Decoupling of charm beyond leading order

We study the effective theory of decoupling of a charm quark at low energies. We do this by simulating a model, QCD with two mass-degenerate charm quarks. At leading order the effective theory is a pure gauge theory. By computing ratios of hadronic scales we have direct access to the power corrections in the effective theory. We show that these corrections follow the expected leading behavior, which is quadratic in the inverse charm quark mass.


Introduction
Several lattice groups are performing simulations of QCD 3 with up, down and strange quarks. They are cheaper and simpler than simulations of QCD 4 which includes a dynamical charm quark. The motivation to neglect the charm quark at low energies E M charm ≡ M c is decoupling. QCD 4 can be described by an effective theory which at leading order is QCD 3 without the charm quark. Neglecting the light quark masses, the effects of decoupling of the charm at low energies are incorporated in the matching of the gauge couplings and in the power corrections stemming from higher order terms in the effective theory. Matching of the gauge couplings of QCD 4 and QCD 3 can be performed in perturbation theory. This is used in the determination of α s from QCD 3 simulations by the ALPHA collaboration [1,2]. Non-perturbative tests of decoupling are important. On one hand to confirm the applicability of perturbation theory, on the other hand to know the size of the power corrections and whether they can be neglected. We address the latter issue here.
In order to avoid a multi-scale problem and control the continuum limit we study a model, QCD 2 with N f = 2 degenerate quarks of mass M. In this case the effective theory for E M is a Yang-Mills theory (N f = 0, M = ∞) at leading order. Beyond leading order there are power corrections starting at M −2 . In a previous work [3] simulation of masses up to M ≈ M c /2 allowed to estimate that the size of the power corrections is at the permille level for M = M c . These estimates were based on interpolations with the pure gauge theory at M = ∞. However a behaviour of the power corrections ∝ M −2 was not seen. In this contribution, which is based on [4], we extend the mass range of the

Effective theory of decoupling
In this work we consider only virtual effects of a heavy quark with mass M. We exclude states with explicit heavy quarks from the discussion. The decoupling of the heavy quark at low energies can be described in terms of an effective Lagrangian [5]. In the case of decoupling of N f = 2 mass-degenerate heavy quarks it reads In eq. (1) L YM is the Yang-Mills Lagrangian which describes the leading order in an expansion in inverse powers of M. Due to gauge invariance there are no dimension 5 operators. Therefore the first correction starts at M −2 with the dimension 6 Lagrangian L 6 defined in eq. (2) which contains two independent terms [6,7]. The Yang-Mills Lagrangian L YM has one free parameter, the gauge coupling. Matching the N f = 2 and Yang-Mills theories means specifying a value of the Yang-Mills coupling at some scale or equivalently its Λ parameter. Matching can be described by the relation which expresses the effective Lambda parameter Λ YM of the Yang-Mills theory after matching as a function of the heavy quark mass and the Lambda parameter of the N f = 2 theory Λ ≡ Λ (N f =2) . 1 For dimensional reasons this function is a dimensionless factor P(M/Λ) times Λ. Consider a low energy hadronic observable m had . It can be a hadronic scale such as 1/ √ t 0 [8] or 1/r 0 [9]. After matching it takes the same value in both theories up to power corrections: Note that the value m had YM of the observable in the Yang-Mills theory depends on M through the matching eq. (4). This mass dependence is described by the factor P in eq. (3) since c = m had YM /Λ YM is a pure number. Consider now two hadronic scales, m had,1 (M) and m had,2 (M), whose values in the Yang-Mills theory are denoted by m had,1 YM and m had,2 YM respectively. A consequence of the matching relation eq. (4) for the ratio of two hadronic scales is Note that the ratio of scales in the Yang-Mills theory is given by the ratio of two pure numbers and is independent of the Lambda parameter Λ YM (or of the gauge coupling). The matching of the couplings is therefore irrelevant for the ratios in eq. (5). An immediate consequence of eq. (5) is where k is a number which depends on the ratio R.

Model calculations with two heavy quarks
The data of Ref. [3] were generated from simulations of N f = 2 O(a) improved Wilson quarks with plaquette gauge action. In this work we use N f = 2 twisted mass [10] Wilson quarks at maximal twist with clover term [11] and plaquette gauge action. We simulated larger masses corresponding to M c /Λ = 4.8700 (charm) and M/Λ = 5.7781. We also simulated the pure gauge theory (N f = 0, M = ∞). The parameters of the new simulations are summarized in Table 1. We use open boundary conditions and the publicly available openQCD simulation package [12]. 2 More details on the simulations can be found in Ref. [4].   The lattice spacing for the N f = 2 theory is determined from the scale L 1 /a [13,14]. For the N f = 0 theory we use the scale r 0 /a. Our spatial box sizes L are such that Lm PS 4 and L/ √ t 0 ≥ 8. In the simulation marked by * we explicitley checked that finite volume effects are negligible. Figure 1 shows the autocorrelation function of t 0 (in units of 16 MDU) for the simulation N f = 2, β = 6.0, M = M c . The autocorrelation function for a derived quantity like t 0 is defined as in Eq. (33) of Ref. [15]. A fit of the form [16] A exp(−t/τ exp ) to the tail between t = 15 und t = 42 (represented by the red line in Fig. 1) gives an estimate of the exponential autocorrelation time τ exp = 477(101) MDU. Considering all our ensembles we find a behaviour which can be parametrized by τ exp = −32(23) + 17.4(2.8) t 0 /a 2 . The scaling τ exp ∝ t 0 /a 2 is expected with open boundary conditions [17].

Results from lattice simulations
On the N f = 2 and N f = 0 ensembles we measure the following ratios of hadronic scales (cf. eq. (5)) The scale t 0 is defined through [8] where E = 1 4 G a µν G a µν is the action density of the gauge field smoothed by the Wilson flow [18][19][20] and t is the flow time whose mass dimension is −2. Similarly, the scale t c is defined by the condition The scale w 0 is defined as [21] The scale r 0 is determined through the condition [9] where the static force F(r) = V (r) is the derivative of the static potential V. For a given value of M/Λ and action A (twisted mass, standard Wilson or for M = ∞ pure gauge) we perform continuum extrapolations of the ratios. From Symanzik's theory we expect O(a 2 ) cut-off effects. We fit our data to where the fit parameters are the continuum values R cont (M/Λ) and the slopes c(M/Λ, A). In the case where simulations at the same mass were performed with two different actions we take a combined continuum limit. We apply a cut, a 2 /t 0 (M) < 0.32, to the data being fitted. The fits of the ratio R = √ t c /t 0 are shown in the left plot of Fig. 2 and the continuum extrapolated values are listed in Table 2. In the right plot of Fig. 2 the continuum values R cont (M/Λ) are plotted against Λ/M. The dashed line in the blue band represents the effective theory prediction eq. (7) fitted through data points from M = ∞ down to M/Λ ≥ 2.5000. It has a good χ 2 /dof = 1.75/2. A linear fit in M −1 is shown by the dashed line in the green band and has a far worse χ 2 /dof = 9.55/2. More fits are discussed in Ref. [4]. They clearly support the onset of the effective theory behavior eq. (7) once data for M M c are included in the analysis.
For a check we also perform a global fit where the slopes are parametrized by the mass-independent coefficient c(A) and by the parameters with the quadratic (dashed line in the blue band) and linear (dashed line in the green band) fits in M −1 . Note that the errors of the continuum values are now correlated and this correlation is taken into account in the fits. The quadratic fit eq. (7) has a correlated χ 2 /dof = 16.0/2 and the linear fit has χ 2 /dof = 7.6/2. None of the fits work well although it seems that the quadratic fit describes better the curvature in the data. The continuum extrapolated values of the ratios R = √ t 0 /w 0 (left plot) and R = r 0 / √ t 0 (right plot) are plotted against Λ/M in Fig. 4. They are obtained using the fit eq. (13). The ratio R = √ t 0 /w 0 strongly favors the M −2 behavior as it is the case for R = √ t c /t 0 . Although a state of the art  determination for r 0 has been used [22], the precision of the ratio R = r 0 / √ t 0 is not enough to resolve the power corrections.

Conclusions and outlook
We studied the decoupling of a charm quark non-perturbatively in QCD with two heavy quarks of mass M. By comparing ratios of hadronic flow scales to their values in the Yang-Mills theory we are able to measure the effects of a dynamical charm quark which are of 2 permille size. Our data can be very well fitted by the effective theory prediction for the power corrections ∝ M −2 down to masses M c /2.
As an outlook, we are computing the effects of a dynamical charm in other observables such as the charm quark mass and charmonium [23] and the strong coupling from the static force [24].