All-plus helicity off-shell gauge invariant multigluon amplitudes at one loop
Etienne Blanco, Andreas van Hameren, Piotr Kotko, Krzysztof Kutak
aa r X i v : . [ h e p - ph ] A ug IFJPAN-IV-2020-6
All-plus helicity off-shell gauge invariant multigluonamplitudes at one loop
Etienne Blanco a , Andreas van Hameren a ,Piotr Kotko b , Krzysztof Kutak aa Institute of Nuclear Physics, Polish Academy of SciencesRadzikowskiego 152, 31-342 Krakow, Poland b AGH University Of Science and Technology, Physics Faculty,Mickiewicza 30, 30-059 Krakow, Poland
Abstract
We calculate one loop scattering amplitudes for arbitrary number of positive helicityon-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 methodis based on embedding the off-shell process, together with contributions needed toretain gauge invariance, in a bigger fully on-shell process with auxiliary quark or gluonline.
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 andexperimental data is still not fully conclusive (here x ∼ / √ s is the longitudinal fraction ofhadron 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 collinearfactorization, supplemented however with parton showers or other type of resummationsand multi-parton interactions. On the other hand, certain types of reactions, for examplethe Mueller-Navalet jet production [2] give strong hints towards the need of inclusion of thesmall- 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, oneneeds higher order corrections for various components of small- x calculations, in particularfor high energy partonic amplitudes. As a matter of fact, in collinear factorization, anypartonic amplitude can be at present calculated at NLO automatically using computersoftware. This is still to be achieved in the small- x domain and our work is a step forwardtowards that goal.The key result in the small- x field is the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation[5, 6], which describes evolution in energy (or x ) of the gluon Green function in the highenergy limit. It can also be converted to energy evolution of so-called unintegrated partondistribution functions, that unlike collinear PDFs, explicitly depend on parton transverse1omentum k T . Other key results in the small- x QCD constitute the k T -factorization (calledalso high energy factorization) [7, 8], as well as further developments that overcome theunitarity bound violation by the BFKL equation and lead to nonlinear evolution of Balitsky-Kovchegov (BK) equation [9, 10], B-JIMWLK equations [9, 11, 12, 13, 14, 15, 16, 17, 18] andColor Glass Condensate (CGC) effective theory (see e.g. [19]). Some key higher order resultsinclude: the next-to-leading order (NLO) BFKL kernel [20, 21, 22], the NLO BK kernel [23],the B-JIMWL equation at NLO [24, 25], the γ ∗ → ¯ qq impact factor at NLO [26, 27, 28, 29]also with heavy quarks [30], partial inclusion of NLO for Higgs + jet [31], single inclusivejet production in CGC at NLO [32], and also the recent calculation of γ ∗ → ¯ qqγ impactfactor at NLO [33]. In addition, there are NLO calculations in the context of the Lipatov’seffective action [34, 35, 36, 37, 38, 39].The concept of k T -factorization is based on analogy with collinear factorization, but hereboth a hard part and a soft hadronic part depend on parton transverse momenta, i.e. wehave explicit higher powers k T /Q present in the hard matrix elements (here, Q is the largestscale present in the process). Thus, instead of the leading twist, the accuracy is set by theleading power in 1 / √ s . The momenta of partons defining the hard amplitude may now beoff-shell, with vector or spinor indices projected into components dominating in the highenergy limit.In the present work we shall consider multigluon amplitudes with a single gluon being offmass 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 theoff-shell gluon has the form k µ = xp µ + k µT , (1)where p µ is the light-like momentum typically associated with the colliding hadron, x isthe fraction of this momentum carried by the scattering parton, and k µT is a transversecomponent satisfying k T · p = 0. The off-shell gluon couples eikonally, i.e. its vector index isprojected onto p µ (the propagator is included in the amplitude), see Fig. 1. The standarddiagrams contributing to off-shell amplitude defined in that fashion are however not gaugeinvariant. The proper definition of such amplitudes can be done either within the Lipatov’shigh energy effective action [50, 51] or by explicitly constructing additional contributionsrequired by the gauge invariance, high energy kinematics and the proper soft and collinearbehavior. The latter method is very useful in automated calculations at tree level and afew approaches exist: using the Ward identities [52], embedding in a bigger on-shell process[53] (see also [54] for earlier application to 2 → n , number of gluons. In particular, we find that for n = 3 our general result coincideswith the existing result obtained from Lipatov’s effective action [39].As a basis for our calculation we shall use existing one-loop results for ( − + · · · +) helicityon-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 corre-sponding external spinors parametrized in a way that – upon taking a proper limit – will2 terms required forgauge invariance p µ xp + k T Figure 1:
In high energy factorization for forward jets (hybrid factorization [58, 40]) the multigluonamplitude has one incoming momentum off mass shell, with the off-shell propagator projected ontolight-like momentum p µ (typically the momentum of the hadron to which the gluon couples). Themomentum 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 itproperly. guarantee both the high energy kinematics (1) and eikonal coupling for the internal off-shellgluon attached to it.We shall focus on the so-called color ordered amplitudes that correspond to planar di-agrams and use the spinor helicity method (see [59] for a review). At tree level, the colordecomposition of a full gluon amplitude into color ordered amplitudes is M a ,...,a n λ ,...,λ n ( k , . . . , k n ) = X perm. (2 ··· n ) Tr ( t a t a . . . t a n ) A (cid:16) ( λ ) , ( λ ) , . . . , n ( λ n ) (cid:17) , (2)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 colorordered amplitudes A . At one-loop level, additional double trace terms are present. Theycan 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-shellleading trace color ordered amplitude. It has a remarkably simple form for arbitrary numberof gluons (conjectured by Z. Bern, G. Chalmers, L. J. Dixon and D. A. Kosower in [60, 61]and demonstrated by G. Mahlon in [62]) : A (1) n = g ns X ≤ i Gauge invariant off-shell amplitudes can be obtained by considering a process with anauxiliary quark-antiquark pair, with momenta parametrized in terms of a parameter Λ in such away, that upon taking the limit Λ → ∞ the coupling to the quark line becomes eikonal and themomentum 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) (calledalso the quasi-multi-Regge kinematics).Let us briefly recall how the method works. The basic idea is to calculate the amplitudewith 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 aredecoupled ensuring gauge invariance of the off-shell amplitude. Schematically, the methodcan be summarized as (see also Fig. 2)lim Λ →∞ (cid:18) x | k T | g s Λ A (¯ q ( k ) q ( k ) X ) (cid:19) = A ∗ ( g ∗ ( k ) X ) , (5)where X stands for other on-shell particles involved in the hard scattering process and Λ is areal parameter parametrizing momenta of auxiliary quarks (see below). The gauge invariantoff-shell amplitude is denoted A ∗ . The momenta of the auxiliary quarks are taken to be thefollowing: p µ = Λ p µ + αq µ + βk µT ,p µ = k µ − p µ , (6)where α = − β k T p · q , β = 11 + p − x/ Λ (7)and q µ is an arbitrary light-like momentum such that q · k T = 0, q · p > 0. Note, that p µ and p µ are light-like and they satisfy p µ + p µ = k µ , where the latter is the momentum of theoff-shell gluon as defined in Eq. (1). In the limit Λ → ∞ the coupling of gluons to the quarkline becomes eikonal, consistent with the high energy limit. The factor 1 /g s in Eq. (5) is tocorrect the power of the coupling, and the factor x | k T | is for the correct matching to k T -dependent PDFs in a cross section. In particular, the factor | k T | makes sure the amplitudeis finite for | k T | → p µ and p µ , we will use their expansionin Λ: p µ = Λ p µ + (cid:18) 12 + x (cid:19) k µT − k T p · q q µ + O (Λ − ) ,p µ = ( x − Λ) p µ + (cid:18) − x (cid:19) k µT + k T p · q q µ + O (Λ − ) . (8)In order to use the helicity method, we need to express k µT in terms of spinors. It can bedecomposed as follows k µT = − ¯ κe µ − ¯ κ ∗ e µ ∗ , (9)with e µ = 12 h p | γ µ | q ] , e µ ∗ = 12 h q | γ µ | p ] (10)4nd ¯ κ = κ [ pq ] = h q | /k | p ]2 p · q , ¯ κ ∗ = κ ∗ h qp i = h p | /k | q ]2 p · q . (11)Realize that k µT is a four-vector with a negative square, and we have k T = − κκ ∗ . (12)The spinors of p µ and p µ can be decomposed into those of p µ and q µ following | i = √ Λ | p i − β ¯ κ ∗ √ Λ | q i , | 1] = √ Λ | p ] − β ¯ κ √ Λ | q ] (13) | i = √ Λ − x | p i + β ¯ κ ∗ √ Λ | q i , | 2] = −√ Λ − x | p ] − β ¯ κ √ Λ | q ] . (14)Realize that p (Λ − x ) / Λ β = 1 − β . We see that spinor products h i = − κ ∗ , [12] = − κ (15)are independent of Λ. Further, the spinors for auxiliary quarks behave for large Λ as | i → √ Λ | p i , | → √ Λ | p ] , | i → √ Λ | p i , | → −√ Λ | p ] . (16)In what follows, we shall call the above kinematics together with the Λ → ∞ the “Λprescription”. Applying it to an amplitude with auxiliary partons gives the gauge invariantoff-shell amplitude.Alternatively, the “embedding” method described above can be used with an auxiliarygluon line, instead of the quark line. Indeed, the color decomposition for ( n − M ij a ,...,a n λ ,λ ,λ ,...,λ n ( k , . . . , k n ) = X perm. (3 ··· n ) ( t a · · · t a n ) ij A (cid:16) q ( λ ) , ¯ q ( λ ) , ( λ ) , . . . , n ( λ n ) (cid:17) , (17)and can be projected onto ( n − t a ∗ ) ji , where a ∗ represents the color index of the off-shell gluon. Now, for an auxiliary gluon pair insteadof quarks, one simply needs to select only those permutations in Eq. (2) that retains theorder of gluons 1 and 2 and substitute t a t a → t a ∗ . At one loop, the color decompositionsget 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 extracta single gluon color from a pair of colors. In [57] it has been shown that at tree level, thepartial amplitudes obtained using different pairs of auxiliary partons are identical. We willsee here that the same holds at one loop for the all-plus amplitudes. In this section we present our results for one loop amplitudes for one off-shell gluon and n − n = 3 (the vertex), n = 4 and n = 5. Then, we will turnto a general result for arbitrary n . For each case, we first present the known amplitude withauxiliary quarks and the amplitude we obtain by applying the Λ prescription on it. We first consider the 3-point vertex with one off-shell gluon and two positive helicity on-shellgluons at one loop. Such vertex has been calculated for arbitrary helicity projection in [39]from the Lipatov’s effective action. 5n 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] : A (1)4 (1 − ¯ q , + q , + , + ) = − ig s π (cid:20) (cid:18) N c (cid:19) + 13 (cid:18) n s − n f N c (cid:19) s s (cid:21) h i [24] h ih i . (18)where n f accounts for the number of Weyl fermions circulating in the loop, n s the numberof complex scalars and ∀ i, j = 1 , . . . , , s ij = 2 k i · k j = h ij i [ ji ] (19)Applying the Λ prescription gives : A ∗ (1)3 ( g ∗ , + , + ) = − ig s (cid:18) n s − n f N c (cid:19) x | k T | π p · k k T κ ∗ [ p h p ih i = − ig s π (cid:18) n s − n f N c (cid:19) x | k T | κ p · k [ p p h p ih p i . (20)We checked that for n s = 0 the above result agrees with the one of [67, 39], up to an overallconstant and a factor xE/ | k T | , where E is the energy component of p µ . The 5-leg amplitude with auxiliary quarks is given by [66] : A (1)5 (1 − ¯ q , + q , + , + , + ) = − (cid:18) N c (cid:19) ig s π h i [23] h i + h i [45] h ih ih ih ih i− (cid:18) n s − n f N c (cid:19) ig s π (cid:18) h i [34] h i h ih i h ih i + h ih i [45] h ih ih ih ih i + [23][25][12] h ih i (cid:19) . (21)Applying the Λ prescription we find that the first term is of the order Λ − and thus vanishes.Further calculation leads to the following result A ∗ (1)4 ( g ∗ , + , + , + ) = − ig s π x | k T | (cid:16) n s − n f N c (cid:17) κ ∗ h p ih ih ih p i× (cid:20) h p i h p i [34] h i + h p i h p i [45] h i − κ ∗ κ s p s p (cid:21) . (22) The amplitude with the auxiliary quarks is given by [68] : A (1)6 (1 − , + , + , + , + , + ) = (cid:18) N c (cid:19) ig s P l =3 h | /K ...l /k l | h ih ih ih ih i + (cid:18) n s − n f N c (cid:19) ig s " h ih | (2 + 3)(3 + 4) | ih ih i h ih ih i + h ih ih | (4 + 5)(5 + 6) | ih ih ih ih i h ih i− h ih | | ih ih ih ih ih i + h | | h i h ih ih | | h | | h | | h ih ih i h | | s − [26] [2 | (3 + 4)(4 + 5)(3 + 4)(4 + 5) | h ih ih | | h | | s , (23)6here ∀ a, b = 1 , . . . , , /K a ··· b = b X i = a /k i , ∀ i, j, k = 1 , . . . , , s ijk = ( k i + k j + k k ) , ∀ a, b, i, j, k, l = 1 , . . . , , h a | ( i + j ) | b ] = h ai i [ ib ] + h aj i [ jb ]and h a | ( i + j )( k + l ) | b i = h ai i [ i | ( k + l ) | b i + h aj i [ j | ( k + l ) | b i . (24)Apply the Λ prescription we find that the term with the factor (cid:16) N c (cid:17) vanishes leadingto A ∗ (1)5 ( g ∗ , + , + , + , + ) = (cid:18) n s − n f N c (cid:19) ×× ix | k T | g s " h p i ( κ ∗ [ p | | p i + h p i [34] h p i ) κ ∗ h i h ih ih p i + h p ih p ih p | (4 + 5)(5 + 6) | p i κ ∗ h p ih ih i h ih p i− h p ih p | | p i κ ∗ h p ih ih ih i + h p | | p ] h i h ih p ih | | p ]+ h p | | κ ∗ h p ih i h | | s − [ p [ p | (3 + 4)(4 + 5)(3 + 4)(4 + 5) | κ h ih ih | | p ] h | | s . (25) Finally, in the following section we shall derive the general expression for one-loop amplitudefor one off-shell gluon and n − n − A (1) n +1 (1 − ¯ q , + q , + , · · · , ( n + 1) + ) = g n +1 s (cid:18) N c (cid:19) i P nl =3 h | /K ··· l /k l | ih i · · · h ( n + 1)1 i + g n +1 s (cid:18) n s − n f N c (cid:19) i S + S h ih i · · · h ( n + 1)1 i , (26)with S = n X j =3 h j ih j + 1 ih | /K j,j +1 /K ( j +1) ··· ( n +1) | ih j ( j + 1) i ,S = n − X j =3 n X l = j +1 h | /K j ··· l /K ( l +1) ··· ( n +1) | i h | /K j ··· l /K ( l +1) ··· ( n +1) | ih | /K ( l +1) ··· ( n +1) /K j ··· l | ( j − ih | /K ( l +1) ··· ( n +1) /K j ··· l | j i× h ( j − j ih l ( l + 1) ih | /K ··· ( j − [ F ( j, l )] /K ( l +1) ··· ( n +1) | ih | /K ··· ( j − /K j ··· l | l ih | /K ··· ( j − /K j ··· l | ( l + 1) i s j ··· l , (27)where F ( j, l ) = l − X i = j l X m = i +1 /k i /k m . (28)After applying the Λ prescription we find that the term with the factor (cid:16) N c (cid:17) is of theorder Λ − , whereas the other term is of order 1 and is the one contributing to the off-shellamplitude. Eventually, we obtain the following expression for the off-shell amplitude: A ∗ (1) n ( g ∗ , + , · · · , ( n + 1) + ) = g ns (cid:18) n s − n f N c (cid:19) ix | k T | U ∗ + U ∗ + U ∗ κ ∗ h p ih i · · · h np i , (29)7ith U ∗ = n X j =3 h pj ih p ( j + 1 ih p | /K j,j +1 /K ( j +1) ··· ( n +1) | p ih j ( j + 1) i ,U ∗ = n − X j =4 n X l = j +1 h p | /K j ··· l /K ( l +1) ··· ( n +1) | p i h p | /K ( l +1) ··· ( n +1) /K j ··· l | ( j − ih p | /K ( l +1) ··· ( n +1) /K j ··· l | j i× h ( j − j ih l ( l + 1) ih p | /K ′ ··· ( j − [ F ( j, l )] /K ( l +1) ··· ( n +1) | p ih p | /K ··· ( j − /K j ··· l | l ih p | /K ··· ( j − /K j ··· l | ( l + 1) i s j ··· l ,U ∗ = n X l =4 h p | /K ··· l /K ( l +1) ··· ( n +1) | p i h p | /K ( l +1) ··· ( n +1) /K ··· l | p ih p | /K ( l +1) ··· ( n +1) /K ··· l | i× h p ih l ( l + 1) i [ p | [ F (3 , l )] /K ( l +1) ··· ( n +1) | p i κ ∗ [ p | /K ··· l | l i [ p | /K ··· l | ( l + 1) i s ··· l . (30)It can be readily checked that the above expression recovers the amplitudes calculatedpreviously for n = 3 , , In the following section we shall verify the off-shell gauge invariant amplitudes we obtainedin the previous section applying the Λ prescription to the corresponding amplitude withauxiliary gluons instead of auxiliary quarks. The 4-point amplitude for one negative helicity gluon and three positive helicity gluons isgiven by [66] A (1)4 (1 − , + , + , + ) = (cid:18) n s − n f N c (cid:19) ig s π h i [24] [12] h ih i [41] . (31)Applying the Λ prescription leads to the same result, as before: A ∗ (1)3 ( g ∗ , + , + ) = − ig s (cid:18) n s − n f N c (cid:19) x | k T | π p · k [ p κ h p ih i = − ig s (cid:18) n s − n f N c (cid:19) x | k T | π p · k k T κ ∗ [ p h p ih i , (32)where we used p · k = − p · k since 0 = p · k = p · ( − k − k ). The amplitude with the auxiliary gluons is [65] A (1)5 (1 − , + , + , + , + ) = ig s π (cid:16) n s − n f N c (cid:17) [12] h ih ih i [51] × (cid:20) ( s + s + s )[25] − [24] h i [35][25] − [12][15] h ih i (cid:18) h i h i [23] h i + h i h i [34] h i + h i h i [45] h i (cid:19) (cid:21) . (33)8his expression leads to the following off-shell amplitude A ∗ (1)4 ( g ∗ , + , + , + ) = ig s π x | k T | (cid:16) n s − n f N c (cid:17) κ ∗ h p ih ih i [5 p ] × (cid:20) s p [ p − κ [ p κ ∗ h p i (cid:18) h p i h p i [34] h i + h p i h p i [45] h i (cid:19) (cid:21) , (34)which turns out to be equal to Eq. (22). Six-point amplitude with auxiliary gluons is given by [68] : A (1)6 (1 − , + , + , + , + , + ) = g s (cid:18) n s − n f N c (cid:19) ×× i " h | | h ih ih i s h | | 6] + h | | h i h ih i s h | | [12][61] s (cid:18) [23][34] h ih | | − [45][56] h ih | | 6] + [35] h ih i (cid:19) − h i [23] h ih i h i h ih ih i + h i h i [56] h ih ih ih i h i − h i h ih | | h ih ih i h i h ih i (35)Applying the Λ prescription to the above on-shell result gives A ∗ (1)5 ( g ∗ , + , + , + , + ) = g s (cid:18) n s − n f N c (cid:19) ×× ix | k T | " ( κ ∗ [ p 6] + h p i [36]) κ ∗ h p ih i s k h | k | 6] + h p | | p ] h i h ih p i ( s p + s p ) h | | p ]+ [ p κ ∗ s (cid:18) [ p h ih | | p ] − [45][56] h ih | k | 6] + [35] h ih i (cid:19) − h p i [ p h p ih i h ih ih p i + h p i h i [56] κ ∗ h p ih ih i h i − h p i h i ( κ ∗ [ p 4] + h p i [34]) κ ∗ h p ih i h i h ih p i . (36)This amplitude turns out to be equal to the one obtained with auxiliary quark line, Eq. (23).The comparison is detailed in Appendix A. For the general case of n -point amplitude, the on-shell gluonic amplitude is taken from [68] A (1) n +1 (1 − , + , + , · · · , ( n + 1) + ) = g n +1 s (cid:18) n s − n f N c (cid:19) i T + T h ih i · · · h n i , (37)9ith T = n X j =2 h j ih j + 1 ih | /K j,j +1 /K ( j +1) ··· ( n +1) | ih j ( j + 1) i ,T = n − X j =3 n X l = j +1 h | /K j ··· l /K ( l +1) ··· ( n +1) | i h | /K ( l +1) ··· ( n +1) /K j ··· l | ( j − ih | /K ( l +1) ··· ( n +1) /K j ··· l | j i× h ( j − j ih l ( l + 1) ih | /K ··· ( j − [ F ( j, l )] /K ( l +1) ··· ( n +1) | ih | /K ··· ( j − /K j ··· l | l ih | /K ··· ( j − /K j ··· l | ( l + 1) i s j ··· l . (38)Applying the Λ prescription to T gives the same result as for S in (27). It turns out that T is equal to S within the Λ description once you realize that the first term in the sumover j in T is of the order Λ − . In the end, applying the Λ prescription to ¯ q − q + g + · · · g + or g − g + g + · · · g + gives the same expression, given in Eq. (29). Now that we obtained an expression for A ∗ (1) n ( g ∗ , + , · · · , ( 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 withmomentum xp µ . We expect that the limit consists of the sum of the amplitudes for whichthe, 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 theamplitude that dominate have a propagator with denominator k T = − κκ ∗ , and have exactlythe form of the first term in Fig. 1. More precisely, they have the form √ p µ x | k T | κκ ∗ J µ (39)where we use the planar Feynman rules as in equation (10) of [69], where J µ represents theoff-shell current, and where we included the factor x | k T | from the Λ-prescription. Usingcurrent conservation k · J = 0, we can see that projecting to p µ is equivalent to projectingto − k µT /x . Secondly, using Eq. (9) to Eq. (11), we see that k µT = − κ √ ε µ − ( p, q ) − κ ∗ √ ε µ + ( p, q ) , (40)with polarization vectors ε µ − ( p, q ) = h p | γ µ | q ] √ pq ] , ε µ + ( p, q ) = h q | γ µ | p ] √ h qp i . (41)Thus we find lim | k T |→ A ∗ (0) n ( g ∗ X ) = | k T | κ ∗ A (0) n ( g − X ) + | k T | κ A (0) n ( g + X ) , (42)where | k T | /κ ∗ = e i φ 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 be-comes an incoherent sum of squared amplitudes in a cross section.When taking the on-shell limits in expressions consisting of spinor products and in-variants involving the momentum p µ , the final step is to interpret this momentum as themomentum of the now on-shell gluon, divided by x . Since the tree amplitudes are homoge-neous in p µ of degree 1, this results in the overall factor 1 /x equivalent to the one comingfrom changing projector p µ → − k µT /x above. The off-shell one-loop all-plus amplitudes caneasily be checked to be homogeneous in p µ of degree 1 too, and the same factor 1 /x willshow up to eat the factor x from the Λ-prescription.10e now verify that the same limit appears for the one-loop n -point all-plus amplitudeswe obtained in Section 3.4. One can notice that U ∗ −−−−−→ | k T |→ T and U ∗ −−−−−→ | k T |→ T , whichimplies lim | k T |→ A ∗ (1) n ( g ∗ , + , · · · , ( n + 1) + ) = | k T | κ ∗ A (1) n ( xp − , + , · · · , ( n + 1) + )+ ix k T → ( U ∗ | k T | /κ ∗ ) h p ih i · · · h ( n + 1) p i . (43)So we already have the contribution from the amplitude with negative helicity gluon (in placeof the off-shell one). We now need to show that the second term is actually the contributionfrom the amplitude with a positive helicity gluon, i.e. A (1) n (1 + , · · · , n + ) = g n i (cid:18) n s − n f N c (cid:19) X ≤ i A 5-point amplitude – detailed calculation In order to compare the off-shell gauge invariant 5-point amplitude obtained from theauxiliary quark line ¯ q − 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 theamplitude with auxiliary quarks (before applying the Λ prescription, see Eq. (23)) h ih | (2 + 3)(3 + 4) | ih ih i h ih ih i = h i ( h i [23] h i + h | | h i ) h ih i h ih ih i = − h i [23] h ih i h ih ih i + h ih ih i h | | h ih ih i h i h ih i = − h i [23] h ih i h ih ih i + h ih ih i h | | h ih ih i h i h ih i− h ih i h | | h ih ih ih i h ih i . (47)Above, we have used the momentum conservation to write h | | 4] = −h | | 4] andthe Schouten identity: h ih i = h ih i + h ih i . It leads to A (1)5 ( g ∗ , + , + , + , + )= g ix | k T | " − h p i [ p h p ih i h ih ih p i + h ih p i h p | | κ ∗ h p ih i h i h ih p i− h p ih p i h p | | κ ∗ h p ih ih i h ih p i + h p | | κ ∗ h p ih i h | | s k + h p ih p i ( h p i [4 | | p i + h p i [56] h p i ) κ ∗ h p ih ih i h ih p i − h p ih p | | p i κ ∗ h p ih ih ih i + h p | | p ] h i h ih p ih | | p ] − [ p [ p | (3 + 4)(4 + 5)(3 + 4)(4 + 5) | κ h ih ih | | p ] h | | s k = g ix | k T | " − h p i [ p h p ih i h ih ih p i + h ih p i h p | | κ ∗ h p ih i h i h ih p i + h p | | κ ∗ h p ih i h | | s k + h p ih p i [56] h p i κ ∗ h p ih ih i h ih p i− h p i [56] h p i κ ∗ h p ih ih ih i + h p | | p ] h i h ih p ih | | p ] − [ p [ p | (3 + 4)(4 + 5)(3 + 4)(4 + 5) | κ h ih ih | | p ] h | | s k . (48)Terms 4 and 5 can be combined using the Schouten identity h p i [56] κ ∗ h p ih ih i h i ( h p ih i − h p ih i ) = h p i h i [56] κ ∗ h p ih ih i h i . (49)12hus, finally, the amplitude reads A (1)5 ( g ∗ , + , + , + , + )= g ix | k T | " h p | | κ ∗ h p ih i h | | s k + h p | | p ] h i h ih p ih | | p ] − [ p [ p | (3 + 4)(4 + 5)(3 + 4)(4 + 5) | κ h ih ih | | p ] h | | s k − h p i [ p h p ih i h ih ih p i + h p i h i [56] κ ∗ h p ih ih i h i + h ih p i h p | | κ ∗ h p ih i h i h ih p i . (50)Let us now rewrite the expression for the amplitude (35). In the second term we use s = s + s + s −−−−−→ Λ prescr. Λ( s p + s p ) + O (1) = Λ( h p i [3 p ] + h p i [4 p ]) = Λ h p | | p ] . (51)In the first term we use h | | 6] = −h | | −−−−−→ Λ prescr. − Λ h p | | 6] + O (1) (52)For the factorized term in the second line, we can use the momentum conservation s = s k . (53)For the last term, before applying Λ prescription, we use : h | | 4] = −h | | −−−−−→ Λ prescr. − Λ h p | | 4] + O (1) (54)In the end, we have A (1)5 ( g ∗ , + , + , + , + )= g ix | k T | " − h p | | κ ∗ h p ih i s k h | k | 6] + h p | | p ] h i h ih p ih | | p ]+ [ p κ ∗ s k (cid:18) [ p h ih | | p ] − [45][56] h ih | k | 6] + [35] h ih i (cid:19) − h p i [ p h p ih i h ih ih p i + h p i h i [56] κ ∗ h p ih ih i h i + h p i h ih p | | κ ∗ h p ih i h i h ih p i . (55)Let us now compare Eq. (50) and Eq. (55). It is clear that the terms 2, 4, 5 and 6 are thesame. The first terms are also equal upon applying h | | 6] = −h | k | p | (3 + 4)(4 + 5)(3 + 4)(4 + 5) | h ih ih | | p ] h | | 6] = [5 | (3 + 4)(4 + 5) | h ih ih | | 6] + [ p | | h ih ih | | p ] h | | h | (4 + 5) | h ih ih | | 6] + [54] h i [56] h ih ih | | 6] + [ p h i [43] h | (4 + 5) | h ih ih | | p ] h | | − [35] h ih i + [45][56] h ih | k | − [ p h ih | | p ] . (56)If we put back the factor − [ p κ ∗ s k (not writen in the calculation for simplicity), we recognizethe second line of Eq. (55). Thus, both approaches give the same result.13 On-shell limit calculation In this appendix we detail the calculation that leads to Eq. (45) which implies the correcton-shell limit for the n -point off-shell amplitude we presented in Eq. (38).In order to rewrite the expression for U ∗ so that the on-shell limit can be utilized, letus come back to the expression for T , see Eq. (38) before applying the Λ prescription. Wefocus on the first term in the sum over j (i.e. for j = 3), since it is the term that leads to U ∗ when applying the Λ prescription. Let us call this term T : T = n X l =4 h | /K ··· l /K ( l +1) ··· ( n +1) | i h | /K ( l +1) ··· ( n +1) /K ··· l | ih | /K ( l +1) ··· ( n +1) /K ··· l | i× h ih l ( l + 1) ih i [2 | [ F (3 , l )] /K ( l +1) ··· ( n +1) | ih i [2 | /K ··· l | l ih i [2 | /K ··· l | ( l + 1) i s ··· l . (57)We have h | /K ...l /K ( l +1) ... ( n +1) | i = −h | /K ...l | i − h | /K ...l | h i −−−−−→ Λ prescr. Λ κ ∗ l X i =3 s pi . (58)Similar, we have h | /K ( l +1) ... ( n +1) /K ...l | i = −h i [2 | /K ...l | i −−−−−→ Λ prescr. − Λ κ ∗ l X i =3 s pi , (59) h | /K ( l +1) ... ( n +1) /K ...l | i = − h | /K ...l | i − h i [2 | /K ...l | i = h i s ...l − h | /K ...l | i − h i [2 | /K ...l | i , (60)which implies h | /K ( l +1) ... ( n +1) /K ...l | i −−−−−→ Λ prescr. −√ Λ (cid:0) κ ∗ [ p | /K ...l | i + h p i s ...l (cid:1) −−−−→ k T → −√ Λ h p i s ...l . (61)We may also notice that, for l = n , we have[2 | /K ...l | ( l + 1) i = [2 | /K ...n | ( n + 1) i = − [21] h n + 1) i −−−−−→ Λ prescr. κ h p ( n + 1) i . (62)This is the only term in the sum over l that has κ in the denominator and that is the onlynon vanishing term when k T tends to 0.Putting all this together leads to T −−−−−→ Λ prescr. U ∗ = κ ∗ n − X l =4 (cid:16)P li =3 s pi (cid:17) h l ( l + 1) i [ p | [ F (3 , l )] /K ( l +1) ··· ( n +1) | p i [ p | /K ··· l | l i [ p | /K ··· l | ( l + 1) i s ··· l + κ ∗ ( P ni =3 s pi ) h n ( n + 1) i [ p | [ F (3 , n )] | ( n + 1)] h ( n + 1) p i [ p ( n + 1)] h ( n + 1) n i κ h p ( n + 1) i s ··· n −−−−→ k T → κ ∗ κ ( P ni =3 s pi ) [ p | [ F (3 , n )] | ( n + 1)][ p ( n + 1)] s ··· n . (63)Notice that s ··· n = s p ( n +1) = h p ( n + 1) i [( n + 1) p ] = − n X i =3 h pi i [ ip ] = − n X i =3 s pi . (64)14ack to U ∗ , we havelim k T → | k T | κ ∗ U ∗ = | k T | κ [ p ( n + 1)] [ p | [ F (3 , n )] | ( n + 1)] . (65)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 amplitudefor n − X ≤ i Quantum chromodynamics at high energy , vol. 33.Cambridge University Press, 2012.[2] A. H. Mueller and H. Navelet, An Inclusive Minijet Cross-Section and the BarePomeron in QCD , Nucl. Phys. B282 (1987) 727–744.[3] B. Duclou´e, L. Szymanowski and S. Wallon, Evidence for high-energy resummationeffects in Mueller-Navelet jets at the LHC , Phys. Rev. Lett. (2014) 082003,[ ].[4] A. van Hameren, P. Kotko, K. Kutak and S. Sapeta, Broadening and saturationeffects in dijet azimuthal correlations in p-p and p-Pb collisions at √ s = , Phys. Lett. B (2019) 511–515, [ ].[5] E. A. Kuraev, L. N. Lipatov and V. S. Fadin, The Pomeranchuk Singularity inNonabelian Gauge Theories , Sov. Phys. JETP (1977) 199–204.[6] I. I. Balitsky and L. N. Lipatov, The Pomeranchuk Singularity in QuantumChromodynamics , Sov. J. Nucl. Phys. (1978) 822–829.[7] S. Catani, M. Ciafaloni and F. Hautmann, High-energy factorization and small xheavy flavor production , Nucl. Phys. B366 (1991) 135–188.[8] J. C. Collins and R. K. Ellis, Heavy quark production in very high-energy hadroncollisions , Nucl. Phys. B360 (1991) 3–30.[9] I. Balitsky, Operator expansion for high-energy scattering , Nucl. Phys. B463 (1996) 99–160, [ hep-ph/9509348 ].[10] Y. V. Kovchegov, Small x F(2) structure function of a nucleus including multiplepomeron exchanges , Phys. Rev. D60 (1999) 034008, [ hep-ph/9901281 ].[11] J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, The BFKL equation fromthe Wilson renormalization group , Nucl. Phys. B504 (1997) 415–431,[ hep-ph/9701284 ].[12] J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, The Wilsonrenormalization group for low x physics: Towards the high density regime , Phys. Rev. D59 (1998) 014014, [ hep-ph/9706377 ].[13] J. Jalilian-Marian, A. Kovner and H. Weigert, The Wilson renormalization group forlow x physics: Gluon evolution at finite parton density , Phys. Rev. D59 (1998) 014015, [ hep-ph/9709432 ].[14] A. Kovner, J. G. Milhano and H. Weigert, Relating different approaches to nonlinearQCD evolution at finite gluon density , Phys. Rev. D62 (2000) 114005,[ hep-ph/0004014 ].[15] A. Kovner and J. G. Milhano, Vector potential versus color charge density in low xevolution , Phys. Rev. D61 (2000) 014012, [ hep-ph/9904420 ].1616] H. Weigert, Unitarity at small Bjorken x , Nucl. Phys. A703 (2002) 823–860,[ hep-ph/0004044 ].[17] E. Iancu, A. Leonidov and L. D. McLerran, Nonlinear gluon evolution in the colorglass condensate. 1. , Nucl. Phys. A692 (2001) 583–645, [ hep-ph/0011241 ].[18] E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran, Nonlinear gluon evolution in thecolor glass condensate. 2. , Nucl. Phys. A703 (2002) 489–538, [ hep-ph/0109115 ].[19] F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, The Color GlassCondensate , Ann. Rev. Nucl. Part. Sci. (2010) 463–489, [ ].[20] V. S. Fadin and L. Lipatov, BFKL pomeron in the next-to-leading approximation , Phys. Lett. B (1998) 127–134, [ hep-ph/9802290 ].[21] M. Ciafaloni and G. Camici, Energy scale(s) and next-to-leading BFKL equation , Phys. Lett. B (1998) 349–354, [ hep-ph/9803389 ].[22] A. Kotikov and L. Lipatov, NLO corrections to the BFKL equation in QCD and insupersymmetric gauge theories , Nucl. Phys. B (2000) 19–43, [ hep-ph/0004008 ].[23] I. Balitsky and G. A. Chirilli, Next-to-leading order evolution of color dipoles , Phys. Rev. D (2008) 014019, [ ].[24] I. Balitsky and G. A. Chirilli, Rapidity evolution of Wilson lines at the next-to-leadingorder , Phys. Rev. D (2013) 111501, [ ].[25] A. Kovner, M. Lublinsky and Y. Mulian, Jalilian-Marian, Iancu, McLerran, Weigert,Leonidov, Kovner evolution at next to leading order , Phys. Rev. D (2014) 061704,[ ].[26] J. Bartels, D. Colferai, S. Gieseke and A. Kyrieleis, NLO corrections to the photonimpact factor: Combining real and virtual corrections , Phys. Rev. D (2002) 094017, [ hep-ph/0208130 ].[27] I. Balitsky and G. A. Chirilli, Photon impact factor and k T -factorization for DIS inthe next-to-leading order , Phys. Rev. D (2013) 014013, [ ].[28] G. Beuf, Dipole factorization for DIS at NLO: Loop correction to the γ ∗ T,L → qq light-front wave functions , Phys. Rev. D (2016) 054016, [ ].[29] R. Boussarie, A. Grabovsky, L. Szymanowski and S. Wallon, On the one loop γ ( ∗ ) → qq impact factor and the exclusive diffractive cross sections for the productionof two or three jets , JHEP (2016) 149, [ ].[30] G. Chachamis, M. Deak and G. Rodrigo, Heavy quark impact factor inkT-factorization , JHEP (2013) 066, [ ].[31] F. G. Celiberto, D. Y. Ivanov, M. M. Mohammed and A. Papa, High-energy resummeddistributions for the inclusive Higgs-plus-jet production at the LHC , .[32] G. A. Chirilli, B.-W. Xiao and F. Yuan, One-loop Factorization for Inclusive HadronProduction in pA Collisions in the Saturation Formalism , Phys. Rev. Lett. (2012) 122301, [ ].[33] K. Roy and R. Venugopalan, NLO impact factor for inclusive photon + dijet productionin e + A DIS at small x , Phys. Rev. D (2020) 034028, [ ].[34] M. Hentschinski and A. Sabio Vera, NLO jet vertex from Lipatov’s QCD effectiveaction , Phys. Rev. D (2012) 056006, [ ].1735] G. Chachamis, M. Hentschinski, J. Madrigal Martinez and A. Sabio Vera, Quarkcontribution to the gluon Regge trajectory at NLO from the high energy effectiveaction , Nucl. Phys. B (2012) 133–144, [ ].[36] G. Chachamis, M. Hentschinski, J. D. Madrigal Mart´ınez and A. Sabio Vera, Next-to-leading order corrections to the gluon-induced forward jet vertex from the highenergy effective action , Phys. Rev. D (2013) 076009, [ ].[37] M. Hentschinski, J. Madrigal Mart´ınez, B. Murdaca and A. Sabio Vera, Thenext-to-leading order vertex for a forward jet plus a rapidity gap at high energies , Phys. Lett. B (2014) 168–172, [ ].[38] M. Nefedov and V. Saleev, On the one-loop calculations with Reggeized quarks , Mod. Phys. Lett. A (2017) 1750207, [ ].[39] M. A. Nefedov, Computing one-loop corrections to effective vertices with two scales inthe EFT for Multi-Regge processes in QCD , Nucl. Phys. B (2019) 114715,[ ].[40] M. Deak, F. Hautmann, H. Jung and K. Kutak, Forward Jet Production at the LargeHadron Collider , JHEP (2009) 121, [ ].[41] A. van Hameren, P. Kotko and K. Kutak, Three jet production and gluon saturationeffects in p-p and p-Pb collisions within high-energy factorization , Phys. Rev. D (2013) 094001, [ ].[42] A. van Hameren, P. Kotko, K. Kutak and S. Sapeta, Small- x dynamics inforward-central dijet decorrelations at the LHC , Phys. Lett. B737 (2014) 335–340,[ ].[43] A. van Hameren, P. Kotko, K. Kutak, C. Marquet and S. Sapeta, Saturation effects inforward-forward dijet production in p + Pb collisions , Phys. Rev. D89 (2014) 094014,[ ].[44] A. van Hameren, P. Kotko, K. Kutak, C. Marquet, E. Petreska and S. Sapeta, Forward di-jet production in p+Pb collisions in the small-x improved TMDfactorization framework , JHEP (2016) 034, [ ].[45] M. Bury, M. Deak, K. Kutak and S. Sapeta, Single and double inclusive forward jetproduction at the LHC at √ s = 7 and 13 TeV , Phys. Lett. B (2016) 594–601,[ ].[46] M. Bury, H. Van Haevermaet, A. Van Hameren, P. Van Mechelen, K. Kutak andM. Serino, Single inclusive jet production and the nuclear modification ratio at veryforward rapidity in proton-lead collisions with √ s NN = 5.02 TeV , Phys. Lett. B (2018) 185–190, [ ].[47] P. Kotko, K. Kutak, S. Sapeta, A. M. Stasto and M. Strikman, Estimating nonlineareffects in forward dijet production in ultra-peripheral heavy ion collisions at the LHC , Eur. Phys. J. C77 (2017) 353, [ ].[48] H. M¨antysaari and H. Paukkunen, Saturation and forward jets in proton-leadcollisions at the LHC , Phys. Rev. D (2019) 114029, [ ].[49] H. Van Haevermaet, A. Van Hameren, P. Kotko, K. Kutak and P. Van Mechelen, Trijets in kt-factorisation: matrix elements vs parton shower , .[50] L. N. Lipatov, Gauge invariant effective action for high-energy processes in QCD , Nucl. Phys. B452 (1995) 369–400, [ hep-ph/9502308 ].1851] E. N. Antonov, L. N. Lipatov, E. A. Kuraev and I. O. Cherednikov, Feynman rulesfor effective Regge action , Nucl. Phys. B721 (2005) 111–135, [ hep-ph/0411185 ].[52] A. van Hameren, P. Kotko and K. Kutak, Multi-gluon helicity amplitudes with oneoff-shell leg within high energy factorization , JHEP (2012) 029, [ ].[53] A. van Hameren, P. Kotko and K. Kutak, Helicity amplitudes for high-energyscattering , JHEP (2013) 078, [ ].[54] A. Leonidov and D. Ostrovsky, Angular and momentum asymmetry in particleproduction at high-energies , Phys. Rev. D (2000) 094009, [ hep-ph/9905496 ].[55] P. Kotko, Wilson lines and gauge invariant off-shell amplitudes , JHEP (2014) 128,[ ].[56] A. van Hameren, KaTie : For parton-level event generation with k T -dependent initialstates , Comput. Phys. Commun. (2018) 371–380, [ ].[57] A. van Hameren, Calculating off-shell one-loop amplitudes for k T -dependentfactorization: a proof of concept , .[58] A. Dumitru, A. Hayashigaki and J. Jalilian-Marian, The Color glass condensate andhadron production in the forward region , Nucl. Phys. A765 (2006) 464–482,[ hep-ph/0506308 ].[59] M. L. Mangano and S. J. Parke, Multiparton amplitudes in gauge theories , Phys. Rept. (1991) 301–367, [ hep-th/0509223 ].[60] Z. Bern, L. Dixon and D. A. Kosower, New qcd results from string theory , tech. rep.,1993.[61] Z. Bern, G. Chalmers, L. Dixon and D. A. Kosower, One-loop n-gluon amplitudes withmaximal helicity violation via collinear limits , Physical review letters (1994) 2134.[62] G. Mahlon, Multigluon helicity amplitudes involving a quark loop , Physical Review D (1994) 4438.[63] Z. Bern and Y. tin Huang, Basics of generalized unitarity , Journal of Physics A: Mathematical and Theoretical (2011) 454003.[64] N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov andJ. Trnka, Grassmannian Geometry of Scattering Amplitudes . Cambridge UniversityPress, 2016, 10.1017/CBO9781316091548.[65] Z. Bern, L. Dixon and D. A. Kosower, One-loop corrections to five-gluon amplitudes , Physical Review Letters (1993) 2677.[66] Z. Bern, L. Dixon and D. A. Kosower, One-loop corrections to two-quark three-gluonamplitudes , Nuclear Physics B (1995) 259–304.[67] M. Nefedov, One-loop corrections to multiscale effective vertices in the eft formulti-regge processes in qcd , arXiv preprint arXiv:1905.01105 (2019) .[68] Z. Bern, L. J. Dixon and D. A. Kosower, Last of the finite loop amplitudes in qcd , Physical Review D (2005) 125003.[69] A. van Hameren, BCFW recursion for off-shell gluons , JHEP (2014) 138,[ ]. 1970] P. Kotko, K. Kutak, C. Marquet, E. Petreska, S. Sapeta and A. van Hameren, Improved TMD factorization for forward dijet production in dilute-dense hadroniccollisions , JHEP (2015) 106, [ ].[71] T. Altinoluk, R. Boussarie and P. Kotko, Interplay of the CGC and TMD frameworksto all orders in kinematic twist , JHEP (2019) 156, [1901.01175