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A parton distribution function specifies a hadron's partonic content as a function of longitudinal momentum fraction and a stated parton factorization scale. In the leading collinear parton model, $f_i(x)dx$ has a number-density interpretation for partons of species $i$ with fractions in $dx$. It is not generally normalized to one.

For a proton, the integrals of the valence differences count two net up quarks and one net down quark. A momentum-weighted sum over all quark, antiquark and gluon species gives total momentum fraction one. These are different sum rules. Beyond the simplest approximation, the definition depends on factorization scheme and renormalized field correlations; a naive positive classical probability interpretation is not universal. The functions enter factorized cross sections and evolve with resolution, rather than being predictions of a free-quark model.^[ref]

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.