CP violation in light neutrino oscillations and heavy neutrino decays:
a general and explicit seesaw-bridged correlation

Zhi-zhong Xing1,2,3 ***E-mail<EMAIL_ADDRESS>
1Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
100049, China

2School of Physical Sciences, University of Chinese Academy of Sciences,
Beijing 100049, China

3Center of High Energy Physics, Peking University, Beijing 100871, China

###### Abstract

With the help of a block parametrization of the canonical seesaw flavor
textures in terms of the Euler-like rotation angles and CP-violaing phases, we
derive a general and explicit expression for the Jarlskog invariant of CP
violation in neutrino oscillations and compare it with the CP-violating
asymmetries of heavy Majorana neutrino decays within the minimal seesaw
framework which contains two right-handed neutrino fields. Two simplified
scenarios are discussed to illustrate how direct or indirect the correlation
between these two types of CP violation can be.

##  1 Motivation

To explain why the three known neutrinos (i.e., the mass eigenstates $\nu_{i}$
versus the flavor eigenstates $\nu_{\alpha\rm L}$ for $i=1,2,3$ and
$\alpha=e,\mu,\tau$) have nonzero but tiny masses, one may go beyond the
standard model (SM) of particle physics by adding some new degrees of freedom.
In this regard the canonical seesaw mechanism [1, 2, 3, 4, 5] is not only most
economical but also most natural 111It is also consistent with the essential
spirit of effective field theories for neutrino mass generation at low
energies [6]., because it just introduces the right-handed neutrino fields
(i.e., the flavor eigenstates $N_{\alpha\rm R}$ versus the mass eigenstates
$N_{i}$ for $\alpha=e,\mu,\tau$ and $i=1,2,3$) and allows for the relevant
Yukawa interactions and lepton number violation [7]) which are both harmless
to the fundamentals of the SM. Moreover, the decays of heavy Majorana
neutrinos $N_{i}$ are CP-violating and thus trigger the leptogenesis mechanism
[8] to naturally interpret the matter-antimatter asymmetry observed in the
Universe [9].

When confronting the seesaw mechanism with today’s experimental and
observational data, however, it is necessary to specify the flavor textures of
active and sterile neutrinos with the help of possible flavor symmetries or
phenomenological assumptions (for the recent reviews with extensive
references, see Refs. [10, 11]). An alternative way, which is independent of
any details of model building, is to fully parametrize the seesaw flavor
textures in terms of the Euler-like rotation angles and CP-violating phases
[12, 13]. In particular, one may adopt two $3\times 3$ unitary submatrices to
describe the respective flavor mixing effects in the presumably isolated
active and sterile neutrino sectors, and utilize a $6\times 6$ non-unitary
matrix to characterize the strengths of the corresponding Yukawa interactions
and hence bridge the gap between the two flavor blocks [13]. Such a block
parametrization can not only reflect the salient dynamical features of the
seesaw mechanism but also make it possible to derive general and explicit
expressions of all the observable quantities of massive neutrinos and thus
reveal their intrinsic correlations in an unprecedentedly transparent manner.

In this work let us show the usefulness of this block parametrization by
deriving a general and explicit expression for the Jarlskog invariant ${\cal
J}_{\nu}$ [14, 15, 16] of leptonic CP violation in neutrino oscillations and
comparing it with the CP-violating asymmetries $\varepsilon_{\alpha i}$
between the lepton-number-violating decays of $N_{i}$ and their CP-conjugated
processes within the minimal seesaw framework which contains only two right-
handed neutrino fields [17, 18, 19]. Our motivation of doing so is two-fold:
(1) the T2K long-baseline neutrino oscillation experiment has recently ruled
out ${\cal J}_{\nu}=0$ at the $2\sigma$ level [20], a very encouraging news
for the unremitting efforts to search for CP violation in the lepton sector;
(2) the number of free flavor parameters in the minimal seesaw mechanism is
seven less than that in the standard seesaw mechanism, making the analytical
results of ${\cal J}_{\nu}$ and $\varepsilon_{\alpha i}$ less lengthy and thus
more instructive. Note that the minimal seesaw mechanism is still a viable
benchmark seesaw framework today and has attracted a lot of attention in
neutrino phenomenology (for a recent review with extensive references, see
Ref. [21]).

The remaining parts of this paper are organized as follows. In section 2 we
briefly describe the block parametrization of the active and sterile flavor
textures in the seesaw mechanism. Section 3 is devoted to deriving the
expression of ${\cal J}_{\nu}$ by taking account of the minimal seesaw
framework, and section 4 compares the dependence of ${\cal J}_{\nu}$ on the
original CP phases with that of $\varepsilon_{\alpha i}$ on the same set of CP
phases. In section 5 we discuss two simplified scenarios of the minimal seesaw
mechanism to illustrate how ${\cal J}_{\nu}$ and $\varepsilon_{\alpha i}$ can
be directly or indirectly correlated with each other in a more transparent
way.

##  2 The block parametrization

Before spontaneous electroweak symmetry breaking, the gauge- and Lorentz-
invariant neutrino mass terms in the canonical seesaw framework can be written
as [22]

$\displaystyle-{\cal L}_{\rm seesaw}$ $\displaystyle=$
$\displaystyle\overline{\ell_{\rm L}}\hskip
1.42271ptY_{\nu}\widetilde{H}N_{\rm R}+\frac{1}{2}\hskip
1.42271pt\overline{(N_{\rm R})^{c}}\hskip 1.42271ptM_{\rm R}N_{\rm R}+{\rm
h.c.}$ (1) $\displaystyle=$ $\displaystyle\frac{1}{2}\hskip
1.42271pt\overline{\big{[}\begin{matrix}\nu_{\rm L}&(N_{\rm
R})^{c}\end{matrix}\big{]}}\left(\begin{matrix}{\bf 0}&Y_{\nu}\phi^{0*}\cr
Y^{T}_{\nu}\phi^{0*}&M_{\rm R}\end{matrix}\right)\left[\begin{matrix}(\nu_{\rm
L})^{c}\cr N_{\rm R}\end{matrix}\right]-\overline{l_{\rm L}}\hskip
1.42271ptY_{\nu}N_{\rm R}\phi^{-}+{\rm h.c.}\;,\hskip 14.22636pt$

in which $\ell_{\rm L}=\big{(}\begin{matrix}\nu_{\rm L}&l_{\rm
L}\end{matrix}\big{)}^{T}$ is the leptonic $\rm SU(2)_{\rm L}$ doublet of the
SM with $\nu_{\rm L}=\big{(}\begin{matrix}\nu_{e\rm L}&\nu_{\mu\rm
L}&\nu_{\tau\rm L}\end{matrix}\big{)}^{T}$ and $l_{\rm
L}=\big{(}\begin{matrix}l_{e\rm L}&l_{\mu\rm L}&l_{\tau\rm
L}\end{matrix}\big{)}^{T}$ standing respectively for the column vectors of the
left-handed neutrino and charged lepton fields, $\widetilde{H}\equiv{\rm
i}\sigma_{2}H^{*}$ with
$H=\big{(}\begin{matrix}\phi^{+}&\phi^{0}\end{matrix}\big{)}^{T}$ denoting the
Higgs doublet of the SM and $\sigma_{2}$ being the second Pauli matrix,
$N_{\rm R}=\big{(}\begin{matrix}N_{e\rm R}&N_{\mu\rm R}&N_{\tau\rm
R}\end{matrix}\big{)}^{T}$ represents the column vector of the right-handed
neutrino fields which are the $\rm SU(2)_{\rm L}$ singlets, $(\nu_{\rm
L})^{c}\equiv{\cal C}\overline{\nu_{\rm L}}^{T}$ and $(N_{\rm
R})^{c}\equiv{\cal C}\overline{N_{\rm R}}^{T}$ with $\cal C$ being the charge-
conjugation matrix, $Y_{\nu}$ denotes the Yukawa coupling matrix of neutrinos,
and $M_{\rm R}$ is the symmetric right-handed neutrino mass matrix. Eq. (1)
tells us that the left- and right-handed neutrino fields are bridged by their
Yukawa couplings to the neutral component of the Higgs doublet at the seesaw
scale. After the Higgs potential of the SM is minimized at $\langle
H\rangle\equiv\langle 0|H|0\rangle=v/\sqrt{2}$ with a special direction
characterized by $\langle\phi^{\pm}\rangle=0$ and
$\langle\phi^{0}\rangle=v/\sqrt{2}$, where $v\simeq 246~{}{\rm GeV}$ is the
vacuum expectation value of the Higgs field, the $\rm SU(2)_{\rm L}\times
U(1)_{\rm Y}$ gauge symmetry is spontaneously broken and consequently all the
particles coupled to $\phi^{0}$ or $\phi^{0*}$ acquire their nonzero masses.
In this case the terms associated with $\phi^{\pm}$ in Eq. (1) will disappear,
and those associated with $\phi^{0}$ or $\phi^{0*}$ will become the so-called
Dirac mass matrix $M_{\rm D}\equiv Y_{\nu}v/\sqrt{2}$ or its transposed form.

To clearly see how nonzero but tiny masses of the three active neutrinos arise
in this mechanism after electroweak symmetry breaking, let us diagonalize the
symmetric $6\times 6$ matrix in Eq. (1) as follows:

$\displaystyle\mathbb{U}^{\dagger}\left(\begin{matrix}{\bf 0}&M_{\rm D}\cr
M^{T}_{\rm D}&M_{\rm
R}\cr\end{matrix}\right)\mathbb{U}^{*}=\left(\begin{matrix}D_{\nu}&{\bf
0}\cr{\bf 0}&D_{N}\cr\end{matrix}\right)\;,$ (2)

where $\mathbb{U}$ is a $6\times 6$ unitary matrix, and the diagonal, real and
positive $3\times 3$ matrices $D_{\nu}$ and $D_{N}$ are defined as
$D_{\nu}\equiv{\rm Diag}\big{\\{}m_{1},m_{2},m_{3}\big{\\}}$ and
$D_{N}\equiv{\rm Diag}\big{\\{}M_{1},M_{2},M_{3}\big{\\}}$ with $m_{i}$ and
$M_{i}$ (for $i=1,2,3$) being the respective masses of light and heavy
Majorana neutrinos. We find that decomposing $\mathbb{U}$ into the product of
the following three $6\times 6$ matrices can help a lot to understand the
salient dynamical features of the canonical seesaw mechanism:

$\displaystyle\mathbb{U}=\left(\begin{matrix}I&{\bf 0}\cr{\bf
0}&U^{\prime}_{0}\cr\end{matrix}\right)\left(\begin{matrix}A&R\cr
S&B\cr\end{matrix}\right)\left(\begin{matrix}U_{0}&{\bf 0}\cr{\bf
0}&I\cr\end{matrix}\right)\;,$ (3)

where $I$ denotes the identity matrix, $U_{0}$ and $U^{\prime}_{0}$ are the
$3\times 3$ unitary matrices responsible respectively for the primary flavor
mixing effects in the active and sterile neutrino sectors, and $A$, $B$, $R$
and $S$ are the $3\times 3$ non-unitary matrices characterizing the interplay
between these two sectors with the help of the Yukawa interactions [13]. So
switching off the Yukawa interactions will immediately lead us to
$D_{\nu}={\bf 0}$ or equivalently $m_{1}=m_{2}=m_{3}=0$, in which case $U_{0}$
will lose its physical meaning. Eqs. (2) and (3) allow us to derive out the
exact seesaw relationship between the light (active) and heavy (sterile)
Majorana neutrino masses:

$\displaystyle\left(AU_{0}\right)D_{\nu}\left(AU_{0}\right)^{T}+RD_{N}R^{T}={\bf
0}\;,$ (4)

in which $A$ and $R$ are also correlated with each other via
$AA^{\dagger}+RR^{\dagger}=I$ as a consequence of the unitarity of
$\mathbb{U}$. Note that $U\equiv AU_{0}$ is just the well-known $3\times 3$
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix describing flavor mixing and CP
violation of the light Majorana neutrinos [23, 24, 25], while $R$ is its
analogue measuring the strengths of weak charged-current interactions of the
heavy Majorana neutrinos in the seesaw framework:

$\displaystyle-{\cal L}_{\rm cc}=\frac{g}{\sqrt{2}}\hskip
2.84544pt\overline{\big{(}\begin{matrix}e&\mu&\tau\end{matrix}\big{)}_{\rm
L}}\hskip
2.84544pt\gamma^{\mu}\left[U\left(\begin{matrix}\nu_{1}\cr\nu_{2}\cr\nu_{3}\cr\end{matrix}\right)_{\rm
L}+R\left(\begin{matrix}N_{1}\cr N_{2}\cr N_{3}\cr\end{matrix}\right)_{\rm
L}\hskip 1.42271pt\right]W^{-}_{\mu}+{\rm h.c.}\;,$ (5)

where the charged lepton fields are in their mass eigenstates too. Of course,
$UU^{\dagger}+RR^{\dagger}=I$ holds, and hence the deviation of $R$ from $\bf
0$ (or the deviation of $A$ from $I$) characterizes the departure of $U$ from
its unitary limit $U_{0}$. One should keep in mind that both $D_{\nu}$ and
$U_{0}$ are the consequences of the seesaw mechanism, simply because of

$\displaystyle U_{0}D_{\nu}U^{T}_{0}=\left({\rm i}\hskip
0.85355ptA^{-1}R\right)D_{N}\left({\rm i}\hskip
0.85355ptA^{-1}R\right)^{T}\;.$ (6)

Namely, the derivational seesaw parameters on the left-hand side of Eq. (6)
can be determined by the original seesaw parameters on the right-handed of Eq.
(6). If $M_{\nu}\equiv U_{0}D_{\nu}U^{T}_{0}$ is defined as the effective mass
matrix for the three active Majorana neutrinos, one may easily arrive at the
more familiar seesaw formula $M_{\nu}\simeq-M_{\rm D}M^{-1}_{\rm R}M^{T}_{\rm
D}$ from Eq. (6) in the approximation of $A\simeq B\simeq I$ (i.e., $M_{\rm
D}\simeq RD_{N}U^{\prime T}_{0}$ and $M_{\rm R}\simeq
U^{\prime}_{0}D_{N}U^{\prime T}_{0}$ hold in this case [22]). So Eq. (6)
provides a unique way to constrain the seesaw parameter space by using the
available experimental data on three active neutrinos.

To write out the explicit expressions of $U_{0}$, $A$ and $R$ in terms of the
Euler-like rotation angles and CP phases, we first introduce fifteen $6\times
6$ unitary matrices of the form $O_{ij}$ (for $1\leq i<j\leq 6$) to realize
the block parametrization of $\mathbb{U}$. The $(i,i)$ and $(j,j)$ elements of
$O_{ij}$ are both equal to $c_{ij}\equiv\cos\theta_{ij}$ with $\theta_{ij}$
being a flavor mixing angle and lying in the first quadrant, its other
diagonal entries are all equal to one, its $(i,j)$ and $(j,i)$ elements are
given respectively by $\hat{s}^{*}_{ij}\equiv e^{-{\rm
i}\delta_{ij}}\sin\theta_{ij}$ and $-\hat{s}_{ij}\equiv-e^{{\rm
i}\delta_{ij}}\sin\theta_{ij}$ with $\delta_{ij}$ being a CP-violating phase,
and its other off-diagonal entries are all equal to zero. Then we group these
matrices as follows [13]:

$\displaystyle\left(\begin{matrix}U_{0}&0\cr 0&I\cr\end{matrix}\right)$
$\displaystyle=$ $\displaystyle O_{23}O_{13}O_{12}\;,$
$\displaystyle\left(\begin{matrix}I&0\cr
0&U^{\prime}_{0}\cr\end{matrix}\right)$ $\displaystyle=$ $\displaystyle
O_{56}O_{46}O_{45}\;,$ $\displaystyle\left(\begin{matrix}A&R\cr
S&B\cr\end{matrix}\right)$ $\displaystyle=$ $\displaystyle
O_{36}O_{26}O_{16}O_{35}O_{25}O_{15}O_{34}O_{24}O_{14}\;,\hskip 19.91684pt$
(7)

so as to describe the active flavor sector, the sterile flavor sector and the
interplay between them, respectively, in a way consistent with the salient
features of the seesaw mechanism. The results of $U_{0}$, $U^{\prime}_{0}$,
$A$, $R$, $S$ and $B$ have been given in Refs. [10, 13]. Here we only quote

$\displaystyle
U_{0}=\left(\begin{matrix}c_{12}c_{13}&\hat{s}^{*}_{12}c_{13}&\hat{s}^{*}_{13}\cr-\hat{s}_{12}c_{23}-c_{12}\hat{s}_{13}\hat{s}^{*}_{23}&c_{12}c_{23}-\hat{s}^{*}_{12}\hat{s}_{13}\hat{s}^{*}_{23}&c_{13}\hat{s}^{*}_{23}\cr\hat{s}_{12}\hat{s}_{23}-c_{12}\hat{s}_{13}c_{23}&-c_{12}\hat{s}_{23}-\hat{s}^{*}_{12}\hat{s}_{13}c_{23}&c_{13}c_{23}\cr\end{matrix}\right)\;,$
(8)

and

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}c_{14}c_{15}c_{16}&0&0\cr\vspace{-0.45cm}\cr\begin{array}[]{l}-c_{14}c_{15}\hat{s}_{16}\hat{s}^{*}_{26}-c_{14}\hat{s}_{15}\hat{s}^{*}_{25}c_{26}\\\
-\hat{s}_{14}\hat{s}^{*}_{24}c_{25}c_{26}\end{array}&c_{24}c_{25}c_{26}&0\cr\vspace{-0.45cm}\cr\begin{array}[]{l}-c_{14}c_{15}\hat{s}_{16}c_{26}\hat{s}^{*}_{36}+c_{14}\hat{s}_{15}\hat{s}^{*}_{25}\hat{s}_{26}\hat{s}^{*}_{36}\\\
-c_{14}\hat{s}_{15}c_{25}\hat{s}^{*}_{35}c_{36}+\hat{s}_{14}\hat{s}^{*}_{24}c_{25}\hat{s}_{26}\hat{s}^{*}_{36}\\\
+\hat{s}_{14}\hat{s}^{*}_{24}\hat{s}_{25}\hat{s}^{*}_{35}c_{36}-\hat{s}_{14}c_{24}\hat{s}^{*}_{34}c_{35}c_{36}\end{array}&\begin{array}[]{l}-c_{24}c_{25}\hat{s}_{26}\hat{s}^{*}_{36}-c_{24}\hat{s}_{25}\hat{s}^{*}_{35}c_{36}\\\
-\hat{s}_{24}\hat{s}^{*}_{34}c_{35}c_{36}\end{array}&c_{34}c_{35}c_{36}\cr\end{matrix}\right)\;,$
$\displaystyle R$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\hat{s}^{*}_{14}c_{15}c_{16}&\hat{s}^{*}_{15}c_{16}&\hat{s}^{*}_{16}\cr\vspace{-0.45cm}\cr\begin{array}[]{l}-\hat{s}^{*}_{14}c_{15}\hat{s}_{16}\hat{s}^{*}_{26}-\hat{s}^{*}_{14}\hat{s}_{15}\hat{s}^{*}_{25}c_{26}\\\
+c_{14}\hat{s}^{*}_{24}c_{25}c_{26}\end{array}&-\hat{s}^{*}_{15}\hat{s}_{16}\hat{s}^{*}_{26}+c_{15}\hat{s}^{*}_{25}c_{26}&c_{16}\hat{s}^{*}_{26}\cr\vspace{-0.45cm}\cr\begin{array}[]{l}-\hat{s}^{*}_{14}c_{15}\hat{s}_{16}c_{26}\hat{s}^{*}_{36}+\hat{s}^{*}_{14}\hat{s}_{15}\hat{s}^{*}_{25}\hat{s}_{26}\hat{s}^{*}_{36}\\\
-\hat{s}^{*}_{14}\hat{s}_{15}c_{25}\hat{s}^{*}_{35}c_{36}-c_{14}\hat{s}^{*}_{24}c_{25}\hat{s}_{26}\hat{s}^{*}_{36}\\\
-c_{14}\hat{s}^{*}_{24}\hat{s}_{25}\hat{s}^{*}_{35}c_{36}+c_{14}c_{24}\hat{s}^{*}_{34}c_{35}c_{36}\end{array}&\begin{array}[]{l}-\hat{s}^{*}_{15}\hat{s}_{16}c_{26}\hat{s}^{*}_{36}-c_{15}\hat{s}^{*}_{25}\hat{s}_{26}\hat{s}^{*}_{36}\\\
+c_{15}c_{25}\hat{s}^{*}_{35}c_{36}\end{array}&c_{16}c_{26}\hat{s}^{*}_{36}\cr\end{matrix}\right)\;,\hskip
14.22636pt$ (9)

which are relevant to the seesaw formula in Eq. (4) or Eq. (6). It is now
clear that the nine flavor parameters associated with the three light Majorana
neutrinos (i.e., $m_{i}$, $\theta_{ij}$ and $\delta_{ij}$ for $i,j=1,2,3$ and
$i<j$) can all be determined by the eighteen original seesaw parameters (i.e.,
$M_{i}$, $\theta_{ij}$ and $\delta_{ij}$ for $i=1,2,3$ and $j=4,5,6$) 222Note
that three of the nine phase parameters (or their independent combinations)
appearing in $A$ and $R$ can always be rotated away through a proper
redefinition of the phases of three charged lepton fields, as one can easily
see from Eq. (5). So we are actually left with six nontrivial CP-violating
phases in the seesaw framework. through Eq. (6).

A careful analysis of those currently available electroweak precision
measurements and neutrino oscillation data has put some rather strong
constraints on possible non-unitarity of the PMNS matrix $U$ — the latter is
expected to be of ${\cal O}\left(10^{-3}\right)$ or smaller in magnitude [26,
27, 28, 29, 30, 31]. In this case we find that the nine active-sterile flavor
mixing angles in $A$ and $R$ must be smaller than ${\cal O}(10^{-1.5})$. Such
a phenomenological observation means that $A\simeq I$ and $U\simeq U_{0}$,
together with

$\displaystyle A^{-1}R$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}\hat{t}^{*}_{14}&c^{-1}_{14}\hat{t}^{*}_{15}&c^{-1}_{14}c^{-1}_{15}\hat{t}^{*}_{16}\cr\vspace{-0.45cm}\cr
c^{-1}_{14}\hat{t}^{*}_{24}&\hat{t}_{14}\hat{t}^{*}_{15}\hat{t}^{*}_{24}+c^{-1}_{15}c^{-1}_{24}\hat{t}^{*}_{25}&\begin{array}[]{l}+\hat{t}_{14}c^{-1}_{15}\hat{t}^{*}_{16}\hat{t}^{*}_{24}+\hat{t}_{15}\hat{t}^{*}_{16}c^{-1}_{24}\hat{t}^{*}_{25}\\\
+c^{-1}_{16}c^{-1}_{24}c^{-1}_{25}\hat{t}^{*}_{26}\end{array}\cr\vspace{-0.45cm}\cr
c^{-1}_{14}c^{-1}_{24}\hat{t}^{*}_{34}&\begin{array}[]{l}+\hat{t}_{14}\hat{t}^{*}_{15}c^{-1}_{24}\hat{t}^{*}_{34}+c^{-1}_{15}\hat{t}_{24}\hat{t}^{*}_{25}\hat{t}^{*}_{34}\\\
+c^{-1}_{15}c^{-1}_{25}c^{-1}_{34}\hat{t}^{*}_{35}\end{array}&\begin{array}[]{l}+\hat{t}_{14}c^{-1}_{15}\hat{t}^{*}_{16}c^{-1}_{24}\hat{t}^{*}_{34}+\hat{t}_{15}\hat{t}^{*}_{16}\hat{t}_{24}\hat{t}^{*}_{25}\hat{t}^{*}_{34}\\\
+\hat{t}_{15}\hat{t}^{*}_{16}c^{-1}_{25}c^{-1}_{34}\hat{t}^{*}_{35}+c^{-1}_{16}\hat{t}_{24}c^{-1}_{25}\hat{t}^{*}_{26}\hat{t}^{*}_{34}\\\
+c^{-1}_{16}\hat{t}_{25}\hat{t}^{*}_{26}c^{-1}_{34}\hat{t}^{*}_{35}+c^{-1}_{16}c^{-1}_{26}c^{-1}_{34}c^{-1}_{35}\hat{t}^{*}_{36}\end{array}\cr\end{matrix}\right)\;\hskip
5.69046pt$ (10) $\displaystyle\simeq$
$\displaystyle\left(\begin{matrix}\hat{s}^{*}_{14}&\hat{s}^{*}_{15}&\hat{s}^{*}_{16}\cr\hat{s}^{*}_{24}&\hat{s}^{*}_{25}&\hat{s}^{*}_{26}\cr\hat{s}^{*}_{34}&\hat{s}^{*}_{35}&\hat{s}^{*}_{36}\cr\end{matrix}\right)+{\cal
O}(s^{3}_{ij})+\cdots$

with $\hat{t}_{ij}\equiv e^{{\rm i}\delta_{ij}}\tan\theta_{ij}$ being defined,
should be a reliable approximation (i.e., $c_{ij}\simeq 1$ for sufficiently
small $\theta_{ij}$ with $i=1,2,3$ and $j=4,5,6$). Hence the flavor texture of
$M_{\nu}$ can be easily reconstructed by using the approximate seesaw relation
$M_{\nu}\equiv U_{0}D_{\nu}U^{T}_{0}\simeq-RD_{N}R^{T}$, from which the three
neutrino masses, three flavor mixing angles and three CP-violating phases in
$D_{\nu}$ and $U_{0}$ are consequently extractable. In particular, one may
calculate the effect of leptonic CP violation in neutrino oscillations by
using the original seesaw parameters $M_{i}$, $\theta_{ij}$ and $\delta_{ij}$
(for $i=1,2,3$ and $j=4,5,6$). We are going to explicitly demonstrate this
interesting point in the next section.

##  3 The Jarlskog invariant ${\cal J}_{\nu}$

If the tiny non-unitarity of the $3\times 3$ PMNS flavor mixing matrix $U$ is
neglected within the canonical seesaw framework (i.e., $U\simeq U_{0}$ in the
safe approximation of $A\simeq I$), then the effect of leptonic CP violation
in neutrino oscillations will be uniquely characterized by a universal
Jarlskog invariant ${\cal J}_{\nu}$ which can be defined through [14, 15, 16]

$\displaystyle{\rm Im}\left[\left(U_{0}\right)_{\alpha
i}\left(U_{0}\right)_{\beta j}\left(U_{0}\right)^{*}_{\alpha
j}\left(U_{0}\right)^{*}_{\beta i}\right]={\cal
J}_{\nu}\sum_{\gamma}\epsilon_{\alpha\beta\gamma}\sum_{k}\epsilon_{ijk}\;,$
(11)

in which $\epsilon_{\alpha\beta\gamma}$ and $\epsilon_{ijk}$ denote the three-
dimensional Levi-Civita symbols, and the Greek and Latin subscripts run
respectively over $(e,\mu,\tau)$ and $(1,2,3)$. Given the unitarity of $U_{0}$
and the definition of $M_{\nu}\equiv U_{0}D_{\nu}U^{T}_{0}$, a lengthy but
straightforward calculation leads us to

$\displaystyle{\cal J}_{\nu}=\prod_{i>j}\Delta^{-1}_{ij}\hskip 2.27626pt{\rm
Im}\left[\left(M_{\nu}M^{\dagger}_{\nu}\right)_{e\mu}\left(M_{\nu}M^{\dagger}_{\nu}\right)_{\mu\tau}\left(M_{\nu}M^{\dagger}_{\nu}\right)_{\tau
e}\right]\;,$ (12)

where $\Delta_{ij}\equiv m^{2}_{i}-m^{2}_{j}$ (for $i,j=1,2,3$) are defined
333Note that the neutrino mass ordering remains undetermined today (i.e.,
either $m_{1}<m_{2}<m_{3}$ or $m_{3}<m_{2}<m_{3}$ may be possible, as
indicated by current neutrino oscillation data), but we are absolutely sure
that $\Delta_{31}$ and $\Delta_{32}$ must have the same sign due to the fact
of $\left|\Delta_{31}\right|\simeq\left|\Delta_{32}\right|\gg\Delta_{21}$ [9].
So the product $\Delta_{21}\Delta_{31}\Delta_{32}$ on the right-hand side of
Eq. (12) is actually insensitive to the sign ambiguity of $\Delta_{31}$ and
$\Delta_{32}$.. A global fit of the present experimental data on solar,
atmospheric, reactor and accelerator neutrino oscillations [32] provides us
with the best-fit values $\Delta_{21}=7.42\times 10^{-5}~{}{\rm eV}^{2}$ and
$\Delta_{31}=2.51\times 10^{-3}~{}{\rm eV}^{2}$ (normal neutrino mass
ordering) or $\Delta_{32}=-2.49\times 10^{-3}~{}{\rm eV}^{2}$ (inverted mass
ordering) with relatively small error bars. That is, the product of
$\Delta_{21}$, $\Delta_{31}$ and $\Delta_{32}$ on the right-hand side of Eq.
(12) is already known.

With the help of Eqs. (6) and (10), the right-hand side of Eq. (12) can be
expressed in terms of the original seesaw parameters $M_{i}$, $\theta_{ij}$
and $\delta_{ij}$ (for $i=1,2,3$ and $j=4,5,6$). First of all, we obtain

$\displaystyle\left(M_{\nu}M^{\dagger}_{\nu}\right)_{e\mu}$ $\displaystyle=$
$\displaystyle\hat{s}^{*}_{14}\hat{s}_{24}AM^{2}_{1}+\hat{s}^{*}_{15}\hat{s}_{25}BM^{2}_{2}+\hat{s}^{*}_{16}\hat{s}_{26}CM^{2}_{3}+\left(\hat{s}^{*}_{14}\hat{s}_{25}X+\hat{s}^{*}_{15}\hat{s}_{24}X^{*}\right)M_{1}M_{2}\hskip
14.22636pt$
$\displaystyle+\left(\hat{s}^{*}_{14}\hat{s}_{26}Y+\hat{s}^{*}_{16}\hat{s}_{24}Y^{*}\right)M_{1}M_{3}+\left(\hat{s}^{*}_{15}\hat{s}_{26}Z+\hat{s}^{*}_{16}\hat{s}_{25}Z^{*}\right)M_{2}M_{3}\;,$
$\displaystyle\left(M_{\nu}M^{\dagger}_{\nu}\right)_{\mu\tau}$
$\displaystyle=$
$\displaystyle\hat{s}^{*}_{24}\hat{s}_{34}AM^{2}_{1}+\hat{s}^{*}_{25}\hat{s}_{35}BM^{2}_{2}+\hat{s}^{*}_{26}\hat{s}_{36}CM^{2}_{3}+\left(\hat{s}^{*}_{24}\hat{s}_{35}X+\hat{s}^{*}_{25}\hat{s}_{34}X^{*}\right)M_{1}M_{2}\hskip
14.22636pt$
$\displaystyle+\left(\hat{s}^{*}_{24}\hat{s}_{36}Y+\hat{s}^{*}_{26}\hat{s}_{34}Y^{*}\right)M_{1}M_{3}+\left(\hat{s}^{*}_{25}\hat{s}_{36}Z+\hat{s}^{*}_{26}\hat{s}_{35}Z^{*}\right)M_{2}M_{3}\;,$
$\displaystyle\left(M_{\nu}M^{\dagger}_{\nu}\right)_{\tau e}$ $\displaystyle=$
$\displaystyle\hat{s}_{14}\hat{s}^{*}_{34}AM^{2}_{1}+\hat{s}_{15}\hat{s}^{*}_{35}BM^{2}_{2}+\hat{s}_{16}\hat{s}^{*}_{36}CM^{2}_{3}+\left(\hat{s}_{14}\hat{s}^{*}_{35}X^{*}+\hat{s}_{15}\hat{s}^{*}_{34}X\right)M_{1}M_{2}\hskip
14.22636pt$ (13)
$\displaystyle+\left(\hat{s}_{14}\hat{s}^{*}_{36}Y^{*}+\hat{s}_{16}\hat{s}^{*}_{34}Y\right)M_{1}M_{3}+\left(\hat{s}_{15}\hat{s}^{*}_{36}Z^{*}+\hat{s}_{16}\hat{s}^{*}_{35}Z\right)M_{2}M_{3}\;,$

where $A=s^{2}_{14}+s^{2}_{24}+s^{2}_{34}$,
$B=s^{2}_{15}+s^{2}_{25}+s^{2}_{35}$, $C=s^{2}_{16}+s^{2}_{26}+s^{2}_{36}$,
and

$\displaystyle X$ $\displaystyle=$
$\displaystyle\hat{s}^{*}_{14}\hat{s}_{15}+\hat{s}^{*}_{24}\hat{s}_{25}+\hat{s}^{*}_{34}\hat{s}_{35}\;,$
$\displaystyle Y$ $\displaystyle=$
$\displaystyle\hat{s}^{*}_{14}\hat{s}_{15}+\hat{s}^{*}_{24}\hat{s}_{26}+\hat{s}^{*}_{34}\hat{s}_{36}\;,$
$\displaystyle Z$ $\displaystyle=$
$\displaystyle\hat{s}^{*}_{15}\hat{s}_{16}+\hat{s}^{*}_{25}\hat{s}_{26}+\hat{s}^{*}_{35}\hat{s}_{36}\;.\hskip
14.22636pt$ (14)

Substituting Eqs. (13) and (14) into Eq. (12), one may calculate the imaginary
part of the product of $\left(M_{\nu}M^{\dagger}_{\nu}\right)_{e\mu}$,
$\left(M_{\nu}M^{\dagger}_{\nu}\right)_{\mu\tau}$ and
$\left(M_{\nu}M^{\dagger}_{\nu}\right)_{\tau e}$. But the analytical result
turns out to be too lengthy to be presented in the present short note.
Instead, we proceed to give a simplified result within the minimal seesaw
mechanism which contains only two right-handed neutrino fields.

In this case the number of the original seesaw parameters is reduced from
eighteen to eleven. Namely, we are left with two heavy Majorana neutrino
masses ($M_{1}$ and $M_{2}$), six active-sterile flavor mixing angles
($\theta_{ij}$ for $i=1,2,3$ and $j=4,5$) and three CP-violating phases (i.e.,
three of $\delta_{ij}$ for $i=1,2,3$ and $j=4,5$, or their independent
combinations) in the minimal seesaw framework. The corresponding derivational
parameters for the light Majorana neutrinos are seven, including two nonzero
masses ($m_{2}$ and $m_{3}$ in the normal mass ordering case, or $m_{1}$ and
$m_{2}$ in the inverted mass ordering case), three flavor mixing angles
($\theta_{ij}$ for $ij=12,13,23$) and two nontrivial CP-violating phases
(i.e., two of $\delta_{ij}$ for $ij=12,13,23$, or their proper combinations).
Here let us simply switch off the parameters $M_{3}$, $\theta_{i6}$ and
$\delta_{i6}$ (for $i=1,2,3$) in Eq. (13) to calculate the Jarlskog invariant
${\cal J}_{\nu}$ within the minimal seesaw scenario. We arrive at the
analytical result

$\displaystyle{\rm
Im}\left[\left(M_{\nu}M^{\dagger}_{\nu}\right)_{e\mu}\left(M_{\nu}M^{\dagger}_{\nu}\right)_{\mu\tau}\left(M_{\nu}M^{\dagger}_{\nu}\right)_{\tau
e}\right]$ $\displaystyle=$ $\displaystyle
C_{0}\left[C_{\alpha\beta}\sin\left(\alpha+\beta\right)+C_{\beta\gamma}\sin\left(\beta+\gamma\right)+C_{\gamma\alpha}\sin\left(\gamma+\alpha\right)\right.\hskip
8.5359pt$ (15) $\displaystyle+\hskip
1.9919ptC^{\prime}_{\alpha\beta}\sin\left(\alpha-\beta\right)+C^{\prime}_{\beta\gamma}\sin\left(\beta-\gamma\right)+C^{\prime}_{\gamma\alpha}\sin\left(\gamma-\alpha\right)$
$\displaystyle+\left.C_{2\alpha}\sin 2\alpha+C_{2\beta}\sin
2\beta+C_{2\gamma}\sin 2\gamma\right]\;,$

where $\alpha\equiv\delta_{14}-\delta_{15}$,
$\beta\equiv\delta_{24}-\delta_{25}$ and
$\gamma\equiv\delta_{34}-\delta_{35}$, together with

$\displaystyle C_{0}$ $\displaystyle=$ $\displaystyle
M^{2}_{1}M^{2}_{2}\left[s^{2}_{14}\left(s^{2}_{25}+s^{2}_{35}\right)+s^{2}_{24}\left(s^{2}_{15}+s^{2}_{35}\right)+s^{2}_{34}\left(s^{2}_{15}+s^{2}_{25}\right)\right.$
(16)
$\displaystyle-\left.2s_{14}s_{15}s_{24}s_{25}\cos\left(\alpha-\beta\right)-2s_{24}s_{25}s_{34}s_{35}\cos\left(\beta-\gamma\right)-2s_{14}s_{15}s_{34}s_{35}\cos\left(\gamma-\alpha\right)\right]\;,\hskip
14.22636pt$

and

$\displaystyle C_{2\alpha}$ $\displaystyle=$ $\displaystyle
M_{1}M_{2}s^{2}_{14}s^{2}_{15}\left(s^{2}_{34}s^{2}_{25}-s^{2}_{24}s^{2}_{35}\right)\;,$
$\displaystyle C_{2\beta}$ $\displaystyle=$ $\displaystyle
M_{1}M_{2}s^{2}_{24}s^{2}_{25}\left(s^{2}_{14}s^{2}_{35}-s^{2}_{34}s^{2}_{15}\right)\;,$
$\displaystyle C_{2\gamma}$ $\displaystyle=$ $\displaystyle
M_{1}M_{2}s^{2}_{34}s^{2}_{35}\left(s^{2}_{24}s^{2}_{15}-s^{2}_{14}s^{2}_{25}\right)\;,$
$\displaystyle C_{\alpha\beta}$ $\displaystyle=$ $\displaystyle
M_{1}M_{2}s_{14}s_{15}s_{24}s_{25}\left(s^{2}_{14}s^{2}_{35}-s^{2}_{34}s^{2}_{15}+s^{2}_{34}s^{2}_{25}-s^{2}_{24}s^{2}_{35}\right)\;,$
$\displaystyle C_{\beta\gamma}$ $\displaystyle=$ $\displaystyle
M_{1}M_{2}s_{24}s_{25}s_{34}s_{35}\left(s^{2}_{14}s^{2}_{35}-s^{2}_{34}s^{2}_{15}+s^{2}_{24}s^{2}_{15}-s^{2}_{14}s^{2}_{25}\right)\;,$
$\displaystyle C_{\gamma\alpha}$ $\displaystyle=$ $\displaystyle
M_{1}M_{2}s_{14}s_{15}s_{34}s_{35}\left(s^{2}_{24}s^{2}_{15}-s^{2}_{14}s^{2}_{25}+s^{2}_{34}s^{2}_{25}-s^{2}_{24}s^{2}_{35}\right)\;,$
$\displaystyle C^{\prime}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\left[M^{2}_{1}\left(s^{2}_{14}+s^{2}_{24}+s^{2}_{34}\right)s^{2}_{34}-M^{2}_{2}\left(s^{2}_{15}+s^{2}_{25}+s^{2}_{35}\right)s^{2}_{35}\right]s_{14}s_{15}s_{24}s_{25}\;,$
$\displaystyle C^{\prime}_{\beta\gamma}$ $\displaystyle=$
$\displaystyle\left[M^{2}_{1}\left(s^{2}_{14}+s^{2}_{24}+s^{2}_{34}\right)s^{2}_{14}-M^{2}_{2}\left(s^{2}_{15}+s^{2}_{25}+s^{2}_{35}\right)s^{2}_{15}\right]s_{24}s_{25}s_{34}s_{35}\;,$
$\displaystyle C^{\prime}_{\gamma\alpha}$ $\displaystyle=$
$\displaystyle\left[M^{2}_{1}\left(s^{2}_{14}+s^{2}_{24}+s^{2}_{34}\right)s^{2}_{24}-M^{2}_{2}\left(s^{2}_{15}+s^{2}_{25}+s^{2}_{35}\right)s^{2}_{25}\right]s_{14}s_{15}s_{34}s_{35}\;.\hskip
14.22636pt$ (17)

It becomes obvious how the Jarlskog invariant ${\cal J}_{\nu}$ depends on the
three original CP-violating phases $\alpha$, $\beta$ and $\gamma$. Since these
three phase parameters actually determine the CP-violating effects in those
lepton-number-violating decays of $N_{1}$ and $N_{2}$, one may therefore
establish a general and explicit correlation between CP violation in light
neutrino oscillations and that in heavy Majorana neutrino decays within the
minimal seesaw framework under discussion.

##  4 The CP asymmetries $\varepsilon_{i\alpha}$

Given that the canonical seesaw scale is expected to be far above the scale of
electroweak symmetry breaking, all the SM particles should be exactly massless
when the heavy Majorana neutrinos $N_{i}$ decay. In this case one only needs
to calculate the lepton-number-violating decays of $N_{i}$ into the leptonic
doublet and the Higgs doublet at the one-loop level, so as to determine the
CP-violating asymmetries $\varepsilon_{i\alpha}$ between
$N_{i}\to\ell_{\alpha}+H$ and their CP-conjugated processes [8, 33, 34, 35,
36]. Namely,

$\displaystyle\varepsilon_{i\alpha}$ $\displaystyle\equiv$
$\displaystyle\frac{\Gamma({N}_{i}\to\ell_{\alpha}+H)-\Gamma({N}_{i}\to\overline{\ell_{\alpha}}+\overline{H})}{\displaystyle\sum_{\alpha}\left[\Gamma({N}_{i}\to\ell_{\alpha}+H)+\Gamma({N}_{i}\to\overline{\ell_{\alpha}}+\overline{H})\right]}$
(18) $\displaystyle=$ $\displaystyle\frac{1}{8\pi\big{(}{\cal
Y}^{\dagger}_{\nu}{\cal Y}_{\nu}\big{)}_{ii}}\sum_{j\neq i}\left\\{{\rm
Im}\left[\big{(}{\cal Y}^{*}_{\nu}\big{)}_{\alpha i}\big{(}{\cal
Y}_{\nu}\big{)}_{\alpha j}\big{(}{\cal Y}^{\dagger}_{\nu}{\cal
Y}_{\nu}\big{)}_{ij}\hskip 1.42271pt\xi(x_{ji})+\big{(}{\cal
Y}^{*}_{\nu}\big{)}_{\alpha i}\big{(}{\cal Y}_{\nu}\big{)}_{\alpha
j}\big{(}{\cal Y}^{\dagger}_{\nu}{\cal Y}_{\nu}\big{)}^{*}_{ij}\hskip
1.42271pt\zeta(x_{ji})\right]\right\\}\;,\hskip 14.22636pt$

where ${\cal Y}_{\nu}\equiv Y_{\nu}U^{\prime*}_{0}$ is defined [22], the Latin
and Greek subscripts run respectively over $(1,2,3)$ and $(e,\mu,\tau)$,
$\xi(x_{ji})=\sqrt{x_{ji}}\left\\{1+1/\left(1-x_{ji}\right)+\left(1+x_{ji}\right)\ln\left[x_{ji}/\left(1+x_{ji}\right)\right]\right\\}$
and $\zeta(x_{ji})=1/\left(1-x_{ji}\right)$ with
$x_{ji}\equiv{M}^{2}_{j}/{M}^{2}_{i}$ [41]. The key point of thermal
leptogenesis is that a net lepton-antilepton asymmetry can arise from nonzero
$\varepsilon_{i\alpha}$ in the early Universe, and later on it can be partly
converted into a net baryon-antibaryon asymmetry via the sphaleron
interactions [8, 42, 43], providing a qualitatively elegant explanation of why
the primordial antibaryons have essentially disappeared in the Universe.

To see how the CP-violating asymmetries $\varepsilon_{i\alpha}$ are explicitly
dependent upon the original CP phases in the minimal seesaw mechanism with
only two right-handed neutrino fields, let us first substitute $Y_{\nu}\simeq
RD_{N}U^{\prime T}_{0}/\langle\phi^{0}\rangle$ into ${\cal Y}_{\nu}\equiv
Y_{\nu}U^{\prime*}_{0}$ and then substitute the expression of ${\cal Y}_{\nu}$
into Eq. (18) in the approximation of $A\simeq B\simeq I$. After switching off
the flavor mixing angles $\theta_{i6}$ (for $i=1,2,3$) associated with $N_{3}$
in Eqs. (9) and (10), we are left with

$\displaystyle\varepsilon_{1e}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{2}s_{14}s_{15}}{8\pi\langle\phi^{0}\rangle^{2}\left(s^{2}_{14}+s^{2}_{24}+s^{2}_{34}\right)}\Big{[}\xi(x_{21})\big{[}s_{14}s_{15}\sin
2\alpha+s_{24}s_{25}\sin\left(\alpha+\beta\right)+s_{34}s_{35}\sin\left(\alpha+\gamma\right)\big{]}$
$\displaystyle+\hskip
1.42271pt\zeta(x_{21})\big{[}s_{24}s_{25}\sin\left(\alpha-\beta\right)+s_{34}s_{35}\sin\left(\alpha-\gamma\right)\big{]}\Big{]}\;,$
$\displaystyle\varepsilon_{1\mu}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{2}s_{24}s_{25}}{8\pi\langle\phi^{0}\rangle^{2}\left(s^{2}_{14}+s^{2}_{24}+s^{2}_{34}\right)}\Big{[}\xi(x_{21})\big{[}s_{14}s_{15}\sin\left(\alpha+\beta\right)+s_{24}s_{25}\sin
2\beta+s_{34}s_{35}\sin\left(\beta+\gamma\right)\big{]}$ $\displaystyle+\hskip
1.42271pt\zeta(x_{21})\big{[}s_{14}s_{15}\sin\left(\beta-\alpha\right)+s_{34}s_{35}\sin\left(\beta-\gamma\right)\big{]}\Big{]}\;,$
$\displaystyle\varepsilon_{1\tau}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{2}s_{34}s_{35}}{8\pi\langle\phi^{0}\rangle^{2}\left(s^{2}_{14}+s^{2}_{24}+s^{2}_{34}\right)}\Big{[}\xi(x_{21})\big{[}s_{14}s_{15}\sin\left(\alpha+\gamma\right)+s_{24}s_{25}\sin\left(\beta+\gamma\right)+s_{34}s_{35}\sin
2\gamma\big{]}$ (19) $\displaystyle+\hskip
1.42271pt\zeta(x_{21})\big{[}s_{14}s_{15}\sin\left(\gamma-\alpha\right)+s_{24}s_{25}\sin\left(\gamma-\beta\right)\big{]}\Big{]}\;,$

for the decays of $N_{1}$; and

$\displaystyle\varepsilon_{2e}$ $\displaystyle=$
$\displaystyle-\frac{M^{2}_{1}s_{14}s_{15}}{8\pi\langle\phi^{0}\rangle^{2}\left(s^{2}_{15}+s^{2}_{25}+s^{2}_{35}\right)}\Big{[}\xi(x_{12})\big{[}s_{14}s_{15}\sin
2\alpha+s_{24}s_{25}\sin\left(\alpha+\beta\right)+s_{34}s_{35}\sin\left(\alpha+\gamma\right)\big{]}$
$\displaystyle+\hskip
1.42271pt\zeta(x_{12})\big{[}s_{24}s_{25}\sin\left(\alpha-\beta\right)+s_{34}s_{35}\sin\left(\alpha-\gamma\right)\big{]}\Big{]}\;,$
$\displaystyle\varepsilon_{2\mu}$ $\displaystyle=$
$\displaystyle-\frac{M^{2}_{1}s_{24}s_{25}}{8\pi\langle\phi^{0}\rangle^{2}\left(s^{2}_{15}+s^{2}_{25}+s^{2}_{35}\right)}\Big{[}\xi(x_{12})\big{[}s_{14}s_{15}\sin\left(\alpha+\beta\right)+s_{24}s_{25}\sin
2\beta+s_{34}s_{35}\sin\left(\beta+\gamma\right)\big{]}$ $\displaystyle+\hskip
1.42271pt\zeta(x_{12})\big{[}s_{14}s_{15}\sin\left(\beta-\alpha\right)+s_{34}s_{35}\sin\left(\beta-\gamma\right)\big{]}\Big{]}\;,$
$\displaystyle\varepsilon_{2\tau}$ $\displaystyle=$
$\displaystyle-\frac{M^{2}_{1}s_{34}s_{35}}{8\pi\langle\phi^{0}\rangle^{2}\left(s^{2}_{15}+s^{2}_{25}+s^{2}_{35}\right)}\Big{[}\xi(x_{12})\big{[}s_{14}s_{15}\sin\left(\alpha+\gamma\right)+s_{24}s_{25}\sin\left(\beta+\gamma\right)+s_{34}s_{35}\sin
2\gamma\big{]}$ (20) $\displaystyle+\hskip
1.42271pt\zeta(x_{12})\big{[}s_{14}s_{15}\sin\left(\gamma-\alpha\right)+s_{24}s_{25}\sin\left(\gamma-\beta\right)\big{]}\Big{]}\;,$

for the decays of $N_{2}$, where the phase parameters
$\alpha\equiv\delta_{14}-\delta_{15}$, $\beta\equiv\delta_{24}-\delta_{25}$
and $\gamma\equiv\delta_{34}-\delta_{35}$ have been defined as in Eqs. (15)
and (16). A sum of $\varepsilon_{i\alpha}$ over $\alpha$ (for
$\alpha=e,\mu,\tau$) immediately leads us to the flavor-independent CP-
violating asymmetries

$\displaystyle\varepsilon_{1}$ $\displaystyle=$
$\displaystyle+\frac{M^{2}_{2}\xi(x_{21})}{8\pi\langle\phi^{0}\rangle^{2}\left(s^{2}_{14}+s^{2}_{24}+s^{2}_{34}\right)}\big{[}s^{2}_{14}s^{2}_{15}\sin
2\alpha+s^{2}_{24}s^{2}_{25}\sin 2\beta+s^{2}_{34}s^{2}_{35}\sin 2\gamma$
$\displaystyle+\hskip
1.9919pt2s_{14}s_{15}s_{24}s_{25}\sin\left(\alpha+\beta\right)+2s_{14}s_{15}s_{34}s_{35}\sin\left(\alpha+\gamma\right)+2s_{24}s_{25}s_{34}s_{35}\sin\left(\beta+\gamma\right)\big{]}\;,$
$\displaystyle\varepsilon_{2}$ $\displaystyle=$
$\displaystyle-\frac{M^{2}_{1}\xi(x_{12})}{8\pi\langle\phi^{0}\rangle^{2}\left(s^{2}_{15}+s^{2}_{25}+s^{2}_{35}\right)}\big{[}s^{2}_{14}s^{2}_{15}\sin
2\alpha+s^{2}_{24}s^{2}_{25}\sin 2\beta+s^{2}_{34}s^{2}_{35}\sin 2\gamma$ (21)
$\displaystyle+\hskip
1.9919pt2s_{14}s_{15}s_{24}s_{25}\sin\left(\alpha+\beta\right)+2s_{14}s_{15}s_{34}s_{35}\sin\left(\alpha+\gamma\right)+2s_{24}s_{25}s_{34}s_{35}\sin\left(\beta+\gamma\right)\big{]}\;,\hskip
11.38092pt$

in which the terms associated with the loop functions $\zeta(x_{21})$ and
$\zeta(x_{12})$ in Eqs. (19) and (20) have been exactly cancelled out. It is
obvious that the three independent CP phases $\alpha$, $\beta$ and $\gamma$
appear both in the formula of ${\cal J}_{\nu}$ and in the expressions of
$\varepsilon_{i\alpha}$ and $\varepsilon_{i}$ (for $i=1,2$ and
$\alpha=e,\mu,\tau$), implying a kind of correlation between CP violation in
light neutrino oscillations (with lepton number conservation) and that in
heavy neutrino decays (with lepton number violation). In general, however,
there is no proportional relation between these two types of CP-violating
effects. This observation means that $\varepsilon_{i\alpha}=0$ (or
$\varepsilon_{i}=0$) does not necessarily result in ${\cal J}_{\nu}=0$, or
vice versa.

Table 1: A brief summary of the dependence of ${\cal J}_{\nu}$, $\varepsilon_{i\alpha}$ and $\varepsilon_{i}$ (for $i=1,2$ and $\alpha=e,\mu,\tau$) on the three CP phases $\alpha$, $\beta$ and $\gamma$ in the minimal seesaw mechanism, where the symbol “$\surd$” means the existence of the $\sin 2\alpha$ term and so on.  | $\sin 2\alpha$ | $\sin 2\beta$ | $\sin 2\gamma$ | $\sin\left(\alpha+\beta\right)$ | $\sin\left(\alpha+\gamma\right)$ | $\sin\left(\beta+\gamma\right)$ | $\sin\left(\alpha-\beta\right)$ | $\sin\left(\beta-\gamma\right)$ | $\sin\left(\gamma-\alpha\right)$
---|---|---|---|---|---|---|---|---|---
${\cal J}_{\nu}$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$
$\varepsilon_{1e}$ | $\surd$ |  |  | $\surd$ | $\surd$ |  | $\surd$ |  | $\surd$
$\varepsilon_{1\mu}$ |  | $\surd$ |  | $\surd$ |  | $\surd$ | $\surd$ | $\surd$ |
$\varepsilon_{1\tau}$ |  |  | $\surd$ |  | $\surd$ | $\surd$ |  | $\surd$ | $\surd$
$\varepsilon_{1}$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ |  |  |
$\varepsilon_{2e}$ | $\surd$ |  |  | $\surd$ | $\surd$ |  | $\surd$ |  | $\surd$
$\varepsilon_{2\mu}$ |  | $\surd$ |  | $\surd$ |  | $\surd$ | $\surd$ | $\surd$ |
$\varepsilon_{2\tau}$ |  |  | $\surd$ |  | $\surd$ | $\surd$ |  | $\surd$ | $\surd$
$\varepsilon_{2}$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ | $\surd$ |  |  |

Table 1 is a brief summary of the dependence of ${\cal J}_{\nu}$,
$\varepsilon_{i\alpha}$ and $\varepsilon_{i}$ (for $i=1,2$ and
$\alpha=e,\mu,\tau$) on the three original CP phases $\alpha$, $\beta$ and
$\gamma$ in the minimal seesaw mechanism. Note that the remarkable result of
${\cal J}_{\nu}$ obtained in Eqs. (12) and (15)—(17) has been expressed in
terms of the original seesaw parameters, and hence the resulting correlation
between ${\cal J}_{\nu}$ and $\varepsilon_{i\alpha}$ (or $\varepsilon_{i}$) is
more straightforward and more transparent than that derived from the basis-
independent flavor invariants in the seesaw-based effective field theory [44,
45, 46, 47]. Our analytical results are expected to be useful for a systematic
analysis of the parameter space of CP violation in the minimal seesaw
framework, once more experimental and observational data are available in the
foreseeable future.

##  5 Two simplified scenarios

We consider two special scenarios of the active-sterile flavor mixing pattern
to simplify the expressions of ${\cal J}_{\nu}$, $\varepsilon_{i\alpha}$ and
$\varepsilon_{i}$ (for $i=1,2$ and $\alpha=e,\mu,\tau$) obtained above, so as
to arrive at a more direct correlation between the effects of CP violation in
light neutrino oscillations and heavy neutrino decays.

###  5.1 Scenario A: two texture zeros

Taking $\theta_{15}=\theta_{34}=0$ leads us to a simple but viable texture of
$M_{\rm D}\propto R$ with the vanishing $(1,2)$ and $(3,1)$ entries, as first
proposed in Ref. [19]. In this case only the coefficients $C_{0}$ and
$C_{2\beta}$ in Eq. (15) are nonzero, and hence we are left with

$\displaystyle{\cal
J}_{\nu}=\frac{M^{3}_{1}M^{3}_{2}}{\Delta_{21}\Delta_{31}\Delta_{32}}s^{2}_{14}s^{2}_{24}s^{2}_{25}s^{2}_{35}\left[s^{2}_{14}s^{2}_{25}+s^{2}_{14}s^{2}_{35}+s^{2}_{24}s^{2}_{35}\right]\sin
2\beta\;,$ (22)

which depends only on a single phase parameter $\beta$. On the other hand, we
obtain

$\displaystyle\varepsilon_{1}$ $\displaystyle=$
$\displaystyle\varepsilon_{1\mu}=+\frac{M^{2}_{2}\xi(x_{21})s^{2}_{24}s^{2}_{25}}{8\pi\langle\phi^{0}\rangle^{2}\left(s^{2}_{14}+s^{2}_{24}\right)}\sin
2\beta\;,$ $\displaystyle\varepsilon_{2}$ $\displaystyle=$
$\displaystyle\varepsilon_{2\mu}=-\frac{M^{2}_{1}\xi(x_{12})s^{2}_{24}s^{2}_{25}}{8\pi\langle\phi^{0}\rangle^{2}\left(s^{2}_{25}+s^{2}_{35}\right)}\sin
2\beta\;,\hskip 11.38092pt$ (23)

which depend on the same phase parameter $\beta$, together with
$\varepsilon_{1e}=\varepsilon_{2e}=\varepsilon_{1\tau}=\varepsilon_{2\tau}=0$.
This scenario can therefore establish a direct link between CP violation in
neutrino oscillations and that in heavy Majorana neutrino decays (i.e., ${\cal
J}_{\nu}\propto\varepsilon_{1}\propto\varepsilon_{2}$). The latter is
responsible for the validity of thermal leptogenesis [19, 21], so as to
interpret the cosmic matter-antimatter asymmetry of the Universe.

One may wonder whether the direct proportionality between $J_{\nu}$ and
$\varepsilon_{1,2}$ obtained above will be lost if the active-sterile flavor
mixing angles are not small and hence the approximation made in Eq. (10) is
not reliable anymore. Within the minimal seesaw framework, the exact
expression of $A^{-1}R$ in the assumption of $\theta_{15}=\theta_{34}=0$
becomes

$\displaystyle A^{-1}R=\left(\begin{matrix}\hat{t}^{*}_{14}&0\cr
c^{-1}_{14}\hat{t}^{*}_{24}&c^{-1}_{24}\hat{t}^{*}_{25}\cr
0&c^{-1}_{25}\hat{t}^{*}_{35}\cr\end{matrix}\right)\;.$ (24)

In this case the results of $J_{\nu}$ and $\varepsilon_{1,2}$ can be directly
obtained from Eqs. (22) and (23) with the replacements
$\hat{s}^{*}_{14}\to\hat{t}^{*}_{14}$, $\hat{s}^{*}_{24}\to
c^{-1}_{14}\hat{t}^{*}_{24}$, $\hat{s}^{*}_{25}\to
c^{-1}_{24}\hat{t}^{*}_{25}$ and $\hat{s}^{*}_{35}\to
c^{-1}_{25}\hat{t}^{*}_{35}$. It turns out that these three CP-violating
quantities are all proportional to $\sin 2\beta$ and thus proportional to one
another. But such a simple proportionality between $J_{\nu}$ and
$\varepsilon_{1,2}$ is highly subject to the two-zero texture of $R$ under
consideration, and hence it should not be expected in general.

###  5.2 Scenario B: flavor mixing democracy

Now we assume that all the six active-sterile flavor mixing angles
$\theta_{ij}$ (for $i=1,2,3$ and $j=4,5$) are equal to $\theta_{0}$ in a
simplified minimal seesaw scenario, but their associated CP phases
$\delta_{ij}$ should be different from one another to assure that $\alpha$,
$\beta$ and $\gamma$ are not all vanishing. In this case only the coefficients
$C_{0}$, $C^{\prime}_{\alpha\beta}$, $C^{\prime}_{\beta\gamma}$ and
$C^{\prime}_{\gamma\alpha}$ in Eq. (15) can be nonzero, leading us to

$\displaystyle{\cal J}_{\nu}$ $\displaystyle=$ $\displaystyle
48\frac{M^{2}_{1}M^{2}_{2}\left(M^{2}_{2}-M^{2}_{1}\right)}{\Delta_{21}\Delta_{31}\Delta_{32}}s^{12}_{0}\sin\frac{\alpha-\beta}{2}\sin\frac{\beta-\gamma}{2}\sin\frac{\gamma-\alpha}{2}$
(25)
$\displaystyle\times\left[\sin^{2}\frac{\alpha-\beta}{2}+\sin^{2}\frac{\beta-\gamma}{2}+\sin^{2}\frac{\gamma-\alpha}{2}\right]\;,$

where $s_{0}\equiv\sin\theta_{0}$ for $\theta_{ij}=\theta_{0}$. In comparison,
the flavor-independent CP-violating asymmetries $\varepsilon_{1}$ and
$\varepsilon_{2}$ for $N_{1}$ and $N_{2}$ decays turn out to be

$\displaystyle\varepsilon_{1}$ $\displaystyle=$
$\displaystyle+\frac{M^{2}_{2}\xi(x_{21})s^{2}_{0}}{12\pi\langle\phi^{0}\rangle^{2}}\left(\sin\alpha+\sin\beta+\sin\gamma\right)\left(\cos\alpha+\cos\beta+\cos\gamma\right)\;,$
$\displaystyle\varepsilon_{2}$ $\displaystyle=$
$\displaystyle-\frac{M^{2}_{1}\xi(x_{12})s^{2}_{0}}{12\pi\langle\phi^{0}\rangle^{2}}\left(\sin\alpha+\sin\beta+\sin\gamma\right)\left(\cos\alpha+\cos\beta+\cos\gamma\right)\;.\hskip
11.38092pt$ (26)

It is obvious that the dependence of $\varepsilon_{1}$ and $\varepsilon_{2}$
on $\alpha$, $\beta$ and $\gamma$ is quite different from that of ${\cal
J}_{\nu}$. The same is true for the dependence of the flavor-dependent CP-
violating asymmetries $\varepsilon_{i\alpha}$ (for $i=1,2$ and
$\alpha=e,\mu,\tau$) on $\alpha$, $\beta$ and $\gamma$, as compared with that
of ${\cal J}_{\nu}$. In other words, it is impossible to achieve a simple
proportional relation like $\varepsilon_{i\alpha}\propto{\cal J}_{\nu}$ in the
flavor mixing democracy case. A special example is
$\alpha=\beta=\gamma\equiv\phi_{0}$, which simply gives rise to
$\varepsilon_{1}\propto\varepsilon_{2}\propto\sin 2\phi_{0}$ and ${\cal
J}_{\nu}=0$.

Scenarios A and B tell us that a careful analysis of the CP-violating
quantities $\varepsilon_{i\alpha}$ and ${\cal J}_{\nu}$ is absolutely
necessary even in the minimal seesaw framework, because all the special
correlations between them proposed in the literature are strongly model-
dependent (see, e.g., Ref. [21]).

A general and explicit seesaw-bridged correlation between CP violation in
light neutrino oscillations and that in heavy Majorana neutrino decays is
certainly more complicated in the canonical case with three right-handed
neutrino fields, but it deserves a comprehensive study in the precision
measurement era of neutrino physics. The relevant results will be presented in
another paper.

The preliminary result of this work was first reported in my talk given at the
International Conference on the Physics of the Two Infinities, which was held
at University of Kyoto from 27 to 30 March 2023. My recent research has been
supported by the National Natural Science Foundation of China under grant No.
12075254 and grant No. 11835013.

## References

  * [1] P. Minkowski, “$\mu\to e\gamma$ at a rate of one out of $10^{9}$ muon decays?,” Phys. Lett. 67B (1977) 421.
  * [2] T. Yanagida, “Horizontal gauge symmetry and masses of neutrinos,” Conf. Proc. C 7902131 (1979) 95.
  * [3] M. Gell-Mann, P. Ramond and R. Slansky, “Complex spinors and unified theories,” Conf. Proc. C 790927 (1979) 315 [arXiv:1306.4669 [hep-th]].
  * [4] S. L. Glashow, “The future of elementary particle physics,” NATO Sci. Ser. B 61 (1980) 687.
  * [5] R. N. Mohapatra and G. Senjanovic, “Neutrino mass and spontaneous parity nonconservation,” Phys. Rev. Lett. 44 (1980) 912.
  * [6] S. Weinberg, “Baryon and Lepton Nonconserving Processes,” Phys. Rev. Lett. 43 (1979), 1566-1570
  * [7] E. Majorana, “Teoria simmetrica dell’elettrone e del positrone,” Nuovo Cim. 14 (1937), 171-184.
  * [8] M. Fukugita and T. Yanagida, “Baryogenesis Without Grand Unification,” Phys. Lett. B 174 (1986), 45-47.
  * [9] R. L. Workman et al. [Particle Data Group], “Review of Particle Physics,” PTEP 2022 (2022), 083C01.
  * [10] Z. z. Xing, “Flavor structures of charged fermions and massive neutrinos,” Phys. Rept. 854 (2020), 1-147 [arXiv:1909.09610 [hep-ph]].
  * [11] F. Feruglio and A. Romanino, “Lepton flavor symmetries,” Rev. Mod. Phys. 93 (2021) no.1, 015007 [arXiv:1912.06028 [hep-ph]].
  * [12] Z. z. Xing, “Correlation between the Charged Current Interactions of Light and Heavy Majorana Neutrinos,” Phys. Lett. B 660 (2008), 515-521 [arXiv:0709.2220 [hep-ph]].
  * [13] Z. z. Xing, “A full parametrization of the 6$\times$6 flavor mixing matrix in the presence of three light or heavy sterile neutrinos,” Phys. Rev. D 85 (2012), 013008 [arXiv:1110.0083 [hep-ph]].
  * [14] C. Jarlskog, “Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal $CP$ Nonconservation,” Phys. Rev. Lett. 55 (1985), 1039.
  * [15] D. d. Wu, “The Rephasing Invariants and CP,” Phys. Rev. D 33 (1986), 860.
  * [16] H. Y. Cheng, “Kobayashi-Maskawa Type of Hard CP Violation Model With Three Generation Majorana Neutrinos,” Phys. Rev. D 34 (1986), 2794.
  * [17] A. Kleppe, “Extending the standard model with two right-handed neutrinos,” in Neutrino physics (Proceedings of the 3rd Tallinn Symposium, Lohusalu, Estonia, October 8-11, 1995), page 118-125.
  * [18] E. Ma, D. P. Roy and U. Sarkar, “A Seesaw model for atmospheric and solar neutrino oscillations,” Phys. Lett. B 444 (1998), 391-396 [arXiv:hep-ph/9810309 [hep-ph]].
  * [19] P. H. Frampton, S. L. Glashow and T. Yanagida, “Cosmological sign of neutrino CP violation,” Phys. Lett. B 548 (2002), 119-121 [arXiv:hep-ph/0208157 [hep-ph]].
  * [20] K. Abe et al. [T2K], “Measurements of neutrino oscillation parameters from the T2K experiment using $3.6\times 10^{21}$ protons on target,” [arXiv:2303.03222 [hep-ex]].
  * [21] Z. z. Xing and Z. h. Zhao, “The minimal seesaw and leptogenesis models,” Rept. Prog. Phys. 84 (2021) no.6, 066201 [arXiv:2008.12090 [hep-ph]].
  * [22] Z. z. Xing, “The formal seesaw mechanism of Majorana neutrinos with unbroken gauge symmetry,” Nucl. Phys. B 987 (2023), 116106 [arXiv:2301.10461 [hep-ph]].
  * [23] B. Pontecorvo, “Mesonium and anti-mesonium,” Sov. Phys. JETP 6 (1957) 429 [Zh. Eksp. Teor. Fiz. 33 (1957) 549].
  * [24] Z. Maki, M. Nakagawa and S. Sakata, “Remarks on the unified model of elementary particles,” Prog. Theor. Phys. 28 (1962) 870.
  * [25] B. Pontecorvo, “Neutrino Experiments and the Problem of Conservation of Leptonic Charge,” Sov. Phys. JETP 26 (1968) 984 [Zh. Eksp. Teor. Fiz. 53 (1967) 1717].
  * [26] S. Antusch, C. Biggio, E. Fernandez-Martinez, M. B. Gavela and J. Lopez-Pavon, “Unitarity of the Leptonic Mixing Matrix,” JHEP 10 (2006), 084 [arXiv:hep-ph/0607020 [hep-ph]].
  * [27] S. Antusch, S. Blanchet, M. Blennow and E. Fernandez-Martinez, “Non-unitary Leptonic Mixing and Leptogenesis,” JHEP 01 (2010), 017 [arXiv:0910.5957 [hep-ph]].
  * [28] M. Blennow, P. Coloma, E. Fernandez-Martinez, J. Hernandez-Garcia and J. Lopez-Pavon, “Non-Unitarity, sterile neutrinos, and Non-Standard neutrino Interactions,” JHEP 04 (2017), 153 [arXiv:1609.08637 [hep-ph]].
  * [29] Z. Hu, J. Ling, J. Tang and T. Wang, “Global oscillation data analysis on the $3\nu$ mixing without unitarity,” JHEP 01 (2021), 124 [arXiv:2008.09730 [hep-ph]].
  * [30] Y. Wang and S. Zhou, “Non-unitary leptonic flavor mixing and CP violation in neutrino-antineutrino oscillations,” Phys. Lett. B 824 (2022), 136797 [arXiv:2109.13622 [hep-ph]].
  * [31] M. Blennow, E. Fernandez-Martinez, J. Hernandez-Garcia, J. Lopez-Pavon, X. Marcano and D. Naredo-Tuero, “Bounds on lepton non-unitarity and heavy neutrino mixing,” [arXiv:2306.01040 [hep-ph]].
  * [32] M. C. Gonzalez-Garcia, M. Maltoni and T. Schwetz, “NuFIT: Three-Flavour Global Analyses of Neutrino Oscillation Experiments,” Universe 7 (2021) no.12, 459 [arXiv:2111.03086 [hep-ph]].
  * [33] M. A. Luty, “Baryogenesis via leptogenesis,” Phys. Rev. D 45 (1992), 455-465.
  * [34] L. Covi, E. Roulet and F. Vissani, “CP violating decays in leptogenesis scenarios,” Phys. Lett. B 384 (1996), 169-174 [arXiv:hep-ph/9605319 [hep-ph]].
  * [35] M. Plumacher, “Baryogenesis and lepton number violation,” Z. Phys. C 74 (1997), 549-559 [arXiv:hep-ph/9604229 [hep-ph]].
  * [36] A. Pilaftsis, “CP violation and baryogenesis due to heavy Majorana neutrinos,” Phys. Rev. D 56 (1997), 5431-5451 [arXiv:hep-ph/9707235 [hep-ph]].
  * [37] M. Drewes, “The Phenomenology of Right Handed Neutrinos,” Int. J. Mod. Phys. E 22 (2013), 1330019 [arXiv:1303.6912 [hep-ph]].
  * [38] L. Canetti, M. Drewes, T. Frossard and M. Shaposhnikov, “Dark Matter, Baryogenesis and Neutrino Oscillations from Right Handed Neutrinos,” Phys. Rev. D 87 (2013), 093006 [arXiv:1208.4607 [hep-ph]].
  * [39] M. Drewes, J. Klarić and P. Klose, “On lepton number violation in heavy neutrino decays at colliders,” JHEP 11 (2019), 032 [arXiv:1907.13034 [hep-ph]].
  * [40] M. Chrzaszcz, M. Drewes, T. E. Gonzalo, J. Harz, S. Krishnamurthy and C. Weniger, “A frequentist analysis of three right-handed neutrinos with GAMBIT,” Eur. Phys. J. C 80 (2020) no.6, 569 [arXiv:1908.02302 [hep-ph]].
  * [41] Z. z. Xing and S. Zhou, “Neutrinos in particle physics, astronomy and cosmology,” Springer-Verlag (2011), ISBN 978-7-308-08024-8.
  * [42] V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, “On the Anomalous Electroweak Baryon Number Nonconservation in the Early Universe,” Phys. Lett. B 155 (1985), 36.
  * [43] J. A. Harvey and M. S. Turner, “Cosmological baryon and lepton number in the presence of electroweak fermion number violation,” Phys. Rev. D 42 (1990), 3344-3349.
  * [44] Y. Wang, B. Yu and S. Zhou, “Flavor invariants and renormalization-group equations in the leptonic sector with massive Majorana neutrinos,” JHEP 09 (2021), 053 [arXiv:2107.06274 [hep-ph]].
  * [45] B. Yu and S. Zhou, “Hilbert series for leptonic flavor invariants in the minimal seesaw model,” JHEP 10 (2021), 017 [arXiv:2107.11928 [hep-ph]].
  * [46] B. Yu and S. Zhou, “Spelling out leptonic CP violation in the language of invariant theory,” Phys. Rev. D 106 (2022) no.5, L051701 [arXiv:2203.00574 [hep-ph]].
  * [47] B. Yu and S. Zhou, “CP violation and flavor invariants in the seesaw effective field theory,” JHEP 08 (2022), 017 [arXiv:2203.10121 [hep-ph]].


[afterskip=-.5em]subsubsection

Copyright for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).

NeSy 2022, 16th International Workshop on Neural-Symbolic Learning and
Reasoning, Cumberland Lodge, Windsor, UK

[orcid=0000-0002-1112-2981<EMAIL_ADDRESS>] [1]

[orcid=0000-0003-3902-2867<EMAIL_ADDRESS>] [1]

[1]Corresponding author.

# CLEVR-Math: A Dataset for Compositional Language, Visual and Mathematical
Reasoning

Adam Dahlgren Lindström  Umeå University, Sweden  Savitha Sam Abraham  Örebro
University, Sweden

(2022)

###### Abstract

We introduce CLEVR-Math, a multi-modal math word problems dataset consisting
of simple math word problems involving addition/subtraction, represented
partly by a textual description and partly by an image illustrating the
scenario. The text describes actions performed on the scene that is depicted
in the image. Since the question posed may not be about the scene in the
image, but about the state of the scene before or after the actions are
applied, the solver envision or imagine the state changes due to these
actions. Solving these word problems requires a combination of language,
visual and mathematical reasoning. We apply state-of-the-art neural and neuro-
symbolic models for visual question answering on CLEVR-Math and empirically
evaluate their performances. Our results show how neither method generalise to
chains of operations. We discuss the limitations of the two in addressing the
task of multi-modal word problem solving.

###### keywords:

Neuro-SymbolicVisual Question Answering Math Word Problem Solving Multimodal
Reasoning

##  1 Introduction

Math word problems are mathematical problems expressed in natural language.
Solving these problems requires one to be able to map the natural language
text to a mathematical expression, identifying the known and unknown
quantities and the functions to be used. Since this is a task that lies at the
intersection of language understanding and reasoning, automatically solving
math word problems has been a research topic of interest in Natural Language
Processing (NLP) community. Consider the following math word problem,

> Problem: _Adam has three apples, and Eve has five. Eve gives Adam all her
> apples. How many apples does Adam have, if he _eats_ one?_
> Equation: $X$ = $3+5-1$

Minor changes in the text may result in large semantic changes, e.g. changing
just one word in the above problem - _eats_ to _finds_ , will change the
equation to $X$ = $3+5+1$. Most of the recent efforts in automatic math word
problem solving treat it as a translation task (from word problem to equation)
and have employed sequence-to-sequence networks or sequence to tree
(generating the expression tree of the equation) networks ([1], [2],[3]).

Figure 1: CLEVR-Math example question _Take away 2 matte cylinders. How many
objects are left?_ with corresponding mathematical equation $X=9-2$.

While text-based math word math problems are a great setting for natural
language understanding, it would also be interesting to consider word problems
which are accompanied by a diagram, and the information required to derive the
solution has to be captured from both its textual and visual representations.
That is, part of the problem scenario description is expressed as text and the
other part is represented in the form of an image. In this paper, we introduce
such a multi-modal math word problem dataset, CLEVR-Math (since it is based on
CLEVR dataset [4]), where each problem has a textual and a visual description
(image). Figure 1 shows an example problem in CLEVR-Math dataset.

While each instance in CLEVR dataset has an image and a natural language query
about the scene depicted in the image, in CLEVR-Math, the natural language
query may not be about the scene represented in the image, but about the state
of the scene after/before a sequence of actions are applied on the scene. The
actions in our case are addition/removal of specific type of objects to/from
the original scene. We believe this is an interesting problem setting as the
ability to envision changes without them being physically manifested is an
important aspect of the human mind.

Our contributions are two-fold, we

  * •

construct an open source multi-modal math word problem dataset, CLEVR-Math and

  * •

analyse the performance of state-of-the-art neural and neuro-symbolic (NeSy)
solutions for solving such multi-modal problems.

Our results and analysis shows how both neural and NeSy methods are unable to
compositionally generalise to chains of operations.

##  2 Related Work

This section gives an overview of the existing datasets in math word problem
solving, multi-modal datasets for visual question answering and existing
neural/neuro-symbolic approaches to the tasks that require a combination of
visual, language and logical reasoning.

#### Math Word Problem Solving - Datasets

Math Word Problem Solving (MAWPS) [5] was one of the earlier datasets
introduced in the domain and collected around $3320$ single/multi equation
word problems involving operators $+$, $-$, $*$, $/$. These word problems were
annotated with equations involved and the answer (solution of the equation).
More recently, larger datasets like Algebra Question Answering with Rationales
(AQuA-RAT) [6] were introduced and it has around $100K$ multiple choice
questions annotated with equations and a textual explanation for the rationale
behind the equations. [7] illustrated the deficiencies in MAWPS dataset by
introducing another dataset named Simple Variations on Arithmetic Math word
Problems (SVAMP). SVAMP is created by making minor variations to problems in
MAWPS. [7] showed that state-of-the-art neural solvers trained on MAWPS
performs poorly on the SVAMP dataset.

#### Visual Question Answering (VQA) and Visual Reasoning - Datasets

One of the first VQA datasets proposed was the DAQUAR dataset [8] based on
real images of indoor scenes. VQA is another widely used dataset [9] with
images from MS-COCO dataset [10]. Questions are manually created and answering
these require commonsense knowledge and reasoning. CLEVR dataset [4] is based
on automatically generated scenes and questions. Such simulated data gives
great control over the distribution of instances. One can decide to generate a
training set with images having only a specific combination of objects (red
cubes and blue cylinders), and a test set with a different combination of
objects (red cylinders and blue cubes), as done in, e.g., CLEVR-Hans [11].
This control allows us to study various aspects like compositional
generalisation of systems. Based on these ideas, CLEVR-Math allows us to test
the ability of the system to generalise to unseen combinations of actions in
the word problem, to e.g. train on single mathematical operations, and test on
chains of operations. Closely related is the CLEVRER (Collision Events for
Video Representation and Reasoning) dataset [12] and CLEVR-Hyp dataset [13].
The questions on videos in CLEVRER requires reasoning about the state of
objects after an video event, instead of after actions in text as in CLEVR-
Math. CLEVR-Hyp focus on VQA where reasoning about effects of actions, and
CLEVR-Math introduces an additional mathematical reasoning dimension to the
problem. GQA is another relevant dataset, where real world images are
annotated with rich scene graphs and a large set of relations and attributes,
and focuses on compositionality in visual reasoning [14]. Graph learning is a
heavily studied area, with applications in multimodal domains such as robotics
[15, 16, 17, 18].

Experiments with Kandinsky patterns [19] show that neural networks are easily
confounded by visual reasoning tasks with shapes, colors, and patterns that
can be difficult to distinguish but follow clear rules. The Winoground dataset
[20] shows similar results, where no state-of-the-art visual reasoning method
is able to distinguish between two confounding captions and images.

#### Existing Approaches to VQA

Most of the earlier approaches in VQA were based on purely neural models that
first encoded the two inputs - the image and the accompanying question into
embeddings using networks like Convolutional Neural Networks (CNN) and Long
Short Term Memory (LSTM) networks and then the two embeddings were forwarded
to a classifier that would then predict the answer to the question ([21],
[22]). Another category of approaches are the attention mechanism-based
approaches that identified the regions in the image that were relevant to
answering the associated question ([23], [24]). Graph neural networks [25]
have also been applied in VQA where both text and the image are represented as
graphs and a multi-modal vectorial representation is learned that captures the
alignment of nodes in the two graphs. [26] introduced the CLIP models where a
representation of the image is learned with natural language supervision by
leveraging the already available huge datasets for image captioning. More
recently, neuro-symbolic approaches have been used in addressing the task of
VQA like Neuro Symbolic Concept Learner (NSCL) [27] and Neuro-Symbolic Visual
Question Answering (NS-VQA) [28]. These approaches convert the input image and
text into an intermediate semantic representation and then employ a quasi-
symbolic program executor to derive an answer from these semantic forms. We
use CLIP and NS-VQA as baselines as they are state-of-the-art on multimodal
language modelling and on the CLEVR dataset, respectively.

##  3 CLEVR-Math

We construct the CLEVR-Math dataset as an extension of CLEVR by introducing
three new functions and 13 templates. Using the codebase provided with CLEVR,
we generate new questions based on the original scenes. We categorise the 13
templates into six types, all based on addition and subtraction. The domain is
restricted to numbers between $0-10$ to conform with CLEVR.

#### New CLEVR functions:

The three functions that we implement are - subtraction and addition to
perform subtraction and addition, and choose to operate on subsets of objects.
Instead of removing all blue spheres, choose allows us to remove a random
number of a specific type of object, e.g. 2 blue spheres out of 4. The random
number generated by choose replaces a questions “X" placeholder during
generation. Figure 2(a) shows three examples of subtraction, and Figure 2(b)
shows a question requiring multihop reasoning. Appendix A includes more
samples from the test set.

(a) (i) _Remove all gray spheres. How many spheres are there? (3)_ , (ii)
_Take away 3 cubes. How many objects are there?_ (7), (iii) _How many blocks
must be removed to get 1 block? (2)_

(b) _Take away all large green metallic spheres. Now remove all cyan objects.
How many objects are left? (4)_

Figure 2: Example image-question pairs from CLEVR-Math, 2(a) showcase addition
and subtraction, and 2(b) shows multihop reasoning. Answers in parenthesis.

#### Question Categories:

The different question categories are shown in Table 1.

  * •

Remove group: All objects belonging to a specific group are removed from the
scene.

  * •

Insertion: A specific number of objects are added to the scene.

  * •

Count backwards: The query is about the change - that is the number of objects
added/removed from the scene to get a goal state.

  * •

Remove subset: A specific number of objects are removed from the scene.

  * •

Adversarial questions: These are trick questions where the actions may be
performed on one object, but the query is about an object that is not affected
by the action. The adversarial actions are always on objects that are seen in
the image.

  * •

Multi-hop: In contrast to the above questions, multi-hop questions perform
sequences of actions (insertion, removal) on the objects. Such questions with
chained functions help us test a model’s ability to generalise to infinite
combinations of operations.

Type | Templates
---|---
Remove group |
| "Remove all <C> <S>s. How many <S>s are there?"
| "Take away all <Z> <C> <M> <S>s. How many <S>s are there?"
| "Take away X <C> <S>s. How many objects are there?"
| "Take away all <C> <S>s. How many objects are there?"
Insertion |
| "Add X <Z> <C> <M> <S>s. How many <Z> <C> <M> <S>s are here?"
| "Add X <Z> <C> <M> <S>s. How many objects are there?"
Count backwards |
| "How many <C> <S>s must be removed to get X <C> <S>s?"
| "Take away <C> <S>s. How many were removed if there are X <C> <S>s left?"
Multi-hop |
| "Take away all <Z> <C> <M> <S>s. Remove all
| <Z2> <C2> <M2> <S2>s. How many objects are left?"
Remove subset |
| "Remove X <S>s. How many <S>s are there?"
Adversarial questions |
| "Remove all <C1> <S1>s. Remove all <C2> <S2>s. How many <S1>s are left?"
| "Remove all <C1> <S1>s. How many <C2> <S2>s are left?"
Table 1: An overview of the different templates implemented by CLEVR-Math.
<Z>, <C>, <M>, <S> are instantiated to size, color, material, and shape during
the question generation.

Each problem in the dataset is also annotated with it’s equivalent functional
program based on the CLEVR functions described in the previous section. For
example, consider the question from insertion category and it’s program (the
arguments of an instruction refer to another instruction - indicating it’s
input is the output of the referred instruction):

> Q: Add 3 blue cylinders. How many cylinders are there?
> Program: 1\. scene, 2. choose[3], 3. count(1), 4. filter_cylinder(1), 5.
> count(4), addition(2, 5)

The program contains the choose function - choose[$i$] operator returns $i$
($i=3$ in this case).

#### Question generation.

To support greater linguistic variation, we add synonyms for addition and
subtraction to the template engine. _Subtract_ can be replaced with _remove,
take away_ and _withdraw_ , and _addition_ with _introduce,_ and _insert_. We
use the same training and validation scenes as CLEVR, and generate 5000 new
scenes as test data. Figure 3 show the distribution of attributes, words,
templates and answers in CLEVR-Math, aggregated over the training, validation,
and test data. The distribution is reflected in each of the splits.

$0$$0.5$$1$$1.5$$2$$\cdot 10^{5}$ShapeMaterialSizeColor (a) Attribute
distribution per category, showing even allocations.

$0$$5$$10$$0$$2$$4$$6$$8$$\cdot 10^{4}$ (b) Answer distribution, from 0 to 10.

$10$$15$$0$$0.5$$1$$1.5$$\cdot 10^{5}$ (c) Distribution of number of words.

$0$$2$$4$$6$$\cdot 10^{4}$SubtractionAdditionAdversarialMultihop (d) Template
distribution over categories of templates. Each bar corresponds to a template
in each respective category. We see that subset subtraction (i.e., _remove 2
blue cubes_) is underrepresented.

Figure 3: The attributes are used evenly throughout the dataset, whereas the
answers are biased towards the smaller numbers. The numbers are aggregated
over all splits.

There are 50 words in the CLEVR-Math vocabulary, where the narrow language
puts focus on the mathematical reasoning rather than advanced language
capabilities. Figure 3(c) show that most questions are 8-9 words long, with a
second peak at 13 for the multihop questions.

Template | Train | Validation | Test
---|---|---|---
Subtraction | 229364 | 49149 | 3281
Addition | 193641 | 41600 | 2752
Adversarial | 65180 | 13900 | 950
Multihop | 67897 | 14553 | 972
Table 2: Distribution of templates in each data split.

Table 2 shows the distribution of templates, with an approximately equal
amount of questions for subtraction and addition, and similarly for
adversarial and multihop questions. The ratios are consistent between splits.
To test multihop reasoning and compositional generalisability we generate
train-validation-test with only singlehop questions in training and
validation, and only multihop questions in the test data. Thus, a model using
the CLEVR-Math-multihop configuration must solve the multihop questions in a
zero-shot fashion.

#### Open sourcing data.

We open source CLEVR-Math as a Huggingface dataset
111https://huggingface.co/datasets/dali-does/clevr-math with two
configurations; CLEVR-Math and CLEVR-Math-multihop. The extended CLEVR source
code is available on Github 222https://github.com/dali-does/clevr-math. Table
3 shows the Huggingface dataset card for CLEVR-Math. The template feature
allows for filtering to perform, e.g., only singlehop training and multihop
testing.

Feature | Type | Example
---|---|---
template | String | subtraction-multihop
id | String | CLEVR_math_test_000010.png
question | String | _Remove 5 spheres. How many objects are there?_
image | image path | CLEVR_v1.0/images/train/CLEVR_new_000010.png
label | int64, 0-10 | 5
Table 3: Huggingface dataset card for CLEVR-Math.

##  4 Experiments

CLIP [26] is used as a neural baseline. Questions and images are embedded
using CLIP, and an additional classification layer is added to predict the
correct answer. Fine tuning CLIP on CLEVR-Math as a masked language task
before adding classification gave no significant improvements, while consuming
significantly more computational resources. CLIP and this classification layer
is trained jointly for $10$ epochs with early stopping using a batch size of
$64$.

NS-VQA [28] is used as the neuro-symbolic baseline. Here, a mask-RCNN [29] is
trained independently to convert an image to a scene graph. In our
experiments, we skip this step and use the actual scene graphs associated with
images. The question is parsed into a functional program by a sequence to
sequence (Seq2Seq) network based on Bi-LSTM. A quasi-symbolic program executor
executes the program generated on the scene graph of the image to return an
answer. The Seq2Seq network is pre-trained in a fully supervised fashion by
providing it a few examples (around 60 examples) of $(question,$ $program)$
pairs. The pre-trained network is then trained further using REINFORCE
algorithm that returns a reward based on whether the program generated could
derive the expected answer or not. Supervised pretraining and REINFORCE were
run for $1000$ and $5000$ iterations, respectively, with a batch size of
$128$. Both CLIP and NS-VQA models were trained on a NVIDIA Tesla P100 GPU
computing processor.

Each model is evaluated on each question category, and are trained on 2500,
5000, 10000, and 20000 samples to see the influence of the amount of data. For
multihop, training and validation sets with and without multihop questions are
used, with the latter named _multihop (0-shot)_.

###  4.1 Results

Table 4 shows the accuracy of CLIP and NS-VQA on the different categories as
well as an aggregated accuracy over the entire dataset. Both the models were
trained on $10,000$ samples. NS-VQA performs better than CLIP models for most
templates apart from multihop. NS-VQA performs better on subtraction and
adversarial problems (both based on ‘subtraction’ CLEVR function) than
addition problems. This could be because the functional programs for addition
problems always contain a choose operator. It is important to identify the
argument to choose operator from the problem statement (which is mostly one of
the numerical quantities in the word problem) to arrive at the correct answer.
Unlike this, there are subtraction and adversarial problems (in remove group)
that do not have a choose operator in the program.

Model | All | Addition | Subtraction | Adversarial | Multihop | Multihop (0-shot)
---|---|---|---|---|---|---
NS-VQA | 0.8840 | 0.9781 | 0.9948 | 0.9957 | 0.286 | 0.267
CLIP | 0.3464 | 0.5699 | 0.3019 | 0.2848 | 0.272 | 0.238
Table 4: Accuracy on the CLEVR-Math dataset, shown for each template group and
aggregated over all templates.

Neither of the methods perform well on the multi-hop questions, with a clear
degradation in the performance for NS-VQA. This is because the question parser
of NS-VQA relies on a Seq2Seq network that does not generalize compositionally
[30]. CLEVR focus on visual attribute compositionality, and the multihop
reasoning introduces higher demands on linguistic compositionality. When
multihop questions are included in the training and validation data, naturally
both methods improve their performance.

To gain further insight into CLIPs’ performance on CLEVR-Math, Appendix C
shows a confusion matrix from training CLIP on 20 000 samples and evaluating
on all question categories. These results show that most errors made by CLIP
is off by ones. This reflects the generative nature of such models, in how
they can get the context correct but sometimes miss out on details. We also
see how CLIP focus on learning in the range 1-5, reflecting that these
problems represent a majority of the problems.

Model | 2500 | 5000 | 10000 | 20000
---|---|---|---|---
NS-VQA | 0.6283 | 0.8840 | 0.6795 | 0.6118
CLIP | 0.2918 | 0.3184 | 0.3528 | 0.3464
Table 5: Accuracy over all templates for different dataset sizes.

Table 5 shows how different training sizes influence the accuracy. We can see
that NS-VQA achieves high accuracy from relatively few examples and plateaus,
which is consistent with the original results on CLEVR. It also seems like NS-
VQA is overfitting with more data given, and one hypothesis is that more
emphasise is put on the program, but that they are similar enough to confound
NS-VQA. In CLEVR, the different questions were more distinguishable from a
program perspective. CLIP scales with the number of samples, but plateaus at a
much lower accuracy. We note that a larger number of samples could lead to
similar performance for CLIP, but at the cost of more computational resources.

(a) Subtract all small purple matte blocks. Subtract all blocks. How many
objects are left? was answered by CLIP with 3 instead of 2.

(b) Subtract all red metallic objects. Subtract all yellow objects. How many
objects are left? was answered with 9 instead of 5 by NS-VQA.

Figure 4: Examples of when CLIP and NS-VQA fails on multihop questions.

We randomly sample 20 correct and 20 incorrect answers from the multihop test
data for both CLIP and NS-VQA. Appendix B contains a subset of those samples,
and Figure 4 illustrates two incorrect answers. There are no clear patterns of
failures, such as only performing one of the actions, but we notice multiple
instances where CLIP fails to perform overlapping subtraction, or subtraction
when no objects match the description. Another observation is that half of the
20 incorrect answers from CLIP, where on images with only three objects.
Scenes with few objects have a much smaller possible action space associated
to it, meaning that there is less room for error. In Figure 4(a), there are no
purple matte blocks to remove, so the corresponding equation is $3-0-1=2$.

##  5 Conclusions and Future Work

We introduced a new dataset, CLEVR-Math, containing word math problems about
visual scenes. Our results show that the state-of-the-art NeSy model, NS-VQA,
achieves higher accuracy on CLEVR-Math with less data and computational
resources, than the neural model, CLIP. This is further evidence that neural
methods, such as CLIP, are lacking in reasoning capabilities, even after fine
tuning. Given that NS-VQA uses perfect scene graphs, the comparison is not
completely fair. We still expect the results of learning end-to-end to be
consistent with the current results in alignment with the original results on
CLEVR for NS-VQA.

CLEVR-Math successfully introduces a focused benchmark for learning and
reasoning in multimodal data. There are a few natural extensions to this work,
both on further development of the dataset and on evaluation. Extending the
benchmark to answers outside of the range 0-10 would provide a more
challenging domain, and providing scene graphs for each step of the reasoning
chain could open up for other methods. The empirical results show that neither
of the models could generalize to chained actions. Hence, it is also of
research interest to design neuro-symbolic models where language perception is
tackled in a more generalizable manner. Focus should lie on the
representations (symbols) that are learned. Other interesting directions is to
introduce a representation that is manipulated internally according to the
actions as they are read. Adding longer chains of operations, or chains with
alternating subtraction and addition, would put even more emphasise on the
reasoning capabilities. Finally, there is an opportunity to add confounding
information to test the robustness, e.g. by associating each shape with a
fixed color during training and randomise it during testing.

##  6 Acknowledgements

This work was partially supported by the Wallenberg AI, Autonomous Systems and
Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation.

## References

  * Luong et al. [2015] M.-T. Luong, M. Kayser, C. D. Manning,  Deep neural language models for machine translation,  in: 19th CoNLL, 2015, pp. 305–309.
  * Xie and Sun [2019] Z. Xie, S. Sun,  A goal-driven tree-structured neural model for math word problems.,  in: IJCAI, 2019, pp. 5299–5305.
  * Zhang et al. [2020] J. Zhang, L. Wang, R. K.-W. Lee, Y. Bin, Y. Wang, J. Shao, E.-P. Lim,  Graph-to-tree learning for solving math word problems,  ACL, 2020.
  * Johnson et al. [2017] J. Johnson, B. Hariharan, L. Van Der Maaten, L. Fei-Fei, C. Lawrence Zitnick, R. Girshick,  Clevr: A diagnostic dataset for compositional language and elementary visual reasoning,  in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp. 2901–2910.
  * Koncel-Kedziorski et al. [2016] R. Koncel-Kedziorski, S. Roy, A. Amini, N. Kushman, H. Hajishirzi,  Mawps: A math word problem repository,  in: Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, 2016, pp. 1152–1157.
  * Ling et al. [2017] W. Ling, D. Yogatama, C. Dyer, P. Blunsom,  Program induction by rationale generation: Learning to solve and explain algebraic word problems,  arXiv preprint arXiv:1705.04146 (2017).
  * Patel et al. [2021] A. Patel, S. Bhattamishra, N. Goyal,  Are nlp models really able to solve simple math word problems?,  arXiv preprint arXiv:2103.07191 (2021).
  * Malinowski and Fritz [2014] M. Malinowski, M. Fritz,  A multi-world approach to question answering about real-world scenes based on uncertain input,  Advances in neural information processing systems 27 (2014).
  * Antol et al. [2015] S. Antol, A. Agrawal, J. Lu, M. Mitchell, D. Batra, C. L. Zitnick, D. Parikh,  Vqa: Visual question answering,  in: Proceedings of the IEEE international conference on computer vision, 2015, pp. 2425–2433.
  * Lin et al. [2014] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, C. L. Zitnick,  Microsoft coco: Common objects in context,  in: European conference on computer vision, Springer, 2014, pp. 740–755.
  * Stammer et al. [2021] W. Stammer, P. Schramowski, K. Kersting,  Right for the right concept: Revising neuro-symbolic concepts by interacting with their explanations,  in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp. 3619–3629.
  * Yi et al. [2019] K. Yi, C. Gan, Y. Li, P. Kohli, J. Wu, A. Torralba, J. B. Tenenbaum,  Clevrer: Collision events for video representation and reasoning,  arXiv preprint arXiv:1910.01442 (2019).
  * Sampat et al. [2021] S. K. Sampat, A. Kumar, Y. Yang, C. Baral,  Clevr_hyp: A challenge dataset and baselines for visual question answering with hypothetical actions over images,  arXiv preprint arXiv:2104.05981 (2021).
  * Hudson and Manning [2019] D. A. Hudson, C. D. Manning,  Gqa: A new dataset for real-world visual reasoning and compositional question answering,  in: Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2019, pp. 6700–6709.
  * Xia et al. [2021] F. Xia, K. Sun, S. Yu, A. Aziz, L. Wan, S. Pan, H. Liu,  Graph learning: A survey,  IEEE Transactions on Artificial Intelligence 2 (2021) 109–127. doi:10.1109/TAI.2021.3076021.
  * Yu et al. [2021] F. Yu, J. Tang, W. Yin, Y. Sun, H. Tian, H. Wu, H. Wang,  Ernie-vil: Knowledge enhanced vision-language representations through scene graphs,  in: Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, 2021, pp. 3208–3216.
  * Wald et al. [2020] J. Wald, H. Dhamo, N. Navab, F. Tombari,  Learning 3d semantic scene graphs from 3d indoor reconstructions,  in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp. 3961–3970.
  * Ji et al. [2020] J. Ji, R. Krishna, L. Fei-Fei, J. C. Niebles,  Action genome: Actions as compositions of spatio-temporal scene graphs,  in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp. 10236–10247.
  * Holzinger et al. [2019] A. Holzinger, M. Kickmeier-Rust, H. Müller,  Kandinsky patterns as iq-test for machine learning,  in: International cross-domain conference for machine learning and knowledge extraction, Springer, 2019, pp. 1–14.
  * Thrush et al. [2022] T. Thrush, R. Jiang, M. Bartolo, A. Singh, A. Williams, D. Kiela, C. Ross,  Winoground: Probing vision and language models for visio-linguistic compositionality,  in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2022, pp. 5238–5248.
  * Ben-Younes et al. [2017] H. Ben-Younes, R. Cadene, M. Cord, N. Thome,  Mutan: Multimodal tucker fusion for visual question answering,  in: Proceedings of the IEEE international conference on computer vision, 2017, pp. 2612–2620.
  * Fukui et al. [2016] A. Fukui, D. H. Park, D. Yang, A. Rohrbach, T. Darrell, M. Rohrbach,  Multimodal compact bilinear pooling for visual question answering and visual grounding,  arXiv preprint arXiv:1606.01847 (2016).
  * Wang et al. [2017] P. Wang, Q. Wu, C. Shen, A. Dick, A. Van Den Hengel,  Fvqa: Fact-based visual question answering,  IEEE transactions on pattern analysis and machine intelligence 40 (2017) 2413–2427.
  * Shih et al. [2016] K. J. Shih, S. Singh, D. Hoiem,  Where to look: Focus regions for visual question answering,  in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp. 4613–4621.
  * Narasimhan et al. [2018] M. Narasimhan, S. Lazebnik, A. Schwing,  Out of the box: Reasoning with graph convolution nets for factual visual question answering,  Advances in neural information processing systems 31 (2018).
  * Radford et al. [2021] A. Radford, J. W. Kim, C. Hallacy, A. Ramesh, G. Goh, S. Agarwal, G. Sastry, A. Askell, P. Mishkin, J. Clark, et al.,  Learning transferable visual models from natural language supervision,  in: International Conference on Machine Learning, PMLR, 2021, pp. 8748–8763.
  * Mao et al. [2019] J. Mao, C. Gan, P. Kohli, J. B. Tenenbaum, J. Wu,  The neuro-symbolic concept learner: Interpreting scenes, words, and sentences from natural supervision,  arXiv preprint arXiv:1904.12584 (2019).
  * Yi et al. [2018] K. Yi, J. Wu, C. Gan, A. Torralba, P. Kohli, J. Tenenbaum,  Neural-symbolic vqa: Disentangling reasoning from vision and language understanding,  Advances in neural information processing systems 31 (2018).
  * He et al. [2017] K. He, G. Gkioxari, P. Dollár, R. Girshick,  Mask r-cnn. corr abs/1703.06870,  arXiv preprint arXiv:1703.06870 (2017).
  * Lake and Baroni [2018] B. Lake, M. Baroni,  Generalization without systematicity: On the compositional skills of sequence-to-sequence recurrent networks,  in: International conference on machine learning, PMLR, 2018, pp. 2873–2882.

##  Appendix A Examples

Figure 5(c) show some examples of questions from different templates in CLEVR-
Math.

(a) Subtract all gray cylinders. Subtract all gray cubes. How many cylinders
are left? (3)

(b) How many cyan cubes must be subtracted to get 1 cyan cubes? (1)

(c) Subtract all purple cylinders. Subtract all yellow blocks. How many
cylinders are left? (1)

(d) Add 2 large cubes. How many objects exist? (5)

(e) Subtract 4 cylinders. How many cylinders are left? (0)

(f) Subtract 0 brown blocks. How many objects are left (8)

Figure 5: Samples from the test set of CLEVR-Math.

##  Appendix B Incorrect answers on multihop questions

###  B.1 CLIP

Figure 6 shows random samples of when CLIP fails to answer multihop questions
correctly.

(a) Subtract all small purple matte blocks. Subtract all blocks. How many
objects are left? was incorrectly answered with 3 instead of 2 by CLIP.

(b) Subtract all tiny balls. Subtract all big matte blocks. How many objects
are left? was incorrectly answered with 3 instead of 2 by CLIP.

(c) Subtract all spheres. Subtract all big green shiny cubes. How many objects
are left? was incorrectly answered with 4 instead of 5 by CLIP.

(d) Subtract all tiny shiny balls. Subtract all purple objects. How many
objects are left? was incorrectly answered with 4 instead of 8 by CLIP.

(e) Subtract all tiny rubber blocks. Subtract all big cyan rubber things. How
many objects are left? was incorrectly answered with 5 instead of 4 by CLIP.

(f) Subtract all gray balls. Subtract all small cylinders. How many objects
are left? was incorrectly answered with 3 instead of 2 by CLIP.

(g) Subtract all gray balls. Subtract all small cylinders. How many objects
are left? was incorrectly answered with 3 instead of 2 by CLIP.

(h) Subtract all large cyan objects. Subtract all small rubber blocks. How
many objects are left? was incorrectly answered with 3 instead of 1 by CLIP.

(i) Subtract all spheres. Subtract all cyan things. How many objects are left?
was incorrectly answered with 3 instead of 2 by CLIP.

(j) Subtract all big green cylinders. Subtract all big blue cubes. How many
objects are left? was incorrectly answered with 5 instead of 8 by CLIP.

(k) Subtract all brown things. Subtract all large red spheres. How many
objects are left? was incorrectly answered with 3 instead of 1 by CLIP.

(l) Subtract all gray objects. Subtract all big green objects. How many
objects are left? was incorrectly answered with 4 instead of 6 by CLIP.

Figure 6: Sampling of incorrect answers by CLIP on multihop.

###  B.2 NS-VQA

Figure 7 shows random samples of when NS-VQA fails to answer multihop
questions correctly.

(a) Subtract all large purple objects. Remove all green metallic objects. How
many objects are left? was incorrectly answered with 7 instead of 8 by NS-VQA.

(b) Subtract all red metallic objects. Subtract all yellow objects. How many
objects are left? was incorrectly answered with 9 instead of 5 by NS-VQA.

(c) Subtract all tiny green metallic cubes. Subtract all large brown blocks.
How many objects are left? was incorrectly answered with 0 instead of 2 by NS-
VQA.

(d) Subtract all small red objects. Subtract all tiny metal cylinders. How
many objects are left? was incorrectly answered with 6 instead of 8 by NS-VQA.

(e) Subtract all cylinders. Subtract all purple objects. How many objects are
left’? was incorrectly answered with 7 instead of 3 by NS-VQA.

(f) Subtract all blue metal cylinders. Subtract all gray objects. How many
objects are left? was incorrectly answered with 1 instead of 2 by NS-VQA.

(g) Subtract all large rubber spheres. Subtract all blue blocks. How many
objects are left? was incorrectly answered with 3 instead of 2 by NS-VQA.

(h) Subtract all big gray blocks. Subtract all large cylinders. How many
objects are left? was incorrectly answered with 4 instead of 6 by NS-VQA.

(i) Subtract all tiny blocks. Subtract all small balls. How many objects are
left? was incorrectly answered with 4 instead of 3 by NS-VQA.

(j) Subtract all small purple blocks. Subtract all matte objects. How many
objects are left? was incorrectly answered with 7 instead of 3 by NS-VQA.

(k) Subtract all tiny red rubber objects. Subtract all small blue balls. How
many objects are left? was incorrectly answered with 2 instead of 1 by NS-VQA.

(l) Subtract all cyan cylinders. Subtract all small rubber objects. How many
objects are left? was incorrectly answered with 6 instead of 5 by NS-VQA.

Figure 7: Sampling of incorrect answers by NS-VQA on multihop.

##  Appendix C CLIP confusion matrix

Figure 8 shows a confusion matrix indicating that CLIP is learning something
for all labels. It also shows that when an answer is wrong, it is off by one.
The confusion matrix also reflects the distribution over answers, showing that
most answers are considered by CLIP to lie in the range 1-5.

Figure 8: Confusion matrix for CLIP trained on 20 000 samples.