Advances in perturbative thermal field theory - Ultra-relativistic ...

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372 U Kraemmer and A Rebhan Figure 6. Strictly perturbative results for the thermal pressure of pure-glue QCD as a function of T/T c (assuming T c / MS = 1.14). The various grey bands bounded by differently dashed lines show the perturbative results to order g 2 , g 3 , g 4 and g 5 , using a two-loop running coupling with MS renormalization point, ¯µ, varied between πT and 4πT. The thick dark-grey line shows the continuum-extrapolated lattice results from [154]; the lighter one behind that of a lattice calculation using an RG-improved action [155]. 4πT. So there seems to be a complete loss of predictive power at any temperature of interest [133, 134, 136, 135]. To alleviate this situation, various mathematical extrapolation techniques have been tried, such as Padé approximants [85, 86, 156], self-similar approximants [157], and Borel resummation [158, 159], though with limited success. While in the scalar toy model of section 3.5 Padé approximants work remarkably well, in the QCD case there is a problems how to handle logarithms of the coupling, and perhaps related to that, the numerical results obtained so far appear less satisfactory. When compared with the perturbative results, it is however remarkable that the next-toleading result to order g 2 performs rather well at temperatures 2T c , though the higherorder results prove that perturbation theory is inconclusive. Moreover, simple quasi-particle models that describe the effective gluonic degrees of freedom by 2N g (N g = N 2 − 1) scalar degrees of freedom with asymptotic masses taken from a HTL approximation can be used quite successfully to model the lattice data by fitting the running coupling [160–163]. In fact, we have seen in section 3.5 that already in the simplest scalar model resummed perturbation theory gives rather poorly convergent results and that fairly simple reorganizations as in (3.13) lead to dramatic improvements of the situation. As has been noted in [137,164], in the calculation as organized through the dimensionally reduced effective theory (4.1), the large-scale dependence of strict perturbation theory can be reduced significantly when the perturbative values of the effective parameters are kept as they appear in (4.4), without expanding and truncating the final result to the considered order in the coupling. What is more, the unexpanded two-loop and three-loop contributions from the soft sector lead to results that no longer exceed the ideal-gas result (as the strictly perturbative results to order g 3 and g 4 do), and their (sizable) scale dependence diminishes by going from two- to three-loop order [164]. At three-loop order, it is in fact possible to eliminate the scale dependence altogether by a principle of minimal sensitivity. The result in fact agrees remarkably well with the 4-d lattice results down to ∼ 2.5T c as shown in figure 7. The four-loop order result depends on the unknown constant c in (4.7). However, it is at least not excluded that c could be such that the four-loop result is also close to the 4-d lattice
Thermal field theory 373 Figure 7. Three-loop pressure in pure-glue QCD with unexpanded effective-field-theory parameters when ¯µ is varied between πT and 4πT (medium-grey band); the dotted lines indicate the position of this band when only the leading-order result for m E is used. The broad light-grey band underneath is the strictly perturbative result to order g 5 with the same scale variations. The full line gives the result upon extremalization (PMS) with respect to ¯µ (which does not have solutions below ∼1.3T c ); the dash-dotted line corresponds to fastest apparent convergence (FAC) in m 2 E , which sets ¯µ ≈ 1.79πT (taken from [164]). Figure 8. Like figure 7, but extended to four-loop order by including the recently determined g 6 ln(1/g) contribution of [137] together with three values for the undetermined constant δ in [g 6 ln(1/g) + δ]. The broad light-grey band underneath is the strictly perturbative result to order g 6 , corresponding to the central value δ = 1 3 , which has a larger scale dependence than the order g 5 result in figure 7; the untruncated results on the other hand show rather small-scale dependence. The full line gives the untruncated result with δ = 1 3 and ¯µ fixed by PMS (which does not have solutions below ∼ 1.9T c ); the dash-dotted line corresponds to FAC in m 2 E , which sets again to ¯µ ≈ 1.79πT (taken from [164]). results [137,165]. Furthermore, while a strictly perturbative treatment leads to increased scale dependence compared with three-loop order, keeping the soft contributions unexpanded in g further diminishes the scale dependence [164] as shown in figure 8. While this goes only minimally beyond a strictly perturbative treatment, it strongly suggests that perturbative QCD at high temperature, when supplemented by appropriate resummation of soft physics, is not limited to T ≫ 10 5 T c as previously thought [133, 135] but seems capable of quantitative predictions at temperatures of possibly only a few times the transition temperature.
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Page 3 and 4: Thermal field theory 353 7.8.1. Ene
Page 5 and 6: Thermal field theory 355 the thermo
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Page 15 and 16: Thermal field theory 365 where φ 0
Page 17 and 18: Thermal field theory 367 where tr d
Page 19 and 20: Thermal field theory 369 m eff ( ¯
Page 21: Thermal field theory 371 determined
Page 25 and 26: Thermal field theory 375 4.4. Low t
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Page 35 and 36: Thermal field theory 385 6. HTL eff
Page 37 and 38: Thermal field theory 387 equations
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Page 47 and 48: Thermal field theory 397 [353, 223,
Page 49 and 50: Thermal field theory 399 and in fac
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Page 65 and 66: Thermal field theory 415 obtained b
Page 67 and 68: Thermal field theory 417 [22] Keldy
Page 69 and 70: Thermal field theory 419 [87] Baym
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Thermal field theory 423 [212] Rebh
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Thermal field theory 425 [278] Gerh
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Thermal field theory 427 [342] Buch
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Thermal field theory 429 [401] Flec
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Thermal field theory 431 [462] Krae
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