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The momentum-differential invariant cross sections of ${π^{0}}$ and $η$ mesons are reported for pp collisions at $\sqrt{s}$ = 13 TeV at midrapidity ($|y|<0.8$). The measurement is performed in a broad transverse-momentum range of $0.2<p_{\rm T}<200$ GeV/$c$ and $0.4 < p_{\rm T} < 60$ GeV/$c$ for the ${π^{0}}$ and $η$, respectively, extending the $p_{\rm T}$ coverage of previous measurements. Transverse-mass-scaling violation of up to 60% at low transverse momentum has been observed, agreeing with measurements at lower collision energies. Transverse Bjorken $x$ ($x_{\rm T}$) scaling of the ${π^{0}}$ cross sections at LHC energies is fulfilled with a power-law exponent of $n = 5.01 \pm 0.05$, consistent with values obtained for charged pions at similar collision energies. The data are compared to predictions from next-to-leading order perturbative QCD calculations, where the ${π^{0}}$ spectrum is best described using the CT18 parton distribution function and the NNFF1.0 or BDSS fragmentation function. Expectations from PYTHIA8 and EPOS LHC overestimate the spectrum for the ${π^{0}}$ and are not able to describe the shape and magnitude of the $η$ spectrum. The charged-particle multiplicity dependent ${π^{0}}$ and $η$ $p_{\rm T}$ spectra show the expected change of the spectral shape, characterized by a flatter slope with increasing multiplicity. This is demonstrated across a broad transverse-momentum range and up to events with a charged-particle multiplicity exceeding five times the mean value in minimum bias collisions. The $η/π^{0}$ ratio depends on the charged-particle multiplicity for $ p_{\rm T} < 4$ GeV/$c$. PYTHIA8 and EPOS LHC qualitatively explain this behavior with an increasing contribution from the feed-down of heavier particles to the ${π^{0}}$ spectrum.
Invariant differential cross section of the $\pi^{0}$ versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV.
Invariant differential cross section of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV.
The $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV.
Ratio of the $\eta/\pi^{0}$ ratio to the $m_{\rm T}$-scaling prediction as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV.
$x_{\rm T}$ scaled $\pi^{0}$ spectrum for pp collisions at $\sqrt{s}$ = 0.9 TeV, with $n = 5.01$.
$x_{\rm T}$ scaled $\pi^{0}$ spectrum for pp collisions at $\sqrt{s}$ = 2.76 TeV, with $n = 5.01$.
$x_{\rm T}$ scaled $\pi^{0}$ spectrum for pp collisions at $\sqrt{s}$ = 7 TeV, with $n = 5.01$.
$x_{\rm T}$ scaled $\pi^{0}$ spectrum for pp collisions at $\sqrt{s}$ = 8 TeV, with $n = 5.01$.
$x_{\rm T}$ scaled $\pi^{0}$ spectrum for pp collisions at $\sqrt{s}$ = 13 TeV, with $n = 5.01$.
$x_{\rm T}$ scaled $\eta$ spectrum for pp collisions at $\sqrt{s}$ = 2.76 TeV, with $n = 4.91$.
$x_{\rm T}$ scaled $\eta$ spectrum for pp collisions at $\sqrt{s}$ = 7 TeV, with $n = 4.91$.
$x_{\rm T}$ scaled $\eta$ spectrum for pp collisions at $\sqrt{s}$ = 8 TeV, with $n = 4.91$.
$x_{\rm T}$ scaled $\eta$ spectrum for pp collisions at $\sqrt{s}$ = 13 TeV, with $n = 4.91$.
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0-100%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 70-100%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 50-70%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 30-50%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 20-30%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 10-20%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 5-10%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 1-5%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0-1%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0.05-0.1%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0-0.1%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0.01-0.05%. (Scaled for visibility in the figure)
Invariant differential yields of the $\pi^{0}$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0-0.01%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0-100%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 70-100%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 50-70%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 30-50%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 20-30%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 10-20%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 5-10%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 1-5%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0-1%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0.05-0.1%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0-0.1%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0.01-0.05%. (Scaled for visibility in the figure)
Invariant differential yields of the $\eta$ meson versus transverse momentum for pp collisions at $\sqrt{s}$ = 13 TeV in the INEL>0 event class in the V0M multiplicity class 0-0.01%. (Scaled for visibility in the figure)
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 70-100% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 50-70% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 30-50% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 20-30% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 10-20% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 5-10% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 1-5% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 0-1% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 0.05-0.1% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 0-0.1% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 0.01-0.05% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\pi^{0}$ meson in the V0M multiplicity class 0-0.01% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 70-100% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 50-70% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 30-50% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 20-30% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 10-20% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 5-10% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 1-5% to the inclusive spectrum in the INEL>0 event class.
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 0-1% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 0.05-0.1% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 0-0.1% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 0.01-0.05% to the inclusive spectrum in the INEL>0 event class (data points not shown in figure 11).
Ratios of the invariant differential yields of $\eta$ meson in the V0M multiplicity class 0-0.01% to the inclusive spectrum in the INEL>0 event class.
Slope parameter (n) of a power-law fit ($f(p_{\rm T}) = \alpha p_{\rm T}^{-n}$) for $5 < p_{\rm T} < 15$ GeV/$c$ as a function the charged-particle multiplicity density in units of the average normalized multiplicity density for the INEL>0 event class for the neutral pion.
Slope parameter (k) of an exponential fit ($f(p_{\rm T}) = \alpha e^{-p_{\rm T}/k}$) for $0.4 < p_{\rm T} < 2$ GeV/$c$ as a function the charged-particle multiplicity density in units of the average normalized multiplicity density for the INEL>0 event class for the neutral pion.
Slope parameter (n) of a power-law fit ($f(p_{\rm T}) = \alpha p_{\rm T}^{-n}$) for $5 < p_{\rm T} < 15$ GeV/$c$ as a function the charged-particle multiplicity density in units of the average normalized multiplicity density for the INEL>0 event class for the eta meson.
Slope parameter (k) of an exponential fit ($f(p_{\rm T}) = \alpha e^{-p_{\rm T}/k}$) for $1.5 < p_{\rm T} < 4$ GeV/$c$ as a function the charged-particle multiplicity density in units of the average normalized multiplicity density for the INEL>0 event class for the eta meson.
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 0-100%.
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 70-100% (data points not shown in figure 13).
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 50-70%.
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 30-50% (data points not shown in figure 13).
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 20-30% (data points not shown in figure 13).
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 10-20% (data points not shown in figure 13).
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 5-10% (data points not shown in figure 13).
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 1-5% (data points not shown in figure 13).
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 0-1% (data points not shown in figure 13).
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 0.05-0.1% (data points not shown in figure 13).
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 0-0.1%.
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 0.01-0.05% (data points not shown in figure 13).
$\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV in INEL>0 collisions in the V0M multiplicity class 0-0.01% (data points not shown in figure 13).
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 70-100% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV (data points not shown in figure 13).
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 50-70% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV.
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 30-50% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV (data points not shown in figure 13).
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 20-30% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV (data points not shown in figure 13).
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 10-20% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV (data points not shown in figure 13).
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 5-10% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV (data points not shown in figure 13).
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 1-5% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV (data points not shown in figure 13).
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 0-1% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV (data points not shown in figure 13).
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 0.05-0.1% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV (data points not shown in figure 13).
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 0-0.1% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV.
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 0.01-0.05% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV (data points not shown in figure 13).
Ratio of $\eta/\pi^{0}$ ratio in INEL>0 collisions in the V0M multiplicity class 0-0.01% to the inclusive $\eta/\pi^{0}$ ratio as a function of $p_{\rm T}$ for pp collisions at $\sqrt{s}$ = 13 TeV (data points not shown in figure 13).
Mean values of the ratio of the $\eta/\pi^{0}$ ratio in a certain multiplicity interval to the $\eta/\pi^{0}$ ratio in the INEL>0 event class for $1 < p_{\rm{ T}} < 4 \rm{~GeV}/c$ as a function of the normalized mean charged-particle multiplicity.
Mean values of the ratio of the $\eta/\pi^{0}$ ratio in a certain multiplicity interval to the $\eta/\pi^{0}$ ratio in the INEL>0 event class for $5 < p_{\rm{ T}} < 15 \rm{~GeV}/c$ as a function of the normalized mean charged-particle multiplicity.
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