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Register a new userThe chiral condensate from renormalization group optimized perturbation
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Our recently developed variant of variationnally optimized perturbation (OPT), in particular consistently incorporating renormalization group properties (RGOPT), is adapted to the calculation of the QCD spectral density of the Dirac operator and the related chiral quark condensate $langle bar q q rangle$ in the chiral limit, for $n_f=2$ and $n_f=3$ massless quarks. The results of successive sequences of approximations at two-, three-, and four-loop orders of this modified perturbation, exhibit a remarkable stability. We obtain $langle bar q qrangle^{1/3}_{n_f=2}(2, {rm GeV}) = -(0.833-0.845) barLambda_2 $, and $ langlebar q qrangle^{1/3}_{n_f=3}(2, {rm GeV}) = -(0.814-0.838) barLambda_3 $ where the range spanned by the first and second numbers (respectively four- and three-loop order results) defines our theoretical error, and $barLambda_{n_f}$ is the basic QCD scale in the $overline{MS}$-scheme. We obtain a moderate suppression of the chiral condensate when going from $n_f=2$ to $n_f=3$. We compare these results with some other recent determinations from other nonperturbative methods (mainly lattice and spectral sum rules).
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Our renormalization group consistent variant of optimized perturbation, RGOPT, is used to calculate the nonperturbative QCD spectral density of the Dirac operator and the related chiral quark condensate $langle bar q q rangle$, for $n_f=2$ and $n_f=3$ massless quarks. Sequences of approximations at two-, three-, and four-loop orders are very stable and give $langle bar q q rangle^{1/3}_{n_f=2}(2, {rm GeV}) = -(0.833-0.845) barLambda_2 $, and $ langle bar q q rangle^{1/3}_{n_f=3}(2, {rm GeV}) = -(0.814-0.838) barLambda_3 $ where the range is our estimated theoretical error and $barLambda_{n_f}$ the basic QCD scale in the $rm bar{MS}$-scheme. We compare those results with other recent determinations (from lattice calculations and spectral sum rules).
We reconsider our former determination of the chiral quark condensate $langle bar q q rangle$ from the related QCD spectral density of the Euclidean Dirac operator, using our Renormalization Group Optimized Perturbation (RGOPT) approach. Thanks to the recently available {em complete} five-loop QCD RG coefficients, and some other related four-loop results, we can extend our calculations exactly to $N^4LO$ (five-loops) RGOPT, and partially to $N^5LO$ (six-loops), the latter within a well-defined approximation accounting for all six-loop contents exactly predictable from five-loops RG properties. The RGOPT results overall show a very good stability and convergence, giving primarily the RG invariant condensate, $langle bar q qrangle^{1/3}_{RGI}(n_f=0) = -(0.840_{-0.016}^{+0.020}) barLambda_0 $, $langlebar q qrangle^{1/3}_{RGI}(n_f=2) = -(0.781_{-0.009}^{+0.019}) barLambda_2 $, $langlebar q qrangle^{1/3}_{RGI}(n_f=3) = -(0.751_{-.010}^{+0.019}) barLambda_3 $, where $barLambda_{n_f}$ is the basic QCD scale in the overline{MS} scheme for $n_f$ quark flavors, and the range spanned is our rather conservative estimated theoretical error. This leads {it e.g.} to $ langlebar q qrangle^{1/3}_{n_f=3}(2, {rm GeV}) = -(273^{+7}_{-4}pm 13)$ MeV, using the latest $barLambda_3$ values giving the second uncertainties. We compare our results with some other recent determinations. As a by-product of our analysis we also provide complete five-loop and partial six-loop expressions of the perturbative QCD spectral density, that may be useful for other purposes.
A recently developed variant of the so-called optimized perturbation theory (OPT), making it perturbatively consistent with renormalization group (RG) properties, RGOPT, was shown to drastically improve its convergence for zero temperature theories. Here the RGOPT adapted to finite temperature is illustrated with a detailed evaluation of the two-loop pressure for the thermal scalar $ lambdaphi^4$ field theory. We show that already at the simple one-loop level this quantity is exactly scale-invariant by construction and turns out to qualitatively reproduce, with a rather simple procedure, results from more sophisticated resummation methods at two-loop order, such as the two-particle irreducible approach typically. This lowest order also reproduces the exact large-$N$ results of the $O(N)$ model. Although very close in spirit, our RGOPT method and corresponding results differ drastically from similar variational approaches, such as the screened perturbation theory or its QCD-version, the (resummed) hard thermal loop perturbation theory. The latter approaches exhibit a sensibly degrading scale dependence at higher orders, which we identify as a consequence of missing RG invariance. In contrast RGOPT gives a considerably reduced scale dependence at two-loop level, even for relatively large coupling values $sqrt{lambda/24}sim {cal O}(1)$, making results much more stable as compared with standard perturbation theory, with expected similar properties for thermal QCD.
We present results for in-medium spectral functions obtained within the Functional Renormalization Group framework. The analytic continuation from imaginary to real time is performed in a well-defined way on the level of the flow equations. Based on this recently developed method, results for the sigma and the pion spectral function for the quark-meson model are shown at finite temperature, finite quark-chemical potential and finite spatial momentum. It is shown how these spectral function become degenreate at high temperatures due to the restoration of chiral symmetry. In addition, results for vector- and axial-vector meson spectral functions are shown using a gauged linear sigma model with quarks. The degeneration of the $rho$ and the $a_1$ spectral function as well as the behavior of their pole masses is discussed.
We explore the influence of the current quark mass on observables in the low energy regime of hadronic interactions within a renormalization group analysis of the Nambu-Jona-Lasinio model in its bosonized form. We derive current quark mass expansions for the pion decay constant and the pion mass, and we recover analytically the universal logarithmic dependence. A numerical solution of the renormalization group flow equations enables us to determine effective low energy constants from the model. We find values consistent with the phenomenological estimates used in chiral perturbation theory.
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