University Quantum Field Theory I · topic UQFTI-15.02
The slope of F_1: F_1(q^2) = 1 + (alpha/(3 pi))(q^2/m^2)[ln(m/mu_gamma) - 3/8]
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.
The slope of F₁ and its infrared logarithm: predict a finite change
The slope of F₁ is the coefficient of q² in the electron's renormalised charge form factor near q² = 0; at one loop with a photon mass μ_γ it is (α/(3πm²))[ln(m/μ_γ) − 3/8], which depends logarithmically on that infrared regulator. Use the supplied topic model Y_out = sqrt(692.75*(-p*ln(10)-0.375)), where Y_out is RMS RADIUS √⟨r²⟩ (fm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; ln(x) means the natural logarithm. p is log₁₀ of photon mass μ_γ/m, dimensionless. Keep znuc (Nuclear charge Z) = 1 fixed. For this exercise use only -7.3625 ≤ p ≤ -2.6375 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 = -6.323 and -5.6615. 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.
RMS RADIUS √⟨r²⟩ against p
Predictions from the supplied The slope of F₁ and its infrared logarithm 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): RMS RADIUS √⟨r²⟩ / fm
Horizontal axis (x): p: log₁₀ of photon mass μ_γ/m
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: log₁₀ of photon mass μ_γ/m | RMS RADIUS √⟨r²⟩ / fm |
|---|---|
| -7.3625 | 107.1646350159468 |
| -6.18125 | 97.97973480255807 |
| -5 | 87.8396144450676 |
| -3.81875 | 76.36469930364844 |
| -2.6375 | 62.82783406751471 |
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 = -6.323, substitute into Y_out = sqrt(692.75*(-p*ln(10)-0.375)): Y₀ = 99.1269 fm.
- At p = -5.6615, the same fixed-parameter model gives Y₁ = 93.6534 fm.
- Signed change: ΔY_out = Y₁ − Y₀ = -5.47343 fm; the input change is Δp = 0.6615.
- Average sensitivity is ΔY_out/Δp = -8.27427 fm 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 = 99.1269 fm, Y₁ = 93.6534 fm; ΔY_out = -5.47343 fm; average sensitivity = -8.27427 fm 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.
The slope of F₁ and its infrared logarithm: infer an input with uncertainty
The slope of F₁ is the coefficient of q² in the electron's renormalised charge form factor near q² = 0; at one loop with a photon mass μ_γ it is (α/(3πm²))[ln(m/μ_γ) − 3/8], which depends logarithmically on that infrared regulator. Use the supplied topic model Y_out = sqrt(692.75*(-p*ln(10)-0.375)), where Y_out is RMS RADIUS √⟨r²⟩ (fm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; ln(x) means the natural logarithm. p is log₁₀ of photon mass μ_γ/m, dimensionless. Keep znuc (Nuclear charge Z) = 1 fixed. For this exercise use only -7.3625 ≤ p ≤ -2.6375 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 = 82.986363 fm, with a bounded reading uncertainty ±3.0238 fm. 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 0.0004725, 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.
RMS RADIUS √⟨r²⟩ against p
Predictions from the supplied The slope of F₁ and its infrared logarithm 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): RMS RADIUS √⟨r²⟩ / fm
Horizontal axis (x): p: log₁₀ of photon mass μ_γ/m
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: log₁₀ of photon mass μ_γ/m | RMS RADIUS √⟨r²⟩ / fm |
|---|---|
| -7.3625 | 107.1646350159468 |
| -6.18125 | 97.97973480255807 |
| -5 | 87.8396144450676 |
| -3.81875 | 76.36469930364844 |
| -2.6375 | 62.82783406751471 |
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(-7.3625) = 107.165 fm and Y_out(-2.6375) = 62.8278 fm. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives p_line = -7.3625 + (82.9864 − 107.165)(-2.6375 − -7.3625)/(62.8278 − 107.165) = -4.78581. 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 = -4.48025; substituting back gives Y_out = 82.986363 fm.
- The interpolation error in the input is |p_line − p| = 0.305557. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 79.962563 fm and 86.010163 fm. Sort the roots even if the response decreases: -4.80061 ≤ p ≤ -4.17135.
- The lower and upper input deviations are 0.32036 and 0.308896. 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 ≈ -4.48025; compatible interval [-4.80061, -4.17135]. Endpoint interpolation gives -4.78581.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.