# Sensitivity to CP Discovery in the Presence of Lorentz Invariance Violating
Potential at T2HK/T2HKK

Kaustav Chakraborty1 0000-0002-1847-8403<EMAIL_ADDRESS>Srubabati
Goswami1 0000-0002-5614-4092<EMAIL_ADDRESS>Supriya Pan1,2
0000-0003-3556-8619<EMAIL_ADDRESS>1 Physical Research Laboratory,
Ahmedabad, Gujarat, 380009, India  2 Indian Institute of Technology,
Gandhinagar, Gujarat, 382355, India

###### Abstract

Investigation of conservation/violation of CP symmetry in the leptonic sector
is very essential in understanding the evolution of the universe. Lorentz
invariance and CPT are fundamental symmetries of nature. The violation of
Lorentz invariance can also lead to CPT violations. The standard three flavour
neutrino oscillation framework presents a scenario to observe the signature of
Lorentz invariance and CP violations. This work focuses on the effect of
Lorentz invariance violating (LIV) parameters on the sensitivity to CP
violation. We investigate the sensitivity in two proposed configurations of
the upcoming T2HK experiment: (i) one detector each placed at 295 km and 1100
km, and (ii) two identical detectors at 295 km. This study probes the effect
of CPT violating parameters $a_{e\mu},a_{e\tau},a_{\mu\tau}$.

††preprint: APS/123-QED

##  I Introduction

The standard model (SM) of particle physics describes the elementary particles
and their interactions except gravity. Its effectiveness has been tested
thoroughly over the years in a successful manner. However, there have been a
few shortcomings that are essential to address. Neutrino oscillation is one
such phenomenon in which there is inter-conversion among three flavors
$\nu_{e},\nu_{\mu},\nu_{\tau}$. This can occur only if the neutrinos are
massive. However, in the SM, neutrinos are massless particles. Incorporation
of neutrino mass requires some new physics beyond the standard model (BSM).
The scale of new physics may be at a higher energy scale and the standard
model can be regarded as an effective low-energetic version of the high energy
theory.

The parameters of the three neutrino oscillation framework are: mixing angles
$\theta_{12},\theta_{13},\theta_{23}$ and Dirac CP phase $\delta_{13}$ along
with the two mass squared differences
$\Delta_{31}=m_{3}^{2}-m_{1}^{2},\Delta_{21}=m_{2}^{2}-m_{1}^{2}$
corresponding to mass eigenstates $m_{1},m_{2},m_{3}$. Currently, the precise
measurement of the octant of $\theta_{23}$, the CP phase $\delta_{13}$, and
the sign of atmospheric mass squared difference $\Delta_{31}$ is still awaited
and probed by current oscillation experiments. Several future experiments with
enhanced capabilities are in the pipeline to measure these parameters with
increased precision. They also open up the possibility of testing different BM
scenarios. Probing new physics beyond the standard model(BSM) like sterile
neutrinos, non-standard interactions, neutrino decay, long range forces, CPT
and Lorentz invariance violations, etc in these experiments is currently a
prominent area of research in neutrino physics.

In this work, the new physics that we focus on is the CPT non-conserving
Lorentz invariance violation. The local relativistic quantum field theories
form the basic structure of nature. The primary assumptions of this theory
consist of Lorentz invariance, locality of the interactions along with the
hermicity of the Hamiltonian. The conservation of CPT symmetry is embedded
into these assumptions. A spontaneous violation of CPT will always be
associated with Lorentz invariance violation but not vice versa. Since both
CPT and Lorentz invariance are fundamental symmetries of nature, any violation
of them will provide us with an indication of new physics governing the laws
of nature.

Lorentz invariance protects isotropy and homogeneity of the local relativistic
QFT in space-time. In the minimal $SU(3)\times SU(2)\times U(1)$ SM, this
symmetry is conserved. However, there are higher dimensional theories (related
to the Plank scale $\sim 10^{19}$ GeV )where Lorentz invariance violation is
generated spontaneously[1, 2, 3, 4, 5]. String theories obeying Lorentz
covariant dynamics are shown to facilitate the spontaneous breaking of Lorentz
symmetry in [1, 2, 3, 4]. At the level of the standard model, Lorentz
invariance violation can be manifested as a new physics effect suppressed by
the Planck scale. There is also a proposed effective field theoretical
description containing the SM in ref. [5] that will reproduce the known
physics phenomena of the SM at low energy. The violation of Lorentz invariance
and CPT have been tested using Kaons [6, 7], neutral $\rm{B_{d}}$ or
$\rm{B_{s}}$ mesons [7, 8], and neutral D mesons [7, 9]. It has been realized
that neutrino oscillations experiments can provide a texting ground for
probing signatures of LIV and there have been several studies of LIV at
various experiments[10, 11, 12, 13, 14, 15, 16, 17].

The presence of LIV will also modify the standard neutrino oscillation
probabilities and this can impact the determination of the standard $3\nu$
parameters in future neutrino oscillation experiments. This aspect and
containing the LIV parameters have been investigated in several recent works
in the context of accelerator and atmospheric neutrino experiments. In ref.
[18], a study has been performed to give bounds on LIV parameters using INO-
ICAL, T2HK, and DUNE. There has been a study on the effect of LIV parameters
in NO$\nu$A and T2K in ref. [19]. Efforts have been made to separately
understand the effects of LIV interactions and non-standard interactions (NSI)
at long baseline experiments in [20, 21]. Other recent studies related to CPT
violation and LIV interactions in neutrinos can be found in [22, 23, 24, 25,
26, 27, 28, 29, 16].

In our study, we focus on the effects of CPT violating LIV parameters on the
determination of the CP phase in the upcoming T2HK[30]/T2HKK[31] detector.
T2HK plans to have two detectors at baseline 295 km (first oscillation maxima)
and T2HKK is proposed to have one detector each at 295 km and 1100 km(second
oscillation maxima). We compare and contrast the CP sensitivities in the
presence of LIV at both these baselines and explore the synergistic effect
between the first and second oscillation maxima.

The structure of our work is as follows. At first, the formalism of LIV in the
neutrino sector is provided in section II. It’s followed by a discussion on
the dependence of the neutrino oscillation probabilities on LIV parameters in
section III. We present the numerical analysis for CP discovery in the
presence of non-diagonal CPT violating LIV parameters in section IV.

##  II Theory of Lorentz invariance violation

Lorentz invariance violation(LIV) can be comprehended through a standard model
extension (SME) framework in the context of a low energy effective theory[32].
The neutrino behavior is contained in the terms

$\mathcal{L}=\frac{1}{2}\iota\bar{L}_{a}\gamma^{\mu}\overleftrightarrow{D_{\mu}}L_{a}-(a_{L})_{\mu
ab}\bar{L}_{a}\gamma^{\mu}L_{b}+\frac{1}{2}(c_{L})_{\mu\nu
ab}\bar{L}_{a}\gamma^{\mu}\overleftrightarrow{D^{\nu}}L_{b}$ (1)

where the first term is the usual Standard-Model kinetic term for the left-
handed doublets $L_{a}$ with index $a$ ranging over the three generations
$e,\mu,\tau$. The coefficients for Lorentz violation are $(a_{L})_{\mu ab}$
which has mass dimension one and controls the CPT violation, and
$(c_{L})_{\mu\nu ab}$ which is dimensionless and is CPT conserving. The
Lorentz-violating terms in (1) modify both interactions and propagation of
neutrinos. Any interaction effects are expected to be tiny and well beyond the
existing sensitivities. In contrast, propagation effects can be substantial if
the neutrinos travel large distances. The time evolution of neutrino states is
controlled by the effective Hamiltonian extracted from (1) as

$(\mathcal{H}_{eff})_{ab}=E\delta_{ab}+\frac{m^{2}_{ab}}{2E}+\frac{1}{E}(a_{L}^{\mu}p_{\mu}-c_{L}^{\mu\nu}p_{\mu}p_{\nu})_{ab}$
(2)

The LIV-induced parameter $a_{L}^{\mu}$ (CPT-violating) will change the sign
in case of anti-neutrinos while $c_{L}^{\mu\nu}$ will remain unchanged. Here
we will focus on only the isotropic component of these parameters in the Sun-
centered celestial-equatorial frame and fix $\mu,\nu$ to zero and redefine
$(a_{L})^{0}_{ab}\equiv a_{ab},(c_{L})^{00}_{ab}\equiv c_{ab}$. The
Hamiltonian due to LIV is given by,

$H_{LIV}=\begin{bmatrix}a_{ee}&a_{e\mu}&a_{e\tau}\\\
a^{\star}_{e\mu}&a_{\mu\mu}&a_{\mu\tau}\\\
a^{\star}_{e\tau}&a^{\star}_{\mu\tau}&a_{\tau\tau}\end{bmatrix}-\frac{4}{3}E\begin{bmatrix}c_{ee}&c_{e\mu}&c_{e\tau}\\\
c^{\star}_{e\mu}&c_{\mu\mu}&c_{\mu\tau}\\\
c^{\star}_{e\tau}&c^{\star}_{\mu\tau}&c_{\tau\tau}\end{bmatrix}$ (3)

The total Hamiltonian for neutrino propagation including the standard MSW
matter effect is given by

$H_{tot}=\frac{1}{2E}\begin{pmatrix}m_{1}^{2}&0&0\\\ 0&m_{2}^{2}&0\\\
0&0&m_{3}^{2}\end{pmatrix}+\begin{pmatrix}\sqrt{2}G_{F}N_{e}&0&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}+H_{LIV}$ (4)

Here we only consider CPT-violating LIV parameters $a_{\alpha\beta}$. The non-
diagonal parameters are complex and given by
$a_{\alpha\beta}=|a_{\alpha\beta}|e^{\iota\phi_{\alpha\beta}}$ where as
diagonal parameters $a_{\alpha\alpha}$ are real. There is an established
correlation between CPT-violating LIV parameters and matter NSI parameters
given by

$\epsilon_{\alpha\beta}^{m}\equiv\frac{a_{\alpha\beta}}{\sqrt{2}G_{F}N_{e}}$
(5)

Irrespective of this mapping, their origins of them are very different as well
as their effect, e.g, matter NSI have similar effects to MSW matter potential
in neutrino propagation whereas CPT-violating LIV has an intrinsic effect on
neutrino propagation even in the vacuum [33]. The current constraints of the
CPT violating LIV parameters are given below in table 1[34, 35, 36].

Parameter | SK Bound | IceCube Bound
---|---|---
$a_{e\mu}$ |  $1.8\times 10^{-23}$ GeV | N.A.
$a_{e\tau}$ |  $4.1\times 10^{-23}$ GeV | N.A.
$a_{\mu\tau}$ |  $0.65\times 10^{-23}$ GeV |  $0.29\times 10^{-23}$ GeV
Table 1: The table depicts $95\%$ C.L. bounds of CPT violating non-diagonal
LIV parameters from SK and IceCube experiments

##  III  Probabilities in presence of LIV parameters

In our study, we probe the effects of the CPT violating LIV parameters
$a_{e\mu},a_{e\tau},a_{\mu\tau}$ on the discovery of CP. For the scope of this
work, we use the proposed long baseline experiment configuration of Tokai to
hyper-Kamiokande (T2HK) and Tokai to hyper-Kamiokande via Korea (T2HKK). The
setup of T2HKK provides an advantage of two detectors; one at hyper Kamiokande
site, 295 Km away from the source, and another at 1100 km away from the source
at Korea. At the leading order of $\alpha=\Delta_{21}/\Delta_{31}$, the
appearance probability $P_{\mu e}$ depends only on parameters
$a_{e\mu},a_{e\tau},\phi_{e\mu},\phi_{e\tau}$ whereas, the disappearance
probability depends on $a_{\mu\tau},\phi_{\mu\tau}$. The probabilities are
calculated in ref. [37, 38] as follows,

$\displaystyle P_{\mu e}=P_{\mu e}^{3\nu}+P_{\mu e}^{a_{e\mu}}+P_{\mu
e}^{a_{e\tau}}$ (6) $\displaystyle
P_{\mu\mu}=P_{\mu\mu}^{3\nu}+P_{\mu\mu}^{a_{\mu\tau}},$ (7)

where $P_{\mu e}^{3\nu},P_{\mu\mu}^{3\nu}$ are the three flavour oscillation
probabilities in the matter and the LIV induced part of the probabilities are
given as,

$P_{\mu
e}^{3\nu}=4s_{13}^{2}s_{23}^{2}\frac{\sin^{2}[(\hat{A}-1)\Delta]}{(\hat{A}-1)^{2}}+2\alpha
s_{13}\sin{2\theta_{12}}\sin{2\theta_{23}}\frac{\sin[\hat{A}\Delta]}{\hat{A}}\frac{\sin[(\hat{A}-1)\Delta]}{\hat{A}-1}\cos(\Delta+\delta_{13})$
(8) $\displaystyle P_{\mu e}^{a_{e\mu}}\simeq\frac{4|a_{e\mu}|\hat{A}\Delta
s_{13}\sin
2\theta_{23}\sin\Delta}{\sqrt{2}G_{F}N_{e}}[Z_{e\mu}\sin(\delta_{13}+\phi_{e\mu})+W_{e\mu}\cos(\delta_{13}+\phi_{e\mu})]$
(9) $\displaystyle P_{\mu e}^{a_{e\tau}}\simeq\frac{4|a_{e\tau}|\hat{A}\Delta
s_{13}\sin
2\theta_{23}\sin\Delta}{\sqrt{2}G_{F}N_{e}}[Z_{e\tau}\sin(\delta_{13}+\phi_{e\tau})+W_{e\tau}\cos(\delta_{13}+\phi_{e\tau})]$
(10) $\displaystyle
P_{\mu\mu}^{a_{\mu\tau}}=\frac{4|a_{\mu\tau}|\hat{A}\Delta\sin
2\theta_{23}\sin\Delta}{\sqrt{2}G_{F}N_{e}}[Z_{\mu\tau}\cos\phi_{\mu\tau}+W_{\mu\tau}\cos\phi_{\mu\tau}]$
(11)

where $\Delta=\frac{\Delta_{31}L}{4E}$, $\alpha=\Delta_{21}/\Delta_{31}$,
$\hat{A}=\frac{2\sqrt{2}G_{F}N_{e}E}{\Delta_{31}}$, $A=2\sqrt{2}G_{F}N_{e}E$,
$s_{ij}=\sin\theta_{ij}$, $c_{ij}=\cos\theta_{ij}$,

$\displaystyle Z_{e\mu}=-\cos\theta_{23}\sin\Delta,\hskip
8.53581ptZ_{e\tau}=\sin\theta_{23}\sin\Delta,\hskip
8.53581ptZ_{\mu\tau}=-\sin^{2}2\theta_{23}\cos\Delta$ (12) $\displaystyle
W_{e\mu}=c_{23}(\frac{s^{2}_{23}\sin\Delta}{\Delta.c^{2}_{23}}+\cos\Delta),W_{e\tau}=s_{23}(\frac{\sin\Delta}{\Delta}-\cos\Delta),W_{\mu\tau}=\frac{-\cos^{2}2\theta_{23}\sin\Delta}{\Delta}$
(13)

###  III.1 Variation in $P_{\mu e}$ with phases at fixed
$a_{e\mu},a_{e\tau},a_{\mu\tau}$

In the presence of the LIV parameters, the appearance channel probability
depends on the parameters $a_{e\mu},a_{e\tau}$, and $a_{\mu\tau}$. This also
shows dependence on the LIV phases $\phi_{e\mu},\phi_{e\tau}$ in conjunction
with $\delta_{13}$. The modifications in $P_{\mu e}$ due to LIV parameters are
probed in this section at 1100 and 295 km baselines. In the following plots,
the values of the oscillation parameters are chosen as,
$\theta_{12}=33.44^{\circ},\theta_{13}=8.57^{\circ},\theta_{23}=49^{\circ},\Delta_{21}=7.42\times
10^{-5}$ eV2, and $|\Delta_{31}|=2.515\times 10^{-3}$ eV2. $P_{\mu e}$ is
plotted as a function of $\delta_{13}$ at 0.6 GeV in fig. 1 for normal (top
panel) and inverted (bottom panel) mass orderings in case of 295 km (red), and
1100 km (blue) baseline while the values of the non-diagonal LIV parameters
are kept fixed at $10^{-23}$ GeV. The bands refer to the variation of LIV
phases. The significant points to be noted are as follows,

  * •

It can be observed from both top and bottom panels that the effect of
$\phi_{e\mu},\phi_{e\tau}$ is larger than $\phi_{\mu\tau}$ as the width of the
red and blue bands are narrower in the right panels than the left and middle
ones. This can be understood from the eq. (9), (10) as $P_{\mu e}$ has no
contribution from $\phi_{\mu\tau}$ at the leading order.

  * •

In the case of NO(upper panels), the variation of $P_{\mu e}$ with
$\delta_{13}$ for 1100 km is sharper as 0.6 GeV is adjacent to the second
oscillation maxima(0.7 GeV). However, in 295 km the variation is less due to
the first oscillation maxima occurring at 0.6 GeV. Thus probabilities at CP
conserving values $0^{\circ},\pm 180^{\circ}$ are more separated from
probabilities at other CP violating values at 1100 km than at 295 km.

  * •

Also, in the case of NO, the maxima and minima of $P_{\mu e}$ happen at
different $\delta_{13}$ values for 295 km and 1100 km. For instance, the
probabilities at $\delta_{13}=\pm 90^{\circ}$ has a maximum difference from
probabilities at CP conserving values for 295 km. However, in the case of 1100
km, the probabilities at $\delta_{13}=\pm 90^{\circ}$ are very close to
probability values at $\pm 180^{\circ}$. Therefore, while evaluating the
sensitivity to CP discovery at $\delta_{13}=\pm 90^{\circ}$, there will be a
higher sensitivity for 295+1100 km configuration than individual 295 km and
1100 km due to the synergy.

  * •

For IO, The variation with $\delta_{13}$ is very flat at 1100 km while the
variation at 295 km remains similar. This leads to poor sensitivity for CP
discovery for the T2HKK configuration for IO.

The disappearance probability $P_{\mu\mu}$ doesn’t depend on the CP phase at
the leading order. Therefore, in the case of $P_{\mu\mu}$, dependence on
$\phi_{\mu\tau}$ isn’t linked with $\delta_{13}$.

Figure 1: $P_{\mu e}$ as a function of $\delta_{13}$ for
$\theta_{23}=49^{\circ}$, $a_{e\mu}=10^{-23}$ GeV (left), $a_{e\tau}=10^{-23}$
GeV (middle), and $a_{\mu\tau}=10^{-23}$ GeV (right) due to variation of
respective phases $\phi_{e\mu},\phi_{e\tau},\phi_{\mu\tau}$ for NO(top),
IO(bottom) at 0.6 GeV in 295 km (red), 1100 km(blue)

Figure 2: $P_{\mu e}$ (left) , $P_{\bar{\mu}\bar{e}}$ (right) as a function of
$\delta_{13}^{true}$ for true values of $\theta_{23}=49^{\circ}$,
$a_{e\mu}=10^{-23}$ GeV. The panels on the top (bottom) two rows refer to 295
km (1100 km) for NO(top) and IO(bottom). Violet, red, green, blue refer to
$\phi_{e\mu}^{true}=-90^{\circ},0^{\circ},90^{\circ},180^{\circ}$
respectively.

In fig. 2, the oscillation probabilities $P_{\mu e},P_{\bar{\mu}\bar{e}}$ are
plotted as a function of $\delta_{13}$ at fixed energy of 0.6 GeV
corresponding to 295 km and 1100 km baselines for NO, IO considering
$\theta_{23}=49^{\circ}$ and $a_{e\mu}=10^{-23}$ GeV. We observe the following
features,

  * •

In 1100 km, the $P_{\mu e}$ probabilities have larger values than
$P_{\bar{\mu}\bar{e}}$. However, in IO that order reverses.

  * •

In NO, the $P_{\mu e}$ curves show a peak at the lower half plane
(LHP)[$-180^{\circ}:0^{\circ}$] in the range $-160^{\circ}:-130^{\circ}$. The
peaks of $P_{\bar{\mu}\bar{e}}$ curves occur at the upper half plane
(UHP)[$0^{\circ}:180^{\circ}$] in the range $130^{\circ}:170^{\circ}$.

  * •

In the case of IO for 1100 km, both the $P_{\mu e}$, $P_{\bar{\mu}\bar{e}}$
has maxima around $0^{\circ}$.

  * •

In 295 km, various probabilities for different values of $\phi_{e\mu}$ vary
over a small region while being very close to each other.

  * •

In 295 km, the maxima of $P_{\mu e}$, $P_{\bar{\mu}\bar{e}}$ curves occur
around $\pm 90^{\circ}$ for both NO and IO.

  * •

In the case of both 295 km and 1100 km, the red (green) curves corresponding
to $\phi_{e\mu}^{tr}=0^{\circ}(180^{\circ})$ give the maximum (minimum)
variation in $P_{\mu e}$. In $P_{\bar{\mu}\bar{e}}$, this order reveres.

##  IV $\chi^{2}$ Analysis of CP discovery

In this section, we study the potential of the experiments T2HKK/T2HK for CP
discovery. The configurations for the proposed experiments are, (i) T2HK
(Tokai to Hyper-Kamiokande): two detectors of 187 kton at 295 km, (ii)T2HKK
(Tokai to Hyper-Kamiokande and Korea): one detector of 187 kton at 295 km and
another similar detector at 1100 km away in Korea[39]. For our study, we
consider the second detector of T2HKK at an off-axis angle of $1.5^{\circ}$
from the source at J-PARC facility in Tokai[31]. The experiment of T2HKK
offers us the advantage of a large matter effect at 1100 km. For our numerical
analysis with GLoBES[40, 41], we use a proposed beam of energy 1.3 MW
considering 2.5 years of neutrino mode and 7.5 years of anti-neutrino mode run
time with an exposure of $27\times 10^{21}$ proton on target (POT). The
detector configuration and systematic errors are taken from [31].

The final value of $\chi^{2}$ is derived after marginalization over pull
variables $\xi$, and variables of oscillation $\omega$ as follows,

$\Delta\chi^{2}={\rm Min}[\chi^{2}_{stat}(\omega,\xi)+\chi^{2}_{pull}(\xi)],$
(14)

where $\chi^{2}_{pull}$ includes the symmetric errors and the Poissonian
$\chi^{2}_{stat}$ is defined in terms of total true no of events
$N_{i}^{true}$ and events generated by theoretical model $N_{i}^{test}$ in the
$i^{th}$ energy beam.

$\chi^{2}_{stat}(\omega,\xi)=2\sum_{i}[N_{i}^{test}-N_{i}^{true}+N_{i}^{true}\text{ln}\frac{N_{i}^{true}}{N_{i}^{test}}]\text{;
}\chi^{2}_{pull}=\sum_{r=1}^{4}\xi_{r}^{2}$ (15)

The systematic uncertainties are included through the method of pull in terms
of variables: signal normalization error, background normalization error,
energy calibration error on signal, and background (tilt). In this work the
test parameters $\omega$ are
$\theta_{23},\delta_{13},|\Delta_{31}|,a_{\alpha\beta},\phi_{\alpha\beta}$.

Parameter | True Value | Marginalization Range
---|---|---
$\theta_{12}$ | $33.4^{\circ}$ | N.A.
$\theta_{13}$ | $8.62^{\circ}$ | N.A.
$\theta_{23}$ | $49^{\circ}$ | $39^{\circ}:51^{\circ}$
$\delta_{13}$ | $-180^{\circ}:180^{\circ}$ | $0^{\circ},180^{\circ}$
$|\Delta_{31}|$ |  $2.5\times 10^{-3}$ eV2 |  $2.4:2.6\times 10^{-3}$ eV2
$a_{\alpha\beta}$ |  $10^{-23}$ GeV |  $10^{-22}:10^{-24}$ GeV
$\phi_{\alpha\beta}$ | $-180^{\circ}:180^{\circ}$ | $0^{\circ},180^{\circ}$
Table 2: True values of all the parameters and their range of marginalization Channel | 295 km | 1100 km
---|---|---
$\nu_{e}$ Appearance |  $3.2\%(5\%)$ |  $3.8\%(5\%)$
$\nu_{\mu}$ Disappearance |  $3.6\%(5\%)$ |  $3.8\%(5\%)$
$\bar{\nu}_{e}$ Appearance |  $3.9\%(5\%)$ |  $4.1\%(5\%)$
$\bar{\nu}_{\mu}$ Disappearance |  $3.6\%(5\%)$ |  $3.8\%(5\%)$
Table 3: The signal (background) normalization uncertainties of the
experiments for different channels

We have seen in table 1 the current bound for NSI parameters are $\sim
10^{-23}$ GeV. Therefore, we have considered true values of
$a_{e\mu},a_{e\tau},a_{\mu\tau}=10^{-23}$ GeV throughout our study. For
numerical analysis for CP discovery in standard three flavour case, the test
values are considered as $\delta_{13}=0^{\circ},180^{\circ}$. Similarly, in
the presence of an extra LIV phase, we consider test values of
$\delta_{13},\phi_{\alpha\beta}$ as combinations of $0^{\circ},180^{\circ}$.
While performing chi-square($\chi^{2}$) analysis in the presence of LIV, we
consider one parameter to be non-zero at a time. Apart from phases, we have
marginalized the chi-square over $\theta_{23}$ and $|\Delta_{31}|$. The true
values[42] and the marginalization ranges of the parameters are given in table
2. The run time in neutrino and anti-neutrino mode is 2.5 years and 7.5 years
respectively.

###  IV.1 Single detector analysis

In this section, the sensitivity to CP discovery is probed with a single
detector at 295 km and 1100 km. This helps in understanding the features of
these individual baselines. The total event rates get equal contributions from
neutrinos and anti-neutrinos because of the chosen run time. Therefore
studying sensitivity for individual channels will help in understanding the
total sensitivity. In this section, we study the effect of only the LIV
parameter $a_{e\mu}$ as a representative case.

The $\chi^{2}$ for appearance channel in neutrino, anti-neutrino mode, along
with the total $\chi^{2}$ for 1100 km is plotted in the top, middle, and
bottom panels of fig. 3. The left(right) panels correspond to the NO (IO). The
different true values of
$\phi_{e\mu}=-90^{\circ},0^{\circ},90^{\circ},180^{\circ}$ have been shown by
violet, red, blue and green curves respectively. The features of significance
in fig. 3 are as follows,

  * •

In the neutrino mode (NO), the red curve $\phi_{e\mu}=0^{\circ}$ has the
maximum sensitivity in both half-planes but in LHP the magnitude at the peak
is significantly larger. The green curve $\phi_{e\mu}=180^{\circ}$ has the
lowest sensitivity. This is consistent with the features seen from the plot of
$P_{\mu e}$ in the left panels of fig. 2.

  * •

In the anti-neutrino mode (NO), the highest sensitivity is achieved for the
green $\phi_{e\mu}=-90^{\circ}$ and blue curve $\phi_{e\mu}=180^{\circ}$ in
both UHP and LHP but the peak value in UHP is higher.

  * •

In the case of total sensitivity (NO), the red curve ($\phi_{e\mu}=0^{\circ}$)
has the highest sensitivity in the UHP and the blue curve
($\phi_{e\mu}=180^{\circ}$) reaches the maximum sensitivity in the LHP. While
marginalizing, the minimum of $\chi^{2}$ occurs at different values of the
parameters for neutrino and anti-neutrino leading to a synergistic effect in
total sensitivity.

  * •

In the case of neutrino mode for IO, the red (green) shows the maximum
(minimum) sensitivity and the value of $\chi^{2}$ is higher in UHP than LHP.

  * •

In the context of anti-neutrino mode for IO, the green (red) curve reaches the
maximum (minimum) value of $\chi^{2}$. The green curve’s maximum value of
$\chi^{2}$ is predominantly the highest in UHP. The other curves also have
maxima of higher value in UHP.

  * •

In the case of IO for the total sensitivity, the green curve
($\phi_{e\mu}=0^{\circ}$) has the maximum $\chi^{2}$ in both LHP and UHP with
the latter case having significantly higher value.

  * •

The sensitivity curves for $\phi_{e\mu}=-90^{\circ},90^{\circ}$ show non-zero
sensitivity at $\delta_{13}^{true}=0^{\circ},\pm 180^{\circ}$. This happens as
the test values $\phi_{e\mu},\delta_{13}$ are a combination of
$0^{\circ},180^{\circ}$ that is different from these true sets of values.

In fig. 4, the value of $\chi^{2}$ is depicted as a function of
$\delta_{13}^{true}$ for a single detector at 295 km considering true ordering
as NO (left) and IO (right). The top, middle, and bottom panels refer to the
sensitivity in neutrino, anti-mode, and the total sensitivity respectively.
The different colors are for various $\phi_{e\mu}$ similar to fig. 3. The main
observations of fig. 4 are as follows,

  * •

In the case both of NO and IO, in the neutrino mode, the red and green curves
show the maximum and minimum sensitivity respectively in LHP but In UHP all
curves have very low and similar sensitivity. In the antineutrino mode, the
order of sensitivity for all the curves is almost similar but the green curve
($\phi_{e\mu}^{tr}=180^{\circ}$) has slightly more sensitivity.

  * •

In both NO and IO we observe higher sensitivity in neutrino mode. This is due
to the fact that $P_{\mu e}$ curves have a higher range of variation than
$P_{\bar{\mu}\bar{e}}$ ones as was seen in fig. 2.

  * •

In the case of total sensitivity, all the curves have similar sensitivity
except for the red curve ($\phi_{e\mu}^{tr}=0^{\circ}$) showing a light higher
value of $\chi^{2}$ in UHP.

The synergy between various channels in the test $\theta_{23}$ is depicted in
fig. 5 for 295 km (left panel) and 1100 km (right panel). We observe that the
minimum of the total chi-square is obtained at one of the minima of the
$\nu_{\mu},\bar{\nu}_{\mu}$ channels around $\theta_{23}=49^{\circ}$. However,
the minimum chi-square value is boosted by the non-zero chi-square
contribution from the $\nu_{e},\bar{\nu}_{e}$ channels.

Figure 3: $\chi^{2}$ in $\nu_{e}$ (top), and $\bar{\nu}_{e}$ (middle) channel
and total $\chi^{2}$ (bottom) as a function of $\delta_{13}^{true}$ for true
values of $\theta_{23}=49^{\circ}$ with $a_{e\mu}=10^{-23}$ GeV at 1100 km for
NO (left), IO(right). Violet, red, green blue refer to
$\phi_{e\mu}=-90^{\circ},0^{\circ},90^{\circ},180^{\circ}$ respectively.
Figure 4: $\chi^{2}$ in $\nu_{e}$ (top), and $\bar{\nu}_{e}$ (middle) channel
and total $\chi^{2}$ (bottom) as a function of $\delta_{13}^{true}$ for true
values of $\theta_{23}=49^{\circ}$ with $a_{e\mu}=10^{-23}$ GeV at 1100 km for
NO (left), IO(right). Violet, red, green blue refer to
$\phi_{e\mu}=-90^{\circ},0^{\circ},90^{\circ},180^{\circ}$ respectively.
Figure 5: $\chi^{2}$ as a function of $\theta_{23}^{test}$ at 295 km (left),
1100 km (right). Green (blue) refers to $\nu(\bar{\nu}$) channels and violet
gives total $\chi^{2}$.

###  IV.2 Comparative analysis between T2HKK and T2HK

In this section, we compare and contrast the CP discovery potential of the
proposed T2HKK and T2HK configurations. This study is performed for the LIV
parameters $a_{e\mu},a_{e\tau},a_{\mu\tau}$ taking one to be non-zero at a
time. In fig. 6, we present the sensitivity as a function of
$\delta_{13}^{true}$ for T2HKK (left) and T2HK (right) for NO (top) and
IO(bottom) for $a_{e\mu}^{true}=10^{-23}$ GeV. Different curves correspond to
the different values of $\phi_{e\mu}^{true}$.

Figure 6: $\chi^{2}$ as a function of $\delta_{13}^{true}$ for true values of
$\theta_{23}=49^{\circ}$, $a_{e\mu}=10^{-23}$ GeV in T2HKK (left) and T2HK
(right) configurations for NO (top), and IO(bottom). Violet, red, green, and
blue curves refer to
$\phi_{e\mu}^{true}=-90^{\circ},0^{\circ},90^{\circ},180^{\circ}$
respectively.

The major points observed from fig. 6 are as follows;

  * •

T2HKK offers the best sensitivity for all values of the LIV phase
$\phi_{e\mu}$. This is due to the synergistic effect between 1100 km and 295
km baselines. This will be explained later in the context of fig. 7.

  * •

The highest sensitivity is obtained at $\delta_{13}=90^{\circ}$ for both T2HK
and T2HKK. The corresponding values of $\phi_{e\mu}$ are
$0^{\circ}$($180^{\circ}$) for NO(IO) case in T2HKK, and $0^{\circ}$ in T2HK.
This can be understood from fig. 3 and 4, which shows that for individual
baseline, the maxima comes at $\phi_{e\mu}=0^{\circ}$ around
$\delta_{13}=90^{\circ}$.

In order to understand the synergy between 295km and 1100 km baselines, in
fig. 7, we have shown the $\chi^{2}$ as a function of test $a_{e\mu}$ (left),
and $\theta_{23}$ (right) for a set of true parameters keeping other test
parameters fixed.

  * •

From the left panel, we show that minimum $\chi^{2}$ for 295 km and 1100 km
occurs at different test values of LIV parameter $a_{e\mu}$. Whereas in T2HKK,
both baselines are analyzed together, the minimum occurs at a different value
of $a_{e\mu}^{ts}$, thus enhancing the $\Delta\chi^{2}$.

  * •

In the right panel, the enhancement in $\chi^{2}$ for T2HKK is due to the
increased statistics. However, when marginalizing the $\chi^{2}$ over other
test parameters, the synergy is also observed in $\theta_{23}$.

Figure 7: $\chi^{2}$ as a function of $a_{e\mu}^{test}$(left),
$\theta_{23}^{test}$(right) for true values of $\theta_{23}=49^{\circ}$,
$\delta_{13}=-90^{\circ}$, $\phi_{e\mu}=180^{\circ}$, $a_{e\mu}=10^{-23}$GeV.

In fig. 8, we present the values of $\chi^{2}$ as a function of
$\delta_{13}^{true}$ for $a_{e\tau}^{true}=10^{-23}$ GeV in the T2HKK and T2HK
configurations corresponding to NO (IO) in the top (bottom) column. The
results for the true values of phase $\phi_{e\tau}$ as $-90^{\circ}$(violet),
$0^{\circ}$(red), $90^{\circ}$(blue), $180^{\circ}$(green) using different
colours as mentioned in the parenthesis. The major observations are as
follows,

  * •

Similar to in fig. 6, the sensitivity at T2HKK is quiet higher than T2HK
configurations.

  * •

We observe the maximum sensitivity in T2HKK around $\delta_{13}=90^{\circ}$
($-90^{\circ}$) which is influenced by the maxima of $P_{\mu e}$
($P_{\bar{\mu}\bar{e}}$) curves in 295 km occurring at $90^{\circ}$
($-90^{\circ}$). Although most of the curves show sensitivity in a similar
range, the red one reaches the highest at UHP of $\delta_{13}^{true}$.

Figure 8: $\chi^{2}$ as a function of $\delta_{13}^{true}$ for true values of
$\theta_{23}=49^{\circ}$ with $a_{e\tau}=10^{-23}$ GeV for T2HKK (left) and
T2HK (right) configurations in NO (top), and IO (bottom). Violet, red, green,
and blue refer to
$\phi_{e\mu}^{true}=-90^{\circ},0^{\circ},90^{\circ},180^{\circ}$
respectively.

We show the $\chi^{2}$ as a function of true $\delta_{13}$ in the fig. 9 for
effects of $a_{\mu\tau}$ for T2HKK and T2HK configurations in NO(top) and
IO(bottom). The noteworthy points from these two figures are as follows,

  * •

The best sensitivity is observed in T2HKK, but the sensitivity of T2HK is also
very close. The reason behind this is no significant effect of $a_{\mu\tau}$
in $P_{\mu e}$.

  * •

Also, there is no significant variation of sensitivity w.r.t phase
$\phi_{\mu\tau}$. This is due to the narrow band of due to $\phi_{\mu\tau}$ as
also seen from probability plots in fig. 1.

Figure 9: $\chi^{2}$ as a function of $\delta_{13}^{true}$ for true values of
$\theta_{23}=49^{\circ}$ with $a_{\mu\tau}=10^{-23}$ GeV for T2HKK (left) and
T2HK (right) configurations for NO (top), and IO (bottom). Violet, red, green
and blue refer to
$\phi_{\mu\tau}^{true}=-90^{\circ},0^{\circ},90^{\circ},180^{\circ}$
respectively.

##  V Precision $\chi^{2}$ analysis of $\delta_{13},\phi_{\alpha\beta}$’s

In this section, we present the precision of $\delta_{13}$ and LIV phases
$\phi_{\alpha\beta}$’s for T2HKK, T2HK in the figures 10 and 11 respectively.
These are presented in terms of contours in $\phi_{\alpha\beta}-\delta_{13}$
test plane of various combinations of true values of
$\phi_{\alpha\beta},\delta_{CP}=0^{\circ},90^{\circ},-90^{\circ}$. We consider
the true values of LIV parameters as $10^{-23}$ GeV, $\theta_{23}=49^{\circ}$.

Figure 10: $1\sigma$(dotted), $2\sigma$(solid), $3\sigma$(dashed) contours[2
d.o.f.] corresponding to three different true values of
$\delta_{13},\phi_{jk}$ for true LIV parameters $a_{e\mu}$ (top), $a_{e\tau}$
(middle) and $a_{\mu\tau}$ (bottom) having value of $10^{-23}$ GeV for T2HKK
configuration

we can observe the following points from fig. 10,

  * •

In the topmost panels corresponding to $a_{e\mu}$, we observe closed $2\sigma$
contours for $\delta_{13},\phi_{e\mu}=90^{\circ},-90^{\circ}$ but not for
$\delta_{13},\phi_{e\mu}=0^{\circ}$(middle panel).

  * •

On the other hand, in the middle panels corresponding to $a_{e\tau}$, the
$2\sigma$ precision is better for $\delta_{13},\phi_{e\tau}=0^{\circ}$ but
worse for $90^{\circ},-90^{\circ}$

  * •

In the lowest panel corresponding to $a_{\mu\tau}$, we observe that $2\sigma$
contours for $\delta_{13}=90^{\circ},\phi_{\mu\tau}=90^{\circ}$ and
$\delta_{13}=-90^{\circ},\phi_{\mu\tau}=-90^{\circ}$ stretch over full range
of $\phi_{\mu\tau}$. However, in the middle panel, very good precision is
obtained for $\delta_{13}=0^{\circ},\phi_{\mu\tau}=0^{\circ}$ with a closed
$3\sigma$ contour.

  * •

The best sensitivity for $\phi_{e\mu}$ is seen for
$\phi_{e\mu}^{tr}=90^{\circ},\delta_{13}^{tr}=90^{\circ}$, whereas best
sensitivity for $\phi_{e\tau}$ is obtained at
$\phi_{e\tau}^{tr}=0^{\circ},\delta_{13}^{tr}=0^{\circ}$.

In fig. 11, we plot similar contours for T2HK. We argue that the precision in
$\delta_{13}$, $\phi_{e\mu}$, $\phi_{e\tau}$ is poorer for T2HK. This is
expected since at 295 km CP sensitivity is less. We can observe that the
contours are similar for T2HK and T2HKK in $\delta_{13}-\phi_{\mu\tau}$ plane,
i.e., precision is similar in both configurations also seen from fig. 9. The
best sensitivity is obtained in T2HK also for
$\delta_{13}=0^{\circ},\phi_{\mu\tau}=0^{\circ}$.

Figure 11: $1\sigma$(dotted), $2\sigma$(solid), $3\sigma$(dashed) contours[2
d.o.f.] corresponding to three different true values of
$\delta_{13},\phi_{jk}$ for true LIV parameters $a_{e\mu}$ (top), $a_{e\tau}$
(middle) and $a_{\mu\tau}$ (bottom) having value of $10^{-23}$ GeV for T2HK
configuration

##  VI Discussions

The main focus of our work is to investigate the CP sensitivity in the future
T2HK/T2HKK experiment in the presence of the CPT violating LIV parameters. We
first study the CP discovery potential for individual baselines of 295km and
1100 km in the presence of LIV phases and ascertain the role played by
neutrino and anti-neutrino contributions to the total $\chi^{2}$. Next, we
obtain the sensitivity for T2HK and T2HKK configurations. We find that T2HKK
gives a better sensitivity because of the synergistic effects of 295 km and
1100 km for LIV in the $e-\mu$ and $e-\tau$ sectors. We have identified
synergy in parameters of
$a_{\alpha\beta},\theta_{23},\phi_{\alpha\beta},\delta_{13}$. However, for LIV
in the $\mu-\tau$ sector, both configurations give similar sensitivity. This
is because of the weak dependence of $P_{\mu e}$ on $\phi_{\mu\tau}$.

We also obtain the precision of $\delta_{CP},\phi_{\alpha,\beta}$ for various
true values of these phases in T2HK, T2HKK. We have found that the sensitivity
of $\delta_{13}$ is better for T2HKK configuration in the presence of
$a_{e\mu},a_{e\tau}$. In the case of $a_{\mu\tau}$, the sensitivity is best
for $\delta_{13}=0^{\circ},\phi_{\mu\tau}=0^{\circ}$.

###### Acknowledgements.

SG acknowledges the J.C. Bose Fellowship (JCB/2020/000011) of the Science and
Engineering Research Board of the Department of Science and Technology,
Government of India.

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The Self-energy of Nucleon for the Pole term of the Axial-vector Current and
the Neutron $\beta$-decay

Susumu Kinpara

Institute for Quantum Medical Science (QST)
Chiba 263-8555, Japan

###### Abstract

The effect of the pion propagator on the $\beta$-decay of neutron is
investigated to account for the ratio of the axial-vector to the vector part
$g_{A}/g_{V}$ by including the pole term. A suggestion is made that the sign
and the size of the vertex correction is appropriate to improve the
discrepancy between the result of the model hamiltonian and the experiment.

## 1 Introduction

For the investigation of the mechanism of the reaction processes the nucleon
is the elementary object in the intermediate energy regions. The idea of the
point-like structure is inevitably modified by the interaction with the other
particle as a probe. Particularly the lightest meson that is pion plays a
decisive role on improvement of the state of nucleon. It is expressed by the
propagator and the effect of the pion degrees of freedom is essential to
evaluate the constants related to each process.
The self-energy of nucleon is the starting point of the pion-nucleon system
and also changes the form of the vertex part of the interaction. The relation
between these functions is obtainable by using the equation of motion
independent of the calculation of the perturbative expansion. Because of the
derivative coupling the pseudovector $\pi$-N interaction appears to be
difficult to control the divergences in the framework of the renormalization
by the counter terms of the mass and the fields. In turn, by virtue of the
non-perturbative term stemmed from the non-perturbative relation the
divergences are removed together with the counter terms mentioned above.
Adding the non-perturbative term the pseudovector coupling is connected to the
pseudoscalar coupling. The additional interaction contains the self-energy and
the vertex part is modified by the term. When the calculations of the
pseudoscalar $\pi$-N system in free space do not give the perfect results the
term of the self-energy is expected to account for the remaining gap.
The method to construct the self-energy is based on the field theoretical
treatment giving the form of the function as the series of powers in the Dirac
operator. Applying it to the scattering processes the structure constants such
as the magnetic moment and the polarizability of nucleon and the $\pi$-N
scattering parameters of the $S$ and $P$ waves are described well by each
model chosen separately for the respective phenomenon.
For the low energy regions below the $\Delta$(1232) resonance the lowest-order
approximation of the perturbative calculation with the self-energy does not
attain to describe the $\pi$-N elastic scattering. The higher-order
calculations could change the form of the self-energy largely so as to provide
the volume of the low energy parameters well. On the other hand the method of
the matrix inversion enables us reach the set of the scattering parameters
which construct the self-energy as the output. The application of it to the
photoproduction of pion is interesting for the pseudoscalar model needs the
additional effects from the threshold to the resonance energy. In the present
study the $\beta$ decay of neutron is chosen to examine the form of the self-
energy by the meson-exchange model of the field theory.

## 2 The vertex function with the axial-vector current

The hadronic part of the lagrangian of the interaction $L_{int}$ is assumed to
consist of the vector $L_{v}$ and the axial-vector $L_{a}$ parts

$\displaystyle
L_{int}=L_{v}+L_{a}=-g\,\bar{\psi}(x)\,\gamma_{\mu}\,(1-\gamma_{5})\,\vec{\tau}\,\psi(x)\,\vec{V}^{\mu}$
(1)

in which the $\psi$ and $\vec{V}_{\mu}$ are the fields of nucleon and the
isovector vector boson respectively. The strength of the coupling constant $g$
is assumed to be comparable with the electric charge and then allows us to
expand the term in powers of $g$ and neglect the terms of the higher-order
practically. The $\vec{\tau}$ is the 2$\times 2$ isospin matrix.
In order to derive the non-perturbative relation and examine the structure of
the vertex part which unites the external lines of some bosons and fermions
the set of two isovector currents that is the vector current $\vec{B}_{\mu}$
and the axial-vector current $\vec{A}_{\mu}$

$\displaystyle(\vec{B}_{\mu},\vec{A}_{\mu})\equiv\bar{\psi}\,(\gamma_{\mu},\gamma_{5}\gamma_{\mu})\,\vec{\tau}\,\psi$
(2)

is indispensable. The sum $\vec{J}\equiv\vec{B}+\vec{A}$ suffices the relation

$\displaystyle\partial\cdot\vec{J}=-2iM\vec{\rho}_{5}+4\,g\,\vec{V}\times\vec{J}$
(3)
$\displaystyle\vec{\rho}_{5}\equiv\bar{\psi}\,\gamma_{5}\,\vec{\tau}\,\psi$
(4)

where the nucleon mass $M$ is the average value of proton and neutron.
The equation of motion for $\vec{V}$ is given as

$\displaystyle(\partial^{2}+m^{2})\vec{V}=g\,(1+m^{-2}\,\partial\partial\cdot)\vec{J}$
(5)

in which the weight of the mass of the boson $m$ is not definite and possibly
simulate the $\rho$-meson or the much heavior boson related to the weak
interaction. In practice the $O(g^{2})$ term is dropped approximately to
derive the relations below.
The generalized relation is derived for the $T$-product which consists of a
set of the field operators

$\displaystyle T[\,V_{\mu}^{i}(x)\cdots\,\varphi_{a}(x_{a})\cdots]$ (6)

in which $\varphi_{a}(x_{a})$ ($\,\equiv\,\psi(x_{a})$ or $\bar{\psi}^{\small
T}(x_{a})\,$) represents the Dirac fields. Operating $\partial^{2}+m^{2}$ on
the quantity (6) and using Eq. (5) without the $O(g^{2})$ term and the
following relation

$\displaystyle\delta(x_{a0}-x_{0})\,[\varphi_{a}(x_{a}),\,\rho_{5}^{i}(x)]=e_{ai}\varphi_{a}(x_{a})\,\delta^{4}(x_{a}-x)$
(7)

with $e_{ai}=\gamma_{0}\gamma_{5}\tau_{i}$ for
$\varphi_{a}(x_{a})=\psi(x_{a})$ or $(\gamma_{0}\gamma_{5}\tau_{i})^{T}$ for
$\varphi_{a}(x_{a})=\bar{\psi}^{\small T}(x_{a})$ the generalized form of the
equation of motion is obtained

$\displaystyle(\partial^{2}+m^{2})T[\,V_{\mu}^{i}(x)\cdots]=g\,T[\,J_{\mu}^{i}(x)\cdots\,]+i\frac{\delta}{\delta
V^{\mu}_{i}(x)}T[\,\cdots\,]$
$\displaystyle-2igm^{-2}M(\,\partial_{\mu}T[\rho_{5}^{i}(x)\cdots]+g_{\mu}^{0}\sum_{a,b,\cdots}e_{ai}\delta(x-x_{a})T[\cdots]\,)$
(8)

The third term of the right-hand side comes from the breaking of the
conservation of the axial-vector current in comparison with the case of the
quantum electrodynamics (QED). Another main difference from the QED is the
part of the functional derivative in Eq. (8) defined as

$\displaystyle\frac{\delta V_{\nu}^{j}(x^{\prime})}{\delta
V^{\mu}_{i}(x)}=\delta_{ij}\,\rho_{\mu\nu}(x-x^{\prime})$ (9)
$\displaystyle\rho_{\mu\nu}(x-x^{\prime})=\int\frac{d^{4}k}{(2\pi)^{4}}\,\rho_{\mu\nu}(k)\,e^{-k\cdot(x-x^{\prime})}$
(10)
$\displaystyle\rho_{\mu\nu}(k)=g_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{m^{2}}-(1-\frac{k^{2}}{m^{2}})\eta_{\mu}\eta_{\nu}$
(11)

where $\eta_{\mu}=(1,0,0,0)$. The additional terms in Eq. (11) are inherent in
the massive vector boson. The delta part
($\sim(m^{2}-k^{2})\eta_{\mu}\eta_{\nu}$) is dropped due to the cancellation
with the normal dependent term of the interacting hamiltonian.
In spite of the breaking of the current conservation the differentiation of
Eq. (8) and the use of the relation of the commutator

$\displaystyle\delta(x_{a0}-x_{0})[\varphi_{a}(x_{a}),J_{0i}(x)]=P_{ai}\,\varphi_{a}(x_{a})\,\delta^{4}(x_{a}-x)$
(12)

with $P_{ai}=(1-\gamma_{5})\tau_{i}$ for $\varphi_{a}(x_{a})=\psi(x_{a})$ or
$-(1+\gamma_{5})\tau_{i}^{\small T}$ for
$\varphi_{a}(x_{a})=\bar{\psi}^{\small T}(x_{a})$ gives the useful form

$\displaystyle(\partial^{2}+m^{2})\partial^{\mu}\,T[\,V_{\mu}^{i}(x)\cdots]=-2igm^{-2}M(\partial^{2}+m^{2})\,T[\,\rho_{5i}(x)\cdots\,]$
$\displaystyle+[\,i\partial^{\mu}\frac{\delta}{\delta
V^{\mu}_{i}(x)}-g\sum_{a,b,\cdots}(P_{ai}+2im^{-2}Me_{ai}\,g_{\mu}^{0}\,\partial^{\mu})\,\delta(x-x_{a})\,]\,T[\cdots]$
(13)

in which the approximate form of the current $\partial\cdot
J_{i}(x)\approx-2iM\rho_{5}^{i}(x)$ is used.
When the $T$-product consists of two boson fields the general relation in Eq.
(13) gives the equation for the boson propagator

$\displaystyle
iD^{ij}_{\mu\nu}(x-y)=\langle\,T[V_{\mu}^{i}(x)\,V_{\nu}^{j}(y)]\,\rangle$
(14)
$\displaystyle(\partial^{2}+m^{2})\partial^{\mu}D^{ij}_{\mu\nu}(x-y)=\partial^{\mu}\rho_{\mu\nu}^{ij}(x-y)$
$\displaystyle-2gm^{-2}M(\partial^{2}+m^{2})\langle\,T[\rho_{5}^{i}(x)\,V_{\nu}^{j}(y)]\,\rangle$
(15)

Using the $T$-product of three fields the vertex function
$\Gamma_{i}^{\mu}(x\,y\,;z)$ is defined so as to give the following relation

$\displaystyle\langle\,T[\psi(x)\,\bar{\psi}(y)\,V_{\mu}^{i}(z)]\,\rangle$
$\displaystyle\equiv-g\int
dx^{\prime}dy^{\prime}dz^{\prime}\,G(x-x^{\prime})\,\Gamma_{j}^{\nu}(x^{\prime}\,y^{\prime}\,;z^{\prime})\,G(y^{\prime}-y)\,D_{\nu\mu}^{ji}(z^{\prime}-z)$
(16)

with the nucleon propagator

$\displaystyle iG(x-y)=\langle\,T[\psi(x)\,\psi(y)]\,\rangle$ (17)

Operating $(\partial_{z}^{2}+m^{2})\partial_{z}^{\mu}$ on Eq. (16) with
respect to $z$ and applying Eq. (13) for three fields and Eq. (15) the
relation between these functions are given as follows

$\displaystyle\int
dx^{\prime}dy^{\prime}dz^{\prime}\,\rho_{\mu\nu}^{ij}(z-z^{\prime})\,G(x-x^{\prime})\,\partial_{z^{\prime}}^{\mu}\Gamma_{j}^{\nu}(x^{\prime}\,y^{\prime}\,;z^{\prime})\,G(y^{\prime}-y)\qquad\qquad\qquad$
$\displaystyle=i\delta(z-x)\,(1-\gamma_{5})\,\tau_{i}\,G(x-y)-i\delta(z-y)\,G(x-y)\,(1+\gamma_{5})\,\tau_{i}$
$\displaystyle+2im^{-2}M(\partial_{z}^{2}+m^{2})[\,\langle\,T[\,\rho_{5i}(z)\psi(x)\,\bar{\psi}(y)\,]\,\rangle$
$\displaystyle-ig\int
dx^{\prime}dy^{\prime}dz^{\prime}\,\langle\,T[\,\rho_{5i}(z)V_{\nu}^{j}(z^{\prime})\,]\,\rangle\,G(x-x^{\prime})\,\Gamma_{j}^{\nu}(x^{\prime}\,y^{\prime}\,;z^{\prime})\,G(y^{\prime}-y)\,]$
$\displaystyle-2m^{-2}M\,\partial_{z}^{0}\,[\,\delta(z-x)\,\gamma_{0}\,\gamma_{5}\,\tau_{i}\,G(x-y)+\delta(z-y)\,G(x-y)\,\gamma_{0}\gamma_{5}\,\tau_{i}\,]$
(18)

By the Fourier transform of Eq. (18) the relation of the current of the vertex
function is obtained

$\displaystyle(p-q)\cdot\Gamma_{i}(p,q)$
$\displaystyle=-A^{-1}[\,(1+\gamma_{5})\tau_{i}G(q)^{-1}-G(p)^{-1}(1-\gamma_{5})\tau_{i}\,]-2M\rho_{5}\tau_{i}\qquad\qquad$
$\displaystyle-2m^{-2}MA^{-1}g_{\mu
0}(p-q)^{\mu}\,[\,G(p)^{-1}\,\gamma_{0}\gamma_{5}\tau_{i}+\gamma_{0}\gamma_{5}\tau_{i}\,G(q)^{-1}\,]$
(19) $\displaystyle A\equiv 1-(p-q)^{2}/m^{2}$ (20)

in the momentum space. Under the on-shell condition of the incoming ($q$) and
the outgoing ($p$) momenta of nucleon such as $\gamma\cdot p\rightarrow M$ and
$\gamma\cdot q\rightarrow M$ Eq. (19) becomes
$(p-q)\cdot\Gamma_{i}(p,q)\rightarrow-2M\rho_{5}\tau_{i}$ and the breaking
term of the conservation of the current remains. The term acts as the pole
term of the vertex $\Gamma_{i}(p,q)$ and contributes to the interaction.

## 3 The effect of the pole term on the $\beta$-decay of neutron

The higher-order correction of the pole term in Eq. (19) by the pion
propagator is essential because the term of the lowest-order cancels with the
other term as seen later. In the present study the vertex correction is
calculated by the perturbative expansion with the pseudovector coupling
interaction lagrangian density

$\displaystyle
L_{pv}=-\frac{f}{m_{\pi}}\bar{\psi}(x)\,\gamma_{5}\gamma_{\mu}\vec{\tau}\,\psi(x)\,\partial^{\mu}\vec{\varphi}(x)$
(21)

where $f$ and $m_{\pi}$ are the coupling constant and the mass of pion
respectively.
Recently we have suggested that the pion-nucleon-nucleon three point vertex
has the non-perturbative term

$\displaystyle\Gamma(p,q)=\gamma_{5}\gamma\cdot(p-q)+G(p)^{-1}\,\gamma_{5}+\gamma_{5}\,G(q)^{-1}$
(22)

in which the perturbative term is represented by the lowest-order
approximation herein. Leaving the self-energy out the vertex is in agreement
with the pseudoscalar coupling irrespective of the on-shell condition.
The exact nucleon propagator $G(p)$ is expressed as

$\displaystyle G(p)=(\gamma\cdot p-M-\Sigma(p))^{-1}$ (23)

along with the self-energy $\Sigma(p)$. Because of the non-perturbative term
in Eq. (22) the quadratic divergent term in the numerator of the fraction
cancels out the denominator. The convergent result of $\Sigma(p)$ is obtained

$\displaystyle\Sigma(p)=Mc_{1}(p^{2})-\gamma\cdot p\,c_{2}(p^{2})$ (24)

in terms of the coefficients $c_{i}(p^{2})$ ($i$=1,2) as a function of
$p^{2}$. These are expanded around the on-shell point $p^{2}=M^{2}$.
Particularly the value of the zeroth order $c\equiv
c_{1}^{(0)}(M^{2})=c_{2}^{(0)}(M^{2})$ is necessary to examine the process of
the $\beta$ decay. The higher-order coefficients are dropped by the on-shell
condition for the momenta of the initial neutron and the final proton.
Using two quantities to characterize the correction that is the self-energy
$c$ and the coefficient of the pole term $a_{n}$ the three-point vertex part
$\Gamma_{i}(p,q)$ is expressed as follows

$\displaystyle\Gamma_{i}(p,q)\rightarrow
A^{-1}(1+c)\,[\,\gamma(1-\gamma_{5})+2M(1+c)^{-1}\hat{c}\,(p-q)^{-2}(p-q)\gamma_{5}\,]\,\tau_{i}$
$\displaystyle+\,\Gamma^{\prime}_{i}(p,q)\qquad\qquad\qquad$ (25)
$\displaystyle\hat{c}\equiv c+(A^{-1}-\sum_{n=0}^{\infty}a_{n})A$ (26)

under the assumption that $\Gamma_{i}(p,q)$ is sandwitched by the Dirac
spinors such as $\bar{u}^{(s^{\prime})}(p)\Gamma_{i}(p,q)u^{(s)}(q)$. The
$\Gamma_{i}^{\prime}(p,q)$ represents the term for which the relation
$(p-q)\Gamma_{i}^{\prime}(p,q)=0$ exists irrespective of the on-shell
condition. At the moment it is not included since the term is related to the
magnetism of the vector current and the part of the axial-vector current is
independent of the strong interaction by the invariance of the charge
conjugation [1]. When the mass of the vector boson is much larger than the
momentum transfer ($m^{2}\gg(p-q)^{2}$), then $A\approx 1$ and the relation in
Eq. (26) is substituted to $\hat{c}=c-\sum_{n=1}a_{n}$ in good approximation.
Here the subscript of the coefficient $a_{n}$ in the pole term specifies the
number of the propagator of pion. The lowest-order $a_{0}\,$(=1) does not
contribute to the pole term.
The process mediated by the pion propagator is a part of $a_{1}$ and it plays
a prominent role. Since the diagram is related to the decay of the $\pi^{-}$
it is free from the result of the perturbative calculation and the strength of
the process depends on the decay constant in the model hamiltonian. To
calculate the correction on the $\beta$-decay we treat the main part of
$a_{1}$ in a particular way different from the other part of the pole term.
The momentum transfer is $(p-q)^{2}\approx 0$ and the divergence of the
current is undetermined as $\sim(p-q)^{2}/((p-q)^{2}-m_{\pi}^{2})\rightarrow
0$ unless the mass of the pion is neglected because of the factor arising from
the loop integral.
The effect of the pion propagator is important for the zeroth order of
$\rho_{5}$ given as $\rho_{5}^{(0)}=a_{0}\gamma_{5}$ is eliminated from the
pole term. Then our interest is to calculate the coefficient of the second-
order $a_{1}\propto(\frac{f}{m_{\pi}})^{2}$ at the limit $(p-q)^{2}\rightarrow
0$ of the on-shell nucleons. The coefficient $a_{1}$ of the vertex correction
$\rho_{5}^{(2)}=a_{1}\gamma_{5}$ is given in the form of the integral as

$\displaystyle\rho_{5}^{(2)}=-(\frac{f}{m_{\pi}})^{2}\int\frac{d^{4}k}{i(4\pi)^{4}}\Delta(k)\Gamma(p,p-k)G(p-k)\gamma_{5}G(q-k)\Gamma(q-k,q)$
(27)

with respect to the momentum $k$ of the virtual pion. To make the evaluation
of the integral in Eq. (27) tractable the pion propagator is approximated by
the free one such as $\Delta(k)\approx\Delta_{0}(k)=(k^{2}-m_{\pi}^{2})^{-1}$
and the self-energy of the nucleon propagator is set equal to zero ($G\approx
G_{0}$). The vertex part is simplified by neglecting the non-perturbative term
($\Gamma(p,q)\approx\gamma_{5}\gamma\cdot(p-q)$). The integral is performed by
using the dimensional reguralization method and it yields as

$\displaystyle a_{1}=\frac{G_{\pi}^{2}}{(4\pi)^{2}}\,(2D-\frac{7}{6}-\,{\rm
log}\frac{M}{m_{\pi}})+O(m_{\pi}^{2}/M^{2})$ (28) $\displaystyle
D\equiv\frac{2}{\epsilon}+1-\gamma-{\rm log}\frac{m_{\pi}^{2}}{4\pi\mu^{2}}$
(29)

where $G_{\pi}\equiv 2Mf/m_{\pi}$ and $\gamma=0.577\cdots$ is the Euler’s
constant. The parameters $\epsilon$ and $\mu$ are ascribed to the shift of the
dimension as ${\rm 4}\rightarrow{\rm 4}-\epsilon$. The calculation of the form
factor is useful to remove the divergence in $a_{1}$ without preparing the
renormalized functions [2]. The relation $D=0$ has been used to obtain the
finite value of the $\gamma$-N-N vertex as a function of the momentum transfer
$Q^{2}$. Subtracting the divergent part by the above relation it yields
$a_{1}=-3.56085$ not including the term of the $O(m_{\pi}^{2}/M^{2})$ order.
Our interest is the pole term in Eq. (25) which possibly has an effect on the
ratio between the vector and the axial-vector coupling constants
$\lambda\equiv g_{A}/g_{V}$. Since there is not a mass term in the denominator
the matrix element gives a non-zero value at the limit $(p-q)^{2}\rightarrow
0$. In the case of the $\beta$-decay the initial state of neutron stands with
the spin directed to the $z$-axis and the final state of baryon is specified
by the proton momentum $\vec{p}$ and the spin. The direction of momentum
enters in the calculation through the parameter $\theta$ which represents the
average value of the polar angle as ${\rm cos}\,\theta=C/2$ using the
asymmetry parameter of proton $C=-0.2377$ [3].
At the limit $(p-q)^{2}\rightarrow 0$ there exists the simple relation for the
elements between the axial-vector part
$\Gamma_{A}\equiv\bar{u}^{(s^{\prime})}(p)\gamma\gamma_{5}u^{(s)}(q)$ and the
pseudoscalar part
$\Gamma_{P}\equiv(p-q)^{-2}(p-q)\bar{u}^{(s^{\prime})}(p)\gamma_{5}u^{(s)}(q)$
as

$\displaystyle\Gamma_{P}\approx(2M)^{-1}{\rm cos}^{2}\theta\,\Gamma_{A}$ (30)

making assumption of the nucleon spin unchanged by the decay ($s^{\prime}=s$).
Then $\Gamma(p,q)$ is modified as

$\displaystyle\Gamma(p,q)\rightarrow(1+c)(\gamma-\gamma\gamma_{5}+\alpha\,\gamma\gamma_{5})$
(31) $\displaystyle\alpha\equiv(1+c)^{-1}\,\hat{c}\;{\rm cos}^{2}\theta$ (32)

The size of $\alpha$ is connected with the assymmetry of the momentum of
proton.
The pion effect of the pole term is interesting to study the $\beta$ decay in
detail and search for the value of $c$ to construct the self-energy. It has
been known that the ratio of the currents $\lambda\equiv g_{A}/g_{V}$ is
related to the decay constant of the charged pion as

$\displaystyle-\lambda=\frac{G_{\pi}F_{\pi}}{\sqrt{2}M}$ (33)

where $G_{\pi}$ is same as the $\pi$-N coupling constant of the pseudoscalar
pion [1]. Making use of the value $f=1$ it yields $G_{\pi}=13.4544$
appropriate to the calculations of the $\pi$-N elastic scattering.
The decay constant $F_{\pi}\,(>0)$ is determined from the relation of the
decay rate $\Gamma_{\pi}$ for the process
$\pi^{-}\rightarrow\mu^{-}+\bar{\nu}_{\mu}$ in which the value of the lifetime
of $\pi^{-}$ is given experimentally as $\Gamma_{\pi}^{-1}=2.6033\times
10^{-8}\,{\rm s}$. Using the Fermi coupling constant $G_{F}=1.16637\times
10^{-5}\,{\rm GeV}^{-2}$ it results in $F_{\pi}=0.12825\,{\rm GeV}$. These
numerical values of the constants are applied to the relation in Eq. (33).
There is a difference of $-\lambda=1.300$ from the current experimental value
$-\lambda_{exp}=1.2695\pm 0.0029$ [4] roughly a few percent. We suggest
modifying the $g_{A}/g_{V}$ ratio as $\lambda\rightarrow\lambda+\alpha$ to
account for the $\beta$ decay correctly. The correction $\alpha$ depends on
the self-energy and it is meaningful to examine the relation with the other
phenomena. Using the value $c=2$ of the $n=2$ model the corrected value
results in $-(\lambda+\alpha)=1.2734$. Around the region $c$ moves $\alpha$ is
positive and the improvement of the numerical value indicates the similarity
to the electromagnetic property of the magnetic moment in favor of the
universality of the electromagnetic and the weak interactions rather than the
$\pi$-N elastic scattering under the strong interaction in the low-energy
region below the resonance.

## 4 Summary and remarks

Concerning the pole term of the vertex in the $\beta$-decay the lowest-order
is suppressed by the cancellation with the other term and required to take
into account the higher-order corrections by the pion propagator. It is
consistent with the hypothesis of the conserved vector current which is not
renormalized by the pion propagators. In spite that the second-order process
mediated by pion is the part of the proper vertex the process is represented
by the decay constant of the charged pion well and we have used the standard
way of the partially conserved axial-vector current to give most amount of the
ratio. The second-order result achieves to correct the excess of the main term
by means of the asymmetry parameter of proton and the self-energy to determine
the electromagnetic interaction. The numerical result is changed by adding the
inversion of the proton spin and then it is compensated by the shift of the
parameter of the self-energy to the lower values.

## References

  * [1] D. Luri$\acute{\rm e}$, Particles and Fields, Interscience Publishers (1968).
  * [2] S. Kinpara, arXiv:nucl-th/1906.08466v1.
  * [3] M. Schumann, M. Kreuz, M. Deissenroth, F. Gl$\ddot{\rm u}$ck, J. Krempel, B. M$\ddot{\rm a}$rkisch, D. Mund, A. Petoukhov, T. Soldner and H. Abele, Phys. Rev. Lett. 100, 151801 (2008).
  * [4] Particle Data Group, Phys. Lett. B592, p864 (2004).