Neutral particle oscillation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Neutral_particle_oscillation 

In particle physics, neutral particle oscillation is the transmutation of a particle with zero electric charge into another neutral particle due to a change of a non-zero internal quantum number, via an interaction that does not conserve that quantum number. Neutral particle oscillations were first investigated in 1954 by Murray Gell-mann and Abraham Pais.

For example, a neutron cannot transmute into an antineutron as that would violate the conservation of baryon number. But in those hypothetical extensions of the Standard Model which include interactions that do not strictly conserve baryon number, neutron–antineutron oscillations are predicted to occur.

Such oscillations can be classified into two types:

• Particle–antiparticle oscillation (for example,
  K⁰
  ⇄
  K⁰
  oscillation,
  B⁰
  ⇄
  B⁰
  oscillation,
  D⁰
  ⇄
  D⁰
  oscillation).
• Flavor oscillation (for example,
  ν
  _e ⇄
  ν
  _μ ⇄
  ν
  _τ oscillation).

In those cases where the particles decay to some final product, then the system is not purely oscillatory, and an interference between oscillation and decay is observed.

History and motivation

CP violation

After the striking evidence for parity violation provided by Wu et al. in 1957, it was assumed that CP (charge conjugation-parity) is the quantity which is conserved. However, in 1964 Cronin and Fitch reported CP violation in the neutral Kaon system. They observed the long-lived K₂ (with CP = −1 ) undergoing decays into two pions (with CP = [−1]·[−1] = +1 ) thereby violating CP conservation.

In 2001, CP violation in the
B⁰
⇄
B⁰
system was confirmed by the BaBar and the Belle experiments. Direct CP violation in the
B⁰
⇄
B⁰
system was reported by both the labs by 2005.

The
K⁰
⇄
K⁰
and the
B⁰
⇄
B⁰
systems can be studied as two state systems, considering the particle and its antiparticle as the two states.

The solar neutrino problem

The pp chain in the sun produces an abundance of
ν
_e. In 1968, R. Davis et al. first reported the results of the Homestake experiment. Also known as the Davis experiment, it used a huge tank of perchloroethylene in Homestake mine (it was deep underground to eliminate background from cosmic rays), South Dakota. Chlorine nuclei in the perchloroethylene absorb
ν
_e to produce argon via the reaction

${\displaystyle {\nu }_{\mathrm{e}}+{}_{17}^{37}\mathrm{C}\mathrm{l}\to {}_{18}^{37}\mathrm{A}\mathrm{r}+{\mathrm{e}}^{-}}$,

which is essentially

${\displaystyle {\nu }_{\mathrm{e}}+\mathrm{n}\to \mathrm{p}+{\mathrm{e}}^{-}}$.

The experiment collected argon for several months. Because the neutrino interacts very weakly, only about one argon atom was collected every two days. The total accumulation was about one third of Bahcall's theoretical prediction.

In 1968, Bruno Pontecorvo showed that if neutrinos are not considered massless, then
ν
_e (produced in the sun) can transform into some other neutrino species (
ν
_μ or
ν
_τ), to which Homestake detector was insensitive. This explained the deficit in the results of the Homestake experiment. The final confirmation of this solution to the solar neutrino problem was provided in April 2002 by the SNO (Sudbury Neutrino Observatory) collaboration, which measured both
ν
_e flux and the total neutrino flux.

This 'oscillation' between the neutrino species can first be studied considering any two, and then generalized to the three known flavors.

Description as a two-state system

A special case: considering mixing only

Main article: Two-state quantum system

Caution: "mixing" discussed in this article is not the type obtained from mixed quantum states. Rather, "mixing" here refers to the superposition of "pure state" energy (mass) eigenstates, described by a "mixing matrix" (e.g. the CKM or PMNS matricies).

Let ${\displaystyle H_0}$ be the Hamiltonian of the two-state system, and ${\displaystyle |1⟩}$ and ${\displaystyle |2⟩}$ be its orthonormal eigenvectors with eigenvalues ${\displaystyle E_1}$ and ${\displaystyle E_2}$ respectively.

Let ${\displaystyle |\mathrm{\Psi }\left(t\right)⟩}$ be the state of the system at time ${\displaystyle t.}$

If the system starts as an energy eigenstate of ${\displaystyle H_0,}$ i.e. say

${\displaystyle |\mathrm{\Psi }\left(0\right)⟩=|1⟩}$

then, the time evolved state, which is the solution of the Schrödinger equation

${\displaystyle {\hat{H}}_0|\mathrm{\Psi }\left(t\right)⟩=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}|\mathrm{\Psi }\left(t\right)⟩}$   (1)

will be,

${\displaystyle |\mathrm{\Psi }\left(t\right)⟩=|1⟩e^{-i\frac{E_{1}t}{\hslash }}}$

But this is physically same as ${\displaystyle |1⟩}$ as the exponential term is just a phase factor and does not produce a new state. In other words, energy eigenstates are stationary eigenstates, i.e. they do not yield physically new states under time evolution.

In the basis ${\displaystyle \left\{|1⟩,|2⟩\right\},}$ ${\displaystyle H_0}$ is diagonal. That is,

${\displaystyle H_0=\left(\begin{array}{cc}{E}_1& 0\\ 0& E_2\end{array}\right)}$

It can be shown, that oscillation between states will occur if and only if off-diagonal terms of the Hamiltonian are non-zero.

Hence let us introduce a general perturbation ${\displaystyle W}$ in ${\displaystyle H_0}$ such that the resultant Hamiltonian ${\displaystyle H}$ is still Hermitian. Then,

${\displaystyle W=\left(\begin{array}{cc}{W}_{11}& W_{12}\\ W_{12}^{\ast }& W_{22}\end{array}\right)}$ where ${\displaystyle W_{11},W_{22}\in \mathbb{R}}$ and ${\displaystyle W_{12}\in \mathbb{C}}$

and,

${\displaystyle H=H_0+W=\left(\begin{array}{cc}{E}_1+W_{11}& W_{12}\\ W_{12}^{\ast }& E_2+W_{22}\end{array}\right)}$   (2)

Then, the eigenvalues of ${\displaystyle H}$ are,

${\displaystyle E_{\pm }=\frac{1}{2}\left[E_1+W_{11}+E_2+W_{22}\pm \sqrt{{\left(E_1+W_{11}-E_{{}^2}-W_{22}\right)}^2+4{\left|W_{12}\right|}^2}\right]}$   (3)

Since ${\displaystyle H}$ is a general Hamiltonian matrix, it can be written as,

${\displaystyle H=\sum _{j=0}^3{a}_j{\sigma }_j=a_0{\sigma }_0+H^{\prime }}$

where,

The following two results are clear:

• ${\displaystyle \left[H,H^{\prime }\right]=0}$

Proof

• ${\displaystyle {H^{\prime }}^2=I}$

Proof

With the following parametrization (this parametrization helps as it normalizes the eigenvectors and also introduces an arbitrary phase ${\displaystyle \varphi }$ making the eigenvectors most general)

${\displaystyle \hat{n}=\left(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta \right)}$,

and using the above pair of results the orthonormal eigenvectors of ${\displaystyle H^{\prime }}$ and hence of ${\displaystyle H}$ are obtained as,

${\displaystyle \begin{array}{rl}|+⟩& =\left(\begin{array}{c}\cos \frac{\theta }{2}{e}^{-i\frac{\varphi }{2}}\\ \sin \frac{\theta }{2}{e}^{i\frac{\varphi }{2}}\end{array}\right)\equiv \cos \frac{\theta }{2}{e}^{-i\frac{\varphi }{2}}|1⟩+\sin \frac{\theta }{2}{e}^{i\frac{\varphi }{2}}|2⟩\\ |-⟩& =\left(\begin{array}{c}-\sin \frac{\theta }{2}{e}^{i\frac{\varphi }{2}}\\ \cos \frac{\theta }{2}{e}^{-i\frac{\varphi }{2}}\end{array}\right)\equiv -\sin \frac{\theta }{2}{e}^{-i\frac{\varphi }{2}}|1⟩+\cos \frac{\theta }{2}{e}^{i\frac{\varphi }{2}}|2⟩\end{array}}$   (4)

where,

Writing the eigenvectors of ${\displaystyle H_0}$ in terms of those of ${\displaystyle H}$ we get,

${\displaystyle \begin{array}{rl}|1⟩& =e^{i\frac{\varphi }{2}}\left(\cos \frac{\theta }{2}|+⟩-\sin \frac{\theta }{2}|-⟩\right)\\ |2⟩& =e^{-i\frac{\varphi }{2}}\left(\sin \frac{\theta }{2}|+⟩+\cos \frac{\theta }{2}|-⟩\right)\end{array}}$   (5)

Now if the particle starts out as an eigenstate of ${\displaystyle H_0}$ (say, ${\displaystyle |1⟩}$), that is,

${\displaystyle |\mathrm{\Psi }\left(0\right)⟩=|1⟩}$

then under time evolution we get,

${\displaystyle |\mathrm{\Psi }\left(t\right)⟩=e^{i\frac{\varphi }{2}}\left(\cos \frac{\theta }{2}|+⟩e^{-i\frac{E_{+}t}{\hslash }}-\sin \frac{\theta }{2}|-⟩e^{-i\frac{E_{-}t}{\hslash }}\right)}$

which unlike the previous case, is distinctly different from ${\displaystyle |1⟩.}$

We can then obtain the probability of finding the system in state ${\displaystyle |2⟩}$ at time ${\displaystyle t}$ as,

${\displaystyle \begin{array}{rl}{P}_{21}\left(t\right)& ={\left|⟨2|\mathrm{\Psi }\left(t\right)⟩\right|}^2={\sin }^2\theta {\sin }^2\left(\frac{E_{+}-E_{-}}{2\hslash }t\right)\\ & =\frac{4{\left|W_{12}\right|}^2}{4{\left|W_{12}\right|}^2+{\left(E_1-E_2\right)}^2}{\sin }^2\left(\frac{\sqrt{4{\left|W_{12}\right|}^2+{\left(E_1-E_2\right)}^2}}{2\hslash }t\right)\end{array}}$   (6)

which is called Rabi's formula. Hence, starting from one eigenstate of the unperturbed Hamiltonian ${\displaystyle H_0,}$ the state of the system oscillates between the eigenstates of ${\displaystyle H_0}$ with a frequency (known as Rabi frequency),

${\displaystyle \omega =\frac{E_{+}-E_{-}}{2\hslash }=\frac{\sqrt{4{\left|W_{12}\right|}^2+{\left(E_1-E_2\right)}^2}}{2\hslash }}$   (7)

From the expression of ${\displaystyle P_{21}(t)}$ we can infer that oscillation will exist only if ${\displaystyle {\left|W_{12}\right|}^2\ne 0.}$ ${\displaystyle W_{12}}$ is thus known as the coupling term as it couples the two eigenstates of the unperturbed Hamiltonian ${\displaystyle H_0}$ and thereby facilitates oscillation between the two.

Oscillation will also cease if the eigenvalues of the perturbed Hamiltonian ${\displaystyle H}$ are degenerate, i.e. ${\displaystyle E_{+}=E_{-}.}$ But this is a trivial case as in such a situation, the perturbation itself vanishes and ${\displaystyle H}$ takes the form (diagonal) of ${\displaystyle H_0}$ and we're back to square one.

Hence, the necessary conditions for oscillation are:

• Non-zero coupling, i.e. ${\displaystyle {\left|W_{12}\right|}^2\ne 0.}$
• Non-degenerate eigenvalues of the perturbed Hamiltonian ${\displaystyle H}$, i.e. ${\displaystyle E_{+}\ne E_{-}.}$

The general case: considering mixing and decay

If the particle(s) under consideration undergoes decay, then the Hamiltonian describing the system is no longer Hermitian. Since any matrix can be written as a sum of its Hermitian and anti-Hermitian parts, ${\displaystyle H}$ can be written as,

${\displaystyle H=M-\frac{i}{2}\mathrm{\Gamma }=\left(\begin{array}{cc}{M}_{11}& M_{12}\\ M_{12}^{\ast }& M_{11}\end{array}\right)-\frac{i}{2}\left(\begin{array}{cc}{\mathrm{\Gamma }}_{11}& {\mathrm{\Gamma }}_{12}\\ {\mathrm{\Gamma }}_{12}^{\ast }& {\mathrm{\Gamma }}_{11}\end{array}\right)}$

where,

The eigenvalues of ${\displaystyle H}$ are,

${\displaystyle \begin{array}{rl}{\mu }_H& =M_{11}-\frac{i}{2}{\mathrm{\Gamma }}_{11}+\frac{1}{2}\left(\mathrm{\Delta }m-\frac{i}{2}\mathrm{\Delta }\mathrm{\Gamma }\right),\\ {\mu }_L& =M_{11}-\frac{i}{2}{\mathrm{\Gamma }}_{11}-\frac{1}{2}\left(\mathrm{\Delta }m-\frac{i}{2}\mathrm{\Delta }\mathrm{\Gamma }\right)\end{array}}$   (8)

where,

The suffixes stand for Heavy and Light respectively (by convention) and this implies that ${\displaystyle \mathrm{\Delta }m}$ is positive.

The normalized eigenstates corresponding to ${\displaystyle {\mu }_L}$ and ${\displaystyle {\mu }_H}$ respectively, in the natural basis ${\displaystyle \left\{|P⟩,|\overline{P}⟩\right\}\equiv \left\{\left(1,0\right),\left(0,1\right)\right\}}$ are,

${\displaystyle \begin{array}{rl}|P_L⟩& =p|P⟩+q|\overline{P}⟩\\ |P_H⟩& =p|P⟩-q|\overline{P}⟩\end{array}}$   (9)

where,

${\displaystyle p}$ and ${\displaystyle q}$ are the mixing terms. Note that these eigenstates are no longer orthogonal.

Let the system start in the state ${\displaystyle |P⟩}$. That is,

${\displaystyle |P\left(0\right)⟩=|P⟩=\frac{1}{2p}\left(|P_L⟩+|P_H⟩\right)}$

Under time evolution we then get,

${\displaystyle |P\left(t\right)⟩=\frac{1}{2p}\left(|P_L⟩e^{-\frac{i}{\hslash }\left(m_L-\frac{i}{2}{\gamma }_L\right)t}+|P_H⟩e^{-\frac{i}{\hslash }\left(m_H-\frac{i}{2}{\gamma }_H\right)t}\right)=g_{+}\left(t\right)|P⟩-\frac{q}{p}{g}_{-}\left(t\right)|\overline{P}⟩}$

where,

Similarly, if the system starts in the state ${\displaystyle |\overline{P}⟩}$, under time evolution we obtain,

${\displaystyle |\overline{P}(t)⟩=\frac{1}{2q}\left(|P_L⟩e^{-\frac{i}{\hslash }\left(m_L-\frac{i}{2}{\gamma }_L\right)t}-|P_H⟩e^{-\frac{i}{\hslash }\left(m_H-\frac{i}{2}{\gamma }_H\right)t}\right)=-\frac{p}{q}{g}_{-}\left(t\right)|P⟩+g_{+}\left(t\right)|\overline{P}⟩}$

CP violation as a consequence

If in a system ${\displaystyle |P⟩}$ and ${\displaystyle |\overline{P}⟩}$ represent CP conjugate states (i.e. particle-antiparticle) of one another (i.e. ${\displaystyle CP|P⟩=e^{i\delta }|\overline{P}⟩}$ and ${\displaystyle CP|\overline{P}⟩=e^{-i\delta }|P⟩}$), and certain other conditions are met, then CP violation can be observed as a result of this phenomenon. Depending on the condition, CP violation can be classified into three types:

CP violation through decay only

Consider the processes where ${\displaystyle \left\{|P⟩,|\overline{P}⟩\right\}}$ decay to final states ${\displaystyle \left\{|f⟩,|\overline{f}⟩\right\}}$, where the barred and the unbarred kets of each set are CP conjugates of one another.

The probability of ${\displaystyle |P⟩}$ decaying to ${\displaystyle |f⟩}$ is given by,

${\displaystyle {\mathrm{\wp }}_{P\to f}\left(t\right)={\left|⟨f|P\left(t\right)⟩\right|}^2={\left|g_{+}\left(t\right)A_f-\frac{q}{p}{g}_{-}\left(t\right){\overline{A}}_f\right|}^2}$,

and that of its CP conjugate process by,

${\displaystyle {\mathrm{\wp }}_{\overline{P}\to \overline{f}}\left(t\right)={\left|⟨\overline{f}|\overline{P}\left(t\right)⟩\right|}^2={\left|g_{+}\left(t\right){\overline{A}}_{\overline{f}}-\frac{p}{q}{g}_{-}\left(t\right)A_{\overline{f}}\right|}^2}$

where,

If there is no CP violation due to mixing, then ${\displaystyle \left|\frac{q}{p}\right|=1}$.

Now, the above two probabilities are unequal if,

${\displaystyle \left|\frac{{\overline{A}}_{\overline{f}}}{A_f}\right|\ne 1}$ and ${\displaystyle \left|\frac{A_{\overline{f}}}{\overline{A_f}}\right|\ne 1}$   (10)

Hence, the decay becomes a CP violating process as the probability of a decay and that of its CP conjugate process are not equal.

CP violation through mixing only

The probability (as a function of time) of observing ${\displaystyle |\overline{P}⟩}$ starting from ${\displaystyle |P⟩}$ is given by,

${\displaystyle {\mathrm{\wp }}_{P\to \overline{P}}\left(t\right)={\left|⟨\overline{P}|P\left(t\right)⟩\right|}^2={\left|\frac{q}{p}{g}_{-}\left(t\right)\right|}^2}$,

and that of its CP conjugate process by,

${\displaystyle {\mathrm{\wp }}_{\overline{P}\to P}\left(t\right)={\left|⟨P|\overline{P}\left(t\right)⟩\right|}^2={\left|\frac{p}{q}{g}_{-}\left(t\right)\right|}^2}$.

The above two probabilities are unequal if,

${\displaystyle \left|\frac{q}{p}\right|\ne 1}$   (11)

Hence, the particle-antiparticle oscillation becomes a CP violating process as the particle and its antiparticle (say, ${\displaystyle |P⟩}$ and ${\displaystyle |\overline{P}⟩}$ respectively) are no longer equivalent eigenstates of CP.

CP violation through mixing-decay interference

Let ${\displaystyle |f⟩}$ be a final state (a CP eigenstate) that both ${\displaystyle |P⟩}$ and ${\displaystyle |\overline{P}⟩}$ can decay to. Then, the decay probabilities are given by,

${\displaystyle \begin{array}{rl}{\mathrm{\wp }}_{P\to f}\left(t\right)& ={\left|⟨f|P\left(t\right)⟩\right|}^2\\ & ={\left|A_f\right|}^2\frac{e^{-\gamma t}}{2}\left[\left(1+{\left|{\lambda }_f\right|}^2\right)\cosh \left(\frac{\mathrm{\Delta }\gamma }{2}t\right)+2\mathrm{Re}\left({\lambda }_f\right)\sinh \left(\frac{\mathrm{\Delta }\gamma }{2}t\right)+\left(1-{\left|{\lambda }_f\right|}^2\right)\cos \left(\mathrm{\Delta }mt\right)+2\mathrm{Im}\left({\lambda }_f\right)\sin \left(\mathrm{\Delta }mt\right)\right]\end{array}}$

and,

${\displaystyle \begin{array}{rl}{\mathrm{\wp }}_{\overline{P}\to f}\left(t\right)& ={\left|⟨f|\overline{P}\left(t\right)⟩\right|}^2\\ & ={\left|A_f\right|}^2{\left|\frac{p}{q}\right|}^2\frac{e^{-\gamma t}}{2}\left[\left(1+{\left|{\lambda }_f\right|}^2\right)\cosh \left(\frac{\mathrm{\Delta }\gamma }{2}t\right)+2\mathrm{Re}\left({\lambda }_f\right)\sinh \left(\frac{\mathrm{\Delta }\gamma }{2}t\right)-\left(1-{\left|{\lambda }_f\right|}^2\right)\cos \left(\mathrm{\Delta }mt\right)-2\mathrm{Im}\left({\lambda }_f\right)\sin \left(\mathrm{\Delta }mt\right)\right]\end{array}}$

where,

From the above two quantities, it can be seen that even when there is no CP violation through mixing alone (i.e. ${\displaystyle \left|q/p\right|=1}$) and neither is there any CP violation through decay alone (i.e. ${\displaystyle \left|{\overline{A}}_f/A_f\right|=1}$) and thus ${\displaystyle \left|{\lambda }_f\right|=1}$, the probabilities will still be unequal provided,

${\displaystyle \mathrm{Im}\left({\lambda }_f\right)=\mathrm{Im}\left(\frac{q}{p}\frac{{\overline{A}}_f}{A_f}\right)\ne 0}$   (12)

The last terms in the above expressions for probability are thus associated with interference between mixing and decay.

An alternative classification

Usually, an alternative classification of CP violation is made:

Direct CP violation	Direct CP violation is defined as, ${\displaystyle \left|{\overline{A}}_f/A_f\right|\ne 1}$	In terms of the above categories, direct CP violation occurs in CP violation through decay only.
Indirect CP violation	Indirect CP violation is the type of CP violation that involves mixing.	In terms of the above classification, indirect CP violation occurs through mixing only, or through mixing-decay interference, or both.

Specific cases

Neutrino oscillation

Main article: Neutrino oscillation

Considering a strong coupling between two flavor eigenstates of neutrinos (for example,
ν
_e–
ν
_μ,
ν
_μ–
ν
_τ, etc.) and a very weak coupling between the third (that is, the third does not affect the interaction between the other two), equation (6) gives the probability of a neutrino of type ${\displaystyle \alpha }$ transmuting into type ${\displaystyle \beta }$ as,

${\displaystyle P_{\beta \alpha }\left(t\right)={\sin }^2\theta {\sin }^2\left(\frac{E_{+}-E_{-}}{2\hslash }t\right)}$

where, ${\displaystyle E_{+}}$ and ${\displaystyle E_{-}}$ are energy eigenstates.

The above can be written as,

${\displaystyle P_{\beta \alpha }\left(x\right)={\sin }^2\theta {\sin }^2\left(\frac{\mathrm{\Delta }{m}^2{c}^3}{4E\hslash }x\right)={\sin }^2\theta {\sin }^2\left(\frac{2\pi }{{\lambda }_{\text{osc}}}x\right)}$   (13)

where,

Proof

Thus, a coupling between the energy (mass) eigenstates produces the phenomenon of oscillation between the flavor eigenstates. One important inference is that neutrinos have a finite mass, although very small. Hence, their speed is not exactly the same as that of light but slightly lower.

Neutrino mass splitting

With three flavors of neutrinos, there are three mass splittings:

${\displaystyle \begin{array}{rl}{\left(\mathrm{\Delta }{m}^2\right)}_{12}& ={m_1}^2-{m_2}^2\\ {\left(\mathrm{\Delta }{m}^2\right)}_{23}& ={m_2}^2-{m_3}^2\\ {\left(\mathrm{\Delta }{m}^2\right)}_{31}& ={m_3}^2-{m_1}^2\end{array}}$

But only two of them are independent, because ${\displaystyle {\left(\mathrm{\Delta }{m}^2\right)}_{12}+{\left(\mathrm{\Delta }{m}^2\right)}_{23}+{\left(\mathrm{\Delta }{m}^2\right)}_{31}=0}$.

For solar neutrinos	${\displaystyle {\left(\mathrm{\Delta }{m}^2\right)}_{\text{sol }}\simeq 8\times {10}^{-5}{\left(eV/c^2\right)}^2}$
For atmospheric neutrinos	${\displaystyle {\left(\mathrm{\Delta }{m}^2\right)}_{\text{atm}}\simeq 3\times {10}^{-3}{\left(eV/c^2\right)}^2}$

This implies that two of the three neutrinos have very closely placed masses. Since only two of the three ${\displaystyle \mathrm{\Delta }{m}^2}$ are independent, and the expression for probability in equation (13) is not sensitive to the sign of ${\displaystyle \mathrm{\Delta }{m}^2}$ (as sine squared is independent of the sign of its argument), it is not possible to determine the neutrino mass spectrum uniquely from the phenomenon of flavor oscillation. That is, any two out of the three can have closely spaced masses.

Moreover, since the oscillation is sensitive only to the differences (of the squares) of the masses, direct determination of neutrino mass is not possible from oscillation experiments.

Length scale of the system

Equation (13) indicates that an appropriate length scale of the system is the oscillation wavelength ${\displaystyle {\lambda }_{\text{osc}}}$. We can draw the following inferences:

• If ${\displaystyle x/{\lambda }_{\text{osc}}\ll 1}$, then ${\displaystyle P_{\beta \alpha }\simeq 0}$ and oscillation will not be observed. For example, production (say, by radioactive decay) and detection of neutrinos in a laboratory.
• If ${\displaystyle x/{\lambda }_{\text{osc}}\simeq n}$, where ${\displaystyle n}$ is a whole number, then ${\displaystyle P_{\beta \alpha }\simeq 0}$ and oscillation will not be observed.
• In all other cases, oscillation will be observed. For example, ${\displaystyle x/{\lambda }_{\text{osc}}\gg 1}$ for solar neutrinos; ${\displaystyle x\sim {\lambda }_{\text{osc}}}$ for neutrinos from nuclear power plant detected in a laboratory few kilometers away.

Neutral kaon oscillation and decay

Main article: Kaon

CP violation through mixing only

The 1964 paper by Christenson et al.^[7] provided experimental evidence of CP violation in the neutral Kaon system. The so-called long-lived Kaon (CP = −1) decayed into two pions (CP = (−1)(−1) = 1), thereby violating CP conservation.

${\displaystyle |K^0⟩}$ and ${\displaystyle |{\overline{K}}^0⟩}$ being the strangeness eigenstates (with eigenvalues +1 and −1 respectively), the energy eigenstates are,

${\displaystyle \begin{array}{rl}|K_{{}^1}^0⟩& =\frac{1}{\sqrt{2}}\left(|K^0⟩+|{\overline{K}}^0⟩\right)\\ |K_2^0⟩& =\frac{1}{\sqrt{2}}\left(|K^0⟩-|{\overline{K}}^0⟩\right)\end{array}}$

These two are also CP eigenstates with eigenvalues +1 and −1 respectively. From the earlier notion of CP conservation (symmetry), the following were expected:

• Because ${\displaystyle |K_{{}^1}^0⟩}$ has a CP eigenvalue of +1, it can decay to two pions or with a proper choice of angular momentum, to three pions. However, the two pion decay is a lot more frequent.
• ${\displaystyle |K_2^0⟩}$ having a CP eigenvalue −1, can decay only to three pions and never to two.

Since the two pion decay is much faster than the three pion decay, ${\displaystyle |K_{{}^1}^0⟩}$ was referred to as the short-lived Kaon ${\displaystyle |K_S^0⟩}$, and ${\displaystyle |K_2^0⟩}$ as the long-lived Kaon ${\displaystyle |K_L^0⟩}$. The 1964 experiment showed that contrary to what was expected, ${\displaystyle |K_L^0⟩}$ could decay to two pions. This implied that the long lived Kaon cannot be purely the CP eigenstate ${\displaystyle |K_2^0⟩}$, but must contain a small admixture of ${\displaystyle |K_{{}^1}^0⟩}$, thereby no longer being a CP eigenstate. Similarly, the short-lived Kaon was predicted to have a small admixture of ${\displaystyle |K_2^0⟩}$. That is,

${\displaystyle \begin{array}{rl}|K_L^0⟩& =\frac{1}{\sqrt{1+{\left|\epsilon \right|}^2}}\left(|K_2^0⟩+\epsilon |K_1^0⟩\right)\\ |K_S^0⟩& =\frac{1}{\sqrt{1+{\left|\epsilon \right|}^2}}\left(|K_1^0⟩+\epsilon |K_2^0⟩\right)\end{array}}$

where, ${\displaystyle \epsilon }$ is a complex quantity and is a measure of departure from CP invariance. Experimentally, ${\displaystyle \left|\epsilon \right|=\left(2.228\pm 0.011\right)\times {10}^{-3}}$.

Writing ${\displaystyle |K_{{}^1}^0⟩}$ and ${\displaystyle |K_2^0⟩}$ in terms of ${\displaystyle |K^0⟩}$ and ${\displaystyle |{\overline{K}}^0⟩}$, we obtain (keeping in mind that ${\displaystyle m_{K_L^0}>m_{K_S^0}}$) the form of equation (9):

${\displaystyle \begin{array}{rl}|K_L^0⟩& =\left(p|K^0⟩-q|{\overline{K}}^0⟩\right)\\ |K_S^0⟩& =\left(p|K^0⟩+q|{\overline{K}}^0⟩\right)\end{array}}$

where, ${\displaystyle \frac{q}{p}=\frac{1-\epsilon }{1+\epsilon }}$.

Since ${\displaystyle \left|\epsilon \right|\ne 0}$, condition (11) is satisfied and there is a mixing between the strangeness eigenstates ${\displaystyle |K^0⟩}$ and ${\displaystyle |{\overline{K}}^0⟩}$ giving rise to a long-lived and a short-lived state.

CP violation through decay only

The
K⁰
_L and
K⁰
_S have two modes of two pion decay:
π⁰

π⁰
or
π⁺

π⁻
. Both of these final states are CP eigenstates of themselves. We can define the branching ratios as,

${\displaystyle \begin{array}{rl}{\eta }_{+-}& =\frac{⟨{\pi }^{+}{\pi }^{-}|K_L^0⟩}{⟨{\pi }^{+}{\pi }^{-}|K_S^0⟩}=\frac{p{A}_{{\pi }^{+}{\pi }^{-}}-q{\overline{A}}_{{\pi }^{+}{\pi }^{-}}}{p{A}_{{\pi }^{+}{\pi }^{-}}+q{\overline{A}}_{{\pi }^{+}{\pi }^{-}}}=\frac{1-{\lambda }_{{\pi }^{+}{\pi }^{-}}}{1+{\lambda }_{{\pi }^{+}{\pi }^{-}}}\\ {\eta }_{00}& =\frac{⟨{\pi }^0{\pi }^0|K_L^0⟩}{⟨{\pi }^0{\pi }^0|K_S^0⟩}=\frac{p{A}_{{\pi }^0{\pi }^0}-q{\overline{A}}_{{\pi }^0{\pi }^0}}{p{A}_{{\pi }^0{\pi }^0}+q{\overline{A}}_{{\pi }^0{\pi }^0}}=\frac{1-{\lambda }_{{\pi }^0{\pi }^0}}{1+{\lambda }_{{\pi }^0{\pi }^0}}\end{array}}$.

Experimentally, ${\displaystyle {\eta }_{+-}=\left(2.232\pm 0.011\right)\times {10}^{-3}}$ and ${\displaystyle {\eta }_{00}=\left(2.220\pm 0.011\right)\times {10}^{-3}}$. That is ${\displaystyle {\eta }_{+-}\ne {\eta }_{00}}$, implying ${\displaystyle \left|A_{{\pi }^{+}{\pi }^{-}}/{\overline{A}}_{{\pi }^{+}{\pi }^{-}}\right|\ne 1}$ and ${\displaystyle \left|A_{{\pi }^0{\pi }^0}/{\overline{A}}_{{\pi }^0{\pi }^0}\right|\ne 1}$, and thereby satisfying condition (10).

In other words, direct CP violation is observed in the asymmetry between the two modes of decay.

CP violation through mixing-decay interference

If the final state (say ${\displaystyle f_{CP}}$) is a CP eigenstate (for example
π⁺

π⁻
), then there are two different decay amplitudes corresponding to two different decay paths:

${\displaystyle \begin{array}{rl}{K}^0& \to f_{CP}\\ K^0& \to {\overline{K}}^0\to f_{CP}\end{array}}$.

CP violation can then result from the interference of these two contributions to the decay as one mode involves only decay and the other oscillation and decay.

Which then is the "real" particle?

The above description refers to flavor (or strangeness) eigenstates and energy (or CP) eigenstates. But which of them represents the "real" particle? What do we really detect in a laboratory? Quoting David J. Griffiths:

The neutral Kaon system adds a subtle twist to the old question, 'What is a particle?' Kaons are typically produced by the strong interactions, in eigenstates of strangeness (
K⁰
and
K⁰
), but they decay by the weak interactions, as eigenstates of CP (K₁ and K₂). Which, then, is the 'real' particle? If we hold that a 'particle' must have a unique lifetime, then the 'true' particles are K₁ and K₂. But we need not be so dogmatic. In practice, it is sometimes more convenient to use one set, and sometimes, the other. The situation is in many ways analogous to polarized light. Linear polarization can be regarded as a superposition of left-circular polarization and right-circular polarization. If you imagine a medium that preferentially absorbs right-circularly polarized light, and shine on it a linearly polarized beam, it will become progressively more left-circularly polarized as it passes through the material, just as a
K⁰
beam turns into a K₂ beam. But whether you choose to analyze the process in terms of states of linear or circular polarization is largely a matter of taste.

The mixing matrix - a brief introduction

Main articles: Cabibbo–Kobayashi–Maskawa matrix and Pontecorvo–Maki–Nakagawa–Sakata matrix

If the system is a three state system (for example, three species of neutrinos
ν
_e ⇄
ν
_μ ⇄
ν
_τ, three species of quarks
d
⇄
s
⇄
b
), then, just like in the two state system, the flavor eigenstates (say ${\displaystyle |{\phi }_{\alpha }⟩}$, ${\displaystyle |{\phi }_{\beta }⟩}$, ${\displaystyle |{\phi }_{\gamma }⟩}$) are written as a linear combination of the energy (mass) eigenstates (say ${\displaystyle |{\psi }_1⟩}$, ${\displaystyle |{\psi }_2⟩}$, ${\displaystyle |{\psi }_3⟩}$). That is,

${\displaystyle \left(\begin{array}{c}|{\phi }_{\alpha }⟩\\ |{\phi }_{\beta }⟩\\ |{\phi }_{\gamma }⟩\end{array}\right)=\left(\begin{array}{ccc}{\mathrm{\Omega }}_{\alpha 1}& {\mathrm{\Omega }}_{\alpha 2}& {\mathrm{\Omega }}_{\alpha 3}\\ {\mathrm{\Omega }}_{\beta 1}& {\mathrm{\Omega }}_{\beta 2}& {\mathrm{\Omega }}_{\beta 3}\\ {\mathrm{\Omega }}_{\gamma 1}& {\mathrm{\Omega }}_{\gamma 2}& {\mathrm{\Omega }}_{\gamma 3}\end{array}\right)\left(\begin{array}{c}|{\psi }_1⟩\\ |{\psi }_2⟩\\ |{\psi }_3⟩\end{array}\right)}$.

In case of leptons (neutrinos for example) the transformation matrix is the PMNS matrix, and for quarks it is the CKM matrix.

The off diagonal terms of the transformation matrix represent coupling, and unequal diagonal terms imply mixing between the three states.

The transformation matrix is unitary and appropriate parameterization (depending on whether it is the CKM or PMNS matrix) is done and the values of the parameters determined experimentally.

Photovoltaic system performance

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Photovoltaic_system_performance 

 

The SR30 pyranometer is an example of an PV monitoring sensor, which can be used in two orientations (horizontal and in plane of array) for measuring irradiance.

Photovoltaic system performance is a function of the climatic conditions, the equipment used and the system configuration. PV performance can be measured as the ratio of actual solar PV system output vs expected values, the measurement being essential for proper solar PV facility's operation and maintenance. The primary energy input is the global light irradiance in the plane of the solar arrays, and this in turn is a combination of the direct and the diffuse radiation.

The performance is measured by PV monitoring systems, which include a data logging device and often also a weather measurement device (on-site device or an independent weather data source). Photovoltaic performance monitoring systems serve several purposes - they are used to track trends in a single photovoltaic (PV) system, to identify faults in or damage to solar panels and inverters, to compare the performance of a system to design specifications or to compare PV systems at different locations. This range of applications requires various sensors and monitoring systems, adapted to the intended purpose. Specifically, there is a need for both electronic monitoring sensors and independent weather sensing (irradiance, temperature and more) in order to normalize PV facility output expectations. Irradiance sensing is very important for the PV industry and can be classified into two main categories - on-site pyranometers and satellite remote sensing; when onsite pyranometers are not available, regional weather stations are also sometimes utilized, but at lower quality of data; the Industrial IoT-powered sensorless measurement approach has recently evolved as the third option.

Sensors and photovoltaic monitoring systems are standardized in IEC 61724-1 and classified into three levels of accuracy, denoted by the letters “A”, “B” or “C”, or by the labels “High accuracy”, “Medium accuracy” and “Basic accuracy”. A parameter called the 'performance ratio' has been developed to evaluate the total value of PV system losses.

Overview

Photovoltaic system performance is generally dependent on incident irradiance in the plane of the solar panels, the temperature of the solar cells, and the spectrum of the incident light. Furthermore, it is dependent upon the inverter, which typically sets the operating voltage of the system. The voltage and current output of the system changes as lighting, temperature and load conditions change, so there is no specific voltage, current, or wattage at which the system always operates. Hence, system performance varies depending on the time of day, amount of solar insolation, direction and tilt of modules, cloud cover, shading, soiling, state of charge, temperature, geographic location, and day of the year.

Performance by system type

Solar PV parks

Further information: Photovoltaic power station

Solar parks of industrial and utility scale may reach high performance figures. In modern solar parks the performance ratio should typically be in excess of 80%. Many solar PV parks utilize advanced performance monitoring solutions, which are supplied by a variety of technology providers.

Distributed solar PV

Further information: Rooftop photovoltaic power station

In rooftop solar systems it typically takes a longer time to identify a malfunction and send a technician, due to lower availability of sufficient photovoltaic system performance monitoring tools and higher costs of human labor. As a result, rooftop solar PV systems typically suffer from lower quality of operation & maintenance and essentially lower levels of system availability and energy output.

Off-grid solar PV

Further information: Stand-alone power system

Most off-grid solar PV facilities lack any performance monitoring tools, due to a number of reasons - including monitoring equipment costs, cloud connection availability and O&M availability.

Performance monitoring

Rbee Solar, PV monitoring with solar irradiance measurement

A number of technical solutions exist to provide performance monitoring for solar photovoltaic installations, differing according to data quality, compatibility with irradiance sensors as well as pricing. In general, monitoring solutions can be classified to inverter manufacturer-provided logger and monitoring software solutions, independent data-logger solutions with custom software and finally agnostic monitoring software-only solutions compatible with different inverters and data-loggers.

Monitoring solutions by inverter manufacturers

Dedicated performance monitoring systems are available from a number of vendors. For solar PV systems that use microinverters (panel-level DC to AC conversion), module power data is automatically provided. Some systems allow setting performance alerts that trigger phone/email/text warnings when limits are reached. These solutions provide data for the system owner and/or the installer. Installers are able to remotely monitor multiple installations, and see at-a-glance the status of their entire installed base. All the major inverter manufacturers provide a data acquisition unit - whether a data logger or a direct means of communication with the portal.

These solutions have the advantage of providing of a maximum information from the inverter and of supplying it on a local display or transmitting it on the internet, in particular alerts from the inverter itself (temperature overload, loss of connection with a network, etc.).

Some of those monitoring solutions are:

• Fronius accessible via Solar.web portal;
• SMA's Webbox/Inverter Manager/Cluster-Controller loggers accessible via Sunnyportal and EnnexOS portals;
• SolarEdge accessible via SolarEdge Monitoring and MySolarEdge (only application) portals;
• Sungrow accessible via the iSolarCloud portal;

Independent data logging solutions connected to inverters

Generic data logging solutions connected to inverters make it possible to overcome the major drawback of inverter-specific manufacturer solutions - being compatible with several different manufacturers. These data acquisition units connect to the serial links of the inverters, complying with each manufacturer’s protocol. Generic data logging solutions are generally more affordable than inverter manufacturer solutions and allow aggregation of solar PV system fleets of varying inverter manufacturers.

Some of those monitoring solutions are:

• AlsoEnergy loggers accessible via the PowerTrack portal;
• Solar-Log loggers accessible via the WEB Enerest™ 4 portal;
• Meteocontrol loggers accessible via the VCOM Cloud portal;
• Solar Analytics' "Smart Solar logger"s accessible via the Solar Analytics portal;

Independent monitoring solutions

The last category is the most recent segment in the solar photovoltaic monitoring domain. Those are software based aggregation portals, able to aggregate information from both inverter-specific portals and data loggers as well as independent data loggers. Such solutions become more widespread as inverter-specific communication to the cloud is done more and more without data loggers, but rather as direct data connections.

• Omnidian residential solar performance insurance partner Omnidian;
• Soltell solar management solution for distributed solar PV, accessible via SysMap portal;
• Solytic generic solar monitoring Solytic portal;
• Sunreport device-agnostic cloud solar monitoring Sunreport platform;

Energy generation data availability and quality

An essential part of PV system performance evaluation is the availability and the quality of energy generation data. Access to the Internet has allowed a further improvement in energy monitoring and communication.

Typically, PV plant data is transmitted via a data logger to a central monitoring portal. Data transmission is dependent on the local cloud connectivity, thus being highly available in OECD countries, but more limited in developed countries. According to Samuel Zhang, vice president of Huawei Smart PV, over 90% of global PV plants will be fully digitilized by 2025.

Weather data sources

On-site irradiance sensors

On-site irradiance measurements are an important part of PV performance monitoring systems. Irradiance can be measured in the same orientation as the PV panels, so-called plane of array (POA) measurements, or horizontally, so-called global horizontal irradiance (GHI) measurements. Typical sensors used for such irradiance measurements include thermopile pyranometers, PV reference devices and photodiode sensors. To conform to a specific accuracy class, each sensor type must meet a certain set of specifications. These specifications are listed in the table below.

Table 5 - Sensor choices and requirements for in-plane and global irradiance cited from IEC 61724-1

Sensor type	Class A High accuracy	Class B Medium accuracy	Class C Basic accuracy
Thermopile pyranometer	Secondary standard per ISO 9060 or High quality per WMO Guide (Uncertainty ≤ 3% for hourly totals)	First class per ISO 9060 or Good quality per WMO Guide (Uncertainty ≤ 8% for hourly totals)	Any
PV reference device	Uncertainty ≤ 3% from 100 W/m2 to 1500 W/m2	Uncertainty ≤ 8% from 100 W/m2 to 1500 W/m2	Any
Photodiode sensors	Not applicable	Not applicable	Any

The VU01 pyranometer ventilation unit with SR20, with heater and ventilation, is an A-class compliant pyranometer according to the IEC 61727-1

If an irradiance sensor is placed in POA, it must be placed at the same tilt angle as the PV module, either by attaching it to the module itself or with an extra platform or arm at the same tilt level. Checking if the sensor is properly aligned can be done with portable tilt sensors or with an integrated tilt sensor.

Sensor maintenance

The standard also specifies a required maintenance schedule per accuracy class. Class C sensors require maintenance per manufacturer's requirement. Class B sensors need to be re-calibrated every 2 years and require a heater to prevent precipitation or condensation. Class A sensors need to be re-calibrated once per year, require cleaning once per week, require a heater and require ventilation (for thermopile pyranometers).

Satellite remote sensing of irradiance

PV performance can also be estimated by satellite remote sensing. These measurements are indirect because the satellites measure the solar radiance reflected off the earth surface. In addition, the radiance is filtered by the spectral absorption of Earth's atmosphere. This method is typically used in non-instrumented class B and class C monitoring systems to avoid costs and maintenance of on-site sensors. If the satellite-derived data is not corrected for local conditions, an error in radiance up to 10% is possible.

Equipment and performance standards

Sensors and monitoring systems are standardized in IEC 61724-1 and classified into three levels of accuracy, denoted by the letters “A”, “B” or “C”, or by the labels “High accuracy”, “Medium accuracy” and “Basic accuracy”.

In California, solar PV performance monitoring has been regulated by the State government. As of 2017, the governmental agency California Solar Initiative (CSI) provided a Performance Monitoring & Reporting Service certificate to eligible companies active in the solar segment and acting in line with CSI requirements.

A parameter called the 'performance ratio' has been developed to evaluate the total value of PV system losses. The performance ratio gives a measure of the output AC power delivered as a proportion of the total DC power which the solar modules should be able to deliver under the ambient climatic conditions.