= Virtuality
{title2=$p^2$}
\b[Virtuality] characterizes the invariant momentum scale of an internal or otherwise off-<mass shell> momentum. One common convention uses $p^2$ itself; for a massive particle, departure from its physical mass shell is measured by $p^2-m^2$ in <natural units>. The convention must be stated: an on-shell massive particle has $p^2=m^2$, not generally zero.
With the <Minkowski metric> signature $(+,-,-,-)$ and $c=1$,
$$
p^2=E^2-|\mathbf p|^2.
$$
It is invariant under <Lorentz transformations>, unlike the energy component by itself. A massless on-shell particle has $p^2=0$ even when its energy is large. Positive and negative values describe timelike and spacelike four-momenta respectively; a spacelike transfer is not the four-momentum of an isolated real massive particle.
An original scattering-kinematics example uses idealized massless incoming and outgoing lepton momenta, in GeV:
$$
k=(5,0,0,5),\qquad k'=(4,2.4,0,3.2).
$$
Both satisfy $k^2=k'^2=0$. Their transferred momentum is $q=k-k'=(1,-2.4,0,1.8)$, giving
$$
q^2=1-2.4^2-1.8^2=-8\,\mathrm{GeV^2}.
$$
In <deep inelastic scattering>, the conventional positive scale is $Q^2=-q^2$, so this example has $Q^2=8\,\mathrm{GeV^2}$. The PDG review defines the transfer and this sign convention when introducing scattering invariants.<Structure Functions in the 2024 Review of Particle Physics>[§18.1, transfer momentum and definition of Q squared]{ref} Measuring the scattered lepton's energy and direction supplies the kinematic information needed for that inference.
The example is not an event measurement or a neglect of four-momentum conservation. An internal momentum can obey the vertex conservation equations while failing a real particle's on-shell mass relation. In <soft-collinear effective theory>, separating large energy from smaller virtuality is a reason to retain energetic nearly lightlike degrees of freedom.<Soft Collinear Effective Theory by Sean Fleming>[PDFp1, collinear momentum and low-invariant-mass splitting discussion]{ref}
Changing the metric signature changes signs in the formula, and different authors may reserve the word virtuality for $p^2$, $|p^2|$ or the displacement from a massive mass shell. Reporting the defining equation prevents these notational conventions from being mistaken for physical disagreement.
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