$\Upsilon$ photo-production on the proton at the Electron-Ion Collider

We present a dispersive analysis with the aim to extract the $\Upsilon$-p scattering length from $\gamma p \to \Upsilon p$ experiments. In this framework, the imaginary part of the $\Upsilon$-p forward scattering amplitude is obtained from $\gamma p \to \Upsilon p$ cross section measurements, and is constrained at high energies from existing HERA and LHC data. Its real part is calculated through a once-subtracted dispersion relation, and the subtraction constant is proportional to the $\Upsilon$-p scattering length. We perform a feasibility study for $\Upsilon$ photo-production experiments at an Electron-Ion Collider and discuss the sensitivity and precision that can be reached in the extraction of the $\Upsilon$-p scattering length.


I. INTRODUCTION
The interaction between heavy quarkonia, such as J/ψ and Υ, and light hadrons or nuclei provides a unique window on the gluonic van der Waals interaction in Quantum ChromoDynamics (QCD). Being a small sized system, the heavy quarkonium QQ can be treated as a color dipole, and the effective two-gluon exchange interaction between the quarkonium and the light hadron or nucleus may be estimated from the knowledge of its chromo-electric polarizability, see Refs. [1][2][3] for reviews and references therein. Provided this effective interaction is strong enough, a bound state between the QQ state and the light hadron or nucleus may be formed [4][5][6]. Early calculations for the chromo-electric polarizability, treating the heavy quarkonium as a Coulombic bound state [7,8], yielded estimates for the quarkonium binding energy in nuclear matter B J/ψ ∼ 10 MeV for J/ψ, and B Υ ∼ 2 − 4 MeV for Υ [6]. Many follow-up calculations have explored the possibility of quarkonium nuclear bound states within different theoretical frameworks [9][10][11][12][13][14].
In recent years, the study of the excitation spectrum in the charmonium and bottomonium sectors above open charm and open bottom thresholds has revealed a plethora of new states, which cannot be explained as conventional QQ bound states, see e.g. [15] for a recent experimental review. Several explanations for the nature of these exotic states have been put forward, ranging, among others, from tetraquark states based on QCD diquarks, QCD hybrids, hadronic molecules, or hadro-charmonium states. In contrast to conventional QQ states above open charm or open bottom thresholds, for which the branching fractions in open flavor decay modes are found to be 2 or 3 orders of magnitudes larger than their hidden flavor decay modes, many of the newly found exotic states have in common that hidden flavor decay modes are discovery channels, and are only suppressed by a factor of 10 or less relative to the open flavor decay modes. The understanding of the nature of these states may therefore shed another light on how hidden charm or hidden bottom systems interact with light quark systems. This is especially prominent in the hadro-quarkonium models for the exotic hadrons [16], in which the charm or bottom QQ pair remains tightly bound while interacting with the light quarks through a van der Waals interaction. Also in the baryon sector, narrow resonances involving two heavy quarks have been discovered in recent years. In the weak decay process Λ b → J/ψpK − , the LHCb Collaboration [17,18] has found evidence for such states in the J/ψp mass spectrum, and interpreted them as hidden-charm pentaquark states. As two of these states were found approximately 5 MeV and 2 MeV below the Σ + cD 0 and Σ + cD * 0 thresholds respectively, these states were interpreted in various studies as loosely bound meson-baryon molecular states through π-or ρexchange interactions (see e.g. [19] among many others). Alternatively, the two narrow pentaquark states near Σ + cD * 0 threshold, P c (4440) + and P c (4457) + were predicted in [20] as the 1/2 − and 3/2 − hyperfine partners of deeply bound hadro-charmonium states of ψ(2S) and the proton, while the narrow pentaquark state P c (4312) + near Σ + cD 0 threshold was interpreted as a 1/2 + hadrocharmonium state of the χ c0 (1P ) and the proton [21]. In such picture, the binding is due to the two-gluon exchange interaction between a compact quarkonium state within a proton, and is proportional to the chromoelectric polarizability of the quarkonium state. The hadro-charmonium framework therefore relates the nature of such exotic states involving heavy quarks to the interaction of the heavy quarkonia with light hadrons.
The interaction of heavy quarkonia with light hadrons can also be studied within lattice QCD. Recently, the HAL QCD Collaboration [22] has performed improved lattice QCD studies of the s-wave effective potentials for the J/ψ-nucleon system (J = 1/2 and J = 3/2), although still for an unphysical pion mass value of m π = 700 MeV. For the J/ψ-nucleon system, the potential was found to be attractive, but not strong enough to allow for bound states. The lattice study extracted J/ψ-nucleon scattering lengths for both spin states: a ψp (J = 1/2) = 0.66 ± 0.07 fm, and a ψp (J = 3/2) = 0.38 ± 0.05 fm, indicating that the J/ψ-N state with J = 1/2 obtains significantly stronger attraction than the J = 3/2 state.
In order to access the J/ψ-nucleon interaction from experiment, a phenomenological analysis of the J/ψ-p forward scattering amplitude within a dispersive framework was performed in [23]. It related the imaginary part of the J/ψ-p forward scattering amplitude to γp → J/ψp and γp → ccX cross section data, and calculated the real part through a once-subtracted dispersion relation, for which the subtraction constant is directly related to the scattering length, and was fitted to the available data in the threshold region. This dispersive framework extracted as value for the spin-averaged s-wave J/ψ-p scattering length a ψp = 0.046 ± 0.005 fm, which can be translated into a J/ψ binding energy in nuclear matter of B ψ = 2.7 ± 0.3 MeV. Such value for the scattering length is at the lower end of the range of values estimated in the literature, ranging from a ψp = 0.05 fm [24] to a ψp = 0.37 fm [25]. However, as the current data base for the J/ψ photo-production in the threshold region is quite scarce, a reliable extraction clearly calls for new high statistics data in that region.
A dedicated experimental program to measure the photo-production of J/ψ near threshold has started in recent years at Jefferson Lab. The GlueX collaboration measured J/ψ photo-production near threshold using the GlueX detector in Hall D and published its first results [26]. Experiment E12-16-007 [27] in Hall C was designed and performed as a direct search of the higher mass narrow width pentaquark state P + c (4450) in photo-production, and will provide good differential cross sections in four-momentum transfer t. It ran during the spring of 2019 and preliminary results are expected soon [28], while experiment E12-12-001 [29] in Hall B has collected data using the CLAS12 detector and a hydrogen target during 2018/2019, with the analysis presently still underway. Furthermore, a more dedicated program of high precision J/ψ electro-and photo-production measurements on the proton using the Solenoidal Large Intensity Device (SoLID) is planned in Hall A. The Jefferson Lab experiment E12-12-006 [30] will be able to measure both the electro-production differential cross sections in the four momentum transfer t, as well as determine the total cross section very close to the threshold region on a nucleon.
One may also consider quarkonium production on a nucleon at the future Electron-Ion Collider (EIC) machine [31]. Here, it is worth noting that access to the threshold region of the J/ψ production is not possible due to the lower limit of the center of mass energy of the machine. Therefore, it is best to consider a higher center of mass energy provided by the threshold production of Υ, hence reachable by the current machine design. With sufficient integrated luminosity a precision measurement of the photo-and electro-production of Υ is possible, and gives a way to address the question of the existence of bottom pentaquarks. As the Υ production probes the gluon fields in the nucleon, such study will also shed light on the origin of the proton mass. Glu-ons are estimated to account for more than half of the proton mass due to the strong gluon chromo-electric and chromo-magnetic fields inside the proton [32]. At an EIC, the mass of the bottom quark as well as the probe resolution in electro-production give two independent knobs in the investigation of the gluonic interaction between the Υ and the nucleon.
To prepare for such a program of Υ production at an EIC, we will extend in the present work our previous dispersive study of J/ψ photo-production to the case of Υ photo-production. We will construct the Υ-proton forward scattering amplitude in Section 2, relating its imaginary part to γp → Υp data. The real part will be calculated from a dispersion relation, involving one subtraction constant, which is directly related to the scattering length. In Section 3, we will constrain the high-energy region from existing HERA and LHC data. As no data are available so far in the threshold region, we will consider several scenarios for the scattering length providing a range of estimates for the subtraction constant. In Section 4, we will show results of a feasibility study for Υ photo-production at the EIC, considering different beam settings, and discuss the sensitivity and precision in the extraction of the Υ-p scattering length from such experiments. In Section 5, we present our conclusions. Some technical details on the EIC simulations are presented in Appendix A.

II. Υ-PROTON FORWARD SCATTERING AMPLITUDE
We consider the spin-averaged Υp → Υp forward elastic scattering process, which is described by an invariant amplitude T Υp , depending on the crossing-symmetry variable ν. The latter is defined in terms of the Mandelstam invariant s = W 2 as: where M and M Υ stand for the masses of the proton and Υ(1S) state, respectively. The forward differential cross section for the Υ p → Υ p scattering process can then be expressed as: where in the forward direction the momentum transfer t = 0, and where q Υp denotes the magnitude of the Υ three-momentum in the c.m. frame, given by: The imaginary part of the amplitude T Υp can be obtained as sum of elastic and inelastic discontinuities: The elastic discontinuity starts from elastic threshold s = s el = (M Υ + M ) 2 = 108.13 GeV 2 , or equivalently ν el = M Υ M = 8.88 GeV 2 , whereas the inelastic discontinuity starts at the BB meson production threshold, corresponding with s inel = (M + 2M B ) 2 = 132.18 GeV 2 , or equivalently ν inel = 20.90 GeV 2 . Analogous to the J/ψ case [23], we will parametrize the elastic and inelastic discontinuities of the Υp forward scattering amplitude by the following 3-parameter forms, for x = el/inel: where the factors ∼ (1 − ν x /ν) bx determine the behavior around the respective threshold ν x , and the factors ∼ ν ax determine the Regge behavior of the amplitude at large ν. In the following we will discuss how we can determine the respective parameters appearing in the elastic and inelastic discontinuities. For the discontinuity across the elastic cut, Disc el T Υp , we use the vector meson dominance (VMD) assumption to relate the Υp elastic cross section σ el Υp to the total γp → Υp photo-production cross section [33,34]: with electric charge e given through α = e 2 /(4π) 1/137, and where f Υ is the Υ decay constant, which is obtained from the Υ → e + e − decay as The experimental value Γ Υ→ee = 1.34 ± 0.02 keV [35] yields f Υ = 0.238 GeV. Furthermore, q γp denotes the magnitude of the photon three-momentum in the c.m. frame of the γp → Υp process: The discontinuity across the inelastic cut, Disc inel T Υp , is related through the optical theorem to the Υp → bbX inelastic cross section σ inel Υp as: Using VMD, one can relate the total γp → bbX photoproduction cross section to the inelastic cross sections for the sum over Υ states (labeled by index i): In contrast to the elastic process, where the final state is fixed to be the Υ(1S)-p state, and for which the elastic photo-production cross section σ(γp → Υp) can be expected to be approximately dominated by its lowest term in the sum over vector bottomonia states, the open bottom bb final state in the inelastic photo-production cross section of Eq. (10) can be expected to get sizeable contributions from several vector bottomonia states. We will therefore refrain from approximating Eq. (10) by its lowest vector bottomonium contribution in the following analysis, and instead constrain the inelastic discontinuity through the forward differential cross section dσ/dt for γp → Υp as discussed in the next section. The real part of the scattering amplitude T Υp is related to its imaginary part through a once-subtracted forward dispersion relation: with T Υp (0) the subtraction constant at ν = 0. In this work, the subtraction constant is suggested to be obtained by performing a fit of the differential γp → Υp photo-production cross section data at t = 0, which is related to T Υp as: The real part of the forward scattering amplitude at threshold T Υp (ν el ) is directly related to the Υ-p scattering length a Υp as: Analogously to the J/ψ case, we may relate a positive Υ-p scattering length, corresponding to an attractive interaction, to an Υ binding energy B Υ in nuclear matter, using a linear density approximation [24]: where ρ nm 0.17 fm −3 denotes the nuclear matter density.
As the aim of our work is to extract a Υp from fitting the subtraction constant T Υp (0) to future Υ photoproduction data in the threshold region, we will consider three scenarios for the subtraction constant in order to explore the data sensitivity to its extraction.
The simplest scenario corresponds with having zero value of the subtraction constant, i.e. T Υp (0) = 0. The real part of the Υ-p scattering amplitude is then fully determined by its imaginary part through the dispersion integral in Eq. (11). The resulting value of the Υ-p scattering length is then extremely small, around a Υp ∼ 10 −3 fm.
A second scenario is to estimate the subtraction constant by a scaling from its value for the J/ψp scattering, which was obtained from a fit to data in [23] as T J/ψp (0) 22.5 ± 2.5. Observing that at high energies the normalizations of both the J/ψp and Υp scattering amplitudes are completely driven by their inelastic discontinuities, as discussed in the following section, and making the strong assumption that the subtraction constants scale in the same way, yields the estimate: With cross section normalization estimate at high energies C Υ inel /C J/ψ inel ≈ 0.9, which we discuss in the next section, this second scenario yields T Υp (0) = 20.5, which corresponds with a scattering length a Υp 0.016 fm.
We also consider a third, theoretically more motivated, scenario, in which an estimate of the Υ-p threshold amplitude is obtained by considering, in the leading approximation, the heavy bottomonium as a Coulombic bound state which interacts with the proton through its chromoelectric polarizability [7] yielding: with α Υ the chromo-electric polarizability. For a Coulombic bound state, α Υ is given by [7,8] α Υ = 28 81 where a 0 is the Bohr radius of the Υ state, given by Using the parameter values for the strong coupling α s ≈ 0.37 and the bottom quark mass m b ≈ 4.76 GeV from a recent potential model calculation for the bottomonium spectrum [36], the Bohr radius for the Υ takes on the value a −1 0 (0.85) −1 GeV. Eq. (17) then yields a chromo-electric polarizability α Υ 0.67 GeV −3 , in good agreement with the recent evaluation given in Ref. [37]: α Υ = 0.5 +0. 42 −0.38 GeV −3 , using color octet intermediate states in the calculation of the polarizability instead of free ones as in [7,8]. For the purpose of providing cross section estimates in our third scenario, we will use an average between both results: α Υ 0.6 GeV −3 , which yields T Υp (ν el ) 88 and for the s-wave scattering length a Υp 0.066 fm. Using the dispersion integral of Eq. (11) in making the small extrapolation between the amplitudes at ν = ν el and ν = 0, this third scenario then yields as value for the subtraction constant: T Υp (0) 87.

III. FIT TO EXISTING DATA FOR Υ PHOTO-PRODUCTION ON THE PROTON
In this section, we discuss the fit of the elastic and inelastic discontinuities, which are parameterized according to the three-parameter forms of Eq. (5), to available Υ-p photo-production data, following similar strategy as our previous analysis for the J/ψ-p system [23].
At present, the γp → Υp photo-production database consists of four data points from HERA [38][39][40], shown in Fig. 1 (upper panel). Furthermore at Large Hadron Collider (LHC) energies, the γp → Υp cross section has been extracted from central pp production data at LHCb [41] and from ultra-peripheral pPb collisions at CMS [42].  Table I. The elastic Υ photo-production data are from HERA: H1 [38] and ZEUS [39,40], and from LHC: LHCb [41] and CMS [42]. Lower panel: W -dependence of the total inelastic photo-production cross section γp → bbX. The curve gives a lower limit arising from the contribution of the Υ(1S) state only in the sum of Eq. (10) for the γp → bbX cross section. The open beauty photo-production data points are from HERA [43] and EMC [44].
The inclusive bb photo-production database is represented by one data point from HERA [43]. Additionally, the lower energy cross section upper limit from EMC [44] is added to guide the low-energy behavior, as shown in Fig. 1 (lower panel).
For high energies √ s = W ∼ 100 GeV, the differential cross section has been measured by the ZEUS Collaboration and follows an exponential t-dependence: with an empirical slope parameter B(W = 100 GeV) = 4.5 ± 0.5 GeV −2 [40]. The exponential dependence of Eq. (19) allows us to express the extrapolated value of the differential cross section at t = 0 as where is the minimum (modulo) physical momentum transfer, corresponding to the forward scattering (θ γΥ = 0). On physical grounds, one may expect the exponential dependence of Eq. (19) to hold in a limited t-range only, turning into a power dependence at larger t values. At high W , this only gives a minor correction to Eq. (20), but at lower W -values one may expect the correction to be more important. In this case, one should apply the fit form of Eq. (20) only in the limited t-range in order to extrapolate to A.
The present database for Υ photo-production is unfortunately insufficient to perform a good fit using the forms of Eq. (5) with all parameters unconstrained. Especially the lack of low-energy data prevents a direct determination of the low-energy slope parameters b x of the cross sections at present. Assuming a similarity in the energy dependence of the cross sections for charm and bottom photo-production, we thus start by simply fixing b el and b inel to the values obtained in the J/ψ analysis [23]. The high-energy elastic slope parameter a el and the elastic normalization constant C el are then fitted to the available data points for the elastic Υ photo-production total cross section, as shown in Fig. 1   To determine the inelastic normalization constant C inel , it was observed in [23] for the J/ψ case that around W = 100 GeV the inelastic discontinuity is around two orders of magnitude larger than its elastic counterpart.
Although there is only one data point for the inclusive γp → bbX cross section, shown in the lower panel of Fig. 1, it also confirms this finding for the Υ-p discontinuities. The amplitude T Υp entering the forward differential cross section A in Eq. (20) is thus dominated by the normalization constant C inel . At high energies around W ∼ 100 GeV, where the subtraction constant makes a negligible contribution to A, we thus solve Eq. (20) and fix the normalization C inel by the available high-energy data for the t-slope parameter B. The W -dependence of the t-slope parameter B in Eq. (19). The curves are obtained by solving Eq. (20) for different values of the subtraction constant T0 in the J/ψp (upper) and Υp (lower) forward scattering amplitudes. The tslope data points are from HERA: for J/ψ from [45], and for Υ from [40]. The red solid curve, corresponding to TΥp(0) = 87, passes through the central value of the data point exactly, according to our constraint on the C inel .
We show this constraint based on the t-slope parameter B in Fig. 2, and illustrate the calculation procedure first for the J/ψ case (upper panel). In solving Eq. (20) for B, we use the total cross section data for σ(γp → J/ψp) which fixes the elastic discontinuity. Furthermore, on the left hand side of Eq. (20), we need the real and imaginary parts of the forward scattering amplitude to determine A, according to Eq. (12). For the imaginary part of the amplitude, which is described by its elastic and inelastic discontinuities, we use C inel and a inel as fit parameters. The real part of the amplitude is calculated from the dispersion relation of Eq. (11) for three values of the subtraction constant T 0 ≡ T ψp (0), considered in Ref. [23]. The solution of Eq. (20) for these three values of T 0 is shown in Fig. 2 and compared with the HERA data for the t-slope. One firstly sees from Fig. 2 that for W ≥ 50 GeV, the sensitivity to the subtraction constant becomes vanishingly small, thus allowing to determine a inel and C inel from a fit to the HERA data as: a inel = 1.20 and C inel = 20.5.
We apply the same procedure to the Υ t-slope parameter in the lower panel of Fig. 2, and show our results for the three scenarios for the subtraction constant T 0 ≡ T Υp (0) discussed above. We again observe that for W ≥ 100 GeV, the sensitivity to the subtraction constant becomes vanishingly small. As there is only one data point at W = 100 GeV in this case, we fix a inel to the J/ψ value and extract C inel by constraining the t-slope to the data point at 100 GeV. The obtained value for C inel is listed in Table I. It is seen that the value of the thus extracted dimensionless parameter C inel is similar within 10 % between the J/ψ and Υ cases. We also note that in the sum of Eq. (10) the Υ contribution to the inelastic open beauty photo-production cross section is around 20 %.
In Fig. 3, we show the W -dependence of real and imaginary parts of the scattering amplitude T Υp in our dispersive formalism, for the three choices of the subtraction constant discussed above. We notice that the real part of the amplitude dominates in the threshold region, whereas the imaginary part dominates at high energies as expected for a diffractive process. For the largest value of the subtraction constant T 0 = 87 considered, the imaginary part overtakes the real part around W ≈ 25 GeV.

IV. RESULTS FOR Υ PHOTO-PRODUCTION AT THE EIC AND DISCUSSION
We investigate in this section how to extract the subtraction constant from a fit to the differential γp → Υp cross section data at an Electron-Ion Collider (EIC).
We consider both a medium-energy and high-energy EIC configuration. The medium-energy configuration (setting 1) has a 10 GeV electron beam incident on a 100 GeV proton beam ( 140 GeV), corresponding to nominal configurations for the EIC design.
For the γp → Υp cross section, we use the dispersive model discussed above with t-dependence as in Eq. (19). We show results for the three scenarios for the subtraction constant T Υp (0), and corresponding s-wave scattering length a Υp in Table II. The uncertainties correspond to the simulated EIC γp → Υp data for the two beam settings.  We use the Argonne l/A-event generator [47] to simulate a realistic event sample for the γp → Υp process at the EIC, and refer to Appendix A for more details on the exact implementation of the simulation. For our simulations, we assumed a total integrated luminosity of 100 fb −1 for each of the settings, which corresponds to 116 days at 10 34 cm −2 s −1 . We simulated both the  [38][39][40] by using the empirically measured slope parameter, using Eq. (20). The bands represent the uncertainty propagated based on the EIC simulated data points, assuming a one-parameter fit of T0. Upper (lower) panels are for EIC beam setting 1 (beam setting 2) respectively. Right panels give a more detailed view of the lower energy region W < 25 GeV.
Υ → e + e − and Υ → µ + µ − decay channels, and only considered events where we fully detect the exclusive final state. We assumed nominal EIC detector parameters in line with the EIC white paper, where we have lepton detection for pseudo-rapidities between −5 < η l < 5, and recoil proton detection for angles θ p > 2 mrad. Furthermore, we assumed we can reconstruct the scattered electron for y = P.q/P.k between 0.01 < y < 0.8, and we ensured a quasi-real regime by requiring that Q 2 < 1 GeV 2 .
In Fig. 4, we show the simulated results for the tdependence of the γp → Υp differential cross sections for different values of W , corresponding to the two EIC beam settings. In each case we consider the three scenarios for the subtraction constant T 0 discussed above. For the generated EIC data, the exponential t-slope B is obtained through solution of Eq. (20). The error bands represent the uncertainty propagated based on the data points, assuming the two-parameter exponential fits of Eq. (19). Only points generated up to −t values of 2.5 GeV are considered in the fits.
In Fig. 5, we show the W -dependence of the γp → Υp differential cross sections extrapolated to t = 0 for the two EIC beam settings considered. We see that in the W < 25 GeV region, the precision that can be reached for beam setting 1 at the EIC will allow to clearly distinguish between the three scenarios for the subtraction constant, and allow to extract this parameter with a statistical precision of the order of a few percent, see Table II. Furthermore, beam setting 1 at the EIC will allow to perform an independent fit of the b inel parameter, which governs the low-energy behavior of the γp → Υp forward differential cross section, according to Eq. (5). For the region W > 25 GeV, beam setting 2 will allow to connect the data with the existing HERA measurement and furthermore allow to perform an independent fit of the parameter a inel , which governs the high-energy behavior of the γp → Υp forward differential cross section.

V. CONCLUSION
In this work, we have extended a previous dispersive study of J/ψ photo-production to the case of Υ photoproduction on a proton target, with the aim to extract the Υ-p scattering length from future γp → Υp experiments. The imaginary part of the Υ-p forward scattering amplitude is constrained at high energies from existing HERA and LHC data for the γp → Υp total and differential cross sections. Its real part is calculated through a once-subtracted dispersion relation, and the subtraction constant is proportional to the Υp scattering length. As no data are available so far in the threshold region, we have considered three scenarios for the subtraction constant: one of them corresponds to a zero value, the other to a value for the Υ study similar to that of the J/ψ case, and a third where we estimate the subtraction constant by considering the Υ as a Coulombic bound state which interacts with the proton through its chromo-electric polarizability. Using these three scenarios, we have performed a feasibility study for Υ quasi-real photon (Q 2 < 1 GeV 2 ) production experiments at an Electron-Ion Collider, and considered a low-energy and a high-energy beam setting. For our simulations, we have assumed a total integrated luminosity of 100 fb −1 for each of the settings, assuming nominal EIC detector parameters in line with the EIC white paper. In both beam settings, the simulated data for the γp → Υp cross section were found to clearly distinguish between the three considered scenarios for the subtraction constant. The low-energy beam setting, accessing the range 12 GeV W 60 GeV, was found to yield the higher statistical precision on the cross section. Furthermore, the high-energy beam setting, accessing the range 15 GeV W 140 GeV, will allow to connect the EIC data with the existing HERA data, and thus provide an independent measurement of the high-energy behavior of the γp → Υp forward differential cross section, further constraining the dispersive formalism. It is worth noting that our projection of the statistical error analysis shows that the Υ-p scattering length can be extracted from such data with a statistical precision of the order of 2% or less. The total experimental uncertainty of this determination will be dominated by the systematic error in the measurement of the absolute value of the differential cross section expected to be in the few percent range, leading to a very potent determination of the scattering length.
Our work shows that an experimental program on Υ quasi-real photo-production at the EIC has the potential to provide a unique view on the gluonic van der Waals interaction in Quantum ChromoDynamics.