Multiparticle azimuthal angular correlations in $pA$ collisions

In the Color Glass Condensate formalism, we evaluate the 3-dipole correlator up to the $\frac{1}{N_c^4}$ order with $N_c$ being the number of colors, and compute the azimuthal cumulant $c_{123}$ for 3-particle productions. In addition, we discuss the patterns appearing in the $n$-dipole formula in terms of $\frac{1}{N_c}$ expansions. This allows us to conjecture the $N_c$ scaling of $c_n\{m\}$, which is crosschecked by our calculation of $c_2\{4\}$ in the dilute limit.


II. MULTIDIPOLE CORRELATORS IN THE MCLERRAN-VENUGOPALAN MODEL
We aim to compute the n-dipole correlator, namely, where U (x ⊥ ) is a Wilson line in the fundamental representation given by where P is x + ordering operator and t a is a color matrix in the fundamental SU(N c ) representation. In the MV model, A − a is the classical color field that obeys the classical Yang-Mills equation where ρ a is the corresponding color charge density inside the nucleus. Deriving A − a from Eq. (3) gives where G 0 is the two-dimensional massless propagator. The angle brackets in Eq. (1) represent the color field average of a given physical quantity f [A], where in the MV model, the average is a Gaussian weighted functional integral in form of where µ 2 (x + ) is the variance of the Gaussian distribution of color field, which represents the squared color charge per unit transverse area at coordinate x + . The µ 2 (x + ) integration over x + is proportional to the saturation momentum square Q 2 s . With the help of Wick's theorem, any correlator of ρ's or A's can be obtained by the most elementary correlators where By using the color transition matrices method developed in Refs. [39][40][41][42], the 2-dipole correlator can be given by where Lii is the so-called tadpole contribution corresponding to the configurations where each gluon link attaches to a single Wilson line. The M 2-dipole is the color transition matrix of the 2-dipole configuration, which is written in terms of color transition factors F 's and L's as where L ij,kl ≡ L ij + L kl , LF ij,kl ≡ C F L ij,kl + 1 2Nc F ijkl , µ 2 ≡ dx + µ 2 (x + ), and the color transition factor The matrix exponential e M 2-dipole in Eq. (10) is the difficult part to calculate, which has been derived in Ref. [41] by using the eigenvalue method. In this paper we calculate the nth power of M 2-dipole in Eq. (9), which gives us a platform to derive the 3-dipole correlator and conjecture a general n-dipole correlator in forms of 1 Nc power expansions. The vector 1 0 in Eq. (9), which represents the initial 2-dipole configuration, picks out the first column of the nth where [ n 2 ] ≡ Floor( n 2 ), namely the rounding down function. Summing over n with the factor 1 n! gives the matrix exponential Substituting Eq. (13) into Eq. (10) gives the 2-dipole correlator: where m 1,3 are of N 0 c order, and m 2,4 are of 1 Nc order. One can also write down the 3-dipole correlator by definition as where M 3-dipole is the corresponding color transition matrix given by . Substituting the block matrix in Eq. (16) into Eq. (15), one obtains the first column of the nth power of the M 3-dipole . In this computation, we only derive the result up to the 1 Summing over n with a factor 1 n! in Eq. (15), one obtains the 3-dipole correlator up to the 1 where each term inside the square brackets has a clear physical pattern, which helps to conjecture the n-dipole correlator in the following sections. For illustration, we focus on the last term which indicates that the original configuration (whose topology is represented by M 11 ) goes through two times of color transitions (represented by M 21 and M 32 ), then becomes another two configurations (whose topologies are represented by M 22 and M 33 ). This term can be graphicly represented by the color transition approach below and in Fig. 1: Actually, the eight terms inside Eq. (18) correspond to eight kinds of color transition approaches, and all of them can be depicted as shown above.
On the other hand, one can check that each term of the multidipole correlators corresponds to an individual approach of color transitions between configurations, and vice versa. It enables us to conjecture an expression for the n-dipole correlator similar to the 2n-point correlator [45], which reads where I k represents summing over all possible permutations a 1 , b 1 , c 1 , d 1 , a 2 , b 2 , c 2 , d 2 , ..., a k , b k , c k , d k , which satisfy the following conditions The recurrence relation between U i and U i+1 is with its first two terms

III. MULTI-PARTICLE AZIMUTHAL CUMULANTS
As the 3-dipole correlator involves such large numbers of color transitions, it is usually difficult to evaluate. As a first attempt, in this section, we calculate the 3-particle cumulant c 123 by using the 3-dipole correlator in Eq. (18) to show how it works. Generally, an m-particle correlation takes the form e i(n1φ1+n2φ2+...+nmφm) ≡ κ{n 1 , n 2 , ..., n m } κ{0, 0, ..., 0} , where n 1 , n 2 , ..., n m are integers which satisfy the azimuthal symmetry: n 1 +n 2 +...+n m = 0. φ i is the azimuthal angle of the ith outgoing particle's momentum p i . κ{n 1 , n 2 , ..., n m } is the mixed harmonic of the m-particle distribution, which is defined by is the m-particle inclusive spectra [26,28,31,32] given by One of the nontrivial 3-particle cumulants is c 123 , which is of the same definition with v 123 in Refs. [43,44] and defined by Within the framework above, the zeroth harmonic of the m-particle inclusive spectra in the MV model is derived by where we used Tr 1 (Nc×Nc) = 1.
where in order to make the expression shorter, each L and F factor absorbs a µ 2 2 , and the integral variables ξ in Eq. (18) are replaced by Using the functional forms [41] of the color transition factors F 's and L's as and replacing the coordinates by one reaches the final expression for order. We substitute this correlator into the definition of c 12 and c 123 in Eq. (29), and integrate over the variables p, b, r by the following five steps: (a) Integrate over φ (the azimuthal angular of p) by order in the MV model, which is plotted as a function of Q 2 s B p in Fig. 2. Some 3-particle charge-dependent azimuthal correlations have been measured at RHIC and the LHC [46,47] to search for the chiral magnetic effect. We see that our −c 123 (with a falloff at large Q 2 s ) has a similar magnitude and trend to the 3-particle correlations cos(φ α + φ β − 2φ c ) (whose magnitude gradually decreases as the event mutiplicity increases) observed in pPb collisions at the CMS [47], where Q s is related to the multiplicity (centrality) by growing (falling) with the multiplicity (centrality). In order to obtain a number of order unity, we scale the mixed cumulant c 123 by the corresponding 2-particle anisotropic flows v n {2} ≡ c n {2} [48], and plot the ratios as in Fig. 2, which also shows a similar magnitude and trend to the result in Ref. [44].

IV. THE Nc SCALING OF MULTIPARTICLE CORRELATIONS
For the 2m-particle harmonic, one usually defines for convenience.
Since the m-particle cumulant c n {m} demonstrates the correlations of these m particles, as a prior guest, its N c scaling should depend on the configurations where all dipoles (particles) connect or ever connected together as in As each gluon exchanging between two unconnected dipoles brings a 1 N 2 c , the cumulants should scale as [14] c n {m} ∼ 1 which is well understood graphicly, and this can also be proved by the matrix method. We consider the 1 and M (1,4), (3,2) = M (1,4) ⊕M (3,2) in the same way, where LF ab,cd;ef,gh ≡ LF ab,cd +LF ef,gh . Thus we see that M (1,2), (3,4) can be decomposed into the Kronecker sum of two individual 2-dipole color transition matrices M (1,2) and M (3,4) , and so can M (1,4), (3,2) . Then the first term of c n {4} gives