### QGP on the Lattice

I was chatting with John Harris, the former spokesman for the STAR Collaboration, a few weeks ago, when he mentioned to me some recent lattice calculations of $\eta/s$, the ratio of shear viscosity to entropy density, in pure $SU(3)$ Yang-Mills.

This sounded really interesting.

The claim-to-fame of AdS/QGP is that it allows you to compute transport coefficients in a strongly-coupled gauge theory, something that lattice techniques are supposed to be no good at.

Most famous of all is the calculation of $\eta/s$, which is exactly $\tfrac{1}{4\pi}$ in conformally-invariant AdS/CFT backgrounds, and $\tfrac{1}{4\pi}\left(1+O(1/\lambda)\right)$ in non-conformally-invariant backgrounds (with the leading $1/\lambda$ correction being positive). The quark gluon plasma studied at RHIC is the least viscous fluid known to man. STAR reports $\eta/s \lesssim 0.1$, more than 100 times smaller than that of water.

It would be pretty cool if lattice calculations could reproduce this. So what’s the trick?

I finally got around to looking at the papers by Harvey Meyer and Sakai and Nakamura, and I’m a little underwhelmed. They *do* report very low values for $\eta/s$ from their measurements. But extracting $\eta$ from the data requires some rather dubious assumptions.

The basic point is that there *are* Kubo formulas which relate transport coefficients to quantities calculable in equilibrium Statistical Mechanics (which is what the lattice *is* good at).

For any gauge-invariant (bosonic) Hermitian local operator, $O$, define the spectral function $\rho_O(\omega,\vec{p}) = \int d^4 x\, e^{i(\omega t -\vec{p}\cdot \vec{x})} {\langle [O(t,\vec{x}), O(0)]\rangle}_{th}$ where ${\langle \dots\rangle}_{th}= \tfrac{1}{Z(T)}Tr\left(e^{-\beta H}\dots \right)$ is the real-time thermal correlation function. The spectral function obeys

A Kubo formula relates the shear viscosity, $\eta$, to the $\rho\to 0$ limit of the spectral function for a component of the traceless part of the stress tensor^{1}

There’s nothing special about $T_{12}$; Meyer actually uses $\tfrac{1}{2}(T_{11}-T_{22})$.

The same spectral function, in turn, is related to the *Euclidean* 2-point function

for $0\leq \tau \lt \beta$ by

via the kernel^{2}
$K(\tau,\omega) = \frac{\cosh(\omega(\tau-\beta/2))}{\sinh(\beta\omega/2)}$

So the program sounds simple:

- Measure the Euclidean 2-point $G(\tau)\equiv \int d^3 x G^E_{T_{12}}(\tau,\vec{x})$, on the lattice.
- Use it to reconstruct the spectral function (4).
- Study the $\omega\to 0$ limit, (2), to recover the viscosity.

Unfortunately, on the lattice, we’re faced with the task of reconstructing a continuous function, $\rho(\omega)$, from a (small number of) discrete measurements of the lattice 2-point function. The lattices used by Meyer and by Sakai and Nakamura have 8 lattice points in the Euclidean time direction. Because of the symmetry $G(\tau)= G(\beta-\tau)$, this means there are only 4 independent data points.

One way to proceed is to assume some functional form for the spectral function, write down a *family* of trial functions with a few free parameters, and try to fit the data by adjusting those parameters.

The large-$\omega$ behaviour of the spectral function may well be captured by perturbation theory (and so we know the functional form). But we’re interested in the small-$\omega$ behaviour and it’s not at all clear what functional form one should assume there.

Sakai and Nakamura fit their data to a Breit-Wigner ansatz $\rho(\omega)\equiv \tfrac{1}{2\pi}\rho_{T_{1 2}}(\omega,0) = \frac{A}{\pi} \left[\frac{\gamma}{(m-\omega)^2 +\gamma^2}-\frac{\gamma}{(m+\omega)^2 +\gamma^2}\right]$ Meyer studies a variety of functional forms (and points out that Sakai and Nakamura’s Breit-Wigner ansatz doesn’t agree with perturbation theory at large-$\omega$).

There is a very fancy technique, called the Maximum Entropy Method for constructing a best-fit for the spectral function from a finite set of lattice measurements, using Bayesian Statistics (see Georg von Hippel’s post). But it’s not what these guys use (apparently, it requires a *much* finer lattice in the Euclidean time direction).

And I really wonder whether even MEM techniques are well-adapted to distinguishing the behaviour of the spectral function near zero.

To see the nature of the difficulty, consider the following very crude method for obtaining an upper bound on $\eta$, by integrating both sides of (4)
$\int_0^{\beta/2} G(\tau) d\tau = \int_0^\infty \frac{\rho(\omega)}{\omega} d\omega$
The LHS involves only the crudest information, namely the sum of our measurements. If you assume you know $\rho(\omega)$ for $\omega\gt \omega_c$ (say, because you expect it to be given accurately by perturbation theory), then you have an upper bound on the *integrated* value of $\frac{\rho(\omega)}{\omega}$ between $0$ and $\omega_c$. Because of positivity (1), this tells you something about the behaviour near zero, but it doesn’t tell you much. You can’t exclude a narrow spike near $\omega=0$, which make a negligible contribution to the integral, but a large correction to $\eta$. Including more independent data points makes the problem better, but doesn’t make it go away.

^{1} The AdS/CFT computation also proceeds via Kubo’s formula. But there, the two-point function of the stress tensor is related to the cross section for absorbing a graviton in the dual gravitational description.

^{2} To relate this form to the more familiar expression involving a sum over Matsubara frequencies, $\omega_n = 2\pi n/\beta$, one uses the identity
$\sum_n \frac{e^{-i\omega_n\tau}}{s - i \omega_n} = \beta \frac{e^{-s\tau}}{1-e^{-\beta s}}$

## Re: QGP on the Lattice

The small euclidean-time extent of many finite-temperature lattices is a well-known problem. Anisotropic lattices with a much smaller lattice spacing in the time direction help, but of course there’s still the question of cost.

Note that what Meyer uses at the end of his paper sounds very much like a version of MEM, even though his description does not mention it.

As for the behaviour of the spectral function near zero frequency, at least in a MEM approach one should presumably use a default model that excludes such a peak. Of course, if the true spectral function should have such a peak (which would not appear to be terribly likely, though), the GIGO (garbage in, garbage out) principle hits, but at least one would obtain a low Bayesian probability for the model in this case (assuming the lattice data would allow to get any information on the peak; if it is beyond reach, it is beyond reach).

More on MEM for transport properties (conductivity in this case) can be found in hep-lat/0703008; more on problems with MEM at small temporal lattice extent in hep-lat/0510026.