To \({d}\) , or not to \({d}\) : recent developments and comparisons of regularization schemes

The European Physical Journal C, Jul 2017

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.

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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)


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C. Gnendiger, A. Signer, D. Stöckinger, A. Broggio, A. L. Cherchiglia, F. Driencourt-Mangin, A. R. Fazio, B. Hiller, P. Mastrolia, T. Peraro, R. Pittau, G. M. Pruna, G. Rodrigo, M. Sampaio, G. Sborlini, W. J. Torres Bobadilla, F. Tramontano, Y. Ulrich, A. Visconti. To \({d}\) , or not to \({d}\) : recent developments and comparisons of regularization schemes, The European Physical Journal C, 2017, pp. 471, Volume 77, Issue 7, DOI: 10.1140/epjc/s10052-017-5023-2