Finite subtraction conditions define different useful coordinates

Use a scalar propagator with inverse denominator s−m²−ΣR(s), where s is Minkowski momentum squared and ΣR is the renormalized self-energy. In an on-shell scheme choose m=M, the stable-particle pole mass, and require both ΣR(M²)=0 and its first derivative at the pole to vanish. The first condition places the pole correctly; the second gives unit residue in the chosen renormalized field. The residue is a normalization convention for that field rather than an additional directly measurable scattering probability.

These elementary conditions assume a real isolated stable pole and sufficiently regular self-energy near it. Unstable particles require a complex-pole treatment and care with gauge invariance. Massless gauge theories can have infrared singularities in on-shell quantities, while colored quark pole masses have a nonperturbative limitation discussed later. Thus an on-shell prescription is useful in an appropriate setting, but it is not a universal assertion that every field corresponds to a stable asymptotic particle with a precisely observable real pole.

With d=4−ε, a typical one-loop scalar integral contains the combination 2/ε−γE+ln4π plus finite logarithms and constants. MS subtracts only the regulator pole. Modified MS subtracts that pole together with the universal Euler-constant and logarithm terms. Equivalently, these prescriptions can be related by a reference-scale convention. The precise factor of two changes when a reference instead defines d=4−2ε, so a quoted subtraction formula is incomplete unless the dimensional convention is stated.

Neither minimal scheme sets all finite self-energy values or derivatives to zero at a particle pole. A finite conversion therefore relates its mass to an on-shell mass. Likewise its field normalization need not give unit residue without the appropriate external-leg factors. Keeping the same numerical mass and coupling in two schemes generally compares two differently specified theories at that accuracy. A valid comparison first matches the same physical inputs, then transforms all parameters and perturbative coefficients through a consistent order.

Minimal schemes often simplify high-energy calculations because their renormalization constants do not depend on masses. On-shell inputs can be convenient when measured stable masses and low-energy charges are natural reference quantities. Both choices leave the exact physical answer unchanged, but their finite-order series can distribute corrections differently. Very large logarithms of momentum over mass can make one organization inconvenient. Choosing a subtraction scale near a relevant hard scale reduces those logarithms without declaring that the scale itself is observable.

The function lab illustrates why a local value subtraction and a local slope subtraction are separate conditions. Reset and set both controls to a=1 and x=0, predicting y=0 and slope zero. Reset and set a=1 and x=1, predicting y=1 and slope two. A tangent subtraction at the chosen point removes a value and a linear variation, analogous only algebraically to pole and residue conditions. The plotted parabola is not a self-energy, and its position control is not a physical momentum-shell prescription.