Neutral particle oscillation

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.{{Cite journal |last1=Wu |first1=C.S. |last2=Ambler |first2=E. |last3=Hayward |first3=R.W. |last4=Hoppes |first4=D.D. |last5=Hudson |first5=R.P. |year=1957 |title=Experimental test of parity conservation in beta decay |journal=Physical Review |volume=105 |issue=4 |pages=1413–1415 |bibcode=1957PhRv..105.1413W |doi=10.1103/PhysRev.105.1413 |doi-access=free}} However, in 1964 Cronin and Fitch reported CP violation in the neutral Kaon system. They observed the long-lived K_L (with {{nowrap| CP {{=}} −1 }}) undergoing decays into two pions (with {{nowrap| CP {{=}} [−1]·[−1] {{=}} +1 }}) thereby violating CP conservation.

In 2001, CP violation in the B–Bbar oscillation was confirmed by the BaBar and the Belle experiments.{{Cite journal |author1-link=Alexander Abashian |last=Abashian |first=A. |display-authors=etal |year=2001 |title=Measurement of the CP violation parameter sin(2φ{{sub|1}}) in B{{su|p=0|b=d}} meson decays |journal=Physical Review Letters |volume=86 |issue=12 |pages=2509–2514 |arxiv=hep-ex/0102018 |bibcode=2001PhRvL..86.2509A |doi=10.1103/PhysRevLett.86.2509|pmid=11289969 |s2cid=12669357 }}{{cite journal |last1=Aubert |first1=B. |display-authors=etal |collaboration=BABAR Collaboration |year=2001 |title=Measurement of CP-violating asymmetries in B⁰ decays to CP eigenstates |journal=Physical Review Letters |volume=86 |issue=12 |pages=2515–2522 |arxiv=hep-ex/0102030 |bibcode=2001PhRvL..86.2515A |doi=10.1103/PhysRevLett.86.2515 |pmid=11289970|s2cid=24606837 }} Direct CP violation in the {{math| {{SubatomicParticle|B0}} ⇄ {{SubatomicParticle|antiB0}} }} system was reported by both the labs by 2005.{{cite journal |last=Aubert |first=B. |display-authors=etal |collaboration=BABAR Collaboration |year=2004 |title=Direct CP violating asymmetry in {{nowrap| B{{sup|0}} → K{{sup|+}}π{{sup|−}} }} decays |journal=Physical Review Letters |volume=93 |issue=13 |pages=131801 |arxiv=hep-ex/0407057 |bibcode=2004PhRvL..93m1801A |doi=10.1103/PhysRevLett.93.131801|pmid=15524703 |s2cid=31279756 }}{{cite journal |last1=Chao |first1=Y. |display-authors=etal |collaboration=Belle Collaboration |year=2005 |title=Improved measurements of the partial rate asymmetry in B → hh decays |journal=Physical Review D |volume=71 |issue=3 |page=031502 |arxiv=hep-ex/0407025 |bibcode=2005PhRvD..71c1502C |doi=10.1103/PhysRevD.71.031502|s2cid=119441257 |url=https://cds.cern.ch/record/777066/files/0407025.pdf }}

The Kaon#Oscillation and the {{math| {{SubatomicParticle|B0}} ⇄ {{SubatomicParticle|antiB0}} }} systems can be studied as two state systems, considering the particle and its antiparticle as two states of a single particle.

The pp chain in the sun produces an abundance of {{math|{{SubatomicParticle|Electron Neutrino}}}}. In 1968, R. Davis et al. first reported the results of the Homestake experiment.{{cite web |last=Bahcall |first=J.N. |date=28 April 2004 |title=Solving the mystery of the missing neutrinos |publisher=The Nobel Foundation |url=https://www.nobelprize.org/nobel_prizes/themes/physics/bahcall/ |access-date=2016-12-08}}{{cite journal |last1=Davis |first1=R. Jr. |last2=Harmer |first2=D.S. |last3=Hoffman |first3=K.C. |year=1968 |title=Search for Neutrinos from the Sun |journal=Physical Review Letters |volume=20 |issue=21 |pages=1205–1209 |bibcode=1968PhRvL..20.1205D |doi=10.1103/PhysRevLett.20.1205}} 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 {{math|{{SubatomicParticle|Electron Neutrino}}}} to produce argon via the reaction

: $\backslash mathrm\{\backslash nu\_e\; +\; \{\{\}^\{37\}\_\{17\}Cl\}\; \backslash rightarrow\; \{\{\}^\{37\}\_\{18\}\}Ar\; +\; e^-\}$,

which is essentially

:$\backslash mathrm\{\backslash nu\_e\; +\; n\; \backslash to\; p\; +\; e^-\}$.{{cite book |last=Griffiths |first=D.J. |year=2008 |title=Elementary Particles |page=390 |edition=Second, revised |publisher=Wiley-VCH |isbn=978-3-527-40601-2}}

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 {{math|{{SubatomicParticle|Electron Neutrino}}}} (produced in the sun) can transform into some other neutrino species ({{math|{{SubatomicParticle|Muon Neutrino}}}} or {{math|{{SubatomicParticle|Tau Neutrino}}}}), 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 {{math|{{SubatomicParticle|Electron Neutrino}}}} flux and the total neutrino flux.{{cite journal |last=Ahmad |first=Q.R. |display-authors=etal |collaboration=SNO Collaboration |year=2002 |title=Direct evidence for neutrino flavor transformation from neutral-current interactions in the Sudbury Neutrino Observatory |journal=Physical Review Letters |volume=89 |issue=1 |page=011301 |arxiv=nucl-ex/0204008 |bibcode=2002PhRvL..89a1301A |doi=10.1103/PhysRevLett.89.011301 |doi-access=free |pmid=12097025}}

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

{{Main|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, prescribed by a "mixing matrix" (e.g. the CKM or PMNS matricies).

Let $\backslash \; H\_0\backslash$ be the Hamiltonian of the two-state system, and $\backslash \; \backslash left|\; 1\; \backslash right\backslash rangle\backslash$ and $\backslash \; \backslash left|\; 2\; \backslash right\backslash rangle\backslash$ be its orthonormal eigenvectors with eigenvalues $\backslash \; E\_1\backslash$ and $\backslash \; E\_2\backslash$ respectively.

Let $\backslash \; \backslash left|\; \backslash Psi(\; t\; )\; \backslash right\backslash rangle\backslash$ be the state of the system at time $\backslash \; t\; ~.$

If the system starts as an energy eigenstate of $\backslash \; H\_0\backslash \; ,$ for example, say

: $\backslash \; \backslash left|\; \backslash Psi(\; 0\; )\; \backslash right\backslash rangle\; =\; \backslash left|\; 1\; \backslash right\backslash rangle\backslash \; ,$

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

{{Equation box 1

| equation = $\backslash hat\; H\_0\backslash left|\backslash Psi(\; t\; )\backslash right\backslash rangle\backslash \; =\backslash \; i\; \backslash hbar\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\backslash left|\backslash Psi\backslash left(\; t\; \backslash right)\; \backslash right\backslash rangle\backslash$

| ref=1

}}

will be

{{cite book

|last=Griffiths |first=D.J.

|year=2005

|title=Introduction to Quantum Mechanics

|publisher=Pearson Education International

|isbn=978-0-13-191175-8

}}

: $\backslash \; \backslash left|\; \backslash Psi(\; t\; )\; \backslash right\backslash rangle\backslash \; =\backslash \; \backslash left|\; 1\; \backslash right\backslash rangle\; e^\{-i\backslash \; \backslash frac\{E\_1\; t\}\{\backslash hbar\}\}\backslash$

But this is physically same as $\backslash \; \backslash left|\; 1\; \backslash right\backslash rangle\backslash \; ,$ since the exponential term is just a phase factor: It does not produce an observable new state. In other words, energy eigenstates are stationary eigenstates, that is, they do not yield observably distinct new states under time evolution.

Define $\backslash \; \backslash left\backslash \{\backslash \; \backslash left|\; 1\; \backslash right\backslash rangle\backslash \; ,\backslash \; \backslash left|\; 2\; \backslash right\backslash rangle\backslash \; \backslash right\backslash \}\backslash \; ,$ to be a basis in which the unperturbed Hamiltonian operator, $\backslash \; H\_0\backslash$, is diagonal:

: $\backslash \; H\_0\; =\; \backslash begin\{pmatrix\}$

E_1 & 0 \\

0 & E_2 \\

\end{pmatrix}\ =\ E_1\ \left| 1 \right\rangle\ +\ E_2\ \left| 2 \right\rangle

\

It can be shown, that oscillation between states will occur if and only if off-diagonal terms of the Hamiltonian are not zero.

Hence let us introduce a general perturbation $\backslash \; W\backslash$ imposed on $\backslash \; H\backslash \; \_0\backslash$ such that the resultant Hamiltonian $\backslash \; H\backslash$ is still Hermitian. Then

: $W\; =\; \backslash begin\{pmatrix\}$

W_{11} & W_{12} \\

W_{12}^* & W_{22} \\

\end{pmatrix}\

where $\backslash \; W\_\{11\},\; W\_\{22\}\; \backslash in\; \backslash mathbb\{R\}\backslash$ and $\backslash \; W\_\{12\}\; \backslash in\; \backslash mathbb\{C\}\backslash$ and

{{Equation box 1

|equation = $\backslash \; H\; =\; H\_0\; +\; W\; =\; \backslash begin\{pmatrix\}$

E_1 + W_{11} & W_{12} \\

W_{12}^* & E_2 + W_{22} \\

\end{pmatrix}\

|ref=2

}}

The eigenvalues of the perturbed Hamiltonian, $\backslash \; H\backslash \; ,$ then change to $\backslash \; E\_+\backslash$ and $\backslash \; E\_-\backslash \; ,$ where

{{cite book

|first1=C. |last1=Cohen-Tannoudji

|first2=B. |last2=Diu

|first3=F. |last3=Laloe

|year=2006

|title=Quantum Mechanics

|publisher=Wiley-VCH

|isbn=978-0-471-56952-7

}}

{{Equation box 1

|equation = $\backslash \; E\_\backslash pm\; =\; \backslash frac\{\; 1\; \}\{\backslash \; 2\backslash \; \}\; \backslash left[$

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]\

|ref=3

}}

Since $\backslash \; H\backslash$ is a general Hamiltonian matrix, it can be written as

{{cite web

|last=Gupta |first=S.

|date=13 August 2013

|title=The mathematics of 2-state systems

|department=course handout 4

|series=Quantum Mechanics I

|website=theory.tifr.res.in/~sgupta

|publisher=Tata Institute of Fundamental Research

|url=http://theory.tifr.res.in/~sgupta/courses/qm2013/hand4.pdf

|access-date=2016-12-08

}}

: $\backslash \; H\; =\; \backslash sum\backslash limits\_\{j=0\}^3\; a\_j\; \backslash sigma\_j\; =\; a\_0\; \backslash sigma\_0\; +\; H\text{'}\backslash$

The following two results are clear:

• $\backslash \; \backslash left[H,\; H\text{'}\backslash right]\; =\; 0\backslash$

:

• $\backslash \; \{H\text{'}\}^2\; =\; I\backslash$

:

With the following parametrization (this parametrization helps as it normalizes the eigenvectors and also introduces an arbitrary phase $\backslash phi$ making the eigenvectors most general)

: $\backslash \; \backslash hat\{n\}\; =\; \backslash left(\backslash \; \backslash sin\backslash theta\; \backslash cos\backslash phi\backslash \; ,\backslash \; \backslash sin\backslash theta\; \backslash sin\backslash phi\backslash \; ,\backslash \; \backslash cos\backslash theta\backslash \; \backslash right)\backslash$

and using the above pair of results the orthonormal eigenvectors of $\backslash \; H\text{'}\backslash$ and consequently those of $\backslash \; H\backslash$ are obtained as

{{Equation box 1

|equation = $\backslash \; \backslash begin\{align\}$

\left| + \right\rangle\ &=

\ \begin{pmatrix}

\; ~\; \cos\tfrac{\theta}{2}\; e^{-i\frac{\phi}{2}} \\

\; ~\; \sin\tfrac{\theta}{2}\; e^{+i\frac{\phi}{2}} \\

\end{pmatrix}\ \equiv

~~~\; \cos\tfrac{\theta}{2}\; e^{-i\frac{\phi}{2}}\ \left| 1 \right\rangle\ +

~\; \sin\tfrac{\theta}{2}\; e^{+i\frac{\phi}{2}}\ \left| 2 \right\rangle \\

\left| - \right\rangle\ &=

\ \begin{pmatrix}

- \sin\frac{\theta}{2}\; e^{+i\frac{\phi}{2}} \\

~ \cos\frac{\theta}{2}\; e^{-i\frac{\phi}{2}} \\

\end{pmatrix}\ \equiv

\ - \sin\frac{\theta}{2}\; e^{-i\frac{\phi}{2}}\ \left| 1 \right\rangle\ +

~ \cos\frac{\theta}{2}\; e^{+i\frac{\phi}{2}}\ \left| 2 \right\rangle \\

\end{align}\

|ref=4

}}

  and

$\backslash \; W\_\{12\}\; =\; \backslash left|\; W\_\{12\}\; \backslash right|\; e^\{i\backslash phi\}\backslash$

|}

Writing the eigenvectors of $\backslash \; H\_0\backslash$ in terms of those of $\backslash \; H\backslash$ we get

{{Equation box 1 |equation = $\backslash \; \backslash begin\{align\}$

\left|\ 1\ \right\rangle\ &=

\ e^{i\frac{\phi}{2}} \left( \cos\tfrac{\theta}{2}\left| + \right\rangle - \sin\tfrac{\theta}{2}\left| - \right\rangle \right) \\

\left|\ 2\ \right\rangle\ &=

\ e^{-i\frac{\phi}{2}} \left( \sin\tfrac{\theta }{2}\left| + \right\rangle + \cos\tfrac{\theta}{2}\left| - \right\rangle \right) \\

\end{align}\

|ref=5

}}

Now if the particle starts out as an eigenstate of $\backslash \; H\_0\backslash$ (say, $\backslash \; \backslash left|\; 1\; \backslash right\backslash rangle\backslash$), that is

: $\backslash \; \backslash left|\backslash \; \backslash Psi(\; 0\; )\backslash \; \backslash right\backslash rangle\backslash \; =\backslash \; \backslash left|\; 1\; \backslash right\backslash rangle\backslash$

then under time evolution we get

: $\backslash ,$

\left|\ \Psi( t )\ \right\rangle\ =\ e^{i\ \frac{\phi}{2}} \left(

\cos\tfrac{\theta}{2}\ \left| + \right\rangle\ e^{-i\ \frac{E_+ t}{\hbar}} -

\sin\tfrac{\theta}{2}\ \left| - \right\rangle\ e^{-i\ \frac{E_- t}{\hbar}}

\right)

\

which unlike the previous case, is distinctly different from $\backslash \; \backslash left|\; 1\; \backslash right\backslash rangle\; ~.$

We can then obtain the probability of finding the system in state $\backslash \; \backslash left|\; 2\; \backslash right\backslash rangle\backslash$ at time $\backslash \; t\backslash$ as

{{Equation box 1

| equation = $\backslash \; \backslash begin\{align\}$

P_{21}\!( t )

&= \Bigl|\ \left\langle\ 2\ |\ \Psi(t)\ \right\rangle\ \Bigr|^2 =

\sin^2\!\theta\ \sin^2\!\!\left( \frac{\ E_+ - E_-\ }{\ 2\ \hbar\ }\ 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\ \hbar\ }\ t

\ \right) \\

\end{align}\

| ref=6

}}

which is called Rabi's formula. Hence, starting from one eigenstate of the unperturbed Hamiltonian $\backslash \; H\_0\backslash \; ,$ the state of the system oscillates between the eigenstates of $\backslash \; H\_0\backslash$ with a frequency (known as Rabi frequency),

{{Equation box 1

|equation = $\backslash ,$

\omega = \frac{\ E_1 - E_2\ }{\ 2\ \hbar\ } =

\frac{\ \sqrt{4\left| W_{12} \right|^2 + \left( E_1 - E_2 \right)^2\ }\ }{\ 2\ \hbar\ }

\

|ref=7

}}

From equation (6), for $\backslash \; P\_\{21\}\backslash !(t)\backslash \; ,$ we can conclude that oscillation will exist only if $\backslash \; \backslash left|\; W\_\{12\}\; \backslash right|^2\; \backslash ne\; 0\; ~.$ So $\backslash \; W\_\{12\}\backslash$ is known as the coupling term as it connects the two eigenstates of the unperturbed Hamiltonian $\backslash \; H\_0\backslash$ and thereby facilitates oscillation between the two.

Oscillation will also cease if the eigenvalues of the perturbed Hamiltonian $\backslash \; H\backslash$ are degenerate, i.e. $\backslash \; E\_+\; =\; E\_-\; ~.$ But this is a trivial case as in such a situation, the perturbation itself vanishes and $\backslash \; H\backslash$ takes the form (diagonal) of $\backslash \; H\_0\backslash$ and we're back to square one.

Hence, the necessary conditions for oscillation are:

• Non-zero coupling, i.e. $\backslash \; \backslash left|\; W\_\{12\}\; \backslash right|^2\; \backslash ne\; 0\; ~.$
• Non-degenerate eigenvalues of the perturbed Hamiltonian $\backslash \; H\backslash \; ,$ i.e. $\backslash \; E\_+\; \backslash ne\; E\_-\; ~.$

If the particle(s) under consideration undergoes decay, then the Hamiltonian describing the system is no longer Hermitian.

{{cite web

|last=Dighe |first=A.

|date=26 July 2011

|title=B physics and CP violation: An introduction

|type=lecture notes

|publisher=Tata Institute of Fundamental Research

|url=http://theory.tifr.res.in/~amol/talks/B-notes.pdf

|access-date=2016-08-12

}}

Since any matrix can be written as a sum of its Hermitian and anti-Hermitian parts, $\backslash \; H\backslash$ can be written as,

: $\backslash \; H\backslash ;\; =\backslash ;\; M\; -\; \backslash frac\{i\}\{2\}\backslash \; \backslash Gamma\backslash ;\; =\backslash ;\; \backslash begin\{pmatrix\}$

M_{11} & M_{12} \\

M_{12}^* & M_{11} \\

\end{pmatrix} - \frac{i}{2}\ \begin{pmatrix}

\Gamma_{11} & \Gamma_{12} \\

\Gamma_{12}^* & \Gamma_{11} \\

\end{pmatrix}

\

Hermiticity of $\backslash \; M\backslash$ and $\backslash \; \backslash Gamma\backslash$ also implies that their diagonal elements are real.

|}

The eigenvalues of $\backslash \; H\backslash$ are

{{Equation box 1

|equation = $\backslash \; \backslash begin\{align\}$

\mu_\mathsf H &= M_{11} - \tfrac{i}{2}\Gamma_{11} + \tfrac{1}{2}\left( \Delta m - \frac{i}{2}\Delta\Gamma \right), \\

\mu_\mathsf L &= M_{11} - \tfrac{i}{2}\Gamma_{11} - \tfrac{1}{2}\left( \Delta m - \frac{i}{2}\Delta\Gamma \right)\end{align}\

|ref=8

}}

The suffixes stand for Heavy and Light respectively (by convention) and this implies that $\backslash Delta\; m$ is positive.

The normalized eigenstates corresponding to $\backslash \; \backslash mu\_\backslash mathsf\; L\backslash$ and $\backslash \; \backslash mu\_\backslash mathsf\; H\backslash$ respectively, in the natural basis $\backslash \; \backslash bigl\backslash \{\; \backslash left|\; P\; \backslash right\backslash rangle\backslash \; ,\backslash \; \backslash left|\; \backslash bar\{P\}\; \backslash right\backslash rangle\; \backslash bigr\backslash \}\; ~\backslash equiv~\; \backslash bigl\backslash \{\backslash \; (1,\; 0)\backslash \; ,\backslash \; (0,\; 1)\backslash \; \backslash bigr\backslash \}\backslash$ are

{{Equation box 1

|equation = $\backslash \; \backslash begin\{align\}$

\left| P_\mathsf L \right\rangle\ &=\ p\ \left| P \right\rangle\ +\ q\left|\ \bar{P} \right\rangle \\

\left| P_\mathsf H \right\rangle\ &=\ p\ \left| P \right\rangle\ -\ q\left|\ \bar{P} \right\rangle

\end{align}\

|ref=9

}}

$\backslash \; p\backslash$ and $\backslash \; q\backslash$ are the mixing terms. Note that these eigenstates are no longer orthogonal.

Let the system start in the state $\backslash \; \backslash left|\; P\; \backslash right\backslash rangle\; ~.$ That is

: $$

\ \left|\ P( 0 )\ \right\rangle\ =

\ \left| P \right\rangle\ =

\ \frac{ 1 }{\ 2\ p\ }\ \Bigl(\ \left| P_\mathsf L \right\rangle\ +\ \left| P_\mathsf H \right\rangle\ \Bigr)

\

Under time evolution we then get

: $$

\ \left|\ P( t )\ \right\rangle\ =

\ \frac{ 1 }{\ 2\ p\ }\ \left(

\ \left| P_\mathsf L \right\rangle\ e^{-\tfrac{i}{\hbar}\ \left( m_L - \tfrac{i}{2}\gamma_L \right)\ t}\ +

\ \left| P_\mathsf H \right\rangle\ e^{-\tfrac{i}{\hbar}\ \left( m_H - \tfrac{i}{2}\gamma_H \right)\ t}

\ \right)\ =

\ g_+( t )\ \left| P \right\rangle\ -\ \frac{\ q\ }{ p }\ g_-( t )\ \left| \bar{P} \right\rangle

\

Similarly, if the system starts in the state $\backslash left|\; \backslash bar\{P\}\; \backslash right\backslash rangle$, under time evolution we obtain

: $\backslash ,$

\left|\ \bar{P}(t)\ \right\rangle =

\frac{ 1 }{\ 2\ q\ }\left(

\left| P_\mathsf L \right\rangle\ e^{-\tfrac{i}{\hbar}\ \left( m_\mathsf L - \tfrac{i}{2}\gamma_\mathsf L \right)\ t} -

\left| P_\mathsf H \right\rangle\ e^{-\tfrac{i}{\hbar}\ \left( m_\mathsf H - \tfrac{i}{2}\gamma_\mathsf H \right)\ t}

\right)\ =

\ -\frac{ p }{\ q\ }\ g_-( t )\ \left| P \right\rangle\ +\ g_+( t )\ \left| \bar{P} \right\rangle

\

If in a system $\backslash left|\; P\; \backslash right\backslash rangle$ and $$

\left| {\bar{P}} \right\rangle represent CP conjugate states (i.e. particle-antiparticle) of one another (i.e. $CP\backslash left|\; P\; \backslash right\backslash rangle\; =\; e^\{i\backslash delta\}\; \backslash left|\; \backslash bar\{P\}\; \backslash right\backslash rangle$ and $CP\backslash left|\; \backslash bar\{P\}\; \backslash right\backslash rangle\; =\; e^\{-i\backslash delta\}\; \backslash left|\; P\; \backslash right\backslash rangle$), 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:{{cite web |last1=Kooijman |first1=P. |last2=Tuning |first2=N. |year=2012 |title=CP violation |url=http://www.nikhef.nl/~h71/Lectures/2012/cp-080212.pdf}}

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

The probability of $\backslash left|\; P\; \backslash right\backslash rangle$ decaying to $\backslash left|\; f\; \backslash right\backslash rangle$ is given by,

: $$

\wp_{P \to f} \left( t \right) =

\left| \left\langle f | P\left( t \right) \right\rangle \right|^2 =

\left| g_+ \left( t \right) A_f - \frac{q}{p} g_- \left( t \right) \bar{A}_f \right|^2

,

and that of its CP conjugate process by,

: $$

\wp_{\bar{P} \to \bar{f}}\left( t \right) =

\left| \left\langle \bar{f} | \bar{P} \left( t \right) \right\rangle \right|^2 =

\left| g_+ \left( t \right) \bar{A}_\bar{f} - \frac{p}{q} g_- \left( t \right) A_\bar{f} \right|^2

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

Now, the above two probabilities are unequal if,

{{Equation box 1

| equation = $\backslash left|\; \backslash frac\{\backslash bar\{A\}\_\backslash bar\{f\}\}\{A\_f\}\; \backslash right|\; \backslash ne\; 1$ and $\backslash left|\; \backslash frac\{A\_\backslash bar\{f\}\}\{\backslash bar\{A\_f\}\}\; \backslash right|\; \backslash ne\; 1$

| ref=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.

The probability (as a function of time) of observing $\backslash left|\; \backslash bar\{P\}\; \backslash right\backslash rangle$ starting from $\backslash left|\; P\; \backslash right\backslash rangle$ is given by,

: $$

\wp_{P \to \bar{P}} \left( t \right) =

\left| \left\langle {\bar{P}} | P\left( t \right) \right\rangle \right|^2 =

\left| \frac{q}{p} g_- \left( t \right) \right|^2

,

and that of its CP conjugate process by,

: $$

\wp_{\bar{P} \to P} \left( t \right) =

\left| \left\langle P | \bar{P}\left( t \right) \right\rangle \right|^2 =

\left| \frac{p}{q} g_- \left( t \right) \right|^2

.

The above two probabilities are unequal if,

{{Equation box 1

|equation = $\backslash left|\; \backslash frac\{q\}\{p\}\; \backslash right|\; \backslash ne\; 1$

|ref=11

}}

Hence, the particle-antiparticle oscillation becomes a CP violating process as the particle and its antiparticle (say, $\backslash left|\; P\; \backslash right\backslash rangle$ and $\backslash left|\; \{\backslash bar\{P\}\}\; \backslash right\backslash rangle$ respectively) are no longer equivalent eigenstates of CP.

Let $\backslash left|\; f\; \backslash right\backslash rangle$ be a final state (a CP eigenstate) that both $\backslash left|\; P\; \backslash right\backslash rangle$ and $\backslash left|\; \backslash bar\{P\}\; \backslash right\backslash rangle$ can decay to. Then, the decay probabilities are given by,

: $\backslash begin\{align\}$

\operatorname{\mathcal P}_{P \to f} \left( t \right)

&= \Bigl| \left\langle f | P( t ) \right\rangle \Bigr|^2 \\

&= \Bigl| A_f \Bigr|^2 \tfrac{1}{2} e^{-\gamma t} \left[

\ \left( 1 + \left| \lambda_f \right|^2 \right) \cosh\!\left( \tfrac{1}{2} \Delta\gamma t \right)

+ 2\ \operatorname\mathcal{R_e}\!\left\{\ \lambda_f\ \right\}\ \sinh\!\left( \tfrac{1}{2}\Delta\gamma t \right)

+ \left( 1 - \left| \lambda_f \right|^2 \right) \cos\!\left( \Delta mt \right)

+ 2\ \operatorname\mathcal{I_m}\!\left\{\ \lambda_f\ \right\}\ \sin\!\left( \Delta mt \right)

\ \right] \\

\end{align}

and,

:$\backslash begin\{align\}$

\operatorname{\mathcal P}_{\bar{P} \to f}( t )

&= \Bigl| \left\langle f | \bar{P}( t ) \right\rangle \Bigr|^2 \\

&= \Bigl| A_f \Bigr|^2 \left| \frac{p}{q} \right|^2 \tfrac{1}{2} e^{-\gamma t} \left[

\ \left( 1 + \left| \lambda_f \right|^2 \right) \cosh\!\left( \tfrac{1}{2}\Delta\gamma t \right)

+ 2\ \operatorname{\mathcal R_e}\!\left\{\ \lambda_f\ \right\}\ \sinh\!\left( \tfrac{1}{2}\Delta\gamma t \right)

- \left( 1 - \left| \lambda_f \right|^2 \right) \cos\left( \Delta mt \right)

- 2\ \operatorname{\mathcal I_m}\!\left\{\ \lambda_f\ \right\}\ \sin\left( \Delta mt \right)

\ \right] \\

\end{align}

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

{{Equation box 1

|equation = $\backslash operatorname\{\backslash mathcal\; I\_m\}\backslash !\backslash left\backslash \{\backslash \; \backslash lambda\_f\backslash \; \backslash right\backslash \}\backslash \; =\backslash \; \backslash operatorname\{\backslash mathcal\; I\_m\}\backslash !\backslash left\backslash \{\backslash \; \backslash frac\{q\}\{p\}\backslash frac\{\backslash bar\{A\}\_f\}\{A\_f\}\backslash \; \backslash right\backslash \}\; \backslash ne\; 0$

|ref=12

}}

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

Usually, an alternative classification of CP violation is made:

{{Main|Neutrino oscillation}}

Considering a strong coupling between two flavor eigenstates of neutrinos (for example, {{SubatomicParticle|Electron Neutrino}}–{{SubatomicParticle|Muon Neutrino}}, {{SubatomicParticle|Muon Neutrino}}–{{SubatomicParticle|Tau Neutrino}}, etc.) and a very weak coupling between the third (that is, the third does not affect the interaction between the other two), equation ({{EquationNote|6}}) gives the probability of a neutrino of type $\backslash alpha$ transmuting into type $\backslash beta$ as,

: $P\_\{\backslash beta\backslash alpha\}\; \backslash left(\; t\; \backslash right)\; =\; \backslash sin^2\backslash theta\; \backslash sin^2\backslash left(\; \backslash frac\{E\_+\; -\; E\_-\}\{2\backslash hbar\}t\; \backslash right)$

where, $E\_+$ and $E\_-$ are energy eigenstates.

The above can be written as,

{{Equation box 1

|equation = $P\_\{\backslash beta\backslash alpha\}\; \backslash left(\; x\; \backslash right)\; =$

\sin^2\theta \sin^2\left( \frac{\Delta m^2 c^3}{4E\hbar}x \right) =

\sin^2\theta \sin^2\left( \frac{2\pi}{\lambda_\text{osc}}x \right)

|ref=13

}}

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.

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

: $\backslash begin\{align\}$

\left( \Delta m^2 \right)_{12} &= {m_1}^2 - {m_2}^2 \\

\left( \Delta m^2 \right)_{23} &= {m_2}^2 - {m_3}^2 \\

\left( \Delta m^2 \right)_{31} &= {m_3}^2 - {m_1}^2

\end{align}

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

This implies that two of the three neutrinos have very closely placed masses. Since only two of the three $\backslash Delta\; m^2$ are independent, and the expression for probability in equation ({{EquationNote|13}}) is not sensitive to the sign of $\backslash Delta\; m^2$ (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.

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

• If $x/\backslash lambda\_\backslash text\{osc\}\; \backslash ll\; 1$, then $P\_\{\backslash beta\backslash alpha\}\; \backslash simeq\; 0$

and oscillation will not be observed. For example, production (say, by radioactive decay) and detection of neutrinos in a laboratory.

• If $x/\backslash lambda\_\backslash text\{osc\}\; \backslash simeq\; n$, where $n$ is a whole number, then $P\_\{\backslash beta\backslash alpha\}\; \backslash simeq\; 0$ and oscillation will not be observed.
• In all other cases, oscillation will be observed. For example, $x/\backslash lambda\_\backslash text\{osc\}\; \backslash gg\; 1$ for solar neutrinos; $x\; \backslash sim\; \backslash lambda\_\backslash text\{osc\}$ for neutrinos from nuclear power plant detected in a laboratory few kilometers away.

{{Main|Kaon}}

The 1964 paper by Christenson et al.{{cite journal |last1=Christenson |first1=J.H. |last2=Cronin |first2=J.W. |last3=Fitch |first3=V.L. |last4=Turlay |first4=R. |date=1964 |title=Evidence for the 2π decay of the K{{su|b=2|p=0}} meson |journal=Physical Review Letters |volume=13 |issue=4 |pages=138–140 |bibcode=1964PhRvL..13..138C |doi=10.1103/PhysRevLett.13.138 |doi-access=free}} 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.

$\backslash left|\; K^0\; \backslash right\backslash rangle$ and $\backslash left|\; \backslash bar\{K\}^0\; \backslash right\backslash rangle$ being the strangeness eigenstates (with eigenvalues +1 and −1 respectively), the energy eigenstates are,

: $\backslash begin\{align\}$

\left| K_{^1}^0 \right\rangle &= \frac{1}{\sqrt{2}} \left(\left| K^0 \right\rangle + \left| \bar{K}^0 \right\rangle\right) \\

\left| K_2^0 \right\rangle &= \frac{1}{\sqrt{2}}\left( \left| K^0 \right\rangle - \left| \bar{K}^0 \right\rangle \right)

\end{align}

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 $\backslash left|\; K\_\{^1\}^0\; \backslash right\backslash rangle$ 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.
• $\backslash left|\; K\_2^0\; \backslash right\backslash rangle$ 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, $\backslash left|\; K\_\{^1\}^0\; \backslash right\backslash rangle$ was referred to as the short-lived Kaon $\backslash left|\; K\_S^0\; \backslash right\backslash rangle$, and $\backslash left|\; K\_2^0\; \backslash right\backslash rangle$ as the long-lived Kaon $\backslash left|\; K\_L^0\; \backslash right\backslash rangle$. The 1964 experiment showed that contrary to what was expected, $\backslash left|\; K\_L^0\; \backslash right\backslash rangle$ could decay to two pions. This implied that the long lived Kaon cannot be purely the CP eigenstate $\backslash left|\; K\_2^0\; \backslash right\backslash rangle$, but must contain a small admixture of $\backslash left|\; K\_\{^1\}^0\; \backslash right\backslash rangle$, thereby no longer being a CP eigenstate. Similarly, the short-lived Kaon was predicted to have a small admixture of $\backslash left|\; K\_2^0\; \backslash right\backslash rangle$. That is,

: $\backslash begin\{align\}$

\left| K_L^0 \right\rangle &= \frac{1}{\sqrt{1 + \left| \varepsilon \right|^2}}

\left( \left| K_2^0 \right\rangle + \varepsilon \left| K_1^0 \right\rangle \right) \\

\left| K_S^0 \right\rangle &= \frac{1}{\sqrt{1 + \left| \varepsilon \right|^2}}

\left( \left| K_1^0 \right\rangle + \varepsilon \left| K_2^0 \right\rangle \right)

\end{align}

where, $\backslash varepsilon$ is a complex quantity and is a measure of departure from CP invariance. Experimentally, $\backslash left|\; \backslash varepsilon\; \backslash right|\; =\; \backslash left(\; 2.228\; \backslash pm\; 0.011\; \backslash right)\backslash times\; 10^\{-3\}$.{{cite journal |last1=Olive |first1=K.A. |display-authors=etal |collaboration=Particle Data Group |year=2014 |title=Review of Particle Physics – Strange mesons |url=http://pdg.lbl.gov/2014/tables/rpp2014-tab-mesons-strange.pdf |journal=Chinese Physics C |volume=38 |issue=9 |page=090001 |bibcode=2014ChPhC..38i0001O |doi=10.1088/1674-1137/38/9/090001|s2cid=260537282 }}

Writing $\backslash left|\; K\_\{^1\}^0\; \backslash right\backslash rangle$ and $\backslash left|\; K\_2^0\; \backslash right\backslash rangle$ in terms of $\backslash left|\; K^0\; \backslash right\backslash rangle$ and $\backslash left|\; \backslash bar\{K\}^0\; \backslash right\backslash rangle$, we obtain (keeping in mind that $m\_\{K\_L^0\}\; >\; m\_\{K\_S^0\}$) the form of equation ({{EquationNote|9}}):

: $\backslash begin\{align\}$

\left| K_L^0 \right\rangle &= \left( p\left| K^0 \right\rangle - q\left| \bar{K}^0 \right\rangle \right) \\

\left| K_S^0 \right\rangle &= \left( p\left| K^0 \right\rangle + q\left| \bar{K}^0 \right\rangle \right)

\end{align}

where, $\backslash frac\{q\}\{p\}\; =\; \backslash frac\{1\; -\; \backslash varepsilon\}\{1\; +\; \backslash varepsilon\}$.

Since $\backslash left|\; \backslash varepsilon\; \backslash right|\backslash ne\; 0$, condition ({{EquationNote|11}}) is satisfied and there is a mixing between the strangeness eigenstates $\backslash left|\; K^0\; \backslash right\backslash rangle$ and $\backslash left|\; \backslash bar\{K\}^0\; \backslash right\backslash rangle$ giving rise to a long-lived and a short-lived state.

The {{SubatomicParticle|K-long0}} and {{SubatomicParticle|K-short0}} have two modes of two pion decay: {{SubatomicParticle|pion0}}{{SubatomicParticle|pion0}} or {{SubatomicParticle|pion+}}{{SubatomicParticle|pion-}}. Both of these final states are CP eigenstates of themselves. We can define the branching ratios as,

: $\backslash begin\{align\}$

\eta_{+-} &=

\frac{\left\langle \pi^+\pi^- | K_L^0 \right\rangle}{\left\langle \pi^+\pi^- | K_S^0 \right\rangle} =

\frac{pA_{\pi^+\pi^-} - q\bar{A}_{\pi^+\pi^-}}{pA_{\pi^+\pi^-} + q\bar{A}_{\pi^+\pi^-}} =

\frac{1 - \lambda_{\pi^+\pi^-}}{1 + \lambda_{\pi^+\pi^-}} \\[3pt]

\eta_{00} &=

\frac{\left\langle \pi^0\pi^0 | K_L^0 \right\rangle}{\left\langle \pi^0\pi^0 | K_S^0 \right\rangle} =

\frac{pA_{\pi^0\pi^0} - q\bar{A}_{\pi^0\pi^0}}{pA_{\pi^0\pi^0} + q\bar{A}_{\pi^0\pi^0}} =

\frac{1 - \lambda_{\pi^0\pi^0}}{1 + \lambda_{\pi^0\pi^0}}

\end{align}.

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

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

If the final state (say $f\_\{CP\}$) is a CP eigenstate (for example {{SubatomicParticle|pion+}}{{SubatomicParticle|pion-}}), then there are two different decay amplitudes corresponding to two different decay paths:{{cite arXiv |last=Pich |first=A. |year=1993 |title=CP violation |eprint=hep-ph/9312297}}

: $\backslash begin\{align\}$

K^0 &\to f_{CP} \\

K^0 &\to \bar{K}^0 \to f_{CP}

\end{align}.

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.

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:{{cite book |last=Griffiths |first=D.J. |year=2008 |title=Elementary Particles |page=147 |edition=2nd, Revised |publisher=Wiley-VCH |isbn=978-3-527-40601-2}}

{{Quotation|

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 ({{SubatomicParticle|Kaon0}} and {{SubatomicParticle|AntiKaon0}}), 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 {{SubatomicParticle|Kaon0}} 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.

}}

{{Main|Cabibbo–Kobayashi–Maskawa matrix|Pontecorvo–Maki–Nakagawa–Sakata matrix}}

If the system is a three state system (for example, three species of neutrinos {{math|{{SubatomicParticle|Electron Neutrino}} ⇄ {{SubatomicParticle|Muon Neutrino}} ⇄ {{SubatomicParticle|Tau Neutrino}}}}, three species of quarks {{math|{{SubatomicParticle|down quark}} ⇄ {{SubatomicParticle|strange quark}} ⇄ {{SubatomicParticle|bottom quark}} }}), then, just like in the two state system, the flavor eigenstates (say $$

\left| {\varphi_\alpha} \right\rangle, $$

\left| {\varphi_\beta} \right\rangle, $$

\left| {\varphi_\gamma} \right\rangle

) are written as a linear combination of the energy (mass) eigenstates (say $$

\left| \psi_1 \right\rangle, $$

\left| \psi_2 \right\rangle, $$

\left| \psi_3 \right\rangle

). That is,

:$$

\begin{pmatrix}

\left| {\varphi_\alpha} \right\rangle \\

\left| {\varphi_\beta} \right\rangle \\

\left| {\varphi_\gamma} \right\rangle \\

\end{pmatrix} = \begin{pmatrix}

\Omega_{\alpha 1} & \Omega_{\alpha 2} & \Omega_{\alpha 3} \\

\Omega_{\beta 1} & \Omega_{\beta 2} & \Omega_{\beta 3} \\

\Omega_{\gamma 1} & \Omega_{\gamma 2} & \Omega_{\gamma 3} \\

\end{pmatrix}\begin{pmatrix}

\left| \psi_1 \right\rangle \\

\left| \psi_2 \right\rangle \\

\left| \psi_3 \right\rangle \\

\end{pmatrix}

.

In case of leptons (neutrinos for example) the transformation matrix is the PMNS matrix, and for quarks it is the CKM matrix.{{cite book |last=Griffiths |first=D.J. |year=2008 |title=Elementary Particles |page=397 |edition=2nd, revised |publisher=Wiley-VCH |isbn=978-3-527-40601-2}}{{efn|N.B.: The three familiar neutrino species {{math|{{SubatomicParticle|Electron Neutrino}}, {{SubatomicParticle|Muon Neutrino}}, and {{SubatomicParticle|Tau Neutrino}}}}, are flavor eigenstates, whereas the three familiar quarks species {{math|{{SubatomicParticle|down quark}}, {{SubatomicParticle|strange quark}}, and {{SubatomicParticle|bottom quark}}}}, are energy eigenstates.}}

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.

• CKM matrix
• CP violation
• CPT symmetry
• Kaon
• PMNS matrix
• Neutral current
• Flavor-changing neutral current
• Rabi cycle

{{notelist}}

{{reflist|25em}}

{{DEFAULTSORT:Neutral Particle Oscillation}}

Category:Particle physics

Category:Standard Model

Category:Murray Gell-Mann