Exploring correlations in the CGC wave function: Odd azimuthal anisotropy

We extend the color glass condensate (CGC) approach to a calculation of the double inclusive gluon production by including the high density effect in the CGC wave function of the projectile (proton). Our main result is that these effects lead to the appearance of odd harmonics in the two particle correlation $C(k,p)$. We find that in the high momentum limit, $|k|,|p|\gg Q_s$, this results in a positive $c_1\{2\}$. Additionally when the magnitudes of the two momenta are approximately equal, $|k|/|p|\approx 1$, the density effects also generate a positive third harmonic $c_3\{2\}$, which translates into a nonvanishing $v_3$ when the momenta of the trigger and an associated particle are in the same momentum bin. The sign of $c_3\{2\}$ becomes negative when $|k|/|p|>1.1$ suggesting an interesting experimental signature.

Figure 1

Schematic representation of the types of diagrams that contribute to the double inclusive production at order $g^6{\rho }^4$. The horizontal lines symbolize the valence gluons constituting the valence color charge density $\rho$. The vertical line denotes the final state. Momenta of two final state gluons are fixed. In the case of the three gluon final state, one of the gluons is summed over inclusively. The blobs symbolize the scatterings on the target. The final state gluons can be emitted either before or after these scatterings, and all the combinations have to be accounted for.

Figure 2

The correlation function as a function of the azimuthal angle, $\varphi$ for different values of $z=p/k$. The correlation functions are defined in the text and normalized by $C(z=1,\varphi =0)$. (Left panel) The correlation function in the projectile wave function. (Right panel) The correlation function in particle productions.

Figure 3

The first and third cumulants as a function of $z=p/k$ in, the projectile wave function (WF) and double inclusive particle production (PP).

Figure 4

The tree level diagrams representing the action of the coherent operator $\mathcal{C}$ on the soft gluon vacuum.

Figure 5

The diagrams representing the action of the Bogoliubov operator on the soft gluon vacuum.