# WHDG

Based on:
Elastic, inelastic, and path length fluctuations in jet tomography
Simon Wicks, William Horowitz, Magdalena Djordjevic, Miklos Gyulassy. Dec 2005. 7pp.
Published in Nucl.Phys.A784:426-442,2007.
e-Print: nucl-th/0512076

Once parton flavor, E, L, and α are fixed the number density of medium scattering centers is the single free parameter of the WHDG model. For the purpose of relating temperature to density, we assume a thermalized medium of ultrarelativistic massless gluons; i.e.

${\displaystyle \textstyle {\rho ={\frac {16\zeta (3)}{\pi ^{2}}}T^{3},}}$

and then take

${\displaystyle \textstyle {\mu =gT,}}$

${\displaystyle \textstyle {\sigma ={\frac {9\pi }{2}}{\frac {\alpha ^{2}}{\mu ^{2}}},}}$

${\displaystyle \textstyle {\lambda ={\frac {1}{\sigma \rho }}.}}$

## Brick Problems

### Temperature Scans

The results presented above come from code altered as slightly as possible from the code used to produce the curves presented in WHDG; the only change made was to make the medium density static instead of Bjorken expanding. All radiative distributions are to first order in opacity only.

To determine the probability of fractional energy loss ${\displaystyle \scriptstyle {P(\epsilon )}}$, where ${\displaystyle \scriptstyle {p_{f}=(1-\epsilon )p_{i}}}$, we use the number distribution of emitted gluons, ${\displaystyle \scriptstyle {dN_{g}/dx}}$, as the kernel for our Poisson convolution (see Eq. 4 of [1]). The probability of losing no energy through radiation is given by ${\displaystyle \scriptstyle {e^{-}}}$; the probability of radiating ${\displaystyle \scriptstyle {\epsilon >1}}$ is integrated over and becomes the coefficient of a ${\displaystyle \scriptstyle {\delta (1-\epsilon )}.}$

The DGLV [2] generalization of GLV [3], which allows for massive quark jets and massive gluon radiation, is used to find the number distribution of radiated gluons. Specifically, we employ Eq. 12 of [2] to find ${\displaystyle \scriptstyle {dN_{g}/dx}}$. We take the upper limit of the $\displaystyle \scriptstyle{k_T}$ integral as $\displaystyle \scriptstyle{k_{\max}=2x(1-x)E}$ . There are other reasonable values for such a cutoff; e.g. in [1], this is taken as $\displaystyle \scriptstyle{\surd [4E^2\min(x^2,(1-x)^2)-\mu^2]}$ . For us the upper limit of the $\displaystyle \scriptstyle{q_T}$ integral in Eq. 12 of [2] is $\displaystyle \scriptstyle{q_{\max}=\sqrt{3\mu E}}$ . Again other reasonable cutoffs exist, such as $\displaystyle \scriptstyle{q_\max = \sqrt{6 E T}}$ . However the final distributions are not particularly sensitive to the details of these cutoffs. Gluons — both radiated and as jets — are given a mass $\displaystyle \scriptstyle{m_g=m_{\infty}=\mu/\sqrt{2}}$ , mimicking the behavior of the full 1-loop HTL dispersion relation [4]. Light quarks are given a mass of $\displaystyle \scriptstyle{\mu/2}$ (this gives the same factor of $\displaystyle \scriptstyle{1/\sqrt{2}}$ difference between glue and quark masses at low-pT).

Note that the derivation of Eq. 12 of [2] assumes the probability distribution of scattering centers is not uniform, $\displaystyle \scriptstyle{\rho(z)=\Theta(L-z)}$ , but rather exponential, ${\displaystyle \scriptstyle {\rho (z)=\exp(z/L_{e})/L_{e}}}$; ${\displaystyle \scriptstyle {L_{e}=L/2}}$ preserves $\displaystyle \scriptstyle{\langle z\rangle=L/2}$ for first order in opacity calculations. This greatly simplifies the numerical problem by converting ${\displaystyle \scriptstyle {\int dz\rho (z)1-\cos(\omega z)\sim }}$oscillatory to $\displaystyle \scriptstyle{1/(1+\omega^2L^2)}$ .

### Elastic Calculation

We take the elastic probability of fractional energy loss to be Gaussian with width given by the Fluctuation-Dissipation theorem: $\displaystyle \scriptstyle{\sigma = (2/p)\int dp/dz T(z) dz}$ . For gluon and light quark jets the mean energy loss comes from [5]. Specifically, for $\displaystyle \scriptstyle{E<2m^2/T}$ , ${\displaystyle \scriptstyle {dp/dz}}$ is given by Eq. 8 of [5], with $\displaystyle \scriptstyle{B(v)=.604}$ assumed constant; for larger energies $\displaystyle \scriptstyle{dp/dz}$ is given by Eq. 12 of [5].

## References

[1] Jet Tomography of Au+Au Reactions Including Multi-gluon Fluctuations
M. Gyulassy (Columbia U.) , P. Levai (Budapest, RMKI) , I. Vitev (Columbia U.) . Dec 2001. 6pp.
Published in Phys.Lett.B538:282-288,2002.
e-Print: nucl-th/0112071

[2] Heavy quark radiative energy loss in QCD matter.
Magdalena Djordjevic, Miklos Gyulassy (Columbia U.) . Oct 2003. 31pp.
Published in Nucl.Phys.A733:265-298,2004.
e-Print: nucl-th/0310076

[3] Reaction operator approach to nonAbelian energy loss.
M. Gyulassy (Columbia U.) , P. Levai (Columbia U. & Budapest, RMKI) , I. Vitev (Columbia U.) . CU-TP-979, Jun 2000. 39pp.
Published in Nucl.Phys.B594:371-419,2001.
e-Print: nucl-th/0006010

[4] The Ter-Mikayelian effect on QCD radiative energy loss.
Magdalena Djordjevic, Miklos Gyulassy (Columbia U.) . May 2003. 18pp.
Published in Phys.Rev.C68:034914,2003.
e-Print: nucl-th/0305062

[5] Energy loss of a heavy quark in the quark - gluon plasma.
Eric Braaten (Northwestern U.) , Markus H. Thoma (LBL, Berkeley) . LBL-30998, NUHEP-TH-91-14, Jul 1991. 14pp.
Published in Phys.Rev.D44:2625-2630,1991.

## Legacy Plots

Below are the plots originally posted. A bug was found in which the implemented $\displaystyle \scriptstyle{q_\max}$ erroneously had the $\displaystyle \scriptstyle{L/2}$ Bjorken-expanding medium approximation hard-coded in; this results in a slight difference in the inelastic single inclusive $\displaystyle \scriptstyle{dN_g/dx}$ distribution ($\displaystyle \scriptstyle{}$ changes by ~0.5%).

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