Hot spots and gluon field fluctuations as causes of eccentricity in small systems

We calculate eccentricities in high energy proton-nucleus collisions, by calculating correlation functions of the energy density field of the glasma immediately after the collision event at proper time $\tau =0^{+}$. We separately consider the effects of color charge and geometrical hot spot fluctuations, analytically performing the averages over both in a dilute-dense limit. We show that geometric fluctuations of hot spots inside the proton are the dominant source of eccentricity whereas color charge fluctuations only give a negligible correction. The size and number of hot spots are the most important parameters characterizing the eccentricities.

Figure 1

Different contributions to the local energy density fluctuations $⟨ϵ(\mathbf{x}{)}^2⟩$, as a function of the position $\mathbf{x}$ relative to the center of mass of the proton ($\mathbf{B}=0$). The bands correspond to varying the UV cutoff $C_0$ by $\pm 50\%$. The bottom plots show the relative contribution of each individual contribution. (a) $N_q=1$ (b) $N_q=3$ (c) $N_q=10$ (d) $N_q=100$.

Figure 2

Different contributions to the energy density correlation functions $⟨ϵ(\mathbf{x})ϵ(\mathbf{y})⟩$, as a function of the coordinate separation $|\mathbf{x}-\mathbf{y}|$. The coordinates $\mathbf{x}=-\mathbf{y}$ are taken to be opposite to each other on a straight line through the center of the proton $(\mathbf{B}=0)$. The different panels show the results for different number of hot spots $N_q=1$, 3, 10, 100. The bands correspond to variations of the UV cutoff $C_0$ by $\pm 50\%$. The bottom plots show the relative contribution of each individual contribution. (a) $N_q=1$ (b) $N_q=3$ (c) $N_q=10$ (d) $N_q=100$.

Figure 3

Different contributions to the local energy density fluctuations $⟨ϵ(\mathbf{x}{)}^2⟩$ (left) and the energy density correlation functions $⟨ϵ(\mathbf{x})ϵ(\mathbf{y})⟩$ (right), for the same configurations of the coordinates $\mathbf{x}$, $\mathbf{y}$ as in Figs. 1 and 2. The bands are obtained by varying the IR regulator mass by $\pm 50\%$, and the results are on a logarithmic scale to illustrate the sizable variation of the long range tails by the variation of the IR regulator mass. (a) $N_q=3$ (b) $N_q=3$.

Figure 4

Eccentricities ${ϵ}_n\{2\}$ with $n=2$, 3, 4 for a proton with three hot spots ($N_q=3$). The eccentricities originating from proton and nucleus fluctuations are shown separately along with the total eccentricity. The bands are obtained by varying the IR regulator mass by $\pm 50\%$.

Figure 5

Eccentricities ${ϵ}_n\{2\}$ as functions of the number of hot spots $N_q$ for $n=2$, 3, 4 (top, middle, bottom). Confidence bands are obtained by varying the IR regulator $m$ by $\pm 50\%$. Numerical results from the our hot spot model, are also compared to the eccentricities in a pointlike energy density model discussed in Appendix pp3. (a) $n=2$ (b) $n=3$ (c) $n=4$.

Figure 6

Elliptic eccentricity ${ϵ}_2\{2\}$ for a proton with $N_q=3$ hot spots, as a function of the hot spot radius $r$, keeping the proton size $R$ fixed. Confidence bands are obtained by varying the IR regulator mass by $\pm 50\%$.

Figure 7

Eccentricities ${ϵ}_n\{2\}$ as functions of the number of hot spots $N_q$ for $n=2$, 3, 4. The bands are obtained by varying the UV cutoff $C_0$ by $\pm 50\%$. (a) $n=2$ (b) $n=3$ (c) $n=4$.

Figure 8

Breakup of different contributions to the proton fluctuation contribution to the two-point function of the energy density $⟨ϵ(\mathbf{x})ϵ(\mathbf{y})⟩$.

Figure 9

Comparison of two different definitions of the eccentricity ${ϵ}_n\{2\}$ and ${ϵ}_n^{\prime }\{2{\}}_{\text{Approx}}$ corresponding to normalizing the eccentricity by energy density two-point function or the square of the one-point function. Numerical results are obtained for $N_q=3$, and error bars indicate the sensitivity to variations of the IR regulator $m$ by $\pm 50\%$.