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1.3 Renormalized QCD Perturbation Theory Durham E-Theses RS-invariant resummations of QCD perturbation theory Tonge, Darrell Graham How to cite: Tonge, Darrell Graham (1997) RS-invariant resummations of QCD perturbation theory, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/4964/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk RS°Invariant Resummatioes of QCD Perturbation Theory A thesis submitted for the degree of Doctor of Philosophy by Darrell Graham Tonge The copyright of this thesis rests with the author. No quotation from it should be published without the written consent of the author and information derived from it should be acknowledged. Centre for Particle Theory University of Durham September 1997 Abstract We propose a renormalon-inspired resummation of QCD perturbation theory based on approximating the renormalization scheme (RS) invariant effective charge (EC) beta- function coefficients by the portion containing the highest power of b = (llJV-2iV/)/6, the first beta-function coefficient, for SU(7V) QCD with Nf quark flavours. This can be accomplished using exact large-iVy all-orders results. The resulting resummation is RS-in variant and the exact next-to-leading order (NLO) and next-to-NLO (NNLO) coef• ficients in any RS are included. This improves on a previously employed naive leading-6 resummation which is RS-dependent. The RS-invariant resummation is used to assess the reliability of fixed-order perturbation + theory for the e e~ i?-ratio, hadronic tau-decay ratio RT, and Deep Inelastic Scattering (DIS) sum rules, by comparing it with the exact NNLO results in the EC RS. For R and RT, where large-order perturbative behaviour is dominated by a leading ultra-violet renormalon singularity, the comparison indicates fixed-order perturbation theory to be very reliable. For DIS sum rules, which have a leading infra-red renormalon singularity, the performance is rather poor. We show that QCD Minkowski observables such as the R and RT are completely de• termined by the EC beta-function, p{x), corresponding to the Euclidean QCD vacuum polarization Adler D-function, together with the NLO perturbative coefficient of D. An efficient numerical algorithm is given for evaluating R, RT from a weighted contour inte• gration of D(.se10) around a circle in the complex squared energy s-plane, with p(x) used to evolve in s around the contour. The difference between the R. RT constructed using the NNLO and leading-6 resummed versions of p(x) provides an estimate of the uncertainty due to the uncalculated higher order corrections. We estimate that at LEP energies ideal data on the i?-ratio could de• termine Qs(M|) to three-significant figures. For RT we estimate a theoretical uncertainty 2 6as(m T) ~ 0.01, corresponding to 6as(M|) ~ 0.002. This encouragingly small uncer• tainty is much less than has recently been deduced from comparison with the analogous naive all-orders resumma.tion, which we demonstrate to be extremely RS dependent and hence misleading. Acknowledgements I would especially like to thank my supervisor, Chris Maxwell, for his guidance and support throughout my three years in Durham. I am very grateful for the privilege and opportunity of working and collaborating with him. My thanks go to Matt for putting up with me in the confined office space over the last three years, and to David 'the complete theoretical physicist' and Andrew 'the complete cynic' for always finding some form of humour (however sick) in the most dire situations. Beyond this I wish to express my thanks to all my fellow students, to numerous to mention, for making the Centre for Particle Theory such a pleasant place to work. My gratitude goes to all the staff in the particle physics group here in Durham. To Nigel Glover, Valya Khoze, Alan Martin, Mike Pennington, James Stirling, and Mike Whalley, thank you all for giving me the opportunity to be part of the group. Can I also thank all my friends outside CPT: the lads from Astrophysics who never let scientific discovery impinge upon valuable drinking time; the girls from archaeological conservation — they know who they are; and the original crew from Palatine House (in alphabetical order) — Dave, Gail, Jason, Peter, Phil and Rob. Most of all can I thank Ania, for her friendship, encouragement and support in my final year at Durham. I would like to acknowledge PPARC for funding this research. Most importantly, I would like to thank my parents and family, for always being there for me, and for their support and encouragement over the years. I dedicate this Ph.D. thesis to them. This thesis is dedicated to my family, to Mum and Dad with all my love. D e c r Si t 1o n I declare that no material presented in this thesis has previously been submitted for a degree at this or any other university. The research described in this thesis has been carried out in collaboration with Dr. C.J. Maxwell and has been published as follows: ® RS-invariant all-orders renormalon resummations for some QCD observables, C.J. Maxwell and D.G. Tonge, Nucl. Phys. B481 (1996) 681. • The uncertainty in as(Af§) determined from hadronic tau decay measurements, C.J. Maxwell and D.G. Tonge, Durham Preprint DTP/97/30 [hep-ph/9705314]. © The copyright of this thesis rests with the author. Contents 1 A theoretical overview of QCD 1 1.1 Introduction 1 1.2 The QCD Lagrangian 1 1.2.1 Colour SU(N) 2 1.2.2 The classical QCD Lagrangian 2 1.2.3 Local gauge invariance 3 1.2.4 Gauge fixing and ghost fields 4 1.3 Renormalized QCD perturbation theory 5 1.3.1 The perturbative expansion of the S-matrix 6 1.3.2 Regularization 8 1.3.3 Renormalization 9 1.4 The renormalization group 11 1.4.1 The renormalization group equation 12 1.4.2 The running coupling 13 1.5 The QCD beta-function 14 1.5.1 Dimensional transmutation 14 1.5.2 The beta-function equation and A parameter 15 1.5.3 RG-improved perturbation theory 16 1.6 Asymptotic freedom and confinement 18 1.7 The operator product expansion (OPE) 19 1.7.1 The Adler D-function 19 1.8 QCD observables of interest 21 1.8.1 The e+e- i?-ratio 22 1.8.2 Hadronic tau decay ratio RT 23 1.8.3 Deep inelastic scattering (DIS) sum rules 27 1.9 Measuring A and the strong coupling as 29 1.9.1 Evolving as through quark mass thresholds 30 1.10 Summary and conclusions 31 CONTENTS ii 2 Renormalization scheme dependence 33 2.1 Introduction , 33 2.2 The scheme dependence problem 35 2.2.1 Parametrizing RS dependence 35 2.2.2 RS dependence at NLO 38 2.2.3 RS dependence at NnLO 40 2.3 Common renormalization schemes 42 2.3.1 Minimal subtraction and variants 42 2.3.2 Momentum space subtraction (MOM) scheme 43 2.3.3 Finite schemes 44 2.4 Proposed solutions to RS dependence 44 2.4.1 Physical scale 44 2.4.2 BLM scale fixing 45 2.4.3 The principle of minimum sensitivity (PMS) 48 2.5 The effective charge (EC) formalism 49 2.5.1 The effective charge beta-function 49 2.5.2 The non-perturbative integrated beta-function 50 2.5.3 The effective charge scheme 52 2.6 Summary and conclusions 53 3 Perturbation theory at large-orders 54 3.1 Introduction 54 3.2 The divergence of the QED perturbation theory 55 3.3 Asymptotic series 56 3.4 Borel summation 59 3.4.1 The Borel transform 60 3.4.2 Exploiting Borel transform technology 62 3.5 The Large-Nj expansion 64 3.6 Renormalons 68 3.6.1 Renormalons in QED 68 3.6.2 Renormalons in QCD 69 3.7 Instanton singularities 71 3.8 IR renormalons and the connection to the OPE 72 3.9 Summary and conclusions 75 4 RS-invariant all-orders leading-b resummations 76 4.1 Introduction 76 CONTENTS iii 4.2 "Naive" leading-6 resummations 80 4.2.1 Leading-6 expansions 80 4.2.2 Borel resummation of the leading-6 expansion 82 4.2.3 RS-dependence of the "naive" resummation 89 4.3 RS-invariant leading-6 Resummations 96 4.3.1 The leading-6 expansion of the EC beta-function 96 4.3.2 The Borel resummation of the EC beta-function 98 4.3.3 Numerical evaluation of the resummation D^L*^ 101 4.3.4 Evaluating the fixed-order approximants D(L*)(n) 102 4.4 Numerical results 104 4.4.1 Comparison of FOPT with leading-6 resummations 104 4.4.2 The reliability of FOPT 107 4.4.3 Energy dependence of the resummation and FOPT 118 4.4.4 Determining the uncertainty in Aj-^ and as(Mz) 118 4.5 Technical issues related to the resummations 123 4.5.1 Analytical continuation between the p® and p^, p^T 123 4.5.2 Analytic inversion of D{L\a) to obt ain p( ' 124 4.5.3 Infra-red properties of p{x) 125 4.5.4 Approximation of the large-N limit by the large- 6 expansion ...
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