# Large-order NSPT for lattice gauge theories with fermions: the plaquette in massless QCD

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## Abstract

Numerical Stochastic Perturbation Theory (NSPT) allows for perturbative computations in quantum field theory. We present an implementation of NSPT that yields results for high orders in the perturbative expansion of lattice gauge theories coupled to fermions. The zero-momentum mode is removed by imposing twisted boundary conditions; in turn, twisted boundary conditions require us to introduce a smell degree of freedom in order to include fermions in the fundamental representation. As a first application, we compute the critical mass of two flavours of Wilson fermions up to order \(O(\beta ^{-7})\) in a \({{\mathrm{{\mathrm {SU}}}}}(3)\) gauge theory. We also implement, for the first time, staggered fermions in NSPT. The residual chiral symmetry of staggered fermions protects the theory from an additive mass renormalisation. We compute the perturbative expansion of the plaquette with two flavours of massless staggered fermions up to order \(O(\beta ^{-35})\) in a \({{\mathrm{{\mathrm {SU}}}}}(3)\) gauge theory, and investigate the renormalon behaviour of such series. We are able to subtract the power divergence in the Operator Product Expansion (OPE) for the plaquette and estimate the gluon condensate in massless QCD. Our results confirm that NSPT provides a viable way to probe systematically the asymptotic behaviour of perturbative series in QCD and, eventually, gauge theories with fermions in higher representations.

## 1 Introduction

The success of perturbation theory in High Energy Physics (HEP) can hardly be denied. In particular, in asymptotically free theories, field correlators at short distances are reliably approximated by perturbative expansions in the running coupling at a large momentum scale. At the same time, even in these (lucky) cases, it is mandatory to have some control on nonperturbative effects, i.e. contributions that scale like powers of the QCD scale \(\Lambda _\mathrm {QCD}\). We will often refer to these as *power corrections*. A tool to take the latter into account was suggested back in the late seventies. This goes under the name of QCD sum rules, or Shifman-Vainshtein-Zakharov (SVZ) sum rules [1, 2]. One of the authors defined the method as “an expansion of the correlation functions in the vacuum condensates” [3]. These condensates are the vacuum expectation value of the operators that emerge in the Operator Product Expansion (OPE) for the relevant correlation function. In the OPE formalism the condensates are fundamental quantities, which are in principle supposed to parametrise power corrections in a universal way. By determining the value of a condensate in one context, one gains insight into different physical processes; this has in turn motivated several approaches to the determination of condensates. Having said all this, the sad news is that not all the condensates have actually the same status. In particular not all the condensates can be defined in a neat way, which ultimately means disentangled from perturbation theory. While this is the case for the chiral condensate, the same cannot be said for the gluon condensate, which is the one we will be concerned with in this work.

Based on a separation of scales, the OPE makes pretty clear what can/must be computed in perturbation theory, i.e. the Wilson coefficients. Still, this does not automatically imply that perturbative and nonperturbative contributions are separated in a clear-cut way. The key issue is that perturbative expansions in HEP are expected to be asymptotic ones on very general grounds. In particular, the series in asymptotically free theories are plagued by ambiguities which are due to so-called infrared renormalons [4, 5]. From a technical point of view, renormalons show up as singularities which are encountered if one tries to Borel resum the perturbative series. All in all, there is a power-like ambiguity in any procedure one can devise in order to sum the series, and this ambiguity unavoidably reshuffles perturbative and nonperturbative contributions in the structure of the OPE. Being the Wilson coefficients affected by ambiguities that are power corrections, the general strategy is to reabsorb the latter in the definition of the condensates. This amounts to a prescription to give a precise meaning both to the perturbative series and to the condensates that appear in the OPE.

The idea of determining the gluon condensate from nonperturbative (Monte Carlo) measurements in lattice gauge theories dates back to the eighties and early nineties [6, 7, 8, 9]. Based on symmetry grounds and dimensional counting, the two leading contributions in the OPE for the basic plaquette are given by the identity operator and the gluon condensate. Both operators appear multiplied by Wilson coefficients that can be computed in perturbation theory, and in particular the coefficient that multiplies the identity operator is simply the perturbative expansion of the plaquette. Other operators that appear in the OPE are of higher dimension, and their contributions are therefore suppressed by powers of \(a \Lambda _\mathrm {QCD}\). Subtracting from a nonperturbative (Monte Carlo) measurement of the plaquette the sum of the perturbative series, and repeating the procedure at different values of the coupling, the signature of asymptotic scaling, i.e. the signature of a quantity of (mass) dimension four, should become visible. With renormalons attracting more and more attention, it eventually became clear that such a procedure must be deeply affected by the ambiguities we discussed above, suggesting that a precise definition of the resummed perturbative expansion is necessary.

In the meantime Numerical Stochastic Perturbation Theory (NSPT) [10] was developed as a new tool for computing high orders in lattice perturbation theory. NSPT paved the way to the evaluation of many more terms in the perturbative expansion of the plaquette, and in turn made it at least conceivable that the behaviour of the series could be understood at the level of pinning down the correct order of magnitude of the ambiguity involved. Results of early investigations [11] were interesting: for the first time, it was clear that very high order contributions can be computed in perturbative series for lattice gauge theories. Unfortunately the pioneering NSPT studies of that time were far away from computing the series up to the orders at which the renormalon growth actually shows up in its full glory. With limited computing power available, a way out was sought in the form of a change of scheme (i.e. a scheme in which the renormalon behaviour is best recognised, possibly at lower orders than in the lattice scheme). Still, the numerical results were in the end puzzling as for consequences, since trying to sum the series from the information available even suggested the idea that an unexpected contribution from a dimension-2 operator was present [12]. Other attempts were made [13], but it eventually took roughly twenty years before the renormalon behaviour was actually captured [14, 15, 16, 17], needless to say, via NSPT.^{1} In \({{\mathrm{{\mathrm {SU}}}}}(3)\) Yang–Mills theory the IR renormalon was indeed directly inspected, and the finite-size effects that are unavoidable on finite lattices assessed. The bottom line is that the victory is twofold. On one side, the renormalon growth is indeed proved to be present as conjectured (ironically, in a scheme – the lattice – which one would have regarded as the very worst to perform the computations). Given this, one has a prescription to sum the series and perform the subtraction (if sufficiently high orders are available, one can look for the inversion point in the series, where contributions start to grow and a minimum indetermination in summing the series can be attained).

The present work is a first attempt at performing the determination of the gluon condensate from the plaquette in full QCD, i.e. with fermionic contributions taken into account. The main focus here is in developing the NSPT technology, and present a first set of results, which allow a definition of the gluon condensate. In particular for the first exploration, we use existing Monte Carlo simulations for the plaquette in full QCD, as detailed below. Having ascertained that the procedure is viable, a precise determination of the condensate in full QCD will require a dedicated Monte Carlo simulation, with a careful choice of the fermionic action. On top of being interesting *per se*, the methodology presented here opens the way to other applications, in which different colour groups and different matter contents can be investigated. The final goal would be to inspect whether in a theory that has an IR fixed point, the renormalon growth is tamed, as one would expect in theories where the condensates vanish. We defer these questions to future investigations, hoping to gain extra insight into the task of identifying the boundaries of the conformal window.

The paper is organised as follows. In Sect. 2 we review briefly how NSPT can be applied to lattice gauge theories. In Sect. 3 twisted boundary conditions for fermions in the fundamental representation are introduced. In Sect. 4 we discuss how to take into account fermions with smell in NSPT. We present our results for the expansion of the critical mass of Wilson fermions in Sect. 5, and for the expansion of the plaquette with staggered fermions in Sect. 6. In Sect. 7 we investigate the asymptotic behaviour of the expansion of the plaquette and extract the gluon condensate in massless QCD. In Sect. 8 we draw our conclusions and present some possible future steps.

## 2 Lattice gauge theories in NSPT

Let us here summarise the main steps in defining NSPT for lattice gauge theories. Rather than trying to give a comprehensive review of the method, we aim here to introduce a consistent notation that will allow us to discuss the new developments in the rest of the paper. For a more detailed discussion of the NSPT formulation, the interested reader can consult e.g. Ref. [18], whose notation we shall try to follow consistently.^{2} In particular, we assume to work with a hypercubic lattice with volume \(L^4=a^4N^4\) and assume the lattice spacing *a* to be 1, unless where stated otherwise. We use *x*, *y*, *z* for position indices, \(\mu ,\nu ,\rho =1,\ldots ,4\) for Lorentz indices and \(\alpha ,\beta ,\gamma =1,\ldots ,4\) for Dirac indices.

*t*, a field \(U_{\mu }(x;t)\) can be defined that satisfies the Langevin equation

*t*distribution of observables built from the solution of the Langevin equation above corresponds to the distribution that defines the path integral of the quantum theory [19, 20]:

*g*, which is given in the lattice formulation by \(\beta ^{-1/2}\):

*perturbative components*of the link variables \(U_{\mu }^{(k)}(x;t)\).

Expanding the solution of Langevin equation in powers of the coupling is a standard approach to proving the equivalence of stochastic and canonical quantisation, i.e. Eq. (5) [21], and was the starting point for stochastic perturbation theory: with this respect NSPT is just the numerical implementation of the latter on a computer. The idea of studying the convergence properties of a stochastic process order by order after an expansion in the coupling is actually quite general. In this spirit different NSPT schemes can be set up, also based on stochastic differential equations different from Langevin [22, 23].

**Euler integrator**Discretising the stochastic time in steps of size \(\epsilon \) allows a numerical integration of the Langevin equation,

*F*in powers of \(\beta ^{-1/2}\),

*U*. Omitting Lorentz and position indices, we get

**Stochastic gauge fixing**The zero modes of the gauge action do not generate a deterministic drift term in the Langevin equation, and therefore their evolution in stochastic time is entirely driven by the stochastic noise, which gives rise to diverging fluctuations. This phenomenon is well known since the early days of NSPT, see e.g. Ref. [24], and is cured by the so-called stochastic gauge fixing procedure [25] applied to the theory formulated on the lattice. The procedure implemented in this work alternates an integration step as described above with a gauge transformation:

*w*(

*x*) is defined in the algebra of the group,

*w*(

*x*). In this work we make the same choice as in Ref. [24], which is slightly different from the one adopted in Ref. [18]: the corresponding gauge transformation does not lead, if iterated, to the Landau gauge. In NSPT the gauge transformation is expanded in powers of the coupling,

**Runge–Kutta integrator** Higher order integrators, in particular Runge–Kutta schemes, have been used for the lattice version of the Langevin equation since the early days [20]. A new, very effective second-order integration scheme for NSPT in lattice gauge theories has been introduced in Ref. [15]. While we have tested Runge–Kutta schemes ourselves for pure gauge NSPT simulations, in this work we adhere to the simpler Euler scheme: when making use of the (standard) stochastic evaluation of the fermionic equations of motion (see Sect. 4), Runge–Kutta schemes are actually more demanding (extra terms are needed [26, 27]).

## 3 Twisted boundary conditions and smell

*smells*, which transform into each other according to the antifundamental representation of \({{\mathrm{{\mathrm {SU}}}}}(N_c)\). The theory has a new global symmetry, but physical observables are singlets under the smell group. Thus, configurations related by a smell transformations are equivalent, and in finite volume we are free to substitute Eq. (19) with

It is worth pointing out that, through a change of variable in the path integral [32, 33], twisted boundary conditions could be equivalently implemented by multiplying particular sets of plaquettes in the action by suitable elements of \(Z_{N_c}\) and considering the fields to be periodic. This change of variable works only in the pure gauge or fermions in the adjoint representation cases. Thus, the explicit transformation of Eq. (20) is required when fermions in the fundamental representation with smell are considered.

## 4 Fermions in NSPT

^{3}

If fermions have smell, then the rescaling \(N_f\rightarrow N_f/N_c\) is required in order to have \(N_f\) flavours in the infinite-volume limit. In other words, this is the same as considering the \(N_c\)th root of the determinant of the fermion operator. In principle such rooted determinant could come from a nonlocal action, because twisted boundary conditions break the invariance under smell transformations. Nevertheless, this rooting procedure is sound since we know in advance that in the infinite-volume limit all the dependence on boundary conditions will be lost and the determinant will factorise as the fermion determinant of a single smell times the identity in smell space. It is also possible to show with arguments similar to those presented in Ref. [35] that, if the theory without smell is renormalisable, this operation leads to a perturbatively renormalisable theory as well. Below we describe in detail Wilson and staggered fermions in the fundamental representation, so we explicitly rescale \(N_f\rightarrow N_f/N_c\). It is also important to remember that the fermion field, seen as a matrix in colour-smell space, is not required to be traceless, thus its Fourier zero-mode does not vanish: we require antiperiodic boundary conditions in time direction not to hit the pole of the free propagator in the massless case. We avoid twisted boundary conditions in time direction because in the massless case it might happen for the free fermion propagator to develop a pole at some particular momenta.

### 4.1 Wilson fermions

### 4.2 Staggered fermions

*v*in momentum space, it is useful to represent \(v(p_\parallel )_{p_\perp }\) as matrices \(N_c\times N_c\) with indices \({{\tilde{n}}_1,{\tilde{n}}_2}\) defined at each \(p_\parallel \) site \((n_1,n_2,n_3,n_4)\) (see again Appendix C). Then the non-diagonal terms become diagonal when shifting iteratively

*v*by

*L*/ 2 in the \(p_\parallel \) space. Incidentally, we must consider

*L*to be even so that at the same time

*L*/ 2 is well defined and (in the massless case) no spurious pole is hit when Eq. (35) is evaluated in finite volume: this stems from the fact that the staggered action is only invariant under translation of two lattice spacings, therefore twisted boundary conditions would be inconsistent for

*L*odd.

## 5 The critical mass of Wilson fermions

*a*is written explicitly. Wilson fermions are not equipped with chiral symmetry when the bare mass

*m*vanishes: the self energy at zero momentum is affected by a power divergence \(a^{-1}\), which has to be cured by an additive renormalisation. In an on-shell renormalisation scheme, the critical value of the bare mass, \(m_c\), for which the lattice theory describes massless fermions, is given by the solution of

*am*. The functions \(\Gamma ^{(n)}(ap,am)\) are matrices in Dirac space; since we are interested in the small momentum region and \(\Gamma ^{(n)}(0,am)\) is proportional to the identity, we consider \(\Gamma ^{(n)}(ap,am)\) as scalar functions: when \(ap\ne 0\) a projection onto the identity is understood. Plugging the perturbative expansion of the critical mass

Both results are familiar from analytical calculations of the critical mass. The first equation encodes the fact that the mass counterterm at first order in perturbation theory is given by the one-loop diagrams computed at zero bare mass. The second equation states that the second-order correction is given by summing two-loop diagrams evaluated at vanishing bare mass, and one-loop diagrams with the insertion of the \(O\left( \beta ^{-1}\right) \) counterterm, see e.g. Ref. [37].

It should also be noted that, when working in finite volume, momenta are quantised. Unless periodic boundary conditions are used, \(p=0\) is not an allowed value for the momentum of the states in a box. Therefore, condition (37) can only be imposed after extrapolating the value of \(\Sigma \) to vanishing momentum. The detailed implementation is discussed below in Sect. 5.1.

*p*is computed by applying the inverse Dirac operator to a point source in momentum space,

*S*(

*p*). We can now extrapolate the stochastic time step to zero and invert the propagator to obtain \(S(p)^{-1}\). Finally, the inverse propagator is projected onto the identity in Dirac space. All these operations are performed order by order in perturbation theory keeping in mind that, after the measure of the propagator, all perturbative orders \(\beta ^{-k/2}\) with an odd

*k*are discarded, since the expansion in powers of \(\beta ^{-1/2}\) is an artefact of NSPT. The errors can be estimated by bootstrapping the whole procedure.

The legacy of this process is the measure of the functions \(\gamma ^{(n)}(ap)\), as it is clear from Eq. (40). The renormalisation condition in Eq. (41) must then be imposed: this can be done iteratively one order after the other. When all the coefficients up to some \(m_c^{(n)}\) are included in the simulation, all the \(\gamma \) functions up to \(\gamma ^{(n)}(ap)\) extrapolate to zero; on the other hand, from \(\gamma ^{(n+1)}(0)\) we can read \(-m_c^{(n+1)}\). In order to move on and compute the following coefficient of the critical mass, a new set of configurations where \(m_c^{(n+1)}\) is taken into account must be generated.

The procedure we described is well defined and even theoretically clean, since it enlightens the status of our \(m_c\) as a perturbative additive renormalisation: once it is plugged in at a given order, the renormalised mass turns out to be zero at the prescribed order. On the other side, it is not at all the only possible procedure. The prescription of the authors of Ref. [23] is to expand the solution of the stochastic process both in the coupling and in the mass counterterm. This is in the same spirit of Ref. [43]: the solution of the stochastic process can be expanded in more than one parameter and once a precise power counting is in place, the resulting hierarchy of equations can be exactly truncated at any given order. There are pros and contras for both approaches, i.e. the one we followed and the double expansion. The latter can provide a better handle on estimating errors due to the critical mass value; on the other side, it is expected to be numerical more demanding. All in all, we did not push Wilson fermions to very high orders: moving to the staggered formulation was by far the most natural option for the purpose of this work.

### 5.1 Zero-momentum extrapolation and valence twist

Since in finite volume it is possible to measure \(\Gamma (ap)\) only for discretised non-zero momenta, the data need to be extrapolated to zero momentum using a suitable functional form. The strategy adopted in the literature – see for example Eqs. (13) and (14) in Ref. [40] – is based on expanding the quantities of interest in powers of *ap*. In the infinite-volume limit, such an expansion leads to a hypercubic symmetric Taylor expansion composed of invariants in *ap*, logarithms of *ap* and ratios of invariants; an explicit one-loop computation to order \(a^2\) is shown e.g. in Eq. (24) of Ref. [44]. The ratios and the logarithms arise because we are expanding a nonanalytic function of the lattice spacing: infrared divergences appear when expanding the integrands in *ap*. On the other hand, working consistently in finite volume does not cause any infrared divergence: expressions for \(\gamma ^{(n)}(ap)\) will be just sums of ratios of trigonometric functions, which we can expand in *ap* obtaining simply a combination of polynomial lattice invariants.^{4}

*ap*-expansion. The hypercubic series becomes just a polynomial in \((ap_4)^2\) by setting all the other components to zero.

### 5.2 A first attempt for high-order critical mass for SU(3), \(N_f = 2\)

^{5}to estimate systematic errors. The total error is the sum in quadrature of half the spread around the central value among the different fits and the largest error from the fits.

The procedure described in Sect. 5.1, even though well-defined, is found to be numerically unstable at high orders. The number of propagators required to reach a clear plateau, like the ones shown in Fig. 2, is beyond what it can be reasonably collected with the current NSPT implementations. Therefore, we decided to proceed with a smaller statistics and to add a new systematic uncertainty for the extrapolated coefficients, as explained below. It has to be emphasised that once a coefficient of the critical mass is determined, only the central value is used as input for the following runs: even if we could collect enough statistics and manage to reduce the error, that is not included in the simulations. This makes the impact of the uncertainty of \(m_c^{(n)}\) on \(m_c^{(n+1)}\) and higher hard to assess; also, performing simulations for several values of each coefficient is not feasible. To be conservative, we adopted the following strategy. Once a critical mass \(m_c^{(n)}\) is determined and put in the next-order simulation, the corresponding \(\gamma ^{(n)}(ap)\) should extrapolate to zero. If it extrapolates to \(\epsilon _n\), we take \(|\epsilon _n/m_c^{(n)}|\) as an estimate of the relative systematic error to be added in quadrature to the determination of all the higher-order critical masses.

Critical masses for \(N_c=3\), \(N_f=2\) Wilson fermions determined with NSPT on a \(16^4\) lattice with twisted boundary condition on a plane, compared with the known values in infinite volume. The \(n=1\) coefficient has been determined analytically in twisted lattice perturbation theory; many digits have been used in the actual simulation

| \(-m_c^{(n)}\) on \(16^4\) | \(-m_c^{(n)}\) in infinite volume |
---|---|---|

1 | \(2.61083\dots \) | \(2.60571\dots \) |

2 | 4.32(3) | |

3 | \(1.21(1)\cdot 10^1\) | |

4 | \(3.9(2)\cdot 10^1\) | \(3.96(4)\cdot 10^1\) [40] |

5 | \(1.7(2) \cdot 10^2\) | – |

6 | \(5(1) \cdot 10^2\) | – |

7 | \(2(1) \cdot 10^3\) | – |

## 6 Perturbative expansion of the plaquette

*P*ranges between 0, when all link variables are equal to the identity, and 1. The plaquette expectation value has the perturbative expansion

### 6.1 Simulation details

Summary of the ensembles for \(N_c=3\) and \(N_f=2\) staggered fermions. The order \(n_{\text {max}}\) is the highest order at which the plaquette \(p_n\) has been measured

| \(\tau \) | \(n_\text {max}\) |
---|---|---|

24 | 0.005 | 35 |

0.0075 | 35 | |

0.01 | 35 | |

28 | 0.005 | 29 |

0.008 | 35 | |

0.01 | 35 | |

32 | 0.005 | 33 |

0.008 | 35 | |

0.01 | 35 | |

48 | 0.005 | 35 |

0.008 | 35 | |

0.01 | 35 |

### 6.2 Numerical instabilities

The study of the NSPT hierarchy of stochastic processes is not trivial. While there are general results for the convergence of the generic correlation function of a finite number of perturbative components of the fields [18, 47], the study of variances is more involved, and many results can only come from direct inspection of the outcome of numerical simulations. In particular, one should keep in mind that in the context of (any formulation of) NSPT, variances are not an intrinsic property of the theory under study; in other words, they are not obtained as field correlators of the underlying theory. Big fluctuations and correspondingly huge variances were observed at (terrifically) high orders in toy models [47]: signals are plagued by several spikes and it is found by inspection that a fluctuation at a given order is reflected and amplified at higher orders. All in all, variances increase with the perturbative order (not surprisingly, given the recursive nature of the equations of motion). Moving to more realistic theories, a robust rule of thumb is that, as expected on general grounds, the larger the number of degrees of freedom, the less severe the problems with fluctuations are. In particular, we have not yet found (nor has anyone else reported) big problems with fluctuations in the computation of high orders in pure Yang–Mills theory.

### 6.3 Determination of the \(p_n\)

^{6}The finite-volume result is \(p_1=1.10317022\dots \) at \(L=8\), therefore we expect finite volume effects to be negligible in the lattices we are employing. In particular, we improved the determination of \(p_{1,f}\) in Eq. (50) using the finite volume calculations at \(L=16\) as the central value, and the variation between \(L=16\) and \(L=14\) as an estimate of its uncertainty, leading to \(p_{1,f}=-0.0587909(3)N_f\) for \(N_c=3\), and hence \(p_1=1.1032139(6)\) for \(N_f=2\). Trying to extract \(p_0\) and \(p_1\) from our data at \(L=48\), we realise that even \(\tau ^2\) effects in the extrapolation must be considered because of the very high precision of the measurements. For these two coefficients, a dedicated study at has been performed, which required new simulations at time steps \(\tau =0.004\) and \(\tau =0.0065\); the agreement with the analytic calculations is found to be excellent, see Fig. 6.

*n*and time step \(\tau \) is computed from the average of the fields generated by the stochastic process, after discarding a number of thermalisation steps. The moving averages result to be stable, as can be seen in the two examples of Fig. 7. In order to exploit all the available data, the thermalisation is set differently at different orders. The covariance \(\text {Cov}(n,m)_\tau \) between \(p_{n,\tau }\) and \(p_{m,\tau }\) is computed taking into account autocorrelations and cross-correlations, as explained in detail in Appendix D. Clearly there is no correlation between different \(\tau \). In order to estimate the covariance when two orders have different thermalisations, we take into account only the largest set of common values where both are thermalised. This pairwise estimation of the covariance matrix does not guarantee positive definiteness, therefore we rely on Higham’s algorithm, which we describe in Appendix G, to find the nearest positive definite covariance matrix; the procedure introduces some dependence on a tolerance \(\delta \). The extrapolation to vanishing time step is performed by minimising

Plaquette coefficients from the combined fit for \(L=24\), 28, 32, 48. The tolerance \(\delta \) is given only when the covariance matrix is found not to be positive definite

\(L=24\) | \(L=28\) | ||||||
---|---|---|---|---|---|---|---|

| \(p_n\) | \(\chi ^2/\text {dof}\) | \(\delta \) | | \(p_n\) | \(\chi ^2/\text {dof}\) | \(\delta \) |

2 | 2.536(1) | 2.178 | − | 2 | 2.537(1) | 0.032 | − |

3 | 7.622(6) | 1.079 | 0.1 | 3 | 7.639(7) | 1.136 | 0.625 |

4 | \(2.626(3) \cdot 10^{1}\) | 0.735 | 0.1 | 4 | \(2.636(3) \cdot 10^{1}\) | 0.648 | 0.5 |

5 | \(9.84(1) \cdot 10^{1}\) | 0.615 | 0.1 | 5 | \(9.89(2) \cdot 10^{1}\) | 0.853 | 0.1 |

6 | \(3.906(6) \cdot 10^{2}\) | 0.828 | 0.01 | 6 | \(3.934(7) \cdot 10^{2}\) | 0.593 | 0.1 |

7 | \(1.615(3) \cdot 10^{3}\) | 0.529 | 0.01 | 7 | \(1.630(4) \cdot 10^{3}\) | 0.480 | 0.1 |

8 | \(6.89(2) \cdot 10^{3}\) | 0.581 | 0.01 | 8 | \(6.97(2) \cdot 10^{3}\) | 0.707 | 0.1 |

9 | \(3.021(9) \cdot 10^{4}\) | 0.421 | 0.01 | 9 | \(3.05(1) \cdot 10^{4}\) | 0.927 | 0.1 |

10 | \(1.357(5) \cdot 10^{5}\) | 0.861 | 0.01 | 10 | \(1.366(5) \cdot 10^{5}\) | 0.753 | 0.1 |

11 | \(6.09(3) \cdot 10^{5}\) | 0.940 | 0.01 | 11 | \(6.21(3) \cdot 10^{5}\) | 0.599 | 0.1 |

12 | \(2.80(2) \cdot 10^{6}\) | 0.753 | 0.01 | 12 | \(2.87(1) \cdot 10^{6}\) | 0.512 | 0.1 |

13 | \(1.302(9) \cdot 10^{7}\) | 0.690 | 0.01 | 13 | \(1.338(7) \cdot 10^{7}\) | 0.443 | 0.1 |

14 | \(6.14(4) \cdot 10^{7}\) | 0.570 | 0.01 | 14 | \(6.31(4) \cdot 10^{7}\) | 0.401 | 0.1 |

15 | \(2.94(2) \cdot 10^{8}\) | 0.652 | 0.01 | 15 | \(3.01(2) \cdot 10^{8}\) | 0.360 | 0.1 |

16 | \(1.41(1) \cdot 10^{9}\) | 0.797 | 0.01 | 16 | \(1.44(1) \cdot 10^{9}\) | 1.012 | 0.01 |

17 | \(6.79(6) \cdot 10^{9}\) | 0.758 | 0.01 | 17 | \(6.96(7) \cdot 10^{9}\) | 0.998 | 0.01 |

18 | \(3.31(3) \cdot 10^{10}\) | 0.730 | 0.01 | 18 | \(3.36(3) \cdot 10^{10}\) | 0.972 | 0.01 |

19 | \(1.65(2) \cdot 10^{11}\) | 0.678 | 0.01 | 19 | \(1.63(2) \cdot 10^{11}\) | 0.953 | 0.01 |

20 | \(8.3(1) \cdot 10^{11}\) | 0.732 | 0.01 | 20 | \(8.0(1) \cdot 10^{11}\) | 0.884 | 0.01 |

21 | \(4.15(7) \cdot 10^{12}\) | 0.755 | 0.01 | 21 | \(3.89(6) \cdot 10^{12}\) | 0.829 | 0.01 |

22 | \(2.08(5) \cdot 10^{13}\) | 0.590 | 0.1 | 22 | \(1.91(3) \cdot 10^{13}\) | 0.821 | 0.01 |

23 | \(10.0(4) \cdot 10^{13}\) | 0.569 | 0.1 | 23 | \(9.5(2) \cdot 10^{13}\) | 0.873 | 0.01 |

24 | \(5.0(2) \cdot 10^{14}\) | 0.543 | 0.1 | 24 | \(4.7(1) \cdot 10^{14}\) | 0.851 | 0.01 |

25 | \(2.5(1) \cdot 10^{15}\) | 0.485 | 0.1 | 25 | \(2.34(6) \cdot 10^{15}\) | 0.764 | 0.01 |

26 | \(1.34(4) \cdot 10^{16}\) | 1.140 | 0.01 | 26 | \(1.14(3) \cdot 10^{16}\) | 0.695 | 0.01 |

27 | \(6.6(2) \cdot 10^{16}\) | 1.054 | 0.01 | 27 | \(5.7(2) \cdot 10^{16}\) | 0.687 | 0.01 |

28 | \(3.2(2) \cdot 10^{17}\) | 0.479 | 0.1 | 28 | \(2.8(1) \cdot 10^{17}\) | 0.671 | 0.01 |

29 | \(1.6(1) \cdot 10^{18}\) | 1.124 | 0.01 | 29 | \(1.5(1) \cdot 10^{18}\) | 0.462 | 0.01 |

30 | \(7.6(7) \cdot 10^{18}\) | 0.836 | 0.01 | 30 | \(7.1(7) \cdot 10^{18}\) | 0.855 | 0.001 |

31 | \(3.6(6) \cdot 10^{19}\) | 0.456 | 0.01 | 31 | \(4.2(7) \cdot 10^{19}\) | 0.663 | 0.001 |

32 | \(1.8(4) \cdot 10^{20}\) | 0.443 | 0.01 | 32 | \(2.0(4) \cdot 10^{20}\) | 0.661 | 0.001 |

33 | \(9(3) \cdot 10^{20}\) | 0.445 | 0.01 | 33 | \(10(3) \cdot 10^{20}\) | 0.651 | 0.001 |

34 | \(5(2) \cdot 10^{21}\) | 0.432 | 0.01 | 34 | \(4(2) \cdot 10^{21}\) | 0.516 | 0.001 |

35 | \(3(1) \cdot 10^{22}\) | 0.425 | 0.01 | 35 | \(2(1) \cdot 10^{22}\) | 0.519 | 0.001 |

\(L=32\) | \(L=48\) | ||||||
---|---|---|---|---|---|---|---|

| \(p_n\) | \(\chi ^2/\text {dof}\) | \(\delta \) | | \(p_n\) | \(\chi ^2/\text {dof}\) | \(\delta \) |

2 | 2.5370(8) | 0.249 | − | 2 | 2.5354(7) | 2.745 | − |

3 | 7.627(4) | 1.182 | − | 3 | 7.615(3) | 1.454 | 0.01 |

4 | \(2.633(2) \cdot 10^{1}\) | 2.412 | − | 4 | \(2.623(1) \cdot 10^{1}\) | 1.428 | 0.1 |

5 | \(9.882(9) \cdot 10^{1}\) | 1.378 | 0.5 | 5 | \(9.826(6) \cdot 10^{1}\) | 1.673 | 0.1 |

6 | \(3.926(5) \cdot 10^{2}\) | 1.015 | 0.1 | 6 | \(3.897(3) \cdot 10^{2}\) | 1.653 | 0.1 |

7 | \(1.626(2) \cdot 10^{3}\) | 0.730 | 0.1 | 7 | \(1.613(2) \cdot 10^{3}\) | 1.338 | 0.1 |

8 | \(6.96(1) \cdot 10^{3}\) | 0.929 | 0.01 | 8 | \(6.88(1) \cdot 10^{3}\) | 1.194 | 0.1 |

9 | \(3.050(6) \cdot 10^{4}\) | 0.772 | 0.01 | 9 | \(3.007(6) \cdot 10^{4}\) | 1.079 | 0.1 |

10 | \(1.367(4) \cdot 10^{5}\) | 0.638 | 0.01 | 10 | \(1.341(3) \cdot 10^{5}\) | 0.998 | 0.1 |

11 | \(6.22(2) \cdot 10^{5}\) | 0.963 | 0.01 | 11 | \(6.08(1) \cdot 10^{5}\) | 0.925 | 0.1 |

12 | \(2.86(1) \cdot 10^{6}\) | 0.645 | 0.1 | 12 | \(2.793(6) \cdot 10^{6}\) | 1.108 | 0.01 |

13 | \(1.337(6) \cdot 10^{7}\) | 0.771 | 0.1 | 13 | \(1.297(3) \cdot 10^{7}\) | 0.978 | 0.01 |

14 | \(6.29(3) \cdot 10^{7}\) | 0.861 | 0.1 | 14 | \(6.08(2) \cdot 10^{7}\) | 0.883 | 0.01 |

15 | \(3.00(2) \cdot 10^{8}\) | 0.952 | 0.1 | 15 | \(2.87(1) \cdot 10^{8}\) | 1.067 | 0.01 |

16 | \(1.438(9) \cdot 10^{9}\) | 1.012 | 0.1 | 16 | \(1.370(5) \cdot 10^{9}\) | 1.013 | 0.01 |

17 | \(6.94(5) \cdot 10^{9}\) | 0.996 | 0.1 | 17 | \(6.57(3) \cdot 10^{9}\) | 0.951 | 0.01 |

18 | \(3.34(3) \cdot 10^{10}\) | 1.000 | 0.1 | 18 | \(3.16(1) \cdot 10^{10}\) | 0.930 | 0.01 |

19 | \(1.63(2) \cdot 10^{11}\) | 0.965 | 0.1 | 19 | \(1.530(6) \cdot 10^{11}\) | 0.938 | 0.01 |

20 | \(7.90(8) \cdot 10^{11}\) | 1.053 | 0.01 | 20 | \(7.45(3) \cdot 10^{11}\) | 0.890 | 0.01 |

21 | \(3.86(4) \cdot 10^{12}\) | 0.995 | 0.01 | 21 | \(3.65(1) \cdot 10^{12}\) | 0.824 | 0.01 |

22 | \(1.90(2) \cdot 10^{13}\) | 0.957 | 0.01 | 22 | \(1.796(9) \cdot 10^{13}\) | 0.748 | 0.01 |

23 | \(9.4(1) \cdot 10^{13}\) | 0.949 | 0.01 | 23 | \(8.88(5) \cdot 10^{13}\) | 0.691 | 0.01 |

24 | \(4.74(9) \cdot 10^{14}\) | 0.979 | 0.01 | 24 | \(4.41(3) \cdot 10^{14}\) | 0.636 | 0.01 |

25 | \(2.39(5) \cdot 10^{15}\) | 0.967 | 0.01 | 25 | \(2.19(2) \cdot 10^{15}\) | 0.575 | 0.01 |

26 | \(1.22(3) \cdot 10^{16}\) | 0.921 | 0.01 | 26 | \(1.09(1) \cdot 10^{16}\) | 0.548 | 0.01 |

27 | \(6.3(2) \cdot 10^{16}\) | 0.871 | 0.01 | 27 | \(5.46(9) \cdot 10^{16}\) | 0.538 | 0.01 |

28 | \(3.2(1) \cdot 10^{17}\) | 0.849 | 0.01 | 28 | \(2.74(6) \cdot 10^{17}\) | 0.523 | 0.01 |

29 | \(1.63(9) \cdot 10^{18}\) | 0.812 | 0.01 | 29 | \(1.38(4) \cdot 10^{18}\) | 0.511 | 0.01 |

30 | \(8.6(7) \cdot 10^{18}\) | 0.779 | 0.01 | 30 | \(7.0(3) \cdot 10^{18}\) | 0.492 | 0.01 |

31 | \(4.5(9) \cdot 10^{19}\) | 0.743 | 0.01 | 31 | \(3.5(2) \cdot 10^{19}\) | 0.494 | 0.01 |

32 | \(1.9(3) \cdot 10^{20}\) | 0.723 | 0.01 | 32 | \(1.7(1) \cdot 10^{20}\) | 0.503 | 0.01 |

33 | \(9(2) \cdot 10^{20}\) | 0.723 | 0.01 | 33 | \(8.3(7) \cdot 10^{20}\) | 1.062 | 0.001 |

34 | \(5(1) \cdot 10^{21}\) | 0.702 | 0.01 | 34 | \(5.2(6) \cdot 10^{21}\) | 1.090 | 0.001 |

35 | \(1(1) \cdot 10^{22}\) | 0.663 | 0.01 | 35 | \(2.3(6) \cdot 10^{22}\) | 0.486 | 0.01 |

## 7 Gluon condensate

*a*and follow the notation of Refs. [16, 17]: the gluon condensate is defined as the vacuum expectation value of the operator

^{7}

*a*, and \(\Lambda _\text {QCD}\). In the limit \(a^{-1}\gg \Lambda _\text {QCD}\), Eq. (57) can be seen as an Operator Product Expansion (OPE) [1, 2, 53], which factorises the dependence on the small scale

*a*. In this framework,

^{8}condensates like \(\mathinner {\langle {O_G}\rangle }\) are process-independent parameters that encode the nonperturbative dynamics, while the Wilson coefficients are defined in perturbation theory,

*Z*and \(C_G\) depend only on the bare coupling \(\beta ^{-1}\), and do not depend on the renormalisation scale \(\mu \), as expected for both coefficients [55, 56]. Nonperturbative contributions to

*Z*, or \(C_G\), originating for example from instantons, would correspond to subleading terms in \(\Lambda _\text {QCD}\). This procedure defines a renormalisation scheme to subtract power divergences: condensates are chosen to vanish in pertubation theory or, in other words, they are normal ordered in the perturbative vacuum. This definition matches the one that is natural in dimensional regularisation, where power divergences do not arise. Nevertheless, it is well known that such a definition of the condensates might lead to ambiguities, since the separation of scales in the OPE does not necessarily correspond to a separation between perturbative and nonperturbative physics (see the interesting discussions in Refs. [3, 57]). For example, the fermion condensate in a massless theory is well-defined since, being the order parameter of chiral symmetry breaking, it must vanish in perturbation theory. The same cannot be said for the gluon condensate [58], and indeed the ambiguity in its definition is reflected in the divergence of the perturbative expansion of the plaquette. For this picture to be consistent, it must be possible to absorb in the definition of the condensate the ambiguity in resumming the perturbative series.

In the following, we are going to study the asymptotic behaviour of the coefficients \(p_n\) determined in the previous section and discuss the implications for the definition of the gluon condensate in massless QCD.

### 7.1 Growth of the coefficients

*n*expansion since the \(\beta _2\) coefficient is scheme-dependent and it is not known for staggered fermions. In Figs. 9 and 10, the comparison between Eq. (59) and our data at different volumes is shown.

*n*. Since the finite size of the lattice provides a natural infrared cutoff, we expect finite-volume effects to be larger at larger perturbative orders. The dependence of \(p_n\) on the lattice size

*N*can be modelled with a finite-volume OPE, exploiting the separation of scales \(a^{-1}\gg (Na)^{-1}\): the leading correction is [16]

*n*. This is shown in the two examples of Fig. 11. A similar behaviour has been observed in Ref. [16], where the data points computed on comparable volumes show little dependence on the lattice size. In that study, a detailed analysis with a large number of volumes was needed in order to be able to fit the finite-volume corrections. The overall effect is found to be an increase of the ratio \(p_n/(n p_{n-1})\), see e.g. Fig. 6 in Ref. [16]. In our case, data in finite volume do cross the theoretical expectation; still, considering the spread between points at different volumes in Fig. 10 as a source of systematic error, we could consider our measurements to be compatible with the asymptotic behaviour of Eq. (59). We also ascertain the existence of an inversion point when resumming the perturbative series, as explained in Sect. 7.3. Despite this encouraging behaviour, any definite conclusion about the existence of the expected renormalon can only be drawn after performing an appropriate infinite-volume study. We emphasise that in this work the discrepancies in the determination of the \(p_ n\) from different volumes must be interpreted as part of our systematic uncertainty, being this an exploratory study. A precise assessment of the finite-volume effects will be sought for a precise determination of the gluon condensate; we are currently planning a set of dedicated simulations in the near future to settle this issue.

### 7.2 Monte Carlo plaquette

Results of the chiral extrapolation for the plaquette and the scale. The order of the polynomials used in the fits is indicated

\(\beta \) | \(\langle 1-P\rangle _\text {MC}\) | Pol. ord. | \(r_0/a\) | Pol. ord. |
---|---|---|---|---|

5.3 | \(0.4951\,(4)\) | 2 | \(2.11\,(7)\) | 3 |

5.35 | \(0.5152\,(9)\) | 3 | \(2.47\,(3)\) | 1 |

5.415 | \(0.5350\,(3)\) | 3 | \(3.30\,(3)\) | 3 |

5.5 | \(0.55128\,(3)\) | 1 | \(4.17\,(2)\) | 1 |

5.6 | \(0.56526\,(5)\) | 1 | \(5.14\,(1)\) | 1 |

Summation up to the minimal term of the perturbative series of the plaquette

\(\beta \) | | \(S_P(\beta )\) | \({\bar{n}}\) | \(p_{\bar{n}} \beta ^{-(\bar{n}+1)}\) |
---|---|---|---|---|

5.3 | 24 | \(0.47515\,(9)\) | 25 | \(3.70 \cdot 10 ^{-4}\) |

28 | \(0.4767\,(1)\) | 30 | \(2.52 \cdot 10 ^{-4}\) | |

32 | \(0.4775\,(4)\) | 35 | \(5.23 \cdot 10 ^{-5}\) | |

48 | \(0.47665\,(7)\) | 33 | \(1.97 \cdot 10 ^{-4}\) | |

5.35 | 24 | \(0.46718\,(8)\) | 25 | \(2.90 \cdot 10 ^{-4}\) |

28 | \(0.46843\,(9)\) | 30 | \(1.88 \cdot 10 ^{-4}\) | |

32 | \(0.4690\,(3)\) | 35 | \(3.73 \cdot 10 ^{-5}\) | |

48 | \(0.46826\,(5)\) | 33 | \(1.43 \cdot 10 ^{-4}\) | |

5.415 | 24 | \(0.4587\,(1)\) | 33 | \(1.06 \cdot 10 ^{-4}\) |

28 | \(0.45844\,(7)\) | 30 | \(1.29 \cdot 10 ^{-4}\) | |

32 | \(0.4588\,(2)\) | 35 | \(2.42 \cdot 10 ^{-5}\) | |

48 | \(0.45822\,(4)\) | 33 | \(9.51 \cdot 10 ^{-5}\) | |

5.5 | 24 | \(0.44663\,(9)\) | 33 | \(6.22 \cdot 10 ^{-5}\) |

28 | \(0.44651\,(6)\) | 30 | \(7.98 \cdot 10 ^{-5}\) | |

32 | \(0.4466\,(1)\) | 35 | \(1.38 \cdot 10 ^{-5}\) | |

48 | \(0.44627\,(4)\) | 33 | \(5.60 \cdot 10 ^{-5}\) | |

5.6 | 24 | \(0.43384\,(6)\) | 34 | \(3.32 \cdot 10 ^{-5}\) |

28 | \(0.43380\,(5)\) | 30 | \(4.57 \cdot 10 ^{-5}\) | |

32 | \(0.43383\,(6)\) | 35 | \(7.21 \cdot 10 ^{-6}\) | |

48 | \(0.43357\,(3)\) | 33 | \(3.03 \cdot 10 ^{-5}\) |

### 7.3 Determination of the minimal term

*L*and \(\beta \) are summarised in Table 5. The order \({\bar{n}}\) at which the series starts to diverge depends only on the central value of the coefficients \(p_n\) and not on their errors: in order to check that the inversion point determined by our procedure is stable, we bootstrapped the procedure by generating an ensemble of sets of coefficients \(\left\{ p_n\right\} \). For each set, the coefficients \(p_n\) are drawn from a Gaussian probability, whose mean and covariance are taken from the fit procedure described in Sect. 6. We then determine \({\bar{n}}\) for each of these sets. The inversion point turns out to be stable, as shown in Fig. 13 for a the case \(L=48\), and \(\beta =5.3\). This particular case is shown for illustration purposes, and the same features are seen in all other combinations of

*L*and \(\beta \).

The result of the subtraction is shown in the left panel of Fig. 14, for the largest volume. Since only a few values of \(\beta \) is available, it is hard to assess unambiguously the presence of a plateau. We decided to discard from the analysis the two values of the coupling corresponding to the coarser lattices, and define our best estimate of the condensate as the weighted average of the values obtained at the remaining \(\beta \)s. Our final results are summarised in the first column of Table 6.

In order to put the choice of fit range on more solid ground, we studied the scaling of \(a^4\mathinner {\langle {O_G}\rangle }\) as a function of \(a^4\), as shown in Fig. 14. The slope of a linear fit of the three finest lattice spacings should give a determination of the condensate compatible with the value extracted from the weighted average. The spread between these two determinations and among the different volumes gives an idea of the magnitude of the systematic uncertainties involved. We also tried to include in the analysis all the available values of \(\beta \) and add a \(a^6\) correction, in the attempt to model the deviations at large values of the coupling; this procedure gives again consistent results (despite a larger \(\chi ^2\)).

*n*is \(\sqrt{\pi {\bar{n}}/2}\,p_{{\bar{n}}}\,\beta ^{-{\bar{n}} -1}\) [5]. In Table 7, the ambiguity associated to the gluon condensate

^{9}

Determination of the gluon condensate at different volumes. The determination labelled with 1 is obtained from the weighted average of the values at the three largest values of \(\beta \). The determinations labelled with 2 and 3 are obtained by studying the scaling of \(a^4\mathinner {\langle {O_G}\rangle }\) with \(a^4\), as in the right panel of Fig. 14; they correspond respectively to the fit without and with \(a^6\) correction (see text for the details)

| \(r_0^4\mathinner {\langle {O_G}\rangle }_1\) | \(r_0^4\mathinner {\langle {O_G}\rangle }_2\) | \(r_0^4\mathinner {\langle {O_G}\rangle }_3\) |
---|---|---|---|

24 | \(2.6\,(1)\) | \(2.9\,(2)\) | \(3.1\,(4)\) |

28 | \(2.8\,(1)\) | \(3.1\,(2)\) | \(3.4\,(4)\) |

32 | \(2.4\,(1)\) | \(2.9\,(2)\) | \(3.2\,(4)\) |

48 | \(3.1\,(1)\) | \(3.1\,(2)\) | \(3.4\,(4)\) |

Ambiguity of the gluon condensate determined from Eq. (64) at the three largest values of \(\beta \)

| \(r_0^4\delta \mathinner {\langle {O_G}\rangle }\) | ||
---|---|---|---|

\(\beta =5.415\) | \(\beta =5.5\) | \(\beta =5.6\) | |

24 | \(0.4\,(2)\) | \(0.5\,(4)\) | \(0.7\,(5)\) |

28 | \(0.4\,(3)\) | \(0.7\,(4)\) | \(0.9\,(5)\) |

32 | \(0.3\,(2)\) | \(0.5\,(3)\) | \(0.3\,(3)\) |

48 | \(0.3\,(2)\) | \(0.5\,(3)\) | \(0.6\,(4)\) |

## 8 Conclusions

We used NSPT to perform for the first time large-order computations in lattice gauge theories coupled to massless fermions. We adopted twisted boundary conditions for the gauge fields to remove the zero-momentum mode. Since our fermions are in the fundamental representation, we consistently provided them with a smell degree of freedom. Both Wilson and (for the first time in NSPT) staggered fermions have been implemented. While for the former we performed an exploratory study of the critical mass up to order \(O(\beta ^{-7})\), the latter are ultimately the best choice to reach very high orders, due to their residual chiral symmetry that bypasses the need of an additive mass renormalisation.

Numerical instabilities were noticed in the study of simple models in NSPT since the early days of the method, but gauge theories have always been reported to stay on a safe side in this respect, even at orders as high as the ones we investigated in this work. With fermions in place, we now found that numerical instabilities arise for lattice gauge theories at high orders. While we plan to investigate the causes and develop a solution to this, the problem did not prevent us to reach order \(O(\beta ^{-35})\) in the expansion of the basic plaquette for \(N_c=3\) and \(N_f=2\).

The plaquette has been for a long time the stage for the determination of the gluon condensate, to which is connected in the continuum limit. The perturbative expansion of the plaquette, which corresponds to the power divergent contribution associated to the identity operator in the relevant OPE, must be subtracted from nonperturbative Monte Carlo lattice computations. This long-standing and tough problem was eventually solved a few years ago in pure gauge [16, 17], thanks to NSPT. Equipped with our high-orders expansions, we tackled once again the problem in the lattice regularisation of full QCD. We computed the perturbative expansion of the plaquette, and subtracted it from Monte Carlo measurements. In this context, NSPT is crucial: it is actually the only tool enabling this procedure, which asks for having the asymptotic behaviour of such series under control. This happens since the perturbative expansion of the plaquette is expected to be plagued by renormalon ambiguities. Under the assumption of considering finite-volume effects as a source of systematic errors, the observed growth of the coefficients in the expansion could be compatible with the leading IR renormalon; nevertheless, the large uncertainties and the lack of a study of finite-volume effects prevent us from drawing any definite conclusion. The IR renormalon forces to absorb the ambiguities attached to the perturbative series into the definition of the condensate itself. All in all, this implies that we needed a prescription to perform the computation. The one we chose amounts to summing the perturbative series up to its minimal term (which means computing the series up to orders that only NSPT can aim at).

We regard this project as a first exploratory study. We could confirm both that the IR renormalon can be directly inspected, and that the series can be computed up to orders where the inversion point beyond which the expansion starts to diverge (at values of the coupling which are the typical ones in lattice simulations) is clearly visible. We performed our simulations at different lattice extents, in order to have a first estimate of finite-size effects (again, in both the study of renormalon behaviour and in the truncation of the series). This is the point which has to be better investigated in a following study. At the moment, finite-size effects are still to be considered as a systematic source of errors in our procedure.

On top of the follow-ups we have already discussed, we plan to extend our study to different number of colours, number of flavours and fermionic representations. It would be of the utmost importance to assess the high-order behaviour of perturbative coefficients in gauge theories different from QCD, to probe regions in the space of theories in which a (quasi-)conformal window can be present. This could be a powerful, alternative method to look for candidate theories for physics beyond the Standard Model.

## Footnotes

- 1.
One should note that one of the reason why the renormalon growth was correctly reproduced and the OPE correctly reconstructed is the adoption of twisted boundary conditions: in this way zero modes are absent and the theoretical picture is clear.

- 2.
For convenience, we summarise our group theory conventions in Appendix A.

- 3.
Obviously \(\xi \) does not have any Dirac structure in the staggered case. The noise can be built from the independent generation of real and imaginary part with zero mean and variance 1 / 2.

- 4.
Expanding in

*ap*and sending the lattice size to infinity are operations that do not commute; in particular this gives rise to different series in the finite- and infinite-volume cases. - 5.
The different subsets are built by varying the number of initial configurations that are excluded in the analysis and by rejecting data at different rates.

- 6.
We are grateful to M. García Pérez and A. González-Arroyo for providing us the gluon contribution in finite volume.

- 7.
We mention that, in a theory with fermions, the operator \(O_G\) must be combined with \(m{\bar{\psi }}\psi \) to give a renormalisation group invariant quantity; moreover mixing with the operators \(m{\bar{\psi }}\psi \) and Open image in new window should also be considered [51, 52]. Clearly such complications are not present in the massless case and the operator Open image in new window can be neglected in the following discussions since it vanishes when the equation of motion are used.

- 8.
It is useful to keep in mind that other definitions of the gluon condensate are possible, see e.g. Ref. [54].

- 9.
Our definition of the ambiguity differs from the one in Ref. [16] by a factor \(\sqrt{\pi /2}\).

- 10.
For recent developments on the code see Ref. [67].

- 11.
Available at https://github.com/gfilaci/GridNSPT.

## Notes

### Acknowledgements

We would like to thank Gunnar Bali and Antonio Pineda for useful comments on the manuscript. A special thought goes to the organiser of the workshop “High-precision QCD at low energies” in Benasque, where these ideas were first discussed. LDD is supported by an STFC Consolidated Grant, ST/P0000630/1, and a Royal Society Wolfson Research Merit Award, WM140078. FDR acknowledges support from INFN under the research project *i.s. QCDLAT*. Access to MARCONI was obtained through a CINECA-INFN agreement. Most of this work was performed using the Cambridge Service for Data Driven Discovery (CSD3), part of which is operated by the University of Cambridge Research Computing on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The DiRAC component of CSD3 was funded by BEIS capital funding via STFC capital grants ST/P002307/1 and ST/R002452/1 and STFC operations grant ST/R00689X/1. DiRAC is part of the National e-Infrastructure.

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