University Physics V · topic UPV-18.07
Colour charge, SU(3)_c, and the QCD Lagrangian
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.
Colour charge, SU(3)_c and the QCD Lagrangian: predict a finite change
Colour charge is the SU(3)_c quantum number: each quark field is a three-component vector rotated by the 3×3 Gell-Mann generators, and the QCD Lagrangian is the renormalisable field theory that stays invariant under that rotation at every point of spacetime. Use the supplied topic model Y_out = (2.2+1.4*max(0,sign(s-3.73))+0.3*max(0,sign(s-10.56)))/((6+4*max(0,sign(s-3.73))+max(0,sign(s-10.56)))/9*(1+as/pi)), where Y_out is N_c IMPLIED BY DATA (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. as is Strong coupling α_s, dimensionless. Keep nc (Colour count N_c) = 3; s (Collision energy √s) = 30 GeV fixed. For this exercise use only 0.053125 ≤ as ≤ 0.446875 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 as = 0.13975 and 0.194875. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δas across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
N_c IMPLIED BY DATA against as
Predictions from the supplied Colour charge, SU(3)_c and the QCD Lagrangian 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): N_c IMPLIED BY DATA
Horizontal axis (x): as: Strong coupling α_s
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| as: Strong coupling α_s | N_c IMPLIED BY DATA |
|---|---|
| 0.053125 | 3.1378474235457587 |
| 0.1515625 | 3.044052311761068 |
| 0.25 | 2.95570181391404 |
| 0.34843749999999996 | 2.8723352283824237 |
| 0.44687499999999997 | 2.793542404721036 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of as separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At as = 0.13975, substitute into Y_out = (2.2+1.4*max(0,sign(s-3.73))+0.3*max(0,sign(s-10.56)))/((6+4*max(0,sign(s-3.73))+max(0,sign(s-10.56)))/9*(1+as/pi)): Y₀ = 3.05501.
- At as = 0.194875, the same fixed-parameter model gives Y₁ = 3.00454.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0504748; the input change is Δas = 0.055125.
- Average sensitivity is ΔY_out/Δas = -0.915642 1 per unit input. Its negative sign means that this response decreases 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 = 3.05501, Y₁ = 3.00454; ΔY_out = -0.0504748; average sensitivity = -0.915642 1 per unit input.
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.
Colour charge, SU(3)_c and the QCD Lagrangian: infer an input with uncertainty
Colour charge is the SU(3)_c quantum number: each quark field is a three-component vector rotated by the 3×3 Gell-Mann generators, and the QCD Lagrangian is the renormalisable field theory that stays invariant under that rotation at every point of spacetime. Use the supplied topic model Y_out = (2.2+1.4*max(0,sign(s-3.73))+0.3*max(0,sign(s-10.56)))/((6+4*max(0,sign(s-3.73))+max(0,sign(s-10.56)))/9*(1+as/pi)), where Y_out is N_c IMPLIED BY DATA (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value. as is Strong coupling α_s, dimensionless. Keep nc (Colour count N_c) = 3; s (Collision energy √s) = 30 GeV fixed. For this exercise use only 0.053125 ≤ as ≤ 0.446875 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 = 2.9184318, with a bounded reading uncertainty ±0.018733. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown as by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000039375, 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.
N_c IMPLIED BY DATA against as
Predictions from the supplied Colour charge, SU(3)_c and the QCD Lagrangian 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): N_c IMPLIED BY DATA
Horizontal axis (x): as: Strong coupling α_s
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| as: Strong coupling α_s | N_c IMPLIED BY DATA |
|---|---|
| 0.053125 | 3.1378474235457587 |
| 0.1515625 | 3.044052311761068 |
| 0.25 | 2.95570181391404 |
| 0.34843749999999996 | 2.8723352283824237 |
| 0.44687499999999997 | 2.793542404721036 |
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(as) = 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(0.053125) = 3.13785 and Y_out(0.446875) = 2.79354. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives as_line = 0.053125 + (2.91843 − 3.13785)(0.446875 − 0.053125)/(2.79354 − 3.13785) = 0.30405. 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 as = 0.293313; substituting back gives Y_out = 2.9184318.
- The interpolation error in the input is |as_line − as| = 0.0107379. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 2.8996988 and 2.9371648. Sort the roots even if the response decreases: 0.271405 ≤ as ≤ 0.315503.
- The lower and upper input deviations are 0.0219075 and 0.0221906. 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 as ≈ 0.293313; compatible interval [0.271405, 0.315503]. Endpoint interpolation gives 0.30405.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.