D-wave effects in heavy quarkonium production in ultraperipheral nuclear collisions

The $D$-wave admixture in quarkonium wave functions is acquired from the photon-like structure of $V\to Q\bar Q$ transition in the light-front frame widely-used in the literature. Such a $D$-wave ballast is not justified by any nonrelativistic model for $Q-\bar Q$ interaction potential and leads to falsified predictions for the cross sections in heavy quarkonium production in ultra-peripheral nuclear collisions. We analyze this negative role of $D$-wave contribution by comparing with our previous studies based on a simple non-photon-like"$S$-wave-only"$V\to Q\bar Q$ transition in the $Q\bar Q$ rest frame.


I. INTRODUCTION
Recent investigations of heavy quarkonium (V = J/ψ(1S), ψ (2S), ψ (3S), ..., Υ(1S), Υ (2S), Υ (3S), ...) production in ultra-peripheral collisions (UPC) at Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) is very effective for theoretical study of various nuclear effects occurring in diffractive photoproduction off nuclei. Although our investigation of the corresponding photoproduction mechanism on the proton target within the light-front (LF) color dipole formalism has a long-standing tradition [1][2][3][4][5][6][7][8] there are still open questions associated mainly with the structure of V → QQ transition. The most of phenomenological studies of diffractive electroproduction of heavy quarkonia are based on an unjustified assumption of a similar structure for γ * → QQ and V → QQ vertices. This leads to an extra D-wave admixture in the photon-like V → QQ vertex in the QQ rest frame. However, any realistic nonrelativistic QQ potential model can not prove the relative weight of such spurious D-wave component contribution. For this reason, in the present paper, we analyze its magnitude using, besides the standard photon-like structure in the LF frame, also our previous studies [9] based on the "S-wave-only" V → QQ transition.
In Ref. [8] we have studied the relative contribution of D-wave component in diffractive electroproduction of heavy quarkonia off proton targets. Here we compared the both structures of V → QQ transitions, the standard photon-like structure in the LF frame with a simple "S-wave-only" structure in the QQ rest frame [6,7]. We have found that for 1S charmonium photoproduction the relative undesirable impact of the D-wave component on the magnitude of production cross section is not large and represents about 5 ÷ 10% depending on the photon energy. Not so for radially excited charmonia where the nodal structure of their wave functions leads to a boosting of the negative role of D-wave admixture in estimations of production cross sections causing their 20 ÷ 30% enhancement.
In the present paper, we extend such a study for heavy quarkonium production in UPC analyzing thus for the first time the relative contribution of undesirable admixture of Dwave component to nuclear cross sections as a function of rapidity and collision energy √ s N . Treating the UPC, besides D-wave effects, the onset of another nuclear phenomena can affect the quarkonium production rate as was analyzed in Ref. [9] within the LF QCD dipole formalism. They concern the higher twist effect related to the lowest QQ Fock component of the photon, as well as the leading twist effect associated with higher photon components containing gluons.
The former effect represents the quark shadowing controlled by the distance called the coherence length (CL) [1,10], which can be expressed in the rest frame of the nucleus as, where q is the photon energy and m N and m V is the nucleon and quarkonium mass, respectively. Following results from our recent paper [9], this phenomenon has been incorporated via the finite-l c correction factors calculated within the rigorous Green function formalism, which naturally includes the CL effects. The latter effect is known as the gluon shadowing (GS) and is treated in terms of the LF QCD dipole approach. The corresponding CL of multi-gluon Fock state is shorter [11] compared to the lowest QQ photon fluctuations and so the higher photon energy is required for manifestation of the corresponding shadowing correction. Similarly as for quarks, in the present paper, we calculate the GS correction adopting the Green function formalism [12] with improvements from Ref. [13].
Besides above shadowing corrections, the nuclear suppression is affected also by the final state absorption of produced quarkonia [1,2,4,10,[14][15][16][17][18][19][20]. It is related to the phenomenon known as the color transparency (CT), where the photon fluctuations with a smaller transverse size associated with a larger quark mass are less absorbed during propagation through the medium. The corresponding evolution of the small-sized QQ photon component to the normal-sized quarkonia is controlled by the length scale known as the formation length. In the rest frame of the nucleus it has the following form [1,10], where m V is the quarkonium mass in 2S state. In our analysis of a negative role of the D-wave component in quarkonium wave functions, we treat only the production of S-wave quarkonia since their wave functions can be simply factorized into radial and spin-dependent parts. The former part can be acquired properly in the QQ rest frame as a solution of the Schrödinger equation for various realistic Q −Q interaction potentials proposed in the literature. As an example, in the present paper we adopt the Buchmüller-Tye (BT) [21] and the power-like (POW) [22,23] potential. The choice of another potentials has practically no impact on the magnitude of analyzed D-wave effects. The same argument concerns to our preference to adopt the KST [12] and GBW [24,25] model for the dipole cross section.
The photon-like structure of the quarkonium vertex is treated directly in the LF frame, what requires only the Lorentz boost of radial components of quarkonium wave functions from the QQ rest frame. Here we adopt a widely used procedure known as the Terentev prescription [26]. However, a simple "S-wave-only" V → QQ structure in the QQ rest frame requires to perform additionally the corresponding boost also for the spin-dependent components known as the Melosh spin rotation [6-8, 18, 27].
The paper is organized as follows. In the next Section we present basic expressions for calculation of nuclear cross sections, separately for coherent (elastic), as well as incoherent (quasi-elastic) heavy quarkonium production in UPC. The Sect. III is devoted to the analysis of the undesirable D-wave admixture, related to the photon-like structure of V → QQ transition, together with estimations of the corresponding impact on magnitudes of nuclear cross sections. Finally, the last Sect. IV contains a summary with the main concluding remarks how the negative role of D-wave component can be identified by the future measurements.

II. BASIC FORMULAS IN THE COLOR-DIPOLE FORMALISM
Within the one-photon-exchange approximation in the rest frame of the target nucleus A, the cross section for the photoproduction of a vector meson V by the Weizsäcker-Williams photons reads where τ is the relative impact parameter of a nuclear collision, b is the impact parameter of the photon-nucleon collision relative to the center of one of the nuclei and the variable n(q, b) represents the photons flux induced by the projectile nucleus, where Z is the ion charge, α em = 1/137 is the fine-structure constant, K 1 is the modified Bessel function and the Lorentz factor γ = s N /2m 2 N . Coherent production (coh). In the LF dipole approach (see Refs. [1, 5-7, 18, 28], for example), assuming large photon energies when l c R A , where R A is the nuclear radius, the corresponding coherent cross section for the process γA → VA (the nucleus remains intact) takes a simple asymptotic form, Here Ψ V (r, α) is the LF wave function for heavy quarkonium and Ψ γ (r, α) is the LF distribution of the QQ Fock component of the quasi-real photon, where the QQ fluctuation (dipole) has the transverse size r and the variable α = p + Q /p + γ is the boostinvariant fraction of the photon momentum carried by a heavy quark (or antiquark). The variable is the nuclear density function of realistic Wood-Saxon form with parameters taken from [29]. The universal dipole-nucleon total cross section σ QQ (r, s), introduced in Ref. [30], depends on the transverse Q −Q separation r and c.m. energy where y is the rapidity.
Incoherent production (inc). Here the vector meson is produced in a quasi-elastic process γA → V A * , where the nucleus is in excited state. In the high energy limit, l c R A , the corresponding nuclear cross section reads [9], where B(s) is the slope parameter in reaction γN → V N . Scenario I. In the conventional standard and frequently used scenario I, corresponding to the photon-like V → QQ transition directly in the LF frame without the Melosh transform, the imaginary part of the γN → V N amplitude has the following structure [8], Here N 1 = Z Q 2N 2 c α em /2 π, the factor N c = 3 represents the number of colors in QCD, Z Q is the electric charge of the heavy quark, J 0,1 and K 0,1 are the Bessel functions of the first kind and the modified Bessel functions of the second kind, respectively.
Scenario II. For a simple "S-wave-only" structure of the V → QQ vertex in the QQ rest frame (scenario II ) [6,7,28,31,32], with subsequent Melosh spin transformation to the LF frame, the corresponding photoproduction amplitude reads, . Our model calculations include also a small real part [4,33,34] of the γN → V N amplitude performing the following replacement in Eqs. (2.5) and (2.6), The expression ( 4). In our analysis, we have included corrections for the finite CL which have been calculated using a rigorous Green function formalism as presented in Ref. [9]. Another nuclear phenomenon incorporated in our study of the negative role of D-wave component is associated with leading twist gluon shadowing. Here we adopt the path integral technique as well with details as described in Refs. [9,13].
In our calculations, we consider that the photo-nuclear reaction can be induced by the photon from the second nucleus of the colliding nuclei in UPC via replacement in Eq. (2.1) y ⇒ −y. The KST and GBW phenomenological models for the dipole cross section used in our analysis are poorly known at small s corresponding to large values of x = m 2 V /s. Here we rely on dipole model modification by an additional factor (1 − x) 7 [36].
Because of a weak sensitivity of charmonium results to a choice of the model for the dipole cross section σ QQ , we analyzed a negative role of the D-wave component in quarkonium wave functions adopting two distinct realistic Q−Q interaction potentials, power-like (POW) and  Buchüller-Tye (BT). These potentials generate the quarkonium wave functions in the QQ rest frame since are inherent in corresponding Schrödinger equation. On the other hand, the production of bottomonium states is more sensitive to the choice of the model for σ QQ as was studied in Ref. [7]. For this reason, the investigation of the corresponding D-wave effects have been realized adopting two distinct models for σ QQ -KST and GBW. In both cases the study of undesired D-wave contributions to magnitudes of nuclear cross sections has been performed including nuclear phenomena, such as the gluon shadowing and the finite-l c corrections described in details in Ref. [9]. The Fig. 1 shows our predictions for the rapidity distributions of coherent (left panels) and incoherent (right panels) charmonium photoproduction in UPC vs. the LHC data from the CMS [37] and ALICE [38][39][40]  Here charmonium wave functions are generated by the POW (thin lines) and BT (thick lines) potential. Scenario I and II, corresponding to the photon-like quarkonium vertex with D-wave admixture and "S-wave-only" V → QQ transition, is depicted by dashed and solid lines, respectively.
One can see from Fig. 1 that the inherence of D-wave component in charmonium wave functions, manifested itself as a difference between dashed and solid lines, is maximal at midrapidity (y = 0) and causes the 7 ÷ 10% undesirable enhancement of dσ/dy depending on the collision energy √ s N . Such a result is not affected by the shape of quarkonium wave functions generated by the BT and POW c −c interaction potentials used in our analysis. The Fig. 1 also demonstrates that a rather weak onset of D-wave effects can be hardly identified by future measurements of J/ψ(1S) production in UPC. However, there is a chance for recognition of such effects in ψ (2S) production. Here the negative role of D-wave admixture is boosted due to a nodal structure of charmonium wave functions for excited states as is presented in Fig. 2 for the 2S state. One can see that undesirable enhancement of dσ/dy is now much larger than for the 1S state and represents ∼ 30 ÷ 35% in the LHC energy range. We have found that this ballast modification of dσ/dy is still higher for ψ (3S) state, due to two-node structure of the corresponding wave function, and reaches almost 50%. Consequently, such a spurious D-wave manifestation is much stronger than other theoretical uncertainties related to different phenomenological models for σ QQ , as well as to different shapes of charmonium wave functions generated by various realistic c −c interaction potentials. More precise future measurements can help to identify and subsequently to eliminate such D-wave component and thus can be effective for the study of the quarkonium vertex structure. The next Fig. 3 clearly shows a weak negative role of D-wave component in 1S (top panels) and 2S (bottom panels) bottomonium production in UPC at √ s N = 5.02 TeV. It causes only ∼ 5% modification of dσ/dy for Υ(1S) production (see differences between dashed and solid lines in top panels). Similarly as for the ψ (2S) state, the nodal structure of the wave function leads to a stronger onset of D-wave effects in Υ (2S) production in UPC compared to Υ(1S) state. This is demonstrated in bottom panels of Fig. 3 where the photon-like structure of Υ (2S) → bb transition causes ∼ 10 ÷ 12% enhancement of dσ/dy with respect to the "S-wave-only" scenario II. We have also estimated that for production of Υ (3S) state this enhancement is still stronger, reaching ∼ 15%. However, such undesired modifications are still smaller than other theoretical uncertainties originated from different models for σ QQ (see thin and thick lines in Fig. 3). For this reason, the production of bottomonia in UPC is not effective for study of the structure of quarkonium V → QQ transition.

IV. CONCLUSIONS
In this paper, we have analyzed how the D-wave admixture in quarkonium wave functions, related to the photon-like structure of V → QQ transition, can falsify the model predictions for distributions dσ/dy in production of various heavy quarkonia in heavy-ion UPC. Our calculations are based on the standard formulas for nuclear cross sections within the LF color dipole approach. The model results include also nuclear phenomena related to higher twist quark shadowing, as well as the leading twist gluon shadowing.
The onset of the D-wave component was quantified by comparing two scenarios. The scenario I corresponds to the photon-like structure of V → QQ transition. Such a structure is imposed in the LF frame and is treated in the most of recent papers. It leads to D-wave component presented in various model predictions with its corresponding role, which is not justified by any nonrelativistic Q −Q interaction potential. The scenario II is based on a simple "S-wave-only" structure of quarkonium wave functions in the QQ rest frame.
We have found that all calculations based on scenario I enhance the magnitude of dσ/dy compared to scenario I, especially at y = 0. Our calculations confirm that the onset of D-wave effects is rather weak in production of J/ψ(1S), Υ(1S) and Υ (2S) states where the corresponding undesirable modification of nuclear cross section does not exceed ∼ 10 ÷ 12% and is smaller than other theoretical uncertainties related mainly to the shape of quarkonium wave functions, as well as to models for the dipole cross section.
However, there is a chance to identify and eventually to eliminate a negative role of Dwave effects in theoretical predictions and so to abandon the unjustified assumption about the photon-like structure of quarkonium vertex. According to this, we propose to investigate the production of higher charmonium states in UPC, such as ψ (2S), ψ (3S), etc. In this case, a spurious enhancement of nuclear cross sections at midrapidities due to scenario I exceeds at least 35% compared to "S-wave-only" scenario II and is larger than other theoretical uncertainties. Besides, such uncertainties can be reduced by studying production of the ψ (2S)-to-J/ψ(1S) ratio. This gives a possibility that more precise photoproduction data at y = 0 from the future measurements at the LHC, as well as at planned electron-ion colliders can shed more light on the structure of V → QQ transition.