Neutrino Astronomy

Abstract

Neutrino astronomy is a lively discipline that has born at the cross road of particle physics, nuclear physics, and astrophysics. Many low-energy neutrino observatories have demonstrated the possibility of investigating the functioning of the Sun, the terrestrial radiactivity and crucial astrophysical phenomena as the gravitational collapse. There are mounting evidences that extraterrestrial high-energy neutrinos are observable. This impacts strongly on our understanding of the high-energy phenomena from the cosmos, and the dataset and the sensitivity continue to improve. In this chapter, the status of the investigations of neutrino radiation is outlined, aiming to provide the reader with a unified description of this discipline that covers its main observational and theoretical aspects. We collect, for a readership at Ph.D. level, the most relevant formulae, expectations, and information concerning neutrino astronomy and astrophysics. Connections with other disciplines and selected applications to particle physics are discussed. Each section begins with an overview of the material included and a brief annotated bibliographical selection to books and review papers, aimed to favor “staged access” into the vast scientific literature.

Appendices

Appendix 1: Neutrinos on Web

Besides the web pages of the main Journals, there are several important repositories of papers online, used by various scientific communities that are interested in neutrinos. They include the arXiv [344] the searchable inSPIRE database [344], and the The SAO/NASA Astrophysics Data System [346].

Carlo Giunti and Marco Laveder provide access to an updated selection of references on neutrinos, mostly aimed at particle physicists, called Neutrino Unbound [347], while Maury Goodman maintains a dedicated newsletter Long Baseline News [348] concerning rumors and references of special interest about neutrinos.

There are several workshops and conference series dedicated to neutrinos, that allow the scientific community to remain informed and updated. The main ones, concerning the themes in which we are interested here, are named,

• Conference on Neutrino Physics and Astrophysics (Neutrino)

• Weak Interaction and Neutrinos (WIN)

• Neutrino Oscillation Workshop (NOW)

• Neutrino Telescopes in Venice (NuTel)

• Neutrino-Nucleus Interactions in Few GeV region (NuInt)

The above names, already, allow one to have an idea of the main thrusts of the discussion—even if they changes a bit in the course of the years, as it is natural. For each conference, typically, useful web sites are arranged where the files of the presentations are made public before proceedings are issued, allowing a fast, wider and more effective diffusion of the information. More complete and updated lists are in the above web pages of Giunti-Laveder [347] and of Goodman [348].

Finally, we recall that the PDG group [349] releases any two year a very useful report on particle physics, including review works on neutrinos. A database with neutrino cross sections measured below 30 GeV is in [350]. There are several useful sites to obtain parton distribution functions in various formats (Fortran, C++, Mathematica), e.g., [351,352,353].

Appendix 2: Wavefunction of the Neutrino

In this section we offer a summary of well-known results concerning the description of neutrinos in theoretical particle physics. This summary does not want to be complete and exhaustive, but rather, aims only to give an idea of the description of relativistic fermions to a generic readership, by using the (supposedly known) concepts of quantum/wave mechanics: wavefunctions, hamiltonian, spin, etc.

Plane waves of fermions

The plane waves that describe a relativistic fermion are eigenstates of the energy, momentum and of the helicity (i.e., the projection of the spin over the momentum). In formal terms, using the units $\hbar =c=1$, so that, e.g., the momentum operator is $\hat{p}^i=-i \partial /\partial x^i$, and denoting the wavefunction with $\psi$ to simplify the notations (or sometimes using the complete notation $\psi (\mathbf {p}\lambda ,x)$) we have,

$$\begin{aligned} \left\{ \begin{array}{lr} \left[ {\alpha }^i \hat{p}^i +\beta m \right] \psi (x) =E\ \psi (x) &{} \text {[energy]} \\ \hat{p}^i \ \psi (x) ={p}^i \ \psi (x) &{} \text {[momentum]} \\ \Sigma ^i \hat{p}^i\ \psi (x) =\lambda \ p\ \psi (x) &{} \text {[helicity]} \end{array} \right. \end{aligned}$$

(164)

where $i=1,2,3$, the contracted indices are summed over. The eigenvalues of the momentum, energy and helicity are, $\mathbf {p}$, $E=\sqrt{p^2+m^2}$ (with $p=|\mathbf {p}|$), $\lambda =\pm 1$.

With a specific choice of basis (Dirac representation) the $4\times 4$ matrices $\alpha ^i$ and $\beta$ of Dirac hamiltonian and the spin matrices $\Sigma ^i$ have the following expressions

$$\begin{aligned} \alpha ^i= \left( \begin{array}{cc} 0 &{} \sigma ^i \\ \sigma ^i &{} 0 \end{array} \right) \ \ \beta = \left( \begin{array}{cc} 1 &{} 0 \\ 0 &{} -1 \end{array} \right) \ \ \Sigma ^i= \left( \begin{array}{cc} \sigma ^i &{} 0 \\ 0 &{} \sigma ^i \end{array} \right) \end{aligned}$$

(165)

where $\sigma ^i$ are the 3 (hermitian, traceless, $2\times 2$ and well-known) Pauli matrices. The normalized wavefunction is given in terms of a spinor u,

$$\begin{aligned} \psi _\alpha (x)=\frac{e^{-i(E t - \mathbf {p}\mathbf {x})}}{\sqrt{2 E V}}\ u_\alpha (\mathbf {p}\lambda ) \text { where } u_\alpha ^*\, u_\alpha =2 E \end{aligned}$$

(166)

where V is the total volume of the space (that of course plays no role in physical quantities). The explicit expression of the spinor is,

$$\begin{aligned} u(\mathbf {p}\lambda )= \left( \begin{array}{c} \sqrt{E+m}\ \chi \\ \lambda \sqrt{E-m}\ \chi \end{array} \right) \text { where } \sigma ^i p^i\, \chi =\lambda p\, \chi \end{aligned}$$

(167)

Again in Dirac representation, the charge conjugation matrix and the chirality matrix are

$$\begin{aligned} C= \left( \begin{array}{cc} 0 &{} i \sigma ^2 \\ i \sigma ^2 &{} 0 \end{array} \right) \ \ \gamma ^5= \left( \begin{array}{cc} 0 &{}1 \\ 1 &{} 0 \end{array} \right) \end{aligned}$$

(168)

The charge conjugated spinor, defined as

$$\begin{aligned} \psi ^c=C \beta ^t \psi ^* \end{aligned}$$

(169)

satisfies the same conditions as $\psi$ but with opposite energy, momentum, and spin (thus, with the same helicity as $\psi$). This wavefunction is used in a very similar manner as one uses progressive and regressive waves in electromagnetism: $\psi$ describes a neutrino, that is ongoing to participate in a reaction, $\psi ^c$ represents an antineutrino, outgoing from the site of the reaction.

Helicity and chirality

At this point, we have a very important remark. The neutrino wavefunctions produced in weak interactions have a well-defined helicity in the ultra relativistic limit. In fact applying the chiral projector $P_L=(1-\gamma ^5)/2$, that is implied by charged currents, it is easy to prove that

$$\begin{aligned} \begin{array}{c} P_L\ \psi (\mathbf {p}+,x)\approx 0 \text { and } P_L\ \psi (\mathbf {p}-,x)\approx \psi (\mathbf {p}-,x) \text { for neutrinos }\\ P_L\ \psi ^c(\mathbf {p}+,x)\approx \psi ^c(\mathbf {p}+,x) \text { and } P_L\ \psi (\mathbf {p}-,x)\approx 0 \text { for antineutrinos } \end{array} \end{aligned}$$

(170)

that can be expressed in words saying that the (ultrarelativistic) neutrinos have negative helicity ($\lambda =-1$) while the antineutrinos have positive helicity ($\lambda =+1$). Therefore, whenever the ultra relativistic limit applies⁴³ we have only two wavefunctions that are produced and that matter: those with wavefunctions $\psi (\mathbf {p}-,x)$ and $\psi ^c(\mathbf {p}+,x)$. For this reason, it is often possible to adopt the same definition given in the standard model of electroweak interactions (where neutrinos are massless), namely: neutral leptons with negative helicity are neutrinos, those with positive helicity are antineutrinos.

A last comment on the nature of neutrino masses and quantum field theory. A Dirac massive neutrino has 4 states, but the only 2 of them matter in the above assumptions, those corresponding to the particle oscillator $\hat{a}(\mathbf {p}-)$ and to the antiparticle oscillator $\hat{b}(\mathbf {p}+)$; a Majorana massive neutrino has only two states, which corresponds to the formal replacement $\hat{b}(\mathbf {p}+)\rightarrow \hat{a}(\mathbf {p}+)$. See Fig. 1.

Appendix 3: Matter Effect, the General Case

Here we collect arguments and technical remarks from [68] that can be useful if one needs to implement oscillations including matter effect numerically, and/or to describe atmospheric neutrino oscillations in the Earth. A web interface, that calls a routine to perform this type of calculations, can be found in [357].

Let us write in full generality the amplitude of three-flavor neutrino oscillations

$$\begin{aligned} \mathscr {A}=\text {Texp}\left[ -i\int dt \mathscr {H}_\nu (t)\right] =R_{23} R_\delta \left( \begin{array}{ccc} a_{11} &{} a_{12} &{} a_{13} \\ a_{21} &{} a_{22} &{} a_{23} \\ a_{31} &{} a_{32} &{} a_{33} \end{array} \right) R_\delta ^* R_{23}^t \end{aligned}$$

(171)

where $a_{ij}$ depend upon $\varDelta m^2_{23}$, $\theta _{13}$, $\varDelta m^2_{12}$, $\theta _{12}$, and $H=\pm 1$ (the type of mass hierarchy, normal/inverted): see in particular Eqs. 1, 3, 5, 6 of [69]. By solving numerically the evolution equations, we calculate the complex numbers $a_{ij}$ and therefore the amplitudes and the probabilities. At this level, there is no approximation, except the numerical ones.

When the “solar” $\varDelta m^2_{12}$ is set to zero—i.e., when its effects are negligible—the only non-zero out-of-diagonal elements $a_{ij}$ in Eq. 171 are $a_{13}$ and $a_{31}$. The CP violating phase $\delta$ drops out from the probabilities $P_{\ell \ell '}=|\mathscr {A}_{\ell '\ell }|^2$, that moreover become symmetric, $P_{\ell \ell '}=P_{\ell '\ell }$ for each $\ell ,\ell '=e,\mu ,\tau$. Therefore, in this approximation we have 3 independent probabilities and all the other ones are fixed. We can chose, e.g.,

$$\begin{aligned} P_{e\mu }=\sin ^2\theta _{23} |a_{13}|^2,\ P_{e\tau }=\cos ^2\theta _{23} |a_{13}|^2, \ P_{\mu \tau }=\sin ^2 \theta _{23} \cos ^2 \theta _{23} |a_{33}-a_{22}|^2, \end{aligned}$$

(172)

so that, e.g., $P_{ee}=1- P_{\mu e}-P_{\tau e}=|a_{11}|^2$. From these formulae we obtain

$$\begin{aligned} P_{\mu e}=\sin ^2\theta _{23}\ (1-P_{ee})\text { and } P_{\mu \tau }=\frac{1}{4}\sin ^2 2 \theta _{23} \left| 1-\sqrt{P_{ee}}\ e^{i\hat{\varphi }}\right| ^2, \end{aligned}$$

(173)

where $\hat{\varphi }$ is a (rapidly varying) phase factor. Two important remarks are in order:

• 1. The last equation shows that $P_{\mu e}$ is large in the region where $P_{ee}$ is small, and that $P_{\mu \tau }$ remains close to zero in the first non-trivial minimum near $\hat{\varphi }=2\pi$, even when $P_{ee}\approx 0.3{-}0.4$ due to matter effect.

• 2. The sign of $\varDelta m^2$ controls the sign of the vacuum hamiltonian; therefore, switching between the two mass hierarchies or switching between neutrinos and antineutrinos has the same effect; e.g., $P_{{e}{\mu }}(\text {IH})=P_{\bar{e}\bar{\mu }}(\text {NH})$.

The first remark is consistent with the numerical findings, that $P_{\mu e}$ is amplified and $P_{\mu \tau }$ does not deviate strongly from its behavior in vacuum in the conditions that are relevant for our discussion.

Proceeding further with the approximations, and considering at this point the case of constant matter density, we obtain simple and closed expressions. For the case of normal mass hierarchy, they read:

$$\begin{aligned} \begin{array}{l} a_{13}=a_{31}=-i \sin \widetilde{\varphi } \sin 2 \widetilde{\theta _{13}} \\ a_{11}=\cos \widetilde{\varphi } + i \sin \widetilde{\varphi } \cos 2 \widetilde{\theta _{13}}=a_{33}^* \\ a_{22}=\cos \widetilde{\varphi }' + i \sin \widetilde{\varphi }' \end{array} \end{aligned}$$

(174)

where

$$\begin{aligned} \widetilde{\varphi }'=\frac{{\varDelta m^2} L}{4 E} (1+\varepsilon ) \end{aligned}$$

(175)

From Eqs. 172 and 174, we recover the expression of Eq. 43, used in the above discussion. In the approximation of constant matter density, the phase $\hat{\varphi }$ entering the expression of the probability $P_{\mu \tau }$ is given by $\sqrt{P_{ee}}\cos \hat{\varphi }\equiv \cos \widetilde{\varphi }\cos \widetilde{\varphi }'- \sin \widetilde{\varphi }\sin \widetilde{\varphi }'\cos 2\widetilde{\theta _{13} }$. This is close to the vacuum phase when $\varepsilon$ is large or small in comparison to 1: in fact, we have $\cos 2\widetilde{\theta _{13} }\sim \pm 1$ and $\widetilde{\varphi }\sim \pm \varDelta m^2 L/(4 E) (1-\varepsilon )$ from Eq. 44, so that $\cos \hat{\varphi }\sim \cos [\varDelta m^2 L/(2 E)]$.

Appendix 4: Treating Cosmological Neutrino Sources

In this appendix we summarize a few important formulae to treat sources of (low or high energy) neutrinos that are located at cosmological distances. A very useful review of practical cosmology is [323]. A recent determination of cosmological parameters is the one due to the Planck Collaboration, [34].

Covariant distance

Cosmological distances are measured by mean of the redhift z, that is connected to the wavelength $\lambda$ and the frequency $\nu$ as follows,

$$\begin{aligned} \frac{\lambda _ {\text {observed}}}{\lambda _ {\text {emitted}}}= \frac{\nu _ {\text {emitted}}}{\nu _ {\text {observed}}}=1+z\end{aligned}$$

(176)

The Hubble parameter $H_0=68\ \frac{\text {km/s}}{\text {Mpc}}$ corresponds to the Hubble scale,

$$\begin{aligned} D_H=\frac{c}{H_0}=1.36\times 10^{27}\text { cm}=4.41\text { Gpc} \end{aligned}$$

(177)

Moreover, we have $\varOmega _\varLambda =0.69$ and $\varOmega _m =0.31$ that fix the covariant distance,

$$\begin{aligned} D_c(z)=D_H\ \int _0^z \frac{dz'}{h(z')}\text { where } h(z)=\sqrt{\varOmega _\varLambda +\varOmega _m (1+z)^3} \end{aligned}$$

(178)

For small z, this gives the usual Hubble law case, where the distance is just proportional to z: $D_c(z)\approx D_H \times z$. E.g., we have $D_c(0.1)=430\text { Mpc}\approx D_H\times 0.1$ within 2.5%. The covariant distance can be calculated numerically and tabulated, or even approximated analytically [356]. The comoving volume $V_c(z)$ is a sphere with radius $D_c(z)$, therefore, we have

$$\begin{aligned} \frac{dV_c}{dz}=4\pi \frac{D_c(z)^2 D_H}{h(z)} \end{aligned}$$

(179)

this relation allows us to convert a distribution over the covariant volume in a distribution over the redshift as in the example given below.

Diffuse flux

If we have a source at redshift z that emits neutrinos (or photons) with a spectrum $dN_\nu /dE$ (differential in the energy), and if the emission is isotropic, the detected fluence is,

$$\begin{aligned} \frac{dF_\nu }{dE}=\frac{1}{4\pi D_c(z)^2}\times \frac{dN_\nu }{dE'}\end{aligned}$$

(180)

where, due to redshift,

$$\begin{aligned} E'=(1+z) E \end{aligned}$$

(181)

Note that $E'$ is the emitted (i.e., initial energy), but it is received as E due to the redshift. If there are many of these sources, distributed uniformly and isotropically with a density per comoving volume and per unit time $\dot{\rho }_ {\text {source}}(z)$, the total diffuse and stationary flux that we will receive a diffuse flux $\varPhi _ {\text {diffuse}}(E)=\int dV_c\ \dot{\rho }_ {\text {source}}(z)\ {dF_\nu }/{dE}$, that is,

$$\begin{aligned} \varPhi _ {\text {diffuse}}(E)= D_H \int _0^\infty dz\ n(z) \ \frac{dN_\nu }{dE'} \text { where }n(z)=\frac{ \dot{\rho }_ {\text {source}}(z) }{h(z)} \end{aligned}$$

(182)

Appendix 5: Principles of High Energy Neutrino Telescopes

The Cherenkov Radiation

Quanta of light are produced when a charged particle travels through a dielectric transparent medium at a speed greater than the speed of light in that medium. The medium may be any liquid or solid provided it is transparent. These photons are referred to as Cherenkov radiation in honor of the Russian physicist P.A. Cherenkov for his basic research on this phenomenon. Cherenkov radiation consists of a continuous spectrum of wavelengths extending from the ultraviolet region into the visible part of the spectrum peaking at about 420 nm. Only a negligible amount of photon emissions is found in the infrared or microwave regions. Cherenkov photon emission is the result of local polarization along the path of travel of the charged particle with the emission of electromagnetic radiation when the polarized molecules return to their original states. The Cherenkov effect is depicted clearly in Fig. 60, where a charged particle traveling along the vertical axis distorts the electron clouds of atoms in close proximity to the high-speed particle traversing a transparent medium.

Fig. 60

Depiction of the production of Cherenkov radiation in a dielectric medium and the resulting wave front expansion.The wave front spreading lengthens the excitation pulse on a time scale that is small in comparison to the fluorescence decay. t is the duration of the light pulse along a line parallel to the axis of the particle at a distance r from the axis. Taken from [19]

The Cherenkov radiation is propagated as a conical wave front, that is, the radiation is emitted as a cone having it’s axis in the direction of particle path. If charged particles move through an isolator of (initially constant) index of refraction n, the latter is always polarized by the electromagnetic field of the flying particle. Usually these polarization superpose destructively and the perturbation can relax while the particle moves on. But if the speed of the particle $\beta$ is higher than the phase velocity of light in the medium c / n the perturbation can no longer decay since the reaction time of the medium is not high enough and the perturbation remains behind the particle.

Fig. 61

Left polarization of an isolator caused by a charged particle traversing it with speed $v<c/n$. Right polarization of an isolator causing by a charged particle traversing it with speed $v>c/n$

Figure 61 shows the two types of polarization of charged particle traversing it with speed $v<c/n$ (left) or speed $v>c/n$ (right). The threshold condition for the production of Cherenkov radiation in a transparent medium is given by $\beta _{thr}=c/n$ where $\beta$ is the relative phase velocity of the particle, that is, the velocity of the particle divided by the speed of light in the vacuum c; n is the refractive index of the medium that is, for definition, the ratio of the velocity of light in the vacuum to its velocity in the medium. Only charged particles moving with velocity $\beta \ge \beta _{thr}=c/n$ produce Cherenkov photons in transparent media. For particles with velocity $\beta \ge \beta _{thr}$ the number of Cherenkov photons emitted per unit photon energy in a medium of particle path length L

$$\begin{aligned} \frac{dN}{dE}=L\frac{\alpha }{\hbar c}sin^{2}\theta \end{aligned}$$

where $\alpha =1/137$ is the fine structure constant, $\hbar =6.582\times 10^{-16} eV^{-1} \,\text {cm}^{-1}$ is Planck’s constant divided by $2\pi$. The number of photons emitted per path length, in the wavelength interval between $\lambda _{1}$ and $\lambda _{2}$, can be calculated according to the formula:

$$\begin{aligned} \frac{dN}{d\ell } = 2\pi \alpha z^{2}\left( \frac{1}{\lambda _{2}}-\frac{1}{\lambda _{1}}\right) sin^{2}\theta \end{aligned}$$

where z indicates the particle charge. Over the visible range of wavelengths from $\lambda _{1}=350$ nm to $\lambda _{2}=650$ nm, the number of photons N per path length L is

$$\begin{aligned} \frac{N}{L}=490 \cdot sin^{2} \theta \,\mathrm {cm^{-1}} \end{aligned}$$

It is easy to evaluate then that in $1\,\mathrm {cm}$ of water, a particle with $\beta = 1$ induces $N\sim 320$ photons in the spectral range of visible light.

Light propagation in the transparent medium

The propagation of light in water is affected by two phenomena: absorption and scattering. Both phenomena depend on the photon wavelength $\lambda$. In the first case the photon is lost, in the second one it changes its direction. Moreover, scattering delays the propagation of photons between the points of emission and detection. The parameters generally chosen to describe the two phenomena are:

• the absorption length $L_{a}(\lambda )$ (or its reciprocal: the absorption coefficient defined as $a(\lambda )=1/L_{a}(\lambda )$) that indicates the distance where the probability that a photon has not been absorbed amount to 1 / e, such that the number N of non-absorbed, photons as a function of photon path length x is given by:

  $$\begin{aligned} N(x)=N_{0} \cdot exp\left( -\frac{x}{L_{a}(\lambda )}\right) \end{aligned}$$

• the scattering length $L_{s}(\lambda )$ (or the scattering coefficient $s(\lambda )=1/L_{s}(\lambda )$), defined as the distance where the probability that a particle has not been scattered amount to 1/e and the number N of no-scattered photons as a function of photon path length x is given by:

  $$\begin{aligned} N(x)=N_{0} \cdot exp\left( -\frac{x}{L_{s}(\lambda )}\right) \end{aligned}$$

• the attenuation length $L_{c}(\lambda )$ (or the attenuation coefficient $c(\lambda )=a(\lambda )+b(\lambda )$), defined as the distance where the probability that a particle has not been neither absorbed nor scattered amount to 1/e and the number N of surviving photons as a function of photon path length x is given by:

  $$\begin{aligned} N(x)=N_{0} \cdot exp\left( -\frac{x}{L_{c}(\lambda )}\right) \end{aligned}$$

• the volume scattering function $\chi (\theta ,\lambda )$, i.e., the distribution in scattering angle $\theta$;

• when the photon detection area is sufficiently large that small scattering angle do not prevent the photon detection is used the geometrical, or effective scattering length defined as

  $$\begin{aligned} L_{s}^{eff}=\frac{L_{s}}{1-\langle cos(\theta ) \rangle } \end{aligned}$$

where $\langle cos(\theta ) \rangle$ is the mean value of the cosine of the scattering angle.

The knowledge of these parameters is very important for a proper comprehension of the Cherenkov propagation in the detector media. For these reasons all the experimental approaches need a direct measurement of these optical properties (Fig. 62).

Fig. 62

Absorption coefficient (left) and effective scattering coefficient (right) in the South Polar ice as functions of depth and wavelength [49]

Also in the water the propagation of light is quantified, for a given wavelength $\lambda$, by the water inherent optical properties (IOP): the absorption $a(\lambda )$, scattering $b(\lambda )$ and attenuation $c(\lambda )= a(\lambda ) + b(\lambda )$ coefficients. Figure 63 shows the absorption and attenuation lengths measured in the Mediterranean deep Sea site where will be located the KM3NeT-Italy future Cherenkov Neutrino Telescope.

Fig. 63

Average absorption and attenuation lengths [324] measured in December 1999 (blue circle), March 2002 (light blue square), May 2002 (purple triangle), August 2002 (red upsidedown triangle) and July 2003 (dark yellow star) and in the site where KM3NeT detector will be constructed, at depths 2850 (left) and $3250\,\mathrm {m}$ (right). Statistical errors are plotted. A solid black line indicates the values of $L_{a}(\lambda )$ and $L_{c}(\lambda )$ for optically pure seawater reported by Smith and Baker [325]

Neutrino interaction detection principle

The detection of the Cherenkov light is made possible by the arrangement of arrays of photo-multiplier tubes (PMTs) housed in transparent pressure spheres which are spread over a large volume in oceans, lakes or glacial ice (Fig. 64).

Fig. 64

Detection principle of high-energy muon neutrinos in an underwater/ice Neutrino Telescope. The incoming neutrino interacts with the material around the detector to create a muon. The muon induces the emission of Cherenkov light in the sea water. Photons are then detected by a matrix of light sensors. The original spectrum of light originated by the muon is attenuated in the water such that the dominant wavelength range detected is between 350 and 500 nm

For each PMT it is usually recorded the arrival time and amplitude, sometimes even the full waveform, of Cherenkov light induced by muons or particle showers. In most designs the spheres are attached to vertical strings. The typical PMT spacing along a string is 10–20 m, and the distance between adjacent strings 60–150 m. An underwater (or under-ice) neutrino telescope in principle is able to detect all neutrino types ($\nu _{\mu }, \nu _{e}, \nu _{\tau }$) CC or NC interacting inside or near the instrumented volume. It is possible to distinguish two main types of signatures: the track-like events, mainly from $\nu _{\mu }$ CC-interaction, and the shower-like ones from $\nu _{\mu , e, \tau }$ NC-interactions and $\nu _{e, \tau }$ CC-interactions.

Muons as high-energy neutrino signals

The CC-interaction of a $\nu _{\mu }$ produce a muon track. The passage of these muons through the seawater induces the emission of Cherenkov light, that can be then detected by a three dimensional array of photo-multiplier tubes (PMTs). The most precise measurement of the neutrino direction is achieved reconstructing the trajectory of the leading muon originated in $\nu _{\mu }$ CC-interactions; this channel is therefore central to all investigations of astrophysical neutrino sources. Using the time and position information of the detected photons, the muon trajectory can be reconstructed, from which the original neutrino direction can be inferred. Muons propagating in water/ice loose energy mainly by ionization, bremsstrahlung, pair production, photonuclear interactions, knock on electrons. Ionization energy losses are dominant at low energies and they are fairly constant and homogeneous over the track. The dE/dx is about 2 MeV/cm at 1 GeV in water. For muon energies above 100 GeV the main energy losses are due to the radiative processes like e+e− pair production, bremsstrahlung and photonuclear processes. These losses are strongly energy dependent and stochastic. Therefore, only an average total energy loss can be calculated. The value and the contributions of different processes to the energy losses for different muon energies are shown in Fig. 65.

Fig. 65

Muon energy loss in pure water as a function of its energy [296]

It is convenient to express the average energy loss as:

$$\begin{aligned} -\langle \frac{dE_{\mu }}{dx}\rangle =a(E_{\mu })+b(E_{\mu })\cdot E_{\mu } \end{aligned}$$

(183)

It is worth to mention that radiative processes may produce hadronic and electromagnetic cascades along the muon track, other relativistic particles that in water can induce erenkov light. As already mentioned underwater/ice telescopes are optimized for the detection of muon tracks at energies of the order TeV or above: muons originated by the CC-interactions of such energetic neutrinos may travel for km in water before to be stopped. The properties of the energy losses make possible the muon energy reconstruction: for low muon energies the track length is proportional to the particle energy, instead, for high energies, the amount of the detected light per track length may be used.

Appendix 6: High Energy $\gamma$-ray-Neutrino Connection

One can estimate an upper limit on the high-energy neutrino flux from a source of $\gamma$-rays, whose $\gamma$-rays are not subject to absorption, by assuming that all $\gamma$-rays are produced in pp collisions. When the $\gamma$-ray obey power-law distributions, we can use the formalism of the Z-factors described in the text [244, 326]. However, as was shown in various works and in particular in [327], more general cases can be treated with relative ease. There are different ways to proceed in the calculations. We mention here two formalisms that are particularly easy-to-use. In both cases, neutrino oscillations are included.

Exponential cut-off

In [328], it was shown that supposing that the primary proton spectrum is given by a power law with index $\alpha$ and an exponential cut-off energy $\varepsilon _p$,

$$ \frac{dN_p}{dE_p}= k_p \left( \frac{E_p}{1\text { TeV}} \right) ^{-\alpha } \exp \left( - \frac{E_p}{\varepsilon _p} \right) $$

the neutrino and gamma-ray spectra are approximatively described by the spectra,

$$ \frac{dN_{\gamma /\nu }}{dE_{\gamma /\nu }}= k_{\gamma /\nu } \left( \frac{E_{\gamma /\nu }}{1\text { TeV}} \right) ^{-\varGamma _{\gamma /\nu }} \exp \left( - \sqrt{ \frac{E_{\gamma /\nu }}{\varepsilon _{\gamma /\nu }}} \right) $$

where the parameters are given by,

$$ \begin{array}{l} k_\nu \approx (0.71-0.16 \alpha ) k_\gamma \\ \varGamma _\nu \approx \varGamma _\gamma \approx \alpha -0.1\\ \varepsilon _\nu \approx 0.59 \varepsilon _\gamma \approx \varepsilon _p/40 \end{array} $$

This method of calculation is based on [329]; it includes pion decays and oscillations in the simplest approximation.

Note the differences among the primary and the secondary spectra: (1) in the region where we have power-law distributions, the secondary spectra are a bit harder, due to scaling violation; (2) the cut-off of the proton spectrum is milder in the spectra of secondary particles, due to the contribution from lower energies particles. The overall coefficient of the secondary particles, however, depends upon the amount of target particles.

General case

Assume that the measured $\gamma$-ray spectrum $\phi _{\gamma } (E)$ is purely hadronic (due to cosmic ray-gas collisions) and that it is not affected by propagation. In [330], based on [327], it was shown that the muon neutrino (resp., antineutrino) spectrum are given by the precise relations

$$\begin{aligned} \phi _{\nu _\mu } (E) =\alpha _\pi \ \phi _{\gamma } \left( \textstyle \frac{E}{1-r_\pi } \right) +\alpha _K\ \phi _{\gamma } \left( \textstyle \frac{E}{1-r_K} \right) +\int _0^1{\frac{dx}{x}\ \mathscr {K}_{\nu _\mu } (x)\ \phi _{\gamma }\! \left( \frac{E}{x} \right) } \end{aligned}$$

(184)

where $\alpha _\pi =0.380\ (0.278)$ and $\alpha _K=0.013\ (0.009)$ for $\nu _\mu$ and $\bar{\nu }_\mu$, respectively, and where $r_x=(m_\mu /m_x)^2$ with $x=\pi ,K$. In each expression, the first two contributions describe neutrinos from the two-body decay by pions and kaons, while the third term accounts for neutrinos from muon decay. The kernels for muon neutrinos $\mathscr {K}_{\nu _\mu }(x)$ and for muon antineutrinos $\mathscr {K}_{\bar{\nu }_\mu }(x)$, which account also for oscillations from the source to the Earth, are

$$ \small { \mathscr {K}_{\nu _\mu }(x)\!=\! \left\{ \begin{array}{ll} x^2(15.34-28.93x) &{} 0<x \le r_K \\ 0.0165+0.1193x+3.747x^2-3.981x^3 &{} r_K<x< r_\pi \\ (1-x)^2(-0.6698+6.588x) &{} r_\pi \le x<1 \\ \end{array} \right. } $$

$$ \small { \mathscr {K}_{\bar{\nu }_\mu }(x)\!=\! \left\{ \begin{array}{ll} x^2(18.48-25.33x) &{} 0<x \le r_K \\ 0.0251+0.0826x+3.697x^2-3.548x^3 &{} r_K<x< r_\pi \\ (1-x)^2(0.0351+5.864x) &{} r_\pi \le x<1 \\ \end{array} \right. } $$

With this procedure, the expected upper bounds on the neutrino spectra are obtained from the measured $\gamma$-ray spectrum in a model-independent way. This method of calculations includes also kaons and it is presumably more accurate than the previous one.

In order to understand the reason why these formulae are valid, we note that the $\gamma$-rays from $\pi ^0$ decay are given by a very simple integral kernel, $F_\gamma (E)=\int _{E}^\infty dE'\, 2 F_{\pi ^0}(E')/E'$. This can be easily inverted [327],

$$ F_{\pi ^0}(E)=-\frac{E}{2} \frac{dF_\gamma }{dE} $$

Then, using the approximate isospin-invariant distribution of pions, $F_{\pi ^+}\approx F_{\pi ^-}\approx F_{\pi ^0}= F_\pi$, for the muon neutrinos from $\pi ^+\rightarrow \mu ^+ \nu _\mu$, we find immediately,

$$ F_{\nu _\mu }(E)=\int _{E/(1-r_\pi )}^\infty \frac{dE'}{1-r_\pi } \frac{F_\pi (E')}{E'}= \frac{F_\gamma (E/(1-r_\pi ))}{2(1-r_\pi )} $$

Some effort is necessary to describe the spectra of the leptons from the muon decay channel in a simple manner (namely to calculate the kernels $\mathscr {K}$ given above). Most other improvements (e.g., oscillations, other two-body decay channels, etc.) can be implemented easily and directly, instead.