Extra Spin Asymmetries From the Breakdown of TMD-Factorization in Hadron-Hadron Collisions
YYITP-SB-13-11
Extra Spin Asymmetries From the Breakdown of TMD-Factorization inHadron-Hadron Collisions
Ted C. Rogers
C.N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York 11794–3840, USA ∗ (Dated: April 15, 2013)We demonstrate that partonic correlations that would traditionally be identified as subleadingon the basis of a generalized TMD-factorization conjecture can become leading-power because ofTMD-factorization breaking that arises in hadron-hadron collisions with large transverse momentumback-to-back hadrons produced in the final state. General forms of TMD-factorization fail forsuch processes because of a previously noted incompatibility between the requirements for TMD-factorization and the Ward identities of non-Abelian gauge theories. We first review the basic stepsfor factorizing the gluon distribution and then show that a conflict between TMD-factorization andthe non-Abelian Ward identity arises already at the level of a single extra soft or collinear gluon whenthe partonic subprocess involves a TMD gluon distribution. Next we show that the resulting TMD-factorization violating effects produce leading-power final state spin asymmetries that would beclassified as subleading in a generalized TMD-factorization framework. We argue that similar extraTMD-factorization breaking effects may be necessary to explain a range of open phenomenologicalQCD puzzles. The potential to observe extra transverse spin or azimuthal asymmetries in futureexperiments is highlighted as their discovery may indicate an influence from novel and unexpectedlarge distance parton correlations. Keywords: perturbative QCD, factorization, nucleon structure
I. INTRODUCTION
This paper examines the consequences of factorization breaking in inclusive high energy cross sections that aredifferential in the transverse momentum of produced particles, with a focus on the region of small transverse mo-mentum where intrinsic motion associated with hadron structure becomes significant. There is at present a widerange of motivations for predicting and measuring intrinsic transverse momentum effects experimentally, so mucheffort continues to be devoted to developing methods to account for them in perturbative QCD (pQCD) treatments.At a phenomenological level, the region of very small transverse momentum is most commonly described within ageneral transverse momentum dependent (TMD) parton model wherein the colliding hadrons are treated as collec-tions of nearly free point-like quark and gluon constituents. Within the parton model description, the TMD partondistribution functions (PDFs) and fragmentation functions are treated much like classical probability densities. Theyare conceptually very similar to the PDFs of the more familiar collinear parton model except that they describe thedistribution of partons in terms of both longitudinal and transverse momentum components.In pQCD, a parton model description should be replaced by a factorization theorem. Demanding that factor-ization be consistent with real pQCD leads to rigid theoretical constraints which, in phenomenology, translate intofirst principles QCD predictions. When factorization theorems accommodate TMD PDFs and TMD fragmentationfunctions, they are usually called “TMD-factorization” or “ k T -factorization” theorems, and the constraints they im-pose are specific to cross sections that are sensitive to intrinsic parton transverse momentum. A derivation or proofof a TMD-factorization theorem for a particular process must show that the transversely differential cross sectionfactorizes into a generalized product of a hard part (perturbatively calculable to fixed order in small coupling), anda collection of well-defined non-perturbative TMD objects corresponding to TMD PDFs and TMD fragmentationfunctions for separate external hadrons. The non-perturbative TMD functions contain detailed information about theinfluence of intrinsic non-perturbative motion of bound state quarks and gluons, so measuring them and explainingthem theoretically is currently the focus of many efforts to formulate a deeper understanding of hadronic bound statesin terms of elementary QCD quark-gluon degrees of freedom. Because of their sensitivity to 3-dimensional intrinsicmotion, TMD PDFs and fragmentation functions are also important in the study of the spin and orbital angularmomentum composition of hadrons in terms of fundamental constituents. For lists of relevant references see, forexample, Refs. [1, 2]. In addition, for high energy collisions at the LHC, certain TMDs may be useful for studying thedetailed properties of the Higgs boson [3, 4]. Generally, TMD-factorization theorems are necessary whenever highly ∗ [email protected] a r X i v : . [ h e p - ph ] A p r accurate calculations of transversely differential cross sections in the region small or zero transverse momentum areneeded. This can include calculations used in the study of hadron structure as well as in searches for new physics.TMD-factorization theorems have been derived in pQCD with a high degree of rigor for a number of processes,including Drell-Yan (DY) scattering, semi-inclusive deep inelastic scattering (SIDIS), and the production of back-to-back hadrons in e + e − annihilation [5–12]. For those processes where TMD-factorization theorems are currentlyunderstood and known to exist, an important next step is to implement unified phenomenological treatments, includingglobal fits with QCD evolution and extractions of the non-perturbative TMD functions. The success or failure of theseefforts will be a crucial test of the validity of small-coupling techniques in studies of hadronic structure and in newphenomenological regimes beyond what are treatable within ordinary collinear factorization.However, the focus of this paper is on processes where normal steps for deriving TMD-factorization in pQCD arenow understood to fail. Recently, standard parton-model-based pictures of TMD-factorization have been found toconflict with pQCD for certain classes of high energy processes, most notably in high energy hadron-hadron collisionswhere a pair of hadrons or jets with large back-to-back transverse momentum is produced in the final state [13–19].Interestingly, the conflict with TMD-factorization arises in kinematical regimes where common partonic intuitionsuggests that factorization should be very reliable. In particular, there is a hard scale, Q , set by the transversemomentum of the produced final state hadrons, which may be of the same order-of-magnitude as the center-of-massenergy. Therefore, the TMD-factorization breaking mechanisms discussed in Refs. [13–19] are distinct from other well-known kinematical complications with factorization such as those expected in the limit of very small Bjorken- x . Theyare instead due to an incompatibility between arguments for leading-power TMD-factorization and the non-Abeliangauge invariance of QCD, which persists even in the limit of a large hard scale.For processes and observables where factorization theorems are valid, Ward identities maintain factorization in thelarge Q limit by ensuring that any leading-power factorization breaking contributions that may appear term-by-termin perturbation theory cancel in the inclusive sum. After all cancellations have occurred, any remaining terms thatviolate factorization must be shown to be suppressed by powers of Λ QCD /Q in order for factorization to be said to bevalid at leading power. The details of these steps of a factorization proof constrain the specific form of factorizationfor the classic hard QCD processes. By contrast, the normally anticipated Ward identity cancellations fail in thescenarios discussed in Refs. [13–19], leaving leftover leading-power TMD-factorization breaking contributions. Thenon-cancellations were first noted in Refs. [13, 14] for their ability to introduce interesting process dependence inTMD-functions, and they were later identified in Refs. [17, 18] as constituting a breakdown in the normal steps ofa factorization proof. Still, it remained common to hypothesize that a more general form of TMD-factorization,called “generalized TMD-factorization” in Refs. [16, 19], holds for these processes so long as the Wilson lines neededfor gauge invariance in TMD definitions are allowed to have non-universaland potentially complicated structures. However, it was later found in [19] that TMD-factorization in the hadro-production of back-to-back hadrons fails evenin this generalized sense. In other words, the TMD-factorization derivations cannot generally be made consistent withgauge invariance even when allowing for non-universal Wilson line topologies in the TMD definitions. Methods foraddressing the resulting non-universality within tree-level or parton-model-like approaches have since been proposedin Refs. [20–23], and methods tailored to the small Bjorken- x limit have been discussed in Ref. [24, 25].A basic goal in the study of hadron structure is to establish a unified theoretical framework, rooted in elementaryQCD quark and gluon concepts, for characterizing the intrinsic partonic correlations associated with QCD bound statesand relating them to experimental observables. In processes where TMD-factorization theorems are known to be valid,the information about intrinsic hadron structure is contained in well-defined non-perturbative correlation functionslike the TMD PDFs and the TMD fragmentation functions. The pattern that emerges is suggestive of a much moregeneral picture of partonic interactions based on descriptions of quark-gluon properties for individual and separateexternal hadrons. With this broad picture as a foundation, the customary TMD classification schemes have usuallyassumed a form of generalized TMD-factorization for all types of interesting or relevant hard hadronic processes. Thevarieties of possible spin and angular behavior are then enumerated by first classifying the separate TMD functionsaccording to their individual intrinsic properties, and then using them in a large set of both conjectured and derivedTMD-factorization formulas. (We will elaborate in more detail on the meaning of “customary TMD classificationschemes” in Sect. IV A.)This very general TMD-factorization picture has intuitive appeal because of the straightforward organizationalmethod it implies and because it has a very direct connection to parton model intuition. In classifications of thepossible spin and angular asymmetries of hard inclusive cross sections, it is almost always taken as an assumption.The only deviations that are usually allowed are those which incorporate non-universal normalization factors such asthe overall sign reversal expected for the Sivers function in comparisons between the Drell-Yan process and SIDIS [26]or an overall color factor normalization for more complicated processes [16]. We will use the terms “gauge link” and “Wilson line” interchangeably throughout this paper.
However, even the loosest forms of a TMD-factorization hypothesis impose strong constraints on the possiblegeneralbehavior of transversely differential and spin dependent cross sections. Those constraints constitute valuablefirst-principles QCD-based predictions where TMD-factorization is expected to hold, but it has become common toalso apply the standard classification scheme outside of the class of processes known to respect TMD-factorization,including in processes like those discussed in Refs. [13–19]. The purpose of this paper is to argue by way of examplethat, for these latter cases, the constraints imposed by a general TMD-factorization framework are too restrictive. It will be shown that factorization breaking partonic correlations can produce unexpected patterns of spin andangular dependence in TMD cross sections that would otherwise be forbidden at leading power if a TMD-factorizationhypothesis is adopted. Hence, the breakdown in TMD-factorization can modify the general landscape of leading-powerspin and angular behavior in hadron-hadron scattering rather than simply giving process dependence to already knownTMD functions.The relevance of TMD-factorization breaking extends beyond hadron or nuclear structure studies and is potentiallyimportant whenever perturbative QCD calculations are sensitive to the details of final states and are intended tohave high precision point-by-point over a wide range of kinematics. For example, because of the detailed accountof final state kinematics in TMD-factorization, it is a potentially valuable tool in the construction of Monte Carloevent generators [27, 28]. Also, it has recently even been found that factorization breaking arises in certain collinearcases [29, 30]. In principle, it should be possible to connect the breakdown of TMD-factorization to the treatment oflarge higher order logarithms in collinear factorization and thus relate the two phenomena.Very generally, factorization theorems for inclusive processes rely on cancellations in the inclusive sums over finalstates. As such, the validity of any factorization theorem is placed in danger whenever extra final state constraints orconditions are imposed. A recent discussion in the context of collinear factorization for top-antitop pair productioncan be found in Ref. [31]. For the TMD-factorization breaking scenarios of Refs. [13–19], it is the specification of asmall total transverse momentum for the final state back-to-back hadron or jet pair that is responsible for breakingTMD-factorization. Violations of standard factorization have been observed to have quite large phenomenological con-sequences, such as in measurements of hard diffraction in hadron-hadron collisions and in dijet projection, particularlywhen compared with measurements of hard deep inelastic diffraction [32–38].The concept of a TMD PDF, and especially of an unintegrated gluon PDF, also appears in extensions of smallcoupling perturbation theory to the limit of small- x where other novel phenomena such as parton saturation becomerelevant. In this context, however, there are varying methods for using TMD gluon PDFs in calculations, and workis still needed to fully reconcile the different approaches with one another [24, 25, 39, 40].A collection of noteworthy QCD-related phenomenological puzzles has developed over the past several decades. (Seealso Ref. [41].) This includes famously large and non-energy-suppressed transverse single spin asymmetries (SSAs)observed in hadron-hadron collision experiments at Argonne National Laboratory (ANL) [42, 43], Fermilab [44,45], CERN [46], Serpukhov [47], and Brookhaven National Laboratory (BNL) [48–50]. For a recent review of thecurrent experimental status of TMDs and spin asymmetries, see Ref. [51]. Very early, large final state Λ-hyperontransverse polarizations were also observed in unpolarized hadron-nucleus collisions at Fermilab [52]. Similar transversepolarizations have not been detected in the inclusive e + e − production of Λ-hyperons in the ALEPH experiment atLEP [53], or in semi-inclusive deep inelastic scattering measurements by ZEUS [54]. More recently, an apparent signdiscrepancy [55] has been identified in comparisons of Sivers asymmetries in SIDIS measurements [56, 57] with Siversasymmetries extracted from h ↑ h → π + X collisions at the BNL Relativistic Heavy Ion Collider (RHIC) [50]. Proposedexplanations include a node in the Sivers function [55, 58], though recent fits disfavor this [59]. It is also possible thatthe apparent h ↑ h → π + X /SIDIS discrepancy is due to contributions beyond the Sivers effect [60, 61]. At present, thefull explanation remains unclear and more analysis is needed. At very high energies, interesting phenomena have alsoemerged, including the CMS ridge structure seen in proton-proton collisions at the LHC [62]. Explanations have beenproposed in terms of Wilson line interactions [63] and the color glass condensate formalism [64–66]. In addition, aforward-backward asymmetry in the production of top-antitop quark pairs has been observed at the Tevatron [67, 68],generating speculation about effects from physics beyond the standard model.Given the collection of observations in last few paragraphs, it is a natural stage now to investigate the possibilitythat TMD-factorization breaking effects lead to unexpected general patterns of behavior at leading power in thehard scale. This paper will demonstrate, by applying power counting to a specific example, that such an extra TMD-factorization breaking spin dependence can be induced by TMD-factorization breaking initial and final state soft gluoninteractions. As our illustrative example, we will use the production of a final state large invariant-mass back-to-backphoton-hadron pair in collisions of unpolarized protons. We will show that a single extra initial state soft gluon, ofthe type more commonly associated with gauge link effects, is sufficient to produce a correlation between the helicity The meaning of “generalized TMD-factorization,” as it is used in this article, is much more general than what was called “generalizedTMD-factorization” in Ref. [19]. The distinction will be explained in detail in Sect. IV of the main text.
FIG. 1. Diagram of the process in Eq. (1) in the coordinate system of frame-2. The final state hadronizing quark has transversepolarization S and the final state prompt photon has helicity λ . of the final state photon and the transverse spin of the exiting final state hadron. We choose this specific examplebecause it provides a relatively simple demonstration of an anomalous spin effect made possible by the breakdownof TMD-factorization while requiring only one extra soft/collinear gluon. However, the discussion is intended toillustrate the more general observation that spin correlations that would not ordinarily be expected to be large fromthe perspective a generalized TMD parton model can become leading-power in TMD-factorization breaking scenarios.The earlier examples of TMD-factorization breaking in Refs. [13–19] focused on the appearance of anomalous colorfactors in low order graphs. In this paper, the TMD-factorization breaking will also involve an anomalous contractionof intrinsic transverse momentum components with Dirac matrices, and so it will appear to come from a differentsource. At the root of all these examples, however, is the breakdown in compatibility between non-Abelian gaugeinvariance cancellations and the factorizability of the transverse momentum dependent cross section at leading powerin the region of small transverse momentum.The paper is organized as follows: In Sect. II, we describe the process to be used throughout the paper to illustrateTMD-factorization breaking and the appearance of an extra TMD-factorization breaking spin asymmetry. Section IIalso provides an overview of notation and conventions. In Sect. III, we give a brief summary of topics needed fordiscussions of TMD-factorization breaking, particularly in cases involving a TMD-gluon distribution in the initial state.In Sect. IV, we precisely state the criteria for what we will call the maximally general form of a TMD-factorizationhypothesis. Also in Sect. IV, we summarize the constraints that are placed on the spin and azimuthal dependenceof differential cross sections by this maximally generalized TMD-factorization hypothesis. In Sect. V we present thedetailed steps necessary to verify TMD-factorization at tree-level, and in Sect. VI we deal with spectator-spectatorinteractions. In Sect. VII we apply direct power counting to demonstrate a failure of TMD-factorization for the caseof a single extra soft gluon attaching to the initial and final state, and we isolate the leftover TMD-factorizationbreaking contributions. We show in Sect. VIII that power counting implies an extra TMD-factorization violating spincorrelation in the final state. In Sect. IX we summarize our main observations and make concluding remarks. II. SET UP AND NOTATIONA. The Process
For concreteness, we will focus on TMD-factorization breaking in the specific example of inclusive production of aback-to-back hadron-photon or jet-photon pair (see Fig. 1):Hadron 1 + Hadron 2 → Jet( S ) + γ ( λ ) + X . (1) P P k k k k Φ q Φ g + Cross Terms FIG. 2. A parton-model level picture showing the 2 → qg → qγ . The large transverse momentum of the final γ and jet/hadron fixes the hard scale. The jet/hadron and the realphoton are nearly back-to-back, so the cross section, which is differential in the total transverse momentum of thefinal state hadron-photon pair, is sensitive to the intrinsic transverse momentum of the constituent partons in thecolliding hadrons. Hence, TMD-factorization is a natural candidate description.The outgoing quark in Fig. 1 is meant to be representative of a final state hadron or jet with a transverse polarization S .Let P and P be the momenta of the incoming colliding hadrons and let k and k be the momenta of the finalstate jet and real photon respectively. The total momentum of the final state pair is defined to be q ≡ k + k , (2)and Q ≡ q is the hard scale. The kinematical region of interest is characterized by final state transverse momentaof order the total center-of-mass energy of the colliding hadrons:( P + P ) ∼ ( P + k ) ∼ ( P + k ) = O ( Q ) . (3)That is, the final state center-of-mass transverse momenta of k and k are large.The hard subprocess includes both q ¯ q → gγ and gq → γq channels, but here we are most interested in the gluon-initiated subprocess since it is this that (as we will later show) most immediately yields a non-trivial extra spinasymmetry from TMD-factorization breaking effects. A parton model level depiction of the gq → γq hard scatteringsubprocess is shown in Fig. 2; the hard collision is the result of a quark inside P scattering off a gluon inside P .For a complete pQCD treatment, the q ¯ q → gγ channel must also be accounted for, but an analysis of the gq → γq subprocess is all that will be needed to obtain our main result.In the treatment of the hard subgraphs, we will often use the kinematical notation:ˆ s ≡ ( k + k ) = ( k + k ) = q = Q , (4)ˆ t ≡ ( k − k ) = ( k − k ) = O ( Q ) . (5)We will also use a momentum variable defined as h ≡ k − k = k − k . (6)The proton mass is M p and the quark mass is m q . We are ultimately interested in spin correlations arising betweenthe final state jet and prompt photon from factorization breaking effects, so we have labeled the jet/hadron and thephoton with a spin vector, S , and a helicity, λ , in Eq. (1) and Fig. 1.A complete kinematical description of the cross section is closely analogous to the usual description of Drell-Yanscattering and e + e − -annihilation into back-to-back jets. Our notation will therefore closely follow the standard Drell-Yan conventions. (See Ref. [12], chapter 13.2.) The modifications to the usual notation needed for the special case ofEq. (1) will be discussed below.First, we define the relevant coordinate reference frames: P , P Center-of-Mass
For categorizing the sizes of momentum components of the incoming partons, it will be most convenient to workin the center-of-mass of the colliding P - P system, where P is moving in the extreme light-cone “+” direction and P is in the extreme “ − ” direction, and the total transverse momentum of the P - P system is zero. We will refer tothis as “frame-1.” Momentum components in frame-1 will be labeled by an f subscript. The positive z -axis in thisframe is along the direction of motion of P so that the four-momentum components of the incoming hadrons are P ,f = (cid:32) P +1 ,f , M p P +1 ,f , t (cid:33) ≈ (cid:16) P +1 ,f , , t (cid:17) , (7) P ,f = (cid:32) M p P − ,f , P − ,f , t (cid:33) ≈ (cid:16) , P − ,f , t (cid:17) . (8)The total four-momentum of the final state jet-photon pair in frame-1 is q f = k ,f + k ,f = ( q + f , q − f , q t,f ) . (9)In frame-1, therefore, small deviations of q t,f from zero are due to intrinsic transverse momentum in the collidinghadrons. The kinematical region of interest for studies of TMD-factorization and TMD-factorization breaking is where q t,f is of order Λ. (For the rest of this paper, Λ refers to a general hadronic mass scale like Λ QCD .)The momentum fractions x and x are defined as: k +1 ,f ≡ x P +1 ; k ,f ≡ x P − . (10)The incoming partons k and k are approximately collinear to their parent hadrons, so in frame-1 their componentsare of size k ,f ∼ (cid:0) Q, Λ /Q, Λ t (cid:1) , k ,f ∼ (cid:0) Λ /Q, Q, Λ t (cid:1) . (11)In frame-1, the components of k and k are all large: k ,f ∼ ( Q, Q, Q t ) , k ,f ∼ ( Q, Q, Q t ) . (12)It will also be convenient to define unit four-vectors characterizing the extreme forward and backward directions inframe-1: n ,f ≡ (1 , , t ) f , (13) n ,f ≡ (0 , , t ) f . (14)Most of the analyses of this paper will be performed in frame-1.At various stages, we will need only the frame-1 transverse components of a parton’s momentum, expressed infour-vector form. Therefore, we define the four-vector notation: v t ≡ (0 , , v t ) . (15)The transverse components of the metric tensor in frame-1 may be conveniently expressed as g µ µ t ≡ g µ µ − n µ , f n µ , f − n µ , f n µ , f . (16) k , k Center-of-Mass
We define “frame-2” to be the center-of-mass of the final state jet-photon system, and the corresponding momentumspace coordinates will be labeled by subscripts f . In this frame the orientation of the interaction planes is clearestand easiest to visualize.If the masses of the incoming hadrons (and the mass of the final state jet) are neglected, then in the center-of-massof the jet-photon system q f = ( Q, ) , (17) k ,f = ( Q/ , k f ) , (18) k ,f = ( Q/ , − k f ) , (19) P ,f = | P ,f | (1 , u ,f ) , (20) P ,f = | P ,f | (1 , u ,f ) , (21)where here we have reverted to standard four-vector notation v = ( v , v , v , v ). The energy of the jet-photon pair is Q , and u ,f , u ,f are unit spatial three-vectors in the directions of P and P respectively. In frame-2, the final statejet and real photon are exactly back-to-back, but the incoming hadrons P and P are slightly away from back-to-back(see Fig. 1). The region of very small intrinsic transverse momentum corresponds to almost exactly back-to-back P and P . The z -axis and x -axis can be covariantly fixed by defining the following four-vectors: Z µ ≡ (0 , u ,f − u ,f ) | u ,f − u ,f | , X µ ≡ (0 , u ,f + u ,f ) | u ,f + u ,f | . (22)The z -axis bisects the angle between P ,f and − P ,f , and the x -axis is orthogonal to the z -axis, so that togetherthey form a “ P , P -plane.” Hence, frame-2 is analogous to the Collins-Soper frame [69] for Drell-Yan scattering, butwith the lepton pair replaced by the jet-photon pair. The other plane, analogous to the lepton plane for Drell-Yanscattering, is formed by the jet-photon pair. We will call it the “ k , k -plane.” An illustration of the process in thecoordinate system of frame-2 is shown in Fig. 1. The polar angle of the jet axis with respect to the z-axis is θ f , andthe azimuthal angle with respect to the P , P -plane is φ f . k Boosted in the z-direction
Finally, it will be useful in some instances to work in the k , k center-of-mass frame with the z -axis lying along thedirection of k , so that k has its plus component boosted to order ∼ Q and k has its minus component boosted toorder ∼ Q . We will call this “frame-3” and denote the components of vectors in this frame with an f . In light-conecoordinates, k ,f ≈ (cid:16) k +4 ,f , , (cid:17) , (23)where k +4 ,f ∼ Q . We also define a vector n ,f = (0 , , t ) f , (24)so that n · k = k +4 ,f ∼ Q , and n ,f = (1 , , t ) f , (25)so that n · k = k − ,f ∼ Q . Frame-3 is simply related to frame-2 by a rotation. Most importantly for us, the transversespatial components of P and P in frame-3 are | ( P ) f | ≈ | ( P ) f | ≈ | P ,f | sin θ f = O ( Q ) , (26)for small intrinsic transverse momentum and wide angle scattering (see Fig. 1). B. Final State Spin Dependence
In later sections, we will be interested in the dependence on the polarization of the final state k in Fig. 1, andthe representation of spin will follow the notation and conventions in appendix A of Ref. [12]. In particular, the spinvector S of the final state quark in frame-3, where k is exactly in the z -direction, is S µ = ( S ,f , S x ,f , S y ,f , S z ,f )= ( λ k z ,f sign( k z ,f ) , m q b xt,f , m q b yt,f , λ k ,f sign( k z ,f )) . (27)The λ and b t,f are the components of the Bloch vector. The largest component of S is of order Q . The spin vectoris normalized with a maximum of − S ≤ m q , and the spin sum is (cid:88) s u ( k , S )¯ u ( k , S ) = ( /k + m q ) (cid:18) γ /S m q (cid:19) . (28)The polarization of the final state photon will be expressed in the usual way; the polarization four-vectors in frame-3are (cid:15) µλ =1 = (cid:18) , − √ , − i √ , (cid:19) f , (cid:15) µλ = − = (cid:18) , √ , − i √ , (cid:19) f , (29)where λ = ± P P k − P nj l ′ j k k Φ q Φ g H L H R l ′ n l ′ n +1 l l m l m +1 l p k − P n +1 j l ′ j FIG. 3. The structure of a general leading region.
C. Leading Regions and the Classification of Subgraphs
In perturbative QCD derivations of factorization in an arbitrary gauge, one must deal with extra soft and collineargluons that are allowed in a general leading region analysis, but which do not factorize topologically graph-by-graph.Where factorization is known to be valid, the derivations show that contributions from these extra gluons eithercancel in the sum over graphs, or factor into contributions corresponding to separate Wilson line operators, with eachWilson line operator belonging to a separately well-defined parton correlation function for each hadron. This standardfactorization applies to sums over all graphs and relies on applications of the QCD Ward identity (gauge invariance) todisentangle the extra gluons that connect different subgraphs with one another. A factorization proof must show that,after the application of the gauge invariance arguments and Ward identity cancellations, any remaining leading-powerfactorization breaking terms cancel in the sum over final states for sufficiently inclusive observables.Standard leading region arguments [72] (see also chapter 13 of [73] and chapter 5 of [12]), show that graphs with thetopology illustrated in Fig. 3 are the dominant ones at leading power for the process in Eq. (1). A generic Feynmangraph may have arbitrarily many soft/collinear gluons (labeled l j , with j running from 1 to m ) entering the hardsubgraph H L from the top gluon subgraph, Φ g . A quark enters H L from the lower quark subgraph, Φ q , along witharbitrarily many extra longitudinally polarized gluons l (cid:48) i (with i running from 1 to n ). Analogous statements applyto the right side of the cut in Fig. 3. We define k ≡ m (cid:88) j =1 l j , (30)so that the total momentum entering the hard subgraphs, H L and H R , matches the routing of momentum in tree-levelgraphs like Fig. 2. There are also important contributions from gluons that directly connect the Φ q and Φ g subgraphs,but to keep the diagram simple we have not shown them explicitly in Fig. 3 (see, however, Sect. VI). The soft andcollinear gluons may cross the final state cut, and this is included in the definitions of the Φ q and Φ g bubbles.From here forward, the subscripts f will be dropped, and it will be assumed to be implicit that we are working inframe-1 except when specified otherwise. In our notation, Lorentz indices denoted by Greek letters are intended toinclude all four components of a four-momentum, while indices denoted with i or j include only the frame-1 transversespatial components.The differential cross section for an arbitrary number of extra soft/collinear gluons is expressed as dσ = C Q (cid:88) X (cid:88) X (cid:88) m,p,n,q (cid:90) d k (2 π ) (cid:90) d l (2 π ) · · · (cid:90) d l m (2 π ) (cid:90) d l m +1 (2 π ) · · · ×× (cid:90) d l p (2 π ) (cid:90) d l (cid:48) (2 π ) · · · (cid:90) d l (cid:48) n (2 π ) (cid:90) d l (cid:48) n +1 (2 π ) · · · ×× (cid:90) d l (cid:48) q (2 π ) | M | × (mom . conserving δ − functions) , (31) P P k − P nj l ′ j k k l ′ n l l m U ( P ; { l j } ) µ ′ ...µ ′ m L ( P , k ; { l i } ) ρ ′ ...ρ ′ n H L ( k , k , k ; { l j } ; { l i } ) µ ...µ m ; ρ ...ρ n FIG. 4. A generic contribution to the amplitude with the different pieces of Eq. (32) labeled. where M is the amplitude in Fig. 4, and the integrals over the momenta entering or exiting the hard subgraphs arewritten explicitly. The inclusive sums and integral over final states in the upper and lower bubbles of Fig. 3 arerepresented by (cid:80) X (cid:80) X . The momentum conserving δ -functions fix k + k = k + k = q , with the definition of k in Eq. (30). For maximum generality, we leave the overall numerical normalization C unspecified in Eq. (31). Theessential requirement is only that the cross section is differential in q t — note that there is no integration over q t inEq. (31).Inspection of the tree-level graphs (see, e.g., Sect. V) verifies that the power-law for M is M ∼ Q . The phasespace integrals of Eq. (31) are Lorentz invariant and so do not contribute extra powers of Q to the cross section whenboosting frames. In later sections we will analyze only the relative sizes, in powers of Q , of the factors that combineto form | M | . Therefore, the right-hand side of Eq. (31) is defined with a normalization factor Q so that its overallpower-law dependence is the same as for the amplitude, dσ ∼ Q (up to logarithmic factors). The overall Q factor isto compensate for the two powers of 1 /Q that come from the momentum conserving delta functions for k +1 and k − .We are mainly interested in the dependence of the cross section on powers of Q , so to keep notation simple we willquote the power-behavior of any factor as a power of Q only, with any other factors of order a hadronic mass scale,necessary for maintaining correct units, left implicit.Since the cross section is order ∼ Q , then when a particular term in the perturbative expansion is of a higherpower of Q than zero it is called “superleading.” In a complete analysis including all graphs, gauge invariance ensuresthat all such terms cancel exactly against other superleading terms.In later sections we will find it simplest to work at the amplitude level and to consider the contribution to the crosssection only in the last step. We will organize the analysis of amplitudes according to Fig. 4. A specific diagrammaticcontribution can be broken into upper and lower subgraphs, U and L , and a hard part, H L , as follows: M = U ( P ; { l j } ) µ (cid:48) ...µ (cid:48) m g µ (cid:48) µ · · · g µ (cid:48) m µ m ×× ¯ u ( k , S ) H L ( k , k , k ; { l j } ; { l i } ) λ ; µ ...µ m ; ρ ...ρ n ×× g ρ (cid:48) ρ · · · g ρ (cid:48) n ρ n L ( P , k ; { l i } ) ρ (cid:48) ...ρ (cid:48) n . (32)The Lorentz indices for gluons that connect U and L to the hard subgraph H L are shown explicitly. In our notation,the final state quark wave function ¯ u ( k , S ) is written separately rather than being included in H L because this willbe convenient for power counting arguments in later sections. H L and L also carry Dirac indices, though to maintainmanageable notation, we do not show these explicitly. The subgraphs U , L and H L are defined to include the sum ofall subgraphs corresponding to a given set of m external gluons for U , n external gluons for L , and m + n externalgluons for H L . However, this over counts the number of graphs by a factor of m ! × n !. To compensate for this,therefore, we define the U subgraph to also include a factor of 1 /m !, and the L subgraph to include a factor of 1 /n !.All of our analysis will be done using Feynman gauge where the analytic properties of Feynman diagrams reflectrelativistic causality and the main issues concerning TMD-factorization and TMD-factorization breaking are clearest.Lorentz indices will by denoted by µ, ν, ρ, σ , transverse components will be denoted by ij , and color indices will bedenoted by α, β, δ, κ .Actual derivations of factorization, where they are valid, involve a number of important points that will not betouched on in this paper, both because they are not directly relevant to the specific issues under consideration andbecause generalized TMD-factorization is anyway known to be violated for processes like Eq. (1). Those other issuesinclude, for example, rapidity divergences and other topological Wilson line issues that must be fully addressed in0situations where TMD-factorization is valid (see, for example, Refs. [74–80]), and their relationship to the evolutionof separate, well-defined TMD PDFs [12]. Furthermore, in the integrations over intrinsic transverse momentum inEq. (31), propagators in the hard part may go on-shell whenever any of the transverse momenta become too large.Therefore, in this paper it will be assumed that all such transverse momentum have upper cutoffs to prevent this,since the main concern is with transverse momenta of the order of hadronic mass scales or smaller. III. SUPERLEADING TERMS, GAUGE INVARIANCE, AND THE GLAUBER REGIONA. Dominant Subdiagrams
Recall that, in frame-1, P is highly boosted in the plus direction while P is highly boosted in the minus direction.The graphs that individually contribute superleading terms to the cross section — i.e., with a power-law behaviorhigher than that of the cross section itself — are those in which all the components of U are in the minus direction: U ( P ; { l j } ) µ (cid:48) ...µ (cid:48) m → U ( P ; { l j } ) − , − , − ,..., − . (33)The situation here is the same as was addressed in Ref. [81] in connection with ordinary collinear factorization andcollinear gluon PDFs. Gauge invariance ensures that, in the sum over all diagrams, any superleading contributionsto a physical cross section cancel. The way this is demonstrated relies on the same basic Ward identity argumentsthat are also needed to identify gauge link contributions and verify normal factorization theorems in processes wherethey apply [6, 82].The observation from Ref. [81] that is most relevant to this paper is that, although the superleading powers of Q cancel after the sum over all graphs, contributions from subgraphs like Eq. (33) generally do not. (See, e.g., thethird term in braces in Eq. (83) of Ref. [81].) To understand this, recall that the minus component of polarization ofthe virtual k -gluon in hadron-2 is not exactly the same as its component along the direction of k because k alsohas order ∼ Λ t transverse components. So a minus polarization is not exactly the same as a polarization along thedirection of k , making the Ward identity cancellations more delicate.In spite of the non-vanishing leading-power contribution that arises from U subgraphs like Eq. (33), the cross sectionfor the collinear case addressed in Ref. [81] does factorize, as expected, into the well-known and well-defined collineargauge-invariant correlation functions. But even in that case the uncanceled leading-power terms from subdiagramslike Eq. (33) remain as contributions from the gauge link, and are therefore critical for maintaining consistency withgauge invariance at leading power.In later sections, we will encounter other complications from subgraphs like Eq. (33) in the treatment of generalizedTMD-factorization breaking scenarios like Eq. (1). B. Grammer-Yennie Method
Factorization in covariant gauges is very conveniently addressed with the Grammar-Yennie approach [83], which webriefly review here. In this method, each extra gluon’s propagator numerator is split into two terms that we call the“G-term” and the “K-term.” For a gluon with momentum l attaching U to the rest of the graph in Fig. 4, we writethe decomposition as g µ (cid:48) µ = K ( l ) µ (cid:48) µ + G ( l ) µ (cid:48) µ . (34)The G and K terms are defined as K ( l ) µ (cid:48) µ ≡ n µ (cid:48) l µ n · l (35) G ( l ) µ (cid:48) µ ≡ g µ (cid:48) µ − n µ (cid:48) l µ n · l . (36)Note the explicit dependence of G ( l ) and K ( l ) on gluon four-momentum. The decomposition breaks the diagram intoterms whose properties have a clear interpretation in terms of Ward identities and gauge invariance. The motivationfor the separation is that it systematizes the identification of dominant and subdominant contributions to a subgraph,while automatically (and exactly) implementing any Ward identity cancellations that come from contracting the gluonmomenta l µ { j } with the rest of the graph in Fig. 4. Notice that in Eqs. (35, 36) it is the exact l µ that appears in the1numerators rather than just l − . After expressing the graphs in terms of G and K terms, it becomes clear how toapply valid approximations while at the same time maintaining gauge invariance.Equation (32) has been written with the metric tensors from the propagator numerators of extra gluons displayedexplicitly. This is in preparation for later sections where we will implement the Grammer-Yennie decomposition oneach extra gluon and examine the power law behavior of all K and G combinations in each Feynman graph.If l ∼ (cid:0) Λ /Q, Q, Λ t (cid:1) in Eq. (35,36) so that it is collinear, then power counting with the G and K terms isstraightforward. For example, it is the g + − component of its propagator numerator in Eq. (32) that gives thedominant power, and the K + − term in Eq. (34) reproduces this exactly. As long as l ∼ (cid:0) Λ /Q, Q, Λ t (cid:1) , the othercomponents of K µ (cid:48) µ and G µ (cid:48) µ are suppressed relative to the K + − components. However, the simple power countingfails to apply directly if l is integrated into different momentum regions such as the Glauber region. C. The Glauber Region
Many issues surrounding factorization and TMD factorization breaking are closely connected to the treatment ofthe Glauber region in the integrations over extra gluons in Fig. 4, so we briefly review the basic issue here.Consider, for example, a virtual gluon l µ attaching U and H L in Fig. 4. The transverse momentum may be ofany size, though for addressing TMD-factorization we are especially interested in the contribution from the regionwhere l t ∼ Λ t . The integral over the l + component is then trapped by spectator poles at sizes of order ∼ Λ /Q . Torecover factorization, l must either be strictly soft (all components of order Λ) or strictly collinear. However, in theintegration over l − there are generally leading regions where | l + l − | << | l t | . This is the Glauber region, where theWard identities arguments that apply to ordinary soft or collinear gluons fail. To justify the steps of a factorizationderivation, it must either be shown that it is possible to deform all contour integrals away from any Glauber regionsso that they can be treated as properly soft or properly collinear, or the cross section must be sufficiently inclusivethat any Glauber pole contributions cancel in the inclusive sum over final states. The potential for Glauber gluonsto spoil factorization derivations has been noted long ago (e.g., Ref. [84]). IV. CONSTRAINTS FROM TMD-FACTORIZATION
To ensure the clarity of later results, we must begin with a precise and detailed statement of what is meant whenwe speak of the usual TMD-factorization-based assumption as applied to Eq. (1). The purpose of this section is tomake a clear statement of what, for this paper, we will call the“maximally general” criteria necessary for a versionof TMD-factorization to be said to hold. We will also summarize the constraints imposed by these assumptions andbriefly discuss what it would mean for them to be violated.
A. TMD-Factorization With Maximally General Criteria
Our initial statement of the TMD-factorization hypothesis must be very general since the goal of later sectionsis to demonstrate that TMD-factorization fails for Eq. (1) even in a very loose sense. As reviewed in Sect. II C, afull derivation of TMD-factorization must account for the extra soft/collinear gluon connections of the type shownin Fig. 3. Thus, our statement of the maximally generalized TMD-factorization hypothesis is that, for N such extragluons, in the sum over all graphs at a given order, the differential cross section in Eq. (31) must be expressible inthe form dσ = M (cid:88) ij (cid:90) d k t (2 π ) Φ g/P (cid:16) P ; ˆ k ( x , q t − k t ) (cid:17) ij (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q Tr (cid:20) ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / ˆ H i, λ L (cid:16) ˆ k ( x , k t ) , ˆ k ( x , q t − k t ) , k , k (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q ×× Φ q/P (cid:16) P ; ˆ k ( x , k t ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ˆ H j, λ R (cid:16) ˆ k ( x , k ) , ˆ k ( x , q t − k t ) , k , k (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / (cid:21) + O ((1 /Q ) a ) , (37)with a > g/P and Φ q/P , need not necessarily correspond to gauge invariant operator definitions, within our maximallygeneral criteria, and they may be highly process dependent. They may also carry any number of non-spacetime indicessuch as color (not shown explicitly). The polarization projections on Lorentz and Dirac indices for partons enteringthe hard subgraphs, ˆ H i, λ L and ˆ H j, λ R , are left as yet unspecified. Dirac indices will not be written explicitly. The M in front of the integration is intended to represent a possible collection of matrices in non-spacetime indices such ascolor, and it includes any overall numerical factors. The power law of Eq. (37) matches that of Eq. (31) for tree-levelgraphs like Fig. 2. With the horizontal braces beneath the equation, we have indicated the powers of Q for eachfactor, and we will use this notation throughout the paper to aid the reader in following the counting of powers of Q . The way that the factors in Eq. (37) arise in a tree-level, parton model treatment will be reviewed in Sect. V.The ordinary, bare-minimum kinematical parton model approximations needed to write down a factorized equationhave been applied to initial state parton momenta in Eq. (37). That is, the minus component of k is ignored outsideits parent subgraph, Φ q/P , and is integrated over within the definition of Φ q/P (whatever that exact definition mightbe). Likewise, the plus component of k is ignored outside Φ g/P and is integrated over in the definition of Φ g/P . Theuse of approximate parton momenta outside Φ q/P and Φ g/P is symbolized by the hats on ˆ k , ˆ k and ˆ H L/R . Thus,the factorization applies after a mapping from exact to approximate k , k → ˆ k ( x , k t ) = (cid:16) ˆ k +1 (cid:0) k +1 , k t (cid:1) , ˆ k − (cid:0) k +1 , k t (cid:1) , ˆk t (cid:0) k +1 , k t (cid:1)(cid:17) , (38)is implemented outside Φ q/P , and a mapping from exact to approximate k , k → ˆ k ( x , q t − k t ) = (cid:16) ˆ k +2 (cid:0) k − , k t (cid:1) , ˆ k − (cid:0) k − , k t (cid:1) , ˆk t (cid:0) k − , k t (cid:1)(cid:17) , (39)is implemented outside Φ g/P . The components of the approximate hatted momentum variables, ˆ k µ and ˆ k µ , are writtenas functions only of k +1 , k t and k − , k t respectively, as demanded by the minimal kinematical approximations. Theonly requirement needed in order for maximally generalized TMD factorization to be said to be valid is that themapping from exact to approximate momenta be independent of k − and k +2 , and that the approximate momentaobey the basic partonic four momentum conservation law, ˆ k + ˆ k = k + k = q . The precise definitions of the ˆ k andˆ k mappings may also depend on physical external hadron momenta, though this is not shown explicitly. The hardsubgraphs in Eq. (37) have been replaced by their approximate versions, which use the hatted momentum variables: H L/R ( k , k , k , k ) → ˆ H L/R (cid:16) ˆ k ( x , k t ) , ˆ k ( x , q t − k t ) , k , k (cid:17) . (40)In a complete derivation of factorization, a basic step in the derivation would be to precisely formulate the exact-to-approximate mapping in Eqs. (38, 39) such that a useful version of factorization is maintained to arbitrary order insmall α s in the hard part. For our purposes, simply dropping the k − and and k +2 components outside their respectiveparent subgraphs would be a sufficient prescription, but we have left the exact transformation laws unspecified inEqs. (38, 39) to ensure that the later demonstration of TMD-factorization breaking is completely general. Our onlyrequirement for the mapping is that k − and k +2 are neglected outside Φ (1 g ) q/P and Φ (1 g ) g/P .To summarize, we say that the maximally general TMD-factorization hypothesis is respected if, for N extrasoft/collinear gluons in Fig. 3, Φ q/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) depends only on the momenta P and ˆ k ( x , k t ) and has the power-lawbehavior in frame-1 of an ordinary quark TMD PDF. It may be any matrix in non-spacetime indices,such as color, that contract with other factors in the factorization formula or with M . The Diracpolarization components of Φ q/P may depend only on the target hadron momentum P and a targetquark momentum k . Φ g/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) ij depends only on the momenta P and ˆ k ( x , k t = q t − k ) and has thepower-law behavior in frame-1 of an ordinary gluon TMD PDF. It may be any matrix in non-spacetime indices that contract with the other factors in the factorization formula or with M . Theframe-1 transverse Lorentz components, ij , may depend only on the total transverse momentum, k t ,of the incident gluon from hadron-2. The way powers of large momentum ∼ Q are partitioned between the hard factors and the effective TMD PDFs differs from someconventions — ours is chosen to simplify the power counting of later sections. ˆ H L/R (cid:16) ˆ k ( x , k t ) , ˆ k ( x , k t ) , k , k (cid:17) ij depends only on the momenta ˆ k ( x , k t ), ˆ k ( x , k t ), k , and k , and has the power-law behavior of the tree-level hard subgraph. It may also be an arbitrary matrixin non-spacetime indices that contract with other factors in the factorization formula or with M . The only exception to the first two criteria is that Φ q/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) may include dependence onan auxiliary unit vector approximately equal to n , to account for possible gauge link operators, andΦ g/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) ij may include dependence on an auxiliary unit vector approximately equal to n .To be consistent with the maximally generally TMD-factorization assumption, the effective TMD PDFs and the hardparts in Eq. (37) may have complicated process dependence and may break gauge invariance as long as they satisfythese general conditions.Finally, note that the gluon TMD PDF in Eq. (37) only includes a sum over the transverse indices, i and j , for thegluon polarization, and not a sum over longitudinal indices. Within generalized TMD-factorization approaches, it isusually assumed from the outset that the basic polarization structure of the gluon subgraph is like that of an ordinarygluon number density, with only transverse components for a single on-shell gluon moving in the large minus directioncontributing. Therefore, we include in the definition of maximally generalized TMD-factorization the assumption thateffective gluon TMD PDFs always have two transverse Lorentz indices and depend on only one small transverse gluonmomentum.Notice that the criteria enumerated above are less constraining than in what was referred to as “generalized TMD-factorization” in Ref. [19]. There it was required at least that separate gauge invariant (albeit process dependent)TMD PDFs be identifiable for generalized TMD-factorization to be said to hold. In this article, we require only ageneral separation into different transverse momentum dependent blocks in Eq. (37) with the properties listed above.
B. Standard Classification Scheme
In this subsection, we review the conventional steps for extracting specific azimuthal angular or spin dependencein a TMD-factorization formalism based on the set of very general criteria laid out in the last subsection. Thetypes of possible angular and spin dependence are attributed to special TMD PDFs that are defined with non-trivial polarization projections. The standard classification strategy, therefore, is to first enumerate the leading-power projections of Dirac structures in the effective unpolarized quark factor, Φ q/P , and the leading projections ontransverse Lorentz components in the effective unpolarized gluon TMD PDF Φ g/P in Eq. (37). The result is a set offunctions that are then identified with the various possible TMD PDFs.For the leading power unpolarized gluon TMD PDF, the only dependence on transverse momentum allowed bycondition 2.) of Sect. IV A is from the total gluon transverse momentum k t , and the only polarization dependenceis from the transverse components i and j . Therefore, the leading power unpolarized gluon TMD PDF may bedecomposed into the sum of a polarization-independent term and a polarization-dependent term: (cid:104) Φ g/P (cid:16) P ; ˆ k ( x , q t − k t ) (cid:17)(cid:105) ij = − g ijt Φ U g/P (cid:16) P ; ˆ k ( x , q t − k t ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q + (cid:32) k i t k j t M P + g ijt k t M P (cid:33) Φ BM g/P (cid:16) P ; ˆ k ( x , q t − k t ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q = P ij U Φ U g/P (cid:16) P ; ˆ k ( x , q t − k t ) (cid:17) + P BM ( k ) ij Φ BM g/P (cid:16) P ; ˆ k ( x , q t − k t ) (cid:17) . (41)Here, Φ U g/P is the gluon TMD PDF for an unpolarized gluon and Φ BM g/P is a gluon Boer-Mulders [85, 86] function. Inthe last line, we have defined projections onto transverse indices:P ij U ≡ − g ijt , (42)P BM ( k t ) ij ≡ (cid:32) k i t k j t M P + g ijt k t M P (cid:33) . (43)An auxiliary gauge-link vector n adds no other structure since it has no transverse components.4The unpolarized quark TMD PDF, Φ q/P , is decomposed into a complete basis of Dirac matrices,Φ q/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) = S + γ P + γ µ V µ + γ µ γ A µ + 12 σ µν T µν ∼ Q , (44)and we keep only the leading power components. Since Φ q/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) depends only on P and ˆ k ( x , k t ),which in frame-1 have large (order Q ) components in the plus direction, then the leading terms in Eq. (44) are thosewith a µ = + contravariant index. The TMD PDFs are therefore obtained from Dirac traces that project those largecomponents. For example, the largest component of the vector term, γ µ V µ , is γ − V + ∼ Q . (45)The ordinary azimuthally symmetric and unpolarized TMD PDF is therefore identified with the projectionΦ[ γ + ] q/P (cid:16) P ; ˆ k ( x , k ) (cid:17) ≡ V + = 14 Tr (cid:104) γ + Φ q/P (cid:16) P ; ˆ k ( x , k ) (cid:17)(cid:105) ∼ Q . (46)In this expression, we have adopted the notation of Ref. [87] by including a [ γ + ] to indicate the specific Dirac projection.The only dependence on external momenta is through P and k , so the largest Dirac components of Φ[ γ + ] q/P areproportional to /P ∼ /k ∼ Q . The trace in Eq. (46) then takes the general form,Φ[ γ + ] q/P (cid:16) P ; ˆ k ( x , k ) (cid:17) ∝
14 Tr (cid:2) γ + /P (cid:3) ∼ Q . (47)An auxiliary gauge-link vector /n introduces no further leading-power projections because it only leads to additionalfactors of γ + in Eq. (47).For a projection with a generic Dirac structure Γ q , the corresponding quark TMD PDF isΦ [Γ q ] q/P (cid:16) P ; ˆ k ( x , k ) (cid:17) = 14 Tr (cid:104) Γ q Φ q/P (cid:16) P ; ˆ k ( x , k ) (cid:17)(cid:105) ∼ Q . (48)The decomposition in Eq. (44) spans the set of possible independent Dirac structures for a quark TMD PDF, andeach Γ q is associated with a characteristic type of angular or spin dependence in the cross section. In the TMD-factorization-based classification scheme, the types of leading-power quark TMD PDFs are categorized according tothe possible Dirac projections Γ q that contribute at leading power. For example, the projection,Γ q → γ + (49)reproduces the ordinary azimuthally symmetric and unpolarized TMD PDF in Eq. (46). Another well-known exampleis the quark Boer-Mulders [85] function, obtained from the projection,Γ q → γ + γ j γ . (50)If hadrons 1 and 2 are also polarized then there is a large collection of additional leading-power TMD PDF structuresinvolving hadron spins. These have been thoroughly classified at least to leading twist [87, 88].So the maximally general statement of the TMD-factorization conjecture of Sect. IV A, when specialized to aparticular set of projections, Γ g and Γ q , is dσ [Γ g , Γ q ] = M (cid:88) ij (cid:90) d k t (2 π ) Φ [Γ g ] g/P (cid:16) P ; ˆ k ( x , q t − k t ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q Φ [Γ q ] q/P (cid:16) P ; ˆ k ( x , k t ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q × P Γ g ( q t − k t ) ij Tr (cid:20) ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / H i, λ L (cid:16) ˆ k ( x , k t ) , ˆ k ( x , q t − k t ) , k , k (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q ×× P Γ q H j, λ R (cid:16) ˆ k ( x , k t ) , ˆ k ( x , q t − k t ) , k , k (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / (cid:21) . (51)This is obtained by using Eqs. (41,44,48) in Eq. (37). The P Γ q on the last line of Eq. (51) is the relevant Dirac matrixfrom Eq. (44) corresponding to projection Γ q . (In the unpolarized azimuthally symmetric case, P Γ q → γ − .) The5Γ g = { U , BM } labels which gluon polarization projection is taken from Eq. (41). The superscript [Γ q , Γ g ] on dσ is toindicate that Eq. (51) is the contribution to the cross section from that particular combination of Γ q and Γ g targetquark and gluon polarization projections.Throughout this section, we have used the language of a generalized parton model by referring to Φ [Γ q ] q/P and Φ [Γ g ] g/P as though they were ordinary TMD PDFs. However, in general pQCD diagrams like Fig. 3, the Φ’s are merely labelsfor the target subgraphs corresponding to hadrons 1 and 2, each of which may be linked to the rest of the process viaarbitrarily many soft and collinear gluon exchanges. Strictly speaking, TMD PDFs acquire a well-defined meaningin pQCD only after a factorization derivation has shown how those extra soft and collinear gluons separate intoindependently defined factors in the sum over all graphs.Because of its very general form, there is a strong temptation to apply the factorized classification scheme ofthis subsection very broadly, even in processes like Eq. (1) where generalized TMD-factorization formally fails [19].But even the maximally general version of the TMD-factorization conjecture from this section imposes significantconstraints on the possible general behavior of the cross section at leading power. For example, the power countinglogic used to identify the leading TMD functions in Eq. (41) and Eq. (44) relies crucially on the assumptions that,a.) [Φ g/P ] ij depends at leading power only on transverse Lorentz indices and the momenta P , k − and k t and,b.) Φ q/P depends only on momenta P , k +1 and k t . When generalized TMD-factorization is broken, we must inprinciple account for the possibility that dependence on extra external momenta leaks into the Φ g/P and Φ q/P fromother subgraphs via the extra soft and collinear gluon exchanges in Fig. 3. However, in confronting this possibility weare no longer justified in assuming from the outset that, for example, the µ = + components in Eq. (44) are the onlydominant ones. Any Dirac structure projected from Eq. (44) must be viewed as a potential candidate leading-powereffect. Analogously, the decomposition of [Φ g/P ] ij may include terms beyond those of Eq. (41), involving polarizationsinduced by internal intrinsic transverse momenta other than k t , and polarization configurations like Eq. (33) mustalso be accounted for. We will discuss the implications of this in more detail in the next subsection.For the sake of clarity, our discussion has so far focused on the TMD PDFs, though the same observations applyto a treatment of final state hadronization, i.e. to the treatment of a quark fragmentation or jet function. Theuse of TMD-factorization, in the very general form summarized in this section, has become standard for classifyingleading-power TMD effects. It is a valid method, of course, in processes where a form of TMD-factorization holds oris expected to hold (such as in SIDIS or Drell-Yan). For TMD-factorization breaking processes, however, we will findthat the structure in Eq. (51) turns out to be overly restrictive. C. Parity, TMD PDFs and TMD-Factorization Breaking
A TMD-factorization conjecture imposes even stronger constraints on the qualitative structure of cross section whenconsidered in combination with discrete symmetries of QCD. Discrete symmetry arguments have long been useful forconstraining the properties of parton correlation functions in pQCD. It is now known, however, that such argumentsare more subtle in processes that involve TMD-factorization than in similar collinear cases, due to the non-trivialrole of gauge links in TMD-factorization. Early on, for example,
T P invariance was applied in Ref. [89] to arguethat the Sivers function should vanish in hard processes. That derivation, however, neglected effects from gauge linksin the definitions of TMD PDFs. As the role of gauge links in the derivation of TMD-factorization became betterunderstood, it was realized that
T P invariance implies not that the Sivers function vanishes, but rather that it reversessign between the Drell-Yan process and SIDIS [26, 90, 91], thus providing a non-trivial prediction for a specific typeof non-universality.Such observations provide motivation to also examine how arguments based on discrete symmetries are affectedwhen TMD-factorization breaks down altogether. As an example, let us return to the quark TMD PDF’s Diracdecomposition in Eq. (44). In addition to Eq. (49), naive power counting would imply another leading-power Diracstructure formed by the axial vector term γ γ − A + ∼ Q . (52)So one should also include in the categorization of leading projections,Γ q → γ γ + . (53)Therefore, by analogy with Eq. (46), a TMD function defined asΦ[ γ γ + ] q/P (cid:16) P ; ˆ k ( x , k ) (cid:17) ≡
14 Tr (cid:104) γ γ + Φ q/P (cid:16) P ; ˆ k ( x , k ) (cid:17)(cid:105) ∼ Q (54)6should be included within the classification of leading-power quark TMD PDFs. The interpretation of this TMD PDFin a generalized parton model/TMD-factorization approach would be that it describes a distribution of quarks withan intrinsic helicity polarization inside an unpolarized hadron. Such TMD PDFs are forbidden by parity invariancefor an unpolarized parent hadron for any TMD PDF definitions that satisfy the basic requirements of the generalizedTMD-factorization hypothesis enumerated in Sect. IV A. In Feynman graph calculations, this appears in the inabilityto construct a pseudo-scalar using only the four-momenta from hadron-1. At least four different four-momentumvectors are needed to give a non-vanishing trace. Specifically, an evaluation of Eq. (54) always leads to a Dirac traceof the form, Φ[ γ γ + ] q/P (cid:16) P ; ˆ k ( x , k ) (cid:17) ∝
14 Tr (cid:104) γ γ + / ˆ k t /P (cid:105) = 0 . (55)There are not enough four momentum vectors involved to give a non-vanishing trace. A gauge link simply introducesmore ∼ n vectors, which only project extra factors of γ + in the Dirac trace:Φ[ γ γ + ] q/P (cid:16) P ; ˆ k ( x , k ) (cid:17) ∝
14 Tr (cid:104) γ γ + / ˆ k t /P /n (cid:105) = 14 Tr (cid:104) γ γ + / ˆ k t /P γ + (cid:105) = 0 . (56)So even including gauge links does not allow for projections of this type within a TMD-factorization conjecture. TMDPDFs like Eq. (55), and any effects that would follow from them, are constrained to vanish at leading power withinthe minimal TMD-factorization criteria of Sect. IV A and IV B. For this reason, they might be referred to as “naivelyparity violating.”To further investigate the consequences of discrete symmetries in a generalized TMD-factorization framework, wewill focus in later sections on the final state parton polarizations — the helicity λ of the prompt photon and thethe transverse spin S , ⊥ of the outgoing quark in Fig. 1. If the initial state partons have no helicity then the hardsubprocess q + g → q ( S , ⊥ ) + γ ( λ ) (57)violates the helicity conservation of massless QCD. For the S , ⊥ − λ correlation to be possible, a helicity must becarried by the q or g inside its parent unpolarized hadron before the hard collision. In a generalized TMD factorizationframework, this would imply a type of naively parity violating TMD PDF.After showing that the criteria of Sect. IV A fail to hold in a more detailed treatment of the extra soft/collineargluons of Fig. 3, we will ultimately demonstrate in Sect. VIII that final state S , ⊥ - λ correlations are actually leadingpower in spite of the apparent helicity non-conservation in the naive 2 → S , in Eq. (44) is at most of order a hadronic mass within the TMD-factorizationframework of Sect. IV A, and so would be counted as subleading. But if, because of the extra gluon connections inFig. 3, Φ q/P (cid:16) P ; ˆ k ( x , k ) (cid:17) were allowed to depend on other external momenta,Φ q/P (cid:16) P ; ˆ k ( x , k ) (cid:17) −→ Φ q/P (cid:16) P ; ˆ k ( x , k ) , q, P , k . . . (cid:17) , (58)then S could acquire dependence on large scales like P · P and q · k t . Then the usual power counting arguments forΦ q/P become invalid.The pseudo-scalar term, γ P , in Eq. (44) is another example that is both subleading and forbidden by parityinvariance in the TMD-factorization framework. It will be the main example that we will use in Sect. VIII todemonstrate a leading-power TMD-factorization breaking final state spin correlation. The projection is simplyΓ q → γ (59)and the substitution in Eq. (51) is P Γ q Γ q → γ = γ . (60)If one works within the TMD-factorization conjecture by following the classification scheme of Sect. IV B, then onewould define a TMD PDF corresponding to Eq. (48) using the projection in Eq. (59):Φ [ γ ] q/P (cid:16) P ; ˆ k ( x , k ) (cid:17) ≡
14 Tr (cid:104) γ Φ q/P (cid:16) P ; ˆ k ( x , k ) (cid:17)(cid:105) ∝ m q Tr [ γ ] = 0 . (61)7So, the possibility of any spin correlation effects arising from such projections would be discounted in the standardTMD-factorization classification scheme.We will return again to the projection in Eq. (59) in Sect. VIII where we will confront TMD-factorization breakingeffects. D. Maximally General TMD-factorization at the Amplitude Level
The conditions enumerated in Sect. IV A are sufficiently weak that it becomes straightforward to separate, at theamplitude level, contributions that are definitely consistent with the maximally general TMD-factorization criteriafrom those which may violate TMD-factorization if left uncanceled. This will be useful for later sections where muchof the analysis is performed at the amplitude level.Assume that for a fixed number m and n of extra gluons in Eq. (32) one is able to sum a set of Feynman diagramsto obtain a contribution to the amplitude of the form M = N (cid:88) i U eff ( P ; k , { l , . . . , l m } ) i (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / H L ( k , k , k , k ) i, λ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L eff ( P ; k , { l (cid:48) , . . . , l (cid:48) n } ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / . (62)Here we have changed variables from l to k ≡ (cid:80) mj l j . The factor N can represent any matrix of non-spacetimeindices such as color. The hard subgraph H L depends, as in Sect. IV A, only on the momenta k , k , k and k .Assume further that the effective U eff and L eff factors take the forms, U eff ( P ; k , { l , . . . , l m } ) i = F U ( P ; k , { l , . . . , l m } ) µ (cid:48) ··· µ (cid:48) m P U ( k t ) iµ (cid:48) ··· µ (cid:48) m , (63)and L eff ( P ; k , { l (cid:48) , . . . , l (cid:48) n } ) = F L ( P ; k , { l (cid:48) , . . . , l (cid:48) n } ) ρ (cid:48) p ··· ρ (cid:48) n P L ( P , k t ) ρ (cid:48) p ··· ρ (cid:48) n . (64)By substituting Eq. (62) into Eq, (31) and applying the minimal kinematical approximations of Eqs. (38-40), oneimmediately recovers the structure in Eq. (37).Note that F U and F L do not share any l -momentum arguments. Since there is no direct dependence on { l , . . . l m } outside U eff , the part of Eq. (63) labeled by F U , which does not depend on transverse polarization components (labeled“ i ”), may have dependence on any of these extra gluon momenta and still maintain consistency with the maximallygeneral TMD-factorization criteria of Sect. IV A. In the squared amplitude, these momenta are simply integrated overinside Φ g/P . However, the polarization-dependent part, labeled by P U ( k t ) i , couples to the transverse “ i ” componentof the hard part in Eq. (62), so it is allowed to depend only on the total transverse momentum, k t , coming out of thegluon subgraph. Without such a requirement, the criteria from Sect. IV A can be violated, and the decomposition ofthe effective gluon TMD PDF into polarization dependent functions in terms of k t , as in Eq. (41), would in generalbe incomplete. The integrations over l t · · · could not be performed independently of the contractions in the hardpart.Similarly, there is no direct dependence on { l (cid:48) , . . . l (cid:48) n } outside L eff , so the non-polarization dependent part ofEq. (64), labeled by F L , may depend on any of these extra gluon momenta. In the squared amplitude, they will beintegrated over inside the definition of Φ q/P . However, the projection onto Dirac components, labeled by P L ( P , k t ),can only depend on P and the total momentum k of the quark coming out of hadron-1.A potential violation of the criteria of Sect. IV A would arise, for example, from a contribution in which one isforced to write Eq. (63) in the form, U eff ( P ; k , { l , . . . , l m } ) i = F U ( P ; k , { l , . . . , l m } ) µ (cid:48) ··· µ (cid:48) m P U ( l t , k t , . . . ) iµ (cid:48) ··· µ (cid:48) m , (65)i.e., where the polarization projection P U depends separately on the transverse components of individual extra gluontransverse momentum, rather than only on their sum. Recall that it is only k t that enters the hard scattering.In Eq. (65) the transverse gluon momentum k t that enters the hard subgraph may differ from the transverse gluonmomentum l t that induces a polarization dependence. If such terms fail to cancel, then they contribute to a breakdownof TMD-factorization, even under the maximally general criteria of Sect. IV A. V. ANALYSIS OF ONE GLUON
Before addressing the case of multiple gluon interactions, we will use this section to give a detailed demonstration ofhow Ward identity arguments apply to the simplest case of just one gluon attaching hadron-2 to the rest of the graph8 P P k k k k L ( P , k ) U ( P ; k ) µ ′ P P k k k k L ( P , k ) U ( P ; k ) µ ′ (a) (b)FIG. 5. The single gluon parton-model-level contributions to Fig. 4. (Fig. 5). In the absence of extra gluons, there are few enough complications that the problems with TMD-factorizationwill not appear — extra soft and collinear gluons of the type normally associated with gauge links are needed for theviolation of TMD-factorization to be apparent. Therefore, the results of this section will be found to be consistentwith the conditions for maximally general TMD-factorization from Sect. IV. Nevertheless, the steps will help establishthe basic framework needed for dealing more generally with factorization issues in Sect. VII. For simplicity, in thissection we will restrict consideration to the case where the final states in Fig. 1 are totally unpolarized. A. Separation into Subgraphs
We begin by applying the analysis of Sect. III to the special case of Fig. 5. The separation into subgraphs accordingto Eq. (32) is M (1 g ) = U ( P ; k ) κ ; µ (cid:48) g µ (cid:48) µ ¯ u ( k ) H L ( k , k , k , k ) κ ; µσ L ( P ; k ) . (66)The Lorentz index for the gluon is µ , that for the photon is σ , and κ is the color-octet index for the gluon. Repeatedcolor indices are summed over. We will drop the explicit σ index on M (1 g ) .In frame-1, the powers of Q for the components of U µ (cid:48) are U − ∼ Q, U i ∼ Q , U + ∼ /Q . (67)As usual, superscripts “ i, j ” denote transverse Lorentz components in frame-1. The final state momenta, k and k ,are at wide angles so all components of H µL are of comparable size: H − L ∼ H + L ∼ H iL ∼ /Q . (68)For this section, the Lorentz index on H L and U always refers to the gluon unless otherwise specified.The final state quark wave function behaves as ¯ u ( k ) ∼ Q / . The dominant contribution from L comes fromleading-power projections onto the Dirac components of Φ q/P , as in Eq. (46), which are at most of order Q . So thedominant component of L also has a power-law behavior L ∼ Q / .Since we are mainly interested the contraction of U with the rest of the graph, let us introduce the notation, R κ ; µσ ≡ ¯ u ( k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / H L ( k , k , k , k ) κ ; µσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / , (69)so that M (1 g ) = U ( P ; k ) κ ; µ (cid:48) g µ (cid:48) µ R κ ; µσ . (70)Then all the components of R κ ; µσ have the power-law behavior, R κ ; − σ ∼ R κ ;+ σ ∼ R κ ; iσ ∼ Q , (71)9and the largest power-law in the full amplitude from the tree-level graphs of Fig. 5 is from the contraction M max(1 g ) ∼ U κ ; − R κ ;+ σ ∼ Q . (72)This is a contribution from the K -gluon term in the Grammer-Yennie decomposition. It is a superleading contributionof the type discussed in Sect. III.We next implement the Grammer-Yennie separation in Eqs. (34-36) in Eq. (70) by writing M (1 g ) = M G (1 g ) + M K (1 g ) (73)with M G (1 g ) = U ( P ; k ) κ ; µ (cid:48) G ( k ) µ (cid:48) µ R κ ; µσ G-term , (74) M K (1 g ) = U ( P ; k ) κ ; µ (cid:48) K ( k ) µ (cid:48) µ R κ ; µσ K-term , (75)and, K ( k ) µ (cid:48) µ ≡ n µ (cid:48) k µ n · k (76) G ( k ) µ (cid:48) µ ≡ g µ (cid:48) µ − n µ (cid:48) k µ n · k . (77)The K -term gives superleading as well as subleading contributions in the amplitude. The G -term, however, has beendeliberately constructed so that its “+ − ” component is removed. The highest power-law contributed by the G -termwhen contracted with U is therefore U κ ; i (cid:124)(cid:123)(cid:122)(cid:125) ∼ Q R κ ; iσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ∼ U κ ; − (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q R κ ; iσ ( k i t /k − ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q ∼ Q . (78)These are the leading-power G -term contributions to the amplitude. The remaining G -term contributions, in thecontraction of U with R , are power suppressed relative to the leading terms: U κ ;+ R κ ; − σ ∼ /Q , (79)in accordance with the power-laws in Eqs. (67) and (71). The explicit expression for R κ ; µσ , from the two graphs inFig. 5, is R κ ; µσ = ig s e q ¯ u ( k ) (cid:26) γ µ ( /k − /k + m q ) γ σ t κ ( k − k ) − m q + i γ σ ( /k + /k + m q ) γ µ t κ ( k + k ) − m q + i (cid:27) L ( P ; k ) . (80) B. Cancellation of Superleading Terms
The superleading K -term in Eq. (70) can be seen to vanish in the sum over graphs in the full amplitude by firstnoting that Eq. (75) can be written as U ( P ; k ) κ ; µ (cid:48) K ( k ) µ (cid:48) µ R κ ; µσ = U ( P ; k ) κ ; µ (cid:48) n µ (cid:48) k µ n · k R κ ; µσ = U ( P ; k ) κ ; − k − (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q × k µ R κ ; µσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ∼ Q , (81)so that attention may be focused solely on k µ R κ ; µσ . (The underbraces in Eq. (81) only denote the maximum powersof Q graph-by-graph . Since they are superleading, the terms of order Q have to cancel in the sum over all graphs.)In order to argue that all the K -gluon contributions in Eq. (81) are power suppressed, it must be shown that boththe order ∼ Q (superleading) and the order ∼ Q (leading) contributions to the k µ R κ ; µσ subgraph cancel in the sumover graphs when contracted with U ( P ; k ) κ ; µ (cid:48) . Including both terms from Eq. (80), k µ R κ ; µσ becomes k µ R κ ; µσ = ig s e q ¯ u ( k + k − k ) × (cid:26) /k ( /k − /k + m q ) γ σ t κ ( k − k ) − m q + i γ σ ( /k + /k + m q ) /k t κ ( k + k ) − m q + i (cid:27) × Term 2 × L ( P ; k ) . (82)0From here forward, the steps for dealing with Fig. 5 are similar to standard arguments for a Ward identity cancellation.For Term 1 in braces in Eq. (82), we apply the Feynman identity replacement, /k = ( /k + /k − /k − m q ) − ( /k − /k − m q ) , (83)while in Term 2 we use /k = ( /k + /k − m q ) − ( /k − m q ) . (84)When the ( /k + /k − /k − m q ) in Eq. (83) acts to the left on ¯ u ( k + k − k ), in Term 1 of Eq. (82), the resultingcontribution vanishes exactly by the action of the Dirac equation on the outgoing quark wavefunction.When the − ( /k − m q ) in Eq. (84) acts to the right on L ( P ; k ), in Term 2 of Eq. (83), the resulting contributionis suppressed by two powers of Q . (It would be exactly zero if the target quark were taken to be exactly on-shell.)To see this, let us rewrite Term 2 from Eq. (82), but with the k -propagator inside L ( P ; k ) explicitly displayed: ig s e q ¯ u ( k + k − k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / γ σ ( /k + /k + m q ) /k t κ ( k + k ) − m q + i (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / = ig s e q ¯ u ( k + k − k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / γ σ ( /k + /k + m q ) t κ ( k + k ) − m q + i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (cid:18) /k ( /k + m q ) k − m q + i (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q · · · (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q − / ∼ Q . (85)In the last line, the “ · · · ” symbolizes the other factors that make up L ( P ; k ), and these are order ∼ Q − / becauseof the overall L ( P ; k ) ∼ Q / power-law. Note that in the second line of Eq. (85), the /k has been moved outsidethe hard propagator and is included in the parentheses with the k -propagator.When the Feynman identity substitution is made for /k , the − ( /k − m q ) from the second term of Eq. (84) gives ig s e q ¯ u ( k + k − k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / γ σ ( /k + /k + m q ) t κ ( k + k ) − m q + i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q × (cid:18) [ − ( /k − m q )] ( /k + m q ) k − m q + i (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q · · · (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q − / ∼ /Q . (86)So the factor in parentheses looses two powers of Q relative to Eq. (85), and the contribution to k µ R κ ; µσ from thesecond term in Eq. (84) has a power law ∼ /Q . That it is suppressed by two powers of Q instead of only one iscrucial because we will ultimately multiply it with the superleading U − in Eq. (81). We will need this result again inSect. VII when we take into account effects from an additional soft/collinear gluon radiated from hadron-2.The remaining K -term contributions are from the second term in Eq. (83) and the first term in Eq. (84) in theFeynman identity substitutions. These exactly cancel against each other when we use quark propagator denominatorcancellations like ( /k + /k + m q )( k + k ) − m q + i × ( /k + /k − m q ) = 1 . (87)The exact cancellation is not put in danger by the i k + k ) is constrained by kinematics to be always oforder Q .Thus, there is no leading-power K -gluon contribution in the one-gluon case, and the superleading contributionscancel as expected. The only remaining contribution is from the G -gluon in Eq. (74). From Eq. (78), it immediatelyfollows that the leading-power G -gluon contributions to M (1 g ) involve only the transverse µ -components of R κ ; µσ .The plus component is already entirely accounted for above in the treatment of the K -term and is, by construction,exactly removed in the G -term. Because of the suppression in Eq. (79), the minus µ -component of R κ ; µσ gives acontribution suppressed by a power of Q relative to the leading power. Therefore, the Lorentz components in the G -term contraction may be restricted to the frame-1 transverse µ -components. That is, we may replace G ( k ) µ (cid:48) µ → G ( k ) µ (cid:48) i = g µ (cid:48) i − n µ (cid:48) k i t n · k , (88)to leading power, which is in agreement with the normal gluon PDF vertex with no gauge link contribution (see, e.g.,Eq. (45) of Ref. [81] and Fig. 3.4 of Ref. [6]). The lowest-order uncanceled contribution to the amplitude is therefore1consistent with typical expectations for the parton model gluon density: M (1 g ) = ig s e q (cid:88) i U ( P ; k ) κµ (cid:48) (cid:32) g µ (cid:48) i − n µ (cid:48) k i t n · k (cid:33) ¯ u ( k ) (cid:26) γ i ( /k − /k + m q ) γ σ t κ ( k − k ) − m q + i γ σ ( /k + /k + m q ) γ i t κ ( k + k ) − m q + i (cid:27) L ( P ; k )= ig s e q (cid:88) i U ( P ; k ) κµ (cid:48) G ( k ) µ (cid:48) i (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ¯ u ( k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / H LO ( k , k , k , k ) κ ; iσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / ∼ Q , (89)where we have used Eqs. (88, 80) in Eq. (74). In the second line, we have defined H LO ( k , k , k , k ) κ ; µσ as the sumof the lowest order contributions to the hard parts in Fig. 5: H LO ( k , k , k , k ) κ ; µσ ≡ γ µ ( /k − /k + m q ) γ σ t κ ( k − k ) − m q + i γ σ ( /k + /k + m q ) γ µ t κ ( k + k ) − m q + i . (90)Checking Eqs. (89,90) with Sect. IV D confirms that the amplitude is now in a form consistent with the TMD-factorization conjecture, Eqs. (62-64), in the maximally general form. Tallying the powers of Q indicated with bracesunderneath Eq. (89) verifies that the combined powers give a leading contribution to the amplitude. C. Tree Level TMD-Factorization
Now nothing obstructs the immediate recovery of the normal TMD-factorization formula characteristic of partonmodel expectations. Using Eq. (89) in Eq. (31) and averaging over initial hadron spins gives dσ = C Q (cid:88) s (cid:88) X (cid:88) X (cid:90) d k (2 π ) d k (2 π ) | M (1 g ) | (2 π ) δ (4) ( k + k − k − k )= C Q e q g s (cid:88) s (cid:88) X (cid:88) X (cid:88) ij (cid:90) d k (2 π ) d k (2 π ) × U κµ (cid:48) U † κ (cid:48) ν (cid:48) G µ (cid:48) i G ν (cid:48) j Tr (cid:104) ¯ u ( k ) H κ ; iσ LO L ¯ L H † κ (cid:48) ; j LO , σ u ( k ) (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (2 π ) δ (4) ( k + k − k − k ) . (91)The sums over X and X represent the sums and integrals over the final states in the L and U subgraphs. The spinsums for the incoming hadrons are included in these sums as well. In Eq. (91), the explicit momentum arguments ofthe various factors have been dropped for convenience. The outgoing quark has spin label s . The minus sign fromthe spin sum on the prompt photon has been absorbed into the definition of C .To put Eq. (91) into a form closer to Eq. (51), we now apply the minimal partonic approximations of Eqs. (38-40). Specifically, we neglect the smallest (order ∼ Λ /Q ) momenta k − and k +2 everywhere except within their ownrespective parent subgraphs, L ( P ; k ) and U ( P ; k ). Then the right side of Eq. (91) becomes C Q e q g s (cid:88) s (cid:88) X (cid:88) X (cid:88) ij (cid:90) d k t (2 π ) (cid:18)(cid:90) dk +2 π U κµ (cid:48) U † κ (cid:48) ν (cid:48) G µ (cid:48) i G ν (cid:48) j (cid:19) (cid:18)(cid:90) dk − π Tr (cid:104) ¯ u ( k ) ˆ H κ ; iσ LO L ¯ L ˆ H † κ (cid:48) ; j LO , σ u ( k ) (cid:105)(cid:19) = C Q e q g s (cid:88) s (cid:88) ij (cid:90) d k t (2 π ) (cid:32)(cid:88) X (cid:90) dk +2 π U κµ (cid:48) U † κ (cid:48) ν (cid:48) (cid:32) g µ (cid:48) i − n µ (cid:48) k i t n · k (cid:33) (cid:32) g ν (cid:48) j − n ν (cid:48) k j t n · k (cid:33)(cid:33) × (cid:32)(cid:88) X (cid:90) dk − π Tr (cid:104) ¯ u ( k ) ˆ H κ ; iσ LO L ¯ L ˆ H † κ (cid:48) ; j LO , σ u ( k ) (cid:105)(cid:33) = C e q g s (cid:88) s (cid:88) ij (cid:90) d k t (2 π ) Φ (1 g ) g/P (cid:16) P ; ˆ k ( x , q t − k t ) (cid:17) κκ (cid:48) ; ij ×× Tr (cid:104) ¯ u ( k ) ˆ H κ ; iσ LO (cid:16) ˆ k ( x , k t ) , ˆ k ( x , q t − k t ) , k , k (cid:17) Φ (1 g ) q/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) ×× ˆ H † κ (cid:48) ; j LO , σ (cid:16) ˆ k ( x , k t ) , ˆ k ( x , q t − k t ) , k , k (cid:17) u ( k ) (cid:105) . (92)2In lines 1-3 we have used the integrals over k t , k +1 , and k − to evaluate the δ -functions in Eq. (91), and the positions ofthe integrals over k − and k +2 have been arranged into factors corresponding to the respective subgraphs for hadron-1and hadron-2, with the factors that correspond to effective TMD PDFs separated by parentheses. The use of thekinematical approximations in Eqs. (38-40) is symbolized by the “hats” on H LO . After the second equality in Eq. (92),we have identified effective tree-level TMD PDFs:Φ (1 g ) q/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) ≡ Q (cid:88) X (cid:90) dk − π L ¯ L ∼
Q , (93)Φ (1 g ) g/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) κκ (cid:48) ; ij ≡ Q (cid:88) X (cid:90) dk +2 π U κµ (cid:48) U † κ (cid:48) ν (cid:48) (cid:32) g µ (cid:48) i − n µ (cid:48) k i t n · k (cid:33) (cid:32) g ν (cid:48) j − n ν (cid:48) k j t n · k (cid:33) ∼ Q . (94)Also, after the last equality of Eq. (92) we have restored the explicit momentum arguments to make clear the connectionwith the statement of TMD-factorization in Eq. (37). Equations (94) are consistent with the basic structure of TMDPDF operator definitions that are typical in a generalized TMD parton model, and they are consistent with criteriaof Sect. IV A. Hence, in Eq. (92) we have recovered a result consistent with TMD-factorization under the maximallygeneral criteria. In these graphs there is not yet sensitivity to gauge links type effects; for that we will need to considerthe extra soft gluons of Fig. 3.With the tree level cross section now in the form of Eq. (37), one may proceed with the classification of quark andgluon polarization dependence following the normal steps in Sect. IV B. The contribution to the cross section thatcorresponds to a generic quark polarization projection, Γ q , and gluon polarization, Γ g , is written as in Eq. (51): dσ [Γ q , Γ g ] = C e q g s (cid:88) s (cid:88) ij (cid:90) d k t (2 π ) Φ (1 g )[Γ g ] g/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) κκ (cid:48) ; ij (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q Φ (1 g )[Γ q ] q/P (cid:16) P ; ˆ k ( x , k t ) (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ×× Tr[ ¯ u ( k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / ˆ H κ ; iσ LO (cid:16) ˆ k ( x , k t ) , ˆ k ( x , q t − k t ) , k , k (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q P Γ q ˆ H † κ (cid:48) ; j LO , σ (cid:16) ˆ k ( x , k t ) , ˆ k ( x , q t − k t ) , k , k (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q u ( k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / ] ∼ Q . (95)For example, the standard azimuthally symmetric contribution is, from Eqs. (41,49): dσ [ γ + , U ] = C e q g s (cid:88) s (cid:88) j (cid:90) d k t (2 π ) Φ (1 g ) , U g/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) κκ (cid:48) Φ (1 g ) [ γ + ] q/P (cid:16) P ; ˆ k ( x , k t ) (cid:17) ×× Tr (cid:104) ¯ u ( k ) ˆ H κ ; iσ LO (cid:16) ˆ k ( x , k t ) , ˆ k ( x , q t − k t ) , k , k (cid:17) γ − ˆ H † κ (cid:48) ; j LO , σ (cid:16) ˆ k ( x , k t ) , ˆ k ( x , q t − k t ) , k , k (cid:17) u ( k ) (cid:105) . (96)As a final step, one may average over quark and gluon color to reproduce the standard expressions for the quark andgluon TMD PDFs with a color averaged hard part, though this is not necessary to satisfy the criteria of Sect. IV.To summarize, in this section we have verified that the generalized parton model picture of TMD-factorizationarises in a detailed tree-level, single-gluon treatment, and we have shown that it is consistent with the maximallygeneral statement of TMD-factorization from Sect. IV. Tallying the powers of Q in the underbraces of Eq. (95) verifiesthat it is a leading-power contribution.Later we will follow similar steps when we include one extra gluon from hadron-2, but there they will fail toreproduce a factorized form, even under the loose criteria of Sect. IV A. VI. SPECTATOR-SPECTATOR INTERACTIONS
The next step is to examine the contribution from one extra soft gluon in Fig. 3. If the generalized criteria for TMD-factorization are to be respected, then the sum over all such graphs must allow contributions from the extra gluon tobe factored into separate contributions in the upper and lower subgraphs. The single extra soft gluon contributionsinclude both spectator attachments as well as attachments to active partons. In this section we will deal separatelywith the spectator-spectator type graphs, arguing that any leading or superleading contributions cancel. Later wewill deal with the active parton attachments and find that they lead to TMD-factorization breaking.3 P P k k k k U αβ ; µ ! µ ! R αβ ; µ µ l L ! µ ; β G ( k ) α β FIG. 6. The amplitude in Eq. (97) with a single spectator-spectator interaction l . P P k k k k l P − k P − k − lP − k + lP − k G ( k ) P P k k k k l P − k P − k − lP − k + lP − k G ( k ) (a) (b)FIG. 7. Spectator-spectator interactions that cancel in the inclusive cross section. The active gluon is a G ( k ) gluon whichyields only leading (non superleading) contributions. (As in Fig. 6, we restrict consideration here to spectator attachments for l in the upper and lower bubbles.) The relevant amplitude for the spectator-spectator case is shown in Fig. 6, and we extend the notation of Sect. V Aby writing it as M spec − spec(2 g ) = U ( P ; k , l ) αβ ; µ (cid:48) µ (cid:48) g µ (cid:48) µ g µ (cid:48) µ R αβ ; µ µ ( k , k , k , k , l, P ) , (97)where R αβ ; µ µ ( k , k , k , k , l, P ) ≡ ¯ u ( k ) H LO ( k , k , k , k ) α ; µ σ L (cid:48) ( P , k , l ) β ; µ (98)includes both the bottom bubble L (cid:48) and the hard subgraph H LO from Eq. (90). (See the labeling in Fig. 6.) Thepartons are nearly on-shell, k ∼ k ∼ l (cid:46) Λ , but now the four-momentum may be shared between k and l . Asusual, we are interested in the region k ,t (cid:46) Λ and k ,t (cid:46) Λ. In Fig. 6, we restrict consideration to graphs withspectator attachments for l inside the upper and lower bubbles.The superleading contributions come from the contraction of R αβ ;++ with U ( P ; k , l ) αβ ; −− . The region k − (cid:28) Q is forbidden because it pushes the quark propagator k , and propagators with momentum P − k inside L (cid:48) , faroff shell. Thus, in the k − (cid:28) Q region, there are at least two extra powers of suppression so that the overallcontribution is suppressed by at least one power of Q relative to the leading power, even in the contraction of R αβ ;++ ( k , k , k , k , l, P ) with U ( P ; k , l ) αβ ; −− . With k − ∼ Q , both the l + and the l − components of the l -gluonare trapped at order Λ /Q by initial and final state poles in U ( P ; k , l ) αβ ; µ (cid:48) µ (cid:48) and in L (cid:48) ( P , k , l ) µ ; β . That is, | l + l − | (cid:28) | l T | (cid:46) Λ . (This is the Glauber region discussed in Sect. III C.) But then the kinematics of the hardsubprocess becomes identical to the single gluon case of Sect. V, with the only difference being the slightly shiftedmomenta P − k − l and P − k + l of the inclusive final state remnants. Therefore, we may once again decomposethe g µ (cid:48) µ in Eq. (97) into G ( k ) and K ( k ) gluons as in Eqs. (76,77) to find that the K ( k ) terms cancel to leadingpower in an argument identical to that of the previous section. Since k − ∼ Q , the remaining G ( k ) term leaves onlyleading, not superleading, contributions.At this stage, the cancellation of the remaining spectator-spectator Glauber interactions in the inclusive sum overthe final states of L and U in the inclusive cross section follows steps familiar from other hadron-hadron processes like4the Drell-Yan process. If a single extra spectator-spectator gluon is exchanged, the graphs that contribute at leadingpower to the cross section are as shown in Fig. 7, where the k gluon is labeled by G ( k ) to emphasize that thesegraphs are leading and not superleading, according to the argument above, and involve only G -gluon attachments inthe hard subgraphs. The extra l momentum is routed entirely through the hadron-1 and hadron-2 bubbles, whichare summed inclusively in the cross section. The cancellation is between the two final state cuts shown in Fig. 7 withdifferent final state momenta inside U and L (cid:48) . VII. ANALYSIS OF TWO GLUONS
The objective for the next two sections is to show, for the process of Sect. II A, that there is a leading-powerviolation of the loose conditions for TMD-factorization from Sect. IV. The demonstration will be for the case of asingle extra soft gluon radiated from hadron-2 and attaching to active partons, with the relevant Feynman graphs atthe amplitude level shown in Fig. 8.Graphs where one gluon couples to the other before the hard collision were already included in the single gluoncase of Sect. V. The possibility of TMD-factorization violation coming from spectator-spectator type interactions wasaddressed in Sect. VI. Spectator attachments in hadron-1 attaching to active partons are sensitive to non-perturbativestructure inside L (cid:48) . Therefore, we now focus only on gluons attaching the hadron-2 subgraph to active partons, asshown in the graphs in Fig. 8. There are only gluon-fermion attachments in the hard part so ghosts do not contributeat this order.The steps in this section are nearly the same as those of Ref. [81], but are tailored to exhibit the role of intrinsictransverse momentum. A. Power Counting
For the two-gluon case, the separation into blocks according to Eq. (32) becomes M (2 g ) = U ( P ; l , l ) αβ ; µ (cid:48) µ (cid:48) g µ (cid:48) µ g µ (cid:48) µ ¯ u ( k ) H L ( k , k , k , k ; l , l ) αβ ; µ µ L ( P ; k )= U ( P ; l , l ) αβ ; µ (cid:48) µ (cid:48) g µ (cid:48) µ g µ (cid:48) µ R αβ ; µ µ ( k , k , k , k ; l , l , P ) . (99)On the second line we have defined an R αβ ; µ µ for the two U -gluon case, analogous to Eq. (69) for the one U -gluon case. With an extra soft/collinear gluon now entering the H L ( k , k , k ; l , l ) αβ ; µ µ subgraph, there is anotherfermion propagator in the unfactorized hard subprocess, so each Lorentz component of H L ( k , k , k ; l , l ) αβ ; µ µ nowacquires an extra power of 1 /Q relative to the one-gluon case in Eq. (68). The lower quark subgraph, L ( P ; k ), isexactly the same as in Sect. V, so all components of R αβ ; µ µ are now of size ∼ /Q (as usual, up to factors of orderΛ and logarithms from renormalization). By contrast, the one-gluon case in Eq. (71) was order ∼ Q .The two-gluon subgraph U αβ ; µ (cid:48) µ (cid:48) now has two Lorentz indices, and in frame-1 its highest power-law behavior isfrom the U −− ∼ Q components. The powers for the individual Lorentz components are U −− ∼ Q , (100) U − j ∼ U j − ∼ Q , (101) U ij ∼ U + − ∼ U − + ∼ Q , (102) U j + ∼ U + j ∼ /Q , (103) U ++ ∼ /Q . (104)The components in Eq. (100) contribute superleading ∼ Q terms graph-by-graph in the amplitude; the U αβ ; −− ∼ Q components of the U αβ ; µ (cid:48) µ (cid:48) subgraph multiply R αβ ; µ µ ∼ /Q components in Eq. (99). Recall that, while overallsuperleading contributions must cancel in the final analysis, superleading subdiagrams can contribute uncanceledleading terms to the amplitude when they multiply subleading factors in the rest of the graph.Next we implement the Grammer-Yennie decomposition on Eq. (99), writing g µ (cid:48) µ = K ( l ) µ (cid:48) µ + G ( l ) µ (cid:48) µ , (105) g µ (cid:48) µ = K ( l ) µ (cid:48) µ + G ( l ) µ (cid:48) µ , (106)5 P P k k k L ( P , k ) U ( P ; l , l ) µ ! µ ! βαl l P P k k k L ( P , k ) β αl l U ( P ; l , l ) µ ! µ ! (a) (b) P P k k k L ( P , k ) β αl l U ( P ; l , l ) µ ! µ ! P P k k L ( P , k ) U ( P ; l , l ) µ ! µ ! βαl l k k (c) (d) P P k k L ( P , k ) U ( P ; l , l ) µ ! µ ! β αl l k k P P k k k L ( P , k ) U ( P ; l , l ) µ ! µ ! βαl l (e) (f)FIG. 8. Graphs with an extra gluon radiated from hadron-2. with the K ’s and G ’s defined as in Sect. III B, but now with the four-momenta that correspond to each gluon: G ( l ) µ (cid:48) µ = g µ (cid:48) µ − n ,µ (cid:48) l ,µ n · l , (107) G ( l ) µ (cid:48) µ = g µ (cid:48) µ − n ,µ (cid:48) l ,µ n · l , (108) K ( l ) µ (cid:48) µ = n ,µ (cid:48) l ,µ n · l , (109) K ( l ) µ (cid:48) µ = n ,µ (cid:48) l ,µ n · l . (110)6Using Eqs. (105, 106) in Eq. (99) separates M (2 g ) into the following set of G / K terms: M (2 g ) = U ( P ; l , l ) αβ ; µ (cid:48) µ (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:0) G ( l ) µ (cid:48) µ G ( l ) µ (cid:48) µ + G ( l ) µ (cid:48) µ K ( l ) µ (cid:48) µ + K ( l ) µ (cid:48) µ G ( l ) µ (cid:48) µ + K ( l ) µ (cid:48) µ K ( l ) µ (cid:48) µ (cid:1) R αβ ; µ µ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q ∼ Q . (111)The power-laws indicated with each underbrace denote maximum powers term-by-term. Thus, the largest contributionto the amplitude from any individual graph is order ∼ Q , which is superleading. As in Sect. V, we must verify thecancelation of such superleading terms in the sum of all graphs by analyzing each of the G / K combinations. B. Basic Expression
We begin by directly writing down the expression for R αβ ; µ µ from the sum of diagrams in Fig. (8). R αβ ; µ µ = ie q g s ¯ u ( k , S ) (cid:40) γ µ ( /k − /l + m q ) γ µ ( /k − /k + m q ) γ σ t β t α (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k − k ) − m q + i (cid:3) + Graph a.+ γ µ ( /k − /l + m q ) γ µ ( /k − /k + m q ) γ σ t α t β (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k − k ) − m q + i (cid:3) + Graph b.+ γ µ ( /k − /l + m q ) γ σ ( /k + /l + m q ) γ µ t α t β (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) + Graph c.+ γ σ ( /k + /k + m q ) γ µ ( /k + /l + m q ) γ µ t β t α (cid:2) ( k + k ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) + Graph d.+ γ σ ( /k + /k + m q ) γ µ ( /k + /l + m q ) γ µ t α t β (cid:2) ( k + k ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) + Graph e.+ γ µ ( /k − /l + m q ) γ σ ( /k + /l + m q ) γ µ t β t α (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) (cid:41) × Graph f. ×L ( P ; k ) . (112)For ease of reference, we have labeled each term in braces with its corresponding diagram from Fig. 8. Note that wehave restored the S argument of the outgoing quark wavefunction. C. Glauber regions
The Glauber regions correspond to intermediate states in the hard subgraph going on shell, so l − and l − must bedeformed far into the complex plane, as reviewed in Sect. III C, to ensure that (cid:12)(cid:12) l − (cid:12)(cid:12) and (cid:12)(cid:12) l − (cid:12)(cid:12) may treated as order ∼ Q .In the two-gluon case of Fig. 8, only one gluon at a time can be Glauber. If both gluons are Glauber, then k − << k t , which is forbidden by the kinematical restriction k − ∼ Q .As an example, assume the case where l − is outside the Glauber and consider the integration on l − . For thefinal state l -interactions in Figs. 8(a,f), the deformations away from the l − -Glauber poles are downward, into thenegative half of the complex plane, while for the initial state l − -interactions in Figs. 8(c,e), the deformations areupward. Therefore, in Figs. 8(a,f) we apply the downward deformation on l − until (cid:12)(cid:12) l − (cid:12)(cid:12) ∼ Q , and we may replace l − by ( l − − i
0) in Eqs. (108, 110): (cid:2) G ( l ) µ (cid:48) µ (cid:3) F . S . = g µ (cid:48) µ − n ,µ (cid:48) l ,µ l − − i , (113) (cid:2) K ( l ) µ (cid:48) µ (cid:3) F . S . = n ,µ (cid:48) l ,µ l − − i . (114)The subscripts “F . S . ” are to symbolize that these are the G ( l ) and K ( l ) expressions we will use in graphs with finalstate interactions and downwardly deformed l contours.7For the initial state l -interactions in Figs. 8(c,e), we implement the upward deformations on l − until | l − | ∼ Q . Inthat case we use (cid:2) G ( l ) µ (cid:48) µ (cid:3) I . S . = g µ (cid:48) µ − n ,µ (cid:48) l ,µ l − + i , (115) (cid:2) K ( l ) µ (cid:48) µ (cid:3) I . S . = n ,µ (cid:48) l ,µ l − + i . (116)The subscripts “I . S . ” are to symbolize that these are the G ( l ) and K ( l ) expressions we will use in graphs with initialstate interactions and upwardly deformed l contours.The requirement is that the signs on the i l − = 0 poles must be such that they do not obstruct therequired deformations on l − away from the Glauber region. Expressions exactly analogous to Eqs. (113, 116) applyfor deformations of l out of the Glauber region.Note that the directions of the deformations are different depending on whether the extra gluon attachments arein the initial or final state. So arguments using Glauber deformations need to be applied separately to each graph. D. G ( l ) G ( l ) Terms
Recall from the one-gluon case in Eq. (78) that the G µ (cid:48) µ factors projected only components of U that were powersuppressed relative to the largest components. In this subsection, we show that this generalizes to the two-gluon case,so the G ( l ) G ( l ) terms are doubly suppressed relative to the largest terms and contribute only subleading terms to M (2 g ) .If we apply Eqs. (113, 115), with the deformed contours, to Eq. (112) and recall Eqs. (100–102), then we mayidentify the leading power-law for each contribution from the G ( l ) G ( l ) projection: (cid:104) U ( P ; l , l ) αβ ; µ (cid:48) µ (cid:48) G ( l ) µ (cid:48) µ G ( l ) µ (cid:48) µ R ( k , k , k , k ; l , l , P ) αβ ; µ µ (cid:105) I . S ./ F . S ∼ U ( P ; l , l ) αβ ; ij (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q R ( k , k , k , k ; l , l , P ) αβ ; ij (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q ∼ U ( P ; l , l ) αβ ; − j (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ( l i / ( l − ± i R ( k , k , k , k ; l , l , P ) αβ ; ij (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q ∼ U ( P ; l , l ) αβ ; i − (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ( l j / ( l − ± i R ( k , k , k , k ; l , l , P ) αβ ; ij (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q ∼ U ( P ; l , l ) αβ ; −− (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ( l i / ( l − ± i l j / ( l − ± i R ( k , k , k , k ; l , l , P ) αβ ; ij (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q ∼ /Q . (117)Here we have written the largest powers, of order ∼ /Q , in the contraction with the G ( l ) G ( l ) term. The “I . S ./ F . S”subscript is included to symbolize that we apply the l -contour deformations upward or downward appropriately ineach term in accordance with whether the gluon attaches to the initial or final quark.The leading powers in M (2 g ) are of order ∼ Q while the largest powers in Eq. (117) are ∼ /Q . So the G ( l ) G ( l )terms are subleading. Notice that the validity of the argument relies on the use of deformed l − and l − contours.8 E. K ( l ) Terms
Next we write the general expression for the case where one gluon is a K -gluon. Applying Eqs. (114,116) toEq. (112) gives (cid:2) K ( l ) µ (cid:48) µ R αβ ; µ µ (cid:3) I . S ./ F . S = ie q g s ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / (cid:40) /l ( /k − /l + m q ) γ µ ( /k − /k + m q ) γ σ t β t α (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k − k ) − m q + i (cid:3) × l − − i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (a.)+ γ µ ( /k − /l + m q ) /l ( /k − /k + m q ) γ σ t α t β (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k − k ) − m q + i (cid:3) × l − (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (b.)+ γ µ ( /k − /l + m q ) γ σ ( /k + /l + m q ) /l t α t β (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − + i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (c.)+ γ σ ( /k + /k + m q ) /l ( /k + /l + m q ) γ µ t β t α (cid:2) ( k + k ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (d.)+ γ σ ( /k + /k + m q ) γ µ ( /k + /l + m q ) /l t α t β (cid:2) ( k + k ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − + i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (e.)+ /l ( /k − /l + m q ) γ σ ( /k + /l + m q ) γ µ t β t α (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − − i (cid:41)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (f.) × n µ (cid:48) L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / . (118)In Figs. 8(b,d), l attaches to an internal hard propagator so no intermediate line in the hard subgraph goes on shellwhen l approaches the Glauber region; there are no Glauber poles and so no need for replacements like Eqs. (113-116). F. G ( l ) K ( l ) Terms
The “ µ ” index in Eq. (118) is ultimately to be contracted with a K ( l ) µ (cid:48) µ and a G ( l ) µ (cid:48) µ in the amplitude inEq. (111). We first consider the G ( l ) µ (cid:48) µ contraction: (cid:2) M (2 g ) (cid:3) G ( l ) K ( l ) = (cid:104) U ( P ; l , l ) αβ ; µ (cid:48) µ (cid:48) G ( l ) µ (cid:48) µ K ( l ) µ (cid:48) µ R αβ ; µ µ (cid:105) I . S ./ F . S . = ie q g s U ( P ; l , l ) αβ ; µ (cid:48) , − ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / × × (cid:40)(cid:18) g µ (cid:48) µ − n ,µ (cid:48) l ,µ l − (cid:19) /l ( /k − /l + m q ) γ µ ( /k − /k + m q ) γ σ t β t α (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k − k ) − m q + i (cid:3) × l − − i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (a.)+ (cid:18) g µ (cid:48) µ − n ,µ (cid:48) l ,µ l − − i (cid:19) × γ µ ( /k − /l + m q ) /l ( /k − /k + m q ) γ σ t α t β (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k − k ) − m q + i (cid:3) × l − (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (b.)+ (cid:18) g µ (cid:48) µ − n ,µ (cid:48) l ,µ l − − i (cid:19) × γ µ ( /k − /l + m q ) γ σ ( /k + /l + m q ) /l t α t β (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − + i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (c.)+ (cid:18) g µ (cid:48) µ − n ,µ (cid:48) l ,µ l − + i (cid:19) × γ σ ( /k + /k + m q ) /l ( /k + /l + m q ) γ µ t β t α (cid:2) ( k + k ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (d.)+ (cid:18) g µ (cid:48) µ − n ,µ (cid:48) l ,µ l − (cid:19) × γ σ ( /k + /k + m q ) γ µ ( /k + /l + m q ) /l t α t β (cid:2) ( k + k ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − + i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (e.)+ (cid:18) g µ (cid:48) µ − n ,µ (cid:48) l ,µ l − + i (cid:19) × /l ( /k − /l + m q ) γ σ ( /k + /l + m q ) γ µ t β t α (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − − i (cid:41)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (f.) × L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / . (119)Here we have completed the contraction with U ( P ; l , l ) αβ ; µ (cid:48) µ (cid:48) in Eq. (111) to get the full G ( l ) K ( l ) contributionto the amplitude.The next step is to exploit the cascade of cancellations that results when /l is substituted with the followingFeynman replacement identities: /l = − ( /k − /l − m q ) + ( /k − m q ) In graphs (a.) & (f.) (120) /l = ( /l + /k − m q ) − ( /k − m q ) In graphs (c.) & (e.) (121) /l = − ( /k − /k − m q ) + ( /k − /l − m q ) In graph (b.) (122) /l = ( /k + /k − m q ) − ( /k + /l − m q ) . In graph (d.) (123)Furthermore, Eq. (119) may be separated into real and imaginary contributions by using the familiar substitution bydistributions convenient for evaluating integrals:1 l − , + i → P . V . l − , − iπδ ( l − , ) , (124)1 l − , − i → P . V . l − , + iπδ ( l − , ) , (125)where “P . V . ” is the symbol for the principal value distribution. Thus, each 1 / ( l − , ± i
0) eikonal factor is rewritten as asum or difference of two distributions, with the relative signs depending on the direction of the contour deformations.For this paper, the main goal is to demonstrate the appearance of a TMD-factorization breaking double spinasymmetry with S and λ in the final state. In the squared the amplitude, this arises at lowest order from the cross0terms between the graphs of Fig. 8 and the lowest order amplitudes in Fig. 5. With just one extra gluon, it is thereal terms in Eqs. (124,125) which yield real terms in the double spin dependent cross section with a violation of theminimal TMD-factorization criteria. Therefore, to get the double spin asymmetry we focus attention on the principalvalue contributions. Using Eqs. (120-123) in Eq. (119), we obtain (cid:2) M (2 g ) (cid:3) G ( l ) K ( l ) , P . V . = − e q g s f αβκ ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / U ( P ; l , l ) αβ ; µ (cid:48) , − (cid:18) g µ (cid:48) ρ − n ,µ (cid:48) l ,ρ P . V . l − (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q P . V . l − (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q × (cid:26) γ ρ ( /k − /k + m q ) γ σ t κ ( k − k ) − m q + i γ σ ( /k + /k + m q ) γ ρ t κ ( k + k ) − m q + i (cid:27)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / + O (1 /Q ) . (126)Here we have combined terms using the non-Abelian commutation relation [ t α , t β ] = if αβκ t κ . In addition, we haveused again the result from Eqs. (85, 86) that the action of /k − m q on L ( P ; k ) only gives terms suppressed by twopowers of Q . Equation (126) is now almost in the form of the leading order result in Eq. (89), with the leading orderhard part, Eq. (90), and L ( P ; k ) factored away from each other on the second line. The only difference now fromEq. (89) is in the details of the factor associated with the upper bubble in the first line of Eq. (126).In the contraction with U ( P ; l , l ) αβ ; µ (cid:48) , − , only the transverse components of µ (cid:48) are non-suppressed. The pluscomponents cancel between the two terms in G ( l ) µ (cid:48) µ , and the minus components are suppressed by an extra powerof Q . Therefore, we may make the replacements l µ (cid:48) → l t µ (cid:48) and g µ (cid:48) ρ → g tµ (cid:48) ρ in Eq. (126) without introducing anyleading-power errors and rewrite it as (cid:2) M (2 g ) (cid:3) G ( l ) K ( l ) , P . V . = − e q g s f αβκ ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / U ( P ; l , l ) αβ ; µ (cid:48) , − (cid:18) g tµ (cid:48) ρ − n ,µ (cid:48) l t ,ρ P . V . l − (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q P . V . l − (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q × H LO ( k , k , k , k ) κ ; ρσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / + O (1 /Q ) . (127)Hence, the G ( l ) K ( l ) terms are leading power. G. K ( l ) G ( l ) Terms
The treatment of the K ( l ) G ( l ) terms is an exact mirror of the previous subsection, but with the roles of l and l switched. Since this also involves an interchange of the α and β color indices, there is a sign reversal relative toEq. (126) due to the anti-symmetry of f αβκ : (cid:2) M (2 g ) (cid:3) K ( l ) G ( l ) , P . V . = e q g s f αβκ ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / U ( P ; l , l ) αβ ; − ,µ (cid:48) (cid:18) g tµ (cid:48) ρ − n ,µ (cid:48) l t ,ρ P . V . l − (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q P . V . l − (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q × H LO ( k , k , k , k ) κ ; ρσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / + O (1 /Q ) . (128) At this order the imaginary parts in Eqs. (124,125) give single spin asymmetries. However, for the one-extra-gluon example there is noimaginary contribution that violates maximally generalized TMD-factorization. H. K ( l ) K ( l ) Terms
Extra caution is needed in treating the K ( l ) K ( l ) contributions because they project terms that are individuallysuperleading. To get the K ( l ) K ( l ) terms, we contract Eq. (118) with K ( l ) µ (cid:48) µ to obtain an explicit expression forthe last term of Eq. (111): (cid:2) M (2 g ) (cid:3) K ( l ) K ( l ) = (cid:104) U ( P ; l , l ) αβ ; µ (cid:48) µ (cid:48) K ( l ) µ (cid:48) µ K ( l ) µ (cid:48) µ R αβ ; µ µ (cid:105) I . S ./ F . S . = ie q g s U ( P ; l , l ) αβ ; − , − (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / ×× (cid:40) l − × /l ( /k − /l + m q ) /l ( /k − /k + m q ) γ σ t β t α (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k − k ) − m q + i (cid:3) × l − − i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (a.)+ 1 l − − i × /l ( /k − /l + m q ) /l ( /k − /k + m q ) γ σ t α t β (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k − k ) − m q + i (cid:3) × l − (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (b.)+ 1 l − − i × /l ( /k − /l + m q ) γ σ ( /k + /l + m q ) /l t α t β (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − + i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (c.)+ 1 l − + i × γ σ ( /k + /k + m q ) /l ( /k + /l + m q ) /l t β t α (cid:2) ( k + k ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (d.)+ 1 l − × γ σ ( /k + /k + m q ) /l ( /k + /l + m q ) /l t α t β (cid:2) ( k + k ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − + i (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (e.)+ 1 l − + i × /l ( /k − /l + m q ) γ σ ( /k + /l + m q ) /l t β t α (cid:2) ( k − l ) − m q + i (cid:3) (cid:2) ( k + l ) − m q + i (cid:3) × l − − i (cid:41)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (f.) × L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / . (129)Counting the powers of Q written underneath Eq. (129) shows the maximum powers to be superleading. Next, werepeat the Feynman identity substitutions of Eqs. (120-123), again using the result from Eqs. (85, 86) that the actionof /k − m q on L ( P ; k ) yields terms suppressed by two powers of Q . In the K ( l ) K ( l ) case above, it is critical thatdropped terms have two powers of suppression because of the potential for extra leading powers of Q in the crosssection coming from the multiplication with a superleading U ( P ; l , l ) αβ ; − , − .The principal value contributions are again obtained using Eqs. (124,125). Thus, after the cascade of cancelations,and neglecting power suppressed terms, the principle value K ( l ) K ( l ) term from Eq. (99) becomes2 (cid:2) M (2 g ) (cid:3) K ( l ) K ( l ) , P . V . = − e q g s f αβκ ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / U ( P ; l , l ) αβ ; − , − (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19) l ρ (cid:48) g ρ (cid:48) ρ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q × (cid:26) γ ρ ( /k − /k + m q ) γ σ t κ ( k − k ) − m q + i γ σ ( /k + /k + m q ) γ ρ t κ ( k + k ) − m q + i (cid:27)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / + O (1 /Q ) . (130)The factors of Q in Eq. (130) still combine to give superleading powers, and the equation does not appear to besymmetric in l and l in spite of it being the K ( l ) K ( l ) contribution. However, Eq. (130) now has the basicstructure of a leading order result like Eq. (89). Therefore, it may be further simplified with another iteration of the G / K Grammer-Yennie decomposition, but now on g ρ (cid:48) ρ : g ρ (cid:48) ρ = G ( k ) ρ (cid:48) ρ + K ( k ) ρ (cid:48) ρ . (131)The K ( k ) term then vanishes up to power suppressed corrections by an argument identical to that of the one-gluoncase in Eqs. (81–87). The G ( k ) term is then the only remaining contribution and it is (cid:2) M (2 g ) (cid:3) K ( l ) K ( l ) , P . V . = − e q g s f αβκ ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / U ( P ; l , l ) αβ ; − , − (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19) l ,ρ (cid:48) (cid:32) g ρ (cid:48) ρ − n ρ (cid:48) k ρ k − (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q × (cid:26) γ ρ ( /k − /k + m q ) γ σ t κ ( k − k ) − m q + i γ σ ( /k + /k + m q ) γ ρ t κ ( k + k ) − m q + i (cid:27)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / + O (1 /Q )= − e q g s f αβκ ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / U ( P ; l , l ) αβ ; − , − (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19) (cid:18) l ρ k − k − − l − k ρ k − (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q × (cid:26) γ ρ ( /k − /k + m q ) γ σ t κ ( k − k ) − m q + i γ σ ( /k + /k + m q ) γ ρ t κ ( k + k ) − m q + i (cid:27)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / + O (1 /Q )= − e q g s f αβκ ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / U ( P ; l , l ) αβ ; − , − (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19) (cid:18) l ρ l − k − − l − l ρ k − (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q × (cid:26) γ ρ ( /k − /k + m q ) γ σ t κ ( k − k ) − m q + i γ σ ( /k + /k + m q ) γ ρ t κ ( k + k ) − m q + i (cid:27)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / + O (1 /Q ) . (132)After the second equality, we have written the result of contracting l ,ρ (cid:48) with the G ( k ) µ (cid:48) µ -factor. To write the lastequality, we have used that k − ≡ l − + l − and k ,ρ ≡ l ,ρ + l ,ρ . By construction, G ( k ) + − = 0 so the super leadingpowers of Q have canceled — notice after the last equality of Eq. (132) that the result vanishes exactly for ρ = “ − ”.We are left after the last equality with a leading-power contribution that is symmetric in a relabeling of l and l . As3a final step we may also make the leading-power replacement l ρ → l ρ t , l ρ → l ρ t : (cid:2) M (2 g ) (cid:3) K ( l ) K ( l ) , P . V . = − e q g s f αβκ ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / U ( P ; l , l ) αβ ; − , − (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19) (cid:18) l t,ρ l − k − − l − l t,ρ k − (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q H LO ( k , k , k , k ) κ ; ρσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / + O (1 /Q ) . (133)Here we have used the notation from Eq. (90) for the leading order hard part. I. Complete Amplitude
Adding Eqs. (127, 128, 133) gives the complete leading-power principal value contribution to the amplitude: (cid:2) M (2 g ) (cid:3) P . V . = − e q g s f αβκ ¯ u ( k , S ) U ( P ; l , l ) αβ ; µ µ (cid:26) g tµ ρ n µ (cid:18) P . V . l − (cid:19) − g tµ ρ n µ (cid:18) P . V . l − (cid:19) ++ n µ n µ (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19) (cid:18) l t,ρ l − k − − l t,ρ l − k − (cid:19) (cid:27) ×× H LO ( k , k , k , k ) κ ; ρσ L ( P ; k ) + O (1 /Q ) . (134)This is the same result found in Eq. (81) of Ref. [81]. It also forms part of the gauge link factor for the ordinarycollinear gluon PDF.The next step is to isolate any terms that can violate the maximally general TMD-factorization criteria of Sect. IV.For this it will be convenient to rewrite Eq. (134) in terms of l t,ρ and k t,ρ using l t,ρ = k t,ρ − l t,ρ : (cid:2) M (2 g ) (cid:3) P . V . = − e q g s f αβκ ¯ u ( k , S ) U ( P ; l , l ) αβ ; µ µ (cid:26) g tµ ρ n µ (cid:18) P . V . l − (cid:19) − g tµ ρ n µ (cid:18) P . V . l − (cid:19) ++ n µ n µ (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19) (cid:18) k t,ρ l − k − − l t,ρ (cid:19)(cid:27) ×× H LO ( k , k , k , k ) κ ; ρσ L ( P ; k ) + O (1 /Q ) . (135)We now organize the terms in Eq. (135) according to their compatibility or non-compatibility with the maximallygeneral TMD-factorization criteria established in Sects. IV A and Sect. IV D. We separate Eq. (135) into the followingterms: (cid:2) M (2 g ) (cid:3) P . V . = (cid:2) M (2 g ) (cid:3) Fact ., + (cid:2) M (2 g ) (cid:3) Fact ., + (cid:2) M (2 g ) (cid:3) Fact ., + (cid:2) M (2 g ) (cid:3) Fact . Viol . + O (1 /Q ) , (136)where we have defined (cid:2) M (2 g ) (cid:3) Fact ., ≡ − e q g s f αβκ (cid:88) i J ( P ; k , l ) αβ ; i (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / H LO ( k , k , k , k ) κ ; iσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / , (137) (cid:2) M (2 g ) (cid:3) Fact ., ≡ − e q g s f αβκ (cid:88) i J ( P ; k , l ) αβ ; i (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / H LO ( k , k , k , k ) κ ; iσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / , (138) (cid:2) M (2 g ) (cid:3) Fact ., ≡ − e q g s f αβκ (cid:88) i J ( P ; k , l ) αβ ; i (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / H LO ( k , k , k , k ) κ ; iσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / , (139) (cid:2) M (2 g ) (cid:3) Fact . Viol . ≡ − e q g s f αβκ (cid:88) i J F . V . ( P ; k , l ) αβ ; i (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / H LO ( k , k , k , k ) κ ; iσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / . (140)4The J factors are defined as J ( P ; k , l ) αβ ; i ≡ (cid:18) U ( P ; l , l ) αβµ µ (cid:18) P . V . l − (cid:19)(cid:19) P µ µ ,i U , ∼ Q , (141) J ( P ; k , l ) αβ ; i ≡ (cid:18) U ( P ; l , l ) αβµ µ (cid:18) P . V . l − (cid:19)(cid:19) P µ µ ,i U , ∼ Q , (142) J ( P ; k , l ) αβ ; i ≡ (cid:18) U ( P ; l , l ) αβµ µ (cid:18) l − k − (cid:19) (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19)(cid:19) P U , ( k t ) µ µ ,i ∼ Q , (143) J F . V . ( P ; k , l ) αβ ; i ≡ (cid:18) U ( P ; l , l ) αβµ µ (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19)(cid:19) P U , F . V . ( l t ) µ µ ,i ∼ Q . (144)The gluon polarization projections are P µ µ ,i U , ≡ n µ g µ it , (145) P µ µ ,i U , ≡ − n µ g µ it , (146) P U , ( k t ) µ µ ,i ≡ n µ n µ k i t , (147) P U , F . V . ( l t ) µ µ ,i ≡ − n µ n µ l i t . (148)Substituting Eqs.(137-140) into Eq. (136) reproduces Eq. (135). Note that Eqs. (145, 146) only project U ( P ; l , l ) − j ∼ Q components, giving an overall Q power when combined with the single eikonal denominators of Eqs. (141,142).Equations (147, 148) project the U ( P ; l , l ) αβ, −− ∼ Q components, giving an overall power of Q when combinedwith the double eikonal denominators in Eqs. (143, 144).Now comparing Eqs. (137–148) with Eqs. (62–65) from Sect. IV D verifies that Eqs. (137, 138, 139) are consistentwith TMD-factorization with the maximally general criteria. However, Eq. (140) is a candidate TMD-factorizationviolating term in the form of Eq. (65). Specifically, the transverse momentum, l t , in the gluon polarization projectionof Eq. (148) differs from the transverse momentum entering the hard subprocess. Therefore, we have used “F . V . ”(super)subscripts in Eqs. (140, 144, 148) to label them as potential TMD-factorization violating contributions. J. Candidate TMD-Factorization Violating Terms in the Amplitude
In Eqs. (137–140) we have categorized each term in the amplitude according to its factorizability, and we haveisolated the term in Eq. (140) that, according to Sect. IV D, can violate the TMD-factorization criteria of Sect. IV A.Writing out this contribution explicitly in terms of its basic factors gives (cid:2) M (2 g ) (cid:3) Fact . Viol . == − e q g s f αβκ (cid:88) i J F . V . ( P ; k , l ) αβ ; i ¯ u ( k , S ) H LO ( k , k , k , k ) κ ; iσ L ( P ; k )= e q g s f αβκ (cid:88) i U ( P ; l , l ) αβ ; − , − (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19) l i t (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / H LO ( k , k , k , k ) κ ; iσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / + O (1 /Q )= − e q g s f αβκ (cid:88) i U ( P ; l , l ) αβ ; µ ,µ (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19) P U , F . V . ( l t ) iµ µ (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / × H LO ( k , k , k , k ) κ ; iσ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q L ( P ; k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / + O (1 /Q ) . (149)5In the last equality, the amplitude has been written as in Eq. (62). The order ∼ Q factor on the first line becomesa U eff , but now it is exactly in the form of a candidate TMD-factorization breaking contribution like Eq. (65). Thegluon transverse momentum, l t , which determines the polarization projection in Eq. (148), differs from the actualtotal transverse gluon momentum, k t , that enters into the calculation of the leading-order hard part. And it is notpossible to shift P U , F . V . ( l t ) i into another factor without H L or L then acquiring dependence on l t .For an Abelian theory, f αβκ → K. Reduction To Factorization in Collinear Case
From the form of Eq. (149), it is evident that any violation of the criteria of Sect. IV A is specific to observables thatare sensitive to intrinsic transverse momentum. The violation of generalized TMD-factorization is due to sensitivityto the difference between the two intrinsic transverse momenta, l t and k t , outside of U . To have this, the observablemust be sensitive to intrinsic transverse momenta to begin with.More specifically, the “F . V . ” transverse polarization projection in Eq. (148) appears at leading order in the crosssection paired with the tree-level projection, G ( k ) µ (cid:48) j , from Eqs. (77, 88, 89): P U , F . V . ( l t ) i G ( k ) j (150)where to simplify notation here we have only written the transverse i, j indices that couple to the hard part.If we were to restrict consideration to observables that are insensitive to a small q t in Fig. 1, i.e. that are azimuthallysymmetric, then the sum over the i and j components is diagonal. Then we could follow Sect. III D of Ref. [81],averaging over transverse gluon polarizations in the hard part while separately tracing over the transverse componentsin Eq. (150) in the upper subgraph, U , for hadron-2. That is, the combination in Eq. (150) could be replaced with (cid:88) i P U , F . V . ( l t ) i G ( k ) i , (151)in which case the sums over transverse polarization components, as well as the integrations over l t and k t , takeplace independently of the rest of the graph. The result is a contribution to the normal integrated collinear gluonPDF identified in Sect. V G of Ref. [81]. Hence, factorization of the gluon distribution is maintained so long as werestrict to collinear observables.In the next section, however, we are specifically interested in observables that are sensitive to the direction of q t ,and therefore we must account for the off-diagonal parts of Eq. (150). VIII. FINAL STATE SPIN DEPENDENCE IN THE PRESENCE OF ONE EXTRA GLUON
The aim of this section is to verify, by directly counting powers of Q , that terms like Eq. (149) result in a breakdownof TMD-factorization with an extra TMD-factorization-violating spin correlation in the final state. Recall the strategyoutlined in Sect. IV C for identifying spin correlations that are specific to TMD-factorization breaking effects. Thereit was noted that a leading-power correlation between the spin S of the final state quark and the helicity λ of theprompt photon does not fit into the standard classification scheme. Thus, the strategy of this section is to isolateterms that give contributions from these specific spin correlations in the cross section.Such extra asymmetries can only arise from contributions that violate the most general version of the TMD-factorization hypothesis, stated in Sect. IV. Therefore, we may focus attention on the only term, isolated in Eq. (149),that violates those criteria. A. TMD-Factorization Breaking Cross Section
The lowest order contribution to the cross section from the term in Eq. (149) is obtained by adding it to the treelevel contribution in Eq. (89), taking the square modulus, and keeping the order g s terms. The steps are exactly6analogous to those of Sect. V C and we write the details here. Using Eq. (31), we find dσ F . V . = C Q (cid:88) X (cid:88) X (cid:88) s (cid:90) d k (2 π ) d (2 π ) d (2 π ) (cid:2) M (2 g ) (cid:3) Fact . Viol . M G † (1 g ) (2 π ) δ (4) ( k + 1 + l − k − k )= C Q (cid:88) X (cid:88) X (cid:88) s (cid:88) ij (cid:90) d k (2 π ) d (2 π ) d (2 π ) × (cid:18) ie q g s f αβκ U ( P ; l , l ) αβµ (cid:48) ,µ (cid:48) U † ( P , k ) κ (cid:48) ν (cid:48) (cid:18) P . V . l − (cid:19) (cid:18) P . V . l − (cid:19) P U , F . V . ( l t ) µ (cid:48) µ (cid:48) ,i G ( k ) ν (cid:48) j (cid:19) × Tr (cid:104) ¯ u ( k , S ) H κ ; i LO σ L ¯ L H † κ (cid:48) ; j LO σ (cid:48) u ( k , S ) (cid:105) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ (2 π ) δ (4) ( k + l + l − k − k )+ H . C . = i C Q e q g s f αβκ (cid:88) X (cid:88) X (cid:88) s (cid:88) ij (cid:90) d k t dk − (2 π ) (cid:90) d l t dl +1 dl − (2 π ) (cid:90) dk +2 π × (cid:18) U ( P ; l , k − l ) αβµ (cid:48) ,µ (cid:48) U † ( P , k ) κ (cid:48) ν (cid:48) (cid:18) P . V . l − (cid:19) (cid:18) P . V . k − − l − ) (cid:19) P U , F . V . ( l t ) µ (cid:48) µ (cid:48) ,i G ( k ) ν (cid:48) j (cid:19) × Tr (cid:104) ¯ u ( k , S ) H κ ; i LO σ L ¯ L H † κ (cid:48) ; j LO , σ (cid:48) u ( k , S ) (cid:105) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ + H . C . = i C Q e q g s f αβκ (cid:88) s (cid:88) ij (cid:90) d k t (2 π ) (cid:90) d l t (2 π ) × (cid:26) (cid:88) X (cid:90) dl +1 dl − dk +2 (2 π ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q × U ( P ; l , k − l ) αβµ (cid:48) ,µ (cid:48) U † ( P , k ) κ (cid:48) ν (cid:48) (cid:18) P . V . l − (cid:19) (cid:18) P . V . k − − l − ) (cid:19) P U , F . V . ( l t ) µ (cid:48) µ (cid:48) ,i G ( k ) ν (cid:48) j (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:27) × Tr (cid:20) ¯ u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / ˆ H κ ; i LO σ (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (cid:40)(cid:88) X (cid:90) dk − π L ¯ L (cid:41)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ˆ H † κ (cid:48) ; j LO , σ (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q u ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q / (cid:21) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ + H . C . ∼ Q . (152)In the second equality, we have written out the explicit factors. In the the third equality, we have changed variablesfrom l to k − l and we have evaluated the momentum conserving δ -functions using the l integration. In the lastequality we have applied the minimal TMD kinematic approximations of Eqs. (38-40), as indicated by the hats placedon ˆ H LO , and we have shifted the positions of the l +1 , l − , k +2 and k − integrals into separate factors reminiscent of thestep in Eq. (92). Also, we have explicitly written the final state photon polarization vectors (cid:15) σλ (cid:15) ∗ σ (cid:48) λ from Eq. (29).Eq. (152) has almost exactly the same structure as the simple tree level result, especially if the factors in braces inthe last equality are identified as special types of effective TMD PDFs. Compare with Eq. (92); the only modificationis that we have isolated the TMD-factorization violating projection from Eq. (150) that comes from a single extrasoft/collinear gluon.A natural strategy would be to attempt to rewrite this as an effective TMD-factorization formula. However, itis clear now from the structure of Eq. (152) that this is impossible without the effective TMD PDFs violating atleast one of the the minimal criteria for generalized TMD-factorization enumerated in Sect. IV A: On one hand, ifthe P U , F . V . ( l t ) µ (cid:48) µ (cid:48) ,i projection is included inside the definition of an effective gluon TMD PDF then criterion 2.) isviolated because the projection of Dirac components on the fermion line then depends on l t . On the other hand, if P U , F . V . ( l t ) µ (cid:48) µ (cid:48) ,i is included in a redefinition of the hard part or in the definition of an effective quark TMD PDFthen criteria 1.) or 3.) are violated respectively. So there is no possible rearrangement of factors that allows for afactorization formula with the properties of Sect. IV A. Hence, the maximally general form of TMD-factorization isbroken.7 B. Effective Factorization Violating TMD Functions
In spite of the problems noted above with TMD-factorization, let us push forward with an identification of effectiveTMD PDFs, but now allowing them to violate the maximally general criteria for TMD-factorization from Sect. IV A.Once we have organized factors into a set of effective (but TMD-factorization violating) TMD PDFs, we may thendirectly examine the consequences for power counting and the classification of final state spin dependence.We use the explicit expression for ˆ H i ; κ LO σ from Eq. (90) and directly apply the expression for the “F . V . ” projectionin Eq. (148). Then, dropping power suppressed terms, the factor inside the Dirac trace on the last line of Eq. (152)becomes P U , F . V . ( l t ) µ (cid:48) µ (cid:48) ,i ˆ s ˆ t (cid:16) ¯ u ( k , S ) γ i / ˆ hγ σ L ( P ; k ) ¯ L ( P ; k ) γ j /qγ σ (cid:48) u ( k , S )+ ¯ u ( k , S ) γ σ /qγ i L ( P ; k ) ¯ L ( P ; k ) γ σ (cid:48) / ˆ hγ j u ( k , S ) (cid:17) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ t κ t κ (cid:48) + P U , F . V . ( l t ) µ (cid:48) µ (cid:48) ,i ˆ s (cid:16) ¯ u ( k , S ) γ σ /qγ i L ( P ; k ) ¯ L ( P ; k ) γ j /qγ σ (cid:48) u ( k , S ) (cid:17) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ t κ t κ (cid:48) + P U , F . V . ( l t ) µ (cid:48) µ (cid:48) ,i ˆ t (cid:16) ¯ u ( k , S ) γ i / ˆ hγ σ L ( P ; k ) ¯ L ( P ; k ) γ σ (cid:48) / ˆ hγ j u ( k , S ) (cid:17) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ t κ t κ (cid:48) = − n µ (cid:48) n µ (cid:48) ˆ s ˆ t ¯ u ( k , S ) /l t F.S.I. / ˆ hγ σ L ( P ; k ) ¯ L ( P ; k ) γ j /qγ σ (cid:48) u ( k , S )++ ¯ u ( k , S ) γ σ /q /l t L ( P ; k ) I.S.I. ¯ L ( P ; k ) γ σ (cid:48) / ˆ hγ j u ( k , S ) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ t κ t κ (cid:48) − n µ (cid:48) n µ (cid:48) ˆ s ¯ u ( k , S ) γ σ /q /l t L ( P ; k ) I.S.I. ¯ L ( P ; k ) γ j /qγ σ (cid:48) u ( k , S ) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ t κ t κ (cid:48) − n µ (cid:48) n µ (cid:48) ˆ t ¯ u ( k , S ) /l t F.S.I. / ˆ h γ σ L ( P ; k ) ¯ L ( P ; k ) γ σ (cid:48) / ˆ hγ j u ( k , S ) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ t κ t κ (cid:48) . (153)After the second equality, we have included horizontal brackets underneath the equation to indicate where the anoma-lous /l t transverse momentum contraction is an extra soft initial state attachment (labeled I.S.I.) and where it is anextra soft final state attachment (labeled F.S.I.).It is in principle possible for soft I.S.I. and F.S.I. attachments to cancel. But any such cancellations cannot be generalbecause they depend on the separate non-perturbative details of /l t L ( P ; k ) in the initial state and ¯ u ( k , S ) /l t inthe final state, including sensitivity to the species of incoming and outgoing hadrons and the quark masses.Therefore, without loss of generality, we may focus attention on the initial state interactions. These correspondto modified TMD-factorization breaking effective TMD PDFs. An analysis similar to what we give below may beapplied to the extra final state interactions, but this will give modified effective TMD-factorization breaking TMDfragmentation or jet functions.It is the second and third terms in Eq. (153) that contain extra initial state Dirac matrix contractions. Thesecorrespond to a cross term and an s -channel term respectively in the squared amplitude. We rewrite them by firstdefining the effective TMD functions for the factors in braces in Eq. (152),Φ (2 g ) q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17) ≡ Qm q (cid:88) X (cid:90) dk − π /q/l t L ¯ L ∼ Q , (154)∆ k /q ( k , S ) ≡ (cid:88) s u ( k , S )¯ u ( k , S ) = ( /k + m q ) (cid:18) γ /S m q (cid:19) ∼ Q , (155)and,8Φ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) αβκ (cid:48) ; j ≡ Q (cid:88) X (cid:90) dl +1 dl − dk +2 (2 π ) U ( P ; l , k − l ) αβ ; − , − U † ( P , k ) κ (cid:48) ν (cid:48) G ( k ) ν (cid:48) j (cid:18) P . V . l − (cid:19) (cid:18) P . V . k − − l − ) (cid:19) ∼ Q . (156)Note the /q/l t in the definition in Eq. (154) as compared with Eq. (93). Equation (154) also has an extra powerof Q relative to Eq. (93) and is defined with an overall factor of 1 /m q to maintain the same units as Eq. (93).Equation (155) is just the spin sum for the final state quark, but represented in notation reminiscent of a jet orfragmentation function.Using Eqs. (153–156) inside Eq. (152) gives dσ I . S . I . X − term = − i C e q g s f αβκ t κ t κ (cid:48) m q ˆ s ˆ t (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (cid:88) j (cid:90) d k t (2 π ) (cid:90) d l t (2 π ) ×× Φ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) αβκ (cid:48) ; j (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q Tr (cid:20) ∆ k /q ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q γ σ Φ (2 g ) q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q γ σ (cid:48) / ˆ hγ j (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:21) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ + H . C . (157)and dσ I . S . I . s − chan = − i C e q g s f αβκ t κ t κ (cid:48) m q ˆ s (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (cid:88) j (cid:90) d k t (2 π ) (cid:90) d l t (2 π ) ×× Φ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) αβκ (cid:48) ; j (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q Tr (cid:20) ∆ k /q ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q γ σ Φ (2 g ) q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q γ j /qγ σ (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:21) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ + H . C . . (158)In Eq. (157), we have pulled the 1 / ˆ t outside the integrals using the leading-power approximation ˆ t ≈ ( k − ˆ k ) ≈− x P +1 k − . The subscripts “X − term” and “s − chan” label the cross-term and s -channel contributions.The above steps for separating the cross section into factors are very similar to those of Sect. V C. However,Eqs. (157, 158) violate the minimal criteria for TMD-factorization from Sect. IV A because of the extra momentumarguments of Φ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) and Φ (2 g ) q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17) . As explained in the last paragraph ofSect. VIII A, at least one such violation of the minimal criteria is required by the pattern of non-Abelian Wardidentity cancellations in Sect. VII.For the sake of demonstrating with explicit color factors, let us now take the target hadrons P and P to be on-shellcolor triplet quarks, and calculate the contribution for averaged initial and summed final state quark color. Also, letus consider the component of Φ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) αβκ (cid:48) ; j proportional to − if αβκ (cid:48)(cid:48) t κ (cid:48)(cid:48) t κ (cid:48) so that we may definean effective colorless gluon TMD function viaΦ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) αβκ (cid:48) ; j → − if αβκ (cid:48)(cid:48) t κ (cid:48)(cid:48) t κ (cid:48) Φ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) j . (159) An equivalent analysis may be performed with /q and/or /l t organized into different factors. We find the above arrangement to be themost convenient for illustrating the basic result. N c − / N c . Equations (157, 158) become: dσ I . S . I . X − term = −C e q g s ( N c − m q N c ˆ s ˆ t (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (cid:88) j (cid:90) d k t (2 π ) (cid:90) d l t (2 π ) ×× Φ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) j (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q Tr (cid:20) ∆ k /q ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q γ σ Tr C (cid:110) Φ (2 g ) q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17)(cid:111)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q γ σ (cid:48) / ˆ hγ j (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:21) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ + H . C . (160)and dσ I . S . I . s − chan = −C e q g s ( N c − m q N c ˆ s (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (cid:88) j (cid:90) d k t (2 π ) (cid:90) d l t (2 π ) ×× Φ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) j (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q Tr (cid:20) ∆ k /q ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q γ σ Tr C (cid:110) Φ (2 g ) q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17)(cid:111)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q γ j /qγ σ (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:21) (cid:15) σλ (cid:15) ∗ σ (cid:48) λ + H . C .. (161)In these equations we have used that L ¯ L is diagonal in triplet color; the Tr C {} denotes a trace over color tripletindices.To obtain an analogue of Eq. (95), we first decompose Φ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) j into its possible transverse indexstructures. The only dependence on transverse indices is from l t and k t so the analogue of Eq. (41) isΦ (2 g ) g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17) j ≡ k j t m q Φ (2 g ) [ k t ] g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q + l j t m q Φ (2 g ) [ l t ] g/P (cid:16) P ; ˆ k ( x , k t ) , l t (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ∼ Q . (162)There may in general be a separate contribution from each effective gluon TMD PDF labeled with the [ k t ] and [ l t ]superscripts.We will focus first on the contribution from the [ k t ]-term in Eq. (162). Using an arbitrary Dirac projection Γ q andfollowing by analogy with Eq. (51), Eqs. (160, 161) then become dσ I . S . I . [Γ q ]X − term = − C e q g s ( N c − N c ˆ s ˆ t (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (cid:90) d k t (2 π ) (cid:90) d l t (2 π ) ×× Φ (2 g ) [ k t ] g/P (cid:16) P ; q − ˆ k ( x , k t ) , l t (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q Φ (2 g ) [Γ q ] q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ×× Tr (cid:104) ∆ k /q ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q /(cid:15) λ P Γ q /(cid:15) ∗ λ / ˆ h ( /q t − /k t ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q (cid:105) + H . C . , (163)0 P P l l k q k /q/l t P P l l k q k /q/l t (a) (b)FIG. 9. Graphs representing the organization of factors in a.) Eq. (163) and b.) Eq. (164). Hermitian conjugate graphs shouldalso be included. The “ ×(cid:13) ” vertex denotes the anomalous factor of /q/l t in Eq. (154), which cannot be associated with a gaugelink operator in a TMD PDF for either hadron alone. and dσ I . S . I . [Γ q ]s − chan = − C e q g s ( N c − N c ˆ s (cid:124) (cid:123)(cid:122) (cid:125) ∼ /Q (cid:90) d k t (2 π ) (cid:90) d l t (2 π ) ×× Φ (2 g ) [ k t ] g/P (cid:16) P ; q − ˆ k ( x , k t ) , l t (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q Φ (2 g ) [Γ q ] q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ×× Tr (cid:104) ∆ k /q ( k , S ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q /(cid:15) λ P Γ q ( /q t − /k t ) /q (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q /(cid:15) ∗ λ (cid:105) + H . C . . (164)Above we have usedΦ (2 g ) [Γ q ] q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17) ≡ Tr C (cid:18)
14 Tr (cid:104) Γ q Φ (2 g ) q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17)(cid:105)(cid:19) , (165)in accordance with Eq. (48).In Eqs. (163, 164) we have reached the analogues of Eq. (51), but now in a form that accounts for the extrainitial state interactions. The effective TMD PDFs now depend on extra momenta external to their parent hadronsand therefore violate the minimal requirements for TMD-factorization. The organization of factors in Eqs. (153)through (165) is represented graphically in Fig 9. The special “ ×(cid:13) ” vertex denotes the TMD-factorization anomalyfactor of /q/l t in Eq. (154) that enters into the trace of Dirac matrices along the lower active quark line. The extra /l t (cid:54) = /k t from hadron-2 is trapped by the /q on the hadron-1 side of the graph. C. Power Counting and TMD-Factorization Violating Spin Asymmetries
In Eqs. (163,164), the initial state interaction contribution to the cross section from Eq. (140) has been writtenin a form analogous to Eq. (51), but with the effective TMD functions now explicitly violating the minimal TMD-factorization criteria of Sect. IV A. We can now examine whether the exclusion of certain leading-power correlationson the basis of parity invariance, as discussed in Sect. IV C, still applies to the effective TMD PDFs in the TMD-factorization breaking scenario of Eqs. (163,164).1In fact, a close inspection of the more general multi-gluon effective TMD objects like Eq. (165) hints at a largevariety of extra angular and spin dependencies. It is beyond the scope of this paper to classify all of them in detail,so we will instead illustrate the general existence of such effects by focusing on the specific case, already discussed inSect. IV C, of a final state correlation between photon helicity λ and transverse quark spin S .
1. Initial State Effective TMD PDF
Applying the projection in Eq. (59) to Eqs. (163, 165) gives the effective (but TMD-factorization violating) TMDPDF: Φ (2 g ) [ γ ] q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17) = Tr C (cid:18)
14 Tr (cid:104) γ Φ (2 g ) q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17)(cid:105)(cid:19) . (166)Now the effective TMD PDF is a function of q and l t as well as P and k , in violation of the minimal criteria forTMD-factorization from Sect. IV A. So Eq. (166) in general gives a non-vanishing leading-power trace,Φ (2 g ) [ γ ] q/P (cid:16) P ; ˆ k ( x , k t ) , q, l t (cid:17) ∝
14 Tr (cid:2) γ /q/l t /k /P (cid:3) ≈ iP · q (cid:16) k i t l j t (cid:15) ij (cid:17) f ∼ Q . (167)The approximation sign is used on the second line because we have neglected subleading terms in the evaluation ofthe trace.
2. Effective Hard Subprocess
There is one more Dirac trace in each of Eq. (163) and Eq. (164). It is the trace for the bottom quark line inFig. 9 involving the final state polarizations λ and S . Since our main interest is in the dependence on the final statetransverse polarization for the quark, we keep only the leading power transverse spin contribution from Eq. (155):∆ k /q ( k , S ) → γ /k /S /m q . (168)Applying Eq. (60), the Dirac trace for the effective hard subprocess in Eq. (163) for the cross-term contributionbecomes Tr (cid:104) /k /S /(cid:15) λ /(cid:15) ∗ λ / ˆ h ( /q t − /k t ) (cid:105) . (169)(Recall the definition of h from Eq. (6).) To get a final state double spin dependence, we must extract the imaginarypart of Eq. (169) to multiply the i in Eq. (167). For this it is useful to note that for arbitrary four-vectors v µ and v ν , v µ v ν Im (cid:0) (cid:15) µλ (cid:15) ∗ νλ (cid:1) = λ v µ v ν Im ( (cid:15) µ (cid:15) ∗ ν ) . (170)Then, the imaginary terms in Eq. (169) are i Im (cid:16) Tr (cid:104) /k /S /(cid:15) λ /(cid:15) ∗ λ / ˆ h ( /q t − /k t ) (cid:105)(cid:17) = iλ Im (cid:16) Tr (cid:104) /k /S /(cid:15) /(cid:15) ∗ / ˆ h ( /q t − /k t ) (cid:105)(cid:17) . (171)This trace may be straightforwardly evaluated and the leading-power terms kept. It is helpful to note thatIm ( v µ v ν (cid:15) µ (cid:15) ∗ ν ) = 12 (cid:16) (cid:15) ij v i v j (cid:17) f , (172)where the f subscript is a reminder that here the transverse indices i, j are defined with respect to frame-3. Evaluatingthe trace in Eq. (171) then gives iλ Im (cid:16) Tr (cid:104) /k /S /(cid:15) /(cid:15) ∗ / ˆ h ( /q t − /k t ) (cid:105)(cid:17) = 4 iλ (cid:16) S i ˆ k j (cid:15) ij (cid:17) f (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ( k · ( q t − k t )) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q + 4 iλ (cid:16) ( q t − k t ) i S j (cid:15) ij (cid:17) f (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ( k · ˆ h ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ∼ Q . (173)2Here we have used the definition of h in Eq. (6). On first line, note that the transverse components of ˆ k are takenwith respect to frame-3 and, recalling Eq. (26), the frame-3 transverse components of the incoming hadrons andpartons are of order ∼ Q .The same steps apply to the s -channel trace of Eq. (164). The result is obtained by replacing / ˆ h ( /q t − /k t ) → ( /q t − /k t ) /q in Eq. (171) . The trace is iλ Im (cid:16) Tr (cid:104) /k /S /(cid:15) /(cid:15) ∗ ( /q t − /k t ) /q (cid:105)(cid:17) = 4 iλ (cid:0) S i ( q t − k t ) j (cid:15) ij (cid:1) f (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ( k · k ) (cid:124) (cid:123)(cid:122) (cid:125) ∼ Q ∼ Q . (174)Analogous steps also apply if we use the [ l t ] effective gluon TMD PDF from second term in Eq. (162) rather thanthe [ k t ] one. This contributions differ from the above only in that q t − k t gets replaced by l t in Eqs. (173, 174) andΦ (2 g ) [ k t ] g/P gets replaced by Φ (2 g ) [ l t ] g/P in Eqs. (163, 164).
3. Double Final State Spin Asymmetry
By counting powers of Q in Eqs. (162, 167, 173, 174) inside Eqs. (163, 164) we are now able to tally the largestpowers contributing to a TMD-factorization violating final state double spin asymmetry: dσ I . S . I . [ γ ] ∝ λ (cid:16) S j (cid:17) f × N c − N c × O ( Q ) j . (175)The imaginary i in Eq. (173) combines with the i in Eq. (171) to give a real contribution to the cross section so thegraphs in Fig. 9 and their Hermitian conjugates add.To summarize, power counting implies a leading contribution to the cross section from a double final state S - λ spin dependence. In Eq. (175), we have explicitly written the λ (cid:16) S j (cid:17) f to emphasize the final state spin-helicitycorrelation, and we have explicitly displayed the color factor to emphasize the non-Abelian nature of the effect — thespin asymmetry exactly vanishes in the Abelian limit of N c → /
9. Allremaining factors from the substitution of Eqs. (162, 167, 173, 174) into Eqs. (163, 164) have been abbreviated by the O ( Q ) j . This corresponds to a convolution integral of effective non-perturbative multiparton correlation functions.Recall that the [ γ ] superscript in Eq. (175) means that we have applied the pseudo-scalar γ projection fromEq. (59) to the effective TMD PDF in Eq. (154). Without the breakdown in maximally generalized TMD-factorization,projections like these would be applied only to ordinary TMD PDFs with the minimal TMD-factorization propertiesfrom Sect. IV A. They would therefore be discarded on the grounds that they violate parity.To totally characterize the final state angular and spin dependence for Eq. (1), all other leading-power projectionsshould also be accounted for, including the axial vector and tensor terms from Eq. (44) for the quark subgraph. Weleave a complete classification of the types of general behavior possible at leading power to future work. We also planto obtain explicit expressions for contributions like Eq. (175), for various initial and final state spin configurations,by using specific non-perturbative models for the incoming hadron wave functions. IX. SUMMARY AND DISCUSSION
By directly counting powers of the hard scale for the process of Sect. II A, we have shown how extra spin asymmetriesmay arise at leading power due to Ward identity non-cancellations that break TMD-factorization. This is sufficientto conclude that TMD-factorization-based classification schemes are too limiting to account for all the qualitativelydistinct types of non-perturbative correlations that are possible in the limit of fixed energy and large Q (cid:29) Λ. The( N c − /N c factor in Eq. (175) reflects the special role of non-Abelian gauge invariance in producing such extra spindependence. That the leftover terms arise from an incomplete Ward identity cancellation is indicative of the role ofTMD-factorization breaking effects in preserving gauge invariance over large distances at leading power in the crosssection. The essential steps for analyzing Feynman diagrams were already presented in Ref. [81] for the collinearcase. In this paper, we have argued that certain terms in the Wilson line factors correspond to the breakdown offactorization when transitioning from the collinear to TMD cases.It should be emphasized that the “extra” soft or collinear gluon attachments here are in the regime of largecoupling, so we are not actually justified in stopping with a fixed number of gluons. Hence, the failure of the extragluons to factorize in perturbation theory corresponds to the influence of extra non-perturbative effects. The types3of interactions are similar to those normally associated with Wilson lines in the definitions of TMD PDFs. However,for the process considered in this paper, they cannot be attributed to Wilson line operators for any separate externalhadron alone. Instead, they describe truly global non-perturbative physical properties of the process.The specific example used in Sects. VII and VIII was chosen to illustrate the existence of TMD-factorizationbreaking correlations in the simplest way possible and with only one extra soft gluon needing to be considered. Thiswas sufficient for the current purpose of demonstrating a non-trivial interplay between transverse momentum, spin,and color degrees of freedom. However, other extra spin and azimuthal asymmetries are also likely to arise followingsimilar mechanisms, especially when going beyond the single extra gluon case. For example, the term in Eq. (140)may contribute to an azimuthal angular dependence in the totally unpolarized cross section. Also, if the imaginaryparts in Eqs. (124, 125) are taken into consideration, then a single final state spin asymmetry may be generated at thesingle extra gluon level. A full classification of spin and angular asymmetries in TMD-factorization breaking scenariosis left for future work.The interactions discussed here and in earlier work on similar topics (e.g., [13–19, 26, 90, 91]) are novel because theytend to give rise to effects outside of the common classical intuition that originates in a parton model picture. Whilethe general appearance of quantum mechanical correlations is not surprising, these specific effects are distinguishableby the fact that they are induced by the kinds of large coupling, long-range interactions that are more commonlyassociated with hadronic binding and confinement in separate hadrons. Even for processes where TMD-factorizationis valid, physical effects from initial and final state soft and collinear interactions are already predicted in, for example,studies of T-odd effects such as the Sivers effect [90, 91]. Namely, the Sivers function [92, 93] (a type of TMD PDF)is predicted to reverse sign between Drell-Yan and SIDIS due to a reversal in the direction of the gauge link betweenthese two processes [26]. (For physical pictures of the sign reversal mechanism see, e.g., Refs. [78, 94].) Interestingconnections between QCD process dependence and a QCD version of the Aharonov-Bohm effect have been suggestedin Refs. [95, 96].The existence of TMD-factorization breaking implies generally that unexpected experimental phenomena are pos-sible. In this context, it is worth recalling again the production of transversely polarized Λ-hyperons in unpolarizedhadron-hadron collisions (spontaneous Λ polarization) [52]. Past TMD-based explanations have focused on effects inthe fragmentation process, such as a polarizing fragmentation function. The production of transversely polarized Λ-hyperons in pp → Λ ↑ + X collisions was described within a hybrid TMD/collinear-factorization formalism in Ref. [97],with the spontaneous transverse polarization being ascribed to the TMD polarizing fragmentation function. It was ar-gued in Ref. [98] (assuming universality for TMD fragmentation functions) that this should imply similar Λ transversepolarizations in the e + e − collisions in the ALEPH experiment at LEP [53], and that the failure to observe such effectssuggests that spontaneous production of Λ polarizations in hadron-hadron collisions might also require contributionsfrom initial state interactions. (Significant Λ polarizations also have not been observed in SIDIS measurements [54].)However, the semi-inclusive pp → Λ ↑ + X and e + e − → Λ ↑ + X processes are not differential in a small q t , and soare not strictly appropriate for treatment within a TMD formalism. Moreover, for totally inclusive e + e − annihilationprocesses, twist-three formalisms [99, 100] are so far consistent with measurements of spontaneous Λ polarization. Atbest, information about the TMDs must be extracted from these measurements indirectly through, for example, hybridformalisms like that of Ref. [97]. Furthermore, the e + e − measurements have been performed with a hard scale near the Z -pole, whereas the measurements of spontaneous transverse Λ-hyperon polarization in the pp → Λ ↑ + X process hasa hard scale of only a few GeV. QCD evolution likely leads to a large suppression in the e + e − measurements relativeto the hadro-production measurements, and this may be sufficient to explain the discrepancy. In addition, currentSIDIS measurements may be small due to effects specific to small- x dynamics. Thus, the status of comparisons ofspontaneous transverse Λ polarization between different processes remains inconclusive, and new studies are needed.Ideally, investigations of TMD universality should be made by comparing processes like pp → hadron(jet) + Λ ↑ + X and e + e − → hadron(jet) + Λ ↑ + X where a TMD-style description is most appropriate, as explained recently inRefs. [101, 102]. New SIDIS measurement may also be useful, as this is a process where TMD-factorization isexpected to hold. (The universality of fragmentation functions between e + e − → hadron(jet) + Λ ↑ + X and SIDIShas been shown in Refs. [103, 104].) Future measurements may be possible, and the above general observationssuggest that spontaneous Λ polarization studies are ideal for comparing TMD-factorization versus TMD-factorizationbreaking scenarios. See also the discussion in Ref. [105]. Alternative explanations for spontaneous Λ polarizationhave proposed in Refs. [98, 106] within the twist-three collinear factorization formalism. It is also interesting to notethat, in Ref. [107], a possible inconstancy was found with universal unpolarized TMD Λ-fragmentation functions in pp → Λ / ¯Λ + X measurements as compared with e + e − → Λ / ¯Λ + X . (See also the discussion in Ref. [108].)Other observables are also likely to be useful for probing the role of TMD-factorization breaking effects. The role ofthree-gluon correlators in a twist-three approach has been recently discussed in Ref. [109]. In Ref. [110], a proposedsolution to the Sivers sign discrepancy between Drell-Yan and SIDIS [55] was also given in the twist-three formalism.Including a treatment of fragmentation in the description of transverse single spin asymmetries was also found to beimportant within the twist-three formalism in Ref. [111]. In principle, it should be possible to relate the TMD and4higher twist collinear formalisms through integrations over transverse momentum. In this way, it may be possible torelate the breakdown of TMD-factorization discussed hear to higher twist collinear correlation functions. Studies of thedetailed properties of final state jets may also help clarify the dynamics responsible for final state asymmetries. Ratherthan the polarizations of final state hadrons, for instance, one may instead investigate the azimuthal distribution ofhadrons around a jet axis [112–115]. The “CMS ridge” of Ref. [62] has also been explained in terms of effects thatare beyond the scope of standard leading twist factorization. These include both Wilson line descriptions [63] andcolor glass condensate descriptions [64–66]. Interestingly, both descriptions incorporate subtle quantum mechanicalentanglement effects.Although a thorough discussion of how TMD-factorization breaking effects might be calculated precisely from firstprinciples QCD is beyond the scope of this paper, it is natural to speculate that a form of TMD perturbative QCD,with a partonic description of the hard scattering, might be recovered by a rearranged treatment of non-perturbativecontributions. Rather than having separate TMD PDFs and TMD fragmentation functions, for example, one mightuse a set of non-perturbative functions wherein the constituents of multiple external hadrons remain entangled, buta hard partonic factor is separated out. We leave the question of whether such a separation is valid to futurestudies. Already, some versions of soft-collinear-effective-theory (SCET) refrain from identifying operator definitionsfor separate hadrons [116]. Other approaches that are likely to be useful in describing TMD-factorization breakingeffects are those that are now more commonly associated with the physics of high energy scattering by large nuclei,where partons are described as propagating through a large size and/or high density gluon field [117–124]. ACKNOWLEDGMENTS
I gratefully acknowledge many conversations with John Collins and George Sterman that influenced the contentof this article. I also thank Christine Aidala, Dani¨el Boer, Maarten Buffing, Aurore Courtoy, Renee Fatemi, BryanField, Leonard Gamberg, Simonetta Liuti, Andreas Metz, Piet Mulders, Alexei Prokudin, Mark Strikman, BowenXiao, and Feng Yuan for useful discussions and comments on the text. Feynman diagrams were produced usingJaxodraw [125, 126]. This work was supported by the National Science Foundation, grant PHY-0969739. [1] D. Boer, M. Diehl, R. Milner, R. Venugopalan, and W. Vogelsang, eds.,
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