A new Wilson line-based action for gluodynamics

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 lowest order vertex is the four-point MHV vertex. Higher point vertices include the MHV and $\overline{\textrm{MHV}}$ vertices, that reduce to the corresponding amplitudes in the on-shell limit. In general, any $n$-leg vertex has $2\leq m \leq n-2$ negative helicity legs. The canonical transformation of fields can be compactly expressed in terms of path-ordered exponentials of fields and their functional derivative. We apply the new action to compute several tree-level amplitudes, up to 8-point NNMHV amplitude, and find agreement with the standard methods. The absence of triple-gluon vertices results in fewer diagrams required to compute amplitudes, when compared to the CSW method and, obviously, considerably fewer than in the standard Yang-Mills action.


Introduction
Although quark and gluon fields are considered to be the most fundamental degrees of freedom of Quantum Chromodynamics (QCD), in a variety of situations they are definitely not the most effective ones. There are several quite distinct aspects of the QCD theory where this becomes manifest. For example, in QCD factorization theorems, which are essential tools to relate the theory to high energy experiments, one needs to account not just for a single quark or gluon exchanged between subprocesses separated by a large energy scale, but also for all collinear and soft gluons; these resummed exchanges of gluons can be included by using Wilson lines (see [1] for a comprehensive review of the factorization theorems). Similar collective degrees of freedom appear naturally in the high energy limit of QCD (see e.g. [2,3]). In the domain of non-perturbative QCD, further example can be provided by lattice QCD, where instead of the gauge fields one uses Wilson lines and Wilson loops. It is also known that, in the limit of large number of colors, the gauge theory can be completely formulated in 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 pure gluonic amplitudes. In that context, it is already well understood that the elementary triple and fourgluon interactions are not the most effective bricks to build the amplitudes. Pictorially, they are way too 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 only type of amplitudes used are the maximally helicity violating (MHV) amplitudes [8]. This type of recursion has been found earlier based on the twistor space formulation of quantum field theory by Cachazo, Svrcek and Witten (CSW) [9], who suggested that the MHV amplitudes continued off-shell are really transformations. In order to reduce the number of degrees of freedom, our starting point will be the Yang-Mills Lagrangian in the light-cone gauge, on the constant light-cone time x + . As we shall recall below, such formulation reduces the four components of the gauge fieldÂ µ = t a A µ a to just two. Here, t a are color generators in the fundamental representation. 1 To this end, we introduce the "plus" and "minus" light-cone coordinates for a four-vector where η = (1, 0, 0, −1) / √ 2, η = (1, 0, 0, 1) / √ 2, and two transverse coordinates defined by the complex null vectors ε ± ⊥ = (0, 1, ±i, 0) / √ 2. The scalar product of two four-vectors in these coordinates reads u · w = u + w − + u − w + − u • w ⋆ − u ⋆ w • . In order to lower the indices one needs to flip + ↔ − and ⋆ ↔ •, where the latter operation also causes a sign change.
The light-cone Yang-Mills action is obtained by setting the light-cone gaugeÂ + = 0 and integrating out theÂ − field from the partition function. The resulting action has only two transverse fieldsÂ • andÂ ⋆ and reads [29] S (LC) where the bold position-space three-vector is defined as x ≡ (x − , x • , x ⋆ ) so that x = (x + , x). The nabla operator reads = 2(∂ + ∂ − − ∂ • ∂ ⋆ ) in these coordinates. The presence of just the physical degrees of freedom in the action leads to a natural identification of the helicity content of the vertices. Assigning the "plus" helicity toÂ • field , and the "minus" helicity to the fieldÂ ⋆ , we see that there are (+ + −), (− − +) and (+ + −−) vertices in the action.
Following the idea of [11], we look for a field transformation 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 the partition function is preserved, up to a field independent factor. Before we present the details on the transformation facilitating the above requirements, let us make some introductory remarks. Suppose we have a set of generalized coordinates and momenta q i , p i . Consider a canonical transformation to a new set Q i , P i . Consider now a particular generating function G for the canonical transformation between {q, p} and {Q, P }, depending only on the generalized coordinates, G(q, Q). Then, the relation between the original and the transformed coordinates is In our context, the role of q i coordinate is played by theÂ • (x) field, and the canonical momentum In the new theory, we identify Q i withẐ ⋆ (x) and P i with ∂ −Ẑ • (x). Therefore, the analogous relations are 1 We use the following normalization of the color generators, common in amplitude-related literature: t a , t b = i √ 2f abc t c and Tr(t a t b ) = δ ab . We re-scale the coupling constant as g → g/ √ 2 to accommodate for the additional factors of √ 2 resulting in that normalization.
where we have explicitly denoted the fact, that the transformation is performed on the hyper-surface of constant light-cone time x + . Although the transformation (4) is rather complicated, we found that, quite amazingly, the generating functional G[A • , Z ⋆ ] can be written in the following simple form: where the functional W (±) [K], for a generic vector field K µ , is directly related to the straight infinite Wilson line in the following way: with The above four vector has the form of a gluon polarization vector. Indeed for α = p · ε ± ⊥ /p + , it is the transverse polarization vector for a gluon with momentum p. This type of functional has been used for the first time in [27] in the context of the MHV Lagrangian. The inverse functional to the Wilson line, W −1 , is defined using the relation W[W −1 [K]] = K. Note, that the functionals W (+) and W (−) are not exactly hermitian conjugates 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 transformation given 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 fieldsÂ • andÂ ⋆ have the following general expansion in terms of the new fields: where Ξ ab1...bi+j i,j (x; y 1 . . . y i+j ) is an apriori unknown kernel for i number of Z • fields and j number of Z ⋆ fields in the expansion ofÂ • , depending on the adjoint color indices a, b 1 . . . b i+j and not depending on the light-cone time. Similarly, Λ ab1...bi+j i,j (x; y 1 . . . y i+j ) is the kernel for i number of Z ⋆ fields and j number of Z • fields in the expansion ofÂ ⋆ . At lowest order we must have In principle, one could find explicitly the kernels Ξ i,j , Λ i,j from Eqs. (6)- (7). However, as we demonstrate in the next section, there is a much better way of doing that, which utilizes the existing results on the MHV Lagrangian [27,28]. 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 the new action: where 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: The above action has the following properties, which we will elaborate on in the next sections: i ) There are no three point interaction vertices.
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 MHV amplitude in the on-shell limit, the new action provides an efficient way to construct tree amplitudes with high multiplicity of legs, as we will demonstrate later in Section 3 by computing several examples.

Derivation
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 a brief summary of this procedure.

MHV action
As explained in detail in [11], the MHV action implementing the CSW rules [9] is obtained by performing a canonical 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: Note that, the two terms on the l.h.s constitute the self-dual sector of the Yang-Mills theory [30,31,32,33,34]. Therefore, as shown in [28] the solution to the required transformation of fields can be expressed in 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 The expressions for the fields in the momentum space have the following form [27,28] Above, the tildes over the fields and the kernels Γ, Υ denote the Fourier transformed quantities with respect to the three momenta p = (p + , p • , p ⋆ ). The curly brackets denote the symmetrization with respect to the pairs of momentum and color indices. The v ij , v ⋆ ij are quantities similar to spinor products ij , [ij], with the following explicit definitions (first introduced in [35] in the context of the gluon wave function): They appear quite naturally in the Wilson line approach, because where ε ± i is the polarization vector for a momentum p i obtained from (9) which appears as the direction of the Wilson line. See [36] for several useful properties of the v ij symbols. We also use a shorthand notation for the sum of momenta p 1 + · · · + p n ≡ p 1...n .
The expressions with Wilson lines (16) (or equivalently (17), (18)) can be inverted to obtain the power expansions for A • , A ⋆ in terms of B • , B ⋆ , which are consistent with [12]. In the momentum space we get: where the kernels are Inserting the above solutions to the Yang-Mills action we can derive the MHV action: where the n-point MHV interaction terms are with the MHV vertices where the color ordered vertex reads Above, we defined the momentum space color ordered vertex without the tilde, as this will not lead to any confusion. Note that, we have written the MHV vertices in a form, where the negative helicity fields are always adjacent, but there is a sum over the color permutations, together with the proper symmetry factor.
The vertices (30) are fully off-shell quantities, but they correspond to the MHV amplitudes in the on-shell limit (which is evident from the fact that the v ij symbols are in one-to-one correspondence to the spinor products). These vertices are in general not gauge invariant for off-shell kinematics, but as shown in [36,27], they do constitute a gauge invariant off-shell amplitude, when only one plus-helicity leg is kept off-shell.

Canonical transformation of the MHV action
In [28] we argued, that the second equation in (16) suggests, that there should exist a more general structure spanning not only on the ε + ⊥ -η plane. The latter should be just a slice of this more general structure. That is, there should exist a functional which path-orders the A ⋆ fields in the plane perpendicular to the ε + ⊥ -η plane. In order to actually introduce such an object, let us consider a canonical transformation of the MHV action itself. We demand that where L −+ is just the kinetic term in either B or Z fields, cf. Eq. (15). (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 transformations are analogous to those leading to the MHV action, but with the replacement • ↔ ⋆. More precisely Let us point out an important feature of the above formula. Unlike the transformation leading to the MHV action, Eq. (16), which involved the Wilson lines W (+) along ε + α , here we have the Wilson line W (−) that have directions ε − α , see the definitions (8). Thus, pictorially, the Z ⋆ field is the Wilson line on the η-ε − ⊥ plane, where the path ordered fields are themselves 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 above transformations. Inserting (7) to (6) we have and Integrating Eq. (33) by parts and using the first equation of (16) we get Comparing this with the canonical transformation rule for the A ⋆ field of [11] which reads we see that or, upon inverting, which gives the left equation of (32). Inserting now (37) into (34) and using the first equation of (16) we get or 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 the solution given by the right equation of (16), we can argue that (40), together with (38), leads to the right 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 both canonical transformations, from A fields to B fields and from B fields to Z fields, simultaneously, as shown in the diagram in Fig. 1.

Solution to the transformations
We have just seen that the transformation from the Yang-Mills action to the new action generated by the functional (7) is equivalent to two canonical transformations: first transforming the self-dual part of the Yang-Mills action to a free action in B-field theory, and then transforming the anti-self-dual part in the latter to a free term in the new Z-field theory. Therefore we can readily write the relations between the Z fields and B fields in momentum space.
In 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 Note that, the above quantity has the same form as (25), however with v ⋆ ij replaced by its complex conjugate v ij . The expansion for the B • field follows from (24) and reads where Ω ab1{b2···bn} Inserting the above expansions to (17)-(18) makes it in principle possible to derive an explicit form of 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 efficient to insert the above expressions into the MHV action, as we shall do in the following.

General form of the vertex
Although, formally, both field transformations A → B and B → Z are such that they remove the triplegluon vertices, it is interesting to see how they actually cancel in the Yang-Mills action. Therefore, in the Appendix A we directly check that the first terms of the expansions (10)-(11) indeed cancel both of the triple-gluon vertices 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 are adjacent. Without loosing the generality we shall focus on the color ordered vertex, defined as where we assumed there are m minus helicity legs. We shall also need the color ordered versions of the kernels in the expansions (41)-(43). We define where for further convenience we explicitly denoted the helicity of the legs in the color ordered kernels. Note, 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 helicity leg and (m − 1) adjacent minus helicity legs. Note also, that similar to (30), we have omitted the tilde signs in the momentum space color-ordered vertex and kernels. We observe, that the most general contribution has the form depicted in Fig. 2. Indeed, substitution of the plus helicity field B • in the MHV vertex by the Z fields, Eq. (43), results in one plus helicity leg in addition to several negative helicity legs. Thus it must be adjacent to other minus helicity legs multiplicated by the Ψ kernels. Therefore, there can be at most two Ω kernels. Also, there can be at most two Ψ kernels, because there are only two negative helicity legs in the MHV vertex. In addition to the above general situation, there are cases when the kernels are trivial, i.e. Ψ 1 = 1 or Ω 1 = 1. Let us note, that the MHV vertex has always outgoing momenta, but the Ψ and Ω kernels have the offshell line incoming. In addition, there is no propagator connecting the kernels with the MHV vertex. Therefore, we can identify the internal lines by a single helicity flow, given by the helicity of the MHV vertex.
Specifically, consider now the vertex for n external legs with the momenta p 1 . . . p n , where p 1 . . . p m correspond to the minus helicity legs. Let us introduce a collective index [i, i + 1, . . . , j] labeling the momentum, p i(i+1)...j = p i + p i+1 + · · · + p j . Using this notation, the general form of the color ordered vertex can be written as: Although the analytic formulas do not seem to collapse, in general, to any simple form, the above expression is operational and can be readily applied in the actual amplitude calculation, as we shall demonstrate in Section 3.

Summary: Feynman rules for the new action
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 have MHV 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 and MHV vertices alone give the corresponding on-shell amplitudes. In addition to these vertices we have a scalar propagator joining two opposite helicity legs.
In the following section we shall demonstrate how various amplitudes are calculated. To this end we introduce the following color-ordered Feynman rules:

Applications
In this section we shall calculate several tree amplitudes using the new action.

4-point and 5-point amplitudes
The lowest non-zero amplitude is the 4-point MHV amplitude. It is simply given by the 4-point vertex in the theory.

6-point amplitudes
The MHV and MHV amplitudes are always given by the single vertices. For the latter we need the U(1 − , 2 − , 3 − , 4 − , 5 + , 6 + ) vertex which is given by the formula (48), see Appendix B. We have verified that it recovers the correct result in the on-shell limit. The remaining amplitude is the NMHV amplitude with helicity configuration (− − − + ++). We have just three contributing diagrams depicted in Fig. 4. First two diagrams connect two MHV vertices, whereas the last one is the NMHV vertex, given by U(1 − , 2 − , 3 − , 4 + , 5 + , 6 + ). We have checked that the sum of those diagrams reproduce the known result in the on-shell limit [37].

7-point amplitudes
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 diagrams connecting 4-point and 5-point MHV vertices and one diagram which consists of the single vertex with NMHV helicity configuration. The corresponding expression is where the individual diagrams are calculated as: In the above, i/p 2 i1...im = i/(p i1 + · · · + p im ) 2 is the scalar propagator. The factor 1/2 multiplying D 5 comes from the fact that there are two possible color orders contributing to diagrams D 1 -D 4 . We have compared the sum of those diagrams, i.e. Eq.(52) with the on-shell result obtained using the GGT Mathematica package [38] together with the S@M package [39] and found an exact match, up to an overall normalization due to the difference in our symbolṽ ij and ij .
For the NNMHV amplitude the number of diagrams stays the same, see Fig. 6. Here, however, we encounter a new feature, namely there are now diagrams that connect the 4-point MHV vertex with 5-point MHV vertex. That is, starting with this amplitude we utilize the new vertices appearing in the theory in a nontrivial way (i.e. by gluing them with other vertices). We again find the on shell limit of the result consistent with the GGT package.

8-point amplitudes
We have also calculated some of the non-trivial 8-point amplitudes, all of which agree with the standard results calculated numerically using the GGT package.

Discussion
In the present work we have constructed a new action for gluodynamics by applying two consecutive canonical transformations on the light-cone Yang-Mills action, see Fig. 1. The same action can be also obtained 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 [27] for the explicit demonstration of that fact for the (+ + −) vertex). Consequently, the number of diagrams needed to calculate the amplitudes is reduced, as compared for example to the CSW method. For example, for the NMHV Z ⋆ (y) Figure 9: Schematic presentation of the geometric structure of the Z ⋆ field (the structure of Z • is quite similar). The vertical planes are the self-dual planes, i.e. the planes spanned by the vectors ε + ⊥ = (0, 1, i, 0) and η = (1, 0, 0, −1). The horizontal plane, in our terminology, is the anti-self-dual plane, i.e. it is spanned by ε − ⊥ = (0, 1, −i, 0) and η. The B ⋆ fields (the minus helicity fields in the MHV action) are the straight infinite Wilson lines on the self-dual plane, integrated over all slopes, and differentiated functionally to replace one plus 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, lying on the anti-self dual plane, and integrated over all slopes.
amplitudes with adjacent helicity, the CSW rules give 2(n − 3) diagrams, whereas the new theory gives 2(n−5)+1 diagrams, n ≥ 5. We give the number of diagrams for various adjacent helicity configurations in Table 1. It is important to stress, that we do not mean here the number of contributing terms, as the vertices in the new theory are not, in general, given by a single term in our representation. The structure of those vertices can be however easily obtained by means of the master equation (48).
One of the very interesting aspects of the field transformations leading to the new action is its incredibly rich geometric structure. Let us recall, that the new minus helicity field Z ⋆ is given by the straight infinite Wilson line on the anti-self-dual plane (i.e. the plane spanned by ε − ⊥ and η), integrated over all directions. This is the Wilson line functional of the minus helicity field of the MHV theory, which itself is given as the analogous Wilson line of the usual gauge fields lying on the self-dual plane, with insertion of the minus helicity field (i.e. the functional derivative, see Eq. (32)). We schematically depict the structure of the Z ⋆ field in Fig. 9. A similar figure can be drawn for the Z • field. We stress, that although the overall picture looks very non-local, it is local in the light-cone time.
In our recent work [28] we have discussed the fields in the MHV theory, where the fields can be expressed as the Wilson line functionals of the standard gauge fields lying exclusively on the self-dual plane. We see that the transformations derived in the present work extend this picture to the whole 3-space, with the light-cone time fixed.
One of the future directions is obviously to find the quantum corrections to the new action. There is an immediate difficulty in that program, namely, the fact that in the quantum MHV action there are contributions evading the S-matrix equivalence theorem [17], and we expect similar contributions in the new action. An alternative approach is based on the world-sheet regularization, which successfully recovered one loop all-plus helicity vertex in the MHV action [19].
Another interesting direction of future study is related to the rich geometric structure of the transformations, sketched in Fig. 9, which has not been fully explored.

A Cancellation of triple-gluon vertices.
In this appendix we show the cancellation of both the triple-gluon vertices when transforming the Yang-Mills fields to the new fields, 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 second path. So the strategy is to express A fields in terms of B fields and then substitute for B fields in terms of Z fields. Then we shall substitute the A fields (expressed already in terms of Z fields up to the second order) into the standard Yang-Mills action (3) to show that the triple-gluon vertices cancel out.
Using the relations (17)- (18) and (41)-(43), it is easy to see that the expansion of A • , A ⋆ fields, to second order in Z fields is the following: and The above kernels represent the momentum space version of Ξ ab1b2 1,1 (x; y 1 , y 2 ), Ξ ab1b2 2,0 (x; y 1 , y 2 ), Λ ab1b2 2,0 (x; y 1 , y 2 ) and Λ ab1b2 1,1 (x; y 1 , y 2 ) respectively, introduced in Eqs. (10)-(11). All the above relations are three dimensional Fourier transforms performed for a fixed light-come time x + . Since the kernels are independent of the minus component of momentum k − , the four dimensional transforms will only account for an extra delta function for the minus component conservation.
In momentum space, the kinetic and triple-gluon terms of the Yang-Mills action (3) read L (LC) with the helicity triple-gluon vertex Using p + ij (p i +p j −p ij ) = − v (i)(j) v * (j)(i) and substituting for Ω and Ψ the kernels from (44) and (42), respectively, after a bit of algebra we obtain T −−+ =ˆdx +ˆd3 p 1 d 3 p 2 d 3 p 3 δ 3 (p 1 + p 2 + p 3 ) where v ij = p + i v ji . Comparing this with (79), we may rewrite the above as This cancels out the triple-gluon vertex coming from the Yang-Mills action (78) when we substitute the first order expansion of A • and A ⋆ in terms of Z • and Z ⋆ fields.
In exactly same fashion, the cancellation of the other triple-gluon vertex V b1b2b3 ++− (p 1 , p 2 , p 3 ) can be shown.

B Six point MHV amplitude.
In this appendix we show the details associated with the calculation of the 6-point MHV (− − − − ++) amplitude. As mentioned previously, the MHV amplitudes are always given by a single vertex in the action (13). For the color ordered amplitude we need the U(1 − , 2 − , 3 − , 4 − , 5 + , 6 + ) vertex which is given by the formula (48). In Fig. 10 we show all the contributing terms. Using the explicit expressions we get: