# 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:

- 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".
- Opacity expansion: Technically, this is an expansion of the integrand of the path integral in powers of (density times pathlength). The GLV opacity result reproduces our expression [1] on the level of the Feynman diagrams and the analytic expression for the - and -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 .) First, we plot the probability that a light quark looses a fraction of its energy. The information about is contained in three pieces:

- "Untouched survival": This is the discrete probability that a parton does not interact with a medium of length and that it looses no energy. This probability is represented by a color-coded dot at .
- "Survival with finite energy loss": Continuous probability: this is the probability that a parton makes it through a medium of length but looses a finite fraction during its passage. This is denoted by the color-coded curve at finite
**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 0 \le \Delta E/E \le 1}**. - "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
**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 L}**. This probability is denoted by the color-coded dot at .

In general, as one increases the average energy loss (i.e. as one increases **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 \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 . The most extreme curve
is that 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 E= 10}**
GeV and fm.
Requiring an energy loss 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 \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 **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 \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 **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 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 **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 L}**
, though with finite probability the parton does not have sufficient initial energy to make it through **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 L}**
, and gets stuck before. In the present calculational framework, finding yield in the spectrum 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 \omega > E}**
, i.e., above the kinematical boundary, signals that one has assumed that the particle propagates through a length

We note that even if **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 dN_g/dx}**
does not have any yield 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 \omega > E}**
, there can be a finite probability of death before arrival, since **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 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 **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 \epsilon}**
, the second one ; line 1 refers to the discrete probability of no energy loss, line 2 to the discrete probability 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 \epsilon=1}**
.

In the energy spectrum of radiated gluons: the first column 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 x}**
, the second **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 dN_g/dx}**
.

- E=10 GeV, L=2 fm:

**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 \Delta E/E=0.05}**:**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 P(\epsilon)}**here,**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 dN_g/dx}**here;**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 \hat{q}=1.0}**GeV**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 ^2}**/fm.**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 \Delta E/E=0.1}**: here,**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 ^2}**/fm.- :
**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 P(\epsilon)}**here,**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 \hat{q}=3.0}**GeV**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 ^2}**/fm. - :
**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 P(\epsilon)}**here, here;**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 \hat{q}=7.5}**GeV

- E=10 GeV, L=5 fm:

**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 \Delta E/E=0.05}**:**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 \Delta E/E=0.1}**: here,**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 \hat{q}=0.2}**GeV/fm.**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 \Delta E/E=0.2}**:**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 \hat{q}=0.4}**GeV/fm.**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 \Delta E/E=0.4}**:**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 \hat{q}=0.9}**GeV

- E=100 GeV, L=2 fm:

**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 \Delta E/E=0.05}**:**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 \hat{q}=5}**GeV**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 \Delta E/E=0.1}**:**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 \Delta E/E=0.2}**:**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 \hat{q}=18}**GeV**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 \Delta E/E=0.4}**:

- E=100 GeV, L=5 fm:

- :
**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 \hat{q}=2.4}**GeV **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 \Delta E/E=0.4}**:**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 \hat{q}=5.5}**GeV/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 **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 \alpha_s}**
= 0.33, not **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 \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].