A Relation between Schwinger-Keldysh and Feynman generating functionals 

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Looking inside a proton, first part

Our story begins with a close look of a proton at rest. This proton at rest is formed by three quarks (2 u and 1 d), as illustrated in figure 2.2.
Those quarks are confined: they can’t be observed traveling freely without their two fellows. This is a specificity of the strong interaction discovered by Wilson in 1974 [72] that is called the « quark confinement ». This means that the attractive force between the quarks is indeed very strong at low energy. Physicists quantify this strength with a coupling constant – called αS for the strong interaction, where the S stands for strong – which is an energy dependent quantity. Unfortunately, the theory that describes the strong force – called quantum chromodynamics, or QCD – is well understood only in its weak coupling sector, which means for αS much smaller than 1. This allows to do a small coupling expansion of the theory and is the basis of pretty much all the perturbative techniques that are one of the main tools on the theoretical treatment of QCD. To compare it with experiment, one therefore has to make such experiments in the weak coupling sector of the strong interaction. Surprisingly, this sector is at very high energy. This could be counter intuitive: When one thinks about gravity, one expects that the further away two object are, the weaker the gravitational interaction is. The opposite happens for the strong force. A good analogy can be done with a string: when a string is pulled, it is subject to a restoring force that will tend to move closer the two endpoints of the string, and this force is stronger when the string is pulled more and more 2. This property is called « asymptotic freedom »: at infinite energy, the confinement property does not hold anymore and the quarks can indeed move freely. This surprising feature of the strong interaction was discovered in 1973-74 by Gross, Wilczek and Politzer [73–76], and is illustrated with recent experimental outputs in figure 2.3.

Looking at a proton, second part

We go back to our proton (the same is true for a heavy ion, but to keep things simple only protons are drawn so far) of figure 2.2, but now consider it highly accelerated, as it is truly in the RHIC and LHC rings. It turns out that in this case, the picture given in figure 2.2 is too 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.

The quark-gluon-plasma: experimental evidences

When two heavy ions that look at figure 2.8 collide at very high energy, one can anticipate that the matter produced out of their collision is extremely complicated to describe. Indeed, gluons from one nucleus can interact with those from the other. The matter formed by this collision is called the Quark-Gluon-Plasma (QGP).
What are the evidences for this new form of matter? The best one is the observation in the experimental detectors of very energetic fluxes, corresponding to very located in space particles. The latter are called « jets ». The reasoning is the following: let’s assume for the time being that the QGP exists. In a heavy ion collision, as already mentioned, products are observed in the detectors of the RHIC and the LHC. But out of these products, some of them are much more energetic than others. Those are the result of the collisions of the most energetic objects in the nuclei – the collisions that imply the valence quarks for instance. When such collisions happen in the center of the QGP, one will usually observe two « back-to-back » jets, which means two very energetic beams of produced particles that cross both half of the QGP and then go hit the detector in opposite directions. Because those jets travel in opposite direction, they are correlated: the probability of finding the second jet at 180 degrees of the first one is very high. Now, consider again a very energetic collision, but happening this time at the edge of the QGP. In this situation one of the jet will only have to cross a tiny region of the QGP before escaping and hitting the detector, while the other has to cross most of the QGP. During this crossing, the jet will interact with the QGP and loose most of its energy 6. So if the QGP exists, one should sometimes observe in heavy ion collisions two correlated back-to-back jets, but with one being much more energetic than the other. Our expectation is illustrated in figure 2.10.

Has the QGP ever existed in the history of the universe?

Since the heavy ion collisions are so energetic, they are sometimes referred to as the « little-bangs ». The different phases of the little-bang are illustrated in figure 2.12.
This analogy in name between the little-bang and its older (and bigger) brother the big-bang is due to the fact that the early stages of a heavy ion collisions can probe the first instants of the life of our own universe. This is why some theoretical physicists study the little-bangs in order to learn more about the big-bang, from which a cartoon picture is given in figure 2.13.
In particular, an hypothetic phase during the creation of the universe called the inflation has attracted a lot of attention recently. Inflation is an expansion of the structure of the uni-verse itself at a speed way faster that the speed of light during a very short time. The inflation scenario has been introduced in the 80’s [97], when people discovered a relic of the formation of the universe called the Cosmic Microwave Background (or CMB). This radiation has the specificity to have a black body temperature of 2.8 Kelvin (with only 10−5 variations) every-where in the universe 7, even in regions that are not causally related today 8. The inflation, and especially the thermalization that occurred at the end of it, is able to explain this paradox. This thermalization shares a lot with the one that happens in the little-bangs.

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Strongly coupled techniques: fast « hydrodynamisation »

The strongly coupled techniques that are used in order to describe the early life of the out of equilibrium QGP rely on the AdS/CFT conjecture [116], illustrated in figure 3.2. This conjecture establishes a link between supersymmetric gauge theories (which QCD is not) and string theories in a five dimensional Anti de-Sitter curved space time. In the limit where the gauge coupling becomes infinite, the string theory reduces to general relativity, and one therefore just needs to solve Einstein’s equations to calculate the quantities of interest in the theory. This is what has been done in [29], where Einstein’s equations have been solved for various initial conditions. The results are reproduced in the figure 3.3.
We see that the out-of-equilibrium AdS/CFT calculations very rapidly coincide with the viscous hydrodynamical simulations. This is illustrated in the left plot (where w plays the role of the time τ), where the agreement between the two theories is compatible with the τ0 ∼ 1 f m/c mentioned in the previous sections. The surprise comes from the right plot, where we see that this agreement already happens when the system is rather far from being isotropic. Does this means that the range of applicability of hydrodynamics is broader than we may expect given the postulates I − IV ′? More realistic initial conditions (colliding shock waves in the AdS space in order to mimic the heavy-ions) are currently investigated in order to answer in a more definitive way this question at strong coupling [34].
Another important result found within the AdS/CFT framework is the fact that hydrody-namics – as perturbation theories – is based on an asymptotic expansion [120]. The radius of convergence of the gradient expansion is in fact 0. This comparison between hydrodynamics and AdS/CFT techniques has been very fruitful. The AdS/CFT framework is an interesting playground to understand what is happening in heavy-ion collisions, but should not be con-sidered as more than a toy model. Indeed, the assumption that the QGP is infinitely strongly coupled seems extreme, as one can see on figure 2.3. At the scale Qs ≈ 1 − 2GeV, the strong coupling constant αs is of the order of 0.3. In addition, QCD is pretty different from a super-symmetric gauge theory. This is what motivates our choice to stay within a weakly coupled description of the QGP in the remaining parts of this manuscript.

The Color Glass Condensate (CGC) effective theory

In the McLerran-Venugopalan model [43, 44] that gave birth to the modern version of the CGC, a momentum separation scale 15 Λ± is introduced between slow and fast gluons. Slow gluons are treated as usual gauge fields Aµa, while fast gluons are described as static color sources Jµa. They are considered to be moving at the speed of light. In light cone coordinates (z being taken to be the direction of the collision axis throughout this manuscript) x± = the two color currents Jnµ modeling the fast partons of the two projectiles (the n = 1, 2 index either standing for projectile one or two) are located on the light-cone axes. Schematically, the CCC picture looks like the figure (3.11).

Table of contents :

1 Introduction 
1.1 Understanding the effectiveness of hydrodynamics in heavy ion collisions
1.2 How to read this thesis?
I Theoretical background 
2 The problem of thermalization in heavy-ion collisions 
2.1 The strong interaction
2.2 Looking inside a proton, first part
2.3 Heavy Ion collisions
2.4 Looking at a proton, second part
2.5 The quark-gluon-plasma: experimental evidences
2.6 Has the QGP ever existed in the history of the universe?
2.7 Quark-gluon-plasma: the puzzle
3 Theoretical tools to deal with the Quark-Gluon-Plasma 
3.1 Kinetic Theory
3.2 Hydrodynamics
3.3 Strongly coupled techniques: fast « hydrodynamisation »
3.4 Quantum Chromodynamics
3.5 Specificities of heavy-ion collisions
3.6 The Color Glass Condensate (CGC) effective theory
3.7 JIMWLK equation
3.8 LO CGC results: Impossible matching with hydrodynamics
3.9 NLO CGC results: Weibel instabilities and secular divergences
3.10 Summary
3.A Bjorken’s law for an ideal fluid
4 Beyond standard perturbation theory 
4.1 Schwinger-Keldysh formalism
4.2 Resummation formula
4.3 The Classical-statistical approximation: a path integral approach
4.A Relation between Schwinger-Keldysh and Feynman generating functionals
II Study of a scalar field theory 
5 Scalar field theory in a fixed volume 
5.1 Setup of the problem, specificities of the scalar model
5.2 The physics of instabilities
5.3 Macroscopic observables: the formation of an EOS
5.4 Microscopic properties of fixed volume scalar field theory
5.5 Summary
5.A Instabilities in the fixed-volume case
5.B Appendix: Effective Hamiltonian
6 Expanding system 
6.1 Expanding scalar theory
6.2 Numerical implementation
6.3 Independence with respect to the initial time
6.4 Resonance band
6.5 Occupation Number
6.6 Energy-momentum tensor
6.7 Hydrodynamical behavior
6.8 Summary
6.A Numerical considerations
7 Non Renormalizability of the Classical Statistical Approximation 
7.1 Renormalization of Green’s functions
7.2 Renormalization of composite operators
7.3 The retarded-advanced basis
7.4 Eliminating the source term
7.5 Ultraviolet power counting in the full theory
7.6 Ultraviolet power counting in the CSA
7.7 Ultraviolet divergences in the CSA
7.8 Impact of the non-renormalizability of the CSA on Tμν
7.9 Cumulative effects of the non-renormalizability
7.10 Possible partial cure
7.11 Could the cure be implemented numerically?
7.12 Summary
7.A Calculation of G1112 and G1222
7.B Calculation of G1122
III Yang-Mills theory 
8 Spectrum of fluctuations above the light cone 
8.1 Spectrum of fluctuations: a new derivation
8.2 Known results for the background field
8.3 The axial gauge
8.4 Going to Fock-Schwinger gauge
8.5 Small fluctuations in the forward light cone
8.6 Summary
8.A Useful formulas to derive (8.35)
8.B Several checks on the step 3
9 Numerical results 
9.1 Numerical implementation of the Yang-Mills Equations
9.2 Matrix multiplication on the lattice
9.3 Leap-frog algorithm
9.4 Initial conditions for the background field
9.5 Discretized form of the energy-momentum tensor
9.6 Numerical checks
9.7 Initial conditions for the small fluctuations
9.8 Monte-Carlo: speed versus storage
9.9 Enforcing the non-linear Gauss’s law
9.10 Renormalization
9.11 Numerical results: isotropization, anomalous viscosity
9.12 Summary
10 Conclusion 


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