Recursion relations for multi-gluon off-shell amplitudes on the light-front and Wilson lines

We analyze the off-shell scattering amplitudes in the framework of the light-front perturbation theory. It is shown that the previously derived recursion relation between tree level off-shell amplitudes in this formalism actually resums whole classes of graphs into a Wilson line. More precisely, we establish a correspondence between the light-front methods for the computation of the off-shell amplitudes and the approach which makes use of the matrix elements of straight infinite Wilson lines, which are manifestly gauge invariant objects. Furthermore, since it is needed to explicitly verify the gauge invariance of light-front amplitudes, it is demonstrated that the Ward identities in this framework need additional instantaneous terms in the light-front graphs.


Introduction
Scattering amplitudes in quantum chromodynamics (QCD) with on-shell initial and final states are basic objects which can be calculated using perturbative methods of quantum field theory. Together with the suitable factorization theorems [1,2] and parametrizations of the non-perturbative parton densities [3,4] and fragmentation functions [5] they are used to evaluate cross sections for various observables at high transverse momenta in the processes that occur in high energy collisions. Over the past two decades there has been an enormous progress in the computational techniques of the scattering amplitudes and their implementation in the computer codes for calculating various processes, see for example [6][7][8][9][10]. Of particular importance are the calculations of the on-shell scattering amplitudes with the fixed helicities [11], for a review see [6,12]. Since the amplitudes with different helicity configurations do not interfere with each other, they can be added incoherently. On-shell helicity scattering amplitudes can be efficiently computed using the Berends-Giele recursion methods [13] which use off-shell currents or the Britto-Cachazo-Feng-Witten (BCFW) recursion relations [14,15] which utilize gauge invariant on-shell amplitudes with shifted complex momenta.
On-shell scattering amplitudes have however some limitations, since in reality the quarks and gluons are never on-shell particles, and thus are never observed as free states in the experiments. The off-shell matrix elements are more general objects which can be used for the construction of the on-shell scattering amplitudes, like for example in the above-mentioned Berends-Giele recursion. Furthermore, the use of the off-shell matrix elements in the phenomenology together with the unintegrated parton densities and appropriate k T factorization approaches is the alternative method for the computation of the cross sections, see for example [16][17][18]. This approach, albeit more theoretically challenging, has the benefit of taking into account kinematics more accurately. This can be essential for example, when computing more exclusive processes which do require information about the details of the kinematics. One complication though in using off-shell matrix elements is the condition of the gauge invariance. Recently, a progress has been made [19][20][21] in the construction of the off-shell amplitudes which do satisfy Ward identities and hence obey gauge invariance. The general method [20] utilizes infinite Wilson line operators corresponding to the off-shell gluons, whose directions are defined by the polarization vectors perpendicular to the momenta of the off-shell gluons. It has been shown that such a definition of the off-shell matrix elements satisfies the corresponding Ward identities with respect to the remaining on-shell states and, as such, is gauge invariant.
In previous works [22][23][24][25] gluon wave functions, fragmentation functions and scattering amplitudes for selected helicity configurations have been derived using the methods of perturbation theory on the light-front [26][27][28]. In particular, certain interesting recursion relations have been proved between the off-shell amplitudes, which enabled in turn to construct all tree level Maximally Helicity Violating (MHV) on-shell amplitudes in this framework [25].
In this paper we shall explain in detail the physical origin of this recursion relation. Namely, we shall show that it is a direct consequence of the gauge invariance, as for any off-shell amplitude one can construct its gauge invariant extension, using for example Wilson lines as in [20]. Those Wilson lines encode certain recursion, which turns out to be identical to the one obtained within the light-front perturbation theory (LFPT). Moreover, since we are interested in the gauge invariance properties of the amplitudes within this formulation of QCD, we need a method to check the Ward identities explicitly on the light-front. To this aim, we shall demonstrate that one needs to redefine the rules for the LFPT in the context of the Ward identities. Namely, in the calculation of the ordinary light-front diagrams, the minus components of the momenta only occur in the energy denominators and as such are not conserved. However, in order to show the Ward identity in the light-front theory the minus components actually appear in the numerators of the expression for the amplitudes, because of the replacement of the polarization vector with the momentum. This results in the additional instantaneous terms which need to be considered in addition to the standard light-front diagrams and which, in fact, restore the full momentum conservation in the vertices, for all light cone components. After this is taken into account, the Ward identities are satisfied on the light-front for gauge invariant objects as expected.
The outline of the paper is as follows: in the next section we introduce the notation and conventions used throughout the paper, in Sec. 3 we recall the recursion relation for the off-shell amplitudes on the light-front which was derived in [25]. This recursion relation was derived starting from the Berends-Giele like relations for the light-front off-shell amplitudes. We also recall an expression for the offshell amplitude with (+ → + · · · +) helicity configuration which was derived in the light-front. This amplitude is non-zero for off-shell states and vanishes in the on-shell limit. In Sec. 4 we discuss the Ward identities within the light-front formalism. We shall enforce ourselves with an explicit example of the lowest nontrivial order amplitude (+ → − + +) on the light-front. Using this low-order example we demonstrate that the light-front recursion relation for off-shell amplitude indeed contains a natural and gauge invariant object, which in turn originates in a straight infinite Wilson line. In Sec. 5 we generalize this picture to the scattering amplitudes with arbitrary large number of external gluons. Finally in Sec. 6 we state our conclusions. Appendices contain useful identities and technical details regarding some formulae discussed in the main text.

Notation and conventions
The decomposition of any four vector u in the light cone basis reads with u µ ⊥ = 0, u 1 , u 2 , 0 ≡ (0, u ⊥ , 0) and the minus and plus components are defined by a projection on the following null four vectors η = (1, 0, 0, 1) , A scalar product in this basis can be written as u · w = 1 2 u + w − + 1 2 u − w + − u ⊥ · w ⊥ . We shall be dealing with gluon amplitudes with definite helicities throughout this paper. A polarization vector ε λ i (q) for a gluon with helicity λ = ± is typically constructed using two null four vectors: a gluon momentum k i and a reference momentum q such that For later use, let us note the following useful properties of the polarization vectors where k j is a momentum of another gluon and q is some other arbitrary null reference four vector. The change of a reference momentum renders a vector proportional to the gluon momentum where β i (q, q ) is a certain function which depends on the actual representation of the polarization vectors (see also Eq. (10) below). If an amplitude is gauge invariant, the second term in (7) does not contribute due to the Ward identity. One of the most convenient representations for the polarization vectors is provided by the spinor formalism (see e.g. [12]). However, for the purpose of this work we choose another representation, given by with Here, the reference momentum is explicitly set to q = η which is especially convenient when one works in the light-like axial gauge with the gauge vector η. The change of the reference momentum can be realized as follows Throughout the paper we shall often encounter the following scalar products where The quantitiesṽ ij and v ij satisfy several useful relations which we list in Appendix A. In the following we shall frequently use the light-front longitudinal momentum fraction z i defined by with P + the total incoming longitudinal momentum. Since P + is the total momentum which is constant and all the objects are boost invariant we shall set P + = 1 for simplicity in the following. The variables v ij are related to the spinor products that are frequently used to express the helicity amplitudes in the literature where the spinor products are defined as and the chiral projections of the spinors for massless particles are defined as We shall use so called dual (or color) decomposition of amplitudes, which is a standard technique to deal with multi-gluon amplitudes. That is, for a gluonic amplitude with external colors a 1 , . . . , a N the expansion has the form

Recursion relation with off-shell light-front amplitudes
The color-ordered amplitude M is a QCD amplitude that is regularly obtained by the addition of the appropriate Feynman diagrams. The focus of this paper is on off-shell amplitudes which have a helicity configuration of (+ → − + . . . +), where the left-most particle is incoming and all the other particles are outgoing. In the on-shell limit of the incoming gluon this is by definition the Maximally Helicity Violating (MHV) amplitude which is the first non-trivial one 1 , see for example [12]. By offshell amplitudes we are referring to amplitudes which have incoming gluons off-shell and outgoing gluons on-shell and, henceforth, it should be assumed that all the processes and diagrams we discuss are off-shell unless otherwise specified. The particular process we will be looking at is depicted in Fig. 1, where the left, incoming gluon with momentum k 1...N is the only off-shell gluon and where the 1 Typically for the helicity amplitudes, the convention used throughout the literature is that all the particles are outgoing. Therefore our amplitude (+ → −+· · · +) in the on-shell limit corresponds to the MHV amplitude (−−+ · · · +) which is indeed the first non-trivial one. The origin of the different convention in this paper stems from the fact that we are performing calculations in the light-front perturbation theory with definite direction of the light-front time in the graphs. Also note that in the convention used in this paper '-' helicity corresponds to the '+' (and vice versa) in the convention used in [12]. single-line-blob represents the sum of all the possible intermediate processes. This off-shell amplitude, M (+→−+···+) (k 1...N ; k 1 , . . . , k N ), has been previously studied using light-front methods [25]. To find this amplitude at the tree level for arbitrary number of external legs a recursion relation has been used which is illustrated in Fig. 2. Each of these graphs gives a contribution to this amplitude and it involves either triple vertex (graphs (a) and (b)), four-gluon vertex (graph (c)) or Coulomb interaction (graph (d)) as well as the amplitudes for lower number of legs and with different helicity configurations: (+ → − + . . . +), (+ → + + . . . +), (− → − + . . . +). Note that the last two amplitudes vanish in the on-shell limit, but, since the objects used here are off-shell, they are not zero, leading to non-trivial contributions to the recursion relation. Finally, a summation over different combinations of the number of external partons is performed, as illustrated in Fig. 2. This recursion has been constructed using the factorization property of the fragmentation functions on the light-front, or the so-called cluster decomposition theorem [29], and it is the light-front analog of the Berends-Giele recursion relation [13]. We note that, to arrive at this recursion, the summation over all possible light-front time orderings of the vertices inside the blobs in Fig. 2 has been performed. The solution to this recursion relation yields the following expression for the off-shell amplitude Here, M is an object that appears when solving the recursion relation. It has the same structure as the on-shell MHV amplitude but cannot be directly calculated from diagrams in the light-front. Given that its kinematics is the same as for M, we refer to it as an off-shell amplitude as well. Later, in Sec. 5, we will see that (18) can also be obtained from a matrix element of a straight infinite Wilson line. It will be then demonstrated that, interestingly, M is gauge invariant. Its explicit expression is 2 .
We should note that, in order to have a fitting correspondence with [20], we use a different notation for amplitudes than [25]. Furthermore, from now on we will be using the following shorthand notation for the arguments of the amplitudes: This amplitude corresponds to a process k 1.
The energy denominator D (1,...,i) is defined as 2 The normalization factor √ 2 j−1 in the MHV amplitude is due to the fact that in our convention the color ordered vertices do not contain 1/ √ 2 factors and therefore they differ from convention used in [12].
One interesting aspect to note about M (+→−+...+) is that it is written as a sum of amplitudes M (+→−+...+) with different number of legs. Looking at Fig. 2, it is not immediately obvious as to why M (+→−+...+) should dominate the expression since only Fig. 2a has a substructure with helicity configuration (+ → − + . . . +). It turns out that the other graphs, even though their substructures do not have the right helicity configuration, do contribute terms proportional to M (+→−+...+) . Writing out the these terms explicitly and through some algebraic manipulation, it can be shown [25] that the following term emerges from Figs. 2b-2d: The energy denominator in this expression is equal to Note that (24) vanishes in the on-shell limit D ((1...i),i+1,...,N ) = 0, which is consistent with the fact that the on-shell amplitude vanishes for this helicity configuration (+ → + · · · +). Comparing with (18), we see that the term inside the sum is very similar to the above off-shell subamplitude with some additional prefactors.  (19)) rather than M which are different objects as explained in the text.
Substituting (24) in (18) and using the identity (63) we finally obtain the following version of the recursion relation We see that the second term on the right hand side of this recursion has a nice factorized form which can be recast diagrammatically as in Fig. 3. It consists of the sum over the factorized products of amplitudes M and M. M, however, is evaluated with a different denominator in the sense that the ratio D (1,...,N ) /D ((1...i),i+1,...,N ) cancels the energy denominator inside of M and replaces it with D (1,...,N ) . We shall prove in Sec. 5 that the above recursion relation has its roots in the recurrence property of the straight infinite Wilson line, which involves gauge invariant amplitude M .

The Ward identity for light-front amplitudes
Since, in the present work, we are interested in the gauge invariance properties of the off-shell amplitudes on the light front, in this section we want to discuss certain issues regarding the Ward identities within this framework. In order to illustrate the issue, we shall first verify the identity for the lowest order amplitude (+ → − + +) on the light-front. It will become clear that one needs to modify the rules for the computation of the Ward identities, in order to guarantee the four-momentum conservation. Standard light-front rules do not involve the minus components in the calculation, except for the denominators, where it is not conserved [26][27][28]. However, for the Ward identity to hold, the minus components need to be taken care of in the vertices as well and, thus, the procedure for the computation of this identity needs to be revised. We shall demonstrate that this results in the additional instantaneous-like component, which needs to be taken into account. Then the result of the calculation is proportional to the energy denominator of the initial state and the Ward identity holds on the light-front. Second, we shall perform the Ward identity check using the recursion relation (18). It turns out that the second term in the r.h.s. of (26), which is a sum of lower order amplitudes in this recursion, gives the expression which is exactly equal to the term previously derived by the explicit calculation of the Ward identity from diagrams. This means that the new amplitude M which appears in the recursion relation is gauge invariant, i.e. the Ward identity gives exactly zero for this object despite the fact that it is off-shell.

Example: the Ward identity check for the lowest order amplitude
Let us recall, that for a generic QCD amplitude M with external momenta k i on-shell and corresponding polarization vectors ε i , the Ward identities read M| εi→ki = 0 for any i .
These identities do not work for light-front amplitudes when applied directly and appropriate modifications must be performed to ensure that they are satisfied. The reason is that one injects into the vertices a minus light cone component when replacing a gluon polarization vector by its momentum, while the actual minus light cone components flowing through the diagram are integrated out prior to this replacement. The problem can be illustrated by the following explicit example. Consider a light-front amplitude for the 1 → 3 process with the helicity configuration (+ → − + +) (first particle is incoming and the rest are outgoing). This is the lowest non-trivial MHV amplitude. We then replace the polarization vector of the third outgoing particle by the corresponding momentum. We have where A 1 -A 5 are the contributions from the diagrams depicted in Fig. 4. Here, and below we use a notation for replacement i ↔ k i in the superscript, i.e. we replace the helicity indication by the corresponding momentum. Using the rules of the LFPT and color-ordered vertices we get Let us note that the above results are obtained without the full four-momentum conservation as, according to LFPT rules, at each vertex there is a Dirac delta for the plus and transverse components, but not for minus components, which for each momentum are fixed by the on-shell condition. Adding the diagrams we get If the Ward identity was satisfied, this result should be proportional to the energy denominator D (1,2,3) , which vanishes for physical on-shell partons. However this is not the case for (34). The above problem stems from the fact that the light-front diagrams in Fig. 4 were obtained assuming that there are no external minus components. Indeed, for an amplitude calculation, the polarization vectors (8)  that flew inside diagrams were integrated out giving energy denominators and instantaneous terms. However, for the above Ward identity check, the triple gluon vertex that appears, for example, in A 1 , reads The problematic term is the first one in the square bracket. Formally, we cannot write k 3 · (k 2 + k 23 ) = 2k 2 · k 3 since we do not have full momentum conservation. We have to consider k 2 · k 3 and k 23 · k 3 as different scalar products and this causes the Ward identity to fail. As already mentioned in the beginning of this section, in the correct procedure one should integrate out all the minus components. In (35) the internal minus component k − 23 appears in the numerator and thus leads to instantaneous-like additional contribution to (28). We give an explicit calculation of this term in Appendix B. It turns out, that this additional term added to (35) gives and it effectively restores the full momentum conservation (in the numerator). Repeating the procedure for A 2 (the other diagrams are not affected) we get which obviously vanishes in the case of the on-shell amplitude, i.e. for D (1,2,3) → 0.

Ward identity and the recursion relation for the lowest order amplitude
In the recursion relation (18) the new amplitude M that actually solves the recurrence, has exactly the form of the MHV amplitude. Therefore, once we impose the on-shell condition for the process, the result is equal to the MHV amplitude as expected. The amplitude M , however, as it stands in the recursion relation, is an off-shell object. As we shall see shortly, it has a remarkable property, namely, it turns out that it is gauge invariant.
This can be explicitly illustrated by taking (26) for n = 3 and checking the Ward identity. However, one needs to take care of the issues discussed in the preceding section. We need to calculate In the above expression we have replaced the polarization vector both in the 1 → 3 amplitude and 1 → 2 subamplitude. The second term can be simplified to where we have used the relations z 12 D (1,2) = 2z 1 z 2 v 12 v * 12 , z 12 D (12,3) = 2z 123 z 3 v (123)3 v * (123)3 and z 1 v (12)1 = −z 2 v 12 . We see that (39) precisely cancels the previously derived term (37), leaving We shall undertake this task in the next section.

Proof of gauge invariance of the amplitude M from Wilson lines
In the previous section it has been claimed that the off-shell amplitude M which appears in the recurrence relation (18) is gauge invariant and thus satisfies the Ward identities (40). Although, in principle, one could show this by arranging a similar recurrence for the Ward identity and showing that M vanishes for arbitrary number of legs, we will study a connection of equation (18) with the matrix element of certain straight infinite Wilson line (or gauge link) operator. For that object one can immediately write a recurrence which resembles (18), which, after a careful derivation turns out to be exactly the same. We will start by reviewing the Wilson line approach for off-shell amplitudes. Later, we will derive the recursion (18) directly from this approach.

Matrix elements with Wilson lines and off-shell amplitudes
Let us consider a tree level gluonic Green's function in momentum space with external momenta k 1...N , k 1 , . . . , k N satisfying momentum conservation (we assume, as before, that k 1...N is incoming and the rest are outgoing). As such, the Green's function is a purely off-shell object, i.e. the external momenta have arbitrary virtuality; moreover, the external gluon Lorentz indices are not contracted. In order to obtain a scattering amplitude, we reduce the Green's function by amputating the external propagators, taking the on-shell limit for the external momenta, and contracting the external legs with appropriate polarization vectors transverse to (on-shell) momenta. Here, we shall consider the Green's function where the legs k 1 , . . . , k N are on-shell and reduced as above, while the leg k 1...N is kept off-shell and is contracted with a vector e 1...N . We shall call this vector a "polarization" vector for the off-shell gluon. At this point, it is only assumed that this vector is transverse to the off-shell momentum, e 1...N · k 1...N = 0. We will call the Green's function reduced in that manner an off-shell amplitude.
The off-shell amplitude constructed according to the above procedure is not gauge invariant, i.e. it does not satisfy the Ward identities with respect to the on-shell legs (for a general choice of e 1,...,N and external polarization vectors). However, one can find a gauge invariant extension of such offshell amplitude. For example, in analysis of scattering at high-energy one encounters similar objects.
There, the e 1...N is set to one of the light-cone components n ± (n 2 ± = 0) of a hadron momentum and k 1...N = xn ± + k T , so that k 1...N · n ± = 0. The gauge invariant vertices corresponding to transitions of such off-shell gluons to a set of on-shell gluons can be derived from the so-called Lipatov's effective action [30,31].
In Ref. [20] the author discussed a more general situation, where e 1...N is arbitrary. In that case the gauge invariant extension of the off-shell amplitude can be found by considering a matrix element of a straight infinite Wilson line operator. More precisely, one defines an object where T is the time-ordering, P is the path-ordering, S Y-M is the Yang-Mills interaction action, and, finally, |k i , λ i , a i are one-gluon on-shell states with momentum k i , helicity λ i and color a i . The color of the off-shell gluon is a 1...N . The subscript c means that we take only connected contributions. The infinite Wilson line operator R a 1...N e 1...N sandwiched in the matrix element is explicitly gauge invariant with respect to small gauge transformations. Actually, in Ref. [20], instead of a straight infinite path in (42), deformed paths were considered in order to regularize the integrals and to show that they form certain generalized functions.
Let us now consider a color-ordered version of the matrix element (41) with order (a 1.. .N , a 1 , . . . , a N ). According to [20] it is proportional to the momentum conservation Dirac delta and the delta assuring In what follows we will denote this amplitude as M (e 1...N →λ1...λ N ) (k 1...N ). It will contain the offshell propagator and and a coupling to the Wilson line (we include an additional i factor for further convenience) ..N →λ1...λ N ) (k 1,...,N ) , where M is the standard QCD amplitude calculated from Feynman diagrams (with, however, off-shell kinematics). Let us underline one more time that We will utilize this recursion throughout the rest of the paper; therefore, we will need its algebraic form:M This is because the Wilson line slope definingM is not perpendicular to the off-shell momentum, e 1...N · k 1...m = 0, as required by (44).

Light-front recursion relation from Wilson lines
We will now relate the recursion with Wilson lines (48) to the recursion (18) obtained within the light-front formalism. To this end, we first have to choose the appropriate "polarization" vector e 1...N for the off-shell gluon. We choose, of course, the same vector as in the formalism to obtain (18), i.e. we choose e µ 1...N = ε +µ 1...N , where ε + 1...N is defined by (8). Note that ε 1...N · k 1...N = 0, despite that k 1...N is off-shell and, thus, M (λ1...λ N ) ε + 1...N (k 1...N ) is gauge invariant. Choosing helicities as λ 1 = −, λ 2 = · · · = λ N = + and the reference momenta for the polarization vectors to be η as in (8) Note that now the sum starts with the index m = 2, as for m = 1 the term vanishes due to (49) and ε + * 1...N (η) · ε − 1 (η) = 0 according to (6). In order to proceed, we have to find an explicit expression for M (+→+···+) . This can be done using the recursion (48) with λ 1 = · · · = λ N = + and observing that The details are given in Appendix C. The result reads Let us note that this is the same as obtained from the light-front approach (24). To show this we set i = 1 in (24), and use the relation (46) with k 2 1...N = z 1...N D 1...N . However, even with this encouraging result, the recursion (52) is different then the one obtained within the light-front formulation. Indeed, this recursion does not involve the same object on the l.h.s and r.h.s of the equation, as we already discussed below equation (48). The recursion relation (52) can be, however, written entirely in terms of gauge invariant off-shell amplitudes. That is, we will look for a kernel K mN such that  (48). There is one more comment in order here. Our assumption of the existence of the kernel K mN is guided by the light-front result discussed in Sec. 3. It is not obvious however if such kernel exists for any helicity configuration.
In order to find K mN , we first have to find the relation betweenM It can be done by observing that the first term on the r.h.s of (52) does not depend on the Wilson line direction (this is the consequence of the gauge we are using). Thus, we can write this equation separately forM .
The desired relation is obtained by inserting the above formula into Eq. (52). However, the result has a very complicated structure containing a tower of sums. Remarkably, it turns out, that these sums satisfy an equation which resembles an old-fashioned propagator theory and the solution to this equation can be found. We relegate all the technical details to Appendix E and here we restrict ourselves to a pictorial description. Namely, the kernel K mN can be thought of as a propagator for certain Hamiltonian which -after algebraic manipulations -is expressed by the tower of sums and the expression withṽ ij in (56). A careful inspection of the sums reveals that they are ordered in a way that resembles time ordering of old fashioned perturbation theory. One can introduce then an object that plays the role of the free propagator (see Eq. (121) in the appendix) and another one which can be interpreted as a vertex (Eq. (122) in the appendix). It becomes then clear that all the sums, except one, form again the full propagator, see diagram (124). One can then write the compact integral equation for the kernel K mN . Finally it is easy to show that its solution gives the desired kernel with the simple form: It is straightforward to check that (55) with (57) and (54) coincides exactly with (18) obtained within the light-front approach. To this end one only needs to redefine M and M in (18) to include the energy denominators with appropriate z i 's forming in fact propagators. This means also, that the MHV off-shell amplitude M in (18) is indeed gauge invariant since it corresponds to the gauge invariantM from the Wilson line approach. Yet another confirmation of this result comes from the work [21] where similar off-shell gauge invariant helicity amplitude was calculated and turned out to have also the MHV form.

Conclusions
In this work we have analyzed gauge invariance properties of the gluon off-shell scattering amplitudes with Maximal Helicity Violating configurations using the light-front formalism. The recurrence relation for such amplitudes, that was first derived in [25], encodes a new object, which is off-shell but has the form of an on-shell MHV amplitude. We demonstrate that this new amplitude is a gauge invariant despite its off-shellness. In order to check the gauge invariance within the light-front formalism, we had to find a way to verify the Ward identities. Unlike in the standard formulation of QCD this is not straightforward on the light front, as the minus light-cone components are integrated out by default within this formalism and the standard QCD prescription has to be modified. The proper treatment of the Ward identities involves additional instantaneous-like terms, which effectively can be taken into account by forcing the full momentum conservation in the vertices. Furthermore, we recognize that the light-front recurrence relation has a very similar form to a recurrence that is encoded in the Wilson line formulation of off-shell amplitudes [20]. In fact, we prove that they are precisely the same. Therefore, remarkably, the new off-shell amplitude that appears in the light-front recursion is gauge invariant.
As far as different helicity configurations are considered, the situation is much more complicated. A recurrence relation, similar to Eq. (52), can be easily written for any helicity configuration using the Wilson line approach, c.f. Eq. (48). However, it appears extremely cumbersome and it remains unknown whether it would be possible to cast it into simple, truly recursive form as in Eq. (55).
Let us, finally, conclude that our study once again stresses the importance of gauge invariance in any QCD computations. As we have shown in the current paper, the complicated resummation of whole classes of light-front diagrams, as performed in Ref. [25], leads precisely to the straight infinite Wilson line, which is a manifestly gauge invariant object.

A Useful identities
In this supplement we summarize certain relations for the quantitiesṽ ij and v ij defined in Eqs. (11), (12).
Straight from the definition, we have an antisymmetry property of For theṽ ij the exchange of indices givesṽ Obviouslyṽ what comes out of the identity (59) or the transversity of polarization vectors. Finally, we have the following decomposition relationṽ The above relations come straight from the definitions and can be easily proved.

B Explicit example for the Ward identity check on the lightfront
In this appendix we will demonstrate that, when checking the Ward identity within the light-front formalism, additional instantaneous-like terms appear. These terms, in fact, recover the full momentum conservation. We will use a specific, yet quite general example. Consider the following diagram where the arrow denotes the contraction with the momentum instead of the corresponding polarization vector. The momenta labels are chosen in such a way that the contact with Fig. 4 can be established.
The amplitude can be expressed as represents the blob andÃ νλ2 (x; k 2 , k 3 ) represents the gluon splitting. The λ 2 is the polarization of the gluon with momentum k 2 . The gluon propagator is included inÃ. For the purpose of this example, the transition to the light-front formalism will be done according to [26]. The basic idea is to integrate the minus components in the propagators. In our example, we will first reintroduce the suitable integration. In order to proceed let us decompose the tensor g µν as follows where λ 23 is the polarization of the intermediate gluon and the dots stand for the gauge terms and the term that will lead to the instantaneous interactions. For the purpose of the present example we will retain only the part containing the polarization vectors and show that the instantaneous-like term appears (in addition to the standard instantaneous term originating from the terms represented by the dots in (70)). The corresponding contribution reads and with the generic color-ordered triple gluon vertex The k 3 instead of a polarization superscript for V 3 in (73) denotes the replacement ε 3 ↔ k 3 . We can rewrite (73) as Above, we have restored the unintegrated propagator (the scalar part). Switching to light-cone variables and using the integral representation for the Dirac delta for the minus component, we havẽ (34) and alone leads to an incorrect result. Clearly, the second term of (81) was missing. Let us now calculate the contribution to the amplitude M coming from the first term in (81). Performing the integrals over the light-cone time we get Above (and below),k − 23 is understood ask Next, the contribution of the instantaneous-like term reads Both contributions have to be added: Θ P + B λ23 (P ) ε λ2 (k 2 ) · ε λ23 (k 23 ) k 23 · k 3 P − −k − 23 + i However, the square bracket can be rewritten aŝ where we have used the delta function δ P − − k − 2 − k − 3 appearing in (87) to write the scalar product on the r.h.s. This is indeed the correct contribution from the vertex (77) as is easily seen by integrating back the delta δ 4 (k 23 − k 2 − k 3 ) in (75).  .
Note that the properties (4)-(6) still hold for ε i (k 1...N ) despite the fact that k 1...N is off-shell (actually, only the last relation of (6) is non-trivial to check). The amplitude (89) is given by the expansion (45).
Consider any blob M (ε + 1...N (η)→ε + i (k 1...N )...ε + j (k 1...N )) , j > i, attached to the Wilson line. Such blob contains terms with at least one scalar product of polarization vectors. This is due to the following standard argument (see e.g. [12])). Since the are at most j − i − 1 triple gluon vertices there may be at most j − i − 1 momentum vectors in the numerator. These vectors are contracted with j − i + 1 polarization vectors, which means that at least two polarization vectors must be contracted together. Due to our choice of reference momenta all such scalar products vanish due to (6). This happens for all the blobs, thereforeM , where we have used (65) to perform the sum. This indeed coincides with (102).

D Derivation of (56)
In order to find the required relation we will use (52). We will subtract two forms of this equation: one with N = m set everywhere except the Wilson line slope (i.e. the subscript ofM), second, with N = m set literally everywhere. We get M (−+···+) ε + Next, for p = 2 we have a (2) i1i2 (N ) − a (2) i1i2 (m) = .
The amplitudes on the r.h.s. depend only on single summation variable i p , and it turns out that one can perform the remaining sums. To this end, let us reorganize the sums as follows