University Physics V · unit UPV-18
The Standard Model
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.
Extra challenging questions
A selection from this pathway. Open a topic below for its complete practice and challenge set.
The gauge principle and non-abelian algebras: infer an input with uncertainty
The gauge principle is the demand that an internal symmetry hold independently at every point of spacetime, which forces the ordinary derivative to be replaced by a covariant derivative carrying one new vector field for every generator of the group. Use the supplied topic model Y_out = 0.04*gc*(k+gc*q*q), where Y_out is PHASE ROUND a = 0.2 GeV⁻¹ (rad). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. q is Amplitude A¹₁ = A²₂, measured in GeV. Keep gc (Coupling g) = 1; k (Curl term ∂₁A³₂) = 0.8 GeV² fixed. For this exercise use only 0.17 GeV ≤ q ≤ 1.43 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 = 0.067238798 rad, with a bounded reading uncertainty ±0.0051124 rad. 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 0.000126 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.
PHASE ROUND a = 0.2 GeV⁻¹ against q
Predictions from the supplied The gauge principle and non-abelian algebras 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): PHASE ROUND a = 0.2 GeV⁻¹ / rad
Horizontal axis (x): q: Amplitude A¹₁ = A²₂ / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| q: Amplitude A¹₁ = A²₂ / GeV | PHASE ROUND a = 0.2 GeV⁻¹ / rad |
|---|---|
| 0.17 | 0.033156000000000005 |
| 0.485 | 0.041409 |
| 0.8 | 0.057600000000000005 |
| 1.115 | 0.08172900000000001 |
| 1.43 | 0.11379600000000001 |
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(0.17) = 0.033156 rad and Y_out(1.43) = 0.113796 rad. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives q_line = 0.17 + (0.0672388 − 0.033156)(1.43 − 0.17)/(0.113796 − 0.033156) = 0.702544 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 = 0.9386 GeV; substituting back gives Y_out = 0.067238798 rad.
- The interpolation error in the input is |q_line − q| = 0.236056 GeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.062126398 rad and 0.072351198 rad. Sort the roots even if the response decreases: 0.867848 GeV ≤ q ≤ 1.00438 GeV.
- The lower and upper input deviations are 0.0707521 GeV and 0.0657804 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 ≈ 0.9386 GeV; compatible interval [0.867848, 1.00438] GeV. Endpoint interpolation gives 0.702544 GeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Interactions, mediators and gravity's exclusion: infer an input with uncertainty
A mediator is the vector field a gauge symmetry demands for each of its generators; the mediator's mass M sets the range of its force through λ = ℏ/Mc and its low-energy strength through g²/M². Use the supplied topic model Y_out = 100000*sqrt(2)*gw*gw/(8*m*m), where Y_out is CONTACT √2 g²/8M² (×10⁻⁵ GeV⁻²). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. m is Mediator mass M, measured in GeV. Keep gw (Coupling g at the vertex) = 0.653; lq (Probe log₁₀(q / GeV)) = 0 fixed. For this exercise use only 32.7687 GeV ≤ m ≤ 268.231 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 = 0.24224226 ×10⁻⁵ GeV⁻², with a bounded reading uncertainty ±0.020621 ×10⁻⁵ GeV⁻². Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown m by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0235462 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.
CONTACT √2 g²/8M² against m
Predictions from the supplied Interactions, mediators and gravity's exclusion 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): CONTACT √2 g²/8M² / ×10⁻⁵ GeV⁻²
Horizontal axis (x): m: Mediator mass M / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: Mediator mass M / GeV | CONTACT √2 g²/8M² / ×10⁻⁵ GeV⁻² |
|---|---|
| 32.76875 | 7.019911277896291 |
| 91.63437499999999 | 0.8977068296928352 |
| 150.5 | 0.33279621136518867 |
| 209.36562499999997 | 0.17196521916805385 |
| 268.23125 | 0.10476895844552363 |
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(m) = 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(32.7687) = 7.01991 ×10⁻⁵ GeV⁻² and Y_out(268.231) = 0.104769 ×10⁻⁵ GeV⁻². The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives m_line = 32.7687 + (0.242242 − 7.01991)(268.231 − 32.7687)/(0.104769 − 7.01991) = 263.55 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 m = 176.401 GeV; substituting back gives Y_out = 0.24224226 ×10⁻⁵ GeV⁻².
- The interpolation error in the input is |m_line − m| = 87.1494 GeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.22162126 ×10⁻⁵ GeV⁻² and 0.26286326 ×10⁻⁵ GeV⁻². Sort the roots even if the response decreases: 169.34 GeV ≤ m ≤ 184.425 GeV.
- The lower and upper input deviations are 7.06041 GeV and 8.0242 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 m ≈ 176.401 GeV; compatible interval [169.34, 184.425] GeV. Endpoint interpolation gives 263.55 GeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
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.
Electroweak mixing and charged currents: infer an input with uncertainty
Electroweak mixing is the orthogonal rotation through θ_W that diagonalises the (W³, B) mass matrix generated by the Higgs vacuum, producing a massless photon and a massive Z, while the charged W± are the SU(2) ladder combinations of W¹ and W². Use the supplied topic model Y_out = 123.11*sqrt(gw^2+gy^2), where Y_out is M_Z = (v/2)√(g²+g′²) (GeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. gy is U(1) coupling g′, dimensionless. Keep gw (SU(2) coupling g) = 0.65; phi (Trial direction φ) = 0 ° fixed. For this exercise use only 0.06375 ≤ gy ≤ 0.53625 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 = 91.000394 GeV, with a bounded reading uncertainty ±1.5892 GeV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown gy by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00004725, 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.
M_Z = (v/2)√(g²+g′²) against gy
Predictions from the supplied Electroweak mixing and charged currents 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): M_Z = (v/2)√(g²+g′²) / GeV
Horizontal axis (x): gy: U(1) coupling g′
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| gy: U(1) coupling g′ | M_Z = (v/2)√(g²+g′²) / GeV |
|---|---|
| 0.06375 | 80.405445627264 |
| 0.18187499999999998 | 83.09501086120319 |
| 0.29999999999999993 | 88.13334755499758 |
| 0.41812499999999997 | 95.14806382790181 |
| 0.5362499999999999 | 103.73900966786267 |
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(gy) = 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(0.06375) = 80.4054 GeV and Y_out(0.53625) = 103.739 GeV. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives gy_line = 0.06375 + (91.0004 − 80.4054)(0.53625 − 0.06375)/(103.739 − 80.4054) = 0.278296. 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 gy = 0.351975; substituting back gives Y_out = 91.000394 GeV.
- The interpolation error in the input is |gy_line − gy| = 0.0736794. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 89.411194 GeV and 92.589594 GeV. Sort the roots even if the response decreases: 0.32399 ≤ gy ≤ 0.378334.
- The lower and upper input deviations are 0.0279855 and 0.0263593. 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 gy ≈ 0.351975; compatible interval [0.32399, 0.378334]. Endpoint interpolation gives 0.278296.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.