# Physics Behind Neutrinos

[TOC]

# Outlines
- Profile:
- Mass: < 0.120 eV (< $2.14 × 10^{−37}$ kg, ~500,000 times smaller than electron)
- Speed: $\lesssim$ Speed of light
- Properties:
- Leptons
- Neutral
- Interact with weak, gravity only
- Oscillation: change flavor periodically with time
- Flavor problem: the origin of the fermion families and of the masses and mixing of quarks and leptons.
- Types (flavour):
- Muon, electron, tau
- Interactions:
- Couple with $Z^0$ boson
- Couple with $W^{+/-}$ boson
- $W^+$ → lepton(+) + neutrino
- $W^-$ → lepton(-) + anti-neutrino
- Productions:
- Electron neutrino (**beta decay**): $$n \rightarrow p + e^- + \bar{\nu}_e$$
- Muon neutrino:
$$\pi^+ \rightarrow \mu^+ + \bar{\nu}_{\mu}$$ $$\mu^+ \rightarrow e^+ + \nu_e + \bar{\nu}_{\mu}$$
- Tau neutrino:
$$\begin{split}
\tau^- &\rightarrow \nu_{\tau} + W^{-} \\
&\rightarrow \nu_{\tau} + e^{-} + \bar{\nu}_e \\
&\rightarrow \nu_{\tau} + \mu^- + \bar{\nu}_{\mu}
\end{split}$$

## Comparison of three flavors of neutrinos

| Property | Electron Neutrino ($\nu_e$) | Muon Neutrino ($\nu_\mu$) | Tau Neutrino ($\nu_\tau$) |
|-------------------------|---------------------------------------------------|------------------------------------------------------|------------------------------------------------------|
| **Symbol** | $\nu_e$ | $\nu_\mu$ | $\nu_\tau$ |
| **Associated Lepton** | Electron ($e^-$) | Muon ($\mu^-$) | Tau ($\tau^-$) |
| **Mass** | ~ $0.07$ eV/c$^2$ | < $0.17$ MeV/c$^2$ | < $18.2$ MeV/c$^2$ |
| **Charge** | 0 | 0 | 0 |
| **Flavor Oscillation** | Participates in neutrino oscillations | Participates in neutrino oscillations | Participates in neutrino oscillations |
| **Interaction** | Weak nuclear force | Weak nuclear force | Weak nuclear force |
| **Discovery** | 1956 (by Clyde Cowan and Frederick Reines) | 1962 (by Leon M. Lederman, Melvin Schwartz, and Jack Steinberger) | 2000 (by DONUT collaboration) |
| **Lifetime of Lepton** | Stable (electron) | $2.2 \times 10^{-6}$ seconds (muon) | $2.9 \times 10^{-13}$ seconds (tau) |
| **Production** | $\beta$ decay, nuclear reactions | Pion and kaon decays, cosmic rays | Heavy meson decays, high-energy processes |
| **Decay Channels** | N/A | $\mu^- \rightarrow e^- + \nu_\mu + \overline{\nu}_e$ | $\tau^- \rightarrow e^- + \nu_\tau + \overline{\nu}_e$, $\tau^- \rightarrow \mu^- + \nu_\tau + \overline{\nu}_\mu$ |

## Decay channels in SBN
In SBN program, the **Booster Neutrino Beam (BNB)** produces **8 GeV protons** that interact with a beryllium (Be) target, primarily generating **charged pions ($\pi^\pm$) and kaons ($K^\pm$)**. These hadrons decay into **muons, electrons, and neutrinos**, which are crucial for neutrino interaction studies. The main decay sequences are:

### 1. Charged-current (CC) interactions
- **Kaon Decays ($K^\pm$) → Pions ($\pi^\pm$)**
- **Charged kaons** primarily decay into pions:
$$
K^\pm \rightarrow \pi^\pm + \pi^0
$$
or directly into **muons and muon neutrinos**:
$$
K^\pm \rightarrow \mu^\pm + \nu_\mu
$$

- **Pion Decays ($\pi^\pm$) → Muons ($\mu^\pm$) and Muon Neutrinos ($\nu_\mu$)**
- The dominant decay mode for **charged pions**:
$$
\pi^\pm \rightarrow \mu^\pm + \nu_\mu
$$
- A **small fraction** (~1.2%) of pions decay directly into **electrons and electron neutrinos** (pion decay-in-flight):
$$
\pi^\pm \rightarrow e^\pm + \nu_e
$$

- **Muon Decays ($\mu^\pm$) → Electrons ($e^\pm$), Muon Neutrinos ($\nu_\mu$), and Electron Neutrinos ($\nu_e$)** i.e. **Michel decay**:
$$
\mu^\pm \rightarrow e^\pm + \nu_e + \bar{\nu}_\mu
$$

### 2. Neutral-current (NC) interactions (background sources)

- **Neutral Kaon Decays ($K^0$) → Multiple Channels**
- **Short-lived ($K_S^0$) and long-lived ($K_L^0$) neutral kaons** can decay into **photons, pions, and leptons**:
$$
K_L^0 \rightarrow \pi^0 + \pi^0 + \pi^0
$$
$$
K_L^0 \rightarrow \pi^\pm + e^\mp + \nu_e
$$

- **Neutral Pion Decays ($\pi^0$) → Photon Pairs**
- Neutral pions decay **electromagnetically**, primarily into two photons:

$$
\pi^0 \rightarrow \gamma + \gamma
$$
This is an important **background source** in electron neutrino appearance searches.

These decays contribute to **Neutral Current (NC) interactions**, where a neutrino scatters but **does not change flavor**.

BNB neutrino production channels:

### **Relevance to SBND and SBN Program**
- **Primary goal:** Detecting **neutrino interactions** from **charged meson decays (CC interactions)**.
- **Neutral meson decays** ($\pi^0$, $K^0$) are significant **backgrounds** for electron neutrino appearance analyses.
- The **SBN program** maximizes the neutrino flux at around **500 MeV**, optimizing for **Charged Current Quasi-Elastic (CCQE) interactions**, while **minimizing $\pi^0$ production** from **Neutral Current (NC) interactions**, which could mimic electron neutrino signals.

In SBND, BNB produces 8 GeV protons on Be target and thus produces primary charged pions and kaons. Given $\rightarrow$ represents the decay, we can simply conclude that:

* Kaons ($K$) $\rightarrow$ Pions ($\pi$)
* Pions ($\pi$) $\rightarrow$ Muons ($\mu$) and muon neutrinos ($\nu_{\mu}$)
* Pions ($\pi$) $\rightarrow$ Electrons ($e$) and electron neutrinos ($\nu_{e}$)
* Muons ($\mu$) $\rightarrow$ Electrons ($e$), muon neutrinos ($\nu_{\mu}$), and electron neutrinos ($\nu_{e}$)

These are the main possible channels, where the primary goal is to detect the neutrino interctions. SBN program typically maximizes neutrino flux at around 500 MeV to reduce $\pi^0$ production from NC interactions.

Feynman diagrams about common decay channels:

## Comparison of common particles in this research

| Property | Electron ($e^-$) | Muon ($\mu^-$) | Pion ($\pi^\pm$, $\pi^0$) | Kaon ($K^\pm$, $K^0$) | Proton ($p^+$) | $\Delta$ Baryons ($\Delta^{++}$, $\Delta^+$, $\Delta^0$, $\Delta^-$) |
|-------------------------|---------------------------------------------------|------------------------------------------------------|----------------------------------------------------|------------------------------------------------------|------------------------------------------------------|-------------------------------------------------------|
| **Mass** | $0.511$ MeV/c$^2$ | $105.66$ MeV/c$^2$ | $139.57$ MeV/c$^2$ ($\pi^\pm$), $134.98$ MeV/c$^2$ ($\pi^0$) | $493.67$ MeV/c$^2$ ($K^\pm$), $497.61$ MeV/c$^2$ ($K^0$) | $938.27$ MeV/c$^2$ | $1232$ MeV/c$^2$ |
| **Charge** | $-1$ | $-1$ | $+1$ ($\pi^+$), $-1$ ($\pi^-$), $0$ ($\pi^0$) | $+1$ ($K^+$), $-1$ ($K^-$), $0$ ($K^0$) | $+1$ | $+2$ ($\Delta^{++}$), $+1$ ($\Delta^+$), $0$ ($\Delta^0$), $-1$ ($\Delta^-$) |
| **Spin** | $\frac{1}{2}$ | $\frac{1}{2}$ | $0$ | $0$ | $\frac{1}{2}$ | $\frac{3}{2}$ |
| **Lifetime** | Stable | $2.2 \times 10^{-6}$ s | $2.6 \times 10^{-8}$ s ($\pi^\pm$), $8.4 \times 10^{-17}$ s ($\pi^0$) | $1.24 \times 10^{-8}$ s ($K^\pm$), $5.1 \times 10^{-8}$ s ($K^0$) | Stable | $\sim 10^{-23}$ s |
| **Decay Modes** | N/A | $\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu$ | $\pi^+ \rightarrow \mu^+ + \nu_\mu$, $\pi^0 \rightarrow \gamma + \gamma$ | $K^+ \rightarrow \mu^+ + \nu_\mu$, $K^0 \rightarrow \pi^+ + \pi^-$ | N/A | $\Delta \rightarrow p + \pi$, $\Delta \rightarrow n + \pi$ |
| **Interaction** | Electromagnetic, Weak | Electromagnetic, Weak | Strong, Electromagnetic, Weak | Strong, Electromagnetic, Weak | Strong, Electromagnetic, Weak | Strong, Electromagnetic, Weak |
| **Quark Composition** | N/A | N/A | $u\overline{d}$ ($\pi^+$), $d\overline{u}$ ($\pi^-$), $u\overline{u}$ or $d\overline{d}$ ($\pi^0$) | $u\overline{s}$ ($K^+$), $s\overline{u}$ ($K^-$), $d\overline{s}$ ($K^0$), $\overline{d}s$ ($\overline{K}^0$) | $uud$ | $uuu$ ($\Delta^{++}$), $uud$ ($\Delta^+$), $udd$ ($\Delta^0$), $ddd$ ($\Delta^-$) |


---

# Cross Section Physics in Particle Physics

## Cross Section

The **cross section** $\sigma$ measures the probability of a specific interaction between particles, with dimensions of area (units: barns, $1 \text{ barn} = 10^{-28} \text{ m}^2$).

### Basic Formula
$$ R = n \Phi \sigma $$
- $R$: rate of interactions
- $n$: number density of target particles
- $\Phi$: incident particle flux

#### Differential Cross Section

The **differential cross section** $\frac{d\sigma}{d\Omega}$ describes the scattering probability as a function of the scattering angle $\theta$ (and azimuthal angle $\phi$).

#### Definition
$$ \frac{d\sigma}{d\Omega} = \frac{1}{\Phi} \frac{dN}{d\Omega} $$
- $\frac{dN}{d\Omega}$: number of particles scattered into solid angle $d\Omega$
- $\Phi$: incident particle flux

#### Solid Angle
$$ d\Omega = \sin \theta \, d\theta \, d\phi $$

#### Scattering Amplitude
$$ \frac{d\sigma}{d\Omega} = |f(\theta, \phi)|^2 $$
- $f(\theta, \phi)$: scattering amplitude

#### Example: Rutherford Scattering
$$ \frac{d\sigma}{d\Omega} = \left( \frac{Z_1 Z_2 e^2}{16 \pi \epsilon_0 E} \right)^2 \frac{1}{\sin^4(\theta/2)} $$
- $Z_1, Z_2$: atomic numbers of projectile and target
- $e$: elementary charge
- $\epsilon_0$: permittivity of free space
- $E$: kinetic energy of incident particle

### Frames of Reference

#### Lab Frame
- The **lab frame** is where the target particle is at rest and the projectile is moving.

#### Center of Mass (COM) Frame
- The **COM frame** is where the total momentum of the system is zero.
- Transformations between lab and COM frames involve Jacobian factors.

### Kinematic Variables

### Mandelstam Variables
- $s$: square of the total energy in the COM frame.
- $t$: square of the momentum transfer.
- $u$: another kinematic variable related to energy and momentum of particles.

### Collider Physics

#### Luminosity
- **Luminosity** $\mathcal{L}$ quantifies the number of particles per unit area per unit time.
- Interaction rate: $R = \mathcal{L} \sigma$

#### Detector Efficiency
- Efficiency of detectors must be accounted for in measuring scattering events.

### Scattering Angles

#### Lab Frame to COM Frame
$$ \cos \theta_{\text{lab}} = \frac{\cos \theta_{\text{cm}} + \beta}{1 + \beta \cos \theta_{\text{cm}}} $$
- $\beta = \frac{v}{c}$ where $v$ is the velocity of the COM frame relative to the lab frame.

#### Transformations Between Frames
$$ \left( \frac{d\sigma}{d\Omega} \right)_{\text{lab}} = \left( \frac{d\sigma}{d\Omega} \right)_{\text{cm}} \left| \frac{d(\cos \theta_{\text{cm}})}{d(\cos \theta_{\text{lab}})} \right| $$

### Practical Considerations

- **Quantum Field Theory (QFT)**: Cross sections are calculated using QFT techniques, like Feynman diagrams.
- **Partial Wave Analysis**: Analyzes scattering process in terms of angular momentum components.
- **Unitarity and Optical Theorem**: Relates total cross section to the imaginary part of the forward scattering amplitude.

### Summary

Understanding cross sections in particle physics involves both theoretical principles and experimental techniques. Key concepts include the differential cross section, frames of reference, kinematic variables, luminosity, and detector efficiency.

### Comments
In particle physics, calculating differential cross section $d\sigma/d\Omega$ is quite often. **Cross section** basically represents the *probability* that the interaction among particles would happen. The rough steps are as the following:
1. Have a general picture about how a particle/particles generated.
2. Sketch possible Feynman diagrams (i.e. the scattering diagram in the CoM frame), and indicate the incoming/outgoing momenta. These will show the initial and final states from possible interactions.
3. Time to use QFT knowledge: write down the Feynman rules, consider the particles and the interactions and combine them together.
4. Above will give the matrix element $\mathcal{M}$, this tells us information about the *transition* between quantum states (i.e. how the interaction happens), and we have to know $|\mathcal{M}|^2$, note that:
- For those diagrams that the final states are the same but we don't know what happened in the intermediate states, we can add together and square i.e. $|\mathcal{M}|^2 = |\mathcal{M_1}+\mathcal{M_2}+...|^2$
- But for those diagrams that the final state particles are different, we will need to consider all possible interactions to add together i.e. $|\mathcal{M}|^2 = |\mathcal{M_1}|^2 + |\mathcal{M_2}|^2+...$
5. The scattering matrix element will be used to determine the differnetial cross section.
$$
\frac{d\sigma}{d\Omega} \propto |\mathcal{M}|^2
$$
E.g. The simplest case: for a 2-to-2 scattering process, $a + b \rightarrow c + d$, the differential cross section with respect to the solid angle $d\Omega$ is:
$$
\frac{d\sigma}{d\Omega} = \frac{1}{64 \pi^2 s} \frac{|\mathbf{p}_c|}{|\mathbf{p}_a|}|\mathcal{M}|^2
$$
6. Then we can calculate the total cross section $\sigma$ by taking an integral. For the 2-to-2 scattering process, this is just
$$
\sigma_{\text{total}} = \frac{1}{64 \pi^2 s} \int |\mathcal{M}|^2 \frac{|\mathbf{p}_c|}{|\mathbf{p}_a|} \, d\Omega
$$
This is basically equivalent to the process that we sum over *all* interaction probabilities.
7. The transition probability is proportional to the absolute square of the scattering amplitude,
$$
P \propto |\mathcal{M}|^2
$$
That is why cross section is a probabilistic concept.

---

# Neutrino-Nucleus Scattering
- Types:
- Energy: Elastic vs. Inelastic
- Phase/fluctuation: Coherent vs. Incoherent
- Types based on current:
- Charged current (CC) cross sections:
$$\nu_l + N \rightarrow l^- + X \quad\quad \bar{\nu}_l + N \rightarrow l^+ + X$$
- Here $N$ represents nucleons (protons or neutrons), $X$ represents hadrons (possible hadronic systems).
- CC exclusive: Specific hadronic final states, i.e. CC $\nu_{\mu}$ with a single proton ($p$), with a single pion ($\pi^+$), etc.
- CC inclusive: Sum the contributions from all exclusive channels.
- Neutral current (NC) cross sections:
$$\nu_l + N \rightarrow \nu_l + X \quad\quad \bar{\nu}_l + N \rightarrow \bar{\nu}_l + X$$
- Three types of neutrino-nucleon interactions:
- **Quasi-elastic (QE)**
- **Hadronic resonance production (RES)**
- **Deep inelastic scattering (DIS)**


In short: neutrino-nucleon interactions under different energy scales result in distinct final states.

## 1. Quasi-Elastic (QE) Scattering
- **Energy scale**: Typically occurs at **low to moderate energies** (up to a few GeV).
- **Interaction**: A neutrino interacts with a **single nucleon** (proton or neutron) inside the nucleus.
- **Final State**: The final state usually involves only a **charged lepton** (e.g., electron or muon) and a **nucleon** (proton or neutron). The nucleus is often left intact, except for the removal of the nucleon.
- **Example**: For neutrino-nucleon QE scattering:
$$\nu_\mu + n \rightarrow \mu^- + p$$

(neutrino converts a neutron to a proton, producing a muon).
- **Signature**: No production of hadrons beyond the outgoing nucleon.
- **Kinematics**: Simplest, with clear lepton and nucleon energy-momentum relations.

## 2. Hadronic Resonance Production (RES)
- **Energy scale**: Typically occurs at **moderate energies**, between QE and DIS (~1 to several GeV).
- **Interaction**: A neutrino excites a nucleon to a higher energy state (a **resonance**, such as the Δ resonance).
- **Final State**: The resonance decays into a **nucleon and a pion** (or other hadrons), resulting in the production of **one or more mesons** (often pions) in addition to the charged lepton.
- **Example**:
$$\nu_\mu + p \rightarrow \mu^- + \Delta^{++} \rightarrow \mu^- + p + \pi^+$$
- **Signature**: Presence of hadronic particles (e.g., pions) in the final state, in addition to the lepton.
- **Kinematics**: More complicated than QE, with the additional hadronic resonance and decay products.

## 3. Deep Inelastic Scattering (DIS)
- **Energy scale**: Occurs at **high energies** (several GeV and above).
- **Interaction**: The neutrino scatters off a quark inside a nucleon (proton or neutron), leading to the **breakup** of the nucleon.
- **Final State**: Multiple hadrons (quarks, gluons) are produced as the nucleon fragments. The final state often includes a **jet of hadrons**, in addition to the outgoing lepton.
- **Example**:
$$\nu_\mu + p \rightarrow \mu^- + X$$

where \(X\) represents the hadronic shower of quarks and gluons.
- **Signature**: Multiple hadrons in the final state, often forming jets, in addition to the lepton.
- **Kinematics**: Complex, with significant energy transfers leading to particle showers.

## Summary of Differences

| **Interaction Type** | **Energy Range** | **Final State** | **Typical Products** | **Nature of Interaction** |
|-------------------------|---------------------------|-------------------------------|-----------------------------------------|-----------------------------------|
| **Quasi-Elastic (QE)** | Low to moderate (few GeV) | Lepton + nucleon | Charged lepton (e.g., $e^-, \mu^-$) + nucleon ($p, n$) | Scattering off a single nucleon |
| **Resonance (RES)** | Moderate (~1 to several GeV) | Lepton + nucleon + meson(s) | Charged lepton (e.g., $e^-, \mu^-$) + nucleon ($p, n$) + mesons ($\pi$) | Excitation of nucleon to resonance |
| **Deep Inelastic (DIS)** | High (>several GeV) | Lepton + hadronic shower | Charged lepton (e.g., $e^-, \mu^-$) + multiple hadrons | Scattering off quarks in nucleon |

Image: Total cross section per nucleon for muon neutrino (left)/antineutrino(right).