All-plus helicity off-shell gauge invariant multigluon amplitudes at one loop

We calculate one loop scattering amplitudes for arbitrary number of positive helicity on-shell gluons and one off-shell gluon treated within the quasi-multi Regge kinematics. The result is fully gauge invariant and possesses the correct on-shell limit. Our method is based on embedding the off-shell process, together with contributions needed to retain gauge invariance, in a bigger fully on-shell process with auxiliary quark or gluon line.


Introduction
Despite the high energy limit of Quantum Chromodynamics (QCD) (see eg. [1] for a review) has been studied for over forty years, the confrontation of various small-x approaches and experimental data is still not fully conclusive (here x ∼ 1/ √ s is the longitudinal fraction of hadron momentum carried by a parton and s is the center-of-mass energy). On one hand, the experimental data relevant to the small-x regime can be often explained by the collinear factorization, supplemented however with parton showers or other type of resummations and multi-parton interactions. On the other hand, certain types of reactions, for example the Mueller-Navalet jet production [2] give strong hints towards the need of inclusion of the small-x effects [3]. In addition, collisions of protons with heavy nuclei provide further hints, as observed for instance in [4] for the forward dijet production case.
In order to provide more solid statements regarding the need of small-x approaches, one needs higher order corrections for various components of small-x calculations, in particular for high energy partonic amplitudes. As a matter of fact, in collinear factorization, any partonic amplitude can be at present calculated at NLO automatically using computer software. This is still to be achieved in the small-x domain and our work is a step forward towards that goal.
The concept of k T -factorization is based on analogy with collinear factorization, but here both a hard part and a soft hadronic part depend on parton transverse momenta, i.e. we have explicit higher powers k T /Q present in the hard matrix elements (here, Q is the largest scale present in the process). Thus, instead of the leading twist, the accuracy is set by the leading power in 1/ √ s. The momenta of partons defining the hard amplitude may now be off-shell, with vector or spinor indices projected into components dominating in the high energy limit.
In the present work we shall consider multigluon amplitudes with a single gluon being off mass shell. Such amplitudes are primarily used in the forward particle production (see eg. [40]) and have large phenomenological impact (see eg. [41,42,43,44,45,46,47,48,49,49] for various application in forward jet production processes at LHC). The momentum of the off-shell gluon has the form where p µ is the light-like momentum typically associated with the colliding hadron, x is the fraction of this momentum carried by the scattering parton, and k µ T is a transverse component satisfying k T · p = 0. The off-shell gluon couples eikonally, i.e. its vector index is projected onto p µ (the propagator is included in the amplitude), see Fig. 1. The standard diagrams contributing to off-shell amplitude defined in that fashion are however not gauge invariant. The proper definition of such amplitudes can be done either within the Lipatov's high energy effective action [50,51] or by explicitly constructing additional contributions required by the gauge invariance, high energy kinematics and the proper soft and collinear behavior. The latter method is very useful in automated calculations at tree level and a few approaches exist: using the Ward identities [52], embedding in a bigger on-shell process [53] (see also [54] for earlier application to 2 → 2 process), using matrix elements of straight infinite Wilson lines [55]. In particular, the embedding method [53] has proved to be very effective in numerical calculations and is implemented in a Monte Carlo generator [56]. Also, it has been generalized to one-loop level with a proof of concept given in [57]. The great advantage of this method is that it can be used to extract the high energy off-shell amplitudes from existing on-shell one loop results. We will review the method in detail in Section 2.
In order to apply the embedding method at one-loop level, and in particular to validate the general concept of [57], it is reasonable to start with simplest one-loop helicity amplitudes. In on-shell case, these are the amplitudes with all helicities being the same, say 'plus' (we use the convention that all momenta are outgoing). Such amplitudes vanish at tree level, but are non-zero at loop level. Thus, in the present work, we shall calculate one-loop amplitudes with all-plus helicity gluons and one off-shell gluon, consistent with gauge invariance and high energy limit of QCD. Our result will be presented for arbitrary, say n, number of gluons. In particular, we find that for n = 3 our general result coincides with the existing result obtained from Lipatov's effective action [39].
As a basis for our calculation we shall use existing one-loop results for (− + · · · +) helicity on-shell amplitudes, where the first pair of particles is either gluon or quark-antiquark pair. The particles with helicity +− will provide an auxiliary quark or gluon line, with corresponding external spinors parametrized in a way that -upon taking a proper limit -will + terms required for gauge invariance p µ xp + k T Figure 1: In high energy factorization for forward jets (hybrid factorization [58,40]) the multigluon amplitude has one incoming momentum off mass shell, with the off-shell propagator projected onto light-like momentum p µ (typically the momentum of the hadron to which the gluon couples). The momentum of the off-shell leg has only one longitudinal component in the high energy kinematics. Such amplitude is in general not gauge invariant and additional terms are required to define it properly.
guarantee both the high energy kinematics (1) and eikonal coupling for the internal off-shell gluon attached to it.
We shall focus on the so-called color ordered amplitudes that correspond to planar diagrams and use the spinor helicity method (see [59] for a review). At tree level, the color decomposition of a full gluon amplitude into color ordered amplitudes is M a1,...,an λ1,...,λn (k 1 , . . . , k n ) = perm.(2···n) where t a are color generators, k i is momentum of i-th gluon with helicity projection λ i and the sum goes over all non-cyclic permutations of the arguments of the trace and color ordered amplitudes A. At one-loop level, additional double trace terms are present. They can be however obtained as linear combinations of the leading trace contributions. It is known that on-shell (± + · · · +) one-loop amplitudes have rather simple structure, given by a rational function of spinor products. Consider for instance the all-plus on-shell leading trace color ordered amplitude. It has a remarkably simple form for arbitrary number of gluons (conjectured by Z. Bern, G. Chalmers, L. J. Dixon and D. A. Kosower in [60,61] and demonstrated by G. Mahlon in [62]) : Above, the spinor products are defined as where u ± (k) are the spinors of helicity ± for an on-shell momentum k. The above result is most easily understood within the unitarity methods (see eg. [63]), or -more generally -the on-shell methods (see [64] for a comprehensive review). The off-shell gauge invariant amplitudes we calculate in the present work inherit the rational structure. Our paper has the following structure. In the following section, we will describe the embedding method in more detail. Next, in Section 3 we will present the main results for the amplitudes. In Section 4 we recalulate the amplitudes using the embedding method with auxiliary gluon line as a verification of our results. In Section 5 we will investigate the on-shell limit of the obtained off-shell amplitudes. Finally, in Section 6 we shall summarize our work and discuss further perspectives.

The method
The method to obtain off-shell amplitudes we are about to use has been covered in [53]. Here we shall apply it to obtain one loop scattering amplitudes for arbitrary number of positive Figure 2: Gauge invariant off-shell amplitudes can be obtained by considering a process with an auxiliary quark-antiquark pair, with momenta parametrized in terms of a parameter Λ in such a way, that upon taking the limit Λ → ∞ the coupling to the quark line becomes eikonal and the momentum of the off-shell gluon has the high energy form (1).
helicity on-shell gluons and one off-shell gluon with the high energy kinematics (1) (called also the quasi-multi-Regge kinematics). Let us briefly recall how the method works. The basic idea is to calculate the amplitude with the off-shell gluon using an on-shell amplitude with an auxiliary quark-antiquark pair, which follows specific kinematics. Ultimately, the auxiliary quark and antiquark spinors are decoupled ensuring gauge invariance of the off-shell amplitude. Schematically, the method can be summarized as (see also Fig. 2 where X stands for other on-shell particles involved in the hard scattering process and Λ is a real parameter parametrizing momenta of auxiliary quarks (see below). The gauge invariant off-shell amplitude is denoted A * . The momenta of the auxiliary quarks are taken to be the following: where and q µ is an arbitrary light-like momentum such that q · k T = 0, q · p > 0. Note, that p µ 1 and p µ 2 are light-like and they satisfy p µ 1 + p µ 2 = k µ , where the latter is the momentum of the off-shell gluon as defined in Eq. (1). In the limit Λ → ∞ the coupling of gluons to the quark line becomes eikonal, consistent with the high energy limit. The factor 1/g s in Eq. (5) is to correct the power of the coupling, and the factor x|k T | is for the correct matching to k Tdependent PDFs in a cross section. In particular, the factor |k T | makes sure the amplitude is finite for |k T | → 0.
In practice, instead of using the above definitions of p µ 1 and p µ 2 , we will use their expansion in Λ: In order to use the helicity method, we need to express k µ T in terms of spinors. It can be decomposed as follows with Realize that k µ T is a four-vector with a negative square, and we have The spinors of p µ 1 and p µ 2 can be decomposed into those of p µ and q µ following Realize that (Λ − x)/Λ β = 1 − β. We see that spinor products are independent of Λ. Further, the spinors for auxiliary quarks behave for large Λ as In what follows, we shall call the above kinematics together with the Λ → ∞ the "Λ prescription". Applying it to an amplitude with auxiliary partons gives the gauge invariant off-shell amplitude.
Alternatively, the "embedding" method described above can be used with an auxiliary gluon line, instead of the quark line. Indeed, the color decomposition for (n − 2)-gluon amplitude with a quark-antiquark pair is given by and can be projected onto (n − 1)-gluon amplitude by a contraction with (t a * ) ji , where a * represents the color index of the off-shell gluon. Now, for an auxiliary gluon pair instead of quarks, one simply needs to select only those permutations in Eq. (2) that retains the order of gluons 1 and 2 and substitute t a1 t a2 → t a * . At one loop, the color decompositions get more complicated and are given by equation (1) in [65] and equation (2.4-5) in [66] respectively. One can however easily see that the same procedure goes through to extract a single gluon color from a pair of colors. In [57] it has been shown that at tree level, the partial amplitudes obtained using different pairs of auxiliary partons are identical. We will see here that the same holds at one loop for the all-plus amplitudes.

All-plus off-shell gauge invariant amplitudes at NLO
In this section we present our results for one loop amplitudes for one off-shell gluon and n − 1 on-shell positive helicity gluons. We begin with several low multiplicity examples, starting with the simplest cases: n = 3 (the vertex), n = 4 and n = 5. Then, we will turn to a general result for arbitrary n. For each case, we first present the known amplitude with auxiliary quarks and the amplitude we obtain by applying the Λ prescription on it.

3-point vertex
We first consider the 3-point vertex with one off-shell gluon and two positive helicity on-shell gluons at one loop. Such vertex has been calculated for arbitrary helicity projection in [39] from the Lipatov's effective action.
In order to calculate it from Λ prescription, we need the 4-point amplitude for quark, anti-quark and two gluons. It has the following form [66] : where n f accounts for the number of Weyl fermions circulating in the loop, n s the number of complex scalars and Applying the Λ prescription gives : We checked that for n s = 0 the above result agrees with the one of [67,39], up to an overall constant and a factor xE/|k T |, where E is the energy component of p µ .

4-point amplitude
The 5-leg amplitude with auxiliary quarks is given by [66] : Applying the Λ prescription we find that the first term is of the order Λ −1 and thus vanishes. Further calculation leads to the following result
Apply the Λ prescription we find that the term with the factor 1 + 1 N 2 c vanishes leading to

n-point amplitude
Finally, in the following section we shall derive the general expression for one-loop amplitude for one off-shell gluon and n − 1 on-shell gluons with all helicities positive. To this end, we need the one loop amplitude for a quark-antiquark pair and n − 1 positive helicity gluons, which has been derived in [68]. It reads with where After applying the Λ prescription we find that the term with the factor 1 + 1 N 2 c is of the order Λ −1 , whereas the other term is of order 1 and is the one contributing to the off-shell amplitude. Eventually, we obtain the following expression for the off-shell amplitude: A * (1) n (g * , 3 + , · · · , (n + 1) + ) = g n with U * 1 = n j=3 pj p(j + 1 p| / K j,j+1 / K (j+1)···(n+1) |p j(j + 1) , It can be readily checked that the above expression recovers the amplitudes calculated previously for n = 3, 4, 5.

Verification with auxiliary gluons
In the following section we shall verify the off-shell gauge invariant amplitudes we obtained in the previous section applying the Λ prescription to the corresponding amplitude with auxiliary gluons instead of auxiliary quarks. .

4-point amplitude
The amplitude with the auxiliary gluons is [65] A This expression leads to the following off-shell amplitude which turns out to be equal to Eq. (22).

5-point amplitude
Six-point amplitude with auxiliary gluons is given by [68] : This amplitude turns out to be equal to the one obtained with auxiliary quark line, Eq. (23). The comparison is detailed in Appendix A.

n-point amplitude
For the general case of n-point amplitude, the on-shell gluonic amplitude is taken from [68] A (1) with , Applying the Λ prescription to T 2 gives the same result as for S 2 in (27). It turns out that T 1 is equal to S 1 within the Λ description once you realize that the first term in the sum over j in T 1 is of the order Λ −1 . In the end, applying the Λ prescription toq − q + g + · · · g + or g − g + g + · · · g + gives the same expression, given in Eq. (29).

On-shell limit
Now that we obtained an expression for A * (1) n (g * , 3 + , · · · , (n + 1) + ), we should verify that, in the on-shell limit i.e. when |k T | → 0, we obtain an on-shell amplitude with a gluon with momentum xp µ . We expect that the limit consists of the sum of the amplitudes for which the, now on-shell, gluon has either negative or positive helicity. For tree-level amplitudes, this can be understood as follows. Firstly, at the on-shell limit, the contributions to the amplitude that dominate have a propagator with denominator k 2 T = −κκ * , and have exactly the form of the first term in Fig. 1. More precisely, they have the form where we use the planar Feynman rules as in equation (10) of [69], where J µ represents the off-shell current, and where we included the factor x|k T | from the Λ-prescription. Using current conservation k · J = 0, we can see that projecting to p µ is equivalent to projecting to −k µ T /x. Secondly, using Eq. (9) to Eq. (11), we see that with polarization vectors Thus we find lim |kT |→0 where |k T |/κ * = e iφ for some angle φ, and |k T |/κ its complex conjugate, and where A (0) n (g ± X ) = ε ± · J. In [69] it is explained how such a coherent sum of amplitudes becomes an incoherent sum of squared amplitudes in a cross section.
When taking the on-shell limits in expressions consisting of spinor products and invariants involving the momentum p µ , the final step is to interpret this momentum as the momentum of the now on-shell gluon, divided by x. Since the tree amplitudes are homogeneous in p µ of degree 1, this results in the overall factor 1/x equivalent to the one coming from changing projector p µ → −k µ T /x above. The off-shell one-loop all-plus amplitudes can easily be checked to be homogeneous in p µ of degree 1 too, and the same factor 1/x will show up to eat the factor x from the Λ-prescription.
We now verify that the same limit appears for the one-loop n-point all-plus amplitudes we obtained in Section 3.4. One can notice that U * A * (1) n (g * , 3 + , · · · , (n + 1) So we already have the contribution from the amplitude with negative helicity gluon (in place of the off-shell one). We now need to show that the second term is actually the contribution from the amplitude with a positive helicity gluon, i.e.
To this end, we have to work on U * 3 . One can show that Injecting it in Eq. (43) leads to lim kT →0 A * (1) n (g * , 3 + , · · · , (n + 1) More details on the above calculation are given in Appendix B. This is exactly what we expect from the on-shell limit of an off-shell amplitude: a contribution from an amplitude where the off-shell gluon is replaced by a positive helicity gluon and another one where it is replaced by a negative helicity gluon.

Summary
In this paper we have calculated expressions for amplitudes in high energy factorization with one off-shell gluon and any number of plus-helicity gluons at one loop level. We also obtained expressions for specific cases: 3, 4 and 5 point amplitudes. To obtain these results we used the embedding method developed in [53,57]. The method relies on identifying pair of on-shell partons as auxiliary lines which can be decoupled in high energy limit, leaving gauge invariant off-shell amplitude with proper high energy kinematics. We find agreement with the existing calculation for the 3-point vertex with a Reggeized gluon in [39]. Furthermore we explicitly demonstrated that we obtain the correct on-shell limit for all calculated amplitudes. Thus, we conclude that the embedding method works at the one-loop level, at least for amplitudes with same helicities.
Our future plans involve calculation of other QCD amplitudes and, in particular, addressing also the real corrections. The ultimate goal is to automatize the NLO calculations in k T -factorization as well as the small-x improved TMD factorization (ITMD) [70,71].

A 5-point amplitude -detailed calculation
In order to compare the off-shell gauge invariant 5-point amplitude obtained from the auxiliary quark lineq − q + g + g + g + g + to the one obtained from the auxiliary gluon line g − g + g + g + g + g + , we will rewrite both expressions. Let's first rewrite the first term of the amplitude with auxiliary quarks (before applying the Λ prescription, see Eq. (23) Let us now rewrite the expression for the amplitude (35). In the second term we use (51) In the first term we use For the factorized term in the second line, we can use the momentum conservation For the last term, before applying Λ prescription, we use : In the end, we have If we put back the factor − [p6] 2 κ * s k6 (not writen in the calculation for simplicity), we recognize the second line of Eq. (55). Thus, both approaches give the same result.

B On-shell limit calculation
In this appendix we detail the calculation that leads to Eq. (45) which implies the correct on-shell limit for the n-point off-shell amplitude we presented in Eq. (38).
In order to rewrite the expression for U * 3 so that the on-shell limit can be utilized, let us come back to the expression for T 2 , see Eq. (38) before applying the Λ prescription. We focus on the first term in the sum over j (i.e. for j = 3), since it is the term that leads to U * 3 when applying the Λ prescription. Let us call this term T 3 : We have κ p(n + 1) .
This is the only term in the sum over l that has κ in the denominator and that is the only non vanishing term when k T tends to 0. Putting all this together leads to This demonstrates the first relation in Eq. (45). We now have to prove the second one i.e. we need to show that the obtained expression corresponds to the numerator of the amplitude for n − 1 gluons with positive helicity (up to some factor Now we can work on U * 3 . Let's first express F in terms of a sum. For a direct comparison, we should also use the expression of U * 3 with the following change in the momenta label : p → 1, ∀i ∈ 3, . . . , n + 1, i → i − 1 (then momentum conservation expresses the same way i.e.
on the other hand, we have The first equality is obtained by momentum conservation on the index l (in the second term only) and the second one also by momentum conservation, on the index i this time (for terms 2 and 3). This finally proves the second relation in Eq. (45) which leads then to the expected on-shell limit for the amplitude in Eq. (29).