One Loop in phi^4: Dimensional Regularisation, Counterterms and the Running Coupling

The superficial degree of divergence D of a Feynman diagram is the power of momentum its loop integral carries when every loop momentum is scaled up together: with a cutoff Λ it grows as Λ^D for positive D, as ln Λ at D = 0, and converges by the count for negative D. Use the supplied topic model Y_out = 4-d, where Y_out is COUPLING DIMENSION [λ] (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is Spacetime dimension d, dimensionless. Keep loops (Loop order marked, L) = 2 fixed. For this exercise use only 3 ≤ d ≤ 6, at the specified test settings separated by 1. 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 = -1, with a bounded reading uncertainty ±0.24. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of d consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Feynman parameters and the Wick rotation are exact rewritings that turn a one-loop Minkowski integral over k into i times a Euclidean integral of (l_E² + Δ)⁻², averaged over a parameter x, with all external momenta inside Δ = m² − x(1 − x)p². Use the supplied topic model Y_out = -sqrt((hypot(q^2+1-xf*(1-xf)*p2,0.4)-(q^2+1-xf*(1-xf)*p2))/2), where Y_out is RIGHT POLE, Im l⁰/m (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; hypot(a,b) means √(a²+b²). p2 is External momentum squared p²/m², dimensionless. Keep xf (Feynman parameter x) = 0.3; q (Spatial loop momentum |l|/m) = 0.5 fixed. For this exercise use only -2.725 ≤ p2 ≤ 6.725 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 = -0.24410371, with a bounded reading uncertainty ±0.014523. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown p2 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000945, 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.

Dimensional regularisation evaluates a loop integral in d dimensions, where it converges, as an analytic function of d, so that a divergence of the four-dimensional integral appears as a pole at d = 4 instead of as a power or logarithm of a cutoff. Use the supplied topic model Y_out = 2*(2^k*cos(pi*(2+k-n))*clamp(3+k-n,0,1)/(max(2+k-n,1)*max(n-1,1)))/(4-d), where Y_out is POLE TERM AT THIS d (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; max(a,b) selects the larger value; clamp(x,a,b) confines x to [a,b]; unmarked trigonometric functions use radians. d is Dimension d, dimensionless. Keep n (Denominator power n) = 2; k (Numerator power k of l_E²) = 0 fixed. For this exercise use only 0.1 ≤ d ≤ 3.9, at the specified test settings separated by 0.2. 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 = 1.3333333, with a bounded reading uncertainty ±1.559. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of d consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Near four dimensions a dimensionally regulated one-loop integral becomes a Laurent series in ε = 4 − d: a pole 2/ε, the constants −γ_E + ln 4π, and a logarithm ln(Δ/μ²) whose scale μ enters because λ has mass dimension ε. Use the supplied topic model Y_out = pi*sqrt((s-4)/s), where Y_out is IM B(s) = πβ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. s is Channel energy s/m² (90° scattering), dimensionless. Keep mu (Scale μ/m) = 1; sch (Scheme: 0 = MS-bar, 1 = MS) = 0 fixed. For this exercise use only 6.125 ≤ s ≤ 21.875 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 = 2.7129781, with a bounded reading uncertainty ±0.019034. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown s by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.001575, 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.