One-loop scalar self-energy

Retain natural units and Minkowski signature $(+---)$, with free propagator $i/(p^2-m^2+i0)$ and interaction $-\lambda\phi^4/4!$. Define the one-particle-irreducible insertion as $-i\Sigma(p^2)$. Summing repeated insertions then gives a denominator $p^2-m^2-\Sigma(p^2)+i0$. This definition fixes the sign of the mass correction; a source defining its insertion with the opposite sign must change the corresponding denominator. A stable isolated pole is assumed when discussing a real pole mass and residue.

The tadpole’s scalar coefficient is one half times the quartic coupling. No external momentum flows through its loop denominator, so its self-energy is constant at this order. Wick rotation takes the loop into Euclidean momentum with a positive denominator; its contour deformation is justified by the Feynman prescription and absence of obstructing external thresholds for this tadpole. The resulting positive cutoff integral is not a classical self-energy of a localized particle. It is a regulated quantum correction in this particular scalar model.

The four-dimensional unit sphere has area $2\pi^2$. Combining that angular measure with the denominator’s $(2\pi)^4$ leaves a radial prefactor $1/(8\pi^2)$. Substitute $u=k_E^2$, so the radial integral contains $u/(u+m^2)$. Splitting it into one minus a simple rational term exposes the quadratic cutoff dependence and the logarithmic mass dependence. The result remains finite for every finite cutoff and positive mass before any subtraction is made.

At large cutoff, the leading term is quadratic and the subleading term contains the mass squared times a logarithm. Other regulators package these contributions differently: dimensional regularization represents the ultraviolet information through meromorphic poles rather than an explicit positive quadratic cutoff term. Comparing bare integral values across regulators is therefore not a comparison of physical masses. Use the same renormalization condition and matched parameters before comparing predictions. A hard momentum cutoff also requires care with momentum shifts and gauge identities in less elementary calculations.

Add a mass counterterm to the self-energy and choose its finite part by a stated condition. In an on-shell prescription, require the corrected denominator to vanish at the selected physical mass. At first order, a constant loop shift can be canceled by an equal and opposite constant counterterm. This does not mean interactions have disappeared: other Green functions and higher-loop momentum dependence remain. It means the mass parameter has been defined through that pole measurement.

The residue depends on the derivative of self-energy with respect to momentum squared. Because this tadpole is constant, it gives no residue correction at this order. A momentum-dependent self-energy in another theory or at a higher loop order can affect both pole and residue. The parabola instrument isolates quadratic algebra only. Its normalized height is not the complete tadpole, whose logarithm, regulator choice, and subtraction condition are essential to the result.