Measurement of the cross-section ratio sigma_{psi(2S)}/sigma_{J/psi(1S)} in deep inelastic exclusive ep scattering at HERA

The exclusive deep inelastic electroproduction of $\psi(2S)$ and $J/\psi(1S)$ at an $ep$ centre-of-mass energy of 317 GeV has been studied with the ZEUS detector at HERA in the kinematic range $2<Q^2<80$ GeV$^2$, $30<W<210$ GeV and $|t|<1$ GeV$^2$, where $Q^2$ is the photon virtuality, $W$ is the photon-proton centre-of-mass energy and $t$ is the squared four-momentum transfer at the proton vertex. The data for $2<Q^2<5$ GeV$^2$ were taken in the HERA I running period and correspond to an integrated luminosity of 114 pb$^{-1}$. The data for $5<Q^2<80$ GeV$^2$ are from both HERA I and HERA II periods and correspond to an integrated luminosity of 468 pb$^{-1}$. The decay modes analysed were $\mu^+\mu^-$ and $J/\psi(1S) \pi^+\pi^-$ for the $\psi(2S)$ and $\mu^+\mu^-$ for the $J/\psi(1S)$. The cross-section ratio $\sigma_{\psi(2S)}/\sigma_{J/\psi(1S)}$ has been measured as a function of $Q^2, W$ and $t$. The results are compared to predictions of QCD-inspired models of exclusive vector-meson production.


Introduction
Exclusive electroproduction of vector mesons in deep inelastic scattering (DIS) at high energies is usually described as a multi-step process, as illustrated in Fig. 1: the electron emits a virtual photon, γ * , with virtuality, Q 2 ; the γ * fluctuates in leading-order QCD into a qq pair with a lifetime which, at large values of the γ * p centre-of-mass energy, W , is long compared to the interaction time; and the qq pair interacts with the proton with momentum transfer squared, t, via a colour-neutral exchange, e.g. through a two-gluon ladder, and then hadronises into the vector meson, V .
In this paper, a measurement of the ratio of the cross sections of the reactions γ * p → ψ(2S) + Y and γ * p → J/ψ(1S) + Y , where Y denotes either a proton or a low-mass proton-dissociative system, is presented. The ψ(2S) and the J/ψ(1S) have the same quark content, different radial distributions of the wave functions, and their mass difference is small compared to the HERA centre-of-mass energy. Therefore, this measurement allows QCD predictions of the wave function dependence of the cc-proton cross section to be tested. A suppression of the ψ(2S) cross section relative to the J/ψ(1S) is expected, as the ψ(2S) wave function has a radial node close to the typical transverse separation of the virtual cc pair, as will be discussed in more detail in Section 7.1.
The luminosity used for the measurement of σ(ep → e ψ(2S) Y )/σ(ep → e J/ψ(1S) Y ) presented in this paper is 468 pb −1 and the kinematic range is 5 < Q 2 < 80 GeV 2 , 30 < W < 210 GeV and |t| < 1 GeV 2 . A sub-sample of 114 pb −1 of HERA I data was used for an additional measurement in the range 2 < Q 2 < 5 GeV 2 . Events were selected with no activity in the central ZEUS detector in addition to signals from the scattered electron and the decay products of the ψ(2S) or J/ψ(1S). The decay channels used were
A detailed description of the ZEUS detector can be found elsewhere [4]. A brief outline of the components that are most relevant for this analysis is given below.
In the kinematic range of the analysis, charged particles were tracked in the central tracking detector (CTD) [5][6][7], which operated in a magnetic field of 1.43 T provided by a thin superconducting solenoid. The CTD consisted of 72 cylindrical drift-chamber layers, organised in nine superlayers covering the polar-angle 2 region 15 • < θ < 164 • . The transverse-momentum resolution for full-length tracks was σ(p T )/p T = 0.0058p T ⊕ 0.0065 ⊕ 0.0014/p T , with p T in GeV. For the HERA II period, the CTD was complemented by a silicon microvertex detector (MVD) [8], which consisted of three cylindrical layers of silicon microstrip sensors in the central region and four planar disks in the forward region.
The muon system consisted of rear, barrel (R/BMUON) and forward (FMUON) tracking detectors. The R/BMUON consisted of limited-streamer (LS) tube chambers placed behind the RCAL (BCAL), inside and outside a magnetised iron yoke surrounding the CAL. The barrel and rear muon chambers covered polar angles from 34 • to 135 • and from 135 • to 171 • , respectively. The FMUON consisted of six trigger planes of LS tubes and four planes of drift chambers covering the angular region from 5 • to 32 • . The muon system exploited the magnetic field of the iron yoke and, in the forward direction, of two iron toroids magnetised to ≈ 1.6 T to provide an independent measurement of the muon momenta.
The iron yoke surrounding the CAL was instrumented with proportional drift chambers to form the backing calorimeter (BAC) [13]. The BAC consisted of 5142 aluminium chambers inserted into the gaps between 7.3 cm thick iron plates (10, 9 and 7 layers in the forward, barrel and rear directions, respectively). The chambers were typically 5 m long and had a wire spacing of 1.5 cm. The anode wires were covered by 50 cm long cathode pads. The BAC was equipped with energy readout and position-sensitive readout for muon tracking. The former was based on 1692 pad towers (50 × 50 cm 2 ), providing an energy resolution of σ(E)/E ≈ 100%/ √ E, with E in GeV. The position information from the wires allowed the reconstruction of muon trajectories in two dimensions (XY in the barrel and Y Z in the endcaps) with a spatial accuracy of a few mm.
The luminosity was measured using the Bethe-Heitler reaction ep → eγp by a luminosity detector which consisted of a lead-scintillator calorimeter [14][15][16] and, additionally in HERA II, a magnetic spectrometer [17].

Monte Carlo simulations
The Diffvm [18] Monte Carlo (MC) programme was used for simulating exclusive vector meson production, ep → eV p, where V denotes the produced vector meson with mass M V . For the event generation, the following cross-section parameterisations were used: for the cross-section ratio of longitudinal to transverse photons; exp(−b |t|), with b = 4 GeV −2 for the dependence on t; s-channel helicity conservation for the production of V → µ + µ − ; and a flat angular distribution for the ψ(2S) → J/ψ(1S)π + π − decay. As described in Section 4.3, the simulated events, after taking into account the acceptance, are reweighted to match the measured distributions. Proton-dissociative ψ(2S) and J/ψ(1S) events were not simulated. In the analysis of the experimental data, events were selected with protondissociative masses M Y 4 GeV, for which the Q 2 and W distributions of ψ(2S) and J/ψ(1S) events are expected to be similar. The fact that proton-dissociative events favour larger |t| values is taken into account by the reweighting procedure, where it has been assumed that the relative contribution is the same for ψ(2S) and J/ψ(1S) production.
Radiative effects were not simulated. The largest contribution is expected to come from the initial-state radiation of the electrons, however, as the ψ(2S) − J/ψ(1S) mass difference is small compared to W , the kinematics of both reactions are similar and radiative effects are expected to cancel in the cross-section ratio.
Non-resonant electroweak dimuon production (Bethe-Heitler background) was simulated using the dimuon programme Grape [19]. The event sample contains both exclusive and proton-dissociative events with a mass of the dissociated proton system, M Y < 25 GeV.
The generated MC events were passed through the ZEUS detector and trigger simulation programmes based on Geant 3 [20]. They were then reconstructed and analysed with the same programmes as used for the data.

Event selection
A three-level trigger system [4,21,22] was used to select events online. For this analysis, DIS events were selected with triggers containing a candidate scattered electron. Further triggers were used with a less stringent electron-candidate selection in coincidence with a muon candidate or with two tracks.
The offline event selection required an electron candidate in the RCAL with an energy E ′ e > 10 GeV as reconstructed using an algorithm based on a neural network [23]. The position of the scattered electron was required to be outside areas with significant inactive material in front of the calorimeter.
To select events containing exclusively produced J/ψ(1S) and ψ(2S) vector mesons, the following additional requirements were imposed: • the Z coordinate of the interaction vertex reconstructed from the tracks was required to be within ±30 cm of the nominal interaction point; • in addition to the scattered electron, two oppositely charged muons were required. Muons were identified using the GMUON algorithm [24] with muon quality ≥ 1. This algorithm required a track with momentum above 1 GeV to be matched with a cluster in the CAL. The cluster was required to be consistent with a muon using an algorithm based on a neural network [25]; • for selecting J/ψ(1S) → µ + µ − and ψ(2S) → µ + µ − events, no additional tracks were allowed; • for selecting ψ(2S) → J/ψ(1S) π + π − events, exactly two oppositely charged tracks were required in addition to the two muons. The tracks had to cross at least three CTD superlayers, produce hits in the first CTD superlayer or in the MVD and the transverse momentum of each track had to exceed 0.12 GeV; • events with calorimeter clusters with energies above 0.4 GeV, not associated with the electron or the decay products of the vector meson, were rejected. The threshold value of this cut was optimised by minimising event loss from calorimeter noise fluctuations and maximising the rejection of non-exclusive vector-meson production with additional energy deposits in the CAL.
The last three requirements significantly reduced the background from non-exclusive charmonium production and also removed proton-dissociative events with diffractive masses M Y 4 GeV [19].

Reconstruction of the kinematic variables and signal extraction
The kinematic variables, Q 2 , W and t were used in the analysis. The constrained method [26] was used to reconstruct Q 2 according to where Here E e denotes the electron beam energy and E ′ e and θ e the energy and the polar angle of the scattered electron, respectively. The quantity (E − p Z ) V denotes the difference of the energy and the Z component of the momentum of V . The momentum components of the vector-meson candidate, V , and its effective mass, M V , were obtained from the momentum vectors of the decay products measured by the tracking detectors. The values of W and t were calculated using Here E p is the proton beam energy, M p is the proton mass and p ′ T,e and p T,V are the transverse-momentum vectors of the scattered electron and of the vector-meson candidate, respectively. The kinematic range for the analysis of the HERA II (HERA I) data is 5 (2) < Q 2 < 80 GeV 2 , 30 < W < 210 GeV and |t| < 1 GeV 2 . The range of Bjorken x, , probed by the measurements is 2 × 10 −4 < x Bj < 10 −2 . Figure 2 shows the µ + µ − mass distribution for the selected events in the region 5 < Q 2 < 80 GeV 2 . Clear J/ψ(1S) and ψ(2S) peaks are seen. No other significant peak is observed. The background was fit by a straight line in the side-bands of the signals: 2.0 < M µµ < 2.62 GeV and 4.05 < M µµ < 5.0 GeV.
The ratio of ψ(2S) to J/ψ(1S) events for the µ + µ − decay channel was obtained from the ratio of the number of events above background in the range 3.59 < M µµ < 3.79 GeV to the corresponding number in the range 3.02 < M µµ < 3.17 GeV. According to a detailed MC study, this choice minimises the systematic uncertainty due to the uncertainties of the mass-resolution function and the shape of the background. The difference in widths of the mass bins chosen takes into account the worsening of the mass resolution with increasing M µµ .
To study the systematic uncertainty related to the background subtraction, a quadratic background function was used and the widths of the M µµ intervals for the signal determination were varied. The Bethe-Heitler MC events provide a good description of the background shape and its absolute normalisation after acceptance correction agrees with the measured rate within the estimated uncertainty of about 20 %. Given this uncertainty, the linear fit described in the previous paragraph was used for the background subtraction. Figure 3 shows a scatter plot of ∆M = M µµππ − M µµ versus M µµ and the ∆M and M µµππ distributions for 3.02 < M µµ < 3.17 GeV. As expected, the ∆M distribution shows a narrow peak with a width of about 5 MeV at the nominal ψ(2S) − J/ψ(1S) mass difference of 589.2 MeV. The mass ranges of 3.02 < M µµ < 3.17 GeV and 0.5 < ∆M < 0.7 GeV were chosen to compute the ratio of ψ(2S) to J/ψ(1S) events. As can be seen from Fig. 3, there is hardly any background in the ψ(2S) signal region; an upper limit of three events at 90 % confidence level was estimated for the background.

Comparison of measured and simulated distributions
In order to determine the acceptance using simulated events, simulated and measured distributions have to agree. To achieve this, the simulated events had to be reweighted. The reweighting functions for the J/ψ(1S) as well as for the ψ(2S) events were obtained by comparing the measured distributions of the J/ψ(1S) → µ + µ − sample to the simulated distributions. For the reweighting of the t and Q 2 distributions, a two-dimensional function was used. No reweighting was required for the W distribution. The following reweighting function was used for the angular distribution for the vector meson decays into µ + µ − : The helicity angle Φ h is the angle between the production and scattering planes, where the production plane is defined by the three-momenta of the photon and the vector meson, and the scattering plane is defined by the three-momenta of the incoming and the scattered electron. The elements of the spin-density matrix, r k ij , were obtained by fitting the weighted simulated events to the measured decay angular distribution of J/ψ(1S). The dominant contribution comes from r 1 00 , compatible with s-channel helicity conservation. The quantity ǫ denotes the ratio of the longitudinal to the transverse virtual-photon flux, which was set to unity in the kinematic range of the measurement. In the MC simulation, the fitted angular distributon was used as reweighting function for both vector-meson decays into µ + µ − , whereas no reweighting was applied for the flatly simulated decay ψ(2S) → J/ψ(1S) π + π − .
The measured |t| distributions for J/ψ(1S) and ψ(2S) have been fitted separately by single exponentials, and by the sum of two exponentials. It is found that the second exponential is not significant, and that the slopes for J/ψ(1S) and ψ(2S) agree within the statistical uncertainties. This confirms the validity of the assumptions made in Section 3: given the limited statistics of the data sample, it is neither necessary to simulate proton dissociative events nor to weight the |t| distributions of the J/ψ(1S) and ψ(2S) differently.
The combined cross-section ratio, R comb , was obtained using the weighted average of the cross sections determined for the two ψ(2S) decay modes. For the weights, the statistical uncertainties were used. The different R values were determined in the full kinematic region as well as in bins of Q 2 , W and |t|. The results are reported in Section 6.

Systematic uncertainties
The systematic uncertainties of the R values were obtained by performing for each source of uncertainty a suitable variation in order to determine the change of R relative to its nominal value. The following sources of systematic uncertainties were considered: • changing the background fit function from linear to quadratic changes the values of R µµ by ≈ −11% and of R J/ψ ππ by ≈ +0.5%; • changing the reconstruction from the constrained to the electron method [28] changes the values of R µµ and of R J/ψ ππ by ≈ +1.5%; • not applying the reweighting of the simulated events discussed in Section 4.3 changes the values of R µµ by ≈ −3% and of R J/ψ ππ by ≈ −1%; • applying different cuts on the total number of tracks, including tracks not associated with the event vertex, changes R µµ by ≈ −5% and R J/ψ ππ by ≈ +3%.
The total systematic uncertainty was obtained from the separate quadratic sums of the positive and negative changes. The estimated total systematic uncertainties are δR µµ = +7 −14 %, δR J/ψ ππ = +4 −5 % and δR comb = +3 −5 %. For the calculation of δR comb , the uncertainties of the two measurements were assumed to be uncorrelated.

Results
The results for the three cross-section ratios σ ψ(2S) /σ J/ψ(1S) : R µµ for ψ(2S) → µ + µ − , R J/ψ ππ for ψ(2S) → J/ψ(1S)π + π − and R comb for the combination, are reported in Table 1 for the kinematic range 5 < Q 2 < 80 GeV 2 , 30 < W < 210 GeV and |t| < 1 GeV 2 for the total integrated luminosity of 468 pb −1 . The cross-section ratios, differential in Q 2 , W and |t|, together with the additional measurement between 2 < Q 2 < 5 GeV 2 , corresponding to an integrated luminosity of 114 pb −1 , are shown in Table 2. The data contain a background of charmonium production with diffractive masses M Y 4 GeV. Assuming that the ψ(2S) to J/ψ(1S) cross-section ratio for exclusive charmonium production is the same as for charmonium production with low M Y , the determination of the R values for exclusive production are not affected. Figure 7 shows the values of R comb as a function of Q 2 , W and |t|. The results for the W and |t| dependence are shown for 5 < Q 2 < 80 GeV 2 . The results for the Q 2 dependence also include the additional measurement for 2 < Q 2 < 5 GeV 2 from the HERA I data. As a function of W and |t|, the values of R comb are compatible with a constant. For the Q 2 dependence, a positive slope is observed with the significance of ≈ 2.5 standard deviations.
As a cross check it was verified that the ratio R J/ψ ππ to R µµ is compatible with 1. For the entire kinematic range of the measurement a value of R ψ(2S) = R J/ψ ππ /R µµ = 1.1 ± 0.2 +0.2 −0.1 ± 0.1 is found. The first error is the statistical uncertainty, the second the systematic uncertainty of the measurement and the third the uncertainty of the ψ(2S) branching fractions. The ratio is consistent with unity for the entire kinematic region and also as a function of the kinematic variables, Q 2 , W and |t|, as shown in Tables 1 and 2.
In Fig. 8, the results are also compared to the previous H1 measurements [2]. The results are compatible. The H1 collaboration has also measured R = σ ψ(2S) /σ J/ψ(1S) in photoproduction (Q 2 ≈ 0), and found a value of R = 0.150 ± 0.035 [3], which is consistent with the trend of the Q 2 dependence presented in this paper. The comparison of the results to various model predictions is presented in the Section 7.

Comparison to model predictions
In this section, the cross-section ratio R = σ ψ(2S) /σ J/ψ(1S) in DIS, presented in this paper, and the results from the H1 collaboration in DIS [2] and photoproduction [3], are compared to model predictions from six different groups labelled by the names of the authors: HIKT, KNNPZZ, AR, LP, FFJS and KMW.

Individual models
In order to calculate the exclusive production of vector charmonium states according to the diagram shown in Fig. 1, the following components need to be determined: • the probability of finding a cc-dipole of transverse size r and impact parameter b in the photon in the infinite momentum frame; • the cc-dipole scattering amplitude or cross section of the proton as a function of r, b and x Bj ≈ (M V + Q 2 )/(W 2 + Q 2 ); • the probability that the cc-dipole forms the vector charmonium state V in the infinite momentum frame.
The probability distribution of cc-dipoles in the photon can be calculated in QED [29,30]. For the probability that the cc-dipole forms the vector charmonium state, its centre-ofmass wave function has to be boosted into the infinite momentum frame, which is done using the boosted Gaussian model [31]. For the evaluation of the cc-dipole scattering amplitude, different QCD calculations are used. All models discussed predict a Q 2 dependent suppression of exclusive ψ(2S) relative to J/ψ(1S) production. For the models, which explicitly use the wave functions of the vector mesons, this is caused by the node of the radial ψ(2S) wave function, which leads to a destructive interference of the contributions to the production amplitude from small and large dipoles.
Hüfner et al. [32] (HIKT) use two phenomenological parameterisations of the cc-dipole cross section, GBW [33] and KST [34], which both describe the low-x inclusive DIS data from HERA. For the centre-of-mass wave functions of the J/ψ(1S) and ψ(2S), they use four different phenomenological potentials, BT, LOG, COR and POW, and cquark masses between 1.48 and 1.84 GeV. However, only the models with c-quark masses around 1.5 GeV, GBW-BT and GBW-LOG, are able to describe the cross sections of exclusive J/ψ(1S) production measured at HERA. For the boost of the charmonium wave functions into the infinite momentum frame, they find the wave function from the Schrödinger equation and then boost the result. A major progress made [32] is the inclusion of the Melosh spin rotation into the boosting procedure, which enhances the ψ(2S) to J/ψ(1S) cross-section ratio by a factor of two to three.
The model of Kopeliovich et al. [35][36][37][38] (KNNPZZ) uses the running gBFKL approach for the cc-dipole cross section and the diffractive slope for its t dependence. The parameterisation of the cc-dipole cross section used gives a quantitative description of the rise of the proton structure function at small x values as well as of the Q 2 and W dependence of diffractive J/ψ(1S) production. KNNPZZ use parametrisations of the vector meson wave functions, inspired by the conventional spectroscopic models and short-distance behaviour driven by hard QCD gluon exchange. For the ψ(2S), an additional parameter is introduced, which controls the position of the node.
Armesto and Rezaeian [39] (AR) calculate the cc-dipole cross section using the Impact-Parameter-dependent Color Glass Condensate model (b-CGC) [40] as well as the Saturation (IP-Sat) [41] dipole model, recently updated with fits to the HERA combined data [42,43]. In the b-CGC model, which is restricted to the gluon sector, saturation is driven by the BFKL evolution, and its validity is therefore limited to x Bj 10 −2 . The IP-Sat model uses DGLAP evolution and smoothly matches the perturbative QCD limit at high values of Q 2 . For the calculation of the light-cone J/ψ(1S) and ψ(2S) wave functions, the boosted Gaussian model and the leptonic decay widths Γ J/ψ(1S) ee and Γ ψ(2S) ee are used.
Lappi and Mäntysaari [44,45] (LM) use the BFKL evolution as well as the IP-Sat model to predict vector-meson production in ep and electron-ion collisions in the dipole picture. The wave functions of the J/ψ(1S) and ψ(2S) have been calculated according to the procedure developed previously [46,47] and the low-x inclusive HERA data have been used to constrain the cc-dipole cross section.
Fazio et al. [48] (FFJS) use a two-component Pomeron model to predict the cross sections for vector-meson production. A normalisation factor of f −1 ψ(2S) = 0.45 ensures that the value of the ψ(2S) cross section is the same as for the other vector mesons at the same values of W , t and Q 2 In this model the Q 2 dependence of the ψ(2S) to J/ψ(1S) suppression is caused by the difference of Q 2 V at a fixed Q 2 due to the ψ(2S) − J/ψ(1S) mass difference.
Kowalski, Motyka and Watt [46] (KMW) assume the universality of the production of vector quarkonia states in the scaling variable Q 2 V in their calculation of R. With the assumptions that the cc → V transition is proportional to the leptonic decay width Γ V ee and that the interaction is mediated by two-gluon exchange and therefore proportional to α s (Q V ) x Bj g(x Bj , Q 2 V ) 2 , R is given in the leading-logarithmic approximation [49][50][51] by The running strong coupling constant, α s (Q), and the gluon density, g(x, Q 2 ), are evaluated at Q V and x Bj . For small x Bj values, the gluon density can be parameterised as with λ(Q V ) ≃λ = 0.25 in the Q V region of this measurement. The parameter δ depends on the choice of the charmonium wave functions. For the nonrelativistic wave functions δ = 0 [49][50][51] and for the relativistic boosted Gaussian model δ ≈ 2 [46]. Note that the Q 2 dependence of the ratio R in this approach is driven by kinematic factors and not by the shapes and nodes of the charmonia wave functions.

Comparison of models and data
In the kinematic region of the measurement, all models predict only a weak W and |t| dependence of R, consistent with the data, as shown in Fig. 7. Therefore, only the comparison of the model calculations with the measurements as a function of Q 2 is presented in Fig. 8. It can be seen that all models predict an increase of R with Q 2 , which is also observed experimentally. In the following discussion, the models are discussed in the sequence from higher to lower predicted R values at high Q 2 .
From the HIKT calculations, the results for R for the two charmonium potentials BT and LOG with c-quark mass around 1.5 GeV and the GBW model for the cc-dipole cross section are shown. The difference of the results when using the KST dipole cross sections are small. For Q 2 values below 24 GeV 2 , the predicted R values for the BT model are significantly larger than the measured values, whereas the values predicted by the LOG model agree with the data.
For the AR calculations, the results for the b-CGC and IP-Sat models of the dipole cross sections are shown in the figure. For Q 2 values below 24 GeV 2 , the b-CGC prediction for R is significantly higher than the data, whereas the IP-Sat model gives a good description of the data for the entire Q 2 range.
The KMW model with δ = 0 provides a good description of the observed Q 2 dependence of R, whereas the prediction for δ = 2 is about 2 standard deviations below the R value measured for Q 2 > 24 GeV 2 .
The predictions of the models FFJS, KNNPZZ and LM, in spite of differences in the values of R at low Q 2 values, also provide fair descriptions of the measurements.
Some discrimination of the different models is possible, although a large spread in the predictions indicates that the uncertainty on the theory is large.

Summary
The cross-section ratio R = σ ψ(2S) /σ J/ψ(1S) in exclusive electroproduction was measured with the ZEUS experiment at HERA in the kinematic range 2 < Q 2 < 80 GeV 2 , 30 < W < 210 GeV and |t| < 1 GeV 2 , with an integrated luminosity of 468 pb −1 . The decay channels used were µ + µ − and J/ψ(1S) π + π − for the ψ(2S) and µ + µ − for the J/ψ(1S). The cross-section ratio has been determined as a function of Q 2 , W and |t|. The results are consistent with a constant value of R as a function of W and t, but show a tendency to increase with increasing Q 2 . A number of model calculations are compared to the measured Q 2 dependence of R. Table 1: Measured cross-section ratios σ ψ(2S) /σ J/ψ(1S) : R J/ψππ for the decay ψ(2S) → J/ψ(1S) π + π − , R µµ for the decay ψ(2S) → µ + µ − , and R comb for the two decay modes combined, for the kinematic range 5 < Q 2 < 80 GeV 2 , 30 < W < 210 GeV and |t| < 1 GeV 2 at an ep centre-of-mass energy of 317 GeV. Also shown is R ψ(2S) = R J/ψππ /R µµ . Statistical and systematic uncertainties are given.  Table 2: Measured cross-section ratios σ ψ(2S) /σ J/ψ(1S) : R J/ψππ for the decay ψ(2S) → J/ψ(1S) π + π − , R µµ for the decay ψ(2S) → µ + µ − , and R comb for the two decay modes combined, as a function of Q 2 , W and |t|. Also shown is R ψ(2S) = R J/ψππ /R µµ . The ratios as a function of W are measured in the range 5 < Q 2 < 80 GeV 2 and |t| < 1 GeV 2 , as a function of Q 2 in the range 30 < W < 210 GeV and |t| < 1 GeV 2 , and as a function of |t| in the range 5 < Q 2 < 80 GeV 2 and 30 < W < 210 GeV. All results are quoted for an ep centre-of-mass energy of 317 GeV, except for the bin 2 < Q 2 < 5 GeV 2 , which refers to 300 GeV. Statistical and systematic uncertainties are given.      Figure 8: Cross-section ratio R = σ ψ(2S) /σ J/ψ(1S) for the combined ψ(2S) decay modes as a function of Q 2 . The kinematic range is 30 < W < 210 GeV and |t| < 1 GeV 2 at an ep centre-of-mass energy of 317 GeV for the ZEUS data with Q 2 > 5 GeV 2 and 300 GeV for the ZEUS data with 2 < Q 2 < 5 GeV 2 . The ZEUS results (solid points) are shown compared to the previous H1 result (open points) [2] measured for 25 < W < 180 GeV and |t| < 1.6 GeV 2 at an ep centre-of-mass energy of 300 GeV. The inner error bars show the statistical and the outer error bars show the quadratic sum of statistical and systematic uncertainties. The ZEUS points are plotted at the average Q 2 of the reweighted simulated ψ(2S) events with the W and t cuts used in the analysis, as recommended elsewhere [52]. The model predictions discussed in Section 7.1 are shown as curves. The sequence of the labelling is in descending order of the R value at the highest Q 2 of each prediction.