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Register a new userChiral Condensate and Spectral Density at full five-loop and partial six-loop orders of Renormalization Group Optimized Perturbation
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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.
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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).
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).
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 analize the impact of two-loop renormalization group equations on the $SU(3)_ctimes SU(2)_wtimes U(1)_Y$ gauge couplings unification in various supersymmetric theories. In general the presence of superfields in higher representation than the doublet spoil the gauge couplings unification at one-loop. The situation is more interesting when the renormalization group equations are calculated at two-loop. In this case we show that the unification of the gauge couplings can be achieved for models with triplet superfield(s). In the analysis of the models with triplet superfield(s) we show that the dimensionless couplings do not have a Landau pole in their evolution at high energies but they run to a nontrivial ultraviolet fixed point.
In this paper, we consider two-flavor QCD at zero temperature and finite isospin chemical potential ($mu_I$) using a model-independent analysis within chiral perturbation theory at next-to-leading order. We calculate the effective potential, the chiral condensate and the pion condensate in the pion-condensed phase at both zero and nonzero pionic source. We compare our finite pionic source results for the chiral condensate and the pion condensate with recent (2+1)-flavor lattice QCD results and find that they are in excellent agreement.
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