= Soft-collinear effective theory
\b[Soft-collinear effective theory] is an <effective field theory> of <quantum chromodynamics> for processes containing energetic particles moving close to lightlike directions together with lower-momentum radiation. It retains the modes needed for that hierarchy and organizes interactions by <power counting>. Fleming's overview describes energetic collinear particles interacting with soft quanta and applies the framework to meson-decay observables.<Soft Collinear Effective Theory by Sean Fleming>[Introduction and Soft Collinear Effective Theory, PDFpp1–3]{ref}
The geometry is clearer in <light-cone coordinates>. Choose the <Minkowski metric> with signature $(+,-,-,-)$ and define
$$
p^+=E-p_z,\qquad p^-=E+p_z,\qquad
p^2=p^+p^- - |\mathbf p_\perp|^2.
$$
A particle moving nearly along the positive $z$ direction has one large component $p^-$, a smaller transverse momentum and a still smaller $p^+$. A common <light-cone power counting> assignment is
$$
(p^+,p^-,|\mathbf p_\perp|)\sim Q(\lambda^2,1,\lambda),\qquad\lambda\ll1.
$$
Individual terms in $p^2$ then scale as $Q^2\lambda^2$; they can cancel for a particle on its <mass shell>. <Virtuality> and energy are therefore different measures of scale.
For an original exact example in <natural units>, take $p^-=100\,\mathrm{GeV}$, $p^+=1\,\mathrm{GeV}$ and $|\mathbf p_\perp|=10\,\mathrm{GeV}$. Then $E=50.5\,\mathrm{GeV}$, $p_z=49.5\,\mathrm{GeV}$ and
$$
p^2=(100)(1)-10^2=0.
$$
The momentum is energetic and exactly lightlike in this example. Removing all energetic fields from a low-virtuality theory would discard this mode. The numerical components illustrate the stated convention and are not data from a collider.
The original2001 construction by <Christian W. Bauer>, <Sean Fleming>, <Dan Pirjol> and <Iain W. Stewart> identifies separated scales and determines effective-theory currents and coefficient evolution.<An effective field theory for collinear and soft gluons: heavy to light decays by Christian W. Bauer and others>[Abstract]{ref} Comparing calculations with the underlying theory fixes <Wilson coefficients>; <renormalization group> evolution then organizes scale dependence. The resulting <effective Lagrangian> is an expansion with a specified domain of validity, rather than a replacement for QCD in every kinematic regime. Precise assignments for lower-energy modes depend on which hierarchy the application requires.
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