Coherent photoproduction of heavy quarkonia on nuclei

Exclusive photo-production of vector mesons on protons and nuclei has been an important playground for strong interaction models since early years of vector dominance model Stodolsky:1966am ; Ross:1965qa based on the pole dominated dispersion relation for the production amplitude (see the comprehensive review Bauer:1977iq ). While for light mesons that assumption was reasonably justified by closeness of the meson pole to the physical region, photo-production of heavy quarkonia, $J/\psi$, $\Upsilon$, etc. (or high $Q^{2}$ electro-production), cannot be dominated by the pole, which is far away from physical region, and other singularities, poles and cuts, become essential Hufner:1995qe ; Hufner:1997jg . Besides, photo-production on nuclei provide unique information about the space-time pattern of interaction Hufner:1996dr .

With advent of the quantum-chromodynamics (QCD) era, an alternative description in terms of color dipoles was proposed in Kopeliovich:1981pz , and applied to exclusive photo-production of heavy quarkonia on protons and nuclei Kopeliovich:1991pu ; Hufner:2000jb ; Ivanov:2002kc .

Experiments with ultra-peripheral collisions (UPC) at the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) (see e.g. Bertulani:2005ru ) offer new opportunities for measurements of exclusive photo-production of vector mesons on protons and nuclei. Corresponding calculations, performed within the color-dipole approach Ivanov:2007ms ; Kopeliovich:2020has , as well as other theoretical descriptions of coherent production of heavy quarkonia in UPC, require improvements. In the present paper, we aim to minimize theoretical uncertainties of the QCD dipole formalism for coherent production on nuclear targets.

The effect, missed in many calculations, is the correlation between the dipole transverse orientation $\vec{r}$ and the impact parameter of the collision $\vec{b}$. Since a colorless $\bar{Q}Q$ dipole interacts only due to the difference between the $Q$ and $\bar{Q}$ scattering amplitudes, the dipole-proton interaction vanishes when dipole separation vector $\vec{r}$ is perpendicular to $\vec{b}$, while reaches maximal strength when they are parallel Kopeliovich:2007fv . The influence of the $\vec{r}$-$\vec{b}$ correlation on the differential cross section of exclusive photo-production of heavy quarkonia on protons was studied recently in Kopeliovich:2021dgx . A significant difference was found with the description based on the conventional $b$-dependent dipole models, lacking the $\vec{b}$-$\vec{r}$ correlation, like b-IPsat Rezaeian:2012ji , b-Sat Kowalski:2003hm , b-CGC Kowalski:2006hc ; Rezaeian:2013tka and b-BK Cepila:2018faq models, for example. We found that the correlation is especially important for photo-production of radially excited charmonium states due to the nodal structure of the wave function. This provides a stringent test of various models for $b$-dependent dipole amplitude. The importance of color dipole orientation in other processes has been discussed in Ref. Kopeliovich:2021dgx (see references therein).

However, on nuclear targets the effect of $\vec{b}$-$\vec{r}$ correlation is diluted, except the nuclear periphery, where the nuclear density steeply varies with $\vec{b}$. Therefore the impact of $\vec{b}$-$\vec{r}$ correlation on the azimuthal asymmetry of photons and pions turns out to be rather small Kopeliovich:2008dy ; Kopeliovich:2007sd ; Kopeliovich:2007fv ; Kopeliovich:2008nx . Nevertheless, in the present paper, we implement the dipole orientation effect to minimize the theoretical uncertainties. The calculations rely on the realistic model for the $\vec{r}$ and $\vec{b}$ dependent partial $\bar{Q}Q$-proton and $\bar{Q}Q$-nucleus amplitudes proposed in Kopeliovich:2021dgx , with parameters corresponding to GBW GolecBiernat:1998js ; GolecBiernat:1999qd and BGBK Bartels:2002cj saturation models for the dipole cross section.

The present paper is organized as follows. The basics of the dipole formalism for photoproduction on a proton target, $\gamma^{*}p\to Vp$, is described in Sec. II. In particular, the effect of $\vec{b}$-$\vec{r}$ correlation, including explicit expression for the partial dipole amplitude is presented in Sec. II.1.

The important part of the calculations, the $V\to\bar{Q}Q$ distribution function in the light-front (LF) frame, is discussed in Sec. II.2, where the procedure of Lorentz boost from the rest frame of the quarkonium is described. It is confronted with the unjustified, but popular photon-like structure for the $V\to\bar{Q}Q$ transition. Its Lorentz-invariant form leads to a huge weight of the $D$-wave component in the quarkonium rest frame, contradicting the solutions of the Schrödinger equation with realistic potentials.

The dipole formalism for coherent photoproduction on nuclei is formulated in Sec. III.

Since the higher-twist nuclear effects for photoproduction of heavy quarkonia are very small, the main nuclear effects are related to gluon shadowing, considered in Sec. IV. In terms of the Fock state expansion, that is related to the higher Fock components of the projectile photon, which contain besides the $\bar{Q}Q$, additional gluons. The lifetime (coherence time) of such components turn out to be rather short, even for the very first state $|\bar{Q}Qg\rangle$, which we calculate in Sec. IV.1 relying on the path-integral formalism. In Sec. IV.2 we evaluate the coherence length for multi-gluon Fock states and demonstrate that it is dramatically reduced with addition of every extra gluon. We conclude that the single-gluon approximation is rather accurate in the available energy range.

Eventually, in Sec. V we present predictions for differential cross sections $d\sigma/dt$ of coherent charmonium photoproduction, which are in a good accord with available results of the ALICE experiment, extracted from data on UPC. Here we present also predictions for other quarkonium states, like $\psi^{\,\prime},\Upsilon$ and $\Upsilon^{\,\prime}$.

The color dipole formalism leads to the following factorized form of the dipole-nucleon photo-production amplitude Kopeliovich:1991pu ,

Here $\vec{q}$ is transverse component of momentum transfer; $\alpha$ is the fractional light-front momentum carried by a heavy quark or antiquark of the $\bar{Q}Q$ Fock component of the photon, with the transverse separation $\vec{r}$. That is the lowest Fock state, while the higher Fock components contribute by default to the dipole-proton amplitude. However, on a nuclear target these higher Fock components will be taken into account separately, due to coherence effect in gluon radiation.

The dipole-proton amplitude $\mathcal{A}^{N}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b}\,)$ in Eq. (1) depends on the transverse dipole size $\vec{r}$ and impact parameter of collision $\vec{b}$. The essential feature of this amplitude is the $\vec{r}$-$\vec{b}$ correlation, explicitly presented below in Sec. II.1. The LF distribution functions $\Psi_{\gamma^{\ast}}(r,\alpha,Q^{2})$ and $\Psi_{V}(r,\alpha)$ correspond to the transitions $\gamma^{\ast}\to\bar{Q}Q$ and $\bar{Q}Q\to V$, respectively. They are specified below in Sect. II.2.

The amplitude (1) depends on Bjorken $x$ evaluated in ryskin in the leading $\log(1/x)$ approximation. Each radiated gluon provides a factor $\int_{x}^{1}dx^{\prime}/x^{\prime}=\ln(1/x)$, where $x$ is the minimal value of the fractional LF gluon momentum related to the longitudinal momentum transfer $q_{L}=(M_{V}^{2}+Q^{2})/2E_{\gamma}$. So,

As far as the production amplitude (1) is known, we are in a position to calculate the differential cross section,

The lowest Fock component of the projectile photon, which contributes to this process, is $\bar{Q}Q$. The transverse dipole size is small, $1/m_{Q}$. Therefore shadowing corrections are as small as $1/m_{Q}^{2}$, so should be treated as a higher twist effect.

For the amplitude of quarkonium photo-production on a nuclear target, $\gamma^{*}A\to VA$, one can employ expression Eq. (1), but replacing the dipole-nucleon by dipole-nucleus amplitude,

In ultra-peripheral collisions at the LHC, the photon virtuality $Q^{2}\sim 0$ and the photon energy in the nuclear rest frame is sufficiently high to make the coherence length $l_{c}$ for $\bar{Q}Q$ photo-production much longer than the nuclear radius, $l_{c}=1/q_{L}=(W^{2}+Q^{2}-m_{N}^{2})/\bigl{(}m_{N}\,(M_{V}^{2}+Q^{2})\approx W^{2}/(m_{N}M_{V}^{2})\bigr{)}\gg R_{A}$. Then Lorentz time delation freezes the fluctuations of the dipole size, and one can rely on the eikonal form for the dipole-nucleus partial amplitude at impact parameter $\vec{b}_{A}$,

where $T_{A}(\vec{b}_{A})=\int_{-\infty}^{\infty}dz\,\rho_{A}(\vec{b}_{A},z)$ is the nuclear thickness function normalized as $\int d^{2}b_{A}\,T_{A}(\vec{b}_{A})=A$, and $\rho_{A}(\vec{b}_{A},z)$ is the nuclear density. For $\rho_{A}(\vec{b}_{A},z)$ we employ the realistic Wood-Saxon form DeJager:1987qc .

Expression for the differential cross sections is analogous to Eq. (3) provided that the coherence length $l_{c}\gg R_{A}$,

We also incorporate the real part Bronzan:1974jh ; Nemchik:1996cw ; forshaw-03 of the nuclear $\gamma^{\ast}A\to VA$ amplitude via a substitution in Eq. (7)

with $\Lambda=\partial\ln(\,{\mathrm{Im}\mathcal{A}^{N}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b}\,)}\,)\huge/\partial\ln(1/x)$.

In order to include the skewness correction Shuvaev:1999ce one should perform the following modification of the partial $\bar{Q}Q$-nucleon amplitude in the eikonal formula (7),

where the skewness factor $R_{S}(\Lambda)=\bigl{(}{2^{2\Lambda+3}}/{\sqrt{\pi}}\,\bigr{)}\,\cdot{\Gamma(\Lambda+5/2)}/{\Gamma(\Lambda+4)}$.

Higher Fock components of the photon contain besides the $\bar{Q}Q$ pair additional gluons, $|\bar{Q}Q\,g\rangle$, $|\bar{Q}Q\,2g\rangle$, etc. In $\gamma^{*}p$ collision they correspond to gluon radiation processes, which should be treated as higher-order corrections to the gluonic exchange, which take part in the building of the Pomeron exchange in the diffractive $\gamma^{*}p$ interaction. Therefore the higher Fock components are included in the $\bar{Q}Q$-dipole interaction with the proton. In the case of photo-production on a nucleus, the higher Fock components contribute to the amplitude $\mathcal{A}^{N}_{\bar{Q}Q}(\vec{r},x,\alpha,\vec{b})$ in Eq. (7). This would correspond to the Bethe-Heitler regime of radiation, when each of multiple interactions produce independent gluon radiation without interferences.

However, the pattern of multiple interactions changes in the regime of long coherence length of gluon radiation $l_{c}^{g}\gg d$, where $d\approx 2\,\mbox{fm}$ is the mean separation between bound nucleons. The gluon radiation length has the form,

where $\alpha_{g}$ is the LF fraction of the photon momentum carried by the gluon, $M_{\bar{Q}Q}$ is the effective mass of the ${\bar{Q}Q}$ pair and the effective gluon mass $m_{g}\approx 0.7\,\mbox{GeV}$ is fixed by data on gluon radiation Kopeliovich:1999am ; Kopeliovich:2007pq . Such a rather large effective gluon mass makes the gluonic coherence length (see Eq. (15) in Sec. IV.2) nearly order of magnitude shorter than for $\bar{Q}Q$ fluctuations Kopeliovich:2000ra . This is why onset of shadowing in DIS, which occur at $x<0.1$ for higher twist $\bar{Q}Q$ shadowing, requires much smaller $x<0.01$ for onset of gluon shadowing Kopeliovich:1999am . That was confirmed by the global analysis of DIS data on nuclei florian .

The dipole model calculations of diffractive photo-production of $J/\psi$ on protons, $d\sigma^{\gamma^{\ast}p\to J\!/\!\psi p}(t)/dt$, were verified in Ref. Kopeliovich:2021dgx by comparing with data from the ZEUS Breitweg:1997rg ; Chekanov:2002xi and H1 Aktas:2005xu ; Alexa:2013xxa experiments at HERA. Predictions for charmonium photo-production on nuclei at high energies were provided in Ivanov:2002kc .

Calculations, performed in the present paper, rely on the quarkonium wave functions generated by the Buchmüller-Tye (BT) Buchmuller:1980su $Q$-$\bar{Q}$ interaction potential. Following Hufner:2000jb we performed a Lorentz boost to the LF frame, including corrections for the effect of Melosh spin rotation. The Lorentz boost procedure was originally proposed in Terentev:1976jk as an educated guess, but proven later in Kopeliovich:2015qna for a heavy $\bar{Q}Q$ system.

Considering the LHC kinematic region, we included gluon shadowing calculated in the single gluon approximation, while multi-gluon Fock states were found to have too short coherence length and neglected. The corresponding suppression factor $R_{g}$ in Eq. (14) provides the main nuclear effect. The results of calculations of the differential cross section for heavy vector meson production relying on the $\vec{b}$-dependent dipole models are presented and compared with available data in Fig. 2.

The photo-production differential cross section $\gamma Pb\to J\!/\!\psi Pb$ was extracted in Acharya:2021bnz from UPC data at c.m. collision energy $\sqrt{s_{N}}=5.02\,\mbox{TeV}$ and at the mid rapidity, which corresponds to the c.m. $\gamma$-$N$ energy $W=\sqrt{M_{V}}\cdot(s_{N})^{1/4}\approx 125\,\,\mbox{GeV}$. They are depicted in the left panel of Fig. 2 together with our predictions, which used two different $\vec{b}$-dependent dipole models. Our calculations are in a good accord with data.

In a wider interval of transverse momentum one should anticipate a specific diffractive behavior of the cross section with intermittent maxima and minima, as was predicted in Ivanov:2002kc , and confirmed by the current calculations depicted in Fig. 2, which are based on more advanced models for the dipole amplitude.

The curves show our predictions, still awaiting for data, for $d\sigma/dt$ of coherent photo-production of other quarkonium states, charmonium $\psi^{\,\prime}(2S)$ (left panel) and bottomonia $\Upsilon(1S)$, $\Upsilon^{\,\prime}(2S)$ (right panel). One can see that the production of the 1S and 2S bottomonium states is more sensitive to the choice of the model for partial dipole-proton amplitude in comparison with charmonium production. The differences in predictions adopting br-BGW and br-BGBK models can be treated as a measure of theoretical uncertainty.

We studied the momentum transfer dependence of differential cross sections for coherent photo-production of heavy quarkonia on nuclei, in the framework of the dipole description.

The LF wave function of a high-energy photon was expanded over Fock states, independently contributing to heavy quarkonium production, $|\bar{Q}Q\rangle$, $|\bar{Q}Qg\rangle$, $|\bar{Q}Q2g\rangle$, etc. The cross section $\gamma A\to VA$ was calculated for every Fock component separately in accordance with the corresponding coherence lengths. At the energies of UPC the photon energy in the nuclear rest frame is so high, that the coherence length, associated with the $\bar{Q}Q$ component is much longer than the nuclear size, so one can eikonalize the dipole $\bar{Q}Q$-N amplitude, as is done in Eq. (7), including the correlation between $\vec{b}$ and dipole orientation $\vec{r}$. Such a dipole amplitude contains Gribov corrections in all orders. The corresponding quark shadowing is a higher twist effect, so it is small at the scale imposed by the heavy quarkonium mass.

For the dipole amplitude $\gamma\to\bar{Q}Q\to V$ we rely on a LF wave function of the vector meson, determined by a solution of the Lorentz boosted Schrödinger equation with realistic potentials. On the other hand, we found that the frequently used unjustified model of photon-like structure $V\to\bar{Q}Q$ leads to an exaggerated weight of the $D$-wave in the rest frame wave function, inconsistent with the solutions of the Schrödinger equation.

The QCD dipole formalism also includes the leading twist gluon shadowing, which is related to the higher Fock components of the photon, $\bar{Q}Qg$, $\bar{Q}Q2g$, etc. The corresponding shadowing effect is a leading twist due to large, nearly scale independent size of the $\bar{Q}Q$-$g$ dipoles. The related nuclear effect is much stronger that the higher twist shadowing, controlled by the small-size of $\bar{Q}Q$ dipoles. However, calculations of the effect of gluon shadowing is more involved, because the dipoles $\bar{Q}Q-g,...$ cannot be treated as ”frozen” even at very high energies, their size fluctuates during propagation through the nucleus, and we took that into account by applying the path-integral technique.

The coherence length of the higher Fock components is much shorter than for the $\bar{Q}Q$ fluctuation of the photon. We demonstrated that radiation of every additional gluon significantly reduces the coherence length. Therefore, at available energies the single gluon approximation was found to be rather accurate, while higher components containing two and more gluons have too short coherence length to produce a sizeable shadowing effect. In particular, the BK equation cannot be applied to nuclear targets, because it treats all dipoles as ”frozen”.

On the contrary to the global data analyses, which provide only $b_{A}$ integrated gluon shadowing, we calculated the $b_{A}$ dependence of shadowing, which is crucial for the differential cross section $d\sigma^{\gamma A\to VA}/dt$.

Our calculations of $d\sigma/dt$ for the coherent process $\gamma Pb\to J\!/\!\psi Pb$ are in a good accord with recent ALICE data at the LHC (see Fig. 2). We also provided predictions for other quarkonium states, $\psi^{\,\prime}(2S)$, $\Upsilon(1S)$ and $\Upsilon^{\,\prime}(2S)$ (see Figs. 2 and 3), that can be verified in the current experiments at the LHC.