Observation of long-range elliptic anisotropies in $\sqrt{s}=$13 and 2.76 TeV $pp$ collisions with the ATLAS detector

ATLAS has measured two-particle correlations as a function of relative azimuthal-angle, $\Delta \phi$, and pseudorapidity, $\Delta \eta$, in $\sqrt{s}$=13 and 2.76 TeV $pp$ collisions at the LHC using charged particles measured in the pseudorapidity interval $|\eta|$<2.5. The correlation functions evaluated in different intervals of measured charged-particle multiplicity show a multiplicity-dependent enhancement at $\Delta \phi \sim 0$ that extends over a wide range of $\Delta\eta$, which has been referred to as the"ridge". Per-trigger-particle yields, $Y(\Delta \phi)$, are measured over 2<$|\Delta\eta|$<5. For both collision energies, the $Y(\Delta \phi)$ distribution in all multiplicity intervals is found to be consistent with a linear combination of the per-trigger-particle yields measured in collisions with less than 20 reconstructed tracks, and a constant combinatoric contribution modulated by $\cos{(2\Delta \phi)}$. The fitted Fourier coefficient, $v_{2,2}$, exhibits factorization, suggesting that the ridge results from per-event $\cos{(2\phi)}$ modulation of the single-particle distribution with Fourier coefficients $v_2$. The $v_2$ values are presented as a function of multiplicity and transverse momentum. They are found to be approximately constant as a function of multiplicity and to have a $p_{\mathrm{T}}$ dependence similar to that measured in $p$+Pb and Pb+Pb collisions. The $v_2$ values in the 13 and 2.76 TeV data are consistent within uncertainties. These results suggest that the ridge in $pp$ collisions arises from the same or similar underlying physics as observed in $p$+Pb collisions, and that the dynamics responsible for the ridge has no strong $\sqrt{s}$ dependence.

The measurements were performed using the ATLAS inner detector (ID), minimum-bias trigger scintillators (MBTS), forward calorimeter (FCal), and the trigger and data acquisition systems [31].The ID detects charged particles within |η|<2.5 using a combination of silicon pixel detectors, silicon micro-strip detectors (SCT), and a straw-tube transition radiation tracker (TRT), all immersed in a 2 T axial magnetic field [32,33].The MBTS system detects charged particles using two hodoscopes of counters positioned at z = ±3.6 m.The FCal covers 3.1<|η|<4.9and uses tungsten and copper absorbers with liquid argon as the active medium.Between Run 1 and Run 2, an additional, innermost pixel layer was added to the ID and the MBTS was replaced.The ATLAS trigger system [34] consists of a Level-1 (L1) trigger implemented using a combination of dedicated electronics and programmable logic, and a software-based high-level trigger (HLT).Chargedparticle tracks were reconstructed in the HLT using methods similar to those applied in the offline analysis, allowing triggers that select on the number of tracks with p T >0. 4 GeV associated with a single vertex.For the 13 TeV measurements, a minimum-bias L1 trigger required one or more signals in the MBTS while the high-multiplicity trigger (HMT) required at least 900 SCT hits and at least 60 HLT-reconstructed tracks.For the 2.76 TeV data the minimum-bias trigger selected random crossings at L1 and applied a threshold to the number of SCT and pixel hits in the HLT, while several HMT triggers were formed by applying thresholds on the total FCal transverse energy at L1 and different thresholds on the number of HLT-reconstructed tracks.HMT triggers are only used where their multiplicity selection is more than 90% efficient.The inefficiency of the HMT triggers does not affect the measurements presented in this paper.This has been checked by comparing the results obtained with and without the HMT-triggered events, over the N rec ch range where the HMT is not fully efficient.Charged-particle tracks and collision vertices are reconstructed in the ID using algorithms that were reoptimized between LHC Runs 1 and 2 [35].Tracks used in the analysis are required to have p T >0.3 GeV, |η|<2.5 and to satisfy additional selection criteria that differ slightly between the 2.76 [4] and 13 TeV [36] data.
Events used in the analysis are required to have at least one reconstructed vertex.For events containing multiple vertices (pileup), only tracks associated with the vertex having the largest p 2 T , where the sum is over all tracks associated with the vertex, are used.The measured charged-particle multiplicity, N rec ch , is defined as the number of tracks having p T >0. 4 GeV associated with this vertex.The distributions of N rec ch are shown in Fig. 1.The structures in the distributions result from the different HMT trigger thresholds.The efficiency, (p T , η), of the track reconstruction and track selection requirements is evaluated using simulated non-diffractive pp events obtained from the pythia 8 [37] event generator (A2 tune [38], MSTW2008LO PDFs [39]) that are passed through a GEANT4 [40] simulation of the ATLAS detector response and reconstructed using the algorithms applied to the data [41].The efficiencies for the two data sets are similar, but differ due to changes in the detector and reconstruction algorithms between Runs 1 and 2. In the simulated events, the efficiency reduces the measured multiplicity relative to the pythia 8 p T >0.4 GeV charged-particle multiplicity by approximately multiplicity-independent factors of 1.18 ± 0.05 and 1.22 ± 0.05 for 13 and 2.76 TeV data, respectively.The uncertainties in these factors result from systematic uncertainties in the tracking efficiencies, which are described in detail in Ref. [36].Those systematic uncertainties vary with pseudorapidity between 1.1% (central) and 6.5% (forward) and result from uncertainties on the material description.
The present analysis follows methods used in previous ATLAS two-particle correlation measurements in Pb+Pb and p+Pb collisions [4,6,[42][43][44].Two-particle correlations for charged particle pairs with transverse momenta p a T and p b T are measured as a function of ∆φ ≡ φ a − φ b and ∆η ≡ η a − η b , with |∆η|≤5, determined by the acceptance of the ID.The particles a and b are conventionally referred to as the "trigger" and "associated" particles, respectively.The correlation function is defined as: where S and B represent the same event and "mixed event" pair distributions respectively [45].When constructing S and B, pairs are weighted by the inverse product of their reconstruction efficiencies 1/( (p a T , η a ) (p b T , η b )).Detector acceptance effects largely cancel in the S /B ratio.Examples of correlation functions in the 13 TeV data are shown in Fig. 2 for N rec ch intervals 0-20 (left) and ≥120 (right), respectively, for 0.5<p a,b T <5 GeV.The C(∆η, ∆φ) distributions have been truncated at different maximum values to suppress a strong peak at ∆η = ∆φ = 0 that arises primarily from jets.The correlation functions also show a ∆η-dependent enhancement centered at ∆φ = π, which is understood to result primarily from dijets.In the higher N rec ch interval, a ridge is observed as the enhancement near ∆φ = 0 that extends over the full ∆η range of the measurement.
One-dimensional correlation functions, C(∆φ), are obtained by integrating the numerator and denominator of Eq. 1 over the long-range part of the correlation function, 2<|∆η|<5.These are converted into "per-trigger-particle yields," Y(∆φ), according to [4,6,45]: where N a denotes the efficiency-corrected total number of trigger particles.Results are shown in Fig. 3 for selected N rec ch intervals in the 13 and 2.76 TeV data, for the p a,b T ranges 0.5<p a,b T <5 GeV.Panel (a) in the figure shows Y(∆φ) for 0≤ N rec ch <20 for both collision energies; these exhibit a minimum at ∆φ = 0 and a broad peak at ∆φ ∼ π that is understood to result primarily from dijets but may also include contributions from low-p T resonance decays and global momentum conservation.The higher Y(∆φ) values for the 2.76 TeV data are due to the relative inefficiency of the 2.76 TeV triggers for the lowest multiplicity events, which results in larger N rec ch for the 2.76 TeV data in this N rec ch interval.Panels (b), (d) and (f) show results from 13 TeV data for the 40-50, 60-70, and ≥ 90 N rec ch intervals, respectively.Panels (c) and (e) show the results from 2.76 TeV data for 50-60 and 70-80 N rec ch intervals, respectively.With increasing N rec ch , the minimum at ∆φ = 0 fills in, and a peak appears and increases in amplitude.To separate the ridge from angular correlations present in low-multiplicity pp collisions, a template fitting procedure is applied to the Y(∆φ) distributions.Motivated by the peripheral subtraction method applied in p+Pb collisions [4], the measured Y(∆φ) distributions are assumed to result from a superposition of a "peripheral" Y(∆φ) distribution, scaled up by a multiplicative factor and a constant modulated by cos(2∆φ).The resulting template fit function, where has two free parameters, F and v 2,2 .The coefficient, G, which represents the magnitude of the combinatoric component of Y ridge (∆φ), is fixed by requiring that π 0 d∆φ Y templ = π 0 d∆φ Y.The peripheral distribution is obtained from the 0≤N rec ch <20 interval.In the fitting procedure, the χ 2 is calculated accounting for statistical uncertainties in both Y(∆φ) and Y periph (∆φ) distributions.Some results of the template fitting procedure are shown in panels (b)-(f) of Fig. 3; a complete set of fit results is provided in Ref. [46].The scaled Y periph (∆φ) distributions shifted up by G are shown with open points; the Y ridge (∆φ) functions shifted up by FY periph (0) are shown with the dashed lines; and the full fit function is shown by the solid curves.The function in Eq. 3 successfully describes the measured Y(∆φ) distributions in all N rec ch intervals.In particular, it simultaneously describes the ridge, which arises from an interplay of the concave Y periph (∆φ) and the cosine function, the height of the peak in the Y(∆φ) at ∆φ ∼ π, and the narrowing of that peak which results from a negative contribution of the 2v 2,2 cos (2∆φ) term in the region near ∆φ = π/2.The agreement between the template functions and the data allows for no significant N rec ch -dependent variation in the width of the dijet peak at ∆φ = π except for that accounted for by the sinusoidal component of the fit function.Including additional cos (3∆φ) and cos (4∆φ) terms in Eq. 4 produces changes in the extracted v 2,2 values that are negligible compared to their statistical uncertainties.
Previous analyses of two-particle angular correlations in pp, p+Pb, and Pb+Pb collisions have traditionally relied on the "zero yield at minimum" (ZYAM) hypothesis to separate the ridge from the dijet peak at ∆φ ∼ π.In the ZYAM method, the ridge is functionally defined to be Y(∆φ) − Y min over the restricted range |∆φ| < φ min , where φ min is the location of the minimum of Y(∆φ) and Y min = Y(φ min ).However, the Y(∆φ) distributions measured in low-N rec ch bins are concave in the region near ∆φ ∼ 0. As a result, if the ridge and dijet correlations add -an assumption that is implicit in all previous analyses using the ZYAM method and is explicit in the template method used here -then the ZYAM method will both underestimate the ridge yield and produce φ min values that vary, unphysically, with the ridge amplitude.In contrast, the template method used here explicitly accounts for the concave shape of the peripheral Y(∆φ).Thus, the template fitting procedure, for example, extracts a non-zero ridge amplitude from the √ s = 2.76 TeV, 50 ≤ N rec ch ≤ 60 Y(∆φ) distribution (middle left panel of Fig. 3) which is approximately flat near ∆φ ∼ 0, and would, as a result, have approximately zero ridge signal using the ZYAM method.
Previous p+Pb analyses used the peripheral-subtraction method, but applied the ZYAM procedure to the peripheral reference and, so, subtracted Y(0) from Y periph (∆φ).Such a subtraction will necessarily change the v 2,2 values, and, when applied to the 13 TeV data, it reduces the measured v 2,2 by a multiplicative factor that varies from 0.4 to 0.8 over 30≤N rec ch <130 [46].However, if, as suggested by the data, Y periph (∆φ) contains not only a hard component, Y hard (∆φ), but also a modulated soft component, the peripheral ZYAM method will subtract 2FG 0 v 0 2,2 cos (2∆φ) as part of the template fit, thereby reducing the extracted v 2,2 .In contrast, the procedure used in this analysis subtracts FG 0 1 + 2v 0 2,2 cos (2∆φ) , which reduces G in Eq. 4 but has less impact on v 2,2 .In particular, if v 0 2,2 is equal to the real v 2,2 in a given N rec ch interval, there will be no bias.Since the measured v 2,2 is approximately N rec ch -independent, the bias resulting from the presence of v 2,2 in the peripheral sample is expected to be small.Thus, the use of the non-subtracted peripheral reference is preferred over the more strongly biased ZYAM-subtracted reference.
If the cos (2∆φ) dependence of Y(∆φ) arises from modulation of the single-particle φ distributions, then v 2,2 should factorize such that v 2,2 (p a T , [42][43][44], where v 2 is the cos(2φ) Fourier coefficient of the single-particle anisotropy.To test this, the analysis was performed using three p b T intervals: 0.5-5, 0.5-1, and 2-3 GeV with 0.5<p a T <5 GeV; results from 2.76 TeV data for the 2-3 GeV interval were obtained using wider N rec ch intervals to improve statistics.Results are shown in the top panels of Fig. 4; the left and right panels show 2.76 and 13 TeV data, respectively.A significant p b T dependence is seen.Separately, the same analysis was applied requiring both p a T and p b T to fall within the above intervals.If factorization holds, the v 2 values calculated using: where p T 1 and p T 2 indicate which of the three intervals, 0.5-5, 0.5-1, and 2-3 GeV, p a T and p b T are required to lie within, should be independent of p T 2 .The v 2 values obtained using Eq.6 are shown in the middle panels of Fig. 4. For both collision energies, the three sets of v 2 values agree within uncertainties, indicating that v 2,2 factorizes.
This analysis is sensitive to potential N rec ch -dependent changes in the shape of the peripheral reference.For example, the pythia 8 sample shows a modest N rec ch -dependent change in the width of the dijet peak for small N rec ch .Also, the v 2,2 could vary with N rec ch over the 0<N rec ch <20 range.To test the sensitivity of the results presented here to such shape changes, the analysis was repeated using 0-5, 0-10, and 10-20 N rec ch intervals to form Y periph (∆φ).The largest resulting change in v 2,2 was taken as a systematic uncertainty.The relative uncertainty varies from 6% at N rec ch =30 to 2% for N rec ch ≥60 in the 13 TeV data, and is less than <6% for all N rec ch for the 2.76 TeV data.When using the 0-5 N rec ch interval for Y periph (∆φ), v 2,2 values consistent with those shown in Fig. 4 are measured in N rec ch intervals 5-10, 10-15 and 15-20.Potential systematic uncertainties on v 2,2 due to a residual ∆φ dependence of the two-particle acceptance that does not cancel in the S /B ratio are evaluated following Ref.[47] and are found to be less than 1%.The effect of the uncertainty on the tracking efficiency on v 2,2 is determined to be less than 1%.A separate systematic on v 2,2 due to the φ and p T resolution of the charged-particle measurement is estimated to be 2% (6%) for p T >0.5 GeV (p T <0.5 GeV).Events with unresolved multiple vertices decrease the measured v 2,2 by increasing the combinatoric pedestal in Y(∆φ) without increasing the modulation.The resulting systematic on v 2,2 increases with N rec ch and is estimated to be less than 0.25% and 5% for the 13 and 2.76 TeV data, respectively.The combined systematic uncertainties on v 2,2 and on v 2 are shown by the shaded boxes in Fig. 4. The total v 2,2 systematic uncertainty for 0.5<p a,b T <5 GeV varies between ∼5% at low N rec ch to ∼3% at high N rec ch in the 13 TeV data, while in the 2.76 TeV data the uncertainty is 8% for all N rec ch .The systematic uncertainty on v 2 is approximately half that for v 2,2 .As shown in Fig. 4, the measured v 2 are independent of N rec ch and are consistent between the two collision energies within uncertainties.The p T dependence of v 2 for the 50-60 N rec ch interval, shown in the bottom left panel of Fig. 4, is similar for both collision energies to that previously measured in p+Pb and Pb+Pb collisions.It increases with p T at low p T , reaches a maximum between 2 and 3 GeV, and then decreases at higher p T .The bottom right panel of Fig. 4 shows the p T dependence of v 2 for different N rec ch intervals; no significant dependence is observed.
In summary, ATLAS has measured the multiplicity and p T dependence of two-charged-particle correlations in √ s=13 and 2.76 TeV pp collisions at the LHC.The correlation functions at both energies show a ridge whose strength increases with multiplicity.A new template fitting procedure shows that the per-trigger-particle yields for |∆η|>2 are described well by a superposition of the yields measured in a low-multiplicity interval and a constant modulated by cos (2∆φ).Thus, as observed in p+Pb collisions [4], the pp data presented here are compatible with both a "near-side" ridge centered at ∆φ = 0 and an "away-side" ridge centered at ∆φ = π that both result from a sinusoidal component of the two-particle correlation.The extracted Fourier coefficients, v 2,2 , exhibit factorization, which is characteristic of a global modulation of the per-event single-particle distributions also seen in p+Pb and Pb+Pb collisions.The amplitudes, v 2 , of the single-particle modulation, are N rec ch -independent and agree between 2.76 and 13 TeV within uncertainties.They increase with p T for p T 3 GeV and then decrease at higher p T , following a trend similar to that observed in p+Pb and Pb+Pb collisions.These results suggest that the ridges in pp and p+Pb collisions may arise from a similar physical mechanism which does not have a strong √ s dependence.

Figure 1 :
Figure 1: Distributions of the multiplicity, N rec ch , of reconstructed charged particles having p T >0.4 GeV for the 2.76 (left) and 13 TeV (right) data used in this analysis.

Figure 4 :
Figure 4: Measured v 2,2 (top) and v 2 (middle) values versus N rec ch for different p a,b T intervals for 2.76 (left) and 13 TeV (right) data.Results are averaged over N rec ch bins of width 10 spanning the range 20≤N rec ch <100 and 20≤N rec ch <130 for 2.76 and 13 TeV data, respectively, except for the 2<p b T <3 GeV results for the 2.76 TeV data which are averaged over bins of width 20.Measured v 2 values versus p a T (bottom) spanning the range 0.3<p a T <5.0 GeV for 13 and 2.76 TeV data for the 50≤N rec ch <60 interval (left) and for three N rec ch intervals in the 13 TeV data (right).Results are averaged over the p a T intervals indicated by horizontal error bars.On all points, the vertical error bars indicate statistical uncertainties.The shaded bands indicate systematic uncertainties.For clarity, they are only shown for the 0.5<p b T <5 GeV case in the middle, for 2.76 TeV data in the lower left, and for the 40≤N rec ch <50 case in the lower right panels.