Electroproduction of heavy quarkonia: significance of dipole orientation
EElectroproduction of heavy quarkonia: significance of dipole orientation
B. Z. Kopeliovich , ∗ M. Krelina , , † and J. Nemchik , ‡ Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Avenida Espa˜na 1680, Valpara´ıso, Chile FNSPE, Czech Technical University in Prague, Bˇrehov´a 7, 11519 Prague, Czech Republic Physikalisches Institut, University of Heidelberg,Im Neuenheimer Feld 226, 69120 Heidelberg, Germany and Institute of Experimental Physics SAS, Watsonova 47, 04001 Koˇsice, Slovakia (Dated: February 12, 2021)The differential cross section dσ/dq of diffractive electroproduction of heavy quarkonia on protonsis a sensitive study tool for the interaction dynamics within the dipole representation. Knowledgeof the transverse momentum transfer (cid:126)q provides a unique opportunity to identify the reaction plane,due to a strong correlation between the directions of (cid:126)q and impact parameter (cid:126)b . On top of that, theelastic dipole-proton amplitude is subject to a strong correlation between (cid:126)b and dipole orientation (cid:126)r .Most of models for b -dependent dipole cross section either completely miss this information, or makeunjustified assumptions. We perform calculations basing on a realistic model for (cid:126)r - (cid:126)b correlation,which significantly affect the q -dependence of the cross section, in particular the ratio of ψ (cid:48) (2 S )to J/ψ yields. We rely on realistic potential models for the heavy quarkonium wave function, andthe Lorentz-boosted Schr¨odinger equation. Good agreement with data on q -dependent diffractiveelectroproduction of heavy quarkonia is achieved. PACS numbers: 14.40.Pq,13.60.Le,13.60.-r
I. INTRODUCTION
Elastic real and virtual photoproduction of heavy quarkonia on protons is an effective tool for study of the space-timepattern of diffraction mechanism, as well as related aspects of quantum-chromodynamics (QCD). The long-standinghistory of our investigation of the photo- and electroproduction of vector mesons [1–18] has provided the foundationsfor theoretical interpretation and has contributed in an essential way to understanding of this process within theQCD color dipole formalism. Such a formalism has been frequently used recently in the literature with only minorimprovements in corresponding model descriptions.The most of experimental data are available only for the cross section integrated over transfer momenta. However,an additional information about the dynamics and properties of the diffraction mechanism itself can be acquired bystudying also the momentum transfer dependence of real and virtual photoproduction cross sections. Knowledge ofthe the transverse momentum transfer (cid:126)q provides a unique opportunity to identify the reaction plane, because theFourier transformation from impact parameters (cid:126)b to momentum representation leads to a strong correlation between (cid:126)b and (cid:126)q .Within the light-front (LF) color dipole formalism, the amplitude for electroproduction of heavy vector mesonswith the transverse momentum transfer (cid:126)q can be expressed in the factorized form, A γ ∗ p → V p ( x, Q , (cid:126)q ) = (cid:10) V | ˜ A| γ ∗ (cid:11) = (cid:90) d r (cid:90) dα Ψ ∗ V ( (cid:126)r, α ) A Q ¯ Q ( (cid:126)r, x, α, (cid:126)q ) Ψ γ ∗ ( (cid:126)r, α, Q ) (1.1)using thus the advantage of the ( (cid:126)r, α ) diagonalization of the scattering matrix ˜ A with the normalization ( dσ/dq ) q =0 = |A| / π . Here A Q ¯ Q ( (cid:126)r, x, α, (cid:126)q ) is the amplitude for elastic scattering of the color dipole on the nucleon target, Ψ V ( r, α ) ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ h e p - ph ] F e b is the LF wave function for heavy quarkonium and Ψ γ ∗ ( r, α, Q ) is the LF distribution of the Q ¯ Q Fock component ofthe real ( Q = 0) or virtual ( Q >
0) photon, where Q is the photon virtuality and the Q ¯ Q fluctuation (dipole) hasthe transverse size (cid:126)r . The variable α is the fractional LF momentum carried by a heavy quark or antiquark from a Q ¯ Q Fock component of the photon and x = ( m V + Q − t ) / ( W + Q ) = ( m V + Q − t ) /s , where m V is the quarkoniummass, W is c.m. energy of the photon-nucleon system and t = − q .In Eq. (1.1) the amplitude A Q ¯ Q ( (cid:126)r, x, α, (cid:126)q ) can be written in terms of the gluon density matrix F ( x, (cid:126)k, (cid:126)k (cid:48) , (cid:126)q ), whichis proportional to the imaginary part of the non-forward gluon-nucleon scattering amplitude. In the limit of (cid:126)q → F ( x, (cid:126)k, (cid:126)k (cid:48) , (cid:126)q = 0) = F ( x, (cid:126)k, (cid:126)k ) ≡ F ( x, k ) = ∂ G ( x, k ) /∂ log k , and the general formula for A Q ¯ Q ( (cid:126)r, x, α, (cid:126)q ) at (cid:126)q = 0 should access the standard expression for thedipole cross section, σ Q ¯ Q ( r, x ) = 4 π (cid:90) d kk (cid:104) − e − i(cid:126)k.(cid:126)r (cid:105) α s ( k ) F ( x, k ) = π r (cid:90) dk k (cid:2) − J ( k r ) (cid:3) ( k r ) α S ( k ) F ( x, k ) , (1.2)which leads for small dipole sizes r (cid:28) r , where r ∼ . σ Q ¯ Q ( r, x ) = π r α S ( r ) G ( x, k s ) , (1.3)where the gluon structure function is scanned at the factorization scale k s ∼ A s /r , with the large factor A s ∼ ÷ A Q ¯ Q ( (cid:126)r, x, α,(cid:126)b ) related to (cid:126)q -dependent amplitude via Fourier transform, A Q ¯ Q ( (cid:126)r, x, α, (cid:126)q ) = (cid:90) d b e − i(cid:126)b · (cid:126)q A Q ¯ Q ( (cid:126)r, x, α,(cid:126)b ) (1.4)with the correct reproduction of the dipole cross section at (cid:126)q = 0, σ Q ¯ Q ( r, x ) = Im A Q ¯ Q ( (cid:126)r, x, α, (cid:126)q = 0) = 2 (cid:90) d b Im A NQ ¯ Q ( (cid:126)r, x, α,(cid:126)b ) , (1.5)where the partial dipole elastic amplitude Im A NQ ¯ Q ( (cid:126)r, x, α,(cid:126)b ) represents the interaction of the Q ¯ Q dipole with a nucleontarget at impact parameter (cid:126)b .Consequently, the next very important step is to determine the partial amplitude Im A NQ ¯ Q ( (cid:126)r, x, α,(cid:126)b ) including aproper correlation between the color dipole orientation (cid:126)r and the impact parameter of a collisions (cid:126)b . There areseveral widely used models for the b -dependent dipole cross sections with additional b -dependent part giving so thepartial dipole amplitude without an adequate (cid:126)b - (cid:126)r correlation, i.e. assuming usually that the scattering amplitudeis independent of the angle between the vectors (cid:126)b and (cid:126)r (see Ref. [23] for b-IPsat model, Ref. [24] for b-Sat model,Ref. [25, 26] for b-CGC model, Ref. [27] for b-BK model, for example).In the present paper we investigate for the first time the momentum transfer dependence of differential cross sectionsin elastic photo- and electroproduction of heavy quarkonia on protons using a proper color dipole orientation withoutany approximation within standard phenomenological models for dipole cross sections of the saturated form. Thecorresponding partial dipole elastic amplitude is based on our previous studies [28–31] and includes a proper correlationbetween vectors (cid:126)b and (cid:126)r as is described in Sec. II. Here we shortly illustrate how the color dipole orientation lookslike within the simplified Born approximation. In the next Section III, we present the explicit form for the partial Q ¯ Q -proton amplitude with parameters corresponding to GBW [32, 33] and BGBK [34] saturation models for thedipole cross section. Section IV contains expressions for calculation of t -dependent differential cross sections as well asforward diffraction slopes within the LF color dipole formalism. In order to exclude a spurious D -wave admixture, herewe treat a simple non-photon-like structure of the V → Q ¯ Q transition in the Q ¯ Q rest frame as in our previous studies[14–18]. This requires to perform, besides the standard Lorentz boost [35] to the LF frame of radial components ofquarkonium wave functions, also transformation of the corresponding spin-dependent parts known as the Melosh spinrotation [36]. The next Sec. V is devoted to comparison of our results for dσ γ ∗ p → J/ψp /dt with available data from H1and ZEUS experiments at HERA. Here we also present predictions for various quarkonium states, as well as for the ψ (cid:48) (2 S )-to- J/ψ (1 S ) and Υ (cid:48) (2 S )-to-Υ(1 S ) ratios R V (cid:48) /V ( t ) = { dσ γ ∗ p → V (cid:48) p /dt } / { dσ γ ∗ p → V p /dt } that can be confirmedby future measurements. Finally, the last Sec. VI contains a summary with the main concluding remarks. II. PARTIAL DIPOLE AMPLITUDE IN BORN APPROXIMATION
It was shown that azimuthal asymmetry of pions [31], as well as direct photons [28–30] in pp and pA collisions isbased on the correlation between the color dipole orientation and the impact parameter of a collision. In the presentpaper we analyse the impact of such correlation on the momentum transfer dependence of differential cross sectionsof heavy quarkonium electroproduction on protons, γ ∗ p → V p .For a colorless heavy quark Q ¯ Q photon fluctuation (dipole) of transverse separation (cid:126)r with the impact parameter (cid:126)b of its center of gravity, the corresponding interaction of the Q ¯ Q dipole can occur due to difference between impactparameters of Q and ¯ Q relative to the scattering center. This eliminates the production of any Q ¯ Q photon componentwith the same impact parameter from the target related to Q and ¯ Q independently of the magnitude of (cid:126)r . Thus thedipole interaction vanishes if (cid:126)r ⊥ (cid:126)b and is maximal if (cid:126)r (cid:107) (cid:126)b . Such a correlation between the vectors (cid:126)r and (cid:126)b can beseen in terms of the dipole partial elastic amplitude A qQ ¯ Q ( (cid:126)r,(cid:126)b ) describing the dipole interaction with a quark in Bornapproximation (two-gluon exchange model),Im A qQ ¯ Q ( (cid:126)r,(cid:126)b ) = N (2 π ) (cid:90) d k d k (cid:48) ( k + m g )( k (cid:48) + m g ) (cid:104) e i(cid:126)k · ( (cid:126)b + (cid:126)r/ − e i(cid:126)k · ( (cid:126)b − (cid:126)r/ (cid:105) (cid:104) e i(cid:126)k (cid:48) · ( (cid:126)b + (cid:126)r/ − e i(cid:126)k (cid:48) · ( (cid:126)b − (cid:126)r/ (cid:105) , = N (cid:20) K (cid:18) m g (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)b + (cid:126)r (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) − K (cid:18) m g (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)b − (cid:126)r (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:21) , (2.1)where m g represents an effective gluon mass accounting for the confinement and other nonperturbative effects; K ( x )is the modified Bessel function of the second kind. Here, for simplicity, we assume the same longitudinal momenta of Q and ¯ Q . The above Eq. (2.1) clearly demonstrates a correlation between (cid:126)r and (cid:126)b , as well as a vanishing of A qQ ¯ Q ( (cid:126)r,(cid:126)b )when (cid:126)b · (cid:126)r = 0.The Born approximation for the dipole partial amplitude is rather crude since does not lead to energy dependentdipole cross section σ Q ¯ Q ( r, x ). For this reason, it is worth switching to a more reliable model for A qQ ¯ Q ( (cid:126)r,(cid:126)b ) without anyapproximation. However, another alternative way how to include the energy dependence is based on an improvementof the Born approximation with corresponding modification of the coefficient N ⇒ N ( x ) in Eq. (2.1) as is describedin Ref. [31]. III. PARTIAL DIPOLE AMPLITUDE IN THE SATURATION MODEL
The partial dipole elastic amplitude has been introduced in Ref. [30] within the standard model for the dipole crosssection σ Q ¯ Q of a conventional saturated form corresponding to various phenomenological parametrizations proposedin the literature (see [8, 23, 25, 32, 34], for example). They are based on the fits to the HERA DIS data and thusincludes contributions from higher order perturbative corrections, as well as nonperturbative effects. For a Q ¯ Q dipoleinteracting with a proton at impact parameter (cid:126)b , such dipole amplitude A NQ ¯ Q reads [30],Im A NQ ¯ Q ( (cid:126)r, x, α,(cid:126)b ) = 112 π (cid:90) d k d k (cid:48) k k (cid:48) (cid:112) α s ( k ) α s ( k (cid:48) ) F ( x, (cid:126)k, (cid:126)k (cid:48) ) e i(cid:126)b · ( (cid:126)k − (cid:126)k (cid:48) ) × (cid:16) e − i(cid:126)k · (cid:126)rα − e i(cid:126)k · (cid:126)r (1 − α ) (cid:17) (cid:16) e i(cid:126)k (cid:48) · (cid:126)r α − e − i(cid:126)k (cid:48) · (cid:126)r (1 − α ) (cid:17) , (3.1)where F ( x, (cid:126)k, (cid:126)k (cid:48) ) is the generalized unintegrated gluon density. In the two-gluon exchange model, Eq. (2.1), weassumed that α = 1 / Q and ¯ Q are equally distant from the dipole center of gravity.The shape of F ( x, (cid:126)k, (cid:126)k (cid:48) ) in Eq. (3.1) has been determined by comparing with the saturated form of the dipole crosssection, σ Q ¯ Q ( r, x ) = σ (cid:18) − exp (cid:104) − r R ( x ) (cid:105)(cid:19) , (3.2)using expression Eq. (1.2). This gives the off-diagonal unintegrated gluon density of the form [30], F ( x, (cid:126)k, (cid:126)k (cid:48) ) = 3 σ π (cid:112) α s ( k ) α s ( k (cid:48) ) k k (cid:48) R ( x ) exp (cid:104) − R ( x ) ( k + k (cid:48) ) (cid:105) exp (cid:104) − R N ( x )( (cid:126)k − (cid:126)k (cid:48) ) (cid:105) , (3.3)with a correct relation to the diagonal gluon density as F ( x, (cid:126)k, (cid:126)k (cid:48) = (cid:126)k ) = F ( x, k ). Adopting the GBW dipole model[32, 33], for example, the above parameters in Eq. (3.2) read: σ = 23 .
03 mb, R ( x ) = 0 . × ( x/x ) . with x = 3 . × − .From Eq. (3.3) one can obtain explicitly the (cid:126)b -dependent partial dipole-proton elastic amplitude performing inte-gration in Eq. (3.1),Im A NQ ¯ Q ( (cid:126)r, x, α,(cid:126)b ) = σ π B ( x ) (cid:40) exp (cid:34) − (cid:2) (cid:126)b + (cid:126)r (1 − α ) (cid:3) B ( x ) (cid:35) + exp (cid:34) − ( (cid:126)b − (cid:126)rα ) B ( x ) (cid:35) − (cid:34) − r R ( x ) − (cid:2) (cid:126)b + (1 / − α ) (cid:126)r (cid:3) B ( x ) (cid:35)(cid:41) , (3.4)where B ( x ) = R N ( x )+ R ( x ) /
8. Consequently, one can calculate the forward ( t = 0) t -slope of the elastic dipole-protoncross section as, B Q ¯ Q ( r, x ) = 12 (cid:104) b (cid:105) = 1 σ Q ¯ Q ( r, x ) (cid:90) d b b Im A NQ ¯ Q ( (cid:126)r, x, α,(cid:126)b ) . (3.5)In Eq. (3.4) the function B ( x ) can be simply determined from Eq. (3.5) assuming α = 1 /
2. Then we have, B Q ¯ Q ( r, x ) = 1 σ Q ¯ Q ( r, x ) (cid:90) d b b Im A NQ ¯ Q ( (cid:126)r, x, α = 1 / ,(cid:126)b ) = B ( x ) + r (cid:2) − e − r /R ( x ) (cid:3) , (3.6)where the magnitude of B Q ¯ Q ( r, x ) is probed at the well known scanning radius [7] r ≈ r S with r S = Y / (cid:112) Q + m V ,and factors Y have been determined in Ref. [14] for electroproduction of various heavy quarkonium states. However,treating electroproduction of heavy quarkonia, one can safely rely on the nonrelativistic limit of α = 1 / Y ≈ Q (cid:29) m V lead to a slow rise of Y arriving at the value Y ≈ Q = 100 GeV .The magnitude of B Q ¯ Q ( r = r S , x, Q ) in Eq. (3.6) can be associated with the diffraction slope B ( γ ∗ → V, x, Q ) ≡ B V ( x, Q ) = B Q ¯ Q ( r S , x, Q = 0) − B ln (cid:2) ( m V + Q ) /m J/ψ (cid:3) , where B Q ¯ Q ( r S , s, Q = 0) with s = ( m V − t ) /x conforms with the standard Regge form, B Q ¯ Q ( r S , s, Q = 0) ≡ B J/ψ ( s ) = B + 2 α (cid:48) (0) ln( s/s ). Here the parameters B = 2 . − , α (cid:48) = 0 .
133 GeV − and s = 1 GeV have been obtained from the fit of the combined H1 [37, 38]and ZEUS [39, 40] data. The value B = 0 .
45 GeV − has been determined in Ref. [15] analyzing electroproductionof J/ψ (see also Ref. [41]).The shape of the dipole amplitude (3.4) correctly reproduces at (cid:126)q = 0 the dipole cross section according to Eq. (1.5).Moreover, such (cid:126)b -dependent amplitude represents a source of an unique and additional information about details of theinteraction mechanism since contains the color dipole orientation, which can be adapted for various phenomenologicalmodels for σ Q ¯ Q ( r, x ) of a realistic saturated form given by Eq. (3.2). However, there is a restriction of a simple GBWmodel related to an absence of the DGLAP evolution at large scales. In order to eliminate this shortcoming, besidesthe GBW dipole model, we take into account also the BGBK dipole model [34] with the following modified parameterin Eq. (3.4), R ( x ) ⇒ R ( x, µ ) = 4 Q s ( x, µ ) = σ N c π α s ( µ ) x g ( x, µ ) , µ = Cr + µ , (3.7)where Q s is the saturation scale and the gluon distribution function x g ( x, µ ) is obtained as a solution of the DGLAPevolution equation acquiring the subsequent parametrization at the initial scale Q = 1 GeV , x g ( x, Q ) = A g x − λ g (1 − x ) . . (3.8)Fitting procedure of the HERA data leads to the following model parameters: A g = 1 . , λ g = 0 . , µ = 0 .
52 GeV C = 0 . σ = 23 . . (3.9)In what follows, the color dipole orientation, corresponding to (cid:126)b - (cid:126)r correlation within the GBW and BGBK dipolemodel will be denoted in our paper as br-GBW and br-BGBK, respectively. IV. ELASTIC ELECTROPRODUCTION OF QUARKONIA ON PROTONS
Treating the elastic electroproduction of heavy quarkonia on proton targets, the corresponding differential crosssection reads [41], dσ γ ∗ p → V p ( s, Q , t = − q ) dt = 116 π (cid:12)(cid:12)(cid:12) A γ ∗ p → V p ( s, Q , (cid:126)q ) (cid:12)(cid:12)(cid:12) = 116 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d r (cid:90) dα Ψ ∗ V ( (cid:126)r, α ) A Q ¯ Q ( (cid:126)r, s, α, (cid:126)q )Ψ γ ∗ ( (cid:126)r, α, Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4.1)where we take into account that the partial dipole amplitude A Q ¯ Q can depend alternatively, instead of the variable x defined in Sect. I, also on c.m. energy squared s = 2 m N p + m N − Q , where m N is the nucleon mass and p is thephoton energy in the rest frame of the target.In Eq. (4.1) the partial amplitude A Q ¯ Q ( (cid:126)r, s, α, (cid:126)q ) is related to that given by Eq. (3.4) through the Fourier fransform, A Q ¯ Q ( (cid:126)r, s, α, (cid:126)q ) = 2 (cid:90) d b e − i(cid:126)b · (cid:126)q Im A NQ ¯ Q ( (cid:126)r, s, α,(cid:126)b ) (4.2)and correctly reproduces the dipole cross section σ Q ¯ Q ( r, s ) at (cid:126)q = 0 as is given by Eq. (1.5).Consequently, the final expression for the electroproduction amplitude A γ ∗ p → V p ( s, Q , (cid:126)q ) in Eq. (4.1) has thefollowing form, A γ ∗ p → V p ( s, Q , (cid:126)q ) = 2 (cid:90) d r (cid:90) dα (cid:90) d b exp (cid:104) − i [ (cid:126)b + (1 / − α ) (cid:126)r ] · (cid:126)q (cid:105) Ψ ∗ V ( (cid:126)r, α ) Im A NQ ¯ Q ( (cid:126)r, s, α,(cid:126)b ) Ψ γ ∗ ( (cid:126)r, α, Q ) , (4.3)where the argument in the exponential function takes into account the matter of fact that the transverse distance fromthe centre of the proton to Q or ¯ Q of the dipole is given by the distance b to the center of gravity of Q ¯ Q dipole andthen by the relative distance of Q or ¯ Q from the Q ¯ Q -center of gravity. The latter distance varies with the fractionalLF momenta as rα or r (1 − α ) and the corresponding α -dependent part of the phase factor should vanish at α → / s -channel helicity conservation, the t -dependentdifferential cross section (4.1) can be expressed as the sum of T and L contributions, dσ γ ∗ p → V p ( s, Q , t = − q ) dt = dσ γ ∗ p → V pT ( x, Q , t ) dt + ˜ ε dσ γ ∗ p → V pL ( x, Q , t ) dt = 116 π (cid:18)(cid:12)(cid:12)(cid:12) A γ ∗ p → V pT ( s, Q , (cid:126)q ) (cid:12)(cid:12)(cid:12) + ˜ ε (cid:12)(cid:12)(cid:12) A γ ∗ p → V pL ( x, Q , (cid:126)q ) (cid:12)(cid:12)(cid:12) (cid:19) , (4.4)where we have taken the photon polarization ˜ ε = 0 . γ ∗ p → V p amplitude performing the followingreplacement in Eq. (4.3), A γ ∗ p → V pT,L ( s, (cid:126)q ) ⇒ A γ ∗ p → V pT,L ( s, (cid:126)q ) · (cid:18) − i π λ T,L (cid:19) , λ T,L = ∂ ln A γ ∗ p → V pT,L ( s, (cid:126)q ) ∂ ln s . (4.5)Since the unintegrated off-diagonal gluon density F ( x, (cid:126)k, (cid:126)k (cid:48) ) can contain gluons with different x (cid:54) = x ≈ x then, assuming a power-like behavior F ( x ) ∝ x − λ T,L and small values of x , the modified skewed gluon distribu-tion ˜ F ( x , x (cid:28) x , (cid:126)k, (cid:126)k (cid:48) ) is related to the conventional one as [44],˜ F ( x , x (cid:28) x , (cid:126)k, (cid:126)k (cid:48) ) = F ( x, (cid:126)k, (cid:126)k (cid:48) ) · R g ( λ T,L ) , (4.6)where the skewness factor R g reads, R g ( λ T,L ) = 2 λ T,L +3 √ π Γ( λ T,L + )Γ( λ T,L + 4) , (4.7)and functions λ T,L are given by Eq. (4.5). Such the skewness effect is included in our calculations via substitution F ( x, (cid:126)k, (cid:126)k (cid:48) ) ⇒ F ( x, (cid:126)k, (cid:126)k (cid:48) ) · R g ( λ T,L ).For a simple “ S -wave-only” structure of the V → Q ¯ Q vertex in the Q ¯ Q rest frame [14, 15, 45–47] the spin-dependentcomponent of the quarkonium wave function undergoes the Melosh spin transformation to the LF frame, what leadsto the following specific form of electroproduction amplitudes given by Eq. (4.3) for T and L polarizations, A γ ∗ p → V pT ( s, Q , (cid:126)q ) = N p (cid:90) d r (cid:90) dα (cid:90) d b exp (cid:104) − i [ (cid:126)b + (1 / − α ) (cid:126)r ] · (cid:126)q (cid:105) × Im A NQ ¯ Q ( (cid:126)r, s, α,(cid:126)b ) (cid:104) Σ (1) T ( r, α, Q ) + Σ (2) T ( r, α, Q ) (cid:105) , Σ (1) T ( r, α, Q ) = K ( εr ) (cid:90) ∞ dp T p T J ( p T r )Ψ V ( α, p T ) (cid:34) m Q ( m L + m T ) + m L p T m T ( m L + m T ) (cid:35) , Σ (2) T ( r, α, Q ) = K ( εr ) (cid:90) ∞ dp T p T J ( p T r )Ψ V ( α, p T ) (cid:34) ε m Q ( m L + 2 m T ) − m T m L m Q m T ( m L + m T ) (cid:35) , (4.8)and A γ ∗ p → V pL ( s, Q , (cid:126)q ) = N p (cid:90) d r (cid:90) dα (cid:90) d b exp (cid:104) − i [ (cid:126)b + (1 / − α ) (cid:126)r ] · (cid:126)q (cid:105) Im A NQ ¯ Q ( (cid:126)r, s, α,(cid:126)b ) Σ L ( r, α, Q ) , Σ L ( r, α, Q ) = K ( εr ) (cid:90) ∞ dp T p T J ( p T r )Ψ V ( α, p T ) (cid:34) Q α (1 − α ) m Q + m L m T m Q ( m L + m T ) (cid:35) , (4.9)where N p = Z Q √ N c α em / π , ε = m Q + α (1 − α ) Q , α em = 1 /
137 is the fine-structure constant, the factor N c = 3 represents the number of colors in QCD, Z Q = 2 / / m T = (cid:113) m Q + p T and m L = 2 m Q (cid:112) α (1 − α ) , and J , and K , are the Bessel functions of the first kind and the modified Bessel functions of the second kind, respectively.The generalization of the color dipole factorization formula (4.3) with Eq. (4.8) for T polarised and with Eq. (4.9)for L polarised photons to the diffraction slope of the process γ ∗ p → V p is straightforward and reads, B T,L ( γ ∗ → V, s, Q ) · A γ ∗ p → V pT,L ( s, Q , (cid:126)q = 0) = (cid:90) d r (cid:90) dα B Q ¯ Q ( r, s ) σ Q ¯ Q ( r, s ) Ψ ∗ V ( (cid:126)r, α ) T,L Ψ γ ∗ ( (cid:126)r, α, Q ) T,L ,B T ( γ ∗ → V, s, Q ) · A γ ∗ p → V pT ( s, Q , (cid:126)q = 0) = N p (cid:90) d r (cid:90) dα (cid:90) d b b Im A NQ ¯ Q ( (cid:126)r, s, α,(cid:126)b ) × (cid:104) Σ (1) T ( r, α, Q ) + Σ (2) T ( r, α, Q ) (cid:105) ,B L ( γ ∗ → V, s, Q ) · A γ ∗ p → V pL ( s, Q , (cid:126)q = 0) = N p (cid:90) d r (cid:90) dα (cid:90) d b b Im A NQ ¯ Q ( (cid:126)r, s, α,(cid:126)b ) Σ L ( r, α, Q ) , (4.10)where the forward dipole t -slope B Q ¯ Q ( r, s ) is given by Eq. (3.5). Consequently, in comparison with the methoddescribed in the previous section, Eq. (4.10) represents more exact way for determination of functions B ( x ) in elasticpartial dipole amplitude Eq. (3.4). However, corresponding differences in acquiring of B ( x ) are very small, ∼ < − B T − B L , for given values of B ( x ) extracted by a more simple method, are tiny as well anddo not exceed 0 .
01 and 0 . − for electroproduction of 1S and 2S charmonium states, respectively, what is inaccordance with results from Ref. [41].Finally, we would like to emphasize that the Lorentz boost of radial components of quarkonium wave functionsto the LF frame has been based on the Terent’ev prescription [35], which represents only ad hoc procedure for sucha transformation. However, its validity has been proven in Ref. [48] by comparing with calculations employing theLorentz-boosted Schr¨odinger equation, which was found rather accurate for heavy quarks V. MODEL PREDICTIONS VS. DATA
In this section, our model calculations for differential cross sections dσ/dt are tested by comparing with availabledata on electroproduction of
J/ψ (1 S ) from experiments at HERA. We present also model predictions for anotherquarkonium states, such as ψ (cid:48) (2 S ), Υ(1 S ) and Υ (cid:48) (2 S ). Although quarkonium wave functions can be generated byseveral distinct models for the Q - ¯ Q interaction potential (see Refs. [49] for the Buchm¨uller-Tye potential (BT), [50]for the effective power-law potential (Pow), [51] for the logarithmic potential (Log), [52, 53] for the Cornell potential(Cor) and [15] for the harmonic oscillatory potential (HO)), in our analysis we adopt only two of them, BT and Pow,since they provide the best description of available data as was shown in Ref. [15].In calculations we include two sets of models. The first group is related to br-GBW and br-BGBK models (seeSec. III) with (cid:126)b -dependent partial dipole amplitude based on realistic color dipole orientation and leading to a propercorrelation between vectors (cid:126)b and (cid:126)r . The second class of models, widely used in the literature, contains b -dependentpart, which is implemented additionally into conventional phenomenological models for the dipole cross section, likeb-IPsat model [23], b-Sat model [24], b-CGC model [25, 26], and/or such (cid:126)b - (cid:126)r correlation is not included properly, likein the b-BK model from Ref. [27] where an angle between these vectors is set to zero. . . . . . . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]10 . . . . . γ p → J/ψ p W = 55 GeV d σ / d t [ nb . G e V − ] | t | [GeV ]H1 (2006)H1 (2013)10 . . . . . γ p → J/ψ p W = 55 GeV . . . . . . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]br-GBWbr-BGBKb-IPsatb-CGCb-BKb-Sat10 . . . . . W = 100 GeV d σ / d t [ nb . G e V − ] | t | [GeV ]10 . . . . . W = 100 GeV . . . . . . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]10 . . . . . W = 251 GeV d σ / d t [ nb . G e V − ] | t | [GeV ]10 . . . . . W = 251 GeV FIG. 1:
Comparison of our predictions for the momentum transfer dependence of the differential cross sections dσ γp → J/ψp ( t ) /dt with HERA data from the H1 [37, 38] collaboration at different c.m. energies W = 55 , and GeV. Different curvescorrespond to our results using br-GBW and br-BGBK models (thick lines) for (cid:126)b -dependent partial dipole amplitude includinga proper (cid:126)b - (cid:126)r correlation (see Eq. (3.4)), as well as b-IPsat, b-CGC, b-Sat and b-BK models (thin lines) with only additionalfactorized b -dependence and with the approximation (cid:126)b · (cid:126)r = br , respectively. The quarkonium wave functions are generated bythe BT potential. Figure 1 shows model predictions for differential cross sections dσ γp → J/ψp /dt in comparison with available HERAdata from the H1 [37, 38] collaboration at several fixed values of c.m. energy W = 55 ,
100 and 251 GeV. Here thecharmonium wave function is determined from the BT potential. Besides, our results based on the br-GBW andbr-BGBK dipole models with a proper (cid:126)b - (cid:126)r correlation are tested by comparing with several conventional b -dependentdipole models where such correlation is not included accurately (denoted as b-IPsat, b-CGC, b-BK and b-Sat).One can see from Fig. 1 that dipole models describing differently the data can be divided into two groups corre-sponding to br-GBW, br-BGBK and b-Sat vs b-IPsat, b-CGC and b-BK models. The models from the former groupexhibit a little bit better agreement with H1 data through the all energy region from W = 55 GeV to 251 GeV.Especially at higher W = 251 GeV the models without a proper (cid:126)b - (cid:126)r correlation substantially underestimate the data. . . . . . . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]10 . . . . . W = 55 GeV d σ / d t [ nb . G e V − ] | t | [GeV ]H1 (2006)H1 (2013)10 . . . . . W = 55 GeV . . . . . . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]br-GBW + BTbr-GBW + Powb-BK + BTb-BK + Pow10 . . . . . γ p → J/ψ p W = 100 GeV d σ / d t [ nb . G e V − ] | t | [GeV ]10 . . . . . γ p → J/ψ p W = 100 GeV . . . . . . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]10 . . . . . W = 251 GeV d σ / d t [ nb . G e V − ] | t | [GeV ]10 . . . . . W = 251 GeV FIG. 2:
The same as Fig. 1 but for the b -dependent partial dipole amplitude generated by br-GBW (thick lines) and b-BK(thin lines) dipole models and for quarkonium wave functions determined from the BT (solid lines) and Pow (dashed lines) c - ¯ c interquark potentials. Our predictions are compared with H1 data [37, 38] at c.m. energy W = 55 GeV (left panel),
GeV(middle panel) and
GeV (right panel).
Besides of a sensitivity of predictions to various b -dependent dipole models analyzed in Fig. 1, we demonstrate thatthe magnitude of dσ γp → J/ψp /dt is strongly correlated with the shape of c -¯ c interaction potential. As an example, weshow in Fig. 2 our results for the BT (solid lines) and Pow (dashed lines) potentials. One can see that calculationswith the charmonium wave function generated by the BT model exhibit a better description of H1 data at larger W .Since both models for partial dipole amplitude, br-GBW and br-BGBK, give similar results for dσ/dt (see Fig. 1),here we confront our predictions using only br-GBW (thick lines) with results based on a popular b-BK (thin lines)dipole model at different c.m. energies W = 55 ,
100 and 251 GeV. Treating the BT potential, in comparison with theformer model, the calculations based on b-BK dipole amplitude show evidently a worse description of data exhibitingtheir underestimation, which is stronger at higher photon energies. − . . . . . − . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]10 − . . . . . γ p → J/ψ p W = 100 GeV Q = 3 . d σ / d t [ nb . G e V − ] | t | [GeV ]H1 (2006)10 − . . . . . γ p → J/ψ p W = 100 GeV Q = 3 . − . . . . . − . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]br-GBWbr-BGBKb-IPsatb-CGCb-BKb-Sat10 − . . . . . W = 100 GeV Q = 7 GeV d σ / d t [ nb . G e V − ] | t | [GeV ]10 − . . . . . W = 100 GeV Q = 7 GeV − . . . . . − . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]10 − . . . . . W = 100 GeV Q = 22 . d σ / d t [ nb . G e V − ] | t | [GeV ]10 − . . . . . W = 100 GeV Q = 22 . FIG. 3:
Comparison of our predictions for differential cross sections dσ γ ∗ p → J/ψp ( t ) /dt with H1 data [37] at different photonvirtualities Q = 3 . GeV (left panels), Q = 7 . GeV (middle panels) and Q = 22 . GeV (right panels). Different linescorrespond to our results at W = 100 GeV using various models for b -dependent partial dipole amplitudes and the quarkoniumwave function generated by the BT potential. Except for the real photoproduction process ( Q = 0), another test for b -dependent dipole models concerns tovirtual electroproduction of charmonia with Q >
0. The corresponding model predictions are compared with H1data [37] in Fig. 3 at c.m. energy W = 100 GeV and at several photon virtualities Q = 3 . (left panels), Q = 7 . (middle panels) and Q = 22 . (right panels). Here all panels show our results adoptingdifferent dipole models and taking the charmonium wave function generated by the BT potential. From Fig. 3 onecannot give any definite conclusion which models provide the best description of H1 data through the all region of Q .The new more precise data from future electron-proton colliders can help us to rule out various models for b -dependentpartial dipole amplitudes, as well as for charmonium wave functions.In the next Fig. 4 we demonstrate how the model results are modified at W = 50 (left panel) and 200 GeV (rightpanel) taking a realistic (cid:126)b - (cid:126)r correlation in the partial dipole amplitude (3.4) (solid lines) in comparison with a simplifiedassumption when (cid:126)b (cid:107) (cid:126)r (dashed lines). Our calculations of dσ γp → J/ψ (1 S ) p ( t ) /dt (top panels) and ψ (cid:48) (2 S )-to- J/ψ (1 S ) ratioof differential cross sections (bottom panels) have been performed with the br-GBW dipole model and charmoniumwave function generated by the BT potential. One can see that incorporation of a proper (cid:126)b - (cid:126)r correlation leads to asizeable modification of the corresponding t -dependence, what has an indispensable impact on all predictions basedon the b-BK model [27] where authors assume that the dipole amplitude is independent of the angle between vectors (cid:126)b and (cid:126)r . Consistently with Fig. 4, if predictions incorporating this b-BK dipole model have provided a good descriptionof data on diffractive electroproduction of vector mesons on protons and nuclei, the subsequent incorporation of arealistic correlation between vectors (cid:126)b and (cid:126)r will spoil such a good agreement with data. In another words, thecorresponding t -slope of dσ γ ∗ p ( A ) → V p ( A ) ( t ) /dt becomes smaller keeping the same model parameters. The Fig. 4 alsodemonstrates that the onset of a proper (cid:126)b - (cid:126)r correlation with respect to a simplified (cid:126)b (cid:107) (cid:126)r -case becomes stronger towardssmaller photon energies.Figure 5 shows our results of dσ/dt for production of various quarkonium states, such as J/ψ (1 S ), ψ (cid:48) (2 S ), Υ(1 S )and Υ (cid:48) (2 S ). The corresponding predictions are depicted at two c.m. energies W = 125 and 220 GeV that can bescanned by recent experiments at the LHC, as well as by the future measurements at electron-proton colliders. Herewe present also a sensitivity of calculations to quarkonium wave functions generated by BT and Pow models for Q - ¯ Q interaction potentials. One can see that corresponding theoretical uncertainties are reduced in production ofbottomonium states due to a smaller variance in determination of the b -quark mass by the BT and Pow b -¯ b potentialmodels used in our analysis. Differences between solid and dashed lines related to various quarkonium wave functionscan be treated as a measure of the theoretical uncertainty.The node effect is demonstrated in charmonium production as an inequality B ( ψ (cid:48) (2 S )) < B ( J/ψ (1 S )) and itsmanifestation can be studied in terms of the t -dependent differential cross section ratio R V (cid:48) (2 S ) /V (1 S ) ( W, t ) = { dσ γp → V (cid:48) (2 S ) p /dt } / { dσ γp → V (1 S ) p /dt } for real photoproduction. One can see from Fig. 6 that, as a consequence . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ] b − r correlation b || r . . . . . γ p → J/ψ p W = 50 GeV . . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]10 . . . . . W = 200 GeV . . . . . .
350 0 . . . . . R V / V ( t ) | t | [GeV ]0 . . . . . .
350 0 . . . . . W = 50 GeV ψ (2 S ) /J/ψ (1 S ) . . . . . .
350 0 . . . . . R V / V ( t ) | t | [GeV ]0 . . . . . .
350 0 . . . . . W = 200 GeV ψ (2 S ) /J/ψ (1 S ) FIG. 4:
Demonstration of importance of the (cid:126)b - (cid:126)r correlation in the partial elastic dipole amplitude (3.4) by performingcalculations of differential cross sections dσ γp → J/ψp ( t ) /dt (top panels) and ψ (cid:48) (2 S ) -to- J/ψ (1 S ) ratio of the differential crosssections R V (cid:48) (2 S ) /V (1 S ) ( W, t ) = { dσ γp → V (cid:48) (2 S ) p /dt } / { dσ γp → V (1 S ) p /dt } (bottom panels) at c.m. energy W = 50 GeV (left panels)and
GeV (right panels). The corresponding results based on the br-GBW dipole model (solid lines) are compared with thecase when vectors (cid:126)b and (cid:126)r are parallel (dashed lines). The charmonium wave function is determined from the BT potential. − − . . . . d σ / d t [ nb . G e V − ] | t | [GeV ] BTPow10 − − . . . . W = 125 GeV J/ψ (1 S ) ψ (2 S ) × . S )Υ (2 S ) × × − − . . . . d σ / d t [ nb . G e V − ] | t | [GeV ]10 − − . . . . W = 220 GeV J/ψ (1 S ) ψ (2 S ) × . S )Υ (2 S ) × × FIG. 5:
The model predictions for t -dependent differential cross sections of photoproduction of different quarkonium statesat c.m. energy W = 125 (left panel) and GeV (right panel). Our calculations have been performed adopting the br-GBWmodel for the partial dipole amplitude taking the BT (solid lines) and Pow (dashed lines) models for c - ¯ c and b - ¯ b interactionpotentials. of the node effect, the ratio R ψ (cid:48) /J/ψ ( t ) rises with t at W = 50 GeV. However, at higher W ∼ >
100 GeV this rise ischanged gradually for a more flat t -behavior of R ψ (cid:48) /J/ψ ( t ) and R Υ (cid:48) / Υ ( t ) as a result of a weaker node effect at largerenergies and for heavier vector mesons, respectively. So such expected scenario is confirmed by our results based onbr-GBW and br-BGBK models and is in correspondence with analysis from Ref. [41].Figure 6 also nicely confirms that the study of t -dependent ψ (cid:48) (2 S ) /J/ψ (1 S ) ratio represents a very effective toolfor ruling out various (cid:126)b -dependent models for the partial elastic dipole amplitude, especially if (cid:126)b - (cid:126)r correlation is notincluded properly. As an example, we discuss here a popular b-BK model where the dipole amplitude is acquiredfor the case (cid:126)b (cid:107) (cid:126)r [27]. The corresponding predictions are plotted by dot-dashed lines. One can see that the rise of R ψ (cid:48) /J/ψ ( t ) is stronger at larger W and is much more intensive in comparison with the flat t -behavior obtained within0 . . . . . .
350 0 . . . . R V / V ( t ) | t | [GeV ]br-GBWbr-BGBKb-BK0 . . . . . .
350 0 . . . . W = 50 GeV ψ (2 S ) /J/ψ (1 S ) . . . . . .
350 0 . . . . R V / V ( t ) | t | [GeV ]0 . . . . . .
350 0 . . . . W = 125 GeV ψ (2 S ) /J/ψ (1 S )Υ (2 S ) / Υ(1 S ) . . . . . .
350 0 . . . . R V / V ( t ) | t | [GeV ]0 . . . . . .
350 0 . . . . W = 220 GeV ψ (2 S ) /J/ψ (1 S )Υ (2 S ) / Υ(1 S ) FIG. 6:
The model predictions for the t -dependent V (cid:48) (2 S ) -to- V (1 S ) ratio of differential cross sections R V (cid:48) /V ( W, t ) at Q = 0 .Ratios ψ (cid:48) (2 S ) /J/ψ (1 S ) and Υ (cid:48) (2 S ) / Υ(1 S ) are depicted at several c.m. energies W = 50 , , GeV and W = 125 , GeV,respectively. The quarkonium wave functions are generated by the BT potential. The solid, dashed and dot-dashed linescorrespond to b -dependent partial dipole amplitude obtained from br-GBW, br-BGBK and b-BK dipole models, respectively. br-GBW and br-BGBK models. Besides, the ratio of forward cross sections R ψ (cid:48) /J/ψ ( t = 0) practically does not dependon energy W . Such results are unexpected, they are in contradiction with our expectations and cannot be provenby any physical reasons. In another words, they correspond to a rise with energy W of a difference between slopeparameters B ( J/ψ (1 S )) − B ( ψ (cid:48) (2 S )), what is not conformed with any physical interpretation. The correct partialexplanation of this puzzle is based on an absence of a proper correlation between vectors (cid:126)b and (cid:126)r in calculations basedon the b-BK model [27]. This means that all related predictions for dσ/dt adopting this model with condition (cid:126)b (cid:107) (cid:126)r arenot accurate. In order to demonstrate this conclusion we have presented in bottom panels of Fig. 4, as an example,also calculations of R ψ (cid:48) /J/ψ ( t ) obtained from our br-GBW model treating also the case (cid:126)b · (cid:126)r = br like in the b-BKmodel. One can see that corresponding results have been changed significantly exhibiting now much stronger rise ofthe ratio R ψ (cid:48) /J/ψ ( t ) with t which is more pronounced at smaller energies. Thus, such an effect allows to concludethat the investigation of t -dependent behavior of R ψ (cid:48) /J/ψ ( t ) at Q = 0 is very suitable for an analysis of a correlationbetween (cid:126)b and (cid:126)r since in this case the larger dipole sizes of 2S states generate more sensitive correlations with theimpact parameter of a collision.Figure 6 also demonstrates that rather large sensitivity of model predictions for R ψ (cid:48) /J/ψ ( t ) to (cid:126)b - (cid:126)r correlation ismelt away in the bottomonium case due to much smaller dipole sizes r (Υ (cid:48) (2 S )) (cid:28) r ( ψ (cid:48) (2 S )) contributing to thecorresponding diffraction process. For this reason the difference between calculations with a proper (cid:126)b - (cid:126)r correlations(br-GBW and br-BGBK models) and results based on a simplified assumption (cid:126)b (cid:107) (cid:126)r (b-BK model) is much smaller.Specifically, results with the br-BGBK dipole amplitude almost coincide with values from the b-BK model. Besides,both types of predictions for R Υ (cid:48) / Υ ( t ) exhibit a similar t -shape, which is also in accordance with expected more flat t -dependence at higher photon energies and for heavier quarkonia as a manifestation of a weaker node effect. VI. CONCLUSIONS
We study the momentum transfer dependence of the differential cross section for diffractive electroproduction ofheavy quarkonia on protons. Our main results are as follows: • Basing on our the previously developed models for the b -dependent partial elastic dipole-proton amplitudeincluding the (cid:126)b - (cid:126)r correlation, we calculated the t -dependent differential cross sections of diffractive productionof various quarkonium states. The results are confronted with the widely used phenomenological models, whicheither miss the (cid:126)b - (cid:126)r correlation or do not incorporate it properly. • The radial component of the quarkonium wave function was generated in the Q ¯ Q rest frame by solving theSchr¨odinger equation with various popular models for the Q - ¯ Q potential. The LF counterpart is then obtainedapplying a Lorentz boost procedure, which validity for heavy dipoles was confirmed in [48]. Here we also includedthe Melosh effect of spin rotation, which significantly affects the production cross section. • Our model predictions for dσ γp → J/ψp /dt were successfully tested comparing with available data from the H1experiment at HERA at different c.m. energies W (Fig. 1) and photon virtualities Q (Fig. 3). The models,labelled as br-GBW and br-BGBK, based on the realistic (cid:126)b - (cid:126)r correlation, exhibit a better description of HERA1data, compared with the conventional b -dependent dipole models, like b-IPsat, b-CGC, b-BK and b-Sat (seealso Fig. 2), which do not include properly such a correlation. Specifically, the calculations performed with thepopular b-BK model, employ a strongly exaggerated strength of the (cid:126)b - (cid:126)r correlation [27] what significantly affectsthe differential cross section, especially for radially excited charmonium states ψ (cid:48) (2 S ) (Fig. 6). In particular,it leads to a larger t -slope of dσ γp → J/ψp /dt , and predicts a much stronger rise of the ψ (2 S ) /J/ψ ratio with t ,especially at smaller photon energies, as was demonstrated in Fig. 4. • We predicted the differential cross sections of photoproduction of various quarkonium states (Fig. 5) that canbe verified in UPC collisions at the LHC, as well as with future experiments at electron-proton colliders. • All expressions for t -dependent differential cross sections for quarkonium electroproduction on protons can begeneralized for nuclear targets. The corresponding predictions for the forthcoming UPC measurements at theLHC, and future electron-ion colliders, will be presented elsewhere [54]. • The effect of (cid:126)b - (cid:126)r correlation leads to a specific polarization of the produced quarkonia, which can be observed inthe polar angle asymmetry of the dileptons from quarkonium decays. This effect will be studied in a separatepaper. 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