= Light-cone power counting
\b[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.<Soft Collinear Effective Theory by Sean Fleming>[Introduction and Soft Collinear Effective Theory, PDFpp1–3]{ref}
Fix <natural units> and metric signature $(+,-,-,-)$, and define $p^+=E-p_z$, $p^-=E+p_z$. A common assignment is
$$
p^-\sim Q,\qquad |\mathbf p_\perp|\sim Q\lambda,\qquad p^+\sim Q\lambda^2,\qquad\lambda\ll1.
$$
The symbol $\sim$ specifies parametric size, not an exact equality. Since
$$
p^2=p^+p^- -|\mathbf p_\perp|^2,
$$
each 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 $p^2=100\,\mathrm{GeV^2}$. 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 $Q\lambda$, so
$$
dp^+\,dp^-\,d^2p_\perp\sim Q^4\lambda^4.
$$
A 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>.
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