# Gluon emission in Quark-Gluon Plasma

## Abstract

Gluon radiation is an important mechanism for parton energy loss as the parton traverses the quark gluon plasma (QGP) medium. We studied the gluon emission in QGP using AMY formalism. In the present work, we obtained gluon emission amplitude F(h,p,k) function , which is a solution of the integral equations describing gluon radiation including Landau-Pomeranchuk-Migdal (LPM) effects, using iterations method. We define a new dynamical scale for gluon emission denoted by . The gluon emission rate is obtained by integrating these amplitude function over h. We show that these obey a simple scaling in terms of this dynamical variable . We define the gluon emission function for gluon radiation for the three processes , and . In terms of this function, the parton energy loss calculations, due to medium induced gluon radiation, may become simplified.

###### pacs:

12.38.Mh ,13.85.Qk , 25.75.-q , 24.85.+pLattice quantum chromo dynamics (LQCD) calculations (1) predict a transition
from confined state in hadrons to a deconfined state of quarks and gluons above a
temperature of 170 MeV or an energy density above 1GeV/fm . In the relativistic heavy ion
collisions at RHIC at BNL with an energy density above 5GeV/fm, experimental
measurements of several observables indicate such a transition to a deconfined state of
matter (2); (3); (4); (5), for details see the reviews
(6); (7); (8); (9). It is currently believed that
this deconfined state consists of a strongly interacting Quark Gluon
Plasma (sQGP), behaving nearly like a perfect liquid (7). Among these
observables, jet-quenching phenomenon is so far an important signal for a
hot dense medium formed. Jet suppression has its origin in parton energy
loss in the quark matter by gluon radiation, which distinctly differs from
energy loss in hadronic matter. For example the suppression of high-
pions, from 3GeV to 10GeV, of BNL experiments can be explained by assuming
a deconfined state.

We studied gluon radiation problem as this has direct application to the
energy loss of partons while traversing the QGP medium due to the gluon bremsstrahlung
processes. In addition to radiation, the elastic energy loss of partons traversing the
QGP medium is important for heavy quark quenching, observed in RHIC experiments.
For an exposition of theoretical and experimental results on parton energy loss
the readers may see an excellent review (10).

In the present work, we study the gluon radiation mechanism. The coherent radiation processes
involve multiple scatterings of the partons in the QGP medium during the gluon formation time.
This leads to interference effects known as Landau-Pomeranchuk-Migdal effect (LPM). Gluon
emission is discussed widely in literature (11)-(17). These works treated
the parton energy loss on the basis of avarage energy loss dependeing on the path length.
As emphasized in (18), the bremsstrahlung (gluon emission), is characteristically
different in the sense that it is a stochastic process. Starting with a group of partons
of fixed energy, the bremsstrahlung process results in a broad spectrum of final partons
of width comparable to its mean energy loss. Further, the LPM effect has different
parametric dependence on energy for soft and hard parts of emitted gluon spectrum.
As compared to the case of bremsstrahlung photon emission, the gluon emission
also involves an enhancement mechanism when the emitted gluon and quark are nearly
collinear, thereby a need to consider ladder diagrams (16).
However, unlike the emission of electromagnetic quanta, the emitted
hard gluon feels the random colored background field. The resummation
of these ladder diagrams leads to Swinger-Dyson type integral equations.
In this work, we follow the formalism given in (16)-(18)
which implements LPM effects by resumming the ladder diagrams.
For calculating parton energy loss arising from gluon radiation, one needs the differential gluon emission rates
and this is given by Eq.1 (18) in terms of the F(h,p,k)
function. The bremsstrahlung integral equations determine the
gluon emission amplitude F(h,p,k) function given by
Eq.2. Here, the two dimensional vector
is of the order of . It is a measure of collinearity and its
magnitude is small compared to . The term in
Eq.2 is the energy differential between initial and final states. Here,
and other quark thermal masses are .
This formalism is very similar to the photon emission integrals, however,
as mentioned before, the emitted gluon has color and therefore interacts
with other scattering centers as well as soft background fields (16).
Accordingly, there are three terms in the integral equations of 2
involving collision kernel . A typical ladder diagram for gluon
emission is shown in Figure 1.
We solve this integral equation for F(h,p,k) function,
by using iterations method as discussed in (19).
The F(h,p,k) distributions for various values of
parton and gluon momenta (p,k) were obtained considering the
mechanism , and processes
using relevant factors (, and ).
Fig.2 shows a few distributions for
these two processes (for high parton momenta) for various values of
parton and gluon momenta (p,k). The real part is shown in figure (a) and negative
of imaginary part in figure (b). Fig.2(c) shows the real
part of the distributions for process for high and low values of
incoming and outgoing gluon momenta (p,k).
The calculations for process are shown in Fig.3.
In all these figures, always stands for the incoming parton momentum.

(1) | |||||

(2) | |||||

(3) | |||||

(4) |

## I Generalized Emission Functions for gluon emission

Before we discuss the emission functions for gluon emission, it is very instructive to recall the emission functions for photon emission. In our previous works (19); (25), we showed generalized photon emission function by integrating (Eq.5) the corresponding distributions (see (19).

(5) |

(6) | |||||

(7) | |||||

(8) |

(9) | |||||

(10) |

(11) |

(12) | |||||

are in general functions of {} and therefore, we defined the generalized emission functions (GEF) in Eq.9, which are functions of only variables. These GEF () are obtained from corresponding values by multiplying with coefficient functions given in (19). As an example, in Figure 4, we show the results for GEF for bremsstrahlung (Fig.4(a)). The solid curve in (a) is the empirical fit to this emission function, given by Eq.11. Consequent to finding the emission functions like those given by Eq.11, we expressed the imaginary part of retarded photon polarisation tensor for any values by using Eq.12. Using this approach, we obtained the phenomenological fits to virtual photon emission rates from QGP for ladder processes with LPM effects (19). We provided simple phenomenological formulae which are useful in model calculations for experimental dilepton yields.

Following the procedure of generalized photon emission function, we now try to obtain
the generalized gluon emission functions. For this, we integrate these distributions,
that have been shown in Figs.2,3 over the variable ,
as given in Eq.13. This quantity is strongly process dependent and is
a function incoming and outgoing parton momenta, plasma temperature and strong
coupling strength as denoted by the variables and . The integrated quantity
is plotted versus in Figure 5 for the process .
Figure shows plotted for different values of labeled on the figure.
As seen in figure, the values are scattered. Therefore, we defined a variable as given
in Eq.14. We show values versus variable in figure 6.
As seen in figure (a), it exhibits a linear behavior on log-log plot, extending over nine orders of magnitude.
This apparently gives an impression that is a good dynamical scale for the process .
In order to examine this, we plot in figure (b). As seen in figure (b), the values are
scattered and exhibit no useful trends, showing that is not a dynamical variable for this
process. Therefore, we now define the dynamical variable and a function ,
for process as given in Eqs.15,16. In the Fig.17, we show the function
versus . As seen in figure, all values for different parton momenta merge. We fit this data with an empirical
curve together with parameters as given in Eqs.17,19. We denote this function in Eq.17 as
gluon emission function for the process .

We carried out this for the processes and . The values for
these two processes also donot exhibit any trends as a function variable, however, remains a good dynamical
variable. We show these results for the process in Fig.8 versus .
The curve in this figure is given by empirical fit and parameters in Eqs.20,23.
Therefore, for , we define gluon emission function as given in Eq.20.
We performed these calculations for the process and these results are shown
in Fig.9. The curve in this figure is given by empirical fit and parameters
in Eq.24,29. Therefore, for the process ,
we define gluon emission function as given in Eq.24.

After obtaining the gluon emission function for these three processes, we can perform
integrations required in Eq.1, i.e. integration in terms of dynamical variables .
This will give us differential gluon emission rates. In the jet-quenching studies, one needs to
estimate the energy loss of high energy partons while traversing the QGP medium. In this
problem, the differential gluon emission rates estimated by integrating over variable, will
have to be coupled in order to determine the differential energy loss. These results were already shown by
(16); (17); (18); (10).

In the following, we examine the integrand of differential gluon emission rate of the Eq.1,
i.e., without integration, and we denote this in short by gluon rate,
which should not be confused with the gluon emission rates after integrating over .
We obtained the values by integrating the distributions
.
The splitting function are as given
in (18). In the Fig.10, we show the values (red curves), splitting
function (blue curves) and the gluon rate (black curves) at three different values
for the process . We show similar results for the process
in Fig.11 and for the process in Fig.12.

(13) | |||||

(14) | |||||

(15) | |||||

(16) |

(17) | |||||

(18) | |||||

(19) |

(20) | |||||

(21) | |||||

(22) | |||||

(23) |

(24) | |||||

(25) | |||||

(26) | |||||

(27) | |||||

(28) | |||||

(29) |

In conclusion, the gluon emission in quark gluon plasma including LPM effects has been
studied at a fixed temperatures and strong coupling strength.
We defined a new dynamical variable for gluon emission. Further, we defined gluon emission functions (GEF)
denoted by for the processes , and .
We have obtained empirical fits to these GEF and provide the functional forms and parameters
for all the three processes. We compared the differential gluon emission rates (without p-integration) for these three processes.

In terms of the GEF, we may calculate the differential gluon emission rates for these processes.
These empirical formulae will be useful in calculations of parton energy loss by medium induced gluon radiation.

###### Acknowledgements.

I am thankful to Drs. A. K. Mohanty, S. Kailas, R. K. Choudhury, S. Ganesan and H. Naik for fruitful discussions. I thank S.V. Ramalakshmi for her kind co-operation during this work.### References

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