W-pair production near threshold in unstable particle effective theory [Elektronische Ressource] / vorgelegt von Pietro Falgari

Sujets

• Physik

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2008
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

W-pair production near threshold in
unstable-particle effective theory
Von der Fakult¨at fu¨r Mathematik, Informatik und
Naturwissenschaften der RWTH Aachen University zur
Erlangung des akademischen Grades eines Doktors der
Naturwissenschaften genehmigte Dissertation
vorgelegt von
Dottore Magistrale in Fisica
Pietro Falgari
aus Bergamo, Italien
Berichter: Universit¨atsprofessor Martin Beneke
Universit¨atsprofessor Werner Bernreuther
Tag der mu¨ndlichen Pru¨fung: 7. November 2008
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek
online verfu¨gbar.Abstract
− +Inthisthesiswepresentadedicated studyofthefour-fermionproductionprocesse e →
− ¯ ν¯ udX near the W-pair production threshold, in view of its importance for a precise
determinationoftheW-bosonmassattheILC.Thecalculation isperformedintheframe-
work of unstable-particle effective theory, which allows for a gauge-invariant inclusion of
instability effects, and for a systematic approximation of the full cross section with an
expansion in the coupling constants, the ratio Γ /M , and the non-relativistic velocityW W
v of the W boson. The effective-theory result, computed to next-to-leading order in the
2expansion parameters Γ /M ∼ α ∼ v , is compared to the full numerical next-to-W W ew
leading order calculation of the four-fermion production cross section, and agreement to
better than 0.5% is found in the region of validity of the effective theory. Furthermore,
we estimate the contributions of missing higher-order corrections to the four-fermion pro-
cess, and how they translate into an error on the W-boson mass determination. We find
that the dominant theoretical uncertainty onM is currently due to an incomplete treat-W
ment of initial-state radiation, while the remaining combined uncertainty of the two NLO
calculations translates into δM ≈ 5 MeV. The latter error is removed by an explicitW
computation of the dominant missing terms, which originate from the expansion in v of
next-to-next-to-leading order Standard Model diagrams. The effect of resummation of
logarithmically-enhanced terms is also investigated, but found to be negligible.
iiiivContents
1 Introduction 1
1.1 Measurement of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1W
1.2 Theoretical status of W-pair production . . . . . . . . . . . . . . . . . . . . 3
1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Effective Field Theory description of unstable-particle production 11
2.1 Unstable-particle effective theory . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Effective Lagrangian for soft and collinear interactions . . . . . . . . 14
2.1.2 Effective vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3 Matching ofL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21EFT
2.2 Unstable-particle effective theory for pair production near threshold . . . . 23
2.2.1 Effective vertices and the leading-order cross section . . . . . . . . . 26
2.3 Classification of corrections up to NLO . . . . . . . . . . . . . . . . . . . . 30
2.3.1 Hard matching coefficients . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.2 Loop calculations in the effective theory . . . . . . . . . . . . . . . . 32
3 The four-fermion Born cross section 35
3.1 Contributions from the potential region . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Threshold expansion of the resummed propagator . . . . . . . . . . 36
3.1.2 Relation between the effective-theory approximation and the fixed-
width prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.3 Expansion of the production squared matrix elements . . . . . . . . 39
3.1.4 NLO potential contributions to the Born cross section . . . . . . . . 40
3.2 Contributions from the hard region . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Vanishing of the leading-order hard correction. . . . . . . . . . . . . 43
1/23.2.2 N LO contribution from the hard region . . . . . . . . . . . . . . . 44
3.3 Comparison of the EFT prediction to the four-fermion Born cross section . 47
3.4 Implementation of kinematical cuts in the EFT formalism . . . . . . . . . . 49
3.4.1 Invariant-mass cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4.2 Angular cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Radiative corrections 55
4.1 Hard corrections to production and decay . . . . . . . . . . . . . . . . . . . 55
4.1.1 Production vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
v4.1.2 Decay corrections and selection of flavour-specific final states at NLO 59
4.2 Coulomb corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3 Soft-photon corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Collinear-photon corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 Initial-state radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.1 The physical next-to-leading order cross section . . . . . . . . . . . . 71
4.5.2 Resummation of initial-state radiation . . . . . . . . . . . . . . . . . 75
5 NLO four-fermion production cross section 77
5.1 Input parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 NLO four-fermion production cross section in the effective theory . . . . . . 79
5.3 Comparison to the full four-fermion calculation . . . . . . . . . . . . . . . . 82
5.4 Theoretical error of theM determination . . . . . . . . . . . . . . . . . . 83W
3/26 N LO corrections to four-fermion production near the WW threshold 89
3/26.1 Relevant momentum regions and classification of N LO corrections . . . . 89
3/26.2 Evaluation of N LO corrections . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2.1 Soft and soft-collinear corrections . . . . . . . . . . . . . . . . . . . . 93
6.2.2 Hard and hard-collinear corrections. . . . . . . . . . . . . . . . . . . 95
6.2.3 The total cross section . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.4 ISR resummation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2.5 Semi-soft and hard corrections to the Coulomb potential . . . . . . . 97
6.2.6 Decay corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2.7 Residue corrections to the W-propagators . . . . . . . . . . . . . . . 100
6.3 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7 Resummation of threshold logarithms 107
7.1 Factorisation and resummation in Drell-Yan . . . . . . . . . . . . . . . . . . 108
7.2 Factorisation of the W-pair production cross section near threshold . . . . . 112
ˆ7.2.1 Comparison of W with the DY soft function . . . . . . . . . . . 122WW
7.3 Resummation of the W-pair production cross section near threshold . . . . 123
7.3.1 Resummation of the hard matching coefficient . . . . . . . . . . . . 123
7.3.2 Resummation of the soft function . . . . . . . . . . . . . . . . . . . . 126
7.3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8 Conclusion 133
A The expansion by regions 135
B Expansion of the Born cross section 143
B.1 Expansion in the potential region . . . . . . . . . . . . . . . . . . . . . . . . 143
B.2 Expansion in the hard region . . . . . . . . . . . . . . . . . . . . . . . . . . 147
viC One-loop electroweak hard corrections to production and decay 153
C.1 Production vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
C.2 Virtual corrections to W decay . . . . . . . . . . . . . . . . . . . . . . . . . 156
C.3 Integrals and auxiliary functions . . . . . . . . . . . . . . . . . . . . . . . . 157
− + − ¯D Soft corrections to e e → ν¯ ud 163
D.1 Virtual corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
D.2 Real Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
D.3 Cancellation between real and virtual corrections . . . . . . . . . . . . . . . 172
E Hard-collinear corrections 177
F Renormalisation of the Coulomb potential by hard corrections 185
F.1 Charge renormalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
F.2 Hard corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
F.3 Formulas for the α(M ) and G schemes . . . . . . . . . . . . . . . . . . . 188Z
Bibliography 191Chapter 1
Introduction
In this chapter we will discuss the phenomenological relevance of the process of W-pair
production near threshold for a precise determination of the W-boson mass M . WeW
will also review the theoretical issues related to the calculation of this process, and give
a short overview of the methods and results available at present. Finally we will present
the outline of this thesis.
1.1 Measurement of MW
ThemassoftheW gaugebosonhasacentralroleintestsoftheStandardModel(SM)and
searches for virtual-particle effects through electroweak precision measurements, and is of
pivotal importance for understanding the physics underlying the electroweak symmetry
breaking. In the Standard ModelM is related to the top quark mass m and the HiggsW t
boson mass M through loop corrections [1], as schematically shown in Figure 1.1, andH
the direct measurements of the first two masses give informations on the third one.