Renormalization and scale evolution of the soft-quark soft function
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Springer
Received: May 8, 2020 Accepted: June 19, 2020 Published: July 16, 2020
Ze Long Liu,a Bianka Mecaj,b Matthias Neubert,b,c Xing Wangb and Sean Flemingb,1 a
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. b PRISMA+ Cluster of Excellence & Mainz Institute for Theoretical Physics, Johannes Gutenberg University, 55099 Mainz, Germany c Department of Physics & LEPP, Cornell University, Ithaca, NY 14853, U.S.A.
E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: Soft functions defined in terms of matrix elements of soft fields dressed by Wilson lines are central components of factorization theorems for cross sections and decay rates in collider and heavy-quark physics. While in many cases the relevant soft functions are defined in terms of gluon operators, at subleading order in power counting soft functions containing quark fields appear. We present a detailed discussion of the properties of the soft-quark soft function consisting of a quark propagator dressed by two finite-length Wilson lines connecting at one point. This function enters in the factorization theorem for the Higgs-boson decay amplitude of the h → γγ process mediated by light-quark loops. We perform the renormalization of this soft function at one-loop order, present a conjecture for its two-loop anomalous dimension and discuss solutions to its renormalization-group evolution equation in momentum space, in Laplace space and in the “diagonal space”, where the evolution is strictly local in the momentum variable. Keywords: Effective Field Theories, Renormalization Group, Resummation, Perturbative QCD ArXiv ePrint: 2005.03013
1
On leave from Department of Physics, University of Arizona, Tucson, AZ 85721, U.S.A. .
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)104
JHEP07(2020)104
Renormalization and scale evolution of the soft-quark soft function
Contents 1
2 One-loop expression for the bare soft function
2
3 Renormalization of the soft function
4
4 Renormalization-group evolution 4.1 Two-loop anomalous dimension 4.2 Exact solution to the RG equation
6 7 8
5 Asymptotic behavior and dynamical scale setting
13
6 Soft function in Laplace space
15
7 Soft function in the diagonal space 7.1 Construction of the diagonal space 7.2 Transformation between momentum space and diagonal space 7.3 Convolution T3 in the diagonal space 7.4 Leading-order soft function in the diagonal space
16 17 19 20 21
8 RG-invariance of the convolution integral T3 8.1 RG invariance and rapidity regularization 8.2 Regularized convolution T3 in the diagonal space
22 23 25
9 Conclusions
27
A Anomalous dimensions and RG functions
28
B Light-cone distribution amplitude of the B-meson
30
C Solution in terms of hypergeometric functions
31
D Soft function in the dual space
32
E Rapidity cutoff scheme
32
–i–
JHEP07(2020)104
1 Introduction
1
Introduction
(4π)1− γE 2 e µ h0| T Tr Sn¯ (0, r1 n ¯ ) qs (r1
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