Solutions of evolution equations for mediuminduced QCD cascades
Abstract
In this paper we present solutions of evolution equations for inclusive distribution of gluons as produced by jet traversing quark–gluon plasma. We reformulate the original equations in such a form that virtual and unresolvedreal emissions as well as unresolved collisions with medium are resummed in a Sudakovtype form factor. The resulting integral equations are then solved most efficiently with use of newly developed Markov Chain Monte Carlo algorithms implemented in a dedicated program called MINCAS. Their results for a gluon energy density are compared with an analytical solution and a differential numerical method. Some results for gluon transversemomentum distributions are also presented. They exhibit interesting patterns not discussed so far in the literature, in particular a departure from the Gaussian behaviour – which does not happen in approximate analytical solutions.
1 Introduction
Quantum chromodynamics (QCD) is the well established theory of strong interactions. However, there are QCD phenomena that still require better understanding. One of such phenomena is jet quenching predicted in [1, 2] and already observed in the context of the RHIC physics [3] (for an overview see [4, 5] and references therein), i.e. stopping of a hadronic jet produced in an early stage of heavy ion collisions and propagating through quarkgluon plasma (QGP) which is formed in a later stage of the collisions. With the LHC being in operation, the jet quenching can be observed at much higher available energies in collisions of lead nuclei [6]. Still one of the open problems is to understand the details of the jet–QGP interaction mechanism and a pattern of energy loss. Various approaches have been proposed which differ in assumptions about properties of plasma and jet–plasma interactions. Examples are: the kinetic theory assuming that the jet–plasma interactions can be described within a weakcoupling regime of QCD [7, 8, 9, 10, 11, 12, 13, 14, 15, 16], the AdS/CFT models where one assumes the plasma to be strongly coupled [17, 18] or the classicalfieldtheorybased approach [19] (for reviews see [7, 20, 21, 22, 23]). Some of the mentioned formalisms are implemented in Monte Carlo event generators [24, 25, 26, 27, 28, 29].
In this paper we look closer at the results obtained in [30, 31] and focus on an analysis of the generation of transverse momenta via cascades of subsequently emitted jets from an energetic jet traversing QGP. In this approach the plasma is modelled by static centres and the jet interacts with it weakly. Using equations for the energy loss of the jet traversing QGP, the authors of [30, 31, 32] found the process to have turbulent properties, i.e. the energy is transported from large values of x to low values of x without being accumulated at intermediate values. In this paper we investigate a more exclusive equation, i.e. the equation which describes time evolution of longitudinal as well as transverse momenta distributions of gluons emitted from the energetic jet. So far this equation has not been solved numerically, and the analytical as well as numerical analyses are limited to some special cases [31, 33, 34] where for instance part the equation leading to broadening of transverse momentum is simplified or it is included as an input distribution. The simplified analysis suggests that the distribution of gluon transverse momenta being a solution of the equation remains Gaussian [33]. We, however, find that the exact numerical solution of the evolution equation given in Ref. [31] is not Gaussian. Furthermore, the numerical method allows to test consequences of the assumptions about properties of the medium for distribution patterns of the jets emitted from the hard jet.
The paper is organised as follows. In Sect. 2, we introduce and overview the equations for the jet energy distribution and for the inclusive gluon distribution. In Sect. 3, we present reformulation of the above equations making use of Sudakovlike resummation, i.e. we resum virtual and unresolvedreal emissions as well as unresolved collisions with the medium of minijets from the highly energetic jet in form of a Sudakovtype form factor. Then, we provide formal iterative solutions of these equations. In Sect. 4, we propose a Markov Chain Monte Carlo (MCMC) algorithms for numerical solutions of the above equations. In Sect. 5, we describe a numerical algorithm for solving the integrodifferential equation for the jet energy distribution which is based on application of the Runge–Kutta method and discuss its limitations in obtaining high accuracy solutions. In Sect. 6, first, we present numerical results from the MCMC method for the jet energy distribution and compare them with an analytical solution as well as with results from the differential Runge–Kuttabased method. Then, we show and discuss some results for the jet transversemomentum distributions obtained with the MCMC method. We summarise our work and present its outlook in Sect. 7. Finally, in Appendix A we provide some further details on the MCMC algorithms, in particular we describe a combination of the branching Monte Carlo method with the importance sampling.
2 Evolution equations
3 Integral equations and iterative solutions
4 Markov Chain Monte Carlo algorithms
The formal solutions given in Eqs. (19) and (21) can be used to develop Markov Chain Monte Carlo algorithms for numerical evaluation of the distribution functions \(D(x,\tau )\) and \(D(x,\mathbf {k},\tau )\), given some initial functions \(D(x_0,\tau _0)\) and \(D(x_0,\mathbf {k}_0,\tau _0)\), respectively.
 with the probability \(p = \Phi (x_{i1})/\Psi (x_{i1})\) generate \(z_i\) according to the density function \(\zeta (z_i)\):and set \(\mathbf {q}_i = \mathbf {0}\),$$\begin{aligned} \zeta (z_i) = \frac{z_i\mathcal{K}(z_i)}{\kappa (\epsilon )}, \quad \kappa (\epsilon ) = \int _0^{1\epsilon } dz\,z\mathcal{K}(z), \end{aligned}$$(26)
 otherwise, i.e. with the probability \(1p\), set \(z_i=1\) and generate \(\mathbf {q}_i\) according to the density function$$\begin{aligned} \omega (\mathbf {q}_i) = \frac{1}{W}\,t^* \, \frac{w(\mathbf {q}_i)}{ (2\pi )^2}, \quad \mathbf {q}_i \ge q_{\mathrm {min}}\, . \end{aligned}$$(27)

Step 1 Start a random walk from the point \(\tau _0\). First generate the variables \(x_0\in [0,1]\) and \(\mathbf {k}_0\) according to the probability density \(\eta (x_0,\mathbf {k}_0)\), then generate \(\tau _1 \in [\tau _0,+\infty )\) according to the probability density \(\uprho (\tau _1)\). If \(\tau _1 > \tau \), set \(x=x_0,\,\mathbf {k}=\mathbf {k}_0\) and stop the random walk, otherwise go to step 2.

Step 2 Generate the variables \(z_1\in [0,1]\) and \(\mathbf {q}_1> \mathbf {q}_{\mathrm {min}}\) according to the probability density \(\xi (z_1,\mathbf {q}_1)\) and calculate \(x_1=z_1x_0,\, \mathbf {k}_1=\mathbf {q}_1 +z_1\mathbf {k}_0\). Then generate \(\tau _2 \in [\tau _1,+\infty )\) according to the probability density \(\uprho (\tau _2)\): if \(\tau _2 > \tau \), set \(x=x_1,\,\mathbf {k}=\mathbf {k}_1\) and stop the random walk, otherwise go to step 3.

...

Step n Generate the variables \(z_n\in [0,1]\) and \(\mathbf {q}_n> \mathbf {q}_{\mathrm {min}}\) according to the probability density \(\xi (z_n,\mathbf {q}_n)\) and calculate \(x_n=z_nx_{n1},\,\mathbf {k}_n=\mathbf {q}_n +z_n\mathbf {k}_{n1}\). Then generate \(\tau _{n+1} \in [\tau _n,+\infty )\) according to the probability density \(\uprho (\tau _{n+1})\): if \(\tau _{n+1} > \tau \), set \(x=x_n,\mathbf {k}=\mathbf {k}_n\) and stop the random walk, otherwise go to step \(\mathbf{n}+1\).

...
One can formally prove that the above algorithm gives a correct solution to Eq. (31), i.e. that the expectation value of a MC weight associated with a random walk trajectory, as described above, is equal to the function \(D(x,\mathbf {k},\tau )\). We skip such a proof here – it will be provided in our future publication dedicated to the MCMC algorithm and its implementation.
In the above MCMC algorithm we have assumed that all random variables can be generated according to the respective probability distribution functions using standard Monte Carlo techniques, preferably the analytical inversetransform method or its combination with the branching method. Among the integration variables in Eq. (31) the most problematic is the variable z because its probability distribution function \(\zeta (z)\) is too complicated to be sampled with the above methods. Details on how to deal with this using a combination of the branching method with the importance sampling are given in Appendix A.
The MCMC algorithm for solving Eq. (21) is analogous to the above – one only needs to set \(w(\mathbf {q}) = 0\) and \(\mathbf {k}_n = \mathbf {0},\, n = 0,1,\ldots \)
5 Differential method
The numerical solution is then advanced in time and reported. Spatial grids used during the simulations have to be very fine to obtain stable and meaningful results. The problem lies in the gain integral (32) where the arguments of \(\mathcal{K}\) and D are reciprocal. When we use the equally distributed fixed grid and try to compute the gain integral with z from the finite subset of gridpoints, we get arguments for D from unequally distributed grid points due to x / z, i.e. a bad approximation of the gain integral as most of the values will be taken from the region close to 0. A substitution does not help here as it will just switch the reciprocal values from one term to the other. To overcome this problematic behaviour, the very fine grid is needed that has an inevitable effect on computational times. One possible solution is the adaptive mesh refinement together with a smart distribution of the gridpoints – this type of approach is still investigated.
6 Numerical results
We have implemented the MCMC algorithms described in Sect. 4 in the Clanguage program called MINCAS (the acronym for MediumINduced CAScades) as two independent MC generators. First, we performed numerical tests of the algorithm for the solution of Eq. (21) by comparing it with the analytical formula of Eq. (8) and with the numerical differential method described in Sect. 5 for the case of the simplified zkernel function, i.e. with \(f(z) = 1\).
In Fig. 2 we show similar results as above, but for the exact zkernel function as given in Eq. (2). The agreement between MINCAS and the differential method is similar as in Fig. 1 which confirms that our numerical solutions of Eq. (7) are also correct for the exact zkernel. Of course, now the analytical solution is away from both of them because it works only for the simplified zkernel – it is shown only for reference. One can see that the x distribution for the exact zkernel differs considerably from the one for the simplified zkernel, particularly in the region of the intermediate x values – the turbulent behaviour of the exact solution is stronger than of the approximate one.
Unfortunately, we could not make comparisons of the \(\mathbf {k}\) distributions with the differential method because it turned out to be inefficient in solving the general evolution Eq. (1). Therefore, in the following we present a few figures with the results from MINCAS only, to show how the the mediuminduced QCD evolution affects transverse gluon momenta.
Finally, in Fig. 6 we show examples of 2D distributions of \(k_x\) vs. \(k_y\) (upper row) and x vs. \(k_T\) (lower row) for the exact z kernel and \(w(\mathbf {q})\) of Eq. (6). The LHS plots present initial distributions, i.e. for \(t = 0\), while the RHS ones the evolved distributions at \(t = 2\,\)fm. One can observe how the initial gluon distributions get ‘diffused’ in x and \(\mathbf {k}\) in the course of the mediuminduced QCD evolution. The apparent departure from the Gaussian \(\mathbf {k}\) distribution can be clearly seen in the upperright plot. In the lower plots the turbulent behaviour of the distribution in the x direction, as discussed above, is also visible.
7 Summary and outlook
In this paper we have obtained numerical solutions of the equations describing the inclusive gluon distribution as produced by a jet the propagating in QGP, given in Ref. [31]. These equations were reformulated as the integral equations which allows for their efficient solution using the newly constructed Markov Chain Monte Carlo algorithms implemented in the dedicated Monte Carlo program MINCAS. The results for the energy density (the x distribution) were crosschecked with algorithm based on a direct numerical solution of the integrodifferential equation by applying the Runge–Kuttabased method, and for the simplified emission kernel also with the exact analytical solution [31]. The MCMC method turns out to be far more efficient in solving the above equations than the differential method.
The resulting distributions of the gluon density as function of the transverse momenta show some new features, not studied so far in the literature on this subject, i.e. the departure, as the evolution time passes, from the initial Gaussian distribution. This is a result of the exact treatment of the gluon transversemomentum broadening due to an arbitrary number of the collisions with the medium together with its shrinking due an arbitrary number of the emission branchings. We observe this behaviour for two different forms of the collision kernel \(w(\mathbf {q})\).
In the future, we plan to study in a more detailed and systematic way a relation of our MCMC solution to the existing approximate solutions as well as to test other possible forms of the collision kernel \(w(\mathbf {q})\) and the quenching parameter \({\hat{q}}\) resulting from them (in the present study, in order to have a correspondence to existing results, we have used the standard value of \({\hat{q}} = 1\, \hbox {GeV}^2\)/fm). This will allow to see how universal the pattern of the gluon distribution in QGP is. For instance, one can use some AdS/CFT models to obtain \(w(\mathbf {q})\). One can also use our MCMCbased method to solve more general versions of Eq. (1) or an even more general kinetic equation (which assumes thermalisation of soft gluons) obtained in Ref. [8], and perform a full partonshower simulation of the final state based on the generated distribution.
Footnotes
Notes
Acknowledgements
We would like to thank Andreas van Hameren, Jacopo Ghiglieri, Bronislav Zakharov for useful comments. KK acknowledges the CERN TH Department for hospitality when a large part of this project was done and for stimulating discussions with Jacopo Ghiglieri, Ulrich Heinz, Alexi Kurkela, Konrad Tywoniuk, Urs Wiedemann, Bin Wu and Korinna Zapp. Furthermore, KK would like to thank Yacine MehtarTani and JeanPaul Blaizot for informative email exchanges.
References
 1.M. Gyulassy, M. Plumer, Jet quenching in dense matter. Phys. Lett. B 243, 432–438 (1990)ADSCrossRefGoogle Scholar
 2.X.N. Wang, M. Gyulassy, Gluon shadowing and jet quenching in A + A collisions at \(\text{ s }**(1/2) = 200\text{GeV }\). Phys. Rev. Lett. 68, 1480–1483 (1992)ADSCrossRefGoogle Scholar
 3.STAR Collaboration, C. Adler et al., Disappearance of backtoback high \(p_{T}\) hadron correlations in central Au+Au collisions at \(\sqrt{s_{NN}} =\) 200GeV. Phys. Rev. Lett. 90, 082302 (2003). arXiv:nuclex/0210033
 4.H. A. Andrews et al., Novel tools and observables for jet physics in heavyion collisions. arXiv:1808.03689
 5.U.A. Wiedemann, Jet quenching in heavy ion collisions. Landolt Bornstein 23, 521 (2010). arXiv:0908.2306 ADSGoogle Scholar
 6.ATLAS Collaboration, G. Aad et al., Observation of a centralitydependent dijet asymmetry in leadlead collisions at \(\sqrt{s_{NN}}=2.77\) TeV with the ATLAS detector at the LHC. Phys. Rev. Lett. 105, 252303 (2010). arXiv:1011.6182
 7.R. Baier, D. Schiff, B.G. Zakharov, Energy loss in perturbative QCD. Ann. Rev. Nucl. Part. Sci. 50, 37–69 (2000). arXiv:hepph/0002198 ADSCrossRefGoogle Scholar
 8.R. Baier, A.H. Mueller, D. Schiff, D.T. Son, ‘Bottom up’ thermalization in heavy ion collisions. Phys. Lett. B 502, 51–58 (2001). arXiv:hepph/0009237 ADSCrossRefGoogle Scholar
 9.S. Jeon, G.D. Moore, Energy loss of leading partons in a thermal QCD medium. Phys. Rev. C 71, 034901 (2005). arXiv:hepph/0309332 ADSCrossRefGoogle Scholar
 10.B.G. Zakharov, Fully quantum treatment of the Landau–Pomeranchuk–Migdal effect in QED and QCD. JETP Lett. 63, 952–957 (1996). arXiv:hepph/9607440 ADSCrossRefGoogle Scholar
 11.B.G. Zakharov, Radiative energy loss of highenergy quarks in finite size nuclear matter and quark–gluon plasma. JETP Lett. 65, 615–620 (1997). arXiv:hepph/9704255 ADSCrossRefGoogle Scholar
 12.B.G. Zakharov, Transverse spectra of radiation processes inmedium. JETP Lett. 70, 176–182 (1999). arXiv:hepph/9906536 ADSCrossRefGoogle Scholar
 13.R. Baier, Y.L. Dokshitzer, S. Peigne, D. Schiff, Induced gluon radiation in a QCD medium. Phys. Lett. B 345, 277–286 (1995). arXiv:hepph/9411409 ADSCrossRefGoogle Scholar
 14.R. Baier, Y.L. Dokshitzer, A.H. Mueller, S. Peigne, D. Schiff, The Landau–Pomeranchuk–Migdal effect in QED. Nucl. Phys. B 478, 577–597 (1996). arXiv:hepph/9604327 ADSCrossRefGoogle Scholar
 15.P.B. Arnold, G.D. Moore, L.G. Yaffe, Photon and gluon emission in relativistic plasmas. JHEP 06, 030 (2002). arXiv:hepph/0204343 ADSCrossRefGoogle Scholar
 16.J. Ghiglieri, G.D. Moore, D. Teaney, Jetmedium interactions at NLO in a weaklycoupled quark–gluon plasma. JHEP 03, 095 (2016). arXiv:1509.07773 ADSCrossRefGoogle Scholar
 17.H. Liu, K. Rajagopal, U.A. Wiedemann, Calculating the jet quenching parameter from AdS/CFT. Phys. Rev. Lett. 97, 182301 (2006). arXiv:hepph/0605178 ADSCrossRefGoogle Scholar
 18.P.M. Chesler, K. Rajagopal, Jet quenching in strongly coupled plasma. Phys. Rev. D 90(2), 025033 (2014). arXiv:1402.6756 ADSCrossRefGoogle Scholar
 19.M.E. Carrington, K. Deja, S. Mrowczynski, Energy loss in unstable quark–gluon plasma. Phys. Rev. C 92(4), 044914 (2015). arXiv:1506.09082 ADSCrossRefGoogle Scholar
 20.Y. MehtarTani, J.G. Milhano, K. Tywoniuk, Jet physics in heavyion collisions. Int. J. Mod. Phys. A 28, 1340013 (2013). arXiv:1302.2579 ADSCrossRefGoogle Scholar
 21.J. Ghiglieri, D. Teaney, Parton energy loss and momentum broadening at NLO in high temperature QCD plasmas. Int. J. Mod. Phys. E 24(11), 1530013 (2015). [,271(2016)], arXiv:1502.03730 ADSCrossRefGoogle Scholar
 22.J.P. Blaizot, Y. MehtarTani, Jet structure in heavy ion collisions. Int. J. Mod. Phys. E 24(11), 1530012 (2015). arXiv:1503.05958 ADSCrossRefGoogle Scholar
 23.W. Busza, K. Rajagopal, W. van der Schee, Heavy ion collisions: the big picture, and the big questions. Ann. Rev. Nucl. Part. Sci. 68, 339–376 (2018). arXiv:1802.04801 ADSCrossRefGoogle Scholar
 24.C.A. Salgado, U.A. Wiedemann, Calculating quenching weights. Phys. Rev. D 68, 014008 (2003). arXiv:hepph/0302184 ADSCrossRefGoogle Scholar
 25.K. Zapp, G. Ingelman, J. Rathsman, J. Stachel, U.A. Wiedemann, A Monte Carlo model for jet quenching. Eur. Phys. J. C 60, 617–632 (2009). arXiv:0804.3568 ADSCrossRefGoogle Scholar
 26.N. Armesto, L. Cunqueiro, C.A. Salgado, QPYTHIA: a mediummodified implementation of final state radiation. Eur. Phys. J. C 63, 679–690 (2009). arXiv:0907.1014 ADSCrossRefGoogle Scholar
 27.B. Schenke, C. Gale, S. Jeon, MARTINI: an event generator for relativistic heavyion collisions. Phys. Rev. C 80, 054913 (2009). arXiv:0909.2037 ADSCrossRefGoogle Scholar
 28.I.P. Lokhtin, A.V. Belyaev, A.M. Snigirev, Jet quenching pattern at LHC in PYQUEN model. Eur. Phys. J. C 71, 1650 (2011). arXiv:1103.1853 ADSCrossRefGoogle Scholar
 29.J. CasalderreySolana, D.C. Gulhan, J.G. Milhano, D. Pablos, K. Rajagopal, A hybrid strong/weak coupling approach to jet quenching. JHEP 10, 019 (2014). [Erratum: JHEP09,175(2015)], arXiv:1405.3864 ADSCrossRefGoogle Scholar
 30.J.P. Blaizot, F. Dominguez, E. Iancu, Y. MehtarTani, Probabilistic picture for mediuminduced jet evolution. JHEP 06, 075 (2014). arXiv:1311.5823 ADSCrossRefGoogle Scholar
 31.J.P. Blaizot, L. Fister, Y. MehtarTani, Angular distribution of mediuminduced QCD cascades. Nucl. Phys. A 940, 67–88 (2015). arXiv:1409.6202 ADSCrossRefGoogle Scholar
 32.L. Fister, E. Iancu, Mediuminduced jet evolution: wave turbulence and energy loss. JHEP 03, 082 (2015). arXiv:1409.2010 CrossRefGoogle Scholar
 33.J.P. Blaizot, Y. MehtarTani, M.A.C. Torres, Angular structure of the inmedium QCD cascade. Phys. Rev. Lett. 114(22), 222002 (2015). arXiv:1407.0326 ADSCrossRefGoogle Scholar
 34.E. Iancu, B. Wu, Thermalization of minijets in a quarkgluon plasma. JHEP 10, 155 (2015). arXiv:1506.07871 ADSCrossRefGoogle Scholar
 35.M. Gyulassy, X.N. Wang, Multiple collisions and induced gluon Bremsstrahlung in QCD. Nucl. Phys. B 420, 583–614 (1994). arXiv:nuclth/9306003 ADSCrossRefGoogle Scholar
 36.S. Jadach, Foam: a general purpose cellular Monte Carlo event generator. Comput. Phys. Commun. 152, 55–100 (2003). arXiv:physics/0203033 ADSCrossRefGoogle Scholar
 37.M. Galassi et al., GNU Scientific Library Reference Manual, 3rd edn. (Network Theory Ltd., Bodmin, 2009)Google Scholar
 38.J. Christiansen, Numerical solution of ordinary simultaneous differential equations of the 1st order using a method for automatic step change. Numer. Math. 14, 317–324 (1970)MathSciNetCrossRefGoogle Scholar
 39.J.P. Blaizot, E. Iancu, Y. MehtarTani, Mediuminduced QCD cascade: democratic branching and wave turbulence. Phys. Rev. Lett. 111, 052001 (2013). arXiv:1301.6102 ADSCrossRefGoogle Scholar
 40.Y.V. Kovchegov, E. Levin, Quantum chromodynamics at high energy, vol. 33 (Cambridge University Press, Cambridge, 2012)CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}