A new Wilson line-based action for gluodynamics
aa r X i v : . [ h e p - t h ] F e b A new Wilson line-based action for gluodynamics
Hiren Kakkad a , Piotr Kotko a , Anna Stasto ba AGH University Of Science and Technology, Physics Faculty,Mickiewicza 30, 30-059 Krakow, Poland b The Pennsylvania State University, Physics Department104 Davey Lab, University Park, PA 16802, USA
Abstract
We perform a canonical transformation of fields that brings the Yang-Mills action in the light-cone gauge to a new classical action, which does not involve any triple-gluon vertices. The lowestorder vertex is the four-point MHV vertex. Higher point vertices include the MHV and MHVvertices, that reduce to corresponding amplitudes in the on-shell limit. In general, any n -leg vertexhas 2 ≤ m ≤ n − Although quark and gluon fields are considered to be the most fundamental degrees of freedom ofQuantum Chromodynamics (QCD), in variety of situation they are definitely not the most effectiveones. There are several quite distinct aspects of the QCD theory where this becomes manifest. Forexample, in QCD factorization theorems, which are essential tools to relate the theory to high energyexperiments, one needs to account not just for a single quark or gluon exchanged between subprocessesseparated by a large energy scale, but also for all collinear and soft gluons; these resummed exchanges ofgluons can be included by using the Wilson lines (see [1] for a comprehensive review of the factorizationtheorems). Similar collective degrees of freedom appear naturally in the high energy limit of QCD (seee.g. [2, 3]). In the domain of the non-perturbative QCD, further example can be provided by thelattice QCD, where instead of the gauge fields one uses Wilson lines and Wilson loops. It is alsoknown that, in the limit of large number of colors, the gauge theory can be completely formulated inthe loop space, i.e. in terms of various contours of the Wilson loop [4] (see also a textbook [5]).The subject of interest of the following work are scattering amplitudes, in particular the puregluonic amplitudes. In that context, it is already well understood that the elementary triple and four-gluon interactions are not the most effective bricks to build the amplitudes. Pictorially, they are waytoo small, so that the number of Feynman diagrams can become intractable for multi gluon processes.Instead, one should rather use the smaller amplitudes (i.e. with fewer legs) as the building blocks,but they must be deformed to the non-physical domain according to the Britto-Cachazo-Feng-Witten(BCFW) method [6, 7]. A particularly interesting example of the BCFW method is when the onlytype of amplitudes used are the maximally helicity violating (MHV) amplitudes [8]. This type ofrecursion has been found earlier based on the twistor space formulation of the quantum field theory byCachazo, Svrcek and Witten (CSW) [9], who suggested that the MHV amplitudes continued off-shell1re really the multileg vertices . Indeed, an explicit action has been found, where the MHV vertices arereproduced as a result of the canonical field transformation on the Yang-Mills action in light-cone gauge[10, 11]. The corresponding action is often called the MHV action. It was also shown that it reproducescorrectly the one-loop same helicity amplitudes [12, 13, 14, 15, 16]. The new fields appearing in theMHV action turn out to be related to the straight infinite Wilson line of the Yang-Mills fields [17, 18].As the above result is central to the present paper, let us describe it in more details. The light-coneYang-Mills action can be expressed in terms of just two transverse gluon fields, that correspond to twopolarization states in the on-shell limit. The MHV action is obtained by transforming both fields to anew pair of fields. In [17] it was found that the plus helicity field in the MHV action is given as thestraight infinite Wilson line along the complex direction determined by the plus helicity polarizationvector. This means that the line lies on the so-called self-dual plane, i.e. the plane on which the tensorsare self-dual. In the recent paper [18] we found that the minus helicity field is on the other hand givenby a similar Wilson line, but with an insertion of the minus helicty gluon field somewhere on the line.Additionally, in the latter paper we postulated, that it should be a part of a bigger structure, extendingbeyond the self-dual plane.Indeed, in the present work we find a more general canonical transformation based on path orderedexponentials of the gauge fields, extending over both the self-dual and anti-self-dual plane. The fieldtransformation can be most easily derived as a subsequent canonical transformation of the anti-self-dualpart of the MHV action, but we also discuss a direct link between the new action and the Yang-Millsaction. The key property of the new action is that it does not have triple-gluon vertices at all. Thereason for this structure is that the triple-gluon vertices have been effectively resummed inside theWilson lines. The absence of the (+ + − ) vertex already occurs at the level of the MHV action, aspreviously demonstrated in [10]. The second canonical transformation of the anti-self-dual part of theMHV action results in the absence of ( − − +) triple-gluon vertex as well. Thus, the lowest multiplicityvertex is the four-point MHV vertex. Higher-point vertices include not only the MHV vertices, butalso other helicity configurations. The number of diagrams needed to obtain amplitudes beyond theMHV level is thus greatly reduced. We perform explicit calculations within the new formulation ofseveral higher multiplicity amplitudes, to verify the consistency of the results.The paper is organized as follows. In Section 2 we introduce the new action, first on a generalground, and then we proceed to a more technical derivation. In Section 3 we apply the new theory toactual amplitude computations. In Section 4 we summarize the work and discuss some aspects of thenew action, in particular the geometric picture behind the field transformation. Finally, in Appendixwe provide more technical details of selected calculations. Motivated by our earlier results [17, 18], we look for a new set of classical fields describing scatteringprocesses with gluons in the simplest possible way. That is, we want to find an action, which hasinteraction vertices as close to the real scattering processes, as possible. It is known, that the lowestnon-zero scattering amplitude in the physical domain (i.e. on-shell real momenta satisfying the mo-mentum conservation) is the four-point amplitude. Therefore, we will look for an action that has notriple-field vertex. In addition, we want the new fields to have a closed form in terms of the ordinaryYang-Mills fields. This is obviously necessary for practical applications of the new action.In general, a field transformation connects four components of fields, thus, in principle we havefour transformations. In order to reduce the number of degrees of freedom, our starting point will bethe Yang-Mills Lagrangian in the light-cone gauge, on the constant light-cone time x + . As we shallrecall below, such formulation reduces the four components of the gauge field ˆ A µ = t a A µa to just two.Here, t a are color generators in the fundamental representation. To this end, we introduce the ”plus” We use the following normalization of the color generators, common in amplitude-related literature: (cid:2) t a , t b (cid:3) = v µ v + = v · η , v − = v · e η , (1)where η = (1 , , , − / √ e η = (1 , , , / √
2, and two transverse coordinates v • = v · ε + ⊥ , v ⋆ = v · ε −⊥ , (2)defined by the complex null vectors ε ±⊥ = (0 , , ± i, / √
2. The scalar product of two four-vectors inthese coordinates reads u · w = u + w − + u − w + − u • w ⋆ − u ⋆ w • . In order to lower the indices one needsto flip + ↔ − and ⋆ ↔ • , where the latter operation also causes a sign change.The light-cone Yang-Mills action is obtained by setting the light-cone gauge ˆ A + = 0 and integratingout the ˆ A − field from the partition function. The resulting action has only two transverse fields ˆ A • and ˆ A ⋆ and reads [19] S (LC)Y − M [ A • , A ⋆ ] = ˆ dx + ˆ d x ( − Tr ˆ A • (cid:3) ˆ A ⋆ − ig Tr ∂ − − ∂ • ˆ A • h ∂ − ˆ A ⋆ , ˆ A • i − ig Tr ∂ − − ∂ ⋆ ˆ A ⋆ h ∂ − ˆ A • , ˆ A ⋆ i − g Tr h ∂ − ˆ A • , ˆ A ⋆ i ∂ − − h ∂ − ˆ A ⋆ , ˆ A • i ) , (3)where the bold position-space three-vector is defined as x ≡ ( x − , x • , x ⋆ ), x = ( x + , x ) and (cid:3) =2( ∂ + ∂ − − ∂ • ∂ ⋆ ). The presence of just the physical degrees of freedom in the action leads to a naturalidentification of the helicity content of the vertices. Assigning the ”plus” helicity to ˆ A • field , and the”minus” helicity to the field ˆ A ⋆ , we see that there are (+ + − ), ( − − +) and (+ + −− ) vertices in theaction.Following the idea of [10], we look for a field transformation n ˆ A • , ˆ A ⋆ o → n ˆ Z • (cid:2) A • , A ⋆ (cid:3) , ˆ Z ⋆ (cid:2) A • , A ⋆ (cid:3)o , (4)which maps the kinetic term and both the triple-gluon vertices into a free term in the new action.In addition, we demand that the transformation is canonical, so that the functional measure in thepartition function is preserved, up to a field independent factor.Before we present the details on the transformation facilitating the above requirements, let usmake some introductory remarks. Suppose we have a set of generalized coordinates and momenta q i , p i and consider a canonical transformation to a new set Q i , P i . Consider now a particular generatingfunction G for the canonical transformation between { q, p } and { Q, P } , depending only on generalizedcoordinates, G ( q, Q ). Then, the relation between the original and the transformed coordinates is p i = ∂ G ( q, Q ) ∂q i , P i = − ∂ G ( q, Q ) ∂Q i . (5)In our context, the role of q i coordinate is played by the ˆ A • ( x ) field, and the canonical momentum p i is ∂ − ˆ A ⋆ ( x ). In the new theory, we identify Q i with ˆ Z ⋆ ( x ) and P i with ∂ − ˆ Z • ( x ). Therefore, theanalogous relations are ∂ − A ⋆a ( x + , y ) = δ G [ A • , Z ⋆ ]( x + ) δA • a ( x + , y ) , ∂ − Z • a ( x + , y ) = − δ G [ A • , Z ⋆ ]( x + ) δZ ⋆a ( x + , y ) , (6)where we have explicitly denoted the fact, that the transformation is performed on the hyper-surfaceof the constant light-cone time x + . Although the transformation (4) is rather complicated, we foundthat, quite amazingly, the generating functional G [ A • , Z ⋆ ] can be written in the following simple form: G [ A • , Z ⋆ ]( x + ) = − ˆ d x Tr ˆ W − − ) [ Z ]( x ) ∂ − ˆ W (+) [ A ]( x ) , (7) i √ f abc t c and Tr( t a t b ) = δ ab . We re-scale the coupling constant as g → g/ √ √ W ( ± ) [ K ], for a generic vector field K µ , is directly related to the straight infiniteWilson line in the following way: W a ( ± ) [ K ]( x ) = ˆ ∞−∞ dα Tr (cid:26) πg t a ∂ − P exp (cid:20) ig ˆ ∞−∞ ds ε ± α · ˆ K (cid:0) x + sε ± α (cid:1)(cid:21)(cid:27) , (8)with ε ± µα = ǫ ± µ ⊥ − αη µ . (9)The above four vector has a form of a gluon polarization vector, indeed for α = p · ε ±⊥ /p + , it is thetransverse polarization vector for a gluon with momentum p . This type of functional has been used forthe first time in [17] in the context of the MHV Lagrangian. The inverse functional to the Wilson line, W − , is defined as W [ W − ] = 1. Note, that the functionals W (+) and W ( − ) are not exactly hermitianconjugates of each other; only the projection on ε + α or ε − α changes inside the path-ordered exponential,but the sign of ig remains unchanged.In the following sections we shall present more details on the implication of the transformationgiven by (6) and the exact form of the vertices in the new action. In the remaining part of this section,we will outline the general structure of the new action.From Eqs. (6)-(7) one can see that the fields ˆ A • and ˆ A ⋆ have the following expansion in terms ofthe new fields: A • a ( x + ; x ) = ∞ X n =1 ˆ d y . . . d y n n X i =1 Ξ ab ...b n i,n − i ( x ; y , . . . , y n ) i Y k =1 Z • b k ( x + ; y k ) n Y l = i +1 Z ⋆b l ( x + ; y l ) , (10) A ⋆a ( x + ; x ) = ∞ X n =1 ˆ d y . . . d y n n X i =1 Λ ab ...b n i,n − i ( x ; y , . . . , y n ) i Y k =1 Z ⋆b k ( x + ; y k ) n Y l = i +1 Z • b l ( x + ; y l ) , (11)where Ξ ab ...b i + j i,j ( x ; y . . . y i + j ) is an apriori unknown kernel for the i number of Z • fields and j num-ber of Z ⋆ fields in the expansion of ˆ A • , depending on the adjoint color indices a, b . . . b i + j and notdepending on the light-cone time. Similarly, Λ ab ...b i + j i,j ( x ; y . . . y i + j ) is the kernel for i number of Z ⋆ fields and j number of Z • fields in the expansion of ˆ A ⋆ . At lowest order we must have A • a ( x + ; x ) = Z • a ( x + ; x ) + . . . , A ⋆a ( x + ; x ) = Z ⋆a ( x + ; x ) + . . . . (12)In principle, one could find explicitly the kernels Ξ i,j , Λ i,j from Eqs. (6)-(7). However, as wedemonstrate in the next section, there is a much better way of doing that, which utilizes the existingresults on the MHV Lagrangian [17, 18]. Since we want to describe a general structure of the action,for the rest of this section we shall assume that the kernels are known.Inserting the solutions (10)-(11) to the Yang-Mills action (3), we find the following structure of thenew action: S (LC)Y − M [ Z • , Z ⋆ ] = ˆ dx + ( − ˆ d x Tr ˆ Z • (cid:3) ˆ Z ⋆ + L (LC) −− ++ + L (LC) −− +++ + L (LC) −− ++++ + . . . + L (LC) −−− ++ + L (LC) −−− +++ + L (LC) −−− ++++ + . . . ...+ L (LC) −−−···− ++ + L (LC) −−−···− +++ + L (LC) −−−···− ++++ + . . . ) , (13)4here the n -point interaction vertex, n ≥
4, that couples m minus helicity fields, m ≥
2, and n − m plus helicity fields, has the following general form: L (LC) − · · · − | {z } m + · · · + | {z } n − m = ˆ d y . . . d y n U b ...b n −···− + ··· + ( y , · · · y n ) m Y i =1 Z ⋆b i ( x + ; y i ) n − m Y j =1 Z • b j ( x + ; y j ) . (14)The above action has the following properties, which we will elaborate on in the next sections: i ) There are no three point interaction vertices. ii ) At the classical level there are no all-plus, all-minus, as well as ( − + · · · +), ( − · · · − +) vertices. iii ) There are MHV vertices, ( − − + · · · +), corresponding to MHV amplitudes in the on-shell limit. iv ) There are MHV vertices, ( − · · · − ++), corresponding to MHV amplitudes in the on-shell limit. v ) All vertices have the form which can be easily calculated.Because the lowest vertex is the single MHV four-point vertex that corresponds to the four-gluon MHVamplitude in the on-shell limit, the new action provides an efficient way to construct tree amplitudeswith high multiplicity of legs, as we will demonstrate later in Section 3 by computing several examples. As we shall see in the following, the easiest way to derive the action (13) from the Yang-Mills action(3) is to first transform the latter into an action containing the MHV vertices. Thus, we start by abrief summary of this procedure.
As explained in detail in [10], the MHV action implementing the CSW rules [9] is obtained by per-forming a canononical field transformation with a requirement that the kinetic term and the (+ + − )triple-gluon vertex is mapped to a free kinetic term in the new action:Tr ˆ A • (cid:3) ˆ A ⋆ + 2 ig Tr ∂ − − ∂ • ˆ A • h ∂ − ˆ A ⋆ , ˆ A • i −→ Tr ˆ B • (cid:3) ˆ B ⋆ . (15)Note, the two terms on the l.h.s constitute the self-dual sector of the Yang-Mills theory [20, 21, 22, 23,24]. Therefore, as shown in [18] the solution to the required transformation of fields can be expressedin terms of the straight infinite Wilson line lying on the self-dual plane, i.e. the plane spanned by the ε + ⊥ and η . It is exactly the Wilson line W (+) introduced in the preceding section. The new fields B ,expressed in terms of the Yang-Mills fields A , read B • a [ A • ]( x ) = W a (+) [ A ]( x ) , B ⋆a [ A • , A ⋆ ]( x ) = ˆ d y " ∂ − ( y ) ∂ − ( x ) δ W a (+) [ A ]( x + ; x ) δA • c ( x + ; y ) A ⋆c ( x + ; y ) . (16)The expressions for the fields in momentum space have the following form [17, 18] e B • a ( x + ; P ) = ∞ X n =1 ˆ d p . . . d p n e Γ a { b ...b n } n ( P ; { p , . . . , p n } ) n Y i =1 e A • b i ( x + ; p i ) , (17) e B ⋆a ( x + ; P ) = ∞ X n =1 ˆ d p . . . d p n e Υ ab { b ...b n } n ( P ; p , { p , . . . , p n } ) e A ⋆b ( x + ; p ) n Y i =2 e A • b i ( x + ; p i ) , (18)5here e Γ a { b ...b n } n ( P ; { p , . . . , p n } ) = ( − g ) n − δ ( p + · · · + p n − P ) Tr (cid:0) t a t b . . . t b n (cid:1)e v ∗ ··· n ) e v ∗ (12)(1 ··· n ) · · · e v ∗ (1 ··· n − ··· n ) , (19) e Υ ab { b ··· b n } n ( P ; p , { p , . . . , p n } ) = n (cid:18) p +1 p +1 ··· n (cid:19) e Γ ab ...b n n ( P ; p , . . . , p n ) . (20)Above, the tildes over the fields and the kernels Γ, Υ denote the Fourier transformed quantities withrespect to the three momenta p = ( p + , p • , p ⋆ ). The curly brackets denote the symmetrization withrespect to the pairs of momentum and color indices. The e v ij , e v ⋆ij are quantities similar to spinorproducts h ij i , [ ij ], with the following explicit definitions (first introduced in [25] in the context of thegluon wave function): e v ij = p + i p ⋆j p + j − p ⋆i p + i ! , e v ∗ ij = p + i p • j p + j − p • i p + i ! . (21)They appear quite naturally in the Wilson line approach, because e v ∗ ij = − ( ε + i · p j ) , e v ij = − ( ε − i · p j ) , (22)where ε ± i is the polarization vector for a momentum p i obtained from (9) which appears as the directionof the Wilson line. See [26] for several useful properties of the e v ij symbols. We also use a shorthandnotation for the sum of momenta p + · · · + p n ≡ p ...n .The expressions with Wilson lines (16) (or equivalently (17), (18)) can be inverted to obtain thepower expansions for A • , A ⋆ in terms of B • , B ⋆ , which are consistent with [11]. In momentum spacewe get: e A • a ( x + ; P ) = ∞ X n =1 ˆ d p . . . d p n e Ψ a { b ...b n } n ( P ; { p , . . . , p n } ) n Y i =1 e B • b i ( x + ; p i ) , (23) e A ⋆a ( x + ; P ) = ∞ X n =1 ˆ d p . . . d p n e Ω ab { b ··· b n } n ( P ; p , { p , . . . , p n } ) e B ⋆b ( x + ; p ) n Y i =2 e B • b i ( x + ; p i ) , (24)where the kernels are e Ψ a { b ··· b n } n ( P ; { p , . . . , p n } ) = − ( − g ) n − e v ⋆ (1 ··· n )1 e v ⋆ ··· n ) δ ( p + · · · + p n − P ) Tr( t a t b · · · t b n ) e v ⋆ e v ⋆ · · · e v ⋆n ( n − , (25) e Ω ab { b ··· b n } n ( P ; p , { p , . . . , p n } ) = n (cid:18) p +1 p +1 ··· n (cid:19) e Ψ ab ··· b n n ( P ; p , . . . , p n ) . (26)Inserting the above solutions to the Yang-Mills action we can derive the MHV action: S (LC)Y − M [ B • , B ⋆ ] = ˆ dx + (cid:18) − ˆ d x Tr ˆ B • (cid:3) ˆ B ⋆ + L (LC) −− + + · · · + L (LC) −− + ··· + + . . . (cid:19) , (27)where the n -point MHV interaction terms are L (LC) −− + ··· + = ˆ d p . . . d p n δ ( p + · · · + p n ) e V b ...b n −− + ··· + ( p , . . . , p n ) e B ⋆b (cid:0) x + ; p (cid:1) e B ⋆b (cid:0) x + ; p (cid:1) e B • b (cid:0) x + ; p (cid:1) . . . e B • b n (cid:0) x + ; p n (cid:1) , (28)with the MHV vertices e V b ...b n −− + ··· + ( p , . . . , p n ) = X noncyclicpermutations Tr (cid:0) t b . . . t b n (cid:1) V (cid:0) − , − , + , . . . , n + (cid:1) , (29)6here the color ordered vertex reads V (cid:0) − , − , + , . . . , n + (cid:1) = ( − g ) n − ( n − (cid:18) p +1 p +2 (cid:19) e v ∗ e v ∗ n e v ∗ n ( n − e v ∗ ( n − n − . . . e v ∗ , (30)Above, we defined the momentum space color ordered vertex without the tilde, as this will not leadto any confusion. Note that, we have written the MHV vertices in a form, where the negative helicityfields are always adjacent, but there is a sum over the color permutations, together with the propersymmetry factor.The vertices (30) are fully off-shell quantities, but they correspond to the MHV amplitudes in theon-shell limit (which is evident from the fact that the e v ij symbols are in one-to-one correspondence tothe spinor products). These vertices are in general not gauge invariant for off-shell kinematics, but asshown in [26, 17], they do constitute a gauge invariant off-shell amplitude, when only one plus-helicityleg is kept off-shell. In [18] we argued, that the second equation in (16) suggests, that there should exist a more generalstructure than the Wilson line on the ε ⊥ - η plane. The latter should be just a slice of this moregeneral structure. That is, there should exist a functional which path-orders the A ⋆ fields in the planeperpendicular to the ε + ⊥ - η plane.In order to actually introduce such an object, let us consider a canonical transformation of theMHV action itself. We demand that L − + [ B • , B ⋆ ] + L −− + [ B • , B ⋆ ] −→ L − + [ Z • , Z ⋆ ] , (31)where L − + is just the kinetic term in either B or Z fields, cf. Eq. (15). The vertex L −− + [ B • , B ⋆ ]appearing in (27) has exactly the same form as the ( − − +) triple-gluon vertex in the original Yang-Mills action (3), but with A fields replaced by B fields. Therefore, the corresponding transformationsare analogous to those leading to the MHV action, but with the replacement • ↔ ⋆ . More precisely Z ⋆a [ B ⋆ ]( x ) = W a ( − ) [ B ]( x ) , Z • a [ B • , B ⋆ ]( x ) = ˆ d y " ∂ − ( y ) ∂ − ( x ) δ W a ( − ) [ B ]( x + ; x ) δB ⋆c ( x + ; y ) B • c ( x + ; y ) . (32)Let us point out the important feature of the above formula. Unlike the transformation leadingto the MHV action, Eq. (16), which involved the Wilson lines W (+) along ε + α , here we have the theWilson lines W ( − ) that have directions ε − α , see the definitions (8). Thus, pictorially, the Z ⋆ field is theWilson line on the η - ε −⊥ plane, where the path ordered fields are itself Wilson lines on the η - ε + ⊥ plane(see also Fig. 9 in Section 4).Already at this stage one can check that the generating functional (7) is consistent with the abovetransformations. Inserting (7) to (6) we have ∂ − A ⋆a (cid:0) x + ; x (cid:1) = − ˆ d y W c − − ) [ Z ]( x + ; y ) ∂ − δδA • a ( x + ; x ) W c (+) [ A ]( x + ; y ) , (33)and ∂ − Z • a (cid:0) x + ; x (cid:1) = ˆ d y (cid:20) δδZ ⋆a ( x + ; x ) W c − − ) [ Z ]( x + ; y ) (cid:21) ∂ − W c (+) [ A ]( x + ; y ) . (34)Integrating Eq. (33) by parts and using the first equation of (16) we get ∂ − A ⋆a (cid:0) x + ; x (cid:1) = ˆ d y h ∂ − W c − − ) [ Z ]( x + ; y ) i δB • c ( x + ; y ) δA • a ( x + ; x ) . (35)7omparing this with the canonical transformation rule for the A ⋆ field of [10] which reads ∂ − A ⋆a (cid:0) x + ; x (cid:1) = ˆ d y δB • c ( x + ; y ) δA • a ( x + ; x ) ∂ − B ⋆c (cid:0) x + ; y (cid:1) , (36)we see that B ⋆c [ Z ⋆ ]( x ) = W c − − ) [ Z ]( x ) , (37)or, upon inverting, Z ⋆c [ B ⋆ ]( x ) = W c ( − ) [ B ]( x ) , (38)which gives the left equation of (32). Inserting now (37) into (34) and using the first equation of (16)we get ∂ − Z • a (cid:0) x + ; x (cid:1) = ˆ d y δB ⋆c ( x + ; y ) δZ ⋆a ( x + ; x ) ∂ − B • c (cid:0) x + ; y (cid:1) . (39)or ∂ − B • a (cid:0) x + ; x (cid:1) = ˆ d y δZ ⋆c ( x + ; y ) δB ⋆a ( x + ; x ) ∂ − Z • c (cid:0) x + ; y (cid:1) . (40)which, is virtually the same as (36), but with the replacement • ↔ ⋆ and the Z fields instead of B fields and B fields instead of A fields. Since (36), together with the left equation of (16), leads to thesolution given by the right equation of (16), we can argue that (40), together with (38), leads to theright equation of (32). Note, that because of the replacement • ↔ ⋆ , the Wilson line W (+) in (16)becomes ↔ W ( − ) in (32).To conclude, we have shown that the generating functional (7) takes care of the chain of bothcanonical transformations, from A fields to B fields and from B fields to Z fields, simultaneously, asshown in the diagram in Fig. 1. Y-M MHV Z
Action ActionAction C a n o n i c a l T r a n s f o r m a t i o n Canonical T r a n s f o r m a t i o n C a n o n i c a l Transformation
Figure 1:
Two ways to derive the new action. First is the direct method which involves the generatingfunctional (7). Second involves two consecutive canonical field transformation.
We have just seen that the transformation from the Yang-Mills action to the new action generated bythe functional (7) is equivalent to two canonical transformations: first transforming the self-dual partof the Yang-Mills action to a free action in B -field theory, and then transforming the anti-self-dualpart in the latter to a free term in the new Z -field theory. Therefore we can readily write the relationsbetween the Z fields and B fields in momentum space.8n order to derive the content of the Z -field action, we need to insert the expansions of B fields in Z fields. For the B ⋆ field we find e B ⋆a ( x + ; P ) = ∞ X n =1 ˆ d p . . . d p n e Ψ a { b ...b n } n ( P ; { p , . . . , p n } ) n Y i =1 e Z ⋆b i ( x + ; p i ) , (41)with e Ψ a { b ··· b n } n ( P ; { p , . . . , p n } ) = − ( − g ) n − e v (1 ··· n )1 e v ··· n ) δ ( p + · · · + p n − P ) Tr( t a t b · · · t b n ) e v e v · · · e v n ( n − . (42)Note that, the above quantity has the same form as (25), however with e v ⋆ij replaced by its complexconjugate e v ij . The expansion for the B • field follows from (24) and reads e B • a ( x + ; P ) = ∞ X n =1 ˆ d p . . . d p n e Ω ab { b ··· b n } n ( P ; p , { p , . . . , p n } ) e Z • b ( x + ; p ) n Y i =2 e Z ⋆b i ( x + ; p i ) , (43)where e Ω ab { b ··· b n } n ( P ; p , { p , . . . , p n } ) = n (cid:18) p +1 p +1 ··· n (cid:19) e Ψ ab ··· b n n ( P ; p , . . . , p n ) . (44)Inserting the above expansions to (17)-(18) makes it in principle possible to derive an explicit formof the expansions between A fields and Z fields. One can thus explicitly find the kernels Ξ i,j , Λ i,j introduced in Eqs. (10)-(11). However, it turns out that this is not necessary. It is much more efficientto insert the above expressions into the MHV action, as we shall do in the following. Although, formally, both field transformations A → B and B → Z are such that they remove thetriple-gluon vertices, this cancellation is actually not immediately visible. Therefore, in Appendix Awe directly check that the first terms of the expansions (10)-(11) indeed cancel both of the triple-gluonvertices in the Yang-Mills action.The remaining terms are obtained by inserting the expansions (41),(43) into the MHV vertices(29), for n ≥
4. We shall find a general expression for the vertex when the negative helicity fields areadjacent. Without loosing the generality we shall focus on the color ordered vertex, defined as U b ...b n −···− + ··· + ( p , . . . , p n ) = X noncyclicpermutations Tr (cid:0) t b . . . t b n (cid:1) U (cid:0) − , . . . , m − , ( m + 1) + , . . . , n + (cid:1) , (45)where we assumed there are m minus helicity legs. We shall also need the color ordered versions ofthe kernels in the expansions (41)-(43). We define e Ψ a { b ...b m } m ( P ; { p , . . . , p m } ) = X noncyclicpermutations Tr (cid:0) t b . . . t b m (cid:1) Ψ (cid:0) − , . . . , m − (cid:1) , (46)and e Ω ab { b ...b m } m ( P ; p , { p , . . . , p m } ) = X noncyclicpermutations Tr (cid:0) t b . . . t b m (cid:1) Ω (cid:0) + , − , . . . , m − (cid:1) , (47)where for further convenience we explicitly denoted helicity of the legs in the color ordered kernels.Above, we have explicitly denoted the fact, that the Ψ m kernel multiplicates the minus helicity leg into m minus helicity legs, whereas the Ω m kernel multiplicates the plus helicity leg into one plus helicity9 − m m + 1 − − + n + V− − − − −− − + + + − + pp + 1 q q + 1 r r + 1 mm + 1 m + 2 n − n − − Figure 2:
Left: color ordered vertex in the Z -field theory with m minus helicity legs. Right: a generalcontribution to the Z -theory vertex. The central blob is the MHV vertex. leg and ( m −
1) adjacent minus helicity legs. Note also, that similar to (30), we have omitted the tildesigns in the momentum space color-ordered vertex and kernels.We observe, that the most general contribution has the form depicted in Fig. 2. Indeed, uponsubstitution for the plus helicity fields of the MHV vertex in terms of the Ω kernel, results in oneplus helicity leg in addition to negative helicity legs. Thus it must be adjacent to other minus helicitylegs multiplicated by the Ψ kernels. Therefore, there can be at most two Ω kernels. Also, there canbe at most two Ψ kernels, because there are only two negative helicity legs in the MHV vertex. Inaddition to the above general situation, there are cases when the kernels are trivial, i.e. Ψ = 1 orΩ = 1. Let us note, that the MHV vertex has always outgoing momenta, but the Ψ and Ω kernelshave the off-shell line incoming. In addition, there is no propagator connecting the kernels with theMHV vertex. Therefore, we can identify the internal lines by a single helicity flow, given by the helicityof the MHV vertex.Specifically, consider now the vertex for n external legs with the momenta p . . . p n , where p . . . p m correspond to the minus helicity legs. Let us introduce a collective index [ i, i + 1 , . . . , j ], labeling themomentum p i ( i +1) ...j = p i + p i +1 + · · · + p j . Using this notation, the general form of the color orderedvertex can be written as: U (cid:0) − , − , . . . , m − , ( m +1) + , . . . , n + (cid:1) = m − X p =0 m − X q = p +1 m X r = q +1 V (cid:0) [ p +1 , . . . , q ] − , [ q +1 , . . . , r ] − , [ r +1 , . . . , m +1] + , ( m +2) + , . . . , ( n − + , [ n, , . . . , p ] + (cid:1) Ω (cid:0) n + , − , . . . , p − (cid:1) Ψ (cid:0) ( p +1) − , . . . , q − (cid:1) Ψ (cid:0) ( q +1) − , . . . , r − (cid:1) Ω (cid:0) ( r +1) − , . . . , m − , ( m +1) + (cid:1) (48)Although the analytic formulas do not seem to collapse, in general, to any simple form, the aboveexpression is operational and can be readily applied in actual amplitude calculation, as we shall demon-strate in Section 3. Let us summarize the content of the new action. It contains a set of vertices with increasing multiplicity,starting at n = 4. Each vertex has at least two minus helicity legs, and at most n −
2. Thus, we have10HV vertices, next-to-MHV (NMHV) vertices, next-to-next-to-MHV (NNMHV) vertices and so on.The vertices with maximal number of minus helicity legs are just MHV vertices. Both the MHV andMHV vertices alone give the corresponding on-shell amplitudes. In addition to these vertices we havea scalar propagator joining two opposite helicity legs.In the following section we shall demonstrate how various amplitudes are calculated. To this endwe introduce the following color-ordered Feynman rules: i ) scalar propagator joining a plus and a minus helicity leg + − p = ip ii ) n -point vertex , n ≥
4, with m negative helicity legs, 2 ≤ m ≤ n − +1 = i U − , . . . , m − , ( m +1) + , . . . , n + ! ++ −− − m m + 2 n m + 1 In this section we shall calculate several tree amplitudes using the new action.
The lowest non-zero amplitude is the 4-point MHV amplitude. It is simply given by the 4-point vertexin the theory.The non-zero 5-point amplitudes are MHV ( − − + + +) and MHV ( − − − + +). Both amplitudesare given just by a single vertex, respectively U (1 − , − , + , + , + ) and U (1 − , − , − , + , + ). For theMHV it is the expression (30), giving for the amplitude A (1 − , − , + , + , + ) = − g (cid:18) p +1 p +2 (cid:19) e v ∗ e v ∗ e v ∗ e v ∗ e v ∗ e v ∗ . (49)For the MHV the expression is easily obtained from (48). In Fig. 3 we show the contributing terms.Using the explicit expressions we have: U (1 − , − , − , + , + ) = g " (cid:18) p +1 p +23 (cid:19) e v ∗ e v ∗ e v ∗ e v ∗ e v ∗ (23)1 × e v (23)2 e v e v + (cid:18) p +12 p +3 (cid:19) e v ∗ e v ∗ (12)5 e v ∗ e v ∗ e v ∗ × e v (12)1 e v e v + (cid:18) p +2 p +3 (cid:19) e v ∗ e v ∗ e v ∗ (15)4 e v ∗ e v ∗ × (cid:18) p +5 p +15 (cid:19) e v (15)5 e v e v + (cid:18) p +1 p +2 (cid:19) e v ∗ e v ∗ e v ∗ e v ∗ (34)2 e v ∗ × (cid:18) p +4 p +34 (cid:19) e v (34)3 e v e v . (50)11 HV Ψ − − − ++ − MHV − ++ − MHV − ++ − Ω + − Ω+ − MHV − ++ − Ψ − − − − − ++ = Figure 3:
The contributions to the color-ordered MHV vertex, with helicity ( − − − + +).
We have checked, that the above expression reduces in the on-shell limit to the known formula for theMHV amplitude: A (1 − , − , − , + , + ) = g (cid:18) p +4 p +5 (cid:19) e v e v e v e v e v e v . (51) The MHV and MHV amplitudes are always given by the single vertices. For the latter we need the U (1 − , − , − , − , + , + ) vertex which is given by the formula (48), see Appendix B. We have verifiedthat it recovers the correct result in the on-shell limit. −−− + + + − + + −−− + ++ − + −−− + + Figure 4:
Diagrams contributiong to 6-point NMHV amplitude ( − − − + ++).
The remaining amplitude is the the NMHV amplitude with helicity configuration ( − − − + ++).We have just three contributing diagrams depicted in Fig. 4. First two diagrams connect two MHVvertices, whereas the last one is the NMHV vertex, given by U (1 − , − , − , + , + , + ). We have checkedthat the sum of those diagrams reproduce in the on-shell limit the known result [27]. In addition to the MHV and MHV amplitudes, which are again calculated through just a single vertex,we have the NMHV and NNMHV amplitudes.The diagrams contributing to the NMHV amplitude are depicted in Fig. 5. We have four diagramsconnecting 4-point and 5-point MHV vertices and one diagram which consists of the single vertex withNMHV helicity configuration. The corresponding expression is D + D + D + D + 12 D , (52)12 − −− ++ + − + − + −− ++ + + − − + − ++ + − + −− −− ++ + + − + + −− + + + − Figure 5:
Diagrams contributiong to 7-point NMHV amplitude ( − − − + + + +). where the individual diagrams are calculated as: D = i U (cid:0) − , − , [3 , , + , + , + (cid:1) × ip × i U (cid:0) [6 , , , − , − , + , + (cid:1) (53) D = i U (cid:0) − , [2 , , − , + , + , + (cid:1) × ip × i U (cid:0) − , − , + , [5 , , , + (cid:1) (54) D = i U (cid:0) − , − , [3 , , , + , + (cid:1) × ip × i U (cid:0) [7 , , − , − , + , + , + (cid:1) (55) D = i U (cid:0) − , [2 , , , − , + , + (cid:1) × ip × i U (cid:0) − , − , + , + , [6 , , + (cid:1) (56) D = i U (cid:0) − , − , − , + , + , + , + (cid:1) . (57)Above, i/p i ...i m = i/ ( p i + · · · + p i m ) is the scalar propagator. The factor 1 / D comes from the fact there are two possible color orders contributing to diagrams D - D . We havecompared the sum of those diagrams, i.e. Eq.(52) with the on-shell result obtained using the GGT
Mathematica package [28] together with the
S@M package [29] and found an exact match, up to anoverall normalization due to the difference in our symbol ˜ v ij and h ij i . −− −− ++ + + − + − −− + − + − + − −− ++ + − + −− −− − + + + − + −−− + + + − Figure 6:
Diagrams contributiong to 7-point NNMHV amplitude ( − − − − + + +).
For the NNMHV amplitude the number of diagrams stays the same, see Fig. 6. Here, however, weencounter a new feature, namely there are now diagrams that connect the 4-point MHV vertex with5-point MHV vertex. That is, starting with this amplitude we utilize the new vertices appearing in13he theory in a nontrivial way (i.e. by gluing them with other vertices). We again find the on shelllimit of the result consistent with the
GGT package.
We have also calculated some of the non-trivial 8-point amplitudes, all of which agree with standardresults calculated numerically using the
GGT package. + − + −− ++ ++ + −−− ++ ++ − −− ++ + − −− ++ + − ++ − + −− ++ + + − + + + − ++ − − + − + − ++ + − ++ − + −− + + + − + Figure 7:
Diagrams contributiong to 8-point NMHV amplitude ( − − − + + + ++).
The 8-point NMHV amplitude turns out to be actually very simple, requiring only 7 diagrams, ofwhich 6 join two MHV vertices, see Fig. 7.In order to really test the new theory we calculated the 8-point NNMHV amplitude. In that case,we encounter only 13 diagrams, shown in Fig. 8, of which some consist of three MHV vertices joinedby the scalar propagators, but there are also diagrams combining the 5-point MHV vertex as well as6-point NMHV vertex appearing in the Z -field theory. As this calculation is less trivial, let us list thestructure of the diagrams (they correspond to the diagrams in Fig. 8 from the top left, going to theright): D = i U (cid:0) [7 , , , , − , − , + , + (cid:1) × ip × i U (cid:0) − , [2 , , , , − , + , + (cid:1) × ip × i U (cid:0) − , − , [4 , , + , [7 , , + (cid:1) , (58) D = i U (cid:0) − , − , + , [6 , , , , + (cid:1) × ip × i U (cid:0) − , [2 , , , , − , + , + (cid:1) × ip × i U (cid:0) − , [3 , , , ] − , + , [7 , , + (cid:1) , (59) D = i U (cid:0) [7 , , , , − , − , + , + (cid:9) × ip × i U (cid:0) − , − , [3 , , , , + , + (cid:1) × ip × i U (cid:0) [8 , , − , − , [4 , , + , + (cid:1) , (60)14 −− + + + − + − + − ++ −− + + + − −
418 7 6 ++ + −−− + + − − + + −−−
32 5 5 + −−−− + ++ + −− + + − −−− ++ ++ + −−− − + ++ − + + −− −−− ++ ++ − + − + −− + − ++ + − − −− ++ + + −− + −− −− ++ + − −− ++ + + − + + − −− ++ + − + − + −− + −−− + + + − + Figure 8:
Diagrams contributiong to 8-point NNMHV amplitude ( − − − − + + ++). D = i U (cid:0) − , − , + , [6 , , , , + (cid:1) × ip × i U (cid:0) − , − , [3 , , , , + , + (cid:1) × ip × i U (cid:0) [8 , , − , [3 , , − , + , + (cid:1) , (61) D = i U (cid:0) − , [2 , , , − , + , + , + (cid:1) × ip × i U (cid:0) − , − , − , + , [6 , , , (cid:1) , (62) D = i U (cid:0) − , − , [3 , , , + , + , + (cid:1) × ip × i U (cid:0) [7 , , , − , − , − , + , + (cid:1) , (63) D = i U (cid:0) − , − , − , [4 , , , + , + (cid:1) × ip × i U (cid:0) [8 , , , − , − , + , + , + (cid:1) , (64) D = i U (cid:0) − , − , [3 , , , − , + , + (cid:1) × iP × i U (cid:0) − , − , + , + , [7 , , , + (cid:1) , (65) D = i U (cid:0) − , − , [3 , , , , + , + (cid:9) × ip × i U (cid:0) [8 , , − , − , − , + , + , + (cid:1) , (66) D = i U (cid:0) − , [2 , , , , − , + , + (cid:1) × ip × i U (cid:0) − , − , − , + , + , [7 , , + (cid:1) , (67) D = i U (cid:0) − , − , + , [6 , , , , + (cid:1) × ip × i U (cid:0) − , − , [3 , , − , + , + , + (cid:1) , (68)15 legs helicity Total number of diagrams for helicity amplitudes of different multiplicities. D = i U (cid:0) [7 , , , , − , − , + , + (cid:1) × ip × i U (cid:0) − , − , − , [4 , , + , + , + (cid:1) , (69) D = i U (cid:0) − , − , − , − , + , + , + , + (cid:1) . (70)The above diagrams have to be combined as follows, due to the additional combinatorial factors asexplained in the previous subsection: D + D + D + D + 12 ( D + D + D + D + D + D + D + D ) + 14 D . (71) In the present work we have constructed a new action for gluodynamics by applying two consecutivecanonical transformations on the light-cone Yang-Mills action, see Fig. 1. The same action can be alsoobtained by a single canonical transformation, whose generating functional is given by Eq. (6).The most striking property of the new action is that it has no triple-gluon vertex. Effectively,the triple-gluon vertices are resummed inside the Wilson lines (see [17] for the explicit demonstrationof that fact for the (+ + − ) vertex). Consequently, the number of diagrams needed to calculate theamplitudes is reduced, as compared for example to the CSW method. For example, for the NMHVamplitudes with adjacent helicity, the CSW rules give 2( n −
3) diagrams, whereas the new theory gives2( n − n ≥
5. We give the number of diagrams for various adjacent helicity configurationsin Table 1. It is important to stress, that we do not mean here the number of contributing terms, asthe vertices in the new theory are not, in general, given by a single term in our representation. Thestructure of those vertices can be however easily obtained by means of the master equation (48).16 A • A ⋆ A • A • x z ǫ + ⊥ - η plane(self-dual plane) ǫ −⊥ - η plane(anti-self-dual plane) B ⋆ ( z ) Z ⋆ ( y ) Figure 9:
Schematic presentation of the geometric structure of the Z ⋆ field (the structure of Z • is quitesimilar). The vertical planes are the self-dual planes, i.e. the planes spanned by the vectors ε + ⊥ = (0 , , i, η = (1 , , , − ε −⊥ = (0 , , − i,
0) and η . The B ⋆ fields (the minus helicity fields in the MHV action) are the straight infiniteWilson lines on the self-dual plane, integrated over all slopes, and differentiated functionally to replace oneplus gluon helicity field by the minus gluon helicity field. This structure is represented by the blue lines. The Z ⋆ field, i.e. the minus helicity field in the new theory, is given by a similar Wilson line of the B ⋆ fields, lyingon the anti-self dual plane, and integrated over all slopes. One of the very interesting aspects of the field transformations leading to the new action is itsincredibly rich geometric structure. Let us recall, that the new minus helicity field Z ⋆ is given bythe straight infinite Wilson line on the anti-self-dual plane (i.e. the plane spanned by ǫ −⊥ and η ),integrated over all directions. Note, this is the Wilson line functional of the minus helicity field of theMHV theory, which itself is given as the analogous Wilson line of the usual gauge fields lying on theself-dual plane, with insertion of the minus helicity field (i.e. the functional derivative, see Eq. (32)).We try to schematically depict the structure of the Z ⋆ field in Fig. 9. A similar figure can be drawnfor the Z • field. We stress, that although the overall picture looks very non-local, it is local in thelight-cone time.In our recent work [18] we have discussed the fields in the MHV theory, where the fields can beexpressed as the Wilson line functionals of the standard gauge fields lying exclusively on the self-dualplane. We see that the transformations derived in the present work extend this picture to the whole3-space, with the light-cone time fixed.One of the future directions is obviously to the quantum corrections to the new action. There isan immediate difficulty in that program, namely, the fact that in the quantum MHV action there arecontributions evading the S-matrix equivalence theorem [12], and we expect similar contributions inthe new action. An alternative approach is based on the world-sheet regularization, which successfullyrecovered one loop all-plus helicity vertex in the MHV action [14].Another interesting direction of future study is related to the rich geometric structure of thetransformations, sketched in Fig. 9, which has not been fully explored. H.K. and P.K. are supported by the National Science Center, Poland grant no. 2018/31/D/ST2/02731.A.M.S. is supported by the U.S. Department of Energy Grant DE-SC-0002145 and in part by National17cience Centre in Poland, grant 2019/33/B/ST2/02588.
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Cancellation of triple-gluon vertices.
In this appendix we show the cancellation of both the triple-gluon vertices when transforming theYang-Mills fields to the new fields, n ˆ A • , ˆ A ⋆ o → n ˆ Z • (cid:2) A • , A ⋆ (cid:3) , ˆ Z ⋆ (cid:2) A • , A ⋆ (cid:3)o . (72)As previously argued, there are two ways of doing this: using directly the generating functional (7),or using two consecutive canonical field transformations. For convenience, we shall follow the secondpath. So the strategy is to express A fields in terms of B fields and then substitute for B fields interms of Z fields. Then we shall substitute the A fields (expressed already in terms of Z fields up tothe second order) into the standard Yang-Mills action (3) to show that the triple-gluon vertices cancelout.Using the relations (17)-(18) and (41)-(43), it is easy to see that the expansion of e A • , e A ⋆ fields, tosecond order in e Z fields is the following: h e A • a ( x + ; P ) i nd = ˆ d p d p e Ω ab { b } ( P ; p , { p } ) e Z • b ( x + ; p ) e Z ⋆b ( x + ; p )+ ˆ d p d p e Ψ a { b b } ( P ; { p , p } ) e Z • b ( x + ; p ) e Z • b ( x + ; p ) , (73)and h e A ⋆a ( x + ; P ) i nd = ˆ d p d p Ψ a { b b } ( P ; { p , p } ) e Z ⋆b ( x + ; p ) e Z ⋆b ( x + ; p )+ ˆ d p d p e Ω ab { b } ( P ; p , { p } ) e Z ⋆b ( x + ; p ) e Z • b ( x + ; p ) . (74)Note that, in the above the kernels represent the momentum space version of Ξ ab b , ( x ; y , y ), Ξ ab b , ( x ; y , y ),Λ ab b , ( x ; y , y ) and Λ ab b , ( x ; y , y ) respectively, introduced in Eqs. (10)-(11).Before we proceed any further, it is worth mentioning that all the above relations are three dimen-sional Fourier transforms performed for a fixed light-come time x + . Since the kernels are independentof minus component of momentum k − , the four dimensional transforms will only account for an extradelta function for the minus component conservation.In momentum space, the kinetic and triple-gluon terms of the Yang-Mills action (3) read L (LC)+ − = ˆ d p d p δ ( p + p ) p e A • a ( p ) e A ⋆a ( p ) , (75) L (LC)++ − = ˆ d p d p d p δ ( p + p + p ) e V abc ++ − ( p , p , p ) e A • a (cid:0) x + ; p (cid:1) e A • b (cid:0) x + ; p (cid:1) e A ⋆c (cid:0) x + ; p (cid:1) , (76)with the helicity triple-gluon vertex e V abc ++ − ( p , p , p ) = − igf abc (cid:18) p ⋆ p +1 − p ⋆ p +2 (cid:19) p +3 (77)and L (LC) −− + = ˆ d p d p d p δ ( p + p + p ) e V abc −− + ( p , p , p ) e A ⋆a (cid:0) x + ; p (cid:1) e A ⋆b (cid:0) x + ; p (cid:1) e A • c (cid:0) x + ; p (cid:1) , (78)20ith e V abc −− + ( p , p , p ) = − igf abc (cid:18) p • p +1 − p • p +2 (cid:19) p +3 . (79)Above, we have used the 3-dimensional Fourier transform at a constant light-cone time x + for theinteraction terms, and the full 4 dimensional Fourier transform for the kinetic term, because theinverse propagator acts also on the light-cone time (i.e. it contains ∂ + operator). Since the triple-gluon vertices are independent of x + , we can directly substitute the A fields in term of Z fields, at aconstant x + time.In order to deal with the kinetic term, we need (73) and (74) written fully in momentum space. Asmentioned, this is straightforward and accounts for a minus momentum component conservation deltain the kernels. We have, up to the third order in Z fields, L + − = ˆ d p d p δ ( p + p ) p (" ˆ d q d q e Ω ac { c } ( p ; q , { q } ) × e Z • c ( q ) e Z ⋆c ( q ) + e Ψ a { c c } ( p ; { q , q } ) e Z • c ( q ) e Z • c ( q ) Z ⋆a ( p )+ e Z • a ( p ) × " ˆ d q d q Ψ a { c c } ( p ; { q , q } ) e Z ⋆c ( q ) e Z ⋆c ( q )+ e Ω ac { c } ( p ; q , { q } ) e Z ⋆c ( q ) e Z • c ( q ) . (80)Let us first consider the terms that have ( e Z ⋆ e Z ⋆ e Z • ) field configuration: T −− + = ˆ d p d p δ ( p + p ) p " ˆ d q d q e Ω ac { c } ( p ; q , { q } ) × e Z • c ( q ) e Z ⋆c ( q ) e Z ⋆a ( p ) + e Z • a ( p ) ˆ d q d q e Ψ a { c c } ( p ; { q , q } ) e Z ⋆c ( q ) e Z ⋆c ( q ) . (81)Integrating the first term and second term over p and p respectively, we see that each term willhave only three momentum variables. Both terms can be combined into one integral by renaming themomentum variables to p , p , p , and color indices to b , b , b . With this we have T −− + = ˆ d p d p d p " p e Ω b b { b } ( − p ; p , { p } )+ p e Ψ b { b b } ( − p ; { p , p } ) Z ⋆b ( p ) e Z ⋆b ( p ) e Z • b ( p ) . (82)In order to bring the above expression to the constant light cone time x + we introduce the followingfields (both for e Z • and e Z ⋆ ) e Z b i ( p i ) = e κ b i ( p i ) p i . (83)Substituting (83) in (82) we obtain T −− + = ˆ d p d p d p " p e Ω b b { b } ( − p ; p , { p } )+ p e Ψ b { b b } ( − p ; { p , p } ) κ ⋆b ( p ) p e κ ⋆b ( p ) p e κ • b ( p ) p . (84)21ext, we integrate out the minus momentum components. To this end, the first term in (84) can berewritten as ˆ d p d p d p ˆ dp − dp − dp − dP − ˆ dz + ˆ dy + e iz + ( P − − p − ) e iy + ( P − + p − + p − ) ( − p +13 )( P − + ˆ p ) e Ω b b { b } ( − p ; p , { p } ) 12 p +1 [ p − − ˆ p + iǫ ] 12 p +2 [ p − − ˆ p + iǫ ]12 p +3 [ p − − ˆ p + iǫ ] e κ ⋆b ( p ) e κ ⋆b ( p ) e κ ⋆b ( p ) . (85)where we introduced the notation ˆ p = p • p ⋆ p + , (86)for any momentum p . This leads to ˆ d p d p d p p +13 (ˆ p + ˆ p − ˆ p ) e Ω b b { b } ( − p ; p , { p } ) ( iπ ) p +1 p +2 p +3 × " Y i =1 Θ( − p + i ) i ˆ p + ˆ p + ˆ p + iǫ + Y i =1 Θ( p + i )( − − i ˆ p + ˆ p + ˆ p + iǫ × e κ ⋆b (ˆ p ; p ) e κ ⋆b (ˆ p ; p ) e κ ⋆b (ˆ p ; p ) . (87)where Θ( p + i ) is Heaviside step function. This may be rewritten as ˆ dx + ˆ d p d p d p p +13 (ˆ p + ˆ p − ˆ p ) e Ω b b { b } ( − p ; p , { p } ) × e Z ⋆b ( x + ; p ) e Z ⋆b ( x + ; p ) e Z • b (cid:0) x + ; p (cid:1) , (88)where in going from (87) to (88), we used the following relation ˆ dx + ˆ d p · · · d p n e f ( p · · · p n ) e Z ⋆b ( x + ; p ) · · · e Z • b n (cid:0) x + ; p n (cid:1) = ˆ d p · · · d p n e f ( p · · · p n ) ( iπ ) n p +1 · · · p + n " n Y i =1 Θ( − p + i ) i ˆ p + · · · + ˆ p n + iǫ + n Y i =1 Θ( p + i )( − n − i ˆ p + · · · + ˆ p n + iǫ κ ⋆b (ˆ p ; p ) · · · e κ ⋆b n (ˆ p n ; p n ) . (89)Above, e f ( p · · · p n ) represents any generic function not depending on the minus momentum compo-nents (or the light-cone time). In a similar way, the second term in (84) gives ˆ dx + ˆ d p d p d p p +12 (ˆ p + ˆ p − ˆ p ) e Ψ b { b b } ( − p ; { p , p } ) × e Z ⋆b ( x + ; p ) e Z ⋆b ( x + ; p ) e Z • b (cid:0) x + ; p (cid:1) . (90)Combining Eq. (88) and (90) we get T −− + = ˆ dx + ˆ d p d p d p " p +13 (ˆ p + ˆ p − ˆ p ) e Ω b b { b } ( − p ; p , { p } )+ p +12 (ˆ p + ˆ p − ˆ p ) e Ψ b { b b } ( − p ; { p , p } ) Z ⋆b ( x + ; p ) e Z ⋆b ( x + ; p ) e Z • b (cid:0) x + ; p (cid:1) . (91)22sing p + ij ( ˆ p i + ˆ p j − ˆ p ij ) = − e v ( i )( j ) e v ∗ ( j )( i ) and substituting for e Ω and e Ψ the kernels from (44) and (42),respectively, after a bit of algebra we obtain T −− + = ˆ dx + ˆ d p d p d p δ ( p + p + p ) × (cid:16) igf b b b p +3 v ∗ (cid:17) × e Z ⋆b ( x + ; p ) e Z ⋆b ( x + ; p ) e Z • b (cid:0) x + ; p (cid:1) , (92)where e v ij = p + i v ji . Comparing this with (79), we may rewrite the above as T −− + = ˆ dx + ˆ d p d p d p δ ( p + p + p ) × (cid:16) − e V b b b −− + ( p , p , p ) (cid:17) × e Z ⋆b ( x + ; p ) e Z ⋆b ( x + ; p ) e Z • b (cid:0) x + ; p (cid:1) . (93)This cancels out the triple-gluon vertex coming from the Yang-Mills action (78) when we substitutethe first order expansion of e A • and e A ⋆ in terms of e Z • and e Z ⋆ fields.In exactly same fashion, the cancellation of the other triple-gluon vertex e V b b b ++ − ( p , p , p ) can beshown. B Six point
MHV amplitude.
In this appendix we show the details associated with the calculation of 6-point MHV ( − − − − ++)amplitude. As mentioned previously, the MHV amplitudes are always given by a single vertex inthe action (13). For the color ordered amplitude we need the U (1 − , − , − , − , + , + ) vertex which isgiven by the formula (48). In Fig. 10 we show all the contributing terms. Using the explicit expressionswe get: 23 (1 − , − , − , − , + , + ) = g " (cid:18) p +12 p +34 (cid:19) e v ∗ e v ∗ (12)6 e v ∗ e v ∗ e v ∗ (34)(12) × e v (12)1 e v e v × e v (34)3 e v e v ! + (cid:18) p +2 p +34 (cid:19) e v ∗ e v ∗ e v ∗ (16)5 e v ∗ e v ∗ (34)2 × e v (34)3 e v e v × (cid:18) p +6 p +16 (cid:19) e v (16)6 e v e v ! + (cid:18) p +1 p +23 (cid:19) e v ∗ e v ∗ e v ∗ e v ∗ (45)(23) e v ∗ (23)1 × e v (23)2 e v e v × (cid:18) p +5 p +45 (cid:19) e v (45)4 e v e v ! + (cid:18) p +23 p +4 (cid:19) e v ∗ e v ∗ (23)(16) e v ∗ (16)5 e v ∗ e v ∗ × e v (23)2 e v e v × (cid:18) p +6 p +16 (cid:19) e v (16)6 e v e v ! + (cid:18) p +12 p +3 (cid:19) e v ∗ e v ∗ (12)6 e v ∗ e v ∗ (45)3 e v ∗ × (cid:18) p +5 p +45 (cid:19) e v (45)4 e v e v × e v (12)1 e v e v ! + (cid:18) p +2 p +3 (cid:19) e v ∗ e v ∗ e v ∗ (16)(45) e v ∗ (45)3 e v ∗ (cid:18) p +5 p +45 (cid:19) e v (45)4 e v e v (cid:18) p +6 p +16 (cid:19) e v (16)6 e v e v ! − (cid:18) p +1 p +234 (cid:19) e v ∗ e v ∗ e v ∗ e v ∗ e v ∗ (234)1 × e v (234)2 e v e v e v ! − (cid:18) p +123 p +4 (cid:19) e v ∗ e v ∗ (123)6 e v ∗ e v ∗ e v ∗ × e v (123)1 e v e v e v ! − (cid:18) p +3 p +4 (cid:19) e v ∗ e v ∗ e v ∗ (612)5 e v ∗ e v ∗ × (cid:18) p +6 p +612 (cid:19) e v (612)6 e v e v e v ! − (cid:18) p +1 p +2 (cid:19) e v ∗ e v ∗ e v ∗ e v ∗ (345)2 e v ∗ × (cid:18) p +5 p +345 (cid:19) e v (345)3 e v e v e v ! . (94)We checked, that the above expression reduces in the on-shell limit to the known expression: A (1 − , − , − , − , + , + ) = g (cid:18) p +5 p +6 (cid:19) e v e v e v e v e v e v e v . (95)24 −−− + + = MHV MHV MHV Ψ − −− − − − − − + + + + + + + + − − MHV Ψ − − ΨΩ ΩΩ 42 1 − + + MHV
Ψ Ω21 −− + + MHV −− + + MHV −− + + MHV
Ω Ω Ψ ΨΨ Ψ − − − − − −− + + − − + − − − −− − −−− + − + + − − −− + + MHV −− + + MHV ΩΩ − − + + − −
12 3 456 1 2 3456
Figure 10:
The contributions to the color-ordered 6 point MHV vertex, with helicity ( − − − − ++).++).