aa r X i v : . [ h e p - t h ] O c t QMUL-PH-07-22CERN-PH-TH/2007-157
On Perturbative Field Theory andTwistor String Theory James Bedford a,ba
Centre for Research in String Theory, Department of PhysicsQueen Mary, University of LondonMile End Road, London E1 4NS, UK b Department of Physics, CERN - Theory Division1211 Geneva 23, Switzerland
ABSTRACT
It is well-known that perturbative calculations in field theory can lead to far simpleranswers than the Feynman diagram approach might suggest. In some cases scatteringamplitudes can be constructed for processes with any desired number of external legsyielding compact expressions which are inaccessible from the point of view of conven-tional perturbation theory. In this thesis we discuss some attempts to address the natureof this underlying simplicity and then use the results to calculate some previously un-known amplitudes of interest. Witten’s twistor string theory is introduced and the CSWrules at tree-level and one-loop are described. We use these techniques to calculate theone-loop gluonic MHV amplitudes in N = 1 super-Yang-Mills as a verification of theirvalidity and then proceed to evaluate the general MHV amplitudes in pure Yang-Millswith a scalar running in the loop. This latter amplitude is a new result in QCD. Inaddition to this, we review some recent on-shell recursion relations for tree-level ampli-tudes in gauge theory and apply them to gravity. As a result we present a new compactform for the n -graviton MHV amplitudes in general relativity. The techniques and re-sults discussed are relevant to the understanding of the structure of field theory andgravity and the non-supersymmetric Yang-Mills amplitudes in-particular are pertinentto background processes at the LHC. The gravitational recursion relations provide newtechniques for perturbative gravity and have some bearing on the ultraviolet propertiesof Einstein gravity. PhD thesis presented in June 2007. o my parents and in loving memory of my grandfather Leonard Rogers. cknowledgements
First and foremost I would like to thank my supervisor Andreas Brandhuber for allhis guidance and support throughout this work and for doing so much to mature myapproach to physics. Similarly, I am especially grateful to Gabriele Travaglini and BillSpence who have also acted as advisors and to Lance Dixon for being an inspirationalmentor.I am greatly indebted to David Mulryne, Constantinos Papageorgakis, John Wardand Konstantinos Zoubos for their unquestioning friendship and support and withoutwhom I would not be the person I am today. I would also like to express my gratitudeto my other collaborators Narit Pidokrajt and Diego Rodr´ıguez-G´omez for putting upwith me. It is likewise a pleasure to thank all these people for greatly helping thedevelopment of my physical understanding.For interesting discussions and help with the subject-matter I am extremely grate-ful to Luis ´Alvarez-Gaum´e, Charalampos Anastasiou, Pascal Anastasopoulos, LilyaAnguelova, Marcos Mari˜no Beiras, David Berman, David Birtles, Emil Bjerrum-Bohr,Vincent Bouchard, Jacob Bourjaily, Ruth Britto, Tom Brown, John Charap, Joe Conlon,Cedric Delaunay, Sara Dobbins, Ellie Dobson, Vittorio Del Duca, David Dunbar, DavidDunstan, Dario Duo, John Ellis, Kazem Bitaghsir Fadafan, Malcolm Fairbairn, Bo Feng,Joshua Friess, Michael Green, Umut G¨ursoy, Stefan Hohenegger, Harald Ita, Bob Jones,Chris Hull, Valya Khoze, Johanna Knapp, David Kosower, Theo Kreouzis, Duc NinhLe, Wolfgang Lerche, Simon McNamara, Alex Morisse, Michele Della Morte, AdeleNasti, Georgios Papageorgiou, Alice Perrett, Roger Penrose, Brian Powell, Are Rak-lev, Sanjaye Ramgoolam, Christian Romelsberger, Rodolfo Rousso, Ricardo Schiappa,Shima Shams, Jonathan Shock, Laura Tadrowski, Alireza Tavanfar, Steve Thomas,David Tong, Nicolaos Toumbas, Massimiliano Vincon, Daniel Waldram, Marlene Weiss,Arne Wiebalck and Giulia Zanderighi.The hospitality of both Queen Mary, University of London & CERN and theirsupport via a Queen Mary studentship and a Marie Curie early stage training grant aregratefully acknowledged, as are the many staff including Terry Arter, Kathy Boydon,Denise Paige, Nanie Perrin, Jeanne Rostant, Kate Shine, Diana De Toth and SuzyVascotto who make these places truly special.3or help with the manuscript I am especially grateful to Andi Brandhuber, Jo-hanna Knapp, Costis Papageorgakis, Jon Shock, Gabriele Travaglini, Marlene Weissand Costas Zoubos.The support of many friends and family members, the number of whom is too greatto list is also gratefully acknowledged. Especially, these include Sam Chamberlain,Jeremy and Sine Pickles, Barbara and Leonard Rogers and Christiane and Varos Shah-bazian. Last, but by no means least I owe a great debt to my parents Andrew andLinda, my brother Simon and to Claire. 4 able of contents
ACKNOWLEDGEMENTS 3INTRODUCTION 11SUMMARY 191 PERTURBATIVE GAUGE THEORY 211.1 Colour ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.2 Spinor helicity formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 271.2.1 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.2.2 Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.3 Variable reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 301.3 Supersymmetric decomposition . . . . . . . . . . . . . . . . . . . . . . . 311.4 Supersymmetric Ward identities . . . . . . . . . . . . . . . . . . . . . . 331.4.1 N = 1 SUSY constraints . . . . . . . . . . . . . . . . . . . . . . . 331.4.2 Amplitude relations . . . . . . . . . . . . . . . . . . . . . . . . . 351.5 Twistor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.5.1 Amplitude localisation . . . . . . . . . . . . . . . . . . . . . . . . 411.6 Twistor string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.6.1 Topological field theory . . . . . . . . . . . . . . . . . . . . . . . 451.6.2 Topological string theory . . . . . . . . . . . . . . . . . . . . . . 461.6.3 The B-model on super-twistor space . . . . . . . . . . . . . . . . 481.6.4 D1-brane instantons . . . . . . . . . . . . . . . . . . . . . . . . . 501.6.5 The MHV amplitudes . . . . . . . . . . . . . . . . . . . . . . . . 511.7 CSW rules (tree-level) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551.7.1 Off-shell continuation . . . . . . . . . . . . . . . . . . . . . . . . 561.7.2 The procedure: An example . . . . . . . . . . . . . . . . . . . . . 565.8 Loop diagrams from MHV vertices . . . . . . . . . . . . . . . . . . . . . 591.8.1 BST rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601.8.2 Integration measure . . . . . . . . . . . . . . . . . . . . . . . . . 601.9 MHV amplitudes in N = 4 super-Yang-Mills . . . . . . . . . . . . . . . . 631.9.1 General integral basis . . . . . . . . . . . . . . . . . . . . . . . . 631.9.2 The N = 4 MHV one-loop amplitudes . . . . . . . . . . . . . . . 641.9.3 MHV vertices at one-loop . . . . . . . . . . . . . . . . . . . . . . 662 MHV AMPLITUDES IN N = 1 SUPER-YANG-MILLS 712.1 The N = 1 MHV amplitudes at one-loop . . . . . . . . . . . . . . . . . . 712.2 MHV one-loop amplitudes in N = 1 SYM from MHV vertices . . . . . . 772.2.1 The procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.2.2 Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802.2.3 The full amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 813 NON-SUPERSYMMETRIC MHV AMPLITUDES 863.1 The scalar amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.2 The scattering amplitude in the adjacent case . . . . . . . . . . . . . . . 883.2.1 Rational terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.2.2 Dispersion integrals for the adjacent case . . . . . . . . . . . . . 893.3 The scattering amplitude in the general case . . . . . . . . . . . . . . . . 933.3.1 Comments on twistor space interpretation . . . . . . . . . . . . . 1013.4 Checks of the general result . . . . . . . . . . . . . . . . . . . . . . . . . 1023.4.1 Adjacent case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.4.2 Five-gluon amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 1033.4.3 Infrared-pole structure . . . . . . . . . . . . . . . . . . . . . . . . 1043.5 The MHV amplitudes in QCD . . . . . . . . . . . . . . . . . . . . . . . 1054 RECURSION RELATIONS IN GRAVITY 1074.1 The recursion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.2 Application to MHV gravity amplitudes . . . . . . . . . . . . . . . . . . 1114.2.1 Four-, five- and six-graviton scattering . . . . . . . . . . . . . . . 1124.2.2 General formula for MHV scattering . . . . . . . . . . . . . . . . 1154.3 Applications to other field theories . . . . . . . . . . . . . . . . . . . . . 1164.4 CSW as BCFW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176 CONCLUSIONS AND OUTLOOK 119A SPINOR AND DIRAC-TRACE IDENTITIES 124A.1 Spinor identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124A.2 The holomorphic delta function . . . . . . . . . . . . . . . . . . . . . . . 126A.3 Dirac traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128B FEYNMAN RULES IN THE SPINOR HELICITY FORMALISM 130B.1 Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130B.2 Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131B.3 Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131B.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132C D-DIMENSIONAL LORENTZ-INVARIANT PHASE SPACE 136C.1 D-spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136C.2 d LIPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136C.3 Overall amplitude normalisation . . . . . . . . . . . . . . . . . . . . . . 138D UNITARITY 140D.1 The optical theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141D.2 Cutting rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142D.2.1 BDDK’s unitarity cuts . . . . . . . . . . . . . . . . . . . . . . . . 143D.3 Dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145E INTEGRALS FOR THE N = 1 AMPLITUDE 147E.1 Passarino-Veltman reduction . . . . . . . . . . . . . . . . . . . . . . . . 147E.2 Box & triangle discontinuities from phase space integrals . . . . . . . . . 150F GAUGE-INVARIANT TRIANGLE RECONSTRUCTION 153F.1 Gauge-invariant dispersion integrals . . . . . . . . . . . . . . . . . . . . 153G INTEGRALS FOR THE NON-SUPERSYMMETRIC AMPLITUDE 156G.1 Passarino-Veltman reduction . . . . . . . . . . . . . . . . . . . . . . . . 156G.2 Evaluating the integral of C ( a, b ) . . . . . . . . . . . . . . . . . . . . . . 158G.3 Phase space integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160H KLT RELATIONS 1627 ist of Figures Feynman rules for
SU( N c ) Yang-Mills theory in Feynman gauge. . . . . 221.2
The number of Feynman diagrams required for tree-level n-gluon scattering.
The MHV amplitudes localise on simple straight lines in twistor space.Here the 5-point MHV amplitude is depicted as an example. . . . . . . . 421.4
Twistor space localisation of tree amplitudes with q = 3 and q = 4 . . . . 431.5 The two MHV diagrams contributing to the + − −− amplitude. All ex-ternal momenta are taken to be outgoing. . . . . . . . . . . . . . . . . . 571.6 A generic MHV diagram contributing to a one-loop MHV scattering am-plitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611.7
Boxes, Triangles and Bubbles. Here P i , K i and Q i are generic momentarepresenting the contribution of one or more external particles. The dif-ferent functions discussed below (1-mass, 2-mass etc.) are all specialcases of these . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631.8 The 2-mass easy box function. . . . . . . . . . . . . . . . . . . . . . . . . 651.9
A one-loop MHV diagram computed using MHV amplitudes as interactionvertices. This diagram has the momentum structure of the cut referredto at the end of § . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.1 The box function F of (2.1.7) , whose finite part B , Eq. (2.1.9) , appears inthe N = 1 amplitude (2.1.2) . The two external gluons with negative helic-ity are labelled by i and j . The legs labelled by s and m correspond to thenull momenta p and q respectively in the notation of (2.1.9) . Moreover,the quantities t [ s − m ] m +1 , t [ s − m ] m , t [ s − m − m +1 , t [ m − s − s +1 appearing in the box func-tion B in (2.1.19) correspond to the kinematical invariants t := ( Q + p ) , s := ( P + p ) , Q , P in the notation of (2.1.9) , with p + q + P + Q = 0 . A triangle function, corresponding to the first term T ǫ ( p m , q a +1 ,m − , q m +1 ,a ) in the second line of (2.1.19) . p , Q and P correspond to p m , q m +1 ,a and q a +1 ,m − in the notation of Eq. (2.1.19) , where j ∈ Q , i ∈ P . In partic-ular, Q → t [ a − m ] m +1 and P → t [ a − m +1] m . . . . . . . . . . . . . . . . . . . . 752.3 This triangle function corresponds to the second term in the second lineof (2.1.19) – where i and j are swapped. As in Figure 2.2, p , Q and P correspond to p m , q m +1 ,a and q a +1 ,m − in the notation of Eq. (2.1.19) ,where now i ∈ Q , j ∈ P . In particular, Q → t [ m − a ] a +1 and P → t [ m − a − a +1 . A one-loop MHV diagram, computed in (2.2.4) using MHV amplitudesas interaction vertices, with the CSW off-shell prescription. The twoexternal gluons with negative helicity are labelled by i and j . . . . . . . . 782.5 A triangle function with massive legs labelled by P and Q , and masslessleg p . This function is reconstructed by summing two dispersion integrals,corresponding to the P z - and Q z -cut. . . . . . . . . . . . . . . . . . . . 832.6 A degenerate triangle function. Here the leg labelled by P is still mas-sive, but that labelled by Q becomes massless. This function is also re-constructed by summing over two dispersion integrals, corresponding tothe P z - and Q z -cut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.1 A one-loop MHV diagram with a complex scalar running in the loop, com-puted in Eq. (3.1.1) . We have indicated the possible helicity assignmentsfor the scalar particle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.2
A triangle function contributing to the amplitude in the case of adja-cent negative-helicity gluons. Here we have defined P := q j,m − , Q := q m +1 ,i = − q j,m (in the text we set i = 1 , j = 2 for definiteness). . . . . 913.3 A box function contributing to the amplitude in the general case. Thenegative-helicity gluons, i and j , cannot be in adjacent positions, as thefigure shows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.4 One type of triangle function contributing to the amplitude in the generalcase, where i ∈ Q , and j ∈ P . . . . . . . . . . . . . . . . . . . . . . . . . 993.5 Another type of triangle function contributing to the amplitude in thegeneral case. By first shifting m − → m , and then swapping m ↔ m ,we convert this into a triangle function as in Figure 3.4 – but with i and j swapped. These are the triangle functions responsible for the i ↔ j swapped terms in (3.3.20) – or (3.3.22) . . . . . . . . . . . . . . . . . . 1004.1 One of the terms contributing to the recursion relation for the MHVamplitude M (1 − , − , + , . . . , n + ) . The gravity scattering amplitude onthe right is symmetric under the exchange of gravitons of the same he-licity. In the recursion relation, we sum over all possible values of k ,i.e. k = 3 , . . . , n . This amounts to summing over cyclical permutationsof (3 , . . . , n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2 This class of diagrams also contributes to the recursion relation for theMHV amplitude M (1 − , − , + , . . . , n + ) ; however, each of these diagramsvanishes if the shifts (4.2.2) are performed. . . . . . . . . . . . . . . . . 1134.3 One of the two diagrams contributing to the recursion relation for theMHV amplitude M (1 − , − , + , + ) . The other is obtained from this bycyclically permuting the labels (3 , – i.e. swapping with . . . . . . . 114B.1 The Vertices of the colour-stripped scheme in terms of spinors. . . . . . 132B.2
The diagrams contributing to the 4-gluon MHV tree-amplitude. All ex-ternal momenta are taken to be outgoing. . . . . . . . . . . . . . . . . . 1329.3
The diagrams for ˜ A (1 − q , +¯ q , − g , + g ) . . . . . . . . . . . . . . . . . . . . . 134B.4 The diagrams for ˆ A (1 − q , +¯ q , − q , +¯ q ) . . . . . . . . . . . . . . . . . . . . . 135D.1 The cut of a bubble diagram in massless λφ theory. . . . . . . . . . . . 143D.2 The cut of a one-loop MHV amplitude in the t [ m − m +1] m channel. . . . . 14410 ntroduction In the realm of high energy physics, the standard model of particle physics is our crown-ing achievement to-date. It describes the fundamental forces of nature - excluding grav-ity - as a quantum (gauge) field theory with gauge symmetry group SU(3) × SU(2) × U(1).In this description, the strong force - described by a gauge theory known as quantumchromodynamics with gauge group SU(3) - is adjoined to electro-weak theory which isitself a unification of quantum electrodynamics and the weak interaction. The standardmodel is well-verified experimentally and will soon be put to even greater tests by thelarge hadron collider at CERN which will start running later in 2007.However, there are a number of features of the standard model (SM) which are notfully understood. Most prominent of these is perhaps that it predicts the existence of ascalar particle called the Higgs boson of mass M H > . hierarchy problem . This problem asks why there is such a large hierarchy of scales forthe interaction strengths of the different forces present. It seems natural to theoriststhat just as the electromagnetic and weak forces are unified into the electro-weak (EW)force at scales M EW ∼
100 GeV, so should EW theory be unified with quantum chro-modynamics (QCD) at some (higher) scale. As such it is generally believed that SMparticles are coming from a grand unified theory (GUT) that spontaneously broke toSU(3) × SU(2) × U(1) at energies M GUT ∼ GeV. Popular gauge groups that mightunify those of the SM include SU(5) and SO(10). See e.g. [1] for an introductory text on the standard model and e.g. [2, 3, 4, 5] for treatises onquantum field theory in general. technicolour [7, 8] which considers all scalar fields in the SM to bebound states of fermions joined by a new set of interactions. Another idea is that a newsymmetry may exist such as supersymmetry - see e.g. [9, 10, 11, 12] for an introduc-tion. Supersymmetry (SUSY) relates bosons and fermions and predicts that many moreparticles exist than are currently observed as each boson/fermion is associated with apartner fermion/boson. It can, however, unify the gauge couplings of the various com-ponent theories of the standard model and thus solve the hierarchy problem. As suchthe SM would be replaced by some supersymmetric version, the minimal realisation ofwhich is usually termed the minimally supersymmetric standard model (MSSM). TheLHC is also geared towards searching for physics beyond the standard model such astechnicolour and supersymmetry.The case for unification with gravity is very much more speculative at present. Thisis not least because its tiny interaction strength compared with the other forces ofnature makes experimental tests of gravity on small length-scales difficult to performwith existing technology. As such there is no accepted quantum theory of gravityat present let alone a unification of quantum gravity with the SM. Currently studiedtheories that address the issue of the quantisation of gravity include causal set theory[15, 16], loop quantum gravity [17, 18] and string theory [19, 20, 21, 22]. Of these,string theory has also emerged as a possible framework for providing a complete unifiedtheory of all the forces of nature or a theory of everything (TOE) as it is sometimescalled.For string theory, the starting point is best understood as a generalisation of theworld-line approach to particle physics as opposed to the spacetime approach of quantumfield theory. In this approach one considers particles from the point of view of theirworld-volume or world-line (as their trajectories are lines in spacetime) and describesthis trajectory using an action of the form S particle = 12 Z dτ (cid:0) e − η µν ∂ τ X µ ∂ τ X ν − em (cid:1) , where τ is a parameter along the world-line which can naturally be taken to be theproper time. e ( τ ) is a function introduced to make the action valid for zero particlemass ( m = 0) as well as m = 0 and X µ ( τ ) represents the position vector of the particlein the ‘target’ space in which it lives. For the sake of generality we may consider thetarget space to be d -dimensional though of course four dimensions is what we’re aimingfor. While S particle describes a free particle, interactions may be included by addingterms such as R dX µ A µ ( X ) for a coupling to the electromagnetic field. See [13, 14] for an overview containing the action and Feynman rules. Actually e ( τ ) is an einbein.
12o go from point-particles to strings we simply replace S particle by an action appro-priate for describing the world-sheet of a string embedded in spacetime. An action whichnaturally incorporates both massive and massless strings is the Brink-Di Vecchia-Howeor Polyakov action S string = − T Z d ξ p det | γ | γ αβ ( η µν ∂ α X µ ∂ β X ν ) . Here the parameters of the world-sheet are ξ = τ, σ , the tension of the string is T and γ αβ can be thought of as a metric on the world-sheet. Consistently quantizing S string leads (eventually) to the many interesting consequences that string theory predicts, notleast of these being that gravity is quantized and the demand that the dimension of thetarget space be 26-dimensional for the bosonic string (the action of which is the onegiven by S string above) or 10-dimensional for any of its supersymmetric extensions.There are 5 of these consistent supersymmetric string theories that are known as typeI, type IIA, type IIB, heterotic SO(32) and heterotic E × E respectively, each of whichhas its use in describing the physics of this 10-dimensional universe in different scenarios.They are, however, intrinsically perturbative constructions and as such it has beenproposed that each of these theories is just a different limit of a unique 11-dimensionaltheory which describes the full non-perturbative range of physics and is known as M-theory [23].The intrinsically higher-dimensional nature of these theories is clearly in contrastwith current experimental results, although such results do not extend down to thePlanck scale M P ∼ GeV where it is believed that the effects of quantum gravitywill be most prevalent. Nonetheless it is hoped by many that a compactification downto four dimensions or a realisation of string theory on a 4-dimensional submanifold suchas a brane [24] may provide a unified description of the standard model plus gravity in3 + 1 dimensions.Aside from quantizing gravity or being a possible TOE, string theory has manyother facets. Not least among these is the capacity to provide alternative or ‘dual’descriptions of many well-known 4-dimensional quantum field theories. In particularthese quantum field theories include highly symmetric gauge theories such as maximallysupersymmetric ( N = 4) Yang-Mills, but also extend to certain aspects of QCD forexample.It has long been thought that gauge theories may be described by string theoriesand the idea goes back at least till ’t Hooft’s diagrammatic proposal [25]. However, itwasn’t until much more recently that this proposal was realised in a concrete way by Note that the string tension is usually written as T = 1 / (2 πα ′ ) where α ′ = l s with l s the stringlength. × S - the product of 5-dimensional anti-de-Sitter space and a 5-sphere -and a certain conformal field theory (CFT), namely N = 4 super-Yang-Mills theory inMinkowski space with gauge group SU( N ). The duality is a ‘weak-strong’ one in thesense that weakly coupled strings are describing the strong coupling regime of a gaugetheory and as such this provides a fascinating perturbative window into non-perturbative4-dimensional physics.In addition to this, the duality provides a concrete realisation of the so-called holo-graphic principle [27, 28] which asserts that physics in d -dimensional spacetimes thatinclude gravity may be describable by degrees of freedom in d − S BH = A/ A is the area of the event horizon. This is in contrast with the fact that entropy is anextensive variable and thus usually scales with the volume of the system concerned. Inthe case of the Maldacena conjecture (also known as the AdS/CFT correspondence),the 5-sphere essentially scales to a point and we are left with gravity ( i.e. closed strings)in 5 dimensions being described by Yang-Mills ( i.e. open strings) in 4 dimensions.In any case, it is not only the non-perturbative aspects of four-dimensional gaugetheory that we would like to understand better. Although weak-coupling perturbationtheory is in-principle well understood for such theories, the complexity is so great as tomake many calculations intractable. The asymptotic freedom of QCD [29, 30] meansadditionally that perturbative results become more important as the energy of interac-tion is increased, and many of these will be necessary input for the discovery of newphysics at colliders such as the LHC. As such it would be very interesting from both atheoretical and a phenomenological perspective if a duality existed that might describea 4-dimensional gauge theory at weak coupling.In fact a key step was taken in this direction by Witten at the end of 2003 [31]. Hediscovered a remarkable new duality between weakly-coupled N = 4 super-Yang-Millstheory in Minkowski space and a weakly-coupled topological string theory (known as theB-model) whose target space is the Calabi-Yau super-manifold CP | . This manifoldhas 6 real bosonic dimensions which are related to the usual 4-dimensional spacetimeof the quantum field theory by the twistor construction of Penrose [32].In [31], it was observed that tree-level gluon-scattering amplitudes in N = 4 super-Yang-Mills localise on holomorphically embedded algebraic curves in twistor space andproposed that they could be calculated from a string theory by integrating over themoduli space of D1-brane instantons in the B-model on (super)-twistor space. The Note that in order to treat the strings perturbatively we must actually take N → ∞ . The same applies to QCD at tree-level due to an effective supersymmetry - see § n − n gluons of positive helicity or 1 gluon of negative helicity and n − A natural question now arises: Can the MHV rules be applied at loop level in any gauge theory? The answer to this is not a priori clear as the duality in [31] applies to N = 4 Yang-Mills which is known to be very special due to its high degree of symmetry.Without the existence of a formal proof of the MHV rules at loop level, one way toproceed is certainly to try a similar method to that in [37] in other theories. To thatend, the present author and the authors of [37] used the MHV rules to calculate the one-loop MHV amplitudes in N = 1 super-Yang-Mills [40] (see also Chapter 2 of this thesis).This was independently confirmed by Quigley and Rosali in [41] and both results foundcomplete agreement with the amplitudes first presented by BDDK [42]. An interesting possibility has recently arisen in [39] where a number of new dualities were con-structed between field theories involving gravity and twistor string theory, One of which is a dualitybetween N = 4 Yang-Mills coupled to Einstein supergravity and a twistor string theory. An interestingfeature of this appears to be the existence of a decoupling limit giving pure Yang-Mills which mightopen the prospect of a twistor string formulation of Yang-Mills at loop-level. § n -gluon MHV amplitudes inQCD are now known. The calculation of the cut-constructible part of the amplitudesin pure Yang-Mills will be the subject of Chapter 3.In a different direction, various results emerging from twistor string theory [46, 47]inspired Britto, Cachazo and Feng to propose certain on-shell recursion relations for tree-level amplitudes in gauge theory [48] which were later proved more rigorously in a paperwith Witten [49]. These represent tree amplitudes as sums over amplitudes containingsmaller numbers of external particles connected by scalar propagators. Starting fromamplitudes with 3 particles one can thus build up all n -point tree-level amplitudesrecursively.Subsequently the present author, together with Brandhuber, Spence and Travaglinishowed that similar on-shell recursion relations for tree-level amplitudes in gravity couldbe constructed [50], where a new form for the n -graviton MHV amplitudes was alsoproposed. Such recursion relations for gravity were independently found by Cachazoand Svrˇcek in [51] which has some overlap with [50]. One striking feature of theserecursion relations is that they require a certain behaviour of the amplitudes ( M ) as afunction of momenta in the ultraviolet (UV) such that when thought of as a function of acomplex parameter z , lim z →∞ M ( z ) = 0. For Yang-Mills amplitudes this was proved tobe the case in [49], but it is a priori less clear how gravity might behave. In [50, 51] theparticular amplitudes in question were shown to have this behaviour and more recentlyit was established for all tree-level gravity amplitudes in [52]. This unambiguouslyestablishes the validity of the recursion relation in gravity, the construction of which isthe subject of Chapter 4, and also lends support to the recent conjectures that gravityas a field theory may not be as divergent as previously thought [53, 54, 55, 56, 57, 58,59, 60].This thesis will be concerned with a few [40, 43, 50] of the many developments arisingfrom twistor string theory [31]. These include the use of MHV vertices to calculate manytree-level (and some one-loop) processes [61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71] as well16s the use of the so-called holomorphic anomaly [72] (which arose to solve a discrepancybetween the twistor space picture of one-loop amplitudes presented in [73] with thederivation in [37]) to evaluate one-loop amplitudes [74, 75, 76]. MHV vertices have alsobeen found at tree-level in gravity [77] (after understanding how to deal with the non-holomorphicity which stalled initial progress [78]) and the CSW rules in gauge theoryat loop level have been more rigorously proved in [79] together with recent advances atelucidating the loop structure in pure Yang-Mills [80, 81, 82, 83].Recent improvements [47, 84, 85] to the unitarity method pioneered in [38, 42, 86,87, 88, 89, 90] use complex momenta (in similarity with the on-shell recursion relationspresented in [48, 49, 50, 51]) which allows, for example, a simple and purely algebraicdetermination of integral coefficients [47, 91]. In [92] Britto, Buchbinder, Cachazo andFeng developed efficient techniques for evaluating generic one-loop unitarity cuts whichhave since been applied in [93] and further developed in [94, 95, 96].Stemming from the on-shell recursion relations written down at tree-level by Britto,Cachazo and Feng [48] (which have been successfully exploited in [97, 98, 99] and un-derstood in terms of twistor-diagram theory in [100, 101, 102]) is the application ofon-shell recursion to one-loop amplitudes which allows for a practical and systematicconstruction of their rational parts. These have been pioneered in [103, 104, 105, 106,107, 108, 109, 110], leading to the full expression for the rational terms of the one-loopMHV amplitudes in QCD in [45]. Some success has also been had with such on-shellrecursion in one-loop gravity [111].Progress on the string theory side has been somewhat more limited after somepromising initial work. Alternative twistor string theories to that introduced by Witten[31] to describe perturbative N = 4 Yang-Mills have been put forward, though thesehave generally seemed to be more formal and less practical than the original proposal.Most notably there is that of Berkovits (and Motl) [112, 113] which was also addressedat loop level in [36], and which has been recently used to calculate loop amplitudes inYang-Mills coupled to conformal supergravity [114]. Other proposals include those of[115, 116, 117, 118].Similarly, dual twistor string theories have been constructed for other field theoriesincluding marginal deformations of N = 4 (and non-supersymmetric theories) [119, 120],orbifolds of Witten’s original proposal to include theories with less supersymmetry andproduct gauge groups [121, 122] as well as twistor string descriptions of supergravitytheories. This latter section of work includes twistor descriptions of N = 1 , N = 4 super-Yang-Mills [127], lower dimensional theories[128, 129, 130, 131, 132] and N = 4 SYM with a chiral mass term [133].17irectly following from [31], it was shown how to construct amplitudes that are morecomplex than the MHV amplitudes from an integral over a suitable moduli space ofcurves in twistor string theory. Some simple 5-point next-to-MHV (NMHV) amplitudeswere addressed in [134] as well as all n-gluon MHV amplitudes in [135] and all 6-gluonamplitudes in [136].Another avenue that has proved illuminating is the study of gauge and gravitytheories in twistor space. This includes [137] where the partition function of N = 4Yang-Mills was examined in twistor space, [138] where the CSW rules were treated froma purely gauge theoretic perspective in twistor space and [139] where loops have beenstudied and other related work including [140, 141]. Furthermore, self-dual supergravitytheories have been investigated from a twistor space perspective in [142, 143], relationsbetween twistors, hidden symmetries and integrability elucidated in [144, 145], and theconnection with string field theory developed in [146]. Finally, twistor string theory hasinspired a great deal of work in understanding supermanifolds and their connectionswith string theory and gauge theory such as that of [147, 148, 149, 150] and referencestherein. i.e. the amplitudes which are MHV amplitudes when the helicities of all particles are reversed.They thus describe the scattering of 2 gluons of positive helicity with n − ummary This thesis is organised as follows:In Chapter 1 we discuss perturbative gauge theory and the unexpectedly simple re-sults that it can produce despite the huge number of Feynman diagrams that have to besummed. We introduce various techniques for explaining this simplicity including colourordering, the spinor helicity formalism, supersymmetric decompositions, supersymmet-ric ward identities and the use of twistor space. We go on to review the twistor stringtheory introduced in [31] and show how it can be used to calculate tree-level scatteringamplitudes of gluons. Finally we describe some key ideas in perturbative gauge theorythat were inspired by the twistor string theory. In particular we present an overview ofthe CSW rules and their application at tree- and loop-level in N = 4 super-Yang-Mills.Chapter 2 is devoted to elucidating the calculation of MHV loop amplitudes in N = 1Yang-Mills using a perturbative expansion in terms of MHV amplitudes as vertices aswas introduced for N = 4 Yang-Mills in [37]. We follow [40] where the calculation wasoriginally performed and use the decomposition of the integration measure advocatedin [37, 79] to reconstruct the n -gluon MHV amplitudes in N = 1 Yang-Mills first givenin [42]. This provides strong evidence that the MHV diagram method is valid in generalsupersymmetric field theories at loop level. Some technical details are relegated toAppendix E.In much the same spirit, Chapter 3 describes the calculation of the MHV amplitudesin pure Yang-Mills with a scalar running in the loop. We take the same approachas in Chapter 2 and closely follow [43]. This produces the first results for the (cut-constructible part of the) n -gluon MHV amplitudes with arbitrary positions for thenegative-helicity particles in pure Yang-Mills. The results obtained are in completeagreement with the previously known special cases in [42, 44] and as with Chapter 2,many technical details to do with the evaluation of integrals are omitted and providedin Appendix G.In Chapter 4 we describe some tree-level on-shell recursion relations in gravity asconstructed in [50] and highlight some of their similarities with the on-shell recursionrelations proposed for gauge theory in [48, 49]. The format followed is that of [50] and19s such we describe a new compact form for the n -graviton MHV amplitudes arisingfrom the recursion relation. We also comment on the existence of recursion relationsin other field theories such as φ theory and mention the connection between the CSWrules at tree-level and these on-shell recursion relations.We conclude and discuss future directions in Chapter 5. Additionally, there areappendices describing the spinor helicity formalism and Feynman rules for masslessgauge theory in such a formalism, d -dimensional Lorentz-invariant phase space, unitarityand the Kawai-Lewellen-Tye (KLT) relations in gravity which relate tree amplitudes ingravity to (products of) tree amplitudes in Yang-Mills.20 HAPTER 1PERTURBATIVE GAUGE THEORY
In the traditional approach to quantum field theory, one writes down a classical La-grangian and can quantise the theory by defining the Feynman path integral. Per-turbative physics can then be studied by drawing Feynman diagrams and using theFeynman rules generated by the path integral to calculate scattering amplitudes. Fora non-Abelian gauge theory the classical theory is well-described by the Yang-MillsLagrangian [151]: L = ¯ ψ ( i∂/ − m ) ψ −
14 ( ∂ µ A aν − ∂ ν A aµ ) + gA aµ ¯ ψγ µ T a ψ − gf abc ( ∂ µ A aν ) A µb A νc − g ( f eab A aµ A bν )( f ecd A µc A νd ) , (1.0.1)where ψ is a fermion field, A the gauge boson field and g is the coupling. Greek indicesare associated with spacetime, while Roman indices describe the structure in gaugegroup space. This can then be used to construct the Feynman rules in the usual way.Although this construction is somewhat technical it is easy so see what these inter-actions will be from a heuristic standpoint. The first two terms in (1.0.1) will give thefermion and gauge boson propagators respectively. The third term involves two ψ s andan A and thus represents a vertex where two fermions interact with a gauge boson. Thefourth term involves 3 A s and represents a 3-boson vertex while the fifth term gives a4-boson vertex.If we work everything out properly then we find that, in Feynman gauge for examplewhere we have set ξ = 1 in a more general gauge boson propagator of the form − ip + iε (cid:18) g µν − (1 − ξ ) p µ p ν p (cid:19) δ ab , (1.0.2)the Feynman rules for an SU( N c ) gauge theory are:21auge Boson Propagator: pa b = − ig µν p + iε δ ab Fermion Propagator: pi ¯ = i ( p/ + m ) p − m + iε δ i ¯ Fermion Vertex: a, µ = igγ µ T a a, µc, ρ b, νp p p = − gf abc [ g µν ( p − p ) ρ + g νρ ( p − p ) µ + g ρµ ( p − p ) ν ]4-Boson Vertex: b, νa, µd, σ c, ρ = 2 ig [ f abe f ecd g µ [ ρ g σ ] ν + f dae f ebc g µ [ ν g σ ] ρ + f cae f ebd g µ [ ν g ρ ] σ ]Figure 1.1: Feynman rules for
SU( N c ) Yang-Mills theory in Feynman gauge.
22n the above rules we have taken all particles to be outgoing and we use the conven-tion that C [ µν ] = ( C µν − C νµ ) / C . We have also ignored thecontributions due to ghost fields and will stick to these choices in what follows unlessotherwise specified. Amplitudes for physical processes are obtained by drawing all theways that the process can occur using the above rules and associating each of these witha specific mathematical expression. They are then evaluated and added up to producethe desired result. Classical results are obtained from diagrams without any closed loopswhile quantum corrections involve an increasing number of loops. For more details seee.g. [2].Even though gauge theories present many technical challenges, the way to proceed(at least perturbatively) is in-principle well understood. In practice, however, the cal-culational complexity grows rapidly with the number of external particles (legs) and thenumber of loops. For example, even at tree-level where there are no loops to consider,the number of Feynman diagrams describing n -particle scattering of external gluons inQCD grows faster than factorially with n [152, 153]: n ,
485 34 ,
300 559 ,
405 10 , , The number of Feynman diagrams required for tree-level n-gluon scattering.
Despite this, the final result is often simple and elegant. A prime example is the so-called Maximally Helicity Violating (MHV) amplitude describing the scattering of 2gluons ( i and j ) of negative helicity with n − A tree n = h i j i h ih i . . . h n − n ih n i , (1.0.3)for any n . We will leave the explanation of the meaning of this expression to later in thechapter, but the reader is nonetheless able to appreciate its simplicity compared withthe ever increasing number of Feynman diagrams needed to produce it.The question then arises of: Why is there this simplicity underlying the apparentlymore complex perturbative expansion and how does it arise. The rest of this chapter isdevoted to setting up a framework in which these questions may be addressed. Note that the following numbers are relevant for the case where one is considering a single colourstructure only. The total number of diagrams after summing over all possible colour structures is evengreater still. For more on this see § .1. COLOUR ORDERING One prominent complication experienced by gauge theories is the extra structure inher-ent in their gauge invariance. This means that fields of the theory do not just carryspacetime indices but also indices relating to their transformation under the gaugegroup. In the standard model it has been found that SU( N c ) groups are the most ap-propriate ones for describing the gauge symmetry and so unless otherwise specified wewill consider gauge groups of this type.As is well-known, gluons carry an adjoint colour index a = 1 , , . . . , N c −
1, whilequarks and antiquarks carry fundamental ( N c ) or anti-fundamental ( ¯N c ) indices i, ¯ =1 , , . . . , N c . The SU( N c ) generators in the fundamental representation are tracelessHermitian N c × N c matrices, ( T a ) i ¯ which we normalise to tr( T a T b ) = δ ab . The Lie-algebra is defined by [ T a , T b ] = if abc T c , where the structure constants f abc satisfy theJacobi Identity: f ade f bcd + f bde f cad + f cde f abd = 0 . (1.1.1)Let us begin by considering a generic tree-level scattering amplitude. It is apparentfrom the Feynman rules given in Figure 1 that each quark-gluon vertex contributes agroup theory factor of ( T a ) i ¯ and each tri-boson vertex a factor of f abc , while four-bosonvertices contribute more complicated contractions involving pairs of structure constantssuch as f abe f cde . The quark and gluon propagators will then contract many of theindices together using their group theory factors of δ ab and δ i ¯ . We can now start toilluminate the general colour structure of the amplitudes if we first use the definition ofthe Lie-algebra to re-write the structure constants as f abc = − i tr( T a [ T b , T c ]) . (1.1.2)Doing this means that all colour factors in the Feynman rules can be replaced by linearcombinations of strings of T a s, e.g. P tr( . . . T a T b . . . ) tr( . . . T b T c . . . ) . . . tr( . . . T d . . . ) ifwe only have external gluons, or . . . ( T a . . . T b ) i ¯ tr( T b . . . T c )( T c T d ... ) k ¯ l . . . - where thestrings are terminated by (anti)-fundamental indices - if external quarks are present.In order to reduce the number of traces we make use of the identity N c − X a =1 ( T a ) i ¯ ( T a ) k ¯ l = δ i ¯ l δ k ¯ − N c δ i ¯ δ k ¯ l , (1.1.3) This is different from the more familiar tr( T a T b ) = δ ab /
2, but is purely a convention used to avoidthe proliferation of factors of 2. Note that the Feynman rules written down at the beginning of thechapter use tr( T a T b ) = δ ab /
2. To rewrite the diagrams in a way that is consistent with these ‘morenatural’ colour ordering conventions one simply has to replace T a → T a / √ f abc → f abc / √
2. Seealso Appendix B. .1. COLOUR ORDERING which is just an algebraic statement of the fact that the generators T a form a completeset of traceless Hermitian matrices. This in turn gives rise to simplifications such as X a tr( T a . . . T a k T a ) tr( T a T a k +1 . . . T a n ) = tr( T a . . . T a k T a k +1 . . . T a n ) − N c tr( T a . . . T a k ) tr( T a k +1 . . . T a n )(1.1.4)and X a tr( T a . . . T a k T a )( T a T a k +1 . . . T a n ) i ¯ = ( T a . . . T a k T a k +1 . . . T a n ) i ¯ − N c tr( T a . . . T a k )( T a k +1 . . . T a n ) i ¯ . (1.1.5)In Eq. (1.1.3) the 1 /N c term corresponds to the subtraction of the trace of the U ( N c ) group in which SU( N c ) is embedded and thus ensures tracelessness of the T a .This trace couples directly only to quarks and commutes with SU( N c ). As such theterms involving it disappear after one sums over all the permutations present - a factwhich is easy to check directly. We can thus see that we are ultimately left with eithersums of single traces of generators if we only have external gluons as in Eq. (1.1.4) orsums of strings of generators terminated by fundamental indices as in Eq. (1.1.5) if wealso have external quarks [153, 154]. In most of what we do we will only be concernedwith gluon scattering and can therefore write the colour decomposition of amplitudesas A tree n ( a i ) = g n − X σ ∈ S n /Z n tr( T a σ (1) T a σ (2) . . . T a σ ( n ) ) A tree n ( σ (1) , σ (2) , . . . , σ ( n )) , (1.1.6)where S n is the set of permutations of n objects and Z n is the subset of cyclic per-mutations. g is the coupling constant of the theory. The A tree n sub-amplitudes arecolour-stripped and depend only on one ordering of the external particles. It is there-fore sufficient to consider A tree n (1 , , . . . , n ) - the ‘reduced colour-ordered amplitude’ -and sum over all ( n − n vectormesons can be written as a sum over non-cyclic permutations of external legs carryingChan-Paton factors [157] multiplied by Koba-Nielsen partial amplitudes [158]. For Note that Eq. (1.1.5) is appropriate for the case where we have just one q ¯ q pair. With more pairsthere will be products of strings with each string terminated by fundamental and anti-fundamentalindices giving terms like ( T a . . . T b ) i ¯ . . . ( T c . . . T d ) k ¯ l . In the final expression, each generator will appearonly once in any given term of course. .1. COLOUR ORDERING the scattering of external gluons that we are interested in we need not worry aboutfundamental matter because at tree-level the Feynman rules forbid it from appearingas internal lines. In the infinite-tension limit ( T → ∞ ; α ′ →
0) the U( N c ) string theoryreduces to a U( N c ) gauge theory and the trace part of this decouples as we have seen. Wecan thus immediately conclude that the gauge theory scattering amplitudes decomposeas Eq. (1.1.6).For one-loop amplitudes a similar colour decomposition exists [156]. In this case,however, there are up to two traces over SU( N c ) generators and one must sum over thespins of the different particles that can circulate in the loop. In an expansion in N c ,the leading (as N c → ∞ ) contributions to the amplitudes are planar and the colourstructure is simply a single trace - in fact it is N c times the tree-level colour factor whenthere are no particles in the fundamental representation propagating in the loop. In thiscase an almost identical formula to (1.1.6) can be written down for a decomposition ofone-loop amplitudes of external gluons [156]: A n ( a i ) = g n h X σ ∈ S n /Z n N c tr( T a σ (1) . . . T a σ ( n ) ) A n ;1 ( σ (1) , . . . , σ ( n ))+ ⌊ n/ ⌋ +1 X c =2 X σ ∈ S n /S n ; c tr( T a σ (1) . . . T a σ ( c − ) tr( T a σ ( c ) . . . T a σ ( n ) ) × A n ; c ( σ (1) , . . . , σ ( n )) i , (1.1.7)and we have left the sum over spins as being implicit in the definitons of the colour-ordered partial amplitudes A n ;1 and A n ; c . ⌊ r ⌋ is the largest integer less than orequal to r and S n ; c is the subset of permutations of n objects leaving the double tracestructure invariant.It is a remarkable result of Bern, Dixon, Dunbar and Kosower that at one-loop,non-planar (multi-trace) amplitudes are simply obtained as a sum over permutationsof the planar (single-trace) ones. This is discussed in Section 7 of [38] where it wasalso noted that this applies to a generic SU( N c ) theory (both supersymmetric andnon-supersymmetric) with external particles and those running in the loop both in theadjoint representation. As far as loop amplitudes go we will only be concerned withparticles that are in the adjoint, so it will be enough for us to consider only one cyclicordering ( i.e. only A n ;1 , which we will generally abbreviate to A n ) and then sumover all the relevant permutations at the very end. We will not actually perform thissummation in what follows but leave it as something which can easily be implementedto obtain the full amplitude.The colour-ordered sub-amplitudes obey a number of identities such as gauge in-variance, cyclicity, order-reversal up to a sign, factorization properties and more. Thismeans that there isn’t a huge proliferation in the number of partial amplitudes that have26 .2. SPINOR HELICITY FORMALISM to be computed. For 5-point gluon scattering for example, there are only 4 independenttree-level sub-amplitudes and it turns out that 2 of these vanish identically because ofa ‘hidden’ supersymmetry (see § So far we have seen that we can reduce some of the complexity of our task by removingthe colour structure and considering only colour-ordered amplitudes. We’ll also onlyconsider massless particles and this restricts us further, though there are still a largenumber of things that A n can depend on. For spinless particles (scalars), the situationis clear and A n = A n ( p i ) δ (4) ( P ni =1 p i ), where the p i are the momenta of the externalparticles obeying p = p µ p µ = 0 and we have written the delta function of momentumconservation explicitly [31, 159]. In fact the momentum dependence only appears interms of Lorentz-invariant quantities such as p i · p j .For massless particles with spin the situation is more complicated and we haveto consider their wavefunctions ψ i , giving A n = A n ( p i , ψ i ) δ (4) ( P ni =1 p i ). Textbookdefinitions have the ψ i being different depending on the spin being considered. Forexample in the case of spin 1 / u ( p ) and v ( p ) and their conjugates (see e.g. Section (3.3)of [2]), while in the case of spin 1 gauge bosons the polarisation vectors ǫ µ in a suitablychosen basis are common. A more unifying description would be highly desirable andin fact one can be found using the so-called spinor helicity formalism [160]. We start with the fact that, when complexified, the Lorentz group is locally isomorphicto SO(1 , , C ) ∼ = SL(2 , C ) × SL(2 , C ) , (1.2.1)and thus the finite-dimensional representations are classified as ( p, q ), where p and q are integers or half-integers. Negative- and positive-chirality spinors transform in the(1 / ,
0) and (0 , /
2) representations respectively. For a generic negative-chirality spinorwe write λ α with α = 1 , λ ˙ α with˙ α = 1 , ǫ αβ and ǫ αβ as λ α = ǫ αβ λ β and λ α = λ β ǫ βα with ǫ = 1 and ǫ αβ ǫ βγ = − δ αγ Note that this section is based largely on the spinor helicity reviews of [31, 159, 161]. See alsoAppendix A for more details and identities and [162] for another good review covering many aspects ofthis chapter. .2. SPINOR HELICITY FORMALISM (and likewise for dotted indices). Given two spinors λ and µ of negative chirality wecan then form a Lorentz-invariant scalar product as h λ, µ i = λ α µ β ǫ βα , (1.2.2)from which it follows that h λ, µ i = −h µ, λ i . Similar formulæ apply for positive-chiralityspinors except that we use square brackets to distinguish the two: [˜ λ, ˜ µ ] = ˜ λ ˙ α ˜ µ ˙ β ǫ ˙ β ˙ α .It is worth noting in-particular that h λ, µ i = 0 implies λ α = cµ α where c is a complexnumber and similarly for ˜ λ and ˜ µ . We will often use even more compact notation forthese scalar products and write h λ, µ i = h λ µ i = −h µ λ i etc .The vector representation of SO(1 , , C ) is the (1 / , /
2) and as such we can representa momentum vector p µ as a bi-spinor p α ˙ α . We can go to such a representation byusing the chiral representation of the Dirac γ -matrices - a process that is well-known insupersymmetric field theories, see e.g. [9, 10]. In signature + − −− the Dirac matricescan then be represented as γ µ = σ µ ¯ σ µ ! , (1.2.3)where ( σ µ ) α ˙ α = (1 , ~σ ) and (¯ σ µ ) ˙ αα = − ( σ µ ) α ˙ α = ǫ ˙ α ˙ β ǫ αβ ( σ µ ) β ˙ β = (1 , − ~σ ) and ~σ =( σ , σ , σ ) are the Pauli matrices as given in Equation (A.1.2). For a given vector p µ we then have p α ˙ α = p µ σ µα ˙ α = p
11 + ~p · ~σ (1.2.4)= p + p p − ip p + ip p − p ! , (1.2.5)from which it follows that p µ p µ = det( p α ˙ α ). Hence p µ is light-like ( p = 0) if det( p α ˙ α ) = 0,which in turn means that massless vectors are those for which p α ˙ α = λ α ˜ λ ˙ α , (1.2.6)for some spinors λ α and ˜ λ ˙ α . These spinors are unique up to the scaling ( λ , ˜ λ ) → ( cλ , c − ˜ λ ) for a complex number c .If we wish p µ to be real in Lorentz signature (in which case p α ˙ α is hermitian) thenwe must take ˜ λ = ± ¯ λ where ¯ λ is the complex conjugate of λ . The sign determineswhether p µ has positive or negative energy. It is also possible (and sometimes useful) toconsider other signatures. In signature ++ −− λ and ˜ λ are real and independent whilein Euclidean signature (+ + ++) the spinor representations are pseudoreal. Light-likevectors cannot be real with Euclidean signature.The formula for p · p = det( p α ˙ α ) generalises for any two momenta p and q and using28 .2. SPINOR HELICITY FORMALISM the fact that ( σ µ ) α ˙ β (¯ σ ν ) ˙ βα = 2 η µν we can write the scalar product for two light-likevectors p α ˙ α = λ α ˜ λ ˙ α and q β ˙ β = µ β ˜ µ ˙ β as2( p · q ) = h λ µ i [˜ µ ˜ λ ] . (1.2.7)This is the standard convention in the perturbative field theory literature and differsfrom the conventions in [31, 161] by a sign that is related to the choice of how to contractindices using ǫ αβ . Once p µ is given, the additional information involved in specifying λ (and hence ˜ λ incomplexified Minkowski space with real p α ˙ α ) is equivalent to a choice of wavefunctionfor a spin 1 / p µ . To see this, we can write the massless Diracequation for a negative-chirality spinor ψ α as i ( σ µ ) α ˙ α ∂ µ ψ α = 0 . (1.2.8)A plane wave ψ α = ρ α e ip · x with constant ρ α obeys this equation iff p α ˙ α ρ α = 0 whichimplies that ρ α = cλ α . Similar considerations apply for positive-chirality spinors andthus we can write fermion wavefunctions of helicity ± / ψ ˙ α = ˜ λ ˙ α e ix β ˙ β λ β ˜ λ ˙ β , ψ α = λ α e ix β ˙ β λ β ˜ λ ˙ β (1.2.9)respectively.For massless particles of spin ± ǫ µ (which we should be careful not to confuse with ǫ αβ ) in addition to their momen-tum and together with the constraint p µ ǫ µ = 0. This constraint is equivalent to theLorentz gauge condition and deals with fixing the gauge invariance inherent in gaugefield theories. It is clear that if we add any multiple of p µ to ǫ µ then this condition isstill satisfied and we have the gauge invarianceˆ ǫ µ = ǫ µ + ω p µ . (1.2.10)If one now has a decomposition of a light-like vector particle with momentum p α ˙ α = λ α ˜ λ ˙ α then one can take the polarisation vectors to be [31] (see also [153, 163] andreferences therein): ǫ + α ˙ α = µ α ˜ λ ˙ α h µ λ i , ǫ − α ˙ α = λ α ˜ µ ˙ α [˜ µ ˜ λ ] , (1.2.11) We will often use the terms chirality and helicity interchangeably. .2. SPINOR HELICITY FORMALISM for positive- and negative-helicity particles respectively. µ and ˜ µ are arbitrary negative-and positive-chirality spinors (not proportional to λ or ˜ λ ) respectively and it is worthnoting that the positive-helicity polarization vector is proportional to the positive-helicity spinor (˜ λ ˙ α ) associated with the momentum vector p µ while the negative-helicitypolarization vector is proportional to the negative-helicity one ( λ α ). These polarizationvectors clearly obey the constraint 0 = p µ ǫ ± µ = p α ˙ α ǫ ± α ˙ α since h λ λ i = [˜ λ ˜ λ ] = 0 and areindependent of µ and ˜ µ up to a gauge transformation [31, 161]. The wavefunctions forpositive and negative-helicity massless vector bosons can thus be written as [161] A + α ˙ α = ǫ + α ˙ α e ix β ˙ β λ β ˜ λ ˙ β , A − α ˙ α = ǫ − α ˙ α e ix β ˙ β λ β ˜ λ ˙ β . (1.2.12)Spinless particles have wavefunction φ = e ix α ˙ α λ α ˜ λ ˙ α as usual. One of the central motivations for all this song and dance is that we can use the resultsto homogenise our description of scattering amplitudes. The plethora of variables thatwe had before can simply be traded for the bi-spinors λ and ˜ λ to yield the compactform of a general scattering amplitude as A n = A n ( λ i , ˜ λ i , h i ) δ (4) n X i =1 λ αi ˜ λ ˙ αi ! , (1.2.13)where h i is the helicity of the i th particle. In this scheme we can therefore calculateamplitudes for the scattering of specific helicity configurations of specific colour order-ings of massless particles. The full amplitude is obtained by summing over all helicityconfigurations and all appropriate colour orderings.As a final remark in this section it is useful to note (and easy to show - see [31,161]) that under the scaling-invariance inherent in the decomposition of Eq. (1.2.6),the wavefunction of a massless particle of helicity h scales as c − h and thus obeys thecondition (cid:18) λ α ∂∂λ α − ˜ λ ˙ α ∂∂ ˜ λ ˙ α (cid:19) ψ ( λ, ˜ λ ) = − hψ ( λ, ˜ λ ) . (1.2.14)Similarly, the amplitude in Eq. (1.2.13) obeys λ αi ∂∂λ αi − ˜ λ ˙ αi ∂∂ ˜ λ ˙ αi ! A n ( λ i , ˜ λ i , h i ) = − h i A n ( λ i , ˜ λ i , h i ) (1.2.15)for each i separately. The full expression A n ( λ i , ˜ λ i , h i ) δ (4) “P ni =1 λ αi ˜ λ ˙ αi ” also obeys (1.2.15) [31]. .3. SUPERSYMMETRIC DECOMPOSITION The interested reader can find the Feynman rules for massless SU( N c ) Yang-Millsgauge theory in the spinor helicity formalism in Appendix B. Supersymmetric field theories are in many ways very similar to the usual Yang-Millstheories whose Feynman rules we wrote down at the start of the chapter. The presence ofthis extra symmetry - supersymmetry - means that the particles of the theories arrangethemselves into supersymmetric multiplets containing equal numbers of bosonic andfermionic degrees of freedom and this can often give rise to great simplifications.Maximally supersymmetric ( N = 4) Yang-Mills for example, which has the maximumamount of supersymmetry consistent with a gauge theory ( i.e. particles with spin lessthan or equal to 1) in four dimensions, contains only 1 multiplet consisting of 1 vectorboson A µ (2 real degrees of freedom (d.o.f.)), 6 real scalars φ I (6 real d.o.f.) and 4Weyl ( i.e. chiral) fermions χ α (8 real d.o.f.) which lives in the adjoint of the gaugegroup. This multiplet is often written in a helicity-basis (the helicities of the particleshere are h = ( − , − / , , / , A − , χ − , φ, χ + , A + ) = (1 , , , ,
1) and is oftenreferred to as the adjoint multiplet of N = 4. The meaning of this notation is that oneof the degrees of freedom of the vector boson is associated with a negative-helicity ( − − / /
2. The scalars are of course spinless and thus associated with helicity0. Other common multiplets in four dimensions include the vector multiplet of N = 2(1 , , , ,
1) - which consists of 1 vector, 2 fermions and 2 scalars - the hyper multipletof N = 2 (0 , , , ,
0) and the vector (1 , , , ,
1) and chiral (0 , , , ,
0) multiplets of N = 1 supersymmetry.The existence of these supersymmetric multiplets generally leads to a better con-trol of the field theory in question, and most-importantly for us a greater control of itsperturbative expansion. Heuristically, fermions propagating in loops give terms whichhave the opposite sign to bosons and the exact matching of the bosonic and fermionicdegrees of freedom leads to cancellations in the ultraviolet divergences that plague non-supersymmetric field theories. In particular, N = 4 super-Yang-Mills is believed to becompletely finite in four dimensions as well as having quantum-mechanical conformal-invariance. Massless QCD on the other hand is classically conformally-invariant, al-though this is broken by quantum effects as is well-known from the existence of itsone-loop (and higher) β -function. QCD is also UV divergent at loop-level and thusmust be renormalised order-by-order in perturbation theory. N = 4 super-Yang-Mills has the most striking features of these four-dimensionalsupersymmetric gauge theories and we will concern ourselves with this theory as well31 .3. SUPERSYMMETRIC DECOMPOSITION as N = 1 super-Yang-Mills. In fact, the results for N = 1 amplitudes in Chapter 2 alsoapply to certain N = 2 amplitudes by virtue of the fact that the N = 2 hyper multipletis twice the N = 1 chiral multiplet and the N = 2 vector multiplet is equal to an N = 1vector multiplet plus an N = 1 chiral multiplet.As we have already mentioned, we will mostly be concerned with gluon scatteringin SU( N c ) Yang-Mills theories (including QCD) and thus will only consider this casehere. At tree-level it is easy to see that gluon scattering amplitudes are the same inQCD as they are in N = 4 super-Yang-Mills theory. This is because vertices connectinggluons to fermions or scalars in these theories couple gluons to pairs of these particles.Thus one cannot create fermions or scalars internally without also creating a loop [164].These QCD scattering amplitudes therefore have a ‘hidden’ N = 4 supersymmetry: A treeQCD = A tree N =4 . (1.3.1)The same can of course be said about any supersymmetric field theory with adjointfields when one is concerned with the scattering of external gluons at tree-level. Wethus have the more general result that A treeQCD = A tree N =4 = A tree N =2 = A tree N =1 . (1.3.2)At one-loop we can of course have other particles propagating in the loop, but wheregluon-scattering only is concerned we can still find a supersymmetric decomposition. Itis: A one-loopQCD = A one-loop N =4 − A one-loop N =1 , chiral + 2 A one-loopscalar . (1.3.3)In words this says that an all-gluon scattering amplitude in QCD at one loop can bedecomposed into 3 terms: Firstly a term where an N = 4 multiplet propagates in theloop. Secondly a term where an N = 1 chiral multiplet propagates in the loop and lastlya term in pure Yang-Mills where we only have 2 real scalars (or one complex scalar)in the loop. This is easily seen due to the multiplicities of the various multiplets inquestion: (1 , , , ,
1) = (1 , , , , − , , , ,
0) + 2(0 , , , , .4. SUPERSYMMETRIC WARD IDENTITIES the case of certain tree-level amplitudes (1.3.2)) of the answer to calculations in theoriessuch as QCD. These supersymmetric decompositions will be of great assistance to us inour quest to understand the hidden simplicity of scattering amplitudes and in order toperform actual calculations.For more information on supersymmetric field theories see any one of a multitudeof books, papers and reviews including [9, 10, 11, 12]. As we can now see, for a large number of scattering amplitudes in gauge theories we canreduce the complexity of our problem by considering an appropriate colour-ordered sub-amplitude that only depends on the positive- and negative-helicity spinors associatedwith the external momenta (we usually drop the h i dependence of (1.2.13) and leave itas being implicit in the definition of the amplitude being considered). Using our ‘hidden’(or not, depending on the theory in question) supersymmetry we are now in a positionto learn something about the scattering amplitudes in question. The following is alsonicely reviewed in a number of places including [153, 154] and was first considered in[164, 165, 166, 167]. See also e.g. [168] for a recent application of supersymmetric Wardidentities to loop amplitudes. N = 1 SUSY constraints Let us consider what is in some ways the simplest possible setup, an adjoint (vector)multiplet in an N = 1 supersymmetric field theory where the SUSY is unbroken. This N = 1 theory has only one supercharge Q ( η ) that generates the supersymmetry with η being the fermionic parameter of the transformation [9]. Because supersymmetry isunbroken we know that Q must annihilate the vacuum: Q ( η ) | i = 0. This in turn givesrise to the following supersymmetric Ward identity (SWI)0 = h | [ Q ( η ) , Ψ . . . Ψ n ] | i = n X i =1 h | Ψ . . . [ Q ( η ) , Ψ i ] . . . Ψ n | i , (1.4.1)for some fields Ψ i . In addition, if we use a suitable helicity basis in which we have amassless vector A ± and a massless spin 1/2 fermion χ ± , then Q ( η ) acts on the doublet( A, χ ) ( i.e. ( A − , χ − , , χ + , A + ) in the notation of the previous subsection) as [166, 167]: (cid:2) Q ( η ) , A ± ( p ) (cid:3) = ∓ Γ ± ( p, η ) χ ± , (cid:2) Q ( η ) , χ ± ( p ) (cid:3) = ∓ Γ ∓ ( p, η ) A ± , (1.4.2)33 .4. SUPERSYMMETRIC WARD IDENTITIES for some momentum p associated with these states. Γ( η, p ) is linear in η and can beconstructed by using the Jacobi identity[[ Q ( η ) , Q ( ζ )] , Ψ( p )] + [[ Q ( ζ ) , Ψ( p )] , Q ( η )] + [[Ψ( p ) , Q ( η )] , Q ( ζ )] = 0 (1.4.3)and the SUSY algebra relation [ Q ( η ) , Q ( ζ )] = − i ¯ ηP/ζ , where P/ = γ µ P µ as usual. Byconsidering (1.4.4) for any of the chiral fields ( A + ( p ) for example), we can readily deducethat Γ + ( p, η )Γ − ( p, ζ ) − Γ + ( p, ζ )Γ − ( p, η ) = − i ¯ ηp/ζ , (1.4.4)which can be solved to give (in the notation of § + ( p, q, ϑ ) = ϑ [ p q ] , Γ − ( p, q, ϑ ) = ϑ h p q i . (1.4.5)In this expression we have written p = λ p ˜ λ p and our parameter η in terms of a Grass-mann parameter ϑ and an arbitrary reference momentum q = λ q ˜ λ q . We have also usedthe shorthand notation h λ p λ q i = h p q i and [˜ λ p ˜ λ q ] = [ p q ] which will often be employedhenceforth.Now consider (1.4.1) with Ψ = χ +1 and Ψ i = A + i for i = 1:0 = h | [ Q ( η ( q, ϑ )) , χ +1 ( p ) A +2 ( p ) . . . A + n ( p n )] | i = − Γ − ( p , q, ϑ ) h | A +1 ( p ) A +2 ( p ) . . . A + n ( p n ) | i + Γ − ( p , q, ϑ ) h | χ +1 ( p ) χ +2 ( p ) . . . A + n ( p n ) | i ...+ Γ − ( p n , q, ϑ ) h | χ +1 ( p ) A +2 ( p ) . . . χ + n ( p n ) | i = − Γ − ( p , q, ϑ ) A n ( A +1 , A +2 , . . . , A + n )+ Γ + ( p , q, ϑ ) A n ( χ +1 , χ +2 , . . . , A + n )...+ Γ + ( p n , q, ϑ ) A n ( χ +1 , A +2 , . . . , χ + n ) . (1.4.6)As all of the couplings of fermions to vectors conserve helicity (you always get onefermion of each helicity coupling to a vector), the n − n − A n ( A +1 , A +2 , . . . , A + n ) = 0. Since supersymmetry commuteswith colour we can write our amplitudes as colour-ordered ones straight away and thenthe relations apply to each colour-ordered amplitude separately.If we consider the case where we have one negative-helicity in our SWI so thatΨ = χ +1 , Ψ = A − and Ψ i = A + i for i = 1 , n − .4. SUPERSYMMETRIC WARD IDENTITIES vanish. This is so both for the case of all gluon scattering and the case of n − = χ +1 , Ψ = A − , Ψ = A − and Ψ i = A + i for i = 1 , ,
3) we can start to relatenon-zero amplitudes to each other. In all of these cases it is useful to remember thatthe reference momentum q is arbitrary and can thus be taken to be one of the externalmomenta ( q = p i ) for example at any given stage in order to simplify the calculationsand deduce useful results. Some of the useful relations that we can obtain are: A SUSY n (1 ± , ± , . . . , n ± ) = 0 , (1.4.7) A SUSY n (1 ∓ , ± , . . . , n ± ) = 0 , (1.4.8)for any spins of the particles involved and [38] A SUSY n ( A − i , . . . , χ − r , . . . , χ + s , . . . ) = h i s ih i r i A SUSY n ( A − i , . . . , A − r , . . . , A + s , . . . ) , (1.4.9) A SUSY n ( A − i , . . . , φ − r , . . . , φ + s , . . . ) = h i s i h i r i A SUSY n ( A − i , . . . , A − r , . . . , A + s , . . . ) , (1.4.10)where we have also played the same game with an N = 2 Vector multiplet in order toinclude scalars φ . These relations hold order-by-order in the loop expansion of super-symmetric field theories as no perturbative approximations were made in deriving them,and by virtue of (1.3.2) they apply directly to tree-level QCD amplitudes involving glu-ons. It turns out that tree-level QCD amplitudes involving fundamental quarks can alsobe obtained from (1.4.9) because of relations between sub-amplitudes involving gluinos( i.e. fermionic superpartners of gluons in an adjoint multiplet such as the χ above) andthose involving fundamental quarks [153, 169].Equations (1.4.7) and (1.4.8) amount to the statement that for any supersymmetrictheory with only adjoint fields, the ‘all-plus’ and ‘all-minus’ helicity amplitudes mustvanish and the amplitudes with one minus and n − n − n − googly MHV (or MHV) amplitudes [31]. Similarly,amplitudes with three negative helicities and n − .5. TWISTOR SPACE (NNMHV) and so on.The tree-level MHV amplitudes for gluon scattering, proposed at n -point in [170]and then proved in [171], are given by (1.0.3) or by A n (1 + , . . . , i − , . . . , j − , . . . , n + ) = h i j i Q nk =1 h k k +1 i , (1.4.11)up to a factor. i and j are the gluons of negative helicity and the amplitude obeys(1.2.15). The amplitude is cyclic in the ordering of the gluons and so the n + 1 th spinorappearing in the denominator of (1.4.11) just denotes the spinor of the 1 st gluon. Notein particular that this function is entirely ‘holomorphic’ in the negative-helicity spinors λ - i.e. it does not depend on any of the ˜ λ s - and this will be important to us presently.We will not discuss NMHV and other amplitudes yet except to mention that they do depend on the ˜ λ s. There is a way in which we can understand some of the properties of amplitudes thatwe have discussed above such as the vanishing of certain helicity configurations andthe simple structure of the MHV amplitudes and that is by going to twistor space [31].This has two primary motivations. One is that the conformal symmetry group has arather exotic representation in terms of the λ and ˜ λ variables and the other is thatthe scaling-invariance mentioned under equation (1.2.6) has an opposite action on theholomorphic spinors λ compared with the anti-holomorphic spinors ˜ λ . It would be niceto put the conformal group into a more standard representation and it may also benice to have the same scaling for the negative and positive-helicity spinors.In terms of the spinors we have already introduced in § The gluon amplitudes at tree-level are invariant under the full conformal group rather than just thePoincar´e group. This is because of the classical conformal invariance of both massless QCD and anyof the other supersymmetric field theories that we have been considering. Amplitudes in some of thesesupersymmetric theories (especially N = 4 Yang-Mills) also have quantum conformal invariance. .5. TWISTOR SPACE are [31] P α ˙ α = λ α ˜ λ ˙ α , (1.5.1) J αβ = i (cid:18) λ α ∂∂λ β + λ β ∂∂λ α (cid:19) , (1.5.2)˜ J ˙ α ˙ β = i (cid:18) ˜ λ ˙ α ∂∂ ˜ λ ˙ β + ˜ λ ˙ β ∂∂ ˜ λ ˙ α (cid:19) , (1.5.3) D = i (cid:18) λ α ∂∂λ α + ˜ λ ˙ α ∂∂ ˜ λ ˙ α + 2 (cid:19) , (1.5.4) K α ˙ α = ∂ ∂λ α ∂ ˜ λ ˙ α , (1.5.5)where P α ˙ α is the momentum operator, J αβ and ˜ J ˙ α ˙ β the Lorentz generators, D thedilatation operator and K α ˙ α the generator of special conformal transformations. Thesegive rise to the algebra of the conformal group as (cid:2) J αβ , P γ ˙ γ (cid:3) = i ǫ βγ P α ˙ γ + ǫ αγ P β ˙ γ ) , (cid:2) ˜ J ˙ α ˙ β , P γ ˙ γ (cid:3) = i (cid:16) ǫ ˙ β ˙ γ P γ ˙ α + ǫ ˙ α ˙ γ P γ ˙ β (cid:17) , (cid:2) J αβ , J γδ (cid:3) = 14 ( ǫ γα J βδ + ǫ γβ J αδ + ǫ δα J βγ + ǫ δβ J αγ ) , (cid:2) ˜ J ˙ α ˙ β , ˜ J ˙ γ ˙ δ (cid:3) = 14 (cid:16) ǫ ˙ γ ˙ α ˜ J ˙ β ˙ δ + ǫ ˙ γ ˙ β ˜ J ˙ α ˙ δ + ǫ ˙ δ ˙ α ˜ J ˙ β ˙ γ + ǫ ˙ δ ˙ β ˜ J ˙ α ˙ γ (cid:17) , (cid:2) D, P α ˙ α (cid:3) = i P α ˙ α , (cid:2) K α ˙ α , D (cid:3) = 2 K α ˙ α , (cid:2) J αβ , K γ ˙ γ (cid:3) = i ǫ γα K β ˙ γ + ǫ γβ K α ˙ γ ) , (cid:2) J ˙ α ˙ β , K γ ˙ γ (cid:3) = i (cid:16) ǫ ˙ γ ˙ α K γ ˙ β + ǫ ˙ γ ˙ β K γ ˙ α (cid:17) , (cid:2) K α ˙ α , P β ˙ β (cid:3) = i (cid:16) ǫ αβ ˜ J ˙ α ˙ β + ǫ ˙ α ˙ β J αβ + ǫ αβ ǫ ˙ β ˙ α D (cid:17) , (1.5.6)with all other commutators being zero. However, as can be seen from (1.5.1)-(1.5.5),the momentum operator is a multiplication operator, the Lorentz generators are firstorder homogeneous differential operators, the dilatation operator an inhomogeneous firstorder differential operator and the special conformal generator a degree two differentialoperator. We have quite a mix.We can in fact reduce these to a more standard representation by performing a thetransformation [31, 32] ˜ λ ˙ α → i ∂∂µ ˙ α ,∂∂ ˜ λ ˙ α → iµ ˙ α . (1.5.7)37 .5. TWISTOR SPACE This breaks the symmetry between λ and ˜ λ as we have chosen to transform one ratherthan the other, but giving the advantage that all the generators become first orderdifferential operators: P α ˙ α = iλ α ∂∂µ ˙ α , (1.5.8) K α ˙ α = iµ ˙ α ∂∂λ α , (1.5.9) J αβ = i (cid:18) λ α ∂∂λ β + λ β ∂∂λ α (cid:19) , (1.5.10)˜ J ˙ α ˙ β = i (cid:18) µ ˙ α ∂∂µ ˙ β + µ ˙ β ∂∂µ ˙ α (cid:19) , (1.5.11) D = i (cid:18) λ α ∂∂λ α − µ ˙ α ∂∂µ ˙ α (cid:19) . (1.5.12)The scaling properties of λ and µ are also changed such that there is an invariance under( λ , µ ) → ( cλ , cµ ) , (1.5.13)for a complex number c , and the amplitude scalings (1.2.15) become (cid:18) λ αi ∂∂λ αi + µ ˙ αi ∂∂µ ˙ αi (cid:19) ˜ A n ( λ i , µ i , h i ) = − (2 h i + 2) ˜ A n ( λ i , µ i , h i ) , (1.5.14)where ˜ A n is the appropriately transformed amplitude.This transformation is perhaps easiest to understand in signature ++ −− . In thiscase one can consider λ α and µ ˙ α to be real and independent and thus they parametrisea copy of R . The scaling (1.5.13) is then a real scaling and reduces the space to real-projective three-space RP and the transform (1.5.7) is implemented by a ‘1/2-Fourier’transform analagous to that encountered in quantum mechanics [31]:˜ f ( µ ) = Z d ˜ λ (2 π ) e iµ ˙ α ˜ λ ˙ α f (˜ λ ) . (1.5.15)In other signatures (such as Minkowski space) it may be more natural to regard λ and µ as being complex and independent. They thus parametrise a copy of C which reducesto CP under the scaling (1.5.13). These spaces - RP and CP - were called twistor spaces by Penrose [32] and we will often use coordinates Z I with I = 1 . . . λ α and µ ˙ α together. One should really refer to ‘real/complex projectivetwistor space’ respectively, but we will denote them all as being twistor space ( T ) andlet the context dictate what we mean by that.In the complex cases, the choice of a contour for the transformation as given by(1.5.15) is not necessarily clear and it seems necessary to take the more sophisticatedapproach of Penrose and use Dolbeault- or sheaf-cohomology [32]. Na¨ıvely, this inter-38 .5. TWISTOR SPACE prets the integrand and measure of (1.5.15) as a (0 , L h of degree − h − L h = O ( − h −
2) for each h . Theamplitudes are thus elements of H (0 , ( CP ′ , O ( − h − The transformation of wavefunctions to twistor space is in some ways more com-plex. One cannot perform such a na¨ıve ‘1/2-Fourier’ transform in essence because thewavefunctions are defined by being solutions to the massless free wave equations and soone must see how one can solve these in twistor space. It turns out that these solutionscan be written as integrals of functions of degree 2 h − ∂ -cohomology group H (0 , ( CP ′ , O (2 h − e.g. [31, 172, 173, 174] for details.In particular these descriptions mean that scattering amplitudes with specific exter-nal states make sense in twistor space. In a usual field theory construction one wouldmultiply a momentum-space scattering amplitude with its momentum-space wavefunc-tions and integrate over all momenta to create a scattering amplitude with specific ex-ternal states in the position-space representation. If the wavefunctions in position-spacesatisfying the appropriate free wave equations are given by ϕ i ( x ) = R d p i δ ( p i ) e ip i · x φ i ( p i ),then we have schematically A ( ϕ i ) = R ( Q d p i δ ( p i ) e ip i · x φ i ( p i )) ˜ A ( p i ).In twistor space, multiplying an amplitude in H (0 , ( CP ′ , O ( − h − H (0 , ( CP ′ , O (2 h − H (0 , ( CP ′ , O ( − CP is a (3 , ′ of (1.6.12)),and so the final integral will be of a (3 , i.e. theintegrand is a top-form on twistor space invariant under (1.5.13)) as an integral over CP ′ . Doing this for each external particle gives the required scattering amplitude inposition-space.Following the original suggestions of Nair [175], there is a similar construction whichis particularly apt for amplitudes in N = 4 Yang-Mills. In this case, particles aredescribed by λ , ˜ λ and an additional spinless fermionic variable η A with A = 1 , . . . , ¯4 representation of the R -symmetry group SU(4) R of N = 4 Yang-Mills. Thespacetime symmetry group in this case is no-longer the usual conformal group, but thesuperconformal group PSU(2 , |
4) and one can write down generators in terms of λ ,˜ λ and η which are again in a somewhat exotic form. After a Penrose transform to Here we follow [31] and write CP ′ instead of CP because H (0 , ( CP , O ( − h − CP (which we denote with a prime) rather then all oftwistor space. .5. TWISTOR SPACE super-twistor space, which just consists of (1.5.7) plus η A → i ∂∂ψ A ∂∂η A → iψ A , (1.5.16)all superconformal generators similarly become first order differential operators and thespace spanned by λ α , µ ˙ α and ψ A is RP | or CP | . The scaling invariance of super-twistor space is: ( Z I , ψ A ) → ( cZ I , c ψ A ) . (1.5.17)In this case, the helicity operator h = 1 − η A ∂∂η A (1.5.18)modifies the scaling relation (1.5.14) so that it becomes (cid:18) Z Ii ∂∂Z Ii + ψ Ai ∂∂ψ Ai (cid:19) ˜ A n ( λ i , µ i , η A , h i ) = 0 , (1.5.19)and so the scattering amplitudes are elements of H (0 , ( CP | ′ , O (0)).On super-twistor space, the wavefunctions are now elements of H (0 , ( CP | ′ , O (0))and can be given explicitly for a particle of helicity h by [31, 36, 120, 161] φ ( λ, µ, h ) = ¯ δ ( h λ , π i ) (cid:18) λπ (cid:19) h − exp (cid:16) i [˜ π , µ ] πλ (cid:17) g h ( ψ ) , (1.5.20)where g h ( ψ ) is simply a factor of 2 − h ψ s. For example, for a positive-helicity gluon g h is 1 while for a negative-helicity gluon it is ψ ψ ψ ψ . (In fact it is just the factor of ψ that the associated state multiplies in the expansion of the superfield A in (1.6.10).)¯ δ is a ‘holomorphic’ delta function which is a (0 , δ ( f ) = δ (2) ( f ) d ¯ f forany holomorphic function f - see Appendix A for a more detailed discussion.In this case, the multiplication of scattering amplitude and wavefunction leads to anelement of H (0 , ( CP | ′ , O (0)) and the volume form is a (3 , CP | ′ and gives the scattering amplitude in position-space.For our treatment of amplitudes, we will generally use the definition (1.5.15) andsignature + + −− and interpret our results in other signatures when necessary. It is This factor of ψ I is precisely what converts the wavefunctions from being of degree 2 h − λ/π is only 2 h − h − CP ( i.e. with the factor of g h omitted) are of degree 2 h −
2. This is because theholomorphic delta function is of degree − .5. TWISTOR SPACE also worth mentioning that we have glossed over many subtleties in the considerationsabove such as the real nature of momenta already alluded to in § i.e. the use of T ′ rather than T ). For moredetails on all these and more detailed discussions of twistor Theory we refer the readerto [31, 32, 172, 173, 174, 176, 177, 178] and related references. Interpreting (1.5.15) as the way to transform amplitudes into twistor space, we are nowready to see what the tree-level MHV amplitudes look like there. If we recall that theseamplitudes depend only on the negative-helicity spinors λ i , the transformed amplitudesare [31]:˜ A MHV n ( λ i , µ i ) = Z n Y j =1 d ˜ λ j (2 π ) e iµ j ˙ α ˜ λ ˙ αj δ (4) n X k =1 λ k ˜ λ k ! A MHV n ( λ i , ˜ λ i )= Z d x Z n Y j =1 d ˜ λ j (2 π ) exp i n X k =1 µ k ˙ α ˜ λ ˙ αk ! exp ix α ˙ α n X k =1 λ αk ˜ λ ˙ αk ! A MHV n ( λ i )= Z d x n Y j =1 δ (2) (cid:0) µ j ˙ α + x α ˙ α λ αj (cid:1) A MHV n ( λ i ) . (1.5.21)In the second line we have used a standard position-space representation for the deltafunction of momentum conservation and then in the third we have similarly interpretedthe ˜ λ integrals as delta functions. The MHV amplitudes are thus supported only when µ j ˙ α + x α ˙ α λ αj = 0 for all j and for ˙ α = 1 ,
2. For each x α ˙ α these equations define acurve of degree one and genus zero in RP or CP (depending on whether the variablesare real or complex) which is in fact an RP or a CP [31]. x α ˙ α is the parameter ormodulus describing any one of these curves and (1.5.21) is thus an integral over themoduli space of degree one genus zero curves in T . As there is a delta function for everyexternal particle, the integral is only non-zero when all n -points ( λ αi , µ i ˙ α ) lie on oneof these curves in twistor space. Thus the MHV amplitudes are localised on simplealgebraic curves in twistor space, which are (projective) straight lines in the real caseand spheres in the complex case.In the maximally supersymmetric case we have an additional localisation from trans-forming the fermionic variables to twistor space. As well as the delta function of momen- Techincally the space is really n copies of twistor space. Recall that S ∼ = CP . .5. TWISTOR SPACE Figure 1.3:
The MHV amplitudes localise on simple straight lines in twistor space. Herethe 5-point MHV amplitude is depicted as an example. tum conservation coming with the amplitudes, we also have a fermionic delta function δ (8) (Θ) = δ (8) n X i =1 λ i η i ! = Z d θ exp iθ Aα n X i =1 λ αi η i A ! , (1.5.22)and the MHV amplitudes for N = 4 Yang-Mills are given by [31, 175] A MHV n ( λ i , ˜ λ i , η i ) = δ (4) ( P ) δ (8) (Θ) 1 Q ni =1 h i i +1 i . (1.5.23)The transform to super-twistor space is a straightforward generalisation of (1.5.21) andthe result is [31]˜ A MHV n ( λ i , µ i , ψ i ) = Z d xd θ n Y j =1 δ (2) (cid:0) µ j ˙ α + x α ˙ α λ αj (cid:1) δ (4) (cid:0) ψ Aj + θ Aα λ αj (cid:1) Q ni =1 h i i +1 i . (1.5.24)The equations µ j ˙ α + x α ˙ α λ αj = 0 and ψ Aj + θ Aα λ αj = 0 then define (for each j ) a CP | oran RP | in CP | or RP | respectively on which the amplitudes lie.The equation µ ˙ α + x α ˙ α λ α = 0 is in fact of central importance in twistor theory and istraditionally taken to be the definition of a twistor. For a given x (as in our case above),it can be regarded as an equation for λ and µ which as we have seen defines a degree onegenus zero curve that is topologically an S . A point in complexified Minkowski space isthus represented by a sphere in twistor space and hence complexified Minkowski spaceis the moduli space of such curves. Alternatively, if λ and µ ( i.e. a point in twistorspace) are given, it can be regarded as an equation for x . The set of solutions is a twocomplex-dimensional subspace of complexified Minkowski space that is null and self-dual called an α -plane. The null condition means that any tangent vector to the planeis null, and the self-duality means that the tangent bi-vector is self-dual in a certainsense. These α -planes can essentially be regarded as being light-rays and twistor spaceis the moduli space of α -planes. 42 .5. TWISTOR SPACE Other amplitudes involving more and more negative helicities can also be treated,though in these cases performing the Penrose transform (1.5.15) explicitly becomesharder. In these cases it has been found that certain differential operators can beconstructed which help to elucidate their localisation properties in twistor space [31, 73].In particular, given three points P i , P j , P k ∈ CP with coordinates Z Ii , Z Ij and Z Ik , thecondition that they lie on a ‘line’ ( i.e. a linearly-embedded copy of CP as discussedabove) is that F ijkL = 0 where F ijkL = ǫ IJKL Z Ii Z Jj Z Kk . (1.5.25)Similarly, the condition that four points in twistor space are ‘coplanar’ ( i.e. lie on alinearly embedded CP ⊂ CP is given by K ijkl = 0 where K ijkl = ǫ IJKL Z Ii Z Jj Z Kk Z Ll . (1.5.26)When these are explicitly used, µ ˙ α is substituted for ∂/∂ ˜ λ ˙ α and then they act onamplitudes as differential operators.The localisation properties of many amplitudes have been checked [31, 43, 47, 53,72, 73, 76, 91, 179, 180, 181, 182, 183, 184], and it has been found that amplitudes withmore and more negative helicities localise on curves of higher and higher degree. Fortree-level amplitudes in particular this means that an amplitude with q negative-helicitygluons localises on a curve of degree q −
1. In general, the twistor version of an n -particlescattering amplitude is supported on an algebraic curve in twistor space whose degreeis given by [31] d = q − l , (1.5.27)where q is the number of negative-helicity gluons and l the number of loops. The curveis not necessarily connected and its genus g is bounded by g ≤ l . ++ + + + + +- -- - - - - Figure 1.4:
Twistor space localisation of tree amplitudes with q = 3 and q = 4Tree-level next-to-MHV amplitudes for example are supported on curves of degree 2,while NNMHV amplitudes are supported on curves of degree 3 as shown in Figure 1.4above. We can also get a geometrical understanding of the vanishing of the all-plus43 .6. TWISTOR STRING THEORY amplitude and the amplitude with one minus and n − d = − d = 0 in twistor space. Inthe first case, there are no algebraic curves of degree −
1, so these amplitudes musttrivially vanish. In the second, a curve of degree 0 is simply a point and so amplitudesof this type are supported by configurations where all the gluons are attached to thesame point ( λ i , µ i ) = ( λ, µ ) ∀ i in twistor space. Recalling from equation (1.2.7) that p i · p j ∝ h λ i λ j i [˜ λ j ˜ λ i ], all these invariants must be zero for these amplitudes. This onthe other hand is impossible for non-trivial scattering amplitudes with n ≥ n = 3 things are a bit more subtle because on-shellness, p i = 0 and momentumconservation, p + p + p = 0, guarantee that for real momenta in Lorentz signature p i · p j = 0. However, for complex momenta and/or other signatures the 3-point amplitudemakes more sense. As 0 = 2 p i · p j = h λ i λ j i [˜ λ j ˜ λ i ], the independence of λ i and ˜ λ i implies that either h λ i λ j i = 0 or [˜ λ j ˜ λ i ] = 0. Thus all λ i are proportional or all ˜ λ i areproportional. As can be read-off from the Yang-Mills Lagrangian (or seen as a specialcase of the googly MHV amplitudes), the − + + amplitude is given by A = [˜ λ , ˜ λ ] [˜ λ , ˜ λ ][˜ λ ˜ λ ][˜ λ ˜ λ ] . (1.5.28)This would vanish identically if all the ˜ λ i are proportional, so we should pick all the λ i to be proportional to ensure momentum conservation. However, SL(4 , R ) invariancein twistor space then implies that the ( λ i , µ i ) all coincide and thus the gluons aresupported at a single point in twistor space as predicted by (1.5.27) [31]. In this section we will give a very brief overview of a string theory that provides a naturalframework for understanding the properties of scattering amplitudes discussed in theprevious sections. We will only describe the original approach (which has also been theone most computationally useful to date) taken by Witten [31] though other approaches,notably by Berkovits [112, 113, 114], have been considered. Further proposals include[115, 116, 117, 118], though these have not so far been used to calculate any amplitudes.A good introduction to the material presented in this section can again be found in [161].It is well known that the usual type I, type II and heterotic string theories livein the critical dimension of d = 10, which is where they really make sense quantummechanically. However, there are other string theories known as topological stringtheories which are typically simpler than ordinary string theories and can make sense SO(3 , ∼ = SL(4 , R ) is the conformal group in signature + + −− . .6. TWISTOR STRING THEORY in other dimensions. They are called topological because they can be obtained fromcertain topological field theories which are field theories whose correlation functionsonly depend on the topological information of their target space and in-particular donot depend on the local information such as the metric of the space. Witten introducedtopological string theory in [185, 186] as a simplified model of string theory, and it hasbeen extensively studied since then. We will only give a ‘lightning’ review here andrefer the reader to the original papers and such excellent introductions as [187] for moredetails. One starts with a field theory in 2-dimensions with N = 2 supersymmetry. The super-symmetry generators usually transform as spin 1 / d this is SO(2) ∼ = U(1) locally and the spin 1 / N = (2 ,
2) with 2 left-moving supercharges and 2 right-movingsupercharges.The symmetries of the theory consist of both the usual Poincar´e algebra as wellas the N = 2 supersymmetry algebra and the R-symmetry of the theory associatedwith the supersymmetry. We will not write all of these down here, but in-particularthe supersymmetry generators and their complex conjugates obey the non-zero anti-commutation relations (in the language of [187]): { Q ± , ¯ Q ± } = P ± H { D ± , ¯ D ± } = − ( P ± H ) , (1.6.1)where H ∼ d/dξ and P ∼ d/dξ are the Hamiltonian and momentum operators of the2- d space with coordinates ξ α .One thing that we can now do is to define new operators Q A and Q B which arelinear combinations of supercharges as Q A = ¯ Q + + Q − Q B = ¯ Q + + ¯ Q − , (1.6.2)and then it follows from (1.6.1) that Q = Q = 0 (1.6.3)and Q A and Q B look like BRST operators. However, Q A / B are not scalars, so we would45 .6. TWISTOR STRING THEORY violate Lorentz invariance by interpreting them as BRST operators straight away. Infact what we can do is to make an additional modification to the Lorentz generator ofthe 2- d space by making linear combinations of it and the R-symmetry generators insuch a way that the Q A / B are scalars under the new generators. This procedure is called twisting and produces two different topological field theories labelled by A and B.Now that we have a BRST operator, we can use the usual definitions for the physicalstates of our theory in terms of BRST cohomology (see for example Chapter 16 of [2]or Chapter 15 of [4] for an introduction). Physical states | ψ i are given by the condition Q A / B | ψ i = 0 with states being equivalent if they differ by something which is BRSTexact such as Q A / B | φ i for some | φ i . Similarly, physical operators are taken to be thosewhich commute with the BRST operator modulo those which can be written as an anti-commutator of Q A / B with some other operator. In particular one can show that thatthe stress-tensors of the twisted theories are BRST exact as they can be written in theform T αβ A / B = { Q A / B , τ αβ } for some τ αβ . This is a general property of topological fieldtheories. What we have so far constructed are two 2-dimensional topological field theories. How-ever, we can promote these to string theories by considering the theories to be livingon the worldsheet of a string and ensuring that we integrate over all metrics of the2-dimensional space in the path integral as well as the other fields appearing in theaction (see e.g.
Chapter 3 of [19] for how this works in the usual string theory settings).The Euclidean path integral Z E = Z D h ( ξ ) D Φ( ξ ) e − S d [ h, Φ] , (1.6.4)where h αβ is the world-sheet metric, Φ are the fields of our 2- d field theory and ξ α are the coordinates of the 2- d space then defines our topological string theory. If wehave re-defined our Lorentz generators to make Q A a scalar then the string theory isknown as the A-model, while if we choose to make Q B a scalar we arrive at the B-model[185, 186].We can also say something about the target spaces of these topological string the-ories. In ‘normal’ string theory settings these target spaces - the spaces in which thestrings live - are known to be 10-dimensional (or 26-dimensional for the purely bosonicstring) in order for them to be quantum-mechanically anomaly-free. The N = (2 , K¨ahler manifolds - even before we performthe topological twisting. These spaces are complex manifolds that are endowed with aHermitian metric ( i.e. a real metric - real in the sense that g ¯ ı ¯ = ( g ij ) ∗ and g ¯ ıj = ( g i ¯ ) ∗ .6. TWISTOR STRING THEORY - with g ij = g ¯ ı ¯ = 0) and which we can write locally as the second derivative of somefunction termed the K¨ahler potential K ( z, ¯ z ): g i ¯ = ∂ K ( z, ¯ z ) ∂z i ∂ ¯ z ¯ . (1.6.5)Here z i and ¯ z ¯ are appropriate complex coordinates on the target space. When we dothe twisting described by (1.6.2) it turns out that the A-model twist can be performedfor any K¨ahler target space, while the B-model twist requires the space to be of a yetmore specialised form known as a Calabi-Yau manifold.There are many different ways to define a Calabi-Yau manifold, but one way thatis good for our purposes is that it is a K¨ahler manifold that is also Ricci-flat, R i ¯ = 0.The moduli (essentially the parameters) describing the variety of such spaces are of twotypes which are termed the K¨ahler moduli and the complex-structure moduli. It can beshown that the space of K¨ahler moduli is locally H (1 , ( M CY ) - that is to say it is locallygiven by the Dolbeault cohomology class of (1 , , H (2 , ( M CY ). BecauseCalabi-Yau manifolds are automatically K¨ahler manifolds to begin with and because oftheir high degree of symmetry, the A-model is often also considered on a Calabi-Yau.Finally, it can be shown that the central charge of the Virasoro algebra of the A- andB-models vanishes identically in any number of dimensions [187], so topological stringsare well-defined in target spaces of any dimension. For more comprehensive discussionsof complex, K¨ahler and Calabi-Yau manifolds see e.g. [187, 188, 189, 190, 191].As for the physical operators in these models, we briefly state without proof that inthe A-model, Q A can be viewed as being Q A ∼ d - the de Rham exterior derivative - andthe local physical operators are in one-to-one correspondence with de Rham cohomologyelements on the target space: O A ∼ A i ...i p ¯ ... ¯ q (Φ) d Φ i · · · d Φ i p d ¯Φ ¯ · · · d ¯Φ ¯ q . (1.6.6)For the B-model on the other hand one can show that Q B ∼ ¯ ∂ - the Dolbeault exteriorderivative - and the local physical operators are now just (0 , p )-forms with values inthe antisymmetrized product of q holomorphic tangent spaces - which we denote by V q T (1 , ( M CY ): O B ∼ B j ...j q ¯ ı ... ¯ ı p (Φ) d ¯Φ ¯ ı · · · d ¯Φ ¯ ı p ∂∂ Φ j · · · ∂∂ Φ j q . (1.6.7)These theories also have the intruiging property of mirror symmetry [192, 193] - see e.g. [194] and references therein for a comprehensive review - that the A-model on oneCalabi-Yau is equivalent to the B-model on a different Calabi-Yau which is known asits Mirror. In the mirror map, the hodge numbers h , and h , are swapped which47 .6. TWISTOR STRING THEORY pertains to the exchange of K¨ahler and complex-structure moduli. This is especiallyuseful as the B-model is generally easier to compute-with than the A-model, while theA-model is more physically interesting in many scenarios. Hard computations in theA-model can often be mapped to easier ones in the B-model. In his original construction [31], Witten considered the B-model and we will do thesame here. The target space on which we will want it to live will be CP | , which is aCalabi-Yau super-manifold (with bosonic and fermionic degrees of freedom) rather thana bosonic manifold as is more common. This is fortunate because CP is not Calabi-Yau,while CP | is . In addition, if we recall that the closed-string sector is where gravitystates arise, we would like to consider the open-string
B-model on twistor space in orderthat we may end up with degrees of freedom with spin 1 or less. In the simplest casethis consists of adding N bosonic-space-filling D5-branes (thus spanning all 6 bosonicdirections of CP | in analogy with the purely bosonic case of [195]. In addition (asWitten did), we take the D5s to wrap the fermionic directions ψ I and ¯ ψ ¯ J in such away that we can set ¯ ψ ¯ J to zero. It is not entirely clear how this should be interpreted,but one might say that the branes wrap the ψ directions while being localised in the¯ ψ directions. The presence of N branes gives rise to a U( N ) gauge symmetry as usualdue to the Chan-Paton factors of the open strings ending on them.So far we have been considering things from a worldsheet perspective. However, foropen strings we also have the spacetime perspective of open-string field theory [196].This has a multiplication law ⋆ , an operator Q obeying Q = 0 and Lagrangian L = 12 Z (cid:18) A ⋆ Q A + 23 A ⋆ A ⋆ A (cid:19) , (1.6.8)where A is the string field. In the presence of D5-branes on a 6-dimensional (bosonic)manifold this has been shown to reduce to holomorphic Chern-Simons theory [195],where the D5-D5 modes of the string field A give a (0 , A = A ¯ ı ( z, ¯ z ) d ¯ z ¯ ı on the branes. On the other hand, when the target space is the super-manifold CP | , A reduces to the (0 , A = A ¯ I ( Z, ¯ Z, ψ, ¯ ψ ) d ¯ Z ¯ I ,while Q becomes the ¯ ∂ operator and ⋆ the usual wedge product operation ∧ . The actiondescends to S = 12 Z CP | Ω ∧ tr (cid:18) A ∧ ¯ ∂ A + 23 A ∧ A ∧ A (cid:19) , (1.6.9) In fact CP |N is Calabi-Yau iff N = 4. .6. TWISTOR STRING THEORY and with ¯ ψ = 0 the superfield A can be expanded as A ( Z, ¯ Z, ψ ) = A + ψ I λ I + 12 ψ I ψ J φ IJ + 13! ǫ IJKL ψ I ψ J ψ K ˜ λ L + 14! ǫ IJKL ψ I ψ J ψ K ψ L G , (1.6.10)where
A, λ I , φ IJ , ˜ λ I , G are all functions of Z and ¯ Z and we have suppressed the (0 , , CP | Ω = 14! ǫ IJKL ǫ MNP Q Z I dZ J dZ K dZ L dψ M dψ N dψ P dψ Q . (1.6.11)Because dZ I and dψ I scale oppositely – as follows from (1.5.17) and the fermionicnature of ψ I ( dψ I → c − dψ I under (1.5.17)) – it is clear that (1.6.11) is invariantunder this scaling and thus the action (1.6.9) is only invariant if A is of degree zero, A ∈ H (0 , ( CP | ′ , O (0)). This means that each component field in the ψ expansion(1.6.10) must be of degree 2 h − h in spacetime- c.f. the twistor description of wavefunctions for particles of helicity h of Eq. (1.5.20)and surrounding paragraphs. In addition, the fermionic nature of the ψ I restricts thenumber of degrees of freedom of the component fields and it can quickly be seen that(1.6.10) describes the N = 4 multiplet, which in the notation of § A − , χ − , φ, χ + , A + ) ≡ ( G, ˜ λ I , φ IJ , λ I , A ), while the action in component form can bewritten as S = Z CP Ω ′ ∧ tr (cid:16) G ∧ ( ¯ ∂A + A ∧ A ) − ˜ λ I ∧ ( ¯ ∂λ I + [ A, λ I ]) (1.6.12)+ 14 ǫ IJKL φ IJ ∧ ( ¯ ∂φ KL + A ∧ φ KL ) − ǫ IJKL λ I ∧ λ J ∧ φ KL (cid:17) , where Ω ′ = ǫ IJKL Z I dZ J dZ K dZ L /
4! is the bosonic reduction of Ω obtained after inte-grating out the ψ I and [ A, λ I ] = A ∧ λ I + λ I ∧ A . The equations of motion followingfrom (1.6.9) are ¯ ∂ A + A ∧ A = 0 and the gauge invariance is δ A = ¯ ∂ω + [ A , ω ].What we have arrived at is ‘half’ of N = 4 super-Yang-Mills. We have all the fields asis apparent from (1.6.10), but it turns out that not all the interactions are present. Oneof the easiest ways to see this is to note that the symmetries of the B-model generallyleave Ω invariant [31]. However there are also interesting transformations of the targetspace that leave the complex structure invariant but transform Ω non-trivially. Onesuch transformation is a U(1) R part of the R-symmetry group U(4) R = SU(4) R × U(1) R that acts as S : Z I → Z I ; ψ I → e iα ψ I (1.6.13) To be more precise it is the twistor transform of the N = 4 multiplet [32, 197]. Recall that for Grassman variables, R dψ ≡ ∂/∂ψ with R dψ I ψ J = δ IJ and δ ( ψ ) = ψ . Note that the I indices on the component fields in (1.6.12) are fundamental indices of this SU(4) R . .6. TWISTOR STRING THEORY with dψ I → e − iα dψ I because of their fermionic nature. Ω → e − iα Ω thus has S = − ψ I inside A arecompensated by equal and opposite transformations of the component fields: A has S = 0, λ I has S = − φ IJ has S = −
2, ˜ λ I has S = − G has S = −
4. In factthe component action (1.6.12) is made up entirely of terms with S = −
4. However, theusual N = 4 Yang-Mills action in component form consists of terms which have S = − and S = −
8. For example the scalar kinetic terms ( ∂ µ φ ) have S = − φ has S = −
8. The holomorphic Chern-Simons action (1.6.9) thus capturesall the fields of maximally supersymmetric Yang-Mills, but not all the interactions.Although we will not discuss it here, the theory described by (1.6.9) is in fact self-dual N = 4 super-Yang-Mills [198] - that is, (super)-Yang-Mills theory for a gauge field A ′ whose field strength appearing in the action is self-dual. A ′ is the spacetime fieldcorresponding to the homogeneity 0 field ( A ) in (1.6.10) and the spacetime action ofthis theory is S = R G ′ ∧ ∗ F ′ ≡ R G ′ ∧ F ′ SD . Here G ′ is a self-dual 2-form whose twistortransform is the homogeneity − G ) in (1.6.10), F ′ SD is the self-dual part of F ′ = dA ′ + A ′ ∧ A ′ and ∗ is the Hodge duality operation. Witten’s solution to the aforementioned problem of the absence of the entire set ofinteractions was to enrich the B-model on CP | with instantons. The ones in questionare Euclidean D1-branes which wrap holomorphic curves in super-twistor space and onwhich the open strings can end. These holomorphic curves are precisely the ones thatwe met earlier on which the scattering amplitudes were found to localise. We won’tgo into much detail here (more can be found in [31]), but the basic idea is that theseinstantons have S -charge − d + 1 − g ) for the connected degree d and genus g case.Thus for the ‘classical’ tree-level MHV case these instantons provide the terms with S = − T . We willthus ignore the D1-D1 strings from now on. Of course we do want to involve theD1-instantons, so we’ll focus on the D1-D5 and D5-D1 strings. Witten argued thatthese strings give rise to fermionic (0 , α i and the D5-D1 modes give a Note that formally we can write the self-dual and anti-self-dual parts of F ′ as F ′ SD = ( F ′ + ∗ F ′ ) / F ′ ASD = ( F ′ − ∗ F ′ ) /
2. Here we have taken ∗ ∗ F ′ = F ′ . We refer to the tree-level MHV amplitudes as being the ‘classical’ case as it turns out that we canre-formulate perturbation theory in terms of ‘MHV-vertices’ - see § S -charge violation at the level of the action. .6. TWISTOR STRING THEORY fermion β ¯ ı , with ¯ ı and i (anti)-fundamental U( N ) indices respectively. The effectiveaction for the low-energy modes is then S eff = Z D1 dz β ( ¯ ∂ ¯ z + A ¯ z d ¯ z ) α , (1.6.14)where z and ¯ z are local complex coordinates on the D1 and A ¯ z (which is a backgroundfield on the D1) is the component of the superfield generated by the D5-D5 strings lyingalong the D1. The first term is the kinetic term of these modes (with ¯ ∂ ¯ z the ¯ ∂ operatorrestricted to the D1), while the second describes their interaction with the gauge field A and can be written as S int = Z D1 J ∧ ( A ¯ z d ¯ z ) , (1.6.15)where we define J to be the current J j ¯ ı = β ¯ ı α j dz .Any particular external state will contribute just one component of this superfield A and therefore its coupling will be V s = Z D1 J s ∧ φ s , (1.6.16)where φ s is the wavefunction of the state in twistor space and thus a (0 , Then if the curve which the D1 wraps were to have no moduli ( i.e. there were only onepossibility for it), one would be able to compute scattering amplitudes by evaluatingthe correlator h V s . . . V s n i . However, we know from the discussion in § do have moduli and thus we should integrate this correlator over their modulispace. Our prescription for computing n -point scattering amplitudes whose externalparticles have wave functions φ s i will then be A n = Z d M d h V s . . . V s n i , (1.6.17)where d M d is an appropriate measure on the moduli space of holomorphic curves ofdegree d (and genus zero for our current purposes). As an example of how (1.6.17) is implemented let us calculate the MHV amplitudesusing this prescription. From § We use subscripts s i etc. to denote the i th particle for the rest of this section in order to avoidconfusion with the gauge indices. .6. TWISTOR STRING THEORY curves that are embedded in CP | via the equations µ s k ˙ α + x α ˙ α λ αs k = 0 ψ As k + θ Aα λ αs k = 0 . (1.6.18) λ αs k are the homogeneous coordinates on the curves (with s k = 1 . . . n denoting the k th particle) and their moduli are x α ˙ α and θ Aα . These are thus the curves that we will takethe D1-instantons to be wrapping. x α ˙ α has 4 (bosonic) degrees of freedom while θ Aα has8 (fermionic) ones and a natural measure on the moduli space is then d M = d xd θ .For clarity let us specialise to the case of 4-particle (gluon) scattering where particles1 and 3 have negative-helicity and particles 2 and 4 positive-helicity. The n -particle caseis an easy generalisation of this. Formally we have A = Z d M h V s V s V s V s i = Z d xd θ (cid:28)Z CP | J s ∧ φ s . . . Z CP | J s ∧ φ s (cid:29) , (1.6.19)where we assume that the wavefunctions φ s k take values in the Lie-algebra of U( N ) andthus contain a generator T a k in addition to (1.5.20). ( J s k ) j ¯ ı = β ¯ ı ( z k ) α j ( z k ) dz k thengives A = Z d xd θ Z dz . . . dz (cid:10) β ¯ ı ( z ) α j ( z ) φ s . . . β ¯ ı ( z ) α j ( z ) φ s (cid:11) (1.6.20)up to a factor. Separating-out the Lie-algebra generators from the rest of the wavefunc-tions ( φ s k = φ ′ s k T a k ) we can re-write the correlator as Z d xd θ Z dz . . . dz φ ′ s . . . φ ′ s D β ¯ ı ( T a ) i ¯ α j . . . β ¯ ı ( T a ) i ¯ α j E . (1.6.21)This correlator has many different contributions (105 in total) coming from thepossible ways of Wick contracting the fermions α and β . Let us consider the cyclic onewhere we contract β ( z ) with α ( z ), β ( z ) with α ( z ) and so on (with β ( z ) contractedwith α ( z )). Because α and β are fermions living on (in this case) CP , their propagatoris the usual one for free fermions on the complex plane h α j ( z k ) β ¯ ı ( z l ) i = δ j ¯ ı z k − z l (1.6.22)52 .6. TWISTOR STRING THEORY and the relevant Wick contraction is W cyclic = ( T a ) i ¯ . . . ( T a ) i ¯ h α j ( z ) β ¯ ı ( z ) i . . . h α j ( z ) β ¯ ı ( z ) i = ( T a ) i ¯ . . . ( T a ) i ¯ δ j ¯ ı z − z . . . δ j ¯ ı z − z = tr( T a . . . T a )( z − z )( z − z )( z − z )( z − z ) . (1.6.23)Dropping the single-trace colour factor for now, (1.6.21) is A = Z d xd θ Z dz . . . dz φ ′ s . . . φ ′ s ( z − z )( z − z )( z − z )( z − z )= Z d xd θ Z h λ s dλ s i . . . h λ s dλ s i φ ′ s . . . φ ′ s h λ s λ s i . . . h λ s λ s i , (1.6.24)where we have changed to homogeneous coordinates λ s k on the CP s by setting z k = λ s k /λ s k with 1 and 2 indicating spinor ‘ α ’ indices here.Now we must introduce the explicit form for the wavefunctions and integrate overthe λ k . For this it is useful to note that with z = λ /λ and making the more specificchoices of λ α = (1 , z ) and ζ α = (1 , b ), (A.2.9) becomes (A.2.10): Z h λ dλ i ¯ δ ( h λ ζ i ) F ( λ ) = − iF ( ζ ) . (1.6.25)Omitting the integral over moduli, (1.6.24) thus gives A = Z CP | h λ s dλ s i ¯ δ ( h λ s π s i ) (cid:18) λ s π s (cid:19) h − e i [˜ π s µ s ]( π s /λ s ) g h ( ψ s )... Z CP | h λ s dλ s i ¯ δ ( h λ s π s i ) (cid:18) λ s π s (cid:19) h − e i [˜ π s µ s ]( π s /λ s ) g h ( ψ s ) H ( λ s i )= ψ s ψ s ψ s ψ s ψ s ψ s ψ s ψ s H ( π s i ) e i P sk =1 [˜ π sk µ sk ] , (1.6.26)where H is the denominator in (1.6.24).We must now perform the integral over the moduli. For this purpose we can recallthe equations describing the embedding (1.6.18) and substitute µ s k ˙ α = − x α ˙ α π αs k and ψ As k = − θ Aα π αs k whereupon the integral over x gives Z d x exp − ix α ˙ α X s k =1 π αs k ˜ π ˙ αs k = δ (4) X s k =1 π s k ˜ π s k , (1.6.27) Recall that the delta functions of (1.6.26) have set λ s k = π s k . .6. TWISTOR STRING THEORY which is just the delta function of momentum conservation. For the fermionic moduliwe have (for example): ψ s ψ s = ( θ π s + θ π s )( θ π s + θ π s )= θ θ π s π s + θ θ π s π s = θ θ ( π s π s − π s π s )= θ θ h π s π s i . (1.6.28)After dealing with all the ψ s in a similar way and then integrating over the eight θ variables gives h π s π s i . Putting all the pieces together we get A (1 − , + , − , + ) = tr( T a . . . T a ) h π s π s i h π s π s i . . . h π s π s i δ (4) X s k =1 π s k ˜ π s k = tr( T a . . . T a ) h i h i . . . h i δ (4) X i =1 π i ˜ π i ! , (1.6.29)which is precisely the formula for the MHV amplitudes that we wrote down before,though we have kept the colour structure explicit here.We should be careful to note that we have simply picked the particular Wick con-traction that we needed in order to get a cyclic colour ordering. All the terms withnon-cyclic colour orderings but a single trace are also present as well as multi-traceterms which in [31, 36] were suggested to be a sign of the presence of closed-string (andthus gravitational) states.We have explicitly described the construction of the MHV amplitudes from the B-model in twistor space. Other amplitudes can be calculated in this way too, though thecomplexity is greater so we will not go into any detail on this. The NMHV amplitudesfor example require one to integrate over the moduli space of degree 2 curves in T andsome simple cases were calculated this way in [134]. Other cases such as the n -pointgoogly MHV amplitudes (with 2 positive-helicity gluons and n − d >
1, one encounters the possibility of describing these as connectedcurves of degree d , or disconnected curves of degree d i with P d i = d . In [134, 135, 136]it was found that the connected prescription alone reproduces the entire amplitudes inthe cases considered (at least up to a factor). However, there is also strong evidencethat the same amplitudes can be computed using the purely disconnected prescription[33]. Indeed, this disconnected prescription led directly to the proposal of new rules fordoing perturbative gauge theory which we will describe in the next section. The authorsof [199] argued that the integrals involved in the connected prescription localised on thesubspace where a connected curve of degree d degenerates to the intersection of curves54 .7. CSW RULES (TREE-LEVEL) of degree d i with P d i = d and thus provided strong evidence that there are ultimately d different prescriptions which are all equivalent. The extreme possibilities are that wehave just one degree d curve to consider, or alternatively d degree one curves. Thislatter case was the inspiration for [33].We have also not said anything about loop diagrams here except for the formal state-ment that they localise on curves of degree d = q − l with g ≤ l . The structure of manyloop diagrams of N = 4 super-Yang-Mills was elucidated in [31, 72, 73, 180, 181, 183],though the situation with their calculation from the B-model is far less clear than thatfor trees unfortunately. In [36] it was shown that closed string modes give rise to statesof N = 4 conformal supergravity describing deformations of the target twistor space aswell as the expected N = 4 Yang-Mills states. Conformal supergravity in 4 dimensionshas a Lagrangian which is the Weyl tensor of gravity squared, S = R d x p det | g µν | W ,and is usually considered to be a somewhat unsavoury theory as it gives rise to fourthorder differential equations which are generally held to lead to a lack of unitarity (see e.g. [201]). One might still hope to decouple these states, but because the couplingconstant is the same in both sectors the amplitudes mix and one ends up with a theoryof N = 4 conformal supergravity coupled to N = 4 super-Yang-Mills, some amplitudesof which were computed in [36] at tree-level and more recently in [114] at loop-level(see also [202]). Despite all this, it was shown by Brandhuber, Spence and Travaglinithat the proposals of [33] can be extended to loop-level and provide a new perturbativeexpansion for field theory which is valid in the quantum regime as well as the classicalone. This discovery will be a central theme in the following chapters of this thesis.As a final remark in this section we point out that twistor string theories have alsobeen constructed to describe other theories with less supersymmetry and/or productgauge groups [119, 120, 121, 122, 124] as well as more recently to describe Einsteinsupergravity [39]. Indeed the proposals in [39] include a twistor description of N = 4SYM coupled to Einstein supergravity which may lead to a resolution of the problemof loops if they can be consistently decoupled. Motivated by the findings we have so far discussed, Cachazo, Svrˇcek and Witten pro-posed a set of alternative graphs for tree-level amplitudes in Yang-Mills theory basedon the MHV vertices [33]. The essential idea is the observation that one can seem-ingly compute tree-level amplitudes from the totally disconnected prescription alludedto above by gluing d disconnected lines together (on each of which there is an MHV am-plitude localised) for an amplitude involving d + 1 negative-helicity gluons. The gluing For a review see e.g. [200]. .7. CSW RULES (TREE-LEVEL) procedure is made concrete by connecting the lines with twistor space propagators. Infield theory terms this corresponds to the use of MHV amplitudes as the fundamentalbuilding blocks - because their localisation properties in twistor space translates to apoint-like interaction in Minkowski space - and gluing these together with simple scalarpropagators 1 /P . The two ends of any propagator must have opposite helicity labelsbecause an incoming gluon of one helicity is equivalent to an outgoing gluon of theopposite helicity. In order to glue MHV vertices together we must continue them off-shell since one or moreof the legs must be connected to the off-shell propagator 1 /P . For this purpose, considera generic off-shell momentum vector, L . On general grounds, it can be decomposed as[65, 179] L = l + zη , (1.7.1)where l = 0, and η is a fixed and arbitrary null vector, η = 0; z is a real number.Equation (1.7.1) determines z as a function of L : z = L L · η ) . (1.7.2)Using spinor notation, we can write l and η as l α ˙ α = l α ˜ l ˙ α , η α ˙ α = η α ˜ η ˙ α . It then followsthat l α = L α ˙ α ˜ η ˙ α [˜ l ˜ η ] , (1.7.3)˜ l ˙ α = η α L α ˙ α h l η i . (1.7.4)We notice that (1.7.3) and (1.7.4) coincide with the CSW prescription proposed in [33]to determine the spinor variables l and ˜ l associated with the non-null, off-shell four-vector L defined in (1.7.1). The denominators on the right hand sides of (1.7.3) and(1.7.4) turn out to be irrelevant for our applications since the expressions we will bedealing with are homogeneous in the spinor variables l α ; hence we will usually discardthem. This defines our off-shell continuation. The CSW rules for joining these MHV amplitudes together are probably best illustratedwith an example. It is clear that a tree diagram with v MHV vertices has 2 v negative-helicity legs, v − .7. CSW RULES (TREE-LEVEL) are left with v + 1 external negative helicities. To put it another way, if we wish tocompute a scattering amplitude with q negative-helicity gluons we will need v = q − A (1 + , − , − , − ). We know from our discussions in § p − p +1 p − p − P − +MHV MHV p +1 p − p − p − P − +MHV MHVFigure 1.5: The two MHV diagrams contributing to the + − −− amplitude. All externalmomenta are taken to be outgoing. As shown in Figure 1.5, there are two diagrams to consider. For each diagramwe should write down the MHV amplitudes corresponding to each vertex and jointhem together with the relevant scalar propagator, remembering to use the off-shellcontinuation of (1.7.3) and (1.7.4) to deal with the spinors associated with the internalparticles. The first (uppermost) diagram gives C = h λ λ P i h λ P λ ih λ λ i P h λ λ i h λ λ P ih λ P λ i , (1.7.5)where the momentum of the propagator is P = − ( p + p ) = ( p + p ). As the externalmomenta are massless, P = 2( p · p ) = h i [2 1] and the off-shell continuation tellsus that λ αP = P α ˙ α ˜ η ˙ α [˜ λ P ˜ η ]= − ( λ α ˜ λ ˙ α + λ α ˜ λ ˙ α )˜ η ˙ α [˜ λ P ˜ η ]= − λ α a b − λ α a b , (1.7.6)57 .7. CSW RULES (TREE-LEVEL) where we have written ˜ λ ˙ αi ˜ η ˙ α = a i and [˜ λ P ˜ η ] = b for clarity and have kept the de-nominators of the off-shell continuation explicit in order to demonstrate that they willdrop out of the expressions. Similarly, an appropriate form for eliminating λ P fromthe MHV vertex on the right is λ αP = λ α a b + λ α a b . (1.7.7)On substituting for λ P and P (1.7.5) then becomes C = ba a b h i h ih i h i [2 1] h i h ih i b a a = a a a a h i [1 2] . (1.7.8)Going through the same procedure for the second contribution in Figure 1.5 gives C = a a a a h i [4 1] , (1.7.9)and the final answer is A = C + C . Momentum conservation is P i =1 λ i ˜ λ i = 0,which can be applied to an expression of the form h i i [ i
1] to give P i =1 h i i [ i
1] = h i [2 1] + h i [4 1] = 0 and means that h i / [1 2] = −h i / [4 1]. Thus C = − C andwe get A = 0 as expected.There are two essential points to note here. The first is that when we performedthe off-shell continuation all the denominators of (1.7.3) cancelled out. This is in factgenerally true for the amplitudes we will be interested in and thus we will discard themfrom now on. The second point is that in C and C , the arbitrary null momentum η ofthe off-shell continuation was still present, lurking as an ˜ η ˙ α in the α i . The contributionscancelled in the end so we didn’t care too much about this, but we might worry aboutthe presence of this arbitrary momentum in the calculation of amplitudes that don’tvanish. In fact it tends to crop up frequently and the expressions that one arrives atseem to depend on η at first sight. However, it can be shown that the amplitudes are η -independent and it can therefore sometimes be of use to set η to be one of the externalmomenta in the problem.This procedure has been implemented both for amplitudes with more external glu-ons and amplitudes with more negative helicities. In both cases the complexity grows, but the number of diagrams grows for large n at most as n [33] which is a markedimprovement on the factorial growth of the number of Feynman diagrams needed tocompute the same processes. Further evidence for the procedure and a heuristic prooffrom twistor string theory can be found in [33], while a proof based on recursive tech-niques was given by Risager in [34] which was then used to give an MHV-vertex approach58 .8. LOOP DIAGRAMS FROM MHV VERTICES to gravity amplitudes [77]. Evidence for the validity of the procedure for tree and loopamplitudes was given in [79]. On the other hand Mansfield found a transformation which takes the usual Yang-Mills Lagrangian and maps it to one where the vertices are explicitly MHV vertices[35] (see also [203]). This involves formulating pure Yang-Mills theory in light-conecoordinates and performing a non-local change of variables which maps the usual 3-and 4-point vertices that arise in Feynman diagram perturbation theory into an infinitesequence of MHV vertices starting with the 3-point − − + vertex. The procedurealso clarifies the origin of the null vector η that we have used to define the off-shellcontinuation. It is just the same null vector as is used to define the light-cone formulationof the theory [35, 81]. For further work related to understanding the CSW rules from aLagrangian approach see [80, 81, 82, 137, 138, 139, 204]. The CSW rules at tree-level provide a new and effective way of re-organising perturba-tion theory and thus lead to more efficient methods for calculating tree-level amplitudeswhich often yield simpler results than more traditional approaches. Naturally we wouldlike to be able to extend this method beyond tree-level and consider quantum cor-rections which are often a substantial contribution to the overall result. However asalready mentioned the picture from twistor string theory is not as clear at loop-leveland one might expect the CSW procedure to fail there due to the presence of conformalsupergravity.Nonetheless Brandhuber, Spence and Travaglini showed that the CSW rules are stillvalid at one-loop and provided a concrete procedure to follow from which they re-derivedthe one-loop n -point MHV gluon scattering amplitudes in N = 4 super-Yang-Mills [37].The answers they obtained are in complete agreement with the original results derivedat 4-point by Green, Schwarz and Brink from the low energy limit of a string theory[205] and then at n -point by BDDK [38]. We will briefly review the method proposedin [37] and outline how it can be used to derive the N = 4 amplitudes. Chapters 2 and3 will then be devoted to applying the same method to the N = 1 amplitudes and thosein pure Yang-Mills with a scalar running in the loop respectively, thus calculating allcut-constructible contributions to the n -point MHV gluon scattering amplitudes inQCD (1.3.3). We will say more about loop amplitudes shortly. It turns out that the CSW approach at loop-level only calculates the cut-containing terms, thusmirroring the cut-constructibility approach of BDDK. The rational terms are inextricably linked tothese in supersymmetric theories but must be obtained in other ways in non-supersymmetric ones. Seealso Appendix D. .8. LOOP DIAGRAMS FROM MHV VERTICES The procedure proposed in [37] can be summarised as follows [40]: Consider only the colour-stripped or partial amplitudes introduced in § Lift the MHV tree-level scattering amplitudes to vertices, by continuing the in-ternal lines off-shell using the prescription described in § Build MHV diagrams with the required external particles at loop level using theMHV tree-level vertices and sum over all independent diagrams obtained in thisfashion for a fixed ordering of external helicity states. Re-express the loop integration measure in terms of the off-shell parametrisationemployed for the loop momenta. Analytically continue to 4 − ǫ dimensions in order to deal with infrared divergencesand perform all loop integrations. The loop legs that we must integrate over are off-shell and in order to proceed we mustwork out the integration measure used in [37]. The details of the measure were moreconcretely worked-out in [79] using the Feynman tree theorem [206, 207, 208] and weuse certain results from there as well as from the original construction of [37] whilefollowing the review of Section 3 of [40].We need to re-express the usual integration measure d L over the loop momentum L in terms of the new variables l and z introduced previously. After a short calculationwe find that [37, 79] d LL + iε = d N ( l ) dzz + iε , (1.8.1)where we define d L := Q i =0 dL i and have introduced the Nair measure [175] d N ( l ) := 14 i (cid:16) h l dl i d ˜ l − [˜ l d ˜ l ] d l (cid:17) = d l l . (1.8.2) The iε prescription in the left- and right-hand sides of (1.8.1) was understood in [37], and, asstressed in [79, 179, 209] it is essential in order to correctly perform loop integrations. .8. LOOP DIAGRAMS FROM MHV VERTICES Eq. (1.8.1) is key to the procedure. It is important to notice that the product of themeasure factor with a scalar propagator d L/ ( L + iε ) in (1.8.1) is independent ofthe reference vector η . In [175], it was noticed that the Lorentz-invariant phase spacemeasure for a massless particle can be expressed precisely in terms of the Nair measure: d l δ (+) ( l ) = d N ( l ) , (1.8.3)where, as before, we write the null vector l as l α ˙ α = l α ˜ l ˙ α , and in Minkowski space weidentify ˜ l = ± ¯ l depending on whether l is positive or negative.Next, we observe that in computing one-loop MHV scattering amplitudes from MHVdiagrams (shown in Figure 1.6) , the four-dimensional integration measure which ap-pears is [37, 79] d M := d L L + iε d L L + iε δ (4) ( L − L + P L ) , (1.8.4)where L and L are loop momenta, and P L is the external momentum flowing outsidethe loop so that L − L + P L = 0. L1L2
MHV MHV P L Figure 1.6:
A generic MHV diagram contributing to a one-loop MHV scattering ampli-tude.
Now we express L and L as in (1.7.1), L i ; α ˙ α = l iα ˜ l i ˙ α + z i η α ˜ η ˙ α , i = 1 , . (1.8.5)Using (1.8.5), we re-write the argument of the delta function as L − L + P L = l − l + P L ; z , (1.8.6)where we have defined P L ; z := P L − zη , (1.8.7) We thank the authors of [79] for allowing the re-production of Figure 17 of that paper. In our conventions all external momenta are outgoing. .8. LOOP DIAGRAMS FROM MHV VERTICES and z := z − z . (1.8.8)Notice that we use the same η for both the momenta L and L . Using (1.8.5), we canthen recast (1.8.4) as [37, 79] d M = dz z + iε dz z + iε (cid:20) d l l d l l δ (4) ( l − l + P L ; z ) (cid:21) , (1.8.9)where ε i := sgn( η l i ) ε = sgn( l i ) ε , i = 1 , η > z and z into an integration over z and z ′ := z + z and with a careful treatment of the integrals [79] we can integrate out z ′ . We also make the replacement d l l d l l δ (4) ( l − l + P L ; z ) → − d LIPS( l − , − l +1 ; P L ; z ) , (1.8.10)where d LIPS( l − , − l +1 ; P L ; z ) := d l δ (+) ( l ) d l δ ( − ) ( l ) δ (4) ( l − l + P L ; z ) (1.8.11)is the two-particle Lorentz-invariant phase space (LIPS) measure and we recall that δ ± ( l ) := θ ( ± l ) δ ( l ). Trading the final integral over z for an integration over P L ; z , theintegration measure finally becomes [37, 79] d M = 2 πi θ ( P L ; z ) dP L ; z P L ; z − P L − iε d LIPS( l ∓ , − l ± ; P L ; z ) . (1.8.12)This can now be immediately dimensionally regularised, which is accomplished bysimply replacing the four-dimensional LIPS measure by its continuation to D = 4 − ǫ dimensions: d D LIPS( l − , − l +1 ; P L ; z ) := d D l δ (+) ( l ) d D l δ ( − ) ( l ) δ ( D ) ( l − l + P L ; z ) . (1.8.13)Eq. (1.8.12) was one of the key results of [37]. It gives a decomposition of the original in-tegration measure into a D -dimensional phase space measure and a dispersive measure.According to Cutkosky’s cutting rules [210], the LIPS measure computes the disconti-nuity of a Feynman diagram across its branch cuts. Which discontinuity is evaluatedis determined by the argument of the delta function appearing in the LIPS measure;in (1.8.12) this is P L ; z (defined in (1.8.7)). Notice that P L ; z always contains a termproportional to the reference vector η , as prescribed by (1.8.7). Finally, discontinuitiesare integrated using the dispersive measure in (1.8.12), thereby reconstructing the fullamplitude. 62 .9. MHV AMPLITUDES IN N = 4 SUPER-YANG-MILLS
As a last remark, notice that in contradistinction with the cut-constructibility ap-proach of BDDK, here we sum over all the cuts – each of which is integrated with theappropriate dispersive measure. N = 4 super-Yang-Mills In this section we will briefly review the one-loop MHV N = 4 super-Yang-Mills am-plitudes and their derivation using MHV vertices. Many more details can be found in[37, 38]. It is known that, at one-loop, all amplitudes in massless gauge field theories can be writ-ten in terms of a certain basis of integral functions termed boxes, triangles and bubblesas well as possible rational contributions ( i.e. contributions which do not contain anybranch cuts) [38, 42]. These functions may involve some number of loop momenta inthe numerator of their integrand, in which case they are termed tensor boxes, trianglesor bubbles, though the basic scalar integrals remain the same and at 4-, 3- and 2-pointrespectively are the basic integrals arising at one-loop in scalar φ theory. P P P P K K K Q Q Figure 1.7:
Boxes, Triangles and Bubbles. Here P i , K i and Q i are generic momentarepresenting the contribution of one or more external particles. The different functionsdiscussed below (1-mass, 2-mass etc.) are all special cases of these .A box integral is characterised by having 4 vertices, a triangle integral by having 3vertices while a bubble has 2. The specific functions that occur are then characterisednot-only by possible powers of loop momenta arising in the numerator, but by thenumber of vertices with more than one external leg. If a vertex has only one externalleg it is called a massless vertex (as the external momentum is massless in the theorieswe are considering), whilst if it has more than one external leg it is termed a massivevertex as the external momentum emanating from it does not square to zero.63 .9. MHV AMPLITUDES IN N = 4 SUPER-YANG-MILLS
There are thus 4 generic types of box integrals: 4-mass boxes where all 4 vertices aremassive; 3-mass boxes; 2-mass ‘easy’ boxes where the massive vertices are opposite eachother; 2-mass ‘hard’ boxes where the massive vertices are adjacent and 1-mass boxes.At 4-point the only possible box integral is a massless box. Similarly one can have3-mass triangles, 2-mass triangles, 1-mass triangles, 2-mass bubbles and 1-mass bubbles(as well as massless triangles and massless bubbles at 3- and 2-point respectively). Explicit forms for all these functions can be found in Appendix I of [42]. N = 4 MHV one-loop amplitudes Concerning the above decomposition, maximally supersymmetric Yang-Mills theory isspecial in that its high degree of symmetry prescribes that its one-loop amplitudes onlycontain scalar box integral functions (up to finite order in the dimensional regularisationparameter ǫ ) [38, 42]. In particular, the MHV amplitudes only depend on the 2-masseasy (2me) box functions. The full one-loop n -point MHV amplitudes are proportionalto the tree-level MHV amplitudes and are given by [38] A N =4 MHV n ;1 = A tree n V gn , (1.9.1)where [38, 73] V gn = n X i =1 [ n ] − X r =1 (cid:16) − δ n − ,r (cid:17) F m en : r ; i . (1.9.2)The basic scalar box integral I is defined by I = − i (4 π ) − ǫ Z d − ǫ p (2 π ) − ǫ p ( p − P ) ( p − P − P ) ( p + P ) , (1.9.3)where dimensional regularisation is used to take care of infrared divergences. Therelevant integrals arising in (1.9.2) are related to I for different choices of the externalmomenta at each vertex P i ( i = 1 . . . I m e r ; i - see Figure 1.8 -and are given in terms of the F m en : r ; i by I m e r ; i = 2 F m en : r ; i t [ r ] i t [ n − r − i + r +1 − t [ r +1] i − t [ r +1] i , (1.9.4) Note that 1-mass and zero-mass bubbles are usually taken to vanish in dimensional regularizationwhich is interpreted as a cancellation of infrared and ultraviolet divergences [42, 211]. .9. MHV AMPLITUDES IN N = 4 SUPER-YANG-MILLS with t [ r ] i = ( k i + k i +1 + · · · + k i + r − ) , r > t [ r ] i = t [ n − r ] i , r < , (1.9.5)where the k i are the external momenta. The explicit form of F m en : r ; i is given by [38] F m en : r ; i = − ǫ h ( − t [ r +1] i − ) − ǫ + ( − t [ r +1] i ) − ǫ − ( − t [ r ] i ) − ǫ − ( − t [ n − r − i + r +1 ) − ǫ i + Li − t [ r ] i t [ r +1] i − ! + Li − t [ r ] i t [ r +1] i ! + Li − t [ n − r − i + r +1 t r +1 i − ! + Li − t n − r − i + r +1 t [ r +1] i ! − Li − t [ r ] i t [ n − r − i + r +1 t [ r +1] i − t [ r +1] i ! + 12 log t [ r +1] i − t [ r +1] i ! , (1.9.6)where Li is Euler’s dilogarithmLi ( z ) := − Z z dt log(1 − t ) t . (1.9.7) k i + r k i − k i k i + r − k i + r +1 k i − I m e r ; i Figure 1.8:
The 2-mass easy box function.
The one-loop MHV amplitudes were constructed in [38] from tree diagrams usingcuts. A given cut results in singularities in the relevant momentum channels and byconsidering all possible cuts one can construct the full set of possible singularities. Fromthis and unitarity one can deduce the amplitude as given in (1.9.1). More explicitly,consider a cut one-loop MHV diagram where the cut separates the external momenta k m & k m − , and k m & k m +1 (i.e. the set of external momenta k m , k m +1 , ..., k m lie to the left of the cut, and the set k m +1 , k m +2 , ..., k m − lie to the right, withmomenta labelled clockwise and outgoing). This separates the diagram into two MHVtree diagrams connected only by two momenta l and l flowing across the cut, with l = l + P L , (1.9.8)65 .9. MHV AMPLITUDES IN N = 4 SUPER-YANG-MILLS where P L = P m i = m k i is the sum of the external momenta on the left of the cut. Themomenta l , l are taken to be null. It is important to note that the resulting integralsare not equal to the corresponding Feynman integrals where l and l would be left offshell; however, the discontinuities in the channel under consideration are identical andthis gives enough information to determine the full amplitude uniquely.However, we will now sketch how to derive the MHV amplitudes using the methodof MHV diagrams. This is quite similar, but not identical to the approach of BDDKusing cut-constructibiliy, a brief review of which can be found in Appendix D. ! L1 MHV MHV L2 m2m1 m1-1m2+1 Figure 1.9:
A one-loop MHV diagram computed using MHV amplitudes as interactionvertices. This diagram has the momentum structure of the cut referred to at the end of §
1. To each MHV vertex we associate the appropriate form of the MHV amplitude forthat vertex, recalling that internal lines must be taken off-shell using the prescrip-tion described in § A = Z d L (2 π ) d L (2 π ) L + iε L + iε A L A R = Z d L L + iε d L L + iε iN L δ (4) ( L − L + P L ) D L iN R δ (4) ( L − L + P R ) D R = δ (4) ( P L + P R ) Z d L L + iε d L L + iε δ (4) ( L − L + P L ) iN L D L iN R D R . (1.9.9)Here L and R denote the left and right vertices respectively and we have P L := k m + k m +1 + . . . + k m and P R := k m +1 + k m +2 + . . . + k m − . N and D denote the functions of spinor variables describing the numerator and denom-66 .9. MHV AMPLITUDES IN N = 4 SUPER-YANG-MILLS inator of each MHV vertex respectively and we have included a factor of i (2 π ) with each vertex in keeping with Nair’s supersymmetric description [31, 37, 175].2. In [37] an approach using Nair super-vertices was used. Here we will just considerthe usual MHV vertices for ease of transition to the later chapters where wewill discuss MHV amplitudes in theories with less supersymmetry. In this casethere are two possibilities to consider. The first is where both external negative-helicity gluons lie on one MHV vertex and the second is where they lie on differentvertices (see e.g. Figure 2.4). After some manipulation (employing the Schoutenidentity stated in Appendix A), they can be shown to give the same contribution.Extracting an overall factor of A tree n := i (2 π ) δ (4) ( P L + P R ) h i j i Q nk =1 h k k + 1 i , (1.9.10)where i and j are the external negative-helicity gluons and regulating by promotingthe integrals to 4 − ǫ dimensions, (1.9.9) becomes A = i (2 π ) A tree n Z d M ˆ R , (1.9.11)where d M is the measure (1.8.12) derived previously andˆ R := h m − m ih l l ih m − l ih− l m i h m m + 1 ih l l ih m l ih− l m + 1 i . (1.9.12)3. Following equations (2.11)-(2.16) of [37] we may finally write ˆ R as a signed sum( i.e. two terms come with plus signs and two with minus signs - see Eq. (2.13) of[37]) of terms of the form R ( i, j ) := h i l ih i l i h j l ih j l i . (1.9.13)Once expressed in terms of momenta by multiplying top and bottom by appropri-ate anti-holomorphic spinor invariants, cancellations arise between different termsof the signed sum and we can schematically write ˆ R = P R → P R eff with[37, 79] R eff = 14 P L ; z ( i j ) − i P L ; z )( j P L ; z )( i l )( j l ) . (1.9.14) Be careful to note that in the following expression i and j refer to the different possibilities m , m , m + 1 and m −
1, and not to the negative-helicity particles of the overall amplitude which now onlyarise in the factor of A tree n . .9. MHV AMPLITUDES IN N = 4 SUPER-YANG-MILLS
The notation ( a b ) here is shorthand for ( a · b ). (1.9.11) then becomes A = i (2 π ) A tree n X Z d MR eff . (1.9.15)It is worth mentioning that the procedure of expressing P R → P R eff is a cleverway of cancelling the triangle and bubble contributions in R to leave only boxfunctions [37, 79] and is equivalent to the usual method of Passarino-Veltmanreduction of [212]. (1.9.15) is then the basic integral that we have to work withand we will consider the specific term R eff ( m , m ) for definiteness.4. Recall that the measure d M involves a dispersive part and an integral overLorentz-invariant phase space ( d LIPS). We wish to begin by performing the in-tegral over this phase space. For this we go to the centre of mass frame for P L ; z - P L ; z = P (1 ,~
0) - and parametrize l = P (1 , ~v ) and l = P ( − , ~v ) with ~v := (sin θ cos θ , sin θ sin θ , cos θ ). In 4 − ǫ dimensions, the LIPS measure(1.8.13) can be written in terms of the angles θ and θ as d − ǫ LIPS = π − ǫ (cid:0) − ǫ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) P (cid:12)(cid:12)(cid:12)(cid:12) − ǫ dθ dθ (sin θ ) − ǫ (sin θ ) − ǫ , (1.9.16)and the denominator of (1.9.14) as( m l )( m l ) = (cid:12)(cid:12)(cid:12)(cid:12) P (cid:12)(cid:12)(cid:12)(cid:12) m (1 − cos θ )( A + B sin θ cos θ + C cos θ ) , (1.9.17)where m := m (1 , , ,
1) and m := ( A, B, , C ) with A = B + C . Thenumerator of (1.9.14) does not involve l or l and we leave it as N ( P L ; z ) for now.We thus haveΛ Z d W Λ Z dθ dθ (sin θ ) − ǫ (sin θ ) − ǫ (1 − cos θ )( A + B sin θ cos θ + C cos θ ) , (1.9.18)where Λ := i (2 π ) π / − ǫ − ǫ Γ(1 / − ǫ ) A tree n , (1.9.19)Λ := N ( P L ; z ) m P ( P ) − ǫ , (1.9.20) d W := (2 πi ) θ ( P L ; z ) dP L ; z P L ; z − P L − iε . (1.9.21)The integral over θ and θ has been performed in [213] and we borrow the resultin a form from [214]. Converting A, B, C, m and P back into Lorentz-invariants See Appendix C for details. .9. MHV AMPLITUDES IN N = 4 SUPER-YANG-MILLS we obtain: 4 π Λ ǫ Z d W ( P L ; z ) − ǫ F (cid:0) , − ǫ, − ǫ ; a z P L ; z (cid:1) . (1.9.22)In Equations (1.9.16), (1.9.19) and (1.9.22) above, Γ is the gamma function and F the Gauss hypergeometric function. They can be defined byΓ( z ) := Z ∞ dt t z − e − t ; ℜ [ z ] > , (1.9.23) F ( a, b, c ; z ) := Γ( c )Γ( b )Γ( c − b ) Z dt t b − (1 − t ) − b + c − (1 − tz ) − a (1.9.24)where the second definition holds when ℜ [ c ] > ℜ [ b ] > | arg(1 − z ) | < π . a z is defined to be a z := ( i j ) /N ( P L ; z ) and so is equal to ( m m ) /N ( P L ; z ) in thiscase.5. Finally, we would like to evaluate this dispersive integral (1.9.22). In [37], this wasdone by combining different terms coming from different R eff to give a convergentintegral. In fact the different R eff that one must combine come not from different i and j in R eff ( i j ) as obtained from ˆ R = P R → P R eff , but from R eff withthe same i and j coming from different terms in the overall summation over allthe cuts of the one-loop integral mentioned at the end of § this section. This summation is just a summation over all cyclicpartitions of the external particles between the two MHV vertices, but at the levelof the integrals we have arrived at in (1.9.22) the summation over R eff ( i j ) withthe same values of i and j from different orderings of the external particles servesto re-construct the 2me box functions from their different cuts.The integrals are explicitly done by expanding the hypergeometric functions abovein an expansion in ǫ in terms of polylogarithms (generalisations of Li ) and thencombining different cuts of the same box function to give a convergent answer. Akey ingredient in all this is the knowledge that the final result will be independentof η . η has already been eliminated from the dispersive integration measure byconverting the integral over z and z ′ into an integral over P L ; z , so one may expectthat even before we evaluate this dispersive integral we should be able to picka particular value for η to simplify the calculation. However, in [37] a strongergauge invariance was proposed; namely that one may choose η separately foreach box function. This was checked numerically in [37] and independently (alsonumerically) in [209] and further evidence was provided in [79]. It means thatone can write N ( P L ; z ) = N ( P L ) if one chooses η = m or η = m in all four R eff ( m m ) which contribute to that particular box function. See also Appendix F for an analytic proof of the same statement for triangle functions. .9. MHV AMPLITUDES IN N = 4 SUPER-YANG-MILLS
The final result (up to finite order in ǫ ) given in Equation (5.16) of [37] is that thecontribution of a particular box function (say a generic box function such as thatin Figure 1.8, which would come from combining the four terms with m = k i + r and m = k i − ) is F m en : r ; i = − ǫ h ( − t [ r +1] i − ) − ǫ + ( − t [ r +1] i ) − ǫ − ( − t [ r ] i ) − ǫ − ( − t [ n − r − i + r +1 ) − ǫ i + Li (cid:16) − a t [ r ] i (cid:17) + Li (cid:16) − a t [ n − r − i + r +1 (cid:17) − Li (cid:16) − a t [ r +1] i − (cid:17) − Li (cid:16) − a t [ r +1] i (cid:17) , (1.9.25)where a = t [ r ] i + t [ n − r − i + r +1 − t [ r +1] i − − t [ r +1] i t [ r ] i t [ n − r − i + r +1 − t [ r +1] i − t [ r +1] i . (1.9.26)Equation (1.9.25) is in fact equal to (1.9.6) but is an alternative form whichwas discovered in [215] and independently derived in [37] and involves one lessdilogarithm and one less logarithm than (1.9.6). After summing over all partitionsof the external particles between the two MHV vertices we recover (1.9.1).The calculation outlined above is essentially what we will follow in Chapters 2 and3 for the N = 1 and N = 0 MHV amplitudes. For full details of the amplitudes in N = 4see [37] and for a short discussion on the overall ǫ -normalisation of the result obtainedthere compared with the one obtained originally in [38] see Appendix C.70 HAPTER 2MHV AMPLITUDES IN N = 1 SUPER-YANG-MILLS
In Chapter 1 we described some of the hidden simplicity of perturbative gauge theory -in particular in the context of maximally supersymmetric Yang-Mills - and saw how itmay be applied to simplifying the calculation of perturbative quantities such as scatter-ing amplitudes. The many techniques available to illuminate the perturbative structureincluded colour stripping, the use of a helicity scheme and supersymmetric decomposi-tions. A perturbative duality with a twistor string theory highlighted the unexpectedcompactness of the MHV amplitudes at tree-level and provided motivation for a newperturbative expansion of gauge theory - the CSW rules.The CSW rules have been shown to be valid even at loop level - despite the failure ofthe duality with twistor string theory - and the MHV amplitudes in N = 4 super-Yang-Mills were derived using these rules in [37] and shown to be identical to the originalderivation of [38] using 2-particle cuts. As a bonus, the CSW rules also gave rise to arepresentation of the 2-mass easy box functions that is simpler to that originally usedin [38]. However, at the time it was far from certain that these remarkable techniqueswould be applicable to other gauge theories. One might not have been surprised if suchresults only held for a theory with an extremely high amount of symmetry such as N = 4SYM.In [40, 41] a first step towards establishing the general validity of the MHV-vertexformalism was taken and it was shown independently by Bedford, Brandhuber, Spence& Travaglini and Quigley & Rosali that the CSW rules correctly calculate the MHVamplitudes in theories with less supersymmetry such as N = 1 and N = 2 super-Yang-Mills. In particular the MHV amplitudes for scattering of external gluons with an N = 1chiral multiplet running in the loop was calculated and it was found that the resultsexactly agree with those originally obtained by BDDK in [42]. This chapter follows [40]and shows how the N = 1 MHV amplitudes may be obtained from MHV vertices. N = 1 MHV amplitudes at one-loop
The expression for the MHV amplitudes at one-loop in N = 1 SYM was obtained forthe first time by BDDK in [42] using the cut-constructibility method. We will shortly71 .1. THE N = 1 MHV AMPLITUDES AT ONE-LOOP give their explicit result and then re-write it by introducing appropriate functions. Thisturns out to be useful when we compare the BDDK result to that which we will deriveby using MHV diagrams.In order to obtain the one-loop MHV amplitudes in N = 1 and N = 2 SYM itis sufficient to compute the contribution A N =1 , chiral n to the one-loop MHV amplitudescoming from a single N = 1 chiral multiplet. This was calculated in [42], and the resultturns out to be proportional to the Parke-Taylor MHV tree amplitude [170] A tree n := h i j i Q nk =1 h k k + 1 i , (2.1.1)as is also the case with the one-loop MHV amplitudes in N = 4 SYM. However, incontradistinction with that case, the remaining part of the N = 1 amplitudes dependsnon-trivially on the position of the negative-helicity gluons i and j . The result obtainedin [42] is: A N =1 , chiral n = A tree n · (cid:26) j − X m = i +1 i − X s = j +1 b i,jm,s B ( t [ s − m ] m +1 , t [ s − m ] m , t [ s − m − m +1 , t [ m − s − s +1 )+ j − X m = i +1 X a ∈D m c i,jm,a log( t [ a − m ] m +1 /t [ a − m +1] m ) t [ a − m ] m +1 − t [ a − m +1] m + i − X m = j +1 X a ∈C m c i,jm,a log( t [ m − a ] a +1 /t [ m − a − a +1 ) t [ m − a ] a +1 − t [ m − a − a +1 + c i,ji +1 ,i − t [2] i K ( t [2] i ) + c i,ji − ,i t [2] i − K ( t [2] i − )+ c i,jj +1 ,j − t [2] j K ( t [2] j ) + c i,jj − ,j t [2] j − K ( t [2] j − ) (cid:27) , (2.1.2)where t [ k ] i := ( p i + p i +1 + · · · + p i + k − ) for k ≥
0, and t [ k ] i = t [ n − k ] i for k <
0. The sumsin the second and third line of (2.1.2) cover the ranges C m and D m defined by C m = { i, i + 1 , . . . , j − } , m = j + 1 , { i, i + 1 , . . . , j − } , j + 2 ≤ m ≤ i − , { i + 1 , i + 2 , . . . , j − } , m = i − , (2.1.3)72 .1. THE N = 1 MHV AMPLITUDES AT ONE-LOOP and D m = { j, j + 1 , . . . , i − } , m = i + 1 , { j, j + 1 , . . . , i − } , i + 2 ≤ m ≤ j − , { j + 1 , j + 2 , . . . , i − } , m = j − . (2.1.4)The coefficients b i,jm,s and c i,jm,a are b i,jm,s := − + ( k/ i k/ j k/ m k/ s ) tr + ( k/ i k/ j k/ s k/ m )[( k i + k j ) ] [( k m + k s ) ] , (2.1.5) c i,jm,a := (cid:20) tr + ( k/ m k/ a +1 k/ j k/ i )( k a +1 + k m ) − tr + ( k/ m k/ a k/ j k/ i )( k a + k m ) (cid:21) tr + ( k/ i k/ j k/ m q/ m,a ) − tr + ( k/ i k/ j q/ m,a k/ m )[( k i + k j ) ] , (2.1.6)where q r,s := P sl = r k l . Notice that both coefficients b i,jm,s and c i,jm,a are symmetric underthe exchange of i and j . In the case of b this is evident; for c it is also manifest as c is expressed as the product of two antisymmetric quantities. The function B in thefirst line of (2.1.2) is the “finite” part of the easy two-mass (2me) scalar box function F ( s, t, P , Q ), with F ( s, t, P , Q ) := − ǫ h ( − s ) − ǫ + ( − t ) − ǫ − ( − P ) − ǫ − ( − Q ) − ǫ i + B ( s, t, P , Q ) . (2.1.7)As in [37] we have introduced the following convenient kinematical invariants: s := ( P + p ) , t := ( P + q ) , (2.1.8)where p and q are null momenta and P and Q are in general massive. We also havemomentum conservation in the form p + q + P + Q = 0. In [37] the following newexpression for B was found: B ( s, t, P , Q ) = Li (1 − aP ) + Li (1 − aQ ) − Li (1 − as ) − Li (1 − at ) , (2.1.9)where a = P + Q − s − tP Q − st . (2.1.10)The expression (2.1.9) contains one less dilogarithm and one less logarithm than the The kinematical invariant s = ( P + p ) should not be confused with the label s which is also usedto label an external leg (as in Figure 2.1 for example). The correct meaning will be clear from thecontext. .1. THE N = 1 MHV AMPLITUDES AT ONE-LOOP ms m+1 j-s-1s+1 i- m-1 QP Figure 2.1:
The box function F of (2.1.7) , whose finite part B , Eq. (2.1.9) , appears in the N = 1 amplitude (2.1.2) . The two external gluons with negative helicity are labelled by i and j . The legs labelled by s and m correspond to the null momenta p and q respectivelyin the notation of (2.1.9) . Moreover, the quantities t [ s − m ] m +1 , t [ s − m ] m , t [ s − m − m +1 , t [ m − s − s +1 appearing in the box function B in (2.1.19) correspond to the kinematical invariants t := ( Q + p ) , s := ( P + p ) , Q , P in the notation of (2.1.9) , with p + q + P + Q = 0 . traditional form used by BDDK, B ( s, t, P , Q ) = Li (cid:16) − P s (cid:17) + Li (cid:16) − P t (cid:17) + Li (cid:16) − Q s (cid:17) + Li (cid:16) − Q t (cid:17) − Li (cid:16) − P Q s t (cid:17) + 12 log (cid:16) st (cid:17) . (2.1.11)The agreement of (2.1.9) with (2.1.11) was discussed and proved in Section 5 of [37]. In Figure 2.1 we give a pictorial representation of the box function F defined in (2.1.7)(with the leg labels identified by s → p , m → q ). More precisely, this agreement holds only in certain kinematical regimes e.g. in the Euclideanregion where all kinematical invariants are negative. More care is needed when analytically continuingthe amplitude to the physical region. The usual prescription of replacing a kinematical invariant s by s + iε and continuing s from negative to positive values gives the correct result only for our form of thebox function (2.1.9), whereas (2.1.11) has to be amended by correction terms [216]. .1. THE N = 1 MHV AMPLITUDES AT ONE-LOOP QP m+ i- j-a+1 am-1 m+1 Figure 2.2:
A triangle function, corresponding to the first term T ǫ ( p m , q a +1 ,m − , q m +1 ,a ) in the second line of (2.1.19) . p , Q and P correspond to p m , q m +1 ,a and q a +1 ,m − inthe notation of Eq. (2.1.19) , where j ∈ Q , i ∈ P . In particular, Q → t [ a − m ] m +1 and P → t [ a − m +1] m . Finally, infrared divergences are contained in the bubble functions K ( t ), defined by K ( t ) := ( − t ) − ǫ ǫ (1 − ǫ ) . (2.1.12)We notice that in order to re-express (2.1.2) in a simpler form, it is useful to introducethe triangle function [73] T ( p, P, Q ) := log( Q /P ) Q − P , (2.1.13)with p + P + Q = 0. A diagrammatic representation of this function is given in Figure 2.2(with m + → p ). We also find it useful to introduce an ǫ -dependent triangle function, T ǫ ( p, P, Q ) := 1 ǫ ( − P ) − ǫ − ( − Q ) − ǫ Q − P . (2.1.14)As long as P and Q are non-vanishing, one haslim ǫ → T ǫ ( p, P, Q ) = T ( p, P, Q ) , P = 0 , Q = 0 . (2.1.15) The function T ǫ ( p, P, Q ) defined in (2.1.14) arises naturally in the twistor-inspired approach whichwill be developed in § .1. THE N = 1 MHV AMPLITUDES AT ONE-LOOP QP m+ j- i-a+1 am-1 m+1 Figure 2.3:
This triangle function corresponds to the second term in the second lineof (2.1.19) – where i and j are swapped. As in Figure 2.2, p , Q and P correspond to p m , q m +1 ,a and q a +1 ,m − in the notation of Eq. (2.1.19) , where now i ∈ Q , j ∈ P . Inparticular, Q → t [ m − a ] a +1 and P → t [ m − a − a +1 . If either of the invariants vanishes, one has a different limit. For example, if Q = 0 onehas T ǫ ( p, P, Q ) | Q =0 −→ − ǫ ( − P ) − ǫ P , ǫ → . (2.1.16)We will call these cases “degenerate triangles”.The usefulness of the previous remark stems from the fact that precisely the quantity(1 /ǫ ) · (cid:2) ( − P ) − ǫ /P (cid:3) appears in the last line of (2.1.2) – the bubble contributions.Therefore, these can be equivalently obtained as degenerate triangles i.e. triangles whereone of the massive legs becomes massless.Specifically, we notice that the four degenerate triangles (bubbles) in the last line of(2.1.2) can be precisely obtained by including the “missing” index assignments in D m and C m : ( m = i + 1 , a = i − , ( m = j − , a = j ) for D m , (2.1.17)which correspond to two degenerate triangles, and( m = j + 1 , a = j − , ( m = i − , a = i ) for C m , (2.1.18)corresponding to two more degenerate triangles.76 .2. MHV ONE-LOOP AMPLITUDES IN N = 1 SYM FROM MHV VERTICES
In conclusion, the previous remarks allow us to rewrite (2.1.2) in a more compactform as follows: A N =1 , chiral n = A tree n · (cid:26) j − X m = i +1 i − X s = j +1 b i,jm,s B ( t [ s − m ] m +1 , t [ s − m ] m , t [ s − m − m +1 , t [ m − s − s +1 ) (2.1.19)+ 11 − ǫ (cid:20) j − X m = i +1 i − X a = j c i,jm,a T ǫ ( p m , q a +1 ,m − , q m +1 ,a ) + ( i ↔ j ) (cid:21)(cid:27) . In this expression it is understood that we only keep terms that survive in the limit ǫ →
0. This means that the factor 1 / (1 − ǫ ) can be replaced by 1 whenever theterm in the sum is finite, i.e. whenever the triangle is non-degenerate. However, inthe case of degenerate triangles, which contain infrared-divergent terms, we have toexpand this factor to linear order in ǫ . In the notation of (2.1.19), q m +1 ,a = t [ a − m ] m +1 and q a +1 ,m − = t [ a − m +1] m ; in Figure 2.2, these invariants correspond to Q and P respectively, where j ∈ Q , i ∈ P . In the sum with i ↔ j , one would have q m +1 ,a = t [ m − a ] a +1 , q a +1 ,m − = t [ m − a − a +1 , corresponding respectively to Q and P in Figure 2.3, with i ∈ Q , j ∈ P . It is the expression (2.1.19) for the N = 1 chiral multiplet amplitude which wewill derive using MHV diagrams. N = 1 SYM from MHV vertices In § N = 4 SYM asreviewed in § N = 1 SYM, in particular to the infinite sequence of MHV one-loop amplitudes, whichwere obtained using the cut-constructibility approach [42], and whose twistor spacepicture has been analysed in [73].Similarly to the N = 4 case, the one-loop amplitude has an overall factor proportionalto the MHV tree-level amplitude, but, as opposed to the N = 4 case, the remaining one-loop factor depends non-trivially on the positions i and j of the two external negative-helicity gluons. This is due to the fact that a different set of fields is allowed to propagatein the loop.The MHV diagrams contributing to MHV one-loop amplitudes consist of two MHVvertices connected by two off-shell scalar propagators. If both negative-helicity glu-ons are on one MHV vertex, only gluons of a particular helicity can propagate in theloop. This is independent of the number of supersymmetries. On the other hand, fordiagrams with one negative-helicity gluon on one MHV vertex and the other negative-77 .2. MHV ONE-LOOP AMPLITUDES IN N = 1 SYM FROM MHV VERTICES helicity gluon on the other MHV vertex, all components of the supersymmetric mul-tiplet propagate in the loop. In the case of N = 4 SYM this corresponds to helicities h = − , − / , , / , , , , ,
1, respectively; for the N = 1 vectormultiplet the multiplicities are 1 , , , ,
1. Hence, we can obtain the N = 1 (vector)amplitude by simply taking the N = 4 amplitude and subtracting three times the con-tribution of an N = 1 chiral multiplet, which has multiplicities 0 , , , , This supersymmetric decomposition of general one-loop amplitudes is useful as itsplits the calculation into pieces of increasing difficulty, and allows one to reduce a one-loop diagram with gluons circulating in the loop to a combination of an N = 4 vectoramplitude, an N = 1 chiral amplitude and finally a non-supersymmetric amplitude witha scalar field running in the loop as in Equation (1.3.3).In our case, the supersymmetric decomposition takes the form A N =1 , vector n = A N =4 n − A N =1 , chiral n , (2.2.1)where n denotes the number of external lines. Since the N = 4 contribution is known,one only needs to determine A N =1 , chiral n using MHV diagrams. To be more precise, weare solely addressing the computation of the planar part of the amplitudes. However,this is sufficient since at one-loop level the non-planar partial amplitudes are obtainedas appropriate sums of permutations of the planar partial amplitudes [38], as discussedin § m1-1m2+1m2m1 ! L1 MHV MHV L2 i- j- Figure 2.4:
A one-loop MHV diagram, computed in (2.2.4) using MHV amplitudes asinteraction vertices, with the CSW off-shell prescription. The two external gluons withnegative helicity are labelled by i and j . We can also obtain the N = 2 amplitude in a completely similar way. .2. MHV ONE-LOOP AMPLITUDES IN N = 1 SYM FROM MHV VERTICES
Our task therefore consists of: Evaluating the class of diagrams where we allow all the helicity states of a chiralmultiplet, h ∈ {− / , , , / } , (2.2.2)to run in the loop. We depict the prototype of such diagrams in Figure 2.4. Summing over all diagrams such that each of the two MHV vertices always hasone external gluon of negative helicity. Assigning i − to the left and j − to theright, the summation range of m and m is determined to be: j + 1 ≤ m ≤ i , i ≤ m ≤ j − . (2.2.3)Hence we get A N =1 , chiral n = X m ,m ,h Z d M A ( − l , m , . . . , i − , . . . , m , l ) · A ( − l , m + 1 , . . . , j − , . . . , m − , l ) , (2.2.4)where the summation ranges of h , m and m are given in (2.2.2), (2.2.3). Notice that,in order to compute the loop amplitude (2.2.4), we make use of the integration measure d M given in (1.8.12).After some spinor algebra and after performing the sum over the helicities h , theintegrand of (2.2.4) becomes − i A tree n · h m ( m +1) i h ( m − m i h i l i h j l ih i l ih j l ih i j i h m l i h ( m − l i h m l ih ( m +1) l i . (2.2.5)The focus of the remainder of this section will be to evaluate the integral in (2.2.4)explicitly. Since − i A tree n factors out completely, we will now drop it and only reinstateit at the very end of the calculation.The integrand (without this factor) can be rewritten in terms of dot products ofmomentum vectors, I = N ( i · j ) ( m · l ) (( m − · l ) ( m · l ) (( m +1) · l ) , (2.2.6)with N = tr + ( l/ k/ m − k/ m l/ k/ j k/ i ) tr + ( l/ k/ m k/ m +1 l/ k/ j k/ i ) . (2.2.7)79 .2. MHV ONE-LOOP AMPLITUDES IN N = 1 SYM FROM MHV VERTICES N is a product of Dirac traces, where the tr + symbol indicates that the projector(1 + γ ) / + ( l/ k/ m − k/ m l/ k/ j k/ i ) = 2( m · l )tr + ( k/ i k/ j k/ m − l/ ) − m − · l )tr + ( k/ i k/ j k/ m l/ ) , (2.2.8)wheretr + ( k/ a k/ b k/ c k/ d ) = 2 (cid:2) ( a · b )( c · d ) − ( a · c )( b · d ) + ( a · d )( b · c ) (cid:3) − iε ( a, b, c, d ) . (2.2.9)The second factor in (2.2.7) takes a similar form. Consequently, the integrand becomesa sum of four terms, one of which istr + ( k/ i k/ j k/ m l/ ) tr + ( k/ i k/ j k/ m l/ )( i · j ) ( m · l ) ( m · l ) . (2.2.10)The other three terms are obtained by replacing m with m − m with m + 1in (2.2.10) and come with alternating signs. Note that the original expression (2.2.5) issymmetric in i , and j , although when we make use of the decomposition (2.2.10) thissymmetry is no longer manifest. We will symmetrize over i and j at the end of thecalculation in order to make this exchange symmetry manifest in the final expression.In the next step we have to perform the phase space integration, which is equivalentto the calculation of a unitarity cut with momentum P L ; z = P m l = m k l − zη flowingthrough the cut. Note that, as explained in § η . The term ( l · m )( l · m ) in the denominatorof (2.2.10) corresponds to two propagators, hence the denominator by itself correspondsto a cut box diagram. However, the numerator of (2.2.10) depends non-trivially on theloop momentum, so that in fact (2.2.10) corresponds to a tensor box diagram, notsimply a scalar box diagram. Using the Passarino-Veltman method [212], we can reducethe expression (2.2.10), integrated with the LIPS measure, to a sum of cuts of scalarbox diagrams, scalar and vector triangle diagrams, and scalar bubble diagrams. Thisprocedure is somewhat technical and details are collected in Appendix E. Luckily, thefinal result takes a less intimidating form than the intermediate expressions. We willnow present the result of these calculations after the LIPS integration. We first observe that loop integrations are performed in 4 − ǫ dimensions. It turns outthat singular 1 /ǫ terms appearing at intermediate steps of the phase space integration80 .2. MHV ONE-LOOP AMPLITUDES IN N = 1 SYM FROM MHV VERTICES cancel out completely. Notice that this does not mean that the final result will be freeof infrared divergences. In fact the dispersion integral can and does give rise to 1 /ǫ divergent terms but there cannot be any 1 /ǫ terms, as expected for the contributionof a chiral multiplet [42]. The 1 /ǫ divergences in the scattering amplitude correspondto the bubble contributions in (2.1.2), or degenerate triangles contributions in (2.1.19),as explained in § C = C ( m − , m ) − C ( m , m ) + C ( m , m + 1) − C ( m − , m + 1) , (2.2.11)with C ( m , m ) = 2 π − ǫ ( P L ; z ) − ǫ ( i · j ) ( m · m ) (cid:20) T ( m , m , P L ; z )( m · P L ; z ) + T ( m , m , P L ; z )( m · P L ; z ) (cid:21) − π T ( m , m , m )( i · j ) ( m · m ) ( P L ; z ) − ǫ log (cid:0) − a z P L ; z (cid:1) , (2.2.12)where T ( m , m , P ) := tr + ( k/ i k/ j k/ m P/ ) tr + ( k/ i k/ j k/ m k/ m ) ,a z := m · m N ( P L ; z ) , (2.2.13)and N ( P ) := ( m · m ) P − m · P )( m · P ) . (2.2.14)A closer inspection of (2.2.12) reveals that the first line of that expression correspondsto two cuts of scalar triangle integrals, up to an ǫ -dependent factor and the explicit z -dependence of the two numerators. The second line is a term familiar from [37],corresponding to the P L ; z -cut of the finite part B of a scalar box function, defined in(2.1.9) (see also (2.1.7)). The full result for the one-loop MHV amplitudes is obtainedby summing over all possible MHV diagrams, as specified in (2.2.4) and (2.2.2), (2.2.3). We begin our analysis by focusing on the box function contributions in (2.2.12), andnotice the following important facts: By taking into account the four terms in (2.2.11) and summing over Feynmandiagrams, we see each fixed finite box function B appears in exactly four phase In (2.2.12) we omit an overall, finite numerical factor that depends on ǫ . This factor, which can beread off from (E.2.12), is irrelevant for our discussion. .2. MHV ONE-LOOP AMPLITUDES IN N = 1 SYM FROM MHV VERTICES space integrals, one for each of its possible cuts, in complete similarity with [37]. Itwas shown in Section 5 of that paper that the corresponding dispersion integrationover z will then yield the finite B part of the scalar box functions F . It was alsonoted in [37] that one can make a particular gauge choice for η such that the z -dependence in N disappears. This happens when η is chosen to be equal to oneof the massless external legs of the box function. The question of gauge invarianceis further discussed in Appendix F. The coefficient multiplying the finite box function is precisely equal to b i,jm ,m defined in (2.1.5). Finally, the functions B generated by summing over all MHV Feynman diagramswith the range dictated by (2.2.3) are precisely those included in the double sumfor the finite box functions in the first line of (2.1.2) (or (2.1.19)) upon identifying m and m with s and m . To be precise, (2.2.3) includes the case where the indices s and/or m (in the notation of (2.1.2) and (2.1.19)) are equal to either i or j ; butfor any of these choices, it is easy to check that the corresponding coefficient b i,jm,s vanishes.This settles the agreement between the result of our computation with MHV verticesand (2.1.19) for the part corresponding to the box functions. Next we have to collectthe cuts contributing to particular triangles, and show that the z -integration reproducesthe expected triangle functions from (2.1.19), each with the correct coefficient.To this end, we notice that for each fixed triangle function T ( p, P, Q ), exactly fourphase space integrals appear, two for each of the two possible cuts of the function.Moreover, a gauge invariance similar to that of the box functions also exists for trianglecuts (see Appendix F), so that we can choose η in a way that the T numerators in(2.2.12) become independent of z . A particularly convenient choice is η = k i , sinceit can be kept fixed for all possible cuts. Choosing this gauge, we see that a sum, T , of terms proportional to cut-triangles is generated from (2.2.11) (up to a commonnormalisation): T := T A + T B + T C + T D , (2.2.15)where T A := (cid:20) S ( i, j, m , m )( m · m ) − S ( i, j, m − , m )(( m − · m ) (cid:21) S ( i, j, m , P L ) ∆ A , (2.2.16) T B := (cid:20) S ( i, j, m , m )( m · m ) − S ( i, j, m + 1 , m )(( m + 1) · m ) (cid:21) S ( i, j, m , P L ) ∆ B ,T C := (cid:20) S ( i, j, m + 1 , m − m + 1) · ( m − − S ( i, j, m , m − m · ( m − (cid:21) S ( i, j, m − , P L ) ∆ C , .2. MHV ONE-LOOP AMPLITUDES IN N = 1 SYM FROM MHV VERTICES T D := (cid:20) S ( i, j, m − , m + 1)(( m − · ( m + 1)) − S ( i, j, m , m + 1)( m · ( m + 1)) (cid:21) S ( i, j, m + 1 , P L ) ∆ D . Here we have defined S ( a, b, c, d ) = tr + ( k/ a k/ b k/ c k/ d ) , (2.2.17)and ∆ I , I = A, . . . , D , are the following cut-triangles, all in the P L ; z -cut :∆ A := 1( m · P L ; z ) = Q -cut of − T (cid:0) m , P L ; z − m , − P L ; z (cid:1) , (2.2.18)∆ B := 1( m · P L ; z ) = P -cut of − T (cid:0) m , − P L ; z , P L ; z − m (cid:1) , ∆ C := 1(( m − · P L ; z ) = Q -cut of T (cid:0) m − , − P L ; z − ( m − , P L ; z (cid:1) , ∆ D := 1(( m + 1) · P L ; z ) = P -cut of T (cid:0) m + 1 , P L ; z , − P L ; z − ( m + 1) (cid:1) . P Q p Figure 2.5:
A triangle function with massive legs labelled by P and Q , and massless leg p . This function is reconstructed by summing two dispersion integrals, corresponding tothe P z - and Q z -cut. Next, we notice that the prefactors multiplying ∆ B , ∆ C become the same, up to aminus sign, upon shifting m − → m in the second prefactor; and so do the prefactorsof ∆ A , ∆ D upon shifting m → m + 1. Doing this, − ∆ B and the shifted ∆ C becomethe two cuts of the same triangle function T ( m , − P L ; z , P L ; z − m ), and similarly, − ∆ A and ∆ D give the two cuts of the function T ( m , P L ; z − m , − P L ; z ). Furthermore, in83 .2. MHV ONE-LOOP AMPLITUDES IN N = 1 SYM FROM MHV VERTICES P Q p Figure 2.6:
A degenerate triangle function. Here the leg labelled by P is still massive,but that labelled by Q becomes massless. This function is also reconstructed by summingover two dispersion integrals, corresponding to the P z - and Q z -cut. Appendix F we will show that summing the two dispersion integrals of the two differentcuts of a triangle indeed generates the triangle function – in fact this procedure givesa novel way of obtaining the triangle functions. Specifically, the result derived inAppendix F is Z dzz (cid:20) ( P z ) − ǫ ( P z p ) + ( Q z ) − ǫ ( Q z p ) (cid:21) = 2 (cid:2) πǫ csc( πǫ ) (cid:3) T ǫ ( p, P, Q ) , (2.2.19)where the ǫ -dependent triangle function T ǫ ( p, P, Q ) (with p + P + Q = 0) was introducedin (2.1.14) and gives, as ǫ →
0, the triangle function (2.1.13) (as well as the bubbles wheneither P or Q vanish). The result (2.2.19) holds for a generic choice of the referencevector η , see (F.1.6)-(F.1.11). We give a pictorial representation of the non-degenerateand degenerate triangle functions in Figures 2.5 and 2.6, respectively. A remark is in order here. In our procedure the momentum appearing in each of the possible cutsis always shifted by an amount proportional to zη ; the triangle is then reproduced by performing theappropriate dispersion integrals. Because of the above mentioned shift, we produce a non-vanishing cut(with shifted momentum) even when the cut includes only one external (massless) leg, say ˜ k , as themomentum flowing in the cut is effectively ˜ k z = ˜ k − zη , so that ˜ k z = 0. .2. MHV ONE-LOOP AMPLITUDES IN N = 1 SYM FROM MHV VERTICES
At this point, it should be noticed that for a gauge choice different from η = k i adopted so far, the numerators T in (2.2.12) do acquire an η -dependence. This gaugedependence should not be present in the final result for the scattering amplitude. Indeed,it is easy to check that, thanks to (F.1.6), the coefficient of the η -dependent termsactually vanishes.Using (2.2.15)-(2.2.19) and collecting terms as specified above, we see that thegeneric term produced by this procedure takes the form (cid:20) S ( i, j, a, p m )( k a · p m ) − S ( i, j, a + 1 , p m )( k a +1 · p m ) (cid:21) S ( i, j, p m , Q ) T ( p m , P, Q ) , (2.2.20)with P = q a +1 ,m − and Q = q m +1 ,a .Finally, we implement the symmetrization of the indices i , j , as explained earlier,and convert (2.2.20) into c i,jm,a T ( p m , P, Q ) , (2.2.21)where the coefficient c i,jm,a is c i,jm,a := 12 (cid:20) S ( i, j, a + 1 , p m )( k a +1 · p m ) − S ( i, j, a, p m )( k a · p m ) (cid:21) S ( i, j, p m , q m,a ) − S ( i, j, q m,a , p m )[( k i + k j ) ] , (2.2.22)which coincides with the definition of c i,jm,a given in (2.1.6). Lastly, it is easy to seethat in summing over the range given by (2.2.3), we produce exactly all the trianglefunctions appearing in the second line of (2.1.19). It is also important to notice thatthe bubbles, which appear in the last line of (2.1.2), are actually obtained as particularcases of triangle functions where one of the massive legs becomes massless, as observedat the end of § i.e. finite box contributionsand triangle contributions - which include the bubbles as special (degenerate) cases -are precisely reproduced in our diagrammatic approach. In writing (2.2.22), we make also use of the fact that S ( i, j, q m − ,a , p m ) = S ( i, j, q m,a , p m ). HAPTER 3NON-SUPERSYMMETRIC MHVAMPLITUDES
Having seen that the CSW rules can be applied at loop level in supersymmetric gaugetheories, the obvious question is whether the same also holds in non-supersymmetricgauge theories. To this end the one-loop MHV amplitudes in pure Yang-Mills with ascalar running in the loop were computed in [43]. This is the last contribution to theMHV amplitudes for gluon scattering in QCD in the supersymmetric decomposition ofEq. (1.3.3) and has only been computed previously in certain special cases in [42, 44].In this chapter we follow [43] and apply the CSW rules to this scalar amplitude inthe general case of n -gluon MHV scattering where the two negative-helicity gluons sit atarbitrary positions. We find that the results agree perfectly with those already obtainedin [42, 44] and we go on to present the general result for the cut-constructible part ofthe one-loop MHV amplitudes in pure Yang-Mills. It turns out that the CSW rulesonly compute this cut-constructible part and the rational terms (which do not containcuts) are not found. This is discussed in [43] and § can and have ,however, been recently computed using an on-shell unitarity bootstrap [45] which thuscompletely determines the one-loop MHV n -gluon amplitudes in QCD. In complete similarity with the N = 4 and N = 1 cases - see Chapters 1 & 2 and e.g. [37, 40] - we can immediately write down the expression for the scalar amplitudein terms of MHV vertices as A scalar n = X m ,m , ± Z d M A ( − l ∓ , m , . . . , i − , . . . , m , l ± ) · A ( − l ∓ , m + 1 , . . . , j − , . . . , m − , l ± ) , (3.1.1)where the ranges of summation of m and m are j + 1 ≤ m ≤ i , i ≤ m ≤ j − . (3.1.2)86 .1. THE SCALAR AMPLITUDE A typical MHV diagram contributing to A scalar n , for fixed m and m , is depicted inFigure 3.1. The off-shell vertices A in (3.1.1) correspond to having complex scalars m1-1m2+1m2m1 ! L1 MHV MHV L2 i- j- +/- +/--/+ -/+ Figure 3.1:
A one-loop MHV diagram with a complex scalar running in the loop, com-puted in Eq. (3.1.1) . We have indicated the possible helicity assignments for the scalarparticle. running in the loop. It follows that there are two possible helicity assignments for thescalar particles in the loop which have to be summed over. These two possibilities aredenoted by ± in (3.1.1) and in the internal lines in Figure 3.1. It turns out that eachof them gives rise to the same integrand for (3.1.1): − i A tree n · h m m +1 i h m − m i h i l i h j l i h i l i h j l i h i j i h m l i h m − l i h m l i h m +1 l i h l l i . (3.1.3)A crucial ingredient in (3.1.1) is (as before in Chapters 1 & 2) the integration measure d M . This measure was constructed in [37, 79] using the decomposition L := l + zη fora non-null four-vector L in terms of a null vector l and a real parameter z as reviewedin § § d M = 2 πi θ ( P L ; z ) dP L ; z P L ; z − P L − iε d − ǫ LIPS( l ∓ , − l ± ; P L ; z ) . (3.1.4)In order to calculate (3.1.1), we will first integrate the expression (3.1.3) over theLorentz-invariant phase space (appropriately regularised to 4 − ǫ dimensions), andthen perform the dispersion integral.For the sake of clarity, we will separate the analysis into two parts. Firstly, we will For scalar fields, the “helicity” simply distinguishes particles from antiparticles (see, for example,[154]). .2. THE SCATTERING AMPLITUDE IN THE ADJACENT CASE present the (simpler) calculation of the amplitude in the case where the two negative-helicity gluons are adjacent. This particular amplitude has already been computed byBern, Dixon, Dunbar and Kosower in [42] using the cut-constructibility approach; theresult we will derive here will be in precise agreement with the result in that approach.Then, in § The adjacent case corresponds to choosing i = m , j = m − m . We will also set i = 2, j = 1 for the sake of definiteness, and m = m .After conversion into traces, the integrand of (3.1.1) takes on the form:tr + ( k/ k/ P/ L ; z l/ ) tr + ( k/ k/ l/ P/ L ; z )2 ( k · k ) ( l · l ) ( tr + ( k/ k/ k/ m +1 l/ )( l · m +1) − tr + ( k/ k/ k/ m l/ )( l · m ) ) , (3.2.1)where we note that ( l · l ) = − P L ; z / I µνρ ( m, P L ; z ) = Z d LIPS( l , − l ; P L ; z ) l µ l ν l ρ ( l · m ) . (3.2.2)This calculation is performed in Appendix G. The result of this procedure gives thefollowing term at O ( ǫ ), which we will later integrate with the dispersive measure:˜ A scalar n = π − P L ; z ) − ǫ [tr + ( k/ k/ k/ m P/ L ; z )] ( k · k ) ( tr + ( k/ k/ P/ L ; z k/ m )( m · P L ; z ) + 2( k · k )( m · P L ; z ) ) − ( m ↔ m + 1) , (3.2.3)and we have dropped a factor of 4 π ˆ λ A tree on the right hand side of (3.2.3), where ˆ λ isdefined in (G.1.11). We can reinstate this factor at the end of the calculation. We alsonotice that (3.2.3) is a finite expression, i.e. it is free of infrared poles. An important remark is in order here. On general grounds, the result of a phase spaceintegral in, say, the P -channel, is of the form I ( ǫ ) = ( − P ) − ǫ · f ( ǫ ) , (3.2.4)88 .2. THE SCATTERING AMPLITUDE IN THE ADJACENT CASE where f ( ǫ ) = f − ǫ + f + f ǫ + · · · , (3.2.5)and f i are rational coefficients. In the case at hand, infrared poles generated by thephase space integrals cancel completely, so that we can in practice replace (3.2.5) by f ( ǫ ) → f + f ǫ + · · · . The amplitude A is then obtained by performing a dispersionintegral, which converts (3.2.4) into an expression of the form A ( ǫ ) = ( − P ) − ǫ ǫ · g ( ǫ ) = g ǫ − g log( − P ) + g + O ( ǫ ) , (3.2.6)where g ( ǫ ) = g + g ǫ + · · · , and the coefficients g i are rational functions, i.e. they are freeof cuts. Importantly, errors can be generated in the evaluation of phase space integralsif one contracts (4 − ǫ )-dimensional vectors with ordinary four-vectors. This does notaffect the evaluation of the coefficient g := g ( ǫ = 0), and hence the part of the amplitudecontaining cuts is reliably computed; but the coefficients g i for i ≥
1, in particular g , are in general affected. This implies that rational contributions to the scatteringamplitude cannot be detected [42] in this construction. A notable exception to this isprovided by the phase space integrals which appear in supersymmetric theories. Theseare “four-dimensional cut-constructible” [42], in the sense that the rational parts areunambiguously linked to the discontinuities across cuts, and can therefore be uniquelydetermined. This occurs, for example, in the calculation of the N = 4 MHV amplitudesat one-loop performed in [37] and reviewed in § N = 1 MHV amplitudes atone-loop in Chapter 2. In the present case, however, the relevant phase space integralsviolate the cut-constructibility criteria given in [42] , since we encounter tensor triangleswith up to three loop momenta in the numerator. Hence, we will be able to computethe part of the amplitude containing cuts, but not the rational terms. In practicethis means that we will compute all phase space integrals up to O ( ǫ ) and discard O ( ǫ )contributions, which would generate rational terms that cannot be determined correctly. After this digression, we now move on to the dispersive integration. In the center ofmass frame, where P L ; z := P L ; z (1 ,~ P L ; z in (3.2.3) cancels out,as there are equal powers of P L ; z in the numerator as in the denominator of any term.As a consequence, the dependence on the arbitrary reference vector η disappears (see[41] for the application of this argument to the N = 1 case). We are thus left with For more details about cut-constructibility, see the detailed analysis in Sections 3-5 of [42] andAppendix D of this thesis for a brief review. An example of an integral violating the power-counting criterion of [42] is provided by (G.1.3). .2. THE SCATTERING AMPLITUDE IN THE ADJACENT CASE dispersion integrals of the form I ( P L ) := Z ds ′ s ′ − P L ( s ′ ) − ǫ = 1 ǫ [ πǫ csc( πǫ )] ( − P L ) − ǫ . (3.2.7)Taking this into account, the dispersion integral of (3.2.3) then gives˜ A scalar n = (cid:2) πǫ csc( πǫ ) (cid:3) π − P L ) − ǫ ǫ [tr + ( k/ k/ k/ m P/ L )] ( k · k ) · " tr + ( k/ k/ P/ L k/ m )( m · P L ) + 2( k · k )( m · P L ) − ( m ↔ m + 1) . (3.2.8)The momentum flow can be conveniently represented as in Figure 3.2, where we define P := q ,m − , Q := q m +1 , = − q ,m , (3.2.9)and q p ,p := P p l = p k l . We also have P L := q ,m = − Q .Now we wish to combine the terms written explicitly in (3.2.8) with those that ariseunder m ↔ m + 1. Since (3.2.8) is summed over m , we simply shift m + 1 → m inthese latter terms. Let us now focus our attention on the second term in (3.2.3) (similarmanipulations will be applied to the first term). Writing the m ↔ m + 1 term explicitly,we obtain a contribution proportional to( − P L ) − ǫ (cid:20) [tr + ( k/ k/ k/ m P/ L )] ( m · P L ) − [tr + ( k/ k/ k/ m +1 P/ L )] (( m + 1) · P L ) (cid:21) . (3.2.10)By shifting m + 1 → m in the second term of (3.2.10), we change its P L so that P L → q ,m − = P (whereas, in the non-shifted term, P L = − Q ). The expression (3.2.10)then reads [tr + ( k/ k/ k/ m Q/ )] ( m · Q ) h ( − Q ) − ǫ − ( − P ) − ǫ i , (3.2.11)where we used tr + ( k/ k/ k/ m Q/ ) = − tr + ( k/ k/ k/ m P/ ) and Q · m = − P · m . Notice also that m · Q = − (1 / Q − P ).Next we re-instate the antisymmetry of the amplitudes under the exchange of theindices 1 ↔ (cid:2) tr + ( k/ k/ k/ m Q/ ) (cid:3) −→ h(cid:0) tr + ( k/ k/ k/ m Q/ ) (cid:1) − (cid:0) tr + ( k/ k/ Q/ k/ m ) (cid:1) i (3.2.12)= 2( k · k )( m · Q ) h tr + ( k/ k/ k/ m Q/ ) − tr + ( k/ k/ Q/ k/ m ) i . Following similar steps for the first term in (3.2.8), we arrive at the following expression90 .2. THE SCATTERING AMPLITUDE IN THE ADJACENT CASE QP m i-j- m-1 m+1 Figure 3.2:
A triangle function contributing to the amplitude in the case of adjacentnegative-helicity gluons. Here we have defined P := q j,m − , Q := q m +1 ,i = − q j,m (inthe text we set i = 1 , j = 2 for definiteness). for the amplitude before taking the ǫ → A ǫ = A ,ǫ + A ,ǫ , (3.2.13)where A ,ǫ = − A tree t [2]1 · π · h tr + ( k/ k/ k/ m q/ m, ) − tr + ( k/ k/ q/ m, k/ m ) i T ǫ ( m, q ,m − , q ,m ) , A ,ǫ = − A tree ( t [2]1 ) · π · h(cid:2) tr + ( k/ k/ k/ m q/ m, ) (cid:3) tr + ( k/ k/ q/ m, k/ m ) −− tr + ( k/ k/ k/ m q/ m, ) (cid:2) tr + ( k/ k/ q/ m, k/ m ) (cid:3) i T (3) ǫ ( m, q ,m − , q ,m ) , (3.2.14)and t [2]1 follows from the definition of equation (1.9.5). In order to write (3.2.14) in acompact from, we have introduced ǫ -dependent triangle functions [40] as in the previouschapter ( c.f. Eq. (2.1.14)) T ( r ) ǫ ( p, P, Q ) := 1 ǫ ( − P ) − ǫ − ( − Q ) − ǫ ( Q − P ) r , (3.2.15)where p + P + Q = 0, and r is a positive integer. For r = 1 we will omit the superscript (1) in T (1) . .2. THE SCATTERING AMPLITUDE IN THE ADJACENT CASE We can now take the ǫ → P and Q are non-vanishing, one haslim ǫ → T ( r ) ǫ ( p, P, Q ) = T ( r ) ( p, P, Q ) , P = 0 , Q = 0 , (3.2.16)where the ǫ -independent triangle functions are defined by T ( r ) ( p, P, Q ) := log( Q /P )( Q − P ) r . (3.2.17)If either of the invariants vanishes, the limit of the ǫ -dependent triangle gives rise toan infrared-divergent term (which we call a “degenerate” triangle - this is one with twomassless legs). For example, if Q = 0, one has T ǫ ( p, P, Q ) | Q =0 −→ − ǫ ( − P ) − ǫ P , ǫ → . (3.2.18)The two possible configurations which give rise to infrared-divergent contributions cor-respond to the following two possibilities: a. q ,m − = k (hence q ,m − = 0). In this case we also have q ,m = t [2]2 . b. − q ,m = k (hence q ,m = 0). Therefore q ,m − = t [2] n .We notice that infrared poles will appear only in terms corresponding to the trian-gle function T . Indeed, whenever one of the kinematical invariants contained in T (3) vanishes, the combination of traces multiplying this function in (3.2.14) vanishes as well.In conclusion we arrive at the following result, where we have explicitly separated-outthe infrared-divergent terms: A scalar n = A poles + A + A , (3.2.19)where A poles = 16 A tree ǫ h ( − t [2]2 ) − ǫ + ( − t [2] n ) − ǫ i , (3.2.20) A = 16 A tree t [2]1 n − X m =4 h tr + ( k/ k/ k/ m q/ m, ) − tr + ( k/ k/ q/ m, k/ m ) i T ( m, q ,m − , q ,m ) , A = 13 A tree t [2]1 ) n − X m =4 h(cid:2) tr + ( k/ k/ k/ m q/ m, ) (cid:3) tr + ( k/ k/ q/ m, k/ m ) − tr + ( k/ k/ k/ m q/ m, ) (cid:2) tr + ( k/ k/ q/ m, k/ m ) (cid:3) i T (3) ( m, q ,m − , q ,m ) . A factor of − π ˆ λ will be understood on the right hand sides of Eqs. (3.2.19), (3.2.21) and (3.2.23),where ˆ λ is defined in (G.1.11). .3. THE SCATTERING AMPLITUDE IN THE GENERAL CASE More compactly, we can recognise that A poles and A reconstruct the contribution ofan N = 1 chiral multiplet, and rewrite (3.2.19) as A scalar n = 13 A N =1 , chiral12 + 13 A tree12 t [2]1 ) n − X m =4 B m T (3) ( m, q ,m − , q ,m ) , (3.2.21)where B m = (cid:2) tr + ( k/ k/ k/ m q/ m, ) (cid:3) tr + ( k/ k/ q/ m, k/ m ) (3.2.22) − (cid:2) tr + ( k/ k/ q/ m, k/ m ) (cid:3) tr + ( k/ k/ k/ m q/ m, ) . and A N =1 , chiral12 = 12 A tree12 t [2]1 n X m =3 n(cid:2) tr + ( k/ k/ k/ m q/ m, ) − tr + ( k/ k/ q/ m, k/ m ) (cid:3) · T ( m, q ,m − , q ,m ) o . (3.2.23)This is our result for the cut-constructible part of the n -gluon MHV scattering amplitudewith adjacent negative-helicity gluons in positions 1 and 2. This expression was firstderived by Bern, Dixon, Dunbar and Kosower in [42], and our result agrees preciselywith this. A remark is in order here. In [42], the final result is expressed in terms of afunction L ( x ) := log x − ( x − /x ) / − x ) , (3.2.24)which contains a rational part − ( x − /x ) / − x ) which removes a spurious third-order pole from the amplitude. With our approach however we did not expect to detectrational terms in the scattering amplitude, and indeed we do not find such terms. Furthermore, we do not find the other rational terms which are known to be present inthe one-loop scattering amplitude [44, 45].
The situation where the negative-helicity gluons are not adjacent is technically morechallenging. Our starting point will be (3.1.3), to which we will apply the Schoutenidentity (see Appendix A for a collection of spinor identities used). Eq. (3.1.3) can then In our notation L corresponds to T (3) , which, however, lacks a rational term. .3. THE SCATTERING AMPLITUDE IN THE GENERAL CASE be written as a sum of four terms: ˆ C = C ( m , m + 1) − C ( m , m ) − C ( m − , m + 1) + C ( m − , m ) (3.3.1)where C ( a, b ) := h i l i h j l i h i l i h j l ih i j i h l l i · h i a i h j b ih l a i h l b i . (3.3.2)The calculation of the phase space integral of this expression is discussed in Appendix G.The result is Z d − ǫ LIPS( l , − l ; P L ; z ) C ( a, b )= 13 tr + ( i/ j/ b/ a/ )( a · b ) " tr + ( i/ j/ P/ L ; z a/ ) (cid:20) tr + ( i/ j/ a/ P/ L ; z )( P L ; z · a ) + 2( i · j )( P L ; z · a ) (cid:21) − ( a ↔ b ) + 12 tr + ( i/ j/ b/ a/ )tr + ( i/ j/ a/ b/ )( a · b ) " tr + ( i/ j/ P/ L ; z a/ ) ( P L ; z · a ) + ( a ↔ b ) − tr + ( i/ j/ a/ b/ )tr + ( i/ j/ b/ a/ )( a · b ) " tr + ( i/ j/ a/ b/ )tr + ( i/ j/ P/ L ; z a/ )( P L ; z · a ) + ( a ↔ b ) + tr + ( i/ j/ a/ b/ ) tr + ( i/ j/ b/ a/ ) ( a · b ) log (cid:18) − ( a · b ) N P L ; z (cid:19) , (3.3.3)where N ≡ N ( P ) := ( a · b ) P − P · a )( P · b ), and we have suppressed a factor of − π ˆ λ ( − P L ) − ǫ · [2 ( i · j ) ] − on the right hand side of (3.3.3), where ˆ λ is defined in(G.1.11). We notice that (3.3.3) is symmetric under the simultaneous exchange of i with j and a with b . This symmetry is manifest in the coefficient multiplying thelogarithm – the last term in (3.3.3); for the remaining terms, nontrivial gamma matrixidentities are required. For instance, consider the terms in the second line of (3.3.3).These terms are present in the adjacent gluon case (3.2.3), and it is therefore naturalto expect that the trace structure of this term is separately invariant when i ↔ j and a ↔ b . Indeed this is the case thanks to the identity32( i · j ) = tr + ( i/ j/ P/ L ; z a/ ) (cid:20) tr + ( i/ j/ a/ P/ L ; z )( P L ; z · a ) + 2( i · j )( P L ; z · a ) (cid:21) (3.3.4)+ tr + ( i/ j/ a/ P/ L ; z ) (cid:20) tr + ( i/ j/ P/ L ; z a/ )( P L ; z · a ) + 2( i · j )( P L ; z · a ) (cid:21) . We drop the factor of − i A tree n from now on and reinstate it at the end of the calculation. .3. THE SCATTERING AMPLITUDE IN THE GENERAL CASE Similar identities show that the third and fourth line of (3.3.3) are invariant under thesimultaneous exchange i ↔ j and a ↔ b .The next step is to perform the dispersion integral of (3.3.3), i.e. the integral overthe variable z which has been converted to an integral over P L ; z . The relevant termsare thus those involving P L ; z in (3.3.3), and in an overall factor ( P L ; z ) − ǫ arising fromthe dimensionally regulated measure.The integral over the term involving the logarithm has been evaluated in [37], withthe result Z dP L ; z P L ; z − P L ( P L ; z ) − ǫ log (cid:18) − ( a · b ) N P L ; z (cid:19) = Li (cid:18) − ( a · b ) N ( P ) P L (cid:19) + O ( ǫ ) . (3.3.5)Notice that these terms were not present in the adjacent negative-helicity gluon caseconsidered in § z -dependence,we see that, in complete similarity with the adjacent case of § P L ; z in the numerator as in the denominator. Hence, in the centreof mass frame in which P L ; z := P L ; z (1 ,~ P L ; z cancels completely. Notethat this also immediately resolves the question of gauge invariance for these terms –this occurs only through the η dependence in P L ; z = P L − zη . Furthermore, the boxfunctions coming from (3.3.5) are separately gauge-invariant [37]. The conclusion isthat our expression for the amplitude below, built from sums over MHV diagrams ofthe dispersion integral of (3.3.3), will be gauge-invariant. Moreover, apart from (3.3.5),the only other dispersion integral we will need is that computed in (3.2.7).It follows from this discussion that the result of the dispersion integral of (3.3.3) is(suppressing a factor of − π ˆ λ ( − P L ) − ǫ · [2 ( i · j ) ] − · [ πǫ csc( πǫ )]): Z dzz Z d − ǫ LIPS( l , − l ; P L ; z ) C ( a, b )= 1 ǫ ( − P L ) − ǫ (
13 tr + ( i/ j/ b/ a/ )( a · b ) " tr + ( i/ j/ P/ L a/ ) (cid:20) tr + ( i/ j/ a/ P/ L )( P L · a ) + 2( i · j )( P L · a ) (cid:21) − ( a ↔ b ) + 12 tr + ( i/ j/ b/ a/ )tr + ( i/ j/ a/ b/ )( a · b ) " tr + ( i/ j/ P/ L a/ ) ( P L · a ) + ( a ↔ b ) − tr + ( i/ j/ a/ b/ )tr + ( i/ j/ b/ a/ )( a · b ) " tr + ( i/ j/ a/ b/ )tr + ( i/ j/ P/ L a/ )( P L · a ) + ( a ↔ b ) + tr + ( i/ j/ a/ b/ ) tr + ( i/ j/ b/ a/ ) ( a · b ) Li (cid:18) − ( a · b ) N ( P L ) P L (cid:19) . (3.3.6)95 .3. THE SCATTERING AMPLITUDE IN THE GENERAL CASE Now, due to the four terms in (3.3.1), the sum over MHV diagrams will include asigned sum over four expressions like (3.3.6). Let us begin by considering the last lineof (3.3.6). This is a term familiar from [37] and [40] and corresponds to one of the fourdilogarithms in the novel expression found in [37] for the finite part B of a scalar boxfunction, B ( s, t, P , Q ) = Li (cid:18) − ( a · b ) N ( P ) P (cid:19) + Li (cid:18) − ( a · b ) N ( P ) Q (cid:19) − Li (cid:18) − ( a · b ) N ( P ) s (cid:19) − Li (cid:18) − ( a · b ) N ( P ) t (cid:19) , (3.3.7)with s := ( P + a ) , t := ( P + b ) , and P + Q + a + b = 0. By taking into account the fourterms in (3.3.1) and summing over MHV diagrams as specified in (3.1.1) and (3.1.2),one sees that each of the four terms in any finite box function B appears exactly once,in complete similarity with [37] and [40]. The final contribution of this term will thenbe i − X m = j +1 j − X m = i +1 (cid:2) b ijm m (cid:3) B ( q m ,m − , q m +1 ,m , q m +1 ,m − , q m +1 ,m − ) , (3.3.8)where t [ k ] i := ( p i + p i +1 + · · · + p i + k − ) for k ≥
0, and t [ k ] i = t [ n − k ] i for k <
0. In writing(3.3.8), we have taken into account that the dilogarithm in (3.3.6) is multiplied by acoefficient proportional to the square of b ijm m , where b ijm m := − + ( k/ i k/ j k/ m k/ m ) tr + ( k/ i k/ j k/ m k/ m )[( k i + k j ) ] [( k m + k m ) ] . (3.3.9)We notice that b ijm m is the coefficient of the box functions in the one-loop N = 1MHV amplitude, originally calculated by Bern, Dixon, Dunbar and Kosower in [42],and derived in [40, 41] using the MHV diagram approach for loops proposed in [37].Furthermore, we observe that b ijm m is holomorphic in the spinor variables, and as suchhas simple localisation properties in twistor space. Indeed, from (3.3.9) it follows that b ijm m = 2 h i m i h i m i h j m i h j m ih i j i h m m i . (3.3.10)Summing over the four terms for the remainder of (3.3.6) can be done in completesimilarity with § We will skip the details of this derivationand now present our result. We multiply our final results by a factor of 2, which takes into account the two possible helicityassignments for the scalars in the loop. In § .3. THE SCATTERING AMPLITUDE IN THE GENERAL CASE m1m2 m1+1 j- m2-1m2+1 i- m1-1 Figure 3.3:
A box function contributing to the amplitude in the general case. Thenegative-helicity gluons, i and j , cannot be in adjacent positions, as the figure shows. In order to do this, we find it convenient to define the following expressions: A ijm m := ( i j m + 1 m )(( m + 1) · m ) − ( i j m m )( m · m ) (3.3.11)= − i j ] h m i ih m j i h m m + 1 ih m + 1 m i h m m i ,S ijm m := ( i j m m + 1)( i j m + 1 m )(( m + 1) · m ) − ( i j m m )( i j m m )( m · m ) , (3.3.12) I ijm m := ( i j m m + 1) ( i j m + 1 m )(( m + 1) · m ) − ( i j m m ) ( i j m m )( m · m ) , (3.3.13)where for notational simplicity we set ( a a a a ) := tr + ( a/ a/ a/ a/ ) in the above. Wealso note the symmetry properties A jim m = − A ijm m , S jim m = S ijm m . (3.3.14)The momentum flow is best described using the triangle diagram in Figure 3.4, where97 .3. THE SCATTERING AMPLITUDE IN THE GENERAL CASE we use the following definitions: P := q m +1 ,m − = − q m ,m , (3.3.15) Q := q m +1 ,m . The triangle in Figure 3.5 also appears in the calculation, and can be converted into atriangle as in Figure 3.4 - but with i and j swapped - if one shifts m − → m , andthen swaps m ↔ m .Next we introduce the coefficients A ijm m := 2 − ( i · j ) − A ijm m h ( i j m Q ) ( i j Q m ) − ( i j m Q )( i j Q m ) i , (3.3.16)˜ A ijm m := 2 − ( i · j ) − A ijm m h ( i j m Q ) − ( i j Q m ) i , (3.3.17) S ijm m = 2 − ( i · j ) − S ijm m h ( i j m Q ) + ( i j Q m ) i , (3.3.18) I ijm m := 2 − ( i · j ) − h I ijm m ( i j Q m ) + I jim m ( i j m Q ) i . (3.3.19)We will also make use of the ǫ -dependent triangle functions introduced in (3.2.15),whose ǫ → We can now present our result for the one-loop MHV amplitude: A scalar A tree = i − X m = j +1 j − X m = i +1 (cid:2) b ijm m (cid:3) B ( q m ,m − , q m +1 ,m , q m +1 ,m − , q m +1 ,m − ) − (cid:18) i − X m = j +1 j − X m = i h A ijm m T (3) ( m , P, Q ) − ( i · j ) ˜ A ijm m T (2) ( m , P, Q ) i + 2 i − X m = j +1 j − X m = i h S ijm m T (2) ( m , P, Q ) + I ijm m T ( m , P, Q ) i + ( i ↔ j ) (cid:19) , (3.3.20)where on the right hand side of (3.3.20) a factor of − π ˆ λ is understood and ˆ λ is defined The infrared-divergent terms will be described below and used to check that our result has thecorrect infrared pole structure. .3. THE SCATTERING AMPLITUDE IN THE GENERAL CASE in (G.1.11). We can also introduce the coefficient c ijm m := 12 (cid:20) ( i j m + 1 m ) (cid:0) ( m + 1) · m (cid:1) − ( i j m m )( m · m ) (cid:21) ( i j m Q ) − ( i j Q m )[( i + j ) ] , (3.3.21)which already appears as the coefficient multiplying the triangle function T in the N = 1amplitude, (see e.g. Eq. (2.1.19)), and rewrite (3.3.20) as F = i − X m = j +1 j − X m = i +1 (cid:2) b ijm m (cid:3) B ( q m ,m − , q m +1 ,m , q m +1 ,m − , q m +1 ,m − ) − (cid:18) i − X m = j +1 j − X m = i c ijm m h ( i j m Q ) ( i j Q m )2( i · j ) T (3) ( m , P, Q ) + T ( m , P, Q ) i + 2 i − X m = j +1 j − X m = i h S ijm m T (2) ( m , P, Q ) + I ijm m T ( m , P, Q ) i + ( i ↔ j ) (cid:19) , (3.3.22)where F = A scalar / A tree . QP m1 i-j- m2+1 m2m1-1 m1+1 Figure 3.4:
One type of triangle function contributing to the amplitude in the generalcase, where i ∈ Q , and j ∈ P . Several remarks are in order. As usual, the variables q m ,m − , q m +1 ,m correspond to the s - and t -channel ofthe finite part of the “easy two-mass” box function with massless legs m and m ,and massive legs q m +1 ,m − , q m +1 ,m − (Figure 3.3).99 .3. THE SCATTERING AMPLITUDE IN THE GENERAL CASE Q'P' m2 j-i- m1 m1-1m2-1 m2+1 Figure 3.5:
Another type of triangle function contributing to the amplitude in the generalcase. By first shifting m − → m , and then swapping m ↔ m , we convert this intoa triangle function as in Figure 3.4 – but with i and j swapped. These are the trianglefunctions responsible for the i ↔ j swapped terms in (3.3.20) – or (3.3.22) . Compared to the ranges of m and m indicated in (3.1.2), we have omitted m = i in the summation of the triangles as for this value the coefficients A , S , I definedin (3.3.16)–(3.3.19) vanish. Notice also that we have i ∈ Q and j ∈ P . In the case of adjacent negative-helicity gluons, the only surviving terms are thosecontaining the coefficient c ijm m , on the second line of (3.3.20) or (3.3.22). Wewill return to this point in § We comment that, in contrast to the adjacent case (see (3.2.21)), in the generalcase the N = 1 chiral amplitude does not separate out naturally in the final result.One can quickly see this from the coefficient of the box function B in (3.3.20) forexample.Next we wish to explicitly separate out the infrared divergences from (3.3.20). Wecan immediately anticipate that there will be four infrared-divergent terms, correspond-ing to the four possible degenerate triangles. Two of these degenerate triangles occurwhen either P or Q happen to vanish. The other two originate from the i ↔ j swappedterms.Let us first consider the terms arising from the summation with i ↔ j unswapped.When Q = 0, it follows that m = i − m = i (see Figure 3.4). When P = 0, it100 .3. THE SCATTERING AMPLITUDE IN THE GENERAL CASE follows that m = j + 1 and m = j − T ( r ) ( p, P, Q ) → ( − ) r ǫ ( − t [2] i − ) − ǫ ( t [2] i − ) r , Q → , (3.3.23) T ( r ) ( p, P, Q ) → − ǫ ( − t [2] j ) − ǫ ( t [2] j ) r , P → . The infrared-divergent terms coming from Q = 0 are then easily extracted, and are − ǫ · ( − t [2] i − ) − ǫ i · j ) ( i j i − i + 1) (cid:0) ( i + 1) · ( i − (cid:1) (3.3.24) · (cid:20)
83 ( i · j ) − i j i + 1 i − (cid:0) ( i + 1) · ( i − (cid:1) ( i · j ) + ( i j i + 1 i − i j i − i + 1) (cid:0) ( i + 1) · ( i − (cid:1) (cid:21) , and from P = 0 − ǫ · ( − t [2] j ) − ǫ i · j ) ( i j j − j + 1) (cid:0) ( j + 1) · ( j − (cid:1) (3.3.25) · (cid:20)
83 ( i · j ) − i j j + 1 j − (cid:0) ( j + 1) · ( j − (cid:1) ( i · j ) + ( i j j + 1 j − i j j − j + 1) (cid:0) ( j + 1) · ( j − (cid:1) (cid:21) . Likewise, from the “swapped” degenerate triangles we obtain the following infrared-divergent terms: − ǫ · ( − t [2] j − ) − ǫ i · j ) ( i j j + 1 j − (cid:0) ( j + 1) · ( j − (cid:1) (3.3.26) · (cid:20)
83 ( i · j ) − i j j − j + 1) (cid:0) ( j + 1) · ( j − (cid:1) ( i · j ) + ( i j j − j + 1)( i j j + 1 j − (cid:0) ( j + 1) · ( j − (cid:1) (cid:21) , and − ǫ · ( − t [2] i ) − ǫ i · j ) ( i j i + 1 i − (cid:0) ( i + 1) · ( i − (cid:1) (3.3.27) · (cid:20)
83 ( i · j ) − i j i − i + 1) (cid:0) ( i + 1) · ( i − (cid:1) ( i · j ) + ( i j i − i + 1)( i j i + 1 i − (cid:0) ( i + 1) · ( i − (cid:1) (cid:21) . We would like to make some brief comments on the interpretation in twistor space ofour result (3.3.22). As noticed earlier, the coefficient b ijm m appears already in the N = 1 chiral mul-tiplet contribution to a one-loop MHV amplitude, where it multiplies the boxfunction. It was noticed in Section 4 of [73] that b ijm m is a holomorphic func-101 .4. CHECKS OF THE GENERAL RESULT tion and hence it does not affect the twistor space localisation of the finite boxfunction. The coefficient c ijm m also appears in the N = 1 amplitude as the coefficient ofthe triangles (see e.g. Eq. (2.19) of [40]). Its twistor space interpretation wasconsidered in Section 4 of [73], where it was found that c ijm m has support ontwo lines in twistor space. Furthermore, it was also found that the correspondingterm in the amplitude has a derivative of a delta function support on coplanarconfigurations. The combination c ijm m ( i j m Q ) ( i j Q m ) / ( i · j ) already appears in the case ofadjacent negative-helicity gluons. The localisation properties of the correspondingterm in the amplitude were considered in Section 5.3 of [73] and found to have,similarly to the previous case, derivative of a delta function support on coplanarconfigurations. On general grounds, we can argue that the remaining terms in the amplitudehave a twistor space interpretation which is similar to that of the terms alreadyconsidered. The gluons whose momenta sum to P are contained on a line; likewise,the gluons whose momenta sum to Q localise on another line.We observe that the rational parts of the amplitude are not generated from theMHV diagram construction presented here. Such rational terms were not present forthe N = 1 and N = 4 amplitudes derived in [37, 40, 41]. However, for the amplitudestudied here, rational terms are required to ensure the correct factorisation properties[42]. These terms have recently been computed using an on-shell unitarity bootstrap[45] which makes use of the cut-constructible part (3.3.20) (or (3.3.22)) as input. In this section we present three consistency checks that we have performed for theresult (3.3.20) (or (3.3.22)) for the one-loop scalar contribution to the MHV scatteringamplitude. These checks are: For adjacent negative-helicity gluons, the general expression (3.3.20) should re-produce the previously calculated form (3.2.21). In the case of five gluons in the configuration (1 − + − + + ), the result (3.3.20)should reproduce the known amplitude given in [44]. The result (3.3.20) should have the correct infrared-pole structure.We next discuss these requirements in turn.102 .4. CHECKS OF THE GENERAL RESULT
The amplitude where the two negative-helicity external gluons are adjacent is given inSection 7 of [42] and was explicitly rederived in § i and j to be adjacent eliminates the sum over m , as we must have m = i and m +1 = j .Moreover, in this case Q = q m +1 ,i , P = q j,m − and one has A m m ij = − i · j ) ,S m m ij = 0 , I m m ij = 0 , (3.4.1)(for m = i , and m + 1 = j ). Similar simplifications occur for the swapped triangle.Hence the only surviving terms are those in the second line of (3.3.20) (or (3.3.22)), andit is then easy to see that they generate the same amplitude (3.2.8) already calculatedin § The other special case is the non-adjacent five-gluon amplitude (1 − + − + + ), givenin Equation (9) of [44]. This amplitude may be written as c Γ A tree times ǫ −
16 log( − s ) + tr + (1 / / / / ) tr + (1 / / / / ) (2 · (1 · B ( s , s , , s ) −
13 tr + (1 / / / / ) tr + (1 / / / / )2 (2 · · (cid:20) tr + (1 / / / / ) log( s /s )( s − s ) + tr + (1 / / / / ) log( s /s )( s − s ) (cid:21) c Γ = r Γ / (4 π ) − ǫ is given in terms of Eq. (C.3.1). The derivation in [44] used string-based methods which affect the coefficient of the pole term. InEq. (3.4.2) we have written the pole coefficient which matches the adjacent case. .4. CHECKS OF THE GENERAL RESULT + 13 12 (1 · (cid:20) tr + (1 / / / / )tr + (1 / / / / ) log( s /s )( s − s ) (cid:21) − tr + (1 / / / / ) tr + (1 / / / / ) (2 · (1 · (cid:20) log( s /s )( s − s ) − log( s /s )( s − s ) (cid:21) + tr + (1 / / / / ) tr + (1 / / / / ) (2 · (1 · (cid:20) log( s /s )( s − s ) + log( s /s )( s − s ) (cid:21) −
13 12 (1 · (cid:20) tr + (1 / / / / ) log( s /s )( s − s ) (cid:21) + (1 , ↔ (3 , , (3.4.2)where the interchange on the last line applies to all terms above it in this equation,including the first two terms. The box function B is defined in (3.3.7). In deriving(3.4.2) from [44], we have used the dilogarithm identityLi (1 − r ) + Li (1 − s ) + log( r ) log( s ) = Li (cid:0) − rs (cid:1) + Li (cid:0) − sr (cid:1) − Li (cid:0) − sr − rs (cid:1) . We have checked explicitly that our expression for the n -gluon non-adjacent amplitude(3.3.20), when specialised to the case with five gluons in the configuration (1 − + − + + ),yields precisely the result (3.4.2) above. For the terms involving dilogarithms, this iseasily done. For the remaining terms, which contain logarithms, a more involved calcu-lation is necessary using various spinor identities from Appendix A. A straightforwardmethod of doing this calculation begins with the explicit sum over MHV diagrams inthis case, isolating the coefficients of each logarithmic function such as e.g. log( s ), andthen checking that these coefficients match those in (3.4.2). The remaining 1 /ǫ termarises from the following discussion. The infrared-divergent terms (poles in 1 /ǫ ) can easily be extracted from (3.3.24)–(3.3.27)by simply replacing ( − t [2] r ) − ǫ → r = i − , i, j − , j ). Consider first the terms in(3.3.25) and (3.3.26). After a little algebra, and using( i j j + 1 j −
1) + ( i j j − i −
1) = 4 ( i · j ) (cid:0) ( j − · ( j + 1) (cid:1) , (3.4.3)one finds that these two contributions add up to − ǫ ( i · j ) . (3.4.4)104 .5. THE MHV AMPLITUDES IN QCD Similarly, the pole contribution arising from (3.3.24) and (3.3.27) gives an additionalcontribution of − (64 / ǫ ) ( i · j ) . Reinstating a factor of − · − ( i · j ) − · A tree , we seethat the pole part of (3.3.20) is simply given by A scalar | ǫ − pole = A tree . (3.4.5)Hence our result (3.3.20) has the expected infrared-singular behaviour. We conclude by mentioning that the full one-loop n -gluon MHV amplitudes (with arbi-trary positions for the negative-helicity gluons) in QCD can now be constructed. Theseare given by: A MHVQCD = A MHV N =4 − A MHV N =1 , chiral + A MHVscalar , (3.5.1)where in contradistinction with (1.3.3) we have written the scalar contribution in termsof a complex scalar rather than a real scalar. The individual pieces (to finite order in ǫ ) can be found as follows: • A MHV N =4 was first computed in [38] and can be found there as Equation (4.1). Al-ternatively it is given as Equation (1.9.1) in Chapter 1 of this thesis. Note that analternative form to Eq. (1.9.6) for the 2me box functions is given by Eq. (1.9.25). • A MHV N =1 , chiral was first computed in [42] and can be found there as Equation (5.12)or more compactly as Equation (2.1.19) in Chapter 2 of this thesis. • In contrast to the N = 4 and N = 1 cases, A MHVscalar is an amplitude in a non-supersymmetric theory and as such its cuts are not uniquely determined by itscut-constructible part ( A MHVs-cut ). A MHVs-cut was first computed in [43] and can be foundthere as Equation (4.20) or Equation (4.22). Alternatively it can be found earlierin this chapter as Equation (3.3.20) or Equation (3.3.22). • Building on the results of [43], the rational part of A MHVscalar ( A MHVs-rational ) was com-puted in [45]. In doing this it was found that it is useful to ‘complete’ the cut-constructible parts obtained in [43] by introducing certain preliminary rationalterms in order to remove spurious singularities. The cut-completion of A MHVs-cut isgiven by Equation (A1) of Appendix A of [45] and the full amplitude is thenobtained by adding the remaining rational terms as given in Equation (5.30) ofthat paper. Explicitly, the full scalar amplitude is given by Equation (5.1) (fornegative-helicity gluons 1 and m ), where ˆ C is given by (A1) and ˆ R by Equation(5.30) of [45]. 105 .5. THE MHV AMPLITUDES IN QCD • A MHVQCD can then be found using the decomposition (3.5.1) and A MHV N =4 = Eq. (4.1) of [38] A MHV N =1 , chiral = Eq. (5.12) of [42] A MHVscalar = Eq. (5.1) of [45] . HAPTER 4RECURSION RELATIONS IN GRAVITY
The proposal of a twistor string dual to perturbative Yang-Mills in [31] led not-onlyto the advances described in Chapters 1-3 of the so-called MHV rules for perturbationtheory, but to many others as well. The support of many quantities such as scatteringamplitudes, their integral functions and the coefficients of these functions in twistorspace has led to many insights [31, 43, 47, 53, 72, 73, 75, 76, 91, 179, 180, 181, 182, 183,184] as has the use of signature + + −− (or equivalently the restriction of momentato be complex rather than real). In particular, this second technique of using complexmomenta has proved very powerful, leading to the idea of generalised unitarity [47, 84]and then to the tree-level on-shell recursion relations [48, 49] which will be central tothis chapter.Recursion relations have been known for some time in field theory since Berends andGiele proposed them in terms of off-shell currents [171]. However, the gluon recursionrelations introduced by Britto, Cachazo and Feng in [48] (stemming from observationsin [46]) and then proved in [49] are in some ways much more powerful. They applydirectly to on-shell scattering amplitudes and are particularly apt when the amplitudesare written in the spinor helicity formalism, which as we have seen in the precedingchapters is a formalism which tends to favour simple and compact expressions.The proof of the on-shell recursion relation for gluons presented in [49] is very simple,only relying (essentially) on the ability to express an amplitude as a function of acomplex variable z and then the asymptotic behaviour of this function as z → ∞ . Assuch, it is natural to ask whether such a recursive structure might persist in other fieldtheories and even in gravity. This question was answered independently in [50] and[51] in the affirmative, where the authors of [50] (including the present author) usedit to present a new compact formula for n -graviton MHV amplitudes at tree-level ingeneral relativity (GR). Such compact formulæ are particularly interesting as gravity isvery-much more complicated than Yang-Mills - the 3-point vertex of GR for examplecontains 171 terms in total, while the 4-point vertex has 2850 altogether [165].In this chapter we will follow [50] and describe the recursion relation in Einsteingravity at tree-level. We will not summarise the proof of the relation in Yang-Mills as Here we mean gravity as a field theory (rather than as a string theory). .1. THE RECURSION RELATION it is almost identical to that in gravity. Any differences between the two are pointedout in what follows.
In this section we closely follow the proof of the recursion relation in Yang-Mills [49],which we will extend to the case of gravity amplitudes. As we shall see, the main newingredient is that gravity amplitudes depend on more kinematical invariants than thecorresponding Yang-Mills amplitudes, namely those which are sums of non-cyclicallyadjacent momenta; hence, more multi-particle channels should be considered.To derive a recursion relation for scattering amplitudes, we start by introducing aone-parameter family of scattering amplitudes, M ( z ) [49], where we choose z in sucha way that M (0) is the amplitude we wish to compute. We work in complexifiedMinkowski space and regard M ( z ) as a complex function of z and the momenta. Onecan then consider the contour integral [103] C ∞ := 12 πi I dz M ( z ) z , (4.1.1)where the integration is taken around the circle at infinity in the complex z plane.Assuming that M ( z ) has only simple poles at z = z i , the integration gives C ∞ = M (0) + X i [Res M ( z )] z = z i z i . (4.1.2)In the important case of Yang-Mills amplitudes, M ( z ) → z → ∞ , and hence C ∞ = 0 [49].Notice that up to this point the definition of the family of amplitudes M ( z ) has notbeen given – we have not even specified the theory whose scattering amplitudes we arecomputing.There are some obvious requirements for M ( z ). The main point is to define M ( z ) insuch a way that poles in z correspond to multi-particle poles in the scattering amplitude M (0). If this occurs then the corresponding residues can be computed from factorisationproperties of scattering amplitudes (see, for example, [3, 154]). In order to accomplishthis, M ( z ) was defined in [48, 49] by shifting the momenta of two of the external particlesin the original scattering amplitude. For this procedure to make sense, we have to makesure that even with these shifts overall momentum conservation is preserved and thatall particle momenta remain on-shell. We are thus led to define M ( z ) as the scatteringamplitude M ( p , . . . , p k ( z ) , . . . , p l ( z ) , . . . , p n ), where the momenta of particles k and l are shifted to p k ( z ) := p k + zη , p l ( z ) := p l − zη . (4.1.3)108 .1. THE RECURSION RELATION Momentum conservation is then maintained. As in [48], we can solve p k ( z ) = p l ( z ) = 0by choosing η = λ l ˜ λ k (or η = λ k ˜ λ l ), which makes sense in complexified Minkowskispace. Equivalently, λ k ( z ) := λ k + zλ l , ˜ λ l ( z ) := ˜ λ l − z ˜ λ k , (4.1.4)with λ l and ˜ λ k unshifted.More general families of scattering amplitudes can also be defined, as pointed outin [103]. For instance, one can single out three particles k , l , m , and define p k ( z ) := p k + zη k , p l ( z ) := p l + zη l , p m ( z ) := p m + zη m , (4.1.5)where η k , η l and η m are null and η k + η l + η m = 0. Imposing p k ( z ) = p l ( z ) = p m ( z ) = 0,one finds the solution η k = − αλ k ˜ λ l − βλ k ˜ λ m , η l = αλ k ˜ λ l , η m = βλ k ˜ λ m , (4.1.6)for arbitrary α and β . This has been used in [103]. In the following we will limitourselves to shifting only two momenta as in [48] and [49].At tree level, scattering amplitudes in field theory can only have simple poles inmulti-particle channels; for M ( z ), these generate poles in z (unless the channel containsboth particles k and l , or none). Indeed, if P ( z ) is a sum of momenta including p l ( z )but not p k ( z ), then P ( z ) = P − z ( P · η ) vanishes at z P = P / P · η ) [49]. In Yang-Mills theory, one considers colour-ordered partial amplitudes which have a fixed cyclicordering of the external legs. This implies that a generic Yang-Mills partial amplitudecan only depend on kinematical invariants made of sums of cyclically adjacent momenta.Hence, tree-level Yang-Mills amplitudes can only have poles in kinematical channelsmade of cyclically adjacent sums of momenta.For gravity amplitudes this is not the case as there is no such notion of ordering forthe external legs. Therefore, the multi-particle poles which produce poles in z are thoseobtained by forming all possible combinations of momenta which include p k ( z ) but not p l ( z ). This is the only modification to the BCFW recursion relation we need to makein order to derive a gravity recursion relation.For any such multi-particle channel P ( z ), we have M ( z ) → X h M hL ( z P ) 1 P ( z ) M − hR ( z P ) , (4.1.7)as P ( z ) → z → z P ). The sum is over the possible helicity assign-ments on the two sides of the propagator which connects the two lower-point tree-level109 .1. THE RECURSION RELATION amplitudes M hL and M − hR . It follows that[Res M ( z )] z = z P = − X h M hL ( z P ) z P P M − hR ( z P ) , (4.1.8)so that finally M (0) = C ∞ + X P,h M hL ( z P ) M − hR ( z P ) P . (4.1.9)The sum is over all possible decompositions of momenta such that p k ∈ P but p l / ∈ P .If C ∞ = 0 there is no boundary term in the recursion relation and M (0) = X P,h M hL ( z P ) M − hR ( z P ) P . (4.1.10)In [49] it was shown that for Yang-Mills amplitudes the boundary terms C YM ∞ alwaysvanish. Two different proofs were presented, the first based on the use of CSW diagrams[33] and the second on Feynman diagrams. An MHV-vertex formulation of gravity onlyrecently appeared [77], so at the time the authors of [50] could only rely on Feynmandiagrams. This is also the case for other field theories we might be interested in (suchas λφ , for example).As we have remarked, C ∞ = 0 if M ( z ) → z → ∞ . M ( z ) is a scattering am-plitude with shifted, z -dependent external null momenta. One can then try to estimatethe behaviour of M ( z ) for large z by using power counting (different theories will ofcourse give different results). In λφ the Feynman vertices are momentum independentand C ∞ = 0 (see § a priori whether or not a boundary term is present.From the previous discussion, it follows that the behaviour of M ( z ) as z → ∞ is related to the high-energy behaviour of the scattering amplitude (and hence to therenormalisability of the theory). The ultraviolet behaviour of quantum gravity, however,is full of surprises (for a summary, see for example Section 2.2 of [217] and also morerecent results of [55, 56, 57, 58, 59, 60]). We may therefore expect a more benignbehaviour of M ( z ) as z → ∞ . Specifically, in the next section we will focus on the n -graviton MHV scattering amplitudes which have been computed by Berends, Gieleand Kuijf (BGK) in [218]. Performing the shifts (4.1.3) explicitly in the BGK formula,one finds the surprising result lim z →∞ M MHV ( z ) = 0 . (4.1.11) We have checked that M ( z ) ∼ O (1 /z ) as z → ∞ , analytically for n ≤ n ≤
11 legs. .2. APPLICATION TO MHV GRAVITY AMPLITUDES
In more general amplitudes one can (at least in principle) use the (field theorylimit of the) KLT relations [219], which connect tree-level gravity amplitudes to tree-level amplitudes in Yang-Mills, to estimate the large- z behaviour of the scatteringamplitude. As an example, we have considered the next-to-MHV gravity amplitude M (1 − , − , − , + , + , + ), and performed the shifts as in (4.1.4), with k = 1 and l = 2.Similarly to the MHV case, we find thatlim z →∞ M (1 − , − , − , + , + , + )( z ) = 0 . (4.1.12)In [51] it was shown that M ( z ) vanishes as z → ∞ for all amplitudes up to eightgravitons and also for all n -point MHV and NMHV amplitudes. Further to this, recentwork [52] provides a proof of this statement for all tree-level n -graviton amplitudes thusestablishing the validity of the recursion relation in gravity unambiguously.In the next section we will apply the recursion relation (4.1.10) to the case of MHVamplitudes in gravity and show that it does generate correct expressions for the am-plitudes. As a bonus, we will derive a new closed-form expression for the n -particlescattering amplitude. In the following we will compute the MHV scattering amplitude M (1 − , − , + , . . . , n + )for n gravitons. We will choose the two negative-helicity gravitons 1 − and 2 − as referencelegs. This is a particularly convenient choice as it reduces the number of terms arisingin the recursion relation to a minimum. The shifts for the momenta of particles 1 and2 are p → p + zλ ˜ λ , p → p − zλ ˜ λ . (4.2.1)In terms of spinors, the shifts are realised as λ → ˆ λ := λ + zλ , ˜ λ → ˆ˜ λ := ˜ λ − z ˜ λ , (4.2.2)with λ and ˜ λ unmodified.Let us consider the possible recursion diagrams that can arise. There are only twopossibilities, corresponding to the two possible internal helicity assignments, (+ − ) and( − +): The amplitude on the left is googly (+ + − ) whereas on the right there is an MHVgravity amplitude with n − See Appendix H for explicit examples of KLT relations for four, five and six legs. .2. APPLICATION TO MHV GRAVITY AMPLITUDES
Figure 4.1:
One of the terms contributing to the recursion relation for the MHV ampli-tude M (1 − , − , + , . . . , n + ) . The gravity scattering amplitude on the right is symmetricunder the exchange of gravitons of the same helicity. In the recursion relation, we sumover all possible values of k , i.e. k = 3 , . . . , n . This amounts to summing over cyclicalpermutations of (3 , . . . , n ) . The amplitude on the right is googly and the amplitude on the left is MHV (seeFigure 4.2).We recall that a gravity amplitude is symmetric under the interchange of identicalhelicity gravitons; this implies that we have to sum n − k , with k = 3 , . . . , n .However, it is easy to see that diagrams of type actually give a vanishing contri-bution. Indeed, they are proportional to[ k ˆ P ] = [ k | ˆ P | ˆ2 ih ˆ P ˆ2 i = [ k | P | ih ˆ P ˆ2 i = 0 , (4.2.3)where the last equality follows from P = p k + p . Hence we will have to computediagrams of type only. We will do this in the following. To show explicitly how our recursion relation generates amplitudes we will now derivethe 4-, 5- and 6-point MHV scattering amplitudes.112 .2. APPLICATION TO MHV GRAVITY AMPLITUDES
Figure 4.2:
This class of diagrams also contributes to the recursion relation for the MHVamplitude M (1 − , − , + , . . . , n + ) ; however, each of these diagrams vanishes if the shifts (4.2.2) are performed. We start with the four point case. There are two diagrams to sum, one of which isrepresented in Figure 4.3; the other is obtained by swapping the labels 4 and 3. For thediagram in Figure 4.3, we have M (4) = M L P M R , (4.2.4)where the superscript denotes the label on the positive-helicity leg in the trivalent googlyMHV vertex, M L = [ ˆ P [4 1][1 ˆ P ] ! , (4.2.5) M R = h ˆ P i h ih P i ! , and P = ( p + p ) . Using h i ˆ P i = h i | P | P , (4.2.6)we find, after a little algebra, M (4) = h i [14] h ih i h i . (4.2.7)113 .2. APPLICATION TO MHV GRAVITY AMPLITUDES Figure 4.3:
One of the two diagrams contributing to the recursion relation for the MHVamplitude M (1 − , − , + , + ) . The other is obtained from this by cyclically permutingthe labels (3 , – i.e. swapping with . The full amplitude is M (1 − , − , + , + ) = M (3) + M (4) . Thus, we conclude that thefour point MHV amplitude generated by our recursion relation is given by M (1 − , − , + , + ) = h i [14] h ih i h i + 3 ↔ . (4.2.8)It is easy to check that this agrees with the conventional formula for this amplitude M (1 − , − , + , + ) = h i [12] N (4) h i , (4.2.9)where N ( n ) := Y ≤ i 5) means that we have to sum over cyclic permutations of the labels114 .2. APPLICATION TO MHV GRAVITY AMPLITUDES , , 5. The conventional formula for the five graviton MHV scattering amplitude is M (1 − , − , + , + , + ) = h i N (5) (cid:16) [12][34] h ih i − [13][24] h ih i (cid:17) . (4.2.12)Using standard spinor identities and momentum conservation, it is straightforward tocheck that our expression (4.2.11) agrees with this (alternatively, one can use the KLTrelation (H.0.3)).For the six graviton scattering amplitude, our recursion relation yields a sum of fourterms, M (1 − , − , + , + , + , + ) = h i [16] h i · h ih ih ih i (4.2.13) · (cid:18) [3 4] h ih i h | | h i + [4 5] h ih i h | | h i + [5 3] h ih i h | | h i (cid:19) + P c (3 , , , . The known formula for this amplitude is M = h i (cid:18) [12][45][3 | | ih ih ih ih ih ih ih ih ih ih i + P (2 , , (cid:19) , (4.2.14)where P (2 , , 4) indicates permutations of the labels 2 , , 4. We have checked numericallythat the formula (4.2.13) agrees with this expression. Recursion relations of the form given in [48], or the graviton recursion relation givenhere, naturally produce general formulæ for scattering amplitudes. For a suitable choiceof reference spinors, these new formulæ can often be simpler than previously knownexamples. For the choice of reference spinors 1 , , which we have made above, thegraviton recursion relation is particularly simple as it produces only one term at eachstep. This immediately suggests that one can use it to generate an explicit expressionfor the n -point amplitude. This turns out to be the case, and experience with the useof our recursion relation leads us to propose the following new general formula for the n -graviton MHV scattering amplitude. This is (labels 1 , M (1 , , i , · · · , i n − ) = h i [1 i n − ] h i n − i G ( i , i , i ) n − Y s =3 h | i + ... + i s − | i s ] h i s i s +1 ih i s +1 i + P ( i , ..., i n − ) , (4.2.15)115 .3. APPLICATIONS TO OTHER FIELD THEORIES where G ( i , i , i ) = 12 [ i i ] h i ih i ih i i ih i i ih i i i . (4.2.16)(For n = 5 the product term is dropped from (4.2.15)). It is straightforward to checkthat this amplitude satisfies the recursion relation with the choice of reference legs 1 − and 2 − .The known general MHV amplitude for two negative-helicity gravitons, 1 and 2, andthe remaining n − M (1 , , , · · · , n ) = h i " [12][ n − n − h n − i N ( n ) n − Y i =1 n − Y j = i +2 h ij i F + P (2 , . . . , n − , (4.2.17)where F = ( Q n − l =3 [ l | ( p l +1 + p l +2 + · · · + p n − ) | n i n ≥ n = 5 (4.2.18)We have checked numerically, up to n = 11, that our formula (4.2.15) gives the sameresults as (4.2.17).It is interesting to note that the very existence of this recursion relation in gravity -described here and in [50, 51] - has something to say about the divergences of quantumgravity. A central feature of the recursion relation is that it requires M ( ∞ ) = 0,and the behaviour of M ( z ) as z → ∞ is related to the high-energy behaviour (andhence the renormalisability) of the theory. It is not a priori clear that gravity has thisbehaviour, though the analyses of [50, 51] and more recently the complete analysis of[52] show that indeed M ( ∞ ) = 0 for any tree-level amplitude in gravity. This meansthat at tree-level, gravity has divergences in the UV that are perhaps better than onemight expect. This supports recent arguments that gravity may not be as divergent aspreviously thought and more specifically that 4-dimensional N = 8 supergravity may befinite [55, 56, 57, 58, 59, 60]. One of the striking features of the BCFW proof of the BCF recursion relations is thatthe specification of the theory with which one is dealing is almost unnecessary. Indeedin [49] the only step where specifying the theory did matter was in the estimate ofthe behaviour of the scattering amplitudes M ( z ) as z → ∞ , which was important toassess the possible existence of boundary terms in the recursion relation. This leads usto conjecture that recursion relations could be a more generic feature of massless (orspontaneously broken) field theories in four dimensions. After all, the BCF recursion This was also suggested in [103]. .4. CSW AS BCFW relations - as well as the recursion relation for gravity amplitudes discussed in thischapter and in [51] - just reconstruct a tree-level amplitude (which is a rational function)from its poles.Let us focus on massless λ ( φφ † ) theory in four dimensions. We use the spinorhelicity formalism, meaning that each momentum will be written as p a ˙ a = λ a ˜ λ ˙ a . Ascalar propagator 1 /P connects states of opposite “helicity”, which here just meansthat the propagator is h φ ( x ) φ † (0) i , with h φ ( x ) φ (0) i = h φ † ( x ) φ † (0) i = 0. Now considera Feynman diagram contributing to an n -particle scattering amplitude, and let us shiftthe momenta of particles k and l as in (4.1.3). As for the Yang-Mills case discussed in[49], there is a unique path of propagators going from particle k to particle l . Each ofthese propagators contributes 1 /z at large z , whereas vertices are independent of z . Wethus expect Feynman diagrams contributing to the amplitude to vanish in the large- z limit.An exception to the above reasoning is represented by those Feynman diagramswhere the shifted legs belong to the same vertex; these diagrams are z -independent,and hence not suppressed as z → ∞ . In order to deal with this problematic situation,and ensure that the full amplitude M ( z ) computed from Feynman diagrams vanishesas z → ∞ we propose two alternatives.Firstly, if one considers ( φφ † ) theory without any group structure, one can removethe problem by performing multiple shifts. This possibility has already been used inthe context of the rational part of one-loop amplitudes in pure Yang-Mills [103]. In ourcase, it is sufficient to shift at least four external momenta.Alternatively, we can consider ( φφ † ) theory with global symmetry group U( N )and φ in the adjoint. In this case we can group the amplitude into colour-orderedpartial amplitudes, as in the Yang-Mills case. Then, for any colour-ordered amplitudeone can always find a choice of shifts such that the shifted legs do not belong to thesame Feynman vertex. The procedure can be repeated for any colour ordering, andthe complete amplitude is obtained by summing over non-cyclic permutations of theexternal legs.In this way, the appearance of a boundary term C ∞ can be avoided, and one canthus derive a recursion relation for scattering amplitudes akin to (4.1.10). A similaranalysis can be carried out in other theories, possibly in the presence of spontaneoussymmetry breaking etc . We expect this to play an important rˆole in future studies. Finally, we would like to point out the connection between the CSW rules at tree-level[33] and the BCFW recursion relation introduced in [48, 49] and discussed for gravity117 .4. CSW AS BCFW in this chapter. This was hinted at in [49] where it was noted that the existence ofBCFW recursion (which can construct any gluon amplitude solely from a knowledge ofits singularities) provides an indirect proof of the CSW rules since the CSW rules provideresults which are Lorentz-invariant, gauge-invariant and have the correct singularities. Itwas also briefly touched on in [50] where some formal observations were made regardingthe relation between the way that the shifts of Eq. (4.1.3) are performed - so as to keepthe corresponding momenta on-shell in the BCFW recursion relation - and the way thatthe internal legs in the CSW rules are shifted (Eq. (1.7.1)).However, Risager showed that the CSW rules are in fact a special case of the BCFWrecursion relation when specific shifts of momenta are made [34]. The most naturalshifts to make when using the recursion relations are those which minimise the numberof terms appearing and thus the work that one has to do. In [34], however, a differentset of shifts was employed which affects every propagator that may appear in a CSWdiagram. The propagators are defined by the momenta that flow through them and thusby a set of consecutive external particles. In the case of CSW diagrams, the verticesare MHV vertices and thus this set of consecutive particles (and its compliment onthe other vertex to-which the propagator is attached) must contain at least one gluonof negative helicity each. Exactly this set of propagators is affected if every externalnegative-helicity gluon is shifted, provided that the sum of any subset of the shifts doesnot vanish. In addition, the shifts must all involve the anti-holomorphic spinors so thatall 3-point googly amplitudes drop out.Using these shifts (see Eq. (5.1) of [34] for an explicit example of the shifts foran NMHV amplitude), Risager used induction to prove the CSW rules directly thushighlighting their connection with the BCFW recursion relation. In [77] these ideaswere then used to construct an MHV-vertex formalism for gravity, thus accentuatingthe remarkable similarities between gauge theory and gravity despite the latter’s morecomplicated structure. 118 HAPTER 5CONCLUSIONS AND OUTLOOK In the previous chapters we have studied gluon scattering amplitudes in perturbativegauge theory and have seen how they can be stripped of colour and written in terms ofspinor variables to illuminate their basic structure in a unified context. Their twistor-space localisation then allows for an understanding of the unexpected simplicity of many n -point processes. The tree-level MHV amplitudes were seen to lie on simple straightlines in twistor space and it was shown how they could be calculated from a topologicalstring theory as an integral over the moduli space of holomorphically embedded, degree1, genus 0 curves. This in turn motivated a new perturbative expansion of Yang-Mills gauge theory where tree-level MHV amplitudes are taken off-shell and joined withscalar propagators to create tree-level amplitudes with successively greater numbersof negative-helicity particles. The MHV vertices effectively combine many Feynmandiagrams into one and thus provide a great simplification which aids calculation andhighlights the underlying geometrical structure.We saw how these techniques could be applied at loop-level to calculate the MHVamplitudes in N = 4 super-Yang-Mills, which is a slightly surprising result as the dualitywith the twistor string theory constructed in [31] (and also that in [112]) fails at loop-level. These string theories contain conformal supergravity states which do not decoupleat one-loop and this suggests that the application of the CSW rules to loops might failor simply calculate amplitudes in some theory of Yang-Mills coupled to conformal su-pergravity. Indeed, a recent calculation of various loop amplitudes in Berkovits’ twistorstring theory appears to give amplitudes in such a theory [114].One might also expect that such a surprising result would only apply to maximallysupersymmetric Yang-Mills. However in Chapter 2 we saw that MHV vertices can beused to calculate amplitudes at loop-level in theories with less supersymmetry suchas N = 1 super-Yang-Mills. There we calculated the one-loop MHV amplitudes andfound complete agreement with the known results in [42]. The calculation itself is moreinvolved than the corresponding one in N = 4 presented in [37] and reviewed in Chapter1 because the reduction in supersymmetry leads to fewer cancellations. Happily though,this does not spoil the technique of using MHV amplitudes as effective vertices.In Chapter 3 we applied the loop-level CSW rules to pure Yang-Mills with a scalarrunning in the loop. Pure Yang-Mills is a non-supersymmetric theory and as such thecalculation is even more involved than before. This still does not invalidate the process,119lthough it was found that the use of MHV vertices only calculates the cut-constructiblepart of the amplitude. The rational parts, which are intrinsically linked to the cuts forsupersymmetric theories, were thus missed. Nonetheless, the results obtained matchperfectly with the known (special) cases [42, 44] and provide the cut-constructible partof the MHV amplitude in pure Yang-Mills with arbitrary positions for the negative-helicity gluons for the first time. The rational part of the amplitude has since beencalculated in [45] building on the results described in Chapter 3.In Chapter 4 we turned our attention to gravity and another interesting develop-ment stemming from twistor string theory, namely that of on-shell recursion relations.Recursion relations have been used before in the construction of amplitudes [171], butit wasn’t until recently that they were used to recursively turn on-shell amplitudes intoamplitudes with a larger number of external legs. They were introduced in [48] at tree-level and have since been used in a bootstrap approach to loop amplitudes which wasin fact one of the techniques applied in [45].We saw that on-shell recursion relations can also be applied at tree-level in gravityand it is a beauty of the proof of these relations in gauge theory [49] which means thatthey can be proved in gravity without too much (!) extra work. The main additionalingredient is a proof of the behaviour of tree-level n -graviton amplitudes as a function ofa complex variable z as z → ∞ . In Chapter 4 we argued the case for many amplitudesof interest, but a recent proof that lim z →∞ M n ( z ) = 0 establishes that the recursionrelation in gravity can construct any tree-level n -graviton amplitude [52].We showed how this recursion relation could be used to construct MHV amplitudeswith successively more external gravitons and as a by-product constructed a new com-pact form for the n -graviton MHV amplitudes which provides an interesting alternativeto the previously-known form in [218]. We finished by commenting on the relation be-tween the tree-level CSW rules and on-shell recursion relations both in field theory andin gravity and also made some observations on the existence of recursion relations inother theories such as scalar φ theory.Unsurprisingly, this is not the end of the story. In the introduction we alreadymentioned some of the directions that have been explored following from and relatedto the material presented here. This includes the construction of twistor string theoriesdescribing N = 4 Yang-Mills as well as ones describing other field theories such as arecent description of Einstein supergravity [39], the use of on-shell recursion relations atloop level in both gauge theory and gravity [111, 220] and improvements to the unitaritymethod [47]. It may be particularly interesting to note that in [39], one of the theoriesfor which a twistor description is found is N = 4 Yang-Mills coupled to N = 4 Einsteinsupergravity. It appears that there exists a decoupling limit for this theory which givespure Yang-Mills and thus opens the door to the possibility of understanding loops inYang-Mills from twistor strings. 120rom the point of view of the MHV diagram formulation of gauge theory there hasalso been some considerable progress. Their use at tree-level is already well-establishedand a Lagrangian formulation now exists [35, 80, 203]. In this scenario, a non-localchange of variables is made to the light-cone Yang-Mills Lagrangian which yields akinetic term describing a scalar propagator connecting positive and negative helicitiesand interaction terms consisting of the infinite sequence of MHV amplitudes.Quantisation of this Lagrangian, however, is still an open problem. One of the mainpoints here is the fact that - as demonstrated in Chapter 3 - the use of MHV diagramsalone is not enough to generate a complete amplitude at the quantum level in non-supersymmetric theories and rational terms are missed. As such, one might ask howone could compute the one-loop all-plus (and − + . . . +) amplitude in pure Yang-Millsfrom MHV diagrams. At tree-level this vanishes, but at one-loop it is a purely rationalfunction - see e.g. Equation (3.4) of [84]. Construction of a one-loop amplitude fromMHV diagrams will always give q negative-helicity gluons that satisfies q ≥ 2, and thusthe all-plus amplitude (and also the − + . . . + amplitude) cannot be constructed fromMHV vertices alone. In [73] it was conjectured that perhaps the all-plus amplitudecould be elevated to the status of a vertex to generate these missing amplitudes, but atthe time an appropriate off-shell continuation for this amplitude could not be found.Recently, however, more progress has been made in this direction [81, 82, 83]. Itappears that the all-plus amplitude is intimately connected with the regularisation pro-cedure needed to evaluate loop diagrams as was initially hinted-at by the fact thatthe parity conjugate of this amplitude, the all-minus amplitude, arises from an ǫ × /ǫ cancellation in dimensional regularisation [81]. Inspired by this, Brandhuber, Spence,Travaglini and Zoubos showed in [82] that a certain one-loop two-point Lorentz-violatingcounterterm is the generating function for the infinite sequence of one-loop all-plus am-plitudes in pure Yang-Mills although there must be another contribution in this story tocorrectly explain the origin of the − + . . . + amplitude. In their approach it was foundthat a certain four-dimensional regularisation scheme (rather than dimensional regular-isation) [221, 222, 223] was most useful. It may be interesting and insightful to see if alight-cone approach and such a regularisation scheme is also helpful for computing thecut-constructible terms of amplitudes using MHV diagrams.Despite these advances, the MHV diagram technique is still practically-speakinglimited to tree-level amplitudes and the cut-constructible part of one-loop MHV ampli-tudes. This is largely because of the intrinsic complexity of loop calculations, thoughthere are other complications. For example, the topologies involved in calculating thecut-constructible part of amplitudes with more than 2 negative-helicity gluons can in-clude (in the case of the NMHV amplitude say) triangle diagrams where each vertex isan MHV vertex. In such diagrams one has 3 different internal particles to take off-shelland it is not clear whether the measure can be found in terms of LIPS integrals and121ispersion integrals such as that described in [37, 79] which has been so instrumentalin the application of the CSW rules at loop-level so far. Such issues are common toone-loop amplitudes which have q > N = 4 Yang-Mills may be expressed (essentially) as an exponential ofcertain one-loop amplitudes [224, 225, 226, 227, 228, 229]. Such expressions are termedcross-order relations and may be remarkably powerful if more generally applicable thanhas been found to date. They could allow the summation of amplitudes in N = 4 Yang-Mills to all orders in perturbation theory and so to non-perturbative information whichmay be connected to perturbative string theory via the AdS/CFT correspondence. It would be interesting to see how the known cross-order relations arise from MHVdiagrams. It is possible that the different terms in the cross-order relations may arisenaturally from MHV diagrams which might then provide a framework for proving theirvalidity more generally.The situation for gravity is in some ways even more exciting, with the possibility thatthere may exist UV-finite field theories of gravity. Such proposals have recently beenmade for N = 8 supergravity [53, 54, 55, 56, 57, 58, 59, 60] and it would be interestingto make contact between this and the twistor approach. One such point of contact maybe the recent proposal of a twistor string theory describing N = 8 supergravity [39].Another possibility is that of loop amplitudes from MHV vertices in ( N = 8 super-)gravity. These have not yet been understood and their explication would provide a newprescriptive method for the calculation of loop amplitudes in gravity which could shedlight on their UV properties. A further possibility that has not been explored so far (in either gauge theory orgravity) is a more direct connection between recursion relations and loop amplitudesthan those already mentioned. Risager [34] showed that the CSW rules at tree-level arereally just a specific case of the on-shell recursion relations proposed by Britto, Cachazoand Feng [48]. As we have seen throughout this thesis, the CSW rules can naturally beextended to loop amplitudes which begs the question of whether the same can be donefor other cases of the on-shell recursion relation in either Yang-Mills or in gravity. A very recent paper [230] by Alday and Maldacena appears to have taken a step in this direction.They show how to calculate gluon scattering amplitudes at strong coupling from a classical stringconfiguration via the AdS/CFT correspondence. As a result the full finite form of the four-gluonscattering amplitude in N = 4 super-Yang-Mills is presented. See also [231] which addresses the n -pointcase. Shortly after the completion of this work [232] appeared which deals with precisely this point. PPENDIX ASPINOR AND DIRAC-TRACE IDENTITIES In this appendix we present some useful identities pertaining to the spinor helicityformalism and also to help in dealing with Dirac traces. A.1 Spinor identities We take the metric to be the usual field-theory one η µν = (1 , − , − , − 1) and the epsilontensors with which we raise and lower indices to be ǫ αβ = ǫ ˙ α ˙ β = iσ = − ! , (A.1.1)with ǫ αβ = ǫ αβ ⇒ ǫ αβ ǫ βγ = − δ γα and σ = ! ,σ = − ii ! ,σ = − ! . (A.1.2)We also have σ µα ˙ α = (1 , ~σ ) (with ~σ = ( σ , σ , σ )), giving P α ˙ α = P µ σ µα ˙ α = P + P P − iP P + iP P − P ! = P − P − P + iP − P − iP P + P ! , (A.1.3)124 .1. SPINOR IDENTITIES and ¯ σ µ ˙ αα = − σ µ α ˙ α = ǫ ˙ α ˙ β ǫ αβ σ µβ ˙ β = (1 , − ~σ ), giving P ˙ αα = P µ ¯ σ µ ˙ αα = P − P − P + iP − P − iP P + P ! = P + P P − iP P + iP P − P ! . (A.1.4)Some useful identities involving σ and ¯ σ are: σ µα ˙ β ¯ σ ν ˙ βα = 2 η µν (A.1.5) σ µα ˙ α ¯ σ β ˙ βµ = 2 δ βα δ ˙ β ˙ α , (A.1.6)which means that we can interpret η µν as acting as 2 ǫ ˙ α ˙ β ǫ αβ and η µν as acting as ǫ ˙ α ˙ β ǫ αβ / η µν η µν = ǫ ˙ α ˙ β ǫ ˙ β ˙ α ǫ αβ ǫ βα = 4 as it should.We are concerned with massless particles for which we can write p α ˙ α = λ α ˜ λ ˙ α = λ λ ! (cid:16) ˜ λ ˙1 ˜ λ ˙2 (cid:17) = λ ˜ λ ˙1 λ ˜ λ ˙2 λ ˜ λ ˙1 λ ˜ λ ˙2 ! , (A.1.7)which implies (by raising indices) that p ˙ αα = − ˜ λ ˙ α λ α = − ˜ λ ˙1 ˜ λ ˙2 ! (cid:16) λ λ (cid:17) = − ˜ λ ˙1 λ ˜ λ ˙1 λ ˜ λ ˙2 λ ˜ λ ˙2 λ ! = ˜ λ ˙2 λ − ˜ λ ˙2 λ − ˜ λ ˙1 λ ˜ λ ˙1 λ ! , (A.1.8)which follows from having λ α = ( ǫ αβ λ β ) T = − λ Tβ ǫ βα and ˜ λ ˙ α = (˜ λ ˙ β ǫ ˙ β ˙ α ) T = − ǫ ˙ α ˙ β ˜ λ T ˙ β .2. THE HOLOMORPHIC DELTA FUNCTION For scalar products we take h λ µ i = λ α µ α = (cid:16) λ λ (cid:17) µ µ ! = ǫ αβ λ Tβ µ α , (A.1.9)and [˜ λ ˜ µ ] = ˜ λ ˙ α ˜ µ ˙ α = (cid:16) ˜ λ ˙1 ˜ λ ˙2 (cid:17) ˜ µ ˙1 ˜ µ ˙2 ! = ˜ λ ˙ α ˜ µ T ˙ β ǫ ˙ β ˙ α (A.1.10)Note that λ α and ˜ λ ˙ α are most naturally associated with column vectors, while ˜ λ ˙ α and λ α are most naturally associated with row vectors.For spinor manipulations, the Schouten identity is very useful: h i j ih k l i = h i k ih j l i + h i l ih k j i , [ i j ] [ k l ] = [ i k ] [ j l ] + [ i l ] [ k j ] . (A.1.11)For other introductions to the spinor helicity formalism see e.g. [153, 154]. A.2 The holomorphic delta function Consider the x − y plane in real coordinates and let ( x, y ) = ( x , x ). Now change tocomplex coordinates by letting z = x + ix (A.2.1)¯ z = x − ix . (A.2.2)Also define derivatives ∂ z = ∂∂z = 12 (cid:18) ∂∂x − i ∂∂x (cid:19) = 12 ( ∂ − i∂ ) (A.2.3) ∂ ¯ z = ∂∂ ¯ z = 12 (cid:18) ∂∂x + i ∂∂x (cid:19) = 12 ( ∂ + i∂ ) , (A.2.4) Recall that we have the shorthand notation h λ i λ j i = h i j i etc . .2. THE HOLOMORPHIC DELTA FUNCTION which have the properties that ∂ z z = 1 ; ∂ z ¯ z = 0 ; ∂ ¯ z ¯ z = 1 ; ∂ ¯ z z = 0 . We take the area element in the x − y plane to be d x = dx dx = | dx ∧ dx | ,where | | just indicates that one picks a plus sign to define the orientation. For the areaelement in the z − ¯ z plane we take d z = i | dz ∧ d ¯ z | so that we have d z = 2 d x .We normalise delta functions in the x − y plane as Z d x δ (2) ( x − a ) = 1 , (A.2.5)where δ (2) ( x − a ) := δ ( x − a ) δ ( x − a ), and after transforming this to the z − ¯ z plane(with b = a + ia and ¯ b = a − ia ) we have Z d z δ (2) ( z − b ) = 1 , (A.2.6)where δ (2) ( z − b ) := δ ( z − b ) δ (¯ z − ¯ b )= 12 δ (2) ( x − a ) . (A.2.7)We now define a holomorphic delta function as¯ δ ( z − b ) = δ (2) ( z − b ) d ¯ z , (A.2.8)which gives us Z dz ¯ δ ( z − b ) = Z dz ∧ d ¯ z δ (2) ( z − b )= − i Z d z δ (2) ( z − b )= − i . (A.2.9)As can be seen the holomorphic delta function is a ¯ ∂ -closed (0 , f by ¯ δ ( f ) = δ (2) ( f ) d ¯ f .A representation of this holomorphic delta function which will be particularly usefulfor us is the following [33]. Consider a momentum-vector described by λ and ˜ λ with˜ λ = ¯ λ in order to ensure that p α ˙ α = λ α ˜ λ ˙ α is real. Go to coordinates where λ α = (1 , z )and choose an arbitrary spinor ζ α = (1 , b ) with b a complex number. The tilded spinors127 .3. DIRAC TRACES are then ˜ λ ˙ α = z ! ˜ ζ ˙ α = b ! , and we have h ζ λ i = z − b and h λ dλ i = dz . So, a more covariant statement of (A.2.9) is Z h λ dλ i ¯ δ ( h ζ λ i ) F ( λ ) = Z dz ¯ δ ( z − b ) F ( z )= − iF ( b )= − iF ( ζ ) . (A.2.10) A.3 Dirac traces Some basic formulæ for converting between spinor invariants and Dirac traces are h i j i [ j i ] = tr + ( k/ i k/ j ) , (A.3.1) h i j i [ j l ] h l m i [ m i ] = tr + ( k/ i k/ j k/ l k/ m ) , (A.3.2) h i j i [ j l ] h l m i [ m n ] h n p i [ p i ] = tr + ( k/ i k/ j k/ l k/ m k/ n k/ p ) , (A.3.3)for momenta k i , k j , k l , k m , k n , k p and where the + sign indicates the insertion of(1 + γ ) / 2: tr + ( k/ i k/ j ) := 12 tr + ((1 + γ ) k/ i k/ j ) . (A.3.4)We also note that tr + ( k/ i k/ j ) = 2( k i · k j ) (A.3.5)tr + ( k/ a k/ b k/ c k/ d ) = 2( k a · k b )( k c · k d ) − k a · k c )( k b · k d )+ 2( k a · k d )( k b · k c ) − iε ( k a , k b , k c , k d ) . (A.3.6)The following identities are additionally useful:tr + ( k/ i k/ j k/ l k/ m ) = tr + ( k/ m k/ l k/ j k/ i ) = tr + ( k/ l k/ m k/ i k/ j ) , (A.3.7)tr + ( k/ i k/ j k/ l k/ m ) = 4( k i · k j )( k l · k m ) − tr + ( k/ j k/ i k/ l k/ m ) , (A.3.8)tr + ( i/ j/ µ P/ ) tr + ( i/ j/ µ m/ ) = 0 , (A.3.9)tr + ( i/ j/ µ P/ ) tr + ( i/ j/ m/ µ ) = 4( i · j ) tr + ( i/ j/ m/ P/ ) (A.3.10)128 .3. DIRAC TRACES for similarly generic momenta and where we use the shorthand tr + ( k/ i k/ j ) = tr + ( i/j/ ) etc .If k i , k j , k m and k m are massless, while P L is not necessarily so, then we have theremarkable identity:2( k m · k m )tr + ( k/ i k/ j k/ m P/ L )tr + ( k/ i k/ j k/ m P/ L )+ P L tr + ( k/ i k/ j k/ m k/ m )tr + ( k/ i k/ j k/ m k/ m ) − k m · P L )tr + ( k/ i k/ j k/ m k/ m )tr + ( k/ i k/ j k/ m P/ L ) − k m · P L )tr + ( k/ i k/ j k/ m P/ L )tr + ( k/ i k/ j k/ m k/ m ) = 0 . (A.3.11)We also have, for null momenta i, j, k, a, b, tr + ( i/ j/ a/ b/ )tr + ( j/ a/ k/ b/ )( j · a ) = − tr + ( i/ j/ b/ a/ )tr + ( i/ a/ k/ b/ )( i · a ) . (A.3.12)129 PPENDIX BFEYNMAN RULES IN THE SPINORHELICITY FORMALISM In this appendix we present the Feynman rules for massless SU( N c ) Yang-Mills theoryin Feynman gauge written in the spinor helicity formalism for comparison with thoselaid out at the start of Chapter 1. As mentioned in a footnote in § T a T b ) = δ ab for the Lie-algebra in order to reduce the proliferation offactors of 2. B.1 Wavefunctions • External Scalar: φ = 1 (B.1.1) • External outgoing fermion i , helicity plus: ψ + i = ˜ λ i ˙ α = [ i | (B.1.2) • External outgoing fermion i , helicity minus: ψ − i = λ αi = h i | (B.1.3) • External outgoing anti-fermion j , helicity plus: ¯ ψ + j = ˜ λ ˙ αj = | j ] (B.1.4) • External outgoing anti-fermion j , helicity minus: ¯ ψ − j = λ j α = | j i (B.1.5) • External outgoing vector p = λ ˜ λ , helicity plus: ǫ + α ˙ α = √ µ α ˜ λ ˙ α h µ λ i = √ | µ i [˜ λ |h µ λ i (B.1.6)130 .2. PROPAGATORS • External outgoing vector p = λ ˜ λ , helicity minus: ǫ − α ˙ α = √ λ α ˜ µ ˙ α [˜ µ ˜ λ ] = √ | λ i [˜ µ | [˜ µ ˜ λ ] , (B.1.7)where q = µ ˜ µ is an arbitrary reference spinor that can be chosen independently for eachexternal particle. All the above wavefunctions are understood to be multiplied by afactor of exp( ix β ˙ β λ β ˜ λ ˙ β ), where p β ˙ β = λ β ˜ λ ˙ β is the momentum of the particle. B.2 Propagators • Scalars with kinetic term ( ∂φ ) / : ip (B.2.1) • Fermions with p = λ ˜ λ and kinetic term ¯ ψi ¯ σ µ ∂ µ ψ : i ¯ σ µ p µ = ip α ˙ α p = i | λ i [˜ λ | p (B.2.2) • Vectors with kinetic term − ( ∂A ) / : − iǫ ˙ α ˙ β ǫ αβ p (B.2.3) B.3 Vertices Fermion Vertex: ˙ ββ ˙ α α = ig √ δ βα δ ˙ β ˙ α αα ˙ γγ ˙ ββp p p = − g √ ǫ ˙ α ˙ β ǫ αβ ( p − p ) ˙ γγ + ǫ ˙ β ˙ γ ǫ βγ ( p − p ) ˙ αα + ǫ ˙ γ ˙ α ǫ γα ( p − p ) ˙ ββ ]131 .4. EXAMPLES ββ ˙ αα ˙ δδ ˙ γγ = ig ǫ ˙ α ˙ γ ǫ αγ ǫ ˙ β ˙ δ ǫ βδ − ǫ ˙ α ˙ δ ǫ αδ ǫ ˙ β ˙ γ ǫ βγ − ǫ ˙ α ˙ β ǫ αβ ǫ ˙ γ ˙ δ ǫ γδ ] .Figure B.1: The Vertices of the colour-stripped scheme in terms of spinors. For more details on how these arise see for example [153]. B.4 Examples Let us consider how we get the A (1 − g , − g , + g , + g ) gluon amplitude. The diagramscontributing to this amplitude are shown in Figure B.2 p − p − p +4 p +3 + p − p − p +4 p +3 + p − p − p +4 p +3 Figure B.2: The diagrams contributing to the 4-gluon MHV tree-amplitude. All externalmomenta are taken to be outgoing. In order to calculate the amplitude we need to specify external wavefunctions asprescribed by the Feynman rules and for gluons this includes a choice of referencemomentum. In order to minimise the number of terms we need to consider we willmake the choices q = q = p and q = q = p . This means that the wavefunctions132 .4. EXAMPLES are ǫ − α ˙ α = √ λ α ˜ λ α [4 1] ǫ − α ˙ α = √ λ α ˜ λ α [4 2] ǫ +3 α ˙ α = √ λ α ˜ λ α h i ǫ +4 α ˙ α = √ λ α ˜ λ α h i , while momentum conservation for the second (going from left to right) two diagramsreads P = − ( p + p ) = p + p and P = − ( p + p ) = p + p where P and P are the momenta of the propagators of the respective diagrams. By writing down theFeynman rules for the different diagrams it can quickly be seen that the contributionsof the 1st and the 3rd diagrams both vanish. The 2nd diagram gives: A = (cid:18) − g √ (cid:19) (cid:16) √ (cid:17) λ α ˜ λ α λ β ˜ λ β λ σ ˜ λ σ λ τ ˜ λ τ [4 1][4 2] h ih i h ǫ ˙ α ˙ β ǫ αβ ( p − p ) ˙ γγ + ǫ ˙ β ˙ γ ǫ βγ ( p − P ) ˙ αα + ǫ ˙ γ ˙ α ǫ γα ( P − p ) ˙ ββ i × − iǫ ˙ γ ˙ δ ǫ γδ P × h ǫ ˙ σ ˙ τ ǫ στ ( p − p ) ˙ δδ + ǫ ˙ τ ˙ δ ǫ τδ ( p + P ) ˙ σσ + ǫ ˙ δ ˙ σ ǫ δσ ( − P − p ) ˙ ττ i = − ig h i [2 1] ( λ α λ α )(˜ λ α ˜ λ ˙ α )( λ τ λ τ )(˜ λ τ ˜ λ ˙ τ )(˜ λ σ ˜ λ β ǫ ˙ β ˙ σ )( ǫ βσ λ σ λ β )[4 1][4 2] h ih i = − ig h i [3 4] h i [1 2][4 1] . (B.4.1)This is our answer, though it is in a rather unfamiliar form! We can convert it intosomething more familiar by multiplying both top and bottom by h ih i . We thenuse momentum conservation in the numerator in the form h i [3 4] = −h i [1 4] andrecognise that s := 2( p · p ) = h i [4 3] = s = h i [2 1] to give A = − ig h i ( h i [3 4])( h i [4 3])[1 2] h ih ih i [41]= − ig h i h ih ih i , (B.4.2)which is the usual form for the Parke-Taylor amplitude at 4-point. MHV q¯q → gg Again we take all momenta to be outgoing. Momentum conservation is the same asfor diagrams 2 and 3 of the previous example and for the gluon wavefunctions we take q = p and q = p . This gives polarisation vectors ǫ − α ˙ α = √ λ α ˜ λ α [4 3] ǫ +4 α ˙ α = √ λ α ˜ λ α h i . .4. EXAMPLES p +2 p − p +4 p − + p +2 p − p +4 p − Figure B.3: The diagrams for ˜ A (1 − q , +¯ q , − g , + g ) . The second diagram can be seen to vanish while the first gives:˜ A = (˜ λ ˙ α ǫ αγ λ γ )( ig √ δ βα δ ˙ β ˙ α ) (cid:18) − iǫ ˙ β ˙ δ ǫ βδ P (cid:19) (cid:18) − g √ (cid:19) (cid:16) ǫ ˙ σ ˙ τ ǫ στ ( p − p ) ˙ δδ + ǫ ˙ τ ˙ δ ǫ τδ ( p + P ) ˙ σσ + ǫ ˙ δ ˙ σ ǫ δσ ( − P − p ) ˙ ττ (cid:17) ( √ λ σ ˜ λ σ λ τ ˜ λ τ [4 3] h i = − g (˜ λ ˙ β ǫ ˙ β ˙ δ ǫ ˙ δ ˙ σ ˜ λ σ )( λ β ǫ βδ ǫ δσ λ σ )( λ τ λ τ )(˜ λ τ ˜ λ ˙ τ ) h i [2 1][4 3] h i = − g h i [4 2] h i [2 1] h i = 4 g h i h ih ih i = i h ih i A (1 − g , + g , − g , + g ) , (B.4.3)thus verifying the relations between amplitudes that we derived from supersymmetricWard identities in § MHV q¯q → q¯q As a final example let us consider the amplitude ˆ A (1 − q , +¯ q , − q , +¯ q ). This time bothof the diagrams are non-zero. The first one givesˆ A = (˜ λ ˙ α ǫ αγ λ γ )( ig √ δ δα δ ˙ δ ˙ α ) (cid:18) − iǫ ˙ δ ˙ σ ǫ δσ P (cid:19) ( ig √ δ σβ δ ˙ σ ˙ β )(˜ λ ˙ β ǫ βτ λ τ )= − ig (˜ λ β ˜ λ ˙ β )( λ β λ β ) h i [2 1]= − ig [2 4] h ih i [2 1]= 4 ig h i h ih i . (B.4.4)134 .4. EXAMPLES p +2 p − p +4 p − + p +2 p − p +4 p − Figure B.4: The diagrams for ˆ A (1 − q , +¯ q , − q , +¯ q ) . A similar calculation - or equivalently the realisation that diagrams two is simply thesame as diagram one with 2 ↔ A = 4 ig h i h ih i (B.4.5)for the second diagram and thus the total isˆ A = ˆ A + ˆ A = 4 ig h i h ih ih ih i ( h ih i + h ih i )= −A (1 − g , + g , − g , + g ) (cid:18) h ih i + h ih ih i (cid:19) . (B.4.6)135 PPENDIX CD-DIMENSIONAL LORENTZ-INVARIANTPHASE SPACE In this appendix we expound on the D -dimensional measure for Lorentz-invariant two-body phase space, ultimately focussing on the case D = 4 − ǫ . C.1 D-spheres One thing that we will need to consider is the volume of a D -dimensional unit sphere V ( S D ). We mean this in the sense of a D -sphere regarded as a manifold. Thus thevolume we are talking about is the volume of that manifold rather than the volumeenclosed by it when it is regarded as being embedded in one-dimension higher. Thus V ( S ) = 2 π - the circumference of a circle - and V ( S ) = 4 π , the surface area of a spheresuch as the Earth.In fact we can parametrize a round D -sphere in terms of D angles θ i . In this casethe volume element of an S D is given by dV ( S D ) = dθ . . . dθ D (sin θ ) D − (sin θ ) D − . . . (sin θ D − ) , (C.1.1)with the result V ( S D ) = Z θ j = π,j = Dθ D =2 πθ i =0 dV ( S D )= 2 π D +12 Γ (cid:0) D +12 (cid:1) . (C.1.2) C.2 d LIPS Recall from Chapter 1, Equation (1.8.13) that d D LIPS( l − , − l +1 ; P ) = d D l d D l δ (+) ( l ) δ ( − ) ( l ) δ ( D ) ( P + l − l ) , (C.2.1)136 .2. D LIPS where δ ( ± ) ( l ) := θ ( ± l ) δ ( l ), θ is the unit step function and l the 0-component (en-ergy) of l . If we also remember that Z dx g ( x ) δ ( f ( x ) − a ) = g ( x ) (cid:12)(cid:12)(cid:12) dfdx (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = x ; f ( x )= a (C.2.2)then we can integrate over the 0-components of l and l to get d D LIPS = d D − ~l | l | d D − ~l | l | δ ( D ) ( P + l − l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l = | ~l | l = −| ~l | , (C.2.3)where ~l represents the spatial components of the D -vector l . Furthermore, going to thecenter of mass frame for the vector P µ , P = ( P ,~ 0) we can use D − d D LIPS = d D − ~l | ~l | d D − ~l | ~l | δ ( D − ( ~l − ~l ) δ ( P − | ~l | )= 12 d D − ~l | ~l | δ (cid:18) | ~l | − P (cid:19) . (C.2.4)Now, for d n ~l we can write d n ~l = d | ~l | | ~l | n − dV ( S n − ) , (C.2.5)so we have d D − | ~l | = d | ~l | | ~l | D − dθ dθ (sin θ ) D − (sin θ ) D − × (cid:0) dθ . . . dθ D − (sin θ ) D − . . . (sin θ D − ) (cid:1) = d | ~l | | ~l | D − dθ dθ (sin θ ) D − (sin θ ) D − dV ( S D − ) . (C.2.6)For our case of a 2-particle phase space in 4 − ǫ dimensions, 2 angles θ and θ aresufficient and none of the momenta will depend on any of the other angles. We can thusintegrate over them to get d D − | ~l | = d | ~l | | ~l | D − dθ dθ (sin θ ) D − (sin θ ) D − V ( S D − )= 2 π D − Γ (cid:0) D − (cid:1) d | ~l | | ~l | D − dθ dθ (sin θ ) D − (sin θ ) D − . (C.2.7) Not to be confused with the angles θ i of (C.1.1) and (C.1.2). .3. OVERALL AMPLITUDE NORMALISATION With D = 4 − ǫ this leads us to d − ǫ LIPS = 12 d − ǫ | ~l | δ (cid:18) | ~l | − P (cid:19) = π − ǫ (cid:0) − ǫ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) P (cid:12)(cid:12)(cid:12)(cid:12) − ǫ dθ dθ (sin θ ) − ǫ (sin θ ) − ǫ = π − ǫ (cid:0) − ǫ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) P (cid:12)(cid:12)(cid:12)(cid:12) − ǫ dθ dθ (sin θ ) − ǫ (sin θ ) − ǫ , (C.2.8)and Z d − ǫ LIPS = π − ǫ (cid:0) − ǫ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) P (cid:12)(cid:12)(cid:12)(cid:12) − ǫ Z πθ =0 Z πθ =0 dθ dθ (sin θ ) − ǫ (sin θ ) − ǫ . (C.2.9) C.3 Overall amplitude normalisation In the original papers of [38, 42], the one-loop amplitudes derived are normalised witha factor of c Γ = r Γ / (4 π ) − ǫ where r Γ = Γ(1 + ǫ )Γ (1 − ǫ )Γ(1 − ǫ ) . (C.3.1)In [37, 40, 43] and this thesis, however, the normalisation most naturally arises as πǫ sin πǫ (cid:0) − ǫ (cid:1) , (C.3.2)where the gamma function comes from the LIPS measure described above and the factorof πǫ csc πǫ comes from performing the dispersion integral (see e.g. Section 5 of [37]).We are mostly interested in the results of these amplitude calculations up to order ǫ ,and as (C.3.2) = 1 / √ π + O ( ǫ ) we have usually dropped it as an uninteresting overallfactor. Nonetheless, the all-orders in ǫ results can be useful and we will here show howthe two are related.To start with there is the product identity for gamma functions:Γ( z )Γ(1 − z ) = π sin πz , (C.3.3)which can be combined with the well-known recurrence relation z Γ( z ) = Γ( z + 1) togive πǫ sin πǫ = Γ(1 + ǫ )Γ(1 − ǫ ) . (C.3.4)There is also the Legendre duplication formula:Γ( z )Γ ( z + 1 / 2) = 2 − z √ π Γ(2 z ) , (C.3.5)138 .3. OVERALL AMPLITUDE NORMALISATION which implies that Γ (1 / − ǫ ) = Γ(1 − ǫ ) √ π ǫ Γ(1 − ǫ ) . (C.3.6)This therefore leads us to πǫ sin πǫ (cid:0) − ǫ (cid:1) = 14 ǫ √ π Γ(1 + ǫ )Γ (1 − ǫ )Γ(1 − ǫ )= r Γ ǫ √ π , (C.3.7)and we can see that the two are the same up to a simple factor.139 PPENDIX DUNITARITY Unitarity is a well-known and useful tool in quantum field theory [210, 233, 234, 235,236, 237]. The unitarity of the S -matrix, S † S = 1, is the basic starting point and leadsto the possibility of being able to reconstruct scattering amplitudes from the knowledgeof their properties as functions of complex momenta. In certain cases this can lead to apurely algebraic construction of amplitudes.It can be checked that each Feynman diagram contributing to an S -matrix element S is purely real unless some denominator vanishes, in which case the iε prescription fortreating poles becomes relevant. We thus get an imaginary part for S only when virtualparticles in a Feynman diagram go on-shell.Consider now S ( s ) as an analytic function of a complex variable s . s is the squareof the centre of mass energy, and while this is physically real we will consider it tobe complex for now. If s is the minimum (square of the) energy for production of thelightest multiparticle state ( i.e. the minimum energy for the creation of an intermediatemultiparticle state such as when a loop is formed in a Feynman diagram), then for real s lying below s , the intermediate state cannot go on-shell. S ( s ) is thus real and wehave S ( s ) = S (¯ s ) . (D.0.1)However, as we are regarding S ( s ) as an analytic function of s , we can analyticallycontinue this equation to anywhere in the complex plane. If we explicitly split S ( s ) intoits real and imaginary parts, S ( s ) = ℜ [ S ( s )] + i ℑ [ S ( s )], then at a point s > s that is ε away from the real line (D.0.1) implies that ℜ [ S ( s + iε )] = ℜ [ S ( s − iε )] , ℑ [ S ( s + iε )] = −ℑ [ S ( s − iε )] . (D.0.2)There is thus a branch cut along the positive real axis starting at s and the discontinuity D of S ( s ) across the cut is D [ S ( s )] = 2 i ℑ [ S ( s + iε )] . (D.0.3) Note that some of this appendix is based on Section 7.3 of [2]. .1. THE OPTICAL THEOREM It turns out that this discontinuity - which only arises because we have intermediatemultiparticle states and thus loop contributions to Feynman diagrams - can be relatedto simpler amplitudes which may be known already or more easily computed. This isthe content of the optical theorem which we review below. D.1 The optical theorem The S -matrix is a unitary operator which evolves the initial states k a so that one maycompute their overlap with the final states p i in a scattering process: out h p i | k a i in = h p i | S | k a i . (D.1.1)It is conventional to split S into the part that describes unimpeded propagation of theinitial particles and a part T due to interactions, S = 1 + iT . The matrix element(D.1.1) taken with the interacting part of S is what then gives a scattering amplitude.More concretely, we can write h p i | T | k a i = (2 π ) δ (4) (cid:16)X ( p i + k a ) (cid:17) S ( k a → p i ) , (D.1.2)where we have taken all particles to be outgoing.Unitarity of S , S † S = 1 implies − i ( T − T † ) = T † T , (D.1.3)and we may extract some useful information by taking the matrix element of this be-tween some particle states p i and k a . The LHS of (D.1.3) gives − i ( h p i | T | k a i − h p i | T † | k a i ) = − i (cid:16) h p i | T | k a i − h k a | T | p i i (cid:17) = − i (2 π ) δ (4) (cid:16)X ( p i + k a ) (cid:17) (cid:0) S ( k a → p i ) − S ( p i → k a ) (cid:1) = − i (2 π ) δ (4) (cid:16)X ( p i + k a ) (cid:17) D [ S ( p i ; k a )] . (D.1.4)On the RHS of (D.1.3) we can insert the identity operator as a sum over a complete set141 .2. CUTTING RULES of intermediate states to obtain h p i | T † T | k a i = X n n Y j =1 Z d l j (2 π ) δ ( l j − m j ) h p i | T † | l j ih l j | T | k a i = (2 π ) X n n Y j =1 Z d l j (2 π ) δ ( l j − m j ) S ( p i → l j ) S ( k a → l j ) × δ (4) (cid:16)X ( p i + l j ) (cid:17) δ (4) (cid:16)X ( k a − l j ) (cid:17) = (2 π ) δ (4) (cid:16)X ( p i + k a ) (cid:17) X n Z d LIPS( n ) S ( p i → l j ) S ( k a → l j ) , (D.1.5)where d LIPS( n ) is the n -body Lorentz-invariant phase space measure. Putting the LHSand RHS of (D.1.3) back together again we find − i D [ S ( p i ; k a )] = X n Z d LIPS( n ) S ( p i → l j ) S ( k a → l j ) . (D.1.6)Equation (D.1.6) says that the discontinuity of a scattering amplitude may be ob-tained as a sum of integrals over the phase spaces of intermediate multiparticle states ofthe amplitudes for scattering of the initial and final states into these intermediate states.In particular, for a one-loop process, the amplitudes arising on the RHS of (D.1.6) aretree-level amplitudes and the phase space is a 2-particle one. D.2 Cutting rules Cutkosky showed that using some cutting rules, one may compute the physical dis-continuity of any Feynman diagram and prove the optical theorem to all orders inperturbation theory [210]. The rules are as follows [2]:1. Cut through a diagram in all possible ways such that the cut propagators may beput on-shell.2. For each cut (massive) propagator replace 1 / ( p − m + iε ) with a delta function − πiδ ( p − m ). This explicitly provides the delta functions which generate the d LIPS measure in (D.1.5). The off-shell vertices that are separated by the cut arethus put on-shell. For massless momenta the replacement is simply 1 / ( p + iε ) →− πiδ ( p ). In fact the optical theorem is usually stated in terms of the forward scattering amplitude, in whichcase we have k a = p i . The theorem is more general than this though and can be applied to genericasymptotic states. .2. CUTTING RULES 3. Sum the contributions of all possible cuts.For example, for a Feynman diagram in massless λφ theory such as Figure D.1, the p p l l Figure D.1: The cut of a bubble diagram in massless λφ theory. Feynman rules would give A ∝ δ (4) ( p + p ) Z d l (2 π ) d l (2 π ) λ l l λδ (4) ( p + l − l ) . (D.2.1)Cutkosky’s rules on the other hand would give D [ A ] ∝ δ (4) ( p + p ) Z d l (2 π ) d l (2 π ) λδ ( l ) δ ( l ) λδ (4) ( p + l − l ) ∝ λ δ (4) ( p + p ) Z d LIPS( l , − l ; p ) , (D.2.2)which allows one to calculate the discontinuity of the diagram concerned. D.2.1 BDDK’s unitarity cuts In [38, 42] Cutkosky’s rules were applied at the level of amplitudes to derive one-loopMHV amplitudes in supersymmetric and non-supersymmetric gauge theories. In thiscase the factors on either side of the cut are not vertices ( e.g. the λ factors of (D.2.2)),but full amplitudes. In fact for the one-loop MHV amplitudes these factors are tree-levelMHV amplitudes.Consider for concreteness the n -point one-loop MHV amplitudes for gluon scatteringin N = 4 super-Yang-Mills as reviewed in § .2. CUTTING RULES l l k − j k − i k + m +1 k + m − MHV MHV++ −− k + m k + m Figure D.2: The cut of a one-loop MHV amplitude in the t [ m − m +1] m channel. procedure which means that these dispersion integrals do not actually need to be done.This involves replacing the delta functions associated with the cuts with propagators(a procedure that is known as ‘reconstruction of the Feynman integral’) which thenproduces Feynman integrals rather than LIPS integrals. These integrals contain cuts inthe channel being considered (as well as cuts in other channels too) and by consideringall channels and avoiding over-counting the amplitude can be re-constructed.When we cut the amplitudes, we must assign helicities to the particles that were inthe loop. Since we use conventions in which all particles are outgoing, the helicities ofthese internal particles are reversed. For the one-loop MHV amplitudes there are twodistinct cases. Case ( a ) is where the negative-helicity external particles i and j are onthe same side of the cut, and case ( b ) is where they are on opposite sides of the cut.Case ( a ), is a priori the simpler of the two as the two internal particles must have thesame helicities and thus amplitude relations of equations (1.4.9) and (1.4.10) mean thatonly gluons can circulate in the loop. This is the situation regardless of the amount ofsupersymmetry present. Case ( b ) involves the entire multiplet circulating in the loopand for maximally supersymmetric Yang-Mills it turns out that this case is the same ascase ( a ) after applying identities such as the Schouten identity (A.1.11). For the casebeing considered of N = 4 Yang-Mills it is thus enough for us to treat case ( a ) only.Consider now a cut in the channel where P L , the momentum on the left of the cut,is given by P L = ( k m + k m +1 + . . . + k m − + k m ) = t [ m − m +1] m and where k i , k j ∈ P L .144 .3. DISPERSION RELATIONS This situation is shown in Figure D.2 and the rules that we have outlined above give D [ A ( t [ m − m +1] m )] = Z d l (2 π ) d l (2 π ) A MHVtree ( − l +1 , m +1 , . . . , i − , . . . , j − , . . . , m +2 , l +2 ) × δ ( l ) δ ( l ) A MHVtree ( − l − , m + 1 + , . . . , m − + , l − ) ≡ i (2 π ) A MHVtree ( i − , j − ) Z d LIPS( l , − l ; P L ) ˆ R (D.2.3) → i (2 π ) A MHVtree ( i − , j − ) Z d l d l l l ˆ R , (D.2.4)where ˆ R := h m − m ih l l ih m − l ih− l m i h m m + 1 ih l l ih m l ih− l m + 1 i (D.2.5)as in (1.9.12) and the MHV amplitudes for negative-helicity gluons l, s are defined as in(1.9.10): A MHVtree ( l − , s − ) := i (2 π ) δ (4) X i k i ! h l s i Q nr =1 h r r + 1 i . (D.2.6)Note that Equation (D.2.4) is a Feynman integral rather than a LIPS integral.Now recall from § e.g. Appendix I of [42]). The Feynman integrals generated in (D.2.4) (and for other channels)can then be compared with the Feynman integrals for the known integral functionsand the amplitude recreated. Since the integral functions are already known one canreconstruct the amplitude in a purely algebraic manner. As a strong check of the finalexpression, the results can be compared with the known behaviour (on general grounds)for the collinear ( p a , p b → p a k p b ) and soft ( p a → 0) limits of such an amplitude.For supersymmetric theories any terms which do not contain cuts are uniquely linkedto the cut-containing terms and thus the entire amplitude is reconstructed. In partic-ular, the N = 4 amplitudes discussed above can be completely constructed in this wayleading to (1.9.1). In non-supersymmetric theories more information is needed to getthe rational (cut-free) terms and thus only the cut-constructible part may be obtainedthis way. D.3 Dispersion relations Imagine now that we stop at (D.2.3) and proceed to do the LIPS integral rather thanuplift to Feynman integrals. If we can actually do this integral we can calculate thediscontinuity of the amplitude directly. However, we would really like to know the wholeamplitude rather than just the imaginary part of it and the natural question is whetherit is possible to arrive at this from what we have so far. For a function with a branchcut, it is in fact possible to reconstruct the real part from the imaginary part and the145 .3. DISPERSION RELATIONS relations which allow one to do this are known as dispersion relations (or sometimesKramers-Kronig relations).By considering a function A ( z ) which is analytic in the complex plane with a branchcut along the positive real axis starting at x , it is possible to show using complexanalysis that ℜ [ A ( x )] = 1 π P Z ∞ x dx ′ x ′ − x ℑ [ A ( x ′ )] + 12 πi I ∞ , (D.3.1)where x ∈ R in the range ( x , ∞ ) and P denotes the Cauchy principal value prescription( i.e. the value of the integral without consideration of the pole at x ′ = x ). I ∞ is thecontribution from the contour at infinity which represents the ambiguity due to possiblerational terms ( i.e. terms which are cut-free functions of the kinematic invariants). I ∞ vanishes in any supersymmetric gauge theory, and while these do contain rationalterms they are fixed uniquely by the supersymmetry once one knows the cut-containingterms [38, 42]. Such theories are said to be cut-constructible (in 4 dimensions). Non-supersymmetric theories are not cut-constructible in 4 dimensions, but are in 4 − ǫ dimensions with ǫ = 0 [86, 87, 213]. While this is a powerful statement, it does meanthat one has to consider the prospect of using amplitudes with particles continued to4 − ǫ dimensions which are not simple.In a sense, the one-loop CSW rules make BDDK’s approach prescriptive for the MHVamplitudes. The imaginary part of the amplitude is constructed as a phase space integraland then the dispersion integral over P L ; z in (1.8.12) performs (D.3.1) with I ∞ absent.For supersymmetric theories this is sufficient to construct the full amplitude, while innon-supersymmetric theories we must find other methods to calculate the rational part. For purely massless theories, x = 0. See e.g. [238] for a fuller explanation of these ideas. PPENDIX EINTEGRALS FOR THE N = 1 AMPLITUDE In this appendix we give details of the integrals needed to compute the discontinuitiesof the N = 1 amplitude discussed in Chapter 2. E.1 Passarino-Veltman reduction In § N = 1 amplitude is the dispersion integral ofthe following phase space integral: C ( m , m ) := Z d LIPS( l , − l ; P L ; z ) tr + ( k/ i k/ j k/ m l/ )tr + ( k/ i k/ j k/ m l/ )( i · j ) ( m · l )( m · l ) . (E.1.1)The full amplitude is then obtained by adding the dispersion integrals of three moreterms similar to (E.1.1) but with m replaced by m − m replaced by m + 1.The goal of this appendix is to perform the Passarino-Veltman reduction [212] of (E.1.1),which will lead us to re-express C ( m , m ) in terms of cut-boxes, cut-triangles and cut-bubbles.The explicit forms for the Dirac traces involve Lorentz contractions over the variousmomenta, so in a short-hand notation we can write these as T ( i, j, m ) µ l µ := tr + ( k/ i k/ j k/ m l/ ) . (E.1.2) C ( m , m ) can then be recast as C ( m , m ) = T ( i, j, m ) µ T ( i, j, m ) ν ( i · j ) I µν ( m , m , P L ; z ) , (E.1.3)where I µν ( m , m , P L ) = Z d LIPS( l , − l ; P L ) l µ l ν ( m · l )( m · l ) . (E.1.4) I µν ( m , m , P L ) contains three independent momenta m , m and P L . On general For the rest of this appendix we drop the subscript z in P L ; z for the sake of brevity. .1. PASSARINO-VELTMAN REDUCTION grounds we can therefore decompose it as I µν = η µν I + m µ m ν I + m µ m ν I + P µL P νL I + m µ m ν I + m µ m ν I + m µ P νL I + P µL m ν I + m µ P νL I + P µL m ν I , (E.1.5)for some coefficients I i , i = 0 , . . . , 9. One can then contract with different combinationsof the independent momenta in order to solve for the I i . For instance, two of theintegrals that we will end up having to do are η µν I µν and m µ m ν I µν . Using momentumconservation l − l + P L = 0 and the identity a · b = ( a + b ) / − ( a − b ) / a , b massless momenta, we can convert these integrals into ones which have the general form˜ I ( a,b ) = Z d LIPS( l , − l ; P L )( l · m ) a ( l · m ) b , (E.1.6)possibly with a kinematical-invariant coefficient, and with a and b ranging over thevalues 1 , , − 1. The results of these integrals are collected in § E.2. As an example, wefind that m µ m ν I µν = Z d LIPS( l , − l ; P L ) ( l · m )( l · m ) − ( m · P L ) Z d LIPS( l , − l ; P L )( l · P L ) . (E.1.7)Considering the values ( a, b ), the case (1 , 1) is a cut scalar box, (1 , 0) and (0 , 1) are cutscalar triangles, (1 , − 1) and ( − , 1) are cut vector triangles, whilst (0 , 0) is a cut scalarbubble.Because of the structure of T ( i, j, m ) µ and T ( i, j, m ) ν , terms with coefficients suchas T ( i, j, m ) µ T ( i, j, m ) ν m µ m ν are zero, and thus some of the I i do not contributeto the final answer. The only contributing terms are found to be I , I , I and I , andwe find that C ( m , m ) = tr + ( k/ i k/ j k/ m P/ L )tr + ( k/ i k/ j k/ m P/ L )( i · j ) I + tr + ( k/ i k/ j k/ m k/ m )tr + ( k/ i k/ j k/ m k/ m )( i · j ) I + tr + ( k/ i k/ j k/ m P/ L )tr + ( k/ i k/ j k/ m k/ m )( i · j ) I + tr + ( k/ i k/ j k/ m k/ m )tr + ( k/ i k/ j k/ m P/ L )( i · j ) I . (E.1.8)The inversion of (E.1.5) in order to find the coefficients is tedious and somewhat lengthy,148 .1. PASSARINO-VELTMAN REDUCTION so we just present the results for the relevant I i in (E.1.8) above: I = 1 N n m · m ) P L ˜ I (0 , − N ( m · P L ) ˜ I (1 , + N ( m · P L ) ˜ I (0 , + 2( m · P L ) ˜ I ( − , + 2( m · P L ) ˜ I (1 , − o , (E.1.9) I = 1( m · m ) N ((cid:20) m · P L ) ( m · P L ) − m · P L )( m · P L )( m · m ) P L + 3( m · m ) (cid:0) P L (cid:1) (cid:21) ˜ I (0 , + (cid:20) m · P L ) ( m · P L ) − 32 ( m · m ) P L (cid:21) N ( m · P L ) ˜ I (1 , − (cid:20) m · P L ) ( m · P L ) − 32 ( m · m ) P L (cid:21) N ( m · P L ) ˜ I (0 , + N I (1 , + 2 (cid:20) ( m · m ) P L − ( m · P L )( m · P L ) (cid:21) ( m · P L ) ˜ I ( − , + 2 (cid:20) ( m · m ) P L − ( m · P L )( m · P L ) (cid:21) ( m · P L ) ˜ I (1 , − ) , (E.1.10) I = 1( m · P L )( m · m ) N ((cid:20) m · P L ) ( m · P L ) − m · P L )( m · P L )( m · m ) P L (cid:21) · ˜ I (0 , + 12 ( m · m ) P L N ( m · P L ) ˜ I (1 , − ( m · P L ) N ( m · P L ) ˜ I (0 , − m · P L )( m · P L ) ˜ I ( − , − ( m · m ) P L ( m · P L ) ˜ I (1 , − ) , (E.1.11) I = 1( m · P L )( m · m ) N ((cid:20) m · P L ) ( m · P L ) − m · P L )( m · P L )( m · m ) P L (cid:21) · ˜ I (0 , + ( m · P L ) N ( m · P L ) ˜ I (1 , − 12 ( m · m ) P L N ( m · P L ) ˜ I (0 , − ( m · m ) P L ( m · P L ) ˜ I ( − , − m · P L ) ( m · P L ) ˜ I (1 , − ) , (E.1.12)where N = ( m · m ) P L − m · P L )( m · P L ). The explicit expressions for the relevant˜ I ( a,b ) are summarised in § E.2.Combining (E.1.8) and (E.1.9)-(E.1.12) with the identity (A.3.11) and the explicitexpressions for the integrals ˜ I ( a,b ) in § E.2, we arrive at the final result (2.2.12).149 .2. BOX & TRIANGLE DISCONTINUITIES FROM PHASE SPACE INTEGRALS E.2 Box & triangle discontinuities from phase space integrals The integrals that arise in the Passarino-Veltman reduction in § E.1 have the generalform: ˜ I ( a,b ) = Z d − ǫ LIPS( l , − l ; P L ; z )( l · m ) a ( l · m ) b , (E.2.1)where we have introduced dimensional regularisation in dimension D = 4 − ǫ [239] inorder to deal with infrared divergences.There are six cases to deal with: ˜ I (0 , , ˜ I (1 , , ˜ I (0 , , ˜ I (1 , , ˜ I ( − , , ˜ I (1 , − , thoughdue to symmetry we can transform ˜ I (1 , into ˜ I (0 , , and ˜ I ( − , into ˜ I (1 , − , so we onlyneed consider four cases overall.Generically we will evaluate these integrals in convenient special frames followingAppendix B of [37], with a convenient choice for m and m . For instance, in the caseof ˜ I (1 , it is convenient to transform to the centre of mass frame of the vector l − l ,so that l = 12 P L ; z (cid:0) , ~v (cid:1) , l = 12 P L ; z (cid:0) − , ~v (cid:1) , (E.2.2)and write ~v = (sin θ cos θ , sin θ sin θ , cos θ ) . (E.2.3)Using a further spatial rotation we write m = ( m , , , m ) , m = ( A, B, , C ) , (E.2.4)with the mass-shell condition A = B + C .After integrating over all angular coordinates except θ and θ , the two-body phasespace measure in 4 − ǫ dimensions becomes (see Appendix C) d − ǫ LIPS( l , − l ; P L ; z ) = π − ǫ (cid:0) − ǫ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) P L ; z (cid:12)(cid:12)(cid:12)(cid:12) − ǫ dθ dθ (sin θ ) − ǫ (sin θ ) − ǫ . (E.2.5)As a result of this and of our parametrizations of l , l , m and m , the integrals takethe form ˜ I ( a,b ) = Λ ( a,b ) π − ǫ (cid:0) − ǫ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) P L ; z (cid:12)(cid:12)(cid:12)(cid:12) − ǫ J ( a,b ) , (E.2.6)150 .2. BOX & TRIANGLE DISCONTINUITIES FROM PHASE SPACE INTEGRALS where Λ (0 , = 1 , (E.2.7)Λ (1 , = 2 P L ; z m , Λ (0 , = − P L ; z m , Λ (1 , = − P L ; z m , Λ ( − , = − m , Λ (1 , − = − m , and J ( a,b ) is the angular integral J ( a,b ) := Z π dθ Z π dθ (sin θ ) − ǫ (sin θ ) − ǫ (1 − cos θ ) a ( A + C cos θ + B sin θ cos θ ) b . (E.2.8)The integrals (E.2.8) have been evaluated in [213] for the values of a and b specifiedabove, and we borrow the results in a form from [214]: J (0 , = 2 π − ǫ , (E.2.9) J (1 , = − πǫ , J (1 , = − πǫ A F (cid:18) , , − ǫ, A − C A (cid:19) , J ( − , = − π (1 − ǫ ) ǫ (1 − ǫ ) F (cid:18) − , , − ǫ, A − C A (cid:19) . Here, A and C will differ depending on which case we are considering and our particularparametrization for it, but in all cases the combinations that arise can be re-expressedin terms of Lorentz-invariant quantities using suitable identities. In the case of J (1 , for example, one uses the easily verified identities N ( P L ; z ) = − P L ; z ( A + C ) m , m · m = m ( A − C ) , (E.2.10)where N ( P L ; z ) was defined in (2.2.14).Eventually, after re-expressing A and C in this way, and upon application of some151 .2. BOX & TRIANGLE DISCONTINUITIES FROM PHASE SPACE INTEGRALS standard hypergeometric identities we find the following: λ − ˜ I (0 , = 2 π − ǫ , (E.2.11) λ − ˜ I (1 , = − ǫ πm · P L ; z ,λ − ˜ I (0 , = 1 ǫ πm · P L ; z ,λ − ˜ I (1 , = − πN ( P L ; z ) ( ǫ + log − ( m · m ) P L ; z N ( P L ; z ) ! + O ( ǫ ) ) ,λ − ˜ I ( − , = π ( m · P L ; z ) (cid:26) − N ( P L ; z ) ǫ + 21 − ǫ (cid:2) ( m · P L ; z )( m · P L ; z ) − ( m · m ) P L ; z (cid:3)(cid:27) ,λ − ˜ I (1 , − = π ( m · P L ; z ) (cid:26) − N ( P L ; z ) ǫ + 21 − ǫ (cid:2) ( m · P L ; z )( m · P L ; z ) − ( m · m ) P L ; z (cid:3)(cid:27) , where λ is the ubiquitous factor λ := π − ǫ (cid:0) − ǫ (cid:1) (cid:12)(cid:12)(cid:12)(cid:12) P L ; z (cid:12)(cid:12)(cid:12)(cid:12) − ǫ . (E.2.12)152 PPENDIX FGAUGE-INVARIANT TRIANGLERECONSTRUCTION In this appendix we find a new representation of the triangle function T ( p, P, Q ) = log( Q /P ) Q − P , (F.0.1)as the dispersion integral of a sum of two cut-triangles. A comment on gauge (in)dependence is in order here. Recall from § η in order to perform loop integrations. The corresponding gauge dependenceshould disappear in the expression for scattering amplitudes. In what follows we willwork in an arbitrary gauge, and show analytically that gauge-dependent terms disappearin the final result for the triangle function. Perhaps unsurprisingly, this gauge invariancewill also hold for the finite- ǫ version of T ( p, P, Q ), which we define in (2.1.14). F.1 Gauge-invariant dispersion integrals To begin with, recall from (2.2.18) that the basic quantity we have to compute reads R := Z dzz (cid:20) ( P z ) − ǫ ( P z p ) + ( Q z ) − ǫ ( Q z p ) (cid:21) , (F.1.1)where P + Q + p = 0. We will work in an arbitrary gauge, where P z := P − zη , Q z := Q + zη . (F.1.2)A short calculation shows that P z p = P p h − b P ( P − P z ) i , (F.1.3) Q z p = Qp h − b Q ( Q − Q z ) i , (F.1.4) For a review of dispersion relations see [237] and Appendix D. .1. GAUGE-INVARIANT DISPERSION INTEGRALS where b P := ηp ηP )( pP ) , b Q := ηp ηQ )( pQ ) . (F.1.5)It is also useful to notice the relation1 b Q = 1 b P + Q − P , (F.1.6)as well as ( P p ) = − ( Qp ) = (1 / Q − P ), which trivially follows from momentumconservation. We can then rewrite (F.1.1) as R = I − I , (F.1.7)where I := 1( P p ) Z ds ′ ( s ′ ) − ǫ s ′ − P ) (cid:2) − b P ( P − s ′ ) (cid:3) (F.1.8)= π csc( πǫ )( P p ) (cid:20) ( − P ) − ǫ − (cid:18) − b P b P P − (cid:19) ǫ (cid:21) , I := 1( P p ) Z ds ′ ( s ′ ) − ǫ s ′ − Q ) (cid:2) − b Q ( Q − s ′ ) (cid:3) (F.1.9)= π csc( πǫ )( P p ) (cid:20) ( − Q ) − ǫ − (cid:18) − b Q b Q Q − (cid:19) ǫ (cid:21) . But (F.1.6) implies − b P b P P − − b Q b Q Q − , (F.1.10)so that we can finally recast (F.1.1) as: R = 2 (cid:2) πǫ csc( πǫ ) (cid:3) ǫ ( − P ) − ǫ − ( − Q ) − ǫ Q − P = 2 (cid:2) πǫ csc( πǫ ) (cid:3) T ǫ ( p, P, Q ) , (F.1.11)where the ǫ -dependent triangle function is T ǫ ( p, P, Q ) := 1 ǫ ( − P ) − ǫ − ( − Q ) − ǫ Q − P . (F.1.12)This is the result we were after. Notice that all the gauge dependence, i.e. any depen-dence on the arbitrary null vector η , has completely cancelled out in (F.1.11).We now discuss the ǫ → § ǫ → R (and hence of T ǫ ( p, P, Q )) we need to distinguish the case where P and Q are both nonvanishing The ǫ -dependent triangle function already appeared in (2.1.14). .1. GAUGE-INVARIANT DISPERSION INTEGRALS from the case where one of the two, say Q , vanishes. In the former case, we get preciselythe triangle function T ( p, P, Q ) defined in (F.0.1):lim ǫ → R = 2 T ( p, P, Q ) , P = 0 , Q = 0 . (F.1.13)In the latter case, where Q = 0, we have insteadlim ǫ → R = − ǫ ( − P ) − ǫ P , P = 0 , Q = 0 , (F.1.14)which corresponds to a degenerate triangle.The final issue is that of the gauge invariance of the contributions to the amplitudefrom the box functions B (this is also relevant to the issue of gauge invariance in the N = 4 calculation of [37], and in that paper a general argument for gauge invariance wasalso given - further evidence can be found in [79]). We expect that an explicit analyticproof of the gauge invariance of the box function contribution to the amplitude couldbe constructed using identities such as those in Appendix B of [37]. In the meantime,numerical tests have shown that gauge invariance is present [209]. Indeed, it would besurprising if this were not the case given that the correct, gauge-invariant, amplitudesare derived with the choices of gauge we have made here and in [37]. We have alsocarried out the MHV diagram analysis of this paper using the alternative gauge choice η = k m ; one obtains (2.1.19). 155 PPENDIX GINTEGRALS FOR THENON-SUPERSYMMETRIC AMPLITUDE In this appendix we give details of the integrals needed to compute the discontinuitiesof the non-supersymmetric amplitude discussed in Chapter 3. G.1 Passarino-Veltman reduction In § C ( m ) := Z d LIPS( l , − l ; P L ; z ) tr + ( k/ k/ P/ L ; z l/ )tr + ( k/ k/ l/ P/ L ; z )tr + ( k/ k/ k/ m l/ )( l · m ) ( k · k ) ( P L ; z ) (G.1.1)The goal of this appendix is to perform the Passarino-Veltman reduction [212] of (G.1.1).To this end, we rewrite C ( m ) as C ( m ) = tr + ( k/ k/ P/ L ; z γ µ ) tr + ( k/ k/ γ ν P/ L ; z ) tr + ( k/ k/ k/ m γ ρ )( k · k ) ( P L ; z ) I µνρ ( m, P L ; z ) , (G.1.2)where I µνρ ( m, P L ) = Z d LIPS( l , − l ; P L ) l µ l ν l ρ ( l · m ) . (G.1.3)On general grounds, I µνρ ( m, P L ) can be decomposed as I µνρ = m µ m ν m ρ J + ( m µ m ν P ρL + m µ P νL m ρ + P µL m ν m ρ ) J + ( m µ P νL P ρL + P µL m ν P ρL + P µL P νL m ρ ) J + P µL P νL P ρL J + ( η µν m ρ + η µρ m ν + η νρ m µ ) J + ( η µν P ρL + η µρ P νL + η νρ P µL ) J (G.1.4) For the rest of this appendix we will generally drop the subscript z in P L ; z for the sake of brevity. .1. PASSARINO-VELTMAN REDUCTION for some coefficients J i , i = 0 , . . . , 6. One can then contract with different combinationsof the independent momenta in order to solve for the J i . Introducing the quantities A := m µ m ν m ρ I µνρ ,B := m µ m ν P Lρ I µνρ ,C := m µ P Lν P Lρ I µνρ ,D := P Lµ P Lν P Lρ I µνρ ,E := η µν m ρ I µνρ = 0 ,F := η µν P Lρ I µνρ = 0 , (G.1.5)the result for the Passarino-Veltman reduction of { J , . . . , J } in the basis { A, . . . , D } is: J = (cid:0) P L ) / (cid:0) m · P L ) (cid:1) , − P L / ( m · P L ) , / ( m · P L ) , (cid:1) , J = (cid:0) − P L / ( m · P L ) , / ( m · P L ) , , (cid:1) , J = (cid:0) / ( m · P L ) , , , (cid:1) J = (cid:0) − ( P L ) / (cid:0) m · P L ) (cid:1) , P L / (cid:0) m · P L ) (cid:1) , − / ( m · P L ) , (cid:1) , J = (cid:0) P L / (cid:0) m · P L ) ) , − / ( m · P L ) , , (cid:1) . (G.1.6)We omit the decomposition for J as the corresponding term in (G.1.4) drops out ofall future expressions due to k m = 0.Finally, using the methods of [40] and the results of § G.3, the integrals in (G.1.5)are found to be, keeping only terms to O ( ǫ ), A = ( m · P L ) π ˆ λ , (G.1.7) B = P L ( m · P L ) π ˆ λ , (G.1.8) C = ( P L ) π ˆ λ , (G.1.9) D = − ( P L ) m · P L ) 4 πǫ ˆ λ , (G.1.10)where ˆ λ := π − ǫ − ǫ Γ (cid:0) − ǫ (cid:1) . (G.1.11)157 .2. EVALUATING THE INTEGRAL OF C ( a, b ) G.2 Evaluating the integral of C ( a, b ) The basic expression which arises in the MHV diagram construction in this paper is C ( a, b ) = h i l i h j l i h i l i h j l ih i j i h l l i h i a i h j b ih l a i h l b i . (G.2.1)We wish to integrate this expression over the Lorentz-invariant phase space. We beginby simplifying it, using multiple applications of the Schouten identity. First note thatusing this identity twice, one deduces that h i l i h j l ih l a i h l b i h a b i = h i a ih b j i + h i a ih a j i h l b ih a l i + h b j ih i b i h l a ih b l i (G.2.2)+ h a j ih i b i − h a j ih i b i h l l ih a b ih a l ih b l i . Now use this identity in C ( a, b ). This generates five terms, which we will label (incorrespondence with the ordering arising from the order of terms in (G.2.2) above) as T i , i = 1 , . . . , 4, and U . The T i have dependence on the loop momenta such that wemay use the phase space integrals of § G.3 to calculate them. The term U is morecomplicated; however, one may again use the identity (G.2.2), generating another fiveterms, which we will label T , . . . , T , and V . Again, the expressions in T i , i = 5 , . . . , § G.3. Finally, the term V may be simplified,here using the identity (G.2.2) with i and j interchanged. This generates a further fiveterms, which we label T , . . . , T . The explicit forms of these terms follow: T = tr + ( i/ j/ b/ a/ ) tr + ( i/ j/ l/ l/ )tr + ( i/ j/ l/ l/ )2 ( i · j ) ( a · b ) ( l · l ) , (G.2.3) T = tr + ( i/ j/ a/ b/ )tr + ( i/ j/ b/ a/ )tr + ( i/ j/ l/ l/ )tr + ( i/ j/ l/ l/ )tr + ( i/ b/ l/ a/ )2 ( i · j ) ( a · b ) ( l · l ) ( i · b )( a · l ) , (G.2.4) T = tr + ( i/ j/ a/ b/ )tr + ( i/ j/ b/ a/ )tr + ( i/ j/ l/ l/ )tr + ( i/ j/ l/ l/ )tr + ( j/ a/ l/ b/ )2 ( i · j ) ( a · b ) ( l · l ) ( j · a )( b · l ) , (G.2.5) T = − tr + ( i/ j/ a/ b/ )tr + ( i/ j/ b/ a/ )tr + ( i/ j/ l/ l/ )tr + ( i/ j/ l/ l/ )2 ( i · j ) ( a · b ) ( l · l ) , (G.2.6)and T = tr + ( i/ j/ b/ a/ ) tr + ( i/ j/ a/ b/ )tr + ( i/ j/ l/ l/ )2 ( i · j ) ( a · b ) ( l · l ) , (G.2.7) T = tr + ( i/ j/ a/ b/ ) tr + ( i/ j/ b/ a/ )tr + ( i/ j/ l/ l/ )tr + ( i/ b/ l/ a/ )2 ( i · j ) ( a · b ) ( l · l )( i · b )( a · l ) , (G.2.8)158 .2. EVALUATING THE INTEGRAL OF C ( a, b ) T = − tr + ( i/ j/ a/ b/ )tr + ( i/ j/ b/ a/ ) tr + ( i/ j/ l/ l/ )tr + ( i/ a/ l/ b/ )2 ( i · j ) ( a · b ) ( l · l )( i · a )( b · l ) , (G.2.9) T = − tr + ( i/ j/ a/ b/ ) tr + ( i/ j/ b/ a/ )tr + ( i/ j/ l/ l/ )2 ( i · j ) ( a · b ) ( l · l ) , (G.2.10)and T = tr + ( i/ j/ a/ b/ ) tr + ( i/ j/ b/ a/ )2 ( i · j ) ( a · b ) , (G.2.11) T = tr + ( i/ j/ a/ b/ ) tr + ( i/ j/ b/ a/ ) tr + ( j/ b/ l/ a/ )2 ( i · j ) ( a · b ) ( j · b )( a · l ) , (G.2.12) T = tr + ( i/ j/ a/ b/ ) tr + ( i/ j/ b/ a/ ) tr + ( i/ a/ l/ b/ )2 ( i · j ) ( a · b ) ( i · a )( b · l ) , (G.2.13) T = − tr + ( i/ j/ a/ b/ ) tr + ( i/ j/ b/ a/ ) ( i · j ) ( a · b ) , (G.2.14) T = tr + ( i/ j/ b/ a/ ) tr + ( i/ j/ a/ b/ ) tr + ( b/ l/ l/ a/ )2 ( i · j ) ( a · b ) ( a · l )( b · l ) , (G.2.15)The expression C ( a, b ) is then the sum of the terms T i , i = 1 , . . . , T + T , T + T , T + T , T + T , T + T and T + T .This leads us to the following decomposition: − C ( a, b ) = tr + ( i/ j/ l/ l/ )tr + ( i/ j/ l/ l/ )tr + ( i/ j/ l/ a/ )tr + ( i/ j/ b/ l/ )2 ( i · j ) ( l · l ) ( l · a )( l · b )= 12 ( i · j ) ( H + · · · + H ) , (G.2.16)where H := tr + ( i/ j/ b/ a/ )tr + ( i/ j/ l/ l/ )tr + ( i/ j/ l/ l/ )( l · l ) ( a · b ) (cid:20) tr + ( i/ j/ l/ a/ )( l · a ) − tr + ( i/ j/ l/ b/ )( l · b ) (cid:21) , H := tr + ( i/ j/ a/ b/ )tr + ( i/ j/ b/ a/ )tr + ( i/ j/ l/ l/ )( l · l ) ( a · b ) (cid:20) tr + ( i/ j/ l/ a/ )( l · a ) − tr + ( i/ j/ l/ b/ )( l · b ) (cid:21) , H := − (tr + ( i/ j/ a/ b/ )) tr + ( i/ j/ b/ a/ )( a · b ) (cid:20) tr + ( i/ j/ l/ a/ )( l · a ) − tr + ( i/ j/ l/ b/ )( l · b ) (cid:21) , H := (tr + ( i/ j/ a/ b/ )) (tr + ( i/ j/ b/ a/ )) tr + ( l/ a/ b/ l/ )4( a · b ) ( l · a ) ( l · b ) . (G.2.17)Finally, we perform the phase space integrals of the above expressions, using the159 .3. PHASE SPACE INTEGRALS formulæ in § G.3 below. One quickly finds that the divergent (as ǫ → 0) part of thetotal expression is zero. The finite part, after further spinor manipulations, becomesthe expression we have given in (3.3.3). G.3 Phase space integrals The basic method which we use for evaluating Lorentz-invariant phase space integralshas been outlined in [37, 40] and also discussed in § R ≡ R d − ǫ LIPS( l , − l ; P L ; z ), and a common factor of 4 π ˆ λ ( − P L ; z ) isunderstood to multiply all expressions, where ˆ λ is the ubiquitous factor of (G.1.11). Wealso define α = ( a · P ), β = ( b · P ), N ( P ) = ( a · b ) P − a · P )( b · P ) and drop the L ; z subscripts from P L ; z for clarity.Firstly we quote the results from Appendix B of [40] up to terms of O ( ǫ ): Z , Z a · l ) = − ǫα , Z b · l ) = 1 ǫβ , (G.3.1) Z a · l )( b · l ) = − N ( P ) (cid:18) ǫ + L (cid:19) , where L = log (cid:18) − ( a · b ) N P (cid:19) . From this, we can recursively derive the following integrals (up to O ( ǫ )): Z l µ = 12 P µ , Z l µ = − P µ , (G.3.2) Z l µ l ν = Z l µ l ν = 13 (cid:18) P µ P ν − η µν P (cid:19) , Z l µ ( a · l ) = − P ǫα a µ + 1 α P µ − P α a µ , Z l µ ( b · l ) = − P ǫβ b µ + 1 β P µ − P β b µ , and Z l µ l ν ( a · l ) = − P ǫα a µ a ν + 12 α P µ P ν + P α P ( µ a ν ) − P α a µ a ν − P α η µν , Z l µ l ν ( b · l ) = P ǫβ b µ b ν − β P µ P ν − P β P ( µ b ν ) + 3 P β b µ b ν + P β η µν , (G.3.3) Z l µ ( a · l )( b · l ) = 1 ǫN (cid:18) P µ − P α a µ + P β b µ (cid:19) + 2 LN (cid:18) P µ − β ( a · b ) a µ + α ( a · b ) b µ (cid:19) . .3. PHASE SPACE INTEGRALS Finally, there are integrals involving cubic powers of loop momenta in the numerator.The first is Z l µ l ν l ρ ( a · l ) = P α P ( µ a ν a ρ ) + P α P ( µ P ν a ρ ) + 13 α P µ P ν P ρ − P α η ( µν a ρ ) − P α η ( µν P ρ ) , (G.3.4)where we have suppressed terms cubic in a as they prove not to contribute when thisintegral is contracted into the products of Dirac traces which appear in the expressionsin § G.2. 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