University Quantum Field Theory I · topic UQFTI-05.01
From |M|^2 to a rate: the squared delta function, wave packets and the flux factor
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.
From |M|² to a rate: squared deltas and the flux factor: predict a finite change
A cross-section is the transition rate per target particle divided by the incident flux; in relativistic normalisation it is dσ = |M|² dΠₙ/(4E₁E₂|v₁ − v₂|), the flux factor 4E₁E₂|v₁ − v₂| undoing the 2E-per-volume normalisation of the two incoming states. Use the supplied topic model Y_out = ((((p+sqrt(p^2+m1^2))/m1)^(2-2*f)-1)/(((p+sqrt(p^2+m1^2))/m1)^(2-2*f)+1)+(((p+sqrt(p^2+m1^2))/m1)^(2*f)-1)/(((p+sqrt(p^2+m1^2))/m1)^(2*f)+1)), where Y_out is |v₁ − v₂| IN THIS FRAME (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. p is Beam momentum in the target frame p, measured in GeV. Keep m1 (Beam particle mass m₁) = 0.14 GeV; f (Observer's boost along the beam, η/Y) = 0 fixed. For this exercise use only 2.57187 GeV ≤ p ≤ 17.9281 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 p = 5.95025 GeV and 8.10012 GeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δp across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
|v₁ − v₂| IN THIS FRAME against p
Predictions from the supplied From |M|² to a rate: squared deltas and the flux factor 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): |v₁ − v₂| IN THIS FRAME
Horizontal axis (x): p: Beam momentum in the target frame p / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Beam momentum in the target frame p / GeV | |v₁ − v₂| IN THIS FRAME |
|---|---|
| 2.571875 | 0.998521700246677 |
| 6.410937499999999 | 0.9997616431197397 |
| 10.249999999999998 | 0.9999067352382695 |
| 14.089062499999999 | 0.9999506337970656 |
| 17.928124999999998 | 0.9999695114710327 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of p separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At p = 5.95025 GeV, substitute into Y_out = ((((p+sqrt(p^2+m1^2))/m1)^(2-2*f)-1)/(((p+sqrt(p^2+m1^2))/m1)^(2-2*f)+1)+(((p+sqrt(p^2+m1^2))/m1)^(2*f)-1)/(((p+sqrt(p^2+m1^2))/m1)^(2*f)+1)): Y₀ = 0.999723.
- At p = 8.10012 GeV, the same fixed-parameter model gives Y₁ = 0.999851.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.000127349; the input change is Δp = 2.14987 GeV.
- Average sensitivity is ΔY_out/Δp = 0.0000592356 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 = 0.999723, Y₁ = 0.999851; ΔY_out = 0.000127349; average sensitivity = 0.0000592356 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.
From |M|² to a rate: squared deltas and the flux factor: infer an input with uncertainty
A cross-section is the transition rate per target particle divided by the incident flux; in relativistic normalisation it is dσ = |M|² dΠₙ/(4E₁E₂|v₁ − v₂|), the flux factor 4E₁E₂|v₁ − v₂| undoing the 2E-per-volume normalisation of the two incoming states. Use the supplied topic model Y_out = ((((p+sqrt(p^2+m1^2))/m1)^(2-2*f)-1)/(((p+sqrt(p^2+m1^2))/m1)^(2-2*f)+1)+(((p+sqrt(p^2+m1^2))/m1)^(2*f)-1)/(((p+sqrt(p^2+m1^2))/m1)^(2*f)+1)), where Y_out is |v₁ − v₂| IN THIS FRAME (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. p is Beam momentum in the target frame p, measured in GeV. Keep m1 (Beam particle mass m₁) = 0.14 GeV; f (Observer's boost along the beam, η/Y) = 0 fixed. For this exercise use only 2.57187 GeV ≤ p ≤ 17.9281 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 = 0.99993126, with a bounded reading uncertainty ±0.0000057377. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown p by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00153562 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.
|v₁ − v₂| IN THIS FRAME against p
Predictions from the supplied From |M|² to a rate: squared deltas and the flux factor 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): |v₁ − v₂| IN THIS FRAME
Horizontal axis (x): p: Beam momentum in the target frame p / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Beam momentum in the target frame p / GeV | |v₁ − v₂| IN THIS FRAME |
|---|---|
| 2.571875 | 0.998521700246677 |
| 6.410937499999999 | 0.9997616431197397 |
| 10.249999999999998 | 0.9999067352382695 |
| 14.089062499999999 | 0.9999506337970656 |
| 17.928124999999998 | 0.9999695114710327 |
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(p) = 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(2.57187) = 0.998522 and Y_out(17.9281) = 0.99997. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives p_line = 2.57187 + (0.999931 − 0.998522)(17.9281 − 2.57187)/(0.99997 − 0.998522) = 17.5224 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 p = 11.9395 GeV; substituting back gives Y_out = 0.99993126.
- The interpolation error in the input is |p_line − p| = 5.58292 GeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.99992552 and 0.999937. Sort the roots even if the response decreases: 11.4703 GeV ≤ p ≤ 12.4714 GeV.
- The lower and upper input deviations are 0.46917 GeV and 0.531881 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 p ≈ 11.9395 GeV; compatible interval [11.4703, 12.4714] GeV. Endpoint interpolation gives 17.5224 GeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.