Dilepton helical production in a vortical quark-gluon plasma

Abstract

In this paper, we propose an observable counting a weighted difference between right-handed and left-handed lepton pairs, which is coined dilepton helical rate. The weight is the momentum difference of the lepton pairs projected onto an auxiliary vector. We derive the helical rate in a quark-gluon plasma with a vorticity in the limit when the quark and lepton masses are ignored. We find the helical rate is maximized when the auxiliary vector is parallel to the vorticity, in which case it has a nearly spherical oblate ellipsoidal distribution. We propose that it can be used as a vortical-meter for quark-gluon plasma.

Limits of \(\mathcal {C}_n\) and \({\chi }_n\) at small q

We are interested in the behavior of $\mathcal {C}_{n}$ in the limit where q tends to zero. We start with the following small q expansions of ${\chi }_n$, which are easily obtained from (16)

$$\begin{aligned} \chi _{2}&=-\frac{q_{0}^{2}e^{q_{0}/2T}}{4T(1+e^{q_{0}/2T})^{3}}\nonumber \\&\quad -\frac{ q^{2}e^{q_{0}/2T}(8+8e^{q_{0}/T}-\frac{8q_{0}}{T}+\frac{q_{0}^{2}}{T^{2}}+e^{q_{0}/2T}(16-\frac{8q_{0}}{T}-\frac{3q_{0}^{2}}{T^{2}}))}{96T(1+e^{q_{0}/2T})^{5}}\nonumber \\&\quad +\mathcal {O}[q]^{4},\nonumber \\ \chi _{1}&=-\frac{q_{0}e^{q_{0}/2T}}{2T(1+e^{q_{0}/2T})^{3}} +\frac{ q^{2}e^{q_{0}/2T}(4-\frac{q_{0}}{T}+e^{q_{0}/2T}(4+\frac{3q_{0}}{T}))}{48T^{2}(1+e^{q_{0}/2T})^{5}}\nonumber \\&\quad +\mathcal {O}[q]^{4},\nonumber \\ \chi _{0}&=-\frac{e^{q_{0}/2T}}{T(1+e^{q_{0}/2T})^{3}} +\frac{ q^{2}e^{q_{0}/2T}(-1+3e^{q_{0}/2T}))}{24T^{3}(1+e^{q_{0}/2T})^{5}} +\mathcal {O}[q]^{4}.\nonumber \\ \end{aligned}$$

(53)

these results in (53) clearly display that all the terms in the expansion are finite and even in q. The finiteness of the expansion is a consequence of cancellation between divergent integrand and vanishing integration domain. We only need to consider the contribution given by the leading order and the next leading order terms of $\chi _{n}$. Plugging (53) into (15), we find the following limits for $\mathcal {C}_n$

$$\begin{aligned}&\mathcal {C}_{1}(q\rightarrow {0})=0,\ \ \ \ \mathcal {C}_{2}(q\rightarrow {0})=\frac{1}{3}\frac{q_{0}^{2}e^{q_{0}/2T}}{T(1+e^{q_{0}/2T})^3}, \nonumber \\&\quad \mathcal {C}_{3}(q\rightarrow {0})=0. \end{aligned}$$

(54)

Plugging (54) into (51), we can obtain

$$\begin{aligned} R_{1}=\frac{q_{0}^{3}e^{q_{0}/2T}}{3T(1+e^{q_{0}/2T})^{3}},\ \ \ R_{2}=0. \end{aligned}$$

(55)

which have been used in Fig. 1 at $q/{\pi }T =0$.