Finite size effects on critical correlations in momentum space

Abstract

The search for the QCD critical end point (CEP) is a major objective of contemporary heavy-ion physics, motivating the study of fluctuation observables that are sensitive to critical dynamics. In particular, baryon-number fluctuations provide a natural probe because the net-baryon density can serve as an effective order parameter in the vicinity of the CEP. Near criticality, long-range correlations and power-law scaling are expected to emerge in the real-space two-point function of the baryon density, yet the finite size and finite lifetime of the fireball created in heavy-ion collisions impose intrinsic cutoffs that regulate the growth of the correlation length. These finite-size constraints significantly modify the observable structure of fluctuations, especially in momentum space, where experiments perform measurements. In this work we present a theoretical analysis of the momentum-space two-point correlation function for a system of finite spatial extent. We show that finite size effects lead to an effective scaling exponent which coincides with that of the infinitely extended system only in a prescribed scaling region within the experimentally accessible momentum range.

Figures (4)

• Figure 1: Plots for fixed value of $\delta = 1 ~ \rm fm$ and $a=1/3$. (a) The momentum space correlations $\mathcal{F}_{\delta}^{d=2}$ as a function of the measure of the two dimensional momentum $k$. The blue dashed line is the infinite domain power law. While the vertical dashed lines are the boundaries $k_{\rm left}$, $k_{\rm right}$. (b) The exponent of $\gamma_{\rm eff}$ for different values of $k$ as derived from $\mathcal{F}_{\delta}^{d=2}$. The blue dashed line is the representative value of the exponent (average) while the black dashed line is the asymptote. Again the vertical lines are the boundaries.
• Figure 2: Contour plots for varying values of $\delta$ and $a=1/3$. (a) The momentum space correlations $\mathcal{F}_{\delta}^{d=2}$ as a function of the measure of the two dimensional momentum $k$ and $\delta$. The dashed lines are the boundaries $k_{\rm left}$, $k_{\rm right}$. (b) The exponent $\gamma_{\rm eff}$ for different values of $k$ and $\delta$ as derived from $\mathcal{F}_{\delta}^{d=2}$. Again the dashed lines are the boundaries.
• Figure 3: Plots for fixed value of $r = 10 ~ \rm fm$, $r_0= 0.3 ~ \rm fm$ and $a=1/3$. (a) The momentum space correlations $\mathcal{F}_{r,r_0}^{d=2}$ as a function of the measure of the two dimensional momentum $k$. The blue dotted line represents the infinite domain power law behavior while the dotted vertical red and green lines the left and right boundaries respectively. The magenta dotted line represents the value of $k$ for which the $-(2-a)$ value for the exponent is attained. (b) The exponent of $\gamma_{\rm eff}$ for different values of $k$ as derived from $\mathcal{F}_{r,r_0}^{d=2}$. The dotted vertical red and green lines the left and right boundaries respectively. The magenta dotted line represents the value of $k$ for which the $-(2-a)$ value for the exponent is attained.
• Figure 4: Contour plots for varying values of $r$ and $a=1/3$. (a) The momentum space correlations $\mathcal{F}_{r,r_0}^{d=2}$ as a function of the measure of the two dimensional momentum $k$ and $r$. The red and cyan dotted lines represent the left and right boundaries respectively while the magenta one represents the value of $k$ for which the $-(2-a)$ value for the exponent is attained. (b) The exponent $\gamma_{\rm eff}$ for different values of $k$ and $r$ as derived from $\mathcal{F}_{r,r_0}^{d=2}$. The red and cyan dotted lines represent the left and right boundaries respectively while the magenta one represents the value of $k$ for which the $-(2-a)$ value for the exponent is attained.