University Quantum Field Theory I · topic UQFTI-09.09
Electron-muon scattering, its Mott limit and the muon's magnetic term
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.
Electron-muon scattering: Mott limit and magnetic term: predict a finite change
Electron-muon scattering at lowest order is single-photon exchange between two point Dirac particles; in the muon's rest frame it becomes Mott scattering when the muon is heavy, and gains a recoil factor and a magnetic term when it is not. Use the supplied topic model Y_out = 5183700*(cosd(th/2)^2 + 2*(eb/(105.66*mt))^2*sind(th/2)^4/(1+2*(eb/(105.66*mt))*sind(th/2)^2))/(eb^2*sind(th/2)^4*(1+2*(eb/(105.66*mt))*sind(th/2)^2)), where Y_out is dσ/dΩ AT θ (nb sr⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. eb is Beam energy E, measured in MeV. Keep mt (Target mass M ÷ m_μ) = 1; th (Marked angle θ) = 90 ° fixed. For this exercise use only 124.125 MeV ≤ eb ≤ 895.875 MeV 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 eb = 293.91 MeV and 401.955 MeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δeb across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
dσ/dΩ AT θ against eb
Predictions from the supplied Electron-muon scattering: Mott limit and magnetic term 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): dσ/dΩ AT θ / nb sr⁻¹
Horizontal axis (x): eb: Beam energy E / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| eb: Beam energy E / MeV | dσ/dΩ AT θ / nb sr⁻¹ |
|---|---|
| 124.125 | 505.76222603720237 |
| 317.0625 | 83.79469145835603 |
| 509.99999999999994 | 34.19260853400902 |
| 702.9374999999999 | 18.59808389389362 |
| 895.8749999999999 | 11.698405586104432 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of eb separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At eb = 293.91 MeV, substitute into Y_out = 5183700*(cosd(th/2)^2 + 2*(eb/(105.66*mt))^2*sind(th/2)^4/(1+2*(eb/(105.66*mt))*sind(th/2)^2))/(eb^2*sind(th/2)^4*(1+2*(eb/(105.66*mt))*sind(th/2)^2)): Y₀ = 96.6723 nb sr⁻¹.
- At eb = 401.955 MeV, the same fixed-parameter model gives Y₁ = 53.5912 nb sr⁻¹.
- Signed change: ΔY_out = Y₁ − Y₀ = -43.0811 nb sr⁻¹; the input change is Δeb = 108.045 MeV.
- Average sensitivity is ΔY_out/Δeb = -0.398733 nb sr⁻¹ per MeV. 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 = 96.6723 nb sr⁻¹, Y₁ = 53.5912 nb sr⁻¹; ΔY_out = -43.0811 nb sr⁻¹; average sensitivity = -0.398733 nb sr⁻¹ per MeV.
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.
Electron-muon scattering: Mott limit and magnetic term: infer an input with uncertainty
Electron-muon scattering at lowest order is single-photon exchange between two point Dirac particles; in the muon's rest frame it becomes Mott scattering when the muon is heavy, and gains a recoil factor and a magnetic term when it is not. Use the supplied topic model Y_out = 5183700*(cosd(th/2)^2 + 2*(eb/(105.66*mt))^2*sind(th/2)^4/(1+2*(eb/(105.66*mt))*sind(th/2)^2))/(eb^2*sind(th/2)^4*(1+2*(eb/(105.66*mt))*sind(th/2)^2)), where Y_out is dσ/dΩ AT θ (nb sr⁻¹). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. eb is Beam energy E, measured in MeV. Keep mt (Target mass M ÷ m_μ) = 1; th (Marked angle θ) = 90 ° fixed. For this exercise use only 124.125 MeV ≤ eb ≤ 895.875 MeV 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 = 25.542977 nb sr⁻¹, with a bounded reading uncertainty ±2.0767 nb sr⁻¹. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown eb by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.077175 MeV, 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.
dσ/dΩ AT θ against eb
Predictions from the supplied Electron-muon scattering: Mott limit and magnetic term 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): dσ/dΩ AT θ / nb sr⁻¹
Horizontal axis (x): eb: Beam energy E / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| eb: Beam energy E / MeV | dσ/dΩ AT θ / nb sr⁻¹ |
|---|---|
| 124.125 | 505.76222603720237 |
| 317.0625 | 83.79469145835603 |
| 509.99999999999994 | 34.19260853400902 |
| 702.9374999999999 | 18.59808389389362 |
| 895.8749999999999 | 11.698405586104432 |
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(eb) = 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(124.125) = 505.762 nb sr⁻¹ and Y_out(895.875) = 11.6984 nb sr⁻¹. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives eb_line = 124.125 + (25.543 − 505.762)(895.875 − 124.125)/(11.6984 − 505.762) = 874.249 MeV. 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 eb = 594.892 MeV; substituting back gives Y_out = 25.542977 nb sr⁻¹.
- The interpolation error in the input is |eb_line − eb| = 279.357 MeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 23.466277 nb sr⁻¹ and 27.619677 nb sr⁻¹. Sort the roots even if the response decreases: 570.872 MeV ≤ eb ≤ 622.067 MeV.
- The lower and upper input deviations are 24.0205 MeV and 27.1748 MeV. 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 eb ≈ 594.892 MeV; compatible interval [570.872, 622.067] MeV. Endpoint interpolation gives 874.249 MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.