(B) Three-neutrino mixing parameters

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

INSPIRE   JSON  (beta) PDGID:
S067DEL
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
$\bf{ 1.21 {}^{+0.19}_{-0.22}}$ OUR AVERAGE  Error includes scale factor of 1.2.
$1.44$ ${}^{+0.24}_{-0.40}$ 1
WESTER
2024
SKAM Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.37$ ${}^{+0.31}_{-0.20}$ 2
ABE
2023F
 T2K Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$0.82$ ${}^{+0.27}_{-0.87}$ 3, 4
ACERO
2022
NOVA Normal mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$, ${{\mathit \theta}_{{{13}}}}$ constrained
• • We do not use the following data for averages, fits, limits, etc. • •
$1.44$ ${}^{+0.23}_{-0.30}$ 5
ABE
2025
SKT2 Both mass orderings, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.52$ ${}^{+0.27}_{-0.30}$ 3, 6
ACERO
2022
NOVA Inverted mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.08$ ${}^{+0.13}_{-0.12}$ 7
SALAS
2021
FIT Normal mass ordering, global fit
$1.58$ ${}^{+0.15}_{-0.16}$ 7
SALAS
2021
FIT Inverted mass ordering, global fit
$1.40$ ${}^{+0.22}_{-0.18}$ 8
ABE
2020F
 T2K Normal mass ordering
$1.09$ ${}^{+0.15}_{-0.13}$ 9
ESTEBAN
2020A
 FIT Normal mass ordering, global fit
$1.57$ ${}^{+0.14}_{-0.17}$ 9
ESTEBAN
2020A
 FIT Inverted mass ordering, global fit
$0.0$ ${}^{+1.3}_{-0.4}$ 10
ACERO
2019
NOVA Normall mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$
$1.33$ ${}^{+0.46}_{-0.53}$ 11
ABE
2018B
 SKAM 3${{\mathit \nu}}$ osc: normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ free
$1.22$ ${}^{+0.76}_{-0.67}$ 11
ABE
2018B
 SKAM 3${{\mathit \nu}}$ osc: inverted mass ordering, ${{\mathit \theta}_{{{13}}}}$ free
$1.33$ ${}^{+0.45}_{-0.51}$ 12
ABE
2018B
 SKAM Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.33$ ${}^{+0.48}_{-0.53}$ 12
ABE
2018B
 SKAM 3${{\mathit \nu}}$ osc: inverted mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.40$ $\pm0.20$ 13
ABE
2018G
 T2K Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.54$ ${}^{+0.14}_{-0.12}$ 95 13
ABE
2018G
 T2K Inverted mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.21$ ${}^{+0.91}_{-0.30}$ 14
ACERO
2018
NOVA Normal mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$
$1.46$ ${}^{+0.56}_{-0.42}$ 14
ACERO
2018
NOVA Normal mass order; octant I for ${{\mathit \theta}_{{{23}}}}$
$1.32$ ${}^{+0.21}_{-0.15}$
DE-SALAS
2018
FIT Normal mass ordering, global fit
$1.56$ ${}^{+0.13}_{-0.15}$
DE-SALAS
2018
FIT Inverted mass ordering, global fit
$1.45$ ${}^{+0.27}_{-0.26}$ 15
ABE
2017F
 T2K Normal mass ordering
$1.54$ ${}^{+0.22}_{-0.23}$ 15
ABE
2017F
 T2K Inverted mass ordering
$1.50$ ${}^{+0.53}_{-0.57}$ 16
ADAMSON
2017B
 NOVA Inverted mass ordering; ${{\mathit \theta}_{{{23}}}}$ in octant II
$0.74$ ${}^{+0.57}_{-0.93}$ 16
ADAMSON
2017B
 NOVA Normal mass ordering; ${{\mathit \theta}_{{{23}}}}$ in octant II
$1.48$ ${}^{+0.69}_{-0.58}$ 16
ADAMSON
2017B
 NOVA Normal mass ordering; ${{\mathit \theta}_{{{23}}}}$ in octant I
$\text{ 0.0 to 0.1, 0.5 to 2.0}$ 90 17, 16
ADAMSON
2016
NOVA Inverted mass ordering
$0.0\text{ to }2.0 $ 90 17
ADAMSON
2016
NOVA Normal mass ordering
$\text{ 0 to 0.15, 0.83 to 2}$ 90
ABE
2015D
 T2K Normal mass ordering
$1.09\text{ to }1.92 $ 90
ABE
2015D
 T2K Inverted mass ordering
$0.05\text{ to }1.2 $ 90 18
ADAMSON
2014
MINS Normal mass ordering
$1.34$ ${}^{+0.64}_{-0.38}$
FORERO
2014
FIT Normal mass ordering
$1.48$ ${}^{+0.34}_{-0.32}$
FORERO
2014
FIT Inverted mass ordering
$1.70$ ${}^{+0.22}_{-0.39}$ 19
GONZALEZ-GARC..
2014
FIT Normal mass ordering; global fit
$1.41$ ${}^{+0.35}_{-0.34}$ 19
GONZALEZ-GARC..
2014
FIT Inverted mass ordering; global fit
$\text{ 0 to 1.5 or 1.9 to 2}$ 90 20
ADAMSON
2013A
 MINS Normal mass ordering
1  WESTER 2024 uses 484.2 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}_{{{13}}}})$ are fixed to $\Delta $m${}^{2}_{21}$ = ($7.53$ $\pm0.18$) $ \times 10^{-5}$ eV${}^{2}$, sin$^2({{\mathit \theta}_{{{12}}}})$ = $0.307$ $\pm0.013$, and sin$^2({{\mathit \theta}_{{{13}}}})$ = $0.0220$ $\pm0.0007$. Supersedes ABE 2018B.
2  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.
3  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.
4  For the octant I (lower octant), the 68$\%$ CL allowed region is discontinuous, and all delta values are allowed at 90$\%$ CL.
5  ABE 2025 reports the results of a joint analysis of Super-Kamiokande atmospheric neutrino data and T2K beam neutrino data, using 3244.4 days of atmospheric data and a (anti)neutrino beam exposure of $1.97 \times 10^{21}$ ($1.63 \times 10^{21}$) protons on target.
6  The inverted mass ordering is rejected at 1.0 $\sigma $. The error bars are reported relative to the global minima in normal mass ordering.
7  SALAS 2021 reports results of a global fit to neutrino oscillation data available at the time of the Neutrino 2020 conference.
8  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.
9  ESTEBAN 2020A reports results of a global fit to neutrino oscillation data available at the time of the Neutrino 2020 conference.
10  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.
11  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$^2{{\mathit \theta}_{{{12}}}}$ = $0.304$ $\pm0.014$. Superseded by WESTER 2024.
12  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$^2{{\mathit \theta}_{{{13}}}}$ = $0.0219$ $\pm0.0012$. Superseded by WESTER 2024.
13  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.
14  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.
15  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.
16  Errors are projections of 68$\%$ C.L. curve of $\delta _{CP}$ vs. sin$^2{{\mathit \theta}_{{{23}}}}$.
17  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.
18  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.
19  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.
20  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
References