sigma\hat{V}^{\rho\sigma} \,+\, \frac{1}{2}\nabla_\rho \hat{N}^\rho \,-\, \hat{U} \bigg\}. \label{Gus4result} \end{align}\tag{16}\] The last expression satisfies the product test of the first calculation [1], i.e., Eq. (16 ) was confirmed in two independent ways and can be regarded correct. In what follows, we obtain one more indirect confirmation of this formula. Using (15 ) in Eq. (16 ), we obtain \[\begin{align} \frac{i}{2}\,Tr\,\,ln\,\hat{F}\Big|_{div, \,tot-der} &=& -\,\frac{1}{\epsilon} \int d^4x\sqrt{-g}\, \bigg\{ \Big(\frac{1}{10}\,+\, \frac{\beta}{36}\Big)\mathop{\mathrm{\square}}R \nonumber \\ && \quad +\,\,\,\Big(\frac{1}{6}\,-\, \frac{\beta}{9}\Big)\mathop{\mathrm{\square}}P \,+\,\frac{5\beta}{18}\nabla_\rho\nabla_\sigma P^{\rho\sigma}\bigg\}. \label{op4result} \end{align}\tag{17}\] Subtracting the contribution of the operator \(\,Tr\,\log \hat{H}(\beta)\) from (7 ), we arrive at \[\begin{align} \frac{i}{2}\,Tr\,\,ln\,\hat{H}(\lambda)\Big|_{div, \,tot-der} &=& -\,\frac{1}{\epsilon} \int d^4x\sqrt{-g}\, \bigg\{ \Big(\frac{2}{15}\,+\, \frac{\beta}{36}\Big)\mathop{\mathrm{\square}}R \nonumber \\ && \quad +\,\Big(\frac{1}{6}\,-\, \frac{\beta}{9}\Big)\mathop{\mathrm{\square}}P \,+\,\frac{5\beta}{18}\nabla_\rho\nabla_\sigma P^{\rho\sigma}\bigg\}. \label{opHresult} \end{align}\tag{18}\]

Now we perform the last decisive test. Setting \(P_{\rho\sigma}= - R_{\rho\sigma}\), we should expect that (17 ) will not depend on \(\lambda\). Indeed, this works. Using (6 ), the \(\beta\)-dependence in (17 ) and, of course, in (18 ) cancels out and we end up with \[\begin{align} \frac{i}{2}\,Tr\,\,ln\,\hat{H} \big(\lambda;\,\,P_{\rho\sigma} \,=\, - \, R_{\rho\sigma}\big) \Big|_{div, \,tot-der} &=& -\,\frac{1}{\epsilon} \int d^4x\sqrt{-g}\, \bigg\{ - \frac{1}{30}\,\mathop{\mathrm{\square}}R \bigg\}, \label{op4-min} \end{align}\tag{19}\] independent of \(\lambda\). Let us remark that the cancellation of \(\beta\)-dependence in formula (18 ) under the condition (6 ) confirms both the correctness of the formula itself, but also the correctness of the expression (16 ), since the last was used in our simple calculation.

The logic presented above looks convincing, but only until it confronts the direct calculation. As we already mentioned in the Introduction, the multiple calculations of the quantity of our interest did not confirm the result (18 ) of our general consideration. In the next sections, we perform one more independent calculation and then discuss the status of the problem.

3 Local momentum representation for the propagator↩︎

Starting from this point, we derive the \(\mathop{\mathrm{\square}}R\) term in the divergences of the \(\,Tr\,\log \hat{H}\) of the nonminimal operator (1 ), by using Riemann normal coordinates and the local momentum representation. To use the dimensional regularization, all the basic elements are defined in the \(D\)-dimensional space⁵. The Green’s function \(G^{\lambda}_{\,\mu^{\prime}} ( x, x^{\prime} )\) is the solution of the equation \[\begin{align} \label{eq:2} H^{\mu}_{\,\,\lambda} \, G^{\lambda}_{\,\mu^{\prime}} ( x, x^{\prime} ) \,=\, - \,\delta^{\mu}_{\,\mu^\prime} \,\delta_c ( x, x^{\prime} ) , \end{align}\tag{20}\] where \(\delta_c ( x, x^{\prime} )\) is the covariant delta function, which satisfies \[\begin{align} \int d^D x \, \sqrt{|g(x')|} \,f(x') \,\delta_c ( x, x^{\prime} ) \,=\, f(x). \end{align}\] Here \(|g(x)|\) is the absolute value of the metric determinant in the point with coordinates \(x\equiv x^\mu\). It is not difficult to see that \[\begin{align} \delta_c ( x, x^{\prime} ) = |g|^{-1/4} \delta ( x - x^{\prime} ) |g'|^{-1/4}, \end{align}\] where \(g = g(x)\), \(g' = g(x')\) and \(\delta ( x - x^{\prime} )\) is the ordinary Dirac delta function. We consider \(G^{\mu}_{\mu^{\prime}} ( x, x^{\prime} )\) a bivector. This means, for a covariant derivatives acting at the point \(x\), the Green function is a vector.

It proves helpful to eliminate the explicit metric dependence on the r.h.s of Eq. (20 ). To this end, we introduce the modified Green’s function \(\bar{G}^{\mu}_{\mu^{\prime}}\) through the relation \[\begin{align} \label{eq:ModProp} G^{\mu}_{\mu^{\prime}} ( x, x^{\prime} ) \,=\, |g|^{-1/4} \, \bar{G}^{\mu}_{\mu^{\prime}} ( x, x^{\prime} ) \, |g'|^{-1/4} . \end{align}\tag{21}\] By doing so, Eq. 20 boils down to \[\begin{align} \label{eq:eqnormal} \bar{H}^{\mu}_{\,\lambda} \, \bar{G}^{\lambda}_{\,\mu^{\prime}} \,=\, -\, \delta^{\mu}_{\mu^{\prime}} \,\delta ( x-x' ) , \end{align}\tag{22}\] where \[\begin{align} \label{eq:4} \bar{H}^{\mu}_{\,\nu} \,=\, \delta^{\mu}_{\,\nu}\, |g|^{1/4} \,\mathop{\mathrm{\square}}\, |g|^{-1/4} \,-\, \lambda\, |g|^{1/4}\, \nabla^{\mu} \nabla_{\nu}\, |g|^{-1/4} \, + \,P^{\mu}_{\,\,\nu}. \end{align}\tag{23}\]

In the local momentum representation formalism, the spacetime metric \(g_{\mu\nu}(x)\) and all related quantities are expanded near the point \(x'\) in the normal coordinates. In this point one can always provide the metric being flat, \(g_{\mu\nu} (x') = \eta_{\mu\nu}\). Then all objects in the vicinity of this point depend on the differences \(y^\mu = x^\mu - x^{\prime\mu}\). Since our goal is the \(\mathop{\mathrm{\square}}R\)-type term, we can restrict our attention to the linear in curvature terms with up to four metric derivatives. Restricting the equations to the corresponding order terms, the relevant expansions have the form [22], [23] \[\begin{align} \label{g95norm} && g_{\mu \nu} (x) \,=\, \eta_{\mu \nu} - \frac{1}{3} R_{\mu \alpha\nu \beta} \, y^{\alpha} y^{\beta} - \frac{1}{6} R_{\mu \alpha\nu \beta; \gamma} \, y^{\alpha} y^{\beta} y^{\gamma} -\frac{1}{20} R_{\mu \alpha\nu \beta; \gamma\delta} \, y^{\alpha} y^{\beta} y^{\gamma} y^{\delta} + \ldots, \\ && g^{\mu \nu} (x) \, = \, \eta^{\mu \nu } + \frac{1}{3} R^{\mu }{}_{\alpha}{}^{\nu }{}_{\beta} \, y^{\alpha} y^{\beta} + \frac{1}{6} R^{\mu }{}_{\alpha}{}^{\nu }{}_{\beta; \gamma} \, y^{\alpha} y^{\beta} y^{\gamma} + \frac{1}{20} R^{\mu }{}_{\alpha}{}^{\nu }{}_{\beta; \gamma\delta } \, y^{\alpha} y^{\beta} y^{\gamma} y^{\delta } + \ldots, \nonumber \\ && \left| g (x) \right|^{d} \, = \, 1 - \frac{d}{3} R_{\alpha\beta} \, y^{\alpha} y^{\beta} - \frac{d}{6} R_{\alpha\beta; \gamma} \, y^{\alpha} y^{\beta} y^{\gamma} - \frac{d}{20} R_{\alpha\beta; \gamma\delta} \, y^{\alpha} y^{\beta} y^{\gamma} y^{\delta} + \ldots, \end{align}\tag{24}\] and also \[\begin{align} \tag{25} && \Gamma^{\lambda}_{\mu \nu} (x) \,=\, - \frac{2}{3} R^{\lambda}{}_{\mu \nu \alpha} \, y^{\alpha} + \Big(\frac{1}{6} R^{\lambda}{}_{\alpha\beta\mu ; \nu} -\frac{1}{3} R^{\lambda}{}_{\mu \nu \alpha; \beta} + \frac{1}{12} R_{\mu \alpha\nu \beta}{}^{; \lambda} \Big) y^{\alpha} y^{\beta} \nonumber \\ && \qquad + \, \Big( \frac{1}{40} R_{\mu \alpha\nu \beta; \gamma}{}^{; \lambda} - \frac{2}{40} R^{\lambda}{}_{\beta\mu \alpha; \nu \gamma} - \frac{2}{40} R^{\lambda}{}_{\beta\mu \alpha; \gamma\nu} - \frac{2}{20} R^{\lambda}{}_{\mu \nu \alpha; \beta\gamma} \Big) y^{\alpha} y^{\beta} y^{\gamma} + \ldots, \\ \tag{26} && P^{\mu}_{\nu} ( x ) \,=\, P^{\mu}_{\nu} \,+\, P^{\mu}_{\nu ; \alpha} \, y^{\alpha} \,+\, \frac{1}{2} P^{\mu}{}_{\nu ; \alpha\beta} \, y^{\alpha} y^{\beta} + \ldots \,. \end{align}\] Here and in the following, the ellipsis stands for higher-order terms in the curvature and its covariant derivatives. The coefficients in the expansions in \(y^\alpha\) in the r.h.s.’s of the above equations are evaluated at the point \(x^{\prime}\). Also, in Eq. 25 , the symmetrization over \((\mu, \nu)\) is assumed, although it is not explicitly written for the compactness of the expression.

Using Eqs. (24 )-(26 ) in (23 ), we obtain \[\begin{align} \label{eq:HessianDec} \bar{H}^{\mu}_{\,\,\nu} \,=\, \bar{H}^{\mu}_{0 \,\nu} \, +\, \bar{H}^{\mu}_{1\,\nu} \,+\, \bar{H}^{\mu}_{2 \,\nu} \,+\, \bar{H}^{\mu}_{3 \,\nu} \,+\, \bar{H}^{\mu}_{4\, \nu} \,+\, \ldots , \end{align}\tag{27}\] where each \(\bar{H}^{\mu}_{k \,\nu}\) has \(k\) derivatives of the metric. The first terms are given by \[\begin{align} && \bar{H}^{\mu}_{0 \nu} \,=\, \delta^{\mu}_{\,\nu} \partial^2 - \lambda\, \partial^{\mu} \partial_{\nu}, \nonumber \\ && \tag{28} \bar{H}^{\mu}_{1 \,\nu} \,=\, 0, \nonumber \\ && \bar{H}^{\mu}_{2 \,\nu} \,= \, P^{\mu}_{\,\nu} + \frac{1}{6} R \delta^{\mu}_{\,\nu} - \frac{1}{3} \bigg( 1 - \frac{\lambda}{2} \bigg) R^{\mu}_{\,\nu} + \frac{1}{6} \big(4 R^{\mu \beta}{}_{\alpha\nu } \partial_\beta + 4 R^{\mu }{}_{\nu \alpha} {}^{ \beta} \partial_{\beta} - 2 \delta^{\mu}_{\nu } R_{\,\alpha}^{\beta} \partial_{\beta} \nonumber \\ && \qquad \qquad + \,\lambda R_{\nu \alpha} \partial^\mu - \lambda R^{\mu }_{\,\alpha} \partial_{\nu } \big) y^{\alpha} + \frac{1}{3} \big( \delta_{\nu }^{\mu } R_{\alpha\rho \beta\sigma} \partial^\rho \partial^\sigma + \lambda R^{\mu }{}_{\alpha\beta} {}^\rho \partial_\rho\partial_\nu \big) y^{\alpha} y^{\beta}, \tag{29} \end{align}\] \[\begin{align} && \bar{H}^{\mu}_{3 \,\nu} \,=\, \Big[ P^\mu_{\,\,\nu ; \,\alpha} + \frac{1}{6} \, \delta_{\nu }^{\mu } R_{; \alpha} - \frac{1}{2} \Big( 1 - \frac{\lambda}{6} \Big) R^{\mu }_{\nu ; \alpha} - \frac{1}{6} \Big( 1 - \frac{\lambda}{2} \Big) R_{\nu \alpha}{}^{; \mu} + \frac{1}{2} \Big(1 + \frac{\lambda}{6} \Big) R^{\mu }_{\alpha;\, \nu } \Big] y^{\alpha} \nonumber \\ && \qquad\quad + \, \Big( \frac{1}{3} R^{\mu }{}_{\nu \alpha} {}^\rho {}_{; \beta} \partial_{\rho } + \frac{1}{3} R^{\mu \rho } {}_{ \alpha\nu ; \beta} \partial_{\rho } + \frac{1}{6} R^{\mu }{}_{\alpha\beta} {}^\rho {}_{; \nu } \partial_{\rho } + \frac{1}{6} R^{\mu }{}_{\alpha\beta\nu} {}^{; \rho } \partial_{\rho } + \frac{1}{6} R_{\alpha\nu \beta} {}^{\rho ; \mu} \partial_{\rho } \nonumber \\ && \qquad\quad - \,\frac{1}{3} \delta^{\mu}_{\nu} R_{\alpha; \beta}^{\rho}\partial_\rho - \frac{\lambda}{12} R^{\mu }{}_{\alpha; \beta} \partial_{\nu } + \frac{\lambda}{12} R_{\nu \alpha; \beta} \partial^{\mu } + \frac{\lambda}{24} R_{\alpha\beta; \nu } \partial^{\mu } - \frac{\lambda}{24} R_{\alpha\beta}{}^{; \mu} \partial_{\nu } \Big) \,y^{\alpha} y^{\beta} \nonumber \\ && \qquad\quad + \,\frac{1}{6} \big( \delta^\mu_{\,\nu} R_{\alpha\rho \beta\sigma; \gamma} \partial^\rho \partial^\sigma + \lambda R^{\mu}{}_{\alpha\beta\rho ; \gamma} \partial^{\rho} \partial_{\nu} \big) \, y^{\alpha} y^{\beta} y^{\gamma}, \label{H3} \end{align}\tag{30}\] and \[\begin{align} \label{H4} \bar{H}^{\mu}_{4 \,\nu} =& \left( \frac{1}{2} P^{\mu }_{\nu; \alpha\beta} - \frac{1}{4} R^{\mu }_{\nu ; \alpha\beta} + \frac{\lambda}{40} R^{\mu }_{\nu ; \alpha\beta} + \frac{3}{40} \delta^{\mu}_{\nu } R_{; \alpha\beta} + \frac{7}{20} R^{\mu }_{\alpha; \nu \beta} + \frac{\lambda}{20} R^{\mu }_{\alpha; \nu \beta} \right. \\ & \left. + \,\frac{1}{40} \delta^{\mu }_{\nu} \mathop{\mathrm{\square}}R_{\alpha\beta} - \frac{1}{20} \mathop{\mathrm{\square}}R^{\mu }{}_{\alpha\nu \beta} - \frac{3}{20} R_{\nu \alpha; \beta}{}^{;\mu} + \frac{1}{20} \lambda R_{\nu \alpha; \beta}{}^{; \mu} + \frac{1}{40} \lambda R_{\alpha\beta; \nu}{}^{; \mu} \right) y^{\alpha} y^{\beta} \\ & + \,\Bigl( \frac{1}{10} R^{\mu }{}_{\rho \alpha\nu ; \beta\gamma} \partial^{\rho } + \frac{1}{10} R^{\mu }{}_{\nu \alpha\rho ; \beta\gamma} \partial^{\rho } + \frac{1}{10} R^{\mu }{}_{\alpha\beta\nu ; \gamma\rho } \partial^{\rho } + \frac{1}{10} R_{\alpha\nu \beta\rho ; \gamma}{}^{; \mu} \partial^{\rho } \\ & + \,\frac{1}{10} R^{\mu }{}_{\alpha\beta\rho ; \gamma\nu } \partial^{\rho } + \frac{1}{10} \delta^{\mu }_{\nu} R_{\alpha\beta; \gamma\rho } \partial^{\rho } - \frac{3}{20} \delta^{\mu }_{\nu} R_{\alpha\rho ; \beta\gamma} \partial^{\rho} - \frac{\lambda}{40} R_{\alpha\beta; \gamma}{}^{; \mu} \partial_{\nu } \\ & + \frac{\lambda}{40} R_{\alpha\nu ; \beta\gamma} \partial^{\mu } - \frac{\lambda}{40} R^{\mu }_{\alpha; \beta\gamma} \partial_{\nu } + \frac{\lambda}{40} R_{\alpha\beta; \gamma\nu } \partial^{\mu } \Bigr) y^{\alpha} y^{\beta} y^{\gamma} + \frac{1}{20} \Bigl( \delta^{\mu }_{\nu} R_{\alpha\rho \beta\sigma; \gamma\delta } \partial^{\rho }\partial^{\sigma} \\ & + \,\lambda R^{\mu }{}_{\alpha\beta\rho ; \gamma\delta } \partial^{\rho }\partial_{\nu } \Bigr) y^{\alpha} y^{\beta} y^{\gamma} y^{\delta} + \ldots\,\,. \end{align}\tag{31}\]

Now we can introduce the local momentum space representation associated with the point \(x'\) by making the Fourier transformation \[\begin{align} \bar{G}^{\mu}_{\,\mu'}(x,x') \,\,=\,\, \int \frac{d^Dk}{(2 \pi)^D} \,\, e^{iky}\, \bar{G}^{\mu}_{\,\mu'}(k), \end{align}\] where we denoted \(ky = \eta^{\alpha\beta} k_{\alpha}y_{\beta}\). Let us find the solution of Eq. (22 ) using an iterative procedure. For this, we write the Green’s function as a series \[\begin{align} \bar{G}^{\mu}_{\mu'}(k) \,=\, \bar{G}^{\mu}_{1\mu'}(k) \,+\, \bar{G}^{\mu}_{2\mu'}(k) \,+\, \bar{G}^{\mu}_{3\mu'}(k) \,+\, \bar{G}^{\mu}_{4\mu'}(k) + \ldots, \end{align}\] where \(\bar{G}^{\mu}_{i\mu'}(k)\) (\(i=0,1,2,3,4\)) has \(i\)’s order derivatives of the metric. These expressions satisfy the following schematic equations: \[\begin{align} & \hat{\bar{H}}_0 \hat{\bar{G}}_0 \,=\, - \hat{1}, \nonumber \\ & \hat{\bar{G}}_1 \,=\, \hat{\bar{G}}_0 \hat{\bar{H}}_1 \hat{\bar{G}}_0, \nonumber \\ & \hat{\bar{G}}_2 \,=\, \hat{\bar{G}}_0 \hat{\bar{H}}_2 \hat{\bar{G}}_0 + \hat{\bar{G}}_0 \hat{\bar{H}}_1 \hat{\bar{G}}_1, \nonumber \\ & \hat{\bar{G}}_3 \,=\,\hat{\bar{G}}_0 \hat{\bar{H}}_3 \hat{\bar{G}}_0 + \hat{\bar{G}}_0 \hat{\bar{H}}_2 \hat{\bar{G}}_1 + \hat{\bar{G}}_0 \hat{\bar{H}}_1 \hat{\bar{G}}_2, \nonumber \\ & \hat{\bar{G}}_4 = \hat{G}_0 \hat{\bar{H}}_4 \hat{\bar{G}}_0 + \hat{G}_0 \hat{\bar{H}}_2 \hat{\bar{G}}_2 + \hat{\bar{G}}_0 \hat{\bar{H}}_3 \hat{\bar{G}}_1 + \hat{\bar{G}}_0 \hat{\bar{H}}_1 \hat{\bar{G}}_3, \label{G4} \end{align}\tag{32}\] where the Fourier transform of \(\bar{H}^{\mu}_{i \,\nu}\) can be directly obtained using the correspondence between the coordinate and momentum spaces \[\begin{align} \partial_{\mu} \longrightarrow i k_{\mu} , \qquad y^{\mu} \longrightarrow - \frac{1}{i} \frac{\partial}{\partial k_{\mu}} \equiv - \frac{1}{i} \, \partial^\mu. \end{align}\] According to Eq. (28 ), we can use \(\hat{H}_1 = 0\) and it follows from the second of Eqs. (32 ) that \(\hat{G}_1 = 0\), which greatly simplify the system (32 ). Also, among the Green’s function, only \(\hat{G}_4\) have terms that are linear in curvature and have four derivatives of the metric. Furthermore, the structure \(\hat{G}_0 \hat{H}_2 \hat{G}_2\) is already \(O(R^2)\) since both \(\hat{H}_2\) and \(\hat{G}_2\) are already linear in the curvatures. Therefore, to obtain the first three coefficients until the desired order (fourth derivatives of the metric, in our case), we need to solve the equations \[\begin{align} \tag{33} && \bar{H}^{\mu}_{0 \,\lambda} ( k )\,\, \bar{G}^{\lambda}_{0\,\mu'} ( k ) \,=\, - \,\delta^\mu_{\mu'}\,, \\ \tag{34} && \bar{G}^\mu_{2\,\mu'} ( k ) \,=\, \bar{G}^{\mu}_{0 \,\rho^{\prime}} ( k ) \,\bar{H}^{\rho}_{2 \,\sigma} ( k ) \,\bar{G}^{\sigma}_{0\, \mu^{\prime}} ( k )\,, \\ && \bar{G}^\mu_{4\,\mu'} ( k ) \,=\, \bar{G}^{\mu}_{0 \,\rho^{\prime}} ( k ) \,\bar{H}^{\rho}_{4 \,\sigma} ( k ) \,\bar{G}^{\sigma}_{0\, \mu^{\prime}} ( k ) \,+\, \ldots, \end{align}\] where \[\begin{align} \label{eqG4} \bar{H}^{\mu}_{0 \nu} (k) \, = \,-\,k^{2} \delta^{\mu}_{\,\nu} + \lambda k^{\mu} k_{\nu}, \qquad \qquad \end{align}\tag{35}\] \[\begin{align} \bar{H}^{\mu}_{2\, \nu} ( k ) = & \,\, P^{\mu }_{\nu } + \frac{1}{3} R^{\mu }_{\nu } + \frac{\lambda}{2} R^{\mu }_{\nu } + \frac{1}{6} \delta_{\nu }^{\mu } R - \frac{1}{3} \delta_{\nu }^{\mu } R^{\alpha\beta} k_{\alpha} \partial_{\beta} + \frac{\lambda}{2} k_{\nu } R^{\mu \lambda} \partial_{\lambda} \\ & - \frac{\lambda}{6} R_{\nu }^{\lambda} k^{\mu} \partial_{\lambda} + \frac{2}{3} R^{\mu \alpha}{}_{\nu }{}^{\beta} k_{\alpha} \partial_{\beta} + \frac{2}{3} R^{\mu }{}_{\nu }{}^{\alpha\beta} k_{\alpha} \partial_{\beta} - \frac{\lambda}{3} R^{\mu \beta\alpha}{}_{\nu } k_{\alpha} \partial_{\beta} \\ & - \frac{\lambda}{3} R^{\mu}{}_{\nu}{}^{\alpha\beta} k_{\alpha}\partial_{\beta} + \frac{1}{3} \delta_{\nu }^{\mu } R^{\lambda\tau \alpha\beta} k_{\alpha} k_{\lambda} \partial_{\beta}\partial_{\tau} - \frac{\lambda}{3} R^{\mu \alpha\beta\lambda} k_{\beta} k_{\nu} \partial_{\alpha}\partial_{\lambda}, \end{align}\] \[\begin{align} \bar{H}^{\mu}_{4 \,\nu} ( k ) \,=\,& \left( \frac{1}{4} R^{\mu }_{\beta ; \nu \alpha} - \frac{\lambda}{4} R^{\mu}_{\beta ; \nu \alpha } - \frac{1}{2} P^{\mu }_{\nu ; \alpha \beta } - \frac{1}{4} R^{\mu }_{\nu ; \alpha \beta } - \frac{7\lambda}{40} R^{\mu }_{\nu ; \alpha \beta } - \frac{3}{40} \delta_{\nu }^{\mu } R_{; \alpha \beta} \right. \\ & \left. + \, \frac{1}{10} R^{\mu }_{\alpha ; \nu \beta } - \frac{\lambda}{10} R^{\mu }_{\alpha ; \nu \beta } - \frac{1}{40} \delta_{\nu }^{\mu } \mathop{\mathrm{\square}}R_{\alpha \beta } - \frac{1}{20} \mathop{\mathrm{\square}}R^{\mu}_{\beta \nu \alpha } - \frac{1}{20} R_{\nu \beta ; \alpha}{}^{; \mu} \right. \\ & \left. + \, \frac{3\lambda}{20} R_{\nu \beta; \alpha}{}^{; \mu} - \frac{1}{10} R_{\nu \alpha; \beta}{}^{; \mu} + \frac{3\lambda}{40} R_{\alpha \beta ; \nu}{}^{; \mu} \right) \partial^\beta\partial^\alpha + \frac{1}{10} \bigg( R^{\mu}{}_{\delta \alpha \gamma; \nu \beta } \\ & \left. - \, R^{\mu }{}_{\delta \nu \gamma ; \alpha \beta } - R^{\mu }{}_{\delta\alpha \gamma ; \nu \beta } - R^{\mu }{}_{\alpha \nu \delta ; \beta \gamma } + R_{\alpha \gamma \nu \delta ; \beta }{}^{; \mu} - R^{\mu}{}_{\nu \alpha \delta ; \beta \gamma } + \frac{\lambda}{2} R^{\mu }{}_{\delta \alpha \nu ; \beta \gamma } \right. \\ & \left. + \,\frac{\lambda}{2} R^{\mu }{}_{\nu \alpha \delta ; \beta \gamma} - \frac{3\lambda}{4} g_{\nu \alpha } R^{\mu}{}_{\delta ; \beta \gamma } + \delta_{\nu }^{\mu } R_{\alpha \gamma ; \beta \delta} + \frac{1}{2}\delta_{\nu }^{\mu } R_{\alpha \delta ; \beta \gamma } - \delta_{\nu }^{\mu } R_{\gamma \delta; \alpha \beta } \right. \\ & \left. - \,\lambda g_{\nu \alpha } R^{\mu}_{\gamma ; \beta \delta} + \frac{3\lambda}{4} g_{\nu \alpha } R_{\gamma \delta ; \beta}{}^{; \mu} \right) k^{\alpha} \partial^\delta\partial^\gamma\partial^\beta + \frac{\lambda}{40} \left( R_{\beta \gamma ; \nu \alpha } + R_{\nu \gamma ; \alpha \beta } \right) k^{\mu} \partial^\gamma\partial^\beta\partial^\alpha \\ & - \,\frac{1}{20} \left( \delta_{\nu}^{\mu } R_{\alpha \zeta \beta \eta ; \gamma \delta} - \lambda g_{\nu \alpha } R^{\mu }{}_{\eta \beta \zeta ; \gamma \delta} \right) k^{\alpha} k^{\beta} \partial^\gamma\partial^\delta\partial^\eta \partial^\zeta + \ldots. \end{align}\]

4 The coincidence limit and divergences↩︎

The Schwinger-DeWitt coefficients can be extracted from the coincidence limit of the Green’s function in the coordinate representation. The coincidence limit of the propagator has a heat kernel expansion \[\label{eq:heatkernel} \begin{align} \bar{G}^{\mu}_{\,\nu} ( x,x ) \,=\, & -\frac{i}{( 4 \pi )^{D/2}} \left[ \frac{\Gamma(1 - D/2)}{( m^2)^{1 - D/2}}\, a^{\,\,\mu}_{0 \,\nu}(x,x) - \frac{\Gamma(2 - D/2)}{(m^2)^{2 - D/2}}\, a^{\,\,\mu}_{1 \,\nu} ( x,x ) \right. \\ & \left. + \frac{\Gamma(3 - D/ 2)}{(m^2)^{3 - D/2}}\,a^{\,\,\mu}_{2\,\nu}(x,x) + \cdots \right]. \end{align}\tag{36}\]

In order to regulate infrared divergences that appear in the integrals over the momentum, let us first introduce a mass parameter \(m\) in the equation for \(\hat{\bar{G}}_0\), \[\begin{align} \label{eq:G0} \big[(-k^2 + m^2 ) \delta^\mu_{\,\nu} \,+\, \lambda k^{\mu} k_{\nu} \big] \, \bar{G}^{\nu}_{0\,\mu^{\prime}} ( k ) \,=\, -\,\delta^\mu_{\,\mu^{\prime}}. \end{align}\tag{37}\] It is not difficult to verify that the solution of Eq. 37 is \[\begin{align} \label{G0sol} \bar{G}^{\mu}_{0 \nu^{\prime}} ( k ) \,=\, \frac{\delta^{\mu}_{\,\,\nu^{\prime}}}{k^2-m^2} \,+\, \frac{ \gamma\,k^{\mu} k_{\nu^{\prime}}}{( k^2 - m^2) \big[ k^2 - \tilde{m}^2( \lambda) \big]}\,, \end{align}\tag{38}\] where \[\begin{align} \label{massLa} \tilde{m} ( \lambda) \,=\, \frac{m}{\sqrt{1-\lambda}} \qquad and \qquad \gamma\,=\, \frac{\lambda}{1-\lambda}. \end{align}\tag{39}\] One can note that the introduction of the mass term provides a new massive parameter \(\tilde{m}( \lambda)\). In case of electromagnetic field and nonminimal gauge, this means the presence of a gauge-fixing dependent massive parameter. This is an important detail, as we will discuss at the end of the consideration.

As a starting point, let us derive the coefficients \(a_{0 \,\nu}^{\,\,\mu} ( x,x )\). At the zeroth order, taking the coincidence limit \(x\to x^{\prime}\) or, equivalently \(y \to 0\), we have \[\begin{align} \label{eq:limitG01} && \bar{G}^{\mu}_{0 \,\nu} ( x,x ) \,=\, \int \frac{d^{D} k}{(2 \pi )^{D}} \bar{G}^{\mu}_{0 \nu} ( k ) \,=\, \int \frac{d^{D} k}{(2 \pi )^{D}} \bigg\{ \frac{\delta^{\mu}_{\,\nu}}{k^2 - m^2} + \gamma\frac{ k^{\mu} k_{\nu}}{( k^{2} - m^{2} ) \big[ k^2 - \tilde{m}^2( \lambda) \big]}\bigg\} \nonumber \\ && \qquad \qquad \,\,\, \,=\, \int \frac{d^D k}{(2\pi)^D} \frac{\delta^\mu_{\,\nu}}{k^2 - m^2} \,+\, \frac{\gamma}{D} \int_0^1 d \xi \int \, \frac{d^{D} k}{( 2 \pi )^{D}} \, \frac{k^{2} \delta^\mu_{\,\nu}}{\big[ k^2 - \xi m^2 - (1 - \xi) \tilde{m}^2 (\lambda) \big]^2}, \qquad \end{align}\tag{40}\] where we introduced a Feynman integration parameter \(\xi\). Using the Feynman prescription for integrals in Minkowski space, we can use standard formulas for the above integrals that can be found in many textbooks, e.g., [24]. The result is \[\begin{align} \label{eq:limitG0} \bar{G}^{\mu}_{0 \nu} ( x,x ) \,=\, -\, \frac{i \Gamma ( 1 - D/ 2 )}{( 4 \pi )^{D/2}\, (m^2)^{1 - D/2}} \left\{ 1+ \frac{1}{D} \big[(1-\lambda)^{-D/2}-1\big] \right\} \delta^\mu_{\,\nu}. \end{align}\tag{41}\] By comparing Eqs. 41 and 36 , we get \[\begin{align} a^{\,\mu}_{0 \,\,\nu} ( x,x ) \,=\, \Big\{ 1 + \frac{1}{D} \big[(1-\lambda)^{-D/2}-1\big] \Big\} \delta^\mu_{\,\nu}, \end{align}\] in agreement with [8].

In the next order, the solution of Eq. (34 ) is \[\begin{align} \label{eq:G2exp} \bar{G}^{\mu}_{2\, \nu^{\prime}} ( k ) = & \,\, \frac{P^{\mu }_{\nu^{\prime} }}{(k^{2} - m^{2} )^2} + \frac{\delta^{\mu }_{\nu^{\prime} } R}{6 (k^{2} - m^{2} )^2} - \frac{\lambda k^{\alpha} k_{\nu^{\prime} } R^{\mu }_{\alpha}}{3 (k^{2} - m^{2} )^3} + \frac{\lambda k^{\alpha} k^{\mu } R_{\nu^{\prime} \alpha}}{3 (k^{2}-m^{2} )^3} \\ & - \frac{2 (\lambda+ 2) k^{\alpha} k^{\beta} R^{\mu }{}_{\alpha\nu^{\prime} \beta}}{3 (k^{2} - m^{2} )^3} + \frac{(3 \lambda+ 2) R^{\mu }_{\nu^{\prime} }}{6 (k^{2} - m^{2} )^2} + \frac{\gamma^2 k^{\alpha} k^{\beta} k^{\mu } k_{\nu^{\prime} } P_{\alpha\beta}}{(k^{2} - m^{2} )^2 [k^{2} - \tilde{m}^{2} ( \lambda) ]^2} \\ & + \frac{\gamma^2 k^{\alpha} k^{\beta} k^{\mu } k_{\nu^{\prime} } R_{\alpha\beta}}{(k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^{2} ( \lambda) ]^2} - \frac{\lambda^2 k^{2} k^{\beta} k_{\nu^{\prime} } R^{\mu }_{\beta}}{3 (1 - \lambda) (k^{2} - m^{2} )^2 [k^{2} - \tilde{m}^{2} ( \lambda) ]^2} \\ & - \frac{\lambda^2 k^{2} k^{\beta} k^{\mu } R_{\nu^{\prime} \beta}}{3 (1 - \lambda) (k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^2(\lambda) ]^2} + \frac{\lambda^2 k^{2} k^{\mu } k_{\nu^{\prime} } R}{6 (1-\lambda) (k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^{2} ( \lambda) ]^2} \\ & - \frac{2 \lambda^2 k^{2} k^{\beta} k^{\gamma} R^{\mu }{}_{\beta\nu^{\prime} \gamma}}{3 (1 - \lambda) (k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^{2} ( \lambda) ]^2} - \frac{\lambda^2 k^{2} k^{\beta} k_{\nu^{\prime} } R^{\mu }_{\beta}}{3 (1 - \lambda) (k^{2} - m^{2} )^3 [ k^{2} - \tilde{m}^{2} ( \lambda) ]} \\ & + \frac{\lambda^2 k^{2} k^{\beta} k^{\mu } R_{\nu^{\prime} \beta}}{3 (1 - \lambda) (k^{2} - m^{2} )^3 [ k^{2} - \tilde{m}^{2} ( \lambda) ]} - \frac{2 \lambda^2 k^{2} k^{\beta} k^{\gamma} R^{\mu }{}_{\beta\nu^{\prime} \gamma}}{3 (1 - \lambda) (k^{2} - m^{2} )^3 [ k^{2} - \tilde{m}^{2} ( \lambda) ]} \\ & + \frac{\gamma k^{\alpha} k_{\nu^{\prime}} P^{\mu }_{\alpha}}{(k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^{2} ( \lambda) ]} + \frac{\gamma k^{\alpha} k^{\mu } P_{\nu^{\prime} \alpha}}{(k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^{2} ( \lambda) ]} \\ & + \frac{\lambda(2 + \lambda) k^{\alpha} k_{\nu^{\prime} } R^{\mu }_{\alpha}}{3 (1 - \lambda) ( k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^{2} ( \lambda) ]} + \frac{\lambda^2 k^{2} R^{\mu }_{\nu^{\prime} }}{2 (1 - \lambda) (k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^{2} ( \lambda) ]} \\ & + \frac{\lambda^2 k^{\alpha} k^{\mu } R_{\nu^{\prime} \alpha}}{3 (1 - \lambda) ( k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^{2} ( \lambda) ]} - \frac{(\lambda-2) \lambda k^{\mu } k_{\nu^{\prime} } R}{6 (1-\lambda) (k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^{2} ( \lambda) ]} \\ & - \frac{2 \lambda k^{\alpha} k^{\beta} R^{\mu }{}_{\alpha\nu^{\prime} \beta}}{3 (k^{2} - m^{2} )^2 [ k^{2} - \tilde{m}^{2} ( \lambda)]}. \end{align}\tag{42}\] In the particular case, when \(\lambda= 0\) and \(P^{\mu}_{\nu^{\prime}} = -R^{\mu}_{\nu^{\prime}}\), we recover the result of Ref. [10], \[\begin{align} \bar{G}^{\mu}_{2 \,\nu^{\prime}} ( k ) \,=\, \frac{\delta^{\mu }_{\,\nu^{\prime} }\, R}{6 (k^{2} - m^{2} )^2} \,-\,\frac{2 R^{\mu }_{\nu^{\prime}}}{3 (k^{2} - m^{2} )^2} \,-\, \frac{4 k^{\alpha} k^{\beta} R^{\mu }{}_{\alpha\nu^{\prime} \beta}}{3 (k^{2} - m^{2} )^3}. \label{papan81} \end{align}\tag{43}\]

Taking the coincidence limit of Eq. 42 and evaluating the momentum integrals, we obtain \[\begin{align} a^{\mu}_{1\, \nu} ( x, x ) \,=\, b_{1} \delta^{\mu}_{\,\nu} P \,+\, b_{2} P^{\mu}_{\,\nu} \,+\, b_{3} \delta^{\mu}_{\,\nu} R \,+\, b_{4} R^{\mu}_{\,\nu}, \end{align}\] where \(P = P^{\rho\sigma} g_{\rho\sigma}\) and \[\begin{align} && b_{1} \,=\, \frac{1}{D (D+2) ( D -2 ) \lambda} \left\{ (D - 2) \lambda + 4 + \left( 1 - \lambda\right)^{-D/2} \big[ D\lambda+ 2\lambda- 4 \big] \right\}, \nonumber \\ && b_{2} \,=\, \frac{1}{D (D+2) ( D -2 ) \lambda} \Big\{ \big(D - 2\big)\big(D^2 - 2\big) \lambda - 4 D - 2 \left( 1 - \lambda\right)^{-D/2} \big[ ( D + 2 ) \lambda- 2 D \big] \Big\}, \nonumber \\ && b_{3} = \frac{1}{6 D (D+2) ( D -2 ) \lambda} \, \Bigl\{ ( D^{3} - D^{2} - 12 ) \lambda+ 24 \nonumber \\ && \qquad \quad - \,( 1 - \lambda)^{-D/2} [ D ( D + 2 ) \lambda^{2} \,-\, \left( D^{2} + 8 D + 12 \right) \lambda+ 24 ] \Bigr\}, \nonumber \\ && b_4 \,= \, \frac{1}{3 D (D+2) ( D -2 ) \lambda} \Bigl\{ \big(12 + 8D - 5 D^2\big) \lambda- 12 D \nonumber \\ && \qquad \quad + \big( 1 - \lambda\big)^{-D/2} \big[ D ( D + 2 ) \lambda^{2}- ( D^{2} + 8 D + 12 ) \lambda+ 12 D \big] \Bigr\}. \end{align}\] For \(\lambda= 0\) we get \[\begin{align} a^{\mu}_{\,1\, \nu} (x, x) \,=\, P^{\mu}_{\,\nu} + \frac{1}{6}\,\delta^{\mu}_{\,\nu}\, R, \end{align}\] which is the well-known result for the minimal operator [10].

Finally, we can evaluate the \(a^{\mu}_{2\,\nu}(x,x)\) coefficient using the coincidence limit \[\begin{align} \label{eq:G4} \lim_{x \to x^{\prime}}\bar{G}^{\mu}_{4 \,\nu^{\prime}}(x,x^{\prime}) \,=\, \int \frac{d^{D} k}{( 2 \pi )^D} \, \bar{G}^{\mu}_{4 \nu^\prime} (k) \,=\, \int \frac{d^{D} k}{(2\pi)^{D}}\,\bar{G}^{\mu}_{0\,\rho} (k) \,\bar{H}^{\rho}_{4 \,\sigma} (k) \,\bar{G}^{\sigma}_{0 \,\nu^{\prime}} ( k ) \,+\, \ldots. \end{align}\tag{44}\] The explicit expression for \(\bar{G}^{\mu}_{4 \,\nu^\prime} (k)\) has been obtained from 35 . Unfortunately, it is too lengthy and will not be displayed here, but can be available upon request. The integrals over momentum in 44 can be dealt with in a straightforward manner by means of Wolfram Mathematica [25], with the tensor algebra package xAct [26]. The evaluation was performed using Feynman parameters, but we have to skip technical details owing to their size. After cumbersome calculations, the results coming from 44 were compared with the heat kernel expansion 36 . This yields the trace of \(\hat{a}_{2}\) as \[\begin{align} a^{\,\,\mu}_{2 \,\mu} ( x,x ) \,=\, c_1 \mathop{\mathrm{\square}}P + c_2 \nabla_{\alpha} \nabla_{\beta} P^{\alpha\beta} + c_3 \mathop{\mathrm{\square}}R + \ldots, \label{a2} \end{align}\tag{45}\] where the coefficients are given by the expressions \[\begin{align} c_{0} \,=\, \frac{1}{6 D ( D + 2 ) ( D - 2 ) ( D - 4 ) \lambda^{2}}, \end{align}\] \[\begin{align} \frac{c_{1}}{c_{0}} \,=\, & \,\, \big( D^4 - 5 D^3 + 2 D^2 + 32 D - 96\big) \lambda^{2} +192 \lambda-96 - (1 - \lambda)^{-D/2} \big[( D^3 + 6 D^{2} \\ & + 8 D ) \lambda^{3} - ( D^{3} + 6 D^{2} + 56 D + 96 ) \lambda^{2} + 48 (D + 4) \lambda- 96 \big], \end{align}\]

\[\begin{align} \frac{c_{2}}{2 c_{0}} \,=\, & \,\, (5 D^{3} - 24 D^2 + 4 D + 48 ) \lambda^2 + 24 \big( D^{2} - 3 D - 2\big) \lambda+ 48 D \qquad \qquad \quad \\ & + (1 - \lambda)^{-\frac{D}{2} + 1} \big[ (D^3 - 4D)\lambda^2+24(D + 2)\lambda-48 D \big] , \end{align}\]

\[\begin{align} \frac{5 c_{3}}{c_{0}} \,=\, & \,\, \big( D^{5} - 5 D^{4} + 15 D^{3} - 70 D^{2} + 104 D - 240\big) \lambda^2 + 120 \big( D^2 - 3 D + 6 \big) \lambda \\ & + 240 ( D - 2 ) - ( 1 - \lambda)^{1 -\frac{D}{2}} \big[ D ( D^{3} + D^2 - 4D - 4 ) \lambda^{3} \\ & - D \big( D^{3} + 11 D^2 + 26 D + 16 ) \lambda^{2} + 120 ( D + 2 ) \lambda + 240 ( D -2 )\big] . \end{align}\]

To obtain the four-dimensional version of the above results, we use the expansion \[\begin{align} \label{laD4} \big(1-\lambda\big)^{-D/2} \,=\, \frac{1}{(1-\lambda)^2} \left[1 - \frac{1}{2} \log(1 - \lambda) (D - 4) + {\mathcal{O}} \big((D - 4)^2\big)\right]. \end{align}\tag{46}\] Then, around \(D=4\) we get \[\begin{align} && c_1 \,=\, \frac{8\lambda^2-21\lambda+6}{36(\lambda-1)\lambda}+\frac{2\lambda-1}{6 \lambda^2}\log (1-\lambda), \qquad \qquad \quad \nonumber \\ && c_{2} \,=\, \frac{13 \lambda^2+6 \lambda-24}{36 (\lambda-1) \lambda}+\frac{\lambda+4}{6 \lambda^2} \log ( 1 - \lambda), \qquad \qquad \quad \nonumber \\ && c_{3} \,=\, - \,\frac{133 \lambda^2-168 \lambda-60}{360\lambda(1 - \lambda)} -\frac{\lambda^2-5 \lambda-2}{12 \lambda^2} \log ( 1 - \lambda), \label{eq:c3} \end{align}\tag{47}\] in the perfect correspondence with [2] and the previous calculations [4], [6]–[8]. In the special case \(\lambda= 0\) (the minimal operator), the result is \[\begin{align} c_{1} \,=\, \frac{1}{6}, \qquad c_{2} \,=\, 0, \qquad c_{3} \,=\, \frac{2}{15}, \end{align}\] as expected. For the Maxwell theory, where \(P^{\,\mu}_{\nu} = - R^{\,\mu}_{\nu}\), we get \[\begin{align} \label{vetor} a^{\,\,\mu}_{2\, \mu} ( x,x ) \,=\, \Big(c_3 - \frac{c_2}{2} -c_1 \Big) \mathop{\mathrm{\square}}R \,+ \ldots \,=\, -\frac{1}{60} \big[ 2 + 5 \log ( 1 - \lambda) \big] \mathop{\mathrm{\square}}R\, + \,\ldots. \end{align}\tag{48}\]

The result (48 ) shows an explicit dependence in the gauge parameter \(\lambda\), in the coefficient of the term \(\mathop{\mathrm{\square}}R\). There are a few aspects of this result to discuss. First of all, the expression (48 ) agrees with the ones obtained in the Refs. [4], [7], [8]. On the other hand, it disagrees with the general proof of the gauge independence for the divergences in the vacuum effective action of Maxwell’s theory given in [15], as we discussed in Sec. 2. One of the explanations is that this dependence, first detected in [4], should cancel with the contribution of the gauge ghosts to anomaly, as discussed in [5]. This is a plausible option, but the linear constant change of variables, on which the arguments of [5] are based, produces the Jacobian proportional to the delta function. This contribution vanishes in the dimensional regularization and since we use the last, the mentioned cancellation seems to be impossible.

One interesting detail concerning the result (47 ) was noted in [27]. Consider again the relation (12 ) from Sec. 2 and assume that \(\hat{H}(\lambda)\) has the condition (6 ) satisfied, i.e., \(P_{\rho \sigma} = - R_{\rho\sigma}\). For the second operator, we assume relation (13 ) to hold, as in Sec. 2. Then, using (48 ), for the first factor of the product we get \[\begin{align} \,tr\,\, \hat{a}_2\big[\hat{H}^*(\lambda)\big] \,=\, -\,\frac{1}{60} \big[2 + 5 \log ( 1 - \lambda) \big] \mathop{\mathrm{\square}}R \,+ \,\ldots\,, \label{1a2} \end{align}\tag{49}\] while for the operator \(\hat{H}^*(\beta)\) we get \[\begin{align} \nonumber \,tr\,\, \hat{a}_2\big[\hat{H}^*(\beta)\big] &=& -\,\frac{1}{60} \big[2 + 5 \log ( 1 - \beta) \big] \mathop{\mathrm{\square}}R \,+ \,\ldots\,, \\ \nonumber &=& -\,\frac{1}{60} \bigg[ 2 + 5 \log \Big( 1 - \frac{\lambda}{\lambda- 1} \Big) \bigg] \mathop{\mathrm{\square}}R \,+\, \ldots \nonumber \\ &=& -\,\frac{1}{60} \big[ 2 - 5 \log ( 1 - \lambda) \big] \mathop{\mathrm{\square}}R \,+\, \ldots. \label{2a2} \end{align}\tag{50}\] Even thought, taken separately, \(\,tr\,\,\hat{a}_2\big[H(\lambda)\big]\) and and \(\,tr\,\, \hat{a}_2\big[H^*(\beta)\big]\) are \(\lambda\)-dependent, the sum \[\begin{align} && \,tr\,\, \hat{a}_2\big[H^(\lambda)\big] \,+\, \,tr\,\, \hat{a}_2\big[H^*(\beta)\big] \nonumber \\ && \qquad \qquad \qquad \,=\, -\,\frac{1}{60}\bigg[4+5\log\Big(\frac{1-\lambda}{1-\lambda}\Big)\bigg] \mathop{\mathrm{\square}}R \,+\, \ldots \,=\, - \,\frac{1}{15}\, \mathop{\mathrm{\square}}R \,+\, \ldots \end{align}\] is independent of \(\lambda\). Let us stress that this remarkable cancellation does not mean that there is no MA because the last requires an independent cancellation of \(\lambda\)-dependence for both individual expressions (49 ) and (50 ).

Another possible source for the \(\lambda\)-dependence of the result (48 ) for the contribution of Abelian vector field may be as follows. The common point of the calculations performed in Refs. [2], [4], [7], [8] and also in the consideration presented above is the introduction of a massive parameter. In our case, the mass in (37 ) has been introduced to regularize infrared divergences. However, it is known that the \(m^2 \to 0\) limit in the vacuum contributions of the electromagnetic field in curved space is discontinuous [3], [28]. The reason is that introducing mass we change the number of degrees of freedom, and these degrees of freedom contribute even in the massless limit. Looking backward, it was precisely the inclusion of the mass that led to the appearance of the massive gauge-dependent parameter \(\tilde{m}^2 (\lambda)\) as introduced in Eq. (39 ).

Any solution of the momentum integral with a propagator corresponding to the mass \(\tilde{m}(\lambda)\) has, by dimensional arguments, an overall coefficient containing the inverse of \[\begin{align} \big[\tilde{m}^2 (\lambda)\big]^{\frac{D}{2}} \,\,=\,\, \Big(\frac{m^2}{1-\lambda} \Big)^{D/2}. \label{mlam} \end{align}\tag{51}\] This factor is, according to (46 ), the source of the \(\log (1-\lambda)\)-terms in the final result.

It is worth discussing an alternative way to deal with the IR problem. Another prescription for dealing with infrared divergences would be to introduce a massive parameter only in the Feynman integrals. In this case, the number of active degrees of freedom does not increase. So, instead of introducing a massive parameter in the theory, let us first obtain the momentum representation for the propagator, make Wick rotation and only afterwards regularize each momentum integral according to the rule \[\begin{align} \label{IR} \int \frac{ d^D k}{(2\pi)^D} \frac{k_{\mu_{1}} k_{\mu_{2}} \ldots k_{\mu_{2p}}}{(k^2)^q} \,\,\, \longrightarrow\,\,\, \int \frac{ d^D k}{(2\pi)^D} \frac{k_{\mu_{1}} k_{\mu_{2}} \ldots k_{\mu_{2p}}}{(k^2 + m^2)^q} \,. \end{align}\tag{52}\] Then, instead of solving (37 ), let us consider the equation for \(\bar{G}^{\nu}_{0\,\mu^{\prime}} ( k )\) in the form \[\begin{align} \label{eq:G0952} \big[-k^2 \delta^\mu_{\,\nu} \,+\, \lambda k^{\mu} k_{\nu} \big] \,\bar{G}^{\nu}_{0\,\mu^{\prime}} ( k ) \,=\, -\,\delta^\mu_{\,\mu^{\prime}} \end{align}\tag{53}\] with the well-known solution \[\begin{align} \label{G0sol2} \bar{G}^{\mu}_{0 \,\nu^{\prime}} ( k ) \,=\, \frac{\delta^{\mu}_{\,\,\nu^{\prime}}}{k^2} \,+\, \frac{\lambda}{1-\lambda} \frac{ \,k^{\mu} k_{\nu^{\prime}}}{k^4}\,. \end{align}\tag{54}\] After that, we can obtain the higher-order curvature correction to the propagator using the procedure described in Sec. 3, through Eqs. (34 ) and (35 ). Since the technical details are very similar to the ones in the last section, we skip the intermediate formulas.

Thus, applying the infrared regularization scheme (52 ), and using the integration formula \[\begin{align} && \int \frac{ d^D k}{(2\pi)^D} \frac{k_{\mu_{1}} k_{\mu_{2}} \ldots k_{\mu_{2p}}}{(k^2 + m^2)^q} = \frac{(m^2)^{D/2-q+p}}{2^p (4\pi)^{D/2}} \, \frac{\Gamma(q-p-D/2)}{\Gamma(q)} \, \delta_{{\mu_{1}}{\mu_{2}} \ldots{\mu_{2p}}}, \end{align}\] \[\begin{align} \qquad \qquad \qquad where \qquad \delta_{{\mu_{1}}{\mu_{2}} \ldots{\mu_{2p}}} \,=\, \delta_{{\mu_{1}}{\mu_{2}}} \dots \delta_{\mu_{2p-1}\mu_{2p}} \,+\, all permutations, \nonumber \end{align}\] we can compare the result for the coincidence limit of the propagator with the heat kernel expansion (36 ). This procedure yields, for the case \(P_{\mu\nu} = - R_{\mu\nu}\), the \(D=4\) result \[\begin{align} a^{\,\,\mu}_{2\, \mu} ( x,x ) \,=\, - \frac{\left(11 \lambda^2 - 35 \lambda+ 10\right)}{300(1-\lambda)} \, \mathop{\mathrm{\square}}R + \ldots, \end{align}\] which still depends on \(\lambda\), but reproduces the \((-1/30) \mathop{\mathrm{\square}}R\) result for the minimal \(\lambda= 0\) theory.

All in all, regardless of several independent calculations of the \(\mathop{\mathrm{\square}}R\)-term in the contribution of the nonminimal vector operator, the issue is not entirely resolved. Owing to the dependence on the IR regulator, the result may be calculation scheme-dependent, i.e., it might be ambiguous.

5 Concluding discussion↩︎

The output (18 ) of the procedure of doubling described in Sec. 2, apparently confirms its consistency and also the correctness of the formula (16 ). However, the logic of our consideration was based on the vanishing MA, i.e., the validity of (12 ). The violation of this condition is not impossible, though. In the published literature, MA was first reported in the paper [29] and subsequent works (see, e.g, [30], [31] and references therein) as the nonzero difference \[\begin{align} \Delta_{MA}\,=\, \log\,Det\,\big(\hat{A}\hat{B}\big) \,-\, \log\,Det\,\hat{A} \,-\, \log\,Det\,\hat{B} , \label{MA} \end{align}\tag{55}\] where \(\hat{A}\) and \(\hat{B}\) are differential operators. The result was obtained by using zeta-regularization [31]. On the other hand, the MA obtained in this way cannot be regarded a physical effect because the difference (55 ) can always be compensated by the change of renormalization conditions [32]–[35]. The conclusion is that the MA is possible only in those parts of effective action that are not subject of an infinite UV renormalization. For instance, this may concern the non-local terms in the effective action, except the leading logarithms since those are controlled by the UV divergences. It is worth noting that the universality of logarithmic divergences was established in [36] and has been confirmed by all known examples since then.

Eventually, the MA has been detected in the sub-leading part of the form factors in the contributions of fermionic operators for the massive fields [37], [38]. The calculations which led to this result were based on the heat kernel solution of [39] and on the more traditional evaluation of the trace of the coincidence limit of the \(\hat{a}_3\) coefficient of the Schwinger-DeWitt expansion. Qualitatively, the conclusion of Refs. [37], [38] was that the MA is directly related to the absence of MA in the divergent part of effective action, i.e., by the universality of the leading logarithms [36].

In the case of massless fields, the logarithmic divergences take control over the leading finite part of the one-loop effective action, and it seems there is no room for the MA. However, the situation becomes different in the theory with local conformal symmetry. In this case, the total derivative divergent terms correspond to the nonconformal local terms in the effective action, which escape the renormalization ambiguity. Recently, the MA of this kind has been reported for the fermionic operators with torsion [40] and antisymmetric tensor field [41].

Thus, one cannot rule out the presence of MA in the formula (12 ). However, the MA, which follows from the results of [2], [6] and was discussed in detail in [27], is different from the mentioned examples. First of all, this is the first example that is not related to the fermionic operators. It is known that fermionic operators are somehow special in the sense they are not Hermitian in curved space [42]. Thus, certain anomalies in this case may be natural. On the contrary, the nonminimal operator (1 ) is Hermitian. Furthermore, this operator does not correspond to either one of the described types of situations when the MA was previously detected.

Finally, there is a subtle point related to the regularization of the infrared divergences by introducing a massive parameter. As we have seen, this operation leads to the effective gauge-dependent mass (51 ), and this \(\lambda\)-dependence of the mass is the source of the MA. In the present article, our aim was to clearly formulate the problem and report on the new calculation that rules out the possibility of the technical mistakes in the known results of [2], [6] and other works.

Acknowledgements↩︎

T.M.S. is grateful to Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) for supporting his MSc project. I.Sh. is grateful to CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil) for the partial support under the grant 305122/2023-1.

References↩︎