^{1}

^{1,2}

^{3}.

Perturbative expansions in many physical systems yield “only” asymptotic series which are not even Borel resummable. Interestingly, the corresponding ambiguities point to nonperturbative physics. We numerically verify this renormalon mechanism for the first time in two-dimensional sigma models, that, like four-dimensional gauge theories, are asymptotically free and generate a strong scale through dimensional transmutation. We perturbatively expand the energy through a numerical version of stochastic quantization. In contrast to the first energy coefficients, the high-order coefficients are independent on the rank of the model. Technically, they require a sophisticated analysis of finite volume effects and the continuum limit of the discretized model. Although the individual coefficients do not grow factorially (yet), but rather decrease strongly, the ratios of consecutive coefficients clearly obey the renormalon asymptotics.

Perturbation theory, the expansion in a small parameter, is a straightforward approach to many physical systems, both classical and quantum. The high-order behavior of the perturbative expansion, however, can be very elaborate: it may not be a convergent, rather an

The typical dependence of an observable

From experience with various models, the emergence of sign-coherent asymptotic expansions is connected to vacuum degeneracy, as labeled by topological quantum numbers. What makes this subject so fascinating is that the

So far, a rigorous mathematical proof of the resurgence picture does not exist for most theories of interest and the calculations rely on certain assumptions (or special features of supersymmetric theories). While these assumptions are physically and mathematically well motivated, it is nevertheless important to check their validity. Moreover, most studies of the renormalon mechanism are based on the summation of a special class of Feynman diagrams. The only known

Renormalons and resurgence in sigma models have been investigated before; see, e.g.,

Since

Correspondingly, the continuum limit

From a statistical physics analogy, the energy density

As

Note that in our approach the energy is formally expanded in half-integer powers, just like the fields, but every second of these coefficients is found to vanish, i.e., to be consistent with zero.

Actually, the first few terms of this weak coupling expansion are known analyticallyTo develop an expectation for the renormalon behavior of the energy expansion, we first of all notice that, although their homotopy groups are trivial,

Where the ’t Hooft coupling kept fixed is

The coefficients

We apply NSPT to calculate the expansion coefficients

In Langevin simulations, the finite stochastic time step

A remarkable feature of NSPT simulations is that neither the lattice spacing

For the results presented in this work, we used the following equation to take finite volume effects into account:

Figure

Expansion coefficients

It is interesting to note that the coefficients

However, the derivation of

To calculate the ratios

The ratios

While our data clearly show the expected asymptotic behavior, the error bars on the ratios are relatively large. It is tempting to look for a plateau in the ratios

The plot shows the results of fitting the ratios

To summarize, we have determined the perturbative coefficients of the energy in particular two-dimensional sigma models by a suitable lattice technique—implicitly summing up factorially many diagrams—including dedicated continuum and infinite volume extrapolations. Using the first few analytically known coefficients as benchmarks, we have determined up to 20 high-order coefficients which spread over many orders of magnitude. Their ratios (divided by the order) clearly approach a constant consistent with the renormalon picture. The latter is based mainly on the coupling dependence of the strong scale (and the mass dimension of the quantity), thus reflecting nonperturbative physics. Our

The NSPT code used in this work

We thank Gerald Dunne, Mithat Ünsal, and in particular Gunnar Bali for useful discussions, and Marcos Mariño for pointing us to Ref.

The framework of NSPT applied in this work is not new, but not well-known outside the lattice community. For the convenience of the reader, we therefore give a brief introduction to NSPT in this section. A more thorough exposition can be found in the referenced original works. NSPT was first developed in the context of QCD in

The idea of numerical perturbation theory is to formally perform a weak coupling expansion of the lattice fields

Algebraic operations with these truncated series are straightforward,

The most expensive operation is the multiplication of fields, which requires

This is only true for a naive implementation of the convolution

An obvious precondition for the use of numerical perturbation theory is that all the functions involved in the computations can be expanded in powers of

One alternative to Metropolis-Hastings type algorithms is stochastic quantization

For numerical calculations, the partial differential equation

It seems counterintuitive that the Euler method with a term proportional to

For completeness, we mention a recently developed new formulation of NSPT, where the Langevin equation is replaced by an stochastic molecular dynamics (SMD) algorithm

So far, we have only discussed the simple case of an unconstrained scalar field

The use of the Euler integrator

Having defined a discrete Langevin equation for fields

We use the Euler method as an example here, but the situation is the same for Runge-Kutta algorithms.

Evidently, this leads to inconsistencies. The solution is to redefine the time stepFor our simulations, we use only symmetric lattices with

Lattice setups for the Langevin runs. The simulations use five different stochastic time step sizes

We emphasize that our lattice fields only fulfill the constraint

Once the lattice is initialized, we perform

In practical calculations, it is important to keep in mind that the NSPT algorithm described in the last section respects the constraint

During our runs, we periodically check if the fields still fulfill the constraints. Calculating

Monitoring the Langevin histories shows that the initialization is in general not sufficient to thermalize the highest order coefficients of the energy density. We do not observe any significant violation of unitarity for the lattices with

For the final computation of the expectation values

Stochastic time history of the coefficient

To be consistent, for a given lattice size, we choose the same

Range of

The choice of which data to include is somewhat arbitrary and one should verify that it does not influence the final results. As a cross-check, we consider two additional datasets. The first consists of all the setups in Table

The Runge-Kutta integrator employed in the simulations is of second order. The extrapolation to

Example for the extrapolation to

The extrapolation to infinite volume is based on OPE methods and we closely follow Ref.

The lattice regularization introduces two distinct scales: the inverse lattice spacing

We are interested in calculating the coefficients

A simple closed expression to generate

Knowing the functional form of the finite volume effects, we can perform fits to our data to extract the infinite volume coefficients. In its most generic form, the fit function we use is given by

The

In all our fits, we neglect higher order terms in the

The vastly different scales of the

The plots in Fig.

Ratios after

Since we cannot resolve the

Like Fig.

The results plotted in Figs.

Like Fig.

Like Fig.

Final results, constant fit to