Why δ_CP Matters: CP Violation in Neutrino Oscillations

There are three CP-violating parameters in the Standard Model of particle physics. One is the strong-CP angle $\bar{\theta }$ in the QCD Lagrangian, observationally consistent with zero to ten decimal places — the unsolved strong-CP problem. One is the phase ${\delta }_{\mathrm{CKM}}$ in the quark mixing matrix, measured at meson factories and known to be about $1.2$ radians, large and unambiguous. The third is δ_CP, the analogous phase in the lepton mixing matrix. Its value is being pinned down right now by accelerator-based oscillation experiments. The current data hint at a large value but do not yet exclude zero. The eventual measurement is one of the central goals of the next generation of long-baseline experiments, and what it tells us touches on questions far broader than neutrino oscillations alone.

CP violation in the lepton sector matters because the universe is made of matter and not antimatter, and the asymmetry has to come from somewhere. CP-violating processes in the early universe are part of any explanation of that asymmetry, and the seesaw mechanism that explains the smallness of neutrino masses naturally hosts CP-violating decays of heavy right-handed neutrinos in leptogenesis. Whether the low-energy phase δ_CP that oscillation experiments measure is directly responsible for the cosmic asymmetry is a separate question — leptogenesis depends on phases in the heavy neutrino sector that need not equal δ_CP — but a measured nonzero δ_CP would at minimum show that the lepton sector breaks CP somewhere.

This post is about what δ_CP is, how it appears in oscillation probabilities, and how experiments turn the measurement into a number.

CP and the PMNS matrix

The lepton mixing matrix $U_{\mathrm{PMNS}}$ relates the flavor states $|{\nu }_e⟩,|{\nu }_{\mu }⟩,|{\nu }_{\tau }⟩$ to the mass eigenstates $|{\nu }_1⟩,|{\nu }_2⟩,|{\nu }_3⟩$ that propagate as definite waves. In the standard PDG parameterization the matrix is written as a product of three rotations and one complex phase, with the phase appearing in the ${\theta }_{13}$ rotation:

$U=\left(\begin{array}{ccc}{c}_{12}{c}_{13}& s_{12}{c}_{13}& s_{13}{e}^{-i{\delta }_{\mathrm{CP}}}\\ -s_{12}{c}_{23}-c_{12}{s}_{23}{s}_{13}{e}^{i{\delta }_{\mathrm{CP}}}& c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}{e}^{i{\delta }_{\mathrm{CP}}}& s_{23}{c}_{13}\\ s_{12}{s}_{23}-c_{12}{c}_{23}{s}_{13}{e}^{i{\delta }_{\mathrm{CP}}}& -c_{12}{s}_{23}-s_{12}{c}_{23}{s}_{13}{e}^{i{\delta }_{\mathrm{CP}}}& c_{23}{c}_{13}\end{array}\right)$

with $c_{ij}=\cos {\theta }_{ij}$ and $s_{ij}=\sin {\theta }_{ij}$. If neutrinos are Dirac, this is the entire phase content; if they are Majorana, two additional Majorana phases appear but contribute only to lepton-number-violating processes such as neutrinoless double beta decay, not to oscillations. The phase ${\delta }_{\mathrm{CP}}$ is therefore the only phase relevant to oscillation physics.

CP conjugation swaps neutrinos for antineutrinos. At the level of the PMNS matrix it means replacing $U$ by its complex conjugate $U^{\ast }$. If ${\delta }_{\mathrm{CP}}$ is zero or $\pi$, then $U$ and $U^{\ast }$ are equal (up to overall phases that drop out of oscillation probabilities), and CP is conserved: neutrinos and antineutrinos oscillate in exactly the same way. For any other value of ${\delta }_{\mathrm{CP}}$, the amplitudes differ and CP is broken.

The asymmetry observable

The cleanest oscillation channel for testing CP is ${\nu }_{\mu }\to {\nu }_e$ appearance at long baselines. Writing the vacuum probability to leading order in the relevant small parameters,

$P({\nu }_{\mu }\to {\nu }_e)\approx {\sin }^{2}2{\theta }_{13}{\sin }^2{\theta }_{23}{\sin }^2\left(\frac{\Delta m_{31}^{2}L}{4E}\right)+(\text{interference term})$

The first term is CP-even and dominates the rate. The second, smaller, term has the structure

$\propto \sin {\delta }_{\mathrm{CP}}\cdot \sin 2{\theta }_{13}\sin 2{\theta }_{12}\sin 2{\theta }_{23}\cdot \sin \left(\frac{\Delta m_{31}^{2}L}{4E}\right)\sin \left(\frac{\Delta m_{21}^{2}L}{4E}\right)\sin \left(\frac{\Delta m_{31}^{2}L}{4E}\right)$

The crucial sign: this interference term flips sign under CP conjugation, while the dominant term does not. So the difference between neutrino and antineutrino appearance probabilities is, to leading order, proportional to $\sin {\delta }_{\mathrm{CP}}$. Define the CP asymmetry

$A_{\mathrm{CP}}=\frac{P({\nu }_{\mu }\to {\nu }_e)-P({\bar{\nu }}_{\mu }\to {\bar{\nu }}_e)}{P({\nu }_{\mu }\to {\nu }_e)+P({\bar{\nu }}_{\mu }\to {\bar{\nu }}_e)}$

If ${\delta }_{\mathrm{CP}}=0$ or $\pi$, $A_{\mathrm{CP}}=0$. If ${\delta }_{\mathrm{CP}}=3\pi /2$ (maximal CP violation in the leptonic direction associated with current data), $A_{\mathrm{CP}}$ can reach $\pm 30\%$ depending on energy and baseline. This is what oscillation experiments measure.

The Jarlskog invariant

Despite the standard form’s apparent simplicity, the value of ${\delta }_{\mathrm{CP}}$ in isolation is not a physical observable — it can be moved around by rephasing redefinitions of the lepton fields. What is invariant is a particular combination of mixing-matrix elements introduced by Cecilia Jarlskog in 1985:

$J=\mathrm{Im}\left[U_{e1}{U}_{\mu 2}{U}_{e2}^{\ast }{U}_{\mu 1}^{\ast }\right]$

For the standard parameterization this evaluates to

$J=\frac{1}{8}\cos {\theta }_{13}\sin 2{\theta }_{12}\sin 2{\theta }_{13}\sin 2{\theta }_{23}\sin {\delta }_{\mathrm{CP}}$

The Jarlskog invariant is what really controls CP violation in oscillations: it appears as a single common prefactor in the CP-odd parts of every oscillation channel. Plugging in the measured values of the mixing angles gives $J\approx 0.033\sin {\delta }_{\mathrm{CP}}$, which can reach about $0.033$ at maximum — substantially larger than the analogous Jarlskog invariant in the quark sector, which sits around $3\times 10^{-5}$.

The leptonic Jarlskog being three orders of magnitude larger than the quark Jarlskog is one of the more striking patterns in the flavor sector. It does not by itself solve the baryogenesis problem — the connection runs through leptogenesis in the heavy-neutrino sector, not directly through δ_CP — but it does mean that the leptonic CP violation, if confirmed, is parametrically robust.

Matter effects and the mass ordering

The vacuum analysis above is a useful starting point but real long-baseline experiments propagate through hundreds of kilometres of Earth, and the matter potential modifies the picture in two ways.

First, the matter potential enhances or suppresses the appearance probability differently for neutrinos and antineutrinos, in a way that depends on the mass ordering. If the ordering is normal ($m_3$ heaviest), the matter effect enhances $P({\nu }_{\mu }\to {\nu }_e)$ and suppresses $P({\bar{\nu }}_{\mu }\to {\bar{\nu }}_e)$; if inverted, the reverse. The matter-induced asymmetry mimics the CP-violating asymmetry to a degree that depends on the baseline.

Second, the size of the matter effect grows with baseline. At T2K’s 295 km baseline, matter effects are modest and the CP-violating piece dominates the observed asymmetry. At NOvA’s 810 km baseline, matter effects are comparable and the two pieces must be disentangled simultaneously. At DUNE’s 1300 km baseline, matter effects are large and the experiment can use them to determine the mass ordering directly while still measuring ${\delta }_{\mathrm{CP}}$.

The strategy long-baseline experiments use is to compare the observed asymmetry to predictions for the four combinations of ${\delta }_{\mathrm{CP}}$ (zero versus $3\pi /2$) and mass ordering (normal versus inverted). With enough statistics and well-controlled systematics the four combinations split into distinct predictions and the data pick one.

Where the measurement stands

T2K and NOvA have been the workhorses of the δ_CP measurement throughout the 2010s and into the 2020s. T2K’s data favours values in the lower half of the circle, close to maximal CP violation around ${\delta }_{\mathrm{CP}}\approx 3\pi /2$. NOvA’s data is less stark, with a best fit somewhat closer to $\pi$ (the CP-conserving boundary) but with errors that overlap with T2K’s preferred region. The global fits that combine both with reactor data on ${\theta }_{13}$ disfavor CP-conservation at the two- to three-sigma level depending on the assumed mass ordering.

This tension between T2K and NOvA is one of the open questions in the field. Whether it is a real statistical fluctuation or a hint of an underlying systematic — for example, in the cross-section modeling of neutrino-nucleus interactions that both experiments rely on — is not yet clear. DUNE and Hyper-Kamiokande, both starting in the late 2020s, will collect an order of magnitude more events than T2K and NOvA combined, with substantially better systematic control. The five-sigma discovery (or exclusion) of CP violation in the lepton sector is one of their central goals.

Why it matters beyond oscillations

Two big-picture reasons make δ_CP one of the most-watched numbers in particle physics.

The first is leptogenesis. The high-scale CP violation that drives the matter-antimatter asymmetry of the universe lives, in the seesaw scenario, in the heavy right-handed neutrino sector. It does not have to equal the low-energy δ_CP, but the two are connected by the underlying flavor structure. A measured δ_CP would not by itself determine the leptogenesis parameters, but a confirmed nonzero value would establish that lepton-sector CP violation exists somewhere — which is a necessary ingredient.

The second is flavor pattern. The lepton mixing pattern, with two large angles and one small angle, is qualitatively different from the quark mixing pattern, with three small angles. Whether δ_CP comes out near $3\pi /2$, $\pi$, or somewhere else is a clue to the underlying flavor symmetry that generates the lepton mass matrix. Models with $\mu$-$\tau$ symmetry, with discrete flavor groups, with anarchy hypotheses about the neutrino mass texture — each predicts a different range for δ_CP. The measurement is, in this sense, a test of flavor model-building.

Summary

The single CP-violating phase ${\delta }_{\mathrm{CP}}$ in the PMNS matrix governs whether neutrinos and antineutrinos oscillate at different rates. At the leading order the asymmetry between ${\nu }_{\mu }\to {\nu }_e$ and ${\bar{\nu }}_{\mu }\to {\bar{\nu }}_e$ appearance probabilities is proportional to $\sin {\delta }_{\mathrm{CP}}$, and the Jarlskog invariant that encodes the same information is parametrically large in the lepton sector compared with the quark sector. Long-baseline accelerator experiments — T2K, NOvA, and the coming generation of DUNE and Hyper-Kamiokande — disentangle the CP effect from matter effects and the mass ordering by comparing neutrino-mode and antineutrino-mode running over hundreds of kilometres. Current data favour values near $3\pi /2$ with two- to three-sigma disagreement with CP conservation; the five-sigma discovery, or definitive null, will come from the next generation. A nonzero ${\delta }_{\mathrm{CP}}$ would establish CP violation in the lepton sector for the first time and constrain the flavor structure that ultimately may explain the matter-antimatter asymmetry of the universe.