Cross-Sections, Decay Rates and Unitarity

A cross-section is the transition rate per target particle divided by the incident flux; in relativistic normalisation it is dσ = |M|² dΠₙ/(4E₁E₂|v₁ − v₂|), the flux factor 4E₁E₂|v₁ − v₂| undoing the 2E-per-volume normalisation of the two incoming states. Use the supplied topic model Y_out = ((((p+sqrt(p^2+m1^2))/m1)^(2-2*f)-1)/(((p+sqrt(p^2+m1^2))/m1)^(2-2*f)+1)+(((p+sqrt(p^2+m1^2))/m1)^(2*f)-1)/(((p+sqrt(p^2+m1^2))/m1)^(2*f)+1)), where Y_out is |v₁ − v₂| IN THIS FRAME (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. p is Beam momentum in the target frame p, measured in GeV. Keep m1 (Beam particle mass m₁) = 0.14 GeV; f (Observer's boost along the beam, η/Y) = 0 fixed. For this exercise use only 2.57187 GeV ≤ p ≤ 17.9281 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 = 0.99993126, with a bounded reading uncertainty ±0.0000057377. 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 0.00153562 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.

Lorentz-invariant phase space dΠₙ is the measure over the momenta of n final particles, d³p/((2π)³ 2E) for each one, constrained by (2π)⁴δ⁴ of total four-momentum, that counts final states with the same weight in every inertial frame. Use the supplied topic model Y_out = sqrt(max(0,(m1+m2+q)^2-(m1+m2)^2)*max(0,(m1+m2+q)^2-(m1-m2)^2))/(2*(m1+m2+q))/(4*pi*(m1+m2+q)), where Y_out is ∫dΠ₂ DISTINCT PAIR (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. q is Energy release Q = √s − m₁ − m₂, measured in GeV. Keep m1 (Mass m₁ of product 1) = 0.6 GeV; m2 (Mass m₂ of product 2) = 0.35 GeV fixed. For this exercise use only 0.310312 GeV ≤ q ≤ 2.23969 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 = 0.036448378, with a bounded reading uncertainty ±0.00021267. 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 0.000192937 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.

A particle's decay rate Γ is its probability per unit time of turning into the final states open to it, Γ = (1/(2E))∫|ℳ|² dΠₙ summed over channels, and at rest its mean life is τ = ħ/Γ. Use the supplied topic model Y_out = 0.197327*16*pi*p/max(sqrt(1-4*mpsi^2/(2*mpsi+q)^2),0.01), where Y_out is LAB DECAY LENGTH (fm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. q is Energy released Q = m − 2M, measured in GeV. Keep mpsi (ψ mass M) = 0.3 GeV; p (Lab momentum of φ, p) = 2 GeV fixed. For this exercise use only 0.2835 GeV ≤ q ≤ 2.2365 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 = 20.722866 fm, with a bounded reading uncertainty ±0.063917 fm. 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 0.0001953 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.

A 2 → 2 differential cross-section is the rate of scatters into a solid angle per target particle per unit incident flux; in the centre-of-mass frame it equals |M|² times |p_f|/|pᵢ|, divided by 64π² s. Use the supplied topic model Y_out = lam^2*389.4*sqrt(max(rs^2/4-mf^2,0))/(32*pi*rs^2*sqrt(rs^2/4-0.0625)), where Y_out is CROSS-SECTION AT MARKER (μb). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. rs is Centre-of-mass energy, measured in GeV. Keep lam (Coupling strength) = 1; mf (Final-pair mass) = 0.25 GeV fixed. For this exercise use only 1.55406 GeV ≤ rs ≤ 8.99594 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 = 0.10431513 μb, with a bounded reading uncertainty ±0.0084678 μb. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown rs by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000744188 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.