As explained above, in thermal equilibrium the functions depend on the difference of two times, t ⌘ t1 t0. As explained above, in thermal equilibrium the functions depend on the difference of two times, t ⌘ t 1 t 0.
The Schwinger-Keldysh contour
Operator orderings
Using the cyclicity of the track, the exponential form of the thermal density operator and the commutation relations of the conserved charge, the two Wightman functions can be related to each other via. In this section we go further into the implications of the Schwinger-.
The ``1-2’’ formalism
Operator orderings and the contour
The r/a basis
We can now go back to our printing problem and see how this appears more transparent in the real-time formalism. In the real-time formalism one works in real-time from the start, no Matsubara sums but a matrix structure.
Main message
Combining a statistical density matrix with real-time evolution naturally gives rise to a 2x2 structure for the propagators and for several types of vertices. The main elements of this 2x2 structure are Wightman functions (measuring correlation) and lagged/advanced functions (measuring causation).
The ``ra’’ basis
The propagation between two a-fields is identically zero for all orders, in and out of equilibrium, due to the ✓ functions in the definitions of the various correlation functions. The propagation between two a-fields is identically zero for all orders, in and out of equilibrium, due to the ✓ functions in the definitions of the various correlation functions.
52) Finally, we recall that the KMS conditions that we introduced for the connected
In-medium generalization of the Cutkosky rules
This is so because on the one hand, if one or more of the bare propagators were an rr propagator Drr, at least one of the native energies would have to be ⇧rr, which vanishes in an identical manner. On the other hand, if one or more of the self-energies were ⇧aa, at least one of the diffusers would have to be a vanishing Daa.
Cutting rules
The ra basis with cutting rules
As such, the optimal technique for its evaluation is in the chapter cutting rules. 55) is easily obtained by changing all occurrences of >. As shown there, the cutout of this diagram corresponds to the square of the tree-level photon emission, which is well known to vanish kinematically for real photons that cannot be emitted by shell quarks. In fact, a straightforward application of Eq. 55) on this diagram results in 61), where the lagging and leading amplitudes in Eq. 55) are Marr( K ; P + where the backward and forward amplitudes in equation 30) show that the propagators S< are proportional to the fermion spectral density ⇢F ( P ), S<( P ) = nF(p0)⇢ F ( P ), which in the bare limit used in conventional perturbation theory in the interaction representation reads ⇢F ( P ) = P / ✏(p0)2⇡ ( P 2).8 Then it is straightforward to check that the integration d4P vanishes over the product of two -functions, which places two quarks per shell as expected.
Normally (see the extended discussion in App. E of [35]) the mostly-plus metric is associated with a factor of i to the matrices, so that the anticommutator maintains a plus sign, as in the case of the mostly-minus - metric . This makes the connection with kinetic theory particularly transparent: notice how the terms in square brackets are nothing but the matrix elements squared for these processes, we see that we have recovered the gain term of a Boltzmann equation for photon production in the case where the photon's distribution fk is negligible, which is exactly the approximation that the derivation of Eq. We conclude this overview of the leading-order calculation by noting that the momentum integration regions that contribute here are best identified by their scale in terms of P.
In addition to the leading order, the total score is the sum of the leading order rate and its O(g) correction. As with the leading-order calculation, the NLO rate is derived from different kinematic ranges, and the NLO correction can be parameterized as The soft and collinear regions are the same kinematic regions as in the leading-order calculation, while the semi-collinear region is an additional kinematic region whose contribution starts at the NLO.
Breakdown of the loop expansion
Soft modes
Exercise: show that it has this form (I might have screwed the prefactors)
Hard Thermal Loops
Sum rules
We now turn to an illustration of sum rules that can be obtained from the analytic properties of the amplitudes, thanks to causality. We start by illustrating the classical sum rules, which can be found in textbooks such as [3].
84) As anticipated, the contour sum rules not only lead us to the simple closed form
In the space-like region, Landau damping also appears for soft quarks, which physically correspond to the scattering of a soft, virtual, space-like quark by the hard components of the medium. The contours of the spectral functions of HTL quarks in the Landau cut are shown in the figure. The result in the second line can be easily obtained from the analytical properties of the spectral function, which is the difference between the backward and forward propagators.
In contrast, upon the disappearance of q we expect to see the three degenerate polarizations of plasmons in the electric case, that is, similar sum rules, again motivated by causality, can also be derived in the fermion case and found in textbooks [3]. This is because the delayed response function is only non-zero in the forward light cone 2x+x x2?.
Motivated by this question, in this paper we address the relation of the two corrections in the case of a weakly coupled quark-gluon plasma. Our conventions, as well as more technical details about the calculations, are collected in the appendices. In the case of QCD, asymptotic freedom would in principle make the integration UV-finite, but this.
Euclideanisation
On the right we draw in blue dashes and red dots the contributions from the transverse and longitudinal poles, respectively. The overall total is plotted in solid black. on the other hand, as we will discuss in Sec. 6.2, the gauge invariance of MQCD – the effective, static theory of chromomagnetic modes on the scale g2T discussed later in Sec. 6.3.1 – prevents the generation of a third degree of freedom for chromomagnetic fields in the IR, and therefore IB is constant as a function of q. These considerations are reflected not only in the integrated results, but also in the integrands: in the limit of small q, the integral of.
In a way, this is a textbook application of causality, as it relies on the basic property of the analyticity of the retarded propagator on the upper half of the complex. However, causality allows for stronger statements that can be used to derive sum rules that apply to the light cone. Thus, the integral in equation 85) has support only for x > 0, and the Fourier integral provides analytical continuation in the upper half of the q+ plane due to the decreasing exponential eiq+x [65].
The proper evaluation of the soft contribution to the photon rate requires HTL resumption for the photon rate. In this case, a different type of light cone sum rule applies, based on the same analytical properties rooted in causality. In the former case, the sum rule allows a simple, analytical evaluation of the leading-order contribution to the photon rate from soft quark momentum;. let's briefly see how this comes about, to link back to our discussion at the beginning of Art. 61) vanishes when both quark propagators are bare, which is appropriate when they are both hard, P K ⇠ T and P ⇠ T , this is no longer the case when one of these two is soft, P ⇠ gT , which, as we have argued, is where a logarithmic IR divergence appears in the naive treatment of Eq.
Collinear physics: LPM resummation
- Introduction and physical picture
So far we have talked about the complications arising from soft kinematics, where (one of) the particles in the discussion have all four momentum components that are soft, i.e. there are the particles hard with momenta of order T , their virtualities are of order g2T 2 and their angular separations are of order g , as we see in Fig. Where the physics is sensitive to this kinematic region, it is necessary to perform a further resummation, other than HTL, to describe the physics correctly.
Such a recurrence scheme begins in the works of Landau, Pomeranchuk [74, 75] and Migdal [76] (LPM) in the context of bremsstrahlung in QED, later generalized to QCD physics by Baier, Dokshitzer, Mueller, Peign' e, and Schi↵ [77, 78] and Zakharov [79, 80]. In the context of Thermal Field Theory, this was introduced by Arnold, Moore and Ya↵e [81, 82], whose formalism we will follow in this review, in its position-space formulation. We are particularly indebted to the heuristic derivation in [83] and the extended and detailed derivation in [84].
A useful Rosetta stone among many different formalisms and related notes can be found in App. The physical origin of entanglement is related to the quantum mechanical formation times of distributions in the medium.
Collinear physics
This Bethe-Heitler process is of the same order as the hard+soft 2<->2 contributions. This is of the same magnitude as the average free time between soft collisions. In the first case, called the bremsstrahlung diagram, the angle between the emitted photon and the outgoing emitting fermion is of order g.
In the second case, called the pair annihilation diagram, it is the angle between the annihilating quark and antiquark that is of order g. In both graphs the gluon is soft and scattered o↵ the hard quarks and gluons of the. These multiple scatterings lead to destructive interference, this is known as the Landau-Pomeranchuk-Migdal (LPM) e↵ek which leads to an O(1) suppression of the collinear rate.
In terms of the two-point function, these processes correspond to diagrams where the two nearly collinear fermion lines are connected to any number of soft, space-like gluons with the same kinematics as Q. The fact that the quark lines are close to the scale makes the diagrams sensitive to the thermal mass m21 ⇠ g2T 2 and the thermal width ⇠ g2T of the quark lines, which must be resumed consistently. 3. We will discuss this comparison in detail in the context of discussing the NLO corrections.
The leading-order photon rate
Conclusions