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 do regularly occur for other neutral particles, and are classified into two types:
Particle–antiparticle oscillation (for example, K0 ⇄ K0 oscillation, B0 ⇄ B0 oscillation, D0 ⇄ D0 oscillation). Flavor oscillation (for example, νe ⇄ νμ ⇄ ντ 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 symmetry that is conserved. However, in 1964 Cronin and Fitch reported CP violation in the neutral kaon system. They observed the long-lived KL (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 two states of a single particle.
Solar neutrino problem The pp chain in the sun produces an abundance of νe. 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 νe to produce argon via the reaction
ν e +
17 37 C l →
18 37 A r + e − {\displaystyle \mathrm {\nu _{e}+{{}_{17}^{37}Cl}\rightarrow {{}_{18}^{37}}Ar+e^{-}} } , which is essentially
ν e + n → p + e − {\displaystyle \mathrm {\nu _{e}+n\to p+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 νe (produced in the sun) can transform into some other neutrino species (νμ or ντ), 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 νe 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
Special case that only considers mixing
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 H 0 {\displaystyle H_{0}} be the Hamiltonian of the two-state system, and | 1 ⟩ {\displaystyle \left|1\right\rangle } and | 2 ⟩ {\displaystyle \left|2\right\rangle } be its orthonormal eigenvectors with eigenvalues E 1 {\displaystyle E_{1}} and E 2 {\displaystyle E_{2}} respectively. Let | Ψ ( t ) ⟩ {\displaystyle \left|\Psi (t)\right\rangle } be the state of the system at time t {\displaystyle t} . If the system starts as an energy eigenstate of H 0 {\displaystyle H_{0}} , for example, say
| Ψ ( 0 ) ⟩ = | 1 ⟩ , {\displaystyle \ \left|\Psi (0)\right\rangle =\left|1\right\rangle \ ,}
then the time evolved state, which is the solution of the Schrödinger equation
will be
| Ψ ( t ) ⟩ = | 1 ⟩ e − i E 1 t ℏ {\displaystyle \ \left|\Psi (t)\right\rangle \ =\ \left|1\right\rangle e^{-i\ {\frac {E_{1}t}{\hbar }}}\ }
But this is physically same as | 1 ⟩ , {\displaystyle \ \left|1\right\rangle \ ,} 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 { | 1 ⟩ , | 2 ⟩ } {\displaystyle \left\{\left\vert 1\right\rangle ,\left\vert 2\right\rangle \right\}} to be a basis in which the unperturbed Hamiltonian operator, H 0 {\displaystyle H_{0}} , is diagonal:
H 0 = ( E 1 0 0 E 2 ) = E 1 | 1 ⟩ + E 2 | 2 ⟩ {\displaystyle \ H_{0}={\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 W {\displaystyle W} imposed on H 0 {\displaystyle H_{0}} such that the resultant Hamiltonian H {\displaystyle H} is still Hermitian. Then
W = ( W 11 W 12 W 12 ∗ W 22 ) {\displaystyle W={\begin{pmatrix}W_{11}&W_{12}\\W_{12}^{*}&W_{22}\\\end{pmatrix}}\ }
where W 11 , W 22 ∈ R {\displaystyle W_{11},W_{22}\in \mathbb {R} } and W 12 ∈ C {\displaystyle W_{12}\in \mathbb {C} } and
The eigenvalues of the perturbed Hamiltonian, H {\displaystyle H} , then change to E + {\displaystyle E_{+}} and E − {\displaystyle E_{-}} , where
Since H {\displaystyle H} is a general Hamiltonian matrix, it can be written as
H = ∑ j = 0 3 a j σ j = a 0 σ 0 + H ′ {\displaystyle H=\sum \limits _{j=0}^{3}a_{j}\sigma _{j}=a_{0}\sigma _{0}+H'}
The following two results are clear:
[ H , H ′ ] = 0 {\displaystyle \left[H,H'\right]=0}
H ′ 2 = I {\displaystyle {H'}^{2}=I}
With the following parametrization (this parametrization helps as it normalizes the eigenvectors and also introduces an arbitrary phase ϕ {\displaystyle \phi } making the eigenvectors most general)
n ^ = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) {\displaystyle {\hat {n}}=\left(\ \sin \theta \cos \phi \ ,\ \sin \theta \sin \phi \ ,\ \cos \theta \ \right)}
and using the above pair of results the orthonormal eigenvectors of H ′ {\displaystyle H'} and consequently those of H {\displaystyle H} are obtained as
Writing the eigenvectors of H 0 {\displaystyle H_{0}} in terms of those of H {\displaystyle H} we get
Now if the particle starts out as an eigenstate of H 0 {\displaystyle H_{0}} (say, | 1 ⟩ {\displaystyle \left\vert 1\right\rangle } ), that is
| Ψ ( 0 ) ⟩ = | 1 ⟩ {\displaystyle \left|\ \Psi (0)\ \right\rangle \ =\ \left|1\right\rangle }
then under time evolution we get
| Ψ ( t ) ⟩ = e i ϕ 2 ( cos θ 2 | + ⟩ e − i E + t ℏ − sin θ 2 | − ⟩ e − i E − t ℏ ) {\displaystyle \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 | 1 ⟩ {\displaystyle \left\vert 1\right\rangle } . We can then obtain the probability of finding the system in state | 2 ⟩ {\displaystyle \left|2\right\rangle } at time t {\displaystyle t} as
which is called Rabi's formula. Hence, starting from one eigenstate of the unperturbed Hamiltonian H 0 {\displaystyle H_{0}} , the state of the system oscillates between the eigenstates of H 0 {\displaystyle H_{0}} with a frequency (known as Rabi frequency),
From equation (6), for P 21 ( t ) {\displaystyle P_{21}\!(t)} , we can conclude that oscillation will exist only if | W 12 | 2 ≠ 0 {\displaystyle \left\vert W_{12}\right\vert ^{2}\neq 0} . So W 12 {\displaystyle \ W_{12}\ } is known as the coupling term as it connects the two eigenstates of the unperturbed Hamiltonian H 0 {\displaystyle H_{0}} and thereby facilitates oscillation between the two. Oscillation will also cease if the eigenvalues of the perturbed Hamiltonian H {\displaystyle H} are degenerate, i.e. E + = E − {\displaystyle E_{+}=E_{-}} . But this is a trivial case as in such a situation, the perturbation itself vanishes and H {\displaystyle H} takes the form (diagonal) of H 0 {\displaystyle H_{0}} and we're back to square one. Hence, the necessary conditions for oscillation are:
Non-zero coupling, i.e. | W 12 | 2 ≠ 0 {\displaystyle \left\vert W_{12}\right\vert ^{2}\neq 0} . Non-degenerate eigenvalues of the perturbed Hamiltonian H {\displaystyle H} , i.e. E + ≠ E − {\displaystyle E_{+}\neq E_{-}} .
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, H {\displaystyle \ H\ } can be written as,
H = M − i 2 Γ = ( M 11 M 12 M 12 ∗ M 11 ) − i 2 ( Γ 11 Γ 12 Γ 12 ∗ Γ 11 ) {\displaystyle \ H\;=\;M-{\frac {i}{2}}\ \Gamma \;=\;{\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}}\ }
The eigenvalues of H {\displaystyle \ H\ } are
The suffixes stand for Heavy and Light respectively (by convention) and this implies that Δ m {\displaystyle \Delta m} is positive. The normalized eigenstates corresponding to μ L {\displaystyle \mu _{\mathsf {L}}} and μ H {\displaystyle \mu _{\mathsf {H}}} respectively, in the natural basis { | P ⟩ , | P ¯ ⟩ } ≡ { ( 1 , 0 ) , ( 0 , 1 ) } {\displaystyle {\bigl \{}\left|P\right\rangle \ ,\ \left|{\bar {P}}\right\rangle {\bigr \}}~\equiv ~{\bigl \{}\ (1,0)\ ,\ (0,1)\ {\bigr \}}} are
p {\displaystyle p} and q {\displaystyle q} are the mixing terms. Note that these eigenstates are no longer orthogonal. Let the system start in the state | P ⟩ {\displaystyle \left\vert P\right\rangle } .That is
| P ( 0 ) ⟩ = | P ⟩ = 1 2 p ( | P L ⟩ + | P H ⟩ ) {\displaystyle \ \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
| P ( t ) ⟩ = 1 2 p ( | P L ⟩ e − i ℏ ( m L − i 2 γ L ) t + | P H ⟩ e − i ℏ ( m H − i 2 γ H ) t ) = g + ( t ) | P ⟩ − q p g − ( t ) | P ¯ ⟩ {\displaystyle \ \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 | P ¯ ⟩ {\displaystyle \left|{\bar {P}}\right\rangle } , under time evolution we obtain
| P ¯ ( t ) ⟩ = 1 2 q ( | P L ⟩ e − i ℏ ( m L − i 2 γ L ) t − | P H ⟩ e − i ℏ ( m H − i 2 γ H ) t ) = − p q g − ( t ) | P ⟩ + g + ( t ) | P ¯ ⟩ {\displaystyle \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 }
CP violation as a consequence If in a system | P ⟩ {\displaystyle \left|P\right\rangle } and | P ¯ ⟩ {\displaystyle \left|{\bar {P}}\right\rangle } represent CP conjugate states (i.e. particle-antiparticle) of one another (i.e. C P | P ⟩ = e i δ | P ¯ ⟩ {\displaystyle CP\left|P\right\rangle =e^{i\delta }\left|{\bar {P}}\right\rangle } and C P | P ¯ ⟩ = e − i δ | P ⟩ {\displaystyle CP\left|{\bar {P}}\right\rangle =e^{-i\delta }\left|P\right\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:
CP violation through decay only Consider the processes where { | P ⟩ , | P ¯ ⟩ } {\displaystyle \left\{\left|P\right\rangle ,\left|{\bar {P}}\right\rangle \right\}} decay to final states { | f ⟩ , | f ¯ ⟩ } {\displaystyle \left\{\left|f\right\rangle ,\left|{\bar {f}}\right\rangle \right\}} , where the barred and the unbarred kets of each set are CP conjugates of one another. The probability of | P ⟩ {\displaystyle \left|P\right\rangle } decaying to | f ⟩ {\displaystyle \left|f\right\rangle } is given by,
℘ P → f ( t ) = | ⟨ f | P ( t ) ⟩ | 2 = | g + ( t ) A f − q p g − ( t ) A ¯ f | 2 {\displaystyle \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,
℘ P ¯ → f ¯ ( t ) = | ⟨ f ¯ | P ¯ ( t )