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(B) Three-neutrino mixing parameters

$\delta $, $\mathit CP$ violating phase

Measurements of $\delta $ come from atmospheric and accelarator experiments looking at ${{\mathit \nu}_{{{e}}}}$ appearance. We encode values between 0 and 2$\pi $, though it is equivalent to use $\text{-}\pi $ to $\pi $.

VALUE (${{\mathit \pi}}$ rad)

CL%

DOCUMENT ID

TECN

COMMENT

$\bf{ 1.19 \pm0.22}$

OUR AVERAGE Error includes scale factor of 1.2.

$1.37$ ${}^{+0.31}_{-0.20}$

T2K

Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained

$0.82$ ${}^{+0.27}_{-0.87}$

NOVA

Normal mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$, ${{\mathit \theta}_{{{13}}}}$ constrained

$1.33$ ${}^{+0.45}_{-0.51}$

SKAM

Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained

• • We do not use the following data for averages, fits, limits, etc. • •

$1.52$ ${}^{+0.30}_{-0.41}$

NOVA

Inverted mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$, ${{\mathit \theta}_{{{13}}}}$ constrained

$1.08$ ${}^{+0.13}_{-0.12}$

FIT

Normal mass ordering, global fit

$1.58$ ${}^{+0.15}_{-0.16}$

FIT

Inverted mass ordering, global fit

$1.40$ ${}^{+0.22}_{-0.18}$

T2K

Normal mass ordering

$1.09$ ${}^{+0.15}_{-0.13}$

FIT

Normal mass ordering, global fit

$1.57$ ${}^{+0.14}_{-0.17}$

FIT

Inverted mass ordering, global fit

$0.0$ ${}^{+1.3}_{-0.4}$

NOVA

Normall mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$

$1.33$ ${}^{+0.46}_{-0.53}$

SKAM

3${{\mathit \nu}}$ osc: normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ free

$1.22$ ${}^{+0.76}_{-0.67}$

SKAM

3${{\mathit \nu}}$ osc: inverted mass ordering, ${{\mathit \theta}_{{{13}}}}$ free

$1.33$ ${}^{+0.48}_{-0.53}$

SKAM

3${{\mathit \nu}}$ osc: inverted mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained

$1.40$ $\pm0.20$

T2K

Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained

$1.54$ ${}^{+0.14}_{-0.12}$

95

T2K

Inverted mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained

$1.21$ ${}^{+0.91}_{-0.30}$

NOVA

Normal mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$

$1.46$ ${}^{+0.56}_{-0.42}$

NOVA

Normal mass order; octant I for ${{\mathit \theta}_{{{23}}}}$

$1.32$ ${}^{+0.21}_{-0.15}$

FIT

Normal mass ordering, global fit

$1.56$ ${}^{+0.13}_{-0.15}$

FIT

Inverted mass ordering, global fit

$1.45$ ${}^{+0.27}_{-0.26}$

T2K

Normal mass ordering

$1.54$ ${}^{+0.22}_{-0.23}$

T2K

Inverted mass ordering

$1.50$ ${}^{+0.53}_{-0.57}$

NOVA

Inverted mass ordering; ${{\mathit \theta}_{{{23}}}}$ in octant II

$0.74$ ${}^{+0.57}_{-0.93}$

NOVA

Normal mass ordering; ${{\mathit \theta}_{{{23}}}}$ in octant II

$1.48$ ${}^{+0.69}_{-0.58}$

NOVA

Normal mass ordering; ${{\mathit \theta}_{{{23}}}}$ in octant I

$\text{ 0.0 to 0.1, 0.5 to 2.0}$

90

NOVA

Inverted mass ordering

$0.0\text{ to }2.0 $

90

NOVA

Normal mass ordering

$\text{ 0 to 0.15, 0.83 to 2}$

90

T2K

Normal mass ordering

$1.09\text{ to }1.92 $

90

T2K

Inverted mass ordering

$0.05\text{ to }1.2 $

90

MINS

Normal mass ordering

$1.34$ ${}^{+0.64}_{-0.38}$

FIT

Normal mass ordering

$1.48$ ${}^{+0.34}_{-0.32}$

FIT

Inverted mass ordering

$1.70$ ${}^{+0.22}_{-0.39}$

GONZALEZ-GARC..

FIT

Normal mass ordering; global fit

$1.41$ ${}^{+0.35}_{-0.34}$

GONZALEZ-GARC..

FIT

Inverted mass ordering; global fit

$\text{ 0 to 1.5 or 1.9 to 2}$

90

MINS

Normal mass ordering

¹

ABE 2023F results are based on data collected between 2010 and 2020 in (anti)neutrino mode and include a neutrino beam exposure of $1.97 \times 10^{21}$ ($1.63 \times 10^{21}$) protons on target. For inverted mass ordering, the quoted result is $1.54$ ${}^{+0.18}_{-0.19}$ $\pi $ rad. Supersedes ABE 2020F.

²	ACERO 2022 uses data from Jun 29, 2016 to Feb 26, 2019 ($12.5 \times 10^{20}$ POT) and Feb 6, 2014 to Mar 20, 2020 ($13.6 \times 10^{20}$ POT). Results for normal and inverted mass ordering, and ${{\mathit \theta}_{{{23}}}}$ octant I and II are presented. Supersedes ACERO 2019.

³	For the octant I (lower octant), the 68$\%$ CL allowed region is discontinuous, and all delta values are allowed at 90$\%$ CL.

⁴

ABE 2018B uses 328 kton$\cdot{}$years of Super-Kamiokande I-IV atmospheric neutrino data to obtain this result. The fit is performed over the three parameters, $\Delta $m${}^{2}_{32}$, sin$^2{{\mathit \theta}_{{{23}}}}$, and $\delta $, while the solar parameters and sin$^2{{\mathit \theta}_{{{23}}}}$ are fixed to $\Delta $m${}^{2}_{21}$= ($7.53$ $\pm0.18$) $ \times 10^{-5}$ eV${}^{2}$, sin$^2{{\mathit \theta}_{{{12}}}}$ = $0.304$ $\pm0.014$, and sin$^2{{\mathit \theta}_{{{13}}}}$ = $0.0219$ $\pm0.0012$.

⁵	The inverted mass ordering is rejected at 1.0 $\sigma $. The quoted error bars are based on the local best-fit point.

⁶	SALAS 2021 reports results of a global fit to neutrino oscillation data available at the time of the Neutrino 2020 conference.

⁷

ABE 2020F results are based on data collected between 2009 and 2018 in (anti)neutrino mode and include a neutrino beam exposure of $1.49 \times 10^{21}$ ($1.64 \times 10^{21}$) protons on target. For inverted mass ordering, the quoted result is $1.56$ ${}^{+0.15}_{-0.17}$ $\pi $ rad. Supersedes ABE 2018G.

⁸	ESTEBAN 2020A reports results of a global fit to neutrino oscillation data available at the time of the Neutrino 2020 conference.

⁹	ACERO 2019 is based on a sample size of $1.33 \times 10^{20}$ protons on target with combined antineutrino and neutrino data. Superseded by ACERO 2022.

¹⁰

ABE 2018B uses 328 kton$\cdot{}$years of Super-Kamiokande I-IV atmospheric neutrino data to obtain this result. The fit is performed over the four parameters, $\Delta $m${}^{2}_{32}$, sin$^2{{\mathit \theta}_{{{23}}}}$, sin$^2{{\mathit \theta}_{{{13}}}}$, and $\delta $, while the solar parameters are fixed to $\Delta $m${}^{2}_{21}$= ($7.53$ $\pm0.18$) $ \times 10^{-5}$ eV${}^{2}$ and sin$^2{{\mathit \theta}_{{{12}}}}$ = $0.304$ $\pm0.014$.

¹¹	ABE 2018G confidence intervals are marginalized over both mass orderings. Normal order preferred with a posterior probability of 87$\%$. The 1-sigma result for normal mass ordering used in the average was provided by the experiment via private communications. Supersedes ABE 2017F.

¹²

ACERO 2018 performs a joint fit to the data for ${{\mathit \nu}_{{{\mu}}}}$ disappearance and ${{\mathit \nu}_{{{e}}}}$ appearance. The overall best fit favors normal mass ordering and ${{\mathit \theta}_{{{23}}}}$ in octant II. No 1$\sigma $ confidence intervals are presented for the inverted mass ordering scenarios. Superseded by ACERO 2019.

¹³	ABE 2017F confidence intervals are obtained using a frequentist analysis including ${{\mathit \theta}_{{{13}}}}$ constraint from reactor experiments. Bayesian intervals based on Markov Chain Monte Carlo method are also provided by the authors. Superseded by ABE 2018G.

¹⁴	Errors are projections of 68$\%$ C.L. curve of $\delta _{CP}$ vs. sin$^2{{\mathit \theta}_{{{23}}}}$.

¹⁵	ADAMSON 2016 result is based on a data sample with $2.74 \times 10^{20}$ protons on target. The likelihood-based analysis observed 6 ${{\mathit \nu}_{{{e}}}}$ events with an expected background of $0.99$ $\pm0.11$ events.

¹⁶	ADAMSON 2014 result is based on three-flavor formalism and ${{\mathit \theta}_{{{23}}}}>{{\mathit \pi}}$/4. Likelihood as a function of $\delta $ is also shown for the other three combinations of hierarchy and ${{\mathit \theta}_{{{23}}}}$ octants; all values of $\delta $ are allowed at 90$\%$ C.L.

¹⁷	GONZALEZ-GARCIA 2014 result comes from a frequentist global fit. The corresponding Bayesian global fit to the same data results are reported in BERGSTROM 2015 as 68$\%$ CL intervals of $1.24 - 1.94$ for normal and $1.15 - 1.77$ for inverted mass ordering.

¹⁸

ADAMSON 2013A result is based on ${{\mathit \nu}_{{{e}}}}$ appearance in MINOS and the calculated sin$^2(2{{\mathit \theta}_{{{23}}}})$ = 0.957,${{\mathit \theta}_{{{23}}}}>{{\mathit \pi}}$/4, and normal mass hierarchy. Likelihood as a function of$\delta $ is also shown for the other three combinations of hierarchy and ${{\mathit \theta}_{{{23}}}}$ octants; all values of $\delta $ are allowed at 90$\%$ C.L.

Conservation Laws:

$\mathit CP$ INVARIANCE

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