Implication of the overlap representation for modelling generalized parton distributions
aa r X i v : . [ h e p - ph ] O c t Implication of the overlap representation for modellinggeneralized parton distributions
D. S. Hwang a and D. M¨uller ba Department of Physics, Sejong UniversitySeoul 143–747, South Korea b Institut f¨ur Theoretische Physik II, Ruhr-Universit¨at BochumD-44780 Bochum, Germany
Abstract
Based on a field theoretically inspired model of light-cone wave functions, we derive valence-likegeneralized parton distributions and their double distributions from the wave function overlapin the parton number conserved s -channel. The parton number changing contributions in the t -channel are restored from duality. In our construction constraints of positivity and polynomialityare simultaneously satisfied and it also implies a model dependent relation between generalizedparton distributions and transverse momentum dependent parton distribution functions. Themodel predicts that the t -behavior of resulting hadronic amplitudes depends on the Bjorken vari-able x Bj . We also propose an improved ansatz for double distributions that embeds this property. Keywords: generalized parton distributions, overlap representation, duality, spectator model
PACS numbers: n µ = (1 , , , − ψ ↑ , ↓ n ( X i , k ⊥ i , λ i ). They are the probability amplitudes for their corresponding n -partonstates | n, p + i , p ⊥ i , λ i i , which build up the proton state with LC helicity S = { +1 / ↑ ) , − / ↓ ) } : | P, S = {↑ , ↓}i = X n Z [ dX d k ] n ψ ↑ , ↓ n ( X i , k ⊥ , λ i ) n Y j =1 p X j | n, X i P + , X i P ⊥ + k ⊥ i , λ i i , (1)where we used a shorthand for the n -parton phase space:[ dx d k ⊥ ] n = n Y i =1 dX i d k ⊥ i π π δ − n X j =1 X i ! δ (2) n X j =1 k ⊥ i ! . (2)The LC wave functions depend on the longitudinal momentum fractions X i = k + i /P + (the pluscomponent of a four-vector V µ is V + = V + V = n · V ), the transverse momenta k ⊥ i , and theLC helicities λ i . They are determined from the eigenvalue problem for the LC Hamiltonian P − : P − | P, S i = M P + | P, S i , with P − = P − P , P + = P + P , P ⊥ = 0 , (3)which can be derived from the QCD Lagrangian. However, in practice the QCD dynamics is notwell understood and it remains very challenging to develop this concept to a stage at which it can beused for quantitative evaluations of physical observables or parton distributions [5]. It is commonto pin down LC wave functions by comparing their resulting model predictions with experimentalobservations. Certainly, so far this concept is extensively elaborated, it connects different non-perturbative quantities, like PDFs, transverse momentum dependent parton distribution functions(TMDs), distribution amplitudes, form factors, and GPDs to a more universal object.1ur study is mainly devoted to GPDs, which are accessible from hard-exclusive reactions,e.g., electroproduction of mesons and photon. They arise from the non-diagonal overlap of LCwave functions, and therefore contain a maximum of information about the proton wave function,compared to other non-perturbative quantities. For instance, they are reducible to elastic formfactors and PDFs. Field theoretically they are defined as off-diagonal matrix elements of two fieldoperators that live on the light cone [6, 7, 8]. Describing the initial and final proton states, withgiven momenta P and P = P − ∆ and LC helicities S and S , respectively, in terms of theLC wave functions (1), one can straightforwardly derive wave function overlap representations ofGPDs for the partonic s -channel exchange [9, 10, 11]. In this partonic process the number ofpartons is conserved and the momentum fraction x of the struck quark is larger than the skewnessparameter η = ( P +1 − P +2 ) / ( P +1 + P +2 ) > outer region of x , the GPDs H and E (Ji’s conventions [12]) read (cid:18) H − η − η E (cid:19) ( x ≥ η, η, t ) = (4)2 − ζ √ − ζ X n X λ i p − ζ − n Z [ dX d k ⊥ ] n δ ( X − X ) ψ ↑∗ n ( X ′ i , k ′⊥ i , λ i ) ψ ↑ n ( X i , k ⊥ i , λ i ) , ∆ − i ∆ M E ( x ≥ η, η, t ) = (5) p − ζ X n X λ i p − ζ − n Z [ dX d k ⊥ ] n δ ( X − X ) ψ ↑∗ n ( X ′ i , k ′⊥ i , λ i ) ψ ↓ n ( X i , k ⊥ i , λ i ) , where ζ = 2 η/ (1 + η ), X = ( x + η ) / (1 + η ), and the momenta of the outgoing partons are X ′ = X − ζ − ζ , k ′⊥ = k ⊥ − − X − ζ ∆ ⊥ for the struck quark , (6) X ′ i = X i − ζ , k ′⊥ i = k ⊥ i + X i − ζ ∆ ⊥ for the spectators i = 2 , . . . , n. (7)Anti-quark GPDs are analogously defined with a negative momentum fraction x ≤ − η . The central region, i.e., − η ≤ x ≤ η , arises from the t -channel process in which the parton numberchanges from n + 2 to n [10, 11]. Viewing this as a mesonic-like t -channel exchange makes contactto Regge phenomenology [13, 14, 15]. Note that positivity constraints, in its most general form[16], should be satisfied in the overlap representations [17], if they are not spoiled by subtractionprocedures. Indeed, this can be easily shown for a two-body LC wave function, as used below.Let us also remind of the constraints of Lorentz covariance for (quark) GPD form factors. Theyare not invariant under general Lorentz transformations, however, they are built by a series of localtwist-two operator matrix elements, belonging to irreducible representations, that are labelled bythe spin J ≥
1. It turns out that Mellin moments of GPDs H + E and E with the weight x J − are polynomials in η of the order J − J , respectively. Time reversal invariance combined2ith hermiticity requires that these polynomials are even [12]. These properties are manifestlyimplemented in the double distribution (DD) representation of GPDs [6, 7]: (cid:26) H + EE (cid:27) ( x, η, t ) = Z dy Z − y − y dz (cid:26) − x (cid:27) δ ( x − y − ηz ) (cid:26) h + ee (cid:27) ( y, z, t ) , (8)where the DDs e and h are symmetric in z . The DD representation for E is not uniquely defined[18, 19]; Eq. (8) shows the one which naturally occurs in our model studies, see below Eq. (20).As explained above, the partonic interpretation of GPDs separates the support in the central( | x | ≤ η ) and outer ( η ≤ | x | ) regions. Residual Lorentz covariance, ensuring the polynomiality ofmoments, requires that both regions are tied to each other and that the functional form of GPDis constrained, e.g., in the outer region the GPD E is given by the integral representation: E ( x ≥ η, η, t ) = (1 − x ) Z x + η ηx − η − η dyη e ( y, ( x − y ) /η, t ) . (9)We consider both regions as dual to each other, i.e., knowing a GPD in one region allows torestore it in the other. Hence, a GPD can be entirely evaluated from the parton number con-served s -channel overlap of LC wave functions. A constructive, however, unwieldy method for therestoration of the central region is based on Mellin moments and its inverse transformation [14].In the following we utilize two-particle LC wave functions. They serve us to describe the protoncontents by a constituent quark and scalar diquark, where the latter plays the role of an collectivespectator. Numerous investigations in this spirit, even much more advanced ones with specificemphases, e.g., Refs. [20, 21, 9, 22, 23, 24, 25, 26], have been made in the past. The new aspect inour study is that we take care of Lorentz constraints for the LC wave functions, cf. Refs. [27, 28].This allows us to evaluate DDs from the parton number conserving overlap representations (4)and (5). We outline the generic features of such model, illuminate their t -dependence, point outits restrictions, and overcome them by hand, yielding improved DD and GPD ansatze.ii. The functional form of LC wave functions is dictated by the underlying Lorentz symmetry,i.e., the longitudinal and transversal variables are tied to each other in a certain but not apparentmanner. Hence, writing down LC wave functions by hand usually results in a violation of the GPDpolynomiality property. Note that this failure can not be fixed by taking into account the particlenumber changing processes. The guidance for an appropriate model comes from a perturbativecalculation in lowest order [30]. We employ the Yukawa theory and have the LC wave functionsfor four helicities, ψ ↑ +1 / ( X, k ⊥ ) = ψ ↓− / ( X, k ⊥ ) = (cid:16) M + mX (cid:17) ϕ ( x, k ⊥ ) , (10) ψ ↓ +1 / ( X, k ⊥ ) = k − ik X ϕ ( X, k ⊥ ) , ψ ↑− / ( X, k ⊥ ) = − k + ik X ϕ ( X, k ⊥ ) , (11) Of course, this task is straightforwardly done in a Lorentz covariant formalism [29].
3n terms of a scalar function ϕ ( X, k ⊥ ). This scalar function arises from the spectator propagatorin a triangle Feynman diagram [30, 31] and so the underlying Lorentz symmetry is respected. Wegeneralize ϕ ( X, k ⊥ ) by an adjustment of its power behavior p : ϕ ( X, k ⊥ ) = gM p √ − X X − p (cid:18) M − k ⊥ + m X − k ⊥ + λ − X (cid:19) − p − , (12)where M , λ and m are the proton, spectator, and quark masses, respectively. Note that the factor X − p takes care of the proper Lorentz behavior and that the Yukawa theory result is for p = 0.The GPDs are now evaluated in the outer region from the overlap representations (4) and (5), (cid:18) H − η − η E (cid:19) ( x ≥ η, η, t ) = (13)2 − ζ √ − ζ Z d k ⊥ π h ψ ↑∗ +1 / ( X ′ , k ′⊥ ) ψ ↑ +1 / ( X, k ⊥ ) + ψ ↑∗− / ( X ′ , k ′⊥ ) ψ ↑− / ( X, k ⊥ ) i , ∆ − i ∆ M E ( x ≥ η, η, t ) = (14) p − ζ Z d k ⊥ π h ψ ↑∗ +1 / ( X ′ , k ′⊥ ) ψ ↓ +1 / ( X, k ⊥ ) + ψ ↑∗− / ( X ′ , k ′⊥ ) ψ ↓− / ( X, k ⊥ ) i , for the two body wave functions (10)–(12). For p > k ⊥ integrals are finite, while for p = 0the representation (13) suffers from an ultraviolet divergence. We find for the GPD E : E ( x ≥ η, η, t ) = g (4 π ) p + 1)Γ( p + 1) Z dα [(1 − X )(1 − X ′ )] p +1 [ αα ] p (cid:0) mM + X − α (1 − X ′ ) ζ (cid:1)(cid:2) f ( X | α ) + f ( X ′ | α ) + αα (1 − X )(1 − X ′ ) t min − tM (cid:3) p +1 , (15)where t min = − ζ M / (1 − ζ ), α = 1 − α , and the mass terms are collected in f ( X | α ) = α (cid:26) (1 − X ) m M + X λ M − X (1 − X ) (cid:27) . (16)The result for H has a similar structure and will not be displayed for shortness.Since our model respects the underlying Lorentz symmetry, there must be now a possibility totransform the overlap result (15) into the form of the DD representation (9). In fact, this can besimply achieved by a linear variable transformation of the integration parameter α = 12 1 − η − x (cid:18) − y + x − yη (cid:19) (17)and removing the residual skewness dependence by using x = y + ηz . We arrive at E ( x ≥ η, η, t ) = (1 − x ) Z x + η ηx − η − η dyη e (cid:18) y, x − yη , t (cid:19) , (18)where the DD is given by e ( y, z, t ) = N (cid:0) mM + y (cid:1) ((1 − y ) − z ) p (cid:2) (1 − y ) m M + y λ M − y (1 − y ) − ((1 − y ) − z ) t M (cid:3) p +1 . (19)4ere we absorbed several factors, including g , in the normalization constant N . From the DD(19) and Eq. (8), we find E for the central region, arising from parton number changing processes: E ( − η ≤ x ≤ η, η, t ) = (1 − x ) Z x + η η dyη e (cid:18) y, x − yη , t (cid:19) . (20)The evaluation of the GPD H in terms of the DD h goes along the same line. The combination( H + E )( x, η, t ) can be written in the form of the DD representation (8) with( h + e )( y, z, t ) = N − p p ((1 − y ) − z ) p (cid:2) (1 − y ) m M + y λ M − y (1 − y ) − ((1 − y ) − z ) t M (cid:3) p + N h mM + y + y λ M + (2 − y ) m M i ((1 − y ) − z ) p (cid:2) (1 − y ) m M + y λ M − y (1 − y ) − ((1 − y ) − z ) t M (cid:3) p +1 . (21)Finally, the overall constant N is fixed by the normalization condition: Z − dx H ( x, η, t = 0) = Z dy Z − y − y dz ( h + y e )( y, z, t = 0) = 1 . (22)iii. We employ now our simple-minded model to evaluate form factors, TMDs, PDFs, and GPDsas net contributions of valence-like quarks in the proton, i.e., we deal with the isospin combination H ( x, η, t ) = H u val ( x, η, t ) − H d val ( x, η, t ) , E ( x, η, t ) = E u val ( x, η, t ) − E d val ( x, η, t ) . (23)We have only three model parameters, i.e., power behavior ( p ) of the LC wave function, spectator( λ ) and quark ( m ) masses. Our goal in doing so is to explore this model and find its restrictions.We start with the electromagnetic form factors, which are obtained from the Drell-Yan-Westformula or, equivalently, by integrating out the momentum fraction dependence in GPDs: F ( t ) = Z − dx H ( x, η, t ) , F ( t ) = Z − dx E ( x, η, t ) . (24)The Dirac form factor is normalized by F ( t = 0) = 1, cf. Eq. (22). The asymptotic drop off forlarge − t is estimated to be 1 / ( − t ) , according to field theoretically inspired [2] and phenomenolog-ical [3] model counting rules and the perturbative analysis [32]. This suggest to set p = 1. We usethe charge radius squared R = 6 dF ( t ) /dt | t =0 and the anomalous magnetic moment κ = F ( t = 0)of the proton to pin down the remaining two parameters. The following plausible mass valuesyield an agreement with experimental measurements, cf. Ref. [20]: p = 1 , λ = 0 .
75 GeV , m = 0 .
45 GeV ⇒ R = 0 .
76 fm , κ = 1 . . (25)The form factor F ( t ) fairly agrees with experimental data as shown in Fig. 1(a) and behaves5 ( t ) (a) − t [GeV ] (b) ( − t ) F ( t ) / κ F ( t ) − t [GeV ]Figure 1: The form factor F ( t ) (a) and the ratio ( − t ) F ( t ) /κF ( t ) with a cut-off for the endpointsingularities (b). Data are taken from Refs. [33, 35, 36]. for − t /
100 GeV as 1 / ( − t ) , however, in the large − t asymptotics it drops faster. We wouldconclude, in agreement with aforementioned counting rules, that F ( t ) decreases as 1 / ( − t ) . Thisfollows from counting the powers of ∆ ⊥ in Eq. (14), arising from k ⊥ -dependence of wave functions(10-12), and the vanishing of the overlap integral for E if spherical symmetry is restored in theasymptotic ∆ ⊥ → ∞ . However, we observe that the limit ∆ ⊥ → ∞ can not be taken before the k ⊥ integration, since divergences appear and render the integral to be infinite. This behavior showsup also in the DD (19), where the limit − t → ∞ causes end-point singularities at z = ± (1 − y ).Hence, we effectively find a F ( t ) ∼ / ( − t ) behavior and thus naive power counting fails. Weremind that the assumptions for the counting rules were carefully spelled out [2, 3, 32].End-point singularities arise also in the perturbative evaluation of F and their regularizationyields a logarithmical modification of the 1 / ( − t ) scaling [37, 38]. The experimental measure-ments indicate a F ( t ) /F ( t ) ∝ / √− t scaling [34, 35, 36], which might be also interpreted asa logarithmical modified F ( t ) /F ( t ) ∝ / ( − t ) scaling. Interestingly, the anomalous break downof the naive power counting in our scalar diquark model leads to a F ( t ) /F ( t ) ∝ const . scal-ing. Applying the recipe of end-point regularization to our result, i.e., imposing the constraint0 . ≤ − y − | z | , we easily can fit the experimental data as demonstrated in Fig. 1(b). Thisexercise should not be considered as a serious attempt to explain experimental data, rather as ademonstration showing that fitting can be done easily. One might wonder whatever this failureof the scalar diquark model is related to the oversimplified spin coupling and could be cured byinclusion of an axial-vector diquark, or it might simply reflect a wrong implementation for thelarge k ⊥ behavior of the LC wave functions.Our model also predicts TMDs and their k ⊥ -integrated PDFs outcome. Note that the TMDconcept has numerous fundamental issues, see, e.g., Ref. [39]. We might define here a unpolarizedvalence-like TMD in terms of the LC wave functions (10-12) overlap, where ζ = 0 and t = 0: q ( x, k ⊥ ) = g M p [( xM + m ) + k ⊥ ] (1 − x ) p +1 [ k ⊥ + (1 − x ) m + xλ − x (1 − x ) M ] p +2 . (26)6t large k ⊥ they fall off with 1 / ( k ⊥ ) p +1 and are suppressed in the large x region by (1 − x ) p +1 .Our model is similar in spirit to the spectator model utilized in Ref. [40] for the evaluation of k ⊥ -(un)integrated parton densities and fragmentation functions. There the fermionic propagatorsare replaced by quark-diquark form factors with a cut-off mass Λ, while in our case they are taketo be on-shell. Replacing m by Λ, we find that Eq. (26) is identical with the result (80) in Ref. [40].The valence-like PDF is obtained from (26) and is related to the GPD H : q ( x ) ≡ H ( x, η = 0 , t = 0) = πg M p +2 p (2 p + 1) (2 p − x +1) m M + pxmM + xλ M + x (2 px + x − xλ + (1 − x ) ( m − M x )) p +1 (1 − x ) p +1 . (27)For p = 1 we obtain the generic behavior of parton densities at large x . Note that the form factor F ( t ) falls at large − t with ( − t ) − p − . Hence, setting p = n − u − d , h x i = Z dx x q ( x ) , q ≡ q u val − q d val , (28)is h x i ≈ .
26 with our parameter specification (25). It nearly agrees with the value in the Gl¨uck,Reya, and Vogt parameterization [41], given at a low input scale µ = 0 . (to perturbativenext-to-leading order accuracy): h x i u val − h x i d val ≈ .
24. This amazing coincidence of momentumfractions should be considered rather as an accident. For x going to zero, the PDF (27) approachesa constant. This behavior we consider as an unrealistic feature of the diquark model. The small x -region, i.e., the high-energy limit might be understood in the Regge picture as an exchangeof mesons in the t − channel which leads to the expectation that the valence-like parton densitiesbehave as x − α (0) , where α (0) is the (effective) intercept of the meson trajectory. From the s -channelview, which we take, the true small x -behavior arises by summing up all Fock state components.We come now to the GPDs. First, we comment on our duality assumption between the partonic s -channel and mesonic like t -channel exchange. We could have added to the central region a term D ( x/η, t ) = θ ( | x | ≤ | η | ) d (cid:18) xη , t (cid:19) , with d (1 , t ) = d ( − , t ) = 0 (29)that entirely lives in the central region, vanishes at the cross over point and is anti-symmetric in x . An explicit GPD evaluation from the parton number changing processes confirms that sucha term is absent and so the underlying field theoretical model respects duality. However, such a D -term is needed to cure the common DD representation for the GPD E or H [42]: (cid:26) HE (cid:27) ( x, η, t ) = Z dy Z − y − y dz δ ( x − y − ηz ) (cid:26) he (cid:27) Rad ( y, z, t ) ± D ( x/η, t ) . (30)Our finings suggest that both e Rad ( y, z, t ) and d ( x/η, t ) have a cross talk. Indeed, they are twodifferent projections of our DD (19); the D -term is extracted from η → ∞ with fixed x/η [43]: d ( x, t ) = x Z −| x | dy N (cid:0) mM + y (cid:1) ((1 − y ) − x ) p (cid:2) (1 − y ) m M + y λ M − y (1 − y ) − ((1 − y ) − x ) t M (cid:3) p +1 . (31)7 ( x , η , t ) x (a) η x η (b)Figure 2: The GPD H ( x, η, t ) in the spectator model (8,19,21) (a) and factorized VGG ansatz (30,33)(b) with the parameter values in (25). The momentum transfer squared is set to t = t min − .
25 GeV . Interestingly, the coefficients in its Gegenbauer expansion [44] are rather small and sign alternating: d ( x, t = 0) ≈ (1 − x ) h . C / ( x ) − . C / ( x ) + 0 . C / ( x ) + · · · i . (32)The first coefficient is in astonishing agreement with the estimate of the chiral soliton model [45],given as 0.33 at a intrinsic scale of µ ∼ .
36 GeV . Note that the flavor singlet quark D -term inthe chiral soliton model is predicted to be negative and large in magnitude [44, 46, 45].In confronting GPDs with experimental data, one usually factorizes the t -dependence in theDD ansatz [47, 48, 44] (VGG refers to the popular code of Vanderhaeghen, Guichon, and Guidal): h VGG ( y, z, t ) = F ( t ) q ( y )1 − y Π VGG ( z/ (1 − y )) , Π VGG ( z ) = Γ (cid:0) b + (cid:1) √ π Γ( b + 1) (1 − z ) b . (33)The even profile function Π VGG ( z ) is chosen to be convex and normalized to R − dz Π VGG ( z ) = 1.Let us compare the spectator model (8,19,21) with the t -factorized VGG ansatz (30,33). Thelatter we make from the PDF (27) and D -term (32), multiplied with the form factor (24). For theprofile function we take Π VGG ( z ) = (3 / − z ), which also appears in our DD (21) with p = 1 at t = 0, cf. Ref. [47], and employ in both models the parameters (25). In Fig. 2 we show the shapeof GPDs versus x and η , where the momentum transfer squared is set to t = t min ( ξ ) − .
25. Thedifferences between the spectator model (a) and the t -factorized (b) GPDs are clearly visible atlarger values of η , where the later ones are broader in their x distribution and have a smaller value[see also below Fig. 3(c)] on the trajectory x = η , compared to the former ones. The importantdifference between the two models is in their t -dependence. In Fig. (3) we display in panel (a)the t -dependence of the ratio H ( η, η, t ) /H ( η, η,
0) on the trajectory x = η for various values of η . Obviously, the t -dependence is flattering out for larger values of η , while in the small η regionit even becomes steeper, compared to the t -factorized GPD ansatz (dashed line). Analogously, Such an ansatz might be improved by replacing the effective Regge intercept in the quark densities by the t − dependent Regge trajectory, which will only partly remove the factorized t -dependence. ( η , η , t ) / H ( η , η , ) − t η = 0 .
7= 0 .
5= 0 .
3= 0 . (a) R d xx n H ( x , , t ) R d xx n H ( x , , ) n = 3= 2= 1= 0 (b) − t ℑ m H ( ξ , t ) ξ (c) ℜ e H ( ξ , t ) ξ (d)Figure 3: The t -dependence of H ( x, η, t ) /H ( x, η, t = 0) at x = η in the t -factorized ansatz (30,33) (thindotted) and the spectator model within η = { . , . , . , . } (a)and for the Mellin moments n = { , , , } at η = 0 in the spectatormodel (b). Imaginary (c) and real (d) part of the amplitude (34) versus ξ , arising from the spectatormodel (solid) and factorized ansatz (dashed), where the dotted curve shows the real part without D -term. we find for the ratio R dx x n H ( x, , t ) / R dx x n H ( x, ,
0) of Mellin moments at η = 0 that the t -dependence becomes flatter with increasing spin n as shown in panel (b). Such a behavior wasalso seen in lattice calculations, see Ref. [49] and references therein.As in the case of PDFs, the small x -behavior of GPDs in a spectator model should be consideredas unrealistic. Since at x = η the momentum fraction X ′ vanishes in the overlap representation(13,14), we realize that on this trajectory realistic GPDs can be obtained only if the small X behavior of the wave functions is understood. This also means that one has to sum up all Fockstate components, see discussion in Refs. [9, 28]. Note that already the evolution to leading orderin the flavor singlet sector knows about the small x behavior, however, in the non-singlet sectorone has to perform a resummation of t -channel exchanges, perhaps, along the line as suggested inRef. [50]. For larger values of x on x = η one is forced to understand at the same time the largeand small X behavior of the wave functions and their interference.Having this warning in mind, we have now a look at the resulting amplitude, which appears9n the hard exclusive photon or ρ electroproduction to leading order of perturbative QCD: H ( ξ, t ) = Z − dx (cid:20) ξ − x − iǫ − ξ + x − iǫ (cid:21) H ( x, η = ξ, t ) . (34)The imaginary and real part of the amplitude versus ξ are shown in Fig. 3 (c) and (d), respectively,where we set t = t min − .
25. Note that the imaginary part is given by πH ( x = ξ, ξ, t ) and providesus the GPD shape on the trajectory x = ξ . The differences between the spectator (solid curve)and t -factorized (dashed curve) models are obvious. Both the imaginary and real part for thespectator model is (much) more enhanced at large ξ . For ξ → − ξ ), while for the t -factorized GPD ansatz an additional suppression factor appear:(1 − ξ )( − t ) − . (1 − ξ )( − t min ) − ∼ (1 − ξ ) . On the other hand, we observe that at smallervalues of ξ the imaginary part of the spectator model is getting smaller. We stress, however, thatsmall ξ -physics is not implemented in the model. The sign change of the real part in panel (d),somewhere in the valence quark region, is a feature that is also observed in other valence-like GPDansatze. The position of the zero, however, is floating and strongly depends on the chosen ansatz.At smaller values of ξ the real parts of both models approach to each other. The dotted curveshows the t -factorized ansatz without D -term, leading only to a slight constant shift.Concerning the results of our spectator model, we conclude that the t -dependence of crosssections for deeply electroproduction of mesons in the large x Bj region is getting flatter and that thecross section could be (much) larger than the prediction from a t -factorized GPD ansatz, cf. Figs. 3(c) and (d). This is consistent with preliminary measurements of the e − p ( P ) → e − p ( P ) ρ (0) crosssection, where the t -slope decreases if larger values of x Bj and Q are approached [51]. We find, forinstance, that the slope for the amplitude square |H ( ξ, t ) | , parameterized as e bt , decreases from b ≈ . − at x Bj = 0 . Q = 2 GeV to b ≈ . − at x Bj = 0 . Q = 5 GeV .iv. In this paper we derived from the overlap representation of LC wave functions the valence-likeGPDs and their relatives: proton form factors, TMDs, and PDFs. We generalized the LC wavefunctions from a scalar diquark spectator model in such a way that the non-manifest Lorentzbehavior of the LC wave functions is respected. This is the key to obtain the DDs from theoverlap representation in the partonic s -channel and then to restore the full GPD support.Our model fairly describes the Dirac form factor F ( t ) and the proton anomalous magneticmoment comes out correctly by a natural choice for the constituent quark and diquark masses.However, the model fails in the description of the t -dependence for the Pauli form factor F ( t ).Namely, its naive power counting for the large t -behavior is spoiled by end-point singularities thatappears at the branch point − t = ∞ . Unpolarized valence-like TMDs and PDFs, also obtainedin Ref. [40], have the expected generic behavior at large momentum fraction x and the averagevalue h x i fairly agrees with the result of Ref. [41], given at a low input scale.10he GPD models satisfy by construction the positivity and polynomiality constraints. Thefound DD representations (8) naturally completes the polynomiality condition and avoids so a‘misleading’ D -term phenomenology. Another important characteristic property of the model isthat the t -dependence of GPDs, resulting from the DDs (21) and (19), is washed out at large x .This behavior is rooted in the fact that the variables t and x are intrinsically correlated because thetransverse and longitudinal degrees of freedom in the wave functions are tied by hidden Lorentzsymmetry. This flattering of t -dependence also appears on the trajectory x = η and therefore itcan be confronted with experimental measurements. Note that the t -behavior of GPDs is onlytested in lattice calculations for η = 0. From the overlap representation it is clear that the t -dependence of GPDs and the k ⊥ -dependence of TMDs are closely related to each other, sinceboth arise from the k ⊥ -dependence of wave functions, a recent discussion is given in Ref. [52].It is in the nature of a spectator model that it fails to describe the small momentum fractionbehavior, which arises from the summation over all partonic Fock state components. Hence, thereis a potential problem for the predicting power of such models for GPDs on the trajectory x = η ,accessible in experiments. Here a GPD arises from the interference of the LC wave function at theextreme limit of vanishing momentum fraction with that of a non-vanishing momentum fraction,controlled by the skewness parameter. Therefore, even the limit x = η → t -dependence in the ansatz for the DD and improve its failure at small y , i.e, small x for the resulting GPD, by hand. Guided by Eq. (19), we would suggest for e , e.g., the ansatz: e ( y, z, t ) = q E ( y, t )(1 − y ) N ( b, p, α ) h (1 − y ) m M + y λ M − y (1 − y ) i P (cid:2) (1 − y ) m M + y λ M − y (1 − y ) − ((1 − y ) − z ) t M (cid:3) P [(1 − y ) − z ] b (1 − y ) b +1 . (35)The Regge improved PDF analog, generically parameterized by its large and small x behavior q E ( x, t ) = κ (1 − α (0))Γ(2 − α ( t ) + β )Γ(2 − α ( t ))Γ(1 + β ) x − α ( t ) (1 − x ) β , Z dx q E ( x, t = 0) = κ , (36)where q E ( x, t ) is normalized at t = 0 to the anomalous magnetic moment κ . The normalization N ( b, P, α ) = Γ (cid:0) b + (cid:1) Γ(2 − α (0) + β )Γ(2 − P − α ( t ) + β ) √ π Γ( b + 1)Γ(2 − P − α (0) + β )Γ(2 − α ( t ) + β ) (37)is introduced in such a way that the t -dependence in the form factor roughly factorizes as 1 / (1 − α ( t )), resulting from the Regge behavior, times an impact form factor, behaving as ( − t ) − P for t → −∞ . The parameter b adjusts the power behavior of E ( ξ, ξ, t ) ∼ (1 − ξ ) b at large ξ andfixed t . The parameters should be fixed from fitting of observables; their generic values read in11ccordance with Regge phenomenology, counting rules [2, 3, 32, 53], and the spectator model: α ( t ) ∼ . . t GeV − , β ∼ , P ∼ , b ∼ , λ ∼ . , m ∼ . . The model features, we spelled out here in momentum fraction representation, are also imple-mented in the Mellin space GPD ansatz [14, 15]. We emphasize that the ansatz (35) is still notoptimal, since a flexible adjustment of the resulting magnitude for the amplitude at given t , inparticular at small ξ , is not incorporated and positivity constraints are no more automaticallysatisfied. A more detailed discussion of improved GPD ansatze should be given somewhere else.D.S.H. thanks the Institute for Theoretical Physics II at the Ruhr-University Bochum, in particularK. Goeke, for the hospitality at the stages of this work. 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