View source for Pontecorvo–Maki–Nakagawa–Sakata matrix

{{Short description|Model of neutrino oscillation}}
{{Flavour quantum numbers}}
In [[particle physics]], the '''Pontecorvo–Maki–Nakagawa–Sakata matrix''' ('''PMNS matrix'''), '''Maki–Nakagawa–Sakata matrix''' ('''MNS matrix'''), '''[[lepton]] mixing matrix''', or '''[[neutrino]] mixing matrix''' is a [[unitary matrix|unitary]]{{efn|
Note however, that the PMNS matrix is ''not'' unitary in the [[seesaw mechanism|seesaw model]].
}}
[[mixing angle|mixing matrix]] that contains information on the mismatch of [[quantum state]]s of [[neutrino]]s when they propagate freely and when they take part in [[weak interaction]]s. It is a model of [[neutrino oscillation]]. This matrix was introduced in 1962 by [[Ziro Maki]], [[Masami Nakagawa]], and [[Shoichi Sakata]],<ref>
{{cite journal
 |first1=Z. |last1=Maki 
 |first2=M. |last2=Nakagawa
 |first3=S. |last3=Sakata
 |year=1962
 |title=Remarks on the unified model of elementary particles
 |journal=[[Progress of Theoretical Physics]]
 |volume=28 |issue=5 |page=870
 |bibcode=1962PThPh..28..870M
 |doi=10.1143/PTP.28.870 |doi-access=free
}}
</ref>
to explain the neutrino oscillations predicted by [[Bruno Pontecorvo]].<ref>
{{cite journal
 |last=Pontecorvo |first=B.
 |year=1957
 |title=Inverse beta processes and nonconservation of lepton charge
 |journal=[[Zhurnal Éksperimental'noĭ i Teoreticheskoĭ Fiziki]]
 |volume=34 |page=247
}}
reproduced and translated in
{{cite journal
 |last=Pontecorvo |first=B.
 |year=1958
 |title=[no title cited]
 |journal=[[Soviet Physics JETP]]
 |volume=7 |page=172
}}
</ref>

== PMNS matrix ==
The [[Standard Model]] of particle physics contains three [[generation (particle physics)|generations]] or "[[Flavour (physics)|flavors]]" of neutrinos, {{tmath|1= \nu_\mathrm{e} }}, {{tmath|1= \nu_\mu }}, and {{tmath|1= \nu_\tau }}, each labeled with a subscript showing the charged [[lepton]] that it partners with in the [[W boson exchange|charged-current weak interaction]]. These three [[eigenstates]] of the weak interaction form a complete, [[orthonormal basis]] for the Standard Model neutrino. Similarly, one can construct an [[eigenbasis]] out of three neutrino states of definite mass, {{tmath|1= \nu_1 }}, {{tmath|1= \nu_2 }}, and {{tmath|1= \nu_3 }}, which diagonalize the neutrino's free-particle [[Hamiltonian (quantum mechanics)|Hamiltonian]]. Observations of neutrino oscillation established experimentally that for neutrinos, as for [[quarks]], these two eigenbases are different – they are 'rotated' relative to each other.

Consequently, each flavor eigenstate can be written as a combination of mass eigenstates, called a "[[Quantum superposition|superposition]]", and vice versa. The PMNS matrix, with components <math>U_{\alpha\,i}</math> corresponding to the amplitude of mass eigenstate {{nowrap|<math>i = </math> 1, 2, 3}} in terms of flavor <math> \alpha = </math> "{{math|e}}", "{{math|μ}}", "{{math|τ}}"; parameterizes the unitary transformation between the two bases:
: <math>\begin{bmatrix} ~ \nu_\mathrm{e} \\ ~ \nu_\mu \\ ~ \nu_\tau ~ \end{bmatrix} 
= \begin{bmatrix} ~ U_{\mathrm{e} 1} ~ & ~ U_{\mathrm{e} 2} ~ & ~ U_{\mathrm{e} 3} \\ ~ U_{\mu 1} & ~ U_{\mu 2} ~ & ~ U_{\mu 3} \\ ~ U_{\tau 1} ~ & ~ U_{\tau 2} ~ & ~ U_{\tau 3} \end{bmatrix} \begin{bmatrix} ~ \nu_1 \\ ~ \nu_2 \\ ~ \nu_3 ~ \end{bmatrix} ~.</math>

The vector on the left represents a generic neutrino expressed in the flavor-eigenstate basis, and on the right is the PMNS matrix multiplied by a vector representing that same neutrino in the mass-eigenstate basis. A neutrino of a given flavor <math>\alpha</math> is thus a "mixed" state of neutrinos with distinct mass: If one could measure directly that neutrino's mass, it would be found to have mass <math>m_i</math> with probability {{tmath|1= \left\vert U_{\alpha\,i}\right\vert^2 }}.

The PMNS matrix for [[antineutrino]]s is identical to the matrix for neutrinos under [[CPT symmetry]].

Due to the difficulties of [[neutrino detector|detecting neutrino]]s, it is much more difficult to determine the individual coefficients than for the equivalent matrix for the quarks (the [[Cabibbo–Kobayashi–Maskawa matrix|CKM matrix]]).

=== Assumptions ===

==== Standard Model ====
In the Standard Model, the PMNS matrix is [[unitarity (physics)|unitary]]. This implies that the sum of the squares of the values in each row and in each column, which represent the probabilities of different possible events given the same starting point, add up to 100%.

In the simplest case, the Standard Model posits three generations of neutrinos with Dirac mass that oscillate between three neutrino mass eigenvalues, an assumption that is made when best fit values for its parameters are calculated.

==== Other models ====
In other models the PMNS matrix is not necessarily unitary, and additional parameters are necessary to describe all possible neutrino mixing parameters in other models of neutrino oscillation and mass generation, such as the see-saw model, and in general, in the case of neutrinos that have [[Majorana fermion|Majorana mass]] rather than [[Dirac fermion|Dirac mass]].

There are also additional mass parameters and mixing angles in a simple extension of the PMNS matrix in which there are more than three flavors of neutrinos, regardless of the character of neutrino mass. As of July&nbsp;2014, scientists studying neutrino oscillation are actively considering fits of the experimental neutrino oscillation data to an extended PMNS matrix with a fourth, light "sterile" neutrino and four mass eigenvalues, although the current experimental data tends to disfavor that possibility.<ref>
{{cite journal
 |first=Boris |last=Kayser
 |title=Are there sterile neutrinos?
 |journal=Dark Matter
 |pages=201–203
 |date=February 13, 2014
 |arxiv=1402.3028 |doi=10.1063/1.4883431
 |series=AIP Conference Proceedings
 |volume=1604
 |issue=1
 |bibcode=2014AIPC.1604..201K
 |citeseerx=10.1.1.761.2915 |s2cid=119182490
 |df=dmy-all
}}
</ref><ref>
{{cite journal
 |first1=Arman  |last1=Esmaili     |first2=Ernesto |last2=Kemp
 |first3=O.L.G. |last3=Peres       |first4=Zahra   |last4=Tabrizi
 |date=30 Oct 2013
 |title=Probing light sterile neutrinos in medium baseline reactor experiments
 |journal=[[Physical Review D]]
 |volume=88 |issue=7 |pages=073012
 |arxiv=1308.6218 |doi=10.1103/PhysRevD.88.073012
 |bibcode=2013PhRvD..88g3012E
|s2cid=119208413 }}
</ref><ref>
{{cite journal
 |first1=F.P. |last1=An
 |display-authors=etal
 |collaboration=Daya Bay collaboration
 |date=27 July 2014
 |title=Search for a light sterile neutrino at Daya Bay
 |journal=Physical Review Letters
 |volume=113 |issue=14 |page=141802
 |arxiv=1407.7259 |doi=10.1103/PhysRevLett.113.141802
 |pmid=25325631
 |bibcode=2014PhRvL.113n1802A  |s2cid=10500157
 |df=dmy-all
}}
</ref>

=== Parameterization ===
In general, there are nine degrees of freedom in any unitary three by three matrix. However, in the case of the PMNS matrix, five of those real parameters can be absorbed as phases of the lepton fields and thus the PMNS matrix can be fully described by four free parameters.<ref>
{{cite journal
 |last=Valle  |first=J.W.F.
 |year=2006
 |title=Neutrino physics overview
 |journal=[[Journal of Physics: Conference Series]]
 |volume=53 |issue=1 |pages=473–505
 |arxiv=hep-ph/0608101 |bibcode=2006JPhCS..53..473V
 |doi=10.1088/1742-6596/53/1/031
 |s2cid=2094005
}}
</ref>
The PMNS matrix is most commonly parameterized by three mixing angles ({{tmath|1= \theta_{12} }}, {{tmath|1= \theta_{23} }}, and {{tmath|1= \theta_{13} }}) and a single phase angle called <math>\delta_{\mathrm{CP}}</math> related to [[CP violation|charge–parity violations]] (i.e. differences in the rates of oscillation between two states with opposite starting points, which makes the order in time in which events take place necessary to predict their oscillation rates), in which case the matrix can be written as:
: <math> \begin{align} & \begin{bmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23} \end{bmatrix}
 \begin{bmatrix} c_{13} & 0 & s_{13}e^{-i\delta_\mathrm{CP}} \\ 0 & 1 & 0 \\ -s_{13}e^{i\delta_\mathrm{CP}} & 0 & c_{13} \end{bmatrix}
 \begin{bmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{bmatrix} \\
 & = \begin{bmatrix} c_{12}c_{13} & s_{12} c_{13} & s_{13}e^{-i\delta_\mathrm{CP}} \\
 -s_{12}c_{23} - c_{12}s_{23}s_{13}e^{i\delta_\mathrm{CP}} & c_{12}c_{23} - s_{12}s_{23}s_{13}e^{i\delta_\mathrm{CP}} & s_{23}c_{13}\\
 s_{12}s_{23} - c_{12}c_{23}s_{13}e^{i\delta_\mathrm{CP}} & -c_{12}s_{23} - s_{12}c_{23}s_{13}e^{i\delta_\mathrm{CP}} & c_{23}c_{13} \end{bmatrix}, \end{align} </math>
where <math>s_{ij}</math> and <math>c_{ij}</math> are used to denote <math>\sin\theta_{ij}</math> and <math>\cos\theta_{ij}</math> respectively. In the case of Majorana neutrinos, two extra complex phases are needed, as the phase of Majorana fields cannot be freely redefined due to the condition {{tmath|1= \nu = \nu^c }}. An infinite number of possible parameterizations exist; one other common example being the [[Cabibbo–Kobayashi–Maskawa matrix#Wolfenstein parameters|Wolfenstein parameterization]].

The mixing angles have been measured by a variety of experiments (see [[neutrino mixing]] for a description). The CP-violating phase <math>\delta_\mathrm{CP}</math> has not been measured directly, but estimates can be obtained by fits using the other measurements.

===Experimentally measured parameter values===
As of November&nbsp;2022, the current best-fit values from Nu-FIT.org, from direct and indirect measurements, using normal ordering, are:<ref name=NuFIT>{{cite web |first1=Ivan |last1=Esteban |first2=Concha |last2=Gonzalez Garcia |first3=Michele |last3=Maltoni |first4=Thomas |last4=Schwetz |first5=Zhou|last5=Albert |edition=NuFIT 5.2 |website=NuFIT.org |url=http://www.nu-fit.org/?q=node/256 |series=Three-neutrino fit |title=Parameter ranges |date=November 2022 |access-date=2023-03-29 |df=dmy-all}}</ref>
For September&nbsp;2024 data, see NuFIT6<ref name=NuFIT6>{{cite web |first1=Ivan |last1=Esteban |first2=Concha |last2=Gonzalez Garcia |first3=Michele |last3=Maltoni |first4=Ivan |last4=Martinez-Soler |first5=João Paulo|last5=Pinheiro |first6=Thomas |last6=Schwetz |edition=NuFIT 6.0 |website=NuFIT.org |url=http://www.nu-fit.org/?q=node/294 |series=Three-neutrino fit |title=Parameter ranges |date=September 2024 |access-date=2024-12-10 |df=dmy-all}}</ref>
: <math>
\begin{align}
\theta_{12} & =  {33.41^\circ}^{+0.75^\circ}_{-0.72^\circ} \\
\theta_{23} & =  {49.1^\circ}^{+1.0^\circ}_{-1.3^\circ}\\
\theta_{13} & =  {8.54^\circ}^{+0.11^\circ}_{-0.12^\circ}  \\
\delta_{\textrm{CP}} & = {197^\circ}^{+42^\circ}_{-25^\circ} \\
\end{align}
</math>

As of November&nbsp;2022, the 3&nbsp;{{mvar|&sigma;}} ranges (99.7% confidence) for the magnitudes of the elements of the matrix were:<ref name=NuFIT/>
<math>
|U| = \begin{bmatrix}
  ~ |U_{\mathrm{e} 1}|    ~ & |U_{\mathrm{e} 2}|    ~ & |U_{\mathrm{e} 3}|    \\
  ~ |U_{\mu 1}|  ~ & |U_{\mu 2}|  ~ & |U_{\mu 3}|  \\
  ~ |U_{\tau 1}| ~ & |U_{\tau 2}| ~ & |U_{\tau 3}|  ~ 
\end{bmatrix} = \left[\begin{array}{rrr}
   ~ 0.803 \sim 0.845 ~~ &  0.514 \sim 0.578 ~~ &  0.142 \sim 0.155 ~ \\
   ~ 0.233 \sim 0.505 ~~ &  0.460 \sim 0.693 ~~ &  0.630 \sim 0.779 ~ \\
   ~ 0.262 \sim 0.525 ~~ &  0.473 \sim 0.702 ~~ &  0.610 \sim 0.762 ~
\end{array}\right]
</math>
<!-- ———————————————— preserved 2014 values ———————————————— 
<ref>
{{cite web
 |last1=Gonzalez-Garcia |first1=M. C. |last2=Maltoni |first2=M.
 |last3=Salvado |first3=J. |last4=Schwetz |first4=T.
 |title=NuFit 1.3
 |date=June 2014
 |url=http://www.nu-fit.org/?q=node/75
 |access-date=2014-07-09
}}
</ref>
:<math>
\begin{align}
\theta_{12} [^\circ] & = 33.36^{+0.81}_{-0.78} \\
\theta_{23} [^\circ] & = 40.0^{+2.1}_{-1.5}~\textrm{or}~50.4^{+1.3}_{-1.3} \\
\theta_{13} [^\circ] & = 8.66^{+0.44}_{-0.46}  \\
\delta_{\textrm{CP}} [^\circ] & = -60^{+66}_{-138} \\
\end{align}
</math>

So the current matrix will be:
<math>
U = \begin{bmatrix}
  |U_{\mathrm{e} 1}|    & |U_{\mathrm{e} 2}|    & |U_{\mathrm{e} 3}| \\
  |U_{\mu 1}|  & |U_{\mu 2}|  & |U_{\mu 3}| \\
  |U_{\tau 1}| & |U_{\tau 2}| & |U_{\tau 3}|
\end{bmatrix} = \left[\begin{array}{rrr}
   0.82 \pm 0.01 &  0.54 \pm 0.02 & -0.15 \pm 0.03 \\
  -0.35 \pm 0.06 &  0.70 \pm 0.06 &  0.62 \pm 0.06 \\
   0.44 \pm 0.06 & -0.45 \pm 0.06 &  0.77 \pm 0.06
\end{array}\right]
 </math>
 ———————————————— end old 2014 values ————————————————  -->

; Notes regarding the best fit parameter values :
* These best fit values imply that there is much more neutrino mixing than there is mixing between the quark flavors in the CKM matrix (in the CKM matrix, the corresponding mixing angles are {{nowrap|<math>\theta_{12} = </math> {{val|13.04|0.05|u=deg}}}}, {{nowrap|<math>\theta_{23} = </math> {{val|2.38|0.06|u=deg}}}}, {{nowrap|<math>\theta_{13} = </math> {{val|0.201|0.011|u=deg}}}}).
* These values are inconsistent with [[tribimaximal mixing|tribimaximal neutrino mixing]] (i.e. {{tmath|1= \theta_{12} \approx }}&nbsp;35.3°, {{tmath|1= \theta_{23} = }}&nbsp;45°, {{tmath|1= \theta_{13} = }}&nbsp;0°) at a statistical significance of more than five standard deviations. Tribimaximal neutrino mixing was a common assumption in theoretical physics papers analyzing neutrino oscillation before more precise measurements were available.
* The value of {{tmath|1= \delta_{\textrm{CP} } = }}&nbsp;{{val|197|42|25|u=deg}} is very difficult to measure, and is the object of ongoing research; however the current constraint 169°&nbsp;{{tmath|1= \le \delta_{\textrm{CP} } \le }}&nbsp;246° in the vicinity of 180° shows a clear bias in favor of charge–parity violation.

== See also ==
* [[Neutrino oscillation]]
* [[Koide formula]]
* [[Cabibbo–Kobayashi–Maskawa matrix]]

== Notes ==
{{notelist}}

== References ==
{{reflist|25em}}

{{cite journal
 |first1=M.C.    |last1=Gonzalez-Garcia
 |first2=Michele |last2=Maltoni
 |first3=Jordi   |last3=Salvado
 |first4=Thomas  |last4=Schwetz
 |date=21 December 2012
 |title=Global fit to three neutrino mixing: Critical look at present precision
 |journal=[[Journal of High Energy Physics]]
 |volume=2012  |issue=12  |page=123
 |arxiv=1209.3023  |bibcode=2012JHEP...12..123G
 |doi=10.1007/JHEP12(2012)123  |citeseerx=10.1.1.762.7366
 |s2cid=118566415
}}

{{DEFAULTSORT:Pontecorvo-Maki-Nakagawa-Sakata matrix}}
[[Category:Neutrinos]]
[[Category:Leptons]]
[[Category:Standard Model]]