1 Erratum to: Eur Phys J C (2012) 72:2100https://doi.org/10.1140/epjc/s10052-012-2100-4

1.1 Nature of the correction

In Ref. [1], the production rate of \(\psi {(2S)}\) mesons in the rapidity range \(2.0< y < 4.5\) was measured for pp collisions at \(\sqrt{s} =7\text {\,TeV} \) using a sample of data corresponding to \(36\text {\,pb} ^{-1} \). Both overall and singly differential \(({\mathrm {d}}\sigma / {\mathrm {d}}p_{\mathrm {T}})\) cross-sections were measured by fitting the invariant-mass spectra to obtain background-subtracted signal yields, which are subsequently efficiency corrected. Two decay modes were used: \({\psi {(2S)}} \rightarrow \mu ^+ \mu ^-\) and \({\psi {(2S)}} \rightarrow {{J /\psi }} ({\mu ^+\mu ^-}) {{\pi } ^+} {{\pi } ^-} \).

Two sources of \(\psi {(2S)}\) production are expected in this environment: mesons produced promptly in the primary interaction (whether directly or through the decay of an intermediate resonance), and those produced via the decays of b hadrons. The vast majority of b hadrons produced in the LHCb acceptance consist of \(B^0\), \(B^+\), \(B_s^0\) mesons and baryons, all with mean lifetimes of approximately 1.5 ps. Consequently, the two classes of production may be separated according to whether the \(\psi {(2S)}\) originates from the primary vertex (PV) or from a downstream secondary vertex. This separation must be done on a statistical level, since some b hadrons will decay close to the PV on the scale of the experimental resolution.

The pseudo-decay-time \(t_z\) was used to distinguish the two sources of production, and is defined as

$$\begin{aligned} t_z=\frac{\left( z_{{\psi {(2S)}}}-z_{\mathrm {PV}}\right) \times M_{{\psi {(2S)}}}}{p_z}, \end{aligned}$$

where \(z_{{\psi {(2S)}}}\) and \(z_{{\mathrm {PV}}}\) are the z coordinates of the reconstructed \({\psi {(2S)}} \) decay vertex and the primary vertex, \(p_z\) is the z-component of the measured \({\psi {(2S)}} \) momentum, \(M_{{\psi {(2S)}}}\) is the known \({\psi {(2S)}} \) mass [2], and the z-axis is the direction of the proton beam pointing downstream into the LHCb acceptance. For a given sample of \({\psi {(2S)}} \) candidates, a fit to the \(t_z\) distribution was used to obtain the prompt fraction \(f_{\mathrm {p}}\), as described in Sec. 4 of Ref. [1].

Fig. 1
figure 1

Pseudo-decay-time \(t_z\) distribution for the \({\psi {(2S)}} \rightarrow \mu ^+ \mu ^-\) decay mode in the range \(4<p_{\mathrm {T}} \le 5\) \(\text {\,GeV\!/}c\), showing the background and prompt contributions

Fig. 2
figure 2

Fraction of prompt \(\psi {(2S)}\), \(f_{\mathrm {p}}\), as a function of \(p_{\mathrm {T}}\). The error bars include statistical and systematic uncertainties added in quadrature

Fig. 3
figure 3

Differential production cross-section of prompt \(\psi {(2S)}\)  as a function of \(p_{\mathrm {T}}\) in the range \(2.0< y < 4.5\). The results are compared with the NRQCD calculations [4]. The error bars include statistical and systematic uncertainties added in quadrature

Fig. 4
figure 4

Differential production cross-section of \(\psi {(2S)}\) from b hadrons as a function of \(p_{\mathrm {T}}\) in the range \(2.0< y < 4.5\). The results are compared with the FONLL calculations [5]. The error bars include statistical and systematic uncertainties added in quadrature

Table 1 Differential cross-sections \(\mathrm {d}\sigma /\mathrm {d}p_{\mathrm {T}} \) (in \(\text {\,nb}\)/(\({\mathrm {GeV\!/}c}\))) of prompt \(\psi {(2S)}\) and \(\psi {(2S)}\)-from-b hadrons at \(\sqrt{s} = 7\text {\,TeV} \), integrated over y between 2.0 and 4.5. The first uncertainty is statistical and the second systematic. The third asymmetric uncertainty for the prompt \(\psi {(2S)}\) mesons is due to the unknown polarisation

Two distinct problems related to the determination of \(f_{\mathrm {p}}\) in Ref. [1] have been identified. The first is that a mathematical mistake was made in calculating the systematic uncertainties on the from-b \(\psi {(2S)}\) production cross-sections that arise due to uncertainties in the \(t_z\) fit; a factor of \(f_{\mathrm {p}} / (1 - f_{\mathrm {p}})\) was omitted. When this mistake is corrected, those systematic uncertainties increase by a factor 3 to 9, depending on the \(p_{\mathrm {T}}\) defined in the range \(0-16\text {\,GeV\!/}c \), with the largest effect at low \(p_{\mathrm {T}}\), where the prompt fraction is close to unity. The correct formula is used in the results below.

The second problem is related to the values of \(f_{\mathrm {p}}\) themselves. A mistake appears to have been made in the measurement of \(f_{\mathrm {p}}\) via the fits to the \(t_z\) distributions used in Ref. [1]. An independent reimplementation of the analysis finds consistently lower values of \(f_{\mathrm {p}}\). This change in \(f_{\mathrm {p}}\) is associated with a change in the mean value of \(t_z\) seen for the from-b component: values of approximately \(1.1\text {\,ps} \) were found in the analysis reported in Ref. [1], compared to approximately \(1.5\text {\,ps} \) (much closer to the mean lifetime of contributing b hadrons) in the reimplementation. This issue has been found using both the original \(t_z\) fit function as described in Ref. [1] and the function used in Ref. [3]. The more sophisticated \(t_z\) fit function used in Ref. [3] achieves a more precise description of experimental data and is thus used in obtaining corrections as described below. An example of a \(t_z\) fit in the \(p_{\mathrm {T}}\) range \(4<p_{\mathrm {T}} \le 5\) \(\text {\,GeV\!/}c\) is shown in Fig. 1.

Fig. 5
figure 5

Ratio of \({\psi {(2S)}} \rightarrow \mu ^+ \mu ^-\) and \({{J /\psi }} \rightarrow {\mu ^+\mu ^-} \) cross-sections for prompt production (a) and for b-hadron decay (b), as a fucntion of \(p_{\mathrm {T}}\)

Because the issue found is limited to the determination of \(f_{\mathrm {p}}\) and does not affect the combined cross-section, and given that the reimplementation uses a sample of \({\psi {(2S)}} \rightarrow {\mu ^+\mu ^-} \) events that is correlated with but not identical to the original analysis, the approach used in this erratum is to use the new and old values of \(f_{\mathrm {p}}\) to determine a correction factor to apply to the results of the prompt and from-b cross-sections of the original analysis. (A separate and statistically independent analysis of the larger \(7\text {\,TeV} \) data sample taken in 2011 has been submitted [3] but is outside the scope of this erratum.) Defining \(f_b \equiv 1 - f_{\mathrm {p}}\) for convenience, the ratio

$$\begin{aligned} {\mathcal {R}}_{b} = \frac{ f_b \text{ obtained } \text{ with } \text{ reimplementation } }{ f_b({\psi {(2S)}} \rightarrow {\mu ^+\mu ^-}) \text{ obtained } \text{ in } \text{ original } \text{ analysis } } \end{aligned}$$

is determined in bins of \(p_{\mathrm {T}}\). The correction is then obtained by fitting a linear function to the individual values of \(R_b\). This also allows the correction to be extrapolated to kinematic regions where data were not available for the reimplementation (\(p_{\mathrm {T}} < 2\text {\,GeV\!/}c \), \(p_{\mathrm {T}} > 11\text {\,GeV\!/}c \)). This correction is applied to the weighted average of \({\psi {(2S)}} \rightarrow {\mu ^+\mu ^-} \) and \({\psi {(2S)}} \rightarrow {{J /\psi }} ({\mu ^+\mu ^-}) {{\pi } ^+} {{\pi } ^-} \) results as reported by Ref. [1].

After applying the correction to \(f_b\), the systematic uncertainties are recomputed. These are unchanged respect to those of the original analysis (other than relative uncertainties being updated for the new central values) except as described below. First, the the mistake in the computation of the uncertainty associated with the \(t_z\) fit is corrected as described above. Second, a new systematic uncertainty associated with the \(f_b\) correction estimate is added, and in particular the extrapolation outside the fit region, which is determined by taking the difference between the correction fitted by a first-order and a second-order polynomial.

1.2 Corrected results

The impact on \(f_{\mathrm {p}}\) itself and on the cross-section for prompt production is modest: they are both reduced by an amount typically of the order of several percent. However the relative impact on \(f_b\) is greater, and the from-b cross-section rises by typically 20–25%.

Corrected versions of all figures and tables in Ref. [1] that were affected by the issue are given in the following. The corrected \(f_{\mathrm {p}}\) distribution as a function of \(p_{\mathrm {T}}\)  is shown in Fig. 2. The singly differential cross-section as a function of \(p_{\mathrm {T}}\)  is shown for prompt production in Fig. 3, and for production from b-hadrons in Fig. 4. In the figures, the updated cross-sections are compared with theory predictions, namely NRQCD calculations [4] for prompt production and FONLL calculations [5] for production of \(\psi {(2S)}\) from b-hadron decays. The integrated cross-sections in the nominal kinematic range for prompt \(\psi {(2S)}\) and \(\psi {(2S)}\)-from-\(b \) are found to be

$$\begin{aligned} \begin{aligned} \sigma _{\mathrm{prompt}}({\psi {(2S)}})&= 1.37 \pm 0.01~(\text {stat}) \pm 0.06~(\text {syst})^{+0.19}_{-0.38}~(\text {pol})\,\upmu \text {b}, \\ \sigma _{b}({\psi {(2S)}})&= 0.31 \pm 0.01~(\text {stat}) \pm 0.02~(\text {syst})\,\upmu \text {b}. \end{aligned} \end{aligned}$$

The numerical results are given in Table 1.

Corrected ratio of \({\psi {(2S)}} \rightarrow \mu ^+ \mu ^-\) and \({{J /\psi }} \rightarrow {\mu ^+\mu ^-} \) cross-sections for prompt production (\({{\mathrm {R}}}_{{\mathrm {p}}}\)) and for b-hadron decay (\({\mathrm {R}}_{b}\)) as a function of \(p_{\mathrm {T}}\) is shown on Fig. 5.

The inclusive \({b} \rightarrow {\psi {(2S)}} X\) branching fraction is computed using the \(\psi {(2S)}\)-from-\(b \) cross-sections reported above and found to be

$$\begin{aligned} {\mathcal {B}}(b \rightarrow {\psi {(2S)}} X) = (3.08 \pm 0.07({\mathrm {stat}}) \pm 0.36({\mathrm {syst}}) \pm 0.27({\mathrm {{{\mathcal {B}}}}})) \times 10^{-3}. \end{aligned}$$

The last uncertainty is due to those of the branching fractions, and is dominated by the \({{\mathcal {B}}} (b \rightarrow {{J /\psi }} X)\) uncertainty.