# SMD-based numerical stochastic perturbation theory

- 507 Downloads
- 5 Citations

## Abstract

The viability of a variant of numerical stochastic perturbation theory, where the Langevin equation is replaced by the SMD algorithm, is examined. In particular, the convergence of the process to a unique stationary state is rigorously established and the use of higher-order symplectic integration schemes is shown to be highly profitable in this context. For illustration, the gradient-flow coupling in finite volume with Schrödinger functional boundary conditions is computed to two-loop (i.e. NNL) order in the SU(3) gauge theory. The scaling behaviour of the algorithm turns out to be rather favourable in this case, which allows the computations to be driven close to the continuum limit.

## 1 Introduction

Numerical stochastic perturbation theory (NSPT) [1, 2, 3] is a powerful tool that allows many interesting calculations in QCD and other quantum field theories to be performed to high order in the interactions. For technical reasons, the computations proceed in the framework of lattice field theory, but results for renormalized quantities in the continuum theory can then be obtained through an extrapolation to vanishing lattice spacing. NSPT can be highly automated and the application of the method in finite volume and to correlation functions of complicated composite fields gives rise to hardly any additional difficulties.

Reliable extrapolations to the continuum limit require accurate data at several lattice spacings in the scaling region. NSPT calculations can therefore rapidly become large-scale projects, where computational efficiency is all-important. Traditionally, NSPT is based on the Langevin equation, but the success of the HMC algorithm [4] in lattice QCD suggests that the inclusion of a molecular-dynamics update step in the underlying stochastic process might be beneficial. Smaller autocorrelation times and an improved scaling behaviour towards the continuum limit could perhaps be achieved in this way. Moreover, through the use of highly efficient symplectic integration schemes, the systematic errors deriving from the discretization of the simulation time may conceivably be reduced.

NSPT based on the SMD (stochastic molecular dynamics, or generalized HMC) algorithm [5, 6, 7] has recently been briefly looked at in Ref. [8] and was found to perform well. Here we establish the convergence of the algorithm to a unique stationary state and study its efficiency in the case of the gradient-flow coupling in the SU(3) gauge theory. Various technical problems are addressed along the way, among them the modifications required to ensure that the stochastic process does not run away in the gauge directions.

## 2 Stochastic molecular dynamics

In order to bring out the basic structure of the SMD-variant of NSPT most clearly, a generic system described by a set \(q=(q_1,\ldots ,q_n)\) of real coordinates and an action *S*(*q*) is considered in this and the following two sections.

### 2.1 Preliminaries

*S*(

*q*) is assumed to be differentiable and to have an expansion in powers of a coupling

*g*of the form

*q*of degree \(d_r\ge 2\). Moreover, it is taken for granted that the leading-order term

### 2.2 SMD algorithm

The SMD algorithm operates in the phase space of the theory and thus updates both the coordinates *q* and their canonical momenta \(p=(p_1,\ldots ,p_n)\). An SMD update cycle consists of a momentum rotation followed by a molecular-dynamics evolution and, optionally, an acceptance–rejection step.

*t*to \(t+\epsilon \) using a reversible symplectic integration scheme (see Sect. 2.3). The algorithm (momentum rotation followed by the molecular-dynamics evolution) simulates the canonical distribution

### 2.3 Integration schemes

*h*proportional to \(\epsilon \). A well-known example is the leapfrog integrator \(I_{p,\epsilon /2}I_{q,\epsilon }I_{p,\epsilon /2}\), and several highly efficient schemes are described in Ref. [10].

*P*stands for the momentum reflection \(p\rightarrow -p\).

## 3 Stochastic perturbation theory

Stochastic perturbation theory [11, 12] is usually derived from the Langevin equation by expanding the stochastic variables and the driving forces in powers of the coupling. In this section, another (although probably closely related) form of stochastic perturbation theory is discussed, which is obtained by expanding the SMD algorithm in the same way.

### 3.1 SMD algorithm at weak coupling

Since the acceptance–rejection step is not smooth in the coupling, its effects would be difficult to take into account in perturbation theory. In the following, the acceptance–rejection step is therefore omitted, without further notice, and one is thus left with an algorithm that simulates the system only up to integration errors.

*p*(

*t*),

*q*(

*t*) of the momenta and coordinates generated by the SMD algorithm depend on the coupling

*g*through the force term in the integration step (2.8). In particular, they are smooth functions of the coupling and may consequently be expanded in the asymptotic series

### 3.2 Perturbation expansion of observables

*t*up to statistical (and integration) errors.

## 4 Convergence to a stationary state

Stochastic processes can run away or do not converge to a stationary distribution for other reasons. In the case of the stochastic perturbation theory described in Sect. 3, the asymptotic stationarity of the underlying process can be rigorously shown if the simulation step size \(\epsilon \) is sufficiently small. The range of step sizes, where convergence is guaranteed, depends on the chosen integration scheme for the molecular-dynamics equations and the matrix \(\Delta \) in the leading-order part (2.2) of the action.

### 4.1 Molecular-dynamics evolution in the free theory

*g*is turned off, the molecular-dynamics equations become linear and their (approximate) integration from time

*t*to \(t+\epsilon \) amounts to a linear transformation

*M*in this equation has a block structure,

### 4.2 Convergence of the leading-order process

*t*. Since the random momenta are normally distributed, the momenta and coordinates at large

*t*are then normally distributed as well, with mean zero and variances equal to their two-point autocorrelation functions. In the large-time limit, the

*pp*,

*pq*and

*qp*blocks of the matrix on the right of Eq. (4.13). The kernel

*K*satisfies

### 4.3 Convergence beyond the leading order

*S*(

*q*) implies that the force in the molecular-dynamics integration step (3.3) is of the form

*r*thus amounts to solving an inhomogeneous linear recursion. A moment of thought then reveals that

Recalling Eq. (4.9) and the fact that \(\tilde{M}\) is a contraction matrix, the convergence of the autocorrelation functions of the momenta and coordinates at large times *t* may now be shown recursively from order 0 to any finite order *r*. Equation (4.18) actually allows the highest-order variables in any correlation function to be expressed through lower-order ones up to an exponentially decaying contribution. Clearly, in the large-time limit, the autocorrelation functions do not depend on the initial distribution of the variables and are stationary, i.e. invariant under time translations.

The expectation values of the coefficients in the perturbation expansion (3.6) of the observables coincide with a sum of autocorrelation functions of the coordinates at equal times. Their convergence at large times is therefore guaranteed as well.

### 4.4 Summary

The discussion in this section shows that the SMD algorithm converges to all orders of perturbation theory if the matrix \(\Delta \) is strictly positive and if \(\epsilon ^2\left\| \Delta \right\| <\kappa \), where \(\kappa \) depends on the molecular-dynamics integrator. In the case of the leapfrog, the second-order OMF and the fourth-order OMF integrators, \(\kappa \) is equal to 4, 6.51 and 9.87, respectively (cf. Appendix A).

## 5 Stochastic perturbation theory in lattice QCD

With respect to the generic system considered so far, the situation in lattice QCD is complicated by the gauge symmetry and the quark fields. In this section, stochastic perturbation theory is first set up for the pure \(\mathrm{SU}(N)\) gauge theory. The modifications required for the damping of the gauge modes are then discussed and the section ends with a brief description of how the quarks can be included in the simulations.

### 5.1 Lattice fields

*x*of the lattice. They vanish at \(x_0=T\) and must, furthermore, be constant at \(x_0=0\) if SF boundary conditions are imposed. In the latter case, the fields may be split in two parts according to

Independently of the boundary conditions imposed on the gauge field, the quark fields are required to vanish at both \(x_0=0\) and \(x_0=T\). No weight factor is included in the scalar product of these fields.

### 5.2 Basic stochastic process

*S*(

*U*) of the gauge field any of the frequently used ones may be taken. The Hamilton function from which the SMD algorithm derives is then given by

### 5.3 Perturbation expansion

### 5.4 Damping of the gauge modes

*A*satisfying \(d^{*}\!A=0\) (\(d^{*}\) is given explicitly in Appendix B).

^{1}

### 5.5 Long-time stationarity of the process

Although the stochastic process now includes variables without associated momenta, the discussion in Sect. 4 applies here too with only minor changes. In particular, the convergence of the process to a unique stationary state is guaranteed to all orders of the coupling, if the matrix \(\tilde{M}\) describing the evolution of the fields \((\hat{\pi }_0,\hat{U}_0,\hat{\omega }_0)\) at leading order is a contraction matrix.

*k*is equal to 1, 5 / 3 and 3.648 in the case of the Wilson [16], tree-level Symanzik improved [17, 18] and Iwasaki [19] gauge action, respectively. In practice, the constraints (5.29) and (5.32) therefore tend to be fairly mild and the main concern is to ensure that the integration errors are sufficiently small at the chosen values of \(\epsilon \).

### 5.6 Inclusion of the quark fields

*R*(

*U*) being some gauge-covariant linear operator (see Ref. [20], for example).

^{2}

When the algorithm is expanded in powers of the coupling \(g_0\), the renormalization of the quark masses should be taken into account so that the masses in the leading-order stochastic equations are the renormalized ones. The pseudo-fermion fields decouple from the other fields at lowest order and are simulated exactly by the random rotation (5.35). Convergence of the stochastic process to a unique stationary state is then again guaranteed to all orders, if the bounds (5.29) and (5.32) are satisfied.

## 6 Computation of the gradient-flow coupling

The gradient-flow coupling in finite volume with SF boundary conditions has recently been used in step scaling studies of three-flavour QCD [22, 23]. Such calculations serve to relate the low-energy properties of the theory to the high-energy regime, where contact with the standard QCD parameters and matrix elements can be made in perturbation theory [24] (see Ref. [25] for a review).

In the following, the perturbation expansion of the gradient-flow coupling is determined in the \(\mathrm{SU(3)}\) Yang–Mills theory up to two-loop order, using the SMD-variant of NSPT described in the previous section. To one-loop order, the expansion coefficient in the \(\overline{\mathrm{MS}}\) scheme of dimensional regularization is known in infinite volume since a while [26], but a huge effort plus the best currently available techniques were required to be able to extend this calculation to the next order [27]. In finite volume with SF boundary conditions, these techniques do not apply and a similar analytical computation may be practically infeasible at present.

### 6.1 Definition of the coupling

*E*(

*t*,

*x*) denotes the Yang–Mills action density at flow time

*t*and position

*x*,

*c*is a parameter of the scheme and the proportionality constant \(k\) is determined by the requirement that \(\bar{g}^2\) coincides with \(g_0^2\) to lowest order of perturbation theory. Most of the time

*c*will be set to 0.3, which implies a localization range of the action density of about \(0.3\times L\).

On the lattice, the gradient flow is implemented as in Ref. [15]. The Wilson action, with boundary terms so as to ensure the absence of O(*a*) lattice effects in the flow equation, is thus used to generate the flow. For the action density *E*(*t*, *x*) in Eq. (6.1) either the Wilson action density or the square of the familiar symmetric “clover” expression for the gauge-field tensor is inserted. Furthermore, alternative couplings, where the full action density is replaced by its spatial or time-like part, are considered as well.

All in all this makes six different action densities and couplings, labeled w, ws, wt, c, cs, and ct, where the letters stand for Wilson, clover, space and time, respectively (\(E^\mathrm{ct}\), for example, denotes the clover expression for the time-like part of the action density).

### 6.2 Expansion in powers of \(\alpha _s\)

*q*, with coefficients \(k_1,k_2,\ldots \) depending on

*q*and

*L*. If

*q*is scaled with

*L*like

*L*in the continuum theory [28].

### 6.3 Computation of the coefficients \(E_k\) in NSPT

*a*) lattice effects in \(E_k\), the action of the theory must include boundary counterterms at \(x_0=0\) and \(x_0=T\) with a tuned coupling [13]

^{3}

*E*(

*t*,

*x*) to be expanded in powers of \(g_0\) for each of these configurations. To this end, the gauge field \(V_t(x,\mu )\) at flow time

*t*is represented by a gauge potential \(B_{\mu }(t,x)\) according to

Simulation parameters

| \(\gamma \) | \(\epsilon \) | \(\Delta t_\mathrm{ms}/\epsilon \) | \(N_\mathrm{ms}\) |
---|---|---|---|---|

10 | 5.0 | 0.168 | 190 | 59400 |

12 | 5.0 | 0.168 | 190 | 59880 |

14 | 5.0 | 0.168 | 190 | 59400 |

16 | 5.0 | 0.168 | 190 | 59880 |

18 | 4.5 | 0.168 | 240 | 58800 |

20 | 4.0 | 0.168 | 290 | 59400 |

24 | 3.0 | 0.190 | 270 | 71880 |

28 | 3.0 | 0.144 | 340 | 79200 |

32 | 3.0 | 0.126 | 480 | 88400 |

40 | 3.0 | 0.100 | 950 | 80100 |

### 6.4 Simulation parameters and tables of results

The parameters of the NSPT runs performed for the “measurement” of \(E_0,E_1,E_2\) are listed in Table 1. In all cases, the gauge-damping parameter \(\lambda _0\) was set to 2 and the fourth-order OMF integrator was employed for the molecular-dynamics equations (cf. Appendix A). Measurements were made after every \(\Delta t_\mathrm{ms}/\epsilon \) update cycles using a third-order Runge–Kutta integrator for the gradient flow [26], with step size varying from 0.002 at small flow times to 0.1 at large times. With this scheme, the gradient-flow integration errors are guaranteed to be completely negligible with respect to the statistical errors. The number \(N_\mathrm{ms}\) of measurements made is listed in the last column of Table 1 (the programs that were used in these simulations can be downloaded from http://luscher.web.cern.ch/luscher/NSPT).

At the chosen values of the parameters, the bounds (5.29) and (5.32) are satisfied by a wide margin so that the convergence of the SMD algorithm is rigorously guaranteed. The results obtained on the simulated lattices for the expansion coefficients \(k_1\) and \(k_2\) and their statistical errors are listed in Appendix D, for \(c\in \{0.2,0.3,0.4\}\) and all choices w,c,ws,\(\ldots \) of the action density (cf. Sect. 6.1).

## 7 Statistical and systematic errors

The values of \(k_1\) and \(k_2\) obtained in NSPT depend on the scheme parameter *c*, the lattice size *L* (in lattice units), the simulation step size \(\epsilon \) and the SMD parameter \(\gamma \). No attempt is made here to determine all these dependencies in detail. Instead some basic facts and empirical results are discussed that helped controlling the errors in the case of the simulations reported in this paper (see Refs. [8, 33] for related complementary studies of the \(\phi ^4\) theory).

### 7.1 Autocorrelations and statistical variances

*r*coefficient \(\hat{\mathcal{O}}_r\) of an observable \(\mathcal{O}\) in general does not coincide with the order-2

*r*coefficient of another observable. The time average of \(\hat{\mathcal{O}}_r^2\) and thus the variance of \(\hat{\mathcal{O}}_r\) are then not necessarily determined by the theory alone. For illustration, the dependence on \(\gamma \) of the variances of the coefficients in

*r*coefficients \(\hat{\mathcal{O}}_r\) of the observables \(\mathcal{O}\) of interest. Empirical studies show that the two factors often work against each other, i.e. algorithms tuned for small autocorrelations tend to give large variances and vice versa.

Autocorrelation times of \(\hat{E}_k^\mathrm{c}\) and associated products (7.2)\(^{\dagger }\)

\(\gamma \) | \(k=0\) | \(k=1\) | \(k=2\) | |||
---|---|---|---|---|---|---|

0.5 | 2.0 | \(1.6\times 10^{-6}\) | 2.0 | \(9.4\times 10^{-6}\) | 2.0 | \(8.8\times 10^{-5}\) |

1.0 | 2.7 | \(2.2\times 10^{-6}\) | 3.0 | \(5.2\times 10^{-6}\) | 2.9 | \(1.0\times 10^{-5}\) |

2.0 | 5.9 | \(4.9\times 10^{-6}\) | 6.7 | \(6.8\times 10^{-6}\) | 7.3 | \(8.1\times 10^{-6}\) |

2.5 | 6.7 | \(5.6\times 10^{-6}\) | 8.4 | \(8.1\times 10^{-6}\) | 9.2 | \(9.1\times 10^{-6}\) |

3.0 | 7.7 | \(6.2\times 10^{-6}\) | 9.4 | \(8.4\times 10^{-6}\) | 11.2 | \(9.8\times 10^{-6}\) |

5.0 | 10.9 | \(8.7\times 10^{-6}\) | 13.0 | \(1.1\times 10^{-5}\) | 16.6 | \(1.3\times 10^{-5}\) |

9.0 | 14.8 | \(1.2\times 10^{-5}\) | 17.7 | \(1.4\times 10^{-5}\) | 21.5 | \(1.5\times 10^{-5}\) |

### 7.2 Critical slowing down

The behaviour of the autocorrelation times and variances near the continuum limit \(L\rightarrow \infty \) depends on the simulation algorithm and the observables considered. When NSPT is based on the Langevin equation, the autocorrelations of the coefficients of multiplicatively renormalizable quantities can be shown to diverge proportionally to \(L^2\) [34, 35], while their variances grow at most logarithmically [36].

At large \(\gamma \), the variant of NSPT studied here is expected to behave similarly, since the SMD algorithm then effectively integrates the Langevin equation. Choosing \(\gamma \) to depend on *L* in some particular way may, however, conceivably lead to an improved scaling behaviour. In the free theory, for example, the autocorrelation times grow proportionally to *L* rather than \(L^2\) if \(\gamma \) is scaled like 1 / *L* [5, 6, 7]. Beyond the leading order, the situation is complicated by the algorithm-dependence of the variances and the effects of the parameter tuning are then not easy to predict.

Considering the fact that the computational cost of the measurements of \(\hat{E}_k\) tends to be larger than the one of the SMD update cycles, the parameters of the runs on the large lattices (those with \(L=24,\ldots ,40\) in Table 1) were chosen so that subsequent measurements are practically uncorrelated. At fixed \(\gamma =3\) the required increase with *L* of the measurement time separation \(\Delta t_\mathrm{ms}\) then turned out to be quite moderate. Moreover, from \(L=24\) to \(L=40\), the variances of \(\hat{E}_k\) grow only slowly (at \(c=0.3\) and for \(k=0\), 1 and 2 by about 2, 19 and 30 percent).

### 7.3 Integration errors

As already noted in Appendix A, the theory is very accurately simulated at leading order if the fourth-order OMF integrator is used for the molecular-dynamics equations. The expectation values of the coefficient \(\hat{E}_0\) calculated in the runs listed in Table 1 in fact all agree with the known analytic formula [15, 21] within a relative statistical precision of about \(2\times 10^{-4}\).

Beyond the leading order, the integration errors remain difficult to detect in empirical studies (see Fig. 2). The stability bounds (5.29), (5.32) are respected in all these runs and the coefficients \(\hat{H}_0,\ldots ,\hat{H}_4\) of the Hamilton function *H* are accurately conserved, with deficits decreasing like \(\epsilon ^5\) (as has to be the case in the asymptotic regime of a fourth-order integrator). It thus seems safe to conclude that the integration errors in the tests reported in Fig. 2 are smaller than the statistical errors.

*L*so that the integration errors fall off like \(1/L^4\) at large

*L*and thus much more rapidly than the leading lattice effects. Since the statistical errors are approximately the same in all runs, the scaling of the step sizes may be a luxury, but was applied here as a safeguard measure against an enhancement of systematic errors through the continuum extrapolation.

### 7.4 Extrapolation to the continuum limit

*L*-dependence of the one-loop coefficient \(k_1^\mathrm{w}\) is asymptotically given by [37, 38]

Values of \(k_1\) and \(k_2\) in the continuum limit

| \(k_1\) | \(k_1^\mathrm{s}\) | \(k_1^\mathrm{t}\) | \(k_2\) | \(k_2^\mathrm{s}\) | \(k_2^\mathrm{t}\) |
---|---|---|---|---|---|---|

0.2 | 1.089(6) | 1.106(7) | 1.066(7) | \(-1.21(6)\) | \(-1.29(6)\) | \(-1.12(7)\) |

0.3 | 1.112(5) | 1.220(6) | 1.005(7) | \(-1.76(4)\) | \(-2.17(5)\) | \(-1.36(6)\) |

0.4 | 1.297(5) | 1.685(6) | 0.935(6) | \(-3.06(6)\) | \(-4.78(7)\) | \(-1.47(8) \) |

In the present context, the goal is not to determine the values of the coefficients \(a_1\), \(b_1\) and \(c_1\), but to perform a short extrapolation of the data to the continuum limit. A sensible way to stabilize fits of the data by the asymptotic expression (7.4) is then to constrain the minimization of the likelihood function to the directions in parameter space orthogonal to its flattest directions (see Fig. 4). The results quoted in Table 3 were obtained in this way and by varying the fit procedure, as in the case of the one-loop coefficients, in order to assess the extrapolation errors.

### 7.5 Miscellaneous remarks

*Lattice effects.* Since the smoothing range of the gradient flow decreases with *c*, it is no surprise that the coefficients \(k_1,k_2\) calculated in NSPT are found to be increasingly sensitive to lattice effects when *c* is lowered. The continuum extrapolation of the data then becomes more and more difficult and eventually requires larger lattices to be simulated.

*Infinite-volume limit.* The gradient-flow coupling in infinite volume runs with the flow time *t* and may be expanded in powers of \(\alpha _s\) at scale \(q=(8t)^{-1/2}\), as in Eq. (6.2), the one- and two-loop coefficients in the continuum limit being \(k_1=1.0978(1)\) [26] and \(k_2=-0.9822(1)\) [27]. In the finite-volume scheme considered in this paper, and after passing to the continuum limit, the infinite-volume limit is reached by sending *c* to zero. The results listed in Table 3 cannot be reliably extrapolated to \(c=0\), but the values of \(k_1,\ldots ,k^\mathrm{t}_2\) at \(c=0.2\) are actually already quite close to the infinite-volume values quoted above.

*Three-loop*\(\beta \)

*-function.*The

*L*-dependence of the gradient-flow coupling \(\alpha =\bar{g}^2/4\pi \) is governed by the renormalization group equation

## 8 Conclusions

Ratio \(\rho _2=\beta _2/\beta _0\) of coefficients of the \(\beta \)-function

| \(\rho _2\) | \(\rho _2^\mathrm{s}\) | \(\rho _2^\mathrm{t}\) |
---|---|---|---|

0.2 | \(-2.38(6)\) | \(-2.51(6)\) | \(-2.22(7)\) |

0.3 | \(-2.99(4)\) | \(-3.74(5)\) | \(-2.29(6)\) |

0.4 | \(-4.88(6)\) | \(-8.04(7)\) | \(-2.21(8)\) |

The SMD algorithm provides a technically attractive basis for NSPT. Compared to the traditional version of NSPT [1, 2, 3], where the starting point is the Langevin equation, a significantly improved efficiency is achieved, particularly so near the continuum limit. Moreover, the available highly accurate integrators for the molecular-dynamics equations allow the integration errors to be easily kept under control. For the reasons mentioned above, some tuning of the friction parameter \(\gamma \) is however required and must take into account the variances of the observables of interest.

The inclusion of the quark fields in the SMD algorithm along the lines of Sect. 5 is straightforward and is not expected to slow down the simulations by a large factor [3]. In general, the cost of NSPT computations very much depends on the observables considered, the order in perturbation theory and the desired precision of the results.

The statistical errors of the expansion coefficients \(k_l\) of the gradient-flow coupling, for example, appear to grow rapidly with the loop order *l*. In practice some loss of precision from one order to the next is tolerable, since the coefficients get multiplied by \(\alpha _s^{l+1}\) in the perturbation series (6.2). An extension of the computations to three-loop order would however only make sense if \(k_1\) and \(k_2\) are recalculated with errors about an order of magnitude smaller than the ones quoted in Table 3. Furthermore, the relation between \(\alpha _s\) and the bare coupling would need to be worked out to three loops.

## Footnotes

- 1.
In the continuous-time limit \(\epsilon \rightarrow 0\), the time-dependence of the gauge-damping field is governed by a first-order differential equation and the modified algorithm integrates a form of the stochastic molecular-dynamics equations, which coincides with the standard one up to a time-dependent gauge transformation (see Appendix C).

- 2.
There is no reason other than simplicity to set the parameter \(\gamma \) that determines the rotation angle to the same value for all fields.

- 3.
In Ref. [15] the improvement coefficient \(c_\mathrm{t}\) is denoted by \(c_\mathrm{G}'\).

## Notes

### Acknowledgements

M.D.B. would like to thank Marco Garofalo, Dirk Hesse, and Tony Kennedy for a pleasant collaboration on related investigations, Alberto Ramos, Stefan Sint, and Rainer Sommer for valuable discussions and the Theoretical Physics Department at CERN for the kind hospitality extended to him. The computations reported in this paper were performed on the SuperMUC machine at the Leibniz Supercomputing Centre in Munich (project ID pr92ci) and dedicated HPC clusters at DESY-Zeuthen and at CERN. We thank all these institutions for the generous support given to this project.

## References

- 1.F. Di Renzo, G. Marchesini, P. Marenzoni, E. Onofri, Lattice perturbation theory on the computer. Nucl. Phys. Proc. Suppl.
**34**, 795 (1994)ADSCrossRefGoogle Scholar - 2.F. Di Renzo, E. Onofri, G. Marchesini, P. Marenzoni, Four loop result in SU(3) lattice gauge theory by a stochastic method: lattice correction to the condensate. Nucl. Phys. B
**426**, 675 (1994)ADSCrossRefGoogle Scholar - 3.F. Di Renzo, L. Scorzato, Numerical stochastic perturbation theory for full QCD. JHEP
**0410**, 073 (2004)CrossRefGoogle Scholar - 4.S. Duane, A.D. Kennedy, B.J. Pendleton, D. Roweth, Hybrid Monte Carlo. Phys. Lett. B
**195**, 216 (1987)ADSCrossRefGoogle Scholar - 5.A.M. Horowitz, Stochastic quantization in phase space. Phys. Lett. B
**156**, 89 (1985)ADSMathSciNetCrossRefGoogle Scholar - 6.A.M. Horowitz, The second order Langevin equation and numerical simulations. Nucl. Phys. B
**280**(FS18), 510 (1987)Google Scholar - 7.A.M. Horowitz, A generalized guided Monte Carlo algorithm. Phys. Lett. B
**268**, 247 (1991)Google Scholar - 8.M. Dalla Brida, M. Garofalo, A. D. Kennedy, Numerical stochastic perturbation theory and gradient flow in \(\phi ^4\) theory. PoS LATTICE2015, 309 (2016)Google Scholar
- 9.K. Jansen, C. Liu, Kramers equation algorithm for simulations of QCD with two flavors of Wilson fermions and gauge group SU(2). Nucl. Phys. B
**453**, 375 (1995). Erratum: Nucl. Phys. B**459**, 437 (1996)Google Scholar - 10.I.P. Omelyan, I.M. Mryglod, R. Folk, Symplectic analytically integrable decomposition algorithms: classification, derivation, and application to molecular dynamics, quantum and celestial mechanics simulations. Comp. Phys. Commun.
**151**, 272 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 11.G. Parisi, Y.-S. Wu, Perturbation theory without gauge fixing. Sci. Sin.
**24**, 483 (1981)MathSciNetGoogle Scholar - 12.P.H. Damgaard, H. Hüffel, Stochastic quantization. Phys. Rep.
**152**, 227 (1987)ADSMathSciNetCrossRefGoogle Scholar - 13.M. Lüscher, R. Narayanan, P. Weisz, U. Wolff, The Schrödinger functional—a renormalizable probe for non-Abelian gauge theories. Nucl. Phys. B
**384**, 168 (1992)ADSCrossRefGoogle Scholar - 14.S. Sint, On the Schrödinger functional in QCD. Nucl. Phys. B
**421**, 135 (1994)ADSCrossRefGoogle Scholar - 15.M. Lüscher, Step scaling and the Yang–Mills gradient flow. JHEP
**1406**, 105 (2014)ADSCrossRefGoogle Scholar - 16.K.G. Wilson, Confinement of quarks. Phys. Rev. D
**10**, 2445 (1974)ADSCrossRefGoogle Scholar - 17.P. Weisz, Continuum limit improved lattice action for pure Yang–Mills theory (I). Nucl. Phys. B
**212**, 1 (1983)ADSMathSciNetCrossRefGoogle Scholar - 18.M. Lüscher, P. Weisz, On-shell improved lattice gauge theories. Commun. Math. Phys.
**97**, 59 (1985). Erratum: Commun. Math. Phys.**98**, 433 (1985)Google Scholar - 19.Y. Iwasaki, Renormalization group analysis of lattice theories and improved lattice action. II—Four-dimensional non-Abelian SU(N) gauge model, preprint UTHEP-118 (1983). arXiv:1111.7054v1
- 20.M. Lüscher, Computational strategies in lattice QCD, in
*Modern perspectives in lattice QCD*, ed. by L. Lellouch et al. (Oxford University Press, New York, 2011)Google Scholar - 21.P. Fritzsch, A. Ramos, The gradient flow coupling in the Schrödinger functional. JHEP
**1310**, 008 (2013)ADSCrossRefzbMATHGoogle Scholar - 22.M. Dalla Brida et al. [ALPHA Collab.], Slow running of the gradient flow coupling from 200 MeV to 4 GeV in \(N_f=3\) QCD. Phys. Rev. D
**95**, 014507 (2017)Google Scholar - 23.I. Campos et al., Non-perturbative running of quark masses in three-flavour QCD, PoS LATTICE2016, 201 (2016)Google Scholar
- 24.M. Lüscher, P. Weisz, U. Wolff, A numerical method to compute the running coupling in asymptotically free theories. Nucl. Phys. B
**359**, 221 (1991)ADSCrossRefGoogle Scholar - 25.M. Lüscher, Advanced lattice QCD, in
*Probing the Standard Model of Particle Interactions (Les Houches 1997)*, ed. by R. Gupta et al. (Elsevier, Amsterdam, 1999)Google Scholar - 26.M. Lüscher, Properties and uses of the Wilson flow in lattice QCD. JHEP
**1008**, 071 (2010). Erratum: JHEP**1403**, 092 (2014)Google Scholar - 27.R.V. Harlander, T. Neumann, The perturbative QCD gradient flow to three loops. JHEP
**1606**, 161 (2016)ADSCrossRefGoogle Scholar - 28.M. Lüscher, P. Weisz, Perturbative analysis of the gradient flow in non-Abelian gauge theories. JHEP
**1102**, 051 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 29.M. Lüscher, P. Weisz, Two-loop relation between the bare lattice coupling and the \({\overline{\rm MS}}\) coupling in pure \({{\rm SU}(N)}\) gauge theories. Phys. Lett. B
**349**, 165 (1995)ADSCrossRefGoogle Scholar - 30.M. Lüscher, R. Sommer, P. Weisz, U. Wolff, A precise determination of the running coupling in the SU(3) Yang–Mills theory. Nucl. Phys. B
**413**, 481 (1994)ADSCrossRefGoogle Scholar - 31.A. Bode, U. Wolff, P. Weisz, Two-loop computation of the Schrödinger functional in pure SU(3) lattice gauge theory. Nucl. Phys. B
**540**, 491 (1999)Google Scholar - 32.A. Bode, P. Weisz, U. Wolff,
*Two-loop computation of the Schrödinger functional in lattice QCD*, Nucl. Phys. B**576**, 517 (2000). [Erratum: Nucl. Phys. B**600**, 453 (2001); Nucl. Phys. B**608**, 481] (2001)Google Scholar - 33.M. Dalla Brida, M. Garofalo, A. D. Kennedy, An investigation of new methods for numerical stochastic perturbation theory in \(\varphi ^4\) theory. arXiv:1703.04406 [hep-lat]
- 34.J. Zinn-Justin, Renormalization and stochastic quantization. Nucl. Phys. B
**275**(FS17), 135 (1986)ADSMathSciNetCrossRefGoogle Scholar - 35.J. Zinn-Justin, D. Zwanziger, Ward identities for the stochastic quantization of gauge fields. Nucl. Phys. B
**295**(FS21), 297 (1988)ADSMathSciNetCrossRefGoogle Scholar - 36.M. Lüscher, Statistical errors in stochastic perturbation theory, notes (2015). http://luscher.web.cern.ch/luscher/notes/enspt.pdf
- 37.K. Symanzik, Cutoff dependence in lattice \(\phi ^4_4\) theory, in
*Recent Developments in Gauge Theories (Cargèse 1979)*, ed. by G. ’t Hooft et al. (Plenum, New York, 1980)Google Scholar - 38.K. Symanzik, Some topics in quantum field theory, in
*Mathematical Problems in Theoretical Physics*, vol. 153, Lecture Notes in Physics, ed. by R. Schrader et al. (Springer, New York, 1982)Google Scholar - 39.D. Zwanziger, Covariant quantization of gauge fields without Gribov ambiguity. Nucl. Phys. B
**192**, 259 (1981)ADSMathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}