Twistor Inspired Methods in Perturbative Field Theory and Fuzzy Funnels Simon McNamara Thesis submitted for the degree of Doctor of Philosophy (PhD) of the University of London Thesis Supervisor Prof. Bill Spence Department of Physics, Queen Mary, University of London, Mile End Road, London, E1 4NS August 2006 Abstract The first part of this thesis contains two new techniques for the calculation of scattering amplitudes in quantum field theories. These methods were inspired by the recent proposal of a correspondence between the weakly coupled regime of the maximally supersymmetric four dimensional gauge theory and a string theory in twistor space. We show how generalised unitarity cuts in D =4 − 2² dimensions can be used to calcu- late efficiently complete one-loop scattering amplitudes in non-supersymmetric Yang-Mills theory. This approach naturally generates the rational terms in the amplitudes, as well as the cut-constructible parts. We then show that the ideas of the Britto, Cachazo, Feng and Witten tree-level on-shell recursion relation can also be applied to the calculation of finite one-loop amplitudes in pure Einstein gravity. The second part of this thesis is a study of the nonabelian phenomena associated with D-branes. Specifically we study the nonabelian bionic brane intersection in which a stack of many coincident D1-branes expand via a non-commutative spherical configuration into a collection of higher dimensional D-branes orthogonal to the original stack of D1-branes. We suggest a construction of monopoles in dimension 2k+1 from fuzzy funnels. We then perform two charge calculations related to this construction. This leads to a new formula for the symmetrised trace quantity. This new formula for the symmetrised trace is then used to study the collapse of a spherical bound state of D0-branes. 2 Declaration I declare that the work in this thesis is my own, unless otherwise stated and resulted from collaborations with Andreas Brandhuber, Costis Papageorgakis, Sanjaye Ramgoolam, Bill Spence and Gabriele Travaglini. Some of the content of this thesis has been published in the papers [1] and [2]. Simon McNamara 3 Acknowledgements It is a pleasure to thank my inspirational supervisor Bill Spence and collaborators Andreas Brandhuber, Costis Papageorgakis, Sanjaye Ramgoolam, and Gabriele Travaglini. Thank you for so many fun hours of doing research. You have taught me a huge number of fascinating things. Thank you for all your help. A big thank you to my friends James Bedford, Samantha Bidwell, John Booth, Peter Brooks, Micheal Coad, Will Creedy, Aneta Dybek, Jannick Guillome, Paul Hafner, David Herbert, Xuan Kroeger, Fanny Lessous, Richard Lewis, Caroline Middleton-Hockin, David Mulryne, Simon Nickerson, Ronald Reid-Edwards, John Richards, John Ward, and Mark Wesker. Thanks also to all the members of Harpenden Cricket Club and the choir of St Martin-in-the-Fields. Finally thank you Mum, Dad and Sarah for looking after me. 4 Table of contents ABSTRACT 2 DECLARATION 3 ACKNOWLEDGEMENTS 4 INTRODUCTION 12 1 PERTURBATIVE FIELD THEORY AND TWISTORS 14 1.1 Introduction . 14 1.2 Amplitudes in perturbative field theory . 17 1.2.1 Feynman rules . 17 1.2.2 Colour stripped amplitudes . 18 1.2.3 The spinor helicity formalism . 19 1.3 Twistors . 22 1.3.1 The twistor transform . 22 1.3.2 The MHV amplitude in twistor space . 23 1.3.3 Twistor string theory . 26 1.4 Field theoretic MHV rules . 27 1.4.1 The tree level CSW rules. 27 1.4.2 The one loop BST rules . 29 1.4.3 Mansfield’s proof of the CSW rules . 30 1.5 BDDK’s two-particle unitarity cuts . 31 1.6 Generalised Unitarity . 34 1.6.1 Quadruple cuts in N =4 Super Yang-Mills . 35 5 1.6.2 Triple cuts in N =1 Super Yang-Mills . 37 1.7 The BCFW recursion relation . 39 1.7.1 A four-point example . 40 1.7.2 Generalisations of BCFW recursion . 42 1.7.3 Risager’s proof of the CSW rules . 43 2 GENERALISED UNITARITY FOR PURE YANG-MILLS 45 2.1 Generalised Unitarity in D = 4−2² Dimensions . 45 2.2 The one-loop ++++ amplitude . 49 2.3 The one-loop −+++ amplitude . 52 2.4 The one-loop −−++ amplitude . 58 2.5 The one-loop −+−+ amplitude . 61 2.6 The one-loop +++++ amplitude . 66 2.7 Future directions . 69 2.7.1 Higher point QCD amplitudes . 69 2.7.2 Higher loops, integrability and the AdS/CFT correspondence . 70 3 ON-SHELL RECURSION RELATIONS FOR ONE-LOOP GRAVITY 73 3.1 Introduction . 73 3.2 The all-plus amplitude . 76 3.2.1 The five-point all-plus amplitude . 77 3.2.2 The six-point all-plus amplitude . 80 3.3 The one-loop − + ++ amplitude . 84 3.3.1 − + ++ in Yang-Mills . 84 3.3.2 − + ++ in Gravity . 85 3.4 The one-loop − + + + + Gravity amplitude . 88 3.5 The description of nonstandard factorisations . 93 3.5.1 − + + + + in Yang-Mills with |1], |3i shifts . 93 3.5.2 − + ++ in gravity with |1], |2i, |3i, |4i shifts . 95 3.6 Avoiding nonstandard factorisations . 96 3.6.1 The − + + + + Yang-Mills amplitude . 97 6 4 THE ADHM CONSTRUCTION AND D-BRANE CHARGES 100 4.1 Introduction . 100 4.2 The ADHM construction . 106 4.2.1 Review of Nahm’s construction for D1 ⊥ D3 . 106 4.2.2 Generalisation to higher dimensions . 109 4.2.3 Construction of the Monopole . 110 4.3 Charge calculations . 111 4.3.1 Monopole charge calculation . 111 4.3.2 Proof of the identity involving N(k, n) and N(k − 1, n + 1) . 112 4.3.3 Fuzzy funnel charge calculation. 113 4.4 Symmetrised trace calculations . 115 4.4.1 Chord diagram calculations . 117 m 4.4.2 The large n expansion of Str(XiXi) . 121 4.4.3 An exact formula for the symmetrised trace . 124 5 FINITE N EFFECTS ON THE COLLAPSE OF FUZZY SPHERES 126 5.1 Introduction . 126 5.2 Lorentz invariance and the physical radius . 127 5.3 The fuzzy S2 at finite n ............................. 129 5.3.1 The D1 ⊥ D3 intersection at finite-n . 131 5.3.2 Finite N dynamics as a quotient of free multi-particle dynamics . 132 5.4 Physical properties of the finite N solutions . 133 5.4.1 Special limits where finite n and large n formulae agree . 133 5.4.2 Finite N effects . 135 5.4.3 Distance to blow-up in D1 ⊥ D3 . 138 5.5 Towards a generalisation to higher even-dimensional fuzzy-spheres . 139 5.6 Summary and Outlook . 141 A INTEGRAL IDENTITIES 143 A.1 Scalar box, triangle and bubble integrals . 143 A.2 Scalar Integral Identities . 144 A.3 PV reduction . 145 7 B GRAVITY AMPLITUDE CHECKS 148 B.1 The VegasShift[n] Mathematica command . 148 B.2 Other shifts of the − + + + + gravity amplitude . 149 B.2.1 The |4], |5i shifts . 149 B.2.2 The |2], |1i shifts . 151 C STR CALCULATIONS USING THE HIGHEST WEIGHT METHOD 152 C.1 Review of spin half for SO(3) . 153 C.2 Derivation of symmetrised trace for minimal SO(2l + 1) representation . 154 C.3 Derivation of spin one symmetrised trace for SO(3) . 155 C.4 Derivation of next-to-minimal representation for SO(2l + 1) . 156 8 List of Figures 1.1 The result that MHV amplitudes are supported on degree one, genus zero curves in twistor space is equivalent to the fact that the MHV amplitude is a function of only the λs and not of the λes. ................... 24 1.2 The twistor space localisation of tree amplitudes. Diagram (a) shows the localisation of an amplitude with three negative helicity gluons and diagram (b) an amplitude with four negative helicity gluons. 25 1.3 The MHV diagrams associated with the CSW construction of the simplest next-to-MHV tree-level amplitude A(1−, 2−, 3−, 4+, 5+). It is conventional to consider amplitudes where all the momenta are outgoing, so an internal leg joining two amplitudes will have different helicities labels at each end. 28 1.4 The BST construction of the MHV one-loop amplitude in N =4 super Yang- Mills by sewing two tree-level MHV amplitudes together. 29 1.5 The s channel cut of a one-loop amplitude. 32 1.6 A simple quadruple cut to evaluate the coefficient of a one-mass box in the five-point MHV amplitude in N =4 Super Yang-Mills. 35 1.7 The diagrams in the triple cut of the one-loop MHV amplitude for N = 1 Super-Yang Mills. ................................. 38 1.8 A diagrammatic representation of the BCFW recursion relation. The sum is over all factorisations into pairs of amplitudes and the possible helicity of the intermediate state. ................................ 40 1.9 The recursive diagram in the BCFW construction of A(1−, 2+, 3−, 4+). 41 2.1 One of the two quadruple-cut diagrams for the amplitude 1+2+3+4+. This di- agram is obtained by sewing tree amplitudes (represented by the blue bubbles) with an external positive-helicity gluon and two internal scalars of opposite ‘helicities’. There are two such diagrams, which are obtained one from the other by flipping all the internal helicities. These diagrams are equal so that the full result is obtained by doubling the contribution from the diagram in this Figure. The same remark applies to all the other diagrams considered in this chapter. ................................... 50 9 2.2 One of the possible three-particle cut diagrams for the amplitude 1+2+3+4+. The others are obtained from this one by cyclic relabelling of the external particles.