arXiv subjects
Publications and source records attributed to J. Rodríguez-Quintero.
This brief note is devoted to reconcile the conclusions from a recent analysis of the IR solutions for the ghost propagator Dyson-Schwinger equations in Coulomb gauge with previous studies in Landau gauge.
We argue from LQCD that there is a non vanishing v.e.v of $A_a^μA^a_μ$ in QCD in the Landau gauge. We use operator product expansion to provide a clear definition of $A_a^μA^a_μ$ and extract a number both in the quenched and unquenched case.
We study the low-momentum behaviour of Yang-Mills propagators obtained from Landau-gauge Dyson-Schwinger equations (DSE) in the PT-BFM scheme. We compare the ghost propagator numerical results with the analytical ones obtained by analyzing the low-momentum behaviour of the ghost propagator DSE in Landau gauge, assuming for the truncation a constant ghost-gluon vertex and a simple model for a massive gluon propagator. The asymptotic expression obtained for the regular or decoupling ghost dressing function up to the order ${\cal O}(q^2)$ is proven to fit pretty well the numerical PT-BFM results. Furthermore, when the size of the coupling renormalized at some scale approaches some critical value, the numerical PT-BFM propagators tend to behave as the scaling ones. We also show that the scaling solution, implying a diverging ghost dressing function, cannot be a DSE solution in the PT-BFM scheme but an unattainable limiting case.
Some very recent computations of $α_{\bar{\rm MS}}(M_Z)$ from $N_f=1+1$ lattice simulations and of the running of the Strong coupling, obtained from the lattice ghost-gluon vertex, over a large momentum window are very briefly reviewed.
Both regular (the zero-momentum ghost dressing function not diverging), also named decoupling, and critical (diverging), also named scaling, Yang-Mills propagators solutions can be obtained by analyzing the low-momentum behaviour of the ghost propagator Dyson-Schwinger equation (DSE) in Landau gauge. The asymptotic expression obtained for the regular or decoupling ghost dressing function up to the order ${\cal O}(q^2)$ fits pretty well the low-momentum ghost propagator obtained through the numerical integration of the coupled gluon and ghost DSE in the PT-BFM scheme. Furthermore, when the size of the coupling renormalized at some scale approaches some critical value, the PT-BFM results seems to tend to the the scaling solution as a limiting case.
We present results concerning the non-perturbative evaluation of the renormalisation constant for the quark field, $Z_q$, from lattice simulations with twisted mass quarks and three values of the lattice spacing. We use the RI'-MOM scheme. $Z_q$ has very large lattice spacing artefacts; it is considered here as a test bed to elaborate accurate methods which will be used for other renormalisation constants. We recall and develop the non-perturbative correction methods and propose tools to test the quality of the correction. These tests are also applied to the perturbative correction method. We check that the lattice spacing artefacts scale indeed as $a^2p^2$. We then study the running of $Z_q$ with particular attention to the non-perturbative effects, presumably dominated by the dimension-two gluon condensate $\VEV{A^2}$ in Landau gauge. We show indeed that this effect is present, and not small. We check its scaling in physical units confirming that it is a continuum effect. It gives a $\sim 4%$ contribution at 2 GeV. Different variants are used in order to test the reliability of our result and estimate the systematic uncertainties. Finally combining all our results and using the known Wilson coefficient of $\VEV{A^2}$ we find $g^2(μ^2) \VEV{A^2}_{μ^2\; CM} = 2.01(11)(^{+0.61}_{- 0.73}) \;\mathrm {GeV}^2$ at $μ=10\, \mathrm{GeV}$, in fair agreement within uncertainties with the value indepently extracted from the strong coupling constant.
A non-perturbative calculation of the ghost-gluon running QCD coupling constant is performed using $N_f=2$ twisted-mass dynamical fermions. The extraction of $Λ_{\bar{MS}}$ in the chiral limit reveals the presence of a non-perturbative OPE contribution that is assumed to be dominated by a dimension-two $\VEV{A^2}$ condensate. In this contest a novel method for calibrating the lattice spacing in lattice simulations is presented.
We present results concerning the non-perturbative evaluation of the ghost-gluon running QCD coupling constant from $N_f=2$ twisted-mass lattice calculations. A novel method for calibrating the lattice spacing, independent of the string tension and hadron spectrum is presented with results in agreement with previous estimates. The value of $Λ_{\overline{MS}}$ is computed from the running of the QCD coupling only after extrapolating to zero dynamical quark mass and after removing a non-perturbative OPE contribution that is assumed to be dominated by the dimension-two $\VEV{A^2}$ gluon condensate. The effect due to the dynamical quark mass in the determination of $\Lams$ is discussed.
We study the low-momentum ghost propagator Dyson-Schwinger equation (DSE) in Landau gauge, assuming for the truncation a constant ghost-gluon vertex, as it is extensively done, and a simple model for a massive gluon propagator. Then, regular DSE solutions (the zero-momentum ghost dressing function not diverging) appear to emerge and we show the ghost propagator to be described by an asymptotic expression reliable up to the order ${\cal O}(q^2)$. That expression, depending on the gluon mass and the zero-momentum Taylor-scheme effective charge, is proven to fit pretty well the low-momentum ghost propagator obtained through big-volume lattice simulations.
We study the predictions for sfermion masses and Lepton Flavour Violation (LFV) for the WMAP preferred parameter space in $b-τ$ Yukawa-unified models with massive neutrinos. A soft term structure as predicted by an Abelian flavour symmetry combined with SU(5) RGEs for scales above $M_{GUT}$, results to an efficient suppression of the off-diagonal terms in the scalar soft matrices, particularly for $m_0< 100$ GeV. Using the WMAP bounds, this implies $35\le\tanβ\le 45$, $350\,\mathrm{GeV}\le m_{1/2}\le 1\,\mathrm{TeV}$, with the higher $\tanβ$ values being favored. Within this framework, SU(5) unification becomes compatible with the current experimental bounds, in contrast to the conventional case where the soft terms are postulated at the GUT scale.
Fundamental quantities of QCD, such as the strong coupling and $Λ_{QCD}$, are studied in the framework of lattice QCD with $N_f=2$ twisted mass fermions. In particular, the contact between lattice and continuous calculations is made by comparing the renormalized ghost-gluon vertex in MOM scheme with 4-loop perturbative results. A power correction is needed in order to have agreement between the two descriptions. This suggests the presence of a dimension-two $\VEV{A^2}$ gluon condensate whose value is found to be higher than in the quanched case.
We discuss the different methods to obtain reliable informations about the deep infra-red behaviour of the gluon and ghost Green functions in QCD. We argue that a clever combination of analytical inputs and numerical ones is necessary. We illustrate this statement about the distinction between two classes of solutions of the ghost propagator Dyson-Schwinger equation (GPDSE). We conclude that the solution II ("decoupling") with a finite renormalised ghost dressing function at zero momentum is strongly favored by lattice QCD, We derive a method to solve numerically the GPDSE using lattice inputs concerning the gluon propagator. We derive an analytical small momentum expansion of the Ghost dressing function. We prove from the large cut-off behaviour of the ghost propagator renormalisation constant, $\widetilde Z_3$, that the bare ghost dressing function is infinite at the infinite cut-off limit.
Beyond the deep IR, the analysis of ghost and gluon propagators still keeps very interesting non-perturbative information. The Taylor-scheme coupling can be computed and applied to obtain the $Λ_{\rm QCD}$ parameter from Landau gauge lattice simulations. Furthermore, a dimension-two gluon condensate, that can be understood in the instanton liquid model, plays an important role in the game.
We study a relation recently derived by K. Kondo at zero momentum between the Zwanziger's horizon function, the ghost dressing function and Kugo's functions $u$ and $w$. We agree with this result as far as bare quantities are considered. However, assuming the validity of the horizon gap equation, we argue that the solution $w(0)=0$ is not acceptable since it would lead to a vanishing renormalised ghost dressing function. On the contrary, when the cut-off goes to infinity, $u(0) \to \infty$, $w(0) \to -\infty$ such that $u(0)+w(0) \to -1$. Furthermore $w$ and $u$ are not multiplicatively renormalisable. Relaxing the gap equation allows $w(0)=0$ with $u(0) \to -1$. In both cases the bare ghost dressing function, $F(0,Λ)$, goes logarithmically to infinity at infinite cut-off. We show that, although the lattice results provide bare results not so different from the $F(0,Λ)=3$ solution, this is an accident due to the fact that the lattice cut-offs lie in the range 1-3 GeV$^{-1}$. We show that the renormalised ghost dressing function should be finite and non-zero at zero momentum and can be reliably estimated on the lattice up to powers of the lattice spacing ; from published data on a $80^4$ lattice at $β=5.7$ we obtain $F_R(0,μ=1.5$ GeV)$\simeq 2.2$.
We compute a formula including OPE power corrections to describe the running of a QCD coupling non-perturbatively defined through the ghost and gluon dressing functions. This turns out to be rather accurate. We propose the ``{\it plateau}''-procedure to compute $Λ_{\bar{\rm MS}}$ from the lattice computation of the running coupling constant. We show a good agreement between the different methods which have been used to estimate $Λ_{\bar{\rm MS}}^{N_f=0}$. We argue that $Λ_{\bar{\rm MS}}$ or the strong coupling constant computed with different lattice spacings may be used to estimate the lattice spacing ratio.
We examine analytically the ghost propagator Dyson-Schwinger Equation (DSE) in the deep IR regime and prove that a finite ghost dressing function at vanishing momentum is an alternative solution (solution II) to the usually assumed divergent one (solution I). We furthermore find that the Slavnov-Taylor identities discriminate between these two classes of solutions and strongly support the solution II. The latter turns out to be also preferred by lattice simulations within numerical uncertainties.
We exploit the Ward-Slavnov-Taylor identity relating the 3-gluons to the ghost-gluon vertices to conclude either that the ghost dressing function is finite and non vanishing at zero momentum while the gluon propagator diverges (although it may do so weakly enough not to be in contradiction with current lattice data) or that the 3-gluons vertex is non-regular when one momentum goes to zero. We stress that those results should be kept in mind when one studies the Infrared properties of the ghost and gluon propagators, for example by means of Dyson-Schwinger equations.
We consider the constraints of the Slavnov-Taylor identity of the IR behaviour of gluon and ghost propagators and their compatibility with solutions of the ghost Dyson-Schwinger equation and with the lattice picture.