To \({d}\) , or not to \({d}\) : recent developments and comparisons of regularization schemes
Eur. Phys. J. C
To d , or not to d : recent developments and comparisons of regularization schemes
C. Gnendiger 7
A. Signer 7 8
D. St?ckinger 9
A. Broggio 10
A. L. Cherchiglia 11
F. Driencourt-Mangin 12
A. R. Fazio 13
B. Hiller 14
P. Mastrolia 15 16
T. Peraro 0
R. Pittau 1
G. M. Pruna 7
G. Rodrigo 12
M. Sampaio 2
G. Sborlini 3 4 12
W. J. Torres Bobadilla 12 15 16
F. Tramontano 5 6
Y. Ulrich 7 8
A. Visconti 7 8
0 Higgs Centre for Theoretical Physics, The University of Edinburgh , Edinburgh EH9 3FD , UK
1 Dep. de Fi?sica Teo?rica y del Cosmos and CAFPE, Universidad de Granada , 18071 Granada , Spain
2 Departamento de Fi?sica , ICEX, UFMG, 30161-970 Belo Horizonte , Brazil
3 Dipartimento di Fisica, Universita? di Milano , 20133 Milan , Italy
4 INFN, Sezione di Milano , 20133 Milan , Italy
5 Dipartimento di Fisica, Universita? di Napoli , 80126 Naples , Italy
6 INFN, Sezione di Napoli , 80126 Naples , Italy
7 Paul Scherrer Institut , 5232 Villigen, PSI , Switzerland
8 Physik-Institut, Universita?t Zu?rich , 8057 Zu?rich , Switzerland
9 Institut fu?r Kernund Teilchenphysik, TU Dresden , 01062 Dresden , Germany
10 Physik Department T31, Technische Universita?t Mu?nchen , 85748 Garching , Germany
11 Centro de Cie?ncias Naturais e Humanas, UFABC , 09210-170 Santo Andre? , Brazil
12 Insituto de Fi?sica Corpuscular, UVEG-CSIC, Universitat de Vale?ncia , 46980 Paterna , Spain
13 Departamento de Fi?sica, Universidad Nacional de Colombia , Bogota? D.C. , Colombia
14 CFisUC, Department of Physics, University of Coimbra , 3004-516 Coimbra , Portugal
15 Dipartimento di Fisica ed Astronomia, Universita? di Padova , 35131 Padua , Italy
16 INFN, Sezione di Padova , 35131 Padua , Italy
We give an introduction to several regularization schemes that deal with ultraviolet and infrared singularities appearing in higher-order computations in quantum field theories. Comparing the computation of simple quantities in the various schemes, we point out similarities and differences between them. 1 Introduction . . . . . . . . . . . . . . . . . . . . . 2 2 DS: dimensional schemes CDR, HV, FDH, DRED . 3 2.1 Integration in d dimensions and dimensional schemes 3 2.2 Application example 1: electron self-energy at NLO . 4 2.3 Application example 2: e+e? ? ? ? ? qq? at NLO . 5 Virtual contributions . . . . . . . . . . . . . . . 6 Real contributions . . . . . . . . . . . . . . . . 7 2.4 Established properties and future developments of DS . . . . . . . . . . . . . . . . . . . . . . 9 3 FDF, SDF: four- and six-dimensional formalism . . 9 3.1 FDF: four-dimensional formulation of FDH . . 10 3.2 Wave functions in FDF . . . . . . . . . . . . . 11 Spinors . . . . . . . . . . . . . . . . . . . . . 11 Polarization vectors . . . . . . . . . . . . . . . 12 3.3 Established properties and future developments of FDF . . . . . . . . . . . . . . . . . . . . . . 13 Equivalence of FDF and FDH at NLO: virtual contributions to e+e? ? ? ? ? qq? . . . 13 Renormalization of the FDF-scalar-fermion coupling . . . . . . . . . . . . . . . . . . . 13 3.4 Automated numerical computation . . . . . . . 15 3.5 SDF: six-dimensional formalism . . . . . . . . 16 Internal degrees of freedom . . . . . . . . . . . 16 Internal states: six-dimensional spinor-helicity formalism . . . . . . . . . . . . . . . . . 16 Applications to integrand reduction via generalized unitarity . . . . . . . . . . . . . . . 17 4 IREG: implicit regularization . . . . . . . . . . . . 18 4.1 Introduction to IREG and electron self-energy at NLO . . . . . . . . . . . . . . . . . . . . . 18 4.2 Application example: e+e? ? ? ? ? qq? at NLO 19 Virtual contributions . . . . . . . . . . . . . . . 20 Real contributions . . . . . . . . . . . . . . . . 21 4.3 Established properties of IREG . . . . . . . . . 22 Gauge invariance . . . . . . . . . . . . . . . . 22
Contents
UV renormalization . . . . . . . . . . . . . . . 23
5 FDR: four-dimensional regularization/renormalization 24
5.1 FDR and UV infinities . . . . . . . . . . . . . 24
5.2 FDR and IR infinities . . . . . . . . . . . . . . 25
5.3 Application example: e+e? ? ? ? ? qq? at NLO 27
Virtual contributions . . . . . . . . . . . . . . . 27
Real contributions . . . . . . . . . . . . . . . . 28
5.4 Established properties and future developments
of FDR . . . . . . . . . . . . . . . . . . . . . 29
Correspondence between integrals in FDR and
DS . . . . . . . . . . . . . . . . . . . . . 29
Gauge invariance, unitarity, and extra integrals . 30
6 FDU: four-dimensional unsubtraction . . . . . . . . 30
6.1 Introduction to LTD . . . . . . . . . . . . . . . 31
6.2 Momentum mapping and IR singularities . . . 31
6.3 Integrand-level renormalization and self-energies 32
6.4 Application example: e+e? ? ? ? ? qq? at NLO 33
6.5 Further considerations and comparison with
other schemes . . . . . . . . . . . . . . . . . . 34
7 Summary and outlook . . . . . . . . . . . . . . . . 34
References . . . . . . . . . . . . . . . . . . . . . . . . 36
1 Introduction
Higher-order calcula (...truncated)