Interplay of the CGC and TMD frameworks to all orders in kinematic twist
aa r X i v : . [ h e p - ph ] J a n Prepared for submission to JHEP
Interplay of the CGC and TMD frameworks to allorders in kinematic twist
Tolga Altinoluk, a Renaud Boussarie b,c and Piotr Kotko c a National Centre for Nuclear Research, 00-681 Warsaw, Poland b Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA c Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Krakow, Poland
E-mail: [email protected] , [email protected] , [email protected] Abstract:
A framework for an improved TMD (iTMD) factorization scheme at small x ,involving off-shell perturbative subamplitudes, was recently developed as an interpolationbetween the TMD k t ≪ Q regime and the BFKL k t ∼ Q regime. In this article, westudy the relation between CGC and iTMD amplitudes. We first show how the dipole-size expansion of CGC amplitudes resembles the twist expansion of a TMD amplitude.Then, by isolating kinematic twists, we prove that iTMD amplitudes are obtained withinfinite kinematic twist accuracy by simply getting rid of all genuine twist contributionsin a CGC amplitude. Finally we compare the amplitudes obtained via a proper kinematictwist expansion to those obtained via a more standard dilute expansion to show the relationbetween the iTMD framework and the dilute low x framework. ontents γ → q ¯ q in the CGC 62.3 TMD power corrections to γ → q ¯ q → process in the CGC 94 Dilute limit of a generic → process in the CGC 135 Small- x Improved TMD factorization (iTMD) 15 qg ∗ → qg channel 205.3 gg ∗ → qq channel 205.4 gg ∗ → gg channel 205.5 γg ∗ → qq channel 205.6 qg ∗ → qγ channel 21 q → qg channel 216.2 g → q ¯ q channel 246.3 g → gg channel 266.4 γ → q ¯ q channel 296.5 q → qγ channel 30 Factorization is one of the most crucial features of QCD: all perturbative QCD studiesrely on this separation between a hard partonic subamplitude and long distance matrixelements. This separation is justified in the presence of a sufficiently large scale Q in the– 1 –bservable, for which α s ( Q ) is small enough for perturbation theory to apply. Howeverlarge logarithms can arise from QCD dynamics and compensate the smallness of α s ( Q ) , which makes the resummation of such logarithms necessary.For most observables, two different factorization schemes can be employed, dependingon the center-of-mass energy s of the process. For processes with the center-of-mass energycomparable to the large scale of the process ( s ∼ Q ), collinear factorization is applied andthe large log( Q ) terms are resummed via the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi(DGLAP) evolution equations [1–3]. On the other hand, processes with the center-of-massenergy much larger than any other scale ( s ≫ Q ) are treated in the so-called low- x regime.In this case, k t -factorization applies and large log( s ) terms are resummed.Several descriptions of k T -factorization for low- x physics have been developed overthe last couple of decades, starting with the well known Balitsky-Fadin-Lipatov-Kuraev(BFKL) framework [4, 5]. The most recent low- x frameworks, namely the dipole model[6–8] and the shockwave framework [9–11] rely on a semi-classical approach, where low x gluon fields are treated as external fields. With such a treatment, all interactions with theexternal field can be resummed into path-ordered Wilson line operators which then consti-tute the building blocks of these low- x formalisms. Remarkably, due to this resummationof all interactions, perturbative results from this framework were found to be compatiblewith previous results for the semi-classical treatment of scattering off dense targets [12–14] which include gluon saturation effects from multiple scatterings. All of these recentframeworks are equivalent, and logarithms are resummed via the Balitsky/Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (B-JIMWLK) hierarchy of evolution equations[15–22], or in the mean field approximation by the Balitsky-Kovchegov (BK) equation[9, 23]. Nowadays, the weak coupling non-perturbative realization of the saturation inQCD is referred to as the Color Glass Condensate (CGC) [21, 22]. Throughout this paper,we refer to CGC as a unified picture (Balitsky formalism/ Mueller’s dipole picture/CGC)of the small- x QCD.The fact that the CGC generalizes the BFKL framework was established early onat Leading-Logarithmic (LL) accuracy [24, 25] and made more explicit in [26], then atNext-to-Leading-Logarithmic accuracy (NLL) in [27] and more explicitly in [28–30]. Thisequivalence relies on the expansion of the path-ordered Wilson lines in powers of the gluonfield for small values of gA , what is known as the dilute limit.Although it is not a true all-order factorization scheme, as opposed to collinear fac-torization for several simple processes [31], the CGC framework applies in principle toany low- x or high-density process regardless of the number of observed scales. In contrast,collinear factorization in its most common form is not valid for processes involving not onlya hard scale Q , but also a second, smaller scale. In the present context the most interestingcase is when that smaller scale is related to the transverse momentum of a parton insidea hadron. The collinear distributions were generalized for such processes, leading to theTransverse Momentum Dependent (TMD) factorization scheme [31–38].For a process with center-of-mass energy s , a hard scale Q , and a hard yet softertransverse momentum scale | k | ? Λ QCD , the respective application ranges of CGC andTMD schemes are s ≫ Q ? | k | and s ∼ Q ≫ | k | . A matching of these schemes in the– 2 –verlapping regime where | k | /Q and Q/s are both small was proven in [39, 40]. Sincethen, gluon TMDs in the CGC have been at stake in many recent studies (see for example[41, 46–48] ). Indeed the measurement of TMD parton distributions offers great insightin the 3D structure of hadrons, yet these distributions are not fully universal and thusthey require case-by-case studies. Studying them at low- x allows one to use standard CGCtools like the McLerran-Venugopalan (MV) model [12–14], Golec-Biernat-W¨usthoff (GBW)parametrization [49] or numerical solutions to the B-JIMWLK hierarchy of equations [41]for the description of these complicated TMD distributions.Notable attention has been drawn to polarized TMDs and to their role in angulardistributions at low- x [42–47], and the relation between process-independence breaking inTMD factorization and the Wilson lines which are natural built-in features of the CGC[48].On the other hand, the CGC framework in the so-called dilute limit also matches BFKLresults, which were built for processes with different kinematics, where s ≫ Q ∼ | k | . Anew scheme for TMD factorization at low- x , which is referred to as the improved TMDscheme (iTMD), was built in [50, 51] as an attempt to interpolate between both | k | ≪ Q and | k | ∼ Q limits. This framework aims at resumming some powers of | k | /Q by takinginto account non-zero k in the hard subamplitude. In practice, as we will show in thisarticle, it resums all kinematic twist corrections to the hard subamplitude which couples tothe leading-twist TMD operator, leaving genuine twist corrections aside. For an alternativeapproach for twist studies in the saturation regime, see [52].The purpose of this paper is to study the relation between CGC and iTMD amplitudes,with a comparison with dilute BFKL amplitudes as well. It is organized as follows. Insection 2, we consider the first corrections to the correlation limit in a CGC amplitude andcompare them to the first power corrections in the TMD factorization, and show how bothexpansions are related to one another. Then in section 3, we start with the most genericform for 1 → This leads to themain result of this article: a completely generic infinite-twist CGC amplitude in Eq. (3.10)in an all-body Wandzura-Wilczek approximation (i.e. where all genuine twist correctionsare neglected) . In section 4, we start again from the generic CGC amplitude and performa more standard dilute expansion, leading to a generic dilute CGC amplitude in Eq. (4.8).Section 5 is devoted to a short review of the iTMD framework and to recalculating theiTMD cross sections in a form that can be compared with the CGC all-kinematic-twistsresult. In section 6, we apply the generic kinematic twist resummed CGC result for differentprocesses and compare them to the iTMD predictions. We find a perfect match betweenthe kinematic twist resummed cross sections for each process and the corresponding iTMDresults. Moreover, we also compare the dilute limit of the generic CGC cross sections withthe kinematic twist ressumed cross sections by simply setting all distributions to the samevalue and find a perfect matching as well. Finally, in section 7 we summarize and discussour findings for this study. – 3 – otations and conventions
We define two lightlike vectors n and n such that n · n = 1, and light cone directions +and − such that n · k = k − , n · k = k + . The projectile (resp. target) is assumed to havea large momentum ∼ √ s along the + (resp. − ) direction. In the CGC calculations we usethe lightcone gauge A + = 0. Transverse components are denoted with a ⊥ subscript inMinkowski space and by bold characters in Euclidean space. Therefore, for two vectors k and x , we write k · x = k + x − + k − x + + k ⊥ · x ⊥ = k + x − + k − x + − k · x (1.1)The CGC part of this paper relies on the separation of the gluon fields in the QCD La-grangian depending on their + momentum between fast fields ( k + > e − Y p + ) and slow fields ( k + < e − Y p + ). In the eikonal approximation, the slow fields have the shockwaveform A µ ( x ) = δ ( x + ) B ( x ⊥ ) n µ + O ( s − / ) , (1.2)where B is a function of x ⊥ only. In the semi-classical approximation for the slow fields,treated as external fields for the projectile, interactions with the target are resummed intopath-ordered Wilson lines[ a + , b + ] x = P exp " ig Z b + a + dz + A − ( z + , , x ⊥ ) , (1.3)and we write U x = [ −∞ , + ∞ ] x . (1.4)CGC Wilson line operators carry a color representation, in which case we define U R x asthe Wilson line obtained from Eq. (1.3) by replacing A − ( x ) → T aR A − a ( x ). Finally, we usethe CGC brackets to describe the normalized forward actions of Wilson line operators ontarget states | P i . For an operator O we define the brackets as: hOi ≡ h P |O| P ih P | P i . (1.5) In this work we study processes that describe the production of a pair of particles with alarge invariant mass from a single particle in an external shockwave field built from thetarget gluons. We consider the case when both outgoing particles are tagged and theirtransverse momenta are fully reconstructed. The produced particles carry longitudinalmomenta p +1 and p +2 , and transverse momenta p and p . The two important combinationsof these momenta are the sum of the two transverse momenta kk ≡ p + p (2.1)– 4 –nd the transverse-boost invariant momentum q which is defined as q ≡ p +2 p − p +1 p p +1 + p +2 . (2.2)The hard scale Q of the process is given by the invariant mass of the outgoing pair whichis directly related to the transverse boost invariant momentum: Q = (cid:0) p +1 + p +2 (cid:1) p +1 p +2 q = q z ¯ z , (2.3)where z ≡ p +1 p +1 + p +2 ≡ − ¯ z . (2.4)As discussed in detail in [40], one can get the gluon TMDs through CGC calculationsin certain limit which is usually referred to as ”back-to-back correlation limit”. In thislimit, the two transverse scales | k | and | q | are well separated, i.e. | q | ≫ | k | . In theCGC framework, the transverse boost invariant momentum q is Fourier conjugate to thetransverse size of the produced pair (dipole size) r and the total transverse momentumis conjugate to the impact parameter b . Therefore, the back-to-back correlation limitcorresponds to the case | r | ≪ | b | in coordinate space allowing a Taylor expansion of theCGC observables in the dipole size r .We start by clarifying the power expansion employed here and in the rest of this sectionwe consider a simple process in the back-to-back correlation limit to utilize the small dipolesize expansion in the CGC framework and compare it with the power expansion in the TMDfactorization framework to clarify the relation between the two procedures. The TMD framework involves gauge invariant light ray operators [53], for which the dis-tinction between kinematic twists and genuine twists is convenient. For a set of gaugeinvariant twist p operators O ( i ) p associated with the hard part H ( i ) p , the n -th power of k ⊥ in the cross section is given by the sum over p ∈ { · · · n } of the p -th power in H ( i ) n − p convoluted with O ( i ) n − p and summed over all i .For inclusive observables, power corrections are split between amplitudes and complexconjugate amplitudes. However for the sake of this article, which aims at comparing CGCand iTMD results, it is actually sufficient to study power corrections at the amplitudelevel. Rather than using full, gauge invariant, inclusive operators, it is also enough for thecomparison to use ”half”-operators at the amplitude level, knowing how they would getcombined into gauge invariant inclusive operators at the cross section level.In the particular cases studied in this article, O ( i ) p will be a set of p -body gluon light ray Note that in a light ray OPE, the gauge links in the operators are not taken into account in the countingof twists. – 5 –alf-operators O ( i ) p ( x , ..., x p ) = [ ±∞ , x ] F − j ( x ) [ x , x ] F − j ( x ) ... [ x p − , x p ] F − j p ( x p ) [ x p , ±∞ ] . (2.5)We refer to O ( i ) p as a p -body operator, with O ( i )1 being the set of leading 1-body operators,which would combine into the leading twist (2-body in the standard counting) TMDs atthe cross section level. Then the n -th power correction is given by the sum of p -th power inthe ( n − p )-body hard part, convoluted with the ( n − p )-body operator. Corrections fromthe hard parts are kinematic twists, while higher-body operators lead to genuine twistcorrections. In particular, fully kinematic twists, that are the main focus of this study, aregiven by successive k ⊥ -derivatives of the 1-body hard part. γ → q ¯ q in the CGC It is informative to start by computing the first few corrections to the correlation limit inthe CGC. As a simple example, let us consider the amplitude for the photoproduction ofa quark-antiquark dijet which reads A γ → q ¯ q = (2 π ) δ (cid:0) p + q + p +¯ q − p + γ (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) r µ ⊥ r h(cid:16) U b +¯ z r U † b − z r (cid:17) − i φ µ (2.6)where the Wilson lines U b +¯ z r are defined in Eq. (1.4) with Eq. (1.3). Here, φ µ is the tensorpart of the amplitude that encodes the Dirac structure for this process and it is defined as φ ( γ → q ¯ q ) µ = i e q π ε σp ⊥ ¯ u p q [2 zg ⊥ µσ − ( γ ⊥ µ γ ⊥ σ )] γ + v p ¯ q . (2.7)In the correlation limit, it is straightforward to expand this amplitude in powers of thesmall dipole size r and keep the first two terms in the expansion. After performing asimple integration by parts, the result can be written as A γ → q ¯ q = i e q π ε lp (2 π ) δ (cid:0) p + q + p +¯ q − p + g (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) × r i r j r ¯ u p q (cid:16) zδ il + γ i γ l (cid:17) γ + v p ¯ q (cid:20) r k (cid:0) ∂ j U b (cid:1) (cid:16) ∂ k U † b (cid:17) (2.8)+ ¯ z (cid:0) ∂ j U b (cid:1) U † b (cid:18) i ¯ z k · r ) (cid:19) − zU b (cid:16) ∂ j U † b (cid:17) (cid:18) −
12 ( iz k · r ) (cid:19)(cid:21) .O (1) terms in Eq.(2.8) give the well-known back-to-back result which reads A ( b b ) γ → q ¯ q = i e q π ε lp (2 π ) δ (cid:0) p + q + p +¯ q − p + g (cid:1) R d b d r e − i ( q · r ) − i ( k · b ) × r i r j r ¯ u p q (cid:0) zδ il + γ i γ l (cid:1) γ + v p ¯ q h ¯ z (cid:0) ∂ j U b (cid:1) U † b − zU b (cid:16) ∂ j U † b (cid:17)i , (2.9)– 6 –hat has been proven to match the leading twist TMD amplitude. The rest of the termsare O ( r ) in Eq.(2.8) that are corrections to the back-to-back result: A ( nb b ) γ → q ¯ q = i e q π ε lp (2 π ) δ (cid:0) p + q + p +¯ q − p + g (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) × r i r j r ¯ u p q (cid:16) zδ il + γ i γ l (cid:17) γ + v p ¯ q (2.10) × h r k (cid:0) ∂ j U b (cid:1) (cid:16) ∂ k U † b (cid:17) + ¯ z (cid:0) ∂ j U b (cid:1) U † b ( i ¯ z k · r ) − zU b (cid:16) ∂ j U † b (cid:17) ( − ( iz k · r )) i . We would like to emphasize that the next-to-back-to-back term, Eq. (2.10), has a veryinteresting form. Noting the fact that a derivative acting on a CGC Wilson line extractsa gluon field, one can immediately conclude that the first term in the brackets is a 2-bodyhalf-operator. On the other hand, one can manipulate the last two terms using the factthat i ¯ z r l e − i ( q · r ) = − ∂∂ p lq e − i ( q · r ) , (2.11) − iz r l e − i ( q · r ) = − ∂∂ p l ¯ q e − i ( q · r ) , (2.12)so that the next-to-back-to-back term can be written as A ( nb b ) γ → q ¯ q = i e q π ε lp (2 π ) δ (cid:0) p + q + p +¯ q − p + g (cid:1) Z d b d r e − i ( k · b ) × r i r j r ¯ u p q (cid:16) zδ il + γ i γ l (cid:17) γ + v p ¯ q (2.13) × (cid:20) r k (cid:0) ∂ j U b (cid:1) (cid:16) ∂ k U † b (cid:17) − ¯ z (cid:0) ∂ j U b (cid:1) U † b (cid:18) k · ∂∂ p q (cid:19) + zU b (cid:16) ∂ j U † b (cid:17) (cid:18) k · ∂∂ p ¯ q (cid:19)(cid:21) e − i ( q · r ) . At this point we can make a diagram-by-diagram correspondence with TMD factorization.Naturally, (cid:0) ∂ j U b (cid:1) U † b terms correspond to the diagram where the TMD gluon hits thequark, while U b (cid:16) ∂ j U † b (cid:17) terms correspond to the diagram where it hits the antiquark. Forsuch diagrams, it is easy to see that the dependance on k and p q (resp. k and p ¯ q ) is only inthe intermediate quark (resp. antiquark) propagator G (cid:0) k + p q (cid:1) (resp. G (cid:0) k − p ¯ q (cid:1) ). Thusfor those diagrams we have k · ∂∂ p q = k · ∂∂ k , (2.14) k · ∂∂ p ¯ q = − k · ∂∂ k . (2.15)Hence, the next-to-back-to-back contribution can be cast into the following form: A ( nb b ) γ → q ¯ q = Z d b e − i ( k · b ) (cid:20)(cid:0) ∂ j U b (cid:1) (cid:16) ∂ k U † b (cid:17) H jk + (cid:0) ∂ j U b (cid:1) U † b (cid:18) k · ∂∂ k (cid:19) H j (cid:21) , (2.16)– 7 –here H jk is a 2-body hard subamplitude, and H j is a 1-body hard subamplitude (givenby the sum of the two diagrams discussed above). γ → q ¯ q For the photoproduction of a quark-antiquark dijet, the 1-body amplitude for TMD fac-torization has the following form: A ( k ) = ig Z d k (2 π ) (2 π ) δ ( k − k ) H i ( k ) Z db +1 d b e − i ( k · b ) (2.17) × (cid:2) −∞ , b +1 (cid:3) b F − i ( b ) (cid:2) b +1 , −∞ (cid:3) b , where H i ( k ) is a hard subamplitude. Power corrections are obtained via the Taylorexpansion of this hard part. Up to the first correction, rewriting the TMD operator as thederivative of a Wilson line, it reads: A ( k ) ≃ Z d b e − i ( k · b ) (cid:20) H i ( ) − (cid:18) k · ∂∂ k H i (cid:19) ( ) (cid:21) (cid:0) ∂ i U b (cid:1) U † b . (2.18)The 2-body amplitude for the same process can be written as A ( k ) = g Z d k (2 π ) d k (2 π ) (2 π ) δ ( k + k − k ) H ij ( k , k ) Z db +1 db +2 d b d b (2.19) × e − i ( k · b ) − i ( k · b ) (cid:2) −∞ , b +1 (cid:3) b (cid:8) F − i ( b ) [ b , b ] F − j ( b ) (cid:9) b − = b − =0 (cid:2) b +2 , −∞ (cid:3) b . Taking the leading term in the Taylor expansion of the hard part yields A ( k ) = g Z db +1 db +2 Z d b d b δ ( b − b ) e − i ( k · b ) H ij ( , ) × (cid:2) −∞ , b +1 (cid:3) b (cid:8) F − i ( b ) [ b , b ] F − j ( b ) (cid:9) b − = b − =0 (cid:2) b +2 , −∞ (cid:3) b . (2.20)Using the δ -function of the impact parameters b and b which sets these two transverse co-ordinates to the same value, one can rewrite the gauge link [ b , b ] as (cid:2) b +1 , + ∞ (cid:3) b (cid:2) + ∞ , b +2 (cid:3) b .This allows us to rewrite the operator as derivatives of Wilson lines and the leading termin the Taylor expansion of the 2-body amplitude for photoproduction of a quark-antiquarkdijet reads A ( k ) = Z d b e − i ( k · b ) H ij ( , ) (cid:0) ∂ i U b (cid:1) (cid:16) ∂ j U † b (cid:17) . (2.21)The comparison between Eqs. (2.18), (2.21) and the CGC result given in Eq. (2.16) shows astrong similarity between the small-dipole expansion in the CGC and the power expansionin the TMD framework. A more general matching could be conjectured. In this paper, weonly focus on kinematic twist corrections and compare the 1-body contributions from the We write the amplitude in an operator form, similarly to what is done in the CGC. The true amplitudeis given by the action of this operator on target states. – 8 –GC to those obtained in the TMD framework with infinite power accuracy via the iTMDscheme developed in [50]. Comparisons for higher-body terms are left for further studies. → process in the CGC In the previous section, we have calculated the next-to-back-to-back corrections for a spe-cific process ( γ → q ¯ q ) in the CGC framework and showed how one can isolate the 1-bodyand 2-body terms in this contribution. Our main goal in this section is to generalize thisprocedure to all orders in the small dipole size expansion. We isolate the 1-body contri-bution from the higher-body contributions, and then resum the 1-body contributions thatappear in higher orders in the small dipole size expansion.We would like to apply our results to several different 1 → → p → p p )process, we use the same longitudinal momentum fractions ( z and ¯ z ) introduced in Eq. (2.4),the total transverse momentum k of the produced particles defined in Eq. (2.1) and thetransverse boost invariant momentum q that is defined in Eq. (2.2). The generic CGCamplitude (see Fig. 1) in this case reads A → = (2 π ) δ (cid:0) p +1 + p +2 − p +0 (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) × r µ ⊥ r h(cid:16) U R b +¯ z r T R U R b − z r (cid:17) − (cid:16) U R b T R U R b (cid:17)i φ µ , (3.1)where φ µ is a Dirac structure which does not depend on coordinates, and ( R , R , R ) p , R b b + ¯ z rb − z r p , R p , R Figure 1 : Generic (0 →
12) process in an external shockwave field. The gray blobsrepresent the dressing of each line crossing it by Wilson line operators, resumming anynumber of eikonal scatterings with the external field.are color representations. This is a well known form in small- x kinematics: the interactionwith the target can be factorized out in the eikonal limit, and it contains all informationon color flow. The spin structure factorizes in the massless case due to transverse boost– 9 –nvariance: the mere topology of a diagram is sufficient to predict its momentum structure,or equivalently in coordinate space its dipole-size dependence. One can easily check thatthe amplitudes listed in Appendix B have the form of Eq. (3.1).The expression for the generic CGC amplitude for a 1 → n -th power of r is obtained by performing a Taylor series expansion of the Wilson lineoperators in A → which can be simply written as A ( n )0 → = (2 π ) δ (cid:0) p +1 + p +2 − p +0 (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) r µ ⊥ φ µ r (3.2) × n ! r α ⊥ ...r α n ⊥ n X m =0 nm ! ¯ z m ( − z ) n − m (cid:16) ∂ α ...∂ α m U R b (cid:17) T R (cid:16) ∂ α m +1 ...∂ α n U R b (cid:17) . The rest of our discussion relies on a symmetry hypothesis based on our experience ofBFKL and CGC amplitudes. In the CGC, diagrams with scattering only on one linegive ( U R − R ) R and R ( U R − R ) contributions, which once summed up withthe symmetric contribution ( U R − R )( U R − R ) lead to the gauge invariant dipole U R U R − R R . In BFKL computations, diagrams with one gluon on each line give theimpact factor ϕ ( k ⊥ , k ⊥ ) + ϕ ( k ⊥ , k ⊥ ) while diagrams with both gluons on one line givecounterterms − ϕ ( k ⊥ + k ⊥ , ⊥ ) and − ϕ (0 ⊥ , k ⊥ + k ⊥ ). The latter insure the cancellationof the full impact factor for k ⊥ = 0 ⊥ and for k ⊥ = 0 ⊥ and thus gauge invariance in theBFKL sense.By analogy, keeping in mind that one derivative equals one gluon in the TMD, we assumethat contributions with no derivative on one line must be a gauge-invariance restoring termfor the 1-body contributions, i.e. a kinematic twist, which we extract with the followingprocedure.We assume that the n -body contribution to the amplitude, for n >
1, does not containthe least symmetric Wilson line operators, in terms of derivatives. In other words, ourstatement is that no U ( ∂ i ...∂ i n U † ) or ( ∂ i ...∂ i n U ) U † term contributes to gauge invariantamplitudes. Operators with the least symmetric derivative structures need to be integratedby parts using Z d b e − i ( k · b ) (cid:16) ∂ α ...∂ α m U R b (cid:17) T R (cid:16) ∂ α m +1 ...∂ α n U R b (cid:17) (3.3)= Z d b e − i ( k · b ) h − ik ⊥ α n (cid:16) ∂ α ...∂ α m U R b (cid:17) T R (cid:16) ∂ α m +1 ...∂ α n − U R b (cid:17) − (cid:16) ∂ α ...∂ α m +1 U R b (cid:17) T R (cid:16) ∂ α m +2 ...∂ α n U R b (cid:17)i or the other way around, depending on which Wilson line has more derivatives acting on it.By employing this procedure, we make sure that the non-symmetric operators are reducedto a more symmetric contribution and a contribution with less derivatives acting on theWilson line operators. One can then proceed recursively in order to isolate all the 1-bodycontributions from the higher-body terms. However, we should emphasize that a strongerhypothesis is required in order to study genuine twist corrections, which are left for future– 10 –tudies. Nevertheless, as mentioned earlier in this study we focus on the kinematic twists.In order to clarify our discussion, let us consider the case for n = 4. The generic CGCamplitude for a 1 → O ( r ), after employing the proceduredescribed above, reads A → = (2 π ) δ (cid:0) p +1 + p +2 − p +0 (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) r µ ⊥ φ µ r × ( r α ⊥ " ¯ z (cid:16) ∂ α U R b (cid:17) T R U R b i ¯ z ( k · r )2! + ( i ¯ z ( k · r ))
3! + ( i ¯ z ( k · r )) ! − zU R b T R (cid:16) ∂ α U R b (cid:17) − iz ( k · r )2! + ( − iz ( k · r ))
3! + ( − iz ( k · r )) ! − r α ⊥ r α ⊥ (cid:16) ∂ α U R b (cid:17) T R (cid:16) ∂ α U R b (cid:17)
12! + − i ( z − ¯ z ) ( k · r )3! + ( − i ( z − ¯ z ) ( k · r )) ! + r α ⊥ r α ⊥ r α ⊥ (cid:20) z (cid:16) ∂ α U R b (cid:17) T R (cid:16) ∂ α ∂ α U R b (cid:17) (cid:18) − iz ( k · r ))4! (cid:19) (3.4) − ¯ z (cid:16) ∂ α ∂ α U R b (cid:17) T R (cid:16) ∂ α U R b (cid:17) (cid:18)
13! + 2 ( i ¯ z ( k · r ))4! (cid:19)(cid:21) + r α ⊥ r α ⊥ r α ⊥ r α ⊥ (cid:16) ∂ α ∂ α U R b (cid:17) T R (cid:16) ∂ α ∂ α U R b (cid:17) (cid:27) . As emphasized multiple times earlier, our aim in thus work is to study the (cid:16) ∂ α U R b (cid:17) T R U R b and U R b T R (cid:16) ∂ α U R b (cid:17) terms and perform an all-order dipole size resummation for them.This amounts to the Wandzura-Wilczek approximation for all twists [54]. Here after, wedenote all the amplitudes and the cross sections obtained from the CGC calculations byadopting the Wandzura-Wilczek approximation with the superscript W W . With our sym-metry argument, it is easy to obtain a generic form for the n -th power in the amplitude,by performing ( n −
1) integrations by parts on the least symmetric terms. Summing upsuch contributions for all n leads to A W W → = (2 π ) δ (cid:0) p +1 + p +2 − p +0 (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) r µ ⊥ φ µ r (3.5) × r α ⊥ " ¯ z (cid:16) ∂ α U R b (cid:17) T R U R b X n [ i ¯ z ( k · r )] n ( n + 1)! − zU R b T R (cid:16) ∂ α U R b (cid:17) X n [ − iz ( k · r )] n ( n + 1)! . It is now straightforward to perform the resummation explicitly which results in the fol-lowing form A W W → = (2 π ) δ (cid:0) p +1 + p +2 − p +0 (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) r µ ⊥ φ µ r (3.6) × r α ⊥ " ¯ z (cid:16) ∂ α U R b (cid:17) T R U R b e i ¯ z ( k · r ) − i ¯ z ( k · r ) − zU R b T R (cid:16) ∂ α U R b (cid:17) e − iz ( k · r ) − − iz ( k · r ) . – 11 –he integral over the dipole size r is factorized from the rest of the expression and can beperformed explicitly by considering the following integral I ij ( p ) ≡ Z d d r r i r j r e − i ( p · r ) − p · r ) e − i ( q · r ) , (3.7)for p = ¯ z k or p = − z k . The details of the calculation can be found in Appendix C andthe result reads I ij ( p ) = − iπ p (cid:16) p i δ jl + p j δ il − p l δ ij (cid:17) (cid:18) q l + p l ( q + p ) − q l q (cid:19) . (3.8)Plugging this result into Eq. (3.6) and reintroducing the transverse momenta of the pro-duced particles ( p , p ) leads to the final expression for the generic CGC amplitude for a1 → A W W → = (2 π ) δ (cid:0) p +1 + p +2 − p +0 (cid:1) Z d b e − i ( k · b ) φ i k (cid:16) k i δ jl + k j δ il − k l δ ij (cid:17) × (cid:20)(cid:18) q l q + p l p (cid:19) (cid:16) ∂ j U R b (cid:17) T R U R b + (cid:18) q l q − p l p (cid:19) U R b T R (cid:16) ∂ j U R b (cid:17)(cid:21) . (3.9)Using the generic CGC amplitude given in Eq. (3.9), the generic cross section can becalculated in a straightforward manner and the result reads dσ W W → dy dy d p d p = (2 π )16 C p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) (cid:16) φ i φ i ′ ∗ (cid:17) Z d b (2 π ) d b ′ (2 π ) e i k · ( b ′ − b ) × k (cid:16) k i δ jl + k j δ il − k l δ ij (cid:17) (cid:16) k i ′ δ j ′ l ′ + k j ′ δ i ′ l ′ − k l ′ δ i ′ j ′ (cid:17) × ((cid:18) q l q + p l p (cid:19) q l ′ q + p l ′ p ! D Tr h(cid:16) ∂ j U R b (cid:17) T R U R b U R † b ′ T R † (cid:16) ∂ j ′ U R † b ′ (cid:17)iE + (cid:18) q l q + p l p (cid:19) q l ′ q − p l ′ p ! D Tr h(cid:16) ∂ j U R b (cid:17) T R U R b (cid:16) ∂ j ′ U R † b ′ (cid:17) T R † U R † b ′ iE (3.10)+ (cid:18) q l q − p l p (cid:19) q l ′ q + p l ′ p ! D Tr h U R b T R (cid:16) ∂ j U R b (cid:17) U R † b ′ T R † (cid:16) ∂ j ′ U R † b ′ (cid:17)iE + (cid:18) q l q − p l p (cid:19) q l ′ q − p l ′ p ! D Tr h U R b T R (cid:16) ∂ j U R b (cid:17) (cid:16) ∂ j ′ U R † b ′ (cid:17) T R † U R † b ′ iE) , where the factor C originates from the spin and color averaging over the incoming stateand h· · · i is defined in Eq. (1.5). The color Fierz factor C is N c for a quark, (cid:0) N c − (cid:1) fora gluon and 1 for a photon.Eq. (3.10) is the main result of this paper. It is the generic CGC cross section for a1 → → → process in the CGC For very high values of the center-of-mass energy s or for dense targets, multiple scatteringsare expected to occur. In practice, for values of | k | of the order of the target saturation scale Q s ∼ ( A/x ) / , it is expected for the target fields A − to scale like 1 /g due to a high gluonoccupation number, so that gA − must be resummed into the path-ordered Wilson lineoperators U R b which are the natural building blocks of the CGC or shockwave formalisms.The regime where | k | ≫ Q s , is referred to as the dilute limit. In this limit gA − isexpected to be small and therefore one is allowed to expand Wilson line operators in gluonfields (or in Reggeon fields for more involved analysis, as [26, 55]) or equivalently to use adilute formalism like BFKL.In this section, we consider the dilute limit of the CGC by expanding the Wilson lineoperators in the generic CGC amplitude for a 1 → g , in arbitrary representation R reads U R x = 1 + igT aR Z dx + A − a (cid:0) x + , , x (cid:1) + O (cid:0) g (cid:1) . (4.1)with T aR being the SU ( N c ) generator in the representation R . Then, in the dilute limit,the generic CGC amplitude given in Eq. (3.1) can be written as A gA ∼ → = ig (2 π ) δ (cid:0) p +1 + p +2 − p +0 (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) r µ ⊥ r φ µ × Z dz + (cid:8) T R T aR (cid:2) A − a (cid:0) z + , , b − z r (cid:1) − A − a (cid:0) z + , , b (cid:1)(cid:3) (4.2)+ T aR T R (cid:2) A − a (cid:0) z + , , b + ¯ z r (cid:1) − A − a (cid:0) z + , , b (cid:1)(cid:3)(cid:9) . After introducing the incoming target state P and the target remnant states X , and usingthe translation invariance of the h X | ( ... ) | P i matrix elements, one can easily integrate overthe impact parameter which yields to the following form of the matrix element: D X (cid:12)(cid:12)(cid:12) A gA ∼ → (cid:12)(cid:12)(cid:12) P E = ig (2 π ) δ ( k + P X − P − p ) Z d r e − i ( q · r ) r µ ⊥ r φ µ (cid:10) X (cid:12)(cid:12) A − a (0) (cid:12)(cid:12) P (cid:11) × h T R T aR (cid:16) e − iz ( k · r ) − (cid:17) + T aR T R (cid:16) e i ¯ z ( k · r ) − (cid:17)i . (4.3)In Eq. (4.3), the integral over the dipole size r can be performed in a straightforward– 13 –anner by using the well known integral Z d r r µ ⊥ r e − i ( ℓ · r ) = − iπ ℓ µ ⊥ ℓ , (4.4)which finally leads to the following form of the dilute amplitude D X (cid:12)(cid:12)(cid:12) A gA ∼ → (cid:12)(cid:12)(cid:12) P E = 2 πg (2 π ) δ ( k + P X − P − p ) (cid:10) X (cid:12)(cid:12) A − a (0) (cid:12)(cid:12) P (cid:11) (4.5) × (cid:20) T R T aR (cid:18) p µ ⊥ p − q µ ⊥ q (cid:19) − T aR T R (cid:18) p µ ⊥ p + q µ ⊥ q (cid:19)(cid:21) φ µ . The cross section in the dilute limit can be easily obtained from Eq. (4.5), and the resultreads dσ gA ∼ → dy dy d p d p = α s s δ (cid:0) p +1 + p +2 − p +0 (cid:1) Z db + d b (2 π ) e − i ( k · b ) (cid:10) P (cid:12)(cid:12) A − c ( b ) A − a (0) (cid:12)(cid:12) P (cid:11) b − =0 (cid:0) φ i φ j ∗ (cid:1) × Tr (cid:26)(cid:20) T R T aR (cid:18) p i p − q i q (cid:19) − T aR T R (cid:18) p i p + q i q (cid:19)(cid:21) (4.6) × " T c † R T † R p j p − q j q ! − T † R T c † R p j p + q j q ! . Finally, it is customary to introduce the unintegrated parton distribution function (uPDF) G ( k ) that is defined as Z db + Z d b (2 π ) e − i ( k · b ) (cid:10) P (cid:12)(cid:12) A − a ( b ) A − c (0) (cid:12)(cid:12) P (cid:11) = (2 π ) P − G ac ( k ) k (4.7)with δ ac G ac ( k ) = G ( k ) . Averaging over the spin and color states of the incoming parton or photon, we arrive tothe generic form of the cross section in the dilute limit: dσ gA ∼ → dy dy d p d p = (2 π )16 C p +0 α s δ (cid:0) p +1 + p +2 − p +0 (cid:1) (cid:0) φ i φ j ∗ (cid:1) G ac ( k ) k × ((cid:18) q i q + p i p (cid:19) q j q + p j p ! Tr (cid:16) T aR T R T † R T c † R (cid:17) + (cid:18) q i q + p i p (cid:19) q j q − p j p ! Tr (cid:16) T aR T R T c † R T † R (cid:17) (4.8)+ (cid:18) q i q − p i p (cid:19) q j q + p j p ! Tr (cid:16) T R T aR T † R T c † R (cid:17) + (cid:18) q i q − p i p (cid:19) q j q − p j p ! Tr (cid:16) T R T aR T c † R T † R (cid:17)) , – 14 –ith C being the factor that one obtains via color averaging, as introduced previously insection 3. We would like to draw attention to the similarity between dilute limit of thegeneric cross section given in Eq. (4.8) and the kinematic-twist-resummed cross sectiongiven in Eq. (3.10). We discuss the implications of this similarity in section 7. x Improved TMD factorization (iTMD)
In the following section we briefly recall the small- x improved TMD factorization con-structed in [50]. Although the framework is more general, here we focus on dijets in pA and γA collisions. This section is organized as follows. We first list and explain the generalform of the formulas for dijets in pA collisions. Next, we shall put the iTMD formulationinto the context of the TMD factorization theorems to better clarify the terminology. Inthe end of this section, we shall give the formulas for the cross section for all channels ina form that can be compared with the CGC framework. The iTMD factorization formula for pA collisions has the form of a hybrid generalized k T -factorization. That is: (i) the incoming dilute projectile is described by the collinearPDF as it is probed at large x – so called hybrid approach [56], (ii) the target is probed atsmall x and is described by a set of process-dependent TMD gluon distributions, (iii) thehard factors are constructed from off-shell gauge invariant matrix elements. Thanks to (i),the formula for the cross section can be written as dσ pA → j + X = X q f q/H ⊗ dσ qA → qg + f g/H ⊗ [ dσ gA → gg + n f dσ gA → qq ] , (5.1)where f a/H is the collinear PDF for parton a = q, g (we can safely neglect antiquarks in thisapproximation), ⊗ denotes the convolution in the longitudinal fraction x p of the protonmomentum carried by parton a , n f is the number of flavors. The remaining objects arecross sections for scattering a parton a off the target to produce the given final states.They can be generically written as follows: dσ aA → bc d p d p dy dy = p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) s X i =1 , ˜ H ( i ) ag ∗ → bc ( p , p , z ) Φ ( i ) ag → bc ( x A , k ) , (5.2)where ¯ s = x p x A s , ˜ H ( i ) ag ∗ → bc are off-shell gauge invariant hard factors and Φ ( i ) ag → bc are un-polarized TMD gluon distributions in the target. The sum over i corresponds to twoinequivalent color flows that exist for each channel.The TMD gluon distributions Φ ( i ) ag → bc are linear combinations [50] (Table 1) of thebasic distributions with the following operator definitions [38]: F (1) qg ( x, | k | ) = 2 Z dξ + d ξ (2 π ) P − e ixP − ξ + − i k · ξ h P | Tr h ˆ F i − ( ξ ) U [ − ] † ˆ F i − (0) U [+] i | P i , (5.3)– 15 – (2) qg ( x, | k | ) = 2 Z dξ + d ξ (2 π ) P − e ixP − ξ + − i k · ξ h P | Tr (cid:2) U [ (cid:3) ] (cid:3) N c Tr h ˆ F i − ( ξ ) U [+] † ˆ F i − (0) U [+] i | P i , (5.4) F (1) gg ( x, | k | ) = 2 Z dξ + d ξ (2 π ) P − e ixP − ξ + − i k · ξ h P | Tr (cid:2) U [ (cid:3) ] † (cid:3) N c Tr h ˆ F i − ( ξ ) U [ − ] † ˆ F i − (0) U [+] i | P i , (5.5) F (2) gg ( x, | k | ) = 2 Z dξ + d ξ (2 π ) P − e ixP − ξ + − i k · ξ N c h P | Tr h ˆ F i − ( ξ ) U [ (cid:3) ] † i Tr h ˆ F i − (0) U [ (cid:3) ] i | P i , (5.6) F (3) gg ( x, | k | ) = 2 Z dξ + d ξ (2 π ) P − e ixP − ξ + − i k · ξ h P | Tr h ˆ F i − ( ξ ) U [+] † ˆ F i − (0) U [+] i | P i , (5.7) F (4) gg ( x, | k | ) = 2 Z dξ + d ξ (2 π ) P − e ixP − ξ + − i k · ξ h P | Tr h ˆ F i − ( ξ ) U [ − ] † ˆ F i − (0) U [ − ] i | P i , (5.8) F (5) gg ( x, | k | ) = 2 Z dξ + d ξ (2 π ) P − e ixP − ξ + − i k · ξ h P | Tr h ˆ F i − ( ξ ) U [ (cid:3) ] † U [+] † ˆ F i − (0) U [ (cid:3) ] U [+] i | P i , (5.9) F (6) gg ( x, | k | ) = 2 Z dξ + d ξ (2 π ) P − e ixP − ξ + − i k · ξ h P | Tr (cid:2) U [ (cid:3) ] (cid:3) N c Tr (cid:2) U [ (cid:3) ] † (cid:3) N c Tr h ˆ F i − ( ξ ) U [+] † ˆ F i − (0) U [+] i | P i , (5.10)with ˆ F ( ξ ) = t a F a ( ξ + , ξ − = 0 , ξ ). The staple-like Wilson lines appearing above are definedas U [ ± ] = (cid:2)(cid:0) + , − , (cid:1) , (cid:0) ±∞ , − , (cid:1)(cid:3)(cid:2)(cid:0) ±∞ , − , (cid:1) , (cid:0) ±∞ , − , ξ (cid:1)(cid:3) (cid:2)(cid:0) ±∞ , − , ξ (cid:1) , (cid:0) ξ + , − , ξ (cid:1)(cid:3) . (5.11)The Wilson loop is made from two staples glued together: U [ (cid:3) ] = U [ − ] † U [+] . (5.12)The off-shell gauge invariant hard factors ˜ H ( i ) involve incoming off-shell gluons withmomentum k = x A P + k ⊥ , k = − k , coupled eikonally to the target via a TMD correlator.In general, such Feynman diagrams are not gauge invariant when calculated using thestandard QCD Feynman rules. There are several ways, to deal with this. First, one coulduse the Lipatov effective action and resulting vertices in the quasi-multi-Regge kinematics[70]. In [71–74] other methods have been developed, based on the spinor helicity formalism,especially convenient to deal with multiparticle processes and to guarantee fast computerimplementation. The method [72] has been recently extended to loop level [75]. The easiestway to understand the diagrammatic content of the hard factors is probably provided bythe method [76] which defines the gauge invariant off-shell amplitudes as partonic matrixelements of straight infinite Wilson line operators. In case of the hard factors involving– 16 – ( i ) gg ∗ → gg N c (cid:0) N c F (1) gg − F (3) gg + F (4) gg + F (5) gg + N c F (6) gg (cid:1) N c (cid:0) N c F (2) gg − F (3) gg + F (4) gg + F (5) gg + N c F (6) gg (cid:1) Φ ( i ) gg ∗ → qq N c − (cid:16) N c F (1) gg − F (3) gg (cid:17) − N c F (2) gg + F (3) gg Φ ( i ) qg ∗ → qg F (1) qg N c − (cid:16) −F (1) qg + N c F (2) qg (cid:17) Φ ( i ) γg ∗ → qq F (3) gg —Φ ( i ) qg ∗ → γq F (1) qg — Table 1 : The TMD gluon distributions corresponding to the hard factors ˜ H ( i ) .one off-shell gluon needed here the Wilson line has a direction along P − . The diagramscontributing to each channel for pA collisions are given in Fig. 2.The form of the generalized factorization (5.2) appears as follows. First the colorstructure is separated from the kinematic part of the amplitude by means of the colordecomposition [77]. The amplitudes with the color structure separated contain only planardiagrams with fixed ordering of the external legs. The TMD gluon distributions Φ ( i ) ag → bc arederived for the color structures (squared) following the general procedure of resummationof collinear gluons constructed in [38]. The color decomposition of amplitudes guaranteesthat each Φ ( i ) ag → bc corresponds to a gauge invariant subset of diagrams. For more detailsand application to multiparticle processes see [78].The iTMD formula was constructed to agree with the k T -factorization for dijet pro-duction [79] in the limit of k ∼ Q ≫ Q s and also with the leading power limit of theCGC expressions [40] for Q ≫ k ∼ Q s . In the present paper we further compare allthe power corrections contained in the framework. To this end, we need the small x limitof the TMD gluon distributions compliant with the CGC theory. They are obtained byneglecting the x dependence in the Fourier transforms and trading the hadronic matrixelements to the averages over the color distributions in the nucleus. In addition, lightconegauge is used, in which for the shockwave approximation the transverse components of the– 17 – g ∗ → qg : k gg ∗ → q ¯ q : k k k gg ∗ → gg : k k kk k k Figure 2 : Diagrams contributing to the gauge invariant hard factors ˜ H ( i ) ag ∗ → bc for variouschannels. We show only planar color-ordered diagrams, i.e. the planar diagrams with fixedordering of external legs, as they are enough to reconstruct the hard factors contributingto the in-equivalent color flows (see Section 6 of [50] on how to reconstruct the hard factorsfrom color-ordered amplitudes and [77] for a general review of color decompositions). Theoff-shell gluon has momentum k . The double line corresponds to the Wilson line propagatorin momentum space, which couples to gluons via the igt a P µ vertex. The double linepropagator with a momentum p is − i/ ( p · P + iǫ ). These diagrams have to be multipliedby k /g – for all the details see [76]. We do not display the diagrams for processes witha photon since they do not require the use of a Wilson line, despite the off-shellnes of thegluon.gauge fields do not contribute due to EOM. This allows to neglect the transverse parts ofthe staple gauge links (5.11). Within the above approximation we have [40, 41]: F (1) qg = 4 g Z d x d y (2 π ) e − i k · ( x − y ) D Tr n ( ∂ i U y ) (cid:16) ∂ i U † x (cid:17)oE , (5.13) F (2) qg = − g Z d x d y (2 π ) e − i k · ( x − y ) N c D Tr n ( ∂ i U x ) U † y (cid:16) ∂ i U † y (cid:17) U † x o Tr n U y U † x oE . (5.14) F (1) gg = 4 g Z d x d y (2 π ) e − i k · ( x − y ) N c D Tr n ( ∂ i U y ) (cid:16) ∂ i U † x (cid:17)o Tr n U x U † y oE , (5.15)– 18 – (2) gg = − g Z d x d y (2 π ) e − i k · ( x − y ) N c D Tr n ( ∂ i U x ) U † y o Tr n ( ∂ i U y ) U † x oE , (5.16) F (3) gg = − g Z d x d y (2 π ) e − i k · ( x − y ) D Tr n ( ∂ i U x ) U † y ( ∂ i U y ) U † x oE , (5.17) F (4) gg = − g Z d x d y (2 π ) e − i k · ( x − y ) D Tr n ( ∂ i U x ) U † x ( ∂ i U y ) U † y oE , (5.18) F (5) gg = − g Z d x d y (2 π ) e − i k · ( x − y ) D Tr n ( ∂ i U x ) U † y U x U † y ( ∂ i U y ) U † x U y U † x oE , (5.19) F (6) gg = − g Z d x d y (2 π ) e − i k · ( x − y ) N c D Tr n ( ∂ i U x ) U † y ( ∂ i U y ) U † x o Tr n U x U † y o Tr n U y U † x oE . (5.20)For completeness, let us now put the iTMD formulation into the context of the formalTMD factorization theorems [31]. First, one should understand that it does not involvean all-order factorization theorem like the ones existing for the Drell-Yan process andsemi-inclusive DIS. These theorems are proved to leading power in the hard scale to anylogarithmic accuracy, while the iTMD framework resums the power corrections, but itsvalidity is limited to leading logarithms of energy. Next, the mentioned TMD factorizationtheorems involve processes with at most two colored partons in the hard process (plussoft/collinear contributions of course) and two TMD correlators (parton distribution orfragmentation function). Because of the simplicity of the color structure, all Wilson linesappearing due to the resummation of collinear gluons can be disentangled and put into thegauge invariant definitions of the TMD objects. For jet production processes in hadron-hadron collision, where formally one has at least two TMD correlators and more than twocolored partons, it is not possible. Thus, formally, even the generalized factorization breaksdown [80]. However, in the iTMD approach, which targets the collisions of a moderate- x projectile and a low- x target, there is only one TMD correlator, thus, at least formally, thisproblem does not occur. On the formal ground there is no all-order proof of the hybridapproach so far.Finally, let us comment on the evolution equations for the TMD gluon distributions.The most adequate treatment would be using the renormalization group equation at smalland moderate x developed in [81, 82]. It however still requires work to derive the completeset of equations, not to mention solving them. An important feature of such procedurewould be that some Sudakov logarithms ln k can be consistently resummed. For exist-ing phenomenological applications using iTMD [51, 83] the evolution was based on BK orB-JIMWLK and some Sudakov resummation effects were estimated by means of a phe-nomenological model. – 19 –elow, we explicitly give formulas for the cross sections (5.2) in a form that can bedirectly compared with the CGC expressions. qg ∗ → qg channel We get dσ qA → qg d p d p dy dy = p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) h H (1) qg → qg Φ (1) qg → qg + H (2) qg → qg Φ (2) qg → qg i , (5.21)with H (1) qg → qg = α s z (cid:0) z (cid:1) q ( z p + 1 N c q − z p z p p ) , (5.22) H (2) qg → qg = α s N A N c z (cid:0) z (cid:1) p p . (5.23)Note, that the above hard factors H ( i ) qg → qg are not exactly the ones in (5.2). The expressionsare however more compact in the above notation. gg ∗ → qq channel dσ gA → qq d p d p dy dy = p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) h H (1) gg → qq Φ (1) gg → qq + H (2) gg → qq Φ (2) gg → qq i , (5.24)where H (1) gg → qq , H (2) gg → qq are the reduced off-shell hard factors. They read H (1) gg → qq = α s N c zz (1 − zz ) p (1 − z ) + p z q p p , (5.25) H (2) gg → qq = α s N c C F ( zz ) (1 − zz ) ( p · p ) q p p . (5.26) gg ∗ → gg channel dσ gA → gg d p d p dy dy = p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) h H (1) gg → gg Φ (1) gg → gg + H (2) gg → gg Φ (2) gg → gg i , (5.27)with H (1) gg → gg = α s N c N A (1 − zz ) p z + p z q p p , (5.28) H (2) gg → gg = α s N c N A (1 − zz ) q − p z − p z q p p . (5.29)Above, an additional symmetry factor of 1 / γg ∗ → qq channel dσ γA → qq d p d p dy dy = p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) H γg ∗ → gg F (3) gg , (5.30)– 20 –ith H γg ∗ → gg = α em α s zz (1 − zz ) p p . (5.31) qg ∗ → qγ channel dσ qA → qγ d p d p dy dy = p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) H qg ∗ → qγ F (1) qg , (5.32)with H qg ∗ → qγ = α em α s C A zz (cid:0) z (cid:1) q p . (5.33) In sections 3 and 4, we have computed both the kinematic-twist-resummed cross sectionand the dilute limit of the CGC cross section for a generic 1 → q → qg channel Let us start our analysis by considering the q → qg channel (see Fig. 3). In this channel,the incoming quark splits into a quark-gluon pair at order g s which then scatters off thetarget via eikonal interaction. The CGC amplitude for this channel is given in Eq. (B.1).In order to be able to use the kinematic twist resummed generic cross section Eq. (3.10),the first thing we need is the tensor part of the amplitude that encodes the Dirac structureof this channel and it is given by φ ( q → qg ) µ = ig s π ε σ ∗ p g ⊥ ¯ u p q [2 zg ⊥ µσ + ¯ z ( γ ⊥ µ γ ⊥ σ )] γ + u p , (6.1)– 21 –hose square for an unpolarized observable can be calculated in a straightforward mannerand the result reads φ i ( q → qg ) φ i ′ ∗ ( q → qg ) = δ ii ′ (cid:16) g s π (cid:17) (cid:0) p +0 (cid:1) z (cid:0) z (cid:1) . (6.2)One can read off the color structure in this channel from Fig. 3 and it is given by setting p , R = F b b + ¯ z rb − z r p , R = Fp , R = Adj Figure 3 : q → qg amplitude in an external shockwave background with the appropriatecolor representations. U R b = U b , U R b = U ab b and T R = T b . This color structure leads to the following TMDoperators O ( q → qg )1 = (cid:0) ∂ j U b (cid:1) T b U ab b U ac † b ′ T c (cid:16) ∂ j ′ U † b ′ (cid:17) O ( q → qg )2 = (cid:0) ∂ j U b (cid:1) T b U ab b (cid:16) ∂ j ′ U ac b ′ (cid:17) T c U † b ′ (6.3) O ( q → qg )3 = U b T b (cid:16) ∂ j U ab b (cid:17) U ac † b ′ T c (cid:16) ∂ j ′ U † b ′ (cid:17) O ( q → qg )4 = U b T b (cid:16) ∂ j U ab b (cid:17) (cid:16) ∂ j ′ U ac b ′ (cid:17) T c U † b ′ . By using the identity that relates the adjoint and fundamental representations of a unitarymatrix U ab ( b ) = 2 Tr (cid:2) t a U ( b ) t b U † ( b ) (cid:3) (6.4)and the Fierz identity t aαβ t aσλ = 12 (cid:20) δ αλ δ βσ − N c δ αβ δ σλ (cid:21) (6.5)one can easily get the following identities T b (cid:16) ∂ j U ab b (cid:17) = (cid:16) ∂ j U † b (cid:17) T a U b + U † b T a (cid:0) ∂ j U b (cid:1) (6.6) (cid:16) ∂ j ′ U ac b ′ (cid:17) T c † = (cid:16) ∂ j ′ U † b ′ (cid:17) T a U b ′ + U † b ′ T a (cid:16) ∂ j ′ U b ′ (cid:17) , The next step is to compute the color trace of the TMD operators that are listed in Eq.(6.3). By using the identities given in Eq. (6.6), these traces can easily be computed and– 22 –he result readsTr h O ( q → qg )1 i = −
12 Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i Tr (cid:16) U b U † b ′ (cid:17) − N c Tr h(cid:0) ∂ j U b (cid:1) (cid:16) ∂ j ′ U † b ′ (cid:17)i Tr h O ( q → qg )2 i = 12 Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i Tr (cid:16) U b U † b ′ (cid:17) (6.7)Tr h O ( q → qg )3 i = 12 Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i Tr (cid:16) U b U † b ′ (cid:17) Tr h O ( q → qg )4 i = −
12 Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i Tr (cid:16) U b U † b ′ (cid:17) + N c h(cid:0) ∂ j U b (cid:1) (cid:16) ∂ j ′ U † b ′ (cid:17)i . Comparing the structure of the trace of the Wilson lines in Eq. (6.7) and the definitionsof the first two gluon TMDs in the quark channel F (1) qg and F (2) qg given in Eqs. (5.13) and(5.14) respectively, one can conclude that these are the two gluon TMDs which appear inthis channel. Moreover, for convenience, we can define the following combinations of thegluon TMDs F (1) qg and F (2) qg :Φ (1) q → qg ( k ) ≡ F (1) qg ( k ) (6.8)Φ (2) q → qg ( k ) ≡ N c F (2) qg ( k ) − F (1) qg ( k ) N c − q → qg channel as dσ W Wq → qg dy dy d p d p = α s z (cid:0) z (cid:1) p p p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) (6.9) × (cid:20)(cid:18) z p q + 1 N c (cid:18) − ¯ z p q (cid:19)(cid:19) Φ (1) q → qg ( k ) + (cid:18) N c − N c (cid:19) Φ (2) q → qg ( k ) (cid:21) which coincides exactly with Eq. (5.21) by using Eqs. (5.22) and (5.23).Our next order of business is to consider the dilute limit in the q → qg channel.Inserting the proper color representations in the dilute limit of the generic cross sectiongiven in Eq. (4.8), we get dσ gA ∼ q → qg dy dy d p d p = α s N c (cid:0) p +0 (cid:1) (2 π ) G ac ( k ) k (cid:0) φ i φ j ∗ (cid:1) p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) (6.10) × Tr (cid:20) if bad t d (cid:18) p i p − q i q (cid:19) − t a t b (cid:18) p i p + q i q (cid:19)(cid:21) " − if bce t e p j p − q j q ! − t b t c p j p + q j q ! , with G ac ( k ) being the unintegrated parton distribution function defined in Eq. (4.7). Using– 23 –he definition of the tensor part of the amplitude that encodes the Dirac structure in the q → qg channel given in Eq. (6.1) and performing some color algebra, one simply gets thedilute limit of the cross section in this channel: dσ gA ∼ q → qg dy dy d p d p = α s z (cid:0) z (cid:1) p p p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) G ( k ) (cid:18) z p q − N c ¯ z p q (cid:19) . (6.11)From Eq. (6.9) and (6.11), we also get a straightforward matching between the improvedTMD scheme and the dilute scheme: σ gA ∼ q → qg = σ W Wq → qg (cid:12)(cid:12) Φ (1) q → qg =Φ (2) q → qg = G . (6.12)The substitution Φ (1) q → qg = Φ (2) q → qg = G in the iTMD scheme in the dilute limit can besimply justified as follows. For | k | ≫ Q s and large, the Fourier transforms in the operatordefinitions force the transverse separation between the fields to be small. In that limitthe gauge links become identical, while the Wilson loops become trivial. This universalbehaviour was tested numerically in [51] and [41]. g → q ¯ q channel The next channel we consider is g → q ¯ q . In this channel, the incoming gluon splits intoa quark-antiquark pair at order g s , then it scatters through the target (see Fig. 4). TheCGC amplitude for this channel is given in Eq. (B.2) and the tensor part of it reads φ ( g → q ¯ q ) µ = − i g s π ε σp ⊥ ¯ u p q [2 zg ⊥ µσ − ( γ ⊥ µ γ ⊥ σ )] γ + v p ¯ q , (6.13)whose square can be computed easily for an unpolarized observable: φ i ( g → q ¯ q ) φ i ′ ∗ ( g → q ¯ q ) = δ ii ′ (cid:16) g s π (cid:17) (cid:0) p +0 (cid:1) z ¯ z (cid:0) z + ¯ z (cid:1) (6.14)The color structure of this channel can be read off from Fig. 4 and it is given by setting p , R = Adj b b + ¯ z rb − z r p , R = Fp , R = F ∗ Figure 4 : g → q ¯ q amplitude in an external shockwave background with the appropriatecolor representations. – 24 – R b = U b , U R b = U † b and T R = T b . This color structure leads to the following gluonTMD operators that appears in the generic kinematic twist resummed cross section givenin Eq. (3.10): O ( g → q ¯ q )1 = (cid:0) ∂ j U b (cid:1) T b U † b U b ′ T b (cid:16) ∂ j ′ U † b ′ (cid:17) O ( g → q ¯ q )2 = (cid:0) ∂ j U b (cid:1) T R U † b (cid:16) ∂ j ′ U b ′ (cid:17) T b U b ′ (6.15) O ( g → q ¯ q )3 = U b T b (cid:16) ∂ j U † b (cid:17) U b ′ T b (cid:16) ∂ j ′ U † b ′ (cid:17) O ( g → q ¯ q )4 = U b T b (cid:16) ∂ j U † b (cid:17) (cid:16) ∂ j ′ U b ′ (cid:17) T b U † b ′ One can easily compute the trace over the color indexes of the operators listed in Eq. (6.15)and the result readsTr h O ( g → q ¯ q )1 i = 12 Tr h(cid:0) ∂ j U b (cid:1) (cid:16) ∂ j ′ U † b ′ (cid:17)i Tr (cid:16) U b ′ U † b (cid:17) + 12 N c Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i Tr h O ( g → q ¯ q )2 i = 12 Tr h(cid:0) ∂ j U b (cid:1) U † b ′ i Tr h U † b (cid:16) ∂ j ′ U b ′ (cid:17)i − N c Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i (6.16)Tr h O ( g → q ¯ q )3 i = 12 Tr h(cid:16) ∂ j U † b (cid:17) U b ′ i Tr h U b (cid:16) ∂ j ′ U † b ′ (cid:17)i − N c Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i Tr h O ( g → q ¯ q )4 i = 12 Tr h(cid:16) ∂ j ′ U b ′ (cid:17) (cid:16) ∂ j U † b (cid:17)i Tr (cid:16) U b U † b ′ (cid:17) + 12 N c Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i . A comparison between the Wilson line structure in this channel given in Eq. (6.16) and thedefinitions of the first three gluon TMDs in the gluon channel F (1) gg , F (2) gg and F (3) gg given inEqs. (5.15), (5.16) and (5.17) suggests that these are the three gluon TMDs that appearin the g → q ¯ q channel. We define the following combinations of the TMDs which are thesame combinations defined in Table 1:Φ (1) g → q ¯ q ( k ) ≡ N c F (1) gg ( k ) − F (3) gg ( k ) N c − (2) g → q ¯ q ( k ) ≡ − N c F (2) gg ( k ) + F (3) gg ( k )Finally, the square of the tensor structure, Eq. (6.14), the Wilson line structure, Eq.(6.16), and the TMD definitions with the combinations given in Eq. (6.17) are plugged inthe generic kinematic twist resummed cross section given in Eq. (3.10). The result cansimply be written as dσ W Wg → q ¯ q dy dy d p d p = α s N c p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) z ¯ z (cid:0) z + ¯ z (cid:1) q (6.18) × " ¯ z p Φ (1) g → q ¯ q ( k ) + z p Φ (1) g → q ¯ q ( − k ) + z ¯ z ( p · p ) p p Φ (2) g → q ¯ q ( k ) + Φ (2) g → q ¯ q ( − k )( N c − which coincides exactly with Eq. (5.24) by using Eqs. (5.25) and (5.26).– 25 –he next step is to consider the dilute limit in the g → q ¯ q channel. Introducing theproper color structure in the generic dilute cross section in Eq. (4.8), we get dσ gA ∼ g → q ¯ q dy dy d p d p = α s
16 ( N c − (cid:0) p +0 (cid:1) (2 π ) p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) G ac ( k ) k (cid:0) φ i φ j ∗ (cid:1) (6.19) × Tr (cid:20) t b t a (cid:18) p i p − q i q (cid:19) + t a t b (cid:18) p i p + q i q (cid:19)(cid:21) " t c t b p j p − q j q ! + t b t c p j p + q j q ! , which after some color algebra leads to dσ gA ∼ g → q ¯ q dy dy d p d p = α s N c ( N c − z ¯ z (cid:0) z + ¯ z (cid:1) p p p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) G ( k ) × (cid:20) N c (cid:18) z p q + ¯ z p q (cid:19) − (cid:21) . (6.20)Finally, a comparison between the kinematic twist resummed cross section Eq. (6.18) andthe dilute limit of the cross section given in Eq. (6.20), again leads to a straightforwardmatching between the iTMD scheme and the dilute scheme for g → q ¯ q channel: σ gA ∼ g → q ¯ q = σ W Wg → q ¯ q (cid:12)(cid:12) Φ (1) g → q ¯ q =Φ (2) g → q ¯ q = G . (6.21) g → gg channel The next channel we consider is g → gg . The CGC amplitude for this channel is given inEq. (B.3). The tensor part for this channel can simply be read off from Eq. (B.3) and itis given as φ ( g → gg ) µ = 2 g s p +0 π ε σ p ⊥ ε σ ∗ p g ⊥ ε σ ∗ q g ⊥ [ zg ⊥ σ σ g ⊥ µσ − z ¯ zg ⊥ σ σ g ⊥ µσ + ¯ zg ⊥ σ σ g ⊥ µσ ] (6.22)Its square can be computed in a straightforward manner with the result being φ i ( g → gg ) φ i ′ ∗ ( g → gg ) = δ ii ′ (cid:16) g s π (cid:17) (cid:0) p +0 (cid:1)
32 (1 − z ¯ z ) (6.23)The color structure of this channel is demonstrated in Fig. 5 and it is given by U R b = U b a b ,U R b = U b a b and T R = f a b b . This leads to the following TMD operators once it isinserted to the Wilson line structure of the generic kinematic twist resummed cross sectionin Eq. (3.10): O ( g → gg )1 = (cid:16) ∂ j U b a b (cid:17) f a b b U b a b U a c b ′ f a c c (cid:16) ∂ j ′ U a c b ′ (cid:17) O ( g → gg )2 = (cid:16) ∂ j U b a b (cid:17) f a b b U b a b (cid:16) ∂ j ′ U a c b ′ (cid:17) f a c c U a c b ′ (6.24) O ( g → gg )3 = U b a b f a b b (cid:16) ∂ j U b a b (cid:17) U a c b ′ f a c c (cid:16) ∂ j ′ U a c b ′ (cid:17) O ( g → gg )4 = U b a b f a b b (cid:16) ∂ j U b a b (cid:17) (cid:16) ∂ j ′ U a c b ′ (cid:17) f a c c U a c b ′ . – 26 – , R = Adj b b + ¯ z rb − z r p , R = Adj p , R = Adj Figure 5 : g → gg amplitude in an external shockwave background with the appropriatecolor representations.After a standard but cumbersome color algebra, the trace over the color indexes of theabove TMD operators can be written in terms of the fundamental Wilson line operators asTr h O ( g → gg )1 i = − Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i Tr (cid:16) U b U † b ′ (cid:17) Tr (cid:16) U b ′ U † b (cid:17) − Tr h(cid:0) ∂ j U b (cid:1) U † b U b ′ U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ U b U † b ′ i + 2Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i − Tr h(cid:16) ∂ j U † b (cid:17) U b (cid:16) ∂ j ′ U † b ′ (cid:17) U b ′ i (6.25)+ N c (cid:26) Tr h(cid:16) ∂ j U † b (cid:17) (cid:16) ∂ j ′ U b ′ (cid:17)i Tr (cid:16) U b U † b ′ (cid:17) + Tr h(cid:0) ∂ j U b (cid:1) (cid:16) ∂ j ′ U † b ′ (cid:17)i Tr (cid:16) U b ′ U † b (cid:17) (cid:27) , Tr h O ( g → gg )2 i = Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i Tr (cid:16) U b U † b ′ (cid:17) Tr (cid:16) U b ′ U † b (cid:17) + Tr h(cid:0) ∂ j U b (cid:1) U † b U b ′ U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ U b U † b ′ i + Tr h(cid:16) ∂ j U † b (cid:17) U b (cid:16) ∂ j ′ U † b ′ (cid:17) U b ′ i − h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i (6.26)+ N c (cid:26) Tr h U † b ′ (cid:0) ∂ j U b (cid:1)i Tr h(cid:16) ∂ j ′ U b ′ (cid:17) U † b i + Tr h U b (cid:16) ∂ j ′ U † b ′ (cid:17)i Tr h(cid:16) ∂ j U † b (cid:17) U b ′ i (cid:27) , Tr h O ( g → gg )3 i = Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i Tr (cid:16) U b U † b ′ (cid:17) Tr (cid:16) U b ′ U † b (cid:17) + Tr h(cid:0) ∂ j U b (cid:1) U † b U b ′ U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ U b U † b ′ i + Tr h(cid:16) ∂ j U † b (cid:17) U b (cid:16) ∂ j ′ U † b ′ (cid:17) U b ′ i − h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i (6.27)+ N c (cid:26) Tr h U b (cid:16) ∂ j ′ U † b ′ (cid:17)i Tr h(cid:16) ∂ j U † b (cid:17) U b ′ i + Tr h(cid:0) ∂ j U b (cid:1) U † b ′ i Tr h(cid:16) ∂ j ′ U b ′ (cid:17) U † b i (cid:27) , – 27 –ndTr h O ( g → gg )4 i = − Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i Tr (cid:16) U b U † b ′ (cid:17) Tr (cid:16) U b ′ U † b (cid:17) − Tr h(cid:0) ∂ j U b (cid:1) U † b U b ′ U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ U b U † b ′ i + 2Tr h(cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ i − Tr h(cid:16) ∂ j U † b (cid:17) U b (cid:16) ∂ j ′ U † b ′ (cid:17) U b ′ i (6.28)+ N c (cid:26) Tr h(cid:16) ∂ j ′ U b ′ (cid:17) (cid:16) ∂ j U † b (cid:17)i Tr (cid:16) U b U † b ′ (cid:17) + Tr h(cid:0) ∂ j U b (cid:1) (cid:16) ∂ j ′ U † b ′ (cid:17)i Tr (cid:16) U † b U b ′ (cid:17) (cid:27) Comparing the Wilson line structures appearing in Eqs. (6.25), (6.26), (6.27) and (6.28)with the TMD definitions given in Eqs. (5.18), (5.19) and (5.20), one can conclude thaton top of the gluon TMDs F (1) gg , F (2) gg and F (3) gg that have already appeared in the g → q ¯ q channel, one also gets new gluon TMDs F (4) gg , F (5) gg and F (6) gg in the g → gg channel. Again,for convenience, we define the following combinations of the TMDsΦ (1) gg → gg ( k ) = 12 N c " N c F (6) gg ( k ) + F (5) gg ( k ) + F (4) gg ( k ) − F (3) gg ( k ) + N c F (1) gg ( k ) + F (1) gg ( − k )2 ! (6.29)Φ (2) gg → gg ( k ) = 1 N c " N c F (6) gg ( k ) + F (5) gg ( k ) + F (4) gg ( k ) − F (3) gg ( k ) + N c F (2) gg ( k ) + F (2) gg ( − k )2 ! , which match exactly the combinations one get from iTMD calculations given in Table 1.After plugging these results into the generic kinematic twist resummed cross section givenin Eq. (3.10), we get the result for the g → gg channel as dσ W Wg → gg dy dy d p d p = 2 α s N c N c − p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) (1 − z ¯ z ) p p (6.30) × (cid:20)(cid:18) z ¯ z ( p · p ) q (cid:19) Φ (1) gg → gg ( k ) − z ¯ z ( p · p ) q Φ (2) gg → gg ( k ) (cid:21) , where a factor 1 / g → gg channel. Oncethe proper color representations of this channel are plugged into the dilute limit of thegeneric cross section given in Eq. (4.8), we get dσ gA ∼ g → gg dy dy d p d p = α s
16 ( N c − (cid:0) p +0 (cid:1) (2 π ) p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) G ac ( k ) k (cid:0) φ i φ j ∗ (cid:1) (6.31) × Tr (cid:26)(cid:20) f a a b f ba a (cid:18) p i p − q i q (cid:19) + f a a b f ba a (cid:18) p i p + q i q (cid:19)(cid:21) × " f a a d f da c p j p − q j q ! + f a a d f da c p j p + q j q ! – 28 –hich after some color algebra leads to dσ gA ∼ g → gg dy dy d p d p = 2 α s N c N c − p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) (1 − z ¯ z ) p p G ( k ) (cid:18) z ¯ z ( p · p ) q (cid:19) , (6.32)with the symmetry factor of 1 /
2. As a last comment for this channel, we would like toemphasize that a comparison between Eqs. (6.30), (6.32) and the iTMD results lead to thesame matching condition between the iTMD scheme and the dilute limit: σ gA ∼ g → gg = σ W Wg → q ¯ q (cid:12)(cid:12) Φ (1) g → gg =Φ (2) g → gg = G . (6.33) γ → q ¯ q channel We have used this channel as an example to study the corrections to the back-to-backcorrelation limit in subsection 2.2. In this subsection, we generalize that study by usingthe generic expressions for the kinemtic twist resummed cross section and the dilute limitof the generic CGC cross section. The amplitude is given by Eq. (2.6) from which we canread off the tensor part: φ ( γ → q ¯ q ) µ = i e q π ε σp ⊥ ¯ u p q [2 zg ⊥ µσ − ( γ ⊥ µ γ ⊥ σ )] γ + v p ¯ q (6.34)The square of the tensor part for an unpolarized observable can be calculated easily andthe result reads φ i ( γ → q ¯ q ) φ i ′ ∗ ( γ → q ¯ q ) = δ ii ′ (cid:16) e q π (cid:17) (cid:0) p +0 (cid:1) z ¯ z (cid:0) z + ¯ z (cid:1) (6.35)The color structure for this channel is demonstrated in Fig. 6 and it is given by setting p , R = + ¯ z rb − z r p , R = Fp , R = F ∗ Figure 6 : γ → q ¯ q amplitude in an external shockwave background with the appropriatecolor representations. U R b = U b , U R b = U † b and T R = 1. This color structure leads to the following TMD– 29 –perators: O ( γ → q ¯ q )1 = − (cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ O ( γ → q ¯ q )2 = (cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ (6.36) O ( γ → q ¯ q )3 = (cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ O ( γ → q ¯ q )4 = − (cid:0) ∂ j U b (cid:1) U † b (cid:16) ∂ j ′ U b ′ (cid:17) U † b ′ . The trace over the color indices can be performed in a straightforward manner and onecan easily conclude that this channel involves only one TMD F (3) gg which is also referred toas the Weizs¨acker-Williams TMD defined in Eq. (5.17). Using this result and the squareof the tensor part given in Eq. (6.35), we can write the kinematic twist resummed crosssection for this channel as dσ W Wγ → q ¯ q dy dy d p d p = α em α s p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) z ¯ z (cid:0) z + ¯ z (cid:1) p p F (3) gg ( k ) , (6.37)which coincides exactly with Eq. (5.30) by using Eq. (5.31).Using the proper color representations for this channel and the dilute limit of thegeneric CGC cross section given in Eq. (4.8), we can simply write the dilute limit of thecross section for the γ → q ¯ q channel as dσ gA ∼ γ → q ¯ q dy dy d p d p = α s (cid:0) p +0 (cid:1) (2 π ) p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) G ( k ) k (cid:0) φ i φ j ∗ (cid:1) (6.38) × Tr (cid:20) t a (cid:18) p i p − q i q (cid:19) + t a (cid:18) p i p + q i q (cid:19)(cid:21) " t c p j p − q j q ! + t c p j p + q j q ! , which, after a simple color algebra and using the result for the square of the tensor partgiven in Eq. (6.35), leads to dσ gA ∼ γ → q ¯ q dy dy d p d p = α em α s p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) z ¯ z (cid:0) z + ¯ z (cid:1) p p G ( k ) . (6.39)Finally, we would like to mention that a comparison between Eq. (6.30) and (6.32) suggestsa similar matching between the iTMD scheme and the dilute limit of the CGC calculation: σ gA ∼ γ → q ¯ q = σ W Wγ → q ¯ q (cid:12)(cid:12) F (3) gg = G . (6.40) q → qγ channel The last channel we consider is the q → qγ one. The CGC amplitude for this channel isgiven by Eq. (B.4) from which we can read off the tensor part as φ ( q → qγ ) µ = − ie q π ε σ ∗ p γ ⊥ ¯ u p q [2 zg ⊥ µσ + ¯ z ( γ ⊥ µ γ ⊥ σ )] γ + u p (6.41)– 30 –ts square for an unpolarized observable can be written as φ i ( q → qγ ) φ i ′ ∗ ( q → qγ ) = δ ii ′ (cid:16) e q π (cid:17) (cid:0) p +0 (cid:1) z (cid:0) z (cid:1) . (6.42)As it can be seen from Fig. 7, the color structure of this channel is quite simple. One gets p , R = F b b + ¯ z rb − z r p , R = Fp , R = Figure 7 : q → qγ amplitude in an external shockwave background with the appropriatecolor representations.the proper color structure by setting U R b = U b , U R b = and T R = 1. With this simplecolor structure, only one TMD operator appears in this channel which reads O ( q → qγ ) = (cid:0) ∂ j U b (cid:1) (cid:16) ∂ j ′ U † b ′ (cid:17) . (6.43)Performing the trace over color indices leads to F (1) qg TMD which has been introduced inEq. (5.13). Plugging these results into Eq. (3.10), we get the cross section for q → qγ channel: dσ W Wq → qγ dy dy d p d p = α em α s N c p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) z ¯ z (cid:0) z (cid:1) p q F (1) qg ( k ) , (6.44)which coincides exactly with Eq. (5.32) by using Eq. (5.33).Before we continue with the dilute limit for this channel we would like to mentionthat Eq.(6.44) is exact and it resums not only the kinematic twists but all twists for thisprocess, i.e. no higher-body twist correction is expected for the q → qγ channel. This isdue to the fact that one of the Wilson line operators is trivial for this process and there isno other TMD operator involved.Inserting the simple color structure of this process into the dilute limit of the genericCGC cross section given in Eq. (4.8), we get the dilute limit of the cross section for the– 31 – → qγ channel: dσ gA ∼ q → qγ dy dy d p d p = α s N c (cid:0) p +0 (cid:1) (2 π ) p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) G ac ( k ) k (cid:0) φ i φ j ∗ (cid:1) (6.45) × Tr (cid:20) t a (cid:18) p i p + q i q (cid:19)(cid:21) " t c p j p + q j q ! , which leads to dσ gA ∼ q → qγ dy dy d p d p = α em α s N c p +0 δ (cid:0) p +1 + p +2 − p +0 (cid:1) z ¯ z (cid:0) z (cid:1) p q G ( k ) . (6.46)As in the case of the other channels, comparing Eq. (6.44) and (6.46), we also get astraightforward matching between the improved TMD scheme and the dilute scheme: σ gA ∼ q → qγ = σ W Wq → qγ (cid:12)(cid:12) F (1) qg = G . (6.47) Earlier studies have shown that for certain observables the small- x limit of the TMD frame-work and the so-called ”correlation limit” of the CGC framework overlap. In particular,two particle production (such as dijet or photon+jet) in forward pp and pA collisions,gluon TMDs can be recovered from the CGC calculations in the correlation limit. Thisspecific limit corresponds to the case when the total transverse momentum of the producedparticles k is much smaller than the hard scale Q . On the other hand, it is also well knownthat in the dilute limit of the CGC framework, that is in the limit when the total trans-verse momentum of the produced particles are of the same order as the hard scale, onerecovers the BFKL results. Recently, the small- x improved TMD (iTMD) formalism hasbeen developed to interpolate between these two limits.In this paper we studied two cases. First, by studying the correlation limit of the CGCamplitude for a generic 1 → → → | k | /Q s when compared to BFKL, bothformalisms rely on the Wandzura-Wilczek approximation in the CGC.With the previous observations, two origins of saturation can be expected. First of all,the difference between BFKL and iTMD is related to the distinction between gauge linkstructures, which account for multiple scattering from low x gluons. As discussed earlier,all distributions are equal at large | k | /Q s and distinct at low | k | /Q s , were saturationis expected. In that sense, the saturation scale Q s is the parameter which controls theimportance of multiple scatterings via gauge links. On the other hand, BFKL and iTMDboth rely on the Wandzura-Wilczek approximation when compared to the CGC. It will bevery instructive to compare predictions from iTMD and full CGC once genuine twists areextracted from the CGC as well [94]. This would probe Q s as the parameter which controlsthe importance of multiple scattering via genuine twists.As a natural extension of this study, we plan to perform a similar analysis for morecomplex observables where not only the unpolarized TMD distributions but their linearlypolarized partners appear. The two immediate observables that we are planning to studyare the heavy quark production [46] and three-particle production such as dijet+photonproduction [47].Last but not least, we would like to mention that recently there have been severalstudies devoted to understand the subeikonal corrections in the CGC framework [84–93].Comparing those to the future moderate- x corrections to the iTMD scheme would be alsoa natural extension of our study. Acknowledgments
We thank P. Taels for stimulating discussions. TA gratefully acknowledges the supportfrom Bourses du Gouvernement Fran¸cais (BGF)-S´ejour de recherche, and expresses hisgratitude to CPHT, Ecole Polytechnique, and to the Institute of Nuclear Physics, PolishAcademy of Sciences, for hospitality when part of this work was done. The work of TA issupported by Grant No. 2017/26/M/ST2/01074 of the National Science Centre, Poland.RB is grateful to NCBJ for hospitality when this project was started. The work of RB issupported by the National Science Centre, Poland, grant No.2015/17/B/ST2/01838, by theU.S. Department of Energy, Office of Nuclear Physics, under Contracts No. DE-SC0012704and by an LDRD grant from Brookhaven Science Associates. The work of PK is partiallysupported by Polish National Science Centre grant no. DEC-2017/27/B/ST2/01985. This– 33 –ork received additional partial support by Polish National Science Centre grant no. DEC-2017/27/B/ST2/01985.
A Effective Feynman rules in the external CGC shockwave field
In this appendix, we list the effective Feynman rules that are used to calculate the CGCamplitudes in Appendix B.
Outgoing quark line: ¯ u ( p q , z ) = 12 p + q π ! d Z d d x e ip + q (cid:18) z − − ( x − z z +0 + i (cid:19) − i ( p q · x ) + i z +02 p + q ( m + i ) (cid:2) U x θ (cid:0) − z +0 (cid:1) + θ (cid:0) z +0 (cid:1)(cid:3) × (cid:18) iz +0 (cid:19) d ¯ u p q γ + (cid:18) γ − − ˆ x ⊥ − ˆ z ⊥ z +0 + mp + q (cid:19) (A.1) Outgoing antiquark line: v ( p ¯ q , z ) = 12 p +¯ q π ! d Z d d x e ip +¯ q (cid:18) z − − ( x − z z +0 + i (cid:19) − i ( p ¯ q · x ) + i z +02 p +¯ q ( m + i ) h U † x θ (cid:0) − z +0 (cid:1) + θ (cid:0) z +0 (cid:1)i × (cid:18) iz +0 (cid:19) d γ − − ˆ x ⊥ − ˆ z ⊥ z +0 − mp +¯ q ! γ + v p ¯ q (A.2) Incoming gluon line: ε b a µ ( p , z ) = (cid:18) p +0 π (cid:19) d Z d d x e − ip +0 (cid:18) z − − ( x − z z +0 − i (cid:19) + i ( p · x ) h U b a x θ (cid:0) z +0 (cid:1) + δ a b θ (cid:0) − z +0 (cid:1)i × (cid:18) − iz +0 (cid:19) d (cid:18) g ⊥ µ σ + x ⊥ σ − z ⊥ σ z +0 n µ (cid:19) ε σ p ⊥ (A.3) Outgoing gluon line: ε ba ∗ µ ( p g , z ) = p + g π ! d Z d d x e ip + g (cid:18) z − − ( x − z − i z +0 (cid:19) − i ( p g · x ) h U ab x θ (cid:0) − z +0 (cid:1) + δ ab θ (cid:0) z +0 (cid:1)i × (cid:18) iz +0 (cid:19) d (cid:18) g ⊥ µσ + x ⊥ σ − z ⊥ σ z +0 n µ (cid:19) ε σ ∗ p g ⊥ (A.4) B CGC amplitudes for all channels
In this appendix we list the CGC amplitudes calculated by using the effective Feynmanrules listed in Appendix A. – 34 – → qg channel for forward dijet production in pp and pA collisions: A q → qg = ig s π ε σ ∗ p g ⊥ (2 π ) δ (cid:0) p + q + p + g − p + (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) × r µ ⊥ r h(cid:16) U b +¯ z r t b U ab b − z r (cid:17) − (cid:16) t b δ ab U b (cid:17)i (B.1) × ¯ u p q [2 zg ⊥ µσ + ¯ z ( γ ⊥ µ γ ⊥ σ )] γ + u p g → q ¯ q channel for forward dijet production in pp and pA collisions: A g → q ¯ q = − i g s π ε σp ⊥ (2 π ) δ (cid:0) p + q + p +¯ q − p + g (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) × r µ ⊥ r h(cid:16) U b +¯ z r t b U † b − z r δ ab (cid:17) − (cid:16) t b U ba b (cid:17)i (B.2) × ¯ u p q [2 zg ⊥ µσ − ( γ ⊥ µ γ ⊥ σ )] γ + v p ¯ q g → gg channel for forward dijet production in pp and pA collisions: A g → gg = 2 g s p + π ε σ p ⊥ ε σ ∗ p g ⊥ ε σ ∗ q g ⊥ (2 π ) δ (cid:0) p + g + q + g − p + (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) × r µ ⊥ r h f b b b δ b a U b a b +¯ z r U b a b − z r − f b b b δ b a δ b a U b a b i (B.3) × [ zg ⊥ σ σ g ⊥ µσ − z ¯ zg ⊥ σ σ g ⊥ µσ + ¯ zg ⊥ σ σ g ⊥ µσ ] Production of a forward photon-jet pair in pp and pA collisions: A q → qγ = ig s π ε σ ∗ p γ ⊥ (2 π ) δ (cid:0) p + q + p + g − p + (cid:1) Z d b d r e − i ( q · r ) − i ( k · b ) × r µ ⊥ r ( U b +¯ z r − U b ) ¯ u p q [2 zg ⊥ µσ + ¯ z ( γ ⊥ µ γ ⊥ σ )] γ + u p (B.4) C The integral
In this appendix, we present the details of the calculation of the following integral I ij ( p ) ≡ Z d d r r i r j r e − i ( p · r ) − p · r ) e − i ( q · r ) . (C.1)This integral is a symmetric tensor, hence we can decompose it in a 3-dimensional basis.Let us choose (cid:18) δ ij , p i q j + q j p i p · q , p i p j p (cid:19) , (C.2)and write I ij ( p ) = I δ ij + I p i q j + q i p j p · q + I p i p j p . (C.3)– 35 –his relations inverts to I = I ii ( p ) − p i p j p I ij ( p ) I = − ( p · q ) p q − ( p · q ) p i p j p I ij ( p ) + ( p · q ) p q − ( p · q ) p i q j I ij ( p ) (C.4) I = − I ii ( p ) + 2 p q p q − ( p · q ) p i p j p I ij ( p ) − p · q ) p q − ( p · q ) p i q j I ij ( p ) . Thus in order to compute I ij , it is sufficient to compute J = δ ij I ij and J j ≡ p i I ij . Onecan actually show that J = 0. This becomes apparent by going to spherical coordinates,integrating | r | out (taking into account the phase regulators i J j is obtained easily with the usual Schwinger representation tricks and reads: J j = − iπ (cid:18) q j + p j ( q + p ) − q j q (cid:19) . (C.5)Finally plugging Eq. (C.5) in Eq. (C.4) then in Eq. (C.3), one obtains I ij ( k ) = − iπ p (cid:20)(cid:18) ( p · q ) q − ( p · q ) + p ( q + p ) (cid:19) δ ij + (cid:18) q + p ) − q (cid:19) (cid:0) p i q j + q i p j (cid:1) + 2 p i p j ( q + p ) (cid:21) , (C.6)which leads to the expression given in Eq. (3.8). References [1] V. N. Gribov and L. N. Lipatov,
Deep inelastic e p scattering in perturbation theory , Sov. J.Nucl. Phys. , 438 (1972) [ Yad. Fiz. , 781 (1972)].[2] G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language , Nucl. Phys. B , 298 (1977).[3] Y. L. Dokshitzer,
Calculation of the Structure Functions for Deep Inelastic Scattering ande+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics , Sov. Phys. JETP (1977) 641 [ Zh. Eksp. Teor. Fiz. (1977) 1216].[4] E. A. Kuraev, L. N. Lipatov and V. S. Fadin, The Pomeranchuk Singularity in NonabelianGauge Theories , Sov. Phys. JETP , 199 (1977) [ Zh. Eksp. Teor. Fiz. , 377 (1977)].[5] I. I. Balitsky and L. N. Lipatov, The Pomeranchuk Singularity in QuantumChromodynamics , Sov. J. Nucl. Phys. , 822 (1978) [ Yad. Fiz. , 1597 (1978)].[6] A. H. Mueller, Small x Behavior and Parton Saturation: A QCD Model , Nucl. Phys. B , 115 (1990).[7] A. H. Mueller,
Soft gluons in the infinite momentum wave function and the BFKL pomeron , Nucl. Phys. B , 373 (1994). – 36 –
8] A. H. Mueller,
Unitarity and the BFKL pomeron , Nucl. Phys. B , 107 (1995),[ hep-ph/9408245 ].[9] I. Balitsky,
Operator expansion for high-energy scattering , Nucl. Phys. B , 99 (1996),[ hep-ph/9509348 ].[10] I. Balitsky,
Factorization for high-energy scattering , Phys. Rev. Lett. , 2024 (1998),[ hep-ph/9807434 ].[11] I. Balitsky, Factorization and high-energy effective action , Phys. Rev. D , 014020 (1999),[ hep-ph/9812311 ].[12] L. D. McLerran and R. Venugopalan, Computing quark and gluon distribution functions forvery large nuclei , Phys. Rev. D , 2233 (1994), [ hep-ph/9309289 ].[13] L. D. McLerran and R. Venugopalan, Gluon distribution functions for very large nuclei atsmall transverse momentum , Phys. Rev. D , 3352 (1994), [ hep-ph/9311205 ].[14] L. D. McLerran and R. Venugopalan, Green’s functions in the color field of a large nucleus , Phys. Rev. D , 2225 (1994), [ hep-ph/9402335 ].[15] J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, The BFKL equation from theWilson renormalization group , Nucl. Phys. B , 415 (1997), [ hep-ph/9701284 ].[16] J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert,
The Wilson renormalizationgroup for low x physics: Towards the high density regime , Phys. Rev. D , 014014 (1998),[ hep-ph/9706377 ].[17] J. Jalilian-Marian, A. Kovner and H. Weigert, The Wilson renormalization group for low xphysics: Gluon evolution at finite parton density , Phys. Rev. D , 014015 (1998),[ hep-ph/9709432 ].[18] A. Kovner and J. G. Milhano, Vector potential versus color charge density in low x evolution , Phys. Rev. D , 014012 (2000), [ hep-ph/9904420 ].[19] A. Kovner, J. G. Milhano and H. Weigert, Relating different approaches to nonlinear QCDevolution at finite gluon density , Phys. Rev. D , 114005 (2000), [ hep-ph/0004014 ].[20] H. Weigert, Unitarity at small Bjorken x , Nucl. Phys. A , 823 (2002), [ hep-ph/0004044 ].[21] E. Iancu, A. Leonidov and L. D. McLerran,
Nonlinear gluon evolution in the color glasscondensate. 1 , Nucl. Phys. A , 583 (2001), [ hep-ph/0011241 ].[22] E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran,
Nonlinear gluon evolution in the colorglass condensate. 2 , Nucl. Phys. A , 489 (2002), [ hep-ph/0109115 ].[23] Y. V. Kovchegov,
Small x F(2) structure function of a nucleus including multiple pomeronexchanges , Phys. Rev. D , 034008 (1999), [ hep-ph/9901281 ].[24] A. H. Mueller and B. Patel, Single and double BFKL pomeron exchange and a dipole pictureof high-energy hard processes , Nucl. Phys. B , 471 (1994), [ hep-ph/9403256 ].[25] Z. Chen and A. H. Mueller,
The Dipole picture of high-energy scattering, the BFKL equationand many gluon compound states , Nucl. Phys. B , 579 (1995).[26] S. Caron-Huot,
When does the gluon reggeize? , JHEP , 093 (2015),[ hep-th/1309.6521 ].[27] I. Balitsky and G. A. Chirilli,
Next-to-leading order evolution of color dipoles , Phys. Rev. D , 014019 (2008), [ hep-ph/ 0710.4330 ]. – 37 –
28] V. S. Fadin, R. Fiore and A. Papa,
The Dipole form of the quark part of the BFKL kernel , Phys. Lett. B , 179 (2007), [ hep-ph/0701075 ].[29] V. S. Fadin, R. Fiore, A. V. Grabovsky and A. Papa,
The Dipole form of the gluon part ofthe BFKL kernel , Nucl. Phys. B , 49 (2007), [ hep-ph/0705.1885 ].[30] V. S. Fadin, R. Fiore, A. V. Grabovsky and A. Papa,
Connection between complete andMoebius forms of gauge invariant operators , Nucl. Phys. B , 111 (2012),[ hep-th/1109.6634 ].[31] J. Collins,
Foundations of perturbative QCD , vol. 32. Cambridge Univ. Press, 2011.[32] J. C. Collins, D. E. Soper and G. F. Sterman,
Relation of Parton Distribution Functions inDrell-Yan Process to Deeply Inelastic Scattering , Phys. Lett. , 275 (1983),[33] S. J. Brodsky, D. S. Hwang and I. Schmidt,
Final state interactions and single spinasymmetries in semiinclusive deep inelastic scattering , Phys. Lett. B , 99 (2002),[ hep-ph/0201296] ].[34] J. C. Collins,
Leading twist single transverse-spin asymmetries: Drell-Yan and deep inelasticscattering , Phys. Lett. B , 43 (2002), [ hep-ph/0204004 ].[35] A. V. Belitsky, X. Ji and F. Yuan,
Final state interactions and gauge invariant partondistributions , Nucl. Phys. B , 165 (2003), [ hep-ph/0208038 ].[36] C. J. Bomhof, P. J. Mulders and F. Pijlman,
Gauge link structure in quark-quark correlatorsin hard processes , Phys. Lett. B , 277 (2004), [ hep-ph/0406099 ].[37] D. Boer and P. J. Mulders,
Color gauge invariance in the Drell-Yan process , Nucl. Phys. B , 505 (2000), [ hep-ph/9906223 ].[38] C. J. Bomhof, P. J. Mulders and F. Pijlman,
The Construction of gauge-links in arbitraryhard processes , Eur. Phys. J. C , 147 (2006), [ hep-ph/0601171 ].[39] F. Dominguez, B. W. Xiao and F. Yuan, k t -factorization for Hard Processes in Nuclei , Phys. Rev. Lett. , 022301 (2011), [ hep-ph/1009.2141 ].[40] F. Dominguez, C. Marquet, B. W. Xiao and F. Yuan,
Universality of Unintegrated GluonDistributions at small x , Phys. Rev. D , 105005 (2011), [ hep-ph/1101.0715 ].[41] C. Marquet, E. Petreska and C. Roiesnel, Transverse-momentum-dependent gluondistributions from JIMWLK evolution , JHEP , 065 (2016), [ hep-ph/1608.02577 ].[42] A. Metz and J. Zhou,
Distribution of linearly polarized gluons inside a large nucleus , Phys. Rev. D , 051503 (2011), [ hep-ph/1105.1991 ].[43] E. Akcakaya, A. Sch¨afer and J. Zhou, Azimuthal asymmetries for quark pair production inpA collisions , Phys. Rev. D , 054010 (2013), [ hep-ph/1208.4965 ].[44] A. Dumitru and V. Skokov, cos(4 ϕ ) azimuthal anisotropy in small- x DIS dijet productionbeyond the leading power TMD limit , Phys. Rev. D , 014030 (2016),[ hep-ph/1605.02739 ].[45] D. Boer, P. J. Mulders, J. Zhou and Y. Zhou, Suppression of maximal linear gluonpolarization in angular asymmetries , JHEP , 196 (2017), [ hep-ph/1702.08195 ].[46] C. Marquet, C. Roiesnel and P. Taels,
Linearly polarized small- x gluons in forwardheavy-quark pair production , Phys. Rev. D , no. 1, 014004 (2018), [ hep-ph/1710.05698 ]. – 38 –
47] T. Altinoluk, R. Boussarie, C. Marquet and P. Taels,
TMD factorization for dijets + photonproduction from the dilute-dense CGC framework , [ hep-ph/1810.11273 ].[48] E. Petreska, TMD gluon distributions at small x in the CGC theory , Int. J. Mod. Phys. E , no. 05, 1830003 (2018), [ hep-ph/1804.04981 ].[49] K. J. Golec-Biernat and M. Wusthoff, Saturation effects in deep inelastic scattering at lowQ**2 and its implications on diffraction , Phys. Rev. D , 014017 (1998), [ hep-ph/9807513 ].[50] P. Kotko, K. Kutak, C. Marquet, E. Petreska, S. Sapeta and A. van Hameren, ImprovedTMD factorization for forward dijet production in dilute-dense hadronic collisions , JHEP , 106 (2015), [ hep-ph/1503.03421 ].[51] A. van Hameren, P. Kotko, K. Kutak, C. Marquet, E. Petreska and S. Sapeta,
Forward di-jetproduction in p+Pb collisions in the small-x improved TMD factorization framework , JHEP , 034 (2016), [ hep-ph/1607.03121 ].[52] J. Bartels, K. Golec-Biernat and L. Motyka,
Phys. Rev. D , 054017 (2010),[ hep-ph/0911.1935 ][53] I. I. Balitsky and V. M. Braun, Evolution Equations for QCD String Operators , Nucl. Phys. B , 541 (1989).[54] S. Wandzura and F. Wilczek,
Sum Rules for Spin Dependent Electroproduction: Test ofRelativistic Constituent Quarks , Phys. Lett. , 195 (1977).[55] T. Altinoluk, C. Contreras, A. Kovner, E. Levin, M. Lublinsky and A. Shulkin,
QCDReggeon Calculus From KLWMIJ/JIMWLK Evolution: Vertices, Reggeization and All , JHEP , 115 (2013), [ hep-ph/1306.2794 ].[56] A. Dumitru, A. Hayashigaki and J. Jalilian-Marian,
The color glass condensate and hadronproduction in the forward region , Nucl. Phys. A (feb, 2006) 464–482, [ hep-ph/0506308 ].[57] T. Altinoluk and A. Kovner,
Particle Production at High Energy and Large TransverseMomentum - ’The Hybrid Formalism’ Revisited , Phys. Rev. D (2011) 105004,[ hep-ph/102.5327 ].[58] 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. , 122301 (2012), [ hep-ph/112.1061 ].[59] G. A. Chirilli, B. W. Xiao and F. Yuan,
Inclusive Hadron Productions in pA Collisions , Phys. Rev. D , 054005 (2012), [ hep-ph/1203.6139 ].[60] A. M. Stasto, B. W. Xiao and D. Zaslavsky, Towards the Test of Saturation Physics BeyondLeading Logarithm , Phys. Rev. Lett. , no. 1, 012302 (2014), [ hep-ph/1307.4057 ].[61] A. M. Stasto, B. W. Xiao, F. Yuan and D. Zaslavsky,
Matching collinear and small x factorization calculations for inclusive hadron production in pA collisions , Phys. Rev. D , no. 1, 014047 (2014), [ hep-ph/1405.6311 ].[62] T. Altinoluk, N. Armesto, G. Beuf, A. Kovner and M. Lublinsky, Single-inclusive particleproduction in proton-nucleus collisions at next-to-leading order in the hybrid formalism , Phys. Rev. D , no. 9, 094016 (2015), [ hep-ph/1411.2869 ].[63] K. Watanabe, B. W. Xiao, F. Yuan and D. Zaslavsky, Implementing the exact kinematicalconstraint in the saturation formalism , Phys. Rev. D , no. 3, 034026 (2015),[ hep-ph/1505.05183 ]. – 39 –
64] B. Duclou, T. Lappi and Y. Zhu,
Single inclusive forward hadron production atnext-to-leading order , Phys. Rev. D , no. 11, 114016 (2016), [ hep-ph/1604.00225 ].[65] E. Iancu, A. H. Mueller and D. N. Triantafyllopoulos, CGC factorization for forward particleproduction in proton-nucleus collisions at next-to-leading order , JHEP , 041 (2016),[ hep-ph/1608.05293 ].[66] B. Duclou, T. Lappi and Y. Zhu,
Implementation of NLO high energy factorization in singleinclusive forward hadron production , Phys. Rev. D , no. 11, 114007 (2017),[ hep-ph/1703.04962 ].[67] T. Altinoluk, N. Armesto, G. Beuf, A. Kovner and M. Lublinsky, Heavy quarks inproton-nucleus collisions - the hybrid formalism , Phys. Rev. D , no. 5, 054049 (2016),[ hep-ph/1511.09415 ].[68] T. Altinoluk, N. Armesto, A. Kovner, M. Lublinsky and E. Petreska, Soft photon and twohard jets forward production in proton-nucleus collisions , JHEP , 063 (2018),[ hep-ph/1802.01398 ].[69] E. Iancu and Y. Mulian,
Forward trijet production in proton-nucleus collisions ,[ hep-ph/1809.05526 ].[70] E. Antonov, I. Cherednikov, E. Kuraev and L. Lipatov, Feynman rules for effective Reggeaction , Nucl. Phys. B (aug, 2005) 111–135, [ ].[71] A. Van Hameren, P. Kotko and K. Kutak,
Multi-gluon helicity amplitudes with one off-shellleg within high energy factorization , J. High Energy Phys. (2012) , [ ].[72] A. Van Hameren, P. Kotko and K. Kutak,
Helicity amplitudes for high-energy scattering , J.High Energy Phys. (2013) .[73] A. Van Hameren,
BCFW recursion for off-shell gluons , J. High Energy Phys. (jul, 2014) 138, [ ].[74] A. van Hameren and M. Serino,
BCFW recursion for TMD parton scattering , J. High Energy Phys. (jul, 2015) 10.[75] A. van Hameren,
Calculating off-shell one-loop amplitudes for k T -dependent factorization: aproof of concept , .[76] P. Kotko, Wilson lines and gauge invariant off-shell amplitudes , J. High Energy Phys. (jul, 2014) 128, [ ].[77] M. L. Mangano and S. J. Parke,
Multi-parton amplitudes in gauge theories , Phys. Rep. (feb, 1991) 301–367, [ hep-th/0509223 ].[78] M. Bury, P. Kotko and K. Kutak,
TMD gluon distributions for multiparton processes , .[79] M. Deak, F. Hautmann, H. Jung and K. Kutak, Forward jet production at the Large HadronCollider , J. High Energy Phys. (sep, 2009) 121–121, [ ].[80] T. C. Rogers and P. J. Mulders,
No generalized transverse momentum dependentfactorization in the hadroproduction of high transverse momentum hadrons , Phys. Rev. D (may, 2010) 094006, [ ].[81] I. Balitsky and A. Tarasov, Rapidity evolution of gluon TMD from low to moderate x , J. High Energy Phys. (oct, 2015) 17. – 40 –
82] I. Balitsky and A. Tarasov,
Gluon TMD in particle production from low to moderate x , J. High Energy Phys. (mar, 2016) 164, [ ].[83] P. Kotko, K. Kutak, S. Sapeta, A. M. Stasto and M. Strikman,
Estimating nonlinear effectsin forward dijet production in ultra-peripheral heavy ion collisions at the LHC , Eur. Phys. J. C (may, 2017) 353, [ ].[84] T. Altinoluk, N. Armesto, G. Beuf, M. Martnez and C. A. Salgado, Next-to-eikonalcorrections in the CGC: gluon production and spin asymmetries in pA collisions , JHEP , 068 (2014), [ hep-ph/404.2219 ].[85] T. Altinoluk, N. Armesto, G. Beuf and A. Moscoso,
Next-to-next-to-eikonal corrections inthe CGC , JHEP , 114 (2016), [ hep-ph/1505.01400 ].[86] T. Altinoluk and A. Dumitru,
Particle production in high-energy collisions beyond theshockwave limit , Phys. Rev. D , no. 7, 074032 (2016), [ hep-ph/1512.00279 ].[87] I. Balitsky and A. Tarasov, Rapidity evolution of gluon TMD from low to moderate x , JHEP , 017 (2015), [ hep-ph/1505.02151 ].[88] I. Balitsky and A. Tarasov,
Gluon TMD in particle production from low to moderate x , JHEP , 164 (2016), [ hep-ph/1603.06548 ].[89] I. Balitsky and A. Tarasov,
Higher-twist corrections to gluon TMD factorization , JHEP , 095 (2017), [ hep-ph/1706.01415 ].[90] G. A. Chirilli,
Sub-eikonal corrections to scattering amplitudes at high energy ,[ hep-ph/1807.11435 ].[91] Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, Helicity Evolution at Small-xJHEP , 072 (2016),
Erratum: [ JHEP , 148 (2016) ], [ hep-ph/1511.06737 ].[92] Y. V. Kovchegov, D. Pitonyak and M. D. Sievert,
Helicity Evolution at Small x : FlavorSinglet and Non-Singlet Observables , Phys. Rev. D , no. 1, 014033 (2017),[ hep-ph/1610.06197 ].[93] Y. V. Kovchegov and M. D. Sievert Small- x Helicity Evolution: an Operator Treatment ,[ hep-ph/1808.09010 ].[94] T. Altinoluk, R. Boussarie, in preparation ..