Heavy Quark Radiation in the Quark-Gluon Plasma in the Moliere Theory: Angular Distribution of the Radiation.

The heavy quark angular distribution in the media is given by ASW

Here $K$ is the propagator of the particle in the media with the two dimensional effective potential due. to the scattering centres, and $K_{0}$ is the corresponding propagator of the free particle in the vacuum. The effective two dimensional potential is given by

Here $d^{2}\sigma_{el}/d^{2}q_{t}$ is the cross section of elastic scattering of high energy particle on the media centre. We assume the static media of the form

where $U=1$ if $s\geq 0$ and 0 if $s<0$ is a conventional step function.

The media is described by Gyulassy-Wang model GW . The effective potential in the momentum space is given by

where the parameter $\mu\sim m_{D}$, and the Debye mass $m_{D}$ is given by

for $N_{f}=3$ light quarks, T is the media/QGP temperature. The density of the scattering centres in the GW model is given by $n=\frac{3}{2}T^{3}$, and the strong coupling is $\alpha_{s}=\frac{g^{2}}{4\pi}$. The effective potential in the coordinate space is

where $\gamma_{E}=0.577$ is the Euler constant, and the bare quenching coefficient is

Note that $\hat{q}$ is fully determined by media properties, and does not depend on the quark mass.

For processes that are dominated by large momentum transfer is enough to take into account only the first terms in the Taylor expansion of $V(\rho)$. The first approximation corresponds to the quadratic term in the expansion 7 and is called the HO (harmonic oscillator ) approximation. In this approximation the effective potential V is given by

Here $\hat{q}_{\rm eff}$ is the effective jet quenching coefficient, given by

and Q is the typical transverse momenta, accumulated by the particle on the scale of the coherence length.

The HO effectively describes the LPM bremsstrahlung BDMPS1 . More precise treatment of the energy loss includes also large Coulomb logarithms and is called in the theory of the Abelian (QED) LPM effect the Moliere theory katkov . In the QCD framework the inclusion of Coulomb interactions can be made using the perturbation theory mehtar ; mehtar1 . Namely, instead of the usual opacity expansion GLV1 ; GLV2 ; GLV3 , we shall consider the perturbation theory around the oscillator potential adding the Coulomb effects as a perturbation. The effective potential in Moliere theory is given by

and includes the short range Coulomb logarithms. In the framework of the perturbation theory this potential is split as

where Q is the typical momenta, defined above, equal to $Q\sim\sqrt{\hat{q}\omega}$ in the HO approximation. We shall need sufficiently large Q, so that

i.e. perturbation theory is applicable meaning that we probe rather small transverse distances.

Then the energy loss is given by Eq. LABEL:e1, where the propagator K is calculated in perturbation theory as mehtar ; mehtar1

Here $K_{HO}$ is the heavy quark propagator in the imaginary two dimensional potential $V_{HO}$ ASW :

and

In the limit when there is no media this propagator reduces to free quark propagator

The expansion written in the form 14 clearly exhibits the formation lengths described in the Introduction: the heavy quark mass leads to the oscillating exponent $\exp(i\theta^{2}\omega/2(t_{1}-t))$ in Eq. LABEL:e2, while the harmonic oscillator part of the propagator LABEL:e2 oscillates with the frequency $\sqrt{\omega/\hat{q}}$. Then it is clear that when $l_{c}^{q}<<l_{c}^{LPM}$ the oscillations due to heavy quark mass cut off the integral for heavy quark energy loss, the oscillating harmonic oscillator part of the propagator is approximately freezed and the LPM effect is not relevant, the energy loss is defined by the induced radiation on the scattering centres-the N=1 GLV. On the other hand, in the opposite case, the heavy quark exponent is close to one, and the integral for energy loss is controlled by the HO multiplier. We have LPM bremsstrahlung plus corrections due to Coulomb logarithms.

We can now choose the subtraction scale $Q$ in the momentum space. As it was explained in arnold1 ; mehtar this scale corresponds to the typical momentum accumulated by the quark along the coherence length propagation. Such momentum squared is $\hat{q}\times\sqrt{\omega/\hat{q}}$ for $\omega<<\omega_{DC}$ and $\sim\theta^{2}\omega^{2}\sim\omega/l^{q}_{c}$ for $\omega>>\omega_{DC}$. Consequently we shall use the interpolation formula

where $U(x)$ is a unit step function:$U(x)=1$ if $x\geq 0$, and $U(x)=0$ if $x\leq 0$.

Alternatively, the dynamics of the heavy quark can be approached using the arguments in arnold1 . Namely , in the LPM (diffusion ) regime the distribution over momentum transfers in the scattering on the media centres is described by a gaussian, peaked in the $Q^{2}_{typ}\sim\sqrt{\hat{q}w}$. The scattering with significantly higher momentum transfers $q_{t}$ is described by the tail of the distribution, which is N=1 GLV, that essentially describes the independent scattering on the media centres. In this region the LPM gaussian is parametrically close to zero, and N=1 GLV dominates. It was explained in Z1 that N=1 term in opacity expansion is a good description of large momentun transfer regime, since such scatterings in the tail occur quite rarely. Since inside dead cone the typical momenta is $k^{2}_{t}\sim\omega/l_{c}^{q}\sim\theta^{2}\omega^{2}\gg\sqrt{\hat{q}\omega}$, inside the dead cone we shall find ourselves in the GLV regime.

We shall also need the explicit expression for N=1 term in the opacity expansion for angular distribution for massive quark. The corresponding result was derived in ASW , and has the form:

where

Here $k_{t}$ is the momentum of the radiated gluon.

Explicitly this term is given by

Substituting the known expressions for $V_{pert}$ and the propagators we obtain

The integral over the transverse momenta azimutal angle can be easily taken using the standard integral AS

Let us introduce two new functions that can be expressed through elementary functions (see appendix A):

So we finally get

We now consider the contribution to the angular distribution due to the perturbative expansion of the propagator in the powers of $V_{pert}.$. We have in the integral over $t_{1}$ two terms: the first is from 0 to L and is called a bulk term, the second corresponds to the case when $t_{1}>L$ and is called a boundary term. Let us consider first the bulk term

After integration by parts we obtain, calculating the gaussian integral over $d^{2}u$

where $R(t_{1},t)$ is given by Eq. 33. The angular integral can be easily taken using the standard formulae AS :

where $J_{0},J_{1}$ are the conv entional Bessel functions. Introducing an additional function

we obtain final answer:

Finally we consider the boundary contribution:

We first do integral over $d^{2}y$ and over $t_{1}-L$,and then gaussian integral over $d^{2}r$. Using the Eqs. LABEL:b3 for angular integration and the definition 37 of the function $F_{4}$ we easily obtain a final answer for the boundary term.

Note that functions $F_{2},F_{3},F_{4}$ can be easily expressed through known special functions, the explicit expressions are given in the Appendix. In addition it is easy to perform integral over the transverse momentum $k_{t}$ between 0 and some scale $\omega_{1}$ analytically.

We shall now present the numerical estimates for the angular distributions of the radiated gluons and compare them with the BDMPS angular spectrum ASW and N=1 GLV angular distributions. We shall depict these angular distributions for typical values of :$\omega=5,10,20$ GeV in Figs. V.1,V.1,V.1. The BDMPS maximum angle is $\theta_{BDMPS}=(\hat{q}/\omega^{3})^{1/4}=0.22$ for $\omega=5$ GeV, 0.13 for $\omega=10$ GeV, and 0.08 for 20 GeV. for $\hat{q}\sim 0.3$ that we use in our calculations.

For $\omega=5$ GeV the BDMPS angle is outside the dead cone for all values of dead cone angle that we we consider,i.e. $\theta\leq 0.2$. The Coulomb corrections to BDMPS are significant and are the biggest for small $k_{t}$, although the calculations for large $k_{t}$ especially of order $\omega$ are not trustworthy, since we use soft gluon approximation in the BDMPS approach. We see that the Coulomb correction is approximately constant at small $k_{t}$ and starts to decrease in parallel with BDMPS contribution.

In Fig. V.1 we consider $\omega=10$GeV. In this case two last values of $\theta=0.15,0.2$ correspond to the BDMPS maximum. $\theta_{BDMPS}$ inside the dead cone. We see that. the Coulomb correction together with BDMPS gluons fill the dead cone.

In Fig. V.1 already three last values of $\theta=0.1,0.15,0.2$ correspond to the situation when $\theta_{BDMPS}$ is inside the dead cone. In all these cases there is no dead cone effect, and BDMPS and Coulomb radiation fills the dead cone region.

It is also interesting to check how the combined effect of the phase space constraints and Coulomb logarithms influence the energy loss. We used the soft gluon approximation, so introducing the explicit boundary for the $k_{t}$ may be beyond the accuracy of our approach ASW , but still introducing the boundary $k_{t}\leq\omega$ will give a good indication of the effect.

It is easy to integrate over $k_{t}$ in arbitrary finite limits, analytically, since the integrands in the expressions for angular distributions in the previous two chapters , since these expressions are gaussian in $k^{2}_{t}$. The remaining integrals are integrals in over $t,t_{1},s$ and are taken numerically using Mathematica, in the same way as the integrals for angular distributions.

We see that compared with the BDMPS spectrum calculated with the same boundary conditions, the corrections increase the energy loss and is rather close to GLV energy loss. Note however that this is just the qualitative estimate since, as we remarked above, the angular distributions were calculated in soft gluon approximation, and precise phase space constraints are beyond the accuracy of this approximation Wiedemann1 ; ASW ..

It will be also interesting to estimate the quenching weights in soft gluon approximation Wiedemann3 ; ADSW but including the finite integration region in the transverse momentum $k_{t}\leq\omega$ and the gluons.

As it is known the jet quenching factor describes the energy loss due to the arbitrary number of Poisson distributed gluons. Indeed, in the previous chapters we calculated the energy loss probability $\omega dId\omega$ in the first order in $\alpha_{s}$. Then we can calculate the quenching factor

where

is determined from the experimental data, ${\it R}\sim 5$. Here $\sigma^{0}$ is the radiation cross section in the vacuum, outside of the media.

The quenching weights for given energy do not change between $\theta=0$ and $\theta\sim 0.06$. We see that the inclusion of Coulomb gluons improves the agreement with experimental data, leading to ratio of quenching weights of massless and heavy quarks with $m_{Q}=5$ GeV (bottom quark). We have $Q=\exp(-S(E)$ of order 1 at 100 GeV, 0.8 at 50 GeV and 0.65- 0.5 for 35 and 25 GeV, There is no difference between massless and charm quarks, at least for jet energies above 25 GeV.