Neutrino oscillation

From Wikipedia, the free encyclopedia

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

Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons

Neutrino oscillation is a quantum mechanical phenomenon in which a neutrino created with a specific lepton family number ("lepton flavor": electron, muon, or tau) can later be measured to have a different lepton family number. The probability of measuring a particular flavor for a neutrino varies between three known states as it propagates through space.

First predicted by Bruno Pontecorvo in 1957, neutrino oscillation has since been observed by a multitude of experiments in several different contexts. Most notably, the existence of neutrino oscillation resolved the long-standing solar neutrino problem.

Neutrino oscillation is of great theoretical and experimental interest, as the precise properties of the process can shed light on several properties of the neutrino. In particular, it implies that the neutrino has a non-zero mass, which requires a modification to the Standard Model of particle physics. The experimental discovery of neutrino oscillation, and thus neutrino mass, by the Super-Kamiokande Observatory and the Sudbury Neutrino Observatories was recognized with the 2015 Nobel Prize for Physics.

Observations

A great deal of evidence for neutrino oscillation has been collected from many sources, over a wide range of neutrino energies and with many different detector technologies. The 2015 Nobel Prize in Physics was shared by Takaaki Kajita and Arthur B. McDonald for their early pioneering observations of these oscillations.

Neutrino oscillation is a function of the ratio L/E, where L is the distance traveled and E is the neutrino's energy (details in § Propagation and interference below). All available neutrino sources produce a range of energies, and oscillation is measured at a fixed distance for neutrinos of varying energy. The limiting factor in measurements is the accuracy with which the energy of each observed neutrino can be measured. Because current detectors have energy uncertainties of a few percent, it is satisfactory to know the distance to within 1%.

Solar neutrino oscillation

The first experiment that detected the effects of neutrino oscillation was Ray Davis' Homestake experiment in the late 1960s, in which he observed a deficit in the flux of solar neutrinos with respect to the prediction of the Standard Solar Model, using a chlorine-based detector. This gave rise to the solar neutrino problem. Many subsequent radiochemical and water Cherenkov detectors confirmed the deficit, but neutrino oscillation was not conclusively identified as the source of the deficit until the Sudbury Neutrino Observatory provided clear evidence of neutrino flavor change in 2001.

Solar neutrinos have energies below 20 MeV. At energies above 5 MeV, solar neutrino oscillation actually takes place in the Sun through a resonance known as the MSW effect, a different process from the vacuum oscillation described later in this article.

Atmospheric neutrino oscillation

Following the theories that were proposed in the 1970s suggesting unification of electromagnetic, weak, and strong forces, a few experiments on proton decay followed in the 1980s. Large detectors such as IMB, MACRO, and Kamiokande II have observed a deficit in the ratio of the flux of muon to electron flavor atmospheric neutrinos (see Muon § Muon decay). The Super-Kamiokande experiment provided a very precise measurement of neutrino oscillation in an energy range of hundreds of MeV to a few TeV, and with a baseline of the diameter of the Earth; the first experimental evidence for atmospheric neutrino oscillations was announced in 1998.

Reactor neutrino oscillation

Illustration of neutrino oscillations.

Many experiments have searched for oscillation of electron anti-neutrinos produced in nuclear reactors. No oscillations were found until a detector was installed at a distance 1–2 km. Such oscillations give the value of the parameter θ₁₃. Neutrinos produced in nuclear reactors have energies similar to solar neutrinos, of around a few MeV. The baselines of these experiments have ranged from tens of meters to over 100 km (parameter θ₁₂). Mikaelyan and Sinev proposed to use two identical detectors to cancel systematic uncertainties in reactor experiment to measure the parameter θ₁₃.

In December 2011, the Double Chooz experiment indicated that θ₁₃ ≠ 0. Then, in 2012, the Daya Bay experiment found that θ₁₃ ≠ 0 with a significance of 5.2 σ; these results have since been confirmed by RENO.

The experiment Neutrino-4 started in 2014 with a detector model and continued with a full-scale detector in 2016–2021 obtained the result of the direct observation of the oscillation effect at parameter region Δm²
₁₄ = (7.3 ± 0.13_st ± 1.16_syst) (eV/c²)² and sin²2θ₁₄ = 0.36 ± 0.12_stat (2.9 σ). The simulation showed the expected detector signal for the case of oscillation detection.

Beam neutrino oscillation

Neutrino beams produced at a particle accelerator offer the greatest control over the neutrinos being studied. Many experiments have taken place that study the same oscillations as in atmospheric neutrino oscillation using neutrinos with a few GeV of energy and several-hundred-kilometre baselines. The MINOS, K2K, and Super-K experiments have all independently observed muon neutrino disappearance over such long baselines.

Data from the LSND experiment appear to be in conflict with the oscillation parameters measured in other experiments. Results from the MiniBooNE appeared in Spring 2007 and contradicted the results from LSND, although they could support the existence of a fourth neutrino type, the sterile neutrino.

In 2010, the INFN and CERN announced the observation of a tauon particle in a muon neutrino beam in the OPERA detector located at Gran Sasso, 730 km away from the source in Geneva.

T2K, using a neutrino beam directed through 295 km of earth and the Super-Kamiokande detector, measured a non-zero value for the parameter θ₁₃ in a neutrino beam. NOνA, using the same beam as MINOS with a baseline of 810 km, is sensitive to the same.

Theory

Neutrino oscillation arises from mixing between the flavor and mass eigenstates of neutrinos. That is, the three neutrino states that interact with the charged leptons in weak interactions are each a different superposition of the three (propagating) neutrino states of definite mass. Neutrinos are produced and detected in weak interactions as flavour eigenstates but propagate as coherent superpositions of mass eigenstates.

As a neutrino superposition propagates through space, the quantum mechanical phases of the three neutrino mass states advance at slightly different rates, due to the slight differences in their respective masses. This results in a changing superposition mixture of mass eigenstates as the neutrino travels; but a different mixture of mass eigenstates corresponds to a different mixture of flavor states. For example, a neutrino born as an electron neutrino will be some mixture of electron, mu, and tau neutrino after traveling some distance. Since the quantum mechanical phase advances in a periodic fashion, after some distance the state will nearly return to the original mixture, and the neutrino will be again mostly electron neutrino. The electron flavor content of the neutrino will then continue to oscillate – as long as the quantum mechanical state maintains coherence. Since mass differences between neutrino flavors are small in comparison with long coherence lengths for neutrino oscillations, this microscopic quantum effect becomes observable over macroscopic distances.

In contrast, due to their larger masses, the charged leptons (electrons, muons, and tau leptons) have never been observed to oscillate. In nuclear beta decay, muon decay, pion decay, and kaon decay, when a neutrino and a charged lepton are emitted, the charged lepton is emitted in incoherent mass eigenstates such as | e⁻
⟩, because of its large mass. Weak-force couplings compel the simultaneously emitted neutrino to be in a "charged-lepton-centric" superposition such as | ν
_e ⟩, which is an eigenstate for a "flavor" that is fixed by the electron's mass eigenstate, and not in one of the neutrino's own mass eigenstates. Because the neutrino is in a coherent superposition that is not a mass eigenstate, the mixture that makes up that superposition oscillates significantly as it travels. No analogous mechanism exists in the Standard Model that would make charged leptons detectably oscillate. In the four decays mentioned above, where the charged lepton is emitted in a unique mass eigenstate, the charged lepton will not oscillate, as single mass eigenstates propagate without oscillation.

The case of (real) W boson decay is more complicated: W boson decay is sufficiently energetic to generate a charged lepton that is not in a mass eigenstate; however, the charged lepton would lose coherence, if it had any, over interatomic distances (0.1 nm) and would thus quickly cease any meaningful oscillation. More importantly, no mechanism in the Standard Model is capable of pinning down a charged lepton into a coherent state that is not a mass eigenstate, in the first place; instead, while the charged lepton from the W boson decay is not initially in a mass eigenstate, neither is it in any "neutrino-centric" eigenstate, nor in any other coherent state. It cannot meaningfully be said that such a featureless charged lepton oscillates or that it does not oscillate, as any "oscillation" transformation would just leave it the same generic state that it was before the oscillation. Therefore, detection of a charged lepton oscillation from W boson decay is infeasible on multiple levels.

Pontecorvo–Maki–Nakagawa–Sakata matrix

Main article: Pontecorvo–Maki–Nakagawa–Sakata matrix

The idea of neutrino oscillation was first put forward in 1957 by Bruno Pontecorvo, who proposed that neutrino–antineutrino transitions may occur in analogy with neutral kaon mixing. Although such matter–antimatter oscillation had not been observed, this idea formed the conceptual foundation for the quantitative theory of neutrino flavor oscillation, which was first developed by Maki, Nakagawa, and Sakata in 1962 and further elaborated by Pontecorvo in 1967. One year later the solar neutrino deficit was first observed, and that was followed by the famous article by Gribov and Pontecorvo published in 1969 titled "Neutrino astronomy and lepton charge".

The concept of neutrino mixing is a natural outcome of gauge theories with massive neutrinos, and its structure can be characterized in general. In its simplest form it is expressed as a unitary transformation relating the flavor and mass eigenbasis and can be written as

${\displaystyle |{\nu }_i⟩=\sum _{\alpha }{U}_{\alpha i}^{\ast }|{\nu }_{\alpha }⟩}$

${\displaystyle |{\nu }_{\alpha }⟩=\sum _i{U}_{\alpha i}|{\nu }_i⟩}$

where

${\displaystyle |{\nu }_{\alpha }⟩}$ is a neutrino with definite flavor ${\displaystyle \alpha }$ = e (electron), μ (muon) or τ (tauon)

${\displaystyle |{\nu }_i⟩}$ is a neutrino with definite mass ${\displaystyle m_i}$ with ${\displaystyle i}$ = 1, 2 or 3

the superscript asterisk (${\displaystyle {}^{\ast }}$) represents a complex conjugate; for antineutrinos, the complex conjugate should be removed from the first equation and inserted into the second.

The symbol ${\displaystyle U_{\alpha i}}$ represents the Pontecorvo–Maki–Nakagawa–Sakata matrix (also called the PMNS matrix, lepton mixing matrix, or sometimes simply the MNS matrix). It is the analogue of the CKM matrix describing the analogous mixing of quarks. If this matrix were the identity matrix, then the flavor eigenstates would be the same as the mass eigenstates. However, experiment shows that it is not.

When the standard three-neutrino theory is considered, the matrix is 3 × 3. If only two neutrinos are considered, a 2 × 2 matrix is used. If one or more sterile neutrinos are added (see later), it is 4 × 4 or larger. In the 3 × 3 form, it is given by

${\displaystyle \begin{array}{rl}U& =\left[\begin{array}{ccc}{U}_{e1}& U_{e2}& U_{e3}\\ U_{\mu 1}& U_{\mu 2}& U_{\mu 3}\\ U_{\tau 1}& U_{\tau 2}& U_{\tau 3}\end{array}\right]\\ & =\left[\begin{array}{ccc}1& 0& 0\\ 0& c_{23}& s_{23}\\ 0& -s_{23}& c_{23}\end{array}\right]\left[\begin{array}{ccc}{c}_{13}& 0& s_{13}{e}^{-i\delta }\\ 0& 1& 0\\ -s_{13}{e}^{i\delta }& 0& c_{13}\end{array}\right]\left[\begin{array}{ccc}{c}_{12}& s_{12}& 0\\ -s_{12}& c_{12}& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}{e}^{i{\alpha }_1/2}& 0& 0\\ 0& e^{i{\alpha }_2/2}& 0\\ 0& 0& 1\end{array}\right]\\ & =\left[\begin{array}{ccc}{c}_{12}{c}_{13}& s_{12}{c}_{13}& s_{13}{e}^{-i\delta }\\ -s_{12}{c}_{23}-c_{12}{s}_{23}{s}_{13}{e}^{i\delta }& c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}{e}^{i\delta }& s_{23}{c}_{13}\\ s_{12}{s}_{23}-c_{12}{c}_{23}{s}_{13}{e}^{i\delta }& -c_{12}{s}_{23}-s_{12}{c}_{23}{s}_{13}{e}^{i\delta }& c_{23}{c}_{13}\end{array}\right]\left[\begin{array}{ccc}{e}^{i{\alpha }_1/2}& 0& 0\\ 0& e^{i{\alpha }_2/2}& 0\\ 0& 0& 1\end{array}\right],\end{array}}$

where c_ij ≡ cos θ_ij, and s_ij ≡ sin θ_ij. The phase factors α₁ and α₂ are physically meaningful only if neutrinos are Majorana particles—i.e. if the neutrino is identical to its antineutrino (whether or not they are is unknown)—and do not enter into oscillation phenomena regardless. If neutrinoless double beta decay occurs, these factors influence its rate. The phase factor δ is non-zero only if neutrino oscillation violates CP symmetry; this has not yet been observed experimentally. If experiment shows this 3 × 3 matrix to be not unitary, a sterile neutrino or some other new physics is required.

Propagation and interference

Since ${\displaystyle |{\nu }_j⟩}$ are mass eigenstates, their propagation can be described by plane wave solutions of the form

${\displaystyle |{\nu }_j(t)⟩=e^{-i\left(E_{j}t-{\overrightarrow{p}}_j\cdot \overrightarrow{x}\right)}|{\nu }_j(0)⟩,}$

where

• quantities are expressed in natural units (${\displaystyle c=1,\hslash =1}$), and ${\displaystyle i^2\equiv -1}$,
• ${\displaystyle E_j}$ is the energy of the mass-eigenstate ${\displaystyle j}$,
• ${\displaystyle t}$ is the time from the start of the propagation,
• ${\displaystyle {\overrightarrow{p}}_j}$ is the three-dimensional momentum,
• ${\displaystyle \overrightarrow{x}}$ is the current position of the particle relative to its starting position

In the ultrarelativistic limit, ${\displaystyle \left|{\overrightarrow{p}}_j\right|=p_j\gg m_j,}$ we can approximate the energy as

${\displaystyle E_j=\sqrt{p_j^2+m_j^2}\simeq p_j+\frac{m_j^2}{2p_j}\approx E+\frac{m_j^2}{2E},}$

where E is the energy of the wavepacket (particle) to be detected.

This limit applies to all practical (currently observed) neutrinos, since their masses are less than 1 eV and their energies are at least 1 MeV, so the Lorentz factor, γ, is greater than 10⁶ in all cases. Using also t ≈ L, where L is the distance traveled and also dropping the phase factors, the wavefunction becomes

${\displaystyle |{\nu }_j(L)⟩=e^{-i{m}_j^{2}L/(2E)}|{\nu }_j(0)⟩.}$

Eigenstates with different masses propagate with different frequencies. The heavier ones oscillate faster compared to the lighter ones. Since the mass eigenstates are combinations of flavor eigenstates, this difference in frequencies causes interference between the corresponding flavor components of each mass eigenstate. Constructive interference causes it to be possible to observe a neutrino created with a given flavor to change its flavor during its propagation. The probability that a neutrino originally of flavor α will later be observed as having flavor β is

${\displaystyle P_{\alpha \to \beta }=|⟨{\nu }_{\beta }|{\nu }_{\alpha }(L)⟩{|}^2={\left|\sum _j{U}_{\alpha j}^{\ast }{U}_{\beta j}{e}^{-i{m}_j^{2}L/(2E)}\right|}^2.}$

This is more conveniently written as

${\displaystyle \begin{array}{rl}{P}_{\alpha \to \beta }={\delta }_{\alpha \beta }& -4\sum _{j>k}{\mathcal{R}}_{\mathcal{e}}\left\{U_{\alpha j}^{\ast }{U}_{\beta j}{U}_{\alpha k}{U}_{\beta k}^{\ast }\right\}{\sin }^2\left(\frac{{\mathrm{\Delta }}_{jk}{m}^{2}L}{4E}\right)\\ & +2\sum _{j>k}{\mathcal{I}}_{\mathcal{m}}\left\{U_{\alpha j}^{\ast }{U}_{\beta j}{U}_{\alpha k}{U}_{\beta k}^{\ast }\right\}\sin \left(\frac{{\mathrm{\Delta }}_{jk}{m}^{2}L}{2E}\right),\end{array}}$

where ${\displaystyle {\mathrm{\Delta }}_{jk}{m}^2\equiv m_j^2-m_k^2.}$

The phase that is responsible for oscillation is often written as (with c and ${\displaystyle \hslash }$ restored)

${\displaystyle \frac{{\mathrm{\Delta }}_{jk}(m{c}^2{)}^{2}L}{4\hslash cE}=\frac{\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{f}\mathrm{m}}{4\hslash c}\times \frac{{\mathrm{\Delta }}_{jk}{m}^2}{{\mathrm{e}\mathrm{V}}^2}\frac{L}{\mathrm{k}\mathrm{m}}\frac{\mathrm{G}\mathrm{e}\mathrm{V}}{E}\approx 1.27\times \frac{{\mathrm{\Delta }}_{jk}{m}^2}{{\mathrm{e}\mathrm{V}}^2}\frac{L}{\mathrm{k}\mathrm{m}}\frac{\mathrm{G}\mathrm{e}\mathrm{V}}{E},}$

where 1.27 is unitless. In this form, it is convenient to plug in the oscillation parameters since:

• The mass differences, Δm², are known to be on the order of 10⁻⁴ (eV/c²)² = (10⁻² eV/c²)²
• Oscillation distances, L, in modern experiments are on the order of kilometres
• Neutrino energies, E, in modern experiments are typically on order of MeV or GeV.

If there is no CP-violation (δ is zero), then the second sum is zero. Otherwise, the CP asymmetry can be given as

${\displaystyle A_{\mathsf{C}\mathsf{P}}^{(\alpha \beta )}=P({\nu }_{\alpha }\to {\nu }_{\beta })-P({\overline{\nu }}_{\alpha }\to {\overline{\nu }}_{\beta })=4\sum _{j>k}{\mathcal{I}}_{\mathcal{m}}\left\{U_{\alpha j}^{\ast }{U}_{\beta j}{U}_{\alpha k}{U}_{\beta k}^{\ast }\right\}\sin \left(\frac{{\mathrm{\Delta }}_{jk}{m}^{2}L}{2E}\right)}$

In terms of Jarlskog invariant

${\displaystyle {\mathcal{I}}_{\mathcal{m}}\left\{U_{\alpha j}^{\ast }{U}_{\beta j}{U}_{\alpha k}{U}_{\beta k}^{\ast }\right\}=J\sum _{\gamma ,\ell }{\epsilon }_{\alpha \beta \gamma }{\epsilon }_{jk\ell },}$

the CP asymmetry is expressed as

${\displaystyle A_{\mathsf{C}\mathsf{P}}^{(\alpha \beta )}=16\sin \left(\frac{{\mathrm{\Delta }}_{21}{m}^{2}L}{4E}\right)\sin \left(\frac{{\mathrm{\Delta }}_{32}{m}^{2}L}{4E}\right)\sin \left(\frac{{\mathrm{\Delta }}_{31}{m}^{2}L}{4E}\right)J\sum _{\gamma }{\epsilon }_{\alpha \beta \gamma }}$

Two-neutrino case

The above formula is correct for any number of neutrino generations. Writing it explicitly in terms of mixing angles is extremely cumbersome if there are more than two neutrinos that participate in mixing. Fortunately, there are several meaningful cases in which only two neutrinos participate significantly. In this case, it is sufficient to consider the mixing matrix

${\displaystyle U=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right).}$

Then the probability of a neutrino changing its flavor is

${\displaystyle P_{\alpha \to \beta ,\alpha \ne \beta }={\sin }^2(2\theta ){\sin }^2\left(\frac{\mathrm{\Delta }{m}^{2}L}{4E}\right)\text{ [natural units] .}}$

Or, using SI units and the convention introduced above

${\displaystyle P_{\alpha \to \beta ,\alpha \ne \beta }={\sin }^2(2\theta ){\sin }^2\left(1.27\frac{\mathrm{\Delta }{m}^{2}L}{E}\frac{[\mathrm{e}{\mathrm{V}}^2][\mathrm{k}\mathrm{m}]}{[\mathrm{G}\mathrm{e}\mathrm{V}]}\right).}$

This formula is often appropriate for discussing the transition ν_μ ↔ ν_τ in atmospheric mixing, since the electron neutrino plays almost no role in this case. It is also appropriate for the solar case of ν_e ↔ ν_x, where ν_x is a mix (superposition) of ν_μ and ν_τ. These approximations are possible because the mixing angle θ₁₃ is very small and because two of the mass states are very close in mass compared to the third.

Classical analogue of neutrino oscillation

Spring-coupled pendulums

Time evolution of the pendulums

Lower frequency normal mode

Higher frequency normal mode

The basic physics behind neutrino oscillation can be found in any system of coupled harmonic oscillators. A simple example is a system of two pendulums connected by a weak spring (a spring with a small spring constant). The first pendulum is set in motion by the experimenter while the second begins at rest. Over time, the second pendulum begins to swing under the influence of the spring, while the first pendulum's amplitude decreases as it loses energy to the second. Eventually all of the system's energy is transferred to the second pendulum and the first is at rest. The process then reverses. The energy oscillates between the two pendulums repeatedly until it is lost to friction.

The behavior of this system can be understood by looking at its normal modes of oscillation. If the two pendulums are identical then one normal mode consists of both pendulums swinging in the same direction with a constant distance between them, while the other consists of the pendulums swinging in opposite (mirror image) directions. These normal modes have (slightly) different frequencies because the second involves the (weak) spring while the first does not. The initial state of the two-pendulum system is a combination of both normal modes. Over time, these normal modes drift out of phase, and this is seen as a transfer of motion from the first pendulum to the second.

The description of the system in terms of the two pendulums is analogous to the flavor basis of neutrinos. These are the parameters that are most easily produced and detected (in the case of neutrinos, by weak interactions involving the W boson). The description in terms of normal modes is analogous to the mass basis of neutrinos. These modes do not interact with each other when the system is free of outside influence.

When the pendulums are not identical the analysis is slightly more complicated. In the small-angle approximation, the potential energy of a single pendulum system is ${\displaystyle \frac{1}{2}\frac{mg}{L}{x}^2}$, where g is standard gravity, L is the length of the pendulum, m is the mass of the pendulum, and x is the horizontal displacement of the pendulum. As an isolated system the pendulum is a harmonic oscillator with a frequency of ${\displaystyle \sqrt{g/L}}$. The potential energy of a spring is ⁠1/2⁠kx², where k is the spring constant and x is the displacement. With a mass attached it oscillates with a period of ${\displaystyle \sqrt{k/m}}$. With two pendulums (labeled a and b) of equal mass but possibly unequal lengths and connected by a spring, the total potential energy is

${\displaystyle V=\frac{m}{2}\left(\frac{g}{L_{\text{a}}}{x}_{\text{a}}^2+\frac{g}{L_{\text{b}}}{x}_{\text{b}}^2+\frac{k}{m}(x_{\text{b}}-x_{\text{a}}{)}^2\right).}$

This is a quadratic form in x_a and x_b, which can also be written as a matrix product:

${\displaystyle V=\frac{m}{2}\left(\begin{array}{cc}{x}_{\text{a}}& x_{\text{b}}\end{array}\right)\left(\begin{array}{cc}\frac{g}{L_{\text{a}}}+\frac{k}{m}& -\frac{k}{m}\\ -\frac{k}{m}& \frac{g}{L_{\text{b}}}+\frac{k}{m}\end{array}\right)\left(\begin{array}{c}{x}_{\text{a}}\\ x_{\text{b}}\end{array}\right).}$

The 2 × 2 matrix is real symmetric and so (by the spectral theorem) it is orthogonally diagonalizable. That is, there is an angle θ such that if we define

${\displaystyle \left(\begin{array}{c}{x}_{\text{a}}\\ x_{\text{b}}\end{array}\right)=\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)}$

then

${\displaystyle V=\frac{m}{2}\left(\begin{array}{c}{x}_1{x}_2\end{array}\right)\left(\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right)\left(\begin{array}{c}{x}_1\\ x_2\end{array}\right)}$

where λ₁ and λ₂ are the eigenvalues of the matrix. The variables x₁ and x₂ describe normal modes which oscillate with frequencies of ${\displaystyle \sqrt{{\lambda }_1}}$ and ${\displaystyle \sqrt{{\lambda }_2}}$. When the two pendulums are identical (L_a = L_b), θ is 45°.

The angle θ is analogous to the Cabibbo angle (though that angle applies to quarks rather than neutrinos).

When the number of oscillators (particles) is increased to three, the orthogonal matrix can no longer be described by a single angle; instead, three are required (Euler angles). Furthermore, in the quantum case, the matrices may be complex. This requires the introduction of complex phases in addition to the rotation angles, which are associated with CP violation but do not influence the observable effects of neutrino oscillation.

Theory, graphically

Two neutrino probabilities in vacuum

In the approximation where only two neutrinos participate in the oscillation, the probability of oscillation follows a simple pattern:

The blue curve shows the probability of the original neutrino retaining its identity. The red curve shows the probability of conversion to the other neutrino. The maximum probability of conversion is equal to sin² 2θ. The frequency of the oscillation is controlled by Δm².

Three neutrino probabilities

If three neutrinos are considered, the probability for each neutrino to appear is somewhat complex. The graphs below show the probabilities for each flavor, with the plots in the left column showing a long range to display the slow "solar" oscillation, and the plots in the right column zoomed in, to display the fast "atmospheric" oscillation. The parameters used to create these graphs (see below) are consistent with current measurements, but since some parameters are still quite uncertain, some aspects of these plots are only qualitatively correct.

Electron neutrino oscillations, long range. Here and in the following diagrams black means electron neutrino, blue means muon neutrino and red means tau neutrino.[27]	Electron neutrino oscillations, short range
Muon neutrino oscillations, long range	Muon neutrino oscillations, short range
Tau neutrino oscillations, long range	Tau neutrino oscillations, short range

The illustrations were created using the following parameter values:

• sin²(2θ₁₃) = 0.10 (Determines the size of the small wiggles.)
• sin²(2θ₂₃) = 0.97
• sin²(2θ₁₂) = 0.861
• δ = 0 (If the actual value of this phase is large, the probabilities will be somewhat distorted, and will be different for neutrinos and antineutrinos.)
• Normal mass hierarchy: m₁ ≤ m₂ ≤ m₃
• Δm²
  ₁₂ = 0.759×10⁻⁴ (eV/c²)²
• Δm²
  ₃₂ ≈ Δm²
  ₁₃ = 23.2×10⁻⁴ (eV/c²)²

Observed values of oscillation parameters

• sin²(2θ₁₃) = 0.093±0.008. PDG combination of Daya Bay, RENO, and Double Chooz results.
• sin²(2θ₁₂) = 0.846±0.021. This corresponds to θ_sol (solar), obtained from KamLand, solar, reactor and accelerator data.
• sin²(2θ₂₃″) > 0.92 at 90% confidence level, corresponding to θ₂₃ ≡ θ_atm = 45±7.1° (atmospheric)
• Δm²
  ₂₁ ≡ Δm²
  _sol = (0.753±0.018)×10⁻⁴ (eV/c²)²
• |Δm²
  ₃₁| ≈ |Δm²
  ₃₂| ≡ Δm²
  _atm = (24.4±0.6)×10⁻⁴ (eV/c²)² (normal mass hierarchy)
• δ, α₁, α₂, and the sign of Δm²
  ₃₂ are currently unknown.

Solar neutrino experiments combined with KamLAND have measured the so-called solar parameters Δm²
_sol and sin² θ_sol. Atmospheric neutrino experiments such as Super-Kamiokande together with the K2K and MINOS long baseline accelerator neutrino experiment have determined the so-called atmospheric parameters Δm²
_atm and sin² θ_atm. The last mixing angle, θ₁₃, has been measured by the experiments Daya Bay, Double Chooz and RENO as sin²(2θ₁₃″).

For atmospheric neutrinos the relevant difference of masses is about Δm² = 24×10⁻⁴ (eV/c²)² and the typical energies are ~ 1 GeV; for these values the oscillations become visible for neutrinos traveling several hundred kilometres, which would be those neutrinos that reach the detector traveling through the earth, from below the horizon.

The mixing parameter θ₁₃ is measured using electron anti-neutrinos from nuclear reactors. The rate of anti-neutrino interactions is measured in detectors sited near the reactors to determine the flux prior to any significant oscillations and then it is measured in far detectors (placed kilometres from the reactors). The oscillation is observed as an apparent disappearance of electron anti-neutrinos in the far detectors (i.e. the interaction rate at the far site is lower than predicted from the observed rate at the near site).

From atmospheric and solar neutrino oscillation experiments, it is known that two mixing angles of the MNS matrix are large and the third is smaller. This is in sharp contrast to the CKM matrix in which all three angles are small and hierarchically decreasing. The CP-violating phase of the MNS matrix is as of April 2020 to lie somewhere between −2° and −178°, from the T2K experiment.

If the neutrino mass proves to be of Majorana type (making the neutrino its own antiparticle), it is then possible that the MNS matrix has more than one phase.

Since experiments observing neutrino oscillation measure the squared mass difference and not absolute mass, one might claim that the lightest neutrino mass is exactly zero, without contradicting observations. This is however regarded as unlikely by theorists.

Origins of neutrino mass

The question of how neutrino masses arise has not been answered conclusively. In the Standard Model of particle physics, fermions only have intrinsic mass because of interactions with the Higgs field (see Higgs boson). These interactions require both left- and right-handed versions of the fermion (see chirality). However, only left-handed neutrinos have been observed so far.

Neutrinos may have another source of mass through the Majorana mass term. This type of mass applies for electrically neutral particles since otherwise it would allow particles to turn into anti-particles, which would violate conservation of electric charge.

The smallest modification to the Standard Model, which only has left-handed neutrinos, is to allow these left-handed neutrinos to have Majorana masses. The problem with this is that the neutrino masses are surprisingly smaller than the rest of the known particles (at least 600000 times smaller than the mass of an electron), which, while it does not invalidate the theory, is widely regarded as unsatisfactory as this construction offers no insight into the origin of the neutrino mass scale.

The next simplest addition would be to add into the Standard Model right-handed neutrinos that interact with the left-handed neutrinos and the Higgs field in an analogous way to the rest of the fermions. These new neutrinos would interact with the other fermions solely in this way and hence would not be directly observable, so are not phenomenologically excluded. The problem of the disparity of the mass scales remains.

Seesaw mechanism

Main article: Seesaw mechanism

The most popular conjectured solution currently is the seesaw mechanism, where right-handed neutrinos with very large Majorana masses are added. If the right-handed neutrinos are very heavy, they induce a very small mass for the left-handed neutrinos, which is proportional to the reciprocal of the heavy mass.

If it is assumed that the neutrinos interact with the Higgs field with approximately the same strengths as the charged fermions do, the heavy mass should be close to the GUT scale. Because the Standard Model has only one fundamental mass scale, all particle masses must arise in relation to this scale.

There are other varieties of seesaw and there is currently great interest in the so-called low-scale seesaw schemes, such as the inverse seesaw mechanism.

The addition of right-handed neutrinos has the effect of adding new mass scales, unrelated to the mass scale of the Standard Model, hence the observation of heavy right-handed neutrinos would reveal physics beyond the Standard Model. Right-handed neutrinos would help to explain the origin of matter through a mechanism known as leptogenesis.

Other sources

There are alternative ways to modify the standard model that are similar to the addition of heavy right-handed neutrinos (e.g., the addition of new scalars or fermions in triplet states) and other modifications that are less similar (e.g., neutrino masses from loop effects and/or from suppressed couplings). One example of the last type of models is provided by certain versions supersymmetric extensions of the standard model of fundamental interactions, where R parity is not a symmetry. There, the exchange of supersymmetric particles such as squarks and sleptons can break the lepton number and lead to neutrino masses. These interactions are normally excluded from theories as they come from a class of interactions that lead to unacceptably rapid proton decay if they are all included. These models have little predictive power and are not able to provide a cold dark matter candidate.

Oscillations in the early universe

During the early universe when particle concentrations and temperatures were high, neutrino oscillations could have behaved differently. Depending on neutrino mixing-angle parameters and masses, a broad spectrum of behavior may arise including vacuum-like neutrino oscillations, smooth evolution, or self-maintained coherence. The physics for this system is non-trivial and involves neutrino oscillations in a dense neutrino gas.

Fine-structure constant

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Fine-structure_constant 

 

Value of α
0.0072973525643(11)
Value of α−1
137.035999177(21)

Quantum field theory
Feynman diagram

In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by ${\displaystyle \alpha }$ (the Greek letter alpha), is a fundamental physical constant that quantifies the strength of the electromagnetic interaction between elementary charged particles.

It is a dimensionless quantity (dimensionless physical constant), independent of the system of units used, which is related to the strength of the coupling of an elementary charge ${\displaystyle e}$ with the electromagnetic field, by the formula ⁠${\displaystyle \alpha =\frac{e^2}{4\pi {\epsilon }_0\hslash c}}$⁠. Its numerical value is approximately 0.0072973525643 ≈ ⁠1/137.035999177⁠, with a relative uncertainty of 1.6×10⁻¹⁰.

The constant was named by Arnold Sommerfeld, who introduced it in 1916 when extending the Bohr model of the atom. ${\displaystyle \alpha }$ quantified the gap in the fine structure of the spectral lines of the hydrogen atom, which had been measured precisely by Michelson and Morley in 1887.

Why the constant should have this value is not understood, but there are a number of ways to measure its value.

Definition

In terms of other physical constants, ${\displaystyle \alpha }$ may be defined as:

$${\displaystyle \alpha =\frac{e^2}{2{\epsilon }_{0}hc}=\frac{e^2}{4\pi {\epsilon }_0\hslash c},}$$

where

${\displaystyle e}$ is the elementary charge (1.602176634×10⁻¹⁹ C);

${\displaystyle h}$ is the Planck constant (6.62607015×10⁻³⁴ J⋅Hz⁻¹);

${\displaystyle \hslash }$ is the reduced Planck constant, ${\displaystyle \hslash =\frac{h}{2\pi }}$ (1.054571817...×10⁻³⁴ J⋅s)

${\displaystyle c}$ is the speed of light (299792458 m⋅s⁻¹);

${\displaystyle {\epsilon }_0}$ is the electrical permittivity of space (8.8541878188(14)×10⁻¹² F⋅m⁻¹).

Since the 2019 revision of the SI, the only quantity in this list that does not have an exact value in SI units is the electric constant (vacuum permittivity).

Alternative systems of units

The electrostatic CGS system implicitly sets ⁠${\displaystyle 4\pi {\epsilon }_0=1}$⁠, as commonly found in older physics literature, where the expression of the fine-structure constant becomes

$${\displaystyle \alpha =\frac{e^2}{\hslash c}.}$$

A normalised system of units commonly used in high energy physics selects artificial units for mass, distance, time, and electrical charge which cause ${\displaystyle {\epsilon }_0=c=\hslash =1}$ in such a system of "natural units" the expression for the fine-structure constant becomes

$${\displaystyle \alpha =\frac{e^2}{4\pi }.}$$

As such, the fine-structure constant is chiefly a quantity determining (or determined by) the elementary charge: $e=\sqrt{4\pi \alpha }\approx$ 0.30282212  in terms of such a natural unit of charge.

In the system of atomic units, which sets ⁠${\displaystyle e=\hslash =4\pi {\epsilon }_0=1}$⁠, the expression for the fine-structure constant becomes

$${\displaystyle \alpha =\frac{1}{c}.}$$

Measurement

Eighth-order Feynman diagrams on electron self-interaction. The arrowed horizontal line represents the electron, the wavy lines are virtual photons, and the circles are virtual electron–positron pairs.

The CODATA recommended value of α is

α = ⁠e²/ 4πε₀ħc⁠ = 0.0072973525643(11).

This has a relative standard uncertainty of 1.6×10⁻¹⁰.

This value for α gives the following value for the vacuum magnetic permeability (magnetic constant): µ₀ = 4π × 0.99999999987(16)×10⁻⁷ H⋅m⁻¹, with the mean differing from the old defined value by only 0.13 parts per billion, 0.8 times the standard uncertainty (0.16 parts per billion) of its recommended measured value.

Historically, the value of the reciprocal of the fine-structure constant is often given. The CODATA recommended value is

⁠1/α⁠ = 137.035999177(21).

While the value of α can be determined from estimates of the constants that appear in any of its definitions, the theory of quantum electrodynamics (QED) provides a way to measure α directly using the quantum Hall effect or the anomalous magnetic moment of the electron. Other methods include the A.C. Josephson effect and photon recoil in atom interferometry. There is general agreement for the value of α, as measured by these different methods. The preferred methods in 2019 are measurements of electron anomalous magnetic moments and of photon recoil in atom interferometry. The theory of QED predicts a relationship between the dimensionless magnetic moment of the electron and the fine-structure constant α (the magnetic moment of the electron is also referred to as the electron g-factor g_e). One of the most precise values of α obtained experimentally (as of 2023) is based on a measurement of g_e using a one-electron so-called "quantum cyclotron" apparatus, together with a calculation via the theory of QED that involved 12672 tenth-order Feynman diagrams:

${\displaystyle \frac{1}{\alpha }=}$ 137.035999166(15) .

This measurement of α has a relative standard uncertainty of 1.1×10⁻¹⁰. This value and uncertainty are about the same as the latest experimental results.

Further refinement of the experimental value was published by the end of 2020, giving the value

⁠1/α⁠ = 137.035999206(11),

with a relative accuracy of 8.1×10⁻¹¹, which has a significant discrepancy from the previous experimental value.

Physical interpretations

The fine-structure constant, α, has several physical interpretations. α is:

• The ratio of two energies:
  1. the energy needed to overcome the electrostatic repulsion between two electrons a distance of d apart, and
  2. the energy of a single photon of wavelength λ = 2πd (or of angular wavelength d; see Planck relation): $${\displaystyle \alpha =\left(\frac{e^2}{4\pi {\epsilon }_{0}d}\right)/\left(\frac{hc}{\lambda }\right)=\frac{e^2}{4\pi {\epsilon }_{0}d}\times \frac{2\pi d}{hc}=\frac{e^2}{4\pi {\epsilon }_{0}d}\times \frac{d}{\hslash c}=\frac{e^2}{4\pi {\epsilon }_0\hslash c}.}$$
• The ratio of the velocity of the electron in the first circular orbit of the Bohr model of the atom, which is ⁠1/4πε₀⁠⁠e²/ħ⁠, to the speed of light in vacuum, c. This is Sommerfeld's original physical interpretation.
• ${\displaystyle {\alpha }^2}$ is the ratio of the potential energy of the electron in the first circular orbit of the Bohr model of the atom and the energy m_ec² equivalent to the mass of an electron. Using the virial theorem in the Bohr model of the atom ⁠${\displaystyle U_{\mathsf{e}\mathsf{l}}=2U_{\mathsf{k}\mathsf{i}\mathsf{n}}}$⁠, which means that ⁠${\displaystyle U_{\mathsf{e}\mathsf{l}}=m_{\mathsf{e}}{v}_{\mathsf{e}}^2=m_{\mathsf{e}}(\alpha c{)}^2={\alpha }^2(m_{\mathsf{e}}{c}^2)}$⁠. Essentially this ratio follows from the electron's velocity being ⁠${\displaystyle v_{\mathsf{e}}=\alpha c}$⁠.
• The two ratios of three characteristic lengths: the classical electron radius r_e, the reduced Compton wavelength of the electron ƛ_e, and the Bohr radius a₀: r_e = αƛ_e = α²a₀.
• In quantum electrodynamics, α is directly related to the coupling constant determining the strength of the interaction between electrons and photons. The theory does not predict its value. Therefore, α must be determined experimentally. In fact, α is one of the empirical parameters in the Standard Model of particle physics, whose value is not determined within the Standard Model.
• In the electroweak theory unifying the weak interaction with electromagnetism, α is absorbed into two other coupling constants associated with the electroweak gauge fields. In this theory, the electromagnetic interaction is treated as a mixture of interactions associated with the electroweak fields. The strength of the electromagnetic interaction varies with the strength of the energy field.
• In the fields of electrical engineering and solid-state physics, the fine-structure constant is one fourth the product of the characteristic impedance of free space, ${\displaystyle Z_0={\mu }_{0}c,}$ and the conductance quantum, ${\displaystyle G_0=2e^2/h:}$ ${\displaystyle \alpha =\frac{1}{4}{Z}_0{G}_0.}$ The optical conductivity of graphene for visible frequencies is theoretically given by ⁠ π /4⁠G₀, and as a result its light absorption and transmission properties can be expressed in terms of the fine-structure constant alone. The absorption value for normal-incident light on graphene in vacuum would then be given by ⁠πα/ (1 + πα/2)²⁠ or 2.24%, and the transmission by ⁠1/(1 + πα/2)²⁠ or 97.75% (experimentally observed to be between 97.6% and 97.8%). The reflection would then be given by ⁠ π² α²/ 4 (1 + πα/2)²⁠.
• The fine-structure constant gives the maximum positive charge of an atomic nucleus that will allow a stable electron-orbit around it within the Bohr model (element feynmanium). For an electron orbiting an atomic nucleus with atomic number Z the relation is ⁠mv²/r⁠ = ⁠1/ 4πε₀⁠ ⁠Ze²/r²⁠ . The Heisenberg uncertainty principle momentum/position uncertainty relationship of such an electron is just mvr = ħ. The relativistic limiting value for v is c, and so the limiting value for Z is the reciprocal of the fine-structure constant, 137.

When perturbation theory is applied to quantum electrodynamics, the resulting perturbative expansions for physical results are expressed as sets of power series in α. Because α is much less than one, higher powers of α are soon unimportant, making the perturbation theory practical in this case. On the other hand, the large value of the corresponding factors in quantum chromodynamics makes calculations involving the strong nuclear force extremely difficult.

Variation with energy scale

In quantum electrodynamics, the more thorough quantum field theory underlying the electromagnetic coupling, the renormalization group dictates how the strength of the electromagnetic interaction grows logarithmically as the relevant energy scale increases. The value of the fine-structure constant α is linked to the observed value of this coupling associated with the energy scale of the electron mass: the electron's mass gives a lower bound for this energy scale, because it (and the positron) is the lightest charged object whose quantum loops can contribute to the running. Therefore, ⁠1/ 137.03600 ⁠ is the asymptotic value of the fine-structure constant at zero energy. At higher energies, such as the scale of the Z boson, about 90 GeV, one instead measures an effective α ≈ 1/127.

As the energy scale increases, the strength of the electromagnetic interaction in the Standard Model approaches that of the other two fundamental interactions, a feature important for grand unification theories. If quantum electrodynamics were an exact theory, the fine-structure constant would actually diverge at an energy known as the Landau pole – this fact undermines the consistency of quantum electrodynamics beyond perturbative expansions.

History

Sommerfeld memorial at University of Munich

Based on the precise measurement of the hydrogen atom spectrum by Michelson and Morley in 1887, Arnold Sommerfeld extended the Bohr model to include elliptical orbits and relativistic dependence of mass on velocity. He introduced a term for the fine-structure constant in 1916. The first physical interpretation of the fine-structure constant α was as the ratio of the velocity of the electron in the first circular orbit of the relativistic Bohr atom to the speed of light in the vacuum. Equivalently, it was the quotient between the minimum angular momentum allowed by relativity for a closed orbit, and the minimum angular momentum allowed for it by quantum mechanics. It appears naturally in Sommerfeld's analysis, and determines the size of the splitting or fine-structure of the hydrogenic spectral lines. This constant was not seen as significant until Paul Dirac's linear relativistic wave equation in 1928, which gave the exact fine structure formula.

With the development of quantum electrodynamics (QED) the significance of α has broadened from a spectroscopic phenomenon to a general coupling constant for the electromagnetic field, determining the strength of the interaction between electrons and photons. The term ⁠α/2π⁠ is engraved on the tombstone of one of the pioneers of QED, Julian Schwinger, referring to his calculation of the anomalous magnetic dipole moment.

History of measurements

Successive values determined for the fine-structure constant

Date	α	1/α	Sources
1969 Jul	0.007297351(11)	137.03602(21)	CODATA 1969
1973	0.0072973461(81)	137.03612(15)	CODATA 1973
1987 Jan	0.00729735308(33)	137.0359895(61)	CODATA 1986
1998	0.007297352582(27)	137.03599883(51)	Kinoshita
2000 Apr	0.007297352533(27)	137.03599976(50)	CODATA 1998
2002	0.007297352568(24)	137.03599911(46)	CODATA 2002
2007 Jul	0.0072973525700(52)	137.035999070(98)	Gabrielse (2007)
2008 Jun	0.0072973525376(50)	137.035999679(94)	CODATA 2006
2008 Jul	0.0072973525692(27)	137.035999084(51)	Gabrielse (2008), Hanneke (2008)
2010 Dec	0.0072973525717(48)	137.035999037(91)	Bouchendira (2010)
2011 Jun	0.0072973525698(24)	137.035999074(44)	CODATA 2010
2015 Jun	0.0072973525664(17)	137.035999139(31)	CODATA 2014
2017 Jul	0.0072973525657(18)	137.035999150(33)	Aoyama et al. (2017)
2018 Dec	0.0072973525713(14)	137.035999046(27)	Parker, Yu, et al. (2018)
2019 May	0.0072973525693(11)	137.035999084(21)	CODATA 2018
2020 Dec	0.0072973525628(6)	137.035999206(11)	Morel et al. (2020)
2022 Dec	0.0072973525643(11)	137.035999177(21)	CODATA 2022
2023 Feb	0.0072973525649(8)	137.035999166(15)	Fan et al. (2023)

The CODATA values in the above table are computed by averaging other measurements; they are not independent experiments.

Potential variation over time

Further information: Time-variation of fundamental constants

Physicists have pondered whether the fine-structure constant is in fact constant, or whether its value differs by location and over time. A varying α has been proposed as a way of solving problems in cosmology and astrophysics. String theory and other proposals for going beyond the Standard Model of particle physics have led to theoretical interest in whether the accepted physical constants (not just α) actually vary.

In the experiments below, Δα represents the change in α over time, which can be computed by α_past − α_now . If the fine-structure constant really is a constant, then any experiment should show that $${\displaystyle \frac{\mathrm{\Delta }\alpha }{\alpha }\stackrel{\underset{\mathsf{d}\mathsf{e}\mathsf{f}}{}}{=}\frac{{\alpha }_{\mathsf{p}\mathsf{a}\mathsf{s}\mathsf{t}}-{\alpha }_{\mathsf{n}\mathsf{o}\mathsf{w}}}{{\alpha }_{\mathsf{n}\mathsf{o}\mathsf{w}}}=0,}$$ or as close to zero as experiment can measure. Any value far away from zero would indicate that α does change over time. So far, most experimental data is consistent with α being constant, up to 10 digits of accuracy.

Past rate of change

The first experimenters to test whether the fine-structure constant might actually vary examined the spectral lines of distant astronomical objects and the products of radioactive decay in the Oklo natural nuclear fission reactor. Their findings were consistent with no variation in the fine-structure constant between these two vastly separated locations and times.

Improved technology at the dawn of the 21st century made it possible to probe the value of α at much larger distances and to a much greater accuracy. In 1999, a team led by John K. Webb of the University of New South Wales claimed the first detection of a variation in α Using the Keck telescopes and a data set of 128 quasars at redshifts 0.5 < z < 3, Webb et al. found that their spectra were consistent with a slight increase in α over the last 10–12 billion years. Specifically, they found that $${\displaystyle \frac{\mathrm{\Delta }\alpha }{\alpha }\stackrel{\underset{\mathsf{d}\mathsf{e}\mathsf{f}}{}}{=}\frac{{\alpha }_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{v}}-{\alpha }_{\mathrm{n}\mathrm{o}\mathrm{w}}}{{\alpha }_{\mathrm{n}\mathrm{o}\mathrm{w}}}=\left(-5.7\pm 1.0\right)\times {10}^{-6}.}$$

In other words, they measured the value to be somewhere between −0.0000047 and −0.0000067. This is a very small value, but the error bars do not actually include zero. This result either indicates that α is not constant or that there is experimental error unaccounted for.

In 2004, a smaller study of 23 absorption systems by Chand et al., using the Very Large Telescope, found no measurable variation: $${\displaystyle \frac{\mathrm{\Delta }\alpha }{{\alpha }_{\mathrm{e}\mathrm{m}}}=\left(-0.6\pm 0.6\right)\times {10}^{-6}.}$$

However, in 2007 simple flaws were identified in the analysis method of Chand et al., discrediting those results.

King et al. have used Markov chain Monte Carlo methods to investigate the algorithm used by the UNSW group to determine ⁠Δα/ α ⁠ from the quasar spectra, and have found that the algorithm appears to produce correct uncertainties and maximum likelihood estimates for ⁠Δα/ α ⁠ for particular models. This suggests that the statistical uncertainties and best estimate for ⁠Δα/ α ⁠ stated by Webb et al. and Murphy et al. are robust.

Lamoreaux and Torgerson analyzed data from the Oklo natural nuclear fission reactor in 2004, and concluded that α has changed in the past 2 billion years by 45 parts per billion. They claimed that this finding was "probably accurate to within 20%". Accuracy is dependent on estimates of impurities and temperature in the natural reactor. These conclusions have yet to be verified.

In 2007, Khatri and Wandelt of the University of Illinois at Urbana-Champaign realized that the 21 cm hyperfine transition in neutral hydrogen of the early universe leaves a unique absorption line imprint in the cosmic microwave background radiation. They proposed using this effect to measure the value of α during the epoch before the formation of the first stars. In principle, this technique provides enough information to measure a variation of 1 part in 10⁹ (4 orders of magnitude better than the current quasar constraints). However, the constraint which can be placed on α is strongly dependent upon effective integration time, going as 1⁄√t . The European LOFAR radio telescope would only be able to constrain ⁠Δα/ α ⁠ to about 0.3%. The collecting area required to constrain ⁠Δα/ α ⁠ to the current level of quasar constraints is on the order of 100 square kilometers, which is economically impracticable at present.

Present rate of change

In 2008, Rosenband et al. used the frequency ratio of Al⁺ and Hg⁺ in single-ion optical atomic clocks to place a very stringent constraint on the present-time temporal variation of α, namely ⁠Δα/ α ⁠ = (−1.6±2.3)×10⁻¹⁷ per year. A present day null constraint on the time variation of alpha does not necessarily rule out time variation in the past. Indeed, some theories that predict a variable fine-structure constant also predict that the value of the fine-structure constant should become practically fixed in its value once the universe enters its current dark energy-dominated epoch.

Spatial variation – Australian dipole

Researchers from Australia have said they had identified a variation of the fine-structure constant across the observable universe.

These results have not been replicated by other researchers. In September and October 2010, after released research by Webb et al., physicists C. Orzel and S.M. Carroll separately suggested various approaches of how Webb's observations may be wrong. Orzel argues that the study may contain wrong data due to subtle differences in the two telescopes. Carroll takes an altogether different approach: he looks at the fine-structure constant as a scalar field and claims that if the telescopes are correct and the fine-structure constant varies smoothly over the universe, then the scalar field must have a very small mass. However, previous research has shown that the mass is not likely to be extremely small. Both of these scientists' early criticisms point to the fact that different techniques are needed to confirm or contradict the results, a conclusion Webb, et al., previously stated in their study.

Other research finds no meaningful variation in the fine-structure constant.

Anthropic explanation

The anthropic principle provides an argument as to the reason the fine-structure constant has the value it does: stable matter, and therefore life and intelligent beings, could not exist if its value were very different. For instance, if modern grand unified theories are correct, then α needs to be between around 1/180 and 1/85 to have proton decay to be slow enough for life to be possible.

Numerological explanations

As a dimensionless constant which does not seem to be directly related to any mathematical constant, the fine-structure constant has long fascinated physicists.

Arthur Eddington argued that the value could be "obtained by pure deduction" and he related it to the Eddington number, his estimate of the number of protons in the universe. This led him in 1929 to conjecture that the reciprocal of the fine-structure constant was not approximately but precisely the integer 137. By the 1940s experimental values for ⁠1/ α ⁠ deviated sufficiently from 137 to refute Eddington's arguments.

Physicist Wolfgang Pauli commented on the appearance of certain numbers in physics, including the fine-structure constant, which he also noted approximates reciprocal of the prime number 137. This constant so intrigued him that he collaborated with psychoanalyst Carl Jung in a quest to understand its significance. Similarly, Max Born believed that if the value of α differed, the universe would degenerate, and thus that α = ⁠1/137⁠ is a law of nature.

Richard Feynman, one of the originators and early developers of the theory of quantum electrodynamics (QED), referred to the fine-structure constant in these terms:

There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.)

Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by humans. You might say the "hand of God" wrote that number, and "we don't know how He pushed His pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out – without putting it in secretly!

— R. P. Feynman

Conversely, statistician I. J. Good argued that a numerological explanation would only be acceptable if it could be based on a good theory that is not yet known but "exists" in the sense of a Platonic Ideal.

Attempts to find a mathematical basis for this dimensionless constant have continued up to the present time. However, no numerological explanation has ever been accepted by the physics community.

In the late 20th century, multiple physicists, including Stephen Hawking in his 1988 book A Brief History of Time, began exploring the idea of a multiverse, and the fine-structure constant was one of several universal constants that suggested the idea of a fine-tuned universe.

Quotes

For historical reasons, α is known as the fine structure constant. Unfortunately, this name conveys a false impression. We have seen that the charge of an electron is not strictly constant but varies with distance because of quantum effects; hence α must be regarded as a variable, too. The value ⁠1/ 137 ⁠ is the asymptotic value of α shown in Fig. 1.5a.

— F. Halzen & A. Martin (1984)

The mystery about α is actually a double mystery: The first mystery – the origin of its numerical value α ≈ ⁠1/ 137 ⁠ – has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized.

— M.H. MacGregor (2007)

When I die my first question to the Devil will be: What is the meaning of the fine structure constant?

— Wolfgang Pauli