Weak bosons as partons below 10 TeV partonic center of momentum

We investigate the modeling of weak boson number densities for leptons and hadrons in practical calculations in the Standard Model. In the framework of the Effective <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>W</a:mi> </a:math> Approximation (EWA) and in the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:msub> <c:mrow> <c:mi>R</c:mi> </c:mrow> <c:mrow> <c:mi>ξ</c:mi> </c:mrow> </c:msub> </c:mrow> </c:math> and axial gauges, we derive the unrenormalized, tree-level parton number densities for weak bosons from massless fermions at next-to-leading power in the collinear expansion. Corrections exhibit various pathologies and properties, including those conjectured but not proven, and parallel heavy quark factorization. We suppress pathologies through a new set of kinematical consistency conditions. When satisfied, shapes and normalizations of full matrix elements for many-leg processes can be well approximated by the EWA and fragmentation contributions at leading power, suggesting the onset of tree-level factorization. Findings also suggest that the EWA is testable at the LHC with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:mi mathvariant="script">L</e:mi> <e:mo>=</e:mo> <e:mn>450</e:mn> <e:mtext> </e:mtext> <e:mtext> </e:mtext> <e:msup> <e:mrow> <e:mi>fb</e:mi> </e:mrow> <e:mrow> <e:mo>−</e:mo> <e:mn>1</e:mn> </e:mrow> </e:msup> </e:mrow> </e:math> of same-sign <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi>W</h:mi> <h:mi>W</h:mi> </h:math> scattering data at <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mrow> <j:msqrt> <j:mrow> <j:mi>s</j:mi> </j:mrow> </j:msqrt> <j:mo>=</j:mo> <j:mn>13.6</j:mn> <j:mtext> </j:mtext> <j:mtext> </j:mtext> <j:mi>TeV</j:mi> </j:mrow> </j:math> .