# Role of dimension-eight operators in an EFT for the 2HDM

Sally<EMAIL_ADDRESS>Department of Physics, Brookhaven National
Laboratory, Upton, New York 11973 U.S.A. Duarte<EMAIL_ADDRESS>Department of Physics, Brookhaven National Laboratory, Upton, New York 11973
U.S.A. Samuel<EMAIL_ADDRESS>Physics Department, Harvard
University, Cambridge, MA, 02138, U.S.A. Matthew<EMAIL_ADDRESS>Department of Physics, Brookhaven National Laboratory, Upton, New York 11973
U.S.A.

[12mm]12mm The Standard Model effective field theory (SMEFT) is the tool of
choice for studying deviations of Higgs couplings from the Standard Model
predictions. The SMEFT is an expansion in an infinite tower of higher
dimension operators, which is typically truncated at dimension-6. We consider
the effective theory including dimension-8 operators and examine the matching
to the 2 Higgs Doublet Model (2HDM) that is assumed to be valid at some high
scale. Both the limits from the direct production of single Higgs bosons and
the indirect limits on the Higgs tri-linear coupling are considered in the
context of the SMEFT matched to the 2HDM, and the importance of the
assumptions about the expansions in powers of the high scale are examined
numerically.

##  1 Introduction

One of the major goals of the LHC program is to extract information about high
scale — or ultraviolet (UV) — physics from precision measurements of Higgs
couplings. In the scenario where the only observed particles are those of the
Standard Model (SM), the measurements can be analyzed in an effective field
theory (EFT) framework, where the Lagrangian $\mathcal{L}$ contains an
infinite tower of higher dimension operators $O_{i}^{n}$ (for a review, see
Ref. [1]),

$\mathcal{L}=\mathcal{L}_{\mathrm{SM}}+\Sigma_{n,i}{C_{i}^{n}O_{i}^{n}\over\Lambda^{n-4}}\,.$
(1)

(Here, $\mathcal{L}_{\mathrm{SM}}$ is the SM Lagrangian, $\Lambda$ is the UV
scale and $C_{i}^{n}$ are coefficient functions.) Considering the Higgs as an
SU(2) doublet defines this as the Standard Model effective field theory
(SMEFT). In this way, all new physics effects are contained in the coefficient
functions $C_{i}^{n}$, also known as Wilson coefficients (WCs). These can be
constrained by the LHC Higgs and top quark data, $W^{+}W^{-}$ data at both
LEP-II and the LHC, and precision electroweak measurements, through analyses
that use increasingly sophisticated global fits[2, 3, 4, 5, 6, 7, 8, 9].

The ultimate goal of this framework is a double one: first, to observe a
pattern of non-zero coefficient functions; then, to associate that pattern
with a specific new physics model. In the absence of any definitive
observation, one can pave the way by focusing on the second task, i.e. by
studying the quality with which one can associate the EFT description to that
of different models. Several studies considered the matching between the two
descriptions at tree-level and truncated Eq. 1 with dimension-6 operators; for
example, simple cases where the UV physics arises from the exchange of a
single weakly coupled particle have been categorized in the pioneering work of
Ref. [10], and the effects in some well known models such as the 2 Higgs
Doublet Model (2HDM) or the Higgs singlet model were studied in Refs. [11, 12,
13, 14, 15, 16, 17].

However, there are many avenues that need to be explored to put this program
on a solid footing. One of them investigates the effects of radiative
corrections to the EFT predictions[18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31]. Furthermore, the one-loop matching between the UV theory and the
EFT at the high scale can potentially have significant numerical effects[32,
33, 34, 35, 36] and was perfomed in the Higgs singlet model [37, 38, 39, 40,
41], where the numerical impact of the one-loop matching ends up being small.
In another relevant avenue, one can explore the effects of higher dimension
operators. The global fits described above typically truncate the expansion
with the dimension-6 operators. The numerical relevance of dimension-8
operators is not known in general, having been clarified in only a few
specific cases. One of them concerns the $WWh$ vertex, where the contribution
from the dimension-8 operator to $Wh$ production at the LHC is
${\cal{O}}(15\%)$ [42]. Yet the size of the dimension-8 effects turns out to
be extremely model dependent: whereas an SU(2) vector triplet model yields
significant dimension-8 effects in the analysis of precision electroweak
data[43], a vector-like top partner induces negligible dimension-8 effections
in $t{\overline{t}}h$ production[44].

In this work, we examine the importance of dimension-8 operators in the EFT
for the 2HDM at the leading order in the loop expansion. A strong motivation
for this study is the fact that at dimension-6 (and for the Type-I version of
the full model), the SMEFT poorly reproduces the predictions of the 2HDM [15],
since the Higgs coupling to gauge bosons first arises at dimension-8. We
organize the paper as follows: in Section 2, we describe our conventions for
the 2HDM in order to set the notation; we then present the matching to the
SMEFT to dimension-8 in Section 3, and some numerical results for fits to
Higgs coupling measurements in the full 2HDM and the SMEFT in Section 5;
finally, we present our conclusions in Section 6.

##  2 2HDM recap

In this section, we review the 2HDM [45, 46, 47] (see also Refs. [15, 48]).
Besides the SM scalar doublet $\Phi_{1}$, the model contains an extra doublet
$\Phi_{2}$; each doublet has in general a non-zero vacuum expectation value
(vev), which we identify as $v_{1}/\sqrt{2}$ and $v_{2}/\sqrt{2}$,
respectively. We assume a softly broken $Z_{2}$ symmetry, according to which
$\Phi_{1}\to\Phi_{1},\Phi_{2}\to-\Phi_{2}$. This symmetry is extended to the
fermion fields; such an extension can be made in four different ways, each one
corresponding to a different type of 2HDM: Type-I, Type-II, Type-L (or Lepton-
Specific) and Type-F (or Flipped); for details, see Ref. [47].

It is convenient to work in the so-called Higgs basis[49, 50, 51, 52], where
the doublets are identified as $H_{1}$ and $H_{2}$, and which is obtained by
rotating the doublets in the original basis according to:

$\left(\begin{array}[]{c}H_{1}\\\
H_{2}\end{array}\right)=\left(\begin{array}[]{cc}c_{\beta}&s_{\beta}\\\
-s_{\beta}&c_{\beta}\end{array}\right)\left(\begin{array}[]{c}\Phi_{1}\\\
\Phi_{2}\end{array}\right),$ (2)

where we introduce the short-hand notation $c_{x}=\cos(x),~{}s_{x}=\sin(x)$,
which we use for any angle $x$. The angle $\beta$ in Eq. 2 obeys the relation
$\tan\beta=v_{2}/v_{1}$, which implies that, in the Higgs basis, the vev is
entirely contained in the first doublet; that is, the vev of $H_{2}$ is zero,
whereas the vev $H_{1}$ is $v/\sqrt{2}$, with
$v=\sqrt{v_{1}^{2}+v_{2}^{2}}=246$ GeV. For our purposes, we can focus on
three sectors of the full Lagrangian:

$\mathcal{L}_{\mathrm{2HDM}}\ni\mathcal{L}_{\mathrm{kin}}+\mathcal{L}_{Y}-V,$
(3)

where the terms in the right-hand side respectively correspond to the scalar
kinetic piece, the Yukawa piece and the potential. In the Higgs basis, these
sectors can be written as:

$\displaystyle\mathcal{L}_{\mathrm{kin}}$ $\displaystyle=$
$\displaystyle\left(D_{\mu}H_{1}\right)^{\dagger}\left(D^{\mu}H_{1}\right)+\left(D_{\mu}H_{2}\right)^{\dagger}\left(D^{\mu}H_{2}\right),$
(4a) $\displaystyle V$ $\displaystyle=$ $\displaystyle
Y_{1}H_{1}^{\dagger}H_{1}+Y_{2}H_{2}^{\dagger}H_{2}+\left(Y_{3}H_{1}^{\dagger}H_{2}+\textrm{h.c.}\right)$
(4b)
$\displaystyle+\frac{Z_{1}}{2}\left(H_{1}^{\dagger}H_{1}\right)^{2}+\frac{Z_{2}}{2}\left(H_{2}^{\dagger}H_{2}\right)^{2}+Z_{3}\left(H_{1}^{\dagger}H_{1}\right)\left(H_{2}^{\dagger}H_{2}\right)+Z_{4}\left(H_{1}^{\dagger}H_{2}\right)\left(H_{2}^{\dagger}H_{1}\right)$
$\displaystyle+\left\\{\frac{Z_{5}}{2}\left(H_{1}^{\dagger}H_{2}\right)^{2}+Z_{6}\left(H_{1}^{\dagger}H_{1}\right)\left(H_{1}^{\dagger}H_{2}\right)+Z_{7}\left(H_{2}^{\dagger}H_{2}\right)\left(H_{1}^{\dagger}H_{2}\right)+\textrm{h.c.}\right\\},$
$\displaystyle\mathcal{L}_{Y}$ $\displaystyle=$
$\displaystyle-\lambda_{u}^{(1)}\bar{u}_{R}\tilde{H}_{1}^{\dagger}q_{L}-\lambda_{u}^{(2)}\bar{u}_{R}\tilde{H}_{2}^{\dagger}q_{L}-\lambda_{d}^{(1)}\bar{d}_{R}H_{1}^{\dagger}q_{L}-\lambda_{d}^{(2)}\bar{d}_{R}H_{2}^{\dagger}q_{L}-\lambda_{l}^{(1)}\bar{e}_{R}H_{1}^{\dagger}l_{L}-\lambda_{l}^{(2)}\bar{e}_{R}H_{2}^{\dagger}l_{L}+\textrm{h.c.}$
(4c) $\displaystyle=$
$\displaystyle-\lambda_{u}^{(1)*}H_{1}^{\dagger}\widehat{q}_{L}u_{R}-\lambda_{u}^{(2)*}H_{2}^{\dagger}\widehat{q}_{L}u_{R}-\lambda_{d}^{(1)}\bar{d}_{R}H_{1}^{\dagger}q_{L}-\lambda_{d}^{(2)}\bar{d}_{R}H_{2}^{\dagger}q_{L}-\lambda_{l}^{(1)}\bar{e}_{R}H_{1}^{\dagger}l_{L}-\lambda_{l}^{(2)}\bar{e}_{R}H_{2}^{\dagger}l_{L}$
$\displaystyle\qquad+\textrm{h.c.},$

where $\tilde{H}_{i}=i\sigma_{2}H_{i}^{*}$ and we introduce
$\widehat{q}_{L}\equiv-i\sigma_{2}(\bar{q}_{L})^{\mathrm{T}}$.111Making the
SU(2) components explicit, $\widehat{q}_{L}=\begin{pmatrix}-\bar{d}_{L}\\\
\bar{u}_{L}\end{pmatrix}$. This implies the relation
$\bar{u}_{R}{\widehat{q}_{L}}^{\dagger}H_{j}=\bar{u}_{R}{\tilde{H}_{j}}^{\dagger}q_{L}$,
as well as its Hermitian conjugate
$H_{j}^{\dagger}\widehat{q}_{L}u_{R}=\bar{q}_{L}\tilde{H}_{j}u_{R}$. We
neglect generation indices on the right-handed SU(2) singlets $u_{R},~{}d_{R}$
and $e_{R}$ and on the left-handed SU(2) doublets, $q_{L}$ and $l_{L}$. The
parameters $Y_{3},Z_{5},Z_{6},Z_{7}$ and the Yukawa parameters
$\lambda_{f}^{(i)}$ are in general complex — for $i=1,2$, and where $f$ can
represent any of the three types of charged fermions: up-type quarks ($u$),
down-type quarks ($d$) and charged leptons ($l$). The remaining parameters are
required to be real by hermiticity. Note, however, that the phases of
$\lambda_{f}^{(i)}$ can always be absorbed by the fermion fields; hence, we
take $\lambda_{f}^{(i)}$ to be real without loss of generality. We thus write:

$\lambda_{f}^{(1)}=\dfrac{\sqrt{2}}{v}m_{f},\qquad\lambda_{f}^{(2)}=\dfrac{\eta_{f}}{\tan\beta}\lambda_{f}^{(1)},$
(5)

where $\eta_{f}$ depends on the particular type of 2HDM, given in Table 1, and
$m_{f}$ is to be understood as a 3x3 diagonal matrix in the flavour space
containing the masses of the three generations of fermion type $f$. Finally,
the minimization equations read:

$Y_{1}=-\dfrac{Z_{1}}{2}v^{2},\qquad Y_{3}=-\dfrac{Z_{6}}{2}v^{2}\,.$ (6)

|  Type-I |  Type-II |  Type-L |  Type-F
---|---|---|---|---
$\eta_{u}$ |  1 |  1 |  1 |  1
$\eta_{d}$ |  1 |  $-\tan^{2}\beta$ |  1 |  $-\tan^{2}\beta$
$\eta_{l}$ |  1 |  $-\tan^{2}\beta$ |  $-\tan^{2}\beta$ |  1
|  |  |  |
Table 1: Values of the parameter $\eta_{f}$ for the different types of 2HDM
and for the different types of charged fermions.

In this work, we are interested in the scenario where the parameters
$Y_{3},Z_{5},Z_{6},Z_{7}$ all take real values, in which case the scalar
sector preserves CP symmetry at the leading order.222It should be clear,
however, that this limit should be seen as a particular solution of the model
where the parameters are in general complex, and not as a model by itself
[53]. Accordingly, we can parameterize $H_{1}$ and $H_{2}$ as:

$\displaystyle H_{1}=\begin{pmatrix}G^{+}\\\
\frac{1}{\sqrt{2}}(v+h_{1}^{\mathrm{H}}+iG_{0})\end{pmatrix},\hskip
8.53581ptH_{2}=\begin{pmatrix}H^{+}\\\
\frac{1}{\sqrt{2}}(h_{2}^{\mathrm{H}}+iA)\end{pmatrix},$ (7)

with $G^{+},H^{+}$ complex fields, and
$h_{1}^{\mathrm{H}},h_{2}^{\mathrm{H}},A,G_{0}$ real fields. All of these
states are already mass eigenstates, except
$h_{1}^{\mathrm{H}},h_{2}^{\mathrm{H}}$; the states $G^{+}$ and $G_{0}$ are
the would-be Goldstone bosons, and $H^{+}$ and $A$ correspond to the charged
scalar and the pseudo-scalar bosons, respectively. The scalar mass states
$h_{125}$ and $H_{0}$ are obtained from $h_{1}^{\mathrm{H}}$ and
$h_{2}^{\mathrm{H}}$ by introducing the mixing angle $\alpha$, such that:

$H_{1}=\left(\begin{array}[]{c}G^{+}\\\
\frac{1}{\sqrt{2}}\left(v+s_{\beta\\!-\\!\alpha}\,h_{125}+c_{\beta\\!-\\!\alpha}\,H_{0}+iG_{0}\right)\end{array}\right),\quad
H_{2}=\left(\begin{array}[]{c}H^{+}\\\
\frac{1}{\sqrt{2}}\left(c_{\beta\\!-\\!\alpha}\,h_{125}-s_{\beta\\!-\\!\alpha}\,H_{0}+iA\right)\end{array}\right),$
(8)

where the field $h_{125}$ is identified with the 125 GeV scalar observed at
the LHC.

Finally, in the CP conserving solution, the $Z_{i}$ parameters can be written
in terms of the masses of the physical scalars, the mixing parameter
$\beta\\!-\\!\alpha$ and the parameters $Y_{2}$ and $v^{2}$ according to:

$\displaystyle Z_{1}$ $\displaystyle=$
$\displaystyle\dfrac{\Big{[}m_{h_{125}}^{2}+m_{H_{0}}^{2}\,\cot(\beta\\!-\\!\alpha)^{2}\Big{]}\,\sin(\beta\\!-\\!\alpha)^{2}}{v^{2}},$
(9a) $\displaystyle Z_{3}$ $\displaystyle=$
$\displaystyle\dfrac{2\,(m_{H^{\pm}}^{2}-Y_{2})}{v^{2}},$ (9b) $\displaystyle
Z_{4}$ $\displaystyle=$
$\displaystyle\dfrac{2\,m_{A}^{2}+m_{h_{125}}^{2}+m_{H_{0}}^{2}-4\,m_{H^{\pm}}^{2}+(m_{h_{125}}^{2}-m_{H_{0}}^{2})\,\cos\Big{[}2\,(\beta\\!-\\!\alpha)\Big{]}}{2\,v^{2}},$
(9c) $\displaystyle Z_{5}$ $\displaystyle=$
$\displaystyle\dfrac{-2\,m_{A}^{2}+m_{h_{125}}^{2}+m_{H_{0}}^{2}+(m_{h_{125}}^{2}-m_{H_{0}}^{2})\,\cos\Big{[}2\,(\beta\\!-\\!\alpha)\Big{]}}{2\,v^{2}},$
(9d) $\displaystyle Z_{6}$ $\displaystyle=$
$\displaystyle\dfrac{(m_{h_{125}}^{2}-m_{H_{0}}^{2})\,\sin\Big{[}2\,(\beta\\!-\\!\alpha)\Big{]}}{2\,v^{2}},$
(9e)

where $m_{h_{125}},m_{H_{0}},m_{A}$ and $m_{H^{\pm}}$ correspond to the masses
of $h_{125},H_{0},A$ and $H^{\pm}$, respectively. In our analysis, we shall
take the following parameters as independent:

$\displaystyle\beta\\!-\\!\alpha,\,m_{h_{125}},\,Y_{2},\,m_{H_{0}},\,m_{A},\,m_{H^{\pm}},\
\ $ (scalar sector), (10a) $\displaystyle\beta,\,m_{f}\ \ $
$\displaystyle\text{(Yukawa sector)}.$ (10b)

##  3 EFT

###  3.1 The effective Lagrangian

We want to obtain the effective Lagrangian that results from integrating out
the doublet $H_{2}$, which is assumed to be heavy.333A detailed explanation of
this procedure up to dimension-8 operators can also be found in Ref. [54]. To
clarify how this is done, let us suppose a generic theory with a generic heavy
field $\Phi$ and generic light fields; then, based on Ref. [55], we can write
the relevant part of the Lagrangian as:

$\mathcal{L}\ni\Big{\\{}\Phi^{\dagger}B+\big{[}\Phi^{\dagger}V_{1}\big{]}\big{[}\Phi^{\dagger}V_{2}\big{]}+\mathrm{h.c.}\Big{\\}}+\Phi^{\dagger}\Big{(}-D^{2}-M_{2}-U\Big{)}\Phi,$
(11)

where $B,V_{1},V_{2}$ and $U$ are generic functions of the light fields,
$D^{2}=D^{\mu}D_{\mu}$, $M_{2}$ is a real parameter (representing the squared
mass of $\Phi$) and we neglect terms that are cubic or higher in $\Phi$.444In
the 2HDM model of interest here, there are no cubic scalar terms consistent
with the gauge symmetry. The equation of motion for $\Phi$ is thus:

$(-D^{2}-M_{2}-U)\Phi-\big{[}\Phi^{\dagger}V_{1}\big{]}V_{2}-\big{[}\Phi^{\dagger}V_{2}\big{]}V_{1}+B=0.$
(12)

Since $\Phi$ is heavy, we assume that the solution for $\Phi$ can be written
as

$\Phi_{c}=\Phi_{c}^{(0)}+\dfrac{\Phi_{c}^{(1)}}{M_{2}}+\dfrac{\Phi_{c}^{(2)}}{{M_{2}}^{2}}+...\,.$
(13)

The different values $\Phi_{c}^{(i)}$ can be determined by inserting Eq. 13 in
Eq. 12 and solving order by order in ${M_{2}}^{-1}$. After this, the effective
Lagrangian can be obtained by replacing $\Phi$ in Eq. 11 by $\Phi_{c}$, up to
the desired order. Now, if we intend to truncate the effective Lagrangian with
${M_{2}}^{-2}$ terms, we do not need $\Phi_{c}^{(2)}$ or higher orders. To see
this, we start by noting that

$\Phi_{c}^{(0)}=0,\hskip 42.67912pt\Phi_{c}^{(1)}=B,$ (14)

which can be trivially obtained by solving Eq. 12 to order ${M_{2}}^{1}$ and
${M_{2}}^{0}$, respectively. Using these results, and inserting the solution
Eq. 13 (up to the $\Phi_{c}^{(2)}$ term) in Eq. 11, the terms up to order
${M_{2}}^{-2}$ are

$\displaystyle\mathcal{L}$ $\displaystyle\ni$
$\displaystyle\Bigg{\\{}\Bigg{[}\dfrac{B}{M_{2}}+\dfrac{\Phi_{c}^{(2)}}{{M_{2}}^{2}}\Bigg{]}^{\dagger}B+\Bigg{[}\dfrac{B}{M_{2}}+\dfrac{\Phi_{c}^{(2)}}{{M_{2}}^{2}}\Bigg{]}^{\dagger}V_{1}\Bigg{[}\dfrac{B}{M_{2}}+\dfrac{\Phi_{c}^{(2)}}{{M_{2}}^{2}}\Bigg{]}^{\dagger}V_{2}+\mathrm{h.c.}\Bigg{\\}}$
(15)
$\displaystyle\qquad\qquad+\Bigg{[}\dfrac{B}{M_{2}}+\dfrac{\Phi_{c}^{(2)}}{{M_{2}}^{2}}\Bigg{]}^{\dagger}\Big{(}-D^{2}-M_{2}-U\Big{)}\Bigg{[}\dfrac{B}{M_{2}}+\dfrac{\Phi_{c}^{(2)}}{{M_{2}}^{2}}\Bigg{]},$
$\displaystyle=$
$\displaystyle\dfrac{|B^{2}|}{M_{2}}+\left\\{\dfrac{(B^{\dagger}V_{1})(B^{\dagger}V_{2})}{{M_{2}}^{2}}+\mathrm{h.c.}\right\\}+\dfrac{B^{\dagger}(D^{2}-U)B}{{M_{2}}^{2}}+\mathcal{O}\left(\dfrac{1}{{M_{2}}^{3}}\right),$

so that the $\Phi_{c}^{(2)}$ contribution cancels, as predicted. Note that we
have assumed that there is only one heavy mass scale, $M_{2}$; if there were
multiple heavy scales, there would be additional contributions.

We now apply this generic procedure to the particular case of the 2HDM. It
should be clear from the start that, in order for the EFT for the 2HDM to be
valid, the heavy degrees of freedom of the 2HDM must decouple, which requires
that the heavy masses defined implicitly in Eqs. 9a to 9e all be much heavier
than the weak scale, $m_{A}^{2}\sim m_{H_{0}}^{2}\sim m_{H^{\pm}}^{2}\gg
v^{2}$. Accordingly, we assume one heavy scale $Y_{2}\equiv\Lambda^{2}$,
obeying $Y_{2}\gg v^{2}$. Following the procedure defined above, then, we can
write:

${H_{2}}_{c}={H_{2}}_{c}^{(0)}+\dfrac{{H_{2}}_{c}^{(1)}}{Y_{2}}+\dfrac{{H_{2}}_{c}^{(2)}}{{Y_{2}}^{2}}+...\,.$
(16)

As in the generic case, we conclude that ${H_{2}}_{c}^{(0)}=0$ and:555Although
we will only present numerical results for the scenario where the parameters
$Y_{3},Z_{5},Z_{6},Z_{7}$ take real values, we consider them in this section
as generally complex. Besides, from here on, we omit the lepton Yukawa
couplings for pedagogical clarity; they follow the contributions from the
down-type quark Yukawa couplings in an obvious fashion.

${H_{2}}_{c}^{(1)}=-Y_{3}^{*}H_{1}-Z_{6}^{*}\left(H_{1}^{\dagger}H_{1}\right)H_{1}-\lambda_{u}^{(2)*}\widehat{q}_{L}u_{R}-\lambda_{d}^{(2)}\bar{d}_{R}q_{L}.$
(17)

Again as in the generic case, we can neglect ${H_{2}}_{c}^{(2)}$ and higher
orders, as we want to truncate the effective Lagrangian with $Y_{2}^{2}$
terms. The effective Lagrangian is then obtained by replacing all occurences
of $H_{2}$ in $\mathcal{L}_{\mathrm{2HDM}}$ by Eq. 16. Now, following the
terminology of Ref. [56], we identify terms with inverse powers of $Y_{2}$ of
0, 1 and 2 as having effective-dimension (EFD) 4, 6 and 8, respectively, and
label them $F_{4}$, $F_{6}$ and $F_{8}$. Then, we split the different $F_{i}$
(with $i$ being the EFD) into several $F_{i,j}$ terms, with $j$ being the
absolute-dimension (ABD). Accordingly, the effective Lagrangian reads:

$\mathcal{L}_{\mathrm{eff}}=F_{4}+\dfrac{F_{6}}{Y_{2}}+\dfrac{F_{8}}{Y_{2}^{2}}+\mathcal{O}\left(\dfrac{1}{Y_{2}^{3}}\right),$
(18)

with:

$\displaystyle F_{4}$ $\displaystyle=$ $\displaystyle F_{4,2}+F_{4,4}\,,$
(19a) $\displaystyle F_{6}$ $\displaystyle=$ $\displaystyle
F_{6,2}+F_{6,4}+F_{6,6}\,,$ (19b) $\displaystyle F_{8}$ $\displaystyle=$
$\displaystyle F_{8,4}+F_{8,6}+F_{8,8}\,.$ (19c)

The terms of EFD 4 are

$\displaystyle F_{4,2}$ $\displaystyle=$ $\displaystyle-
Y_{1}H_{1}^{\dagger}H_{1},$ (20a) $\displaystyle F_{4,4}$ $\displaystyle=$
$\displaystyle\left(D_{\mu}H_{1}\right)^{\dagger}\left(D^{\mu}H_{1}\right)-\frac{Z_{1}}{2}\left(H_{1}^{\dagger}H_{1}\right)^{2}-\Big{(}\lambda_{u}^{(1)}\bar{u}_{R}{\widehat{q}_{L}}^{\dagger}H_{1}+\lambda_{d}^{(1)}\bar{d}_{R}H_{1}^{\dagger}q_{L}+\textrm{h.c.}\Big{)},$
(20b)

and the terms of EFD 6

$\displaystyle F_{6,2}$ $\displaystyle=$
$\displaystyle{|Y_{3}|}^{2}(H_{1}^{\dagger}H_{1}),$ (21a) $\displaystyle
F_{6,4}$ $\displaystyle=$ $\displaystyle
Y_{3}\lambda_{u}^{(2)*}H_{1}^{\dagger}\widehat{q}_{L}u_{R}+Y_{3}\lambda_{d}^{(2)}\bar{d}_{R}H_{1}^{\dagger}q_{L}+Y_{3}Z_{6}^{*}(H_{1}^{\dagger}H_{1})^{2}+\mathrm{h.c.},$
(21b) $\displaystyle F_{6,6}$ $\displaystyle=$
$\displaystyle(H_{1}^{\dagger}H_{1})\bigg{[}|Z_{6}|^{2}(H_{1}^{\dagger}H_{1})^{2}+\Big{\\{}Z_{6}\lambda_{u}^{(2)*}H_{1}^{\dagger}\widehat{q}_{L}u_{R}+Z_{6}\lambda_{d}^{(2)}\bar{d}_{R}H_{1}^{\dagger}q_{L}+\mathrm{h.c.}\Big{\\}}\bigg{]}+\mathrm{4F},$
(21c)

where 4F represents operators involving four fermion fields; we consistently
ignore them in the following, as they do not affect the lowest order Higgs
couplings. Finally, for the terms of EFD 8,

$\displaystyle F_{8,4}$ $\displaystyle=$
$\displaystyle{|Y_{3}|}^{2}(D_{\mu}H_{1})^{\dagger}(D^{\mu}H_{1})-(H_{1}^{\dagger}H_{1})^{2}\left[{|Y_{3}|}^{2}Z_{34}+\dfrac{1}{2}(Y_{3})^{2}Z_{5}^{*}+\dfrac{1}{2}(Y_{3}^{*})^{2}Z_{5}\right]\,,$
(22a) $\displaystyle F_{8,6}$ $\displaystyle=$
$\displaystyle\left\\{Y_{3}Z_{6}^{*}+Y_{3}^{*}Z_{6}\right\\}(H_{1}^{\dagger}H_{1})(D_{\mu}H_{1})^{\dagger}(D^{\mu}H_{1})+\left\\{Y_{3}Z_{6}^{*}(D_{\mu}H_{1})^{\dagger}H_{1}+\mathrm{h.c.}\right\\}\partial^{\mu}(H_{1}^{\dagger}H_{1})$
(22b)
$\displaystyle+\bigg{\\{}Y_{3}^{*}\lambda_{u}^{(2)}\Big{(}D_{\mu}(\widehat{q}_{L}u_{R})\Big{)}^{\dagger}(D^{\mu}H_{1})+Y_{3}^{*}\lambda_{d}^{(2)*}\Big{(}D_{\mu}(\bar{d}_{R}q_{L})\Big{)}^{\dagger}(D^{\mu}H_{1})+\mathrm{h.c.}\bigg{\\}}$
$\displaystyle-(H_{1}^{\dagger}H_{1})^{3}\left[Y_{3}Z_{34}Z_{6}^{*}+Y_{3}Z_{5}^{*}Z_{6}+\mathrm{h.c.}\right]$
$\displaystyle-(H_{1}^{\dagger}H_{1})\bigg{[}H_{1}^{\dagger}\widehat{q}_{L}u_{R}\Big{(}Y_{3}Z_{34}\lambda_{u}^{(2)*}+Y_{3}^{*}Z_{5}\lambda_{u}^{(2)*}\Big{)}+\bar{d}_{R}H_{1}^{\dagger}q_{L}\Big{(}Y_{3}Z_{34}\lambda_{d}^{(2)}+Y_{3}^{*}Z_{5}\lambda_{d}^{(2)}\Big{)}+\mathrm{h.c.}\bigg{]}\,,$
$\displaystyle F_{8,8}$ $\displaystyle=$
$\displaystyle{|Z_{6}|}^{2}(H_{1}^{\dagger}H_{1})^{2}(D_{\mu}H_{1})^{\dagger}(D^{\mu}H_{1})+2{|Z_{6}|}^{2}(H_{1}^{\dagger}H_{1})\partial_{\mu}(H_{1}^{\dagger}H_{1})\partial^{\mu}(H_{1}^{\dagger}H_{1})$
(22c)
$\displaystyle-(H_{1}^{\dagger}H_{1})^{4}\left[Z_{34}{|Z_{6}|}^{2}+\dfrac{1}{2}Z_{5}^{*}Z_{6}^{2}+\dfrac{1}{2}Z_{5}(Z_{6}^{*})^{2}\right]$
$\displaystyle-(H_{1}^{\dagger}H_{1})^{2}\bigg{[}H_{1}^{\dagger}\widehat{q}_{L}u_{R}\Big{(}Z_{34}Z_{6}\lambda_{u}^{(2)*}+Z_{5}Z_{6}^{*}\lambda_{u}^{(2)*}\Big{)}+\bar{d}_{R}H_{1}^{\dagger}q_{L}\Big{(}Z_{34}Z_{6}\lambda_{d}^{(2)}+Z_{5}Z_{6}^{*}\lambda_{d}^{(2)}\Big{)}+\mathrm{h.c.}\bigg{]}$
$\displaystyle+\Bigg{\\{}\Big{[}Z_{6}^{*}\lambda_{u}^{(2)}\Big{(}D_{\mu}(\widehat{q}_{L}u_{R})\Big{)}^{\dagger}+Z_{6}^{*}\lambda_{d}^{(2)*}\Big{(}D_{\mu}(\bar{d}_{R}q_{L})\Big{)}^{\dagger}\Big{]}\Big{[}\partial^{\mu}(H_{1}^{\dagger}H_{1})H_{1}+(H_{1}^{\dagger}H_{1})(D^{\mu}H_{1})\Big{]}+\mathrm{h.c.}\Bigg{\\}}$
$\displaystyle+\mathrm{4F},$

where $Z_{34}\equiv Z_{3}+Z_{4}$.

###  3.2 Introduction to the conversion to SMEFT

The theory described in Eqs. 18 to 22 is an EFT for the 2HDM, resulting from
integrating out $H_{2}$. It contains an implicit matching between the EFT and
the 2HDM — that is, a correspondence between the coefficient of an operator in
the EFT and the parameters of the 2HDM. We now want to write the theory in the
SMEFT format — meaning, write it as the SM Lagrangian plus operators of ABD 6
(in the Warsaw basis[57, 58]) plus operators of ABD 8 (in the basis of Ref.
[59]) —, and render the matching explicit. To that end, we start by noting
that the terms in SMEFT containing operators of ABD 2 and 4 are those of the
SM, and only those; to see this explicitly, the relevant terms in the SMEFT
containing operators of ABD 2 and 4 can be written as follows:

$\mathcal{L}_{\mathrm{SMEFT}}\ni S_{4,2}+S_{4,4},$ (23)

where we are following the same convention for the subscripts, and where

$\displaystyle S_{4,2}$ $\displaystyle=$
$\displaystyle\mu^{2}\mathcal{H}^{\dagger}\mathcal{H},$ (24a) $\displaystyle
S_{4,4}$ $\displaystyle=$
$\displaystyle\left(D_{\mu}\mathcal{H}\right)^{\dagger}\left(D^{\mu}\mathcal{H}\right)-\lambda\left(\mathcal{H}^{\dagger}\mathcal{H}\right)^{2}-\Big{(}Y_{u}\,\bar{u}_{R}{\widehat{q}_{L}}^{\dagger}\mathcal{H}+Y_{d}\,\bar{d}_{R}\mathcal{H}^{\dagger}q_{L}+\textrm{h.c.}\Big{)},$
(24b)

with $\mathcal{H}$ being the SMEFT Higgs doublet. This implies that Eqs. 18 to
22 do not allow a direct comparison with the SMEFT, despite appearances. In
fact, if one would naively identify $H_{1}$ with $\mathcal{H}$, the comparison
would seem straightforward (given the similarity between $S_{4,2}$ and
$F_{4,2}$, on the one hand, and between $S_{4,4}$ and $F_{4,4}$, on the other
hand). So, for example, $Y_{d}$ would be identified with $\lambda_{d}^{(1)}$.
A closer look, however, reveals that such an identification is not correct, as
there are other terms with the operator $\bar{d}_{R}H_{1}^{\dagger}q_{L}$ in
Eqs. 21 and 22 besides the one in $F_{4,4}$. Therefore, in order for the EFT
of Section 3.1 to be interpreted in the SMEFT environment, a careful
identification needs to be done. Such an identification is rendered even less
direct due to the need for basis conversions; indeed, as mentioned in passing,
we want to write all the operators of ABD 6 in the Warsaw basis and all those
of ABD 8 in the basis of Ref. [59].666Although the basis of Ref. [59] is
equivalent to that of Ref. [60], we stick to the notation of the former. This
conversion will generate operators of ABD 4, which need to be taken into
account for a proper identification.

In what follows, we perform the basis conversions; we identify $H_{1}$ as $H$,
to simplify the notation.

###  3.3 Operators of ABD 6

We begin with the terms with ABD 6 operators; only those in the first two
lines of $F_{8,6}$ have operators which do not belong to the Warsaw basis. Let
us start with the second term of the first line;

$\displaystyle\left\\{Y_{3}Z_{6}^{*}(D_{\mu}H)^{\dagger}H+\mathrm{h.c.}\right\\}\partial^{\mu}(H^{\dagger}H)=\left\\{Y_{3}Z_{6}^{*}(D_{\mu}H)^{\dagger}H+\mathrm{h.c.}\right\\}D^{\mu}(H^{\dagger}H)$
$\displaystyle=$
$\displaystyle-(H^{\dagger}H)\left\\{Y_{3}Z_{6}^{*}D^{\mu}\Big{[}(D_{\mu}H)^{\dagger}H\Big{]}+\mathrm{h.c.}\right\\}$
$\displaystyle=$ $\displaystyle-
Y_{3}Z_{6}^{*}(H^{\dagger}H)\Big{[}(D^{2}H)^{\dagger}H+(D_{\mu}H)^{\dagger}(D^{\mu}H)\Big{]}+\mathrm{h.c.}\,.$
(25)

We see that the term with $(D_{\mu}H)^{\dagger}(D^{\mu}H)$ in Eq. 25 precisely
cancels the first term of $F_{8,6}$. As for the term with $D^{2}H$, we can use
the Equation of Motion (EoM) for $H$,777When including operators beyond
dimension 6, one should generally use field redefinitions to match onto a
given basis of dimension-6 operators[61]. In this particular case, though, the
operators that we are rewriting are EFD 8. Thus, the difference between using
field redefinitions and using equations of motion will be
$\mathcal{O}(\Lambda^{-6})$, which is beyond the order to which we are
matching.

$D^{2}H=-Y_{1}H-Z_{1}(H^{\dagger}H)H-\lambda_{u}^{(1)}\widehat{q}_{L}u_{R}-\lambda_{d}^{(1)}\bar{d}_{R}q_{L}.$
(26)

Inserting this equality in the term with $D^{2}H$ in Eq. 3.3 gives:

$Y_{3}Z_{6}^{*}\Bigg{[}Y_{1}(H^{\dagger}H)^{2}+Z_{1}(H^{\dagger}H)^{3}+\lambda_{u}^{(1)}(H^{\dagger}H)\bar{u}_{R}\widehat{q}_{L}^{\dagger}H+\lambda_{d}^{(1)}(H^{\dagger}H)\bar{q}_{L}Hd_{R}\Bigg{]}+\mathrm{h.c.}\,.$
(27)

Here, the operator of the first term is ABD 4, while the operators in the
other three terms which have ABD 6 belong to the Warsaw basis. Finally, for
the two terms in the second line of $F_{8,6}$, we can use Integration by Parts
(IbP) and Eq. 26 to conclude that they are respectively given by:

$\displaystyle
Y_{3}^{*}\lambda_{u}^{(2)}\Bigg{[}Y_{1}\bar{u}_{R}\widehat{q}_{L}^{\dagger}H+Z_{1}(H^{\dagger}H)\bar{u}_{R}\widehat{q}_{L}^{\dagger}H+\mathrm{4F}\Bigg{]}+\mathrm{h.c.},$
(28a) $\displaystyle
Y_{3}^{*}\lambda_{d}^{(2)*}\Bigg{[}Y_{1}\bar{q}_{L}Hd_{R}+Z_{1}(H^{\dagger}H)\bar{q}_{L}Hd_{R}+\mathrm{4F}\Bigg{]}+\mathrm{h.c.}\,.$
(28b)

As before, the operators in the first terms have ABD 4, whereas the other
operators belong to the Warsaw basis.

###  3.4 Operators of ABD 8

We now consider the terms with operators of ABD 8; using IbP and the EoM, we
find that the second term of $F_{8,8}$ can be rewritten as

$\displaystyle
2{|Z_{6}|}^{2}(H^{\dagger}H)\partial_{\mu}(H^{\dagger}H)\partial^{\mu}(H^{\dagger}H)=2Y_{1}{|Z_{6}|}^{2}(H^{\dagger}H)^{3}+{|Z_{6}|}^{2}\Big{\\{}\lambda_{u}^{(1)}(H^{\dagger}H)^{2}\bar{q}_{L}u_{R}\tilde{H}$
$\displaystyle\qquad+\lambda_{d}^{(1)}(H^{\dagger}H)^{2}\bar{d}_{R}H^{\dagger}q_{L}+\mathrm{h.c.}\Big{\\}}+2Z_{1}{|Z_{6}|}^{2}(H^{\dagger}H)^{4}-2{|Z_{6}|}^{2}(H^{\dagger}H)^{2}(D_{\mu}H_{1})^{\dagger}(D^{\mu}H_{1})\,,$
(29)

where the first term on the right of Eq. 29 has ABD 6. As for the terms inside
the curly brackets in the third line of $F_{8,8}$, we can use SU(2) identities
to see that:888Here, $\tau^{I}$ are the Pauli matrices. Besides, as in the
case of $\tilde{H}_{i}$, we have $\widetilde{D^{\mu}H}=i\sigma_{2}D^{\mu}H$.

$\displaystyle\Big{[}(D_{\mu}H)^{\dagger}\tau^{I}(D^{\mu}H)\Big{]}\Big{[}\bar{q}_{L}u_{R}\tau^{I}\tilde{H}\Big{]}=\Big{[}(D_{\mu}H)^{\dagger}(D^{\mu}H)\Big{]}\Big{[}\bar{q}_{L}u_{R}\tilde{H}\Big{]}-2\Big{[}H^{\dagger}(D_{\mu}H)\Big{]}\Big{[}\bar{q}_{L}u_{R}\widetilde{D^{\mu}H}\Big{]},$
(30a)
$\displaystyle\Big{[}(D_{\mu}H)^{\dagger}\tau^{I}(D^{\mu}H)\Big{]}\Big{[}\bar{q}_{L}d_{R}\tau^{I}H\Big{]}=2\Big{[}(D_{\mu}H)^{\dagger}H\Big{]}\Big{[}\bar{q}_{L}d_{R}D^{\mu}H\Big{]}-\Big{[}(D_{\mu}H)^{\dagger}(D^{\mu}H)\Big{]}\Big{[}\bar{q}_{L}d_{R}H\Big{]},$
(30b)

which, together with IbP and the EoM, yields

$\displaystyle
Z_{6}^{*}\lambda_{u}^{(2)}\Big{(}D_{\mu}(\widehat{q}_{L}u_{R})\Big{)}^{\dagger}\Big{[}\partial^{\mu}(H_{1}^{\dagger}H_{1})H_{1}+(H_{1}^{\dagger}H_{1})(D^{\mu}H_{1})\Big{]}+\mathrm{h.c.}$
$\displaystyle=Z_{6}\lambda_{u}^{(2)*}\bigg{\\{}3Y_{1}(H^{\dagger}H)\bar{q}_{L}u_{R}\tilde{H}+3Z_{1}(H^{\dagger}H)^{2}\bar{q}_{L}u_{R}\tilde{H}-3(D_{\mu}H)^{\dagger}(D^{\mu}H)\bar{q}_{L}u_{R}\tilde{H}$
$\displaystyle\qquad+\Big{[}(D_{\mu}H)^{\dagger}\tau^{I}(D^{\mu}H)\Big{]}\Big{[}\bar{q}_{L}u_{R}\tau^{I}\tilde{H}\Big{]}-2\Big{[}(D_{\mu}H)^{\dagger}H\Big{]}\Big{[}\bar{q}_{L}u_{R}\widetilde{D^{\mu}H}\Big{]}+\mathrm{4F}\bigg{\\}}+\mathrm{h.c.}\,,$
(31a) $\displaystyle
Z_{6}^{*}\lambda_{d}^{(2)*}\Big{(}D_{\mu}(\bar{d}_{R}\,q_{L})\Big{)}^{\dagger}\Big{[}\partial^{\mu}(H_{1}^{\dagger}H_{1})H_{1}+(H_{1}^{\dagger}H_{1})(D^{\mu}H_{1})\Big{]}+\mathrm{h.c.}$
$\displaystyle=Z_{6}^{*}\lambda_{d}^{(2)*}\bigg{\\{}3Y_{1}(H^{\dagger}H)\bar{q}_{L}d_{R}H+3Z_{1}(H^{\dagger}H)^{2}\bar{q}_{L}d_{R}H-3(D_{\mu}H)^{\dagger}(D^{\mu}H)\bar{q}_{L}d_{R}H$
$\displaystyle\qquad-\Big{[}(D_{\mu}H)^{\dagger}\tau^{I}(D^{\mu}H)\Big{]}\Big{[}\bar{q}_{L}d_{R}\tau^{I}H\Big{]}-2(H^{\dagger}D_{\mu}H)\bar{q}_{L}d_{R}D^{\mu}H+\mathrm{4F}\bigg{\\}}+\mathrm{h.c.}\,.$
(31b)

###  3.5 Results

Putting the pieces together, we have

$\mathcal{L}_{\mathrm{eff}}=F_{4}+\dfrac{F_{6}}{Y_{2}}+\dfrac{F^{\prime}_{8}}{Y_{2}^{2}}+\mathcal{O}\left(\dfrac{1}{Y_{2}^{3}}\right),$
(32)

where, for the EFD 4 and 6 terms, we have the same result as previously (Eqs.
20 and 21, respectively), while for the EFD 8 contribution we have

$F^{\prime}_{8}=F^{\prime}_{8,4}+F^{\prime}_{8,6}+F^{\prime}_{8,8}\,,$ (33)

with:

$\displaystyle F^{\prime}_{8,4}$ $\displaystyle=$
$\displaystyle{|Y_{3}|}^{2}(D_{\mu}H)^{\dagger}(D^{\mu}H)-(H^{\dagger}H)^{2}\Big{[}{|Y_{3}|}^{2}Z_{34}+\dfrac{1}{2}(Y_{3})^{2}Z_{5}^{*}+\dfrac{1}{2}(Y_{3}^{*})^{2}Z_{5}-Y_{1}Y_{3}Z_{6}^{*}$
(34a) $\displaystyle-
Y_{1}Y_{3}^{*}Z_{6}\Big{]}+\Big{\\{}Y_{1}Y_{3}\lambda_{u}^{(2)*}H^{\dagger}\widehat{q}_{L}u_{R}+Y_{1}Y_{3}\lambda_{d}^{(2)}\bar{d}_{R}H^{\dagger}q_{L}+\mathrm{h.c.}\Big{\\}}\,,$
$\displaystyle F^{\prime}_{8,6}$ $\displaystyle=$
$\displaystyle-(H^{\dagger}H)^{3}\left[Y_{3}Z_{34}Z_{6}^{*}+Y_{3}Z_{5}^{*}Z_{6}-Y_{3}Z_{1}Z_{6}^{*}-Y_{1}|Z_{6}|^{2}+\mathrm{h.c.}\right]$
(34b)
$\displaystyle-(H^{\dagger}H)\bigg{\\{}H^{\dagger}\widehat{q}_{L}u_{R}\Big{(}Y_{3}Z_{34}\lambda_{u}^{(2)*}+Y_{3}^{*}Z_{5}\lambda_{u}^{(2)*}-Y_{3}^{*}Z_{6}\lambda_{u}^{(1)}-Y_{3}Z_{1}\lambda_{u}^{(2)*}-3Y_{1}Z_{6}\lambda_{u}^{(2)*}\Big{)}$
$\displaystyle\hskip
36.98857pt+\bar{d}_{R}H^{\dagger}q_{L}\Big{(}Y_{3}Z_{34}\lambda_{d}^{(2)}+Y_{3}^{*}Z_{5}\lambda_{d}^{(2)}-Y_{3}^{*}Z_{6}\lambda_{d}^{(1)}-Y_{3}Z_{1}\lambda_{d}^{(2)}-3Y_{1}Z_{6}\lambda_{d}^{(2)}\Big{)}+\mathrm{h.c.}\bigg{\\}}$
$\displaystyle+~{}\mathrm{4F}\,,$ $\displaystyle F^{\prime}_{8,8}$
$\displaystyle=$
$\displaystyle-{|Z_{6}|}^{2}(H^{\dagger}H)^{2}(D_{\mu}H)^{\dagger}(D^{\mu}H)$
(34c)
$\displaystyle-(H^{\dagger}H)^{4}\left[Z_{34}{|Z_{6}|}^{2}+\dfrac{1}{2}Z_{5}^{*}Z_{6}^{2}+\dfrac{1}{2}Z_{5}(Z_{6}^{*})^{2}-2Z_{1}|Z_{6}|^{2}\right]$
$\displaystyle-(H^{\dagger}H)^{2}\bigg{\\{}H^{\dagger}\widehat{q}_{L}u_{R}\Big{(}Z_{34}Z_{6}\lambda_{u}^{(2)*}+Z_{5}Z_{6}^{*}\lambda_{u}^{(2)*}-|Z_{6}|^{2}\lambda_{u}^{(1)}-3Z_{1}Z_{6}\lambda_{u}^{(2)*}\Big{)}$
$\displaystyle\hskip
42.67912pt+\bar{d}_{R}H^{\dagger}q_{L}\Big{(}Z_{34}Z_{6}\lambda_{d}^{(2)}+Z_{5}Z_{6}^{*}\lambda_{d}^{(2)}-|Z_{6}|^{2}\lambda_{d}^{(1)}-3Z_{1}Z_{6}\lambda_{d}^{(2)}\Big{)}+\mathrm{h.c.}\bigg{\\}}$
$\displaystyle-3(D_{\mu}H)^{\dagger}(D^{\mu}H)\Big{\\{}Z_{6}\lambda_{u}^{(2)*}H^{\dagger}\widehat{q}_{L}u_{R}+Z_{6}^{*}\lambda_{d}^{(2)*}\bar{q}_{L}d_{R}H+\mathrm{h.c.}\Big{\\}}$
$\displaystyle+\Bigg{\\{}Z_{6}\lambda_{u}^{(2)*}\bigg{(}\Big{[}(D_{\mu}H)^{\dagger}\tau^{I}(D^{\mu}H)\Big{]}\Big{[}\bar{q}_{L}u_{R}\tau^{I}\tilde{H}\Big{]}-2\Big{[}(D_{\mu}H)^{\dagger}H\Big{]}\Big{[}\bar{q}_{L}u_{R}\widetilde{D^{\mu}H}\Big{]}\bigg{)}$
$\displaystyle\hskip 8.53581pt-
Z_{6}^{*}\lambda_{d}^{(2)*}\bigg{(}\Big{[}(D_{\mu}H)^{\dagger}\tau^{I}(D^{\mu}H)\Big{]}\Big{[}\bar{q}_{L}d_{R}\tau^{I}H\Big{]}+2(H^{\dagger}D_{\mu}H)\bar{q}_{L}d_{R}D^{\mu}H\bigg{)}+\mathrm{h.c.}\Bigg{\\}}$
$\displaystyle+\mathrm{4F}.$

###  3.6 The SMEFT Lagrangian

We are now ready to write the effective Lagrangian in the SMEFT formalism,
which involves determining the matching coefficients up to EFD 8. In
particular, we can determine all the coefficients involved in Eq. 23; for
example, by equating all the terms of ABD 2 of Eq. 32 (namely, $F_{4,2}$ and
$F_{6,2}$) with all the terms of ABD 2 of Eq. 23 (namely, $S_{4,2}$), we find:

$\mu^{2}=-Y_{1}+\dfrac{|Y_{3}|^{2}}{Y_{2}}.$ (35)

A similar exercise can be done for the remaining parameters of Eq. 23. This
exercise is nothing but the matching between SMEFT and the 2HDM for the
operators of ABD 2 and 4. From the point of view of the SMEFT, however, only
the matching for the operators of ABD greater than 4 is relevant; the reason
is that relations like Eq. 35 correspond to redefinitions of SM parameters, so
that they have no observable effect beyond the SM. On the other hand, care
should be taken with the kinetic terms of the Higgs doublet; in fact, in order
to obtain a proper correspondence between Eqs. 23 and 32 for the operators of
ABD 4, we must have:

$\left(D_{\mu}\mathcal{H}\right)^{\dagger}\left(D^{\mu}\mathcal{H}\right)=\left(D_{\mu}H\right)^{\dagger}\left(D^{\mu}H\right)\Bigg{(}1+\dfrac{{|Y_{3}|}^{2}}{Y_{2}^{2}}\Bigg{)},$
(36)

which leads to,

$H=\mathcal{H}\Bigg{(}1-\dfrac{{|Y_{3}|}^{2}}{2Y_{2}^{2}}\Bigg{)}.$ (37)

This relation is the last element required to write the SMEFT Lagrangian and
perform the matching of operators of ABD 6 and 8; replacing $Y_{2}$ by the
heavy (squared) scale $\Lambda^{2}$, we write it as:

$\mathcal{L}_{\mathrm{SMEFT}}=\mathcal{L}_{\mathrm{SM}}+\dfrac{S_{\text{all},6}}{\Lambda^{2}}+\dfrac{S_{\text{all},8}}{\Lambda^{4}}+\mathcal{O}\left(\dfrac{1}{\Lambda^{6}}\right),$
(38)

where the subscript ‘all’ means that all EFDs are being included; moreover,
$\mathcal{L}_{\mathrm{SM}}$ is given by Eqs. 23 and 24, and:

$S_{\text{all},6}=C_{\mathcal{H}}(\mathcal{H}^{\dagger}\mathcal{H})^{3}+\left\\{C_{u\mathcal{H}}\,(\mathcal{H}^{\dagger}\mathcal{H})\,\bar{q}_{L}u_{R}\tilde{\mathcal{H}}+C_{d\mathcal{H}}\,(\mathcal{H}^{\dagger}\mathcal{H})\,\bar{q}_{L}d_{R}\mathcal{H}+\mathrm{h.c.}\right\\}+\mathrm{4F},$
(39)

with:

$\displaystyle C_{\mathcal{H}}$ $\displaystyle=$ $\displaystyle
C_{\mathcal{H}}^{[6]}+\dfrac{C_{\mathcal{H}}^{[8]}}{\Lambda^{2}}$ (40a)
$\displaystyle=$
$\displaystyle|Z_{6}|^{2}+\dfrac{1}{\Lambda^{2}}\Big{(}Y_{3}Z_{1}Z_{6}^{*}+Y_{3}^{*}Z_{1}Z_{6}-Y_{3}Z_{34}Z_{6}^{*}-Y_{3}^{*}Z_{34}Z_{6}-Y_{3}Z_{5}^{*}Z_{6}-Y_{3}^{*}Z_{5}Z_{6}^{*}+2Y_{1}{|Z_{6}|}^{2}\Big{)},\hskip
14.22636pt$ $\displaystyle C_{u\mathcal{H}}$ $\displaystyle=$ $\displaystyle
C_{u\mathcal{H}}^{[6]}+\dfrac{C_{u\mathcal{H}}^{[8]}}{\Lambda^{2}}$ (40b)
$\displaystyle=$ $\displaystyle
Z_{6}\lambda_{u}^{(2)*}+\dfrac{1}{\Lambda^{2}}\Big{(}Y_{3}^{*}Z_{6}\lambda_{u}^{(1)}+Y_{3}Z_{1}\lambda_{u}^{(2)*}-Y_{3}Z_{34}\lambda_{u}^{(2)*}-Y_{3}^{*}Z_{5}\lambda_{u}^{(2)*}+3Y_{1}Z_{6}\lambda_{u}^{(2)*}\Big{)},$
$\displaystyle C_{d\mathcal{H}}$ $\displaystyle=$ $\displaystyle
C_{d\mathcal{H}}^{[6]}+\dfrac{C_{d\mathcal{H}}^{[8]}}{\Lambda^{2}}$ (40c)
$\displaystyle=$ $\displaystyle
Z_{6}^{*}\lambda_{d}^{(2)*}+\dfrac{1}{\Lambda^{2}}\Big{(}Y_{3}Z_{6}^{*}\lambda_{d}^{(1)}+Y_{3}^{*}Z_{1}\lambda_{d}^{(2)*}-Y_{3}^{*}Z_{34}\lambda_{d}^{(2)*}-Y_{3}Z_{5}^{*}\lambda_{d}^{(2)*}+3Y_{1}Z_{6}^{*}\lambda_{d}^{(2)*}\Big{)},$

where the superscript $[i]$ indicates that the EFD at stake is $i$, and

$\displaystyle S_{\text{all},8}$ $\displaystyle=$ $\displaystyle
C_{\mathcal{H}^{8}}(\mathcal{H}^{\dagger}\mathcal{H})^{4}+C_{\mathcal{H}^{6}}^{(1)}(\mathcal{H}^{\dagger}\mathcal{H})^{2}\left(D_{\mu}\mathcal{H}\right)^{\dagger}\left(D^{\mu}\mathcal{H}\right)+\Big{\\{}C_{qu\mathcal{H}^{5}}(\mathcal{H}^{\dagger}\mathcal{H})^{2}\bar{q}_{L}u_{R}\tilde{\mathcal{H}}$
(41)
$\displaystyle+C_{qu\mathcal{H}^{3}D^{2}}^{(1)}(D_{\mu}\mathcal{H})^{\dagger}(D^{\mu}\mathcal{H})\bar{q}_{L}u_{R}\tilde{\mathcal{H}}+C_{qu\mathcal{H}^{3}D^{2}}^{(2)}\Big{[}(D_{\mu}\mathcal{H})^{\dagger}\tau^{I}(D^{\mu}\mathcal{H})\Big{]}\Big{[}\bar{q}_{L}u_{R}\tau^{I}\tilde{\mathcal{H}}\Big{]}$
$\displaystyle+C_{qu\mathcal{H}^{3}D^{2}}^{(5)}\Big{[}(D_{\mu}\mathcal{H})^{\dagger}\mathcal{H}\Big{]}\Big{[}\bar{q}_{L}u_{R}\widetilde{D^{\mu}\mathcal{H}}\Big{]}+C_{qd\mathcal{H}^{5}}(\mathcal{H}^{\dagger}\mathcal{H})^{2}\bar{q}_{L}d_{R}\mathcal{H}$
$\displaystyle+C_{qd\mathcal{H}^{3}D^{2}}^{(1)}(D_{\mu}\mathcal{H})^{\dagger}(D^{\mu}\mathcal{H})\bar{q}_{L}d_{R}\mathcal{H}+C_{qd\mathcal{H}^{3}D^{2}}^{(2)}\Big{[}(D_{\mu}\mathcal{H})^{\dagger}\tau^{I}(D^{\mu}\mathcal{H})\Big{]}\Big{[}\bar{q}_{L}d_{R}\tau^{I}\mathcal{H}\Big{]}$
$\displaystyle+C_{qd\mathcal{H}^{3}D^{2}}^{(5)}(\mathcal{H}^{\dagger}D_{\mu}\mathcal{H})(\bar{q}_{L}d_{R}D^{\mu}\mathcal{H})+\mathrm{h.c.}\Big{\\}}+\mathrm{4F},$

where:

$\displaystyle C_{\mathcal{H}^{8}}$ $\displaystyle=$ $\displaystyle-
Z_{34}{|Z_{6}|}^{2}-\dfrac{1}{2}Z_{5}^{*}Z_{6}^{2}-\dfrac{1}{2}Z_{5}(Z_{6}^{*})^{2}+2Z_{1}{|Z_{6}|}^{2},$
(42a) $\displaystyle C_{\mathcal{H}^{6}}^{(1)}$ $\displaystyle=$
$\displaystyle-{|Z_{6}|}^{2},$ (42b) $\displaystyle C_{qu\mathcal{H}^{5}}$
$\displaystyle=$ $\displaystyle-
Z_{34}Z_{6}\lambda_{u}^{(2)*}-Z_{5}Z_{6}^{*}\lambda_{u}^{(2)*}+{|Z_{6}|}^{2}\lambda_{u}^{(1)}+3Z_{1}Z_{6}\lambda_{u}^{(2)*},$
(42c) $\displaystyle C_{qu\mathcal{H}^{3}D^{2}}^{(1)}$ $\displaystyle=$
$\displaystyle-3Z_{6}\lambda_{u}^{(2)*},$ (42d) $\displaystyle
C_{qu\mathcal{H}^{3}D^{2}}^{(2)}$ $\displaystyle=$ $\displaystyle
Z_{6}\lambda_{u}^{(2)*},$ (42e) $\displaystyle
C_{qu\mathcal{H}^{3}D^{2}}^{(5)}$ $\displaystyle=$
$\displaystyle-2Z_{6}\lambda_{u}^{(2)*},$ (42f) $\displaystyle
C_{qd\mathcal{H}^{5}}$ $\displaystyle=$ $\displaystyle-
Z_{34}Z_{6}^{*}\lambda_{d}^{(2)*}-Z_{5}^{*}Z_{6}\lambda_{d}^{(2)*}+{|Z_{6}|}^{2}\lambda_{d}^{(1)}+3Z_{1}Z_{6}^{*}\lambda_{d}^{(2)*},$
(42g) $\displaystyle C_{qd\mathcal{H}^{3}D^{2}}^{(1)}$ $\displaystyle=$
$\displaystyle-3Z_{6}^{*}\lambda_{d}^{(2)*},$ (42h) $\displaystyle
C_{qd\mathcal{H}^{3}D^{2}}^{(2)}$ $\displaystyle=$ $\displaystyle-
Z_{6}^{*}\lambda_{d}^{(2)*},$ (42i) $\displaystyle
C_{qd\mathcal{H}^{3}D^{2}}^{(5)}$ $\displaystyle=$
$\displaystyle-2Z_{6}^{*}\lambda_{d}^{(2)*}.$ (42j)

###  3.7 The decoupling limit

As already suggested, in order for the EFT for the 2HDM to be valid (either in
the form of $\mathcal{L}_{\mathrm{eff}}$ of Eq. 32 or in the form of
$\mathcal{L}_{\mathrm{SMEFT}}$ of Eq. 38), the heavy degrees of freedom of the
2HDM must decouple. In the scenario where CP is conserved in the scalar
sector, we characterize the decoupling limit [62] by the assumptions:

$m_{A}^{2}\sim m_{H_{0}}^{2}\sim m_{H^{\pm}}^{2}\sim Y_{2}\equiv\Lambda^{2}\gg
v^{2},\qquad m_{h}^{2}\simeq v^{2},$ (43)

as well as the assumption of perturbative unitarity. The latter is ensured by
requiring $Z_{i}/(4\pi)\simeq\mathcal{O}(1)$ [63]. To ensure this condition,
in turn, Eqs. 9 can be used to conclude that the following relation must hold:

$|\cos(\beta-\alpha)|\propto\dfrac{v^{2}}{\Lambda^{2}}.$ (44)

The decoupling limit thus implies the alignment limit, which is characterized
by $\cos(\beta-\alpha)\to 0$ and corresponds to the SM prediction.999Using
Eqs. 6, 9, it is easy to see that all the terms with operators of EFD 6 and 8
vanish in the limit $\cos(\beta-\alpha)\to 0$. The relation 44 means that,
since we are working up EFD 8 (i.e. up to quadratic order in the expansion in
powers of $1/\Lambda^{2}$), we can always expand the results to quadratic
order in $\cos(\beta-\alpha)$. Performing this expansion, we find:

$\displaystyle{C_{u\mathcal{H}}\over\Lambda^{2}}$ $\displaystyle=$
$\displaystyle-\dfrac{2}{\sqrt{2}}\,(\sqrt{2}G_{F})^{3/2}\,\dfrac{\cos(\beta-\alpha)}{\tan\beta}\,\eta_{u}\,{m_{u}}\,$
(45a)
$\displaystyle\qquad-\dfrac{\cos(\beta-\alpha)\,{m_{u}}\,(\sqrt{2}G_{F})^{3/2}}{\sqrt{2}\,{\Lambda}^{2}}\bigg{[}\cos(\beta-\alpha)\,{\Lambda}^{2}-\dfrac{6\,m_{h}^{2}\,\eta_{u}}{\tan\beta}\bigg{]},$
$\displaystyle{C_{d\mathcal{H}}\over\Lambda^{2}}$ $\displaystyle=$
$\displaystyle-\dfrac{2}{\sqrt{2}}\,(\sqrt{2}G_{F})^{3/2}\,\dfrac{\cos(\beta-\alpha)}{\tan\beta}\,\eta_{d}\,{m_{d}}\,$
(45b)
$\displaystyle\qquad-\dfrac{\cos(\beta-\alpha)\,{m_{d}}\,(\sqrt{2}G_{F})^{3/2}}{\sqrt{2}\,{\Lambda}^{2}}\bigg{[}\cos(\beta-\alpha)\,{\Lambda}^{2}-\dfrac{6\,m_{h}^{2}\,\eta_{d}}{\tan\beta}\bigg{]},$
$\displaystyle{C_{\mathcal{H}}\over\Lambda^{2}}$ $\displaystyle=$
$\displaystyle\cos(\beta-\alpha)^{2}\,(\sqrt{2}G_{F})^{2}\left({\Lambda}^{2}-4\,m_{h}^{2}\right)$
(45c)

and

$\displaystyle{C_{\mathcal{H}^{8}}\over\Lambda^{4}}$ $\displaystyle=$
$\displaystyle 2\,\cos(\beta-\alpha)^{2}\,m_{h}^{2}\,(\sqrt{2}G_{F})^{3},$
(46a) $\displaystyle{C_{\mathcal{H}^{6}}^{(1)}\over\Lambda^{4}}$
$\displaystyle=$ $\displaystyle-\cos(\beta-\alpha)^{2}\,(\sqrt{2}G_{F})^{2},$
(46b) $\displaystyle{C_{qu\mathcal{H}^{5}}\over\Lambda^{4}}$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{2}\,\cos(\beta-\alpha)\,{m_{u}}\,(\sqrt{2}G_{F})^{5/2}}{{\Lambda}^{2}}\,\bigg{[}\cos(\beta-\alpha)\,{\Lambda}^{2}-\dfrac{3\,m_{h}^{2}\,\eta_{u}}{\tan\beta}\bigg{]},$
(46c) $\displaystyle{C_{qu\mathcal{H}^{3}D^{2}}^{(1)}\over\Lambda^{4}}$
$\displaystyle=$
$\displaystyle\dfrac{3\,\sqrt{2}\,\cos(\beta-\alpha)\,{m_{u}}\,(\sqrt{2}G_{F})^{3/2}\,\eta_{u}}{{\tan\beta\,\Lambda}^{2}},$
(46d) $\displaystyle{C_{qu\mathcal{H}^{3}D^{2}}^{(2)}\over\Lambda^{4}}$
$\displaystyle=$
$\displaystyle-\dfrac{\sqrt{2}\,\cos(\beta-\alpha)\,{m_{u}}\,(\sqrt{2}G_{F})^{3/2}\,\eta_{u}}{{\tan\beta\,\Lambda}^{2}},$
(46e) $\displaystyle{C_{qu\mathcal{H}^{3}D^{2}}^{(5)}\over\Lambda^{4}}$
$\displaystyle=$
$\displaystyle\dfrac{2\,\sqrt{2}\,\cos(\beta-\alpha)\,{m_{u}}\,(\sqrt{2}G_{F})^{3/2}\,\eta_{u}}{{\tan\beta\,\Lambda}^{2}},$
(46f) $\displaystyle{C_{qd\mathcal{H}^{5}}\over\Lambda^{4}}$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{2}\,\cos(\beta-\alpha)\,{m_{d}}\,(\sqrt{2}G_{F})^{5/2}}{{\Lambda}^{2}}\,\bigg{[}\cos(\beta-\alpha)\,{\Lambda}^{2}-\dfrac{3\,m_{h}^{2}\,\eta_{d}}{\tan\beta}\bigg{]},$
(46g) $\displaystyle{C_{qd\mathcal{H}^{3}D^{2}}^{(1)}\over\Lambda^{4}}$
$\displaystyle=$
$\displaystyle\dfrac{3\,\sqrt{2}\,\cos(\beta-\alpha)\,{m_{d}}\,(\sqrt{2}G_{F})^{3/2}\,\eta_{d}}{{\tan\beta\,\Lambda}^{2}},$
(46h) $\displaystyle{C_{qd\mathcal{H}^{3}D^{2}}^{(2)}\over\Lambda^{4}}$
$\displaystyle=$
$\displaystyle\dfrac{\sqrt{2}\,\cos(\beta-\alpha)\,{m_{d}}\,(\sqrt{2}G_{F})^{3/2}\,\eta_{d}}{{\tan\beta\,\Lambda}^{2}},$
(46i) $\displaystyle{C_{qd\mathcal{H}^{3}D^{2}}^{(5)}\over\Lambda^{4}}$
$\displaystyle=$
$\displaystyle\dfrac{2\,\sqrt{2}\,\cos(\beta-\alpha)\,{m_{d}}\,(\sqrt{2}G_{F})^{3/2}\,\eta_{d}}{{\tan\beta\,\Lambda}^{2}},$
(46j)

where, as before, $m_{u}$ and $m_{d}$ are a compact way to accomodate the
three generations of masses of up-type and down-type quarks,
respectively.101010We continue to omit the terms for the leptons, which are
trivially equivalent to the ones for the down-type quarks.

###  3.8 Relations in SMEFT

Finally, in order to derive results in SMEFT, we need to consider two aspects.
The first one involves the SMEFT Higgs doublet, which can be parametrized as

$\mathcal{H}=\begin{pmatrix}G^{+}_{\text{\tiny S}}\\\
\frac{1}{\sqrt{2}}(v_{T}+h_{\text{\tiny S}}+iG_{0,\text{\tiny
S}})\end{pmatrix},$ (47)

where $v_{T}$ is the minimum of the SMEFT potential. Replacing Eq. 47 in
$\mathcal{L}_{\mathrm{SMEFT}}$, one realizes that $h_{\text{\tiny S}}$ has
non-canonical kinetic energy. We thus use the field $h$, defined by:

$h\equiv
h_{s}\bigg{[}1+{v_{T}^{4}C_{\mathcal{H}^{6}}^{(1)}\over\Lambda^{4}}\bigg{]},$
(48)

whose propagator is canonically normalized and whose mass we term $m_{h}$. The
second aspect is that we need to write the relevant dependent parameters in
terms of independent ones.111111These relations are in general different from
those in the SM, due to terms proportional to powers of $1/\Lambda$. We take
the following set of parameters as independent:

$G_{F},M_{W},M_{Z},m_{h},m_{f},$ (49)

besides all the WCs contained in Eqs. 39 and 41. The relevant dependent
parameters are those occuring in our calculations, namely
${\overline{g}}_{2},v_{T},Y_{f}$ and $\lambda$. Now, ${\overline{g}}_{2}$ is
the SU(2) gauge coupling occuring in the Lagrangian; we can determine it via
the Fermi constant, $G_{F}$, which in turn can be obtained from muon decay;
ignoring operators involving four fermion fields, we find:121212The relevant
operators involving four fermion fields are proportional to the Yukawa
parameters of the muon and of the electron, so that their contribution can be
safely neglected for our purposes.

${G_{F}\over\sqrt{2}}={{\overline{g}}_{2}^{2}\over 8M_{W}^{2}},$ (50)

which can be inverted to yield:

${\overline{g}}_{2}=(\sqrt{2}G_{F})^{1/2}\,2\,M_{W}.$ (51)

To write $v_{T}$, one can see that the squared mass of the W boson in the
Lagrangian of Eq. 38 is given by:

$M_{W}^{2}={{\overline{g}}_{2}^{2}v_{T}^{2}\over
4}\biggl{[}1+{v_{T}^{4}C_{{\mathcal{H}}^{6}}^{(1)}\over
4\Lambda^{4}}\biggr{]},$ (52)

so that, combining this result with Eq. 51, we find:

$v_{T}=(\sqrt{2}G_{F})^{-1/2}\bigg{[}1-\dfrac{C_{{\mathcal{H}}^{6}}^{(1)}}{16\,G_{F}^{2}\,\Lambda^{4}}\bigg{]}.$
(53)

$Y_{f}$ can be determined via the fermion masses:

$m_{f}=\dfrac{v_{T}}{\sqrt{2}}\Big{[}Y_{f}-\dfrac{C_{f\mathcal{H}}v_{T}^{2}}{2\,\Lambda^{2}}-\dfrac{C_{qf\mathcal{H}^{5}}v_{T}^{4}}{4\Lambda^{4}}\Big{]};$
(54)

combining this result with Eq. 53, we find:

$Y_{f}=\sqrt{2}(\sqrt{2}G_{F})^{1/2}\,m_{f}+\frac{C_{f\mathcal{H}}}{2\sqrt{2}\,G_{F}\Lambda^{2}}+\frac{C_{qf\mathcal{H}^{5}}}{8\,G_{F}^{2}\Lambda^{4}}+\frac{C_{\mathcal{H}6}^{(1)}m_{f}}{4\sqrt{2}\,(\sqrt{2}G_{F})^{3/2}\Lambda^{4}}.$
(55)

Finally, to determine $\lambda$, we minimize the potential to find the squared
Higgs mass:

$m_{h}^{2}=2\lambda
v_{T}^{2}\left(1-\frac{3v_{T}^{2}C_{\mathcal{H}}}{2\lambda\Lambda^{2}}-\frac{v_{T}^{4}C_{\mathcal{H}^{6}}^{(1)}}{4\Lambda^{4}}-\frac{3v_{T}^{4}C_{\mathcal{H}^{8}}}{2\lambda\Lambda^{4}}\right),$
(56)

so that, combining this result with Eq. 53, we find:

$\lambda=\frac{G_{F}m_{h}^{2}}{\sqrt{2}}+\frac{3C_{\mathcal{H}}}{2\sqrt{2}G_{F}\Lambda^{2}}+\frac{3C_{\mathcal{H}^{8}}}{4G_{F}^{2}\Lambda^{4}}+\frac{C_{\mathcal{H}^{6}}^{(1)}m_{h}^{2}}{4\sqrt{2}G_{F}\Lambda^{4}}.$
(57)

Using this relation, we can also conclude that the coefficient of the tri-
linear Higgs coupling, which we term $\lambda_{3}$ (we have
$V\ni{m_{h}^{2}\over 2}h^{2}+\lambda_{3}h^{3}+\lambda_{4}h^{4}$), is given by:

$\lambda_{3}=\frac{m_{h}^{2}}{2}(\sqrt{2}G_{F})^{1/2}-\frac{C_{\mathcal{H}}(\sqrt{2}G_{F})^{-3/2}}{\Lambda^{2}}-\frac{2C_{\mathcal{H}^{8}}(\sqrt{2}G_{F})^{-5/2}}{\Lambda^{4}},$
(58)

which shall also be useful below.

##  4 Fit procedure and SMEFT results

In this section, we describe our fit procedure. We perform $\chi^{2}$ fits to
the global Higgs signal strengths — not only in the context of SMEFT, but also
in that of the 2HDM. This allows us to compare each of these two approaches
with the experimental results, as well as with each other; the last aspect
thus enables us to ascertain the quality with which the EFT presented in the
previous section can describe the 2HDM introduced in Section 2. In what
follows, we specify the fermion flavors whenever it is relevant.

###  4.1 Direct determination of Higgs couplings

In Fig. 1, we schematically show which coefficients contribute to the tree
level Higgs couplings at ${\cal{O}}({1\over\Lambda^{2}})$ and
${\cal{O}}({1\over\Lambda^{4}})$. It is apparent that, at dimension-6, there
is no contribution from the Higgs decays to vector bosons. We perform our fit
to Higgs decays including only the coefficients generated in the 2HDM. In
practice, it could be that a global fit to data (Higgs, top, electroweak
precision, and di-boson production) measures some pattern of non-zero
coefficients, which might suggest an underlying 2HDM model. In this scenario,
a subsequent fit could be done just to the coefficients of the 2HDM.

(a)  (b)  (c)

Figure 1: Feynman diagrams and the coefficients that contribute to tree level
Higgs decays at ${\cal{O}}({1\over\Lambda^{2}})$ and
${\cal{O}}({1\over\Lambda^{4}})$.

The Higgs signal strengths may generically be represented as $\mu^{P}_{pp\to
h\to f}$, where $P$ represents a production mode and $f$ a final state. Using
the narrow width approximation, we write:

$\mu^{P}_{pp\to h\to f}=\mu_{P}\times\mu_{f}={\sigma^{P}(pp\rightarrow
h)\over\sigma^{P}(pp\rightarrow
h)_{\mathrm{SM}}}\times{\mathrm{BR}(h\rightarrow
f)\over\mathrm{BR}(h\rightarrow f)_{\mathrm{SM}}}.$ (59)

We then compare the different signal strengths evaluated theoretically (both
for the SMEFT and the 2HDM) with those measured experimentally. We consider
the $ggh$, VBF, $Wh$, $Zh$ and $t{\overline{t}}h$ production modes, and the
$\gamma\gamma$, $b{\overline{b}}$, $\tau^{+}\tau^{-}$, $W^{+}W^{-}$ and $ZZ$
final states. For the SM production modes, we use the masses and couplings
from Ref. [64] (with $m_{h}=125$ GeV); for the SM branching ratios (BRs), we
use the results computed by the LHC Higgs cross section working group [65].
For the experimental signal strengths, we include the combined 7/8 TeV
ATLAS/CMS data [66], the $13$ TeV and $139~{}\mathrm{fb}^{-1}$ data from ATLAS
[67], and the $13$ TeV and $137~{}\mathrm{fb}^{-1}$ data from CMS [68]
(including known correlations).

We begin by computing the $95\%$ confidence level (CL) limits on the WCs that
are generated in the 2HDM. Fig. 2 compares the limits when Eq. 59 is expanded
consistently to ${\cal{O}}({1\over\Lambda^{2}})$ with those resulting from an
expansion to ${\cal{O}}({1\over\Lambda^{4}})$. We note that for poorly
constrained coefficients, for example $C_{\mathcal{H}}$ and
$C_{\mu{\mathcal{H}}}$, the limits are quite sensitive to the expansion.

(a)  (b)

Figure 2: Single parameter limits on WCs occuring in the 2HDM at
${\cal{O}}({1\over\Lambda^{2}})$ and ${\cal{O}}({1\over\Lambda^{4}})$ with
$\Lambda=1$ TeV.

Moreover, in the case of $C_{b{\mathcal{H}}}$ and $C_{\tau{\mathcal{H}}}$,
there are two separate solutions when ${\cal{O}}({1\over\Lambda^{4}})$ effects
are included, but only one when these effects are excluded; the extra solution
shall correspond to the so-called wrong-sign solution in the context of the
2HDM, as shall be discussed below. In Fig. 3, we consider the limits in the
$C_{t\mathcal{H}}-C_{b\mathcal{H}}$ plane, with other coefficients set to
zero. As before, we find two disconnected solutions for $C_{b{\mathcal{H}}}$
only when ${\cal{O}}({1\over\Lambda^{4}})$ effects are included. In both
solutions, there is a large correlation between $C_{t\mathcal{H}}$ and
$C_{b\mathcal{H}}$. It is also clear that $C_{t\mathcal{H}}$ is much less
contrained in any of the solutions than it was in Fig. 2.

(a)  Figure 3: Two parameter fit to $C_{t\mathcal{H}}-C_{b\mathcal{H}}$ from
single Higgs production with $\Lambda=1~{}$TeV. The diamond represents the SM
value.

###  4.2 Indirect determination of Higgs tri-linear coupling

Finally, we are interested in ascertaining the importance of the Higgs self-
interactions. At leading order, direct experimental constraints on those
interactions can only be obtained from double Higgs production; the cross
section for this type of process, however, is much smaller than that for
single Higgs production. The authors of Ref. [69, 70] thus propose to
indirectly determine the Higgs self-interactions by calculating the
contribution of such interactions using higher order corrections to single
Higgs production and decays. These contributions depend on the cubic self-
interaction, which in our SMEFT framework is given by Eq. 58 and depends on
the WCs $C_{\mathcal{H}}$ and $C_{\mathcal{H}^{8}}$. For each signal strength
$\mu_{j}$, we adopt the prescription of Ref. [69] to calculate the effects of
the Higgs tri-linear self-interactions on $\mu_{j}$; we term those effects
$\delta\hat{\mu}_{j}$. The combination of $\mu_{j}$ and $\delta\hat{\mu}_{j}$
is ambiguous at ${\cal{O}}({1\over\Lambda^{4}})$, since we do not know the
interference effects between the two types of contributions. We combine them
as

$\mu_{j}^{\mathrm{com}}=\mu_{j}+\delta{\hat{\mu_{j}}}.$ (60)

It should be clear that this is only a rough estimation of the combined
contribution of the Higgs signal strenth of Eq. 59 and the Higgs self-
interactions.

##  5 Results

We now turn to our main results. We start by discussing in section 5.1 two
relevant aspects related to the validity of the EFT expansion; after that, we
present our fits in section 5.2. In both these sections, we ignore the effects
on the signal strengths coming from the Higgs self-interactions; these shall
be investigated in section 5.3. In what follows, and wherever the results
explicitly depend on the scale $\Lambda$, we take by default $\Lambda=1$ TeV.

###  5.1 Preliminary aspects

The first aspect we discuss here concerns the existence of different methods
of expanding a BR. Let us consider the BR of $h\rightarrow b\bar{b}$ in the
Type-II 2HDM, given by $\mathrm{BR}(h\rightarrow
b\bar{b})=\Gamma({h\rightarrow b\bar{b}})/\Gamma({h\rightarrow\mathrm{all}})$.
When expanding the EFT results in powers of $\frac{1}{\Lambda^{2}}$, one may
choose either to expand separately $\Gamma({h\rightarrow b\bar{b}})$ and
$\Gamma({h\rightarrow\mathrm{all}})$, or to expand $\mathrm{BR}(h\rightarrow
b\bar{b})$ as a whole. The crucial difference between the two approaches is
that the former expands $\Gamma({h\rightarrow\mathrm{all}})$ itself, whereas
the latter expands the inverse of $\Gamma({h\rightarrow\mathrm{all}})$. Now,
when the contributions from higher order operators to
$\Gamma({h\rightarrow\mathrm{all}})$ are sizable compared to the SM
contribution, the expansion of $1/\Gamma({h\rightarrow\mathrm{all}})$ does not
converge.131313This is not a statement about a lack of convergence of the EFT;
rather, it is a general statement about the radius of convergence of the power
series of rational functions. As a consequence, the expansion of
$\mathrm{BR}(h\rightarrow b\bar{b})$ as a whole also does not converge. This
is what can be seen on the right panel of Fig. 4, for almost the entire range
of values of $\cos({\beta-\alpha})$. In order to properly investigate these
regions, then, one should expand $\Gamma({h\rightarrow\mathrm{all}})$ itself
and leave it in the denominator, as done on the left panel.

Figure 4: Calculations of $\mathrm{BR}(h\rightarrow bb)$ in the Type-II 2HDM
for $\tan\beta=5$ with two different expansions: on the left, the partial
width of $h\rightarrow b\bar{b}$ and the total width of $h$ are expanded
separately to the given order in the EFT; on the right, the ratio of the
partial width to the total width is expanded.

Still concerning Fig. 4, it is clear in both plots that there are truncations
which lead to BRs larger than 1 or smaller than 0 (which is clearly
unphysical, signalling that the expansion is not valid). In particular, the
green curve on the left plot — that includes linear effects from EFD 8
operators, but neglects the squared effects – describes negative BRs for a
certain range of values of $\cos(\beta-\alpha)$.141414This is a general
behaviour in SMEFT truncations whenever there is a removal of squared effects.
On the other hand, and as shall be seen below, such range of values (in Type-
II) is excluded by experiment, as $|\cos({\beta-\alpha})|$ is required to be
very close to zero. Therefore, in what follows, we consistently ignore squared
effects from EFD 8 operators, which are of order
${\cal{O}}({1\over\Lambda^{8}})$.

The second preliminary aspect is related to the range of validity for the EFT
expansion of the 2HDM. As discussed in section 3.7, the EFT requires the
decoupling limit, which in turn requires the different $Z_{i}$ parameters of
eq. 9 to obey $Z_{i}/(4\pi)\simeq\mathcal{O}(1)$. Let us then define
$|Z_{i}|_{\mathrm{max}}$ as the maximum absolute value among the different
$Z_{i}$, for a certain point in the parameter space. In Fig. 5, we show the
$|Z_{i}|_{\mathrm{max}}$ as a function of $|\cos(\beta-\alpha)|$, for three
different values of the scale $\Lambda$.

Figure 5: The maximum absolute value among the different $Z_{i}$ against
$\cos(\beta-\alpha)$, for three different scales. The dashed lines highlight
the values 1 and $4\pi$.

It is clear that the green and orange curves are always below the upper dashed
line, which corresponds to $|Z_{i}|_{\mathrm{max}}=4\pi$. In other words, for
the two lower scales (500 GeV and 1 TeV), the entire range of
$\cos(\beta-\alpha)$ shown provides a valid description of the EFT expansion
with regards to perturbative unitarity, since it is compatible with the
requirement $Z_{i}/(4\pi)\simeq\mathcal{O}(1)$. The heavier scale $\Lambda=2$
TeV starts yielding $Z_{i}/(4\pi)>1$ for $\cos(\beta-\alpha)\gtrsim 0.2$.

###  5.2 Extraction of Higgs couplings and matching to 2HDM

(a)  (b)

(c)  (d)

Figure 6: 95$\%$ CL constraints from the LHC data on the $\tan\beta$ vs
$\cos({\beta-\alpha})$ plane, for Type-I (top left), Type-II (top right),
Type-L (bottom left) and Type-F (bottom right). The blue lines represent
limits on the full UV model, whereas the remaining lines describe limits
within the SMEFT framework in different approximations made at the level of
the Higgs signal strengths.

In Fig. 6, we present the fits to both the exact 2HDM and the SMEFT matched to
the 2HDM, for the four types of 2HDM. In order to ascertain the importance of
higher EFD operators in the EFT description, we consider three different
approximations for the calculation of the Higgs signal strengths in the SMEFT:
with EFD 6 operators and excluding squared contributions (orange), with EFD 6
operators but including squared contributions (green), and with EFD 8
operators (red). In what follows, we discuss in detail each type of 2HDM.

Let us begin with Type-I. Here, the results for the SMEFT up to EFD 6 are
weakly constrained, especially for higher values of $\tan\beta$. As already
noted in Ref. [15], this is due to a combination of two reasons: first, up to
EFD 6 (and excluding self-interactions), the only WCs contributing to the
SMEFT fit are those modifying the Yukawa couplings, namely $C_{f\mathcal{H}}$
(recall Eq. 39); second, the Yukawa parameters $\eta_{f}$ in Type-I are equal
to 1 for all types of fermions (see Table 1), which implies that the
$C_{f\mathcal{H}}$ up to EFD 6 (e.g. in the first term of the right-hand side
of Eq. 45a) are suppressed by $\tan\beta$. As a consequence, for high values
of $\tan\beta$, the EFT for the 2HDM Type-I truncated with EFD 6 operators has
no relevant information that can be restricted by experiment. This is in clear
contrast with the full Type-I 2HDM, which obviously contains more predictions
than simply the Yukawa interactions (in particular, it contains those related
to the interactions between the Higgs and gauge bosons), and thus ends up
being contrained by the experimental results. Therefore, the region of
parameter space of the Type-I 2HDM where $\tan\beta$ takes high values
constitutes a scenario for which the EFT truncated with EFD 6 operators is
utterly unable to provide a correct description of the full UV model. The
inclusion of quadratic effects does not change this picture, as it does not
alter the two reasons given above.

What does change the picture — and quite significantly — is the inclusion of
EFD 8 operators. Indeed, the EFT now has information besides the Yukawa
couplings; in particular, it contains a prediction for the cubic Higgs-gauge
interactions (via the coefficient $C_{\mathcal{H}^{6}}^{(1)}$, cf. Eq. 41),
which is not supressed with $\tan\beta$ (see Eq. 46b). Accordingly, the SMEFT
framework is now able to be experimentally constrained for the entire spectrum
of $\tan\beta$. Not surprisingly, then, it now provides a very accurate
description of the full UV model, as can be seen in the figure. Finally, we
note that, although there is an explicit scale dependence when the EFD 8
operators are included, the dependence is not numerically significant.

Concerning the other types of 2HDM, the figure shows that, for Type-II and
Type-F, the SMEFT approach provides an excellent description of the
corresponding full UV model, even if one truncates the expansion with EFD 6
operators and neglects quadratic effects in the signal strengths. The
difference between these models and Type-I is that, in the former, there is
always at least one fermion type whose $\eta_{f}$ parameter is proportional to
$\tan^{2}\beta$, so that the respective $C_{f\mathcal{H}}$ no longer scales
with $\tan^{-1}\beta$ — which in turn implies that the SMEFT is constrained by
experiment for high values of $\tan\beta$, even if it only contains the
Yukawa-related WCs. For these two types of 2HDM, therefore, the inclusion of
higher-order effects in the SMEFT expansion is irrelevant.

Finally, Type-L stands out among the other types, as it is still compatible
with the interesting scenario of the wrong-sign solution, corresponding to the
isolated band centered around $\cos({\beta-\alpha})=0.3$.151515Although Type-
II and Type-L would in principle allow such a solution, it is ruled out by the
most recent experimental data[71]. Here, we verify what was already observed
in Ref. [15], namely: the wrong-sign solution cannot be captured by the SMEFT
description if only the linear effects of the EFD 6 operators are included in
the SMEFT predictions for the Higgs signal strengths. This is because the
likelihood that results from such an approximation is a Gaussian one, which
contains only one minimum; this minimum corresponds to the solution where the
WCs are close to zero, which is the solution favored by the experimental data.
So, in order to obtain different minima, the likelihood must be non-Gaussian,
which in turn can only be obtained by including the higher order quadratic
effects. Accordingly, the wrong-sign solution in Type-L constitutes a scenario
where a SMEFT approach that consistently includes only the $1/\Lambda^{2}$
terms in the Higgs signal strengths completely misses the description of the
full model. In order to capture that region, though, one does not necessarily
need to include the effects of EFD 8 operators: the figure shows that the
squared terms of EFD 6 operators already leads to the generation of the wrong-
sign solution. On the other hand, it is also clear from the figure that such a
solution is far from being a faithful reproduction of the band of the full UV
model; in fact, whereas the latter only reaches values of
$\cos({\beta-\alpha})$ slightly larger than $0.2$, the green band extends to
values larger than $0.4$. The reason is that, in the full UV model, the large
values of $\cos({\beta-\alpha})$ are ruled out by measurements of the Higgs
couplings to gauge bosons; but since the green band does not have information
about such couplings (they only show up with $C_{\mathcal{H}^{6}}^{(1)}$, at
higher order), it is not constrained to small values of
$\cos({\beta-\alpha})$. This simple reasoning is confirmed by the orange band,
which shows that the SMEFT result is already restricted to smaller values of
$\cos({\beta-\alpha})$ when the SMEFT expansion is truncated with EFD 8
operators, thus reproducing quite well the solution of the full model.

As a final note, we should stress that the wrong-sign solution does not spoil
the convergence of the EFT expansion [15]. This is not obvious, since that
solution requires the EFT effects to be twice as large (in modulus) as those
of the SM. On the other hand, the validity of the EFT expansion in Eq. 1 does
not necessarily require dimension-6 operators to have smaller effects than
those with dimension 4, but only that the subsequent orders do not become
relevant. Actually, there are many examples where dimension-6 operators can be
much larger than dimension-4 operators, without ruining the EFT expansion
[72]. As we discussed above, the wrong-sign solution described by SMEFT in
Type-L reproduces quite well the full model, which implies the convergence of
the EFT expansion.

###  5.3 Inclusion of Higgs tri-linear couplings

We now investigate the effects of including the Higgs self-interactions in the
SMEFT predictions for the Higgs signal strengths, as described in section 4.2.
In Fig. 7, we show again the fits for Type-I and Type-L, but now explicitly
comparing the results with and without the Higgs self-interactions. The
differences are striking and motivate a complete calculation which would
include the neglected interference effects in the SMEFT predictions.

(a)  (b)

Figure 7: 95$\%$ CL constraints from the LHC data on the $\tan\beta$ vs
$\cos({\beta-\alpha})$ plane, for Type-I (left) and Type-L (right) and
including the effects of the Higgs tri-linear coupling. The different types of
lines represent different approximations through which the Higgs signal
strengths are calculated: by keeping only linear effects of the EFD 6
operators (dashed lines) or by including up to EFD 8 operators (solid lines).

In Type-I up to EFD 6, the curves that include the self-interactions no longer
have the problem of describing a prediction that is globally suppressed with
$\tan\beta$; precisely due to the self-interactions, indeed, such a prediction
also includes $C_{\mathcal{H}}$, which does not depend on $\tan\beta$ (cf. Eq.
45c). It does depend, however, very strongly on the scale $\Lambda$, as can
also be seen in the figure. In the case of Type-L, the inclusion of the self-
interaction excludes the wrong-sign solution, and there is no relevant
dependence on $\Lambda$. We do not compare these results with the full 2HDM,
because this would require comparing with the one-loop predictions for the
2HDM in the heavy mass limit, which is beyond the scope of this work. Fig. 7
motivates such a comparison. Finally, for the entire range of
$\cos(\beta-\alpha)$ considered, all the results in Fig. 7 respect the
requeriment of perturbative unitarity. In fact, and as can be seen comparing
to Fig. 5, never does one scale reach the region of values of
$\cos(\beta-\alpha)$ where $Z_{i}/(4\pi)>1$.

##  6 Conclusions

This work considers the SMEFT as an effective low-energy theory for the 2HDM.
Whereas the usual approach to describe a full UV model with the SMEFT
truncates the EFT expansion of the Lagrangian with
$\mathcal{O}({1\over\Lambda^{2}})$ terms, we truncate including the
dimension-8 $\mathcal{O}({1\over\Lambda^{4}})$ terms and investigate the
importance of these terms. Assuming the decoupling limit, we integrate out the
heavy scalar doublet $H_{2}$ of the 2HDM, which allows us to consistently
derive the matching between the SMEFT and the 2HDM up to
$\mathcal{O}({1\over\Lambda^{4}})$. This matching is used to convert the LHC
constraints on the SMEFT Wilson coefficientss into constraints on the
parameter space of the 2HDM, and thus evaluate the quality with which the
SMEFT is able to reproduce the LHC direct constraints on the 2HDM and the
importance of the $\mathcal{O}({1\over\Lambda^{4}})$ terms.

Focusing on the scenario where CP is conserved in the scalar sector, we
analyze the four types of 2HDM models that result from different applications
of the $Z_{2}$ symmetry to the fermion sector, using the latest experimental
data to perform fits for the UV complete version. We find two situations where
the inclusion of the $\mathcal{O}({1\over\Lambda^{4}})$ terms is crucial: the
region in Type-I with moderate values of $\tan\beta$, and the wrong-sign
solution found in Type-L. In both cases, the SMEFT approach provides a very
poor description if truncated at $\mathcal{O}({1\over\Lambda^{2}})$, but a
very good one if the $\mathcal{O}({1\over\Lambda^{4}})$ terms are included.
The reason for this behaviour is that the Higgs-gauge interactions that
constrain the full 2HDM only show up in the SMEFT expansion with the
$\mathcal{O}({1\over\Lambda^{4}})$ terms. For the remaining 2HDM models, the
SMEFT truncated at $\mathcal{O}({1\over\Lambda^{2}})$ provides a good
description of the UV complete model, so that the inclusion of
$\mathcal{O}({1\over\Lambda^{4}})$ terms becomes irrelevant.

We also discuss the effects of the self-Higgs interactions that arise beyond
tree level in single Higgs production and decay in the SMEFT. These lead to
strong constraints on both Type-I and Type-L models in the SMEFT, along with a
numerically significant scale dependence. It would be of interest to compute
the SMEFT predictions for the signal strengths to one-loop order (two loop for
gluon fusion) to assess the relevance of the Higgs-self couplings in a
consistent manner including the interference between the contributions. The
loop suppression factor of $1/(16\,\pi^{2})$ could have similar size effects
as the dimension-8 contributions which scale as ${v^{4}\over\Lambda^{4}}$.
Also of interest would be to compare our results with those of a different
approach, namely the Higgs Effective Field Theory (HEFT); there, instead of
the doublet $H_{2}$, one integrates out the heavy mass states of the 2HDM. The
validity of HEFT is expected to be more general than that of SMEFT.

## Acknowledgements

We thank Adam Falkowski, Xiaochuan Lu, Christopher Murphy and Robert Szafron
for discussions. SD, DF, and MS are supported by the United States Department
of Energy under Grant Contract DE- SC0012704. SH is supported in part by the
DOE Grant DE-SC0013607, and in part by the Alfred P. Sloan Foundation Grant
No. G-2019-12504. Digital data is posted at
https://quark.phy.bnl.gov/Digital_Data_Archive/dawson/dim8_22.

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