∎

e1e-mail<EMAIL_ADDRESS>e2e-mail:
<EMAIL_ADDRESS>e3e-mail<EMAIL_ADDRESS>e4e-mail<EMAIL_ADDRESS>11institutetext: Departamento de Física and
Centro de Física Teórica de Partículas (CFTP),
Instituto Superior Técnico (IST), U. de Lisboa (UL),
Av. Rovisco Pais, P-1049-001 Lisboa, Portugal. 22institutetext: Institute for
Theoretical Physics, Karlsruhe Institute for Technology,
Wolfgang-Gaede-Straße 1, 76131 Karlsruhe, Germany

# Leaks of CP violation in the real two-Higgs-doublet model

Duarte Fontese1,addr1  Maximilian Löschnere2,addr2  Jorge C. Romãoe3,addr1
João P. Silvae4,addr1

(August 27, 2024)

###### Abstract

We discuss the $Z_{2}$ symmetric two-Higgs-dou- blet model with a real soft-
breaking term (real 2HDM). We explain in detail why it is not tenable to
assume CP conservation in the scalar sector to keep the dimension two term
real, while CP is violated by the dimension four Yukawa couplings. We propose
the calculation of the infinite tadpole of the (would-be) pseudoscalar neutral
scalar. We construct a simple toy model with the same flaws, where the
unrenormalizable infinity is easier to calculate. We then consider the same
tadpole in the real 2HDM. We spearhead this effort focusing on diagrams
involving solely bare quantities. This involves hundreds of Feynman three-loop
diagrams that could feed the CP violation from the quark into the scalar
sector, and is only possible with state of the art automatic computation
tools. Remarkably, some intermediate results agree when using three
independent derivations, including the peculiar cancellation of the leading
pole divergence due to a subtle interplay between masses and the Jarlskog
invariant, which we calculate analytically. The calculation is not complete
however, since the full two-loop renormalization of the real 2HDM is not yet
available in the literature. Still, we argue convincingly that there is an
irremovable infinity.

##  1 Introduction

The exact implementation of the symmetry breaking sector of the gauge theory
describing fundamental particles is one of the most interesting open problems.
In particular, the recent discovery at LHC of a $125\,\textrm{GeV}$
fundamental scalar ($h_{125}$) Aad:2012tfa ; Chatrchyan:2012xdj  begs the
question of how many fundamental scalars there are in Nature. The Standard
Model (SM) contains one single scalar doublet, but there is no fundamental
requirement for this choice.

Indeed, there have been countless studies on models with two or more Higgs
doublets; for reviews see hhg ; Branco:2011iw ; Ivanov:2017dad  and references
therein. The most studied model includes two Higgs doublets (2HDM) with a
$Z_{2}$ symmetry, softly broken by a real parameter, with the additional
requirement that the vacuum expectation values (vev) are taken as real. As
such, the scalar sector by itself is CP conserving. The $Z_{2}$ symmetry is
carried over to the fermions in such a way that only one scalar couples to all
fermions of a given charge. The crucial point is that the experimentally
observed CP violation Zyla:2020zbs  is accounted for by complex Yukawa
couplings. This is sometimes referred to as the “real 2HDM”, because the soft
$Z_{2}$ breaking parameter is taken as real. Remarkably, this most prevalent
model can suffer from from theoretical inconsistencies regarding
renormalization, a fact that is mostly ignored.111We know of only one
published exception, appearing in one line on page 4 of Altenkamp:2017ldc .
This is the problem we address in detail here.

The paper is organized as follows. In section 2, we discuss the inconsistency
of requiring CP conservation in the potential of the real 2HDM, while allowing
for CP violation elsewhere. We argue that, at sufficient high order in
perturbation theory, there could be divergences in CP-violating one-point and
two-point functions that one can not remove by the counterterms provided by
the theory. We show in section 3 that this is precisely what happens in a toy
model suffering from the same inconsistency as the real 2HDM. Then, in section
4, we present the result for the leading pole of the three-loop one-point
function of the alleged CP-odd physical field of the real 2HDM. We describe in
section 5 the details of the different steps involved in the three-loop
calculation. In section 6, we summarize our conclusions.

##  2 Shortcomings of the real 2HDM

Let us consider a $SU(2)_{L}\otimes U(1)_{Y}$ gauge theory with two Higgs-
doublets $\Phi_{a}$, with the same hypercharge $1/2$, and with real vacuum
expectation values (vevs)

$\langle\Phi_{a}\rangle=\left(\begin{array}[]{c}0\\\
v_{a}/\sqrt{2}\end{array}\right)\,,\ \ \ (a=1,2),$ (1)

with $v=\sqrt{v_{1}^{2}+v_{2}^{2}}=246\textrm{GeV}$. Our definition for the
charge is $Q=T_{3}+Y$, and we introduce the angle $\beta$ through

$\tan{\beta}=v_{2}/v_{1}\,.$ (2)

The most general 2HDM scalar potential may be written as

$\displaystyle V_{H}$
$\displaystyle=\,m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-\left[m_{12}^{2}\Phi_{1}^{\dagger}\Phi_{2}+\mathrm{H.c.}\right]$
(3)
$\displaystyle+\,\tfrac{1}{2}\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\tfrac{1}{2}\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})$
$\displaystyle+\,\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\left[\tfrac{1}{2}\lambda_{5}(\Phi_{1}^{\dagger}\Phi_{2})^{2}+\mathrm{H.c.}\right]$
$\displaystyle+\left[\lambda_{6}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{1}^{\dagger}\Phi_{2})+\lambda_{7}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{1}^{\dagger}\Phi_{2})+\mathrm{H.c.}\right],$

where “H.c.” stands for Hermitian conjugation. The coefficients $m_{11}^{2}$,
$m_{22}^{2}$, and $\lambda_{1},\cdots,\lambda_{4}$ are real while
$m_{12}^{2}$, $\lambda_{5}$, $\lambda_{6}$ and $\lambda_{7}$ may be complex.

When extending this model to the fermion sector, one finds flavour changing
neutral scalar interactions, which are very strongly constrained by
experiments on neutral meson systems. This problem can be solved by imposing a
$Z_{2}$ symmetry: $\Phi_{1}\rightarrow\Phi_{1}$;
$\Phi_{2}\rightarrow-\Phi_{2}$ Glashow:1976nt ; Paschos:1976ay . If the
symmetry is exact, then the quadratic term $m_{12}^{2}=0$, and the quartic
terms $\lambda_{6}=\lambda_{7}=0$. (The quartic term $\lambda_{5}$ can then be
made real by a simple rephasing of $\Phi_{2}$.) This has the consequence that
the model has no decoupling limit. That is, one cannot make arbitrarily large
the masses of the new particles resultant from the presence of the second
scalar doublet, thus approaching smoothly the SM limit. Such a decoupling is a
desirable feature, especially since the couplings probed by current LHC data
are consistent with the SM predictions, within errors of order 20%
Khachatryan:2016vau . Decoupling is recovered by reintroducing $m_{12}^{2}\neq
0$ Gunion:2002zf , which, because it breaks softly the $Z_{2}$ symmetry, does
not affect the renormalizability of the theory.

Most articles addressing this model then make the choice that $m_{12}^{2}$ and
$\lambda_{5}$ are both real and the vevs are real, arguing that CP
conservation in the scalar sector has been imposed (choice 1). Then one would
proceed to discuss the various implementations of the $Z_{2}$ symmetry in the
fermion sector, and perform a variety of fits to experiment. Among these, one
must fit the well measured CP violation with origin in the Cabibbo-Kobayashi-
Maskawa (CKM) matrix Cabibbo:1963yz ; Kobayashi:1973fv , accommodated by the
complex Yukawa couplings (choice 2). The problem is that choice 1 and choice 2
are incompatible.

Indeed, either one applies the CP symmetry to the whole Lagrangian, in which
case the Yukawa couplings are real and one cannot account for the observed CP
violation; or, else, one does not apply CP symmetry anywhere, allowing the
Yukawa couplings to be complex, but then allowing also for a complex
$m_{12}^{2}$.222Recall that $\lambda_{5}$ can always be made real through a
convenient rephasing of $\Phi_{2}$. In fact, one could instead use the
rephasing to make $m_{12}^{2}$ real, at the price of getting a complex
$\lambda_{5}$. What really matters is the rephasing invariant quantity
$\textrm{Im}\left[\lambda_{5}^{\ast}\left(m_{12}^{2}\right)^{2}\right]$. For
simplicity, we will make the discussion in the basis where $\lambda_{5}$ is
real. Said otherwise, requiring complex CKM and excluding the parameter
$\textrm{Im}\left(m_{12}^{2}\right)$ leads to a non-renormalizable theory. At
sufficiently high loop level, the CP violation in the quark sector will leak
into the scalar sector, through a divergent contribution that cannot be
absorbed by a $\textrm{Im}\left(m_{12}^{2}\right)$ counterterm, which was
absent from the theory from the start.

So why do all articles fitting the real 2HDM “model” to experiment ignore this
problem? Because the divergent contribution can only be shown to happen in at
least three loops. However, precisely because they are divergent, the problem
cannot be ignored if one wishes to use a theoretically sound model.

Given the fact that the problem seems to occur due to (the lack of)
$\textrm{Im}\left(m_{12}^{2}\right)$, one could be tempted to assume that such
a dimension two operator could not affect renormalizability. And indeed, it
cannot affect renormalizability due to its soft-breaking of the $Z_{2}$
symmetry. But the problem with CP symmetry being invoked is not that it is
broken by $m_{12}^{2}$ (real or complex); rather, it is (hard) broken by the
dimension four Yukawa couplings.

It is true that one can look at the real 2HDM as a limiting case of the
$Z_{2}$ 2HDM, softly broken by a complex $m_{12}^{2}$. This model is known as
the complex 2HDM (C2HDM) and has been studied in detail; see, for example,
Ginzburg:2004vp ; ElKaffas:2006gdt ; Arhrib:2010ju ; Barroso:2012wz ;
Inoue:2014nva ; Fontes:2014xva ; Grzadkowski:2014ada ; Fontes:2017zfn ;
Boto:2020wyf ; Cheung:2020ugr . In that case, one can choose any tree-level
values for the parameters, and, in particular, set
$\textrm{Im}\left(m_{12}^{2}\right)=0$ at tree level. In that context, setting
$\textrm{Im}\left(m_{12}^{2}\right)=0$ at tree level, does not constitute a
problem, since the theory does have its counterterm and is renormalizable. Is
this the same as the real 2HDM? No: setting
$\textrm{Im}\left(m_{12}^{2}\right)=0$ in the C2HDM means that we are studying
a very specific corner of tree-level parameter space of a more general model.
The real 2HDM, where there is no $\textrm{Im}\left(m_{12}^{2}\right)$ nor its
counterterm, is not a consistent model.

There is a more physical way to state the non renormalization problem. In any
2HDM there are three neutral scalars ($h_{1}$, $h_{2}$, and $h_{3}$), and a
pair of charged scalars $H^{\pm}$. Typically, it is assumed that the lightest
neutral scalar ($h_{1}$) coincides with the 125GeV particle found at
LHC.333This is not mandatory. One can accommodate the possibility that the
125GeV particle is not the lightest neutral scalar, but we shall not concern
ourselves here with that case. See for example Ferreira:2012my ;
Bernon:2015wef . In the real 2HDM, the (proclaimed) lack of CP violation in
the scalar sector, leads to the separation of the three neutral scalars into
one single CP odd scalar ($A$) and two CP even (usually denoted by $h$ for the
lightest and $H$ for the heaviest). If the CKM CP violation indeed seeps into
the scalar sector, then there should be divergent contributions to the $h$-$A$
and $H$-$A$ two-point functions. There will also be divergent contributions to
the A tadpole. Since such terms are absent from the real scalar sector at tree
level, there are no counterterms to absorb those infinities, and the theory is
formally inconsistent.

###  2.1 The scalar sector

We start from the scalar sector of the real 2HDM

$\displaystyle V_{r}$
$\displaystyle=\,m_{11}^{2}\Phi_{1}^{\dagger}\Phi_{1}+m_{22}^{2}\Phi_{2}^{\dagger}\Phi_{2}-m_{12}^{2}\left[\Phi_{1}^{\dagger}\Phi_{2}+\Phi_{2}^{\dagger}\Phi_{1}\right]$
(4)
$\displaystyle+\,\tfrac{1}{2}\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}+\tfrac{1}{2}\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})$
$\displaystyle+\,\lambda_{4}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})+\tfrac{1}{2}\lambda_{5}\left[(\Phi_{1}^{\dagger}\Phi_{2})^{2}+(\Phi_{2}^{\dagger}\Phi_{1})^{2}\right],$

with all parameters real, and we parametrize the fields in the original basis
as

$\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}c_{\beta}G^{+}-s_{\beta}H^{+}\\\\[2.84526pt]
\tfrac{1}{\sqrt{2}}\left[vc_{\beta}+(c_{\alpha}H-s_{\alpha}h)+i(c_{\beta}G^{0}-s_{\beta}A)\right]\end{array}\right),$
(7) $\displaystyle\Phi_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}s_{\beta}G^{+}+c_{\beta}H^{+}\\\\[2.84526pt]
\tfrac{1}{\sqrt{2}}\left[vs_{\beta}+(s_{\alpha}H+c_{\alpha}h)+i(s_{\beta}G^{0}+c_{\beta}A)\right]\end{array}\right).$
(10)

Throughout $c_{\theta}=\cos{\theta}$ and $s_{\theta}=\sin{\theta}$, for
whatever angle $\theta$ appears as a sub-index. Comparing eq. (1) and eq.
(10), we recognize the choice $v_{1}=vc_{\beta}$ and $v_{2}=vs_{\beta}$. With
that choice, $G^{+}$ and $G^{0}$ will be massless and $H^{+}$ is the physical
charged scalar of mass $m_{H^{\pm}}^{2}$.444This is in fact a feature of the
general 2HDM (which can be extended to multiple doublets) related to the
existence of a “Higgs basis” Botella:1994cs  –
$H_{1}=v_{1}^{*}\Phi_{1}+v_{2}^{*}\Phi_{2}$,
$H_{2}=-v_{2}\Phi_{1}+v_{1}\Phi_{2}$ – where all the vev is in the first
doublet Georgi:1978ri ; Donoghue:1978cj . These facts are confirmed by
substituting eq. (10) in eq. (4). Performing the expansion, one sees that
there is no linear term in $A$ (and, thus, no possibility to absorb any
infinities that might appear in A tadpoles at loop level), nor are there any
quadratic $hA$ or $HA$ terms (and, similarly, no possibility to absorb any
infinities that might appear in the corresponding two-point functions at loop
level).

The expansion does contain linear (tadpole) terms for H ($t_{H}$) and h
($t_{h}$). Equating these tadpoles to zero, one obtains the same conditions
that one would obtain by finding the stationarity conditions $\partial
V_{r}/\partial v_{1}=0$ and $\partial V_{r}/\partial v_{2}=0$ directly from
eq. (4). Those two equations can be solved for $m_{11}^{2}$ and $m_{22}^{2}$,
which are then substituted back into the expression for the potential. After
this substitution using the vacuum conditions, there are no quadratic
$G^{0}G^{0}$ and $G^{+}G^{-}$ terms. There are also no mixed $G^{0}A$ or
$G^{\pm}H^{\mp}$ terms, meaning that as expected $G^{0}$ and $G^{+}$ are
Goldstone bosons, while $A$ and $H^{+}$ are already mass eigenstates. One
finds quadratic terms of the type $HH$, $hh$, and $hH$. The angle $\alpha$ is
chosen to kill the latter, meaning that $h$ and $H$ are the physical fields.
Using $v=246\textrm{GeV}$ and $m_{h}=125\textrm{GeV}$, the scalar sector of
the real 2HDM is parametrized by six further parameters: the mixing angles
$\alpha$ and $\beta$; the masses $m_{H}$, $m_{A}$, $m_{H^{\pm}}$; and the
soft-breaking parameter $m_{12}^{2}=\textrm{Re}\left(m_{12}^{2}\right)$.

###  2.2 CP violation from the CKM matrix

In the SM and in the real 2HDM, CP violation arises from the complex Yukawa
couplings. When the quark fields are rotated into their mass basis, all CP
violation phases are contained in the CKM matrix Cabibbo:1963yz ;
Kobayashi:1973fv . But, each quark field can still be rephased at will, thus
moving the CP violating phase around the various entries of the CKM matrix.
The only rephasing invariant quantity is Jarlskog:1985ht ; Jarlskog:1985cw ;
Dunietz:1985uy

$I^{\alpha i}_{\beta j}=\textrm{Im}\left(V_{\alpha i}V_{\beta j}V_{\alpha
j}^{\ast}V_{\beta i}^{\ast}\right)\,,$ (11)

where $\alpha\neq\beta$ and $i\neq j$. We use the notation of Branco:1999fs ,
where Greek letters $\alpha,\beta,\gamma,\dots$ refer to up-type quarks
$u_{\alpha}=u,c,t$, while Roman letters $i,j,k,\dots$ refer to down-type
quarks $d_{i}=d,s,b$. There are nine distinct four quark combinations with
different flavours: $(ds)$, $(db)$, and $(sb)$ for the down-type quarks times
the three for up-type quarks: $(uc)$, $(ut)$, and $(ct)$. Using the unitarity
of the CKM matrix, the following symmetries hold

$I^{\alpha i}_{\beta j}=I^{\beta j}_{\alpha i}=-I^{\alpha j}_{\beta
i}=-I^{\beta i}_{\alpha j},$ (12)

showing that indeed there is only one independent CP violating quantity. The
antisymmetry with respect to interchange of same quark-type indices is easiest
to see in the form

$I^{\alpha i}_{\beta
j}=J\,\sum_{\gamma,k}\epsilon_{\alpha\beta\gamma}\,\epsilon_{ijk}\,,$ (13)

where $J$ is the Jarlskog invariant Jarlskog:1985ht ; Jarlskog:1985cw ;
Dunietz:1985uy , defined for example as $J=I^{ud}_{cs}$. Notice that
$I^{ud}_{cs}\neq 0$ even though only quarks from the first two families are
involved. This does not contradict the fact that there would be no CP
violating phase in the SM if there existed only two families of quarks. The
fact that the CKM matrix is $3\times 3$ unitary (and, thus, has one
irremovable complex phase) is built into eq. (13). Nevertheless, in CP
violating processes with no external quarks (and, thus, a quark loop), the
appearance of four CKM ($V$) factors can only occur in diagrams at the three
loop level and above.

As emphasized by Khriplovich and Pospelov Pospelov:1991zt  and by Booth
Booth:1993af  in the context of the electric dipole moments (edm) of the $W$
and the electron, the antisymmetry of $I^{\alpha i}_{\beta j}$ is very
powerful. Any CP violating amplitude from a fermion box diagram will appear as
the product $I^{\alpha i}_{\beta j}$ with some amplitude

$\mathcal{A}(m_{u_{\alpha}},m_{u_{\beta}},m_{d_{i}},m_{d_{j}})\,.$ (14)

When all contributions are summed over ($\alpha$, $\beta$, $i$, and $j$), all
terms in $\mathcal{A}(m_{u_{\alpha}},m_{u_{\beta}},m_{d_{i}},m_{d_{j}})$
symmetric under $\alpha\leftrightarrow\beta$, or $i\leftrightarrow j$ will not
contribute. A much more involved analysis along these lines was used in
Pospelov:1991zt ; Booth:1993af  to show that the SM electroweak contributions
to the electric dipole moments of the $W$ and the electron vanish to two-loop
and three-loop approximation, respectively.

##  3 A theoretically unsound toy model

To better illustrate our claim, we consider here a toy model that suffers from
the same inconsistency as the real 2HDM. In both models, CP conservation is
enforced in a particular sector of an otherwise CP violating theory. As a
consequence, CP violating radiative effects end up contaminating the alleged
CP conserving sector, thus leading to divergences that cannot be absorbed by
the counterterms. The major feature of our toy model is that such divergences
show up immediately at one-loop order. Therefore, it constitutes a simple
materialization of the same theoretical pathology that we claim to be present
at three-loop or above in the real 2HDM.

The present model is inspired in a model by Pilaftsis Pilaftsis:1998pe ,
which, however, does not suffer from the flaw we wish to point out. Consider
two Abelian gauge symmetries $U(1)_{Q}$ and $U(1)_{B}$, with gauge bosons
$A_{\mu}$ and $B_{\mu}$, respectively. Suppose also four complex scalars,
$\Phi_{1}$, $\Phi_{2}$, $\chi_{L}$ and $\chi_{R}$, with charges

$Q(\Phi_{1})=0,\,Q(\Phi_{2})=0,\,Q(\chi_{L})=1,\,Q(\chi_{R})=1,$ (15)
$B(\Phi_{1})=1,\,B(\Phi_{2})=1,\,B(\chi_{L})=-\dfrac{1}{5},\,B(\chi_{R})=\dfrac{4}{5},$
(16)

where $Q$ and $B$ represent the conserved charges of $U(1)_{Q}$ and
$U(1)_{B}$, respectively. A discrete symmetry $D$ is imposed on the fields,
under which:

$\Phi_{1}\stackrel{{\scriptstyle D}}{{\to}}-\Phi_{1},\ \
\Phi_{2}\stackrel{{\scriptstyle D}}{{\to}}\Phi_{2},\ \
\chi_{L}\stackrel{{\scriptstyle D}}{{\to}}-\chi_{L},\ \
\chi_{R}\stackrel{{\scriptstyle D}}{{\to}}\chi_{R}.$ (17)

However, $D$ is allowed to be softly broken. The complete renormalizable
Lagrangian can then be written in four terms,

$\mathcal{L}=\mathcal{L}_{kin}+\mathcal{L}_{\Phi}+\mathcal{L}_{\chi}+\mathcal{L}_{\Phi\chi},$
(18)

where $\mathcal{L}_{kin}$ represents the kinetic terms555We assume no
$A_{\mu}$-$B_{\mu}$ kinetic mixing. and

$\displaystyle-\mathcal{L}_{\Phi}$ $\displaystyle=$
$\displaystyle\mu_{1}^{2}\Phi_{1}^{*}\Phi_{1}+\mu_{2}^{2}\Phi_{2}^{*}\Phi_{2}+\mu^{2}\Phi_{1}^{*}\Phi_{2}+(\mu^{2})^{*}\,\Phi_{2}^{*}\Phi_{1}$
(19a)
$\displaystyle+\lambda_{1}{\left(\Phi_{1}^{*}\Phi_{1}\right)}^{2}+\lambda_{2}{\left(\Phi_{2}^{*}\Phi_{2}\right)}^{2}+\lambda_{34}\,\Phi_{1}^{*}\Phi_{1}\Phi_{2}^{*}\Phi_{2}$
$\displaystyle+\lambda_{5}{\left(\Phi_{1}^{*}\Phi_{2}\right)}^{2}+\lambda_{5}^{*}{\left(\Phi_{2}^{*}\Phi_{1}\right)}^{2},$
$\displaystyle-\mathcal{L}_{\chi}$ $\displaystyle=$ $\displaystyle
m_{L}^{2}\,\chi_{L}\chi_{L}^{*}+m_{R}^{2}\,\chi_{R}\chi_{R}^{*}+\rho_{1}(\chi_{L}^{*}\chi_{L})^{2}$
(19b)
$\displaystyle+\rho_{2}(\chi_{R}^{*}\chi_{R})^{2}+\rho_{34}\,\chi_{L}^{*}\chi_{L}\chi_{R}^{*}\chi_{R},$
$\displaystyle-\mathcal{L}_{\Phi\chi}$ $\displaystyle=$ $\displaystyle
f_{1}\,\Phi_{1}\chi_{L}\chi_{R}^{*}+f_{1}^{*}\,\Phi_{1}^{*}\chi_{L}^{*}\chi_{R}+f_{2}\,\Phi_{2}\chi_{L}\chi_{R}^{*}$
(19c)
$\displaystyle+f_{2}^{*}\,\Phi_{2}^{*}\chi_{L}^{*}\chi_{R}+g_{1}\,\Phi_{1}^{*}\Phi_{1}\chi_{L}^{*}\chi_{L}+g_{2}\,\Phi_{2}^{*}\Phi_{2}\chi_{L}^{*}\chi_{L}$
$\displaystyle+\,g_{3}\,\Phi_{1}^{*}\Phi_{1}\chi_{R}^{*}\chi_{R}+\,g_{4}\,\Phi_{2}^{*}\Phi_{2}\chi_{R}^{*}\chi_{R}.$

The parameters $\mu^{2}$, $\lambda_{5}$, $f_{1}$ and $f_{2}$ are in general
complex, while the remaining ones are real by construction. The terms
involving $\mu^{2}$ and $f_{2}$ break the symmetry $D$ softly. It is easy to
show that the conditions for CP conservation are:

$\displaystyle\text{Im}\left[\mu^{2}\,f_{1}\,f_{2}^{*}\right]$
$\displaystyle=$ $\displaystyle 0,$ (20a)
$\displaystyle\text{Im}\Big{[}\lambda_{5}\,f_{1}^{2}\,(f_{2}^{*})^{2}\Big{]}$
$\displaystyle=$ $\displaystyle 0,$ (20b)
$\displaystyle\text{Im}\Big{[}\lambda_{5}^{*}\,(\mu^{2})^{2}\Big{]}$
$\displaystyle=$ $\displaystyle 0.$ (20c)

Mimicking the usual real 2HDM treatment, we take $\langle\Phi_{1}\rangle$ and
$\langle\Phi_{2}\rangle$ real and parametrize

$\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{2}}\left(v_{1}+H_{1}+iA_{1}\right),$ (21a)
$\displaystyle\Phi_{2}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{2}}\left(v_{2}+H_{2}+iA_{2}\right),$ (21b)

where $v_{1}$, $v_{2}$ are real and non-negative, and $H_{1}$, $H_{2}$,
$A_{1}$, and $A_{2}$ are real fields. The vevs $v_{1}$ and $v_{2}$ break
spontaneously the gauge symmetry $U(1)_{B}$. Recall that $\mu^{2}$,
$\lambda_{5}$, $f_{1}$, $f_{2}$ are in general complex.

But suppose we force CP to be conserved in $\mathcal{L}_{\Phi}$ by the ad-hoc
imposition that $\mu^{2}$ and $\lambda_{5}$ are real. This we will do in the
following. It will lead to irremovable divergences at one-loop, as we now
show.

We start by determining the minimization (or tadpole) equations. These
are:666The tadpole equations for $A_{1}$ and $A_{2}$ are trivially zero.

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
t_{H_{1}}:=\dfrac{\partial\mathcal{L}_{\Phi}}{\partial H_{1}}\bigg{|}_{<>=0}$
(22a) $\displaystyle=$ $\displaystyle-
v_{1}\left(\mu_{1}^{2}+\dfrac{v_{2}}{v_{1}}\mu^{2}+v_{1}^{2}\lambda_{1}+\dfrac{1}{2}v_{2}^{2}\lambda_{34}+v_{2}^{2}\,\lambda_{5}\right),$
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
t_{H_{2}}:=\dfrac{\partial\mathcal{L}_{\Phi}}{\partial H_{2}}\bigg{|}_{<>=0}$
(22b) $\displaystyle=$ $\displaystyle-
v_{2}\left(\mu_{2}^{2}+\dfrac{v_{1}}{v_{2}}\mu^{2}+v_{2}^{2}\lambda_{2}+\dfrac{1}{2}v_{1}^{2}\lambda_{34}+\,v_{1}^{2}\,\lambda_{5}\right),$

where $t_{H_{1}},t_{H_{2}}$ represent the tree-level tadpoles for $H_{1}$,
$H_{2}$, respectively, and $<>=0$ means that the expectation values of all
fields on the right hand side of eqs. (21) are set to zero. Recall that we are
taking $\mu^{2}$ and $\lambda_{5}$ as real parameters. For that reason, the
mass matrices for $H_{1}$ and $H_{2}$, on the one hand, and $A_{1}$ and
$A_{2}$, on the other, can be separately diagonalized. We thus define the
angles $\theta$ and $\beta$ such that:

$\displaystyle\left(\begin{array}[]{c}H_{1}\\\ H_{2}\end{array}\right)$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}c_{\theta}&-s_{\theta}\\\
s_{\theta}&c_{\theta}\end{array}\right)\left(\begin{array}[]{l}h\\\
H\end{array}\right),$ (23g) $\displaystyle\left(\begin{array}[]{c}A_{1}\\\
A_{2}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}c_{\beta}&-s_{\beta}\\\
s_{\beta}&c_{\beta}\end{array}\right)\left(\begin{array}[]{c}G^{0}\\\
A\end{array}\right),$ (23n)

where $h$ and $H$ are the CP even states, and $A$ and $G^{0}$ are the CP odd
states, where $G^{0}$ is the massless would-be Goldstone boson.777$G^{0}$ is
eaten by the longitudinal component of $B_{\mu}$ in the unitary gauge through
the Higgs mechanism. The fact that $G^{0}$ is massless forces $\beta$ to obey
the relation $\tan\beta=\frac{v_{2}}{v_{1}}$. We continue using the short
notation $s_{x}\equiv\sin(x)$, $c_{x}\equiv\cos(x)$, for a generic angle $x$.
As for the $\mathcal{L}_{\chi}$ sector, the mass matrix is given by

$-\mathcal{L}^{\chi}_{\text{mass}}=\left(\begin{array}[]{cc}\chi_{L}^{*}&\chi_{R}^{*}\end{array}\right)\left(\begin{array}[]{cc}a&b^{*}\\\
b&c\end{array}\right)\left(\begin{array}[]{c}\chi_{L}\\\
\chi_{R}\end{array}\right),$ (24)

with

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{1}v_{1}^{2}+\frac{1}{2}g_{2}v_{2}^{2}+m_{L}^{2},$
(25a) $\displaystyle b$ $\displaystyle=$
$\displaystyle\dfrac{f_{1}v_{1}+f_{2}v_{2}}{\sqrt{2}},$ (25b) $\displaystyle
c$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{3}v_{1}^{2}+\frac{1}{2}g_{4}v_{2}^{2}+m_{R}^{2}\,,$
(25c)

where $a$ and $c$ are real, while $b$ is in principle complex. However, we can
rephase $\chi_{R}$ through $\chi_{R}\to e^{i\,\text{arg}(b)}\chi_{R}$ so that
it absorbs the phase of $b$. In the new basis, then, $b$ is real, which
implies that the mass matrix in eq. (24) is symmetric.888If it were hermitian,
one would need a unitary matrix to diagonalize it, instead of an orthogonal
one. Moreover, note that $b$ real forces the relation
$f_{1}^{*}=f_{1}+\left(f_{2}-f_{2}^{*}\right)\tan{\beta}$. We thus need an
orthogonal matrix with a new angle $\phi$ to diagonalize the states:

$\left(\begin{array}[]{c}\chi_{L}\\\
\chi_{R}\end{array}\right)=\left(\begin{array}[]{cc}c_{\phi}&-s_{\phi}\\\
s_{\phi}&c_{\phi}\end{array}\right)\left(\begin{array}[]{c}\chi_{1}\\\
\chi_{2}\end{array}\right),$ (26)

where $\chi_{1}$ and $\chi_{2}$ are the (complex) diagonalized states with
(real) masses $M_{1}$ and $M_{2}$.

When considering the theory up to one-loop level, one should renormalize it in
order to obtain finite $S$-matrix elements. This is done through the usual
procedure: taking an independent set of parameters, identifying them with bare
quantities (represented in the following with the index $0$) and relating them
to their renormalized equivalents through a counterterm. Tadpoles can be taken
care of through the tadpole scheme identified by PRTS in Denner:2019vbn . It
then follows from the set of Eqs. (22) that:

$\displaystyle\delta t_{H_{1}}$ $\displaystyle:=$ $\displaystyle-
v_{1}\bigg{(}\mu_{1,0}^{2}+\dfrac{v_{2}}{v_{1}}\mu_{0}^{2}+v_{1}^{2}\lambda_{1,0}+\dfrac{1}{2}v_{2}^{2}\lambda_{34,0}$
(27a) $\displaystyle\hskip 56.9055pt+v_{2}^{2}\,\lambda_{5,0}\bigg{)},$
$\displaystyle\delta t_{H_{2}}$ $\displaystyle:=$ $\displaystyle-
v_{2}\bigg{(}\mu_{2,0}^{2}+\dfrac{v_{1}}{v_{2}}\mu_{0}^{2}+v_{2}^{2}\lambda_{2,0}+\dfrac{1}{2}v_{1}^{2}\lambda_{34,0}$
(27b) $\displaystyle\hskip 56.9055pt+v_{1}^{2}\,\lambda_{5,0}\bigg{)},$

where the tadpole counterterms $\delta t_{H_{1}}$ and $\delta t_{H_{2}}$ are
such that $\delta t_{H_{1}}=-T_{H_{1}}$ and $\delta t_{H_{2}}=-T_{H_{2}}$,
with $T_{H_{1}}$ and $T_{H_{2}}$ being the one-loop tadpole for $H_{1}$ and
$H_{2}$, respectively. The set of eq. (27) fixes the values for $v_{1}$ and
$v_{2}$ at one-loop level. Note that, since we imposed CP conservation in
$\mathcal{L}_{\Phi}$, there are no tadpole counterterms for the CP odd fields.
Specifically, in the mass basis,

$\delta t_{A}=0.$ (28)

But it is easy to see that this is inconsistent. Indeed, there is a one-loop
tadpole for $A$, whose diagrams are represented in Fig. 1.

200 (70,40) arrow_len3mm arrow_ang20 nJ1 $A$nJ1 nJ2
dashes,tension=2nJ1,nJ1J1J4 phantom,tension=3nJ2,nJ2J3J2
scalar,label=$\chi_{1}$,right=1nJ2J3J2,nJ1J1J4
scalar,label=$\chi_{1}$,right=1nJ1J1J4,nJ2J3J2

201 (70,40) arrow_len3mm arrow_ang20 nJ1 $A$nJ1 nJ2
dashes,tension=2nJ1,nJ1J1J4 phantom,tension=3nJ2,nJ2J2J3
scalar,label=$\chi_{2}$,right=1nJ2J2J3,nJ1J1J4
scalar,label=$\chi_{2}$,right=1nJ1J1J4,nJ2J2J3

Figure 1: Feynman diagrams contributing to the one-loop tadpole of the CP odd
state.

The sum of diagrams is divergent. In fact,

$\left(T_{A}\right)\Big{|}_{\text{div}}=-\dfrac{1}{\varepsilon}\dfrac{{c_{\phi}}\,{s_{\phi}}\,\left(M_{1}^{2}-M_{2}^{2}\right)\,\text{Im}[f_{2}]}{8\,\sqrt{2}\,\pi^{2}\,c_{\beta}},$
(29)

in $d=4-2\varepsilon$ dimensions, where $T_{A}$ represents the one-loop
tadpole for $A$ and $\big{|}_{\text{div}}$ means that we consider only
divergent parts.

The origin of the problem lies in the fact that we imposed $\mu^{2}$ and
$\lambda_{5}$ to be real. To clarify this point, let us provisionally take
these parameters to be complex, as they originally were. By rewriting eq.
(19a) in terms of bare quantities, and separating the real and imaginary parts
of $\mu_{0}^{2}$ and $\lambda_{5,0}$, the terms proportional to these
parameters are:

$\begin{split}&-\mathcal{L}_{\Phi_{0}}\ni\mu_{0}^{2}\,\Phi_{1,0}^{*}\Phi_{2,0}+(\mu_{0}^{2})^{*}\,\Phi_{2,0}^{*}\Phi_{1,0}\\\
&\hskip
28.45274pt+\lambda_{5,0}\,\left(\Phi_{1,0}^{*}\Phi_{2,0}\right)^{2}+\lambda_{5,0}^{*}\,\left(\Phi_{2,0}^{*}\Phi_{1,0}\right)^{2}\\\
&=\text{Re}[\mu_{0}^{2}](\Phi_{1,0}^{*}\Phi_{2,0}+\Phi_{2,0}^{*}\Phi_{1,0})\\\
&\hskip
14.22636pt+\text{Re}[\lambda_{5,0}]\Big{\\{}\left(\Phi_{1,0}^{*}\Phi_{2,0}\right)^{2}+\left(\Phi_{2,0}^{*}\Phi_{1,0}\right)^{2}\Big{\\}}\\\
&\hskip
14.22636pt+i\,\text{Im}[\mu_{0}^{2}]\left(\Phi_{1,0}^{*}\Phi_{2,0}-\Phi_{2,0}^{*}\Phi_{1,0}\right)\\\
&\hskip
14.22636pt+i\,\text{Im}[\lambda_{5,0}]\Big{\\{}\left(\Phi_{1,0}^{*}\Phi_{2,0}\right)^{2}-\left(\Phi_{2,0}^{*}\Phi_{1,0}\right)^{2}\Big{\\}}.\end{split}$
(30)

As a consequence, when we set
$\text{Im}[\mu_{0}^{2}]=\text{Im}[\lambda_{5,0}]=0$, we are not including in
the model the terms of the two last lines of eq. (30). Naturally, since such
terms are not in the model, there is no counterterm for the parameters
involved therein. That is, there is neither $\text{Im}[\delta\mu^{2}]$ nor
$\text{Im}[\delta\lambda_{5}]$.999The situation would not be different if we
decided to exclude any other term from the theory. For example, had we decided
not to include the term proportional to $\lambda_{1}$ in the model, there
would be no counterterm $\delta\lambda_{1}$. Now, it is a matter of course
that this would not be a problem if the fact that we did not include the terms
in the last two lines of eq. (30) would follow from a symmetry that forbade
them. In other words, should there be a symmetry in the theory that proscribed
those terms, they could logically not be included; and since the symmetry
would prevent any Green’s functions generated by such terms from showing up,
there would never be divergences involved therein, so that the absence of
counterterms for them would never be a problem. So, for example, if CP was a
symmetry of the theory, it would preclude those terms, in which case the
absence of $\text{Im}[\delta\mu^{2}]$ and $\text{Im}[\delta\lambda_{5}]$ would
not be inconsistent.

However, CP is not a symmetry of theory: even if we try to impose it in the
$\mathcal{L}_{\Phi}$ sector, it still is violated in the
$\mathcal{L}_{\Phi\chi}$ sector through the phases of $f_{1}$ and $f_{2}$, as
Eqs. (20a) and (20b) show. So, there is no CP symmetry forbidding the terms in
the last two lines of eq. (30). As a consequence, even if we exclude them, CP
violating radiative effects can nonetheless contribute to the Green’s
functions involved therein. Such Green’s functions will in general be
divergent; but since the last two lines of eq. (30) were not included in the
theory, there will in general not be enough counterterms to absorb them.

We have already seen one example of Green’s function that indeed cannot be
renormalized: the one-loop 1-point function $T_{A}$. Other examples are the
one-loop CP violating 2-point functions $\Sigma^{G^{0}h}$, $\Sigma^{G^{0}H}$,
$\Sigma^{Ah}$ and $\Sigma^{AH}$ for the scalar-pseudoscalar mixing of
$G^{0}\,h$, $G^{0}\,H$, $A\,h$ and $A\,H$, respectively. Their Feynman
diagrams are represented in Fig. 2.

3 (60,30) arrow_len3mm arrow_ang20 nJ1 $A,G_{0}$nJ1 nJ2 $h,H$nJ2
dashes,tension=3nJ1,nJ1J2J3 dashes,tension=3nJ2,nJ2J1J4
scalar,label=$\chi_{1}$,right=1nJ1J2J3,nJ2J1J4
scalar,label=$\chi_{2}$,right=1nJ2J1J4,nJ1J2J3

4 (60,39) arrow_len3mm arrow_ang20 nJ1 $A,G_{0}$nJ1 nJ2 $h,H$nJ2
dashes,tension=3nJ1,nJ1J1J4 dashes,tension=3nJ2,nJ2J3J2
scalar,label=$\chi_{2}$,right=1nJ2J3J2,nJ1J1J4
scalar,label=$\chi_{2}$,right=1nJ1J1J4,nJ2J3J2

1 (60,30) arrow_len3mm arrow_ang20 nJ1 $A$nJ1 nJ2 $h,H$nJ2
dashes,tension=3nJ1,nJ1J1J4 dashes,tension=3nJ2,nJ2J3J2
scalar,label=$\chi_{1}$,right=1nJ2J3J2,nJ1J1J4
scalar,label=$\chi_{1}$,right=1nJ1J1J4,nJ2J3J2

2 (60,30) arrow_len3mm arrow_ang20 nJ1 $A$nJ1 nJ2 $h,H$nJ2
dashes,tension=3nJ1,nJ1J1J4 dashes,tension=3nJ2,nJ2J2J3
scalar,label=$\chi_{1}$,right=1nJ2J2J3,nJ1J1J4
scalar,label=$\chi_{2}$,right=1nJ1J1J4,nJ2J2J3

Figure 2: Feynman diagrams contributing to the one-loop CP violating 2-point
functions.

Just like in the case of $T_{A}$, there simply is no counterterm for these
functions, which nonetheless are divergent. Their divergent parts are:

$\displaystyle\Sigma^{G^{0}h}(k^{2})\Big{|}_{\text{div}}=\Sigma^{AH}(k^{2})\Big{|}_{\text{div}}$
$\displaystyle\hskip
19.91692pt=-\dfrac{1}{\varepsilon}\,\sin(\beta-\theta)\,\frac{\text{Im}\left[f_{2}\right]\left(f_{1}+f_{2}\tan\beta\right)}{16\pi^{2}},$
(31a)
$\displaystyle\Sigma^{Ah}(k^{2})\Big{|}_{\text{div}}=-\Sigma^{G^{0}H}(k^{2})\Big{|}_{\text{div}}$
$\displaystyle\hskip
19.91692pt=-\dfrac{1}{\varepsilon}\,\cos(\beta-\theta)\,\frac{\text{Im}\left[f_{2}\right]\left(f_{1}+f_{2}\tan\beta\right)}{16\pi^{2}}.$
(31b)

In conclusion, the fact that we imposed $\mu^{2}$ and $\lambda_{5}$ to be real
leads to several divergences that cannot be removed by counterterms.

There are two ways to heal this model: either CP is imposed as a whole, or it
is not imposed at all. In the first case, all the three relations in eqs. (20)
should be verified, which implies that there is a basis where $\mu^{2}$,
$\lambda_{5}$, $f_{1}$ and $f_{2}$ are all real. In this scenario, therefore,
CP violating Green’s functions are precluded, which implies, in particular,
that no divergent CP violating Green’s functions will appear in any order.
This is consistent with what we obtained in eqs. (29) and (31), which vanish
in the limit of real $f_{1}$ and $f_{2}$. In the second case, $\mu^{2}$,
$\lambda_{5}$, $f_{1}$ and $f_{2}$ are in general complex parameters, which
implies that their counterterms are also in general complex. Since CP is
violated, there are no scalar states with well-defined CP, and Green’s
functions will in general be CP violating. The model is renormalizable as long
as all the terms compatible with the symmetries are included. Finally, note
that, in such a CP violating scenario, there may be regions of the parameter
space in which $\lambda_{5}$ and $\mu^{2}$ are real, and $f_{1}$ and $f_{2}$
complex. But this is a completely different situation from that where one
builds a theory taking ab initio $\lambda_{5}$ and $\mu^{2}$ real, while
$f_{1}$ and $f_{2}$ in general complex. In fact, while the former situation
corresponds to a particular solution of a consistent, renormalizable theory,
the latter suffers from the inconsistencies we have shown.

##  4 Three-loop tadpole for $A$ in the real 2HDM

Our goal is to check whether the complex phases of the fermion mixing matrices
introduce CP violating effects into the otherwise CP conserving scalar sector
of the real 2HDM via radiative corrections. For this purpose, we focus on the
effects of quark-mixing. The quantity that signifies quark-induced CP
violation in a convention independent way is the Jarlskog invariant $J$ in eq.
(13). So, we are looking for radiative corrections to the 2HDM which contain
this quantity. As the simplest check, we have looked for diagrams proportional
to $J$, contributing to the $A$ tadpole. As argued above, this can only happen
in amplitudes with at least four vertices, each containing a factor of
$V_{u_{\alpha}d_{j}}$ and additionally, a vertex to couple to $A$ (none of the
$A$-couplings have CKM-factors). Therefore, the first possible appearance of
$J$ in $A$-tadpoles is at three loops. This is indeed what we find at the
amplitude-level.

An example of a pair of diagrams yielding the Jarlskog invariant is shown in
Fig. 3.

three-loop-tadpole-J1 thin.7pt dash_len1.5mm arrow_len2.5mm wiggly_len2.2mm
arrow_len2.5mm (70,50) l $A$l r t b phantomvrb,r,vrt phantomvlb,l,vlt
phantom,tension=1.5vlt,t,vrt phantom,tension=1.5vlb,b,vrb phantomr,vr,vl,l
dashes,tension=2l,vl
fermion,label.side=left,label.dist=3,label=$u_{\alpha}$vlb,vl,vlt
fermionvlt,vrt fermion,left=0.5vrt,vrb fermionvrb,vlb
dashes_arrow,right=0.5,label.side=left,label.dist=2,label=$H^{-}$vlb,vlt
wiggly,right=0.5,label.side=left,label.dist=2,label=$W^{\vphantom{-}}$vrt,vrb
label=$d_{j}$,label.angle=-90,label.dist=0.t
label=$d_{i}$,label.angle=90,label.dist=0.b
label=$u_{\beta}$,label.angle=180,label.dist=0.05r

(a)

three-loop-tadpole-J2 thin.7pt dash_len1.5mm arrow_len2.5mm wiggly_len2.2mm
arrow_len2.5mm (70,50) l $A$l r t b phantomvrt,r,vrb phantomvlt,l,vlb
phantom,tension=1.5vlb,t,vrb phantom,tension=1.5vlt,b,vrt phantomr,vr,vl,l
dashes,tension=2l,vl
fermion,label.side=right,label.dist=3,label=$u_{\alpha}$vlb,vl,vlt
fermionvlt,vrt fermion,right=0.5vrt,vrb fermionvrb,vlb
dashes_arrow,left=0.5,label.side=right,label.dist=2,label=$H^{-}$vlb,vlt
wiggly,left=0.5,label.side=right,label.dist=2,label=$W^{\vphantom{-}}$vrt,vrb
label=$d_{j}$,label.angle=-90,label.dist=0.t
label=$d_{i}$,label.angle=90,label.dist=0.b
label=$u_{\beta}$,label.angle=180,label.dist=0.05r

(b)

Figure 3: Example of pair of Feynman diagrams where $J$ factorizes (for fixed
$\alpha,\beta,i,j$). They differ only by the direction of fermion flow (or
equivalently by the exchange of the two down-type quarks,
$d_{i}\leftrightarrow d_{j}$).

If contributions of this kind were divergent, then one would lack the
respective counterterm needed to absorb these divergences within the real
2HDM. Therefore, we checked for the existence of a leading
$1/\varepsilon^{3}$-pole in said contributions, which would be a strong
indication for the necessity of a genuine three-loop $A$-tadpole counterterm.
For this purpose, we generated all three-loop $A$-tadpole amplitudes $T_{A}$
with the only condition being that a fixed set of quarks
$\\{u_{\alpha},u_{\beta},d_{i},d_{j}\\}$ must be contained. Other
contributions are CP conserving operators and, thus, irrelevant to our
discussion.

Our calculation was carried out in three independent ways (two numeric; one
analytical), fully explained in section 5. The result is101010Notice that the
angle $\beta$ in $s_{\beta}c_{\beta}$ is the angle in eq. (2), while in all
other instances of eq. (32), $\beta$ refers to the up-type quark being
considered. Here and henceforth, which $\beta$ is meant should be clear from
the context.  tadpole-A0 thin.7pt dash_len1.5mm

$\displaystyle\left(T_{A}\right)^{\alpha i}_{\beta j}$
$\displaystyle=-i\,\Big{(}\;\begin{gathered}\vspace{-4pt}\fmfgraph*(30,40)\fmfright{t}\fmfleft{b}\fmf{dashes,label=\footnotesize{$A$},label.dist=2}{b,t}\fmfblob{14}{t}\end{gathered}\quad\Big{)}^{u_{\alpha}d_{i}}_{u_{\beta}d_{j}}$
$\displaystyle=\frac{g^{5}}{8\varepsilon^{3}m_{W}^{3}s_{\beta}c_{\beta}}M^{\alpha
i}_{\beta j}\,I^{\alpha i}_{\beta j}+\mathcal{O}(\varepsilon^{-2}),$ (32)

where there is no sum over repeated indices, and

$\displaystyle M^{\alpha i}_{\beta j}=$
$\displaystyle(m_{u_{\alpha}}^{2}-m_{u_{\beta}}^{2})(m_{d_{i}}^{2}-m_{d_{j}}^{2})$
$\displaystyle\times(m_{u_{\alpha}}^{2}-m_{d_{i}}^{2}+m_{u_{\beta}}^{2}-m_{d_{j}}^{2})\,.$
(33)

The fact that such different calculational techniques yielded the same result
is truly significant.

Remarkably, when summing over all different sets of up- and down-type quark
contributions, the leading pole vanishes exactly. Indeed, it is easy to show
that summing the combination $M^{\alpha i}_{\beta j}I^{\alpha i}_{\beta j}$
over all the nine distinct sets of four different quarks (two up-type and two
down-type), the result vanishes. Notice that both $M^{\alpha i}_{\beta j}$ and
$I^{\alpha i}_{\beta j}$ are antisymmetric under $\alpha\leftrightarrow\beta$
(or $i\leftrightarrow j$). Thus, the vanishing of eq. (32) is not due to the
simple symmetry reasons mentioned in connection with eq. (14). It is the
specific form of the mass term $M^{\alpha i}_{\beta j}$ in eq. (33) which
makes this possible. We cannot see how one would have guessed from the start
this rather peculiar mass combination. We resonate with Khriplovich and
Pospelov’s remark in the context of edm that: “We cannot get rid of the
feeling that this simple result (…) should have a simple transparent
explanation. Unfortunately, we have not been able to find it.”

But the physical consequence is quite clear:

$\sum_{\alpha<\beta}\sum_{i<j}\left(T_{A}\right)^{\alpha i}_{\beta
j}=\mathcal{O}(\varepsilon^{-2}).$ (34)

It remains uncertain whether this cancellation has a physical origin or it is
to be interpreted as _accidental_. There is the possibility that the next
order $1/\varepsilon^{2}$-poles would be non-vanishing. Otherwise, a genuine
CP violating tadpole counterterm for $A$ would only become relevant at the
four-loop level.

##  5 Details of the calculation

In this section, we discuss our derivation of eq. (32). First note that a
complete calculation of the renormalized three-loop tadpole for $A$ would
require the full renormalization of the model at both one-loop and two-loop
order. Although unlikely, one cannot exclude the possibility that combinations
of the one- and two-loop counterterms of a CP-conserving scalar sector
conspire to cancel the divergences of a CP-violating three-loop tadpole.
Secondly, one caveat in our calculation is the treatment of amplitudes with an
uneven number of $\gamma$-matrices together with $\gamma_{5}$. We chose to
work in _naive dimensional regularisation_ with the expectation that the
leading $\varepsilon$-poles do not depend on the choice of a
$\gamma_{5}$-scheme. This claim is supported by the findings in Belusca-
Maito:2020ala  at the one-loop level.

As mentioned before, at least three generations of quarks are necessary to
generate a CP-violating tadpole. Therefore, we focus on a particular set of
diagrams. Let us then define $S^{\\{dcbt\\}}$ as the set of all the three-loop
tadpole diagrams for $A$ containing the quarks $d$, $c$, $b$, $t$. We started
by generating the amplitudes for $S^{\\{dcbt\\}}$ in an $R_{\xi}$-gauge. We
did this through two independent softwares: FeynMaster Fontes:2019wqh  (which
makes use of FeynRules Christensen:2008py  and QGRAF Nogueira:1991ex ) and
FeynArts Hahn:2000kx .111111It is worth emphasizing that the three-loop
tadpole amplitudes generated with FeynArts and FeynMaster coincide. At three
loops, there are 360 amplitudes containing the quarks $d,c,b$ and $t$.
However, 120 among them involve two closed loops of fermions, which means that
they can never factorize the Jarlskog invariant $J$; and since a) the tadpole
for $A$ violates $CP$ and b) all the $CP$-violation in the real 2HDM must be
proportional to $J$ (with $J$ being the only rephasing-invariant quantity
signifying CP-violation), those diagrams must sum up to zero. We checked this
explicitly using FeynCalc Mertig:1990an ; Shtabovenko:2016sxi ;
Shtabovenko:2020gxv .

We then focused on the remaining 240 diagrams. After simplifying the Lorentz
and Dirac algebra of the 240 diagrams with FeynCalc, another 32 diagrams, such
as the ones with two internal W-boson loops, vanish immediately (in naive
dimensional regularisation) due to the chirality of the interactions involved.
This eventually left us with 208 diagrams which can be categorized as follows:

  1. 1.

The first group of diagrams can be generated from connecting $A$ to any
fermion line in Fig. 4 and the corresponding ones with reversed fermion flow
giving 16 diagrams. The same goes for all possible vector boson and scalar
insertions, namely

$\\{HW,WH,HG,GH,HH,GW,WG,GG\\}.$

Diagrams where we connect $A$ to a line with an attached $W$-loop vanish
though, leaving us with $8\times 16-4\times 4=112$ diagrams.

three-loop-tadpole-C1a thin.7pt dash_len1.5mm arrow_len2.5mm wiggly_len2.2mm
arrow_len2.5mm (0,10)(0,10) (60,50) lb,lt rb,rt
fermion,label.side=left,label=$c$lb,lt fermion,label.side=left,label=$d$lt,rt
fermion,label.side=left,label=$t$rt,rb fermion,label.side=left,label=$b$rb,lb
dashes_arrow,right=0.6,label.side=right,label.dist=2,label=$H^{-}$lb,lt
wiggly,right=0.6,label.side=left,label.dist=2,label=$W^{\vphantom{-}}$rt,rb

(a)

three-loop-tadpole-C1b thin.7pt dash_len1.5mm arrow_len2.5mm wiggly_len2.2mm
arrow_len2.5mm (0,10)(0,10) (60,50) lb,lt rb,rt
fermion,label.side=left,label=$d$lb,lt fermion,label.side=left,label=$t$lt,rt
fermion,label.side=left,label=$b$rt,rb fermion,label.side=left,label=$c$rb,lb
dashes_arrow,right=0.6,label.side=right,label.dist=2,label=$H^{+}$lb,lt
wiggly,right=0.6,label.side=left,label.dist=2,label=$W^{\vphantom{-}}$rt,rb

(b)

Figure 4: Attaching $A$ to the fermion lines of these diagrams and the ones
with reversed fermion flow generates the first category of relevant diagrams.

  2. 2.

An example of the second group is shown in Fig. 5. From this diagram and the
one with reversed fermion flow, we get 8 diagrams by cyclic permutations of
the fermions. The $A$ can be connected to either $\\{WH,HW,GH,HG\\}$ and we
can have either a $W$-, $H$-, or $G$-loop in the diagram. This gives $3\times
4\times 8=96$ diagrams.

three-loop-tadpole-C2a thin.7pt dash_len1.5mm arrow_len2.5mm wiggly_len2.2mm
arrow_len2.5mm (0,10)(0,10) (80,60) l $A$l r t b phantomvrb,r,vrt
phantomvlb,l,vlt phantom,tension=1.5vlt,t,vrt phantom,tension=1.5vlb,b,vrb
phantomr,vr,vl,l dashes,tension=2l,vl wiggly,label.side=left,label=$W$vl,vlt
dashes_arrow,label.side=left,label=$H^{-}$vlb,vl
fermion,label.side=left,label=$d$vlt,vrt
fermion,label.side=right,label=$t$vrt,vrb
fermion,label.side=left,label=$b$vrb,vlb
fermion,label.side=right,label=$c$vlb,vlt
wiggly,left=0.5,label.side=left,label=$W$vrt,vrb

Figure 5: The second set of relevant diagrams is generated from permutations
of the fermions in this diagram and by replacing the $W$-insertions with a
Goldstone boson.

We proceeded to numerically evaluate the most divergent part of the 208
diagrams using FIESTA Smirnov:2015mct  in Feynman gauge, i.e. at $\xi_{W}=1$.
In order to generate input integrals, the FeynCalc function ApartFF was
essential for decomposing the diagrams via partial fractioning. This
decomposition yielded scalar integrals for which we could easily get an
accurate result from FIESTA. One integral type needed additional attention
though, namely the one with a scalar product in the numerator and five
different propagator factors (see A for a discussion). This type of integral
yielded large error estimates in FIESTA, such that the results could no be
trusted. In order to obtain an exact result for those, we used integration-by-
parts identities with FIRE Smirnov:2019qkx  to decompose this integral type
into a set of scalar integrals. The intermediate steps required to link
FeynCalc, FIRE and FIESTA were performed by two independent sets of private
codes.

For the numerical input values of the scalar sector, one should choose a point
in parameter space which does not violate any theoretical or experimental
constraints. The theoretical bounds include boundedness from below,
perturbative unitarity Kanemura:1993hm ; Akeroyd:2000wc ; Ginzburg:2005dt  as
well as electroweak precision measurements using the oblique parameters S, T
and U Branco:2011iw . The experimental constraints include the exclusion
bounds from Higgs searches at LHC that were verified using HiggsBounds
Bechtle:2013wla ; Bechtle:2020pkv  and the signal strengths for the SM-like
Higgs boson were forced to be within 2$\sigma$ of the fits given in
Khachatryan:2016vau ; Aad:2019mbh . Among the points that pass all
constraints, we pick the following one:

$\begin{split}\alpha&=-0.83797\\\ \beta&=0.73908\\\
m_{H^{\pm}}&=581.18\,\,\text{GeV},\\\ m_{H}&=592.81\,\,\text{GeV},\\\
m_{A}&=597.44\,\,\text{GeV},\\\ m_{12}^{2}&=19.458\,\,\text{TeV}.\end{split}$
(35)

Using these together with eq. (40), the result for the most divergent part of
$S^{\\{dcbt\\}}$ is:

$\left(iT_{A}\right)^{cd}_{tb}=2392.6\,\,(\text{GeV})^{3}\times\dfrac{1}{\varepsilon^{3}}J+\mathcal{O}(\varepsilon^{-2}).$
(36)

We tried to ascertain whether sets of diagrams with different combinations of
quarks could possibly cancel with each other by checking the result for
another set of quarks. For example, consider the set of diagrams
$S^{\\{dcst\\}}$, defined as identical to $S^{\\{dcbt\\}}$ except that the $b$
quark is replaced by an $s$ quark. Using the same point in parameter space
(eq. 35), the result for the most divergent part of $S^{\\{dcst\\}}$ is

$\left(iT_{A}\right)^{cd}_{ts}=-0.91341\,\,(\text{GeV})^{3}\times\dfrac{1}{\varepsilon^{3}}\,J+\mathcal{O}(\varepsilon^{-2}).$
(37)

Clearly, the numbers differ, which might lead one to believe that summing over
the results for all quarks combinations would likely yield a non-zero result.

Only later, our attention was drawn to TVID Bauberger:2017nct ;
Bauberger:2019heh , a software package for the numerical evaluation of
arbitrary three-loop vacuum integrals – see also Martin:2016bgz . The authors
discuss a set of three master integrals into which any three-loop tadpole
diagram can be decomposed. Using the same amplitude decomposition as for our
numerical evaluation, we were then able to map to this set of master-integrals
and use the analytic pole expressions of TVID (or equivalently of Ref.
Freitas:2016zmy ) to acquire the result for the $1/\varepsilon^{3}$-pole shown
in eq. (32).121212This calculation also showed that the evaluation of scalar
three-loop integrals with up to five propagators and different mass scales via
FIESTA yields accurate results for the leading poles. This might be the first
such stress test on this package. After confirming that the analytic results
coincided with our previous numerical findings, we were able to sum over all
possible quark combinations using eq. (32) to find the surprising result of
the poles cancelling, as shown in eq. (34).

##  6 Conclusions

We argued that the real 2HDM can suffer from theoretical inconsistencies, as
the simultaneous enforcement of CP conservation in the potential and allowance
of CP violation in another sector may lead to divergences that cannot be
removed by the counterterms. Because such divergences cannot show up at two-
loop level and below, the unsoundness of the model has been by and large
ignored in the literature. But the problem cannot be dismissed. In order to
highlight its potential theoretical unsoundness we introduced a simple toy
model, characterized by the same inconsistency as the real 2HDM. There, and as
we showed, the irremovable divergences (that are expected at least at three-
loops in the real 2HDM) show up immediately at one-loop level. This simple
example ought to make the point: the real 2HDM could suffer from the same kind
of pathology. We addressed this claim by calculating the leading pole of the
three-loop one-point function of the $A$ field in the real 2HDM. We showed
that, surprisingly, the pole vanishes exactly after summing all contributions.
This does not mean that the model is sound after all, but only that its
unsoundness is likely to be found either at lower order in $1/\varepsilon$ or
upon two-loop renormalization, or possibly at four-loop order. A complete
discussion would require the full one- and two-loop renormalization as well as
a discussion of the $\gamma_{5}$-scheme beyond naive dimensional
regularisation though.

We hope that our work spurs further interest in this subject and that a full
calculation will become possible in the future.

###### Acknowledgements.

We are grateful to Francisco Botella, Renato Fonseca, Paulo Nogueira and
Howard Haber for discussions, as well as to Vladyslav Shtabovenko and
Alexander Smirnov for useful suggestions concerning softwares for multi-loop
evaluation. This work is supported in part by the Portuguese Fundação para a
Ciência e Tecnologia (FCT) under contracts UIDB/ 00777/2020, UIDP/00777/2020,
CERN/FIS-PAR/0004/2017, and PTDC/FIS-PAR/29436/2017. D.F. is also supported by
the Portuguese Fundação para a Ciência e Tecnologia under the project
SFRH/BD/135698/2018. M.L. is supported partially by the DFG Collaborative
Research Center TRR 257 “Particle Physics Phenomenology after the Higgs
Discovery”.

##  Appendix A Integral decomposition

As mentioned in section 5, one integral type in our amplitude decomposition
needed special attention. The one in question is

$\displaystyle U_{5}^{(1,2)}(m_{1},m_{2},m_{3},m_{4},m_{5})$
$\displaystyle=i\frac{e^{3\gamma_{E}\varepsilon}}{\pi^{3d/2}}\int\\!\\!\mathrm{d}^{d}q_{1}\,\mathrm{d}^{d}q_{2}\,\mathrm{d}^{d}q_{3}\frac{q_{1}\cdot
q_{2}}{(q_{1}^{2}-m_{1}^{2})(q_{2}^{2}-m_{2}^{2})(q_{3}^{2}-m_{3}^{2})}$
$\displaystyle\qquad\times\frac{1}{\big{(}(q_{1}-q_{3})^{2}-m_{4}^{2}\big{)}\big{(}(q_{2}-q_{3})^{2}-m_{5}^{2}\big{)}}.$
(38)

Evaluating integrals of this kind with FIESTA yields small yet non-zero error
estimates already for the leading pole which is why the numerical results
could not be trusted. Therefore, we made use of the integration by parts
routines of FIRE to decompose $U_{5}^{(1,2)}$ into a set of scalar integrals,
which in turn yield vanishing error estimates for the leading poles when
evaluated with FIESTA. Using analytic expressions for the leading poles of the
resulting integral decomposition also made it possible to recombine everything
into a joint expression for the leading pole of $U_{5}^{(1,2)}$, viz.

$\displaystyle U_{5}^{(1,2)}$ $\displaystyle(m_{1},m_{2},m_{3},m_{4},m_{5})$
$\displaystyle=$
$\displaystyle\frac{1}{24\varepsilon^{3}}\Big{[}(m_{1}^{2}+m_{2}^{2})^{2}+(m_{2}^{2}+m_{5}^{2})^{2}+(m_{3}^{2}+m_{4}^{2})^{2}$
$\displaystyle\quad\quad+(m_{3}^{2}+m_{5}^{2})^{2}-4(m_{2}^{2}-m_{3}^{2})^{2}+m_{2}^{4}+m_{3}^{4}\Big{]}$
$\displaystyle+\mathcal{O}(\varepsilon^{-2}).$ (39)

##  Appendix B Numerical input values

$\begin{split}m_{W}&=80.358\,\,\text{GeV},\\\ m_{u}&=2.2\times
10^{-3}\,\,\text{GeV},\\\ m_{d}&=4.8\times 10^{-3}\,\,\text{GeV},\\\
m_{c}&=1.4464\,\,\text{GeV},\\\ m_{s}&=0.093\,\,\text{GeV},\\\
m_{t}&=172.5\,\,\text{GeV},\\\ m_{b}&=4.8564\,\,\text{GeV},\\\ e&=0.30812,\\\
\sin\theta_{w}&=0.47206.\end{split}$ (40)

## References

  * (1) G. Aad et al. [ATLAS Collaboration], “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,” Phys. Lett. B 716 (2012) 1, doi:10.1016/j.physletb.2012.08.020 [arXiv:1207.7214 [hep-ex]].
  * (2) S. Chatrchyan et al. [CMS Collaboration], “Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC,” Phys. Lett. B 716 (2012) 30, doi:10.1016/j.physletb.2012.08.021 [arXiv:1207.7235 [hep-ex]].
  * (3) J.F. Gunion, H.E. Haber, G. Kane and S. Dawson, The Higgs Hunter’s Guide (Westview Press, Boulder, CO, 2000).
  * (4) G.C. Branco, P.M. Ferreira, L. Lavoura, M.N. Rebelo, M. Sher and J.P. Silva, “Theory and phenomenology of two-Higgs-doublet models,” Phys. Rept. 516, 1 (2012) [arXiv:1106.0034 [hep-ph]].
  * (5) I. P. Ivanov, “Building and testing models with extended Higgs sectors,” Prog. Part. Nucl. Phys. 95 (2017), 160-208 doi:10.1016/j.ppnp.2017.03.001 [arXiv:1702.03776 [hep-ph]].
  * (6) P. A. Zyla et al. [Particle Data Group], “Review of Particle Physics,” PTEP 2020 (2020) no.8, 083C01 doi:10.1093/ptep/ptaa104
  * (7) L. Altenkamp, S. Dittmaier and H. Rzehak, “Renormalization schemes for the Two-Higgs-Doublet Model and applications to $h\rightarrow WW/ZZ\rightarrow 4\textrm{fermions}$,” JHEP 09 (2017), 134 doi:10.1007/JHEP09(2017)134 [arXiv:1704.02645 [hep-ph]].
  * (8) S. L. Glashow and S. Weinberg, “Natural Conservation Laws for Neutral Currents,” Phys. Rev. D 15 (1977) 1958, doi:10.1103/PhysRevD.15.1958
  * (9) E. A. Paschos, “Diagonal Neutral Currents,” Phys. Rev. D 15 (1977), 1966 doi:10.1103/PhysRevD.15.1966
  * (10) G. Aad et al. [ATLAS and CMS Collaborations], “Measurements of the Higgs boson production and decay rates and constraints on its couplings from a combined ATLAS and CMS analysis of the LHC pp collision data at $\sqrt{s}=7$ and 8 TeV,” JHEP 1608 (2016) 045, doi:10.1007/JHEP08(2016)045 [arXiv:1606.02266 [hep-ex]].
  * (11) J. F. Gunion and H. E. Haber, “The CP conserving two Higgs doublet model: The Approach to the decoupling limit,” Phys. Rev. D 67 (2003) 075019, doi:10.1103/PhysRevD.67.075019 [hep-ph/0207010].
  * (12) N. Cabibbo, “Unitary Symmetry and Leptonic Decays,” Phys. Rev. Lett. 10 (1963), 531-533 doi:10.1103/PhysRevLett.10.531
  * (13) M. Kobayashi and T. Maskawa, “CP Violation in the Renormalizable Theory of Weak Interaction,” Prog. Theor. Phys. 49 (1973), 652-657 doi:10.1143/PTP.49.652
  * (14) I.F. Ginzburg and M. Krawczyk, “Symmetries of two Higgs doublet model and CP violation,” Phys. Rev. D 72, 115013 (2005) [hep-ph/0408011].
  * (15) A.W. El Kaffas, W. Khater, O.M. Ogreid and P. Osland, “Consistency of the two Higgs doublet model and CP violation in top production at the LHC,” Nucl. Phys. B 775, 45 (2007) [hep-ph/0605142].
  * (16) A. Arhrib, E. Christova, H. Eberl and E. Ginina, “CP violation in charged Higgs production and decays in the Complex Two Higgs Doublet Model,” JHEP 1104, 089 (2011) [arXiv:1011.6560 [hep-ph]].
  * (17) A. Barroso, P.M. Ferreira, R. Santos and J.P. Silva, “Probing the scalar-pseudoscalar mixing in the 125 GeV Higgs particle with current data,” Phys. Rev. D 86, 015022 (2012) [arXiv:1205.4247 [hep-ph]].
  * (18) S. Inoue, M.J. Ramsey-Musolf and Y. Zhang, “CP violating phenomenology of flavor conserving two Higgs doublet models,” Phys. Rev. D 89, no. 11, 115023 (2014) [arXiv:1403.4257 [hep-ph]].
  * (19) D. Fontes, J.C. Romão and J.P. Silva, “$h\rightarrow Z\gamma$ in the complex two Higgs doublet model,” JHEP 1412, 043 (2014) [arXiv:1408.2534 [hep-ph]].
  * (20) B. Grzadkowski, O.M. Ogreid and P. Osland, “Measuring CP violation in Two-Higgs-Doublet models in light of the LHC Higgs data,” JHEP 1411, 084 (2014) [arXiv:1409.7265 [hep-ph]].
  * (21) D. Fontes, M. Mühlleitner, J.C. Romão, R. Santos, J.P. Silva and J. Wittbrodt, “The C2HDM revisited,” JHEP 1802, 073 (2018) [arXiv:1711.09419 [hep-ph]].
  * (22) R. Boto, T. V. Fernandes, H. E. Haber, J. C. Romão and J. P. Silva, “Basis-independent treatment of the complex 2HDM,” Phys. Rev. D 101 (2020) no.5, 055023 doi:10.1103/PhysRevD.101.055023 [arXiv:2001.01430 [hep-ph]].
  * (23) K. Cheung, A. Jueid, Y. N. Mao and S. Moretti, Phys. Rev. D 102 (2020) no.7, 075029 doi:10.1103/PhysRevD.102.075029 [arXiv:2003.04178 [hep-ph]].
  * (24) P. M. Ferreira, R. Santos, M. Sher and J. P. Silva, “Could the LHC two-photon signal correspond to the heavier scalar in two-Higgs-doublet models?,” Phys. Rev. D 85 (2012), 035020 doi:10.1103/PhysRevD.85.035020 [arXiv:1201.0019 [hep-ph]].
  * (25) J. Bernon, J. F. Gunion, H. E. Haber, Y. Jiang and S. Kraml, “Scrutinizing the alignment limit in two-Higgs-doublet models. II. mH=125 GeV,” Phys. Rev. D 93 (2016) no.3, 035027 doi:10.1103/PhysRevD.93.035027 [arXiv:1511.03682 [hep-ph]].
  * (26) F. J. Botella and J. P. Silva, “Jarlskog - like invariants for theories with scalars and fermions,” Phys. Rev. D 51 (1995), 3870-3875 doi:10.1103/PhysRevD.51.3870 [arXiv:hep-ph/9411288 [hep-ph]].
  * (27) H. Georgi and D. V. Nanopoulos, “Suppression of Flavor Changing Effects From Neutral Spinless Meson Exchange in Gauge Theories,” Phys. Lett. B 82 (1979), 95-96 doi:10.1016/0370-2693(79)90433-7
  * (28) J. F. Donoghue and L. F. Li, “Properties of Charged Higgs Bosons,” Phys. Rev. D 19 (1979), 945 doi:10.1103/PhysRevD.19.945
  * (29) C. Jarlskog, “Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Violation,” Phys. Rev. Lett. 55 (1985), 1039 doi:10.1103/PhysRevLett.55.1039
  * (30) C. Jarlskog, “A Basis Independent Formulation of the Connection Between Quark Mass Matrices, CP Violation and Experiment,” Z. Phys. C 29 (1985), 491-497 doi:10.1007/BF01565198
  * (31) I. Dunietz, O. W. Greenberg and D. d. Wu, “A Priori Definition of Maximal CP Violation,” Phys. Rev. Lett. 55 (1985), 2935 doi:10.1103/PhysRevLett.55.2935
  * (32) G. C. Branco, L. Lavoura and J. P. Silva, “CP Violation,” Int. Ser. Monogr. Phys. 103 (1999), 1-536.
  * (33) M. E. Pospelov and I. B. Khriplovich, “Electric dipole moment of the W boson and the electron in the Kobayashi-Maskawa model,” Sov. J. Nucl. Phys. 53 (1991), 638-640.
  * (34) M. J. Booth, “The Electric dipole moment of the W and electron in the Standard Model,” [arXiv:hep-ph/9301293 [hep-ph]].
  * (35) A. Pilaftsis, “CP odd tadpole renormalization of Higgs scalar - pseudoscalar mixing,” Phys. Rev. D 58 (1998) 096010 doi:10.1103/PhysRevD.58.096010 [hep-ph/9803297].
  * (36) A. Denner and S. Dittmaier, Phys. Rept. 864 (2020), 1-163 doi:10.1016/j.physrep.2020.04.001 [arXiv:1912.06823 [hep-ph]].
  * (37) H. Bélusca-Maïto, A. Ilakovac, M. Mađor-Božinović and D. Stöckinger, “Dimensional regularization and Breitenlohner-Maison/’t Hooft-Veltman scheme for $\gamma_{5}$ applied to chiral YM theories: full one-loop counterterm and RGE structure,” JHEP 08 (2020) no.08, 024 doi:10.1007/JHEP08(2020)024 [arXiv:2004.14398 [hep-ph]].
  * (38) D. Fontes and J. C. Romão, “FeynMaster: a plethora of Feynman tools,” Comput. Phys. Commun. 256 (2020), 107311 doi:10.1016/j.cpc.2020.107311 [arXiv:1909.05876 [hep-ph]].
  * (39) N. D. Christensen and C. Duhr, “FeynRules - Feynman rules made easy,” Comput. Phys. Commun. 180 (2009), 1614-1641 doi:10.1016/j.cpc.2009.02.018 [arXiv:0806.4194 [hep-ph]].
  * (40) P. Nogueira, “Automatic Feynman graph generation,” J. Comput. Phys. 105 (1993), 279-289 doi:10.1006/jcph.1993.1074
  * (41) T. Hahn, “Generating Feynman diagrams and amplitudes with FeynArts 3,” Comput. Phys. Commun. 140 (2001), 418-431 doi:10.1016/S0010-4655(01)00290-9 [arXiv:hep-ph/0012260 [hep-ph]].
  * (42) R. Mertig, M. Bohm and A. Denner, “FEYN CALC: Computer algebraic calculation of Feynman amplitudes,” Comput. Phys. Commun. 64 (1991), 345-359 doi:10.1016/0010-4655(91)90130-D
  * (43) V. Shtabovenko, R. Mertig and F. Orellana, “New Developments in FeynCalc 9.0,” Comput. Phys. Commun. 207 (2016), 432-444 doi:10.1016/j.cpc.2016.06.008 [arXiv:1601.01167 [hep-ph]].
  * (44) V. Shtabovenko, R. Mertig and F. Orellana, “FeynCalc 9.3: New features and improvements,” Comput. Phys. Commun. 256 (2020), 107478 doi:10.1016/j.cpc.2020.107478 [arXiv:2001.04407 [hep-ph]].
  * (45) A. V. Smirnov, “FIESTA4: Optimized Feynman integral calculations with GPU support,” Comput. Phys. Commun. 204 (2016), 189-199 doi:10.1016/j.cpc.2016.03.013 [arXiv:1511.03614 [hep-ph]].
  * (46) A. V. Smirnov and F. S. Chuharev, “FIRE6: Feynman Integral REduction with Modular Arithmetic,” doi:10.1016/j.cpc.2019.106877 [arXiv:1901.07808 [hep-ph]].
  * (47) S. Kanemura, T. Kubota and E. Takasugi, “Lee-Quigg-Thacker bounds for Higgs boson masses in a two doublet model,” Phys. Lett. B 313 (1993), 155-160 doi:10.1016/0370-2693(93)91205-2 [arXiv:hep-ph/9303263 [hep-ph]].
  * (48) A. G. Akeroyd, A. Arhrib and E. M. Naimi, “Note on tree level unitarity in the general two Higgs doublet model,” Phys. Lett. B 490 (2000), 119-124 doi:10.1016/S0370-2693(00)00962-X [arXiv:hep-ph/0006035 [hep-ph]].
  * (49) I. F. Ginzburg and I. P. Ivanov, “Tree-level unitarity constraints in the most general 2HDM,” Phys. Rev. D 72 (2005), 115010 doi:10.1103/PhysRevD.72.115010 [arXiv:hep-ph/0508020 [hep-ph]].
  * (50) P. Bechtle, O. Brein, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein and K. E. Williams, “$\mathsf{HiggsBounds}-4$: Improved Tests of Extended Higgs Sectors against Exclusion Bounds from LEP, the Tevatron and the LHC,” Eur. Phys. J. C 74 (2014) no.3, 2693 doi:10.1140/epjc/s10052-013-2693-2 [arXiv:1311.0055 [hep-ph]].
  * (51) P. Bechtle, D. Dercks, S. Heinemeyer, T. Klingl, T. Stefaniak, G. Weiglein and J. Wittbrodt, “HiggsBounds-5: Testing Higgs Sectors in the LHC 13 TeV Era,” doi:10.1140/epjc/s10052-020-08557-9 [arXiv:2006.06007 [hep-ph]].
  * (52) G. Aad et al. [ATLAS], “Combined measurements of Higgs boson production and decay using up to $80$ fb-1 of proton-proton collision data at $\sqrt{s}=$ 13 TeV collected with the ATLAS experiment,” Phys. Rev. D 101 (2020) no.1, 012002 doi:10.1103/PhysRevD.101.012002 [arXiv:1909.02845 [hep-ex]].
  * (53) S. Bauberger and A. Freitas, “TVID: Three-loop Vacuum Integrals from Dispersion relations,” [arXiv:1702.02996 [hep-ph]].
  * (54) S. Bauberger, A. Freitas and D. Wiegand, “TVID 2: Evaluation of planar-type three-loop self-energy integrals with arbitrary masses,” JHEP 01 (2020), 024 doi:10.1007/JHEP01(2020)024 [arXiv:1908.09887 [hep-ph]].
  * (55) S. P. Martin and D. G. Robertson, “Evaluation of the general 3-loop vacuum Feynman integral,” Phys. Rev. D 95 (2017) no.1, 016008 doi:10.1103/PhysRevD.95.016008 [arXiv:1610.07720 [hep-ph]].
  * (56) A. Freitas, “Three-loop vacuum integrals with arbitrary masses,” JHEP 11 (2016), 145 doi:10.1007/JHEP11(2016)145 [arXiv:1609.09159 [hep-ph]].


# Notes on an Elementary Proof for the Stability of Persistence Diagrams

Primoz Skraba and Katharine Turner

##  1\. Setup

These notes provide a self contained algorithmic proof bottleneck stability of
persistence diagrams. The only assumption is familiarity with the standard
persistence algorithm. The proof technique itself is a special case of the
proof of $p$-Wasserstein stability for cellular complexes in [1, Section 3].
The proof is further simplified due to the use of bottleneck stability.
Similar ideas can also be found in [2].

As input, fix a finite simplicial complex $K$, endowed with two functions
$f_{0},f_{1},:K\rightarrow\mathbb{R}$, which we assume satisfy the following
conditions:

  1. (1)

The functions are piecewise constant, i.e. the function assign a function
value to each simplex – so for any simplex $\sigma\in K$, the notion
$f_{0}(\sigma)$ and $f_{1}(\sigma)$ make sense.

  2. (2)

The functions are bounded, i.e. for all $\sigma\in K$,
$|f_{0}(\sigma)|<\infty$ and $|f_{1}(\sigma)|<\infty$

  3. (3)

For any $\alpha\in\mathbb{R}$, the sublevel sets $f_{0}^{-1}(-\infty,\alpha]$
and $f_{1}^{-1}(-\infty,\alpha]$ are simplicial complexes.

Notice that the above conditions are just needed for the standard persistence
algorithm from [3] to be well-defined. The function defines an ordering on the
simplices. If each simplex has a unique function value, then the ordering is
total (a linear order). Otherwise, we can extend the partial order to a total
order. If two simplices have the same function value, they are ordered
according to increasing dimension (to ensure that at each step in the ordering
is a valid simplicial complex). If they have the same dimension, an arbitrary
ordering can be chosen (e.g. lexicographical ordering). Our statement will
involve the persistence diagrams of the sub-level set filtrations of $f_{0}$
and $f_{1}$, which we denote $\mathrm{Dgm}(f_{0})$ and $\mathrm{Dgm}(f_{1})$
respectively.

We recall the following definitions:

######  Definition 1.1.

For every point in a diagram $p\in\mathrm{Dgm}$,

  * •

$b(p):$ birth time ($x$-coordinate)

  * •

$d(p):$ death time ($y$-coordinate)

There are several distances between diagrams which have been defined. but the
_bottleneck distance_ is perhaps the best known. The distances are all defined
in terms of a matching. A matching in our context is a set map, $\pi$ between
the between the points in the two diagrams.

######  Definition 1.2.

The bottleneck distance between two diagrams is given by

$d_{B}(\mathrm{Dgm}(f_{0}),\mathrm{Dgm}(f_{1}))=\inf\limits_{\pi\in\mathrm{bijections}}\max\limits_{p\in\mathrm{Dgm}(f_{0})}\max\\{b(p)-d(\pi(p)),b(p)-d(\pi(p))\\}$

In the above theorem, we take the infimum over all bijections between the
diagrams. It is important to observe that we are taking the algorithmic
definition of the diagram where the points in the diagram are precisely the
pairs of simplices returned by the algorithm. This may include points with
multiplicity as well as points on the diagonal, i.e. cycles which immediately
become bounded. In the general case, we must allow points to get matched to an
arbitrary point on the diagonal of the diagram. However, in this restricted
setting, the number of points is always equal and we can consider the distance
over all bijections without the addition of the diagonal – although some of
the points may lie on the diagonal (see the following exercises).

######  Exercise 1.1.

Prove that for a fixed simplicial complex, the number of points in the
persistence diagram is always the same (regardless of the filtration
function).

######  Exercise 1.2.

Prove that the bottleneck distance where we allow points to map to the
diagonal is smaller than the distance defined above.

######  Definition 1.3.

$||f_{0}-f_{1}||_{\infty}=\max\limits_{\sigma\in
K}|f_{0}(\sigma)-f_{1}(\sigma)|$

We will prove the following theorem:

######  Theorem 1.1.

$d_{B}(\mathrm{Dgm}(f_{0}),\mathrm{Dgm}(f_{1}))\leq||f_{0}-f_{1}||_{\infty}$

where $d_{B}(\cdot)$ is the bottleneck distance.

We will use the following three facts about the persistence algorithm:

  1. (1)

The persistence algorithm returns the diagram in the form of a list of pairs
of simplices $(\sigma,\tau)$, i.e. one pair per point in the diagram. In fact,
we take the pairs as the persistent diagrams.

  2. (2)

If the ordering of simplices does not change, the persistence algorithm
returns the same pairs.

  3. (3)

Any extension of a partial order to a total order produces the same diagram.

##  2\. Proof

After running the persistence algorithm on the filtration induced by $f$. For
each point in the computed persistence diagram $p\in\mathrm{Dgm}(f)$, we have
the corresponding pivots. This defines the following map:

$\mathrm{piv}(p)=(\sigma,\tau)$

where $\sigma$ and $\tau$ are the birth and death simplices respectively, i.e.
$f(\sigma)=b(p)$ and $f(\tau)=d(p)$. One way of interpreting this map is that
it returns the pivots in the reduced boundary matrix in the algorithm. For
convenience we use $\mathrm{piv}_{b}(p)$ and $\mathrm{piv}_{d}(p)$ to denote
the simplices corresponding to birth and death times respectively.

To prove the result, we will linearly interpolate between $f_{0}$ and $f_{1}$.
For $t\in[0,1]$ for each $\sigma\in K$

$f_{t}(\sigma)=(1-t)\cdot f_{0}(\sigma)+t\cdot f_{1}(\sigma)$

######  Exercise 2.1.

Show that for each $t$, the sublevel sets each form a simplicial complex.

For two functions $f_{\alpha}$ and $f_{\beta}$ with $\alpha<\beta\in[0,1]$, we
say that the ordering of the simplices does not change if for any two
simplices $\sigma,\tau\in K$, either

$f_{t}(\sigma)\leq f_{t}(\tau)$

for all $t\in[\alpha,\beta]$, or

$f_{t}(\sigma)\geq f_{t}(\tau)$

for all $t\in[\alpha,\beta]$.

######  Assumption 2.1.

We assume that both $f_{0}$ and $f_{1}$ assign a unique value to each simplex,
i.e. the ordering in both filtrations is unique.

######  Lemma 2.1.

There are a finite number of values of $0\leq t\leq 1$, where there exist
simplices $\sigma$ and $\tau$ such that

$f_{t}(\sigma)=f_{t}(\tau)$

###### Proof.

By assumption, $t=0$ and $t=1$, there are no such simplices. Therefore, the
only time two simplices have equivalent values is when their linear
interpolations cross as in the following Figure.

$f_{0}$$f_{1}$$f_{0}(\sigma)$$f_{0}(\tau)$$f_{1}(\sigma)$$f_{1}(\tau)$

So for every pair of simplices, there is at most one intersection. Assuming
there are $n$ simplices, there are at most $\frac{n^{2}-n}{2}$ intersections.
At all other points, the function values will be unique. ∎

######  Lemma 2.2.

Let $\alpha<\beta\in[0,1]$ such that

  1. (1)

There is an $s\in[\alpha,\beta]$ such that $f_{s}$ induces a total order.

  2. (2)

The ordering of the simplices in $[\alpha,\beta]$ does not change .

Then

$d_{B}(\mathrm{Dgm}(f_{\alpha}),\mathrm{Dgm}(f_{\beta}))\leq||f_{\alpha}-f_{\beta}||_{\infty}$

###### Proof.

First we must show that there exists a consistent ordering for the persistence
algorithm in the interval, i.e. for all points in the diagram, the map
$\mathrm{piv}(\cdot)$ is constant. Choose the (unique) ordering induced at
$f_{s}$. Since the ordering does not change throughout the interval, this is a
valid ordering for all $t\in[\alpha,\beta]$. As this ordering does not change,
the persistence pairs returned by the algorithm are the same throughout the
interval.

Now we construct a matching between the diagrams $\mathbf{M}$ which maps
points in $\mathrm{Dgm}(f_{\alpha})$ to points in $\mathrm{Dgm}(f_{\beta})$.
$\mathbf{M}$ maps a point $p\in\mathrm{Dgm}(f_{a})$ to a point
$q\in\mathrm{Dgm}(f_{b})$ if and only if

$\mathrm{piv}(p)=\mathrm{piv}(q)$

Since the bottleneck distance is infimum over all matchings

$d_{B}(\mathrm{Dgm}(f_{\alpha}),\mathrm{Dgm}(f_{\beta}))\leq\max\limits_{p\in\mathrm{Dgm}(f_{\alpha})}\max\\{|b(p)-b(\mathbf{M}(p))|,|d(p)-d(\mathbf{M}(p))\\}$

By definition we have that

$b(p)=f_{\alpha}(\mathrm{piv}_{b}(p)),\quad\quad
d(p)=f_{\alpha}(\mathrm{piv}_{d}(p)),$

and

$b(\mathbf{M}(p))=f_{\beta}(\mathrm{piv}_{b}(\mathbf{M}(p))),\quad\quad
d(\mathbf{M}(p))=f_{\beta}(\mathrm{piv}_{d}(\mathbf{M}(p))),$

Substituting in we conclude the proof.

$\displaystyle d_{B}(\mathrm{Dgm}(f_{a}),\mathrm{Dgm}(f_{b}))$
$\displaystyle\leq\max\limits_{p\in\mathrm{Dgm}(f_{a})}\max\\{|f_{a}(\mathrm{piv}_{b}(p))-f_{b}(\mathrm{piv}_{b}(\mathbf{M}(p)))|,f_{a}(\mathrm{piv}_{d}(p))-f_{b}(\mathrm{piv}_{d}(\mathbf{M}(p)))|\\}$
$\displaystyle=\max\limits_{\sigma\in
K}|f_{a}(\sigma)-f_{b}(\sigma)|=||f_{a}-f_{b}||_{\infty}$

∎

######  Exercise 2.2.

Prove Lemma 2.2 without the assumption of uniqueness on $f_{0}$ and $f_{1}$,
i.e. prove Lemma 2.2 without Assumption 2.1.

We can now complete the proof.

###### Proof of Theorem 1.1.

Recall by Lemma 2.1, that there are only finitely many values of $t\in[0,1]$,
where the values of any two simplices coincide. We denote these values

$t(1)<t(2)<\ldots<t(k)$

and we set $t(0)=0$ and $t(k+1)=1$.

We now observe two facts. First, notice that since we are considering a linear
interpolation so for any $\alpha<\beta\in[0,1]$

$||f_{\alpha}-f_{\beta}||_{\infty}=(\beta-\alpha)||f_{0}-f_{1}||_{\infty}$

Second, if we restrict to each of the intervals $[t(i),t(i+1)]$ for
$i=0,\ldots k$, the conditions of Lemma 2.2 apply. So we have

$\displaystyle
d_{B}(\mathrm{Dgm}(f_{t(i)}),\mathrm{Dgm}(f_{t(i+1)}))\leq||f_{t(i)}-f_{t(i+1)}||_{\infty}$

Using the triangle inequality,

$\displaystyle d_{B}(\mathrm{Dgm}(f_{0}),\mathrm{Dgm}(f_{1}))$
$\displaystyle\leq\sum_{i=0}^{k}d_{B}(\mathrm{Dgm}(f_{t(i)}),\mathrm{Dgm}(f_{t(i+1)}))$
$\displaystyle\leq\sum_{i=0}^{k}||f_{t(i)}-f_{t(i+1)}||_{\infty}$
$\displaystyle=\sum_{i=0}^{k}(t(i+1)-t(i))||f_{0}-f_{1}||_{\infty}$
$\displaystyle=(t(k+1)-t(0))||f_{0}-f_{1}||_{\infty}=||f_{0}-f_{1}||_{\infty}$

which completes the proof. ∎

######  Remark 2.1.

One of the important things to realize in this proof is that although we can
choose a consistent ordering in each interval, there is necessarily a
discontinuity in the choice at the intersection of two intervals (when the
ordering is not consistent). The reason we can do this in the proof is that at
that each of those points in the interpolation, there is consistent ordering
from below and a different consistent ordering from above. At exactly the
function value, there are multiple valid orderings, which by Fact (3), yields
the same diagram.

## References

  * [1] Primoz Skraba and Katharine Turner.  Wasserstein stability for persistence diagrams.  arXiv preprint arXiv:2006.16824, 2020.
  * [2] David Cohen-Steiner, Herbert Edelsbrunner, and Dmitriy Morozov.  Vines and vineyards by updating persistence in linear time.  In Proceedings of the twenty-second annual symposium on Computational geometry, pages 119–126, 2006.
  * [3] Afra Zomorodian and Gunnar Carlsson.  Computing persistent homology.  Discrete & Computational Geometry, 33(2):249–274, 2005.