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 *******
 Physics
 *******
+
+Neutrinos
+=========
+
+Introduction
+------------
+The name *neutrino* was coined to describe a then hypothetical particle
+suggested by Wolfgang Pauli in the 1930's [1]_. Pauli's idea was that of a
+neutral, weakly interacting particle having a very small mass. He proposed this
+particle could be used to explain the spectrum of electrons emitted by
+:math:`\beta`-decays of atomic nuclei, which was of considerable controversy at
+the time. The observed spectrum of these electrons is continuous [2]_, which is
+incompatible with the two-body decay description successfully used to describe
+discrete spectral lines in :math:`\alpha`- and :math:`\gamma`- decay of atomic
+nuclei. As Pauli observed, by describing :math:`\beta`-decay as a three-body
+decay instead, releasing both an electron and his proposed particle, one can
+explain the continuous :math:`\beta`-decay spectrum.  Soon after discovery of
+the neutron [3]_, Enrico Fermi used Pauli's light neutral particle as an
+essential ingredient in his successful theory of :math:`\beta`-decay, giving
+the neutrino it's name as a play on words of *little neutron* in Italian [4]_.
+
+It was not until some 20 years later that the discovery of the neutrino was
+realised. It was eventually understood that neutrinos came in three distinct
+*flavours* :math:`\left (\nu_e,\nu_\mu,\nu_\tau\right )` along with their
+associated antiparticles :math:`\left
+(\bar{\nu}_e,\bar{\nu}_\mu,\bar{\nu}_\tau\right)`.
+
+Neutrino Mixing
+---------------
+For the three massive neutrinos, the flavour eigenstates of the neutrino
+:math:`\mid{\nu_\alpha}>`, :math:`\alpha\in\{e,\mu,\tau\}`, are related to the
+mass eigenstates :math:`\mid{\nu_i}>`, :math:`i\in\{1,2,3\}` via a unitary
+mixing matrix :math:`U_{\alpha i}` known as the PMNS matrix [5]_, [6]_:
+
+.. math::
+
+  \mid{\nu_\alpha}>=\sum^3_{i=1}U^*_{\alpha i}\mid{\nu_i}>
+
+This relationship can be seen better in this image:
+
+.. figure:: _static/mixing.png
+  :width: 500px
+  :align: center
+
+  Graphical representation of the relationship between the neutrino flavour and
+  mass eigenstates. The three mass eigenstates are depicted as three boxes,
+  coloured such that the relative area gives the probability of finding the
+  corresponding flavour neutrino in that given mass state.
+
+The time evolution of the flavour eigenstate as the neutrino propagates is
+given by:
+
+.. math::
+
+  \mid{\nu_\alpha\left(t\right)}>=
+  \sum^3_{i=1}U^*_{\alpha i}\mid{\nu_i\left(t\right)}>
+
+The oscillation probability gives the probability that a neutrino produced in a
+flavour state :math:`\alpha` is then detected in a flavour state :math:`\beta`
+after a propagation distance :math:`L`:
+
+.. math::
+
+  \begin{align}
+    P_{\nu_\alpha\rightarrow\nu_\beta}\left(L\right) &=
+    \mid{<{\nu_\beta\left(L\right)}|{\nu_\alpha\left(0\right)}>}\mid^2\\
+    &=\mid{\sum_{i=1}^3\sum_{j=1}^3<{\nu_j\left(L\right)}|
+    U_{\beta j}U_{\alpha i}^*|{\nu_i\left(0\right)}>}\mid^2\\
+    &=\mid{\sum_{i=1}^3\sum_{j=1}^3U_{\beta j}U_{\alpha i}^*
+    <{\nu_j\left(0\right)}|e^{-i\frac{m_j^2L}{2E}}
+    e^{-i\frac{m_i^20}{2E}}|{\nu_i\left(0\right)}>}\mid^2\\
+    &=\mid{\sum_{i=1}^3U_{\beta i}U_{\alpha i}^*
+    e^{-i\frac{m_i^2L}{2E}}}\mid^2
+  \end{align}
+
+where the relation :math:`<{\nu_i}|{\nu_j}>=\delta_{ij}` was used. Expanding
+this expression gives [7]_:
+
+.. math::
+
+  \begin{align}
+    P_{\nu_\alpha\rightarrow\nu_\beta}\left(L, E\right) =
+    \delta_{\alpha\beta}&-4\sum_{i>j}\text{Re}\left(U_{\alpha i}^*U_{\beta i}
+    U_{\alpha j}U_{\beta j}^*\right)\sin^2
+    \left(\frac{\Delta m^2_{ij}L}{4E}\right)\\
+    &+2\sum_{i>j}\text{Im}\left(U_{\alpha i}^*U_{\beta i}U_{\alpha j}
+    U_{\beta j}^*\right)\sin\left(\frac{\Delta m^2_{ij}L}{2E}\right)
+  \end{align}
+
+where :math:`\Delta m_{ij}^2=m_i^2-m_j^2`. Note that for neutrino oscillations
+to occur, there must be at least one non-zero :math:`\Delta m_{ij}^2` and
+therefore there must exist at least one non-zero neutrino mass state.
+
+The mixing matrix can be parameterised using the standard factorisation [8]_:
+
+.. math::
+
+  \begin{align}
+    U=
+    \begin{pmatrix}
+      1    & 0       & 0      \\
+      0    & c_{23}  & s_{23} \\
+      0    & -s_{23} & c_{23} \\
+    \end{pmatrix}
+    \begin{pmatrix}
+      c_{13}             & 0 & s_{13}e^{-i\delta} \\
+      0                  & 1 & 0                  \\
+      -s_{13}e^{i\delta} & 0 &c_{13}              \\
+    \end{pmatrix}
+    \begin{pmatrix}
+      c_{12}  & s_{12} & 0 \\
+      -s_{12} & c_{12} & 0 \\
+      0       & 0      & 1 \\
+    \end{pmatrix}
+  \end{align}
+
+where :math:`s_{ij}\equiv\sin\theta_{ij}`, :math:`c_{ij}\equiv\cos\theta_{ij}`,
+:math:`\theta_{ij}` are the three mixing angles and :math:`\delta` is the CP
+violating phase. Overall phases in the mixing matrix do not affect neutrino
+oscillations, which only depend on quartic products, and so they have been
+omitted.  Therefore, this gives a total of six independent free parameters
+describing neutrino oscillations for three neutrino flavours in a vacuum. This
+table outlines the current knowledge of these parameters determined by a fit to
+global data [9]_:
+
+.. figure:: _static/osc_params.png
+  :width: 500px
+  :align: center
+
+  Three neutrino flavour oscillation parameters from a fit to global data
+  [9]_.
+
+This table shows two columns of values, *normal ordering* and *inverted
+ordering* corresponding to the case where the mass of :math:`\nu_3` is greater
+than the mass of :math:`\nu_1` or the mass of :math:`\nu_1` is greater than the
+mass of :math:`\nu_3`, respectively. The experimental determination of this
+mass ordering is ongoing.
+
+Astrophysical Neutrinos
+-----------------------
+The origin and acceleration mechanism of ultra-high-energy cosmic rays is still
+unknown. The difficulty comes from the fact that the cosmic rays are bent by
+interstellar magnetic fields, and so their arrival direction on Earth does not
+point back to their sources. The observation of these ultra-high-energy cosmic
+rays supports the existence of neutrino production at the sources of a similar
+energy range - an astrophysical neutrino flux. Neutrinos are electrically
+neutral, so are not perturbed by interstellar magnetic fields, and they also
+have a small enough interaction cross-section to escape from dense regions.
+This makes them ideal messengers to help identify the sources of cosmic rays:
+
+.. figure:: _static/nu_messengers.png
+  :align: center
+
+  Neutrinos as messengers of astrophysical objects. Exotic astrophysical
+  objects produce high-energy cosmic rays, photons and neutrinos, which can be
+  detected on Earth. Credit: IceCube, NSF.
+
+Ultra-high-energy cosmic rays detected on Earth manifestly succeed in
+escaping their sources, therefore these sources must be optically thin
+compared to the Earth's atmosphere. Thus, the following interactions of the
+accelerated protons are expected to be more important than lengthy shower
+processes. High-energy protons can interact with photons as such:
+
+.. math::
+
+  p+\gamma\rightarrow\Delta^+\rightarrow 
+  \begin{cases}
+    p+\pi^0\\
+    n+\pi^+
+  \end{cases}
+
+
+They can also interact with other hadrons:
+
+.. math::
+
+  p+p\rightarrow
+  \begin{cases}
+    p+p+\pi^0\\
+    p+n+\pi^+
+  \end{cases}\\
+  p+n\rightarrow
+  \begin{cases}
+    p+n+\pi^0\\
+    p+p+\pi^-
+  \end{cases}
+
+Importantly, final states here tend to produce pions which decay into either
+photons if neutral, :math:`\pi^0\rightarrow\gamma\gamma`, or if they are charged
+they decay into charged leptons and neutrinos. The neutral and charged pions are
+produced in similar amounts, meaning that the neutrino and photon fluxes are
+related. Indeed, the diffuse astrophysical neutrino flux can be estimated
+through :math:`\gamma`-ray astronomy [10]_.
+
+Point source searches of neutrinos are also being pursued. In 2017, a
+multi-messenger approach which searched for :math:`\gamma`-ray observations in
+coincidence with neutrinos coming from a particular source has successfully
+been able to identify for the very first time, a source of high-energy
+astrophysical neutrinos [11]_, [12]_.
+
+Of particular interest is the composition of flavours produced at the source.
+In the simple pion decay model described above, the *neutrino flavour
+composition* (sometimes referred to as the *neutrino flavour ratio*)
+produced at the source is:
+
+.. math::
+
+  \pi\text{ decay}\rightarrow
+  \left(f_e:f_\mu:f_\tau\right)_\text{S}=\left(1:2:0\right)_\text{S}
+
+For all discussions on the astrophysical neutrino flavour composition, the
+neutrino and antineutrino fluxes will been summed over as it is not yet
+experimentally possible to distinguish between the two. In the case that the
+muon interacts in the source before it has a chance to decay, e.g.\@ losing
+energy rapidly in strong magnetic fields or being absorbed in matter, only the
+:math:`\nu_\mu` from the initial pion decay escapes and so the source flavour
+composition is simply:
+
+.. math::
+  \mu\text{ suppressed }\rightarrow
+  \left(f_e:f_\mu:f_\tau\right)_\text{S}=\left(0:1:0\right)_\text{S}
+
+Another popular model is one in which the produced flux is dominated by neutron
+decay, :math:`n\rightarrow p+e^-+\bar{\nu}_e`, which gives rise to a purely
+:math:`\nu_e` component:
+
+.. math::
+
+  n\text{ decay}\rightarrow
+  \left(f_e:f_\mu:f_\tau\right)_\text{S}=\left(1:0:0\right)_\text{S}
+
+Production of :math:`\nu_\tau` at the source is not expected in standard
+astrophysics models. However, even in the standard construction, the
+composition could vary between any of the three idealised models above, which
+can be represented as a source flavour composition of :math:`(x:1-x:0)`, where
+:math:`x` is the fraction of :math:`\nu_e` and can vary between
+:math:`0\rightarrow1`.
+
+Once the neutrinos escape the source, they are free to propagate in the vacuum.
+As discussed above, neutrinos can transform from one flavour to another.
+Astrophysical neutrinos have :math:`\mathcal{O}(\text{Mpc})` or higher
+baselines, large enough that the mass eigenstates completely decouple. The
+astrophysical neutrinos detected on Earth are decoherent and are propagating in
+pure mass eigenstates. Taking this assumption greatly simplifies the transition
+probability as all the interference terms between the three mass eigenstates
+can be dropped, and all that is left is to convert from the propagating mass
+state to the flavour states:
+
+.. math::
+
+  \phi_{i,\oplus}&=\sum_\alpha\phi_{\alpha,\text{S}}\mid{U_{\alpha i}}\mid^2\\
+  \phi_{\alpha,\oplus}&=\sum_{i,\beta}
+    \mid{U_{\alpha i}}\mid^2\mid{U_{\beta i}}\mid^2\phi_{\beta,\text{S}}
+
+where :math:`\phi_\alpha` is the flux for a neutrino flavour :math:`\nu_\alpha`
+and :math:`\phi_i` is the flux for a neutrino mass state :math:`\nu_i`. The
+subscript :math:`\text{S}` denotes the source and :math:`\oplus` denotes as
+measured on Earth. The same result can be obtained in the plane wave picture of
+the neutrino mixing equations above and taking the limit
+:math:`L\rightarrow\infty`, thus this type of decoherent mixing is also known
+as oscillation-averaged neutrino mixing. From this, the flavour composition on
+Earth is defined as
+:math:`f_{\alpha,\oplus}=\phi_{\alpha,\oplus}/\sum_\alpha\phi_{\alpha,\oplus}`
+and this can be calculated using the mixing matrix parameters the table above.
+For the three source models discussed above:
+
+.. math::
+
+  \begin{align}
+    \left(1:2:0\right)_\text{S}&\rightarrow\left(0.31:0.35:0.34\right)_\oplus\\
+    \left(0:1:0\right)_\text{S}&\rightarrow\left(0.18:0.44:0.38\right)_\oplus\\
+    \left(1:0:0\right)_\text{S}&\rightarrow\left(0.55:0.18:0.27\right)_\oplus
+  \end{align}
+
+This can be visualised in a ternary plot, which you can make yourself by
+checking out the :doc:`examples` section! The axes here are the fraction of
+each neutrino flavour as shown below. The coloured circle, square and triangle
+show the source flavour compositions. The arrows show the effect of neutrino
+mixing on the flavour composition. The unfilled circle, square and triangle
+show the corresponding measured flavour composition. Neutrino mixing during
+propagation has the effect of averaging out the flavour contributions, which is
+why the arrows point towards the centre of the triangle. This effect is more
+pronounced for :math:`\nu_\mu\leftrightarrow\nu_\tau` due to the their larger
+mixings. Also shown on this figure in the hatched *Standard Model* area, is the
+region of measured flavour compositions containing all source models of
+:math:`\left(x:1-x:0\right)`, using Gaussian priors on the standard mixing
+angles. Therefore, this hatched area is the region in which all standard
+astrophysical models live.
+
+.. figure:: _static/fr.png
+  :width: 700px
+  :align: center
+
+  Astrophysical neutrino flavour composition ternary plot. Axes show the
+  fraction of each neutrino flavour. Coloured shapes show 3 models for the
+  source flavour composition. The arrows indicate the effect of neutrino mixing
+  during propagation and the unfilled shapes show the corresponding measured
+  flavour compositions. The hatched area shows the region in measured flavour
+  space in which all standard astrophysical models live.
+
+IceCube
+=======
+
+Introduction
+----------------
+The `IceCube Neutrino Obervatory <https://icecube.wisc.edu/>`_ is a cubic
+kilometre photomultiplier array embedded in the extremely thick and clear
+glacial ice located near the geographic South Pole in Antarctica. The IceCube
+array is made up of 5160 purpose built *Digital Optical Modules* (DOMs) which
+are deployed on 86 cables between 1450 and 2450 m below the ground. The
+interaction of a neutrino releases a burst of light in the detector, which is
+detected by this array of DOMs. The timing and intensity of these photons form
+the raw data set at IceCube. This data is analysed so that we can learn more
+about the properties of neutrinos. You can checkout some cool animations of how
+an event looks like in IceCube on `this website
+<https://www.nsf.gov/news/mmg/mmg_disp.jsp?med_id=184062>`_.
+
+A schematic layout of IceCube is shown below. The IceCube *In-Ice* array is
+made up of 5160 purpose built *Digital Optical Modules* (DOMs) which are
+deployed on 86 *strings* (or cables) between 1450 and 2450 m below the ground.
+The inner string separation is 125 m with a vertical DOM separation of 17 m.
+Eight of the centrally located strings make up the subarray *DeepCore* which
+are sensitive to lower energy neutrinos. It achieves this through denser
+instrumentation, having an inner string separation of 60 m and a vertical DOM
+separation of 7 m. A surface air shower array, IceTop, is instrumented on the
+surface and consists of a set of frozen water tanks which act as a veto against
+the background cosmic rays.
+
+.. figure:: _static/icecube.png
+  :width: 600px
+  :align: center
+
+  The IceCube neutrino observatory with the In-Ice array, its subarray DeepCore
+  and the cosmic ray shower array IceTop.
+
+Event Signatures
+----------------
+Cherenkov telescope arrays such as IceCube are able to classify the properties
+of a neutrino event by looking at the morphology of photon hits across its PMT
+array. There are two main types of neutrino event signatures at IceCube -
+*tracks* and *cascades*.
+
+Tracks are predominantly made by muons which are directly produced by
+neutrinos in the *charged current* :math:`\nu_\mu` interaction channel. Muons
+have a long lifetime, :math:`\sim` 2 :math:`\mu` s at rest, and in ice they
+have relatively low energy losses. Therefore, as shown in in the figure below,
+a high-energy muon travelling through the IceCube array will leave a long trail
+of hits. These features, along with the timing information of hits across the
+DOMs, help in determining the directionality of the muon, giving an angular
+resolution typically around :math:`0.5-1^\circ` [13]_. At energies of concern
+here, there is little deviation between the direction of the neutrino and the
+induced muon as they are both heavily boosted. Therefore, this pointing ability
+of tracks makes them the most attractive events to use for point source
+searches.  Energy reconstruction is more complicated, however. At the lower
+energies (:math:`\lesssim100` GeV), the muon's range is short enough that it is
+able to deposit all its energy inside the detector. This is the ideal situation
+for a good energy reconstruction as the IceCube array acts as a calorimeter, so
+the total deposited charge is proportional to the energy of the muon. At higher
+energies, the range of the muon is typically greater than the length of the
+detector. Therefore the energy of muon must be extrapolated from the portion of
+energy deposited inside the detector. This is particularly challenging for
+muons which are not produced inside the detector, for which only a lower bound
+can be made. The typical approach taken to reconstruct the muon is to segment
+the reconstruction along the track. In this way, biases from stochastic
+variations of the energy loss can be minimised by applying some averaging over
+each segment. In each segment, the mean :math:`\text{d} E/\text{d} x` is
+determined, which is then roughly proportional to the muon momentum. The energy
+resolution improves with the muon energy up to an uncertainty of a factor of 2
+[14]_. For more details on this see the IceCube energy reconstruction methods
+publication [15]_.
+
+.. figure:: _static/track.jpg
+  :width: 700px
+  :align: center
+
+  A track event initiated by a CC muon neutrino interaction in the detector.
+  The muon deposits 74 TeV before escaping.
+
+Cascades are created as a result of hadronic cascades and/or EM cascades.
+*Neutral current* interactions and CC :math:`\nu_e` interactions are the
+channels in which a pure cascade is created, and an example of one is shown in
+the figure below. However, this does not mean that neutrino events produce
+exclusively one type of signature, in fact all high-energy neutrino-nucleon
+events produce at least a hadronic cascade at the interaction vertex.
+Characteristic of a cascade is the isotropic deposition of energy in a
+localised region near the neutrino vertex. Contrary to tracks, cascade events
+have much shorter typical lengths and so the entire energy deposition is easily
+contained within the detector array. This is ideal for energy reconstruction
+giving a deposited energy resolution of :math:`\sim` 15% at neutrino energies
+above 10 TeV [15]_. Inferring the true neutrino energy is more difficult,
+however, as IceCube is not capable of resolving the difference between EM
+showers and hadronic showers, which potentially have a large amount of missing
+energy, leading to :math:`\sim` 15% lower light yield compared to an equivalent
+EM shower. Deposited energy is reconstructed using an EM shower hypothesis, and
+therefore this quantity gives the lower limit of the neutrino energy.
+Directional reconstruction is more challenging than for tracks and is done by
+looking for timing/light intensity anisotropies around the interaction vertex.
+The deviations are small, but it is expected that the light deposition in the
+forward direction is greater.  Typical angular resolutions are
+:math:`10-15^\circ` [16]_.
+
+.. figure:: _static/cascade.jpg
+  :width: 700px
+  :align: center
+
+  A cascade event initiated by a neutrino interaction in the detector. The
+  cascade deposits 1070 TeV in the detector.
+
+Not mentioned so far are the charged current :math:`\nu_\tau` interactions,
+which for energies :math:`\gtrsim` 1 PeV, can produce :math:`\tau` which
+travels a detectable distance before decaying. This provides a unique signature
+for such events. The initial :math:`\nu_\tau` interaction produces a hadronic
+cascade, followed by a track by the :math:`\tau` itself, in turn followed by
+either a track from the :math:`\tau`'s muonic decay
+(:math:`\tau^-\rightarrow\mu^-\bar{\nu}_\mu\nu_\tau` with branching ratio
+:math:`\sim` 17%), or a cascade from its other decays. Because of their
+distinctive signatures, such events are called *double bangs* or *double
+cascades*. See [17]_, [18]_ for more details.
+
+.. [1] Pauli, W. *Letter to Tübingen conference participants* Web document. 1930.
+.. [2] Chadwick, J. Intensitätsverteilung im magnetischen Spectrum der :math:`\beta`-Strahlen von radium B + C. Verhandl. Dtsc. Phys. Ges. 16, 383 (1914).
+.. [3] Chadwick, J. Possible Existence of a Neutron. Nature 129, 312 (1932).
+.. [4] Fermi, E. An attempt of a theory of beta radiation. 1. Z. Phys. 88, 161–177 (1934).
+.. [5] Pontecorvo, B. Neutrino Experiments and the Problem of Conservation of Leptonic Charge. Sov. Phys. JETP 26. [Zh. Eksp. Teor. Fiz.53,1717(1967)], 984–988 (1968).
+.. [6] Maki, Z., Nakagawa, M. & Sakata, S. Remarks on the unified model of elementary particles. Prog. Theor. Phys. 28. [,34(1962)], 870–880 (1962).
+.. [7] Giunti, C. & Kim, C. W. Fundamentals of Neutrino Physics and Astrophysics isbn: 9780198508717 (2007).
+.. [8] Beringer, J. et al. Review of Particle Physics (RPP). Phys. Rev. D86, 010001 (2012).
+.. [9] Esteban, I., Gonzalez-Garcia, M. C., Maltoni, M., Martinez-Soler, I. & Schwetz, T.  Updated fit to three neutrino mixing: exploring the accelerator-reactor complemen- tarity. JHEP 01, 087 (2017).
+.. [10] Kappes, A., Hinton, J., Stegmann, C. & Aharonian, F. A. Potential Neutrino Signals from Galactic Gamma-Ray Sources. Astrophys. J. 656. [Erratum: Astrophys. J.661,1348(2007)], 870–896 (2007).
+.. [11] Aartsen, M. G. et al. Multimessenger observations of a flaring blazar coincident with high-energy neutrino IceCube-170922A. Science 361, eaat1378 (2018).
+.. [12] Aartsen, M. G. et al. Neutrino emission from the direction of the blazar TXS 0506+056 prior to the IceCube-170922A alert. Science 361, 147–151 (2018).
+.. [13] Aartsen, M. G. et al. All-sky Search for Time-integrated Neutrino Emission from Astrophysical Sources with 7 yr of IceCube Data. Astrophys. J. 835, 151 (2017).
+.. [14] Weaver, C. Evidence for Astrophysical Muon Neutrinos from the Northern Sky PhD thesis (Wisconsin U., Madison, 2015). https://docushare.icecube.wisc.edu/dsweb/Get/Document-73829/.
+.. [15] Aartsen, M. G. et al. Energy Reconstruction Methods in the IceCube Neutrino Telescope. JINST 9, P03009 (2014).
+.. [16] Aartsen, M. G. et al. Evidence for High-Energy Extraterrestrial Neutrinos at the IceCube Detector. Science 342, 1242856 (2013).
+.. [17] Hallen, P. On the Measurement of High-Energy Tau Neutrinos with IceCube PhD thesis (RWTH Aachen University, 2013). https://www.institut3b.physik.rwth-aachen.de/global/show_document.asp?id=aaaaaaaaaapwhzq.
+.. [18] Xu, D. L. Search for astrophysical tau neutrinos in three years of IceCube data PhD thesis (The University of Alabama, 2015). http://acumen.lib.ua.edu/content/u0015/0000001/0001906/u0015_0000001_0001906.pdf