Musings

Liu, Rajagopol and Wiedemann looked at the jet-quenching parameter, $\hat{q}$, a measure of the energy-loss of high-$p_T$ partons as they move through the quark-gluon plasma. In the so-called dipole approximation, valid for small transverse distances, $L$, it is related to the expectation-value of a Wilson loop in the adjoint representation, $$\langle W_A(C)\rangle \sim e^{-\hat{q} L^- L^2/4}$$ where $C$ is a light-like rectangle, of extent $L^-$ in the $x^-$ direction and length $L$ in the transverse direction. Beyond the dipole approximation, they take this as the definition of $\hat{q}$: the coefficient of $L^- L^2/4$, for small $L$, in the expansion of $-\log\langle W_A(C)\rangle$.

For $\mathcal{N}=4$ SYM, in the large-$N$, large $\lambda= g^2N$ limit, this expectation value can be computed using AdS/CFT in an AdS₅ blackhole background and the large-$N$ relation, $\log \langle W_A(C)\rangle =2 \log \langle W_F(C)\rangle$. The result is $$\hat{q}_{\mathcal{N}=4}= \frac{\pi^{3/2}\Gamma(3/4)}{\sqrt{2}\Gamma(5/4)}\sqrt{\lambda} T^3$$

What’s measured in experiments is some time-averaged value of the jet-quenching parameter, as the plasma cools. Putting in the parameters relevant to RHIC ($N=3$, $\alpha_s\sim 1/2$), $\hat{q}_{\mathcal{N}=4}$ turns out rather too small compared to the experimentally-measured value.

Alex Buchel decided to look at the same calculation in the $\mathcal{N}=1$ supersymmetric cascading gauge theory dual to the Klebanov-Strassler background¹. He found $$\frac{\hat{q}_{\text{KS}}}{\hat{q}_{\mathcal{N}=4}}= 1+ c\frac{M^2}{N_{\text{eff}}(T)} + O\left(\frac{M^4}{N_{\text{eff}}^2(T)}\right)$$ where $$N_{\text{eff}}(E)\sim 2 M^2 \log(E/\Lambda),\quad\text{for}\, E\gg\lambda$$ and the constant, $c\simeq -1.388$. At least for $T\gg \Lambda$, the ratio increases with increasing temperature.

The speed of sound in the plasma has, for the cascading gauge theory, has a similar expansion in powers of $M^2/N_{\text{eff}}(T)$ $$v_s^2 = \frac{1}{3} + \frac{4}{9}\frac{M^2}{N_{\text{eff}}(T)} + O\left(\frac{M^4}{N_{\text{eff}}^2(T)}\right)$$ Buchel conjectures² the relation $$\frac{\hat{q}_{\text{KS}}}{\hat{q}_{\mathcal{N}=4}}=1 +\textstyle{\frac{9c}{4}}\left(\textstyle{\frac{1}{3}} -v_s^2\right)$$ and proposes to apply this to QCD, by plugging in the QCD sound speed in the regime relevant to RHIC.