Spinning gluons from the QCD light-ray OPE

Abstract

A bstract We study the transverse spin structure of the squeezed limit of the three-point energy correlator, $$ \left\langle \mathrm{\mathcal{E}}\left({\overrightarrow{n}}_1\right)\mathrm{\mathcal{E}}\left({\overrightarrow{n}}_2\right)\mathrm{\mathcal{E}}\left({\overrightarrow{n}}_3\right)\right\rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mrow> <mml:mi>ℰ</mml:mi> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:mfenced> <mml:mi>ℰ</mml:mi> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>2</mml:mn> </mml:msub> </mml:mfenced> <mml:mi>ℰ</mml:mi> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>3</mml:mn> </mml:msub> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> . To describe its all orders perturbative behavior, we develop the light-ray operator product expansion (OPE) in QCD. At leading twist the iterated OPE of $$ \mathrm{\mathcal{E}}\left({\overrightarrow{n}}_i\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℰ</mml:mi> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mi>i</mml:mi> </mml:msub> </mml:mfenced> </mml:math> operators closes onto light-ray operators $$ {\mathbbm{O}}^{\left[J\right]}\left(\overrightarrow{n}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mi>J</mml:mi> </mml:mfenced> </mml:msup> <mml:mfenced> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:mfenced> </mml:math> with spin J , and transverse spin j = 0 , 2. We compute the $$ \mathrm{\mathcal{E}}\left({\overrightarrow{n}}_1\right)\mathrm{\mathcal{E}}\left({\overrightarrow{n}}_2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℰ</mml:mi> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:mfenced> <mml:mi>ℰ</mml:mi> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>2</mml:mn> </mml:msub> </mml:mfenced> </mml:math> , $$ \mathrm{\mathcal{E}}\left({\overrightarrow{n}}_1\right){\mathbbm{O}}^{\left[J\right]}\left({\overrightarrow{n}}_2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℰ</mml:mi> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:mfenced> <mml:msup> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mi>J</mml:mi> </mml:mfenced> </mml:msup> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>2</mml:mn> </mml:msub> </mml:mfenced> </mml:math> and $$ {\mathbbm{O}}^{\left[{J}_1\right]}\left({\overrightarrow{n}}_1\right){\mathbbm{O}}^{\left[{J}_2\right]}\left({\overrightarrow{n}}_2\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mfenced> </mml:msup> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msub> </mml:mfenced> <mml:msup> <mml:mi>O</mml:mi> <mml:mfenced> <mml:msub> <mml:mi>J</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mfenced> </mml:msup> <mml:mfenced> <mml:msub> <mml:mover> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mn>2</mml:mn> </mml:msub> </mml:mfenced> </mml:math> OPEs as analytic functions of J , which allows for the description of arbitrary squeezed limits of N -point correlators in QCD. We use these results with J = 3 to reproduce the perturbative expansion in the squeezed limit of the three-point correlator, as well as to resum the leading twist singular structure for both quark and gluon jets, including transverse spin contributions, as required for phenomenological applications. Finally, we briefly comment on the transverse spin structure at higher twists, and show that to all orders in the twist expansion the highest transverse spin contributions are universal between quark and gluon jets, and are descendants of the leading twist transverse spin-2 operator, allowing their resummation into a simple two-dimensional Euclidean conformal block. Due to the general applicability of our results to arbitrary correlation functions of energy flow operators, we anticipate that they can be widely applied to improving our understanding of jet substructure at the LHC.

Topics & Concepts