More about the physics

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RHIC

The RHIC facility at Brookhaven National Laboratory includes four experiments: BRAHMS, PHENIX, PHOBOS, and STAR.

The QCD coupling

Siggi Bethke is one of the modern experts on the running coupling of QCD. here is a talk he gave on the subject; see in particular the figure on this page, reproduced here, showing αs as a function of scale. An often-used value in RHIC physics is αs = 1 / 2.

Ian Hinchliffe has a summary page on αs, including world-average values at the Z peak and a Java script that tells you αs(Q) for any Q you enter.

Black hole temperature

A calculation which comes up often in black hole physics is the Hawking temperature. When the black hole metric has the form

ds^2 = g_{tt} dt^2 + g_{rr} dr^2 + \ldots \,,

where the \ldots represents some additional terms that are static and radially symmetric, this is quite an easy computation. One first rotates to Euclidean time, upon which

ds_E^2 = -g_{tt} dt_E^2 + g_{rr} dr^2 + \ldots \,,

and then demands that the horizon, where gtt and grr have simple zeroes (simple, at least, in the case where the temperature is non-zero), should become a non-singular point in the Euclidean geometry. The \ldots part of the calculation doesn't affect the calculation because it's supposed to remain non-singular at the horizon. Assuming the horizon is at r = 0, and expanding gtt and grr to first order in r, one finds

ds_E^2 = -{\partial g_{tt} \over \partial r} r dt_E^2 + {dr^2 \over {\partial g^{rr} \over \partial r} r} = dy^2 + y^2 d\theta^2 \,,

where I have introduced near-horizon coordinates

y = 2 \sqrt{r \over \partial g^{rr}/\partial r} \qquad\qquad \theta = {t \over 2} \sqrt{-{\partial g_{tt} \over \partial r} {\partial g^{rr} \over \partial r}}

Imposing that θ should be periodic with period 2π, and that tE should have period β = 1 / T, one immediately finds

T = {1 \over 4\pi} \sqrt{ -{\partial g_{tt} \over \partial r} {\partial g^{rr} \over \partial r}} \,.

For example, if − gtt = grr = 1 − 2M / r, as for the four-dimensional Schwarzschild black hole, then

T = {1 \over 8\pi M} \,.

It's also nice to know that T = κ / 2π, where κ is the surface gravity at the horizon. This allows for a calculation of T that doesn't involve Wick rotation, but it's usually not the fastest method. Finally, it's important to recall that T is computed with respect to a particular asymptotic Killing time: in the current context that time of course is t, and its normalization determines the normalization of T.


Conformal Fourier integrals

A useful result in dealing with conformal correlators is (in Euclidean signature)

\int d^4 x {e^{i p \cdot x} \over x^{2\Delta}} = {4^{2-\Delta} \pi^2 \Gamma(2-\Delta) \over \Gamma(\Delta)} p^{2\Delta-4} + \ldots

Here \ldots represent terms associated with regularization. For a limited range of Δ, the integral converges, and these terms aren't there. A typical way to deal with them is to say that 1 / x2Δ has some distribution added to it which is supported at x = 0; correspondingly, p2Δ − 4 has some terms analytic in p2 added to it. In short, these are contact terms.

A Bessel function identity

K_\nu(z) = {\pi \over 2} {I_{-\nu}(z) - I_\nu(z) \over \sin\pi\nu} \,.

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