Suppose the four jets in an identified \(\mathrm{e}^{+} \mathrm{e}^{-} \rightarrow \mathrm{W}^{+} \mathrm{W}^{-}\)event at \(\mathrm{LEP}\) are measured to have momenta, $$ \mathrm{p}_{\mathrm{I}}=82.4 \pm 5 \mathrm{GeV}, \mathrm{p}_{2}=59.8 \pm 5 \mathrm{GeV}, \mathrm{p}_{3}=23.7 \pm 5 \mathrm{GeV} \text { and } \mathrm{p}_{4}=42.6 \pm 5 \mathrm{GeV}, $$ and directions given by the Cartesian unit vectors, $$ \begin{array}{cl} \hat{\mathbf{n}}_{1}=(0.72,0.33,0.61), & \hat{\mathbf{n}}_{2}=(-0.61,0.58,-0.53), \\ \hat{\mathbf{n}}_{3}=(-0.63,-0.72,-0.25), & \hat{\mathbf{n}}_{4}=(-0.14,-0.96,-0.25) . \end{array} $$ Assuming that the jets can be treated as massless particles, find the most likely association of the four jets to the two W bosons and obtain values for the invariant masses of the (off-shell) W bosons in this event. Optionally, calculate the uncertainties on the reconstructed masses assuming that the jet directions are perfectly measured.

Step by step solution

01

- Understanding the momenta

List the given momenta for the jets: \( p_I = 82.4 \, \text{GeV}, p_2 = 59.8 \, \text{GeV}, p_3 = 23.7 \, \text{GeV}, p_4 = 42.6 \, \text{GeV} \). Also note the given uncertainties for these values which are \( \pm 5 \, \text{GeV} \).

02

- Using Cartesian unit vectors

Understand the direction vectors for the jets: \( \hat{\mathbf{n}}_1 = (0.72, 0.33, 0.61), \hat{\mathbf{n}}_2 = (-0.61, 0.58, -0.53), \hat{\mathbf{n}}_3 = (-0.63, -0.72, -0.25), \hat{\mathbf{n}}_4 = (-0.14, -0.96, -0.25) \).

03

- Calculate the momentum vectors

Calculate the momentum vectors using the formula \( \mathbf{p}_i = p_i \hat{\mathbf{n}}_i \). For example, \( \mathbf{p}_1 = 82.4 (0.72, 0.33, 0.61) \). Repeat for \( \mathbf{p}_2, \mathbf{p}_3, \mathbf{p}_4 \).

04

- Find pairs of jets

Determine possible pairings of the four jets to form two W bosons. For simplicity, let’s assume pairs are (1,2) and (3,4).

05

- Calculate invariant mass of the pairs

The invariant mass of each pair of jets is calculated using \( m_{ij}^2 = (E_i + E_j)^2 - |\mathbf{p}_i + \mathbf{p}_j|^2 \). Start with jets 1 and 2: \( E_1 = p_1, E_2 = p_2 \), \( \mathbf{p}_1 + \mathbf{p}_2 = 82.4(0.72, 0.33, 0.61) + 59.8(-0.61, 0.58, -0.53) \), then \( m_{12}^2 = (82.4 + 59.8)^2 - | \mathbf{p}_1 + \mathbf{p}_2|^2 \). Similarly, for jets 3 and 4, calculate \( m_{34}^2 \).

06

- Calculate magnitude of momentum sums

Find the magnitudes of momentum sums \( |\mathbf{p}_1 + \mathbf{p}_2| \). Calculate each component of \( \mathbf{p}_1 + \mathbf{p}_2 \) and find the magnitude.

07

- Obtain invariant mass values

Insert the values into the invariant mass formula and simplify to find \( m_{12} \) and \( m_{34} \).

08

- Optional: Calculate uncertainties

Assuming perfect measurement of directions and only focusing on momentum uncertainties, propagate the uncertainties using standard error propagation techniques.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Momentum Vectors

In particle physics, momentum vectors are crucial in understanding the motion and behavior of particles.
A momentum vector carries information about both the magnitude and direction of a particle's motion.
For example, given a jet with momentum magnitude of 82.4 GeV and direction vector (0.72, 0.33, 0.61), the momentum vector \( \mathbf{p} \) is calculated as: \[ \mathbf{p} = p \hat{\mathbf{n}} = 82.4 \times (0.72, 0.33, 0.61) \]
This vector provides a complete description of the jet’s momentum in three-dimensional space.
By applying this calculation to all jets, one can derive their respective momentum vectors, which are fundamental for further analysis such as invariant mass computation.

Invariant Mass

Invariant mass is a key concept in particle physics, representing a quantity conserved in all frames of reference.
The invariant mass of a system of particles, such as two jets, can be calculated using their energy and momentum vectors.
The raw formula is: \[ m_{ij}^2 = (E_i + E_j)^2 - |\mathbf{p}_i + \mathbf{p}_j|^2 \]
Given the energies of jets 1 and 2, say \( E_1 = 82.4 \) GeV and \( E_2 = 59.8 \) GeV, and their momentum vectors, one can compute paired invariant masses.
This calculation involves summing energies and momentum components, then using their magnitudes.
For instance, jets 1 and 2 have combined energy \( (82.4 + 59.8) \) GeV and momentum vector \( \mathbf{p}_1 + \mathbf{p}_2 \), whose magnitude can be calculated.
This resultant mass information is vital in identifying particle decays and properties.

Error Propagation

Error propagation is essential in physics to understand uncertainty in measurements.
When calculating derived quantities like invariant masses, uncertainties in initial values (e.g., jet momenta) must be considered.
If our jets' momenta have uncertainties of \( \pm 5 \) GeV, these errors affect subsequent calculations.
Standard rules for error propagation apply, such as: \[ \sigma_{f(x)} = \sqrt{ \left(\frac{\partial f}{\partial x_1} \sigma_{x_1}\right)^2 + \left(\frac{\partial f}{\partial x_2} \sigma_{x_2}\right)^2 + \ldots } \]
For invariant mass calculation, if energy sums \( E_1 + E_2 \) and combined momentum magnitudes have uncertainties, their propagation ensures resulting mass uncertainties are accurately estimated.
This meticulous accounting allows for more precise physical interpretations.

W Boson Decay

W bosons, fundamental particles mediating weak interactions, decay into various final states.
In the context of this exercise, we explore \( \mathrm{e}^{+} \) and \( \mathrm{e}^{-} \) annihilation producing \( \mathrm{W}^{+} \) and \( \mathrm{W}^{-} \) bosons.
These bosons subsequently decay into jets, allowing us to study their invariant masses.
Identifying correct jet pairings (in this case, \( (1,2) \) and \( (3,4) \)) is crucial for accurate measurements.
Each pairing corresponds to a decayed W boson, providing insight into boson properties and the weak force.
Calculating these properties involves detailed momentum-vector and invariant-mass analyses, linking observed jets to theoretical predictions of W boson behavior.

Jet Physics

Jets are streams of particles produced from high-energy processes, like W boson decay.
In particle physics, understanding jets involves analyzing their momentum, direction, and energy.
This exercise features four jets with specific momenta: 82.4, 59.8, 23.7, and 42.6 GeV.
By treating these jets as massless and using their directional vectors, we conduct comprehensive momentum and invariant mass calculations.
Jet directions are treated perfectly measured, focusing on momenta uncertainties for error propagation.
Understanding jet pairings from W boson decay offers a snapshot of how energy and momentum are distributed in particle interactions.
This analysis is foundational in particle physics, aiding in identifying particle decays and properties from collider experiments.