Renormalising QED: the Electron Self-Energy and the Ward-Takahashi Identity

The superficial degree of divergence D of a QED amplitude is the power of the cutoff its loop integral grows as when every loop momentum is scaled up together; for B external photons and F external fermions it is D = 4 − B − (3/2)F at every order. Use the supplied topic model Y_out = 2*nl-2+nb+nf, where Y_out is VERTICES V, ORDER e^V (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. nb is External photons B, dimensionless. Keep nf (External fermions F) = 2; nl (Loops L) = 1 fixed. For this exercise use only 0 ≤ nb ≤ 4, 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 = 4, with a bounded reading uncertainty ±0.32. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of nb 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.

A QED renormalisation condition defines a measured quantity and so fixes one counterterm: the electron propagator has its pole at the measured mass with residue 1, the photon propagator has residue 1 at q² = 0, and the vertex equals γ^μ at zero momentum transfer. Use the supplied topic model Y_out = 137.036*(1-0.0035657*uv), where Y_out is BARE 1/α₀ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. uv is Regulator mass, log₁₀(Λ/m), dimensionless. Keep ir (Photon mass, log₁₀(m/λ)) = 9 fixed. For this exercise use only 2 ≤ uv ≤ 22, 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 = 130.19519, with a bounded reading uncertainty ±0.78181. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of uv 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.

The one-loop electron self-energy Σ₂(p) is the one-particle-irreducible diagram in which the electron emits and reabsorbs one virtual photon: a matrix of the form A(p²) + B(p²)p̸ that, inside the propagator i/(p̸ − m₀ − Σ), moves its pole and its residue. Use the supplied topic model Y_out = 1/t0+2/t0^2+2*ln(1-t0)/t0^3, where Y_out is SLOPE db/d(p²/m²) (× α/4π). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; ln(x) means the natural logarithm. t0 is Marker at p²/m², dimensionless. Keep k (O(ε) Dirac terms: 1 kept, 0 dropped) = 1; mu (Scale μ in units of m) = 1 fixed. For this exercise use only -2.14938 ≤ t0 ≤ 0.449375 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 = -0.47146294 × α/4π, with a bounded reading uncertainty ±0.03121 × α/4π. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t0 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000259875, 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.

The self-mass δm = m − m₀ is how far the electron's interaction with its own photon field moves the pole of its propagator; in QED it is proportional to the mass itself and grows only as the logarithm of the cutoff. Use the supplied topic model Y_out = 510.99895*10^k*0.0017421*(4.60517*(lam-k)+15.6583), where Y_out is SELF-MASS δm (keV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. k is Fermion mass log₁₀(m/mₑ), dimensionless. Keep lam (Cutoff log₁₀(Λ/GeV)) = 19 fixed. For this exercise use only -1.575 ≤ k ≤ 1.575 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 = 200.7795 keV, with a bounded reading uncertainty ±29.725 keV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown k by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315, 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.