University Physics V · topic UPV-18.03
Fermions: chirality, generations, weak isospin, and hypercharge
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.
Chirality, isospin and hypercharge: predict a finite change
A chiral fermion is one Weyl component of a Dirac field, selected by P_L or P_R; the Standard Model puts every left-handed component in an SU(2)_L doublet and every right-handed one in a singlet, each with a hypercharge Y fixed by Q = T_3 + Y/2. Use the supplied topic model Y_out = (nc*yq+3*yl)/3, where Y_out is DOUBLET SUM N_c Y_Q + Y_L (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. yl is Lepton doublet hypercharge Y_L, dimensionless. Keep nc (Number of colours N_c) = 3; yq (Quark doublet 3Y_Q (Y_Q in thirds)) = 1 fixed. For this exercise use only -1.7875 ≤ yl ≤ -0.2125 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 yl = -1.441 and -1.2205. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δyl across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
DOUBLET SUM N_c Y_Q + Y_L against yl
Predictions from the supplied Chirality, isospin and hypercharge 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): DOUBLET SUM N_c Y_Q + Y_L
Horizontal axis (x): yl: Lepton doublet hypercharge Y_L
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| yl: Lepton doublet hypercharge Y_L | DOUBLET SUM N_c Y_Q + Y_L |
|---|---|
| -1.7875 | -0.7875000000000002 |
| -1.39375 | -0.3937500000000001 |
| -1 | 0 |
| -0.6062500000000002 | 0.3937499999999998 |
| -0.21250000000000013 | 0.7875 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of yl separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At yl = -1.441, substitute into Y_out = (nc*yq+3*yl)/3: Y₀ = -0.441.
- At yl = -1.2205, the same fixed-parameter model gives Y₁ = -0.2205.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.2205; the input change is Δyl = 0.2205.
- Average sensitivity is ΔY_out/Δyl = 1 1 per unit input. 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 = -0.441, Y₁ = -0.2205; ΔY_out = 0.2205; average sensitivity = 1 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.
Chirality, isospin and hypercharge: infer an input with uncertainty
A chiral fermion is one Weyl component of a Dirac field, selected by P_L or P_R; the Standard Model puts every left-handed component in an SU(2)_L doublet and every right-handed one in a singlet, each with a hypercharge Y fixed by Q = T_3 + Y/2. Use the supplied topic model Y_out = (nc*yq+3*yl)/3, where Y_out is DOUBLET SUM N_c Y_Q + Y_L (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. yl is Lepton doublet hypercharge Y_L, dimensionless. Keep nc (Number of colours N_c) = 3; yq (Quark doublet 3Y_Q (Y_Q in thirds)) = 1 fixed. For this exercise use only -1.7875 ≤ yl ≤ -0.2125 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 = 0.17325, with a bounded reading uncertainty ±0.092137. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown yl by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575, 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.
DOUBLET SUM N_c Y_Q + Y_L against yl
Predictions from the supplied Chirality, isospin and hypercharge 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): DOUBLET SUM N_c Y_Q + Y_L
Horizontal axis (x): yl: Lepton doublet hypercharge Y_L
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| yl: Lepton doublet hypercharge Y_L | DOUBLET SUM N_c Y_Q + Y_L |
|---|---|
| -1.7875 | -0.7875000000000002 |
| -1.39375 | -0.3937500000000001 |
| -1 | 0 |
| -0.6062500000000002 | 0.3937499999999998 |
| -0.21250000000000013 | 0.7875 |
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(yl) = 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(-1.7875) = -0.7875 and Y_out(-0.2125) = 0.7875. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives yl_line = -1.7875 + (0.17325 − -0.7875)(-0.2125 − -1.7875)/(0.7875 − -0.7875) = -0.82675. 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 yl = -0.82675; substituting back gives Y_out = 0.17325.
- The interpolation error in the input is |yl_line − yl| = 1.11022e-16. This is within the requested numerical tolerance.
- Invert both limits Y_out = 0.081113 and 0.265387. Sort the roots even if the response decreases: -0.918887 ≤ yl ≤ -0.734613.
- The lower and upper input deviations are 0.092137 and 0.092137. They agree on this branch; this does not establish linearity for changes in other inputs or outside the interval. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer yl ≈ -0.82675; compatible interval [-0.918887, -0.734613]. Endpoint interpolation gives -0.82675.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.