# One-loop Effective Action up to Dimension Eight: Integrating out Heavy
Scalar(s)

Upalaparna Banerjee, Joydeep Chakrabortty, Shakeel Ur Rahaman, Kaanapuli
Ramkumar<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the complete 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 effects of only
heavy scalar propagators in the loops. We also verify part of the results
using the covariant diagram technique.

††preprint: ††institutetext: Indian Institute of Technology Kanpur, Kalyanpur,
Kanpur 208016, Uttar Pradesh, India

##  1 Introduction

The precision era in particle physics began to dawn following the discovery of
the Higgs boson. Since then, various experiments indicate that the Standard
Model (SM) and the new physics are separated on the energy scale. In this
situation, the Effective Field Theory (EFT) Weinberg:1980wa ; Georgi:1994qn ;
Manohar:2018aog ; Cohen:2019wxr  emerges as the practical and economical
working framework. Instead of dealing with all of the numerous feasible
options for beyond Standard Model (BSM) scenarios, we can put them on the same
footing by treating the SM as a low-energy effective theory. The Standard
Model effective field theory (SMEFT) Brivio:2017vri ; Isidori:2023pyp  works
as a bridge between the unknown territory of new physics and the SM.

EFT is primarily divided into two categories. The model-independent approach,
also known as the bottom-up approach, involves augmenting the low-energy
Lagrangian with higher mass dimensional operators while respecting the
symmetry of the theory. Because of the redundancy caused by algebraic
relations such as Fierz identities and integration by parts (IBP) satisfied by
the operators, listing the higher dimensional operators becomes a challenging
endeavour. The operators that are free from these redundancies are known to
form Green’s set Gherardi:2020det ; Chala:2021cgt . When considering the
external states to be on-shell, the equation of motion or the freedom of field
redefinition Georgi:1991ch ; Barzinji:2018xvu ; Criado:2018sdb ;
Banerjee:2022thk  must also be taken into account, imposing more constraints
on the number of independent operators that form a basis. As we go higher and
higher dimension the number of operators proliferates making it impossible to
keep track of these redundancies. Several techniques and automated tools
Henning:2015alf ; Lehman:2015coa ; Henning:2017fpj ; Lehman:2015via ;
Fonseca:2019yya ; Fonseca:2017lem ; Gripaios:2018zrz ; Criado:2019ugp ;
Marinissen:2020jmb ; Banerjee:2020bym ; Harlander:2023psl ; Li:2021tsq ;
Li:2022tec  have been developed in order to construct higher dimensional
operators BUCHMULLER1986621 ; Grzadkowski:2010es ; Lehman:2014jma ;
Murphy:2020rsh ; Li:2020gnx ; Li:2020xlh ; Liao:2020jmn ; Anisha:2019nzx ;
Banerjee:2020jun . In this bottom-up approach, the deviation from the low
energy theory predictions is parametrised by the Wilson coefficients (WCs) of
these effective operators by treating them as free parameters Hartland:2019bjb
; Brivio:2019ius ; Brivio:2017btx ; Ellis:2020unq ; Bagnaschi:2022whn ;
Ellis:2018gqa . If some anomaly in the data is well explained by the effective
operators, we can find UV completion using the straightforward symmetry base
arguments, which aids in the model selection DasBakshi:2021xbl ;
Naskar:2022rpg ; Cepedello:2022pyx ; Guedes:2023azv ; Gargalionis:2020xvt ;
Li:2022abx ; Chakrabortty:2020mbc .

The alternate is the model dependent or the top-down approach. In this
framework, we begin with an action $S[\Phi,\phi]$ for a UV complete model
consisting of both heavy and light fields, which we denote as $\Phi$ and
$\phi$ respectively. The corresponding low-energy effective action is obtained
by expanding $\Phi$ around its classical minima ($\Phi_{c}$) and evaluating
the path integral over the dynamic fluctuations ($\eta$), so the heavy field
is written as, $\Phi=\Phi_{c}+\eta$. The effective action is obtained by,

$\displaystyle e^{i\,S_{\text{eff}}[\phi]}$ $\displaystyle=$
$\displaystyle\int[\mathcal{D}\Phi]e^{i\,S[\Phi,\phi]}=\int[\mathcal{D}\eta]\,\exp\left[i\,\left(S[\Phi_{c},\phi]\,+\,\frac{1}{2}\frac{\delta^{2}S}{\delta\Phi^{2}}\bigg{|}_{\Phi=\Phi_{c}}\,\eta^{2}\,+\,\mathcal{O}(\eta^{3})\right)\right],$
$\displaystyle S_{\text{eff}}$ $\displaystyle\approx$ $\displaystyle
S[\Phi_{c}]\,+\,\frac{i}{2}\text{Tr}\log\left(\frac{\delta^{2}S}{\delta\Phi^{2}}\bigg{|}_{\Phi=\Phi_{c}}\right)=S[\Phi_{c}]\,+\,\frac{i}{2}\text{Tr}\log\Big(D^{2}+M^{2}+U\Big{missing}),$
(1)

where, $D$ is the covariant derivative, $M$ is the mass corresponding to the
heavy field $\Phi$ and $U$ is the field-dependent term, functional of the
light fields $\phi$. The term $S[\Phi_{c}]$ in Eq. (1) arises when the heavy
fields are integrated out at the tree-level and can be simply obtained by
substituting the heavy field with its classical solution in the action. The
other term in the equation corresponds to the part of the effective action
generated at the one-loop level. This procedure of integrating out the heavy
field(s) is also known as the matching of the UV theory and the low energy
effective theory at an energy scale $M$. In this approach, the WCs
accompanying the effective operators are functions of the UV parameters. This
framework provides the means to bring together a variety of UV models under
the same umbrella and conduct a comparative analysis Anisha:2020ggj ;
Anisha:2021hgc ; Bakshi:2020eyg .

As one can infer from Eq. (1), the structure of the UV action is quite general
and it does not depend on the specific choice of the UV theory. So computing
the effective operators in terms of $D$ and $U$ with proper factors of $1/M$
should provide a universal formula for integrating out Henning:2014wua . This
is indeed the case and computation of the one-loop term in Eq. (1), has
resulted in a master formula known as the universal one-loop effective action
(UOLEA) Drozd:2015rsp ; Ellis:2016enq ; delAguila:2016zcb ; Ellis:2017jns ;
Kramer:2019fwz ; Angelescu:2020yzf ; Ellis:2020ivx . In addition, some
automated tools have been devised to aid this process Bakshi:2018ics ;
Fuentes-Martin:2022jrf ; Carmona:2021xtq ; Cohen:2020qvb ; Fuentes-
Martin:2020udw ; Criado:2017khh . The UOLEA described in the aforementioned
literature has been limited to computing operators up to mass dimension six
(D6). In spite of the fact that it can be calculated up to any mass dimension,
the process becomes very complicated as we go for the higher dimension.

There are multiple theoretical ways to integrate out the heavy fields to
compute the low-energy effective Lagrangian: using functional methods
Gaillard:1985uh ; Cheyette:1987qz ; Chan1985 ; Henning:2014wua ;
Henning:2016lyp ; Dittmaier:2021fls ; Fuentes-Martin:2016uol ; Cohen:2020fcu ,
using Feynman diagrams, and covariant diagram techniques Vandeven:1985 ;
Zhang:2016pja . Recently, in Ref. Gersdorff:2023  the Heat-Kernel (HK) method
is used to find covariant Feynman rules in the context of EFT. The HK method
Minakshisundaram:1949xg ; Minakshisundaram:1953xh ; Hadamard2003 ;
DeWitt:1964mxt ; Seeley:1969 ; Schwinger:1951nm ; Vassilevich:2003xt ;
Avramidi:2001ns ; Avramidi:2015ch5 ; Kirsten:2001wz ; Fulling:1989nb  has been
instrumental to compute effective actions in quantum field theory and quantum
gravity Avramidi:1994fx . Using its intrinsic properties and background field
method, the higher-order corrections in different space-time dimensions are
computed in Avramidi:1991 ; Belkov:1996CA ; Fliegner:1993wh ; Fliegner:1997rk
.

As we commence to make progressively precise measurements of low energy
observable, the dimension six terms will no longer be enough to explain the
data, and much emphasis has been directed to mass dimension eight (D8)
operators to keep up with the experimental precision Dawson:2021xei ;
Hays:2018zze ; Corbett:2021eux ; DasBakshi:2023htx ; DasBakshi:2022mwk ;
Chala:2021pll ; Alioli:2020kez ; Degrande:2013kka ; Hays:2020scx ;
Dawson:2023ebe ; Dawson:2022cmu ; Ellis:2019zex ; Corbett:2023qtg ;
Ellis:2023zim ; Degrande:2023iob ; Banerjee:2023bzl . Additionally, at the
cross-section level, the interference term between the dimension eight and the
renormalisable terms may become equivalent with the dimension six squared
terms. It also provides leading order contributions to some physical processes
such as light-by-light scattering that involves the neutral quartic gauge
couplings. In the study of electroweak precision observable (EWPO), the
$\widehat{U}$-parameter receives its first contribution at dimension eight
Grinstein:1991cd .

The primary goal of our paper is to extend the UOLEA to generate terms up to
mass dimension eight. In this regard, we would also wish to advocate the
advantages of the HK method. The application of the Heat-Kernel method in this
context is relatively new. We provide an extensive overview of the computation
of the effective action and limit ourselves to the one-loop diagrams that
involve either one heavy scalar or multiple degenerate heavy scalars. We also
use the conventional covariant diagram approach to validate some of our
findings. We want to emphasise that the HK prescription is not just limited to
heavy scalar loops and can be applied to mixed loops regardless of the spin or
mass (heavy or light) of the particles involved. We are aware that in order to
complete the UOLEA, one needs also to take into consideration these
contributions, so we have set this aside for future study. Heat-Kernel has the
capacity to expand beyond one-loop, making it more applicable to the study of
precision physics.

The paper is organised as follows. To start with we briefly review the
relevant working principle of the Heat-Kernel method in Sec. 2. Then, in the
following Sec. 3, we discuss the computation of HK coefficients with one
detailed example. In Sec. 4, we define the connection between the one-loop
effective action with the HK coefficients. In the following Sec. 5, we discuss
all possible independent structures of operators having up to mass dimension
eight, accompanied by suitable numerical factors, that appear after
integrating out heavy scalars from one-loop diagrams. In the next Sec. 6, we
employ a new method to compute the effective operators, starting from the same
UV action, based on the covariant diagram technique. This allows us to
independently cross-check some of our results computed using the HK method. In
Sec. 7, we write down the universal one-loop effective Lagrangian after
collecting all the contributions computed in the earlier section and adding
them up suitably. We also discuss a toy model to highlight the failure of
naive power counting and the importance of different terms in our effective
action. Then, we conclude in Sec. 8.

##  2 Heat Kernel: a brief review

The one-loop effective action in Eq. (1) can be written as a spectral function
of a second-order elliptic differential operator after Wick’s rotation. This
allows us to study the one-loop effective action using spectral analysis
methods, one such example is the computation of the Heat-Kernel coefficients
(HKC).

Any second-order elliptic partial differential operators can be written in
terms of a covariant derivative and a scalar function as Avramidi:2015ch5 ,

$\Delta=D^{2}+U+M^{2},$ (2)

where $D_{\mu}\equiv\partial_{\mu}-iA_{\mu}$ is the covariant derivative with
$A_{\mu}$ being the connection, $U$ is a space dependent scalar function, and
$M$ is a space independent scalar function. Later, $M$ will be mapped to the
mass term of the heavy field(s) in the process of one-loop effective action
computation using HKCs.

Let us consider $\lambda_{n}$ as the eigenvalues of the operator $\Delta$
corresponding to eigenvectors $\phi_{n}$. If $\Delta$ is a self-adjoint
operator, the Heat-Kernel can be written as Kirsten:2001wz ;
Vassilevich:2003xt

$K(t,x,y,\Delta)=\bra{y}e^{-t\Delta}\ket{x}=\sum_{n}e^{-t\Delta}\phi_{n}(x)\phi^{\dagger}_{n}(y).$
(3)

This HK satisfies the heat equation Avramidi:2015ch5 ; Vassilevich:2003xt

$\left(\partial_{t}+\Delta_{x}\right)K(t,x,y,\Delta)=0,$ (4)

along with the initial condition

$K(0,x,y,\Delta)=\delta(x-y).$ (5)

In passing we would like to mention that $t>0$ is a parameter, not to be
confused with time, while defining the heat equation. In the absence of any
interaction, i.e., for the free operator,
$\Delta_{0}=\partial_{\mu}\partial^{\mu}+M^{2}$, the HK can be written as
Avramidi:2015ch5 ; Fulling:1989nb

$K_{0}(t,x,y)=(4\pi t)^{-d/2}\
\mathrm{Exp}\left[\frac{z^{2}}{4t}-t\,M^{2}\right],$ (6)

where $z_{\mu}=(x-y)_{\mu}$ and $d$ is the dimension we are working in. From
here on, we assume four-dimensional Euclidean space, i.e., $d=4$.

For any general Laplace type operator with a potential term within a gauge
theory, see the operator in Eq. (2), the HK can be written in terms of the
free operator HK and an interaction part ($H$) as DeWitt:1964mxt ;
Avramidi:2015ch5

$K(t,x,y,\Delta)=K_{0}(t,x,y)\ H(t,x,y,\Delta).$ (7)

In case of this potential term is bounded from below, the operator $\Delta$ is
positive, and $H(t,x,y,\Delta)$ converges as $t\rightarrow\infty$. This allows
one to write the interaction part in terms of a power law expansion in $t$ as
DeWitt:1964mxt ; Seeley:1969

$H(t,x,y,\Delta)=\sum_{k}\frac{(-t)^{k}}{k\,!}b_{k}(x,y),$ (8)

where $b_{k}$ are the Heat-Kernel coefficients 111Also known as the Hadamard -
Minackshisundaram - De Witt - Seeley (HMDS) coefficients Hadamard2003 ;
Minakshisundaram:1953xh ; DeWitt:1964mxt ; Seeley:1969 . (HKCs). These $b_{k}$
are polynomials of covariant derivative ($D_{\mu}$) and the scalar operator
$U$. This method has no restrictions on the dimensions of the covariant
derivative and scalar function in the elliptic operator and can be generalised
to matrix-valued $U$ and $D_{\mu}$ Fulling:1989nb .

##  3 Computation of Heat-Kernel Coefficients: general method

Blending the ansatz depicted in Eq. (7) in heat equation, Eq. (4), the
interaction term satisfies Avramidi:2015ch5

$\left(\partial_{t}+\frac{1}{t}z_{\mu}D^{\mu}+D^{2}+U\right)H(t,x,y,\Delta)=0.$
(9)

Then, further invoking the power law expansion in $t$, see Eq. (9), we obtain
a recursion relation for the HKCs Belkov:1996CA

$\left(k+z\cdot D\right)b_{k}=k\left(U+D^{2}\right)b_{k-1}.$ (10)

Here, $b_{k}$s for $k<0$ are considered to be 0. The combination of the free
heat kernel part Eq. (6), and the initial condition Eq. (5) leads to
$b_{0}(x,x)=I$, where $I$ is the identity matrix Avramidi:2015ch5 ;
Fulling:1989nb . This allows us to start the recursion relation with $k\geq
0$.

In a later section, Sec. 4, we will see that the trace over the HKC at the
coinciding limit, i.e., $\tr\ [b_{k}(x,x)]$, are related to the operators of
the one-loop effective action where the scalar functions $U$ and $M^{2}$ will
be mapped in to functional of light-fields and the mass term of the heavy
scalar field. Thus, our primary focus is to compute the HKCs at the coincident
limit $[b_{k}(x,x)]$. To do so, we use the following relations Belkov:1996CA ,

$\displaystyle
D_{\mu_{1}}D_{\mu_{2}}...D_{\mu_{m}}b_{k}\Big{|}_{z=0}=\frac{1}{m+k}\Big{\\{}k\,D_{\mu_{1}}D_{\mu_{2}}...D_{\mu_{m}}\
(U+D^{2})\ b_{k-1}-T_{\mu_{1}\mu_{2}...\mu_{m}}b_{k}\Big{\\}}\Big{|}_{z=0},$
(11) $\displaystyle\text{where,
}T_{\mu_{1}\mu_{2}...\mu_{m}}=\\{D_{\mu_{1}}D_{\mu_{2}}...D_{\mu_{m}}(z\cdot
D)\\}\big{|}_{z=0}-m\,D_{\mu_{1}}D_{\mu_{2}}...D_{\mu_{m}},$ (12)
$\displaystyle\hskip
14.22636pt\Rightarrow\;T_{\mu_{1}\mu_{2}...\mu_{m}}=D_{\mu_{1}}T_{\mu_{2}...\mu_{m}}+R_{\mu_{2}...\mu_{m},\mu_{1}},$
(13) $\displaystyle\text{in the above equation,
}R_{\mu_{2}...\mu_{m},\mu_{1}}=[D_{\mu_{2}}...D_{\mu_{m}},D_{\mu_{1}}],$ (14)
$\displaystyle\hskip
85.35826pt\Rightarrow\;R_{\mu_{2}\mu_{3}...\mu_{m},\mu_{1}}=G_{\mu_{2}\mu_{1}}D_{\mu_{3}}...D_{\mu_{m}}+D_{\mu_{2}}D_{\mu_{3}...\mu_{m},\mu_{1}},$
(15)

where $G_{\mu\nu}=[D_{\mu},D_{\nu}]$ is the stress tensor.

While solving the HKCs, the recursion relation Eq. (10) leads to terms
involving derivatives of the HKCs, e.g.,
$D_{\mu}D_{\nu}...b_{k}(x,y)|_{z=0}$222It is important to note that for HKC
computation at coinciding point, one must not set $z=0$ in Eq. (10) as
$\\{D_{\mu}(z\cdot D)\,b_{k}(x,y)\\}|_{z=0}=D_{\mu}\,b_{k}(x,y)|_{z=0}\neq
0$.. To compute of one-loop effective Lagrangian systematically, we organise
the operator classes of the form $\mathcal{O}(D^{r}U^{s})$ and arrange them in
a polynomial with different non-negative integer powers of the covariant
derivative $(D)$ and the field-dependent term $(U)$.

Operators of the form $\mathcal{O}(D^{r}U^{s})$ appear in the HKC $b_{n}(x,y)$
where $n=r/2+s$. It is possible to extract the relevant part of the HKC
$b_{n}$ for the operators of a specific class using Eqs. (11)-(15) as

$\mathcal{O}(D^{r}U^{s})\equiv[b_{r/2+s}]\llbracket
U^{s}\rrbracket=\sum_{k=0}^{n=\frac{r}{2}+s}\frac{n!\,(n-1)!}{k!\,(2n-k)!}\Big{\\{}k\
D^{2(n-k)}\\{U\,b_{k-1}\llbracket
U^{s-1}\rrbracket\\}-T_{2(n-k)}\,b_{k}\llbracket
U^{s}\rrbracket\Big{\\}}_{z=0}.$ (16)

In the next sections, we will further use the following notations to express
our results in a compact form

$\displaystyle b_{k}\llbracket U^{s}\rrbracket\equiv\text{terms of order}\
U^{s}\ \text{in the HKC}\ b_{k};\qquad[b_{k}]=b_{k}(x,x);\qquad
G_{\mu\rho;\mu}\equiv J_{\rho};$ $\displaystyle T_{2(k)}\equiv
T_{\mu_{1}\mu_{1}...\mu_{k}\mu_{k}};\qquad D_{\mu_{1}...\mu_{n}}\equiv
D_{\mu_{1}}...D_{\mu_{n}};\qquad D^{2(k)}\equiv
D_{\mu_{1}\mu_{1}...\mu_{k}\mu_{k}};$ $\displaystyle
D_{\mu_{1}...\mu_{n}}(U)\equiv U_{;\mu_{n}...\mu_{1}},\qquad
D_{\mu_{1}...\mu_{n}}(G_{\rho\sigma})\equiv
G_{\rho\sigma;\mu_{n}...\mu_{1}};\qquad D_{\mu_{1}...\mu_{n}}(J_{\rho})\equiv
J_{\rho;\mu_{n}...\mu_{1}}.$

Here, for the sake of the readers, we systematically chalk out the necessary
instructions to calculate the operators of a particular class,
$\mathcal{O}(D^{r}U^{s})\subset$ HKC $b_{n}(x,y)$.

  * •

First, start with Eq. (16) and note down all the equations for
$\mathcal{O}(D^{r}U^{s-i})$ in a recursive way till $\mathcal{O}(D^{r}U^{0})$.
Each of these equations contains operators of the form $D^{2(k)}(U\,b_{j})$,
and $T_{2(k)}$ with $k\in[0,r/2]$.

  * •

Then, expand the operator $T_{2(k)}$, using Eqs. (13)-(15), recursively until
one is left with only field tensor $G_{\mu\nu}$ and the covariant derivatives
$(D)$.

  * •

Similarly expand the operator $D^{2(k)}(U\,b_{j})$ such that the derivative
acts either on $U$ or $b_{j}$.

  * $\ast$

Following these steps, one ends up with terms that contain either HKCs of
lower order or derivatives of HKCs.

  * •

It is advised to start with operators of $\mathcal{O}(D^{r}U^{0})$. Then,
employing the initial condition $[b_{0}]=I$, evaluate the necessary
derivatives of $b_{0}$ using Eq. (11) and substitute them in the relation
obtained for $\mathcal{O}(D^{r}U^{0})$.

  * •

Next work out another related class of operator $\mathcal{O}(D^{r}U^{i+1})$
and calculate the required derivatives of HKCs.

  * $\ast\;\ast$

Repeat these previously mentioned two steps until one reaches the desired
class of operators $\mathcal{O}(D^{r}U^{s})$. Then, use the trace properties
to note down the HKCs on a minimal basis.

Here, we want to explicitly mention a few subtleties while following the
above-mentioned state of the art of calculation:

  * •

The order in which the coincidence limit is used is important especially when
derivative operators are involved as

$D_{\mu_{1}...\mu_{n}}b_{k}|_{z=0}\neq D_{\mu_{1}...\mu_{n}}([b_{k}]).$

  * •

Usage of the trace properties in intermediate steps of the recursive procedure
is strictly forbidden.

  * •

As HKCs ($b_{k}$) and its $n$th derivative ($D^{n}b_{k}$) can be expressed in
terms of the lower order HKCs $b_{k-i}$ and lower order derivatives
$D^{(n-j)}b_{k}$, and $D^{(n-j)}b_{k-i}$. Thus, one may avoid reaching out to
$b_{0}$ every time while calculating the HKC or its derivatives, and instead,
it is advised to use results for the intermediate HKCs, if available already.

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

Based on the methodology discussed in the previous section, we perform, here,
an explicit calculation of the operators class $\mathcal{O}(D^{4}U^{3})$ for
the sake of detailed demonstration. To compute $\mathcal{O}(D^{4}U^{3})$, we
obtain the following necessary relations using Eq. (16).

$\displaystyle[b_{2}]\llbracket
U^{0}\rrbracket=\mathcal{O}(D^{4}U^{0})=-\frac{1}{12}\,\left\\{T_{(4)}b_{0}\right\\}_{z=0},$
(17) $\displaystyle[b_{3}]\llbracket
U^{1}\rrbracket=\mathcal{O}(D^{4}U^{1})=U\,[b_{2}]\llbracket
U^{0}\rrbracket+\left\\{\frac{1}{2}D^{2}\left(U\,b_{1}\right)+\frac{1}{10}\left(D^{4}\left(U\,b_{0}\right)-T_{(4)}b_{1}\right)\right\\}_{z=0}\llbracket
U^{1}\rrbracket,$ (18) $\displaystyle[b_{4}]\llbracket
U^{2}\rrbracket=\mathcal{O}(D^{4}U^{2})=U\,[b_{3}]\llbracket
U^{1}\rrbracket+\left\\{\frac{3}{5}D^{2}\left(U\,b_{2}\right)+\frac{1}{10}\left(2\,D^{4}\left(U\,b_{1}\right)-T_{(4)}b_{2}\right)\right\\}_{z=0}\llbracket
U^{2}\rrbracket,$ (19) $\displaystyle[b_{5}]\llbracket
U^{3}\rrbracket=\mathcal{O}(D^{4}U^{3})=U\,[b_{4}]\llbracket
U^{2}\rrbracket+\left\\{\frac{2}{3}D^{2}\left(U\,b_{3}\right)+\frac{2}{21}\left(3\,D^{4}\left(U\,b_{2}\right)-T_{(4)}b_{3}\right)\right\\}_{z=0}\llbracket
U^{3}\rrbracket.$ (20)

Here, we set

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

that are quite evident from Eq. (13). 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. (18)-(20), as follows

$D^{2}(U\,b_{k})=U_{;\mu\mu}\,b_{k}+2\,U_{;\mu}D_{\mu}b_{k}+U\,D^{2}b_{k},$
(21)
$\begin{split}D^{4}(U\,b_{k})=&U_{;\mu\mu\nu\nu}\,b_{k}+2\,U_{;\mu\nu\nu}D_{\mu}b_{k}+2\,U_{;\nu\nu\mu}D_{\mu}b_{k}+2\,U_{;\mu}D_{\mu\nu\nu}b_{k}\\\
&+2\,U_{;\mu}D_{\nu\nu\mu}b_{k}+4\,U_{;\nu\mu}D_{\mu\nu}b_{k}+2\,U_{;\mu\mu}D^{2}b_{k}+U\,D^{4}b_{k},\end{split}$
(22)

Action of $T_{(4)}$ can be addressed in the following form derived from Eqs.
(13)-(15)

$\begin{split}T_{(4)}=T_{\mu\mu\nu\nu}&=D_{\mu}T_{\mu\nu\nu}+R_{\mu\nu\nu,\mu}=D_{\mu\mu}T_{\nu\nu}+D_{\mu}R_{\nu\nu,\mu}+R_{\mu\nu\nu,\mu}\\\
&=D_{\mu}R_{\nu\nu,\mu}+D_{\mu}R_{\nu\nu,\mu}+G_{\mu\mu}D_{\nu\nu}=2\,D_{\mu\nu}G_{\nu\mu}+2\,D_{\mu}G_{\nu\mu}D_{\nu}\\\
&=G_{\mu\nu}G_{\nu\mu}-2\,G_{\mu\nu;\mu}D_{\nu}-2\,G_{\mu\nu}D_{\mu\nu}=-2\,(G_{\mu\nu})^{2}-2\,J_{\nu}D_{\nu}.\end{split}$
(23)

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

$X_{;\nu\mu}=X_{;\mu\nu}+G_{\mu\nu}X-X\,G_{\mu\nu},$ (24)

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

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

Now, we are ready to demonstrate the explicit computation of operators
$\mathcal{O}(D^{4}U^{3})$ belonging to the HKC $b_{5}$.

####  $\mathcal{O}(D^{4}U^{0})$

$T_{(4)}$ 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. (11) we find

$\begin{split}D_{\nu}b_{0}|_{z=0}=-T_{\nu}b_{0}|_{z=0}=0.\end{split}$ (25)

This provides $\mathcal{O}(D^{4}U^{0})$, directly from Eq. (17), as

$\begin{split}[b_{2}]\llbracket
U^{0}\rrbracket=\mathcal{O}(D^{4}U^{0})&=\frac{1}{6}\,\Big{\\{}(G_{\mu\nu})^{2}b_{0}+J_{\nu}D_{\nu}b_{0}\Big{\\}}_{z=0}=\frac{1}{6}\,(G_{\mu\nu})^{2}.\end{split}$
(26)

####  $\mathcal{O}(D^{4}U^{1})$

Next, to compute $\mathcal{O}(D^{4}U^{1})$ we calculate the necessary
derivatives of HKCs using Eqs. (11)-(15).

$\begin{split}D^{2}b_{0}\Big{|}_{z=0}=D_{\mu\mu}b_{0}\Big{|}_{z=0}&=-\frac{1}{2}T_{\mu\mu}b_{0}\Big{|}_{z=0}=0.\end{split}$
(27)

$D_{\mu\nu}b_{0}\Big{|}_{z=0}=-\frac{1}{2}T_{\mu\nu}b_{0}\Big{|}_{z=0}=-\frac{1}{2}\,\\{D_{\mu}T_{\nu}+R_{\nu,\mu}\\}b_{0}\Big{|}_{z=0}=\frac{1}{2}\,G_{\mu\nu}.$
(28)

$D_{\mu\nu\nu}b_{0}\Big{|}_{z=0}=-\frac{1}{3}T_{\mu\nu\nu}b_{0}\Big{|}_{z=0}=-\frac{1}{3}\,\\{D_{\mu}T_{\nu\nu}+R_{\nu\nu,\mu}\\}b_{0}\Big{|}_{z=0}=-\frac{1}{3}\,\\{D_{\nu}G_{\nu\mu}+G_{\nu\mu}D_{\nu}\\}b_{0}\Big{|}_{z=0},\\\
=-\frac{1}{3}\,\\{G_{\nu\mu;\nu}+2\,G_{\nu\mu}D_{\nu}\\}b_{0}\Big{|}_{z=0}-\frac{1}{3}\,J_{\mu}.\qquad\qquad\quad$
(29)

$D_{\nu\nu\mu}b_{0}\Big{|}_{z=0}=-\frac{1}{3}T_{\nu\nu\mu}b_{0}\Big{|}_{z=0}=-\frac{1}{3}\,\\{D_{\nu}T_{\nu\mu}+R_{\nu\mu,\nu}\\}b_{0}\Big{|}_{z=0}=-\frac{2}{3}\,D_{\nu}G_{\mu\nu}b_{0}\Big{|}_{z=0},\\\
=\frac{2}{3}\,\\{G_{\nu\mu;\nu}+\,G_{\nu\mu}D_{\nu}\\}b_{0}\Big{|}_{z=0}=\frac{2}{3}\,J_{\mu}.\qquad\qquad$
(30)

$D^{4}b_{0}\Big{|}_{z=0}=D_{\mu\mu\nu\nu}b_{0}\Big{|}_{z=0}=-\frac{1}{4}T_{(4)}b_{0}\Big{|}_{z=0}=\frac{1}{2}\,(G_{\mu\nu})^{2}.$
(31)

$[b_{1}]=\\{U+D^{2}\\}b_{0}\Big{|}_{z=0}=U.$ (32)

$\begin{split}D_{\mu}b_{1}\Big{|}_{z=0}=\frac{1}{2}\\{D_{\mu}(U+D^{2})b_{0}-T_{\mu}b_{1}\\}\Big{|}_{z=0}&=\frac{1}{2}\Big{\\{}U_{;\mu}b_{0}+U\,D_{\mu}b_{0}+D_{\mu\nu\nu}b_{0}\Big{\\}}_{z=0},\\\
&=\frac{1}{2}U_{;\mu}-\frac{1}{6}J_{\mu}.\end{split}$ (33)

$\begin{split}D_{\mu\mu}b_{1}\Big{|}_{z=0}\llbracket
U^{0}\rrbracket&=\frac{1}{3}\\{D_{\mu\mu}(U+D^{2})b_{0}-T_{\mu\mu}b_{1}\\}\Big{|}_{z=0}\llbracket
U^{0}\rrbracket=\frac{1}{3}D_{\mu\mu\nu\nu}b_{0}\Big{|}_{z=0}\llbracket
U^{0}\rrbracket=\frac{1}{6}(G_{\mu\nu})^{2}.\end{split}$ (34)

Assembling all the contributions, computed here, in Eq. (18) we find

$\begin{split}[b_{3}]\llbracket
U^{0}\rrbracket=\mathcal{O}(D^{4}U^{1})=&\frac{3}{10}U\,(G_{\mu\nu})^{2}+\frac{1}{5}(G_{\mu\nu})^{2}U-\frac{1}{10}U_{;\mu}J_{;\mu}+\frac{1}{10}J_{;\mu}U_{;\mu}\\\
&+\frac{1}{10}U_{;\mu\mu\nu\nu}+\frac{2}{10}U_{;\nu\mu}G_{\mu\nu}.\end{split}$
(35)

####  $\mathcal{O}(D^{4}U^{2})$

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

$\begin{split}D_{\mu\mu}b_{1}|_{z=0}\llbracket
U^{1}\rrbracket&=\frac{1}{3}\\{D^{2}(U+D^{2})b_{0}-T_{\mu\mu}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{3}\\{U_{;\mu\mu}b_{0}+2\,U_{;\mu}D_{\mu}b_{0}+U\,D^{2}b_{0}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\ &=\frac{1}{3}U_{;\mu\mu}.\end{split}$ (36)

$\begin{split}D_{\mu\nu}b_{1}|_{z=0}\llbracket
U^{1}\rrbracket&=\frac{1}{3}\\{D_{\mu\nu}(U+D^{2})b_{0}-T_{\mu\nu}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{3}\\{U_{;\nu\mu}b_{0}+U_{;\mu}D_{\nu}b_{0}+U_{;\nu}D_{\mu}b_{0}+U\,D_{\mu\nu}b_{0}-G_{\nu\mu}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{3}\\{U_{;\nu\mu}+\frac{1}{2}U\,G_{\mu\nu}+G_{\mu\nu}U\\},\\\
\end{split}$ (37)

$\begin{split}D_{\mu\nu\nu}b_{1}|_{z=0}\llbracket
U^{1}\rrbracket&=\frac{1}{4}\\{D_{\mu\nu\nu}(U+D^{2})b_{0}-T_{\mu\nu\nu}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{4}\\{D_{\mu}(U_{;\nu\nu}b_{0}+2\,U_{;\nu}D_{\nu}b_{0}+U\,D^{2}b_{0})-D_{\nu}G_{\nu\mu}b_{1}-G_{\nu\mu}D_{\nu}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{4}\\{U_{;\nu\nu\mu}b_{0}+U_{;\nu\nu}D_{\mu}b_{0}+2\,U_{;\nu\mu}D_{\nu}b_{0}+2\,U_{;\nu}D_{\mu\nu}b_{0}+U_{;\mu}\,D^{2}b_{0}\\\
&\quad+U\,D_{\mu\nu\nu}b_{0}-G_{\nu\mu;\nu}b_{1}-2\,G_{\nu\mu}D_{\nu}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{4}\\{U_{;\nu\nu\mu}+U_{;\nu}G_{\mu\nu}-\frac{1}{3}U\,J_{\mu}-J_{\mu}U+G_{\mu\nu}U_{;\nu}\\}.\end{split}$
(38)

$\begin{split}D_{\nu\nu\mu}b_{1}|_{z=0}\llbracket
U^{1}\rrbracket&=\frac{1}{4}\\{D_{\nu\nu\mu}(U+D^{2})b_{0}-T_{\nu\nu\mu}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{4}\\{D_{\nu\nu}(U_{\mu}b_{0}+U\,D_{\mu}b_{0})-2\,D_{\nu}G_{\mu\nu}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{4}\\{U_{;\mu\nu\nu}b_{0}+U_{\mu}D^{2}b_{0}+2\,U_{\mu\nu}D_{\nu}b_{0}+U_{\nu\nu}\,D_{\mu}b_{0}+U\,D_{\nu\nu\mu}b_{0}\\\
&\quad+2\,U_{;\nu}\,D_{\nu\mu}b_{0}+2\,G_{\nu\mu;\nu}b_{1}+2\,G_{\nu\mu}D_{\nu}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{4}\\{U_{;\mu\nu\nu}+\frac{2}{3}U\,J_{\mu}+U_{;\nu}G_{\nu\mu}+2\,J_{\mu}U+G_{\nu\mu}U_{;\nu}\\}.\end{split}$
(39)

$\begin{split}D^{4}\,b_{1}|_{z=0}\llbracket
U^{1}\rrbracket&=\frac{1}{5}\\{D_{\mu\mu\nu\nu}(U+D^{2})b_{0}-T_{\mu\mu\nu\nu}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{5}\\{U_{;\mu\mu\nu\nu}b_{0}+2\,U_{;\mu\nu\nu}D_{\mu}b_{0}+2\,U_{;\nu\nu\mu}D_{\mu}b_{0}+2\,U_{;\mu}D_{\mu\nu\nu}b_{0}+2\,U_{;\mu}D_{\nu\nu\mu}b_{0}\\\
&\quad+4\,U_{;\nu\mu}D_{\mu\nu}b_{0}+2\,U_{;\mu\mu}D^{2}b_{0}+U\,D^{4}b_{0}+2\,J_{\mu}D_{\mu}b_{1}+2\,(G_{\mu\nu})^{2}b_{1}\\}|_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{5}\\{U_{;\mu\mu\nu\nu}+\frac{2}{3}\,U_{;\mu}J_{\mu}+2\,U_{;\nu\mu}G_{\mu\nu}+\frac{1}{2}U\,(G_{\mu\nu})^{2}+J_{\mu}U_{;\mu}+2\,(G_{\mu\nu})^{2}U\\}.\end{split}$
(40)

$\begin{split}[b_{2}]\llbracket
U^{1},U^{2}\rrbracket&=\\{U+D^{2}\\}b_{1}|_{z=0}\llbracket
U^{1},U^{2}\rrbracket=U^{2}+\frac{1}{3}U_{;\mu\mu}.\end{split}$ (41)

$\begin{split}D_{\mu}b_{2}|_{z=0}\llbracket
U^{1},U^{2}\rrbracket&=\frac{1}{3}\\{2\,D_{\mu}(U+D^{2})b_{1}-T_{\mu}b_{2}\\}_{z=0}\llbracket
U^{1},U^{2}\rrbracket,\\\
&=\frac{2}{3}\\{U_{;\mu}b_{1}+U\,D_{\mu}b_{1}+D_{\mu\nu\nu}b_{1}\\}_{z=0}\llbracket
U^{1},U^{2}\rrbracket,\\\
&=\frac{2}{3}\\{U_{;\mu}U+\frac{1}{2}U\,U_{;\mu}-\frac{1}{4}U\,J_{\mu}+\frac{1}{4}U_{;\nu\nu\mu}+\frac{1}{4}U_{;\nu}G_{\mu\nu}-\frac{1}{4}J_{\mu}U+\frac{1}{4}G_{\mu\nu}U_{;\nu}\\}.\end{split}$
(42)

$\begin{split}D_{\mu\mu}b_{2}|_{z=0}\llbracket
U^{1}\rrbracket&=\frac{1}{4}\\{2\,D_{\mu\mu}(U+D^{2})b_{1}-T_{\mu\mu}b_{2}\\}_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{2}\\{U_{;\mu\mu}b_{1}+2\,U_{;\mu}D_{\mu}b_{1}+U\,D^{2}b_{1}+D_{\mu\mu\nu\nu}b_{1}\\}_{z=0}\llbracket
U^{1}\rrbracket,\\\
&=\frac{1}{2}\\{-\frac{1}{5}\,U_{;\mu}J_{\mu}+\frac{4}{15}U\,(G_{\mu\nu})^{2}+\frac{1}{5}U_{;\mu\mu\nu\nu}+\frac{2}{5}\,U_{;\nu\mu}G_{\mu\nu}+\frac{2}{5}\,(G_{\mu\nu})^{2}U\\}.\end{split}$
(43)

Again, we collect all these contributions and with the help of Eq. (19), we
note the following equation.

$\begin{split}[b_{4}]\llbracket
U^{2}\rrbracket=\mathcal{O}(D^{4}U^{2})=&\frac{1}{5}U_{;\mu\mu\nu}U_{;\nu}+\frac{3}{10}U_{;\mu}U_{;\nu\nu\mu}+\frac{1}{5}U_{;\mu\nu\nu}U_{;\mu}+\frac{4}{15}(U_{;\mu\nu})^{2}+\frac{1}{3}(U_{;\mu\mu})^{2}\\\
&+\frac{1}{5}U_{;\mu\mu\nu\nu}U+\frac{1}{10}U_{;\mu}U_{;\mu\nu\nu}+\frac{1}{5}U\,U_{;\mu\mu\nu\nu}+\frac{2}{15}J_{\mu}U_{;\mu
U}\\\
&-\frac{2}{15}U\,U_{;\mu}J_{\mu}+\frac{1}{5}U\,J_{\mu}U_{;\mu}-\frac{1}{6}U_{;\mu}U\,J_{\mu}-\frac{1}{10}U_{;\mu}J_{\mu}U\\\
&+\frac{1}{15}J_{\mu}U\,U_{;\mu}+\frac{1}{5}U^{2}(G_{\mu\nu})^{2}+\frac{4}{15}U(G_{\mu\nu})^{2}U+\frac{2}{15}(U\,G_{\mu\nu})^{2}\\\
&+\frac{1}{15}G_{\mu\nu}U^{2}G_{\mu\nu}+\frac{2}{15}(G_{\mu\nu}U)^{2}+\frac{1}{5}U_{;\mu}U_{;\nu}G_{\mu\nu}+\frac{1}{5}U_{;\mu}G_{\mu\nu}U_{;\nu}\\\
&+\frac{1}{5}(G_{\mu\nu})^{2}U^{2}.\end{split}$ (44)

####  $\mathcal{O}(D^{4}U^{3})$

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

$\begin{split}D_{\mu\mu}b_{2}|_{z=0}\llbracket
U^{2}\rrbracket&=\frac{1}{4}\\{2\,D^{2}(U+D^{2})b_{1}-T_{\mu\mu}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{1}{2}\\{U_{;\mu\mu}b_{1}+2\,U_{;\mu}D_{\mu}b_{1}+U\,D^{2}b_{1}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{1}{2}\\{U_{;\mu\mu}U+U_{;\mu}U_{;\mu}+\frac{1}{3}U\,U_{;\mu\mu}\\}.\end{split}$
(45)

$\begin{split}D_{\mu\nu}b_{2}|_{z=0}\llbracket
U^{2}\rrbracket&=\frac{1}{4}\\{2\,D_{\mu\nu}(U+D^{2})b_{1}-T_{\mu\nu}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{1}{4}\\{2\,U_{;\nu\mu}b_{1}+2\,U_{;\mu}D_{\nu}b_{1}+2\,U_{;\nu}D_{\mu}b_{1}+2\,U\,D_{\mu\nu}b_{1}-G_{\nu\mu}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{1}{4}\\{2\,U_{;\nu\mu}U+U_{;\mu}U_{;\nu}+U_{;\nu}U_{;\mu}+\frac{2}{3}U\,U_{;\nu\mu}+\frac{1}{3}U^{2}\,G_{\mu\nu}+\frac{2}{3}U\,G_{\mu\nu}U\\\
&\quad+G_{\mu\nu}U^{2}\\}.\end{split}$ (46)

$\begin{split}D_{\mu\nu\nu}b_{2}|_{z=0}\llbracket
U^{2}\rrbracket&=\frac{1}{5}\\{2\,D_{\mu\nu\nu}(U+D^{2})b_{1}-T_{\mu\nu\nu}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{2}{5}\\{D_{\mu}(U_{;\nu\nu}b_{1}+2\,U_{;\nu}D_{\nu}b_{1}+U\,D^{2}b_{1})-\frac{1}{2}D_{\nu}G_{\nu\mu}b_{2}\\\
&\quad-\frac{1}{2}G_{\nu\mu}D_{\nu}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{2}{5}\\{U_{;\nu\nu\mu}b_{1}+U_{;\nu\nu}D_{\mu}b_{1}+2\,U_{;\nu\mu}D_{\nu}b_{1}+2\,U_{;\nu}D_{\mu\nu}b_{1}+U_{;\mu}\,D^{2}b_{1}\\\
&\quad+U\,D_{\mu\nu\nu}b_{1}-\frac{1}{2}G_{\nu\mu;\nu}b_{2}-G_{\nu\mu}D_{\nu}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{2}{5}\\{U_{;\nu\nu\mu}U+\frac{1}{2}U_{;\nu\nu}U_{;\mu}+U_{;\nu\mu}U_{;\nu}+\frac{2}{3}\,U_{;\nu}U_{;\nu\mu}+\frac{1}{3}\,U_{;\nu}U\,G_{\mu\nu}\\\
&\quad+\frac{2}{3}\,U_{;\nu}G_{\mu\nu}U+\frac{1}{3}U_{;\mu}U_{;\nu\nu}+\frac{1}{4}U\,U_{;\nu\nu\mu}+\frac{1}{4}U\,U_{;\nu}G_{\mu\nu}-\frac{1}{12}U^{2}\,J_{\mu}\\\
&\quad-\frac{1}{4}U\,J_{\mu}U+\frac{1}{4}U\,G_{\mu\nu}U_{;\nu}-\frac{1}{2}J_{\mu}U^{2}+\frac{2}{3}G_{\mu\nu}U_{;\nu}U+\frac{1}{3}G_{\mu\nu}U\,U_{;\nu}\\}.\end{split}$
(47)

$\begin{split}D_{\nu\nu\mu}b_{2}|_{z=0}\llbracket
U^{2}\rrbracket&=\frac{1}{5}\\{2\,D_{\nu\nu\mu}(U+D^{2})b_{1}-T_{\nu\nu\mu}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{2}{5}\\{D_{\nu\nu}(U_{\mu}b_{1}+U\,D_{\mu}b_{1})-D_{\nu}G_{\mu\nu}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{2}{5}\\{U_{;\mu\nu\nu}b_{1}+U_{\mu}D^{2}b_{1}+2\,U_{\mu\nu}D_{\nu}b_{1}+U_{\nu\nu}\,D_{\mu}b_{1}\\\
&\quad+U\,D_{\nu\nu\mu}b_{1}+2\,U_{;\nu}\,D_{\nu\mu}b_{1}+G_{\nu\mu;\nu}b_{2}+G_{\nu\mu}D_{\nu}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{2}{5}\\{U_{;\mu\nu\nu}U+\frac{1}{3}U_{\mu}U_{;\nu\nu}+U_{\mu\nu}U_{;\nu}+\frac{1}{2}U_{\nu\nu}U_{;\mu}+\frac{1}{4}U\,U_{;\mu\nu\nu}+J_{\mu}U^{2}\\\
&\quad+\frac{1}{4}U\,U_{;\nu}G_{\nu\mu}+\frac{1}{6}U^{2}J_{\mu}+\frac{1}{2}U\,J_{\mu}U+\frac{1}{4}U\,G_{\nu\mu}U_{;\nu}+\frac{2}{3}\,U_{;\nu}\,U_{;\mu\nu}\\\
&\quad+\frac{1}{3}\,U_{;\nu}U\,G_{\nu\mu}+\frac{2}{3}\,U_{;\nu}G_{\nu\mu}U+\frac{2}{3}G_{\nu\mu}U_{;\nu}U+\frac{1}{3}G_{\nu\mu}U\,U_{;\nu}\\}.\end{split}$
(48)

$\begin{split}D^{4}\,b_{2}|_{z=0}\llbracket
U^{2}\rrbracket&=\frac{1}{6}\\{2\,D_{\mu\mu\nu\nu}(U+D^{2})b_{1}-T_{\mu\mu\nu\nu}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{1}{3}\\{U_{;\mu\mu\nu\nu}b_{1}+2\,U_{;\mu\nu\nu}D_{\mu}b_{1}+2\,U_{;\nu\nu\mu}D_{\mu}b_{1}+2\,U_{;\mu}D_{\mu\nu\nu}b_{1}+2\,U_{;\mu}D_{\nu\nu\mu}b_{1}\\\
&\quad+4\,U_{;\nu\mu}D_{\mu\nu}b_{1}+2\,U_{;\mu\mu}D^{2}b_{1}+U\,D^{4}b_{1}+J_{\mu}D_{\mu}b_{2}+(G_{\mu\nu})^{2}b_{2}\\}|_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{1}{3}\\{U_{;\mu\mu\nu\nu}U+\frac{1}{5}U\,U_{;\mu\mu\nu\nu}+\frac{2}{3}\,U_{;\mu\mu}U_{;\nu\nu}+U_{;\nu\nu\mu}U_{;\mu}+U_{;\mu\nu\nu}U_{;\mu}\\\
&\quad+\frac{2}{5}U\,U_{;\mu}J_{\mu}-\frac{4}{15}U\,(G_{\mu\nu})^{2}U+(G_{\mu\nu})^{2}U^{2}+\frac{1}{5}U\,J_{\mu}U_{;\mu}-\frac{2}{15}(U\,G_{\mu\nu})^{2}\\\
&\quad-\frac{1}{10}U^{2}(G_{\mu\nu})^{2}+\frac{1}{2}U_{;\mu}U_{;\nu\nu\mu}+\frac{1}{2}U_{;\mu}J_{\mu}U+\frac{1}{6}U_{;\mu}U\,J_{\mu}+\frac{1}{2}U_{;\mu}U_{;\mu\nu\nu}\\\
&\quad+\frac{4}{3}(U_{;\mu\nu})^{2}+\frac{1}{3}G_{\mu\nu}U^{2}\,G_{\mu\nu}+\frac{2}{3}(G_{;\mu\nu}U)^{2}+\frac{2}{3}J_{\mu}U_{;\mu}U+\frac{1}{3}J_{\mu}UU_{;\mu}\\}.\end{split}$
(49)

$\begin{split}[b_{3}]\llbracket
U^{2},U^{3}\rrbracket&=\\{U+D^{2}\\}b_{2}|_{z=0}\llbracket
U^{2},U^{3}\rrbracket,\\\
&=U^{3}+\frac{1}{2}U\,U_{;\mu\mu}+\frac{1}{2}U_{;\mu\mu}U+\frac{1}{2}(U_{;\mu})^{2}.\\\
\end{split}$ (50)

$\begin{split}D_{\mu}b_{3}|_{z=0}\llbracket
U^{2},U^{3}\rrbracket&=\frac{1}{4}\\{3\,D_{\mu}(U+D^{2})b_{2}-T_{\mu}b_{3}\\}_{z=0}\llbracket
U^{2},U^{3}\rrbracket,\\\
&=\frac{3}{4}\\{U_{;\mu}b_{2}+U\,D_{\mu}b_{2}+D_{\mu\nu\nu}b_{2}\\}_{z=0}\llbracket
U^{2},U^{3}\rrbracket,\\\
&=\frac{3}{4}\\{U_{;\mu}U^{2}+\frac{2}{3}U\,U_{;\mu}U+\frac{1}{3}U^{2}U_{;\mu}-\frac{1}{5}U^{2}J_{\mu}-\frac{4}{15}UJ_{\mu}U\\\
&\quad+\frac{4}{15}U\,U_{;\nu\nu\mu}+\frac{4}{15}U\,U_{;\nu}G_{\mu\nu}+\frac{4}{15}U\,G_{\mu\nu}U_{;\nu}+\frac{7}{15}U_{;\mu}U_{;\nu\nu}\\\
&\quad+\frac{2}{5}U_{;\nu\nu\mu}U+\frac{1}{5}U_{;\nu\nu}U_{;\mu}+\frac{2}{5}U_{;\nu\mu}U_{;\nu}+\frac{4}{15}\,U_{;\nu}U_{;\nu\mu}-\frac{1}{5}J_{\mu}U^{2}\\\
&\quad+\frac{2}{15}\,U_{;\nu}U\,G_{\mu\nu}+\frac{4}{15}\,U_{;\nu}G_{\mu\nu}U+\frac{4}{15}G_{\mu\nu}U_{;\nu}U+\frac{2}{15}G_{\mu\nu}U\,U_{;\nu}\\}.\end{split}$
(51)

$\begin{split}D_{\mu\mu}b_{3}|_{z=0}\llbracket
U^{2}\rrbracket&=\frac{1}{5}\\{3\,D_{\mu\mu}(U+D^{2})b_{2}-T_{\mu\mu}b_{3}\\}_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{3}{5}\\{U_{;\mu\mu}b_{2}+2\,U_{;\mu}D_{\mu}b_{2}+U\,D^{2}b_{2}+D_{\mu\mu\nu\nu}b_{2}\\}_{z=0}\llbracket
U^{2}\rrbracket,\\\
&=\frac{1}{5}\\{\frac{1}{10}\,U\,U_{;\mu}J_{\mu}+\frac{1}{2}U\,U_{;\mu\mu\nu\nu}+\frac{1}{6}\,(U\,G_{\mu\nu})^{2}+\frac{1}{3}G_{\mu\nu}U^{2}\,G_{\mu\nu}\\\
&\quad+\frac{1}{2}U\,J_{\mu}U_{;\mu}+\frac{3}{2}U_{;\mu}U_{;\nu\nu\mu}+U_{;\mu}G_{\mu\nu}U_{;\nu}+U_{;\mu}U_{;\nu}G_{\mu\nu}+\frac{4}{3}(U_{;\mu\nu})^{2}\\\
&\quad+U_{;\mu\mu\nu\nu}U+\frac{2}{3}\,U_{;\mu\mu}U_{;\nu\nu}+U_{;\nu\nu\mu}U_{;\mu}+U_{;\mu\nu\nu}U_{;\mu}+\frac{1}{3}\,U(G_{\mu\nu})^{2}U\\\
&\quad+(G_{\mu\nu})^{2}U^{2}-\frac{1}{2}U_{;\mu}J_{\mu}U-\frac{5}{6}U_{;\mu}U\,J_{\mu}+\frac{1}{2}U_{;\mu}U_{;\mu\nu\nu}+\frac{1}{3}J_{\mu}UU_{\mu}\\\
&\quad+(U_{;\mu\mu})^{2}+\frac{2}{3}(G_{;\mu\nu}U)^{2}+\frac{2}{3}J_{\mu}U_{\mu}U\\}.\end{split}$
(52)

Finally, we combine all the above-computed expressions and put them in Eq.
(20) to find the operators of the class $\mathcal{O}(D^{4}U^{3})$. Note that,
at this stage, all the evaluated operator structures are not independent. We
employ the trace properties and a few identities333Under trace, as cyclic
permutations are equivalent, a commutator is zero. Total derivatives can be
written as a commutator, i.e., $(D_{\mu}\;U)=[D_{\mu},U]$ and hence are zero
under a trace. Along with the identity given in Eq. (24) we further use the
Bianchi identity,
$G_{\rho\sigma;\mu}+G_{\sigma\mu;\rho}+G_{\mu\rho;\sigma}=0$, to simplify
terms. that simplify the HKCs and allow us to write them in terms of
independent operators. Finally, we find the independent operators of the form
$\mathcal{O}(D^{4}U^{3})$ as

$\begin{split}[b_{5}]\llbracket U^{3}\rrbracket=\mathcal{O}(D^{4}U^{3})=&\
U^{3}(G_{\mu\nu})^{2}+\frac{2}{3}U^{2}G_{\mu\nu}U\,G_{\mu\nu}+\frac{1}{3}U^{2}J_{\mu}U_{;\mu}+\frac{1}{3}U\,G_{\mu\nu}U_{;\mu}U_{;\nu}\\\
&+\frac{1}{3}U\,U_{;\mu}U_{;\nu}\,G_{\mu\nu}-\frac{1}{3}U^{2}U_{;\mu}J_{\mu}+U\,U_{;\mu\mu}U_{;\nu\nu}+\frac{2}{3}U_{;\mu\mu}(U_{;\nu})^{2}.\end{split}$
(53)

###  3.2 Relevant Coefficients at Coincidence point

The necessary and relevant HKCs ($[b_{k}]$) computed at the coincidence point
can be written in a compact form as

(54) (55) (56) (57) (58) (59) (60) (61)

##  4 One-loop effective Lagrangian and Heat-Kernel Coefficients

The one-loop effective Lagrangian obtained from Eq. (1) in the Euclidean space
is given by

$\mathcal{L}_{\text{eff}}=c_{s}\tr\log(-P^{2}+U+M^{2}),$ (62)

where $P_{\mu}=iD^{E}_{\mu}$, with $D_{\mu}^{E}$ being the derivative operator
in the Euclidean signature444From now onwards we will drop the superscript $E$
and will use $D$ uniformly., and $c_{s}=+1/2$ and $+1$ for $\phi$ being a real
scalar and complex scalar background respectively. In the Minkowski signature,
the d’Alembertian operator is a hyperbolic second-order partial differential
operator for which the Heat-Kernel expansion (HKE) is not convergent. Hence,
by performing Wick’s rotation to Euclidean space, the second-order partial
differential operator is transformed into an elliptical one for which a
convergent HKE is well-defined. The following identity,

$\ln\lambda=-\int_{0}^{\infty}\frac{dt}{t}e^{-t\lambda},$ (63)

helps to recast the one-loop effective action in terms of the HK as

$\mathcal{L}_{\text{eff}}=c_{s}\tr\int_{0}^{\infty}\frac{dt}{t}K(t,x,x,\Delta).$
(64)

Employing the ansatz, noted in Eq. (7), the $\mathcal{L}_{\text{eff}}$ can be
expressed in terms of the coincident limit HKCs ($[b_{k}]$) as

$\begin{split}\mathcal{L}_{\text{eff}}&=c_{s}\int_{0}^{\infty}\frac{dt}{t}(4\pi
t)^{-d/2}\ e^{-t\,M^{2}}\sum_{k}\frac{(-t)^{k}}{k\,!}\tr[b_{k}]\\\
&=\frac{c_{s}}{(4\pi)^{d/2}}\sum_{k}\frac{(-1)^{k}}{k!}\int_{0}^{\infty}dt\
t^{k-1-d/2}\ e^{-t\,M^{2}}\tr[b_{k}].\end{split}$ (65)

A suitable change in variable $t\,M^{2}\to\tau^{2}$, the above integral reads
as

$\mathcal{L}_{\text{eff}}=\frac{c_{s}}{(4\pi)^{d/2}}\sum_{k}M^{d-2k}\frac{(-1)^{k}}{k!}\
2\int_{0}^{\infty}d\tau\ \tau^{2(k-d/2)-1}\ e^{-\tau^{2}}\tr[b_{k}].$ (66)

Note that the integral over $\tau$ mimics the integral representation of gamma
function $\Gamma[z]$

$\Gamma[z]=2\int_{0}^{\infty}d\tau\ \tau^{2z-1}\ e^{-\tau^{2}},$ (67)

and that eases out writing down a compact form of the one-loop effective
Lagrangian as

$\mathcal{L}_{\text{eff}}=\frac{c_{s}}{(4\pi)^{d/2}}\sum_{k=0}^{\infty}M^{d-2k}\frac{(-1)^{k}}{k!}\
\Gamma[k-d/2]\tr[b_{k}].$ (68)

###  4.1 Effective Contributions to Renormalisable Lagrangian

It is evident from Eq. (68), that for $k\leq d/2$ the Gamma function has
simple poles. Thus, for such cases, we need to renormalise the theory
employing dimensional regularisation, and $\overline{MS}$ renormalisation
scheme.

We are working with 4-dim Euclidean space. Assuming $d=4-\epsilon$, we find

$\Gamma[k-d/2]=\frac{(\epsilon/2-3+k)!}{(\epsilon/2-1)!}\,\Gamma[\epsilon/2].$
(69)

In case of 4-dim, $d\to 4$, i.e., $\epsilon\rightarrow 0$, the Gamma function
has simple poles as
$\Gamma[\epsilon/2]=2/\epsilon-\gamma_{E}+\mathcal{O}(\epsilon)$. In that
scenario, the divergent part of the one-loop effective Lagrangian can be
written as

$\mathcal{L}^{(k)}_{div}=\frac{c_{s}}{(4\pi)^{2-\epsilon/2}}M^{d-2k}\frac{(-1)^{k}}{k!}\
\frac{(\epsilon/2-3+k)!}{(\epsilon/2-1)!}\,\Gamma[\epsilon/2]\ \tr[b_{k}],$
(70)

with $k=0,1,2$. These three cases are explicitly demonstrated below.

#### $k=0$

$\begin{split}\mathcal{L}^{(0)}_{div}&=\frac{c_{s}}{(4\pi)^{2-\epsilon/2}}M^{d}\
\frac{(\epsilon/2-3)!}{(\epsilon/2-1)!}\,\Gamma[\epsilon/2]\ \tr[b_{0}]\\\
&=\frac{c_{s}}{(4\pi)^{2-\epsilon/2}}M^{4-\epsilon}\
\frac{1}{(\epsilon/2-1)(\epsilon/2-2)}\,\Gamma[\epsilon/2]\ \tr[b_{0}]\\\
&=c_{s}\left(\frac{M^{2}}{4\pi}\right)^{2}\left(\frac{4\pi}{M^{2}}\right)^{\epsilon/2}\
\frac{1}{(\epsilon/2-1)(\epsilon/2-2)}\,\Gamma[\epsilon/2]\ \tr[b_{0}].\\\
\end{split}$ (71)

Taylor expansion in limit $\epsilon\rightarrow 0$ leads to

$\mathcal{L}^{(0)}_{div}=\frac{c_{s}}{(4\pi)^{2}}\ M^{4}\
\frac{1}{2}\,\left(\frac{2}{\epsilon}-\gamma_{E}-\ln\left[\frac{M^{2}}{4\pi}\right]+3/2\right)\
\tr[b_{0}].$ (72)

Employing $\overline{MS}$ scheme, we can write the finite part as

$\mathcal{L}^{(0)}_{\text{eff}}=\frac{c_{s}}{(4\pi)^{2}}\ M^{4}\
\left[-\frac{1}{2}\,\left(\ln\left[\frac{M^{2}}{\mu^{2}}\right]-3/2\right)\,\tr[b_{0}]\right].$
(73)

#### $k=1$

$\begin{split}\mathcal{L}^{(1)}_{div}&=-\frac{c_{s}}{(4\pi)^{2-\epsilon/2}}M^{2-\epsilon}\
\frac{(\epsilon/2-2)!}{(\epsilon/2-1)!}\,\Gamma[\epsilon/2]\ \tr[b_{1}]\\\
&=-c_{s}\,\left(\frac{m}{4\pi}\right)^{2}\left(\frac{4\pi}{M^{2}}\right)^{\epsilon/2}\
\frac{1}{(\epsilon/2-1)}\,\Gamma[\epsilon/2]\ \tr[b_{1}].\\\ \end{split}$ (74)

Taylor expansion in limit $\epsilon\rightarrow 0$ leads to

$\mathcal{L}^{(1)}_{div}=-\frac{c_{s}}{(4\pi)^{2}}\ M^{2}\
(-1)\,\left(\frac{2}{\epsilon}-\gamma_{E}-\ln\left[\frac{M^{2}}{4\pi}\right]+1\right)\
\tr[b_{1}].$ (75)

Employing $\overline{MS}$ scheme, we can write the finite part as

$\mathcal{L}^{(1)}_{\text{eff}}=\frac{c_{s}}{(4\pi)^{2}}\ M^{2}\
\left[-\left(\ln\left[\frac{M^{2}}{\mu^{2}}\right]-1\right)\,\tr[b_{1}]\right].$
(76)

#### $k=2$

$\begin{split}\mathcal{L}^{(2)}_{div}=\frac{c_{s}}{(4\pi)^{2-\epsilon/2}}M^{-\epsilon}\frac{1}{2}\
\frac{(\epsilon/2-1)!}{(\epsilon/2-1)!}\,\Gamma[\epsilon/2]\
\tr[b_{2}]=\frac{c_{s}}{(4\pi)^{2}}\left(\frac{4\pi}{M^{2}}\right)^{\epsilon/2}\frac{1}{2}\,\Gamma[\epsilon/2]\
\tr[b_{2}].\end{split}$ (77)

Taylor expansion in limit $\epsilon\rightarrow 0$ leads to

$\mathcal{L}^{(2)}_{div}=\frac{c_{s}}{(4\pi)^{2}}\
M^{0}\,\frac{1}{2}\left(\frac{2}{\epsilon}-\gamma_{E}-\ln\left[\frac{M^{2}}{4\pi}\right]\right)\
\tr[b_{2}].$ (78)

Employing $\overline{MS}$ scheme, we can write the finite part as

$\mathcal{L}^{(1)}_{\text{eff}}=\frac{c_{s}}{(4\pi)^{2}}\
M^{0}\,\frac{1}{2}\left[-\left(\ln\left[\frac{M^{2}}{\mu^{2}}\right]\right)\,\tr[b_{2}]\right].$
(79)

### Renormalised one-loop effective Lagrangian

After renormalising the effective Lagrangian for three cases, $k=0,1,2$ we
collect all the finite parts, and the one-loop effective contributions to
renormalisable part of the Lagrangian that contains operators up to mass
dimension four can be written as

$\mathcal{L}_{\text{eff}}^{\text{ren}}=\frac{c_{s}}{(4\pi)^{2}}\tr\\{M^{4}\
\left[-\frac{1}{2}\,\left(\ln\left[\frac{M^{2}}{\mu^{2}}\right]-3/2\right)\,[b_{0}]\right]+M^{2}\
\left[-\left(\ln\left[\frac{M^{2}}{\mu^{2}}\right]-1\right)\,[b_{1}]\right]\\\
+M^{0}\
\left[-\frac{1}{2}\left(\ln\left[\frac{M^{2}}{\mu^{2}}\right]\right)\,[b_{2}]\right]\Bigg{\\}}.$
(82)

##  5 Pure Heavy Scalar loop UOLEA up to D8

In order to facilitate the readability of our result, we organise the
effective operators according to the number of appearances of its
constituents, i.e., covariant derivatives $(P)$, and the light field-dependent
functional $(U)$. Here, if an operator is composed of $i$ number of covariant
derivatives and $j$ number of $U$, the corresponding quantities are
represented by the integrer-superscripts $(i,j)$. The Lorentz invariance
allows $i$ to be only even integer. The effective Lagrangian up to mass
dimension eight can be written as

$\displaystyle\mathcal{L}_{\text{eff}}=\frac{c_{s}}{(4\pi)^{2}}\sum_{i,j}\mathcal{L}^{(i,j)}_{\text{eff}}=\frac{c_{s}}{(4\pi)^{2}}\sum_{i,j,k}\mathcal{C}_{k}^{(i,j)}O_{k}(P^{i}\,U^{j}).$

Here, $\mathcal{C}_{k}^{(i,j)}$ is the coefficient associated with different
operator $O_{k}(P^{i}\,U^{j})$. The $k$ sums over the number of operators in
each class with $i,j=[0,8]$. To keep in agreement with the literature, we use
the following notation $P_{\mu}=iD_{\mu}$. With the help of the HKCs computed
at the coincident point, see Subsec. 3.2, and using Eq. (7), here, we
catalogue the operators associated with the one-loop effective Lagrangian.

###  5.1 $O(P^{8}\;U^{j})$

$\mathcal{L}^{(8,0)}_{\text{eff}}=\frac{1}{M^{4}}\frac{1}{24}\tr[\frac{17}{210}[P_{\mu},P_{\nu}][P_{\mu},P_{\nu}][P_{\rho},P_{\sigma}][P_{\rho},P_{\sigma}]\\\
+\frac{2}{35}[P_{\mu},P_{\rho}][P_{\rho},P_{\nu}][P_{\mu},P_{\sigma}][P_{\sigma},P_{\nu}]+\frac{1}{105}[P_{\mu},P_{\nu}][P_{\nu},P_{\rho}][P_{\rho},P_{\sigma}][P_{\sigma},P_{\mu}]\\\
+\frac{1}{420}[P_{\mu},P_{\nu}][P_{\rho},P_{\sigma}][P_{\mu},P_{\nu}][P_{\rho},P_{\sigma}]+\frac{1}{35}[P_{\mu},[P_{\rho},[P_{\rho},P_{\nu}]]][P_{\mu},[P_{\sigma},[P_{\sigma},P_{\nu}]]]\\\
+\frac{16}{105}[P_{\mu},[P_{\rho},[P_{\rho},P_{\nu}]][P_{\nu},P_{\sigma}][P_{\sigma},P_{\mu}]+\frac{16}{105}[P_{\mu},P_{\nu}][P_{\sigma},[P_{\sigma},P_{\mu}]][P_{\rho},[P_{\rho},P_{\nu}]]\bigg{]},$
$\displaystyle\mathcal{C}^{(8,0)}_{1}=\frac{1}{M^{4}}\frac{17}{5040},\quad\mathcal{C}^{(8,0)}_{2}=\frac{1}{M^{4}}\frac{1}{420},\quad\mathcal{C}^{(8,0)}_{3}=\frac{1}{M^{4}}\frac{1}{2520},\quad\mathcal{C}^{(8,0)}_{4}=\frac{1}{M^{4}}\frac{1}{10080},$
$\displaystyle\mathcal{C}^{(8,0)}_{5}=\frac{1}{M^{4}}\frac{1}{840},\quad\mathcal{C}^{(8,0)}_{6}=\frac{1}{M^{4}}\frac{2}{315},\quad\mathcal{C}^{(8,0)}_{7}=\frac{1}{M^{4}}\frac{2}{315}.$
(84)

###  5.2 $O(P^{6}\;U^{j})$

$\displaystyle\mathcal{L}^{(6,0)}_{\text{eff}}=\frac{1}{M^{2}}\frac{1}{6}\tr[\frac{1}{15}[P_{\mu},P_{\nu}][P_{\nu},P_{\rho}][P_{\rho},P_{\mu}]-\frac{1}{10}[P_{\mu},[P_{\mu},P_{\nu}]][P_{\rho},[P_{\rho},P_{\nu}]\bigg{]},$
$\displaystyle\mathcal{C}^{(6,0)}_{1}=\frac{1}{M^{2}}\frac{1}{90},\quad\mathcal{C}^{(6,0)}_{2}=-\frac{1}{M^{2}}\frac{1}{60}.$
(85)

$\mathcal{L}^{(6,1)}_{\text{eff}}=\frac{1}{M^{4}}\frac{1}{24}\tr[-\frac{4}{15}U\,[P_{\mu},P_{\nu}][P_{\nu},P_{\rho}][P_{\rho},P_{\mu}]+\frac{2}{5}U[P_{\rho},[P_{\rho},P_{\mu}]][P_{\sigma},[P_{\sigma},P_{\mu}]]\\\
-\frac{2}{15}[P_{\mu},[P_{\mu},U]][P_{\rho},P_{\sigma}][P_{\rho},P_{\sigma}]+\frac{8}{15}[P_{\mu},[P_{\nu},U]]\,[P_{\rho},P_{\mu}][P_{\nu},P_{\rho}]\bigg{]},$
$\mathcal{C}^{(6,1)}_{1}=-\frac{1}{M^{4}}\frac{1}{90},\mathcal{C}^{(6,1)}_{2}=+\frac{1}{M^{4}}\frac{1}{60},\mathcal{C}^{(6,1)}_{3}=-\frac{1}{M^{4}}\frac{1}{180},\mathcal{C}^{(6,1)}_{4}=+\frac{1}{M^{4}}\frac{1}{45}.$
(88)

$\begin{split}\mathcal{L}^{(6,2)}_{\text{eff}}=\
\tr[&\mathcal{C}^{(6,2)}_{1}\,U^{2}\,[P_{\mu},P_{\nu}]\,[P_{\nu},P_{\alpha}][P_{\alpha},P_{\mu}]\,+\,\mathcal{C}^{(6,2)}_{2}\,U\,[P_{\mu},P_{\nu}]\,U\,[P_{\nu},P_{\alpha}][P_{\alpha},P_{\mu}]\\\
&+\,\mathcal{C}^{(6,2)}_{3}\,U^{2}\,[P_{\mu},[P_{\mu},P_{\nu}]]\,[P_{\alpha},[P_{\alpha},P_{\nu}]]\,+\,\mathcal{C}^{(6,2)}_{4}\,U\,[P_{\mu},[P_{\mu},P_{\nu}]]\,U\,[P_{\alpha},[P_{\alpha},P_{\nu}]]\\\
&+\,\mathcal{C}^{(6,2)}_{5}\,U\,[P_{\mu},[P_{\mu},U]]\,[P_{\nu},P_{\alpha}]\,[P_{\nu},P_{\alpha}]+\,\mathcal{C}^{(6,2)}_{6}[P_{\mu},[P_{\mu},U]]\,U\,[P_{\nu},P_{\alpha}]\,[P_{\nu},P_{\alpha}]\\\
&+\,\mathcal{C}^{(6,2)}_{7}\,U\,[P_{\mu},U]\,[P_{\nu}[P_{\nu},P_{\alpha}]\,[P_{\mu},P_{\alpha}]+\,\mathcal{C}^{(6,2)}_{8}U\,[P_{\mu},P_{\alpha}]\,[P_{\nu}[P_{\nu},P_{\alpha}]\,[P_{\mu},U]\\\
&+\,\mathcal{C}^{(6,2)}_{9}\,U\,[P_{\mu},U]\,[P_{\mu},P_{\nu}]\,[P_{\alpha}[P_{\alpha},P_{\nu}]+\,\mathcal{C}^{(6,2)}_{10}U\,[P_{\alpha}[P_{\alpha},P_{\nu}]\,[P_{\mu},P_{\nu}]\,[P_{\mu},U]\\\
&+\mathcal{C}^{(6,2)}_{11}\,U\,[P_{\mu},P_{\nu}]\,[P_{\alpha},[P_{\alpha},U]\,[P_{\mu},P_{\nu}]\,+\,\mathcal{C}^{(6,2)}_{12}\,\,[P_{\mu},U]\,[P_{\mu},U]\,[P_{\nu},P_{\alpha}]\,[P_{\nu},P_{\alpha}]\\\
&+\,\mathcal{C}^{(6,2)}_{13}\,\,[P_{\mu},U]\,[P_{\nu},[P_{\nu},P_{\alpha}]]\,U\,[P_{\mu},P_{\alpha}]+\,\mathcal{C}^{(6,2)}_{14}\,[P_{\mu},P_{\alpha}]\,U\,[P_{\nu},[P_{\nu},P_{\alpha}]]\,[P_{\mu},U]\\\
&+\,\mathcal{C}^{(6,2)}_{15}\,\,[P_{\mu},U]\,[P_{\nu},U]\,[P_{\mu},P_{\alpha}]\,[P_{\alpha},P_{\nu}]+\,\mathcal{C}^{(6,2)}_{16}\,\,[P_{\mu},U]\,[P_{\nu},U]\,[P_{\nu},P_{\alpha}]\,[P_{\alpha},P_{\mu}]\\\
&+\,\mathcal{C}^{(6,2)}_{17}\,\,[P_{\mu},U]\,[P_{\mu},P_{\nu}]\,[P_{\alpha},U]\,[P_{\alpha},P_{\nu}]+\,\mathcal{C}^{(6,2)}_{18}\,[P_{\mu},U]\,[P_{\alpha},P_{\nu}]\,[P_{\alpha},U]\,[P_{\mu},P_{\nu}]\\\
&+\,\mathcal{C}^{(6,2)}_{19}\,\,[P_{\mu},U]\,[P_{\nu},P_{\alpha}]\,[P_{\mu},U]\,[P_{\nu},P_{\alpha}]\,+\,\mathcal{C}^{(6,2)}_{20}\,\,[P_{\mu},[P_{\mu},U]]\,[P_{\nu},U]\,[P_{\alpha},[P_{\alpha},P_{\nu}]]\\\
&+\,\mathcal{C}^{(6,2)}_{21}\,[P_{\mu},[P_{\mu},U]]\,[P_{\alpha},[P_{\alpha},P_{\nu}]]\,[P_{\nu},U]\,+\,\mathcal{C}^{(6,2)}_{22}\,\,[P_{\mu},U]\,[P_{\nu},U]\,[P_{\mu},[P_{\alpha},[P_{\alpha},P_{\nu}]]]\\\
&+\,\mathcal{C}^{(6,2)}_{23}\,[P_{\mu},U]\,[P_{\mu},[P_{\alpha},[P_{\alpha},P_{\nu}]]]\,[P_{\nu},U]+\,\mathcal{C}^{(6,2)}_{24}\,[P_{\mu},[P_{\nu},[P_{\nu},U]]]\,[P_{\mu},[P_{\alpha},[P_{\alpha},U]]]\\\
&+\,\mathcal{C}^{(6,2)}_{25}[P_{\alpha},P_{\mu}][P_{\mu},P_{\beta}]U\,[P_{\alpha},[P_{\beta},U]]+\,\mathcal{C}^{(6,2)}_{26}[P_{\alpha},P_{\mu}][P_{\mu},P_{\beta}][P_{\alpha},[P_{\beta},U]]U\,\Big{]},\end{split}$
(89)

###  5.3 $O(P^{4}\;U^{j})$

$\displaystyle\mathcal{L}^{(4,0)}_{\text{eff}}=-M^{0}\frac{1}{12}\ln\left[\frac{M^{2}}{\mu^{2}}\right]\tr[\,[P_{\mu},P_{\nu}][P_{\mu},P_{\nu}]\bigg{]},\;\Rightarrow\;\mathcal{C}^{(4,0)}_{1}=-M^{0}\frac{1}{12}\ln\left[\frac{M^{2}}{\mu^{2}}\right].$
(90)

$\displaystyle\mathcal{L}^{(4,1)}_{\text{eff}}=-\frac{1}{M^{2}}\frac{1}{12}\tr[U[P_{\mu},P_{\nu}][P_{\mu},P_{\nu}]\bigg{]},\;\Rightarrow\;\mathcal{C}^{(4,1)}_{1}=-\frac{1}{M^{2}}\frac{1}{12}.$
(91)

$\mathcal{L}^{(4,2)}_{\text{eff}}=\frac{1}{M^{4}}\frac{1}{24}\tr[\frac{4}{5}U^{2}[P_{\mu},P_{\nu}][P_{\mu},P_{\nu}]+\frac{1}{5}U\,[P_{\mu},P_{\nu}]\,U\,[P_{\mu},P_{\nu}]\\\
+\frac{1}{5}[P_{\mu},[P_{\mu},U]][P_{\nu},[P_{\nu},U]]-\frac{2}{5}U\,[P_{\nu},U]\,[P_{\rho},[P_{\rho},P_{\nu}]]\bigg{]},$
$\mathcal{C}^{(4,2)}_{1}=\frac{1}{M^{4}}\frac{1}{30},\mathcal{C}^{(4,2)}_{2}=\frac{1}{M^{4}}\frac{1}{120},\mathcal{C}^{(4,2)}_{3}=\frac{1}{M^{4}}\frac{1}{120},\mathcal{C}^{(4,2)}_{4}=-\frac{1}{M^{4}}\frac{1}{60}.$
(94)

$\mathcal{L}^{(4,3)}_{\text{eff}}=\frac{1}{M^{6}}\frac{1}{60}\tr[-U^{3}[P_{\mu},P_{\nu}][P_{\mu},P_{\nu}]-\frac{2}{3}U^{2}[P_{\mu},P_{\nu}]\,U\,[P_{\mu},P_{\nu}]\\\
+\frac{1}{3}U^{2}[P_{\mu},U][P_{\rho},[P_{\rho},P_{\mu}]]-\frac{1}{3}U^{2}[P_{\rho},[P_{\rho},P_{\nu}]][P_{\nu},U]\\\
-\frac{1}{3}U\,[P_{\mu},P_{\nu}][P_{\mu},U][P_{\nu},U]-\frac{1}{3}U\,[P_{\mu},U][P_{\nu},U][P_{\mu},P_{\nu}]\\\
-U\,[P_{\mu},[P_{\mu},U]][P_{\nu},[P_{\nu},U]]-\frac{2}{3}[P_{\mu},[P_{\mu},U]][P_{\nu},U][P_{\nu},U]\bigg{]},$
$\displaystyle\mathcal{C}^{(4,3)}_{1}$
$\displaystyle=-\frac{1}{M^{6}}\frac{1}{60},$
$\displaystyle\mathcal{C}^{(4,3)}_{2}$
$\displaystyle=-\frac{1}{M^{6}}\frac{1}{90},$
$\displaystyle\mathcal{C}^{(4,3)}_{3}$
$\displaystyle=\frac{1}{M^{6}}\frac{1}{180},$
$\displaystyle\mathcal{C}^{(4,3)}_{4}$
$\displaystyle=-\frac{1}{M^{6}}\frac{1}{180},$
$\displaystyle\mathcal{C}^{(4,3)}_{5}$
$\displaystyle=-\frac{1}{M^{6}}\frac{1}{180},$
$\displaystyle\mathcal{C}^{(4,3)}_{6}$
$\displaystyle=-\frac{1}{M^{6}}\frac{1}{180},$
$\displaystyle\mathcal{C}^{(4,3)}_{7}$
$\displaystyle=-\frac{1}{M^{6}}\frac{1}{60},$
$\displaystyle\mathcal{C}^{(4,3)}_{8}$
$\displaystyle=-\frac{1}{M^{6}}\frac{1}{90}.$ (99)

$\begin{split}\mathcal{L}^{(4,4)}_{\text{eff}}=\frac{1}{M^{8}}\frac{1}{120}\tr&\bigg{[}\frac{12}{7}U^{2}[P_{\mu},[P_{\nu},U]][P_{\nu},[P_{\mu},U]]+\frac{9}{7}U[P_{\mu},[P_{\nu},U]]U[P_{\nu},[P_{\mu},U]]\\\
&+\frac{26}{7}[P_{\mu},[P_{\nu},U]][P_{\mu},U][P_{\nu},U]U+\frac{18}{7}[P_{\mu},[P_{\nu},U]][P_{\mu},U]U[P_{\nu},U]\\\
&+\frac{26}{7}[P_{\mu},[P_{\nu},U]]U[P_{\mu},U][P_{\nu},U]+\frac{17}{14}[P_{\mu},U][P_{\nu},U][P_{\mu},U][P_{\nu},U]\\\
&+\frac{9}{7}[P_{\mu},U][P_{\mu},U][P_{\nu},U][P_{\nu},U]+\frac{5}{7}U^{4}[P_{\mu},P_{\nu}][P_{\mu},P_{\nu}]\\\
&+\frac{8}{7}U^{3}[P_{\mu},P_{\nu}]U[P_{\mu},P_{\nu}]+\frac{9}{14}U^{2}[P_{\mu},P_{\nu}]U^{2}[P_{\mu},P_{\nu}]\\\
&+\frac{18}{7}[P_{\mu},P_{\nu}][P_{\mu},U]U^{2}[P_{\nu},U]+\frac{18}{7}[P_{\mu},P_{\nu}]U[P_{\mu},U][P_{\nu},U]U\\\
&+\frac{8}{7}[P_{\mu},P_{\nu}]U[P_{\mu},U]U[P_{\nu},U]+\frac{26}{7}[P_{\mu},P_{\nu}][P_{\mu},U]U[P_{\nu},U]U\\\
&+\frac{24}{7}[P_{\mu},P_{\nu}][P_{\mu},U][P_{\nu},U]U^{2}-\frac{2}{7}[P_{\mu},P_{\nu}]U^{2}[P_{\mu},U][P_{\nu},U]\bigg{]},\end{split}$
$\displaystyle\mathcal{C}^{(4,4)}_{1}$
$\displaystyle=\frac{1}{M^{8}}\frac{1}{70},$
$\displaystyle\mathcal{C}^{(4,4)}_{2}$
$\displaystyle=\frac{1}{M^{8}}\frac{3}{280},$
$\displaystyle\mathcal{C}^{(4,4)}_{3}$
$\displaystyle=\frac{1}{M^{8}}\frac{13}{420},$
$\displaystyle\mathcal{C}^{(4,4)}_{4}$
$\displaystyle=\frac{1}{M^{8}}\frac{3}{140},$
$\displaystyle\mathcal{C}^{(4,4)}_{5}$
$\displaystyle=\frac{1}{M^{8}}\frac{13}{420},$
$\displaystyle\mathcal{C}^{(4,4)}_{6}$
$\displaystyle=\frac{1}{M^{8}}\frac{17}{1680},$
$\displaystyle\mathcal{C}^{(4,4)}_{7}$
$\displaystyle=\frac{1}{M^{8}}\frac{3}{280},$
$\displaystyle\mathcal{C}^{(4,4)}_{8}$
$\displaystyle=\frac{1}{M^{8}}\frac{1}{168},$
$\displaystyle\mathcal{C}^{(4,4)}_{9}$
$\displaystyle=\frac{1}{M^{8}}\frac{1}{105},$
$\displaystyle\mathcal{C}^{(4,4)}_{10}$
$\displaystyle=\frac{1}{M^{8}}\frac{3}{560},$
$\displaystyle\mathcal{C}^{(4,4)}_{11}$
$\displaystyle=\frac{1}{M^{8}}\frac{3}{140},$
$\displaystyle\mathcal{C}^{(4,4)}_{12}$
$\displaystyle=\frac{1}{M^{8}}\frac{3}{140},$
$\displaystyle\mathcal{C}^{(4,4)}_{13}$
$\displaystyle=\frac{1}{M^{8}}\frac{1}{105},$
$\displaystyle\mathcal{C}^{(4,4)}_{14}$
$\displaystyle=\frac{1}{M^{8}}\frac{13}{420},$
$\displaystyle\mathcal{C}^{(4,4)}_{15}$
$\displaystyle=\frac{1}{M^{8}}\frac{1}{35},$
$\displaystyle\mathcal{C}^{(4,4)}_{16}$
$\displaystyle=-\frac{1}{M^{8}}\frac{1}{420}.$ (100)

###  5.4 $O(P^{2}\;U^{j})$

$\displaystyle\mathcal{L}^{(2,2)}_{\text{eff}}=\frac{1}{M^{2}}\frac{1}{12}\tr[-[P_{\mu},U][P_{\mu},U]]\bigg{]},\;\Rightarrow\;\mathcal{C}^{(2,2)}_{1}=-\frac{1}{M^{2}}\frac{1}{12}.$
(101)

$\displaystyle\mathcal{L}^{(2,3)}_{\text{eff}}=\frac{1}{M^{4}}\frac{1}{24}\tr[-U^{2}[P_{\mu},[P_{\mu},U]]\bigg{]},\;\Rightarrow\;\mathcal{C}^{(2,3)}_{1}=-\frac{1}{M^{4}}\frac{1}{24}.$
(102)

$\mathcal{L}^{(2,4)}_{\text{eff}}=\frac{1}{M^{6}}\frac{1}{60}\tr[2\,U^{3}[P_{\mu},[P_{\mu},U]]+U^{2}[P_{\mu},U][P_{\mu},U]\bigg{]},$
$\mathcal{C}^{(2,4)}_{1}=-\frac{1}{M^{8}}\frac{1}{40},\quad\mathcal{C}^{(2,4)}_{2}=-\frac{1}{M^{8}}\frac{4}{60}.$
(103)

$\mathcal{L}^{(2,5)}_{\text{eff}}=\frac{1}{M^{8}}\frac{1}{120}\tr[-3\,U^{4}[P_{\mu},[P_{\mu},U]]-2\,U^{3}[P_{\mu},U][P_{\mu},U]\bigg{]},$
$\mathcal{C}^{(2,5)}_{1}=-\frac{1}{M^{8}}\frac{1}{40},\quad\mathcal{C}^{(2,5)}_{2}=-\frac{1}{M^{8}}\frac{4}{60}.$
(104)

$\mathcal{L}^{(2,6)}_{\text{eff}}=\frac{1}{M^{10}}\frac{1}{210}\tr[-5\,U^{4}[P_{\mu},U][P_{\mu},U]-8\,U^{3}[P_{\mu},U]\,U\,[P_{\mu},U]\\\
-\frac{9}{2}U^{2}[P_{\mu},U]U^{2}[P_{\mu},U]\bigg{]},$
$\mathcal{C}^{(2,6)}_{1}=-\frac{1}{M^{10}}\frac{1}{42},\quad\mathcal{C}^{(2,6)}_{2}=-\frac{1}{M^{10}}\frac{4}{105},\quad\mathcal{C}^{(2,6)}_{3}=-\frac{1}{M^{10}}\frac{3}{140}.$
(107)

###  5.5 $O(P^{0}\;U^{j})$

$\displaystyle\mathcal{L}^{(0,0)}_{\text{eff}}=M^{4}\
\frac{1}{2}\,\tr\left[\frac{3}{2}-\ln\left[\frac{M^{2}}{\mu^{2}}\right]\right],\;\Rightarrow\;\mathcal{C}^{(0,0)}_{1}=M^{4}\
\frac{1}{2}\,\tr\left[\frac{3}{2}-\ln\left[\frac{M^{2}}{\mu^{2}}\right]\right].$
(108)

$\displaystyle\mathcal{L}^{(0,1)}_{\text{eff}}=M^{2}\,\left(1-\ln\left[\frac{M^{2}}{\mu^{2}}\right]\right)\tr\left[\,U\right],\;\Rightarrow\;\mathcal{C}^{(0,1)}_{1}=M^{2}\
\left(1-\ln\left[\frac{M^{2}}{\mu^{2}}\right]\right).$ (109)

$\displaystyle\mathcal{L}^{(0,2)}_{\text{eff}}=-M^{0}\,\ln\left[\frac{M^{2}}{\mu^{2}}\right]\,\tr\left[\,U^{2}\right],\;\Rightarrow\;\mathcal{C}^{(0,2)}_{1}=-M^{0}\
\ln\left[\frac{M^{2}}{\mu^{2}}\right].$ (110)

$\displaystyle\mathcal{L}^{(0,3)}_{\text{eff}}=-\frac{1}{M^{2}}\frac{1}{6}\,\text{tr}\,\left[U^{3}\right],\;\Rightarrow\;\mathcal{C}^{(0,3)}_{1}=-\frac{1}{M^{2}}\frac{1}{6}.$
(111)

$\displaystyle\mathcal{L}^{(0,4)}_{\text{eff}}=\frac{1}{M^{4}}\frac{1}{24}\,\text{tr}\,\left[U^{4}\right],\;\Rightarrow\;\mathcal{C}^{(0,4)}_{1}=\frac{1}{M^{4}}\frac{1}{24}.$
(112)

$\displaystyle\mathcal{L}^{(0,5)}_{\text{eff}}=-\frac{1}{M^{6}}\frac{1}{60}\,\text{tr}\,\left[U^{5}\right],\;\Rightarrow\;\mathcal{C}^{(0,5)}_{1}=-\frac{1}{M^{6}}\frac{1}{60}.$
(113)

$\displaystyle\mathcal{L}^{(0,6)}_{\text{eff}}=\frac{1}{M^{8}}\frac{1}{120}\,\text{tr}\,\left[U^{6}\right],\;\Rightarrow\;\mathcal{C}^{(0,6)}_{1}=\frac{1}{M^{8}}\frac{1}{120}.$
(114)

$\displaystyle\mathcal{L}^{(0,7)}_{\text{eff}}=-\frac{1}{M^{10}}\frac{1}{210}\,\text{tr}\,\left[U^{7}\right],\;\Rightarrow\;\mathcal{C}^{(0,7)}_{1}=-\frac{1}{M^{10}}\frac{1}{210}.$
(115)

$\displaystyle\mathcal{L}^{(0,8)}_{\text{eff}}=\frac{1}{M^{12}}\frac{1}{336}\,\text{tr}\,\left[U^{8}\right]\;\Rightarrow\;\mathcal{C}^{(0,8)}_{1}=\frac{1}{M^{12}}\frac{1}{336}.$
(116)

##  6 Validation using Covariant Diagram

We present the one-loop effective action up to dimension eight after
integrating out heavy degenerate scalars in Sec. 5. We agree with the results
so far available in the existing literature Henning:2014wua ; Drozd:2015rsp ;
Ellis:2016enq . To cross-check the new results computed in the earlier
section, we employ a new method of computation based on covariant diagram
techniques, see Refs. Vandeven:1985 ; Zhang:2016pja  for details. A brief
review of this method has been given in Appendix A. We focus on computing only
relevant one-loop diagrams that can generate the results depicted in the
previous section.

###  6.1 Method of Covariant diagram

Here, we illustrate some aspects of the covariant diagram method that are
pertinent to our objectives. Eq. (A), given in Appendix A, reduces to Eq.
(117) up to additive constant while integrating out heavy scalar field or
multiple degenerate heavy scalar fields Zhang:2016pja .

$\displaystyle\mathcal{L}_{\text{eff}}\,[\phi]=-ic_{s}\,\text{tr}\,\sum_{n=1}^{\infty}\,\frac{1}{n}\int\frac{d^{d}\,q}{(2\pi)^{d}}\,\big{[}(q^{2}-M^{2})^{-1}\,(2q.P-P^{2}+U)\big{]}^{n}.$
(117)

This method allows one to map each integral of order $n$ into a number of
covariant diagrams consisting of $n$ number of heavy propagators
$1/(q^{2}-M^{2})$ and along with all permissible combinations of $2q.P$,
$-P^{2}$, and $U$ as vertex insertions. This automatically respects the
covariant nature of the functional matching. The most generic form of the loop
integrals at $n^{th}$ order with $2n_{c}$ number of $2q.P$ vertices that can
appear in the process, is given as

$\displaystyle\int\frac{d^{d}q}{(2\pi)^{d}}\,\frac{q^{\mu_{1}}\cdots
q^{\mu_{2n_{c}}}}{(q^{2}-M^{2})^{n}}\,\equiv\,g^{\mu_{1}\cdots\mu_{2n_{c}}}\,\mathcal{I}[q^{2n_{c}}]^{n}.$
(118)

The completely symmetric tensor $g^{\mu_{1}\cdots\mu_{2n_{c}}}$, in Eq. (118),
takes care of all possible contractions among the $P_{\mu}$’s. We present the
explicit expressions for all the relevant and necessary master integrals “
$\mathcal{I}$ ” for our calculation in the Appendix A.2.

###  6.2 A detailed case analysis: $O(P^{2}\,U^{6})$

The covariant diagram method maintains a record of all CDE terms and maps the
integral at each order to one-loop diagrams. To exemplify we focus on the
$O(P^{2}U^{6})$ that appears at $n=8$. Adhering to the ideas discussed in Ref.
Zhang:2016pja , we provide all possible diagrams for this operator class in
Table 1.

Diagram $O(P^{2}U^{6})$ structure Values
tr$\big{(}P_{\mu}UP_{\mu}UUUUU\big{)}$
$-ic_{s}\,2^{2}\,\mathcal{I}[q^{2}]^{8}$
tr$\big{(}P_{\mu}UUP_{\mu}UUUU\big{)}$
$-ic_{s}\,2^{2}\,\mathcal{I}[q^{2}]^{8}$
tr$\big{(}P_{\mu}UUUP_{\mu}UUU\big{)}$
$-i\frac{c_{s}}{2}\,2^{2}\,\mathcal{I}[q^{2}]^{8}$

Table 1: All possible diagrams at the level of $\mathcal{O}(P^{2}U^{6})$
containing two $P_{\mu}$’s that are contracted among themselves and six $U$’s.
In the second column, we present their corresponding operator structures with
open covariant derivatives. Their values are given in the third column.

In order to discuss how to compute these diagrams, we will concentrate on Fig.
1. Since in this category only two $P_{\mu}$’s are present, they must be
contracted among themselves to form a Lorentz-invariant structure. Following
the conventions of Zhang:2016pja , we denote the vertices containing
$P_{\mu}$’s and $U$’s with black and white blobs on the loop diagrams
respectively and the contraction of the $P_{\mu}$’s is shown with a dotted
line. The structure that is unique to the diagram then can be read off
starting from one particular blob and going clockwise until we exhaust all the
vertices present in the diagram. Following the rule, the structure
corresponding to this diagram can then be written as
$\text{tr}\,\big{(}P_{\mu}UUUP_{\mu}UUU\big{)}$.

Figure 1: Representative diagram at $\mathcal{O}(P^{2}U^{6})$.

Each of the black blobs corresponds to a $2q.P$ vertex factor, which leaves an
additional factor of $2$ along with $P_{\mu}$’s when the loop momentum
$q_{\mu}$’s are taken inside the integral $\mathcal{I}[q^{2n_{c}}]$. When the
diagram exhibits an $N$-fold rotational symmetry, we divide the total value of
the loop-integral with a factor of $N$555This division is necessary to
eliminate the overcounting when different operator structures under trace give
rise to the same diagram.. Thus, the contribution from this diagram reads as:
$-i\frac{c_{s}}{2}\,2^{2}\,\mathcal{I}[q^{2}]^{8}=i\frac{c_{s}}{2}\,2^{2}\times\big{(}\frac{i}{16\pi^{2}}\frac{1}{M^{10}}\frac{12}{7!}\big{)}$666It
should be noted that if a specific diagram and its mirror image cannot be
superimposed onto each other even after rotation under trace (see e.g. the
second diagram in Table 6), these diagrams are connected via Hermitian
conjugation. The conjugate diagrams receive exactly similar contributions as
their parent diagrams after the expansion of the covariant structures, so we
avoid writing them separately..

Along with this, two more independent diagrams can arise at this level. Table
1 contains all the diagrams, their corresponding structures with open
covariant derivatives, and individual contributions. The number of possible
structures appearing at each class implies that the same number of independent
covariant structures must be present where $P_{\mu}$’s only appear through
commutators. To verify the results obtained in Eq. (107), first, we start with
three distinct forms of the effective operators given in that equation. Then
we expand the commutators to encompass all the diagrams within this class as

$C^{(2,6)}_{1}\,\text{tr}\big{(}U^{4}\,[P_{\mu},U]\,[P_{\mu},U]\big{)}\,+\,C^{(2,6)}_{2}\,\text{tr}\big{(}U^{3}\,[P_{\mu},U]\,U\,[P_{\mu},U]\big{)}\\\
+\,C^{(2,6)}_{3}\,\text{tr}\big{(}U^{2}\,[P_{\mu},U]\,U^{2}\,[P_{\mu},U]\big{)}=\,\,(2C^{(2,6)}_{1}-C^{(2,6)}_{2})\,\text{tr}\,\big{(}P_{\mu}UP_{\mu}UUUUU\big{)}\\\
+\,(2C^{(2,6)}_{2}-2C^{(2,6)}_{3}-C^{(2,6)}_{1})\,\text{tr}\,\big{(}P_{\mu}UUP_{\mu}UUUU\big{)}\\\
+\,(2C^{(2,6)}_{3}-C^{(2,6)}_{2})\,\text{tr}\,\big{(}P_{\mu}UUUP_{\mu}UUU\big{)}\,+\,\text{tr}\big{(}\cdots
P^{2}\cdots\big{)}\text{terms}.$ (127)

The contraction of two adjacent $2q.P$ vertices and the contribution from
$(-P^{2})$ vertices can produce $\text{tr}\big{(}\cdots P^{2}\cdots\big{)}$
terms with the diagrams where the adjacent $P_{\mu}$’s are contracted. It is
not necessary to consider these diagrams separately since their coefficients
are functions of the same $C_{k}^{(i,j)}$’s which can be determined from other
diagrams. The coefficients of the structures in the RHS of Eq. (127),
correspond to values of the diagrams given in the third column of Table 1,

$\displaystyle 2C^{(2,6)}_{1}-C^{(2,6)}_{2}$ $\displaystyle=$
$\displaystyle-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{10}}\frac{48}{7!},$
$\displaystyle 2C^{(2,6)}_{2}-C^{(2,6)}_{1}-2C^{(2,6)}_{3}$ $\displaystyle=$
$\displaystyle-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{10}}\frac{48}{7!},$
$\displaystyle 2C^{(2,6)}_{3}-C^{(2,6)}_{2}$ $\displaystyle=$
$\displaystyle-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{10}}\frac{24}{7!}.$ (128)

Finally, we find the coefficients associated with the operators given in Eq.
(127) as

$C^{(2,6)}_{1}=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{10}}\frac{1}{42},\,\hskip
19.91684ptC^{(2,6)}_{2}=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{10}}\frac{4}{105},\hskip
19.91684ptC^{(2,6)}_{3}=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{10}}\frac{3}{140},$
(129)

comparing with Eq. (107), we can infer
$\mathcal{C}^{(2,6)}_{k}=(c_{s}/(16\pi^{2}))C^{(2,6)}_{k}$, which validates
our findings.

###  6.3 Coefficients obtained using Covaraint diagrams

####  $\bullet$ $O(P^{8})$

Starting with the covariant structures derived in Eq. (5.1), we expand them to
find the contributions to each of the diagrams of this class. The diagrams and
their corresponding values are given in Table 3,

$\displaystyle
C^{(8,0)}_{1}\,\text{tr}\Big{(}[P_{\mu},P_{\nu}]\,[P_{\mu},P_{\nu}][P_{\alpha},P_{\beta}][P_{\alpha},P_{\beta}]\Big{)}\,+\,C^{(8,0)}_{2}\,\text{tr}\Big{(}[P_{\mu},P_{\nu}]\,[P_{\nu},P_{\rho}][P_{\mu},P_{\sigma}][P_{\sigma},P_{\rho}]\Big{)}$
$\displaystyle+$
$\displaystyle\,C^{(8,0)}_{3}\,\text{tr}\Big{(}[P_{\mu},P_{\nu}]\,[P_{\nu},P_{\rho}][P_{\rho},P_{\sigma}][P_{\sigma},P_{\mu}]\Big{)}+\,C^{(8,0)}_{4}\,\text{tr}\Big{(}[P_{\mu},P_{\nu}]\,[P_{\rho},P_{\sigma}][P_{\mu},P_{\nu}][P_{\rho},P_{\sigma}]\Big{)}$
$\displaystyle+$
$\displaystyle\,C^{(8,0)}_{5}\,\text{tr}\Big{(}[P_{\mu},[P_{\alpha},[P_{\alpha},P_{\nu}]]]\,[P_{\mu},[P_{\rho},[P_{\rho},P_{\nu}]]]\Big{)}$
$\displaystyle+$
$\displaystyle\,C^{(8,0)}_{6}\,\text{tr}\Big{(}[P_{\mu},[P_{\alpha},[P_{\alpha},P_{\nu}]]]\,[P_{\nu},P_{\rho}][P_{\rho},P_{\mu}]\Big{)}$
$\displaystyle+$
$\displaystyle\,C^{(8,0)}_{7}\,\text{tr}\Big{(}[P_{\alpha},[P_{\alpha},P_{\mu}]]\,P_{\beta},[P_{\beta},P_{\nu}]]\,[P_{\mu},P_{\nu}]\Big{)}$
$\displaystyle\supset$
$\displaystyle\quad\Big{(}4C^{(8,0)}_{1}\,+\,2C^{(8,0)}_{3}\,+\,2C^{(8,0)}_{6}\,-\,4C^{(8,0)}_{7}\Big{)}\,\text{tr}\Big{(}P_{\mu}P_{\nu}P_{\mu}P_{\rho}P_{\sigma}P_{\nu}P_{\rho}P_{\sigma}\Big{)}$
$\displaystyle+$
$\displaystyle\,\Big{(}2C^{(8,0)}_{2}\,-\,2C^{(8,0)}_{6}\,+\,8C^{(8,0)}_{5}\Big{)}\,\text{tr}\Big{(}P_{\mu}P_{\nu}P_{\rho}P_{\nu}P_{\mu}P_{\sigma}P_{\rho}P_{\sigma}\Big{)}$
$\displaystyle+$
$\displaystyle\,\Big{(}2C^{(8,0)}_{2}\,-\,4C^{(8,0)}_{3}\,-\,4C^{(8,0)}_{6}\,+\,4C^{(8,0)}_{7}\Big{)}\,\text{tr}\Big{(}P_{\mu}P_{\nu}P_{\mu}P_{\rho}P_{\nu}P_{\sigma}P_{\rho}P_{\sigma}\Big{)}$
$\displaystyle-$
$\displaystyle\,\Big{(}4C^{(8,0)}_{2}\,-\,2C^{(8,0)}_{6}\Big{)}\,\text{tr}\Big{(}P_{\mu}P_{\nu}P_{\mu}P_{\rho}P_{\sigma}P_{\nu}P_{\rho}P_{\sigma}\Big{)}$
$\displaystyle+$
$\displaystyle\,\Big{(}C^{(8,0)}_{2}\,-\,8C^{(8,0)}_{4}\Big{)}\,\text{tr}\Big{(}P_{\mu}P_{\nu}P_{\rho}P_{\sigma}P_{\mu}P_{\nu}P_{\sigma}P_{\rho}\Big{)}$
$\displaystyle+$
$\displaystyle\,\Big{(}C^{(8,0)}_{3}\,+\,4C^{(8,0)}_{4}\Big{)}\,\text{tr}\,\Big{(}P_{\mu}P_{\nu}P_{\rho}P_{\mu}P_{\sigma}P_{\rho}P_{\nu}P_{\sigma}\Big{)}+\,4C^{(8,0)}_{4}\,\text{tr}\,\Big{(}P_{\mu}P_{\nu}P_{\rho}P_{\sigma}P_{\mu}P_{\nu}P_{\rho}P_{\sigma}\Big{)}.$
(130)

Equating the coefficients of the structures on the RHS of the Eq. (6.3) with
the values in Table 3, we find,

$\displaystyle C^{(8,0)}_{1}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{17}{5040},$
$\displaystyle C^{(8,0)}_{2}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{1}{420},$
$\displaystyle C^{(8,0)}_{3}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{1}{2520},$
$\displaystyle C^{(8,0)}_{4}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{1}{10080},$
$\displaystyle C^{(8,0)}_{5}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{1}{840},$
$\displaystyle C^{(8,0)}_{6}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{2}{315},$
$\displaystyle C^{(8,0)}_{7}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{2}{315}.$ (131)

From Eq. (5.1), one can check
$\mathcal{C}^{(8,0)}_{k}=(c_{s}/(16\pi^{2}))C^{(8,0)}_{k}$, which validates
our findings.

####  $\bullet$ $O(P^{6}\,U)$

The structures derived in Eq. (88) are expanded to map them back to the
diagrams of this class given in Table. 6,

$\displaystyle
C^{(6,1)}_{1}\,\text{tr}\Big{(}U\,[P_{\mu},P_{\nu}]\,[P_{\nu},P_{\rho}]\,[P_{\rho},P_{\mu}]\Big{)}\,+\,C^{(6,1)}_{2}\,\text{tr}\Big{(}U\,[P_{\nu},[P_{\nu},P_{\mu}]]\,[P_{\rho},[P_{\rho},P_{\mu}]]\Big{)}$
$\displaystyle+$
$\displaystyle\,C^{(6,1)}_{3}\,\text{tr}\Big{(}[P_{\alpha},[P_{\alpha},U]\,[P_{\mu},P_{\nu}]\,[P_{\mu},P_{\nu}]]\Big{)}+\,C^{(6,1)}_{4}\,\text{tr}\Big{(}[P_{\beta},[P_{\gamma},U]]\,[P_{\gamma},P_{\alpha}]\,[P_{\alpha},P_{\beta}]\Big{)}$
$\displaystyle\supset$
$\displaystyle\,\,-\Big{(}C^{(6,1)}_{1}+4C^{(6,1)}_{3}+C^{(6,1)}_{4}\Big{)}\,\text{tr}\Big{(}P_{\mu}P_{\nu}P_{\rho}P_{\mu}P_{\rho}P_{\nu}U\Big{)}\,$
$\displaystyle+$
$\displaystyle\,\Big{(}C^{(6,1)}_{1}+C^{(6,1)}_{4}\Big{)}\,\text{tr}\Big{(}P_{\mu}P_{\nu}P_{\rho}P_{\mu}P_{\rho}P_{\nu}U\Big{)}$
$\displaystyle+$
$\displaystyle\,\Big{(}C^{(6,1)}_{1}+4C^{(6,1)}_{2}-2C^{(6,1)}_{4}\Big{)}\,\text{tr}\Big{(}P_{\mu}P_{\nu}P_{\mu}P_{\rho}P_{\nu}P_{\rho}U\Big{)}\,-\,C^{(6,1)}_{1}\,\text{tr}\Big{(}P_{\mu}P_{\nu}P_{\rho}P_{\mu}P_{\nu}P_{\rho}U\Big{)}.$
(132)

We equate the coefficients of the structures with the open covariant
derivatives with their corresponding values and reproduce the results given in
Eq. (88),

$\displaystyle C^{(6,1)}_{1}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{1}{90},$
$\displaystyle C^{(6,1)}_{2}$
$\displaystyle=+\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{1}{60},$
$\displaystyle C^{(6,1)}_{3}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{1}{180},$
$\displaystyle C^{(6,1)}_{4}$
$\displaystyle=+\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{4}}\frac{1}{45}.$ (133)

With an appropriate scale factor, one can compare these coefficients to the
previously obtained coefficients shown in Eq. (88), the relation reads,
$\mathcal{C}^{(6,1)}_{k}=(c_{s}/(16\pi^{2}))C^{(6,1)}_{k}$.

####  $\bullet$ $O(P^{4}\,U^{3})$

We start with the structures obtained in Eq. (5.3). From Table 2, it can be
seen that there are six independent diagrams that can arise in this category.
Therefore, we assume that the covariant structures that have distinct
Hermitian conjugates (h.c.’s), both of them should have the same coefficients
to maintain the overall Hermiticity of the Lagrangian. In this way, we
encounter exactly six independent variables that can be solved from six
associated diagrams.

$\displaystyle
C^{(4,3)}_{1}\,\text{tr}\Big{(}U^{3}\,[P_{\mu},P_{\nu}]\,[P_{\mu},P_{\nu}]\,\Big{)}\,+\,C^{(4,3)}_{2}\,\text{tr}\Big{(}U^{2}\,[P_{\mu},P_{\nu}]\,U\,[P_{\mu},P_{\nu}]\Big{)}$
$\displaystyle+C^{(4,3)}_{3}\,\Bigg{\\{}\,\text{tr}\Big{(}U^{2}\,[P_{\mu},U]\,[P_{\nu},[P_{\nu},P_{\mu}]]\Big{)}-\text{tr}\Big{(}[P_{\mu},U]\,U^{2}\,[P_{\nu},[P_{\nu},P_{\mu}]]\Big{)}\Bigg{\\}}$
$\displaystyle+\,C^{(4,3)}_{4}\,\Bigg{\\{}\text{tr}\Big{(}[P_{\mu},U]\,[P_{\nu},U]\,U\,[P_{\mu},P_{\nu}]\Big{)}+\text{tr}\Big{(}U\,[P_{\mu},U]\,[P_{\nu},U]\,[P_{\mu},P_{\nu}]\Big{)}\Bigg{\\}}$
$\displaystyle+\,C^{(4,3)}_{5}\,\text{tr}\Big{(}U\,[P_{\mu}[P_{\mu},U]]\,[P_{\nu}[P_{\nu},U]]\Big{)}+\,C^{(4,3)}_{6}\,\text{tr}\Big{(}[P_{\mu}[P_{\mu},U]]\,[P_{\nu},U][P_{\nu},U]\Big{)}$
$\displaystyle\supset$
$\displaystyle\,\,\Big{(}2C^{(4,3)}_{1}+4C^{(4,3)}_{3}\Big{)}\,\text{tr}\Big{(}UUUP_{\nu}P_{\mu}P_{\nu}P_{\mu}\Big{)}\,+\,\Big{(}2C^{(4,3)}_{2}-2C^{(4,3)}_{4}\Big{)}\,\text{tr}\Big{(}UUP_{\mu}P_{\nu}UP_{\mu}P_{\nu}\Big{)}$
$\displaystyle+$
$\displaystyle\Big{(}-2C^{(4,3)}_{2}+2C^{(4,3)}_{4}+2C^{(4,3)}_{6}\Big{)}\,\text{tr}\Big{(}UUP_{\mu}P_{\nu}UP_{\nu}P_{\mu}\Big{)}\,-\,2C^{(4,3)}_{3}\,\text{tr}\Big{(}UUP_{\mu}UP_{\nu}P_{\mu}P_{\nu}\Big{)}$
$\displaystyle+$ $\displaystyle
2C^{(4,3)}_{4}\,\text{tr}\Big{(}UP_{\mu}UP_{\nu}UP_{\mu}P_{\nu}\Big{)}-\Big{(}2C^{(4,3)}_{4}-4C^{(4,3)}_{5}+4C^{(4,3)}_{6}\Big{)}\,\text{tr}\Big{(}UP_{\mu}UP_{\mu}P_{\nu}UP_{\nu}\Big{)}.$
(134)

It is evident from the covariant structures that the operators that have
distinct h.c.’s, both yield identical contributions to the mirror-symmetric
diagrams, while the conjugate operators produce the mirror images of the
diagrams that are not mirror-symmetric.

$\displaystyle C^{(4,3)}_{1}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\,\mathcal{C}^{(4,3)}_{1}=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{60},$
$\displaystyle C^{(4,3)}_{2}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\,\mathcal{C}^{(4,3)}_{2}=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{90},$
$\displaystyle C^{(4,3)}_{3}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\,\mathcal{C}^{(4,3)}_{3}=-\frac{c_{s}}{16\pi^{2}}\,\mathcal{C}^{(4,3)}_{4}=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{180},$
$\displaystyle C^{(4,3)}_{4}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\,\mathcal{C}^{(4,3)}_{5}=\frac{c_{s}}{16\pi^{2}}\,\mathcal{C}^{(4,3)}_{6}=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{180}$
$\displaystyle C^{(4,3)}_{5}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\,\mathcal{C}^{(4,3)}_{7}=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{60},$
$\displaystyle C^{(4,3)}_{6}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\,\mathcal{C}^{(4,3)}_{8}=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{90}.$
(135)

The coefficients match exactly with those obtained in Eq. (5.3).

####  $\bullet$ $O(P^{6}\,U^{2})$

Here, to begin with, we consider the operator structures given in Eq. (89)
derived using the Heat-Kernel method. Noting the possible independent
covariant diagrams, allowed for this class (see Tables 4 and 5), we anticipate
that there should be at most seventeen independent covariant operators
(excluding the h.c.’s). Regarding the h.c.’s, we follow the similar
prescription discussed in the case of $O(P^{4}\,U^{3})$. We express the last
two operators in Eq. (89) (i.e.,
$[P_{\alpha},P_{\mu}][P_{\mu},P_{\beta}]U\,[P_{\alpha},[P_{\beta},U]]$, and
$[P_{\alpha},P_{\mu}][P_{\mu},P_{\beta}][P_{\alpha},[P_{\beta},U]]U$) in terms
of other ones.

$\displaystyle
C^{(6,2)}_{1}\,\text{tr}\Big{(}U^{2}\,[P_{\mu},P_{\nu}]\,[P_{\nu},P_{\alpha}][P_{\alpha},P_{\mu}]\Big{)}\,+\,C^{(6,2)}_{2}\,\text{tr}\Big{(}U\,[P_{\mu},P_{\nu}]\,U\,[P_{\nu},P_{\alpha}][P_{\alpha},P_{\mu}]\Big{)}$
$\displaystyle+$
$\displaystyle\,C^{(6,2)}_{3}\,\text{tr}\Big{(}U^{2}\,[P_{\mu},[P_{\mu},P_{\nu}]]\,[P_{\alpha},[P_{\alpha},P_{\nu}]]\Big{)}\,+\,C^{(6,2)}_{4}\,\text{tr}\Big{(}U\,[P_{\mu},[P_{\mu},P_{\nu}]]\,U\,[P_{\alpha},[P_{\alpha},P_{\nu}]]\Big{)}$
$\displaystyle+$
$\displaystyle\,C^{(6,2)}_{5}\,\Bigg{\\{}\text{tr}\Big{(}U\,[P_{\mu},[P_{\mu},U]]\,[P_{\nu},P_{\alpha}]\,[P_{\nu},P_{\alpha}]\Big{)}+\text{tr}\Big{(}[P_{\mu},[P_{\mu},U]]\,U\,[P_{\nu},P_{\alpha}]\,[P_{\nu},P_{\alpha}]\Big{)}\Bigg{\\}}$
$\displaystyle+$
$\displaystyle\,C^{(6,2)}_{6}\,\Bigg{\\{}\text{tr}\Big{(}U\,[P_{\mu},U]\,[P_{\nu}[P_{\nu},P_{\alpha}]\,[P_{\mu},P_{\alpha}]\Big{)}+\text{tr}\Big{(}U\,[P_{\mu},P_{\alpha}]\,[P_{\nu}[P_{\nu},P_{\alpha}]\,[P_{\mu},U]\Big{)}\Bigg{\\}}$
$\displaystyle+$
$\displaystyle\,C^{(6,2)}_{7}\,\Bigg{\\{}\text{tr}\Big{(}U\,[P_{\mu},U]\,[P_{\mu},P_{\nu}]\,[P_{\alpha}[P_{\alpha},P_{\nu}]\Big{)}+\text{tr}\Big{(}U\,[P_{\alpha}[P_{\alpha},P_{\nu}]\,[P_{\mu},P_{\nu}]\,[P_{\mu},U]\Big{)}\Bigg{\\}}$
$\displaystyle+$ $\displaystyle
C^{(6,2)}_{8}\,\text{tr}\Big{(}U\,[P_{\mu},P_{\nu}]\,[P_{\alpha},[P_{\alpha},U]\,[P_{\mu},P_{\nu}]\Big{)}\,+\,C^{(6,2)}_{9}\,\text{tr}\Big{(}\,[P_{\mu},U]\,[P_{\mu},U]\,[P_{\nu},P_{\alpha}]\,[P_{\nu},P_{\alpha}]\Big{)}$
$\displaystyle+$
$\displaystyle\,C^{(6,2)}_{10}\,\Bigg{\\{}\text{tr}\Big{(}\,[P_{\mu},U]\,[P_{\nu},[P_{\nu},P_{\alpha}]]\,U\,[P_{\mu},P_{\alpha}]\Big{)}+\text{tr}\Big{(}\,[P_{\mu},P_{\alpha}]\,U\,[P_{\nu},[P_{\nu},P_{\alpha}]]\,[P_{\mu},U]\Big{)}\Bigg{\\}}$
$\displaystyle+$
$\displaystyle\,C^{(6,2)}_{11}\,\text{tr}\Big{(}\,[P_{\mu},U]\,[P_{\nu},U]\,[P_{\mu},P_{\alpha}]\,[P_{\alpha},P_{\nu}]\Big{)}+\,C^{(6,2)}_{12}\,\text{tr}\Big{(}\,[P_{\mu},U]\,[P_{\nu},U]\,[P_{\nu},P_{\alpha}]\,[P_{\alpha},P_{\mu}]\Big{)}$
$\displaystyle+$
$\displaystyle\,C^{(6,2)}_{13}\,\Bigg{\\{}\text{tr}\Big{(}\,[P_{\mu},U]\,[P_{\mu},P_{\nu}]\,[P_{\alpha},U]\,[P_{\alpha},P_{\nu}]\Big{)}+\text{tr}\Big{(}\,[P_{\mu},U]\,[P_{\alpha},P_{\nu}]\,[P_{\alpha},U]\,[P_{\mu},P_{\nu}]\Big{)}\Bigg{\\}}$
$\displaystyle+$
$\displaystyle\,C^{(6,2)}_{14}\,\text{tr}\Big{(}\,[P_{\mu},U]\,[P_{\nu},P_{\alpha}]\,[P_{\mu},U]\,[P_{\nu},P_{\alpha}]\Big{)}\,+\,C^{(6,2)}_{15}\,\Bigg{\\{}\text{tr}\Big{(}\,[P_{\mu},[P_{\mu},U]]\,[P_{\nu},U]\,[P_{\alpha},[P_{\alpha},P_{\nu}]]\Big{)}$
$\displaystyle-$
$\displaystyle\text{tr}\Big{(}\,[P_{\mu},[P_{\mu},U]]\,[P_{\alpha},[P_{\alpha},P_{\nu}]]\,[P_{\nu},U]\Big{)}\Bigg{\\}}\,+\,C^{(6,2)}_{16}\,\Bigg{\\{}\text{tr}\Big{(}\,[P_{\mu},U]\,[P_{\nu},U]\,[P_{\mu},[P_{\alpha},[P_{\alpha},P_{\nu}]]]\Big{)}$
$\displaystyle-$
$\displaystyle\text{tr}\Big{(}\,[P_{\mu},U]\,[P_{\mu},[P_{\alpha},[P_{\alpha},P_{\nu}]]]\,[P_{\nu},U]\Big{)}\Bigg{\\}}+2\,C^{(6,2)}_{17}\,\text{tr}\Big{(}\,[P_{\mu},[P_{\nu},[P_{\nu},U]]]\,[P_{\mu},[P_{\alpha},[P_{\alpha},U]]]\Big{)}$
$\displaystyle\supset$
$\displaystyle-(C_{1}+C_{11})\,\text{tr}\,\Big{(}P_{\mu}P_{\nu}P_{\rho}UUP_{\mu}P_{\nu}P_{\rho}\Big{)}+\,(C_{1}\,+\,4C_{3}\,-\,4C_{7}\,-C_{12})\,\text{tr}\,\Big{(}P_{\mu}P_{\nu}P_{\mu}UUP_{\rho}P_{\nu}P_{\rho}\Big{)}$
$\displaystyle-$
$\displaystyle(C_{1}\,+\,2C_{9}\,+\,4C_{6})\,\text{tr}\,\Big{(}P_{\mu}P_{\nu}P_{\rho}UUP_{\rho}P_{\mu}P_{\nu}\Big{)}$
$\displaystyle+$
$\displaystyle(C_{1}+2C_{6}+C_{11}+2C_{16})\,\text{tr}\,\Big{(}P_{\mu}P_{\nu}P_{\rho}UUP_{\nu}P_{\mu}P_{\rho}\Big{)}$
$\displaystyle+$
$\displaystyle(4C_{4}-4C_{10}+2C_{13})\,\text{tr}\,\Big{(}UP_{\mu}P_{\nu}P_{\mu}UP_{\rho}P_{\nu}P_{\rho}\Big{)}+4C_{14}\,\text{tr}\,\Big{(}UP_{\mu}P_{\nu}P_{\rho}UP_{\mu}P_{\nu}P_{\rho}\Big{)}$
$\displaystyle-$
$\displaystyle(2C_{14}-C_{13})\,\text{tr}\,\Big{(}UP_{\mu}P_{\nu}P_{\rho}UP_{\nu}P_{\mu}P_{\rho}\Big{)}-(4C_{8}+4C_{13})\,\text{tr}\,\Big{(}UP_{\mu}P_{\nu}P_{\rho}UP_{\rho}P_{\mu}P_{\nu}\Big{)}$
$\displaystyle+$
$\displaystyle(4C_{8}+2C_{13}+8C_{17})\,\text{tr}\,\Big{(}UP_{\mu}P_{\nu}P_{\rho}UP_{\rho}P_{\nu}P_{\mu}\Big{)}$
$\displaystyle-$
$\displaystyle(\frac{1}{2}C_{2}+\frac{1}{2}C_{11}+2C_{14})\,\text{tr}\,\Big{(}UP_{\mu}P_{\nu}UP_{\rho}P_{\mu}P_{\nu}P_{\rho}\Big{)}$
$\displaystyle+$
$\displaystyle(C_{2}-C_{12}+4C_{14})\,\text{tr}\,\Big{(}UP_{\mu}P_{\nu}UP_{\rho}P_{\nu}P_{\mu}P_{\rho}\Big{)}$
$\displaystyle+$
$\displaystyle(C_{2}+C_{11}+2C_{10}+2C_{16}-2C_{13})\,\text{tr}\,\Big{(}UP_{\mu}P_{\nu}UP_{\mu}P_{\rho}P_{\nu}P_{\rho}\Big{)}$
$\displaystyle-$
$\displaystyle(C_{2}+2C_{10}-C_{12}-2C_{13}-4C_{15}+2C_{16})\,\text{tr}\,\Big{(}UP_{\mu}P_{\nu}UP_{\nu}P_{\rho}P_{\mu}P_{\rho}\Big{)}$
$\displaystyle+$ $\displaystyle
2C_{11}\,\text{tr}\,\Big{(}P_{\mu}P_{\nu}P_{\rho}UP_{\mu}UP_{\nu}P_{\rho}\Big{)}$
$\displaystyle+$
$\displaystyle(2C_{6}+2C_{9}-C_{12}-4C_{15}+2C_{16}-4C_{5}-2C_{7})\,\text{tr}\,\Big{(}P_{\mu}UP_{\mu}UP_{\nu}P_{\rho}P_{\nu}P_{\rho}\Big{)}$
$\displaystyle-$
$\displaystyle(2C_{6}+C_{11}-C_{12}-2C_{7}+2C_{16})\,\text{tr}\,\Big{(}P_{\mu}UP_{\nu}UP_{\rho}P_{\mu}P_{\rho}P_{\nu}\Big{)}$
$\displaystyle-$
$\displaystyle(2C_{11}+4C_{16})\,\text{tr}\,\Big{(}P_{\mu}P_{\nu}P_{\rho}P_{\nu}P_{\mu}UP_{\rho}U\Big{)}$
(136)

The coefficients in the LHS of the Eq. (6.3) can be obtained from the values
of the loop diagrams,

$\displaystyle C^{(6,2)}_{1}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{210},$
$\displaystyle C^{(6,2)}_{2}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{2}{315},$
$\displaystyle C^{(6,2)}_{3}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{105},$
$\displaystyle C^{(6,2)}_{4}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{140},$
$\displaystyle C^{(6,2)}_{5}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{105},$
$\displaystyle C^{(6,2)}_{6}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{210},$
$\displaystyle C^{(6,2)}_{7}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{105},$
$\displaystyle C^{(6,2)}_{8}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{315},$
$\displaystyle C^{(6,2)}_{9}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{11}{1260},$
$\displaystyle C^{(6,2)}_{10}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{126},$
$\displaystyle C^{(6,2)}_{11}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{630},$
$\displaystyle C^{(6,2)}_{12}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{126},$
$\displaystyle C^{(6,2)}_{13}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{420},$
$\displaystyle C^{(6,2)}_{14}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{2520},$
$\displaystyle C^{(6,2)}_{15}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{315},$
$\displaystyle C^{(6,2)}_{16}$
$\displaystyle=\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{630},$
$\displaystyle C^{(6,2)}_{17}$
$\displaystyle=-\frac{c_{s}}{16\pi^{2}}\frac{1}{M^{6}}\frac{1}{840}.$ (137)

##  7 Universal One-loop Effective Lagrangian up to $D8$

Relying on the computation based on the Heat-Kernel method and supported by
the covariant diagram technique, we exhaustively compute all possible operator
structures that can emerge after integrating out degenerate heavy scalars at
the one-loop considering only heavy propagators in the loop. We collect all
such terms and provide the universal one-loop effective Lagrangian up to
dimension eight. This effective action does not necessarily depend on either
the specific UV or low energy theories, and in that sense it is universal.

$\displaystyle\mathcal{L}_{\text{eff}}^{d\leq 8}=$
$\displaystyle\mathcal{L}_{\text{eff}}^{\text{ren}}+\cfrac{c_{s}}{(4\pi)^{2}}\Bigg{[}\mathcal{L}^{(8,0)}_{\text{eff}}+\mathcal{L}^{(6,0)}_{\text{eff}}+\mathcal{L}^{(6,1)}_{\text{eff}}+\mathcal{L}^{(6,2)}_{\text{eff}}+\mathcal{L}^{(4,0)}_{\text{eff}}+\mathcal{L}^{(4,1)}_{\text{eff}}+\mathcal{L}^{(4,2)}_{\text{eff}}+\mathcal{L}^{(4,3)}_{\text{eff}}+\mathcal{L}^{(4,4)}_{\text{eff}}$
$\displaystyle+\mathcal{L}^{(2,0)}_{\text{eff}}+\mathcal{L}^{(2,1)}_{\text{eff}}+\mathcal{L}^{(2,2)}_{\text{eff}}+\mathcal{L}^{(2,3)}_{\text{eff}}+\mathcal{L}^{(2,4)}_{\text{eff}}+\mathcal{L}^{(2,5)}_{\text{eff}}+\mathcal{L}^{(2,6)}_{\text{eff}}+\mathcal{L}^{(0,0)}_{\text{eff}}+\mathcal{L}^{(0,1)}_{\text{eff}}$
$\displaystyle+\mathcal{L}^{(0,2)}_{\text{eff}}+\mathcal{L}^{(0,3)}_{\text{eff}}+\mathcal{L}^{(0,4)}_{\text{eff}}+\mathcal{L}^{(0,5)}_{\text{eff}}+\mathcal{L}^{(0,6)}_{\text{eff}}+\mathcal{L}^{(0,7)}_{\text{eff}}+\mathcal{L}^{(0,8)}_{\text{eff}}\Bigg{]}$
$\displaystyle=$ $\displaystyle\cfrac{c_{s}}{(4\pi)^{2}}M^{4}\
\left[-\frac{1}{2}\,\left(\ln\left[\frac{M^{2}}{\mu^{2}}\right]-\frac{3}{2}\right)\right]+\cfrac{c_{s}}{(4\pi)^{2}}\tr\,\Bigg{\\{}M^{2}\
\Bigg{[}-\left(\ln\left[\frac{M^{2}}{\mu^{2}}\right]-1\right)\,U\Bigg{]}$
$\displaystyle+M^{0}\
\frac{1}{2}\Bigg{[}-\ln\left[\frac{M^{2}}{\mu^{2}}\right]\,U^{2}-\frac{1}{6}\ln\left[\frac{M^{2}}{\mu^{2}}\right]\,(G_{\mu\nu})^{2}\Bigg{]}$
$\displaystyle+\frac{1}{M^{2}}\frac{1}{6}\,\Bigg{[}-U^{3}-\frac{1}{2}(P_{\mu}U)^{2}-\frac{1}{2}U\,(G_{\mu\nu})^{2}-\frac{1}{10}(J_{\nu})^{2}+\frac{1}{15}\,G_{\mu\nu}\,G_{\nu\rho}\,G_{\rho\mu}\Bigg{]}$
$\displaystyle+\frac{1}{M^{4}}\frac{1}{24}\,\Bigg{[}U^{4}-U^{2}(P^{2}U)+\frac{4}{5}U^{2}(G_{\mu\nu})^{2}+\frac{1}{5}(U\,G_{\mu\nu})^{2}+\frac{1}{5}(P^{2}U)^{2}$
$\displaystyle\hskip
56.9055pt-\frac{2}{5}U\,(P_{\mu}U)\,J_{\mu}+\frac{2}{5}U(J_{\mu})^{2}-\frac{2}{15}(P^{2}U)(G_{\rho\sigma})^{2}+\frac{1}{35}(P_{\nu}J_{\mu})^{2}$
$\displaystyle\hskip
56.9055pt-\frac{4}{15}U\,G_{\mu\nu}G_{\nu\rho}G_{\rho\mu}-\frac{8}{15}(P_{\mu}P_{\nu}U)\,G_{\rho\mu}G_{\rho\nu}+\frac{16}{105}G_{\mu\nu}J_{\mu}J_{\nu}$
$\displaystyle\hskip
56.9055pt+\frac{1}{420}(G_{\mu\nu}G_{\rho\sigma})^{2}+\frac{17}{210}(G_{\mu\nu})^{2}(G_{\rho\sigma})^{2}+\frac{2}{35}(G_{\mu\nu}G_{\nu\rho})^{2}$
$\displaystyle\hskip
56.9055pt+\frac{1}{105}G_{\mu\nu}G_{\nu\rho}G_{\rho\sigma}G_{\sigma\mu}+\frac{16}{105}(P_{\mu}J_{\nu})G_{\nu\sigma}G_{\sigma\mu}\Bigg{]}$
$\displaystyle+\frac{1}{M^{6}}\frac{1}{60}\,\Bigg{[}-U^{5}+2\,U^{3}(P^{2}U)+U^{2}(P_{\mu}U)^{2}-\frac{2}{3}U^{2}G_{\mu\nu}U\,G_{\mu\nu}-U^{3}(G_{\mu\nu})^{2}$
$\displaystyle\hskip
56.9055pt+\frac{1}{3}U^{2}(P_{\mu}U)J_{\mu}-\frac{1}{3}U\,(P_{\mu}U)(P_{\nu}U)\,G_{\mu\nu}-\frac{1}{3}U^{2}J_{\mu}(P_{\mu}U)$
$\displaystyle\hskip
56.9055pt-\frac{1}{3}U\,G_{\mu\nu}(P_{\mu}U)(P_{\nu}U)-U\,(P^{2}U)^{2}-\frac{2}{3}(P^{2}U)(P_{\nu}U)^{2}-\frac{1}{7}((P_{\mu}U)G_{\mu\alpha})^{2}$
$\displaystyle\hskip
56.9055pt+\frac{2}{7}U^{2}G_{\mu\nu}G_{\nu\alpha}G_{\alpha\mu}+\frac{8}{21}U\,G_{\mu\nu}U\,G_{\nu\alpha}G_{\alpha\mu}-\frac{4}{7}U^{2}(J_{\mu})^{2}-\frac{3}{7}(U\,J_{\mu})^{2}$
$\displaystyle\hskip
56.9055pt+\frac{4}{7}U\,(P^{2}U)(G_{\mu\nu})^{2}+\frac{4}{7}(P^{2}U)U(G_{\mu\nu})^{2}-\frac{2}{7}U\,(P_{\mu}U)J_{\nu}G_{\mu\nu}$
$\displaystyle\hskip
56.9055pt-\frac{2}{7}(P_{\mu}U)U\,G_{\mu\nu}J_{\nu}-\frac{4}{7}U\,(P_{\mu}U)G_{\mu\nu}J_{\nu}-\frac{4}{7}(P_{\mu}U)U\,J_{\nu}G_{\mu\nu}$
$\displaystyle\hskip
56.9055pt+\frac{4}{21}U\,G_{\mu\nu}(P^{2}U)G_{\mu\nu}+\frac{11}{21}(P_{\alpha}U)^{2}(G_{\mu\nu})^{2}-\frac{10}{21}(P_{\mu}U)J_{\nu}U\,G_{\mu\nu}$
$\displaystyle\hskip
56.9055pt-\frac{10}{21}(P_{\mu}U)G_{\mu\nu}U\,J_{\nu}-\frac{2}{21}(P_{\mu}U)(P_{\nu}U)G_{\mu\alpha}G_{\alpha\nu}+\frac{10}{21}(P_{\nu}U)(P_{\mu}U)G_{\mu\alpha}G_{\alpha\nu}$
$\displaystyle\hskip
56.9055pt-\frac{1}{7}(G_{\alpha\mu}(P_{\mu}U))^{2}-\frac{1}{42}((P_{\alpha}U)G_{\mu\nu})^{2}-\frac{1}{14}(P_{\mu}P^{2}U)^{2}-\frac{4}{21}(P^{2}U)(P_{\mu}U)J_{\mu}$
$\displaystyle\hskip
56.9055pt+\frac{4}{21}(P_{\mu}U)(P^{2}U)J_{\mu}+\frac{2}{21}(P_{\mu}U)(P_{\nu}U)(P_{\mu}J_{\nu})-\frac{2}{21}(P_{\nu}U)(P_{\mu}U)(P_{\mu}J_{\nu})\Bigg{]}$
$\displaystyle+\frac{1}{M^{8}}\frac{1}{120}\,\Bigg{[}U^{6}-3\,U^{4}(P^{2}U)-2\,U^{3}(P_{\nu}U)^{2}+\frac{12}{7}U^{2}(P_{\mu}P_{\nu}U)(P_{\nu}P_{\mu}U)$
$\displaystyle\hskip
14.22636pt+\frac{26}{7}(P_{\mu}P_{\nu}U)U\,(P_{\mu}U)(P_{\nu}U)+\frac{26}{7}(P_{\mu}P_{\nu}U)(P_{\mu}U)(P_{\nu}U)U+\frac{9}{7}(P_{\mu}U)^{2}(P_{\nu}U)^{2}$
$\displaystyle\hskip
56.9055pt+\frac{9}{7}U\,(P_{\mu}P_{\nu}U)U\,(P_{\nu}P_{\mu}U)+\frac{17}{14}((P_{\mu}U)(P_{\nu}U))^{2}+\frac{8}{7}U^{3}G_{\mu\nu}U\,G_{\mu\nu}$
$\displaystyle\hskip
56.9055pt+\frac{5}{7}U^{4}(G_{\mu\nu})^{2}+\frac{18}{7}G_{\mu\nu}(P_{\mu}U)U^{2}(P_{\nu}U)+\frac{9}{14}(U^{2}G_{\mu\nu})^{2}$
$\displaystyle\hskip
56.9055pt+\frac{18}{7}G_{\mu\nu}U\,(P_{\mu}U)(P_{\nu}U)U+\frac{18}{7}(P_{\mu}P_{\nu}U)(P_{\mu}U)U\,(P_{\nu}U)$
$\displaystyle\hskip
56.9055pt+\Bigg{(}\frac{8}{7}G_{\mu\nu}U\,(P_{\mu}U)U\,(P_{\nu}U)+\frac{26}{7}G_{\mu\nu}(P_{\mu}U)U\,(P_{\nu}U)U\Bigg{)}$
$\displaystyle\hskip
56.9055pt+\Bigg{(}\frac{24}{7}G_{\mu\nu}(P_{\mu}U)(P_{\nu}U)U^{2}-\frac{2}{7}G_{\mu\nu}U^{2}(P_{\mu}U)(P_{\nu}U)\Bigg{)}\Bigg{]}$
$\displaystyle+\frac{1}{M^{10}}\frac{1}{210}\,\Bigg{[}-U^{7}-5\,U^{4}(P_{\nu}U)^{2}-8\,U^{3}(P_{\mu}U)U(P_{\mu}U)-\frac{9}{2}(U^{2}(P_{\mu}U))^{2}\Bigg{]}$
$\displaystyle+\frac{1}{M^{12}}\frac{1}{336}\,\Bigg{[}U^{8}\Bigg{]}\Bigg{\\}}.$
(138)

Here, we consider that the tensors $G_{\mu\nu}$ and $J_{\mu}$ are functions of
$P$: $G_{\mu\nu}=[P_{\mu},P_{\nu}]$, and
$J_{\mu}=P_{\nu}G_{\nu\mu}=[P_{\nu},[P_{\nu},P_{\mu}]]$. Please note that the
hermitian conjugates are already fed in the above expression such that the
effective Lagrangian is self-hermitian. We agree with the effective action up
to dimension six computed using the functional method Henning:2014wua  and
covariant diagram Zhang:2016pja .

###  7.1 Dimension of $U$ vs Dimension of Operator

We note that the dimension of scalar functional $U$ does not always reflect
the dimension of the emerged operator consisting of light fields $(\phi)$. As
$U$ is defined as a double functional derivative of action w.r.to. the heavy
fields $(\Phi_{i})$, its mass dimension (in a four-dimensional space-time
case) is always $+2$. Thus, $U$ may contain a single light scalar field
accompanied by a non-zero mass-dimensional coupling. In that case,
identification of the operator’s mass dimension by naive power counting of $U$
may be misleading. For example, the structure $\mathcal{O}(P^{2m}U^{n})$ has
mass dimension $2(m+n)$, with $m,n\in\mathbb{Z}^{+}$. But an operator of the
mass dimension $2(m+n)$ can also be generated from all the structures up to
$\mathcal{O}(P^{2m}U^{2n})$. We demonstrate this notion through a simple toy
example where the scalar potential takes the following form

$V_{\text{scalar}}\supset\lambda\,(\phi^{\dagger}\phi)^{2}+\lambda_{1}\,(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\lambda_{2}\,(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}\,(\phi^{\dagger}\phi)(\Phi_{1}^{\dagger}\Phi_{1})\\\
+\lambda_{4}\,(\phi^{\dagger}\phi)(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{5}\,(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\big{(}\kappa\,\phi^{\dagger}(\Phi_{1}^{\dagger}\Phi_{2})+\text{h.c.}\big{)}.$
(141)

The explicit structure of $U$ can be obtained through,

$U[\phi]=\begin{bmatrix}\cfrac{\delta^{2}\,V_{\text{scalar}}}{\delta\Phi_{1}^{\dagger}\delta\Phi_{1}}&\cfrac{\delta^{2}\,V_{\text{scalar}}}{\delta\Phi_{1}^{\dagger}\delta\Phi_{2}}\\\
\cfrac{\delta^{2}\,V_{\text{scalar}}}{\delta\Phi_{2}^{\dagger}\delta\Phi_{1}}&\cfrac{\delta^{2}\,V_{\text{scalar}}}{\delta\Phi_{2}^{\dagger}\delta\Phi_{2}}\end{bmatrix}\rule[-30.0pt]{1.0pt}{70.0pt}_{\,\begin{subarray}{c}\Phi_{1}=\Phi_{1,c}[\phi],\\\
\Phi_{2}=\Phi_{2,c}[\phi]\end{subarray}}.$ (142)

As the Lagrangian does not contain terms linear in each of the heavy fields,
their classical solutions correspond to $\Phi_{1,c}=\Phi_{2,c}=0$. So the
final form of $U$ can be written as

$U[\phi]=\begin{bmatrix}\lambda_{3}\,(\phi^{\dagger}\phi)&\kappa\,\phi^{\dagger}\\\
\kappa^{*}\,\phi&\lambda_{4}\,(\phi^{\dagger}\phi)\end{bmatrix}.$ (143)

Now, we compute the total contributions to the dimension eight operator
$(\phi^{\dagger}\phi)^{4}$

$\displaystyle\mathcal{L}_{\text{eff}}^{(0,4)}$ $\displaystyle\supset$
$\displaystyle\frac{1}{24\,M^{4}}\,(\lambda_{3}^{4}+\lambda_{4}^{4})\,(\phi^{\dagger}\phi)^{4}\sim\mathcal{O}(U^{4}),$
$\displaystyle\mathcal{L}_{\text{eff}}^{(0,5)}$ $\displaystyle\supset$
$\displaystyle-\frac{1}{12\,M^{6}}\,(|\kappa|^{2}\lambda_{3}^{3}+|\kappa|^{2}\lambda_{3}^{2}\lambda_{4}+|\kappa|^{2}\lambda_{3}\lambda_{4}^{2}+|\kappa|^{2}\lambda_{4}^{3})\,(\phi^{\dagger}\phi)^{4}\sim\mathcal{O}(U^{5}),$
$\displaystyle\mathcal{L}_{\text{eff}}^{(0,6)}$ $\displaystyle\supset$
$\displaystyle\frac{1}{40\,M^{8}}\,(3|\kappa|^{4}\lambda_{3}^{2}+4|\kappa|^{4}\lambda_{3}\lambda_{4}+3|\kappa|^{4}\lambda_{4}^{2})\,(\phi^{\dagger}\phi)^{4}\sim\mathcal{O}(U^{6}),$
$\displaystyle\mathcal{L}_{\text{eff}}^{(0,7)}$ $\displaystyle\supset$
$\displaystyle-\frac{1}{30\,M^{10}}\,(|\kappa|^{6}\lambda_{3}+|\kappa|^{6}\lambda_{4})\,(\phi^{\dagger}\phi)^{4}\sim\mathcal{O}(U^{7}),$
$\displaystyle\mathcal{L}_{\text{eff}}^{(0,8)}$ $\displaystyle\supset$
$\displaystyle\frac{1}{168\,M^{12}}\,|\kappa|^{8}\,(\phi^{\dagger}\phi)^{4}\sim\mathcal{O}(U^{8}).$
(144)

As the mass dimension of $U$ is $+2$, it was expected to have a solitary
contribution from $U^{4}$ to the dimension eight operators
$(\phi^{\dagger}\phi)^{4}$. But, that is not certainly true and we find
contributions to the same operator from $U^{5,6,7,8}$ all the structures. As
an example, one can find a similar effect when the SM is extended by another
Higgs Doublet (2HDM) and a gauge singlet scalar and low energy theory is
SMEFT.

##  8 Conclusions

Effective Field Theory (EFT) has drawn much attention in recent times by its
virtue. We are working to achieve more and more precision in EFT calculations.
Also, higher dimensional effective operators have their own signatures and
impact to crack the degeneracy among different UV theories as well. At this
point, the computation of dimension eight effective operators will have a very
large impact on our ongoing analysis. In this paper, we have stepped in that
direction. We have enabled the Heat-Kernel (HK) method to compute the
dimension eight one-loop effective Lagrangian after integrating out either a
heavy or multiple degenerate heavy scalars. Our results do not depend on the
form of either the UV or low energy theories. In this paper, we have computed
the contributions from the loops that consist of only heavy scalar
propagators. We have also employed the covariant diagram method to cross-check
part of our results. These two methods complement each other to validate our
results. We are in the process of extending this result by including heavy
fermions and light-heavy propagator mixing in future works.

## Acknowledgements

We acknowledge the useful discussions with Diptarka Das and Nilay Kundu. The
authors would also like to acknowledge the initial discussions with Priyank
Kaushik.

##  Appendix A 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.
Zhang:2016pja ; Vandeven:1985 . As shown in Eq. (1), the one-loop part of the
effective action for a field can be given as,

$\Delta S_{\text{eff}}=ic_{s}\,\text{Tr}\log(-P^{2}+M^{2}+U),$ (145)

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,

$\displaystyle\int d^{d}x\,\mathcal{L}_{\text{eff}}[\phi]$ $\displaystyle=$
$\displaystyle
ic_{s}\,\int\,\frac{d^{d}q}{(2\pi)^{d}}\bra{q}\text{tr}\log(-\widehat{P}^{2}+M^{2}+U)\ket{q}$
(146) $\displaystyle=$ $\displaystyle ic_{s}\int
d^{d}x\int\frac{d^{d}q}{(2\pi)^{d}}\innerproduct{q}{x}\bra{x}\text{tr}\log(-\widehat{P}^{2}+M^{2}+U)\ket{q}$
$\displaystyle=$ $\displaystyle ic_{s}\int
d^{d}x\int\frac{d^{d}q}{(2\pi)^{d}}\,e^{i\,q.x}\,\text{tr}\log(-P^{2}+M^{2}+U)\,e^{-i\,q.x}.$

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,

$\displaystyle\mathcal{L}_{\text{eff}}[\phi]$
$\displaystyle=ic_{s}\int\frac{d^{d}q}{(2\pi)^{d}}\,\text{tr}\log(-P^{2}+M^{2}+U)_{P\to
P-q}$
$\displaystyle=ic_{s}\int\frac{d^{d}q}{(2\pi)^{d}}\text{tr}\log(-P^{2}-q^{2}+2q.P+M^{2}+U)$
$\displaystyle\begin{split}&=ic_{s}\int\frac{d^{d}q}{(2\pi)^{d}}\text{tr}\bigg{\\{}\log(-q^{2}+M^{2})\\\
&\qquad\qquad\qquad+\log[1-(q^{2}-M^{2})^{-1}(-P^{2}+2q.P+U)\big{]}\bigg{\\}}.\end{split}$
(147)

After performing the momentum integral, the first term in Eq. (A) reduces to a
constant, while the second term can be expanded in an infinite series as shown
in Eq. (117).

###  A.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.

#### $\mathbf{O(P^{4}\,U^{3})}$

Diagram $O(P^{4}U^{3})$ structure Value
tr$\big{(}P_{\mu}UP_{\nu}P_{\mu}P_{\nu}UU\big{)}$
$-ic_{s}\,2^{4}\mathcal{I}[q^{4}]^{7}$
tr$\big{(}UUUP_{\mu}P_{\nu}P_{\mu}P_{\nu}\big{)}$
$-ic_{s}\,2^{4}\mathcal{I}[q^{4}]^{7}$
tr$\big{(}UP_{\mu}P_{\nu}UUP_{\nu}P_{\mu}\big{)}$
$-ic_{s}\,2^{4}\mathcal{I}[q^{4}]^{7}$
tr$\big{(}UP_{\mu}P_{\nu}UUP_{\mu}P_{\nu}\big{)}$
$-ic_{s}\,2^{4}\mathcal{I}[q^{4}]^{7}$
tr$\big{(}P_{\mu}UP_{\nu}UP_{\mu}P_{\nu}U\big{)}$
$-ic_{s}\,2^{4}\mathcal{I}[q^{4}]^{7}$
tr$\big{(}P_{\mu}UP_{\nu}UP_{\nu}P_{\mu}U\big{)}$
$-ic_{s}\,2^{4}\mathcal{I}[q^{4}]^{7}$

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

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

Diagram $O(P^{8})$ structure Value
tr$\big{(}P_{\mu}P_{\nu}P_{\mu}P_{\nu}P_{\rho}P_{\sigma}P_{\rho}P_{\sigma}\big{)}$
$-i\frac{c_{s}}{2}\,2^{8}\mathcal{I}[q^{8}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\mu}P_{\rho}P_{\sigma}P_{\nu}P_{\rho}P_{\sigma}\big{)}$
$-ic_{s}\,2^{8}\mathcal{I}[q^{8}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\mu}P_{\rho}P_{\nu}P_{\sigma}P_{\rho}P_{\sigma}\big{)}$
$-ic_{s}\,2^{8}\mathcal{I}[q^{8}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}P_{\sigma}P_{\mu}P_{\nu}P_{\sigma}P_{\rho}\big{)}$
$-i\frac{c_{s}}{2}\,2^{8}\mathcal{I}[q^{8}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}P_{\nu}P_{\mu}P_{\sigma}P_{\rho}P_{\sigma}\big{)}$
$-i\frac{c_{s}}{2}\,2^{8}\mathcal{I}[q^{8}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}P_{\mu}P_{\sigma}P_{\rho}P_{\nu}P_{\sigma}\big{)}$
$-i\frac{c_{s}}{4}\,2^{8}\mathcal{I}[q^{8}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}P_{\sigma}P_{\mu}P_{\nu}P_{\rho}P_{\sigma}\big{)}$
$-i\frac{c_{s}}{8}\,2^{8}\mathcal{I}[q^{8}]^{8}$

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.

#### $\mathbf{O(P^{6}\,U^{2})}$

Diagram $O(P^{6}\,U^{2})$ structure Value
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}UUP_{\mu}P_{\nu}P_{\rho}\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\mu}UUP_{\rho}P_{\nu}P_{\rho}\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}UUP_{\rho}P_{\mu}P_{\nu}\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}UUP_{\mu}P_{\rho}P_{\nu}\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\mu}UP_{\rho}P_{\nu}P_{\rho}U\big{)}$
$-i\frac{c_{s}}{2}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}UP_{\mu}P_{\nu}P_{\rho}U\big{)}$
$-i\frac{c_{s}}{2}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}UP_{\nu}P_{\mu}P_{\rho}U\big{)}$
$-i\frac{c_{s}}{2}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}UP_{\rho}P_{\mu}P_{\nu}U\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$

Table 4: All possible diagrams at the level of $O(P^{6}U^{2})$ 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.

Diagram $O(P^{6}U^{2})$ structure Value
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}UP_{\rho}P_{\nu}P_{\mu}U\big{)}$
$-i\frac{c_{s}}{2}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}UP_{\rho}P_{\nu}P_{\mu}P_{\rho}U\big{)}$
$-i\frac{c_{s}}{2}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}UP_{\rho}P_{\mu}P_{\nu}P_{\rho}U\big{)}$
$-i\frac{c_{s}}{2}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}UP_{\mu}P_{\rho}P_{\nu}P_{\rho}U\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}UP_{\nu}P_{\rho}P_{\mu}P_{\rho}U\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}UP_{\rho}P_{\mu}P_{\nu}P_{\rho}U\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\mu}UP_{\rho}UP_{\nu}P_{\rho}\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}UP_{\mu}UP_{\rho}P_{\nu}\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$
tr$\big{(}P_{\mu}P_{\nu}UP_{\rho}UP_{\rho}P_{\mu}P_{\nu}\big{)}$
$-ic_{s}\,2^{6}\,\mathcal{I}[q^{6}]^{8}$

Table 5: Table 4 continued.

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

Diagram $O(P^{6}U)$ structure Value
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}P_{\nu}P_{\rho}P_{\mu}U\big{)}$
$-ic_{s}\,2^{6}\mathcal{I}[q^{6}]^{7}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}P_{\mu}P_{\rho}P_{\nu}U\big{)}$
$-ic_{s}\,2^{6}\mathcal{I}[q^{6}]^{7}$
tr$\big{(}P_{\mu}P_{\nu}P_{\mu}P_{\rho}P_{\nu}P_{\rho}U\big{)}$
$-ic_{s}\,2^{6}\mathcal{I}[q^{6}]^{7}$
tr$\big{(}P_{\mu}P_{\nu}P_{\rho}P_{\mu}P_{\nu}P_{\rho}U\big{)}$
$-ic_{s}\,2^{6}\mathcal{I}[q^{6}]^{7}$

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

###  A.2 Master integrals for heavy loops

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

$\displaystyle\int\frac{d^{d}q}{(2\pi)^{d}}\,\frac{q^{\mu_{1}}\cdots
q^{\mu_{2n_{c}}}}{(q^{2}-M^{2})^{n}}\,\equiv\,g^{\mu_{1}\cdots\mu_{2n_{c}}}\,\mathcal{I}[q^{2n_{c}}]^{n},$
(187)

We have used Package-X Patel:2015tea ; Patel:2016fam  to compute the loop
integrals. In Table 7, we have listed the results for the loop integrals
discussed in Sec. 6.

Integral Value Mentioned in $\mathcal{I}[q^{2}]^{8}$
$-\frac{i}{16\pi^{2}}\frac{1}{420\,\text{M}^{10}}$ Table 1
$\mathcal{I}[q^{4}]^{7}$ $-\frac{i}{16\pi^{2}}\frac{1}{1440\,\text{M}^{6}}$
Table 2 $\mathcal{I}[q^{6}]^{7}$
$\frac{i}{16\pi^{2}}\frac{1}{5760\,\text{M}^{4}}$ Table 6
$\mathcal{I}[q^{6}]^{8}$ $-\frac{i}{16\pi^{2}}\frac{1}{20160\,\text{M}^{6}}$
Table 4 and Table 5 $\mathcal{I}[q^{8}]^{8}$
$\frac{i}{16\pi^{2}}\frac{1}{80640\,\text{M}^{4}}$ Table 3

Table 7: The values corresponding to the master integrals associated with the
covariant diagrams.

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