Exploring straight infinite Wilson lines in the Self Dual and the MHV Lagrangians

We investigate the appearance of straight infinite Wilson lines lying on the self-dual plane in the context of the Self Dual sector of the Yang Mills theory and in a connection to the Lagrangian implementing the MHV vertices (MHV Lagrangian) according to the Cachazo-Svrcek-Witten method. It was already recognized in the past by two of the authors, that such Wilson line functional provides the field transformation of positive helicity fields between the Yang-Mills theory on the light-cone and the MHV Lagrangian. Here we discuss in detail the connection to the Self Dual sector and we provide a new insight into the solution for the minus helicity field transformation, which can be expressed in terms of a functional derivative of the straight infinite Wilson line on the self-dual plane.


Introduction
Since the publication of the famous paper on Maximally Helicity Violating (MHV) amplitudes by Tomasz Taylor and Stephen Parke [1], there has been an enormous development in methods of calculating scattering amplitudes in Quantum Chromodynamics (QCD), without use of the Feynman diagrams. Indeed, this direct technique, which is more than a half of century old, is extremely inefficient when confronted with today's needs. Proliferation of diagrams is huge even for small multiplicity processes. Moreover, individual diagrams do not respect gauge invariance leading to large cancellations between distinct sets of tices used in the CSW method and straight infinite Wilson lines on certain complex plane, that has been started in [27]. These Wilson lines emerge as the transformation of the positive helicity Yang-Mills field appearing in the light cone action, to a new action where the MHV vertices are explicit [28,29,30,31]. As a first objective we explore the connection of this Wilson line to the Self Dual sector of the Yang-Mills theory. The scattering amplitudes in that sector have been discussed in detail in the literature [32,33,34,35,36], as well as the integrability of the self dual equations of motion [37,38,39,40]. Despite the fact that there exists rich and mathematically sophisticated literature on the subject, there is no direct discussion of the infinite Wilson line discovered in [27] in that context. The second objective of the paper is to extend the discussion beyond the self dual sector. As mentioned, the straight infinite Wilson line functional considered in [27] transforms the positive helicity fields to new fields that constitute the MHV action. Here, we present an analogous solution for the minus helicity field, which turns out to be transformed by certain functional derivative of the Wilson line.
The structure of the paper is the following. In Section 2 we discuss the self dual sector. We recall the equations of motion and structure of the tree level currents existing in the theory. Next, we discuss how the straight infinite Wilson line emerges as an inverse functional to the generating function for the solutions. In Section 3 we extend the discussion of Wilson lines beyond the Self Dual sector. We find a direct relation of the Wilson line from the self dual sector to the transformation of the minus helicity fields that leads to the MHV action. The last Section is devoted to a summary and description of future prospects.

Classical EOM and scattering amplitudes
Our starting point is the full Yang-Mills action in the light-cone gauge A + = 0, where the A − fields (appearing quadratically) were integrated out from the theory [41] leaving only two complex fields A • , A ⋆ that correspond to plushelicity and minus-helicity gluon fields. We use here very convenient so-called 'double-null' coordinates defined as ⊥ with the two light-like basis four-vectors η = (1, 0, 0, −1) / √ 2, η = (1, 0, 0, 1) / √ 2, and two space like complex four-vectors spanning the transverse plane ε ± ⊥ = 1 √ 2 (0, 1, ±i, 0). With the above definitions one can easily raise and lower indices, taking care of the flip + ↔ − and ⋆ ↔ •, where the latter causes also the sign change. The action can be written as where the individual terms in the Lagrangian read: . Above, the bold position vector is defined as x ≡ (x − , x • , x ⋆ ). We denoteÂ = A a t a as the color algebra element and use the normalization of color generators that fulfills t a , t b = i √ 2f abc t c . With this normalization, our coupling constant is re-scaled as g → g/ √ 2 comparing to 'standard' normalization of the action.
The fully covariant form of the self dual equations iŝ whereF µν = ∂ µÂν − ∂ νÂµ − ig Â µ ,Â ν and the Hodge dual is defined as * F µν = −iǫ µναβF αβ . The corresponding self dual equation in light-cone gauge and upon eliminating the A − field reads which can be obtained from the following truncation of the full action We are interested in the classical solution to the self-dual EOM (7) that give us information about scattering amplitudes. The tree-level Green functions can be extracted by coupling the classical action to an external current and postulating the power series solution: Above, the current J has been replaced by a current that accommodates the on-shell pole j a (x) = −1 J a (x) .
Assuming the currents j a are supported on the light-cone, the momentum space off-shell currentsΨ ab1...bn generated by the solution (10) correspond to the offshell currents similar to the Berends-Giele currents [42], i.e. to an amplitude of an off-shell gluon with momentum P and plus helicty (momentum incoming) scattering into n on-shell gluons with momenta k 1 , . . . , k n . The solution can be obtained iteratively, see Fig. 1 and reads [34,43] Ψ a{b1...bn} n (P ; {p 1 , . . . , p n }) = −(−g) n−1 δ 4 (p 1 + · · · + p n − P ) Tr t a t b1 · · · t bn ṽ * where we introduced spinor-like variables following [44] v We use a shorthand notation for the sum of momenta p 1 + . . . p n ≡ p 1...n . The curly braces denote the symmetrization with respect to color and momentum variables, i.e. pairs {b i , p i }.
Here and in what follows we always skip factors of 2π accompanying conservation delta functions. Theṽ ij ,ṽ * ij symbols have the property that in addition to being directly proportional to the spinor products ij and [ij] (see. e.g. [44]), they can be defined in terms of the polarization vectors as followsṽ * This turns out to be important in the context of the inverse functional to (10) as we discuss in the next subsection.
The off-shell currentsΨ ab1...bn n can be turned into on-shell −, +, . . . , + scattering amplitude by putting the incoming off-shell gluon on mass shell and amputating the propagator, or equivalently, the energy denominator Above, the the light-cone energy is equal to the 'minus' momentum component obtained from the on-shell condition k 2 i = 0. Since the solution (12) does not have the 1/D 1...n factor (it gets canceled -see Fig. 1) we get zero upon amputation and imposing the onshell limit, i.e. when the minus component is conserved. Thus the on-shell −, +, . . . , + tree amplitude is zero in agreement with well known general analysis.

The straight infinite Wilson line
It is very interesting how the Self Dual equation encodes the infinite Wilson line spanning over the transverse complex plane. Obviously, due to the integrability property of the theory the relation to Wilson lines should not be a surprise [37], but the following exposition is new and quite elementary.
Let us consider an inverse functional to (10). First note, that the kernels Ψ n are functions exclusively of the three momenta p ≡ (p + , p • , p ⋆ ), up to the 'minus' component momentum conservation delta function δ(p − 1 +· · ·+p − n −P − ). Therefore we define the 3D kernelsΨ a{b1...bn} n (P; {p 1 , . . . , p n }) using We can get rid of the 'minus' component delta function in (10) by Fourier transforming it with respect to the 'minus' momentum component: We assume that the inverse functional has the expansioñ Inserting (20) to (19) .
(21) Noticing that each of the factors in the denominator is actually a scalar product of a sum of subsequent momenta and the same polarization vector (14): it is not difficult to prove [27] that (21) are the momentum space expansion coefficients of the functional directly related to the following straight infinite Wilson line along ε + P (in the light-cone gauge): where is a generic polarization vector. Thus, from the position space point of view the functional (23) is the (derivative of) Wilson line integrated over all possible directions. However, after passing to momentum space and integrating over α, the generic polarization vector (24) becomes ε + P . One may worry that the functional (23) is in contradiction with the EOM with the current (compare (7)): as it contains terms at most quadratic in the fields, whereas (23) contains all powers. However, the current j in (25) is supported on the light-cone and only the second term in the expansion of (23),Γ 2 , contains implicit pole 1/P 2 . We discuss this point in Appendix A. In view of the above, the functional (23) is a more general object then the external current j and therefore we rename it as follows: The Wilson line (23) has been already recognized in [27] as the relation between the positive helicity Yang-Mills field and positive helicity field in the MHV action -we shall come back to this point in the next section. It is also worth mentioning that interestingly the kernelsΓ n contain the same diagrams asΨ n , but different energy denominators, see [27] for details.
Let us now discuss the origin of the Wilson line (23) in more geometrical terms.
Notice first, that this Wilson line lies within the plane spanned by η and ǫ + ⊥ . It is a null plane, i.e. any tangent vector is light-like. It is interesting, that this plane is also a self-dual plane, i.e. any bivector w µν = u µ v ν − u ν v µ with tangent vectors u,v is self-dual (such plane is also called α-plane or β-plane in the twistor space formulation [38]). Therefore, any Wilson line lying within the plane is the functional of the self-dual connection A and one can attempt to reconstruct the connection from the Wilson line.
A generic Wilson line functional U C [A](x, y) depends on two endpoints and a path C, and thus cannot be used to solve for the self-dual gauge field. The trick is now, to choose the simplest possible path crossing arbitrary point and extending over the whole plane, i.e. the straight line (see Fig. 2). Let us assume we pick up the line along the vector ε α defined in (24). In addition to being independent on the end-points, the resulting infinite Wilson line U α [A](x) is gauge invariant with respect to the small gauge transformations. Finally, integration over α reduces the functional U α [A](x) to a functional independent on the path details U [A](x). One can now find an inverse functional A[U ](x) which is the generating functional for the off-shell amplitudes. Notice the functional U can be treated here as the fundamental representation field as it depends only on color and position. It is the origin of the B • field defined in (26).
Although the procedure outlined above is quite intuitive, the detailed mathematical treatment and connection to integrability of the self-dual sector is beyond the scope of the present paper.

The MHV Lagrangian
In the previous section we recalled the self dual Yang-Mills theory and showed how straight infinite Wilson line functional, integrated over all directions, emerges as an inverse functional to the solution of the EOM. It can be interpreted as a new field B • a in the fundamental color representation, see Eq. (26). It turns out [27] (23) is one of the two Mansfield transformations taking the full Yang-Mills action to the action with MHV vertices [28]. The second transformation relates the starred fields A ⋆ → B ⋆ [A • , A ⋆ ] and will be discussed in more details below.

Recap of the MHV action
The action involving MHV vertices reads [28]: where the form of the kinetic term is analogous to the one in (1) but with A fields replaced by the B fields Figure 2: The inverse functional to the generating functional for the self dual solutions is given by the straight infinite Wilson line lying on the plane spanned by with the MHV vertices The Wilson line functional (23) appears here naturally as a consequence of mapping the − + + helicity vertex of the Yang-Mills theory, which is not present in the CSW method, to a new free theory of B fields, as proposed in [28]. This vertex constitutes the self-dual theory as discussed in the previous section.
Because there are at most two B ⋆ fields for arbitrarily large vertex, we see that the MHV action is almost entirely occupied by the Wilson lines. Of course it is the 'impurity' injected by the B ⋆ fields that gives the rich structure of the full Yang-Mills theory.

Relation of B ⋆ field to infinite Wilson line
The structure of the B ⋆ field has not been discussed in the literature in a more geometrical manner. Actually, only the functional A ⋆ [B • , B ⋆ ] and not its inverse has been so far calculated [29], as this is what is needed to derive the MHV action. Here we are interested in exploring the structure of the new field B ⋆ as seen in the original theory.
In order to find an expression for B ⋆ [A • , A ⋆ ] similar to (23), we first derive the functional in momentum space, as a power series. This is done in a similar way as for the B • field (see Appendix B for details). We obtaiñ Note, that the kernels (31) are very similar to the Wilson line kernels (21), modulo the prefactor. Indeed, it turns out that the solution for the B ⋆ field has the following compact functional form in the position space: The proof of this formula is given in Appendix B. Because the infinite Wilson line in (32) is the same as in (23) we can write the above formula in terms of the functional derivative of B • field: where ∂ − (x) = ∂/∂x − . From the above formula it is easy to see that the momentum space expression (30) with the kernels (31) indeed satisfies (32). The differential operator ∂ 2 − (y)/∂ 2 − (x) will give the momentum dependent prefactor in (31), whereas the functional derivative will replace one A • by A ⋆ in the Wilson line B • . For n-th term in the expansion there are n possibilities to do so, thus the factor of n in (31).
Geometrically, the functional (32) is just the infinite Wilson line patched with one A ⋆ field -see Fig. 3. It is interesting to note, that it is natural to think about the A ⋆ fields as belonging to Wilson lines living within the antiself-dual plane spanned by ε − α (recall thet the B • lives on the plane spanned by ε + α ). Therefore, the solution (32) looks a bit like a cut through a bigger structure, spanning over both planes. Such possible extension of the Wilson line application in MHV formalism is beyond the scope of the present paper and is left for future study.
In the end of this subsection, let us note that from the equation (33) one can derive another interesting formula for the B • field. Differentiating (33) over ∂ − (x) and using the Mansfield's transformation rule for the A ⋆ field [28]: we easily obtain the following equation: Integrating by parts over y − we see that merely the operator ∂ − (y)/∂ − (x) is responsible for inverting the functional derivative of the Wilson line.

Summary
In this paper we have explored structures in the Yang-Mills theory that give rise to the MHV vertices in the CSW formulation. These structures turn out to be functionals of the gauge fields, directly related to the straight infinite Wilson lines extending over a complex self-dual plane spanned by ε + α defined in Eq. (24). Similar Wilson lines appear in the literature in the context of the gauge invariant amplitudes with some external partons being kept off-shell.
The functionals that involve complexified, straight, infinite Wilson lines transform the original fields appearing in the light cone Yang-Mills action to the fields which enter the MHV action. The light cone action involves just two components of the gauge field that correspond to plus and minus helicity gluons. The tranformation of the plus helicity field is given solely by the solution to the Self Dual equation of motion. Therefore, we explored in details the connection between the Wilson line and the Self Dual sector of the Yang-Mills theory. To be precise we showed that the Wilson line expression satisfies the self-dual EOM, when the currents are restricted to the support on the light-cone. Thus the Wilson line itself represents a more general object than the self-dual current. We have also found the explicit compact form of the transformation for the minus helicity field in position space. This turns out to be given by the functional derivative of the Wilson line solution, which effectively replaces one of the plus fields along the line by the minus field.
There are further possible avenues to follow. One of them is to investigate further the geometry of the solutions. The Wilson lines corresponding to the positive helicity fields live on the plane spanned by the ε + α . On the other hand one could think of the minus helicity fields as belonging to the gauge links which live in the plane spanned by ε − α . Since the transformation for the minus helicity field that takes the Yang-Mills theory to the MHV action involves both plus and minus helicity fields, it appears to be a certain cut through a bigger geometrical object which is spanned over both planes. Investigation of such possible extensions of Yang-Mills field transformations is left for the future.
(40) Above is the 4D version of the Wilson line kernel (21), cf. (18). In what follows we skip the color indices for more compact formulae. For further use we can write P 2Γ n (p 1 , p 2 , · · · p n ) = −P + D 1···nΓn (p 1 , p 2 , · · · p n ) , where where we have used an identity similar to the Schouten identity for spinor products in the on-shell case. The above expression holds because the momenta p 1 , . . . , p n are on-shell.
Let us now proceed term by term in (38). Since the first term in (39) is A • a (P ), it cancels out the first term of (38). For the second term in (39), we haveΓ Tr(t a t b1 t b2 ) .
It is easy to show that the above expression can be written as Using (42) we get Comparing this with the second term of (38), we see that they cancel out. An important point to note is that equation (45) represents a 3-gluon vertex where the incoming gluon is off-shell and the outgoing gluons are on-shell. When we impose the on-shell condition, P 2 → 0, the quantity on the right side of (45) represents 3-gluon vertex with all gluons on-shell. This we know must be zero. Hence we have,ṽ Since we have canceled the two terms present in (37) (the g 0 and g 1 terms), we now need to show that ∞ n=3 ˆd 4 p 1 · · · d 4 p n P 2Γ n (p 1 , p 2 , · · · p n )Ã • (p 1 )Ã • (p 2 ) · · ·Ã • (p n ) (47) From equation (42) we have Above, we see that each term in the expansion of P 2 is positive definite. Hence under the limit P 2 → 0, each term in (48) must independently go to zero. Thus The implication of the expression above (which will be useful later) is the following. Since the three gluon vertex for splitting P → p i p j , with i, j on-shell, is proportional toṽ ij (cf. 45), we must conclude that the on-shell limit is approached byṽ This follows from the fact that for n = 2 Eq. (48) is just ∼ṽ 12ṽ * 21 . On the other hand the fully on-shell 3-gluon vertex is ∼ṽ 12 and must be zero due to momentum conservation. Thus, in the on-shell limitṽ 12 → 0. This holds for any 3-gluon vertex with on-shell legs.
In [27] it was shown thatΓ n (p 1 , p 2 , · · · p n ) can be written as . (51) where Using this, we can write . (53) It can be rewritten as Using Eqs. (46),(52) we see that each term in the above expression is zero under the limit P 2 → 0. Therefore the whole expression vanishes in that limit.

B Prove of the B ⋆ field expansion in momentum space
In this appendix we derive the momentum space power series forB ⋆ (x + ; P) which we postulate to have the following expansioñ where, throughout the derivation, the curly braces represent that the enclosed momenta and color indices are symmetrized. The goal is to find the coefficient functionsΥ n which we shall derive for a first few terms and generalize the result.
where the sum is over all permutations of (1 · · · n). Above, we have introduced the following notation for convenience Tr(a1 · · · n) = Tr(t a t b1 · · · t bn ) , The same notations will be used for the coefficient functions of the power series forB • in the momentum space,Γ ab1···bn n (P; p 1 · · · p n ). To obtainΥ n we substitute equations (58) and (56) in (55) and equate the terms of equal order. To differentiate the color indices and momentum variables, we use c 1 , c 2 , · · · c n as the color indices and q 1 , q 2 , · · · q n as momentum for (55) and respectively d 1 , d 2 , · · · d n and s 1 , s 2 , · · · s n for (56) and b 1 , b 2 , · · · b n and p 1 , p 2 , · · · p n for (58).
The first order term is trivial. For the second order, we get For the third order term, we proceed analogously. We have Integrating the third term with respect to q 1 and the fourth term with respect to q 2 we see each term will have only three momentum variables. All the terms can be combined into one integral by renaming the momentum variables to t 1 , t 2 , t 3 , and color indices to e 1 , e 2 , e 3 . With this we have For convenience, let us rewrite (73) as follows where The functional derivative of the path ordered integral will give a series containing in each term a Dirac delta (δ 4 (x + s i ε α − y)) for the position space variables and a Kronecker delta (δ c bi ) for the color indices. Integrating with respect to y, we get ⋆ (x + s 1 ε α )Â • (x + s 2 ε α )Â • (x + s 3 ε α ) + · · · (76) The above expression can be rewritten as where e α ≡ (−α, 0, −1) are the (x − , x • , x ⋆ ) coordinates of ε α . For the ordered integrals we havê this we get B ⋆ a (x + ; P) n th = (n!)ˆ+ Tr(a1 · · · n)Γ n (1 · · · n) . (91) Compactly we have ab1···bn n (P; p 1 · · · p n ) . (92) The expression in square bracket is exactly (31). Thus we have proven that the position space functional (73) corresponds to (30) with (31), upon transforming to momentum space.