(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 $.
S067DEL
VALUE (${{\mathit \pi}}$ rad) CL% DOCUMENT ID TECN  COMMENT
$\bf{ 1.69 {}^{+0.11}_{-0.08}}$ OUR AVERAGE  Error includes scale factor of 1.1.
$1.72$ ${}^{+0.11}_{-0.07}$ 1
ABUBAKAR
2025
 
FIT Normal mass ordering, joint T2K and NOvA fit
$1.44$ ${}^{+0.24}_{-0.40}$ 2
WESTER
2024
 
SKAM Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
• • We do not use the following data for averages, fits, limits, etc. • •
$1.44$ ${}^{+0.23}_{-0.30}$ 3
ABE
2025
 
SKT2 Both mass orderings, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.31$ ${}^{+0.39}_{-0.15}$ 4
ABE
2025B
 
T2K Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.37$ ${}^{+0.31}_{-0.20}$ 5
ABE
2023F
 
T2K Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$0.82$ ${}^{+0.27}_{-0.87}$ 6, 7
ACERO
2022
 
NOVA Normal mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.52$ ${}^{+0.27}_{-0.30}$ 6, 8
ACERO
2022
 
NOVA Inverted mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.08$ ${}^{+0.13}_{-0.12}$ 9
SALAS
2021
 
FIT Normal mass ordering, global fit
$1.58$ ${}^{+0.15}_{-0.16}$ 9
SALAS
2021
 
FIT Inverted mass ordering, global fit
$1.40$ ${}^{+0.22}_{-0.18}$ 10
ABE
2020F
 
T2K Normal mass ordering
$1.09$ ${}^{+0.15}_{-0.13}$ 11
ESTEBAN
2020A
 
FIT Normal mass ordering, global fit
$1.57$ ${}^{+0.14}_{-0.17}$ 11
ESTEBAN
2020A
 
FIT Inverted mass ordering, global fit
$0.0$ ${}^{+1.3}_{-0.4}$ 12
ACERO
2019
 
NOVA Normall mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$
$1.33$ ${}^{+0.46}_{-0.53}$ 13
ABE
2018B
 
SKAM 3${{\mathit \nu}}$ osc: normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ free
$1.22$ ${}^{+0.76}_{-0.67}$ 13
ABE
2018B
 
SKAM 3${{\mathit \nu}}$ osc: inverted mass ordering, ${{\mathit \theta}_{{{13}}}}$ free
$1.33$ ${}^{+0.45}_{-0.51}$ 14
ABE
2018B
 
SKAM Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.33$ ${}^{+0.48}_{-0.53}$ 14
ABE
2018B
 
SKAM 3${{\mathit \nu}}$ osc: inverted mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.40$ $\pm0.20$ 15
ABE
2018G
 
T2K Normal mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.54$ ${}^{+0.14}_{-0.12}$ 95 15
ABE
2018G
 
T2K Inverted mass ordering, ${{\mathit \theta}_{{{13}}}}$ constrained
$1.21$ ${}^{+0.91}_{-0.30}$ 16
ACERO
2018
 
NOVA Normal mass ordering, octant II for ${{\mathit \theta}_{{{23}}}}$
$1.46$ ${}^{+0.56}_{-0.42}$ 16
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
$1.45$ ${}^{+0.27}_{-0.26}$ 17
ABE
2017F
 
T2K Normal mass ordering
$1.54$ ${}^{+0.22}_{-0.23}$ 17
ABE
2017F
 
T2K Inverted mass ordering
$1.50$ ${}^{+0.53}_{-0.57}$ 18
ADAMSON
2017B
 
NOVA Inverted mass ordering; ${{\mathit \theta}_{{{23}}}}$ in octant II
$0.74$ ${}^{+0.57}_{-0.93}$ 18
ADAMSON
2017B
 
NOVA Normal mass ordering; ${{\mathit \theta}_{{{23}}}}$ in octant II
$1.48$ ${}^{+0.69}_{-0.58}$ 18
ADAMSON
2017B
 
NOVA Normal mass ordering; ${{\mathit \theta}_{{{23}}}}$ in octant I
$\text{ 0.0 to 0.1, 0.5 to 2.0}$ 90 19, 18
ADAMSON
2016
 
NOVA Inverted mass ordering
$0.0\text{ to }2.0 $ 90 19
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 20
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
$1.70$ ${}^{+0.22}_{-0.39}$ 21
GONZALEZ-GARC..
2014
 
FIT Normal mass ordering; global fit
$1.41$ ${}^{+0.35}_{-0.34}$ 21
GONZALEZ-GARC..
2014
 
FIT Inverted mass ordering; global fit
$\text{ 0 to 1.5 or 1.9 to 2}$ 90 22
ADAMSON
2013A
 
MINS Normal mass ordering
1  ABUBAKAR 2025 results are based on the joint analysis of data sets from NOvA and T2K experiments. The reactor constraint on ${{\mathit \theta}_{{{13}}}}$ is applied in the fits. Supersedes ABE 2025B and ACERO 2022.
2  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.
3  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.
4  ABE 2025B 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 2023F.
5  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.
6  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.
7  For the octant I (lower octant), the 68$\%$ CL allowed region is discontinuous, and all delta values are allowed at 90$\%$ CL.
8  The inverted mass ordering is rejected at 1.0 $\sigma $. The error bars are reported relative to the global minima in normal mass ordering.
9  SALAS 2021 reports results of a global fit to neutrino oscillation data available at the time of the Neutrino 2020 conference.
10  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.
11  ESTEBAN 2020A reports results of a global fit to neutrino oscillation data available at the time of the Neutrino 2020 conference.
12  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.
13  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.
14  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.
15  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.
16  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.
17  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.
18  Errors are projections of 68$\%$ C.L. curve of $\delta _{CP}$ vs. sin$^2{{\mathit \theta}_{{{23}}}}$.
19  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.
20  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.
21  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.
22  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