---
# System prepended metadata

title: Neutrino Physics

---

# 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:
![Screenshot 2025-02-10 at 1.42.59 PM](https://hackmd.io/_uploads/Hy8a_TPYkx.png)


### **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:
![image](https://hackmd.io/_uploads/Bk-PB6DYye.png)
![image](https://hackmd.io/_uploads/Hkosravtkl.png)
![image](https://hackmd.io/_uploads/rkxKBTDFJe.png)




## 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 $d\overline{d}$ ($\pi^0$) | $u\overline{s}$ ($K^+$), $s\overline{u}$ ($K^-$), $d\overline{s}$ ($K^0$), $\overline{d}s$ ($\overline{K}^0$) | $uud$                                                | $uuu$ ($\Delta^{++}$), $uud$ ($\Delta^+$), $udd$ ($\Delta^0$), $ddd$ ($\Delta^-$) |

![Standard_Model_of_Elementary_Particles](https://hackmd.io/_uploads/BkSROytQA.svg)



---


# 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)**

![Screenshot 2024-08-15 at 2.05.38 PM](https://hackmd.io/_uploads/r1N9ZCj50.png)

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  |

![Screenshot 2024-09-18 at 7.21.53 PM](https://hackmd.io/_uploads/HyR9lktaA.png)
Image: Total cross section per nucleon for muon neutrino (left)/antineutrino(right).




