The fully corrected invariant cross sections of \(\omega \) production were obtained for each reconstruction technique using
$$\begin{aligned} E\frac{\hbox {d}^3{\sigma ^{pp\rightarrow \omega + X}}}{{\hbox {d}p^{3}}}= & {} \frac{1}{2\pi } \frac{1}{{p_{\mathrm {T}}}}\cdot \frac{1}{\mathscr {L}_{\text {int}}}\nonumber \\&\cdot \frac{1}{A\cdot \varepsilon _{\text {rec.}}}\cdot \frac{1}{\text {BR}_{\omega \rightarrow \pi ^+\pi ^-\pi ^0}}\cdot \frac{N^{\omega }}{\Delta y\Delta {p_{\mathrm {T}}}}. \end{aligned}$$
(1)
Here, \(\mathscr {L}_{\text {Int}}\) is the integrated luminosity given in Sect. 3, \(\varepsilon _{\text {rec.}}\) and A are the reconstruction efficiency and acceptance of the corresponding method and \(\text {BR}=(89.3\pm 0.6)\%\) is the branching ratio of the \(\omega \rightarrow \pi ^+\pi ^-\pi ^0\) decay [10]. Moreover, \(N^{\omega }\) denotes the number of reconstructed \(\omega \) mesons in the transverse momentum range \(\Delta \) \(p_{\mathrm {T}}\) and the given rapidity range \(\Delta y\).
The production cross sections were measured individually for each reconstruction method and then combined using \(p_{\mathrm {T}}\)-dependent weights that are calculated according to the best linear unbiased estimate (BLUE) algorithm [42], which uses concepts that are routinely applied in statistical fields. The combination took into account statistical and systematic uncertainties. For the systematic uncertainties, the individual measurements are found to be correlated by about 30%, dominantly originating from the charged-pion selection and the material budget uncertainties. These correlations were taken into account in the combination procedure. The statistical and systematic uncertainties of the combined measurement are given in Table 1.
The cross section of \(\omega \) meson production for \(2 < \) \(p_{\mathrm {T}}\) \(<17\,\hbox {GeV}/c\) at midrapidity in pp collisions at \(\sqrt{s}=7\,\hbox {TeV}\) is shown in Fig. 3. It was fitted using a Levy–Tsallis function [46] given by
$$\begin{aligned} E\frac{\text{ d}^3\sigma }{\text{ d }p^3} =\frac{C}{2\pi } \frac{(n-1)(n-2)}{nT [nT + m(n-2)]} \left( 1 + \frac{m_{T}-m}{nT}\right) ^{-n}, \end{aligned}$$
(2)
which describes the cross section over the whole measured transverse momentum range, as demonstrated in the lower panel of the figure. The parameters m and \(m_{\text {T}} = \sqrt{m^2 + p_{\text {T}}^2}\) correspond to the particle mass and the transverse mass, respectively, while C, T and n are the free parameters of the Levy–Tsallis function.
Table 2 Parameters and \(\chi ^2\)/NDF of the fit to the \(\omega \) invariant cross section using the Levy–Tsallis function [46] from Eq. 2 The values of the fit parameters and the reduced \(\chi ^2\) of the fit are given in Table 2, where the fit was obtained using only statistical uncertainties, and using the systematic and statistical uncertainties of the measurement added in quadrature. To account for finite \(p_{\mathrm {T}}\)-interval width, the combined cross section points were assigned to \(p_{\mathrm {T}}\) values shifted from the bin centre of the \(p_{\mathrm {T}}\) intervals according to the underlying spectrum [47] described by a Levy–Tsallis function. This correction resulted in a shift below 2% in each \(p_{\mathrm {T}}\) interval.
Figure 4, which shows the ratios of the cross sections for the individual reconstruction methods to the Levy–Tsallis fit of the combined measurement, demonstrates the agreement between all methods within the statistical and systematic uncertainties, justifying the combination of the individual results as discussed earlier.
The measured differential cross section of \(\omega \) production is compared to several calculations in Fig. 3. The ratio of each prediction to the Levy–Tsallis fit of the measurement is shown in the bottom panel of the figure. Two PYTHIA 8.2 [43] Monte Carlo event generator calculations were considered for comparison, which are based on the Monash 2013 [44] and the 4C [45] tunes, respectively. The Monash 2013 tune describes the measurement over the full reported \(p_{\mathrm {T}}\) range within the uncertainties, while the Tune 4C overestimates the data by about 50%. The Monash 2013 tune includes more recent experimental results than Tune 4C and thus a more refined set of parameters. In particular, the rate of light flavor vector meson production used in hadronisation process was revised and lowered, improving the description of \(\omega \) meson yields [44].
The measurement is also compared to a next-to-leading order (NLO) calculation using a model with broken SU(3) symmetry to describe vector meson production [14], where the model parameters have been constrained using \(\omega \) production data measured by PHENIX in pp collisions at \(\sqrt{s}=200\,\hbox {GeV}\) [16]. The same scale \(\mu ={p_{\mathrm {T}}}\) was used for factorisation, renormalisation and fragmentation for the calculation and the shaded band reported in Fig. 3 denotes the scale variation of \({p_{\mathrm {T}}}^2/2\le \mu ^2\le 2{p_{\mathrm {T}}}^2\). The calculation describes the measurement within the uncertainties below \(6\,\hbox {GeV}/c\), and overestimates the data by up to 50% for higher \(p_{\mathrm {T}}\).
The ratio of \(\omega \) relative to \(\pi ^0\) meson production is shown as a function of \(p_{\mathrm {T}}\) in Fig. 5, where data points for the \(\pi ^0\) measurement were taken from Ref. [6]. The ratio is observed to be constant above \(2.5\,\hbox {GeV}/c\) with a value of \(C^{\omega /\pi ^{0}}= 0.69 \pm 0.03 \text {~(stat)~} \pm 0.04 \text {~(sys)}\). Within the uncertainties, the \(\omega /\pi ^0\) ratio is described by the PYTHIA predictions. Even though the Tune 4C overestimates the \(\omega \) production, it describes the \(\omega /\pi ^0\) ratio due to a similar overestimation of \(\pi ^0\) production, which was reported in Ref. [8].
The measured \(\omega /\pi ^0\) ratio at \(\sqrt{s}=7\,\hbox {TeV}\) is compared to data from lower collision energies at \(\sqrt{s} = 62\) [15] and 200 GeV [16,17,18]. The \(\omega /\pi ^0\) ratios measured at the different collision energies agree within the uncertainties. In order to test the validity of \(m_{\mathrm {T}}\)-scaling, the Levy–Tsallis parametrisation \(f_{\pi ^0}(p_{\text {T,}\pi ^0})\) of the \(\pi ^0\) spectrum reported in Ref. [6] was scaled using the ratio \(C^{\omega /\pi ^{0}}= 0.67\), following the procedure discussed in detail in Ref. [20]. The scaled parametrisation \(f_\omega (p_{\text {T,}\omega })\) was used to calculate the \(\omega /\pi ^0\) ratio via \(f_\omega (p_{\text {T,}\omega })/f_{\pi ^0}(p_{\text {T,}\omega })\), where the relation \(p_{\text {T,}\omega }^2 + m_{0,\omega }^2 = p_{\text {T,}\pi ^0}^2 + m_{0,\pi ^0}^2\) was used to ensure the evaluation of both spectra at the same transverse mass. The obtained \(m_{\mathrm {T}}\)-scaling prediction of the \(\omega /\pi ^0\) ratio is shown in Fig. 5 and found to be consistent with the measurement. Unlike in the case of the \(\eta /\pi ^0\) ratio measured at \(\sqrt{s} = 2.76\), 7 and 8 TeV [6, 8, 33], where a violation of \(m_{\mathrm {T}}\)-scaling was observed below \(3.5\,\hbox {GeV}\), no such violation is observed within the uncertainties for the \(\omega \) meson in the entire measured momentum range. However, while the measurement is compatible with the \(m_{\mathrm {T}}\)-scaling prediction at low-\(p_{\mathrm {T}}\), the sensitivity of the measurement to a possible \(m_{\mathrm {T}}\)-scaling violation is limited by the uncertainties and \(p_{\mathrm {T}}\) reach. Here, future studies with increased precision could provide further insights and more stringent tests of \(m_{\mathrm {T}}\)-scaling for low-\(p_{\mathrm {T}}\) \(\omega \) mesons. Interestingly, the PYTHIA calculations and the \(m_{\mathrm {T}}\)-scaled prediction both describe the \(\omega /\pi ^0\) ratio at lower collision energies even below \(p_{\mathrm {T}}=2\,\hbox {GeV}/c\), suggesting a universal feature of meson production.