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Collectivity in large and small systems formed in ultrarelativistic collisions


Collective flow of the final-state hadrons observed in ultrarelativistic heavy-ion collisions or even in smaller systems formed in high-multiplicity pp and p/d/\(^3\)He-nucleus collisions is one of the most important diagnostic tools to probe the initial state of the system and to shed light on the properties of the short-lived, strongly interacting many-body state formed in these collisions. Limited, in the initial years, to the study of mainly the directed and elliptic flows—the first two Fourier harmonics of the single-particle azimuthal distribution—this field has evolved in recent years into a much richer area of activity. This includes not only higher Fourier harmonics and multiparticle cumulants, but also a variety of other related observables, such as the ridge seen in two-particle correlations, flow decorrelation, symmetric cumulants and event-plane correlators which measure correlations between the magnitudes or phases of the complex flows in different harmonics, coefficients that measure the nonlinear hydrodynamic response, statistical properties, e.g. the non-Gaussianity of the flow fluctuations, etc. We present a Tutorial Review of the modern flow picture and the various aspects of the collectivity—an emergent phenomenon in quantum chromodynamics.

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Fig. 1

Figure adapted from [3]

Fig. 2
Fig. 3

Figure from [22]

Fig. 4

Figure from [22]

Fig. 5

Figure from [24]

Fig. 6
Fig. 7
Fig. 8

Figure from [38]

Fig. 9
Fig. 10

Figure from [30]

Fig. 11

Figure from [62]

Fig. 12

Figure from [64]

Fig. 13

Figure from [77]

Fig. 14


  1. 1.

    See Eqs. (1) and (4) for definitions of PP (\(\varPhi _n\)) and EP (\(\varPsi _n\)) specific to harmonic n.

  2. 2.

    \(n=1\) forms a special case [5].

  3. 3.

    \(V_n\) is sometimes called a flow vector, and ‘flow’ and ‘azimuthal anisotropy’ are often used synonymously.

  4. 4.

    These names are suggestive of the shapes of the polar plots \(r=1+2v_n \cos n\phi \), for \(0 < v_n \ll 1\).

  5. 5.

    Nonflow correlations are not related to the initial-state geometry and hence not associated with the symmetry plane \(\varPsi _n\), but arise due to jets, particle decays, etc. They are of short range.

  6. 6.

    \(v_n\{m\}\) are also called multiparticle cumulants of order m of the flow \(v_n\).

  7. 7.

    Recall the discussion in the last paragraph of Sect. 2.3.

  8. 8.

    ATLAS collaboration denotes the event plane by \(\varPhi _n\), which is different from our convention here. In Ref. [43], we have used the ATLAS convention.

  9. 9.

    Note also that \(\left\langle V_{n\mathrm{{L}}} \right\rangle =0\), because \(V_{n\mathrm{{L}}}\), like \(V_n\), is expected to carry a random phase factor depending on the reaction-plane angle \(\varPhi _{\mathrm{RP}}\).

  10. 10.

    A recent experiment [12] has shown that the ordering persists up to \(n=7\), with some enhancement for \(n=8,9\).

  11. 11.

    For simplicity of notation, we use the same symbols \(\rho \) and C to denote 1-, 2-, 3- and multiparticle correlation functions and cumulants, respectively.


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I am very thankful to Jean-Yves Ollitrault for helpful comments on the manuscript. I also thank him for our long-term collaboration which allowed me to learn many things. I acknowledge the award of the Core Research Grant, by the Science and Engineering Research Board, Department of Science and Technology, Government of India. I thank Bhavya Bhatt for drawing Fig. 14.

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Correspondence to Rajeev S. Bhalerao.


Appendix A

Moments and cumulants of a probability distribution

The nth moment of a (real, continuous) function f(x), about a constant a, is defined as

$$\begin{aligned} \mu _n(a) \equiv \int _{-\infty }^\infty (x-a)^n f(x) \mathrm{{d}}x. \end{aligned}$$

We shall assume f(x) to be the probability density function (PDF), normalized to unity. The two most interesting values of a are 0 and \(\mu \equiv \left\langle x \right\rangle \), the mean of the distribution. Usually one refers to \(\mu _n(a=0)\) simply as the “moment” and \(\mu _n(a=\mu )\) as the “central moment”. Henceforth, we denote moments \(\mu _n(a=0)\) by \(\mu '_n\) and central moments \(\mu _n(a=\mu )\) by \(\mu _n\). Obviously, \(\mu '_0=1=\mu _0\) and \(\mu '_n=\left\langle x^n \right\rangle \). The first four central moments are

$$\begin{aligned} \begin{aligned} \mu _1&=0, \\ \mu _2&=\left\langle x^2 \right\rangle -\mu ^2 \equiv {{\mathrm{variance}}} \,(\sigma ^2) \equiv \,\, ({\mathrm{standard}} {\mathrm{deviation}} = \sigma )^2, \\ \mu _3&= \left\langle x^3 \right\rangle -3\mu \left\langle x^2 \right\rangle +2\mu ^3, \\ \mu _4&= \left\langle x^4 \right\rangle -4\mu \left\langle x^3 \right\rangle +6\mu ^2 \left\langle x^2 \right\rangle -3\mu ^4. \end{aligned}\nonumber \\ \end{aligned}$$

It is often convenient to define standardized central moments which are scale-invariant or dimensionless quantities: \(\mu _n/\sigma ^n\). The first four standardized central moments are

$$\begin{aligned} \begin{aligned} \mu _1/\sigma&= 0, \\ \mu _2/\sigma ^2&= 1, \\ \mu _3/\sigma ^3&\equiv {{\mathrm{skewness}}} \,\, (\gamma ), \\ \mu _4/\sigma ^4&\equiv {{\mathrm{kurtosis}}} \,\, (\kappa ). \end{aligned}\end{aligned}$$

Excess kurtosis is defined as \(\kappa -3\). However, it is not uncommon to find the excess kurtosis itself termed as kurtosis. The reason for subtracting 3 will become clear when we discuss the Gaussian (or normal) distribution (Appendix C).

\(\bullet \) Variance is a measure of the spread of the random numbers about their mean value.

\(\bullet \) Skewness is a measure of the lopsidedness or asymmetry of the distribution about its mean. It is clear from the definition that a distribution that is symmetric about its mean has vanishing skewness. In general, skewness can be positive or negative. If the left (right) tail is drawn out, or in other words, is longer than the right (left) tail, the distribution is said to be left(right)-skewed and has a negative (positive) skewness; see Fig. 14.

\(\bullet \) Kurtosis is a measure of the heaviness of the tails of the distribution as compared to the normal distribution with the same variance. It is clear from the definition that kurtosis is a nonnegative number. (Excess kurtosis may be positive or negative.) Kurtosis is a measure of the “tailedness” and not the “peakedness” of a distribution (Fig. 14), because the proportion of the kurtosis that is determined by the central \(\mu {\pm } \sigma \) range is usually quite small [88].

Moment-generating function M(t)

$$\begin{aligned} M(t)\equiv \left\langle e^{tx} \right\rangle =1+t\left\langle x \right\rangle +\frac{t^2}{2!}\left\langle x^2 \right\rangle +\cdots = \sum _{n=0}^\infty \frac{t^n}{n!}\left\langle x^n \right\rangle . \end{aligned}$$

Observe that the moments \(\mu '_n=\left\langle x^n \right\rangle \) appear in the above expansion. To isolate the mth moment \(\left\langle x^m \right\rangle \), for example, one uses

$$\begin{aligned} \left[ \frac{\mathrm{{d}}^m}{\mathrm{{d}}t^m} M(t) \right] _{t=0} =\left\langle x^m \right\rangle . \end{aligned}$$

Differentiating M(t)   m times removes the first m terms, i.e., the terms containing \(1, \left\langle x \right\rangle , \ldots , \left\langle x^{m-1} \right\rangle \). Further, setting \(t=0\) removes terms containing \(\left\langle x^{m+1} \right\rangle ,\left\langle x^{m+2} \right\rangle , \ldots \), leaving behind only \(\left\langle x^m \right\rangle \). The moment-generating function provides an alternative description of the PDF. The central moment generating function is \(e^{-\mu t}M(t)\).

Cumulant-generating function K(t)

$$\begin{aligned} K(t)\equiv \ln M(t) = \ln \left\langle e^{tx} \right\rangle =t\kappa _1+\frac{t^2}{2!}\kappa _2+\frac{t^3}{3!}\kappa _3+\cdots = \sum _{n=1}^\infty \frac{t^n}{n!}\kappa _n, \end{aligned}$$

where \(\kappa _n\) are the cumulants. To isolate the mth cumulant \(\kappa _m\), for example, one uses

$$\begin{aligned} \left[ \frac{\mathrm{{d}}^m}{\mathrm{{d}}t^m} K(t) \right] _{t=0} =\kappa _m. \end{aligned}$$

Differentiating K(t)   m times removes the first \(m-1\) terms, i.e., the terms containing \(\kappa _1, \ldots , \kappa _{m-1}\). Further, setting \(t=0\) removes terms containing \(\kappa _{m+1},\kappa _{m+2}, \ldots \), leaving behind only \(\kappa _m\). It is straightforward to show that the first few cumulants are

$$\begin{aligned} \begin{aligned} \kappa _1&= \left\langle x \right\rangle = {{\mathrm{mean}}} \,\, \mu , \\ \kappa _2&= \left\langle x^2 \right\rangle -\left\langle x \right\rangle ^2 = \left\langle (x-\left\langle x \right\rangle )^2 \right\rangle ={\mathrm{variance}} \,\, (\sigma ^2)\\&= {\mathrm{second \,\,central\,\, moment}}, \\ \kappa _3&= \left\langle x^3 \right\rangle -3\left\langle x \right\rangle \left\langle x^2 \right\rangle +2\left\langle x \right\rangle ^3 \\&= \left\langle (x-\left\langle x \right\rangle )^3 \right\rangle = {{\mathrm{third \,\, central \,\, moment}}}, \\ \kappa _4&= \left\langle x^4 \right\rangle -4\left\langle x \right\rangle \left\langle x^3 \right\rangle -3\left\langle x^2 \right\rangle ^2+12\left\langle x \right\rangle ^2\left\langle x^2 \right\rangle -6\left\langle x \right\rangle ^4 \\&\ne {{\mathrm{fourth \,\, central \,\, moment}}} \,\,\left\langle (x-\left\langle x \right\rangle )^4 \right\rangle , \\ \kappa _5&= \left\langle x^5 \right\rangle -5\left\langle x \right\rangle \left\langle x^4 \right\rangle -10\left\langle x^2 \right\rangle \left\langle x^3 \right\rangle +20\left\langle x \right\rangle ^2\left\langle x^3 \right\rangle \\&+30\left\langle x \right\rangle \left\langle x^2 \right\rangle ^2 -60\left\langle x \right\rangle ^3\left\langle x^2 \right\rangle +24\left\langle x \right\rangle ^5, \\ \kappa _6&= \left\langle x^6 \right\rangle -6\left\langle x \right\rangle \left\langle x^5 \right\rangle -15\left\langle x^2 \right\rangle \left\langle x^4 \right\rangle +30\left\langle x \right\rangle ^2\left\langle x^4 \right\rangle \\&\quad -10\left\langle x^3 \right\rangle ^2 +120\left\langle x \right\rangle \left\langle x^2 \right\rangle \left\langle x^3 \right\rangle \\&\quad -120\left\langle x \right\rangle ^3\left\langle x^3 \right\rangle +30\left\langle x^2 \right\rangle ^3 - 270\left\langle x \right\rangle ^2\left\langle x^2 \right\rangle ^2\\&\quad +360\left\langle x \right\rangle ^4\left\langle x^2 \right\rangle -120\left\langle x \right\rangle ^6. \end{aligned} \end{aligned}$$

Fourth- and higher order cumulants are not identical to the corresponding central moments. Thus cumulants are certain polynomial functions of the moments and provide an alternative to the moments of the distribution.

The above equations can be inverted to express moments in terms of cumulants:

$$\begin{aligned} \begin{aligned} \left\langle x \right\rangle&= \kappa _1, \\ \left\langle x^2 \right\rangle&= \kappa _1^2+\kappa _2, \\ \left\langle x^3 \right\rangle&= \kappa _1^3+3\kappa _1\kappa _2+\kappa _3, \\ \left\langle x^4 \right\rangle&= \kappa _1^4+6\kappa _1^2\kappa _2+4\kappa _1\kappa _3+3\kappa _2^2+\kappa _4, \\ \left\langle x^5 \right\rangle&= \kappa _1^5+10\kappa _1^3\kappa _2+10\kappa _1^2\kappa _3 +15\kappa _1\kappa _2^2+5\kappa _1\kappa _4\\&+10\kappa _2\kappa _3+\kappa _5, \\ \left\langle x^6 \right\rangle&= \kappa _1^6+6\kappa _1\kappa _5+15\kappa _1^2\kappa _4 +20\kappa _1^3\kappa _3+15\kappa _1^4\kappa _2\\&\quad +60\kappa _1\kappa _2\kappa _3 +45\kappa _1^2\kappa _2^2+15\kappa _2\kappa _4 \\&\quad +10\kappa _3^2+ 15\kappa _2^3+\kappa _6. \end{aligned} \end{aligned}$$

Appendix B

Correlation functions and cumulants

The n-particle correlation function (or simply a correlator) \(\rho (1,2,3,\ldots ,n)\) consists of terms that represent combinations of lower order correlations and a term that represents a genuine or “true” n-particle correlation \(C(1,2,3,\ldots ,n)\) which is called a cumulant. For example, \(\rho (1,2)=\rho (1)\rho (2)+C(1,2)\), so that \(C(1,2)=\rho (1,2)-\rho (1)\rho (2)\). If the two particles are statistically independent, \(\rho (1,2)\) simply reduces to \(\rho (1)\rho (2)\), whereas C(1, 2) vanishes by construction.Footnote 11 The first few correlation functions are [89]

$$\begin{aligned} \rho (1)= & {} C(1), \nonumber \\ \rho (1,2)= & {} \rho (1)\rho (2)+C(1,2), \nonumber \\ \rho (1,2,3)= & {} \rho (1)\rho (2)\rho (3)+\rho (1)C(2,3)+\rho (2)C(3,1)\nonumber \\&+ \rho (3)C(1,2)+C(1,2,3),\nonumber \\\equiv & {} \rho (1)\rho (2)\rho (3)+\sum _{(3)}\rho (1)C(2,3)+C(1,2,3), \nonumber \\ \rho (1,2,3,4)= & {} \rho (1)\rho (2)\rho (3)\rho (4)+\sum _{(6)}\rho (1)\rho (2)C(3,4)\nonumber \\&+\sum _{(4)}\rho (1)C(2,3,4) \nonumber \\&+\sum _{(3)}C(1,2)C(3,4)+C(1,2,3,4), \nonumber \\ \rho (1,2,3,4,5)= & {} \rho (1)\rho (2)\rho (3)\rho (4)\rho (5) +\sum _{(10)}\rho (1)\rho (2)\rho (3)C(4,5) \nonumber \\&+\sum _{(10)}\rho (1)\rho (2)C(3,4,5)\nonumber \\&+\sum _{(15)}\rho (1)C(2,3)C(4,5) +\sum _{(5)}\rho (1)C(2,3,4,5)\nonumber \\&+\sum _{(10)}C(1,2)C(3,4,5) +C(1,2,3,4,5), \nonumber \\ \rho (1,2,3,4,5,6)= & {} \rho (1)\rho (2)\rho (3)\rho (4)\rho (5)\rho (6)\nonumber \\&+\sum _{(6)}\rho (1)C(2,3,4,5,6) \nonumber \\&+\sum _{(15)}\rho (1)\rho (2)C(3,4,5,6)\nonumber \\&+\sum _{(20)}\rho (1)\rho (2)\rho (3)C(4,5,6) \nonumber \\&+\sum _{(15)}\rho (1)\rho (2)\rho (3)\rho (4)C(5,6)\nonumber \\&+\sum _{(60)}\rho (1)C(2,3)C(4,5,6) \nonumber \\&+\sum _{(45)}\rho (1)\rho (2)C(3,4)C(5,6)\nonumber \\&+\sum _{(15)}C(1,2)C(3,4,5,6) \nonumber \\&+\sum _{(10)}C(1,2,3)C(4,5,6)\nonumber \\&+\sum _{(15)}C(1,2)C(3,4)C(5,6) \nonumber \\&+C(1,2,3,4,5,6), \end{aligned}$$

where the numbers in parentheses under the summation signs indicate the number of possible permutations of the indices.

The above equations can be inverted to get the expressions for the cumulants in terms of the correlation functions:

$$\begin{aligned} C(1)= & {} \rho (1), \nonumber \\ C(1,2)= & {} \rho (1,2)-\rho (1)\rho (2), \nonumber \\ C(1,2,3)= & {} \rho (1,2,3)-\sum _{(3)}\rho (1)\rho (2,3)+2\rho (1)\rho (2)\rho (3), \nonumber \\ C(1,2,3,4)= & {} \rho (1,2,3,4)-\sum _{(4)}\rho (1)\rho (2,3,4)\nonumber \\&-\sum _{(3)}\rho (1,2)\rho (3,4) \nonumber \\&+2\sum _{(6)}\rho (1)\rho (2)\rho (3,4) -6\rho (1)\rho (2)\rho (3)\rho (4), \nonumber \\ C(1,2,3,4,5)= & {} \rho (1,2,3,4,5) -\sum _{(5)}\rho (1)\rho (2,3,4,5)\nonumber \\&-\sum _{(10)}\rho (1,2)\rho (3,4,5)\nonumber \\&+2\sum _{(10)}\rho (1)\rho (2)\rho (3,4,5)\nonumber \\&+2\sum _{(15)}\rho (1)\rho (2,3)\rho (4,5) \nonumber \\&-6\sum _{(10)}\rho (1)\rho (2)\rho (3)\rho (4,5)\nonumber \\&+24\rho (1)\rho (2)\rho (3)\rho (4)\rho (5), \nonumber \\ C(1,2,3,4,5,6)= & {} \rho (1,2,3,4,5,6)\nonumber \\&-\sum _{(6)}\rho (1)\rho (2,3,4,5,6)\nonumber \\&-\sum _{(15)}\rho (1,2)\rho (3,4,5,6) \nonumber \\&+2\sum _{(15)}\rho (1)\rho (2)\rho (3,4,5,6)\nonumber \\&-\sum _{(10)}\rho (1,2,3)\rho (4,5,6) \nonumber \\&+4\sum _{(30)}\rho (1)\rho (2,3)\rho (4,5,6)\nonumber \\&-6\sum _{(20)}\rho (1)\rho (2)\rho (3)\rho (4,5,6) \nonumber \\&+2\sum _{(15)}\rho (1,2)\rho (3,4)\rho (5,6)\nonumber \\&-3\sum _{(90)}\rho (1)\rho (2)\rho (3,4)\rho (5,6) \nonumber \\&+24\sum _{(15)}\rho (1)\rho (2)\rho (3)\rho (4)\rho (5,6)\nonumber \\&-\sum _{(120)}\rho (1)\rho (2)\rho (3)\rho (4)\rho (5)\rho (6). \end{aligned}$$

It is easy to verify that the cumulant vanishes if any one or more particles is statistically independent of the others. Thus the n-particle cumulant measures the statistical dependence of the entire n-particle set. For this reason, cumulants are also called connected correlation functions.


Appendix C

Gaussian or normal distribution in 1D

One-dimensional Gaussian (centered at \(\mu \)) is

$$\begin{aligned} f(x)=\frac{1}{\sigma \sqrt{2\pi }}\exp \left[ -\frac{(x-\mu )^2}{2\sigma ^2}\right] , \end{aligned}$$

where \(\mu =\left\langle x \right\rangle \) is the mean, \(\sigma ^2\) is the variance and f(x) is normalized to unity. The odd central moments are obviously zero. The even central moments are given by \((n-1)!! \,\sigma ^n \,\, (n ~{\mathrm{even}})\). The first few even central moments are given in Table 1. The skewness (\(\gamma \)) is zero, kurtosis (\(\kappa \)) is 3 and excess kurtosis (\(\kappa -3\)) is zero. A probability distribution with tails heavier (lighter) than those of the normal distribution shows a higher (lower) propensity to produce outliers and has a positive (negative) excess kurtosis (Fig 14).

Table 1 First few central and noncentral moments of the Gaussian distribution
Table 2 First few moments (\(\mu _n\)) of f(r), Eq. (C3)

For the normal distribution, the moment-generating function is \(M(t)= \exp \,(\mu t + (\sigma ^2 t^2 /2))\) and the cumulant-generating function is \(K(t) = \ln M(t)= \mu t + (\sigma ^2 t^2 /2)\). Since this is a quadratic in t, only the first two cumulants survive, namely the mean \(\mu \) and the variance \(\sigma ^2\). It can be shown that the normal distribution is the only one for which the third and higher cumulants vanish.


Gaussian or normal distribution in 2D

Two-dimensional Gaussian centered at the origin and normalized to unity is

$$\begin{aligned} f(x,y)=\frac{1}{2 \pi \sigma _x \sigma _y} \exp \left[ -\frac{x^2}{2\sigma _x^2}-\frac{y^2}{2\sigma _y^2} \right] , \end{aligned}$$

where \(\sigma _x^2\) and \(\sigma _y^2\) are the variances.


It is often convenient to introduce the symmetric case (\(\sigma _x=\sigma _y\)) and write it in terms of \(r=\sqrt{x^2+y^2}\):

$$\begin{aligned} f(r) =\frac{1}{\pi \sigma ^2}\exp \left[ -\frac{r^2}{\sigma ^2}\right] , \end{aligned}$$

where \(\sigma ^2\equiv \sigma _x^2+\sigma _y^2\). f(r) is also centered at the origin and normalized to unity. Note, however, that unlike \(\left\langle x \right\rangle \) and \(\left\langle y \right\rangle \), \(\left\langle r \right\rangle \) is not zero. The first few moments of f(r), \(\mu _n=\left\langle r^n \right\rangle =\sigma ^n \, \varGamma ((n/2)+1)\), are given in Table 2. Note also that \(\left\langle r^2 \right\rangle =\left\langle x^2 \right\rangle +\left\langle y^2 \right\rangle \); using \(\left\langle r^2 \right\rangle \) from Table 2, we get \(\sigma ^2=\sigma _x^2+\sigma _y^2\). The kurtosis in this case is 2, unlike the case of a 1D Gaussian discussed earlier where it was 3.

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Bhalerao, R.S. Collectivity in large and small systems formed in ultrarelativistic collisions. Eur. Phys. J. Spec. Top. 230, 635–654 (2021).

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