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Thesis Presented for the Degree of Doctor of Philosophy Durham E-Theses Hadronic productions of a Higgs boson in association with two jets at next-to-leading order WILLIAMS, CIARAN How to cite: WILLIAMS, CIARAN (2010) Hadronic productions of a Higgs boson in association with two jets at next-to-leading order , Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/414/ 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 2 Hadronic production of a Higgs boson in association with two jets at next-to-leading order Ciaran Williams A Thesis presented for the degree of Doctor of Philosophy Institute for Particle Physics Phenomenology Department of Physics University of Durham England August 2010 Dedicated to My Parents Hadronic production of a Higgs boson in association with two jets at next-to-leading order Ciaran Williams Submitted for the degree of Doctor of Philosophy August 2010 Abstract We present the calculation of hadronic production of a Higgs boson in association with two jets at next-to-leading order in perturbation theory. We consider ampli- tudes in an effective theory in which the Higgs couples to gluons in the limit of a large top quark mass. We treat the Higgs as the real part of the complex field φ that couples to the self-dual field strengths. We use modern on-shell inspired methods to calculate helicity amplitudes and we give a detailed review of unitarity based and on-shell methods. Using these unitarity methods we derive the cut-constructible pieces of the general φ-MHV amplitudes in which the positions of the two negative gluons are arbitrary. We then generate the cut-constructible pieces of the φ-NMHV (1) + − − − (1) − + − − four parton amplitudes A4 (φ, 1 , 2 , 3 , 4 ) and A4 (φ, 1q , 2q , 3 , 4 ). We gen- erate the rational pieces of these amplitudes and the four-gluon φ-MHV amplitude (1) − + − + A4 (φ, 1 , 2 , 3 , 4 ), using Feynman diagrams. For the φ-MHV amplitude we also use the unitarity-boostrap method to calculate the rational pieces. We then imple- ment these, and analytic results from previous calculations, into MCFM. Using this program we are able to perform some phenomenological studies at the Tevatron and LHC. Declaration The work in this thesis is based on research carried out at the Institute for Particle Physics Phenomenology, University of Durham, England. No part of this thesis has been submitted elsewhere for any other degree or qualification. The research described in this thesis was carried out in collaboration with, Simon D. Badger, John M. Campbell, R. Keith Ellis, E. W. Nigel Glover and Pierpaolo Mastrolia. Aspects of chapters 2, 3, 4, 5 and 6 are based on the following published works: J. M. Campbell, R. K. Ellis and C. Williams, Hadronic production of a Higgs • boson and two jets at next-to-leading-order, Phys. Rev. D 81, 074023 (2010). S. D. Badger, J. M. Campbell, R. K. Ellis and C. Williams, Analytic results • for the one-loop NMHV Hqqgg amplitude, JHEP 12 (2009) 035. S. D. Badger, E. W. N. Glover, P. Mastrolia and C. Williams, One-Loop Higgs • plus four gluon Amplitudes: Full analytic results, JHEP 01 (2010) 036 E. W. N. Glover and C. Williams, One-Loop Gluonic Amplitudes from Single • Unitarity Cuts, JHEP 12 (2008) 067 E. W. N. Glover, P. Mastrolia and C. Williams, One-loop phi-MHV amplitudes • using the unitarity bootstrap: the general helicity case, JHEP 08 (2008) 017 Copyright c 2010 by Ciaran Williams. iv Acknowledgements First and foremost I would like to thank my supervisor Nigel Glover, whose help and guidance has been amazing. Your advice and support has been invaluable. I would like to thank my collaborators from whom I have learnt a great deal, Simon Badger, John Campbell, Keith Ellis, Valya Khoze and Pierpaolo Mastrolia. I would like to thank all at the IPPP who have made my four years here extremely enjoyable, I would like to thank Linda Wilkinson and Trudy Forster, who have helped me countless times with various problems (usually of my own making). I would like to thank Daniel Maitre for hours of free Mathematica tutorials, and David Ambrose-Griffith and Peter Richardson for helping me get started on the GRID. I would also like to thank my office mates Nehir Ikizlerli, Joao Pires and newbie James Currie who have provided great banter, together we made one great Nigel tracking team. Special mention should also go to Jamie “Tatts” Tattersall possibly the second greatest TuxKart racer that ever lived. Thanks to all the other postdocs, PhD students and staff that made the IPPP so fun, there’s too many to list explicitly here but you know who you are! I would like to thank my friends and family for their love and support, especially Mum, Dad, Ryan, Steve, Dave, Tim, Phil and Chelle. Finally I would like to thank Heather, for more than can ever be written down, but especially for the abstract reading! This work was funded by an STFC studentship. v Contents Abstract iii Declaration iv Acknowledgements v 1 Introduction 1 1.1 The Standard Model of Particle Physics . 2 1.1.1 Standard Model: Yang-Mills theories and gauge invariance. 2 1.1.2 Regularisation of UV and IR divergences . 7 1.1.3 An overview of a hadronic collision . 9 1.2 Electroweak Symmetry Breaking . 12 1.2.1 Spontaneous breaking of O(N) symmetries, Goldstone bosons 12 1.2.2 SpontaneousbreakingofscalarQED . 14 1.2.3 TheHiggsmechanism . 15 1.3 Higgssearchesatcolliders . 17 1.3.1 HiggssearchesatLEP . 17 1.3.2 Higgssearches attheTevatronandtheLHC . 18 1.4 Effective coupling between gluons and a Higgs in the limit of a heavy topquark ................................. 23 vi Contents vii 1.4.1 EffectiveLagrangian . 23 1.4.2 φ,φ† splitting oftheEffective Lagrangian. 27 1.5 Higgs plus two jets: Its phenomenological role and an overview of this thesis.................................... 28 2 Unitarity and on-shell methods 31 2.1 Unitarity.................................. 31 2.1.1 Unitarity: AnIntroduction. 31 2.2 Quadruplecuts .............................. 34 2.2.1 Thequadruplecutmethod. 34 2.2.2 Aquadruplecutexample. 37 2.3 Forde’s Laurent expansion method . 40 2.3.1 Thetriplecutmethod . 40 2.3.2 Atriplecutexample . 44 2.3.3 Thedoublecutmethod. 44 2.4 Spinorintegration............................. 45 2.4.1 Spinor integration via the holomorphic anomaly . ... 45 2.4.2 Spinor integration via Stokes’ Theorem . 48 2.4.3 Adoublecutexample .. .. 51 2.5 MHV rules and BCFW recursion relations . 52 2.5.1 TheMHV/CSWrules .. .. 52 2.5.2 TheBCFWrecursionrelations . 54 2.6 Theunitarity-bootstrap . 56 2.6.1 BCFWatone-loop ........................ 57 2.6.2 Cut-constructible, cut-completion, rational and overlap terms 59 2.6.3 Techniques for general helicity amplitudes . ... 62 Contents viii 2.7 D-dimensional techniques . 65 2.8 Recent progress: One-loop automatisation . .... 66 2.9 Summary ................................. 68 3 One-loop φ-MHV amplitudes: the general helicity case 69 3.1 Introduction................................ 69 3.2 Thecut-constructibleparts. 70 3.2.1 Boxcoefficientsfromfour-cuts . 71 3.2.2 TriangleCoefficients . 83 −2 3.2.3 Cancellation of Nf ǫ poles ................... 90 3.2.4 φ-MHVDoublecuts ....................... 92 3.2.5 Combined cuts: The cut-constructible pieces of the φ-MHV amplitude ............................. 96 3.3 Therationalpieces ............................ 98 3.3.1 Thecut-completionterms . 99 3.3.2 Therecursiveterms. .100 3.3.3 Theoverlapterms. .102 3.3.4 The large z behaviour of the completion terms . 103 3.3.5 Combinedrationalpieces. .105 3.4 CrossChecksandLimits . .106 3.4.1 Collinearlimits . .106 3.4.2 Collinear factorisation of the cut-constructible contributions . 107 3.4.3 The cancellation of unphysical singularities . .110 3.4.4 Collinear factorisation of the rational pieces . .112 (1) − + − + 3.4.5 Soft limit of A4 (φ, 1 , 2 , 3 , 4 )................113 3.5 Summary .................................114 Contents ix 4 One-loop Higgs plus four-gluon amplitudes: full analytic results 116 4.1 Introduction................................116 4.2 Cut-Constructible Contributions . .117 4.2.1 BoxIntegralCoefficients . .117 4.2.2 TriangleIntegralCoefficients. .119 4.2.3 Bubble Integral Coefficients . 121 4.2.4 TheCut-Completionterms. .123 4.3 RationalTerms ..............................123 4.4 Higgsplusfourgluonamplitudes . 125 (1) − − − − 4.4.1 The all-minus amplitude A4 (H, 1 , 2 , 3 , 4 ) ........126 (1) − − + + 4.4.2 The MHV amplitude A4 (H, 1 , 2 , 3 , 4 ) ..........126 (1) − + − + 4.4.3 The MHV amplitude A4 (H, 1 , 2 , 3 , 4 ) ..........127 (1) + − − − 4.4.4 The NMHV amplitude A4 (H, 1 , 2 , 3 , 4 ) .........128 4.5 NumericalEvaluation. .129 4.6 Summary .................................130 5 The φqqgg- NMHV amplitude 132 5.1 Introduction................................132 5.1.1 Definition of colour ordered amplitudes . 133 5.1.2 Known analytic results for Hqqjj amplitudes . .134 5.2 One-loopresults .............................137 − − 5.2.1 Results for A4;1(φ, 1q¯, 2q, 3g , 4g ).................137 − − 5.2.2 Results for A4;3(φ, 1q¯, 2q, 3g , 4g ).................142 5.3 Numericalresults .............................145 5.4 Summary .................................145 Contents x 6 Phenomenological Studies 147 6.1 Introduction................................147 6.2 Improvements from the semi-numeric code .
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