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\begin{document}
\title{Collectivity and thermalization in \pp, $p$$+$A, \dau and \heau collisions}

\author{Shengli Huang}

\address{Department of Physics and Astronomy, Station B 1807, Vanderbilt University,
Nashville, TN 37235, USA}

\ead{shengli.huang@vanderbilt.edu}

\begin{abstract}
In this proceeding, the recent measurements of the
anisotropy($v_2$, $v_3$) in small collision systems such as \pp,
$p$$+$A, \dau and \heau collisions at the LHC and RHIC energies
are presented. In \pp collisions, the $v_2$ are measured by two
and multi-particle \dphi correlations at \sqsn~=~2.76, 5, 7, and
13 TeV. In 200 GeV \pau, \dau and \heau collisions. the $v_2$ are
measured with event plane methods. I also present the measurements
of $v_2$ for $\PKzS$ and $\PgL/\PagL$ in high multiplicity \pp
collisions, in which a particle mass ordering is observed. The
$v_2$ of identified $\pi^{\pm}$, $K^{\pm}$ and(anti)protons in
central $^{3}$He+Au collisions are also measured and quark-number
scaling is observed. These observations are similar to those seen
in A$+$A collisions, and support the interpretation of a
collective origin.


\end{abstract}

\section{Introduction}
Recently, in the high-multiplicity small collision systems such as
$p$+$p$ and $p$/$d$/$^{3}$He+A collisions, the near-side,
long-range two particle angular correlation (so called 'ridge')
and anisotropic flow ($v_n$) have been observed at the
LHC~\cite{CMS:2012qk,Aad:2012,
Abelev:2012cya,Atlas:2013cm,CMS:2013cm} and
RHIC~\cite{Adare:2013piz,Adare:2014keg,PhysRevLett.115.142301}.
Several physics models, which include initial-state gluon
saturation~\cite{Dusling:2013oia}, and hydrodynamic flow from a
mini-QGP(Quark Gluon Plasma)~\cite{Bozek:2011if} are potential
explanations for these observations. While the former will not
depend on the initial geometry, the hydrodynamic models have
strong initial geometry dependence. Therefore, measuring the $v_2$
and $v_3$ in $p$/$d$/$^{3}$He+Au collisions will provide direct
testing for these explanations since the initial geometry is quite
different for these collisions~\cite{Nagle:2013lja}. If a mini-QGP
exists in the small collision system, studying it will help us to
further understand: 1)the condition for the QGP thermalization;
2)the role of the internal structure of proton on the initial
geometry eccentricity~\cite{eccproton}; and 3)the contributions
from pre-equilibrium stage to the system evolution~\cite{sonic}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%  Experimental section
\section{The $v_2$ measurements in \pau, \dau and \heau collisions at \sqsn~=~200 GeV at RHIC}
The $v_{2}$ for inclusive charged hadrons produced at mid-rapidity
$|\eta|<0.35$ are measured in 0-5\% central $p$+Au in the PHENIX
experiment and compared with that from $d$+Au and $^{3}$He+Au
collisions, as shown in the Fig.~\ref{fig:v2_pdheau}. The $v_2$ in
the $p$+Au is found to be smaller than that of $d$+Au and
$^{3}$He+Au. The eccentricity of $p$+Au is also smaller than that
of $d$+Au and $^{3}$He+Au. It indicates the anisotropy in the
small collisions system depends on the initial geometry, which is
supported by the hydrodynamic model calculation~\cite{sonic} as
shown in the figure.

The $v_3$ for inclusive charged hadrons produced at mid-rapidity
$|\eta|<0.35$ in high-multiplicity \heau collisions at
\sqsn~=~200~GeV are also measured in the PHENIX experiment with
respect to the $\Psi_3$ event planes.
%%The observed event plane angles $\Psi_{n}^{{\rm Obs}}$ are
%%measured by the Forward Silicon Vertex~\cite{Aidala:2013vna}
%%(FVTX) detectors by the reconstructed clusters in the south,
%%Au-going endcap (FVTX-S), which covers $-3.0<\eta<-1.0$. The
%%contribution from nonflow correlations at each \pT is estimated
%%from pp collisions. It reaches a maximum of 7\%(15\%) for
%%$v_2$($v_3$) and are treated as a systematic uncertainty.
The $v_3$ results are shown in Fig.~\ref{fig:v2v3_pdheAu} with
$v_2$ and different theory calculations. Four such predictions
shown in Fig.~\ref{fig:v2v3_pdheAu} employ viscous hydrodynamics
with $\eta/s$ at or near the conjectured lower bound
$1/4\pi$~\cite{Kovtun:2004de} and one is from the {\sc ampt}
(A-Multi-Phase-Transport-Model) framework~\cite{Lin:2004en}. The
{\sc sonic} calculation~\cite{Nagle:2013lja} employs Glauber
initial conditions, viscous hydrodynamics, and then at $T=170$~MeV
a transition to a hadronic cascade. The {\sc supersonic}
calculation~\cite{sonic} additionally includes pre-equilibrium
dynamics that boosts the initial velocity fields at the earliest
times. The impact of pre-equilibrium is modest on the $v_{2}$
values and the data agree with both calculations within
uncertainties. The effect of pre-equilibrium on $v_{3}$ is
significantly larger as the triangular flow takes longer to
develop~\cite{Nagle:2013lja}. The {\sc supersonic} prediction
agrees well with the experimental data for $p_{T} < 1.2$ GeV/$c$,
and then the data trends towards the {\sc sonic} prediction at
higher \pT.



%%%%%%%%%%%%%%%%%  Section on p/d/He+Au and PID v2
\begin{figure}[htbp]
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figure_4_QM15.pdf}
\caption{\label{fig:v2_pdheau} Measured $v_2$ for mid-rapidity
charged tracks in 0-5\% central $p$+Au,$d$+Au and $^{3}$He+Au at
\sqsn~=~200~GeV using the event plane method. Also shown are SONIC
model calculations~\cite{sonic}}
\end{minipage}\hspace{1pc}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figure_4_final.pdf}
\caption{\label{fig:v2v3_pdheAu} $v_{2}$ (circles) and $v_{3}$
(squares) as a function of \pT for inclusive charged hadrons at
mid-rapidity in 0-5\% central \heau collisions at \sqsn~=~200~GeV;
Also shown are various theoretical calculations.}
\end{minipage}
\end{figure}




\section{The $v_2$ measurements in \pp collisions at \sqsn~=~2.76, 5, 7, and
13 TeV at the LHC}

The $v_2$ in \pp collisions at \sqsn~=~2.76 and 13 TeV has been
measured with long-range($|\deta|>2$) two particle \dphi
correlations by ATLAS~\cite{Aad:2016}. A new method, called
template fitting has been developed to separate the ridge and the
back-to-back jet correlation in this long-rang two particles
correlation, by assuming that the shape of the jet-induced
correlations is invariant with event multiplicity and can be
extracted from low-multiplicity events. The second order harmonic
coefficients $v_{2,2}$ from template fitting and the $v_2$
extracted by assuming factorization are shown in the
Fig~\ref{fig:v2_ATLAS}. Both $v_{2,2}$ and $v_2$ show a very week
multiplicity dependence and are nearly identical for collision
energy from 2.76 to 7 TeV.

The anisotropy of charged particles are also measured in \pp
collisions at 13 TeV by CMS~\cite{cmspidpp}. A different
subtraction method~\cite{CMS:2013cm} is employed by CMS. The
second-order Fourier coefficients $V_{2\Delta}$ extracted from
long-range two-particle \dphi correlations in the
higher-multiplicity region are subtracted from the $V_{2\Delta}$
coefficients from $10 \leq \noff < 20$ with
\begin{linenomath}
\begin{equation}
\label{eq:vnsubperiph}
V^\text{sub}_{2\Delta}=V_{2\Delta}-V_{2\Delta}(10\leq\noff<20)\,\frac{N_\text{assoc}(10\leq\noff<20)}{N_\text{assoc}}\,\frac{Y_\text{jet}}{Y_\text{jet}(10\leq\noff<20)}.
\end{equation}
\end{linenomath}

Here, $Y_\text{jet}$ represents the near-side jet yield obtained
by integrating the difference of the short- and long-range
event-normalized associated yields for each multiplicity class as
shown for $105 \leq \noff < 150$ over $|\Delta\phi| < 1.2$. The
ratio, $Y_\text{jet}/Y_\text{jet}(10\leq\noff<20)$, is introduced
to account for the enhanced jet correlations resulting from the
selection of higher-multiplicity events.

The $V_{2\Delta}$ is shown in the Fig~\ref{fig:v2_CMS} as a
function of \noff for charged particles. Before subtraction, the
$V_{2\Delta}$ coefficients are found to be nearly constant as a
function of multiplicity. After subtraction, $V_{2\Delta}$
exhibits an increase with multiplicity for $\noff \lesssim 100$.
This jet subtraction procedure is also tested in \PYTHIA
simulations, where no jet modification from initial or final state
effects is present. The $V_{2\Delta}$ after subtraction is found
to be consistent with zero as shown in the Fig~\ref{fig:v2_CMS}.

\begin{figure}[htbp]
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figure_ATLAS_ppv2.pdf}
\caption{ The integral $v_{2,2}$ and $v_2$ from template fitting
as a function of multiplicity in in \pp collisions at
\sqsn~=~2.76~TeV(top left) and ~7~TeV(top right). The $v_2$ as a
function of \pT are also shown in these two collision
energy(bottom left) and different multiplicity bins at 7
TeV.(bottom right) \label{fig:v2_ATLAS}}
\end{minipage}\hspace{1pc}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figure_CMS_ppv2.pdf}
\caption{\label{fig:v2_CMS} $V_{2\Delta}$ as a function of \noff
for charged particles, averaged over $0.3 < \pT < 3.0 GeV/c$, in
\pp collisions at \sqsn~=~13~TeV, before (open) and after (filled)
subtraction of jet correlations, estimated from the $10 \leq \noff
< 20$ range. Results from \PYTHIA are shown as curves.}
\end{minipage}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fig_4
%%%%%%  Main figure with final results and theory curves
\section{Fourier harmonics from multi-particle correlations}
Different methods of jet subtraction will lead to quite different
multiplicity dependence for $v_2$ measurements, as shown by
results above. To avoid this uncertainty, CMS also measures the
$v_2$ in \pp collisions by using a multi-particle cumulant
method~\cite{Bilandzic:2010jr}. This technique has the advantage
of suppressing short-range two-particle correlations such as jets
and resonance decays.
\begin{figure}[htbp]
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figure_CMS_Four.pdf}
\caption{$\cn{4}$ as a function of \noff for charged particles,
averaged over $0.3 < \pT < 3.0 GeV/c$, in \pp collisions at
\sqsn~=~5, 7, and 13 TeV. The $\cn{4}$ from \ppb collisions at
5~TeV are also shown for comparison \label{fig:v2_four}}
\end{minipage}\hspace{1pc}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figure_CMS_Six.pdf}
\caption{\label{fig:v2_six} $\cn{6}$ as a function of \noff for
charged particles, averaged over $0.3 < \pT < 3.0 GeV/c$, in \pp
collisions at \sqsn~=~13~TeV and compare with that measured in
\ppb collisions at 5~TeV.}
\end{minipage}
\end{figure}
The corresponding cumulants, $\cn{4}$ and $\cn{6}$, are calculated
as follows~\cite{Bilandzic:2010jr}:
\begin{equation}
\begin{split}
\cn{4} =& \dmean{4} - 2 \times \dmean{2}^2,
\\
\cn{6} =& \dmean{6} - 9 \times \dmean{4}\dmean{2} + 12 \times
\dmean{2}^3. \label{eq:cn}
\end{split}
\end{equation}
Fig.~\ref{fig:v2_four} shows the four-particle cumulant
$c_{2}\{4\}$ values for charged particles($0.3 < \pT < 3.0 GeV/c$)
as a function of \noff for \pp collisions at \sqsn~=~5, 7, and 13
TeV. The \ppb data at \sqsn~=~5~TeV~\cite{CMS:2013cm} are also
plotted for comparison. The six-particle cumulant $c_{2}\{6\}$
values for \pp collisions at \sqsn~=~13~TeV are shown in
Fig.~\ref{fig:v2_six}, comparing with \ppb data at \sqsn~=~
5~TeV~\cite{CMS:2013cm}.

Similar to that found for \ppb collisions, the $c_{2}\{4\}$ values
decrease as a function of increasing multiplicity in \pp
collisions for all three collision energies. In \pp collisions at
13~TeV, the $c_{2}\{4\}$ switches sign from positive to negative
at \noff above 60, and indicates a collective $v_2\{4\}$
signal~\cite{Borghini:2000sa}. An indication of energy dependence
of $c_{2}\{4\}$ values is seen in Fig.~\ref{fig:v2_four}, where
$c_{2}\{4\}$ tends to be more positive for a given \noff range at
lower collision energies. It may be due to different average \pT
values at different collisions energies or different multiplicity
fluctuations. The positive $c_{2}\{6\}$ values are also observed
in the \pp collisions at 13~TeV which is similar to what was
observed in \ppb collisions



\section{The $v_2$ of identified particles in \pp and \heau collisions}
The $v_2$ of $\PKzS$ and \PgL/\PagL\ particles are measured in the
high multiplicity \pp collisions at 13 TeV by CMS~\cite{cmspidpp}.
After correction for jet correlations estimated from
low-multiplicity data, the $v_{2}^\text{sub}$ results as a
function of \pT for $105 \leq \noff < 150$ are shown in
Fig.~\ref{fig:v2_cms_pid} (top).  The particle mass ordering of
$v_{2}$ values is observed in the lower \pT region, while at
higher \pT the ordering is reversed. The number of quark scaling
as a function of $KE_{T}/n_{q}$ is shown in
Fig.~\ref{fig:v2_cms_pid} (bottom) with a dashed curve
corresponding to a polynomial fitting to the $\PKzS$ data. The
ratio of $v_{2}^\text{sub}/n_{q}$ results for $\PKzS$ and
\PgL/\PagL\ particles divided by this polynomial function fitting
is also shown in Fig.~\ref{fig:v2_cms_pid} (bottom). An
approximate scaling is seen for $KE_{T}/n_{q} \gtrsim 0.2 GeV$
within about $\pm$10\%.
\begin{figure}[htbp]
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figure_CMS_PID.pdf}
\caption{Top: the $v_{2}^\text{sub}$ results of inclusive charged
particles, $\PKzS$ and
     \PgL/\PagL\ particles as a function of \pT in 13 TeV \pp for multiplicity as $105 \leq \noff <
     150$, after subtracting jet correlations
     estimated from low-multiplicity data. Bottom: the $v_{2}^\text{sub}/n_{q}$
     for $\PKzS$ and \PgL/\PagL\ as a function of $KE_{T}/n_{q}$.
     Ratios to a smooth fit function of data for $\PKzS$ particles are also
shown.
  \label{fig:v2_cms_pid}}
\end{minipage}\hspace{1pc}
\begin{minipage}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figure_5_QM15.pdf}
\caption{\label{fig:v2_phenix_pid} The scaling of number of quark
of $v_2(p_T)$ for $\pi^{\pm}$, $K^{\pm}$ and (anti)protons in
0-5\% central $^{3}$He+Au collisions. }
\end{minipage}
\end{figure}

PHENIX measured the $v_2$ of $\pi^{\pm}$, $K^{\pm}$ and
(anti)protons in the central \heau collisions at 200 GeV.
Fig.~\ref{fig:v2_phenix_pid} shows the number of quark scaling for
$v_{2}$ of $\pi^{\pm}$, $K^{\pm}$ and (anti)protons as a function
of $KE_{T}/n_{q}$ (GeV). The number of quark scaling is found to
hold in the small collisions system, which is consistent with
viscous hydro calculations or quark coalescence
models~\cite{Hwa:2002tu, Fries:2003vb}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Section  Conclusion/Summary
\section{Summary}
I summarize the recent measurements of anisotropy($v_2$, $v_3$) in
small collision systems which include \pp, $p$$+$A, \dau and \heau
collisions. The $v_2$ and $v_3$ are measured with several
different ways which include the long-range two particles \dphi
correlations, event plane with large $\eta$ gap and
multi-particles cumulant methods. A sizable $v_2$ and $v_3$ are
observed in the small collision systems and found to be related to
the initial geometry. A particle mass ordering of $v_2$ values are
also observed for the identified particles in small collisions.
All these measurements are similar to what were observed in large
A$+$A collisions system and indicate a mini-QGP has been generated
in small collision systems.


\section*{References}
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\end{document}
