One-loop effective action up to dimension eight: integrating out heavy scalar(s)

Abstract

We present the one-loop effective action up to dimension eight after integrating out degenerate scalars using the Heat-Kernel method. The result is provided without assuming any specific form of either UV or low energy theories, i.e., universal. In this paper, we consider the complete effects of only heavy scalar propagators in the loops. We also verify part of the results using the covariant diagram technique.

Appendices

Appendix A: Computation of operators ${\mathcal {O}}(D^4 U^3)$: a detailed example

Based on the methodology discussed in Sect. 3, we perform, here, an explicit calculation of the operators class ${\mathcal {O}}\left( D^4 U^3\right)$ for the sake of detailed demonstration. To compute ${\mathcal {O}}\left( D^4 U^3\right)$, we obtain the following necessary relations using Eq. (3.8).

$$\begin{aligned}&[b_2] \left[\!\left[ U^0 \right]\!\right] ={\mathcal {O}}\left( D^4 U^0\right) = -\frac{1}{12}\, \left\{ T_{(4)}b_0\right\} _{z=0}, \end{aligned}$$

(A.1)

$$\begin{aligned}&[b_3] \left[\!\left[ U^1 \right]\!\right] ={\mathcal {O}}\left( D^4 U^1\right) = U\,[b_2] \left[\!\left[ U^0 \right]\!\right] + \left\{ \frac{1}{2} D^2 \left( U\,b_1 \right) + \frac{1}{10} \left( D^4 \left( U\,b_0 \right) - T_{(4)}b_1 \right) \right\} _{z=0}\left[\!\left[ U^1 \right]\!\right] , \end{aligned}$$

(A.2)

$$\begin{aligned}&[b_4] \left[\!\left[ U^2 \right]\!\right] = {\mathcal {O}}\left( D^4 U^2\right) = U\,[b_3] \left[\!\left[ U^1 \right]\!\right] + \left\{ \frac{3}{5} D^2 \left( U\,b_2 \right) + \frac{1}{10} \left( 2\,D^4 \left( U\,b_1 \right) - T_{(4)}b_2 \right) \right\} _{z=0} \left[\!\left[ U^2 \right]\!\right] , \end{aligned}$$

(A.3)

$$\begin{aligned}&[b_5] \left[\!\left[ U^3 \right]\!\right] = {\mathcal {O}}\left( D^4 U^3\right) = U\,[b_4] \left[\!\left[ U^2 \right]\!\right] + \left\{ \frac{2}{3} D^2 \left( U\,b_3 \right) + \frac{2}{21} \left( 3\,D^4 \left( U\,b_2 \right) - T_{(4)}b_3\right) \right\} _{z=0} \left[\!\left[ U^3 \right]\!\right] . \end{aligned}$$

(A.4)

Here, we set

$$\begin{aligned} T = 0\,,\qquad T_{\mu } = 0\,,\qquad T_{(2)}=T_{\mu \mu } = 0, \end{aligned}$$

that are quite evident from Eq. (3.5). It is important to note that in the above equations, we have two different kinds of structures: (i) $D^4$ and $D^2$ act on $U\,b_k$, and (ii) $T_{(4)}$ acts on $b_k$. Thus, our initial aim is to calculate the explicit form of these operators, first.

The actions of $D^4$ and $D^2$ operators on $(U\,b_k)$ are defined from Eqs. (A.2)–(A.4), as follows:

$$\begin{aligned} D^2(U\,b_k){} & {} = U_{;\mu \mu }\, b_k + 2\,U_{;\mu } D_\mu b_k + U\,D^2b_k, \end{aligned}$$

(A.5)

$$\begin{aligned} D^4(U\,b_k){} & {} = U_{;\mu \mu \nu \nu }\, b_k + 2\, U_{;\mu \nu \nu } D_\mu b_k + 2\, U_{;\nu \nu \mu } D_\mu b_k + 2\,U_{;\mu } D_{\mu \nu \nu } b_k \nonumber \\{} & {} \quad + 2\,U_{;\mu } D_{\nu \nu \mu } b_k + 4\,U_{;\nu \mu } D_{\mu \nu } b_k + 2\, U_{;\mu \mu } D^2 b_k + U\,D^4 b_k, \end{aligned}$$

(A.6)

Action of $T_{(4)}$ can be addressed in the following form derived from Eqs. (3.5)–(3.7)

$$\begin{aligned} \begin{aligned} T_{(4)}=T_{\mu \mu \nu \nu }&= D_\mu T_{\mu \nu \nu } + R_{\mu \nu \nu ,\mu } =D_{\mu \mu } T_{\nu \nu } + D_\mu R_{\nu \nu ,\mu }+ R_{\mu \nu \nu ,\mu }\\&=D_\mu R_{\nu \nu ,\mu }+ D_\mu R_{\nu \nu ,\mu } + G_{\mu \mu }D_{\nu \nu } =2\,D_{\mu \nu } G_{\nu \mu }+ 2\,D_\mu G_{\nu \mu }D_\nu \\&= G_{\mu \nu } G_{\nu \mu } - 2\,G_{\mu \nu ;\mu } D_\nu -2\,G_{\mu \nu }D_{\mu \nu } = -2\,(G_{\mu \nu })^2 - 2\,J_\nu D_\nu . \end{aligned} \end{aligned}$$

(A.7)

In this derivation, we use the anti-symmetric nature of $G_{\mu \nu }$ and the following identity

$$\begin{aligned} X_{;\nu \mu } = X_{;\mu \nu } + G_{\mu \nu }X - X\, G_{\mu \nu }, \end{aligned}$$

(A.8)

where X is any arbitrary tensor. This leads to our finding

$$\begin{aligned} 2\,D_{\mu \nu } G_{\mu \nu } = (G_{\mu \nu })^2. \end{aligned}$$

Now, we are ready to demonstrate the explicit computation of operators ${\mathcal {O}}\left( D^4 U^3\right)$ belonging to the HKC $b_5$.

$\underline{{\mathcal {O}}\left( D^4 U^0\right) }$

$T_{(4)}$ operator contains a derivative acting on HKCs. Hence, to calculate $[b_2]_{U^0}$, we, first, need to calculate $D_\nu b_0 |_{z=0}$. From Eq. (3.3), we find

$$\begin{aligned} \begin{aligned} D_\nu b_0|_{z=0} = -T_\nu b_0|_{z=0} = 0. \end{aligned} \end{aligned}$$

(A.9)

This provides ${\mathcal {O}}\left( D^4U^0\right)$, directly from Eq. (A.1), as

$$\begin{aligned} \begin{aligned}{}[b_2] \left[\!\left[ U^0 \right]\!\right] ={\mathcal {O}}\left( D^4 U^0\right)&= \frac{1}{6}\, \Big \{(G_{\mu \nu })^2 b_0 + J_\nu D_\nu b_0\Big \}_{z=0} = \frac{1}{6}\, (G_{\mu \nu })^2. \end{aligned} \end{aligned}$$

(A.10)

$\underline{{\mathcal {O}}\left( D^4 U^1\right) }$

Next, to compute ${\mathcal {O}} (D^4 U^1),$ we calculate the necessary derivatives of HKCs using Eqs. (3.3)-(3.7).

$$\begin{aligned}{} & {} D^2 b_0\Big |_{z=0} = D_{\mu \mu } b_0\Big |_{z=0} = -\frac{1}{2} T_{\mu \mu } b_0\Big |_{z=0} = 0. \end{aligned}$$

(A.11)

$$\begin{aligned}{} & {} D_{\mu \nu } b_0\Big |_{z=0} = -\frac{1}{2} T_{\mu \nu } b_0\Big |_{z=0} = -\frac{1}{2}\, \{D_\mu T_{\nu } + R_{\nu ,\mu }\} b_0 \Big |_{z=0} = \frac{1}{2}\, G_{\mu \nu }. \end{aligned}$$

(A.12)

$$\begin{aligned}{} {} D_{\mu \nu \nu } b_0\Big |_{z=0}& = -\frac{1}{3} T_{\mu \nu \nu } b_0\Big |_{z=0} = -\frac{1}{3}\, \{D_\mu T_{\nu \nu } + R_{\nu \nu ,\mu }\} b_0 \Big |_{z=0} = -\frac{1}{3}\, \{D_\nu G_{\nu \mu } + G_{\nu \mu } D_\nu \} b_0 \Big |_{z=0},\nonumber \\{} & {} = -\frac{1}{3}\, \{G_{\nu \mu ;\nu } + 2\,G_{\nu \mu } D_\nu \} b_0 \Big |_{z=0} -\frac{1}{3}\,J_\mu . \qquad \qquad \quad \end{aligned}$$

(A.13)

$$\begin{aligned}{} {} D_{\nu \nu \mu } b_0\Big |_{z=0} &= -\frac{1}{3} T_{\nu \nu \mu } b_0\Big |_{z=0} = -\frac{1}{3}\, \{D_\nu T_{\nu \mu } + R_{\nu \mu ,\nu }\} b_0 \Big |_{z=0} = -\frac{2}{3}\, D_\nu G_{\mu \nu } b_0 \Big |_{z=0},\nonumber \\{} & {} = \frac{2}{3}\, \{G_{\nu \mu ;\nu } + \,G_{\nu \mu } D_\nu \} b_0 \Big |_{z=0} = \frac{2}{3}\,J_\mu . \qquad \qquad \end{aligned}$$

(A.14)

$$\begin{aligned}{} & {} D^4 b_0\Big |_{z=0} = D_{\mu \mu \nu \nu } b_0\Big |_{z=0} = -\frac{1}{4} T_{(4)} b_0\Big |_{z=0} = \frac{1}{2}\, (G_{\mu \nu })^2. \end{aligned}$$

(A.15)

$$\begin{aligned}{} & {} [b_1] = \{U+D^2\}b_0\Big |_{z=0} = U. \end{aligned}$$

(A.16)

$$\begin{aligned}{} {} D_\mu b_1\Big |_{z=0} &= \frac{1}{2}\{D_\mu \left( U+D^2\right) b_0-T_\mu b_1\}\Big |_{z=0} = \frac{1}{2} \Big \{U_{;\mu }b_0 + U\,D_\mu b_0 + D_{\mu \nu \nu } b_0\Big \}_{z=0}, \nonumber \\{} & {} = \frac{1}{2} U_{;\mu } - \frac{1}{6} J_\mu . \end{aligned}$$

(A.17)

$$\begin{aligned}{} & {} D_{\mu \mu } b_1\Big |_{z=0} \left[\!\left[ U^0 \right]\!\right] = \frac{1}{3} \{D_{\mu \mu }\left( U+D^2\right) b_0 - T_{\mu \mu } b_1\}\Big |_{z=0} \left[\!\left[ U^0 \right]\!\right] = \frac{1}{3} D_{\mu \mu \nu \nu }b_0\Big |_{z=0} \left[\!\left[ U^0 \right]\!\right] = \frac{1}{6} (G_{\mu \nu })^2. \end{aligned}$$

(A.18)

Assembling all the contributions, computed here, in Eq. (A.2) we find

$$\begin{aligned}[b_3] \left[\!\left[ U^0 \right]\!\right] = {\mathcal {O}}\left( D^4 U^1\right) &=\frac{3}{10} U\,(G_{\mu \nu })^2 +\frac{1}{5}(G_{\mu \nu })^2 U - \frac{1}{10} U_{;\mu }J_{;\mu } + \frac{1}{10} J_{;\mu }U_{;\mu }\\&\quad+ \frac{1}{10} U_{;\mu \mu \nu \nu } + \frac{2}{10} U_{;\nu \mu }G_{\mu \nu }. \end{aligned}$$

(A.19)

$\underline{{\mathcal {O}}\left( D^4 U^2\right) }$

Following the similar path, we calculate the derivatives of HKCs required for the computation of operators ${\mathcal {O}}\left( D^4 U^2\right)$.

$$\begin{aligned} D_{\mu \mu } b_1|_{z=0} \left[\!\left[ U^1 \right]\!\right]{} & {} = \frac{1}{3} \left\{ D^2\left( U+D^2\right) b_0 - T_{\mu \mu } b_1\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{3} \left\{ U_{;\mu \mu } b_0 + 2\,U_{;\mu } D_\mu b_0 + U\,D^2 b_0\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{3} U_{;\mu \mu }. \end{aligned}$$

(A.20)

$$\begin{aligned} D_{\mu \nu } b_1|_{z=0} \left[\!\left[ U^1 \right]\!\right]{} & {} = \frac{1}{3} \left\{ D_{\mu \nu }\left( U+D^2\right) b_0 - T_{\mu \nu } b_1\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{3} \left\{ U_{;\nu \mu } b_0 + U_{;\mu } D_\nu b_0 + U_{;\nu } D_\mu b_0 + U\,D_{\mu \nu } b_0 - G_{\nu \mu }b_1\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{3}\left\{ U_{;\nu \mu } + \frac{1}{2}U\,G_{\mu \nu } + G_{\mu \nu }U\right\} , \end{aligned}$$

(A.21)

$$\begin{aligned} D_{\mu \nu \nu } b_1|_{z=0} \left[\!\left[ U^1 \right]\!\right]{} & {} = \frac{1}{4} \left\{ D_{\mu \nu \nu }\left( U+D^2\right) b_0 - T_{\mu \nu \nu } b_1\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{4} \left\{ D_{\mu }\left( U_{;\nu \nu } b_0 + 2\,U_{;\nu }D_\nu b_0 + U\,D^2 b_0\right) - D_\nu G_{\nu \mu } b_1 - G_{\nu \mu } D_\nu b_1\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{4} \left\{ U_{;\nu \nu \mu } b_0 + U_{;\nu \nu } D_\mu b_0 + 2\,U_{;\nu \mu }D_\nu b_0 + 2\,U_{;\nu }D_{\mu \nu } b_0 + U_{;\mu }\,D^2 b_0\right. \nonumber \\{} & {} \left. \quad + U\,D_{\mu \nu \nu } b_0 - G_{\nu \mu ;\nu } b_1 - 2\, G_{\nu \mu }D_\nu b_1\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{4} \left\{ U_{;\nu \nu \mu } + U_{;\nu }G_{\mu \nu } -\frac{1}{3} U\,J_{\mu } -J_\mu U + G_{\mu \nu } U_{;\nu }\right\} . \end{aligned}$$

(A.22)

$$\begin{aligned} D_{\nu \nu \mu } b_1|_{z=0} \left[\!\left[ U^1 \right]\!\right]{} & {} = \frac{1}{4} \{D_{\nu \nu \mu }\left( U+D^2\right) b_0 - T_{\nu \nu \mu } b_1\}|_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{4} \left\{ D_{\nu \nu }(U_{\mu }b_0 + U\,D_\mu b_0) - 2\,D_\nu G_{\mu \nu } b_1\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{4} \left\{ U_{;\mu \nu \nu }b_0 + U_{\mu } D^2 b_0 + 2\,U_{\mu \nu } D_\nu b_0 + U_{\nu \nu }\,D_\mu b_0 + U\,D_{\nu \nu \mu } b_0 \right. \nonumber \\{} & {} \left. \quad +\,2\, U_{;\nu }\,D_{\nu \mu } b_0 + 2\, G_{\nu \mu ;\nu } b_1 + 2\, G_{\nu \mu } D_\nu b_1\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{4} \left\{ U_{;\mu \nu \nu } + \frac{2}{3}U\,J_{\mu } + U_{;\nu }G_{\nu \mu } + 2\, J_\mu U + G_{\nu \mu } U_{;\nu }\right\} . \end{aligned}$$

(A.23)

$$\begin{aligned} D^4\,b_1|_{z=0} \left[\!\left[ U^1 \right]\!\right]{} & {} = \frac{1}{5} \left\{ D_{\mu \mu \nu \nu }\left( U+D^2\right) b_0 - T_{\mu \mu \nu \nu } b_1\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{5} \left\{ U_{;\mu \mu \nu \nu } b_0 + 2\, U_{;\mu \nu \nu } D_\mu b_0 + 2\, U_{;\nu \nu \mu } D_\mu b_0 + 2\,U_{;\mu } D_{\mu \nu \nu } b_0 + 2\,U_{;\mu } D_{\nu \nu \mu } b_0 \right. \nonumber \\{} & {} \left. \quad +\, 4\,U_{;\nu \mu } D_{\mu \nu } b_0 + 2\, U_{;\mu \mu } D^2 b_0 + U\,D^4 b_0 + 2\, J_\mu D_\mu b_1 + 2\, (G_{\mu \nu })^2 b_1\right\} |_{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{5} \left\{ U_{;\mu \mu \nu \nu } + \frac{2}{3}\,U_{;\mu } J_{\mu } + 2\,U_{;\nu \mu } G_{\mu \nu } + \frac{1}{2} U\,(G_{\mu \nu })^2 + J_\mu U_{;\mu } + 2\, (G_{\mu \nu })^2 U\right\} . \end{aligned}$$

(A.24)

$$\begin{aligned} \left[\!\left[ U^1,U^2 \right]\!\right]{} & {} = \{U+D^2\}b_1|_{z=0} \left[\!\left[ U^1,U^2 \right]\!\right] = U^2 + \frac{1}{3} U_{;\mu \mu }. \end{aligned}$$

(A.25)

$$\begin{aligned} D_\mu b_2|_{z=0} \left[\!\left[ U^1,U^2 \right]\!\right]{} & {} = \frac{1}{3}\left\{ 2\,D_\mu \left( U+D^2\right) b_1-T_\mu b_2\right\} _{z=0} \left[\!\left[ U^1,U^2 \right]\!\right] ,\nonumber \\{} & {} = \frac{2}{3} \left\{ U_{;\mu }b_1 + U\,D_\mu b_1 + D_{\mu \nu \nu } b_1\right\} _{z=0} \left[\!\left[ U^1,U^2 \right]\!\right] ,\nonumber \\{} & {} = \frac{2}{3} \left\{ U_{;\mu } U + \frac{1}{2} U\,U_{;\mu } -\frac{1}{4} U\,J_\mu + \frac{1}{4} U_{;\nu \nu \mu } + \frac{1}{4}U_{;\nu }G_{\mu \nu } - \frac{1}{4}J_\mu U + \frac{1}{4}G_{\mu \nu } U_{;\nu }\right\} . \end{aligned}$$

(A.26)

$$\begin{aligned} D_{\mu \mu } b_2|_{z=0} \left[\!\left[ U^1 \right]\!\right]{} & {} = \frac{1}{4}\left\{ 2\,D_{\mu \mu }\left( U+D^2\right) b_1-T_{\mu \mu } b_2\right\} _{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{2} \left\{ U_{;\mu \mu } b_1 + 2\,U_{;\mu } D_\mu b_1 + U\,D^2 b_1 + D_{\mu \mu \nu \nu } b_1\right\} _{z=0} \left[\!\left[ U^1 \right]\!\right] ,\nonumber \\{} & {} = \frac{1}{2} \left\{ -\frac{1}{5}\,U_{;\mu } J_\mu + \frac{4}{15}U\,(G_{\mu \nu })^2 + \frac{1}{5}U_{;\mu \mu \nu \nu } + \frac{2}{5}\,U_{;\nu \mu } G_{\mu \nu }+ \frac{2}{5}\, (G_{\mu \nu })^2 U\right\} . \end{aligned}$$

(A.27)

Again, we collect all these contributions and with the help of Eq. (A.3), we note the following equation.

$$\begin{aligned}{}[b_4] \left[\!\left[ U^2 \right]\!\right] ={\mathcal {O}}\left( D^4 U^2\right) =&\frac{1}{5} U_{;\mu \mu \nu } U_{;\nu } +\frac{3}{10} U_{;\mu } U_{;\nu \nu \mu } + \frac{1}{5} U_{;\mu \nu \nu } U_{;\mu } + \frac{4}{15} (U_{;\mu \nu })^2 +\frac{1}{3} (U_{;\mu \mu })^2 \\&+\frac{1}{5} U_{;\mu \mu \nu \nu } U +\frac{1}{10} U_{;\mu } U_{;\mu \nu \nu } +\frac{1}{5} U\,U_{;\mu \mu \nu \nu } +\frac{2}{15} J_\mu U_{;\mu U } \\&-\frac{2}{15} U\,U_{;\mu }J_\mu +\frac{1}{5}U\,J_\mu U_{;\mu } - \frac{1}{6} U_{;\mu }U\,J_\mu -\frac{1}{10} U_{;\mu }J_\mu U \\&+ \frac{1}{15} J_\mu U\, U_{;\mu } +\frac{1}{5} U^2 (G_{\mu \nu })^2 +\frac{4}{15} U(G_{\mu \nu })^2U+\frac{2}{15} (U\, G_{\mu \nu })^2 \\&+\frac{1}{15} G_{\mu \nu } U^2 G_{\mu \nu } +\frac{2}{15} (G_{\mu \nu } U)^2 + \frac{1}{5} U_{;\mu } U_{;\nu } G_{\mu \nu } + \frac{1}{5} U_{;\mu } G_{\mu \nu } U_{;\nu }\\ {}&+\frac{1}{5} (G_{\mu \nu })^2 U^2. \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\end{aligned}$$

(A.28)

$\underline{{\mathcal {O}}\left( D^4 U^3\right) }$

We repeat the same task, one more time. We focus on the computation of the relevant derivatives of HKCs to calculate the operators ${\mathcal {O}}\left( D^4 U^3\right)$.

$$\begin{aligned} D_{\mu \mu } b_2|_{z=0} \left[\!\left[ U^2 \right]\!\right]&= \frac{1}{4} \left\{ 2\,D^2\left( U+D^2\right) b_1 - T_{\mu \mu } b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{1}{2} \left\{ U_{;\mu \mu } b_1 + 2\,U_{;\mu } D_\mu b_1 + U\,D^2 b_1\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{1}{2} \left\{ U_{;\mu \mu } U + U_{;\mu } U_{;\mu } + \frac{1}{3}U\,U_{;\mu \mu }\right\} . \end{aligned}$$

(A.29)

$$\begin{aligned} D_{\mu \nu } b_2|_{z=0} \left[\!\left[ U^2 \right]\!\right]&= \frac{1}{4} \left\{ 2\,D_{\mu \nu }\left( U+D^2\right) b_1 - T_{\mu \nu } b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{1}{4} \left\{ 2\,U_{;\nu \mu } b_1 + 2\,U_{;\mu } D_\nu b_1 + 2\,U_{;\nu } D_\mu b_1 +2\, U\,D_{\mu \nu } b_1 - G_{\nu \mu }b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{1}{4} \left\{ 2\,U_{;\nu \mu } U + U_{;\mu } U_{;\nu } + U_{;\nu } U_{;\mu } + \frac{2}{3} U\,U_{;\nu \mu } + \frac{1}{3} U^2\,G_{\mu \nu }+ \frac{2}{3} U\,G_{\mu \nu }U + G_{\mu \nu }U^2\right\} . \end{aligned}$$

(A.30)

$$\begin{aligned} D_{\mu \nu \nu } b_2|_{z=0} \left[\!\left[ U^2 \right]\!\right]&= \frac{1}{5} \left\{ 2\,D_{\mu \nu \nu }\left( U+D^2\right) b_1 - T_{\mu \nu \nu } b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{2}{5} \left\{ D_{\mu }\left( U_{;\nu \nu } b_1 + 2\,U_{;\nu }D_\nu b_1 + U\,D^2 b_1\right) - \frac{1}{2}D_\nu G_{\nu \mu } b_2\right. \\&\left. \quad - \frac{1}{2} G_{\nu \mu } D_\nu b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{2}{5} \left\{ U_{;\nu \nu \mu } b_1 + U_{;\nu \nu } D_\mu b_1 + 2\,U_{;\nu \mu }D_\nu b_1 + 2\,U_{;\nu }D_{\mu \nu } b_1 + U_{;\mu }\,D^2 b_1 \right. \\&\left. \quad + U\,D_{\mu \nu \nu } b_1 - \frac{1}{2} G_{\nu \mu ;\nu } b_2 - G_{\nu \mu }D_\nu b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{2}{5} \left\{ U_{;\nu \nu \mu } U + \frac{1}{2} U_{;\nu \nu } U_{;\mu } + U_{;\nu \mu }U_{;\nu } + \frac{2}{3}\,U_{;\nu }U_{;\nu \mu } + \frac{1}{3}\,U_{;\nu }U\,G_{\mu \nu }\right. \\&\quad + \frac{2}{3}\,U_{;\nu }G_{\mu \nu }U + \frac{1}{3}U_{;\mu }U_{;\nu \nu } + \frac{1}{4}U\,U_{;\nu \nu \mu } + \frac{1}{4}U\,U_{;\nu }G_{\mu \nu } - \frac{1}{12}U^2\,J_{\mu }\\&\left. \quad - \frac{1}{4}U\,J_{\mu }U + \frac{1}{4}U\,G_{\mu \nu }U_{;\nu } -\frac{1}{2} J_\mu U^2 + \frac{2}{3} G_{\mu \nu }U_{;\nu }U + \frac{1}{3} G_{\mu \nu }U\,U_{;\nu } \right\} . \end{aligned}$$

(A.31)

$$\begin{aligned} D_{\nu \nu \mu } b_2|_{z=0} \left[\!\left[ U^2 \right]\!\right]&= \frac{1}{5} \left\{ 2\,D_{\nu \nu \mu }\left( U+D^2\right) b_1 - T_{\nu \nu \mu } b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{2}{5} \left\{ D_{\nu \nu }(U_{\mu }b_1 + U\,D_\mu b_1) - D_\nu G_{\mu \nu } b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{2}{5} \left\{ U_{;\mu \nu \nu }b_1 + U_{\mu } D^2 b_1 + 2\,U_{\mu \nu } D_\nu b_1 + U_{\nu \nu }\,D_\mu b_1\right. \\&\left. \quad + U\,D_{\nu \nu \mu } b_1 +2\, U_{;\nu }\,D_{\nu \mu } b_1 + G_{\nu \mu ;\nu } b_2 + G_{\nu \mu } D_\nu b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{2}{5} \left\{ U_{;\mu \nu \nu } U + \frac{1}{3}U_{\mu } U_{;\nu \nu } + U_{\mu \nu } U_{;\nu } + \frac{1}{2}U_{\nu \nu } U_{;\mu }+ \frac{1}{4}U\,U_{;\mu \nu \nu } + J_\mu U^2\right. \\&\quad +\frac{1}{4} U\,U_{;\nu }G_{\nu \mu } + \frac{1}{6} U^2 J_\mu + \frac{1}{2}U\,J_\mu U +\frac{1}{4} U\,G_{\nu \mu }U_{;\nu } +\frac{2}{3}\, U_{;\nu }\,U_{;\mu \nu }\\&\left. \quad +\frac{1}{3}\, U_{;\nu }U\,G_{\nu \mu } +\frac{2}{3}\, U_{;\nu }G_{\nu \mu }U + \frac{2}{3} G_{\nu \mu } U_{;\nu }U + \frac{1}{3} G_{\nu \mu }U\, U_{;\nu }\right\} . \end{aligned}$$

(A.32)

$$\begin{aligned} D^4\,b_2|_{z=0} \left[\!\left[ U^2 \right]\!\right]&= \frac{1}{6} \left\{ 2\,D_{\mu \mu \nu \nu }\left( U+D^2\right) b_1 - T_{\mu \mu \nu \nu } b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{1}{3} \left\{ U_{;\mu \mu \nu \nu } b_1 + 2\, U_{;\mu \nu \nu } D_\mu b_1 + 2\, U_{;\nu \nu \mu } D_\mu b_1 + 2\,U_{;\mu } D_{\mu \nu \nu } b_1 + 2\,U_{;\mu } D_{\nu \nu \mu } b_1\right. \\&\left. \quad + 4\,U_{;\nu \mu } D_{\mu \nu } b_1 + 2\, U_{;\mu \mu } D^2 b_1 + U\,D^4 b_1 + J_\mu D_\mu b_2 + (G_{\mu \nu })^2 b_2\right\} |_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{1}{3} \left\{ U_{;\mu \mu \nu \nu } U + \frac{1}{5} U\,U_{;\mu \mu \nu \nu } + \frac{2}{3}\,U_{;\mu \mu }U_{;\nu \nu } + U_{;\nu \nu \mu } U_{;\mu } + U_{;\mu \nu \nu } U_{;\mu } \right. \\&\quad + \frac{2}{5} U\,U_{;\mu }J_\mu -\frac{4}{15}U\,(G_{\mu \nu })^2 U + (G_{\mu \nu })^2 U^2 +\frac{1}{5}U\,J_\mu U_{;\mu } - \frac{2}{15}(U\,G_{\mu \nu })^2 \\&\quad - \frac{1}{10}U^2 (G_{\mu \nu })^2 + \frac{1}{2} U_{;\mu } U_{;\nu \nu \mu } +\frac{1}{2} U_{;\mu }J_\mu U +\frac{1}{6} U_{;\mu }U\,J_\mu + \frac{1}{2}U_{;\mu }U_{;\mu \nu \nu }\\&\left. \quad +\frac{4}{3}(U_{;\mu \nu })^2 +\frac{1}{3}G_{\mu \nu }U^2\,G_{\mu \nu } + \frac{2}{3}(G_{;\mu \nu }U)^2 +\frac{2}{3} J_\mu U_{;\mu } U +\frac{1}{3} J_\mu U U_{;\mu } \right\} . \end{aligned}$$

(A.33)

$$\begin{aligned}{}[b_3] \left[\!\left[ U^2,U^3 \right]\!\right]&= \left\{ U+D^2\right\} b_2|_{z=0} \left[\!\left[ U^2,U^3 \right]\!\right] ,\\&= U^3 + \frac{1}{2} U\,U_{;\mu \mu } + \frac{1}{2} U_{;\mu \mu } U + \frac{1}{2} (U_{;\mu })^2.\\ \end{aligned}$$

(A.34)

$$\begin{aligned} D_\mu b_3|_{z=0} \left[\!\left[ U^2,U^3 \right]\!\right]&= \frac{1}{4}\left\{ 3\,D_\mu \left( U+D^2\right) b_2-T_\mu b_3\right\} _{z=0} \left[\!\left[ U^2,U^3 \right]\!\right] ,\\&= \frac{3}{4} \left\{ U_{;\mu }b_2 + U\,D_\mu b_2 + D_{\mu \nu \nu } b_2\right\} _{z=0} \left[\!\left[ U^2,U^3 \right]\!\right] ,\\&= \frac{3}{4} \left\{ U_{;\mu } U^2 + \frac{2}{3} U\,U_{;\mu }U + \frac{1}{3} U^2U_{;\mu } -\frac{1}{5} U^2J_\mu - \frac{4}{15} UJ_\mu U\right. \\&\quad + \frac{4}{15} U\,U_{;\nu \nu \mu } + \frac{4}{15}U\,U_{;\nu }G_{\mu \nu } + \frac{4}{15} U\,G_{\mu \nu } U_{;\nu } + \frac{7}{15} U_{;\mu } U_{;\nu \nu }\\&\quad + \frac{2}{5} U_{;\nu \nu \mu } U + \frac{1}{5} U_{;\nu \nu } U_{;\mu } + \frac{2}{5} U_{;\nu \mu }U_{;\nu } + \frac{4}{15}\,U_{;\nu }U_{;\nu \mu } -\frac{1}{5} J_\mu U^2\\&\left. \quad + \frac{2}{15}\,U_{;\nu }U\,G_{\mu \nu } + \frac{4}{15}\,U_{;\nu }G_{\mu \nu }U + \frac{4}{15} G_{\mu \nu }U_{;\nu }U + \frac{2}{15} G_{\mu \nu }U\,U_{;\nu } \right\} . \end{aligned}$$

(A.35)

$$\begin{aligned} D_{\mu \mu } b_3|_{z=0} \left[\!\left[ U^2 \right]\!\right]&= \frac{1}{5}\left\{ 3\,D_{\mu \mu }\left( U+D^2\right) b_2 -T_{\mu \mu } b_3\right\} _{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{3}{5} \{U_{;\mu \mu } b_2 + 2\,U_{;\mu } D_\mu b_2 + U\,D^2 b_2 + D_{\mu \mu \nu \nu } b_2\}_{z=0} \left[\!\left[ U^2 \right]\!\right] ,\\&= \frac{1}{5} \left\{ \frac{1}{10}\,U\,U_{;\mu } J_\mu + \frac{1}{2}U\,U_{;\mu \mu \nu \nu } + \frac{1}{6}\,(U\,G_{\mu \nu })^2 +\frac{1}{3}G_{\mu \nu }U^2\,G_{\mu \nu }\right. \\&\quad + \frac{1}{2}U\,J_\mu U_{;\mu } + \frac{3}{2}U_{;\mu }U_{;\nu \nu \mu } + U_{;\mu }G_{\mu \nu } U_{;\nu } + U_{;\mu } U_{;\nu } G_{\mu \nu } +\frac{4}{3}(U_{;\mu \nu })^2 \\&\quad + U_{;\mu \mu \nu \nu } U + \frac{2}{3}\,U_{;\mu \mu }U_{;\nu \nu } + U_{;\nu \nu \mu } U_{;\mu } + U_{;\mu \nu \nu } U_{;\mu } + \frac{1}{3}\, U(G_{\mu \nu })^2 U \\&\quad + (G_{\mu \nu })^2 U^2 -\frac{1}{2} U_{;\mu }J_\mu U -\frac{5}{6} U_{;\mu }U\,J_\mu + \frac{1}{2}U_{;\mu }U_{;\mu \nu \nu } +\frac{1}{3} J_\mu U U_\mu \\&\left. \quad + (U_{;\mu \mu })^2 + \frac{2}{3}(G_{;\mu \nu }U)^2 +\frac{2}{3} J_\mu U_\mu U \right\} . \end{aligned}$$

(A.36)

Finally, we combine all the above-computed expressions and put them in Eq. (A.4) to find the operators of the class ${\mathcal {O}}\left( D^4 U^3\right)$. Note that, at this stage, all the evaluated operator structures are not independent. We employ the trace properties and a few identities⁶ that simplify the HKCs and allow us to write them in terms of independent operators. Finally, we find the independent operators of the form ${\mathcal {O}}\left( D^4 U^3\right)$ as

$$\begin{aligned} \begin{aligned}{}[b_5] \left[\!\left[ U^3 \right]\!\right] = {\mathcal {O}}\left( D^4 U^3\right) =&\ U^3 (G_{\mu \nu })^2 + \frac{2}{3} U^2 G_{\mu \nu } U\,G_{\mu \nu } + \frac{1}{3} U^2 J_\mu U_{;\mu } + \frac{1}{3} U\,G_{\mu \nu }U_{;\mu }U_{;\nu } \\&+ \frac{1}{3} U\,U_{;\mu }U_{;\nu }\,G_{\mu \nu } - \frac{1}{3} U^2 U_{;\mu }J_\mu + U\,U_{;\mu \mu }U_{;\nu \nu } + \frac{2}{3} U_{;\mu \mu } (U_{;\nu })^2. \end{aligned} \end{aligned}$$

(A.37)

Appendix B: Review of covariant diagram method

In this section, we present a brief review of the development of the covariant diagram representation starting from the original gauge-covariant functional form of the effective action. The topic has been greatly discussed in Refs. [72, 73]. As shown in Eq. (1.1), the one-loop part of the effective action for a field can be given as follows:

$$\begin{aligned} \Delta S_{\text {eff}} = ic_s\, \text {Tr} \log \left( -P^2+M^2+U\right) . \end{aligned}$$

(B.1)

here, $c_s = +1/2$, or $+1$ depending on whether the heavy field is a real scalar or complex scalar. The trace “Tr” can then be evaluated by taking an integral over the momentum eigenstate basis,

$$\begin{aligned} \int d^dx\,{\mathcal {L}}_{\text {eff}}[\phi ] & = ic_s\,\int \,\frac{d^d q}{(2\pi )^d}\langle {q}\vert \text {tr} \log \left( -{\widehat{P}}^2+M^2+U\right) \vert {q} \rangle \nonumber \\&= ic_s\int d^dx\int \frac{d^d q}{(2\pi )^d}{\langle q\vert x\rangle }\langle {x}\vert \text {tr} \log \left( -{\widehat{P}}^2+M^2+U\right) \vert {q} \rangle \nonumber \\&= ic_s\int d^dx\int \frac{d^d q}{(2\pi )^d}\,e^{i\,q.x}\,\text {tr} \log \left( -P^2+M^2+U\right) \,e^{-i\,q.x}. \end{aligned}$$

(B.2)

By following a straightforward manipulation of introducing the completeness relation for the basis of the spatial eigenstates, we find the following form of the effective one-loop Lagrangian,

$$\begin{aligned} {\mathcal {L}}_{\text {eff}}[\phi ]{} & {} = ic_s\int \frac{d^d q}{(2\pi )^d}\,\text {tr} \log \left( -P^2+M^2+U\right) _{P\rightarrow P-q}\nonumber \\{} & {} = ic_s\int \frac{d^d q}{(2\pi )^d}\text {tr} \log \left( -P^2-q^2+2q.P+M^2+U\right) \nonumber \\{} & {} = ic_s\int \frac{d^d q}{(2\pi )^d}\text {tr} \bigg \{\log \left( -q^2+M^2\right) +\log \big [1-\left( q^2-M^2\right) ^{-1}\left( -P^2+2q.P+U\right) \big ]\bigg \}. \end{aligned}$$

(B.3)

After performing the momentum integral, the first term in Eq. (B.3) reduces to a constant, while the second term can be expanded in an infinite series as shown in Eq. (6.1).

1.1 B.1 Covariant loop diagrams, their structures and values

In this subsection, we present all the diagrams that can contribute to dimension eight interactions at each order of $P^{2n}U^{m}$ with possible contractions among $P_{\mu }$’s and note down the corresponding operator structures containing open covariant derivatives ($P_{\mu }$’s) and value for the loops.

1.1.1 $\mathbf {O\left( P^4\,U^3\right) }$

See Table 2.

Table 2 All possible diagrams at the level of $O\left( P^4U^3\right)$. In the second column, we present their corresponding operator structures with open covariant derivatives. Their values are given in the third column

1.1.2 $\mathbf {O(P^8)}$

See Table 3.

Table 3 All possible diagrams at the level of $O(P^8)$ containing eight $P_{\mu }$’s that are contracted among themselves. In the second column, we present their corresponding operator structures with open covariant derivatives. Their values are given in the third column

1.1.3 $\mathbf {O\left( P^6\,U^2\right) }$

See Table 4.

Table 4 All possible diagrams at the level of $O(P^6U^2)$ containing six $P_{\mu }$’s that are contracted among themselves and two U’s. In the second column, we present their corresponding operator structures with open covariant derivatives. Their values are given in the third column

1.1.4 $\mathbf {O(P^6\,U)}$

See Table 5.

Table 5 All possible diagrams at the level of $O(P^6U)$. In the second column, we present their corresponding operator structures with open covariant derivatives. Their values are given in the third column

1.2 B.2 Master integrals for heavy loops

Each of the covariant diagrams mentioned in Sect. 6 corresponds to a loop integral with n heavy propagators and $2n_c$ contractions which can be generalised in the following form

$$\begin{aligned} \int \frac{d^d q}{(2\pi )^d}\,\frac{q^{\mu _1}\cdots q^{\mu _{2n_c}}}{(q^2-M^2)^{n}}\,\equiv \,g^{\mu _1\cdots \mu _{2n_c}} \,{\mathcal {I}}[q^{2n_c}]^{n}, \end{aligned}$$

(B.4)

We have used Package-X [109, 110] to compute the loop integrals. In Table 6, we have listed the results for the loop integrals discussed in Sect. 6.

Table 6 The values corresponding to the master integrals associated with the covariant diagrams