Anomalous dimension of subleading-power N-jet operators. Part II

Published for SISSA by Springer Received: August Revised: October Accepted: October Published: November 16, 24, 31, 19, 2018 2018 2018 2018 Martin Beneke, Mathias Garny, Robert Szafron and Jian Wang Physik Department T31, Technische Universität München, James-Franck-Straße 1, D-85748 Garching, Germany E-mail: , , Abstract: We continue the investigation of the anomalous dimension of subleading-power N -jet operators. In this paper, we focus on the operators with fermion number one in each collinear direction, corresponding to quark (antiquark) initiated jets in QCD. We investigate the renormalization effects induced by the soft loop and compute the one-loop mixing of time-ordered products involving power-suppressed SCET Lagrangian insertions into N -jet currents through soft loops. We discuss fermion number conservation in collinear directions and provide explicit results for the collinear anomalous dimension matrix of the currents. The Feynman rules for the power-suppressed SCET interactions in the positionspace formalism are collected in an appendix. Keywords: Effective Field Theories, Perturbative QCD ArXiv ePrint: 1808.04742 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP11(2018)112 JHEP11(2018)112 Anomalous dimension of subleading-power N -jet operators. Part II Contents 1 Introduction 1 2 Set-up of notation and conventions 2.1 Operator basis 2.2 Anomalous dimension matrix 2 2 4 7 11 13 13 14 20 25 4 Collinear sector 4.1 O(λ) 4.2 O(λ2 ), overview 4.3 Mixing of B-type currents into B-type currents 4.4 Mixing of B-type currents into C-type currents B2 (x) → J C2 (y , y ) 4.4.1 Mixing JA∂χ AAχ 1 2 B2 (x) → J C2 (y , y ) 4.4.2 Mixing JA∂χ χχ̄χ 1 2 4.5 Mixing of C-type currents into C-type currents 26 26 27 28 30 31 36 39 5 Summary 41 A SCET Feynman rules A.1 Preliminaries A.2 Derivative operators and Wilson lines A.3 Notation for Yang-Mills Feynman rules A.4 Fermionic Feynman rules A.4.1 Purely collinear or purely soft vertices A.4.2 Soft-collinear interaction vertices A.5 Three gluon vertices A.5.1 Purely collinear or purely soft vertices A.5.2 Soft-collinear interaction vertices A.6 Four gluon vertices A.6.1 Purely collinear or purely soft vertices A.6.2 Soft-collinear interaction vertices A.7 Ghost vertices A.8 Collinear building blocks 43 43 45 46 47 48 48 50 50 51 52 52 53 55 55 –i– JHEP11(2018)112 3 Soft sector 3.1 Single insertion of L(1) 3.2 Double insertion of L(1) 3.2.1 Double insertion in a single collinear direction 3.2.2 Double insertion in different collinear directions 3.3 Single insertion of L(2) 3.4 Soft-quark exchange B Soft master integral 56 C Auxiliary functions entering the collinear anomalous dimension C.1 B-to-B mixing C.2 B-to-C mixing B2 (x) → J C2 (y , y ) C.2.1 JA∂χ AAχ 1 2 B2 C.2.2 JA∂χ (x) → JχC2 χ̄χ (y1 , y2 ) 57 57 58 58 62 D Anomalous dimension of hermitian conjugated operators 63 Introduction The analysis of infrared (IR) divergences in QCD and gauge theories in general has always been a fertile field for exposing the universal structure of high-energy scattering amplitudes and performing all-order resummations of the perturbative expansion in the gauge coupling. What is commonly called the soft anomalous dimension of an amplitude of N widely separated energetic particles refers in the framework of soft-collinear effective theory (SCET) to the simplest N -jet operator, where every jet is sourced by a single collinear gauge-invariant quark or gluon field [1, 2]. The increasing sophistication of multi-loop calculations and the corresponding advance in precision has also triggered recent interest in subleading-power effects in the expansion in the scale 1/Q of the hard scattering [3–13]. In a recent paper [14, 15] we began the systematic investigation of the one-loop anomalous dimension matrix of these subleading power operators. Previous relevant work on anomalous dimensions of power-suppressed operators has been done in the context of heavy-quark decay [16, 17] and for thrust [18, 19]. All-order resummations of subleading-power logarithms can be found in refs. [16, 17, 20–22] covering cases with one or two collinear directions at the leading logarithmic order (next-to-leading for heavy quark decay to one jet). The purpose of our investigation is the complete analysis of one-loop infrared divergences of an arbitrary subleading-power N -jet operator. This provides one of the ingredients in resumming or generating at fixed order subleading-power next-to-leading order logarithms for amplitudes with any number of collinear directions or jets. In the present paper, which follows upon ref. [14], we extend the calculation of the oneloop anomalous dimension matrix from the case |F | = 2 to those with odd F , where F refers to the fermion number of the product of collinear fields in a given collinear direction. This contains as its simplest realizations the quark-antiquark initiated two-jet operators relevant to the subleading-power resummation of thrust and other event shape variables in e+ e− annihilation, and the threshold resummation of Drell-Yan type processes in hadron-hadron collisions. In addition to the collinear renormalization kernels for the odd-fermion number operators in a collinear sector, we discuss and calculate for the first time the subleadingpower soft contributions to the anomalous dimension matrix, which did not appear for the |F | = 2 operators. This involves a new contribution, which is not of the eikonal –1– JHEP11(2018)112 1 2 Set-up of notation and conventions To make the paper self-contained we first review some notation from ref. [14] and then discuss the structure of the N -jet operator basis and the anomalous dimension matrix relevant to the present work. 2.1 Operator basis We consider N copies of the collinear SCET Lagrangian Li [24], i = 1, . . . , N , furnished with corresponding collinear fields ψi , as well as one set of soft fields ψs that interact with all collinear fields and with themselves according to the soft Lagrangian Ls , in total LSCET = N X i=1 Li (ψi , ψs ) + Ls (ψs ) . (2.1) The collinear fields are characterized by N pairs of light-like reference vectors ni± with ni− · ni+ = 2, ni− · nj− = O(1), defining N widely separated directions. We are interested in current operators of the form Z J= dt C({tik }) Js (0) N Y Ji (ti1 , ti2 , . . . ) , (2.2) i=1 characterized by one soft and N collinear contributions with certain transformation properties under soft- and collinear gauge transformations [14]. The collinear contributions are composed of ni collinear building blocks ψik , Ji (ti1 , ti2 , . . . ) = ni Y k=1 –2– ψik (tik ni+ ) , (2.3) JHEP11(2018)112 type, and arises instead from the mixing of power-suppressed soft-collinear interactions in the SCET Lagrangian into power-suppressed N -jet operators with additional transverse derivatives or collinear fields. The anomalous dimensions discussed here can b (...truncated)