(a) Distributions of $\cos\theta_{\mathrm{GJ}}$ for $\Upsilon(1\mathrm{S})$ (red circles), $\Upsilon(2\mathrm{S})$ (blue squares) and $\Upsilon(3\mathrm{S})$ (green diamonds) signal candidates and the background component (histogram) in the region $6< p^{\Upsilon}_{\mathrm{T}} <8 {\mathrm{\,GeV\!/}c} $, $2.2< y^{\Upsilon} <3.0$ for data accumulated at $\sqrt{s}=7\,\mathrm{TeV}$. To improve visibility, the distributions for the $\Upsilon(1\mathrm{S})$ and $\Upsilon(2\mathrm{S})$ signals are scaled by factors of 3 and 2, respectively. (b) Dimuon mass distribution in the region $6< p^{\Upsilon}_{\mathrm{T}} <8 {\mathrm{\,GeV\!/}c} $, $2.2< y^{\Upsilon} <3.0$ for data accumulated at $\sqrt{s}=7\,\mathrm{TeV}$. The thick dark yellow solid curve shows the result of the fit, as described in the text. The three peaks, shown with thin red solid lines, correspond to the $\Upsilon(1\mathrm{S})$ , $\Upsilon(2\mathrm{S})$ and $\Upsilon(3\mathrm{S})$ signals\,(left to right). The background component is indicated with a dashed blue line.

Background-subtracted distributions of (a) $\cos\theta$ and (b) $\phi$ for the $\Upsilon(1\mathrm{S})$ signal candidates in the HX frame, measured in the kinematic region $6< p^{\Upsilon}_{\mathrm{T}} <8 {\mathrm{\,GeV\!/}c} $ and $2.2< y^{\Upsilon} <3.0$, for data accumulated at $\sqrt{s}=7\,\mathrm{TeV}$.

The polarization parameters (top) $\uplambda_{\theta}$, (middle) $\uplambda_{\theta\phi}$ and (bottom) $\uplambda_{\phi}$, measured in the HX frame for the $\Upsilon(1\mathrm{S})$ state in different bins of $ p^{\Upsilon}_{\mathrm{T}}$ and three rapidity ranges, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the rapidity ranges $2.2< y^{\Upsilon} <3.0$, $3.0< y^{\Upsilon} <3.5$ and $3.5< y^{\Upsilon} <4.5$ are shown with red circles, blue squares and green diamonds, respectively. The vertical inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. The horizontal error bars indicate the bin width. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters (top) $\uplambda_{\theta}$, (middle) $\uplambda_{\theta\phi}$ and (bottom) $\uplambda_{\phi}$, measured in the CS frame for the $\Upsilon(1\mathrm{S})$ state in different bins of $ p^{\Upsilon}_{\mathrm{T}}$ and three rapidity ranges, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the rapidity ranges $2.2< y^{\Upsilon} <3.0$, $3.0< y^{\Upsilon} <3.5$ and $3.5< y^{\Upsilon} <4.5$ are shown with red circles, blue squares and green diamonds, respectively. The vertical inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. The horizontal error bars indicate the bin width. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters (top) $\uplambda_{\theta}$, (middle) $\uplambda_{\theta\phi}$ and (bottom) $\uplambda_{\phi}$, measured in the GJ frame for the $\Upsilon(1\mathrm{S})$ state in different bins of $ p^{\Upsilon}_{\mathrm{T}}$ and three rapidity ranges, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the rapidity ranges $2.2< y^{\Upsilon} <3.0$, $3.0< y^{\Upsilon} <3.5$ and $3.5< y^{\Upsilon} <4.5$ are shown with red circles, blue squares and green diamonds, respectively. The vertical inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. The horizontal error bars indicate the bin width. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters (top) $\uplambda_{\theta}$, (middle) $\uplambda_{\theta\phi}$ and (bottom) $\uplambda_{\phi}$, measured in the HX frame for the $\Upsilon(2\mathrm{S})$ state in different bins of $ p^{\Upsilon}_{\mathrm{T}}$ and three rapidity ranges, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the rapidity ranges $2.2< y^{\Upsilon} <3.0$, $3.0< y^{\Upsilon} <3.5$ and $3.5< y^{\Upsilon} <4.5$ are shown with red circles, blue squares and green diamonds, respectively. The vertical inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. The horizontal error bars indicate the bin width. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters (top) $\uplambda_{\theta}$, (middle) $\uplambda_{\theta\phi}$ and (bottom) $\uplambda_{\phi}$, measured in the CS frame for the $\Upsilon(2\mathrm{S})$ state in different bins of $ p^{\Upsilon}_{\mathrm{T}}$ and three rapidity ranges, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the rapidity ranges $2.2< y^{\Upsilon} <3.0$, $3.0< y^{\Upsilon} <3.5$ and $3.5< y^{\Upsilon} <4.5$ are shown with red circles, blue squares and green diamonds, respectively. The vertical inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. The horizontal error bars indicate the bin width. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters (top) $\uplambda_{\theta}$, (middle) $\uplambda_{\theta\phi}$ and (bottom) $\uplambda_{\phi}$, measured in the GJ frame for the $\Upsilon(2\mathrm{S})$ state in different bins of $ p^{\Upsilon}_{\mathrm{T}}$ and three rapidity ranges, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the rapidity ranges $2.2< y^{\Upsilon} <3.0$, $3.0< y^{\Upsilon} <3.5$ and $3.5< y^{\Upsilon} <4.5$ are shown with red circles, blue squares and green diamonds, respectively. The vertical inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. The horizontal error bars indicate the bin width. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters (top) $\uplambda_{\theta}$, (middle) $\uplambda_{\theta\phi}$ and (bottom) $\uplambda_{\phi}$, measured in the HX frame for the $\Upsilon(3\mathrm{S})$ state in different bins of $ p^{\Upsilon}_{\mathrm{T}}$ and three rapidity ranges, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the rapidity ranges $2.2< y^{\Upsilon} <3.0$, $3.0< y^{\Upsilon} <3.5$ and $3.5< y^{\Upsilon} <4.5$ are shown with red circles, blue squares and green diamonds, respectively. The vertical inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. The horizontal error bars indicate the bin width. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters $\uplambda_{\theta}$ (top), $\uplambda_{\theta\phi}$ (middle) and $\uplambda_{\phi}$ (bottom), measured in the CS frame for the $\Upsilon(3\mathrm{S})$ state in different bins of $ p^{\Upsilon}_{\mathrm{T}}$ and three rapidity ranges, for data collected at $\sqrt{s} =7\,\mathrm{TeV}$\,(left) and $\sqrt{s} =8\,\mathrm{TeV}$\,(right). The results for the rapidity ranges $2.2< y^{\Upsilon} <3.0$, $3.0< y^{\Upsilon} <3.5$ and $3.5< y^{\Upsilon} <4.5$ are shown with red circles, blue squares and green diamonds, respectively. The vertical inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. The horizontal error bars indicate the bin width. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters $\uplambda_{\theta}$ (top), $\uplambda_{\theta\phi}$ (middle) and $\uplambda_{\phi}$ (bottom), measured in the GJ frame for the $\Upsilon(3\mathrm{S})$ state in different bins of $ p^{\Upsilon}_{\mathrm{T}}$ and three rapidity ranges, for data collected at $\sqrt{s} =7\,\mathrm{TeV}$\,(left) and $\sqrt{s} =8\,\mathrm{TeV}$\,(right). The results for the rapidity ranges $2.2< y^{\Upsilon} <3.0$, $3.0< y^{\Upsilon} <3.5$ and $3.5< y^{\Upsilon} <4.5$ are shown with red circles, blue squares and green diamonds, respectively. The vertical inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. The horizontal error bars indicate the bin width. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters (top) $\uplambda_{\theta}$, (middle) $\uplambda_{\theta\phi}$ and (bottom) $\uplambda_{\phi}$, for $\Upsilon(1\mathrm{S})$ mesons as a function of $ p^{\Upsilon}_{\mathrm{T}}$ , for the rapidity range $2.2< y^{\Upsilon} <4.5$, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the HX, CS and GJ frames are shown with red circles, blue squares and green diamonds, respectively. The inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters (top) $\uplambda_{\theta}$, (middle) $\uplambda_{\theta\phi}$ and (bottom) $\uplambda_{\phi}$, for $\Upsilon(2\mathrm{S})$ mesons as a function of $ p^{\Upsilon}_{\mathrm{T}}$ , for the rapidity range $2.2< y^{\Upsilon} <4.5$, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the HX, CS and GJ frames are shown with red circles, blue squares and green diamonds, respectively. The inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. Some data points are displaced from the bin centers to improve visibility.

The polarization parameters (top) $\uplambda_{\theta}$, (middle) $\uplambda_{\theta\phi}$ and (bottom) $\uplambda_{\phi}$, for $\Upsilon(3\mathrm{S})$ mesons as a function of $ p^{\Upsilon}_{\mathrm{T}}$ , for the rapidity range $2.2< y^{\Upsilon} <4.5$, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the HX, CS and GJ frames are shown with red circles, blue squares and green diamonds, respectively. The inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. Some data points are displaced from the bin centers to improve visibility.

The polarization parameter $\tilde\uplambda$ for (top) $\Upsilon(1\mathrm{S})$ mesons, (middle) $\Upsilon(2\mathrm{S})$ mesons and (bottom) $\Upsilon(3\mathrm{S})$ mesons as a function of $ p^{\Upsilon}_{\mathrm{T}}$ , for the rapidity range $2.2< y^{\Upsilon} <4.5$, for data collected at (left) $\sqrt{s} =7\,\mathrm{TeV}$ and (right)\, $\sqrt{s} =8\,\mathrm{TeV}$. The results for the HX, CS and GJ frames are shown with red circles, blue squares and green diamonds, respectively. The inner error bars indicate the statistical uncertainty, whilst the outer error bars indicate the sum of the statistical and systematic uncertainties added in quadrature. Some data points are displaced from the bin centers to improve visibility.

Animated gif made out of all figures.

Ranges of the absolute systematic uncertainties of the parameters $\pmb{\uplambda}$ and $\tilde{\uplambda}$. The ranges indicate variations depending on the $\left( p^{\Upsilon}_{\mathrm{T}} , y^{\Upsilon} \right)$ bin and frame.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the HX frame for the $\Upsilon(1\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the HX frame for the $\Upsilon(1\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the CS frame for the $\Upsilon(1\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the CS frame for the $\Upsilon(1\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the GJ frame for the $\Upsilon(1\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the GJ frame for the $\Upsilon(1\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$ and $\uplambda_{\phi}$ measured in the HX, CS and GJ frames for the $\Upsilon(1\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$ in the rapidity range $2.2< y^{\Upsilon} <4.5$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$ and $\uplambda_{\phi}$ measured in the HX, CS and GJ frames for the $\Upsilon(1\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$ in the rapidity range $2.2< y^{\Upsilon} <4.5$. The first uncertainty is statistical and the second systematic.

Values of $\tilde \uplambda$ measured in the HX, CS and GJ frames for the $\Upsilon(1\mathrm{S})$ produced at $\sqrt{s}=7$ and $8\,\,{\mathrm{TeV}}$ in the rapidity range $2.2< y^{\Upsilon} <4.5$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the HX frame for the $\Upsilon(2\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the HX frame for the $\Upsilon(2\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the CS frame for the $\Upsilon(2\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the CS frame for the $\Upsilon(2\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the GJ frame for the $\Upsilon(2\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the GJ frame for the $\Upsilon(2\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$ and $\uplambda_{\phi}$ measured in the HX, CS and GJ frames for the $\Upsilon(2\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$ in the rapidity range $2.2< y^{\Upsilon} <4.5$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$ and $\uplambda_{\phi}$ measured in the HX, CS and GJ frames for the $\Upsilon(2\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$ in the rapidity range $2.2< y^{\Upsilon} <4.5$. The first uncertainty is statistical and the second systematic.

Values of $\tilde \uplambda$ measured in the HX, CS and GJ frames for the $\Upsilon(2\mathrm{S})$ produced at $\sqrt{s}=7$ and $8\,\,{\mathrm{TeV}}$ in the rapidity range $2.2< y^{\Upsilon} <4.5$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the HX frame for the $\Upsilon(3\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the HX frame for the $\Upsilon(3\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the CS frame for the $\Upsilon(3\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the CS frame for the $\Upsilon(3\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the GJ frame for the $\Upsilon(3\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$, $\uplambda_{\phi}$ and $\tilde \uplambda$ measured in the GJ frame for the $\Upsilon(3\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$ and $\uplambda_{\phi}$ measured in the HX, CS and GJ frames for the $\Upsilon(3\mathrm{S})$ produced at $\sqrt{s}=7\,\,{\mathrm{TeV}}$ in the rapidity range $2.2< y^{\Upsilon} <4.5$. The first uncertainty is statistical and the second systematic.

Values of $\uplambda_{\theta}$, $\uplambda_{\theta\phi}$ and $\uplambda_{\phi}$ measured in the HX, CS and GJ frames for the $\Upsilon(3\mathrm{S})$ produced at $\sqrt{s}=8\,\,{\mathrm{TeV}}$ in the rapidity range $2.2< y^{\Upsilon} <4.5$. The first uncertainty is statistical and the second systematic.

Values of $\tilde \uplambda$ measured in the HX, CS and GJ frames for the $\Upsilon(3\mathrm{S})$ produced at $\sqrt{s}=7$ and $8\,\,{\mathrm{TeV}}$ in the rapidity range $2.2< y^{\Upsilon} <4.5$. The first uncertainty is statistical and the second systematic.

Supplementary material full pdf