# WHDG

## Radiative, Elastic, and Convolved Rad + El Energy Loss

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.

and then take

## Brick Problems

### Original Brick

**Q = up, E = 10 GeV, L = 5 fm, T = 300 MeV, α = .3****Q = up, E = 10 GeV, L = 2 fm, T = 300 MeV, α = .3****Q = up, E = 10 GeV, L = 5 fm, T = 200 MeV, α = .3**

### Wiedemann Brick

**Q = up, E = 10 GeV, L = 2 fm, T = 150 MeV, α = .3, <ε> = .05****Q = up, E = 10 GeV, L = 2 fm, T = 215 MeV, α = .3, <ε> = .1****Q = up, E = 10 GeV, L = 2 fm, T = 315 MeV, α = .3, <ε> = .2****Q = up, E = 10 GeV, L = 2 fm, T = 485 MeV, α = .3, <ε> = .4**

### Temperature Scans

**Q = up, E = 10 GeV, L = 2 fm****Q = up, E = 10 GeV, L = 5 fm****Q = up, E = 100 GeV, L = 2 fm****Q = up, E = 100 GeV, L = 5 fm**

## Additional Details

### Radiative Calculation

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 , where , we use the number distribution of emitted gluons, , as the kernel for our Poisson convolution (see Eq. 4 of [1]). The probability of losing no energy through radiation is given by ; the probability of radiating is integrated over and becomes the coefficient of a

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 . We take the upper limit of the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{k_T}}**
integral as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{\surd [4E^2\min(x^2,(1-x)^2)-\mu^2]}}**
. For us the upper limit of the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{q_T}}**
integral in Eq. 12 of [2] is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{q_{\max}=\sqrt{3\mu E}}}**
. Again other reasonable cutoffs exist, such as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{\mu/2}}**
(this gives the same factor of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{\rho(z)=\Theta(L-z)}}**
, but rather exponential, ; preserves **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{\langle z\rangle=L/2}}**
for first order in opacity calculations. This greatly simplifies the numerical problem by converting oscillatory to **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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: **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{E<2m^2/T}}**
, is given by Eq. 8 of [5], with **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{B(v)=.604}}**
assumed constant; for larger energies **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{q_\max}}**
erroneously had the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{L/2}}**
Bjorken-expanding medium approximation hard-coded in; this results in a slight difference in the inelastic single inclusive **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{dN_g/dx}}**
distribution (**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle{<x>}}**
changes by ~0.5%).

**Q = up, E = 10 GeV, L = 5 fm, T = 300 MeV, α = .3****Q = up, E = 10 GeV, L = 2 fm, T = 300 MeV, α = .3****Q = up, E = 10 GeV, L = 5 fm, T = 200 MeV, α = .3**

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