Neutrinos

INPP FS24

Damian Goeldi

Standard Model

Standard Model \(\nu\)

  • Fermions
  • Three generations
  • Electrically neutral
  • Massless
  • Only weakly interacting
  • Only left-handed

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1930: \(\beta\)-decay puzzle

Pauli predicts the neutrino (but calls it neutron!)

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1956: Cowan and Reines: First \(\nu\) detection

Chadwick discovered the actual neutron in the meantime

P Reactor, Savannah River Plant, Aiken, SC, USA

  • \(\ce{Cd}\)-doped water sandwiched between scintillators
  • Signature:
    • 2 prompt \(\pu{511 keV}\) \(\gamma\) from \(e^{+}\) annihilation
    • Late \(\gamma\) from deexcitation of \(\ce{Cd}\) after \(n\) capture
  • 3 \(\bar{\nue}\) interactions per hour

νee+uduuddW±pn

5

Why did it take 25 years to find Pauli’s predicted particle?

\(\approx \pu{8 au}\) of liquid argon (or even more water)

would be required to reliably detect a single \(\nu\).

Key requirements:

  • Large (\(\pu{kt}\)) detectors
  • High \(\nu\) flux

Nuclear reactors were needed first.

Large-scale detectors took even longer.

\[ \sigma \propto \left|\mathcal{M}_{f,i}\right| ^ 2 \propto \frac{g_{W} ^ 2}{q ^ 2 - m_{W} ^ 2} \]

6

1960s: Davis tries to violate lepton flavour conservation

But finds a deficiency in the solar \(\nue\) flux

Homestake Gold Mine, Lead, SD, USA

  • Reactor \(\bar{\nue} \ce{^{37}Cl} \rightarrow e^{-} \ce{^{37}Ar} \Rightarrow\) Failure
  • Solar \(\nue \ce{^{37}Cl} \rightarrow e^{-} \ce{^{37}Ar} \Rightarrow\) Success
  • \(\pu{380 m3}\) tank of perchloroethylene (dry-cleaning fluid)
  • Offline counting experiment:
    • Measure activity of produced \(\ce{^{37}Ar}\)
  • Only 1/3 of flux predicted by theory

For 30 years, people thought that
this experiment had been flawed!

7

1998: Super-Kamiokande: Atmospheric \(\num\) flux deficiency

\(\pu{50 kt}\) of water and 11146 PMTs of \(\pu{0.5 m}\) diameter

\(\nue n \rightarrow e^{-} p\)

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\(\num n \rightarrow \mu^{-} p\)

https://www-sk.icrr.u-tokyo.ac.jp/realtimemonitor/

2002: Sudbury Neutrino Observatory (SNO): Davis was right!

\(\pu{1 kt}\) of heavy water (\(\ce{^{2}H_{2}O}\))—See script for more details on detection scheme

Total \(\nu\) flux (neutral current)

ννdduuddZ0nn

  • Also possible on \(p\) and \(e^{-}\)

9

\(\nue\) flux (charged current)

νee−duuuddW±np

  • Impossible for \(\num\)/\(\nut\)
  • Solar \(\nu\) energy too low
  • 1/3 of total flux

2012: Daya Bay Reactor \(\nu\) Experiment: \(\bar{\nue}\) disappearance

6 nuclear reactors and 6 \(\pu{20 t}\) liquid scintillator detectors—Detection scheme of Cowan and Reines

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2013 T2K: \(\nue\) appearance in an accelerator \(\num\) beam

Tokai to Kamioka: J-PARC accelerator to Super-Kamiokande detector

\(\num\) beam production

  • Same principle as PSI’s HIPA facility
  • Much higher \(p\) energy of \(\pu{30 GeV}\)
  • Need high-energy \(\num\) to create \(\mu^{-}\) in detectors
  • Off-axis beam to optimise \(\num\) energy

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What does it all mean?

\(\nu\) mixing and oscillation

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Pontecorvo-Maki-Nakagawa-Sakata matrix

\(s_{ij} = \sin\theta_{ij} \quad c_{ij} = \cos\theta_{ij}\)

\[ \begin{pmatrix} \nue \\ \num \\ \nut \end{pmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23} \end{bmatrix} \begin{bmatrix} c_{13} & 0 & s_{13} e ^ {-i \delta} \\ 0 & 1 & 0 \\ -s_{13} e ^ {i \delta} & 0 & c_{13} \end{bmatrix} \begin{bmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{pmatrix} \nu_{1} \\ \nu_{2} \\ \nu_{3} \end{pmatrix} = U_{\text{PMNS}} \begin{pmatrix} \nu_{1} \\ \nu_{2} \\ \nu_{3} \end{pmatrix} \]

\(\theta_{23}\)

\(\theta_{13}\)

\(\theta_{12}\)

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How do these mixed states propagate?

Production/Detection: Flavour eigenstates—Propagation: Mass eigenstates

Assumptions

  • Only two weak eigenstates:
    \(\ket{\nu_{\alpha}}\) with \(\alpha = e, \mu\)
  • Only two mass eigenstates:
    \(\ket{\nu_{k}}\) with \(k = 1, 2\)
  • \(\Delta m_{12} ^ 2 = m_{1} ^ 2 - m_{2} ^ 2 > \pu{0 eV}c^{-2}\)
  • Mixing is parametrised by a single angle \(\theta_{12}\):

\[ \begin{pmatrix} \nue \\ \num \end{pmatrix} = \begin{bmatrix} \cos\theta_{12} & \sin\theta_{12} \\ -\sin\theta_{12} & \cos\theta_{12} \end{bmatrix} \begin{pmatrix} \nu_{1} \\ \nu_{2} \end{pmatrix} \]

  • \(E_{k} = \sqrt{\left|\vec{p}\right| ^ 2 + m_{k} ^ 2} \approx \left|\vec{p}\right| \left(1 + \frac{m_{k} ^ 2}{2 \left|\vec{p}\right| ^ 2}\right)\)

Plane wave approximation

\[ \begin{aligned} \ket{\nu_{k}(t)} &= \exp\left(-\frac{i}{\hbar} H t\right) \ket{\nu_{k}(0)} \\ H \ket{\nu_{k}(0)} &= E_{k} \ket{\nu_{k}(0)} \quad \hbar = 1 \\ \ket{\nu_{k}(t)} &= \exp\left(-i E_{k} t\right) \ket{\nu_{k}(0)} \\ \ket{\nue(t)} &= \cos\theta_{12} \ket{\nu_{1}(t)} + \sin\theta_{12} \ket{\nu_{2}(t)} \\ &= \cos\theta_{12} e ^ {-i E_{1} t} \ket{\nu_{1}} + \sin\theta_{12} e ^ {-i E_{2} t} \ket{\nu_{2}} \end{aligned} \]

Example: Solar \(\nue\) (see script for intermediate steps)

\[ \begin{aligned} \begin{pmatrix} \nue(0) \\ \num(0) \end{pmatrix} &= \begin{pmatrix} 1 \\ 0 \end{pmatrix} \\ P \left(\nue \rightarrow \nue, t\right) &= \left|\braket{\nue(t) | \nue(0)}\right| ^ 2 \\ P \left(\nue \rightarrow \nue, t\right) &= 1 - \sin ^ 2 \left(2 \theta_{12}\right) \sin ^ 2 \left(\frac{\left(E_{1} - E_{2}\right) t}{2}\right) \\ \left|\vec{p}\right| &= c \left|\vec{p}\right| \approx E \quad t = c t \approx L \\ \left(E_{1} - E_{2}\right) t &= \frac{m_{1} ^ 2 - m_{2} ^ 2}{2 \left|\vec{p}\right|} t = \frac{\Delta m_{12} ^ 2}{2 E} L \\ P \left(\nue \rightarrow \nue, L\right) &= 1 - \sin ^ 2 \left(2 \theta_{12}\right) \sin ^ 2 \left(\frac{\Delta m_{12} ^ 2}{4 E} L\right) \leq 1 \end{aligned} \]

A real-world example

\(\pu{4 MeV}\) reactor \(\bar{\nue}\)

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\(\nu\) do have mass!

\(P \left(\nue \rightarrow \nue, L\right) = 1 - \sin ^ 2 \left(2 \theta_{12}\right) \sin ^ 2 \left(\frac{\Delta m_{12} ^ 2}{4 E} L\right) \leq 1\)

Oscillation properties

  • Amplitude \(\propto \sin ^ 2 \left(2 \theta_{12}\right)\)
  • Frequency (oscillation length) \(\propto \frac{\Delta m_{12} ^ 2}{4 E}\)
  • Measure different \(L/E \Rightarrow \theta_{12}, \Delta m_{12} ^ 2\)
  • Impossible to determine absolute masses

Real world: Packetised instead of plane wave

  • Slight difference in mass eigenstate speeds
  • Loss of coherence over long distances
  • Masses separate in spacetime (supernova \(\nu\))

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KATRIN: Determining the absolute \(\nu\) masses

What does KATRIN measure exactly?

  • \(\ce{^{3}H}\) \(\beta\)-decay energy spectrum end point
  • \(\nue\) eigenstate at production
    \(\rightarrow\) superposition of mass eigenstates
  • Average mass of \(\nue\)

An extremely challenging measurement

  • Trying to measure absence of events
  • Rate approaches 0 at point of interest
  • Finite detector resolution introduces fake events

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KATRIN: Determining the absolute \(\nu\) masses

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DUNE: A next-generation experiment testing CP-symmetry

Comparing \(\num\) to \(\bar{\num}\) oscillation using \(\pu{40 kt}\) of liquid \(\ce{Ar}\)

\[ \begin{pmatrix} \nue \\ \num \\ \nut \end{pmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & c_{23} & s_{23} \\ 0 & -s_{23} & c_{23} \end{bmatrix} \begin{bmatrix} c_{13} & 0 & s_{13} e ^ {-i {\color{red} \delta}} \\ 0 & 1 & 0 \\ -s_{13} e ^ {i {\color{red} \delta}} & 0 & c_{13} \end{bmatrix} \begin{bmatrix} c_{12} & s_{12} & 0 \\ -s_{12} & c_{12} & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{pmatrix} \nu_{1} \\ \nu_{2} \\ \nu_{3} \end{pmatrix} = U_{\text{PMNS}} \begin{pmatrix} \nu_{1} \\ \nu_{2} \\ \nu_{3} \end{pmatrix} \]

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DUNE: Precise measurements require advanced detectors

Liquid \(\ce{Ar}\) time projection chambers allow full reconstruction of event topology and energy

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\(\nu\) sources from \(\pu{eV}\) to \(\pu{EeV}\)

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Footnotes

  1. Title image adapted from https://commons.wikimedia.org/wiki/File:FirstNeutrinoEventAnnotated.jpg

  2. Adapted from https://commons.wikimedia.org/wiki/File:Standard_Model_of_Elementary_Particles_dark.svg

  3. Adapted from https://commons.wikimedia.org/wiki/File:RaE1.jpg

  4. https://cds.cern.ch/record/83282

  5. https://commons.wikimedia.org/wiki/File:Clyde_Cowan.jpg

  6. Adapted from https://doi.org/10.1103/RevModPhys.84.1307

  7. https://commons.wikimedia.org/wiki/File:U.S._Department_of_Energy_-_Science_-_390_002_007_(9952118384).jpg

  8. Adapted from https://www-sk.icrr.u-tokyo.ac.jp/en/sk/about/detector/ and https://www-sk.icrr.u-tokyo.ac.jp/en/sk/experience/gallery/

  9. https://en.wikipedia.org/wiki/File:Sudbury_Neutrino_Observatory.detector_outside.jpg

  10. https://doi.org/10.1016/j.xinn.2022.100332

  11. https://commons.wikimedia.org/wiki/File:The_Daya_Bay_Antineutrino_Detector_(8056998030).jpg

  12. Adapted from https://doi.org/10.1088/1367-2630/16/7/075015

  13. Adapted from https://www-sk.icrr.u-tokyo.ac.jp/en/sk/neutrino/kajita/vibration/

  14. https://commons.wikimedia.org/wiki/File:Top_of_Atmosphere.jpg

  15. https://commons.wikimedia.org/wiki/File:Akw_goesgenmai2010.JPG

  16. https://commons.wikimedia.org/wiki/File:Giant_prominence_on_the_sun_erupted.jpg

  17. Adapted from https://commons.wikimedia.org/wiki/File:41467_2015_Article_BFncomms7935_Fig1_HTML-en.svg

  18. Adapted from https://doi.org/10.1016/j.ppnp.2017.01.003

  19. Adapted from https://commons.wikimedia.org/wiki/File:KATRIN_Spectrum.svg

  20. Adapted from https://commons.wikimedia.org/wiki/File:NeutrinoMassTimeline2022.webp

  21. https://www.katrin.kit.edu/702.php

  22. https://www.dunescience.org/

  23. Adapted from https://doi.org/10.3390/instruments3020028

  24. Adapted from https://doi.org/10.48550/arXiv.2403.03212

  25. Adapted from https://doi.org/10.1103/RevModPhys.84.1307