The crosssections of $\psi(2S)$ meson production in protonproton collisions at $\sqrt{s}=13 \mathrm{TeV}$ are measured with a data sample collected by the LHCb detector corresponding to an integrated luminosity of $275 p\mathrm{b}^{1}$. The production crosssections for prompt $\psi(2S)$ mesons and those for $\psi(2S)$ mesons from $b$hadron decays ($\psi{(2S)}\mathrm{from}b$) are determined as functions of the transverse momentum, $p_{\mathrm{T}}$, and the rapidity, $y$, of the $\psi(2S)$ meson in the kinematic range $2<p_{\mathrm{T}}<20 \mathrm{GeV}/c$ and $2.0<y<4.5$. The production crosssections integrated over this kinematic region are \begin{equation*} \begin{split} \sigma( prompt \psi(2S),13 \mathrm{TeV}) &= {1.430 \pm 0.005(\mathrm{stat}) \pm 0.099 (\mathrm{syst})\mu\mathrm{b}},\\ \sigma(\psi(2S)\mathrm{from}b,13 \mathrm{TeV})&={0.426 \pm 0.002(\mathrm{stat}) \pm0.030 (\mathrm{syst})\mu\mathrm{b}}. \end{split} \end{equation*} A new measurement of $\psi(2S)$ production crosssections in $pp$ collisions at $\sqrt{s}=7 \mathrm{TeV}$ is also performed using data collected in 2011, corresponding to an integrated luminosity of $614 {p\mathrm{b}^{1}}$.The integrated production crosssections in the kinematic range $3.5<p_{\mathrm{T}}<14 \mathrm{GeV}/c$ and $2.0<y<4.5$ are \begin{equation*} \begin{split} \sigma( prompt \psi(2S),7 \mathrm{TeV}) &={0.471 \pm0.001 (\mathrm{stat}) \pm 0.025 (\mathrm{syst})\mu\mathrm{b}},\\ \sigma(\psi(2S)\mathrm{from}b,7 \mathrm{TeV}) &={0.126\pm0.001 (\mathrm{stat}) \pm0.008 (\mathrm{syst})\mu\mathrm{b}}. \end{split} \end{equation*} All results show reasonable agreement with theoretical calculations.
Distributions of (left) the invariant mass $m_{\mu ^+\mu ^ }$ and (right) pseudo decay time $t_z$ of selected $\psi {(2S)}$ candidates in the kinematic bin of $5< p_{\mathrm{T}} <6\text{ Ge V /}c $ and $2.5<y<3.0$ in the $13\text{ Te V} $ data sample. Projections of the twodimensional fit result are also shown. The solid (red) line is the total fit function, the shaded (green) area corresponds to the background component. The prompt $\psi {(2S)}$ contribution is shown in crosshatched (blue) area, $\psi {(2S)}$ from$b$ in a solid (black) line and the tail contribution due to the association of $\psi {(2S)}$ with the wrong PV is shown in filled (magenta) area. The tail contribution is invisible in the invariantmass plot. 
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Doubledifferential production crosssections of prompt $\psi {(2S)}$ as functions of $ p_{\mathrm{T}}$ in bins of $y$ at (left) $13\text{ Te V} $ and (right) $7\text{ Te V} $. The statistical and systematic uncertainties are added in quadrature. The $\psi {(2S)}$ meson is assumed to be produced unpolarised. 
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Doubledifferential production crosssections of $\psi {(2S)}$ from $ b $ as functions of $ p_{\mathrm{T}} $ in bins of $y$ at (left) $13\text{ Te V} $ and (right) $7\text{ Te V} $. The statistical and systematic uncertainties are added in quadrature. The $\psi {(2S)}$ meson is assumed to be produced unpolarised. 
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Differential production crosssections as functions of $ p_{\mathrm{T}}$ in the range $2.0<y<4.5$ for the (top) $13\text{ Te V} $ and (bottom) $7\text{ Te V} $ samples. The lefthand figures are for prompt $\psi {(2S)}$ and the results are compared with the NRQCD calculations \cite{Shao:2014yta}; the righthand figures are for $\psi {(2S)}$ from $ b $ and the results are compared with the FONLL calculations \cite{Cacciari:1998it}. 
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Differential production crosssections as functions of $y$ in the range $2< p_{\mathrm{T}} <20\text{ Ge V /}c $ for the $13\text{ Te V} $ sample (top) and in the range $3.5< p_{\mathrm{T}} <14\text{ Ge V /}c $ for the $7\text{ Te V} $ sample (bottom). The left figures are for prompt $\psi {(2S)}$ , the right figures are for $\psi {(2S)}$ from $ b $ compared with the FONLL calculations \cite{Cacciari:1998it}. 
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Fractions of $\psi {(2S)} $from$ b $ in bins of $ p_{\mathrm{T}}$ and $y$ for the (left) $13\text{ Te V} $ and (right) $7\text{ Te V} $ samples. The error bars represent the statistical uncertainties. 
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Ratios of differential crosssections between prompt $\psi {(2S)}$ and prompt $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ mesons at $13\text{ Te V} $ as functions of (left) $ p_{\mathrm{T}}$ and (right) $y$. The NRQCD predicted ratio \cite{Shao:2014yta} is shown in the left panel for comparison. 
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Ratios of differential crosssections between $\psi {(2S)}$ from $ b $ and $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ mesons from $ b $ hadron decays at $13\text{ Te V} $ as functions of (left) $ p_{\mathrm{T}}$ and (right) $y$. The FONLL calculations \cite{Cacciari:2015fta} are shown for comparison. 
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Ratio of differential production crosssections between the $13\text{ Te V} $ and $7\text{ Te V} $ measurements as a function of (left) $ p_{\mathrm{T}}$ integrated over $y$ and (right) $y$ integrated over $ p_{\mathrm{T}} $ for prompt $\psi {(2S)}$ production. Theoretical calculations of NRQCD \cite{Shao:2014yta} are compared to the data on the left side. 
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Ratio of differential production crosssections between the $13\text{ Te V} $ and the $7\text{ Te V} $ measurements as a function of (left) $ p_{\mathrm{T}}$ integrated over $y$ and (right) $y$ integrated over $ p_{\mathrm{T}} $ for $\psi {(2S)}$ from $ b $ . Theoretical FONLL calculations \cite{Cacciari:2015fta} are compared to the data. 
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Animated gif made out of all figures. 
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Systematic uncertainties on the $\psi {(2S)}$ crosssection measurements. The uncertainty from the $t_z$ fit only affects the $\psi {(2S)} $from$b$ result. Uncertainties labelled with "$\ast$" are correlated between kinematic bins. 
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Doubledifferential production crosssections (in $\text{ nb}$ /( $ {\mathrm{Ge V /}c}$ )) of prompt $\psi {(2S)}$ mesons at $13\text{ Te V} $ in bins of ( $ p_{\mathrm{T}}$ , $y$). The first uncertainties are statistical, the second are the uncorrelated systematic uncertainties between bins, and the last are the correlated systematic uncertainties between bins. Adjacent bins with large statistical uncertainties have been merged. 
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Doubledifferential production crosssections (in $\text{ nb} /( {\mathrm{Ge V /}c} )$) of $\psi {(2S)}$ from$b$ mesons at $13\text{ Te V} $ in bins of ( $ p_{\mathrm{T}}$ ,$y$). The first uncertainties are statistical, the second are the uncorrelated systematic uncertainties between bins, and the last are the correlated systematic uncertainties between bins. Adjacent bins with large statistical uncertainties have been merged. 
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Doubledifferential production crosssection in $\text{ nb}$ /( $ {\mathrm{Ge V /}c}$ ) of prompt $\psi {(2S)}$ mesons at $7\text{ Te V} $ in bins of ( $ p_{\mathrm{T}}$ ,$y$). The first uncertainty is statistical, the second is the uncorrelated systematic uncertainties shared between bins and the last is the correlated systematic uncertainties. 
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Doubledifferential production crosssection in $\text{ nb}$ /( $ {\mathrm{Ge V /}c}$ ) of $\psi {(2S)}$ from$b$ mesons at $7\text{ Te V} $ in bins of ( $ p_{\mathrm{T}}$ ,$y$). The first uncertainties are statistical, the second are the uncorrelated systematic uncertainties shared between bins and the last are the correlated systematic uncertainties. 
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Differential production crosssections $\mathrm{d}\sigma/\mathrm{d} p_{\mathrm{T}} $ (in $\text{ nb}$ /( $ {\mathrm{Ge V /}c}$ )) of prompt $\psi {(2S)}$ and $\psi {(2S)}$ from$ b $ mesons at $13\text{ Te V} $. The first uncertainties are statistical and the second (third) are uncorrelated (correlated) systematic uncertainties amongst bins. 
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Differential production crosssections $\mathrm{d}\sigma/\mathrm{d} y$ (in $\text{ nb}$ ) of prompt $\psi {(2S)}$ and $\psi {(2S)}$ from$ b $ mesons at $13\text{ Te V} $ per rapidity unit. The first uncertainties are statistical and the second (third) are uncorrelated (correlated) systematic uncertainties amongst bins. 
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Differential crosssections $\mathrm{d}\sigma/\mathrm{d} p_{\mathrm{T}} $ (in $\text{ nb}$ /( $ {\mathrm{Ge V /}c}$ )) of prompt $\psi {(2S)}$ and $\psi {(2S)}$ from$b$ mesons at $7\text{ Te V} $, integrated over $y$ between 2.0 and 4.5. The first uncertainties are statistical and the second (third) are uncorrelated (correlated) systematic uncertainties amongst bins. 
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Differential crosssections $\mathrm{d}\sigma/\mathrm{d} y$ (in $\text{ nb}$ ) of prompt $\psi {(2S)}$ and $\psi {(2S)}$ from$b$ mesons at $7\text{ Te V} $, integrated over $ p_{\mathrm{T}}$ between 3.5 and 14 $\text{ Ge V /}c$ . The first uncertainties are statistical and the second (third) are uncorrelated (correlated) systematic uncertainties amongst bins. 
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Fractions of $\psi {(2S)}$ from$ b $ (in %) at $13\text{ Te V} $ in bins of $( p_{\mathrm{T}} ,y)$ of $\psi {(2S)}$ mesons. The uncertainties are statistical only. The systematic uncertainties are negligible. Adjacent bins with large statistical uncertainty have been merged. 
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Fractions of $\psi {(2S)}$ from$ b $ (in %) at $7\text{ Te V} $ in bins of $( p_{\mathrm{T}} ,y)$ of $\psi {(2S)}$ mesons. The uncertainties are statistical only. The systematic uncertainties are negligible. 
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Ratios of production crosssections at $13\text{ Te V} $ between $\psi {(2S)}$ mesons and $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ mesons in bins of $ p_{\mathrm{T}}$ for prompt production and for those from $ b $hadron decays integrated in the rapidity range $2.0<y<4.5$. The statistical and systematic uncertainties are added in quadrature. 
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Ratios of production crosssections between $\psi {(2S)}$ mesons and $ { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu}$ mesons at $13\text{ Te V} $ in bins of $y$ for prompt production and for those from $ b $hadron decays integrated in the transverse momentum range $2< p_{\mathrm{T}} <14\text{ Ge V /}c $. The statistical and systematic uncertainties are added in quadrature. 
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Ratios of production crosssections between $13\text{ Te V} $ and $7\text{ Te V} $ in bins of $ p_{\mathrm{T}}$ for prompt $\psi {(2S)}$ and $\psi {(2S)}$ from$ b $ mesons integrated in the rapidity range $2.0<y<4.5$. The statistical and systematic uncertainties are added in quadrature. 
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Ratios of crosssections between measurements at $13\text{ Te V} $ and $7\text{ Te V} $ in different bins of $y$ for prompt $\psi {(2S)}$ and $\psi {(2S)}$ from$ b $ mesons integrated in the transverse momentum range $3.5< p_{\mathrm{T}} <14\text{ Ge V /}c $. The statistical and systematic uncertainties are added in quadrature. 
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Multiplicative scaling factors needed to obtain the prompt $\psi {(2S)}$ differential crosssections from unpolarised crosssection measurements at $13\text{ Te V} $ as reported in Table ??? under the assumption of fully transverse polarisation ($\alpha = +1$). 
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Multiplicative scaling factors needed to obtain the prompt $\psi {(2S)}$ differential crosssections from unpolarised crosssection measurements at $13\text{ Te V} $ as reported in Table ??? under the assumption of fully longitudinal polarisation ($\alpha = 1$). 
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Multiplicative scaling factors needed to obtain the prompt $\psi {(2S)}$ differential crosssections from unpolarised crosssection measurements at $13\text{ Te V} $ as reported in Table ??? under the assumption of 20 % longitudinal polarisation ($\alpha = 0.2$), corresponding to a conservative limit of the $\psi {(2S)}$ polarisation measured at $7\text{ Te V} $ \cite{LHCbPAPER2013067}. 
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Multiplicative scaling factors needed to obtain the prompt $\psi {(2S)}$ differential crosssections from unpolarised crosssection measurements at $7\text{ Te V} $ as reported in Table ??? under the assumption of fully transverse polarisation ($\alpha = +1$). 
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Multiplicative scaling factors needed to obtain the prompt $\psi {(2S)}$ differential crosssections from unpolarised crosssection measurements at $7\text{ Te V} $ as reported in Table ??? under the assumption of fully longitudinal polarisation ($\alpha = 1$). 
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Multiplicative scaling factors needed to obtain the prompt $\psi {(2S)}$ differential crosssections from unpolarised crosssection measurements at $7\text{ Te V} $ as reported in Table ??? under the assumption of 20 % longitudinal polarisation ($\alpha = 0.2$), corresponding to a conservative limit of the $\psi {(2S)}$ polarisation measured at $7\text{ Te V} $ \cite{LHCbPAPER2013067} 
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Created on 22 January 2021.