Medium-induced cascade in expanding media

A detailed insight into the interplay between parton energy loss and the way how deconfined medium created in heavy-ion collisions expands is of a great importance for improving the understanding of the jet quenching phenomenon. In this paper we study the impact of the expansion of deconfined medium on single-gluon emission spectra and the jet suppression factor ($Q_{AA}$) within the BDMPS-Z formalism. We calculate these quantities for three types of media, namely static medium, exponentially decaying medium and Bjorken expanding medium. The distribution of medium-induced gluons and the jet $Q_{AA}$ are calculated using the numerical evaluation of in-medium evolution with splitting kernels derived from the gluon emission spectra. The distinctive features in the medium-induced distributions for the low-$x$ and high-$x$ regimes are discussed and the impact of the medium expansion on the jet $Q_{AA}$ is highlighted. It is shown that the impact of the medium expansion can be scaled out from the jet $Q_{AA}$ by an effective quenching parameter which is however different from the time-averaged quenching parameter that can be derived for each type of expanding media. We find sizable differences among the values of the effective quenching parameter that point to the importance of the medium expansion for precise modeling of the jet quenching phenomenon.


Introduction
Measurements of jets in heavy-ion collisions at RHIC and the LHC revealed many interesting results. The production of inclusive jets was found to be strongly suppressed in central heavy-ion collisions with respect to proton-proton collisions [1][2][3][4] as a direct consequence of the parton energy loss. It was shown that the fragmentation patterns of jets are significantly modified in heavy-ion collisions and that the lost energy is transferred to soft particles, predominantly emitted away from the jet axis [5][6][7][8][9][10]. These are only few of important results (for review see e.g. Ref. [11]) which however clearly demonstrate that the rich phenomenon of jet quenching calls for an accurate theoretical description.
In this paper we study one particular aspect of the jet quenching, namely the impact of the medium expansion on the rate of stimulated radiation and the related in-medium medium-induced branching. The starting point of the calculations presented here is the formalism for propagation and radiation in a dense medium within the BDMPS-Z framework [12][13][14][15]. This allows to resum multiple interactions with the medium through a Schrödinger equation for the relevant in-medium correlator, see [16,17]. The solution can be obtained via direct numerical evaluation [18][19][20] or as an expansion in terms of the medium opacity [21][22][23]. Currently, we work within the approximation of multiple-soft scattering, also referred to as the "dipole" or "harmonic oscillator" approximation [24,25], when the resummation for dense media can be performed analytically and that describes well the regime of typical gluon emissions [26]. 1 Multiple scattering in expanding media was analyzed in [29] and later in [24,30,31] (see also [32] for a numerical solution in this case). These calculations indicate an important impact of the finite expanding medium on the observable quantities such as the nuclear modification factor of hadrons. Interesting features, such as the scaling of gluon energy spectra in expanding media with average transport coefficient, were early identified [24,30]. This scaling indicates that some of the main features of the medium-induced spectra remain unchanged no matter the underlying density profile of the background medium.
These approaches have been successfully confronted with experimental data on jet and single-inclusive hadron suppression [32][33][34] with the aim to reliably extract properties of the dense medium created in heavy-ion collisions. Phenomenological studies aim ultimately at establishing the relation between the jet quenching parameter and the energy density of the quark gluon plasma [35] which, according to perturbative estimates, should scale likeq/T 3 ∼ 2( /T 4 ) 3/4 [36]. 2 Since the energy density is expected to change dramatically during the life-time of the system, jet modifications carry an imprint of this evolution. This prompts us to improve the theoretical description of jet quenching in expanding media.
In this work, rather than attempting a full phenomenological description of experimental data, we focus on shedding light on the universal features of radiative energy loss, and deviations from them. We extend previous studies to obtain single-inclusive gluon spectra and related in-medium emission rates, and use these to obtain the jet suppression factor for three different types of expanding medium. However, we do not attempt to model the fluctuations related to the production point of the jet or its substructure. The in-medium distributions are found using numerical solution of the evolution equation for gluon emission spectra introduced in [38,39] with the important input from a unified treatment of expanding media derived in [40]. This allows us to study specific properties and scaling of single-inclusive gluon emission spectra and jet suppression factor which can be compared to recent measurements done at the LHC. While many analyses of in-medium evolution so far have focused on static media [38,41], it is also important to establish whether the qualitative features observed there, such as the rapid transfer of energy to low-energy modes [42,43], can be carried over to expanding cases.
The paper is organized as follows. Section 2 introduces emission spectra and rate of emissions in an expanding medium for three different types of media and discusses their properties. Section 3 provides calculations of medium evolved gluon emission spectra obtained using the evolution equation with input rates from Section 2. In Section 4, the moments of gluon spectra are calculated allowing to obtain the jet suppression factor, Q AA , for different types of expanding media. The scaling properties of the jet Q AA with respect to the transport properties of the expanding media are discussed. Section 5 provides a summary and outlook.

Emission spectrum and rate in an expanding medium
Calculations of medium-induced gluon radiation in the evolving media presented in this paper are done in the limit of multiple soft scatterings and follow the BDMPS-Z formalism [12][13][14][15]. The starting point is the gluon emission spectrum radiated from an initial massless parton with energy p (we only consider gluon splitting at the moment). The final expression can be cast in a general form as [40] dI dz = α s π P (z) ln |c(0)| , The strong coupling constant α s runs with the typical transverse momentum accumulated during the emission, k ⊥ ∼ (z(1 − z)pq) 1/4 , but in the remainder of the paper we will treat it as a constant, α s = 0.3. In Eq. (2.1), c(t) is a function that encodes information about the medium and its expansion [40]. It is the solution of a differential equation where Ω(t) is a time-dependent, complex frequency. For our purposes (gluon splitting), this frequency is simply given by where the effective jet quenching parameter is given byq The boundary conditions are such that c(t) approaches 1 at t → ∞; this realizes the fact that the particle ends up in a vacuum state, i.e.q → 0 and therefore Ω(t) → 0 as t → ∞. On the other hand, t = 0 corresponds to the position of the hard scattering that produces the hard particle sourcing the splitting. We can also derive an emission rate, defined as per unit "time" τ . This is a dimensionless number defined as τ = q 0 p L , (2.5) where L is the distance the initial parton travels trough the medium. The parameter q 0 =q(t 0 ) is the initial value of the jet quenching parameter. The rate K(z, τ ) is an input to calculations of medium evolved gluon spectra, which will be discussed in Section 3. For expanding media, the quenching parameter is time dependent,q =q(t). The average quenching parameter for a given type of the expanding medium is where t 0 corresponds to the time-scale for the onset of quenching effects, i.e.q(t < t 0 ) = 0. In this work we will consider three examples of medium evolution, differing byq(t) profiles and therefore with different c(0). These are the static medium, exponentially decaying medium and the Bjorken expanding medium. The time dependence ofq for these different media is shown in Fig. 1. For the former two examples we can safely put t 0 = 0 while for the Bjorken scenario, where the energy density and therefore alsoq(t) diverges at small times, we have to use a finite t 0 = 0.1 fm. Moreover, the exponentially decreasing spectrum is automatically regularized at late times, and it turns out to be natural to define the average jet quenching parameter as in this case. The reference values for the jet quenching parameter at initial time and the size of the medium used in this section areq 0 = 1.5 GeV 2 /fm and L = 4 fm, respectively. Emission spectra and rates for these examples as well as their properties are detailed in the remainder of this section.

Static medium
For a static medium,q(t) =q 0 for t < L and vanishes at later times, and obviously q =q 0 as well. In this case, Ω 2 (t) = Ω 2 0 , see below, at t < L and Ω 2 (t) = 0 at t > L. The spectrum is given by [12][13][14][15] where and Focusing on the small-z limit, z 1, and defining the gluon frequency ω = zp, we see that the spectrum has two regimes, namely for ω ω c , 1 12 ωc ω 2 for ω ω c , where the characteristic (hard) gluon frequency is The ω −1/2 behavior at low energies is a consequence of the LPM interference effect, and applies for gluons with formation times shorter than the medium length, t f ∼ ω/q < L. This essential feature fundamentally impacts the resulting distribution of medium-induced parton cascade [38]. Furthermore, in this regime the spectrum is proportional to the inmedium path length, ωdI/dω ∝ L. At long formation times, t f ∼ ω/q > L, or ω > ω c , the spectrum is strongly suppressed. Returning now to finite z, in terms of the evolution variable τ , the rate then becomes It is useful also to recall the "soft" limit of this spectrum that will be used for comparison later on. We are interested in the regime κ( where the rate K(z, τ ) is constant in "time". In order to highlight the features of the rate, and corresponding distribution of gluons emitted in the medium, we will further simplify this expression by neglecting all z-dependence in the numerators. This leaves the enhanced soft divergence at z → 0 apparent. In this case, the rate reads whereᾱ = α s N c /π. This can also be found by considering the limit of large times τ , conversely small z, directly in Eq. (2.12), where lim x→∞ tan(1 − i)x = −i and hence the rate tends to constant, time-independent value at large times. It turns out that the medium evolution of the gluon distribution is exactly solvable using Eq. (2.13) [38], which makes it an interesting limiting case.

Exponentially decaying medium
For exponentially decaying media the profile of the jet quenching parameter is given bŷ Note that in this case the average parameter, according to Eq. (2.7), is q exp /q 0 ≈ 2, i.e. twice as big as for the static medium. This is a consequence of the fact that, although exponentially suppressed, the quenching is allowed to take place over very long distances. The solution of c(t) satisfying the boundary conditions at t → ∞ is readily found, and in this case the spectrum is given by where J 0 (z) is a Bessel function of the first kind and Ω 0 L is given in Eq. (2.9). We point out the factor 2 appearing inside the Bessel function, that highlights some of the peculiar features of this particular scenario. The rate then becomes We notice again the ratio of Bessel functions tend to a constant value at large times However, given the profile defined in Eq. (2.15), this limiting value is twice as large as for the static case, This mismatch can be remedied by substituting L → L/2 in Eq. (2.15).

Bjorken expanding medium
This type of the medium is motivated by the Bjorken expansion, which leads to the drop of energy density ε(t) with proper time as ε(t) = ε(t 0 )(t 0 /t) 4/3 for massless relativistic particles. Sinceq ∝ ε 3/4 , one can therefore model the time dependence of the jet quenching parameter as [29],q The average value of q is in this case dependent on the ratio t 0 /L. For the typical values t 0 = 0.1 fm and L = 4 fm, we find q Bjork /q 0 ≈ 0.05, i.e. the expansion reduces the average quenching parameter by a factor of 20.
The spectra for generic power-law expansions characterized by α were analyzed in [24,29,40]. The spectrum is given by where τ 0 = q 0 /pt 0 . In what follows α = 1 (and ν = 1) implying , (2.23) and the rate becomes We point out that this rate depends implicitly on τ 0 as well as on τ . The long-time behavior of this scenario stands out compared to the other two cases analyzed above. While the factor inside the square brackets in Eq. (2.24) goes to a constant, i.e. lim τ →∞ Re[. . .] = 1, the square root in front leads to a power-like decay of the rate at large times, i.e.
However, this can be also obtained for sufficiently small z, i.e. z z c ≡ τ 0 τ . In fact, for these small z values the properties of the Bjorken expanding and the static case, where z c ≡ τ 2 are quite similar, i.e. K static (z c , τ ) ≈ K Bjork (z c , τ ) ∝ τ −1 .

Properties of the emission spectrum and rate
We compare the spectra of medium-induced gluon in Fig. 2. We have plotted the spectrum ω dI/dω versus ω/ ω c , i.e. the energy rescaled by the maximal available gluon energy in the medium ω c ≡ q L 2 /2. We see that the Bjorken model (red, dot-dashed curve in  Fig. 1. We have also plotted the rate from the static, soft approximation, resulting from Eq. (2.13), and additionally scaled by a factor 2 to reproduce the long-time limit of Eq. (2.17). Fig. 2) respects the scaling, as first discussed in [24,30]. The exponential profile, with q defined as in Eq. (2.7), also obeys the scaling, cf. the (orange) dashed curve in Fig. 2. This confirms that the properties of the spectra for different expanding profiles are well understood in terms of a proper re-scaling of the medium parameters.
In Fig. 3, we compare the resulting rates K(z, τ ) for two values of z, z = 0.001 and z = 0.4 (however, note that K(1−z, τ ) = K(z, τ )). The splitting rate for the static medium in the soft limit is constant, see (grey) dotted curves. For the exponential and full static cases, the splitting rate starts to grow from zero at τ = 0, then it saturates at τ ∼ √ z. For the exponentially decaying medium, the rate saturates at slightly larger times compared to the static case, which is a consequence of limiting behavior of the ratio of Bessel's function in Eq. (2.17). In contrast, the rate for Bjorken expanding medium converges to zero at large times. While the ratio of Bessel's function in Eq. (2.24) converges to unity for τ ∼ z/τ 0 , the presence of the factor τ 0 /(τ 0 + τ ) leads to the dumping of the splitting rate for τ > τ 0 . The values of K(z, τ ) at large τ are larger for the exponential case than for the Bjorken case due to a consistent treatment of length of the medium (see definition Eq. (2.15)). As discussed above, this mismatch can in principle be corrected by the proper redefinition of the profile defined in Eq. (2.15).
Finally, Fig. 4 shows a comparison of K(z, τ ) for two values of τ , τ = 0.05 and τ = 0.5, plotted as a function of the momentum fraction z. While at high values of z the rates for different profiles differ significantly, at the low-z values they all have the same, universal slope which is a consequence of the P (z)κ(z) factor present in splitting rates of all the profiles which diverges for z → 0 as z −3/2 . We therefore expect to recover a universal behavior of the resulting parton branching evolution for expanding media in the soft gluon regime.

Rate equation for expanding medium
Equipped with the rate of emissions, we can now turn to the task of resumming multiple gluon emissions in the medium. In a large medium, possible interference terms are suppressed [44,45], and the resummation is enforced via a kinetic rate equation. The evolution equation for the distribution of medium-induced gluons, D(x, τ ) = x dN/dx, is given by [38,39] ∂D(x, τ ) ∂τ = The initial value of the D(x, τ ) is a δ-function at x = 1 which characterizes the initial single color charge entering the evolving medium. Furthermore, conservation of energy implies While this is formally violated due to the soft singularity at x = 0, this can be reinstated by assuming the accumulation of energy at the thermal scale x ∼ T /p where elastic rescattering leads to thermalization [42]. The rate equation Eq. (3.1) was solved numerically for the static medium, exponentially decaying medium, and the Bjorken case introduced in Sec. 2. In this section and onwards, the medium parameters areq 0 = 1 GeV 2 /fm and L = 4 fm if not indicated otherwise. The resulting distributions of D(x) are shown for three representative values of τ in Fig. 5, see figure caption for further details. Despite the differences observed in the rates, at low x, all the D(x) distributions converge to a universal scaling with 1/ √ x which is a consequence of the low-x behavior discussed in Sec. 2.4 driven by a presence of factors P (z)κ(z) in all the splitting rates. The magnitude of the effects is different and is expected to scale with the average parameter q which is hierarchically q exp > q static > q Bjork . At high-x, which predominantly drives the jet suppression factor, see Sec. 4, the reduction of D(x, τ ) is the strongest for the exponentially decaying medium and the weakest for the Bjorken Medium induced gluon distribution D(x) for three different values of τ and four types of medium expansion calculated numerically, static soft (dashed black), static (solid blue), exponential (dashed orange), and Bjorken (dashed-dotted red), and for the soft limit of the static medium calculated analytically (solid black). Left panels show the full distribution, right panels zoom in the high-x region. case, according to the established hierarchy of q for the parameters used here. We have not been able to extract any further scaling properties.
As a check of our numerical routine, we have also evaluated the distribution for the static soft limit, i.e. where we use Eq. (2.14) as the splitting rate. These results can be compared with the known analytical solution [38], whereτ =ᾱτ . The numerical and analytical results are plotted in Fig. 5 as the solid (black) and dashed (black) curves and we see a good agreement over a wide range in x. In this situation, the energy stored in the spectrum decreases exponentially with τ , E(τ ) = 1 0 dx D(x, τ ) = e −πτ 2 [38]. Small deviation between the numerical and analytical results impacting the physics observables discussed in the next section are covered by systematic uncertainties obtained from varying the discreet parameterization in the numerical calculation.

Moments of D(x, τ ) and the jet suppression factor
One of key observables quantifying the inclusive jet suppression is the jet nuclear modification factor, measured by LHC experiments [1][2][3][4]. The yield for the inclusive jet suppression can be obtained as a convolution of the D(x, τ ) distribution with the initial parton spectra, see e.g. [24,33,46]. Note that the evolution time τ now depends on the unknown initial energy of the parton. The initial parton spectra can be approximated by a power law, dσ 0 /dp T ∝ p −n T . In this case, the jet suppression factor Q AA (p T ) = dσ AA /dp T dσ 0 /dp T , is where now τ = q 0 /p T L, as before. In the case of the Bjorken model, the distribution has additionally a dependence on the initial time τ 0 , D(x, τ 0 , τ ), that is also rescaled by the (unknown) initial p T . However, in our current implementation, we have not implemented this exact dependence and choose to show an uncertainty band related to the starting time τ 0 . 3 For our phenomenological applications, we include only one parton species (gluons) in the hard spectrum with n = 5.6 [47]. The distribution D(x, τ ) is found by a numerical solution to Eq. (3.1), as described in Sec. 3. We start our discussion by fixing a common reference value of the jet quenching parameter at initial timeq 0 = 1 GeV 2 /fm = 0.2 GeV 3 for all three profiles. We show the resulting Q AA distributions in Fig. 6, plotted for two values of ω c = 60 GeV (left) and ω c = 100 GeV (right), where ω c =q 0 L 2 /2. From this we extract the path-length in the medium to be L = 5 fm and 6.3 fm, respectively. Apart from the common value ofq 0 at initial time, a large difference can be seen for different media due to the varying rate of expansion in Fig. 6. In order to guide the eye, we have also plotted experimental data for high-p T (anti-k t , R = 0.4) jet suppression [2]. As an illustration, we will use the ω c = 100 GeV andq 0 = 0.2 GeV 3 values extracted from the static medium, see Fig. 6 (right), as reference values for the remaining studies. 3 Error bands shown on the plots in this section were obtained by varying the parameters of numerical procedures such as the number of points on a discreet grid or the resolution parameter, and, for the Bjorken case, by varying τ0 by a factor of 4 up and down starting from τ0 = 0.03 at p T 50 GeV.  Figure 7. The jet suppression factor for four types of medium expansion calculated numerically, static soft (dashed line), static (blue), exponential (orange), and Bjorken (red), and the soft limit of the static medium calculated analytically (full line). On the left: Q AA calculated for q static = q exp = q Bjork . On the right: Q AA for valuesq 0 =q 0,optimal fitted to experimental data from ATLAS [2], keeping ω c = constant. See legend or Tab. 1 for reference values.
The impact of the difference in the type of the medium expansion on the spectrum can to a certain degree be scaled out by replacing theq 0 parameter by the average q , given by Eq. (2.6), for a given medium profile. While for the static medium we trivially have that the average jet quenching parameter equals the initial one, q =q 0 , this does not hold for the case of expanding media where q exp ≈ 2q 0 for an exponential medium profile and q Bjork ≈ 0.05q 0 for the Bjorken-like profile. The left panel of Fig. 7 shows the resulting suppression for the same q for the three cases (corresponding toq 0,scaled = 0.2 (static), 0.4 (exponential), and 0.01 (Bjorken) GeV 3 , respectively, resulting in q static = q exp = q Bjork ). Since the dependence of dI/dz on τ is different for different types of the medium expansion, kernels of the evolution as well as resulting D(x, τ ) distribution may differ even if the dI/dz have the same dependence on z. This is indeed seen in the left Quenching parameter [GeV 3 ] static exponential Bjorken q 0 0.2 0.2 0.2 q 0,scaled 0.2 0.1 4 q 0,optimal 0.2 0.05 8 Table 1. Table showing a comparison of the values of the jet quenching coefficients at initial time t 0 ,q 0 , for the different medium profiles. The first row corresponds to the reference values used in Fig. 6. The second row corresponds to the re-scaled values ofq 0 given the same average value of q for the three cases, see Fig. 7 (left). Finally, the values in the third row corresponds to the optimal initial densities found through a χ 2 minimization to fit the data on jet suppression, see Fig. 7 (right).
panel of Fig. 7 where the Q AA factors for different types of the medium expansion differ by roughly a factor ∼ 1.3 − 2 (at high-and low-p T ) between the largest and the smallest quenching factors.
On the other hand, it is possible to find an effective quenching parameter,q 0,optimal independently in the three scenarios that minimizes the difference among the jet Q AA factors. The values ofq 0,optimal were obtain by a χ 2 minimization of the difference between the jet suppression data and theory calculations by keeping ω c =q 0 L 2 /2 = 100 GeV fixed. The resulting distributions of jet Q AA are shown in the right panel of Fig. 7 withq 0,optimal being approximately 0.2, 0.05, and 8 GeV 3 for static soft, exponential and Bjorken type of the expansion, respectively. Note that the values ofq 0,scaled andq 0,optimal are quite different for expanding media, see Tab. 1. We suspect that this difference could also be affected by the steepness of the hard spectrum. This exercise shows that, having fixed the relevant medium parameters, the p T -behavior is quite similar for the different medium expansion scenarios (small differences can perhaps be seen at high-p T ).

Conclusions & outlook
Three types of the medium expansion were studied within the framework of multiple soft scattering, namely the static medium, exponentially decaying medium and Bjorken-like expanding medium. The spectra and rates of induced gluon emissions were evaluated and single-inclusive gluon distributions were calculated by means of numerical solutions of the evolution equation, see Eq. (3.1). One of the main results of our study is that the evolved distributions obey a 1/ √ x scaling for all the studied types of expansion signaling a universal behavior with reduced sensitivity to the details of the medium expansion.
Single-inclusive gluon distributions were then used to calculate the jet suppression factor Q AA , see Eq. (4.2). It was demonstrated that the impact of the medium expansion cannot be scaled out by using the average quenching parameter, see Fig. 7 (left). Hence, although many features of the medium-induced spectrum, see Fig. 2, and the single-inclusive distributions at low-x, see Fig. 5, are common to all studied medium expansion scenarios, the details of the high-x behavior of D(x, τ ), that ultimately drives the quenching factor Q AA , is still relatively sensitive to the details of medium expansion. Values of quenching parameter at initial timeq 0,optimal that minimize the differences in the jet Q AA among the different types of the expansion were also found. Large difference between the theoretically motivated scaledq 0 values andq 0,optimal , see also Tab. 1, indicate the importance of taking into account the medium expansion in precise modeling of the jet quenching phenomenon.
The extracted values of the initialq 0 obtained in this paper cannot be taken at face value, given the fitting procedure described above. Several improvements, such as including proper quark and gluon jet fractions [47], using a comprehensive emission rate [28,32], accounting for quark and gluon coupled induced branching [41] and including the effect of in-medium jet fragmentation (Sudakov suppression) [48], are planned to be included for future phenomenological applications of other observables, such as v 2 at high-p T .