LMU-ASC 40/23
December 2023

Two-Higgs Doublet Model Matched to
Nonlinear Effective Theory

G. Buchalla, F. König, Ch. Müller-Salditt and F. Pandler

Ludwig-Maximilians-Universität München, Fakultät für Physik,

Arnold Sommerfeld Center for Theoretical Physics, D–80333 München, Germany

We use functional methods to match the Two-Higgs Doublet Model with heavy
scalars in the nondecoupling regime to the appropriate nonlinear effective
field theory, which takes the form of an electroweak chiral Lagrangian (HEFT).
The effective Lagrangian is derived to leading order in the chiral counting.
This includes the loop induced $h\to\gamma\gamma$ and $h\to Z\gamma$ local
terms, which enter at the same chiral order as their counterparts in the
Standard Model. An algorithm is presented that allows us to compute the
coefficient functions to all orders in $h$. Some of the all-orders results are
given in closed form. The parameter regimes for decoupling, nondecoupling and
alignment scenarios in the effective field theory context and some
phenomenological implications are briefly discussed.

##  1 Introduction

Indirect effects of New Physics (NP) at colliders can be consistently
described with effective field theories (EFTs), where the new heavy particles
are integrated out. Applying this approach to electroweak symmetry breaking
and Higgs-boson properties, the nonlinear EFT in the form of an electroweak
chiral Lagrangian (EWChL, also refered to as nonlinear Higgs-sector EFT, or
HEFT) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] provides
us with the most natural framework [19]. It is economic and general and
properly accounts for nondecoupling effects in the scalar sector. While the
EFT is model independent, matching its parameters to a specific scenario
connects the EFT coefficients to a given UV theory. Recently, there has been
renewed interest in the Two-Higgs Doublet Model (2HDM) [20] and the treatment
of its properties at the electroweak scale in an EFT approximation [21, 22,
23, 24]. Our motivation for addressing this topic is essentially twofold.
First, we would like to investigate the description of the 2HDM in the
nondecoupling regime, which corresponds to interesting regions of parameter
space. Second, our analysis exemplifies the structure of the Higgs-EWChL in
the context of the 2HDM as a prototypical extension of the Higgs sector. In
addition, we use functional methods throughout, which make the calculations
rather efficient and transparent. Exploiting the advantages of the functional
approach, we go beyond the existing literature in computing higher terms in
the Higgs functions, including some all-orders results in powers of the Higgs
field $h$, and an algorithmic prescription for their general derivation. The
method used in the present study has been developed in detail in [25], where
it was applied to the matching of a singlet extension of the SM to the
nonlinear EFT.

The paper is organized as follows. In Section 2 we introduce polar coordinates
for the scalar sector of the 2HDM, which are especially convenient for the
matching to the nonlinear EFT. In Section 3 we perform the matching of the
2HDM in the nondecoupling regime to the leading-order (LO) chiral Lagrangian
at tree level, integrating out the heavy scalars by functional methods. The
matching calculation is extended to the one-loop induced $h\to\gamma\gamma$
and $h\to Z\gamma$ local EFT operators in Section 4. Section 5 summarizes
important aspects of the 2HDM parameter space with heavy-scalar masses (of
order TeV), including the decoupling, nondecoupling and alignment regimes.
Some phenomenological implications are discussed in Section 6, before we
conclude in Section 7. An Appendix contains the solution $H_{0}(h)$ of the LO
equations of motion (eom) for the heavy scalar field $H_{0}$ to all orders in
$h$ (Appendix A), the one-loop matching for $h\to\gamma\gamma$ and $h\to
Z\gamma$ to all orders in $h$ (Appendix B), and explicit expressions for the
parameters of the 2HDM scalar potential (Appendix C).

##  2 2HDM scalar sector in polar coordinates

The scalar sector of the 2HDM consists of two complex doublets $H_{1}$,
$H_{2}$, both in the fundamental representation of the weak gauge group
$SU(2)$ and with weak hypercharge ${\cal Y}=1/2$. It is convenient to define
the conjugate doublets $\tilde{H}_{n}\equiv i\sigma_{2}H^{*}_{n}$, with
$n=1,2$, and the matrix fields

$S_{n}\equiv(\tilde{H}_{n},H_{n})$ (1)

The Lagrangian of the scalar sector can then be expressed as

$\displaystyle{\cal L}_{S}=\frac{1}{2}\langle
D_{\mu}S^{\dagger}_{n}D^{\mu}S_{n}\rangle-V$ (2)

where $\langle\ldots\rangle$ denotes the trace, a sum over $n$ is understood,
and $V$ is the potential to be discussed below. Following [26], the matrix
fields $S_{n}$ can be written in polar coordinates as

$\displaystyle S_{n}\equiv UR_{n}\,,\qquad
R_{n}=\frac{1}{\sqrt{2}}\Bigl{[}(v_{n}+h_{n})\boldsymbol{1}+iC_{n}\sigma_{a}\rho_{a}\Bigr{]}$
(3)

Here $\sigma_{a}\equiv 2T_{a}$, $a=1,2,3$, are the Pauli matrices, and
$U\equiv\exp(2i\varphi_{a}T_{a}/v)$ is the matrix of the electroweak Goldstone
bosons, where $v=246\,{\rm GeV}$ is the electroweak vacuum expectation value
(vev). The vevs of the two Higgs doublet fields are $v_{1}$ and $v_{2}$,
respectively, with $v^{2}_{1}+v^{2}_{2}=v^{2}$, and

$\displaystyle C_{1}=-\frac{v_{2}}{v}\equiv-\sin\beta\,,\qquad
C_{2}=\frac{v_{1}}{v}\equiv\cos\beta$ (4)

Using the decomposition in (3), the eight real degrees of freedom in the
complex doublets $H_{n}$ are expressed through the eight real fields
$\varphi_{a}$, $\rho_{a}$ and $h_{n}$. The electroweak quantum numbers of
$S_{n}$ and $U$ imply that the covariant derivative reads

$\displaystyle D_{\mu}\Phi=\partial_{\mu}\Phi+igW_{\mu}\Phi-
ig^{\prime}B_{\mu}\Phi T_{3}\qquad{\rm for}\quad\Phi=S_{n},\,U$ (5)

where $W^{\mu}=W^{\mu}_{a}T_{a}$ and $B^{\mu}$ are the gauge fields of
$SU(2)_{L}$ and $U(1)_{Y}$. It follows from (3) that

$\displaystyle
D_{\mu}R_{n}=\partial_{\mu}R_{n}+ig^{\prime}B_{\mu}[T_{3},R_{n}]$ (6)

Consequently, $h_{1,2}$ and $\rho_{3}$ are electroweak singlets, whereas
$\rho_{1,2}$ are singlets of $SU(2)_{L}$, but charged under $U(1)_{Y}$. Hence,

$\displaystyle D_{\mu}h_{n}=\partial_{\mu}h_{n}\qquad{\rm and}\qquad
D_{\mu}\rho_{a}=\partial_{\mu}\rho_{a}+g^{\prime}B_{\mu}\,\varepsilon_{ab3}\,\rho_{b}$
(7)

For $a=1,2$, this can also be written in terms of the eigenstates $\rho^{\pm}$
of charge and hypercharge (with $Q={\cal Y}=\pm 1$) as

$\displaystyle D_{\mu}\rho^{\pm}=\partial_{\mu}\rho^{\pm}\pm
ig^{\prime}B_{\mu}\rho^{\pm}\,,\qquad\rho^{\pm}=\frac{1}{\sqrt{2}}(\rho_{1}\mp
i\rho_{2})$ (8)

Inserting (3) into (2), the kinetic term becomes

$\displaystyle{\cal L}_{S,kin}$ $\displaystyle=\frac{1}{2}\langle
D_{\mu}S^{\dagger}_{n}D^{\mu}S_{n}\rangle=$ $\displaystyle=\frac{1}{4}\langle
D_{\mu}U^{\dagger}D^{\mu}U\rangle\Bigl{[}(v_{n}+h_{n})^{2}+\rho_{a}\rho_{a}\Bigr{]}+\frac{1}{2}\partial_{\mu}h_{n}\partial^{\mu}h_{n}+\frac{1}{2}D_{\mu}\rho_{a}D^{\mu}\rho_{a}$
$\displaystyle+\langle
iU^{\dagger}D_{\mu}UT_{a}\rangle\Bigl{[}\varepsilon_{abc}\,\rho_{b}D^{\mu}\rho_{c}+C_{n}(\rho_{a}\partial^{\mu}h_{n}-D^{\mu}\rho_{a}h_{n})\Bigr{]}$
(9)

The potential in (2) can be written as [26]

$\displaystyle V=$ $\displaystyle\frac{m^{2}_{11}}{2}\langle
S^{\dagger}_{1}S_{1}\rangle+\frac{m^{2}_{22}}{2}\langle
S^{\dagger}_{2}S_{2}\rangle-m^{2}_{12}\langle S^{\dagger}_{1}S_{2}\rangle$
$\displaystyle+\frac{\lambda_{1}}{8}\langle
S^{\dagger}_{1}S_{1}\rangle^{2}+\frac{\lambda_{2}}{8}\langle
S^{\dagger}_{2}S_{2}\rangle^{2}+\frac{\lambda_{3}}{4}\langle
S^{\dagger}_{1}S_{1}\rangle\langle S^{\dagger}_{2}S_{2}\rangle$
$\displaystyle+\lambda_{4}\langle S^{\dagger}_{1}S_{2}P_{+}\rangle\langle
S^{\dagger}_{1}S_{2}P_{-}\rangle+\frac{\lambda_{5}}{2}\Bigl{(}\langle
S^{\dagger}_{1}S_{2}P_{+}\rangle^{2}+\langle
S^{\dagger}_{1}S_{2}P_{-}\rangle^{2}\Bigr{)}$ (10)

in terms of the matrix fields $S_{n}$ from (1). $P_{\pm}=(1\pm\sigma_{3})/2$
are projection operators. Here we assume invariance under $S_{1}\to-S_{1}$,
$S_{2}\to S_{2}$, softly broken by $m^{2}_{12}$, and CP invariance, so that
all parameters in (2) are real.

When $S_{n}$ is expressed as in (3), the Goldstone field $U$ disappears from
the potential $V$ in (2), which becomes a function of $h_{n}$ and $\rho_{a}$.
The vevs $v_{1,2}$ are defined such that terms linear in $h_{1,2}$ vanish. The
terms quadratic in the fields are diagonalised by $\rho^{\pm}$, $\rho_{3}$,
and by $h$ and $H$, which are related to $h_{1,2}$ by

$\left(\begin{array}[]{c}h_{1}\\\
h_{2}\end{array}\right)=\left(\begin{array}[]{cc}c_{\alpha}&-s_{\alpha}\\\
s_{\alpha}&c_{\alpha}\end{array}\right)\left(\begin{array}[]{c}H\\\
h\end{array}\right)$ (11)

Here and in the following, we define $\cos\phi\equiv c_{\phi}$ and
$\sin\phi\equiv s_{\phi}$ for generic angles $\phi$. The mass eigenstates of
the scalar sector are then given by $h$, identified as the observed Higgs at
$m_{h}=125\,{\rm GeV}$, and the additional scalars $H\equiv H_{0}$,
$H^{\pm}\equiv\pm i\rho^{\pm}$ and $A_{0}\equiv-\rho_{3}$.

The eight parameters of $V$ in (2), $m^{2}_{11}$, $m^{2}_{22}$, $m^{2}_{12}$,
$\lambda_{1},\ldots,\lambda_{5}$, can be traded for the vevs, the particle
masses, the Higgs mixing angle and the soft breaking term:

$\displaystyle v_{1},\quad v_{2},\quad m_{h},\quad M_{0}\equiv M_{H_{0}},\quad
M_{H}\equiv M_{H^{\pm}},\quad M_{A}\equiv M_{A_{0}},\quad s_{\alpha},\quad
m^{2}_{12}$ (12)

or, equivalently,

$\displaystyle v,\quad\tan\beta\equiv t_{\beta}=v_{2}/v_{1},\quad m_{h},\quad
M_{0},\quad M_{H},\quad M_{A},\quad
c_{\beta-\alpha},\quad\overline{m}^{2}\equiv\frac{m_{12}^{2}}{s_{\beta}c_{\beta}}$
(13)

Dropping an irrelevant additive constant, the potential finally takes the form

$\displaystyle V=$
$\displaystyle\frac{1}{2}m^{2}_{h}h^{2}+\frac{1}{2}M^{2}_{0}H^{2}+M^{2}_{H}H^{+}H^{-}+\frac{1}{2}M^{2}_{A}A^{2}_{0}$
$\displaystyle-
d_{1}h^{3}-d_{2}h^{2}H-d_{3}hH^{2}-d_{4}H^{3}-d_{5}hH^{+}H^{-}-d_{6}hA^{2}_{0}-d_{7}HH^{+}H^{-}-d_{8}HA^{2}_{0}$
$\displaystyle-z_{1}h^{4}-z_{2}h^{3}H-z_{3}h^{2}H^{2}-z_{4}hH^{3}-z_{5}H^{4}$
$\displaystyle-
z_{6}h^{2}H^{+}H^{-}-z_{7}hHH^{+}H^{-}-z_{8}H^{2}H^{+}H^{-}-z_{9}(H^{+}H^{-})^{2}-z_{10}h^{2}A^{2}_{0}$
$\displaystyle-
z_{11}hHA^{2}_{0}-z_{12}H^{2}A^{2}_{0}-z_{13}H^{+}H^{-}A^{2}_{0}-z_{14}A^{4}_{0}$
(14)

The coefficients can be found in Appendix C.

The scalar sector couples to fermions through Yukawa interactions. We assume a
type-II Yukawa sector given by the Lagrangian [20]

$\displaystyle\mathcal{L}_{Y}=-\bar{q}_{L}H_{1}Y_{d}d_{R}-\bar{q}_{L}\tilde{H}_{2}Y_{u}u_{R}-\bar{\ell}_{L}H_{1}Y_{e}e_{R}+\text{h.c.}$
(15)

where $q_{L}=(u_{L},d_{L})^{T}$ and $\ell_{L}=(\nu_{L},e_{L})^{T}$ are the
left-handed doublets, and $u_{R}$, $d_{R}$, $e_{R}$ the right-handed singlets.
The latter may be collected into $q_{R}=(u_{R},d_{R})^{T}$ and
$\ell_{R}=(\nu_{R},e_{R})^{T}$. We suppress generation indices, which are
understood for the fermion fields in (15). $\mathcal{L}_{Y}$ can be written in
terms of the matrix fields $S_{n}$ in (3) as

$\displaystyle\mathcal{L}_{Y}=-\bar{q}_{L}Y_{d}S_{1}P_{-}q_{R}-\bar{q}_{L}Y_{u}S_{2}P_{+}q_{R}-\bar{\ell}_{L}Y_{e}S_{1}P_{-}\ell_{R}+\text{h.c.}$
(16)

##  3 Tree-level matching in the nondecoupling regime

The full Lagrangian of the 2HDM can be written as

$\displaystyle\mathcal{L}_{2HDM}=\mathcal{L}_{0}+\mathcal{L}_{S,kin}-V+\mathcal{L}_{Y}$
(17)

where the scalar sector is represented by $\mathcal{L}_{S,kin}$, $V$ and
$\mathcal{L}_{Y}$ from (2), (2) and (16), and $\mathcal{L}_{0}$ denotes the
unbroken Standard Model (SM),

$\displaystyle\mathcal{L}_{0}=$ $\displaystyle-\frac{1}{2}\langle
G_{\mu\nu}G^{\mu\nu}\rangle-\frac{1}{2}\langle
W_{\mu\nu}W^{\mu\nu}\rangle-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}$
$\displaystyle+\bar{q}_{L}i\\!\not\\!\\!Dq_{L}+\bar{\ell}_{L}i\\!\not\\!\\!D\ell_{L}+\bar{u}_{R}i\\!\not\\!\\!Du_{R}+\bar{d}_{R}i\\!\not\\!\\!Dd_{R}+\bar{e}_{R}i\\!\not\\!\\!De_{R}$
(18)

In terms of the model parameters in (12), (13) the nondecoupling limit is
defined by the hierarchy

$\displaystyle v\sim m_{h}\sim\overline{m}\ll M_{0}\,,M_{H}\,,M_{A}\sim M_{S}$
(19)

with $t_{\beta}$ and $c_{\beta-\alpha}$ of order unity in general. To leading
order, all terms in the effective Lagrangian that are unsuppressed by the
heavy scale $M_{S}$ (of order $(M_{S})^{0}$) have to be retained.

The procedure of integrating out the heavy scalars at tree level in the
nondecoupling scenario has been described in detail in [25]. It consists of
the following steps:

  * •

The equation of motion (eom) is solved to obtain the heavy field $H_{0}(h)$ to
LO in the heavy-mass limit, ${\cal O}(M^{0}_{S})$. This requires the LO terms
in the full-theory Lagrangian of order $M^{2}_{S}$. A closed-form solution for
$H_{0}(h)$ is derived in Appendix A. The ${\cal O}(M^{2}_{S})$-Lagrangian
contains the heavy fields $A_{0}$ and $H^{\pm}$ only at quadratic order or
higher. Contributions with only internal lines from these fields, therefore,
cannot arise at tree level. Integrating them out at tree level and to LO then
implies $A_{0}=H^{\pm}=0$.

  * •

The eom solutions $H_{0}=H_{0}(h)$ and $A_{0}=H^{\pm}=0$ are inserted into the
Lagrangian (17). The ${\cal O}(M^{2}_{S})$-terms cancel and an expression of
${\cal O}(M^{0}_{S})$ in the heavy-mass expansion is obtained.

  * •

The field redefinition

$\displaystyle\tilde{h}=\int^{h}_{0}\sqrt{1+\left(\frac{dH_{0}(s)}{ds}\right)^{2}}\,ds$
(20)

is performed to achieve a canonically normalized kinetic term for the Higgs
field $\tilde{h}$. For notational convenience we will drop the tilde in the
end, taking $\tilde{h}\to h$.

Proceeding in this way, the effective theory takes the form of an electroweak
chiral Lagrangian at chiral dimension two, ${\cal L}={\cal L}_{0}+{\cal
L}_{Uh,2}$, with

$\displaystyle\mathcal{L}_{Uh,2}=$ $\displaystyle\frac{v^{2}}{4}\langle
D_{\mu}U^{{\dagger}}D^{\mu}U\rangle\left(1+F_{U}(h)\right)+\frac{1}{2}\partial_{\mu}h\partial^{\mu}h-V(h)$
$\displaystyle-\Bigl{[}\bar{q}_{L}\left({\cal M}_{u}+\sum_{n=1}^{\infty}{\cal
M}_{u}^{(n)}\left(\frac{h}{v}\right)^{n}\right)UP_{+}q_{R}+\bar{q}_{L}\left({\cal
M}_{d}+\sum_{n=1}^{\infty}{\cal
M}_{d}^{(n)}\left(\frac{h}{v}\right)^{n}\right)UP_{-}q_{R}$
$\displaystyle+\bar{\ell}_{L}\left({\cal M}_{e}+\sum_{n=1}^{\infty}{\cal
M}_{e}^{(n)}\left(\frac{h}{v}\right)^{n}\right)UP_{-}\ell_{R}+\mathrm{h.c.}\Bigr{]}$
(21)

where

$\displaystyle
F_{U}(h)=2s_{\beta-\alpha}\frac{h}{v}+\left(1-\frac{s_{2\alpha}}{s_{2\beta}}c^{2}_{\beta-\alpha}\right)\left(\frac{h}{v}\right)^{2}-\frac{4}{3}\frac{s^{2}_{2\alpha}}{s^{2}_{2\beta}}c^{2}_{\beta-\alpha}s_{\beta-\alpha}\left(\frac{h}{v}\right)^{3}+\ldots$
(22)

$\displaystyle V(h)$
$\displaystyle=\frac{v^{2}m_{h}^{2}}{2}\bigg{\\{}\bigg{(}\frac{h}{v}\bigg{)}^{2}+\bigg{[}s_{\beta-\alpha}+\frac{2c_{\beta-\alpha}^{2}c_{\beta+\alpha}}{s_{2\beta}}\bigg{(}1-\frac{\overline{m}^{2}}{m_{h}^{2}}\bigg{)}\bigg{]}\bigg{(}\frac{h}{v}\bigg{)}^{3}$
$\displaystyle+\bigg{[}\frac{1}{4}-\frac{c_{\beta-\alpha}^{2}}{4s_{2\beta}^{2}}\bigg{(}\frac{1}{6}(7-12c_{2(\beta+\alpha)}-19c_{4\alpha})-(1-2c_{2\alpha}c_{2\beta}-3c_{4\alpha})\frac{\overline{m}^{2}}{m_{h}^{2}}\bigg{)}\bigg{]}\bigg{(}\frac{h}{v}\bigg{)}^{4}$
$\displaystyle-\frac{c_{\beta-\alpha}^{2}s_{2\alpha}^{2}}{2s_{2\beta}^{3}}\bigg{[}c_{\beta+\alpha}+3c_{\beta-3\alpha}-(2c_{\beta+\alpha}+3c_{3\beta-\alpha}+11c_{\beta-3\alpha})\frac{\overline{m}^{2}}{4m_{h}^{2}}\bigg{]}\bigg{(}\frac{h}{v}\bigg{)}^{5}+\dots\bigg{\\}}$
(23)

$\displaystyle{\cal M}_{u}+\sum_{n=1}^{\infty}{\cal
M}_{u}^{(n)}\left(\frac{h}{v}\right)^{n}=$ $\displaystyle{\cal
M}_{u}\left[1+\frac{c_{\alpha}}{s_{\beta}}\frac{h}{v}-\frac{c_{\beta-\alpha}}{2}\frac{s^{2}_{\alpha}c_{\alpha}}{s^{2}_{\beta}c_{\beta}}\left(\frac{h}{v}\right)^{2}-\frac{c_{\beta-\alpha}}{6}\frac{s^{2}_{2\alpha}}{s^{2}_{2\beta}}(2s_{2\alpha}-(1-2c_{2\alpha})t^{-1}_{\beta})\left(\frac{h}{v}\right)^{3}+\ldots\right]$
(24) $\displaystyle{\cal M}_{d}+\sum_{n=1}^{\infty}{\cal
M}_{d}^{(n)}\left(\frac{h}{v}\right)^{n}=$ $\displaystyle{\cal
M}_{d}\left[1-\frac{s_{\alpha}}{c_{\beta}}\frac{h}{v}-\frac{c_{\beta-\alpha}}{2}\frac{s_{\alpha}c^{2}_{\alpha}}{s_{\beta}c^{2}_{\beta}}\left(\frac{h}{v}\right)^{2}+\frac{c_{\beta-\alpha}}{6}\frac{s^{2}_{2\alpha}}{s^{2}_{2\beta}}(2s_{2\alpha}-(1+2c_{2\alpha})t_{\beta})\left(\frac{h}{v}\right)^{3}+\ldots\right]$
(25)

The mass matrices ${\cal M}_{q}$ are related to the Yukawa matrices in (16)
through

$\displaystyle{\cal M}_{u}=\frac{v}{\sqrt{2}}Y_{u}s_{\beta}\,,\qquad{\cal
M}_{d}=\frac{v}{\sqrt{2}}Y_{d}c_{\beta}$ (26)

The expressions for the charged leptons, ${\cal M}_{e}$, ${\cal M}^{(n)}_{e}$,
are similar to those for the down-quark case.

Our method reproduces the results of [24] and gives several new expressions,
the cubic coefficient of $F_{U}$, the coefficient of $h^{5}$ in $V(h)$ and the
fermionic couplings. More generally, the procedure summarized at the beginning
of Section 3, together with the all-orders expression for $H_{0}(h)$ in
Appendix A, defines an algorithm to extend the tree-level matching to all
orders in $h$.

### Other Yukawa interactions

Besides the type-II Yukawa interactions discussed above, there are three other
possibilities without tree-level flavour changing neutral currents.
Conventionally, these are given by

  * •

Type-I

$\mathcal{L}=-\bar{q}_{L}Y_{d}S_{2}P_{-}q_{R}-\bar{q}_{L}Y_{u}S_{2}P_{+}q_{R}-\bar{l}_{L}Y_{e}S_{2}P_{-}l_{R}$
(27)

  * •

Type-X (lepton-specific)

$\mathcal{L}=-\bar{q}_{L}Y_{d}S_{2}P_{-}q_{R}-\bar{q}_{L}Y_{u}S_{2}P_{+}q_{R}-\bar{l}_{L}Y_{e}S_{1}P_{-}l_{R}$
(28)

  * •

Type-Y (flipped)

$\mathcal{L}=-\bar{q}_{L}Y_{d}S_{1}P_{-}q_{R}-\bar{q}_{L}Y_{u}S_{2}P_{+}q_{R}-\bar{l}_{L}Y_{e}S_{2}P_{-}l_{R}$
(29)

Using the results of the matching for the type-II 2HDM, it is straightforward
to find the matching for the other Yukawa structures. For example, in the
type-I model, all terms depend only on $S_{2}$, so the matching will have the
same form as the up-type terms of the type-II 2HDM. As a result, we find for
the type-I 2HDM

$\displaystyle\mathcal{M}_{u,d,e}+\sum_{n=1}^{\infty}\mathcal{M}_{u,d,e}^{(n)}\bigg{(}\frac{h}{v}\bigg{)}^{n}=$
$\displaystyle\mathcal{M}_{u,d,e}\bigg{[}1+\frac{c_{\alpha}}{s_{\beta}}\frac{h}{v}-\frac{c_{\beta-\alpha}}{2}\frac{s_{\alpha}^{2}c_{\alpha}}{s_{\beta}^{2}c_{\beta}}\bigg{(}\frac{h}{v}\bigg{)}^{2}-\frac{c_{\beta-\alpha}}{6}\frac{s_{2\alpha}^{2}}{s_{2\beta}^{2}}(2s_{2\alpha}-(1-2c_{2\alpha})t_{\beta}^{-1})\bigg{(}\frac{h}{v}\bigg{)}^{3}+\dots\bigg{]}$
(30)

with $\mathcal{M}_{u,d,e}=vY_{u,d,e}s_{\beta}/\sqrt{2}$. It is straightforward
to obtain similar expressions for the type-X and type-Y models.

##  4 Nondecoupling effects at one loop

The procedure of integrating out the heavy scalars can be extended to one loop
using functional methods [27]. The most important effects at this order are
the local operators inducing $h\to\gamma\gamma$ and $h\to Z\gamma$
transitions, because they are loop suppressed in the SM. The EFT corrections
are then at the same loop order as the leading contributions. In the 2HDM, the
contributions to $h\to\gamma\gamma$ and $h\to Z\gamma$ due to the heavy sector
come from charged scalars $H^{\pm}$ within the loop, or equivalently from the
real fields $\rho_{1,2}$ in (3). To obtain the one-loop contributions with
internal $\rho_{i}$ from functional integration, the Lagrangian in (2) has to
be expanded to quadratic order in these fields. The quadratic piece takes the
form

$\displaystyle{\cal
L}^{(2)}_{\rho_{1,2}}=\frac{1}{2}\rho_{i}\Delta_{ij}\rho_{j}\,,\qquad\Delta=-D^{2}-M^{2}_{H}-\hat{Y}$
(31)

where

$\displaystyle
D^{\mu}_{ij}=\partial^{\mu}\,\delta_{ij}+X^{\mu}\varepsilon_{ij}\equiv(\partial^{\mu}+\hat{X}^{\mu})_{ij}\,,\qquad\hat{Y}_{ij}=Y\,\delta_{ij}$
(32)

with $i,j\in\\{1,2\\}$, $\varepsilon_{ij}$ the two-dimensional Levi-Civita
symbol, and

$\displaystyle X^{\mu}$
$\displaystyle=eA^{\mu}+\frac{g}{2c_{W}}(1-2s^{2}_{W})Z^{\mu}$
$\displaystyle-Y$ $\displaystyle=d_{5}h+d_{7}H+z_{6}h^{2}+z_{7}hH+z_{8}H^{2}$
(33)

Here $e$ is the electromagnetic coupling, $s_{W}=\sin\theta_{W}$,
$c_{W}=\cos\theta_{W}$ with the Weinberg angle $\theta_{W}$, $A$ the photon
and $Z$ the $Z$-boson field. Performing the Gaussian integration of
$\exp(i\int d^{4}x\,{\cal L}^{(2)}_{\rho_{12}})$ over $\rho_{i}$ gives the
effective Lagrangian [27]

$\displaystyle{\cal L}_{\rm
eff}=-\frac{i}{2}\sum_{n=1}^{\infty}\frac{1}{n}\int\frac{d^{4}p}{(2\pi)^{4}}\,\Big{\langle}\left(\frac{2ip\cdot
D+D^{2}+\hat{Y}}{p^{2}-M^{2}_{H}}\right)^{n}\Big{\rangle}$ (34)

In a weakly-coupled model of the heavy sector, a generic matrix $\hat{Y}$
scales at most with the first power of $M_{H}$. The series in (34) will then
converge and only a finite number of terms will contribute to any given order
in the $1/M_{H}$ expansion. By contrast, in the present nondecoupling scenario
we have $\hat{Y}\sim M^{2}_{H}$, which is of the same order as the denominator
$p^{2}-M^{2}_{H}$. Therefore, an infinite number of terms in the sum over $n$
contributes at a given order in the $1/M_{H}$ expansion. However, higher
powers of $\hat{Y}$ come with higher powers of $h$, since $\hat{Y}^{n}={\cal
O}(h^{n})$ in the field expansion. As a consequence, the infinite series
generates a Higgs-function $F_{O}(h)$ that accompanies an EFT operator $O$, as
it is characteristic for the Higgs-electroweak chiral Lagrangian. At any given
order in $h^{n}$, the operator coefficient is well-defined and calculable.

Following this reasoning, we can extract the terms of interest here from (34).
These contain two factors of the field strength $\hat{X}_{\mu\nu}$,

$\displaystyle\hat{X}^{\mu\nu}_{ij}\equiv\left[D^{\mu},D^{\nu}\right]_{ij}=X^{\mu\nu}\varepsilon_{ij}$
(35)

corresponding to four covariant derivatives $D$, along with powers of
$\hat{Y}$. Neglecting contributions with three or more powers of $h$, we need
to include terms of order $\hat{Y}$ and $\hat{Y}^{2}$. The result is given by

$\displaystyle 32\pi^{2}\,{\cal L}_{\rm
eff}=-\frac{1}{12M^{2}_{H}}\langle\hat{Y}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}\rangle+\frac{1}{40M^{4}_{H}}\langle\hat{Y}^{2}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}\rangle+\frac{1}{60M^{4}_{H}}\langle(\hat{Y}\hat{X}_{\mu\nu})^{2}\rangle$
(36)

which simplifies to

$\displaystyle{\cal L}_{\rm eff}\equiv{\cal
L}_{X,4}=\frac{X_{\mu\nu}X^{\mu\nu}}{192\pi^{2}}\,\left[\frac{Y}{M^{2}_{H}}-\frac{Y^{2}}{2M^{4}_{H}}+{\cal
O}(h^{3})\right]$ (37)

Using (4) and eliminating $H$ in favour of $h$, we obtain

$\displaystyle{\cal L}_{X,4}$
$\displaystyle=\frac{e^{2}}{16\pi^{2}}\left(A_{\mu\nu}A^{\mu\nu}+\frac{1-2s^{2}_{W}}{s_{W}c_{W}}A_{\mu\nu}Z^{\mu\nu}\right)\,F_{X}(h)$
(38) $\displaystyle F_{X}(h)$
$\displaystyle=\frac{s_{\beta-\alpha}}{6}\frac{h}{v}-\frac{1}{12}\left(s^{2}_{\beta-\alpha}+\frac{s_{2\alpha}}{s_{2\beta}}c^{2}_{\beta-\alpha}\right)\left(\frac{h}{v}\right)^{2}+{\cal
O}(h^{3})$ (39)

with $A_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$,
$Z_{\mu\nu}=\partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu}$. We note that the
field redefinition of $h$, needed to make its kinetic term canonically
normalized, plays no role for $F_{X}$ through order $h^{2}$.

The first term $\sim h$ in $F_{X}(h)$ agrees with the result of [24], the term
$\sim h^{2}$ is new. Employing the procedure described above, it is
straightforward to extend the calculation of $F_{X}$ to higher orders in $h$.
As discussed in Appendix B, in the alignment limit $c_{\beta-\alpha}=0$ the
function $F_{X}$, to all orders in $h$, takes the simple form

$\displaystyle F_{X}(h)=\frac{1}{6}\ln\left(1+\frac{h}{v}\right)$ (40)

corresponding to the well-known low-energy theorems [28].

##  5 Parameter space and the decoupling limit

For the construction of a low-energy EFT, we consider the phenomenologically
viable scenario where the masses of the BSM scalar degrees of freedom in the
2HDM are taken to be much larger than the electroweak scale, i.e.

$M_{S}\sim M_{0},M_{H},M_{A}\gg m_{h}\sim v$ (41)

Depending on the numerical values of the parameters, we can discern two basic
scenarios, corresponding to weak and strong coupling, respectively. They are
given by

  * •

Nondecoupling regime111Although not stated explicitly, this limit was used to
derive nondecoupling effects in [24]. (strong coupling, nonlinear EFT)

$1\ll|\lambda_{i}|\lesssim 16\pi^{2},\quad m_{h}\sim v\sim\overline{m}\ll
M_{S}\quad\Longrightarrow\quad c_{\beta-\alpha}=\mathcal{O}(1)$ (42)

While $c_{\beta-\alpha}$ is a priori unconstrained in this regime, we will
also consider the case $c_{\beta-\alpha}\ll 1$, referred to as the
nondecoupling regime with (quasi-)alignment. We also note that the model with
$\overline{m}=0$, the $\mathbb{Z}_{2}$ symmetric 2HDM without soft breaking,
has no decoupling limit [29, 30, 31, 32].

  * •

Decoupling regime222This limit has been studied extensively in [29]. The model
we consider in this work is simpler because of an additional, softly broken,
$\mathbb{Z}_{2}$ symmetry $S_{1}\rightarrow-S_{1}$. (weak coupling, linear
EFT)

$\lambda_{i}=\mathcal{O}(1),\quad m_{h}\sim v\ll\overline{m}\sim
M_{S}\quad\Longrightarrow\quad c_{\beta-\alpha}\ll 1$ (43)

In the strong-coupling case, we require the $\lambda_{i}$ to be somewhat below
the nominal strong-coupling limit $M_{S}\approx 4\pi v$ corresponding to
$|\lambda_{i}|\approx 16\pi^{2}$. Otherwise a description of the heavy scalar
dynamics in terms of resonances would no longer be valid. To be more precise,
the magnitude of the couplings is constrained by perturbative unitarity [33,
34, 35, 36, 37, 38, 39]. For loop corrections to the constraints, see [40, 41,
42]. Generally speaking, these give much stronger bounds, namely
$|\lambda_{i}|\lesssim 4\pi$. Furthermore, the couplings are constrained such
that the potential is bounded from below and that the symmetry breaking vacuum
is the global minimum of the potential. For the 2HDM with (softly broken)
$\mathbb{Z}_{2}$ symmetry, the necessary and sufficient conditions on the
couplings read [43, 44, 45, 46, 47]

$\lambda_{1}\geq 0\,,\quad\lambda_{2}\geq
0\,,\quad\lambda_{3}\geq-\sqrt{\lambda_{1}\lambda_{2}}\,,\quad\lambda_{3}+\lambda_{4}-|\lambda_{5}|\geq-\sqrt{\lambda_{1}\lambda_{2}}$
(44)

To satisfy these bounds, the absolute values of the couplings have to be taken
large uniformly, which limits the possible mass splitting between the heavy
scalars. Especially the perturbative unitarity constraints severely restrict
the possible parameter space of the nondecoupling regime. Nevertheless, masses
of $M_{S}\lesssim 1$ TeV are still possible for $\overline{m}\sim v$, which
clearly fulfills the power counting of the nondecoupling scenario.

In the decoupling regime, all NP effects are suppressed by powers of the heavy
mass scale $M_{S}$ as formalised by the Appelquist-Carazzone decoupling
theorem [48]. Several EFT matching calculations have been performed in the
decoupling limit, see e.g. [23, 49, 50]. A decoupling regime automatically
implies the alignment limit $c_{\beta-\alpha}=0$, where the $h$-couplings
approach their SM values [29]. An explicit calculation gives

$c_{\beta-\alpha}^{2}=\frac{v^{4}}{16\overline{m}^{4}}s_{2\beta}^{2}\big{[}\lambda_{1}-\lambda_{2}+c_{2\beta}(\lambda_{1}+\lambda_{2}-2\lambda_{345})\big{]}^{2}+\mathcal{O}(v^{6}/\overline{m}^{6})$
(45)

with $\lambda_{345}\equiv\lambda_{3}+\lambda_{4}+\lambda_{5}$. When
$\overline{m}\gg v$, this indeed approaches zero. As mentioned above, there is
no similar relation in the nondecoupling regime, and thus, $c_{\beta-\alpha}$
is unconstrained a priori.

To illustrate the two regimes discussed above, we take the
$hH_{0}^{2}$-coupling $d_{3}$, given in Appendix C, as an example. In the
nondecoupling regime, $M_{0}\sim M_{S}\gg m_{h},\overline{m}$, so
$d_{3}=\mathcal{O}(M_{S}^{2})$, whereas in the decoupling regime, the masses
and parameters of the model scale as

$M_{0}^{2},\,M^{2}_{H},\,M^{2}_{A}=\overline{m}^{2}+\mathcal{O}(v^{2})\,,\qquad
m_{h}^{2}=\mathcal{O}(v^{2})\,,\qquad
c_{\beta-\alpha}=\mathcal{O}(v^{2}/\overline{m}^{2})$ (46)

leading to

$d_{3}=-\frac{m_{h}^{2}}{2v}-vs_{\beta}^{2}c_{\beta}^{2}(\lambda_{1}+\lambda_{2}-2\lambda_{345})+\mathcal{O}(v^{3}/\overline{m}^{2})$
(47)

Evidently, all heavy mass dependence has cancelled. Similar calculations show
that this cancellation works for all $d_{i}$ and $z_{i}$. It is now easy to
see that all nondecoupling effects vanish in the decoupling-regime. Obviously,
all tree-level nondecoupling effects vanish in the decoupling limit, since
they are all proportional to $c_{\beta-\alpha}$. Also the anomalous
$h\gamma\gamma$\- and $hZZ$-couplings disappear as the ratios
$d_{i}/M^{2}_{S}$ and $z_{i}/M_{S}^{2}$ go to zero in the limit
$M_{S}\to\infty$.

##  6 Phenomenological considerations

The simplest way to confront the nondecoupling effects of the 2HDM with
experiment is by using a global HEFT fit. Such a fit has been performed using
LHC run 1 and 2 data [51], where the authors fit the couplings of the HEFT
Lagrangian in the form

$\displaystyle\mathcal{L}_{\text{fit}}$
$\displaystyle=2c_{V}(m_{W}W_{\mu}^{+}W^{-\mu}+\tfrac{1}{2}m_{Z}^{2}Z_{\mu}Z^{\mu})\frac{h}{v}-\sum_{\psi}c_{\psi}m_{\psi}\bar{\psi}\psi\frac{h}{v}$
$\displaystyle+\frac{e^{2}}{16\pi^{2}}c_{\gamma}A_{\mu\nu}A^{\mu\nu}\frac{h}{v}+\frac{e^{2}}{16\pi^{2}}c_{\gamma
Z}A_{\mu\nu}Z^{\mu\nu}\frac{h}{v}+\frac{g_{s}^{2}}{16\pi^{2}}c_{g}\langle
G_{\mu\nu}G^{\mu\nu}\rangle\frac{h}{v}$ (48)

with $\psi\in\\{t,b,c,\tau,\mu\\}$333The couplings to the lighter fermions are
so small that they are not included in the fit. . Our matching results are
given in Table 1.

Tree Level | Loop Level
---|---
$c_{V}$ | $s_{\beta-\alpha}$ | $c_{\gamma}$ | $\frac{s_{\beta-\alpha}}{6}$
$c_{u}$ | $s_{\beta-\alpha}+c_{\beta-\alpha}t^{-1}_{\beta}$ | $c_{\gamma Z}$ | $\frac{1-2s_{W}^{2}}{s_{W}c_{W}}\frac{s_{\beta-\alpha}}{6}$
$c_{d}$ | $s_{\beta-\alpha}-c_{\beta-\alpha}t_{\beta}$ | $c_{g}$ | $0$
Table 1: LO matching results for the 2HDM. $c_{u}$ is the same for all up-type
quarks ($u,c,t$) and $c_{d}$ is the same for all down-type quarks ($d,s,b$)
and charged leptons ($e,\mu,\tau$).

The strongest constraint is derived from the Higgs–vector boson coupling

$c_{V}=1.01\pm 0.06\quad\Longrightarrow\quad s_{\beta-\alpha}\gtrsim 0.95$
(49)

where the given error corresponds to the $68\%$ probability interval. This
motivates the (quasi-)alignment limit as it constrains $c_{\beta-\alpha}\ll
1$.

Applying the above bound to the anomalous Higgs–photon coupling, we find

$c_{\gamma}\in[0.16,0.17]$ (50)

This coupling is particularly important, as it is bounded from below in the
alignment limit. We see here that this is a direct consequence of the bound on
the Higgs–vector boson coupling. From the global HEFT fit, the bound on
$c_{\gamma}$ is given by

$c_{\gamma}=0.05\pm 0.20$ (51)

which is consistent with the matching prediction. Nevertheless, with more data
from the LHC, it is plausible that the limits on $c_{\gamma}$ could be
sufficiently improved to exclude the nondecoupling regime experimentally.
Local couplings with more than one Higgs in (38) and (39), such as
$h^{2}A_{\mu\nu}A^{\mu\nu}$, could in principle be probed at a photon collider
[52] in a process like $\gamma\gamma\to hh$.

Aside from using an EFT approach, there is a large amount of literature using
global fits for the 2HDM directly. Depending on the structure of the Yukawa
interactions, these fits can give much stronger bounds on $s_{\beta-\alpha}$
than the global HEFT fit (see e.g. [53, 54]). However, a detailed analysis
lies beyond the scope of this work.

##  7 Conclusions

We presented a systematic derivation of the EFT at the electroweak scale for
the 2HDM in the nondecoupling regime. In this regime, the EFT takes the form
of an electroweak chiral Lagrangian (nonlinear EFT). Our discussion follows
closely the detailed discussion given in [25] for the nondecoupling regime of
the SM extension with a heavy scalar singlet. The scalar sector of the 2HDM is
written in polar coordinates, with a nonlinear representation of the Goldstone
fields, which facilitates the use of functional methods that we employ
throughout. An algorithmic procedure is given, by which the LO EFT Lagrangian
can be worked out to arbitrary order in the Higgs field $h$. We confirm
previous results for the EFT Higgs couplings and extend the derivation to
additional terms. The main results are displayed in (3) – (25) and (38), (39).
Some all-orders expressions are given in closed form (Appendix A and B). We
derive the LO EFT Lagrangian, including the fermionic Yukawa interactions and
the loop-induced local terms for $h\to\gamma\gamma$ and $h\to Z\gamma$. As
already pointed out in [24], the latter terms have interesting nondecoupling
effects that survive in the alignment limit. Those are still compatible with
present data. They could be discovered or ruled out in future measurements of
anomalous Higgs-boson couplings.

## Note added

While this paper was being finalized, the article [55] appeared on arXiv. It
also addresses the HEFT matching of models with extended scalar sectors and
partially overlaps with our results on the 2HDM.

## Acknowledgements

This work is supported in part by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy – EXC-2094 –
390783311.

##  Appendix A Exact solution for $H_{0}(h)$

The LO term for $H$ is calculated from the equations of motion at
$\mathcal{O}(M^{2}_{S})$. In particular, we can set $H^{\pm}=A=0$ in this
approximation. As a result, retaining only $\mathcal{O}(M^{2}_{S})$ terms the
Lagrangian simplifies to

$\mathcal{L}_{M}=-m_{11}^{2}\phi_{1}^{2}-m_{22}^{2}\phi_{2}^{2}-\frac{\lambda_{1}}{2}\phi_{1}^{4}-\frac{\lambda_{2}}{2}\phi_{2}^{4}-\lambda_{345}\phi_{1}^{2}\phi_{2}^{2}$
(A.1)

where $\phi_{n}^{2}\equiv(v_{n}+h_{n})^{2}/2$ and

$\displaystyle\lambda_{1}=\frac{M^{2}_{0}}{v^{2}}\frac{c_{\alpha}^{2}}{c_{\beta}^{2}}\,,\qquad\lambda_{2}=\frac{M^{2}_{0}}{v^{2}}\frac{s_{\alpha}^{2}}{s_{\beta}^{2}}\,,\qquad\lambda_{345}\equiv\lambda_{3}+\lambda_{4}+\lambda_{5}=\frac{M^{2}_{0}}{v^{2}}\frac{s_{\alpha}c_{\alpha}}{s_{\beta}c_{\beta}}$
$\displaystyle
m_{11}^{2}=-\frac{M^{2}_{0}}{2}\bigg{(}c_{\alpha}^{2}+\frac{s_{\alpha}c_{\alpha}s_{\beta}}{c_{\beta}}\bigg{)}\,,\qquad
m_{22}^{2}=-\frac{M^{2}_{0}}{2}\bigg{(}s_{\alpha}^{2}+\frac{s_{\alpha}c_{\alpha}c_{\beta}}{s_{\beta}}\bigg{)}$
(A.2)

After expressing $h_{1}$ and $h_{2}$ through $h$ and $H$, the two fields take
the form

$\displaystyle\phi_{1}$
$\displaystyle=\frac{1}{\sqrt{2}}(c_{\beta}v+c_{\alpha}H-s_{\alpha}h)$
$\displaystyle\phi_{2}$
$\displaystyle=\frac{1}{\sqrt{2}}(s_{\beta}v+s_{\alpha}H+c_{\alpha}h)$ (A.3)

By defining the combination

$R^{2}=\frac{c_{\alpha}}{c_{\beta}}\phi_{1}^{2}+\frac{s_{\alpha}}{s_{\beta}}\phi_{2}^{2}$
(A.4)

the Lagrangian (A.1) can be rewritten as

$\mathcal{L}_{M}=\frac{M^{2}_{0}}{2}(s_{\alpha}s_{\beta}+c_{\alpha}c_{\beta})R^{2}-\frac{M^{2}_{0}}{2v^{2}}R^{4}$
(A.5)

At this point we can solve the eom by analogy to the heavy singlet model
studied in [25]. By direct comparison to the results in the Appendix of [25],
we can identify

$\phi_{1}\rightarrow S\,,\quad\phi_{2}\rightarrow\phi\,,\quad v\rightarrow
f\,,\quad s_{\beta}\rightarrow\sqrt{\xi}\,,\quad
s_{\alpha}\rightarrow\sqrt{\omega}\,,\quad M_{0}\rightarrow M$ (A.6)

which exactly reproduces the corresponding terms of the heavy singlet model.
The solution to the eom is then given by

$H_{0}(h)=\frac{v+\Big{(}\frac{s_{\alpha}^{2}c_{\alpha}}{s_{\beta}}-\frac{s_{\alpha}c_{\alpha}^{2}}{c_{\beta}}\Big{)}h}{\frac{s_{\alpha}^{3}}{s_{\beta}}+\frac{c_{\alpha}^{3}}{c_{\beta}}}\left[\sqrt{1-\frac{\Big{(}\frac{s_{\alpha}^{3}}{s_{\beta}}+\frac{c_{\alpha}^{3}}{c_{\beta}}\Big{)}\Big{(}\frac{s_{\alpha}c_{\alpha}^{2}}{s_{\beta}}+\frac{s_{\alpha}^{2}c_{\alpha}}{c_{\beta}}\Big{)}h^{2}}{\Big{(}v+\Big{(}\frac{s_{\alpha}^{2}c_{\alpha}}{s_{\beta}}-\frac{s_{\alpha}c_{\alpha}^{2}}{c_{\beta}}\Big{)}h\Big{)}^{2}}}-1\right]$
(A.7)

This expression fulfills

$R^{2}=\frac{v^{2}}{2}(s_{\alpha}s_{\beta}+c_{\alpha}c_{\beta})=\frac{v^{2}}{2}c_{\beta-\alpha}$
(A.8)

which, when inserted back into the Lagrangian (A.5), shows that the
$\mathcal{O}(M^{2}_{S})$-terms cancel up to a constant. Furthermore, the
solution starts at $\mathcal{O}(h^{2})$ with coefficients that are functions
of $s_{\alpha}$, $c_{\alpha}$, $s_{\beta}$ and $c_{\beta}$. Note that the
combination

$\frac{s_{\alpha}c_{\alpha}^{2}}{s_{\beta}}+\frac{s_{\alpha}^{2}c_{\alpha}}{c_{\beta}}=-c_{\beta-\alpha}[1-2c_{\beta-\alpha}^{2}+2c_{\beta-\alpha}s_{\beta-\alpha}\cot{(2\beta)}]$
(A.9)

vanishes in the alignment limit. Then, the square root in (A.7) reduces to 1,
which gives $H_{0}(h)=0$. As a result, all tree level nondecoupling effects
vanish in the alignment limit.

##  Appendix B One-loop matching of $h^{n}X_{\mu\nu}X^{\mu\nu}$ to all orders
in $n$

When calculating the one-loop EFT contributions of the form
$h^{n}X_{\mu\nu}X^{\mu\nu}$, we noted that, since
$\hat{Y}=\mathcal{O}(M_{H}^{2})$, the series does not converge. Therefore, in
order to calculate the full Higgs function associated with the operator
$X_{\mu\nu}X^{\mu\nu}$, we need all coefficients $C_{n}$ of the expression

$\mathcal{L}_{\text{eff}}\supset\sum_{n=1}^{\infty}C_{n}\langle\hat{Y}^{n}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}\rangle$
(B.1)

In writing the above, we made essential use of the fact that
$\hat{Y}\propto\boldsymbol{1}$ and thus commutes with $\hat{X}^{\mu\nu}$. This
is, however, a special case. In general, $\hat{Y}$ does not commute, giving
more possible operator structures for each $n$.

To derive an all-orders result for the $C_{n}$, we start from the general
expression for the one-loop effective Lagrangian given in (34). We now use a
slightly adapted form of a trick explained in the Appendix of [56]: We
evaluate expression (34) in the special configuration
$\partial_{\mu}\hat{X}_{\nu}=\partial_{\mu}\hat{Y}=0$, allowing us to drop all
derivatives. In this case

$D_{\mu}\hat{G}\rightarrow[\hat{X}_{\mu},\hat{G}]\,,\qquad\hat{X}_{\mu\nu}\rightarrow[\hat{X}_{\mu},\hat{X}_{\nu}]$
(B.2)

where $\hat{G}$ is any matrix valued function of $\hat{Y}$ and
$\hat{X}_{\mu}$. In the final expressions, we can express everything through
$D_{\mu}$ and $\hat{X}_{\mu\nu}$, regaining the general result.

In our special case, we are only interested in the terms of the form
$\hat{Y}^{n}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}$, which contribute to the
$h^{n}\gamma\gamma$ nondecoupling effects. Setting $[\hat{X}_{\mu},\hat{Y}]=0$
automatically removes all terms of the form $D_{\mu}\hat{Y}$.

In this way, it is easy to evaluate all terms from (34) with 4 derivatives (4
factors of $X_{\mu}$) and $n$ factors of $Y$, which reduce to the terms of
interest due to the formal gauge invariance of the functional integral.
Finally we obtain

$\displaystyle\mathcal{L}_{\text{eff}}=\frac{1}{32\pi^{2}}\sum_{n=1}^{\infty}\frac{(-1)^{n}}{12nM_{H}^{2n}}\langle\hat{Y}^{n}\hat{X}_{\mu\nu}\hat{X}^{\mu\nu}\rangle=\frac{X_{\mu\nu}X^{\mu\nu}}{192\pi^{2}}\ln\left(1+\frac{Y(h)}{M^{2}_{H}}\right)$
(B.3)

The first two terms in the sum over $n$ agree with those given in [27] and the
third and fourth terms agree with those given in [57], in the special case
that $\hat{Y}$ and $\hat{X}_{\mu\nu}$ commute. The second expression in (B.3)
represents the full Higgs function after taking

$\displaystyle
Y(h)=-d^{(0)}_{5}h-d^{(0)}_{7}H_{0}(h)-z^{(0)}_{6}h^{2}-z^{(0)}_{7}hH_{0}(h)-z^{(0)}_{8}H_{0}(h)^{2}$
(B.4)

and expressing $h$ in terms of the canonically normalized Higgs field
$\tilde{h}$, $h=h(\tilde{h})$, through the inverse of the function
$\tilde{h}(h)$ defined in (20). Here $d^{(0)}_{5}$ is the ${\cal
O}(M^{2}_{S})$-part of $d_{5}$, and similarly for the other coefficients, and
$H_{0}(h)$ is the function derived in Appendix A.

In the alignment limit $c_{\beta-\alpha}=0$, the Lagrangian in (B.3) can be
given in closed form to all orders in $h$. In this case, $H_{0}(h)$ vanishes
and $Y$ in (4) reduces to $Y/M^{2}_{H}=2h/v+(h/v)^{2}$. Then (B.3) takes the
form of (38) with the function $F_{X}(h)$ given by (40).

##  Appendix C Parameters of the 2HDM potential

The full scalar potential is given in (2). In this section we give all
coefficients of the potential in terms of the input parameters

$\displaystyle v,\quad m_{h},\quad M_{0},\quad M_{H},\quad
M_{A},\quad\overline{m},\quad t_{\beta},\quad c_{\beta-\alpha}$ (C.1)

The cubic couplings read

$\displaystyle v\,d_{1}$
$\displaystyle=c_{\beta-\alpha}^{2}\left(\overline{m}^{2}-m_{h}^{2}\right)\left(s_{\beta-\alpha}+c_{\beta-\alpha}\cot(2\beta)\right)-\frac{m_{h}^{2}}{2}s_{\beta-\alpha}$
(C.2) $\displaystyle v\,d_{2}$
$\displaystyle=\frac{c_{\beta-\alpha}}{2}\left[\left(2m_{h}^{2}+M_{0}^{2}-3\overline{m}^{2}\right)\left(1-2c_{\beta-\alpha}^{2}+2s_{\beta-\alpha}c_{\beta-\alpha}\cot(2\beta)\right)-\overline{m}^{2}\right]$
(C.3) $\displaystyle v\,d_{3}$
$\displaystyle=-\frac{s_{\beta-\alpha}}{2}\left[\left(2M_{0}^{2}+m_{h}^{2}-3\overline{m}^{2}\right)\left(1-2c_{\beta-\alpha}^{2}+2s_{\beta-\alpha}c_{\beta-\alpha}\cot(2\beta)\right)+\overline{m}^{2}\right]$
(C.4) $\displaystyle v\,d_{4}$
$\displaystyle=s_{\beta-\alpha}^{2}\left(\overline{m}^{2}-M_{0}^{2}\right)\left(c_{\beta-\alpha}-s_{\beta-\alpha}\cot(2\beta)\right)-\frac{M_{0}^{2}}{2}c_{\beta-\alpha}$
(C.5) $\displaystyle v\,d_{5}$
$\displaystyle=2s_{\beta-\alpha}\left(\overline{m}^{2}-M_{H}^{2}-\frac{m_{h}^{2}}{2}\right)+2c_{\beta-\alpha}\cot(2\beta)\left(\overline{m}^{2}-m_{h}^{2}\right)$
(C.6) $\displaystyle v\,d_{6}$
$\displaystyle=s_{\beta-\alpha}\left(\overline{m}^{2}-M_{A}^{2}-\frac{m_{h}^{2}}{2}\right)+c_{\beta-\alpha}\cot(2\beta)\left(\overline{m}^{2}-m_{h}^{2}\right)$
(C.7) $\displaystyle v\,d_{7}$
$\displaystyle=2c_{\beta-\alpha}\left(\overline{m}^{2}-M_{H}^{2}-\frac{M_{0}^{2}}{2}\right)+2s_{\beta-\alpha}\cot(2\beta)\left(M_{0}^{2}-\overline{m}^{2}\right)$
(C.8) $\displaystyle v\,d_{8}$
$\displaystyle=c_{\beta-\alpha}\left(\overline{m}^{2}-M_{A}^{2}-\frac{M_{0}^{2}}{2}\right)+s_{\beta-\alpha}\cot(2\beta)\left(M_{0}^{2}-\overline{m}^{2}\right)$
(C.9)

and the quartic couplings are given by

$\displaystyle v^{2}z_{1}=$
$\displaystyle-\frac{m_{h}^{2}}{8}+\frac{c_{\beta-\alpha}^{2}}{8}\left[4s_{\beta-\alpha}^{2}\overline{m}^{2}+(-3+4c_{\beta-\alpha}^{4})m_{h}^{2}-\left(1-2c_{\beta-\alpha}^{2}\right)^{2}M_{0}^{2}\right.$
$\displaystyle+4c_{\beta-\alpha}s_{\beta-\alpha}\cot(2\beta)\left(2\overline{m}^{2}-\left(1+2c_{\beta-\alpha}^{2}\right)m_{h}^{2}-\left(1-2c_{\beta-\alpha}^{2}\right)M_{0}^{2}\right)$
$\displaystyle\left.+4c_{\beta-\alpha}^{2}\cot^{2}(2\beta)\left(\overline{m}^{2}-c_{\beta-\alpha}^{2}m_{h}^{2}-s_{\beta-\alpha}^{2}M_{0}^{2}\right)\right]$
(C.10) $\displaystyle v^{2}z_{2}=$
$\displaystyle\frac{s_{\beta-\alpha}c_{\beta-\alpha}}{2}\left(1-2c_{\beta-\alpha}^{2}\right)\left[m_{h}^{2}+M_{0}^{2}-2c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)-2\overline{m}^{2}\right]$
$\displaystyle+c_{\beta-\alpha}^{2}\cot(2\beta)\left[m_{h}^{2}\left(1+2c_{\beta-\alpha}^{2}-4c_{\beta-\alpha}^{4}\right)+2M_{0}^{2}\left(1-3c_{\beta-\alpha}^{2}+2c_{\beta-\alpha}^{4}\right)\right.$
$\displaystyle+\left.\overline{m}^{2}\left(-3+4c_{\beta-\alpha}^{2}\right)\right]+2c_{\beta-\alpha}^{3}s_{\beta-\alpha}\cot^{2}(2\beta)\left[M_{0}^{2}-\overline{m}^{2}-c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right]$
(C.11) $\displaystyle v^{2}z_{3}=$
$\displaystyle\frac{1}{4}\left[\left(2-12c_{\beta-\alpha}^{2}+12c_{\beta-\alpha}^{4}\right)\overline{m}^{2}\right.$
$\displaystyle+\left(1-2c_{\beta-\alpha}^{2}\right)\left(\left(-1-3c_{\beta-\alpha}^{2}+6c_{\beta-\alpha}^{4}\right)m_{h}^{2}+\left(-2+9c_{\beta-\alpha}^{2}-6c_{\beta-\alpha}^{4}\right)M_{0}^{2}\right)$
$\displaystyle+2c_{\beta-\alpha}s_{\beta-\alpha}\cot(2\beta)\left(\left(6-12c_{\beta-\alpha}^{2}\right)\overline{m}^{2}+\left(-1-6c_{\beta-\alpha}^{2}+12c_{\beta-\alpha}^{4}\right)m_{h}^{2}\right.$
$\displaystyle\left.+\left(-5+18c_{\beta-\alpha}^{2}-12c_{\beta-\alpha}^{4}\right)M_{0}^{2}\right)$
$\displaystyle\left.+12c_{\beta-\alpha}^{2}s_{\beta-\alpha}^{2}\cot^{2}(2\beta)\left(\overline{m}^{2}-s_{\beta-\alpha}^{2}M_{0}^{2}-c_{\beta-\alpha}^{2}m_{h}^{2}\right)\right]$
(C.12) $\displaystyle v^{2}z_{4}=$
$\displaystyle\frac{s_{\beta-\alpha}c_{\beta-\alpha}}{2}\left(1-2c_{\beta-\alpha}^{2}\right)\left[m_{h}^{2}-3M_{0}^{2}+2c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)+2\overline{m}^{2}\right]$
$\displaystyle+s_{\beta-\alpha}^{2}\cot(2\beta)\left[m_{h}^{2}\left(2c_{\beta-\alpha}^{2}-4c_{\beta-\alpha}^{4}\right)+M_{0}^{2}\left(1-6c_{\beta-\alpha}^{2}+4c_{\beta-\alpha}^{4}\right)\right.$
$\displaystyle+\left.\overline{m}^{2}\left(-1+4c_{\beta-\alpha}^{2}\right)\right]+2c_{\beta-\alpha}s_{\beta-\alpha}^{3}\cot^{2}(2\beta)\left[M_{0}^{2}-\overline{m}^{2}-c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right]$
(C.13) $\displaystyle v^{2}z_{5}=$
$\displaystyle\frac{1}{8}\left[4s_{\beta-\alpha}^{2}c_{\beta-\alpha}^{2}\overline{m}^{2}-s_{\beta-\alpha}^{2}\left(1-2c_{\beta-\alpha}^{2}\right)^{2}m_{h}^{2}-c_{\beta-\alpha}^{2}\left(3-2c_{\beta-\alpha}^{2}\right)^{2}M_{0}^{2}\right.$
$\displaystyle+4c_{\beta-\alpha}s_{\beta-\alpha}^{3}\cot(2\beta)\left(-2\overline{m}^{2}+\left(-1+2c_{\beta-\alpha}^{2}\right)m_{h}^{2}+\left(3-2c_{\beta-\alpha}^{2}\right)M_{0}^{2}\right)$
$\displaystyle\left.+4s_{\beta-\alpha}^{4}\cot^{2}(2\beta)\left(\overline{m}^{2}-M_{0}^{2}+c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)\right]$
(C.14) $\displaystyle v^{2}z_{6}=$
$\displaystyle\frac{1}{2}\left[2s_{\beta-\alpha}^{2}\left(\overline{m}^{2}-M_{H}^{2}\right)-m_{h}^{2}+c_{\beta-\alpha}^{2}\left(1-2c_{\beta-\alpha}^{2}\right)\left(M_{0}^{2}-m_{h}^{2}\right)\right.$
$\displaystyle+2c_{\beta-\alpha}s_{\beta-\alpha}\cot(2\beta)\left(2\overline{m}^{2}-m_{h}^{2}-M_{0}^{2}+3c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)$
$\displaystyle\left.+4c_{\beta-\alpha}^{2}\cot^{2}(2\beta)\left(\overline{m}^{2}-s_{\beta-\alpha}^{2}M_{0}^{2}-c_{\beta-\alpha}^{2}m_{h}^{2}\right)\right]$
(C.15) $\displaystyle v^{2}z_{7}=$ $\displaystyle
c_{\beta-\alpha}s_{\beta-\alpha}\left(2\overline{m}^{2}-2M_{H}^{2}-\left(1-2c_{\beta-\alpha}^{2}\right)\left(M_{0}^{2}-m_{h}^{2}\right)\right)$
$\displaystyle+$ $\displaystyle
2\cot(2\beta)\left(M_{0}^{2}-\overline{m}^{2}+2c_{\beta-\alpha}^{2}\overline{m}^{2}+c_{\beta-\alpha}^{2}M_{0}^{2}\left(-4+3c_{\beta-\alpha}^{2}\right)\right.$
$\displaystyle+$ $\displaystyle
c_{\beta-\alpha}^{2}\left.m_{h}^{2}\left(2-3c_{\beta-\alpha}^{2}\right)\right)+4c_{\beta-\alpha}s_{\beta-\alpha}\cot^{2}(2\beta)\left(M_{0}^{2}-\overline{m}^{2}-c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)$
(C.16) $\displaystyle v^{2}z_{8}=$
$\displaystyle\frac{1}{2}\left[-m_{h}^{2}+c_{\beta-\alpha}^{2}\left(2\left(\overline{m}^{2}-M_{H}^{2}\right)+\left(M_{0}^{2}-m_{h}^{2}\right)\left(-3+2c_{\beta-\alpha}^{2}\right)\right)\right.$
$\displaystyle+2c_{\beta-\alpha}s_{\beta-\alpha}\cot(2\beta)\left(-2\overline{m}^{2}-2m_{h}^{2}+4M_{0}^{2}-3c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)$
$\displaystyle\left.+4s_{\beta-\alpha}^{2}\cot^{2}(2\beta)\left(\overline{m}^{2}-M_{0}^{2}+c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)\right]$
(C.17) $\displaystyle v^{2}z_{9}=$
$\displaystyle\frac{1}{2}\left[-s_{\beta-\alpha}^{2}m_{h}^{2}-c_{\beta-\alpha}^{2}M_{0}^{2}+4\cot(2\beta)c_{\beta-\alpha}s_{\beta-\alpha}\left(M_{0}^{2}-m_{h}^{2}\right)\right.$
$\displaystyle\left.+4\cot^{2}(2\beta)\left(\overline{m}^{2}-M_{0}^{2}+c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)\right]$
(C.18) $\displaystyle v^{2}z_{10}=$
$\displaystyle\frac{1}{4}\left[2s_{\beta-\alpha}^{2}\left(\overline{m}^{2}-M_{A}^{2}\right)-m_{h}^{2}\left(1+c_{\beta-\alpha}^{2}-2c_{\beta-\alpha}^{4}\right)+c_{\beta-\alpha}^{2}M_{0}^{2}\left(1-2c_{\beta-\alpha}^{2}\right)\right.$
$\displaystyle+2c_{\beta-\alpha}s_{\beta-\alpha}\cot(2\beta)\left(2\overline{m}^{2}-m_{h}^{2}-M_{0}^{2}+3c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)$
$\displaystyle\left.+4c_{\beta-\alpha}^{2}\cot^{2}(2\beta)\left(\overline{m}^{2}-s_{\beta-\alpha}^{2}M_{0}^{2}-c_{\beta-\alpha}^{2}m_{h}^{2}\right)\right]$
(C.19) $\displaystyle v^{2}z_{11}=$
$\displaystyle\frac{1}{2}\left[c_{\beta-\alpha}s_{\beta-\alpha}\left(2\overline{m}^{2}-2M_{A}^{2}+\left(2c_{\beta-\alpha}^{2}-1\right)\left(M_{0}^{2}-m_{h}^{2}\right)\right)\right.$
$\displaystyle+$ $\displaystyle
2\cot(2\beta)\left(M_{0}^{2}-\overline{m}^{2}+2c_{\beta-\alpha}^{2}\overline{m}^{2}+c_{\beta-\alpha}^{2}M_{0}^{2}\left(-4+3c_{\beta-\alpha}^{2}\right)+c_{\beta-\alpha}^{2}m_{h}^{2}\left(2-3c_{\beta-\alpha}^{2}\right)\right)$
$\displaystyle\left.+4c_{\beta-\alpha}s_{\beta-\alpha}\cot^{2}(2\beta)\left(M_{0}^{2}-\overline{m}^{2}-c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)\right]$
(C.20) $\displaystyle v^{2}z_{12}=$
$\displaystyle\frac{1}{4}\left[-m_{h}^{2}+c_{\beta-\alpha}^{2}\left(2\overline{m}^{2}+3m_{h}^{2}-2M_{A}^{2}-3M_{0}^{2}\right)+2c_{\beta-\alpha}^{4}\left(M_{0}^{2}-m_{h}^{2}\right)\right.$
$\displaystyle+2c_{\beta-\alpha}s_{\beta-\alpha}\cot(2\beta)\left(4M_{0}^{2}-2m_{h}^{2}-2\overline{m}^{2}-3c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)$
$\displaystyle\left.+4s_{\beta-\alpha}^{2}\cot^{2}(2\beta)\left(\overline{m}^{2}-M_{0}^{2}+c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)\right]$
(C.21) $\displaystyle v^{2}z_{13}=$ $\displaystyle
4v^{2}z_{14}=\frac{1}{2}\left[-s_{\beta-\alpha}^{2}m_{h}^{2}-c_{\beta-\alpha}^{2}M_{0}^{2}+4\cot(2\beta)c_{\beta-\alpha}s_{\beta-\alpha}\left(M_{0}^{2}-m_{h}^{2}\right)\right.$
$\displaystyle\left.+4\cot^{2}(2\beta)\left(\overline{m}^{2}-M_{0}^{2}+c_{\beta-\alpha}^{2}\left(M_{0}^{2}-m_{h}^{2}\right)\right)\right]$
(C.22)

In the alignment limit ($c_{\beta-\alpha}\to 0$) the coupling constants
simplify to

$\displaystyle vd_{1}$ $\displaystyle=-\frac{m_{h}^{2}}{2},\quad
vd_{2}=0,\quad vd_{3}=\overline{m}^{2}-M_{0}^{2}-\frac{m_{h}^{2}}{2}$
$\displaystyle vd_{5}$
$\displaystyle=2\overline{m}^{2}-2M_{H}^{2}-m_{h}^{2},\quad
vd_{6}=\overline{m}^{2}-M_{A}^{2}-\frac{m_{h}^{2}}{2},$ $\displaystyle d_{4}$
$\displaystyle=\frac{1}{2}d_{7}=d_{8}=\cot(2\beta)\frac{\left(M_{0}^{2}-\overline{m}^{2}\right)}{v}$
(C.23) $\displaystyle v^{2}z_{1}$ $\displaystyle=-\frac{m_{h}^{2}}{8},\quad
v^{2}z_{2}=0,\quad
v^{2}z_{3}=\frac{1}{2}\left(\overline{m}^{2}-M_{0}^{2}-\frac{m_{h}^{2}}{2}\right),\quad
v^{2}z_{6}=\overline{m}^{2}-M_{H}^{2}-\frac{m_{h}^{2}}{2}$ $\displaystyle
z_{4}$
$\displaystyle=\frac{z_{7}}{2}=z_{11}=\cot(2\beta)\frac{\left(M_{0}^{2}-\overline{m}^{2}\right)}{v^{2}},\quad
v^{2}z_{10}=-\frac{1}{2}\left(M_{A}^{2}-\overline{m}^{2}\right)-\frac{m_{h}^{2}}{4}$
$\displaystyle 4z_{5}$
$\displaystyle=z_{8}=z_{9}=2z_{12}=z_{13}=4z_{14}=-\frac{m_{h}^{2}}{2v^{2}}-2\cot^{2}(2\beta)\frac{\left(M_{0}^{2}-\overline{m}^{2}\right)}{v^{2}}$
(C.24)

It is also useful to express the parameters of the potential in the original
form of (2) in terms of the physical parameters in (C.1). The relations read

$\displaystyle m^{2}_{11}$
$\displaystyle=s^{2}_{\beta}\overline{m}^{2}-\frac{1}{2}(c^{2}_{\alpha}M^{2}_{0}+s^{2}_{\alpha}m^{2}_{h})-\frac{s^{2}_{\beta}}{2}\frac{s_{2\alpha}}{s_{2\beta}}(M^{2}_{0}-m^{2}_{h})$
(C.25) $\displaystyle m^{2}_{22}$
$\displaystyle=c^{2}_{\beta}\overline{m}^{2}-\frac{1}{2}(s^{2}_{\alpha}M^{2}_{0}+c^{2}_{\alpha}m^{2}_{h})-\frac{c^{2}_{\beta}}{2}\frac{s_{2\alpha}}{s_{2\beta}}(M^{2}_{0}-m^{2}_{h})$
(C.26) $\displaystyle\lambda_{1}$
$\displaystyle=\frac{1}{c^{2}_{\beta}v^{2}}(c^{2}_{\alpha}M^{2}_{0}+s^{2}_{\alpha}m^{2}_{h}-s^{2}_{\beta}\overline{m}^{2})$
(C.27) $\displaystyle\lambda_{2}$
$\displaystyle=\frac{1}{s^{2}_{\beta}v^{2}}(s^{2}_{\alpha}M^{2}_{0}+c^{2}_{\alpha}m^{2}_{h}-c^{2}_{\beta}\overline{m}^{2})$
(C.28) $\displaystyle\lambda_{3}$
$\displaystyle=\frac{1}{v^{2}}\left(2M^{2}_{H}-\overline{m}^{2}+\frac{s_{2\alpha}}{s_{2\beta}}(M^{2}_{0}-m^{2}_{h})\right)$
(C.29) $\displaystyle\lambda_{4}$
$\displaystyle=\frac{1}{v^{2}}(M^{2}_{A}-2M^{2}_{H}+\overline{m}^{2})$ (C.30)
$\displaystyle\lambda_{5}$
$\displaystyle=\frac{1}{v^{2}}(\overline{m}^{2}-M^{2}_{A})$ (C.31)

The absence of a decoupling limit for $\overline{m}=0$ is obvious from these
formulas.

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acmlicensed 978-1-4503-6708-0/20/04

# A proxemics game between festival visitors and an industrial robot

Brigitte Krenn
Stephanie Gross

Bernhard Dieber
Horst Pichler

Kathrin Meyer

Austrian Research Institute for Artificial Intelligence  Vienna, Austria
<EMAIL_ADDRESS>Joanneum Research - Robotics  Klagenfurt, Austria
<EMAIL_ADDRESS>LIT Robopsychology Lab, Johannes Kepler
University  Linz, Austria<EMAIL_ADDRESS>
(2021)

###### Abstract

With increased applications of collaborative robots (cobots) in industrial
workplaces, behavioural effects of human-cobot interactions need to be further
investigated. This is of particular importance as nonverbal behaviours of
collaboration partners in human-robot teams significantly influence the
experience of the human interaction partners and the success of the
collaborative task. During the Ars Electronica 2020 Festival for Art,
Technology & Society in Linz, we invited visitors to exploratively interact
with an industrial robot, exhibiting restricted interaction capabilities:
extending and retracting its arm, depending on the movements of the volunteer.
The movements of the arm were pre-programmed and telecontrolled for safety
reasons (which was not obvious to the participants). We recorded video data of
these interactions and investigated general nonverbal behaviours of the humans
interacting with the robot, as well as nonverbal behaviours of people in the
audience. Our results showed that people were more interested in exploring the
robot’s action and perception capabilities than just reproducing the
interaction game as introduced by the instructors. We also found that the
majority of participants interacting with the robot approached it up to a
distance which would be perceived as threatening or intimidating, if it were a
human interaction partner. Regarding bystanders, we found examples where
people made movements as if trying out variants of the current participant’s
behaviour.

###### doi:

https://doi.org/10.1145/3313831.XXXXXXX

###### keywords:

non-verbal human-robot interaction; cobots; data collection experiment;
qualitative study

††conference: HRI’21,Workshop ”Exploring Applications for Autonomous Non-
Verbal Human-Robot Interactions” March 8, 2021

##  1 Introduction

Industrial robotics is a field of increasing relevance. Sensor systems of
robots are continuously improved, opening new possibilities to allow safe
collaborations with humans. In interactions with collaborative robots
(cobots), multimodal signals from the interaction partner (human or robot)
need to be identified and interpreted, giving insights in their respective
intentions. These communicative aspects are of great importance to facilitate
successful joint task completion and to foster user acceptance and optimize
the user’s experience [11], and define requirements for robot engineers to
design trustworthy and transparent robotic AI [9].

In this paper, we present a data collection experiment where we brought
together naive users with
CHIMERA111https://www.joanneum.at/en/robotics/infrastructure/mobile-
manipulation (Figure 1), a custom made sensitive robot for industrial and
academic research. CHIMERA combines an UR10 robot arm and a MiR-100 mobile
platform to an overall versatile system. UR10 is a collaborative industrial
robot arm with long reach capability, 10 stands for 10kg
payload.222https://www.universal-robots.com/products/ur10-robot/ MiR-100 is a
mobile robot platform, designed with a focus on safety, as it is able to
navigate around obstacles and people, or to stop if the system finds no
bypass.333https://www.mobile-industrial-robots.com/en/solutions/robots/mir100/
This characteristics make the two robotic components well suited for
collaborative robotics applications in industry, and CHIMERA a convenient
robot for researching close encounters of humans with cobots.

The explorative study is part of a larger research project "CoBot Studio –
Crossing Realities for Mutual Understanding in Human-Robot
Teams"444https://www.cobotstudio.org/ which researches human-robot interaction
from a collaborative point of view where the mutual ability of human and robot
to understand each other’s signals is core to the successful completion of
joint tasks, and also plays a major role for human acceptance of and trust in
the robot as a workmate. In the project, we study close collaborations of
humans with cobots employing both Virtual Reality (VR) scenarios and real
world encounters. The advantage of VR is that it allows for very close
physical proximity between human and robot, and for robot action speeds that
would be impossible near humans in reality due to safety and security
regulations. However, to ensure comparability with real world interactions,
both environments are employed in user studies where CHIMERA is used as a
comparatively safe and agile cobot.

The goal of the study presented in this paper was to collect qualitative data
on how naive users react to, interact with and interpret an apparently
interactive industrial robot. The cobot was staged at the Ars Electronica 2020
Festival for Art, Technology &
Society,555https://ars.electronica.art/keplersgardens/en/ which provided us
with a constant flow of visitors. Human and robot were to play a kind of
proxemics game where the robot in a one-to-one encounter with the human tries
to keep its arm length distance from the human. The robot arm could move into
three positions: center, forward, and backward. The setting is derived from
psychology research on proxemics, i.e., how people, typically subconsciously,
organize the space between themselves and their interaction partners [4, 5].

Figure 1: The CHIMERA robot, a combination of UR10 robot arm and MiR100 mobile
platform. Here CHIMERA is depicted in the lab carrying a box.

As CHIMERA is a custom made robot and its actions were programmed particularly
for the presented data collection experiment, we can be very sure that the
visitors had not seen or interacted with the robot before. This was exactly
what we aimed at: collect video data from first encounters of naive users with
a cobot making use of very limited and strictly non-verbal communicative
signals (movements of the platform and the Tool Center Point of the robotic
arm). Having manually analysed the video recordings from one day of human-
robot interactions, capturing both the robot’s direct interaction partner and
bystanders, we can summarize our main findings as follows: (i) there was a
remarkable lack of spatial distance from the human interaction partner towards
the robot, (ii) people showed a tendency for explorative play with the robot,
(iii) some people assumed that there is a motion sensor below the gripper,
(iv) some of the onlookers moved as if trying out variants of the current
participant’s behaviour.

In the next sections, we first present related work. Subsequently, the
technical realisation of the cobot behaviour is described as well as the
scenario and context where the data collection took place. Qualitative
analysis and findings are then presented. This is followed by a conclusion and
discussion of future directions.

##  2 Related Work

The distance between two agents during an interaction is a fundamental
principle of social interaction and known as proxemics. Proxemics was first
described by anthropologist Eduard Hall, who emphasized the impact of the use
of space, i.e. proxemic behavior, on interpersonal communication [4]. Hall
defined four primary proximity zones [3]:666The units are translated from
inch/feet in the original work to cm/m. One should also keep in mind that the
borders between the zones are more guidelines than strict numbers.

  * •

intimate (contact - 0.46m): two agents can touch, smell, talk in whisper etc.,
but cannot see each other well.

  * •

personal (0.46m - 1.22m): personal distance corresponds to personal space,
where discomfort can be felt if that space is penetrated by someone whom the
individual is unfamiliar with and it is possible to touch each other by
reaching.

  * •

social (1.22-3.66m): social distances are used in formal business purposes,
e.g. across a desk with the need to speak louder.

  * •

public (3.66m and beyond): public distance is the distance kept from public
figures, facial expressions are difficult to see, a louder voice is needed and
bodily movements need to be exaggerated.

Proxemics has been shown to apply to human-robot interactions (see e.g. [11]),
which is consistent with several studies demonstrating that individuals
respond socially to computers [8]. In human-robot interaction, the robot’s
effective use of proxemic behaviors can be crucial for the engagement with
humans as it allows for intuitive interaction between humans and robots [11].

In a study on comfortable approach distances, Walters et al. [14] investigated
both the robot approach perspective and the human approach perspective, using
the PeopleBot telepresence robot. They found that 60% of the participants were
comfortable with a social distance compatible with human-human interaction
distances. However, 40% actually approached the robot up to a distance which
would be perceived as either threatening or intimate, depending if it involves
strangers or close friends. When investigating human personality traits, the
authors found that timidity and nervousness increased the comfortable approach
distance, whereas it was decreased by traits such as proactiveness. In a
subsequent long-term study, Walters et al. [15] showed reductions in
comfortable human-robot approach distances in the first one to two interaction
sessions. For the remaining sessions, approach distance preferences remained
relatively steady.

Koay et al. [7] found that humans allowed robots to approach more closely
during physical interactions (e.g. hand-over tasks) than during verbal
interactions or no interaction conditions (e.g. when the robot moves through
the same room). In their study, participants also exhibited a strong tendency
to allow the robot with mechanoid appearance to approach closer during their
first encounter than the robot with humanoid appearance. However, results
showed that this tendency faded away as the participants habituated to the
robots. In a subsequent study, Koay et al. [6] found evidence that
participants preferred to be approached from the front versus the side. This
preference increased with closer distances.

Shi et al. [13] investigated perceived safety levels associated with distance
and velocity, using a Segway robotic platform. Results of their study showed
that distance itself did not effect participants. However, for the fast
velocity condition participants felt slightly unsafe and some even moved away
from the robot’s path as it approached them.

In our study, the robot was placed at a certain location in the room. After a
participant had approached the robot, the interaction started. The robot
remained in its position during the interaction for safety reasons, while the
robot arm moved back and forth depending on the human’s movements.

##  3 Robot Technical Set-up

CHIMERA (Figure 1) is a combination of two commercial robots: MiR-100 (the
mobile base) and Universal Robots UR-10 (the robot arm). They are integrated
into CHIMERA’s (henceforth the robot’s) internal network. Mobile base and arm
have their own programming methods supplied by the respective vendors but can
be interconnected for a limited set of functions (e.g., triggering a program
on the robot arm as part of a mission of the robot base). However, running on
CHIMERA’s internal industrial PC, there is a software stack that enables joint
programs of both MiR and UR when the robot performs autonomous operations.

For safety and security reasons in a public space, however, both, mobile
platform and robot arm were human operated by two researchers from our team
(which was not obvious to the participants). The robot arm was pre-programmed
with a reactive application that responds to a two-button remote. Using this,
the robot arm could be moved into a center (idle), a forward (approach) or a
rear (retract) position. The mobile base first recorded a map of the
surrounding area. Based on that, missions were programmed which guided the
robot through the exhibition area. Also direct manual remote control of the
mobile base was used where appropriate.

##  4 Data Collection Scenario

The users were visitors to Kepler’s Gardens at the Ars Electronica Festival
2020. Figure 2 shows the exhibition space of the LIT Robopsychology
Lab777https://www.jku.at/en/lit-robopsychology-lab/ where the human-robot
encounters took place. The room was big enough (105 m2) for the robot platform
to move and the arm to extend to its maximum length, and at the same time give
the human plenty of space to maneuver. Due to COVID-19 restrictions, visitors
had to book slots in advance and came in groups of approximately 20 people or
less. Visiting groups were introduced to the robot by one of the researchers
present. They were told that the robot reacts to proximity, i.e., when a
person stands in front of the robot and s/he gives way to the robot, the arm
will move forward in the human’s direction. If s/he leans forward towards the
robot, the arm will move back. In other words, the robot wants to keep its
"arm length" distance to the human. The visitors were also told that they
should make sure that they kept their heads away from the robot arm so that
they cannot be harmed. The interaction was then demonstrated to the visitors
by the researcher in charge and they were invited to try it out themselves,
one by one. For data collection, video recordings were taken from the visits
during 12 September 2020. Videos were taken with (i) a statically mounted
GoPro directed at the robot and the human interaction partner from the right
side, and covering the background in a wide angle from a distance of
approximately 5 meters; (ii) a camera mounted on the end effector of the robot
arm providing a frontal view on the human interaction partner and the scene in
the background whenever the human was not too close to the robot’s camera;
(iii) in addition, some of the scenes were also captured by videos taken by
one or two members of the research team moving around with mobile phone
cameras. This provided us with plenty of material for studying the behaviours
of the human directly interacting with the robot and gave us impressions of
bystanders’ reactions.

Figure 2: Location at Ars Electronica 2020 where human-robot encounters took
place. View from statically mounted Gopro.

##  5 Qualitative Analysis and Findings

The videos were sifted with respect to what is the non-verbal behaviour of the
human interacting with the robot, and what do people in the audience do? First
of all, we analysed the recordings from the Gopro and the mobile phone
cameras. In a first round, we were interested in the behaviours of the robot’s
interaction partners. Aspects of interest were: does the human more or less
copy the behaviour of the presenter; to which extent does the human reproduce
the proxemics game, i.e., does the human give way or take space so that the
robot arm can move forward towards the human or backs off; what kinds of
explorative behaviour does the human show; what are the spatial distances the
human maintains towards the robot. In a second round, we concentrated on the
bystanders in our video material. As regards the audience, we did not have any
detailed assumptions, we were generally interested in typical behaviours that
would occur. In a third round, we looked into the videos from the camera
mounted at the robot’s end effector, to get a first person perspective from
the robot’s side. These data are particularly interesting, as they vividly
depict how close to the robot human interaction partners tend to come, either
with their faces or at least their hands.

###  5.1 Behaviours of Humans Interacting with the Robot

#### Human reproduction of game play

Interestingly, there were only a few visitors who just interacted with the
robot in the way we had demonstrated the interaction. There was also a small
number who seemed to be waiting for the robot to make the first step or even
more they seemed to expect the robot to guide them through the interaction. If
this did not happen, they lost interest. The majority, however, were much more
interested in exploring the robot.

#### Human explorative behaviours

Quite a number of people were much more interested in exploring the robot than
just interacting in the demonstrated manner. People seemed to suspect that the
end effector had some kind of sensor as people tended to hold their hands very
close below the end effector and tried to make the robot follow their hand
movements. As if the end effector were the robot’s snout. This may possibly be
due to the slight up and down movements of the end effector when moving the
robot arm back and forth, which were a side effect of keeping the camera in
forward orientation. This made the end part of the robot arm the most vivid
part of the robot during the interaction. Besides, it was also the smallest
part of the robot and fitted very well with the size of the human palm. Other
frequent human behaviours were gesturing in front of the camera mounted on the
end effector, or getting very close to it with one’s face. For the former, as
if the humans wanted to test if the robot’s arm would follow their moving hand
or index finger. Just like in gaze following games where one person gets very
close to the face of the other person and makes the other person’s eyes follow
their moving hand or index finger. As regards the latter, we had the
impression as if the humans wanted to provoke the robot to show a reaction by
them massively invading the robot’s intimate space. In general, we had the
impression that the human interaction partners were seeking for the robot’s
sensors, typically by gesticulating and coming very close. Sensors were
expected (i) at the end of the end effector; (ii) the camera mounted on the
end effector was identified; and (iii) the frontal sides (left and right) of
the mobile platform were searched for further sensors.

#### Proxemics

Even though the visitors were instructed to keep at least an arm length
distance from the robot, putting one’s arm forward and at the same time
keeping one’s head well away from the robot arm, the volunteers came very
close to the robot during interaction. Distances ranged from close personal to
intimate distance. In this respect our data provide further evidence for Koay
et al. and Walters et al., that people during first encounters tend to come
rather close to robots [14] with mechanoid appearance [7, 6]. Walters et al.
[14] hypothesized that in these very close interactions, humans do not treat
the robot as a social entity.

We did not see anybody approaching the robot from behind. This may be due to
the fact that the robot had a clear front to rear orientation. It was
positioned in the room opposite of the entrance with the arm facing the
incoming visitors, and the arm going backward and forward in the proxemics
game supported this orientation, too.

Another observation was that when children and adults stepped forward together
to interact with the robot, it happened that when one started the interaction
with the robot, the other one quite quickly started to push the interacting
one aside, in order to try out the interaction by themselves.

###  5.2 Behaviours of Bystanders

In the behaviours of the onlookers, we found the following typicalities: (i)
the audience uniformly kept a wide social up to public distance during the
presentation by the instructor and while someone else was interacting with the
robot, (ii) attention according to head gaze was clearly oriented towards the
scene of human-robot interaction. In other words, the bystanders were
observing the scene from a distance. (iii) In some cases, we found dry runs of
onlookers imitating the person currently interacting with the robot. Moreover,
it seemed that people were trying variants of the behaviour the robot’s
current interaction partner displayed. Such behaviours were displayed in cases
where the robot did not seem to respond to the human, even though the human
displayed a behaviour similar to the one demonstrated by the instructor. It
seemed as if the dry runs were aimed at exploring potentially more successful
variants of the behaviour. It may also be that those individuals who made
their dry runs were interested in the interaction but were too shy to step
forward and actually interact with the robot.

##  6 Conclusion and Future Work

We presented a qualitative study comprising a simple proxemics game between a
cobot and walk-in visitors of a festival for arts, technology and society.
From the data we found that people had a high interest in exploring the robots
action and perception capabilities, much more than just reproducing the
interaction game as they were told by the instructors. We also found that
people who interacted with the robot did this from very close proximity,
especially when they seemed to explore the robot. In addition, we found
examples of behavioural dry runs by bystanders who seemed to test out variants
of the interaction behaviour which was presented by the instructors and did
not seem to immediately trigger the expected reaction from the robot by the
human currently interacting with the robot.

Overall, our qualitative study was a first step towards further investigating
the interplay of robot communicative signals and human non-verbal behaviours
in interactions with cobots. From the findings, we derive, on the one hand,
technical requirements for robot action and perception capabilities including
the necessity of a cobot (i) to check the attention of its human collaboration
partner, in particular, which part of the robot the human looks at or gets
close to with their hands or faces; the interpretation of which also requires
(ii) a model of proxemics; (iii) to signal to the human where the robot’s
current focus of attention is, e.g. by change of orientation of the robot arm;
(iv) to use movement (approaching, moving away, facing away and turning
around) as communicative signals for proxemics and attention to the human; (v)
to be able to identify and correctly interpret human gesturing such as
beckoning vs. pointing, emblems e.g. stop.

For autonomous or semi-autonomous robots interacting with humans, perceiving
and interpreting human motions correctly and subsequently following some
social behaviors is considered as one of the most important capabilities [16].
Therefore, the findings are very valuable inputs for engineers, as they will
add significantly to the robot’s transparency towards humans in order to
increase the trustworthiness of autonomous robots.

On the other hand, we derive questions for strands of further experimentation,
both qualitative and quantitative which include: studying human behaviour
changes in long term interactions with cobots, or studying human-cobot
interaction behaviours for specific industrial interacion scenarios, ideally
with industrial workers/trainees.

Another research question requiring further investigations is to what extent
an artifical agent is perceived as a social agent, especially when proximity
zones of the artificial agent are frequently violated.

With regards to behavioural effects of humans interacting with cobots, an
important aspect of future work is to more closely investigate differences in
behaviour and their origin. They might occur due to specific requirements of a
certain collaboration task, or be influenced by the coworker’s personality and
other psychologically motivated aspects. As regards the former, in some tasks
mutual gaze behaviour between the collaboration partners is essential for task
success; in others joint attention is heavily directed by which objects are
currently manipulated by one or both collaboration partners; posture shifts
can be strong indicators for whose turn it is to do the next working step [12,
2]. As regards the latter, the effects of personality traits [14], trust [10]
and habituation [7], as well as other socio-emotional aspects [13] or
perceived job security may have significant effects on human behaviours in the
interaction with cobots.

##  7 Acknowledgments

This work was supported by the Austrian Research Promotion Agency (FFG), Ideen
Lab 4.0 project CoBot Studio (872590), and the Vienna Science and Technology
Fund (WWTF), project "Human tutoring of robots in industry" (NXT19-005).

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