Virtuality ($p^2$)

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 $c=1$,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:Both satisfy $k^2=k^{\prime 2}=0$. Their transferred momentum is $q=k-k^{\prime }=(1,-2.4,0,1.8)$, givingIn deep inelastic scattering, the conventional positive scale is $Q^2=-q^2$, so this example has $Q^2=8\mathrm{Ge}{\mathrm{V}}^2$. The PDG review defines the transfer and this sign convention when introducing scattering invariants.^[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.^[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.