How to Measure Specific Heat Using Event-by-Event Average $p_T$ Fluctuations

A simple way to visualize event-by-event average $p_T$ fluctuations is by assuming that each collision has a different temperature parameter (inverse $p_T$ slope) and that the ensemble of events has a temperature distribution about the mean, $<T>$, with standard deviation $\sigma_T$. PHENIX characterizes the non-random fluctuation of $M_{p_T}$, the event-by-event average $p_T$, by $F_{p_T}$, the fractional difference of the standard deviation of the data from that of a random sample obtained with mixed events. This can be related to the temperature fluctuation: $$F_{p_T}=\sigma^{\rm data}_{M_{p_T}}/\sigma^{\rm random}_{M_{p_T}}-1\simeq(< n > -1) \sigma^2_{T}/< T>^2$$ Combining this with the Gavai, {\it et al.},\cite{Gavai05} and Korus, {\it et al.},\cite{Korus} definitions of the specific heat per particle, a simple relationship is obtained: $$c_v/T^3={\mean{n}\over \mean{N_{tot}}} {1\over F_{p_T}}$$ $F_{p_T}$ is measured with a fraction $\mean{n}/\mean{N_{tot}}$ of the total particles produced, a purely geometrical factor representing the fractional acceptance, $\sim 1/33$ in PHENIX. Gavai, {\it et al.} predict that $c_v/T^3=15$, which corresponds to $F_{p_T}\sim 0.20$% in PHENIX, which may be accessible by measurements of $M_{p_T}$ in the range $0.2\leq p_T\leq 0.6$ GeV/c. In order to test the Gavai, {\it et al.} prediction that $c_v/T^3$ is reduced in a QGP compared to the ideal gas value (15 compared to 21), precision measurements of $F_{p_T}$ in the range 0.20% for $0.2\leq p_T\leq 0.6$ GeV/c may be practical.