#  Functional Matching and Renormalization Group Equations
at Two-Loop Order

Javier Fuentes-Martín<EMAIL_ADDRESS>Departamento de Física Teórica y
del Cosmos, Universidad de Granada, Campus de Fuentenueva, E–18071 Granada,
Spain  Ajdin Palavrić<EMAIL_ADDRESS>Department of Physics,
University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland  Anders
Eller Thomsen<EMAIL_ADDRESS>Department of Physics, University of
Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland  Albert Einstein
Center for Fundamental Physics, Institute for Theoretical Physics, University
of Bern, CH-3012 Bern, Switzerland

###### Abstract

We present a systematic method for determining the two-loop effective
Lagrangian resulting from integrating out a set of heavy particles in an
ultraviolet scalar theory. We prove that the matching coefficients are
entirely determined from the (double-)hard region of the loop integrals and
present a master formula for matching, applicable to both diagrammatic and
functional approaches. We further employ functional methods to determine
compact expressions for the effective Lagrangian that do not rely on any
previous knowledge of its structure or symmetries. The same methods are also
applicable to the computation of renormalization group equations. We
demonstrate the application of the functional approach by computing the two-
loop matching coefficients and renormalization group equations in a scalar toy
model.

##  1  Introduction

The use of Effective Field Theories (EFTs) in beyond the Standard Model (BSM)
searches is ubiquitous. Given the current experimental bounds, it seems
increasingly likely that there is a large gap between the electroweak scale
and the next energy threshold. In this event, EFTs are the ideal tool to
capture the possible low-energy effects of new physics (NP), whatever it might
be. With them, we can look for subtle NP effects, opening the door to the
exploration of energy scales orders of magnitude beyond what can be reached
on-shell at the LHC.

There is an ongoing community effort to automate the steps that go into EFT
computations [1, 2]. Starting with [3] (see also [4, 5, 6, 7, 8, 9, 10, 11])
and further refined in [12, 13], functional methods have been put forward as a
handy way to organize one-loop EFT matching and Renormalization Group (RG)
computations in BSM physics. They are used in the development of the so-called
Universal One-Loop Effective Action (UOLEA) [14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24], and have seen continual development [25, 26, 27]. It is only
recently that the first fairly general tools for automated one-loop matching
were introduced, one following diagrammatic amplitude matching [28] and the
other based on functional methods [29]. Both approaches rely on a master
formula for matching that identifies the one-loop EFT action with the hard
momenta region of the one-loop contributions in the underlying theory [12,
13].

It is now a decade since the computation of the one-loop RG equations in the
Standard Model Effective Theory (SMEFT) [30, 31, 32], and we wonder if one-
loop corrections are sufficient for the current precision requirements.
Indeed, certain low-energy effects that are generated only at two-loop order
[33]. Furthermore, both the strong and the top Yukawa couplings are large
enough that they might generate considerable running contributions, such as in
[34]. On top of this, the inclusion of two-loop RG effects becomes mandatory
if one wants to restore the scheme independence in one-loop matching
calculations [35, 36], making them an important ingredient in the automated
one-loop matching endeavor.

With this letter, we take the first step towards efficient, functional two-
loop RG and matching calculations. It should come as no surprise that such a
step is possible, as other variations of functional methods have been used for
the calculation of the quantum effective potential [37, 38] and the
counterterms of chiral perturbation theory [39] with heat-kernel methods. Both
of these calculations have also been performed at two-loop order [40, 41].
What is perhaps more remarkable is that the functional formalism lets us prove
a generic master formula for two-loop matching, where we directly identify the
two-loop EFT action with the hard part of the ultraviolet (UV) quantum
effective action. With this formula, there is no need to identify
cancellations between EFT and UV contributions on a case-by-case basis. It
also opens up the future possibility of automating (functional) matching
beyond one-loop order.

In this letter, we outline the basic extension of functional methods to two-
loop order in the simple case of scalar theories. In Section 2, we present our
functional method, with a proof to the generic matching formula supplied in
Appendix A. Next, in Section 3, we demonstrate the practical application of
the method by calculating the two-loop matching of a scalar toy model to its
low-energy EFT and the two-loop RG equations of the resulting theory. We leave
further details of the method along with extensions to fermionic degrees of
freedom and gauge theories to a forthcoming paper.

##  2  Functional Methods

Through path-integral manipulations in a UV theory, one can determine the EFT
Lagrangian describing the dynamics of the light particles at low energies
compared to the heavy-particles masses. We discuss here how this can be done
at the two-loop level.

###  2.1  The vacuum functional and quantum effective action

The vacuum functional, $\mathcal{W}[J]$, is the generating functional of all
connected Green’s functions of a theory and, therefore, contains all physical
information. If we take the fields (including their conjugates) to be
collectively denoted by $\eta_{a}(x)$ and the action by $S[\eta]$, the vacuum
functional is defined by the path integral:

$\displaystyle
e^{i\hbar^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\mathcal{W}[J]}=\int\mathcal{D}\eta\,e^{i\hbar^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\left(S[\eta]+J_{I}\eta_{I}\right)}\,,$
(1)

where the subindices with capital Latin letters $I,J,\ldots$ correspond to
DeWitt notation, where the spacetime dependence is included as part of the
label, e.g. $I=(x,a)$. Thus, the contraction of repeated indices denotes not
only an implicit summation in the field labels but also an integration over
spacetime.111In the case at hand,
$J_{I}\eta_{I}=\int_{x}\,J_{a}(x)\eta_{a}(x)$ with
$\int_{x}\;\equiv\int\;\mathop{}\\!\mathrm{d}^{d}x$.

The path integral can be evaluated perturbatively, using a saddle-point
approximation around the classical background-field configuration
$\overline{\eta}$, which is the solution to the tree-level equations of motion
(EOMs), that is

$\displaystyle\dfrac{\delta
S^{(0)}}{\delta\eta_{I}}[\overline{\eta}]+J_{I}=0\,,$ (2)

where the superindex denotes loop order, with $(0)$ indicating tree level. We
parameterize the expansion of the action around the background-field
configuration as

$\displaystyle S[\eta+\overline{\eta}]$
$\displaystyle\\!=\\!\sum_{\ell=0}^{\infty}\hbar^{\ell}\Big{[}\overline{S}^{(\ell)}\\!\\!+\overline{\mathcal{A}}^{(\ell)}_{I}\\!\eta_{I}\\!+\overline{\mathcal{B}}^{(\ell)}_{IJ}\dfrac{\eta_{I}\eta_{J}}{2}\\!+\overline{\mathcal{C}}^{(\ell)}_{IJK}\dfrac{\eta_{I}\eta_{J}\eta_{K}}{3!}$
$\displaystyle\quad+\overline{\mathcal{D}}^{(\ell)}_{IJKL}\dfrac{\eta_{I}\eta_{J}\eta_{K}\eta_{L}}{4!}+\mathcal{O}(\eta^{5})\Big{]}\,,$
(3)

with $\overline{\mathcal{A}}^{(0)}_{I}=-J_{I}$, and the bar being shorthand
for exclusive dependence on $\overline{\eta}$, e.g. $\overline{S}\equiv
S[\overline{\eta}]$. It is convention to denote dressed propagator of the
quantum field (also known as the fluctuation operator) by
$\mathcal{Q}_{IJ}\equiv\mathcal{B}^{(0)}_{IJ}$. For renormalizable theories,
all higher-loop vertices
$\mathcal{A}_{I}^{(\ell)},\mathcal{B}_{IJ}^{(\ell)},\ldots$ for $\ell\geq 1$
stem from the counterterms of the renormalized action; however, this could be
different if our UV theory is itself an EFT.

Using the defining relation of the vacuum functional given by (1) together
with the expansion of the action around the background field, the vacuum
functional can be written in the form

$\displaystyle\mathcal{W}[J]$
$\displaystyle=\overline{S}^{(0)}\\!+J_{I}\overline{\eta}_{I}+\hbar\,\overline{S}^{(1)}\\!+\frac{i\hbar}{2}\big{(}\log\overline{\mathcal{Q}}\big{)}_{II}\\!+\hbar^{2}\,\overline{S}^{(2)}+\frac{i\hbar^{2}}{2}\overline{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{IJ}\overline{\mathcal{B}}^{(1)}_{JI}+\frac{\hbar^{2}}{12}\overline{\mathcal{C}}^{(0)}_{IJK}\overline{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{IL}\overline{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{JM}\overline{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{KN}\overline{\mathcal{C}}^{(0)}_{LMN}$
$\displaystyle\quad-\frac{\hbar^{2}}{8}\overline{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{IJ}\,\overline{\mathcal{D}}^{(0)}_{IJKL}\overline{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{KL}-\frac{\hbar^{2}}{2}\left(\overline{\mathcal{A}}^{(1)}_{K}+\frac{i}{2}\overline{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{IJ}\overline{\mathcal{C}}^{(0)}_{IJK}\right)\overline{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{KL}\left(\overline{\mathcal{A}}^{(1)}_{L}+\frac{i}{2}\overline{\mathcal{C}}^{(0)}_{LMN}\overline{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{MN}\right)+\mathcal{O}(\hbar^{3})\,.$
(4)

The expansion of the vacuum functional reproduces the usual loop expansion
(formally in $\hbar$). Therefore, the terms of $\mathcal{O}(\hbar^{2})$ in
(2.1) correspond to the two-loop topologies, while the terms of higher order
are dropped.

Figure 1: Graphical representation of the $\mathcal{O}(\hbar^{2})$ terms
appearing in the effective action in (2.1).

It is convenient to relate this functional to the quantum effective action,
$\Gamma$, as we will ultimately relate $\Gamma$ to the EFT action. The quantum
effective action is the generating functional of all one-particle-irreducible
(1PI) Green’s functions of the theory and is defined by the Legendre transform
of $\mathcal{W}$:

$\displaystyle\Gamma[\hat{\eta}]$
$\displaystyle=\mathcal{W}[J]-J_{I}\hat{\eta}_{I}\,,$
$\displaystyle\hat{\eta}_{I}$
$\displaystyle\equiv\frac{\delta\mathcal{W}}{\delta J_{I}}\,.$ (5)

The background field $\hat{\eta}$ and the sources then satisfy the quantum EOM

$\displaystyle\dfrac{\delta\Gamma}{\delta\eta_{I}}[\hat{\eta}]+J_{I}=0\,.$ (6)

Plugging this back into definition (5) along with the saddle-point
approximation of $\mathcal{W}$, we obtain

$\displaystyle\Gamma[\hat{\eta}]$
$\displaystyle=\widehat{S}^{(0)}+\hbar\,\widehat{S}^{(1)}+\dfrac{i\hbar}{2}\big{(}\\!\log\widehat{\mathcal{Q}}\big{)}_{II}+\hbar^{2}\widehat{S}^{(2)}$
$\displaystyle\quad+\dfrac{i\hbar^{2}}{2}\widehat{\mathcal{Q}}_{IJ}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widehat{\mathcal{B}}_{JI}^{(1)}-\dfrac{\hbar^{2}}{8}\widehat{\mathcal{Q}}_{IJ}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\,\widehat{\mathcal{D}}^{(0)}_{IJKL}\widehat{\mathcal{Q}}_{KL}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}$
$\displaystyle\quad+\dfrac{\hbar^{2}}{12}\,\widehat{\mathcal{C}}^{(0)}_{IJK}\widehat{\mathcal{Q}}_{IL}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widehat{\mathcal{Q}}_{JM}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widehat{\mathcal{Q}}_{KN}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widehat{\mathcal{C}}^{(0)}_{LMN}+\mathcal{O}(\hbar^{3})\,,$
(7)

where, analogously to the bar notation, the hat is used as shorthand for
exclusive dependence on $\hat{\eta}$, e.g.,
$\widehat{\mathcal{Q}}_{IJ}\equiv\mathcal{Q}_{IJ}[\hat{\eta}]$. This is an
extension of the well-known expression for the effective action at one-loop
order, where all one-loop contributions are contained in a functional
(super)trace. Interestingly, the terms in the expression above can be
understood as vacuum graphs with
$\widehat{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}$ acting as a
quantum-field propagator, dressed with arbitrary insertions of the background
fields $\hat{\eta}$, and $\mathcal{B}^{(1)}$, $\mathcal{C}^{(0)}$ and
$\mathcal{D}^{(0)}$ corresponding to two-, three-, and four-point quantum
field interactions (again in the presence of background fields). This is
represented in Figure 1. Each closed loop in these dressed graphs can be
associated to an integration over a loop momentum, in a similar manner to
traditional Feynman graphs.

In general, evaluating the effective action (2.1) can be very complicated
because of the need for inverting $\widehat{\mathcal{Q}}$ and evaluating
$\widehat{\mathcal{C}}$ at different spacetime points. Nonetheless, when all
loop momenta are restricted to a hard region,222The method of regions [42, 43]
describes how loop integrals in dimensional regularization can be decomposed
in momentum regions, which are identified by the singularities of the loop-
integral propagators (see Appendix A.2), and in each region the propagators
are expanded accordingly. The relevant regions for this discussion are hard
and soft, with the loop momentum $k$ satisfying $k\sim\Lambda$ (with $\Lambda$
being a heavy scale) and $k\sim E\ll\Lambda$, respectively. these quantities
can be evaluated directly in terms of an operator-product expansion around the
hard scale. As we will now discuss, the hard-momenta region of $\Gamma$ is all
that is needed for EFT matching and RG evolution, also at two-loop order.

###  2.2  Master formula for EFT matching

Let us consider a weakly-coupled UV theory,
$S_{\scriptscriptstyle\mathrm{{\scriptscriptstyle\mathrm{UV}}}}[\eta]$, where
$\eta_{I}=(\Phi_{\alpha},\,\phi_{i})$ denotes the collection of all fields,
heavy and light, respectively. We seek to determine an EFT action
$S_{\scriptscriptstyle\mathrm{EFT}}[\phi]$ that reproduces the physics of the
full theory at energies much below the masses of the heavy fields, which we
assume to lie around a generic scale $\Lambda$.

In off-shell matching computations, we begin with an even stronger requirement
for the EFT matching condition: the EFT should reproduce all low-energy
Green’s functions of the full theory. This stronger requirement lets us
consider the generating functional of the theories rather than the $S$-matrix.
Thus, our aim is to determine $S_{\scriptscriptstyle\mathrm{EFT}}[\phi]$ such
that

$\displaystyle\mathcal{W}_{\scriptscriptstyle\mathrm{EFT}}[J_{\phi}]=\mathcal{W}_{\scriptscriptstyle\mathrm{UV}}\big{[}J_{\Phi}=0,\,J_{\phi}\big{]}\,.$
(8)

That is, we enforce equality of all connected Green’s functions involving the
light fields. Each side of (8) is to be understood in terms of power series in
$1/\Lambda$.

To make matters simpler, we proceed with a Legendre transformation of the
light-field sources in order to frame the matching condition in terms of the
quantum effective actions:

$\displaystyle\Gamma_{\scriptscriptstyle\mathrm{EFT}}[\hat{\phi}]$
$\displaystyle=\Gamma_{\scriptscriptstyle\mathrm{UV}}\big{[}\widehat{\Phi}[\hat{\phi}],\,\hat{\phi}\big{]}\,,$
$\displaystyle\dfrac{\delta\Gamma_{\scriptscriptstyle\mathrm{UV}}}{\delta\Phi_{\alpha}}\big{[}\widehat{\Phi}[\hat{\phi}],\,\hat{\phi}\big{]}=0\,.$
(9)

The heavy fields $\widehat{\Phi}[\hat{\phi}]$ are solutions to the quantum
EOMs in the presence of the light background fields. In diagrammatic terms,
condition (9) equates all one-light-particle-irreducible (1LPI) Green’s
functions of the UV theory with the 1PI Green’s functions of the EFT. It was
demonstrated in [12, 13] that, at one-loop order, there is a cancellation
between the loop contributions in the EFT and the soft-region of the loops in
the UV theory. This enables a very direct computation of
$S^{(1)}_{\scriptscriptstyle\mathrm{EFT}}[\phi]$ in terms of the hard region
of the UV quantum effective action.

We are now ready to generalize these considerations and present a master
formula for perturbative EFT matching at multi-loop order: the off-shell EFT
action is determined by

$\displaystyle S_{\scriptscriptstyle\mathrm{EFT}}$
$\displaystyle=\Gamma_{\scriptscriptstyle\mathrm{UV}}\big{[}\widehat{\Phi}[\hat{\phi}],\,\hat{\phi}\big{]}\Big{|}_{\mathrm{hard}}\,,$
$\displaystyle\dfrac{\delta\Gamma_{\scriptscriptstyle\mathrm{UV}}\big{|}_{\mathrm{hard}}}{\delta\Phi_{\alpha}}\big{[}\widehat{\Phi}[\hat{\phi}],\,\hat{\phi}\big{]}$
$\displaystyle=0\,.$ (10)

Here the ‘hard’ part is taken to include all contributions without _any_ soft
loops. It includes tree-level contributions as well as loops where _all_ loop
momenta are hard. In many ways, this an intuitive leap: the hard, local part
of the UV theory is identified with the EFT action, while the long-distance
physics is captured by loops in the EFT. Nevertheless, we have never seen an
explicit formulation of this notion, much less a matching formula applicable
to practical computations. A constructive proof of the matching formula (10)
at two-loop order is provided in Appendix A, where we show that there is a
one-to-one correspondence between (partially) soft loops in the UV and loops
in the EFT. We postpone the discussion on a possible extension of this proof
to higher-loop orders to a more comprehensive follow-up paper. The end result
is that all loop integrals needed to perform the EFT matching (i.e., those in
the hard limit) reduce to vacuum integrals, for which expressions are known up
to three loops [44].

The matching formula (10) is a $d$-dimensional off-shell relation. A
complication associated to this kind of matching is that it does not produce
the EFT Lagrangian directly in a four-dimensional on-shell basis. Rather, one
has to apply field redefinitions to reduce the output to an on-shell basis.
Likewise, the matching result may also produce EFT operators that are not
present in a four-dimensional basis. As a result, one has to separate out
evanescent operators and, preferably, convert the EFT action to an
evanescence-free scheme [45, 46, 47, 48, 49, 50, 51]. A related consideration
to be aware of beyond one-loop order is that lower-order matching coefficients
may contain $\mathcal{O}(\epsilon)$ contributions. These cannot be ignored, as
their insertion in an EFT loop can lead to finite contributions. Therefore,
one has to carefully remove the $\mathcal{O}(\epsilon)$ terms and absorb them
into finite coefficients at higher-loop order, similarly to what is done for
evanescent contributions [48].

###  2.3  Functional approach for RG evolution

The MS (or $\overline{\text{\small MS}}$) counterterms of a theory, $S[\eta]$,
can also be determined from the effective action from the observation that it
must be free of UV divergences. If we use dimensional regularization to
regularize the loop integrals, finiteness of the renormalized generating
functional translates to the condition

$\displaystyle K_{\epsilon}\,\Gamma[\hat{\eta}]=0\,,$ (11)

where $K_{\epsilon}$ is an operator that extracts all $1/\epsilon$ poles of UV
origin. This equation establishes a relation between the MS counterterms of
the theory, identified with the UV poles of $\widehat{S}^{(\ell)}$, and the
other terms in the effective action. Denoting the MS counterterms by
$\delta\widehat{S}^{(\ell)}_{\rm MS}$, we have

$\displaystyle\delta\widehat{S}^{(\ell)}_{\rm
MS}=K_{\epsilon}\,\widehat{S}^{(\ell)}\,,$ (12)

which, together with the expression of $\Gamma$ and condition (11),
establishes a direct relation to determine $\delta\widehat{S}^{(\ell)}_{\rm
MS}$ functionally. Other renormalization schemes of the subtraction family can
also be obtained by appropriately adapting the definitions of the counterterms
in the expression above.

Restricting to the two-loop expression of the effective action in (2.1), we
obtain the following MS counterterms up to two-loop order:

$\displaystyle\delta\widehat{S}^{(1)}_{\rm MS}$
$\displaystyle=-\dfrac{i}{2}K_{\epsilon}\big{(}\\!\log\widehat{\mathcal{Q}}\big{)}_{II}\,,$
$\displaystyle\delta\widehat{S}^{(2)}_{\rm MS}$
$\displaystyle=K_{\epsilon}\Big{[}-\frac{i}{2}\widehat{\mathcal{Q}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{IJ}\widehat{\mathcal{B}}^{(1)}_{JI}+\dfrac{1}{8}\widehat{\mathcal{Q}}_{IJ}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widehat{\mathcal{D}}^{(0)}_{IJKL}\widehat{\mathcal{Q}}_{KL}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}$
$\displaystyle\quad-\dfrac{1}{12}\,\widehat{\mathcal{C}}^{(0)}_{IJK}\widehat{\mathcal{Q}}_{IL}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widehat{\mathcal{Q}}_{JM}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widehat{\mathcal{Q}}_{KN}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widehat{\mathcal{C}}^{(0)}_{LMN}\Big{]}\,.$
(13)

A convenient prescription for extracting the UV poles in theories with
massless states consists in introducing a common mass, $\Lambda$, in all
propagators of the loop integrals, which acts as a hard scale (assumed to be
much larger than any other scales in the loop integrals) and serves as an
infrared regulator. The overall UV divergence of a loop integral is identified
with the part where all loop momenta are large. Hence, it is easy to show that
the UV divergences can be extracted from the hard-region (defined by all loop
momenta being of order $\Lambda$) of the loop integrals [52]. This effectively
establishes a power-counting on $\Lambda$, around which
$\widehat{\mathcal{Q}}^{-1}$ can be expanded. As in the matching case, the
resulting loop integrals are just vacuum integrals (in this case with a single
mass $\Lambda$) for which results are known up to three loops [44]. The only
drawback of this method for UV-pole extraction is that one also needs to
consider spurious counterterms with positive powers of $\Lambda$. These
spurious counterterms can break the symmetries of the original Lagrangian and
are needed for the cancellation of subdivergences. After the counterterms have
been determined, the RG equations can be readily obtained via standard
techniques.

###  2.4  Functional evaluation of the effective action

The functional evaluation of the effective action requires manipulating the
inverse dressed propagator, $\hat{\mathcal{Q}}$, which can be generically
parametrized by

$\displaystyle\widehat{\mathcal{Q}}_{IJ}\equiv\widehat{\mathcal{Q}}_{ab}(x,y)=Q_{ab}(x,\,P_{x})\,\delta(x-y)\,,$
(14)

with $P^{\mu}_{x}=i\partial^{\mu}_{x}$ denoting the momentum operator. Since
the action is local, it is always possible to factor out a delta function also
when dealing with more complicated functional forms involving
$\widehat{\mathcal{Q}}$, such as $\ln\widehat{\mathcal{Q}}$ or
$\widehat{\mathcal{Q}}^{-1}$. It is well-known that the one-loop contribution
to the quantum effective action, c.f. (2.1), is given by the functional
(super)trace

$\displaystyle\widehat{\Gamma}^{(1)}$
$\displaystyle=\widehat{S}^{(1)}+\frac{i}{2}(\ln\widehat{\mathcal{Q}})_{II}$
$\displaystyle=\widehat{S}^{(1)}+\frac{i}{2}\int_{x,k}\;\,[\ln
Q(x,P_{x}-k)]_{aa}\,,$ (15)

with
$\int_{x,k}\equiv\int\,\mathop{}\\!\mathrm{d}^{d}x\int\,\mathop{}\\!\mathrm{d}^{d}k/(2\pi)^{d}$.
This expression is a non-local function of $Q$ and is difficult to evaluate in
general. However, as we are only interested in the hard region, where the loop
momentum is taken to be of the order of the heavy scales, we can perform an
operator-product expansion. For scalar theories, the inverse dressed
propagator takes the generic form

$\displaystyle
Q_{ab}(x,\,P_{x})=(P_{x}^{2}-M_{a}^{2})\,\delta_{ab}-X_{ab}(x,P_{x})\,,$ (16)

with $M_{a}$ being a possible hard mass that would be zero for light fields.
Denoting

$\displaystyle\Delta^{-1}_{ab}$
$\displaystyle\equiv(k^{2}-M_{a}^{2})\,\delta_{ab}\,,$ $\displaystyle U_{ab}$
$\displaystyle\equiv(-P_{x}^{2}+2k\cdot
P_{x})\,\delta_{ab}+X_{ab}(x,P_{x}-k)\,,$ (17)

the operator-product expansion of the logarithm reads

$\displaystyle\int_{x,k}\ln
Q(x,P_{x}-k)=\int_{x,k}\ln\Delta^{-1}-\int_{x,k}\;\,\sum_{n=1}^{\infty}\frac{1}{n}(\Delta
U)^{n}\,,$ (18)

where the first term contributes to an unphysical constant that is subtracted
when normalizing the path integral. Likewise, the dressed propagator,
necessary for evaluating the two-loop contributions, admits the expansion

$\displaystyle Q^{-1}(x,P_{x}-k)=\sum_{n=0}^{\infty}(\Delta U)^{n}\Delta\,.$
(19)

The hard-region evaluation guarantees that subsequent terms in this series are
further suppressed, so only a finite number of terms need to be retained to a
given order in the EFT expansion. A manifestly local result is obtained only
in the hard-region limit.

Locality of the action likewise ensures that delta functions can be factored
out of the remaining functional objects. We write the quantum-field
interactions as series in the momenta operator:333For compactness, we use a
power-like notation with underlined superscripts for Lorentz indices, e.g.,
$B_{ab}^{(1)\,\underline{m}}P_{x}^{\underline{m}}=B_{ab}^{(1)\,\mu_{1}\ldots\mu_{m}}P_{x}^{\mu_{1}}\cdots
P_{x}^{\mu_{m}}$.

$\displaystyle\widehat{\mathcal{B}}^{(1)}_{IJ}$
$\displaystyle=\sum_{m=0}^{\infty}B^{(1)\,\underline{m}}_{ab}(w)P_{x}^{\underline{m}}\,\delta(x-w)\,,$
$\displaystyle\widehat{\mathcal{C}}^{(0)}_{IJK}$
$\displaystyle=\sum_{m,n=0}^{\infty}C^{\underline{m},\underline{n}}_{abc}(w)P_{x}^{\underline{m}}P_{y}^{\underline{n}}\,\delta(x-w)\delta(y-w)\,,$
$\displaystyle\widehat{\mathcal{D}}^{(0)}_{IJKL}$
$\displaystyle=\sum_{m,n,r=0}^{\infty}D^{\underline{m},\underline{n},\underline{r}}_{abcd}(w)P_{x}^{\underline{m}}P_{y}^{\underline{n}}P_{z}^{\underline{r}}$
$\displaystyle\qquad\qquad\times\delta(x-w)\delta(y-w)\delta(z-w)\,.$ (20)

In most practical applications, only a small number of terms from this
momentum operator expansions are present. In fact, for renormalizable
theories, only the terms with at most two momenta in $\widehat{\mathcal{B}}$,
one in $\widehat{\mathcal{C}}$, and none in $\widehat{\mathcal{D}}$ are
nonzero.

Having made these definitions, we can now evaluate the two-loop contributions
to the effective action in (2.1). We parameterize them as

$\displaystyle\widehat{\Gamma}^{(2)}=\widehat{S}^{(2)}+\dfrac{i}{2}G_{\mathrm{ct}}-\dfrac{1}{8}G_{\mathrm{f8}}+\dfrac{1}{12}G_{\mathrm{ss}}\,,$
(21)

where the loop contributions $G_{i}$ are identified with the counterterm,
figure-8, and sunset topologies, respectively (as depicted in Figure 1). The
main subtlety in the functional evaluation of the two-loop effective action is
related to the sunset topology, where it is necessary to power expand one of
the two vertices around the location of the other. We obtain integral formulas
for the functional contractions and, using a momentum-space representation for
the delta functions, we find the expressions

$\displaystyle G_{\mathrm{ct}}$
$\displaystyle=\sum_{m}\\!\int_{x,k}B^{(1)\,\underline{m}}_{ab}(x)\big{[}(P_{x}-k)^{\underline{m}}\,Q_{ab}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}(x,\,P_{x}-k)\big{]}\,,$
$\displaystyle G_{\mathrm{f8}}$
$\displaystyle=\sum_{m,n,r}(\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1)^{m+n+r}\\!\\!\int_{x,k,\ell}D^{\underline{m},\underline{n},\underline{r}}_{abcd}(x)$
$\displaystyle\quad\times\big{[}k^{\underline{n}}\,(P_{x}-k)^{\underline{m}}\,Q_{ab}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}(x,\,P_{x}-k)\big{]}$
$\displaystyle\quad\times\big{[}(P_{x}-\ell)^{\underline{r}}\,Q_{cd}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}(x,\,P_{x}-\ell)\big{]}\,,$
$\displaystyle G_{\mathrm{ss}}$
$\displaystyle=\\!\\!\\!\\!\sum_{m^{(\prime)},\,n^{(\prime)},\,s}\\!\\!\\!\\!\\!\\!(\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1)^{m+n+m^{\prime}+n^{\prime}}\dfrac{i^{s}}{s!}\int_{x,k,l}C^{\underline{m},\underline{n}}_{abc}(x)\partial_{x}^{\underline{s}}\,C^{\underline{m}^{\prime},\underline{n}^{\prime}}_{def}(x)$
$\displaystyle\quad\times\big{[}\partial_{k}^{\underline{s}}\,Q_{cf}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}(x,\,P_{x}+k+\ell)\big{]}$
$\displaystyle\quad\times\big{[}k^{\underline{m}^{\prime}}(P_{x}-k)^{\underline{m}}\,Q_{ad}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}(x,\,P_{x}-k)\big{]}$
$\displaystyle\quad\times\big{[}\ell^{\underline{n}^{\prime}}(P_{x}-\ell)^{\underline{n}}\,Q_{be}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}(x,\,P_{x}-\ell)\big{]}\,.$
(22)

with
$\int_{x,k,\ell}\equiv\int\,\mathop{}\\!\mathrm{d}^{d}x\int\,\mathop{}\\!\mathrm{d}^{d}k/(2\pi)^{d}\int\,\mathop{}\\!\mathrm{d}^{d}\ell/(2\pi)^{d}$.
In the equations above, any $P_{x}$ acting to the rightmost of a bracket
yields a null contribution. This would not be the case anymore in the gauge
non-singlet case, where the derivative in $P_{x}$ would be promoted to a
covariant derivative. A manifestly covariant generalization of these
expressions will be presented in a follow-up paper. While the sum in $s$ in
the last expression run to infinity, only a finite number of terms need to be
retained at a given order in the EFT counting when considering the hard-
momentum limit. In particular, only terms up to $s=4$ contributes at EFT
dimension six.

##  3  A toy-model example

We illustrate the functional method described in the previous section with a
concrete example: a toy-model, consisting of one heavy and one light real
scalar fields, $\Phi$ and $\phi$, respectively. For simplicity, we assume that
the theory possesses a $\mathbb{Z}_{2}^{\phi}\times\mathbb{Z}_{2}^{\Phi}$
symmetry, with $\mathbb{Z}_{2}^{\Phi}$ softly-broken by a trilinear term with
coupling $\kappa\ll M$. This soft-breaking term is included to allow for a
non-trivial quantum EOM of the heavy-field, cf. (10). The UV Lagrangian of
this theory is given by

$\displaystyle\mathcal{L}_{\scriptscriptstyle\mathrm{UV}}$
$\displaystyle=\frac{1}{2}(\partial_{\mu}\phi)^{2}+\frac{1}{2}(\partial_{\mu}\Phi)^{2}-\frac{1}{2}m_{\phi}^{2}\phi^{2}-\frac{1}{2}M^{2}\Phi^{2}-\frac{\lambda_{\phi}}{4!}\phi^{4}$
$\displaystyle\quad-\frac{\lambda_{\Phi}}{4!}\Phi^{4}-\frac{\lambda_{\Phi\phi}}{4}\Phi^{2}\phi^{2}-\frac{\kappa}{3!}\Phi^{3}+\mathcal{L}^{\rm
ct}_{\scriptscriptstyle\mathrm{UV}}\,,$ (23)

with the corresponding UV counterterms $\mathcal{L}^{\rm ct}_{\rm UV}$ defined
as

$\displaystyle\mathcal{L}^{\rm ct}_{\scriptscriptstyle\mathrm{UV}}$
$\displaystyle=\\!\frac{\delta_{\phi}}{2}(\partial_{\mu}\phi)^{2}\\!+\frac{\delta_{\Phi}}{2}(\partial_{\mu}\Phi)^{2}-\frac{\delta_{m_{\phi}}}{2}\phi^{2}-\frac{\delta_{M}}{2}\Phi^{2}-\frac{\delta_{\lambda_{\phi}}}{4!}\phi^{4}$
$\displaystyle\quad-\frac{\delta_{\lambda_{\Phi}}}{4!}\Phi^{4}-\frac{\delta_{\lambda_{\Phi\phi}}}{4}\Phi^{2}\phi^{2}-\frac{\delta_{\kappa}}{3!}\Phi^{3}-\frac{\delta_{\kappa^{\prime}}}{2}\Phi\phi^{2}-\delta_{\kappa_{T}}\Phi\,,$
(24)

where we anticipate that new $\mathbb{Z}_{2}^{\Phi}$-breaking interactions are
generated radiatively and need to be renormalized.444The trilinear coupling
$\Phi\phi^{2}$ is generated radiatively in the RG and should properly be
included. As we are not attempting to build a realistic model, we simply
ignore it.

###  3.1  One-loop UV counterterms

Our calculations are done in a tadpole-free $\overline{\text{\small MS}}$
scheme (for both UV theory and the EFT), such that the $\Phi$ tadpole is
removed with a finite counterterm. Given this scheme choice, the determination
of the finite parts of the two-loop matching conditions requires only the
calculation of the one-loop counterterms. From (2.3) and with the functional
evaluation described in Section 2.4, it follows that555Not only the
$1/\epsilon_{\scriptscriptstyle\mathrm{UV}}$ pole but also the finite pieces
are retained for $\Phi$ tadpole counterterms. Functional techniques are still
appropriate for evaluating the finite tadpole, as it is independent of
external momenta.

$\displaystyle\mathcal{L}^{\rm ct\,(1)}_{\scriptscriptstyle\mathrm{UV}}$
$\displaystyle=\frac{i}{2}\,K_{\epsilon}\int_{k}\;\;\sum_{n=1}^{\infty}\frac{1}{n}[(\Delta
U)^{n}]_{aa}\,,$ (25)

where $a=\Phi,\phi$ and the functional objects relevant for this calculation
are given by

$\displaystyle\Delta_{\Phi\Phi}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}(k)$
$\displaystyle=k^{2}-M^{2}\,,\qquad\Delta_{\phi\phi}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}(k)=k^{2}-m_{\phi}^{2}\,,$
$\displaystyle U_{\Phi\Phi}(k)$ $\displaystyle=-P_{x}^{2}+2k\cdot
P_{x}+\frac{\lambda_{\Phi}}{2}\Phi^{2}+\frac{\lambda_{\Phi\phi}}{2}\phi^{2}+\kappa\Phi\,,$
$\displaystyle U_{\phi\phi}(k)$ $\displaystyle=-P_{x}^{2}+2k\cdot
P_{x}+\frac{\lambda_{\phi}}{2}\phi^{2}+\frac{\lambda_{\Phi\phi}}{2}\Phi^{2}\,,$
$\displaystyle U_{\Phi\phi}(k)$
$\displaystyle=U_{\phi\Phi}(k)=\lambda_{\Phi\phi}\,\Phi\phi\,.$ (26)

As the UV theory in our example contains no massless states, no IR divergences
appear and the UV divergences are readily obtained. Only the terms with $n\leq
2$ in the sum above contribute to the UV divergences. Denoting
$\delta_{i}\equiv\frac{\hbar}{16\pi^{2}}\,\delta_{i}^{(1)}+\mathcal{O}(\hbar^{2})$,
we find

$\displaystyle\delta_{\phi}^{(1)}$ $\displaystyle=\delta_{\Phi}^{(1)}=0\,,$
$\displaystyle\delta_{\kappa}^{(1)}$
$\displaystyle=\frac{3\lambda_{\Phi}\kappa}{2\epsilon}\,,$
$\displaystyle\delta_{m_{\phi}}^{(1)}$
$\displaystyle=\frac{\lambda_{\phi}m_{\phi}^{2}+\lambda_{\Phi\phi}M^{2}}{2\epsilon}\,,$
$\displaystyle\delta_{\lambda_{\Phi}}^{(1)}$
$\displaystyle=\frac{3\lambda_{\Phi}^{2}+3\lambda_{\Phi\phi}^{2}}{2\epsilon}\,,$
$\displaystyle\delta_{M}^{(1)}$
$\displaystyle=\frac{\lambda_{\Phi}M^{2}+\lambda_{\Phi\phi}m_{\phi}^{2}+\kappa^{2}}{2\epsilon}\,,$
$\displaystyle\delta_{\lambda_{\phi}}^{(1)}$
$\displaystyle=\frac{3\lambda_{\phi}^{2}+3\lambda_{\Phi\phi}^{2}}{2\epsilon}\,,$
$\displaystyle\delta_{\lambda_{\phi\Phi}}^{(1)}$
$\displaystyle=\frac{\lambda_{\phi}\lambda_{\Phi\phi}+\lambda_{\Phi}\lambda_{\Phi\phi}+4\lambda_{\Phi\phi}^{2}}{2\epsilon}\,,$
$\displaystyle\delta_{\kappa^{\prime}}^{(1)}$
$\displaystyle=\frac{\lambda_{\Phi\phi}\kappa}{2\epsilon}\,,$
$\displaystyle\delta_{\kappa_{T}}^{(1)}$
$\displaystyle=\frac{M^{2}\kappa}{2}\left(\frac{1}{\epsilon}-\overline{\ln}\,M^{2}+1\right)\,.$
(27)

where $\overline{\ln}\,M^{2}\equiv\ln M^{2}/\bar{\mu}^{2}$ and $\bar{\mu}$ is
the $\overline{\text{\small MS}}$ renormalization scale. The value of these
counterterms has been cross-checked against the RG functions obtained with
RGBeta [53].

###  3.2  EFT matching at two-loop order

For the present example, it is easy to see that there is no tree-level
contribution to the EFT action, as the Lagrangian has no linear dependence on
the heavy field $\Phi$. The one-loop part of the EFT Lagrangian is calculated
as usual from the hard region of the functional (super)trace which, after
doing the expansion (18), yields

$\displaystyle\mathcal{L}^{(1)}_{\scriptscriptstyle\mathrm{EFT}}$
$\displaystyle=-\frac{i}{2}\int_{k}\;\;\sum_{n=1}^{\infty}\frac{1}{n}[(\Delta
U)^{n}]_{aa}\bigg{|}_{\rm hard}\,,$ (28)

with $\Delta$ and $U$ provided in in (3.1). However, differently from the
counterterm evaluation, we are now interested in the hard-momentum region
defined by $k\sim M\gg m_{\phi}$, with $k$ being the loop momentum. In
practice, this implies that, for this calculation, the mass term in
$\Delta^{-1}_{\phi\phi}(k)$ needs to be power-expanded before loop
integration, that is

$\displaystyle\Delta^{-1}_{\phi\phi}(k)\big{|}_{\rm
hard}=\sum_{n=0}^{\infty}\frac{m_{\phi}^{2n}}{k^{2(n+1)}}\,.$ (29)

One difference to keep in mind with respect to the usual one-loop matching
evaluations is the need to retain $\mathcal{O}(\epsilon)$ terms. As described
in Section 2.2, they can be shifted into the two-loop matching coefficients
with an appropriate modification of the renormalization conditions. The one-
loop EFT Lagrangian resulting from the functional trace is a function of both
heavy and light fields. The former are removed by means of the heavy-field
EOMs defined in (10). In our case we find that

$\displaystyle\Phi=-\frac{\hbar}{16\pi^{2}}\frac{\lambda_{\Phi\phi}\kappa}{2M^{2}}\,\overline{\ln}\,M^{2}\,\frac{\phi^{2}}{2}+\mathcal{O}(M^{-4},\hbar^{2})\,.$
(30)

In our example, the contributions from $\Phi$ yield two-loop effects when
inserted back into the one-loop (and tree-level) EFT Lagrangian. No terms of
two-loop order are needed in our toy-model calculation, as these would start
contributing only at the three-loop level.

The two-loop EFT Lagrangian follows directly from the evaluation of
expressions (2.4) in the double-hard region, defined by $k,\ell\gtrsim M\gg
m$. In this example, only terms without powers of $P_{x,y,z}$ appear in
$\widehat{\mathcal{B}}^{(1)}$, $\widehat{\mathcal{C}}$, and
$\widehat{\mathcal{D}}$, and thus, only the terms with
$m^{(\prime)}=n^{(\prime)}=r=0$ in (2.4) contribute. We have

$\displaystyle B^{(1)}_{\Phi\Phi}$
$\displaystyle=-\delta_{M}^{(1)}-\frac{\delta_{\lambda_{\Phi\phi}}^{(1)}}{2}\phi^{2}\,,$
$\displaystyle\\!\\!\\!B^{(1)}_{\phi\phi}$
$\displaystyle=-\delta_{m_{\phi}}^{(1)}-\frac{\delta_{\lambda_{\phi}}^{(1)}}{2}\phi^{2}\,,$
$\displaystyle B^{(1)}_{\Phi\phi}$
$\displaystyle=-\delta_{\kappa_{\Phi\phi}}^{(1)}\phi\,,$ $\displaystyle
C_{\Phi\Phi\Phi}$ $\displaystyle=-\kappa\,,$
$\displaystyle\\!\\!\\!C_{\phi\phi\phi}$
$\displaystyle=-\lambda_{\phi}\phi\,,$ $\displaystyle C_{\Phi\Phi\phi}$
$\displaystyle=-\lambda_{\Phi\phi}\phi\,,$
$\displaystyle\\!\\!\\!C_{\Phi\phi\phi}$ $\displaystyle=0\,,$ $\displaystyle
D_{\Phi\Phi\Phi\Phi}$ $\displaystyle=-\lambda_{\Phi}\,,$
$\displaystyle\\!\\!\\!D_{\phi\phi\phi\phi}$
$\displaystyle=-\lambda_{\phi}\,,$ $\displaystyle D_{\Phi\Phi\phi\phi}$
$\displaystyle=-\lambda_{\Phi\phi}\,,$
$\displaystyle\\!\\!\\!D_{\Phi\Phi\Phi\phi}$
$\displaystyle=D_{\Phi\phi\phi\phi}=0\,,$ (31)

where we dropped terms containing $\Phi$, as these contribute only at higher
loop orders. The final (off-shell) EFT Lagrangian is obtained by combining the
results from (2.4) with the contributions from the $\Phi$-field EOM and the
two-loop shift from removing the one-loop $\mathcal{O}(\epsilon)$ terms. The
resulting EFT Lagrangian can be written in terms of an on-shell operator basis
via appropriate $\phi$-field redefinitions. For concreteness, we present here
this final on-shell result:

$\displaystyle\mathcal{L}_{\scriptscriptstyle\mathrm{EFT}}$
$\displaystyle=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4}-\frac{c_{6}}{6!}\phi^{6}+\mathcal{L}_{\scriptscriptstyle\mathrm{EFT}}^{\mathrm{ct}}\,.$
(32)

with the two-loop matching conditions between the EFT and UV Lagrangians
taking the form

$\displaystyle m^{2}$
$\displaystyle=m_{\phi}^{2}-\frac{\lambda_{\Phi\phi}}{32\pi^{2}}\left(1-\overline{\ln}\,M^{2}\right)M^{2}+\frac{1}{(16\pi^{2})^{2}}\left[\left(5-4\,\overline{\ln}\,M^{2}+\overline{\ln}^{\,2}M^{2}\right)\frac{\lambda_{\Phi\phi}^{2}}{2}M^{2}-\left(\overline{\ln}\,M^{2}-\overline{\ln}^{2}M^{2}\right)\frac{\lambda_{\Phi}\lambda_{\Phi\phi}}{4}M^{2}\right.$
$\displaystyle\quad\left.+\left(1-2\sqrt{3}\,\mathrm{Cl}_{2}-2\,\overline{\ln}\,M^{2}+\overline{\ln}^{2}M^{2}\right)\frac{\lambda_{\Phi\phi}}{8}\kappa^{2}-\left(\frac{11}{4}+\overline{\ln}\,M^{2}+\overline{\ln}^{\,2}M^{2}\right)\frac{\lambda_{\Phi\phi}^{2}}{4}m_{\phi}^{2}+\left(5+6\,\overline{\ln}\,M^{2}\right)\frac{\lambda_{\Phi\phi}^{2}m_{\phi}^{4}}{36M^{2}}\right]\,,$
$\displaystyle\lambda$
$\displaystyle=\lambda_{\phi}+\frac{1}{16\pi^{2}}\left[\frac{3\lambda_{\Phi\phi}^{2}}{2}\,\overline{\ln}\,M^{2}-\frac{\lambda_{\Phi\phi}^{2}m_{\phi}^{2}}{3M^{2}}\right]+\frac{1}{(16\pi^{2})^{2}}\left[\left(19-37\,\overline{\ln}\,M^{2}+18\,\overline{\ln}^{2}\,M^{2}\right)\frac{\lambda_{\Phi\phi}^{3}}{6}-\left(11+26\,\overline{\ln}\,M^{2}\right)\frac{\lambda_{\Phi\phi}^{3}m_{\phi}^{2}}{6M^{2}}\right.$
$\displaystyle\quad\left.-\left(35+20\,\overline{\ln}\,M^{2}+12\,\overline{\ln}^{2}M^{2}\right)\frac{\lambda_{\Phi\phi}^{2}\lambda_{\phi}}{8}+\left(1+\overline{\ln}\,M^{2}\right)\frac{4\lambda_{\Phi\phi}^{2}\lambda_{\phi}m_{\phi}^{2}}{3M^{2}}-\left(1-\overline{\ln}\,M^{2}-\overline{\ln}^{2}\,M^{2}\right)\frac{3\lambda_{\Phi}\lambda_{\Phi\phi}^{2}}{4}\right.$
$\displaystyle\quad\left.-\left(1+\overline{\ln}\,M^{2}\right)\frac{\lambda_{\Phi}\lambda_{\Phi\phi}^{2}m_{\phi}^{2}}{6M^{2}}-\left(1-\overline{\ln}\,M^{2}+\overline{\ln}^{2}\,M^{2}\right)\frac{3\kappa^{2}\lambda_{\Phi\phi}^{2}}{4M^{2}}\right]\,,$
$\displaystyle c_{6}$
$\displaystyle=\frac{1}{16\pi^{2}}\left[\frac{15\lambda_{\Phi\phi}^{3}}{2M^{2}}-\frac{5\lambda_{\Phi\phi}^{2}\lambda_{\phi}}{3M^{2}}\right]+\frac{1}{(16\pi^{2})^{2}}\left[-\bigg{(}18-17\,\overline{\ln}\,M^{2}\bigg{)}\frac{5\lambda_{\Phi\phi}^{4}}{2M^{2}}-\bigg{(}47+62\,\overline{\ln}\,M^{2}\bigg{)}\frac{5\lambda_{\Phi\phi}^{3}\lambda_{\phi}}{6M^{2}}\right.$
$\displaystyle\quad\left.+\bigg{(}13+10\,\overline{\ln}\,M^{2}\bigg{)}\frac{5\lambda_{\Phi\phi}^{2}\lambda_{\phi}^{2}}{6M^{2}}+\left(1+2\,\overline{\ln}\,M^{2}\right)\frac{15\lambda_{\Phi}\lambda_{\Phi\phi}^{3}}{4M^{2}}-\left(1+\overline{\ln}\,M^{2}\right)\frac{5\lambda_{\Phi}\lambda_{\Phi\phi}^{2}\lambda_{\phi}}{6M^{2}}\right]\,,$
(33)

where $\mathrm{Cl}_{2}(x)$ is the Clausen function of order 2 with
$\mathrm{Cl}_{2}\equiv\mathrm{Cl}_{2}(2\pi/3)\approx 0.6766277$. To our
knowledge, this is the first two-loop matching computation performed with
functional methods. As a crosscheck of our result, we verified that all
single- and double-logarithmic contributions are consistent with the RG
functions in both the UV theory666The two-loop RG functions in the UV theory
necessary for this crosscheck can easily be determined using RGBeta [53]. and
the EFT such as to ensure matching scale independence. The determination of
the RG functions in the EFT using functional methods is discussed in the next
section.

###  3.3  Renormalization group functions at two-loop order

The EFT counterterms are calculated functionally using expressions (2.3).
Analogously to the UV counterterms, the EFT counterterms at one-loop order are
given by

$\displaystyle\mathcal{L}^{\rm ct\,(1)}_{\scriptscriptstyle\mathrm{EFT}}$
$\displaystyle=\frac{i}{2}\,K_{\epsilon}\\!\int_{k}\;\;\sum_{n=1}^{\infty}\frac{1}{n}[(\Delta_{\scriptscriptstyle\mathrm{EFT}}U_{\scriptscriptstyle\mathrm{EFT}})^{n}]_{aa}$
(34)

with the EFT functional objects

$\displaystyle\Delta_{\scriptscriptstyle\mathrm{EFT}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}(k)$
$\displaystyle=k^{2}-m^{2}\,,$ $\displaystyle
U_{\scriptscriptstyle\mathrm{EFT}}(k)$ $\displaystyle=-P_{x}^{2}+2k\cdot
P_{x}+\frac{\lambda}{2}\phi^{2}+\frac{c_{6}}{4!}\phi^{4}\,.$ (35)

The counterterm Lagrangian from the expression above is obtained in an off-
shelf basis, and can be reduced to an on-shell Lagrangian by means of
appropriate field redefinitions. Parameterizing the on-shell EFT counterterms
by

$\displaystyle\mathcal{L}_{\scriptscriptstyle\mathrm{EFT}}^{\mathrm{ct}}$
$\displaystyle=\frac{1}{2}\delta_{\phi}(\partial_{\mu}\phi)^{2}-\frac{1}{2}\delta_{m^{2}}\phi^{2}-\frac{\delta_{\lambda}}{4!}\phi^{4}-\frac{\delta_{c_{6}}}{6!}\phi^{6}\,,$
(36)

and separating the couplings by loop order and power of the $\epsilon$-pole as

$\displaystyle\delta_{i}=\frac{\hbar}{16\pi^{2}}\frac{\delta_{i}^{(1)}}{\epsilon}+\frac{\hbar^{2}}{(16\pi^{2})^{2}}\left[\frac{\delta_{i,1}^{(2)}}{\epsilon}+\frac{\delta_{i,2}^{(2)}}{\epsilon^{2}}\right]+\mathcal{O}(\hbar^{3})\,,$
(37)

we find

$\displaystyle\delta^{(1)}_{\phi}$ $\displaystyle=0\,,$
$\displaystyle\delta^{(1)}_{m^{2}}$ $\displaystyle=\frac{m^{2}\lambda}{2}\,,$
$\displaystyle\delta^{(1)}_{\lambda}$
$\displaystyle=\frac{3\lambda^{2}+m^{2}c_{6}}{2}\,,$
$\displaystyle\delta_{c_{6}}^{(1)}$ $\displaystyle=\frac{15\lambda
c_{6}}{2}\,,$ (38)

at one-loop order.

The two-loop part of the counterterms is then obtained from the second line of
(2.3). Once more, only terms with no powers of $P_{x,y,z}$ in
$\widehat{\mathcal{B}}^{(1)}_{\scriptscriptstyle\mathrm{EFT}}$,
$\widehat{\mathcal{C}}_{\scriptscriptstyle\mathrm{EFT}}$, and
$\widehat{\mathcal{D}}_{\scriptscriptstyle\mathrm{EFT}}$ are present, and we
have that

$\displaystyle B_{\scriptscriptstyle\mathrm{EFT}}^{(1)}$
$\displaystyle=-\frac{1}{16\pi^{2}}\frac{1}{\epsilon}\left[\delta^{(1)}_{m^{2}}+\frac{\delta_{\lambda}^{(1)}}{2}\phi^{2}+\frac{\delta_{c_{6}}^{(1)}}{4!}\phi^{4}\right]\,,$
$\displaystyle C_{\scriptscriptstyle\mathrm{EFT}}$
$\displaystyle=-\lambda\phi-\frac{c_{6}}{3!}\phi^{3}\,,\qquad
D_{\scriptscriptstyle\mathrm{EFT}}=-\lambda-\frac{c_{6}}{2}\phi^{2}\,,$ (39)

where we used that $\delta_{\phi}^{(1)}=0$ in this example. Inserting these
operators into the expressions in (2.4) (where, again, only the terms with
$m^{(\prime)}=n^{(\prime)}=r=0$ contributes) together with the expansion of
$Q^{-1}$ in (19) and evaluating the loop integrals, we find the two-loop
contributions to the counterterms. As before, those are obtained in an off-
shell basis, which reduces to the on-shell result

$\displaystyle\delta_{m^{2},1}^{(2)}$
$\displaystyle=-\frac{\lambda^{2}m^{2}}{4}\,,$
$\displaystyle\delta_{m^{2},2}^{(2)}$
$\displaystyle=\frac{4m^{2}\lambda^{2}+m^{4}c_{6}}{8}\,,$
$\displaystyle\delta_{\lambda,1}^{(2)}$
$\displaystyle=-\frac{9\lambda^{3}+5m^{2}\lambda c_{6}}{6}\,,$
$\displaystyle\delta_{\lambda,2}^{(2)}$
$\displaystyle=\frac{9\lambda^{3}+11m^{2}\lambda c_{6}}{4}\,,$
$\displaystyle\delta_{c_{6},1}^{(2)}$
$\displaystyle=-\frac{215\lambda^{2}c_{6}}{12}\,,$
$\displaystyle\delta_{c_{6},2}^{(2)}$
$\displaystyle=\frac{135\lambda^{2}c_{6}}{4}\,,$
$\displaystyle\delta_{\phi,1}^{(2)}$
$\displaystyle=-\frac{\lambda^{2}}{24}\,,$
$\displaystyle\delta_{\phi,2}^{(2)}$ $\displaystyle=0$ (40)

after appropriate field redefinitions. As a cross-check, we have verified that
these counterterms satisfy the consistency conditions on the double-poles
necessary for finite RG functions (see, e.g., [54]).

Having obtained the two-loop counterterms, we readily determine the anomalous
dimension $\gamma_{\phi}$ of the field $\phi$ along with the beta functions
for $m^{2}$, $\lambda$ and $c_{6}$ couplings:

$\displaystyle\gamma_{\phi}$
$\displaystyle=\frac{1}{2}\frac{\mathop{}\\!\mathrm{d}\ln(1+\delta_{\phi})}{\mathop{}\\!\mathrm{d}\ln\mu}=-\frac{\partial\delta_{\phi,1}^{(2)}}{\partial\lambda}=\frac{1}{(16\pi^{2})^{2}}\frac{\lambda^{2}}{12}\,,$
$\displaystyle\beta_{m^{2}}$
$\displaystyle=\frac{m^{2}\lambda}{16\pi^{2}}-\frac{5}{6}\frac{m^{2}\lambda^{2}}{(16\pi^{2})^{2}}\,,$
$\displaystyle\beta_{\lambda}$
$\displaystyle=\frac{3\lambda^{2}+m^{2}c_{6}}{16\pi^{2}}-\frac{1}{3}\frac{17\lambda^{3}+10\lambda
m^{2}c_{6}}{(16\pi^{2})^{2}},$ $\displaystyle\beta_{c_{6}}$
$\displaystyle=\frac{15\lambda
c_{6}}{16\pi^{2}}-\frac{427}{6}\frac{\lambda^{2}c_{6}}{(16\pi^{2})^{2}}\,.$
(41)

Those terms of the beta functions that involve only renormalizable couplings
have been cross-checked with RGBeta [53] while the contributions with the
$c_{6}$ coupling are found to agree with [55, 56, 57].

##  4  Conclusions and outlook

The discovery of physics beyond the SM is proving more challenging than
initially anticipated. Given the precision increase associated with upcoming
experimental searches and the absence of clear indications of the possible
shape of NP, the use of novel and more precise EFT approaches becomes more
important than ever.

In this letter, we have presented the initial steps toward functional EFT
matching and RG evolution at two-loop order, so far restricted to the case of
scalar theories. We have explicitly demonstrated that the _hard part_ of the
UV effective action is all that is needed for such calculations. To our
knowledge, this is the first time that an explicit proof of this statement has
been presented. While this result is applicable for both diagrammatic and
functional approaches, it becomes particularly useful for the latter since it
enables a power counting around which to perform an operator-product expansion
of the functional results. Building on this, we have provided closed-form
expressions for the evaluation of the two-loop effective action in the hard
limit. In this way, and analogously to the one-loop functional result, our
calculation of the two-loop EFT Lagrangian does not require the determination
of the target EFT basis and essentially amounts to simple algebraic
manipulations, making it particularly suitable for automation. We have also
presented a toy-model example that illustrates the main rationale behind the
application of our functional approach and, as a byproduct, we have verified
recent literature results concerning the determination of two-loop RG
equations in the geometric approach [57].

The extension of these methods to the more general case, including theories
with fermions and/or gauge bosons, remains non-trivial and will be discussed
in a forthcoming publication. In particular, we observe that the standard
techniques to make the functional evaluations manifestly covariant [10, 9] are
no longer applicable at two-loop order, and new strategies are required.
Likewise, the generalization of these results to higher-loop orders will also
be explored in the future.

##  Acknowledgments

We thank Julie Pagès and Supratim Das Bakshi for helpful discussions. The
authors are grateful to the Mainz Institute for Theoretical Physics (MITP) of
the Cluster of Excellence PRISMA+ (Project ID 39083149) for its hospitality
and support. The work of JFM is supported by the Spanish Ministry of Science
and Innovation (MCIN) and the European Union NextGenerationEU/PRTR under grant
IJC2020-043549-I, by the MCIN and State Research Agency (SRA) projects
PID2019-106087GB-C22 and PID2022-139466NB-C21 (ERDF), and by the Junta de
Andalucía projects P21_00199 and FQM101. The work of AP has received funding
from the Swiss National Science Foundation (SNSF) through the Eccellenza
Professorial Fellowship “Flavor Physics at the High Energy Frontier” project
number 186866. AET is funded by the Swiss National Science Foundation (SNSF)
through the Ambizione grant “Matching and Running: Improved Precision in the
Hunt for New Physics,” project number 209042.

##  Appendix A Matching formula at two-loop order

In this appendix, we use the functional expansion of the effective action to
demonstrate the explicit cancellation between soft-region loops in the UV and
loops in the EFT and prove the master formula for EFT matching (10) up to two-
loop order.

Our starting point is the off-shell matching condition (9). On the UV side, we
are dealing with the generating functional

$\displaystyle\Gamma^{\phi}_{\scriptscriptstyle\mathrm{UV}}[\hat{\phi}]$
$\displaystyle\equiv\Gamma_{\scriptscriptstyle\mathrm{UV}}\big{[}\widehat{\Phi}[\hat{\phi}],\,\hat{\phi}\big{]}\,,$
$\displaystyle\dfrac{\delta\Gamma_{\scriptscriptstyle\mathrm{UV}}}{\delta\Phi_{\alpha}}\big{[}\widehat{\Phi}[\hat{\phi}],\,\hat{\phi}\big{]}=0\,,$
(42)

which generates all 1LPI Green’s functions. We need an explicit expression for
$\Gamma^{\phi}_{\scriptscriptstyle\mathrm{UV}}$ order-by-order in the loop
expansion, and, so, we have to explicitly isolate each loop contribution in
the heavy-field EOMs. To this end, let
$\tilde{\eta}_{I}=\big{(}\overline{\Phi}_{\alpha}[\hat{\phi}],\,\hat{\phi}_{i}\big{)}$
solve the tree-level heavy field EOMs in the presence of $\hat{\phi}$, that is

$\displaystyle\dfrac{\delta
S_{\scriptscriptstyle\mathrm{UV}}^{(0)}}{\delta\Phi_{\alpha}}[\tilde{\eta}]=0\,.$
(43)

Up to one-loop order, the quantum-EOM solution reads

$\displaystyle\widehat{\Phi}_{\alpha}[\hat{\phi}]=\overline{\Phi}_{\alpha}[\hat{\phi}]-\hbar\,\widetilde{\mathcal{Q}}_{\alpha\beta}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\left(\widetilde{\mathcal{A}}^{(1)}_{\beta}+\dfrac{i}{2}\,\widetilde{\mathcal{C}}^{\,(0)}_{\beta
IJ}\widetilde{\mathcal{Q}}_{JI}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\right)+\mathcal{O}(\hbar^{2})\,,$
(44)

where we employ the same compact notation as in the main text, i.e.,
$\widetilde{\mathcal{Q}}_{IJ}=\mathcal{Q}_{IJ}[\tilde{\eta}]$ and likewise for
other functions exclusively dependent of $\widetilde{\eta}_{I}$. Explicitly,
the UV generating functional is then

$\displaystyle\Gamma_{\scriptscriptstyle\mathrm{UV}}^{\phi}[\hat{\phi}]$
$\displaystyle=\widetilde{S}_{\scriptscriptstyle\mathrm{UV}}^{(0)}+\hbar\,\widetilde{S}_{\scriptscriptstyle\mathrm{UV}}^{(1)}+\dfrac{i\hbar}{2}\big{(}\log\widetilde{\mathcal{Q}}\big{)}_{II}+\hbar^{2}\,\widetilde{S}_{\scriptscriptstyle\mathrm{UV}}^{(2)}+\dfrac{i\hbar^{2}}{2}\widetilde{\mathcal{Q}}_{IJ}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widetilde{\mathcal{B}}_{JI}^{(1)}+\dfrac{\hbar^{2}}{12}\widetilde{\mathcal{C}}_{IJK}\widetilde{\mathcal{Q}}_{IL}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widetilde{\mathcal{Q}}_{JM}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widetilde{\mathcal{Q}}_{KN}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\widetilde{\mathcal{C}}_{LMN}$
$\displaystyle\quad-\dfrac{\hbar^{2}}{8}\widetilde{\mathcal{Q}}_{IJ}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\,\widetilde{\mathcal{D}}_{IJKL}\widetilde{\mathcal{Q}}_{KL}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}-\dfrac{\hbar^{2}}{2}\bigg{(}\widetilde{\mathcal{A}}^{(1)}_{\alpha}+\dfrac{i}{2}\widetilde{\mathcal{Q}}_{IJ}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\,\widetilde{\mathcal{C}}^{(0)}_{IJ\alpha}\bigg{)}\widetilde{\mathcal{Q}}_{\alpha\beta}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\bigg{(}\widetilde{\mathcal{A}}^{(1)}_{\beta}+\dfrac{i}{2}\,\widetilde{\mathcal{C}}^{(0)}_{\beta
KL}\widetilde{\mathcal{Q}}_{KL}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\bigg{)}+\mathcal{O}(\hbar^{3})\,.$
(45)

This expression is the same we would have gotten directly from (2.1) by
setting $J_{\Phi}=0$ and performing the Legendre transform on the light fields
only. We will also need the expression

$\displaystyle\dfrac{\delta\overline{\Phi}_{\alpha}[\hat{\phi}]}{\delta\hat{\phi}_{i}}=-\widetilde{\mathcal{Q}}_{i\beta}\widetilde{\mathcal{Q}}_{\beta\alpha}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}$
(46)

for our proof, which is obtained by taking a functional derivative of the
light classical fields in (43). Finally, it is useful to define the upper-
triangular transformation $V_{IJ}$ that diagonalizes the inverse dressed
propagator $\mathcal{Q}$. It takes the form

$\displaystyle V_{IJ}$
$\displaystyle=\begin{pmatrix}\delta_{\alpha\beta}&-\mathcal{Q}_{\alpha\gamma}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}\mathcal{Q}_{\gamma
j}\\\ 0&\delta_{ij}\end{pmatrix}\,,$ (47)

such that

$\displaystyle\mathds{Q}_{IJ}$ $\displaystyle\equiv
V^{\intercal}_{IK}\mathcal{Q}_{KL}V_{LJ}=\begin{pmatrix}\mathcal{Q}_{\alpha\beta}&0\\\
0&\mathcal{Q}_{ij}-\mathcal{Q}_{i\alpha}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}_{\alpha\beta}\mathcal{Q}_{\beta
j}\end{pmatrix}\,.$ (48)

In the next sections, we proceed to establish the matching formula order by
order in the loop expansion.

###  A.1  Tree level

It is important to discriminate between non-local objects such as
$\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1}$ and their EFT
operator-product expansions, as they have vastly different behaviors when
performing the loop integrals. We denote the finite-order expansion of the
heavy-field EOM solution by

$\displaystyle\Phi^{\mathrm{s}}=\overline{\Phi}[\hat{\phi}]+\mathcal{O}(1/\Lambda^{N+1})\,,$
(49)

which is sufficient to determine the $N$-order matching. In what follows, the
superscript ‘s’ in any other quantities will be used to denote the dependence
on $\Phi^{\mathrm{s}}[\hat{\phi}]$, e.g.,
$\mathcal{Q}^{\mathrm{s}}=\mathcal{Q}[\eta^{\mathrm{s}}]=\mathcal{Q}\big{[}\Phi^{\mathrm{s}}[\hat{\phi}],\hat{\phi}\big{]}$.

The tree-level matching criteria follows directly from the matching condition
(9), understood as a series in $\Lambda$. At leading order, we simply find

$\displaystyle
S^{(0)}_{\scriptscriptstyle\mathrm{EFT}}[\hat{\phi}]=S_{\scriptscriptstyle\mathrm{UV}}^{(0)}[\eta^{\mathrm{s}}]=S_{\scriptscriptstyle\mathrm{UV}}^{(0)\mathrm{s}}\,.$
(50)

This is a familiar result: the tree-level EFT is obtained by substituting in
the UV theory the classic EOMs of all heavy fields expanded in the heavy
masses.

###  A.2  Expansion by regions

Figure 2: Region decomposition of the sunset graph, with red and violet lines
denoting soft and hard momenta, respectively.

Loop integrals in dimensional regularization can, under certain conditions, be
written as a sum over different momentum regions [42, 43]. This decomposition
is known as expansion by regions, and in each region the propagators are
series-expanded according to the magnitude of the loop momenta in that region.
Working as we are, with a generic heavy scale set by $\Lambda$, we
discriminate only between whether (Euclidean) loop momenta are soft
($\ll\Lambda$) or hard ($\gtrsim\Lambda$). Given a topology $G$, we can
decompose it in regions as

$\displaystyle G=\sum_{\begin{subarray}{c}\gamma\subseteq G\\\
\mathrm{loops}\end{subarray}}R_{\mathrm{hard}}^{(\gamma)}(G)\,.$ (51)

The sum runs over all (sub)loops $\gamma$ of $G$ and the operator
$R_{\mathrm{hard}}^{(\gamma)}$ sets the corresponding loop momenta hard, while
the remaining ones are taken to be soft. For instance, $\gamma$ runs over 5
subloops in the case of the sunset topology: no loops, 3 distinct one-loop
graphs, and the full two-loop graph (see Figure 2). We have verified that this
region decomposition is valid up to two-loop order, following the criteria in
[43]. To keep track of the regions in DeWitt notation, we introduce colored
indices to distinguish soft and hard momentum modes:

$\displaystyle{\color[rgb]{0.68,0.25648,0.30984}I},\,{\color[rgb]{0.68,0.25648,0.30984}\alpha},\,{\color[rgb]{0.68,0.25648,0.30984}i}:\quad$
$\displaystyle
p_{\color[rgb]{0.68,0.25648,0.30984}I},\,p_{\color[rgb]{0.68,0.25648,0.30984}\alpha},\,p_{\color[rgb]{0.68,0.25648,0.30984}i}\ll\Lambda\,,$
$\displaystyle{\color[rgb]{0.415,0.20692,0.61507}I},\,{\color[rgb]{0.415,0.20692,0.61507}\alpha},\,{\color[rgb]{0.415,0.20692,0.61507}i}:\quad$
$\displaystyle
p_{\color[rgb]{0.415,0.20692,0.61507}I},\,p_{\color[rgb]{0.415,0.20692,0.61507}\alpha},\,p_{\color[rgb]{0.415,0.20692,0.61507}i}\gtrsim\Lambda\,,$
(52)

All dressed propagators are understood as being expanded according to the
color of their indices, cf. (2.4).

In the EFT, there are no propagators with heavy masses, so all hard-momenta
loops are scaleless and vanish in dimensional regularization. As a result,
_all indices in the EFT are soft type_. The series expansion in
$\Phi^{\mathrm{s}}[\hat{\phi}]$ and in the soft regions are identical, so in
general

$\displaystyle\dfrac{\delta\Phi^{\mathrm{s}}_{\alpha}[\hat{\phi}]}{\delta\hat{\phi}_{{\color[rgb]{0.68,0.25648,0.30984}i}}}=\left.\dfrac{\delta\overline{\Phi}^{(0)}_{\alpha}[\hat{\phi}]}{\delta\hat{\phi}_{{\color[rgb]{0.68,0.25648,0.30984}i}}}\right|_{\overline{\Phi}^{(0)}=\Phi^{\mathrm{s}}}\,,$
(53)

which, however, is not the case in the hard region (or for generic indices).
Having defined this new notation, we turn to the one-loop case.

###  A.3  One-loop level

We use expansion by regions to decompose the one-loop UV effective action as

$\displaystyle\Gamma_{\scriptscriptstyle\mathrm{UV}}^{\phi(1)}[\hat{\phi}]=S_{\scriptscriptstyle\mathrm{UV}}^{(1)\mathrm{s}}+\dfrac{i}{2}\big{(}\\!\log\mathcal{Q}^{\mathrm{s}}\big{)}_{\\!{\color[rgb]{0.415,0.20692,0.61507}II}}+\dfrac{i}{2}\big{(}\\!\log\mathds{Q}^{\mathrm{s}}\big{)}_{\\!{\color[rgb]{0.68,0.25648,0.30984}ii}}\,.$
(54)

This follows from using the identity

$\displaystyle\mathrm{Tr}\log\mathcal{Q}=\log\mathrm{Det}\,\mathcal{Q}\,,$
(55)

and block-diagonalizing the inverse dressed propagator with $V^{\mathrm{s}}$
satisfying $\mathrm{Det}\,V=1$. The equality is then due to
$(\log\mathds{Q}^{\mathrm{s}})_{{\color[rgb]{0.68,0.25648,0.30984}\alpha\alpha}}$
being a scaleless loop, as can be seen from
$\mathds{Q}^{\mathrm{s}}_{\alpha\beta}=\mathcal{Q}^{\mathrm{s}}_{\alpha\beta}$,
indicating that heavy masses are present in all propagators.

Next, we determine the inverse dressed propagator in the EFT, as we need it
for the EFT loop contributions. Using (53) and (46), one can show that

$\displaystyle\mathcal{Q}^{{\scriptscriptstyle\mathrm{EFT}}}_{{\color[rgb]{0.68,0.25648,0.30984}ij}}[\hat{\phi}]=\dfrac{\delta^{2}S_{\scriptscriptstyle\mathrm{UV}}^{(0)}[\eta^{\mathrm{s}}]}{\delta\hat{\phi}_{\color[rgb]{0.68,0.25648,0.30984}i}\delta\hat{\phi}_{\color[rgb]{0.68,0.25648,0.30984}j}}=\mathds{Q}^{\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}ij}\,.$
(56)

We can now determine the one-loop EFT action by inserting (56) and (54) into
matching condition (9):

$\displaystyle S_{\scriptscriptstyle\mathrm{EFT}}^{(1)}[\hat{\phi}]$
$\displaystyle=\Gamma_{\scriptscriptstyle\mathrm{UV}}^{\phi(1)}[\hat{\phi}]-\dfrac{i}{2}\big{(}\\!\log\mathcal{Q}^{{\scriptscriptstyle\mathrm{EFT}}}\big{)}_{{\color[rgb]{0.68,0.25648,0.30984}ii}}$
$\displaystyle=S_{\scriptscriptstyle\mathrm{UV}}^{(1)\mathrm{s}}+\dfrac{i}{2}\big{(}\\!\log\mathcal{Q}^{\mathrm{s}}\big{)}_{\\!{\color[rgb]{0.415,0.20692,0.61507}II}}\,.$
(57)

This reproduces the well-known one-loop matching formula [12, 13], and we can
finally proceed to generalize to the two-loop case.

###  A.4  Two-loop level

It will be useful to work in the basis where the inverse dressed propagator is
block-diagonal also at two-loop order. To this end, we introduce the notation

$\displaystyle\mathds{A}^{(\ell)}_{I}$
$\displaystyle=\mathcal{A}^{(\ell)}_{J}V_{JI}\,,$
$\displaystyle\mathds{B}^{(\ell)}_{IJ}$
$\displaystyle=\mathcal{B}^{(\ell)}_{KL}V_{KI}V_{LJ}\,,$
$\displaystyle\mathds{C}^{(\ell)}_{IJK}$
$\displaystyle=\mathcal{C}^{(\ell)}_{LMN}V_{LI}V_{MJ}V_{NK}\,,$
$\displaystyle\mathds{D}^{(\ell)}_{IJKL}$
$\displaystyle=\mathcal{D}^{(\ell)}_{MNOP}V_{MI}V_{NJ}V_{OK}V_{PL}\,.$ (58)

Applying the expansion by regions (51) to the 1LPI UV effective action (A)
yields

$\displaystyle\Gamma_{\scriptscriptstyle\mathrm{UV}}^{\mathrm{\phi(2)}}[\hat{\phi}]=$
$\displaystyle\phantom{+}S_{\scriptscriptstyle\mathrm{UV}}^{(2)\mathrm{s}}+\dfrac{i}{2}\mathcal{Q}_{{\color[rgb]{0.415,0.20692,0.61507}IJ}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\mathcal{B}^{(1)\mathrm{s}}_{{\color[rgb]{0.415,0.20692,0.61507}JI}}+\dfrac{i}{2}\mathds{Q}_{{\color[rgb]{0.68,0.25648,0.30984}IJ}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\mathds{B}_{{\color[rgb]{0.68,0.25648,0.30984}JI}}^{(1)\mathrm{s}}-\dfrac{1}{8}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}IJ}\mathcal{D}^{(0)\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}IJKL}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}KL}-\dfrac{1}{4}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}IJ}\mathds{D}^{(0)\mathrm{s}}_{{\color[rgb]{0.415,0.20692,0.61507}IJ}{\color[rgb]{0.68,0.25648,0.30984}KL}}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}KL}$
$\displaystyle-\dfrac{1}{8}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IJ}\mathds{D}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}IJKL}}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}KL}+\dfrac{1}{12}\mathcal{C}^{(0)\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}IJK}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}IL}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}JM}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}KN}\mathcal{C}^{(0)\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}LMN}+\dfrac{1}{4}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}I}{\color[rgb]{0.415,0.20692,0.61507}KL}}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IJ}\mathds{Q}^{\mathrm{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}}_{{\color[rgb]{0.415,0.20692,0.61507}KM}}\mathds{Q}^{\mathrm{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}}_{{\color[rgb]{0.415,0.20692,0.61507}LN}}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}J}{\color[rgb]{0.415,0.20692,0.61507}MN}}$
$\displaystyle+\dfrac{1}{12}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IJK}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IL}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}JM}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}KN}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}LMN}-\dfrac{1}{2}\bigg{(}\mathcal{A}^{(1)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\alpha}}+\dfrac{i}{2}\mathcal{Q}_{{\color[rgb]{0.415,0.20692,0.61507}IJ}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\mathcal{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.415,0.20692,0.61507}JI}{\color[rgb]{0.68,0.25648,0.30984}\alpha}}\bigg{)}\mathcal{Q}_{{\color[rgb]{0.68,0.25648,0.30984}\alpha\beta}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\bigg{(}\mathcal{A}^{(1)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\beta}}+\dfrac{i}{2}\mathcal{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\beta}{\color[rgb]{0.415,0.20692,0.61507}KL}}\mathcal{Q}_{{\color[rgb]{0.415,0.20692,0.61507}LK}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\bigg{)}$
$\displaystyle-\dfrac{i}{2}\bigg{(}\mathds{A}^{(1)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\alpha}}+\dfrac{i}{2}\mathds{Q}_{{\color[rgb]{0.415,0.20692,0.61507}IJ}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.415,0.20692,0.61507}JI}{\color[rgb]{0.68,0.25648,0.30984}\alpha}}\bigg{)}\mathds{Q}_{{\color[rgb]{0.68,0.25648,0.30984}\alpha\beta}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\beta}{\color[rgb]{0.68,0.25648,0.30984}KL}}\mathds{Q}_{{\color[rgb]{0.68,0.25648,0.30984}LK}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}+\dfrac{1}{8}\mathds{Q}_{{\color[rgb]{0.68,0.25648,0.30984}IJ}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}JI\alpha}}\mathds{Q}_{{\color[rgb]{0.68,0.25648,0.30984}\alpha\beta}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\beta}{\color[rgb]{0.68,0.25648,0.30984}KL}}\mathds{Q}_{{\color[rgb]{0.68,0.25648,0.30984}LK}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\,.$
(59)

Consistently with the low-energy expansion, we have taken the soft limit in
all tree-level propagators (present in the last three terms of the expression
above). Building on the tree-level and one-loop EFT matching results, we can
now construct the genuine loop contributions to the two-loop EFT effective
action. EFT vertices are determined by the application of consecutive
derivatives to the dressed propagator (56) together with (46):

$\displaystyle\mathcal{C}^{(0){\scriptscriptstyle\mathrm{EFT}}}_{\color[rgb]{0.68,0.25648,0.30984}ijk}[\hat{\phi}]$
$\displaystyle=\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}ijk}\,,$
$\displaystyle\mathcal{D}^{(0){\scriptscriptstyle\mathrm{EFT}}}_{\color[rgb]{0.68,0.25648,0.30984}ijk\ell}[\hat{\phi}]$
$\displaystyle=\mathds{D}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}ijk\ell}-\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}ij\alpha}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}\alpha\beta}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}\beta
k\ell}-\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}ik\alpha}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}\alpha\beta}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}\beta
j\ell}$
$\displaystyle\quad-\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}i\ell\alpha}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}\alpha\beta}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}\beta
jk}\,.$ (60)

Hence, the EFT 4-point vertex is the 4-point function of the UV theory, with
the heavy legs put on-shell with EOMs, supplemented by tree-level connected
4-point functions from a propagating heavy state in the UV. There are also
contributions from the insertion of the one-loop EFT action in a one-loop
topology. We find that

$\displaystyle\widehat{\mathcal{B}}^{(1){\scriptscriptstyle\mathrm{EFT}}}_{\color[rgb]{0.68,0.25648,0.30984}ij}$
$\displaystyle=\mathds{B}^{(1)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}ij}-\left(\\!\mathds{A}^{(1)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\alpha}}+\dfrac{i}{2}\mathds{Q}^{\mathrm{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}}_{{\color[rgb]{0.415,0.20692,0.61507}IJ}}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.415,0.20692,0.61507}JI}{\color[rgb]{0.68,0.25648,0.30984}\alpha}}\\!\right)\\!\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}_{\color[rgb]{0.68,0.25648,0.30984}\alpha\beta}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}\beta
ij}$
$\displaystyle\quad+\dfrac{i}{2}\mathds{Q}^{\mathrm{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}}_{{\color[rgb]{0.415,0.20692,0.61507}IJ}}\mathds{D}^{(0)\mathrm{s}}_{{\color[rgb]{0.415,0.20692,0.61507}JI}{\color[rgb]{0.68,0.25648,0.30984}ij}}-\dfrac{i}{2}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}i}{\color[rgb]{0.415,0.20692,0.61507}KL}}\mathds{Q}^{\mathrm{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}}_{{\color[rgb]{0.415,0.20692,0.61507}KM}}\mathds{Q}^{\mathrm{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}}_{{\color[rgb]{0.415,0.20692,0.61507}LN}}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.415,0.20692,0.61507}MN}{\color[rgb]{0.68,0.25648,0.30984}j}}\,,$
(61)

based on the matching result (A.3). Inserting these results into the
expression for the two-loop effective action (2.1) and, by extending sums over
light indices to run over all indices (recognizing that scaleless loop
integrals vanish), we find the following result for the EFT effective action:

$\displaystyle\Gamma^{(2)}_{\scriptscriptstyle\mathrm{EFT}}[\hat{\phi}]=$
$\displaystyle\phantom{+}\widehat{S}_{\scriptscriptstyle\mathrm{EFT}}^{(2)}+\dfrac{i}{2}\mathds{B}^{(1)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IJ}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}JI}-\dfrac{i}{2}\left(\\!\mathds{A}^{(1)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\alpha}}+\dfrac{i}{2}\mathds{Q}^{\mathrm{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}}_{{\color[rgb]{0.415,0.20692,0.61507}IJ}}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.415,0.20692,0.61507}JI}{\color[rgb]{0.68,0.25648,0.30984}\alpha}}\\!\right)\\!\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}_{\color[rgb]{0.68,0.25648,0.30984}\alpha\beta}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}\beta
IJ}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}JI}-\dfrac{1}{4}\mathds{Q}^{\mathrm{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}}_{{\color[rgb]{0.415,0.20692,0.61507}IJ}}\mathds{D}^{(0)\mathrm{s}}_{{\color[rgb]{0.415,0.20692,0.61507}JI}{\color[rgb]{0.68,0.25648,0.30984}KL}}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}LK}$
$\displaystyle+\dfrac{1}{4}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}I}{\color[rgb]{0.415,0.20692,0.61507}KL}}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IJ}\mathds{Q}^{\mathrm{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}}_{{\color[rgb]{0.415,0.20692,0.61507}KM}}\mathds{Q}^{\mathrm{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,s}}_{{\color[rgb]{0.415,0.20692,0.61507}LN}}\mathds{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}J}{\color[rgb]{0.415,0.20692,0.61507}MN}}-\dfrac{1}{8}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IJ}\mathds{D}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IJKL}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}KL}+\dfrac{1}{8}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IJ}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IJ\alpha}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}\alpha\beta}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}\beta
KL}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}KL}$
$\displaystyle+\dfrac{1}{12}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IJK}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}IL}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}JM}\mathds{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}KN}\mathds{C}^{(0)\mathrm{s}}_{\color[rgb]{0.68,0.25648,0.30984}LMN}\,.$
(62)

We can now determine the master formula for two-loop matching directly from
the matching condition (9) by canceling terms between (A.4) and (A.4):

$\displaystyle S_{\scriptscriptstyle\mathrm{EFT}}^{(2)}[\hat{\phi}]$
$\displaystyle=\Gamma_{\scriptscriptstyle\mathrm{UV}}^{\phi(2)}[\hat{\phi}]-\big{(}\Gamma_{\scriptscriptstyle\mathrm{EFT}}^{(2)}[\hat{\phi}]-\widehat{S}_{\scriptscriptstyle\mathrm{EFT}}^{(2)}\big{)}$
$\displaystyle=S_{\scriptscriptstyle\mathrm{UV}}^{(2)\mathrm{s}}+\dfrac{i}{2}\mathcal{Q}_{{\color[rgb]{0.415,0.20692,0.61507}IJ}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\mathcal{B}_{{\color[rgb]{0.415,0.20692,0.61507}JI}}^{(1)\,\mathrm{s}}-\dfrac{1}{8}\mathcal{D}^{(0)\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}IJKL}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}IJ}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}KL}+\dfrac{1}{12}\mathcal{C}^{(0)\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}IJK}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}IL}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}JM}\mathcal{Q}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}KN}\mathcal{C}^{(0)\mathrm{s}}_{\color[rgb]{0.415,0.20692,0.61507}LMN}$
$\displaystyle\quad-\dfrac{1}{2}\bigg{(}\mathcal{A}^{(1)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\alpha}}+\dfrac{i}{2}\mathcal{Q}_{{\color[rgb]{0.415,0.20692,0.61507}IJ}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\mathcal{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.415,0.20692,0.61507}JI}{\color[rgb]{0.68,0.25648,0.30984}\alpha}}\bigg{)}\mathcal{Q}_{{\color[rgb]{0.68,0.25648,0.30984}\alpha\beta}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\bigg{(}\mathcal{A}^{(1)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\beta}}+\dfrac{i}{2}\mathcal{C}^{(0)\mathrm{s}}_{{\color[rgb]{0.68,0.25648,0.30984}\beta}{\color[rgb]{0.415,0.20692,0.61507}KL}}\mathcal{Q}_{{\color[rgb]{0.415,0.20692,0.61507}LK}}^{\vbox{\hbox{\scalebox{0.6}[1.0]{$-$}}}1\,\mathrm{s}}\bigg{)}\,.$
(63)

We identify the master formula for two-loop matching (A.4) as the two-loop
case of the generic master formula (10). The first line of (A.4) is the pure
hard part of the two-loop 1PI topologies, whereas the second line comes from
two insertion of the hard one-loop contribution to the heavy-field EOM.

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