The concept of phenomenological light-front wave functions -- Regge improved diquark model predictions
aa r X i v : . [ h e p - ph ] J u l The concept of phenomenological light-front wave functions– Regge improved diquark model predictions –
D. M¨uller a and D. S. Hwang ba Institut f¨ur Theoretische Physik II, Ruhr-Universit¨at BochumD-44780 Bochum, Germany b Department of Physics, Sejong UniversitySeoul 143–747, South Korea
Abstract
We introduce a classification scheme for parton distribution models and we model generalized par-ton distributions (GPDs), their form factors, and parton distribution functions (PDFs), integratedand unintegrated ones, in terms of unintegrated double distributions that are obtained from theparton number conserved overlap of effective light-front wave functions. For a so-called “spheri-cal” model we present general expressions for all twist-two related non-perturbative quantities interms of one effective light-front wave function, including chiral-odd GPDs. We also discuss theRegge improvement of such quark models from the s -channel point of view and study the rela-tions between zero-skewness GPDs and unintegrated PDFs on a more general ground. Finally, weprovide a few phenomenological applications that emphasize the role of orbital angular momenta. Keywords: generalized parton distributions, overlap representation, duality, spectator model
PACS numbers: ontents L z = 0 . . . . . . . . . . . . . . . . . . . 373.3.2 Mismatches and new model relations in the | ∆ L z | = 1 sector . . . . . . . . 393.3.3 Does a “pretzelosity” sum rule exist? . . . . . . . . . . . . . . . . . . . . . 42 k ⊥ -dependent LFWF . . . . . . . . . . . . . . . . . . . . . . . 641.1.2 Exponentially k ⊥ -dependent LFWF . . . . . . . . . . . . . . . . . . . . . . 655.2 Some GPD model insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2.1 Recovering Radyushkin‘s double distribution ansatz . . . . . . . . . . . . . 675.2.2 How solid are inclusive-exclusive relations? . . . . . . . . . . . . . . . . . . 675.3 Models versus phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.3.1 Scalar diquark sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.3.2 Comments about the axial-vector sector . . . . . . . . . . . . . . . . . . . 795.3.3 Modeling pomeron like behavior . . . . . . . . . . . . . . . . . . . . . . . . 81 B.1 Chiral even and parity even GPDs: H and E . . . . . . . . . . . . . . . . . . . . . 91B.2 Chiral even and parity odd GPDs: e H and e E GPDs . . . . . . . . . . . . . . . . . 93B.3 Chiral odd GPDs: H T , E T , e H T and e E T . . . . . . . . . . . . . . . . . . . . . . . . 932 Introduction
During the last two decades we witnessed the development of new phenomenological approachesto study hadronic physics, in particular the proton. On one hand a renaissance of transversemomentum dependent parton distribution functions (PDFs), so-called unintegrated PDFs [1, 2](uPDFs ), occurred in the description of semi-inclusive reactions and on the other hand generalparton distributions (GPDs) have been proposed to describe hard exclusive processes on amplitudelevel [3, 4, 5, 6]. Certainly, both suggestions are based on the partonic picture and partially onfactorization theorems [6, 7] which state that hard physics can be systematically evaluated whilesoft physics, encoded in parton distributions, is universal, i.e., independent of the consideredprocess. Unfortunately, such a strict factorization of hard and soft physics does not hold for theuPDF description of semi-inclusive processes [8, 9, 10, 11]. Triggered by the promise that one canaddress with GPDs and to some extend with uPDFs both the so-called “nucleon spin puzzle”, i.e.,the mismatch of the small quark spin contribution that is measured in polarized deep inelasticscattering with the large quark spin contribution predictions from constituent quark models [12],and the transverse distribution of partons, enormous efforts have been made to measure both hardexclusive processes and semi-inclusive processes.The question arises: What can we learn from such measurements? The generic answer to thisquestion is that we will get a deeper insight into the nucleon as a composite system of partonicdegrees of freedom. Unfortunately, Quantum Chromodynamics (QCD) remains an unsolved theoryand the phenomenological approach is based on factorization theorems, the parametrization ofnon-perturbative quantities, and fitting is finally used to access them. It is clear that for theinterpretation of such fitting results one needs a model based framework. Early attempts tointerpret or predict non-perturbative quantities within pure constituent quark degrees of freedomdo not only imply the ‘nucleon spin puzzle’, however, were also not fully successful even in thecase of unpolarized PDFs, cf. Refs. [13] and [14]. In this circumstance it should be useful to havea fresh look, and parameterize partonic quantities and observables in a more universal way, whichoffer the possibility to have a deeper understanding of both the reliability of QCD tools, one isusing, and the hadronic structure.Numerous model studies for uPDFs and GPDs were performed in various quark models, e.g.,such as the MIT bag model [15], chiral quark soliton model [16], and various constituent quark These functions are often called transverse momentum dependent parton distributions and they have theacronym TMDs. Since we will also consider other unintegrated quantities such as GPDs and double distributions(DDs) that are related to “Wigner” or “phase-space” distributions or to so-called “mother” functions or generalizedtransverse momentum dependent parton distributions (GTMDs), we adopt a simple naming scheme by adding theprefix “u”, staying for unintegrated, to the common acronym for the various parton distributions. k ⊥ and momentumtransfer square t dependence might be indirectly linked and the role of quark orbital angularmomentum plays a complementary role. Alternatively, in the Hamilton approach to QCD bothof them probe different aspects of the nucleon wave function. An interesting possible attempt forit is to employ light-front wave functions (LFWFs), where it seems to be a rather straightforwardtask to build (unintegrated) PDFs, however, it is more intricate for GPDs. In a series of papers,e.g., Refs. [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31], wave function models have beenemployed to provide information on non-perturbative quantities. Moreover, such studies are onlya subset of investigations, which are also performed in other frameworks. One might wonder thatsome uPDF studies, done in different model frameworks, provide rather equivalent results [32].To reveal the reason for such findings, a classification scheme for models is needed.Unfortunately, since the underlying Poincar´e symmetry is not manifestly implemented, see,e.g., Ref. [33, 34, 35, 36], GPD modeling in terms of LFWFs is not straightforward and might yieldinconsistent results [37]. The GPDs are governed by both polynomiality, arising from Poincar´einvariance, and positivity constraints in the so–called outer region in which the modulus of themomentum fraction is larger than the skewness parameter. In this region a GPD may be consideredas an overlap of LFWFs which can be interpreted as the exchange of a quark in the s -channel,while in the central region we can view it as an exchange of a mesonic like parton state. UsingGPD duality among these two regions, see Ref. [38] and references therein, will open the roadfor unifying the QCD phenomenologies of hard exclusive and inclusive measurements in termsof phenomenological LFWFs. Working along the line of Ref. [39], we will follow this idea andchoose an effective two-body LFWF as an appropriate tool to parameterize different aspects ofthe nucleon in a universal manner.In the following section, we introduce our notation, give the LFWF overlap representation forthe eight quark uPDFs that appear in the leading–power description of semi-inclusive deep inelasticscattering. We introduce a classification scheme for uPDFs models and illustrate then that someknown results for uPDFs can be represented in terms of one effective LFWF, considered to beunspecified. Therefore, our model represents a certain class of models. In Sec. 3 we introduce theeight GPDs at twist-two level with zero longitudinal momentum transfer in the t -channel and givean analog classification scheme as for uPDFs models. Utilizing the LFWF overlap representation,we discuss then the correspondence of uPDFs and these GPDs on general ground. In Sec. 4 wethen turn to the non-trivial part, namely, the construction of a whole set of twist-two chiral even4nd odd GPDs from the parton number conserving overlap of LFWFs in the outer region. Weintroduce a k ⊥ –dependent double distribution (DD) as a tool that allows us to restore Lorentzcovariance. We note that in a LFWF longitudinal and transversal momenta are tied to each otherby the underlying Lorentz symmetry, which is explicitly given by the quantization procedure,and present a representation that allows to restore the t dependence from its transverse part ∆ ⊥ .However, there will be still left non-trivial constraints on the longitudinal momentum dependence,for which we provide one solution that allows to build up rather flexible LFWFs. In Sec. 5 weintroduce both power-likely and exponentially k ⊥ -dependent LFWF models and confront themextensively in the scalar diquark sector with phenomenological and lattice estimate findings. Wealso shortly comment on LFWF modeling in the axial-vector sector and present a “pomeron”inspired model for sea quarks, from which we extract a negative and sizeable D -term contribution[40]. Finally, we give the conclusions. Two appendices are devoted to technicalities for theconstruction of proper two-body LFWFs and the overlap representation of GPDs. According to Dirac one may set up a Hamiltonian approach to relativistic Quantum field theoryin three different forms which are characterized by the form of the hypersphere at some given“time” [41]. Instead of initializing the fields in the three dimensional space at time t = 0 onecan alternatively take the tangential plane of the light-cone at light-cone time τ ≡ t + z/c = 0.This so-called front form is perhaps more suited for quantifying the partonic picture in which wethink of the proton as a bunch of partons that move nearly on the light-cone, e.g., specified by n µ = (1 , , , −
1) [42, 43, 44, 45, 46]. This leads to the concept of LFWFs ψ Sm ( X i , k ⊥ i , s i ). Theyare the probability amplitudes for their corresponding n -parton states | n, p + i , p ⊥ i , s i i , which buildup the proton state with spin projection S = { +1 / ⇒ ) , − / ⇐ ) } on the z -axis: | P, S i = X n Z [ dX d k ⊥ ] n n Y j =1 p X j ψ Sn ( X i , k ⊥ i , s i ) | n, X i P + , X i P ⊥ + k ⊥ i , s i i , (2.1)where we used a shorthand notation for the n -parton phase space:[ dX d k ⊥ ] n = n Y i =1 dX i d k ⊥ i π π δ − n X i =1 X i ! δ (2) n X i =1 k ⊥ i ! . (2.2)The n -parton states are normalized as following h s ′ i , p ′⊥ i , p ′ + i , n | n, p + i , p ⊥ i , s i i = n Y i =1 π p + i δ ( p + i − p ′ + i ) δ (2) ( p ⊥ i − p ′⊥ i ) δ s i ,s ′ i . (2.3)5 transversely polarized proton state with projection S = { +1 / ⇑ ) , − / ⇓ ) } on the x -axis isbuilt from the longitudinal ones by the corresponding linear combination, i.e., | P, ⇑i = 1 √ | P, ⇒i + | P, ⇐i ] , | P, ⇓i = 1 √ | P, ⇒i − | P, ⇐i ] . (2.4)The LFWFs depend on the longitudinal momentum fractions X i = k + i /P + (the plus componentof a four-vector V µ is V + = V + V = n · V ), the transverse momenta k ⊥ i , and the LF-helicities s i . In principle, they are determined from the eigenvalue problem, P − | P, S i = M P + | P, S i , with P − = P − P , P + = P + P , P ⊥ = 0 , (2.5)for the LF-Hamiltonian P − that is given in terms of quark and gluon creation and annihilationoperators.Let us emphasize that in any Hamiltonian approach the time coordinate, in our case the light-cone time τ = t + z/c , is singled out and so Poincar´e symmetry is not explicitly implemented.In other words some generators of the Poincar´e transformations depend on interaction terms andso the behavior of the LFWFs under Poincar´e transformations remains unknown as long as thebound state problem is unsolved. For us it is important that both the boost along the z -axis andthe rotation around the z -axis is independent on the interaction [41, 47] and, hence, each LFWFtransforms simultaneously, i.e., ψ Sn ( X i , k ⊥ i , s i ) → ψ Sn ( X ′ i , k ′⊥ i , s i ) , where the momentum fractions X ′ i = X i remain invariant and k ′⊥ i arise from the rotation of k ⊥ i . This may be used to label thepartonic states with respect to the angular momentum projections L zi on the z -axis. The phaseof the corresponding LFWF is then given byarg ψ Sn ( X i , k ⊥ i , s i ) = ± exp ( n − X i =1 L zi ϕ i ) , (2.6)where ϕ i is the polar angle, appearing in the parametrization of the vector k ⊥ i . Angular momen-tum conservation tells us that the longitudinal nucleon spin is decomposed as following [48] S = n − X i =1 L zi + n X i =1 s i . (2.7)Obviously, if we boost the proton to the rest frame these partonic quantum numbers do not alterand so they are appropriate for the discussion of the nucleon “spin puzzle”. Let us also addthat it might be not so obvious how a LFWF behaves under discrete parity and time-reversaltransformations. There are two further kinematical Lorentz transformations, namely, the “Galilean” boost in x - and y -directions,which arise from a combined boost and rotation transformation. A Field theoretical definition of a uPDF depends on the choice of both the reference frame andthe gauge link [1, 2, 49]. We will here not enter a discussion about the gauge link, which has atransversal part that induces a nontrivial phase [49]. We generically define a uPDF as Φ ( x, k ⊥ ) = Z Z d y ⊥ (2 π ) e − i y ⊥ · k ⊥ Z dy − π e ixP + y − / h P | ˆ φ (0) { gauge link } ˆ φ ( y ) (cid:12)(cid:12) y + =0 | P i . (2.8)Apart from singularities, caused by the choice of the gauge link path [2, 49], the operator is welldefined for y ⊥ = 0. To avoid any confusion here, we stress that the operator in Eq. (2.8) mightbe formally expanded in a series of operators with well defined geometrical twist [50], however,it is expected that such an expansion has mathematical problems that might be overcome withan appropriate renormalization prescription. Integrating over k ⊥ sets then the operator on thelight-cone and establishes the formal definition of a PDF: Φ ( x, µ ) = Z Z d k ⊥ Φ ( x, k ⊥ ) = Z dy − π e ixP + y − / h P | ˆ φ (0) ˆ φ ( y ) (cid:12)(cid:12) y + =0 , y ⊥ =0 | P i , (2.9)where we adopted the light-cone gauge in which the gauge link is equated to one. The resultinglight-ray operator has now well defined twist and possesses singularities that are removed bya renormalization procedure. Intuitively, we might connect the scale µ with a cut-off for the k ⊥ integration. This scale dependence can be evaluated perturbatively and is governed by theDoskshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [51, 52, 53].The quark uPDFs that appear, e.g., in the description of semi-inclusive deep inelastic scatteringat leading power in the inverse photon virtuality, are given by Eq. (2.8) in terms of three quark We use the term twist in the original manner, namely, as a classification scheme of operators with respect totheir irreducible representations of the Lorentz group. ˆ φ (0) ˆ φ ( y ) → ˆ¯ ψ (0)Γ ˆ ψ ( y ) with Γ ∈ { γ + , γ + γ , iσ j + γ } , σ j + = i (cid:2) γ j γ + − γ + γ j (cid:3) , where the chiral even ones have parity even or odd. In the following we adopt the commonnomenclator, see e.g., Refs. [1, 54], Φ [ γ + (1 ± γ ) / ( x, k ⊥ ) = 12 f ( x, k ⊥ ) ± S L g ( x, k ⊥ ) (2.10) − ǫ jl ⊥ k j ⊥ S l ⊥ M f ⊥ ( x, k ⊥ ) ± k ⊥ · S ⊥ M g ⊥ ( x, k ⊥ ) , Φ [ iσ j + γ ] ( x, k ⊥ ) = S j ⊥ h ( x, k ⊥ ) + S L k j ⊥ M h ⊥ ( x, k ⊥ ) (2.11)+ 2 k j ⊥ k l ⊥ − k ⊥ δ jl M S l ⊥ h ⊥ ( x, k ⊥ ) + ǫ jl ⊥ k l ⊥ M h ⊥ ( x, k ⊥ ) , where S L and S ⊥ are the longitudinal and transverse component of the polarization vector S µ ,respectively, and the sign of the two-dimensional Levi-Civita tensor is fixed by ǫ ⊥ = 1. In thechiral even sector (2.10) the Dirac structure γ + (1 ± γ ) / → ( ← )for + ( − ). In the chiral odd quark sector (2.11) the transversity combination i ( σ ± iσ ) γ / γ + ± iσ γ ) / − )on a transversely polarized incoming quark in x ( − x ) direction. The spin density matrix, havingelements e Φ ab ( x, k ⊥ ) with a = Λ ′ λ ′ , b = Λ λ ∈ {⇒→ , ⇒← , ⇐→ , ⇐←} , can be straightforwardly calculated from the definitions (2.10,2.11). Thereby, a transversely po-larized proton ⇑ , aligned to the x -axis, may be represented as superposition (2.4), where thetransverse polarization vector is then given as (1 , i ) / √
2. Parameterizing the transverse momen-tum in terms of polar coordinates k ≡ k ⊥ = | k ⊥ | cos ϕ , k ≡ k ⊥ = | k ⊥ | sin ϕ , one finds the hermitian 4 × e Φ ( x, k ⊥ ) = f + g | k ⊥ | e iϕ M h ⊥ − ih ⊥ | k ⊥ | e − iϕ M g ⊥ + if ⊥ h | k ⊥ | e − iϕ M h ⊥ + ih ⊥ f − g k ⊥ e − i ϕ M h ⊥ −| k ⊥ | e − iϕ M g ⊥ − if ⊥ | k ⊥ | e iϕ M g ⊥ − if ⊥ k ⊥ e i ϕ M h ⊥ f − g −| k ⊥ | e iϕ M h ⊥ + ih ⊥ h −| k ⊥ | e iϕ M g ⊥ + if ⊥ −| k ⊥ | e − iϕ M h ⊥ − ih ⊥ f + g ( x, k ⊥ ) . (2.12)Since parity and time-reversal invariance is already implemented, it is not surprising that thespin-density matrix can be decomposed into four 2 × f . As we willsee below from its LFWF overlap representation (2.34), it is also a semi-positive definite matrix.Hence, one can straightforwardly derive positivity bounds [56], e.g., the analog of the Soffer boundfor PDFs [57] reads: 2 | h ( x, k ⊥ ) | ≤ f ( x, k ⊥ ) + g ( x, k ⊥ ) . (2.13)If we utilize the definitions of the matrix elements Φ [ ··· ] , cf. Eq. (2.8), and neglect the gaugelink, we can easily write down the LFWF overlap representation for the uPDFs. For instance,there are two of them that are related to chiral even twist-two PDFs , namely, the unpolarized( f ) and longitudinal polarized ( g ) ones. They are obtained from Eq. (2.10) and read as f ( x, k ⊥ ) = X n ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇒→ ,n + ψ ∗ ⇒← ,n ψ ⇒← ,n (cid:3) ( X i , k ⊥ i , s i ) , (2.14) g ( x, k ⊥ ) = X n ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇒→ ,n − ψ ∗ ⇒← ,n ψ ⇒← ,n (cid:3) ( X i , k ⊥ i , s i ) . (2.15)Here, we use for the ( n − ZX ( n − · · · ≡ X s ,...,s n Z [ dX d k ⊥ ] n δ ( x − X ) δ (2) ( k ⊥ − k ⊥ ) · · · (2.16)with the phase space (2.2) and ψ Ss ,n ( X i , k ⊥ i , s , . . . , s n ) = ψ Sn ( X i , k ⊥ i , s i ) is an n -parton Fock-stateLFWF with the struck quark LF-spin projection s ∈ {→ , ←} on the z -axis, and the momentumfraction and transverse momentum of the struck parton are denoted as x ≡ X and k ⊥ ≡ k , ⊥ = − n X i =2 k i, ⊥ , respectively.As one clearly realizes in Eqs. (2.14) and (2.15), the unpolarized and polarized uPDFs arisefrom the sum and difference of diagonal LFWF overlaps in which the struck quark spin is lon-gitudinally aligned and opposite to the proton one, respectively. If the quark spin is pointing inthe same direction as the proton spin, the phase of the LFWF (2.6) that is related to the struckquark is given by e i ¯ L z ϕ , where angular momentum conservation (2.7) tells us that the orbital an-gular momentum of the struck quark is given by the sums of spin and orbital angular momentum We emphasize that our twist definition differs from the notion “twist”, often used in the literature withoutquotation mark, in which people denote all eight uPDFs (2.10,2.11) “twist-two”, partially, not realizing that powerand twist counting in semi-inclusive processes do not match. L z = − P ni =2 s i − P n − i =2 L zi . In the case that the struck quarkspin ← [ → ] is pointing in the opposite direction of the proton spin ⇒ [ ⇐ ], the phase is given by e i (¯ L z +1) ϕ [ e i (¯ L z − ϕ ]. It is obvious that the overall phase e i ¯ L z ϕ is not accessible in leading–poweruPDFs, and so only the difference of proton spin and struck quark projections, l z ≡ S − s = L z − ¯ L z ∈ {− , , } , S ∈ {⇒ , ⇐} , s ≡ s ∈ {→ , ←} , (2.17)is the relevant partonic quantum number. Within this definition we say that f and g containcontributions from both diagonal l z = 0 and l z = 1 LFWF overlaps. We can certainly evaluate f and g for proton spin ⇐ rather than ⇒ , as used in Eqs. (2.14) and (2.15). Although it is notso obvious how a certain n -parton LFWF behaves under parity and time-reversal transformations[46], we can state that for the LFWF overlaps the following identities must be satisfied: ZX ( n − ψ ∗ ⇒→ ,n ( X i , k ⊥ i , s i ) ψ ⇒→ ,n ( X i , k ⊥ i , s i ) = ZX ( n − ψ ∗ ⇐← ,n ( X i , k ⊥ i , s i ) ψ ⇐← ,n ( X i , k ⊥ i , s i ) , (2.18) ZX ( n − ψ ∗ ⇒← ,n ( X i , k ⊥ i , s i ) ψ ⇒← ,n ( X i , k ⊥ i , s i ) = ZX ( n − ψ ∗ ⇐→ ,n ( X i , k ⊥ i , s i ) ψ ⇐→ ,n ( X i , k ⊥ i , s i ) . (2.19)Formally, integrating (2.14) and (2.15) over k ⊥ provides the overlap representation for unpolarizedand polarized twist-two PDFs.The two twist-tree related chiral even uPDFs, the Sivers function ( f ⊥ ) [58] and transversallypolarized uPDF ( g ⊥ ), arise for a transversally polarized proton (2.4). We take the polarizationalong the x -axis and we find then from Eqs. (2.8,2.10) the LFWF overlap representations: f ⊥ ( x, k ⊥ ) = M ik X n ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇐→ ,n − ψ ∗ ⇐← ,n ψ ⇒← ,n − c.c. (cid:3) ( X i , k ⊥ i , s i ) , (2.20) g ⊥ ( x, k ⊥ ) = M k X n ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇐→ ,n − ψ ∗ ⇐← ,n ψ ⇒← ,n + c.c. (cid:3) ( X i , k ⊥ i , s i ) , (2.21)where we also employed the identities (2.18,2.19) to simplify the overlaps of the LFWF superpo-sitions ψ ⇑ ( n ) ( · · · ) = 1 √ (cid:2) ψ ⇒ ( n ) ( · · · ) + ψ ⇐ ( n ) ( · · · ) (cid:3) . As one sees both twist-three related uPDFs arise from the differences of a l z = 0 with l z = 1 and l z = 0 with l z = − T -odd) Sivers function and T -even transversally polarized uPDF is given by theimaginary and real part of these overlaps, respectively. Note that there exist alternative overlap10epresentations, in which one might define f ⊥ and g ⊥ as the real and imaginary part, respectively: f ⊥ ( x, k ⊥ ) = M k X n ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇐→ ,n + ψ ∗ ⇐← ,n ψ ⇒← ,n + c.c. (cid:3) ( X i , k ⊥ i , s i ) , (2.22) g ⊥ ( x, k ⊥ ) = − M ik X n ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇐→ ,n + ψ ∗ ⇐← ,n ψ ⇒← ,n − c.c. (cid:3) ( X i , k ⊥ i , s i ) . (2.23)To derive the overlap representation in the chiral odd sector, we employ again LFWFs thatare labeled by longitudinal spin projections. By means of Eqs. (2.8,2.11) we find then that for alongitudinal polarized target the LFWFs for a transversally polarized struck quark, projected onthe x -axis, are given as superpositions ψ S ↑ ,n = 1 √ (cid:2) ψ S → ,n + ψ S ← ,n (cid:3) , ψ S ↓ ,n = 1 √ (cid:2) ψ S → ,n − ψ S ← ,n (cid:3) , S ∈ {⇒ , ⇐} . Moreover, the twist-three related uPDF h ⊥ ( x, k ⊥ ) = M k X n ZX ( n − (cid:2) ψ ∗⇒→ ,n ψ ⇒← ,n − ψ ∗⇐← ,n ψ ⇐→ ,n + c.c. (cid:3) ( X i , k ⊥ i , s i ) (2.24)is given by the real part of the differences of a l z = 0 with l z = 1 and l z = 0 with l z = − T -even uPDF also use theequivalent representation h ⊥ ( x, k ⊥ ) = M ik X n ZX ( n − (cid:2) ψ ∗⇒→ ,n ψ ⇒← ,n + ψ ∗⇐← ,n ψ ⇐→ ,n − c.c. (cid:3) ( X i , k ⊥ i , s i ) . (2.25)The other three transversity uPDFs are naturally defined for a transversely polarized proton,where, e.g., the spin points in the x direction, in terms of LFWFs with transverse quark spinprojection s ∈ {↑ , ↓} on the x -axis δq ( x, k ⊥ ) ≡ Φ [ iσ γ ] ( x, k ⊥ ) (cid:12)(cid:12)(cid:12) S ⊥ =(1 , = X n ZX ( n − h ψ ⇑∗↑ ,n ψ ⇑↑ ,n − ψ ⇑∗↓ ,n ψ ⇑↓ ,n i ( X i , k ⊥ i , s i ) . (2.26)Representing these LFWFs as superposition of those with longitudinal spin projections, ψ ⇑↑ ,n = 12 (cid:2) ψ ⇒→ ,n + ψ ⇒← ,n + ψ ⇐→ ,n + ψ ⇐← ,n (cid:3) , ψ ⇑↓ ,n = 12 (cid:2) ψ ⇒→ ,n − ψ ⇒← ,n + ψ ⇐→ ,n − ψ ⇐← ,n (cid:3) , we find that δq ( x, k ⊥ ) = h ( x, k ⊥ ) + ( k ) − ( k ) M h ⊥ ( x, k ⊥ ) + k M h ⊥ ( x, k ⊥ ) , (2.27)where the three scalar functions h , h ⊥ , and h ⊥ depend on x and k ⊥ , i.e., are invariant underrotation around the z -axis. As said the transverse polarization vector is here along the x -axis;11he result for the y -direction follows by the interchange of k → k and k → − k . The twist-tworelated transversity uPDF h ( x, k ⊥ ) = X n ZX ( n − ψ ∗ ⇒→ ,n ( X i , k ⊥ i , s i ) ψ ⇐← ,n ( X i , k ⊥ i , s i ) (2.28)is given as the overlap of l z = 0 LFWFs, the twist-four related “pretzelosity” distribution h ⊥ ( x, k ⊥ ) = − X n ZX ( n − M ik k (cid:2) ψ ∗ ⇒← ,n ψ ⇐→ ,n − c.c. (cid:3) ( X i , k ⊥ i , s i ) (2.29)is the only leading–power quark uPDF that arises from off-diagonal | l z | = 1 overlaps, and thetwist-three related Boer-Mulders function h ⊥ ( x, k ⊥ ) = − M ik X n ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇒← ,n − ψ ∗ ⇐← ,n ψ ⇐→ ,n − c.c. (cid:3) ( X i , k ⊥ i , s i ) , (2.30)which is T -odd [54], is the imaginary part of differences of l z = 0 with l z = 1 and l z = 0 with l z = − ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ( · · · ) ψ ⇐← ,n ( · · · ) − ψ ∗ ⇐← ,n ( · · · ) ψ ⇒→ ,n ( · · · ) (cid:3) were neglected. These identities guarantees that the transversity (2.28) has a second representationin which both the target and struck quark spin projections are flipped. As in the case of the Siversfunction (2.20,2.22), we might also define the Boer-Mulders function as the real part of the sumof the l z = 0 with | l z | = 1 LFWF overlaps, h ⊥ ( x, k ⊥ ) = M k X n ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇒← ,n + ψ ∗ ⇐← ,n ψ ⇐→ ,n + c.c. (cid:3) ( X i , k ⊥ i , s i ) . (2.31)Finally, we add that also the “pretzelosity” distribution (2.29) has an equivalent representationas real part of off-diagonal | l z | = 1 LFWF overlaps h ⊥ ( x, k ⊥ ) = X n ZX ( n − M ( k ) − ( k ) (cid:2) ψ ∗ ⇒← ,n ψ ⇐→ ,n + c.c. (cid:3) ( X i , k ⊥ i , s i ) . (2.32)Let us shortly comment on the interpretation of the transversity functions. If a transversedirection is selected, the rotation invariance around the z -axis is distorted for the l z = ± δq uPDF (2.27) an off-diagonal overlap of the two | l z | = 1 LFWFs or a chiral-odd quark spin flip contribution enters, yielding a ( k ) − ( k ) proportional term, which is the“pretzelosity” distribution (2.29). Certainly, this distortion is in the concept of operator product12xpansion a geometric twist-four (= dimension − spin) effect, which can be also seen from thefact that if one integrates over k ⊥ , even weighted with k ⊥ , the distortion vanishes. Furthermore,we also have in δq a distortion that is pointing in the k -direction and proportional to the Boer-Mulders function (2.30), which is induced by the overlap of l z = 0 and | l z | = 1 LFWFs. If naivetime reversal invariance holds true, as it should be in a pure quark model, this twist-three relatedeffect is absent.To have a more compact overlap representation at hand for all leading–power uPDFs, weintroduce a LFWF “spinor” for each n -parton contribution, which we write as ψ ( n ) ( X i , k ⊥ i , s i ) = ψ ⇒→ ,n ψ ⇒← ,n ψ ⇐→ ,n ψ ⇐← ,n ( X i , k ⊥ i , s i ) . The spin-density matrix e Φ ( x, k ⊥ ) = X n e Φ ( n ) ( x, k ⊥ ) , e Φ ( n ) ( x, k ⊥ ) = ψ ∗ ( n ) ( X i , k ⊥ i , s i ) ( n − ⊗ ψ ( n ) ( X i , k ⊥ i , s i ) (2.33)is then defined as the infinite sum of overlap spin-density matrices that are given by the Cartesianproduct of the n -parton LFWF “spinors”, where the symbol ( n − ⊗ also stays for the n − e Φ ( n ) ( x, k ⊥ ) = ZX ( n − ψ ∗ ⇒→ ,n ψ ⇒→ ,n ψ ∗ ⇒→ ,n ψ ⇒← ,n ψ ∗ ⇒→ ,n ψ ⇐→ ,n ψ ∗ ⇒→ ,n ψ ⇐← ,n ψ ∗ ⇒← ,n ψ ⇒→ ,n ψ ∗ ⇒← ,n ψ ⇒← ,n ψ ∗ ⇒← ,n ψ ⇐→ ,n ψ ∗ ⇒← ,n ψ ⇐← ,n ψ ∗ ⇐→ ,n ψ ⇒→ ,n ψ ∗ ⇐→ ,n ψ ⇒← ,n ψ ∗ ⇐→ ,n ψ ⇐→ ,n ψ ∗ ⇐→ ,n ψ ⇐← ,n ψ ∗ ⇐← ,n ψ ⇒→ ,n ψ ∗ ⇐← ,n ψ ⇒← ,n ψ ∗ ⇐← ,n ψ ⇐→ ,n ψ ∗ ⇐← ,n ψ ⇐← ,n ( X i , k ⊥ i , s i ) . (2.34)It is straightforward to see that all LFWF overlap representations for uPDFs, we have discussed,can be easily obtained by equating this equation (2.34) with the spin-density matrix (2.12). Since the infinite number of LFWFs in the overlap representations (2.14,2.15,2.20,2.21,2.24,2.28–2.30) remains unknown, it is rather popular to introduce effective degrees of freedom and employsome model assumptions. This has been done in different representations and with various dy-namical assumptions. Thereby, surprisingly, linearly and even quadratically algebraic relationsamong uPDFs have been found in rather different quark models [59, 60, 61, 62, 63]. Certainly,one might wonder how general such relations are. To understand that these findings are partially13rivial and in some cases rather surprising, we should introduce a classification scheme for uPDFmodels. To do so, we simply ask for the spectrum of the spin-density matrix e Φ ( x, k ⊥ ) = X m e Φ ( m ) ( x, k ⊥ ) , where m is a generic label for the LFWF overlaps, given in terms of some n − e Φ e . v . = f − h + k ⊥ M h ⊥ − q(cid:0) g − h − k M h ⊥ (cid:1) + k ⊥ M (cid:0) g ⊥ + h ⊥ (cid:1) + k ⊥ M (cid:0) f ⊥ − h ⊥ (cid:1) f − h + k ⊥ M h ⊥ + q(cid:0) g − h − k M h ⊥ (cid:1) + k ⊥ M (cid:0) g ⊥ + h ⊥ (cid:1) + k ⊥ M (cid:0) f ⊥ − h ⊥ (cid:1) f + h − k ⊥ M h ⊥ − q(cid:0) g + h + k M h ⊥ (cid:1) + k ⊥ M (cid:0) g ⊥ − h ⊥ (cid:1) + k ⊥ M (cid:0) f ⊥ + h ⊥ (cid:1) f + h − k ⊥ M h ⊥ + q(cid:0) g + h + k M h ⊥ (cid:1) + k ⊥ M (cid:0) g ⊥ − h ⊥ (cid:1) + k ⊥ M (cid:0) f ⊥ + h ⊥ (cid:1) . (2.35)of the spin-density matrix (2.12). Note that as a consequence of parity invariance the upper twoeigenvalues are related with the two lower ones by the substitution h ...... → − h ...... . Models in whichthe spin-density matrix possesses degenerate triplet and duplet states are called “spherical” and “axial-symmetric”, respectively. We also take the rank of the spin-density matrix, i.e., thenumber of zero modes is given by 4 − rank e Φ , as a further model characteristic. On the other hand,knowing that such model relations are satisfied in a specific model, one can read off to which classit belongs and one might give an equivalent representation in terms of effective two-body LFWFs. The simplest quark models consist of a struck quark and one collective spectator. Hence, we havethen only one scalar two-body LFWF and the spin-spin coupling of the struck quark and spectatoris fixed. It is clear that all uPDFs (or other non-perturbative quantities) are expressed by oneLFWF overlap and we have seven model dependent constraints among the eight quark uPDFs,entering the spin-density matrix (2.12). However, among these constraints there are those whichdo not depend on the specific choice of the spin-spin coupling. To understand the origin of these This notion has been introduced without quotation marks in Ref. [64], where the authors wrote the spin-densitymatrix as a SO(3,1) ≃ SU(2) ⊗ SU(2) representation and suggested that the degeneration of the spin-density matrixis connected with a SO(3) symmetry in three dimensional space. two-body spin-density matrix e Φ rank − ( x, k ⊥ ) = e Φ (1) ( x, k ⊥ ) = ( ψ ⇒→ , ψ ⇒← , ψ ⇐→ , ψ ⇐← ) ∗ ⊗ ψ ⇒→ ψ ⇒← ψ ⇐→ ψ ⇐← ( X i , k ⊥ i ) (2.36)has rank-one, i.e., it contains three zero modes.Let us first derive the conditions for having (degenerate) triplet states. From the eigenvalues(2.35), given in terms of uPDFs, it is not hard to realize that for three identical eigenvalues oneof the two square roots has to vanish, which implies one set of three linear constraints: g ⊥ ± h ⊥ = 0 , f ⊥ ∓ h ⊥ = 0 , g ∓ h ∓ k ⊥ M h ⊥ = 0 , (2.37)where the upper or lower sign has to be taken consistently. Furthermore, the quadratic constraint (cid:0) h ⊥ (cid:1) + (cid:0) h ⊥ (cid:1) + 2 h h ⊥ = 0 (2.38)guarantees that a third degenerate eigenvalue exists and that the forth one is already fixed (since P e Φ e . v . = 2 f ). The three degenerate eigenvalues are given by ( f + g ∓ h ) / T -odd functions areidentically zero and, since the spin-density matrix has rank-one, the Soffer bound is automaticallysaturated, i.e., altogether we have g ⊥ + h ⊥ = 0 , g − h − k ⊥ M h ⊥ = 0 , (cid:0) h ⊥ (cid:1) + 2 h h ⊥ = 0 , h ⊥ = f ⊥ = 0 , (2.39)2 h = f + g or equivalently f − h + k ⊥ M h ⊥ = 0 , (2.40)where the last equation simply follows from combining linear constraints. As we realize now thetransversity related twist-two and twist-tree uPDFs, h ( x, k ⊥ ) = (cid:2) ψ ∗⇒→ ,n ψ ⇐← ,n + c.c. (cid:3) ( x, k ⊥ ) , h ⊥ ( x, k ⊥ ) = Mk (cid:2) ψ ∗⇒→ ,n ψ ⇒← ,n + c.c. (cid:3) ( x, k ⊥ ) , may be considered as independent functions and the four remaining ones can be uniquely obtainedfrom the constraints (2.39) and (2.40). The ratio of these both functions is just determined by theratio of the l z = 0 and l z = 1 LFWFs. Specifying it, means that all uPDFs are fixed in terms ofone two-body LFWF. Let us add that for a non-trivial gauge-link, expanded to leading order or15reated in the eikonal approximation, the constraint for the T -odd functions h ⊥ = f ⊥ holds true,too, however, the quadratic equation (2.38) is only satisfied if these T -odd functions are neglected[60, 65, 66, 67, 68].A more non-trivial example for a rank-one model is provided by the 2 u/ − d/ necessary condition: e Φ “spheric” ( x, k ⊥ ) = m min . X m =1 e Φ ( m ) ( x, k ⊥ ) , with m min . ≥ . A simple, however, highly non-trivial example is the u/ d/ u - and d -quarkthe spin-density matrix is then the sum of a rank-one and a diagonal matrix, e Φ u ( x, k ⊥ ) sph&SU(4) = 13 × f u/ d/ ( x, k ⊥ ) + 43 e Φ u/ − d/ ( x, k ⊥ ) , rank e Φ u/ − d/ = 1 , (2.41) e Φ d ( x, k ⊥ ) sph&SU(4) = 23 × f u/ d/ ( x, k ⊥ ) − e Φ u/ − d/ ( x, k ⊥ ) , (2.42)where × is the four dimensional identity matrix. It is obvious that the spin-density matrices(2.41,2.42) together with e Φ u/ − d/ can be simultaneously diagonalized and that for the formertwo a degenerated triplet state with non-vanishing eigenvalue exist, which are given by the entryof the corresponding diagonal matrix. Consequently, we have a “spherical” model and the con-straints (2.39) for the scalar diquark model hold true, where the Soffer bound for uPDFs is nowunsaturated. We add that the axial-vector diquark model in the version of Ref. [59], in whichthe sum over the diquark polarization vectors provides − g µν + P µ P ν /M A with the diquark mass M A and proton momentum P µ , yields an equivalent result, of course, the effective scalar LFWFoverlap is different.Also the bag [69], the chiral quark soliton [70], and the covariant parton [63] model are “spher-ical”. So far we did not check if all these models have the same relative l z = 1 to l z = 0 LFWFcoupling and the same flavor-spin content. Nevertheless, we can make the conjecture that this16lass of models can be represented by a parity even “spinor” times an effective two-body LFWF, ψ sca ( x, k ⊥ ) = 1 M M − g | k ⊥ | e iϕ g | k ⊥ | e − iϕ M φ sca ( x, k ⊥ ) , (2.43)and a positive definite uPDF f q . Here, g is the relative strength of the l z = 1 LFWF comparedto the l z = 0 one. This coupling might depend on k ⊥ and x , however, the underlying Lorentzsymmetry tells us that this function cannot be ambiguously chosen. We will come back to thispoint below in Sec. 2.3 and Sec. 4.1. The spin-density matrix can then be written as e Φ q ( x, k ⊥ ) sph = 12 × f q ( x, k ⊥ ) + g sca q (cid:20) ( ψ ∗ sca ⊗ ψ sca ) − × Tr ( ψ ∗ sca ⊗ ψ sca ) (cid:21) ( x, k ⊥ ) , (2.44)where g sca q is the “coupling” of the quark q to the scalar diquark sector. Note that g sca q mightbe negative. Since the non-trivial part of the matrix is written as a rank-zero matrix and f q is a positive function, the spin-density matrix (2.44) of our ”spherical” model is always positivedefinite. We also emphasize that an identity proportional part of the spin-density matrix arisesfrom the overlap contributions of four LFWF “spinors” that belong to (rather) different states.To provide a toy example, we might for instance choose the following complete set of “spinors”:12 − e iϕ e − iϕ , e iϕ e − iϕ − , e − iϕ √ e iϕ , e iϕ √ − e − iϕ . (2.45)Here, the first “spinor” mimics a scalar diquark coupling, cf. Eq. (2.43), the second one thecoupling to a pseudo-scalar diquark, and the remaining two might be considered as coupling toan axial-vector diquark that has spin projection +1 and −
1, respectively. Of course, the choiceof such a basis is not unique rather we can utilize any unitary transformation to obtain another“spinor” basis. Hence, from this point of view it is not surprising that quark models with differentorbital angular momentum content, e.g., with [61] and without [59] D -waves, provide equivalentuPDF models. Diquark models that are based on two orthogonal states are build from the sum, e Φ rank − ( x, k ⊥ ) = X m =1 e Φ rank − m ) ( x, k ⊥ ) ,
17f two rank-one spin-density matrices. We suppose that the LFWF “spinors” in these two con-tributions are linearly independent and so the spin-density matrix has rank two and possessestwo zero modes. There are now two plus four possibilities to distribute these two zero eigen-values among the four ones (2.35). More generally, we might ask for “axial-symmetric” models,possessing a degenerated duplet state with non-negative eigenvalue.The first kind of a “axial-symmetrical” model is realized if one set of the linear relations (2.37)holds true, i.e., the upper or lower two eigenvalues (2.35) are degenerated. Consequently, we havethe degenerated eigenvalues ( f + g ∓ h ) /
2, which collapses to zero for a rank-two model, i.e.,the Soffer bound (2.13) is saturated. An example of such a model is the scalar diquark modelthat contains a gauge link that implies in a perturbative leading order expansion non-vanishingBoer-Mulders and Sivers distributions that are equal [60, 68]. This equality holds also true if thegauge link contributions are summed by means of the eikonal approximation [71]. More generally,if we extend a given “spherical” model in the T -even sector by two T -odd functions which areequal to each other (or differ by a sign if the third and forth eigenvalues (2.35) are degenerated),we will end up with an “axial-symmetrical” model in which the linear relations (2.37) hold trueand the Soffer bound (2.13) is unsaturated. It is known that the condition f ⊥ = h ⊥ is violatedfor a gauged bag model [72, 73, 74, 75, 76]. One might wonder if a gauged “spherical” axial-vectordiquark or a three quark LFWF model will become an “axial-symmetrical” model. To our bestknowledge such model calculations are not performed yet.The other four possibilities for the realization of a rank-two model are that one eigenvalue outof the first two eigenvalues (2.35) coincides with one of the last two eigenvalues (2.35). We mightwrite this condition for an “axial-symmetrical” model of the second kind in the following form: g − h + k ⊥ M (cid:20)(cid:0) g ⊥ (cid:1) + (cid:0) h ⊥ (cid:1) + (cid:0) f ⊥ (cid:1) + (cid:0) h ⊥ (cid:1) − k ⊥ M (cid:0) h ⊥ (cid:1) + 3 h h ⊥ (cid:21) = ± s(cid:18) g − h − k M h ⊥ (cid:19) + k ⊥ M (cid:0) g ⊥ + h ⊥ (cid:1) + k ⊥ M (cid:0) f ⊥ − h ⊥ (cid:1) (2.46) × s(cid:18) g + h + k M h ⊥ (cid:19) + k ⊥ M (cid:0) g ⊥ − h ⊥ (cid:1) + k ⊥ M (cid:0) f ⊥ + h ⊥ (cid:1) . In the case that the two eigenvalues vanish both quadratic relations( f + g − h ) (cid:18) f − g + k ⊥ M h ⊥ (cid:19) − k ⊥ M h(cid:0) g ⊥ + h ⊥ (cid:1) + (cid:0) f ⊥ − h ⊥ (cid:1) i = 0 , (2.47)( f + g + 2 h ) (cid:18) f − g − k ⊥ M h ⊥ (cid:19) − k ⊥ M h(cid:0) g ⊥ − h ⊥ (cid:1) + (cid:0) f ⊥ + h ⊥ (cid:1) i = 0 (2.48)18ust be satisfied. Combining them, we might also write the two constraints as f − g − k ⊥ M (cid:20)(cid:0) g ⊥ (cid:1) + (cid:0) h ⊥ (cid:1) + (cid:0) f ⊥ (cid:1) + (cid:0) h ⊥ (cid:1) − k ⊥ M (cid:0) h ⊥ (cid:1) + 2 h h ⊥ (cid:21) = 0 , (2.49)( f − g ) h − k ⊥ M (cid:2) ( f + g ) h ⊥ − g ⊥ h ⊥ − f ⊥ h ⊥ ) (cid:3) = 0 . (2.50)An example of such a rank-two model is the diquark model with axial-vector coupling, whereonly the transverse polarization of the diquark is taken into account [68]. Such a model willbreak in general the underlying Lorentz symmetry and, therefore, it is not applicable for GPDmodeling, however, it might be still used for uPDF modeling. In this model the linear equations(2.37) are not satisfied, however, the quadratic ones (2.49,2.50) hold true. Obviously, if we replacethe axial-vector coupling by a vector coupling, we will find rather analogous results. In the case ofa gauge field theory, gauge invariance ensures then that the result also respects Lorentz symmetry.The so-called quark-target model [60], calculated to leading order, belongs to this rank-two modelclass . We are not aware of an “axial-symmetrical” model of the second kind that has a rank fourspin-density matrix. Diquark models which are based on three orthogonal states have the spin-density matrix e Φ rank − ( x, k ⊥ ) = X m =1 e Φ rank − m ) ( x, k ⊥ )and they possess at least one zero mode. Hence, the determinant of the spin density matrix,given by the product of eigenvalues, vanish and we find two possibilities two satisfy the nonlinearconstraints, which we already wrote down in Eqs. (2.47) and (2.48). Thereby, if the spin-densitymatrix has truly rank-three, only one root of one of these quadratic relations vanishes.An example of such a model is an axial-vector diquark model of Ref. [68] where the diquarkposses two transversal and one longitudinal polarization, however, the polarization tensor differsfrom that of the “spherical” axial-vector diquark model [59]. Even if the time-like polarization istaken into account, i.e., the polarization tensor is − g µν , one still has a rank-three model and thesame constraint ( f + g + 2 h ) (cid:18) f − g − k ⊥ M h ⊥ (cid:19) − k ⊥ M (cid:0) g ⊥ − h ⊥ (cid:1) = 0is satisfied. Of course, the proton mass has now to be identified with the target quark mass. Alternatively, if one setsthe quark mass to zero and keep the proton mass, the model constraints (2.49,2.50) for a rank-two model are stillsatisfied. Thereby, the non-vanishing uPDFs are f = g and h . .3 “Spherical” diquark models in terms of effective LFWFs In the following we will utilize effective two-particle LFWFs of an effective struck quark withmass m and spin projection s = {− / , / } , where the collective spectator system has mass λ .Formally, we might introduce a Hilbert space and write the proton wave function in terms of these new degrees of freedom as | P, S i = / X s = − / Z dλ Z [ dX d k ] √ X X Ψ Sh ( X i , k ⊥ i , s i | λ ) | λ , X i P + , X i P ⊥ + k ⊥ i , s i . (2.51)The normalization condition for the states is taken to be the following h s ′ , P ⊥ + k ′⊥ i , X ′ i P + , λ ′ | λ , X i P + , X i P ⊥ + k ⊥ i , s i (2.52)= Y i =1 π X i δ ( X ′ i − X i ) δ (2) ( k ′⊥ i − k ⊥ i ) δ ( λ ′ − λ ) δ s ′ s . If we restrict ourselves to a scalar–diquark spectator, we have four LFWFs [39],Ψ ⇒→ ( X, k ⊥ | λ ) = m + XMM √ − X φ ( X, k ⊥ | λ ) , Ψ ⇒← ( X, k ⊥ | λ ) = − k − ik M √ − X φ ( X, k ⊥ | λ ) , (2.53)Ψ ⇐← ( X, k ⊥ | λ ) = m + XMM √ − X φ ( X, k ⊥ | λ ) , Ψ ⇐→ ( X, k ⊥ | λ ) = k − ik M √ − X φ ( X, k ⊥ | λ ) , (2.54)in terms of one effective LFWF φ ( X, k ⊥ | λ ), which we might also write in terms of a LFWF“spinor” (2.43): ψ sca ( X, k ⊥ ) = 1 M m + XM −| k ⊥ | e iϕ | k ⊥ | e − iϕ m + XM φ ( X, k ⊥ | λ ) √ − X . (2.55)In our model (2.51) the proton is described as a superposition of struck quark states with alignedand opposite spin projection on the z -axis. The former and latter have for a longitudinallypolarized proton ( ⇒ ) an orbital angular momentum projection L z = 0 and L z = 1, respectively,where the phase is exp { iL z ϕ } . Since we have a scalar spectator, we can equate the orbital angularmomenta projection L z = l z with the spin difference projection (2.17), which is as discussedabove the relevant quantum number for our partonic quantities. We emphasize that the relativenormalization in (2.53–2.54) cannot be changed otherwise we will be not able to satisfy the GPDconstraints that arise from Lorentz covariance. 20o obtain a spectral representation with respect to the spectator quark mass one might intro-duce a λ dependent coupling between the struck quark and the spectator system, which has to beindependent on the spin in our model, φ ( X, k ⊥ | λ ) = p ρ ( λ ) φ ( X, k ⊥ | λ ) , (2.56)where ρ ( λ ) is considered as the spectral mass density. Since in our model the spin-spin coupling isfixed, it is obvious that all non-perturbative quantities are expressed by the overlap of one effectiveLFWF, which spectral representation follows from the ansatz (2.51,2.52,2.56), Φ ( x, k ⊥ ) = Z dλ ρ ( λ ) (cid:12)(cid:12) φ ( X = x, k ⊥ | λ ) (cid:12)(cid:12) − x . (2.57)Note that such a representation will be essential to recover phenomenological PDF parameteri-zations, which cannot be obtained from ‘pure’ quark models. The unintegrated parton density,related to the unpolarized twist-two PDF, read then in the quark spin averaged case (2.14) asfollowing and the T -odd Sivers function f ⊥ vanishes: f ( x, k ⊥ ) = ( m + xM ) + k ⊥ M Φ ( x, k ⊥ ) , f ⊥ ( x, k ⊥ ) = 0 . (2.58)The overall normalization of the wave functions (2.53,2.54) is fixed for valence like quarks by thequark number n and for sea-quarks by the momentum fraction average h x i : Z dx Z d k ⊥ (cid:26) x (cid:27) f ( x, k ⊥ ) = (cid:26) n h x i (cid:27) . (2.59)For the spin difference of longitudinally polarized quarks in an longitudinally ( g ) and transversally( g ⊥ ) polarized hadron we find a twist-two and twist-three related unintegrated parton density,respectively: g ( x, k ⊥ ) = ( m + xM ) − k ⊥ M Φ ( x, k ⊥ ) , g ⊥ ( x, k ⊥ ) = 2 (cid:16) mM + x (cid:17) Φ ( x, k ⊥ ) . (2.60)Analogously, if we rotate the quark spin the transversity for a transversally ( h , h ⊥ ), longitudinally( h ⊥ ), and un-( h ⊥ ) polarized target is related to twist-two, twist-four, and two twist-three PDFs: h ( x, k ⊥ ) = (cid:16) mM + x (cid:17) Φ ( x, k ⊥ ) , h ⊥ ( x, k ⊥ ) = − Φ ( x, k ⊥ ) , (2.61) h ⊥ ( x, k ⊥ ) = − (cid:16) mM + x (cid:17) Φ ( x, k ⊥ ) , h ⊥ ( x, k ⊥ ) = 0 . (2.62)As we have shown in Sec. 2.2, the overlap representation in terms of only one LFWF “spinor”implies that we have a “spherical” model of rank-one and so one quadratic and three linearconstraints (2.39,2.40) hold true for the six T -even uPDFs, which one might easily verify from the21xplicit expressions (2.58,2.60,2.61,2.62). The relative strength of the l z = 1 and l z = 0 LFWFsis fixed by the ratio h ⊥ ( x, k ⊥ )2 h ( x, k ⊥ ) = − Mm + xM and, hence, the transversity uPDF (2.61) might be chosen as independent function. Alternatively,we might consider the unpolarized uPDF as the independent function. Indeed, it is trivial torealize that the concept of a (scalar) diquark model has immediately the consequence that alluPDFs, not only twist-two related ones, are expressed by the unpolarized PDF: g ( x, k ⊥ ) = ( m + xM ) − k ⊥ ( m + xM ) + k ⊥ f ( x, k ⊥ ) , g ⊥ ( x, k ⊥ ) = 2 M ( m + xM )( m + xM ) + k ⊥ f ( x, k ⊥ ) , (2.63) h ( x, k ⊥ ) = ( m + xM ) ( m + xM ) + k ⊥ f ( x, k ⊥ ) , h ⊥ ( x, k ⊥ ) = − M ( m + xM ) + k ⊥ f ( x, k ⊥ ) , (2.64) h ⊥ ( x, k ⊥ ) = − M ( m + xM )( m + xM ) + k ⊥ f ( x, k ⊥ ) . (2.65)It is also not surprising that within another spin-spin coupling, e.g., in an axial-vector diquarkmodel of Ref. [68], the set of such relations is altering or getting smaller. We also emphasizethat our chosen spectral representation of the LFWF overlap (2.57) preserves both the constraints(2.39,2.40) and the specific model relations (2.63–2.65). In the case we would have also introduceda spectral representation w.r.t. the struck quark mass, the strength of the spin coupling for the l z = 0 LFWF, given by m + xM , becomes “dynamical” in the quantities (2.58,2.60,2.61,2.62).Since the model constraints (2.39,2.40) are independent on the struck quark mass m , they wouldstill be satisfied. On the other hand the algebraic equations (2.63–2.65) would not hold anymore,giving an example that interference effects will break uPDF model relations. However, employingthe mean value theorem one would recover Eqs. (2.63–2.65) within an effective quark mass thatbecomes a function of both longitudinal and transversal degrees of freedom x and k ⊥ .We may have also evaluated in our model the remaining other unintegrated parton densities,which are of higher twist in the sense of power counting, and then one may also express the findingsin terms of the unpolarized one. We will skip this exercise here, since first some clarification aboutthe classification scheme of TMDs is needed [77], which is far beyond the scope of the presentstudy.The above set of formulae (2.58,2.60,2.61,2.62) clearly illuminate the role of orbital angularmomentum L z within a scalar diquark spectator. The ( m + xM ) and k ⊥ proportional terms arisefrom the (diagonal) overlap of L z = l z = 0 and L z = l z = 1 LFWFs, respectively. In particularthe l z = 1 LFWF overlap reduces the amount of the polarized g PDF (2.60). We note that thedifference in sign of the k ⊥ proportional terms in the unpolarized and polarized uPDFs, given inEqs. (2.58) and (2.60), indicates an inconsistency of such models with perturbative QCD. Namely,22he leading log behavior for PDFs arises just from these l z = 1 overlaps and according to theevolution equations in leading order approximation they have the same sign, which contradictsthe scalar diquark model. Similarly, we find also an inconsistency from the evolution equation forthe transversity PDF h . Namely, in any scalar diquark model the transversity h saturates theSoffer bound (2.40) for the unintegrated quantities and, hence, this is also true for the integratedones, h ( x, µ ) = 12 (cid:2) f ( x, µ ) + g ( x, µ ) (cid:3) . (2.66)However, this equation is again in conflict with leading order evolution equations, since they aredifferent in the chiral odd and even sectors. Let us also note that QCD equations-of-motion [78]were utilized to relate twist-two with twist-three related PDFs [79, 80], e.g., g , h , and g ⊥ , wherethe latter is accessible in transversally polarized deep inelastic scattering and results from theoverlap of l z = 0 and l z = 1 LFWFs. In such kind of dynamic QCD relations quark-gluon-quarkcorrelations enters, too, and one might wonder to which extend such interaction dependent termsare mimicked in a given quark model, e.g., the chiral quark soliton or the bag model [69].If we ignore our critics on the scalar diquark model rather rely on it, the various uPDFs provideus a model dependent handle on the quark orbital angular momenta. As we realized, the quarkorbital angular momenta can be also traced within PDFs. Furthermore, we might reverse thelogic and ask what we can learn about the k ⊥ dependence. To realize that even the integratedPDFs remember their k ⊥ dependence, it is instructive to express them in terms of the integratedLFWF overlap (2.57), Φ ( x, µ ) = Z d k ⊥ Φ ( x, k ⊥ ) , (2.67)and the k ⊥ average h k ⊥ i ( x, µ ) = R d k ⊥ k ⊥ Φ ( x, k ⊥ ) R d k ⊥ Φ ( x, k ⊥ ) . (2.68)These equations have to be taken with care. In QCD one should find after integration over k ⊥ a logarithmical scale dependence, which should show up in the definition (2.67), hence, the k ⊥ –average (2.68) is expected to be a divergent quantity. Let us ignore here the challenge tounderstand QCD dynamics, and let us proceed with the scalar diquark model in the commonfashion. Formally, we can express then the unpolarized and polarized PDFs by means of theintegrated LFWF overlap (2.67) and k ⊥ –average (2.68): f ( x, µ ) = ( m + xM ) + h k ⊥ i ( x, µ ) M Φ ( x, µ ) , (2.69) g ( x, µ ) = ( m + xM ) − h k ⊥ i ( x, µ ) M Φ ( x, µ ) , (2.70) h ( x, µ ) = ( m + xM ) M Φ ( x, µ ) , (2.71)23espectively. Hence, we read off that in a scalar diquark model the k ⊥ –average ratio h k ⊥ i ( x, µ )( m + xM ) = f ( x, µ ) − g ( x, µ ) f ( x, µ ) + g ( x, µ ) (2.72)is given by the ratio of parton densities with helicities opposite and aligned to the longitudinalproton spin. According to large x counting rules [81], this ratio should vanish as (1 − x ) for x →
1. In the limit x → h k ⊥ i ( x, µ ) approaches the struck quark mass square m .From what was said in Sec. 2.2.1 we can easily extend our scalar diquark model by meansof Eq. (2.43) and the LFWF “spinor” (2.55) to a “spherical” model of rank-four by adding anunpolarized uPDF, which might be ambiguously chosen for u and d quarks. If we like to adoptSU(4) flavor-spin symmetry to dress our uPDFs with flavor, we can immediately take the formulae(2.41) and (2.42), where then only the unpolarized uPDF in the u/ d/ f u/ d/ ( x, k ⊥ ) sph&SU(4) = 2 f u/ − d/ ( x, k ⊥ ) , where f u/ − d/ is taken from Eq. (2.58). Field theoretically GPDs are defined as off-diagonal matrix elements of two field operators, con-nected by a gauge link, that lives on the light cone [3, 4, 5]. Generically, in the light-cone gaugea GPD definition looks like as following F ( x, η, t, µ ) = Z dy − π e ixP + y − / h P ′ | ˆ φ (0) ˆ φ ( y ) (cid:12)(cid:12) y + =0 , y ⊥ =0 | P i , (3.1)where the skewness parameter is defined as η = P + − P ′ + P + + P ′ + ≥ . (3.2)As already stated above in Sec. 2.1, the operator on the light cone possesses divergencies which areremoved within a renormalization procedure and the scale dependence is governed by evolutionequations, see Refs. [82, 83] and references therein.GPDs are accessible in hard exclusive electroproduction [3, 4, 5] of photons or mesons [6] andtheir moments are measurable in QCD Lattice simulations [84]. The support of a GPD might be Up to a minus sign we use the variable conventions of [3]. − ≤ x ≤ − η , − η ≤ x ≤ η , and η ≤ x ≤ ≤ η. In the outer region η ≤ x and x ≤ − η a GPD is the probability amplitude for a s -channel exchangeof a parton and anti-parton, respectively. We will adopt the common habit and map anti-partondistributions to the region η ≤ x . The GPD in the central region − η ≤ x ≤ η , interpreted as amesonic-like t -channel exchange, is the dual counterpart of the GPD in the outer regions [85, 38],see also explanations given below in Sec. 4, in particular Sec. 4.1, and in Sec. 5.2.The zero-skewness case η = 0 is of specific partonic interest, since it allows for a probabilisticinterpretation and allow so to provide a three dimensional picture of the nucleon. In particular inthe infinite momentum frame a GPD in the impact parameter space [1] is the probability for findinga parton in dependence on the momentum fraction x and the transverse distance of the protoncenter [86], see also Ref. [87, 88]. Since they generalize the common probabilistic interpretationof PDFs to non-forward kinematics, we will denote them in the following as non-forward PDFs(nfPDFs).In Sec. 3.1 we give our twist-two GPD conventions, present their LFWF overlap representa-tions, and discuss the parameterization of uGPDs. In Sec. 3.2 we introduce then in analogy touPDF models a classification scheme for GPD models. Finally, in Sec. 3.3 we utilize the LFWFoverlap representation to have a closer look to the model dependent cross talks of zero-skewnessGPDs and uPDFs. In the following we will employ the standard notation of GPDs, appearing in the form factordecomposition of matrix elements such as generically given in (3.1), where the bilocal operatorsread as ˆ φ (0) ˆ φ ( y ) → ˆ¯ ψ (0)Γ ˆ ψ ( y ) with Γ ∈ { γ + , γ + γ , iσ + j } , σ + j = i (cid:2) γ + γ j − γ j γ + (cid:3) . In the chiral even and parity even [odd] sector we adopt the form factor decomposition of Ji [5].Here, we deal with the GPDs H and E [ e H and e E ], where the latter one is related to a targethelicity-flip contribution, F [ γ + ] = 12 ¯ P + ¯ U ( P ′ , S ′ ) (cid:20) H ( x, η, t ) γ + + E ( x, η, t ) σ + α ∆ α iM (cid:21) U ( P, S ) , (3.3) F [ γ + γ ] = 12 ¯ P + ¯ U ( P ′ , S ′ ) (cid:20) e H ( x, η, t ) γ + γ + e E ( x, η, t ) − ∆ + M γ (cid:21) U ( P, S ) , (3.4)25here ¯ P = ( P + P ′ ) / H T , E T , e H T , and e E T , defined as F [ iσ + j ] = 12 ¯ P + ¯ U ( P ′ , S ′ ) " H T ( x, η, t ) iσ + j + E T ( x, η, t ) ∆ + γ j − γ + ∆ j M (3.5)+ e H T ( x, η, t ) ∆ + ¯ P j − ¯ P + ∆ j M + e E T ( x, η, t ) γ + ¯ P j − ¯ P + γ j M U ( P, S ) . All GPDs are real valued functions and combining hermiticity and time-reversal invariance tellsus that they are even in η , except for the chiral odd function e E T ( x, η, t ) that is odd in η : F ( x, η, t ) = F ( x, − η, t ) for F ∈ { H, E, e H, e E, H T , E T , e H T } and e E T ( x, η, t ) = − e E T ( x, − η, t ) . Hence, the zero-skewness GPD e E T ( x, η = 0 , t ) vanishes, however, the limit lim η → e E T ( x, η, t ) /η will generally exist and proOf course, we have might also defined unintegrated GPDs (uGPDs) in terms of quark operators,which make contact to the “phase space” distributions of Ref. [17]. However, this will increasethe number of our considered quark GPDs from eight to sixteen [90]. Moreover, such uGPDs arehardly to access in experiment. Let us only emphasis here that we might decorate the commoneight twist-two GPDs with k ⊥ dependence, introduce other five uGPDs that generalize the set of f ⊥ , g ⊥ , h ⊥ , h ⊥ and h ⊥ uPDFs and in addition we are left with three uGPDs that die out in theforward limit and by k ⊥ integration. We find it rather natural to adopt such a parameterizationfor uGPDs, e.g., for the unpolarized quark uGPD, F [ γ + ] = ¯ U ( P ′ , S ′ ) " H γ + P + + E σ + α ∆ α iM ¯ P + + if ⊥ iσ + j k j ⊥ M ¯ P + + f , k j ⊥ iσ jα ∆ α M U ( P, S ) , (3.6)we have two k ⊥ dependent GPDs H and E , one skewed and with t -dependence decorated Siversfunction, and one new function f , .As outlined in Appendix B, the LFWF overlap representation for all twist-two GPDs can bestraightforwardly derived [24, 91, 92]. We will focus here on the region η ≤ x in which GPDs areobtained from the parton number conserved LFWF overlaps. We find it convenient to group theresults in a spin-correlation matrix F ( n ) ( x, η, t | ϕ ) = ZX ( n ) ψ ∗ ⇒→ ,n ψ ⇒→ ,n ψ ∗ ⇒→ ,n ψ ⇒← ,n ψ ∗ ⇒→ ,n ψ ⇐→ ,n ψ ∗ ⇒→ ,n ψ ⇐← ,n ψ ∗ ⇒← ,n ψ ⇒→ ,n ψ ∗ ⇒← ,n ψ ⇒← ,n ψ ∗ ⇒← ,n ψ ⇐→ ,n ψ ∗ ⇒← ,n ψ ⇐← ,n ψ ∗ ⇐→ ,n ψ ⇒→ ,n ψ ∗ ⇐→ ,n ψ ⇒← ,n ψ ∗ ⇐→ ,n ψ ⇐→ ,n ψ ∗ ⇐→ ,n ψ ⇐← ,n ψ ∗ ⇐← ,n ψ ⇒→ ,n ψ ∗ ⇐← ,n ψ ⇒← ,n ψ ∗ ⇐← ,n ψ ⇐→ ,n ψ ∗ ⇐← ,n ψ ⇐← ,n ( X i , k ′⊥ i , s i | X i , k ⊥ i , s i ) , (3.7)26here ϕ denotes now the polar angle in transverse momentum space (cid:0) ∆ , ∆ (cid:1) = | ∆ ⊥ | (cos ϕ, sin ϕ ) . The phase space integral (B.16), which might be written as ZX ( n ) · · · ≡ X s ,...,s n Z [ dX d k ⊥ ] n (cid:18) η − η (cid:19) n − p − η δ (cid:18) x + η η − X (cid:19) · · · , (3.8)includes now also the integral over the transverse momenta of the struck quark. The transversemomenta of the outgoing LFWFs are fixed by momentum conservation and read in Leipzig con-vention as X ′ = x − η − η , k ′⊥ = k ⊥ − − x − η ∆ ⊥ for the struck parton, X ′ i = 1 + η − η X i , k ′⊥ i = k ⊥ i + 1 + η − η X i ∆ ⊥ for the spectator i ∈ { , · · · , n } , (3.9)cf. Eqs. (B.6–B.10).Rather analog to the uPDF spin-density matrix (2.12), the GPD spin-correlation matrix resultsfrom summing up the individual LFWF overlaps (3.7), F ab ( x, η, t | ϕ ) = X n F ( n ) ab ( x, η, t | ϕ ) with a = Λ ′ λ ′ , b = Λ λ ∈ {⇒→ , ⇒← , ⇐→ , ⇐←} . The entries in this matrix might be easily read off from the GPD definitions (3.3–3.5) or theLFWF overlap representations (B.14,B.15,B.23,B.24,B.28–B.31), F ( x ≥ η, η, t | ϕ ) = (3.10)= H + t M E + e H + t M e E −| ¯∆ ⊥ | e iϕ M E T − η ˆ E T | ¯∆ ⊥ | e − iϕ M E − η e E H T + t ( e H T + ˆ E T ) M | ¯∆ ⊥ | e − iϕ M E T + η ˆ E T H + t M E − e H − t M e E − ¯∆ ⊥ (1 − η ) e − i ϕ M e H T | ¯∆ ⊥ | e − iϕ M E + η e E −| ¯∆ ⊥ | e iϕ M E + η e E − ¯∆ ⊥ (1 − η ) e i ϕ M e H T H + t M E − e H − t M e E −| ¯∆ ⊥ | e iϕ M E T + η ˆ E T H T + t ( e H T + ˆ E T ) M −| ¯∆ ⊥ | e iϕ M E − η e E | ¯∆ ⊥ | e − iϕ M E T − η ˆ E T H + t M E + e H + t M e E , where we use the shorthands ¯∆ ⊥ = ∆ ⊥ − η = √ t − t p − η and t = − M η − η that are symmetric functions under η → − η , for kinematics see Eqs. (B.1–B.6). With our choice ofvariables − t is the exact expression for the minimal value of the negative square of the momentumtransfer − t = − ∆ . Furthermore, we introduced the chiral odd GPD combinations H T = H T − t M e H T , E T = E T + 2 e H T − η e E T , ˆ E T = E T − η e E T . (3.11)27he spin-correlation matrix (3 .
10) has eight non-vanishing real valued GPD entries and we canread off the following property, implied by time-reversal and parity invariance, F † ( x, η, t | ϕ ) = F ( x, − η, t | ϕ + π ) . (3.12)The trace of the spin correlation matrix, given by 2 H ( x, η, t ) − η E ( x, η, t ) / (1 − η ) , is for vanishingskewness expressed by the unpolarized GPD H , see (2.12).As alluded above, zero skewness GPDs, called nfPDFs, in the impact space have their owninterest. Such densities can be obtained from a two-dimensional Fourier transform of certain GPDcombinations. In particular, their spin density matrix is obtained from the GPD spin-correlationmatrix (3.10) for η = 0 and reads as e F ( x, b ) = Z Z d ∆ ⊥ (2 π ) e − i b · ∆ ⊥ F ( x, ∆ ⊥ ) with F ( x, ∆ ⊥ ) = F ( x, η = 0 , t = − ∆ ⊥ | ϕ ) , (3.13)where the two dimensional vector b = b (cos ϕ, sin ϕ ) with b = | b | is the distance from the nucleoncenter and ϕ denotes now the polar angle in the two-dimensional impact space. From the properties(3.12) of the spin-correlation matrix in momentum space it is easy to realize that the Fouriertransformation provides a hermitian spin-density matrix in impact parameter space, e F † ( x, b ) = e F ( x, b ) . (3.14)Its trace, Tr e F ( x, b ) = 2 H ( x, b ), is expressed by the unpolarized nfPDF, which depends besidesthe momentum fraction x only on the distance b . Furthermore, this matrix is also semi-positivedefinite and it can be used to derive positivity constraints [93], even for the general η = 0 case[94]. The explicit expression of the spin-density matrix is straightforwardly obtained by means ofEq. (3.13): e F ( x, b ) = H + e H i e iϕ E ′ T − i e − iϕ E ′ H T − i e − iϕ E ′ T H − e H e − i ϕ e H ′′ T − i e − iϕ E ′ i e iϕ E ′ e i ϕ e H ′′ T H − e H i e iϕ E ′ T H T i e iϕ E ′ − i e − iϕ E ′ T H + e H ( x, b ) , (3.15)where the (dimensionless) derivatives of GPDs are denoted as E ′ ( x, b ) = bM ∂∂b E ( x, b ) , E ′ T ( x, b ) = bM ∂∂b h E T + 2 e H T i ( x, b ) , (3.16) e H ′′ T ( x, b ) = b M ∂∂b ∂∂b e H T ( x, b ) , H T ( x, b ) = H T ( x, b ) + e H ′′ T ( x, b ) , and ϕ is now the polar angle in impact parameter space. Note that we do not introduce newsymbols for the GPDs in impact parameter space rather we implicitly distinguish them from28hose in momentum space by specifying the argument. We also emphasize that in the forwardcase the “ T -odd” GPDs e E and ˆ E T drop out and so the spin-density matrix is only parameterizedby six real valued GPDs and their first and second order derivatives. We add that after reorderingand redefinition, our spin-density matrix (3.15) coincides with the result (34) of Ref. [93].As we saw in Sec. 2.1 time-reversal and parity invariance allows us to represent uPDFs bydifferent LFWF overlaps. This is also the case for our eight twist-two GPDs. We relate them hereto the LFWF overlaps in such a manner that we can easily realize formal relations among GPDsand uPDFs. Alternatively, such correspondences between these quantities would also follow if weintroduce uGPD definitions as mentioned above. From the spin-correlation matrix (3.10) we canread off for x ≥ η in the chiral even sector the diagonal n -parton LFWF overlaps H ( n ) ( x, η, t ) = ZX ( n ) (cid:2) ψ ∗ ⇒→ ,n ψ ⇒→ ,n + ψ ∗ ⇒← ,n ψ ⇒← ,n (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) − t M E ( n ) ( x, η, t ) , (3.17) E ( n ) ( x, η, t ) = M ¯∆ ZX ( n ) (cid:2) ψ ∗ ⇒→ ,n ψ ⇐→ ,n + ψ ∗ ⇒← ,n ψ ⇐← ,n + c.c. (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) , (3.18) e H ( n ) ( x, η, t ) = ZX ( n ) (cid:2) ψ ∗ ⇒→ ,n ψ ⇒→ ,n − ψ ∗ ⇒← ,n ψ ⇒← ,n (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) − t M e E ( n ) ( x, η, t ) , (3.19) e E ( n ) ( x, η, t ) = M ¯∆ η ZX ( n ) (cid:2) − ψ ∗ ⇒→ ,n ψ ⇐→ ,n + ψ ∗ ⇒← ,n ψ ⇐← ,n + c.c. (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) . (3.20)The target helicity-conserved unpolarized quark GPD H and polarized quark GPD e H essentiallyarise from the sum and difference of diagonal l z = 0 and diagonal l z = 1 LFWF overlaps, respec-tively, where the orbital angular momentum transfer∆ L z = ∆ l z = L ′ z − L z = l ′ z − l z (3.21)vanishes. Directly, from the uPDF overlaps (2.14) and (2.15) one realizes that they generalize f and g PDFs. However, both of these GPDs contain a t proportional E ( n ) and e E ( n ) addendathat presumable will cancel some contributions in the ∆ L z = 0 LFWF overlaps. The targethelicity-flip unpolarized quark GPD E and polarized quark GPD e E are given by l z = 0 and l z = 1 LFWF overlaps and so | ∆ L z | = 1. Note that the similarities of | ∆ L z | = 1 uPDF andGPD overlaps (2.22,2.21) and (3.18,3.20) suggest that E and e E GPDs have a cross talk with f ⊥ and g ⊥ uPDFs, respectively. As already said above and explained below in Sec. 3.3.2, such auPDF/GPD relation in the | ∆ L z | = 1 sector is presumable lost in QCD. In the chiral odd sector29e write the parton number conserved LFWF overlaps for x ≥ η in terms of the basis (3.11), H ( n )T ( x, η, t ) = 12 ZX ( n ) (cid:2) ψ ∗ ⇒→ ,n ψ ⇐← ,n + c.c. (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) − t M he H ( n )T + ˆ E ( n )T i ( x, η, t ) , (3.22) E ( n )T ( x, η, t ) = − M ¯∆ ZX ( n ) (cid:2) ψ ∗ ⇒→ ,n ψ ⇒← ,n + ψ ∗ ⇐→ ,n ψ ⇐← ,n + c.c. (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) , (3.23)ˆ E ( n )T ( x, η, t ) = Mη ¯∆ ZX ( n ) (cid:2) ψ ∗ ⇒→ ,n ψ ⇒← ,n − ψ ∗ ⇐→ ,n ψ ⇐← ,n + c.c. (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) , (3.24) e H ( n )T ( x, η, t ) = M i (1 − η ) ¯∆ ¯∆ ZX ( n ) (cid:2) ψ ∗ ⇐→ ,n ψ ⇒← ,n − ψ ∗ ⇒← ,n ψ ⇐→ ,n (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) . (3.25)All these GPDs probe a longitudinal quark spin transfer (of one unit), where H T essentially arisefrom off-diagonal l z = 0 LFWF overlaps, supplemented by t proportional correction terms andgeneralizes the h PDF, see Eq. (2.28). In the | ∆ L z | = 1 sector we have E and ˆ E GPDs, accessiblewith unpolarized and longitudinally polarized target, respectively. Again the similarities with theuPDF overlap representations (2.31) and (2.24) suggest that E and ˆ E GPDs are somehow relatedto h ⊥ and h ⊥ uPDFs, respectively. Finally, we have in the transversity sector the | ∆ L z | = 2 GPD e H ( n )T that stems from l z = 1 and l z = − h ⊥ , compare Eqs. (2.29) and (3.25). As for | ∆ L z | = 1 sector also for the | ∆ L z | = 2distributions the relation is only suggestive, however, as shown below in Sec. 3.3.3 might not beexist in QCD. The classification scheme for uPDF models might be also taken for uGPD ones. However, thenumber of model constraints among GPDs in momentum space will in generally be smaller thanfor uPDFs or in other words an eventual degeneracy of the spin-density matrix for uPDFs ispartially removed or even lost in those for GPDs. This is easy to understand, e.g., for the scalardiquark model the uPDFs arise from the Cartesian product of a LFWF “spinor”, see Eq. (2.36),while the GPD spin-correlation matrix additionally involves a k ⊥ -convolution: F (1) ( x, η, t | ϕ ) = ZZ d k ⊥ η ψ † (cid:18) x − η − η , k ⊥ − − x − η ∆ ⊥ (cid:19) ⊗ ψ (cid:18) x + η η , k ⊥ (cid:19) , ψ = ψ ⇒→ ψ ⇒← ψ ⇐→ ψ ⇐← . (3.26)Hence, we expect that for a “spherical” model of rank-one and for an “axial-symmetrical” one ofthe first kind with rank-two the four linear uPDF constraints, i.e., (2.37) and the saturated Soffer30ound (2.13), will reduce in the GPD case to three algebraic constraints, where the GPD analogsof the uPDF constraints g ∓ (cid:20) h + k ⊥ M h ⊥ (cid:21) sph = 0 and f ∓ (cid:20) h − k ⊥ M h ⊥ (cid:21) sph = 0 or f − g ± k ⊥ M h ⊥
1T sph = 0 (3.27)do not hold true anymore, however, their sum, i.e., the GPD analog of a saturated Soffer bound(2.13) exist. The fifth “spherical” uPDF constraint, the quadratic one in Eq. (2.38), is independenton k ⊥ , however, we do not expect that the corresponding GPD analog holds true, see next section.Contrarily to the momentum space representation, the spin-density matrices of zero-skewnessnfPDFs in impact parameter space (3.13), might have the same degeneracy as the correspondinguPDFs spin-density matrices. This is obvious for models that are build within effective two-bodyLFWFs, since the convolution integral in momentum space, e.g., Eq. (3.26), turns then into theproduct of the Fourier transformed two-body LFWFs e ψ Ss ( x, b ) in impact parameter space, e.g., e F (1) ( x, b ) ∝ e ψ † (cid:18) x, b − x (cid:19) ⊗ e ψ (cid:18) x, b − x (cid:19) , e ψ ( x, b ) = ZZ d k ⊥ (2 π ) e − i b · k ⊥ ψ ⇒→ ( x, k ⊥ ) ψ ⇒← ( x, k ⊥ ) ψ ⇐→ ( x, k ⊥ ) ψ ⇐← ( x, k ⊥ ) . (3.28)Hence, for such nfPDF models in impact parameter space the analogous situation appears as for“spherical” uPDF models (2.36), discussed in Sec. 2.2.1. More generally, both nfPDF modelsin impact parameter space and uPDF model in momentum space the n -parton LFWF overlapcontain only a ( n − n -parton one as it appear forGPDs in momentum space. Since the impact parameter representation plays a certain role forboth the partonic interpretation of GPDs and for the description of partonic processes in hadron-hadron collision, we will classify the nfPDF models in Sec. 3.2.2. Indeed, from this analysis wewill gain some new insights even for GPD models in momentum space. After these heuristic thoughts, let us derive algebraic GPD model constraints, where the classi-fication scheme is adopted from the uPDF models, given in Sec. 2.2. To do so we employ the31igenvalues of the spin-correlation matrix (3.10), F e . v . ( x ≥ η, η, t ) = (3.29)= H − H T − η e E T + t e H T M + t ∆ E M − r(cid:16) e H − H T + t M ∆ e E (cid:17) − ¯∆ ⊥ M (cid:16) ∆ E − η ∆ e E (cid:17) H − H T − η e E T + t e H T M + t ∆ E M + r(cid:16) e H − H T + t M ∆ e E (cid:17) − ¯∆ ⊥ M (cid:16) ∆ E − η ∆ e E (cid:17) H + H T + η e E T − t e H T M + t Σ E M − r(cid:16) e H + H T + t M Σ e E (cid:17) − ¯∆ ⊥ M (cid:16) Σ E − η Σ e E (cid:17) H + H T + η e E T − t e H T M + t Σ E M + r(cid:16) e H + H T + t M Σ e E (cid:17) − ¯∆ ⊥ M (cid:16) Σ E − η Σ e E (cid:17) ( x, η, t ) . Here, we denote the differences and sums of chiral even and odd quantities as (cid:26) ∆ E Σ E (cid:27) = E ∓ E T , E T = E T + 2 e H T − η e E T , ( ∆ e E Σ e E ) = e E ∓ ˆ E T , ˆ E T = E T − η e E T . For a “spherical” uPDF model of rank-one and an “axial-symmetrical” uPDF model of the firstkind with rank-two we expect that the corresponding GPD spin-correlation matrix is of rank lessthan four. Hence, we have at least one vanishing eigenvalue and so one of the following twoequations are satisfied2 (cid:18) H + e H − H T + t M e H T − η e E T (cid:19) (cid:18) H − e H + t M e H T − η e E T (cid:19) = (3.30)= t M (cid:20) e H − H T + t M ∆ e E (cid:21) ∆ e E − t M (cid:20) H − H T + t M e H T − η e E T + t + 4 η M η M ∆ E (cid:21) ∆ E , (cid:18) H + e H + 2 H T − t M e H T + η e E T (cid:19) (cid:18) H − e H − t M e H T + η e E T (cid:19) = (3.31)= t M (cid:20) e H + H T + t M Σ e E (cid:21) Σ e E − t M (cid:20) H + H T − t M e H T + η e E T − t + 4 η M η M Σ E (cid:21) Σ E .
As argued above, in the constraint that holds true each of the three terms separately vanishes,where the t -dependent combination H − e H ± (cid:20) t M e H T − η e E T (cid:21) = 0 , as analog of the third relation in Eq. (3.27), might not be necessarily equated to zero, however,one of the combinations H + e H ∓ H T ± t M e H T ∓ η e E T = H + e H ∓ (2 H T + η e E T )might vanish. Consequently, with the requirement that the r.h.s. of the constraint (3.30) [or (3.31)],i.e., ∆ E = 0 and ∆ ˜ E = 0 [or Σ E = 0 and Σ ˜ E = 0] we find the following tree linear GPD model32onstraints that should be valid for a “spherical” model of rank-one or an “axial-symmetrical”one of the first kind with rank-two H + e H ∓ h H T + η e E T i sph3ax12 = 0 , E ∓ h E T + 2 e H T − η e E T i sph3ax12 = 0 , e E ∓ (cid:20) E T − η e E T (cid:21) sph3ax12 = 0 . (3.32)We note that a “hidden” quadratic relation among transversity GPDs4 H T ( x, η, t ) e H T ( x, η, t ) + h E T ( x, η, t ) + 2 e H T ( x, η, t ) i − h e E T ( x, η, t ) i = 0 (3.33)does also not necessarily imply a degeneration of the eigenvalues. As we will see below in Sec. 4.4.1a quadratic constraint holds true in the DD representation. However, this relation would necessar-ily arise in the case that the GPD spin-correlation matrix (3.10) of our “spherical” model possessesthree zero modes, i.e., the linear combinations e H ∓ (cid:20) H T + t M e H T (cid:21) and H ∓ (cid:20) H T − t M e H T + η e E T (cid:21) or H − e H ± (cid:20) t M e H T − η e E T (cid:21) as analog of the “spherical” uPDF model constraints (3.27) vanish. However, as argued above,this is rather unlikely.In summary, a “spherical” uPDF model of rank one might imply only one vanishing eigenvalue(3.29) of the GPD spin-correlation matrix (3.10). The same happens for an “axial-symmetrical”model of the first kind with a rank-two. In both cases the three linear constraints (3.32) hold true,where also for the “spherical” model a possible quadratic relation (3.33) is unlikely to hold. Ifthese two models have rank-four the third GPD model constraint in Eq. (3.32), the analog of thesaturated Soffer bound (2.13), is not valid and so we are left with the first two linear constraints inEq. (3.32). For other uPDF models that we have discussed in Sec. 2.2, i.e., the “axial-symmetrical”models of the second kind and rank-three ones, we do not expect that general algebraic GPD modelrelations in momentum space representations are valid. However, if the number of LFWFs utilizedin a GPD model is limited and the various couplings are fixed, specific algebraic model relationsamong GPDs might be found.
In full analogy to our uPDFs model treatment in Sec. 2.2, we can now introduce a classificationscheme for nfPDFs models in impact space. Indeed, comparing the uPDF and GPD spin-densitymatrices (2.12) and (3.15), we can read off the replacement rules: H ( x, b ) ↔ f ( x, k ⊥ ) , e H ( x, b ) ↔ g ( x, k ⊥ ) , H T ( x, b ) ↔ h ( x, k ⊥ ) , (3.34) E ′ ( x, b ) ↔ − | k ⊥ | M f ⊥ ( x, k ⊥ ) , E ′ T ( x, b ) ↔ − | k ⊥ | M h ⊥ ( x, k ⊥ ) , e H ′′ T ( x, b ) ↔ k ⊥ M h ⊥ ( x, k ⊥ ) . H T ( x, b ) = H T ( x, b ) + e H ′′ T ( x, b ) ↔ h ( x, k ⊥ ) + k ⊥ M h ⊥ ( x, k ⊥ ). Setting g ⊥ and h ⊥ to zero in the model constraints of Sec. 2.2 will then provide us the one-to-one correspondencewith the nfPDF constraints, which we present now. The eigenvalues e F e . v . ( x, b ) = H − H T + e H ′′ T − r(cid:16) e H − H T (cid:17) + (cid:16) E ′ − E ′ T (cid:17) H − H T + e H ′′ T + r(cid:16) e H − H T (cid:17) + (cid:16) E ′ − E ′ T (cid:17) H + H T − e H ′′ T − r(cid:16) e H + H T (cid:17) + (cid:16) E ′ + E ′ T (cid:17) H + H T − e H ′′ T + r(cid:16) e H + H T (cid:17) + (cid:16) E ′ + E ′ T (cid:17) ( x, b ) (3.35)of the spin-density matrix (3.15) might be also obtained by means of the replacement rules (3.34)from those for uPDFs (2.12). It is also useful to have the formal correspondence to the eigenvalues(3.29) of the GPD spin-correlation matrix in momentum space for the η = 0 case, H ( x, b ) ↔ H ( x, , t ) , e H ( x, b ) ↔ e H ( x, , t ) , H T ( x, b ) ↔ H T ( x, , t ) , (3.36) E ′ ( x, b ) ↔ − √− t M E ( x, , t ) , E ′ T ( x, b ) ↔ − √− t M E T ( x, , t ) , e H ′′ T ( x, b ) ↔ t M e H T ( x, , t ) , where | ¯∆ ⊥ | = −√− t and η proportional terms such as η e E , η ˆ E T , and t × · · · drop out.For a “spherical” or “axial-symmetrical” model of the first kind one set of the linear constraints h e H ∓ H T i ( x, b ) sphax1 = 0 , (cid:2) E ∓ E T (cid:3) ( x, b ) sphax1 = 0 , lim η → (cid:20) e E ∓ E T ± η e E T (cid:21) ( x, b ) sphax1 = 0 (3.37)is valid. Here, the last equation does not follow from the degeneracy of the eigenvalues rather it isthe analog of the third formula in Eq. (3.32). We also conclude from Eq. (3.32) that the equalitiesamong first derivative quantities, e.g., E ′ = ± E ′ T , are valid for the quantities itself. In the casethat a “spherical” or an “axial-symmetrical” model of the first kind possesses a rank-one or -twospin-density matrix, respectively, the three constraints (3.37) are supplemented by H ( x, b ) ∓ h H T ( x, b ) − e H ′′ T ( x, b ) i sph3ax12 = 0 or H ( x, b ) + e H ( x, b ) ∓ H T ( x, b ) sph3ax12 = 0 (3.38)with H T ( x, b ) = H T ( x, b ) − e H ′′ T ( x, b ). Here, the latter constraint results from the combination ofthe former with the first model equality of the set (3.37). It is the analog of the saturated Sofferbound (2.13) for uPDFs and corresponds to the first algebraic condition in Eq. (3.32) for a GPDmodel in momentum space with rank less than four. For a “spherical” nfPDF model the quadraticrelation 4 e H ′′ T ( x, b ) H T ( x, b ) + h E ′ T ( x, b ) + 2 e H ′ T ( x, b ) i = 0 (3.39)34ust hold true in addition to the constraints (3.37) and eventually to the rank-one condition(3.38).Analogously to Eq. (2.46), for an “axial-symmetrical” model of the second kind one of theconditions h e H ( x, b ) i − [ H T ( x, b )] + 8 e H ′′ T ( x, b ) h H T − e H ′′ T i ( x, b ) + [ E ′ ( x, b )] + h E ′ T ( x, b ) i = ax2 = ± "r(cid:16) e H − H T (cid:17) + (cid:16) E ′ − E ′ T (cid:17) r(cid:16) e H + H T (cid:17) + (cid:16) E ′ + E ′ T (cid:17) ( x, b ) (3.40)is satisfied. If such a model has rank-two, the two quadratic constraints[ H ( x, b )] − h e H ( x, b ) i − e H ′′ T ( x, b ) h H T − e H ′′ T i ( x, b ) − [ E ′ ( x, b )] − h E ′ T ( x, b ) i = 0 , (3.41) h H ( x, b ) − e H ( x, b ) i H T ( x, b ) − H ( x, b ) e H ′′ T ( x, b ) − E ′ ( x, b ) E ′ T ( x, b ) ax2 = 0 (3.42)are separately valid, cf. Eqs. (2.49,2.50). Finally, in the case of a rank-three model only oneeigenvalue (3.29) vanishes, i.e., one of the following two quadratic conditions holds true: h H + e H ∓ H T ± e H T i h H − e H ± e H ′′ T i ( x, b ) − h E ′ ∓ E ′ T i ( x, b ) mod = 0 , (3.43)cf. Eqs. (2.47,2.48).Let us finally have a closer look to a “spherical” model or “axial-symmetrical” model of the firstkind. The main difference to the GPD model classification in momentum space is the appearanceof the first equality in the model constraints (3.37). Utilizing the representation e H ′′ T ( x, b ) = 14 M ∂ ∂ b e H T ( x, b ) − M b b · ∂∂ b e H T ( x, b ) and H T ( x, b ) = H T ( x, b ) + e H ′′ T ( x, b ) , we can convert this model relation into the momentum space and we might also generalize it tothe η = 0 case e H ( x, η, t ) ∓ (cid:20) H T ( x, η, t ) + Z t −∞ dt ′ M e H T ( x, η, t ′ ) (cid:21) sph3ax12 = 0 , (3.44)where we assumed that e H ( x, b ) and H T ( x, b ) corresponds to e H ( x, η, t ) and H T ( x, η, t ), respectively,and the GPDs vanish at − t → ∞ . Certainly, this integral relation differs from the analog algebraiccondition for a “spherical” model of rank-one in momentum space, e H ( x, η, t ) ∓ H T ( x, η, t ) , which one would read off with ∆ E = 0 or Σ E = 0 from the eigenvalues (3.29) of the spin-correlation matrix (3.10). In the case the Soffer bound is saturated, we can combine Eq. (3.44)with Eq. (3.32) to express the GPD H by the chiral odd ones, H ( x, η, t ) ∓ (cid:20) H T ( x, η, t ) − Z t −∞ dt ′ M e H T ( x, η, t ′ ) + η e E T ( x, η, t ) (cid:21) sph3ax12 = 0 . (3.45)35 ∆ L z | parity even parity odd chiral oddGPD uPDF GPD uPDF GPD uPDF0 H f e H g H T h e E − g ˆ E T h ⊥ E if ⊥ – – E T ih ⊥ e H T − h ⊥ / Classification of chiral even and odd twist-two GPDs (3.3–3.5) as well as leading–power uPDFs(2.10, 2.11) with respect to the orbital angular momentum transfer ∆ L z in the LFWF overlap represen-tation. Both GPDs and uPDFs allow for the resolution of transverse degrees of freedom and since both ofthem are embedded in uGPDs one might wonder whether GPDs and uPDFs can be more directlyrelated to each other. From the Field theoretical definitions such as (2.8) and (3.1), one canformally write down sum rules for twist-two related uPDFs and GPDs, H ( x, η = 0 , t = 0 , µ ) = ZZ d k ⊥ f ( x, k ⊥ ) , (3.46) e H ( x, η = 0 , t = 0 , µ ) = ZZ d k ⊥ g ( x, k ⊥ ) , (3.47) H T ( x, η = 0 , t = 0 , µ ) = ZZ d k ⊥ h ( x, k ⊥ ) , (3.48)which suggest that GPDs and uPDFs have somehow a cross talk. There are attempts to findsuch cross-talks on a more generic basis [93] or within models [60]. Thereby, often one employsin such studies the impact parameter space for GPDs and the momentum space for uPDFs. Inour opinion, see below the considerations in Sec. 4, it is more useful to consider both sets ofdistributions only in one space rather two different ones. To get some insights in the expectedcross talks we employ the LFWFs overlap representations in momentum space. Thereby, we willrestrict ourselves to nfPDFs (zero-skewness GPDs).Comparing the spin-density matrices (3.10) and (2.12) we read off besides the substitution rules k ⊥ → ∆ ⊥ / L z orbital angular momentum transfer (3.21). Note that∆ L z = 0 LFWF overlap quantities are those that connect directly twist-two distributions, while | ∆ L z | = 1 relate twist-two nucleon helicity-flip nfPDFs with twist-three PDFs. The | ∆ L z | = 2orbital angular momentum transfer can only appear in the chiral odd sector and relates the twist-two nfPDF e H T with the “pretzelosity” distribution, related to a twist-four PDF. To have a closer36ook to how nfPDFs and uPDFs are related to each other, we employ the overlap representationsof these parton distributions, where we distinguish between the three cases | ∆ L z | ∈ { , , } . ∆ L z = 0As mentioned above, the three twist-two zero-skewness GPDs and PDFs can be embedded intwist-two related uGPDs: H ( x, , t ) ← H ( x, , ∆ ⊥ , k ⊥ ) → f ( x, k ⊥ ) , e H ( x, , t ) ← e H ( x, , ∆ ⊥ , k ⊥ ) → g ( x, k ⊥ ) ,H T ( x, , t ) ← H T ( x, , ∆ ⊥ , k ⊥ ) → h ( x, k ⊥ ) . (3.49)To find the LFWF overlap contributions for the twist-two related zero-skewness uGPDs, we simplydrop in the LFWFs overlaps (3.17), (3.19) and (3.22) the k ⊥ integration and set η = 0, H ( n ) ( x, η = 0 , ∆ ⊥ , k ⊥ ) = ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇒→ ,n + ψ ∗ ⇒← ,n ψ ⇒← ,n (cid:3) ( X i , k ′⊥ i , s i | X i , k ⊥ i , s i ) , (3.50) e H ( n ) ( x, η = 0 , ∆ ⊥ , k ⊥ ) = ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇒→ ,n − ψ ∗ ⇒← ,n ψ ⇒← ,n (cid:3) ( X i , k ′⊥ i , s i | X i , k ⊥ i , s i ) , (3.51) H ( n )T ( x, η = 0 , ∆ ⊥ , k ⊥ ) = 12 ZX ( n − ψ ∗ ⇒→ ,n ( X i , k ′⊥ i , s i ) ψ ⇐← ,n ( X i , k ⊥ i , s i ) + c.c. . (3.52)Here, we use the shorthand (2.16) for the ( n −
1) particle phase space integration and the outgoingtransverse momenta (3.9) simplify to k ′⊥ = k ⊥ − (1 − x ) ∆ ⊥ , k ′⊥ i = k ⊥ i + X i ∆ ⊥ for i ∈ { , · · · , n } . (3.53)Compared to the incoming transverse momenta, they are shifted by ∆ ⊥ , weighted with a longitu-dinal momentum fraction. If we now integrate over k ⊥ or drop the ∆ ⊥ dependence, we establishthe definitions of nfPDFs or uPDFs, see definitions (2.14,2.15,2.28), respectively, F ( x, η = 0 , t ) = X n ZZ d k ⊥ F ( n ) ( x, η = 0 , ∆ ⊥ , k ⊥ ) for F ∈ { H, e H, H T } , (3.54) q ( x, k ⊥ ) = X n F ( n ) ( x, η = 0 , ∆ ⊥ = 0 , k ⊥ ) for q ∈ { f , g , h } . (3.55)Obviously, from these representations one easily obtains the sum rules (3.49).We realize also from the LFWF overlaps (3.50-3.52) that the t -dependency is induced by the k ⊥ -dependency, however, it seems to be impossible to find a model independent one-to-one mapof zero-skewness GPDs and uPDFs. Indeed, LFWFs overlaps in GPDs, e.g., ∝ (cid:20) e − iϕ − (1 − x ) | ∆ ⊥ || k ⊥ | e − iφ (cid:21) ¯ L z e iL z ϕ | ∆ ⊥ |→ = ⇒ ∝ L z , see Eq. (2.17), which imply specificcontributions to the t -dependence of GPDs, while in uPDFs these phase differences disappear.Hence, we conclude that in QCD a general map of GPDs to uPDFs or reversely might not exist.However, let us illustrate that for any model which is based on an effective two-body LFWFthat is real valued and free of nodes we can map a generic scalar “uPDF” to the generic scalar“nfPDF”. For doing so we switch to the impact parameter space, where the convolution integralof our “nfPDF” F ( x, η = 0 , t ) = 11 − x ZZ d k ⊥ φ ∗ ( x, k ⊥ − (1 − x ) ∆ ⊥ ) φ ( x, k ⊥ ) , (3.56)e.g., the scalar version of the spin-correlation function (3.26), turns into the product of the corre-sponding LFWFs : e F ( x, η = 0 , b ) = ZZ d ∆ ⊥ (2 π ) e − i b · ∆ ⊥ F ( x, η = 0 , t = − ∆ ⊥ )= (2 π ) (1 − x ) e φ ∗ ( x, b / (1 − x )) e φ ( x, b / (1 − x )) , (3.57)where the LFWF in the impact parameter space is the Fourier transform e φ ( x, b ) = ZZ d k ⊥ (2 π ) e − i b · k ⊥ φ ( x, k ⊥ )= √ − x ZZ d k ⊥ (2 π ) e − i b · k ⊥ p Φ ( x, k ⊥ ) . (3.58)of the square root of the diagonal LFWF overlap, which is the corresponding “uPDF” Φ ( x, k ⊥ ) = φ ( x, k ⊥ )1 − x . (3.59)Hence, within our assumptions the zero-skewness “GPD” in impact parameter space can be ob-tained from the “uPDF”: e F ( x, η = 0 , b ) = (2 π ) (1 − x ) (cid:12)(cid:12)(cid:12)(cid:12)ZZ d k ⊥ (2 π ) e − i b · k ⊥ / (1 − x ) p Φ ( x, k ⊥ ) (cid:12)(cid:12)(cid:12)(cid:12) . (3.60)Mapping back this “nfPDF” into momentum space, F ( x, η = 0 , t = − ∆ ⊥ ) = ZZ d k ⊥ p Φ ( x, k ⊥ − (1 − x ) ∆ ⊥ ) p Φ ( x, k ⊥ ) , (3.61)is nothing but taking the square root of the LFWF overlap (3.59) and putting it into the convolu-tion (3.56). Obviously, if we have overlap contributions from different states, even a superpositionof different spectator quark models, the conversion formulae (3.60,3.61) are not valid. Hence, evenin models the information that is encoded in a uPDF is not sufficient to restore the t -dependenceof the corresponding nfPDF. Here e F denotes the Fourier transform of the generic scalar “nfPDF” F and the symbol should not be confusedwith, e.g., parity odd GPDs. .3.2 Mismatches and new model relations in the | ∆ L z | = 1 sector We now consider the | ∆ L z | = 1 contributions, arising from the off-diagonal LFWF overlaps of l z = 0 and | l z | = 1 LFWFs. Here, the four helicity non-conserved twist-two nfPDFs might besomehow related with four twist-three related uPDFs, see Tab. 1. We already stated above, seeEq. (3.6), that these nfPDFs and uPDFs are parameterized by different sets of uGPDs. The firstset of such relations E ( x, , t ) ← E ( x, , ∆ ⊥ , k ⊥ ) ? ↔ if ⊥ ( x, , ∆ ⊥ , k ⊥ ) → if ⊥ ( x, k ⊥ ) , (3.62) E T ( x, , t ) ← E T ( x, , ∆ ⊥ , k ⊥ ) ? ↔ ih ⊥ ( x, , ∆ ⊥ , k ⊥ ) → ih ⊥ ( x, k ⊥ ) , refers to the correspondence of T -even nfPDF E and the T -odd Sivers function f ⊥ , in both thequarks are unpolarized and the nucleon is polarized, and to the correspondence of transverity T -even nfPDF E T and the T -odd Boer-Mulders function h ⊥ , where now the quarks are transversallypolarized and the nucleon is unpolarized. It has been intensively argued in the literature, see,e.g. Refs. [95], that T -even nfPDF E is related with the T -odd Sivers function. However, ina pure quark model the Sivers function simply vanishes, telling us that the connection to thetwist-two nfPDF E is lost. Including a transverse gauge link in the quark correlators one mightestablish a model dependent nfPDF/uPDF connection, where it has been suggested to write thisas a convolution integral with a so-called lensing function.The second set of relations e E ( x, , t ) ← e E ( x, , ∆ ⊥ , k ⊥ ) ? ↔ − g ⊥ ( x, , ∆ ⊥ , k ⊥ ) → − g ⊥ ( x, k ⊥ ) , (3.63)ˆ E T ( x, , t ) ← ˆ E T ( x, , ∆ ⊥ , k ⊥ ) ? ↔ h ⊥ ( x, , ∆ ⊥ , k ⊥ ) → h ⊥ ( x, k ⊥ ) , where ˆ E T ( x, , t ) = lim η → E T ( x, η, t ) − e E T ( x, η, t ) /η , tells us that the nfPDF e E and ˆ E T might berelated to the transverse polarized uPDF g and the longitudinal polarized one h ⊥ , respectively.Since of the T -parity mismatch these nfPDFs drop out in the spin-density matrix and so a possiblerelation among them is to our best knowledge not discussed in the literature.To understand that nfPDF/uPDF relations (3.62,3.63) can be found in model studies, however,might not exist in QCD we define for η ≥ k ⊥ unintegrated | ∆ L z | = 1 LFWF overlaps as E ( n ) ( x, η, ∆ ⊥ , k ⊥ ) = ZX ( n − (cid:2) ψ ∗⇒← ,n ψ ⇐← ,n + ψ ∗ ⇒→ ,n ψ ⇐→ ,n (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) + c.c. , (3.64) e E ( n ) ( x, η, ∆ ⊥ , k ⊥ ) = ZX ( n − (cid:2) ψ ∗⇒← ,n ψ ⇐← ,n − ψ ∗ ⇒→ ,n ψ ⇐→ ,n (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) + c.c. , (3.65) E ( n ) ( x, η, ∆ ⊥ , k ⊥ ) = ZX ( n − (cid:2) − ψ ∗ ⇒→ ,n ψ ⇒← ,n − ψ ∗ ⇐→ ,n ψ ⇐← ,n (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) + c.c. (3.66)ˆ E ( n ) ( x, η, ∆ ⊥ , k ⊥ ) = ZX ( n − (cid:2) ψ ∗ ⇒→ ,n ψ ⇒← ,n − ψ ∗ ⇐→ ,n ψ ⇐← ,n (cid:3) ( X ′ i , k ′⊥ i , s i | X i , k ⊥ i , s i ) + c.c. (3.67)39here now the ( n −
1) parton phase space integral is defined as ZX ( n − · · · ≡ X s ,...,s n Z [ dX d k ⊥ ] n (cid:18) η − η (cid:19) n − p − η δ (cid:18) x + η η − X (cid:19) δ (2) ( k ⊥ − k ⊥ ) · · · , (3.68)see Eq. (3.8), and the outgoing momenta are specified in Eq. (3.9). All these four unintegrated | ∆ L z | = 1 LFWF overlaps enter in both nfPDF and uPDF definitions, however, with differentprojections on the phase. According to Eqs. (2.22,3.18), (2.21,3.20), (2.32,3.23), and (2.24,3.24)we decompose the zero-skewness uGPDs as E ( n ) ( x, , ∆ ⊥ , k ⊥ ) = 2 k M f ( n ) ⊥ ( x, η = 0 , ∆ ⊥ , k ⊥ ) + ∆ M E ( n ) ( x, η = 0 , ∆ ⊥ , k ⊥ ) , (3.69) e E ( n ) ( x, η, ∆ ⊥ , k ⊥ ) = − k M g ( n ) ⊥ ( x, η = 0 , ∆ ⊥ , k ⊥ ) + η ∆ M e E ( n ) ( x, η = 0 , ∆ ⊥ , k ⊥ ) + O ( η ) , (3.70) E ( n ) ( x, , ∆ ⊥ , k ⊥ ) = − k M h ( n ) ⊥ ( x, η = 0 , ∆ ⊥ , k ⊥ ) + ∆ M E ( n )T ( x, η = 0 , ∆ ⊥ , k ⊥ ) , (3.71)ˆ E ( n ) ( x, η, ∆ ⊥ , k ⊥ ) = 2 k M h ( n ) ⊥ ( x, η = 0 , ∆ ⊥ , k ⊥ ) + η ∆ M ˆ E ( n )T ( x, η = 0 , ∆ ⊥ , k ⊥ ) + O ( η ) . (3.72)Here specific care has to be taken on the definition of η proportional T -odd GPD overlaps e E ( n ) and ˆ E ( n )T . Since the two projections are induced by QCD dynamics, which is a priory unknown,we should consider them as independent. Consequently, even nfPDF/uPDF sum rules such asgiven in Eqs. (3.46–3.48) for ∆ L z = 0 LFWF overlap quantities (or twist-two related ones) cannotbe derived. Hence, we can only state that one can obtain from | ∆ L z | = 1 uGPDs zero-skewnesstwist-two GPDs at t=0 and twist-three related PDFs, F ( x, η = 0 , t = 0) = X n ZZ d k ⊥ F ( n ) ( x, η = 0 , ∆ ⊥ = 0 , k ⊥ ) , F ∈ { E, e E, E T , ˆ E T } , (3.73) f ( x ) = X n ZZ d k ⊥ f ( n ) ( x, η = 0 , ∆ ⊥ = 0 , k ⊥ ) , f ∈ { f ⊥ , g ⊥ , h ⊥ , h ⊥ } , (3.74)which are a priory not tied to each other. Of course, in any model the projections are “fixed”and so one will discover some model dependent relations, where however, in all four nfPDF/uPDFcorrespondences a mismatch under time inversion appears.Let us point out that one can also sacrifice a mismatch under parity reflection rather timeinversion. Then one might ask for the model correspondences of E ( x, , t ) ← E ( x, , ∆ ⊥ , k ⊥ ) ? ↔ g ⊥ ( x, , ∆ ⊥ , k ⊥ ) → g ⊥ ( x, k ⊥ ) , (3.75) E T ( x, , t ) ← E T ( x, , ∆ ⊥ , k ⊥ ) ? ↔ h ⊥ ( x, , ∆ ⊥ , k ⊥ ) → h ⊥ ( x, k ⊥ ) . Let us suppose that the T -odd Sivers and Boer-Mulders functions vanish. Then the zero-skewnessnfPDF E and E T as well as the uPDFs g ⊥ and h ⊥ can be obtained from the following ∆ L z = 140FWF overlaps, respectively:2 ZX ( n − ψ ∗ ⇒← ,n ( X i , k ′⊥ i , s i ) ψ ⇐← ,n ( X i , k ⊥ i , s i ) + c.c. = E ( n ) ( x, , ∆ ⊥ , k ⊥ ) + e E ( n ) ( x, , ∆ ⊥ , k ⊥ ) , (3.76) − ZX ( n − ψ ∗ ⇐→ ,n ( X i , k ′⊥ i , s i ) ψ ⇐← ,n ( X i , k ⊥ i , s i ) + c.c. = E ( n ) ( x, , ∆ ⊥ , k ⊥ ) + ˆ E ( n ) ( x, , ∆ ⊥ , k ⊥ ) , (3.77)which might be expressed by uGPD combinations (3.64–3.67). Indeed, from the uGPD parma-terizations (3.69–3.72) we read off that nfPDF E and uPDF g ⊥ are two different projections of auGPD combination E ( x, η = 0 , t = 0) = X n lim ∆ → M ∆ ZZ d k ⊥ h E ( n ) + e E ( n ) i ( x, η = 0 , ∆ ⊥ , k ⊥ ) , (3.78) g ⊥ ( x ) = − X n ZZ d k ⊥ M k h E ( n ) + e E ( n ) i ( x, η = 0 , ∆ ⊥ = 0 , k ⊥ ) , (3.79)while for transversity nfPDF E and uPDF h ⊥ we find the analog result E ( x, η = 0 , t = 0) = X n lim ∆ → M ∆ ZZ d k ⊥ h E ( n ) + ˆ E ( n ) i ( x, η = 0 , ∆ ⊥ , k ⊥ ) , (3.80) h ⊥ ( x ) = X n ZZ d k ⊥ M k h E ( n ) + ˆ E ( n ) i ( x, η = 0 , ∆ ⊥ = 0 , k ⊥ ) . (3.81)Let us have a closer look to the two uGPD projections, e.g., E and g ⊥ . Angular momentumconservation tells us that for a contribution with given ¯ L z , see Eq. (2.17) and discussion above,we have h E ( n, ¯ L z ) + e E ( n, ¯ L z ) i ( x, η = 0 , ∆ ⊥ , k ⊥ ) ∝ (cid:2) | k ⊥ | e − iϕ − (1 − x ) | ∆ ⊥ | e − iϕ (cid:3) ¯ L z +1 | k ⊥ | ¯ L z e iL z ϕ + c.c.As one realizes for ¯ L z = 0 the nfPDF on the r.h.s. of Eqs. (3.78,3.80) depends on the angular mo-mentum ¯ L z , even its first Taylor coefficient in the vicinity of ∆ ⊥ = 0, while for the correspondinguPDF (3.79,3.81) it does not. Consequently, in QCD we would consider both partonic quantitiesas independent. For quark models with a limited number of LFWFs one can easily establish modelrelations, e.g., for a two-body ¯ L z = 0 LFWF model the following two sum rules show up: E ( x, η = 0 , t = 0 , µ ) ¯ L z =0 = (1 − x ) ZZ d k ⊥ g ⊥ ( x, k ⊥ ) for f ⊥ = 0 , (3.82) E T ( x, η = 0 , t = 0 , µ ) ¯ L z =0 = − (1 − x ) ZZ d k ⊥ h ⊥ ( x, k ⊥ ) for h ⊥ = 0 . (3.83)Note that due to the k ⊥ -integration the linear k ⊥ -term in GPD E and E T yields ZZ d k ⊥ [ k ⊥ − (1 − x ) ∆ ⊥ ] · · · = − − x ∆ ⊥ ZZ d k ⊥ · · · , k ′⊥ = [ k ⊥ − (1 − x ) ∆ ⊥ ] and k ⊥ and symmetric under the simultaneous reflection k ′⊥ → − k ′⊥ and k ⊥ → − k ⊥ . Finally, that means that the integrand has definite symmetryw.r.t. the shifted integration variable k ⊥ = k ⊥ − (1 − x ) ∆ ⊥ / It has been somehow speculated in Ref. [60] that a sum rule, such as for unpolarized quantities(3.46–3.48), might also exist among the “pretzelosity” uPDF h ⊥ ( x ), given in Eq. (2.29), and thetransversity nfPDF e H T , see Eq. (3.25), where both arise from a | ∆ L z | = 2 LFWF overlap. Toshow that such a relation is not generally justified, let us define the embedding uGPD in terms ofLFWF overlaps e H ( n )T ( x, η = 0 , ∆ ⊥ , k ⊥ ) = ZX ( n − ψ ∗ ⇒← ,n ( X i , k ′⊥ i , s i ) ψ ⇐→ ,n ( X i , k ⊥ i , s i ) − c.c. . (3.84)Both quantities of interest are then again obtained by two different projections e H T ( x, η = 0 , t = 0) = X n lim ∆ → M i ∆ ∆ ZZ d k ⊥ e H ( n )T ( x, η = 0 , ∆ ⊥ , k ⊥ ) , (3.85) h ⊥ ( x ) = − X n ZZ d k ⊥ M ik k e H ( n )T ( x, η = 0 , ∆ ⊥ = 0 , k ⊥ ) , (3.86)see uPDF and GPD definitions (2.29) and (3.25), respectively. As discussed in the precedingsection, also here the projection on the nfPDF depends on the overall partonic angular momenta¯ L z while the projection on uPDF is free of it. Introducing the shifted integration variable k ⊥ = k ⊥ − (1 − x ) ∆ ⊥ /
2, we find that e H ( n, ¯ L z )T ( x, η = 0 , ∆ ⊥ , k ⊥ ) is proportional to (cid:20) | k ⊥ | e − iϕ − − x | ∆ ⊥ | e − iϕ (cid:21) ¯ L z +1 (cid:20) | k ⊥ | e iϕ + 1 − x | ∆ ⊥ | e iϕ (cid:21) ¯ L z − − c.c. . Obviously, also the second Taylor coefficient in the expansion around the vicinity ∆ ⊥ = 0 dependon ¯ L z and, thus, in general the nfPDF e H T at t = 0 differs from the twist-four uPDF h ⊥ . Evenfor the scalar diquark model, i.e., ¯ L z = 0, we find that the uGPD is proportional to e H ( n, ¯ L z =0)T ( x, η = 0 , ∆ ⊥ , k ⊥ ) ∝ i (cid:16) k k − (1 − x ) ∆ ∆ (cid:17) . Consequently, the k ⊥ -integration in the transversity nfPDF (3.85) and the “pretzelosity” distri-bution (3.86) possesses also for this simple model a different weight. Hence, both distributionsare proportional to each other, e H T ( x, η = 0 , t = 0) diquark ∝ (1 − x ) h ⊥ ( x ) . (3.87)42here, however, the proportionality factor depends on the functional form of the LFWF. Thus,even in a model that contains different ¯ L z = 0 states a sum rule that connects the twist-twonfPDF e H T with the twist-four uPDF h ⊥ ( x ) does not exist in general. Let us now apply the model procedure to GPDs, which are accessible from hard-exclusive reactions,e.g., electroproduction of mesons and photon. Generically, a GPD definition looks like as given inEq. (3.1) from which follows that the x -moments of the GPD F are polynomials in the η skewnessparameter (3.2). This polynomiality property is manifestly implemented in the so-called doubledistribution (DD) representation, e.g., for the “quark” part of F GPD F ( x, η, t ) = Z dy Z − y − y dz δ ( x − y − zη ) f ( y, z, t ) , (4.1)which lives in the region − η ≤ x ≤
1. On the other hand, describing the initial and finalproton states in terms of the LFWFs (2.1), we can straightforwardly write down LFWF overlaprepresentations of GPDs [24, 91, 92], see Sec. 3.1 and Appendix B, in which positivity constraintsare manifestly implemented, however, not the polynomiality property. Thereby, the GPD in theregion η ≤ x ≤ s -channel exchange). In the region − η ≤ x ≤ η the GPD is given by an overlap ofLFWFs in which the parton numbers change from n + 1 to n − t -channel exchange)[91, 92].Both the central and outer regions are tied to each other by polynomiality or Poincar`e co-variance, see e.g., Ref. [38]. Employing this GPD duality, we can simplify the GPD modelingin terms of LFWFs, in particular, we are interested to obtain the GPD from the parton numberconserved LFWF overlap, which generically reads for a effective two-body LFWF, cf. the GPDspin-correlation matrix (3.26): F ( x ≥ η, η, t ) = 11 − x ZZ d k ⊥ φ ∗ (cid:18) x − η − η , k ⊥ − − x − η ∆ ⊥ (cid:19) φ (cid:18) x + η η , k ⊥ (cid:19) . (4.2)Note that the normalization of this scalar LFWF differs from those of the “spinor” ones (2.55) bya common factor 1 / p − X = p (1 + η ) / (1 − x ) , where X = x + η η , which we also took into account in the definition of the unintegrated scalar LFWF overlap for η = 0, see, e.g., Eq. (2.57). Having such a scalar LFWF at hand, one might transform the overlap43epresentation (4.2) into the DD representation F ( x ≥ η, η, t ) = Z x + η ηx − η − η dyη f ( y, ( x − y ) /η, t ) , (4.3)from which one can read off the DD f ( y, z, t ) [39]. Hence, one can so obtain the GPD F in thewhole region. In the case that such a procedure fails, we might conclude that the employed LFWFdoes not respect the underlying Lorentz symmetry and should not be used in model estimates. Ofcourse, there are plenty of LFWF models in the literature that are ruled out by this requirement.In the following we implement in such a LFWF modeling procedure k ⊥ -dependence by con-sidering unintegrated DDs (uDDs), which also makes contact to the spectral properties of bothuGPDs and phase space distributions [96]. Neglecting the spin content, in Sec. 4.1 we spell out theconstraints from Poincar´e covariance for the functional form of a scalar effective LFWF and showthat the DD representation for a GPD can be found from the parton number conserved overlapof a two-body LFWF. Within these restrictions, in Sec. 4.2 we discuss then the implementationof Regge behavior from the s -channel point of view. In Sec. 4.3 we present the DD representationfor all twist-two nucleon GPDs in terms of a scalar diquark LFWF. Finally, in Sec. 4.4 we discussthe GPD model constraints among the eight twist-two GPDs and the model dependent cross talkof nfPDs and uPDFs. Let us restrict to the region x ≥ η in which partons are probed within a s -channel exchange. Asit turned out for uPDFs in Sec. 2.1, it is rather elegant to introduce a scalar LFWF overlap inthe outer region, which we write as Φ ( x ≥ η, η, ∆ ⊥ , k ⊥ ) = 11 − x φ ∗ (cid:18) x − η − η , k ⊥ − − x − η ∆ ⊥ (cid:19) φ (cid:18) x + η η , k ⊥ (cid:19) , (4.4)which might be also directly viewed in a scalar toy theory as a “uGPD” definition. A “uGPD”definition in terms of a scalar LFWF overlap for antiparticles can be analogously defined with anegative momentum fraction x ≤ − η and for convenience they might be mapped to the η ≤ x region, decorated with a sign according to its charge conjugation parity. Positivity constraints, inits most general form, should be satisfied in the overlap representations, if they are not spoiled bya subtraction procedure. For a deeper discussion we refer to the work of P. Pobylitsa [94, 97].As explained above, the residual Lorentz covariance, the so-called polynomiality conditionsfor GPD form factors (GFFs), is manifestly implemented in the DD representation and it ties φ and consequently Φ are considered as functions of the spectator quark mass λ , i.e., in full notation we wouldwrite φ ( . . . | λ ) and Φ ( . . . | λ ), see Eq. (2.56). For shortness this dependence is not indicated in this Section. φ ( X, k ⊥ ) = Z ∞ dα ϕ ( X, α ) exp (cid:26) − α k ⊥ − X (1 − X ) M (1 − X ) M (cid:27) , (4.5)the restoration of t -dependence can be achieved. Note that the existence of the integral requiresthat lim α →∞ exp { αX } ϕ ( X, α ) sufficiently fast vanishes, which is ensured by the stability condi-tion for the spectator system. This allows us to transform the LFWF overlap (4.4) into a uDDrepresentation, see Appendix A, Φ ( x, η, ∆ ⊥ , k ⊥ ) = Z dy Z − y − y dz δ ( x − y − zη ) ˆΦ (cid:0) y, z, t, k ⊥ (cid:1) , (4.6)where k ⊥ = k ⊥ − (1 − y + z ) ∆ ⊥ /
2. We also find that the uDD is represented as a Laplace transform ˆΦ ( y, z, t, k ⊥ ) = 12 Z ∞ dA A ϕ ∗ (cid:18) y − η (1 − z )1 − η , A − y + z (cid:19) ϕ (cid:18) y + η (1 + z )1 + η , A − y − z (cid:19) × exp ( A " y (1 − y ) + (cid:2) (1 − y ) − z (cid:3) t M − k ⊥ M . (4.7)However, it remains the (sufficient) condition that the product of Laplace kernels in (4.7) mustbe independent on η , i.e., ddη (cid:20) ϕ ∗ (cid:18) y − η (1 − z )1 − η , A − y + z (cid:19) ϕ (cid:18) y + η (1 + z )1 + η , A − y − z (cid:19)(cid:21) = 0 . (4.8)A rather simple example which provides a non-trivial solution of Eq. (4.8) reads: ϕ ( X, α ) = ϕ ( α ) exp (cid:26) − α m M − α X − X λ M (cid:27) , (4.9)which yields in Eq. (4.7) to the replacement ϕ ∗ ( X ′ , α ) ϕ ( X, α ) → ϕ ∗ (cid:18) A − y + z (cid:19) ϕ (cid:18) A − y − z (cid:19) exp (cid:26) − A (1 − y ) m M − Ay λ M (cid:27) . (4.10)Hence, we find for the uDD (4.7) a rather general representation ˆΦ ( y, z, t, k ⊥ ) = 12 Z ∞ dA A ϕ ∗ (cid:18) A − y + z (cid:19) ϕ (cid:18) A − y − z (cid:19) (4.11) × exp ( − A " (1 − y ) m M + y λ M − y (1 − y ) − (cid:2) (1 − y ) − z (cid:3) t M + k ⊥ M , which depends besides the set { m, λ, M } of mass parameters from the reduced LFWF ϕ ( α ).A few comments are in order. 45 In the unintegrated DD (4.11) the ∆ ⊥ dependence has been absorbed in the t and k ⊥ variables, and we see that they are tied to each other. This allows us to rewrite the k ⊥ average, needed for the evaluation of DDs and associated GPDs, as an integral over t : ˆΦ ( y, z, t ) = ZZ d k ⊥ ˆΦ (cid:0) y, z, t, k (cid:1) = (1 − y ) − z π Z t −∞ dt ′ ˆΦ (cid:0) y, z, t ′ , k ⊥ = 0 (cid:1) . (4.12)Analogously, the k ⊥ moment of the unintegrated DD (4.11), needed below, can be under-stood as an integral of the DD (4.12) over t : ˆΦ (2) ( y, z, t ) = ZZ d k ⊥ k ⊥ M ˆΦ (cid:0) y, z, t, k ⊥ (cid:1) = (cid:2) (1 − y ) − z (cid:3) Z t −∞ dt ′ M ˆΦ ( y, z, t ′ ) . (4.13) • We went here along the line of Ref. [39], i.e., switching to the DD representation (4.6), werestore the whole GPD from the parton number conserved LFWF overlap (4.4). Integratingover k ⊥ in Eq. (4.6), we find the common form of the DD representation Φ ( x, η, t ) = Z dy Z − y − y dz δ ( x − y − zη ) ˆΦ ( y, z, t ) , ˆΦ ( y, z, t ) = ZZ d k ⊥ ˆΦ (cid:0) y, z, t, k ⊥ (cid:1) , (4.14)ensuring that GPD polynomiality constraints are satisfied. • The t (or k ⊥ ) dependency has also a cross talk to the skewness one. For instance, if onetakes a reduced LFWF ϕ ( α ) that is concentrated in some point A one obtains exponential k ⊥ or t dependent quantities and a η -independent GPD. On the other hand if one choose apower-like dependence ϕ ( α ) ∝ α p , one will recover the model of Ref. [39]. It is well known that spectator quark models entirely fail in the small x region, where theypredict that PDFs have a constant behavior rather a x − α Regge behavior, which is with α ∼ . y → − η ≤ x ≤ η as a mesonic-like t -channel exchange makes directly contact to Regge phenomenology[98, 85, 99]. Alternatively, we will adopt here the s -channel point of view.Namely, one realizes in Eq. (4.11) that the variable y and the spectator mass square λ areconjugate to each other. We follow now the proposal [100] that the large spectator mass λ behavior might be considered as “dual” to the Reggeon exchange in the t -channel. In this manner,see Eq. (2.52), we might cure the small x failure by an incoherent sum over the spectator mass λ : ˆΦ ( y, z, t, k ⊥ ) = X λ Z ρ ( λ ) ˆΦ ( y, z, t, k ⊥ | λ ) . (4.15)46he λ integration is restricted from below by the cut-off mass λ c ≤ λ , which is considered as aparameter that is constrained by the stability criteria for the spectator system, M − m ≤ λ c . Toobtain a y − α behavior, the spectator mass spectral function must behave at large λ aslim λ →∞ ρ α ( λ, λ c ) ∝ M − α λ α − , (4.16)where α is the Regge intercept. For the spectral function we use a simple functional form ρ α ( λ, λ c ) = θ ( λ − λ c ) ( λ − λ c ) α − Γ( α ) M α , (4.17)which in the limit α → ρ ( λ, λ c ) = δ ( λ − λ c ) . Convolutingthe unintegrated DD (4.11) with this spectral density (4.23) and renaming the variable λ c → λ ,we obtain a uDD with the desired small y behavior: ˆΦ ( y, z, t, k ⊥ ) = 12 y α Z ∞ dA A − α ϕ ∗ (cid:18) A − y + z (cid:19) ϕ (cid:18) A − y − z (cid:19) (4.18) × exp ( − A " (1 − y ) m M + y λ M − y (1 − y ) − (cid:2) (1 − y ) − z (cid:3) t M + k ⊥ M , which we will utilize as a “master” formula for modeling. Note that up to the factor y − α A − α this uDD has the same functional form as the original one in Eq. (4.11). One might include some t -dependence in α by hand, e.g., following the common wisdom, by taking a linear trajectory α → α ( t ) = α + α ′ t . However, we emphasize that then positivity constraints are not guaranteed byconstruction. A rather non-trivial t -dependence might be introduced by a spectral representationw.r.t. the struck quark mass m , which needs some further investigation. We also point out that itis not allowed in our effective scalar LFWFs (2.53) and (2.53) to change the relative normalization.Hence, the implemented Regge behavior appears in all quantities in a unique manner, which mightinduce a contradiction with common Regge phenomenology. Employing the standard GPD definitions in terms of bilocal quark operators and the LFWF over-lap representations, listed in Eqs. (3.17–3.25), it is now straightforward to find DD representationsfor nucleon GPDs in terms of the “DD” (4.12). Thereby, we employ first the scalar diquark model,where the spin-couplings (2.53,2.54) are fixed, the effective LFWF (4.5) within the Laplace kernel(4.9). For instance, if we take the GPD E from the parton number conserved LFWF overlaprepresentation, we find then with one effective spectator ( n = 1) from Eq. (3.18): E ( x ≥ η, η, t ) = (1 − x ) ZZ d k ⊥ Φ ( x ≥ η, η, ∆ ⊥ , k ⊥ ) , (4.19)47here the unintegrated LFWF overlap is defined in Eq. (4.4). To convert the GPD E (4.19) intothe DD representation, we can now employ the formalism of Sec. 4.1 and we obtain E ( x, η, t ) = (1 − x ) Z dy Z − y − y dz δ ( x − y − ηz ) 2 (cid:16) mM + y (cid:17) ZZ d k ⊥ ˆΦ ( y, z, t, k ⊥ ) , (4.20)where the scalar uDD ˆΦ ( y, z, t, k ⊥ ) is given in Eq. (4.7) and the factor 2 (cid:0) mM + y (cid:1) reminds usthat the GPD E stems from an overlap of L z = 0 and L z = 1 LFWFs. Employing the Reggeimproved representation (4.18) rather the scalar one (4.7) with fixed diquark mass λ , we find after k ⊥ -integration the scalar DD in terms of a reduced LFWF ˆΦ ( y, z, t ) = π M y α Z ∞ dA A − α ϕ ∗ (cid:18) A − y + z (cid:19) ϕ (cid:18) A − y − z (cid:19) (4.21) × exp (cid:26) − A (cid:20) (1 − y ) m M + y λ M − y (1 − y ) − (cid:2) (1 − y ) − z (cid:3) t M (cid:21)(cid:27) . In the following two sections we give the DD representations for a scalar diquark and axial-vectordiquark model, respectively.
Analogously, for our model LFWFs all other chiral even (3.17,3.19,3.20) and odd (3.22–3.25)twist-two GPD overlap representations can be straightforwardly converted into DD representa-tions, providing us a set of formulae in which both Lorentz covariance and positivity constraintsare implemented. Alternatively, one might simply use the definition of the spin-correlation func-tion (3.10) and the overlap representation (3.26) with the scalar diquark LFWF “spinor” that isborrowed from Yukawa theory, i.e., it is build from the LFWFs (2.53,2.54): ψ S ( X, k ⊥ ) = 1 M m + XM −| k ⊥ | e iϕ | k ⊥ | e − iϕ m + XM φ ( X, k ⊥ | λ ) √ − X . (4.22)Thereby, we will replace the k ⊥ moments, entering in H and e H GPDs, by the integral (4.13) over t . Note, however, that the DD representations are not uniquely defined [101, 102]; we will showthose that naturally occur in the scalar diquark model. We now provide explicit representationsfor all eight twist-two GPDs and discuss some particularities and model constraints among them.In the chiral even sector we find it convenient to write the DD representation in a model48ependent manner, giving up a GPD/DD correspondence F ∈ { H, E, e H, e E } ↔ f ∈ { h, e, e h, e e } , HE e H e E ( x, η, t ) = Z dy Z − y − y dz δ ( x − y − ηz ) h + ( x − y ) e (1 − x ) e e h e e + (1 − y − z/η ) e ( y, z, t ) , (4.23)where all DDs are symmetric under z → − z reflection. In our GPD H and E representations(4.23) a first order polynomial in x = y + ηz appears. Thus, the odd x n -moments of these GPDsare even polynomials in η of order n + 1 [103] and so polynomiality is completed . In the GPD E a 1 − x = 1 − y − ηz factor appears while the GPD H is decorated with ( x − y ) e = ηz e addenda.Thus, the “magnetic” GPD combination H + E is given in terms of the DD h + (1 − y ) e andit has a common DD representation in accordance with the fact that its x n -moments are evenpolynomials of order n . The GPD e H has a common DD representation, too. In the representationof GPD e E appears a z/η proportional term, which is commonly not utilized. For shortness weexpressed the addenda in terms of the DD e , which is in general not the case. Since the DD e iseven in z , this term does not contradict polynomiality nor time reversal invariance, i.e., the GPD e E is symmetric under the exchange η → − η . The DDs are expressible by the k ⊥ -moments (4.12)and (4.13) of the Regge improved “uDD” (4.18): h = (cid:16) mM + y (cid:17) ˆΦ ( y, z, t ) + (cid:2) (1 − y ) − z (cid:3) (cid:20) t M ˆΦ ( y, z, t ) + Z t −∞ dt ′ M ˆΦ ( y, z, t ′ ) (cid:21) , (4.24) e = 2 (cid:16) mM + y (cid:17) ˆΦ ( y, z, t ) , (4.25) e h = (cid:16) mM + y (cid:17) ˆΦ ( y, z, t ) − (cid:2) (1 − y ) − z (cid:3) (cid:20) t M ˆΦ ( y, z, t ) + Z t −∞ dt ′ M ˆΦ ( y, z, t ′ ) (cid:21) , (4.26) e e = 2 (cid:0) (1 − y ) − z (cid:1) ˆΦ ( y, z, t ) . (4.27)As already emphasized for uPDFs in Sec. 2.3, we see that in a scalar diquark model thedynamical information is contained in one scalar LFWF overlap, i.e., now in a scalar “DD”. Tomake this property explicit, we expressed the k ⊥ -moment (4.13) by a t -integral over this “DD”.This t -integral arises from the diagonal L z = 1 LFWF overlap and, hence, it only enters in thetarget helicity conserved DDs h and e h within different signs. The various prefactors in Eqs. (4.24–4.27) arise from the overlap of the spin parts, labeled by orbital angular momenta L z = 0 and L z = ±
1. The diagonal L z = 0 overlap, proportional to ( m + yM ) /M , enters in both h (4.24)and e h (4.26) with a positive sign, while the diagonal L z = 1 one, proportional to (1 − y ) − z , In the original suggested or common DD representation [3, 104], where a first order polynomial in x is absent,all x n moments are only even polynomials of order n . Meanwhile, it is clarified that different DD representationsare equivalent, a short presentation on this subject can be found in Ref. [105] h and e h with a positive and negative sign, respectively. Moreover, both diagonaloverlaps possess different t -dependence. The target spin non-conserved DDs e (4.25) and e e (4.27)arise from the L z = 0 with L z = 1 LFWF overlap respectively, cf. Eqs. (3.18) and (3.20), andposses a relative simple structure. Hence, we have three constraints among the four chiral evenGPDs, which might be written in the following form: h ( y, z, t ) + e h ( y, z, t ) = (cid:16) mM + y (cid:17) e ( y, z, t ) , (4.28) h ( y, z, t ) − e h ( y, z, t ) = t M e e ( y, z, t ) + Z t −∞ dt ′ M e e ( y, z, t ′ ) M ≫− t ≈ Z t −∞ dt ′ M e e ( y, z, t ′ ) , (4.29) e e ( y, z, t ) = (1 − y ) − z mM + y e ( y, z, t ) . (4.30)Adopting the convention of Ref. [89], we find for the four chiral odd twist-two GPDs H T E T e H T e E T ( x, η, t ) = Z dy Z − y − y dz δ ( x − y − ηz ) h T e T e h T e e T ( y, z, t ) , (4.31)a common DD representation, where h T = (cid:20)(cid:16) mM + y (cid:17) − (cid:0) (1 − y ) − z (cid:1) t M (cid:21) ˆΦ ( y, z, t ) , (4.32) e T = 2 h(cid:16) mM + y (cid:17) (1 − y ) + (1 − y ) − z i ˆΦ ( y, z, t ) , (4.33) e h T = − (cid:2) (1 − y ) − z (cid:3) ˆΦ ( y, z, t ) , (4.34) e e T = 2 (cid:16) mM + y (cid:17) z ˆΦ ( y, z, t ) . (4.35)Compared to our findings (4.23–4.27) in the chiral even sector, we see that the chiral odd ones lookrather simple. Although it is not immediately obvious from the overlap representations (3.22–3.25)in the GPD basis (3.11), we can now easily identify the origin of the various DDs. Namely, thefactor ( m + yM ) /M in h T arises from the diagonal L z = 0 LFWF overlap, while the t -dependentpart comes from an (off-)diagonal | L z | = 1 overlap. It is also clear that e h T and e e T entirely arisesfrom a | L z | = 1 and a L z = 0 with | L z | = 1 off-diagonal LFWF overlap, respectively, where e T contains both of these two overlaps. Obviously, we can write down various relations among thedifferent DDs or we might express the chiral odd ones simply by chiral even DDs, e.g., h T ( y, z, t ) = 12 h h + e h i ( y, z, t ) − t M e e ( y, z, t ) , e h T ( y, z, t ) = − e e ( y, z, t ) ,e T ( y, z, t ) = (1 − y ) e ( y, z, t ) + e e ( y, z, t ) , e e T ( y, z, t ) = z e ( y, z, t ) . (4.36)Since H , E , and e E GPDs have very specific DD representations, a few comments are in order.50 . D-term
To complete polynomiality in a common DD representation, it has been suggested to add (sub-tract) a so-called D -term to the H ( E ) GPD, which entirely lives in the central region [40], (cid:26) HE (cid:27) = Z dy Z − y − y dz δ ( x − y − zη ) (cid:26) h new e new (cid:27) ( y, z, t ) ± θ ( | x | ≤ | η | ) d ( x/η, t ) , (4.37)where d (1) = d ( −
1) = 0 vanishes at the boundaries. Indeed, the form of the DD representation isnot unique [106, 102, 107, 108] and our DD representations (4.23–4.25) might be put in the form(4.37). Note that the form (4.37) suggest that the D -term is an independent GPD contribution,which in our model is not true (see discussion in Sec. 3.2.2 of Ref. [99]). We might project on the D -term in different ways [99, 38], taking the “low-energy” limit η → ∞ with fixed x/η provides[106]: d ( z, t ) = z Z −| z | dy e ( y, z, t ) , with e ( y, z, t ) = 2 (cid:16) mM + y (cid:17) ˆΦ ( y, z, t ) . (4.38)Finding the representations for the new DDs h new and e new is a straightforward task, which hasbeen considered in Ref. [106, 102, 105]. ii. η → limit Mostly the skewness-zero case can be easily taken in Eqs. (4.23,4.31) by setting η = 0. Forinstance, the overall normalization condition for the LFWFs (2.53,2.54), fixed in Eq. (2.59), mightbe directly expressed in terms of the DD h : Z dx (cid:26) x (cid:27) f ( x ) = Z dy Z − y − y dz (cid:26) y (cid:27) h ( y, z, t = 0) = (cid:26) n h x i (cid:27) . (4.39)In the skewness-zero case the six GPDs F ∈ n H, e H, H T , E T , e H T , e E T o have the standard rep-resentation F ( x, η = 0 , t ) = Z − x − x dz f ( x, z, t ) for f ∈ n h, e h, h T , e T , e h T , e e T o , (4.40)while for the GPD E a extra factor (1 − x ) appears E ( x, η = 0 , t ) = (1 − x ) Z − x − x dz e ( x, z, t ) . (4.41)Setting in addition t = 0, the H , e H , and H T GPDs are reduced to the f , g , and h PDFs,respectively.More care is required for the z/η term in e E . Its x -moments can be straightforwardly evaluatedfrom the DD representation (4.23), e.g., the two lowest ones read ( e E e E ) ( t ) ≡ Z − η dx (cid:26) x (cid:27) e E ( x, η, t ) = Z dy Z − y − y dz (cid:26) e e + (1 − y ) ey e e + y (1 − y ) e − z e (cid:27) ( y, z, t ) . (4.42)51e might now remember that GPDs are generalized functions in the mathematical sense and thuswe might formally writelim η → τ ( x ) zη δ ( x − y − zη ) = τ ( x ) lim η → zη δ ( x − y ) + dτ ( x ) dx z δ ( x − y ) , where τ ( x ) is a test function. One immediately finds for our GPD e E the non-standard represen-tation e E ( x, η = 0 , t ) = Z − x − x dz [ e e + (1 − x ) e ] ( x, z, t ) − ←− ddx Z − x − x dz z e ( x, z, t ) . (4.43)This equation, considered in the sense of a generalized function, is consistent with the Mellinmoments evaluated from the DD representation, see, e.g., the two lowest ones (4.42). For a testfunction that sufficiently vanish on the boundary x = 0, e.g. x , we can write Eq. (4.43) afterpartial integration also as x e E ( x, η = 0 , t ) = x Z − x − x dz [ e e + (1 − x ) e ] ( x, z, t ) + x ddx Z − x − x dz z e ( x, z, t ) , (4.44)where we assumed that e ( x, z, t ) vanishes at z = ± (1 − x ). iii. GPDs on the cross-over line The imaginary parts of hard-exclusive amplitudes to leading order accuracy, e.g., for Comptonform factors in deeply virtual Compton scattering F ( ξ, · · · ) LO = X q = u,d, ··· e q Z − dx (cid:20) ξ − x − iǫ ∓ ξ + x − iǫ (cid:21) F q ( x, η = ξ, · · · ) for F ↔ F ∈ (cid:26) H , E e H , e E , (4.45)are given by the corresponding GPDs on the cross-over line η = x , ℑ m F ( ξ, · · · ) LO = π X q = u,d, ··· e q [ F q ( x = ξ, η = ξ, · · · ) ∓ F q ( x = − ξ, η = ξ, · · · )] . (4.46)Here, e q are the quark charges and according to the discussion at the beginning of Sec. 3 the GPD F q can be decomposed in a quark and anti-quark GPD. One can simplify the DD-representationfor quark (or anti-quark) GPDs at the cross-over line, e.g., they can be written as an integral overthe variable w = z/ (1 − y ). Let us consider here only the GPDs E and x e E for which we find fromEq. (4.23) the integral representations E ( x, x, t ) = Z − dw (1 − x ) (1 − x w ) e (cid:18) x (1 − w )1 − x w , (1 − x ) w, t (cid:19) , (4.47) x e E ( x, x, t ) = Z − dw − x (1 − x w ) (cid:20) x e e + ( x − w )(1 − x )1 − x w e (cid:21) (cid:18) x (1 − w )1 − x w , (1 − x ) w, t (cid:19) , (4.48)52here e and e e are specified in Eqs. (4.25,4.27). It is instructive to compare the asymptoticallysmall x -behavior of E ( x, x, t ) and x e E ( x, x, t ):lim x → E ( x, x, t ) = x − α Z − dw (1 − w ) − α e res ( w, t ) , (4.49)lim x → x e E ( x, x, t ) = − x − α Z − dw w (1 − w ) − α e res ( w, t ) , (4.50)where the residue function e res ( w, t ) = lim x → x α e res ( x, w, t ). Other twist-two GPDs possess ananalogous small- x behavior as GPD E and we clearly realize that even GPD x e E has the samesmall x asymptotics. Hence, it could provide an important contribution, too, and one might beconcerned about the assumption, often employed in GPD phenomenology, that only the pion pole[109, 110] is essential in the Compton form factor e E , cf. Ref. [111]. iv. “dispersion relations” The real parts of hard-exclusive amplitudes to leading order accuracy can be alternatively obtainedfrom a “dispersion relation” [112, 113] (for a treatment beyond leading order accuracy see alsoRefs. [99, 114]). The appearance of a subtraction constant in the “dispersion relation” is tied tothe kind of DD representation. For the Compton form factors H and E , given in Eq. (4.45), wehave the “dispersion relation” ℜ e (cid:26) HE (cid:27) ( ξ, t ) LO = PV Z xξ − x (cid:26) HE (cid:27) ( x, x, t ) ± D ( t ) , D ( t ) = Z − dz z − z d ( z, t ) , (4.51)where the subtraction constant D is expressed in terms of the D -term (4.38) and H , E , and d contain here both quark and anti-quark GPDs, weighted with the corresponding charge factors.For e H and transversity GPDs, which possess the common DD representation, a subtraction con-stant is absent. Some attention should be given to the GPD e E . Since of the x − − α behavior, it ismore appropriate to work with an over-subtracted dispersion relation [115]. The real part of thecorresponding Compton form factor reads then as following ℜ e e E ( ξ, t ) LO = 1 ξ PV Z dx x ξ − x e E ( x, x, t ) + 1 ξ e C ( t ) , e C ( t ) = Z − dz − z e d ( z, t ) , (4.52)where the subtraction constant can be evaluated from a e d function that is commonly associatedwith the pion pole. As the d -term, the e d one can be obtained from the limit η → ∞ with fixed x/η , where the z/η -term in the DD representation (4.23) drops out: e d ( z, t ) = Z −| z | dy [ e + (1 − y ) e e ] ( y, z, t ) (4.53)= 2 Z −| z | dy h mM + y + (1 − y ) − (1 − y ) z i ˆΦ ( y, z, t ) .
53t is obvious that in our model the subtraction constant inherits the rather smooth t -dependenceof GPD e E and so the steep t -dependence of the pion pole will be generally absent. As already emphasized above, utilizing an arbitrary LFWF “spinor” could yield GPDs that donot satisfy the polynomiality conditions. To find appropriate LFWF “spinors”, we employ thebuilding blocks mM + X , | k ⊥ | M e iϕ , | k ⊥ | M e − iϕ for a L z = 0, L z = +1, and L z = − L z = 0 LFWF overlap contributions in the spin-correlation matrix (3.10), i.e., the diagonal entries F = F , F = F , and F = F , containa t / M = − η / (1 − η ) proportional addenda that it expressed in terms ∆ L z = 0 GPDs. Apartfrom the parity/time-reversal constraints, this ∆ L z = 0 and ∆ L z = 0 cross talk provides us therestrictions for the LFWF “spinors”.For a scalar spectator it turns out that there exist only one more LFWF “spinor”, ψ A0 ( X, k ⊥ ) = 1 √ M m + XM | k ⊥ | e iϕ | k ⊥ | e − iϕ − m − XM φ ( X, k ⊥ | λ ) √ − X , (4.54)with four non-vanishing entries that meets the GPD requirements. It might be interpreted as thecoupling of quarks to an axial-vector diquark spectator with zero spin projection s on the z -axis.Compared to the scalar diquark LFWF “spinor” (4.22), the relative sign of the first/third andsecond/forth entries in the “spinor” (4.54) is changed and, moreover, the normalization is alteredby a factor 1 / √
3. The corresponding GPDs are obtained from the scalar diquark ones (4.23–4.27,4.31–4.35), given in Sec. 4.3.1, by multiplying them with a factor 1 / s = 0 axial-vector diquark spectatormodel and forming a positive semi-definite linear combination with them, we can obtain a modelin which the normalization of the chiral odd GPDs can be easily tuned. Obviously, we can evencombine these models in such a manner that the GPDs (of course, also the corresponding uPDFs)54n the chiral odd sector vanish. Such a model can be also build from a pair of LFWF “spinors”, Ψ S+ = 1 M m + XM | k ⊥ | e − iϕ φ ( X, k ⊥ | λ ) √ − X , Ψ S − = 1 M −| k ⊥ | e iϕ m + XM φ ( X, k ⊥ | λ ) √ − X , (4.55)that belong to states in which only struck quarks with spin projection +1 / − /
2, respectively,occur. Again the chiral even GPDs can be taken from Sec. 4.3.1 while the chiral odd ones aresimply zero.Finally, we like to add that one can easily obtain the results for a pseudo-scalar or s = 0 vectordiquark model. Essentially, we have to replace the L z = 0 coupling m/M + X by − m/M + X andadopt the relative sign of the first/third and second/forth entries. For a pseudo-scalar target wefind then in accordance with an explicit calculation (up to an overall minus sign) ψ PS ( X, k ⊥ ) = 1 M − m + XM | k ⊥ | e iϕ | k ⊥ | e − iϕ m − XM φ ( X, k ⊥ | λ ) √ − X , (4.56)and for the s = 0 state of a vector diquark we have ψ V0 ( X, k ⊥ ) = 1 √ M − m + XM −| k ⊥ | e iϕ | k ⊥ | e − iϕ − m + XM φ ( X, k ⊥ | λ ) √ − X . (4.57)Since this ± m/M + X coupling is the only term in which the quark mass enters linearly in ourGPD models, we can again obtain the GPD expressions from the scalar diquark ones in Sec. 4.3.1,namely, by the replacement m → − m . Thereby, for pseudo-scalar diquark GPDs the overall signof the chiral odd GPDs changes in addition and the overall normalization of all vector diquarkGPDs is conventionally modified by a factor 1 /
3. We emphasize that for smaller values of x thesign of the L z = 0 and | L z | = 1 coupling, given by the factor y − m/M , becomes in such modelsnegative. In particular the GPD E possesses now a node and is for smaller x values naturallynegative, see the DD e expression (4.25). For an (axial-)vector diquark model it is perhaps not an entirely trivial task to find the LFWFsthat respect the underlying Lorentz symmetry and possess the required frame independent form552.1), i.e., they are in particular independent on the transverse proton momentum p ⊥ . Assumingfrom the beginning that LFWFs possess the frame independent form (2.1) and constructing themin a frame where the nucleon transverse momentum vanishes might yield erroneous results. Itis beyond the scope of this paper to construct (axial-)vector diquark LFWFs from explicitlycovariant models rather we propose now a minimal model for an axial-vector diquark spectatorwhich is relatively simple, cf. Ref. [116]. The guidance for finding such a model we take from theobservation that the axial-vector diquark model of Ref. [59] is a “spherical” uPDF model, whichspin-density matrix (2.41,2.42) can be represented by some unpolarized f uPDF, proportional tothe identity matrix, and a non-trivial part that can be taken from a scalar diquark model. If weemploy the LFWF “spinors” (4.54) and Ψ A+1 = √ √ M m + XM | k ⊥ | e − iϕ φ ( X, k ⊥ | λ ) √ − X , Ψ A − = √ √ M | k ⊥ | e iϕ − m − XM φ ( X, k ⊥ | λ ) √ − X , (4.58)where the subscripts { , +1 , − } label the polarization states of the axial-vector diquark [comparewith the generic “spinors” (2.45) ], we can easily verify that these “spinors” yield by an appropriatechoice of the effective LFWF the leading–power uPDFs of Ref. [59]. Forming the GPD spin-correlation matrix (3.26) for each axial-vector diquark state and summing up the findings, ouraxial-vector diquark GPD spin-correlation matrix can be straightforwardly obtained: F av ( x, η, t | ϕ ) = 43 × H ( x, η, t ) + 43 e × E ( x, η, t ) − F sca ( x, η, t | ϕ ) . (4.59)Here, × is again the identity matrix, we introduced the matrix e × = ∆ ⊥ (1 − η )4 M e − iϕ
00 0 0 e − iϕ − e iϕ − e iϕ , F sca is the spin-correlation matrix (3.10) of the scalar diquark model, where all GPDs are definedby the double distribution representations (4.23–4.27,4.31–4.32), given in Sec. 4.3.1. Our axial-vector diquark H and E GPDs are the same as in the scalar diquark model. The renaming sixGPDs can be also taken from the scalar diquark model, however, they are decorated with anadditional − / s = +1 and s = − e H and e E have an additional overall minus sign. Moreover, a vector diquark56odel for GPDs and uPDFs can be easily found by replacing the struck quark mass m → − m : Ψ V+1 = √ √ M − m + XM | k ⊥ | e − iϕ φ ( X, k ⊥ | λ ) √ − X , Ψ V − = √ √ M | k ⊥ | e iϕ m − XM φ ( X, k ⊥ | λ ) √ − X , (4.60)where the s = 0 vector diquark LFWF “spinor” is given in Eq. (4.57).According to SU(6) symmetry, we might associate the axial-vector diquark GPDs with d quarks, while the u -quark GPDs are given as combination of scalar diquark ones 2 u/ − d/ e F u sph&SU(4) = 23 × H u/ d/ ( x, η, t ) + 23 e × E u/ d/ ( x, η, t ) + 43 e F u/ − d/ ( x, η, t ) , (4.61) e F d sph&SU(4) = 43 × H u/ d/ ( x, η, t ) + 43 e × E u/ d/ ( x, η, t ) − e F u/ − d/ ( x, η, t ) , (4.62)where we might also impose the conditions H u/ d/ = H u/ − d/ and E u/ d/ = E u/ − d/ . Weemphasize that in the uPDF analog SU(6) symmetric “spherical” model constructions (2.41,2.42)no analog of GPD E appears, where, of course, the nfPDF/uPDF correspondence H ↔ f holds.Contrarily to a “spherical” uPDF model, we have according to Eq. (4.59) in the linear combination F av + 13 F sc − × H ( x, η, t ) = 43 e × E ( x, η, t )polarization effects that arise from the GPD E . This GPD E appearance can be traced back tothe restoration of Lorentz symmetry. Namely, in the diagonal entries of the GPD spin-correlationmatrix (3.10), which are now identically given by12 H ( x, η, t ) + t M E ( x, η, t ) = 12 H ( x, η, t ) − η − η ) E ( x, η, t ) , a t / M addenda appears in our model calculation that is absorbed by the non-vanishing E GPD. If this absorption does not occur, an additional η / (1 − η ) proportional term exist in theGPD H and, thus, polynomiality is broken.To set up more flexible GPD and uPDF models that respect Lorentz symmetry and positivity,one can relax the SU(6) symmetry conditions and one might write the spin-correlation and -density matrices for each quark flavor as a linear combination of those we have discussed, seeTab. 2. In each of these six independent sectors (4.22,4.56,4.54, 4.58,4.57,4.60) we might evenchoose a separate effective two-body LFWF. In this way one can mostly eliminate all modeldependent relations. 57iquark coupling LFWFs rel. presign parity even parity odd chiral oddEqs. of quark (4.23,4.24,4.25) (4.23,4.26,4.27) (4.31–4.35)mass term norm. norm. norm.scalar (4.22) + +1 +1 +1pseudo-scalar (4.56) − +1 +1 − − ) +1 +1 0lon. axial-vector (4.54) + +1/3 +1 / − / − / − / − / − +1/3 +1 / / − +2/3 − / − +1 − / / Diquark models built from various LFWF “spinors”. The resulting uPDF and GPD expressionsare taken from the scalar diquark model, given in Eqs. (2.58,2.60–2.62) and Eqs. (4.23–4.27,4.31–4.35),respectively, where they are modified corresponding to the formal (relative) presign setting of the struckquark mass term m in the LFWF “spinors” and the overall normalization factors in the parity even,parity odd, and chiral odd sectors. Our (pseudo-)scalar and longitudinal (axial-)vector LFWF models are “spherical” GPD modelswhere their spin-correlation matrices (3.10) have rank-three and so the model constraints, derivedin Sec. 3.2, must be valid. In analogy to the uPDF relations of a “spherical” model, given inSec. 2.3, we collect now the “spherical” GPD model relations and the GPD/uPDF cross talksfor the scalar diquark model. Some of these relations are also valid for the class of “spherical”models of rank-four and so they are supposed to hold true for the bag model [69], chiral quarksoliton model [70], covariant parton model [63], and the axial-vector diquark model in the versionof Ref. [59]. Apart from the purpose of model estimates such relations might be also directlyconfronted with phenomenological findings, which in principle allow to judge on the quark-diquarkcoupling without further specification of the effective two-body LFWF.
Using the DD results (4.23–4.27,4.31–4.35) of our “spherical” model, one can conveniently checkthat within the upper sign the three linear constraints (3.32), i.e., also the quadratic relation(3.30), and the integral equation (3.44) hold true. These four constraints allow us in analogy to a58spherical” uPDF model of rank-one, see Sec. 2.2.1 and Sec. 2.3, to express the chiral even GPDsentirely by the chiral odd ones: H ( x, η, t ) sph = ± (cid:20) H T ( x, η, t ) − t M e H T ( x, η, t ) − Z t −∞ dt ′ M e H T ( x, η, t ′ ) + η e E T ( x, η, t ) (cid:21) , (4.63) E ( x, η, t ) sph = ± h E T ( x, η, t ) + 2 e H T ( x, η, t ) − η e E T ( x, η, t ) i , (4.64) e H ( x, η, t ) sph = ± (cid:20) H T ( x, η, t ) + t M e H T ( x, η, t ) + Z t −∞ dt ′ M e H T ( x, η, t ′ ) (cid:21) , (4.65) e E ( x, η, t ) sph = ± (cid:20) E T ( x, η, t ) − η e E T ( x, η, t ) (cid:21) , (4.66)where the upper (lower) sign applies for the (pseudo-)scalar and longitudinal (axial-)vector diquarkmodels. Here, the transversity GPD H T , arising from L z = 0 LFWF overlaps, is in our modelsgiven as H T ≡ H T − t M e H T sph = ± Z dy Z − y − y dz δ ( x − y − ηz ) (cid:16) ± mM + y (cid:17) ˆΦ ( y, z, t ) . Moreover, the analog of the “hidden” quadratic constraint (3.33), connecting the four chiral oddGPDs, is not satisfied in general, however, it is easily to see that the analog equation is valid inthe DD representation. Hence, it can be employed to evaluate one transversity DD in terms ofthe three remaining chiral odd ones. In addition to this five constraints for a “spherical” GPDmodel, further model constraints (4.28–4.30, 4.36,4.36) can be formulated in terms of DDs. Inanalogy to a “spherical” uPDF model a “spherical” GPD model of rank-four might be obtained byadding a GPD H addenda, see analog discussion in Sec. 2.2.1, and so only the first relation (4.63)will alter. However, we emphasize that we do not have a LFWF overlap representation for sucha model at hand and that the linear combination of “spherical” scalar and axial-vector diquarkmodels, see Eq. (4.59), will alter the H and E GPD relations (4.63) and (4.64) as well. Utilizingthe constraints (4.63–4.66) and the prefactors in Tab. 2, one might write down the correspondingGPD relations for our minimal version of an axial-vector diquark model.Let us emphasize once more that in the scalar diquark model the analogy to the saturationof the Soffer bound (2.66) holds. Combining the model constraints (4.63,4.65), the transversityGPD H T is essentially given by the average of GPDs H and e H , H T ( x, η, t ) sca = 12 h H + e H − η e E T i ( x, η, t ) ⇒ H T ( x, η = 0 , t ) sca = 12 h H + e H i ( x, , t ) , (4.67)where in the non-forward case also a η proportional | L z | = 1 with | L z | = 0 LFWF overlap, givenin terms of the GPD e E T , appears. Since the corresponding DD (4.35) is odd in z , this chiral oddGPD e E T vanish already by itself in the zero-skewness case. Thus, we expect that for small ormoderate η values the e E T term in Eq. (4.67) is negligible.59ote also that the model relation (4.64) tells us that the chiral odd GPD E T is given by thechiral even E GPD E T ( x, η, t ) ≡ h E T + 2 e H T − η e E T i ( x, η, t ) sph = ± E ( x, η, t ) for (cid:26) scalarlon. axial-vector . (4.68)Since from the DDs (4.34) and (4.35) it follows that − e H T and e E T are positive (negative) in ourscalar (axial-vector) diquark model, we can also read off from the relation (4.68) that the absolutevalue of E T is even larger as the positive E , see corresponding DD (4.25): ± E T ( x, η, t ) sph > E ( x, η, t ) for (cid:26) scalarlon. axial-vector . (4.69)Analogously, we can derive from the relation (4.66) a model dependent inequality for the GPD e E , ± e E ( x, η, t ) sph > ± E T ( x, η, t ) sph > E ( x, η, t ) for (cid:26) scalarlon. axial-vector . (4.70)We emphasize that these inequalities (4.69) and (4.70) hold only as long as the scalar LFWFoverlap ˆΦ is positive definite for a given kinematical point. In Sec. 3.3 we explained that the appearance of orbital angular momentum implies that in QCDa one-to-one map of uPDFs and nfPDFs does not exist. In our scalar diquark model the quarkangular momentum is restricted to | L z | ≤ L z = 0 LFWF overlap, i.e., to the transversity nfPDF H T which is given in terms of uPDF h H T ( x, η = 0 , t ) sca = ZZ d ¯ k ⊥ q h ( x, ¯ k ⊥ − (1 − x ) ∆ ⊥ / q h ( x, ¯ k ⊥ + (1 − x ) ∆ ⊥ / , (4.71)where we use here the shifted variable ¯ k ⊥ = k ⊥ + (1 − x ) ∆ ⊥ /
2. We might also switch to diagonal L z = 0 LFWF overlap, i.e., we replace in (4.71) the nfPDF H T by H + e H and the uPDF h by f + g . The diagonal L z = 1 overlap in the ∆ L z = 0 sector is given by the difference H − e H and f − g , respectively, which in the scalar diquark model includes an extra k ⊥ factor. Hence, wefind now from the effective LFWF in impact parameter space, cf. Eqs. (2.58,2.60,3.58), the map h H − e H i ( x, η = 0 , t ) sca = ZZ d ¯ k ⊥ ¯ k ⊥ + (1 − x ) ∆ ⊥ p (¯ k ⊥ − (1 − x ) ∆ ⊥ / p (¯ k ⊥ + (1 − x ) ∆ ⊥ / (4.72) × q [ f − g ]( x, ¯ k ⊥ − (1 − x ) ∆ ⊥ / q [ f − g ]( x, ¯ k ⊥ + (1 − x ) ∆ ⊥ / .
60e also concluded in Sec. 3.3.1 that for a model, which is build from a superposition of effectivetwo-body LFWFs, the nfPDF/uPDF convolution formulae (4.71,4.72) do not apply. In our Reggeimproved two-body LFWF model the t -dependence of nfPDF arises from the asymmetric k ⊥ overlap (4.2) of two effective LFWFs, while the k ⊥ dependency of uPDF is simply given bythe unintegrated diagonal LFWF overlap. We like now to illustrate how k ⊥ , t , and skewnessdependence are talking to each other and we like to provide simple conversion formulae, whichare only approximately valid. Thereby, the missing information for the desired model dependentnfPDF/uPDF map is apart from the spin-spin coupling contained in the uDD (4.11).A generic scalar “uPDF” is generally obtained from the scalar “uDD” by integrating out the z -dependency, Φ ( x, k ⊥ ) = 2(1 − x ) Z dz ˆΦ ( x, (1 − x ) z, , k ⊥ ) , (4.73)where we used that the “uDD” for ∆ ⊥ = 0 is even in z . Relying on the functional form of the“uDD” (4.11), arising from the implementation of Lorentz invariance, the scalar “nfPDF” can bewritten as F ( x, η = 0 , t ) = 2 π (1 − x ) Z dz Z ∞− (1 − x ) (1 − z ) t/ d k ⊥ ˆΦ ( x, (1 − x ) z, , k ⊥ ) . (4.74)From this equation it also follows that the t -slope of the “nfPDF” provides us some certain z -moment of the “uDD”: Z dz (1 − z ) ˆΦ ( x, (1 − x ) z, , √ − z k ⊥ ) = 2 π (1 − x ) ddt F ( x, η = 0 , t ) (cid:12)(cid:12)(cid:12) t = − k ⊥ / (1 − x ) (4.75)If we employ the mean value theorem in Eqs. (4.73) and (4.75) and assume that the two meanvalues are approximately given by the common value z , we can state that the “nfPDF” and“uPDF” can be approximately mapped to each other: F ( x, η = 0 , t ) ≈ π Z ∞− (1 − x ) (1 − z ) t/ d k ⊥ Φ ( x, k ⊥ ) , (4.76) Φ ( x, k ⊥ ) ≈ π (1 − x ) (1 − z ) ddt F ( x, η = 0 , t ) (cid:12)(cid:12)(cid:12) t = − k ⊥ / (1 − x ) (1 − z ) . (4.77)These two conversion formulae become exact if the scalar “uDD” (4.11) is concentrated in somespecific z point, e.g., z = 0.In the case that our quantities contain a diagonal | L z | = 1 overlap, e.g., h ( y, z, t, k ⊥ ) − e h ( y, z, t, k ⊥ ) = (cid:2) (1 − y ) − z (cid:3) t M + k ⊥ M ! ˆΦ ( y, z, t, k ⊥ ) , h H − e H i ( x, η = 0 , t ) ≈ π Z ∞− (1 − x )2(1 − z t d k ⊥ (cid:18) − x ) (1 − z ) t k ⊥ (cid:19) [ f − g ] ( x, k ⊥ ) , (4.78)[ f − g ] ( x, k ⊥ ) ≈ k ⊥ π (1 − x ) (1 − z ) d dt F ( x, η = 0 , t ) (cid:12)(cid:12)(cid:12) t = − k ⊥ / (1 − x ) (1 − z ) . (4.79)We conclude that in a “spherical” model of rank-one/three an approximate (eventually anexact) uPDF/nfPDF mapping might be employed: H T ( x, η = 0 , t ) (4.76,4.77) ≈←→ (cid:16) ←− (cid:17) h ( x, k ⊥ ) , (4.80) H ( x, η = 0 , t ) + e H ( x, η = 0 , t ) (4.76,4.77) ←→ (cid:16) (4.71) ←− (cid:17) f ( x, k ⊥ ) + g ( x, k ⊥ ) (4.81) H ( x, η = 0 , t ) − e H ( x, η = 0 , t ) (4.78,4.79 ) ←→ (cid:16) (4.72) ←− (cid:17) f ( x, k ⊥ ) − g ( x, k ⊥ ) . (4.82)In a “spherical” rank-four model, i.e., adding a nfPDF H and the corresponding uPDF f , thesecond and third relations will be in general broken. In the case that the f addenda are (not)decorated with an additional k ⊥ factor, arising from a diagonal L z = 1 LFWF overlap, the (second)third relation holds still true. In Sec. 3.3.2 we pointed out that also model dependent relations (3.75) among the target helicityflip E and E T nfPDFs and twist-tree related g ⊥ and h ⊥ uPDFs exist. Using the uPDF and nfPDFrepresentations (2.60,2.62) and (4.25,4.64), respectively, one can easily verify that the relations(3.82) and (3.83) are valid, i.e., they hold true in a scalar diquark model, (cid:26) EE T (cid:27) ( x, η = 0 , t = 0) sca = (1 − x ) ZZ d k ⊥ (cid:26) g ⊥ − h ⊥ (cid:27) ( x, k ⊥ ) . (4.83)Thereby, we employed the fact that the forward GPD is given by both the k ⊥ integral of theuPDF and the z -integral of the DD at t = 0, Φ ( x ) = ZZ d k ⊥ Φ ( x, k ⊥ ) = Z − y − y dz ˆΦ ( x, z, t = 0) , (4.84)which simply follows from the DD definition in Sec. 4.1. From Tab. 2 we read off that the scalardiquark relation (4.83) among E and g ⊥ is modified in a minimal axial-vector diquark model bya factor − E T and h ⊥ still holds: (cid:26) EE T (cid:27) ( x, η = 0 , t = 0) av = (1 − x ) ZZ d k ⊥ (cid:26) − g ⊥ − h ⊥ (cid:27) ( x, k ⊥ ) . (4.85)62et us finally consider the model dependent sum rule (3.87) for the integrated “pretzelosity”distribution h ⊥ T ( x ) = Z Z d k ⊥ h ⊥ T ( x, k ⊥ ) sca = − Φ ( x ) , (4.86)cf. uPDF (2.61), which provides the GPD e H T (4.31,4.34) in the forward kinematics. As explainedin Sec. 3.3.3, in our effective two-body LFWF models different k ⊥ -weights enter in the definitionsof these functions and determine the proportionality factor in the sum rule (3.87). In Sec. 4.1we saw that the z -dependency of the DD originates from the functional form of the LFWF andso we might expect that the relation among h ⊥ T and e H T depends finally on the skewness effect.Indeed, plugging the DD representation (4.84) into Eq. (4.86), we find the following formula forthe “pretzelosity” distribution h ⊥ T ( x ) sca = − − x ) Z − dw ˆΦ ( x, (1 − x ) w, t = 0) , (4.87)while the DD e h T contains an additional factor (1 − y ) (1 − w ), see Eq. (4.34), which tells us thatthe GPD e H T in forward kinematics is given by e H T ( x, η = 0 , t = 0) sca = − (1 − x ) Z − dw (1 − w ) ˆΦ ( x, (1 − x ) w, t = 0) (4.88)arises from a different w -moment. At t = 0 the y and w = z/ (1 − y ) dependency factorizes in ˆΦ , see below Eq. (5.19) in Sec. 5.2, and so we can also write the model relation among thesedistributions as e H T ( x, η = 0 , t = 0) sca = R − dw (1 − w )Π( w )2 R − dw Π( w ) (1 − x ) h ⊥ T ( x ) , (4.89)where the proportionality factor is determined by the specific w -moment of a so-called profilefunction Π( w ). The profile function of the model [60] is just a constant. For such a choice we findon the r.h.s. of Eq. (4.89) a proportionality factor 1 /
3, which confirms the very specific relationof Ref. [60], given there in Eq. (110). We add that the model dependent relation (4.89) holds truein all our diquark models, cf. Tab. 2.
In the previous sections (2.3) and (4.3) we constructed (u)PDFs and GPDs from LFWF “spinors”,giving emphasize to the scalar diquark “spinor” (4.22) [or its components (2.53,2.54)]. We likenow to confront such models with experimental/phenomenological results and expectations from63attice simulations. In Sec. 5.1 we set up two concrete models from a power-likely and exponen-tially k ⊥ -dependent effective two-body LFWF. In Sec. 5.2 we provide some general insights intothe resulting GPD models. In Sec. 5.3.1 we compare then the scalar diquark model predictions inthe flavor sector (2 u − d ) / x . Choosing an appropriate LFWF “spinor”, we can consistently build GPDs, uPDFs, and PDFs interms of a scalar “DD” ˆΦ ( y, z, t ), “uPDF” ˆΦ ( x, k ⊥ ), and “PDF” ˆΦ ( x ), respectively. The “DD”and “(u)PDF” are obtained from a “uDD” ˆΦ ( y, z, t, k ⊥ ), which is given as overlap of an effectivetwo-body LFWF φ . Our representation for this scalar LFWF arises from Eqs. (2.56,4.5,4.9,4.17), φ ( X, k ⊥ | λ ) = θ ( λ − λ c ) s ( λ − λ c ) α − Γ( α ) M α φ ( X, k ⊥ | λ ) , (5.1) φ ( X, k ⊥ | λ ) = Z ∞ d ¯ α ϕ ( ¯ α ) exp (cid:26) − ¯ α k ⊥ + m (1 − X ) + Xλ − X (1 − X ) M (1 − X ) M (cid:27) , (5.2)and it guarantees that both Regge behavior and Lorentz symmetry are implemented. To evaluatethe “uDD”, we utilize the Regge improved master formula (4.18). Specifying the reduced Laplacetransform ϕ ( ¯ α ), we utilize in the following Sec. 5.1.1 and Sec. 5.1.2 a power-like and an exponential k ⊥ -dependent LFWF, respectively. ⊥ -dependent LFWF The ansatz (5.2) within the reduced Laplace transform ϕ ( α ) = g ¯ α p + α/ M Γ(1 + p + α/
2) (5.3)provides a power-likely scalar LFWF φ ( X, k ⊥ | λ ) = gM (cid:20) k ⊥ + m (1 − X ) + Xλ − X (1 − X ) M (1 − X ) M (cid:21) − − p − α/ , (5.4)which is nothing but the generalized LFWF from the Yukawa theory ( α = p = 0) where g isthe coupling [48]. For p = 1 and α = 0 we have the model that we utilized in Ref. [39]. The A ˆΦ ( y, z, t, k ⊥ ) = N (2 p + 1) πM y − α ((1 − y ) − z ) p + α/ h (1 − y ) m M + y λ M − y (1 − y ) − ((1 − y ) − z ) t M + k ⊥ M i p +2 , (5.5)where k ⊥ = k ⊥ − (1 − y + z ) ∆ ⊥ / g : N = πg Γ(2 p + 1)2 p +1+ α Γ ( p + 1 + α/ . The k ⊥ -integration of this “uDD” (5.5) leads to the “DD” ˆΦ ( y, z, t ) = N y − α ((1 − y ) − z ) p + α/ (cid:2) (1 − y ) m M + y λ M − y (1 − y ) − ((1 − y ) − z ) t M (cid:3) p +1 , (5.6)which provides with the representations (4.23–4.27) and (4.31–4.35) the chiral even and odd GPDs,respectively. The k ⊥ -moment (4.13) of the “uDD” might be alternatively obtained by t -integration: ˆΦ (2) ( y, z, t ) = 12 p (cid:20) (1 − y ) m M + y λ M − y (1 − y ) − ((1 − y ) − z ) t M (cid:21) ˆΦ ( y, z, t ) . (5.7)Integrating over z , we find from the “uDD” (5.5) at t = 0 the “uPDF”, Φ ( x, k ⊥ ) = g Γ(2 + 2 p ) M Γ(2 + 2 p + α ) x − α (1 − x ) p +1+ α h (1 − x ) m M + x λ M − x (1 − x ) + k ⊥ M i p +2 , (5.8)entering in all uPDFs (2.58,2.60–2.62). From the “DDs” (5.6) and (5.7) we can now obtain the“PDF” (2.67) and the corresponding k ⊥ -moment, Φ ( x ) = πg Γ(1 + 2 p )Γ(2 + 2 p + α ) x − α (1 − x ) p +1+ α (cid:2) (1 − x ) m M + x λ M − x (1 − x ) (cid:3) p +1 , (5.9) Φ (2) ( x ) = h k ⊥ i ( x ) M Φ ( x ) , h k ⊥ i ( x ) = 12 p (cid:2) (1 − x ) m + xλ − x (1 − x ) M (cid:3) , (5.10)entering in all PDFs (2.69–2.71). Here we used the definition (2.68) to find the expression for h k ⊥ i . Alternatively, these k ⊥ –integrated results can be also directly derived from the “uPDF”(5.8). ⊥ -dependent LFWF Let us now illustrate that one can also utilize LFWFs with exponential k ⊥ -dependence. To do sowe take the reduced Laplace transform ϕ ( α ) = gM δ ( α − ¯ A ) , (5.11)65here Eq. (5.2) provides us the scalar LFWF φ ( X, k ⊥ | λ ) = gM exp (cid:26) − ¯ A k ⊥ + m (1 − X ) + Xλ − X (1 − X ) M (1 − X ) M (cid:27) . (5.12)Hence, the “uDD” (4.18) is concentrated in z = 0 and the A integration in this function isconstrained by A = ¯ A/ (1 − y ), which finally yields ˆΦ ( y, z, t, k ⊥ ) = g M (cid:18) Ay − y (cid:19) − α δ ( z )1 − y (5.13) × exp ( − ¯ A − y " (1 − y ) m M + y λ M − y (1 − y ) − (1 − y ) t M + k ⊥ M with k ⊥ = k ⊥ − (1 − y ) ∆ ⊥ /
2. As we realize from the Radon transforms (4.23), with our rather ex-treme ansatz (5.11) we lose any skewness dependence in GPDs. Obviously, an exponential LFWFthat provides a skewness depended DDs can be obtained from a smeared Dirac δ -function. Hence,polynomiality constraints and exponential k ⊥ fall-off do not contradict each other. Integrating inEq. (5.13) over k ⊥ provides immediately “DDs” that have an exponential t -dependence, ˆΦ ( y, z, t ) = πg ¯ A (cid:18) Ay − y (cid:19) − α δ ( z ) (5.14) × exp (cid:26) − ¯ A − y (cid:20) (1 − y ) m M + y λ M − y (1 − y ) − (1 − y ) t M (cid:21)(cid:27) , ˆΦ (2) ( y, z, t ) = 1 − y ¯ A ˆΦ ( y, z, t ) . (5.15)Integrating Eq. (5.13) over z , we trivially find in the forward case the “uPDF” Φ ( x, k ⊥ ) = g M (cid:18) Ax − x (cid:19) − α − x exp ( − ¯ A − x " (1 − x ) m M + x λ M − x (1 − x ) + k ⊥ M , (5.16)Analogously, from Eqs. (5.14,5.15) the “PDFs” can be obtained Φ ( x ) = πg ¯ A (cid:18) Ax − x (cid:19) − α exp (cid:26) − ¯ A − x (cid:20) (1 − x ) m M + x λ M − x (1 − x ) (cid:21)(cid:27) , (5.17) Φ (2) ( x ) = h k ⊥ i ( x ) M Φ ( x ) , h k ⊥ i ( x ) = (1 − x ) M ¯ A . (5.18)As described in Sec. 5.1.1, from the given “DD” and “(u)PDFs” expressions one can easilyfind the corresponding chiral even (4.23–4.27) and chiral odd (4.31–4.35) twist-two GPDs, leading–power uPDFs (2.58,2.60–2.62), and twist-two PDFs (2.69–2.71). According to Tab. 2 we can thenset up scalar, pseudo-scalar, minimal axial-vector, and minimal vector models.66 .2 Some GPD model insights
It is rather popular to build GPD models by utilizing Radyushkin’s DD ansatz (RDDA) [104]that has been given for t = 0, e.g., utilized in Refs. [117, 118], used in the so-called VGG code[119], and in the Goloskokov/Kroll model [120, 121]. Since this ansatz was inspired from a trianglediagram, too, it is not surprising that our scalar “DDs”, e.g., Eq. (5.6), reduces at t = 0 to RDDA.This ansatz is written as product of the “PDF” (2.67) and a normalized profile function Π, ˆΦ ( y, z, t = 0) = Φ ( y )1 − y Π (cid:18) z − y (cid:19) , (5.19)Π( w ) = Γ( b + 3 / √ π Γ( b + 1) (1 − w ) b , where Z − dw Π( w ) = 1 . Thereby, the additional factor 1 / (1 − y ) in RDDA drops out if we change to the new integrationvariable w = z/ (1 − y ). In our powerlike LFWF model (5.6) the parameter b = p + α/ p and α , while the δ ( z ) function in our exponential LFWF model (5.14) arises now from thelimit b → ∞ . The “PDFs” for these models are given in Eq. (5.9) and (5.17), respectively.For t = 0 the factorization of y - and w -dependency is broken for power-like LFWF model, seeEq. (5.6), however, we might write the “DD” in analogy to RDDA as ˆΦ ( y, z, t ) = Φ ( y )1 − y Π( z/ (1 − y ) | y, t ) , (5.20)Π( w | y, t ) = Γ( b + 3 / √ π Γ( b + 1) (1 − w ) b h − (1 − w ) tM ( y ) i p +1 , b = p + α/ . (5.21)Here, the t -dependency is included in the profile function and is it governed by a y -dependentcut-off mass M ( y ) = m + yλ − y (1 − y ) M (1 − y ) . (5.22)Note, however, that our DDs h and e h (4.24,4.26) are more intricate and additional t -dependentterms appear. Even for t = 0, h and e h models will consist of two factorizable profile functions,which arise from a diagonal L z = 0 and L z = 1 LFWF overlaps, cf. Eqs. (4.24) and (4.26). Counting rules for the large − t behavior of form factors [122, 123] or the large x behavior ofPDFs [123, 81] are a useful guidance for model builders. They predict a power-like behavior in thecorresponding asymptotic, and, thus, our power-like LFWF model, set up in Sec. 5.1.1, is morefavorable than the exponential LFWF model, given in Sec. 5.1.2. The power-like LFWF model67ith p = 1 and α = 0 was studied in Ref. [39], and we found that the following power behaviorfor PDF f and elastic form factors at large x and − t , respectively,lim x → f ( x ) ∼ (1 − x ) , lim − t →∞ F ( t ) ∼ ( − t ) − , lim − t →∞ F ( t ) ∼ ( − t ) − , generically expected from counting rules [122, 123], holds only approximately true for F and evenfails for F . Thus, the inclusive-exclusive relationlim x → f ( x ) ∼ (1 − x ) n − ⇐⇒ lim − t →∞ F ( t ) ∼ ( − t ) − n (5.23)of Drell-Yan [43], also obtained by West [44], is only to some extent respected. It is worthy tomention that Drell and Yan employed LFWF overlap representations, general arguments, however,also some “empirical” findings to get rid of some contributions, see Ref. [43], to conjecture theinclusive-exclusive relation (5.23) between unpolarized deep inelastic scattering structure function(taken here as PDF combination) and Dirac form factor. Utilizing our LFWF models, we willnow have a closer look at the inclusive-exclusive relation, which reveals that the conjecture (5.23)is indeed based on specific assumptions.In our power-like LFWF model the large x behavior of PDF f is inherited from the “PDF”(5.9), see f definition (2.69) and, consequently, it is given by (1 − x ) p +1+ α . The large t -behaviorof the Dirac form factor can be calculated from the lowest x moment of GPD H (4.23,4.24), wherewe can set η = 0. As one realizes the leading − t behavior stems from the diagonal L z = 1 LFWFoverlaps, and reads in terms of the ansatz (5.20–5.22) as F ( t ) − t ≫ ∝ Z dx Φ ( x ) Z − dw (1 − w ) p + α/ h − (1 − w ) tM ( x ) i p +1 (cid:20) · · · + · · · (1 − w ) tM ( x ) (cid:21) , (5.24)where the omissions denote some x -dependent terms. Some care is needed to find the large − t asymptotics. The integral over w provides us a Hypergeometric functions and their expansion inthe vicinity of − t = ∞ gives us after integration over x the following asymptotic behaviorlim t →−∞ F ( t ) ∼ · · · ( − t ) − p − α/ − (1 + O (1 /t )) + · · · ( − t ) − p (1 + O (1 /t )) . The first term on the r.h.s. reflects the end-point behavior | w | ∼ | w | <
1. If we take 1 < α/ < p , the first term,proportional to ( − t ) − p − α/ − , is the leading one and we recover with n = p + α/ > n − p + α + 1 > n = 2, obtained from (dimensional) counting rules. Choosing the parameter p < α/ − t ) − p , governs the form factor asymptotics and the inclusive-exclusive relation turns with n = 2 p intolim x → f ( x ) ∼ (1 − x ) n +1+ α ⇐⇒ lim − t →∞ F ( t ) ∼ ( − t ) − n . (5.25)For p = α/ − t ) − p − α/ − behavior might be accompanied by a logarithmical modification, too, see alsothe numerical discussion for the α = 0 case in Ref. [39].For the exponential LFWF model the “PDF” (5.17) and the momentum faction integrated“DD” (5.14) are exponentially suppressed in the large x region and − t region, respectively. Apartfrom power-like modifications of these exponential suppressions, we can state our exponentialLFWF model possesses the following inclusive-exclusive relationlim x → f ( x ) ∼ e − ¯ A/ (1 − x ) ⇐⇒ lim − t →∞ F ( t ) ∼ e − ¯ A (1 − ¯ x ) t M , (5.26)where ¯ x denotes the mean value.Let us spell out that Drell and Yan took it for granted that the unpolarized deep inelasticscattering structure function possesses a power-like behavior. In the covariant wave functionmodel study of West [44] an exponential scenario was considered, too, however, with a differentmodel assumption, lim x → f ( x ) ∼ e − a ′ / √ − x ⇐⇒ lim − t →∞ F ( t ) ∼ e − a √− t , compare with our model findings (5.26). We are not aware that the power-like relation (5.25),contradicting the inclusive-exclusive conjecture (5.23), was mentioned somewhere else. It arisesin the DD representation from the dominance of the region | w | <
1. A direct link of our DDconsiderations and those of Drell and Yan or West might be hardly visible, however, we notethat introducing a ultraviolet cut-off, as assumed in Ref. [43] to get rid of contributions from thevalence region 0 < c < − x <
1, will certainly also weaker the | w | < / ( − t ).Let us conclude that even in a simple diquark model the intricate interplay of Regge, ultra-violet, large − t , and large x behavior shows up. The underlying Lorentz symmetry, which weimplemented in our LFWF and West in his covariant model tells us that the functional form ofa valence PDF at large x and of the Dirac form factor at large − t should match each other. Wealso found a simple counter example for which the Drell-Yan conjecture (5.23) does not hold true,even for a certain power-like LFWF. 69 .3 Models versus phenomenology The models presented above are suitable to describe the scalar diquark content of the nucleon,which is according to SU(6) symmetry related to the flavor combination 2 u/ − d/
3. We willconsider the valence quark sector, where we use for PDFs and GPDs the following notation q sca = 23 q u val − q d val , q ∈ { f , g , h } , (5.27) F sca = 23 F u val − F d val , F ∈ { H, E, e H, e E, H T , E T , e H T , e E T } . (5.28)In the first place we do not consider our model as a constituent quark model, given at an intrinsiclow scale of few hundred MeVs or so, rather as an effective one that matches partonic and effectivequark degrees of freedom. Hence, we might it even employ at a larger input scale, e.g., µ = 4 GeV .This scale setting, which is quite convenient to compare with phenomenological and Lattice results,avoids also the problem that perturbative QCD evolution is utilized at very low scales, where theapplication of perturbation theory might be questionable.Certainly, various aspects of the scalar diquark model has been widely studied in the literature,however, we are not aware of a critical and consistent analysis versus the full set of available data.Utilizing the formulae of Sec. 5.1.1 and Sec. 5.1.2, we are now in the position to confront ourRegge improved diquark models with phenomenological findings for PDFs, (generalized) formfactors, GPDs, and to some extend with uPDFs. After fixing the proton mass M = 0 .
938 GeV,and the normalization (4.39), i.e., n = 1, our two models depend on four parameters α, λ, m , and p or A . To fix these parameters, we require a good description of the valence-like unpolarizedPDF f sca1 . Parameterizations from global fits incorporate a α ∼ . ρ/ω trajectory. We take here Alekhin‘s parameterization asa phenomenological reference PDF at µ = 4 GeV , which has build in α = 0 .
47. Moreover, thisPDF possesses the averaged momentum fraction: h x i sca ( µ = 4 GeV ) ≡ Z dx x f sca1 ( x, µ = 4 GeV ) ≈ . . (5.29)To match the x -shape with more precision, we employ also the second x -moment, h x i sca ( µ = 4 GeV ) ≡ Z dx x f sca1 ( x, µ = 4 GeV ) ≈ . , (5.30)and the PDF value at x = 0 . f sca1 ( x = 0 . , µ = 4 GeV ) ≈ . . (5.31)70e also require that the anomalous magnetic moment is well described. It is given by the zerothmoment of E sca GPD, h κ i sca ≡ Z dx E sca ( x, η = 0 , t = 0) = 23 κ u − κ d = κ p ≈ . , (5.32)and it can be directly expressed by the proton anomalous magnetic moment. Furthermore, weutilize the averaged longitudinal quark spin, h s i sca ( µ ) = 12 Z dx g sca1 ( x, µ ) , (5.33)which, however, is not well constrained from phenomenology nor from Lattice simulations. Wemight take some value that lies between a fully-broken ‘valence’ ( ≈ .
19) and a standard flavorsymmetric sea ( ≈ .
37) scenario [125]:0 . . h s i sca . . , (5.34)while the NLO analysis of Ref. [124] provides a value h s i sca ( µ = 4 GeV ) ≈ . h κ i sca = 1 .
79 an anti-correlation between matching the x -shape of unpolarized PDFs and the h s i sca value, namely,a better shape matching requires a lower h s i sca value. Here we present a result in which the shapeof the unpolarized PDF is fairly matched, see dash-dotted curve in the right upper panel of Fig. 1:LFWF pow : α = 0 . , p = 0 . , m = 0 .
415 GeV , λ = 0 .
858 GeV , (5.35)LFWF exp : α = 0 . , A = 4 . , m = 0 .
404 GeV , λ = 0 .
880 GeV . (5.36)Mostly independent on the shape of the reduced LFWF our models provide h x i sca ≈ . h s i sca ≈ .
13, and h κ i sca ≈ .
79. Compared to our spectator diquark model [39] (dotted curve),
HM07 : α = 0 , p = 1 , m = 0 .
45 GeV , λ = 0 .
75 GeV , (5.37)which was “eye fitted” to the t -dependence of the Dirac form factor and κ p , the p parameter (5.35)has a rather low value, where the quark and spectator masses are compatible with our new ones(5.35) and (5.36).In the upper panels of Fig. 1 we illustrate that the shape and normalization of the PDFsin the Regge improved models are now much more realistic than in the pure spectator model(dotted), where the exponential LFWF (dashed) is more pronounced in the valence region andexponentially suppressed in the large x region, cf. Eq. (5.17). As alluded above, this is exemplified In deep inelastic scattering (DIS) only the sum of polarized valence and sea quark PDFs can be accessed; aseparation of them is possible by taking into account semi-inclusive DIS data [124]. f x x q H x , Μ = G e V L x g x x q H x , Μ = G e V L x h x x q H x , Μ = G e V L x E sca x x F H x , Η = , t = , Μ = G e V L x E Ž sca x x q H x , Μ = G e V L powerexpon.HM07 x E Tsca x x q H x , Μ = G e V L Figure 1:
Our Regge improved power-like (dash-dotted), exponential (dashed) LFWFs, and
HM07 (dotted) [39] models are confronted with unpolarized (left), polarized (middle), and transversity (right)valence quark PDF combinations (5.27) from leading order fits [126], [127], and [128], shown at Q =4 GeV as grayed areas (solid line is from the polarized DIS fit [129]). In the lower panels the nfPDFs xE , x e E , xE T at t = 0 are presented from left to right. in the left panel for the unpolarized PDF f sca1 ( x ) = H sca ( x, η = 0 , t = 0) where the power-likeansatz (dash-dotted) provides the best matching with a phenomenological PDF (grayed errorband). Our polarized PDF g sca1 ( x ) = e H ( x, η = 0 , t = 0) possesses not the steep small- x behaviorthat is commonly included in standard PDF parameterizations, cf. middle panel in the upper rowof Fig. 1. Note that we confronted our valence model with global DIS fits [129, 127], which do notallow discriminating between valence and sea quarks. The global DSSV fit [124], which includesalso semi-inclusive DIS and polarized proton-proton collision data and so it allows to disentanglethe different quark contributions, lies somewhere between the Gehrmann/Stirling (solid curve)and the Bl¨umlein/Bottcher (grayed area) parameterization. We emphasize that in our model theRegge behavior is in the parity odd sector inherited from the parity even sector ( ρ/ω exchanges),while on the phenomenological side there is no clear understanding, even not for on-shell highenergy scattering processes. In the upper right panel we show that the model prediction fortransversity h sca1 ( x ) = H scaT ( x, η = 0 , t = 0), saturating the Soffer bound, is compatible with thoseextracted from semi-inclusive measurements [130, 128].In the lower panels of Fig. 1 we display the resulting model predictions for other nfPDFs at t = 0 that decouple in the forward limit, i.e., they drop out in the form factor decompositionof matrix elements (3.3–3.5). The partonic anomalous magnetic moment h κ i sca = 1 .
79 matchesthe measurements of the nucleon anomalous magnetic moments and our nfPDFs E sca , having72o nodes, is even more sizeable than the unpolarized PDF, shown in the upper row (left). The x e E ( x, η = 0 , t = 0) nfPDF (4.44) is shown in the lower middle panel, which has a relative smallcontribution and a x − α +1 behavior at small x . The chiral odd nfPDF e H T (not displayed) decouplesin the forward limit, too, however, apart from the sign and a larger suppression at large x , seeDD (4.34), its size is compatible with the unpolarized PDF. Hence, the nfPDF E scaT at t = 0,cf. (4.69), is even larger as the nfPDF E (right). In the zero-skewness case the chiral odd nfPDF E scaT is given by E sca , cf. the constraint (4.68), and e E scaT vanishes.Let us now have a closer look to our model predictions for the quark orbital angular momentum,which plays a central role in the so-called nucleon spin puzzle. Unfortunately, its definitionis controversially discussed, namely, one can define it in terms of an interaction independentoperator [131], which contains a partial derivative, or in a gauge invariant manner by using thecovariant derivative [132]. In a quark model, where the gauge field is absent, both definitionscoincide with each other. To calculate the projection of the quark orbital angular momentumon the z -axis, one might either use the definition of the quark angular momentum operator inits overlap representation, which is expressed by the (relative) orbital angular momentum of theproton [133, 48] that is weighted with the momentum fraction 1 − x of the spectator [134]: hLi sca sca = Z dx (1 − x ) ZZ d k ⊥ Z dλ | ψ ⇒← ( x, k ⊥ , λ ) | = Z dx − x f sca1 ( x ) − g sca1 ( x )] , (5.38)or Ji‘s sum rule [132], h L i EOM = h J i − h s i , h J i = 12 h x i + 12 Z dx xE ( x, η = 0 , t = 0) . (5.39)Note that equations-of-motion (EOM) play an essential role in the Field theoretical derivationof the angular momentum operator and that LFWFs enter into both orbital angular momentumdefinitions (5.38) and (5.39) in different combinations. Therefore, in Yukawa theory these bothformulae provide the same value, i.e., h L i = hLi [134] . But in a quark model the various LFWFsare not related by equations-of-motion and so one might find two different values for the quarkorbital angular momentum, i.e., h L i 6 = hLi .Our spin average for the Regge improved LFWF models, h s i sca ≈ . . . for LFWF pow
LFWF exp
HM07 , (5.40)is compared to standard PDF parameterizations (5.34) rather low, see also the upper panel in themiddle of Fig. 1. Hence, in these both models the quark orbital angular momentum, evaluated Choosing improper renormalization conditions, one might also render an inequality in Yukawa theory. .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.8 x ` k T p H x L x ` k T p H x L (cid:144) H m + x M L pheno.powerexpon.HM07 Figure 2:
Transverse momentum square average h k ⊥ ( x ) i , defined in Eq. (2.68), (left) and the ratio h k ⊥ ( x ) i / ( m + xM ) (right) versus the momentum fraction x for a power-like LFWF (dash-dotted),exponential LFWF (dashed), and the HM07 (dotted) model. The grayed area (solid curve) shows thephenomenological findings, evaluated from Eq. (2.72), for Alekhin‘s [126] and Bl¨umlein/Bottcher [127](Gehrmann/Stirling [129]) parameterizations. from Ji‘s sum rule (5.39), is a relative small positive value, h L i sca = . . − . , h J i sca = . . . for LFWF pow
LFWF exp
HM07 , (5.41)while the HM07 model has a large longitudinal spin component and a small negative orbitalangular momentum. Utilizing Ji‘s sum rule in lattice simulations, one finds that the orbitalangular momentum of sea and valence quarks is estimated to be L u ∼ − L d ∼ − .
15 [84], whichprovides in the scalar sector again a negative quark orbital angular momentum of L sca ∼ − . J sca ∼ .
15. The direct calculation of the quarkorbital angular momentum (5.38) provides us much larger numbers hLi sca = . . . , hJ i sca = . . . for LFWF pow
LFWF exp
HM07 , (5.42)where the quark angular momentum hJ i sca = h s i sca + hLi sca has now for all three models (almost)the same large value. We conclude that a quark model calculation, where usually the dynamicsof LFWFs is ignored, can provide two values, which might largely differ from each other, for boththe orbital and total quark angular momentum.The h k ⊥ i ( x ) average (2.68) possesses in our two LFWF models rather different behavior, seeleft panel in Fig. 2. As one easily realizes from Eq. (5.10), for our powerlike LFWF (dash-dotted and dotted curves) h k ⊥ i ( x ) takes at the endpoints x = 0 and x = 1 the values m / p λ / p , respectively, where a non-vanishing minima might exist in the range x ∈ [0 ,
1] for( M − m ) < λ < m + M : h k ⊥ i min = 12 p [ λ − ( M − m ) ][( M + m ) − λ ]4 M , x min = m + M − λ M . For the exponential LFWF (dashed) the k ⊥ -average (5.18) takes at x = 0 the value M/A ,monotonously decreases with growing x , and vanishes at x = 0. In the HM07 model we find arather low h k ⊥ i ∼ . GeV in the small x and valence region, which slightly increases to 0 . GeV in the large x region. Both of our Regge improved models provides a larger h k ⊥ i ∼ . GeV outside the large x region and reaching at x → ∼ . GeV and zero for the powerlikeand exponential LFWF, respectively. In the right panel we show the quantity h k ⊥ i ( x ) / ( m + xM ) ,which might be expressed by the PDF ratio (2.72). The grayed area displays the phenomenologicalfindings, evaluated from Alekhin’s [126] and Bl¨umlein/Bottcher [127] parameterizations. Clearly,for 0 . . x the polarized PDFs form Bl¨umlein/Bottcher overwhelm the unpolarized PDFs, whichyields a negative h k ⊥ i ( x ) value. On the other hand, if we take the Gehrmann/Stirling [129]parameterization we find a rather reasonable value for the ratio (2.72), shown as solid curve. Asimilar curve, which is not displayed, we obtain with the DSSV parameterization [124]. Surely, thephenomenological uncertainties are so large that a serious confrontation with the scalar diquarkpicture is impossible.Elastic form factors and GFFs are presented in Fig. 3. Here, for our powerlike LFWF modelwe took into account α ′ = 0 .
75 GeV , which compensates the relative small p value. Hence, in thismodel (dash-dotted) we have almost the same − t dependence as in the HM07 model (dotted),which slightly under- and overestimate the experimental data (circles) for the Pauli (upper, left)and Dirac (down, left) form factor, respectively. For the exponential LFWF model (dashed) wefind that a positive value of α ′ cannot cure the larger discrepancy to experimental data. Comparingthe thick and thin curves in the upper right and lower panels, it is clearly to realize that the t -dependence of GFFs is dying out with increasing n (thin). This simply reflects the fact that inthe zero skewness case the t -dependence is accompanied by a factor (1 − x ) , cf. Eqs. (5.6,5.7) and(5.15,5.15), or enters via x − α ( t ) .Our powerlike LFWF model is also supported by lattice estimates, which give a positive largeratio (upper, right panel) H , ( t ) /H , ( t ) = d dη Z − dx x H ( x, η, t ) (cid:12)(cid:12)(cid:12) η =0 , Z − dx x H ( x, η = 0 , t ) , which quantifies the skewness effect. The sizeable ratio, displayed in the upper right panel bythe dash-dotted and dotted curves, are easily understood from the explicit formulae in the DD In lattice nomenclature this ratio is given by 4 A , /A , . owerexpon.HM07 èèèèèèèèèèèèè è èè è è è è è è è è è Ÿ ŸŸŸ ŸŸŸŸ ŸŸ ŸŸŸ ŸŸŸ ì ììì ìììì ìì ìì ììì - t @ GeV D H n0 s c a H t , Μ = G e V L ô ô ô ô - t @ GeV D H s c a (cid:144) H s c a H t , Μ = G e V L è experiment Ÿ n = H ´ L ì n = H ´ L èèèèèèèèèèèèè è èè è è è è è è è è è ŸŸŸ ŸŸŸŸ ŸŸ ŸŸŸ ŸŸŸ ìì ììì ìì ììì ììì - t @ GeV D E n0 s c a H t , Μ = G e V L Ÿ n = H ´ L ò n = H ´ L ŸŸ ŸŸ Ÿ Ÿ Ÿ ŸŸ Ÿ Ÿ òò òò ò ò ò òò ò ò - t @ GeV D E T , n0 s c a H t , Μ = G e V L Figure 3:
Our Regge improved power-like (dash-dotted), exponential (dashed) LFWFs, and
HM07 (dotted) [39] models are confronted versus experimental data [135] and Lattice estimates for the flavorcombination 2 u val / − d val /
3. The upper [lower] left panel displays the Dirac (Paule) FF combination F [ F ] from experiment (circle) [135], H [ E ] (squares,thick curves) and GFF H [ H ] (diamonds, thincurves ×
5) from the data set VI ( m π = 352 . H /H with m π = 292 . ×
2) moments of zero-skewness GPD E T versus Lattice measurementfrom the heavy pion world m π = 600 MeV [138]. representation. Roughly spoken, in the DD the moments H , and H , are multiplied with y and z , respectively, where y provides a larger suppression effect. Our exponential LFWF modelpossesses a zero-skewness effect and, therefore, this ratio is zero. The GFF H , = 4 C has beenmeasured in Lattice calculations, too, and it was found that its value is small in the iso-vectorsector [136]. Our powerlike LFWF models provide also a small value H , ( t = 0) = . . . for LFWF pow
LFWF exp
HM07 , (5.43)qualitative consistent with the statement from the chiral quark soliton model that the valencecontributions are positive and (relatively) small [139]. Transversity GFFs have been also studiedin lattice measurements [138]. The E T = E T + 2 e H T GPD, given in our model by E GPD, cf. Eqs.(4.36), which is qualitative consistent with lattice data from the heavy pion world, see the lower76 H sca X x F H x , x , t , Μ = G e V L x H Ž sca X x F H x , x , t , Μ = G e V L x H Tsca X x F H x , x , t , Μ = G e V L - x H Ž Tsca X x F H x , x , t , Μ = G e V L x E sca X x F H x , x , t , Μ = G e V L x E Ž sca X x F H x , x , t , Μ = G e V L powerexpon.HM07 x2 E Tsca X x F H x , x , t , Μ = G e V L x E Ž Tsca X x F H x , x , t , Μ = G e V L Figure 4:
Our Regge improved power-like (dash-dotted), exponential (dashed) LFWFs, and
HM07 (dotted) [39] models are displayed for valence quark GPDs F sca = 2 F u val / − F d val / η = x and t = − . versus X = 2 x/ (1 + x ). Upper [lower] panels from left to right: xH [ xE ] , e H [ x e E ], H T [ xE T / − e H T [ x e E T ]. right panel.Let us now consider GPD predictions, relevant for the phenomenology of hard exclusive pro-cesses. In a leading order (LO) description of such processes the imaginary part of the amplitudeis proportional to the GPDs on the cross-over line η = x , while the real part can be obtainedfrom a (subtracted) ”dispersion relation”, see Eqs. (4.46,4.51,4.52). In Fig. 4 we illustrate that allGPDs might be sizeable, where the pronounced one is the chiral odd GPD E sca T , which we rescaledby a factor x/ x , while the polarized GPD e H sca and the chiral odd GPD e E scaT are lesspronounced. We note that transversity contribution are utilized to describe the hard exclusive π + and π electroproduction in a handbag model analysis [140, 141, 142]. The GPD x e E sca is a smallquantity, however, it is clearly different from zero.A possibility to quantity the skewness effect of H , e H , and H T GPDs is provided by the ratios r ( x ) = lim t → H ( x, x, t ) f ( x ) , e r ( x ) = lim t → e H ( x, x, t ) g ( x ) , r T ( x ) = lim t → H T ( x, x, t ) h ( x ) , (5.44)which are in principle accessible from experimental data. Phenomenologically, it was found fromdeeply virtual Compton scattering measurements that at small x the r ratio is compatible withone [143]. As shown in the left panels of Fig. 5, this feature is implemented in all our models.Thereby, the e r ratio for the Regge improved power-like LFWF model (dash-dotted) has a singularbehavior in the vicinity of X ∼ . x ∼ . e H ( x, x,
0) and e H ( x, ,
0) that are slightly shifted. These nodes (sign changes) are caused by the competition of To make contact with the kinematics in experimental measurements, we show here the ratios versus themomentum fraction X = 2 x/ (1 + x ), which can be equated to the Bjorken variable x Bj , rather x . owerexpon.HM07 X H s c a H x , x , L (cid:144) H s c a H x , , L < X = < X = - t @ GeV D H s c a H x , x , t L (cid:144) H s c a H x , x , L - t @ GeV D H s c a H x , , t L (cid:144) H s c a H x , , L X H Ž s c a H x , x , L (cid:144) H Ž s c a H x , , L - t @ GeV D H Ž s c a H x , x , t L (cid:144) H Ž s c a H x , x , L - t @ GeV D H Ž s c a H x , , t L (cid:144) H Ž s c a H x , , L X H T s c a H x , x , L (cid:144) H T s c a H x , , L - t @ GeV D H T s c a H x , x , t L (cid:144) H T s c a H x , x , L - t @ GeV D H T s c a H x , , t L (cid:144) H T s c a H x , , L Figure 5:
Ratios of our Regge improved power-like (dash-dotted), exponential (dashed) LFWFs, and
HM07 (dotted) [39] models are displayed for valence quark GPD combinations F sca = 2 F u val / − F d val / F = { H sca (top) , e H sca (middle) , H sca T (bottom } : r ratios (5.44) versus X = 2 x/ (1 + x ) (left), F sca ( x, η = x, t ) /F sca ( x, η = x, t = 0) (middle) and F sca ( x, η = 0 , t ) /F sca ( x, η = 0 , t = 0) (right) for X = 0 . X = 0 . − t . the diagonal L z = 0 and L z = 1 LFWF overlaps, see discussion below Eq. (4.27). Note that inthe limit x → α and the parameter p . For the Regge improved model we have values that areslightly larger than one. For the exponential LFWF model, which has no skewness dependence,the skewness ratios are one (dashed curves). For the power-like LFWF models the GPDs at thecross-over line have a weaker large- x fall-off than the corresponding PDFs, where such a behavioris expected from large- x counting rules [144]. Hence, the ratios increase drastically in the large- x region.In the middle (right) column of Fig. 5 we display the t -dependence of GPDs F ( x, η = x, t ) F ( x, η = x, t = 0) (middle) and F ( x, η = 0 , t ) F ( x, η = 0 , t = 0) (right) for F ∈ { H, e H, H T } , (5.45)normalized to one at t = 0, at the cross-over ( η = 0) line for fixed Bjorken-like momentum fraction X = 0 . X = 0 . H (top) and H T (bottom)GPDs one clearly realizes that the t -dependence flatters out at large x = X/ (2 − X ). Comparing78he panels in the middle and right columns, one realizes that for the power-like LFWFs models(dash-dotted, dotted) this effect is more pronounced in the zero-skewness case. Of course, theexponential LFWF model (dashed) does not dependent on the skewness parameter and so the t -dependence is the same in the panels of the middle and right columns. Note that in this model H T becomes negative at large − t , e.g., for x ∼ .
05 at − t & . This arises from the L z = 1LFWF which induces a t -dependence in the h T DD, given in Eq. (4.33). For e H GPD (middle) theinterplay of diagonal L z = 0 and L z = 1 LFWF overlap induces for the Regge improved models(dash-dotted, dashed) a more intricate pattern as for H or H T GPD. Here, we observe now thatat small − t the ratios (5.45) increase with growing − t . Interestingly, in Ref. [145] it was arguedthat deeply virtual Compton scattering data indicate that GPD e H has a weaker t -dependencethan GPD H .As already pointed out in Ref. [39], both features that valence-like GPDs are dying out sloweras PDFs at large x and that the t -dependence vanishes at large x are in qualitative agreementwith hard exclusive ρ electroproduction cross section measurements at very large x Bj & . − t also increases with growing x Bj , the condition − t ≪ Q withina rather small value of the photon virtuality 2 GeV . Q < might be not satisfied forthe longitudinal part of the t -integrated cross section and so it might be doubtful to employ thecollinear factorization analysis within the GPD framework to the t -integrated cross section itself. As we have discussed in Sec. 2.2.1, by adding an identity proportional spin density matrix a scalardiquark model for uPDFs can be easily extended to a “spherical” model of rank-four. If we relyon SU(6) flavor-spin symmetry, then all off-diagonal entries of the uPDF spin-density matrix inthe u/ d/ h s axi i = 16 h s u val i + 23 h s d val i ∼ − . , where its modulus is indeed smaller than in the scalar sector, cf. Eq. (5.34). For the lowest momentof the transversity PDF h , often called “tensor charge”, we even find from the fit results [128] aterm that is compatible with zero Z dx (cid:20) h u val ( x ) + 23 h d val ( x ) (cid:21) ≈ − . +0 . − . , Z dx h sca1 ( x ) ≈ . +0 . − . . Hence, a “spherical” quark model of rank-four that relies on SU(6) flavor-spin symmetry, suchas the three-quark LFWF [61], axial-vector diquark [59], bag [69], the chiral quark soliton [70],79nd the covariant parton [63] model, might be tuned somehow compatible w.r.t. to (u)PDF phe-nomenological findings.However, for GPDs the situation is more intricate. Taking our minimal axial-vector diquarkGPD model, which respect the underlying Lorentz symmetry, and associating it with d -quark, asdictated by SU(6) symmetry, leads to a complete failure w.r.t. the anomalous magnetic momentof the nucleon. The GPD spin-correlation matrix (4.59) suggest that κ d is positive while itsexperimental value κ d = F p ( t = 0) + 2 F n ( t = 0) ≈ − . d -quark GPD E , however, this modelhas not been developed to the stage where it respect the underlying Lorentz symmetry. Alsothe bag model calculation of Ref. [148] provides a negative d -quark GPD E . Interestingly, thecovariant axial-vector diquark model of Ref. [116], where the polarization sum of the axial-vectordiquark is taken to be − g µν , provides a d -quark GPD E which possesses a node. According toEq. (38) of Ref. [116] the L z = 0 and | L z | = 1 coupling reads in our notation e d [116] ∝ (cid:20) mM (2 x − y ) − (1 − x + y ) y (cid:21) ⇒ E d ( x, η = 0 , t ) [116] ∝ (cid:20) mM − (1 − x ) (cid:21) x For zero-skewness this GPD is positive at large x and might become slightly negative at small x , however, the d -quark anomalous magnetic moment remains positive. It is interesting to notethat the chiral quark soliton model of Ref. [149] gives for the iso-singlet combination a qualitativedifferent behavior, namely, its GPD E is negative and positive at large and small x , respectively.Obviously, this contradicts both the axial-vector diquark [116] and pseudo-scalar diquark modelpredictions, where the latter is given by e d PS ∝ − (1 − x ) h mM − y i ⇒ E d ( x, η = 0 , t ) PS ∝ − (1 − x ) h mM − x i , see Sec. 4.3.2. We add that a LFWF model for nfPDFs in the axial-vector sector predicts a negative E for d -quarks without nodes [150], where, unfortunately, the Gaussian form of the scalar two-body LFWF does not respect the underlying Lorentz symmetry. To our best knowledge it isnot investigated whether the GPD results of the equivalent “spherical” uPDF models, i.e., thethree-quark LFWF [61], axial-vector diquark [59], bag [69], the chiral quark soliton [70], and thecovariant parton [63] model are still equivalent and can be partially represented by a “spherical”GPD model, too. Let us conclude we are not aware on a LFWF model in the axial-vectorsector that has been shown to be consistent with Lorentz symmetry and is in accordance withphenomenological findings. 80 .3.3 Modeling pomeron like behavior We employ now effective two-body LFWF modeling in the sea quark sector, which we considerhere as flavor symmetric. Large N c counting rules might suggest that polarized T -even partondistributions in the iso-singlet sector are relatively suppressed, compared to the iso-vector channel[151, 152, 117, 153]. While for polarized PDFs this is consistent with polarized DIS measurements,not much is known about the behavior of GPD E , which might be addressed in deeply virtualCompton scattering off a transversal polarized proton. It is important to note that at large N c the iso-singlet GPD E is relatively suppressed to the iso-vector GPD E , however, it is on thesame footing as the iso-singlet GPD H [117]. Moreover, GPDs E and H are rather loosely tied toeach other by Lorentz invariance and so we might wonder how GPD E , associated with a protonhelicity non-conserved “pomeron” exchange that plays no role in Regge phenomenology, behavesat small x .According to Table 2, a diquark model which provides only GPDs H and E can be build fromthe pair of scalar LFWFs (4.55) and the two transverse ones (4.58) for a minimal axial-vectordiquark coupling, F sea = X λ Z ρ ( λ ) X σ = − , + ZZ d k ⊥ η " ψ † S σ (cid:18) x − η − η , k ⊥ − − x − η ∆ ⊥ (cid:12)(cid:12)(cid:12) λ (cid:19) ⊗ ψ S σ (cid:18) x + η η , k ⊥ (cid:12)(cid:12)(cid:12) λ (cid:19) (5.46)+ 34 ψ † A σ (cid:18) x − η − η , k ⊥ − − x − η ∆ ⊥ (cid:12)(cid:12)(cid:12) λ (cid:19) ⊗ ψ A σ (cid:18) x + η η , k ⊥ (cid:12)(cid:12)(cid:12) λ (cid:19) . This spin-correlation matrix F sea has indeed only GPD H and GPD E entries that are given bythe DD representation (4.23), ( H sea E sea ) ( x, η, t ) = Z dy Z − y − y dz δ ( x − y − ηz ) ( h sea + ( x − y ) e sea (1 − x ) e sea ) ( y, z, t ) , (5.47)where the DDs are defined in Eqs. (4.24, 4.25). We recall that because of the explicit x dependencein the DD representation polynomiality is completed in our model and so we can extract by meansof Eq. (4.38) the D -term. Note that the corresponding spin-density matrix for uPDFs is an identitymatrix, proportional to f sea1 ( x, k ⊥ ) = (cid:20)(cid:16) mM + x (cid:17) + k ⊥ M (cid:21) Φ ( x, k ⊥ ) , (5.48)where the “uPDF” Φ is specified for the power-like and exponential LFWF model in (5.8) and(5.16), respectively.To adjust the parameters of our model, we take for our convenience now the input scale tobe Q = 2 GeV . From hard exclusive electroproduction of mesons and photons it is known that81 - - x H H x , , t = , Q = G e V L pheno.powerexpon. - - x H H x , x , t = , Q = G e V L - - x E H x , , t = , Q = G e V L - - x E H x , x , t = , Q = G e V L - t @ GeV D H H x , x , t , Q = G e V L - t @ GeV D E H x , x , t , Q = G e V L Figure 6:
Upper panels: the unpolarized sea quark PDF (left) and x -dependence for t = 0 (middle) and t -dependence for x = 10 − (right) of GPD H ( x, x, t ) at the input scale Q = 2 GeV , extracted fromLO fits [126] and [143], are shown as grayed area and solid curves. Our adjusted sea quark GPD modelswith power-like (5.50) and exponential (5.51) LFWF are displayed as dash-dotted and dashed curves,respectively. In the three lower panels we show from left to right the analog predictions for the GPD E .The thin solid line in the lower third panel shows for comparison the H fit result [143]. the Regge slope parameter is smaller than the soft pomeron one, α ′ = 0 .
25, and we will simplyset it here to zero. Thus, our GPD models satisfy positivity constraints. The effective pomeronintercept α = 1 .
13, slightly larger than one, is consistent with global PDF and GPD fit results[126, 143]. We set in agreement with phenomenological findings p = 3, i.e., our sea quark PDFvanishes at large x as (1 − x ) . and it is slightly stronger suppressed as (1 − x ) , stated bycounting rules. The normalization (4.39) is fixed by the averaged sea quark momentum fraction A sea = Z dx x H sea ( x, η = 0 , t = 0) = X q = u,d,s Z dx x ¯ q ( x ) = 0 . . (5.49)The mass parameters m and λ are taken from phenomenological PDF [126] and GPD [143] find-ings, extracted at leading order, where for the later we utilized the KM09a parameterization.Analogously, we also adjust the exponential LFWF model. Altogether, our sea quark GPD modelis for Q = 2 GeV fixed by the following parametersLFWF pow : α = 1 . , p = 3 , m = 0 .
63 GeV , λ = 1 .
00 GeV , (5.50)LFWF exp : α = 1 . , A = 3 , m = 0 .
63 GeV , λ = 1 .
45 GeV . (5.51)We illustrate in the upper left and middle panel of Fig. 6 that our both sea quark modelscan fairly describe the PDF [126] and GPD [143] fit results at Q = 2 GeV , respectively. Also82he t -dependence of the power-like and exponential ansatz, shown in the upper right panel, arehardly to distinguish in the experimentally accessible region | t | < from the fit result tothe deeply virtual Compton scattering cross section [143]. The r -ratio for the power-like LFWFmodel is slightly larger than one, however, we expect that both of our GPD models will grow toofast with increasing Q and so the r -ratio will increase [154], while the experimental data requirea constant r ≈ Q lever arm [143].Our models might now serve to estimate the E GPDs and in particular they allow us tocalculate the associated D -term (4.38). In the lower panels of Fig. 6 we display the E GPDs at η = 0 (left) and η = x (middle) versus x at t = 0. As we realize with the specific spin-spin coupling(5.46) the resulting E GPDs is for small x about twice times bigger than the corresponding H GPDs, while the t -dependency, shown in the right panel, differs for | t | ≤ only slightly fromthose of H GPDs. Note that all these findings respect positivity constraints, e.g., − H ( x, b ) ≤ M ∂∂b E ( x, b ) ≤ H ( x, b ) , a special case of those in Refs. [155, 156] that follow from the positivity of spin-correlation matrixeigenvalues (3.15) in impact parameter space.We can also easily determine the sea quark angular momentum, J sea = 12 [ A sea + B sea ] , B sea = Z dx x E sea ( x, η = 0 , t = 0) , (5.52)in terms of the anomalous gravitomagnetic moment B sea . Its estimation from Lattice simulationsor deeply virtual Compton scattering measurements has been not reached at present [143]. Ourmodels provide a rather large value B sea ≈ (cid:26) . . (cid:27) ⇒ J sea ≈ (cid:26) . . (cid:27) for (cid:26) LFWF pow
LFWF exp (cid:27) at Q = 2 GeV . (5.53)Hence, we find from the angular momentum sum rule and the phenomenological values of A val and A G J val = 12 (cid:2) A val + B val (cid:3) ≈ .
22 + 12 B val , J G = 12 (cid:2) A G + B G (cid:3) ≈ . B G , (5.54)where angular momentum conservation requires that the anomalous gravitomagnetic momentvanishes, i.e., B u val + B d val + B G = − B sea ≈ (cid:26) − . − . (cid:27) for (cid:26) LFWF pow
LFWF exp (cid:27) . (5.55)Assuming that the valence GPDs E u val and E d val have a similar functional form than GPDs H u val lattice è Χ QSM Χ QSM LFWF pow
LFWF exp
ŸŸŸ ŸŸŸŸ ŸŸ ŸŸ Ÿ Ÿ ŸŸ - t @ GeV D E , H t , Μ L Χ QSM LFWF pow
LFWF exp - - - - z d H z , t = , Μ L è Χ QSM KM09aKM09bLFWF pow
LFWF exp - t @ GeV D - D H t , Μ L Figure 7:
The GFF E u + d , versus − t (left), D -term versus z (middle), and subtraction constant −D versus − t (right). The squares are from the data set VI ( m π = 352 . t = 0. The thin( KM09a ) and thick (
KM09b ) solid lines in the right panel show the subtraction constant extracted fromdispersion relation fits [160]. Our adjusted sea quark GPD models with power-like (5.50) and exponential(5.51) LFWF are displayed as dash-dotted and dashed curves, respectively. and H d val , one finds that the anomalous gravitomagnetic moment in the valence quark sector iscompatible with zero. Hence, we have in our model a large gluonic anomalous gravitomagneticmoment B G ≈ − B sea ∼ − .
28 and so the gluonic angular momentum J G ∼ .
05 is rather small.This angular momentum scenario is to our best knowledge not much discussed in the literature[157, 158], which might be related to the fact that sea quarks and gluon degrees of freedom arenaturally ignored by quark model builders and that they are also so far not (directly) accessiblefrom Lattice simulations, see, e.g., Ref. [84].Let us also consider the first Mellin moments of GPDs H and E that belong to the highestpossible power in η and so they differ by sign, H , ( t ) = − E , ( t ) , E , ( t ) = 12 d dη Z − dx x E ( x, η, t ) . (5.56)This GFF appears in the form factor decomposition (of the quark part) of the energy momentumtensor and should be somehow been governed by the chiral dynamics [40]. We recall that the partof the Mellin moments that complete polynomiality is in our model incorporated in both GPDregions and so the duality between the partonic s -channel and mesonic like t -channel exchanges isfully incorporated in the model, see, e.g., in Ref. [99]. Loosely spoken the GFF (5.56) is associatedwith the “pomeron” exchange, which dictate in particular the sign. From the point of view ofchiral dynamics it appears more natural to interpret such terms as meson-like t -channel exchanges The valence anomalous gravitomagnetic moment B val = P u val κ u + P d val κ d ≈ . P u val − . P d val might beexpressed by the partonic anomalous magnetic momentsand the Mellin-moment fractions P i = R dx x E i ( x, η =0 , t = 0) . R dx E i ( x, η = 0 , t = 0) . If we assume that P u val /P d val nearly coincides with the averaged momentumfraction ratio h x i u val / h x i d val ≈ .
5, we might conclude that B val is compatible with zero. J = 0 σ -exchange. The physical interpretation of the energy-momentum form factor (5.56) has been illuminated in Ref. [159]. Namely, the absence of a poleat t = 0 in H , ( t ),lim t → t H , ( t ) = 6 M Z d r p ( r ) = 0 , H , ( t = 0) = M Z d r r p ( r ) , is nothing but the proton stability criteria which tells us that the net pressure R d r p ( r ) vanishes.From that it appears to be plausible that p ( r ) is negative at large distances | r | and so it follows thatthe partonic net contribution to the GFF H , ( t ) [ E , ( t )] at t = 0 should be negative [positive].Our sea quark model, build from a power-like LFWF, is in accordance with this expectation andwe find H sea1 , ( t = 0) = − E sea1 , ( t = 0) ≈ (cid:26) − . (cid:27) for (cid:26) LFWF pow
LFWF exp (cid:27) . (5.57)This value quantitatively agrees with the estimate of both the chiral quark soliton model [159],given as − (4 / × . ≈ − . and lattice simulations [136], quoted thereas − × . ≈ − . Q = 4 GeV . In the left panel of Fig. 7 we show the GFF E , versus t , where its t -dependency is compatible with both the chiral quark soliton model result [159] andLattice simulations [136].Let us finally have a look to the complete D -term (4.38), which for our power-like LFWFmodel reads d ( z, t ) = z Z −| z | (0) dy N (cid:0) mM + y (cid:1) y − α ((1 − y ) − z ) p + α/ (cid:2) (1 − y ) m M + y λ M − y (1 − y ) − ((1 − y ) − z ) t M (cid:3) p +1 , (5.58)and identically vanishes for our exponential one. To evaluate the y -integral for 1 < α <
2, weemploy analytic (or canonical) regularization [162]. Let us adopt the Gegenbauer expansion ofRef. [117], d ( z, t = 0) ≈ (1 − z ) h − . C / ( z ) + 1 . C / ( z ) − . C / ( z ) + · · · i , (5.59)where the first coefficient is given by 5 H , /
4. Interestingly, in contrast to the chiral quark solitonmodel result of Ref. [117], d ( z, t = 0) ≈ (1 − z ) h − . C / ( z ) − . C / ( z ) − . C / ( z ) + · · · i , (5.60)our model has a sign alternating series and so our d -term is end-point suppressed while the chiralquark soliton model result is rather end-point enhanced, cf. middle panel in Fig. 7. Note thatour first coefficient is closer to the chiral quark soliton model result of Ref. [159] and that the Note that a variety of chiral model estimates for this quantity [117, 161, 159, 139] exist. H and E , can be straightforwardly evaluated from the D -term and reads within our flavor decompositionas D ( t ) = 29 Z − dz z − z d ( z, t ) , where the additional factor 2 / D -term has been estimated in the chiralquark soliton model [117] and in Lattice simulations to be negative and sizeable, which is inagreement with the negative subtraction constant found in “dispersion relation” fits of deeplyvirtual Compton scattering data [143]. In our reggezised models a large negative D -term mightbe related to an effective ‘pomeron’ exchange with α slightly larger than one. The goal of this article, which contains various new results, was to provide a start up for aconsistent description of uPDFs and GPDs in terms of LFWFs and to illuminate the challengesfor the modeling procedure and LFWF phenomenology. Thereby, we followed the diquark pictureand so we restricted ourselves to effective two-body LFWFs. The model topics, we have areaddressed, are the following: • model classification based on the symmetry of the spin-density or -correlation matrices • model dependent cross talks of GPDs and uPDFs • implementing of the underlying Lorentz symmetry in the effective LFWF approach • Regge improvement as a collective phenomena from the s -channel point of view • emphasizing GPD dualityWe introduced a new classification scheme for quark models and clearly realized that variousuPDF models, including a three quark LFWF model, can be mapped to “spherical” diquarkmodels, which supports the diquark picture. We also clearly spelled out that although uPDFsand GPDs can be obtained from the proton LFWFs the cross talk among them might be washed86ut in QCD. This we illustrated in the axial-vector channel, where one might even have variousLFWF models that provide the same results for uPDFs, however, different predictions for GPDs.We stress that this sector can be modeled in various manner, that it is not well understood,and models consistent with phenomenology are known to be inconsistent with Lorentz symmetryrequirements and reverse. We emphasize that the confrontation of highly symmetric models withphenomenological findings are a useful method to provide insights into the validity of “dynamical”QCD symmetries. However, to set up consistent, simple, and more flexible models, the spin-spincoupling of the struck quark with the spectator system has to be studied in more detail. We onlygave here some rather simple examples that are known to provide consistent models.We also like to emphasize that the quark orbital angular momentum is only a useful partonicquantum number for simple models, however, in QCD a partonic distribution is build from aninfinite number of states with various quark orbital angular momenta that contributions are in-distinguishable. On the other hand, it is just the orbital angular momenta which destroys thecross talk between the three twist-two related uPDFs f , g , and h with the corresponding zero-skewness GPDs H , e H , and H T . Hence, nucleon tomography with GPDs and uPDFs might provideus two complementary three-dimensional pictures. Their (qualitative) comparison might provideus in the future some handle on the collective phenomena of orbital angular momentum. We alsodemonstrated that in a model one naturally renders two numbers for the quark orbital angularmomentum, namely, one from its direct definition and the other one from utilizing Ji‘s sum rule.The reason for this “puzzle” is obvious, namely, in static LFWF modeling the equations-of-motionare violated or even do not exist.Let us summarize our GPD modeling attempts. We derived a relatively general representa-tion for GPDs in terms of LFWFs, which satisfies both polynomiality and positivity constraintsby construction. The uPDFs can be easily given in terms of LFWF overlaps, too. Here, werestricted ourselves for shortness to the twist-two sector where the extension to the twist-threeone is straightforward. We showed how t and k ⊥ dependencies are tied to each other in a modeldependent manner. We also illustrated that a rather simple LFWF parameterization is able todescribe experimentally measured form factors, polarized and unpolarized PDFs from global fits,Lattice estimates, and is also compatible with phenomenological GPD features.Certainly, LFWF modeling, considered as an attempt to address QCD aspects from the Fieldtheoretical side, is challenging and we consider our work, presented here, as partially simpleminded. Various rather old problems were not addressed here. For instance, the k ⊥ and t in-terplay yields to a paradox that the large k ⊥ behavior predicted by perturbative QCD is ratherdifferent from the t -behavior from dimensional and perturbative counting rules. Of course, such aparadox challenges for a deeper understanding of QCD dynamics, which might be obtained during87xploration of effective LFWFs. Let us list a few of the problems, starting with the simple ones,which must be addressed along this road: • Is the implementation of Regge behavior in highly symmetric quark models realistic? • Can one utilize (simple) spin-spin parton spectator couplings to set up flexible models? • How t -dependence of Regge trajectories can be consistently implemented? • How to set up a flexible parameterization of the skewness effect in term of LFWFs? • How one can understand the interplay of k ⊥ , t , and renormalization scale dependence?Even if the list can easily be extended, we finally conclude that a phenomenological LFWF ap-proach might be a feasible alternative to the unsolved QCD bound state problem.D.S.H. thanks the Institute for Theoretical Physics II at the Ruhr-University Bochum for the hos-pitality at the stages of this work. We are thankful for discussions and comments to B. Pasquini,A. Prokudin, A. Radyushkin, Ch. Weiss, and F. Yuan. This work was supported in part by theBMBF (Federal Ministry for Education and Research), contract FKZ 06 B0 103, by the Inter-national Cooperation Program of the KICOS (Korea Foundation for International Cooperationof Science & Technology), by Basic Science Research Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0011034). A Unintegrated double distributions
To derive the unintegrated double distribution representation (4.6) we start from the definition(4.4), valid for x ≥ η , and the Laplace transform (4.5), which yields Φ ( x, η, t, k ⊥ ) = 11 − x Z ∞ dα Z ∞ dα ϕ ∗ ( X ′ , α ) ϕ ( X, α ) (A.1) × exp (cid:26) − α k ′ ⊥ − X ′ (1 − X ′ ) M (1 − X ′ ) M − α k ⊥ − X (1 − X ) M (1 − X ) M (cid:27) . Setting α i = x i A , the scalar LFWF overlap is given by Φ ( x, η, ∆ ⊥ , k ⊥ ) = Z ∞ dAA Z dx Z dx δ (1 − x − x ) ϕ ∗ ( X ′ , Ax ) ϕ ( X, Ax ) (A.2) × exp (cid:26) − A (cid:20) x k ′ ⊥ − X ′ (1 − X ′ ) M (1 − X ′ ) M + x k ⊥ − X (1 − X ) M (1 − X ) M (cid:21)(cid:27) . y and z by setting x = − y − z − y ) and x = − y + z − y ) with theconstraint x = y + zη . This leads to the expression Φ ( x, η, ∆ ⊥ , k ⊥ ) = 12 Z ∞ dAA (1 − y ) Z dy Z − y − y dz δ ( x − y − zη ) × ϕ ∗ (cid:18) x − η − η , A − y + z − y ) (cid:19) ϕ (cid:18) x + η η , A − y − z − y ) (cid:19) (A.3) × exp ( − A − y " k ⊥ M − (cid:2) (1 − y ) − z (cid:3) t M − y (1 − y ) , where k ⊥ = k ⊥ − (1 − y + z ) ∆ ⊥ /
2. We can now relax the constraint for the momentum fraction x and consider the function as defined in the central region − η ≤ x ≤ η , too. After rescaling theintegration variable, A → A (1 − y ), we arrive at the DD representation (4.6), where the DD isgiven in (4.7). B LFWF overlap representation for nucleon GPDs
We specify the frame by choosing a convenient parameterization of the light-cone coordinates forthe initial and final proton: P = (cid:18) P + , ⊥ , M P + (cid:19) , (B.1) P ′ = (cid:18) (1 − ζ ) P + , − ∆ ⊥ , M + ∆ ⊥ (1 − ζ ) P + (cid:19) , (B.2)where M is the proton mass. We use the component notation V = ( V + , V ⊥ , V − ), and our metricis specified by V ± = V ± V z and V = V + V − − V ⊥ . The four-momentum transfer from thetarget is ∆ = P − P ′ = (cid:18) ζ P + , ∆ ⊥ , t + ∆ ⊥ ζ P + (cid:19) , (B.3)where the Lorentz invariant Mandelstam variable t ≡ ∆ and energy-momentum conservationrequires ∆ − ≡ P − − P ′− = ( t + ∆ ⊥ ) /ζ P + . The latter equation connects the transverse momentumtransfer, parameterized in the following as ∆ ⊥ ≡ (∆ , ∆ ) = | ∆ ⊥ | (cos ϕ, sin ϕ ) , (B.4)the skewness variable ζ = ∆ + /P + , and the Mandelstam variable t according to t ≡ ∆ = − ζ M + ∆ ⊥ − ζ (B.5)= t − − ζ )(2 − ζ ) ¯∆ ⊥ with t = − ζ M − ζ and ¯∆ ⊥ = 2 − ζ − ζ ) | ∆ ⊥ | (cos ϕ, sin ϕ ) . − t is the kinematical allowed minimal value of − t and ¯∆ ⊥ is a convenient definition thatabsorbs a ζ -dependent prefactor.We note that w.r.t. symmetries, e.g., s and u -channel or time reflection, it is more appropriateto use the symmetric skewness variable η , defined in Eq. (3.2), rather than ζ . Both scaling variablesare related to each other by a SL(2 | R ) transformation ζ = 2 η η or η = ζ − ζ . (B.6)The momentum transfer square (B.5) reads now in an explicit symmetric manner as t = t − (1 − η ) ¯∆ ⊥ , where t = − M η − η and ¯∆ ⊥ = √ t − t p − η (cos ϕ, sin ϕ ) (B.7)are symmetric functions w.r.t. η → − η reflection.In the following we list the LFWF overlap representations for all twist-two GPD combinations,where the initial and final nucleon state, having momenta P and P ′ as well as spin projections S and S ′ , are described by n -parton LFWFs ψ S ( n ) ( X i , k ⊥ i , s i ) and ψ ∗ S ′ ( n ) ( X ′ i , k ′⊥ i , s ′ i ) , respectively. Here and in the following we use common LF variables, namely, the momentumfraction X i and k ⊥ i for the incoming partons. We have to deal with the parton number conservedand changing LFWF overlaps. In the former case the momenta of the outgoing partons are givenby X ′ = X − ζ − ζ , k ′⊥ = k ⊥ − − X − ζ ∆ ⊥ for the struck parton, X ′ i = X i − ζ , k ′⊥ i = k ⊥ i + X i − ζ ∆ ⊥ for the spectator i ∈ { , · · · , n } . (B.8)In the latter case the struck partons, appearing in the initial state, are labeled by i = 1 and n + 1and their parton momenta are related by X n +1 = ζ − X , k ⊥ n +1 = ∆ ⊥ − k ⊥ for the struck partons, (B.9)where the outgoing momenta of the spectator system ( i = 2 , · · · , n ) are given in Eq. (B.8). Weadd that we alternatively might use the skewness variable η together with the momentum fractions x i = 2 X i − ζ − ζ or X i = x i + η η . (B.10)For the spinors of a free spin-1 / U ( p, s = 1 /
2) = 1 √ p + p + + mp + ip p + − mp + ip , U ( p, s = − /
2) = 1 √ p + − p + ip p + + mp − ip − p + + m , (B.11)90hich can be represented as linear combination of Bjorken and Drell spinors. The definition ofDirac matrices can be found in [164]. B.1 Chiral even and parity even GPDs: H and E The H and E GPDs appear as form factors in the matrix elements of a vector operator (3.3),Using the LFWF (2.1) for the initial and final nucleon state and the one-parton matrix element Z dy − π e ixP + y − / (cid:10) X ′ P ′ + , p ′⊥ , s ′ (cid:12)(cid:12) ¯ ψ (0) γ + ψ ( y ) (cid:12)(cid:12) X P + , p ⊥ , s (cid:11) (cid:12)(cid:12)(cid:12) y + =0 ,y ⊥ =0 = p X X ′ p − ζ δ ( X − X ) δ s ′ s , (B.12)where for definiteness we have labeled the struck quark with the index i = 1, we can now evaluatethe n → n diagonal LFWF overlap to the matrix element on the l.h.s. of Eq. (3.3). Evaluating theform factor decomposition on the r.h.s. of Eq. (3.3) in terms of helicity amplitudes, convenientlywritten as 12 ¯ P + ¯ U ( P ′ , S ′ ) γ + U ( P, S ) = 2 √ − ζ − ζ δ S, S ′ , (B.13)12 ¯ P + ¯ U ( P ′ , S ′ ) σ + α ∆ α iM U ( P, S ) = 2 √ − ζ − ζ (cid:20) t M δ S, S ′ − | ¯∆ ⊥ | e iSϕ M S δ S, − S ′ (cid:21) , we find from the target spin conserved ( S ′ = S = 1 /
2) and flip ( S ′ = − S = 1 /
2) cases thefollowing linear combinations in the outer GPD region η ≤ x ≤ H ( x, η, t ) + t M E ( x, η, t ) = X n,s ZZ d k ⊥ ZX ( n − ψ ∗ ⇒ ( n ) ( X ′ i , k ′⊥ i , s i ) ψ ⇒ ( n ) ( X i , k ⊥ i , s i ) (B.14)and | ¯∆ ⊥ | e − iϕ M E ( x, η, t ) = X n,s ZZ d k ⊥ ZX ( n − ψ ∗ ⇒ ( n ) ( X ′ i , k ′⊥ i , s i ) ψ ⇐ ( n ) ( X i , k ⊥ i , s i ) , (B.15)respectively. Here, ¯∆ ⊥ is given in Eq. (B.5) and the integral measure is defined as ZX ( n − · · · ≡ X s ,...,s n Z [ dX d k ⊥ ] n √ − ζ n − − ζ √ − ζ δ ( X − X ) δ (2) ( k ⊥ − k ⊥ ) · · · , (B.16)see also Eq. (2.2), and the arguments of the final-state wavefunctions are given in Eq. (B.8). InEqs. (B.14) and (B.15) one has to sum over all possible combinations of spins s i and over allparton numbers n in the Fock states. We also imply a sum over all possible ways of numberingthe partons in the n -particle Fock state so that the struck quark has the index i = 1. Note a flip ofthe target spin projection leaves Eq. (B.14) invariant, however, it will change the phase ϕ → − ϕ and the sign on the l.h.s. of Eq. (B.15). 91nalogous formulae hold in the domain − < x < − η , where the struck parton in the target isan antiquark instead of a quark. Some care has to be taken regarding overall signs arising becausefermion fields anticommute. For details we refer to [24, 91, 92].For the n + 1 → n − n + 1 of the initial wavefunction annihilate into the current leaving n − x n +1 = ζ − x and k ⊥ n +1 = ~ ∆ ⊥ − k ⊥ . The current matrix element is Z dy − π e iXP + y − / (cid:10) (cid:12)(cid:12) ¯ ψ (0) γ + ψ ( y ) (cid:12)(cid:12) x P + , x n +1 P + , p ⊥ , p ⊥ n +1 , s , s n +1 (cid:11) (cid:12)(cid:12)(cid:12) y + =0 ,y ⊥ =0 (B.17)= p X X n +1 δ ( X − X ) δ s − s n +1 , and we thus obtain the formulae for the off-diagonal contributions to H and E in the domain | x | ≤ η : H ( x, η, t ) + t M E ( x, η, t ) (B.18)= X n,s ZZ d k ⊥ ZX ( n − p − ζ ψ ∗ ⇒ ( n − ( X ′ i , k ′⊥ i , s i ) ψ ⇒ ( n +1) ( X i , k ⊥ i , s i ) δ s − s n +1 , | ¯∆ ⊥ | e − iϕ M E ( x, η, t ) (B.19)= X n,s ZZ d k ⊥ ZX ( n − p − ζ ψ ∗ ⇒ ( n − ( X ′ i , k ′⊥ i , s i ) ψ ⇐ ( n +1) ( X i , k ⊥ i , s i ) δ s − s n +1 , where the momenta are specified in Eq. (B.9).A few comments are in order. Comparing Eqs. (B.14,B.15) and Eqs. (B.18,B.19), we realizethat the parton number off-diagonal expression can be obtained from the diagonal one by thesubstitution: ψ ∗ ⇒ ( n ) ( X ′ i , k ′⊥ i , s i ) ψ S ( n ) ( X i , k ⊥ i , s i ) ⇒ p − ζ ψ ∗ ⇒ ( n − ( X ′ i , k ′⊥ i , λ i ) ψ S ( n +1) ( X i , k ⊥ i , s i ) δ s − s n +1 , (B.20)where the momenta are specified in Eqs. (B.8,B.9). The powers of √ − ζ in (B.14), (B.15) and(B.18), (B.19) have their origin in the integration measures in the Fock state decomposition (2.1)for the outgoing proton. The fractions X ′ i appearing there refer to the light-cone momentum P ′ + = (1 − ζ ) P + , whereas the fractions X i in the incoming proton wavefunction refer to P + .Transforming all fractions so that they refer to P + as in our final formulae thus gives factors of √ − ζ . Different powers appear in the n → n and n + 1 → n − .2 Chiral even and parity odd GPDs: e H and e E GPDs
The e H and e E GPDs are defined through matrix elements of the axial-vector operator (3.4). Theevaluation of the LFWF overlap is analogously performed as in Sec. B.1. For the n → n diagonalterm, the relevant one-parton matrix element at quark level is Z dy − π e ixP + y − / (cid:10) X ′ P ′ + , p ′⊥ , s ′ (cid:12)(cid:12) ¯ ψ (0) γ + γ ψ ( y ) (cid:12)(cid:12) X P + , p ⊥ , s (cid:11) (cid:12)(cid:12)(cid:12) y + =0 ,y ⊥ =0 (B.21)= p X X ′ p − ζ δ ( X − X ) 2 s δ s ′ s and the helicity amplitudes of the form factors read:12 ¯ P + ¯ U ( P ′ , S ′ ) γ + γ U ( P, S ) = 2 √ − ζ − ζ S δ
S, S ′ , (B.22)12 ¯ P + ¯ U ( P ′ , S ′ ) − ∆ + M γ U ( P, S ) = 2 √ − ζ − ζ h t M S δ
S, S ′ − ζ | ¯∆ ⊥ | e i Sϕ (2 − ζ )2 M δ S, − S ′ i . Considering the target spin conserved ( S ′ = S = 1 /
2) and flip ( S ′ = − S = 1 /
2) cases, we obtainfor the n → n diagonal overlap contributions to the linear combinations of e H and e E GPDs in thedomain η ≤ x ≤ e H ( x, η, t ) + t M e E ( x, η, t ) = X n,s s ZZ d k ⊥ ZX ( n − ψ ∗ ⇒ ( n ) ( X ′ i , k ′⊥ i , s i ) ψ ⇒ ( n ) ( X i , k ⊥ i , s i ) (B.23)and | ¯∆ ⊥ | e − iϕ M ( − η ) e E ( x, η, t ) = X n,s s ZZ d k ⊥ ZX ( n − ψ ∗ ⇒ ( n ) ( X ′ i , k ′⊥ i , s i ) ψ ⇐ ( n ) ( X i , k ⊥ i , s i ) . (B.24)Flipping the target spin changes the sign on the l.h.s. of Eq. (B.23) and the sign of the phase inEq. (B.24)For the n + 1 → n − → Z dy − π e ixP + y − / (cid:10) (cid:12)(cid:12) ¯ ψ (0) γ + γ ψ ( y ) (cid:12)(cid:12) X P + , X n +1 P + , p ⊥ , p ⊥ n +1 , s , s n +1 (cid:11) (cid:12)(cid:12)(cid:12) y + =0 ,y ⊥ =0 (B.25)= p X X n +1 δ ( X − X ) 2 s δ s − s n +1 . We thus obtain that the formulae for the off-diagonal contributions to e H and e E GPDs in thedomain | x | ≤ η are obtained from Eqs. (B.23, B.24) by the substitution (B.20). B.3 Chiral odd GPDs: H T , E T , e H T and e E T The H T , E T , e H T , and e E T GPDs are defined trough the matrix elements of the tensor operator(3.5). The evaluation of the LFWF overlap is analogously performed as in Sec. B.1. For the n → n Z dy − π e ixP + y − / (cid:10) X ′ P ′ + , p ′⊥ , s ′ (cid:12)(cid:12) ¯ ψ (0) (cid:26) iσ +1 iσ +2 (cid:27) ψ ( y ) (cid:12)(cid:12) X P + , p ⊥ , s (cid:11)(cid:12)(cid:12)(cid:12) y + =0 ,y ⊥ =0 (B.26)= p X X ′ p − ζ δ ( X − X ) (cid:26) − s − i (cid:27) δ − s ′ s . The helicity amplitudes of the bilinear spinors, which we need in the calculation, are the vectorone in Eq. (B.13) and12 M ¯ U ( P ′ , S ′ ) U ( P, S ) = 2 − ζ √ − ζ δ S, S ′ + 2 √ − ζ − ζ | ¯∆ ⊥ | e i Sϕ M S δ S, − S ′ , (B.27)12 M ¯ U ( P ′ , S ′ ) (cid:26) γ γ (cid:27) U ( P, S ) = 2 √ − ζ − ζ (cid:26) − S − i (cid:27) (cid:20) | ¯∆ ⊥ | e − i Sϕ M S δ
S, S ′ − (2 − ζ ) ζ − ζ ) δ S, − S ′ (cid:21) ,
12 ¯ P + ¯ U ( P ′ , S ′ ) (cid:26) iσ +1 iσ +2 (cid:27) U ( P, S ) = 2 √ − ζ − ζ (cid:26) − S − i (cid:27) δ S, − S ′ . The n → n diagonal LFWF overlap contributions to H T , E T , e H T , and e E T GPD combinations inthe domain η ≤ x ≤ i = 1 and i = 2 as | ¯∆ ⊥ | e − iϕ M h e H T ( x, η, t ) + (1 + η ) E T ( x, η, t ) − (1 + η ) e E T ( x, η, t ) i (B.28)= X n,s s ZZ d k ⊥ ZX ( n − ψ ∗ ⇒ ( n ) ( X ′ i , k ′⊥ i , − s , s i =2 , ··· ,n ) ψ ⇒ ( n ) ( X i , k ⊥ i , s i ) , −| ¯∆ ⊥ | e iϕ M h e H T ( x, η, t ) + (1 − η ) E T ( x, η, t ) + (1 − η ) e E T ( x, η, t ) i (B.29)= X n,s − s ZZ d k ⊥ ZX ( n − ψ ∗ ⇒ ( n ) ( X ′ i , k ′⊥ i , − s , s i =2 , ··· ,n ) ψ ⇒ ( n ) ( X i , k ⊥ i , s i ) , and H T ( x, η, t ) + t − t M e H T ( x, η, t ) + t M (cid:20) E T − η e E T (cid:21) ( x, η, t ) (B.30)= X n,s − s ZZ d k ⊥ ZX ( n − ψ ∗ ⇒ ( n ) ( X ′ i , k ′⊥ i , − s , s i =2 , ··· ,n ) ψ ⇐ ( n ) ( X i , k ⊥ i , s i ) , − ¯∆ ⊥ (1 − η )4 M e − i ϕ e H T ( x, η, t ) (B.31)= X n,s s ZZ d k ⊥ ZX ( n − ψ ∗ ⇒ ( n ) ( X ′ i , k ′⊥ i , − s , s i =2 , ··· ,n ) ψ ⇐ ( n ) ( X i , k ⊥ i , s i ) , respectively. 94rom the comparison of the diagonal one-parton matrix elements (B.28) with Z dy − π e iXP + y − / (cid:10) (cid:12)(cid:12) ¯ ψ (0) (cid:26) iσ +1 iσ +2 (cid:27) ψ ( y ) (cid:12)(cid:12) X P + , X n +1 P + , p ⊥ , p ⊥ n +1 , s , s n +1 (cid:11)(cid:12)(cid:12)(cid:12) y + =0 ,y ⊥ =0 = p X X n +1 δ ( X − X ) (cid:26) − s − i (cid:27) δ s s n +1 , we read off that the formulae for the off-diagonal contributions to the chiral odd GPD combinationsin the central region | x | ≤ η are obtained from Eqs. (B.28–B.31) by the substitution: ψ ∗ ⇒ ( n ) ( X ′ i , k ′⊥ i , − s , s i =2 , ··· ,n ) ψ S ( n ) ( X i , k ⊥ i , s i ) (B.32) ⇒ p − ζ ψ ∗ ⇒ ( n − ( X ′ i , k ′⊥ i , s i ) ψ S ( n +1) ( X i , k ⊥ i , s i ) δ s s n +1 , where the momenta are specified in Eqs. (B.8,B.9). References [1] J. P. Ralston and D. E. Soper, Nucl.Phys.
B152 , 109 (1979).[2] J. C. Collins and D. E. Soper, Nucl.Phys.
B194 , 445 (1982).[3] D. M¨uller, D. Robaschik, B. Geyer, F.-M. Dittes, and J. Hoˇrejˇsi, Fortschr. Phys. , 101(1994), hep-ph/9812448.[4] A. V. Radyushkin, Phys. Lett. B380 , 417 (1996), hep-ph/9604317.[5] X. Ji, Phys. Rev.
D55 , 7114 (1997), hep-ph/9609381.[6] J. Collins, L. Frankfurt, and M. Strikman, Phys. Rev.
D56 , 2982 (1997), hep-ph/9611433.[7] J. Collins and A. Freund, Phys. Rev.
D59 , 074009 (1999), hep-ph/9801262.[8] J. C. Collins and D. E. Soper, Nucl.Phys.
B193 , 381 (1981), Erratum-ibid. B213 (1983)545.[9] J. C. Collins, D. E. Soper, and G. F. Sterman, Nucl.Phys.
B250 , 199 (1985).[10] X.-d. Ji, J.-p. Ma, and F. Yuan, Phys.Rev.
D71 , 034005 (2005), hep-ph/0404183.[11] J. Collins,
Foundations of Perturbative QCD (Cambridge Univ. Press, Cambridge, 2011).[12] EMC, J. Ashman et al. , Nucl. Phys.
B328 , 1 (1989).[13] M. Gluck and E. Reya, Nucl. Phys.
B130 , 76 (1977).9514] M. Gluck, E. Reya, and A. Vogt, Z.Phys.
C67 , 433 (1995).[15] A. Chodos, R. Jaffe, K. Johnson, C. B. Thorn, and V. Weisskopf, Phys.Rev. D9 , 3471(1974).[16] D. Diakonov, V. Petrov, and P. Pobylitsa, Nucl.Phys. B306 , 809 (1988).[17] A. V. Belitsky, X. Ji, and F. Yuan, Phys. Rev.
D69 , 074014 (2004), hep-ph/0307383.[18] V. Berestetsky and M. Terentev, Sov.J.Nucl.Phys. , 547 (1976).[19] V. Berestetsky and M. Terentev, Sov.J.Nucl.Phys. , 347 (1977).[20] I. Aznaurian, A. Bagdasaryan, and N. Ter-Isaakian, Yad.Fiz. , 1278 (1982).[21] F. Schlumpf, Relativistic constituent quark model for baryons , PhD thesis, Uinv. Z¨urich,1992, hep-ph/9211255, Ph.D. Thesis.[22] F. Schlumpf, J. Phys.
G20 , 237 (1994), hep-ph/9301233.[23] J. Bolz, R. Jakob, P. Kroll, M. Bergmann, and N. Stefanis, Z. Phys.
C66 , 267 (1995).[24] M. Diehl, T. Feldmann, R. Jakob, and P. Kroll, Eur. Phys. J. C8 , 409 (1999), hep-ph/9811253.[25] H.-M. Choi, C.-R. Ji, and L. S. Kisslinger, Phys. Rev. D64 , 093006 (2001), hep-ph/0104117.[26] B. C. Tiburzi and G. A. Miller, Phys. Rev.
C64 , 065204 (2001), hep-ph/0104198.[27] B. C. Tiburzi and G. A. Miller, Phys. Rev.
D65 , 074009 (2002), hep-ph/0109174.[28] S. Boffi, B. Pasquini, and M. Traini, Nucl. Phys.
B649 , 243 (2003), hep-ph/0207340.[29] S. Ahmad, H. Honkanen, S. Liuti, and S. K. Taneja, Phys. Rev.
D75 , 094003 (2006),hep-ph/0611046.[30] A. Mukherjee, I. V. Musatov, H. C. Pauli, and A. V. Radyushkin, Phys. Rev.
D67 , 073014(2003), hep-ph/0205315.[31] C.-R. Ji, Y. Mishchenko, and A. Radyushkin, Phys. Rev.
D73 , 114013 (2006), hep-ph/0603198.[32] H. Avakian et al. , Mod.Phys.Lett.
A24 , 2995 (2009), 0910.3181.[33] H. Leutwyler and J. Stern, Phys.Lett.
B73 , 75 (1978).9634] B. Keister, Phys.Rev.
D49 , 1500 (1994), hep-ph/9303264.[35] W. Jaus, Phys.Rev.
D60 , 054026 (1999).[36] A. Krutov and V. Troitsky, Phys.Part.Nucl. , 136 (2009).[37] S. Boffi and B. Pasquini, Generalized parton distributions and the structure of the nucleon,[0711.2625 [hep-ph]], 2007.[38] K. Kumeriˇcki, D. M¨uller, and K. Passek-Kumeriˇcki, Eur. Phys. J. C58 , 193 (2008),0805.0152.[39] D. S. Hwang and D. M¨uller, Phys. Lett.
B660 , 350 (2008), 0710.1567.[40] M. V. Polyakov and C. Weiss, Phys. Rev.
D60 , 114017 (1999), hep-ph/9902451.[41] P. A. Dirac, Rev.Mod.Phys. , 392 (1949).[42] S. D. Drell, D. J. Levy, and T.-M. Yan, Phys. Rev. Lett. , 744 (1969).[43] S. D. Drell and T.-M. Yan, Phys. Rev. Lett. , 181 (1970).[44] G. B. West, Phys. Rev. Lett. , 1206 (1970).[45] S. J. Brodsky and S. D. Drell, Phys. Rev. D22 , 2236 (1980).[46] S. J. Brodsky, H.-C. Pauli, and S. S. Pinsky, Phys. Rept. , 299 (1998), hep-ph/9705477.[47] J. Kogut and D. Soper, Phys. Rev. D1 , 2901 (1970).[48] S. J. Brodsky, D. S. Hwang, B.-Q. Ma, and I. Schmidt, Nucl.Phys. B593 , 311 (2001),hep-th/0003082.[49] A. V. Belitsky, X. Ji, and F. Yuan, Nucl. Phys.
B656 , 165 (2003), hep-ph/0208038.[50] B. Geyer, M. Lazar, and R. Robaschik, Nucl. Phys.
B559 , 339 (1999).[51] Y. Dokshitzer, Sov. Phys. JETP , 641 (1977).[52] V. Gribov and L. Lipatow, Sov. J. Nucl. Phys. , 438 (1972).[53] G. Altarelli and G. Parisi, Nucl. Phys. B 126 , 298 (1977).[54] D. Boer and P. J. Mulders, Phys. Rev.
D57 , 5780 (1998), hep-ph/9711485.[55] R. L. Jaffe, (1996), hep-ph/9602236. 9756] A. Bacchetta, M. Boglione, A. Henneman, and P. Mulders, Phys.Rev.Lett. , 712 (2000),hep-ph/9912490.[57] J. Soffer, Phys.Rev.Lett. , 1292 (1995), hep-ph/9409254.[58] D. W. Sivers, Phys.Rev. D41 , 83 (1990).[59] R. Jakob, P. J. Mulders, and J. Rodrigues, Nucl. Phys.
A626 , 937 (1997), hep-ph/9704335.[60] S. Meissner, A. Metz, and K. Goeke, Phys. Rev.
D76 , 034002 (2007), hep-ph/0703176.[61] B. Pasquini, S. Cazzaniga, and S. Boffi, Phys. Rev.
D78 , 034025 (2008), 0806.2298.[62] H. Avakian, A. V. Efremov, P. Schweitzer, and F. Yuan, Phys. Rev.
D78 , 114024 (2008),0805.3355.[63] A. V. Efremov, P. Schweitzer, O. V. Teryaev, and P. Zavada, Phys. Rev.
D80 , 014021(2009), 0903.3490.[64] C. Lorce and B. Pasquini, Phys.Rev.
D84 , 034039 (2011), 1104.5651.[65] S. J. Brodsky, D. S. Hwang, and I. Schmidt, Phys.Lett.
B530 , 99 (2002), hep-ph/0201296.[66] D. Boer, S. J. Brodsky, and D. S. Hwang, Phys.Rev.
D67 , 054003 (2003), hep-ph/0211110.[67] L. P. Gamberg, G. R. Goldstein, and K. A. Oganessyan, Phys.Rev.
D67 , 071504 (2003),hep-ph/0301018.[68] A. Bacchetta, F. Conti, and M. Radici, Phys. Rev.
D78 , 074010 (2008), 0807.0323.[69] H. Avakian, A. V. Efremov, P. Schweitzer, and F. Yuan, Phys. Rev.
D81 , 074035 (2010),1001.5467.[70] C. Lorce, B. Pasquini, and M. Vanderhaeghen, JHEP , 041 (2011), 1102.4704.[71] L. Gamberb, Private Communication.[72] F. Yuan, Phys.Lett. B575 , 45 (2003), hep-ph/0308157.[73] I. Cherednikov, U. D’Alesio, N. Kochelev, and F. Murgia, Phys.Lett.
B642 , 39 (2006),hep-ph/0606238.[74] A. Courtoy, S. Scopetta, and V. Vento, Phys.Rev.
D79 , 074001 (2009), 0811.1191.[75] A. Courtoy, S. Scopetta, and V. Vento, Phys.Rev.
D80 , 074032 (2009), 0909.1404.9876] A. Courtoy, PoS
LC2010 , 055 (2010), 1010.2175.[77] K. Goeke, A. Metz, and M. Schlegel, Phys. Lett.
B618 , 90 (2005), hep-ph/0504130.[78] E. V. Shuryak and A. Vainshtein, Nucl.Phys.
B201 , 141 (1982).[79] P. Mulders and R. Tangerman, Nucl.Phys.
B461 , 197 (1996), hep-ph/9510301.[80] K. Goeke, A. Metz, P. Pobylitsa, and M. Polyakov, Phys.Lett.
B567 , 27 (2003), hep-ph/0302028.[81] S. J. Brodsky, M. Burkardt, and I. Schmidt, Nucl. Phys.
B441 , 197 (1995), hep-ph/9401328.[82] F.-M. Dittes, D. M¨uller, D. Robaschik, B. Geyer, and J. Hoˇrejˇsi, Phys. Lett. , 325(1988).[83] A. V. Belitsky, A. Freund, and D. M¨uller, Nucl. Phys.
B574 , 347 (2000), hep-ph/9912379.[84] P. Hagler, Phys. Rept. , 49 (2010), 0912.5483.[85] D. M¨uller and A. Sch¨afer, Nucl. Phys.
B739 , 1 (2006), hep-ph/0509204.[86] M. Burkardt, Int. J. Mod. Phys.
A18 , 173 (2003), hep-ph/0207047.[87] J. P. Ralston and B. Pire, Phys. Rev.
D66 , 111501 (2002), hep-ph/0110075.[88] M. Diehl, Eur. Phys. J.
C25 , 223 (2002), hep-ph/0205208, Erratum-ibid. C31 (2003) 277.[89] M. Diehl, Eur. Phys. J.
C19 , 485 (2001), hep-ph/0101335.[90] S. Meissner, A. Metz, and M. Schlegel, JHEP , 056 (2009), 0906.5323.[91] S. J. Brodsky, M. Diehl, and D. S. Hwang, Nucl. Phys.
B596 , 99 (2001), hep-ph/0009254.[92] M. Diehl, T. Feldmann, R. Jakob, and P. Kroll, Nucl. Phys.
B596 , 33 (2001), hep-ph/0009255, Erratum-ibid. B605 (2001) 647.[93] M. Diehl and P. Hagler, Eur. Phys. J.
C44 , 87 (2005), hep-ph/0504175.[94] P. Pobylitsa, Phys.Rev.
D66 , 094002 (2002), hep-ph/0204337.[95] M. Burkardt, Phys.Rev.
D66 , 114005 (2002), hep-ph/0209179.[96] A. V. Belitsky, X.-d. Ji, and F. Yuan, Phys. Rev. Lett. , 092003 (2003), hep-ph/0212351.[97] P. V. Pobylitsa, Phys. Rev. D67 , 034009 (2003), hep-ph/0210150.9998] M. V. Polyakov and A. G. Shuvaev, On ’dual’ parametrizations of generalized partondistributions, hep-ph/0207153, 2002.[99] K. Kumeriˇcki, D. M¨uller, and K. Passek-Kumeriˇcki, Nucl. Phys. B , 244 (2008), hep-ph/0703179.[100] P. V. Landshoff and J. C. Polkinghorne, Phys. Rept. , 1 (1972).[101] A. Belitsky, A. Kirchner, D. M¨uller, and A. Sch¨afer, Phys.Lett. B510 , 117 (2001), hep-ph/0103343.[102] O. V. Teryaev, Phys. Lett.
B510 , 125 (2001), hep-ph/0102303.[103] X. Ji, J. Phys.
G24 , 1181 (1998), hep-ph/9807358.[104] A. V. Radyushkin, Phys. Rev.
D56 , 5524 (1997), hep-ph/9704207.[105] D. Mueller, (2014), 1405.2817.[106] A. V. Belitsky, D. M¨uller, A. Kirchner, and A.Sch¨afer, Phys. Rev.
D64 , 116002 (2001),hep-ph/0011314.[107] A. Radyushkin, Phys.Rev.
D83 , 076006 (2011), 1101.2165.[108] A. Radyushkin, Phys.Rev.
D87 , 096017 (2013), 1304.2682.[109] L. Mankiewicz, G. Piller, and A. Radyushkin, Eur. Phys. J.
C 10 , 307 (1999), hep-ph/9812467.[110] L. L. Frankfurt, P. V. Pobylitsa, M. V. Polyakov, and M. Strikman, Phys. Rev.
D60 ,0140010 (1999), hep-ph/9901429.[111] M. Vanderhaeghen, P. A. M. Guichon, and M. Guidal, Phys. Rev.
D60 , 094017 (1999),hep-ph/9905372.[112] O. V. Teryaev, Analytic properties of hard exclusive amplitudes, 2005, hep-ph/0510031,hep-ph/0510031.[113] I. V. Anikin and O. V. Teryaev, Phys. Rev.
D76 , 056007 (2007), 0704.2185.[114] M. Diehl and D. Y. Ivanov, Eur. Phys. J.
C52 , 919 (2007), 0707.0351.[115] C. Bechler and D. M¨uller, (2009), 0906.2571.100116] B. C. Tiburzi, W. Detmold, and G. A. Miller, Phys. Rev.
D70 , 093008 (2004), hep-ph/0408365.[117] K. Goeke, M. V. Polyakov, and M. Vanderhaeghen, Prog. Part. Nucl. Phys. , 401 (2001),hep-ph/0106012.[118] A. V. Belitsky, D. M¨uller, and A. Kirchner, Nucl. Phys. B629 , 323 (2002), hep-ph/0112108.[119] M. Vanderhaeghen, P. A. M. Guichon, and M. Guidal, Phys. Rev. Lett. , 5064 (1998).[120] S. V. Goloskokov and P. Kroll, Eur. Phys. J. C42 , 281 (2005), hep-ph/0501242.[121] S. V. Goloskokov and P. Kroll, Eur. Phys. J.
C53 , 367 (2008), 0708.3569.[122] V. Matveev, R. Muradyan, and V. Tavkhelidze, Lett. Nuovo Cimento , 719 (1973).[123] S. Brodsky and G. Farrar, Phys. Rev. D11 , 1309 (1975).[124] D. de Florian, R. Sassot, M. Stratmann, and W. Vogelsang, (2009), 0904.3821.[125] M. Gluck, E. Reya, M. Stratmann, and W. Vogelsang, Phys. Rev.
D63 , 094005 (2001),hep-ph/0011215.[126] S. Alekhin, JETP Lett. , 628 (2005), hep-ph/0508248.[127] J. Bl¨umlein and H. Bottcher, Nucl. Phys. B636 , 225 (2002), hep-ph/0203155.[128] M. Anselmino et al. , Nucl. Phys. Proc. Suppl. , 98 (2009), 0812.4366.[129] T. Gehrmann and W. J. Stirling, Phys. Rev.
D53 , 6100 (1996), hep-ph/9512406.[130] M. Anselmino et al. , Phys. Rev.
D75 , 054032 (2007), hep-ph/0701006.[131] R. Jaffe and A. Manohar, Nucl.Phys.
B337 , 509 (1990), Revised version.[132] X. Ji, Phys. Rev. Lett. , 610 (1997), hep-ph/9603249.[133] A. Harindranath and R. Kundu, Phys.Rev. D59 , 116013 (1999), hep-ph/9802406.[134] M. Burkardt and H. BC, Phys.Rev.
D79 , 071501 (2009), 0812.1605.[135] G. Hohler et al. , Nucl. Phys.
B114 , 505 (1976).[136] LHPC, P. H¨agler et al. , Phys. Rev.
D77 , 094502 (2008), 0705.4295.[137] LHPC, J. Bratt et al. , (2010). 101138] QCDSF, M. Gockeler et al. , Phys. Rev. Lett. , 222001 (2007), hep-lat/0612032.[139] M. Wakamatsu, Phys. Lett. B648 , 181 (2007), hep-ph/0701057.[140] S. Ahmad, G. R. Goldstein, and S. Liuti, Phys.Rev.
D79 , 054014 (2009), 0805.3568.[141] S. V. Goloskokov and P. Kroll, Eur. Phys. J.
C65 , 137 (2010), 0906.0460.[142] S. Goloskokov and P. Kroll, Eur.Phys.J.
A47 , 112 (2011), 1106.4897.[143] K. Kumeriˇcki and D. M¨uller, Nucl. Phys.
B841 , 1 (2010), 0904.0458.[144] F. Yuan, Phys. Rev.
D69 , 051501 (2004), hep-ph/0311288.[145] M. Guidal, Phys. Lett.
B693 , 17 (2010), 1005.4922.[146] M. Guidal and S. Morrow, Exclusive ρ electroproduction on the proton: GPDs or notGPDs?, 0711.3743 [hep-ph], 2007.[147] CLAS, S. A. Morrow et al. , Eur. Phys. J. A39 , 5 (2009), 0807.3834.[148] X. Ji, W. Melnitchouk, and X. Song, Phys. Rev.
D56 , 5511 (1997), hep-ph/9702379.[149] J. Ossmann, M. V. Polyakov, P. Schweitzer, D. Urbano, and K. Goeke, Phys. Rev.
D71 ,034011 (2005), hep-ph/0411172.[150] Z. Lu and I. Schmidt, (2010), 1008.2684.[151] D. Diakonov, V. Petrov, P. Pobylitsa, M. V. Polyakov, and C. Weiss, Nucl. Phys.
B480 ,341 (1996), hep-ph/9606314.[152] P. Pobylitsa and M. V. Polyakov, Phys.Lett.
B389 , 350 (1996), hep-ph/9608434.[153] P. Pobylitsa, Transverse momentum dependent parton distributions in large N(c) QCD,2003.[154] M. Diehl and W. Kugler, Phys. Lett.
B660 , 202 (2008), 0711.2184 [hep-ph].[155] M. Polyakov, Phys. Lett.
B555 , 57 (2002), hep-ph/0210165.[156] M. Burkardt, Phys.Lett.
B582 , 151 (2004), hep-ph/0309116.[157] D. M¨uller, K. Kumeriˇcki, and K. Passek-Kumeriˇcki, GPD sum rules: a tool to reveal thequark angular momentum, 0807.0170, 2008.102158] M. Diehl, How large can the distributions e q and e g be?, in Gluons and the quark sea athigh energies: Distributions, polarization, tomography. , 2011, 1108.1713.[159] K. Goeke et al. , Phys. Rev.
D75 , 094021 (2007), hep-ph/0702030.[160] D. M¨uller and K. Kumeriˇcki, Mod. Phys .Lett.
A24 , 2838 (2009).[161] P. Schweitzer, S. Boffi, and M. Radici, Phys. Rev.
D66 , 114004 (2002), hep-ph/0207230.[162] I. M. Gelfand and G. E. Shilov
Generalized Functions
Vol. I (Academic Press, New York,1964).[163] G. Lepage and S. Brodsky, Phys. Rev.
D22 , 2157 (1980).[164] J. Bjorken and S. Drell,