Branch cuts and threshold discontinuities

Use the finite channel function already defined, with real $m>0$ and Minkowski invariant s. Its mass combination is $m^2-sx(1-x)$. For real s below $4m^2$, this expression stays nonnegative over the unit interval and is positive away from the threshold endpoint case. The first zero occurs at the maximum parameter product, x equal to one half. At larger s, two real roots enclose an interval in which the mass combination is negative.

Introduce the dimensionless two-body velocity factor beta as the square root of one minus four mass squared over s. The two roots are one half times one plus or minus beta, so their separation is beta. This calculation identifies a normal two-particle threshold for this particular bubble. More complicated graphs can have additional singularities and sheets; solving one quadratic does not establish a complete singularity classification. Positive mass also keeps this example separate from a massless soft or collinear infrared limit.

Use the principal logarithm with its cut on the negative real axis. For a positive number a, the value just below that axis is $\ln(-a-i0)=\ln a-i\pi$. The channel function contains minus the logarithm, so each point in its negative-argument interval contributes positive pi to its imaginary part. Integrating over an interval of length beta gives positive pi times beta. Both the minus sign in the function definition and the sign of the infinitesimal are needed.

Define discontinuity as the upper-side value minus the lower-side value as the external invariant crosses its physical cut. With real parameters and Schwarz reflection on this cut, it equals twice i times the imaginary part on the upper physical side. The sign would reverse if discontinuity were defined in the opposite order. Report the convention alongside the answer rather than comparing bare imaginary numbers across sources with different amplitude factors or logarithm definitions.

A real local ultraviolet counterterm has no two-particle discontinuity in this channel. It can shift a real subtraction constant or polynomial without altering the bubble’s absorptive part. That separation provides a useful consistency check when changing renormalization schemes. The full scattering amplitude still multiplies the channel function by its diagram coefficient and adds other channels. An imaginary part of F is not itself a total cross section or a decay probability.

Near threshold, beta tends to zero like the square root of the excess invariant energy squared. Its vanishing reflects the shrinking available two-body phase space. Far above threshold it tends to one in this massive example, but other logarithms and channels remain in the full amplitude. The root-finding lab can verify a selected square root after its dimensionless target is supplied by hand. It does not continue a logarithm, implement the Feynman prescription, or determine a threshold from a particle model.