# ASW

## Overview

The 'ASW-formalism' calculates parton energy loss based on a path-integral formalism [1-3]. The path-integral can be evaluated in two different approximations:

1. Multiple soft scattering limit: Technically, this is a saddle-point approximation of the path integral. For the case of infinite in-medium pathlength, the result coincides with the BDMPS expression for parton energy loss [2]. For this reason, we refer to this limit sometimes as "BDMPS-limit".
2. Opacity expansion: Technically, this is an expansion of the integrand of the path integral in powers of (density times pathlength). The GLV ${\displaystyle N=1}$ opacity result reproduces our expression [1] on the level of the Feynman diagrams and the analytic expression for the ${\displaystyle \omega }$- and ${\displaystyle k_{T}}$-differential gluon energy distribution.

## Results for the brick problem

### Probability distribution of energy loss

The following figures show results from the ASW multiple soft scattering limit. (All results have been computed with ${\displaystyle \alpha _{s}=0.3}$.) First, we plot the probability ${\displaystyle P(\Delta E/E)}$ that a light quark looses a fraction of its energy. The information about ${\displaystyle P(\Delta E/E)}$ is contained in three pieces:

1. "Untouched survival": This is the discrete probability that a parton does not interact with a medium of length ${\displaystyle L}$ and that it looses no energy. This probability is represented by a color-coded dot at ${\displaystyle \Delta E/E=-0.05}$.
2. "Survival with finite energy loss": Continuous probability: this is the probability that a parton makes it through a medium of length ${\displaystyle L}$ but looses a finite fraction ${\displaystyle \Delta E/E}$ during its passage. This is denoted by the color-coded curve at finite $\displaystyle 0 \le \Delta E/E \le 1$ .
3. "Death before arrival": In general, if one shoots a particle into a wall of thickness $L$, it can get stopped on its journey before reaching the length $\displaystyle L$ . This probability is denoted by the color-coded dot at ${\displaystyle \Delta E/E=1.05}$.

In general, as one increases the average energy loss (i.e. as one increases $\displaystyle \hat{q}$ ),

• the probability of untouched survival decreases,
• the probability of survival with finite energy loss shifts to larger values of $\Delta E/E$,
• the probability of death before arrival increases.

This is seen clearly in all the plots shown for ${\displaystyle P(\Delta E/E)}$. The most extreme curve is that for $\displaystyle E= 10$ GeV and ${\displaystyle L=2}$ fm. Requiring an energy loss of $\displaystyle \Delta E = 4$ GeV in this case amounts to > 40 percent probability of untouched survival but 30 percent probability of death before arrival. This comes close to an all-or-nothing scenario, where a particle either goes through without medium-modification or gets stuck, but emerges with relatively small probability as an object with reduced energy.

### Energy spectra of radiated gluons

We now turn to the corresponding spectra: The multiple soft scattering limit suppresses the production of infrared gluons by a destructive interference effect. As a consequence, all spectra are peaked at finite gluon energies. In general, the radiated gluons become harder as one increases the average energy loss (i.e. as one increases $\displaystyle \hat{q}$ ). If the projectile energy is sufficiently large and the in-medium pathlength is sufficiently small, then the radiated gluons carry small fractions of the projectile energy. This is the case for a projectile quark energy $\displaystyle E=100$ GeV.

However, if the projectile energy is too small, one faces a particular problem: One calculates the radiated gluon spectrum as if the parton would propagate through a medium of path length $\displaystyle L$ , though with finite probability the parton does not have sufficient initial energy to make it through $\displaystyle L$ , and gets stuck before. In the present calculational framework, finding yield in the spectrum for $\displaystyle \omega > E$ , i.e., above the kinematical boundary, signals that one has assumed that the particle propagates through a length $\displaystyle L$ though its probability of "death before arrival" is finite.

We note that even if $\displaystyle dN_g/dx$ does not have any yield for $\displaystyle \omega > E$ , there can be a finite probability of death before arrival, since $\displaystyle P(\Delta E/E)$ contains information about multiple gluon emissions.

### Data files

The corresponding data files can be found here.

In the probability distributions: the first column is $\displaystyle \epsilon$ , the second one ${\displaystyle P(\epsilon )}$; line 1 refers to the discrete probability of no energy loss, line 2 to the discrete probability of $\displaystyle \epsilon=1$ .

In the energy spectrum of radiated gluons: the first column is $\displaystyle x$ , the second $\displaystyle dN_g/dx$ .

• E=10 GeV, L=2 fm:
1. $\displaystyle \Delta E/E=0.05$ : $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; $\displaystyle \hat{q}=1.0$ GeV$\displaystyle ^2$ /fm.
2. $\displaystyle \Delta E/E=0.1$ : ${\displaystyle P(\epsilon )}$ here, $\displaystyle dN_g/dx$ here; ${\displaystyle {\hat {q}}=1.6}$ GeV$\displaystyle ^2$ /fm.
3. ${\displaystyle \Delta E/E=0.2}$: $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; $\displaystyle \hat{q}=3.0$ GeV$\displaystyle ^2$ /fm.
4. ${\displaystyle \Delta E/E=0.4}$: $\displaystyle P(\epsilon)$ here, ${\displaystyle dN_{g}/dx}$ here; $\displaystyle \hat{q}=7.5$ GeV$\displaystyle ^2$ /fm.
• E=10 GeV, L=5 fm:
1. $\displaystyle \Delta E/E=0.05$ : $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; ${\displaystyle {\hat {q}}=0.1}$ GeV$\displaystyle ^2$ /fm.
2. $\displaystyle \Delta E/E=0.1$ : ${\displaystyle P(\epsilon )}$ here, $\displaystyle dN_g/dx$ here; $\displaystyle \hat{q}=0.2$ GeV${\displaystyle ^{2}}$/fm.
3. $\displaystyle \Delta E/E=0.2$ : $\displaystyle P(\epsilon)$ here, ${\displaystyle dN_{g}/dx}$ here; $\displaystyle \hat{q}=0.4$ GeV${\displaystyle ^{2}}$/fm.
4. $\displaystyle \Delta E/E=0.4$ : $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; $\displaystyle \hat{q}=0.9$ GeV$\displaystyle ^2$ /fm.
• E=100 GeV, L=2 fm:
1. $\displaystyle \Delta E/E=0.05$ : $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; $\displaystyle \hat{q}=5$ GeV$\displaystyle ^2$ /fm.
2. $\displaystyle \Delta E/E=0.1$ : $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; ${\displaystyle {\hat {q}}=9}$ GeV$\displaystyle ^2$ /fm.
3. $\displaystyle \Delta E/E=0.2$ : $\displaystyle P(\epsilon)$ here, ${\displaystyle dN_{g}/dx}$ here; $\displaystyle \hat{q}=18$ GeV$\displaystyle ^2$ /fm.
4. $\displaystyle \Delta E/E=0.4$ : $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; ${\displaystyle {\hat {q}}=41}$ GeV$\displaystyle ^2$ /fm.
• E=100 GeV, L=5 fm:
1. $\displaystyle \Delta E/E=0.05$ : $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; ${\displaystyle {\hat {q}}=0.7}$ GeV$\displaystyle ^2$ /fm.
2. $\displaystyle \Delta E/E=0.1$ : $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; ${\displaystyle {\hat {q}}=1.2}$ GeV$\displaystyle ^2$ /fm.
3. ${\displaystyle \Delta E/E=0.2}$: $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; $\displaystyle \hat{q}=2.4$ GeV$\displaystyle ^2$ /fm.
4. $\displaystyle \Delta E/E=0.4$ : $\displaystyle P(\epsilon)$ here, $\displaystyle dN_g/dx$ here; $\displaystyle \hat{q}=5.5$ GeV${\displaystyle ^{2}}$/fm.

Quenching weights code

The code for calculating the (parametrised) Quenching Weights in the ASW-BDMPS formalism and the opacity expansion, N=1 case, was published with ref [2]. Note that this code uses $\displaystyle \alpha_s$ = 0.33, not $\displaystyle \alpha_s$ = 0.3 as used for the brick problem. The published code is in FORTRAN, a small wrapper package for interactive use in ROOT/C++ is available.

## References

[1] U. A. Wiedemann, Nucl. Phys. B588 (2000) 303 [hep-ph/0005129].

[2] C. A. Salgado and U. A. Wiedemann, Phys. Rev. D68 (2003) 014008 [hep-ph/0302184].

[3] N. Armesto, C. A. Salgado and U. A. Wiedemann, Phys. Rev. D69 (2004) 114003 [hep-ph/0312106].