Comment on 'Index-free heat kernel coefficients'

Spectral techniques are one of the preferred tools in high energy physics, because of their efficiency in the computation of effective actions and vacuum expectation values in nontrivial backgrounds. In particular, the heat kernel [1] offers a direct, efficient way to analyze the ultraviolet behaviour of a theory, carrying this information in its propertime expansion in terms of the Gilkey–Seeley–DeWitt (GSDW) coefficients [2]. Alternatively, nonperturbative phenomena can be analyzed employing resummations of the heat kernel.

The computation of higher-order GSDW coefficients for a general situation is rather involved and few references tackle this problem, van de Ven [3] being one of them. The importance of having at disposal reliable expressions for these coefficients resides not only in their possible direct application, but also in the necessity to have solid cross-checks for analytical developments, such as resummations. van de Ven [3] presents a method for obtaining the heat kernel coefficients of Laplace-type operators in an index-free notation. In this elegant way, its author was able to give the first (implicit) complete expression of the fifth GSDW coefficient for a Laplace operator on a curved manifold, as well as the sixth-order GSDW coefficient in flat space, including a general (non-)Abelian gauge connection in both cases.

Anyone who has ever attempted this type of computation is aware of how tedious it becomes at such orders; thus, the reader would not be surprised by the presence of some errors in the literature. In fact, in redoing some of the calculations present in [3], we have found two errors and ambiguities that, while in some sense minor, easily turn into headaches for the interested reader.

The first comment concerns the penultimate factor in the expression for $\mathsf{a}_5$, [3, equation (6.1)], i.e. $\frac{4}{21} {\mathsf{Z}}_{(3)}{}^{\dagger} \mathsf{Z}_{(3)}$. After a reading of the article, it becomes clear that the expression is nonsensical, given that, according to the notation of the article, parentheses in the subindices are used to denote a symmetrization followed by a contraction with a product of metric factors, which should be done pair by pair ¹ . One could think that this is just a typographical error, the author meaning ${\mathsf{Z}}_{(3}{}^{\dagger} \mathsf{Z}_{3)}$; however, this is not the case. The abovementioned factor should be replaced with

$$\begin{align} \frac{4}{21} {\mathsf{Z}}_{(3)}{}^{\dagger} \mathsf{Z}_{(3)} \to \frac{4}{21} {\mathsf{Z}}_{(3}{}^{\dagger} \mathsf{Z}_{1)(2)}, \end{align} \tag{ 2.1 }$$

i.e. the indices in the last $\mathsf{Z}$ factor are subject to two different types of symmetrizations and contractions.

The second one, concerns the validity of equation (4.27) of [3], which defines the action of the $\hat{\mathsf{Z}}$ operators in flat space, including a nontrivial gauge connection. Note in particular that the term $\hat{\mathsf{Z}}_0\hat{\mathsf{Z}}_0\mathsf{Z}_{1)}$ is involved in the calculation of the $\mathsf{a}_5$ GSDW coefficient. Applying the second line in [3, equation (4.27)], one obtains

$$\begin{align} \textrm{`}\hat{\mathsf{Z}}_0\hat{\mathsf{Z}}_0\mathsf{Z}_{1)} = {\mathsf{Z}}_0\hat{\mathsf{Z}}_0\mathsf{Z}_{1)}+ 2 \mathsf{Y}_{1}{}^{\nu} \left[ 0 \times \hat{\mathsf{Z}}_{-1}{}^{\nu} \mathsf{Z}_{1)}+ \hat{\mathsf{Z}}_{0)}\mathsf{Z}_{\nu}\right]\textrm{'}. \end{align} \tag{ 2.2 }$$

An inattentive reader might wonder why we have explicitly written the product of zero times the $\mathsf{Z}$-factors: The problem resides in the fact that $\hat{\mathsf{Z}}_{-1}\mathsf{Z}_{1)}$ is not defined in the manuscript. In fact, [3, equation (4.27)] gives a definition only for $n\unicode{x2A7E}0$; in particular, for $n = -1$ one obtains a divergent quantity. A direct computation shows that, in this case, the second line in [3, equation (4.27)] should be intended in the limit n → 0 after replacing every $\hat{\mathsf{Z}}$ factor present in the expression. In this process a new contribution is created, which for the term needed in the $\mathsf{a}_5$ calculation reads

$$\begin{align} \hat{\mathsf{Z}}_0\hat{\mathsf{Z}}_0\mathsf{Z}_{1)} = {\mathsf{Z}}_0\hat{\mathsf{Z}}_0\mathsf{Z}_{1)}+ 4 \mathsf{Y}_{1)}{}^{\nu} \mathsf{Y}_{\nu}{}^{\mu} \mathsf{Z}_{\mu}+2 \mathsf{Y}_{1)}{}^{\nu} \hat{\mathsf{Z}}_{0}\mathsf{Z}_{\nu}. \end{align} \tag{ 2.3 }$$

A similar remark applies to the expression that is tacitly assumed in equation (5.42), when a general curved manifold is considered.

Of course, the validity of both these comments can be checked by direct computation of the $\mathsf{a}_5$ GSDW coefficient. Using the resummation in [4], its explicit expression greatly simplifies, as is shown in appendix. Alternatively, after using integration by parts one can readily compare for example with [5] or using the resummed expressions in [6].

The author is indebted to F D Mazzitelli for several crucial discussions. The author acknowledges the support from the Helmholtz-Zentrum Dresden-Rossendorf, from Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) through the Project PIP 11220200101426CO and from UNLP through the Project 11/X748.

All data that support the findings of this study are included within the article (and any supplementary files).

Consider the heat kernel $K(x,x^{^{\prime}};\tau)$ that satisfies the following equation in a Euclidean, d-dimensional space with an Abelian gauge connection A_µ and an arbitrary scalar potential V:

$$\begin{align} \left[\partial_\tau - \nabla^2 +2A_\mu\left(x\right) \nabla^\mu + V\right] K\left(x,x^{^{\prime}};\tau\right)& = 0, \end{align} \tag{ A.1 }$$

$$\begin{align} K\left(x,x^{^{\prime}},0^+\right)& = \delta\left(x-x^{^{\prime}}\right). \end{align} \tag{ A.2 }$$

According to [4], its diagonal can be cast as

$$\begin{align} \begin{split} K\left(x,x;\tau\right) & = \frac{e^{-\tau V +\nabla^{\alpha }V \left[ \gamma^{-3} \left({\gamma \tau - 2 \tanh\left(\tfrac{1}{2} \gamma \tau\right)}\right)\right]_{\alpha \beta } \nabla^{\beta }V }} {\left(4\pi\right)^{d/2} \;{\det} ^{1/2}\left(\left(\gamma \tau\right)^{-1} \sinh\left(\gamma \tau\right)\right) } \Omega\left(x,x;\tau\right)\Bigg\vert_\mathrm{coincidence}, \end{split} \end{align} \tag{ A.3 }$$

where we have defined ² $\gamma^2_{\mu\nu}: = 2\nabla_{\mu\nu}V$. In addition, the gauge potential is assumed to be in the Fock–Schwinger gauge, so that the scalar potential has both an intrinsic scalar contribution, $\tilde V$, and one resulting from the Abelian gauge connection, V_EM ; in formulae, we have

$$\begin{align} \begin{split} V:& = \tilde V+V_{EM}\left(x\right) \\ V_{EM}\left(x\right):& = \nabla_\mu A^\mu-A_\mu A^\mu. \end{split} \end{align} \tag{ A.4 }$$

The subindex 'coincidence' in equation (A.3) means that every single instance of the gauge field or its derivatives should be replaced by its coincidence limit (in the Fock–Schwinger gauge [4]), to wit

$$\begin{align} A^{\nu}{}_{;\mu_1\cdots\mu_n}\left(x\right) \to \frac{n}{n+1} F_{(\mu_1}{}^{\nu}{}_{;\mu_2\cdots \mu_n )}\left(x\right), \end{align} \tag{ A.5 }$$

where the parentheses in the subindex indicate the standard idempotent symmetrization. Note that the replacement in equation (A.5) is also intended in the prefactor.

As always, we can perform an expansion of Ω in the propertime τ, thus defining the modified heat kernel coefficients c_j :

$$\begin{align} \Omega\left(x,x^{^{\prime}};\tau\right) = :\sum_{j = 0}^\infty c_j\left(x,x^{^{\prime}}\right) \tau^{j-d/2}. \end{align} \tag{ A.6 }$$

To enable future comparisons, we write down the explicit expressions for the diagonal of the first five modified heat kernel coefficients; all the quantities in the RHS are evaluated at x and equations (A.4) and (A.5) are intended:

$$\begin{align} c_0\left(x,x\right)& = 1, \end{align} \tag{ A.7 }$$

$$\begin{align} c_1\left(x,x\right)& = 0, \end{align} \tag{ A.8 }$$

$$\begin{align} c_2\left(x,x\right)& = 0, \end{align} \tag{ A.9 }$$

$$\begin{align} c_3\left(x,x\right)& = - \tfrac{1}{18} \nabla^{\alpha }V\, \nabla_{\beta }F_{\alpha }{}^{\beta } - \tfrac{1}{60} \nabla_{\beta }{}^{\beta }{}_{\alpha }{}^{\alpha }V, \end{align} \tag{ A.10 }$$

$$\begin{align}\begin{split} c_4\left(x,x\right)& = - \tfrac{1}{90} F_{\alpha }{}^{\beta } \nabla^{\alpha }V\, \nabla_{\gamma }F_{\beta }{}^{\gamma } - \tfrac{1}{90} \nabla^{\beta \alpha }V\, \nabla_{\gamma \beta }F_{\alpha }{}^{\gamma } \\ &\quad\, + \tfrac{1}{30} \nabla^{\alpha }V\, \nabla_{\beta }{}^{\beta }{}_{\alpha }V - \tfrac{1}{150} \nabla^{\alpha }V\, \nabla_{\gamma }{}^{\gamma }{}_{\beta }F_{\alpha }{}^{\beta } \\ &\quad\, + \tfrac{1}{90} \nabla_{\alpha }F^{\alpha \beta } \nabla_{\gamma }{}^{\gamma }{}_{\beta }V - \tfrac{1}{840} \nabla_{\gamma }{}^{\gamma }{}_{\beta }{}^{\beta }{}_{\alpha }{}^{\alpha }V, \end{split} \end{align} \tag{ A.11 }$$

$$\begin{align}\begin{split} c_5\left(x,x\right)& = - \tfrac{1}{360} F_{\alpha }{}^{\gamma } F_{\beta \gamma } \nabla^{\alpha }V\, \nabla^{\beta }V \\ &\quad\, - \tfrac{1}{540} F_{\alpha }{}^{\beta } F_{\beta }{}^{\gamma } \nabla^{\alpha }V\, \nabla_{\delta }F_{\gamma }{}^{\delta } + \tfrac{1}{90} \nabla^{\alpha }V\, \nabla_{\gamma }F_{\beta }{}^{\gamma } \nabla^{\beta }{}_{\alpha }V \\ &\quad\, - \tfrac{2}{945} \nabla_{\gamma }F_{\alpha }{}^{\gamma } \nabla_{\delta }F_{\beta }{}^{\delta } \nabla^{\beta \alpha }V - \tfrac{2}{945} \nabla_{\beta }F_{\alpha }{}^{\gamma } \nabla_{\delta }F_{\gamma }{}^{\delta } \nabla^{\beta \alpha }V \\ &\quad\, + \tfrac{1}{120} \nabla^{\alpha }V\, \nabla^{\beta }V\, \nabla_{\gamma \beta }F_{\alpha }{}^{\gamma } - \tfrac{1}{630} \nabla^{\alpha }V\, \nabla^{\gamma }F_{\alpha }{}^{\beta } \nabla_{\delta \beta }F_{\gamma }{}^{\delta } \\ &\quad\, - \tfrac{1}{630} F^{\alpha \beta } \nabla^{\gamma }{}_{\alpha }V\, \nabla_{\delta \beta }F_{\gamma }{}^{\delta } + \tfrac{1}{420} \nabla^{\alpha }V\, \nabla_{\beta }F^{\beta \gamma } \nabla_{\delta \gamma }F_{\alpha }{}^{\delta } \\ &\quad\, - \tfrac{1}{630} \nabla^{\alpha }V\, \nabla^{\gamma }F_{\alpha }{}^{\beta } \nabla_{\delta \gamma }F_{\beta }{}^{\delta } - \tfrac{1}{630} F^{\alpha \beta } \nabla^{\gamma }{}_{\alpha }V\, \nabla_{\delta \gamma }F_{\beta }{}^{\delta } \\ &\quad\, + \tfrac{1}{840} \nabla^{\alpha }V\, \nabla_{\beta }F^{\beta \gamma } \nabla_{\delta }{}^{\delta }F_{\alpha \gamma } + \tfrac{17}{5040} \nabla^{\beta }{}_{\alpha }{}^{\alpha }V\, \nabla_{\gamma }{}^{\gamma }{}_{\beta }V \\ &\quad\, + \tfrac{1}{840} \nabla_{\gamma \beta \alpha }V\, \nabla^{\gamma \beta \alpha }V - \tfrac{1}{3500} \nabla^{\gamma \beta \alpha }V\, \nabla_{\delta \beta \alpha }F_{\gamma }{}^{\delta } \\ &\quad\, - \tfrac{1}{875} \nabla^{\gamma \beta \alpha }V\, \nabla_{\delta \gamma \beta }F_{\alpha }{}^{\delta } - \tfrac{1}{3500} \nabla^{\gamma \beta \alpha }V\, \nabla_{\delta }{}^{\delta }{}_{\beta }F_{\alpha \gamma } \\ &\quad\, + \tfrac{2}{945} F^{\alpha \beta } \nabla_{\gamma }F_{\alpha }{}^{\gamma } \nabla_{\delta }{}^{\delta }{}_{\beta }V - \tfrac{2}{1575} F_{\alpha }{}^{\beta } \nabla^{\alpha }V\, \nabla_{\delta }{}^{\delta }{}_{\gamma }F_{\beta }{}^{\gamma } \\ &\quad\, - \tfrac{17}{12600} \nabla^{\beta }{}_{\alpha }{}^{\alpha }V\, \nabla_{\delta }{}^{\delta }{}_{\gamma }F_{\beta }{}^{\gamma } + \tfrac{1}{210} \nabla^{\beta \alpha }V\, \nabla_{\gamma }{}^{\gamma }{}_{\beta \alpha }V \\ &\quad\, - \tfrac{1}{630} \nabla^{\beta \alpha }V\, \nabla_{\delta }{}^{\delta }{}_{\gamma \beta }F_{\alpha }{}^{\gamma } + \tfrac{1}{420} \nabla^{\gamma }{}_{\alpha }F^{\alpha \beta } \nabla_{\delta }{}^{\delta }{}_{\gamma \beta }V \\ &\quad\, + \tfrac{1}{280} \nabla^{\alpha }V\, \nabla_{\gamma }{}^{\gamma }{}_{\beta }{}^{\beta }{}_{\alpha }V - \tfrac{1}{1960} \nabla^{\alpha }V\, \nabla_{\delta }{}^{\delta }{}_{\gamma }{}^{\gamma }{}_{\beta }F_{\alpha }{}^{\beta } \\ &\quad\, + \tfrac{1}{840} \nabla_{\alpha }F^{\alpha \beta } \nabla_{\delta }{}^{\delta }{}_{\gamma }{}^{\gamma }{}_{\beta }V - \tfrac{1}{15120} \nabla_{\delta }{}^{\delta }{}_{\gamma }{}^{\gamma }{}_{\beta }{}^{\beta }{}_{\alpha }{}^{\alpha }V. \end{split} \end{align} \tag{ A.12 }$$