Discretization effects on renormalized gauge-field Green’s functions,
scale setting and gluon mass

We have produced the $SU(3)$ lattice gauge field configurations $U_{\mu}(x)$ from the Monte Carlo sampling using the standard Wilson gauge action,

where $\beta\equiv 6/g_{0}^{2}(a)$; and next gauge fixed them to the minimal Landau gauge as explained, for instance, in Ref. Ayala et al. (2012). The set-up’s parameters can be found in Tab. 1. Then, the gauge field is defined as

with $\hat{\mu}$ indicating the unit lattice vector in the $\mu$ direction. The two-point gluon Green function is then computed in momentum space through the following Monte-Carlo average

with

where $\lambda^{a}$ stands for the Gell-Mann matrices and the trace is evaluated in color space. Then, the gluon dressing function results from taking the appropriate trace of the propagator,

On the other hand, the Landau gauge ghost propagator result from the Monte-Carlo averages of the inverse of the Faddeev-Popov operator, $M$. Namely,

Thus, Eq. (14) and Eq. (15) define the bare gluon and ghost dressing function which are obtained as the appropriated projection of the lattice propagators. Statistical errors have been derived by applying the Jackknife procedure. More details of the computations can be found in Refs. Boucaud et al. (2005b); Ayala et al. (2012).

Before applying the MOM prescription to get the renormalized dressing functions, as above stated, the $H(4)$-extrapolation is applied to deal with the $O(4)$-breaking artifacts. The dressing functions, being scalar form factors of two-point Green’s functions, computed in lattice QCD, are not invariants under $O(4)$ but under $H(4)$ transformations. Therefore, the prescribed recipe implies the average of results obtained for momenta corresponding to the same $H(4)$ orbit (all the lattice four-momenta with the same $k^{[n]}$ invariants) and, next, an extrapolation towards the continuum limit by the subtraction of the $O(4)$-breaking contributions, fitted as smooth functions of $k^{[n]}$ for all orbits sharing the same $k^{2}$. More details for the $H(4)$-extrapolation procedure can be found in Refs. Becirevic et al. (1999, 2000); de Soto and Roiesnel (2007). On top of this, for all the set-up’s, we will apply an upper cut in lattice momenta: $ka(\beta)\leq\pi/2$.

Finally, the scale setting is a key issue for the combined analysis of data resulting from different set-up’s with several $\beta$’s. Specially, such a combined analysis relies on an accurate relative lattice calibration; i.e., the knowledge of the ratios of lattice spacings for any pair of $\beta$’s. Indeed, a thorough discussion about how deviations in the lattice calibration might impact on the scaling of renormalized propagators has been the object of a Comment Boucaud et al. (2017b) to Ref. Duarte et al. (2016) and its corresponding Reply Duarte et al. (2017). With this in mind, we can follow Ref. Guagnelli et al. (1998) and write

where the coefficients $a_{j}$, obtained by a fit of Eq. (16) to lattice spacings obtained from applying the Sommer’s parameter method for $5.7\leq\beta\leq 6.92$ Necco and Sommer (2002), appear gathered in Tab. 2. Nevertheless, cut-off effects borrowed by the scale-setting procedure, the determination of the heavy quark potential at intermediate distances in the case of the Sommer’s parameter, can induce significative deviations at low $\beta$ for the lattice spacings obtained with two different procedures. The effect of these deviations will be anyhow cured by the extrapolation to the continuum limit when is properly made. However, as suggested in Ref. Boucaud et al. (2017b), the scaling of a renormalized Green’s function obtained for different $\beta$’s, when it exists, provides with a strong criterion guaranteeing the negligible impact on them of cut-effects from the lattice scale setting. This would be an ace for our analysis and we can thus, as done in Ref. Boucaud et al. (2017b), assume that $O(4)$-breaking artifacts dominate the cut-off deviations for the gluon propagator so that, after applying $H(4)$-extrapolation, the scaling of the gluon dressing function can be imposed as the condition in order to fix the ratios of lattice spacings. As it will be seen below, this assumption underlying the validity of the relative scale setting, i.e. the scaling of the gluon dressing function after renormalization and $H(4)$-extrapolation, can be explicitly confirmed a posteriori. In so doing, we obtain the results of Tab. 2 for the ratios $a(\beta)/a(\beta_{0})$ and fit Eq. (16) to them, thus obtaining a refined set of coefficients $a_{j}$, also collected in Tab. 2, reliable only from $5.6\leq\beta\leq 6.2$. We have then displayed the results from Eq. (16) with the two sets of coefficients $a_{j}$, from Necco and Sommer (2002) and the one herein obtained, in Fig. 1 and show that both agree pretty well when $\beta,\beta_{0}\geq 5.9$ and that deviations appear only if the ratios involve lower values of the gauge-coupling parameter, distancing a simulation set-up from the continuum limit; the lower is $\beta$ the larger the deviation. Still, if $\beta_{0}$=6.2, the deviation amounts only a 3.6 % for ratios with $\beta$=5.7, in the lower border of the validity range of Ref. Necco and Sommer (2002)’s results, and increases up to a 5.3 % with $\beta$=5.6, outside this range.

In what follows, we will apply the ratios of lattice spacings from Tab. 2 obtained by requiring the scaling of the gluon dressing functions and will thus express all momentum- and mass-dimension quantities in units of $1/a(5.8)$. A conversion to usual physical units can be done by implementing an absolute calibration for one of the lattices at a particular $\beta$ and applying next the ratios from Tab. 2. When needed, we will take $r_{0}\Lambda_{\overline{\rm MS}}$=0.586 and $\ln{(a(\beta)/r_{0})}$ from Eq.(2.6) in Ref. Necco and Sommer (2002), $\Lambda_{\overline{\rm MS}}$=0.224 GeV from Ref. Boucaud et al. (2009) and the ratio $a(\beta)/a(5.8)$ from Tab.2 to obtain $1/a(5.8)=1.372$ GeV and make the conversion to physical units (this particular result is obtained for $\beta=6.2$ but, as it is apparent from Fig. 1, results derived from any other $\beta$ between 6.0 and 6.2 only differ by less than a 0.5 %). Now, we are in good position of placing Eq. (9) for the lattice Taylor coupling under scrutiny.

In order to need no further assumption concerning the nonperturbative running for the continuum $\alpha_{T}(k^{2})$, we proceed by analyzing ratios of $\alpha_{T}^{L}(k^{2};a)$ estimated from different lattice set-up’s. Indeed, according to Eq. (9), these ratios differ from 1 by

where the overlined quantities denote that they are expressed in units of $1/a(\beta_{0})$; i.e., $\overline{\#}\equiv\#\;a(\beta_{0})$. Therefore, the deviations from $1$ for the ratios of $\alpha_{T}^{L}(k^{2};a)$ appear to be described by an expression, Eq. (III.2)’s second line, not depending on the lattice parameter’s set-up, weighted by the factor $a^{2}(\beta)/a^{2}(\beta_{0})-1$. Specially, according to our conjecture in Eq. (5), the parameters $m_{g}$ or $\Lambda$ should presumably result from a nonperturbative resummmation up to all orders in $g_{R}$, this is the physical interaction and they will thus be universal (except for their possible borrowing of higher-order $o(a^{2})$-corrections by practical fitting); while $c_{\alpha}$ and $d_{\alpha}$ rely on the discretization and will generally depend on the details of the lattice action, the gauge-field definition or the gauge-fixing. However, once these details are fixed and remain the same for all the simulations, as we did, the results obtained for different choices of the bare gauge coupling, $g^{2}_{0}(a)=6/\beta$ (see Tab. 1), can only differ by the effect of the factor $a^{2}(\beta)/a^{2}(\beta_{0})-1$. This is a main feature that is made strikingly apparent by Fig. 2. To produce it, we first compute $\alpha_{T}^{L}(k^{2};a)$ after Eq. (7) for all the set-up’s indicated in Tab. 1. In particular, we did it for $\beta=5.6,5.7,5.8,6.02$ and the larger volume of $5.9$, and evaluate next Eq. (III.2)’s l.h.s. with $\beta_{0}=5.8$, over the momentum intervals where the data for $\beta$ and $\beta_{0}$ overlap. In order to compute the ratios, the lattice running couplings for $\beta_{0}$ have been estimated at the same momenta as those for each $\beta$ by performing an interpolation with a Legendre polynomial and propagating errors. Thus, the data displayed in Fig. 2 have been obtained directly from the bare gluon and ghost Green’s functions without any renormalization other than the multiplication by $6/\beta$, which introduces no additional freedom for data rescaling. As it can be clearly seen,

• (i)

  Eq. (III.2) fits well the data (obtained for five different values of $\beta$), explaining satisfactorily their structure and dispersion in terms of $\beta$, only controlled by the factor $a^{2}(\beta)/a^{2}(\beta_{0})-1$;

• (ii)

  data clearly deviates from a linear behavior on $k^{2}a^{2}(\beta_{0})$, strongly supporting the introduction of the Eq. (6)’s nonperturbative logarithmic term, and

• (iii)

  are consistent with the emergence of a mass scale preventing from the zero-momentum logarithmic divergence.

For the fit, we have taken $m_{g}$ from Ref. Binosi et al. (2017a) and are so left with the mass-dimension $\Lambda$ and the dimensionless $c_{\alpha}$ and $d_{\alpha}$ as free parameters that take the best-fit values gathered in Tab. 3.

It is worthwhile to highlight a striking feature shown by Fig. 2 (specially in the lower panel): the data for all the different $\beta$ happen to cross zero fairly at the same momentum, thus implying that $\alpha_{T}^{L}(\zeta_{0}^{2};a(\beta))=\alpha_{T}(\zeta_{0}^{2},a(\beta_{0}))$, where $\zeta_{0}$ is the same physical momentum for all $\beta$. In other words, the $O(a^{2})$-corrections for $\alpha_{T}^{L}(k^{2};a)$ in Eq. (9) become quenched at the same non-zero physical momentum, for which the physical running coupling $\alpha_{T}(k^{2})$ is recovered, irrespective of the values of the gauge coupling and lattice spacing that are used for the lattice set-up. This momentum can be estimated as

from Eq. (III.2) and Tab. 3. Actually, Fig. 2 tells us that the Taylor coupling directly computed from the bare gluon and ghost Green’s functions obtained from four different lattice set-up’s ($\beta$ ranging from 5.6 to 5.9) only coincides with each other very much near $\overline{k}=\overline{\zeta}_{0}$. A fifth simulation at $\beta$=6.02 is performed in a lattice volume so small in physical units that it cannot reach the zero-crossing low-momentum region without being significantly affected by volume artifacts (as will be discussed below). Its data obtained for $\overline{k}>0.6$ appear however to behave well according to Eq. (III.2) with the parameters displayed in Tab. 3.

After the painstaking scrutiny we have made in the previous subsection for a reliable description of discretization lattice artifacts from the Taylor coupling, Eq. (9) can be applied to take the continuum limit and thus obtain the nonperturbative physical running of the coupling, $\alpha_{T}(k^{2})$, from its corresponding lattice estimate, $\alpha_{T}^{L}(k^{2})$. Still, for so to do, one needs to make sure that finite volume effects are under control and do not appear entangled with discretization artifacts in the lattice estimates of the lattice coupling. As it happens in the analysis of an amplitude in spectroscopy, corrections of the order of $\exp{(-mL)}$ are expected for any lattice correlation function, where $L$ is the physical size of the hypercubic lattice and $m$ the mass of the physical bound states which propagates all over the lattice. However, in a quenched theory, even the dominant contribution is negligible when the lattice size is of a few fm’s, as it should come from the lightest glueball state, for which $mL\sim O(10)$. On the other hand, when computing the gauge-field correlations functions wich take part in Eq. (7), a sizeable effect should appear when the associated gauge-field wavelength is, at least, of the same order as the lattice size, i.e. when $Lp\raise 1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}1$. Thus, the larger is the momentum for which the correlation function is evaluated and the shorter the associated gauge-field wavelength, the less impact the volume effects have. In practice, this impact can be estimated in Fig. 3, where we display the results for the continuum $\alpha_{T}(k^{2})$, obtained with Eq. (9) and the parameters of Tab. 3, for the lattice estimates from three simulations made at $\beta$=5.9 in three different lattice volumes, $V$=$3.48^{4}$ ($L/a$=30), $5.56^{4}$ ($L/a$=48) and $7.42^{4}$ fm⁴ ($L/a$=64) and a fourth one at $\beta$=6.02 and $V$=$(3.35\;\mbox{\rm fm})^{4}$ (see Tab. 1). We can there clearly appreciate that

• (i)

  the two simulations made in lattices of $L$=5.56 fm and $L$=7.42 fm at $\beta$=5.9 exhibit results plainly compatible within the errors for all comparable momenta, while the one for the same $\beta$ and $L$=3.48 fm shows a significant volume effect only for $\overline{k}\raise 1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}0.6$;

• (ii)

  the results from the two simulations made in the same smaller physical volume and different $\beta$ appear also to fully agree over the whole range of momenta.

We can thus conclude that the finite-volume artifacts affecting the Taylor coupling via gauge-field correlation functions are controlled by the lattice physical volume; and that, in quenched QCD, they are negligible in practice when this physical volume is above $(6\;\mbox{\rm fm})^{4}$. The latter agrees with the findings of Ref. Duarte et al. (2016).

Therefore, the four simulations at $\beta$=5.6,5.7,5.8 and 5.9 we dealt with in Subsec. III.2, made in lattices of more than 6 fm, can be taken, in very good approximation, as free of finite-volume artifacts. In Fig. 4, it is shown how the results from these simulations look like, before (upper panel) and after (lower) the continuum extrapolation made through Eq. (9). For the sake of comparison, the results obtained at $\beta$=6.02 and in a volume $V$=$(3.35\;\mbox{\rm fm})^{4}$ lattice appear also displayed and, after extrapolation, all data for $\overline{k}\raise 1.29167pt\hbox{$<$\kern-7.5pt\raise-4.73611pt\hbox{$\sim$}}0.6$ dropped. The scaling found for five different simulations, with five different values of $\beta$, is extremely good, all data lying strikingly on top of each other after applying Eq. (9) with the gluon mass borrowed from literature and the three other fitted parameters shown in Tab. 3.

To obtain the results displayed in the previous subsection for the Taylor running coupling, one essentially needs to deal with bare gauge-field two-point Green’s functions and the bare coupling, $g^{2}(a)=6/\beta$, properly combined as Eq. (7) reads; and the relative lattice calibration described in Subsec. III.1, which left us with the ratios of lattice spacings of Tab. 2. We thus apply the $H(4)$-extrapolation to cure from $O(4)$-breaking artifacts and express all the Green’s functions, so obtained from different lattice set-up’s, in terms of the momentum expressed in the same non-standard but physical units, $1/a(5.8)$; and produce then the plots of Fig. 2. They make strikingly apparent that the remaining cut-off corrections behave as Eq. (9) dictates. We have employed a relative lattice calibration which agrees well with that in Ref. Necco and Sommer (2002) but introduces marginal deviations at low $\beta$ of as much as about 3.6 %. This calibration is anyhow grounded on the assumption that $O(4)$-invariant artifacts have a negligible impact on the gluon propagator, implying thereupon that its dressing function should exhibit a physical scaling after renormalization and $H(4)$-extrapolation. The latter is a factual statement¹¹1One may argue that the tree-level gluon dressing function in lattice perturbation theory reads $D_{L}(k^{2},a)=1+1/12\;a^{2}k^{[4]}/k^{2}+o(a^{2})$ and should be so cured only from a $O(4)$-breaking artifact. Furthermore, the examination of the effective operators improving the gauge action à la Symanzik at the ${\cal O}(a^{2})$-order seems not to justify the need of curing in addition from $O(4)$-invariant artifacts. However, it is hard to exclude that such artifacts might result in a pure nonperturbative approach. discussed in Ref. Boucaud et al. (2017b) and, precisely, applied therein to refine the lattice scale setting. Here, it merely implies that, after MOM renormalization and $H(4)$-extrapolation, both gluon and ghost propagators should behave as dictated by Eqs. (5)-(6) with

$c_{\alpha}$ and $d_{\alpha}$ given in Tab. 3. This is, strikingly again, confirmed by Figs. 5 and 6. In them, and owing to a sampling of gauge-field configurations of $O(1000)$, we displayed data for the renormalized gluon and ghost propagators with statistical errors of the order of, respectively, one and five per mil. Even at this impressive level of statistical accuracy, the ratios of gluon dressing functions obtained in large-volume lattices at four different $\beta$’s do not differ from 1 within the errors, except for deeply low momenta (see upper panel of Fig. 5). There, for $\overline{k}<0.2$, results at $\beta$=5.6 and $V^{1/4}$=11.3 fm deviate by around a 2 % from those at $\beta$=5.7,5.8 and 5.9 and $V^{1/4}\simeq$ 7 fm which, on their side, remain compatible with each other. This strongly suggest that this slight deviation is caused by a still-remaining finite-volume effect. This systematic effect is very nearly negligible, made only apparent by the huge statistics here employed, and only happens at very low momenta: namely, there is no impact from it at $\overline{k}\simeq 0.29$ (highlighted by a red dashed line in Figs. 5 and 6), the momentum for which all the lattice estimates for the Taylor coupling coincide in Fig. 2. The same is shown in the lower panel of Fig. 5, where the gluon propagators from the four simulations appear plotted and lie, very accurately, on top of each other. This is a non-obvious result firmly confirming the starting hypothesis of the relative scale setting and the efficiency of the H4-extrapolation.

In Fig. 6, the ratios of ghost propagator dressing functions from the four lattice simulations are clearly proven to behave according to

obtained from Eqs. (5)-(6), and with Eq. (19) (upper panel); and, after the appropriate continuum extrapolation, the dressing functions are shown to exhibit a nearly perfect physical scaling (lower panel). The renormalization point is chosen here to be $\overline{\zeta}$=0.8, lying thus within the momentum range, and far from its borders, for the four simulations. This is why, precisely, we do not include here the data from the simulation at $\beta$=6.02 in the small volume ($V^{1/4}$= 3.35 fm): they are significantly affected by finite-volume artifacts at this renormalization point and, therefore, the MOM renormalization prescription will contaminate with these artifacts the whole momentum range. Its momentum range and $\beta$=5.8 simulation’s offer anyhow a reliable overlap which makes possible the determination of the ratio $a(6.02)/a(5.8)$ in Tab. 2.

Furthermore, as explained in Fig. 1’s caption, we have also computed the gluon propagator dressing function for a simulation at $\beta$=6.202 and $V^{1/4}$=3.35 fm, exploited the overlap with $\beta$=5.8 and so extracted the ratio $r=a(6.202)/a(5.8)$ in Tab. 2. Then, we calculated $a(\beta)/a(6.202)=r^{-1}a(\beta)/a(5.8)$ and plotted the results in Fig. 1. Indeed, as can be seen in the plot, our estimates for $\beta>$ 5.9 appear to be in nearly perfect agreement with those of Ref. Necco and Sommer (2002).

It should be reminded that no fit is made here to produce the results displayed in Figs. 5 and 6. After the scale setting, we merely apply a MOM renormalization prescription, compute the ratios of dressing functions and, when needed, use Eq. (20) with the parameters of Tab. 3, obtained for the Taylor coupling, and Eq. (19). It is important to highlight that the connection between the lattice Taylor coupling and the dressing functions relies on the field Theory and its renormalization: Eq. (7) for $\alpha_{T}^{L}(k^{2};a)$ involves the bare two-point Green functions and the bare gauge coupling which is directly given by the parameter $\beta$ in the lattice gauge action. Their dependence on the lattice spacing is singular for each but cancels in Eq. (7), remaining only a residual one which vanishes in the continuum limit. Such a residual dependence is also related to the one from the MOM renormalized Green’s functions. On the other hand, the way in which the lattice spacing relates to $\beta$ depends on the lattice action and determines the physical scaling of quantities from lattice simulations; namely, the dressing functions in the continuum limit. How all this takes place and match is highly non-trivial. This is what we have exposed here.