In
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix), Maki–Nakagawa–Sakata matrix (MNS matrix),
lepton mixing matrix, or
neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
mixing matrix is a
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigrou ...
mixing matrix which contains information on the mismatch of
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
s of
neutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass ...
s when they propagate freely and when they take part in
weak interaction
In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction ...
s. It is a model of
neutrino oscillation. This matrix was introduced in 1962 by
Ziro Maki,
Masami Nakagawa, and
Shoichi Sakata
was a Japanese physicist and Marxist who was internationally known for theoretical work on the subatomic particles.Nussbaum, Louis-Frédéric. (2005). "''Sakata Shōichi''" in ; n.b., Louis-Frédéric is pseudonym of Louis-Frédéric Nussbaum, '' ...
,
to explain the neutrino oscillations predicted by
Bruno Pontecorvo
Bruno Pontecorvo (; russian: Бру́но Макси́мович Понтеко́рво, ''Bruno Maksimovich Pontecorvo''; 22 August 1913 – 24 September 1993) was an Italian and Soviet nuclear physicist, an early assistant of Enrico Fermi and ...
.
The PMNS matrix
The
Standard Model of particle physics contains three
generations or "
flavors" of neutrinos,
,
, and
,
each labeled with a subscript showing the charged
lepton that it partners with in the
charged-current weak interaction. These three
eigenstates
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
of the weak interaction form a complete,
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
for the Standard Model neutrino. Similarly, one can construct an
eigenbasis
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
out of three neutrino states of definite mass,
,
, and
, which diagonalize the neutrino's free-particle
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
. Observations of neutrino oscillation established experimentally that for neutrinos, as for
quarks
A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly ...
, 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 "
superposition", and vice versa. The PMNS matrix, with components
corresponding to the amplitude of mass eigenstate
in terms of flavor
"", "", ""; parameterizes the unitary transformation between the two bases:
:
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
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
with probability
.
The PMNS matrix for
antineutrino
A neutrino ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is ...
s is identical to the matrix for neutrinos under
CPT symmetry
Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and ...
.
Due to the difficulties of
detecting neutrinos, it is much more difficult to determine the individual coefficients than in the equivalent matrix for the quarks (the
CKM matrix).
Assumptions
Standard Model
In the Standard Model, the PMNS matrix is
unitary
Unitary may refer to:
Mathematics
* Unitary divisor
* Unitary element
* Unitary group
* Unitary matrix
* Unitary morphism
* Unitary operator
* Unitary transformation
* Unitary representation
* Unitarity (physics)
* ''E''-unitary inverse semigrou ...
. 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 mass rather than
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 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.
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.
The PMNS matrix is most commonly parameterized by three mixing angles (
,
, and
) and a single phase angle called
related to
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:
::
where
and
are used to denote
and
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 An infinite number of possible parameterizations exist; one other common example being the
Wolfenstein parameterization.
The mixing angles have been measured by a variety of experiments (see
neutrino mixing for a description). The CP-violating phase
has not been measured directly, but estimates can be obtained by fits using the other measurements.
Experimentally measured parameter values
As of October 2021, the current best-fit values from , from direct and indirect measurements, using normal ordering, are:
:
As of October 2021, the 3 ranges (99.7% confidence) for the magnitudes of the elements of the matrix were:
;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 ).
* These values are inconsistent with
tribimaximal neutrino mixing (i.e.
) 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
is very difficult to measure, and is the object of ongoing research; however the current constraint
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
References
{{DEFAULTSORT:Pontecorvo-Maki-Nakagawa-Sakata matrix
Neutrinos
Leptons
Standard Model