# Top PDF Fluctuations in High-Energy Particle Collisions

### Fluctuations in High-Energy Particle Collisions

In chapter 3 we continue the study of fluctuations in nucleus-nucleus collisions. Here, however, we step away from the quantum world of the previous chapters and consider classical hydrodynamics. As already noted, the medium created in nucleus- nucleus collisions is known to behave like a fluid during the first few fm/c after the collision. This fluid is very hot and very dense, and also strongly interacting. It flows with very little dissipation, meaning that the viscosity of the fluid is small. This is why the fluid produced in nucleus-nucleus collisions is called nearly ideal or perfect. Due to the very high density and temperature of this fluid, it starts expanding and cooling down immediately after it has been produced. This three-dimensional expansion of a viscous fluid is quite complicated to describe. Indeed, for many years, a solution of a fluid expanding only in the longitudinal direction, that is, the direction of the beam axis, was only known to exist. In recent years, fully three-dimensional numerical simulations have been developed. Furthermore, and of great interest to us in chapter 3, an analytical solution has been published. This solution has the three-dimensional fluid expansion that we already mentioned, with the symmetry of the original one-dimensional solution taken into account as well. This three- dimensional solution relies on conformal symmetry, meaning scale invariance. What is typical for conformal transformations is that they preserve angles but not sizes. The fluid produced in nucleus-nucleus collisions is known to be conformal to a good approximation. Therefore, we come close to describing reality during (the beginning of) the hydrodynamic phase. What will be lacking in our analysis, however, are the stages where the fluid has already cooled down quite a bit, and the hydrodynamic description breaks down. This is the domain of hadronization and freeze-out.

### High resolution probe of coherence in low-energy charge exchange collisions with oriented targets

Apprehending the dynamics of nonadiabatic processes in atomic and molecular physics is not an easy task. The initial and final states have generally complex fine struc- tures and even in the case of well-defined boundary con- ditions, multiple pathways can drive the dynamics from the beginning to the end of the process. Beyond the iden- tification of the reaction routes, their relative coherence must explicitely be considered since it monitors the even- tual occurence of interferences and determines the final outcome of the reaction. The degree of coherence, that our (basically classical) intuition cannot easily grasp, is generally revealed by quantum mechanical or semiclas- sical calculations. However continuous advances in co- incidence and probe techniques have opened the way to ’quantum mechanical complete’ coherence experiments [1] which are able to measure not only the reaction prob- abilities but also the relative phase of the transition am- plitudes. Such experiments have been performed for low- energy ion-atom collisions [2–7], electron impact excita- tion of helium [8, 9] and photon-induced ionization of krypton atoms [10]. Nevertheless, in most of the cases involving heavy particle impact, the experimental reso- lution did not allow to scrutinize the details of the scat- tering patterns predicted theoretically.

### Relative particle yield fluctuations in $\text{ Pb-Pb }$ collisions at $\sqrt{s_\mathrm{{NN}}} =2.76\hbox { TeV}$

Abstract First results on K/π, p/π and K/p fluctuations are obtained with the ALICE detector at the CERN LHC as a function of centrality in Pb–Pb collisions at √ s NN = 2.76 TeV. The observable ν dyn , which is defined in terms of the moments of particle multiplicity distributions, is used to quantify the magnitude of dynamical fluctuations of relative particle yields and also provides insight into the correlation between particle pairs. This study is based on a novel experimental technique, called the Identity Method, which allows one to measure the moments of multiplicity distributions in case of incomplete particle identification. The results for p/π show a change of sign in ν dyn from positive

### Measurement of density correlations in pseudorapidity via charged particle multiplicity fluctuations in Au+Au collisions at $\sqrt{s_{NN}}$=200 GeV

Further, one has to take into account the fact that the damping behavior in Eq. (A8) is caused only by the spatial inhomogeneity of the system at a fixed tempera- ture. In realistic collisions and event samples there is no single relevant temperature. For instance, finite central- ity binning adds together a range of fluctuations origi- nating from collisions with different N part . However, in principle these centrality-correlated fluctuations are in- dependent of the thermally-induced spatial fluctuations. In addition, although the self correlation at the zero dis- tance between two sub-volumes in Eq. (A6) was excluded, the self correlation cannot be excluded in the integrated two particle correlation function contained in Eq. (3). We have tried various kind of functional forms for C 2 which contained power terms and also plural correlation lengths. However, we found empirically that just adding the constant term in Eq. (7) produced the best fit results to all data points.

### Measurement of Event Background Fluctuations for Charged Particle Jet Reconstruction in Pb-Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV

Jet quenching has been observed at RHIC [12, 13, 14, 15] and at the LHC [16] via the measurement of high p t hadron inclusive production and correlations, which are observed to be strongly suppressed in central A–A collisions compared to a scaled pp reference. These high p t hadron observables have been the major tool for measuring the energy loss of hard scattered partons and thereby the properties of the medium, but they provide only indirect and biased information on the parton evolution in the medium. The aim of full jet reconstruction is to measure jet modifications due to energy loss in an unbiased way [17, 18]. Already first measurements of reconstructed jets in heavy-ion collisions at the LHC showed an energy imbalance between back-to-back dijets, which is attributed to jet quenching [9, 19].

### Approach to equilibrium in high energy heavy ion collisions

2.4. LOOKING AT A PROTON, SECOND PART 23 simplistic: it is in fact only valid when the proton is at rest. When the proton is moving at almost the speed of light, the picture is drastically changed. There are indeed other partons (a generic term regrouping the gluons and the quarks) that appear inside the proton. They are the sea quarks 3 and the gluons, which are the mediators of the strong interaction, as the photons are the mediators of the electromagnetic force. Why is it so? This is related to one of the most important principles in quantum theory called the Heisenberg uncertainty principle. What this principle essentially states is that it is not possible to know exactly at the same time the position and the speed of a particle, nor is it possible to know exactly its energy at a given time. This last uncertainty implies that on sufficiently small time scales - and in a heavy ion collisions, we are talking about a few times 10 − 24 seconds - the uncertainty on the energy triggers incredibly high fluctuations of its intensity at any point of space so that sometimes, the vacuum can acquire sufficiently high energy so that a pair of particles is created. This is called vacuum fluctuations, and is one of the explanation for such a rich content of a proton at high energy. Figure 2.7 illustrates the part of momentum which is carried by the different constituents of the protons in function of the momentum scale.

### First Measurement of the $\rho$ Spectral Function in High-Energy Nuclear Collisions

The excess mass spectra for all 4 multiplicity bins, resulting from subtraction of the conservative hadron de- cay cocktail from the measured data, are shown in Fig. 3. The cocktail  and the level of charm decays, found in the 3 upper centrality bins to be about 1=3 of the measured yield in the mass interval 1:2 < M < 1:4 GeV=c 2 [15], are shown for comparison. The qualitative features of the spectra are striking: a peaked structure is seen in all cases, broadening strongly with centrality, but remaining essen- tially centered around the position of the nominal  pole. At the same time, the total yield increases relative to the cocktail , their ratio reaching values above 4 for M < 0:9 GeV=c 2 in the most central bin. Such values are con- sistent with the results found by CERES [12], if the latter are also referred to the cocktail  and rescaled according to the different multiplicity density. The errors shown are purely statistical. The dominant sources of systematic er- rors are connected to the uncertainties in the levels of the combinatorial background (1%) and fake matches (5%). On the basis of these values and the signal-to-background ratios, the systematic errors in the broad continuum region 0:4 < M < 0:6 and 0:8 < M < 1:0 GeV=c 2 are estimated

### High Energy Collisions of Dense Hadrons in Quantum Chromodynamics : LHC Phenomenology and Universality of Parton Distributions

5 Chapter 3 is the rst one dealing with new results. Especially the di-gluon production cross-section is the main result of our paper [38] with Edmond Iancu. Many physical ideas introduced in the previous chapter are used in this one. I spent some pages to motivate why the di-gluon production has a special importance in p-A collisions at the LHC for nding quantitative evidences of saturation physics. To get the formal background, I rst deal with the simplest case : the single quark production. Indeed it shows the emergence of color operators and playing with it enables us to introduce almost everything we will need for the di-gluon production. Then when I discuss the di-gluon case, I will not have to set plenty of denitions all along the discussion. Appendices C and D will be used in this chapter for cross-sections and Feynman rules respectively. These appendices are the derivation of results which are intuitive in the sense that the structure of the cross-sections and the Feynman rules can be more or less guessed by a familiar reader who is more interested by an overall understanding rather than a careful check of prefactors. Chapter 4 is more theoretical. It deals with the small x evolution in nucleus-nucleus collisions that shows a universal character encoded into the factorization property. I fully detail calculations leading to the LO to NLO recursion relations both for gluons and quarks. The former is the new result, mentioned in our paper with Francois Gelis [39]. The starting point of this chapter will make use of the Schwinger-Keldysh formalism treated in appendix E. There are also gauge xing questions whose deep existence are justied in appendix F.

### Azimuthal Anisotropy Distributions in High-Energy Collisions

(Dated: September 25, 2018) Elliptic flow in ultrarelativistic heavy-ion collisions results from the hydrodynamic response to the spatial anisotropy of the initial density profile. A long-standing problem in the interpretation of flow data is that uncertainties in the initial anisotropy are mingled with uncertainties in the response. We argue that the non-Gaussianity of flow fluctuations in small systems with large fluctuations can be used to disentangle the initial state from the response. We apply this method to recent measurements of anisotropic flow in Pb+Pb and p+Pb collisions at the LHC, assuming linear response to the initial anisotropy. The response coefficient is found to decrease as the system becomes smaller and is consistent with a low value of the ratio of viscosity over entropy of η/s ' 0.19. Deviations from linear response are studied. While they significantly change the value of the response coefficient they do not change the rate of decrease with centrality. Thus, we argue that the estimate of η/s is robust against non-linear effects.

### Multiparticle correlations and intermittency in high energy collisions

V \.. singular multiparticle distributions in 3D momentum space [48]. They have observed that the effect of the dimensional projection depends on the range of the transverse momentum, ov[r]

### Nuclear effects in high-energy proton-nucleus collisions : transverse momentum broadening of energetic parton systems and soft anomalous dimension matrices

There are some useful limits that can be used to compensate this growth. First, I will look at the operator B when two arguments are set to the same value. This will naturally lead to the pointlike limit where a pair in the amplitude (and the conjugate pair in the conjugate amplitude) is set to have a vanishing transverse separation. Within this pointlike limit of B, one is able to find how the distribution f simplifies into simpler distributions involving less partons. Another approach is to consider a system with a large number of partons but an observable that requires to observe only a few of them. In such a situation, the complicated evolution of the 2m-parton system in the medium relates to the simpler 2n-parton system evolution (with n < m). This evolution is described by a thus a convolution in momentum space of an initial distribution h with a broadening distribution f n encoding the evolution in the medium.

### Critical fluctuations of the proton density in A+A collisions at 158A GeV

a signature of the CP. Certainly, within both approaches, pre- cision measurements are needed in order to reach conclusive results. Acknowledgments This work was supported by the US Department of Energy Grant DE-FG03-97ER41020/A000, the Bundesministerium fur Bildung und Forschung (06F 137), Germany, the German Research Foundation (Grant GA 1480/2-2), the National Science Centre, Poland (Grants DEC-2011/03/B/ST2/02617 and 2014/14/E/ST2/00018), the Hungarian Scientific Research Foundation (Grants OTKA 68506, 71989, A08-77719 and A08-77815), the Bolyai Research Grant, the Bulgarian National Science Fund (Ph-09/05), the Croatian Ministry of Science, Education and Sport (Project 098-0982887-2878), Stichting FOM, the Netherlands and the “Demokritos” National Center for Sci- entific Research, Greece.

### Fluctuations in heavy-ion collisions generated by QCD interactions in the color glass condensate effective theory

INTRODUCTION Relativistic heavy-ion collisions are performed at the BNL Relativistic Heavy Ion Collider (RHIC) and at the CERN Large Hadron Collider (LHC) with the aim of creating the quark-gluon plasma, the high-temperature state of strongly-interacting matter. The nuclear-sized droplet of quark-gluon plasma formed in a collision ex- pands like a low-viscosity fluid [1], whose properties are studied through characteristic azimuthal anisotropies generated during the expansion [2–6]. In a hydrodynamic framework, azimuthal anisotropy in the final state is en- gendered by the spatial anisotropy that characterizes the energy-density profile at the onset of the hydrodynamic evolution [1, 7]. This primordial spatial anisotropy has, in a heavy-ion collision, a twofold origin: First, it is due to the almond shape of the overlap area between two nuclei for noncentral collisions, that generates elliptic flow [8]; Second, it originates from event-to-event density fluctuations [9], that yield an elliptic deformation even in central collisions [10], and a triangular anisotropy [11]. The role of primordial fluctuations for heavy-ion phe- nomenology draws, hence, an interesting parallel [12] with the physics of primordial fluctuations in cosmology, where the observed anisotropies of the Cosmic Microwave Background [13] originate from quantum fluctuations in the early Universe [14].

### Multifractal analysis and simulation of wind energy fluctuations

1.4 Summary Of Chapter 1 Our goal is to reduce uncertainties in wind resource assessment. We claim that, based on the state of the art in wind energy, poor approximations to the high- number of degrees of freedom that arise in a bounded, turbulent, atmosphere are the main cause of uncertainty. Numerical attempts to model these complex and highly non-linear processes typically require a truncation of scales and more often than not complex parameterisations. Even if all of the scales were resolved (i.e., if we were given an infinitely powerful computer so that we could resolve all scales down to the dissipation scale) this will not prevent the uncertainties that would occur from the upscaled initial/boundary conditions at infinitely small scales. A statistical understanding of the corresponding simulations of the Navier-Stokes equations is therefore still required. We argue that due to the symmetries of the governing equations of fluid motion for a high-Reynolds number flow, the statistics of the wind are scaling and multifractal. It is therefore unnecessary to truncate the scales of the process.

### Jet evolution in a dense medium: event-by-event fluctuations and multi-particle correlations

Abstract 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 jet evolution is a Markovian stochastic process, which is exactly solvable. For this process, we construct exact analytic solutions for all the n-point correlation functions describing the gluon distribution in the space of energy [1, 2]. Using these results, we study the event-by-event distribution of the energy lost by the jet at large angles and of the multiplicities of the soft particles which carry this energy. We ﬁnd that the event-by-event ﬂuctuations are huge: the standard deviation in the energy loss is parametrically as large as its mean value [1]. This has important consequences for the phenomenology of di-jet asymmetry in Pb +Pb collisions at the LHC: it implies that the ﬂuctuations in the branching process can contribute to the measured asymmetry on an equal footing with the geometry of the di-jet event (i.e. as the di ﬀerence between the in-medium path lengths of the two jets). We compute the higher moments of the multiplicity distribution and identify a remarkable regularity known as Koba-Nielsen-Olesen (KNO) scaling [2]. These predictions could be tested via event- by-event measurements of the di-jet asymmetry.

### Charged-Particle Nuclear Modification Factors in XeXe Collisions at √s[subscript NN] = 5.44 TeV

radial flow [ 44 ], and the Cronin effect [ 45 ]. At higher p T , parton energy loss also becomes a significant effect. Generally, R AA and R ∗ AA agree with each other in the range p T < 4 GeV. However, the data may indicate a slight difference in suppression levels at higher p T . As the centrality range examined becomes more peripheral, the oscillating shape of R AA ∗ becomes less pronounced. In the most peripheral collisions examined, the XeXe data are relatively flat, indicating that the spectral shape for peripheral centrality ranges is similar to that of pp collisions. Although there is a large normalization uncertainty, the R ∗ AA is significantly below unity in this centrality range. Such a suppression in peripheral events is not expected to be caused by strong energy loss effects, but might be related to correlations between the charged-particle yields in the mid-rapidity region with event activity in the range 3 < |η| < 5.2 that is used to determine the event centrality [ 46 ]. Recent measurements of R AA in peripheral PbPb collisions by the ALICE Collaboration show a similar effect

### Particle number fluctuations and correlations in transfer reactions obtained using the Balian-Vénéroni variational principle

nature. Fig. 1(a) shows the scattering angle θ c.m. as a function of the angular momentum L. For L ≥ 90, the trajectories are close to Rutherford scattering (dashed line) and correspond to quasi-elastic reactions with small total kinetic energy loss (TKEL) as shown by the solid line in Fig. 1(b). At lower L, deviations from the Ruther- ford formula occur because of nuclear attraction, lead- ing to rotation of the di-nuclear system formed by the two fragments in contact. For instance, L ≤ 72 de- fines (arbitrarily) the orbiting region where the fragments have crossed their incoming trajectory, i.e., with n ≥ 4 crossings of the x (collision axis) or y−axis [see inset in Fig. 1(a)]. Following [24], damped events are defined by a TKEL≥ 30 MeV, corresponding to L < 82 in Fig. 1. We see that a wide range of scattering angles may oc- cur for these damped events, which is a known feature of DIC. For L ≤ 66, capture occurs. It corresponds to a fusion cross section of ∼ 1140 mb with the sharp cut- off formula [13], in good agreement with a fit on fusion- evaporation measurements at lower energies [26].

### Associated production of one particle and a Drell-Yan pair in hadronic collisions

Keywords: Collinear factorization, fracture functions, Drell-Yan process I. INTRODUCTION The description of particle production in hadronic collisions is interesting and challenging in many aspects. Perturbation theory can be applied whenever a sufficiently hard scale characterizes the scattering process. The comparison of early LHC charged particle spectra with next-to-leading order perturbative QCD predictions [1] shows that the theory offers a rather good description of data at sufficiently high hadronic transverse momentum, of the order of a few GeV. For inelastic scattering processes at even lower transverse momentum, the theoretical description in terms of perturbative QCD breaks down since both the coupling and partonic matrix elements diverge as the transverse momenta of final state parton vanish. In this paper we will study the semi-inclusive version of the Drell-Yan process, H 1 + H 2 → H + γ ∗ + X, in which one particle is tagged in the final