University Quantum Field Theory I · topic UQFTI-05.02
Lorentz-invariant phase space, and two bodies in the centre-of-mass frame
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.
Lorentz-invariant phase space and two-body final states: predict a finite change
Lorentz-invariant phase space dΠₙ is the measure over the momenta of n final particles, d³p/((2π)³ 2E) for each one, constrained by (2π)⁴δ⁴ of total four-momentum, that counts final states with the same weight in every inertial frame. Use the supplied topic model Y_out = sqrt(max(0,(m1+m2+q)^2-(m1+m2)^2)*max(0,(m1+m2+q)^2-(m1-m2)^2))/(2*(m1+m2+q))/(4*pi*(m1+m2+q)), where Y_out is ∫dΠ₂ DISTINCT PAIR (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. q is Energy release Q = √s − m₁ − m₂, measured in GeV. Keep m1 (Mass m₁ of product 1) = 0.6 GeV; m2 (Mass m₂ of product 2) = 0.35 GeV fixed. For this exercise use only 0.310312 GeV ≤ q ≤ 2.23969 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 q = 0.734775 GeV and 1.00489 GeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δq across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
∫dΠ₂ DISTINCT PAIR against q
Predictions from the supplied Lorentz-invariant phase space and two-body final states 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): ∫dΠ₂ DISTINCT PAIR
Horizontal axis (x): q: Energy release Q = √s − m₁ − m₂ / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| q: Energy release Q = √s − m₁ − m₂ / GeV | ∫dΠ₂ DISTINCT PAIR |
|---|---|
| 0.3103125 | 0.02562662415303772 |
| 0.7926562499999998 | 0.03301153201880001 |
| 1.275 | 0.03575182521782777 |
| 1.7573437499999998 | 0.037099533534971786 |
| 2.2396874999999996 | 0.03786617489029977 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of q separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At q = 0.734775 GeV, substitute into Y_out = sqrt(max(0,(m1+m2+q)^2-(m1+m2)^2)*max(0,(m1+m2+q)^2-(m1-m2)^2))/(2*(m1+m2+q))/(4*pi*(m1+m2+q)): Y₀ = 0.0324962.
- At q = 1.00489 GeV, the same fixed-parameter model gives Y₁ = 0.034489.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.00199285; the input change is Δq = 0.270112 GeV.
- Average sensitivity is ΔY_out/Δq = 0.00737785 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 = 0.0324962, Y₁ = 0.034489; ΔY_out = 0.00199285; average sensitivity = 0.00737785 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.
Lorentz-invariant phase space and two-body final states: infer an input with uncertainty
Lorentz-invariant phase space dΠₙ is the measure over the momenta of n final particles, d³p/((2π)³ 2E) for each one, constrained by (2π)⁴δ⁴ of total four-momentum, that counts final states with the same weight in every inertial frame. Use the supplied topic model Y_out = sqrt(max(0,(m1+m2+q)^2-(m1+m2)^2)*max(0,(m1+m2+q)^2-(m1-m2)^2))/(2*(m1+m2+q))/(4*pi*(m1+m2+q)), where Y_out is ∫dΠ₂ DISTINCT PAIR (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. q is Energy release Q = √s − m₁ − m₂, measured in GeV. Keep m1 (Mass m₁ of product 1) = 0.6 GeV; m2 (Mass m₂ of product 2) = 0.35 GeV fixed. For this exercise use only 0.310312 GeV ≤ q ≤ 2.23969 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 = 0.036448378, with a bounded reading uncertainty ±0.00021267. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown q by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000192937 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.
∫dΠ₂ DISTINCT PAIR against q
Predictions from the supplied Lorentz-invariant phase space and two-body final states 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): ∫dΠ₂ DISTINCT PAIR
Horizontal axis (x): q: Energy release Q = √s − m₁ − m₂ / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| q: Energy release Q = √s − m₁ − m₂ / GeV | ∫dΠ₂ DISTINCT PAIR |
|---|---|
| 0.3103125 | 0.02562662415303772 |
| 0.7926562499999998 | 0.03301153201880001 |
| 1.275 | 0.03575182521782777 |
| 1.7573437499999998 | 0.037099533534971786 |
| 2.2396874999999996 | 0.03786617489029977 |
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(q) = 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(0.310312) = 0.0256266 and Y_out(2.23969) = 0.0378662. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives q_line = 0.310312 + (0.0364484 − 0.0256266)(2.23969 − 0.310312)/(0.0378662 − 0.0256266) = 2.01619 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 q = 1.48723 GeV; substituting back gives Y_out = 0.036448378.
- The interpolation error in the input is |q_line − q| = 0.528963 GeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.036235708 and 0.036661048. Sort the roots even if the response decreases: 1.41575 GeV ≤ q ≤ 1.56599 GeV.
- The lower and upper input deviations are 0.0714853 GeV and 0.0787626 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 q ≈ 1.48723 GeV; compatible interval [1.41575, 1.56599] GeV. Endpoint interpolation gives 2.01619 GeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.