Unique Prediction for High-Energy J/\psi Photoproduction: Color Transparency and Saturation

Based on the color-dipole picture, we present a successful parameter-free prediction for the recent high-energy $J/\psi$ photoproduction data from the Large Hadron Collider. The experimental data provide empirical evidence for the transition from color transparency to saturation.

In standard notation, |ψ T,L (r ⊥ , z(1 − z), Q 2 )| 2 denotes the probability for the photon Figure 1: The forward Compton amplitude Figure 2: One of the 16 diagrams for diffractive production. The vertical line indicates the unitarity cut corresponding to the diffractively produced final states, (qq) J . Production of (discrete or continuum) vector states corresponds to (qq) J production with J = 1.
of virtuality Q 2 to couple to a (qq) J=1 T,L state specified by the transverse size r ⊥ and the longitudinal momentum partition 0 ≤ z ≤ 1, and σ (qq) J =1 T,L p r ⊥ z(1 − z), W 2 denotes the (qq)-dipole-proton cross section at the total γ * p center-of-mass energy W . The two-gluon coupling of the qq dipole state requires a representation of the (qq)-color-dipole cross section of the form [3,4] Upon introducing the mass, M, of the (qq) J=1 states, and upon applying the optical theorem to relate (qq) J=1 p forward scattering to the (qq) J=1 p total cross section, we identically reformulate the total photoabsorption cross section (1) in terms of the (qq)p forward-production cross section (2), obtaining [5] The representations (4) and (5) explicitly relate the transverse and longitudinal total photoabsorption cross sections to the diffractive forward-production cross sections, of a continuum of (qq) J=1 states of mass M. The lower limit m 2 0 in (4) and (5), via quarkhadron duality 3 , for the (dominant) contribution due to light quarks fulfills m 0 < m ρ , where m ρ is the ρ-meson mass, and R e + e − = 3 q Q 2 q , where the sum runs over the actively contributing quark flavors, and Q q denotes the quark charge.
The representations (4) and (5) for the photoabsorption cross section explicitly demonstrate that low-x photoabsorption is due to the coupling of the photon to (qq) J=1 (vector) dipole states that subsequently propagate and interact via coupling to two gluons with the gluon field in the nucleon. The massive qq continuum contains the low-lying vector mesons, ρ 0 , ω, φ, J/ψ, Υ etc. followed by more massive J = 1 true continuum states, qq , also diffractively produced via γ * p → X J=1 (qq) p. For an appropriate magnitude of the kinematic variables, the X J=1 (qq) states appear as two-jet states with angular distribution characteristic for spin J = 1.
For quantitative predictions,σ (qq) J =1 T,L p ( l ′2 ⊥ , W 2 ) in (3) must be specified. Agreement with experiment for the total photoabsorption cross section in (1), and, equivalently, in (4) and (5), then determines the diffractive (forward) production of vector states and allows for a parameter-free prediction of their photoproduction cross section (6). Indeed, for vector-meson production, specifically for J/ψ production 4 , we have [2,3] where In (7), m c denotes the mass of the charm quark, and M 2 ≡ M 2 cc denotes the mass squared of the cc state in the interval ∆M 2 J/ψ determined by the level spacing of the cc bound-state spectrum of states observed in e + e − annihilation.
Replacing the cross section for the open-charm continuum on the right-hand side in (7) by the cross section for cc production at threshold, the integration in (7) simplifies to become [3] where The generalization to Υ-production is straight forward 4 With (9) to (11), the cross section (7) becomes The cross section for J/ψ electroproduction in (12) is accordingly represented by charmquark pair production at threshold multiplied by the factor ∆F 2 (m 2 , [3] we have evaluated the J/ψ-production cross section (12) upon specifying the dipole cross section in (3) by the ansatz previously employed [4] in our (successful) representation of the body of experimental data on the total photoabsorption cross section (1), or equivalently, (4) and (5). The (asymp- and of the "saturation scale" Λ 2 sat (W 2 ) will be given below. With (13), one finds that the cross section (12) takes the simple explicit form [2] where R (J/ψ) = 4 3 . We stress again that (14) provides a parameter-free prediction for J/ψ photo-and electroproduction, once the dipole cross section, σ (∞) (W 2 ), and the saturation scale, Λ 2 sat (W 2 ), in (13) are specified from agreement with experiment of the CDP prediction for the total photoabsorption cross section σ γ * p (W 2 , Q 2 ) in (1). The total J/ψ production cross section is obtained from (14) by multiplication [2] with the inverse of the experimentally known slope parameter of the J/ψ production diffraction peak.
In refs. [2,3] we compared the threshold-approximation result (14) with the (numerical) results obtained without relying on the threshold approximation (9). We found that (14), upon multiplication by an overall factor of magnitude 2/3, corresponding to 5 W (GeV)

Conclusions
The addition of new scintillators in the forward region has resulted in lower backgrounds at = 13 TeV compared to the previous measurement at = 7 TeV. Both the J/ψ and (2 ) results are in better agreement with the JMRT 'NLO' rather than LO predictions. The derived photoproduction cross-section for J/ψ production shows a deviation from a pure power-law extrapolation of HERA data while the (2 results are consistent although more data is required in this channel to make a critical comparison.  (15)) given in the last column of Table 1, are seen to agree with the experimental results from the LHCb collaboration. (14), is in good agreement with the results that do not rely on the threshold approximation (9). In addition, in ref. [2], the cross section in (14) was modified by the replacement of σ (∞) (W 2 ) by σ ′(∞) (W 2 ) = const σ (∞) (W 2 ), where the constant factor takes care of a so-called "skewing correction" and a correction for a small real part in the J/ψ production amplitude. The agreement with the ZEUS experimental data [6] for the Q 2 and the W dependence of J/ψ production was elaborated upon in detail in ref. [2].

3/2 → 1 in
In the present note, we concentrate on photoproduction, Q 2 = 0, and in particular we concentrate on the experimental results on the high-energy dependence of σ γp→J/ψp (W ), for W > ∼ 100GeV , recently presented by the LHCb collaboration [1]. The experimental data are reproduced in fig. 3, taken from ref. [1]. At the energy of W = W 1 = 100GeV , the LHCb result of σ γp→J/ψp (W = W 1 = 100GeV ) ∼ = 80nb is consistent with our theoretical prediction [2] and with the ZEUS experimental results [6]. With respect to the LHCb results in fig. 3, we can accordingly restrict ourselves to an analysis of the energy variation of the measured cross section relative to the cross section at W = W 1 = 100GeV . From (14), at Q 2 = 0, for σ γp→J/ψp (W 2 ) in terms of σ γp→J/ψp (W 2 1 ), we find where σ γp→J/ψp (W 2 1 = (100GeV ) 2 ) = 80nb is to be inserted on the right-hand side in (15). The dominant energy dependence of J/ψ photoproduction is due to the first factor, sat (W 2 )), on the right-hand side in (15). Inserting M J/ψ = 3.1GeV , and [2] where B ′ = 0.340GeV 2 , we find the results for F A (Λ 2 sat (W 2 )) shown in the fourth column of Table. 1. Concerning the magnitude of F B (W 2 ) in (15), we note that the (logarithmic) variation of σ (∞) (W 2 ) with energy in the CDP is determined [4] by consistency of the Q 2 → 0 limit of the total cross section σ γ * p (W 2 , Q 2 ) with the experimental results on photoproduction, σ γp (W 2 ). The photoproduction limit in the CDP is given by [4] where m 2 0 = 0.15GeV 2 . For σ γp (W 2 ) ∼ (ln W 2 ) 2 , the dipole cross section σ (∞) (W 2 ) increases as a single power of ln W . According to (18) for F B (W 2 ) in (15), we have We adopt a simple asymptotic Pomeron dependence for photoproduction, σ γp ≃ (W 2 ) 0.08 [7], and from (19), we obtain the results for F B (W 2 ) given in the fifth column of Table 1.
Our results for the energy dependence of the J/ψ photoproduction cross section following from (14), according to (15) and (19), are presented in the last column of Table   1. A comparison to the experimental data in fig. 3 shows agreement of our results from the CDP in Table 1 with the experimental results [1] from the LHCb collaboration. The failure of the power-law fit in fig. 1 at large values of the energy W is an obvious consequence of the fact that in the CDP, due to the denominator in (14), the cross section is not proportional to the square of the saturation scale, (Λ 2 sat (W 2 )) 2 . This behaviour may be traced back to the different dependence on the dipole cross section in (1) and (2), respectively. For a discussion on the connection between Λ 2 sat (W 2 ) and the gluon distribution in connection with J/ψ production, compare ref. [2]. As pointed out [2], the W dependence of Λ 2 sat (W 2 ) ∼ (Q 2 /x) C 2 measured in Q 2 = 0 photoproduction (nevertheless) determines the x dependence of the gluon distribution function (gluon PDF) at sufficiently large values of Q 2 > ∼ M 2 J/ψ , even though the simple proportionality of the cross section to Λ 4 (W 2 ) and the square of the gluon distribution function does not hold in general.
This range corresponds to a region in the vicinity of η cc (W 2 , Q 2 ) = 1 that determines the transition from color transparency to saturation. 5 The (approximate) Λ 4 sat (W 2 ) power law dependence, with increasing energy, breaks down to turn into a weak (ln W 2 ) 2 dependence.
The deviation of the LHCb experimental data in fig. 3 from the H1 power-law fit at W > ∼ 500GeV provides empirical evidence for the transition from color transparency to saturation. An extension [8] of a simple proportionality, Λ 4 sat (W 2 ) ∝ (α s (Q 2 )xg(x, Q 2 )) 2 , of the cross section of J/ψ production, σ γ * p→J/ψp (W 2 , Q 2 ), at fixed Q 2 ≥ 0 to large values of W does not lead to a decent high-energy behaviour, such as (ln W 2 ) 2 , of the cross section in the true limit of W → ∞ that is required to hold, however, and in fact is guaranteed by saturation as contained in the color-dipole picture.