Measurements of long-range azimuthal anisotropies and associated Fourier coefficients for $pp$ collisions at $\sqrt{s}=5.02$ and 13 TeV and $p+\text{Pb}$ collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV with the ATLAS detector

ATLAS measurements of two-particle correlations are presented for $\sqrt{s}=5.02$ and 13 TeV $pp$ collisions and for $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV $p+\text{Pb}$ collisions at the LHC. The correlation functions are measured as a function of relative azimuthal angle $\mathrm{\Delta }\varphi$, and pseudorapidity separation $\mathrm{\Delta }\eta$, using charged particles detected within the pseudorapidity interval $\left|\eta \right|<2.5$. Azimuthal modulation in the long-range component of the correlation function, with $\left|\mathrm{\Delta }\eta \right|>2$, is studied using a template fitting procedure to remove a “back-to-back” contribution to the correlation function that primarily arises from hard-scattering processes. In addition to the elliptic, $\cos \left(2\mathrm{\Delta }\varphi \right)$, modulation observed in a previous measurement, the $pp$ correlation functions exhibit significant $\cos \left(3\mathrm{\Delta }\varphi \right)$ and $\cos \left(4\mathrm{\Delta }\varphi \right)$ modulation. The Fourier coefficients $v_{n,n}$ associated with the $\cos \left(n\mathrm{\Delta }\varphi \right)$ modulation of the correlation functions for $n=2–4$ are measured as a function of charged-particle multiplicity and charged-particle transverse momentum. The Fourier coefficients are observed to be compatible with $\cos \left(n\varphi \right)$ modulation of per-event single-particle azimuthal angle distributions. The single-particle Fourier coefficients $v_n$ are measured as a function of charged-particle multiplicity, and charged-particle transverse momentum for $n=2–4$. The integrated luminosities used in this analysis are, 64 ${\mathrm{nb}}^{-1}$ for the $\sqrt{s}=13$ TeV $pp$ data, 170 ${\mathrm{nb}}^{-1}$ for the $\sqrt{s}=5.02$ TeV $pp$ data, and 28 ${\mathrm{nb}}^{-1}$ for the $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV $p+\text{Pb}$ data.

Figure 1

Distributions of the multiplicity, $N_{\mathrm{ch}}^{\mathrm{rec}}$, of reconstructed charged particles having $p_{\mathrm{T}}>0.4$ GeV in the 13 $\text{TeV} pp$ (left), 5.02 $\text{TeV} pp$ (middle), and 5.02 $\text{TeV} p+\text{Pb}$ (right) data used in this analysis. The discontinuities in the distributions correspond to different high-multiplicity trigger thresholds.

Figure 2

Two-particle correlation functions $C(\mathrm{\Delta }\eta ,\mathrm{\Delta }\varphi )$ in 13 $\text{TeV} pp$ collisions (top panels), 5.02 $\text{TeV} pp$ collisions (middle panels), and in 5.02 $\text{TeV} p+\text{Pb}$ collisions (bottom panels). The left panels correspond to a lower-multiplicity range of $0\le N_{\mathrm{ch}}^{\mathrm{rec}}<20$. The right panels correspond to higher multiplicity ranges of $N_{\mathrm{ch}}^{\mathrm{rec}}\ge 120$ for 13 $\text{TeV} pp, 90\le N_{\mathrm{ch}}^{\mathrm{rec}}<100$ for the 5.02 $\text{TeV} pp$, and $N_{\mathrm{ch}}^{\mathrm{rec}}\ge 220$ for the 5.02 $\text{TeV} p+\text{Pb}$. The plots are for charged particles having $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5 \text{GeV}$. The distributions have been truncated to suppress the peak at $\mathrm{\Delta }\eta =\mathrm{\Delta }\varphi =0$ and are plotted over $\left|\mathrm{\Delta }\eta \right|<4.6$ ($\left|\mathrm{\Delta }\eta \right|<4.0$ for middle row) to avoid statistical fluctuations at larger $\left|\mathrm{\Delta }\eta \right|$. For the middle-right panel, the peak at $\mathrm{\Delta }\varphi =\pi$ has also been truncated.

Figure 3

Left panel: template fit to the per-trigger particle yields $Y\left(\mathrm{\Delta }\varphi \right)$ in 13 $\text{TeV} pp$ collisions for charged-particle pairs with $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5$ $\text{GeV}$ and $2<\left|\mathrm{\Delta }\eta \right|<5$. This plot corresponds to the $N_{\mathrm{ch}}^{\mathrm{rec}}\ge 90$ multiplicity range. The template fitting includes only the second-order harmonic, $v_{2,2}$. The solid points indicate the measured $Y\left(\mathrm{\Delta }\varphi \right)$, the open points and curves show different components of the template (see legend) that are shifted along the $y$ axis by $G$ or by $F{Y}^{\mathrm{periph}}\left(0\right)$, where necessary, for presentation. Right panel: The difference between the $Y\left(\mathrm{\Delta }\varphi \right)$ and the template fit, showing the presence of $v_{3,3}$ and $v_{4,4}$ components. The vertical error bars indicate statistical uncertainties.

Figure 4

Template fits to the per-trigger particle yields $Y\left(\mathrm{\Delta }\varphi \right)$, in 13 $\text{TeV}$ (left panels) and in 5.02 $\text{TeV}$ (right panels) $pp$ collisions for charged-particle pairs with $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5 \text{GeV}$ and $2<\left|\mathrm{\Delta }\eta \right|<5$. The template fitting includes second-order, third-order, and fourth-order harmonics. From top to bottom, each panel represents a different $N_{\mathrm{ch}}^{\mathrm{rec}}$ range. The solid points indicate the measured $Y\left(\mathrm{\Delta }\varphi \right)$, the open points and curves show different components of the template (see legend) that are shifted along the $y$ axis by $G$ or by $F{Y}^{\mathrm{periph}}\left(0\right)$, where necessary, for presentation.

Figure 5

Template fits to the per-trigger particle yields $Y\left(\mathrm{\Delta }\varphi \right)$ in 5.02 $\text{TeV} p+\text{Pb}$ collisions for charged-particle pairs with $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5$ $\text{GeV}$ and $2<\left|\mathrm{\Delta }\eta \right|<5$. The template fitting includes second-order, third-order, and fourth-order harmonics. Each panel represents a different $N_{\mathrm{ch}}^{\mathrm{rec}}$ range. The solid points indicate the measured $Y\left(\mathrm{\Delta }\varphi \right)$, the open points and curves show different components of the template (see legend) that are shifted along the $y$ axis by $G$ or by $F{Y}^{\mathrm{periph}}\left(0\right)$, where necessary, for presentation.

Figure 6

The $v_{n,n}$ obtained from the template fitting procedure in the 13 $\text{TeV} pp$ data, as a function of $N_{\mathrm{ch}}^{\mathrm{rec}}$ (left panels), and as a function of $E_{\mathrm{T}}^{\mathrm{FCal}}$ (right panels). The top, middle, and bottom panels correspond to $v_{2,2}, v_{3,3}$, and $v_{4,4}$, respectively. The results are for $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5$ $\text{GeV}$. The error bars indicate statistical uncertainties. The $v_{n,n}$ obtained from a direct Fourier transform of $Y\left(\mathrm{\Delta }\varphi \right)$ and from the ZYAM-based template fits are also shown for comparison.

Figure 7

The $v_{n,n}$ obtained from the template fitting procedure in the 5.02 $\text{TeV} pp$ data, as a function of $N_{\mathrm{ch}}^{\mathrm{rec}}$. The three panels correspond to $n=2$, 3, and 4, respectively. The results are for $1<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5 \text{GeV}$. The error bars indicate statistical uncertainties. The $v_{n,n}$ obtained from a direct Fourier transform of $Y\left(\mathrm{\Delta }\varphi \right)$ and from the ZYAM-based template fits are also shown for comparison.

Figure 8

The $v_{n,n}$ obtained from the template fitting procedure in the 5.02 $\text{TeV} p+\text{Pb}$ data, as a function of $N_{\mathrm{ch}}^{\mathrm{rec}}$ (left panels), and as a function of the Pb-fragmentation side FCal-$E_{\mathrm{T}}$(right panels). The top, middle, and bottom panels correspond to $v_{2,2}, v_{3,3}$, and $v_{4,4}$, respectively. The results are for $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5 \text{GeV}$. The error bars indicate statistical uncertainties. The $v_{n,n}$ obtained from a direct Fourier transform of $Y\left(\mathrm{\Delta }\varphi \right)$, the peripheral subtraction procedure, and from the ZYAM-based template fits are also shown for comparison.

Figure 9

The left panel shows $v_{2,2}$ as a function of $N_{\mathrm{ch}}^{\mathrm{rec}}$ in the 13 $\text{TeV} pp$ data, for $0.5<p_{\mathrm{T}}^{\mathrm{a}} <5 \text{GeV}$ and for different choices of the $p_{\mathrm{T}}^{\mathrm{b}}$ interval. The right panel shows the corresponding $v_2$ values obtained using Eq. (4). The error bars indicate statistical uncertainties only.

Figure 10

The left panels show $v_{2,2}$ (top) and $v_{3,3}$ (bottom) as a function of $N_{\mathrm{ch}}^{\mathrm{rec}}$ in the 5.02 $\text{TeV} p+\text{Pb}$ data, for $0.5<p_{\mathrm{T}}^{\mathrm{a}}<5 \text{GeV}$ and for different choices of the $p_{\mathrm{T}}^{\mathrm{b}}$ interval. The right panels shows the corresponding $v_2$ (top) and $v_3$ (bottom) values obtained using Eq. (4). The error bars indicate statistical uncertainties only.

Figure 11

The left panels show $v_{2,2}$ (top) and $v_{3,3}$ (bottom) as a function of $p_{\mathrm{T}}^{\mathrm{a}}$ in the 13 $\text{TeV} pp$ data, for $N_{\mathrm{ch}}^{\mathrm{rec}} \ge 90$ and for different choices of the $p_{\mathrm{T}}^{\mathrm{b}}$ interval. The right panels shows the corresponding $v_2$ (top) and $v_3$ (bottom) values obtained using Eq. (4). The error bars indicate statistical uncertainties only. The $p_{\mathrm{T}}^{\mathrm{a}}$ intervals plotted are 0.4–0.5, 0.5–1, 1–2, 2–3, and 3–5 $\text{GeV}$. In some cases, the data points have been slightly shifted along the $x$ axis, for clarity.

Figure 12

The left panels show the $v_{n,n}$ as a function of $p_{\mathrm{T}}^{\mathrm{a}}$ in the 5.02 $\text{TeV} p+\text{Pb}$ data, for $N_{\mathrm{ch}}^{\mathrm{rec}} \ge 140$ and for different choices of the $p_{\mathrm{T}}^{\mathrm{b}}$ interval. From top to bottom, the three rows correspond to $n=2,3$, and 4. The right panels shows the corresponding $v_n$ values obtained using Eq. (4). The error bars indicate statistical uncertainties only. The $p_{\mathrm{T}}^{\mathrm{a}}$ intervals plotted are 0.4–0.5, 0.5–1, 1–2, 2–3, and 3–5 $\text{GeV}$. In some cases, the data points have been slightly shifted along the $x$ axis, for clarity.

Figure 13

The $\left|\mathrm{\Delta }\eta \right|$ dependence of the $v_{n,n}$ (left panels) and $v_n$ (right panels) in the 13 $\text{TeV} pp$ data. From top to bottom the rows correspond to $n=2$, 3, and 4, respectively. The ZYAM-template and Fourier-$v_{n,n}$ values are also shown for comparison. Only the range $\left|\mathrm{\Delta }\eta \right|>1$ is shown to suppress the large Fourier $v_{n,n}$ at $\left|\mathrm{\Delta }\eta \right|\sim 0$ that arise due to the near-side jet peak. Plots are for the $N_{\mathrm{ch}}^{\mathrm{rec}}\ge 90$ multiplicity range and for $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5 \text{GeV}$. The error bars indicate statistical uncertainties only. For points where $v_{3,3}$ is negative, $v_3$ is not defined and hence not plotted.

Figure 14

Dependence of $v_2$ on the peripheral bin chosen for the default (left panels) and ZYAM-based (right panels) template fitting methods. The top panels correspond to 13 $\text{TeV} pp$ collisions, middle panels correspond to 5.02 $\text{TeV} pp$ collisions, and the lower panels correspond to 5.02 $\text{TeV} p+\text{Pb}$ collisions. The results are plotted as a function of $N_{\mathrm{ch}}^{\mathrm{rec}}$ and for $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5 \text{GeV}$. The error bars indicate statistical uncertainties.

Figure 15

Left panel: comparison of the $v_2$ for same-charge and opposite-charge pairs in the 13 $\text{TeV} pp$ data. Also shown are the $v_2$ values for the two types of pairs combined. Right panel: ratio of the same-charge to the opposite-charge $v_2$. The results are for $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5\text{GeV}$. For clarity, the data points for the same-charge and opposite-charge $v_n$ are slightly shifted along the $x$ axis. The error bars indicate statistical uncertainties.

Figure 16

Left panels: comparison of the $v_n$ obtained from the template fitting procedure in the 13 $\text{TeV} pp$, 5.02 $\text{TeV} pp$, and 5.02 $\text{TeV} p+\text{Pb}$ data, as a function of $N_{\mathrm{ch}}^{\mathrm{rec}}$. The results are for $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5 \text{GeV}$. Right panels: the $p_{\mathrm{T}}$ dependence of the $v_n$ for the $N_{\mathrm{ch}}^{\mathrm{rec}}\ge 60$ multiplicity range. From top to bottom the rows correspond to $n=2$, 3, and 4, respectively. The error bars and shaded bands indicate statistical and systematic uncertainties, respectively.

Figure 17

Comparison of the shapes of the $v_2\left(p_{\mathrm{T}}\right)$ in the 13 $\text{TeV} pp$ and 5.02 $\text{TeV} p+\text{Pb}$ data. The $pp v_2$ has been scaled by a factor of 1.51 along the $y$ axis in order to match the maximum of the $v_2$ in the two data sets. The results are for $0.5<p_{\mathrm{T}}^{\mathrm{b}}<5 \text{GeV}$ and $N_{\mathrm{ch}}^{\mathrm{rec}}\ge 60$. The error bars indicate statistical uncertainties.

Figure 18

Comparison of the $v_n$ obtained from the template fitting procedure in the 13 $\text{TeV} pp$, 5.02 $\text{TeV} pp$, and 5.02 $\text{TeV} p+\text{Pb}$ data, as a function of $N_{\mathrm{ch}}^{\mathrm{rec}}$. The results are for $1<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5 \text{GeV}$. The three panels correspond to $n=2$, 3, and 4. The error bars and shaded bands indicate statistical and systematic uncertainties, respectively.

Figure 19

Ratio of $v_4$ to $v_2^2$ as a function of $N_{\mathrm{ch}}^{\mathrm{rec}}$ in the 13 $\text{TeV} pp$ and 5.02 $\text{TeV} p+\text{Pb}$ data. The results are for $0.5<p_{\mathrm{T}}^{\mathrm{a},\mathrm{b}}<5 \text{GeV}$. The error bars and shaded bands indicate statistical and systematic uncertainties, respectively.