S-matrix equivalence restored
Chih-Hao Fu, Jonathan Fudger, Paul R.W. Mansfield, Tim R. Morris, Zhiguang Xiao
aa r X i v : . [ h e p - t h ] M a y Preprint typeset in JHEP style - HYPER VERSION
SHEP 09-06,DCPT-09/31
S-matrix equivalence restored
Chih-Hao Fu, † Jonathan Fudger, ‡ Paul R.W. Mansfield, † Tim R. Morris ‡ andZhiguang Xiao ‡ † Department of Mathematical Sciences, University of DurhamSouth Road, Durham, DH1 3LE, U.K. ‡ School of Physics and Astronomy, University of SouthamptonHighfield, Southampton, SO17 1BJ, U.K.E-mails: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
The canonical transformation that maps light-cone Yang-Mills theory to aLagrangian description of the MHV rules is non-local, consequently the two sets of fieldsdo not necessarily generate the same S-matrix. By deriving a new recursion relation forthe canonical transformation expansion coefficients, we find a direct map between thesecoefficients and tree level light-cone diagrams. We use this to show that, at least up toone-loop with dimensionally regularised MHV vertices, the only difference is the omissionof the one-loop amplitudes in which all gluons have positive helicity.
Keywords:
Gauge symmetry, QCD. ontents
1. Introduction 12. Dimensional Regularisation 53. Canonical transformation in D dimensions 6
4. ‘Missing’ amplitudes from equivalence theorem evasion reviewed 125. Equivalence theorem evasion in general. 14
6. One-loop ( − + ++) amplitude 227. One-loop (+ + − ) amplitude with external tadpole dressing propagators 238. Conclusion and higher loops 25A. Some remaining thoughts on translation kernels 27B. Proof of recursion relation (3.9) 32
1. Introduction
The MHV rules of Cachazo, Svrˇcek and Witten [1] are equivalent to a new set of Feynmanrules for QCD tree-level scattering amplitudes that are particularly efficient. Initiallyconjectured on the basis of an analogy with strings moving on twistor space [2] they wereproven by recursion relations [3]. They emerge from gauge fixing a twistor space actionfor Yang-Mills [4]-[10] and can also be derived using a canonical transformation appliedto the light-cone gauge Yang-Mills Lagrangian [11, 12]. To generalise these rules to looplevel requires the introduction of a regulator, for example some variant of dimensionalregularisation. Although much of the mathematical structure underlying this approach toYang-Mills theory, such as conformal invariance and twistor space, is broken by the passageto arbitrary dimension there is some cause for optimism that progress towards formulating– 1 –HV rules for loop processes can still be made [13]. One of the features that emerges atloop order is that the new fields do not generate the same S-matrix as the original onesbecause of the non-locality of the canonical transformation. This effect accounts for theone-loop amplitudes for gluons of purely positive helicity which would otherwise appearto be absent from the theory. However, it is potentially damaging for the efficiency ofthe MHV rules because it would seem to require an extra ingredient in the calculation ofamplitudes to describe the translation between the two sets of fields. It is the purpose ofthis paper to show that although extra structure is required to translate between the twosets of fields, these ‘translation kernels’ are required only for all plus amplitudes at one-loop, so that for the calculation of general amplitudes we are free to use Green functionsfor either set of fields, thus partially regaining the simplicity of the CSW rules for theregulated theory. In theories with exact supersymmetry these problems are absent and itis known that four dimensional MHV vertices and MHV rules may be used to recover allamplitudes at one loop [26].We begin by describing the canonical transformation as it is constructed in four di-mensions. Using light-cone co-ordinates in Minkowski spaceˆ x = √ ( t − x ) , ˇ x = √ ( t + x ) , z = √ ( x + ix ) , ¯ z = √ ( x − ix ) . (1.1)and the gauge condition ˆ A = 0 allows the Yang-Mills action to be written in terms ofpositive and negative helicity fields A ≡ A z and ¯ A ≡ A ¯ z (after elimination of unphysicaldegrees of freedom) as the light-cone action S = 4 g Z d ˆ x Z Σ d x ( L − + + L − ++ + L −− + + L ′ −− ++ ) , (1.2)where L − + = tr ¯ A (cid:16) ˇ ∂ ˆ ∂ − ∂ ¯ ∂ (cid:17) A , (1.3) L − ++ = − tr ( ¯ ∂ ˆ ∂ − A ) [ A , ˆ ∂ ¯ A ] , (1.4) L −− + = − tr [ ¯ A , ˆ ∂ A ] ( ∂ ˆ ∂ − ¯ A ) , (1.5) L ′ −− ++ = − tr [ ¯ A , ˆ ∂ A ] ˆ ∂ − [ A , ˆ ∂ ¯ A ] , (1.6)and Σ is a constant-ˆ x quantisation surface and d x = d ˇ x dz d ¯ z .The combination L − + + L − ++ by itself describes self-dual gauge theory [14]. Attree-level this is a free theory because the only connected scattering amplitudes that canbe constructed involve one negative helicity particle and an arbitrary number of positivehelicity particles. The Feynman diagrams contributing to this are the same as in the fullYang-Mills theory, for which such amplitudes are known to vanish. (Bizarrely, the one-loopamplitudes for processes involving only positive helicity particles are non-zero, and theseare the only non-vanishing amplitudes in the theory.) This encourages us to find a new field B that is a non-local functional of A on the surface of constant ˆ x such that L − + + L − ++ can be written as a free theory, i.e. L − + [ A , ¯ A ] + L − ++ [ A , ¯ A ] = L − + [ B , ¯ B ] , (1.7)– 2 –here ¯ B is determined by the requirement that the transformation be canonical:ˆ ∂ ¯ A a (ˆ x, x ) = Z Σ d y δ B b (ˆ x, y ) δ A a (ˆ x, x ) ˆ ∂ ¯ B b (ˆ x, y ) ⇔ ˆ ∂ ¯ B a (ˆ x, x ) = Z Σ d y δ A b (ˆ x, y ) δ B a (ˆ x, x ) ˆ ∂ ¯ A b (ˆ x, y ) . (1.8)This transformation is readily expressed in terms of the fields after taking the Fouriertransform with respect to position within the quantisation surface B (ˆ x, p ) = A (ˆ x, p ) + ∞ X n =2 Z d k (2 π ) . . . d k n (2 π ) ˆ p n − (2 π ) δ ( p − P k i )( p, k ) ( p, k + k ) . . . ( p, k + · · · + k n − ) A (ˆ x, k ) . . . A (ˆ x, k n )(1.9)where ( k , k ) ≡ ˆ k k − ˆ k k . (1.10)The transformation is therefore local in ˆ x and the coefficients of the products A . . . A areindependent of both ˆ x and ¯ k . (1.8) shows that ¯ A is a linear functional of ¯ B , which we writeas ¯ A (ˆ x, p ) = ¯ B (ˆ x, p ) + ∞ X m =3 m X s =2 Z d k (2 π ) . . . d k n (2 π ) ˆ k s ˆ p Ξ s − ( p , − k , . . . , − k m ) × (2 π ) δ ( p − X k i ) B (ˆ x, k ) . . . ¯ B (ˆ x, k s ) . . . B (ˆ x, k m ) (1.11)so that when the remaining terms in the action are written in the new variables we obtainan infinite series, each term of which contains two powers of ¯ B . Labelling these terms bytheir helicities gives L [ A , ¯ A ] = L − + [ B , ¯ B ] + L −− + [ B , ¯ B ] + L −− ++ [ B , ¯ B ] + L −− +++ [ B , ¯ B ] + · · · . (1.12)The coefficients of the fields in the interaction terms can be shown [16], by explicit calcu-lation, to consist of the Parke-Taylor amplitudes [17] (continued off-shell).The LSZ procedure gives scattering amplitudes in terms of the momentum space Greenfunctions (suitably normalised) for A and ¯ A fields by cancelling each external leg using afactor p and then taking each momentum on-shell, p →
0. The equivalence theorem forS-matrix elements seems to allow us to use Green functions for the B and ¯ B fields insteadof the A and ¯ A , provided we include a multiplicative wave-function renormalisation. Thisis because, to leading order in the fields, A is the same as B . In any Feynman diagramcontributing to a Green function these fields are attached to the rest of the diagram by apropagator ∼ /p which cancels the LSZ factor of p and so survives the on-shell limit.In the higher order terms in (1.9) the momentum p is shared between the A fields, so thepropagators that attach these to diagrams cannot directly cancel p . The cancellation canoccur if the diagram forces just these momenta to flow together through some internal line,because by momentum conservation this line will contribute ∼ /p . The effect of suchdiagrams is to renormalise the field, and this will cancel in the computation of scattering– 3 –mplitudes. Another source of 1 /p could be the kernels in (1.9). These kernels are non-local within the quantisation surface, and a requirement of the equivalence theorem is thatthe transformation be local. However our transformation is still local in light-cone ‘time’ˆ x which means that the kernels are independent of ˇ p (and also, for other reasons, ¯ p ) so itis hard (but not impossible, as we will see) to imagine how the kernels can generate the1 /p needed to stop us generalising the theorem to the case in hand.So it would seem safe to invoke the S-matrix equivalence theorem and use the B fieldsto calculate scattering amplitudes, expecting to get physical gluon amplitudes. It is clearthat the new Lagrangian would then generate the CSW (or MHV) rules of [1], and, once wehave a Lagrangian we are much closer to being able to generalise the rules beyond tree-level.However, this cannot be correct as the rules cannot generate the one-loop amplitudes forprocesses in which the gluons all have positive helicity. These amplitudes have long beenconsidered to be related to an anomaly [18]. In the context of the change of variables from A to B this anomaly could be related to the Jacobian which ought to be unity since thetransformation is canonical. However, in [15] it was shown instead that these amplitudesresult from an evasion of the equivalence theorem when the theory is formulated usingdimensional regularisation. This implies a flaw in the argument we have just presented.Specifically, it was shown that in the case of the four-point all-plus amplitude the changeof variables can be implemented with unit Jacobian by directly comparing both sides of:lim p i → Z D ( A , ¯ A ) e iS lc p ¯ A a ( p ) . . . p ¯ A a ( p ) =lim p i → Z D ( B , ¯ B ) e iS MHV p { ¯ B a ( p ) + . . . } . . . p { ¯ B a ( p ) + . . . } (1.13)where the dots in ¯ B ( p ) + . . . represent the extra terms involving the Ξ in (1.11). If weignored these extra terms, as the S-matrix equivalence theorem implies we should, then theright-hand side would vanish because there are no interactions in S MHV that would allowus to contract all the ¯ B together. Since it is known that this amplitude is in fact non-zerothe extra terms must contribute and the equivalence theorem is not directly applicable.These extra terms appear to spoil the efficiency of our approach. If we have to includethe details of the transformation in computing scattering amplitudes then we are unlikelyto be able to profit from any gains resulting from the simplicity of the MHV Lagrangian.It is the purpose of this paper to investigate just how damaging this is. We will see thatactually the problem is quite contained and the equivalence theorem is only spoilt for aclass of known amplitudes.To simplify our discussion we will regulate using Four-Dimensional-Helicity regular-isation [19] in which the external helicity are in four dimensions and only the internalmomenta are in D dimensions. It is not essential to use this scheme, and in our earlierpaper [15] we used standard dimensional regularisation, but it will simplify our expressionsconsiderably. In section 2, we will describe this. Then in section 3, we examine the canon-ical transformation using it. We will find that the effect of regularisation is to make onlyminor changes to the recursion relations for the expansion coefficients. In order to avoid– 4 –purious poles in the recursion expansion of Ξ s , we also establish a new recursion relationof Ξ s which involves only true singularities in each term of the expansion. As a byproductwe also find a relation between the tree-level light-cone diagrams and the canonical ex-pansion coefficients, which will facilitate the singularity analysis in the translation kernelcontribution later. We also review the tree-level evasion of the equivalence theorem for the( − + +) amplitude in section 4.After this preparation, in section 5, we will discuss systematically the different waysthat the S-matrix equivalence theorem can be evaded. We first argue that at tree-levelevasion will not occur in higher point amplitudes. Then we discuss the three ways that thetheorem can potentially be evaded at one-loop: by dressing propagators, in tadpoles, andby infrared divergences. We will conclude that only tadpoles can evade the equivalencetheorem at one-loop. During this discussion, we find that there is a puzzle in the (+ + − )amplitude with an external leg dressed by a tadpole. By examining the calculation of the(+ + + − ) amplitude in section 6, we find that the one-minus-helicity amplitudes shouldcome just from tadpoles made out of MHV vertices, but when we cut the diagrams thereappear to be additional contributions from equivalence theorem evading tadpoles whichcan dress external legs. In section 7, we resolve this double-counting puzzle by choosinga suitable limiting order in the LSZ procedure and show that these extra terms do notcontribute to the on-shell amplitude. Section 8 is the conclusion.
2. Dimensional Regularisation
We will regulate the ultra-violet divergences of pure Yang-Mills by working in arbitraryspace-time dimension, D , and using co-ordinates which replace the pair z, ¯ z of complexspace-like co-ordinates by D/ − z ( i ) , ¯ z ( i ) . In [15] we used standard dimen-sional regularisation in which the gauge-field A µ has D space-time components. We couldinstead use four-dimensional-helicity regularization (FDH) [19] and keep µ four dimen-sional. Consequently polarisation vectors would remain four dimensional, so we retain justtwo helicities, and the gauge invariance of the action is four dimensional. Just as in theusual dimensional regularisation the momenta of ‘physical’ gluons which appear in asymp-totic states of scattering processes also remain in four dimensions, but the momenta ofvirtual gluons that appear as internal lines in Feynman diagrams will be D dimensional.The advantage of FDH is that the light-cone gauge action is very similar to the four di-mensional version, the only change being in the free part which becomes L − + = tr ¯ A ˇ ∂ ˆ ∂ − D/ − X i =1 ∂ ( i ) ¯ ∂ ( i ) A . Tree-level amplitudes are unchanged when the external legs all have four dimensional mo-menta, however when the external legs are allowed to have D dimensional momenta thenthey are modified. In particular the amplitudes in which all but one of the scattered glu-ons have the same helicity no longer vanish. This is responsible for the non-vanishing ofthe one-loop amplitude in which all the scattered gluons have the same helicity because– 5 –he optical theorem relates the imaginary part of this latter amplitude to the product oftree-level amplitudes of the former. The one-loop four-gluon all positive helicity reducedamplitude is [20] − ig π { p , p } { p , p } ( p , p ) ( p , p ) (2.1)where p , . . . , p are the momenta of the gluons and { p , p } ≡ ˆ p ¯ p − ˆ p ¯ p . The all-plusone-loop amplitudes are missing from a na¨ıve application of the MHV rules at one-loopbecause if we are limited to vertices of Parke-Taylor type then we cannot construct suchamplitudes. (In [15] it was shown that such amplitudes originate in a failure of the S-matrixequivalence of the A and B fields, we shall enlarge on this later.)The failure of the one minus rest plus helicity tree-level amplitudes to vanish hassignificant consequences for the attempt to construct an MHV Lagrangian in D dimensions.Firstly it means that the theory described by the truncated Lagrangian L − + + L − ++ thatgenerates these amplitudes is not free. Secondly it means that the Parke-Taylor verticesare likely to be much more complicated in D dimensions because their simplicity in fourdimensions can be explained within the BCFW recursion method [3] as deriving from thevanishing of the one minus rest plus tree-level amplitude. We will now investigate howdamaging these facts are.
3. Canonical transformation in D dimensions Perhaps surprisingly we can still construct a canonical transformation in D dimensions sothat (1.7) holds. Using FDH regularization, and given (1.8) we have to solve ω A ( x ) + A ( x ) (cid:18) ¯ ∂ ˆ ∂ A ( x ) (cid:19) − (cid:18) ¯ ∂ ˆ ∂ A ( x ) (cid:19) A ( x ) = Z ˆ x = const . ω ′ B ( x ′ ) δ A ( x ) δ B ( x ′ ) d D − x ′ (3.1)where ω = D/ − X i =1 ∂ ( i ) ¯ ∂ ( i ) / ˆ ∂ . Re-arranging: ω A ( x ) − Z ˆ x = const . ω B ( x ′ ) δ A ( x ) δ B ( x ′ ) d D − x ′ = −A ( x ) (cid:18) ¯ ∂ ˆ ∂ A ( x ) (cid:19) + (cid:18) ¯ ∂ ˆ ∂ A ( x ) (cid:19) A ( x ) . (3.2)We make the basic assumption, appropriate to perturbation theory, that we can expandthe Fourier transform of A in powers of the transform of B , with kernels Υ. (Note we usethe same symbol for the fields and their Fourier transforms) A p = ∞ X n =1 Z Υ( p, p , . . . , p n ) δ ( p + n X i =1 p i ) B ¯1 . . . B ¯ n d D p . . . d D p n , (3.3)where we adopt the notation that the subscripts of the fields label the momenta: A p ≡ A ( p )and B ¯ ı ≡ B ( − p i ). Then the first term on the left-hand-side of (3.2) multiplies each term in– 6 –he expansion by the Fourier transform of ω , i Ω ≡ i Ω( p ), whereas the second replaces each B ¯ ı by i Ω i B ¯ ı , and the right-hand-side glues two expansions together using what is essentiallythe three-point vertex corresponding to helicities + + − (and which we are attempting toeliminate from the theory by performing the canonical transformation to new variables).This is most easily represented graphically. Let us denote the expansion (3.3) by A = 1 B + 2 BB + B BB + .... (3.4)and the Fourier transform of the right-hand-side of (3.2) by i jk = ¯ V ( p j , p k , p i ) / ˆ p i , where ¯ V ( p , p , p ) = i (¯1 / ˆ1 − ¯2 / ˆ2)ˆ3 is the factor from the three-point (+ + − ) vertex ofthe lagrangian (1.12). The small black dots in the diagram denote the minus-helicity endof the propagators. Then the terms in (3.2) with n B fields give (cid:16) P n Ω i (cid:17) n B .... B = − P r + s = n r B .... B s B .... B If we were to use usual dimensional regularisation rather than FDH, we would have arrivedat the same graphical equation, but with indices attached to the lines and ¯ V ( p j , p k , p i ) = i ( { p i , p j } K δ IJ / ˆ p k + { p k , p i } J δ KI / ˆ p j ), in the notation of [15]. We can divide through by P Ω when it is non-zero and obtain the recursion relation for Υ in momentum spaceΥ(¯1 · · · ¯ n ) = 1ˆ1(Ω + · · · + Ω n ) n − X j =2 ¯ V ( P j , P j +1 ,n , − , ¯2 , . . . , ¯ j )Υ( − , j + 1 , . . . , ¯ n ) , (3.5)where we use the notation P i,j = p i + p i +1 + · · · + p j , for j > i , P i,j = p i + p i +1 + · · · + p n + p + · · · + p j for j < i , ¯ n = − p n and the − in the bracket of Υ denotes the minus of thesum of all the other momenta in Υ. This can be represented graphically n B .... B = P n Ω i P r + s = n r B .... B s B .... B We will encounter situations when P Ω vanishes, and then we need a prescription fordealing with this singularity. We will address this in the appendices.– 7 –f we denote − / ( P n Ω i ) by a closed broken curve cutting each line whose momentumappears in the sum, each order of the expansion of A can be represented as n B ... B n = − P r + s = n r B ... B s B n ... B (3.6)where ¯ p n B ... B n = R ··· n Υ(¯ p ¯1 · · · ¯ n ) B · · · B n . This can be easily iterated, starting with the leading term A = B : A = B − BB + B BB + B BB + . . . Similarly we can expand ¯ A in terms of B , and ¯ B in which it is linear. It is more convenientto expand ˆ ∂ ¯ A in terms of B , and ˆ ∂ ¯ B , and we denote this graphically byˆ ∂ ¯ A = ˆ ∂ ¯ B + ˆ ∂ ¯ BB + B ˆ ∂ ¯ B + ˆ ∂ ¯ BBB + B ˆ ∂ ¯ BB + BB ˆ ∂ ¯ B + .... and in momentum space we use Ξ to denote the expansion coefficients ¯ p ˆ ı ¯ B i n B ...... B n = R ··· n ˆ ı Ξ i (¯ p ¯1 · · · ¯ n ) B · · · ¯ B i · · · B n . Using this and (3.4) allows us to depict the second of (1.8) as– 8 – ∂ ¯ B = P BB .... ˆ ∂ ¯ BBB ....Since there are no B -fields on the left-hand-side we can equate to zero the sum of terms onthe right that contain precisely n B -fields, when n > P BB .... ˆ ∂ ¯ BBB ....The term in which there are no B -fields in the left-hand factor is the kernel we are lookingfor, so ˆ ∂ ¯ BBB .... = − P ′ BB .... ˆ ∂ ¯ BBB .... (3.7)where the prime on the sum indicates that we sum over terms in which there is at leastone B -field in the left-hand factor, and the ordering of fields matches on both sides of theequation. This is iterated to yieldˆ ∂ ¯ A = ˆ ∂ ¯ B + B ˆ ∂ ¯ B + ˆ ∂ ¯ BB − ˆ ∂ ¯ B BB− B B ˆ ∂ ¯ B + B B ˆ ∂ ¯ B + ... (3.8)However, the broken curves in the above diagrams do not denote the real singularities in theexpansion of ¯ A . Some singularities are cancelled out. For example, by explicit calculation,one finds that the singularity represented by the inner broken curves around the left bigblack dot in the fifth and sixth terms are cancelled out. In fact, by induction, one can– 9 –rove another recursion relation of Ξ s :Ξ i − ,..., ¯ n = − + · · · + Ω n (cid:16) i − X l =2 P l +1 ,n ¯ V ( p , P ,l , P l +1 ,n )Υ( − , ¯2 , . . . , ¯ l )Ξ i − l ( − , l + 1 , . . . , ¯ n )+ n − X l = i P ,l ¯ V ( P l +1 ,n , p , P ,l )Ξ i − ( − , ¯2 , . . . , ¯ l )Υ( − , l + 1 , . . . , ¯ n ) (cid:17) . (3.9)Using this we can represent each order of the expansion of ˆ ∂ ¯ A by diagrams:ˆ ∂ ¯ B n B ...... B = P r + s = n r B ... B ... ˆ ∂ ¯ B ... s BB + ... ˆ ∂ ¯ B ... r BB ... s BB ! (3.10)The proof of the new recursion relation starts with the old one (3.7) and uses relation ll + r + s +11 ...... r l +1 ... l + ri s l + r + s ...... l + r +1 − ...... r ... s ...... = − ...... r ... s ......(3.11)repeatedly (see appendix B for a sketch of the proof). The relation above is simply a resultof the equation 1Ω l + r + s +1 ,l + P l + r + si = l +1 Ω i − P ni =1 Ω i = Ω l +1 ,l + r + s + P li = l + r + s +1 Ω i (Ω l + r + s +1 ,l + P l +1 i =1 Ω l + r + s ) P ni =1 Ω i (3.12)where Ω i,j = P i,j ¯ P i,j / ˆ P i,j . The numerator on the right-hand-side of (3.12) will cancel thedenominator of the left Υ blob in the diagrams. We denote this cancellation by filling inthe left-hand blob. In fact, there is an easy way to prove this recursion relation in fourdimensions where we do not care about regularization: If we use the relation obtained in– 10 –16] Ξ i − (1 · · · n ) = − ˆ ı ˆ1 Υ(1 · · · n ) , (3.13)this recursion relation recovers that of (3.6) for Υ. From (3.5) we observe that the expansion terms of A can be constructed as follows: foreach term of the expansion, draw all the tree-level Feynman diagrams with an A as oneend of an external propagator and all B s in the term as amputated external lines using only(+ + − ) vertices; then calculate this diagram using ¯ V as vertices and 1 / (ˆ p (Ω p + P Ω)) ascorresponding propagators. Notice that the light-cone Feynman rule for vertex (+ + − ) is¯ V (1 , ,
3) = i g ¯ V (¯1 , ¯2 , ¯3) = − i g ¯ V (1 , ,
3) (3.14)and the light-cone propagator is hA p A ¯ p i = − i g p . (3.15)So hA A ¯3 i ¯ V (1 , ,
3) = − p ¯ V (1 , ,
3) (3.16)is consistent with the coefficient of each term in the recursion relation if we make thereplacement − /p → / (ˆ p (Ω + Ω + Ω )). As a result, we can reconstruct the terms of A from light-cone tree-level calculations by replacing the light-cone propagators using1 P ij → −
12 ˆ P j +1 ,i − (Ω j +1 ,i − + Ω i + Ω i +1 + · · · + Ω j ) . (3.17)Here P j +1 ,i − should be understood as the sum of all momenta except those labelled from i to j . The momentum in each term in the bracket of the denominators corresponds to theoutgoing momentum of the external line of the sub-tree diagram not involving A when thepropagator is cut. For example, for terms with B B B : A ¯1 ∼ Υ(¯1¯2¯3¯4) B B B = 1ˆ1(Ω + · · · + Ω ) (cid:18) ¯ V (2 , ,
1) ¯ V (3 , , P (Ω + Ω + Ω ) + ¯ V (23 , ,
1) ¯ V (2 , , P (Ω + Ω + Ω ) (cid:19) ×B B B . (3.18)The corresponding diagrams are: A ¯1 B B B A ¯1 B B B – 11 –rom the light-cone calculation of these Feynman diagrams, we have:2 p (cid:18) ¯ V (2 , ,
1) ¯ V (3 , , P + ¯ V (23 , ,
1) ¯ V (2 , , P (cid:19) (3.19)in which the two terms correspond to the two tree-level Feynman diagrams. We can seethat (3.18) and (3.19) only differ by the change1 /p → − / (2ˆ1(Ω + · · · + Ω )) , (3.20)1 /P → − / (2 ˆ P (Ω + Ω + Ω )) , (3.21)1 /P → − / (2 ˆ P (Ω + Ω + Ω )) . (3.22)If we put p , p , p on shell, the → in the above equations can be replaced by =, thus (3.18)is equal to (3.19) which gives the translation kernel contribution to the amplitude as itshould.For ¯ A , the same rule also holds allowing us to reconstruct the expansion of ¯ A fromlight-cone calculations: one needs to first draw the tree-level diagrams with one ¯ A as anexternal propagator, all the B , ¯ B in the term as amputated legs using (+ + − ) vertices, andthen calculate the diagram using the light-cone Feynman rules with the replacement (3.17).This can be justified from the recursion relation (3.9) with a similar discussion to that forΥ: First, in (3.9) all the Υ’s already obey this rule. The − / ˆ P l +1 ,n = 1 / ˆ P ,l in the firstterm in the bracket will combine with the 1 / P Ω factor in the expansion of the next Ξ inthis term to be 1 / ( P ,l (Ω ,l + P ni = l +1 Ω i )) which is just what we need to be consistent withthe rule. It is the same for the second term in the bracket. We only need to consider thefactor of the Ξ in the first iteration and the last iteration. We should divide the expansionof ˆ ∂ ¯ A by the corresponding i ˆ p in the momentum space, to obtain the expansion of ¯ A . Thisfactor 1 / ˆ p will combine with the factor of the first Ξ to be 1 / (ˆ p ( P ni =1 Ω i )) in (3.9). Thelast iteration corresponds to the right-most grey blob adjacent to ˆ ∂ B in each term of thefull iteratively expanded diagrams in (3.10). The extra factor 1 / ˆ ı in the Ξ of the last step ofthe iteration will cancel the ˆ ı in the ˆ ı B i from ˆ ∂ B . So just as in the case of Υ, the expansionof ¯ A requires calculating tree-level diagrams using ¯ V as vertices and 1 / (ˆ p (Ω p + P Ω)) asthe propagators, and so obeys the same rule.
4. ‘Missing’ amplitudes from equivalence theorem evasion reviewed
In [15] we explained how the tree-level ( − + +) and the one-loop (+ + · · · + +) amplitudesare obtained from the B , ¯ B theory, despite there being no vertices in this theory that couldcontribute. The amplitudes are non-zero because the equivalence theorem is not directlyapplicable to our non-local transformation. Thus A and ¯ A do not create the same particlesas B , ¯ B . This would appear to drastically complicate the calculation of amplitudes withinthe B , ¯ B theory. It is the main purpose of this paper to show that only certain amplitudesare affected by this, and that in the general case we can use either set of fields to generateamplitudes. In this section we briefly review the ‘missing’ tree-level amplitude.In light-cone gauge Yang-Mills theory the tree-level contribution to the Green function h A ( p ) ¯ A ( p ) ¯ A ( p ) i comes from the vertex in L − ++ , so to this order, and taking account– 12 –f the iǫ -prescription in propagators (and suppressing Lie algebra indices on the under-standing that we deal with colour-ordered amplitudes)( p + iǫ ) ( p + iǫ ) ( p + iǫ ) h A ( p ) ¯ A ( p ) ¯ A ( p ) i = ! ˆ p , p p p and as all three momenta go on-shell this becomes the three-point amplitude (which van-ishes in four dimensional Minkowski space, but is non-zero in other signatures and dimen-sions.) Clearly h B ( p ) ¯ B ( p ) ¯ B ( p ) i = 0 at tree-level due to the helicity assignment of theParke-Taylor vertices. To compute the Green function in the B , ¯ B theory we must use thetranslation kernels: h A ( p ) ¯ A ( p ) ¯ A ( p ) i = −h B ( p ) + p BB + ... ! × p ˆ ∂ ¯ B ( p ) − p B ˆ ∂ ¯ B − p ˆ ∂ ¯ BB + ... ! × p ˆ ∂ ¯ B ( p ) − p B ˆ ∂ ¯ B − p ˆ ∂ ¯ BB + ... ! i since no vertices contribute to leading order this can be computed by contracting the B , ¯ B fields using the free propagator, which we denote by h A ( p ) ¯ A ( p ) ¯ A ( p ) i =+ p p p ! ˆ p ˆ p + p p p ! ˆ p ˆ p p p p = ! ˆ p ( p + iǫ ) ( p + iǫ ) ( p + iǫ ) (cid:16) p + iǫ ˆ p + p + iǫ ˆ p + p + iǫ ˆ p (cid:17) , p p p The broken line cutting the three lines denotes division by − X j Ω( p j ) = − X j D/ − X i =1 p j ( i ) ¯ p j ( i ) / ˆ p j , which does not depend on the ˇ p j . However, if we add P j ˇ p j , which vanishes by momentumconservation, this becomes P j p j / ˆ p j . If we also include iǫ terms to match the last factorthen we reproduce the light-cone Yang-Mills amplitude. This tells us how to treat 1 / P Ω– 13 –hen the denominator is singular, so in general the broken lines in our diagrams will denote1 P j p j + iǫ ˆ p j . It is of course not surprising that we reproduce the usual Green function, as all we havedone is transform to new variables to do the calculation. It will be useful, for what comeslater, to examine how the equivalence theorem has been evaded. Note that the combinedlimit p + iǫ, p + iǫ, p + iǫ → p + iǫ ,p + iǫ,p + iǫ → p + iǫ p + iǫ ˆ1 + p + iǫ ˆ2 + p + iǫ ˆ3 depends on the order in which the limits are taken, but it is valid to take the limit of thesum of the three terms because the factor ( p + iǫ ) / ˆ1 + ( p + iǫ ) / ˆ2 + ( p + iǫ ) / ˆ3 in thedenominator is cancelled out. Consequently we can take the limit of the sum in any order.Suppose we take the legs on-shell one after another, beginning with p and p . We include ǫ in the mass-shell condition because it enters the propagators for external legs that haveto be cancelled by the LSZ factors. Since p + iǫ and p + iǫ cancel the propagators inthe first diagram, but not in the other two, it is clear that for general p the contributionsfrom the last two diagrams are wiped out in the limit leavinglim p + iǫ → lim p + iǫ → ( p + iǫ ) ( p + iǫ ) h A ( p ) ¯ A ( p ) ¯ A ( p ) i =lim p + iǫ → lim p + iǫ → ( p + iǫ ) ( p + iǫ ) p p p ! =lim p + iǫ → lim p + iǫ → p iǫ ˆ p + p iǫ ˆ p + p iǫ ˆ p p p p ! = ˆ p p + iǫ p p p ! . So the 1 / ( p + iǫ ) needed to cancel the p + iǫ coming from the LSZ prescription is generatedas part of the translation kernels, even though these appeared to be independent of the ˇ p components of momenta. We should point out that ‘missing amplitudes’ can be generatedin different ways if the theory is formulated differently such as in the gauge fixing of thetwistor action [22] or in the light-cone friendly regularisation of [23].
5. Equivalence theorem evasion in general.
When the equivalence theorem holds we can ignore all except the leading translation ker-nels. However the theorem will be evaded whenever the translation kernels that express– 14 – ( p ) or ¯ A ( p ) in terms of B and ¯ B produce a 1 / ( p + iǫ ) that can cancel the LSZ factors. Wewill now list all the types of process in which this can occur. The singular terms originatein the 1 / P Ω represented by the broken lines in our diagrams. These must cut the linewith momentum p if we are to end up with 1 / ( p + iǫ ). Suppose that the other lines cutcarry momenta p , . . . , p n , then 1 P Ω = 1 p + iǫ ˆ p + P nj =1 p j + iǫ ˆ p j , so we have to examine the conditions under which P nj =1 ( p j + iǫ ) / ˆ p j = 0. Notice that herewe actually take P nj =1 ( p j + iǫ ) / ˆ p j → p → In the absence of loops there are two ways that P nj =1 ( p j + iǫ ) / ˆ p j = 0. The first is thateach of the legs cut by the broken line are external and so their momenta must be put onshell. For example, in the four-particle process with one − helicity and three + helicitygluons we need the Green function h A ( p ) ¯ A ( p ) ¯ A ( p ) ¯ A ( p ) i . Contributing to this aretranslation kernels for A ( p ), ¯ A ( p ), ¯ A ( p ), and ¯ A ( p ) which give rise to diagrams like p p p p in which four external legs are cut by the broken line. Since all the external lines will becut by the broken curve we cannot include any Parke-Taylor vertices. Consequently, inthe general case we can only ever have a contribution to a tree-level amplitude with one − helicity external gluon and n + helicity external gluons. For each light-cone tree-leveldiagram of such an amplitude, there are terms from the translation kernels that contribute.For example, for the four-point diagram: p p p p using the method in section (3.2), we can construct the translation kernel contribution tothis diagram from the canonical expansion of A , ¯ A , ¯ A , ¯ A which can be representedgraphically: – 15 – p p p p p p p p p p p p p p p The difference between these translation kernels and the light-cone contribution is only inthe denominators. Examining these:lim p ,p ,p ,p → p p p p (cid:16) p ˆ1 + p ˆ2 + p ˆ3 + p ˆ4 (cid:17) (ˆ1 + ˆ4) (cid:16) p ˆ1+ˆ4 + p ˆ2 + p ˆ3 (cid:17) p p p + 1ˆ2 (cid:16) p ˆ1 + p ˆ2 + p ˆ3 + p ˆ4 (cid:17) (ˆ2 + ˆ3) (cid:16) p ˆ2+ˆ3 + p ˆ1 + p ˆ4 (cid:17) p p p + 1ˆ3 (cid:16) p ˆ1 + p ˆ2 + p ˆ3 + p ˆ4 (cid:17) (ˆ2 + ˆ3) (cid:16) p ˆ2+ˆ3 + p ˆ1 + p ˆ4 (cid:17) p p p + 1ˆ4 (cid:16) p ˆ1 + p ˆ2 + p ˆ3 + p ˆ4 (cid:17) (ˆ1 + ˆ4) (cid:16) p ˆ1+ˆ4 + p ˆ2 + p ˆ3 (cid:17) p p p = lim p ,p ,p ,p → − p ˆ1+ˆ4 + p ˆ1 + p ˆ4 − p ˆ2 − p ˆ3 (ˆ1 + ˆ4) (cid:16) p ˆ1+ˆ4 + p ˆ2 + p ˆ3 (cid:17) (cid:16) − p ˆ1+ˆ4 + p ˆ1 + p ˆ4 (cid:17) = 1 p , (5.1)(We omit the + iǫ accompanying each p here since it is not important in our discussion.)we see that the factor p / ˆ1 + p / ˆ2 + p / ˆ3 + p / ˆ4 in the denominator is cancelled out and thelimit procedure is valid at last. The denominator provides the propagator needed in thelight-cone computation. Since the combined limit is valid, like in the (++ − ) case, we couldtake the limit in any order, for example take the p , p , p → p → p / ˆ1 + p / ˆ2 + p / ˆ3 + p / ˆ4) becomes p to be cancelled with p inthe numerator from the LSZ procedure. This reproduces the light-cone computation of theamplitude. One can imagine that the same thing happens for general multileg one-minus-helicity amplitudes. Fortunately these amplitudes vanish at tree-level in four dimensionalMinkowski space, (and for n > P nj =1 ( p j + iǫ ) / ˆ p j = 0 without all of the p j being external legs– 16 –s if some of the terms in the sum cancel against each other, or if the translation kernelis connected to a vertex by a momentum that is on-shell. At tree-level this can onlyoccur for special choices of the momenta of the external particles, and cannot contributeto an amplitude with generic values of external momenta. So at tree-level the equivalencetheorem can be used for non-trivial generic amplitudes, which is why the MHV rulescorrectly reproduce tree-level amplitudes without having to take account of the translationbetween A , ¯ A and B , ¯ B fields. There are several processes that can occur at one-loop order that give rise to evasions of S-matrix equivalence. The first is that loops can dress the propagators that occur in tree-leveldiagrams. Secondly, we can have tadpole diagrams in which two legs of a translation kernelare contracted with each other. These diagrams are responsible for the all positive helicityamplitudes ‘missing’ from a straightforward application of the MHV rules. Thirdly we canhave more general processes in which the loop integration has an infra-red divergence thatmight cancel the LSZ factor.
Loops can dress propagators, so, at one-loop, as for tree-level P nj =1 ( p j + iǫ ) / ˆ p j can vanishwhen each of the p j is the momentum of an on-shell gluon. For example, the first interac-tion in (1.12), L −− + which we denote by B ¯ B ¯ B can be contracted with the fifth term in the expansion of ˆ ∂ ¯ A equation (3.8) B B ˆ ∂ ¯ B to give B ¯ B This will contribute to the Green function h A ( p ) ¯ A ( p ) ¯ A ( p ) i , for example by contract-ing B with the leading term in the expansion of ¯ A ( p ) and ¯ B with that of A ( p ). Thepropagators cancel two of the LSZ factors for the + + − amplitude. Taking p + iǫ = 0and p + iǫ = 0 causes the 1 / P Ω factor denoted by the inner broken curve to reduce– 17 –o 1 / ( p + iǫ ), which will cancel the remaining LSZ factor, thus evading the equivalencetheorem and producing a contribution to the three-point amplitude that is the same asthe tree-level diagram with a self-energy insertion on the p leg. In Minkowski space thethree-point amplitude vanishes on-shell anyway, so this evasion appears inconsequential.For complex on-shell momenta, of the kind used in the BCFW rules, this amplitude doesnot vanish, so it is worthwhile considering this further. We noted earlier that the relations(3.9) enable us to re-write the series for ¯ A in a way that moves the position of the dottedlines so that the singularity 1 / P i Ω i corresponding to the dotted lines around the left bigblack dot is cancelled out after we sum the fifth and sixth term in (3.8). These combine togive ¯ A B ¯ B Since the contribution of the internal line to the denominator P p i / ˆ p i represented by theinner broken curve can not be zero, it is obvious that there is no 1 /p generated in thisdiagram. So this diagram can not contribute to the amplitude. The same is true for thecase of a dressed propagator on a B leg:¯ A ¯ B B
The three-point interaction can dress a propagator either in the way just described,or, potentially by two such vertices being glued together¯ B ¯ B An insertion of this kind into a diagram effectively changes a B -field into a ¯ B -field, howeverexplicit calculation shows that this vanishes. At one-loop the only other vertices that cancontribute to dressing propagators are contained in L −− ++ :¯ BB B ¯ B and these produce insertions that connect B with ¯ B – 18 – ¯ B Dressing propagators can produce diagrams that evade the equivalence theorem, but only ifthe corresponding tree-level diagrams do so already, in which case the result is proportionalto the tree-level amplitude. As we have seen this only happens for amplitudes that vanishin the physical dimension, so this source of equivalence theorem evasion has no physicalconsequence. However there is a subtlety involved in the one-loop (+ + − ) amplitudein (+ + −− ) signature. In section 6, we will find that the tadpoles formed from MHVvertices already include the diagrams with external leg corrections. Including translationkernel contributions in the amplitude would appear to count the diagrams with externalleg corrections twice. We will solve this puzzle in section 7. At tree-level we dismissed the second way that P nj =1 ( p j + iǫ ) / ˆ p j could vanish because itcould only apply to special configurations of external momenta. When we integrate overloop momenta such special configurations can easily arise, and so we must analyse them.The simplest way that two of the terms in P nj =1 ( p j + iǫ ) / ˆ p j could cancel without eachbeing on-shell occurs in the translation kernel for ¯ A when a B and ¯ B field are contracted,because then their lines carry equal and opposite momenta. These are ‘tadpoles’ whendrawn in terms of the translation kernels, e.g. B B but are rather more complicated when drawn in terms of the graphical solution. Forexample, one of the terms contributing to this tadpole originates in the following termwhich appears in the expansion of ¯ A : ˆ ∂ ¯ B BBB
Contracting ¯ B with a B and the remaining fields with external gluons gives– 19 – p p When the p and p are put on-shell, p + iǫ = 0 and p + iǫ = 0, so the dotted line cuttingthe three external momenta and the gluon propagator reduces to P nj =1 ( p j + iǫ ) / ˆ p j =( p + iǫ ) / ˆ p resulting in an evasion of S-matrix equivalence. Because the contraction usedto make a tadpole removes a B and a ¯ B field from the translation kernels for ¯ A they cancontribute to one-loop amplitudes involving only positive helicity gluons. In [15] it wasfound that it is this mechanism that is responsible for generating the one-loop all plusfour-point amplitude (2.1) that a na¨ıve application of the MHV rules cannot account for. Evasion of S-matrix equivalence might arise in a more general situation when a vertex isattached to a translation kernel. For illustration we focus on one term in the expansion of¯ A and contract two of the legs with those of some arbitrary subgraph denoted by the opencircle: p q − j − qp ¯ BB ....... If we take p + iǫ = 0, (having cancelled the corresponding LSZ factor with the propagator,)the loop integration is Z d D q p + iǫ ˆ p + q + iǫ ˆ q − ( q + j ) + iǫ ˆ q +ˆ j j + iǫ ˆ + q + iǫ ˆ q − ( q + j ) + iǫ ˆ q +ˆ j q + iǫ j + q ) + iǫ f ( j, q )(5.2)with j = p + p .We need to investigate whether this integral can generate a factor of 1 / ( p + i ǫ ). Todo so it would have to be divergent as p goes on-shell. The integrand has a number ofsingularities as a function of the components of loop momentum q µ but by deforming theintegration contours into the complex q µ -planes the surfaces where the integrand divergescan typically be avoided so that the integral is well-defined. We are aided in identifyingthe directions in which to deform the contours by the iǫ prescription. (We can ignore whathappens as q → ∞ as the ultra-violet behaviour is regulated). However, as we vary p thepositions of these singularities move, and it is possible that our integration surface maylie between several singularity surfaces that approach each other for some values of p andpinch the contours so that they can no longer be deformed to avoid the singularity. As thishappens the value of the integral itself diverges as a function of p . Prior to taking the on-shell limit we can deform the integration surface so that it consists of a piece surrounding– 20 –he singularities and a piece that we can move well away from either singularity. In the on-shell limit we can ignore this last piece because of the LSZ factor, p + iǫ . We now focus onthe contribution from the piece surrounding the singularity, which means that in the loopintegral we take f ( j, q ) as constant. We begin by integrating out the ˇ q component. Thefirst two factors of the integrand come from the translation kernels and so do not dependon ˇ q . As we close the ˇ q contour in the complex plane we pick up singularities from thepropagators. Using conservation of momentum the residue can be put into a form similarto that of the kernel, but without ( p + iǫ ) / ˆ p : θ (ˆ q ) θ ( − ˆ q − ˆ j ) − θ ( − ˆ q ) θ (ˆ q + ˆ j )ˆ q (ˆ q + ˆ j ) 2 πi q + iǫ ˆ q − ( q + j ) + iǫ ˆ q +ˆ j (5.3)which, of course, does not depend on ˇ q . This allows us to extract 1 / ( p + iǫ ) explicitlyfrom the integral (5.2) which becomes − πi ˆ pp + iǫ Z D/ − Y i =1 dq ( i ) d ¯ q ( i ) d ˆ q θ (ˆ q ) θ ( − ˆ q − ˆ j ) − θ ( − ˆ q ) θ (ˆ q + ˆ j )ˆ q (ˆ q + ˆ j ) (5.4) × p + iǫ ˆ p + q + iǫ ˆ q − ( q + j ) + iǫ ˆ q +ˆ j − q + iǫ ˆ q − ( q + j ) + iǫ ˆ q +ˆ j f ( j, q ) . (5.5)Since the second factor in the integrand of (5.2) is finite when the first factor is singular,it is irrelevant to our discussion and we absorb it into f .The LSZ factor is cancelled by the 1 / ( p + iǫ ). If we now take the on-shell limit p + iǫ → q . However, for certainvalues of j and q ( i ) both terms in the square brackets are divergent close to the real axis,so we have to investigate the location of these singularities. The first diverges forˆ q = − b ± q b − a ( c + iǫ ˆ j )2 a (5.6)with a = φ − j , φ = p + iǫ ˆ p , b = ˆ jφ − j + 2 X i (cid:0) q ( i ) ¯ j ( i ) + ¯ q ( i ) j ( i ) (cid:1) , c = − j X i q ( i ) ¯ q ( i ) , (5.7)whilst the location of the pole in the second term is given by the above expression with φ set to zero. For the moment treat φ as being real. Then for b > ac the poles are close tothe real axis, with an imaginary piece ∓ ǫ ˆ j √ b − ac . (5.8)Since these are on the same side of the real axis for both terms in square brackets it is clearthat the contribution to the integral of these two terms cancels even when the singularitiesare close to the real axis. Consequently there is no S-matrix equivalence evasion in thiscase, provided that we keep φ real as we take the on-shell limit for external legs.– 21 – . One-loop ( − + ++) amplitude In [15] we described how the one-loop (+ + ++) amplitude arises in this approach asa tadpole-like diagram constructed from translation kernel. By contrast the one-minushelicity amplitude is constructed from the tadpole diagram of a MHV vertex.Let us now look at a box diagram, A (1 − + + + ) with 1 − attached to MHV ( − − +)vertex. The integrand of the light-cone amplitude is A (1) (1 − + + + ) = 2 V (¯ a , , a ) ¯ V (4 , a, ¯ a
23) ¯ V (3 , a, ¯ a
2) ¯ V (2 , a, ¯ a ) p a p a p a p a . (6.1)It must come from the tadpole diagram in the CSW method by connecting two lines ofsix-point MHV vertices. We can identify the tadpole contributions to this amplitude in thefollowing way. First, we can cut any one of the four internal lines and get four tree-levelMHV diagrams. − + + + p a −→ − + + + − + + + − + + + − + + + MHV vertices are generated by expanding the A and ¯ A in the lagrangian L −− + . We firstidentify the three point MHV ( − − +) vertex in the tree-level diagrams and the three partsin the diagrams corresponding to the expansion of A and ¯ A in L −− + . By comparing withthe three parts of the diagram, we can find out the corresponding three parts in the lightcone amplitude (6.1). Then by replacing the propagators in the light-cone amplitude using(3.17) we can reconstruct the contribution to the one-loop box diagram of the tadpole.The four tree-level diagram contributions are (we label the internal line between leg 1 and2 as a ): A (1 , = 2 V (¯ a , , a ) ¯ V (4 , a, ¯ a
23) ¯ V (3 , a, ¯ a
2) ¯ V (2 , a, ¯ a ) p a ˆ P ¯ a ˆ P ¯ a ˆ P a ( P a ˆ P a + P a ˆ P ¯ a )( P a ˆ P a + P a ˆ P ¯ a )( P a ˆ P a + P a ˆ P ¯ a ) , (6.2) A (1 , = 2 V (¯ a , , a ) ¯ V (4 , a, ¯ a
23) ¯ V (3 , a, ¯ a
2) ¯ V (2 , a, ¯ a ) P a ˆ P ¯ a ˆ P ¯ a ˆ P ¯ a ( P a ˆ P a + P a ˆ P ¯ a )( P a ˆ P ¯ a + P a ˆ P a )( P a ˆ P ¯ a + P a ˆ P a ) , (6.3) A (1 , = 2 V (¯ a , , a ) ¯ V (4 , a, ¯ a
23) ¯ V (3 , a, ¯ a
2) ¯ V (2 , a, ¯ a ) P a ˆ P ¯ a ˆ P ¯ a ˆ P a ( P a ˆ P a + P a ˆ P ¯ a )( P a ˆ P a + P a ˆ P ¯ a )( P a ˆ P a + P a ˆ P ¯ a ) , (6.4) A (1 , = 2 V (¯ a , , a ) ¯ V (4 , a, ¯ a
23) ¯ V (3 , a, ¯ a
2) ¯ V (2 , a, ¯ a ) P a ˆ P a ˆ P ¯ a ˆ P a ( P a ˆ P a + P a ˆ P ¯ a )( P a ˆ P a + P a ˆ P ¯ a )( P a ˆ P a + P a ˆ P ¯ a ) . (6.5)We have already set the p i in the denominator of the external particles to zero, since thereis no singularity when we put the external particles on-shell. It makes no difference if wetake the on-shell limit before or after the LSZ procedure. It is easy to check that thesefour terms add up to the integrand of the light-cone amplitude for the box diagram (6.1).The other box diagrams, triangle, bubble diagrams of light-cone amplitude can be checkedin the same way. There are some subtle problems with diagrams including corrections to– 22 –xternal propagators which we will address in next section. So, in general, one can believethat the one loop one-minus helicity amplitudes should all come just from the tadpoles ofMHV vertices.
7. One-loop (+ + − ) amplitude with external tadpole dressing propagators At first sight, for three point (+ + − ) amplitude, there could also be contributions tothe cut diagrams considered in the previous section from translation kernels with dressedexternal propagators, since they yield contributions proportional to tree-level amplitudeswhich are not zero in (++ −− ) signature. This seems to count diagrams with corrections toexternal propagators twice. This problem arises from the order of limits in LSZ procedure.In the example of the previous section it does not matter when we take the on shell limitbecause no singularities are encountered in this limit. But we must be more careful with thediagrams with dressed propagators on external legs because there will then be singularitiesfrom 1 / P Ω. We should first calculate the off-shell Green function and then apply theLSZ procedure. Also from the discussion in section (5.2.1) the Green function receivescontributions should not just from tadpoles of MHV five point vertices, but also fromtranslation kernels with dressed propagators. Let us look at the example of a light-conediagram for h ¯ A ¯1 ¯ A ¯2 A ¯3 i : ¯ A ¯1 ll + p ¯ A ¯2 A ¯3 The integrand of the light-cone computation of the diagram for the Green function is A (1 + + − ) = ig ¯ V (1 , ,
3) ¯ V (¯3 , l + p , ¯ l ) V ( − l − p , , l ) p p ( p ) l ( l + p ) . (7.1)We have omitted the iǫ in the propagators. According to the method of the last section, thecontribution from the tadpole of MHV five-point vertices to this diagram can be constructedby replacing the corresponding 1 /p → − / (2ˆ p P Ω):¯ A ¯1 ¯ A ¯2 A ¯3 A (1) ¯ A ¯1 ¯ A ¯2 A ¯3 A (2) A (1) = ig ¯ V (1 , ,
3) ¯ V (¯3 , l + p , ¯ l ) V ( − l − p , , l ) p p p ˆ3 (cid:16) p ˆ1 + p ˆ2 + p ˆ3 (cid:17) (ˆ l + ˆ3) (cid:16) ( l + p ) ˆ l +ˆ3 − l ˆ l + p ˆ1 + p ˆ2 (cid:17) l , (7.2) A (2) = ig ¯ V (1 , ,
3) ¯ V (¯3 , l + p , ¯ l ) V ( − l − p , , l ) p p p ˆ3 (cid:16) p ˆ1 + p ˆ2 + p ˆ3 (cid:17) ( − ˆ l ) (cid:16) ( l + p ) ˆ l +ˆ3 − l ˆ l + p ˆ1 + p ˆ2 (cid:17) ( l + p ) . (7.3)– 23 –he contribution from translation kernels with dressed propagators can be represented asfour diagrams: ¯ A ¯1 ¯ A ¯2 A ¯3 A (3) ¯ A ¯1 ¯ A ¯2 A ¯3 A (4) ¯ A ¯1 ¯ A ¯2 A ¯3 A (5) ¯ A ¯1 ¯ A ¯2 A ¯3 A (6) A (3) = ig ¯ V (1 , ,
3) ¯ V (¯3 , l + p , ¯ l ) V ( − l − p , , l ) p ( p ) ˆ1 (cid:16) p ˆ1 + p ˆ2 + p ˆ3 (cid:17) (ˆ l + ˆ3) (cid:16) ( l + p ) ˆ l +ˆ3 − l ˆ l − p ˆ3 (cid:17) l , (7.4) A (4) = ig ¯ V (1 , ,
3) ¯ V (¯3 , l + p , ¯ l ) V ( − l − p , , l ) p ( p ) ˆ1 (cid:16) p ˆ1 + p ˆ2 + p ˆ3 (cid:17) ( − ˆ l ) (cid:16) ( l + p ) ˆ l +ˆ3 − l ˆ l − p ˆ3 (cid:17) ( l + p ) , (7.5) A (5) = ig ¯ V (1 , ,
3) ¯ V (¯3 , l + p , ¯ l ) V ( − l − p , , l ) p ( p ) ˆ2 (cid:16) p ˆ1 + p ˆ2 + p ˆ3 (cid:17) (ˆ l + ˆ3) (cid:16) ( l + p ) ˆ l +ˆ3 − l ˆ l − p ˆ3 (cid:17) l , (7.6) A (6) = ig ¯ V (1 , ,
3) ¯ V (¯3 , l + p , ¯ l ) V ( − l − p , , l ) p ( p ) ˆ2 (cid:16) p ˆ1 + p ˆ2 + p ˆ3 (cid:17) ( − ˆ l ) (cid:16) ( l + p ) ˆ l +ˆ3 − l ˆ l − p ˆ3 (cid:17) ( l + p ) . (7.7)After summing over A (1) to A (6) one finds that the factor p / ˆ1 + p / ˆ2 + p / ˆ3 in the denom-inator is cancelled and we can apply the LSZ procedure:lim p ,p ,p → ( p p p ) Z d l X i =1 A ( i ) = lim p ,p ,p → ig p Z d l ¯ V (1 , ,
3) ¯ V (¯3 , l + p , ¯ l ) V ( − l − p , , l ) l ( l + p ) × (cid:16) ( l + p ) ˆ l +ˆ3 − l ˆ l + p ˆ1 + p ˆ2 − p ˆ3 (cid:17) (cid:16) ( l + p ) ˆ l +ˆ3 − l ˆ l (cid:17)(cid:16) ( l + p ) ˆ l +ˆ3 − l ˆ l − p ˆ3 (cid:17) (cid:16) ( l + p ) ˆ l +ˆ3 − l ˆ l + p ˆ1 + p ˆ2 (cid:17) = lim p ,p ,p → ig f ( p , p , p ) p = lim p ,p ,p → ig ∂f ( p , p , p ) ∂p . (7.8)Since the integration is uniformly convergent after regularization, we can take the limitbefore integration and differentiation which will give the same on-shell integral as the– 24 –ight-cone calculation. So we have reproduced the light-cone computation. For the otherdiagrams with dressed propagators a similar situation happens and it can be checked thatthey give the same amplitudes as light-cone calculations.From this example, we see that we should first collect the diagrams with the sameinternal helicity configurations and with tadpoles on the same legs and then impose thelimit p , p , p → p at the last step after integration, we choosethe limit p , p → p p p and take p , p → p / ˆ1 + p / ˆ2 + p / ˆ3simply contributes to the propagator i/p needed in the amplitude. So the result is thatwe do not need to consider the translation kernel contribution in this case.In section 5.2.1, we have argued that since the sum of one-loop diagrams in whichthe external legs are dressed are proportional to tree-level amplitudes, their contributionsto higher point one-minus-helicity amplitudes vanish. But it is also instructive to applythe above arguments to these higher point amplitudes. In fact, a similar situation occurs.For these amplitudes there are also 1 / ( P i p i / ˆ ı ) factors both from the tadpoles of MHVvertices and the translation kernels, where i enumerates all the external momenta. If wecollect the diagrams with the same internal helicity configuration and with tadpoles onthe same legs first, (including tadpoles of MHV vertices and translation kernels,) then the P i p i / ˆ ı in the denominator is cancelled and we can take the on-shell limits in any order. Ifwe first set all the external legs on-shell except that with the tadpole then the translationkernel contributions vanish leaving just the tadpoles of MHV vertices. So we come to theconclusion that we do not need to consider external propagators dressed by tadpoles fromtranslation kernel.
8. Conclusion and higher loops
We have seen that the S-matrix equivalence theorem is not immediately applicable to thechange of variables from A and ¯ A to B and ¯ B because of the non-locality of the translationkernels, and this accounts for the one-loop all plus helicity amplitudes apparently missingfrom the CSW rules. However, by analysing the mechanisms that generate singularities inthe external momenta that are able to cancel the LSZ factors we have seen that the typesof amplitude in which S-matrix equivalence is violated are very restricted. At tree-levelthe amplitudes that might have displayed this violation actually vanish. At one-loop theequivalence violating amplitudes that do not vanish are ones in which all the gluons havepositive helicity, and these have a known form, e.g. (2.1). Because the only non-zeroone-loop amplitudes that show S-matrix equivalence violation are given by the tadpolediagrams in which the single ¯ B field of an ¯ A translation kernel is contracted with a B fieldit follows that higher loops can only contribute to violating processes by dressing the legsof these one-loop diagrams. So, apart from this class of known amplitudes we are free tocalculate S-matrix elements using the B and ¯ B fields directly.– 25 –ince one-minus-helicity diagrams can not be constructed from more than one MHVvertex or from completion vertices, they can only arise as tadpoles of MHV vertices. Byanalysing an example we saw how the light-cone amplitudes really can be reconstructedfrom tadpoles of MHV vertices.We found a new recursion relation for the expansion coefficients Ξ s of ¯ A , which encodeda cancellation of certain singularities that would otherwise have contributed to furtherevasion of the S-matrix equivalence theorem. Using this recursion relation for Ξ togetherwith the one for Υ we were led to a better understanding of the canonical transformation:they can be reconstructed from the light-cone tree level diagrams built with only (+ + − ) vertices by replacing the propagator using (3.17). This was useful in discussing therelationship between light-cone and MHV methods.A few remarks about the rational parts of one-loop diagrams is in order. The CSWor MHV rules, although initially conjectured and proven at tree-level have been studiedat one-loop level. It has been shown that they give supersymmetric amplitudes correctly[13, 24, 25, 26], but when applied to non-supersymmetric amplitudes, the rational partscan not be correctly reproduced [27], not only in all-plus diagrams. Our discussion inthe present paper has focussed on the (limited) breakdown of the equivalence theoremthat is responsible, in our approach, for the rational one-loop all plus amplitude in non-supersymmetric Yang-Mills. Our conclusion is that only these amplitudes require the useof the translation kernels, and so all other one-loop amplitudes can be calculated directlyfrom the Green functions of the B fields. One may then ask where the missing rationalparts of the other diagrams might come from. Here, we should point out that we haveformulated the transformation from light-cone Yang-Mills to the new MHV Lagrangian in D dimensions. Consequently our canonical transformation coefficients Υ and Ξ are formulatedin D dimensions ( D = 4 − ǫ ) and the MHV vertices derived from these coefficients are alsoin D dimensions, whether one uses standard dimensional regularisation as in [15] or FDH.This is different from the usual analyses of MHV one-loop calculations in [13, 24, 25, 26, 27]which use four dimensional MHV vertices. In the FDH procedure the ǫ dependence entersthe transformation coefficients only through P Ω = P p / ˆ p where p is the D-dimensionalmomentum, rather than the four dimensional momentum, in recursion relations (3.5) and(3.9), but this is enough to make the vertices of our Lagrangian different from the ordinaryfour dimensional Parke-Taylor vertices. In ordinary dimensional regularisation the verticeswould, in addition, acquire indices relating to the extra dimensions. In either formalismthe vertices differ from the four dimensional ones. One would expect that, in general, thesemodifications would produce terms proportional to ǫ which would cancel the divergence1 /ǫ from the loop integration resulting in rational pieces missing in the ordinary MHVcalculation.Our arguments can easily be extended to super Yang-Mills theory using the supersym-metry transformation in [28]. We expect the supersymmety transformation is not affectedin D dimensions, and the results in [28] can be directly used here after setting the chiralfields to zero. The A transformation is not changed. From equation (B.7) and (C.14) in– 26 –28], ¯ A has an additional term which involves gluino Π:¯ A B Π q = − √ q ∞ X n =2 Z ··· n n X s =1 (cid:20) Ξ sq, ¯1 ··· ¯ n n X l =1 ,l = s ( − δ ls B · · · Π l · · · ¯Π s · · · B n (cid:21) δ q ¯1 ··· ¯ n , (8.1)Ξ s is just the coefficient appearing in the pure bosonic expansion. The fermion propagatorand bosonic propagators are h Π ¯Π i = i g ˆ p √ p , hA ¯ Ai = i g p . (8.2)Considering the √ p factor, when connected to a gluino propagator the coefficient is thesame as the one in the pure bosonic expansion up to a sign. So all the foregoing discussioncan be applied to diagrams with inner gluinos. Therefore one would expect that onlythe tadpole would evade the equivalence theorem. One can easily check that the all-plustranslation kernel contribution to the amplitude is cancelled using above expansion (8.1).We also expect that our MHV calculation should reproduce the light-cone super Yang-Millscalculation, so the one-minus-helicity amplitude in supersymmetric Yang-Mills should alsobe zero. As is well known [21], in supersymmetric theories the rational parts of amplitudesare determined uniquely by their (four dimensional) cut-constructible parts. It follows thatall the remaining rational parts discussed in the previous paragraph should be cancelled inthe supersymmetric theory. Acknowledgements
Tim, Xiao, and Paul thank STFC for support under the rolling grants ST/G000557/1 andST/G000433/1, and Jonathan thanks the Richard Newitt bursary scheme, for financialsupport. It is a pleasure to acknowledge useful conversations with James Ettle.
A. Some remaining thoughts on translation kernels
The careful reader may have noticed that the translation kernel can become ill-defineddue to the symmetry of graphs. For instance in the tadpole graph below arising fromself-contraction of the Ξ ¯ BBB term the gluons flowing in and out of the kernel must carryequal and opposite momenta as required by conservation of momentum. As a result thefactors ( p j + iǫ ) / ˆ p j which appear in the denominator of the kernel cancel in pairs. Thesame cancellation can also occur for special values of momentum. Note that in this casethe standard iǫ prescription fails to prevent P ( p j + iǫ ) / ˆ p j from vanishing.This problem can be fixed by adding a small correction to the definition of translationkernels. To break symmetry we distinguish the iǫ associated with A fields and B fields.Υ(123) is now modified as Υ(123) = i (cid:16) ¯ p ˆ p − ¯ p ˆ p (cid:17) p + iǫ A ˆ p + p + iǫ B ˆ p + p + iǫ B ˆ p (A.1)– 27 – qp p Figure 1:
Translation kernels diverge in symmetrical graphs
Higher order terms in the A field expansion can all be redefined following the samespirit and the coefficients for the ¯ A expansion are in turn determined from the canonicaltransformation condition (3.8). However a small price is to be paid for getting around thedivergences. By substituting the modified kernels back into (1.7) which used to define Υwe find two sides of the equation slightly mismatch. The differences generate new verticescarrying infinitesimal corrections. p + iǫ A ˆ p A ( p ) + i Z d D − q (cid:20) ¯ q ˆ q A ( q ) , A ( p − q ) (cid:21) (A.2)= Z d D − q q + iǫ A ˆ q B ( q ) δA ( p ) δB ( q )+ ∞ X n =2 Z n Y i =2 d D − q ( i ) ! n X j =2 i ( ǫ A − ǫ B )ˆ q j Υ ...n B ( q ) . . . B ( q n ) (A.3)= Z d D − q q + iǫ A ˆ q B ( q ) δA ( p ) δB ( q ) + i ( ǫ A − ǫ B )ˆ q B ( q ) δA ( p ) δB ( q ) − i ( ǫ A − ǫ B )ˆ p B ( p ) (A.4)Equivalently this can be written as L − + [ A , ¯ A ] + L − ++ [ A , ¯ A ] = L − + [ B , ¯ B ] + L ǫ [ B , ¯ B ] (A.5)where L ǫ represents the new vertex terms. L ǫ [ B , ¯ B ] = − ¯ A i ( ǫ A − ǫ B ) B + ¯ B ( ǫ A − ǫ B ) B = ∞ X m =2 m X s =2 Z ...m ˆ s ˆ p Ξ s − B . . . ¯ B . . . B ! i ( ǫ A − ǫ B ) B (A.6)Introducing double circles to denote the factor ˆ s ˆ p i ( ǫ A − ǫ B ), these terms are expressedgraphically asIn most cases these corrections do not really enter into our calculations because of theinfinitesimal nature of the vertices, except for extremely divergent graphs such as (Fig.1).Because of the asymmetry treatment the factor P ( p j + iǫ ) / ˆ p j in the denominator of thekernel do not cancel completely. A factor of i ( ǫ A − ǫ B ) / ˆ p in the translation kernel is leftto cancel the infinitesimal factor brought by the new vertex, resulting a finite contributionto the loop integral. It is straightforward to show the following four graphs (Fig.3(a) to– 28 – ¯ B B ¯ BB BB ¯ BBB B ¯ B ... B Figure 2:
Infinitesimal vertex terms
Fig.3(d)) constructed from the new vertex restore the h ¯ A ¯ Ai self-energy bubble integral inthe LCYM theory (Fig.4(a)). ppq (a) ppq (b) p p p − q (c) p p p − q (d) Figure 3:
Contributions to the (cid:10) ¯ A ¯ A (cid:11) symmetric loop graph + − − + −− ++¯ A ¯ A Figure 4: h ¯ A ¯ Ai self-energy graph in the LCYM theory – 29 –nother issue regarding iǫ prescription arises if we wish to apply (3.12) to simplify¯ A expansions. In the example illustrated below (Fig.5) the first two graphs are combinedaccording to the identity (A.7).1 p + iǫ B ˆ p + p + iǫ A ˆ p + ( p + p ) + iǫ B ˆ p +ˆ p − p + iǫ B ˆ p + p + iǫ B ˆ p + p + iǫ B ˆ p + p + iǫ A ˆ p = 1 p + iǫ B ˆ p + p + iǫ A ˆ p + ( p + p ) + iǫ B ˆ p +ˆ p p + iǫ B ˆ p + p + iǫ B ˆ p + ( p + p ) + iǫ B ˆ p +ˆ p p + iǫ B ˆ p + p + iǫ B ˆ p + p + iǫ B ˆ p + p + iǫ A ˆ p (A.7)12 3 4 −
12 3 4= ¯ A , B , B , B , Figure 5:
Simplification of the ¯ A expansion – 30 –owever we see in (A.7) the numerator generated from subtraction has a different iǫ as-sociated with line ( p + p ) and does not exactly cancel the factor 1 / (cid:16) p + iǫ B ˆ p + p + iǫ B ˆ p + ( p + p ) + iǫ A ˆ p +ˆ p (cid:17) represented by the small dash line circle on the left. The difference can be accounted for ifwe introduce even more correction graphs carrying infinitesimal vertices.¯ A , B , B , B , Figure 6:
Correction term to the ¯ A expansion i ( ǫ A − ǫ B )ˆ p + ˆ p p + iǫ B ˆ p + p + iǫ A ˆ p + ( p + p ) + iǫ B ˆ p +ˆ p × p + iǫ B ˆ p + p + iǫ B ˆ p + ( p + p ) + iǫ A ˆ p +ˆ p p + iǫ B ˆ p + p + iǫ B ˆ p + p + iǫ B ˆ p + p + iǫ A ˆ p (A.8)Again these corrections can generally be neglected except for symmetrical tadpolessuch as the graph constructed by contracting leg p and p .Another way to deal with this problem without bothering with the iǫ is to change theorders of the LSZ procedure and the overall delta function. Let us look at diagrams: p p p p p p p p We can impose δ (ˆ p + ˆ p ) and the momentum conservation on the right vertex, then applyLSZ procedure and impose the δ (¯ p + ¯ p ) δ (˜ p + ˜ p ) at last. In the LSZ procedure we impose– 31 – → p → lim p → p p Z ˆ3ˆ1(Ω + Ω ) (cid:20) (¯ ζ − ¯ ζ )(¯ ζ − ¯ ζ ) − Ω + Ω + Ω (cid:21) p p (A.9)= lim p → lim p → p p Z ˆ3ˆ1( p ˆ1 + p ˆ2 ) (cid:20) (¯ ζ − ¯ ζ )(¯ ζ − ¯ ζ ) − P ˆ P + p ˆ2 + p ˆ3 (cid:21) p p (A.10)=4 Z ˆ3 p (cid:20) (¯ ζ − ¯ ζ )(¯ ζ − ¯ ζ ) − P ˆ P + p ˆ3 (cid:21) (A.11)=2 Z ˆ3 p (cid:20) (¯ ζ − ¯ ζ )(¯ ζ − ¯ ζ ) − P ˆ P + p ˆ3 (cid:21) − Z ˆ P P (cid:20) (¯ ζ − ¯ ζ )(¯ ζ − ¯ ζ ) − P ˆ P + p ˆ3 (cid:21) (A.12)= − Z ˆ3(ˆ2 + ˆ3) p P (¯ ζ − ¯ ζ )(¯ ζ − ¯ ζ ) (A.13)+ ˆ P P (cid:16) − P ˆ P + p ˆ3 (cid:17) (cid:20) (¯ ζ − ¯ ζ )(¯ ζ − ¯ ζ ) − (¯ ζ − ¯ ζ )(¯ ζ − ¯ ζ ) (cid:21) (A.14)= − Z ¯ V (4 , ,
23) ¯ V (41 , , p P , (A.15)where ¯ ζ ij = ¯ P ij / ˆ P ij . This recovers the light-cone integral. From (A.9) to (A.10) we impose δ (ˆ p + ˆ p ) and the momentum conservation on the right vertex. From (A.10) to (A.11)we apply the LSZ procedure. The would-be singularity of 1 / ( p / ˆ1 + p / ˆ2) is cancelled bythe p factor from LSZ. From (A.11) to (A.12) we split the integrand into two parts andchange the integration variable to one part. (A.12) to (A.14) is simply algebra and from(A.14) to (A.15) we impose the last delta functions. The second diagram can be worked outsimilarly. In fact, this integral is zero after integration as required by helicity conservation.So these kind of diagrams do not contribute to the amplitude. B. Proof of recursion relation (3.9)
We start with the old recursion relation in momentum space:¯1 ¯ ı n − ¯2¯ n .... = − P ≤ l ≤ n − ¯1.... ¯ ıl .... . (B.1)It is easy to see that for n = 3¯1 ¯2¯32 = − ¯2¯1¯3 2 = , – 32 –1 ¯2¯32 = − ¯3¯1¯2 2 = . For n = 4, Ξ (¯1¯2¯3¯4):¯1 ¯33 ¯2¯4 = − (cid:20) (cid:21) (B.2)= − " − (cid:20) (cid:21) (B.3)= − " − − (cid:21) (B.4)= 2 + 2 . (B.5)Equation (B.2) is just the old recursion relation. From (B.2) to (B.3) we expand the secondterm. From (B.3) to (B.4) we combine the first two terms using (3.11). Then the recursionrelation for Ξ (¯1¯2¯3¯4) is proven. Similarly, one can also prove that Ξ (¯1¯2¯3¯4), Ξ (¯1¯2¯3¯4)satisfy the recursion relation.For Ξ s with general n arguments, we suppose that for Ξ s with less than n argumentsthe recursion relation is already proven. Then at the first step we combine the followingterms from the old recursion relation − " ¯1 ...... n − ¯ ıi − ¯1 ...... n − ¯ ıi +1 ¯1 ...... n − . (B.6)By expanding the third term using recursion for Υ and combining terms, using the relation– 33 –3.11), we obtain ¯1 ...... n − ¯ ıi − ¯1 ...... n − ¯ ıi +1 P m ≥ ¯1 ...... ¯ ı ... m + P m ≥ ¯1 ...... ¯ ı ... m (B.7)The first two terms come from the first two terms in (B.6) combined with two terms fromthe expansion of the last terms in (B.6). The two sums are what is left from the expansionof the last term in (B.6).For step l −
1, 3 ≤ l ≤ n −
3, we combine terms − " P ¯1 ...... n − l ¯ ı ...... l − P ¯1 ...... ¯ ı ...... l − − P m ≥ ¯1 ...... ... l − m · · · m − P m ≥ ¯1 ...... ... l − m · · · m (B.8)where the first term is from the old recursion relation, the second term and the m = 2terms in the last two sums come from step l − l − m . After expanding the grey blob in the first term and the black blob in thesecond terms, collecting terms using the relation (3.11) and counting in the other termsleft from step l − m , we obtain P ¯1 ...... ¯ ı ...... l + P m ≥ P r ≥ m +1 " ¯1 ...... ... l − m · · · r + ¯1 ...... ... l − m · · · r . (B.9)Iterate this procedure from (B.8), and at the last step l = n −
2, one can find the result(B.9) is just the right hand side of the recursion relation to be proved.
References [1] F. Cachazo, P. Svrcek and E. Witten, JHEP , 006 (2004) [arXiv:hep-th/0403047]. – 34 –
2] E. Witten, Commun. Math. Phys. (2004) 189 [arXiv:hep-th/0312171].[3] R. Britto, F. Cachazo, B. Feng and E. Witten, Phys. Rev. Lett. (2005) 181602[arXiv:hep-th/0501052].[4] R. Boels, L. Mason and D. Skinner, JHEP (2007) 014 [arXiv:hep-th/0604040].[5] L. J. Mason and D. Skinner, Phys. Lett. B (2006) 60 [arXiv:hep-th/0510262].[6] R. Boels, L. Mason and D. Skinner, Phys. Lett. B (2007) 90 [arXiv:hep-th/0702035].[7] R. Boels, Phys. Rev. D (2007) 105027 [arXiv:hep-th/0703080].[8] R. Boels, C. Schwinn and S. Weinzierl, arXiv:0712.3506 [hep-ph].[9] R. Boels and C. Schwinn, JHEP (2008) 007 [arXiv:0805.1197 [hep-th]].[10] R. Boels and C. Schwinn, arXiv:0805.4577 [hep-th].[11] A. Gorsky and A. Rosly, JHEP (2006) 101 [arXiv:hep-th/0510111].[12] P. Mansfield, JHEP (2006) 037 [arXiv:hep-th/0511264].[13] A. Brandhuber, B. J. Spence and G. Travaglini, Nucl. Phys. B (2005) 150[arXiv:hep-th/0407214].[14] G. Chalmers and W. Siegel, Phys. Rev. D (1996) 7628 [arXiv:hep-th/9606061].[15] J.H. Ettle, C.-H. Fu, J.P. Fudger, P.R. W. Mansfield, and T.R. Morris, JHEP , 011(2007) [arXiv:hep-th/0703286].[16] J. H. Ettle and T. R. Morris, JHEP , 003 (2006) [arXiv:hep-th/0605121].[17] S. J. Parke and T. R. Taylor, Phys. Rev. Lett. (1986) 2459.[18] W. A. Bardeen, Prog. Theor. Phys. Suppl. (1996) 1.[19] Z. Bern and D. A. Kosower, Nucl. Phys. B (1992) 451.[20] Z. Bern and A. G. Morgan, Nucl. Phys. B (1996) 479 [arXiv:hep-ph/9511336].[21] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Nucl. Phys. B (1995) 59[arXiv:hep-ph/9409265].[22] Wen Jiang, D.Phil Thesis. Oxford University, 2008.[23] A. Brandhuber, B. Spence, G. Travaglini and K. Zoubos, JHEP (2007) 002[arXiv:0704.0245 [hep-th]].[24] C. Quigley and M. Rozali, JHEP (2005) 053 [arXiv:hep-th/0410278].[25] J. Bedford, A. Brandhuber, B. J. Spence and G. Travaglini, Nucl. Phys. B (2005) 100[arXiv:hep-th/0410280].[26] A. Brandhuber, B. Spence and G. Travaglini, JHEP (2006) 142 [arXiv:hep-th/0510253].[27] J. Bedford, A. Brandhuber, B. J. Spence and G. Travaglini, Nucl. Phys. B (2005) 59[arXiv:hep-th/0412108].[28] T. R. Morris and Z. Xiao, JHEP (2008) 028 [arXiv:0810.3684 [hep-th]].(2008) 028 [arXiv:0810.3684 [hep-th]].