aa r X i v : . [ h e p - ph ] N ov GPDs with ζ = 0 Matthias Burkardt
Department of Physics, New Mexico State University, Las Cruces, NM 88003-0001, U.S.A. (Dated: November 8, 2018)We revisit the light-cone wave function representation for generalized parton distributions with ζ = 0. After translating the t -slope into a ∆ ⊥ -slope, the two-dimensional Fourier transform of GPDsis interpreted as the transition matrix element as a function of the separation between the activequark and the center of momentum of the spectators. In the limit x → ζ it is discussed how thisinformation can be used to learn about the dependence of the mean separation between the activequark and the spectators on the momentum fraction carried by the active quark. PACS numbers:
I. INTRODUCTION
Hard exclusive processes, such as Deeply Virtual Compton Scattering (DVCS), γ ∗ p −→ γp , where γ ∗ is a virtualphoton with virtuality q = − Q <
0, have emerged as a novel probe for hadron structure. In the Bjorken limit, andfor a momentum transfer t to the proton that is much less than Q , the DVSC amplitude factorizes into a convolutionof the Compton amplitude off a quark, constituting the hard part, and a quark correlation function, constituting thesoft part [1, 2, 3]. The latter is parameterized by Generalized Parton Distributions GPDs, through their dependenceon the Bjorken variable ζ = Q p · q , and the momentum fraction x of the active quark before being struck by the virtualphoton. A physical interpretation for GPDs is most easily available in the light-cone framework [4] where, for x > ζ they represent the probability amplitude that the proton remains intact after a quark carrying momentum fraction x absorbs a longitudinal momentum − ζ (in units of the initial proton momentum) and an invariant momentumtransfer t . As in Deep-Inelastic Scattering (DIS), the variable Q has the interpretation of the spatial resolution.However, since GPDs provide information about the distribution of partons in impact parameter space [5, 6, 7],the Q dependence in DVCS not only provides the scale dependence, but also the ‘pixel-size’ for the spatial imagesobtained from Fourier transforming GPDs. Unfortunately, a probabilistic interpretation for the Fourier transformsof GPDs is restricted to ζ = 0 [8, 9]. Since the probabilistic interpretation facilitates the development of intuitivemodels for GPDs, most phenomenological models for GPDs are more reliable for ζ = 0, and utilizing these models for ζ = 0 gives rise to uncertainties that are difficult to quantify. This is very unfortunate since DVCS typically providesconstraints only for GPDs with ζ = 0. In particular, the imaginary part of the DVCS amplitude is only sensitive to x = ζ ℑ (cid:8) T DV CS (cid:9) ∝ GP D ( x = ζ, ζ, t, Q ) , (1)The real part appears in a convolution integral ℜ (cid:8) T DV CS (cid:9) ∝ Z dx GP D ( x, ζ, t, Q ) x ± ζ , (2)where the factor x ± ζ emphasizes the regions x ≈ ± ζ . In either case, understanding the vicinity of x ≈ ± ζ appearsto be crucial for understanding the DVCS amplitude. The goal of this note is to develop some intuition about GPDsin this important regime. More specifically, we will consider the t -slopes of GPDs for x ≈ ζ and what can be learnedfrom them. II. LIGHT-CONE WAVE FUNCTION REPRESENTATION FOR GPDS
Although the primary focus of this work is x ≈ ζ , we consider first x > ζ , where GPDs are diagonal in Fock space.The regime x = ζ is then approached through a limiting procedure. For x > ζ simple overlap representations forGPDs in terms of light-cone wave functions exist that resemble overlap integrals for form factors in non-relativisticsystems [4, 10] GP D ( x, ζ, t ) = X n,λ i (1 − ζ ) − n Z n Y i =1 d x i d k ⊥ ,i π π δ − n X j =1 x j δ n X j =1 k ⊥ j δ ( x − x ) (3) × ψ s ′ ( n ) ( x ′ i , k ′⊥ i , λ i ) ∗ ψ s ( n ) ( x i , k ⊥ i , λ i ) , (4)where GP D ( x, ζ, t ) = √ − ζ − ζ H ( x, ζ, t ) − ζ (cid:16) − ζ (cid:17) √ − ζ E ( x, ζ, t ) (5)for s ′ = s , and GP D ( x, ζ, t ) = 1 √ − ζ ∆ − i ∆ M E ( x, ζ, t ) (6)for s ′ = ↑ and s = ↓ , and ∆ is the transverse momentum transfer. The arguments of the final state wave function are x ′ = x − ζ − ζ and k ′⊥ = k ⊥ − − x − ζ ∆ ⊥ for the active quark, and x ′ i = x i − ζ and k ′⊥ = k ⊥ + x i − ζ ∆ ⊥ for the spectators i = 2 , ..., n .In order to elucidate the essential steps, we study first the simple case of a two-particle system (e.g. quark plusdiquark), where we consider light-cone wave functions as a function of the distance between the active quark and thespectator ˜ ψ s ( x, r ⊥ ) = Z d k ⊥ π ψ s ( x, k ⊥ ) e i k ⊥ · r ⊥ . (7)Inserting the position space wave function (7) diagonalizes the transverse part of the overlap integral in Eq. (4),yielding Z d k ⊥ ψ s ′ ( x ′ , k ′⊥ ) ∗ ψ s ( x, k ⊥ ) = Z d r ⊥ ˜ ψ s ′ ( x ′ , r ⊥ ) ∗ ˜ ψ s ( x, r ⊥ ) e − i − x − ζ r ⊥ · ∆ ⊥ (8)For ζ → b ⊥ = (1 − x ) r ⊥ = r ⊥ − R ⊥ is the separation of the active quark fromthe center of momentum R ⊥ ≡ x r ⊥ + (1 − x ) r ⊥ .For the general case, we also switch to transverse position ψ s ( n ) ( x i , k ⊥ i , λ i ) = Z n Y i =1 d r ⊥ i π e − i k ⊥ i · r ⊥ i ˜ ψ s ( n ) ( x i , r ⊥ i , λ i ) . (9)Since we are dealing with plane wave states, one needs to be careful with the normalization of these states and a morecareful treatment should involve working with wave packets. Here we will skip these tedious steps that have beenstudied carefully in Refs. [5, 6] and immediately insert (9) into (4), yielding GP D ( x, ζ, t ) = X n,λ i (1 − ζ ) − n Z n Y i =1 d r ⊥ i π ˜ ψ s ′ ( n ) ( x ′ i , r ⊥ i , λ i ) ∗ ˜ ψ s ( n ) ( x i , r ⊥ i , λ i ) e − i − ζ ( r ⊥ − R ⊥ ) · ∆ ⊥ . (10)where R ⊥ = P i x i r ⊥ i is the transverse center of momentum of all partons in the initial state.Since the transverse center of momentum changes in the process [6], it is useful to replace it by the separationbetween the active quark and the center of momentum of the spectators R ⊥ s , using r ⊥ ≡ r ⊥ − R ⊥ s = 11 − x ( r ⊥ − R ⊥ ) (11)and one finds GP D ( x, ζ, t ) = X n,λ i (1 − ζ ) − n Z n Y i =1 d r ⊥ i π ˜ ψ s ′ ( n ) ( x ′ i , r ⊥ i , λ i ) ∗ ˜ ψ s ( n ) ( x i , r ⊥ i , λ i ) e − i − x − ζ ( r ⊥ − R ⊥ s ) · ∆ ⊥ . (12)While GPDs for x > ζ > ⊥ center of momentum [6]. However, as the momentumcarried by the active quark changes between initial and final state, so does the location of the transverse center ofmomentum [7]. Therefore, even though the (absolute) ⊥ positions of the active quark/spectators remain unchanged,their separation from the ⊥ center of momentum changes since the latter does. For the physical interpretation ofGPDs in the case of ζ = 0, working with relative ⊥ position coordinates (i.e. relative to each other) rather than impactparameter (measured relative to the ⊥ center of momentum may thus be preferable. Indeed, the discussion aboveillustrates that, for nonzero ζ , the Fourier transform of GPDs w.r.t. the transverse momentum transfer ∆ ⊥ yieldsinformation about the transition matrix element between the initial and final state, when the ⊥ distance between theactive quark and the center of momentum of the spectators is r ⊥ . More precisely, − ζ − x r ⊥ is Fourier conjugate to ∆ ⊥ ,and for x = ζ , the variable conjugate to the ∆ ⊥ is just r ⊥ . III. GPDS FOR x → ζ When x = ζ , the coefficient multiplying r ⊥ · ∆ ⊥ in the exponent in Eq. (12) becomes equal to one, i.e. in that limitthe Fourier transform of GPDs w.r.t. ∆ ⊥ yields the dependence of the overlap matrix element on the separation r ⊥ between the active quark and the center of momentum R ⊥ s of the spectators. While for ζ = 0 it is the separation fromthe center of momentum of the whole hadron that sets the scale, it is the separation from then center of momentumof the spectators that matters for x = ζ . In order to utilise the above observations in the interpretation of GPDs, wenote that [4] − t = ζ M + ∆ ⊥ − ζ . (13)Therefore, if the t -dependence of GPDs is parameterized in the form GP D ∝ e Bt (14)one finds for the ∆ ⊥ -dependence GP D ∝ e − B ⊥ ∆ ⊥ (15)with B ⊥ = − ζ B . Thus, even if the ∆ ⊥ -slope (described by B ⊥ ) remains finite as ζ →
1, the t -slope (described by B ) goes to zero. This purely kinematical effect arises from the relation between t and ∆ ⊥ (13) with ζ = 1 − p + ′ p + fixed.Since ∆ ⊥ is the momentum space variable conjugate to r ⊥ = r ⊥ − R ⊥ s (for x = ζ ), it is thus important totranslate the t -dependence of GPDs first into a ∆ ⊥ -dependence before attempting to interpret the data.What should one thus expect for the ζ -dependence of GPDs at x = ζ ? The relevant GPDs are proportional to thethe overlap between on initial state where the active quark carries momentum fraction ζ and a final state where theactive quark carries almost no momentum. Intuitively one would expect that the average separation between activequark and the spectators increases as the momentum fraction of the active quark decreases, i.e. in this case the finalstate wave function should be smaller than the initial state wave function.In general, the overlap integral describing the GPDs (4) depends not only on the distribution of the active quarkbut also on that of the spectators. However, it appears reasonable to assume that the spectator wave function (for agiven position of the spectator center of momentum) does not depend very strongly on the position of the active quarkwhen the active quark is far away from the spectators. In the following we will thus make the simplifying assumptionthat the overlap integral for the spectators (at fixed x and ζ ) does not depend on the separation of the active quarkfrom the spectators. This does not mean that the spectators wave function is point-like!In order to qualitatively understand how the above overlap integrals depend on ζ (for , we rescale all momentumfractions in units of the final state momentum, i.e. the initial state hadron carries momentum 1 / (1 − ζ ) and thefinal state hadron carries momentum 1. As the active quark carries momentum fraction 0, nothing in the final statedepends on ζ and hence the ζ -dependence arises from the change in the initial state wave function, and the resultingchange in the overlap integrals. This observation suggests the following interpretation for the ζ dependence of the∆ ⊥ -slope of GPDs. For instance, if the ∆ ⊥ -slope decreases with increasing ζ , that would be an indication that themean separation between the active quark and the spectators decreases with the momentum fraction carried by theactive quark.If one neglects the ∆ ⊥ -dependence of the overlap integral for the spectators, one can use this reasoning to extractthe ‘size’ (mean separation of the active quarks from the spectators) as a function of the momentum fraction carriedby the active quark. For example, when the initial and final state wave function are proportional to e − r ⊥ R and e − r ⊥ R then the effective radius appearing in the product is the harmonic mean of the rms radii of the individual wavefunctions squared R eff = (cid:16) R + R (cid:17) . IV. SUMMARY
For ζ = 0, the two dimensional Fourier transform of GPDs is more easily interpretable if one introduces theseparation r ⊥ between the active quark and the center of momentum of the spectators, as this variable is the samein the initial and final state of the hadronic matrix element defining the GPDs. The r ⊥ dependence of the matrixelement is obtained by Fourier transforming GPDs with a factor e − i − x − ζ r · ∆ ⊥ , i.e. for x = ζ the variable r is Fourierconjugate to ∆ ⊥ .The mean r ⊥ , and hence the ∆ ⊥ -slope of GPDs should be a typical hadronic scale. Therefore the t -slope, which isrelated to the ∆ ⊥ -slope by a kinematic factor of 1 − ζ , should go to zero as ζ →
1, even if the wave function does notbecome point-like. The t -slope divided by 1 − ζ can be used to study how the mean separation of the active quarkfrom the center of momentum of the spectators varies with ζ . Intuitively, one would expect this ‘size’ to decreasewith ζ . Application of the above procedure to deeply-virtual meson production indeed yields a size that decreaseswith increasing ζ [11]. DVCS data for the t -slope [12] also shows a decrease with increasing ζ . Acknowledgements:
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