University Physics V · topic UPV-19.06
Lepton number, family number, and where family number fails
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Lepton number and family number: predict a finite change
Lepton family number L_α (α = e, μ, τ) counts each lepton of that family as +1 and each antilepton as −1, summed over every particle in a state; the Standard Model with massless neutrinos conserves each L_α separately, and therefore total L = L_e + L_μ + L_τ as well. Use the supplied topic model Y_out = 1-s*sin(0.0031675*b/en)^2, where Y_out is P(ν_μ → ν_μ) AT L (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; unmarked trigonometric functions use radians. en is Beam energy E, measured in GeV. Keep s (Mixing sin²2θ) = 1; b (Detector baseline L) = 295 km fixed. For this exercise use only 0.65775 GeV ≤ en ≤ 2.71312 GeV on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at en = 1.10993 GeV and 1.39769 GeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δen across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
P(ν_μ → ν_μ) AT L against en
Predictions from the supplied Lepton number and family number model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): P(ν_μ → ν_μ) AT L
Horizontal axis (x): en: Beam energy E / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| en: Beam energy E / GeV | P(ν_μ → ν_μ) AT L |
|---|---|
| 0.6577500000000001 | 0.022384030459218107 |
| 1.17159375 | 0.48784261468771073 |
| 1.6854375 | 0.7228648572236664 |
| 2.19928125 | 0.8300879047235048 |
| 2.713125 | 0.8860017822108844 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of en separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At en = 1.10993 GeV, substitute into Y_out = 1-s*sin(0.0031675*b/en)^2: Y₀ = 0.443654.
- At en = 1.39769 GeV, the same fixed-parameter model gives Y₁ = 0.615794.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.17214; the input change is Δen = 0.287753 GeV.
- Average sensitivity is ΔY_out/Δen = 0.598224 1 per GeV. Its positive sign means that this response increases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 0.443654, Y₁ = 0.615794; ΔY_out = 0.17214; average sensitivity = 0.598224 1 per GeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Lepton number and family number: infer an input with uncertainty
Lepton family number L_α (α = e, μ, τ) counts each lepton of that family as +1 and each antilepton as −1, summed over every particle in a state; the Standard Model with massless neutrinos conserves each L_α separately, and therefore total L = L_e + L_μ + L_τ as well. Use the supplied topic model Y_out = 1-s*sin(0.0031675*b/en)^2, where Y_out is P(ν_μ → ν_μ) AT L (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; unmarked trigonometric functions use radians. en is Beam energy E, measured in GeV. Keep s (Mixing sin²2θ) = 1; b (Detector baseline L) = 295 km fixed. For this exercise use only 0.65775 GeV ≤ en ≤ 2.71312 GeV on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.77948223, with a bounded reading uncertainty ±0.015978. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown en by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000205537 GeV, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
P(ν_μ → ν_μ) AT L against en
Predictions from the supplied Lepton number and family number model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): P(ν_μ → ν_μ) AT L
Horizontal axis (x): en: Beam energy E / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| en: Beam energy E / GeV | P(ν_μ → ν_μ) AT L |
|---|---|
| 0.6577500000000001 | 0.022384030459218107 |
| 1.17159375 | 0.48784261468771073 |
| 1.6854375 | 0.7228648572236664 |
| 2.19928125 | 0.8300879047235048 |
| 2.713125 | 0.8860017822108844 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(en) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(0.65775) = 0.022384 and Y_out(2.71312) = 0.886002. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives en_line = 0.65775 + (0.779482 − 0.022384)(2.71312 − 0.65775)/(0.886002 − 0.022384) = 2.45961 GeV. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives en = 1.91153 GeV; substituting back gives Y_out = 0.77948223.
- The interpolation error in the input is |en_line − en| = 0.548084 GeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.76350423 and 0.79546023. Sort the roots even if the response decreases: 1.8399 GeV ≤ en ≤ 1.99108 GeV.
- The lower and upper input deviations are 0.0716264 GeV and 0.0795548 GeV. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer en ≈ 1.91153 GeV; compatible interval [1.8399, 1.99108] GeV. Endpoint interpolation gives 2.45961 GeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.