Long-baseline experiments

A long-baseline experiment is neutrino oscillation done on purpose. The atmospheric and solar deficits that first revealed oscillation were gifts of nature — beams no one built, over baselines no one chose. A long-baseline experiment takes the same physics into the laboratory: an accelerator manufactures an intense, almost pure muon-neutrino beam of known composition and tunable energy and aims it through the Earth at a detector hundreds of kilometres away. The survival probability $P(\nu_\mu\to\nu_\mu)$ a two-flavour analysis hands us — recalled here from aq-pp-two-flavour-oscillations — becomes a quantity you dial into resonance by choosing the beam energy and the detector distance. The defining trick is not the beam but the pair of detectors. Neither the absolute flux nor the cross section is known better than about ten per cent, so predicting the far rate from first principles is hopeless. Instead a near detector, a kilometre from the source, measures the beam before it oscillates; a far detector, at the oscillation maximum, measures it after. Both see the identical beam striking identical nuclei, so the flux and cross-section factors divide out in the near-to-far ratio, and what survives is the oscillation probability itself. Because the baseline is fixed, oscillation does not flicker in time or dim the beam uniformly; it carves an energy-dependent notch whose position fixes the splitting $\Delta m^2_{32}$ and whose depth fixes the mixing $\sin^2 2\theta_{23}$ . MINOS, T2K, NOvA and now DUNE are all, at heart, machines for reading one number: the phase $\Delta m^2 L/4E$ .

Key takeaways

An accelerator makes a pure, tunable $\nu_\mu$ beam (π⁺/K⁺ decay) and fires it hundreds of km through the Earth to a far detector on the oscillation maximum.
A near detector measures the unoscillated spectrum and a far detector the oscillated one, so flux and cross-section systematics cancel in the ratio.
With $L$ fixed, oscillation appears as an energy-dependent dip (not a time variation or uniform flux loss): the dip position fixes $\Delta m^2_{32}$ , the dip depth fixes $\sin^2 2\theta_{23}$ .
MINOS ( $L=735$ km) measures $\nu_\mu$ disappearance, since most oscillated $\nu_\tau$ fall below tau-production threshold: $|\Delta m^2_{32}|\approx 2.4\times10^{-3}\,\mathrm{eV}^2$ , $\sin^2 2\theta_{23}\approx 1$ .
An off-axis beam ( $E_\nu=0.43\,E_\pi/(1+\gamma^2\theta^2)$ ) clamps the neutrino energy nearly independent of pion energy, giving a narrow band parked on the first maximum — the T2K/NOvA design.
Key idea: by measuring the same beam at a near and a far detector, a long-baseline experiment turns oscillation into a measurable energy-dependent deficit while cancelling flux and cross-section systematics; the position of the spectral minimum fixes the mass-squared splitting and its depth fixes the mixing angle.

Where this lives today

Long-baseline experiments are the workhorses of precision neutrino physics. T2K (295 km, off-axis 2.5°, peak $\sim0.6$ GeV) and NOvA (810 km, off-axis 0.8°, $\sim2$ GeV) both measure $\nu_\mu$ disappearance and $\nu_e$ appearance in neutrino and antineutrino modes, comparing the two to hunt for CP violation through $\delta_{\rm CP}$ . Because appearance depends on the mass ordering through the MSW matter effect accrued crossing the Earth (owned by aq-pp-matter-effects-msw), NOvA's longer baseline breaks degeneracies T2K alone cannot; combined fits currently disfavour CP conservation while leaving the ordering and $\delta_{\rm CP}$ unresolved. The next-generation flagship is DUNE — a wide-band, on-axis 1.2 MW beam sent 1300 km from Fermilab to a 40 kton liquid-argon detector in South Dakota, whose long baseline fixes the ordering while its wide band spans the first and second maxima to over-constrain $\delta_{\rm CP}$ , with the 260 kton Hyper-Kamiokande as its counterpart on the T2K line. Reactor experiments (owned by aq-pp-reactor-neutrinos) supply the independent $\theta_{13}$ the appearance measurement needs, and every one of these machines is, at bottom, still reading the phase $1.267\,\Delta m^2 L/E$ off a spectrum.

The physical picture

Follow the beam from the proton up. A high-energy proton beam — 120 GeV at Fermilab's NuMI, 30 GeV at J-PARC — slams into a graphite target and sprays out pions and kaons. A magnetic horn, a pulsed toroidal lens, focuses the positive mesons forward and lets the negatives diverge, sign-selecting a $\nu_\mu$ beam (flip the current for antineutrinos). The $\pi^+$ decay in flight down an evacuated pipe, $\pi^+\to\mu^+\nu_\mu$ ; a downstream absorber stops the muons and hadrons, and only neutrinos emerge to cross the Earth. The first figure lays out this one-line machine and the two detectors that read it.

Figure

A long-baseline experiment in one line: 120 GeV protons strike a graphite target, a magnetic horn sign-selects and focuses the π⁺ whose in-flight decays π⁺ → μ⁺ νμ fill a νμ beam down the decay pipe, a near detector 1 km away records the unoscillated flux, and a far detector 735 km away records the oscillated spectrum — the beam is tuned so the first oscillation maximum 1.267 Δm² L/E = π/2 falls at E ≈ 1.4 GeV

Now the payoff. With the baseline fixed at 735 km, the phase $1.267\,\Delta m^2 L/E$ depends only on energy: high-energy neutrinos barely oscillate, while those near $E=1.4$ GeV hit the first maximum and vanish almost completely. Plotting the far spectrum against the no-oscillation expectation from the near detector reveals a clean energy-dependent notch. The second figure is that observable.

Figure

The measured far-detector observable: the νμ survival probability P(νμ → νμ) = 1 − sin²2θ₂₃ · sin²(1.267 Δm² L/E) reconstructed against neutrino energy for MINOS at L = 735 km with Δm² = 2.4×10⁻³ eV² and sin²2θ₂₃ = 1. A sharp disappearance dip sits at E = 1.4 GeV — the first oscillation maximum — where the beam almost vanishes; its position fixes Δm² and its depth fixes the mixing angle, while above the dip the beam is too energetic to have oscillated

The core equations

The physics the far detector measures is the muon-neutrino survival probability. Its derivation belongs to aq-pp-two-flavour-oscillations; here it is recalled in laboratory form — mass splitting in eV², baseline in km, energy in GeV — so $1.267$ absorbs all of $\hbar$ , $c$ , and the unit conversions.

P(\nu_\mu\to\nu_\mu)\simeq 1-\sin^2 2\theta_{23}\,\sin^2\!\left(1.267\,\frac{\Delta m^2_{32}\,[\mathrm{eV}^2]\;L\,[\mathrm{km}]}{E\,[\mathrm{GeV}]}\right) \qquad (1)

where: $\theta_{23}$ — atmospheric mixing angle; $\Delta m^2_{32}=m_3^2-m_2^2$ — atmospheric mass-squared splitting; $L$ — baseline; $E$ — reconstructed neutrino energy. The two-flavour amplitude $\sin^2 2\theta_{23}$ is the leading form; the exact three-flavour amplitude is $4|U_{\mu 3}|^2(1-|U_{\mu 3}|^2)$ with $|U_{\mu 3}|^2=\sin^2\theta_{23}\cos^2\theta_{13}$ , reducing to $\sin^2 2\theta_{23}$ as $\theta_{13}\to0$ . Reading it: the beam energy scans the phase, so at fixed $L$ the probability is a spectral shape, not a rate. The two-detector method is the article's own equation. Neither the flux $\Phi(E)$ nor the cross section $\sigma(E)$ is known to better than $\sim10\%$ , but both appear identically at near and far, so the probability is read as a pure ratio of the observed far spectrum to the far spectrum expected from the near measurement.

P(\nu_\mu\to\nu_\mu;E)=\frac{N^{\rm obs}_{\rm far}(E)}{N^{\rm exp}_{\rm far}(E)},\qquad N^{\rm exp}_{\rm far}(E)=\mathcal{F}\!\left[N^{\rm obs}_{\rm near}(E)\right] \qquad (2)

where: $N^{\rm obs}_{\rm near},N^{\rm obs}_{\rm far}$ — measured charged-current counts per energy bin; $N^{\rm exp}_{\rm far}$ — the no-oscillation far prediction; $\mathcal{F}$ — the near-to-far extrapolation, dominated by the geometric flux ratio $(L_{\rm near}/L_{\rm far})^2$ and the known detector masses and efficiencies. Reading it: the uncertain product $\Phi(E)\,\sigma(E)$ cancels top and bottom, so a $10\%$ flux error does not become a $10\%$ error on $\Delta m^2$ . Everything hinges on knowing $E$ event by event, since equation (1) is a function of energy. Two techniques dominate — calorimetric for a tracking calorimeter like MINOS, quasi-elastic for a water-Cherenkov detector like Super-Kamiokande.

E^{\rm cal}_\nu=E_\mu+E_{\rm had},\qquad E^{\rm QE}_\nu=\frac{2M E_\mu-m_\mu^2}{2\,(M-E_\mu+p_\mu\cos\theta_\mu)} \qquad (3)

where: $E_\mu,p_\mu,\theta_\mu$ — the muon's energy, momentum, and angle to the beam; $E_{\rm had}$ — hadronic-shower energy; $M$ — nucleon mass; $m_\mu$ — muon mass. The calorimetric form sums the visible energy of $\nu_\mu N\to\mu^- X$ ; the quasi-elastic form reconstructs $E_\nu$ from the muon kinematics alone in $\nu_\mu n\to\mu^- p$ . Reading it: the far detector must be a spectrometer — a mis-measured energy moves an event into the wrong bin. Finally, the beam itself is engineered. Pion decay is two-body, so in the pion rest frame the neutrino is monoenergetic; boosting to the lab at a small angle $\theta$ off the axis gives the narrow-band off-axis relation.

E_\nu=\frac{\big(1-m_\mu^2/m_\pi^2\big)\,E_\pi}{1+\gamma^2\theta^2}\;\approx\;\frac{0.43\,E_\pi}{1+\gamma^2\theta^2},\qquad \gamma=\frac{E_\pi}{m_\pi} \qquad (4)

where: $E_\pi$ — parent pion energy; $\gamma$ — pion Lorentz factor; $\theta$ — detector angle from the beam axis; the factor $0.43=1-m_\mu^2/m_\pi^2$ follows from the rest-frame neutrino energy $E^\ast=(m_\pi^2-m_\mu^2)/2m_\pi=29.8$ MeV. Reading it: on-axis the energy tracks $E_\pi$ (broad beam); off-axis it is clamped nearly independent of $E_\pi$ , a narrow band on the oscillation maximum.

Derivation AThe two-detector method and why the systematics cancel

The goal is to show why a long-baseline experiment needs two detectors, and how the near-to-far comparison turns an absolute flux-and-cross-section measurement — hopeless at the ten-per-cent level — into a ratio in which those uncertainties divide out, leaving the oscillation probability itself. We follow one energy bin from near to far, tracking what is known, unknown, and cancelling, to reach equation (2).

Step 1. Write the near count. In an energy bin at $E$ the near detector records $N_{\rm near}(E)=\Phi(E)\,\sigma(E)\,\varepsilon_{\rm near}(E)\,n_{\rm near}$ , with $\Phi$ the flux, $\sigma$ the charged-current cross section, $\varepsilon$ the efficiency and $n$ the target nucleons. Why: the product $\Phi\sigma$ is the whole problem — both the flux and the neutrino–nucleus cross section are uncertain at the $\sim10\%$ level, so no first-principles prediction of the rate is trustworthy.

Step 2. Write the far no-oscillation expectation, $N^{\rm exp}_{\rm far}(E)=\Phi(E)\,\sigma(E)\,\varepsilon_{\rm far}(E)\,n_{\rm far}\,(L_{\rm near}/L_{\rm far})^2$ . The identical $\Phi(E)\,\sigma(E)$ appears, because both detectors sample the same neutrinos on the same nuclei. Why: the far flux is the near flux geometrically diluted by $1/L^2$ — a computable factor — while $\Phi\sigma$ is literally common to the two rates.

Step 3. Extrapolate. Divide to form $N^{\rm exp}_{\rm far}=N_{\rm near}\,(\varepsilon_{\rm far}n_{\rm far}/\varepsilon_{\rm near}n_{\rm near})(L_{\rm near}/L_{\rm far})^2\equiv\mathcal{F}[N_{\rm near}]$ . The uncertain $\Phi\sigma$ cancels; what remains, $\mathcal{F}$ , is built only from detector masses, efficiencies, and geometry. Why: a $10\%$ flux error is now a common factor top and bottom, so it does not propagate into the prediction at all.

Step 4. Insert oscillation: $N^{\rm obs}_{\rm far}(E)=N^{\rm exp}_{\rm far}(E)\,P(\nu_\mu\to\nu_\mu;E)$ , so the ratio returns the probability directly, $P(E)=N^{\rm obs}_{\rm far}(E)/N^{\rm exp}_{\rm far}(E)$ — equation (2). Why: the signal is read as a clean bin-by-bin ratio with flux and cross-section systematics already removed, which is what lets MINOS quote $\Delta m^2_{32}$ to a few per cent. *Why this works as a limit: as $L_{\rm far}\to L_{\rm near}$ , $\mathcal{F}\to$ (mass and efficiency ratio) and $P\to1$ , so the far detector reproduces the scaled near spectrum; the residual systematic is the small near-point-source versus far-line-source acceptance difference, not the $10\%$ flux scale.*

Go deeper · Derivation B · 1 minThe off-axis beam and how it makes a narrow band at the oscillation maximum

Derivation BThe off-axis beam and how it makes a narrow band at the oscillation maximum

The goal is to derive the off-axis relation $E_\nu=0.43\,E_\pi/(1+\gamma^2\theta^2)$ from two-body pion decay and a Lorentz boost, and expose the consequence that made T2K and NOvA possible: at a fixed angle $\theta$ off the axis the neutrino energy becomes almost independent of the parent pion momentum, so a broad pion spectrum collapses into a narrow-band beam parked exactly on the first oscillation maximum — the single most important design choice in the modern experiments.

Step 1. Fix the rest-frame energy. The decay $\pi^+\to\mu^+\nu_\mu$ is two-body, so in the pion rest frame the neutrino is monoenergetic, $E^\ast=(m_\pi^2-m_\mu^2)/2m_\pi=(139.57^2-105.66^2)/(2\cdot139.57)\ \mathrm{MeV}=29.8$ MeV, emitted isotropically. Why: two-body kinematics fixes the CM neutrino energy uniquely from the masses — there is no spread to begin with.

Step 2. Boost to the lab. With $\gamma=E_\pi/m_\pi\gg1$ and $\beta\approx1$ , the lab energy is $E_\nu=E^\ast/[\gamma(1-\beta\cos\theta)]$ ; expanding for small lab angle $\theta$ ( $1-\beta\cos\theta\approx(1+\gamma^2\theta^2)/2\gamma^2$ ) gives $E_\nu=2\gamma E^\ast/(1+\gamma^2\theta^2)=(1-m_\mu^2/m_\pi^2)E_\pi/(1+\gamma^2\theta^2)$ , equation (4). Why: the boost squeezes the isotropic emission into a forward cone of half-angle $\sim1/\gamma$ ; at angle $\theta$ the energy is degraded by $1/(1+\gamma^2\theta^2)$ .

Step 3. On-axis ( $\theta=0$ ): $E_\nu=0.43\,E_\pi$ is strictly proportional to the pion energy, so a broad pion spectrum makes a broad, wide-band beam smeared across many phases. Why: on-axis you keep the full boosted energy, passing all the pion-momentum spread straight to the neutrinos.

Step 4. Off-axis: $E_\nu(E_\pi)$ at fixed $\theta$ rises, peaks where $\gamma^2\theta^2=1$ ( $E_\pi=m_\pi/\theta$ ), then falls, so $\mathrm{d}E_\nu/\mathrm{d}E_\pi=0$ at the peak and a wide range of pion energies piles up at nearly one energy $E_\nu^{\max}=0.43\,m_\pi/2\theta$ . For $\theta=2.5^\circ=0.0436$ rad this gives $E_\nu^{\max}\approx0.69$ GeV. Why: this stationarity is the whole trick — the energy is set by the angle alone, a narrow band whose peak you tune onto the first maximum by choosing $\theta$ . *As a check on the numbers: $E^\ast=29.8$ MeV; on-axis $E_\nu=0.43E_\pi$ ; the off-axis peak at $\gamma\theta=1$ gives $E_\nu^{\max}=0.215\,m_\pi/\theta\approx0.69$ GeV for $\theta=2.5^\circ$ , consistent with T2K's measured $\sim0.6$ GeV peak and the first-maximum energy $0.60$ GeV at $L=295$ km.*

Go deeper · Derivation C · 1 minReading the mass splitting and mixing angle off the disappearance spectrum

Derivation CReading the mass splitting and mixing angle off the disappearance spectrum

The goal is to show how the two parameters of the atmospheric sector are read from a single measured curve — the far-detector survival spectrum of figure 2. The dip's position on the energy axis fixes $\Delta m^2_{32}$ , its depth fixes $\sin^2 2\theta_{23}$ ; the two are nearly orthogonal features of one curve, which is why a single spectrum determines both at once.

Step 1. Locate the dip. The deepest disappearance sits at phase $\pi/2$ , $1.267\,\Delta m^2_{32}L/E_{\rm dip}=\pi/2$ , which inverts to $\Delta m^2_{32}=(\pi/2)\,E_{\rm dip}/(1.267\,L)$ . For the MINOS dip at $E_{\rm dip}=1.4$ GeV, $L=735$ km, this returns $(1.5708\cdot1.4)/(1.267\cdot735)=2.36\times10^{-3}\,\mathrm{eV}^2$ . Why: the dip energy is a calibration-free readout of $\Delta m^2$ given only the surveyed baseline.

Step 2. Measure the depth. At the dip $\sin^2(\cdot)=1$ , so equation (1) gives $P_{\min}=1-\sin^2 2\theta_{23}$ ; the fractional depth is the amplitude $\sin^2 2\theta_{23}$ . A dip reaching zero means $\sin^2 2\theta_{23}=1$ , maximal mixing $\theta_{23}=45^\circ$ , which the atmospheric data very nearly show. Why: the mixing angle sets how deep the beam disappears at the maximum, separately from where the maximum sits.

Step 3. Exploit orthogonality. Sliding $\Delta m^2_{32}$ moves the dip along the energy axis; changing $\theta_{23}$ deepens or fills it in place. Because position and depth are independent features of one curve, a $\chi^2$ fit over energy bins pins both at once — and the near detector removes the flux normalization otherwise degenerate with the depth. Why: this is why one disappearance spectrum yields both $|\Delta m^2_{32}|$ and $\sin^2 2\theta_{23}$ , not a single combined number. *As a consistency check: $E_{\rm dip}=1.4$ GeV at $L=735$ km gives $\Delta m^2_{32}=2.36\times10^{-3}\,\mathrm{eV}^2$ ; a full-depth dip gives $\sin^2 2\theta_{23}\approx1$ ; both are consistent with the MINOS results $|\Delta m^2_{32}|=(2.4\pm0.1)\times10^{-3}\,\mathrm{eV}^2$ and $\sin^2 2\theta_{23}>0.9$ at 90% C.L.*

Limiting cases

Case	Result	Meaning
Near detector, $L\to1$ km	$P\to1$	measures the unoscillated flux × cross section that anchors the extrapolation
First maximum, phase $=\pi/2$	$P=1-\sin^2 2\theta_{23}$	the design point — deepest disappearance, dip depth reads off the mixing
Maximal mixing, $\theta_{23}=45^\circ$	dip $\to0$	beam disappears completely at the maximum; atmospheric $\nu_\mu\to\nu_\tau$ is near this
High energy, phase $\ll1$	$P\to1-\sin^2 2\theta_{23}(1.267\Delta m^2 L/E)^2$	too fast to have oscillated — far spectrum matches the near expectation
Broad on-axis beam, phase $\gg1$	$\langle P\rangle=1-\tfrac12\sin^2 2\theta_{23}$	rapid oscillations wash out under averaging; only the mean survives
Off-axis, $\gamma\theta=1$	$E_\nu\approx0.43\,m_\pi/2\theta$	pion energies pile up at one neutrino energy — a narrow band on the maximum

Common trapThe far detector just sees fewer neutrinos, so oscillation is a flux loss

It is not a loss, and it is not uniform. Three mistakes hide here. First, oscillation is not a time variation — the survival probability depends only on $L$ and $E$ , so nothing flickers; a longer run just accumulates more of the same spectrum. Second, it is not a uniform suppression — with $L$ fixed, oscillation carves an energy-dependent notch, deep where $1.267\,\Delta m^2 L/E\approx\pi/2$ (near $1.4$ GeV for MINOS) and shallow at high energy; a single-number rate deficit throws away almost all the information, which lives in the shape. Third, the far/near ratio is not simply the geometric $1/L^2$ — that factor is divided out, and the residual deviation of the far spectrum from the near-based expectation is the signal. The missing neutrinos have not vanished either: they have become $\nu_\tau$ (and a little $\nu_e$ ), undetected only because they fall below the tau-production threshold, not because probability is lost.

Worked examples

1 · The MINOS disappearance spectrum, end to end (fully worked)

MINOS fires a $\nu_\mu$ beam peaking near $3$ GeV to a far detector $L=735$ km away in the Soudan mine. Using $\Delta m^2_{32}=2.4\times10^{-3}\,\mathrm{eV}^2$ and $\sin^2 2\theta_{23}=1$ , find the energy of the deepest disappearance, the survival probability at the $3$ GeV beam peak, and check that the observed dip returns $\Delta m^2$ .

Step 1. Deepest disappearance sits at phase $\pi/2$ : $E_{\rm dip}=1.267\,\Delta m^2 L/(\pi/2)=(1.267)(2.4\times10^{-3})(735)/1.5708=2.235/1.5708=\mathbf{1.42\ GeV}$ . This is the notch in figure 2.

Step 2. Survival at the beam peak $E=3$ GeV: $\Phi=1.267(2.4\times10^{-3})(735)/3=2.235/3=0.745$ rad, so $\sin^2\Phi=0.461$ and $P=1-0.461=\mathbf{0.54}$ — about 46% of the muon neutrinos have disappeared even at the spectral peak.

Step 3. Survival at the dip $E=1.42$ GeV: $\Phi=2.235/1.42=1.574\approx\pi/2$ , so $\sin^2\Phi=0.9998$ and $P=\mathbf{2\times10^{-4}}$ — the beam essentially vanishes at $1.4$ GeV.

Step 4. Invert the measured dip: reading $E_{\rm dip}=1.42$ GeV off the far spectrum, $\Delta m^2_{32}=(\pi/2)E_{\rm dip}/(1.267\,L)=(1.5708)(1.42)/(931.2)=\mathbf{2.40\times10^{-3}\ eV^2}$ , recovering the input. The dip position is the measurement of $\Delta m^2_{32}$ ; its full depth says $\sin^2 2\theta_{23}\approx1$ .

2 · Why T2K went off-axis (narrow-band beam)

T2K aims the J-PARC beam at Super-Kamiokande, $L=295$ km away, but places the far detector $\theta=2.5^\circ$ off the axis. Show this clamps the neutrino energy nearly independent of pion energy, and that the peak lands on the first maximum. Use $\Delta m^2_{32}=2.5\times10^{-3}\,\mathrm{eV}^2$ .

Step 1. Off-axis energy at $E_\pi=3$ GeV: $\gamma=E_\pi/m_\pi=3000/139.57=21.5$ , $\theta=2.5^\circ=0.0436$ rad, so $\gamma^2\theta^2=(21.5\times0.0436)^2=0.938^2=0.880$ . Then $E_\nu=0.43\times3/(1+0.880)=1.29/1.88=\mathbf{0.686\ GeV}$ .

Step 2. Vary the pion energy. For $E_\pi=2$ GeV: $\gamma^2\theta^2=0.391$ , $E_\nu=0.86/1.391=0.618$ GeV. For $E_\pi=5$ GeV: $\gamma^2\theta^2=2.44$ , $E_\nu=2.15/3.44=0.625$ GeV. Pion energies from 2 to 5 GeV all give $E_\nu\approx0.62$ – $0.69$ GeV — a narrow band clustered near the peak at $\gamma\theta=1$ ( $E_\nu^{\max}\approx0.69$ GeV).

Step 3. First oscillation maximum at $L=295$ km: $E_{\max}=1.267(2.5\times10^{-3})(295)/1.5708=\mathbf{0.60\ GeV}$ . The off-axis peak ( $\sim0.65$ GeV) sits right on the first maximum ( $0.60$ GeV) — the design principle of T2K.

Step 4. Contrast on-axis: there $E_\nu=0.43\,E_\pi$ tracks the pion energy ( $2$ GeV $\to0.86$ GeV, $5$ GeV $\to2.15$ GeV), smearing the beam across energies and diluting the dip. Off-axis trades raw intensity for a nearly monochromatic beam parked on the maximum.

Physics of the portrait

Long-baseline physics needed one person to insist that a neutrino oscillation could be manufactured — build the beam, choose the distance, place the detector on the maximum. That was Koichiro Nishikawa, who built K2K and then T2K.

Origins · 1 minKoichiro Nishikawa and how this was discovered

Koichiro Nishikawa

1949–2018 · Japanese physicist, founder of accelerator long-baseline neutrino experiments K2K and T2K

Koichiro Nishikawa spent his career at KEK and Kyoto turning neutrino oscillation from an observation into an instrument. In the early 1990s, when Kamiokande's atmospheric-neutrino anomaly was suggestive but uncontrolled — a beam nobody made, over baselines nobody chose — Nishikawa proposed the audacious alternative: fire an accelerator-made $\nu_\mu$ beam from the KEK proton synchrotron 250 km across Japan to Super-Kamiokande, and watch the same disappearance under laboratory control. The result, K2K (1999–2004), was the first long-baseline accelerator neutrino experiment, confirming with a man-made beam what atmospheric neutrinos had only hinted at. He then led its far more powerful successor, T2K, which from 2010 fired a high-intensity, deliberately off-axis beam from J-PARC 295 km to Super-Kamiokande. It was Nishikawa's off-axis strategy — placing the far detector 2.5° from the axis for a narrow-band beam on the oscillation maximum — that let T2K observe electron-neutrino appearance in a muon-neutrino beam (2011, definitively 2013), the first sighting of a neutrino turning into a specific different flavour and the opening move in the search for leptonic CP violation. He shared the 2016 Breakthrough Prize with the K2K and T2K collaborations and died in November 2018, having built the experiments that turned neutrino oscillation into precision measurement.

History

1962

— the first accelerator neutrino beam (Lederman, Schwartz, Steinberger) proves νμ is distinct from νe

1998

— Super-Kamiokande discovers atmospheric νμ disappearance, the effect long-baseline experiments set out to control

1999

— K2K sends the first accelerator neutrino beam 250 km, KEK to Super-Kamiokande

2006

— MINOS measures Δm²₃₂ from the NuMI beam over a 735 km baseline

2013

— T2K establishes νe appearance in a νμ beam, opening the study of leptonic CP violation

Going further

Texts. Thomson, Modern Particle Physics, §13.8 (long-baseline experiments) and §13.8.1 (MINOS) · Griffiths, Introduction to Elementary Particles, 2nd ed., §11.6 (neutrino oscillations and experiments) · Halzen & Martin, Quarks and Leptons, Ch. 12 (weak mixing and the lepton sector) · Schwartz, Quantum Field Theory and the Standard Model, §29 (neutrino masses and mixing) · Peskin & Schroeder, An Introduction to Quantum Field Theory, §20.3 (fermion masses and mixing). Key numerical anchors for this article: (1) MINOS dip $E_{\rm dip}=1.267\,\Delta m^2 L/(\pi/2)=1.42$ GeV at $L=735$ km, $\Delta m^2=2.4\times10^{-3}\,\mathrm{eV}^2$ ; (2) survival $P=0.54$ at $E=3$ GeV; (3) off-axis peak $E_\nu^{\max}=0.43\,m_\pi/2\theta\approx0.69$ GeV at $\theta=2.5^\circ$ , $E^\ast=29.8$ MeV; (4) T2K first maximum $E=0.60$ GeV at $L=295$ km. Standard parameters used throughout: $|\Delta m^2_{32}|\approx2.4$ – $2.5\times10^{-3}\,\mathrm{eV}^2$ , $\sin^2\theta_{23}\approx0.5$ , $\sin^2\theta_{13}\approx0.022$ .

Prerequisites. Three-Flavour PMNS Mixing · Two-Flavour Oscillations · Oscillation Experiments · Neutrino Scattering. Leads to. Matter Effects and the MSW Resonance · CP Violation in the Standard Model · Reactor Neutrinos.

Ask the tutor: Ask me to walk through why T2K, NOvA and DUNE chose baselines of 295, 810 and 1300 km — and how each choice trades matter effect, flux, and CP sensitivity against the others.

Why not just build one very good far detector and skip the near detector?

Because the number you would need to predict — the far-detector rate without oscillation — depends on the absolute flux and the cross section, neither known to better than about ten per cent. That error would swamp the few-per-cent oscillation signal. The near detector solves it by measuring the beam before it oscillates: the identical flux and cross section appear in both detectors and cancel in the ratio, leaving the oscillation probability with the systematics divided out. The near detector is not a luxury; it is what makes the measurement quantitative rather than merely suggestive.

Why send the beam through the Earth — isn't the matter (MSW) effect a nuisance?

It is both a complication and a tool. Neutrinos crossing the Earth pick up a potential from coherent forward scattering on electrons, which shifts the effective mixing with opposite sign for neutrinos and antineutrinos and a size that grows with baseline. For disappearance the effect is small and simply included in the fit. For appearance it is a feature: because the matter term depends on the mass ordering, a long enough baseline (DUNE's 1300 km) turns the Earth into part of the apparatus, deciding whether the third mass state is heaviest or lightest — something a vacuum experiment cannot do. Shorter baselines like T2K's keep the matter term small on purpose, so the CP measurement is less entangled with the ordering.

How is the neutrino energy reconstructed if the neutrino leaves no track?

From what it leaves behind in the charged-current interaction $\nu_\mu N\to\mu^- X$ . In a magnetized tracking calorimeter like MINOS the muon momentum comes from its curvature and the hadronic recoil $E_{\rm had}$ from the scintillation light, so $E_\nu=E_\mu+E_{\rm had}$ . In a water-Cherenkov detector like Super-Kamiokande, where the interaction is quasi-elastic $\nu_\mu n\to\mu^- p$ , two-body kinematics reconstruct $E_\nu$ from the muon energy and angle alone. Both methods have resolution limits — nuclear effects and unseen neutrons smear the energy — and that smearing is what keeps a real disappearance dip from reaching exactly zero.

What is the difference between a disappearance and an appearance measurement?

A disappearance experiment starts with a $\nu_\mu$ beam and counts how many survive — $P(\nu_\mu\to\nu_\mu)$ as a deficit. An appearance experiment looks for a different flavour not in the beam — $\nu_e$ in a $\nu_\mu$ beam, $P(\nu_\mu\to\nu_e)$ . MINOS does disappearance because the dominant oscillation is $\nu_\mu\to\nu_\tau$ and most of those $\nu_\tau$ carry too little energy to make the heavy tau that would tag them, so you infer the oscillation from the missing $\nu_\mu$ . Appearance is harder — a handful of events over background — but it carries $\theta_{13}$ and the CP phase, which is why T2K and NOvA invest so heavily in it.

How does a long-baseline experiment measure CP violation?

By running the beam in two modes — neutrino and antineutrino — and comparing $P(\nu_\mu\to\nu_e)$ with $P(\bar\nu_\mu\to\bar\nu_e)$ . If the CP phase $\delta_{\rm CP}$ is nonzero these differ even after accounting for the CP-even Earth-matter effect, which must be subtracted carefully because it also makes them differ. That subtraction is why baseline matters: T2K's short 295 km keeps the matter term small so the residual asymmetry is more cleanly CP; NOvA's 810 km has a larger matter term that helps fix the ordering but must be modelled out; DUNE is built to measure both at once. Current global fits hint at large CP violation but are not yet decisive.

What actually sets the choice of baseline — why 295, 735, or 1300 km?

The baseline is chosen so the first oscillation maximum, $1.267\,\Delta m^2_{32}L/E=\pi/2$ , falls at a neutrino energy the accelerator and detector handle well. With $\Delta m^2_{32}\approx2.4\times10^{-3}\,\mathrm{eV}^2$ tying $L$ and $E$ , 295 km lands the maximum at $\sim0.6$ GeV (T2K), 735 km at $\sim1.4$ GeV (MINOS), 1300 km at a few GeV (DUNE). Longer baselines buy a larger matter effect for the ordering and CP program at the cost of lower flux (the beam spreads as $1/L^2$ ). The choice negotiates between the oscillation physics you want to reach, the beam energy your accelerator makes efficiently, and the event rate you can afford to lose to distance.