Light-cone power counting

Light-cone power counting organizes terms in a multiscale field theory by assigning powers of a small parameter to light-cone momentum components. In soft-collinear effective theory, energetic nearly lightlike momenta can have one large longitudinal component, smaller transverse components and a still smaller opposite longitudinal component. Fleming's review explains why these collinear degrees of freedom must remain in a theory that also describes lower-momentum radiation.^[ref]

Fix natural units and metric signature $(+,-,-,-)$, and define $p^{+}=E-p_z$, $p^{-}=E+p_z$. A common assignment isThe symbol $\sim$ specifies parametric size, not an exact equality. Sinceeach term is generically of order $Q^2{\lambda }^2$. Their difference can be smaller, including zero on the mass shell. The assignment therefore concerns momentum regions and generic virtuality, not a claim that every momentum in the region has the same invariant mass.

For an original example, take $Q=200\mathrm{GeV}$ and $\lambda =0.05$. The assigned component scales are200,10and0.5GeV. Values $p^{-}=200$, $|{\mathbf{p}}_{\perp }|=10$ and $p^{+}=0.5$ give $p^2=0$; changing the last value to1 gives $p^2=100\mathrm{Ge}{\mathrm{V}}^2$. Both small components have the same parametric order, differing by a factor of2, while the exact cancellation changes.

Power assignments also organize integration measures. Two transverse components each scale as $Q\lambda$, soA generic scalar factor $1/p^2$ in this region scales as $Q^{-2}{\lambda }^{-2}$. Together these contribute $Q^2{\lambda }^2$ before any numerator, couplings or further denominators are included. This illustrative count is not by itself a complete scattering amplitude.

Precise assignments depend on the scale hierarchy under study and the chosen light-cone directions. An isotropic mode with all components of order $Q{\lambda }^2$ has a different generic virtuality scale, $Q^2{\lambda }^4$. Explicit counting keeps such contributions distinct and states what it means to retain a given order in an effective Lagrangian.