††institutetext: Department of Physics and Siyuan Laboratory, Jinan
University,
Guangzhou 510632, P.R. China

# Matter Effects of Sterile Neutrino in Light of Renormalization-Group
Equations

Shuge Zeng  Fanrong Xu 111Corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The renormalization-group equation (RGE) approach to neutrino matter effects
is further developed in this work. We derive a complete set of differential
equations for effective mixing elements, masses and Jarlskog-like invariants
in presence of a light sterile neutrino. The evolutions of mixing elements as
well as Jarlskog-like invariants are obtained by numerically solving these
differential equations. We calculate terrestrial matter effects in long-
baseline (LBL) experiments, taking NOvA, T2K and DUNE as examples. In both
three-flavor and four-flavor frameworks, electron-neutrino survival
probabilities as well as the day-night asymmetry of solar neutrino are also
evaluated as a further examination of the RGE approach.

###### Keywords:

Sterile neutrino, Matter effects, Renormalization-group equations

##  1 Introduction

Now it is well established that active neutrinos are massive with tiny masses
and oscillate among different flavor states. The measurement of oscillation
parameters are improved noticeabely in the last two decades. On the other
hand, as hypothetical particles, sterile neutrinos do not couple directly to
the gauge bosons and can only participate in weak interactions through mixing
with active neutrinos. Theoretically their mass and number are unconstrained,
and have been quested in experiments for a long time. The existence of eV mass
scale sterile neutrinos has been motivated by several low energy anomalies
which cannot be accounted for by the standard three-flavor active neutrino
framework. These involve short-baseline (SBL) experiments with neutrinos from
accelerators, reactors and radioactive sources.

Two most well-known accelerator-based experiments are LSND and MiniBooNE. The
LSND anomaly is the $3.8\sigma$ excess of events compatible to $\bar{\nu}_{e}$
appearance in a $\bar{\nu}_{\mu}$ beam observed by the LSND experiment
LSND:2001aii , while the MiniBooNE anomaly is the $4.8\sigma$ excess of
electron-like events in the MiniBooNE experiment observed in both $\nu_{\mu}$
and $\bar{\nu}_{\mu}$ beams MiniBooNE:2018esg . As a direct test of LSND
anomaly, the JSNS2 experiment has started data taking Hino:2021uwz  at J-PARC
in Japan. With the same neutrino beam as MiniBooNE but superior event
reconstruction capabilities, the MicroBooNE collaboration recently has
released results scrutinizing the MiniBooNE anomaly, known as the MiniBooNE
low-energy excess (MBLEE). A first MicroBooNE analysis disfavors that the
MBLEE is due to underestimated production of $\Delta$ baryons followed by
decays to photons at a significance of $94.8\%$ C.L. MicroBooNE:2021zai  and
concluded that “no excess of $\nu_{e}$ events is observed” MicroBooNE:2021rmx
. However, a recent work shows quantitatively that the MicroBooNE
collaboration “unquestionably do not probe the full parameter space of sterile
neutrino models hinted at by MiniBooNE, nor do they probe the $\nu_{e}$
interpretation of the MiniBooNE excess in a model-independent way”
Arguelles:2021meu , which indicates that it is still early to exclude eV scale
sterile neutrinos. The reactor antineutrino anomaly (RAA), is about the $6\%$
deficit (a $3\sigma$ effect) of $\bar{\nu}_{e}$ in reactor VSBL (very short-
baseline, $<100$ m) experiments resulting from the re-evaluation of the
reactor antineutrino flux Huber:2011wv ; Mention:2011rk , compared to the new
prediction in 2011 Mueller:2011nm . Many VSBL experiments have been created
and progresses have been made, including NEOSNEOS:2016wee ,
DANSSDanilov:2019aef , STEREOSTEREO:2020hup , Neutrino-4Serebrov:2020kmd ,
PROSPECTPROSPECT:2020sxr  and so on. The gallium anomaly occurred in the two
experiments GALLEX Hampel:1997fc ; Kaether:2010ag  and SAGE
Abdurashitov:1998ne ; Abdurashitov:2005tb , where a deficit of $16\%$ events
between observations and theoretical calculation Schwetz:2013naa  was found.
Though the latest estimation carried out in 2019 using new nuclear shell-model
wave functions diminish the significance from $3\sigma$ to $2.3\sigma$
Kostensalo:2019vmv , recently BEST Barinov:2021asz  reaffirmed the gallium
anomaly with a larger deficit $20\\!-\\!24\%$. More can be referred to the
recent review papers Dasgupta:2021ies ; Athar:2021xsd .

Regardless of the sources, baselines and energy ranges of these experiments,
all of the results involving above experiments can be understood individually
via SBL neutrino oscillations driven by $\Delta m^{2}\sim 1\textrm{eV}^{2}$,
requiring a fourth neutrino mass eigenstate to account for this higher mass-
squared difference. Hence the existence of eV mass scale light sterile
neutrinos is still an open question to be answered at this stage.

The Mikheyev-Smirnov-Wolfenstein (MSW) matter effects Wolfenstein:1977ue ;
Mikheev:1986wj  plays an essential role in the determination of neutrino mass
ordering and leptonic CP-violating phase via the long-baseline (LBL)
accelerator neutrino experiments. When the neutrino beam propagates in the
Earth matter for a long distance, the MSW effect becomes crucially important.
In presence of a light sterile neutrino, more degrees of freedom enter the MSW
effect even in the propagation of active flavor neutrinos. For the calculation
of matter effects involving a light sterile neutrino, different methods
Klop:2014ima ; Li:2018ezt ; Yue:2019qat  have been developed in recent years.

The idea of renormalization-group equations (RGEs) has been widely applied in
quantum fields theoriesStueckelbergdeBreidenbach:1952pwl ; Gell-Mann:1954yli ,
solid-state physicsWilson:1971bg ; Wilson:1971dh , and other fields of modern
physics. There have been attempts to make a connection between RGE and
neutrino matter effects in the past few yearsChiu:2010da ; Chiu:2017ckv .
Recently, more progress has been made Xing:2018lob ; Zhou:2020iei ;
Wang:2019yfp . By taking the matter term $a$ (see Section 2) as the role of
renormalization energy scale, a close set of differential equations describing
the evolution of effective mixing matrix and effective mass-square differences
in three-flavor case has been established Xing:2018lob . The analytical
solutions for the mixing and masses are further given in Wang:2019yfp . One
may naturally wonder whether it is accessible to extend the RGE framework to
include the sterile neutrino contribution. In addition to developing the form
theoretically, it is also interesting to realize the practical application of
this new method in neutrino phenomenology. These topics are supposed to be
investigated in current work. Although it does not participate weak
interactions directly, the sterile neutrino contribution in Hamiltonian is
brought in through subtracting a global identical term produced via charged
lepton neutral-current, denoted as mass term $a^{\prime}$ (see Section 2). In
the scenario of constant density matter, two mass terms of electron-type and
sterile neutrino is connected by a constant $k$. Then we can continue working
in the framework of RGE approach Xing:2018lob , and extend the evolution of
mixing matrix and mass-square differences with respect to mass term $a$ (or
equivalently neutrino beam energy $E$), including the sterile neutrino
contribution. After analytically deriving the completed set of differential
equations, we further explore their solutions numerically. In principle,
oscillation probabilities should be calculable in the RGE approach, which now
is realized here for the first time in both cases with purely three active
neutrinos as well as with one more sterile neutrino. We take three LBL
accelerator neutrino experiments, NOvA, T2K and DUNE, as typic examples to
test the validity of RGE approach in practical numerical analysis and find
that the results perfectly agree with the analytical approach Li:2018ezt . In
the adiabatic approximation of solar matter and slab approximation of Earth
matter, we also calculate the day-night asymmetry observed in the surface of
Earth, as a further application of RGE method.

The remaining part of this paper is organized as follows. In Section 2, we
present the derivation details and exact forms of differential equations of
masses and mixing matrix, as well as oscillation probabilities. After
exhibiting the explicit evolution of mixing matrix elements involving a light
sterile neutrino in first part of Section 3, we further apply the methodology
to Earth matter effects in three LBL experiments and solar matter effects in
name of day-night asymmetry. We conclude the validity of RGE approach in
Section 4 and related input parameters are summarized in Appendix A.

##  2 Analytical formulae with RGE

It is known that physical observables are independent of energy scale while
parameters describing the observables depend on scale themselves, and
renormalization-group equations (RGEs) depict the evolutions of these
parameters with respect to the energy scale. With the general idea of RGE, a
new method is developed to describe flavor mixing parameters evolve when
neutrinos propagate in a medium in the framework of three-flavor neutrino
oscillationXing:2018lob . As introduced in Sec.1, the existence of eV-scale
light sterile neutrinos is still an open question and the calculation of
matter effects with sterile neutrino is meaningful, especially in the
measurement of mass ordering and leptonic CP-violating phase(es). In this
section, we will make an extension of the RGE approach to contain the case in
presence of a light sterile neutrino in addition to three active flavors.

###  2.1 Master equations for effective masses and mixing matrix

Similar to the standard three-flavor case, the effective Hamiltonian with
sterile neutrino contribution is given as

$H_{m}=\frac{1}{2E}\left[U\left(\begin{array}[]{cccc}m_{1}^{2}&0&0&0\\\
0&m_{2}^{2}&0&0\\\ 0&0&m_{3}^{2}&0\\\
0&0&0&m_{4}^{2}\end{array}\right)U^{\dagger}+\left(\begin{array}[]{cccc}a&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\
0&0&0&a^{\prime}\end{array}\right)\right]\equiv\frac{1}{2E}V\left(\begin{array}[]{cccc}\tilde{m}_{1}^{2}&0&0&0\\\
0&\tilde{m}_{2}^{2}&0&0\\\ 0&0&\tilde{m}_{3}^{2}&0\\\
0&0&0&\tilde{m}^{2}_{4}\\\ \end{array}\right)V^{\dagger},$ (1)

where $a\equiv 2\sqrt{2}G_{F}N_{e}E$ with $N_{e}$ being the net electron
number density and $E$ the neutrino beam energy, mass terms $m_{i}^{2}$ and
$\tilde{m}_{i}^{2}$ in vacuum and matter, respectively. For the mixing
matrices, the mixing in vacuum (PMNS matrix) is denoted as $U$ 222 The
convention and inputs to parameterize PMNS matrix $U$ is introduced in
Appendix A.  and $V$ stands for the effective mixing incorporating matter
effects. Though it does not take part in weak interaction itself, the sterile
neutrino has an matter effects term $a^{\prime}$ here due to the subtraction
of a global identical term produced via neutral-current interaction of active
neutrinos. In general, $a^{\prime}$ dependents on the medium neutrino beam
propagating. In the scenario of constant density matter, we can further write
down $a^{\prime}=ka$ and the medium information is encoded in the matter
parameter $k$.

We take $a$ as the role of energy scale in RGE similar as Xing:2018lob .
Differentiating both sides of Eq.(1) with respect to $a$, we have

$\displaystyle\dot{D}+\left[V^{\dagger}\dot{V},D\right]$ $\displaystyle=$
$\displaystyle V^{\dagger}\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ 0&0&0&k\end{array}\right)V$ (6) $\displaystyle=$
$\displaystyle{\scriptsize{\left(\begin{array}[]{cccc}|V_{e1}|^{2}+k|V_{s1}|^{2}&V_{e1}^{*}V_{e2}+kV_{s1}^{*}V_{s2}&V_{e1}^{*}V_{e3}+kV_{s1}^{*}V_{s3}&V_{e1}^{*}V_{e4}+kV_{s1}^{*}V_{s4}\\\
V_{e2}^{*}V_{e1}+kV_{s2}^{*}V_{s1}&|V_{e2}|^{2}+k|V_{s2}|^{2}&V_{e2}^{*}V_{e3}+kV_{s2}^{*}V_{s3}&V_{e2}^{*}V_{e4}+kV_{s2}^{*}V_{s4}\\\
V_{e3}^{*}V_{e1}+kV_{s3}^{*}V_{s1}&V_{e3}^{*}V_{e2}+kV_{s3}^{*}V_{s2}&|V_{e3}|^{2}+k|V_{s3}|^{2}&V_{e3}^{*}V_{e4}+kV_{s3}^{*}V_{s4}\\\
V_{e4}^{*}V_{e1}+kV_{s4}^{*}V_{s1}&V_{e4}^{*}V_{e2}+kV_{s4}^{*}V_{s2}&V_{e4}^{*}V_{e3}+kV_{s4}^{*}V_{s3}&|V_{e4}|^{2}+k|V_{s4}|^{2}\end{array}\right),}}$
(11)

where
$D\equiv{\textrm{diag}}\left\\{\tilde{m}_{1}^{2},\tilde{m}_{2}^{2},\tilde{m}_{3}^{2},\tilde{m}_{4}^{2}\right\\}$.
Apparentely, the first term in the left-handed side of Eq. (6) is diagonal
while diagonal entries of the second term vanish. Extracting the diagonal
entries in both sides, we can firstly get a relation

$\frac{{\text{d}}\tilde{m}_{i}^{2}}{{\text{d}}a}=|V_{ei}|^{2}+k|V_{si}|^{2},$

which further leads to the first order differential equations

$\frac{{\text{d}}}{{\text{d}}a}\Delta\tilde{m}^{2}_{ij}=(|V_{ei}|^{2}-|V_{ej}|^{2})+k(|V_{si}|^{2}-|V_{sj}|^{2})$
(12)

with the effective mass square difference
$\Delta\tilde{m}^{2}_{ij}\equiv\tilde{m}_{i}^{2}-\tilde{m}_{j}^{2}$.
Similarly, by comparing the off-diagonal entries in Eq. (6), one has

$\sum_{\alpha}V_{\alpha i}^{*}\dot{V}_{\alpha
j}=(\Delta\tilde{m}^{2}_{ji})^{-1}\left(V_{ei}^{*}V_{ej}+kV_{si}^{*}V_{sj}\right).$
(13)

Multiply both sides of the orthogonal relation

$\sum_{j\neq i}V_{\alpha j}^{*}V_{\beta j}=\delta_{\alpha\beta}-V_{\alpha
i}^{*}V_{\beta i},$

with $\dot{V}_{\alpha i}$, and take a summation over $\alpha$ (so that
$\sum\limits_{\alpha}\dot{V}_{\alpha i}V_{\alpha i}^{*}=0$ ), we can write
down the differential equation for effective mixing matrix as

$\dot{V}_{\beta i}=\sum_{j\neq i}V_{\beta
j}(V_{ej}^{*}V_{ei}+kV_{sj}^{*}V_{si})(\Delta\tilde{m}^{2}_{ij})^{-1},$ (14)

in which Eq. (13) is applied and summation is not taken for index $i$. Due to
the fact

$\sum_{\beta}\dot{V}^{*}_{\beta i}V_{\beta i}V^{*}_{\alpha i}V_{\alpha
i}+\sum_{\beta}V^{*}_{\alpha i}V_{\alpha i}\dot{V}_{\beta i}V^{*}_{\beta
i}=|V_{\alpha i}|^{2}\sum_{\beta}(\dot{V}^{*}_{\beta i}V_{\beta
i}+\dot{V}_{\beta i}V^{*}_{\beta i})=0,$

the differential equation of mixing matrix can also be written in terms of the
square of mixing matrix,

$\frac{{\text{d}}}{{\text{d}}a}|V_{\alpha
i}|^{2}=\left(\frac{{\text{d}}}{{\text{d}}a}V_{\alpha i}^{*}\right)V_{\alpha
i}+V^{*}_{\alpha i}\frac{{\text{d}}}{{\text{d}}a}V_{\alpha i}=2\sum_{j\neq
i}{\text{Re}}\left[V_{\alpha j}V^{*}_{\alpha
i}(V^{*}_{ej}V_{ei}+kV^{*}_{sj}V_{si})\right](\Delta\tilde{m}^{2}_{ij})^{-1},$
(15)

this compact form is phase-independent and brings convenience in some parts of
following calculation.

###  2.2 The detailed differential equations

The master equations (12) and (15) (or (14)) form a closed and complete
equation group, which contains 19 independent differential equations. To have
a global impression, it is helpful to exhibit some necessary differential
equations explicitly. Three independent equations on mass square differences
can be detailed from equation (12),

$\begin{split}&\frac{{\text{d}}}{{\text{d}}a}\Delta\tilde{m}_{12}^{2}=(|V_{e1}|^{2}-|V_{e2}|^{2})+k(|V_{s1}|^{2}-|V_{s2}|^{2}),\\\
&\frac{{\text{d}}}{{\text{d}}a}\Delta\tilde{m}_{13}^{2}=(|V_{e1}|^{2}-|V_{e3}|^{2})+k(|V_{s1}|^{2}-|V_{s3}|^{2}),\\\
&\frac{{\text{d}}}{{\text{d}}a}\Delta\tilde{m}_{14}^{2}=(|V_{e1}|^{2}-|V_{e4}|^{2})+k(|V_{s1}|^{2}-|V_{s4}|^{2}).\end{split}$
(16)

The right handed side of equation (16) indicates that the evolution property
of mass square difference relies on square of mixing matrix entries together
with the matter parameter $k$. As part of the closed equation group, the
detailed differential equations for $\alpha=e$ can be read from equation (15),

$\begin{split}&\frac{{\text{d}}}{{\text{d}}a}|V_{e1}|^{2}=2|V_{e1}|^{2}\left[|V_{e2}|^{2}(\Delta\tilde{m}_{12}^{2})^{-1}-|V_{e3}|^{2}(\Delta\tilde{m}_{31}^{2})^{-1}-|V_{e4}|^{2}(\Delta\tilde{m}_{41}^{2})^{-1}\right]\\\
&\hskip
42.67912pt+2k{\text{Re}}\left[V_{e1}^{*}V_{s1}\Big{(}V_{e2}V_{s2}^{*}(\Delta\tilde{m}_{12}^{2})^{-1}-V_{e3}V_{s3}^{*}(\Delta\tilde{m}_{31}^{2})^{-1}-V_{e4}V_{s4}^{*}(\Delta\tilde{m}_{41}^{2})^{-1}\Big{)}\right],\\\
&\frac{{\text{d}}}{{\text{d}}a}|V_{e2}|^{2}=2|V_{e2}|^{2}\left[|V_{e3}|^{2}(\Delta\tilde{m}_{23}^{2})^{-1}-|V_{e4}|^{2}(\Delta\tilde{m}_{42}^{2})^{-1}-|V_{e1}|^{2}(\Delta\tilde{m}_{12}^{2})^{-1}\right]\\\
&\hskip
42.67912pt+2k{\text{Re}}\left[V_{e2}^{*}V_{s2}\Big{(}V_{e1}V_{s1}^{*}(\Delta\tilde{m}_{21}^{2})^{-1}-V_{e3}V_{s3}^{*}(\Delta\tilde{m}_{32}^{2})^{-1}-V_{e4}V_{s4}^{*}(\Delta\tilde{m}_{42}^{2})^{-1}\Big{)}\right],\\\
&\frac{{\text{d}}}{{\text{d}}a}|V_{e3}|^{2}=2|V_{e3}|^{2}\left[|V_{e4}|^{2}(\Delta\tilde{m}_{34}^{2})^{-1}-|V_{e1}|^{2}(\Delta\tilde{m}_{13}^{2})^{-1}-|V_{e2}|^{2}(\Delta\tilde{m}_{23}^{2})^{-1}\right]\\\
&\hskip
42.67912pt+2k{\text{Re}}\left[V_{e3}^{*}V_{s3}\Big{(}V_{e4}V_{s4}^{*}(\Delta\tilde{m}_{34}^{2})^{-1}-V_{e1}V_{s1}^{*}(\Delta\tilde{m}_{13}^{2})^{-1}-V_{e2}V_{s2}^{*}(\Delta\tilde{m}_{23}^{2})^{-1}\Big{)}\right],\\\
&\frac{{\text{d}}}{{\text{d}}a}|V_{e4}|^{2}=2|V_{e4}|^{2}\left[|V_{e1}|^{2}(\Delta\tilde{m}_{41}^{2})^{-1}-|V_{e2}|^{2}(\Delta\tilde{m}_{24}^{2})^{-1}-|V_{e3}|^{2}(\Delta\tilde{m}_{34}^{2})^{-1}\right]\\\
&\hskip
42.67912pt+2k{\text{Re}}\left[V_{e4}^{*}V_{s4}\Big{(}V_{e1}V_{s1}^{*}(\Delta\tilde{m}_{41}^{2})^{-1}-V_{e2}V_{s2}^{*}(\Delta\tilde{m}_{24}^{2})^{-1}-V_{e3}V_{s3}^{*}(\Delta\tilde{m}_{34}^{2})^{-1}\Big{)}\right].\end{split}$
(17)

The evolution equations of $|V_{\alpha i}|^{2}$, taking $|V_{ei}|^{2}$ as an
example, contain more information than the ones of $\Delta\tilde{m}^{2}_{ij}$
for their entanglement with mass-square differences, as well as individual
mixing matrix elements. From the known the general expression for the
evolution of $V_{\alpha i}$ with respect to $a$ equation (14)), we can further
write down explicitly those related to electron-type neutrino and sterile
neutrino, giving

$\begin{split}\frac{{\text{d}}V_{e1}}{{\text{d}}a}=\,&(V_{e1}|V_{e2}|^{2}+kV_{e2}V_{s1}V_{s2}^{\ast})(\Delta\tilde{m}_{12}^{2})^{-1}-(V_{e1}|V_{e3}|^{2}+kV_{e3}V_{s1}V_{s3}^{\ast})(\Delta\tilde{m}_{31}^{2})^{-1}\\\
&-(V_{e1}|V_{e4}|^{2}+kV_{e4}V_{s1}V_{s4}^{\ast})(\Delta\tilde{m}_{41}^{2})^{-1},\\\
\frac{{\text{d}}V_{e2}}{{\text{d}}a}=\,&(V_{e2}|V_{e1}|^{2}+kV_{e1}V_{s2}V_{s1}^{\ast})(\Delta\tilde{m}_{21}^{2})^{-1}-(V_{e2}|V_{e3}|^{2}+kV_{e3}V_{s2}V_{s3}^{\ast})(\Delta\tilde{m}_{32}^{2})^{-1}\\\
&-(V_{e2}|V_{e4}|^{2}+kV_{e4}V_{s2}V_{s4}^{\ast})(\Delta\tilde{m}_{42}^{2})^{-1},\\\
\frac{{\text{d}}V_{e3}}{{\text{d}}a}=\,&(V_{e3}|V_{e1}|^{2}+kV_{e1}V_{s3}V_{s1}^{\ast})(\Delta\tilde{m}_{31}^{2})^{-1}-(V_{e3}|V_{e2}|^{2}+kV_{e2}V_{s3}V_{s2}^{\ast})(\Delta\tilde{m}_{23}^{2})^{-1}\\\
&-(V_{e3}|V_{e4}|^{2}+kV_{e4}V_{s3}V_{s4}^{\ast})(\Delta\tilde{m}_{43}^{2})^{-1},\\\
\frac{{\text{d}}V_{e4}}{{\text{d}}a}=\,&(V_{e4}|V_{e1}|^{2}+kV_{e1}V_{s4}V_{s1}^{\ast})(\Delta\tilde{m}_{41}^{2})^{-1}+(V_{e4}|V_{e2}|^{2}+kV_{e2}V_{s4}V_{s2}^{\ast})(\Delta\tilde{m}_{42}^{2})^{-1}\\\
&+(V_{e4}|V_{e3}|^{2}+kV_{e3}V_{s4}V_{s3}^{\ast})(\Delta\tilde{m}_{43}^{2})^{-1},\end{split}$
(18)

and

$\begin{split}\frac{{\text{d}}V_{s1}}{{\text{d}}a}=\,&(V_{s2}V_{e2}^{\ast}V_{e1}+k|V_{s2}|^{2}V_{s1})(\Delta\tilde{m}_{12}^{2})^{-1}-(V_{s3}V_{e3}^{\ast}V_{e1}+k|V_{s3}|^{2}V_{s1})(\Delta\tilde{m}_{31}^{2})^{-1}\\\
&-(V_{s4}V_{e4}^{\ast}V_{e1}+k|V_{s4}|^{2}V_{s1})(\Delta\tilde{m}_{41}^{2})^{-1},\\\
\frac{{\text{d}}V_{s2}}{{\text{d}}a}=\,&(V_{s1}V_{e1}^{\ast}V_{e2}+k|V_{s1}|^{2}V_{s2})(\Delta\tilde{m}_{21}^{2})^{-1}-(V_{s3}V_{e3}^{\ast}V_{e2}+k|V_{s3}|^{2}V_{s2})(\Delta\tilde{m}_{32}^{2})^{-1}\\\
&-(V_{s4}V_{e4}^{\ast}V_{e2}+k|V_{s4}|^{2}V_{s2})(\Delta\tilde{m}_{42}^{2})^{-1},\\\
\frac{{\text{d}}V_{s3}}{{\text{d}}a}=\,&(V_{s1}V_{e1}^{\ast}V_{e3}+k|V_{s1}|^{2}V_{s3})(\Delta\tilde{m}_{31}^{2})^{-1}-(V_{s2}V_{e2}^{\ast}V_{e3}+k|V_{s2}|^{2}V_{s3})(\Delta\tilde{m}_{23}^{2})^{-1}\\\
&-(V_{s4}V_{e4}^{\ast}V_{e3}+k|V_{s4}|^{2}V_{s3})(\Delta\tilde{m}_{43}^{2})^{-1},\\\
\frac{{\text{d}}V_{s4}}{{\text{d}}a}=\,&(V_{s1}V_{e1}^{\ast}V_{e4}+k|V_{s1}|^{2}V_{s4})(\Delta\tilde{m}_{41}^{2})^{-1}+(V_{s2}V_{e2}^{\ast}V_{e4}+k|V_{s2}|^{2}V_{s4})(\Delta\tilde{m}_{42}^{2})^{-1}\\\
&-(V_{s3}V_{e3}^{\ast}V_{e4}+k|V_{s3}|^{2}V_{s4})(\Delta\tilde{m}_{43}^{2})^{-1}.\\\
\end{split}$ (19)

Although equations (18) and (17) are not independent ones, here for the
purpose of solving the tangled differential equations a clear exhibition is
useful and helpful. Combining above equations together, the masses and mixings
involving electron-type and sterile neutrino are in principle calculable,
which further provide input to solve the mixings involving muon and tau
neutrinos,

$\begin{split}&\frac{{\text{d}}}{{\text{d}}a}|V_{\mu
1}|^{2}=2{\text{Re}}\left[V_{\mu 1}^{*}V_{e1}\Big{(}V_{\mu
2}V_{e2}^{*}(\Delta\tilde{m}_{12}^{2})^{-1}-V_{\mu
3}V_{e3}^{*}(\Delta\tilde{m}_{31}^{2})^{-1}-V_{\mu
4}V_{e4}^{*}(\Delta\tilde{m}_{41}^{2})^{-1}\Big{)}\right]\\\ &\hskip
42.67912pt+2k{\text{Re}}\left[V_{\mu 1}^{*}V_{s1}\Big{(}V_{\mu
2}V_{s2}^{*}(\Delta\tilde{m}_{12}^{2})^{-1}-V_{\mu
3}V_{s3}^{*}(\Delta\tilde{m}_{31}^{2})^{-1}-V_{\mu
4}V_{s4}^{*}(\Delta\tilde{m}_{41}^{2})^{-1}\Big{)}\right],\\\
&\frac{{\text{d}}}{{\text{d}}a}|V_{\mu 2}|^{2}=2{\text{Re}}\left[V_{\mu
2}^{*}V_{e2}\Big{(}V_{\mu 1}V_{e1}^{*}(\Delta\tilde{m}_{21}^{2})^{-1}-V_{\mu
3}V_{e3}^{*}(\Delta\tilde{m}_{32}^{2})^{-1}-V_{\mu
4}V_{e4}^{*}(\Delta\tilde{m}_{42}^{2})^{-1}\Big{)}\right]\\\ &\hskip
42.67912pt+2k{\text{Re}}\left[V_{\mu 2}^{*}V_{s2}\Big{(}V_{\mu
1}V_{s1}^{*}(\Delta\tilde{m}_{21}^{2})^{-1}-V_{\mu
3}V_{s3}^{*}(\Delta\tilde{m}_{32}^{2})^{-1}-V_{\mu
4}V_{s4}^{*}(\Delta\tilde{m}_{42}^{2})^{-1}\Big{)}\right],\\\
&\frac{{\text{d}}}{{\text{d}}a}|V_{\mu 3}|^{2}=2{\text{Re}}\left[V_{\mu
3}^{*}V_{e3}\Big{(}V_{\mu 2}V_{e2}^{*}(\Delta\tilde{m}_{32}^{2})^{-1}+V_{\mu
1}V_{e1}^{*}(\Delta\tilde{m}_{31}^{2})^{-1}-V_{\mu
4}V_{e4}^{*}(\Delta\tilde{m}_{43}^{2})^{-1}\Big{)}\right]\\\ &\hskip
42.67912pt+2k{\text{Re}}\left[V_{\mu 3}^{*}V_{s3}\Big{(}V_{\mu
2}V_{s2}^{*}(\Delta\tilde{m}_{32}^{2})^{-1}+V_{\mu
1}V_{s1}^{*}(\Delta\tilde{m}_{31}^{2})^{-1}-V_{\mu
4}V_{s4}^{*}(\Delta\tilde{m}_{43}^{2})^{-1}\Big{)}\right],\\\
&\frac{{\text{d}}}{{\text{d}}a}|V_{\mu 4}|^{2}=2{\text{Re}}\left[V_{\mu
4}^{*}V_{e4}\Big{(}V_{\mu 2}V_{e2}^{*}(\Delta\tilde{m}_{42}^{2})^{-1}+V_{\mu
3}V_{e3}^{*}(\Delta\tilde{m}_{43}^{2})^{-1}+V_{\mu
1}V_{e1}^{*}(\Delta\tilde{m}_{41}^{2})^{-1}\Big{)}\right]\\\ &\hskip
42.67912pt+2k{\text{Re}}\left[V_{\mu 4}^{*}V_{s4}\Big{(}V_{\mu
2}V_{s2}^{*}(\Delta\tilde{m}_{42}^{2})^{-1}+V_{\mu
3}V_{s3}^{*}(\Delta\tilde{m}_{43}^{2})^{-1}+V_{\mu
1}V_{s1}^{*}(\Delta\tilde{m}_{41}^{2})^{-1}\Big{)}\right].\\\ \end{split}$
(20)

The differential equations on tau flavor has been neglected here as they take
the similar form as muon. Among all the equations in the whole equation group,
equations (16) and (17) play a key role to solve all the equations. The
sterile neutrino contributions are characterized both by matter parameter $k$
and the mixings involving the fourth row and column. By vanishing the sterile
neutrino related mixing matrix elements, $V_{\alpha 4}(\alpha=e,\ldots,s)$ and
$V_{si}(i=1,\ldots 4)$ , equations (16) and (17) return to standard three-
flavor case Xing:2018lob .

###  2.3 Effective Jarlskog-like invariants

The Jarlskog invariant Jarlskog:1985ht , defined as the imaginary part of
products of mixing matrix elements, is an important measurement to
characterize the CP violation in fermion sector. It is known that there is
only one independent invariant in models with fermion family number being
three. Previously there have been some discussions on Jarlskog-like invariants
beyond three generation of quarks Hou:2010wf  and leptons Xing:2001bg . Here
in this work by incorporating a light sterile neutrino, we define the
effective Jarlskog-like invariants

$\mathcal{J}^{ij}_{\alpha\beta}={\rm{Im}}(U_{\alpha i}U_{\beta j}U^{*}_{\alpha
j}U^{*}_{\beta
i}),\qquad\tilde{\mathcal{J}}^{ij}_{\alpha\beta}={\rm{Im}}(V_{\alpha
i}V_{\beta j}V^{*}_{\alpha j}V^{*}_{\beta i})$ (21)

in vacuum as $\mathcal{J}$ and its counterpart $\tilde{\mathcal{J}}$ in
matter, in which $\alpha,\beta=e,\mu,\tau,s$ and $i,j=1,2,3,4$. From the fact
$\mathcal{J}_{\alpha\alpha}^{ij}=\mathcal{J}_{\alpha\beta}^{ii}=0$ and

$\sum_{i}\mathcal{J}^{ij}_{\alpha\beta}=\sum_{j}\mathcal{J}^{ij}_{\alpha\beta}=\sum_{\alpha}\mathcal{J}^{ij}_{\alpha\beta}=\sum_{\beta}\mathcal{J}^{ij}_{\alpha\beta}=0$
(22)

due to the unitary condition of the mixing matrix, the number of independent
Jarlskog-like invariants increases to $9$ compared with $1$ in the three-
flavor case, which holds for the counterpart in matter.

The differential equation for the Jarlskog-like invariants can be derived by
using equation (14) after a straightforward algebraic calculation,

$\displaystyle\frac{{\text{d}}}{{\text{d}}a}\tilde{\mathcal{J}}^{ij}_{\alpha\beta}$
$\displaystyle=$ $\displaystyle\sum_{k\neq i}{\rm{Im}}\left[V^{*}_{\alpha
j}V^{*}_{\beta i}V_{\alpha k}V_{\beta
j}(V^{*}_{ek}V_{ei}+kV^{*}_{sk}V_{si})\right](\Delta\tilde{m}^{2}_{ik})^{-1}$
$\displaystyle+\sum_{k\neq i}{\rm{Im}}\left[V_{\alpha i}V_{\beta
j}V^{*}_{\alpha j}V^{*}_{\beta
k}(V_{ek}V^{*}_{ei}+kV_{sk}V^{*}_{si})\right](\Delta\tilde{m}^{2}_{ik})^{-1}+(\alpha\leftrightarrow\beta,i\leftrightarrow
j).$

As an example, we take $(\alpha\beta,ij)=(e\mu,12)$ and show explicitly the
evolution equation of one of the Jarlskog-like invariants in matter,

$\begin{split}\frac{d}{da}\tilde{\mathcal{J}}_{e\mu}^{12}=&\quad\;\tilde{\mathcal{J}}_{e\mu}^{12}\left(\frac{|V_{e2}|^{2}}{\Delta\tilde{m}^{2}_{12}}+\frac{|V_{e3}|^{2}}{\Delta\tilde{m}^{2}_{13}}+\frac{|V_{e4}|^{2}}{\Delta\tilde{m}^{2}_{14}}\right)+\frac{|V_{e1}|^{2}}{\Delta\tilde{m}^{2}_{13}}\tilde{\mathcal{J}}_{e\mu}^{32}+\frac{|V_{e1}|^{2}}{\Delta\tilde{m}^{2}_{14}}\tilde{\mathcal{J}}_{e\mu}^{42}\\\
&+\tilde{\mathcal{J}}_{e\mu}^{12}\left(\frac{|V_{e1}|^{2}}{\Delta\tilde{m}^{2}_{21}}+\frac{|V_{e3}|^{2}}{\Delta\tilde{m}^{2}_{23}}+\frac{|V_{e4}|^{2}}{\Delta\tilde{m}^{2}_{24}}\right)+\frac{|V_{e2}|^{2}}{\Delta\tilde{m}^{2}_{23}}\tilde{\mathcal{J}}_{e\mu}^{13}+\frac{|V_{e2}|^{2}}{\Delta\tilde{m}^{2}_{24}}\tilde{\mathcal{J}}_{e\mu}^{14}\\\
&+k\left(\frac{|V_{e2}|^{2}\tilde{\mathcal{J}}_{\mu
s}^{21}}{\Delta\tilde{m}^{2}_{12}}+\frac{{\textrm{Im}}\big{[}V_{\mu
2}V_{e2}^{\ast}V_{\mu
1}^{\ast}V_{e3}V_{s1}V_{s3}^{\ast}\big{]}}{\Delta\tilde{m}^{2}_{13}}+\frac{{\textrm{Im}}\big{[}V_{\mu
2}V_{e2}^{\ast}V_{\mu
1}^{\ast}V_{e4}V_{s1}V_{s4}^{\ast}\big{]}}{\Delta\tilde{m}^{2}_{14}}\right)\\\
&+k\left(\frac{|V_{\mu
2}|^{2}\tilde{\mathcal{J}}_{es}^{12}}{\Delta\tilde{m}^{2}_{12}}+\frac{{\textrm{Im}}\big{[}V_{e1}V_{\mu
2}V_{e2}^{\ast}V_{\mu
3}^{\ast}V_{s3}V_{s1}^{\ast}\big{]}}{\Delta\tilde{m}^{2}_{13}}+\frac{{\textrm{Im}}\big{[}V_{e1}V_{\mu
2}V_{e2}^{\ast}V_{\mu
4}^{\ast}V_{s4}V_{s1}^{\ast}\big{]}}{\Delta\tilde{m}^{2}_{14}}\right)\\\
&+k\left(\frac{|V_{e1}|^{2}\tilde{\mathcal{J}}_{\mu
s}^{21}}{\Delta\tilde{m}^{2}_{21}}+\frac{{\textrm{Im}}\big{[}V_{e1}V_{\mu
2}V_{\mu
1}^{\ast}V_{e3}^{\ast}V_{s3}V_{s2}^{\ast}\big{]}}{\Delta\tilde{m}^{2}_{23}}+\frac{{\textrm{Im}}\big{[}V_{e1}V_{\mu
2}V_{\mu
1}^{\ast}V_{e4}^{\ast}V_{s4}V_{s2}^{\ast}\big{]}}{\Delta\tilde{m}^{2}_{24}}\right)\\\
&+k\left(\frac{|V_{\mu
1}|^{2}\tilde{\mathcal{J}}_{es}^{12}}{\Delta\tilde{m}^{2}_{21}}+\frac{{\textrm{Im}}\big{[}V_{e1}V_{e2}^{\ast}V_{\mu
1}^{\ast}V_{\mu
3}V_{s2}V_{s3}^{\ast}\big{]}}{\Delta\tilde{m}^{2}_{23}}+\frac{{\textrm{Im}}\big{[}V_{e1}V_{e2}^{\ast}V_{\mu
1}^{\ast}V_{\mu
4}V_{s2}V_{s4}^{\ast}\big{]}}{\Delta\tilde{m}^{2}_{24}}\right),\end{split}$
(24)

which recovers the standard three-active-flavor case in the limit of
$\tilde{m}_{4}\to\infty$ and $k\to 0$. More Jarlskog-like invariants enter
equation (24) compared with the general form, equation (2.3), since the
special roles of electron and sterile neutrino play. In the following
numerical analysis, we will present evolutions of all the $9$ Jarlskog-like
invariants completely.

In addition, it can be verified that the relation

$\sum_{i=1}^{4}\frac{{\text{d}}}{{\text{d}}a}\ln|V_{ei}|^{2}+2\sum_{j>k}\frac{{\text{d}}}{{\text{d}}a}\ln\Delta\tilde{m}^{2}_{jk}=0$
(25)

holds in the limit of $k=0$, corresponding to equation (16) of Xing:2018lob .
On the other hand, the Naumov-like Naumov:1991ju  and Toshev-like
Toshev:1991ku  relation in RGE approach will be explored in a separate work
Naumov4  due to its complication.

###  2.4 General formulae for oscillation probabilities

The mixing matrix and mass square differences calculated above are not
physical observables themselves. In fact, they are encoded in neutrino
oscillations. By replacing the mixing and mass in the vacuum with
corresponding ones in matter, the general form of neutrino oscillation
probability can be obtain

$P(\nu_{\alpha}\to\nu_{\beta})=\sum_{i}|V_{\alpha i}|^{2}|V_{\beta
i}|^{2}+2\sum_{i<j}\left[{\text{Re}}(V_{\alpha i}V_{\beta j}V_{\alpha
j}^{*}V_{\beta i}^{*})\cos\tilde{\Delta}_{ij}-{\textrm{Im}}(V_{\alpha
i}V_{\beta j}V_{\alpha j}^{*}V_{\beta i}^{*})\sin\tilde{\Delta}_{ij}\right]$
(26)

with $\tilde{\Delta}_{ij}\equiv\frac{\Delta\tilde{m}^{2}_{ij}L}{2E}$. We will
see equation (26) is the most generic expression to incorporate all situations
of the neutrino oscillation supported by the following facts:

  * •

Making use of the replacement rule $V_{ij}\leftrightarrow V^{*}_{ij}$, the
oscillations among antineutrinos can also be calculated by equation (26).

  * •

Apparently oscillations occurring in both vacuum and matter are applicable,
which depends on the choice of mixing and mass parameters, $(U,m_{i})$ or
$(V,\tilde{m}_{i})$.

  * •

It can be utilized to describe oscillations among both in three pure active
flavor case and three active together with one sterile case. The former one
can be easily trended to by vanishing $V_{si}$ ($i=1,\ldots,4)$ and $V_{\alpha
4}$ ($\alpha=e,\ldots,s)$.

So far analytical tools for further exploring neutrino phenomenology have been
prepared. We will examine this method in practical numerical study in the
following part.

##  3 Applications and numerical results

To establish the connection between RGE and neutrino matter effects is more
than a concept formally. In practical qualitative analysis, the RGE approach
is also valuable and we will realize it here. In this section, we first
numerically analyze the evolution of effective mixing matrix explicitly . Then
we utilize the general oscillation eq. (26) together with RGE approach to
effective mixing and mass in Earth and solar matters to calculate
corresponding observables and the numerical results are presented,
respectively.

###  3.1 The evolution of mixing matrix elements

The propagation of neutrino in various mediums depends on effective mixing
matrix in different materials, hence it is necessary to perform an explicit
analysis of detailed mixing matrix elements firstly. In constant density
matters, the features of matter effects are controlled by the matter parameter
$k$. For example, it is known that $k=-0.5$ depicts earth matter. By adopting
the input parameters in Appendix A, we show in Fig.1 the evolution of
effective mixing matrix elements with particular $k$ in normal hierarchy (NH)
case with respect to energy scale $a/\Delta m^{2}_{21}$, where neutrinos and
antineutrinos are characterized as solid red and dashed blue curves,
respectively.

$\begin{array}[]{c}(a):k=0\\\
\includegraphics[width=381.5877pt]{4neu_k0.eps}\\\ (b):k=-0.5\\\
\includegraphics[width=381.5877pt]{4neu_km05.eps}\\\ \end{array}$

Figure 1: The evolution of $|V_{\alpha i}|^{2}$ (for $\alpha=e,\mu,\tau,s$ and
$i=1,2,3,4$) in NH case with respect to energy scale $a/\Delta m^{2}_{21}$, in
which: $k=0$ in $(a)$ and $k=-0.5$ in $(b)$. The red solid and blue dashed
curves correspond to the results of neutrino and antineutrino oscillations,
respectively.

The situation for $k=0$ is special. For an illustration, we take the first row
of Fig.1(a) as an example to interpret the evolutions of $|V_{\alpha i}|^{2}$
according to equation (17). The negative (positive) sign of the right hand
side of $|V_{e1}|^{2}$ ($|V_{e4}|^{2}$) evolution equation indicates a
monotonic increasing (decreasing) behavior for $|V_{e1}|^{2}$
($|V_{e4}|^{2}$). Due to a competition between positive and negative parts in
the RGE equations, combining their particular initial values, the maximums
appear for the evolution of $|V_{e2}|^{2}$ and $|V_{e3}|^{2}$. In the large
$\Delta\tilde{m}_{41}^{2}$ limit, the sterile neutrino contribution is
decoupled and the evolutions return to three pure active flavor case. 333The
standard three-flavor limit can also be obtained by setting $V_{\alpha 4}=0$
and $V_{si}=0$ as aforementioned. Here we understand $3\nu$ picture in this
way is for the purpose of examining corresponding results in Xing:2018lob .
Comparing the left three columns of the rows except the last one (the mixings
among three active flavors) in Fig.1(a) with the plot Fig.1 of Xing:2018lob ,
one can observe the behaviors of evolution curves are consistent with each
other. Both the evolution behaviors and initial values of $|V_{\mu i}|^{2}$,
standing for vacuum matrix elements $|U_{\mu i}|^{2}$, are qualitatively
identical to $|V_{\tau i}|^{2}$ ($i=1,2,3$). The approximate $\mu-\tau$
symmetry is kept and well understood as the flavors are indistinguishable for
muon and tau for the universal neutral-current interactions of the two flavors
in ordinary matter. Mathematically, this can also be explained by setting zero
values to $k$ and $(\Delta\tilde{m}^{2}_{41})^{-1}$ in the first three
equations of (17) and (20). Among all the six active-sterile mixing matrix
elements for neutrino case (red solid curves), only two of them $|V_{e4}|^{2}$
and $|V_{s3}|^{2}$ manifest themselves as unity when energy scale $a$ reaches
around $m_{4}^{2}$ (or $a/\Delta m^{2}_{21}\sim\Delta{m}^{2}_{41}/\Delta
m^{2}_{21}\sim 1000$ ). In contrast, there is no mixing among active and
sterile flavors for antineutrinos. For the pure sterile neutrino,
$|V_{s4}|^{2}$ keeps unity and drops to small value near to zero when $a$
approaches $\Delta{m}_{41}^{2}\sim 10^{3}\Delta m^{2}_{21}$, which guarantees
the unitary conditions $\sum\limits_{i=1}^{4}|V_{\alpha i}|^{2}=1$ for a
particular flavor $\alpha$ and $\sum\limits_{\alpha=e}^{s}|V_{\alpha
i}|^{2}=1$ for a particular mass index $i$.

The case of $k=-0.5$, corresponding to the terrestrial matter, contains the
contribution from non-vanishing $k$ terms, which reflects complete sterile
neutrino contributions in constant density matter. In addition to the well
respected unitary conditions, some obvious changes occur in Fig.1(b):

  * •

The effective $\mu-\tau$ symmetry is partially breaking for $i=1,2,3$ in
neutrino case and $i=3,4$ for antineutrino case.

  * •

While keeping the active-sterile mixings unchanged from the $k=0$ scenario of
antineutrino case, two of the six mixing matrix elements of neutrino case
($|V_{s1}|^{2}$ and $|V_{s3}|^{2}$) change their behaviors.

  * •

The value of $|V_{s4}|^{2}$ is stably kept around one since the mixing among
sterile neutrino and other ones is close to zero for the antineutrino case.
The evolution for neutrino case differs as a sudden drop from unity to zero
occurs around $10^{3}$, which is consistent with the behaviors of mixing among
sterile neutrino and the active ones.

Similarly, the inverse mass hierarchy (IH) case as well as the evolution of
effective masses, described in Eq. (16), can also be analyzed and we have
neglected the corresponding plots. Now with the effective mixing matrix and
mass square prepared, we may apply these results to further analyze neutrino
propagation in typic mediums.

###  3.2 The evolution of Jarlskog-like invariants

Figure 2: The evolution of the Jarlskog-like invariants
$\tilde{\mathcal{J}}^{ij}_{\alpha\beta}$ in NH case, normalized by
$\mathcal{J}^{12}_{e\mu}$, with respect to $a/\Delta m^{2}_{21}$. The red
solid and blue dashed curves correspond to the results of neutrino and
antineutrino oscillations, respectively.

For the nine Jarlskog-like invariants, the choice of the basis is not unique.
Here we fix one of the flavor and mass indices as electron and $1$ and let the
remaining indices run over all the paramete,sr space. In Fig. 2, within NH
framework by setting $k=0$ we show the evolution behaviors of these nine
invariants normalized to the vacuum value of one invariant
$\mathcal{J}_{e\mu}^{12}$ with respect to $a/\Delta m^{2}_{21}$.

Typically the behavior of the Jarlskog-like invariant in the first plot of
Fig. 2 can be interpreted from equation (2.3). Among all the terms in the
right hand side, terms containing $\Delta\tilde{m}^{2}_{21}$ dominate the
evolution trend. During the initial stage, the increase of
$\tilde{\mathcal{J}}_{e\mu}^{12}$ is determined by positivity of
$(|V_{e1}|^{2}-|V_{e2}|^{2})\Delta\tilde{m}^{2}_{21}$. At around $a/\Delta
m^{2}_{21}\sim 1$, the first resonance is approached for the sign flip of
$(|V_{e1}|^{2}-|V_{e2}|^{2})$, see the first two plots in Fig. 1(a). And then
a second peak appears at around $a/\Delta m^{2}_{21}\sim 10$, due to the
change of the relative size between $|V_{e2}|^{2}$ and $|V_{e3}|^{2}$, which
is consistent with the pure three-flavor situation. Analogically the other
eight invariants can be analyzed and here we exhibit their evolutions as the
remaining parts of Fig. 2, in which the sum rules in equation (22) are
respected in each row or column.

###  3.3 The matter effects in Earth

Generally speaking, terrestrial matter effects should be taken into account in
all the experiments taken place in Earth. Practically, there is few difference
in the propagation between vacuum and matter in short-baseline (SBL) and
medium-baselined (MBL) experiments. These experiments are usually built around
sites nearby nuclear power plants with electron-type antineutrino as their
source, given

$P(\bar{\nu}_{e}\to\bar{\nu}_{e})=\sum_{i}|V_{ei}|^{4}+2\sum_{i<j}|V_{ei}|^{2}|V_{ej}|^{2}\cos\tilde{\Delta}_{ij}.$
(27)

Since the matter effects are not significant, the mixing matrix and mass
difference in above Eq. (27) can be safely replaced by the corresponding ones
in vacuum.

To further study neutrino mass ordering and measure CP-violating phase in the
mixing matrix, taking the neutrino produced from accelerators with high energy
as the experiment source is a good choice. The more energetic neutrino is
supposed to propagate in a longer distance and hence longer baselines are
required as one of the experimental conditions. In the LBL experiments, the
matter effects are no longer negligible. The baselines for two ongoing LBL
experiments, NOvA (NuMI Off-axis electron Neutrino Appearance) and T2K (Tokai
to Kamioka), are given as $L=810\,{\text{km}}$ and $L=295\,{\text{km}}$, while
the baseline is designed to be $L=1300\,{\text{km}}$ for the near-future
Fermilab experiment DUNE (The Deep Underground Neutrino Experiment). In these
LBL experiments with accelerator neutrino source, both flavor changing and
conserving modes can be observed and more physical goals, such as the
measurement of leptonic CP-violating phase, are supposed to be achieved. In
presence of a light sterile neutrino, the number of CP-violating phases
enhances to $3$ and hence the reliance on phases is more complicated. To make
use of previous numerical results of mixing matrix and mass square difference
in RGE approach, the general expression for neutrino oscillation is preferred,
giving

$P({\nu}_{\mu}\to{\nu}_{e})=\sum_{i}|V_{\mu
i}|^{2}|V_{ei}|^{2}+2\sum_{i<j}\left[{\text{Re}}(V_{\mu i}V_{ej}V_{\mu
j}^{*}V_{ei}^{*})\cos\tilde{\Delta}_{ij}-{\textrm{Im}}(V_{\mu i}V_{ej}V_{\mu
j}^{*}V_{ei}^{*})\sin\tilde{\Delta}_{ij}\right].$ (28)

For the three LBL experiments, though with different baselines, they share the
common mixing matrix since the neutrinos received by detectors all travel
through the crust. In the following analysis, we adopt the appearance mode as
an illustration.

$\begin{array}[]{cc}\includegraphics[width=216.81pt]{T2K_130_140.eps}&\includegraphics[width=216.81pt]{T2K_130_140-1.eps}\\\
(1)&(2)\\\
\includegraphics[width=216.81pt]{Nova_130_140.eps}&\includegraphics[width=216.81pt]{Nova_130_140-1.eps}\\\
(3)&(4)\\\
\includegraphics[width=216.81pt]{DUNE1.eps}&\includegraphics[width=216.81pt]{DUNE2.eps}\\\
(5)&(6)\end{array}$

Figure 3: The oscillation probabilities of appearance mode
${\nu}_{\mu}\to{\nu}_{e}$ in LBL accelerator neutrino experiments T2K, NOvA
and DUNE for NH case with
$(\delta_{13},\delta_{14})=(0,\frac{\pi}{2}),(\frac{\pi}{2},\frac{\pi}{2})$.

In the numerical calculations, the vacuum mixing (PMNS) matrix as well as
particular mass-square differences are treated as the input. We adopt the
values in Zyla:2020zbs  for three-flavor situation and a global fit result
Gariazzo:2017fdh  for the four-flavor case, summarized in Appendix A. To
produce the curve $P(E)$, the $E$ dependence of $|V_{\alpha i}|^{2},V_{\alpha
i}$ and $\tilde{\Delta}_{ij}$ are required. These quantities are calculable by
integrating related differential equations derived in Section 2.2.

In addition to plotting the oscillation curves of the three LBL experiments in
RGE approach, we also make use of the exact analytical formulae derived in
Li:2018ezt  to guarantee the effectiveness of this quantitative analysis based
on RGE, shown in Fig. 3. We find that:

  * •

All the curves plotted in two different ways coincide perfectly, indicating
that the RGE approach works well in numerical study.

  * •

The formulae developed in this work, including sterile neutrino contribution,
can be utilized to describe four different situations ($3\nu$ in vacuum and
matter, $4\nu$ in vacuum and matter) by opting for proper parameters.

  * •

The matter effects are manifest for all the three LBL experiments. Although
the orders of magnitude are of percentage level, with sterile neutrino the
values of probability have a global enhancement. Taking NOvA with
$(\delta_{13},\delta_{14})=(0.5\pi,0.5\pi)$ as an example (see Fig. 3(4)), the
enhancement can reach as high as $100\%$ when the energy of neutrino beam is
around $1.5{\textrm{GeV}}$. Obviously the future precise measurement from LBL
accelerator neutrino experiments will provide complementary information of
light sterile neutrino.

Now we have verified that the RGE approach is workable in quantitative study
of neutrino oscillation in single constant density matter. We will explore the
wider application with more complicated structure in the following part.

###  3.4 The day-night asymmetry of solar neutrino

Recently there have been some progresses in the study of solar neutrino.
Neutrinos produced in the CNO fusion cycle have been observed BOREXINO:2020aww
, and the potential to observe 8B solar neutrino has been discussed
JUNO:2020hqc . In this part, we continue to study solar neutrino in the RGE
approach as an extension of pure terrestrial matter effects.

Before being observed on the surface of earth during daytime, the electron-
type neutrinos produced in solar core have travelled through solar matter, the
vacuum between sun and earth, giving the $\nu_{e}$ survival probabilities in
the adiabatic approximation

$P_{D}=\sum_{i=1}^{4}P_{ei}^{S}P_{ie}^{0},$ (29)

where $P_{ei}^{S}$ denotes the $\nu_{i}$ component of $\nu_{e}$ produced in
solar core and its propagation to earth surface while $P_{ie}^{0}$ stands for
the probabilities of electron-type neutrino contained in the $i$-th mass
eigenstate. The equation (29) is an extension of three-flavor framework
Akhmedov:2004rq , including the contribution from a light sterile neutrino
contribution. Before arriving at the detectors on the other side of the
incident point, solar neutrinos pass some distance inside the earth at night
and the extra earth matter effects should be taken into account. The nighttime
$\nu_{e}$ survival probability is then

$P_{N}=\sum_{i=1}^{4}P_{ei}^{S}P_{ie}^{E},$ (30)

where $P_{ie}^{E}$ denotes the probability of $\nu_{i}$ propagation in earth
matter and finally captured by detector in $\nu_{e}$ state. The day-night
asymmetry stands for the probability difference between the day and night
time, giving

$\Delta P=P_{D}-P_{N}=\sum_{i=1}^{4}P_{ei}^{S}(P_{ie}^{0}-P_{ie}^{E}).$ (31)

The calculation of $\Delta P$ requires the information of
$P_{ei}^{S},P_{ie}^{0}$ and $P_{ie}^{E}$. In general, we have

$P_{ie}^{0}=|U_{ei}|^{2},\quad P_{ei}^{S}=|V^{S}_{ei}|^{2},\quad
P_{ei}^{E}=|V^{E}_{ei}|^{2}$ (32)

with vacuum PMNS matrix $U_{ei}$, solar matter effective mixing matrix
$V^{S}_{ei}$ and earth matter effective mixing matrix $V^{E}_{ei}$. In the
practical calculation, after producing in solar core 444In solar core, taking
$N_{e}=102N_{A}/{\textrm{cm}}^{3}$ for electron density Bahcall:2000nu  and
$N_{n}=0.1639N_{e}$ for neutron density Asplund:2009fu , we obtain the solar
core matter parameter $k=-0.5\times 0.1639\approx-0.08$. , we adopt adiabatic
approximation to describe neutrino propagation through other part of the sun
and the vacuum between sun and the Earth. For the Earth model, we take the
slab approximation to describe the terrestrial matterDziewonski:1981xy .
Explicitly, the effective mixing matrix in Earth is in the form of

$V^{E}=\left(\prod_{i=1}^{N}V_{i}e^{-iH_{i}L_{i}}V_{i}^{\dagger}\right)U,$
(33)

where $N$ is the number of slabs, $H_{i}$ is the effective Hamiltonian after
diagonalization in $i$-th layer with corresponding neutrino traveling distance
$L_{i}$ and effective mixing $V_{i}$. Here in 8-Layer Earth Model
Dziewonski:1981xy , including the structure of Inner Core, Outer Core, Mantel
and Crust, the value of $N$ depends on the zenith angle.

$\begin{array}[]{cc}\includegraphics[width=216.81pt]{dna-1_4th.eps}&\includegraphics[width=216.81pt]{dna-2_4th.eps}\\\
(1)&(2)\end{array}$

Figure 4: Electron-neutrino survival probabilities from solar neutrino in the
three-flavor (red) and four-flavor (blue) pictures, in which the daytime
behaviors are marked as dashed curves while nighttime ones vary in their
corresponding colored ranges.

Although the mixing in Earth, equation (33), is complicated due to the
multiple structures of the Earth, RGE approach provides a proper choice for
realizing the calculation. Combining all the necessary mixings both in Earth
and the sun, the electron-neutrino survival probabilities from solar neutrino
in various conditions are numerically presented in Fig. 4. Some features can
be drawn:

  * •

In general, the survival probability in $4\nu$ picture decreases compared with
the corresponding one in $3\nu$ picture. This is understandable as the sterile
neutrino takes up part of the components of electron-neutrino produced in
solar core, diminishing the survival probability.

  * •

Though input parameter dependence exists, the difference is tiny between
different parameter choices in $4\nu$ framework.

  * •

The probability at nighttime, which depends on the altitude of incident beam,
vary in a range due to the terrestrial matter effects.

  * •

For the maxima of day-night asymmetry $\Delta P$, it can be reached around
$13.5\textrm{MeV}$ for three-flavor case while $15\textrm{MeV}$ in the four-
flavor framework. This could be further discriminated by future solar neutrino
experiments.

As demonstrated in above quantitative analysis, the RGE approach to neutrino
matter provides a proper tool in exploring objects with richer structures.

##  4 Concluding remarks

The RGE has a wide application in many fields of modern physics. In the
pioneering work, a connection between RGEs and neutrino matter effects in
standard three-flavor picture has been established and a set of differential
equations have been obtainedXing:2018lob . To demonstrate the validity of RGE
approach to matter effects in quantitative study is a worthy topic.
Furthermore, to explore an extension of the entire methodology incorporating a
light sterile neutrino is also valuable for the recent fast development of
sterile neutrino-related experiments. These tasks have been achieved in
current work.

In presence of a light sterile neutrino, we have derived a completed set of
differential equations on effective mixing matrix elements, mass-square
differences and Jarlskog-like invariants in RGE approach. By opting for proper
input parameters (mixings and mass differences), these equations as well as
their solutions can describe both three-flavor and four-flavor neutrino
oscillation. The mixing parameters $V_{\alpha i}$ and $\tilde{m}_{i}^{2}$
depend on the matter parameter $k$ in the constant density matter. Solving
these differential equations, we carry out numerical analysis for the
evolution of $|V_{\alpha i}|^{2}$ and Jarlskog-like invariants. The
consistence of three-flavor case with Xing:2018lob  is verified, which further
guarantees the correctness for further study. By combining three LBL neutrino
experiments, we calculate their corresponding oscillation probabilities in NH
scenario as examples of terrestrial matter effects. The study of electron-
neutrino survival probabilities and hence day-night asymmetry of solar
neutrino observed on the Earth surface is more complicated for types of
mediums involved. Taking the adiabatic approximation for solar matter and slab
approximation for terrestrial matter, we numerically calculate the survival
probabilities at daytime and nighttime. The day-night asymmetry maximizes
itself at around $13.5\textrm{MeV}$ in the standard three active flavor
framework and $15\textrm{MeV}$ in four-flavor framework, which could be
discriminated in future precise measurement.

The RGE approach developed in this work provides a complementary way to study
neutrino phenomenology involving matter effects. On the other hand, the
correctness of its application shown in this work helps us to have a deep
understanding of RGE.

###### Acknowledgements.

We would like to thank Prof. Z.-z. Xing for his encouragement that produces
the current work. Discussions with Porf. Jiajie Ling and Prof. Benda Xu are
also acknowledged. This work is supported by NSFC under Grant No. U1932104 and
No. 12142502, by Guangdong Provincial Key Laboratory of Nuclear Science with
No. 2019B121203010.

##  Appendix A Input parameters

Two types of input parameters, mixing matrix and masses in both three-flavor
and four-flavor pictures, are summarize in Table 1.

Table 1: A summary of the involved input parameters in the numerical analysis of this work. $\sin^{2}\theta_{13}$ | $\sin^{2}\theta_{12}$ | $\sin^{2}\theta_{23}$ | $\Delta{m}^{2}_{21}$ | $\Delta{m}^{2}_{31}$ | $\delta_{13}$
---|---|---|---|---|---
$0.0220$ | $0.307$ | $0.546$ | $7.53\times 10^{-5}\textrm{eV}^{2}$ | $2.526\times 10^{-3}\textrm{eV}^{2}$ | $1.36\pi$
$\sin^{2}\theta_{14}$ | $\sin^{2}\theta_{24}$ | $\sin^{2}\theta_{34}$ | $\Delta{m}^{2}_{41}$ | $\delta_{14}$ | $\delta_{34}$
$0.019$ | $0.015$ | $0$ | $0.1\textrm{eV}^{2}$ | $0.5\pi$ | $0$

  * •

Three-flavor framework

We adopt the standard parameterization of PMNS matrix and the inputs are taken
from PDG Zyla:2020zbs .

  * •

Four-flavor framework

We follow the convention in Appendix A of Li:2018ezt  to parameterize the
four-dimensional mixing matrix with three leptonic CP-violating phases
$(\delta_{13},\delta_{14},\delta_{34})$. A global fit results Gariazzo:2017fdh
are taken as input here, in which $\delta_{13}$ is kept as a free parameter.

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11institutetext: Georgetown University,
11email<EMAIL_ADDRESS>

# Auction Type Resolution on Smart Derivatives

Yusuke Ikeno 11 James Angel 11 Shin’ichiro Matsuo 11 Ryosuke Ushida 11

###### Abstract

This paper proposes an auction type resolution for smart derivatives. It has
been discussed to migrate derivatives contracts to smart contracts (smart
derivatives). Automation is often discussed in this context. It is also
important to prepare to avoid disputes from practical perspectives. There are
controversial issues to terminate the relationship at defaults.In OTC
derivative markets, master agreements define a basic policy for the
liquidation process but there happened some disputes over these processes. We
propose to define an auction type resolution in smart derivatives, which each
participant would find beneficial.

###### Keywords:

OTC derivatives Dispute resolutions Auction.

##  1 Introduction

###  1.1 Background

OTC derivatives are traded in the financial market for the purpose of hedging
or transforming market risk and speculations. According to BIS statistics,
notional amounts outstanding in the first half of 2020 is 606,810 billion US
dollars 111https://www.bis.org/statistics/derstats.htm?m=6%7C32%7C71. This
figure is significant since the global GDP in 2019 is 87,799 billion (the
World Bank data222https://data.worldbank.org/indicator/ny.gdp.mktp.cd).

After the financial crisis, the regulation for OTC derivatives had got harder
to stabilize financial markets. This made the post trading process more
complicated. Not only financial institutions but also non financial
derivatives users are suffering from the regulation cost.

The financial industry finds that distributed ledger technology and smart
contracts could improve their business. The purpose is not only to make their
operations more efficient but also pioneer new area. For example, JPMorgan
developed ”Quorum” and Barclays held some hackasons called ”DerivHack”. There
are many research cases globally. Recently many central banks are studying
digital currencies, which would have a significant impact on OTC derivatives
as one of the use cases.[5]

Though there are many types of products, it is well known that they are
composites of conditional cash flows or settlements.There are some studies
that show they can be expressed in a functional programming framework.[8] This
feature also inspires it is appropriate to bring these products into smart
contracts.

###  1.2 Related works

Morini discussed re-design of the buisiness model noting that ”Blockchain
technology was created to change some trust-based business processes to make
them less reliant on trust”. [17] He also investigated the implementation of
collateral settlement on public blockchain. [18]

ISDA had published a series of guidelines for smart derivative contracts from
legal perspectives .[15]

Fries and Kohl-Landgraf thought regulatory implementations are inefficient due
to historic infrastructures and lack of standardization and automation of a
trade life cycle processes.[11] They also implemented prototypes under some
assumptions of digital currencies. [9, 10]

Clack and McGonalge discussed how smart contract code might process and
automate payments-related and deliveries-related events and proposed we should
understand derivatives contracts at different levels, in terms of both the
legal documentation and the work-flow. [4]

###  1.3 Contributions

It is important to process close-out effectively because events of defaults
affect significantly. Though master agreements define basic processes to close
out, calculation methodology is vague to some extent.

Clack and Datoo recently discussed how we can make smart close-out efficiently
from legal perspectives. [7] We discussed this issue from operational
viewpoint. As the liquidation process at default contains conflicts between
involve parties by nature, the stakeholders are likely to take actions to
protect their profit. The current method investigates the mid point under
possibly transparent manner. We proposed an auction method to calculate the
termination amounts and look for new trades so that the involved parties can
process more algorithmically and each participant can find merits.

##  2 Practice in OTC derivatives

###  2.1 Model of OTC derivatives over blockchain

Our discussion focuses on uncleared collateralized OTC derivatives. When two
parties start to trade OTC derivatives, they usually sign a ”master
agreement”, which governs all derivatives contracts between them. ISDA
(International Swap and Derivatives Association) provides templates for the
master agreement. Master agreements treat governed individual transactions as
one contract, which make netting settlements possible. The definition of
jurisdiction and basic policies for resolving disputes are also described in
master agreements. Schedule and credit support anex (so called ”CSA”) are the
related documents for precise information and the rule for collateral
management, which complement the master agreement.

Figure 1: Trading scheme

From here, we review the basic process after events of default. See [16] for
more details. Master agreements define events of default as below.

  * $\diamondsuit$

Events of Defaults

  * •

Failure to pay or deliver

  * •

Breach/ repudiation of agreement

  * •

Credit support default

  * •

Misrepresentation

  * •

Default under specified Transaction

  * •

Cross-default

  * •

Bankruptcy

  * •

Merger without Assumption

Generally speaking, an event of default occurs if one of the parties is at
fault.

If an event of default occurs (the party at fault is called ”defaulting
party”), the other party called ”non defaulting party” has a right to
terminate the contracts under their master agreement. The legal enforceability
of this right depends on the jurisdiction. The party usually request legal
opnions. This point is discussed in [7].

When the right is exercised, both parties enter into ”Early Termination” as
the agreement describes.

  1. 1.

Designation of early termination date

  2. 2.

Cessation of payment and delivery obligations

  3. 3.

Occurence of early termination

  4. 4.

Calculation of early termination amounts

  5. 5.

Delivery of statement detailing net termination amount payable

  6. 6.

Payment of termination amount

As described precisely below, the termination amount is the summation of
defined payments and present values of defaulted transactions. Since someone
has to calculate the present value, it is usually expected that the non
defaulting party asks ”Reference Market Makers” to quote from neutral
positions.

If an event of default occurs (and the right is exercised), the non defaulting
party loses its risk exposure coming from those defaulted transactions. The
non defaulting party will cover the exposure in the market. If there are only
a few defaulted transactions, it is easy to cover at once. Otherwise it takes
time to reconstruct the exposure as there might be concerns on market impact
and liquidity. This is why initial margin is required. IM is theoretically
designed to absorb such timing and liquidity risk. But we should understand
that the cost might exceed the margin depending on market conditions even
though the model explains we rarely face excess loss.

Figure 2: Covering the defaulted exposures

Especially for early termination, the parties should

  * •

terminate transactions under the agreed value.

  * •

close the process as soon as possible

  * •

reduce the effect to the financial system

The master agreements give a policy for these points but the exact way is up
to the involved parties. There could be conflicts at termination amounts
between both parties because the profit of one party means the loss of the
other. Unfortunately there were some cases the defaulting party started
disputes even though the quoted amount should have been accepted according to
the agreement.

Since the market circumstance might be stressed after events of default, it is
helpful for stakeholders to reconsider the process depending on the situation.
We should investigate the way satisfying

  1. 1.

the predefined process will be executed algorithmically within expectation.

  2. 2.

the process can be paused if unexpected loss would happen.

  3. 3.

each participant can cooperate in a constructive manner.

###  2.2 Credit risk in uncleared trades

####  2.2.1 Mechanism

The price movement in financial markets affects the valuation of contracts
like other financial products. Unrealized profit (or loss) would be lost if
the counterparty fails to continue. In other words, derivative contracts
include counterparty’s credit risk.

If markets move significantly, we should take care of not only our profit or
loss but also how counterparties are working.

####  2.2.2 Margin

New York Fed describes the history of the regulation of derivatives as below;
In 2009, the G20 Leaders agreed to reforms in the OTC derivatives market to
achieve central clearing and, where appropriate, exchange or electronic
trading of standardized OTC derivatives; reporting of all transactions to
trade repositories; and higher capital as well as margin requirements for non-
centrally cleared transactions 333New York Fed ”Over-The-Counter
Derivatives”https://www.newyorkfed.org/financial-services-and-
infrastructure/financial-market-infrastructure-and-reform/over-the-counter-
derivatives”. The margin requirement on uncleared OTC derivatives is yet on
the way.

There are two types of margin. One is called ”variable margin”(VM), which
covers present values of contracts. If your position has profit, you will
receive some assets as collateral whose value matches your profit. If loss,
you will have to transfer to your counterparty.

The other one is called ”initial margin”(IM), which is the cost for
reconstructing the positions. Under the margin requirement for uncleared OTC
derivatives, IM are required to be deposited into segregated accounts from
each party, such as custodians. We need statistical models to calculate this
amount. The SIMM, which is now regarded as a standard margin model for
uncleared OTC derivatives, is designed to ”meet a 99% confidence level of
cover over a 10-day standard margin period of risk”. [6]

###  2.3 Issues for calculation of termination amount

The calculation includes defined unpaid amounts that were due before the early
termination date and present values of contracts at the early termination
date. The early termination amount is usually calculated by the non defaulting
party. (There are some other cases which both parties agreed.

The ”Market Quotation” is used for calculation of the amount to protect each
profit . The master agreement defines that “Market Quotation” is an amount
determined on the basis of quotations from Reference Market-makers. ”Reference
Market Makers” is also defined as following; “Reference Market-makers” means
four leading dealers in the relevant market selected by the party determining
a Market Quotation in good faith (a) from among dealers of the highest credit
standing which satisfy all the criteria that such party applies generally at
the time in deciding whether to offer or to make an extension of credit and
(b) to the extent practicable, from among such dealers having an office in the
same city. [1] In short, ”Reference Market Makers” are those trusted third
parties who can provide fair calculation of derivative transactions. We can
imagine which companies are suitable for this role but it is uncertain that
those companies are willing to join the liquidation process.

Even though the process does not permit claims from default parties, there
were some cases that a defaulting party had disagreed with the amount, which
caused long lasting disputes. Since the defaulting party can be motivated to
raise their value at the events of default, it is effective to provide the
party another incentive so that they can close smoothly.

###  2.4 CCP cases

In case of centrally cleared trades, the derivatives transactions of defaulted
members could be transferred to other clearing members via the auction process
in a few days.

For example, LCH, one of the largest clearing house, maintains the default
process as below444https://www.lch.com/services/swapclear/risk-management ;

  1. (1)

Risk neutralisation & client porting:
After a Clearing Member default, LCH immediately begins porting non-defaulting
clients to non-defaulting Clearing Members. Alongside this, we hedge the
defaulter’s trade portfolio with the assistance of our Default Management
Group (DMG), senior executives with appropriate skills and expertises from its
clearing members and certain other members as the DMG considers appropriate,
that are seconded to LCH in the event of default.

  2. (2)

Portfolio auction:
Once the risk of the portfolio is substantially reduced by the DMG, LCH splits
the defaulter’s portfolio by product and currency. An auction is then
conducted for each portfolio. The ability to receive and price an auctioned
portfolio is one of the criteria we verify prior to granting firms membership
to SwapClear.

  3. (3)

Loss attribution:
In the event that losses are greater than the financial resources of the
defaulting member and of LCH, the funded Default Fund contributions of non-
defaulting SwapClear members are used.

##  3 Auction resolution

There are several types of auctions implemented on the smart contracts.[2, 12]
The main purpose in this case is to investigate the fair mid value of
transactions and liquidity in the market. We use the same algorithm for the
mid value as the ”Market Quotation”. A second price sealed bid auction is
applied to choose the winner of trade value. We require a stopping condition
about the trade cost and IM, which each participant finds beneficial to
cooperate.

Figure 3: Initiate an auction

###  3.1 Workflow

Based on the current workflow, we would like to open an auction to investigate
the fair values for termination amounts.

  * •

When the non defaulting party exercises the right, the auction contract is
executed.

  * •

The non defaulting party asks tradable entities from its relationship to join
the auction,

  * •

Bidders are required to show the mid valuation. If possible to trade, they can
also show the tradable value.

  * •

After bidding, the smart contract calculates the mid valuation (=MQ) in the
predefined manner. (For example, averaging excluding outliers for the mid
values.)

  * •

The defaulted trades would be terminated at MQ.

  * •

As for the trade value, the second highest bid is chosen.(second-price sealed-
bid auctions)

  * •

If the trade cost ($:=|\mathrm{the\ mid\ value}-\mathrm{the\ trade\ value}|$)
is more than IM, the non defaulting party has a right to avoid to trade.

    * –

As described before, IM is supposed to compensate for the potential loss,
which can be understood as a cost to reconstruct the exposure.

    * –

In this auction, the difference between the mid value and the trade costs is
the cost that the non defaulting party has to pay for.

    * –

If the cost is more than IM, the non defaulting party might find it
advantageous to cover in the market directly.

    * –

If the cost is less then IM, the residual can be reverted to the defaulting
party, which could incentivises the defaulting party to agree with this
auction process.

Since the optimal strategy in second bid auctions is showing the maximum value
which they are willing to pay, the non defaulting party can redeem the
realistic value for package trades.

It looks curious to revert the residual to the defaulting party because IM is
supposed to transfer to the non defaulting party by nature. As described
before IM is the potential loss to cover exposures arising from defaulted
trades. If the new trades are given without costs, the non defaulting party
does not require this margin for that purposes. We can think it to revert the
residual to the defaulting party if the termination amounts are payable.

Figure 4: Settlement of termination amounts

The following Table 1 shows some sample cases. Each value represents the
transactions value from A (the defaulting party). There are four or five
bidders depending on the cases. The ”mid” value is quoted for the ”Market
Quotation” and the ”trade” value is the price at which the bidder is willing
to trade. The ”aggregate” value means the ”Market Quotation” for mid values
and the second price auction winner and the price for trade values, leaving IM
amount aside.

Table 1: Sample auction scenarios. |  |  |  | bidders |
---|---|---|---|---|---
No | A | B |  value | 1 | 2 | 3 | 4 | 5 |  aggregate
1 |  100 |  80 |  mid |  90.00 |  95.00 |  85.00 |  100.00 |  90.00 |  91.67
|  |  | trade | 85.00 | - | 80.00 | 90.00 | - |  ($\sharp$ 4, 85.00)
2 |  100 |  80 |  mid |  90.00 |  95.00 |  85.00 |  100.00 |  - |  92.50
|  |  | trade | - | - | 80.00 | - | - | ( $\sharp$ 3, 80.00)
3 |  100 |  80 |  mid |  95.00 |  95.00 |  75.00 |  105.00 |  - |  95.00
|  |  | trade | 90.00 | - | - | 95.00 | - | ($\sharp$ 4, 90.00)
4 |  100 |  80 |  mid |  90.00 |  90.00 |  100.00 |  100.00 |  90.00 |  93.33
|  |  | trade | - | 75.00 | 80.00 | - | - | ($\sharp$ 3, 75.00)
5 |  100 |  80 |  mid |  90.00 |  90.00 |  90.00 |  90.00 |  90.00 |  90.00
|  |  | trade | - | - | 88.00 | - | - | ($\sharp$ 3, 88.00)

Depending on the market liquidity, B might want immediate counterparties to
cover its exposure. In such a case, B could ask the bidder after the process
because the cancel is a result of algorithm. Though there could be some
critics that this is not fair for the defaulting party, IM covers these
covering risk by its definition. The residual is a bonus for the defaulting
party.

###  3.2 Requirements

####  3.2.1 Standardization of information

Since bidders have to calculate the value of transactions by themselves, it is
important to share the transactions and related information.

  * •

Both parties names

  * •

Master agreement and related documents
The content of master agreement and related documents depends individually.
For example, collateral information affects valuation directly.

  * •

Details of each transaction
Exotic trades might be included among the defaulted trades as they are not
cleared.

If we could record this information in the same manner, it would be helpful to
share. There are some formats such as FIX and FpML to describe the
transactions. The booking system records the transactions in its original
format based on these common formats. Sharing details of transactions require
some kind of transformation.

When two parties start to trade OTC derivatives, they negotiate the details of
master agreement based on the template. The amendment of template causes a
potentially infinite pattern of master agreement and the customization makes
the initial negotiation annoying.

Under the digitization of derivatives, ISDA develops ”Common Domain Model”
(so-called ”CDM”) and ”Clause Library”.[3, 13, 14] CDM helps the users to
describe the transactions in a unified manner and Clause Library will reduce
the cost of legal negotiations.

####  3.2.2 Invitation of bidders

Bidders with good faith are essential to make these auctions successful. We
can not exclude the case of no bidder cases and manipulations by bidders.

In the current process, reference market makers are assumed to quote market
quotations. We can assume they will join to quote the mid values faithfully.
In other words, the existence of reference market makers is the trust point of
this process. If both parties could not find out such parties, they would have
to negotiate their termination amounts by themselves. They might need
enforceable authority such as a court.

##  4 Evaluation

If both parties involved in OTC derivatives As this auction satisfies the
current price decision process ( ”Market Quotation”), this can be understood
as the improvement.

If the tradeable value is available, the non defaulting party can avoid
liquidity concerns and the defaulting party may be able to retrieve some of
the IM.

Bidders may find it attractive to take part in this auction if they can reduce
their risk exposures at lower cost. Though this depends on the market
situation, their stance could send a signal of liquidity to the non defaulting
party.

It is basically beneficial for each participant to open an auction as a part
of the liquidatuin.

##  5 Discussions

###  5.1 Auction style

We select the second price sealed auction for trade value to clarify the
meaning of bids. The policy is bit different from calculation of the mid
value, which investigates the average value. Those participants whose
calculate relatively higher are likely chosen as the winners. Higher price is
nice as the defaulting party gains more residual and it costs less for the non
defaulting party to cover the exposure at once. The difference of the
valuation could cause another problem, for example collateral process. Under
the collateral exchange, two parties are required to exchange VM reflecting
each exposure calculation. If the valuation is far, they meet disputes. There
are some temporal resolution such as averaging or undisputed amount.[19]

Since CCP can not keep the defaulted trades, they have to immediately transfer
the portfolio to another member chosen by an auction. In this case, VM and IM
are alos transferred to the new party at the same time. If we totally imitate
this, the situation is very simple but we could not find out the merit from
the defaulting party. CCP scheme is strongly supported by the duty of clearing
members.

OTC trades are basically closed bilaterally. It is difficult to define the
”fair value” in OTC markets because the contract data is private by nature so
that we have to rely on the market participants at both entry and exit points.
As the master agreement assumes reference market makers, it is important to
manage risk in the community.

###  5.2 Electronic platforms

Electronic platforms (e.g., TradeWeb) support efficient executions, on which
users can request for a quote of not only one trade but also several trades.
Some may wonder if the non defaulting party could request for ”Market
Quotation” via these platforms efficiently instead of opening an auction. What
we emphasise is the predefined process to gain the fair value on the smart
contracts. Using smart contracts do not depend on a going concern principle,
which could eliminate a kind of trust. Of course there may exist the more
efficient way to combine the execution platform with smart contracts.

###  5.3 Governance of the auction process

We assume that the process is governed by the smart contract. Inputs and
decisions are given by participants. We must rely on the bidders to succeed in
the auction. As the current procedure assumes there are referenced market
makers, we also assume trusted quotes here.

###  5.4 Privacy

If eligible tradeable prices do not appear in the auction at all, the non
defaulting party has to trade in the market directly. In this case bidders
know the defaulted risk exposure, which might make it difficult for the party
to trade efficiently. As we assume the non defaulting party invites the
bidders in this scheme those parties are unlikely do so. Though this is a kind
of morality, we should take some measure against such immoral players in
future.

On the other hand, this information might have a kind of signaling effects, so
that some faithful market makers support to close the position.

Pricing OTC derivatives can be expressed as a real-valued mapping whose
domains are transactions, valuation models and market data. If there were a
contract which could calculate the value from transactions (provided by the
non defaulting party) and the pair of a valuation model and market data
(provided by the bidders), we would not concern the privacy. From this point,
standarzation of valuation matters.

##  6 Conclusion and future works

Since we discussed the concept to improve the process in this paper, we should
implement this idea to investigate more practically.

For efficient valuation, it is more convenient to assume trusted valuation
agents independent from dealers; in other words, oracles pricing the
transactions. If it is difficult to set up agents which calculate
automatically, we may prepare for the shared calculation methodologies in
advance. Fries et al [9] also pointed out this point.

Decentralised finances (so called ”DeFi”) provide derivatives for crypto
assets. They are usually overcollateralized and lock the potential cash flow
at first to avoid any disputes. If users request more flexibility as
traditional derivatives, the similar issues including dispute resolutions are
required. In this area, the algorithmic methodology would be prefered.

We assume some parties would quote the tradable value because the current
process also assumes ”reference market makers”. As financial markets might
face less liquidity at events of defaults, the non defaulting party would not
find out the candidates for its auction. In such a case, the IM could not
compensate for the cover cost. We have to continue monitoring how the
situation will change in future. Even though we could define reasonable
resolutions, the involved parties might change their mind at the liquidation.
We would also rely on other enforceable resolutions which work with the
algorithm. Under the context of digital transformation, smart contracts would
play important roles to automate and redefine many workflows. It is important
to not only enjoy its efficiencies but also scrutinize its potential disputes
and resolutions from practical viewpoints.

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