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We consider the kinetic wave equation, or phonon Boltzmann equation, set on the torus (physical system set on the lattice). We describe entropy maximizers for fixed mass and energy; our framework is very general, being valid in any dimension, for any dispersion relation, and even including the quantum kinetic wave equation. Of particular interest is the presence of condensation in certain regimes which we characterize., Comment: 10 pages, 1 figure

QTRAJ is a computer code that simulates the propagation of quarkonium in the quark-gluon plasma (QGP) based on the quantum-trajectory algorithm. This algorithm solves a master equation in which the quarkonium is treated as an open quantum system (OQS). A major advantage of this approach is that it turns a 3D spatial evolution for a density matrix into a 1D Schr\"odinger equation for a wavefunction with a non-hermitian Hamiltonian, drastically reducing the computational cost. So far, the interaction implemented in the master equation was obtained within the framework of potential non-relativistic QCD (pNRQCD), and restricted to the regime $rT \ll 1$, where $r$ is the size of the color dipole and $T$ is the temperature. In the environment produced in heavy-ion collisions (HIC's) this limit is accurate for $\Upsilon(1S)$, but the applicability to other quarkonium states is dubious. In the present study we generalize the above approach, extending it to the regime $rT\!\sim\! 1$ in the one-gluon exchange approximation, with proper Hard Thermal Loop (HTL) resummation of medium effects. This is done by implementing new jump operators connecting different color states of the $Q\bar Q$ pair and expanding them in plane waves, giving rise to a variation of the algorithm present in QTRAJ 1.0. Here we provide an overview of this approach comparing the $rT \ll 1$ and $rT\sim 1$ cases, and we discuss prospects for phenomenological application to excited states of bottomonium., Comment: Proceedings of the Conference QNP 2024

We consider the singular Rayleigh-Jeans equilibrium of the $4$-waves kinetic turbulence equation for the three dimensional Schr\"{o}dinger equation. We first show the formation in finite time of a Dirac measure at zero frequency in the solution of the wave kinetic equation when the initial data has the form of Rayleigh-Jeans, truncated at large values of the energy. The initial value problem for the linearization around the singular Rayleigh-Jeans equilibria is then solved in several functional spaces. Then, long time convergence to a Dirac measure at the origin is described in detail for some of the solutions. This determines a basin of attraction of the Dirac measure., Comment: 43 pages, 2 figures

We study the potential of $X(3872)$ at finite temperature in the Born-Oppenheimer approximation under the assumption that it is a tetraquark. We argue that, at large number of colors, it is a good approximation to assume that the potential consists in a real part plus a constant imaginary term. The real part is then computed adapting an approach by Rothkopf and Lafferty and using as input lattice QCD determinations of the potential for hybrids. This model allows us to qualitatively estimate at which temperature range the formation of a heavy tetraquark is possible, and to propose a qualitative picture for the dissociation of the state in a medium. Our approach can be applied to other suggested internal structures for the $X(3872)$ and to other exotic states., Comment: 12 pages, 7 figures

We describe regularizing effects in the linearization of a kinetic equation that arises in study of a system of nonlinear waves satisfying the Schr\"odinger equation in terms of weak turbulence and condensate. The problem is first considered in spaces of bounded functions with weights, where existence of solutions and some first regularity properties are proved. After a suitable change of variables the equation is written in terms of a pseudo differential operator. Homogeneity of the equation and classical arguments of freezing of coefficients may then be used to prove regularizing effect in local Sobolev type spaces.

The evolution of hard probes in a medium is a complex multiscale problem that significantly benefits from the use of Effective Field Theories (EFTs). Within the EFT framework, we aim to define a series of EFTs in a way that addresses each energy scale individually in separate steps. However, studying hard probes in a medium presents challenges. This is because an EFT is typically constructed by formulating the most general Lagrangian compatible with the problem's symmetries. Nevertheless, medium effects may not always be encoded adequately in an effective action. In this paper, we construct an EFT that is valid for studying the evolution of a heavy quark in a QCD plasma containing few other heavy quarks, where degrees of freedom with an energy of the order of the temperature scale are integrated out. Through this example, we explicitly demonstrate how to handle the doubling of degrees that arise in non-equilibrium field theory. As a result, we derive a Fokker-Planck equation using only symmetry and power counting arguments. The methods introduced in this paper will pave the way for future developments in the study of quarkonium suppression., Comment: 34 pages. Matches published version

sPHENIX is a next-generation detector experiment at the Relativistic Heavy Ion Collider, designed for a broad set of jet and heavy-flavor probes of the Quark-Gluon Plasma created in heavy ion collisions. In anticipation of the commissioning and first data-taking of the detector in 2023, a RIKEN-BNL Research Center (RBRC) workshop was organized to collect theoretical input and identify compelling aspects of the physics program. This paper compiles theoretical predictions from the workshop participants for jet quenching, heavy flavor and quarkonia, cold QCD, and bulk physics measurements at sPHENIX., Comment: 60 pages, summary paper of the "Predictions for sPHENIX" RBRC workshop held at BNL in July 2022, published version

We demonstrate the importance of quantum jumps in the nonequilibrium evolution of bottomonium states in the quark-gluon plasma. Based on nonrelativistic effective field theory and the open quantum system framework, we evolve the density matrix of color singlet and octet pairs. We show that quantum regeneration of singlet states from octet configurations is necessary to understand experimental results for the suppression of both bottomonium ground and excited states. The values of the heavy-quarkonium transport coefficients used are consistent with recent lattice QCD determinations., Comment: 5 pages, 7 figures, 1 page supplement; v2 - minor changes, updated references, published version

This paper presents a comprehensive review of both the theory and experimental successes of Quantum Chromodynamics, starting with its emergence as a well defined theory in 1972-73 and following developments and results up to the present day. Topics include a review of the earliest theoretical and experimental foundations; the fundamental constants of QCD; an introductory discussion of lattice QCD, the only known method for obtaining exact predictions from QCD; methods for approximating QCD, with special focus on effective field theories; QCD under extreme conditions; measurements and predictions of meson and baryon states; a special discussion of the structure of the nucleon; techniques for study of QCD at high energy, including treatment of jets and showers; measurements at colliders; weak decays and quark mixing; and a section on the future, which discusses new experimental facilities or upgrades currently funded. The paper is intended to provide a broad background for Ph.D. students and postdocs starting their career. Some contributions include personal accounts of how the ideas or experiments were developed., Comment: Invited volume for the EJPC; 567 pages if text/figures and 4783 references occupying about 160 additional pages. arXiv abstract abridged, for the complete abstract please see the full text

Short time existence of classical solutions is proved for a system of equations that involves a three excitations kinetic operator. The system is related to the description of a gas of bosons below but close to the critical temperature, where the three excitations integral aims at describing the interaction between the particles in the condensate and the excitations in the normal gas. Some qualitative properties of the solutions are obtained., Comment: Some of the results are not correct. Revision is needed

Using the potential non-relativistic quantum chromodynamics (pNRQCD) effective field theory, we derive a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix that is accurate to next-to-leading order (NLO) in the ratio of the binding energy of the state to the temperature of the medium. The resulting NLO Lindblad equation can be used to more reliably describe heavy-quarkonium evolution in the quark-gluon plasma at low temperatures compared to the leading-order truncation. For phenomenological application, we numerically solve the resulting NLO Lindblad equation using the quantum trajectories algorithm. To achieve this, we map the solution of the three-dimensional Lindblad equation to the solution of an ensemble of one-dimensional Schr\"odinger evolutions with Monte-Carlo sampled quantum jumps. Averaging over the Monte-Carlo sampled quantum jumps, we obtain the solution to the NLO Lindblad equation without truncation in the angular momentum quantum number of the states considered. We also consider the evolution of the system using only the complex effective Hamiltonian without stochastic jumps and find that this provides a reliable approximation for the ground state survival probability at LO and NLO. Finally, we make comparisons with our prior leading-order pNRQCD results and experimental data available from the ATLAS, ALICE, and CMS collaborations., Comment: 40 pages, 8 figures

The Cauchy problem for the linearization of a system of equations arising in the kinetic theory of a condensed gas of bosons near the critical temperature around one of its equilibria is solved for radially symmetric initial data. It is proved that the linearized system has global classical solutions that satisfy the natural conservation laws for a large set of initial data. Some regularity properties of the solutions and their long time asymptotic behavior are described.

The linearization around one of its equilibrium of a system that describes the correlations between the superfluid component and the normal fluid part of a condensed Bose gas in the approximation of very low temperature and small condensate density, is studied. A simple and transparent argument gives a necessary and sufficient condition on the initial data for the existence of global solutions satisfying the conservation of the total number of particles and energy. Their convergence to a suitable stationary state is also shown and rates of convergence for the normal fluid and superfluid components are obtained.

The present paper provides a new representation of the solution to the fragmentation equation as a power series in the Banach space of Radon measures endowed with the total variation norm. This representation is used to justify how the fragmentation kernel, which is one of the two key parameters of the fragmentation equation, can be recovered from short-time experimental measurements of the particle size distributions when the initial condition is a delta function. A new stability result for this equation is also provided using a Wasserstein-type norm. We exploit this stability to prove the robustness of our reconstruction formula with respect to noise and initial data.

Quarkonium suppression is one of the more useful observables to obtain information about the hot medium created in ultrarelativistic heavy-ion collisions. In this manuscript, we discuss a simple way to implement both the initial-state effects and the hot-medium evolution, and to compute the quarkonium nuclear modification factor if the survival probability for a bound state at a given energy density is known. Based on the Glauber model, the temperature of the evolving medium and the centrality dependence of the nuclear modification factor will be analysed., Comment: 22 pages, 4 figure. Updated to match published version

We report predictions for the suppression and elliptic flow of the $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ as a function of centrality and transverse momentum in ultra-relativistic heavy-ion collisions. We obtain our predictions by numerically solving a Lindblad equation for the evolution of the heavy-quarkonium reduced density matrix derived using potential nonrelativistic QCD and the formalism of open quantum systems. To numerically solve the Lindblad equation, we make use of a stochastic unraveling called the quantum trajectories algorithm. This unraveling allows us to solve the Lindblad evolution equation efficiently on large lattices with no angular momentum cutoff. The resulting evolution describes the full 3D quantum and non-abelian evolution of the reduced density matrix for bottomonium states. We expand upon our previous work by treating differential observables and elliptic flow; this is made possible by a newly implemented Monte-Carlo sampling of physical trajectories. Our final results are compared to experimental data collected in $\sqrt{s_{NN}} = 5.02$ TeV Pb-Pb collisions by the ALICE, ATLAS, and CMS collaborations., Comment: 14 pages, 11 figures

We introduce an open-source package called QTraj that solves the Lindblad equation for heavy-quarkonium dynamics using the quantum trajectories algorithm. The package allows users to simulate the suppression of heavy-quarkonium states using externally-supplied input from 3+1D hydrodynamics simulations. The code uses a split-step pseudo-spectral method for updating the wave-function between jumps, which is implemented using the open-source multi-threaded FFTW3 package. This allows one to have manifestly unitary evolution when using real-valued potentials. In this paper, we provide detailed documentation of QTraj 1.0, installation instructions, and present various tests and benchmarks of the code., Comment: 35 pages, 10 figures; v2 - minor typos fixed; published version

The study of heavy quarkonium suppression in heavy-ion collisions represents an important source of information about the properties of the quark-gluon plasma produced in such collisions. In a previous paper, we have considered how to model the evolution of a quarkonium in such a way that the solution of the resulting equations evolves toward the correct thermal equilibrium distribution for an homogeneous and static medium. We found that it is crucial to take into account the energy gap between singlet and octet configurations when the temperature is not much greater than this energy gap. In this manuscript, we explore in more detail the phenomenological consequences of this observation in the more realistic situation of an expanding system. We consider two different scenarios, based on the same approximation scheme, but on different choices of parameters. In the first case, we rely on a Hard Thermal Loop approximation, while the second case is based on a recent determination of the static potential in lattice QCD. In both cases, we compute the decay width and the nuclear modification factor, both taking the energy gap into account and ignoring it. We find that the impact on the predictions is as large in the expanding medium as it is in the static case. Our conclusion is that this energy gap should be taken into account in phenomenological studies., Comment: Published version. 25 pages, 11 figures

We solve the Lindblad equation describing the Brownian motion of a Coulombic heavy quark-antiquark pair in a strongly coupled quark-gluon plasma using the highly efficient Monte Carlo wave-function method. The Lindblad equation has been derived in the framework of pNRQCD and fully accounts for the quantum and non-Abelian nature of the system. The hydrodynamics of the plasma is realistically implemented through a 3+1D dissipative hydrodynamics code. We compute the bottomonium nuclear modification factor and compare with the most recent LHC data. The computation does not rely on any free parameter, as it depends on two transport coefficients that have been evaluated independently in lattice QCD. Our final results, which include late-time feed down of excited states, agree well with the available data from LHC 5.02 TeV PbPb collisions., Comment: 47 pages, 20 figures, corrections to feed down matrix leading to slight change in phenomenological predictions, journal version

We study the transitions between the different color states of a static quark-antiquark pair, singlet and octet, in a thermal medium. This is done non-perturbatively exploiting the infinite mass limit of QCD. This study is interesting because it can be used for future developments within the framework of Effective Field Theories (EFTs) and because it can be combined with other techniques, like lattice QCD or AdS/CFT, to gain non-perturbative information about the evolution of quarkonium in a medium. We also study the obtained expressions in the large $N_{c}$ limit. This allows us to learn lessons that are useful to simplify phenomenological models of quarkonium in a plasma., Comment: 32 pages, 14 figures. Matches published version

The fundamental solution of the classical approximation of a three waves kinetic equation that happens in the kinetic theory of a condensed gas of bosons near the critical temperature is obtained. It is also proved to be unique in a suitable space of distributions and several of its properties are described. The fundamental solution is used to solve the initial value problem for a general class of initial data., Comment: 54 pages. One Proposition and two references have been added. Some misprints have been corrected

We study exclusive quarkonium production in the dipole picture at next-to-leading order (NLO) accuracy, using the non-relativistic expansion for the quarkonium wavefunction. This process offers one of the best ways to obtain information about gluon distributions at small $x$, in ultraperipheral heavy ion collisions and in deep inelastic scattering. The quarkonium light cone wave functions needed in the dipole picture have typically been available only at tree level, either in phenomenological models or in the nonrelativistic limit. In this paper, we discuss the compatibility of the dipole approach and the non-relativistic expansion and compute NLO relativistic corrections to the quarkonium light-cone wave function in light-cone gauge. Using these corrections we recover results for the NLO decay width of quarkonium to $e^{+}e^{-}$ and we check that the non-relativistic expansion is consistent with ERBL evolution and with B-JIMWLK evolution of the target. The results presented here will allow computing the exclusive quarkonium production rate at NLO once the one loop photon wave function with massive quarks, currently under investigation, is known., Comment: 33 pages. 9 figures. Updated to incorporate the changes made to the published version

The in medium dynamics of heavy particles are governed by transport coefficients. The heavy quark momentum diffusion coefficient, $\kappa$, is an object of special interest in the literature, but one which has proven notoriously difficult to estimate, despite the fact that it has been computed by weak-coupling methods at next-to-leading order accuracy, and by lattice simulations of the pure SU(3) gauge theory. Another coefficient, $\gamma$, has been recently identified. It can be understood as the dispersive counterpart of $\kappa$. Little is known about $\gamma$. Both $\kappa$ and $\gamma$ are, however, of foremost importance in heavy quarkonium physics as they entirely determine the in and out of equilibrium dynamics of quarkonium in a medium, if the evolution of the density matrix is Markovian, and the motion, quantum Brownian; the medium could be a strongly or weakly coupled plasma. In this paper, using the relation between $\kappa$, $\gamma$ and the quarkonium in medium width and mass shift respectively, we evaluate the two coefficients from existing 2+1 flavor lattice QCD data. The resulting range for $\kappa$ is consistent with earlier determinations, the one for $\gamma$ is the first non-perturbative determination of this quantity., Comment: 8 pages, 3 figures, journal version

Hard Probes are an essential tool to discover the properties of the quark-gluon plasma created in heavy-ion collisions. The study of hard probes always involves taking into account very different energy scales, and this is precisely the situation in which Effective Fields Theories (EFTs) are useful. EFTs can be used to separate the short-distance and perturbative physics from the long-distance and non-perturbative. This method combined with Lattice QCD evaluations of the long-distance effects can provide accurate and first principles results. In this proceeding, I will report recent advances in this direction. Results from an EFT computation of quarkonium $R_{AA}$ at $\sqrt{s_{NN}}=5.02\,\textit{TeV}$ are shown for the first time here., Comment: 8 pages, 2 figures, for the International Conference on Hard and Electromagnetic Probes of High-Energy Nuclear Collisions, 30 September - 5 October 2018 in Aix-Les-Bains, Savoie, France

We consider the incompressible Navier-Stokes equations in a bounded domain with $\mathcal{C}^{1,1}$ boundary, completed with slip boundary condition. Apart from studying the general semigroup theory related to the Stokes operator with Navier boundary condition where the slip coefficient $\alpha$ is a non-smooth scalar function, our main goal is to obtain estimate on the solutions, independent of $\alpha$. We show that for $\alpha$ large, the weak and strong solutions of both the linear and non-linear system are bounded uniformly with respect to $\alpha$. This justifies mathematically that the solution of the Navier-Stokes problem with slip condition converges in the energy space to the solution of the Navier-Stokes with no-slip boundary condition as $\alpha \to \infty$., Comment: 45 pages

We consider the fragmentation equation $\dfrac{\partial}{\partial t}f (t, x) = --B(x)f (t, x) + \int\_{ y=x}^{ y=\infty} k(y, x)B(y)f (t, y)dy,$ and address the question of estimating the fragmentation parameters-i.e. the division rate $B(x)$ and the fragmentation kernel $k(y, x)$-from measurements of the size distribution $f (t, $\times$)$ at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance (Xue, Radford, Biophys. Journal, 2013) for amyloid fibril breakage. Under the assumption of a polynomial division rate $B(x) = \alpha x^{\gamma}$ and a self-similar fragmentation kernel $k(y, x) = \frac{1}{y} k\_0 (x/ y)$, we use the asymptotic behaviour proved in (Escobedo, Mischler, Rodriguez-Ricard, Ann. IHP, 2004) to obtain uniqueness of the triplet $(\alpha, \gamma, k \_0)$ and a representation formula for $k\_0$. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.

The existence of global solutions for a system of differential equations is proved, and some of their properties are described. The system involves a kinetic equation for quantum particles. It is a simplified version of a mathematical description of a weakly interacting dilute gas of bosons in the presence of a condensate near the critical temperature., Comment: 71 pages

We derive equations of motion for the reduced density matrix of a heavy quarkonium in contact with a quark-gluon plasma in thermal equilibrium. These equations allow in particular a proper treatment of the regime when the temperature of the plasma is comparable to the binding energy of the quarkonium. These equations are used to study how the quarkonium approaches equilibrium with the plasma, and we discuss the corresponding entropy increase, or free energy decrease, depending on the temperature regime. The effect of collisions can be accounted for by the generalization of the imaginary potential introduced in previous studies, and from which collision rates are derived. An important outcome of the present analysis is that this imaginary potential has a sizeable dependence on the energy of the relevant transitions., Comment: 47 pages, 8 figures

We derive equations for the time evolution of the reduced density matrix of a collection of heavy quarks and antiquarks immersed in a quark gluon plasma. These equations, in their original form, rely on two approximations: the weak coupling between the heavy quarks and the plasma, the fast response of the plasma to the perturbation caused by the heavy quarks. An additional semi-classical approximation is performed. This allows us to recover results previously obtained for the abelian plasma using the influence functional formalism. In the case of QCD, specific features of the color dynamics make the implementation of the semi-classical approximation more involved. We explore two approximate strategies to solve numerically the resulting equations in the case of a quark-antiquark pair. One involves Langevin equations with additional random color forces, the other treats the transition between the singlet and octet color configurations as collisions in a Boltzmann equation which can be solved with Monte Carlo techniques., Comment: 62 pages, 14 figures

We perform a comprehensive study of the time evolution of heavy-quarkonium states in an expanding hot QCD medium by implementing effective field theory techniques in the framework of open quantum systems. The formalism incorporates quarkonium production and its subsequent evolution in the fireball including quarkonium dissociation and recombination. We consider a fireball with a local temperature that is much smaller than the inverse size of the quarkonium and much larger than its binding energy. The calculation is performed at an accuracy that is leading-order in the heavy-quark density expansion and next-to-leading order in the multipole expansion. Within this accuracy, for a smooth variation of the temperature and large times, the evolution equation can be written as a Lindblad equation. We solve the Lindblad equation numerically both for a weakly-coupled quark-gluon plasma and a strongly-coupled medium. As an application, we compute the nuclear modification factor for the $\Upsilon(1S)$ and $\Upsilon(2S)$ states. We also consider the case of static quarks, which can be solved analytically. Our study fulfils three essential conditions: it conserves the total number of heavy quarks, it accounts for the non-Abelian nature of QCD and it avoids classical approximations., Comment: Figure 8 added and improved bibliography. Journal version

A growth fragmentation equation with constant dislocation density measure is considered, in which growth and division rates balance each other. This leads to a simple example of equation where the so called Malthusian hypothesis $(M_+)$ of J. Bertoin and A. Watson (2016) is not necessarily satisfied. It is proved that when that happens, and as it was first suggested by these authors, no global non negative weak solution, satisfying some boundedness condition on several of its moments, exist. Non existence of local non negative solutions satisfying a similar condition, is proved to happen also. When a local non negative solution exists, the explicit expression is given.

Maximal $L^p$-$L^q$ regularity is proved for the strong, weak and very weak solutions of the inhomogeneous Stokes problem with Navier-type boundary conditions in a bounded domain $\Omega$, not necessarily simply connected. This extends previous results of the authors (2017).

In the early stages of heavy-ion collisions, the hot QCD matter expands more longitudinally than transversely. This imbalance causes the system to become rapidly colder in the longitudinal direction and a local momentum anisotropy appears. In this paper, we study the heavy-quarkonium spectrum in the presence of a small plasma anisotropy. We work in the framework of pNRQCD at finite temperature. We inspect arrangements of non-relativistic and thermal scales complementary to those considered in the literature. In particular, we consider temperatures larger and Debye masses smaller than the binding energy, which is a temperature range relevant for presently running LHC experiments. In this setting we compute the leading thermal corrections to the binding energy and the thermal width induced by quarkonium gluo-dissociation., Comment: 6 pages, 5 figures

We address the evolution of heavy-quarkonium states in an expanding quark-gluon plasma by implementing effective field theory techniques in the framework of open quantum systems. In this setting we compute the nuclear modification factors for quarkonia that are $S$-wave Coulombic bound states in a strongly-coupled quark-gluon plasma. The calculation is performed at an accuracy that is leading-order in the heavy-quark density expansion and next-to-leading order in the multipole expansion. The quarkonium density-matrix evolution equations can be written in the Lindblad form, and, hence, they account for both dissociation and recombination. Thermal mass shifts, thermal widths and the Lindblad equation itself depend on only two non-perturbative parameters: the heavy-quark momentum diffusion coefficient and its dispersive counterpart. Finally, by numerically solving the Lindblad equation, we provide results for the $\Upsilon(1S)$ and $\Upsilon(2S)$ suppression., Comment: 6 pages, 3 figures

We discuss the evolution of an energetic jet which propagates through a dense quark-gluon plasma and radiates gluons due to its interactions with the medium. Within perturbative QCD, this evolution can be described as a stochastic branching process, that we have managed to solve exactly. We present exact, analytic, results for the gluon spectrum (the average gluon distribution) and for the higher n-point functions, which describe correlations and fluctuations. Using these results, we construct the event-by-event picture of the gluon distribution produced via medium-induced gluon branching. In contrast to what happens in a usual QCD cascade in vacuum, the medium-induced branchings are quasi-democratic, with offspring gluons carrying sizable fractions of the energy of their parent parton. We find large fluctuations in the energy loss and in the multiplicity of soft gluons. The multiplicity distribution is predicted to exhibit KNO (Koba-Nielsen-Olesen) scaling. These predictions can be tested in Pb+Pb collisions at the LHC, via event-by-event measurements of the di-jet asymmetry. Based on JHEP 1605 (2016) 008 and arXiv:1609.06104, Comment: 4 pages, no figures, Hard Probes 2016 proceedings. Based on JHEP 1605 (2016) 008 and arXiv:1609.06104

We study the gluon distribution produced via successive medium-induced branchings by an energetic jet propagating through a weakly-coupled quark-gluon plasma. We show that under suitable approximations the evolution of the jet can be described as a classical stochastic process, which is exactly solvable. For this process, we construct exact analytic solutions for all the n-point correlation functions (the n-body densities in the space of energy). The corresponding results for the one-point and the two-point functions were already known, but those for the higher-point functions are new. These results demonstrate strong correlations associated with the existence of common ancestors in the branching process. By integrating these n-point functions over the gluon energies, we deduce the mean gluon multiplicity $\langle N\rangle$ as well as the higher moments $\langle N^p\rangle$ with $p\ge 2$. We find that the multiplicities of the soft gluons are parametrically large and show a remarkable regularity, known as Koba-Nielsen-Olesen (KNO) scaling: the reduced moments $\langle N^p\rangle/\langle N\rangle^p$ are pure numbers, independent of any of the physical parameters of the problem. We recognize a special negative binomial distribution which is characterized by large statistical fluctuations. These predictions can be tested in Pb+Pb collisions at the LHC, via event-by-event measurements of the di-jet asymmetry., Comment: 31 pages, 7 figures

In this article we consider the Stokes problem with Navier-type boundary conditions on a domain $\Omega$, not necessarily simply connected. Since under these conditions the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stokes operator considered in different functional spaces. We obtain strong, weak and very weak solutions for the time dependent Stokes problem with the Navier-type boundary condition under different hypothesis on the initial data $\boldsymbol{u}_0$ and external force $\boldsymbol{f}$. Then, we study the fractional and pure imaginary powers of several operators related with our Stokes operators. Using the fractional powers, we prove maximal regularity results for the homogeneous Stokes problem. On the other hand, using the boundedness of the pure imaginary powers we deduce maximal $L^{p}-L^{q}$ regularity for the inhomogeneous Stokes problem.

We develop the event-by-event picture of the gluon distribution produced via medium-induced gluon branching by an energetic jet which propagates through a dense QCD medium. A typical event is characterized by the production of a large number of soft gluons which propagate at large angles with respect to the jet axis and which collectively carry a substantial amount of energy. By explicitly computing 2-gluon correlations, we demonstrate the existence of large event-by-event fluctuations, which reflect the stochastic nature of the branching process. For the two quantities that we have investigated -- the energy loss at large angles and the soft gluon multiplicity --, the dispersion is parametrically as large as the respective expectation value. We identify interesting scaling laws, which suggest that the multiplicity distribution should exhibit KNO (Koba-Nielsen-Olesen) scaling. A similar scaling is known to hold for a jet branching in the vacuum, but the medium-induced distribution is found to be considerably broader. We predict that event-by-event measurements of the di-jet asymmetry in Pb+Pb collisions at the LHC should observe large fluctuations in the number of soft hadrons propagating at large angles and also in the total energy carried by these hadrons., Comment: 39 pages, 7 figures

In a model where Majorana neutrinos heavier than the electroweak scale couple to Standard Model Higgs bosons and leptons, we compute systematically thermal corrections to the direct and indirect CP asymmetries in the Majorana neutrino decays. These are key ingredients entering the equations that describe the thermodynamic evolution of the induced lepton-number asymmetry eventually leading to the baryon asymmetry in the universe. We compute the thermal corrections in an effective field theory framework that assumes the temperature smaller than the masses of the Majorana neutrinos and larger than the electroweak scale, and we provide the leading corrections in an expansion of the temperature over the mass. In this work, we consider the case of two Majorana neutrinos with nearly degenerate masses., Comment: Equations (B.25), (B.26) and (B.27) corrected in the parts depending on the SU(2) gauge coupling; final results modified accordingly

Fragmentation and growth-fragmentation equations is a family of problems with varied and wide applications. This paper is devoted to description of the long time time asymptotics of two critical cases of these equations, when the division rate is constant and the growth rate is linear or zero. The study of these cases may be reduced to the study of the following fragmentation equation:$$\frac{\partial}{\partial t} u(t,x) + u(t,x)=\int\limits\_x^\infty k\_0(\frac{x}{y}) u(t,y) dy.$$Using the Mellin transform of the equation, we determine the long time behavior of the solutions. Our results show in particular the strong dependence of this asymptotic behavior with respect to the initial data.

We consider an approximation of the linearised equation of the homogeneous Boltzmann equation that describes the distribution of quasiparticles in a dilute gas of bosons at low temperature. The corresponding collision frequency is neither bounded from below nor from above. We prove the existence and uniqueness of solutions satisfying the conservation of energy. We show that these solutions converge to the corresponding stationary state, at an algebraic rate as time tends to infinity.

In the present article we consider several issues concerning the doubly parabolic Keller-Segel system in the plane, when the initial data belong to critical scaling-invariant Lebesgue spaces. More specifically, we analyze the global existence of integral solutions, their optimal time decay, uniqueness and positivity, together with the uniqueness of self-similar solutions. In particular, we prove that there exist integral solutions of any mass, provided that $\e>0$ is sufficiently large. With those results at hand, we are then able to study the large time behavior of global solutions and prove that in the absence of the degradation term the solutions behave like self-similar solutions, while in presence of the degradation term global solutions behave like the heat kernel.

Heavy quarkonium suppression was proposed long ago as a signal of the formation of a deconfined phase in heavy ion collisions. Originally the mechanism responsible for this suppression was thought to be color screening. However perturbative computations and recent lattice studies suggest the existence of an imaginary part of the potential which could have a more important role for suppression than screening. In this work we review some general aspects of effective field theories for heavy quarkonium in the medium and discuss the physical phenomena behind the imaginary part of the potential and the decay width of heavy quarkonium and their corresponding cross sections., Comment: 5 pages, no figures; Contribution for FAIRNESS 2013 proceedings

Heavy Majorana neutrinos enter in many scenarios of physics beyond the Standard Model: in the original seesaw mechanism they provide a natural explanation for the small masses of the Standard Model neutrinos and in the simplest leptogenesis framework they are at the origin of the baryonic matter of the universe. In this paper, we develop an effective field theory for non-relativistic Majorana particles, which is analogous to the heavy-quark effective theory. Then, we apply it to the case of a heavy Majorana neutrino decaying in a hot and dense plasma of Standard Model particles, whose temperature is much smaller than the mass of the Majorana neutrino but still much larger than the electroweak scale. The neutrino width gets zero-temperature contributions that can be computed from in-vacuum matrix elements, and thermal corrections. Only the latter will be addressed. Symmetry and power counting arguments made manifest by the effective field theory restrict the form of the thermal corrections and simplify their calculation. The final result agrees with recent determinations obtained with different methods. The effective field theory presented here is suitable to be used for a variety of different models involving non-relativistic Majorana fermions., Comment: Published version. 25 pages, 7 figures, references added

We study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schr\"odinger equation. We define suitable concepts of weak and mild solutions and prove local and global well posedness results. Several qualitative properties of the solutions, including long time asymptotics, blow up results and condensation in finite time are obtained. We also prove the existence of a family of solutions that exhibit pulsating behavior., Comment: 3 figures, 82 pages