Parton Energy Loss in the Extremely Prolate Quark-Gluon Plasma

# Parton Energy Loss in the Extremely Prolate Quark-Gluon Plasma

## Abstract

The energy loss per unit path length of a highly energetic parton scattering elastically in a weakly coupled quark-gluon plasma is studied as an initial value problem. The approach is designed to study unstable plasmas but in the case of an equilibrium plasma the well known result is reproduced. As an example of an unstable system, an extremely prolate plasma, where the momentum distribution is infinitely elongated along one direction, is considered here. The energy loss is shown to be strongly time and directionally dependent and its magnitude can much exceed the energy loss in equilibrium plasma.

\ShortTitle

Parton Energy Loss in the Extremely Prolate QGP \FullConferenceXth Quark Confinement and the Hadron Spectrum,
October 8-12, 2012
TUM Campus Garching, Munich, Germany

## 1 Introduction

When a highly energetic parton travels through a quark-gluon plasma (QGP), it losses its energy due to, in particular, elastic interactions with plasma constituents. This is called collisional energy loss and was computed for equilibrium QGP twenty years ago, see the review [1] and the handbook [2]. The quark-gluon plasma produced in relativistic heavy-ion collisions, however, reaches a state of local equilibrium only after a short but finite time interval, and during this period the momentum distribution of plasma partons is anisotropic. It is important to note that a plasma with an anisotropic momentum distribution is unstable (for a review see [4]). Collisional energy loss has been computed for an anisotropic QGP in Ref. [3], but the fact that unstable systems are explicitly time dependent as unstable modes exponentially grow in time was not taken into account.

We have developed an approach, see [5, 6] for a preliminary account, where energy loss is studied as an initial value problem. The approach is applicable to plasma systems evolving quickly in time. We compute the energy loss by treating the parton as an energetic classical particle with color charge. For an equilibrium plasma the known result is recovered and for an unstable plasma the energy loss is shown to have contributions which exponentially grow in time. In Refs. [5, 6] we have calculated the energy loss in a two-stream system which is unstable due to longitudinal chromoelectric modes and found that it manifests strong time and directional dependence. In this paper we focus on an extremely prolate quark-gluon plasma with momentum distribution infinitely elongated in one (beam) direction. Such a system is unstable due to transverse chromomagnetic modes and the spectrum of collective excitations can be obtained in explicit analytic form. The system has thus nontrivial dynamics but the computation of energy loss is relatively simple. After a brief presentation of our approach, we show some of our results. The energy loss grows exponentially and after some time its magnitude is much bigger than in equilibrium plasma.

## 2 Formalism

Our approach is classical and thus we start with the Wong equations [7] describing the motion of classical parton in a chromodynamic field.

### 2.1 General energy-loss formula

 dxμ(τ)dτ = uμ(τ), (1) dpμ(τ)dτ = gQa(τ)Fμνa(x(τ))uν(τ), (2) dQa(τ)dτ = −gfabcuμ(τ)Aμb(x(τ))Qc(τ), (3)

where , , and are, respectively, the parton’s proper time, its trajectory, four-velocity and four-momentum; and denote the chromodynamic field strength tensor and four-potential along the parton’s trajectory and is the classical color charge of the parton; is the coupling constant and is assumed to be small. We also assume that the potential vanishes along the parton’s trajectory i.e. our gauge condition is . Due to Eq. (3) the classical parton’s charge is a constant of motion within the chosen gauge.

The energy loss is given directly by Eq. (2) with . Using the time instead of the proper time and replacing the strength tensor by the chromoelectric and chromomagnetic fields, Eq. (2) gives

 dE(t)dt=gQaEa(t,r(t))⋅v, (4)

where is the parton’s velocity. Since we consider a parton which is very energetic, is assumed to be constant and , but the parton’s momentum and energy vary.

Since we deal with an initial value problem, we apply to the field and current not the usual Fourier transformation but the one-sided Fourier transformation defined as

 f(ω,k) = ∫∞0dt∫d3rei(ωt−k⋅r)f(t,r), (5) f(t,r) = ∫∞+iσ−∞+iσdω2π∫d3k(2π)3e−i(ωt−k⋅r)f(ω,k), (6)

where the real parameter is chosen is such a way that the integral over is taken along a straight line in the complex plane, parallel to the real axis, above all singularities of . Introducing the current generated by the parton , and using Eqs. (5, 6), Eq. (4) can be rewritten:

 dE(t)dt=gQa∫∞+iσ−∞+iσdω2π∫d3k(2π)3e−i(ω−¯ω)tEa(ω,k)⋅v, (7)

where .

The next step is to compute the chromoelectric field . Applying the one-sided Fourier transformation to the linearized Yang-Mills equations, which represent QCD in the Hard Loop approximation, we get the chromoelectric field given as

 Eia(ω,k)=−i(Σ−1)ij(ω,k)[ωjja(ω,k)+ϵjklkkBl0a(k)−ωDj0a(k)], (8)

where and are the initial values of the chromomagnetic field and the chromoelectric induction, and with being chromodielectric tensor which equals

 εij(ω,k)=δij+g22ω∫d3p(2π)3viω−k⋅v+i0+∂f(p)∂pk[(1−k⋅vω)δkj+kkvjω],

where is the momentum distribution of plasma constituents. The color indices are dropped as is a unit matrix in color space. The matrix in Eq. (8) is defined

 Σij(ω,k)≡−k2δij+kikj+ω2εij(ω,k). (9)

Substituting the expression (8) into Eq. (7), we get the formula

 dE(t)dt = gQavi∫∞+iσ−∞+iσdω2πi∫d3k(2π)3e−i(ω−¯ω)t(Σ−1)ij(ω,k) (10) × [iωgQavjω−¯ω+ϵjklkkBl0a(k)−ωDj0a(k)].

As seen, the integral over is controlled by the poles of the matrix which represent the collective modes of the system.

### 2.2 Equilibrium plasma

When the plasma is stable, all modes are damped and the poles of are located in the lower half-plane of complex . Consequently, the contributions to the energy loss corresponding to the poles of exponentially decay in time. The only stationary contribution is given by the pole . Therefore, the terms in Eq. (10), which depend on the initial values of the fields, are neglected and Eq. (10) provides

 dEdt=−ig2CR∫d3k(2π)3¯ωk2[1εL(¯ω,k)+k2v2−¯ω2¯ω2εT(¯ω,k)−k2], (11)

where the color factor , which equals for a quark and for a gluon, results from the averaging over colors of the test parton. The formula (11) agrees with the standard energy loss due to soft collisions in equilibrium QGP [2].

To compare the energy loss in an unstable plasma to that in an equilibrium one, we have computed the integral in Eq. (11) numerically using cylindrical coordinates, which will also be used for the prolate system. Since the integral is known to be logarithmically divergent, it has been taken over a finite domain such that and . The energy loss in an equilibrium plasma of massless constituents can be expressed through the Debye mass, which we write as

 μ2≡g2∫d3p(2π)3f(p)|p|. (12)

We define dimensionless variables by scaling dimensionful quantities by the Debye mass. In Fig. 1 we show the energy loss in equilibrium QGP divided by as a function of computed for and .

### 2.3 Unstable plasma

When the plasma is unstable, the matrix has poles in the upper half-plane of complex , and the contributions to the energy loss from these poles grow exponentially in time. The terms in Eq. (10) which depend on the initial values of the fields and cannot be neglected, as they exponentially grow in time. Using the linearized Yang-Mills equations, the initial values and are expressed through the current and we obtain

 dE(t)dt=g2CRvivl∫∞+iσ−∞+iσdω2π∫d3k(2π)3e−i(ω−¯ω)t(Σ−1)ij(ω,k) (13) ×[ωδjlω−¯ω−(kjkk−k2δjk)(Σ−1)kl(¯ω,k)+ω¯ωεjk(¯ω,k)(Σ−1)kl(¯ω,k)],

which gives the energy loss of a parton in an unstable quark-gluon plasma.

### 2.4 Inversion of Σ

When the anisotropy of the momentum distribution of plasma constituents is controlled by a single (unit) vector , it is not difficult to invert the matrix . Following [9], we introduce the vector defined as

 niT≡(δij−kikjk2)nj (14)

and we use the basis of four symmetric tensors

 Aij(k)=δij−kikjk2,Bij(k)=kikjk2,Cij(k,n)=niTnjTn2T,Dij(k,n)=kinjT+kjniT. (15)

Since the matrix is symmetric, it can be decomposed as , where the coefficients , , and are found from the equations

 kiΣijkj=k2b,niTΣijnjT=n2T(a+c),niTΣijkj=n2Tk2d,TrΣ=2a+b+c. (16)

The inverse matrix is found to be

and consequently, the poles of the matrix are given by the following dispersion equations

 a=0, d2k2n2T−b(a+c)=0. (18)

Writing down the energy-loss formula (13) in terms of the projectors we obtain the form

 dE(t)dt = Missing or unrecognized delimiter for \Big (19) × [ωω−¯ω+ω¯ω+ω+¯ω¯ωk2(1¯aA+(−¯d2k2n2T+¯b¯c)C+¯a¯dAD¯a(¯d2k2n2T−¯b(¯a+¯c)))]v,

where the coefficients are computed at and the coefficients at . The formula (19) will be used in the subsequent section to compute the energy loss in an extremely prolate QGP.

## 3 Extremely prolate plasma

Anisotropy is a generic feature of the parton momentum distribution in heavy-ion collisions. At the early stage, when partons emerge from the incoming nucleons, the momentum distribution is strongly elongated along the beam - it has a prolate shape with the average transverse momentum being much smaller than the average longitudinal one. Due to free streaming, the distribution evolves in the local rest frame to a form which is squeezed along the beam - it has oblate shape with the average transverse momentum being much larger than the average longitudinal one. We consider here the extremely prolate momentum distribution which is either infinitely elongated in the longitudinal direction (defined be the unit vector ) or infinitely squeezed in the transverse directions. In this case the spectrum of collective excitations can be found analytically.

### 3.1 Momentum distribution

The extremely prolate momentum distribution can be written as

 f(p)=δ(p2−(p⋅n)2)h(p⋅n), (20)

where is any positive even function such that is finite. The integral in Eq. (12) can be used to define a mass parameter for either isotropic or anisotropic momentum distributions, although in the later case cannot be interpreted as the screening length. Applying the prolate distribution a different mass parameter denoted as naturally appears. It is related to as .

Since the velocity of a massless parton as given by the distribution (20) is for and for , the matrix defined by Eq. (9) is found to be

 Σij(ω,k)=(ω2−m2−k2)δij+kikj − m2k⋅nω2−(k⋅n)2(kinj+nikj) (21) − m2(ω2+(k⋅n)2)(k2−ω2)(ω2−(k⋅n)2)2ninj,

and the coefficients are

 a(ω,k) = ω2−m2−k2, (22) b(ω,k) = ω2−m2−2m2(k⋅n)2ω2−(k⋅n)2−m2(ω2+(k⋅n)2)(k2−ω2)(ω2−(k⋅n)2)2(k⋅n)2k2, (23) c(ω,k) = m2(ω2+(k⋅n)2)(k2−ω2)(ω2−(k⋅n)2)2((k⋅n)2k2−1), (24) d(ω,k) = −m2(k⋅n)ω2−(k⋅n)2−m2(ω2+(k⋅n)2)(k2−ω2)(ω2−(k⋅n)2)2(k⋅n)k2. (25)

### 3.2 Collective excitations

The dispersion relations are obtained from Eqs. (18) with the coefficients given by Eqs. (22-25). The first of these equations provides . Although the second equation looks rather complicated, it has three relatively simple solutions

 ω21(k) = m2+(k⋅n)2, (26) ω2±(k) = 12(k2+(k⋅n)2±√k4+(k⋅n)4+4m2k2−4m2(k⋅n)2−2k2(k⋅n)2), (27)

which hold under the condition . The modes , and , are always stable. The solution is negative when . Writing , , the solutions are . The first is the Weibel unstable mode and the second is an overdamped mode. Denoting the angle between and as , the condition for the existence of an instability is . One can show that for fixed the instability growth is maximal when . Collective excitations in the extremely prolate QGP were earlier studied in [10] using a method different than ours.

Using Eqs. (26) and (27) one can write

 a(d2k2n2T−b(a+c))=−ω2(ω2−ω21(k))(ω2−ω22(k))(ω2−ω2+(k))(ω2−ω2−(k))ω2−(k⋅n)2 (28)

and from Eq. (19) one therefore sees that the energy loss in the extremely prolate system is controlled by the double pole at and 8 single poles: , , , . Since the collective modes are known analytically, the integral over in Eq.  (19) can be computed analytically as well. The remaining integrals are performed numerically.

### 3.3 Numerical results

To compute the integral over in Eq.  (19), we use cylindrical coordinates with the axis along the vector . Since the integral is divergent (as is the case in equilibrium (11)), we choose a finite domain such that and with . The values of remaining parameters are: , . In Fig. 2 we show the parton’s energy loss per unit time as a function of time for three different orientations of the parton’s velocity with respect to the axis. The energy loss manifests a strong directional dependence and it exponentially grows in time, which indicates the effect of the unstable modes. After a sufficiently long time, the magnitude of energy loss much exceeds that in equilibrium plasma which, as shown in Fig. 1, equals 0.18 for .

## 4 Conclusions

We have developed a formalism where the energy loss of a fast parton in a plasma medium is found as the solution of an initial value problem. The formalism, which allows one to obtain the energy loss in an unstable plasma, is applied to an extremely prolate quark-gluon plasma with momentum distribution infinitely elongated in the direction. This system is unstable due to chromomagnetic transverse modes. The energy loss per unit length of a highly energetic parton is not a constant, as in the equilibrium plasma, but it exponentially grows in time and exhibits a strong directional dependence. The magnitude of the energy loss can much exceed the equilibrium value.

## Acknowledgment

This work was partially supported by the Polish National Science Centre under grant 2011/03/B/ST2/00110.

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