A toolbox for $$q_{T}$$ and 0-jettiness subtractions at $$\hbox {N}^3\hbox {LO}$$
Georgios Billis, Markus A. Ebert, Johannes K. L. Michel, Frank J. Tackmann
Abstract
Abstract We derive the leading-power singular terms at three loops for both $$q_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> and 0-jettiness, $$\mathcal {T}_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , for generic color-singlet processes. Our results provide the complete set of differential subtraction terms for $$q_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> and $$\mathcal {T}_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> subtractions at $$\hbox {N}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mtext>N</mml:mtext> <mml:mn>3</mml:mn> </mml:msup> <mml:mtext>LO</mml:mtext> </mml:mrow> </mml:math> , which are an important ingredient for matching $$\hbox {N}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mtext>N</mml:mtext> <mml:mn>3</mml:mn> </mml:msup> <mml:mtext>LO</mml:mtext> </mml:mrow> </mml:math> calculations with parton showers. We obtain the full three-loop structure of the relevant beam and soft functions, which are necessary ingredients for the resummation of $$q_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> and $$\mathcal {T}_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> at $$\hbox {N}^3\hbox {LL}'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mtext>N</mml:mtext> <mml:mn>3</mml:mn> </mml:msup> <mml:msup> <mml:mtext>LL</mml:mtext> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:math> and $$\hbox {N}^4\hbox {LL}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mtext>N</mml:mtext> <mml:mn>4</mml:mn> </mml:msup> <mml:mtext>LL</mml:mtext> </mml:mrow> </mml:math> order, and which constitute important building blocks in other contexts as well. The nonlogarithmic boundary coefficients of the beam functions, which contribute to the integrated subtraction terms, are not yet fully known at three loops. By exploiting consistency relations between different factorization limits, we derive results for the $$q_T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>q</mml:mi> <mml:mi>T</mml:mi> </mml:msub> </mml:math> and $$\mathcal {T}_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> beam function coefficients at $$\hbox {N}^3\hbox {LO}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mtext>N</mml:mtext> <mml:mn>3</mml:mn> </mml:msup> <mml:mtext>LO</mml:mtext> </mml:mrow> </mml:math> in the $$z\rightarrow 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>z</mml:mi> <mml:mo>→</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> threshold limit, and we also estimate the size of the unknown terms beyond threshold.